diff --git a/lean/OAI/Probability/ThorpRouting/Dense/Main.lean b/lean/OAI/Probability/ThorpRouting/Dense/Main.lean
index 1a27bc3d393b7e49b97570094305f4b11c58c167..a550c07fd29d7504f14661961b5a664333afb20f 100644
--- a/lean/OAI/Probability/ThorpRouting/Dense/Main.lean
+++ b/lean/OAI/Probability/ThorpRouting/Dense/Main.lean
@@ -1,1125 +1,1125 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Dense.SingleSlotsEquiv
-
-namespace OAI
-
-namespace ThorpNine.Dense
-
-open scoped BigOperators
-namespace Thorp
-
-lemma sharedPartnerMean_nonneg (r s : ℕ) (A : Finset (Card ((s+1)+r))) :
-    0 ≤ sharedPartnerMean r s A := by
-  unfold sharedPartnerMean
-  positivity
-
-lemma sum_suffix_pair_counts (r s : ℕ) (A : Finset (Card ((s+1)+r))) :
-    (∑ y : Card s, ((suffixMarks r (s+1) A (Fin.cons false y)).card +
-      (suffixMarks r (s+1) A (Fin.cons true y)).card)) = A.card := by
-  classical
-  have hh := sum_card_blockMarks ((cardSplit r (s+1)).trans (Equiv.prodComm _ _)) A
-  rw [←Equiv.sum_comp (headTailEquiv s).symm,Fintype.sum_prod_type,Fintype.sum_bool] at hh
-  simpa [suffixMarks,headTailEquiv,Fin.consEquiv,Finset.sum_add_distrib,add_comm] using hh
-
-lemma sharedPartnerMean_twice_le (r s : ℕ) (A : Finset (Card ((s+1)+r))) :
-    2*sharedPartnerMean r s A ≤ A.card := by
-  have hcap (y : Card (s+1)) : (suffixMarks r (s+1) A y).card ≤ 2^r := by
-    simpa only [card_positions] using Finset.card_le_univ (suffixMarks r (s+1) A y)
-  have hp : (0:ℝ) < (2:ℝ)^r := by positivity
-  rw [sharedPartnerMean,←mul_div_assoc,div_le_iff₀ hp,Finset.mul_sum]
-  have hs : (A.card:ℝ) = ∑ y : Card s,
-      ((suffixMarks r (s+1) A (Fin.cons false y)).card +
-        (suffixMarks r (s+1) A (Fin.cons true y)).card : ℝ) := by
-    exact_mod_cast (sum_suffix_pair_counts r s A).symm
-  rw [hs,Finset.sum_mul]
-  apply Finset.sum_le_sum
-  intro y _
-  have h₀ : (suffixMarks r (s+1) A (Fin.cons false y)).card ≤ (2:ℝ)^r := by
-    exact_mod_cast hcap (Fin.cons false y)
-  have h₁ : (suffixMarks r (s+1) A (Fin.cons true y)).card ≤ (2:ℝ)^r := by
-    exact_mod_cast hcap (Fin.cons true y)
-  nlinarith [mul_nonneg (sub_nonneg.mpr h₀)
-    (Nat.cast_nonneg (suffixMarks r (s+1) A (Fin.cons true y)).card),
-    mul_nonneg (sub_nonneg.mpr h₁)
-    (Nat.cast_nonneg (suffixMarks r (s+1) A (Fin.cons false y)).card)]
-
-
-lemma palindrome_level_dense_mgf (r s : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card ((s+1)+r)) (t : ℝ) (ht : 0 ≤ t) :
-    finiteMean (fun ω : BenesCoins ((s+1)+r) =>
-      Real.exp (t * palindromeLevelCost (s+1) ((s+1)+r) e ω)) ≤
-        Real.exp ((Real.exp (denseLevelCoefficient s t)-1) *
-          sharedPartnerMean r s (Finset.univ.image e)) := by
-  classical
-  rw [finiteMean_prod,finiteMean_comm]
-  calc
-    _ ≤ finiteMean (fun X : SwitchIndex ((s+1)+r) → Bool =>
-        Real.exp (denseLevelCoefficient s t * sharedLevel (s+1) ((s+1)+r) e X)) := by
-      apply finiteMean_mono
-      intro X
-      exact palindrome_level_dense_conditional s ((s+1)+r) e X t ht
-    _ = finiteMean (fun hi : Card (s+1) × SwitchIndex r → Bool =>
-        Real.exp (denseLevelCoefficient s t *
-          gridShared r s (inverseButterflyMarks ((s+1)+r) (Finset.univ.image e)
-            (assembleBits r (s+1) (fun _ => false) hi)))) := by
-      rw [finiteMean_split]
-      simp_rw [sharedLevel_partial_marks]
-      rw [finiteMean_const]
-    _ ≤ _ := partial_grid_mgf r s (Finset.univ.image e) _
-        (denseLevelCoefficient_nonneg s ht)
-
-
-lemma cardSplit_snd_apply (r s : ℕ) (x : Card (s+r)) (i : Fin s) :
-    (cardSplit r s x).2 i = x ⟨r+i,by omega⟩ := by
-  induction r with
-  | zero => simp [cardSplit]
-  | succ r ih =>
-    change (cardSplit r s (Fin.tail x)).2 i = _
-    rw [ih]
-    simp only [Fin.tail]
-    congr 1
-    apply Fin.ext
-    simp only [Fin.val_succ]
-    omega
-
-def PairSeparated {d : ℕ} (i : Fin d) (x y : Card d) : Prop :=
-  x i = false ∧ y i = true ∧ ∀ j : Fin d, i < j → x j = y j
-
-lemma PairSeparated.unique {d : ℕ} {i j : Fin d} {x y : Card d}
-    (hi : PairSeparated i x y) (hj : PairSeparated j x y) : i = j := by
-  by_contra h
-  rcases lt_or_gt_of_ne h with h | h
-  · have hh := hi.2.2 j h
-    rw [hj.1,hj.2.1] at hh
-    exact Bool.false_ne_true hh
-  · have hh := hj.2.2 i h
-    rw [hi.1,hi.2.1] at hh
-    exact Bool.false_ne_true hh
-
-lemma PairSeparated.not_reverse {d : ℕ} {i j : Fin d} {x y : Card d}
-    (hi : PairSeparated i x y) : ¬ PairSeparated j y x := by
-  intro hj
-  rcases lt_trichotomy i j with h | h | h
-  · have hh := hi.2.2 j h
-    rw [hj.2.1,hj.1] at hh
-    exact Bool.noConfusion hh
-  · subst j
-    exact Bool.false_ne_true (hi.1.symm.trans hj.2.1)
-  · have hh := hj.2.2 i h
-    rw [hi.1,hi.2.1] at hh
-    exact Bool.noConfusion hh
-
-lemma pairSeparated_split_iff (r s : ℕ) (x y : Card ((s+1)+r)) :
-    PairSeparated ⟨r,by omega⟩ x y ↔
-      ∃ z : Card s, (cardSplit r (s+1) x).2 = Fin.cons false z ∧
-        (cardSplit r (s+1) y).2 = Fin.cons true z := by
-  constructor
-  · intro h
-    refine ⟨Fin.tail (cardSplit r (s+1) x).2, ?_, ?_⟩
-    · have h₀ : (cardSplit r (s+1) x).2 0 = false := by
-        simpa only [cardSplit_snd_apply,Fin.val_zero,Nat.add_zero] using h.1
-      rw [←h₀,Fin.cons_self_tail]
-    · funext k
-      refine Fin.cases ?_ (fun k => ?_) k
-      · simpa only [cardSplit_snd_apply,Fin.cons_zero,Fin.val_zero,Nat.add_zero] using h.2.1
-      · simp only [Fin.cons_succ,Fin.tail,cardSplit_snd_apply]
-        exact (h.2.2 _ (by change r < r + (k.val+1); omega)).symm
-  · rintro ⟨z,hx,hy⟩
-    refine ⟨?_,?_,?_⟩
-    · have hh := congrFun hx 0
-      simpa only [cardSplit_snd_apply,Fin.cons_zero,Fin.val_zero,Nat.add_zero] using hh
-    · have hh := congrFun hy 0
-      simpa only [cardSplit_snd_apply,Fin.cons_zero,Fin.val_zero,Nat.add_zero] using hh
-    · intro j hj
-      change r < j.val at hj
-      have hbound : j.val-r-1 < s := by omega
-      let k : Fin s := ⟨j.val-r-1,hbound⟩
-      have hjk : j = ⟨r+(k.val+1),by omega⟩ := Fin.ext (by dsimp [k]; omega)
-      rw [hjk]
-      have hx' := congrFun hx k.succ
-      have hy' := congrFun hy k.succ
-      simp only [cardSplit_snd_apply,Fin.cons_succ] at hx' hy'
-      exact hx'.trans hy'.symm
-
-
-abbrev SeparatedPair {d : ℕ} (A : Finset (Card d)) (i : Fin d) :=
-  {p : A × A // PairSeparated i p.1.val p.2.val}
-
-noncomputable def separatedCount {d : ℕ} (A : Finset (Card d)) (i : Fin d) : ℕ := by
-  classical
-  exact Fintype.card (SeparatedPair A i)
-
-lemma separated_budget_strong {d : ℕ} (A : Finset (Card d)) :
-    2 * ∑ i : Fin d, separatedCount A i ≤ A.card * (A.card-1) := by
-  classical
-  let F : Bool × (Σ i : Fin d, SeparatedPair A i) → A × A :=
-    fun ⟨b,i,p⟩ => if b then (p.val.2,p.val.1) else p.val
-  have hF : Function.Injective F := by
-    rintro ⟨b,i,p⟩ ⟨c,j,q⟩ h
-    cases b <;> cases c
-    · change p.val = q.val at h
-      have hij : i = j := p.property.unique (by rw [h]; exact q.property)
-      subst j
-      have hpq : p = q := Subtype.ext h
-      subst q
-      rfl
-    · change p.val = (q.val.2,q.val.1) at h
-      have hp := p.property
-      rw [h] at hp
-      exact False.elim (hp.not_reverse q.property)
-    · change (p.val.2,p.val.1) = q.val at h
-      have hq := q.property
-      rw [←h] at hq
-      exact False.elim (hq.not_reverse p.property)
-    · change (p.val.2,p.val.1) = (q.val.2,q.val.1) at h
-      have hpq : p.val = q.val := Prod.ext (congrArg Prod.snd h) (congrArg Prod.fst h)
-      have hij : i = j := p.property.unique (by rw [hpq]; exact q.property)
-      subst j
-      have hpqq : p = q := Subtype.ext hpq
-      subst q
-      rfl
-  have hneq (v : Bool × (Σ i : Fin d, SeparatedPair A i)) : (F v).1 ≠ (F v).2 := by
-    obtain ⟨b,i,p⟩ := v
-    have hp : p.val.1 ≠ p.val.2 := by
-      intro he
-      have hx := p.property.1
-      have hy := p.property.2.1
-      rw [he] at hx
-      exact Bool.false_ne_true (hx.symm.trans hy)
-    cases b <;> simp [F,hp,Ne.symm hp]
-  let G : Bool × (Σ i : Fin d, SeparatedPair A i) → ↥((Finset.univ : Finset A).offDiag) :=
-    fun v => ⟨F v,Finset.mem_offDiag.mpr ⟨Finset.mem_univ _,Finset.mem_univ _,hneq v⟩⟩
-  have hG : Function.Injective G := fun _ _ he => hF (congrArg Subtype.val he)
-  have hh := Fintype.card_le_of_injective G hG
-  simpa only [Fintype.card_prod,Fintype.card_bool,Fintype.card_sigma,
-    Fintype.card_coe,separatedCount,Finset.offDiag_card,Finset.card_univ,
-    Nat.mul_sub_left_distrib,mul_one] using hh
-
-lemma separated_budget {d : ℕ} (A : Finset (Card d)) :
-    2 * ∑ i : Fin d, separatedCount A i ≤ A.card * A.card :=
-  (separated_budget_strong A).trans (Nat.mul_le_mul_left _ (Nat.sub_le _ _))
-
-lemma mem_suffixMarks (r s : ℕ) (A : Finset (Card (s+r))) (y : Card s) (x : Card r) :
-    x ∈ suffixMarks r s A y ↔ (cardSplit r s).symm (x,y) ∈ A := by
-  exact mem_blockMarks _ _ _ _
-
-lemma separatedCount_eq (r s : ℕ) (A : Finset (Card ((s+1)+r))) :
-    separatedCount A ⟨r,by omega⟩ =
-      ∑ y : Card s, (suffixMarks r (s+1) A (Fin.cons false y)).card *
-        (suffixMarks r (s+1) A (Fin.cons true y)).card := by
-  classical
-  let D := Σ y : Card s, ↥(suffixMarks r (s+1) A (Fin.cons false y)) ×
-    ↥(suffixMarks r (s+1) A (Fin.cons true y))
-  let F : D → SeparatedPair A ⟨r,by omega⟩ := fun ⟨y,u,v⟩ =>
-    ⟨(⟨(cardSplit r (s+1)).symm (u.val,Fin.cons false y),
-          (mem_suffixMarks _ _ _ _ _).mp u.property⟩,
-        ⟨(cardSplit r (s+1)).symm (v.val,Fin.cons true y),
-          (mem_suffixMarks _ _ _ _ _).mp v.property⟩),
-      (pairSeparated_split_iff r s _ _).mpr
-        ⟨y,by simp only [Equiv.apply_symm_apply],by simp only [Equiv.apply_symm_apply]⟩⟩
-  have hF : Function.Bijective F := by
-    constructor
-    · rintro ⟨y,u,v⟩ ⟨z,u',v'⟩ he
-      have h₀ := congrArg (fun p => cardSplit r (s+1) p.val.1.val) he
-      have h₁ := congrArg (fun p => cardSplit r (s+1) p.val.2.val) he
-      change cardSplit r (s+1) ((cardSplit r (s+1)).symm (u.val,Fin.cons false y)) =
-        cardSplit r (s+1) ((cardSplit r (s+1)).symm (u'.val,Fin.cons false z)) at h₀
-      change cardSplit r (s+1) ((cardSplit r (s+1)).symm (v.val,Fin.cons true y)) =
-        cardSplit r (s+1) ((cardSplit r (s+1)).symm (v'.val,Fin.cons true z)) at h₁
-      simp only [Equiv.apply_symm_apply] at h₀ h₁
-      have hyz : y = z := by
-        have hh := congrArg (fun p => Fin.tail p.2) h₀
-        simpa only [Fin.tail_cons] using hh
-      subst z
-      have hu : u = u' := Subtype.ext (congrArg Prod.fst h₀)
-      have hv : v = v' := Subtype.ext (congrArg Prod.fst h₁)
-      subst u'; subst v'; rfl
-    · intro p
-      obtain ⟨y,hx,hy⟩ := (pairSeparated_split_iff r s p.val.1.val p.val.2.val).mp p.property
-      have hx' : (cardSplit r (s+1)).symm ((cardSplit r (s+1) p.val.1.val).1,Fin.cons false y) =
-          p.val.1.val := by rw [←hx,Prod.mk.eta,Equiv.symm_apply_apply]
-      have hy' : (cardSplit r (s+1)).symm ((cardSplit r (s+1) p.val.2.val).1,Fin.cons true y) =
-          p.val.2.val := by rw [←hy,Prod.mk.eta,Equiv.symm_apply_apply]
-      refine ⟨⟨y,⟨(cardSplit r (s+1) p.val.1.val).1,?_⟩,
-          ⟨(cardSplit r (s+1) p.val.2.val).1,?_⟩⟩,?_⟩
-      · rw [mem_suffixMarks,hx']; exact p.val.1.property
-      · rw [mem_suffixMarks,hy']; exact p.val.2.property
-      · apply Subtype.ext
-        exact Prod.ext (Subtype.ext hx') (Subtype.ext hy')
-  have hh := Fintype.card_congr (Equiv.ofBijective F hF)
-  simpa only [D,Fintype.card_sigma,Fintype.card_prod,Fintype.card_coe,separatedCount] using hh.symm
-
-
-noncomputable def partnerMean (d : ℕ) (A : Finset (Card d)) (j : ℕ) : ℝ :=
-  if hj : j < d then (separatedCount A ⟨d-j-1,by omega⟩ : ℝ) / (2:ℝ)^(d-j-1) else 0
-
-lemma partnerMean_nonneg (d : ℕ) (A : Finset (Card d)) (j : ℕ) :
-    0 ≤ partnerMean d A j := by
-  unfold partnerMean
-  split <;> positivity
-
-lemma partnerMean_eq (r s : ℕ) (A : Finset (Card ((s+1)+r))) :
-    partnerMean ((s+1)+r) A s = sharedPartnerMean r s A := by
-  classical
-  have hs : s < (s+1)+r := by omega
-  have hr : (s+1+r)-s-1 = r := by omega
-  simp only [partnerMean,dite_eq_left hs,hr,separatedCount_eq,sharedPartnerMean,
-    Nat.cast_sum,Nat.cast_mul]
-
-lemma partnerMean_twice_le (d : ℕ) (A : Finset (Card d)) (j : ℕ) :
-    2*partnerMean d A j ≤ A.card := by
-  by_cases hj : j < d
-  · obtain ⟨r,rfl⟩ := Nat.exists_eq_add_of_le (Nat.succ_le_of_lt hj)
-    rw [partnerMean_eq]
-    exact sharedPartnerMean_twice_le r j A
-  · simp only [partnerMean,dite_eq_right hj,mul_zero,Nat.cast_nonneg]
-
-lemma partnerMean_rate (d : ℕ) (A : Finset (Card d)) (j : Fin d) :
-    partnerMean d A j / (2:ℝ)^(j.val+1) =
-      (separatedCount A j.rev : ℝ) / (2:ℝ)^d := by
-  have hj : j.val < d := j.isLt
-  have hd : (d-j.val-1)+(j.val+1) = d := by omega
-  simp only [partnerMean,dite_eq_left hj,div_div,←pow_add,hd]
-  rfl
-
-lemma partnerMean_budget (d : ℕ) (A : Finset (Card d)) :
-    2 * (∑ j ∈ Finset.range d, partnerMean d A j / (2:ℝ)^(j+1)) ≤
-      (A.card:ℝ)^2/(2:ℝ)^d := by
-  classical
-  rw [←Fin.sum_univ_eq_sum_range]
-  simp_rw [partnerMean_rate]
-  rw [←Finset.sum_div]
-  have he : (∑ j : Fin d, (separatedCount A j.rev:ℝ)) = ∑ j : Fin d, (separatedCount A j:ℝ) := by
-    exact Equiv.sum_comp (Fin.revPerm : Equiv.Perm (Fin d)) (fun j : Fin d => (separatedCount A j : ℝ))
-  rw [he,←mul_div_assoc]
-  apply div_le_div_of_nonneg_right _ (by positivity)
-  exact_mod_cast (by simpa only [pow_two] using separated_budget A)
-
-noncomputable def partnerAllocation (d : ℕ) (A : Finset (Card d)) (j : ℕ) : ℝ :=
-  2*partnerMean d A j/A.card
-
-lemma partnerAllocation_admissible (d : ℕ) (A : Finset (Card d)) (hA : 0 < A.card) :
-    DenseTruncation.AdmissibleAllocation ((A.card:ℝ)/(2:ℝ)^d) (partnerAllocation d A) := by
-  classical
-  have hA' : (0:ℝ) < A.card := by exact_mod_cast hA
-  constructor
-  · intro j
-    constructor
-    · exact div_nonneg (mul_nonneg (by norm_num) (partnerMean_nonneg d A j)) hA'.le
-    · rw [partnerAllocation,div_le_one hA']
-      exact partnerMean_twice_le d A j
-  · have hz (j : ℕ) (hj : j ∉ Finset.range d) :
-        partnerAllocation d A j/(2:ℝ)^(j+1) = 0 := by
-      have h : ¬ j < d := by simpa only [Finset.mem_range] using hj
-      simp only [partnerAllocation,partnerMean,dite_eq_right h,mul_zero,zero_div]
-    rw [tsum_eq_sum (s := Finset.range d) hz]
-    have he (j : ℕ) : partnerAllocation d A j/(2:ℝ)^(j+1) =
-        2/A.card * (partnerMean d A j/(2:ℝ)^(j+1)) := by
-      unfold partnerAllocation
-      ring
-    simp_rw [he]
-    rw [←Finset.mul_sum]
-    have hh := partnerMean_budget d A
-    apply (show 2/(A.card:ℝ) * ∑ j ∈ Finset.range d, partnerMean d A j/(2:ℝ)^(j+1) =
-      (2*(∑ j ∈ Finset.range d,partnerMean d A j/(2:ℝ)^(j+1)))/A.card by ring).le.trans
-    apply (div_le_div_of_nonneg_right hh hA'.le).trans
-    have hne : (A.card:ℝ) ≠ 0 := ne_of_gt hA'
-    exact le_of_eq (by field_simp)
-
-
-lemma partner_weighted_sum_le (d H : ℕ) (A : Finset (Card d)) (hA : 0 < A.card) :
-    (∑ j ∈ Finset.range H, DenseTruncation.h (j+1)*partnerMean d A j) ≤
-      A.card * DenseTruncation.H ((A.card:ℝ)/(2:ℝ)^d) := by
-  have ha := partnerAllocation_admissible d A hA
-  have hh := DenseTruncation.allocation_le_H ha
-  have hA' : (0:ℝ) < A.card := by exact_mod_cast hA
-  have he (j : ℕ) : DenseTruncation.h (j+1)*partnerMean d A j =
-      (A.card:ℝ)/2*(DenseTruncation.h (j+1)*partnerAllocation d A j) := by
-    unfold partnerAllocation
-    field_simp
-  simp_rw [he]
-  rw [←Finset.mul_sum]
-  calc
-    _ ≤ (A.card:ℝ)/2 * ∑' j, DenseTruncation.h (j+1)*partnerAllocation d A j := by
-      apply mul_le_mul_of_nonneg_left _ (by positivity)
-      exact (DenseTruncation.allocation_value_summable ha).sum_le_tsum _
-        (fun j _ => mul_nonneg (DenseTruncation.h_nonneg _) (ha.1 j).1)
-    _ = (A.card:ℝ)*((1/2:ℝ)*∑' j, DenseTruncation.h (j+1)*partnerAllocation d A j) := by ring
-    _ ≤ _ := mul_le_mul_of_nonneg_left hh (Nat.cast_nonneg _)
-
-lemma finiteMean_rpow_le {Ω : Type*} [Fintype Ω] [Nonempty Ω]
-    (f : Ω → ℝ) (hf : ∀ ω, 0 ≤ f ω) {p : ℝ} (hp : 0 ≤ p) (hp1 : p ≤ 1) :
-    finiteMean (fun ω => (f ω)^p) ≤ (finiteMean f)^p := by
-  have hN : (0:ℝ) < Fintype.card Ω := Nat.cast_pos.mpr Fintype.card_pos
-  have hh := (Real.concaveOn_rpow hp hp1).le_map_sum
-    (t := Finset.univ) (w := fun _ : Ω => 1/(Fintype.card Ω:ℝ))
-    (p := f) (fun _ _ => by positivity)
-    (by simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]; field_simp)
-    (fun ω _ => hf ω)
-  simpa only [smul_eq_mul,one_div,←Finset.mul_sum,mul_comm _ (∑ _ : Ω,_),
-    ←div_eq_mul_inv,finiteMean] using hh
-
-lemma finiteMean_exp_scale {Ω : Type*} [Fintype Ω] [Nonempty Ω]
-    (f : Ω → ℝ) (a b t : ℝ) (hb : 0 < b) (ht : 0 ≤ t) (htb : t ≤ b)
-    (h : finiteMean (fun ω => Real.exp (b*f ω)) ≤ Real.exp (b*a)) :
-    finiteMean (fun ω => Real.exp (t*f ω)) ≤ Real.exp (t*a) := by
-  have hp : 0 ≤ t/b := div_nonneg ht hb.le
-  have hp1 : t/b ≤ 1 := (div_le_one hb).mpr htb
-  have hh := (finiteMean_rpow_le (fun ω => Real.exp (b*f ω))
-    (fun _ => (Real.exp_pos _).le) hp hp1).trans
-      (Real.rpow_le_rpow (finiteMean_nonneg (fun _ => (Real.exp_pos _).le)) h hp)
-  have he (x : ℝ) : (Real.exp (b*x))^(t/b) = Real.exp (t*x) := by
-    rw [←Real.exp_mul]
-    congr 1
-    field_simp
-  simpa only [he] using hh
-
-lemma finiteMean_exp_sum_le {Ω I : Type*} [Fintype Ω] [Nonempty Ω]
-    [Fintype I] [Nonempty I] (f : I → Ω → ℝ) (a : I → ℝ) (t : ℝ)
-    (h : ∀ i, finiteMean (fun ω => Real.exp ((Fintype.card I:ℝ)*t*f i ω)) ≤
-      Real.exp ((Fintype.card I:ℝ)*t*a i)) :
-    finiteMean (fun ω => Real.exp (t*∑ i, f i ω)) ≤ Real.exp (t*∑ i, a i) := by
-  have hN : (0:ℝ) < Fintype.card I := Nat.cast_pos.mpr Fintype.card_pos
-  let N : ℝ := Fintype.card I
-  have hc (i : I) : finiteMean (fun ω => Real.exp (N*t*(f i ω-a i))) ≤ 1 := by
-    have he (ω : Ω) : Real.exp (N*t*(f i ω-a i)) =
-        Real.exp (N*t*f i ω) * Real.exp (-(N*t*a i)) := by
-      rw [←Real.exp_add]; congr 1; ring
-    simp_rw [he]
-    rw [finiteMean_mul_const]
-    calc
-      _ ≤ Real.exp (N*t*a i)*Real.exp (-(N*t*a i)) :=
-          mul_le_mul_of_nonneg_right (h i) (Real.exp_pos _).le
-      _ = 1 := by rw [←Real.exp_add,add_neg_cancel,Real.exp_zero]
-  have hj (ω : Ω) : Real.exp (t*(∑ i, (f i ω-a i))) ≤
-      ∑ i, (1/N)*Real.exp (N*t*(f i ω-a i)) := by
-    have hh := convexOn_exp.map_sum_le
-      (t := Finset.univ) (w := fun _ : I => 1/N)
-      (p := fun i => N*t*(f i ω-a i))
-      (fun _ _ => by dsimp [N]; positivity)
-      (by simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]; dsimp [N]; field_simp)
-      (fun _ _ => Set.mem_univ _)
-    have he (i : I) : (1/N)*(N*t*(f i ω-a i)) = t*(f i ω-a i) := by
-      dsimp [N]; field_simp
-    simpa only [smul_eq_mul,he,←Finset.mul_sum] using hh
-  have hm : finiteMean (fun ω => Real.exp (t*(∑ i, (f i ω-a i)))) ≤ 1 := by
-    apply (finiteMean_mono hj).trans
-    rw [finiteMean_sum]
-    calc
-      _ ≤ ∑ _ : I, 1/N := by
-        apply Finset.sum_le_sum
-        intro i _
-        have he (ω : Ω) : 1/N*Real.exp (N*t*(f i ω-a i)) =
-          Real.exp (N*t*(f i ω-a i))*(1/N) := mul_comm _ _
-        simp_rw [he]
-        rw [finiteMean_mul_const]
-        simpa only [one_mul] using mul_le_mul_of_nonneg_right (hc i)
-          (by dsimp [N]; positivity : 0 ≤ 1/N)
-      _ = 1 := by simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]; dsimp [N]; field_simp
-  have he (ω : Ω) : Real.exp (t*∑ i,f i ω) =
-      Real.exp (t*(∑ i, (f i ω-a i)))*Real.exp (t*∑ i,a i) := by
-    rw [←Real.exp_add,Finset.sum_sub_distrib]
-    congr 1; ring
-  simp_rw [he]
-  rw [finiteMean_mul_const]
-  simpa only [one_mul] using mul_le_mul_of_nonneg_right hm (Real.exp_pos _).le
-
-
-open Filter Topology
-
-lemma dense_coefficient_hasDerivAt (j : ℕ) :
-    HasDerivAt (fun t : ℝ => Real.exp (denseLevelCoefficient j t)-1)
-      (DenseTruncation.h (j+1)) 0 := by
-  have hp : (2:ℝ)^j ≠ 0 := by positivity
-  have h₀ := (hasDerivAt_id (0:ℝ)).mul_const ((2:ℝ)^j)
-  have h₁ := h₀.exp.sub_const 1
-  have h₂ := h₁.div_const ((2:ℝ)^j)
-  have hh := h₂.const_mul (DenseTruncation.h (j+1))
-  simpa [denseLevelCoefficient, hp] using hh.exp.sub_const 1
-
-lemma dense_coefficient_small (H : ℕ) (δ : ℝ) (hδ : 0 < δ) :
-    ∃ τ : ℝ, 0 < τ ∧ τ ≤ 1/500 ∧ ∀ j < H,
-      Real.exp (denseLevelCoefficient j τ)-1 ≤ τ*(DenseTruncation.h (j+1)+δ) := by
-  have hj (j : Fin H) : ∀ᶠ t : ℝ in 𝓝[>] 0,
-      (Real.exp (denseLevelCoefficient j t)-1)/t < DenseTruncation.h (j+1)+δ := by
-    have hh := (dense_coefficient_hasDerivAt j).tendsto_slope_zero_right
-    simp only [zero_add,denseLevelCoefficient,zero_mul,Real.exp_zero,sub_self,zero_div,
-      mul_zero,smul_eq_mul,sub_zero] at hh
-    have hh' : Tendsto (fun t : ℝ => (Real.exp (denseLevelCoefficient j t)-1)/t)
-        (𝓝[>] 0) (𝓝 (DenseTruncation.h (j+1))) := by
-      simpa only [denseLevelCoefficient,div_eq_mul_inv,mul_comm] using hh
-    exact (tendsto_order.mp hh').2 _ (by linarith)
-  have ha : ∀ᶠ t : ℝ in 𝓝[>] 0, ∀ j : Fin H,
-      (Real.exp (denseLevelCoefficient j t)-1)/t < DenseTruncation.h (j+1)+δ :=
-    Filter.eventually_all.mpr hj
-  have hb : ∀ᶠ t : ℝ in 𝓝[>] 0, t < 1/500 :=
-    (eventually_lt_nhds (by norm_num : (0:ℝ) < 1/500)).filter_mono nhdsWithin_le_nhds
-  have hc : ∀ᶠ t : ℝ in 𝓝[>] 0, 0 < t := self_mem_nhdsWithin
-  obtain ⟨τ,ht,hb,ha⟩ := (hc.and (hb.and ha)).exists
-  refine ⟨τ,ht,hb.le,fun j hj => ?_⟩
-  exact ((div_lt_iff₀ ht).mp (ha ⟨j,hj⟩)).le.trans_eq (mul_comm _ _)
-
-lemma palindrome_level_dense_mgf_all (d j : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (t : ℝ) (ht : 0 ≤ t) :
-    finiteMean (fun ω : BenesCoins d => Real.exp (t*palindromeLevelCost (j+1) d e ω)) ≤
-      Real.exp ((Real.exp (denseLevelCoefficient j t)-1)*partnerMean d (Finset.univ.image e) j) := by
-  classical
-  by_cases hj : j < d
-  · obtain ⟨r,rfl⟩ := Nat.exists_eq_add_of_le (Nat.succ_le_of_lt hj)
-    rw [partnerMean_eq]
-    exact palindrome_level_dense_mgf r j e t ht
-  · simp only [palindromeLevelCost_gt (j+1) d (by omega) e, Nat.cast_zero,mul_zero,
-      Real.exp_zero,finiteMean_const,partnerMean,dite_eq_right hj,le_refl]
-
-lemma palindrome_low_dense_mgf (H : ℕ) (hH : 0 < H) (δ τ : ℝ) (hδ : 0 ≤ δ) (ht : 0 ≤ τ)
-    (hτ : ∀ j < H, Real.exp (denseLevelCoefficient j τ)-1 ≤ τ*(DenseTruncation.h (j+1)+δ))
-    (d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (hι : 0 < Fintype.card ι) :
-    finiteMean (fun ω : BenesCoins d => Real.exp ((τ/H)*palindromeLowCost H d e ω)) ≤
-      Real.exp ((τ/H)*Fintype.card ι*
-        (DenseTruncation.H ((Fintype.card ι:ℝ)/(2:ℝ)^d)+H*δ/2)) := by
-  classical
-  let A := Finset.univ.image e
-  have hA : A.card = Fintype.card ι := by simp [A, Finset.card_image_of_injective _ e.injective]
-  have hn : (H:ℝ) ≠ 0 := by exact_mod_cast (ne_of_gt hH)
-  have he : (H:ℝ)*(τ/H) = τ := by field_simp
-  let : Nonempty (Fin H) := Fin.pos_iff_nonempty.mp hH
-  have hh := finiteMean_exp_sum_le
-    (fun j : Fin H => fun ω : BenesCoins d => (palindromeLevelCost (j+1) d e ω:ℝ))
-    (fun j : Fin H => (DenseTruncation.h (j+1)+δ)*partnerMean d A j) (τ/H) (by
-      intro j
-      simp only [Fintype.card_fin,he]
-      apply (palindrome_level_dense_mgf_all d j e τ ht).trans
-      apply Real.exp_le_exp.mpr
-      simpa only [mul_assoc] using mul_le_mul_of_nonneg_right (hτ j j.isLt)
-        (partnerMean_nonneg d A j))
-  have hsum : (∑ j : Fin H, (DenseTruncation.h (j+1)+δ)*partnerMean d A j) ≤
-      Fintype.card ι*(DenseTruncation.H ((Fintype.card ι:ℝ)/(2:ℝ)^d)+H*δ/2) := by
-    rw [Fin.sum_univ_eq_sum_range (fun j => (DenseTruncation.h (j+1)+δ)*partnerMean d A j)]
-    simp_rw [add_mul]
-    rw [Finset.sum_add_distrib]
-    have hm := partner_weighted_sum_le d H A (by omega)
-    rw [hA] at hm
-    have hs : (∑ j ∈ Finset.range H, δ*partnerMean d A j) ≤ (H:ℝ)*δ*(Fintype.card ι)/2 := by
-      calc
-        _ ≤ ∑ _j ∈ Finset.range H, δ*((Fintype.card ι:ℝ)/2) := by
-          apply Finset.sum_le_sum
-          intro j _
-          apply mul_le_mul_of_nonneg_left _ hδ
-          have hg := partnerMean_twice_le d A j
-          rw [hA] at hg
-          linarith
-        _ = _ := by simp only [Finset.sum_const,Finset.card_range,nsmul_eq_mul]; ring
-    nlinarith
-  have hh' := hh.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left hsum (by positivity)))
-  have hl (ω : BenesCoins d) :
-      (∑ j : Fin H, (palindromeLevelCost (j+1) d e ω : ℝ)) =
-        ∑ j ∈ Finset.range H, (palindromeLevelCost (j+1) d e ω : ℝ) :=
-    Fin.sum_univ_eq_sum_range (fun j => (palindromeLevelCost (j+1) d e ω : ℝ)) H
-  simpa only [palindromeLowCost_sum,Nat.cast_sum,hl,mul_assoc] using hh'
-
-
-lemma heightDecay_tendsto : Tendsto heightDecay atTop (𝓝 0) := by
-  have hr : 0 ≤ Real.exp (-(1:ℝ)/64) := (Real.exp_pos _).le
-  have hr1 : Real.exp (-(1:ℝ)/64) < 1 := Real.exp_lt_one_iff.mpr (by norm_num)
-  have hh := ((tendsto_pow_atTop_nhds_zero_of_lt_one hr hr1).const_mul 68).div_const
-    (1-Real.exp (-(1:ℝ)/64))
-  have he (H : ℕ) : (Real.exp (-(1:ℝ)/64))^H = Real.exp (-(H:ℝ)/64) := by
-    rw [←Real.exp_nat_mul]
-    congr 1; ring
-  change Tendsto (fun H : ℕ => 68*Real.exp (-(H:ℝ)/64)/(1-Real.exp (-(1:ℝ)/64))) _ _
-  simpa only [he,mul_zero,zero_div] using hh
-
-lemma heightDecay_exists {ε : ℝ} (hε : 0 < ε) :
-    ∃ H : ℕ, 0 < H ∧ 500*heightDecay H < ε := by
-  have hh := heightDecay_tendsto.const_mul 500
-  simp only [mul_zero] at hh
-  have he := (tendsto_order.mp hh).2 ε hε
-  exact ((eventually_gt_atTop 0).and he).exists
-
-lemma palindrome_high_exp_mgf (H n d : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (t : ℝ) (ht : 0 ≤ t) (htb : t ≤ 1/500) :
-    finiteMean (fun ω : BenesCoins d => Real.exp (t*
-      (palindromeFamilyCost H n d e ω+palindromeFamilyCost (2*H) n d e ω))) ≤
-      Real.exp (t*Fintype.card ι*(500*heightDecay H)) := by
-  have hh := palindrome_high_full_mgf H n d hH e (Real.exp (1/500))
-    (Real.one_le_exp_iff.mpr (by norm_num)) (by rw [Real.log_exp])
-  have hp : finiteMean (fun ω : BenesCoins d => Real.exp ((1/500:ℝ)*
-      (palindromeFamilyCost H n d e ω+palindromeFamilyCost (2*H) n d e ω))) ≤
-      Real.exp ((1/500:ℝ)*(Fintype.card ι*(500*heightDecay H))) := by
-    have he (k : ℕ) : (Real.exp (1/500:ℝ))^k = Real.exp ((1/500:ℝ)*k) := by
-      rw [←Real.exp_nat_mul]; congr 1; ring
-    simpa only [he,Nat.cast_add,show (1/500:ℝ)*(Fintype.card ι*(500*heightDecay H)) =
-      Fintype.card ι*heightDecay H by ring] using hh
-  have hs := finiteMean_exp_scale
-    (fun ω : BenesCoins d => ((palindromeFamilyCost H n d e ω+
-      palindromeFamilyCost (2*H) n d e ω):ℝ))
-    (Fintype.card ι*(500*heightDecay H)) (1/500) t (by norm_num) ht htb
-    (by simpa only [Nat.cast_add] using hp)
-  simpa only [Nat.cast_add,mul_assoc] using hs
-
-end Thorp
-
-namespace Thorp.DenseTruncation
-
-lemma harmonic_le_h (j : ℕ) :
-    (harmonic (2^j):ℝ)/(2:ℝ)^j ≤ h (j+1) := by
-  classical
-  let M := 2^j
-  have hM : 0 < M := by dsimp [M]; positivity
-  have hM' : (0:ℝ) < M := Nat.cast_pos.mpr hM
-  have hMc : (M:ℝ) = (2:ℝ)^j := by simp [M]
-  let p : ℕ → ℝ := fun l => if l < M then 1/M else 0
-  have hp : AdmissibleCycleLaw (j+1) p := by
-    refine ⟨fun l => by dsimp [p]; split_ifs <;> positivity, ?_, ?_⟩
-    · have hs : HasSum p (∑ l ∈ Finset.range M,p l) :=
-        hasSum_sum_of_ne_finset_zero (s := Finset.range M) (f := p)
-        (fun l hl => by simp only [Finset.mem_range] at hl; simp [p,hl])
-      have he : (∑ l ∈ Finset.range M,p l) = 1 := by
-        simp only [p]
-        rw [Finset.sum_congr rfl (fun l hl => ite_eq_left (Finset.mem_range.mp hl))]
-        simp only [Finset.sum_const,Finset.card_range,nsmul_eq_mul]
-        field_simp
-      exact he ▸ hs
-    · intro l
-      simp only [Nat.add_sub_cancel]
-      by_cases hl : l < M
-      · simp only [p,ite_eq_left hl,←hMc]
-        apply div_le_div_of_nonneg_right _ hM'.le
-        exact_mod_cast Nat.succ_pos l
-      · simp only [p,ite_eq_right hl]
-        positivity
-  have he : (∑' l, p l / ((l+1:ℕ):ℝ)) = (harmonic M:ℝ)/(M:ℝ) := by
-    rw [tsum_eq_sum (s := Finset.range M) (fun l hl => by
-      simp only [Finset.mem_range] at hl; simp [p,hl])]
-    simp only [harmonic,Rat.cast_sum,Rat.cast_inv,Rat.cast_natCast,Finset.sum_div]
-    apply Finset.sum_congr rfl
-    intro l hl
-    simp only [p,ite_eq_left (Finset.mem_range.mp hl)]
-    ring
-  have hh := reciprocal_le_h hp
-  rw [he,hMc] at hh
-  exact hh
-
-lemma harmonic_lower (m : ℕ) (hm : 0 < m) :
-    Real.log m+1/2 ≤ (harmonic m:ℝ) := by
-  have hh := Real.one_half_lt_eulerMascheroniConstant.trans
-    (Real.eulerMascheroniConstant_lt_eulerMascheroniSeq' m)
-  simp only [Real.eulerMascheroniSeq',ite_eq_right (ne_of_gt hm)] at hh
-  linarith
-
-lemma h_log_lower (j : ℕ) :
-    ((j:ℝ)*Real.log 2+1/2)/(2:ℝ)^j ≤ h (j+1) := by
-  have hh := harmonic_lower (2^j) (by positivity)
-  rw [Nat.cast_pow,Nat.cast_ofNat,Real.log_pow] at hh
-  exact (div_le_div_of_nonneg_right hh (by positivity)).trans (harmonic_le_h j)
-
-noncomputable def unusedTail (d : ℕ) : ℝ :=
-  (1/2:ℝ)*∑' j : ℕ, h (d+j+1)
-
-lemma unusedTail_nonneg (d : ℕ) : 0 ≤ unusedTail d := by
-  apply mul_nonneg (by norm_num)
-  exact tsum_nonneg (fun j => h_nonneg _)
-
-lemma unusedTail_lower (d : ℕ) :
-    ((d+1:ℝ)*Real.log 2+1/2)/(2:ℝ)^d ≤ unusedTail d := by
-  have hg := hasSum_geometric_of_norm_lt_one (show ‖(1/2:ℝ)‖ < 1 by norm_num)
-  have hj := hasSum_coe_mul_geometric_of_norm_lt_one (show ‖(1/2:ℝ)‖ < 1 by norm_num)
-  norm_num at hg hj
-  have hs : HasSum (fun j : ℕ => (((d+j:ℕ):ℝ)*Real.log 2+1/2)/(2:ℝ)^(d+j))
-      (2*((d+1:ℝ)*Real.log 2+1/2)/(2:ℝ)^d) := by
-    convert! ((hg.mul_left ((d:ℝ)*Real.log 2+1/2)).add
-      (hj.mul_left (Real.log 2))).div_const ((2:ℝ)^d) using 1
-    · funext j
-      simp only [Nat.cast_add,pow_add,div_pow,one_pow]
-      ring
-    · ring
-  have hsum : Summable (fun j : ℕ => h (d+j+1)) := by
-    simpa only [Nat.add_comm d] using (summable_nat_add_iff d).mpr h_summable
-  have hh := hs.summable.tsum_le_tsum (fun j => h_log_lower (d+j)) hsum
-  rw [hs.tsum_eq] at hh
-  unfold unusedTail
-  calc
-    _ = (1/2:ℝ)*(2*((d+1:ℝ)*Real.log 2+1/2)/(2:ℝ)^d) := by ring
-    _ ≤ _ := mul_le_mul_of_nonneg_left hh (by norm_num)
-
-end Thorp.DenseTruncation
-
-namespace Thorp
-
-lemma partnerMean_budget_strong (d : ℕ) (A : Finset (Card d)) (hA : 0 < A.card) :
-    2 * (∑ j ∈ Finset.range d, partnerMean d A j / (2:ℝ)^(j+1)) ≤
-      (A.card:ℝ)*(A.card-1)/(2:ℝ)^d := by
-  classical
-  rw [←Fin.sum_univ_eq_sum_range]
-  simp_rw [partnerMean_rate]
-  rw [←Finset.sum_div]
-  have he : (∑ j : Fin d, (separatedCount A j.rev:ℝ)) = ∑ j : Fin d, (separatedCount A j:ℝ) := by
-    exact Equiv.sum_comp (Fin.revPerm : Equiv.Perm (Fin d)) (fun j : Fin d => (separatedCount A j : ℝ))
-  rw [he,←mul_div_assoc]
-  apply div_le_div_of_nonneg_right _ (by positivity)
-  have hh : (2:ℝ)*(∑ i : Fin d,(separatedCount A i:ℝ)) ≤
-      (A.card:ℝ)*((A.card-1:ℕ):ℝ) := by exact_mod_cast separated_budget_strong A
-  simpa only [Nat.cast_sub (show 1 ≤ A.card by omega),Nat.cast_one] using hh
-
-lemma partnerAllocation_range_budget (d : ℕ) (A : Finset (Card d)) (hA : 0 < A.card) :
-    (∑ j ∈ Finset.range d, partnerAllocation d A j/(2:ℝ)^(j+1)) ≤
-      ((A.card:ℝ)-1)/(2:ℝ)^d := by
-  have hp : (0:ℝ) < A.card := Nat.cast_pos.mpr hA
-  have he (j : ℕ) : partnerAllocation d A j/(2:ℝ)^(j+1) =
-      (2*partnerMean d A j/(2:ℝ)^(j+1))/A.card := by unfold partnerAllocation; ring
-  simp_rw [he]
-  rw [←Finset.sum_div]
-  simp_rw [mul_div_assoc]
-  rw [←Finset.mul_sum]
-  apply (div_le_div_of_nonneg_right (partnerMean_budget_strong d A hA) hp.le).trans
-  exact le_of_eq (by field_simp)
-
-lemma dyadic_weights_hasSum (d : ℕ) :
-    HasSum (fun j : ℕ => 1/(2:ℝ)^(d+j+1)) (1/(2:ℝ)^d) := by
-  have hg := hasSum_geometric_of_norm_lt_one (show ‖(1/2:ℝ)‖ < 1 by norm_num)
-  norm_num at hg
-  convert! hg.div_const ((2:ℝ)^(d+1)) using 1
-  · funext j
-    simp only [pow_add,div_pow,one_pow,pow_one]
-    ring
-  · simp only [pow_succ]
-    field_simp
-
-noncomputable def augmentedAllocation (d : ℕ) (A : Finset (Card d)) (j : ℕ) : ℝ :=
-  if j < d then partnerAllocation d A j else 1
-
-lemma augmentedAllocation_bounds (d : ℕ) (A : Finset (Card d)) (hA : 0 < A.card) (j : ℕ) :
-    0 ≤ augmentedAllocation d A j ∧ augmentedAllocation d A j ≤ 1 := by
-  unfold augmentedAllocation
-  split_ifs
-  · exact (partnerAllocation_admissible d A hA).1 j
-  · norm_num
-
-lemma augmentedAllocation_admissible (d : ℕ) (A : Finset (Card d)) (hA : 0 < A.card) :
-    DenseTruncation.AdmissibleAllocation ((A.card:ℝ)/(2:ℝ)^d) (augmentedAllocation d A) := by
-  classical
-  refine ⟨augmentedAllocation_bounds d A hA,?_⟩
-  have hs : Summable (fun j => augmentedAllocation d A j/(2:ℝ)^(j+1)) := by
-    apply Summable.of_nonneg_of_le (fun j => div_nonneg (augmentedAllocation_bounds d A hA j).1
-      (by positivity)) (fun j => div_le_div_of_nonneg_right (augmentedAllocation_bounds d A hA j).2
-      (by positivity))
-    simpa only [Nat.zero_add] using (dyadic_weights_hasSum 0).summable
-  have hh := hs.sum_add_tsum_nat_add d
-  have ht : (∑' j, augmentedAllocation d A (j+d)/(2:ℝ)^(j+d+1)) = 1/(2:ℝ)^d := by
-    have he (j : ℕ) : augmentedAllocation d A (j+d)/(2:ℝ)^(j+d+1) = 1/(2:ℝ)^(d+j+1) := by
-      rw [augmentedAllocation,ite_eq_right (show ¬j+d<d by omega)]
-      simp only [Nat.add_comm j d]
-    simp_rw [he]
-    exact (dyadic_weights_hasSum d).tsum_eq
-  have hf : (∑ j ∈ Finset.range d, augmentedAllocation d A j/(2:ℝ)^(j+1)) =
-      ∑ j ∈ Finset.range d, partnerAllocation d A j/(2:ℝ)^(j+1) := by
-    exact Finset.sum_congr rfl (fun j hj => by rw [augmentedAllocation,ite_eq_left (Finset.mem_range.mp hj)])
-  rw [←hh,ht,hf]
-  apply (add_le_add (partnerAllocation_range_budget d A hA) (le_refl ((1:ℝ)/2^d))).trans
-  exact le_of_eq (by ring)
-
-lemma partner_weighted_sum_gap (d H : ℕ) (A : Finset (Card d)) (hA : 0 < A.card) :
-    (∑ j ∈ Finset.range H, DenseTruncation.h (j+1)*partnerMean d A j) ≤
-      A.card * (DenseTruncation.H ((A.card:ℝ)/(2:ℝ)^d)-DenseTruncation.unusedTail d) := by
-  classical
-  have ha := augmentedAllocation_admissible d A hA
-  have hs := DenseTruncation.allocation_value_summable ha
-  have HH := DenseTruncation.allocation_le_H ha
-  have he := hs.sum_add_tsum_nat_add d
-  have ht : (∑' j, DenseTruncation.h (j+d+1)*augmentedAllocation d A (j+d)) =
-      ∑' j, DenseTruncation.h (d+j+1) := by
-    congr 1
-    funext j
-    rw [augmentedAllocation,ite_eq_right (show ¬j+d<d by omega)]
-    simp only [mul_one,Nat.add_comm j d]
-  rw [←he,ht] at HH
-  have hf : (∑ j ∈ Finset.range d, DenseTruncation.h (j+1)*augmentedAllocation d A j) =
-      (2/(A.card:ℝ)) * ∑ j ∈ Finset.range d, DenseTruncation.h (j+1)*partnerMean d A j := by
-    rw [Finset.mul_sum]
-    apply Finset.sum_congr rfl
-    intro j hj
-    rw [augmentedAllocation,ite_eq_left (Finset.mem_range.mp hj),partnerAllocation]
-    ring
-  rw [hf] at HH
-  have hz (j : ℕ) (hj : j ∉ Finset.range d) :
-      DenseTruncation.h (j+1)*partnerMean d A j = 0 := by
-    have hj' : ¬j<d := by simpa only [Finset.mem_range] using hj
-    simp only [partnerMean,dite_eq_right hj',mul_zero]
-  have hsum : Summable (fun j => DenseTruncation.h (j+1)*partnerMean d A j) :=
-    (hasSum_sum_of_ne_finset_zero hz : HasSum _
-      (∑ j ∈ Finset.range d, DenseTruncation.h (j+1)*partnerMean d A j)).summable
-  have hlow : (∑ j ∈ Finset.range H, DenseTruncation.h (j+1)*partnerMean d A j) ≤
-      ∑ j ∈ Finset.range d, DenseTruncation.h (j+1)*partnerMean d A j := by
-    calc
-      _ ≤ ∑' j, DenseTruncation.h (j+1)*partnerMean d A j :=
-        hsum.sum_le_tsum _ (fun j _ => mul_nonneg (DenseTruncation.h_nonneg _)
-          (partnerMean_nonneg _ _ _))
-      _ = _ := tsum_eq_sum hz
-  have hp : (0:ℝ) < A.card := Nat.cast_pos.mpr hA
-  have HH' := mul_le_mul_of_nonneg_left HH hp.le
-  have HE : (A.card:ℝ)*((1/2:ℝ)*((2/(A.card:ℝ))*
-      (∑ j ∈ Finset.range d, DenseTruncation.h (j+1)*partnerMean d A j)+
-      ∑' j, DenseTruncation.h (d+j+1))) =
-      (∑ j ∈ Finset.range d, DenseTruncation.h (j+1)*partnerMean d A j)+
-      (A.card:ℝ)*((1/2:ℝ)*∑' j, DenseTruncation.h (d+j+1)) := by
-    field_simp
-  rw [HE] at HH'
-  unfold DenseTruncation.unusedTail
-  exact hlow.trans (by nlinarith [HH'])
-
-
-lemma palindrome_low_dense_mgf_gap (H : ℕ) (hH : 0 < H) (δ τ : ℝ) (hδ : 0 ≤ δ) (ht : 0 ≤ τ)
-    (hτ : ∀ j < H, Real.exp (denseLevelCoefficient j τ)-1 ≤ τ*(DenseTruncation.h (j+1)+δ))
-    (d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (hι : 0 < Fintype.card ι) :
-    finiteMean (fun ω : BenesCoins d => Real.exp ((τ/H)*palindromeLowCost H d e ω)) ≤
-      Real.exp ((τ/H)*Fintype.card ι*
-        (DenseTruncation.H ((Fintype.card ι:ℝ)/(2:ℝ)^d)-DenseTruncation.unusedTail d+H*δ/2)) := by
-  classical
-  let A := Finset.univ.image e
-  have hA : A.card = Fintype.card ι := by simp [A, Finset.card_image_of_injective _ e.injective]
-  have hn : (H:ℝ) ≠ 0 := by exact_mod_cast (ne_of_gt hH)
-  have he : (H:ℝ)*(τ/H) = τ := by field_simp
-  let : Nonempty (Fin H) := Fin.pos_iff_nonempty.mp hH
-  have hh := finiteMean_exp_sum_le
-    (fun j : Fin H => fun ω : BenesCoins d => (palindromeLevelCost (j+1) d e ω:ℝ))
-    (fun j : Fin H => (DenseTruncation.h (j+1)+δ)*partnerMean d A j) (τ/H) (by
-      intro j
-      simp only [Fintype.card_fin,he]
-      apply (palindrome_level_dense_mgf_all d j e τ ht).trans
-      apply Real.exp_le_exp.mpr
-      simpa only [mul_assoc] using mul_le_mul_of_nonneg_right (hτ j j.isLt)
-        (partnerMean_nonneg d A j))
-  have hsum : (∑ j : Fin H, (DenseTruncation.h (j+1)+δ)*partnerMean d A j) ≤
-      Fintype.card ι*(DenseTruncation.H ((Fintype.card ι:ℝ)/(2:ℝ)^d)-DenseTruncation.unusedTail d+H*δ/2) := by
-    rw [Fin.sum_univ_eq_sum_range (fun j => (DenseTruncation.h (j+1)+δ)*partnerMean d A j)]
-    simp_rw [add_mul]
-    rw [Finset.sum_add_distrib]
-    have hm := partner_weighted_sum_gap d H A (by omega)
-    rw [hA] at hm
-    have hs : (∑ j ∈ Finset.range H, δ*partnerMean d A j) ≤ (H:ℝ)*δ*(Fintype.card ι)/2 := by
-      calc
-        _ ≤ ∑ _j ∈ Finset.range H, δ*((Fintype.card ι:ℝ)/2) := by
-          apply Finset.sum_le_sum
-          intro j _
-          apply mul_le_mul_of_nonneg_left _ hδ
-          have hg := partnerMean_twice_le d A j
-          rw [hA] at hg
-          linarith
-        _ = _ := by simp only [Finset.sum_const,Finset.card_range,nsmul_eq_mul]; ring
-    nlinarith
-  have hh' := hh.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left hsum (by positivity)))
-  have hl (ω : BenesCoins d) :
-      (∑ j : Fin H, (palindromeLevelCost (j+1) d e ω : ℝ)) =
-        ∑ j ∈ Finset.range H, (palindromeLevelCost (j+1) d e ω : ℝ) :=
-    Fin.sum_univ_eq_sum_range (fun j => (palindromeLevelCost (j+1) d e ω : ℝ)) H
-  simpa only [palindromeLowCost_sum,Nat.cast_sum,hl,mul_assoc] using hh'
-
-
-lemma palindrome_dense_mgf (ε : ℝ) (hε : 0 < ε) :
-    ∃ t : ℝ, 0 < t ∧ ∀ d : ℕ, ∀ {ι : Type} [Fintype ι],
-      ∀ e : ι ↪ Card d, 0 < Fintype.card ι →
-      finiteMean (fun ω : BenesCoins d => Real.exp (t*palindromeCost d e ω)) ≤
-        Real.exp (t*Fintype.card ι*(DenseTruncation.H ((Fintype.card ι:ℝ)/(2:ℝ)^d)-
-          DenseTruncation.unusedTail d+ε)) := by
-  obtain ⟨H,hH,hdecay⟩ := heightDecay_exists (show 0 < ε/2 by positivity)
-  have hHr : (0:ℝ) < H := Nat.cast_pos.mpr hH
-  let δ := ε/H
-  have hδ : 0 < δ := div_pos hε hHr
-  obtain ⟨τ,ht,hτb,hτ⟩ := dense_coefficient_small H δ hδ
-  let t := (τ/H)/2
-  have htp : 0 < t := by dsimp [t]; positivity
-  have hσb : τ/H ≤ 1/500 := by
-    apply (div_le_self ht.le (by exact_mod_cast hH)).trans hτb
-  refine ⟨t,htp,?_⟩
-  intro d ι _ e he
-  have hlow := palindrome_low_dense_mgf_gap H hH δ τ hδ.le ht.le hτ d e he
-  have hhigh := palindrome_high_exp_mgf H d d hH e (τ/H) (by positivity) hσb
-  have hex (k : ℝ) : (Real.exp (t*k))^2 = Real.exp ((τ/H)*k) := by
-    rw [←Real.exp_nat_mul]
-    congr 1
-    dsimp [t]
-    ring
-  have hh := finiteMean_mul_exp_le
-    (fun ω : BenesCoins d => Real.exp (t*palindromeLowCost H d e ω))
-    (fun ω : BenesCoins d => Real.exp (t*(palindromeFamilyCost H d d e ω+
-      palindromeFamilyCost (2*H) d d e ω)))
-    ((τ/H)*Fintype.card ι*(DenseTruncation.H ((Fintype.card ι:ℝ)/(2:ℝ)^d)-
-      DenseTruncation.unusedTail d+H*δ/2))
-    ((τ/H)*Fintype.card ι*(500*heightDecay H))
-    (by simpa only [hex] using hlow)
-    (by simpa only [hex,Nat.cast_add] using hhigh)
-  have hd : d ≤ H*4^d := by
-    have hg := pow_four_ge d
-    nlinarith
-  have hcost (ω : BenesCoins d) : palindromeCost d e ω = palindromeLowCost H d e ω+
-      (palindromeFamilyCost H d d e ω+palindromeFamilyCost (2*H) d d e ω) := by
-    have hc := palindromeCost_family H d d e ω
-    rw [palindromeLowCost_eq _ _ hd] at hc
-    simpa only [Nat.add_assoc] using hc.symm
-  have hf (ω : BenesCoins d) : Real.exp (t*palindromeCost d e ω) =
-      Real.exp (t*palindromeLowCost H d e ω)*
-        Real.exp (t*(palindromeFamilyCost H d d e ω+palindromeFamilyCost (2*H) d d e ω)) := by
-    rw [hcost,Nat.cast_add,Nat.cast_add,mul_add,Real.exp_add]
-  simp_rw [←hf] at hh
-  apply hh.trans
-  apply Real.exp_le_exp.mpr
-  have heδ : (H:ℝ)*δ/2=ε/2 := by dsimp [δ]; field_simp
-  rw [heδ]
-  have hc : (ε/2+500*heightDecay H) ≤ ε := by linarith
-  have hm := mul_le_mul_of_nonneg_left hc (show 0 ≤ t*(Fintype.card ι) by positivity)
-  dsimp [t] at hm ⊢
-  nlinarith [hm]
-
-end Thorp
-
-open scoped BigOperators
-namespace Thorp.DenseTruncation
-
-noncomputable def stirlingRemainder (n : ℕ) : ℝ :=
-  Real.log (n.factorial:ℝ) - n*Real.log n+n
-
-lemma stirlingRemainder_zero : stirlingRemainder 0 = 0 := by
-  simp [stirlingRemainder]
-
-lemma stirlingRemainder_step (n : ℕ) :
-    stirlingRemainder (n+1)-stirlingRemainder n =
-      1-(n:ℝ)*(Real.log (n+1)-Real.log n) := by
-  have hn : (0:ℝ) < n+1 := by positivity
-  have hf : (0:ℝ) < n.factorial := Nat.cast_pos.mpr (Nat.factorial_pos n)
-  rw [stirlingRemainder,stirlingRemainder,Nat.factorial_succ,Nat.cast_mul,Nat.cast_add,Nat.cast_one,
-    Real.log_mul (ne_of_gt hn) (ne_of_gt hf)]
-  ring
-
-lemma log_increment_mul_monotone :
-    Monotone (fun n : ℕ => (n:ℝ)*(Real.log (n+1)-Real.log n)) := by
-  intro m n hmn
-  have hm : (0:ℝ) ≤ m := Nat.cast_nonneg _
-  have hn : (0:ℝ) ≤ n := Nat.cast_nonneg _
-  have hmn' : (m:ℝ) ≤ n := by exact_mod_cast hmn
-  by_cases hm0 : m = 0
-  · subst m
-    simp only [Nat.cast_zero,zero_mul]
-    by_cases hn0 : n = 0
-    · simp [hn0]
-    · have hnpos : (0:ℝ) < n := Nat.cast_pos.mpr (Nat.pos_of_ne_zero hn0)
-      exact mul_nonneg hn (sub_nonneg.mpr (Real.log_le_log hnpos (by linarith)))
-  have hmpos : (0:ℝ) < m := Nat.cast_pos.mpr (Nat.pos_of_ne_zero hm0)
-  have hnpos : (0:ℝ) < n := lt_of_lt_of_le hmpos hmn'
-  have ha : 0 ≤ (m:ℝ)/n := div_nonneg hm hn
-  have hb : 0 ≤ 1-(m:ℝ)/n := by rw [sub_nonneg,div_le_one hnpos]; exact hmn'
-  have hc := strictConcaveOn_log_Ioi.concaveOn.2
-    (show (m+1:ℝ)/m ∈ Set.Ioi 0 by exact div_pos (by positivity) hmpos)
-    (show (1:ℝ) ∈ Set.Ioi 0 by norm_num) ha hb
-    (show (m:ℝ)/n+(1-(m:ℝ)/n) = 1 by ring)
-  have he : (m:ℝ)/n*((m+1)/m)+(1-(m:ℝ)/n)*1 = (n+1)/n := by
-    field_simp
-    ring
-  simp only [smul_eq_mul,Real.log_one,mul_zero,add_zero,he] at hc
-  rw [Real.log_div (by positivity : (m+1:ℝ) ≠ 0) (ne_of_gt hmpos),
-    Real.log_div (by positivity : (n+1:ℝ) ≠ 0) (ne_of_gt hnpos)] at hc
-  have hc' := mul_le_mul_of_nonneg_left hc hn
-  have he' : (n:ℝ)*((m:ℝ)/n*(Real.log (m+1)-Real.log m)) =
-      m*(Real.log (m+1)-Real.log m) := by field_simp
-  simpa only [he'] using hc'
-
-lemma stirlingRemainder_increment_antitone :
-    Antitone (fun n : ℕ => stirlingRemainder (n+1)-stirlingRemainder n) := by
-  intro m n hmn
-  simp only [stirlingRemainder_step]
-  linarith [log_increment_mul_monotone hmn]
-
-lemma stirlingRemainder_upper {n : ℕ} (hn : 0 < n) :
-    stirlingRemainder n ≤ 1+Real.log n/2 := by
-  obtain ⟨m,rfl⟩ := Nat.exists_eq_succ_of_ne_zero (ne_of_gt hn)
-  have hm : (0:ℝ) < m+1 := by positivity
-  have he : Real.log (2*((m+1):ℝ)) = Real.log 2+Real.log (m+1) :=
-    Real.log_mul (by norm_num) (ne_of_gt hm)
-  have hh := Stirling.log_stirlingSeq'_antitone (show 0 ≤ m by omega)
-  change Real.log (Stirling.stirlingSeq (m+1)) ≤ Real.log (Stirling.stirlingSeq 1) at hh
-  rw [Stirling.log_stirlingSeq_formula,Stirling.log_stirlingSeq_formula] at hh
-  simp only [Nat.cast_add,Nat.cast_one,he,Nat.factorial_one,Real.log_one,mul_one,
-    Real.log_div (ne_of_gt hm) (ne_of_gt (Real.exp_pos 1)),Real.log_exp,
-    Real.log_div (by norm_num : (1:ℝ) ≠ 0) (ne_of_gt (Real.exp_pos 1))] at hh
-  dsimp [stirlingRemainder]
-  push_cast
-  linarith
-
-lemma stirlingRemainder_sum (n : ℕ) :
-    (∑ i ∈ Finset.range n, (stirlingRemainder (i+1)-stirlingRemainder i)) =
-      stirlingRemainder n := by
-  induction n with
-  | zero => simp [stirlingRemainder_zero]
-  | succ n ih => rw [Finset.sum_range_succ,ih]; ring
-
-lemma stirlingRemainder_step_bound (n : ℕ) :
-    (n:ℝ)*(stirlingRemainder (n+1)-stirlingRemainder n) ≤ stirlingRemainder n := by
-  calc
-    _ = ∑ _i ∈ Finset.range n, (stirlingRemainder (n+1)-stirlingRemainder n) := by
-      simp only [Finset.sum_const,Finset.card_range,nsmul_eq_mul]
-    _ ≤ ∑ i ∈ Finset.range n, (stirlingRemainder (i+1)-stirlingRemainder i) := by
-      exact Finset.sum_le_sum fun i hi => stirlingRemainder_increment_antitone
-        (Nat.le_of_lt (Finset.mem_range.mp hi))
-    _ = _ := stirlingRemainder_sum n
-
-lemma stirlingRemainder_average_antitone :
-    Antitone (fun n : ℕ => stirlingRemainder (n+1)/(n+1)) := by
-  apply antitone_nat_of_succ_le
-  intro n
-  have h₁ : (0:ℝ) < n+1 := by positivity
-  have h₂ : (0:ℝ) < n+1+1 := by positivity
-  push_cast
-  rw [div_le_div_iff₀ h₂ h₁]
-  have hh := stirlingRemainder_step_bound (n+1)
-  push_cast at hh ⊢
-  nlinarith
-
-lemma stirlingRemainder_difference_bound {n r : ℕ} (hn : 0 < n) (hr : r ≤ n) :
-    stirlingRemainder n-stirlingRemainder (n-r) ≤ (r:ℝ)/n*stirlingRemainder n := by
-  have hn' : (0:ℝ) < n := Nat.cast_pos.mpr hn
-  have he : (n-r:ℕ)+r=n := Nat.sub_add_cancel hr
-  by_cases hm : n-r=0
-  · have hn_eq : n=r := by omega
-    subst n
-    simp [stirlingRemainder_zero,div_self (ne_of_gt hn')]
-  · obtain ⟨m,hm'⟩ := Nat.exists_eq_succ_of_ne_zero hm
-    obtain ⟨q,hq⟩ := Nat.exists_eq_succ_of_ne_zero (ne_of_gt hn)
-    have hmq : m ≤ q := by omega
-    have hh := stirlingRemainder_average_antitone hmq
-    have hposm : (0:ℝ) < m+1 := by positivity
-    have hposq : (0:ℝ) < q+1 := by positivity
-    rw [div_le_div_iff₀ hposq hposm] at hh
-    have he' : (m:ℝ)+1+r=q+1 := by exact_mod_cast (show m+1+r=q+1 by omega)
-    rw [hm',hq]
-    push_cast
-    rw [div_mul_eq_mul_div,le_div_iff₀ hposq]
-    simp only [Nat.succ_eq_add_one] at *
-    have heq : (m:ℝ)+1 = (q+1:ℝ)-r := by linarith
-    rw [heq] at hh
-    nlinarith
-
-lemma stirlingRemainder_dyadic (d : ℕ) (hd : 1 ≤ d) :
-    stirlingRemainder (2^d)/Real.log 2 ≤ (d+1:ℝ)*Real.log 2+1/2 := by
-  have hl : 0 < Real.log 2 := Real.log_pos (by norm_num)
-  have hlog : (6931/10000:ℝ) < Real.log 2 := lt_trans (by norm_num) Real.log_two_gt_d9
-  rw [div_le_iff₀ hl]
-  by_cases hd1 : d=1
-  · subst d
-    norm_num [stirlingRemainder]
-    nlinarith [sq_nonneg (Real.log 2-6931/10000)]
-  · have hd2 : (2:ℝ) ≤ d := by exact_mod_cast (show 2 ≤ d by omega)
-    have hh := stirlingRemainder_upper (n := 2^d) (by positivity)
-    rw [Nat.cast_pow,Nat.cast_ofNat,Real.log_pow] at hh
-    have hq : 0 ≤ Real.log 2*Real.log 2-Real.log 2/2 := by
-      nlinarith [sq_nonneg (Real.log 2-6931/10000)]
-    have hprod := mul_nonneg (sub_nonneg.mpr hd2) hq
-    nlinarith [sq_nonneg (Real.log 2-6931/10000)]
-
-lemma factorial_error_identity {n r : ℕ} (hn : 0 < n) (hr : r ≤ n) :
-    Real.log ((n.descFactorial r:ℝ)/(n:ℝ)^r) =
-      stirlingRemainder n-stirlingRemainder (n-r)-(r:ℝ)-
-        (n-r:ℕ)*Real.log (1-(r:ℝ)/n) := by
-  have hn' : (0:ℝ) < n := Nat.cast_pos.mpr hn
-  have hm : (0:ℝ) < (n-r).factorial := Nat.cast_pos.mpr (Nat.factorial_pos _)
-  have hd : (0:ℝ) < n.descFactorial r := Nat.cast_pos.mpr (Nat.descFactorial_pos.mpr hr)
-  have hf : ((n-r).factorial:ℝ)*(n.descFactorial r:ℝ) = n.factorial :=
-    by exact_mod_cast Nat.factorial_mul_descFactorial hr
-  have hh := congrArg Real.log hf
-  rw [Real.log_mul (ne_of_gt hm) (ne_of_gt hd)] at hh
-  rw [Real.log_div (ne_of_gt hd) (ne_of_gt (pow_pos hn' r)),Real.log_pow]
-  unfold stirlingRemainder
-  by_cases hz : n-r=0
-  · have hrn : r=n := by omega
-    subst r
-    simp only [Nat.sub_self,Nat.factorial_zero,Nat.cast_one,Real.log_one,Nat.cast_zero,
-      zero_mul,add_zero,sub_zero,Nat.descFactorial_self]
-    ring
-  · have hm' : (0:ℝ) < (n-r:ℕ) := Nat.cast_pos.mpr (Nat.pos_of_ne_zero hz)
-    have he : ((n-r:ℕ):ℝ) = (n:ℝ)*(1-(r:ℝ)/n) := by
-      rw [Nat.cast_sub hr]
-      field_simp
-    have hρ : 0 < 1-(r:ℝ)/n := by
-      rw [←mul_pos_iff_of_pos_left hn',←he]
-      exact hm'
-    have hl : Real.log ((n-r:ℕ):ℝ) = Real.log n+Real.log (1-(r:ℝ)/n) := by
-      rw [he, Real.log_mul (ne_of_gt hn') (ne_of_gt hρ)]
-    rw [hl,Nat.cast_sub hr]
-    nlinarith [hh]
-
-
-lemma normalization_le (d r : ℕ) (hd : 1 ≤ d) (hr : 1 ≤ r) (hrn : r ≤ 2^d) :
-    Real.log (((2^d).descFactorial r:ℝ)/((2:ℝ)^d)^r)/Real.log 2 ≤
-      (r:ℝ)*(entropyCorrection ((r:ℝ)/(2:ℝ)^d)+unusedTail d) := by
-  have hnp : 0 < (2:ℕ)^d := by positivity
-  have hnr : (0:ℝ) < (2:ℝ)^d := by positivity
-  have hr' : (0:ℝ) < r := by exact_mod_cast hr
-  have hl : 0 < Real.log 2 := Real.log_pos (by norm_num)
-  have hh := stirlingRemainder_difference_bound hnp hrn
-  simp only [Nat.cast_pow,Nat.cast_ofNat] at hh
-  have hg := (stirlingRemainder_dyadic d hd).trans
-    ((div_le_iff₀ hnr).mp (unusedTail_lower d))
-  have he : (stirlingRemainder (2^d)-stirlingRemainder (2^d-r))/Real.log 2 ≤
-      (r:ℝ)*unusedTail d := by
-    calc
-      _ ≤ ((r:ℝ)/(2:ℝ)^d*stirlingRemainder (2^d))/Real.log 2 :=
-        div_le_div_of_nonneg_right hh hl.le
-      _ = ((r:ℝ)/(2:ℝ)^d)*(stirlingRemainder (2^d)/Real.log 2) := by ring
-      _ ≤ ((r:ℝ)/(2:ℝ)^d)*(unusedTail d*(2:ℝ)^d) :=
-        mul_le_mul_of_nonneg_left hg (by positivity)
-      _ = _ := by field_simp
-  have hid := factorial_error_identity hnp hrn
-  simp only [Nat.cast_pow,Nat.cast_ofNat] at hid
-  have heq : Real.log (((2^d).descFactorial r:ℝ)/((2:ℝ)^d)^r)/Real.log 2 =
-      (r:ℝ)*entropyCorrection ((r:ℝ)/(2:ℝ)^d)+
-      (stirlingRemainder (2^d)-stirlingRemainder (2^d-r))/Real.log 2 := by
-    rw [hid,entropyCorrection,Nat.cast_sub hrn]
-    simp only [Nat.cast_pow,Nat.cast_ofNat]
-    field_simp
-    ring
-  rw [heq]
-  nlinarith
-
-
-theorem dense_truncation : MainStatement := by
-  classical
-  intro ρ hρ ε hε
-  obtain ⟨t,ht,hm⟩ := palindrome_dense_mgf (ε/2) (by positivity)
-  refine ⟨t*ε/2, by positivity, ?_⟩
-  intro d r hd hr hρr x
-  have hrn : r ≤ 2^d := by
-    have hh := Fintype.card_le_of_injective x x.injective
-    simpa only [Fintype.card_fin,card_positions] using hh
-  have hm' := hm d x (by simpa only [Fintype.card_fin] using (show 0 < r by omega))
-  simp only [Fintype.card_fin,hρr] at hm'
-  have hn := normalization_le d r hd hr hrn
-  rw [hρr] at hn
-  let a : ℝ := r*(H ρ+entropyCorrection ρ+ε)
-  let b : ℝ := Real.log (((2^d).descFactorial r:ℝ)/((2:ℝ)^d)^r)/Real.log 2
-  have hp (ω : BenesCoins d) :
-      (if a < densityExponent d r x ω then (1:ℝ) else 0) ≤
-      Real.exp (t*palindromeCost d x ω)*Real.exp (t*(b-a)) := by
-    split_ifs with h
-    · rw [←Real.exp_add]
-      apply Real.one_le_exp_iff.mpr
-      have hh : 0 ≤ t*((palindromeCost d x ω:ℝ)+b-a) := by
-        apply mul_nonneg ht.le
-        exact sub_nonneg.mpr (le_of_lt h)
-      nlinarith
-    · positivity
-  have he : t*(r:ℝ)*(H ρ-unusedTail d+ε/2)+t*(b-a) ≤ -(t*ε/2)*r := by
-    have hh := mul_le_mul_of_nonneg_left hn ht.le
-    dsimp [b,a]
-    nlinarith [hh]
-  calc
-    _ ≤ finiteMean (fun ω : BenesCoins d =>
-        Real.exp (t*palindromeCost d x ω)*Real.exp (t*(b-a))) := finiteMean_mono hp
-    _ = finiteMean (fun ω : BenesCoins d => Real.exp (t*palindromeCost d x ω))*
-        Real.exp (t*(b-a)) := finiteMean_mul_const _ _
-    _ ≤ Real.exp (t*(r:ℝ)*(H ρ-unusedTail d+ε/2))*Real.exp (t*(b-a)) :=
-      mul_le_mul_of_nonneg_right hm' (Real.exp_nonneg _)
-    _ ≤ Real.exp (-(t*ε/2)*r) := by rw [←Real.exp_add]; exact Real.exp_le_exp.mpr he
-
-end Thorp.DenseTruncation
-
-end ThorpNine.Dense
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Dense.SingleSlotsEquiv
+
+namespace OAI
+
+namespace ThorpNine.Dense
+
+open scoped BigOperators
+namespace Thorp
+
+lemma sharedPartnerMean_nonneg (r s : ℕ) (A : Finset (Card ((s+1)+r))) :
+    0 ≤ sharedPartnerMean r s A := by
+  unfold sharedPartnerMean
+  positivity
+
+lemma sum_suffix_pair_counts (r s : ℕ) (A : Finset (Card ((s+1)+r))) :
+    (∑ y : Card s, ((suffixMarks r (s+1) A (Fin.cons false y)).card +
+      (suffixMarks r (s+1) A (Fin.cons true y)).card)) = A.card := by
+  classical
+  have hh := sum_card_blockMarks ((cardSplit r (s+1)).trans (Equiv.prodComm _ _)) A
+  rw [←Equiv.sum_comp (headTailEquiv s).symm,Fintype.sum_prod_type,Fintype.sum_bool] at hh
+  simpa [suffixMarks,headTailEquiv,Fin.consEquiv,Finset.sum_add_distrib,add_comm] using hh
+
+lemma sharedPartnerMean_twice_le (r s : ℕ) (A : Finset (Card ((s+1)+r))) :
+    2*sharedPartnerMean r s A ≤ A.card := by
+  have hcap (y : Card (s+1)) : (suffixMarks r (s+1) A y).card ≤ 2^r := by
+    simpa only [card_positions] using Finset.card_le_univ (suffixMarks r (s+1) A y)
+  have hp : (0:ℝ) < (2:ℝ)^r := by positivity
+  rw [sharedPartnerMean,←mul_div_assoc,div_le_iff₀ hp,Finset.mul_sum]
+  have hs : (A.card:ℝ) = ∑ y : Card s,
+      ((suffixMarks r (s+1) A (Fin.cons false y)).card +
+        (suffixMarks r (s+1) A (Fin.cons true y)).card : ℝ) := by
+    exact_mod_cast (sum_suffix_pair_counts r s A).symm
+  rw [hs,Finset.sum_mul]
+  apply Finset.sum_le_sum
+  intro y _
+  have h₀ : (suffixMarks r (s+1) A (Fin.cons false y)).card ≤ (2:ℝ)^r := by
+    exact_mod_cast hcap (Fin.cons false y)
+  have h₁ : (suffixMarks r (s+1) A (Fin.cons true y)).card ≤ (2:ℝ)^r := by
+    exact_mod_cast hcap (Fin.cons true y)
+  nlinarith [mul_nonneg (sub_nonneg.mpr h₀)
+    (Nat.cast_nonneg (suffixMarks r (s+1) A (Fin.cons true y)).card),
+    mul_nonneg (sub_nonneg.mpr h₁)
+    (Nat.cast_nonneg (suffixMarks r (s+1) A (Fin.cons false y)).card)]
+
+
+lemma palindrome_level_dense_mgf (r s : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card ((s+1)+r)) (t : ℝ) (ht : 0 ≤ t) :
+    finiteMean (fun ω : BenesCoins ((s+1)+r) =>
+      Real.exp (t * palindromeLevelCost (s+1) ((s+1)+r) e ω)) ≤
+        Real.exp ((Real.exp (denseLevelCoefficient s t)-1) *
+          sharedPartnerMean r s (Finset.univ.image e)) := by
+  classical
+  rw [finiteMean_prod,finiteMean_comm]
+  calc
+    _ ≤ finiteMean (fun X : SwitchIndex ((s+1)+r) → Bool =>
+        Real.exp (denseLevelCoefficient s t * sharedLevel (s+1) ((s+1)+r) e X)) := by
+      apply finiteMean_mono
+      intro X
+      exact palindrome_level_dense_conditional s ((s+1)+r) e X t ht
+    _ = finiteMean (fun hi : Card (s+1) × SwitchIndex r → Bool =>
+        Real.exp (denseLevelCoefficient s t *
+          gridShared r s (inverseButterflyMarks ((s+1)+r) (Finset.univ.image e)
+            (assembleBits r (s+1) (fun _ => false) hi)))) := by
+      rw [finiteMean_split]
+      simp_rw [sharedLevel_partial_marks]
+      rw [finiteMean_const]
+    _ ≤ _ := partial_grid_mgf r s (Finset.univ.image e) _
+        (denseLevelCoefficient_nonneg s ht)
+
+
+lemma cardSplit_snd_apply (r s : ℕ) (x : Card (s+r)) (i : Fin s) :
+    (cardSplit r s x).2 i = x ⟨r+i,by omega⟩ := by
+  induction r with
+  | zero => simp [cardSplit]
+  | succ r ih =>
+    change (cardSplit r s (Fin.tail x)).2 i = _
+    rw [ih]
+    simp only [Fin.tail]
+    congr 1
+    apply Fin.ext
+    simp only [Fin.val_succ]
+    omega
+
+def PairSeparated {d : ℕ} (i : Fin d) (x y : Card d) : Prop :=
+  x i = false ∧ y i = true ∧ ∀ j : Fin d, i < j → x j = y j
+
+lemma PairSeparated.unique {d : ℕ} {i j : Fin d} {x y : Card d}
+    (hi : PairSeparated i x y) (hj : PairSeparated j x y) : i = j := by
+  by_contra h
+  rcases lt_or_gt_of_ne h with h | h
+  · have hh := hi.2.2 j h
+    rw [hj.1,hj.2.1] at hh
+    exact Bool.false_ne_true hh
+  · have hh := hj.2.2 i h
+    rw [hi.1,hi.2.1] at hh
+    exact Bool.false_ne_true hh
+
+lemma PairSeparated.not_reverse {d : ℕ} {i j : Fin d} {x y : Card d}
+    (hi : PairSeparated i x y) : ¬ PairSeparated j y x := by
+  intro hj
+  rcases lt_trichotomy i j with h | h | h
+  · have hh := hi.2.2 j h
+    rw [hj.2.1,hj.1] at hh
+    exact Bool.noConfusion hh
+  · subst j
+    exact Bool.false_ne_true (hi.1.symm.trans hj.2.1)
+  · have hh := hj.2.2 i h
+    rw [hi.1,hi.2.1] at hh
+    exact Bool.noConfusion hh
+
+lemma pairSeparated_split_iff (r s : ℕ) (x y : Card ((s+1)+r)) :
+    PairSeparated ⟨r,by omega⟩ x y ↔
+      ∃ z : Card s, (cardSplit r (s+1) x).2 = Fin.cons false z ∧
+        (cardSplit r (s+1) y).2 = Fin.cons true z := by
+  constructor
+  · intro h
+    refine ⟨Fin.tail (cardSplit r (s+1) x).2, ?_, ?_⟩
+    · have h₀ : (cardSplit r (s+1) x).2 0 = false := by
+        simpa only [cardSplit_snd_apply,Fin.val_zero,Nat.add_zero] using h.1
+      rw [←h₀,Fin.cons_self_tail]
+    · funext k
+      refine Fin.cases ?_ (fun k => ?_) k
+      · simpa only [cardSplit_snd_apply,Fin.cons_zero,Fin.val_zero,Nat.add_zero] using h.2.1
+      · simp only [Fin.cons_succ,Fin.tail,cardSplit_snd_apply]
+        exact (h.2.2 _ (by change r < r + (k.val+1); omega)).symm
+  · rintro ⟨z,hx,hy⟩
+    refine ⟨?_,?_,?_⟩
+    · have hh := congrFun hx 0
+      simpa only [cardSplit_snd_apply,Fin.cons_zero,Fin.val_zero,Nat.add_zero] using hh
+    · have hh := congrFun hy 0
+      simpa only [cardSplit_snd_apply,Fin.cons_zero,Fin.val_zero,Nat.add_zero] using hh
+    · intro j hj
+      change r < j.val at hj
+      have hbound : j.val-r-1 < s := by omega
+      let k : Fin s := ⟨j.val-r-1,hbound⟩
+      have hjk : j = ⟨r+(k.val+1),by omega⟩ := Fin.ext (by dsimp [k]; omega)
+      rw [hjk]
+      have hx' := congrFun hx k.succ
+      have hy' := congrFun hy k.succ
+      simp only [cardSplit_snd_apply,Fin.cons_succ] at hx' hy'
+      exact hx'.trans hy'.symm
+
+
+abbrev SeparatedPair {d : ℕ} (A : Finset (Card d)) (i : Fin d) :=
+  {p : A × A // PairSeparated i p.1.val p.2.val}
+
+noncomputable def separatedCount {d : ℕ} (A : Finset (Card d)) (i : Fin d) : ℕ := by
+  classical
+  exact Fintype.card (SeparatedPair A i)
+
+lemma separated_budget_strong {d : ℕ} (A : Finset (Card d)) :
+    2 * ∑ i : Fin d, separatedCount A i ≤ A.card * (A.card-1) := by
+  classical
+  let F : Bool × (Σ i : Fin d, SeparatedPair A i) → A × A :=
+    fun ⟨b,i,p⟩ => if b then (p.val.2,p.val.1) else p.val
+  have hF : Function.Injective F := by
+    rintro ⟨b,i,p⟩ ⟨c,j,q⟩ h
+    cases b <;> cases c
+    · change p.val = q.val at h
+      have hij : i = j := p.property.unique (by rw [h]; exact q.property)
+      subst j
+      have hpq : p = q := Subtype.ext h
+      subst q
+      rfl
+    · change p.val = (q.val.2,q.val.1) at h
+      have hp := p.property
+      rw [h] at hp
+      exact False.elim (hp.not_reverse q.property)
+    · change (p.val.2,p.val.1) = q.val at h
+      have hq := q.property
+      rw [←h] at hq
+      exact False.elim (hq.not_reverse p.property)
+    · change (p.val.2,p.val.1) = (q.val.2,q.val.1) at h
+      have hpq : p.val = q.val := Prod.ext (congrArg Prod.snd h) (congrArg Prod.fst h)
+      have hij : i = j := p.property.unique (by rw [hpq]; exact q.property)
+      subst j
+      have hpqq : p = q := Subtype.ext hpq
+      subst q
+      rfl
+  have hneq (v : Bool × (Σ i : Fin d, SeparatedPair A i)) : (F v).1 ≠ (F v).2 := by
+    obtain ⟨b,i,p⟩ := v
+    have hp : p.val.1 ≠ p.val.2 := by
+      intro he
+      have hx := p.property.1
+      have hy := p.property.2.1
+      rw [he] at hx
+      exact Bool.false_ne_true (hx.symm.trans hy)
+    cases b <;> simp [F,hp,Ne.symm hp]
+  let G : Bool × (Σ i : Fin d, SeparatedPair A i) → ↥((Finset.univ : Finset A).offDiag) :=
+    fun v => ⟨F v,Finset.mem_offDiag.mpr ⟨Finset.mem_univ _,Finset.mem_univ _,hneq v⟩⟩
+  have hG : Function.Injective G := fun _ _ he => hF (congrArg Subtype.val he)
+  have hh := Fintype.card_le_of_injective G hG
+  simpa only [Fintype.card_prod,Fintype.card_bool,Fintype.card_sigma,
+    Fintype.card_coe,separatedCount,Finset.offDiag_card,Finset.card_univ,
+    Nat.mul_sub_left_distrib,mul_one] using hh
+
+lemma separated_budget {d : ℕ} (A : Finset (Card d)) :
+    2 * ∑ i : Fin d, separatedCount A i ≤ A.card * A.card :=
+  (separated_budget_strong A).trans (Nat.mul_le_mul_left _ (Nat.sub_le _ _))
+
+lemma mem_suffixMarks (r s : ℕ) (A : Finset (Card (s+r))) (y : Card s) (x : Card r) :
+    x ∈ suffixMarks r s A y ↔ (cardSplit r s).symm (x,y) ∈ A := by
+  exact mem_blockMarks _ _ _ _
+
+lemma separatedCount_eq (r s : ℕ) (A : Finset (Card ((s+1)+r))) :
+    separatedCount A ⟨r,by omega⟩ =
+      ∑ y : Card s, (suffixMarks r (s+1) A (Fin.cons false y)).card *
+        (suffixMarks r (s+1) A (Fin.cons true y)).card := by
+  classical
+  let D := Σ y : Card s, ↥(suffixMarks r (s+1) A (Fin.cons false y)) ×
+    ↥(suffixMarks r (s+1) A (Fin.cons true y))
+  let F : D → SeparatedPair A ⟨r,by omega⟩ := fun ⟨y,u,v⟩ =>
+    ⟨(⟨(cardSplit r (s+1)).symm (u.val,Fin.cons false y),
+          (mem_suffixMarks _ _ _ _ _).mp u.property⟩,
+        ⟨(cardSplit r (s+1)).symm (v.val,Fin.cons true y),
+          (mem_suffixMarks _ _ _ _ _).mp v.property⟩),
+      (pairSeparated_split_iff r s _ _).mpr
+        ⟨y,by simp only [Equiv.apply_symm_apply],by simp only [Equiv.apply_symm_apply]⟩⟩
+  have hF : Function.Bijective F := by
+    constructor
+    · rintro ⟨y,u,v⟩ ⟨z,u',v'⟩ he
+      have h₀ := congrArg (fun p => cardSplit r (s+1) p.val.1.val) he
+      have h₁ := congrArg (fun p => cardSplit r (s+1) p.val.2.val) he
+      change cardSplit r (s+1) ((cardSplit r (s+1)).symm (u.val,Fin.cons false y)) =
+        cardSplit r (s+1) ((cardSplit r (s+1)).symm (u'.val,Fin.cons false z)) at h₀
+      change cardSplit r (s+1) ((cardSplit r (s+1)).symm (v.val,Fin.cons true y)) =
+        cardSplit r (s+1) ((cardSplit r (s+1)).symm (v'.val,Fin.cons true z)) at h₁
+      simp only [Equiv.apply_symm_apply] at h₀ h₁
+      have hyz : y = z := by
+        have hh := congrArg (fun p => Fin.tail p.2) h₀
+        simpa only [Fin.tail_cons] using hh
+      subst z
+      have hu : u = u' := Subtype.ext (congrArg Prod.fst h₀)
+      have hv : v = v' := Subtype.ext (congrArg Prod.fst h₁)
+      subst u'; subst v'; rfl
+    · intro p
+      obtain ⟨y,hx,hy⟩ := (pairSeparated_split_iff r s p.val.1.val p.val.2.val).mp p.property
+      have hx' : (cardSplit r (s+1)).symm ((cardSplit r (s+1) p.val.1.val).1,Fin.cons false y) =
+          p.val.1.val := by rw [←hx,Prod.mk.eta,Equiv.symm_apply_apply]
+      have hy' : (cardSplit r (s+1)).symm ((cardSplit r (s+1) p.val.2.val).1,Fin.cons true y) =
+          p.val.2.val := by rw [←hy,Prod.mk.eta,Equiv.symm_apply_apply]
+      refine ⟨⟨y,⟨(cardSplit r (s+1) p.val.1.val).1,?_⟩,
+          ⟨(cardSplit r (s+1) p.val.2.val).1,?_⟩⟩,?_⟩
+      · rw [mem_suffixMarks,hx']; exact p.val.1.property
+      · rw [mem_suffixMarks,hy']; exact p.val.2.property
+      · apply Subtype.ext
+        exact Prod.ext (Subtype.ext hx') (Subtype.ext hy')
+  have hh := Fintype.card_congr (Equiv.ofBijective F hF)
+  simpa only [D,Fintype.card_sigma,Fintype.card_prod,Fintype.card_coe,separatedCount] using hh.symm
+
+
+noncomputable def partnerMean (d : ℕ) (A : Finset (Card d)) (j : ℕ) : ℝ :=
+  if hj : j < d then (separatedCount A ⟨d-j-1,by omega⟩ : ℝ) / (2:ℝ)^(d-j-1) else 0
+
+lemma partnerMean_nonneg (d : ℕ) (A : Finset (Card d)) (j : ℕ) :
+    0 ≤ partnerMean d A j := by
+  unfold partnerMean
+  split <;> positivity
+
+lemma partnerMean_eq (r s : ℕ) (A : Finset (Card ((s+1)+r))) :
+    partnerMean ((s+1)+r) A s = sharedPartnerMean r s A := by
+  classical
+  have hs : s < (s+1)+r := by omega
+  have hr : (s+1+r)-s-1 = r := by omega
+  simp only [partnerMean,dite_eq_left hs,hr,separatedCount_eq,sharedPartnerMean,
+    Nat.cast_sum,Nat.cast_mul]
+
+lemma partnerMean_twice_le (d : ℕ) (A : Finset (Card d)) (j : ℕ) :
+    2*partnerMean d A j ≤ A.card := by
+  by_cases hj : j < d
+  · obtain ⟨r,rfl⟩ := Nat.exists_eq_add_of_le (Nat.succ_le_of_lt hj)
+    rw [partnerMean_eq]
+    exact sharedPartnerMean_twice_le r j A
+  · simp only [partnerMean,dite_eq_right hj,mul_zero,Nat.cast_nonneg]
+
+lemma partnerMean_rate (d : ℕ) (A : Finset (Card d)) (j : Fin d) :
+    partnerMean d A j / (2:ℝ)^(j.val+1) =
+      (separatedCount A j.rev : ℝ) / (2:ℝ)^d := by
+  have hj : j.val < d := j.isLt
+  have hd : (d-j.val-1)+(j.val+1) = d := by omega
+  simp only [partnerMean,dite_eq_left hj,div_div,←pow_add,hd]
+  rfl
+
+lemma partnerMean_budget (d : ℕ) (A : Finset (Card d)) :
+    2 * (∑ j ∈ Finset.range d, partnerMean d A j / (2:ℝ)^(j+1)) ≤
+      (A.card:ℝ)^2/(2:ℝ)^d := by
+  classical
+  rw [←Fin.sum_univ_eq_sum_range]
+  simp_rw [partnerMean_rate]
+  rw [←Finset.sum_div]
+  have he : (∑ j : Fin d, (separatedCount A j.rev:ℝ)) = ∑ j : Fin d, (separatedCount A j:ℝ) := by
+    exact Equiv.sum_comp (Fin.revPerm : Equiv.Perm (Fin d)) (fun j : Fin d => (separatedCount A j : ℝ))
+  rw [he,←mul_div_assoc]
+  apply div_le_div_of_nonneg_right _ (by positivity)
+  exact_mod_cast (by simpa only [pow_two] using separated_budget A)
+
+noncomputable def partnerAllocation (d : ℕ) (A : Finset (Card d)) (j : ℕ) : ℝ :=
+  2*partnerMean d A j/A.card
+
+lemma partnerAllocation_admissible (d : ℕ) (A : Finset (Card d)) (hA : 0 < A.card) :
+    DenseTruncation.AdmissibleAllocation ((A.card:ℝ)/(2:ℝ)^d) (partnerAllocation d A) := by
+  classical
+  have hA' : (0:ℝ) < A.card := by exact_mod_cast hA
+  constructor
+  · intro j
+    constructor
+    · exact div_nonneg (mul_nonneg (by norm_num) (partnerMean_nonneg d A j)) hA'.le
+    · rw [partnerAllocation,div_le_one hA']
+      exact partnerMean_twice_le d A j
+  · have hz (j : ℕ) (hj : j ∉ Finset.range d) :
+        partnerAllocation d A j/(2:ℝ)^(j+1) = 0 := by
+      have h : ¬ j < d := by simpa only [Finset.mem_range] using hj
+      simp only [partnerAllocation,partnerMean,dite_eq_right h,mul_zero,zero_div]
+    rw [tsum_eq_sum (s := Finset.range d) hz]
+    have he (j : ℕ) : partnerAllocation d A j/(2:ℝ)^(j+1) =
+        2/A.card * (partnerMean d A j/(2:ℝ)^(j+1)) := by
+      unfold partnerAllocation
+      ring
+    simp_rw [he]
+    rw [←Finset.mul_sum]
+    have hh := partnerMean_budget d A
+    apply (show 2/(A.card:ℝ) * ∑ j ∈ Finset.range d, partnerMean d A j/(2:ℝ)^(j+1) =
+      (2*(∑ j ∈ Finset.range d,partnerMean d A j/(2:ℝ)^(j+1)))/A.card by ring).le.trans
+    apply (div_le_div_of_nonneg_right hh hA'.le).trans
+    have hne : (A.card:ℝ) ≠ 0 := ne_of_gt hA'
+    exact le_of_eq (by field_simp)
+
+
+lemma partner_weighted_sum_le (d H : ℕ) (A : Finset (Card d)) (hA : 0 < A.card) :
+    (∑ j ∈ Finset.range H, DenseTruncation.h (j+1)*partnerMean d A j) ≤
+      A.card * DenseTruncation.H ((A.card:ℝ)/(2:ℝ)^d) := by
+  have ha := partnerAllocation_admissible d A hA
+  have hh := DenseTruncation.allocation_le_H ha
+  have hA' : (0:ℝ) < A.card := by exact_mod_cast hA
+  have he (j : ℕ) : DenseTruncation.h (j+1)*partnerMean d A j =
+      (A.card:ℝ)/2*(DenseTruncation.h (j+1)*partnerAllocation d A j) := by
+    unfold partnerAllocation
+    field_simp
+  simp_rw [he]
+  rw [←Finset.mul_sum]
+  calc
+    _ ≤ (A.card:ℝ)/2 * ∑' j, DenseTruncation.h (j+1)*partnerAllocation d A j := by
+      apply mul_le_mul_of_nonneg_left _ (by positivity)
+      exact (DenseTruncation.allocation_value_summable ha).sum_le_tsum _
+        (fun j _ => mul_nonneg (DenseTruncation.h_nonneg _) (ha.1 j).1)
+    _ = (A.card:ℝ)*((1/2:ℝ)*∑' j, DenseTruncation.h (j+1)*partnerAllocation d A j) := by ring
+    _ ≤ _ := mul_le_mul_of_nonneg_left hh (Nat.cast_nonneg _)
+
+lemma finiteMean_rpow_le {Ω : Type*} [Fintype Ω] [Nonempty Ω]
+    (f : Ω → ℝ) (hf : ∀ ω, 0 ≤ f ω) {p : ℝ} (hp : 0 ≤ p) (hp1 : p ≤ 1) :
+    finiteMean (fun ω => (f ω)^p) ≤ (finiteMean f)^p := by
+  have hN : (0:ℝ) < Fintype.card Ω := Nat.cast_pos.mpr Fintype.card_pos
+  have hh := (Real.concaveOn_rpow hp hp1).le_map_sum
+    (t := Finset.univ) (w := fun _ : Ω => 1/(Fintype.card Ω:ℝ))
+    (p := f) (fun _ _ => by positivity)
+    (by simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]; field_simp)
+    (fun ω _ => hf ω)
+  simpa only [smul_eq_mul,one_div,←Finset.mul_sum,mul_comm _ (∑ _ : Ω,_),
+    ←div_eq_mul_inv,finiteMean] using hh
+
+lemma finiteMean_exp_scale {Ω : Type*} [Fintype Ω] [Nonempty Ω]
+    (f : Ω → ℝ) (a b t : ℝ) (hb : 0 < b) (ht : 0 ≤ t) (htb : t ≤ b)
+    (h : finiteMean (fun ω => Real.exp (b*f ω)) ≤ Real.exp (b*a)) :
+    finiteMean (fun ω => Real.exp (t*f ω)) ≤ Real.exp (t*a) := by
+  have hp : 0 ≤ t/b := div_nonneg ht hb.le
+  have hp1 : t/b ≤ 1 := (div_le_one hb).mpr htb
+  have hh := (finiteMean_rpow_le (fun ω => Real.exp (b*f ω))
+    (fun _ => (Real.exp_pos _).le) hp hp1).trans
+      (Real.rpow_le_rpow (finiteMean_nonneg (fun _ => (Real.exp_pos _).le)) h hp)
+  have he (x : ℝ) : (Real.exp (b*x))^(t/b) = Real.exp (t*x) := by
+    rw [←Real.exp_mul]
+    congr 1
+    field_simp
+  simpa only [he] using hh
+
+lemma finiteMean_exp_sum_le {Ω I : Type*} [Fintype Ω] [Nonempty Ω]
+    [Fintype I] [Nonempty I] (f : I → Ω → ℝ) (a : I → ℝ) (t : ℝ)
+    (h : ∀ i, finiteMean (fun ω => Real.exp ((Fintype.card I:ℝ)*t*f i ω)) ≤
+      Real.exp ((Fintype.card I:ℝ)*t*a i)) :
+    finiteMean (fun ω => Real.exp (t*∑ i, f i ω)) ≤ Real.exp (t*∑ i, a i) := by
+  have hN : (0:ℝ) < Fintype.card I := Nat.cast_pos.mpr Fintype.card_pos
+  let N : ℝ := Fintype.card I
+  have hc (i : I) : finiteMean (fun ω => Real.exp (N*t*(f i ω-a i))) ≤ 1 := by
+    have he (ω : Ω) : Real.exp (N*t*(f i ω-a i)) =
+        Real.exp (N*t*f i ω) * Real.exp (-(N*t*a i)) := by
+      rw [←Real.exp_add]; congr 1; ring
+    simp_rw [he]
+    rw [finiteMean_mul_const]
+    calc
+      _ ≤ Real.exp (N*t*a i)*Real.exp (-(N*t*a i)) :=
+          mul_le_mul_of_nonneg_right (h i) (Real.exp_pos _).le
+      _ = 1 := by rw [←Real.exp_add,add_neg_cancel,Real.exp_zero]
+  have hj (ω : Ω) : Real.exp (t*(∑ i, (f i ω-a i))) ≤
+      ∑ i, (1/N)*Real.exp (N*t*(f i ω-a i)) := by
+    have hh := convexOn_exp.map_sum_le
+      (t := Finset.univ) (w := fun _ : I => 1/N)
+      (p := fun i => N*t*(f i ω-a i))
+      (fun _ _ => by dsimp [N]; positivity)
+      (by simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]; dsimp [N]; field_simp)
+      (fun _ _ => Set.mem_univ _)
+    have he (i : I) : (1/N)*(N*t*(f i ω-a i)) = t*(f i ω-a i) := by
+      dsimp [N]; field_simp
+    simpa only [smul_eq_mul,he,←Finset.mul_sum] using hh
+  have hm : finiteMean (fun ω => Real.exp (t*(∑ i, (f i ω-a i)))) ≤ 1 := by
+    apply (finiteMean_mono hj).trans
+    rw [finiteMean_sum]
+    calc
+      _ ≤ ∑ _ : I, 1/N := by
+        apply Finset.sum_le_sum
+        intro i _
+        have he (ω : Ω) : 1/N*Real.exp (N*t*(f i ω-a i)) =
+          Real.exp (N*t*(f i ω-a i))*(1/N) := mul_comm _ _
+        simp_rw [he]
+        rw [finiteMean_mul_const]
+        simpa only [one_mul] using mul_le_mul_of_nonneg_right (hc i)
+          (by dsimp [N]; positivity : 0 ≤ 1/N)
+      _ = 1 := by simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]; dsimp [N]; field_simp
+  have he (ω : Ω) : Real.exp (t*∑ i,f i ω) =
+      Real.exp (t*(∑ i, (f i ω-a i)))*Real.exp (t*∑ i,a i) := by
+    rw [←Real.exp_add,Finset.sum_sub_distrib]
+    congr 1; ring
+  simp_rw [he]
+  rw [finiteMean_mul_const]
+  simpa only [one_mul] using mul_le_mul_of_nonneg_right hm (Real.exp_pos _).le
+
+
+open Filter Topology
+
+lemma dense_coefficient_hasDerivAt (j : ℕ) :
+    HasDerivAt (fun t : ℝ => Real.exp (denseLevelCoefficient j t)-1)
+      (DenseTruncation.h (j+1)) 0 := by
+  have hp : (2:ℝ)^j ≠ 0 := by positivity
+  have h₀ := (hasDerivAt_id (0:ℝ)).mul_const ((2:ℝ)^j)
+  have h₁ := h₀.exp.sub_const 1
+  have h₂ := h₁.div_const ((2:ℝ)^j)
+  have hh := h₂.const_mul (DenseTruncation.h (j+1))
+  simpa [denseLevelCoefficient, hp] using hh.exp.sub_const 1
+
+lemma dense_coefficient_small (H : ℕ) (δ : ℝ) (hδ : 0 < δ) :
+    ∃ τ : ℝ, 0 < τ ∧ τ ≤ 1/500 ∧ ∀ j < H,
+      Real.exp (denseLevelCoefficient j τ)-1 ≤ τ*(DenseTruncation.h (j+1)+δ) := by
+  have hj (j : Fin H) : ∀ᶠ t : ℝ in 𝓝[>] 0,
+      (Real.exp (denseLevelCoefficient j t)-1)/t < DenseTruncation.h (j+1)+δ := by
+    have hh := (dense_coefficient_hasDerivAt j).tendsto_slope_zero_right
+    simp only [zero_add,denseLevelCoefficient,zero_mul,Real.exp_zero,sub_self,zero_div,
+      mul_zero,smul_eq_mul,sub_zero] at hh
+    have hh' : Tendsto (fun t : ℝ => (Real.exp (denseLevelCoefficient j t)-1)/t)
+        (𝓝[>] 0) (𝓝 (DenseTruncation.h (j+1))) := by
+      simpa only [denseLevelCoefficient,div_eq_mul_inv,mul_comm] using hh
+    exact (tendsto_order.mp hh').2 _ (by linarith)
+  have ha : ∀ᶠ t : ℝ in 𝓝[>] 0, ∀ j : Fin H,
+      (Real.exp (denseLevelCoefficient j t)-1)/t < DenseTruncation.h (j+1)+δ :=
+    Filter.eventually_all.mpr hj
+  have hb : ∀ᶠ t : ℝ in 𝓝[>] 0, t < 1/500 :=
+    (eventually_lt_nhds (by norm_num : (0:ℝ) < 1/500)).filter_mono nhdsWithin_le_nhds
+  have hc : ∀ᶠ t : ℝ in 𝓝[>] 0, 0 < t := self_mem_nhdsWithin
+  obtain ⟨τ,ht,hb,ha⟩ := (hc.and (hb.and ha)).exists
+  refine ⟨τ,ht,hb.le,fun j hj => ?_⟩
+  exact ((div_lt_iff₀ ht).mp (ha ⟨j,hj⟩)).le.trans_eq (mul_comm _ _)
+
+lemma palindrome_level_dense_mgf_all (d j : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (t : ℝ) (ht : 0 ≤ t) :
+    finiteMean (fun ω : BenesCoins d => Real.exp (t*palindromeLevelCost (j+1) d e ω)) ≤
+      Real.exp ((Real.exp (denseLevelCoefficient j t)-1)*partnerMean d (Finset.univ.image e) j) := by
+  classical
+  by_cases hj : j < d
+  · obtain ⟨r,rfl⟩ := Nat.exists_eq_add_of_le (Nat.succ_le_of_lt hj)
+    rw [partnerMean_eq]
+    exact palindrome_level_dense_mgf r j e t ht
+  · simp only [palindromeLevelCost_gt (j+1) d (by omega) e, Nat.cast_zero,mul_zero,
+      Real.exp_zero,finiteMean_const,partnerMean,dite_eq_right hj,le_refl]
+
+lemma palindrome_low_dense_mgf (H : ℕ) (hH : 0 < H) (δ τ : ℝ) (hδ : 0 ≤ δ) (ht : 0 ≤ τ)
+    (hτ : ∀ j < H, Real.exp (denseLevelCoefficient j τ)-1 ≤ τ*(DenseTruncation.h (j+1)+δ))
+    (d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (hι : 0 < Fintype.card ι) :
+    finiteMean (fun ω : BenesCoins d => Real.exp ((τ/H)*palindromeLowCost H d e ω)) ≤
+      Real.exp ((τ/H)*Fintype.card ι*
+        (DenseTruncation.H ((Fintype.card ι:ℝ)/(2:ℝ)^d)+H*δ/2)) := by
+  classical
+  let A := Finset.univ.image e
+  have hA : A.card = Fintype.card ι := by simp [A, Finset.card_image_of_injective _ e.injective]
+  have hn : (H:ℝ) ≠ 0 := by exact_mod_cast (ne_of_gt hH)
+  have he : (H:ℝ)*(τ/H) = τ := by field_simp
+  let : Nonempty (Fin H) := Fin.pos_iff_nonempty.mp hH
+  have hh := finiteMean_exp_sum_le
+    (fun j : Fin H => fun ω : BenesCoins d => (palindromeLevelCost (j+1) d e ω:ℝ))
+    (fun j : Fin H => (DenseTruncation.h (j+1)+δ)*partnerMean d A j) (τ/H) (by
+      intro j
+      simp only [Fintype.card_fin,he]
+      apply (palindrome_level_dense_mgf_all d j e τ ht).trans
+      apply Real.exp_le_exp.mpr
+      simpa only [mul_assoc] using mul_le_mul_of_nonneg_right (hτ j j.isLt)
+        (partnerMean_nonneg d A j))
+  have hsum : (∑ j : Fin H, (DenseTruncation.h (j+1)+δ)*partnerMean d A j) ≤
+      Fintype.card ι*(DenseTruncation.H ((Fintype.card ι:ℝ)/(2:ℝ)^d)+H*δ/2) := by
+    rw [Fin.sum_univ_eq_sum_range (fun j => (DenseTruncation.h (j+1)+δ)*partnerMean d A j)]
+    simp_rw [add_mul]
+    rw [Finset.sum_add_distrib]
+    have hm := partner_weighted_sum_le d H A (by omega)
+    rw [hA] at hm
+    have hs : (∑ j ∈ Finset.range H, δ*partnerMean d A j) ≤ (H:ℝ)*δ*(Fintype.card ι)/2 := by
+      calc
+        _ ≤ ∑ _j ∈ Finset.range H, δ*((Fintype.card ι:ℝ)/2) := by
+          apply Finset.sum_le_sum
+          intro j _
+          apply mul_le_mul_of_nonneg_left _ hδ
+          have hg := partnerMean_twice_le d A j
+          rw [hA] at hg
+          linarith
+        _ = _ := by simp only [Finset.sum_const,Finset.card_range,nsmul_eq_mul]; ring
+    nlinarith
+  have hh' := hh.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left hsum (by positivity)))
+  have hl (ω : BenesCoins d) :
+      (∑ j : Fin H, (palindromeLevelCost (j+1) d e ω : ℝ)) =
+        ∑ j ∈ Finset.range H, (palindromeLevelCost (j+1) d e ω : ℝ) :=
+    Fin.sum_univ_eq_sum_range (fun j => (palindromeLevelCost (j+1) d e ω : ℝ)) H
+  simpa only [palindromeLowCost_sum,Nat.cast_sum,hl,mul_assoc] using hh'
+
+
+lemma heightDecay_tendsto : Tendsto heightDecay atTop (𝓝 0) := by
+  have hr : 0 ≤ Real.exp (-(1:ℝ)/64) := (Real.exp_pos _).le
+  have hr1 : Real.exp (-(1:ℝ)/64) < 1 := Real.exp_lt_one_iff.mpr (by norm_num)
+  have hh := ((tendsto_pow_atTop_nhds_zero_of_lt_one hr hr1).const_mul 68).div_const
+    (1-Real.exp (-(1:ℝ)/64))
+  have he (H : ℕ) : (Real.exp (-(1:ℝ)/64))^H = Real.exp (-(H:ℝ)/64) := by
+    rw [←Real.exp_nat_mul]
+    congr 1; ring
+  change Tendsto (fun H : ℕ => 68*Real.exp (-(H:ℝ)/64)/(1-Real.exp (-(1:ℝ)/64))) _ _
+  simpa only [he,mul_zero,zero_div] using hh
+
+lemma heightDecay_exists {ε : ℝ} (hε : 0 < ε) :
+    ∃ H : ℕ, 0 < H ∧ 500*heightDecay H < ε := by
+  have hh := heightDecay_tendsto.const_mul 500
+  simp only [mul_zero] at hh
+  have he := (tendsto_order.mp hh).2 ε hε
+  exact ((eventually_gt_atTop 0).and he).exists
+
+lemma palindrome_high_exp_mgf (H n d : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (t : ℝ) (ht : 0 ≤ t) (htb : t ≤ 1/500) :
+    finiteMean (fun ω : BenesCoins d => Real.exp (t*
+      (palindromeFamilyCost H n d e ω+palindromeFamilyCost (2*H) n d e ω))) ≤
+      Real.exp (t*Fintype.card ι*(500*heightDecay H)) := by
+  have hh := palindrome_high_full_mgf H n d hH e (Real.exp (1/500))
+    (Real.one_le_exp_iff.mpr (by norm_num)) (by rw [Real.log_exp])
+  have hp : finiteMean (fun ω : BenesCoins d => Real.exp ((1/500:ℝ)*
+      (palindromeFamilyCost H n d e ω+palindromeFamilyCost (2*H) n d e ω))) ≤
+      Real.exp ((1/500:ℝ)*(Fintype.card ι*(500*heightDecay H))) := by
+    have he (k : ℕ) : (Real.exp (1/500:ℝ))^k = Real.exp ((1/500:ℝ)*k) := by
+      rw [←Real.exp_nat_mul]; congr 1; ring
+    simpa only [he,Nat.cast_add,show (1/500:ℝ)*(Fintype.card ι*(500*heightDecay H)) =
+      Fintype.card ι*heightDecay H by ring] using hh
+  have hs := finiteMean_exp_scale
+    (fun ω : BenesCoins d => ((palindromeFamilyCost H n d e ω+
+      palindromeFamilyCost (2*H) n d e ω):ℝ))
+    (Fintype.card ι*(500*heightDecay H)) (1/500) t (by norm_num) ht htb
+    (by simpa only [Nat.cast_add] using hp)
+  simpa only [Nat.cast_add,mul_assoc] using hs
+
+end Thorp
+
+namespace Thorp.DenseTruncation
+
+lemma harmonic_le_h (j : ℕ) :
+    (harmonic (2^j):ℝ)/(2:ℝ)^j ≤ h (j+1) := by
+  classical
+  let M := 2^j
+  have hM : 0 < M := by dsimp [M]; positivity
+  have hM' : (0:ℝ) < M := Nat.cast_pos.mpr hM
+  have hMc : (M:ℝ) = (2:ℝ)^j := by simp [M]
+  let p : ℕ → ℝ := fun l => if l < M then 1/M else 0
+  have hp : AdmissibleCycleLaw (j+1) p := by
+    refine ⟨fun l => by dsimp [p]; split_ifs <;> positivity, ?_, ?_⟩
+    · have hs : HasSum p (∑ l ∈ Finset.range M,p l) :=
+        hasSum_sum_of_ne_finset_zero (s := Finset.range M) (f := p)
+        (fun l hl => by simp only [Finset.mem_range] at hl; simp [p,hl])
+      have he : (∑ l ∈ Finset.range M,p l) = 1 := by
+        simp only [p]
+        rw [Finset.sum_congr rfl (fun l hl => ite_eq_left (Finset.mem_range.mp hl))]
+        simp only [Finset.sum_const,Finset.card_range,nsmul_eq_mul]
+        field_simp
+      exact he ▸ hs
+    · intro l
+      simp only [Nat.add_sub_cancel]
+      by_cases hl : l < M
+      · simp only [p,ite_eq_left hl,←hMc]
+        apply div_le_div_of_nonneg_right _ hM'.le
+        exact_mod_cast Nat.succ_pos l
+      · simp only [p,ite_eq_right hl]
+        positivity
+  have he : (∑' l, p l / ((l+1:ℕ):ℝ)) = (harmonic M:ℝ)/(M:ℝ) := by
+    rw [tsum_eq_sum (s := Finset.range M) (fun l hl => by
+      simp only [Finset.mem_range] at hl; simp [p,hl])]
+    simp only [harmonic,Rat.cast_sum,Rat.cast_inv,Rat.cast_natCast,Finset.sum_div]
+    apply Finset.sum_congr rfl
+    intro l hl
+    simp only [p,ite_eq_left (Finset.mem_range.mp hl)]
+    ring
+  have hh := reciprocal_le_h hp
+  rw [he,hMc] at hh
+  exact hh
+
+lemma harmonic_lower (m : ℕ) (hm : 0 < m) :
+    Real.log m+1/2 ≤ (harmonic m:ℝ) := by
+  have hh := Real.one_half_lt_eulerMascheroniConstant.trans
+    (Real.eulerMascheroniConstant_lt_eulerMascheroniSeq' m)
+  simp only [Real.eulerMascheroniSeq',ite_eq_right (ne_of_gt hm)] at hh
+  linarith
+
+lemma h_log_lower (j : ℕ) :
+    ((j:ℝ)*Real.log 2+1/2)/(2:ℝ)^j ≤ h (j+1) := by
+  have hh := harmonic_lower (2^j) (by positivity)
+  rw [Nat.cast_pow,Nat.cast_ofNat,Real.log_pow] at hh
+  exact (div_le_div_of_nonneg_right hh (by positivity)).trans (harmonic_le_h j)
+
+noncomputable def unusedTail (d : ℕ) : ℝ :=
+  (1/2:ℝ)*∑' j : ℕ, h (d+j+1)
+
+lemma unusedTail_nonneg (d : ℕ) : 0 ≤ unusedTail d := by
+  apply mul_nonneg (by norm_num)
+  exact tsum_nonneg (fun j => h_nonneg _)
+
+lemma unusedTail_lower (d : ℕ) :
+    ((d+1:ℝ)*Real.log 2+1/2)/(2:ℝ)^d ≤ unusedTail d := by
+  have hg := hasSum_geometric_of_norm_lt_one (show ‖(1/2:ℝ)‖ < 1 by norm_num)
+  have hj := hasSum_coe_mul_geometric_of_norm_lt_one (show ‖(1/2:ℝ)‖ < 1 by norm_num)
+  norm_num at hg hj
+  have hs : HasSum (fun j : ℕ => (((d+j:ℕ):ℝ)*Real.log 2+1/2)/(2:ℝ)^(d+j))
+      (2*((d+1:ℝ)*Real.log 2+1/2)/(2:ℝ)^d) := by
+    convert! ((hg.mul_left ((d:ℝ)*Real.log 2+1/2)).add
+      (hj.mul_left (Real.log 2))).div_const ((2:ℝ)^d) using 1
+    · funext j
+      simp only [Nat.cast_add,pow_add,div_pow,one_pow]
+      ring
+    · ring
+  have hsum : Summable (fun j : ℕ => h (d+j+1)) := by
+    simpa only [Nat.add_comm d] using (summable_nat_add_iff d).mpr h_summable
+  have hh := hs.summable.tsum_le_tsum (fun j => h_log_lower (d+j)) hsum
+  rw [hs.tsum_eq] at hh
+  unfold unusedTail
+  calc
+    _ = (1/2:ℝ)*(2*((d+1:ℝ)*Real.log 2+1/2)/(2:ℝ)^d) := by ring
+    _ ≤ _ := mul_le_mul_of_nonneg_left hh (by norm_num)
+
+end Thorp.DenseTruncation
+
+namespace Thorp
+
+lemma partnerMean_budget_strong (d : ℕ) (A : Finset (Card d)) (hA : 0 < A.card) :
+    2 * (∑ j ∈ Finset.range d, partnerMean d A j / (2:ℝ)^(j+1)) ≤
+      (A.card:ℝ)*(A.card-1)/(2:ℝ)^d := by
+  classical
+  rw [←Fin.sum_univ_eq_sum_range]
+  simp_rw [partnerMean_rate]
+  rw [←Finset.sum_div]
+  have he : (∑ j : Fin d, (separatedCount A j.rev:ℝ)) = ∑ j : Fin d, (separatedCount A j:ℝ) := by
+    exact Equiv.sum_comp (Fin.revPerm : Equiv.Perm (Fin d)) (fun j : Fin d => (separatedCount A j : ℝ))
+  rw [he,←mul_div_assoc]
+  apply div_le_div_of_nonneg_right _ (by positivity)
+  have hh : (2:ℝ)*(∑ i : Fin d,(separatedCount A i:ℝ)) ≤
+      (A.card:ℝ)*((A.card-1:ℕ):ℝ) := by exact_mod_cast separated_budget_strong A
+  simpa only [Nat.cast_sub (show 1 ≤ A.card by omega),Nat.cast_one] using hh
+
+lemma partnerAllocation_range_budget (d : ℕ) (A : Finset (Card d)) (hA : 0 < A.card) :
+    (∑ j ∈ Finset.range d, partnerAllocation d A j/(2:ℝ)^(j+1)) ≤
+      ((A.card:ℝ)-1)/(2:ℝ)^d := by
+  have hp : (0:ℝ) < A.card := Nat.cast_pos.mpr hA
+  have he (j : ℕ) : partnerAllocation d A j/(2:ℝ)^(j+1) =
+      (2*partnerMean d A j/(2:ℝ)^(j+1))/A.card := by unfold partnerAllocation; ring
+  simp_rw [he]
+  rw [←Finset.sum_div]
+  simp_rw [mul_div_assoc]
+  rw [←Finset.mul_sum]
+  apply (div_le_div_of_nonneg_right (partnerMean_budget_strong d A hA) hp.le).trans
+  exact le_of_eq (by field_simp)
+
+lemma dyadic_weights_hasSum (d : ℕ) :
+    HasSum (fun j : ℕ => 1/(2:ℝ)^(d+j+1)) (1/(2:ℝ)^d) := by
+  have hg := hasSum_geometric_of_norm_lt_one (show ‖(1/2:ℝ)‖ < 1 by norm_num)
+  norm_num at hg
+  convert! hg.div_const ((2:ℝ)^(d+1)) using 1
+  · funext j
+    simp only [pow_add,div_pow,one_pow,pow_one]
+    ring
+  · simp only [pow_succ]
+    field_simp
+
+noncomputable def augmentedAllocation (d : ℕ) (A : Finset (Card d)) (j : ℕ) : ℝ :=
+  if j < d then partnerAllocation d A j else 1
+
+lemma augmentedAllocation_bounds (d : ℕ) (A : Finset (Card d)) (hA : 0 < A.card) (j : ℕ) :
+    0 ≤ augmentedAllocation d A j ∧ augmentedAllocation d A j ≤ 1 := by
+  unfold augmentedAllocation
+  split_ifs
+  · exact (partnerAllocation_admissible d A hA).1 j
+  · norm_num
+
+lemma augmentedAllocation_admissible (d : ℕ) (A : Finset (Card d)) (hA : 0 < A.card) :
+    DenseTruncation.AdmissibleAllocation ((A.card:ℝ)/(2:ℝ)^d) (augmentedAllocation d A) := by
+  classical
+  refine ⟨augmentedAllocation_bounds d A hA,?_⟩
+  have hs : Summable (fun j => augmentedAllocation d A j/(2:ℝ)^(j+1)) := by
+    apply Summable.of_nonneg_of_le (fun j => div_nonneg (augmentedAllocation_bounds d A hA j).1
+      (by positivity)) (fun j => div_le_div_of_nonneg_right (augmentedAllocation_bounds d A hA j).2
+      (by positivity))
+    simpa only [Nat.zero_add] using (dyadic_weights_hasSum 0).summable
+  have hh := hs.sum_add_tsum_nat_add d
+  have ht : (∑' j, augmentedAllocation d A (j+d)/(2:ℝ)^(j+d+1)) = 1/(2:ℝ)^d := by
+    have he (j : ℕ) : augmentedAllocation d A (j+d)/(2:ℝ)^(j+d+1) = 1/(2:ℝ)^(d+j+1) := by
+      rw [augmentedAllocation,ite_eq_right (show ¬j+d<d by omega)]
+      simp only [Nat.add_comm j d]
+    simp_rw [he]
+    exact (dyadic_weights_hasSum d).tsum_eq
+  have hf : (∑ j ∈ Finset.range d, augmentedAllocation d A j/(2:ℝ)^(j+1)) =
+      ∑ j ∈ Finset.range d, partnerAllocation d A j/(2:ℝ)^(j+1) := by
+    exact Finset.sum_congr rfl (fun j hj => by rw [augmentedAllocation,ite_eq_left (Finset.mem_range.mp hj)])
+  rw [←hh,ht,hf]
+  apply (add_le_add (partnerAllocation_range_budget d A hA) (le_refl ((1:ℝ)/2^d))).trans
+  exact le_of_eq (by ring)
+
+lemma partner_weighted_sum_gap (d H : ℕ) (A : Finset (Card d)) (hA : 0 < A.card) :
+    (∑ j ∈ Finset.range H, DenseTruncation.h (j+1)*partnerMean d A j) ≤
+      A.card * (DenseTruncation.H ((A.card:ℝ)/(2:ℝ)^d)-DenseTruncation.unusedTail d) := by
+  classical
+  have ha := augmentedAllocation_admissible d A hA
+  have hs := DenseTruncation.allocation_value_summable ha
+  have HH := DenseTruncation.allocation_le_H ha
+  have he := hs.sum_add_tsum_nat_add d
+  have ht : (∑' j, DenseTruncation.h (j+d+1)*augmentedAllocation d A (j+d)) =
+      ∑' j, DenseTruncation.h (d+j+1) := by
+    congr 1
+    funext j
+    rw [augmentedAllocation,ite_eq_right (show ¬j+d<d by omega)]
+    simp only [mul_one,Nat.add_comm j d]
+  rw [←he,ht] at HH
+  have hf : (∑ j ∈ Finset.range d, DenseTruncation.h (j+1)*augmentedAllocation d A j) =
+      (2/(A.card:ℝ)) * ∑ j ∈ Finset.range d, DenseTruncation.h (j+1)*partnerMean d A j := by
+    rw [Finset.mul_sum]
+    apply Finset.sum_congr rfl
+    intro j hj
+    rw [augmentedAllocation,ite_eq_left (Finset.mem_range.mp hj),partnerAllocation]
+    ring
+  rw [hf] at HH
+  have hz (j : ℕ) (hj : j ∉ Finset.range d) :
+      DenseTruncation.h (j+1)*partnerMean d A j = 0 := by
+    have hj' : ¬j<d := by simpa only [Finset.mem_range] using hj
+    simp only [partnerMean,dite_eq_right hj',mul_zero]
+  have hsum : Summable (fun j => DenseTruncation.h (j+1)*partnerMean d A j) :=
+    (hasSum_sum_of_ne_finset_zero hz : HasSum _
+      (∑ j ∈ Finset.range d, DenseTruncation.h (j+1)*partnerMean d A j)).summable
+  have hlow : (∑ j ∈ Finset.range H, DenseTruncation.h (j+1)*partnerMean d A j) ≤
+      ∑ j ∈ Finset.range d, DenseTruncation.h (j+1)*partnerMean d A j := by
+    calc
+      _ ≤ ∑' j, DenseTruncation.h (j+1)*partnerMean d A j :=
+        hsum.sum_le_tsum _ (fun j _ => mul_nonneg (DenseTruncation.h_nonneg _)
+          (partnerMean_nonneg _ _ _))
+      _ = _ := tsum_eq_sum hz
+  have hp : (0:ℝ) < A.card := Nat.cast_pos.mpr hA
+  have HH' := mul_le_mul_of_nonneg_left HH hp.le
+  have HE : (A.card:ℝ)*((1/2:ℝ)*((2/(A.card:ℝ))*
+      (∑ j ∈ Finset.range d, DenseTruncation.h (j+1)*partnerMean d A j)+
+      ∑' j, DenseTruncation.h (d+j+1))) =
+      (∑ j ∈ Finset.range d, DenseTruncation.h (j+1)*partnerMean d A j)+
+      (A.card:ℝ)*((1/2:ℝ)*∑' j, DenseTruncation.h (d+j+1)) := by
+    field_simp
+  rw [HE] at HH'
+  unfold DenseTruncation.unusedTail
+  exact hlow.trans (by nlinarith [HH'])
+
+
+lemma palindrome_low_dense_mgf_gap (H : ℕ) (hH : 0 < H) (δ τ : ℝ) (hδ : 0 ≤ δ) (ht : 0 ≤ τ)
+    (hτ : ∀ j < H, Real.exp (denseLevelCoefficient j τ)-1 ≤ τ*(DenseTruncation.h (j+1)+δ))
+    (d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (hι : 0 < Fintype.card ι) :
+    finiteMean (fun ω : BenesCoins d => Real.exp ((τ/H)*palindromeLowCost H d e ω)) ≤
+      Real.exp ((τ/H)*Fintype.card ι*
+        (DenseTruncation.H ((Fintype.card ι:ℝ)/(2:ℝ)^d)-DenseTruncation.unusedTail d+H*δ/2)) := by
+  classical
+  let A := Finset.univ.image e
+  have hA : A.card = Fintype.card ι := by simp [A, Finset.card_image_of_injective _ e.injective]
+  have hn : (H:ℝ) ≠ 0 := by exact_mod_cast (ne_of_gt hH)
+  have he : (H:ℝ)*(τ/H) = τ := by field_simp
+  let : Nonempty (Fin H) := Fin.pos_iff_nonempty.mp hH
+  have hh := finiteMean_exp_sum_le
+    (fun j : Fin H => fun ω : BenesCoins d => (palindromeLevelCost (j+1) d e ω:ℝ))
+    (fun j : Fin H => (DenseTruncation.h (j+1)+δ)*partnerMean d A j) (τ/H) (by
+      intro j
+      simp only [Fintype.card_fin,he]
+      apply (palindrome_level_dense_mgf_all d j e τ ht).trans
+      apply Real.exp_le_exp.mpr
+      simpa only [mul_assoc] using mul_le_mul_of_nonneg_right (hτ j j.isLt)
+        (partnerMean_nonneg d A j))
+  have hsum : (∑ j : Fin H, (DenseTruncation.h (j+1)+δ)*partnerMean d A j) ≤
+      Fintype.card ι*(DenseTruncation.H ((Fintype.card ι:ℝ)/(2:ℝ)^d)-DenseTruncation.unusedTail d+H*δ/2) := by
+    rw [Fin.sum_univ_eq_sum_range (fun j => (DenseTruncation.h (j+1)+δ)*partnerMean d A j)]
+    simp_rw [add_mul]
+    rw [Finset.sum_add_distrib]
+    have hm := partner_weighted_sum_gap d H A (by omega)
+    rw [hA] at hm
+    have hs : (∑ j ∈ Finset.range H, δ*partnerMean d A j) ≤ (H:ℝ)*δ*(Fintype.card ι)/2 := by
+      calc
+        _ ≤ ∑ _j ∈ Finset.range H, δ*((Fintype.card ι:ℝ)/2) := by
+          apply Finset.sum_le_sum
+          intro j _
+          apply mul_le_mul_of_nonneg_left _ hδ
+          have hg := partnerMean_twice_le d A j
+          rw [hA] at hg
+          linarith
+        _ = _ := by simp only [Finset.sum_const,Finset.card_range,nsmul_eq_mul]; ring
+    nlinarith
+  have hh' := hh.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left hsum (by positivity)))
+  have hl (ω : BenesCoins d) :
+      (∑ j : Fin H, (palindromeLevelCost (j+1) d e ω : ℝ)) =
+        ∑ j ∈ Finset.range H, (palindromeLevelCost (j+1) d e ω : ℝ) :=
+    Fin.sum_univ_eq_sum_range (fun j => (palindromeLevelCost (j+1) d e ω : ℝ)) H
+  simpa only [palindromeLowCost_sum,Nat.cast_sum,hl,mul_assoc] using hh'
+
+
+lemma palindrome_dense_mgf (ε : ℝ) (hε : 0 < ε) :
+    ∃ t : ℝ, 0 < t ∧ ∀ d : ℕ, ∀ {ι : Type} [Fintype ι],
+      ∀ e : ι ↪ Card d, 0 < Fintype.card ι →
+      finiteMean (fun ω : BenesCoins d => Real.exp (t*palindromeCost d e ω)) ≤
+        Real.exp (t*Fintype.card ι*(DenseTruncation.H ((Fintype.card ι:ℝ)/(2:ℝ)^d)-
+          DenseTruncation.unusedTail d+ε)) := by
+  obtain ⟨H,hH,hdecay⟩ := heightDecay_exists (show 0 < ε/2 by positivity)
+  have hHr : (0:ℝ) < H := Nat.cast_pos.mpr hH
+  let δ := ε/H
+  have hδ : 0 < δ := div_pos hε hHr
+  obtain ⟨τ,ht,hτb,hτ⟩ := dense_coefficient_small H δ hδ
+  let t := (τ/H)/2
+  have htp : 0 < t := by dsimp [t]; positivity
+  have hσb : τ/H ≤ 1/500 := by
+    apply (div_le_self ht.le (by exact_mod_cast hH)).trans hτb
+  refine ⟨t,htp,?_⟩
+  intro d ι _ e he
+  have hlow := palindrome_low_dense_mgf_gap H hH δ τ hδ.le ht.le hτ d e he
+  have hhigh := palindrome_high_exp_mgf H d d hH e (τ/H) (by positivity) hσb
+  have hex (k : ℝ) : (Real.exp (t*k))^2 = Real.exp ((τ/H)*k) := by
+    rw [←Real.exp_nat_mul]
+    congr 1
+    dsimp [t]
+    ring
+  have hh := finiteMean_mul_exp_le
+    (fun ω : BenesCoins d => Real.exp (t*palindromeLowCost H d e ω))
+    (fun ω : BenesCoins d => Real.exp (t*(palindromeFamilyCost H d d e ω+
+      palindromeFamilyCost (2*H) d d e ω)))
+    ((τ/H)*Fintype.card ι*(DenseTruncation.H ((Fintype.card ι:ℝ)/(2:ℝ)^d)-
+      DenseTruncation.unusedTail d+H*δ/2))
+    ((τ/H)*Fintype.card ι*(500*heightDecay H))
+    (by simpa only [hex] using hlow)
+    (by simpa only [hex,Nat.cast_add] using hhigh)
+  have hd : d ≤ H*4^d := by
+    have hg := pow_four_ge d
+    nlinarith
+  have hcost (ω : BenesCoins d) : palindromeCost d e ω = palindromeLowCost H d e ω+
+      (palindromeFamilyCost H d d e ω+palindromeFamilyCost (2*H) d d e ω) := by
+    have hc := palindromeCost_family H d d e ω
+    rw [palindromeLowCost_eq _ _ hd] at hc
+    simpa only [Nat.add_assoc] using hc.symm
+  have hf (ω : BenesCoins d) : Real.exp (t*palindromeCost d e ω) =
+      Real.exp (t*palindromeLowCost H d e ω)*
+        Real.exp (t*(palindromeFamilyCost H d d e ω+palindromeFamilyCost (2*H) d d e ω)) := by
+    rw [hcost,Nat.cast_add,Nat.cast_add,mul_add,Real.exp_add]
+  simp_rw [←hf] at hh
+  apply hh.trans
+  apply Real.exp_le_exp.mpr
+  have heδ : (H:ℝ)*δ/2=ε/2 := by dsimp [δ]; field_simp
+  rw [heδ]
+  have hc : (ε/2+500*heightDecay H) ≤ ε := by linarith
+  have hm := mul_le_mul_of_nonneg_left hc (show 0 ≤ t*(Fintype.card ι) by positivity)
+  dsimp [t] at hm ⊢
+  nlinarith [hm]
+
+end Thorp
+
+open scoped BigOperators
+namespace Thorp.DenseTruncation
+
+noncomputable def stirlingRemainder (n : ℕ) : ℝ :=
+  Real.log (n.factorial:ℝ) - n*Real.log n+n
+
+lemma stirlingRemainder_zero : stirlingRemainder 0 = 0 := by
+  simp [stirlingRemainder]
+
+lemma stirlingRemainder_step (n : ℕ) :
+    stirlingRemainder (n+1)-stirlingRemainder n =
+      1-(n:ℝ)*(Real.log (n+1)-Real.log n) := by
+  have hn : (0:ℝ) < n+1 := by positivity
+  have hf : (0:ℝ) < n.factorial := Nat.cast_pos.mpr (Nat.factorial_pos n)
+  rw [stirlingRemainder,stirlingRemainder,Nat.factorial_succ,Nat.cast_mul,Nat.cast_add,Nat.cast_one,
+    Real.log_mul (ne_of_gt hn) (ne_of_gt hf)]
+  ring
+
+lemma log_increment_mul_monotone :
+    Monotone (fun n : ℕ => (n:ℝ)*(Real.log (n+1)-Real.log n)) := by
+  intro m n hmn
+  have hm : (0:ℝ) ≤ m := Nat.cast_nonneg _
+  have hn : (0:ℝ) ≤ n := Nat.cast_nonneg _
+  have hmn' : (m:ℝ) ≤ n := by exact_mod_cast hmn
+  by_cases hm0 : m = 0
+  · subst m
+    simp only [Nat.cast_zero,zero_mul]
+    by_cases hn0 : n = 0
+    · simp [hn0]
+    · have hnpos : (0:ℝ) < n := Nat.cast_pos.mpr (Nat.pos_of_ne_zero hn0)
+      exact mul_nonneg hn (sub_nonneg.mpr (Real.log_le_log hnpos (by linarith)))
+  have hmpos : (0:ℝ) < m := Nat.cast_pos.mpr (Nat.pos_of_ne_zero hm0)
+  have hnpos : (0:ℝ) < n := lt_of_lt_of_le hmpos hmn'
+  have ha : 0 ≤ (m:ℝ)/n := div_nonneg hm hn
+  have hb : 0 ≤ 1-(m:ℝ)/n := by rw [sub_nonneg,div_le_one hnpos]; exact hmn'
+  have hc := strictConcaveOn_log_Ioi.concaveOn.2
+    (show (m+1:ℝ)/m ∈ Set.Ioi 0 by exact div_pos (by positivity) hmpos)
+    (show (1:ℝ) ∈ Set.Ioi 0 by norm_num) ha hb
+    (show (m:ℝ)/n+(1-(m:ℝ)/n) = 1 by ring)
+  have he : (m:ℝ)/n*((m+1)/m)+(1-(m:ℝ)/n)*1 = (n+1)/n := by
+    field_simp
+    ring
+  simp only [smul_eq_mul,Real.log_one,mul_zero,add_zero,he] at hc
+  rw [Real.log_div (by positivity : (m+1:ℝ) ≠ 0) (ne_of_gt hmpos),
+    Real.log_div (by positivity : (n+1:ℝ) ≠ 0) (ne_of_gt hnpos)] at hc
+  have hc' := mul_le_mul_of_nonneg_left hc hn
+  have he' : (n:ℝ)*((m:ℝ)/n*(Real.log (m+1)-Real.log m)) =
+      m*(Real.log (m+1)-Real.log m) := by field_simp
+  simpa only [he'] using hc'
+
+lemma stirlingRemainder_increment_antitone :
+    Antitone (fun n : ℕ => stirlingRemainder (n+1)-stirlingRemainder n) := by
+  intro m n hmn
+  simp only [stirlingRemainder_step]
+  linarith [log_increment_mul_monotone hmn]
+
+lemma stirlingRemainder_upper {n : ℕ} (hn : 0 < n) :
+    stirlingRemainder n ≤ 1+Real.log n/2 := by
+  obtain ⟨m,rfl⟩ := Nat.exists_eq_succ_of_ne_zero (ne_of_gt hn)
+  have hm : (0:ℝ) < m+1 := by positivity
+  have he : Real.log (2*((m+1):ℝ)) = Real.log 2+Real.log (m+1) :=
+    Real.log_mul (by norm_num) (ne_of_gt hm)
+  have hh := Stirling.log_stirlingSeq'_antitone (show 0 ≤ m by omega)
+  change Real.log (Stirling.stirlingSeq (m+1)) ≤ Real.log (Stirling.stirlingSeq 1) at hh
+  rw [Stirling.log_stirlingSeq_formula,Stirling.log_stirlingSeq_formula] at hh
+  simp only [Nat.cast_add,Nat.cast_one,he,Nat.factorial_one,Real.log_one,mul_one,
+    Real.log_div (ne_of_gt hm) (ne_of_gt (Real.exp_pos 1)),Real.log_exp,
+    Real.log_div (by norm_num : (1:ℝ) ≠ 0) (ne_of_gt (Real.exp_pos 1))] at hh
+  dsimp [stirlingRemainder]
+  push_cast
+  linarith
+
+lemma stirlingRemainder_sum (n : ℕ) :
+    (∑ i ∈ Finset.range n, (stirlingRemainder (i+1)-stirlingRemainder i)) =
+      stirlingRemainder n := by
+  induction n with
+  | zero => simp [stirlingRemainder_zero]
+  | succ n ih => rw [Finset.sum_range_succ,ih]; ring
+
+lemma stirlingRemainder_step_bound (n : ℕ) :
+    (n:ℝ)*(stirlingRemainder (n+1)-stirlingRemainder n) ≤ stirlingRemainder n := by
+  calc
+    _ = ∑ _i ∈ Finset.range n, (stirlingRemainder (n+1)-stirlingRemainder n) := by
+      simp only [Finset.sum_const,Finset.card_range,nsmul_eq_mul]
+    _ ≤ ∑ i ∈ Finset.range n, (stirlingRemainder (i+1)-stirlingRemainder i) := by
+      exact Finset.sum_le_sum fun i hi => stirlingRemainder_increment_antitone
+        (Nat.le_of_lt (Finset.mem_range.mp hi))
+    _ = _ := stirlingRemainder_sum n
+
+lemma stirlingRemainder_average_antitone :
+    Antitone (fun n : ℕ => stirlingRemainder (n+1)/(n+1)) := by
+  apply antitone_nat_of_succ_le
+  intro n
+  have h₁ : (0:ℝ) < n+1 := by positivity
+  have h₂ : (0:ℝ) < n+1+1 := by positivity
+  push_cast
+  rw [div_le_div_iff₀ h₂ h₁]
+  have hh := stirlingRemainder_step_bound (n+1)
+  push_cast at hh ⊢
+  nlinarith
+
+lemma stirlingRemainder_difference_bound {n r : ℕ} (hn : 0 < n) (hr : r ≤ n) :
+    stirlingRemainder n-stirlingRemainder (n-r) ≤ (r:ℝ)/n*stirlingRemainder n := by
+  have hn' : (0:ℝ) < n := Nat.cast_pos.mpr hn
+  have he : (n-r:ℕ)+r=n := Nat.sub_add_cancel hr
+  by_cases hm : n-r=0
+  · have hn_eq : n=r := by omega
+    subst n
+    simp [stirlingRemainder_zero,div_self (ne_of_gt hn')]
+  · obtain ⟨m,hm'⟩ := Nat.exists_eq_succ_of_ne_zero hm
+    obtain ⟨q,hq⟩ := Nat.exists_eq_succ_of_ne_zero (ne_of_gt hn)
+    have hmq : m ≤ q := by omega
+    have hh := stirlingRemainder_average_antitone hmq
+    have hposm : (0:ℝ) < m+1 := by positivity
+    have hposq : (0:ℝ) < q+1 := by positivity
+    rw [div_le_div_iff₀ hposq hposm] at hh
+    have he' : (m:ℝ)+1+r=q+1 := by exact_mod_cast (show m+1+r=q+1 by omega)
+    rw [hm',hq]
+    push_cast
+    rw [div_mul_eq_mul_div,le_div_iff₀ hposq]
+    simp only [Nat.succ_eq_add_one] at *
+    have heq : (m:ℝ)+1 = (q+1:ℝ)-r := by linarith
+    rw [heq] at hh
+    nlinarith
+
+lemma stirlingRemainder_dyadic (d : ℕ) (hd : 1 ≤ d) :
+    stirlingRemainder (2^d)/Real.log 2 ≤ (d+1:ℝ)*Real.log 2+1/2 := by
+  have hl : 0 < Real.log 2 := Real.log_pos (by norm_num)
+  have hlog : (6931/10000:ℝ) < Real.log 2 := lt_trans (by norm_num) Real.log_two_gt_d9
+  rw [div_le_iff₀ hl]
+  by_cases hd1 : d=1
+  · subst d
+    norm_num [stirlingRemainder]
+    nlinarith [sq_nonneg (Real.log 2-6931/10000)]
+  · have hd2 : (2:ℝ) ≤ d := by exact_mod_cast (show 2 ≤ d by omega)
+    have hh := stirlingRemainder_upper (n := 2^d) (by positivity)
+    rw [Nat.cast_pow,Nat.cast_ofNat,Real.log_pow] at hh
+    have hq : 0 ≤ Real.log 2*Real.log 2-Real.log 2/2 := by
+      nlinarith [sq_nonneg (Real.log 2-6931/10000)]
+    have hprod := mul_nonneg (sub_nonneg.mpr hd2) hq
+    nlinarith [sq_nonneg (Real.log 2-6931/10000)]
+
+lemma factorial_error_identity {n r : ℕ} (hn : 0 < n) (hr : r ≤ n) :
+    Real.log ((n.descFactorial r:ℝ)/(n:ℝ)^r) =
+      stirlingRemainder n-stirlingRemainder (n-r)-(r:ℝ)-
+        (n-r:ℕ)*Real.log (1-(r:ℝ)/n) := by
+  have hn' : (0:ℝ) < n := Nat.cast_pos.mpr hn
+  have hm : (0:ℝ) < (n-r).factorial := Nat.cast_pos.mpr (Nat.factorial_pos _)
+  have hd : (0:ℝ) < n.descFactorial r := Nat.cast_pos.mpr (Nat.descFactorial_pos.mpr hr)
+  have hf : ((n-r).factorial:ℝ)*(n.descFactorial r:ℝ) = n.factorial :=
+    by exact_mod_cast Nat.factorial_mul_descFactorial hr
+  have hh := congrArg Real.log hf
+  rw [Real.log_mul (ne_of_gt hm) (ne_of_gt hd)] at hh
+  rw [Real.log_div (ne_of_gt hd) (ne_of_gt (pow_pos hn' r)),Real.log_pow]
+  unfold stirlingRemainder
+  by_cases hz : n-r=0
+  · have hrn : r=n := by omega
+    subst r
+    simp only [Nat.sub_self,Nat.factorial_zero,Nat.cast_one,Real.log_one,Nat.cast_zero,
+      zero_mul,add_zero,sub_zero,Nat.descFactorial_self]
+    ring
+  · have hm' : (0:ℝ) < (n-r:ℕ) := Nat.cast_pos.mpr (Nat.pos_of_ne_zero hz)
+    have he : ((n-r:ℕ):ℝ) = (n:ℝ)*(1-(r:ℝ)/n) := by
+      rw [Nat.cast_sub hr]
+      field_simp
+    have hρ : 0 < 1-(r:ℝ)/n := by
+      rw [←mul_pos_iff_of_pos_left hn',←he]
+      exact hm'
+    have hl : Real.log ((n-r:ℕ):ℝ) = Real.log n+Real.log (1-(r:ℝ)/n) := by
+      rw [he, Real.log_mul (ne_of_gt hn') (ne_of_gt hρ)]
+    rw [hl,Nat.cast_sub hr]
+    nlinarith [hh]
+
+
+lemma normalization_le (d r : ℕ) (hd : 1 ≤ d) (hr : 1 ≤ r) (hrn : r ≤ 2^d) :
+    Real.log (((2^d).descFactorial r:ℝ)/((2:ℝ)^d)^r)/Real.log 2 ≤
+      (r:ℝ)*(entropyCorrection ((r:ℝ)/(2:ℝ)^d)+unusedTail d) := by
+  have hnp : 0 < (2:ℕ)^d := by positivity
+  have hnr : (0:ℝ) < (2:ℝ)^d := by positivity
+  have hr' : (0:ℝ) < r := by exact_mod_cast hr
+  have hl : 0 < Real.log 2 := Real.log_pos (by norm_num)
+  have hh := stirlingRemainder_difference_bound hnp hrn
+  simp only [Nat.cast_pow,Nat.cast_ofNat] at hh
+  have hg := (stirlingRemainder_dyadic d hd).trans
+    ((div_le_iff₀ hnr).mp (unusedTail_lower d))
+  have he : (stirlingRemainder (2^d)-stirlingRemainder (2^d-r))/Real.log 2 ≤
+      (r:ℝ)*unusedTail d := by
+    calc
+      _ ≤ ((r:ℝ)/(2:ℝ)^d*stirlingRemainder (2^d))/Real.log 2 :=
+        div_le_div_of_nonneg_right hh hl.le
+      _ = ((r:ℝ)/(2:ℝ)^d)*(stirlingRemainder (2^d)/Real.log 2) := by ring
+      _ ≤ ((r:ℝ)/(2:ℝ)^d)*(unusedTail d*(2:ℝ)^d) :=
+        mul_le_mul_of_nonneg_left hg (by positivity)
+      _ = _ := by field_simp
+  have hid := factorial_error_identity hnp hrn
+  simp only [Nat.cast_pow,Nat.cast_ofNat] at hid
+  have heq : Real.log (((2^d).descFactorial r:ℝ)/((2:ℝ)^d)^r)/Real.log 2 =
+      (r:ℝ)*entropyCorrection ((r:ℝ)/(2:ℝ)^d)+
+      (stirlingRemainder (2^d)-stirlingRemainder (2^d-r))/Real.log 2 := by
+    rw [hid,entropyCorrection,Nat.cast_sub hrn]
+    simp only [Nat.cast_pow,Nat.cast_ofNat]
+    field_simp
+    ring
+  rw [heq]
+  nlinarith
+
+
+theorem dense_truncation : MainStatement := by
+  classical
+  intro ρ hρ ε hε
+  obtain ⟨t,ht,hm⟩ := palindrome_dense_mgf (ε/2) (by positivity)
+  refine ⟨t*ε/2, by positivity, ?_⟩
+  intro d r hd hr hρr x
+  have hrn : r ≤ 2^d := by
+    have hh := Fintype.card_le_of_injective x x.injective
+    simpa only [Fintype.card_fin,card_positions] using hh
+  have hm' := hm d x (by simpa only [Fintype.card_fin] using (show 0 < r by omega))
+  simp only [Fintype.card_fin,hρr] at hm'
+  have hn := normalization_le d r hd hr hrn
+  rw [hρr] at hn
+  let a : ℝ := r*(H ρ+entropyCorrection ρ+ε)
+  let b : ℝ := Real.log (((2^d).descFactorial r:ℝ)/((2:ℝ)^d)^r)/Real.log 2
+  have hp (ω : BenesCoins d) :
+      (if a < densityExponent d r x ω then (1:ℝ) else 0) ≤
+      Real.exp (t*palindromeCost d x ω)*Real.exp (t*(b-a)) := by
+    split_ifs with h
+    · rw [←Real.exp_add]
+      apply Real.one_le_exp_iff.mpr
+      have hh : 0 ≤ t*((palindromeCost d x ω:ℝ)+b-a) := by
+        apply mul_nonneg ht.le
+        exact sub_nonneg.mpr (le_of_lt h)
+      nlinarith
+    · positivity
+  have he : t*(r:ℝ)*(H ρ-unusedTail d+ε/2)+t*(b-a) ≤ -(t*ε/2)*r := by
+    have hh := mul_le_mul_of_nonneg_left hn ht.le
+    dsimp [b,a]
+    nlinarith [hh]
+  calc
+    _ ≤ finiteMean (fun ω : BenesCoins d =>
+        Real.exp (t*palindromeCost d x ω)*Real.exp (t*(b-a))) := finiteMean_mono hp
+    _ = finiteMean (fun ω : BenesCoins d => Real.exp (t*palindromeCost d x ω))*
+        Real.exp (t*(b-a)) := finiteMean_mul_const _ _
+    _ ≤ Real.exp (t*(r:ℝ)*(H ρ-unusedTail d+ε/2))*Real.exp (t*(b-a)) :=
+      mul_le_mul_of_nonneg_right hm' (Real.exp_nonneg _)
+    _ ≤ Real.exp (-(t*ε/2)*r) := by rw [←Real.exp_add]; exact Real.exp_le_exp.mpr he
+
+end Thorp.DenseTruncation
+
+end ThorpNine.Dense
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Dense/Model.lean b/lean/OAI/Probability/ThorpRouting/Dense/Model.lean
index 9b350e85dab8886f84a942b405f1b8aeaa3b3ef8..9c6a6c189c8eb6d70cbb8b3f0db609f7a06e7f13 100644
--- a/lean/OAI/Probability/ThorpRouting/Dense/Model.lean
+++ b/lean/OAI/Probability/ThorpRouting/Dense/Model.lean
@@ -1,267 +1,267 @@
-import Mathlib
-
-namespace OAI
-
-namespace ThorpNine.Dense
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-
-abbrev Card (d : ℕ) := Fin d → Bool
-
-def switchFun {d : ℕ} (ξ : Card d → Bool) (x : Card (d + 1)) : Card (d + 1) :=
-  Fin.cons (Bool.xor (x 0) (ξ (Fin.tail x))) (Fin.tail x)
-
-theorem switchFun_involutive {d : ℕ} (ξ : Card d → Bool) :
-    Function.Involutive (switchFun ξ) := by
-  intro x
-  funext i
-  refine Fin.cases ?_ (fun j => ?_) i
-  · simp only [switchFun, Fin.cons_zero, Fin.tail_cons]
-    cases x 0 <;> cases ξ (Fin.tail x) <;> rfl
-  · simp [switchFun, Fin.tail]
-
-def pairSwitch {d : ℕ} (ξ : Card d → Bool) : Equiv.Perm (Card (d + 1)) :=
-  { toFun := switchFun ξ
-    invFun := switchFun ξ
-    left_inv := switchFun_involutive ξ
-    right_inv := switchFun_involutive ξ }
-
-def Butterfly : ℕ → Type
-  | 0 => Unit
-  | d + 1 => (Card d → Bool) × (Bool → Butterfly d)
-
-def childLift {d : ℕ} (p : Bool → Equiv.Perm (Card d)) : Equiv.Perm (Card (d + 1)) where
-  toFun x := Fin.cons (x 0) (p (x 0) (Fin.tail x))
-  invFun x := Fin.cons (x 0) ((p (x 0)).symm (Fin.tail x))
-  left_inv x := by simp only [Fin.cons_zero, Fin.tail_cons, Equiv.symm_apply_apply, Fin.cons_self_tail]
-  right_inv x := by simp only [Fin.cons_zero, Fin.tail_cons, Equiv.apply_symm_apply, Fin.cons_self_tail]
-
-def butterflyPerm : (d : ℕ) → Butterfly d → Equiv.Perm (Card d)
-  | 0, _ => 1
-  | d + 1, b => pairSwitch b.1 * childLift (fun ε => butterflyPerm d (b.2 ε))
-
-noncomputable def finiteMean {Ω : Type*} [Fintype Ω] (f : Ω → ℝ) : ℝ :=
-  (∑ ω, f ω) / Fintype.card Ω
-
-def SwitchIndex : ℕ → Type
-  | 0 => Empty
-  | d + 1 => Sum (Card d) (Bool × SwitchIndex d)
-
-instance (d : ℕ) : Fintype (SwitchIndex d) := by
-  induction d with
-  | zero => exact inferInstanceAs (Fintype Empty)
-  | succ d ih => exact inferInstanceAs (Fintype (Sum (Card d) (Bool × SwitchIndex d)))
-
-instance (d : ℕ) : DecidableEq (SwitchIndex d) := by
-  induction d with
-  | zero => exact inferInstanceAs (DecidableEq Empty)
-  | succ d ih => exact inferInstanceAs (DecidableEq (Sum (Card d) (Bool × SwitchIndex d)))
-
-def decodeButterfly : (d : ℕ) → (SwitchIndex d → Bool) → Butterfly d
-  | 0, _ => ()
-  | d + 1, ω => (fun y => ω (Sum.inl y), fun ε =>
-      decodeButterfly d (fun i => ω (Sum.inr (ε,i))))
-
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-noncomputable def orbitSet (p : Equiv.Perm α) (x : α) : Finset α :=
-  Finset.univ.filter (p.SameCycle x)
-
-end Thorp
-
-namespace Thorp.RoutingNetwork
-open scoped BigOperators
-open Filter
-variable {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι] [DecidableEq ι]
-variable {A : ℕ}
-
-abbrev SelectedCycle (p : Equiv.Perm (α)) (S : Finset (α)) :=
-  {C : Finset (α) // (∃ x, C = orbitSet p x) ∧ C ⊆ S}
-
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) :
-    Fintype (SelectedCycle p S) := Fintype.ofFinite _
-
-end Thorp.RoutingNetwork
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-
-section
-variable {β : Type*} [Fintype β] [DecidableEq β]
-
-def pairLayer (ξ : β → Bool) : Equiv.Perm (Bool × β) where
-  toFun x := (Bool.xor x.1 (ξ x.2),x.2)
-  invFun x := (Bool.xor x.1 (ξ x.2),x.2)
-  left_inv x := by
-    rcases x with ⟨b,x⟩
-    change (Bool.xor (Bool.xor b (ξ x)) (ξ x),x) = (b,x)
-    cases b <;> cases ξ x <;> rfl
-  right_inv x := by
-    rcases x with ⟨b,x⟩
-    change (Bool.xor (Bool.xor b (ξ x)) (ξ x),x) = (b,x)
-    cases b <;> cases ξ x <;> rfl
-
-end
-
-def coinStepEquiv (d : ℕ) : (SwitchIndex (d+1) → Bool) ≃
-    ((SwitchIndex d → Bool) × (SwitchIndex d → Bool)) × (Card d → Bool) where
-  toFun ω := ((fun i => ω (Sum.inr (false,i)),fun i => ω (Sum.inr (true,i))),
-    fun x => ω (Sum.inl x))
-  invFun ω := Sum.elim ω.2 (fun bi => if bi.1 then ω.1.2 bi.2 else ω.1.1 bi.2)
-  left_inv ω := by
-    funext i
-    cases i with
-    | inl x => rfl
-    | inr bi => cases bi with | mk b i => cases b <;> rfl
-  right_inv ω := by
-    rcases ω with ⟨⟨ω₀,ω₁⟩,ξ⟩
-    rfl
-
-def headTailEquiv (d : ℕ) : Card (d+1) ≃ Bool × Card d :=
-  (Fin.consEquiv (fun _ : Fin (d+1) => Bool)).symm
-
-end Thorp
-
-namespace Thorp.CycleColoring
-open scoped BigOperators
-open Filter
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-def alternatingPerm (p : Equiv.Perm α) : Equiv.Perm (Bool × α) where
-  toFun x := if x.1 then (false,p x.2) else (true,x.2)
-  invFun x := if x.1 then (false,x.2) else (true,p.symm x.2)
-  left_inv x := by rcases x with ⟨b,x⟩; cases b <;> simp
-  right_inv x := by rcases x with ⟨b,x⟩; cases b <;> simp
-
-end Thorp.CycleColoring
-
-namespace Thorp.PairRouting
-open scoped BigOperators
-open Filter
-variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
-
-def required (x : ι → Bool × α) (c : ι → Bool) (i : ι) : Bool := Bool.xor (x i).1 (c i)
-
-def Compatible (x : ι → Bool × α) (c : ι → Bool) : Prop :=
-  ∀ i j, (x i).2 = (x j).2 → required x c i = required x c j
-
-def colors (x : ι → Bool × α) (ξ : α → Bool) (i : ι) : Bool :=
-  Bool.xor (x i).1 (ξ (x i).2)
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma required_colors (x : ι → Bool × α) (ξ : α → Bool) (i : ι) :
-    required x (colors x ξ) i = ξ (x i).2 := by
-  simp only [required,colors]
-  cases (x i).1 <;> cases ξ (x i).2 <;> rfl
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma colors_compatible (x : ι → Bool × α) (ξ : α → Bool) : Compatible x (colors x ξ) := by
-  intro i j he
-  simp only [required_colors,he]
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma Compatible.tail_injective {x : ι ↪ Bool × α} {c : ι → Bool} (hc : Compatible x c) (b : Bool) :
-    Function.Injective (fun i : {i : ι // c i = b} => (x i.val).2) := by
-  intro i j he
-  apply Subtype.ext
-  apply x.injective
-  apply Prod.ext
-  · have hh := hc i j he
-    simp only [required,i.property,j.property] at hh
-    cases h₀ : (x i.val).1 <;> cases h₁ : (x j.val).1 <;> cases b <;> simp_all
-  · exact he
-
-def childEmbedding (x : ι ↪ Bool × α) (c : ι → Bool) (hc : Compatible x c) (b : Bool) :
-    {i : ι // c i = b} ↪ α := ⟨fun i => (x i.val).2,hc.tail_injective b⟩
-
-noncomputable def labelSet (e : ι ↪ Bool × α) : Finset (Bool × α) := Finset.univ.image e
-
-def alternatingProjection (p : Bool → Equiv.Perm α) : Equiv.Perm α := (p false).symm * p true
-
-noncomputable def alternatingCycles (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) : ℕ :=
-  Fintype.card (RoutingNetwork.SelectedCycle
-    (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e))
-
-def switchedEmbedding (e : ι ↪ Bool × α) (ξ : α → Bool) : ι ↪ Bool × α :=
-  e.trans (pairLayer ξ).toEmbedding
-
-end Thorp.PairRouting
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-
-abbrev BenesCoins (d : ℕ) := (SwitchIndex d → Bool) × (SwitchIndex d → Bool)
-
-def palindromePerm (d : ℕ) (ω : BenesCoins d) : Equiv.Perm (Card d) :=
-  butterflyPerm d (decodeButterfly d ω.1) * (butterflyPerm d (decodeButterfly d ω.2)).symm
-
-def sandwichShuffle {A B C D E F : Type*} :
-    ((A × B) × C) × ((D × E) × F) ≃ ((A × D) × (B × E)) × (F × C) where
-  toFun x := (((x.1.1.1,x.2.1.1),(x.1.1.2,x.2.1.2)),(x.2.2,x.1.2))
-  invFun x := (((x.1.1.1,x.1.2.1),x.2.2),((x.1.1.2,x.1.2.2),x.2.1))
-  left_inv _ := rfl
-  right_inv _ := rfl
-
-def benesStepEquiv (d : ℕ) : BenesCoins (d+1) ≃
-    (BenesCoins d × BenesCoins d) × ((Card d → Bool) × (Card d → Bool)) :=
-  (Equiv.prodCongr (coinStepEquiv d) (coinStepEquiv d)).trans sandwichShuffle
-
-noncomputable def palindromeCost : (d : ℕ) → {ι : Type*} → [Fintype ι] →
-    (ι ↪ Card d) → BenesCoins d → ℕ
-  | 0, _, _, _, _ => 0
-  | d+1, _, _, e, ω =>
-      let σ := benesStepEquiv d ω
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x σ.2.1
-      let hc := PairRouting.colors_compatible x σ.2.1
-      PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
-          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) +
-        palindromeCost d (PairRouting.childEmbedding x c hc false) σ.1.1 +
-        palindromeCost d (PairRouting.childEmbedding x c hc true) σ.1.2
-
-end Thorp
-
-namespace Thorp.DenseTruncation
-open scoped BigOperators
-
-def AdmissibleCycleLaw (j : ℕ) (p : ℕ → ℝ) : Prop :=
-  (∀ l, 0 ≤ p l) ∧ HasSum p 1 ∧
-    ∀ l, p l ≤ ((l + 1 : ℕ) : ℝ) / (2 : ℝ) ^ (j - 1)
-
-noncomputable def h (j : ℕ) : ℝ :=
-  sSup {z : ℝ | ∃ p : ℕ → ℝ, AdmissibleCycleLaw j p ∧
-    z = ∑' l : ℕ, p l / ((l + 1 : ℕ) : ℝ)}
-
-def AdmissibleAllocation (ρ : ℝ) (x : ℕ → ℝ) : Prop :=
-  (∀ j, 0 ≤ x j ∧ x j ≤ 1) ∧
-    (∑' j : ℕ, x j / (2 : ℝ) ^ (j + 1)) ≤ ρ
-
-noncomputable def H (ρ : ℝ) : ℝ :=
-  sSup {z : ℝ | ∃ x : ℕ → ℝ, AdmissibleAllocation ρ x ∧
-    z = (1 / 2 : ℝ) * ∑' j : ℕ, h (j + 1) * x j}
-
-noncomputable def entropyCorrection (ρ : ℝ) : ℝ :=
-  (-ρ - (1 - ρ) * Real.log (1 - ρ)) / (ρ * Real.log 2)
-
-noncomputable def densityExponent (d r : ℕ) (x : Fin r ↪ Card d)
-    (ω : BenesCoins d) : ℝ :=
-  palindromeCost d x ω +
-    Real.log (((2 ^ d).descFactorial r : ℝ) / ((2 : ℝ) ^ d) ^ r) / Real.log 2
-
-def MainStatement : Prop :=
-  ∀ ρ : ℝ, 0 < ρ → ∀ ε : ℝ, 0 < ε →
-    ∃ c : ℝ, 0 < c ∧ ∀ d r : ℕ, 1 ≤ d → 1 ≤ r →
-      (r : ℝ) / (2 : ℝ) ^ d = ρ → ∀ x : Fin r ↪ Card d,
-      finiteMean (fun ω : BenesCoins d =>
-        if (r : ℝ) * (H ρ + entropyCorrection ρ + ε) < densityExponent d r x ω
-        then (1 : ℝ) else 0) ≤ Real.exp (-c * r)
-
-end Thorp.DenseTruncation
-
-end ThorpNine.Dense
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+namespace ThorpNine.Dense
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+
+abbrev Card (d : ℕ) := Fin d → Bool
+
+def switchFun {d : ℕ} (ξ : Card d → Bool) (x : Card (d + 1)) : Card (d + 1) :=
+  Fin.cons (Bool.xor (x 0) (ξ (Fin.tail x))) (Fin.tail x)
+
+theorem switchFun_involutive {d : ℕ} (ξ : Card d → Bool) :
+    Function.Involutive (switchFun ξ) := by
+  intro x
+  funext i
+  refine Fin.cases ?_ (fun j => ?_) i
+  · simp only [switchFun, Fin.cons_zero, Fin.tail_cons]
+    cases x 0 <;> cases ξ (Fin.tail x) <;> rfl
+  · simp [switchFun, Fin.tail]
+
+def pairSwitch {d : ℕ} (ξ : Card d → Bool) : Equiv.Perm (Card (d + 1)) :=
+  { toFun := switchFun ξ
+    invFun := switchFun ξ
+    left_inv := switchFun_involutive ξ
+    right_inv := switchFun_involutive ξ }
+
+def Butterfly : ℕ → Type
+  | 0 => Unit
+  | d + 1 => (Card d → Bool) × (Bool → Butterfly d)
+
+def childLift {d : ℕ} (p : Bool → Equiv.Perm (Card d)) : Equiv.Perm (Card (d + 1)) where
+  toFun x := Fin.cons (x 0) (p (x 0) (Fin.tail x))
+  invFun x := Fin.cons (x 0) ((p (x 0)).symm (Fin.tail x))
+  left_inv x := by simp only [Fin.cons_zero, Fin.tail_cons, Equiv.symm_apply_apply, Fin.cons_self_tail]
+  right_inv x := by simp only [Fin.cons_zero, Fin.tail_cons, Equiv.apply_symm_apply, Fin.cons_self_tail]
+
+def butterflyPerm : (d : ℕ) → Butterfly d → Equiv.Perm (Card d)
+  | 0, _ => 1
+  | d + 1, b => pairSwitch b.1 * childLift (fun ε => butterflyPerm d (b.2 ε))
+
+noncomputable def finiteMean {Ω : Type*} [Fintype Ω] (f : Ω → ℝ) : ℝ :=
+  (∑ ω, f ω) / Fintype.card Ω
+
+def SwitchIndex : ℕ → Type
+  | 0 => Empty
+  | d + 1 => Sum (Card d) (Bool × SwitchIndex d)
+
+instance (d : ℕ) : Fintype (SwitchIndex d) := by
+  induction d with
+  | zero => exact inferInstanceAs (Fintype Empty)
+  | succ d ih => exact inferInstanceAs (Fintype (Sum (Card d) (Bool × SwitchIndex d)))
+
+instance (d : ℕ) : DecidableEq (SwitchIndex d) := by
+  induction d with
+  | zero => exact inferInstanceAs (DecidableEq Empty)
+  | succ d ih => exact inferInstanceAs (DecidableEq (Sum (Card d) (Bool × SwitchIndex d)))
+
+def decodeButterfly : (d : ℕ) → (SwitchIndex d → Bool) → Butterfly d
+  | 0, _ => ()
+  | d + 1, ω => (fun y => ω (Sum.inl y), fun ε =>
+      decodeButterfly d (fun i => ω (Sum.inr (ε,i))))
+
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+noncomputable def orbitSet (p : Equiv.Perm α) (x : α) : Finset α :=
+  Finset.univ.filter (p.SameCycle x)
+
+end Thorp
+
+namespace Thorp.RoutingNetwork
+open scoped BigOperators
+open Filter
+variable {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι] [DecidableEq ι]
+variable {A : ℕ}
+
+abbrev SelectedCycle (p : Equiv.Perm (α)) (S : Finset (α)) :=
+  {C : Finset (α) // (∃ x, C = orbitSet p x) ∧ C ⊆ S}
+
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) :
+    Fintype (SelectedCycle p S) := Fintype.ofFinite _
+
+end Thorp.RoutingNetwork
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+
+section
+variable {β : Type*} [Fintype β] [DecidableEq β]
+
+def pairLayer (ξ : β → Bool) : Equiv.Perm (Bool × β) where
+  toFun x := (Bool.xor x.1 (ξ x.2),x.2)
+  invFun x := (Bool.xor x.1 (ξ x.2),x.2)
+  left_inv x := by
+    rcases x with ⟨b,x⟩
+    change (Bool.xor (Bool.xor b (ξ x)) (ξ x),x) = (b,x)
+    cases b <;> cases ξ x <;> rfl
+  right_inv x := by
+    rcases x with ⟨b,x⟩
+    change (Bool.xor (Bool.xor b (ξ x)) (ξ x),x) = (b,x)
+    cases b <;> cases ξ x <;> rfl
+
+end
+
+def coinStepEquiv (d : ℕ) : (SwitchIndex (d+1) → Bool) ≃
+    ((SwitchIndex d → Bool) × (SwitchIndex d → Bool)) × (Card d → Bool) where
+  toFun ω := ((fun i => ω (Sum.inr (false,i)),fun i => ω (Sum.inr (true,i))),
+    fun x => ω (Sum.inl x))
+  invFun ω := Sum.elim ω.2 (fun bi => if bi.1 then ω.1.2 bi.2 else ω.1.1 bi.2)
+  left_inv ω := by
+    funext i
+    cases i with
+    | inl x => rfl
+    | inr bi => cases bi with | mk b i => cases b <;> rfl
+  right_inv ω := by
+    rcases ω with ⟨⟨ω₀,ω₁⟩,ξ⟩
+    rfl
+
+def headTailEquiv (d : ℕ) : Card (d+1) ≃ Bool × Card d :=
+  (Fin.consEquiv (fun _ : Fin (d+1) => Bool)).symm
+
+end Thorp
+
+namespace Thorp.CycleColoring
+open scoped BigOperators
+open Filter
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+def alternatingPerm (p : Equiv.Perm α) : Equiv.Perm (Bool × α) where
+  toFun x := if x.1 then (false,p x.2) else (true,x.2)
+  invFun x := if x.1 then (false,x.2) else (true,p.symm x.2)
+  left_inv x := by rcases x with ⟨b,x⟩; cases b <;> simp
+  right_inv x := by rcases x with ⟨b,x⟩; cases b <;> simp
+
+end Thorp.CycleColoring
+
+namespace Thorp.PairRouting
+open scoped BigOperators
+open Filter
+variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
+
+def required (x : ι → Bool × α) (c : ι → Bool) (i : ι) : Bool := Bool.xor (x i).1 (c i)
+
+def Compatible (x : ι → Bool × α) (c : ι → Bool) : Prop :=
+  ∀ i j, (x i).2 = (x j).2 → required x c i = required x c j
+
+def colors (x : ι → Bool × α) (ξ : α → Bool) (i : ι) : Bool :=
+  Bool.xor (x i).1 (ξ (x i).2)
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma required_colors (x : ι → Bool × α) (ξ : α → Bool) (i : ι) :
+    required x (colors x ξ) i = ξ (x i).2 := by
+  simp only [required,colors]
+  cases (x i).1 <;> cases ξ (x i).2 <;> rfl
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma colors_compatible (x : ι → Bool × α) (ξ : α → Bool) : Compatible x (colors x ξ) := by
+  intro i j he
+  simp only [required_colors,he]
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma Compatible.tail_injective {x : ι ↪ Bool × α} {c : ι → Bool} (hc : Compatible x c) (b : Bool) :
+    Function.Injective (fun i : {i : ι // c i = b} => (x i.val).2) := by
+  intro i j he
+  apply Subtype.ext
+  apply x.injective
+  apply Prod.ext
+  · have hh := hc i j he
+    simp only [required,i.property,j.property] at hh
+    cases h₀ : (x i.val).1 <;> cases h₁ : (x j.val).1 <;> cases b <;> simp_all
+  · exact he
+
+def childEmbedding (x : ι ↪ Bool × α) (c : ι → Bool) (hc : Compatible x c) (b : Bool) :
+    {i : ι // c i = b} ↪ α := ⟨fun i => (x i.val).2,hc.tail_injective b⟩
+
+noncomputable def labelSet (e : ι ↪ Bool × α) : Finset (Bool × α) := Finset.univ.image e
+
+def alternatingProjection (p : Bool → Equiv.Perm α) : Equiv.Perm α := (p false).symm * p true
+
+noncomputable def alternatingCycles (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) : ℕ :=
+  Fintype.card (RoutingNetwork.SelectedCycle
+    (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e))
+
+def switchedEmbedding (e : ι ↪ Bool × α) (ξ : α → Bool) : ι ↪ Bool × α :=
+  e.trans (pairLayer ξ).toEmbedding
+
+end Thorp.PairRouting
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+
+abbrev BenesCoins (d : ℕ) := (SwitchIndex d → Bool) × (SwitchIndex d → Bool)
+
+def palindromePerm (d : ℕ) (ω : BenesCoins d) : Equiv.Perm (Card d) :=
+  butterflyPerm d (decodeButterfly d ω.1) * (butterflyPerm d (decodeButterfly d ω.2)).symm
+
+def sandwichShuffle {A B C D E F : Type*} :
+    ((A × B) × C) × ((D × E) × F) ≃ ((A × D) × (B × E)) × (F × C) where
+  toFun x := (((x.1.1.1,x.2.1.1),(x.1.1.2,x.2.1.2)),(x.2.2,x.1.2))
+  invFun x := (((x.1.1.1,x.1.2.1),x.2.2),((x.1.1.2,x.1.2.2),x.2.1))
+  left_inv _ := rfl
+  right_inv _ := rfl
+
+def benesStepEquiv (d : ℕ) : BenesCoins (d+1) ≃
+    (BenesCoins d × BenesCoins d) × ((Card d → Bool) × (Card d → Bool)) :=
+  (Equiv.prodCongr (coinStepEquiv d) (coinStepEquiv d)).trans sandwichShuffle
+
+noncomputable def palindromeCost : (d : ℕ) → {ι : Type*} → [Fintype ι] →
+    (ι ↪ Card d) → BenesCoins d → ℕ
+  | 0, _, _, _, _ => 0
+  | d+1, _, _, e, ω =>
+      let σ := benesStepEquiv d ω
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x σ.2.1
+      let hc := PairRouting.colors_compatible x σ.2.1
+      PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
+          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) +
+        palindromeCost d (PairRouting.childEmbedding x c hc false) σ.1.1 +
+        palindromeCost d (PairRouting.childEmbedding x c hc true) σ.1.2
+
+end Thorp
+
+namespace Thorp.DenseTruncation
+open scoped BigOperators
+
+def AdmissibleCycleLaw (j : ℕ) (p : ℕ → ℝ) : Prop :=
+  (∀ l, 0 ≤ p l) ∧ HasSum p 1 ∧
+    ∀ l, p l ≤ ((l + 1 : ℕ) : ℝ) / (2 : ℝ) ^ (j - 1)
+
+noncomputable def h (j : ℕ) : ℝ :=
+  sSup {z : ℝ | ∃ p : ℕ → ℝ, AdmissibleCycleLaw j p ∧
+    z = ∑' l : ℕ, p l / ((l + 1 : ℕ) : ℝ)}
+
+def AdmissibleAllocation (ρ : ℝ) (x : ℕ → ℝ) : Prop :=
+  (∀ j, 0 ≤ x j ∧ x j ≤ 1) ∧
+    (∑' j : ℕ, x j / (2 : ℝ) ^ (j + 1)) ≤ ρ
+
+noncomputable def H (ρ : ℝ) : ℝ :=
+  sSup {z : ℝ | ∃ x : ℕ → ℝ, AdmissibleAllocation ρ x ∧
+    z = (1 / 2 : ℝ) * ∑' j : ℕ, h (j + 1) * x j}
+
+noncomputable def entropyCorrection (ρ : ℝ) : ℝ :=
+  (-ρ - (1 - ρ) * Real.log (1 - ρ)) / (ρ * Real.log 2)
+
+noncomputable def densityExponent (d r : ℕ) (x : Fin r ↪ Card d)
+    (ω : BenesCoins d) : ℝ :=
+  palindromeCost d x ω +
+    Real.log (((2 ^ d).descFactorial r : ℝ) / ((2 : ℝ) ^ d) ^ r) / Real.log 2
+
+def MainStatement : Prop :=
+  ∀ ρ : ℝ, 0 < ρ → ∀ ε : ℝ, 0 < ε →
+    ∃ c : ℝ, 0 < c ∧ ∀ d r : ℕ, 1 ≤ d → 1 ≤ r →
+      (r : ℝ) / (2 : ℝ) ^ d = ρ → ∀ x : Fin r ↪ Card d,
+      finiteMean (fun ω : BenesCoins d =>
+        if (r : ℝ) * (H ρ + entropyCorrection ρ + ε) < densityExponent d r x ω
+        then (1 : ℝ) else 0) ≤ Real.exp (-c * r)
+
+end Thorp.DenseTruncation
+
+end ThorpNine.Dense
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Dense/PalindromeFamilyCost.lean b/lean/OAI/Probability/ThorpRouting/Dense/PalindromeFamilyCost.lean
index e69f682dff43d8b5c6a667ce93dcb114f28920f1..0c6a7816b51c092ee2ba7a27a6db998fcdc89052 100644
--- a/lean/OAI/Probability/ThorpRouting/Dense/PalindromeFamilyCost.lean
+++ b/lean/OAI/Probability/ThorpRouting/Dense/PalindromeFamilyCost.lean
@@ -1,834 +1,834 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Dense.CutoffMass
-
-namespace OAI
-
-namespace ThorpNine.Dense
-
-namespace Thorp
-
-lemma familyEnd_two (L n : ℕ) : 2*familyEnd L n ≤ L*4^n := by
-  cases n with
-  | zero => simp [familyEnd]
-  | succ n => simp only [familyEnd,Nat.succ_ne_zero,ite_false,Nat.add_one_sub_one,pow_succ]; nlinarith
-
-lemma familyEnd_ge (L n j : ℕ) (hj : j < n) : 2*(L*4^j) ≤ familyEnd L n := by
-  cases n with
-  | zero => omega
-  | succ n =>
-    simp only [familyEnd,Nat.succ_ne_zero,ite_false,Nat.add_one_sub_one]
-    apply Nat.mul_le_mul_left
-    exact Nat.mul_le_mul_left L (Nat.pow_le_pow_right (by decide) (by omega : j ≤ n))
-
-noncomputable def palindromeFamilyCost (L n d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) : ℕ :=
-  ∑ j ∈ Finset.range n, palindromeSlabCost (L*4^j) d e ω
-
-lemma palindromeFamilyCost_adapted (L n d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) :
-    LayerAdapted d (familyEnd L n) (fun Y => palindromeFamilyCost L n d e (Y,X)) := by
-  apply layerAdapted_sum
-  intro j hj
-  exact layerAdapted_mono (palindromeSlabCost_adapted (L*4^j) d e X)
-    (familyEnd_ge L n j (Finset.mem_range.mp hj))
-
-theorem palindrome_family_mgf (L n d : ℕ) (hL : 0 < L) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) ≤
-      Real.exp (Fintype.card ι * (68*∑ j ∈ Finset.range n,
-        Real.exp (-((L*4^j:ℕ):ℝ)/64))) := by
-  induction n with
-  | zero => simp only [palindromeFamilyCost,Finset.range_zero,Finset.sum_empty,pow_zero,
-      finiteMean_const,mul_zero,Real.exp_zero,le_refl]
-  | succ n ih =>
-    let l := L*4^n
-    have hl : 0 < l := Nat.mul_pos hL (pow_pos (by decide) _)
-    have hs : 2*familyEnd L n ≤ l := familyEnd_two L n
-    have hc : ∀ Y, palindromeFamilyCost L (n+1) d e (Y,X) =
-        palindromeFamilyCost L n d e (Y,X)+palindromeSlabCost l d e (Y,X) := by
-      intro Y
-      exact Finset.sum_range_succ _ _
-    have hm : finiteMean (fun Y => q^(palindromeFamilyCost L (n+1) d e (Y,X))) ≤
-        finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) *
-          Real.exp (Fintype.card ι*(68*Real.exp (-(l:ℝ)/64))) := by
-      by_cases hld : l ≤ d
-      · simp_rw [hc,pow_add]
-        exact palindrome_slab_adapted_mul_le d (familyEnd L n) l (by omega) hl hs e X _
-          (fun _ => pow_nonneg (by linarith) _)
-          (layerAdapted_comp (palindromeFamilyCost_adapted L n d e X) (fun c => q^c))
-          q hq hlog
-      · simp_rw [hc,palindromeSlabCost_gt l d (by omega) e,Nat.add_zero]
-        exact le_mul_of_one_le_right
-          (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
-          (Real.one_le_exp_iff.mpr (by positivity))
-    refine (hm.trans (mul_le_mul_of_nonneg_right ih (Real.exp_nonneg _))).trans_eq ?_
-    rw [←Real.exp_add,Finset.sum_range_succ]
-    congr 1
-    dsimp [l]
-    ring
-
-lemma pow_four_ge (j : ℕ) : j+1 ≤ 4^j := by
-  induction j with
-  | zero => norm_num
-  | succ j ih => rw [pow_succ]; omega
-
-lemma sum_family_decay (L n : ℕ) (hL : 0 < L) :
-    (∑ j ∈ Finset.range n, Real.exp (-((L*4^j:ℕ):ℝ)/64)) ≤
-      Real.exp (-(L:ℝ)/64)/(1-Real.exp (-(1:ℝ)/64)) := by
-  let r := Real.exp (-(1:ℝ)/64)
-  have hr : 0 < r := Real.exp_pos _
-  have hr1 : r < 1 := Real.exp_lt_one_iff.mpr (by norm_num)
-  have hg : (∑ j ∈ Finset.range n, r^j) ≤ 1/(1-r) := by
-    apply (le_div_iff₀ (by linarith : (0:ℝ)<1-r)).mpr
-    have hh := geom_sum_mul r n
-    nlinarith [pow_nonneg (le_of_lt hr) n]
-  calc
-    _ ≤ ∑ j ∈ Finset.range n, Real.exp (-(L:ℝ)/64)*r^j := by
-      apply Finset.sum_le_sum
-      intro j _
-      have hj := pow_four_ge j
-      have hlj : L+j ≤ L*4^j := by nlinarith
-      have hlj' : (L:ℝ)+j ≤ (L*4^j:ℕ) := by exact_mod_cast hlj
-      rw [←Real.exp_nat_mul,←Real.exp_add]
-      exact Real.exp_le_exp.mpr (by linarith)
-    _ = Real.exp (-(L:ℝ)/64)*(∑ j ∈ Finset.range n, r^j) := by rw [Finset.mul_sum]
-    _ ≤ Real.exp (-(L:ℝ)/64)*(1/(1-r)) :=
-      mul_le_mul_of_nonneg_left hg (Real.exp_nonneg _)
-    _ = _ := by ring
-
-
-lemma palindromeLevelCost_step_add (t d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (d+1)) (ω : BenesCoins (d+1)) :
-    palindromeLevelCost t (d+1) e ω =
-      let σ := benesStepEquiv d ω
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x σ.2.1
-      let hc := PairRouting.colors_compatible x σ.2.1
-      (if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
-          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) else 0) +
-        palindromeLevelCost t d (PairRouting.childEmbedding x c hc false) σ.1.1 +
-        palindromeLevelCost t d (PairRouting.childEmbedding x c hc true) σ.1.2 := by
-  classical
-  rw [palindromeLevelCost]
-  dsimp only
-  split_ifs with h
-  · simp only [palindromeLevelCost_gt t d (by omega),Nat.add_zero]
-  · simp only [Nat.zero_add]
-
-lemma palindromeLowCost_sum (H d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) :
-    palindromeLowCost H d e ω = ∑ i ∈ Finset.range H, palindromeLevelCost (i+1) d e ω := by
-  classical
-  induction d generalizing ι with
-  | zero => simp only [palindromeLowCost,palindromeLevelCost,Finset.sum_const_zero]
-  | succ d ih =>
-    rw [palindromeLowCost]
-    simp only [palindromeLevelCost_step_add,Finset.sum_add_distrib]
-    rw [←ih,←ih]
-    congr 2
-    simp only [Nat.add_right_cancel_iff,Finset.sum_ite_eq',Finset.mem_range,Nat.succ_le_iff]
-
-lemma palindromeLowCost_slab (L d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) :
-    palindromeLowCost L d e ω + palindromeSlabCost L d e ω = palindromeLowCost (2*L) d e ω := by
-  rw [palindromeLowCost_sum,palindromeLowCost_sum,show 2*L=L+L by omega,Finset.sum_range_add]
-  congr 1
-  exact Fin.sum_univ_eq_sum_range (fun i => palindromeLevelCost (L+i+1) d e ω) L
-
-lemma palindromeCost_family (L n d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) :
-    palindromeLowCost L d e ω + palindromeFamilyCost L n d e ω +
-      palindromeFamilyCost (2*L) n d e ω = palindromeLowCost (L*4^n) d e ω := by
-  induction n with
-  | zero => simp only [palindromeFamilyCost,Finset.range_zero,Finset.sum_empty,Nat.add_zero,
-      pow_zero,mul_one]
-  | succ n ih =>
-    have he : (2*L)*4^n = 2*(L*4^n) := by ring
-    have he' : L*4^(n+1) = 2*(2*(L*4^n)) := by rw [pow_succ]; ring
-    simp only [palindromeFamilyCost,Finset.sum_range_succ] at ih ⊢
-    rw [he,he',←palindromeLowCost_slab,←palindromeLowCost_slab,←ih]
-    omega
-
-
-lemma finiteMean_mul_sq_le {Ω : Type*} [Fintype Ω] (f g : Ω → ℝ) :
-    (finiteMean (fun ω => f ω*g ω))^2 ≤ finiteMean (fun ω => (f ω)^2)*finiteMean (fun ω => (g ω)^2) := by
-  have h := Finset.sum_mul_sq_le_sq_mul_sq Finset.univ f g
-  simp only [finiteMean,div_pow,div_mul_div_comm,←pow_two]
-  exact div_le_div_of_nonneg_right h (sq_nonneg _)
-
-lemma finiteMean_mul_exp_le {Ω : Type*} [Fintype Ω] (f g : Ω → ℝ) (A B : ℝ)
-    (hf : finiteMean (fun ω => (f ω)^2) ≤ Real.exp A)
-    (hg : finiteMean (fun ω => (g ω)^2) ≤ Real.exp B) :
-    finiteMean (fun ω => f ω*g ω) ≤ Real.exp ((A+B)/2) := by
-  have hc := finiteMean_mul_sq_le f g
-  have hn : 0 ≤ finiteMean (fun ω => (g ω)^2) := finiteMean_nonneg (fun _ => sq_nonneg _)
-  have hu := mul_le_mul hf hg hn (Real.exp_nonneg _)
-  have he : Real.exp A*Real.exp B = (Real.exp ((A+B)/2))^2 := by
-    rw [←Real.exp_add,←Real.exp_nat_mul]
-    congr 1
-    push_cast
-    ring
-  rw [he] at hu
-  have hh := hc.trans hu
-  have hep := Real.exp_pos ((A+B)/2)
-  nlinarith
-
-
-noncomputable def heightDecay (H : ℕ) : ℝ :=
-  68*Real.exp (-(H:ℝ)/64)/(1-Real.exp (-(1:ℝ)/64))
-
-lemma heightDecay_nonneg (H : ℕ) : 0 ≤ heightDecay H := by
-  have h : Real.exp (-(1:ℝ)/64) < 1 := Real.exp_lt_one_iff.mpr (by norm_num)
-  apply div_nonneg (by positivity) (by linarith)
-
-lemma heightDecay_antitone : Antitone heightDecay := by
-  intro L H h
-  have hden : 0 ≤ 1-Real.exp (-(1:ℝ)/64) := by
-    have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
-    linarith
-  apply div_le_div_of_nonneg_right _ hden
-  apply mul_le_mul_of_nonneg_left _ (by norm_num)
-  apply Real.exp_le_exp.mpr
-  have h' : (L:ℝ) ≤ H := by exact_mod_cast h
-  linarith
-
-lemma palindrome_family_mgf_decay (L n d : ℕ) (hL : 0 < L) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) ≤
-      Real.exp (Fintype.card ι*heightDecay L) := by
-  apply (palindrome_family_mgf L n d hL e X q hq hlog).trans
-  apply Real.exp_le_exp.mpr
-  apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
-  have h := mul_le_mul_of_nonneg_left (sum_family_decay L n hL) (by norm_num : (0:ℝ)≤68)
-  simpa only [heightDecay,mul_div_assoc] using h
-
-theorem palindrome_high_mgf (H n d : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/500) :
-    finiteMean (fun Y => q^(palindromeFamilyCost H n d e (Y,X)+
-      palindromeFamilyCost (2*H) n d e (Y,X))) ≤ Real.exp (Fintype.card ι*heightDecay H) := by
-  have hq2 := one_le_pow₀ (n:=2) hq
-  have hlog2 : Real.log (q^2) ≤ 1/250 := by rw [Real.log_pow]; norm_num; linarith
-  have h₀ := palindrome_family_mgf_decay H n d hH e X (q^2) hq2 hlog2
-  have h₁ := palindrome_family_mgf_decay (2*H) n d (by omega) e X (q^2) hq2 hlog2
-  have h₁' := h₁.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left
-    (heightDecay_antitone (show H ≤ 2*H by omega)) (Nat.cast_nonneg _)))
-  have hh := finiteMean_mul_exp_le (fun Y => q^(palindromeFamilyCost H n d e (Y,X)))
-    (fun Y => q^(palindromeFamilyCost (2*H) n d e (Y,X)))
-    (Fintype.card ι*heightDecay H) (Fintype.card ι*heightDecay H)
-    (by simpa only [pow_right_comm] using h₀) (by simpa only [pow_right_comm] using h₁')
-  simpa only [←pow_add,add_self_div_two] using hh
-
-lemma palindrome_high_full_mgf (H n d : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/500) :
-    finiteMean (fun ω : BenesCoins d => q^(palindromeFamilyCost H n d e ω+
-      palindromeFamilyCost (2*H) n d e ω)) ≤ Real.exp (Fintype.card ι*heightDecay H) := by
-  rw [finiteMean_prod,finiteMean_comm]
-  calc
-    _ ≤ finiteMean (fun _X : SwitchIndex d → Bool => Real.exp (Fintype.card ι*heightDecay H)) := by
-      apply finiteMean_mono
-      intro X
-      exact palindrome_high_mgf H n d hH e X q hq hlog
-    _ = _ := finiteMean_const _
-
-theorem palindrome_cost_mgf_cutoff (r H : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (H+r)) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/1000)
-    (hr : (2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*(q^2)^H) ≤ 1/2) :
-    finiteMean (fun ω : BenesCoins (H+r) => q^(palindromeCost (H+r) e ω)) ≤
-      Real.exp (((2:ℝ)^r*(2*((2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*(q^2)^H))^2)+
-        Fintype.card ι*heightDecay H)/2) := by
-  let d := H+r
-  have hd : d ≤ H*4^d := by
-    have hh := pow_four_ge d
-    nlinarith
-  have hcost (ω : BenesCoins d) : palindromeCost d e ω = palindromeLowCost H d e ω+
-      (palindromeFamilyCost H d d e ω+palindromeFamilyCost (2*H) d d e ω) := by
-    have hh := palindromeCost_family H d d e ω
-    rw [palindromeLowCost_eq _ _ hd] at hh
-    simpa only [Nat.add_assoc] using hh.symm
-  have hq2 := one_le_pow₀ (n:=2) hq
-  have hlog2 : Real.log (q^2) ≤ 1/500 := by rw [Real.log_pow]; norm_num; linarith
-  have hlow := palindrome_low_mgf r H e (q^2) hq2 hr
-  have hhigh := palindrome_high_full_mgf H d d hH e (q^2) hq2 hlog2
-  have hh := finiteMean_mul_exp_le (fun ω : BenesCoins d => q^(palindromeLowCost H d e ω))
-    (fun ω : BenesCoins d => q^(palindromeFamilyCost H d d e ω+palindromeFamilyCost (2*H) d d e ω))
-    _ _ (by simpa only [pow_right_comm] using hlow) (by simpa only [pow_right_comm] using hhigh)
-  change finiteMean (fun ω : BenesCoins d => q^(palindromeCost d e ω)) ≤ _
-  simpa only [hcost,pow_add,pow_right_comm] using hh
-
-
-lemma cutoffMass_le (H d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (ω : BenesCoins d) :
-    cutoffMass H d e ω ≤ Fintype.card ι := by
-  classical
-  induction d generalizing ι with
-  | zero => exact Nat.zero_le _
-  | succ d ih =>
-    rw [cutoffMass]
-    split_ifs
-    · exact crowdedMass_le _
-    · exact (Nat.add_le_add (ih _ _) (ih _ _)).trans_eq (PairRouting.color_card_add _)
-
-lemma palindromeLowCost_le (H d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) : palindromeLowCost H d e ω ≤ H*Fintype.card ι :=
-  (palindromeLowCost_le_cutoff H d e ω).trans (Nat.mul_le_mul_left H (cutoffMass_le H d e ω))
-
-theorem palindrome_cost_mgf_coarse (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/1000) :
-    finiteMean (fun ω : BenesCoins d => q^(palindromeCost d e ω)) ≤
-      Real.exp (Fintype.card ι*(1+heightDecay 1)) := by
-  have hd : d ≤ 1*4^d := by have hh := pow_four_ge d; omega
-  have hcost (ω : BenesCoins d) : palindromeCost d e ω = palindromeLowCost 1 d e ω+
-      (palindromeFamilyCost 1 d d e ω+palindromeFamilyCost 2 d d e ω) := by
-    have hh := palindromeCost_family 1 d d e ω
-    rw [palindromeLowCost_eq _ _ hd] at hh
-    simpa only [Nat.add_assoc] using hh.symm
-  have hq2 := one_le_pow₀ (n:=2) hq
-  have hlog2 : Real.log (q^2) ≤ 1/500 := by rw [Real.log_pow]; norm_num; linarith
-  have hlow : finiteMean (fun ω : BenesCoins d => (q^2)^(palindromeLowCost 1 d e ω)) ≤
-      Real.exp (Fintype.card ι) := by
-    calc
-      _ ≤ finiteMean (fun _ω : BenesCoins d => (q^2)^(Fintype.card ι)) := by
-        apply finiteMean_mono
-        intro ω
-        apply pow_le_pow_right₀ hq2
-        simpa only [one_mul] using palindromeLowCost_le 1 d e ω
-      _ = (q^2)^(Fintype.card ι) := finiteMean_const _
-      _ = Real.exp (Fintype.card ι * Real.log (q^2)) := by
-        rw [Real.exp_nat_mul,Real.exp_log (by positivity : 0<q^2)]
-      _ ≤ _ := Real.exp_le_exp.mpr (by
-        have hh := mul_le_mul_of_nonneg_left hlog2 (Nat.cast_nonneg (Fintype.card ι) : (0:ℝ)≤_)
-        nlinarith [Nat.cast_nonneg (α:=ℝ) (Fintype.card ι)])
-  have hhigh := palindrome_high_full_mgf 1 d d (by decide) e (q^2) hq2 hlog2
-  have hh := finiteMean_mul_exp_le (fun ω : BenesCoins d => q^(palindromeLowCost 1 d e ω))
-    (fun ω : BenesCoins d => q^(palindromeFamilyCost 1 d d e ω+palindromeFamilyCost 2 d d e ω))
-    (Fintype.card ι) (Fintype.card ι*heightDecay 1)
-    (by simpa only [pow_right_comm] using hlow) (by simpa only [pow_right_comm] using hhigh)
-  simp only [←pow_add,←hcost] at hh
-  apply hh.trans (Real.exp_le_exp.mpr _)
-  have hk : (0:ℝ) ≤ Fintype.card ι := Nat.cast_nonneg _
-  have hD := heightDecay_nonneg 1
-  nlinarith
-
-noncomputable def momentBase : ℝ := (2:ℝ)^(1/1000:ℝ)
-
-lemma momentBase_one_le : 1 ≤ momentBase := by
-  exact Real.one_le_rpow (by norm_num) (by norm_num)
-
-lemma momentBase_log : Real.log momentBase ≤ 1/1000 := by
-  rw [momentBase,Real.log_rpow (by norm_num)]
-  have hh := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
-  norm_num at hh ⊢
-  linarith
-
-lemma momentBase_pow (c : ℕ) : momentBase^c = (2:ℝ)^((c:ℝ)*(1/1000:ℝ)) := by
-  rw [mul_comm,Real.rpow_mul_natCast (by norm_num)]
-  rfl
-
-lemma palindromeRowMoment_le_cost (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) :
-    palindromeRowMoment d e (1/1000) ≤
-      finiteMean (fun ω : BenesCoins d => momentBase^(palindromeCost d e ω)) := by
-  have hn : (0:ℝ) < ((2^d:ℕ):ℝ)^(Fintype.card ι) := by positivity
-  have hn0 : (0:ℝ) ≤ ((2^d).descFactorial (Fintype.card ι):ℝ) /
-      ((2^d:ℕ):ℝ)^(Fintype.card ι) := by positivity
-  have hn1 : ((2^d).descFactorial (Fintype.card ι):ℝ) /
-      ((2^d:ℕ):ℝ)^(Fintype.card ι) ≤ 1 := by
-    apply (div_le_one hn).mpr
-    exact_mod_cast Nat.descFactorial_le_pow (2^d) (Fintype.card ι)
-  have hpow := Real.rpow_le_one hn0 hn1 (by norm_num : (0:ℝ)≤1/1000)
-  have hm := palindrome_row_density_moment d e (1/1000) (by norm_num)
-  simp_rw [←momentBase_pow] at hm
-  exact hm.trans (mul_le_of_le_one_left (finiteMean_nonneg (fun _ =>
-    pow_nonneg (le_trans (by norm_num) momentBase_one_le) _)) hpow)
-
-theorem palindrome_row_moment_coarse (d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) :
-    palindromeRowMoment d e (1/1000) ≤ Real.exp (Fintype.card ι*(1+heightDecay 1)) :=
-  (palindromeRowMoment_le_cost d e).trans
-    (palindrome_cost_mgf_coarse d e momentBase momentBase_one_le momentBase_log)
-
-
-noncomputable def densityCutoff (p : ℝ) : ℕ := ⌊-Real.log p/(4*Real.log 2)⌋₊
-
-lemma log_small_density {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
-    Real.log p ≤ -8*Real.log 2 := by
-  have he : Real.log (1/256:ℝ) = -8*Real.log 2 := by
-    rw [show (1/256:ℝ) = 1/2^8 by norm_num,
-      Real.log_div (by norm_num) (by positivity),Real.log_one,Real.log_pow]
-    ring
-  exact (Real.log_le_log hp hs).trans_eq he
-
-lemma densityCutoff_bounds {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
-    0 < densityCutoff p ∧
-    (densityCutoff p:ℝ)*(4*Real.log 2) ≤ -Real.log p ∧
-    -Real.log p < ((densityCutoff p:ℝ)+1)*(4*Real.log 2) := by
-  have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
-  have hd : 0 < 4*Real.log 2 := by positivity
-  have hh := log_small_density hp hs
-  have hx : 1 ≤ -Real.log p/(4*Real.log 2) := by
-    apply (le_div_iff₀ hd).mpr
-    nlinarith
-  refine ⟨Nat.floor_pos.mpr hx,?_,?_⟩
-  · exact (le_div_iff₀ hd).mp (Nat.floor_le (by linarith : 0 ≤ -Real.log p/(4*Real.log 2)))
-  · exact (div_lt_iff₀ hd).mp (Nat.lt_floor_add_one (-Real.log p/(4*Real.log 2)))
-
-lemma densityCutoff_decay {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
-    Real.exp (-(densityCutoff p:ℝ)/64) ≤ Real.exp (1/64) * p^(1/512:ℝ) := by
-  have hh := (densityCutoff_bounds hp hs).2.2
-  have hb : Real.log 2 ≤ 2 := by
-    have := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
-    linarith
-  have hH : 0 ≤ (densityCutoff p:ℝ)+1 := by positivity
-  have hu := mul_le_mul_of_nonneg_left hb hH
-  rw [Real.rpow_def_of_pos hp,←Real.exp_add]
-  apply Real.exp_le_exp.mpr
-  nlinarith
-
-lemma densityCutoff_power {p q : ℝ} (hp : 0 < p) (hs : p ≤ 1/256)
-    (hq : 0 < q) (hlog : Real.log q ≤ Real.log 2/4) (n : ℕ) (hn : n ≤ 4)
-    (hq1 : 1 ≤ q) :
-    p*(2:ℝ)^(densityCutoff p)*q^(n*densityCutoff p) ≤ p^(1/2:ℝ) := by
-  have hcut := (densityCutoff_bounds hp hs).2.1
-  have hH : 0 ≤ (densityCutoff p:ℝ) := Nat.cast_nonneg _
-  have hn' : (n:ℝ) ≤ 4 := by exact_mod_cast hn
-  have hln := Real.log_nonneg hq1
-  have hl := mul_le_mul_of_nonneg_right hn' hln
-  have hnl : (n:ℝ)*Real.log q ≤ Real.log 2 := by linarith
-  have hm := mul_le_mul_of_nonneg_left hnl hH
-  have heq : p*(2:ℝ)^(densityCutoff p)*q^(n*densityCutoff p) =
-      Real.exp (Real.log p+(densityCutoff p:ℝ)*Real.log 2+
-        (n*densityCutoff p:ℕ)*Real.log q) := by
-    rw [Real.exp_add,Real.exp_add,Real.exp_nat_mul,Real.exp_nat_mul,
-      Real.exp_log hp,Real.exp_log hq,Real.exp_log (by norm_num : (0:ℝ)<2)]
-  rw [heq,Real.rpow_def_of_pos hp]
-  apply Real.exp_le_exp.mpr
-  push_cast
-  nlinarith
-
-lemma densityCutoff_sparse {p q : ℝ} (hp : 0 < p) (hs : p ≤ 1/256)
-    (hq : 0 < q) (hlog : Real.log q ≤ Real.log 2/4) (hq1 : 1 ≤ q) :
-    (2:ℝ)^(densityCutoff p)*(p*(q^2)^(densityCutoff p)) ≤ 1/2 := by
-  have hh := densityCutoff_power hp hs hq hlog 2 (by decide) hq1
-  rw [←pow_mul] at ⊢
-  have hle : p^(1/2:ℝ) ≤ 1/2 := by
-    have hlogp := log_small_density hp hs
-    have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
-    rw [Real.rpow_def_of_pos hp]
-    calc
-      _ ≤ Real.exp (-Real.log 2) := by apply Real.exp_le_exp.mpr; nlinarith
-      _ = 1/2 := by rw [Real.exp_neg,Real.exp_log (by norm_num : (0:ℝ)<2)]; norm_num
-  nlinarith [hh.trans hle]
-
-
-lemma momentBase_log_exact : Real.log momentBase = Real.log 2/1000 := by
-  rw [momentBase,Real.log_rpow (by norm_num)]
-  ring
-
-lemma densityCutoff_le_height (d k : ℕ) (hk : 0 < k) :
-    densityCutoff ((k:ℝ)/(2:ℝ)^d) ≤ d := by
-  have hk' : (1:ℝ) ≤ k := by exact_mod_cast hk
-  have hp : 0 < (k:ℝ)/(2:ℝ)^d := by positivity
-  have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
-  have hlogk := Real.log_nonneg hk'
-  by_cases hx : 0 ≤ -Real.log ((k:ℝ)/(2:ℝ)^d)/(4*Real.log 2)
-  · have hh := (le_div_iff₀ (by positivity : 0 < 4*Real.log 2)).mp (Nat.floor_le hx)
-    have hd : (0:ℝ) ≤ d := Nat.cast_nonneg _
-    have hH : (0:ℝ) ≤ densityCutoff ((k:ℝ)/(2:ℝ)^d) := Nat.cast_nonneg _
-    change (densityCutoff ((k:ℝ)/(2:ℝ)^d):ℝ)*(4*Real.log 2) ≤ -Real.log ((k:ℝ)/(2:ℝ)^d) at hh
-    rw [Real.log_div (by positivity) (by positivity),Real.log_pow] at hh
-    have hle : (densityCutoff ((k:ℝ)/(2:ℝ)^d):ℝ) ≤ d := by nlinarith
-    exact_mod_cast hle
-  · have hz := Nat.floor_of_nonpos (le_of_not_ge hx)
-    simp only [densityCutoff,hz,Nat.zero_le]
-
-noncomputable def densityTailConstant : ℝ :=
-  68*Real.exp (1/64)/(1-Real.exp (-(1:ℝ)/64))
-
-lemma densityTailConstant_nonneg : 0 ≤ densityTailConstant := by
-  have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
-  apply div_nonneg (by positivity) (by linarith)
-
-lemma heightDecay_cutoff {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
-    heightDecay (densityCutoff p) ≤ densityTailConstant*p^(1/512:ℝ) := by
-  have hden : 0 ≤ 1-Real.exp (-(1:ℝ)/64) := by
-    have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
-    linarith
-  have hh := div_le_div_of_nonneg_right
-    (mul_le_mul_of_nonneg_left (densityCutoff_decay hp hs) (by norm_num : (0:ℝ)≤68)) hden
-  simpa only [heightDecay,densityTailConstant,mul_assoc,div_mul_eq_mul_div] using hh
-
-lemma sparse_low_algebra (r H k : ℕ) (q : ℝ) :
-    (2:ℝ)^r*(2*((2:ℝ)^H*(((k:ℝ)/(2:ℝ)^(H+r))*(q^2)^H))^2) =
-    2*k*(((k:ℝ)/(2:ℝ)^(H+r))*(2:ℝ)^H*q^(4*H)) := by
-  rw [pow_add,←pow_mul,show 4*H = (2*H)*2 by omega,pow_mul]
-  have hH : (2:ℝ)^H ≠ 0 := by positivity
-  have hr : (2:ℝ)^r ≠ 0 := by positivity
-  field_simp
-  ring
-
-lemma palindrome_row_moment_small (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι)
-    (hs : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1/256) :
-    palindromeRowMoment d e (1/1000) ≤
-      Real.exp (Fintype.card ι*(1+densityTailConstant)*
-        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ)) := by
-  generalize hpdef : ((Fintype.card ι:ℝ)/(2:ℝ)^d) = p at hs ⊢
-  have hp : 0 < p := by rw [←hpdef]; positivity
-  have hle := densityCutoff_le_height d (Fintype.card ι) hk
-  rw [hpdef] at hle
-  obtain ⟨r,hr⟩ := Nat.exists_eq_add_of_le hle
-  subst d
-  have hH := (densityCutoff_bounds hp hs).1
-  have hq := momentBase_one_le
-  have hqp : 0 < momentBase := by linarith
-  have hlog : Real.log momentBase ≤ Real.log 2/4 := by
-    rw [momentBase_log_exact]
-    have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
-    linarith
-  have hsp := densityCutoff_sparse hp hs hqp hlog hq
-  rw [←hpdef] at hsp
-  have hh := palindrome_cost_mgf_cutoff r (densityCutoff p) hH e momentBase hq momentBase_log
-    (by simpa only [hpdef] using hsp)
-  apply (palindromeRowMoment_le_cost _ e).trans (hh.trans (Real.exp_le_exp.mpr _))
-  rw [sparse_low_algebra,hpdef]
-  have hlow := densityCutoff_power hp hs hqp hlog 4 (by decide) hq
-  have hp1 : p ≤ 1 := by linarith
-  have he := Real.rpow_le_rpow_of_exponent_ge hp hp1 (by norm_num : (1:ℝ)/512 ≤ 1/2)
-  have hlow' := mul_le_mul_of_nonneg_left (hlow.trans he)
-    (by positivity : (0:ℝ)≤2*Fintype.card ι)
-  have hhigh := mul_le_mul_of_nonneg_left (heightDecay_cutoff hp hs)
-    (Nat.cast_nonneg (Fintype.card ι): (0:ℝ)≤_)
-  have hnon : 0 ≤ Fintype.card ι*densityTailConstant*p^(1/512:ℝ) :=
-    mul_nonneg (mul_nonneg (Nat.cast_nonneg _) densityTailConstant_nonneg)
-      (Real.rpow_nonneg (le_of_lt hp) _)
-  nlinarith
-
-noncomputable def densityConstant : ℝ :=
-  256*(1+heightDecay 1)+1+densityTailConstant
-
-lemma densityConstant_pos : 0 < densityConstant := by
-  dsimp [densityConstant]
-  have hh := heightDecay_nonneg 1
-  have ht := densityTailConstant_nonneg
-  linarith
-
-theorem palindrome_row_moment_bound (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι) :
-    palindromeRowMoment d e (1/1000) ≤ Real.exp (densityConstant*Fintype.card ι*
-        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ)) := by
-  have hp : 0 < (Fintype.card ι:ℝ)/(2:ℝ)^d := by positivity
-  have hp1 : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1 := by
-    apply (div_le_one (by positivity)).mpr
-    have h := Fintype.card_le_of_injective e e.injective
-    rw [card_positions] at h
-    exact_mod_cast h
-  have hk0 : (0:ℝ) ≤ Fintype.card ι := Nat.cast_nonneg _
-  have hpow := Real.rpow_nonneg (le_of_lt hp) (1/512:ℝ)
-  by_cases hs : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1/256
-  · apply (palindrome_row_moment_small d e hk hs).trans (Real.exp_le_exp.mpr _)
-    have hD : 1+densityTailConstant ≤ densityConstant := by
-      dsimp [densityConstant]
-      have := heightDecay_nonneg 1
-      linarith
-    have hh := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hD hk0) hpow
-    nlinarith
-  · have hlow : (1:ℝ)/256 ≤ ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ) := by
-      have he := Real.rpow_le_rpow_of_exponent_ge hp hp1 (by norm_num : (1:ℝ)/512 ≤ 1)
-      rw [Real.rpow_one] at he
-      exact (le_of_not_ge hs).trans he
-    have hD : 256*(1+heightDecay 1) ≤ densityConstant := by
-      dsimp [densityConstant]
-      have := densityTailConstant_nonneg
-      linarith
-    apply (palindrome_row_moment_coarse d e).trans (Real.exp_le_exp.mpr _)
-    have hD0 : 0 ≤ 1+heightDecay 1 := by have := heightDecay_nonneg 1; linarith
-    have ha := mul_le_mul_of_nonneg_left hlow (mul_nonneg (by norm_num : (0:ℝ)≤256) hD0)
-    have hb := mul_le_mul_of_nonneg_right hD hpow
-    have hc := mul_le_mul_of_nonneg_right (ha.trans hb) hk0
-    nlinarith
-
-lemma palindromeRowMoment_nonneg (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (a : ℝ) : 0 ≤ palindromeRowMoment d e a := by
-  apply finiteMean_nonneg
-  intro f
-  exact Real.rpow_nonneg (mul_nonneg (Nat.cast_nonneg _)
-    (PairRouting.tupleProbability_nonneg _ _ _)) _
-
-theorem palindrome_log_row_moment (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι) :
-    Real.log (palindromeRowMoment d e (1/1000)) ≤ densityConstant*Fintype.card ι*
-        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ) := by
-  by_cases hz : palindromeRowMoment d e (1/1000) = 0
-  · rw [hz,Real.log_zero]
-    exact mul_nonneg (mul_nonneg (le_of_lt densityConstant_pos) (Nat.cast_nonneg _))
-      (Real.rpow_nonneg (by positivity) _)
-  · exact (Real.log_le_iff_le_exp (lt_of_le_of_ne (palindromeRowMoment_nonneg _ _ _) (Ne.symm hz))).mpr
-      (palindrome_row_moment_bound d e hk)
-
-end Thorp
-
-open scoped BigOperators
-namespace Thorp.RoutingNetwork
-variable {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι] [DecidableEq ι]
-  {d : ℕ} (R : RoutingNetwork α ι d)
-
-omit [DecidableEq α] in
-lemma single_min_eq_within (o : α → Option (Fin 1)) (u v : α) :
-    min (pathWeight o u) (pathWeight o v) = withinWeight o u v := by
-  cases hu : o u with
-  | none =>
-    cases hv : o v with
-    | none => simp [pathWeight, withinWeight, hu, hv]
-    | some j =>
-      have hj : 0 ≤ slotWeight o j := by unfold slotWeight; positivity
-      simp [pathWeight, withinWeight, hu, hv, min_eq_left hj]
-  | some i =>
-    cases hv : o v with
-    | none =>
-      have hi : 0 ≤ slotWeight o i := by unfold slotWeight; positivity
-      simp [pathWeight, withinWeight, hu, hv, min_eq_right hi]
-    | some j =>
-      have hij : i = j := Subsingleton.elim _ _
-      simp [pathWeight, withinWeight, hu, hv, hij]
-
-lemma single_rotationCharge_mean_bound {o : α → Option (Fin 1)}
-    [∀ i : Fin 1, Nonempty {x : α // o x = some i}]
-    (b : ι → Bool) :
-    finiteMean (rotationCharge R o b) ≤
-      Real.log (cycleFiber o 0).card / Real.log 2 := by
-  classical
-  have hn : ∀ i, (cycleFiber o i).Nonempty := by
-    intro i
-    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : α // o x = some i})
-    exact ⟨x, (mem_cycleFiber _ _ _).mpr hx⟩
-  have hm : Monotone (fun i => (cycleFiber o i).card) := by
-    intro i j _
-    have : i = j := Subsingleton.elim _ _
-    simp [this]
-  unfold rotationCharge
-  rw [finiteMean_sum]
-  simp only [rotationMeeting_mean hm, single_min_eq_within, ← two_mul]
-  rw [← Finset.mul_sum]
-  have h := withinWeight_sum_bound R o hn b
-  simp only [Fin.sum_univ_one] at h
-  rw [mul_comm 2 (Real.log 2), div_mul_eq_div_div] at h
-  linarith
-
-lemma single_rotation_exp_bound {o : α → Option (Fin 1)}
-    [∀ i : Fin 1, Nonempty {x : α // o x = some i}]
-    (b : ι → Bool) :
-    1 / (cycleFiber o 0).card ≤
-      finiteMean (fun z : Terminal o => Real.exp (-Real.log 2 * rotationCharge R o b z)) := by
-  have h := single_rotationCharge_mean_bound R (o := o) b
-  have hlog : 0 < Real.log 2 := Real.log_pos (by norm_num)
-  have hcard : (0 : ℝ) < (cycleFiber o 0).card := by
-    apply Nat.cast_pos.mpr
-    apply Finset.card_pos.mpr
-    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : α // o x = some 0})
-    exact ⟨x, (mem_cycleFiber _ _ _).mpr hx⟩
-  have hm : -Real.log (cycleFiber o 0).card ≤
-      finiteMean (fun z : Terminal o => -Real.log 2 * rotationCharge R o b z) := by
-    have he := finiteMean_mul_const (rotationCharge R o b) (-Real.log 2)
-    simp only [mul_comm] at he
-    rw [he]
-    have hh := mul_le_mul_of_nonpos_left h (neg_nonpos.mpr hlog.le)
-    calc
-      _ = -Real.log 2 * (Real.log (cycleFiber o 0).card / Real.log 2) := by
-        field_simp
-      _ ≤ _ := hh
-  have he := (Real.exp_le_exp.mpr hm).trans (exp_finiteMean_le _)
-  simpa only [Real.exp_neg, Real.exp_log hcard, one_div] using he
-
-lemma single_collection_rotation_lower (S : Finset α) (r : ℕ)
-    (ω : ι → Bool) (c : CycleSlots (R.perm ω) S (fun _ : Fin 1 => r)) :
-    (2:ℝ)^d ≤ ∑ z : Terminal c.val,
-      (2:ℝ)^d * Real.exp (-Real.log 2 * rotationCharge R c.val ω z) := by
-  classical
-  let (i : Fin 1) : Nonempty {x : α // c.val x = some i} := c.terminal_nonempty i
-  have hb := single_rotation_exp_bound R ω (o := c.val)
-  have hcard : (Fintype.card (Terminal c.val) : ℝ) = (cycleFiber c.val 0).card := by
-    rw [Fintype.card_pi]
-    simp only [Fin.prod_univ_one, card_cycleFiber]
-  have hp : (0:ℝ) < Fintype.card (Terminal c.val) := Nat.cast_pos.mpr Fintype.card_pos
-  have hh := (le_div_iff₀ hp).mp hb
-  rw [hcard, one_div_mul_cancel (ne_of_gt (by simpa only [hcard] using hp))] at hh
-  rw [← Finset.mul_sum]
-  simpa only [mul_one] using mul_le_mul_of_nonneg_left hh (show 0 ≤ (2:ℝ)^d by positivity)
-
-lemma single_cycleSlots_weighted_moment (S : Finset α) (r : ℕ) :
-    finiteMean (fun ω : ι → Bool =>
-      (Fintype.card (CycleSlots (R.perm ω) S (fun _ : Fin 1 => r)) : ℝ)) *
-      (2:ℝ)^d ≤ S.card := by
-  classical
-  let rs : Fin 1 → ℕ := fun _ => r
-  let g (a : Fin 1 → S) (ω : ι → Bool) : ℝ :=
-    if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,rs i))) ∅).succeeds ω then
-      ((2:ℝ)^d)^1 / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,rs i))) ∅).staleCharge ∅ ω
-    else 0
-  have hω (ω : ι → Bool) :
-      (Fintype.card (CycleSlots (R.perm ω) S rs) : ℝ) * (2:ℝ)^d ≤
-        ∑ a : Fin 1 → S, g a ω := by
-    calc
-      _ = ∑ _c : CycleSlots (R.perm ω) S rs, (2:ℝ)^d := by simp
-      _ ≤ ∑ c : CycleSlots (R.perm ω) S rs,
-          ∑ z : Terminal c.val, ((2:ℝ)^d)^1 * Real.exp (-Real.log 2 * rotationCharge R c.val ω z) := by
-        apply Finset.sum_le_sum
-        intro c _
-        simpa only [pow_one] using single_collection_rotation_lower R S r ω c
-      _ = ∑ c : RootedSlots (R.perm ω) S rs,
-          ((2:ℝ)^d)^1 * Real.exp (-Real.log 2 * rotationCharge R c.1.val ω c.2) :=
-        (Fintype.sum_sigma (fun c : RootedSlots (R.perm ω) S rs =>
-          ((2:ℝ)^d)^1 * Real.exp (-Real.log 2 * rotationCharge R c.1.val ω c.2))).symm
-      _ ≤ _ := rooted_weight_sum_le R S rs ω
-  have hb := finiteMean_mono hω
-  rw [finiteMean_mul_const, finiteMean_sum] at hb
-  calc
-    _ ≤ ∑ a : Fin 1 → S, finiteMean (g a) := hb
-    _ ≤ ∑ _a : Fin 1 → S, (1:ℝ) := by
-      apply Finset.sum_le_sum
-      intro a _
-      have hh := QueryTree.cycles_weighted_success R S (List.ofFn (fun i : Fin 1 => ((a i).val,rs i)))
-      simpa only [List.length_ofFn] using hh
-    _ = _ := by simp
-
-noncomputable def singleRootedEquiv (p : Equiv.Perm α) (r : ℕ) :
-    RootedSlots p Finset.univ (fun _ : Fin 1 => r) ≃
-      {x : α // (orbitSet p x).card = r + 1} := by
-  classical
-  let f : RootedSlots p Finset.univ (fun _ : Fin 1 => r) →
-      {x : α // (orbitSet p x).card = r + 1} := fun c =>
-    ⟨(c.2 0).val, by rw [c.1.property.2.2 0 _ (c.2 0).property, c.1.property.1]⟩
-  apply Equiv.ofBijective f
-  constructor
-  · intro c c' h
-    apply rootedStart_injective
-    funext i
-    have hi : i = 0 := Subsingleton.elim _ _
-    subst i
-    apply Subtype.ext
-    exact congrArg p (congrArg Subtype.val h)
-  · intro x
-    let C : SelectedCycle p Finset.univ := ⟨orbitSet p x, ⟨⟨x,rfl⟩, Finset.subset_univ _⟩⟩
-    let a : SlotAssignment p Finset.univ (fun _ : Fin 1 => r) :=
-      ⟨fun _ => C, Function.injective_of_subsingleton _, fun _ => x.property⟩
-    let c := assignmentToSlots a
-    have hx : ∀ i : Fin 1, c.val x = some i := by
-      intro i
-      change assignmentOwner a x = some i
-      rw [assignmentOwner_eq_some]
-      exact (mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.refl p x)
-    exact ⟨⟨c, fun i => ⟨x, hx i⟩⟩, Subtype.ext rfl⟩
-
-lemma single_root_count (p : Equiv.Perm α) (r : ℕ) :
-    (Fintype.card {x : α // (orbitSet p x).card = r+1} : ℝ) =
-      (Fintype.card (CycleSlots p Finset.univ (fun _ : Fin 1 => r)) : ℝ) * (r+1) := by
-  classical
-  rw [← Fintype.card_congr (singleRootedEquiv p r), Fintype.card_sigma]
-  simp only [Fintype.card_pi, Fin.prod_univ_one, card_cycleFiber]
-  simp only [Nat.cast_sum]
-  have he (c : CycleSlots p Finset.univ (fun _ : Fin 1 => r)) :
-      ((cycleFiber c.val 0).card : ℝ) = (r : ℝ) + 1 := by
-    rw [c.property.1]; norm_cast
-  simp only [he, Finset.sum_const, Finset.card_univ, nsmul_eq_mul]
-
-lemma uniform_cycle_length_cap [Nonempty α] (r : ℕ) :
-    finiteMean (fun ω : ι → Bool => finiteMean (fun x : α =>
-      if (orbitSet (R.perm ω) x).card = r+1 then (1:ℝ) else 0)) ≤
-      ((r : ℝ)+1) / (2:ℝ)^d := by
-  classical
-  have hn : 0 < (Fintype.card α : ℝ) := Nat.cast_pos.mpr Fintype.card_pos
-  have hinner (ω : ι → Bool) :
-      finiteMean (fun x : α => if (orbitSet (R.perm ω) x).card = r+1 then (1:ℝ) else 0) =
-        (Fintype.card (CycleSlots (R.perm ω) Finset.univ (fun _ : Fin 1 => r)) : ℝ) *
-          (((r:ℝ)+1) / Fintype.card α) := by
-    rw [← mul_div_assoc, ← single_root_count (R.perm ω) r]
-    simp [finiteMean, Fintype.card_subtype]
-  simp only [hinner, finiteMean_mul_const]
-  have h := single_cycleSlots_weighted_moment R Finset.univ r
-  simp only [Finset.card_univ] at h
-  have h' := (le_div_iff₀ (show 0 < (2:ℝ)^d by positivity)).mpr h
-  calc
-    _ ≤ ((Fintype.card α : ℝ) / (2:ℝ)^d) * (((r:ℝ)+1) / Fintype.card α) :=
-      mul_le_mul_of_nonneg_right h' (by positivity)
-    _ = _ := by field_simp
-
-end Thorp.RoutingNetwork
-
-
-namespace Thorp.DenseTruncation
-open scoped BigOperators
-
-lemma law_reciprocal_summable {j : ℕ} {p : ℕ → ℝ}
-    (hp : AdmissibleCycleLaw j p) :
-    Summable (fun l => p l / ((l + 1 : ℕ) : ℝ)) := by
-  apply Summable.of_nonneg_of_le (fun l => div_nonneg (hp.1 l) (by positivity))
-    (fun l => ?_) hp.2.1.summable
-  exact div_le_self (hp.1 l) (by exact_mod_cast Nat.succ_pos l)
-
-lemma law_reciprocal_nonneg {j : ℕ} {p : ℕ → ℝ}
-    (hp : AdmissibleCycleLaw j p) :
-    0 ≤ ∑' l : ℕ, p l / ((l + 1 : ℕ) : ℝ) :=
-  tsum_nonneg (fun l => div_nonneg (hp.1 l) (by positivity))
-
-lemma law_reciprocal_le_one {j : ℕ} {p : ℕ → ℝ}
-    (hp : AdmissibleCycleLaw j p) :
-    (∑' l : ℕ, p l / ((l + 1 : ℕ) : ℝ)) ≤ 1 := by
-  rw [← hp.2.1.tsum_eq]
-  exact Summable.tsum_le_tsum (fun l => div_le_self (hp.1 l)
-    (by exact_mod_cast Nat.succ_pos l)) (law_reciprocal_summable hp) hp.2.1.summable
-
-lemma reciprocal_values_bdd (j : ℕ) : BddAbove
-    {z : ℝ | ∃ p : ℕ → ℝ, AdmissibleCycleLaw j p ∧
-      z = ∑' l : ℕ, p l / ((l + 1 : ℕ) : ℝ)} :=
-  ⟨1, fun _ ⟨_,hp,hz⟩ => hz ▸ law_reciprocal_le_one hp⟩
-
-lemma reciprocal_le_h {j : ℕ} {p : ℕ → ℝ}
-    (hp : AdmissibleCycleLaw j p) :
-    (∑' l : ℕ, p l / ((l + 1 : ℕ) : ℝ)) ≤ h j :=
-  le_csSup (reciprocal_values_bdd j) ⟨p,hp,rfl⟩
-
-lemma subprob_reciprocal_le_h {j : ℕ} {q : ℕ → ℝ}
-    (hq : ∀ l, 0 ≤ q l) (hs : Summable q) (ht : (∑' l, q l) ≤ 1)
-    (hc : ∀ l, q l ≤ ((l+1 : ℕ) : ℝ) / (2:ℝ)^(j-1)) :
-    (∑' l, q l / ((l+1 : ℕ) : ℝ)) ≤ h j := by
-  let t := 2^(j-1)-1
-  have hpow : 1 ≤ 2^(j-1) := Nat.one_le_pow _ _ (by norm_num)
-  have htlen : (t+1 : ℕ) = 2^(j-1) := Nat.sub_add_cancel hpow
-  have hreal : ((t+1 : ℕ) : ℝ) = (2:ℝ)^(j-1) := by rw [htlen]; norm_cast
-  let m := 1 - ∑' l, q l
-  have hm : 0 ≤ m := sub_nonneg.mpr ht
-  let a : ℕ → ℝ := fun l => if l=t then m else 0
-  have ha : HasSum a m := by
-    simpa only [a,ite_eq_left rfl] using (hasSum_single t (f := a) (fun l hl => by simp [a,hl]))
-  let p : ℕ → ℝ := fun l => q l+a l
-  have hp : AdmissibleCycleLaw j p := by
-    refine ⟨fun l => add_nonneg (hq l) (by dsimp [a]; split_ifs <;> positivity), ?_, ?_⟩
-    · convert hs.hasSum.add ha using 1; dsimp [p,m]; ring
-    · intro l
-      by_cases hl : l=t
-      · subst l
-        have hqt := hs.le_tsum t (fun l _ => hq l)
-        have hp2 : (2:ℝ)^(j-1) ≠ 0 := by positivity
-        simp only [p,a,hreal,div_self hp2]
-        dsimp [m]
-        linarith
-      · simpa only [p,a,ite_eq_right hl,add_zero] using hc l
-  have hqs : Summable (fun l => q l / ((l+1 : ℕ) : ℝ)) := by
-    apply Summable.of_nonneg_of_le (fun l => div_nonneg (hq l) (by positivity))
-      (fun l => div_le_self (hq l) (by exact_mod_cast Nat.succ_pos l)) hs
-  calc
-    _ ≤ ∑' l, p l / ((l+1 : ℕ) : ℝ) := by
-      apply Summable.tsum_le_tsum _ hqs (law_reciprocal_summable hp)
-      intro l
-      apply div_le_div_of_nonneg_right _ (by positivity)
-      dsimp [p,a]
-      split_ifs <;> linarith
-    _ ≤ _ := reciprocal_le_h hp
-
-end Thorp.DenseTruncation
-
-
-end ThorpNine.Dense
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Dense.CutoffMass
+
+namespace OAI
+
+namespace ThorpNine.Dense
+
+namespace Thorp
+
+lemma familyEnd_two (L n : ℕ) : 2*familyEnd L n ≤ L*4^n := by
+  cases n with
+  | zero => simp [familyEnd]
+  | succ n => simp only [familyEnd,Nat.succ_ne_zero,ite_false,Nat.add_one_sub_one,pow_succ]; nlinarith
+
+lemma familyEnd_ge (L n j : ℕ) (hj : j < n) : 2*(L*4^j) ≤ familyEnd L n := by
+  cases n with
+  | zero => omega
+  | succ n =>
+    simp only [familyEnd,Nat.succ_ne_zero,ite_false,Nat.add_one_sub_one]
+    apply Nat.mul_le_mul_left
+    exact Nat.mul_le_mul_left L (Nat.pow_le_pow_right (by decide) (by omega : j ≤ n))
+
+noncomputable def palindromeFamilyCost (L n d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) : ℕ :=
+  ∑ j ∈ Finset.range n, palindromeSlabCost (L*4^j) d e ω
+
+lemma palindromeFamilyCost_adapted (L n d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) :
+    LayerAdapted d (familyEnd L n) (fun Y => palindromeFamilyCost L n d e (Y,X)) := by
+  apply layerAdapted_sum
+  intro j hj
+  exact layerAdapted_mono (palindromeSlabCost_adapted (L*4^j) d e X)
+    (familyEnd_ge L n j (Finset.mem_range.mp hj))
+
+theorem palindrome_family_mgf (L n d : ℕ) (hL : 0 < L) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) ≤
+      Real.exp (Fintype.card ι * (68*∑ j ∈ Finset.range n,
+        Real.exp (-((L*4^j:ℕ):ℝ)/64))) := by
+  induction n with
+  | zero => simp only [palindromeFamilyCost,Finset.range_zero,Finset.sum_empty,pow_zero,
+      finiteMean_const,mul_zero,Real.exp_zero,le_refl]
+  | succ n ih =>
+    let l := L*4^n
+    have hl : 0 < l := Nat.mul_pos hL (pow_pos (by decide) _)
+    have hs : 2*familyEnd L n ≤ l := familyEnd_two L n
+    have hc : ∀ Y, palindromeFamilyCost L (n+1) d e (Y,X) =
+        palindromeFamilyCost L n d e (Y,X)+palindromeSlabCost l d e (Y,X) := by
+      intro Y
+      exact Finset.sum_range_succ _ _
+    have hm : finiteMean (fun Y => q^(palindromeFamilyCost L (n+1) d e (Y,X))) ≤
+        finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) *
+          Real.exp (Fintype.card ι*(68*Real.exp (-(l:ℝ)/64))) := by
+      by_cases hld : l ≤ d
+      · simp_rw [hc,pow_add]
+        exact palindrome_slab_adapted_mul_le d (familyEnd L n) l (by omega) hl hs e X _
+          (fun _ => pow_nonneg (by linarith) _)
+          (layerAdapted_comp (palindromeFamilyCost_adapted L n d e X) (fun c => q^c))
+          q hq hlog
+      · simp_rw [hc,palindromeSlabCost_gt l d (by omega) e,Nat.add_zero]
+        exact le_mul_of_one_le_right
+          (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
+          (Real.one_le_exp_iff.mpr (by positivity))
+    refine (hm.trans (mul_le_mul_of_nonneg_right ih (Real.exp_nonneg _))).trans_eq ?_
+    rw [←Real.exp_add,Finset.sum_range_succ]
+    congr 1
+    dsimp [l]
+    ring
+
+lemma pow_four_ge (j : ℕ) : j+1 ≤ 4^j := by
+  induction j with
+  | zero => norm_num
+  | succ j ih => rw [pow_succ]; omega
+
+lemma sum_family_decay (L n : ℕ) (hL : 0 < L) :
+    (∑ j ∈ Finset.range n, Real.exp (-((L*4^j:ℕ):ℝ)/64)) ≤
+      Real.exp (-(L:ℝ)/64)/(1-Real.exp (-(1:ℝ)/64)) := by
+  let r := Real.exp (-(1:ℝ)/64)
+  have hr : 0 < r := Real.exp_pos _
+  have hr1 : r < 1 := Real.exp_lt_one_iff.mpr (by norm_num)
+  have hg : (∑ j ∈ Finset.range n, r^j) ≤ 1/(1-r) := by
+    apply (le_div_iff₀ (by linarith : (0:ℝ)<1-r)).mpr
+    have hh := geom_sum_mul r n
+    nlinarith [pow_nonneg (le_of_lt hr) n]
+  calc
+    _ ≤ ∑ j ∈ Finset.range n, Real.exp (-(L:ℝ)/64)*r^j := by
+      apply Finset.sum_le_sum
+      intro j _
+      have hj := pow_four_ge j
+      have hlj : L+j ≤ L*4^j := by nlinarith
+      have hlj' : (L:ℝ)+j ≤ (L*4^j:ℕ) := by exact_mod_cast hlj
+      rw [←Real.exp_nat_mul,←Real.exp_add]
+      exact Real.exp_le_exp.mpr (by linarith)
+    _ = Real.exp (-(L:ℝ)/64)*(∑ j ∈ Finset.range n, r^j) := by rw [Finset.mul_sum]
+    _ ≤ Real.exp (-(L:ℝ)/64)*(1/(1-r)) :=
+      mul_le_mul_of_nonneg_left hg (Real.exp_nonneg _)
+    _ = _ := by ring
+
+
+lemma palindromeLevelCost_step_add (t d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (d+1)) (ω : BenesCoins (d+1)) :
+    palindromeLevelCost t (d+1) e ω =
+      let σ := benesStepEquiv d ω
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x σ.2.1
+      let hc := PairRouting.colors_compatible x σ.2.1
+      (if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
+          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) else 0) +
+        palindromeLevelCost t d (PairRouting.childEmbedding x c hc false) σ.1.1 +
+        palindromeLevelCost t d (PairRouting.childEmbedding x c hc true) σ.1.2 := by
+  classical
+  rw [palindromeLevelCost]
+  dsimp only
+  split_ifs with h
+  · simp only [palindromeLevelCost_gt t d (by omega),Nat.add_zero]
+  · simp only [Nat.zero_add]
+
+lemma palindromeLowCost_sum (H d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) :
+    palindromeLowCost H d e ω = ∑ i ∈ Finset.range H, palindromeLevelCost (i+1) d e ω := by
+  classical
+  induction d generalizing ι with
+  | zero => simp only [palindromeLowCost,palindromeLevelCost,Finset.sum_const_zero]
+  | succ d ih =>
+    rw [palindromeLowCost]
+    simp only [palindromeLevelCost_step_add,Finset.sum_add_distrib]
+    rw [←ih,←ih]
+    congr 2
+    simp only [Nat.add_right_cancel_iff,Finset.sum_ite_eq',Finset.mem_range,Nat.succ_le_iff]
+
+lemma palindromeLowCost_slab (L d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) :
+    palindromeLowCost L d e ω + palindromeSlabCost L d e ω = palindromeLowCost (2*L) d e ω := by
+  rw [palindromeLowCost_sum,palindromeLowCost_sum,show 2*L=L+L by omega,Finset.sum_range_add]
+  congr 1
+  exact Fin.sum_univ_eq_sum_range (fun i => palindromeLevelCost (L+i+1) d e ω) L
+
+lemma palindromeCost_family (L n d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) :
+    palindromeLowCost L d e ω + palindromeFamilyCost L n d e ω +
+      palindromeFamilyCost (2*L) n d e ω = palindromeLowCost (L*4^n) d e ω := by
+  induction n with
+  | zero => simp only [palindromeFamilyCost,Finset.range_zero,Finset.sum_empty,Nat.add_zero,
+      pow_zero,mul_one]
+  | succ n ih =>
+    have he : (2*L)*4^n = 2*(L*4^n) := by ring
+    have he' : L*4^(n+1) = 2*(2*(L*4^n)) := by rw [pow_succ]; ring
+    simp only [palindromeFamilyCost,Finset.sum_range_succ] at ih ⊢
+    rw [he,he',←palindromeLowCost_slab,←palindromeLowCost_slab,←ih]
+    omega
+
+
+lemma finiteMean_mul_sq_le {Ω : Type*} [Fintype Ω] (f g : Ω → ℝ) :
+    (finiteMean (fun ω => f ω*g ω))^2 ≤ finiteMean (fun ω => (f ω)^2)*finiteMean (fun ω => (g ω)^2) := by
+  have h := Finset.sum_mul_sq_le_sq_mul_sq Finset.univ f g
+  simp only [finiteMean,div_pow,div_mul_div_comm,←pow_two]
+  exact div_le_div_of_nonneg_right h (sq_nonneg _)
+
+lemma finiteMean_mul_exp_le {Ω : Type*} [Fintype Ω] (f g : Ω → ℝ) (A B : ℝ)
+    (hf : finiteMean (fun ω => (f ω)^2) ≤ Real.exp A)
+    (hg : finiteMean (fun ω => (g ω)^2) ≤ Real.exp B) :
+    finiteMean (fun ω => f ω*g ω) ≤ Real.exp ((A+B)/2) := by
+  have hc := finiteMean_mul_sq_le f g
+  have hn : 0 ≤ finiteMean (fun ω => (g ω)^2) := finiteMean_nonneg (fun _ => sq_nonneg _)
+  have hu := mul_le_mul hf hg hn (Real.exp_nonneg _)
+  have he : Real.exp A*Real.exp B = (Real.exp ((A+B)/2))^2 := by
+    rw [←Real.exp_add,←Real.exp_nat_mul]
+    congr 1
+    push_cast
+    ring
+  rw [he] at hu
+  have hh := hc.trans hu
+  have hep := Real.exp_pos ((A+B)/2)
+  nlinarith
+
+
+noncomputable def heightDecay (H : ℕ) : ℝ :=
+  68*Real.exp (-(H:ℝ)/64)/(1-Real.exp (-(1:ℝ)/64))
+
+lemma heightDecay_nonneg (H : ℕ) : 0 ≤ heightDecay H := by
+  have h : Real.exp (-(1:ℝ)/64) < 1 := Real.exp_lt_one_iff.mpr (by norm_num)
+  apply div_nonneg (by positivity) (by linarith)
+
+lemma heightDecay_antitone : Antitone heightDecay := by
+  intro L H h
+  have hden : 0 ≤ 1-Real.exp (-(1:ℝ)/64) := by
+    have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
+    linarith
+  apply div_le_div_of_nonneg_right _ hden
+  apply mul_le_mul_of_nonneg_left _ (by norm_num)
+  apply Real.exp_le_exp.mpr
+  have h' : (L:ℝ) ≤ H := by exact_mod_cast h
+  linarith
+
+lemma palindrome_family_mgf_decay (L n d : ℕ) (hL : 0 < L) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) ≤
+      Real.exp (Fintype.card ι*heightDecay L) := by
+  apply (palindrome_family_mgf L n d hL e X q hq hlog).trans
+  apply Real.exp_le_exp.mpr
+  apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
+  have h := mul_le_mul_of_nonneg_left (sum_family_decay L n hL) (by norm_num : (0:ℝ)≤68)
+  simpa only [heightDecay,mul_div_assoc] using h
+
+theorem palindrome_high_mgf (H n d : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/500) :
+    finiteMean (fun Y => q^(palindromeFamilyCost H n d e (Y,X)+
+      palindromeFamilyCost (2*H) n d e (Y,X))) ≤ Real.exp (Fintype.card ι*heightDecay H) := by
+  have hq2 := one_le_pow₀ (n:=2) hq
+  have hlog2 : Real.log (q^2) ≤ 1/250 := by rw [Real.log_pow]; norm_num; linarith
+  have h₀ := palindrome_family_mgf_decay H n d hH e X (q^2) hq2 hlog2
+  have h₁ := palindrome_family_mgf_decay (2*H) n d (by omega) e X (q^2) hq2 hlog2
+  have h₁' := h₁.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left
+    (heightDecay_antitone (show H ≤ 2*H by omega)) (Nat.cast_nonneg _)))
+  have hh := finiteMean_mul_exp_le (fun Y => q^(palindromeFamilyCost H n d e (Y,X)))
+    (fun Y => q^(palindromeFamilyCost (2*H) n d e (Y,X)))
+    (Fintype.card ι*heightDecay H) (Fintype.card ι*heightDecay H)
+    (by simpa only [pow_right_comm] using h₀) (by simpa only [pow_right_comm] using h₁')
+  simpa only [←pow_add,add_self_div_two] using hh
+
+lemma palindrome_high_full_mgf (H n d : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/500) :
+    finiteMean (fun ω : BenesCoins d => q^(palindromeFamilyCost H n d e ω+
+      palindromeFamilyCost (2*H) n d e ω)) ≤ Real.exp (Fintype.card ι*heightDecay H) := by
+  rw [finiteMean_prod,finiteMean_comm]
+  calc
+    _ ≤ finiteMean (fun _X : SwitchIndex d → Bool => Real.exp (Fintype.card ι*heightDecay H)) := by
+      apply finiteMean_mono
+      intro X
+      exact palindrome_high_mgf H n d hH e X q hq hlog
+    _ = _ := finiteMean_const _
+
+theorem palindrome_cost_mgf_cutoff (r H : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (H+r)) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/1000)
+    (hr : (2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*(q^2)^H) ≤ 1/2) :
+    finiteMean (fun ω : BenesCoins (H+r) => q^(palindromeCost (H+r) e ω)) ≤
+      Real.exp (((2:ℝ)^r*(2*((2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*(q^2)^H))^2)+
+        Fintype.card ι*heightDecay H)/2) := by
+  let d := H+r
+  have hd : d ≤ H*4^d := by
+    have hh := pow_four_ge d
+    nlinarith
+  have hcost (ω : BenesCoins d) : palindromeCost d e ω = palindromeLowCost H d e ω+
+      (palindromeFamilyCost H d d e ω+palindromeFamilyCost (2*H) d d e ω) := by
+    have hh := palindromeCost_family H d d e ω
+    rw [palindromeLowCost_eq _ _ hd] at hh
+    simpa only [Nat.add_assoc] using hh.symm
+  have hq2 := one_le_pow₀ (n:=2) hq
+  have hlog2 : Real.log (q^2) ≤ 1/500 := by rw [Real.log_pow]; norm_num; linarith
+  have hlow := palindrome_low_mgf r H e (q^2) hq2 hr
+  have hhigh := palindrome_high_full_mgf H d d hH e (q^2) hq2 hlog2
+  have hh := finiteMean_mul_exp_le (fun ω : BenesCoins d => q^(palindromeLowCost H d e ω))
+    (fun ω : BenesCoins d => q^(palindromeFamilyCost H d d e ω+palindromeFamilyCost (2*H) d d e ω))
+    _ _ (by simpa only [pow_right_comm] using hlow) (by simpa only [pow_right_comm] using hhigh)
+  change finiteMean (fun ω : BenesCoins d => q^(palindromeCost d e ω)) ≤ _
+  simpa only [hcost,pow_add,pow_right_comm] using hh
+
+
+lemma cutoffMass_le (H d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (ω : BenesCoins d) :
+    cutoffMass H d e ω ≤ Fintype.card ι := by
+  classical
+  induction d generalizing ι with
+  | zero => exact Nat.zero_le _
+  | succ d ih =>
+    rw [cutoffMass]
+    split_ifs
+    · exact crowdedMass_le _
+    · exact (Nat.add_le_add (ih _ _) (ih _ _)).trans_eq (PairRouting.color_card_add _)
+
+lemma palindromeLowCost_le (H d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) : palindromeLowCost H d e ω ≤ H*Fintype.card ι :=
+  (palindromeLowCost_le_cutoff H d e ω).trans (Nat.mul_le_mul_left H (cutoffMass_le H d e ω))
+
+theorem palindrome_cost_mgf_coarse (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/1000) :
+    finiteMean (fun ω : BenesCoins d => q^(palindromeCost d e ω)) ≤
+      Real.exp (Fintype.card ι*(1+heightDecay 1)) := by
+  have hd : d ≤ 1*4^d := by have hh := pow_four_ge d; omega
+  have hcost (ω : BenesCoins d) : palindromeCost d e ω = palindromeLowCost 1 d e ω+
+      (palindromeFamilyCost 1 d d e ω+palindromeFamilyCost 2 d d e ω) := by
+    have hh := palindromeCost_family 1 d d e ω
+    rw [palindromeLowCost_eq _ _ hd] at hh
+    simpa only [Nat.add_assoc] using hh.symm
+  have hq2 := one_le_pow₀ (n:=2) hq
+  have hlog2 : Real.log (q^2) ≤ 1/500 := by rw [Real.log_pow]; norm_num; linarith
+  have hlow : finiteMean (fun ω : BenesCoins d => (q^2)^(palindromeLowCost 1 d e ω)) ≤
+      Real.exp (Fintype.card ι) := by
+    calc
+      _ ≤ finiteMean (fun _ω : BenesCoins d => (q^2)^(Fintype.card ι)) := by
+        apply finiteMean_mono
+        intro ω
+        apply pow_le_pow_right₀ hq2
+        simpa only [one_mul] using palindromeLowCost_le 1 d e ω
+      _ = (q^2)^(Fintype.card ι) := finiteMean_const _
+      _ = Real.exp (Fintype.card ι * Real.log (q^2)) := by
+        rw [Real.exp_nat_mul,Real.exp_log (by positivity : 0<q^2)]
+      _ ≤ _ := Real.exp_le_exp.mpr (by
+        have hh := mul_le_mul_of_nonneg_left hlog2 (Nat.cast_nonneg (Fintype.card ι) : (0:ℝ)≤_)
+        nlinarith [Nat.cast_nonneg (α:=ℝ) (Fintype.card ι)])
+  have hhigh := palindrome_high_full_mgf 1 d d (by decide) e (q^2) hq2 hlog2
+  have hh := finiteMean_mul_exp_le (fun ω : BenesCoins d => q^(palindromeLowCost 1 d e ω))
+    (fun ω : BenesCoins d => q^(palindromeFamilyCost 1 d d e ω+palindromeFamilyCost 2 d d e ω))
+    (Fintype.card ι) (Fintype.card ι*heightDecay 1)
+    (by simpa only [pow_right_comm] using hlow) (by simpa only [pow_right_comm] using hhigh)
+  simp only [←pow_add,←hcost] at hh
+  apply hh.trans (Real.exp_le_exp.mpr _)
+  have hk : (0:ℝ) ≤ Fintype.card ι := Nat.cast_nonneg _
+  have hD := heightDecay_nonneg 1
+  nlinarith
+
+noncomputable def momentBase : ℝ := (2:ℝ)^(1/1000:ℝ)
+
+lemma momentBase_one_le : 1 ≤ momentBase := by
+  exact Real.one_le_rpow (by norm_num) (by norm_num)
+
+lemma momentBase_log : Real.log momentBase ≤ 1/1000 := by
+  rw [momentBase,Real.log_rpow (by norm_num)]
+  have hh := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
+  norm_num at hh ⊢
+  linarith
+
+lemma momentBase_pow (c : ℕ) : momentBase^c = (2:ℝ)^((c:ℝ)*(1/1000:ℝ)) := by
+  rw [mul_comm,Real.rpow_mul_natCast (by norm_num)]
+  rfl
+
+lemma palindromeRowMoment_le_cost (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) :
+    palindromeRowMoment d e (1/1000) ≤
+      finiteMean (fun ω : BenesCoins d => momentBase^(palindromeCost d e ω)) := by
+  have hn : (0:ℝ) < ((2^d:ℕ):ℝ)^(Fintype.card ι) := by positivity
+  have hn0 : (0:ℝ) ≤ ((2^d).descFactorial (Fintype.card ι):ℝ) /
+      ((2^d:ℕ):ℝ)^(Fintype.card ι) := by positivity
+  have hn1 : ((2^d).descFactorial (Fintype.card ι):ℝ) /
+      ((2^d:ℕ):ℝ)^(Fintype.card ι) ≤ 1 := by
+    apply (div_le_one hn).mpr
+    exact_mod_cast Nat.descFactorial_le_pow (2^d) (Fintype.card ι)
+  have hpow := Real.rpow_le_one hn0 hn1 (by norm_num : (0:ℝ)≤1/1000)
+  have hm := palindrome_row_density_moment d e (1/1000) (by norm_num)
+  simp_rw [←momentBase_pow] at hm
+  exact hm.trans (mul_le_of_le_one_left (finiteMean_nonneg (fun _ =>
+    pow_nonneg (le_trans (by norm_num) momentBase_one_le) _)) hpow)
+
+theorem palindrome_row_moment_coarse (d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) :
+    palindromeRowMoment d e (1/1000) ≤ Real.exp (Fintype.card ι*(1+heightDecay 1)) :=
+  (palindromeRowMoment_le_cost d e).trans
+    (palindrome_cost_mgf_coarse d e momentBase momentBase_one_le momentBase_log)
+
+
+noncomputable def densityCutoff (p : ℝ) : ℕ := ⌊-Real.log p/(4*Real.log 2)⌋₊
+
+lemma log_small_density {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
+    Real.log p ≤ -8*Real.log 2 := by
+  have he : Real.log (1/256:ℝ) = -8*Real.log 2 := by
+    rw [show (1/256:ℝ) = 1/2^8 by norm_num,
+      Real.log_div (by norm_num) (by positivity),Real.log_one,Real.log_pow]
+    ring
+  exact (Real.log_le_log hp hs).trans_eq he
+
+lemma densityCutoff_bounds {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
+    0 < densityCutoff p ∧
+    (densityCutoff p:ℝ)*(4*Real.log 2) ≤ -Real.log p ∧
+    -Real.log p < ((densityCutoff p:ℝ)+1)*(4*Real.log 2) := by
+  have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
+  have hd : 0 < 4*Real.log 2 := by positivity
+  have hh := log_small_density hp hs
+  have hx : 1 ≤ -Real.log p/(4*Real.log 2) := by
+    apply (le_div_iff₀ hd).mpr
+    nlinarith
+  refine ⟨Nat.floor_pos.mpr hx,?_,?_⟩
+  · exact (le_div_iff₀ hd).mp (Nat.floor_le (by linarith : 0 ≤ -Real.log p/(4*Real.log 2)))
+  · exact (div_lt_iff₀ hd).mp (Nat.lt_floor_add_one (-Real.log p/(4*Real.log 2)))
+
+lemma densityCutoff_decay {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
+    Real.exp (-(densityCutoff p:ℝ)/64) ≤ Real.exp (1/64) * p^(1/512:ℝ) := by
+  have hh := (densityCutoff_bounds hp hs).2.2
+  have hb : Real.log 2 ≤ 2 := by
+    have := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
+    linarith
+  have hH : 0 ≤ (densityCutoff p:ℝ)+1 := by positivity
+  have hu := mul_le_mul_of_nonneg_left hb hH
+  rw [Real.rpow_def_of_pos hp,←Real.exp_add]
+  apply Real.exp_le_exp.mpr
+  nlinarith
+
+lemma densityCutoff_power {p q : ℝ} (hp : 0 < p) (hs : p ≤ 1/256)
+    (hq : 0 < q) (hlog : Real.log q ≤ Real.log 2/4) (n : ℕ) (hn : n ≤ 4)
+    (hq1 : 1 ≤ q) :
+    p*(2:ℝ)^(densityCutoff p)*q^(n*densityCutoff p) ≤ p^(1/2:ℝ) := by
+  have hcut := (densityCutoff_bounds hp hs).2.1
+  have hH : 0 ≤ (densityCutoff p:ℝ) := Nat.cast_nonneg _
+  have hn' : (n:ℝ) ≤ 4 := by exact_mod_cast hn
+  have hln := Real.log_nonneg hq1
+  have hl := mul_le_mul_of_nonneg_right hn' hln
+  have hnl : (n:ℝ)*Real.log q ≤ Real.log 2 := by linarith
+  have hm := mul_le_mul_of_nonneg_left hnl hH
+  have heq : p*(2:ℝ)^(densityCutoff p)*q^(n*densityCutoff p) =
+      Real.exp (Real.log p+(densityCutoff p:ℝ)*Real.log 2+
+        (n*densityCutoff p:ℕ)*Real.log q) := by
+    rw [Real.exp_add,Real.exp_add,Real.exp_nat_mul,Real.exp_nat_mul,
+      Real.exp_log hp,Real.exp_log hq,Real.exp_log (by norm_num : (0:ℝ)<2)]
+  rw [heq,Real.rpow_def_of_pos hp]
+  apply Real.exp_le_exp.mpr
+  push_cast
+  nlinarith
+
+lemma densityCutoff_sparse {p q : ℝ} (hp : 0 < p) (hs : p ≤ 1/256)
+    (hq : 0 < q) (hlog : Real.log q ≤ Real.log 2/4) (hq1 : 1 ≤ q) :
+    (2:ℝ)^(densityCutoff p)*(p*(q^2)^(densityCutoff p)) ≤ 1/2 := by
+  have hh := densityCutoff_power hp hs hq hlog 2 (by decide) hq1
+  rw [←pow_mul] at ⊢
+  have hle : p^(1/2:ℝ) ≤ 1/2 := by
+    have hlogp := log_small_density hp hs
+    have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
+    rw [Real.rpow_def_of_pos hp]
+    calc
+      _ ≤ Real.exp (-Real.log 2) := by apply Real.exp_le_exp.mpr; nlinarith
+      _ = 1/2 := by rw [Real.exp_neg,Real.exp_log (by norm_num : (0:ℝ)<2)]; norm_num
+  nlinarith [hh.trans hle]
+
+
+lemma momentBase_log_exact : Real.log momentBase = Real.log 2/1000 := by
+  rw [momentBase,Real.log_rpow (by norm_num)]
+  ring
+
+lemma densityCutoff_le_height (d k : ℕ) (hk : 0 < k) :
+    densityCutoff ((k:ℝ)/(2:ℝ)^d) ≤ d := by
+  have hk' : (1:ℝ) ≤ k := by exact_mod_cast hk
+  have hp : 0 < (k:ℝ)/(2:ℝ)^d := by positivity
+  have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
+  have hlogk := Real.log_nonneg hk'
+  by_cases hx : 0 ≤ -Real.log ((k:ℝ)/(2:ℝ)^d)/(4*Real.log 2)
+  · have hh := (le_div_iff₀ (by positivity : 0 < 4*Real.log 2)).mp (Nat.floor_le hx)
+    have hd : (0:ℝ) ≤ d := Nat.cast_nonneg _
+    have hH : (0:ℝ) ≤ densityCutoff ((k:ℝ)/(2:ℝ)^d) := Nat.cast_nonneg _
+    change (densityCutoff ((k:ℝ)/(2:ℝ)^d):ℝ)*(4*Real.log 2) ≤ -Real.log ((k:ℝ)/(2:ℝ)^d) at hh
+    rw [Real.log_div (by positivity) (by positivity),Real.log_pow] at hh
+    have hle : (densityCutoff ((k:ℝ)/(2:ℝ)^d):ℝ) ≤ d := by nlinarith
+    exact_mod_cast hle
+  · have hz := Nat.floor_of_nonpos (le_of_not_ge hx)
+    simp only [densityCutoff,hz,Nat.zero_le]
+
+noncomputable def densityTailConstant : ℝ :=
+  68*Real.exp (1/64)/(1-Real.exp (-(1:ℝ)/64))
+
+lemma densityTailConstant_nonneg : 0 ≤ densityTailConstant := by
+  have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
+  apply div_nonneg (by positivity) (by linarith)
+
+lemma heightDecay_cutoff {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
+    heightDecay (densityCutoff p) ≤ densityTailConstant*p^(1/512:ℝ) := by
+  have hden : 0 ≤ 1-Real.exp (-(1:ℝ)/64) := by
+    have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
+    linarith
+  have hh := div_le_div_of_nonneg_right
+    (mul_le_mul_of_nonneg_left (densityCutoff_decay hp hs) (by norm_num : (0:ℝ)≤68)) hden
+  simpa only [heightDecay,densityTailConstant,mul_assoc,div_mul_eq_mul_div] using hh
+
+lemma sparse_low_algebra (r H k : ℕ) (q : ℝ) :
+    (2:ℝ)^r*(2*((2:ℝ)^H*(((k:ℝ)/(2:ℝ)^(H+r))*(q^2)^H))^2) =
+    2*k*(((k:ℝ)/(2:ℝ)^(H+r))*(2:ℝ)^H*q^(4*H)) := by
+  rw [pow_add,←pow_mul,show 4*H = (2*H)*2 by omega,pow_mul]
+  have hH : (2:ℝ)^H ≠ 0 := by positivity
+  have hr : (2:ℝ)^r ≠ 0 := by positivity
+  field_simp
+  ring
+
+lemma palindrome_row_moment_small (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι)
+    (hs : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1/256) :
+    palindromeRowMoment d e (1/1000) ≤
+      Real.exp (Fintype.card ι*(1+densityTailConstant)*
+        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ)) := by
+  generalize hpdef : ((Fintype.card ι:ℝ)/(2:ℝ)^d) = p at hs ⊢
+  have hp : 0 < p := by rw [←hpdef]; positivity
+  have hle := densityCutoff_le_height d (Fintype.card ι) hk
+  rw [hpdef] at hle
+  obtain ⟨r,hr⟩ := Nat.exists_eq_add_of_le hle
+  subst d
+  have hH := (densityCutoff_bounds hp hs).1
+  have hq := momentBase_one_le
+  have hqp : 0 < momentBase := by linarith
+  have hlog : Real.log momentBase ≤ Real.log 2/4 := by
+    rw [momentBase_log_exact]
+    have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
+    linarith
+  have hsp := densityCutoff_sparse hp hs hqp hlog hq
+  rw [←hpdef] at hsp
+  have hh := palindrome_cost_mgf_cutoff r (densityCutoff p) hH e momentBase hq momentBase_log
+    (by simpa only [hpdef] using hsp)
+  apply (palindromeRowMoment_le_cost _ e).trans (hh.trans (Real.exp_le_exp.mpr _))
+  rw [sparse_low_algebra,hpdef]
+  have hlow := densityCutoff_power hp hs hqp hlog 4 (by decide) hq
+  have hp1 : p ≤ 1 := by linarith
+  have he := Real.rpow_le_rpow_of_exponent_ge hp hp1 (by norm_num : (1:ℝ)/512 ≤ 1/2)
+  have hlow' := mul_le_mul_of_nonneg_left (hlow.trans he)
+    (by positivity : (0:ℝ)≤2*Fintype.card ι)
+  have hhigh := mul_le_mul_of_nonneg_left (heightDecay_cutoff hp hs)
+    (Nat.cast_nonneg (Fintype.card ι): (0:ℝ)≤_)
+  have hnon : 0 ≤ Fintype.card ι*densityTailConstant*p^(1/512:ℝ) :=
+    mul_nonneg (mul_nonneg (Nat.cast_nonneg _) densityTailConstant_nonneg)
+      (Real.rpow_nonneg (le_of_lt hp) _)
+  nlinarith
+
+noncomputable def densityConstant : ℝ :=
+  256*(1+heightDecay 1)+1+densityTailConstant
+
+lemma densityConstant_pos : 0 < densityConstant := by
+  dsimp [densityConstant]
+  have hh := heightDecay_nonneg 1
+  have ht := densityTailConstant_nonneg
+  linarith
+
+theorem palindrome_row_moment_bound (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι) :
+    palindromeRowMoment d e (1/1000) ≤ Real.exp (densityConstant*Fintype.card ι*
+        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ)) := by
+  have hp : 0 < (Fintype.card ι:ℝ)/(2:ℝ)^d := by positivity
+  have hp1 : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1 := by
+    apply (div_le_one (by positivity)).mpr
+    have h := Fintype.card_le_of_injective e e.injective
+    rw [card_positions] at h
+    exact_mod_cast h
+  have hk0 : (0:ℝ) ≤ Fintype.card ι := Nat.cast_nonneg _
+  have hpow := Real.rpow_nonneg (le_of_lt hp) (1/512:ℝ)
+  by_cases hs : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1/256
+  · apply (palindrome_row_moment_small d e hk hs).trans (Real.exp_le_exp.mpr _)
+    have hD : 1+densityTailConstant ≤ densityConstant := by
+      dsimp [densityConstant]
+      have := heightDecay_nonneg 1
+      linarith
+    have hh := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hD hk0) hpow
+    nlinarith
+  · have hlow : (1:ℝ)/256 ≤ ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ) := by
+      have he := Real.rpow_le_rpow_of_exponent_ge hp hp1 (by norm_num : (1:ℝ)/512 ≤ 1)
+      rw [Real.rpow_one] at he
+      exact (le_of_not_ge hs).trans he
+    have hD : 256*(1+heightDecay 1) ≤ densityConstant := by
+      dsimp [densityConstant]
+      have := densityTailConstant_nonneg
+      linarith
+    apply (palindrome_row_moment_coarse d e).trans (Real.exp_le_exp.mpr _)
+    have hD0 : 0 ≤ 1+heightDecay 1 := by have := heightDecay_nonneg 1; linarith
+    have ha := mul_le_mul_of_nonneg_left hlow (mul_nonneg (by norm_num : (0:ℝ)≤256) hD0)
+    have hb := mul_le_mul_of_nonneg_right hD hpow
+    have hc := mul_le_mul_of_nonneg_right (ha.trans hb) hk0
+    nlinarith
+
+lemma palindromeRowMoment_nonneg (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (a : ℝ) : 0 ≤ palindromeRowMoment d e a := by
+  apply finiteMean_nonneg
+  intro f
+  exact Real.rpow_nonneg (mul_nonneg (Nat.cast_nonneg _)
+    (PairRouting.tupleProbability_nonneg _ _ _)) _
+
+theorem palindrome_log_row_moment (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι) :
+    Real.log (palindromeRowMoment d e (1/1000)) ≤ densityConstant*Fintype.card ι*
+        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ) := by
+  by_cases hz : palindromeRowMoment d e (1/1000) = 0
+  · rw [hz,Real.log_zero]
+    exact mul_nonneg (mul_nonneg (le_of_lt densityConstant_pos) (Nat.cast_nonneg _))
+      (Real.rpow_nonneg (by positivity) _)
+  · exact (Real.log_le_iff_le_exp (lt_of_le_of_ne (palindromeRowMoment_nonneg _ _ _) (Ne.symm hz))).mpr
+      (palindrome_row_moment_bound d e hk)
+
+end Thorp
+
+open scoped BigOperators
+namespace Thorp.RoutingNetwork
+variable {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι] [DecidableEq ι]
+  {d : ℕ} (R : RoutingNetwork α ι d)
+
+omit [DecidableEq α] in
+lemma single_min_eq_within (o : α → Option (Fin 1)) (u v : α) :
+    min (pathWeight o u) (pathWeight o v) = withinWeight o u v := by
+  cases hu : o u with
+  | none =>
+    cases hv : o v with
+    | none => simp [pathWeight, withinWeight, hu, hv]
+    | some j =>
+      have hj : 0 ≤ slotWeight o j := by unfold slotWeight; positivity
+      simp [pathWeight, withinWeight, hu, hv, min_eq_left hj]
+  | some i =>
+    cases hv : o v with
+    | none =>
+      have hi : 0 ≤ slotWeight o i := by unfold slotWeight; positivity
+      simp [pathWeight, withinWeight, hu, hv, min_eq_right hi]
+    | some j =>
+      have hij : i = j := Subsingleton.elim _ _
+      simp [pathWeight, withinWeight, hu, hv, hij]
+
+lemma single_rotationCharge_mean_bound {o : α → Option (Fin 1)}
+    [∀ i : Fin 1, Nonempty {x : α // o x = some i}]
+    (b : ι → Bool) :
+    finiteMean (rotationCharge R o b) ≤
+      Real.log (cycleFiber o 0).card / Real.log 2 := by
+  classical
+  have hn : ∀ i, (cycleFiber o i).Nonempty := by
+    intro i
+    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : α // o x = some i})
+    exact ⟨x, (mem_cycleFiber _ _ _).mpr hx⟩
+  have hm : Monotone (fun i => (cycleFiber o i).card) := by
+    intro i j _
+    have : i = j := Subsingleton.elim _ _
+    simp [this]
+  unfold rotationCharge
+  rw [finiteMean_sum]
+  simp only [rotationMeeting_mean hm, single_min_eq_within, ← two_mul]
+  rw [← Finset.mul_sum]
+  have h := withinWeight_sum_bound R o hn b
+  simp only [Fin.sum_univ_one] at h
+  rw [mul_comm 2 (Real.log 2), div_mul_eq_div_div] at h
+  linarith
+
+lemma single_rotation_exp_bound {o : α → Option (Fin 1)}
+    [∀ i : Fin 1, Nonempty {x : α // o x = some i}]
+    (b : ι → Bool) :
+    1 / (cycleFiber o 0).card ≤
+      finiteMean (fun z : Terminal o => Real.exp (-Real.log 2 * rotationCharge R o b z)) := by
+  have h := single_rotationCharge_mean_bound R (o := o) b
+  have hlog : 0 < Real.log 2 := Real.log_pos (by norm_num)
+  have hcard : (0 : ℝ) < (cycleFiber o 0).card := by
+    apply Nat.cast_pos.mpr
+    apply Finset.card_pos.mpr
+    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : α // o x = some 0})
+    exact ⟨x, (mem_cycleFiber _ _ _).mpr hx⟩
+  have hm : -Real.log (cycleFiber o 0).card ≤
+      finiteMean (fun z : Terminal o => -Real.log 2 * rotationCharge R o b z) := by
+    have he := finiteMean_mul_const (rotationCharge R o b) (-Real.log 2)
+    simp only [mul_comm] at he
+    rw [he]
+    have hh := mul_le_mul_of_nonpos_left h (neg_nonpos.mpr hlog.le)
+    calc
+      _ = -Real.log 2 * (Real.log (cycleFiber o 0).card / Real.log 2) := by
+        field_simp
+      _ ≤ _ := hh
+  have he := (Real.exp_le_exp.mpr hm).trans (exp_finiteMean_le _)
+  simpa only [Real.exp_neg, Real.exp_log hcard, one_div] using he
+
+lemma single_collection_rotation_lower (S : Finset α) (r : ℕ)
+    (ω : ι → Bool) (c : CycleSlots (R.perm ω) S (fun _ : Fin 1 => r)) :
+    (2:ℝ)^d ≤ ∑ z : Terminal c.val,
+      (2:ℝ)^d * Real.exp (-Real.log 2 * rotationCharge R c.val ω z) := by
+  classical
+  let (i : Fin 1) : Nonempty {x : α // c.val x = some i} := c.terminal_nonempty i
+  have hb := single_rotation_exp_bound R ω (o := c.val)
+  have hcard : (Fintype.card (Terminal c.val) : ℝ) = (cycleFiber c.val 0).card := by
+    rw [Fintype.card_pi]
+    simp only [Fin.prod_univ_one, card_cycleFiber]
+  have hp : (0:ℝ) < Fintype.card (Terminal c.val) := Nat.cast_pos.mpr Fintype.card_pos
+  have hh := (le_div_iff₀ hp).mp hb
+  rw [hcard, one_div_mul_cancel (ne_of_gt (by simpa only [hcard] using hp))] at hh
+  rw [← Finset.mul_sum]
+  simpa only [mul_one] using mul_le_mul_of_nonneg_left hh (show 0 ≤ (2:ℝ)^d by positivity)
+
+lemma single_cycleSlots_weighted_moment (S : Finset α) (r : ℕ) :
+    finiteMean (fun ω : ι → Bool =>
+      (Fintype.card (CycleSlots (R.perm ω) S (fun _ : Fin 1 => r)) : ℝ)) *
+      (2:ℝ)^d ≤ S.card := by
+  classical
+  let rs : Fin 1 → ℕ := fun _ => r
+  let g (a : Fin 1 → S) (ω : ι → Bool) : ℝ :=
+    if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,rs i))) ∅).succeeds ω then
+      ((2:ℝ)^d)^1 / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,rs i))) ∅).staleCharge ∅ ω
+    else 0
+  have hω (ω : ι → Bool) :
+      (Fintype.card (CycleSlots (R.perm ω) S rs) : ℝ) * (2:ℝ)^d ≤
+        ∑ a : Fin 1 → S, g a ω := by
+    calc
+      _ = ∑ _c : CycleSlots (R.perm ω) S rs, (2:ℝ)^d := by simp
+      _ ≤ ∑ c : CycleSlots (R.perm ω) S rs,
+          ∑ z : Terminal c.val, ((2:ℝ)^d)^1 * Real.exp (-Real.log 2 * rotationCharge R c.val ω z) := by
+        apply Finset.sum_le_sum
+        intro c _
+        simpa only [pow_one] using single_collection_rotation_lower R S r ω c
+      _ = ∑ c : RootedSlots (R.perm ω) S rs,
+          ((2:ℝ)^d)^1 * Real.exp (-Real.log 2 * rotationCharge R c.1.val ω c.2) :=
+        (Fintype.sum_sigma (fun c : RootedSlots (R.perm ω) S rs =>
+          ((2:ℝ)^d)^1 * Real.exp (-Real.log 2 * rotationCharge R c.1.val ω c.2))).symm
+      _ ≤ _ := rooted_weight_sum_le R S rs ω
+  have hb := finiteMean_mono hω
+  rw [finiteMean_mul_const, finiteMean_sum] at hb
+  calc
+    _ ≤ ∑ a : Fin 1 → S, finiteMean (g a) := hb
+    _ ≤ ∑ _a : Fin 1 → S, (1:ℝ) := by
+      apply Finset.sum_le_sum
+      intro a _
+      have hh := QueryTree.cycles_weighted_success R S (List.ofFn (fun i : Fin 1 => ((a i).val,rs i)))
+      simpa only [List.length_ofFn] using hh
+    _ = _ := by simp
+
+noncomputable def singleRootedEquiv (p : Equiv.Perm α) (r : ℕ) :
+    RootedSlots p Finset.univ (fun _ : Fin 1 => r) ≃
+      {x : α // (orbitSet p x).card = r + 1} := by
+  classical
+  let f : RootedSlots p Finset.univ (fun _ : Fin 1 => r) →
+      {x : α // (orbitSet p x).card = r + 1} := fun c =>
+    ⟨(c.2 0).val, by rw [c.1.property.2.2 0 _ (c.2 0).property, c.1.property.1]⟩
+  apply Equiv.ofBijective f
+  constructor
+  · intro c c' h
+    apply rootedStart_injective
+    funext i
+    have hi : i = 0 := Subsingleton.elim _ _
+    subst i
+    apply Subtype.ext
+    exact congrArg p (congrArg Subtype.val h)
+  · intro x
+    let C : SelectedCycle p Finset.univ := ⟨orbitSet p x, ⟨⟨x,rfl⟩, Finset.subset_univ _⟩⟩
+    let a : SlotAssignment p Finset.univ (fun _ : Fin 1 => r) :=
+      ⟨fun _ => C, Function.injective_of_subsingleton _, fun _ => x.property⟩
+    let c := assignmentToSlots a
+    have hx : ∀ i : Fin 1, c.val x = some i := by
+      intro i
+      change assignmentOwner a x = some i
+      rw [assignmentOwner_eq_some]
+      exact (mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.refl p x)
+    exact ⟨⟨c, fun i => ⟨x, hx i⟩⟩, Subtype.ext rfl⟩
+
+lemma single_root_count (p : Equiv.Perm α) (r : ℕ) :
+    (Fintype.card {x : α // (orbitSet p x).card = r+1} : ℝ) =
+      (Fintype.card (CycleSlots p Finset.univ (fun _ : Fin 1 => r)) : ℝ) * (r+1) := by
+  classical
+  rw [← Fintype.card_congr (singleRootedEquiv p r), Fintype.card_sigma]
+  simp only [Fintype.card_pi, Fin.prod_univ_one, card_cycleFiber]
+  simp only [Nat.cast_sum]
+  have he (c : CycleSlots p Finset.univ (fun _ : Fin 1 => r)) :
+      ((cycleFiber c.val 0).card : ℝ) = (r : ℝ) + 1 := by
+    rw [c.property.1]; norm_cast
+  simp only [he, Finset.sum_const, Finset.card_univ, nsmul_eq_mul]
+
+lemma uniform_cycle_length_cap [Nonempty α] (r : ℕ) :
+    finiteMean (fun ω : ι → Bool => finiteMean (fun x : α =>
+      if (orbitSet (R.perm ω) x).card = r+1 then (1:ℝ) else 0)) ≤
+      ((r : ℝ)+1) / (2:ℝ)^d := by
+  classical
+  have hn : 0 < (Fintype.card α : ℝ) := Nat.cast_pos.mpr Fintype.card_pos
+  have hinner (ω : ι → Bool) :
+      finiteMean (fun x : α => if (orbitSet (R.perm ω) x).card = r+1 then (1:ℝ) else 0) =
+        (Fintype.card (CycleSlots (R.perm ω) Finset.univ (fun _ : Fin 1 => r)) : ℝ) *
+          (((r:ℝ)+1) / Fintype.card α) := by
+    rw [← mul_div_assoc, ← single_root_count (R.perm ω) r]
+    simp [finiteMean, Fintype.card_subtype]
+  simp only [hinner, finiteMean_mul_const]
+  have h := single_cycleSlots_weighted_moment R Finset.univ r
+  simp only [Finset.card_univ] at h
+  have h' := (le_div_iff₀ (show 0 < (2:ℝ)^d by positivity)).mpr h
+  calc
+    _ ≤ ((Fintype.card α : ℝ) / (2:ℝ)^d) * (((r:ℝ)+1) / Fintype.card α) :=
+      mul_le_mul_of_nonneg_right h' (by positivity)
+    _ = _ := by field_simp
+
+end Thorp.RoutingNetwork
+
+
+namespace Thorp.DenseTruncation
+open scoped BigOperators
+
+lemma law_reciprocal_summable {j : ℕ} {p : ℕ → ℝ}
+    (hp : AdmissibleCycleLaw j p) :
+    Summable (fun l => p l / ((l + 1 : ℕ) : ℝ)) := by
+  apply Summable.of_nonneg_of_le (fun l => div_nonneg (hp.1 l) (by positivity))
+    (fun l => ?_) hp.2.1.summable
+  exact div_le_self (hp.1 l) (by exact_mod_cast Nat.succ_pos l)
+
+lemma law_reciprocal_nonneg {j : ℕ} {p : ℕ → ℝ}
+    (hp : AdmissibleCycleLaw j p) :
+    0 ≤ ∑' l : ℕ, p l / ((l + 1 : ℕ) : ℝ) :=
+  tsum_nonneg (fun l => div_nonneg (hp.1 l) (by positivity))
+
+lemma law_reciprocal_le_one {j : ℕ} {p : ℕ → ℝ}
+    (hp : AdmissibleCycleLaw j p) :
+    (∑' l : ℕ, p l / ((l + 1 : ℕ) : ℝ)) ≤ 1 := by
+  rw [← hp.2.1.tsum_eq]
+  exact Summable.tsum_le_tsum (fun l => div_le_self (hp.1 l)
+    (by exact_mod_cast Nat.succ_pos l)) (law_reciprocal_summable hp) hp.2.1.summable
+
+lemma reciprocal_values_bdd (j : ℕ) : BddAbove
+    {z : ℝ | ∃ p : ℕ → ℝ, AdmissibleCycleLaw j p ∧
+      z = ∑' l : ℕ, p l / ((l + 1 : ℕ) : ℝ)} :=
+  ⟨1, fun _ ⟨_,hp,hz⟩ => hz ▸ law_reciprocal_le_one hp⟩
+
+lemma reciprocal_le_h {j : ℕ} {p : ℕ → ℝ}
+    (hp : AdmissibleCycleLaw j p) :
+    (∑' l : ℕ, p l / ((l + 1 : ℕ) : ℝ)) ≤ h j :=
+  le_csSup (reciprocal_values_bdd j) ⟨p,hp,rfl⟩
+
+lemma subprob_reciprocal_le_h {j : ℕ} {q : ℕ → ℝ}
+    (hq : ∀ l, 0 ≤ q l) (hs : Summable q) (ht : (∑' l, q l) ≤ 1)
+    (hc : ∀ l, q l ≤ ((l+1 : ℕ) : ℝ) / (2:ℝ)^(j-1)) :
+    (∑' l, q l / ((l+1 : ℕ) : ℝ)) ≤ h j := by
+  let t := 2^(j-1)-1
+  have hpow : 1 ≤ 2^(j-1) := Nat.one_le_pow _ _ (by norm_num)
+  have htlen : (t+1 : ℕ) = 2^(j-1) := Nat.sub_add_cancel hpow
+  have hreal : ((t+1 : ℕ) : ℝ) = (2:ℝ)^(j-1) := by rw [htlen]; norm_cast
+  let m := 1 - ∑' l, q l
+  have hm : 0 ≤ m := sub_nonneg.mpr ht
+  let a : ℕ → ℝ := fun l => if l=t then m else 0
+  have ha : HasSum a m := by
+    simpa only [a,ite_eq_left rfl] using (hasSum_single t (f := a) (fun l hl => by simp [a,hl]))
+  let p : ℕ → ℝ := fun l => q l+a l
+  have hp : AdmissibleCycleLaw j p := by
+    refine ⟨fun l => add_nonneg (hq l) (by dsimp [a]; split_ifs <;> positivity), ?_, ?_⟩
+    · convert hs.hasSum.add ha using 1; dsimp [p,m]; ring
+    · intro l
+      by_cases hl : l=t
+      · subst l
+        have hqt := hs.le_tsum t (fun l _ => hq l)
+        have hp2 : (2:ℝ)^(j-1) ≠ 0 := by positivity
+        simp only [p,a,hreal,div_self hp2]
+        dsimp [m]
+        linarith
+      · simpa only [p,a,ite_eq_right hl,add_zero] using hc l
+  have hqs : Summable (fun l => q l / ((l+1 : ℕ) : ℝ)) := by
+    apply Summable.of_nonneg_of_le (fun l => div_nonneg (hq l) (by positivity))
+      (fun l => div_le_self (hq l) (by exact_mod_cast Nat.succ_pos l)) hs
+  calc
+    _ ≤ ∑' l, p l / ((l+1 : ℕ) : ℝ) := by
+      apply Summable.tsum_le_tsum _ hqs (law_reciprocal_summable hp)
+      intro l
+      apply div_le_div_of_nonneg_right _ (by positivity)
+      dsimp [p,a]
+      split_ifs <;> linarith
+    _ ≤ _ := reciprocal_le_h hp
+
+end Thorp.DenseTruncation
+
+
+end ThorpNine.Dense
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Dense/ParallelFollow.lean b/lean/OAI/Probability/ThorpRouting/Dense/ParallelFollow.lean
index 5ec706ee83cb3f54d6a6b1fadd20224eb28932d9..c5252c972ab1f94aae9b6f90f7acc6436e30f599 100644
--- a/lean/OAI/Probability/ThorpRouting/Dense/ParallelFollow.lean
+++ b/lean/OAI/Probability/ThorpRouting/Dense/ParallelFollow.lean
@@ -1,854 +1,854 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Dense.PriorPaths
-
-namespace OAI
-
-namespace ThorpNine.Dense
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-section ParallelButterfly
-variable {α B : Type*} [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] {d : ℕ}
-omit [DecidableEq B] in
-lemma parallel_entropy (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool) (S : Finset α) :
-    (∑ i : B × SwitchIndex d,
-      if (parallelUsers pre ω i).1 ∈ S ∧ (parallelUsers pre ω i).2 ∈ S then (1:ℝ) else 0) ≤
-      (S.card : ℝ) * Real.log S.card / (2 * Real.log 2) := by
-  classical
-  rw [Fintype.sum_prod_type]
-  calc
-    _ ≤ ∑ b : B, (blockMarks pre S b).card * Real.log (blockMarks pre S b).card / (2*Real.log 2) := by
-      apply Finset.sum_le_sum
-      intro b _
-      simpa only [←mem_blockMarks,parallelUsers] using
-        marked_switch_entropy_bound d (decodeButterfly d (fun i => ω (b,i))) (blockMarks pre S b)
-    _ ≤ ∑ b : B, (blockMarks pre S b).card * Real.log S.card / (2*Real.log 2) := by
-      apply Finset.sum_le_sum
-      intro b _
-      by_cases hz : (blockMarks pre S b).card = 0
-      · simp [hz]
-      · apply div_le_div_of_nonneg_right _ (by positivity)
-        apply mul_le_mul_of_nonneg_left _ (by positivity)
-        exact Real.log_le_log (by exact_mod_cast Nat.pos_of_ne_zero hz)
-          (by exact_mod_cast card_blockMarks_le pre S b)
-    _ = _ := by rw [←Finset.sum_div,←Finset.sum_mul,←Nat.cast_sum,sum_card_blockMarks]
-
-noncomputable def parallelFollow (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x : α) (k : α → QueryTree (B × SwitchIndex d)) : QueryTree (B × SwitchIndex d) :=
-  QueryTree.followPath d (fun i => ((pre x).1,i)) (pre x).2 (fun y => k (post ((pre x).1,y)))
-
-omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
-lemma parallelFollow_succeeds (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
-    (parallelFollow pre post x k).succeeds ω = (k (relabeledPerm pre post ω x)).succeeds ω := by
-  exact QueryTree.followPath_succeeds _ _ _ _ _
-
-omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
-lemma parallelFollow_raw_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
-    (parallelFollow pre post x k).charge ω = (k (relabeledPerm pre post ω x)).charge ω := by
-  exact QueryTree.followPath_raw_charge _ _ _ _ _
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma parallelFollow_stale_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
-    (ω : B × SwitchIndex d → Bool) :
-    (parallelFollow pre post x k).staleCharge R ω =
-      (k (relabeledPerm pre post ω x)).staleCharge (R ∪ parallelPath pre ω x) ω := by
-  exact QueryTree.followPath_stale_charge _ _ _ _ _ _
-
-noncomputable def closingDomain (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) : Finset (B × SwitchIndex d) :=
-  (routeDomain d (pre x).2 (post.symm y).2).image (fun i => ((pre x).1,i))
-
-def closingValue (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (i : B × SwitchIndex d) : Bool := routeValue d (pre x).2 (post.symm y).2 i.2
-
-omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
-lemma relabeled_endpoint_pair (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (ω : B × SwitchIndex d → Bool) (x y : α) :
-    relabeledPerm pre post ω x = y ↔
-      (pre x).1 = (post.symm y).1 ∧
-      butterflyPerm d (decodeButterfly d (fun i => ω ((pre x).1,i))) (pre x).2 = (post.symm y).2 := by
-  change post ((pre x).1,_) = y ↔ _
-  rw [←post.eq_symm_apply,Prod.mk.injEq]
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma relabeled_endpoint_iff (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (ω : B × SwitchIndex d → Bool) (x y : α) :
-    relabeledPerm pre post ω x = y ↔
-      (pre x).1 = (post.symm y).1 ∧
-        ∀ i ∈ closingDomain pre post x y, ω i = closingValue pre post x y i := by
-  classical
-  rw [relabeled_endpoint_pair,butterfly_endpoint_iff]
-  simp only [closingDomain,Finset.forall_mem_image,closingValue]
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma closingDomain_eq (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (ω : B × SwitchIndex d → Bool) (x y : α) (h : relabeledPerm pre post ω x = y) :
-    closingDomain pre post x y = parallelPath pre ω x := by
-  rw [closingDomain,parallelPath,(relabeled_endpoint_pair _ _ _ _ _).mp h |>.2]
-
-noncomputable def parallelClose (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (k : QueryTree (B × SwitchIndex d)) : QueryTree (B × SwitchIndex d) :=
-  if (pre x).1 = (post.symm y).1 then
-    QueryTree.forceList (closingDomain pre post x y).toList (closingValue pre post x y) k
-  else QueryTree.reject
-
-omit [Fintype α] in
-lemma parallelClose_succeeds (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (k : QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
-    (parallelClose pre post x y k).succeeds ω =
-      if relabeledPerm pre post ω x = y then k.succeeds ω else false := by
-  classical
-  simp only [relabeled_endpoint_iff,parallelClose]
-  by_cases hb : (pre x).1 = (post.symm y).1
-  · rw [ite_eq_left hb,QueryTree.forceList_succeeds]
-    simp only [Finset.mem_toList,hb,true_and]
-  · simp [hb,QueryTree.succeeds]
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma parallelClose_raw_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (k : QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool)
-    (h : relabeledPerm pre post ω x = y) :
-    (parallelClose pre post x y k).charge ω = k.charge ω + d := by
-  have hh := (relabeled_endpoint_iff _ _ _ _ _).mp h
-  rw [parallelClose,ite_eq_left hh.1,QueryTree.forceList_raw_charge _ _ _ _
-    (by simpa only [Finset.mem_toList] using hh.2),Finset.length_toList,
-    closingDomain_eq _ _ _ _ _ h,card_parallelPath]
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma parallelClose_fresh_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (k : QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
-    (ω : B × SwitchIndex d → Bool) (h : relabeledPerm pre post ω x = y) :
-    (parallelClose pre post x y k).freshCharge R ω =
-      k.freshCharge (R ∪ parallelPath pre ω x) ω + (parallelPath pre ω x \ R).card := by
-  have hh := (relabeled_endpoint_iff _ _ _ _ _).mp h
-  rw [parallelClose,ite_eq_left hh.1,QueryTree.forceList_charge _ _ _ _ _
-    (by simpa only [Finset.mem_toList] using hh.2),Finset.toList_toFinset,
-    closingDomain_eq _ _ _ _ _ h]
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma parallelClose_stale_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (k : QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
-    (ω : B × SwitchIndex d → Bool) (h : relabeledPerm pre post ω x = y) :
-    (parallelClose pre post x y k).staleCharge R ω =
-      k.staleCharge (R ∪ parallelPath pre ω x) ω + (parallelPath pre ω x ∩ R).card := by
-  have h₀ := QueryTree.fresh_add_stale (parallelClose pre post x y k) R ω
-  have h₁ := QueryTree.fresh_add_stale k (R ∪ parallelPath pre ω x) ω
-  rw [parallelClose_raw_charge _ _ _ _ _ _ h,parallelClose_fresh_charge _ _ _ _ _ _ _ h] at h₀
-  have h₂ : (parallelPath pre ω x \ R).card + (parallelPath pre ω x ∩ R).card = d := by
-    rw [Finset.card_sdiff_add_card_inter,card_parallelPath]
-  omega
-
-noncomputable def parallelNetwork (pre : α ≃ B × Card d) (post : B × Card d ≃ α) :
-    RoutingNetwork α (B × SwitchIndex d) d where
-  perm := relabeledPerm pre post
-  path := parallelPath pre
-  users := parallelUsers pre
-  users_ne := parallelUsers_ne pre
-  mem_path := mem_parallelPath pre
-  card_path := card_parallelPath pre
-  entropy := parallel_entropy pre
-  follow := parallelFollow pre post
-  close := parallelClose pre post
-  follow_succeeds := parallelFollow_succeeds pre post
-  follow_raw_charge := parallelFollow_raw_charge pre post
-  follow_stale_charge := parallelFollow_stale_charge pre post
-  close_succeeds := parallelClose_succeeds pre post
-  close_raw_charge := parallelClose_raw_charge pre post
-  close_stale_charge := parallelClose_stale_charge pre post
-
-lemma parallel_cycle_log_mgf (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (S : Finset α) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    Real.log (finiteMean (fun ω : B × SwitchIndex d → Bool =>
-      q^(Fintype.card (RoutingNetwork.SelectedCycle (relabeledPerm pre post ω) S)))) ≤
-      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
-  exact (parallelNetwork pre post).butterfly_cycle_log_mgf S J hJ q hq
-
-end ParallelButterfly
-
-def cardSplit : (r s : ℕ) → Card (s+r) ≃ Card r × Card s
-  | 0, s =>
-    { toFun := fun x => ((fun i => Fin.elim0 i), x)
-      invFun := fun x => x.2
-      left_inv := fun _ => rfl
-      right_inv := fun x => by
-        apply Prod.ext
-        · exact Subsingleton.elim _ _
-        · rfl }
-  | r+1, s =>
-    { toFun := fun x => (Fin.cons (x 0) ((cardSplit r s (Fin.tail x)).1),
-        (cardSplit r s (Fin.tail x)).2)
-      invFun := fun x => Fin.cons (x.1 0) ((cardSplit r s).symm (Fin.tail x.1,x.2))
-      left_inv := by
-        intro x
-        simp only [Fin.cons_zero, Fin.tail_cons, Prod.mk.eta, Equiv.symm_apply_apply, Fin.cons_self_tail]
-      right_inv := by
-        intro x
-        simp only [Fin.cons_zero, Fin.tail_cons, Equiv.apply_symm_apply, Fin.cons_self_tail] }
-
-def assembleBits : (r s : ℕ) → (Card r × SwitchIndex s → Bool) →
-    (Card s × SwitchIndex r → Bool) → SwitchIndex (s+r) → Bool
-  | 0, _, lo, _, i => lo ((fun j => Fin.elim0 j), i)
-  | r+1, s, _, hi, Sum.inl y => hi ((cardSplit r s y).2,Sum.inl (cardSplit r s y).1)
-  | r+1, s, lo, hi, Sum.inr (b,i) => assembleBits r s
-      (fun j => lo (Fin.cons b j.1,j.2)) (fun j => hi (j.1,Sum.inr (b,j.2))) i
-
-lemma butterfly_split (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
-    (hi : Card s × SwitchIndex r → Bool) (x : Card (s+r)) :
-    cardSplit r s (butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi)) x) =
-      let y := cardSplit r s x
-      let z := butterflyPerm s (decodeButterfly s (fun i => lo (y.1,i))) y.2
-      (butterflyPerm r (decodeButterfly r (fun i => hi (z,i))) y.1,z) := by
-  induction r with
-  | zero =>
-    apply Prod.ext
-    · funext i
-      exact Fin.elim0 i
-    · rfl
-  | succ r ih =>
-    let lo' : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons (x 0) j.1,j.2)
-    let hi' : Card s × SwitchIndex r → Bool := fun j => hi (j.1,Sum.inr (x 0,j.2))
-    have hh := ih lo' hi' (Fin.tail x)
-    let u := butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo' hi')) (Fin.tail x)
-    let y := cardSplit r s (Fin.tail x)
-    let z := butterflyPerm s (decodeButterfly s (fun i => lo' (y.1,i))) y.2
-    let w := butterflyPerm r (decodeButterfly r (fun i => hi' (z,i))) y.1
-    change cardSplit r s u = (w,z) at hh
-    change ((Fin.cons (Bool.xor (x 0) (hi ((cardSplit r s u).2,Sum.inl (cardSplit r s u).1)))
-        ((cardSplit r s u).1) : Card (r+1)), (cardSplit r s u).2) =
-      ((Fin.cons (Bool.xor (x 0) (hi (z,Sum.inl w))) w : Card (r+1)),z)
-    rw [hh]
-
-def splitSwitch : (r s : ℕ) → SwitchIndex (s+r) ≃
-    (Card r × SwitchIndex s) ⊕ (Card s × SwitchIndex r)
-  | 0, _ =>
-    { toFun := fun i => Sum.inl ((fun j => Fin.elim0 j),i)
-      invFun := fun i => match i with
-        | Sum.inl (_,j) => j
-        | Sum.inr (_,j) => nomatch j
-      left_inv := fun _ => rfl
-      right_inv := by
-        intro i
-        rcases i with ⟨x,j⟩ | ⟨x,j⟩
-        · exact congrArg Sum.inl (Prod.ext (Subsingleton.elim _ _) rfl)
-        · exact Empty.elim j }
-  | r+1, s =>
-    { toFun := fun i => match i with
-        | Sum.inl y => Sum.inr ((cardSplit r s y).2,Sum.inl (cardSplit r s y).1)
-        | Sum.inr (b,j) => match splitSwitch r s j with
-          | Sum.inl (p,j) => Sum.inl (Fin.cons b p,j)
-          | Sum.inr (q,j) => Sum.inr (q,Sum.inr (b,j))
-      invFun := fun i => match i with
-        | Sum.inl (p,j) => Sum.inr (p 0,(splitSwitch r s).symm (Sum.inl (Fin.tail p,j)))
-        | Sum.inr (q,Sum.inl p) => Sum.inl ((cardSplit r s).symm (p,q))
-        | Sum.inr (q,Sum.inr (b,j)) => Sum.inr (b,(splitSwitch r s).symm (Sum.inr (q,j)))
-      left_inv := by
-        intro i
-        rcases i with y | ⟨b,j⟩
-        · dsimp only
-          rw [Prod.mk.eta, Equiv.symm_apply_apply]
-        · dsimp only
-          cases hz : splitSwitch r s j with
-          | inl z =>
-            rcases z with ⟨p,k⟩
-            simp only [Fin.cons_zero, Fin.tail_cons]
-            rw [←hz,Equiv.symm_apply_apply]
-          | inr z =>
-            rcases z with ⟨q,k⟩
-            dsimp only
-            rw [←hz,Equiv.symm_apply_apply]
-      right_inv := by
-        intro i
-        rcases i with ⟨p,j⟩ | ⟨q,j⟩
-        · simp only [Equiv.apply_symm_apply, Fin.cons_self_tail]
-        · rcases j with p | ⟨b,j⟩ <;> simp only [Equiv.apply_symm_apply] }
-
-lemma assembleBits_eq (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
-    (hi : Card s × SwitchIndex r → Bool) (i : SwitchIndex (s+r)) :
-    assembleBits r s lo hi i = Sum.elim lo hi (splitSwitch r s i) := by
-  induction r with
-  | zero => rfl
-  | succ r ih =>
-    rcases i with y | ⟨b,j⟩
-    · rfl
-    · simp only [assembleBits,splitSwitch,Equiv.coe_fn_mk]
-      rw [ih]
-      cases splitSwitch r s j <;> rfl
-
-def splitCoinEquiv (r s : ℕ) :
-    ((Card r × SwitchIndex s → Bool) × (Card s × SwitchIndex r → Bool)) ≃
-      (SwitchIndex (s+r) → Bool) :=
-  (Equiv.sumArrowEquivProdArrow _ _ Bool).symm.trans
-    (Equiv.arrowCongr (splitSwitch r s).symm (Equiv.refl Bool))
-
-lemma splitCoinEquiv_apply (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
-    (hi : Card s × SwitchIndex r → Bool) :
-    splitCoinEquiv r s (lo,hi) = assembleBits r s lo hi := by
-  funext i
-  exact (assembleBits_eq r s lo hi i).symm
-
-lemma finiteMean_prod {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] (f : Ω × Ω' → ℝ) :
-    finiteMean f = finiteMean (fun x => finiteMean (fun y => f (x,y))) := by
-  simp only [finiteMean, Fintype.sum_prod_type, Fintype.card_prod, Nat.cast_mul,
-    ←Finset.sum_div, div_div, mul_comm]
-
-lemma finiteMean_split (r s : ℕ) (f : (SwitchIndex (s+r) → Bool) → ℝ) :
-    finiteMean f = finiteMean (fun lo : Card r × SwitchIndex s → Bool =>
-      finiteMean (fun hi : Card s × SwitchIndex r → Bool => f (assembleBits r s lo hi))) := by
-  rw [finiteMean_equiv (splitCoinEquiv r s).symm]
-  simp only [Equiv.symm_symm]
-  rw [finiteMean_prod]
-  apply finiteMean_congr
-  intro lo
-  apply finiteMean_congr
-  intro hi
-  rw [splitCoinEquiv_apply]
-
-section PartiallyExposed
-variable {α : Type*} {r s : ℕ}
-
-def freshPre (pre : α ≃ Card (s+r)) (lo : Card r × SwitchIndex s → Bool) :
-    α ≃ Card s × Card r :=
-  ((pre.trans (cardSplit r s)).trans (parallelPerm lo)).trans (Equiv.prodComm _ _)
-
-def freshPost (post : Card (s+r) ≃ α) : Card s × Card r ≃ α :=
-  ((Equiv.prodComm _ _).trans (cardSplit r s).symm).trans post
-
-lemma exposed_perm_eq (pre : α ≃ Card (s+r)) (post : Card (s+r) ≃ α)
-    (lo : Card r × SwitchIndex s → Bool) (hi : Card s × SwitchIndex r → Bool) :
-    (pre.trans (butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi)))).trans post =
-      relabeledPerm (freshPre pre lo) (freshPost post) hi := by
-  ext x
-  simpa only [relabeledPerm, freshPre, freshPost, parallelPerm, Equiv.trans_apply,
-    Equiv.coe_fn_mk, Equiv.prodComm_apply, Prod.swap, Equiv.symm_apply_apply] using
-    congrArg (fun y => post ((cardSplit r s).symm y)) (butterfly_split r s lo hi (pre x))
-
-lemma partially_exposed_cycle_log_mgf [Fintype α] [DecidableEq α]
-    (pre : α ≃ Card (s+r)) (post : Card (s+r) ≃ α)
-    (lo : Card r × SwitchIndex s → Bool) (S : Finset α)
-    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    Real.log (finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(Fintype.card (RoutingNetwork.SelectedCycle
-        ((pre.trans (butterflyPerm (s+r)
-          (decodeButterfly (s+r) (assembleBits r s lo hi)))).trans post) S)))) ≤
-      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^r) := by
-  simp only [exposed_perm_eq]
-  exact parallel_cycle_log_mgf (freshPre pre lo) (freshPost post) S J hJ q hq
-
-end PartiallyExposed
-
-lemma inclusionMoment_eq {α Ω : Type*} [DecidableEq α] [Fintype Ω]
-    (F : Ω → Finset α) (S : Finset α) :
-    inclusionMoment F S = finiteMean (fun ω => if S ⊆ F ω then (1:ℝ) else 0) := by
-  unfold inclusionMoment
-  apply finiteMean_congr
-  intro ω
-  split_ifs <;> rfl
-
-lemma finiteMean_bool (f : Bool → ℝ) : finiteMean f = (f false+f true)/2 := by
-  simp only [finiteMean, Fintype.sum_bool, Fintype.card_bool, Nat.cast_ofNat]
-  ring
-
-lemma finiteMean_const {Ω : Type*} [Fintype Ω] [Nonempty Ω] (c : ℝ) :
-    finiteMean (fun _ : Ω => c) = c := by
-  simp only [finiteMean,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
-  exact mul_div_cancel_left₀ c (Nat.cast_ne_zero.mpr Fintype.card_ne_zero)
-
-lemma finiteMean_pi_prod {ι : Type*} [Fintype ι] [DecidableEq ι] {Ω : ι → Type*}
-    [∀ i, Fintype (Ω i)] (f : (i : ι) → Ω i → ℝ) :
-    finiteMean (fun ω : (i : ι) → Ω i => ∏ i, f i (ω i)) = ∏ i, finiteMean (f i) := by
-  simp only [finiteMean,←Fintype.prod_sum,Fintype.card_pi,Nat.cast_prod,Finset.prod_div_distrib]
-
-section ParallelFairLayer
-variable {β : Type*} [Fintype β] [DecidableEq β]
-
-noncomputable def pairFiber (S : Finset (Bool × β)) (b : β) : Finset Bool :=
-  Finset.univ.filter (fun ε => (ε,b) ∈ S)
-
-lemma prod_pairFiber (S : Finset (Bool × β)) (f : Bool × β → ℝ) :
-    (∏ x ∈ S, f x) = ∏ b : β, ∏ ε ∈ pairFiber S b, f (ε,b) := by
-  classical
-  simp only [pairFiber,Finset.prod_filter]
-  rw [Finset.prod_comm,←Fintype.prod_prod_type (fun x : Bool × β => if x ∈ S then f x else 1)]
-  rw [←Finset.prod_filter]
-  congr 1
-  ext x
-  simp
-
-lemma fiber_average_bound (S : Finset Bool) (p : Bool → ℝ) (hp : ∀ b, 0 ≤ p b) :
-    finiteMean (fun c : Bool => ∏ ε ∈ S, p (Bool.xor ε c)) ≤
-      ∏ ε ∈ S, (p ε + p (!ε))/2 := by
-  rw [finiteMean_bool]
-  have hxor : ∀ ε : Bool, Bool.xor ε true = Equiv.swap false true ε := by decide
-  simp only [Bool.xor_false,hxor]
-  calc
-    _ ≤ ∏ ε ∈ S, (p ε+p (Equiv.swap false true ε))/2 :=
-      swap_average_product S p hp false true
-    _ = _ := Finset.prod_congr rfl (fun ε _ => by cases ε <;> rfl)
-
-lemma CylinderBound.fair_parallel {m : Finset (Bool × β) → ℝ} {p : Bool × β → ℝ}
-    (h : CylinderBound m p) (hp : ∀ x, 0 ≤ p x) :
-    CylinderBound (fun S => finiteMean (fun ξ : β → Bool => m (S.image (pairLayer ξ))))
-      (fun x => (p x+p (!x.1,x.2))/2) := by
-  intro S
-  calc
-    _ ≤ finiteMean (fun ξ : β → Bool => ∏ x ∈ S, p (pairLayer ξ x)) := by
-      apply finiteMean_mono
-      intro ξ
-      simpa only [Finset.prod_image (pairLayer ξ).injective.injOn] using h (S.image (pairLayer ξ))
-    _ = ∏ b : β, finiteMean (fun c : Bool => ∏ ε ∈ pairFiber S b, p (Bool.xor ε c,b)) := by
-      calc
-        _ = finiteMean (fun ξ : β → Bool => ∏ b : β,
-            ∏ ε ∈ pairFiber S b, p (Bool.xor ε (ξ b),b)) :=
-          finiteMean_congr (fun ξ => prod_pairFiber S (fun x => p (pairLayer ξ x)))
-        _ = _ := finiteMean_pi_prod (fun b c => ∏ ε ∈ pairFiber S b, p (Bool.xor ε c,b))
-    _ ≤ ∏ b : β, ∏ ε ∈ pairFiber S b, (p (ε,b)+p (!ε,b))/2 := by
-      apply Finset.prod_le_prod₀
-      · intro b _
-        exact div_nonneg (Finset.sum_nonneg (fun c _ =>
-          Finset.prod_nonneg (fun ε _ => hp (Bool.xor ε c,b)))) (Nat.cast_nonneg _)
-      · intro b _
-        exact fiber_average_bound (pairFiber S b) (fun ε => p (ε,b)) (fun ε => hp (ε,b))
-    _ = _ := (prod_pairFiber S (fun x => (p x+p (!x.1,x.2))/2)).symm
-
-omit [Fintype β] in
-lemma subset_pairLayer_image_iff (S A : Finset (Bool × β)) (ξ : β → Bool) :
-    S ⊆ A.image (pairLayer ξ) ↔ S.image (pairLayer ξ) ⊆ A := by
-  have hi (x : Bool × β) : pairLayer ξ (pairLayer ξ x) = x := (pairLayer ξ).left_inv x
-  constructor
-  · intro h
-    have h' := Finset.image_subset_image h (f := pairLayer ξ)
-    simpa only [Finset.image_image,Function.comp_def,hi,Finset.image_id'] using h'
-  · intro h
-    have h' := Finset.image_subset_image h (f := pairLayer ξ)
-    simpa only [Finset.image_image,Function.comp_def,hi,Finset.image_id'] using h'
-
-noncomputable def fairSetLayer {Ω : Type*} (F : Ω → Finset (Bool × β)) :
-    Ω × (β → Bool) → Finset (Bool × β) := fun ω => (F ω.1).image (pairLayer ω.2)
-
-lemma finiteMean_comm {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] (f : Ω → Ω' → ℝ) :
-    finiteMean (fun x => finiteMean (f x)) = finiteMean (fun y => finiteMean (fun x => f x y)) := by
-  simp only [finiteMean,←Finset.sum_div,div_div]
-  rw [Finset.sum_comm,mul_comm]
-
-lemma inclusionMoment_fairSetLayer {Ω : Type*} [Fintype Ω]
-    (F : Ω → Finset (Bool × β)) (S : Finset (Bool × β)) :
-    inclusionMoment (fairSetLayer F) S =
-      finiteMean (fun ξ : β → Bool => inclusionMoment F (S.image (pairLayer ξ))) := by
-  classical
-  simp only [inclusionMoment_eq]
-  have he : (fun ω : Ω × (β → Bool) => if S ⊆ fairSetLayer F ω then (1:ℝ) else 0) =
-      (fun ω : Ω × (β → Bool) => if S.image (pairLayer ω.2) ⊆ F ω.1 then (1:ℝ) else 0) := by
-    funext ω
-    simp only [fairSetLayer,subset_pairLayer_image_iff]
-  apply (congrArg finiteMean he).trans
-  exact (finiteMean_prod (fun ω : Ω × (β → Bool) =>
-    if S.image (pairLayer ω.2) ⊆ F ω.1 then (1:ℝ) else 0)).trans
-      (finiteMean_comm (fun ω ξ => if S.image (pairLayer ξ) ⊆ F ω then (1:ℝ) else 0))
-
-lemma inclusionMoment_fairSetLayer_singleton {Ω : Type*} [Fintype Ω]
-    (F : Ω → Finset (Bool × β)) (x : Bool × β) :
-    inclusionMoment (fairSetLayer F) {x} =
-      (inclusionMoment F {x}+inclusionMoment F {(!x.1,x.2)})/2 := by
-  rw [inclusionMoment_fairSetLayer]
-  simp only [Finset.image_singleton,pairLayer,Equiv.coe_fn_mk]
-  rw [finiteMean_pi_eval (A := fun _ : β => Bool) x.2 (fun c : Bool => inclusionMoment F {(Bool.xor x.1 c,x.2)})]
-  rw [finiteMean_bool]
-  rcases x with ⟨b,x⟩
-  cases b <;> simp only [Bool.xor_false,Bool.false_xor,Bool.true_xor, Bool.not_false,Bool.not_true]
-
-lemma cylinder_fairSetLayer {Ω : Type*} [Fintype Ω]
-    (F : Ω → Finset (Bool × β))
-    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
-    CylinderBound (inclusionMoment (fairSetLayer F))
-      (fun x => inclusionMoment (fairSetLayer F) {x}) := by
-  have hh := h.fair_parallel (fun x => inclusionMoment_nonneg F {x})
-  intro S
-  rw [inclusionMoment_fairSetLayer]
-  simp only [inclusionMoment_fairSetLayer_singleton]
-  exact hh S
-
-end ParallelFairLayer
-
-section RelabelMarks
-variable {α β Ω Ω' : Type*} [DecidableEq α] [DecidableEq β] [Fintype Ω] [Fintype Ω']
-
-omit [Fintype Ω] [Fintype Ω'] in
-lemma subset_image_equiv_iff (e : α ≃ β) (A : Finset α) (S : Finset β) :
-    S ⊆ A.image e ↔ S.image e.symm ⊆ A := by
-  constructor
-  · intro h x hx
-    obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
-    obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp (h hy)
-    simpa only [←hxy,Equiv.symm_apply_apply] using hx
-  · intro h y hy
-    exact Finset.mem_image.mpr ⟨e.symm y,h (Finset.mem_image_of_mem _ hy),e.apply_symm_apply y⟩
-
-omit [Fintype Ω'] in
-lemma inclusionMoment_image_equiv (e : α ≃ β) (F : Ω → Finset α) (S : Finset β) :
-    inclusionMoment (fun ω => (F ω).image e) S = inclusionMoment F (S.image e.symm) := by
-  classical
-  simp only [inclusionMoment_eq]
-  apply finiteMean_congr
-  intro ω
-  simp only [subset_image_equiv_iff]
-
-omit [Fintype Ω'] in
-lemma cylinder_image_equiv (e : α ≃ β) (F : Ω → Finset α)
-    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
-    CylinderBound (inclusionMoment (fun ω => (F ω).image e))
-      (fun x => inclusionMoment (fun ω => (F ω).image e) {x}) := by
-  intro S
-  simp only [inclusionMoment_image_equiv,Finset.image_singleton]
-  have hh := h (S.image e.symm)
-  simpa only [Finset.prod_image e.symm.injective.injOn] using hh
-
-omit [DecidableEq β] in
-lemma inclusionMoment_comp_equiv (e : Ω ≃ Ω') (F : Ω' → Finset α) (S : Finset α) :
-    inclusionMoment (fun ω => F (e ω)) S = inclusionMoment F S := by
-  classical
-  simp only [inclusionMoment_eq]
-  simpa only [Equiv.apply_symm_apply] using
-    finiteMean_equiv e (fun ω => if S ⊆ F (e ω) then (1 : ℝ) else 0)
-
-omit [DecidableEq β] in
-lemma cylinder_comp_equiv (e : Ω ≃ Ω') (F : Ω' → Finset α)
-    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
-    CylinderBound (inclusionMoment (fun ω => F (e ω)))
-      (fun x => inclusionMoment (fun ω => F (e ω)) {x}) := by
-  intro S
-  simpa only [inclusionMoment_comp_equiv] using h S
-end RelabelMarks
-
-lemma finiteMean_prod_mul {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
-    (f : Ω → ℝ) (g : Ω' → ℝ) :
-    finiteMean (fun ω : Ω × Ω' => f ω.1*g ω.2) = finiteMean f * finiteMean g := by
-  simp only [finiteMean,Fintype.sum_prod_type,Fintype.card_prod,Nat.cast_mul,
-    ←Finset.mul_sum,←Finset.sum_mul]
-  ring
-
-section JoinChildren
-variable {β : Type*} [DecidableEq β]
-
-noncomputable def childJoin (A B : Finset β) : Finset (Bool × β) :=
-  A.image (fun x => (false,x)) ∪ B.image (fun x => (true,x))
-
-lemma mem_childJoin (A B : Finset β) (b : Bool) (x : β) :
-    (b,x) ∈ childJoin A B ↔ (if b then x ∈ B else x ∈ A) := by
-  cases b <;> simp [childJoin]
-
-noncomputable def boolFiber (S : Finset (Bool × β)) (b : Bool) : Finset β :=
-  (S.filter (fun x => x.1 = b)).image Prod.snd
-
-lemma mem_boolFiber (S : Finset (Bool × β)) (b : Bool) (x : β) :
-    x ∈ boolFiber S b ↔ (b,x) ∈ S := by
-  constructor
-  · intro h
-    obtain ⟨⟨c,y⟩,hy,hxy⟩ := Finset.mem_image.mp h
-    obtain ⟨hy,hc⟩ := Finset.mem_filter.mp hy
-    change c = b at hc
-    change y = x at hxy
-    simpa only [hc,hxy] using hy
-  · intro h
-    exact Finset.mem_image.mpr ⟨(b,x),Finset.mem_filter.mpr ⟨h,rfl⟩,rfl⟩
-
-lemma childJoin_boolFiber (S : Finset (Bool × β)) :
-    childJoin (boolFiber S false) (boolFiber S true) = S := by
-  ext ⟨b,x⟩
-  rw [mem_childJoin]
-  cases b <;> simp only [Bool.false_eq_true,↓reduceIte,mem_boolFiber]
-
-lemma subset_childJoin_iff (S : Finset (Bool × β)) (A B : Finset β) :
-    S ⊆ childJoin A B ↔ boolFiber S false ⊆ A ∧ boolFiber S true ⊆ B := by
-  constructor
-  · intro h
-    constructor
-    · intro x hx
-      exact (mem_childJoin A B false x).mp (h ((mem_boolFiber S false x).mp hx))
-    · intro x hx
-      exact (mem_childJoin A B true x).mp (h ((mem_boolFiber S true x).mp hx))
-  · rintro ⟨h₀,h₁⟩ ⟨b,x⟩ hx
-    rw [mem_childJoin]
-    cases b
-    · exact h₀ ((mem_boolFiber S false x).mpr hx)
-    · exact h₁ ((mem_boolFiber S true x).mpr hx)
-
-lemma prod_childJoin (A B : Finset β) (f : Bool × β → ℝ) :
-    (∏ x ∈ childJoin A B, f x) = (∏ x ∈ A, f (false,x)) * ∏ x ∈ B, f (true,x) := by
-  classical
-  have hd : Disjoint (A.image (fun x => (false,x))) (B.image (fun x => (true,x))) := by
-    apply Finset.disjoint_left.mpr
-    rintro x hx hy
-    obtain ⟨a,ha,rfl⟩ := Finset.mem_image.mp hx
-    obtain ⟨b,hb,hba⟩ := Finset.mem_image.mp hy
-    exact Bool.false_ne_true (congrArg Prod.fst hba).symm
-  rw [childJoin,Finset.prod_union hd,
-    Finset.prod_image (fun _ _ _ _ h => congrArg Prod.snd h),
-    Finset.prod_image (fun _ _ _ _ h => congrArg Prod.snd h)]
-
-lemma prod_boolFibers (S : Finset (Bool × β)) (f : Bool × β → ℝ) :
-    (∏ x ∈ S, f x) = (∏ x ∈ boolFiber S false, f (false,x)) *
-      ∏ x ∈ boolFiber S true, f (true,x) := by
-  rw [←prod_childJoin,childJoin_boolFiber]
-
-lemma inclusionMoment_childJoin {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
-    (F : Ω → Finset β) (G : Ω' → Finset β) (S : Finset (Bool × β)) :
-    inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) S =
-      inclusionMoment F (boolFiber S false) * inclusionMoment G (boolFiber S true) := by
-  classical
-  simp only [inclusionMoment_eq]
-  rw [←finiteMean_prod_mul]
-  apply finiteMean_congr
-  intro ω
-  simp only [subset_childJoin_iff]
-  split_ifs <;> simp_all
-
-lemma cylinder_childJoin {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] [Nonempty Ω] [Nonempty Ω']
-    (F : Ω → Finset β) (G : Ω' → Finset β)
-    (hF : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x}))
-    (hG : CylinderBound (inclusionMoment G) (fun x => inclusionMoment G {x})) :
-    CylinderBound (inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)))
-      (fun x => inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {x}) := by
-  intro S
-  rw [inclusionMoment_childJoin,prod_boolFibers]
-  have hf (b : Bool) (x : β) : boolFiber {(b,x)} b = {x} := by
-    ext y
-    simp only [mem_boolFiber,Finset.mem_singleton,Prod.mk.injEq,true_and]
-  have hn (b : Bool) (x : β) : boolFiber {(!b,x)} b = ∅ := by
-    ext y
-    cases b <;> simp [mem_boolFiber]
-  have h₀ (x : β) : inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {(false,x)} =
-      inclusionMoment F {x} * inclusionMoment G ∅ := by
-    rw [inclusionMoment_childJoin,hf]
-    have hh := hn true x
-    exact congrArg (fun T => inclusionMoment F {x} * inclusionMoment G T) hh
-  have h₁ (x : β) : inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {(true,x)} =
-      inclusionMoment F ∅ * inclusionMoment G {x} := by
-    rw [inclusionMoment_childJoin,hf]
-    have hh := hn false x
-    exact congrArg (fun T => inclusionMoment F T * inclusionMoment G {x}) hh
-  have emF : inclusionMoment F ∅ = 1 := by simp [inclusionMoment,finiteMean_const]
-  have emG : inclusionMoment G ∅ = 1 := by simp [inclusionMoment,finiteMean_const]
-  simp only [h₀,h₁,emF,emG,mul_one,one_mul]
-  exact mul_le_mul (hF _) (hG _) (inclusionMoment_nonneg _ _)
-    (Finset.prod_nonneg (fun x _ => inclusionMoment_nonneg F {x}))
-end JoinChildren
-
-lemma childMarks_join (d : ℕ) (A : Finset (Card (d+1))) :
-    A = (childMarks A false).image (Fin.cons false) ∪
-      (childMarks A true).image (Fin.cons true) := by
-  classical
-  ext x
-  constructor
-  · intro hx
-    cases hb : x 0
-    · exact Finset.mem_union_left _ (Finset.mem_image.mpr
-        ⟨Fin.tail x,(mem_childMarks _ _ _).mpr (by simpa only [←hb,Fin.cons_self_tail] using hx),
-          by simp only [←hb,Fin.cons_self_tail]⟩)
-    · exact Finset.mem_union_right _ (Finset.mem_image.mpr
-        ⟨Fin.tail x,(mem_childMarks _ _ _).mpr (by simpa only [←hb,Fin.cons_self_tail] using hx),
-          by simp only [←hb,Fin.cons_self_tail]⟩)
-  · intro hx
-    rcases Finset.mem_union.mp hx with hx | hx
-    · obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
-      exact (mem_childMarks _ _ _).mp hy
-    · obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
-      exact (mem_childMarks _ _ _).mp hy
-
-noncomputable def butterflyMarks (d : ℕ) (A : Finset (Card d))
-    (ω : SwitchIndex d → Bool) : Finset (Card d) :=
-  A.image (butterflyPerm d (decodeButterfly d ω))
-
-lemma butterflyMarks_succ (d : ℕ) (A : Finset (Card (d+1))) (ω : SwitchIndex (d+1) → Bool) :
-    butterflyMarks (d+1) A ω =
-      (fairSetLayer (fun p : (SwitchIndex d → Bool) × (SwitchIndex d → Bool) =>
-        childJoin (butterflyMarks d (childMarks A false) p.1)
-          (butterflyMarks d (childMarks A true) p.2)) (coinStepEquiv d ω)).image
-            (headTailEquiv d).symm := by
-  classical
-  unfold butterflyMarks fairSetLayer childJoin
-  rw [Finset.image_union,Finset.image_union]
-  simp only [Finset.image_image]
-  conv_lhs => rw [childMarks_join d A]
-  rw [Finset.image_union]
-  simp only [Finset.image_image]
-  apply congrArg₂ (· ∪ ·)
-  · apply Finset.image_congr
-    intro x _
-    simp only [Function.comp_def,butterflyPerm,decodeButterfly,Equiv.Perm.coe_mul,
-      childLift,pairSwitch,switchFun,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
-      coinStepEquiv,headTailEquiv,Equiv.symm_symm,pairLayer]
-    rfl
-  · apply Finset.image_congr
-    intro x _
-    simp only [Function.comp_def,butterflyPerm,decodeButterfly,Equiv.Perm.coe_mul,
-      childLift,pairSwitch,switchFun,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
-      coinStepEquiv,headTailEquiv,Equiv.symm_symm,pairLayer]
-    rfl
-
-lemma butterflyMarks_cylinder (d : ℕ) (A : Finset (Card d)) :
-    CylinderBound (inclusionMoment (butterflyMarks d A))
-      (fun x => inclusionMoment (butterflyMarks d A) {x}) := by
-  induction d with
-  | zero =>
-    have he : butterflyMarks 0 A = fun _ : SwitchIndex 0 → Bool => A := by
-      funext ω
-      simp [butterflyMarks,butterflyPerm]
-    rw [he]
-    exact cylinder_const A
-  | succ d ih =>
-    have h := cylinder_childJoin (butterflyMarks d (childMarks A false))
-      (butterflyMarks d (childMarks A true)) (ih _) (ih _)
-    have hh := cylinder_image_equiv (headTailEquiv d).symm _ (cylinder_fairSetLayer _ h)
-    have hs := cylinder_comp_equiv (coinStepEquiv d) _ hh
-    simpa only [←butterflyMarks_succ] using hs
-
-lemma indicator_mem_image_perm {α : Type*} [DecidableEq α]
-    (A : Finset α) (p : Equiv.Perm α) (y : α) :
-    (if y ∈ A.image p then (1 : ℝ) else 0) = ∑ x ∈ A, if p x = y then 1 else 0 := by
-  classical
-  by_cases h : y ∈ A.image p
-  · obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp h
-    rw [ite_eq_left (Finset.mem_image.mpr ⟨x,hx,hxy⟩)]
-    symm
-    calc
-      _ = (if p x = y then (1:ℝ) else 0) := Finset.sum_eq_single_of_mem x hx (by
-        intro z _ hzx
-        exact ite_eq_right (fun hz => hzx (p.injective (hz.trans hxy.symm))))
-      _ = 1 := ite_eq_left hxy
-  · rw [ite_eq_right h]
-    symm
-    apply Finset.sum_eq_zero
-    intro x hx
-    exact ite_eq_right (fun hxy => h (Finset.mem_image.mpr ⟨x,hx,hxy⟩))
-
-lemma butterflyMarks_singleton (d : ℕ) (A : Finset (Card d)) (y : Card d) :
-    inclusionMoment (butterflyMarks d A) {y} = (A.card : ℝ)/(2:ℝ)^d := by
-  classical
-  unfold inclusionMoment
-  simp only [Finset.singleton_subset_iff,butterflyMarks,indicator_mem_image_perm]
-  rw [finiteMean_sum]
-  simp only [butterfly_single_card_uniform,Finset.sum_const,nsmul_eq_mul]
-  ring
-
-lemma butterflyMarks_inclusion (d : ℕ) (A S : Finset (Card d)) :
-    inclusionMoment (butterflyMarks d A) S ≤ ((A.card : ℝ)/(2:ℝ)^d)^S.card := by
-  have h := butterflyMarks_cylinder d A S
-  simpa only [butterflyMarks_singleton,Finset.prod_const] using h
-
-namespace CycleColoring
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-abbrev Coloring (p : Equiv.Perm α) (S : Finset α) :=
-  {f : α → Bool // (∀ x, x ∉ S → f x = false) ∧
-    ∀ x ∈ S, p x ∈ S → f (p x) = f x}
-
-noncomputable instance (p : Equiv.Perm α) (S : Finset α) : Fintype (Coloring p S) :=
-  Fintype.ofFinite _
-
-noncomputable def boundary (p : Equiv.Perm α) (S : Finset α) : Finset α :=
-  S.filter (fun x => p.symm x ∉ S)
-
-noncomputable def root {p : Equiv.Perm α} {S : Finset α}
-    (C : RoutingNetwork.SelectedCycle p S) : α := C.nonempty.choose
-
-lemma root_mem {p : Equiv.Perm α} {S : Finset α} (C : RoutingNetwork.SelectedCycle p S) :
-    root C ∈ C.val := C.nonempty.choose_spec
-
-abbrev Seed (p : Equiv.Perm α) (S : Finset α) :=
-  {x // x ∈ boundary p S} ⊕ RoutingNetwork.SelectedCycle p S
-
-noncomputable def record {p : Equiv.Perm α} {S : Finset α}
-    (f : Coloring p S) : Seed p S → Bool :=
-  Sum.elim (fun x => f.val x.val) (fun C => f.val (root C))
-
-lemma orbit_subset_of_predecessor_closed (p : Equiv.Perm α) (D : Finset α)
-    (hD : ∀ x ∈ D, p.symm x ∈ D) {x : α} (hx : x ∈ D) : orbitSet p x ⊆ D := by
-  intro y hy
-  have hh := ((mem_orbitSet p x y).mp hy).inv
-  obtain ⟨n,hn⟩ := hh.exists_nat_pow_eq
-  have hp (k : ℕ) : (p⁻¹ ^ k) x ∈ D := by
-    induction k with
-    | zero => simpa using hx
-    | succ k ih => simpa only [pow_succ',Equiv.Perm.mul_apply,Equiv.Perm.coe_inv] using hD _ ih
-  exact hn ▸ hp n
-
-lemma record_injective (p : Equiv.Perm α) (S : Finset α) :
-    Function.Injective (record (p := p) (S := S)) := by
-  classical
-  intro f g hfg
-  apply Subtype.ext
-  funext x
-  by_contra hne
-  have hx : x ∈ S := by
-    by_contra h
-    exact hne ((f.property.1 x h).trans (g.property.1 x h).symm)
-  let D := S.filter (fun y => f.val y ≠ g.val y)
-  have hxD : x ∈ D := Finset.mem_filter.mpr ⟨hx,hne⟩
-  have hprev (y : α) (hy : y ∈ D) : p.symm y ∈ D := by
-    obtain ⟨hyS,hyne⟩ := Finset.mem_filter.mp hy
-    have hpS : p.symm y ∈ S := by
-      by_contra hn
-      have hseed : y ∈ boundary p S := Finset.mem_filter.mpr ⟨hyS,hn⟩
-      exact hyne (congrFun hfg (Sum.inl ⟨y,hseed⟩))
-    refine Finset.mem_filter.mpr ⟨hpS,?_⟩
-    have hf := f.property.2 (p.symm y) hpS (by simpa only [Equiv.apply_symm_apply] using hyS)
-    have hg := g.property.2 (p.symm y) hpS (by simpa only [Equiv.apply_symm_apply] using hyS)
-    simp only [Equiv.apply_symm_apply] at hf hg
-    intro he
-    exact hyne (hf.trans (he.trans hg.symm))
-  have horb := orbit_subset_of_predecessor_closed p D hprev hxD
-  let C : RoutingNetwork.SelectedCycle p S :=
-    ⟨orbitSet p x,⟨x,rfl⟩,fun y hy => (Finset.mem_filter.mp (horb hy)).1⟩
-  have hrD : root C ∈ D := horb (root_mem C)
-  have hrne := (Finset.mem_filter.mp hrD).2
-  exact hrne (congrFun hfg (Sum.inr C))
-
-lemma card_coloring_le (p : Equiv.Perm α) (S : Finset α) :
-    Fintype.card (Coloring p S) ≤
-      2 ^ ((boundary p S).card + Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
-  have h := Fintype.card_le_of_injective (record (p := p) (S := S)) (record_injective p S)
-  simpa only [Fintype.card_fun,Fintype.card_sum,Fintype.card_coe,Fintype.card_bool] using h
-
-omit [Fintype α] in
-lemma boundary_card (p : Equiv.Perm α) (S : Finset α) :
-    (boundary p S).card = S.card - (S.filter (fun x => p.symm x ∈ S)).card := by
-  classical
-  have h := Finset.card_filter_add_card_filter_not (s := S) (fun x => p.symm x ∈ S)
-  unfold boundary
-  omega
-
-lemma card_coloring_le_edges (p : Equiv.Perm α) (S : Finset α) :
-    Fintype.card (Coloring p S) ≤
-      2 ^ (S.card - (S.filter (fun x => p.symm x ∈ S)).card +
-        Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
-  simpa only [boundary_card] using card_coloring_le p S
-
-
-abbrev OppositeColoring (p : Equiv.Perm α) (S : Finset α) :=
-  {f : α → Bool // (∀ x, x ∉ S → f x = false) ∧
-    ∀ x ∈ S, p x ∈ S → f (p x) = !(f x)}
-
-noncomputable instance (p : Equiv.Perm α) (S : Finset α) : Fintype (OppositeColoring p S) :=
-  Fintype.ofFinite _
-
-noncomputable def bipartitionFlip (p : Equiv.Perm α) (S : Finset α)
-    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x))
-    (f : OppositeColoring p S) : Coloring p S := by
-  classical
-  refine ⟨(fun x => if x ∈ S then Bool.xor (f.val x) (χ x) else false),?_,?_⟩
-  · intro x hx
-    exact ite_eq_right hx
-  · intro x hx hpx
-    simp only [ite_eq_left hx,ite_eq_left hpx,f.property.2 x hx hpx,hχ x]
-    cases f.val x <;> cases χ x <;> rfl
-
-end CycleColoring
-end Thorp
-
-end ThorpNine.Dense
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Dense.PriorPaths
+
+namespace OAI
+
+namespace ThorpNine.Dense
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+section ParallelButterfly
+variable {α B : Type*} [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] {d : ℕ}
+omit [DecidableEq B] in
+lemma parallel_entropy (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool) (S : Finset α) :
+    (∑ i : B × SwitchIndex d,
+      if (parallelUsers pre ω i).1 ∈ S ∧ (parallelUsers pre ω i).2 ∈ S then (1:ℝ) else 0) ≤
+      (S.card : ℝ) * Real.log S.card / (2 * Real.log 2) := by
+  classical
+  rw [Fintype.sum_prod_type]
+  calc
+    _ ≤ ∑ b : B, (blockMarks pre S b).card * Real.log (blockMarks pre S b).card / (2*Real.log 2) := by
+      apply Finset.sum_le_sum
+      intro b _
+      simpa only [←mem_blockMarks,parallelUsers] using
+        marked_switch_entropy_bound d (decodeButterfly d (fun i => ω (b,i))) (blockMarks pre S b)
+    _ ≤ ∑ b : B, (blockMarks pre S b).card * Real.log S.card / (2*Real.log 2) := by
+      apply Finset.sum_le_sum
+      intro b _
+      by_cases hz : (blockMarks pre S b).card = 0
+      · simp [hz]
+      · apply div_le_div_of_nonneg_right _ (by positivity)
+        apply mul_le_mul_of_nonneg_left _ (by positivity)
+        exact Real.log_le_log (by exact_mod_cast Nat.pos_of_ne_zero hz)
+          (by exact_mod_cast card_blockMarks_le pre S b)
+    _ = _ := by rw [←Finset.sum_div,←Finset.sum_mul,←Nat.cast_sum,sum_card_blockMarks]
+
+noncomputable def parallelFollow (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x : α) (k : α → QueryTree (B × SwitchIndex d)) : QueryTree (B × SwitchIndex d) :=
+  QueryTree.followPath d (fun i => ((pre x).1,i)) (pre x).2 (fun y => k (post ((pre x).1,y)))
+
+omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
+lemma parallelFollow_succeeds (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
+    (parallelFollow pre post x k).succeeds ω = (k (relabeledPerm pre post ω x)).succeeds ω := by
+  exact QueryTree.followPath_succeeds _ _ _ _ _
+
+omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
+lemma parallelFollow_raw_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
+    (parallelFollow pre post x k).charge ω = (k (relabeledPerm pre post ω x)).charge ω := by
+  exact QueryTree.followPath_raw_charge _ _ _ _ _
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma parallelFollow_stale_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
+    (ω : B × SwitchIndex d → Bool) :
+    (parallelFollow pre post x k).staleCharge R ω =
+      (k (relabeledPerm pre post ω x)).staleCharge (R ∪ parallelPath pre ω x) ω := by
+  exact QueryTree.followPath_stale_charge _ _ _ _ _ _
+
+noncomputable def closingDomain (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) : Finset (B × SwitchIndex d) :=
+  (routeDomain d (pre x).2 (post.symm y).2).image (fun i => ((pre x).1,i))
+
+def closingValue (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (i : B × SwitchIndex d) : Bool := routeValue d (pre x).2 (post.symm y).2 i.2
+
+omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
+lemma relabeled_endpoint_pair (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (ω : B × SwitchIndex d → Bool) (x y : α) :
+    relabeledPerm pre post ω x = y ↔
+      (pre x).1 = (post.symm y).1 ∧
+      butterflyPerm d (decodeButterfly d (fun i => ω ((pre x).1,i))) (pre x).2 = (post.symm y).2 := by
+  change post ((pre x).1,_) = y ↔ _
+  rw [←post.eq_symm_apply,Prod.mk.injEq]
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma relabeled_endpoint_iff (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (ω : B × SwitchIndex d → Bool) (x y : α) :
+    relabeledPerm pre post ω x = y ↔
+      (pre x).1 = (post.symm y).1 ∧
+        ∀ i ∈ closingDomain pre post x y, ω i = closingValue pre post x y i := by
+  classical
+  rw [relabeled_endpoint_pair,butterfly_endpoint_iff]
+  simp only [closingDomain,Finset.forall_mem_image,closingValue]
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma closingDomain_eq (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (ω : B × SwitchIndex d → Bool) (x y : α) (h : relabeledPerm pre post ω x = y) :
+    closingDomain pre post x y = parallelPath pre ω x := by
+  rw [closingDomain,parallelPath,(relabeled_endpoint_pair _ _ _ _ _).mp h |>.2]
+
+noncomputable def parallelClose (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (k : QueryTree (B × SwitchIndex d)) : QueryTree (B × SwitchIndex d) :=
+  if (pre x).1 = (post.symm y).1 then
+    QueryTree.forceList (closingDomain pre post x y).toList (closingValue pre post x y) k
+  else QueryTree.reject
+
+omit [Fintype α] in
+lemma parallelClose_succeeds (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (k : QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
+    (parallelClose pre post x y k).succeeds ω =
+      if relabeledPerm pre post ω x = y then k.succeeds ω else false := by
+  classical
+  simp only [relabeled_endpoint_iff,parallelClose]
+  by_cases hb : (pre x).1 = (post.symm y).1
+  · rw [ite_eq_left hb,QueryTree.forceList_succeeds]
+    simp only [Finset.mem_toList,hb,true_and]
+  · simp [hb,QueryTree.succeeds]
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma parallelClose_raw_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (k : QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool)
+    (h : relabeledPerm pre post ω x = y) :
+    (parallelClose pre post x y k).charge ω = k.charge ω + d := by
+  have hh := (relabeled_endpoint_iff _ _ _ _ _).mp h
+  rw [parallelClose,ite_eq_left hh.1,QueryTree.forceList_raw_charge _ _ _ _
+    (by simpa only [Finset.mem_toList] using hh.2),Finset.length_toList,
+    closingDomain_eq _ _ _ _ _ h,card_parallelPath]
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma parallelClose_fresh_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (k : QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
+    (ω : B × SwitchIndex d → Bool) (h : relabeledPerm pre post ω x = y) :
+    (parallelClose pre post x y k).freshCharge R ω =
+      k.freshCharge (R ∪ parallelPath pre ω x) ω + (parallelPath pre ω x \ R).card := by
+  have hh := (relabeled_endpoint_iff _ _ _ _ _).mp h
+  rw [parallelClose,ite_eq_left hh.1,QueryTree.forceList_charge _ _ _ _ _
+    (by simpa only [Finset.mem_toList] using hh.2),Finset.toList_toFinset,
+    closingDomain_eq _ _ _ _ _ h]
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma parallelClose_stale_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (k : QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
+    (ω : B × SwitchIndex d → Bool) (h : relabeledPerm pre post ω x = y) :
+    (parallelClose pre post x y k).staleCharge R ω =
+      k.staleCharge (R ∪ parallelPath pre ω x) ω + (parallelPath pre ω x ∩ R).card := by
+  have h₀ := QueryTree.fresh_add_stale (parallelClose pre post x y k) R ω
+  have h₁ := QueryTree.fresh_add_stale k (R ∪ parallelPath pre ω x) ω
+  rw [parallelClose_raw_charge _ _ _ _ _ _ h,parallelClose_fresh_charge _ _ _ _ _ _ _ h] at h₀
+  have h₂ : (parallelPath pre ω x \ R).card + (parallelPath pre ω x ∩ R).card = d := by
+    rw [Finset.card_sdiff_add_card_inter,card_parallelPath]
+  omega
+
+noncomputable def parallelNetwork (pre : α ≃ B × Card d) (post : B × Card d ≃ α) :
+    RoutingNetwork α (B × SwitchIndex d) d where
+  perm := relabeledPerm pre post
+  path := parallelPath pre
+  users := parallelUsers pre
+  users_ne := parallelUsers_ne pre
+  mem_path := mem_parallelPath pre
+  card_path := card_parallelPath pre
+  entropy := parallel_entropy pre
+  follow := parallelFollow pre post
+  close := parallelClose pre post
+  follow_succeeds := parallelFollow_succeeds pre post
+  follow_raw_charge := parallelFollow_raw_charge pre post
+  follow_stale_charge := parallelFollow_stale_charge pre post
+  close_succeeds := parallelClose_succeeds pre post
+  close_raw_charge := parallelClose_raw_charge pre post
+  close_stale_charge := parallelClose_stale_charge pre post
+
+lemma parallel_cycle_log_mgf (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (S : Finset α) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    Real.log (finiteMean (fun ω : B × SwitchIndex d → Bool =>
+      q^(Fintype.card (RoutingNetwork.SelectedCycle (relabeledPerm pre post ω) S)))) ≤
+      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
+  exact (parallelNetwork pre post).butterfly_cycle_log_mgf S J hJ q hq
+
+end ParallelButterfly
+
+def cardSplit : (r s : ℕ) → Card (s+r) ≃ Card r × Card s
+  | 0, s =>
+    { toFun := fun x => ((fun i => Fin.elim0 i), x)
+      invFun := fun x => x.2
+      left_inv := fun _ => rfl
+      right_inv := fun x => by
+        apply Prod.ext
+        · exact Subsingleton.elim _ _
+        · rfl }
+  | r+1, s =>
+    { toFun := fun x => (Fin.cons (x 0) ((cardSplit r s (Fin.tail x)).1),
+        (cardSplit r s (Fin.tail x)).2)
+      invFun := fun x => Fin.cons (x.1 0) ((cardSplit r s).symm (Fin.tail x.1,x.2))
+      left_inv := by
+        intro x
+        simp only [Fin.cons_zero, Fin.tail_cons, Prod.mk.eta, Equiv.symm_apply_apply, Fin.cons_self_tail]
+      right_inv := by
+        intro x
+        simp only [Fin.cons_zero, Fin.tail_cons, Equiv.apply_symm_apply, Fin.cons_self_tail] }
+
+def assembleBits : (r s : ℕ) → (Card r × SwitchIndex s → Bool) →
+    (Card s × SwitchIndex r → Bool) → SwitchIndex (s+r) → Bool
+  | 0, _, lo, _, i => lo ((fun j => Fin.elim0 j), i)
+  | r+1, s, _, hi, Sum.inl y => hi ((cardSplit r s y).2,Sum.inl (cardSplit r s y).1)
+  | r+1, s, lo, hi, Sum.inr (b,i) => assembleBits r s
+      (fun j => lo (Fin.cons b j.1,j.2)) (fun j => hi (j.1,Sum.inr (b,j.2))) i
+
+lemma butterfly_split (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
+    (hi : Card s × SwitchIndex r → Bool) (x : Card (s+r)) :
+    cardSplit r s (butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi)) x) =
+      let y := cardSplit r s x
+      let z := butterflyPerm s (decodeButterfly s (fun i => lo (y.1,i))) y.2
+      (butterflyPerm r (decodeButterfly r (fun i => hi (z,i))) y.1,z) := by
+  induction r with
+  | zero =>
+    apply Prod.ext
+    · funext i
+      exact Fin.elim0 i
+    · rfl
+  | succ r ih =>
+    let lo' : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons (x 0) j.1,j.2)
+    let hi' : Card s × SwitchIndex r → Bool := fun j => hi (j.1,Sum.inr (x 0,j.2))
+    have hh := ih lo' hi' (Fin.tail x)
+    let u := butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo' hi')) (Fin.tail x)
+    let y := cardSplit r s (Fin.tail x)
+    let z := butterflyPerm s (decodeButterfly s (fun i => lo' (y.1,i))) y.2
+    let w := butterflyPerm r (decodeButterfly r (fun i => hi' (z,i))) y.1
+    change cardSplit r s u = (w,z) at hh
+    change ((Fin.cons (Bool.xor (x 0) (hi ((cardSplit r s u).2,Sum.inl (cardSplit r s u).1)))
+        ((cardSplit r s u).1) : Card (r+1)), (cardSplit r s u).2) =
+      ((Fin.cons (Bool.xor (x 0) (hi (z,Sum.inl w))) w : Card (r+1)),z)
+    rw [hh]
+
+def splitSwitch : (r s : ℕ) → SwitchIndex (s+r) ≃
+    (Card r × SwitchIndex s) ⊕ (Card s × SwitchIndex r)
+  | 0, _ =>
+    { toFun := fun i => Sum.inl ((fun j => Fin.elim0 j),i)
+      invFun := fun i => match i with
+        | Sum.inl (_,j) => j
+        | Sum.inr (_,j) => nomatch j
+      left_inv := fun _ => rfl
+      right_inv := by
+        intro i
+        rcases i with ⟨x,j⟩ | ⟨x,j⟩
+        · exact congrArg Sum.inl (Prod.ext (Subsingleton.elim _ _) rfl)
+        · exact Empty.elim j }
+  | r+1, s =>
+    { toFun := fun i => match i with
+        | Sum.inl y => Sum.inr ((cardSplit r s y).2,Sum.inl (cardSplit r s y).1)
+        | Sum.inr (b,j) => match splitSwitch r s j with
+          | Sum.inl (p,j) => Sum.inl (Fin.cons b p,j)
+          | Sum.inr (q,j) => Sum.inr (q,Sum.inr (b,j))
+      invFun := fun i => match i with
+        | Sum.inl (p,j) => Sum.inr (p 0,(splitSwitch r s).symm (Sum.inl (Fin.tail p,j)))
+        | Sum.inr (q,Sum.inl p) => Sum.inl ((cardSplit r s).symm (p,q))
+        | Sum.inr (q,Sum.inr (b,j)) => Sum.inr (b,(splitSwitch r s).symm (Sum.inr (q,j)))
+      left_inv := by
+        intro i
+        rcases i with y | ⟨b,j⟩
+        · dsimp only
+          rw [Prod.mk.eta, Equiv.symm_apply_apply]
+        · dsimp only
+          cases hz : splitSwitch r s j with
+          | inl z =>
+            rcases z with ⟨p,k⟩
+            simp only [Fin.cons_zero, Fin.tail_cons]
+            rw [←hz,Equiv.symm_apply_apply]
+          | inr z =>
+            rcases z with ⟨q,k⟩
+            dsimp only
+            rw [←hz,Equiv.symm_apply_apply]
+      right_inv := by
+        intro i
+        rcases i with ⟨p,j⟩ | ⟨q,j⟩
+        · simp only [Equiv.apply_symm_apply, Fin.cons_self_tail]
+        · rcases j with p | ⟨b,j⟩ <;> simp only [Equiv.apply_symm_apply] }
+
+lemma assembleBits_eq (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
+    (hi : Card s × SwitchIndex r → Bool) (i : SwitchIndex (s+r)) :
+    assembleBits r s lo hi i = Sum.elim lo hi (splitSwitch r s i) := by
+  induction r with
+  | zero => rfl
+  | succ r ih =>
+    rcases i with y | ⟨b,j⟩
+    · rfl
+    · simp only [assembleBits,splitSwitch,Equiv.coe_fn_mk]
+      rw [ih]
+      cases splitSwitch r s j <;> rfl
+
+def splitCoinEquiv (r s : ℕ) :
+    ((Card r × SwitchIndex s → Bool) × (Card s × SwitchIndex r → Bool)) ≃
+      (SwitchIndex (s+r) → Bool) :=
+  (Equiv.sumArrowEquivProdArrow _ _ Bool).symm.trans
+    (Equiv.arrowCongr (splitSwitch r s).symm (Equiv.refl Bool))
+
+lemma splitCoinEquiv_apply (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
+    (hi : Card s × SwitchIndex r → Bool) :
+    splitCoinEquiv r s (lo,hi) = assembleBits r s lo hi := by
+  funext i
+  exact (assembleBits_eq r s lo hi i).symm
+
+lemma finiteMean_prod {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] (f : Ω × Ω' → ℝ) :
+    finiteMean f = finiteMean (fun x => finiteMean (fun y => f (x,y))) := by
+  simp only [finiteMean, Fintype.sum_prod_type, Fintype.card_prod, Nat.cast_mul,
+    ←Finset.sum_div, div_div, mul_comm]
+
+lemma finiteMean_split (r s : ℕ) (f : (SwitchIndex (s+r) → Bool) → ℝ) :
+    finiteMean f = finiteMean (fun lo : Card r × SwitchIndex s → Bool =>
+      finiteMean (fun hi : Card s × SwitchIndex r → Bool => f (assembleBits r s lo hi))) := by
+  rw [finiteMean_equiv (splitCoinEquiv r s).symm]
+  simp only [Equiv.symm_symm]
+  rw [finiteMean_prod]
+  apply finiteMean_congr
+  intro lo
+  apply finiteMean_congr
+  intro hi
+  rw [splitCoinEquiv_apply]
+
+section PartiallyExposed
+variable {α : Type*} {r s : ℕ}
+
+def freshPre (pre : α ≃ Card (s+r)) (lo : Card r × SwitchIndex s → Bool) :
+    α ≃ Card s × Card r :=
+  ((pre.trans (cardSplit r s)).trans (parallelPerm lo)).trans (Equiv.prodComm _ _)
+
+def freshPost (post : Card (s+r) ≃ α) : Card s × Card r ≃ α :=
+  ((Equiv.prodComm _ _).trans (cardSplit r s).symm).trans post
+
+lemma exposed_perm_eq (pre : α ≃ Card (s+r)) (post : Card (s+r) ≃ α)
+    (lo : Card r × SwitchIndex s → Bool) (hi : Card s × SwitchIndex r → Bool) :
+    (pre.trans (butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi)))).trans post =
+      relabeledPerm (freshPre pre lo) (freshPost post) hi := by
+  ext x
+  simpa only [relabeledPerm, freshPre, freshPost, parallelPerm, Equiv.trans_apply,
+    Equiv.coe_fn_mk, Equiv.prodComm_apply, Prod.swap, Equiv.symm_apply_apply] using
+    congrArg (fun y => post ((cardSplit r s).symm y)) (butterfly_split r s lo hi (pre x))
+
+lemma partially_exposed_cycle_log_mgf [Fintype α] [DecidableEq α]
+    (pre : α ≃ Card (s+r)) (post : Card (s+r) ≃ α)
+    (lo : Card r × SwitchIndex s → Bool) (S : Finset α)
+    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    Real.log (finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(Fintype.card (RoutingNetwork.SelectedCycle
+        ((pre.trans (butterflyPerm (s+r)
+          (decodeButterfly (s+r) (assembleBits r s lo hi)))).trans post) S)))) ≤
+      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^r) := by
+  simp only [exposed_perm_eq]
+  exact parallel_cycle_log_mgf (freshPre pre lo) (freshPost post) S J hJ q hq
+
+end PartiallyExposed
+
+lemma inclusionMoment_eq {α Ω : Type*} [DecidableEq α] [Fintype Ω]
+    (F : Ω → Finset α) (S : Finset α) :
+    inclusionMoment F S = finiteMean (fun ω => if S ⊆ F ω then (1:ℝ) else 0) := by
+  unfold inclusionMoment
+  apply finiteMean_congr
+  intro ω
+  split_ifs <;> rfl
+
+lemma finiteMean_bool (f : Bool → ℝ) : finiteMean f = (f false+f true)/2 := by
+  simp only [finiteMean, Fintype.sum_bool, Fintype.card_bool, Nat.cast_ofNat]
+  ring
+
+lemma finiteMean_const {Ω : Type*} [Fintype Ω] [Nonempty Ω] (c : ℝ) :
+    finiteMean (fun _ : Ω => c) = c := by
+  simp only [finiteMean,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
+  exact mul_div_cancel_left₀ c (Nat.cast_ne_zero.mpr Fintype.card_ne_zero)
+
+lemma finiteMean_pi_prod {ι : Type*} [Fintype ι] [DecidableEq ι] {Ω : ι → Type*}
+    [∀ i, Fintype (Ω i)] (f : (i : ι) → Ω i → ℝ) :
+    finiteMean (fun ω : (i : ι) → Ω i => ∏ i, f i (ω i)) = ∏ i, finiteMean (f i) := by
+  simp only [finiteMean,←Fintype.prod_sum,Fintype.card_pi,Nat.cast_prod,Finset.prod_div_distrib]
+
+section ParallelFairLayer
+variable {β : Type*} [Fintype β] [DecidableEq β]
+
+noncomputable def pairFiber (S : Finset (Bool × β)) (b : β) : Finset Bool :=
+  Finset.univ.filter (fun ε => (ε,b) ∈ S)
+
+lemma prod_pairFiber (S : Finset (Bool × β)) (f : Bool × β → ℝ) :
+    (∏ x ∈ S, f x) = ∏ b : β, ∏ ε ∈ pairFiber S b, f (ε,b) := by
+  classical
+  simp only [pairFiber,Finset.prod_filter]
+  rw [Finset.prod_comm,←Fintype.prod_prod_type (fun x : Bool × β => if x ∈ S then f x else 1)]
+  rw [←Finset.prod_filter]
+  congr 1
+  ext x
+  simp
+
+lemma fiber_average_bound (S : Finset Bool) (p : Bool → ℝ) (hp : ∀ b, 0 ≤ p b) :
+    finiteMean (fun c : Bool => ∏ ε ∈ S, p (Bool.xor ε c)) ≤
+      ∏ ε ∈ S, (p ε + p (!ε))/2 := by
+  rw [finiteMean_bool]
+  have hxor : ∀ ε : Bool, Bool.xor ε true = Equiv.swap false true ε := by decide
+  simp only [Bool.xor_false,hxor]
+  calc
+    _ ≤ ∏ ε ∈ S, (p ε+p (Equiv.swap false true ε))/2 :=
+      swap_average_product S p hp false true
+    _ = _ := Finset.prod_congr rfl (fun ε _ => by cases ε <;> rfl)
+
+lemma CylinderBound.fair_parallel {m : Finset (Bool × β) → ℝ} {p : Bool × β → ℝ}
+    (h : CylinderBound m p) (hp : ∀ x, 0 ≤ p x) :
+    CylinderBound (fun S => finiteMean (fun ξ : β → Bool => m (S.image (pairLayer ξ))))
+      (fun x => (p x+p (!x.1,x.2))/2) := by
+  intro S
+  calc
+    _ ≤ finiteMean (fun ξ : β → Bool => ∏ x ∈ S, p (pairLayer ξ x)) := by
+      apply finiteMean_mono
+      intro ξ
+      simpa only [Finset.prod_image (pairLayer ξ).injective.injOn] using h (S.image (pairLayer ξ))
+    _ = ∏ b : β, finiteMean (fun c : Bool => ∏ ε ∈ pairFiber S b, p (Bool.xor ε c,b)) := by
+      calc
+        _ = finiteMean (fun ξ : β → Bool => ∏ b : β,
+            ∏ ε ∈ pairFiber S b, p (Bool.xor ε (ξ b),b)) :=
+          finiteMean_congr (fun ξ => prod_pairFiber S (fun x => p (pairLayer ξ x)))
+        _ = _ := finiteMean_pi_prod (fun b c => ∏ ε ∈ pairFiber S b, p (Bool.xor ε c,b))
+    _ ≤ ∏ b : β, ∏ ε ∈ pairFiber S b, (p (ε,b)+p (!ε,b))/2 := by
+      apply Finset.prod_le_prod₀
+      · intro b _
+        exact div_nonneg (Finset.sum_nonneg (fun c _ =>
+          Finset.prod_nonneg (fun ε _ => hp (Bool.xor ε c,b)))) (Nat.cast_nonneg _)
+      · intro b _
+        exact fiber_average_bound (pairFiber S b) (fun ε => p (ε,b)) (fun ε => hp (ε,b))
+    _ = _ := (prod_pairFiber S (fun x => (p x+p (!x.1,x.2))/2)).symm
+
+omit [Fintype β] in
+lemma subset_pairLayer_image_iff (S A : Finset (Bool × β)) (ξ : β → Bool) :
+    S ⊆ A.image (pairLayer ξ) ↔ S.image (pairLayer ξ) ⊆ A := by
+  have hi (x : Bool × β) : pairLayer ξ (pairLayer ξ x) = x := (pairLayer ξ).left_inv x
+  constructor
+  · intro h
+    have h' := Finset.image_subset_image h (f := pairLayer ξ)
+    simpa only [Finset.image_image,Function.comp_def,hi,Finset.image_id'] using h'
+  · intro h
+    have h' := Finset.image_subset_image h (f := pairLayer ξ)
+    simpa only [Finset.image_image,Function.comp_def,hi,Finset.image_id'] using h'
+
+noncomputable def fairSetLayer {Ω : Type*} (F : Ω → Finset (Bool × β)) :
+    Ω × (β → Bool) → Finset (Bool × β) := fun ω => (F ω.1).image (pairLayer ω.2)
+
+lemma finiteMean_comm {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] (f : Ω → Ω' → ℝ) :
+    finiteMean (fun x => finiteMean (f x)) = finiteMean (fun y => finiteMean (fun x => f x y)) := by
+  simp only [finiteMean,←Finset.sum_div,div_div]
+  rw [Finset.sum_comm,mul_comm]
+
+lemma inclusionMoment_fairSetLayer {Ω : Type*} [Fintype Ω]
+    (F : Ω → Finset (Bool × β)) (S : Finset (Bool × β)) :
+    inclusionMoment (fairSetLayer F) S =
+      finiteMean (fun ξ : β → Bool => inclusionMoment F (S.image (pairLayer ξ))) := by
+  classical
+  simp only [inclusionMoment_eq]
+  have he : (fun ω : Ω × (β → Bool) => if S ⊆ fairSetLayer F ω then (1:ℝ) else 0) =
+      (fun ω : Ω × (β → Bool) => if S.image (pairLayer ω.2) ⊆ F ω.1 then (1:ℝ) else 0) := by
+    funext ω
+    simp only [fairSetLayer,subset_pairLayer_image_iff]
+  apply (congrArg finiteMean he).trans
+  exact (finiteMean_prod (fun ω : Ω × (β → Bool) =>
+    if S.image (pairLayer ω.2) ⊆ F ω.1 then (1:ℝ) else 0)).trans
+      (finiteMean_comm (fun ω ξ => if S.image (pairLayer ξ) ⊆ F ω then (1:ℝ) else 0))
+
+lemma inclusionMoment_fairSetLayer_singleton {Ω : Type*} [Fintype Ω]
+    (F : Ω → Finset (Bool × β)) (x : Bool × β) :
+    inclusionMoment (fairSetLayer F) {x} =
+      (inclusionMoment F {x}+inclusionMoment F {(!x.1,x.2)})/2 := by
+  rw [inclusionMoment_fairSetLayer]
+  simp only [Finset.image_singleton,pairLayer,Equiv.coe_fn_mk]
+  rw [finiteMean_pi_eval (A := fun _ : β => Bool) x.2 (fun c : Bool => inclusionMoment F {(Bool.xor x.1 c,x.2)})]
+  rw [finiteMean_bool]
+  rcases x with ⟨b,x⟩
+  cases b <;> simp only [Bool.xor_false,Bool.false_xor,Bool.true_xor, Bool.not_false,Bool.not_true]
+
+lemma cylinder_fairSetLayer {Ω : Type*} [Fintype Ω]
+    (F : Ω → Finset (Bool × β))
+    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
+    CylinderBound (inclusionMoment (fairSetLayer F))
+      (fun x => inclusionMoment (fairSetLayer F) {x}) := by
+  have hh := h.fair_parallel (fun x => inclusionMoment_nonneg F {x})
+  intro S
+  rw [inclusionMoment_fairSetLayer]
+  simp only [inclusionMoment_fairSetLayer_singleton]
+  exact hh S
+
+end ParallelFairLayer
+
+section RelabelMarks
+variable {α β Ω Ω' : Type*} [DecidableEq α] [DecidableEq β] [Fintype Ω] [Fintype Ω']
+
+omit [Fintype Ω] [Fintype Ω'] in
+lemma subset_image_equiv_iff (e : α ≃ β) (A : Finset α) (S : Finset β) :
+    S ⊆ A.image e ↔ S.image e.symm ⊆ A := by
+  constructor
+  · intro h x hx
+    obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
+    obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp (h hy)
+    simpa only [←hxy,Equiv.symm_apply_apply] using hx
+  · intro h y hy
+    exact Finset.mem_image.mpr ⟨e.symm y,h (Finset.mem_image_of_mem _ hy),e.apply_symm_apply y⟩
+
+omit [Fintype Ω'] in
+lemma inclusionMoment_image_equiv (e : α ≃ β) (F : Ω → Finset α) (S : Finset β) :
+    inclusionMoment (fun ω => (F ω).image e) S = inclusionMoment F (S.image e.symm) := by
+  classical
+  simp only [inclusionMoment_eq]
+  apply finiteMean_congr
+  intro ω
+  simp only [subset_image_equiv_iff]
+
+omit [Fintype Ω'] in
+lemma cylinder_image_equiv (e : α ≃ β) (F : Ω → Finset α)
+    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
+    CylinderBound (inclusionMoment (fun ω => (F ω).image e))
+      (fun x => inclusionMoment (fun ω => (F ω).image e) {x}) := by
+  intro S
+  simp only [inclusionMoment_image_equiv,Finset.image_singleton]
+  have hh := h (S.image e.symm)
+  simpa only [Finset.prod_image e.symm.injective.injOn] using hh
+
+omit [DecidableEq β] in
+lemma inclusionMoment_comp_equiv (e : Ω ≃ Ω') (F : Ω' → Finset α) (S : Finset α) :
+    inclusionMoment (fun ω => F (e ω)) S = inclusionMoment F S := by
+  classical
+  simp only [inclusionMoment_eq]
+  simpa only [Equiv.apply_symm_apply] using
+    finiteMean_equiv e (fun ω => if S ⊆ F (e ω) then (1 : ℝ) else 0)
+
+omit [DecidableEq β] in
+lemma cylinder_comp_equiv (e : Ω ≃ Ω') (F : Ω' → Finset α)
+    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
+    CylinderBound (inclusionMoment (fun ω => F (e ω)))
+      (fun x => inclusionMoment (fun ω => F (e ω)) {x}) := by
+  intro S
+  simpa only [inclusionMoment_comp_equiv] using h S
+end RelabelMarks
+
+lemma finiteMean_prod_mul {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
+    (f : Ω → ℝ) (g : Ω' → ℝ) :
+    finiteMean (fun ω : Ω × Ω' => f ω.1*g ω.2) = finiteMean f * finiteMean g := by
+  simp only [finiteMean,Fintype.sum_prod_type,Fintype.card_prod,Nat.cast_mul,
+    ←Finset.mul_sum,←Finset.sum_mul]
+  ring
+
+section JoinChildren
+variable {β : Type*} [DecidableEq β]
+
+noncomputable def childJoin (A B : Finset β) : Finset (Bool × β) :=
+  A.image (fun x => (false,x)) ∪ B.image (fun x => (true,x))
+
+lemma mem_childJoin (A B : Finset β) (b : Bool) (x : β) :
+    (b,x) ∈ childJoin A B ↔ (if b then x ∈ B else x ∈ A) := by
+  cases b <;> simp [childJoin]
+
+noncomputable def boolFiber (S : Finset (Bool × β)) (b : Bool) : Finset β :=
+  (S.filter (fun x => x.1 = b)).image Prod.snd
+
+lemma mem_boolFiber (S : Finset (Bool × β)) (b : Bool) (x : β) :
+    x ∈ boolFiber S b ↔ (b,x) ∈ S := by
+  constructor
+  · intro h
+    obtain ⟨⟨c,y⟩,hy,hxy⟩ := Finset.mem_image.mp h
+    obtain ⟨hy,hc⟩ := Finset.mem_filter.mp hy
+    change c = b at hc
+    change y = x at hxy
+    simpa only [hc,hxy] using hy
+  · intro h
+    exact Finset.mem_image.mpr ⟨(b,x),Finset.mem_filter.mpr ⟨h,rfl⟩,rfl⟩
+
+lemma childJoin_boolFiber (S : Finset (Bool × β)) :
+    childJoin (boolFiber S false) (boolFiber S true) = S := by
+  ext ⟨b,x⟩
+  rw [mem_childJoin]
+  cases b <;> simp only [Bool.false_eq_true,↓reduceIte,mem_boolFiber]
+
+lemma subset_childJoin_iff (S : Finset (Bool × β)) (A B : Finset β) :
+    S ⊆ childJoin A B ↔ boolFiber S false ⊆ A ∧ boolFiber S true ⊆ B := by
+  constructor
+  · intro h
+    constructor
+    · intro x hx
+      exact (mem_childJoin A B false x).mp (h ((mem_boolFiber S false x).mp hx))
+    · intro x hx
+      exact (mem_childJoin A B true x).mp (h ((mem_boolFiber S true x).mp hx))
+  · rintro ⟨h₀,h₁⟩ ⟨b,x⟩ hx
+    rw [mem_childJoin]
+    cases b
+    · exact h₀ ((mem_boolFiber S false x).mpr hx)
+    · exact h₁ ((mem_boolFiber S true x).mpr hx)
+
+lemma prod_childJoin (A B : Finset β) (f : Bool × β → ℝ) :
+    (∏ x ∈ childJoin A B, f x) = (∏ x ∈ A, f (false,x)) * ∏ x ∈ B, f (true,x) := by
+  classical
+  have hd : Disjoint (A.image (fun x => (false,x))) (B.image (fun x => (true,x))) := by
+    apply Finset.disjoint_left.mpr
+    rintro x hx hy
+    obtain ⟨a,ha,rfl⟩ := Finset.mem_image.mp hx
+    obtain ⟨b,hb,hba⟩ := Finset.mem_image.mp hy
+    exact Bool.false_ne_true (congrArg Prod.fst hba).symm
+  rw [childJoin,Finset.prod_union hd,
+    Finset.prod_image (fun _ _ _ _ h => congrArg Prod.snd h),
+    Finset.prod_image (fun _ _ _ _ h => congrArg Prod.snd h)]
+
+lemma prod_boolFibers (S : Finset (Bool × β)) (f : Bool × β → ℝ) :
+    (∏ x ∈ S, f x) = (∏ x ∈ boolFiber S false, f (false,x)) *
+      ∏ x ∈ boolFiber S true, f (true,x) := by
+  rw [←prod_childJoin,childJoin_boolFiber]
+
+lemma inclusionMoment_childJoin {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
+    (F : Ω → Finset β) (G : Ω' → Finset β) (S : Finset (Bool × β)) :
+    inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) S =
+      inclusionMoment F (boolFiber S false) * inclusionMoment G (boolFiber S true) := by
+  classical
+  simp only [inclusionMoment_eq]
+  rw [←finiteMean_prod_mul]
+  apply finiteMean_congr
+  intro ω
+  simp only [subset_childJoin_iff]
+  split_ifs <;> simp_all
+
+lemma cylinder_childJoin {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] [Nonempty Ω] [Nonempty Ω']
+    (F : Ω → Finset β) (G : Ω' → Finset β)
+    (hF : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x}))
+    (hG : CylinderBound (inclusionMoment G) (fun x => inclusionMoment G {x})) :
+    CylinderBound (inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)))
+      (fun x => inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {x}) := by
+  intro S
+  rw [inclusionMoment_childJoin,prod_boolFibers]
+  have hf (b : Bool) (x : β) : boolFiber {(b,x)} b = {x} := by
+    ext y
+    simp only [mem_boolFiber,Finset.mem_singleton,Prod.mk.injEq,true_and]
+  have hn (b : Bool) (x : β) : boolFiber {(!b,x)} b = ∅ := by
+    ext y
+    cases b <;> simp [mem_boolFiber]
+  have h₀ (x : β) : inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {(false,x)} =
+      inclusionMoment F {x} * inclusionMoment G ∅ := by
+    rw [inclusionMoment_childJoin,hf]
+    have hh := hn true x
+    exact congrArg (fun T => inclusionMoment F {x} * inclusionMoment G T) hh
+  have h₁ (x : β) : inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {(true,x)} =
+      inclusionMoment F ∅ * inclusionMoment G {x} := by
+    rw [inclusionMoment_childJoin,hf]
+    have hh := hn false x
+    exact congrArg (fun T => inclusionMoment F T * inclusionMoment G {x}) hh
+  have emF : inclusionMoment F ∅ = 1 := by simp [inclusionMoment,finiteMean_const]
+  have emG : inclusionMoment G ∅ = 1 := by simp [inclusionMoment,finiteMean_const]
+  simp only [h₀,h₁,emF,emG,mul_one,one_mul]
+  exact mul_le_mul (hF _) (hG _) (inclusionMoment_nonneg _ _)
+    (Finset.prod_nonneg (fun x _ => inclusionMoment_nonneg F {x}))
+end JoinChildren
+
+lemma childMarks_join (d : ℕ) (A : Finset (Card (d+1))) :
+    A = (childMarks A false).image (Fin.cons false) ∪
+      (childMarks A true).image (Fin.cons true) := by
+  classical
+  ext x
+  constructor
+  · intro hx
+    cases hb : x 0
+    · exact Finset.mem_union_left _ (Finset.mem_image.mpr
+        ⟨Fin.tail x,(mem_childMarks _ _ _).mpr (by simpa only [←hb,Fin.cons_self_tail] using hx),
+          by simp only [←hb,Fin.cons_self_tail]⟩)
+    · exact Finset.mem_union_right _ (Finset.mem_image.mpr
+        ⟨Fin.tail x,(mem_childMarks _ _ _).mpr (by simpa only [←hb,Fin.cons_self_tail] using hx),
+          by simp only [←hb,Fin.cons_self_tail]⟩)
+  · intro hx
+    rcases Finset.mem_union.mp hx with hx | hx
+    · obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
+      exact (mem_childMarks _ _ _).mp hy
+    · obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
+      exact (mem_childMarks _ _ _).mp hy
+
+noncomputable def butterflyMarks (d : ℕ) (A : Finset (Card d))
+    (ω : SwitchIndex d → Bool) : Finset (Card d) :=
+  A.image (butterflyPerm d (decodeButterfly d ω))
+
+lemma butterflyMarks_succ (d : ℕ) (A : Finset (Card (d+1))) (ω : SwitchIndex (d+1) → Bool) :
+    butterflyMarks (d+1) A ω =
+      (fairSetLayer (fun p : (SwitchIndex d → Bool) × (SwitchIndex d → Bool) =>
+        childJoin (butterflyMarks d (childMarks A false) p.1)
+          (butterflyMarks d (childMarks A true) p.2)) (coinStepEquiv d ω)).image
+            (headTailEquiv d).symm := by
+  classical
+  unfold butterflyMarks fairSetLayer childJoin
+  rw [Finset.image_union,Finset.image_union]
+  simp only [Finset.image_image]
+  conv_lhs => rw [childMarks_join d A]
+  rw [Finset.image_union]
+  simp only [Finset.image_image]
+  apply congrArg₂ (· ∪ ·)
+  · apply Finset.image_congr
+    intro x _
+    simp only [Function.comp_def,butterflyPerm,decodeButterfly,Equiv.Perm.coe_mul,
+      childLift,pairSwitch,switchFun,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
+      coinStepEquiv,headTailEquiv,Equiv.symm_symm,pairLayer]
+    rfl
+  · apply Finset.image_congr
+    intro x _
+    simp only [Function.comp_def,butterflyPerm,decodeButterfly,Equiv.Perm.coe_mul,
+      childLift,pairSwitch,switchFun,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
+      coinStepEquiv,headTailEquiv,Equiv.symm_symm,pairLayer]
+    rfl
+
+lemma butterflyMarks_cylinder (d : ℕ) (A : Finset (Card d)) :
+    CylinderBound (inclusionMoment (butterflyMarks d A))
+      (fun x => inclusionMoment (butterflyMarks d A) {x}) := by
+  induction d with
+  | zero =>
+    have he : butterflyMarks 0 A = fun _ : SwitchIndex 0 → Bool => A := by
+      funext ω
+      simp [butterflyMarks,butterflyPerm]
+    rw [he]
+    exact cylinder_const A
+  | succ d ih =>
+    have h := cylinder_childJoin (butterflyMarks d (childMarks A false))
+      (butterflyMarks d (childMarks A true)) (ih _) (ih _)
+    have hh := cylinder_image_equiv (headTailEquiv d).symm _ (cylinder_fairSetLayer _ h)
+    have hs := cylinder_comp_equiv (coinStepEquiv d) _ hh
+    simpa only [←butterflyMarks_succ] using hs
+
+lemma indicator_mem_image_perm {α : Type*} [DecidableEq α]
+    (A : Finset α) (p : Equiv.Perm α) (y : α) :
+    (if y ∈ A.image p then (1 : ℝ) else 0) = ∑ x ∈ A, if p x = y then 1 else 0 := by
+  classical
+  by_cases h : y ∈ A.image p
+  · obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp h
+    rw [ite_eq_left (Finset.mem_image.mpr ⟨x,hx,hxy⟩)]
+    symm
+    calc
+      _ = (if p x = y then (1:ℝ) else 0) := Finset.sum_eq_single_of_mem x hx (by
+        intro z _ hzx
+        exact ite_eq_right (fun hz => hzx (p.injective (hz.trans hxy.symm))))
+      _ = 1 := ite_eq_left hxy
+  · rw [ite_eq_right h]
+    symm
+    apply Finset.sum_eq_zero
+    intro x hx
+    exact ite_eq_right (fun hxy => h (Finset.mem_image.mpr ⟨x,hx,hxy⟩))
+
+lemma butterflyMarks_singleton (d : ℕ) (A : Finset (Card d)) (y : Card d) :
+    inclusionMoment (butterflyMarks d A) {y} = (A.card : ℝ)/(2:ℝ)^d := by
+  classical
+  unfold inclusionMoment
+  simp only [Finset.singleton_subset_iff,butterflyMarks,indicator_mem_image_perm]
+  rw [finiteMean_sum]
+  simp only [butterfly_single_card_uniform,Finset.sum_const,nsmul_eq_mul]
+  ring
+
+lemma butterflyMarks_inclusion (d : ℕ) (A S : Finset (Card d)) :
+    inclusionMoment (butterflyMarks d A) S ≤ ((A.card : ℝ)/(2:ℝ)^d)^S.card := by
+  have h := butterflyMarks_cylinder d A S
+  simpa only [butterflyMarks_singleton,Finset.prod_const] using h
+
+namespace CycleColoring
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+abbrev Coloring (p : Equiv.Perm α) (S : Finset α) :=
+  {f : α → Bool // (∀ x, x ∉ S → f x = false) ∧
+    ∀ x ∈ S, p x ∈ S → f (p x) = f x}
+
+noncomputable instance (p : Equiv.Perm α) (S : Finset α) : Fintype (Coloring p S) :=
+  Fintype.ofFinite _
+
+noncomputable def boundary (p : Equiv.Perm α) (S : Finset α) : Finset α :=
+  S.filter (fun x => p.symm x ∉ S)
+
+noncomputable def root {p : Equiv.Perm α} {S : Finset α}
+    (C : RoutingNetwork.SelectedCycle p S) : α := C.nonempty.choose
+
+lemma root_mem {p : Equiv.Perm α} {S : Finset α} (C : RoutingNetwork.SelectedCycle p S) :
+    root C ∈ C.val := C.nonempty.choose_spec
+
+abbrev Seed (p : Equiv.Perm α) (S : Finset α) :=
+  {x // x ∈ boundary p S} ⊕ RoutingNetwork.SelectedCycle p S
+
+noncomputable def record {p : Equiv.Perm α} {S : Finset α}
+    (f : Coloring p S) : Seed p S → Bool :=
+  Sum.elim (fun x => f.val x.val) (fun C => f.val (root C))
+
+lemma orbit_subset_of_predecessor_closed (p : Equiv.Perm α) (D : Finset α)
+    (hD : ∀ x ∈ D, p.symm x ∈ D) {x : α} (hx : x ∈ D) : orbitSet p x ⊆ D := by
+  intro y hy
+  have hh := ((mem_orbitSet p x y).mp hy).inv
+  obtain ⟨n,hn⟩ := hh.exists_nat_pow_eq
+  have hp (k : ℕ) : (p⁻¹ ^ k) x ∈ D := by
+    induction k with
+    | zero => simpa using hx
+    | succ k ih => simpa only [pow_succ',Equiv.Perm.mul_apply,Equiv.Perm.coe_inv] using hD _ ih
+  exact hn ▸ hp n
+
+lemma record_injective (p : Equiv.Perm α) (S : Finset α) :
+    Function.Injective (record (p := p) (S := S)) := by
+  classical
+  intro f g hfg
+  apply Subtype.ext
+  funext x
+  by_contra hne
+  have hx : x ∈ S := by
+    by_contra h
+    exact hne ((f.property.1 x h).trans (g.property.1 x h).symm)
+  let D := S.filter (fun y => f.val y ≠ g.val y)
+  have hxD : x ∈ D := Finset.mem_filter.mpr ⟨hx,hne⟩
+  have hprev (y : α) (hy : y ∈ D) : p.symm y ∈ D := by
+    obtain ⟨hyS,hyne⟩ := Finset.mem_filter.mp hy
+    have hpS : p.symm y ∈ S := by
+      by_contra hn
+      have hseed : y ∈ boundary p S := Finset.mem_filter.mpr ⟨hyS,hn⟩
+      exact hyne (congrFun hfg (Sum.inl ⟨y,hseed⟩))
+    refine Finset.mem_filter.mpr ⟨hpS,?_⟩
+    have hf := f.property.2 (p.symm y) hpS (by simpa only [Equiv.apply_symm_apply] using hyS)
+    have hg := g.property.2 (p.symm y) hpS (by simpa only [Equiv.apply_symm_apply] using hyS)
+    simp only [Equiv.apply_symm_apply] at hf hg
+    intro he
+    exact hyne (hf.trans (he.trans hg.symm))
+  have horb := orbit_subset_of_predecessor_closed p D hprev hxD
+  let C : RoutingNetwork.SelectedCycle p S :=
+    ⟨orbitSet p x,⟨x,rfl⟩,fun y hy => (Finset.mem_filter.mp (horb hy)).1⟩
+  have hrD : root C ∈ D := horb (root_mem C)
+  have hrne := (Finset.mem_filter.mp hrD).2
+  exact hrne (congrFun hfg (Sum.inr C))
+
+lemma card_coloring_le (p : Equiv.Perm α) (S : Finset α) :
+    Fintype.card (Coloring p S) ≤
+      2 ^ ((boundary p S).card + Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
+  have h := Fintype.card_le_of_injective (record (p := p) (S := S)) (record_injective p S)
+  simpa only [Fintype.card_fun,Fintype.card_sum,Fintype.card_coe,Fintype.card_bool] using h
+
+omit [Fintype α] in
+lemma boundary_card (p : Equiv.Perm α) (S : Finset α) :
+    (boundary p S).card = S.card - (S.filter (fun x => p.symm x ∈ S)).card := by
+  classical
+  have h := Finset.card_filter_add_card_filter_not (s := S) (fun x => p.symm x ∈ S)
+  unfold boundary
+  omega
+
+lemma card_coloring_le_edges (p : Equiv.Perm α) (S : Finset α) :
+    Fintype.card (Coloring p S) ≤
+      2 ^ (S.card - (S.filter (fun x => p.symm x ∈ S)).card +
+        Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
+  simpa only [boundary_card] using card_coloring_le p S
+
+
+abbrev OppositeColoring (p : Equiv.Perm α) (S : Finset α) :=
+  {f : α → Bool // (∀ x, x ∉ S → f x = false) ∧
+    ∀ x ∈ S, p x ∈ S → f (p x) = !(f x)}
+
+noncomputable instance (p : Equiv.Perm α) (S : Finset α) : Fintype (OppositeColoring p S) :=
+  Fintype.ofFinite _
+
+noncomputable def bipartitionFlip (p : Equiv.Perm α) (S : Finset α)
+    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x))
+    (f : OppositeColoring p S) : Coloring p S := by
+  classical
+  refine ⟨(fun x => if x ∈ S then Bool.xor (f.val x) (χ x) else false),?_,?_⟩
+  · intro x hx
+    exact ite_eq_right hx
+  · intro x hx hpx
+    simp only [ite_eq_left hx,ite_eq_left hpx,f.property.2 x hx hpx,hχ x]
+    cases f.val x <;> cases χ x <;> rfl
+
+end CycleColoring
+end Thorp
+
+end ThorpNine.Dense
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Dense/PriorPaths.lean b/lean/OAI/Probability/ThorpRouting/Dense/PriorPaths.lean
index 169541c18d1f6a74692834c6ac9bec7e814a87ee..f9e6c587760ace0bf47c17ef737dc4d710b5e316 100644
--- a/lean/OAI/Probability/ThorpRouting/Dense/PriorPaths.lean
+++ b/lean/OAI/Probability/ThorpRouting/Dense/PriorPaths.lean
@@ -1,876 +1,876 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Dense.AssignmentSlots
-
-namespace OAI
-
-namespace ThorpNine.Dense
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-namespace RoutingNetwork
-variable {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι] [DecidableEq ι]
-  {d : ℕ} (R : RoutingNetwork α ι d)
-lemma mem_exposedPaths (b : ι → Bool) (V : Finset (α)) (s : ι) :
-    s ∈ exposedPaths R b V ↔ (R.users b s).1 ∈ V ∨ (R.users b s).2 ∈ V := by
-  classical
-  simp only [exposedPaths,Finset.mem_biUnion,R.mem_path]
-  constructor
-  · rintro ⟨x,hx,h | h⟩
-    · exact Or.inl (h ▸ hx)
-    · exact Or.inr (h ▸ hx)
-  · rintro (h | h)
-    · exact ⟨_,h,Or.inl rfl⟩
-    · exact ⟨_,h,Or.inr rfl⟩
-
-lemma terminalCharge_sum (b : ι → Bool) (z : α) (V : Finset (α))
-    (hz : z ∉ V) :
-    (terminalCharge R b z V : ℝ) = ∑ s : ι,
-      ((if (R.users b s).1 = z ∧ (R.users b s).2 ∈ V then (1:ℝ) else 0) +
-      (if (R.users b s).2 = z ∧ (R.users b s).1 ∈ V then (1:ℝ) else 0)) := by
-  classical
-  have he : R.path b z ∩ exposedPaths R b V =
-      Finset.univ.filter (fun s =>
-        ((R.users b s).1 = z ∧ (R.users b s).2 ∈ V) ∨
-        ((R.users b s).2 = z ∧ (R.users b s).1 ∈ V)) := by
-    ext s
-    simp only [Finset.mem_inter,R.mem_path,mem_exposedPaths,
-      Finset.mem_filter,Finset.mem_univ,true_and]
-    by_cases h₁ : (R.users b s).1 = z <;>
-      by_cases h₂ : (R.users b s).2 = z <;> simp [h₁,h₂,hz]
-  rw [terminalCharge,he]
-  simp only [← Finset.sum_boole]
-  apply Finset.sum_congr rfl
-  intro s _
-  by_cases h₁ : (R.users b s).1 = z <;>
-    by_cases h₂ : (R.users b s).2 = z <;> simp [h₁,h₂,hz]
-
-noncomputable def priorPaths {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) : Finset (α) :=
-  Finset.univ.filter (fun x => ∃ i : Fin A, o x = some i ∧ i.val < t)
-
-omit [DecidableEq α] in
-@[simp] lemma mem_priorPaths {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) (x : α) :
-    x ∈ priorPaths o t ↔ ∃ i : Fin A, o x = some i ∧ i.val < t := by
-  classical
-  simp [priorPaths]
-
-lemma terminalCharge_prior_sum {A : ℕ} (o : α → Option (Fin A))
-    (b : ι → Bool) (z : Terminal o) (i : Fin A) :
-    (terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) =
-      ∑ s : ι,
-        ((if (z i).val = (R.users b s).1 ∧ (R.users b s).2 ∈ priorPaths o (i.val+1)
-          then (1:ℝ) else 0) +
-        (if (z i).val = (R.users b s).2 ∧ (R.users b s).1 ∈ priorPaths o (i.val+1)
-          then (1:ℝ) else 0)) := by
-  classical
-  rw [terminalCharge_sum _ _ _ _ (Finset.notMem_erase _ _)]
-  apply Finset.sum_congr rfl
-  intro s _
-  have hn := R.users_ne b s
-  simp only [Finset.mem_erase]
-  by_cases h₁ : (z i).val = (R.users b s).1 <;>
-    by_cases h₂ : (z i).val = (R.users b s).2 <;> simp_all [eq_comm]
-
-lemma rotationMeeting_eq_slot_sum {A : ℕ} (o : α → Option (Fin A))
-    (z : Terminal o) (u v : α) :
-    rotationMeeting o z u v = ∑ i : Fin A,
-      ((if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) +
-      (if (z i).val = v ∧ u ∈ priorPaths o (i.val+1) then (1:ℝ) else 0)) := by
-  classical
-  have only_owner (u : α) (a : Fin A) (hou : o u = some a)
-      (i : Fin A) (hia : i ≠ a) : (z i).val ≠ u := by
-    intro h
-    have hi := (z i).property
-    rw [h,hou] at hi
-    exact hia (Option.some.inj hi).symm
-  have sum_one (u v : α) (a : Fin A) (hou : o u = some a) :
-      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) =
-      if (z a).val = u ∧ v ∈ priorPaths o (a.val+1) then 1 else 0 := by
-    apply Finset.sum_eq_single a
-    · intro i _ hia
-      simp [only_owner u a hou i hia]
-    · simp
-  have sum_none (u v : α) (hou : o u = none) :
-      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) = 0 := by
-    apply Finset.sum_eq_zero
-    intro i _
-    have hn : (z i).val ≠ u := by
-      intro h
-      have hi := (z i).property
-      rw [h,hou] at hi
-      contradiction
-    simp [hn]
-  rw [Finset.sum_add_distrib]
-  cases hou : o u with
-  | none =>
-    rw [sum_none u v hou]
-    simp [rotationMeeting,hou,mem_priorPaths]
-  | some a =>
-    rw [sum_one u v a hou]
-    cases hov : o v with
-    | none =>
-      rw [sum_none v u hov]
-      simp [rotationMeeting,hou,hov,mem_priorPaths]
-    | some b =>
-      rw [sum_one v u b hov]
-      simp only [rotationMeeting,hou,hov,mem_priorPaths,Option.some.injEq]
-      rcases lt_trichotomy a b with hab | rfl | hba
-      · have hn : ¬ b.val < a.val+1 := by omega
-        have hy : a.val < b.val+1 := by omega
-        simp [hab,hn,hy]
-      · simp
-      · have hn : ¬ a.val < b.val+1 := by omega
-        have hy : b.val < a.val+1 := by omega
-        simp [hba,not_lt_of_ge (le_of_lt hba),hn,hy]
-
-lemma rotationCharge_eq_terminal_sum {A : ℕ} (o : α → Option (Fin A))
-    (b : ι → Bool) (z : Terminal o) :
-    rotationCharge R o b z = ∑ i : Fin A,
-      (terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) := by
-  simp only [rotationCharge,rotationMeeting_eq_slot_sum,terminalCharge_prior_sum]
-  exact Finset.sum_comm
-
-omit [DecidableEq α] in
-@[simp] lemma priorPaths_zero {A : ℕ} (o : α → Option (Fin A)) : priorPaths o 0 = ∅ := by
-  classical
-  ext x
-  simp
-
-lemma priorPaths_succ {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) (ht : t < A) :
-    priorPaths o (t+1) = priorPaths o t ∪ cycleFiber o ⟨t,ht⟩ := by
-  classical
-  ext x
-  simp only [mem_priorPaths,Finset.mem_union,mem_cycleFiber]
-  constructor
-  · rintro ⟨i,hi,hit⟩
-    by_cases h : i.val < t
-    · exact Or.inl ⟨i,hi,h⟩
-    · have he : i = ⟨t,ht⟩ := Fin.ext (by change i.val = t; omega)
-      exact Or.inr (he ▸ hi)
-  · rintro (⟨i,hi,hit⟩ | hi)
-    · exact ⟨i,hi,by omega⟩
-    · exact ⟨⟨t,ht⟩,hi,by change t < t+1; omega⟩
-
-omit [DecidableEq α] in
-lemma priorPaths_disjoint {A : ℕ} (o : α → Option (Fin A)) (i : Fin A) :
-    Disjoint (priorPaths o i.val) (cycleFiber o i) := by
-  classical
-  apply Finset.disjoint_left.mpr
-  intro x hx hx'
-  obtain ⟨j,hj,hji⟩ := (mem_priorPaths _ _ _).mp hx
-  have he : j = i := Option.some.inj (hj.symm.trans ((mem_cycleFiber _ _ _).mp hx'))
-  subst j
-  exact (lt_irrefl _ hji)
-
-namespace QueryTree
-open Thorp.QueryTree
-
-lemma seekCycles_owned {A : ℕ} (S : Finset (α)) (r : Fin A → ℕ)
-    (o : α → Option (Fin A)) (z : Terminal o) (ω : ι → Bool)
-    (hS : ∀ i, cycleFiber o i ⊆ S)
-    (hO : ∀ i, orbitSet (R.perm ω) (z i).val = cycleFiber o i)
-    (hC : ∀ i, (cycleFiber o i).card = r i+1) :
-    let p := R.perm ω
-    let L := List.ofFn (fun i : Fin A => (p (z i).val,r i))
-    (seekCycles R S L ∅).succeeds ω = true ∧
-      ((seekCycles R S L ∅).staleCharge ∅ ω : ℝ) = rotationCharge R o (ω) z := by
-  classical
-  let b := ω
-  let p := R.perm b
-  let f (i : Fin A) := (p (z i).val,r i)
-  let c (i : Fin A) : ℝ := terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val)
-  have hc (i : Fin A) : Function.minimalPeriod p (z i).val - 1 = r i := by
-    have he := orbitSet_card p (z i).val
-    rw [hO,hC] at he
-    omega
-  have hs (t : ℕ) (ht : t ≤ A) :
-      (seekCycles R S ((List.ofFn f).drop t) (priorPaths o t)).succeeds ω = true ∧
-        ((seekCycles R S ((List.ofFn f).drop t) (priorPaths o t)).staleCharge (exposedPaths R b (priorPaths o t)) ω : ℝ) =
-          ((List.ofFn c).drop t).sum := by
-    induction ht using Nat.decreasingInduction with
-    | self => simp only [List.length_ofFn,le_refl,List.drop_eq_nil_of_le,List.sum_nil,seekCycles,Thorp.QueryTree.succeeds,staleCharge,Nat.cast_zero,and_self]
-    | of_succ t ht ih =>
-      have htf : t < (List.ofFn f).length := by simpa using ht
-      have htc : t < (List.ofFn c).length := by simpa using ht
-      rw [List.drop_eq_getElem_cons htf,List.drop_eq_getElem_cons htc]
-      simp only [List.getElem_ofFn,List.sum_cons,seekCycles]
-      have h := seekCycle_orbit R S (priorPaths o t) (z ⟨t,ht⟩).val
-        (fun V => seekCycles R S ((List.ofFn f).drop (t+1)) V) ω
-        (by rw [hO]; exact hS _) (by rw [hO]; exact priorPaths_disjoint o ⟨t,ht⟩)
-      dsimp only at h
-      rw [hc,hO,←priorPaths_succ o t ht] at h
-      dsimp only [f]
-      constructor
-      · exact h.1.trans ih.1
-      · rw [h.2,Nat.cast_add,ih.2]
-        exact add_comm _ _
-  have h := hs 0 (Nat.zero_le _)
-  simp only [List.drop_zero,priorPaths_zero,exposedPaths_empty,List.sum_ofFn] at h
-  dsimp only
-  exact ⟨h.1,h.2.trans (rotationCharge_eq_terminal_sum R o b z).symm⟩
-
-end QueryTree
-section Collections
-variable {A : ℕ}
-
-def ValidCycleSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ)
-    (o : α → Option (Fin A)) : Prop :=
-  (∀ i, (cycleFiber o i).card = r i + 1) ∧
-  (∀ i, cycleFiber o i ⊆ S) ∧
-  (∀ i x, o x = some i → orbitSet p x = cycleFiber o i)
-
-abbrev CycleSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
-  {o : α → Option (Fin A) // ValidCycleSlots p S r o}
-
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
-    Fintype (CycleSlots p S r) := Fintype.ofFinite _
-
-lemma CycleSlots.terminal_nonempty {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (c : CycleSlots p S r) (i : Fin A) : Nonempty {x : α // c.val x = some i} := by
-  have hn : (cycleFiber c.val i).Nonempty := Finset.card_pos.mp (by rw [c.property.1]; omega)
-  obtain ⟨x,hx⟩ := hn
-  exact ⟨x,(mem_cycleFiber _ _ _).mp hx⟩
-
-omit [DecidableEq α] in
-lemma owner_ext {o o' : α → Option (Fin A)}
-    (h : ∀ i, cycleFiber o i = cycleFiber o' i) : o = o' := by
-  funext x
-  have he (i : Fin A) : o x = some i ↔ o' x = some i := by
-    simpa only [mem_cycleFiber] using (Finset.ext_iff.mp (h i) x)
-  cases ho : o x with
-  | some i => exact ((he i).mp ho).symm
-  | none =>
-    cases ho' : o' x with
-    | none => rfl
-    | some i => have hn := (he i).mpr ho'; rw [ho] at hn; contradiction
-
-abbrev RootedSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
-  Σ c : CycleSlots p S r, Terminal c.val
-
-noncomputable def rootedStart {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (c : RootedSlots p S r) : Fin A → S := fun i =>
-  ⟨p (c.2 i).val,c.1.property.2.1 i (by
-    rw [←c.1.property.2.2 i (c.2 i).val (c.2 i).property]
-    exact (mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.apply_right (Equiv.Perm.SameCycle.refl p _)))⟩
-
-lemma rootedStart_injective {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ} :
-    Function.Injective (rootedStart (p:=p) (S:=S) (r:=r)) := by
-  rintro ⟨c,z⟩ ⟨c',z'⟩ h
-  have hz (i : Fin A) : (z i).val = (z' i).val :=
-    p.injective (congrArg Subtype.val (congrFun h i))
-  have hc : c = c' := Subtype.ext (owner_ext (fun i => by
-    rw [←c.property.2.2 i (z i).val (z i).property,
-      ←c'.property.2.2 i (z' i).val (z' i).property,hz]))
-  subst c'
-  have hzz : z = z' := funext (fun i => Subtype.ext (hz i))
-  subst z'
-  rfl
-
-lemma rooted_weight_sum_le (S : Finset (α)) (r : Fin A → ℕ) (ω : ι → Bool) :
-    (∑ c : RootedSlots (R.perm ω) S r,
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2)) ≤
-    ∑ a : Fin A → S,
-      if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-        ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-      else 0 := by
-  classical
-  let p := R.perm ω
-  let g (a : Fin A → S) : ℝ :=
-    if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-    else 0
-  have he (c : RootedSlots p S r) :
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2) =
-      g (rootedStart c) := by
-    have hs := QueryTree.seekCycles_owned R S r c.1.val c.2 ω c.1.property.2.1
-      (fun i => c.1.property.2.2 i (c.2 i).val (c.2 i).property) c.1.property.1
-    dsimp only at hs
-    unfold g
-    rw [ite_eq_left (show (QueryTree.seekCycles R S (List.ofFn (fun i => ((rootedStart c i).val,r i))) ∅).succeeds ω = true from hs.1)]
-    rw [← hs.2,neg_mul,Real.exp_neg,mul_comm (Real.log 2),Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
-    rfl
-  simp_rw [he]
-  exact sum_comp_injective_le rootedStart rootedStart_injective g (fun a => by dsimp [g]; split_ifs <;> positivity)
-
-lemma collection_rotation_lower (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r)
-    (ω : ι → Bool) (c : CycleSlots (R.perm ω) S r) :
-    collectionWeight d r ≤ ∑ z : Terminal c.val,
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z) := by
-  classical
-  let (i : Fin A) : Nonempty {x : α // c.val x = some i} := c.terminal_nonempty i
-  have hm : Monotone (fun i => (cycleFiber c.val i).card) := by
-    intro i j hij
-    dsimp only
-    rw [c.property.1,c.property.1]
-    exact Nat.add_le_add_right (hr hij) 1
-  have hb := rotation_exp_bound R hm (ω)
-  have hcard : (Fintype.card (Terminal c.val) : ℝ) = ∏ i : Fin A, (r i+1 : ℝ) := by
-    rw [Fintype.card_pi,Nat.cast_prod]
-    apply Finset.prod_congr rfl
-    intro i _
-    rw [card_cycleFiber,c.property.1,Nat.cast_add,Nat.cast_one]
-  have hp : (0:ℝ) < Fintype.card (Terminal c.val) := Nat.cast_pos.mpr Fintype.card_pos
-  have hl : rotationLower d r ≤ finiteMean (fun z : Terminal c.val => Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z)) := by
-    simpa only [rotationLower,c.property.1,Nat.cast_add,Nat.cast_one] using hb
-  have hh := (le_div_iff₀ hp).mp hl
-  rw [hcard] at hh
-  unfold collectionWeight
-  rw [← Finset.mul_sum]
-  calc
-    _ = ((2:ℝ)^d)^A * (rotationLower d r * ∏ i : Fin A, (r i+1 : ℝ)) := by ring
-    _ ≤ _ := mul_le_mul_of_nonneg_left hh (by positivity)
-
-lemma cycleSlots_weighted_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : ι → Bool =>
-      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) *
-      collectionWeight d r ≤ (S.card : ℝ)^A := by
-  classical
-  let g (a : Fin A → S) (ω : ι → Bool) : ℝ :=
-    if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-    else 0
-  have hω (ω : ι → Bool) :
-      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ) * collectionWeight d r ≤
-        ∑ a : Fin A → S, g a ω := by
-    calc
-      _ = ∑ _c : CycleSlots (R.perm ω) S r, collectionWeight d r := by simp
-      _ ≤ ∑ c : CycleSlots (R.perm ω) S r,
-          ∑ z : Terminal c.val, ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z) :=
-        Finset.sum_le_sum (fun c _ => collection_rotation_lower R S r hr ω c)
-      _ = ∑ c : RootedSlots (R.perm ω) S r,
-          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2) :=
-        (Fintype.sum_sigma (fun c : RootedSlots (R.perm ω) S r =>
-          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2))).symm
-      _ ≤ _ := rooted_weight_sum_le R S r ω
-  have hb := finiteMean_mono hω
-  rw [finiteMean_mul_const,finiteMean_sum] at hb
-  calc
-    _ ≤ ∑ a : Fin A → S, finiteMean (g a) := hb
-    _ ≤ ∑ _a : Fin A → S, (1:ℝ) := by
-      apply Finset.sum_le_sum
-      intro a _
-      have hh := QueryTree.cycles_weighted_success R S (List.ofFn (fun i : Fin A => ((a i).val,r i)))
-      simpa only [List.length_ofFn] using hh
-    _ = _ := by simp
-
-lemma cycleSlots_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : ι → Bool =>
-      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
-        (S.card : ℝ)^A / collectionWeight d r := by
-  exact (le_div_iff₀ (collectionWeight_pos d r)).mpr (cycleSlots_weighted_moment R S r hr)
-
-lemma butterfly_cycleSlots_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : ι → Bool =>
-      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
-        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
-  have h := cycleSlots_moment R S r hr
-  rw [collectionWeight_eq_sqrt,div_mul_eq_div_mul_one_div] at h
-  rw [div_pow]
-  simpa only [one_div,Finset.prod_inv_distrib] using h
-
-end Collections
-
-section CycleCounting
-variable {A : ℕ}
-
-lemma SelectedCycle.nonempty {p : Equiv.Perm (α)} {S : Finset (α)}
-    (C : SelectedCycle p S) : C.val.Nonempty := by
-  obtain ⟨x,hx⟩ := C.property.1
-  rw [hx]
-  exact orbitSet_nonempty p x
-
-lemma SelectedCycle.orbit_eq {p : Equiv.Perm (α)} {S : Finset (α)}
-    (C : SelectedCycle p S) {x : α} (hx : x ∈ C.val) : orbitSet p x = C.val := by
-  obtain ⟨y,hy⟩ := C.property.1
-  rw [hy] at hx ⊢
-  exact (orbitSet_eq p ((mem_orbitSet _ _ _).mp hx)).symm
-
-lemma SelectedCycle.eq_of_mem {p : Equiv.Perm (α)} {S : Finset (α)}
-    (C D : SelectedCycle p S) {x : α} (hC : x ∈ C.val) (hD : x ∈ D.val) : C = D := by
-  apply Subtype.ext
-  exact (C.orbit_eq hC).symm.trans (D.orbit_eq hD)
-
-abbrev SlotAssignment (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
-  {f : Fin A → SelectedCycle p S // Function.Injective f ∧ ∀ i, (f i).val.card = r i+1}
-
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
-    Fintype (SlotAssignment p S r) := Fintype.ofFinite _
-
-noncomputable def slotsToAssignment {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (c : CycleSlots p S r) : SlotAssignment p S r := by
-  classical
-  let f (i : Fin A) : SelectedCycle p S := ⟨cycleFiber c.val i,by
-    obtain ⟨x,hx⟩ := c.terminal_nonempty i
-    exact ⟨⟨x,(c.property.2.2 i x hx).symm⟩,c.property.2.1 i⟩⟩
-  refine ⟨f,?_,fun i => c.property.1 i⟩
-  intro i j hij
-  have he : cycleFiber c.val i = cycleFiber c.val j := congrArg Subtype.val hij
-  obtain ⟨x,hx⟩ := c.terminal_nonempty i
-  have hxi : x ∈ cycleFiber c.val i := (mem_cycleFiber _ _ _).mpr hx
-  rw [he] at hxi
-  exact Option.some.inj (hx.symm.trans ((mem_cycleFiber _ _ _).mp hxi))
-
-noncomputable def assignmentOwner {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) (x : α) : Option (Fin A) := by
-  classical
-  exact if h : ∃ i, x ∈ (f.val i).val then some (Classical.choose h) else none
-
-lemma assignmentOwner_eq_some {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) (x : α) (i : Fin A) :
-    assignmentOwner f x = some i ↔ x ∈ (f.val i).val := by
-  classical
-  unfold assignmentOwner
-  split_ifs with h
-  · constructor
-    · intro hi
-      have hx := Classical.choose_spec h
-      rw [Option.some.inj hi] at hx
-      exact hx
-    · intro hx
-      have he := (f.val (Classical.choose h)).eq_of_mem (f.val i) (Classical.choose_spec h) hx
-      exact congrArg some (f.property.1 he)
-  · constructor
-    · intro hi; contradiction
-    · intro hx; exact (h ⟨i,hx⟩).elim
-
-lemma assignmentOwner_fiber {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) (i : Fin A) : cycleFiber (assignmentOwner f) i = (f.val i).val := by
-  ext x
-  simp only [mem_cycleFiber,assignmentOwner_eq_some]
-
-noncomputable def assignmentToSlots {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) : CycleSlots p S r :=
-  ⟨assignmentOwner f,by
-    refine ⟨?_,?_,?_⟩
-    · intro i
-      rw [assignmentOwner_fiber]
-      exact f.property.2 i
-    · intro i
-      rw [assignmentOwner_fiber]
-      exact (f.val i).property.2
-    · intro i x hx
-      rw [assignmentOwner_fiber]
-      exact (f.val i).orbit_eq ((assignmentOwner_eq_some f x i).mp hx)⟩
-
-noncomputable def cycleSlotsEquivAssignments (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
-    CycleSlots p S r ≃ SlotAssignment p S r where
-  toFun := slotsToAssignment
-  invFun := assignmentToSlots
-  left_inv c := by
-    apply Subtype.ext
-    apply owner_ext
-    intro i
-    rw [assignmentToSlots,assignmentOwner_fiber]
-    rfl
-  right_inv f := by
-    apply Subtype.ext
-    funext i
-    apply Subtype.ext
-    exact assignmentOwner_fiber f i
-
-end CycleCounting
-
-section GroupedCounting
-variable {A : ℕ}
-
-abbrev LengthClass (r : Fin A → ℕ) := {j : ℕ // j ∈ Finset.univ.image (fun i => r i + 1)}
-
-def slotClass (r : Fin A → ℕ) (i : Fin A) : LengthClass r :=
-  ⟨r i+1, Finset.mem_image.mpr ⟨i, Finset.mem_univ _, rfl⟩⟩
-
-abbrev SlotsOfLength (r : Fin A → ℕ) (j : LengthClass r) := {i : Fin A // slotClass r i = j}
-abbrev CyclesOfLength (p : Equiv.Perm (α)) (S : Finset (α)) (j : ℕ) :=
-  {C : SelectedCycle p S // C.val.card = j}
-
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (j : ℕ) :
-    Fintype (CyclesOfLength p S j) := Fintype.ofFinite _
-
-abbrev GroupedAssignments (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
-  ∀ j : LengthClass r, SlotsOfLength r j ↪ CyclesOfLength p S j.val
-
-noncomputable def assignmentToGroups {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) : GroupedAssignments p S r := fun _j =>
-  { toFun := fun i => ⟨f.val i.val, (f.property.2 i.val).trans (congrArg Subtype.val i.property)⟩
-    inj' := fun _i _k h => Subtype.ext (f.property.1 (congrArg Subtype.val h)) }
-
-noncomputable def groupsEmbedding {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (F : GroupedAssignments p S r) : Fin A ↪ SelectedCycle p S :=
-  ((Equiv.sigmaFiberEquiv (slotClass r)).symm.toEmbedding.trans
-    (Function.Embedding.sigmaMap (Function.Embedding.refl _) F)).trans
-    ((Function.Embedding.sigmaMap (Function.Embedding.subtype _) (fun _ => Function.Embedding.refl _)).trans
-      (Equiv.sigmaFiberEquiv (fun C : SelectedCycle p S => C.val.card)).toEmbedding)
-
-noncomputable def groupsToAssignment {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (F : GroupedAssignments p S r) : SlotAssignment p S r :=
-  ⟨groupsEmbedding F, (groupsEmbedding F).injective,
-    fun i => (F (slotClass r i) ⟨i,rfl⟩).property⟩
-
-noncomputable def assignmentEquivGroups (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
-    SlotAssignment p S r ≃ GroupedAssignments p S r where
-  toFun := assignmentToGroups
-  invFun := groupsToAssignment
-  left_inv f := by
-    apply Subtype.ext
-    rfl
-  right_inv F := by
-    funext j
-    apply Function.Embedding.ext
-    intro i
-    rcases i with ⟨i,hi⟩
-    subst j
-    rfl
-
-lemma card_cycleSlots_factorial (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
-    Fintype.card (CycleSlots p S r) =
-      ∏ j : LengthClass r, (Fintype.card (CyclesOfLength p S j.val)).descFactorial
-        (Fintype.card (SlotsOfLength r j)) := by
-  classical
-  rw [Fintype.card_congr (cycleSlotsEquivAssignments p S r),
-    Fintype.card_congr (assignmentEquivGroups p S r), Fintype.card_pi]
-  simp only [Fintype.card_embedding_eq]
-
-lemma butterfly_factorial_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : ι → Bool =>
-       (∏ j : LengthClass r,
-         (Fintype.card (CyclesOfLength (R.perm ω) S j.val)).descFactorial
-           (Fintype.card (SlotsOfLength r j)) : ℕ)) ≤
-      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-        ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
-  simpa only [←card_cycleSlots_factorial] using butterfly_cycleSlots_moment R S r hr
-
-end GroupedCounting
-
-section AllLengths
-variable {A : ℕ}
-
-noncomputable def assignmentRelabel (p : Equiv.Perm (α)) (S : Finset (α))
-    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
-    SlotAssignment p S r ≃ SlotAssignment p S (r ∘ e) where
-  toFun f := ⟨f.val ∘ e, f.property.1.comp e.injective, fun i => f.property.2 (e i)⟩
-  invFun f := ⟨f.val ∘ e.symm, f.property.1.comp e.symm.injective, fun i => by
-    simpa only [Function.comp_apply,Equiv.apply_symm_apply] using f.property.2 (e.symm i)⟩
-  left_inv f := by apply Subtype.ext; funext i; simp
-  right_inv f := by apply Subtype.ext; funext i; simp
-
-lemma card_cycleSlots_relabel (p : Equiv.Perm (α)) (S : Finset (α))
-    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
-    Fintype.card (CycleSlots p S r) = Fintype.card (CycleSlots p S (r ∘ e)) := by
-  exact Fintype.card_congr ((cycleSlotsEquivAssignments p S r).trans
-    ((assignmentRelabel p S r e).trans (cycleSlotsEquivAssignments p S (r ∘ e)).symm))
-
-lemma butterfly_cycleSlots_moment_all (S : Finset (α)) (r : Fin A → ℕ) :
-    finiteMean (fun ω : ι → Bool =>
-      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
-        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
-  have h := butterfly_cycleSlots_moment R S (r ∘ Tuple.sort r) (Tuple.monotone_sort r)
-  simp_rw [←card_cycleSlots_relabel] at h
-  have hp := Equiv.prod_comp (Tuple.sort r) (fun i : Fin A => (1 / Real.sqrt (r i+1) : ℝ))
-  simpa only [Function.comp_apply,hp] using h
-
-abbrev ShortCycle (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :=
-  {C : SelectedCycle p S // C.val.card ≤ J}
-
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    Fintype (ShortCycle p S J) := Fintype.ofFinite _
-
-noncomputable def shortCycleLength {p : Equiv.Perm (α)} {S : Finset (α)} {J : ℕ}
-    (C : ShortCycle p S J) : Fin J :=
-  ⟨C.val.val.card-1, by have := C.val.nonempty.card_pos; have := C.property; omega⟩
-
-lemma shortCycleLength_add_one {p : Equiv.Perm (α)} {S : Finset (α)} {J : ℕ}
-    (C : ShortCycle p S J) : (shortCycleLength C : ℕ)+1 = C.val.val.card := by
-  have := C.val.nonempty.card_pos
-  simp only [shortCycleLength]
-  omega
-
-noncomputable def shortEmbeddingEquiv (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    (Fin A ↪ ShortCycle p S J) ≃
-      Σ r : Fin A → Fin J, SlotAssignment p S (fun i => (r i : ℕ)) where
-  toFun f := ⟨fun i => shortCycleLength (f i),
-    ⟨fun i => (f i).val, fun i j h => f.injective (Subtype.ext h),
-      fun i => (shortCycleLength_add_one (f i)).symm⟩⟩
-  invFun c :=
-    { toFun := fun i => ⟨c.2.val i, by rw [c.2.property.2]; exact (c.1 i).isLt⟩
-      inj' := fun _i _j h => c.2.property.1 (congrArg Subtype.val h) }
-  left_inv f := by
-    apply Function.Embedding.ext
-    intro i
-    rfl
-  right_inv c := by
-    rcases c with ⟨r,f⟩
-    have he : (fun i => shortCycleLength (⟨f.val i, by rw [f.property.2]; exact (r i).isLt⟩ : ShortCycle p S J)) = r := by
-      funext i
-      apply Fin.ext
-      simp only [shortCycleLength,f.property.2,Nat.add_sub_cancel]
-    apply Sigma.ext he
-    apply (Subtype.heq_iff_coe_eq (by
-      intro x
-      change (Function.Injective x ∧ ∀ i, (x i).val.card = (f.val i).val.card - 1 + 1) ↔ _
-      simp only [f.property.2,Nat.add_sub_cancel])).mpr
-    rfl
-
-lemma shortCycle_factorial_eq (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    (Fintype.card (ShortCycle p S J)).descFactorial A =
-      ∑ r : Fin A → Fin J, Fintype.card (CycleSlots p S (fun i => (r i : ℕ))) := by
-  classical
-  have hc := Fintype.card_embedding_eq (α := Fin A) (β := ShortCycle p S J)
-  simp only [Fintype.card_fin] at hc
-  rw [←hc, Fintype.card_congr (shortEmbeddingEquiv p S J),Fintype.card_sigma]
-  apply Finset.sum_congr rfl
-  intro r _
-  exact (Fintype.card_congr (cycleSlotsEquivAssignments p S (fun i => (r i : ℕ)))).symm
-
-end AllLengths
-
-section CycleMGF
-
-lemma butterfly_shortCycle_moment (S : Finset (α)) (J A : ℕ) :
-    finiteMean (fun ω : ι → Bool =>
-      ((Fintype.card (ShortCycle (R.perm ω) S J)).descFactorial A : ℝ)) ≤
-      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))^A := by
-  classical
-  simp_rw [shortCycle_factorial_eq,Nat.cast_sum,finiteMean_sum]
-  calc
-    _ ≤ ∑ r : Fin A → Fin J,
-        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-          ∏ i : Fin A, (1 / Real.sqrt ((r i : ℕ)+1)) := by
-      apply Finset.sum_le_sum
-      intro r _
-      exact butterfly_cycleSlots_moment_all R S (fun i => (r i : ℕ))
-    _ = _ := by
-      rw [←Finset.mul_sum,mul_pow]
-      congr 1
-      rw [←Fintype.prod_sum (fun (_ : Fin A) (j : Fin J) => (1 / Real.sqrt ((j:ℕ)+1) : ℝ))]
-      simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fin]
-
-noncomputable def cycleMass (p : Equiv.Perm (α)) (S : Finset (α)) : ℕ :=
-  ∑ C : SelectedCycle p S, C.val.card
-
-lemma cycleMass_le (p : Equiv.Perm (α)) (S : Finset (α)) : cycleMass p S ≤ S.card := by
-  classical
-  let f : (Σ C : SelectedCycle p S, {x : α // x ∈ C.val}) → S :=
-    fun c => ⟨c.2.val,c.1.property.2 c.2.property⟩
-  have hi : Function.Injective f := by
-    intro c e h
-    have hx : c.2.val = e.2.val := congrArg Subtype.val h
-    have hC : c.1 = e.1 := c.1.eq_of_mem e.1 c.2.property (hx ▸ e.2.property)
-    rcases c with ⟨c,x⟩
-    rcases e with ⟨e,y⟩
-    dsimp only at hC
-    subst e
-    have he : x=y := Subtype.ext hx
-    cases he
-    rfl
-  have hh := Fintype.card_le_of_injective f hi
-  simpa only [Fintype.card_sigma,Fintype.card_coe,cycleMass] using hh
-
-lemma shortCycle_card_le (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    Fintype.card (ShortCycle p S J) ≤ S.card := by
-  classical
-  calc
-    _ ≤ Fintype.card (SelectedCycle p S) := Fintype.card_subtype_le _
-    _ = ∑ _ : SelectedCycle p S, 1 := by simp
-    _ ≤ cycleMass p S := Finset.sum_le_sum (fun C _ => C.nonempty.card_pos)
-    _ ≤ _ := cycleMass_le p S
-
-lemma butterfly_shortCycle_mgf (S : Finset (α)) (J : ℕ) (q : ℝ) (hq : 1 ≤ q) :
-    finiteMean (fun ω : ι → Bool =>
-      q^(Fintype.card (ShortCycle (R.perm ω) S J))) ≤
-      Real.exp ((q-1)*((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))) := by
-  apply factorial_mgf _ S.card _ q (fun ω => shortCycle_card_le _ _ _)
-  · positivity
-  · exact hq
-  · exact butterfly_shortCycle_moment R S J
-
-end CycleMGF
-
-
-abbrev LongCycle (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :=
-  {C : SelectedCycle p S // ¬ C.val.card ≤ J}
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    Fintype (LongCycle p S J) := Fintype.ofFinite _
-
-lemma selectedCycle_card_split (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    Fintype.card (SelectedCycle p S) =
-      Fintype.card (ShortCycle p S J) + Fintype.card (LongCycle p S J) := by
-  classical
-  rw [←Fintype.card_congr (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J)),Fintype.card_sum]
-
-lemma longCycle_card_bound (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    J * Fintype.card (LongCycle p S J) ≤ S.card := by
-  classical
-  have hs := Equiv.sum_comp (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J))
-    (fun C : SelectedCycle p S => C.val.card)
-  rw [Fintype.sum_sum_type] at hs
-  have h0 : J * Fintype.card (LongCycle p S J) ≤
-      ∑ C : LongCycle p S J, C.val.val.card := by
-    calc
-      _ = ∑ _C : LongCycle p S J, J := by simp [mul_comm]
-      _ ≤ _ := Finset.sum_le_sum (fun C _ => Nat.le_of_lt (Nat.lt_of_not_ge C.property))
-  have hh := cycleMass_le p S
-  unfold cycleMass at hh
-  rw [←hs] at hh
-  exact h0.trans ((Nat.le_add_left _ _).trans hh)
-
-lemma longCycle_pow_bound (p : Equiv.Perm (α)) (S : Finset (α))
-    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    q^(Fintype.card (LongCycle p S J)) ≤ Real.exp ((S.card : ℝ)/J * Real.log q) := by
-  have hq0 : 0 < q := lt_of_lt_of_le zero_lt_one hq
-  have hc : (Fintype.card (LongCycle p S J) : ℝ) ≤ (S.card : ℝ)/J := by
-    apply (le_div_iff₀ (by exact_mod_cast hJ)).mpr
-    have hh := longCycle_card_bound p S J
-    exact_mod_cast (by simpa only [mul_comm] using hh)
-  calc
-    _ = Real.exp ((Fintype.card (LongCycle p S J) : ℝ)*Real.log q) := by
-      rw [Real.exp_nat_mul,Real.exp_log hq0]
-    _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right hc (Real.log_nonneg hq))
-
-lemma butterfly_cycle_log_mgf (S : Finset (α)) (J : ℕ) (hJ : 0 < J)
-    (q : ℝ) (hq : 1 ≤ q) :
-    Real.log (finiteMean (fun ω : ι → Bool =>
-      q^(Fintype.card (SelectedCycle (R.perm ω) S)))) ≤
-      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
-  let lam : ℝ := (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1)
-  let c : ℝ := (S.card : ℝ)/J * Real.log q
-  have hnum : (q-1)*lam ≤ 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
-    have hsum := sum_inv_sqrt J
-    have hp : 0 ≤ (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) := by positivity
-    have h0 : 0 ≤ 2*Real.sqrt J := by positivity
-    have hh := mul_le_mul_of_nonneg_left hsum hp
-    have hq1 : 0 ≤ q-1 := sub_nonneg.mpr hq
-    calc
-      _ ≤ (q-1)*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
-        mul_le_mul_of_nonneg_left hh hq1
-      _ ≤ q*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
-        mul_le_mul_of_nonneg_right (by linarith) (mul_nonneg hp h0)
-      _ = _ := by ring
-  have hm : finiteMean (fun ω : ι → Bool =>
-      q^(Fintype.card (SelectedCycle (R.perm ω) S))) ≤
-      Real.exp (c+(q-1)*lam) := by
-    calc
-      _ ≤ finiteMean (fun ω : ι → Bool =>
-          q^(Fintype.card (ShortCycle (R.perm ω) S J))*Real.exp c) := by
-        apply finiteMean_mono
-        intro ω
-        rw [selectedCycle_card_split _ _ J,pow_add]
-        exact mul_le_mul_of_nonneg_left (longCycle_pow_bound _ S J hJ q hq) (by positivity)
-      _ = finiteMean (fun ω : ι → Bool =>
-          q^(Fintype.card (ShortCycle (R.perm ω) S J))) * Real.exp c :=
-        finiteMean_mul_const _ _
-      _ ≤ Real.exp ((q-1)*lam) * Real.exp c :=
-        mul_le_mul_of_nonneg_right (butterfly_shortCycle_mgf R S J q hq) (Real.exp_nonneg _)
-      _ = _ := by rw [←Real.exp_add,add_comm]
-  have hpos : 0 < finiteMean (fun ω : ι → Bool =>
-      q^(Fintype.card (SelectedCycle (R.perm ω) S))) := by
-    apply finiteMean_pos
-    intro ω
-    positivity
-  have hlog := Real.log_le_log hpos hm
-  rw [Real.log_exp] at hlog
-  dsimp [c] at hlog
-  linarith
-
-
-end RoutingNetwork
-
-variable {α B : Type*} [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] {d : ℕ}
-
-def parallelPerm (ω : B × SwitchIndex d → Bool) : Equiv.Perm (B × Card d) where
-  toFun x := (x.1,butterflyPerm d (decodeButterfly d (fun i => ω (x.1,i))) x.2)
-  invFun y := (y.1,(butterflyPerm d (decodeButterfly d (fun i => ω (y.1,i)))).symm y.2)
-  left_inv x := by simp
-  right_inv y := by simp
-
-def relabeledPerm (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (ω : B × SwitchIndex d → Bool) : Equiv.Perm α :=
-  (pre.trans (parallelPerm ω)).trans post
-
-noncomputable def parallelPath (pre : α ≃ B × Card d)
-    (ω : B × SwitchIndex d → Bool) (x : α) : Finset (B × SwitchIndex d) :=
-  (routeDomain d (pre x).2
-    (butterflyPerm d (decodeButterfly d (fun i => ω ((pre x).1,i))) (pre x).2)).image
-      (fun i => ((pre x).1,i))
-
-def parallelUsers (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
-    (i : B × SwitchIndex d) : α × α :=
-  let u := switchUsers d (decodeButterfly d (fun j => ω (i.1,j))) i.2
-  (pre.symm (i.1,u.1),pre.symm (i.1,u.2))
-
-omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
-lemma parallelUsers_ne (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
-    (i : B × SwitchIndex d) : (parallelUsers pre ω i).1 ≠ (parallelUsers pre ω i).2 := by
-  intro h
-  exact switchUsers_ne d _ _ (congrArg Prod.snd (pre.symm.injective h))
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma mem_parallelPath (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
-    (x : α) (i : B × SwitchIndex d) :
-    i ∈ parallelPath pre ω x ↔ (parallelUsers pre ω i).1 = x ∨ (parallelUsers pre ω i).2 = x := by
-  classical
-  obtain ⟨b,j⟩ := i
-  simp only [parallelPath,Finset.mem_image,Prod.mk.injEq]
-  constructor
-  · rintro ⟨k,hk,hb,rfl⟩
-    subst b
-    rcases (mem_routeDomain_iff_user d _ _ _).mp hk with h | h
-    · left
-      apply pre.injective
-      change pre (pre.symm ((pre x).1,_)) = pre x
-      rw [Equiv.apply_symm_apply]
-      exact Prod.ext rfl h
-    · right
-      apply pre.injective
-      change pre (pre.symm ((pre x).1,_)) = pre x
-      rw [Equiv.apply_symm_apply]
-      exact Prod.ext rfl h
-  · intro h
-    have hh : (b,(switchUsers d (decodeButterfly d (fun j => ω (b,j))) j).1) = pre x ∨
-        (b,(switchUsers d (decodeButterfly d (fun j => ω (b,j))) j).2) = pre x := by
-      rcases h with h | h
-      · exact Or.inl (by simpa only [parallelUsers,Equiv.apply_symm_apply] using congrArg pre h)
-      · exact Or.inr (by simpa only [parallelUsers,Equiv.apply_symm_apply] using congrArg pre h)
-    have hb : b = (pre x).1 := hh.elim (congrArg Prod.fst) (congrArg Prod.fst)
-    subst b
-    refine ⟨j,?_,rfl,rfl⟩
-    apply (mem_routeDomain_iff_user d _ _ _).mpr
-    exact hh.elim (fun h => Or.inl (congrArg Prod.snd h)) (fun h => Or.inr (congrArg Prod.snd h))
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma card_parallelPath (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool) (x : α) :
-    (parallelPath pre ω x).card = d := by
-  classical
-  rw [parallelPath,Finset.card_image_of_injective _ (fun _ _ h => Prod.mk.inj h |>.2),card_routeDomain]
-
-noncomputable def blockMarks (pre : α ≃ B × Card d) (S : Finset α) (b : B) : Finset (Card d) :=
-  Finset.univ.filter (fun x => pre.symm (b,x) ∈ S)
-
-omit [Fintype α] [Fintype B] [DecidableEq B] in
-@[simp] lemma mem_blockMarks (pre : α ≃ B × Card d) (S : Finset α) (b : B) (x : Card d) :
-    x ∈ blockMarks pre S b ↔ pre.symm (b,x) ∈ S := by
-  classical
-  simp [blockMarks]
-
-omit [DecidableEq B] in
-lemma sum_card_blockMarks (pre : α ≃ B × Card d) (S : Finset α) :
-    ∑ b : B, (blockMarks pre S b).card = S.card := by
-  classical
-  calc
-    _ = ∑ b : B, ∑ x : Card d, if pre.symm (b,x) ∈ S then 1 else 0 := by
-      simp only [blockMarks,Finset.card_filter]
-    _ = ∑ u : B × Card d, if pre.symm u ∈ S then 1 else 0 := (Fintype.sum_prod_type (fun u : B × Card d => if pre.symm u ∈ S then (1:ℕ) else 0)).symm
-    _ = ∑ x : α, if x ∈ S then 1 else 0 := Equiv.sum_comp pre.symm (fun x : α => if x ∈ S then (1:ℕ) else 0)
-    _ = _ := by simp
-
-omit [DecidableEq B] in
-lemma card_blockMarks_le (pre : α ≃ B × Card d) (S : Finset α) (b : B) :
-    (blockMarks pre S b).card ≤ S.card := by
-  rw [←sum_card_blockMarks pre S]
-  exact Finset.single_le_sum (f := fun b : B => (blockMarks pre S b).card) (fun _ _ => Nat.zero_le _) (Finset.mem_univ b)
-
-end Thorp
-
-end ThorpNine.Dense
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Dense.AssignmentSlots
+
+namespace OAI
+
+namespace ThorpNine.Dense
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+namespace RoutingNetwork
+variable {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι] [DecidableEq ι]
+  {d : ℕ} (R : RoutingNetwork α ι d)
+lemma mem_exposedPaths (b : ι → Bool) (V : Finset (α)) (s : ι) :
+    s ∈ exposedPaths R b V ↔ (R.users b s).1 ∈ V ∨ (R.users b s).2 ∈ V := by
+  classical
+  simp only [exposedPaths,Finset.mem_biUnion,R.mem_path]
+  constructor
+  · rintro ⟨x,hx,h | h⟩
+    · exact Or.inl (h ▸ hx)
+    · exact Or.inr (h ▸ hx)
+  · rintro (h | h)
+    · exact ⟨_,h,Or.inl rfl⟩
+    · exact ⟨_,h,Or.inr rfl⟩
+
+lemma terminalCharge_sum (b : ι → Bool) (z : α) (V : Finset (α))
+    (hz : z ∉ V) :
+    (terminalCharge R b z V : ℝ) = ∑ s : ι,
+      ((if (R.users b s).1 = z ∧ (R.users b s).2 ∈ V then (1:ℝ) else 0) +
+      (if (R.users b s).2 = z ∧ (R.users b s).1 ∈ V then (1:ℝ) else 0)) := by
+  classical
+  have he : R.path b z ∩ exposedPaths R b V =
+      Finset.univ.filter (fun s =>
+        ((R.users b s).1 = z ∧ (R.users b s).2 ∈ V) ∨
+        ((R.users b s).2 = z ∧ (R.users b s).1 ∈ V)) := by
+    ext s
+    simp only [Finset.mem_inter,R.mem_path,mem_exposedPaths,
+      Finset.mem_filter,Finset.mem_univ,true_and]
+    by_cases h₁ : (R.users b s).1 = z <;>
+      by_cases h₂ : (R.users b s).2 = z <;> simp [h₁,h₂,hz]
+  rw [terminalCharge,he]
+  simp only [← Finset.sum_boole]
+  apply Finset.sum_congr rfl
+  intro s _
+  by_cases h₁ : (R.users b s).1 = z <;>
+    by_cases h₂ : (R.users b s).2 = z <;> simp [h₁,h₂,hz]
+
+noncomputable def priorPaths {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) : Finset (α) :=
+  Finset.univ.filter (fun x => ∃ i : Fin A, o x = some i ∧ i.val < t)
+
+omit [DecidableEq α] in
+@[simp] lemma mem_priorPaths {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) (x : α) :
+    x ∈ priorPaths o t ↔ ∃ i : Fin A, o x = some i ∧ i.val < t := by
+  classical
+  simp [priorPaths]
+
+lemma terminalCharge_prior_sum {A : ℕ} (o : α → Option (Fin A))
+    (b : ι → Bool) (z : Terminal o) (i : Fin A) :
+    (terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) =
+      ∑ s : ι,
+        ((if (z i).val = (R.users b s).1 ∧ (R.users b s).2 ∈ priorPaths o (i.val+1)
+          then (1:ℝ) else 0) +
+        (if (z i).val = (R.users b s).2 ∧ (R.users b s).1 ∈ priorPaths o (i.val+1)
+          then (1:ℝ) else 0)) := by
+  classical
+  rw [terminalCharge_sum _ _ _ _ (Finset.notMem_erase _ _)]
+  apply Finset.sum_congr rfl
+  intro s _
+  have hn := R.users_ne b s
+  simp only [Finset.mem_erase]
+  by_cases h₁ : (z i).val = (R.users b s).1 <;>
+    by_cases h₂ : (z i).val = (R.users b s).2 <;> simp_all [eq_comm]
+
+lemma rotationMeeting_eq_slot_sum {A : ℕ} (o : α → Option (Fin A))
+    (z : Terminal o) (u v : α) :
+    rotationMeeting o z u v = ∑ i : Fin A,
+      ((if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) +
+      (if (z i).val = v ∧ u ∈ priorPaths o (i.val+1) then (1:ℝ) else 0)) := by
+  classical
+  have only_owner (u : α) (a : Fin A) (hou : o u = some a)
+      (i : Fin A) (hia : i ≠ a) : (z i).val ≠ u := by
+    intro h
+    have hi := (z i).property
+    rw [h,hou] at hi
+    exact hia (Option.some.inj hi).symm
+  have sum_one (u v : α) (a : Fin A) (hou : o u = some a) :
+      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) =
+      if (z a).val = u ∧ v ∈ priorPaths o (a.val+1) then 1 else 0 := by
+    apply Finset.sum_eq_single a
+    · intro i _ hia
+      simp [only_owner u a hou i hia]
+    · simp
+  have sum_none (u v : α) (hou : o u = none) :
+      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) = 0 := by
+    apply Finset.sum_eq_zero
+    intro i _
+    have hn : (z i).val ≠ u := by
+      intro h
+      have hi := (z i).property
+      rw [h,hou] at hi
+      contradiction
+    simp [hn]
+  rw [Finset.sum_add_distrib]
+  cases hou : o u with
+  | none =>
+    rw [sum_none u v hou]
+    simp [rotationMeeting,hou,mem_priorPaths]
+  | some a =>
+    rw [sum_one u v a hou]
+    cases hov : o v with
+    | none =>
+      rw [sum_none v u hov]
+      simp [rotationMeeting,hou,hov,mem_priorPaths]
+    | some b =>
+      rw [sum_one v u b hov]
+      simp only [rotationMeeting,hou,hov,mem_priorPaths,Option.some.injEq]
+      rcases lt_trichotomy a b with hab | rfl | hba
+      · have hn : ¬ b.val < a.val+1 := by omega
+        have hy : a.val < b.val+1 := by omega
+        simp [hab,hn,hy]
+      · simp
+      · have hn : ¬ a.val < b.val+1 := by omega
+        have hy : b.val < a.val+1 := by omega
+        simp [hba,not_lt_of_ge (le_of_lt hba),hn,hy]
+
+lemma rotationCharge_eq_terminal_sum {A : ℕ} (o : α → Option (Fin A))
+    (b : ι → Bool) (z : Terminal o) :
+    rotationCharge R o b z = ∑ i : Fin A,
+      (terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) := by
+  simp only [rotationCharge,rotationMeeting_eq_slot_sum,terminalCharge_prior_sum]
+  exact Finset.sum_comm
+
+omit [DecidableEq α] in
+@[simp] lemma priorPaths_zero {A : ℕ} (o : α → Option (Fin A)) : priorPaths o 0 = ∅ := by
+  classical
+  ext x
+  simp
+
+lemma priorPaths_succ {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) (ht : t < A) :
+    priorPaths o (t+1) = priorPaths o t ∪ cycleFiber o ⟨t,ht⟩ := by
+  classical
+  ext x
+  simp only [mem_priorPaths,Finset.mem_union,mem_cycleFiber]
+  constructor
+  · rintro ⟨i,hi,hit⟩
+    by_cases h : i.val < t
+    · exact Or.inl ⟨i,hi,h⟩
+    · have he : i = ⟨t,ht⟩ := Fin.ext (by change i.val = t; omega)
+      exact Or.inr (he ▸ hi)
+  · rintro (⟨i,hi,hit⟩ | hi)
+    · exact ⟨i,hi,by omega⟩
+    · exact ⟨⟨t,ht⟩,hi,by change t < t+1; omega⟩
+
+omit [DecidableEq α] in
+lemma priorPaths_disjoint {A : ℕ} (o : α → Option (Fin A)) (i : Fin A) :
+    Disjoint (priorPaths o i.val) (cycleFiber o i) := by
+  classical
+  apply Finset.disjoint_left.mpr
+  intro x hx hx'
+  obtain ⟨j,hj,hji⟩ := (mem_priorPaths _ _ _).mp hx
+  have he : j = i := Option.some.inj (hj.symm.trans ((mem_cycleFiber _ _ _).mp hx'))
+  subst j
+  exact (lt_irrefl _ hji)
+
+namespace QueryTree
+open Thorp.QueryTree
+
+lemma seekCycles_owned {A : ℕ} (S : Finset (α)) (r : Fin A → ℕ)
+    (o : α → Option (Fin A)) (z : Terminal o) (ω : ι → Bool)
+    (hS : ∀ i, cycleFiber o i ⊆ S)
+    (hO : ∀ i, orbitSet (R.perm ω) (z i).val = cycleFiber o i)
+    (hC : ∀ i, (cycleFiber o i).card = r i+1) :
+    let p := R.perm ω
+    let L := List.ofFn (fun i : Fin A => (p (z i).val,r i))
+    (seekCycles R S L ∅).succeeds ω = true ∧
+      ((seekCycles R S L ∅).staleCharge ∅ ω : ℝ) = rotationCharge R o (ω) z := by
+  classical
+  let b := ω
+  let p := R.perm b
+  let f (i : Fin A) := (p (z i).val,r i)
+  let c (i : Fin A) : ℝ := terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val)
+  have hc (i : Fin A) : Function.minimalPeriod p (z i).val - 1 = r i := by
+    have he := orbitSet_card p (z i).val
+    rw [hO,hC] at he
+    omega
+  have hs (t : ℕ) (ht : t ≤ A) :
+      (seekCycles R S ((List.ofFn f).drop t) (priorPaths o t)).succeeds ω = true ∧
+        ((seekCycles R S ((List.ofFn f).drop t) (priorPaths o t)).staleCharge (exposedPaths R b (priorPaths o t)) ω : ℝ) =
+          ((List.ofFn c).drop t).sum := by
+    induction ht using Nat.decreasingInduction with
+    | self => simp only [List.length_ofFn,le_refl,List.drop_eq_nil_of_le,List.sum_nil,seekCycles,Thorp.QueryTree.succeeds,staleCharge,Nat.cast_zero,and_self]
+    | of_succ t ht ih =>
+      have htf : t < (List.ofFn f).length := by simpa using ht
+      have htc : t < (List.ofFn c).length := by simpa using ht
+      rw [List.drop_eq_getElem_cons htf,List.drop_eq_getElem_cons htc]
+      simp only [List.getElem_ofFn,List.sum_cons,seekCycles]
+      have h := seekCycle_orbit R S (priorPaths o t) (z ⟨t,ht⟩).val
+        (fun V => seekCycles R S ((List.ofFn f).drop (t+1)) V) ω
+        (by rw [hO]; exact hS _) (by rw [hO]; exact priorPaths_disjoint o ⟨t,ht⟩)
+      dsimp only at h
+      rw [hc,hO,←priorPaths_succ o t ht] at h
+      dsimp only [f]
+      constructor
+      · exact h.1.trans ih.1
+      · rw [h.2,Nat.cast_add,ih.2]
+        exact add_comm _ _
+  have h := hs 0 (Nat.zero_le _)
+  simp only [List.drop_zero,priorPaths_zero,exposedPaths_empty,List.sum_ofFn] at h
+  dsimp only
+  exact ⟨h.1,h.2.trans (rotationCharge_eq_terminal_sum R o b z).symm⟩
+
+end QueryTree
+section Collections
+variable {A : ℕ}
+
+def ValidCycleSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ)
+    (o : α → Option (Fin A)) : Prop :=
+  (∀ i, (cycleFiber o i).card = r i + 1) ∧
+  (∀ i, cycleFiber o i ⊆ S) ∧
+  (∀ i x, o x = some i → orbitSet p x = cycleFiber o i)
+
+abbrev CycleSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
+  {o : α → Option (Fin A) // ValidCycleSlots p S r o}
+
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
+    Fintype (CycleSlots p S r) := Fintype.ofFinite _
+
+lemma CycleSlots.terminal_nonempty {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (c : CycleSlots p S r) (i : Fin A) : Nonempty {x : α // c.val x = some i} := by
+  have hn : (cycleFiber c.val i).Nonempty := Finset.card_pos.mp (by rw [c.property.1]; omega)
+  obtain ⟨x,hx⟩ := hn
+  exact ⟨x,(mem_cycleFiber _ _ _).mp hx⟩
+
+omit [DecidableEq α] in
+lemma owner_ext {o o' : α → Option (Fin A)}
+    (h : ∀ i, cycleFiber o i = cycleFiber o' i) : o = o' := by
+  funext x
+  have he (i : Fin A) : o x = some i ↔ o' x = some i := by
+    simpa only [mem_cycleFiber] using (Finset.ext_iff.mp (h i) x)
+  cases ho : o x with
+  | some i => exact ((he i).mp ho).symm
+  | none =>
+    cases ho' : o' x with
+    | none => rfl
+    | some i => have hn := (he i).mpr ho'; rw [ho] at hn; contradiction
+
+abbrev RootedSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
+  Σ c : CycleSlots p S r, Terminal c.val
+
+noncomputable def rootedStart {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (c : RootedSlots p S r) : Fin A → S := fun i =>
+  ⟨p (c.2 i).val,c.1.property.2.1 i (by
+    rw [←c.1.property.2.2 i (c.2 i).val (c.2 i).property]
+    exact (mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.apply_right (Equiv.Perm.SameCycle.refl p _)))⟩
+
+lemma rootedStart_injective {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ} :
+    Function.Injective (rootedStart (p:=p) (S:=S) (r:=r)) := by
+  rintro ⟨c,z⟩ ⟨c',z'⟩ h
+  have hz (i : Fin A) : (z i).val = (z' i).val :=
+    p.injective (congrArg Subtype.val (congrFun h i))
+  have hc : c = c' := Subtype.ext (owner_ext (fun i => by
+    rw [←c.property.2.2 i (z i).val (z i).property,
+      ←c'.property.2.2 i (z' i).val (z' i).property,hz]))
+  subst c'
+  have hzz : z = z' := funext (fun i => Subtype.ext (hz i))
+  subst z'
+  rfl
+
+lemma rooted_weight_sum_le (S : Finset (α)) (r : Fin A → ℕ) (ω : ι → Bool) :
+    (∑ c : RootedSlots (R.perm ω) S r,
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2)) ≤
+    ∑ a : Fin A → S,
+      if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+        ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+      else 0 := by
+  classical
+  let p := R.perm ω
+  let g (a : Fin A → S) : ℝ :=
+    if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+    else 0
+  have he (c : RootedSlots p S r) :
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2) =
+      g (rootedStart c) := by
+    have hs := QueryTree.seekCycles_owned R S r c.1.val c.2 ω c.1.property.2.1
+      (fun i => c.1.property.2.2 i (c.2 i).val (c.2 i).property) c.1.property.1
+    dsimp only at hs
+    unfold g
+    rw [ite_eq_left (show (QueryTree.seekCycles R S (List.ofFn (fun i => ((rootedStart c i).val,r i))) ∅).succeeds ω = true from hs.1)]
+    rw [← hs.2,neg_mul,Real.exp_neg,mul_comm (Real.log 2),Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
+    rfl
+  simp_rw [he]
+  exact sum_comp_injective_le rootedStart rootedStart_injective g (fun a => by dsimp [g]; split_ifs <;> positivity)
+
+lemma collection_rotation_lower (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r)
+    (ω : ι → Bool) (c : CycleSlots (R.perm ω) S r) :
+    collectionWeight d r ≤ ∑ z : Terminal c.val,
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z) := by
+  classical
+  let (i : Fin A) : Nonempty {x : α // c.val x = some i} := c.terminal_nonempty i
+  have hm : Monotone (fun i => (cycleFiber c.val i).card) := by
+    intro i j hij
+    dsimp only
+    rw [c.property.1,c.property.1]
+    exact Nat.add_le_add_right (hr hij) 1
+  have hb := rotation_exp_bound R hm (ω)
+  have hcard : (Fintype.card (Terminal c.val) : ℝ) = ∏ i : Fin A, (r i+1 : ℝ) := by
+    rw [Fintype.card_pi,Nat.cast_prod]
+    apply Finset.prod_congr rfl
+    intro i _
+    rw [card_cycleFiber,c.property.1,Nat.cast_add,Nat.cast_one]
+  have hp : (0:ℝ) < Fintype.card (Terminal c.val) := Nat.cast_pos.mpr Fintype.card_pos
+  have hl : rotationLower d r ≤ finiteMean (fun z : Terminal c.val => Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z)) := by
+    simpa only [rotationLower,c.property.1,Nat.cast_add,Nat.cast_one] using hb
+  have hh := (le_div_iff₀ hp).mp hl
+  rw [hcard] at hh
+  unfold collectionWeight
+  rw [← Finset.mul_sum]
+  calc
+    _ = ((2:ℝ)^d)^A * (rotationLower d r * ∏ i : Fin A, (r i+1 : ℝ)) := by ring
+    _ ≤ _ := mul_le_mul_of_nonneg_left hh (by positivity)
+
+lemma cycleSlots_weighted_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : ι → Bool =>
+      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) *
+      collectionWeight d r ≤ (S.card : ℝ)^A := by
+  classical
+  let g (a : Fin A → S) (ω : ι → Bool) : ℝ :=
+    if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+    else 0
+  have hω (ω : ι → Bool) :
+      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ) * collectionWeight d r ≤
+        ∑ a : Fin A → S, g a ω := by
+    calc
+      _ = ∑ _c : CycleSlots (R.perm ω) S r, collectionWeight d r := by simp
+      _ ≤ ∑ c : CycleSlots (R.perm ω) S r,
+          ∑ z : Terminal c.val, ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z) :=
+        Finset.sum_le_sum (fun c _ => collection_rotation_lower R S r hr ω c)
+      _ = ∑ c : RootedSlots (R.perm ω) S r,
+          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2) :=
+        (Fintype.sum_sigma (fun c : RootedSlots (R.perm ω) S r =>
+          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2))).symm
+      _ ≤ _ := rooted_weight_sum_le R S r ω
+  have hb := finiteMean_mono hω
+  rw [finiteMean_mul_const,finiteMean_sum] at hb
+  calc
+    _ ≤ ∑ a : Fin A → S, finiteMean (g a) := hb
+    _ ≤ ∑ _a : Fin A → S, (1:ℝ) := by
+      apply Finset.sum_le_sum
+      intro a _
+      have hh := QueryTree.cycles_weighted_success R S (List.ofFn (fun i : Fin A => ((a i).val,r i)))
+      simpa only [List.length_ofFn] using hh
+    _ = _ := by simp
+
+lemma cycleSlots_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : ι → Bool =>
+      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
+        (S.card : ℝ)^A / collectionWeight d r := by
+  exact (le_div_iff₀ (collectionWeight_pos d r)).mpr (cycleSlots_weighted_moment R S r hr)
+
+lemma butterfly_cycleSlots_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : ι → Bool =>
+      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
+        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
+  have h := cycleSlots_moment R S r hr
+  rw [collectionWeight_eq_sqrt,div_mul_eq_div_mul_one_div] at h
+  rw [div_pow]
+  simpa only [one_div,Finset.prod_inv_distrib] using h
+
+end Collections
+
+section CycleCounting
+variable {A : ℕ}
+
+lemma SelectedCycle.nonempty {p : Equiv.Perm (α)} {S : Finset (α)}
+    (C : SelectedCycle p S) : C.val.Nonempty := by
+  obtain ⟨x,hx⟩ := C.property.1
+  rw [hx]
+  exact orbitSet_nonempty p x
+
+lemma SelectedCycle.orbit_eq {p : Equiv.Perm (α)} {S : Finset (α)}
+    (C : SelectedCycle p S) {x : α} (hx : x ∈ C.val) : orbitSet p x = C.val := by
+  obtain ⟨y,hy⟩ := C.property.1
+  rw [hy] at hx ⊢
+  exact (orbitSet_eq p ((mem_orbitSet _ _ _).mp hx)).symm
+
+lemma SelectedCycle.eq_of_mem {p : Equiv.Perm (α)} {S : Finset (α)}
+    (C D : SelectedCycle p S) {x : α} (hC : x ∈ C.val) (hD : x ∈ D.val) : C = D := by
+  apply Subtype.ext
+  exact (C.orbit_eq hC).symm.trans (D.orbit_eq hD)
+
+abbrev SlotAssignment (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
+  {f : Fin A → SelectedCycle p S // Function.Injective f ∧ ∀ i, (f i).val.card = r i+1}
+
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
+    Fintype (SlotAssignment p S r) := Fintype.ofFinite _
+
+noncomputable def slotsToAssignment {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (c : CycleSlots p S r) : SlotAssignment p S r := by
+  classical
+  let f (i : Fin A) : SelectedCycle p S := ⟨cycleFiber c.val i,by
+    obtain ⟨x,hx⟩ := c.terminal_nonempty i
+    exact ⟨⟨x,(c.property.2.2 i x hx).symm⟩,c.property.2.1 i⟩⟩
+  refine ⟨f,?_,fun i => c.property.1 i⟩
+  intro i j hij
+  have he : cycleFiber c.val i = cycleFiber c.val j := congrArg Subtype.val hij
+  obtain ⟨x,hx⟩ := c.terminal_nonempty i
+  have hxi : x ∈ cycleFiber c.val i := (mem_cycleFiber _ _ _).mpr hx
+  rw [he] at hxi
+  exact Option.some.inj (hx.symm.trans ((mem_cycleFiber _ _ _).mp hxi))
+
+noncomputable def assignmentOwner {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) (x : α) : Option (Fin A) := by
+  classical
+  exact if h : ∃ i, x ∈ (f.val i).val then some (Classical.choose h) else none
+
+lemma assignmentOwner_eq_some {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) (x : α) (i : Fin A) :
+    assignmentOwner f x = some i ↔ x ∈ (f.val i).val := by
+  classical
+  unfold assignmentOwner
+  split_ifs with h
+  · constructor
+    · intro hi
+      have hx := Classical.choose_spec h
+      rw [Option.some.inj hi] at hx
+      exact hx
+    · intro hx
+      have he := (f.val (Classical.choose h)).eq_of_mem (f.val i) (Classical.choose_spec h) hx
+      exact congrArg some (f.property.1 he)
+  · constructor
+    · intro hi; contradiction
+    · intro hx; exact (h ⟨i,hx⟩).elim
+
+lemma assignmentOwner_fiber {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) (i : Fin A) : cycleFiber (assignmentOwner f) i = (f.val i).val := by
+  ext x
+  simp only [mem_cycleFiber,assignmentOwner_eq_some]
+
+noncomputable def assignmentToSlots {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) : CycleSlots p S r :=
+  ⟨assignmentOwner f,by
+    refine ⟨?_,?_,?_⟩
+    · intro i
+      rw [assignmentOwner_fiber]
+      exact f.property.2 i
+    · intro i
+      rw [assignmentOwner_fiber]
+      exact (f.val i).property.2
+    · intro i x hx
+      rw [assignmentOwner_fiber]
+      exact (f.val i).orbit_eq ((assignmentOwner_eq_some f x i).mp hx)⟩
+
+noncomputable def cycleSlotsEquivAssignments (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
+    CycleSlots p S r ≃ SlotAssignment p S r where
+  toFun := slotsToAssignment
+  invFun := assignmentToSlots
+  left_inv c := by
+    apply Subtype.ext
+    apply owner_ext
+    intro i
+    rw [assignmentToSlots,assignmentOwner_fiber]
+    rfl
+  right_inv f := by
+    apply Subtype.ext
+    funext i
+    apply Subtype.ext
+    exact assignmentOwner_fiber f i
+
+end CycleCounting
+
+section GroupedCounting
+variable {A : ℕ}
+
+abbrev LengthClass (r : Fin A → ℕ) := {j : ℕ // j ∈ Finset.univ.image (fun i => r i + 1)}
+
+def slotClass (r : Fin A → ℕ) (i : Fin A) : LengthClass r :=
+  ⟨r i+1, Finset.mem_image.mpr ⟨i, Finset.mem_univ _, rfl⟩⟩
+
+abbrev SlotsOfLength (r : Fin A → ℕ) (j : LengthClass r) := {i : Fin A // slotClass r i = j}
+abbrev CyclesOfLength (p : Equiv.Perm (α)) (S : Finset (α)) (j : ℕ) :=
+  {C : SelectedCycle p S // C.val.card = j}
+
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (j : ℕ) :
+    Fintype (CyclesOfLength p S j) := Fintype.ofFinite _
+
+abbrev GroupedAssignments (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
+  ∀ j : LengthClass r, SlotsOfLength r j ↪ CyclesOfLength p S j.val
+
+noncomputable def assignmentToGroups {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) : GroupedAssignments p S r := fun _j =>
+  { toFun := fun i => ⟨f.val i.val, (f.property.2 i.val).trans (congrArg Subtype.val i.property)⟩
+    inj' := fun _i _k h => Subtype.ext (f.property.1 (congrArg Subtype.val h)) }
+
+noncomputable def groupsEmbedding {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (F : GroupedAssignments p S r) : Fin A ↪ SelectedCycle p S :=
+  ((Equiv.sigmaFiberEquiv (slotClass r)).symm.toEmbedding.trans
+    (Function.Embedding.sigmaMap (Function.Embedding.refl _) F)).trans
+    ((Function.Embedding.sigmaMap (Function.Embedding.subtype _) (fun _ => Function.Embedding.refl _)).trans
+      (Equiv.sigmaFiberEquiv (fun C : SelectedCycle p S => C.val.card)).toEmbedding)
+
+noncomputable def groupsToAssignment {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (F : GroupedAssignments p S r) : SlotAssignment p S r :=
+  ⟨groupsEmbedding F, (groupsEmbedding F).injective,
+    fun i => (F (slotClass r i) ⟨i,rfl⟩).property⟩
+
+noncomputable def assignmentEquivGroups (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
+    SlotAssignment p S r ≃ GroupedAssignments p S r where
+  toFun := assignmentToGroups
+  invFun := groupsToAssignment
+  left_inv f := by
+    apply Subtype.ext
+    rfl
+  right_inv F := by
+    funext j
+    apply Function.Embedding.ext
+    intro i
+    rcases i with ⟨i,hi⟩
+    subst j
+    rfl
+
+lemma card_cycleSlots_factorial (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
+    Fintype.card (CycleSlots p S r) =
+      ∏ j : LengthClass r, (Fintype.card (CyclesOfLength p S j.val)).descFactorial
+        (Fintype.card (SlotsOfLength r j)) := by
+  classical
+  rw [Fintype.card_congr (cycleSlotsEquivAssignments p S r),
+    Fintype.card_congr (assignmentEquivGroups p S r), Fintype.card_pi]
+  simp only [Fintype.card_embedding_eq]
+
+lemma butterfly_factorial_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : ι → Bool =>
+       (∏ j : LengthClass r,
+         (Fintype.card (CyclesOfLength (R.perm ω) S j.val)).descFactorial
+           (Fintype.card (SlotsOfLength r j)) : ℕ)) ≤
+      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+        ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
+  simpa only [←card_cycleSlots_factorial] using butterfly_cycleSlots_moment R S r hr
+
+end GroupedCounting
+
+section AllLengths
+variable {A : ℕ}
+
+noncomputable def assignmentRelabel (p : Equiv.Perm (α)) (S : Finset (α))
+    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
+    SlotAssignment p S r ≃ SlotAssignment p S (r ∘ e) where
+  toFun f := ⟨f.val ∘ e, f.property.1.comp e.injective, fun i => f.property.2 (e i)⟩
+  invFun f := ⟨f.val ∘ e.symm, f.property.1.comp e.symm.injective, fun i => by
+    simpa only [Function.comp_apply,Equiv.apply_symm_apply] using f.property.2 (e.symm i)⟩
+  left_inv f := by apply Subtype.ext; funext i; simp
+  right_inv f := by apply Subtype.ext; funext i; simp
+
+lemma card_cycleSlots_relabel (p : Equiv.Perm (α)) (S : Finset (α))
+    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
+    Fintype.card (CycleSlots p S r) = Fintype.card (CycleSlots p S (r ∘ e)) := by
+  exact Fintype.card_congr ((cycleSlotsEquivAssignments p S r).trans
+    ((assignmentRelabel p S r e).trans (cycleSlotsEquivAssignments p S (r ∘ e)).symm))
+
+lemma butterfly_cycleSlots_moment_all (S : Finset (α)) (r : Fin A → ℕ) :
+    finiteMean (fun ω : ι → Bool =>
+      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
+        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
+  have h := butterfly_cycleSlots_moment R S (r ∘ Tuple.sort r) (Tuple.monotone_sort r)
+  simp_rw [←card_cycleSlots_relabel] at h
+  have hp := Equiv.prod_comp (Tuple.sort r) (fun i : Fin A => (1 / Real.sqrt (r i+1) : ℝ))
+  simpa only [Function.comp_apply,hp] using h
+
+abbrev ShortCycle (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :=
+  {C : SelectedCycle p S // C.val.card ≤ J}
+
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    Fintype (ShortCycle p S J) := Fintype.ofFinite _
+
+noncomputable def shortCycleLength {p : Equiv.Perm (α)} {S : Finset (α)} {J : ℕ}
+    (C : ShortCycle p S J) : Fin J :=
+  ⟨C.val.val.card-1, by have := C.val.nonempty.card_pos; have := C.property; omega⟩
+
+lemma shortCycleLength_add_one {p : Equiv.Perm (α)} {S : Finset (α)} {J : ℕ}
+    (C : ShortCycle p S J) : (shortCycleLength C : ℕ)+1 = C.val.val.card := by
+  have := C.val.nonempty.card_pos
+  simp only [shortCycleLength]
+  omega
+
+noncomputable def shortEmbeddingEquiv (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    (Fin A ↪ ShortCycle p S J) ≃
+      Σ r : Fin A → Fin J, SlotAssignment p S (fun i => (r i : ℕ)) where
+  toFun f := ⟨fun i => shortCycleLength (f i),
+    ⟨fun i => (f i).val, fun i j h => f.injective (Subtype.ext h),
+      fun i => (shortCycleLength_add_one (f i)).symm⟩⟩
+  invFun c :=
+    { toFun := fun i => ⟨c.2.val i, by rw [c.2.property.2]; exact (c.1 i).isLt⟩
+      inj' := fun _i _j h => c.2.property.1 (congrArg Subtype.val h) }
+  left_inv f := by
+    apply Function.Embedding.ext
+    intro i
+    rfl
+  right_inv c := by
+    rcases c with ⟨r,f⟩
+    have he : (fun i => shortCycleLength (⟨f.val i, by rw [f.property.2]; exact (r i).isLt⟩ : ShortCycle p S J)) = r := by
+      funext i
+      apply Fin.ext
+      simp only [shortCycleLength,f.property.2,Nat.add_sub_cancel]
+    apply Sigma.ext he
+    apply (Subtype.heq_iff_coe_eq (by
+      intro x
+      change (Function.Injective x ∧ ∀ i, (x i).val.card = (f.val i).val.card - 1 + 1) ↔ _
+      simp only [f.property.2,Nat.add_sub_cancel])).mpr
+    rfl
+
+lemma shortCycle_factorial_eq (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    (Fintype.card (ShortCycle p S J)).descFactorial A =
+      ∑ r : Fin A → Fin J, Fintype.card (CycleSlots p S (fun i => (r i : ℕ))) := by
+  classical
+  have hc := Fintype.card_embedding_eq (α := Fin A) (β := ShortCycle p S J)
+  simp only [Fintype.card_fin] at hc
+  rw [←hc, Fintype.card_congr (shortEmbeddingEquiv p S J),Fintype.card_sigma]
+  apply Finset.sum_congr rfl
+  intro r _
+  exact (Fintype.card_congr (cycleSlotsEquivAssignments p S (fun i => (r i : ℕ)))).symm
+
+end AllLengths
+
+section CycleMGF
+
+lemma butterfly_shortCycle_moment (S : Finset (α)) (J A : ℕ) :
+    finiteMean (fun ω : ι → Bool =>
+      ((Fintype.card (ShortCycle (R.perm ω) S J)).descFactorial A : ℝ)) ≤
+      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))^A := by
+  classical
+  simp_rw [shortCycle_factorial_eq,Nat.cast_sum,finiteMean_sum]
+  calc
+    _ ≤ ∑ r : Fin A → Fin J,
+        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+          ∏ i : Fin A, (1 / Real.sqrt ((r i : ℕ)+1)) := by
+      apply Finset.sum_le_sum
+      intro r _
+      exact butterfly_cycleSlots_moment_all R S (fun i => (r i : ℕ))
+    _ = _ := by
+      rw [←Finset.mul_sum,mul_pow]
+      congr 1
+      rw [←Fintype.prod_sum (fun (_ : Fin A) (j : Fin J) => (1 / Real.sqrt ((j:ℕ)+1) : ℝ))]
+      simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fin]
+
+noncomputable def cycleMass (p : Equiv.Perm (α)) (S : Finset (α)) : ℕ :=
+  ∑ C : SelectedCycle p S, C.val.card
+
+lemma cycleMass_le (p : Equiv.Perm (α)) (S : Finset (α)) : cycleMass p S ≤ S.card := by
+  classical
+  let f : (Σ C : SelectedCycle p S, {x : α // x ∈ C.val}) → S :=
+    fun c => ⟨c.2.val,c.1.property.2 c.2.property⟩
+  have hi : Function.Injective f := by
+    intro c e h
+    have hx : c.2.val = e.2.val := congrArg Subtype.val h
+    have hC : c.1 = e.1 := c.1.eq_of_mem e.1 c.2.property (hx ▸ e.2.property)
+    rcases c with ⟨c,x⟩
+    rcases e with ⟨e,y⟩
+    dsimp only at hC
+    subst e
+    have he : x=y := Subtype.ext hx
+    cases he
+    rfl
+  have hh := Fintype.card_le_of_injective f hi
+  simpa only [Fintype.card_sigma,Fintype.card_coe,cycleMass] using hh
+
+lemma shortCycle_card_le (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    Fintype.card (ShortCycle p S J) ≤ S.card := by
+  classical
+  calc
+    _ ≤ Fintype.card (SelectedCycle p S) := Fintype.card_subtype_le _
+    _ = ∑ _ : SelectedCycle p S, 1 := by simp
+    _ ≤ cycleMass p S := Finset.sum_le_sum (fun C _ => C.nonempty.card_pos)
+    _ ≤ _ := cycleMass_le p S
+
+lemma butterfly_shortCycle_mgf (S : Finset (α)) (J : ℕ) (q : ℝ) (hq : 1 ≤ q) :
+    finiteMean (fun ω : ι → Bool =>
+      q^(Fintype.card (ShortCycle (R.perm ω) S J))) ≤
+      Real.exp ((q-1)*((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))) := by
+  apply factorial_mgf _ S.card _ q (fun ω => shortCycle_card_le _ _ _)
+  · positivity
+  · exact hq
+  · exact butterfly_shortCycle_moment R S J
+
+end CycleMGF
+
+
+abbrev LongCycle (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :=
+  {C : SelectedCycle p S // ¬ C.val.card ≤ J}
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    Fintype (LongCycle p S J) := Fintype.ofFinite _
+
+lemma selectedCycle_card_split (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    Fintype.card (SelectedCycle p S) =
+      Fintype.card (ShortCycle p S J) + Fintype.card (LongCycle p S J) := by
+  classical
+  rw [←Fintype.card_congr (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J)),Fintype.card_sum]
+
+lemma longCycle_card_bound (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    J * Fintype.card (LongCycle p S J) ≤ S.card := by
+  classical
+  have hs := Equiv.sum_comp (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J))
+    (fun C : SelectedCycle p S => C.val.card)
+  rw [Fintype.sum_sum_type] at hs
+  have h0 : J * Fintype.card (LongCycle p S J) ≤
+      ∑ C : LongCycle p S J, C.val.val.card := by
+    calc
+      _ = ∑ _C : LongCycle p S J, J := by simp [mul_comm]
+      _ ≤ _ := Finset.sum_le_sum (fun C _ => Nat.le_of_lt (Nat.lt_of_not_ge C.property))
+  have hh := cycleMass_le p S
+  unfold cycleMass at hh
+  rw [←hs] at hh
+  exact h0.trans ((Nat.le_add_left _ _).trans hh)
+
+lemma longCycle_pow_bound (p : Equiv.Perm (α)) (S : Finset (α))
+    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    q^(Fintype.card (LongCycle p S J)) ≤ Real.exp ((S.card : ℝ)/J * Real.log q) := by
+  have hq0 : 0 < q := lt_of_lt_of_le zero_lt_one hq
+  have hc : (Fintype.card (LongCycle p S J) : ℝ) ≤ (S.card : ℝ)/J := by
+    apply (le_div_iff₀ (by exact_mod_cast hJ)).mpr
+    have hh := longCycle_card_bound p S J
+    exact_mod_cast (by simpa only [mul_comm] using hh)
+  calc
+    _ = Real.exp ((Fintype.card (LongCycle p S J) : ℝ)*Real.log q) := by
+      rw [Real.exp_nat_mul,Real.exp_log hq0]
+    _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right hc (Real.log_nonneg hq))
+
+lemma butterfly_cycle_log_mgf (S : Finset (α)) (J : ℕ) (hJ : 0 < J)
+    (q : ℝ) (hq : 1 ≤ q) :
+    Real.log (finiteMean (fun ω : ι → Bool =>
+      q^(Fintype.card (SelectedCycle (R.perm ω) S)))) ≤
+      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
+  let lam : ℝ := (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1)
+  let c : ℝ := (S.card : ℝ)/J * Real.log q
+  have hnum : (q-1)*lam ≤ 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
+    have hsum := sum_inv_sqrt J
+    have hp : 0 ≤ (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) := by positivity
+    have h0 : 0 ≤ 2*Real.sqrt J := by positivity
+    have hh := mul_le_mul_of_nonneg_left hsum hp
+    have hq1 : 0 ≤ q-1 := sub_nonneg.mpr hq
+    calc
+      _ ≤ (q-1)*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
+        mul_le_mul_of_nonneg_left hh hq1
+      _ ≤ q*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
+        mul_le_mul_of_nonneg_right (by linarith) (mul_nonneg hp h0)
+      _ = _ := by ring
+  have hm : finiteMean (fun ω : ι → Bool =>
+      q^(Fintype.card (SelectedCycle (R.perm ω) S))) ≤
+      Real.exp (c+(q-1)*lam) := by
+    calc
+      _ ≤ finiteMean (fun ω : ι → Bool =>
+          q^(Fintype.card (ShortCycle (R.perm ω) S J))*Real.exp c) := by
+        apply finiteMean_mono
+        intro ω
+        rw [selectedCycle_card_split _ _ J,pow_add]
+        exact mul_le_mul_of_nonneg_left (longCycle_pow_bound _ S J hJ q hq) (by positivity)
+      _ = finiteMean (fun ω : ι → Bool =>
+          q^(Fintype.card (ShortCycle (R.perm ω) S J))) * Real.exp c :=
+        finiteMean_mul_const _ _
+      _ ≤ Real.exp ((q-1)*lam) * Real.exp c :=
+        mul_le_mul_of_nonneg_right (butterfly_shortCycle_mgf R S J q hq) (Real.exp_nonneg _)
+      _ = _ := by rw [←Real.exp_add,add_comm]
+  have hpos : 0 < finiteMean (fun ω : ι → Bool =>
+      q^(Fintype.card (SelectedCycle (R.perm ω) S))) := by
+    apply finiteMean_pos
+    intro ω
+    positivity
+  have hlog := Real.log_le_log hpos hm
+  rw [Real.log_exp] at hlog
+  dsimp [c] at hlog
+  linarith
+
+
+end RoutingNetwork
+
+variable {α B : Type*} [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] {d : ℕ}
+
+def parallelPerm (ω : B × SwitchIndex d → Bool) : Equiv.Perm (B × Card d) where
+  toFun x := (x.1,butterflyPerm d (decodeButterfly d (fun i => ω (x.1,i))) x.2)
+  invFun y := (y.1,(butterflyPerm d (decodeButterfly d (fun i => ω (y.1,i)))).symm y.2)
+  left_inv x := by simp
+  right_inv y := by simp
+
+def relabeledPerm (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (ω : B × SwitchIndex d → Bool) : Equiv.Perm α :=
+  (pre.trans (parallelPerm ω)).trans post
+
+noncomputable def parallelPath (pre : α ≃ B × Card d)
+    (ω : B × SwitchIndex d → Bool) (x : α) : Finset (B × SwitchIndex d) :=
+  (routeDomain d (pre x).2
+    (butterflyPerm d (decodeButterfly d (fun i => ω ((pre x).1,i))) (pre x).2)).image
+      (fun i => ((pre x).1,i))
+
+def parallelUsers (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
+    (i : B × SwitchIndex d) : α × α :=
+  let u := switchUsers d (decodeButterfly d (fun j => ω (i.1,j))) i.2
+  (pre.symm (i.1,u.1),pre.symm (i.1,u.2))
+
+omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
+lemma parallelUsers_ne (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
+    (i : B × SwitchIndex d) : (parallelUsers pre ω i).1 ≠ (parallelUsers pre ω i).2 := by
+  intro h
+  exact switchUsers_ne d _ _ (congrArg Prod.snd (pre.symm.injective h))
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma mem_parallelPath (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
+    (x : α) (i : B × SwitchIndex d) :
+    i ∈ parallelPath pre ω x ↔ (parallelUsers pre ω i).1 = x ∨ (parallelUsers pre ω i).2 = x := by
+  classical
+  obtain ⟨b,j⟩ := i
+  simp only [parallelPath,Finset.mem_image,Prod.mk.injEq]
+  constructor
+  · rintro ⟨k,hk,hb,rfl⟩
+    subst b
+    rcases (mem_routeDomain_iff_user d _ _ _).mp hk with h | h
+    · left
+      apply pre.injective
+      change pre (pre.symm ((pre x).1,_)) = pre x
+      rw [Equiv.apply_symm_apply]
+      exact Prod.ext rfl h
+    · right
+      apply pre.injective
+      change pre (pre.symm ((pre x).1,_)) = pre x
+      rw [Equiv.apply_symm_apply]
+      exact Prod.ext rfl h
+  · intro h
+    have hh : (b,(switchUsers d (decodeButterfly d (fun j => ω (b,j))) j).1) = pre x ∨
+        (b,(switchUsers d (decodeButterfly d (fun j => ω (b,j))) j).2) = pre x := by
+      rcases h with h | h
+      · exact Or.inl (by simpa only [parallelUsers,Equiv.apply_symm_apply] using congrArg pre h)
+      · exact Or.inr (by simpa only [parallelUsers,Equiv.apply_symm_apply] using congrArg pre h)
+    have hb : b = (pre x).1 := hh.elim (congrArg Prod.fst) (congrArg Prod.fst)
+    subst b
+    refine ⟨j,?_,rfl,rfl⟩
+    apply (mem_routeDomain_iff_user d _ _ _).mpr
+    exact hh.elim (fun h => Or.inl (congrArg Prod.snd h)) (fun h => Or.inr (congrArg Prod.snd h))
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma card_parallelPath (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool) (x : α) :
+    (parallelPath pre ω x).card = d := by
+  classical
+  rw [parallelPath,Finset.card_image_of_injective _ (fun _ _ h => Prod.mk.inj h |>.2),card_routeDomain]
+
+noncomputable def blockMarks (pre : α ≃ B × Card d) (S : Finset α) (b : B) : Finset (Card d) :=
+  Finset.univ.filter (fun x => pre.symm (b,x) ∈ S)
+
+omit [Fintype α] [Fintype B] [DecidableEq B] in
+@[simp] lemma mem_blockMarks (pre : α ≃ B × Card d) (S : Finset α) (b : B) (x : Card d) :
+    x ∈ blockMarks pre S b ↔ pre.symm (b,x) ∈ S := by
+  classical
+  simp [blockMarks]
+
+omit [DecidableEq B] in
+lemma sum_card_blockMarks (pre : α ≃ B × Card d) (S : Finset α) :
+    ∑ b : B, (blockMarks pre S b).card = S.card := by
+  classical
+  calc
+    _ = ∑ b : B, ∑ x : Card d, if pre.symm (b,x) ∈ S then 1 else 0 := by
+      simp only [blockMarks,Finset.card_filter]
+    _ = ∑ u : B × Card d, if pre.symm u ∈ S then 1 else 0 := (Fintype.sum_prod_type (fun u : B × Card d => if pre.symm u ∈ S then (1:ℕ) else 0)).symm
+    _ = ∑ x : α, if x ∈ S then 1 else 0 := Equiv.sum_comp pre.symm (fun x : α => if x ∈ S then (1:ℕ) else 0)
+    _ = _ := by simp
+
+omit [DecidableEq B] in
+lemma card_blockMarks_le (pre : α ≃ B × Card d) (S : Finset α) (b : B) :
+    (blockMarks pre S b).card ≤ S.card := by
+  rw [←sum_card_blockMarks pre S]
+  exact Finset.single_le_sum (f := fun b : B => (blockMarks pre S b).card) (fun _ _ => Nat.zero_le _) (Finset.mem_univ b)
+
+end Thorp
+
+end ThorpNine.Dense
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Dense/ProjectCycle.lean b/lean/OAI/Probability/ThorpRouting/Dense/ProjectCycle.lean
index 65249176e1f4d21d1ffa2f481a5a2ea7ba8e0f5c..fbcb11b37419d61386fbc48e870c9207807c5167 100644
--- a/lean/OAI/Probability/ThorpRouting/Dense/ProjectCycle.lean
+++ b/lean/OAI/Probability/ThorpRouting/Dense/ProjectCycle.lean
@@ -1,811 +1,811 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Dense.ParallelFollow
-
-namespace OAI
-
-namespace ThorpNine.Dense
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-namespace CycleColoring
-variable {α : Type*} [Fintype α] [DecidableEq α]
-omit [Fintype α] in
-lemma bipartitionFlip_injective (p : Equiv.Perm α) (S : Finset α)
-    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x)) :
-    Function.Injective (bipartitionFlip p S χ hχ) := by
-  classical
-  intro f g he
-  apply Subtype.ext
-  funext x
-  by_cases hx : x ∈ S
-  · have hh := congrFun (congrArg Subtype.val he) x
-    change (if x ∈ S then Bool.xor (f.val x) (χ x) else false) =
-      (if x ∈ S then Bool.xor (g.val x) (χ x) else false) at hh
-    simp only [ite_eq_left hx] at hh
-    cases hfx : f.val x <;> cases hgx : g.val x <;> cases hcx : χ x <;> simp_all
-  · exact (f.property.1 x hx).trans (g.property.1 x hx).symm
-
-lemma card_opposite_le (p : Equiv.Perm α) (S : Finset α)
-    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x)) :
-    Fintype.card (OppositeColoring p S) ≤
-      2 ^ ((boundary p S).card + Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
-  exact (Fintype.card_le_of_injective _ (bipartitionFlip_injective p S χ hχ)).trans
-    (card_coloring_le p S)
-
-omit [Fintype α] [DecidableEq α] in
-lemma alternatingPerm_flip (p : Equiv.Perm α) (x : Bool × α) :
-    (alternatingPerm p x).1 = !x.1 := by
-  rcases x with ⟨b,x⟩
-  cases b <;> rfl
-
-omit [Fintype α] [DecidableEq α] in
-lemma alternating_step_sameCycle (p : Equiv.Perm α) (x : Bool × α) :
-    p.SameCycle x.2 (alternatingPerm p x).2 := by
-  rcases x with ⟨b,x⟩
-  cases b
-  · exact Equiv.Perm.SameCycle.refl p x
-  · exact (Equiv.Perm.SameCycle.refl p x).apply_right
-
-omit [DecidableEq α] in
-lemma alternating_sameCycle_snd (p : Equiv.Perm α) {x y : Bool × α}
-    (h : (alternatingPerm p).SameCycle x y) : p.SameCycle x.2 y.2 := by
-  obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
-  have hp (k : ℕ) : p.SameCycle x.2 (((alternatingPerm p)^k) x).2 := by
-    induction k with
-    | zero => exact Equiv.Perm.SameCycle.refl p x.2
-    | succ k ih =>
-      rw [pow_succ',Equiv.Perm.mul_apply]
-      exact ih.trans (alternating_step_sameCycle p _)
-  exact hn ▸ hp n
-
-omit [DecidableEq α] in
-lemma alternating_zero_sameCycle (p : Equiv.Perm α) {x y : α}
-    (h : p.SameCycle x y) : (alternatingPerm p).SameCycle (false,x) (false,y) := by
-  obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
-  have hp (k : ℕ) : (alternatingPerm p).SameCycle (false,x) (false,(p^k) x) := by
-    induction k with
-    | zero => exact Equiv.Perm.SameCycle.refl _ _
-    | succ k ih =>
-      have hs := ih.apply_right.apply_right
-      simpa only [pow_succ',Equiv.Perm.mul_apply,alternatingPerm,Equiv.coe_fn_mk,
-        Bool.false_eq_true,↓reduceIte] using hs
-  simpa only [hn] using hp n
-
-omit [Fintype α] [DecidableEq α] in
-lemma alternating_sameCycle_zero (p : Equiv.Perm α) (x : Bool × α) :
-    (alternatingPerm p).SameCycle x (false,x.2) := by
-  rcases x with ⟨b,x⟩
-  cases b
-  · exact Equiv.Perm.SameCycle.refl _ _
-  · exact ((Equiv.Perm.SameCycle.refl (alternatingPerm p) (false,x)).apply_right).symm
-
-omit [DecidableEq α] in
-lemma alternating_sameCycle_iff (p : Equiv.Perm α) (x y : Bool × α) :
-    (alternatingPerm p).SameCycle x y ↔ p.SameCycle x.2 y.2 := by
-  constructor
-  · exact alternating_sameCycle_snd p
-  · intro h
-    exact (alternating_sameCycle_zero p x).trans
-      ((alternating_zero_sameCycle p h).trans (alternating_sameCycle_zero p y).symm)
-
-lemma orbit_alternating (p : Equiv.Perm α) (x : Bool × α) :
-    orbitSet (alternatingPerm p) x = childJoin (orbitSet p x.2) (orbitSet p x.2) := by
-  ext ⟨b,y⟩
-  simp only [mem_orbitSet,alternating_sameCycle_iff,mem_childJoin,ite_self]
-
-omit [Fintype α] in
-lemma boolFiber_childJoin (A B : Finset α) (b : Bool) :
-    boolFiber (childJoin A B) b = if b then B else A := by
-  ext x
-  cases b <;> simp only [mem_boolFiber,mem_childJoin,Bool.false_eq_true,↓reduceIte]
-
-noncomputable def projectCycle (p : Equiv.Perm α) (A B : Finset α)
-    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) :
-    RoutingNetwork.SelectedCycle p A := by
-  refine ⟨boolFiber C.val false,?_,?_⟩
-  · obtain ⟨x,hx⟩ := C.property.1
-    refine ⟨x.2,?_⟩
-    rw [hx,orbit_alternating,boolFiber_childJoin]
-    rfl
-  · intro x hx
-    exact (mem_childJoin A B false x).mp (C.property.2 ((mem_boolFiber _ _ _).mp hx))
-
-lemma selected_alternating_eq (p : Equiv.Perm α) (A B : Finset α)
-    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) :
-    C.val = childJoin (projectCycle p A B C).val (projectCycle p A B C).val := by
-  change C.val = childJoin (boolFiber C.val false) (boolFiber C.val false)
-  obtain ⟨x,hx⟩ := C.property.1
-  rw [hx,orbit_alternating,boolFiber_childJoin]
-  rfl
-
-lemma projectCycle_injective (p : Equiv.Perm α) (A B : Finset α) :
-    Function.Injective (projectCycle p A B) := by
-  intro C D he
-  apply Subtype.ext
-  rw [selected_alternating_eq p A B C,selected_alternating_eq p A B D,he]
-
-lemma card_alternating_cycles_le (p : Equiv.Perm α) (A B : Finset α) :
-    Fintype.card (RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) ≤
-      Fintype.card (RoutingNetwork.SelectedCycle p A) :=
-  Fintype.card_le_of_injective _ (projectCycle_injective p A B)
-
-end CycleColoring
-
-section PairAssignmentCount
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-omit [Fintype α] in
-lemma mem_image_perm_iff (p : Equiv.Perm α) (A : Finset α) (y : α) :
-    y ∈ A.image p ↔ p.symm y ∈ A := by
-  constructor
-  · intro h
-    obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp h
-    simpa only [←hxy,Equiv.symm_apply_apply] using hx
-  · intro h
-    exact Finset.mem_image.mpr ⟨p.symm y,h,p.apply_symm_apply y⟩
-
-omit [Fintype α] in
-lemma card_childJoin (A B : Finset α) : (childJoin A B).card = A.card+B.card := by
-  have hd : Disjoint (A.image (fun x => (false,x))) (B.image (fun x => (true,x))) := by
-    apply Finset.disjoint_left.mpr
-    rintro x hx hy
-    obtain ⟨a,ha,rfl⟩ := Finset.mem_image.mp hx
-    obtain ⟨b,hb,hba⟩ := Finset.mem_image.mp hy
-    exact Bool.false_ne_true (congrArg Prod.fst hba).symm
-  have hi (b : Bool) : Function.Injective (fun x : α => (b,x)) :=
-    fun _ _ h => congrArg Prod.snd h
-  rw [childJoin,Finset.card_union_of_disjoint hd,
-    Finset.card_image_of_injective A (hi false),Finset.card_image_of_injective B (hi true)]
-
-omit [Fintype α] in
-lemma alternating_internal (p : Equiv.Perm α) (A B : Finset α) :
-    ((childJoin A B).filter (fun x => (CycleColoring.alternatingPerm p).symm x ∈ childJoin A B)) =
-      childJoin (A ∩ B.image p) (B ∩ A) := by
-  ext ⟨b,x⟩
-  rw [Finset.mem_filter]
-  cases b <;> simp only [mem_childJoin,CycleColoring.alternatingPerm,
-    Equiv.symm_mk,Equiv.coe_fn_mk,Bool.false_eq_true,↓reduceIte,Finset.mem_inter,mem_image_perm_iff]
-
-omit [Fintype α] in
-lemma alternating_boundary_card (p : Equiv.Perm α) (A B : Finset α) :
-    (CycleColoring.boundary (CycleColoring.alternatingPerm p) (childJoin A B)).card =
-      (A ∪ B).card + (A ∪ B.image p).card - (A.card+B.card) := by
-  rw [CycleColoring.boundary_card,alternating_internal,card_childJoin,card_childJoin]
-  have h₀ := Finset.card_union_add_card_inter A B
-  have h₁ := Finset.card_union_add_card_inter A (B.image p)
-  rw [Finset.card_image_of_injective _ p.injective] at h₁
-  have h₂ := Finset.inter_comm B A
-  rw [h₂]
-  omega
-
-lemma compatible_assignment_bound (p : Equiv.Perm α) (A B : Finset α) :
-    Fintype.card (CycleColoring.OppositeColoring (CycleColoring.alternatingPerm p) (childJoin A B)) ≤
-      2 ^ ((A ∪ B).card + (A ∪ B.image p).card - (A.card+B.card) +
-        Fintype.card (RoutingNetwork.SelectedCycle p A)) := by
-  have h := CycleColoring.card_opposite_le (CycleColoring.alternatingPerm p) (childJoin A B)
-    Prod.fst (CycleColoring.alternatingPerm_flip p)
-  rw [alternating_boundary_card] at h
-  exact h.trans (Nat.pow_le_pow_right (by norm_num)
-    (Nat.add_le_add_left (CycleColoring.card_alternating_cycles_le p A B) _))
-
-end PairAssignmentCount
-
-lemma finiteMean_nonneg {Ω : Type*} [Fintype Ω] {f : Ω → ℝ}
-    (h : ∀ ω, 0 ≤ f ω) : 0 ≤ finiteMean f := by
-  exact div_nonneg (Finset.sum_nonneg (fun ω _ => h ω)) (Nat.cast_nonneg _)
-
-lemma boolFiber_headTail (d : ℕ) (S : Finset (Card (d+1))) (b : Bool) :
-    boolFiber (S.image (headTailEquiv d)) b = childMarks S b := by
-  classical
-  ext x
-  rw [mem_boolFiber,mem_childMarks]
-  constructor
-  · intro h
-    obtain ⟨y,hy,he⟩ := Finset.mem_image.mp h
-    have hh := congrArg (headTailEquiv d).symm he
-    rw [Equiv.symm_apply_apply] at hh
-    change y = Fin.cons b x at hh
-    exact hh ▸ hy
-  · intro h
-    exact Finset.mem_image.mpr ⟨Fin.cons b x,h,by rfl⟩
-
-lemma prod_childMarks (d : ℕ) (S : Finset (Card (d+1))) (f : Card (d+1) → ℝ) :
-    (∏ x ∈ S, f x) =
-      (∏ x ∈ childMarks S false, f (Fin.cons false x)) *
-      ∏ x ∈ childMarks S true, f (Fin.cons true x) := by
-  classical
-  have h := prod_boolFibers (S.image (headTailEquiv d)) (fun x => f ((headTailEquiv d).symm x))
-  rw [Finset.prod_image (headTailEquiv d).injective.injOn,
-    boolFiber_headTail,boolFiber_headTail] at h
-  simp only [Equiv.symm_apply_apply] at h
-  exact h
-
-noncomputable def headAverage {d : ℕ} (p : Card (d+1) → ℝ) (y : Card d) : ℝ :=
-  (p (Fin.cons false y)+p (Fin.cons true y))/2
-
-lemma finiteMean_headAverage (d : ℕ) (p : Card (d+1) → ℝ) :
-    finiteMean (headAverage p) = finiteMean p := by
-  rw [finiteMean_equiv (headTailEquiv d),finiteMean_prod,finiteMean_comm]
-  apply finiteMean_congr
-  intro y
-  rw [finiteMean_bool]
-  rfl
-
-lemma pairSwitch_product_bound (d : ℕ) (S : Finset (Card (d+1)))
-    (p : Card (d+1) → ℝ) (hp : ∀ x, 0 ≤ p x) :
-    finiteMean (fun ξ : Card d → Bool => ∏ x ∈ S, p (pairSwitch ξ x)) ≤
-      ∏ x ∈ S, headAverage p (Fin.tail x) := by
-  classical
-  have h : CylinderBound (fun T => ∏ x ∈ T, p ((headTailEquiv d).symm x))
-      (fun x => p ((headTailEquiv d).symm x)) := fun _ => le_rfl
-  have hh := h.fair_parallel (fun x => hp _) (S.image (headTailEquiv d))
-  simp only [Finset.prod_image (pairLayer _).injective.injOn,
-    Finset.prod_image (headTailEquiv d).injective.injOn] at hh
-  convert hh using 1
-  · apply finiteMean_congr
-    intro ξ
-    apply Finset.prod_congr rfl
-    intro x _
-    rfl
-  · apply Finset.prod_congr rfl
-    intro x _
-    change (p (Fin.cons false (Fin.tail x))+p (Fin.cons true (Fin.tail x)))/2 =
-      (p (Fin.cons (x 0) (Fin.tail x))+p (Fin.cons (!(x 0)) (Fin.tail x)))/2
-    cases x 0
-    · rfl
-    · simp only [Bool.not_true]
-      ring
-
-lemma butterflyPerm_step (d : ℕ)
-    (c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (ξ : Card d → Bool) :
-    butterflyPerm (d+1) (decodeButterfly (d+1) ((coinStepEquiv d).symm (c,ξ))) =
-    pairSwitch ξ * childLift (fun b => butterflyPerm d (decodeButterfly d (if b then c.2 else c.1))) := by
-  change pairSwitch (fun y => (coinStepEquiv d).symm (c,ξ) (Sum.inl y)) *
-    childLift (fun b => butterflyPerm d (decodeButterfly d
-      (fun i => (coinStepEquiv d).symm (c,ξ) (Sum.inr (b,i))))) = _
-  apply congrArg₂ (· * ·)
-  · rfl
-  · apply congrArg childLift
-    funext b
-    apply congrArg (fun ω => butterflyPerm d (decodeButterfly d ω))
-    funext i
-    cases b <;> rfl
-
-lemma butterfly_product_bound (d : ℕ) (S : Finset (Card d))
-    (p : Card d → ℝ) (hp : ∀ x, 0 ≤ p x) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      ∏ x ∈ S, p (butterflyPerm d (decodeButterfly d ω) x)) ≤ (finiteMean p)^S.card := by
-  classical
-  induction d with
-  | zero =>
-    have he : p = fun _ => p (fun i => Fin.elim0 i) := by
-      funext x
-      exact congrArg p (Subsingleton.elim _ _)
-    rw [he]
-    simp only [Finset.prod_const,finiteMean_const,le_refl]
-  | succ d ih =>
-    let g : Card d → ℝ := headAverage p
-    have hg : ∀ x, 0 ≤ g x := fun x => div_nonneg (add_nonneg (hp _) (hp _)) (by norm_num)
-    rw [finiteMean_equiv (coinStepEquiv d),finiteMean_prod]
-    calc
-      _ ≤ finiteMean (fun c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool) =>
-        (∏ x ∈ childMarks S false, g (butterflyPerm d (decodeButterfly d c.1) x)) *
-        ∏ x ∈ childMarks S true, g (butterflyPerm d (decodeButterfly d c.2) x)) := by
-        apply finiteMean_mono
-        intro c
-        let f : Equiv.Perm (Card (d+1)) := childLift (fun b =>
-          butterflyPerm d (decodeButterfly d (if b then c.2 else c.1)))
-        have hh := pairSwitch_product_bound d (S.image f) p hp
-        simp only [Finset.prod_image f.injective.injOn] at hh
-        convert hh using 1
-        · apply finiteMean_congr
-          intro ξ
-          apply Finset.prod_congr rfl
-          intro x _
-          rw [butterflyPerm_step]
-          rfl
-        · rw [prod_childMarks]
-          simp only [f,childLift,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
-            Bool.false_eq_true,↓reduceIte]
-          rfl
-      _ = (finiteMean (fun c : SwitchIndex d → Bool =>
-        ∏ x ∈ childMarks S false, g (butterflyPerm d (decodeButterfly d c) x))) *
-        finiteMean (fun c : SwitchIndex d → Bool =>
-        ∏ x ∈ childMarks S true, g (butterflyPerm d (decodeButterfly d c) x)) :=
-          finiteMean_prod_mul
-            (fun c : SwitchIndex d → Bool => ∏ x ∈ childMarks S false,
-              g (butterflyPerm d (decodeButterfly d c) x))
-            (fun c : SwitchIndex d → Bool => ∏ x ∈ childMarks S true,
-              g (butterflyPerm d (decodeButterfly d c) x))
-      _ ≤ (finiteMean g)^(childMarks S false).card * (finiteMean g)^(childMarks S true).card :=
-        mul_le_mul (ih (childMarks S false) g hg) (ih (childMarks S true) g hg)
-          (finiteMean_nonneg fun c => Finset.prod_nonneg (fun x _ => hg _))
-          (pow_nonneg (finiteMean_nonneg hg) _)
-      _ = (finiteMean p)^S.card := by
-        rw [←pow_add,card_childMarks_add,finiteMean_headAverage]
-
-noncomputable def inverseButterflyMarks (d : ℕ) (A : Finset (Card d))
-    (ω : SwitchIndex d → Bool) : Finset (Card d) :=
-  A.image (butterflyPerm d (decodeButterfly d ω)).symm
-
-lemma inverseButterflyMarks_inclusion (d : ℕ) (A S : Finset (Card d)) :
-    inclusionMoment (inverseButterflyMarks d A) S ≤
-      ((A.card : ℝ)/(2:ℝ)^d)^S.card := by
-  classical
-  let p : Card d → ℝ := fun x => if x ∈ A then 1 else 0
-  have hm : finiteMean p = (A.card : ℝ)/(2:ℝ)^d := by
-    simp only [finiteMean,p,card_positions,Nat.cast_pow,Nat.cast_ofNat]
-    congr 1
-    simp
-  have he : inclusionMoment (inverseButterflyMarks d A) S =
-      finiteMean (fun ω : SwitchIndex d → Bool =>
-        ∏ x ∈ S, p (butterflyPerm d (decodeButterfly d ω) x)) := by
-    rw [inclusionMoment_eq]
-    apply finiteMean_congr
-    intro ω
-    simp only [inverseButterflyMarks,subset_image_equiv_iff,Equiv.symm_symm,indicator_subset_prod,
-      Finset.prod_image (butterflyPerm d (decodeButterfly d ω)).injective.injOn,p]
-  rw [he]
-  simpa only [hm] using butterfly_product_bound d S p (fun x => by dsimp [p]; split_ifs <;> norm_num)
-
-namespace PairRouting
-
-section
-variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
-
-noncomputable def occupied (x : ι → Bool × α) : Finset α :=
-  Finset.univ.image (fun i => (x i).2)
-
-omit [Fintype α] in
-lemma mem_occupied (x : ι → Bool × α) (y : α) : y ∈ occupied x ↔ ∃ i, (x i).2 = y := by
-  simp [occupied]
-
-noncomputable def chosenCoin (x : ι → Bool × α) (c : ι → Bool) (y : α) : Bool := by
-  classical
-  exact if h : ∃ i, (x i).2 = y then required x c (Classical.choose h) else false
-
-omit [Fintype α] in
-lemma chosenCoin_apply (x : ι → Bool × α) (c : ι → Bool) (hc : Compatible x c) (i : ι) :
-    chosenCoin x c (x i).2 = required x c i := by
-  classical
-  simp only [chosenCoin,dite_eq_left (show ∃ j, (x j).2 = (x i).2 from ⟨i,rfl⟩)]
-  exact hc _ i (Classical.choose_spec (show ∃ j, (x j).2 = (x i).2 from ⟨i,rfl⟩))
-
-omit [Fintype α] in
-lemma colors_eq_iff (x : ι → Bool × α) (c : ι → Bool) (hc : Compatible x c) (ξ : α → Bool) :
-    colors x ξ = c ↔ ∀ y ∈ occupied x, ξ y = chosenCoin x c y := by
-  constructor
-  · intro he y hy
-    obtain ⟨i,rfl⟩ := (mem_occupied x y).mp hy
-    rw [chosenCoin_apply x c hc i]
-    rw [←he,required_colors]
-  · intro he
-    funext i
-    apply (xor_eq_iff _ _ _).mpr
-    exact (he _ ((mem_occupied x _).mpr ⟨i,rfl⟩)).trans (chosenCoin_apply x c hc i)
-
-noncomputable instance (x : ι → Bool × α) (c : ι → Bool) : Decidable (Compatible x c) :=
-  Classical.propDecidable _
-
-lemma colors_probability (x : ι → Bool × α) (c : ι → Bool) :
-    finiteMean (fun ξ : α → Bool => if colors x ξ = c then (1 : ℝ) else 0) =
-      if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0 := by
-  classical
-  by_cases hc : Compatible x c
-  · rw [ite_eq_left hc]
-    have he := fun ξ => colors_eq_iff x c hc ξ
-    simp only [he]
-    exact mean_coinCylinder (occupied x) (chosenCoin x c)
-  · rw [ite_eq_right hc]
-    have he : ∀ ξ, colors x ξ ≠ c := fun ξ hh => hc (hh ▸ colors_compatible x ξ)
-    simp only [he,ite_false,finiteMean_const]
-
-
-def lift (p : Bool → Equiv.Perm α) : Equiv.Perm (Bool × α) where
-  toFun x := (x.1,p x.1 x.2)
-  invFun x := (x.1,(p x.1).symm x.2)
-  left_inv x := by simp
-  right_inv x := by simp
-
-def sandwich (ξ η : α → Bool) (p : Bool → Equiv.Perm α) : Equiv.Perm (Bool × α) :=
-  pairLayer η * lift p * pairLayer ξ
-
-def ChildMatches (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) : Prop :=
-  ∀ i, p (c i) (x i).2 = (y i).2
-
-noncomputable instance (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
-    Decidable (ChildMatches p x y c) := Classical.propDecidable _
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma sandwich_matches_iff (ξ η : α → Bool) (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) :
-    (∀ i, sandwich ξ η p (x i) = y i) ↔
-      colors x ξ = colors y η ∧ ChildMatches p x y (colors x ξ) := by
-  have he (i : ι) : sandwich ξ η p (x i) = y i ↔
-      colors x ξ i = colors y η i ∧ p (colors x ξ i) (x i).2 = (y i).2 := by
-    change pairLayer η (lift p (pairLayer ξ (x i))) = y i ↔ _
-    rw [←Equiv.eq_symm_apply]
-    change (colors x ξ i,p (colors x ξ i) (x i).2) = (colors y η i,(y i).2) ↔ _
-    exact ⟨Prod.mk.inj,fun h => Prod.ext h.1 h.2⟩
-  simp only [he,forall_and,←funext_iff,ChildMatches]
-
-lemma sandwich_probability [DecidableEq ι] (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) :
-    finiteMean (fun coins : (α → Bool) × (α → Bool) =>
-      if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then (1:ℝ) else 0) =
-    ∑ c : ι → Bool,
-      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
-      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) *
-      (if ChildMatches p x y c then 1 else 0) := by
-  classical
-  have he (coins : (α → Bool) × (α → Bool)) :
-      (if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then (1:ℝ) else 0) =
-      ∑ c : ι → Bool, (if colors x coins.1 = c then (1:ℝ) else 0) *
-        (if colors y coins.2 = c then 1 else 0) * (if ChildMatches p x y c then 1 else 0) := by
-    rw [Finset.sum_eq_single (colors x coins.1)]
-    · by_cases hc : colors x coins.1 = colors y coins.2
-      · simp only [sandwich_matches_iff,hc,true_and,↓reduceIte,one_mul]
-      · simp only [sandwich_matches_iff,hc,false_and,↓reduceIte,one_mul,Ne.symm hc,zero_mul]
-    · intro c _ hc
-      simp only [Ne.symm hc,ite_false,zero_mul]
-    · simp
-  simp_rw [he]
-  rw [finiteMean_sum]
-  apply Finset.sum_congr rfl
-  intro c _
-  rw [finiteMean_mul_const]
-  rw [finiteMean_prod_mul
-    (fun ξ : α → Bool => if colors x ξ = c then (1:ℝ) else 0)
-    (fun η : α → Bool => if colors y η = c then (1:ℝ) else 0),
-    colors_probability,colors_probability]
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma childMatches_split (p₀ p₁ : Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
-    ChildMatches (fun b => if b then p₁ else p₀) x y c ↔
-      (∀ i : {i // c i = false}, p₀ (x i.val).2 = (y i.val).2) ∧
-      (∀ i : {i // c i = true}, p₁ (x i.val).2 = (y i.val).2) := by
-  constructor
-  · intro h
-    constructor
-    · intro i
-      simpa only [i.property,Bool.false_eq_true,↓reduceIte] using h i.val
-    · intro i
-      simpa only [i.property,↓reduceIte] using h i.val
-  · rintro ⟨h₀,h₁⟩ i
-    cases hi : c i
-    · simpa only [hi,Bool.false_eq_true,↓reduceIte] using h₀ ⟨i,hi⟩
-    · simpa only [hi,↓reduceIte] using h₁ ⟨i,hi⟩
-
-noncomputable def tupleProbability {Ω β : Type*} [Fintype Ω] [DecidableEq β]
-    (P : Ω → Equiv.Perm β) (x y : ι → β) : ℝ := by
-  classical
-  exact finiteMean (fun ω => if ∀ i, P ω (x i) = y i then 1 else 0)
-
-omit [Fintype α] in
-lemma childMatches_mean {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
-    (P : Ω → Equiv.Perm α) (Q : Ω' → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
-    finiteMean (fun ω : Ω × Ω' =>
-      if ChildMatches (fun b => if b then Q ω.2 else P ω.1) x y c then (1:ℝ) else 0) =
-      tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
-        (fun i : {i // c i = false} => (y i.val).2) *
-      tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
-        (fun i : {i // c i = true} => (y i.val).2) := by
-  classical
-  let f : Ω → ℝ := fun ω => if ∀ i : {i // c i = false}, P ω (x i.val).2 = (y i.val).2 then 1 else 0
-  let g : Ω' → ℝ := fun ω => if ∀ i : {i // c i = true}, Q ω (x i.val).2 = (y i.val).2 then 1 else 0
-  have he (ω : Ω × Ω') :
-      (if ChildMatches (fun b => if b then Q ω.2 else P ω.1) x y c then (1:ℝ) else 0) = f ω.1*g ω.2 := by
-    simp only [childMatches_split,f,g]
-    split_ifs <;> simp_all
-  simp_rw [he]
-  exact finiteMean_prod_mul f g
-
-lemma sandwich_mean [DecidableEq ι] {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
-    (P : Ω → Equiv.Perm α) (Q : Ω' → Equiv.Perm α) (x y : ι → Bool × α) :
-    finiteMean (fun ω : Ω × Ω' => finiteMean (fun coins : (α → Bool) × (α → Bool) =>
-      if ∀ i, sandwich coins.1 coins.2 (fun b => if b then Q ω.2 else P ω.1) (x i) = y i
-      then (1:ℝ) else 0)) =
-    ∑ c : ι → Bool,
-      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
-      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) *
-      (tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
-        (fun i : {i // c i = false} => (y i.val).2) *
-      tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
-        (fun i : {i // c i = true} => (y i.val).2)) := by
-  classical
-  simp_rw [sandwich_probability]
-  rw [finiteMean_sum]
-  apply Finset.sum_congr rfl
-  intro c _
-  simp only [mul_comm ((if Compatible x c then _ else _) * _),finiteMean_mul_const,
-    childMatches_mean]
-
-
-abbrev BothAssignments (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :=
-  {c : ι → Bool // Compatible e c ∧ Compatible (fun i => lift p (e i)) c}
-
-noncomputable instance (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    Fintype (BothAssignments e p) := Fintype.ofFinite _
-
-noncomputable def assignmentColoring (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α)
-    (c : BothAssignments e p) : CycleColoring.OppositeColoring
-      (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e) := by
-  classical
-  refine ⟨Function.extend e c.val (fun _ => false),?_,?_⟩
-  · intro z hz
-    apply Function.extend_apply'
-    rintro ⟨i,rfl⟩
-    exact hz (Finset.mem_image.mpr ⟨i,Finset.mem_univ _,rfl⟩)
-  · intro z hz hnext
-    obtain ⟨i,_,hi⟩ := Finset.mem_image.mp hz
-    obtain ⟨j,_,hj⟩ := Finset.mem_image.mp hnext
-    have hfi := e.injective.extend_apply c.val (fun _ => false) i
-    have hfj := e.injective.extend_apply c.val (fun _ => false) j
-    rw [hi] at hfi
-    rw [hj] at hfj
-    rw [hfi,hfj]
-    rcases z with ⟨b,a⟩
-    cases b
-    · have hj' : e j = (true,a) := hj
-      have ht : (e i).2 = (e j).2 := by rw [hi,hj']
-      have hh := c.property.1 i j ht
-      simp only [required,hi,hj'] at hh
-      revert hh
-      generalize c.val i = bi
-      generalize c.val j = bj
-      cases bi <;> cases bj <;> decide
-    · have hj' : e j = (false,alternatingProjection p a) := hj
-      have ht : (lift p (e i)).2 = (lift p (e j)).2 := by
-        simp only [hi,hj',lift,Equiv.coe_fn_mk,alternatingProjection,Equiv.Perm.coe_mul,
-          Function.comp_apply,Equiv.apply_symm_apply]
-      have hh := c.property.2 i j ht
-      simp only [required,hi,hj',lift,Equiv.coe_fn_mk] at hh
-      revert hh
-      generalize c.val i = bi
-      generalize c.val j = bj
-      cases bi <;> cases bj <;> decide
-
-omit [Fintype α] in
-lemma assignmentColoring_injective (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    Function.Injective (assignmentColoring e p) := by
-  intro c c' he
-  apply Subtype.ext
-  funext i
-  have hh := congrFun (congrArg Subtype.val he) (e i)
-  change Function.extend e c.val (fun _ => false) (e i) =
-    Function.extend e c'.val (fun _ => false) (e i) at hh
-  simpa only [e.injective.extend_apply] using hh
-
-lemma bothAssignments_card (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    Fintype.card (BothAssignments e p) ≤
-      2 ^ ((boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true).card +
-        (boolFiber (labelSet e) false ∪ (boolFiber (labelSet e) true).image
-          (alternatingProjection p)).card -
-        ((boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card) +
-        Fintype.card (RoutingNetwork.SelectedCycle (alternatingProjection p) (boolFiber (labelSet e) false))) := by
-  have h := compatible_assignment_bound (alternatingProjection p)
-    (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
-  rw [childJoin_boolFiber] at h
-  exact (Fintype.card_le_of_injective _ (assignmentColoring_injective e p)).trans h
-
-omit [Fintype α] in
-lemma occupied_labelSet (e : ι ↪ Bool × α) : occupied e = (labelSet e).image Prod.snd := by
-  classical
-  simp only [occupied,labelSet,Finset.image_image,Function.comp_def]
-
-omit [Fintype ι] [Fintype α] in
-lemma image_snd_childJoin (A B : Finset α) : (childJoin A B).image Prod.snd = A ∪ B := by
-  classical
-  simp only [childJoin,Finset.image_union,Finset.image_image,Function.comp_def,Finset.image_id']
-
-omit [Fintype ι] [Fintype α] in
-lemma image_lift_childJoin (p : Bool → Equiv.Perm α) (A B : Finset α) :
-    (childJoin A B).image (fun x => (lift p x).2) = A.image (p false) ∪ B.image (p true) := by
-  classical
-  simp only [childJoin,Finset.image_union,Finset.image_image,Function.comp_def,lift,Equiv.coe_fn_mk]
-
-omit [Fintype α] in
-lemma occupied_lift_card (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    (occupied (fun i => lift p (e i))).card =
-      (boolFiber (labelSet e) false ∪ (boolFiber (labelSet e) true).image (alternatingProjection p)).card := by
-  classical
-  have he : occupied (fun i => lift p (e i)) = (labelSet e).image (fun x => (lift p x).2) := by
-    simp only [occupied,labelSet,Finset.image_image,Function.comp_def]
-  rw [he,←childJoin_boolFiber (labelSet e),image_lift_childJoin]
-  simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
-  let A := boolFiber (labelSet e) false
-  let B := boolFiber (labelSet e) true
-  change (A.image (p false) ∪ B.image (p true)).card = (A ∪ B.image (alternatingProjection p)).card
-  have hh : A.image (p false) ∪ B.image (p true) =
-      (A ∪ B.image (alternatingProjection p)).image (p false) := by
-    simp only [Finset.image_union,Finset.image_image,Function.comp_def,
-      alternatingProjection,Equiv.Perm.coe_mul,Equiv.apply_symm_apply]
-  rw [hh,Finset.card_image_of_injective _ (p false).injective]
-
-lemma bothAssignments_occupied_bound (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    Fintype.card (BothAssignments e p) ≤
-      2 ^ ((occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι +
-        Fintype.card (RoutingNetwork.SelectedCycle (alternatingProjection p) (boolFiber (labelSet e) false))) := by
-  have he : (occupied e).card =
-      (boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true).card := by
-    rw [occupied_labelSet,←childJoin_boolFiber (labelSet e),image_snd_childJoin]
-    simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
-  have hc : (boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card = Fintype.card ι := by
-    rw [←card_childJoin,childJoin_boolFiber,labelSet,Finset.card_image_of_injective _ e.injective,
-      Finset.card_univ]
-  simpa only [he,occupied_lift_card,←hc] using bothAssignments_card e p
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma compatible_pairLayer_iff (x : ι → Bool × α) (c : ι → Bool) (ξ : α → Bool) :
-    Compatible (fun i => pairLayer ξ (x i)) c ↔ Compatible x c := by
-  have he (i j : ι) (ht : (x i).2 = (x j).2) :
-      required (fun i => pairLayer ξ (x i)) c i = required (fun i => pairLayer ξ (x i)) c j ↔
-      required x c i = required x c j := by
-    simp only [required,pairLayer,Equiv.coe_fn_mk,ht]
-    cases (x i).1 <;> cases (x j).1 <;> cases c i <;> cases c j <;> cases ξ (x j).2 <;> decide
-  constructor
-  · intro h i j ht
-    exact (he i j ht).mp (h i j ht)
-  · intro h i j ht
-    exact (he i j ht).mpr (h i j ht)
-
-
-omit [Fintype α] in
-lemma occupied_eq_fibers (e : ι ↪ Bool × α) :
-    occupied e = boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true := by
-  rw [occupied_labelSet,←childJoin_boolFiber (labelSet e),image_snd_childJoin]
-  simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
-
-omit [Fintype α] in
-lemma card_labelSet (e : ι ↪ Bool × α) : (labelSet e).card = Fintype.card ι := by
-  classical
-  exact (Finset.card_image_of_injective _ e.injective).trans (Finset.card_univ)
-
-omit [Fintype α] in
-lemma occupied_card_double (e : ι ↪ Bool × α) : Fintype.card ι ≤ 2*(occupied e).card := by
-  have h₀ : (boolFiber (labelSet e) false).card ≤ (occupied e).card := by
-    rw [occupied_eq_fibers]
-    exact Finset.card_le_card Finset.subset_union_left
-  have h₁ : (boolFiber (labelSet e) true).card ≤ (occupied e).card := by
-    rw [occupied_eq_fibers]
-    exact Finset.card_le_card Finset.subset_union_right
-  have he : (boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card = Fintype.card ι := by
-    rw [←card_childJoin,childJoin_boolFiber,card_labelSet]
-  omega
-
-lemma bothAssignments_exact_bound (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    Fintype.card (BothAssignments e p) ≤
-      2 ^ ((occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι +
-        alternatingCycles e p) := by
-  have h := CycleColoring.card_opposite_le
-    (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e)
-    Prod.fst (CycleColoring.alternatingPerm_flip (alternatingProjection p))
-  have hb : (CycleColoring.boundary (CycleColoring.alternatingPerm (alternatingProjection p))
-      (labelSet e)).card =
-      (occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι := by
-    have hh := alternating_boundary_card (alternatingProjection p)
-      (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
-    rw [childJoin_boolFiber] at hh
-    rw [occupied_eq_fibers,occupied_lift_card]
-    have he := card_childJoin (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
-    rw [childJoin_boolFiber,card_labelSet] at he
-    simpa only [he] using hh
-  rw [hb] at h
-  exact (Fintype.card_le_of_injective _ (assignmentColoring_injective e p)).trans h
-
-abbrev Assignments (x y : ι → Bool × α) :=
-  {c : ι → Bool // Compatible x c ∧ Compatible y c}
-
-noncomputable instance (x y : ι → Bool × α) : Fintype (Assignments x y) := Fintype.ofFinite _
-
-def routedEmbedding (e : ι ↪ Bool × α) (ξ η : α → Bool) (p : Bool → Equiv.Perm α) :
-    ι ↪ Bool × α := e.trans (sandwich ξ η p).toEmbedding
-
-omit [Fintype α] in
-lemma occupied_pairLayer (x : ι → Bool × α) (ξ : α → Bool) :
-    occupied (fun i => pairLayer ξ (x i)) = occupied x := rfl
-
-noncomputable def assignmentsEquiv (e : ι ↪ Bool × α) (ξ η : α → Bool)
-    (p : Bool → Equiv.Perm α) :
-    Assignments e (routedEmbedding e ξ η p) ≃ BothAssignments (switchedEmbedding e ξ) p := by
-  apply Equiv.subtypeEquivRight
-  intro c
-  change Compatible e c ∧ Compatible (fun i => pairLayer η (lift p (pairLayer ξ (e i)))) c ↔ _
-  rw [compatible_pairLayer_iff]
-  exact and_congr (compatible_pairLayer_iff e c ξ).symm Iff.rfl
-
-lemma assignments_actual_bound (e : ι ↪ Bool × α) (ξ η : α → Bool) (p : Bool → Equiv.Perm α) :
-    Fintype.card (Assignments e (routedEmbedding e ξ η p)) ≤
-      2 ^ ((occupied e).card + (occupied (routedEmbedding e ξ η p)).card - Fintype.card ι +
-        alternatingCycles (switchedEmbedding e ξ) p) := by
-  rw [Fintype.card_congr (assignmentsEquiv e ξ η p)]
-  exact bothAssignments_exact_bound (switchedEmbedding e ξ) p
-
-omit [Fintype α] in
-lemma assignments_actual_exponent_nonneg (e : ι ↪ Bool × α) (ξ η : α → Bool)
-    (p : Bool → Equiv.Perm α) :
-    Fintype.card ι ≤ (occupied e).card + (occupied (routedEmbedding e ξ η p)).card := by
-  have h₀ := occupied_card_double e
-  have h₁ := occupied_card_double (routedEmbedding e ξ η p)
-  omega
-
-end
-variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α] [DecidableEq ι]
-
-noncomputable def compatibleSet (x y : ι → Bool × α) : Finset (ι → Bool) := by
-  classical
-  exact Finset.univ.filter (fun c => Compatible x c ∧ Compatible y c)
-
-omit [Fintype α] [DecidableEq α] in
-lemma mem_compatibleSet (x y : ι → Bool × α) (c : ι → Bool) :
-    c ∈ compatibleSet x y ↔ Compatible x c ∧ Compatible y c := by
-  classical
-  simp only [compatibleSet,Finset.mem_filter,Finset.mem_univ,true_and]
-
-omit [Fintype α] [DecidableEq α] in
-lemma card_compatibleSet (x y : ι → Bool × α) :
-    (compatibleSet x y).card = Fintype.card (Assignments x y) := by
-  classical
-  exact (Fintype.card_subtype _).symm
-
-noncomputable def outerWeight (x y : ι → Bool × α) : ℝ :=
-  1/(2:ℝ)^((occupied x).card+(occupied y).card)
-
-omit [Fintype α] [DecidableEq ι] in
-lemma outerWeight_pos (x y : ι → Bool × α) : 0 < outerWeight x y := by
-  unfold outerWeight
-  positivity
-
-omit [Fintype α] in
-lemma sum_colors_factor (x y : ι → Bool × α) (f : (ι → Bool) → ℝ) :
-    (∑ c : ι → Bool,
-      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
-      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) * f c) =
-      outerWeight x y * ∑ c ∈ compatibleSet x y, f c := by
-  classical
-  rw [compatibleSet,Finset.sum_filter,Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro c _
-  by_cases hx : Compatible x c <;> by_cases hy : Compatible y c
-  · simp only [hx,hy,true_and,↓reduceIte,outerWeight,pow_add,one_div_mul_one_div]
-  · simp only [hx,hy,and_false,↓reduceIte,mul_zero,zero_mul]
-  · simp only [hx,hy,false_and,↓reduceIte,mul_zero,zero_mul]
-  · simp only [hx,hy,false_and,↓reduceIte,mul_zero,zero_mul]
-
-lemma sandwich_weighted_probability (p : Bool → Equiv.Perm α) (x y : ι → Bool × α)
-    (W : (ι → Bool) → ℝ) :
-    finiteMean (fun coins : (α → Bool) × (α → Bool) =>
-      if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then W (colors x coins.1) else 0) =
-    outerWeight x y * ∑ c ∈ compatibleSet x y, if ChildMatches p x y c then W c else 0 := by
-  classical
-  have he (coins : (α → Bool) × (α → Bool)) :
-      (if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then W (colors x coins.1) else 0) =
-      ∑ c : ι → Bool, (if colors x coins.1 = c then (1:ℝ) else 0) *
-        (if colors y coins.2 = c then 1 else 0) * (if ChildMatches p x y c then W c else 0) := by
-    rw [Finset.sum_eq_single (colors x coins.1)]
-    · by_cases hc : colors x coins.1 = colors y coins.2
-      · simp only [sandwich_matches_iff,hc,true_and,↓reduceIte,one_mul]
-      · simp only [sandwich_matches_iff,hc,false_and,↓reduceIte,one_mul,Ne.symm hc,zero_mul]
-    · intro c _ hc
-      simp only [Ne.symm hc,ite_false,zero_mul]
-    · simp
-  simp_rw [he]
-  rw [finiteMean_sum,←sum_colors_factor]
-  apply Finset.sum_congr rfl
-  intro c _
-  rw [finiteMean_mul_const,finiteMean_prod_mul
-    (fun ξ : α → Bool => if colors x ξ = c then (1:ℝ) else 0)
-    (fun η : α → Bool => if colors y η = c then (1:ℝ) else 0),
-    colors_probability,colors_probability]
-
-omit [Fintype α] [DecidableEq ι] in
-lemma tupleProbability_nonneg {Ω β : Type*} [Fintype Ω] [DecidableEq β]
-    (P : Ω → Equiv.Perm β) (x y : ι → β) : 0 ≤ tupleProbability P x y := by
-  classical
-  exact finiteMean_nonneg (fun ω => by split_ifs <;> norm_num)
-
-end PairRouting
-end Thorp
-
-end ThorpNine.Dense
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Dense.ParallelFollow
+
+namespace OAI
+
+namespace ThorpNine.Dense
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+namespace CycleColoring
+variable {α : Type*} [Fintype α] [DecidableEq α]
+omit [Fintype α] in
+lemma bipartitionFlip_injective (p : Equiv.Perm α) (S : Finset α)
+    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x)) :
+    Function.Injective (bipartitionFlip p S χ hχ) := by
+  classical
+  intro f g he
+  apply Subtype.ext
+  funext x
+  by_cases hx : x ∈ S
+  · have hh := congrFun (congrArg Subtype.val he) x
+    change (if x ∈ S then Bool.xor (f.val x) (χ x) else false) =
+      (if x ∈ S then Bool.xor (g.val x) (χ x) else false) at hh
+    simp only [ite_eq_left hx] at hh
+    cases hfx : f.val x <;> cases hgx : g.val x <;> cases hcx : χ x <;> simp_all
+  · exact (f.property.1 x hx).trans (g.property.1 x hx).symm
+
+lemma card_opposite_le (p : Equiv.Perm α) (S : Finset α)
+    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x)) :
+    Fintype.card (OppositeColoring p S) ≤
+      2 ^ ((boundary p S).card + Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
+  exact (Fintype.card_le_of_injective _ (bipartitionFlip_injective p S χ hχ)).trans
+    (card_coloring_le p S)
+
+omit [Fintype α] [DecidableEq α] in
+lemma alternatingPerm_flip (p : Equiv.Perm α) (x : Bool × α) :
+    (alternatingPerm p x).1 = !x.1 := by
+  rcases x with ⟨b,x⟩
+  cases b <;> rfl
+
+omit [Fintype α] [DecidableEq α] in
+lemma alternating_step_sameCycle (p : Equiv.Perm α) (x : Bool × α) :
+    p.SameCycle x.2 (alternatingPerm p x).2 := by
+  rcases x with ⟨b,x⟩
+  cases b
+  · exact Equiv.Perm.SameCycle.refl p x
+  · exact (Equiv.Perm.SameCycle.refl p x).apply_right
+
+omit [DecidableEq α] in
+lemma alternating_sameCycle_snd (p : Equiv.Perm α) {x y : Bool × α}
+    (h : (alternatingPerm p).SameCycle x y) : p.SameCycle x.2 y.2 := by
+  obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
+  have hp (k : ℕ) : p.SameCycle x.2 (((alternatingPerm p)^k) x).2 := by
+    induction k with
+    | zero => exact Equiv.Perm.SameCycle.refl p x.2
+    | succ k ih =>
+      rw [pow_succ',Equiv.Perm.mul_apply]
+      exact ih.trans (alternating_step_sameCycle p _)
+  exact hn ▸ hp n
+
+omit [DecidableEq α] in
+lemma alternating_zero_sameCycle (p : Equiv.Perm α) {x y : α}
+    (h : p.SameCycle x y) : (alternatingPerm p).SameCycle (false,x) (false,y) := by
+  obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
+  have hp (k : ℕ) : (alternatingPerm p).SameCycle (false,x) (false,(p^k) x) := by
+    induction k with
+    | zero => exact Equiv.Perm.SameCycle.refl _ _
+    | succ k ih =>
+      have hs := ih.apply_right.apply_right
+      simpa only [pow_succ',Equiv.Perm.mul_apply,alternatingPerm,Equiv.coe_fn_mk,
+        Bool.false_eq_true,↓reduceIte] using hs
+  simpa only [hn] using hp n
+
+omit [Fintype α] [DecidableEq α] in
+lemma alternating_sameCycle_zero (p : Equiv.Perm α) (x : Bool × α) :
+    (alternatingPerm p).SameCycle x (false,x.2) := by
+  rcases x with ⟨b,x⟩
+  cases b
+  · exact Equiv.Perm.SameCycle.refl _ _
+  · exact ((Equiv.Perm.SameCycle.refl (alternatingPerm p) (false,x)).apply_right).symm
+
+omit [DecidableEq α] in
+lemma alternating_sameCycle_iff (p : Equiv.Perm α) (x y : Bool × α) :
+    (alternatingPerm p).SameCycle x y ↔ p.SameCycle x.2 y.2 := by
+  constructor
+  · exact alternating_sameCycle_snd p
+  · intro h
+    exact (alternating_sameCycle_zero p x).trans
+      ((alternating_zero_sameCycle p h).trans (alternating_sameCycle_zero p y).symm)
+
+lemma orbit_alternating (p : Equiv.Perm α) (x : Bool × α) :
+    orbitSet (alternatingPerm p) x = childJoin (orbitSet p x.2) (orbitSet p x.2) := by
+  ext ⟨b,y⟩
+  simp only [mem_orbitSet,alternating_sameCycle_iff,mem_childJoin,ite_self]
+
+omit [Fintype α] in
+lemma boolFiber_childJoin (A B : Finset α) (b : Bool) :
+    boolFiber (childJoin A B) b = if b then B else A := by
+  ext x
+  cases b <;> simp only [mem_boolFiber,mem_childJoin,Bool.false_eq_true,↓reduceIte]
+
+noncomputable def projectCycle (p : Equiv.Perm α) (A B : Finset α)
+    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) :
+    RoutingNetwork.SelectedCycle p A := by
+  refine ⟨boolFiber C.val false,?_,?_⟩
+  · obtain ⟨x,hx⟩ := C.property.1
+    refine ⟨x.2,?_⟩
+    rw [hx,orbit_alternating,boolFiber_childJoin]
+    rfl
+  · intro x hx
+    exact (mem_childJoin A B false x).mp (C.property.2 ((mem_boolFiber _ _ _).mp hx))
+
+lemma selected_alternating_eq (p : Equiv.Perm α) (A B : Finset α)
+    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) :
+    C.val = childJoin (projectCycle p A B C).val (projectCycle p A B C).val := by
+  change C.val = childJoin (boolFiber C.val false) (boolFiber C.val false)
+  obtain ⟨x,hx⟩ := C.property.1
+  rw [hx,orbit_alternating,boolFiber_childJoin]
+  rfl
+
+lemma projectCycle_injective (p : Equiv.Perm α) (A B : Finset α) :
+    Function.Injective (projectCycle p A B) := by
+  intro C D he
+  apply Subtype.ext
+  rw [selected_alternating_eq p A B C,selected_alternating_eq p A B D,he]
+
+lemma card_alternating_cycles_le (p : Equiv.Perm α) (A B : Finset α) :
+    Fintype.card (RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) ≤
+      Fintype.card (RoutingNetwork.SelectedCycle p A) :=
+  Fintype.card_le_of_injective _ (projectCycle_injective p A B)
+
+end CycleColoring
+
+section PairAssignmentCount
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+omit [Fintype α] in
+lemma mem_image_perm_iff (p : Equiv.Perm α) (A : Finset α) (y : α) :
+    y ∈ A.image p ↔ p.symm y ∈ A := by
+  constructor
+  · intro h
+    obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp h
+    simpa only [←hxy,Equiv.symm_apply_apply] using hx
+  · intro h
+    exact Finset.mem_image.mpr ⟨p.symm y,h,p.apply_symm_apply y⟩
+
+omit [Fintype α] in
+lemma card_childJoin (A B : Finset α) : (childJoin A B).card = A.card+B.card := by
+  have hd : Disjoint (A.image (fun x => (false,x))) (B.image (fun x => (true,x))) := by
+    apply Finset.disjoint_left.mpr
+    rintro x hx hy
+    obtain ⟨a,ha,rfl⟩ := Finset.mem_image.mp hx
+    obtain ⟨b,hb,hba⟩ := Finset.mem_image.mp hy
+    exact Bool.false_ne_true (congrArg Prod.fst hba).symm
+  have hi (b : Bool) : Function.Injective (fun x : α => (b,x)) :=
+    fun _ _ h => congrArg Prod.snd h
+  rw [childJoin,Finset.card_union_of_disjoint hd,
+    Finset.card_image_of_injective A (hi false),Finset.card_image_of_injective B (hi true)]
+
+omit [Fintype α] in
+lemma alternating_internal (p : Equiv.Perm α) (A B : Finset α) :
+    ((childJoin A B).filter (fun x => (CycleColoring.alternatingPerm p).symm x ∈ childJoin A B)) =
+      childJoin (A ∩ B.image p) (B ∩ A) := by
+  ext ⟨b,x⟩
+  rw [Finset.mem_filter]
+  cases b <;> simp only [mem_childJoin,CycleColoring.alternatingPerm,
+    Equiv.symm_mk,Equiv.coe_fn_mk,Bool.false_eq_true,↓reduceIte,Finset.mem_inter,mem_image_perm_iff]
+
+omit [Fintype α] in
+lemma alternating_boundary_card (p : Equiv.Perm α) (A B : Finset α) :
+    (CycleColoring.boundary (CycleColoring.alternatingPerm p) (childJoin A B)).card =
+      (A ∪ B).card + (A ∪ B.image p).card - (A.card+B.card) := by
+  rw [CycleColoring.boundary_card,alternating_internal,card_childJoin,card_childJoin]
+  have h₀ := Finset.card_union_add_card_inter A B
+  have h₁ := Finset.card_union_add_card_inter A (B.image p)
+  rw [Finset.card_image_of_injective _ p.injective] at h₁
+  have h₂ := Finset.inter_comm B A
+  rw [h₂]
+  omega
+
+lemma compatible_assignment_bound (p : Equiv.Perm α) (A B : Finset α) :
+    Fintype.card (CycleColoring.OppositeColoring (CycleColoring.alternatingPerm p) (childJoin A B)) ≤
+      2 ^ ((A ∪ B).card + (A ∪ B.image p).card - (A.card+B.card) +
+        Fintype.card (RoutingNetwork.SelectedCycle p A)) := by
+  have h := CycleColoring.card_opposite_le (CycleColoring.alternatingPerm p) (childJoin A B)
+    Prod.fst (CycleColoring.alternatingPerm_flip p)
+  rw [alternating_boundary_card] at h
+  exact h.trans (Nat.pow_le_pow_right (by norm_num)
+    (Nat.add_le_add_left (CycleColoring.card_alternating_cycles_le p A B) _))
+
+end PairAssignmentCount
+
+lemma finiteMean_nonneg {Ω : Type*} [Fintype Ω] {f : Ω → ℝ}
+    (h : ∀ ω, 0 ≤ f ω) : 0 ≤ finiteMean f := by
+  exact div_nonneg (Finset.sum_nonneg (fun ω _ => h ω)) (Nat.cast_nonneg _)
+
+lemma boolFiber_headTail (d : ℕ) (S : Finset (Card (d+1))) (b : Bool) :
+    boolFiber (S.image (headTailEquiv d)) b = childMarks S b := by
+  classical
+  ext x
+  rw [mem_boolFiber,mem_childMarks]
+  constructor
+  · intro h
+    obtain ⟨y,hy,he⟩ := Finset.mem_image.mp h
+    have hh := congrArg (headTailEquiv d).symm he
+    rw [Equiv.symm_apply_apply] at hh
+    change y = Fin.cons b x at hh
+    exact hh ▸ hy
+  · intro h
+    exact Finset.mem_image.mpr ⟨Fin.cons b x,h,by rfl⟩
+
+lemma prod_childMarks (d : ℕ) (S : Finset (Card (d+1))) (f : Card (d+1) → ℝ) :
+    (∏ x ∈ S, f x) =
+      (∏ x ∈ childMarks S false, f (Fin.cons false x)) *
+      ∏ x ∈ childMarks S true, f (Fin.cons true x) := by
+  classical
+  have h := prod_boolFibers (S.image (headTailEquiv d)) (fun x => f ((headTailEquiv d).symm x))
+  rw [Finset.prod_image (headTailEquiv d).injective.injOn,
+    boolFiber_headTail,boolFiber_headTail] at h
+  simp only [Equiv.symm_apply_apply] at h
+  exact h
+
+noncomputable def headAverage {d : ℕ} (p : Card (d+1) → ℝ) (y : Card d) : ℝ :=
+  (p (Fin.cons false y)+p (Fin.cons true y))/2
+
+lemma finiteMean_headAverage (d : ℕ) (p : Card (d+1) → ℝ) :
+    finiteMean (headAverage p) = finiteMean p := by
+  rw [finiteMean_equiv (headTailEquiv d),finiteMean_prod,finiteMean_comm]
+  apply finiteMean_congr
+  intro y
+  rw [finiteMean_bool]
+  rfl
+
+lemma pairSwitch_product_bound (d : ℕ) (S : Finset (Card (d+1)))
+    (p : Card (d+1) → ℝ) (hp : ∀ x, 0 ≤ p x) :
+    finiteMean (fun ξ : Card d → Bool => ∏ x ∈ S, p (pairSwitch ξ x)) ≤
+      ∏ x ∈ S, headAverage p (Fin.tail x) := by
+  classical
+  have h : CylinderBound (fun T => ∏ x ∈ T, p ((headTailEquiv d).symm x))
+      (fun x => p ((headTailEquiv d).symm x)) := fun _ => le_rfl
+  have hh := h.fair_parallel (fun x => hp _) (S.image (headTailEquiv d))
+  simp only [Finset.prod_image (pairLayer _).injective.injOn,
+    Finset.prod_image (headTailEquiv d).injective.injOn] at hh
+  convert hh using 1
+  · apply finiteMean_congr
+    intro ξ
+    apply Finset.prod_congr rfl
+    intro x _
+    rfl
+  · apply Finset.prod_congr rfl
+    intro x _
+    change (p (Fin.cons false (Fin.tail x))+p (Fin.cons true (Fin.tail x)))/2 =
+      (p (Fin.cons (x 0) (Fin.tail x))+p (Fin.cons (!(x 0)) (Fin.tail x)))/2
+    cases x 0
+    · rfl
+    · simp only [Bool.not_true]
+      ring
+
+lemma butterflyPerm_step (d : ℕ)
+    (c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (ξ : Card d → Bool) :
+    butterflyPerm (d+1) (decodeButterfly (d+1) ((coinStepEquiv d).symm (c,ξ))) =
+    pairSwitch ξ * childLift (fun b => butterflyPerm d (decodeButterfly d (if b then c.2 else c.1))) := by
+  change pairSwitch (fun y => (coinStepEquiv d).symm (c,ξ) (Sum.inl y)) *
+    childLift (fun b => butterflyPerm d (decodeButterfly d
+      (fun i => (coinStepEquiv d).symm (c,ξ) (Sum.inr (b,i))))) = _
+  apply congrArg₂ (· * ·)
+  · rfl
+  · apply congrArg childLift
+    funext b
+    apply congrArg (fun ω => butterflyPerm d (decodeButterfly d ω))
+    funext i
+    cases b <;> rfl
+
+lemma butterfly_product_bound (d : ℕ) (S : Finset (Card d))
+    (p : Card d → ℝ) (hp : ∀ x, 0 ≤ p x) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      ∏ x ∈ S, p (butterflyPerm d (decodeButterfly d ω) x)) ≤ (finiteMean p)^S.card := by
+  classical
+  induction d with
+  | zero =>
+    have he : p = fun _ => p (fun i => Fin.elim0 i) := by
+      funext x
+      exact congrArg p (Subsingleton.elim _ _)
+    rw [he]
+    simp only [Finset.prod_const,finiteMean_const,le_refl]
+  | succ d ih =>
+    let g : Card d → ℝ := headAverage p
+    have hg : ∀ x, 0 ≤ g x := fun x => div_nonneg (add_nonneg (hp _) (hp _)) (by norm_num)
+    rw [finiteMean_equiv (coinStepEquiv d),finiteMean_prod]
+    calc
+      _ ≤ finiteMean (fun c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool) =>
+        (∏ x ∈ childMarks S false, g (butterflyPerm d (decodeButterfly d c.1) x)) *
+        ∏ x ∈ childMarks S true, g (butterflyPerm d (decodeButterfly d c.2) x)) := by
+        apply finiteMean_mono
+        intro c
+        let f : Equiv.Perm (Card (d+1)) := childLift (fun b =>
+          butterflyPerm d (decodeButterfly d (if b then c.2 else c.1)))
+        have hh := pairSwitch_product_bound d (S.image f) p hp
+        simp only [Finset.prod_image f.injective.injOn] at hh
+        convert hh using 1
+        · apply finiteMean_congr
+          intro ξ
+          apply Finset.prod_congr rfl
+          intro x _
+          rw [butterflyPerm_step]
+          rfl
+        · rw [prod_childMarks]
+          simp only [f,childLift,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
+            Bool.false_eq_true,↓reduceIte]
+          rfl
+      _ = (finiteMean (fun c : SwitchIndex d → Bool =>
+        ∏ x ∈ childMarks S false, g (butterflyPerm d (decodeButterfly d c) x))) *
+        finiteMean (fun c : SwitchIndex d → Bool =>
+        ∏ x ∈ childMarks S true, g (butterflyPerm d (decodeButterfly d c) x)) :=
+          finiteMean_prod_mul
+            (fun c : SwitchIndex d → Bool => ∏ x ∈ childMarks S false,
+              g (butterflyPerm d (decodeButterfly d c) x))
+            (fun c : SwitchIndex d → Bool => ∏ x ∈ childMarks S true,
+              g (butterflyPerm d (decodeButterfly d c) x))
+      _ ≤ (finiteMean g)^(childMarks S false).card * (finiteMean g)^(childMarks S true).card :=
+        mul_le_mul (ih (childMarks S false) g hg) (ih (childMarks S true) g hg)
+          (finiteMean_nonneg fun c => Finset.prod_nonneg (fun x _ => hg _))
+          (pow_nonneg (finiteMean_nonneg hg) _)
+      _ = (finiteMean p)^S.card := by
+        rw [←pow_add,card_childMarks_add,finiteMean_headAverage]
+
+noncomputable def inverseButterflyMarks (d : ℕ) (A : Finset (Card d))
+    (ω : SwitchIndex d → Bool) : Finset (Card d) :=
+  A.image (butterflyPerm d (decodeButterfly d ω)).symm
+
+lemma inverseButterflyMarks_inclusion (d : ℕ) (A S : Finset (Card d)) :
+    inclusionMoment (inverseButterflyMarks d A) S ≤
+      ((A.card : ℝ)/(2:ℝ)^d)^S.card := by
+  classical
+  let p : Card d → ℝ := fun x => if x ∈ A then 1 else 0
+  have hm : finiteMean p = (A.card : ℝ)/(2:ℝ)^d := by
+    simp only [finiteMean,p,card_positions,Nat.cast_pow,Nat.cast_ofNat]
+    congr 1
+    simp
+  have he : inclusionMoment (inverseButterflyMarks d A) S =
+      finiteMean (fun ω : SwitchIndex d → Bool =>
+        ∏ x ∈ S, p (butterflyPerm d (decodeButterfly d ω) x)) := by
+    rw [inclusionMoment_eq]
+    apply finiteMean_congr
+    intro ω
+    simp only [inverseButterflyMarks,subset_image_equiv_iff,Equiv.symm_symm,indicator_subset_prod,
+      Finset.prod_image (butterflyPerm d (decodeButterfly d ω)).injective.injOn,p]
+  rw [he]
+  simpa only [hm] using butterfly_product_bound d S p (fun x => by dsimp [p]; split_ifs <;> norm_num)
+
+namespace PairRouting
+
+section
+variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
+
+noncomputable def occupied (x : ι → Bool × α) : Finset α :=
+  Finset.univ.image (fun i => (x i).2)
+
+omit [Fintype α] in
+lemma mem_occupied (x : ι → Bool × α) (y : α) : y ∈ occupied x ↔ ∃ i, (x i).2 = y := by
+  simp [occupied]
+
+noncomputable def chosenCoin (x : ι → Bool × α) (c : ι → Bool) (y : α) : Bool := by
+  classical
+  exact if h : ∃ i, (x i).2 = y then required x c (Classical.choose h) else false
+
+omit [Fintype α] in
+lemma chosenCoin_apply (x : ι → Bool × α) (c : ι → Bool) (hc : Compatible x c) (i : ι) :
+    chosenCoin x c (x i).2 = required x c i := by
+  classical
+  simp only [chosenCoin,dite_eq_left (show ∃ j, (x j).2 = (x i).2 from ⟨i,rfl⟩)]
+  exact hc _ i (Classical.choose_spec (show ∃ j, (x j).2 = (x i).2 from ⟨i,rfl⟩))
+
+omit [Fintype α] in
+lemma colors_eq_iff (x : ι → Bool × α) (c : ι → Bool) (hc : Compatible x c) (ξ : α → Bool) :
+    colors x ξ = c ↔ ∀ y ∈ occupied x, ξ y = chosenCoin x c y := by
+  constructor
+  · intro he y hy
+    obtain ⟨i,rfl⟩ := (mem_occupied x y).mp hy
+    rw [chosenCoin_apply x c hc i]
+    rw [←he,required_colors]
+  · intro he
+    funext i
+    apply (xor_eq_iff _ _ _).mpr
+    exact (he _ ((mem_occupied x _).mpr ⟨i,rfl⟩)).trans (chosenCoin_apply x c hc i)
+
+noncomputable instance (x : ι → Bool × α) (c : ι → Bool) : Decidable (Compatible x c) :=
+  Classical.propDecidable _
+
+lemma colors_probability (x : ι → Bool × α) (c : ι → Bool) :
+    finiteMean (fun ξ : α → Bool => if colors x ξ = c then (1 : ℝ) else 0) =
+      if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0 := by
+  classical
+  by_cases hc : Compatible x c
+  · rw [ite_eq_left hc]
+    have he := fun ξ => colors_eq_iff x c hc ξ
+    simp only [he]
+    exact mean_coinCylinder (occupied x) (chosenCoin x c)
+  · rw [ite_eq_right hc]
+    have he : ∀ ξ, colors x ξ ≠ c := fun ξ hh => hc (hh ▸ colors_compatible x ξ)
+    simp only [he,ite_false,finiteMean_const]
+
+
+def lift (p : Bool → Equiv.Perm α) : Equiv.Perm (Bool × α) where
+  toFun x := (x.1,p x.1 x.2)
+  invFun x := (x.1,(p x.1).symm x.2)
+  left_inv x := by simp
+  right_inv x := by simp
+
+def sandwich (ξ η : α → Bool) (p : Bool → Equiv.Perm α) : Equiv.Perm (Bool × α) :=
+  pairLayer η * lift p * pairLayer ξ
+
+def ChildMatches (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) : Prop :=
+  ∀ i, p (c i) (x i).2 = (y i).2
+
+noncomputable instance (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
+    Decidable (ChildMatches p x y c) := Classical.propDecidable _
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma sandwich_matches_iff (ξ η : α → Bool) (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) :
+    (∀ i, sandwich ξ η p (x i) = y i) ↔
+      colors x ξ = colors y η ∧ ChildMatches p x y (colors x ξ) := by
+  have he (i : ι) : sandwich ξ η p (x i) = y i ↔
+      colors x ξ i = colors y η i ∧ p (colors x ξ i) (x i).2 = (y i).2 := by
+    change pairLayer η (lift p (pairLayer ξ (x i))) = y i ↔ _
+    rw [←Equiv.eq_symm_apply]
+    change (colors x ξ i,p (colors x ξ i) (x i).2) = (colors y η i,(y i).2) ↔ _
+    exact ⟨Prod.mk.inj,fun h => Prod.ext h.1 h.2⟩
+  simp only [he,forall_and,←funext_iff,ChildMatches]
+
+lemma sandwich_probability [DecidableEq ι] (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) :
+    finiteMean (fun coins : (α → Bool) × (α → Bool) =>
+      if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then (1:ℝ) else 0) =
+    ∑ c : ι → Bool,
+      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
+      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) *
+      (if ChildMatches p x y c then 1 else 0) := by
+  classical
+  have he (coins : (α → Bool) × (α → Bool)) :
+      (if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then (1:ℝ) else 0) =
+      ∑ c : ι → Bool, (if colors x coins.1 = c then (1:ℝ) else 0) *
+        (if colors y coins.2 = c then 1 else 0) * (if ChildMatches p x y c then 1 else 0) := by
+    rw [Finset.sum_eq_single (colors x coins.1)]
+    · by_cases hc : colors x coins.1 = colors y coins.2
+      · simp only [sandwich_matches_iff,hc,true_and,↓reduceIte,one_mul]
+      · simp only [sandwich_matches_iff,hc,false_and,↓reduceIte,one_mul,Ne.symm hc,zero_mul]
+    · intro c _ hc
+      simp only [Ne.symm hc,ite_false,zero_mul]
+    · simp
+  simp_rw [he]
+  rw [finiteMean_sum]
+  apply Finset.sum_congr rfl
+  intro c _
+  rw [finiteMean_mul_const]
+  rw [finiteMean_prod_mul
+    (fun ξ : α → Bool => if colors x ξ = c then (1:ℝ) else 0)
+    (fun η : α → Bool => if colors y η = c then (1:ℝ) else 0),
+    colors_probability,colors_probability]
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma childMatches_split (p₀ p₁ : Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
+    ChildMatches (fun b => if b then p₁ else p₀) x y c ↔
+      (∀ i : {i // c i = false}, p₀ (x i.val).2 = (y i.val).2) ∧
+      (∀ i : {i // c i = true}, p₁ (x i.val).2 = (y i.val).2) := by
+  constructor
+  · intro h
+    constructor
+    · intro i
+      simpa only [i.property,Bool.false_eq_true,↓reduceIte] using h i.val
+    · intro i
+      simpa only [i.property,↓reduceIte] using h i.val
+  · rintro ⟨h₀,h₁⟩ i
+    cases hi : c i
+    · simpa only [hi,Bool.false_eq_true,↓reduceIte] using h₀ ⟨i,hi⟩
+    · simpa only [hi,↓reduceIte] using h₁ ⟨i,hi⟩
+
+noncomputable def tupleProbability {Ω β : Type*} [Fintype Ω] [DecidableEq β]
+    (P : Ω → Equiv.Perm β) (x y : ι → β) : ℝ := by
+  classical
+  exact finiteMean (fun ω => if ∀ i, P ω (x i) = y i then 1 else 0)
+
+omit [Fintype α] in
+lemma childMatches_mean {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
+    (P : Ω → Equiv.Perm α) (Q : Ω' → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
+    finiteMean (fun ω : Ω × Ω' =>
+      if ChildMatches (fun b => if b then Q ω.2 else P ω.1) x y c then (1:ℝ) else 0) =
+      tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
+        (fun i : {i // c i = false} => (y i.val).2) *
+      tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
+        (fun i : {i // c i = true} => (y i.val).2) := by
+  classical
+  let f : Ω → ℝ := fun ω => if ∀ i : {i // c i = false}, P ω (x i.val).2 = (y i.val).2 then 1 else 0
+  let g : Ω' → ℝ := fun ω => if ∀ i : {i // c i = true}, Q ω (x i.val).2 = (y i.val).2 then 1 else 0
+  have he (ω : Ω × Ω') :
+      (if ChildMatches (fun b => if b then Q ω.2 else P ω.1) x y c then (1:ℝ) else 0) = f ω.1*g ω.2 := by
+    simp only [childMatches_split,f,g]
+    split_ifs <;> simp_all
+  simp_rw [he]
+  exact finiteMean_prod_mul f g
+
+lemma sandwich_mean [DecidableEq ι] {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
+    (P : Ω → Equiv.Perm α) (Q : Ω' → Equiv.Perm α) (x y : ι → Bool × α) :
+    finiteMean (fun ω : Ω × Ω' => finiteMean (fun coins : (α → Bool) × (α → Bool) =>
+      if ∀ i, sandwich coins.1 coins.2 (fun b => if b then Q ω.2 else P ω.1) (x i) = y i
+      then (1:ℝ) else 0)) =
+    ∑ c : ι → Bool,
+      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
+      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) *
+      (tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
+        (fun i : {i // c i = false} => (y i.val).2) *
+      tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
+        (fun i : {i // c i = true} => (y i.val).2)) := by
+  classical
+  simp_rw [sandwich_probability]
+  rw [finiteMean_sum]
+  apply Finset.sum_congr rfl
+  intro c _
+  simp only [mul_comm ((if Compatible x c then _ else _) * _),finiteMean_mul_const,
+    childMatches_mean]
+
+
+abbrev BothAssignments (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :=
+  {c : ι → Bool // Compatible e c ∧ Compatible (fun i => lift p (e i)) c}
+
+noncomputable instance (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    Fintype (BothAssignments e p) := Fintype.ofFinite _
+
+noncomputable def assignmentColoring (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α)
+    (c : BothAssignments e p) : CycleColoring.OppositeColoring
+      (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e) := by
+  classical
+  refine ⟨Function.extend e c.val (fun _ => false),?_,?_⟩
+  · intro z hz
+    apply Function.extend_apply'
+    rintro ⟨i,rfl⟩
+    exact hz (Finset.mem_image.mpr ⟨i,Finset.mem_univ _,rfl⟩)
+  · intro z hz hnext
+    obtain ⟨i,_,hi⟩ := Finset.mem_image.mp hz
+    obtain ⟨j,_,hj⟩ := Finset.mem_image.mp hnext
+    have hfi := e.injective.extend_apply c.val (fun _ => false) i
+    have hfj := e.injective.extend_apply c.val (fun _ => false) j
+    rw [hi] at hfi
+    rw [hj] at hfj
+    rw [hfi,hfj]
+    rcases z with ⟨b,a⟩
+    cases b
+    · have hj' : e j = (true,a) := hj
+      have ht : (e i).2 = (e j).2 := by rw [hi,hj']
+      have hh := c.property.1 i j ht
+      simp only [required,hi,hj'] at hh
+      revert hh
+      generalize c.val i = bi
+      generalize c.val j = bj
+      cases bi <;> cases bj <;> decide
+    · have hj' : e j = (false,alternatingProjection p a) := hj
+      have ht : (lift p (e i)).2 = (lift p (e j)).2 := by
+        simp only [hi,hj',lift,Equiv.coe_fn_mk,alternatingProjection,Equiv.Perm.coe_mul,
+          Function.comp_apply,Equiv.apply_symm_apply]
+      have hh := c.property.2 i j ht
+      simp only [required,hi,hj',lift,Equiv.coe_fn_mk] at hh
+      revert hh
+      generalize c.val i = bi
+      generalize c.val j = bj
+      cases bi <;> cases bj <;> decide
+
+omit [Fintype α] in
+lemma assignmentColoring_injective (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    Function.Injective (assignmentColoring e p) := by
+  intro c c' he
+  apply Subtype.ext
+  funext i
+  have hh := congrFun (congrArg Subtype.val he) (e i)
+  change Function.extend e c.val (fun _ => false) (e i) =
+    Function.extend e c'.val (fun _ => false) (e i) at hh
+  simpa only [e.injective.extend_apply] using hh
+
+lemma bothAssignments_card (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    Fintype.card (BothAssignments e p) ≤
+      2 ^ ((boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true).card +
+        (boolFiber (labelSet e) false ∪ (boolFiber (labelSet e) true).image
+          (alternatingProjection p)).card -
+        ((boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card) +
+        Fintype.card (RoutingNetwork.SelectedCycle (alternatingProjection p) (boolFiber (labelSet e) false))) := by
+  have h := compatible_assignment_bound (alternatingProjection p)
+    (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
+  rw [childJoin_boolFiber] at h
+  exact (Fintype.card_le_of_injective _ (assignmentColoring_injective e p)).trans h
+
+omit [Fintype α] in
+lemma occupied_labelSet (e : ι ↪ Bool × α) : occupied e = (labelSet e).image Prod.snd := by
+  classical
+  simp only [occupied,labelSet,Finset.image_image,Function.comp_def]
+
+omit [Fintype ι] [Fintype α] in
+lemma image_snd_childJoin (A B : Finset α) : (childJoin A B).image Prod.snd = A ∪ B := by
+  classical
+  simp only [childJoin,Finset.image_union,Finset.image_image,Function.comp_def,Finset.image_id']
+
+omit [Fintype ι] [Fintype α] in
+lemma image_lift_childJoin (p : Bool → Equiv.Perm α) (A B : Finset α) :
+    (childJoin A B).image (fun x => (lift p x).2) = A.image (p false) ∪ B.image (p true) := by
+  classical
+  simp only [childJoin,Finset.image_union,Finset.image_image,Function.comp_def,lift,Equiv.coe_fn_mk]
+
+omit [Fintype α] in
+lemma occupied_lift_card (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    (occupied (fun i => lift p (e i))).card =
+      (boolFiber (labelSet e) false ∪ (boolFiber (labelSet e) true).image (alternatingProjection p)).card := by
+  classical
+  have he : occupied (fun i => lift p (e i)) = (labelSet e).image (fun x => (lift p x).2) := by
+    simp only [occupied,labelSet,Finset.image_image,Function.comp_def]
+  rw [he,←childJoin_boolFiber (labelSet e),image_lift_childJoin]
+  simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
+  let A := boolFiber (labelSet e) false
+  let B := boolFiber (labelSet e) true
+  change (A.image (p false) ∪ B.image (p true)).card = (A ∪ B.image (alternatingProjection p)).card
+  have hh : A.image (p false) ∪ B.image (p true) =
+      (A ∪ B.image (alternatingProjection p)).image (p false) := by
+    simp only [Finset.image_union,Finset.image_image,Function.comp_def,
+      alternatingProjection,Equiv.Perm.coe_mul,Equiv.apply_symm_apply]
+  rw [hh,Finset.card_image_of_injective _ (p false).injective]
+
+lemma bothAssignments_occupied_bound (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    Fintype.card (BothAssignments e p) ≤
+      2 ^ ((occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι +
+        Fintype.card (RoutingNetwork.SelectedCycle (alternatingProjection p) (boolFiber (labelSet e) false))) := by
+  have he : (occupied e).card =
+      (boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true).card := by
+    rw [occupied_labelSet,←childJoin_boolFiber (labelSet e),image_snd_childJoin]
+    simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
+  have hc : (boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card = Fintype.card ι := by
+    rw [←card_childJoin,childJoin_boolFiber,labelSet,Finset.card_image_of_injective _ e.injective,
+      Finset.card_univ]
+  simpa only [he,occupied_lift_card,←hc] using bothAssignments_card e p
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma compatible_pairLayer_iff (x : ι → Bool × α) (c : ι → Bool) (ξ : α → Bool) :
+    Compatible (fun i => pairLayer ξ (x i)) c ↔ Compatible x c := by
+  have he (i j : ι) (ht : (x i).2 = (x j).2) :
+      required (fun i => pairLayer ξ (x i)) c i = required (fun i => pairLayer ξ (x i)) c j ↔
+      required x c i = required x c j := by
+    simp only [required,pairLayer,Equiv.coe_fn_mk,ht]
+    cases (x i).1 <;> cases (x j).1 <;> cases c i <;> cases c j <;> cases ξ (x j).2 <;> decide
+  constructor
+  · intro h i j ht
+    exact (he i j ht).mp (h i j ht)
+  · intro h i j ht
+    exact (he i j ht).mpr (h i j ht)
+
+
+omit [Fintype α] in
+lemma occupied_eq_fibers (e : ι ↪ Bool × α) :
+    occupied e = boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true := by
+  rw [occupied_labelSet,←childJoin_boolFiber (labelSet e),image_snd_childJoin]
+  simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
+
+omit [Fintype α] in
+lemma card_labelSet (e : ι ↪ Bool × α) : (labelSet e).card = Fintype.card ι := by
+  classical
+  exact (Finset.card_image_of_injective _ e.injective).trans (Finset.card_univ)
+
+omit [Fintype α] in
+lemma occupied_card_double (e : ι ↪ Bool × α) : Fintype.card ι ≤ 2*(occupied e).card := by
+  have h₀ : (boolFiber (labelSet e) false).card ≤ (occupied e).card := by
+    rw [occupied_eq_fibers]
+    exact Finset.card_le_card Finset.subset_union_left
+  have h₁ : (boolFiber (labelSet e) true).card ≤ (occupied e).card := by
+    rw [occupied_eq_fibers]
+    exact Finset.card_le_card Finset.subset_union_right
+  have he : (boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card = Fintype.card ι := by
+    rw [←card_childJoin,childJoin_boolFiber,card_labelSet]
+  omega
+
+lemma bothAssignments_exact_bound (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    Fintype.card (BothAssignments e p) ≤
+      2 ^ ((occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι +
+        alternatingCycles e p) := by
+  have h := CycleColoring.card_opposite_le
+    (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e)
+    Prod.fst (CycleColoring.alternatingPerm_flip (alternatingProjection p))
+  have hb : (CycleColoring.boundary (CycleColoring.alternatingPerm (alternatingProjection p))
+      (labelSet e)).card =
+      (occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι := by
+    have hh := alternating_boundary_card (alternatingProjection p)
+      (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
+    rw [childJoin_boolFiber] at hh
+    rw [occupied_eq_fibers,occupied_lift_card]
+    have he := card_childJoin (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
+    rw [childJoin_boolFiber,card_labelSet] at he
+    simpa only [he] using hh
+  rw [hb] at h
+  exact (Fintype.card_le_of_injective _ (assignmentColoring_injective e p)).trans h
+
+abbrev Assignments (x y : ι → Bool × α) :=
+  {c : ι → Bool // Compatible x c ∧ Compatible y c}
+
+noncomputable instance (x y : ι → Bool × α) : Fintype (Assignments x y) := Fintype.ofFinite _
+
+def routedEmbedding (e : ι ↪ Bool × α) (ξ η : α → Bool) (p : Bool → Equiv.Perm α) :
+    ι ↪ Bool × α := e.trans (sandwich ξ η p).toEmbedding
+
+omit [Fintype α] in
+lemma occupied_pairLayer (x : ι → Bool × α) (ξ : α → Bool) :
+    occupied (fun i => pairLayer ξ (x i)) = occupied x := rfl
+
+noncomputable def assignmentsEquiv (e : ι ↪ Bool × α) (ξ η : α → Bool)
+    (p : Bool → Equiv.Perm α) :
+    Assignments e (routedEmbedding e ξ η p) ≃ BothAssignments (switchedEmbedding e ξ) p := by
+  apply Equiv.subtypeEquivRight
+  intro c
+  change Compatible e c ∧ Compatible (fun i => pairLayer η (lift p (pairLayer ξ (e i)))) c ↔ _
+  rw [compatible_pairLayer_iff]
+  exact and_congr (compatible_pairLayer_iff e c ξ).symm Iff.rfl
+
+lemma assignments_actual_bound (e : ι ↪ Bool × α) (ξ η : α → Bool) (p : Bool → Equiv.Perm α) :
+    Fintype.card (Assignments e (routedEmbedding e ξ η p)) ≤
+      2 ^ ((occupied e).card + (occupied (routedEmbedding e ξ η p)).card - Fintype.card ι +
+        alternatingCycles (switchedEmbedding e ξ) p) := by
+  rw [Fintype.card_congr (assignmentsEquiv e ξ η p)]
+  exact bothAssignments_exact_bound (switchedEmbedding e ξ) p
+
+omit [Fintype α] in
+lemma assignments_actual_exponent_nonneg (e : ι ↪ Bool × α) (ξ η : α → Bool)
+    (p : Bool → Equiv.Perm α) :
+    Fintype.card ι ≤ (occupied e).card + (occupied (routedEmbedding e ξ η p)).card := by
+  have h₀ := occupied_card_double e
+  have h₁ := occupied_card_double (routedEmbedding e ξ η p)
+  omega
+
+end
+variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α] [DecidableEq ι]
+
+noncomputable def compatibleSet (x y : ι → Bool × α) : Finset (ι → Bool) := by
+  classical
+  exact Finset.univ.filter (fun c => Compatible x c ∧ Compatible y c)
+
+omit [Fintype α] [DecidableEq α] in
+lemma mem_compatibleSet (x y : ι → Bool × α) (c : ι → Bool) :
+    c ∈ compatibleSet x y ↔ Compatible x c ∧ Compatible y c := by
+  classical
+  simp only [compatibleSet,Finset.mem_filter,Finset.mem_univ,true_and]
+
+omit [Fintype α] [DecidableEq α] in
+lemma card_compatibleSet (x y : ι → Bool × α) :
+    (compatibleSet x y).card = Fintype.card (Assignments x y) := by
+  classical
+  exact (Fintype.card_subtype _).symm
+
+noncomputable def outerWeight (x y : ι → Bool × α) : ℝ :=
+  1/(2:ℝ)^((occupied x).card+(occupied y).card)
+
+omit [Fintype α] [DecidableEq ι] in
+lemma outerWeight_pos (x y : ι → Bool × α) : 0 < outerWeight x y := by
+  unfold outerWeight
+  positivity
+
+omit [Fintype α] in
+lemma sum_colors_factor (x y : ι → Bool × α) (f : (ι → Bool) → ℝ) :
+    (∑ c : ι → Bool,
+      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
+      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) * f c) =
+      outerWeight x y * ∑ c ∈ compatibleSet x y, f c := by
+  classical
+  rw [compatibleSet,Finset.sum_filter,Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro c _
+  by_cases hx : Compatible x c <;> by_cases hy : Compatible y c
+  · simp only [hx,hy,true_and,↓reduceIte,outerWeight,pow_add,one_div_mul_one_div]
+  · simp only [hx,hy,and_false,↓reduceIte,mul_zero,zero_mul]
+  · simp only [hx,hy,false_and,↓reduceIte,mul_zero,zero_mul]
+  · simp only [hx,hy,false_and,↓reduceIte,mul_zero,zero_mul]
+
+lemma sandwich_weighted_probability (p : Bool → Equiv.Perm α) (x y : ι → Bool × α)
+    (W : (ι → Bool) → ℝ) :
+    finiteMean (fun coins : (α → Bool) × (α → Bool) =>
+      if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then W (colors x coins.1) else 0) =
+    outerWeight x y * ∑ c ∈ compatibleSet x y, if ChildMatches p x y c then W c else 0 := by
+  classical
+  have he (coins : (α → Bool) × (α → Bool)) :
+      (if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then W (colors x coins.1) else 0) =
+      ∑ c : ι → Bool, (if colors x coins.1 = c then (1:ℝ) else 0) *
+        (if colors y coins.2 = c then 1 else 0) * (if ChildMatches p x y c then W c else 0) := by
+    rw [Finset.sum_eq_single (colors x coins.1)]
+    · by_cases hc : colors x coins.1 = colors y coins.2
+      · simp only [sandwich_matches_iff,hc,true_and,↓reduceIte,one_mul]
+      · simp only [sandwich_matches_iff,hc,false_and,↓reduceIte,one_mul,Ne.symm hc,zero_mul]
+    · intro c _ hc
+      simp only [Ne.symm hc,ite_false,zero_mul]
+    · simp
+  simp_rw [he]
+  rw [finiteMean_sum,←sum_colors_factor]
+  apply Finset.sum_congr rfl
+  intro c _
+  rw [finiteMean_mul_const,finiteMean_prod_mul
+    (fun ξ : α → Bool => if colors x ξ = c then (1:ℝ) else 0)
+    (fun η : α → Bool => if colors y η = c then (1:ℝ) else 0),
+    colors_probability,colors_probability]
+
+omit [Fintype α] [DecidableEq ι] in
+lemma tupleProbability_nonneg {Ω β : Type*} [Fintype Ω] [DecidableEq β]
+    (P : Ω → Equiv.Perm β) (x y : ι → β) : 0 ≤ tupleProbability P x y := by
+  classical
+  exact finiteMean_nonneg (fun ω => by split_ifs <;> norm_num)
+
+end PairRouting
+end Thorp
+
+end ThorpNine.Dense
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Dense/RotationCharge.lean b/lean/OAI/Probability/ThorpRouting/Dense/RotationCharge.lean
index d78c92cd83675d66a3f811b2046eae8ea65aecbd..f34b15488e9077e3738ddeca269fa2b90cec4ecc 100644
--- a/lean/OAI/Probability/ThorpRouting/Dense/RotationCharge.lean
+++ b/lean/OAI/Probability/ThorpRouting/Dense/RotationCharge.lean
@@ -1,888 +1,888 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Dense.ExposedCoins
-
-namespace OAI
-
-namespace ThorpNine.Dense
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-section Rotation
-variable {d A : ℕ}
-
-lemma card_cycleFiber (o : Card d → Option (Fin A)) (i : Fin A) :
-    Fintype.card {x : Card d // o x = some i} = (cycleFiber o i).card := by
-  classical
-  simp [cycleFiber,Fintype.card_subtype]
-
-lemma terminal_mean {o : Card d → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
-    (i : Fin A) (x : Card d) (hx : o x = some i) :
-    finiteMean (fun z : Terminal o => if (z i).val = x then (1:ℝ) else 0) = slotWeight o i := by
-  classical
-  have hev := finiteMean_pi_eval (A := fun i : Fin A => {x : Card d // o x = some i})
-    i (fun z => if z.val = x then (1:ℝ) else 0)
-  rw [hev]
-  have he : (fun z : {x : Card d // o x = some i} => if z.val = x then (1:ℝ) else 0) =
-      (fun z => if z = ⟨x,hx⟩ then 1 else 0) := by
-    funext z
-    simp only [Subtype.ext_iff]
-  rw [he,finiteMean_eq_indicator,card_cycleFiber,slotWeight]
-
-lemma slotWeight_antitone (o : Card d → Option (Fin A))
-    (hn : ∀ i, (cycleFiber o i).Nonempty)
-    (hm : Monotone (fun i => (cycleFiber o i).card)) : Antitone (slotWeight o) := by
-  intro i j hij
-  unfold slotWeight
-  apply one_div_le_one_div_of_le
-  · exact_mod_cast Finset.card_pos.mpr (hn i)
-  · exact_mod_cast hm hij
-
-lemma rotationMeeting_mean {o : Card d → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
-    (hm : Monotone (fun i => (cycleFiber o i).card)) (u v : Card d) :
-    finiteMean (fun z : Terminal o => rotationMeeting o z u v) =
-      min (pathWeight o u) (pathWeight o v) + withinWeight o u v := by
-  classical
-  have hn : ∀ i, (cycleFiber o i).Nonempty := by
-    intro i
-    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : Card d // o x = some i})
-    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
-  have hw := slotWeight_antitone o hn hm
-  cases hu : o u with
-  | none =>
-    cases hv : o v with
-    | none => simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean]
-    | some j =>
-      have hj : 0 ≤ slotWeight o j := by unfold slotWeight; positivity
-      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_left hj]
-  | some i =>
-    cases hv : o v with
-    | none =>
-      have hi : 0 ≤ slotWeight o i := by unfold slotWeight; positivity
-      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_right hi]
-    | some j =>
-      simp only [rotationMeeting,pathWeight,withinWeight,hu,hv]
-      rcases lt_trichotomy i j with hij | rfl | hji
-      · simp only [hij,↓reduceIte,ne_of_lt hij]
-        rw [terminal_mean j v hv,min_eq_right (hw (le_of_lt hij)),add_zero]
-      · simp only [lt_self_iff_false,↓reduceIte,min_self]
-        rw [finiteMean_add,terminal_mean i u hu,terminal_mean i v hv]
-      · simp only [not_lt_of_ge (le_of_lt hji),hji,↓reduceIte,ne_of_gt hji]
-        rw [terminal_mean i u hu,min_eq_left (hw (le_of_lt hji)),add_zero]
-
-lemma pathWeight_sum (o : Card d → Option (Fin A)) (hn : ∀ i, (cycleFiber o i).Nonempty) :
-    ∑ x : Card d, pathWeight o x = (A : ℝ) := by
-  classical
-  have he (x : Card d) : pathWeight o x =
-      ∑ i : Fin A, if o x = some i then slotWeight o i else 0 := by
-    cases hx : o x with
-    | none => simp [pathWeight,hx]
-    | some j => simp [pathWeight,hx]
-  simp only [he]
-  rw [Finset.sum_comm]
-  have hf (i : Fin A) : (∑ x : Card d, if o x = some i then slotWeight o i else 0) = 1 := by
-    rw [← Finset.sum_filter]
-    change (∑ _x ∈ cycleFiber o i, slotWeight o i) = 1
-    simp only [Finset.sum_const,nsmul_eq_mul,slotWeight]
-    exact mul_one_div_cancel (by exact_mod_cast (Finset.card_pos.mpr (hn i)).ne')
-  simp only [hf,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,mul_one]
-
-lemma withinWeight_sum_bound (o : Card d → Option (Fin A))
-    (hn : ∀ i, (cycleFiber o i).Nonempty) (b : Butterfly d) :
-    (∑ s : SwitchIndex d, withinWeight o (switchUsers d b s).1 (switchUsers d b s).2) ≤
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
-  classical
-  have he (u v : Card d) : withinWeight o u v =
-      ∑ i : Fin A, (if u ∈ cycleFiber o i ∧ v ∈ cycleFiber o i then (1:ℝ) else 0) * slotWeight o i := by
-    simp only [mem_cycleFiber]
-    cases hu : o u <;> cases hv : o v <;> simp [withinWeight,hu,hv,ite_and,eq_comm]
-  simp only [he]
-  rw [Finset.sum_comm]
-  apply Finset.sum_le_sum
-  intro i _
-  rw [← Finset.sum_mul]
-  have hi := marked_switch_entropy_bound d b (cycleFiber o i)
-  have hpos : (0:ℝ) < (cycleFiber o i).card := by exact_mod_cast Finset.card_pos.mpr (hn i)
-  have hmul := mul_le_mul_of_nonneg_right hi (show 0 ≤ slotWeight o i by unfold slotWeight; positivity)
-  calc
-    _ ≤ ((cycleFiber o i).card : ℝ) * Real.log (cycleFiber o i).card /
-        (2 * Real.log 2) * slotWeight o i := hmul
-    _ = _ := by unfold slotWeight; field_simp
-
-noncomputable def rotationCharge (o : Card d → Option (Fin A)) (b : Butterfly d)
-    (z : Terminal o) : ℝ :=
-  ∑ s : SwitchIndex d, rotationMeeting o z (switchUsers d b s).1 (switchUsers d b s).2
-
-lemma rotationCharge_mean_bound {o : Card d → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
-    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : Butterfly d) :
-    finiteMean (rotationCharge o b) ≤ (A:ℝ) * d / 2 +
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
-  classical
-  have hn : ∀ i, (cycleFiber o i).Nonempty := by
-    intro i
-    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : Card d // o x = some i})
-    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
-  unfold rotationCharge
-  rw [finiteMean_sum]
-  simp only [rotationMeeting_mean hm,Finset.sum_add_distrib]
-  have h₁ := weighted_switch_min_bound d b (pathWeight o)
-  rw [pathWeight_sum o hn] at h₁
-  have h₂ := withinWeight_sum_bound o hn b
-  linarith
-
-lemma rotation_exp_bound {o : Card d → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
-    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : Butterfly d) :
-    Real.exp (-Real.log 2 * ((A:ℝ)*d/2 +
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2))) ≤
-      finiteMean (fun z : Terminal o => Real.exp (-Real.log 2 * rotationCharge o b z)) := by
-  have hb := rotationCharge_mean_bound hm b
-  have hlog : 0 ≤ Real.log 2 := Real.log_nonneg (by norm_num)
-  have hm' : -Real.log 2 * ((A:ℝ)*d/2 +
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2)) ≤
-      finiteMean (fun z : Terminal o => -Real.log 2 * rotationCharge o b z) := by
-    have he := finiteMean_mul_const (rotationCharge o b) (-Real.log 2)
-    simp only [mul_comm] at he
-    rw [he]
-    exact mul_le_mul_of_nonpos_left hb (neg_nonpos.mpr hlog)
-  exact (Real.exp_le_exp.mpr hm').trans (exp_finiteMean_le _)
-
-end Rotation
-
-section OrbitCore
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-@[simp] lemma mem_orbitSet (p : Equiv.Perm α) (x y : α) :
-    y ∈ orbitSet p x ↔ p.SameCycle x y := by
-  classical
-  simp [orbitSet]
-
-lemma orbitSet_nonempty (p : Equiv.Perm α) (x : α) : (orbitSet p x).Nonempty :=
-  ⟨x,(mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.refl _ _)⟩
-
-lemma orbitSet_eq (p : Equiv.Perm α) {x y : α} (h : p.SameCycle x y) :
-    orbitSet p x = orbitSet p y := by
-  ext z
-  simp only [mem_orbitSet]
-  exact ⟨fun hz => h.symm.trans hz,fun hz => h.trans hz⟩
-
-@[simp] lemma orbitSet_apply (p : Equiv.Perm α) (x : α) : orbitSet p (p x) = orbitSet p x := by
-  ext y
-  simp only [mem_orbitSet,Equiv.Perm.sameCycle_apply_left]
-
-lemma orbitSet_eq_or_disjoint (p : Equiv.Perm α) (x y : α) :
-    orbitSet p x = orbitSet p y ∨ Disjoint (orbitSet p x) (orbitSet p y) := by
-  classical
-  by_cases h : p.SameCycle x y
-  · exact Or.inl (orbitSet_eq p h)
-  · right
-    apply Finset.disjoint_left.mpr
-    intro z hz hz'
-    exact h (((mem_orbitSet _ _ _).mp hz).trans ((mem_orbitSet _ _ _).mp hz').symm)
-
-omit [DecidableEq α] in
-lemma perm_minimalPeriod_pos (p : Equiv.Perm α) (x : α) :
-    0 < Function.minimalPeriod p x :=
-  Function.minimalPeriod_pos_of_mem_periodicPts (p.injective.mem_periodicPts x)
-
-lemma orbitSet_card (p : Equiv.Perm α) (x : α) :
-    (orbitSet p x).card = Function.minimalPeriod p x := by
-  classical
-  let f : Fin (Function.minimalPeriod p x) → {y // p.SameCycle x y} := fun n =>
-    ⟨p^[n.val] x,by rw [Equiv.Perm.iterate_eq_pow]; exact ⟨(n.val:ℤ),by simp⟩⟩
-  have hf : Function.Bijective f := by
-    constructor
-    · intro i j hij
-      apply Fin.ext
-      exact Function.iterate_injOn_Iio_minimalPeriod i.isLt j.isLt (congrArg Subtype.val hij)
-    · intro y
-      obtain ⟨n,hn⟩ := y.property.exists_nat_pow_eq
-      refine ⟨⟨n % Function.minimalPeriod p x, Nat.mod_lt _ (perm_minimalPeriod_pos p x)⟩,?_⟩
-      apply Subtype.ext
-      change p^[n % Function.minimalPeriod p x] x = y.val
-      rw [Function.iterate_mod_minimalPeriod_eq,Equiv.Perm.iterate_eq_pow,hn]
-  have hc := Fintype.card_congr (Equiv.ofBijective f hf)
-  simpa [orbitSet,Fintype.card_subtype] using hc.symm
-
-def walkSet (p : Equiv.Perm α) (r : ℕ) (x : α) : Finset α :=
-  (Finset.range r).image (fun n => p^[n] x)
-
-omit [Fintype α] in
-@[simp] lemma mem_walkSet (p : Equiv.Perm α) (r : ℕ) (x y : α) :
-    y ∈ walkSet p r x ↔ ∃ n < r, p^[n] x = y := by
-  simp [walkSet]
-
-omit [Fintype α] in
-@[simp] lemma walkSet_zero (p : Equiv.Perm α) (x : α) : walkSet p 0 x = ∅ := by
-  simp [walkSet]
-
-omit [Fintype α] in
-lemma walkSet_succ_last (p : Equiv.Perm α) (r : ℕ) (x : α) :
-    walkSet p (r+1) x = insert (p^[r] x) (walkSet p r x) := by
-  simp [walkSet,Finset.range_add_one]
-
-omit [Fintype α] in
-lemma walkSet_succ_first (p : Equiv.Perm α) (r : ℕ) (x : α) :
-    walkSet p (r+1) x = insert x (walkSet p r (p x)) := by
-  ext y
-  simp only [mem_walkSet,Finset.mem_insert]
-  constructor
-  · rintro ⟨n,hn,hny⟩
-    cases n with
-    | zero => exact Or.inl (by simpa using hny.symm)
-    | succ n => exact Or.inr ⟨n,by omega,by simpa only [Function.iterate_succ_apply] using hny⟩
-  · rintro (rfl | ⟨n,hn,hny⟩)
-    · exact ⟨0,by omega,rfl⟩
-    · exact ⟨n+1,by omega,by simpa only [Function.iterate_succ_apply] using hny⟩
-
-omit [Fintype α] in
-lemma not_mem_short_walk (p : Equiv.Perm α) (x : α) {r : ℕ}
-    (hr : r < Function.minimalPeriod p x) : x ∉ walkSet p r (p x) := by
-  rintro hx
-  obtain ⟨n,hn,hnx⟩ := (mem_walkSet _ _ _ _).mp hx
-  have hperiod : Function.IsPeriodicPt p (n+1) x := by
-    simpa only [Function.IsPeriodicPt,Function.IsFixedPt,Function.iterate_succ_apply] using hnx
-  have hp := hperiod.minimalPeriod_le (by omega : 0 < n+1)
-  omega
-
-lemma walkSet_period (p : Equiv.Perm α) (x : α) :
-    walkSet p (Function.minimalPeriod p x) x = orbitSet p x := by
-  ext y
-  simp only [mem_walkSet,mem_orbitSet]
-  constructor
-  · rintro ⟨n,_,rfl⟩
-    rw [Equiv.Perm.iterate_eq_pow]
-    exact ⟨(n:ℤ),by simp⟩
-  · intro h
-    obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
-    refine ⟨n % Function.minimalPeriod p x,Nat.mod_lt _ (perm_minimalPeriod_pos p x),?_⟩
-    rw [Function.iterate_mod_minimalPeriod_eq,Equiv.Perm.iterate_eq_pow,hn]
-
-omit [Fintype α] in
-lemma walkSet_card (p : Equiv.Perm α) (x : α) {r : ℕ}
-    (hr : r ≤ Function.minimalPeriod p x) : (walkSet p r x).card = r := by
-  rw [walkSet,Finset.card_image_of_injOn]
-  · exact Finset.card_range _
-  · intro i hi j hj hij
-    exact Function.iterate_injOn_Iio_minimalPeriod
-      (lt_of_lt_of_le (Finset.mem_range.mp hi) hr)
-      (lt_of_lt_of_le (Finset.mem_range.mp hj) hr) hij
-
-lemma walkSet_subset_orbit (p : Equiv.Perm α) (r : ℕ) (x : α) :
-    walkSet p r x ⊆ orbitSet p x := by
-  intro y hy
-  obtain ⟨n,_,rfl⟩ := (mem_walkSet _ _ _ _).mp hy
-  rw [mem_orbitSet,Equiv.Perm.iterate_eq_pow]
-  exact ⟨(n:ℤ),by simp⟩
-
-def cycleTrace (p : Equiv.Perm α) (S : Finset α) (a : α) :
-    ℕ → α → Finset α → Option (Finset α × α)
-  | 0, x, V => if x ∈ S ∧ x ∉ V ∧ p x = a then some (insert x V,x) else none
-  | r+1, x, V => if x ∈ S ∧ x ∉ V then cycleTrace p S a r (p x) (insert x V) else none
-
-lemma cycleTrace_walk (p : Equiv.Perm α) (S : Finset α) (r : ℕ) (x : α) (V : Finset α)
-    (hr : r < Function.minimalPeriod p x) (hS : walkSet p (r+1) x ⊆ S)
-    (hV : Disjoint V (walkSet p (r+1) x)) :
-    cycleTrace p S (p^[r+1] x) r x V = some (V ∪ walkSet p (r+1) x,p^[r] x) := by
-  induction r generalizing x V with
-  | zero =>
-    have hxS : x ∈ S := hS (by simp [walkSet_succ_first])
-    have hxV : x ∉ V := fun h => Finset.disjoint_left.mp hV h (by simp [walkSet_succ_first])
-    simp [cycleTrace,hxS,hxV,walkSet_succ_first]
-  | succ r ih =>
-    have hxS : x ∈ S := hS (by rw [walkSet_succ_first]; exact Finset.mem_insert_self _ _)
-    have hxV : x ∉ V := fun h => Finset.disjoint_left.mp hV h
-      (by rw [walkSet_succ_first]; exact Finset.mem_insert_self _ _)
-    have hr' : r < Function.minimalPeriod p (p x) := by
-      rw [Function.minimalPeriod_apply (p.injective.mem_periodicPts x)]
-      omega
-    have hS' : walkSet p (r+1) (p x) ⊆ S := by
-      intro y hy
-      apply hS
-      rw [walkSet_succ_first]
-      exact Finset.mem_insert_of_mem hy
-    have hV' : Disjoint (insert x V) (walkSet p (r+1) (p x)) := by
-      apply Finset.disjoint_left.mpr
-      intro y hy hy'
-      rcases Finset.mem_insert.mp hy with hy | hy
-      · subst y
-        exact not_mem_short_walk p x hr hy'
-      · apply Finset.disjoint_left.mp hV hy
-        rw [walkSet_succ_first]
-        exact Finset.mem_insert_of_mem hy'
-    rw [cycleTrace,ite_eq_left ⟨hxS,hxV⟩,Function.iterate_succ_apply p (r+1) x,
-      ih _ _ hr' hS' hV',walkSet_succ_first p (r+1) x,Function.iterate_succ_apply]
-    simp only [Finset.insert_union,Finset.union_insert]
-
-lemma cycleTrace_orbit (p : Equiv.Perm α) (S V : Finset α) (z : α)
-    (hS : orbitSet p z ⊆ S) (hV : Disjoint V (orbitSet p z)) :
-    cycleTrace p S (p z) (Function.minimalPeriod p z - 1) (p z) V =
-      some (V ∪ orbitSet p z,z) := by
-  have hpos := perm_minimalPeriod_pos p z
-  have he : Function.minimalPeriod p (p z) = Function.minimalPeriod p z :=
-    Function.minimalPeriod_apply (p.injective.mem_periodicPts z)
-  have hp : Function.minimalPeriod p z - 1 + 1 = Function.minimalPeriod p z := by omega
-  have hw : walkSet p (Function.minimalPeriod p z - 1 + 1) (p z) = orbitSet p z := by
-    rw [hp,←he,walkSet_period,orbitSet_apply]
-  have hr : Function.minimalPeriod p z - 1 < Function.minimalPeriod p (p z) := by rw [he]; omega
-  have h := cycleTrace_walk p S (Function.minimalPeriod p z-1) (p z) V hr
-    (by simpa only [hw] using hS) (by simpa only [hw] using hV)
-  have ht : p^[Function.minimalPeriod p z-1] (p z) = z := by
-    rw [← Function.iterate_succ_apply,show (Function.minimalPeriod p z-1).succ =
-      Function.minimalPeriod p z by omega,Function.iterate_minimalPeriod]
-  have ha : p^[Function.minimalPeriod p z-1+1] (p z) = p z := by
-    rw [hp,←he,Function.iterate_minimalPeriod]
-  simpa only [ha,ht,hw] using h
-
-end OrbitCore
-
-noncomputable def exposedPaths (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) :
-    Finset (SwitchIndex d) :=
-  V.biUnion (fun x => routeDomain d x (butterflyPerm d b x))
-
-@[simp] lemma exposedPaths_empty (d : ℕ) (b : Butterfly d) : exposedPaths d b ∅ = ∅ := by
-  classical
-  simp [exposedPaths]
-
-lemma exposedPaths_insert (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) (x : Card d) :
-    exposedPaths d b (insert x V) = exposedPaths d b V ∪ routeDomain d x (butterflyPerm d b x) := by
-  classical
-  simp [exposedPaths,Finset.union_comm]
-
-noncomputable def terminalCharge (d : ℕ) (b : Butterfly d) (z : Card d) (V : Finset (Card d)) : ℕ :=
-  (routeDomain d z (butterflyPerm d b z) ∩ exposedPaths d b V).card
-
-namespace QueryTree
-
-lemma followPath_stale_charge {ι : Type*} [DecidableEq ι]
-    (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
-    (k : Card d → QueryTree ι) (R : Finset ι) (ω : ι → Bool) :
-    staleCharge (followPath d e x k) R ω =
-      staleCharge (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x))
-        (R ∪ (routeDomain d x (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).image e) ω := by
-  have h₀ := fresh_add_stale (followPath d e x k) R ω
-  have h₁ := fresh_add_stale (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x))
-    (R ∪ (routeDomain d x (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).image e) ω
-  rw [followPath_charge,followPath_raw_charge] at h₀
-  omega
-
-lemma seekCycle_semantics (d : ℕ) (S : Finset (Card d)) (a : Card d) (r : ℕ)
-    (x : Card d) (V W : Finset (Card d)) (z : Card d)
-    (k : Finset (Card d) → QueryTree (SwitchIndex d)) (ω : SwitchIndex d → Bool)
-    (h : cycleTrace (butterflyPerm d (decodeButterfly d ω)) S a r x V = some (W,z)) :
-    (seekCycle d S a r x V k).succeeds ω = (k W).succeeds ω ∧
-      (seekCycle d S a r x V k).staleCharge (exposedPaths d (decodeButterfly d ω) V) ω =
-        (k W).staleCharge (exposedPaths d (decodeButterfly d ω) W) ω +
-          terminalCharge d (decodeButterfly d ω) z (W.erase z) := by
-  classical
-  induction r generalizing x V with
-  | zero =>
-    simp only [cycleTrace] at h
-    split_ifs at h with hv
-    obtain ⟨hxS,hxV,hxa⟩ := hv
-    obtain ⟨rfl,rfl⟩ := Prod.mk.inj (Option.some.inj h)
-    simp only [seekCycle,ite_eq_left (show x ∈ S ∧ x ∉ V from ⟨hxS,hxV⟩),closePath_succeeds,ite_eq_left hxa]
-    constructor
-    · trivial
-    · rw [closePath_stale_charge _ _ _ _ _ _ hxa,exposedPaths_insert,
-        Finset.erase_insert hxV]
-      simp only [terminalCharge,hxa]
-  | succ r ih =>
-    simp only [cycleTrace] at h
-    split_ifs at h with hv
-    obtain ⟨hxS,hxV⟩ := hv
-    rw [seekCycle,ite_eq_left ⟨hxS,hxV⟩,followPath_succeeds,followPath_stale_charge]
-    simp only [Function.comp_id,Finset.image_id]
-    rw [← exposedPaths_insert]
-    exact ih _ _ h
-
-lemma seekCycle_orbit (d : ℕ) (S V : Finset (Card d)) (z : Card d)
-    (k : Finset (Card d) → QueryTree (SwitchIndex d)) (ω : SwitchIndex d → Bool)
-    (hS : orbitSet (butterflyPerm d (decodeButterfly d ω)) z ⊆ S)
-    (hV : Disjoint V (orbitSet (butterflyPerm d (decodeButterfly d ω)) z)) :
-    let p := butterflyPerm d (decodeButterfly d ω)
-    let W := V ∪ orbitSet p z
-    (seekCycle d S (p z) (Function.minimalPeriod p z-1) (p z) V k).succeeds ω = (k W).succeeds ω ∧
-      (seekCycle d S (p z) (Function.minimalPeriod p z-1) (p z) V k).staleCharge
-          (exposedPaths d (decodeButterfly d ω) V) ω =
-        (k W).staleCharge (exposedPaths d (decodeButterfly d ω) W) ω +
-          terminalCharge d (decodeButterfly d ω) z (W.erase z) := by
-  exact seekCycle_semantics _ _ _ _ _ _ _ _ _ _ (cycleTrace_orbit _ _ _ _ hS hV)
-
-end QueryTree
-
-lemma mem_exposedPaths (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) (s : SwitchIndex d) :
-    s ∈ exposedPaths d b V ↔ (switchUsers d b s).1 ∈ V ∨ (switchUsers d b s).2 ∈ V := by
-  classical
-  simp only [exposedPaths,Finset.mem_biUnion,mem_routeDomain_iff_user]
-  constructor
-  · rintro ⟨x,hx,h | h⟩
-    · exact Or.inl (h ▸ hx)
-    · exact Or.inr (h ▸ hx)
-  · rintro (h | h)
-    · exact ⟨_,h,Or.inl rfl⟩
-    · exact ⟨_,h,Or.inr rfl⟩
-
-lemma terminalCharge_sum (d : ℕ) (b : Butterfly d) (z : Card d) (V : Finset (Card d))
-    (hz : z ∉ V) :
-    (terminalCharge d b z V : ℝ) = ∑ s : SwitchIndex d,
-      ((if (switchUsers d b s).1 = z ∧ (switchUsers d b s).2 ∈ V then (1:ℝ) else 0) +
-      (if (switchUsers d b s).2 = z ∧ (switchUsers d b s).1 ∈ V then (1:ℝ) else 0)) := by
-  classical
-  have he : routeDomain d z (butterflyPerm d b z) ∩ exposedPaths d b V =
-      Finset.univ.filter (fun s =>
-        ((switchUsers d b s).1 = z ∧ (switchUsers d b s).2 ∈ V) ∨
-        ((switchUsers d b s).2 = z ∧ (switchUsers d b s).1 ∈ V)) := by
-    ext s
-    simp only [Finset.mem_inter,mem_routeDomain_iff_user,mem_exposedPaths,
-      Finset.mem_filter,Finset.mem_univ,true_and]
-    by_cases h₁ : (switchUsers d b s).1 = z <;>
-      by_cases h₂ : (switchUsers d b s).2 = z <;> simp [h₁,h₂,hz]
-  rw [terminalCharge,he]
-  simp only [← Finset.sum_boole]
-  apply Finset.sum_congr rfl
-  intro s _
-  by_cases h₁ : (switchUsers d b s).1 = z <;>
-    by_cases h₂ : (switchUsers d b s).2 = z <;> simp [h₁,h₂,hz]
-
-noncomputable def priorPaths {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) : Finset (Card d) :=
-  Finset.univ.filter (fun x => ∃ i : Fin A, o x = some i ∧ i.val < t)
-
-@[simp] lemma mem_priorPaths {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) (x : Card d) :
-    x ∈ priorPaths o t ↔ ∃ i : Fin A, o x = some i ∧ i.val < t := by
-  classical
-  simp [priorPaths]
-
-lemma terminalCharge_prior_sum {d A : ℕ} (o : Card d → Option (Fin A))
-    (b : Butterfly d) (z : Terminal o) (i : Fin A) :
-    (terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) =
-      ∑ s : SwitchIndex d,
-        ((if (z i).val = (switchUsers d b s).1 ∧ (switchUsers d b s).2 ∈ priorPaths o (i.val+1)
-          then (1:ℝ) else 0) +
-        (if (z i).val = (switchUsers d b s).2 ∧ (switchUsers d b s).1 ∈ priorPaths o (i.val+1)
-          then (1:ℝ) else 0)) := by
-  classical
-  rw [terminalCharge_sum _ _ _ _ (Finset.notMem_erase _ _)]
-  apply Finset.sum_congr rfl
-  intro s _
-  have hn := switchUsers_ne d b s
-  simp only [Finset.mem_erase]
-  by_cases h₁ : (z i).val = (switchUsers d b s).1 <;>
-    by_cases h₂ : (z i).val = (switchUsers d b s).2 <;> simp_all [eq_comm]
-
-lemma rotationMeeting_eq_slot_sum {d A : ℕ} (o : Card d → Option (Fin A))
-    (z : Terminal o) (u v : Card d) :
-    rotationMeeting o z u v = ∑ i : Fin A,
-      ((if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) +
-      (if (z i).val = v ∧ u ∈ priorPaths o (i.val+1) then (1:ℝ) else 0)) := by
-  classical
-  have only_owner (u : Card d) (a : Fin A) (hou : o u = some a)
-      (i : Fin A) (hia : i ≠ a) : (z i).val ≠ u := by
-    intro h
-    have hi := (z i).property
-    rw [h,hou] at hi
-    exact hia (Option.some.inj hi).symm
-  have sum_one (u v : Card d) (a : Fin A) (hou : o u = some a) :
-      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) =
-      if (z a).val = u ∧ v ∈ priorPaths o (a.val+1) then 1 else 0 := by
-    apply Finset.sum_eq_single a
-    · intro i _ hia
-      simp [only_owner u a hou i hia]
-    · simp
-  have sum_none (u v : Card d) (hou : o u = none) :
-      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) = 0 := by
-    apply Finset.sum_eq_zero
-    intro i _
-    have hn : (z i).val ≠ u := by
-      intro h
-      have hi := (z i).property
-      rw [h,hou] at hi
-      contradiction
-    simp [hn]
-  rw [Finset.sum_add_distrib]
-  cases hou : o u with
-  | none =>
-    rw [sum_none u v hou]
-    simp [rotationMeeting,hou,mem_priorPaths]
-  | some a =>
-    rw [sum_one u v a hou]
-    cases hov : o v with
-    | none =>
-      rw [sum_none v u hov]
-      simp [rotationMeeting,hou,hov,mem_priorPaths]
-    | some b =>
-      rw [sum_one v u b hov]
-      simp only [rotationMeeting,hou,hov,mem_priorPaths,Option.some.injEq]
-      rcases lt_trichotomy a b with hab | rfl | hba
-      · have hn : ¬ b.val < a.val+1 := by omega
-        have hy : a.val < b.val+1 := by omega
-        simp [hab,hn,hy]
-      · simp
-      · have hn : ¬ a.val < b.val+1 := by omega
-        have hy : b.val < a.val+1 := by omega
-        simp [hba,not_lt_of_ge (le_of_lt hba),hn,hy]
-
-lemma rotationCharge_eq_terminal_sum {d A : ℕ} (o : Card d → Option (Fin A))
-    (b : Butterfly d) (z : Terminal o) :
-    rotationCharge o b z = ∑ i : Fin A,
-      (terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) := by
-  simp only [rotationCharge,rotationMeeting_eq_slot_sum,terminalCharge_prior_sum]
-  exact Finset.sum_comm
-
-@[simp] lemma priorPaths_zero {d A : ℕ} (o : Card d → Option (Fin A)) : priorPaths o 0 = ∅ := by
-  classical
-  ext x
-  simp
-
-lemma priorPaths_succ {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) (ht : t < A) :
-    priorPaths o (t+1) = priorPaths o t ∪ cycleFiber o ⟨t,ht⟩ := by
-  classical
-  ext x
-  simp only [mem_priorPaths,Finset.mem_union,mem_cycleFiber]
-  constructor
-  · rintro ⟨i,hi,hit⟩
-    by_cases h : i.val < t
-    · exact Or.inl ⟨i,hi,h⟩
-    · have he : i = ⟨t,ht⟩ := Fin.ext (by change i.val = t; omega)
-      exact Or.inr (he ▸ hi)
-  · rintro (⟨i,hi,hit⟩ | hi)
-    · exact ⟨i,hi,by omega⟩
-    · exact ⟨⟨t,ht⟩,hi,by change t < t+1; omega⟩
-
-lemma priorPaths_disjoint {d A : ℕ} (o : Card d → Option (Fin A)) (i : Fin A) :
-    Disjoint (priorPaths o i.val) (cycleFiber o i) := by
-  classical
-  apply Finset.disjoint_left.mpr
-  intro x hx hx'
-  obtain ⟨j,hj,hji⟩ := (mem_priorPaths _ _ _).mp hx
-  have he : j = i := Option.some.inj (hj.symm.trans ((mem_cycleFiber _ _ _).mp hx'))
-  subst j
-  exact (lt_irrefl _ hji)
-
-namespace QueryTree
-
-lemma seekCycles_owned {d A : ℕ} (S : Finset (Card d)) (r : Fin A → ℕ)
-    (o : Card d → Option (Fin A)) (z : Terminal o) (ω : SwitchIndex d → Bool)
-    (hS : ∀ i, cycleFiber o i ⊆ S)
-    (hO : ∀ i, orbitSet (butterflyPerm d (decodeButterfly d ω)) (z i).val = cycleFiber o i)
-    (hC : ∀ i, (cycleFiber o i).card = r i+1) :
-    let p := butterflyPerm d (decodeButterfly d ω)
-    let L := List.ofFn (fun i : Fin A => (p (z i).val,r i))
-    (seekCycles d S L ∅).succeeds ω = true ∧
-      ((seekCycles d S L ∅).staleCharge ∅ ω : ℝ) = rotationCharge o (decodeButterfly d ω) z := by
-  classical
-  let b := decodeButterfly d ω
-  let p := butterflyPerm d b
-  let f (i : Fin A) := (p (z i).val,r i)
-  let c (i : Fin A) : ℝ := terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val)
-  have hc (i : Fin A) : Function.minimalPeriod p (z i).val - 1 = r i := by
-    have he := orbitSet_card p (z i).val
-    rw [hO,hC] at he
-    omega
-  have hs (t : ℕ) (ht : t ≤ A) :
-      (seekCycles d S ((List.ofFn f).drop t) (priorPaths o t)).succeeds ω = true ∧
-        ((seekCycles d S ((List.ofFn f).drop t) (priorPaths o t)).staleCharge (exposedPaths d b (priorPaths o t)) ω : ℝ) =
-          ((List.ofFn c).drop t).sum := by
-    induction ht using Nat.decreasingInduction with
-    | self => simp only [List.length_ofFn,le_refl,List.drop_eq_nil_of_le,List.sum_nil,seekCycles,succeeds,staleCharge,Nat.cast_zero,and_self]
-    | of_succ t ht ih =>
-      have htf : t < (List.ofFn f).length := by simpa using ht
-      have htc : t < (List.ofFn c).length := by simpa using ht
-      rw [List.drop_eq_getElem_cons htf,List.drop_eq_getElem_cons htc]
-      simp only [List.getElem_ofFn,List.sum_cons,seekCycles]
-      have h := seekCycle_orbit d S (priorPaths o t) (z ⟨t,ht⟩).val
-        (fun V => seekCycles d S ((List.ofFn f).drop (t+1)) V) ω
-        (by rw [hO]; exact hS _) (by rw [hO]; exact priorPaths_disjoint o ⟨t,ht⟩)
-      dsimp only at h
-      rw [hc,hO,←priorPaths_succ o t ht] at h
-      dsimp only [f]
-      constructor
-      · exact h.1.trans ih.1
-      · rw [h.2,Nat.cast_add,ih.2]
-        exact add_comm _ _
-  have h := hs 0 (Nat.zero_le _)
-  simp only [List.drop_zero,priorPaths_zero,exposedPaths_empty,List.sum_ofFn] at h
-  dsimp only
-  exact ⟨h.1,h.2.trans (rotationCharge_eq_terminal_sum o b z).symm⟩
-
-end QueryTree
-
-section Collections
-variable {d A : ℕ}
-
-def ValidCycleSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ)
-    (o : Card d → Option (Fin A)) : Prop :=
-  (∀ i, (cycleFiber o i).card = r i + 1) ∧
-  (∀ i, cycleFiber o i ⊆ S) ∧
-  (∀ i x, o x = some i → orbitSet p x = cycleFiber o i)
-
-abbrev CycleSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
-  {o : Card d → Option (Fin A) // ValidCycleSlots p S r o}
-
-noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
-    Fintype (CycleSlots p S r) := Fintype.ofFinite _
-
-lemma CycleSlots.terminal_nonempty {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (c : CycleSlots p S r) (i : Fin A) : Nonempty {x : Card d // c.val x = some i} := by
-  have hn : (cycleFiber c.val i).Nonempty := Finset.card_pos.mp (by rw [c.property.1]; omega)
-  obtain ⟨x,hx⟩ := hn
-  exact ⟨x,(mem_cycleFiber _ _ _).mp hx⟩
-
-lemma owner_ext {o o' : Card d → Option (Fin A)}
-    (h : ∀ i, cycleFiber o i = cycleFiber o' i) : o = o' := by
-  funext x
-  have he (i : Fin A) : o x = some i ↔ o' x = some i := by
-    simpa only [mem_cycleFiber] using (Finset.ext_iff.mp (h i) x)
-  cases ho : o x with
-  | some i => exact ((he i).mp ho).symm
-  | none =>
-    cases ho' : o' x with
-    | none => rfl
-    | some i => have hn := (he i).mpr ho'; rw [ho] at hn; contradiction
-
-abbrev RootedSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
-  Σ c : CycleSlots p S r, Terminal c.val
-
-noncomputable def rootedStart {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (c : RootedSlots p S r) : Fin A → S := fun i =>
-  ⟨p (c.2 i).val,c.1.property.2.1 i (by
-    rw [←c.1.property.2.2 i (c.2 i).val (c.2 i).property]
-    exact (mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.apply_right (Equiv.Perm.SameCycle.refl p _)))⟩
-
-lemma rootedStart_injective {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ} :
-    Function.Injective (rootedStart (p:=p) (S:=S) (r:=r)) := by
-  rintro ⟨c,z⟩ ⟨c',z'⟩ h
-  have hz (i : Fin A) : (z i).val = (z' i).val :=
-    p.injective (congrArg Subtype.val (congrFun h i))
-  have hc : c = c' := Subtype.ext (owner_ext (fun i => by
-    rw [←c.property.2.2 i (z i).val (z i).property,
-      ←c'.property.2.2 i (z' i).val (z' i).property,hz]))
-  subst c'
-  have hzz : z = z' := funext (fun i => Subtype.ext (hz i))
-  subst z'
-  rfl
-
-lemma sum_comp_injective_le {α β : Type*} [Fintype α] [Fintype β]
-    (e : α → β) (he : Function.Injective e) (f : β → ℝ) (hf : ∀ b, 0 ≤ f b) :
-    ∑ a, f (e a) ≤ ∑ b, f b := by
-  classical
-  calc
-    _ = ∑ b ∈ Finset.univ.image e, f b := (Finset.sum_image (fun _ _ _ _ h => he h)).symm
-    _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _) (fun b _ _ => hf b)
-
-lemma rooted_weight_sum_le (S : Finset (Card d)) (r : Fin A → ℕ) (ω : SwitchIndex d → Bool) :
-    (∑ c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r,
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2)) ≤
-    ∑ a : Fin A → S,
-      if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-        ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-      else 0 := by
-  classical
-  let p := butterflyPerm d (decodeButterfly d ω)
-  let g (a : Fin A → S) : ℝ :=
-    if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-    else 0
-  have he (c : RootedSlots p S r) :
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2) =
-      g (rootedStart c) := by
-    have hs := QueryTree.seekCycles_owned S r c.1.val c.2 ω c.1.property.2.1
-      (fun i => c.1.property.2.2 i (c.2 i).val (c.2 i).property) c.1.property.1
-    dsimp only at hs
-    unfold g
-    rw [ite_eq_left (show (QueryTree.seekCycles d S (List.ofFn (fun i => ((rootedStart c i).val,r i))) ∅).succeeds ω = true from hs.1)]
-    rw [← hs.2,neg_mul,Real.exp_neg,mul_comm (Real.log 2),Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
-    rfl
-  simp_rw [he]
-  exact sum_comp_injective_le rootedStart rootedStart_injective g (fun a => by dsimp [g]; split_ifs <;> positivity)
-
-noncomputable def rotationLower (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : ℝ :=
-  Real.exp (-Real.log 2 * ((A:ℝ)*d/2 + ∑ i : Fin A, Real.log (r i+1) / (2 * Real.log 2)))
-
-noncomputable def collectionWeight (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : ℝ :=
-  ((2:ℝ)^d)^A * (∏ i : Fin A, (r i+1 : ℝ)) * rotationLower d r
-
-lemma collectionWeight_pos (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : 0 < collectionWeight d r := by
-  unfold collectionWeight rotationLower
-  positivity
-
-lemma collection_rotation_lower (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r)
-    (ω : SwitchIndex d → Bool) (c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) :
-    collectionWeight d r ≤ ∑ z : Terminal c.val,
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z) := by
-  classical
-  let (i : Fin A) : Nonempty {x : Card d // c.val x = some i} := c.terminal_nonempty i
-  have hm : Monotone (fun i => (cycleFiber c.val i).card) := by
-    intro i j hij
-    dsimp only
-    rw [c.property.1,c.property.1]
-    exact Nat.add_le_add_right (hr hij) 1
-  have hb := rotation_exp_bound hm (decodeButterfly d ω)
-  have hcard : (Fintype.card (Terminal c.val) : ℝ) = ∏ i : Fin A, (r i+1 : ℝ) := by
-    rw [Fintype.card_pi,Nat.cast_prod]
-    apply Finset.prod_congr rfl
-    intro i _
-    rw [card_cycleFiber,c.property.1,Nat.cast_add,Nat.cast_one]
-  have hp : (0:ℝ) < Fintype.card (Terminal c.val) := Nat.cast_pos.mpr Fintype.card_pos
-  have hl : rotationLower d r ≤ finiteMean (fun z : Terminal c.val => Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z)) := by
-    simpa only [rotationLower,c.property.1,Nat.cast_add,Nat.cast_one] using hb
-  have hh := (le_div_iff₀ hp).mp hl
-  rw [hcard] at hh
-  unfold collectionWeight
-  rw [← Finset.mul_sum]
-  calc
-    _ = ((2:ℝ)^d)^A * (rotationLower d r * ∏ i : Fin A, (r i+1 : ℝ)) := by ring
-    _ ≤ _ := mul_le_mul_of_nonneg_left hh (by positivity)
-
-lemma cycleSlots_weighted_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) *
-      collectionWeight d r ≤ (S.card : ℝ)^A := by
-  classical
-  let g (a : Fin A → S) (ω : SwitchIndex d → Bool) : ℝ :=
-    if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-    else 0
-  have hω (ω : SwitchIndex d → Bool) :
-      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ) * collectionWeight d r ≤
-        ∑ a : Fin A → S, g a ω := by
-    calc
-      _ = ∑ _c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r, collectionWeight d r := by simp
-      _ ≤ ∑ c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r,
-          ∑ z : Terminal c.val, ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z) :=
-        Finset.sum_le_sum (fun c _ => collection_rotation_lower S r hr ω c)
-      _ = ∑ c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r,
-          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2) :=
-        (Fintype.sum_sigma (fun c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r =>
-          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2))).symm
-      _ ≤ _ := rooted_weight_sum_le S r ω
-  have hb := finiteMean_mono hω
-  rw [finiteMean_mul_const,finiteMean_sum] at hb
-  calc
-    _ ≤ ∑ a : Fin A → S, finiteMean (g a) := hb
-    _ ≤ ∑ _a : Fin A → S, (1:ℝ) := by
-      apply Finset.sum_le_sum
-      intro a _
-      have hh := QueryTree.cycles_weighted_success d S (List.ofFn (fun i : Fin A => ((a i).val,r i)))
-      simpa only [List.length_ofFn] using hh
-    _ = _ := by simp
-
-lemma cycleSlots_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) ≤
-        (S.card : ℝ)^A / collectionWeight d r := by
-  exact (le_div_iff₀ (collectionWeight_pos d r)).mpr (cycleSlots_weighted_moment S r hr)
-
-lemma collectionWeight_eq_sqrt (d : ℕ) {A : ℕ} (r : Fin A → ℕ) :
-    collectionWeight d r = (Real.sqrt ((2:ℝ)^d))^A * ∏ i : Fin A, Real.sqrt (r i+1) := by
-  have hn : ((2:ℝ)^d)^A = Real.exp ((A:ℝ)*d*Real.log 2) := by
-    rw [mul_assoc,Real.exp_nat_mul,Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
-  have hp : (∏ i : Fin A, (r i+1 : ℝ)) = Real.exp (∑ i : Fin A, Real.log (r i+1)) := by
-    rw [Real.exp_sum]
-    apply Finset.prod_congr rfl
-    intro i _
-    exact (Real.exp_log (by positivity)).symm
-  have hs : Real.sqrt ((2:ℝ)^d) = Real.exp ((d:ℝ)*Real.log 2/2) := by
-    rw [Real.sqrt_eq_rpow,Real.rpow_def_of_pos (by positivity),Real.log_pow]
-    congr 1
-    ring
-  have hj (i : Fin A) : Real.sqrt (r i+1) = Real.exp (Real.log (r i+1)/2) := by
-    rw [Real.sqrt_eq_rpow,Real.rpow_def_of_pos (by positivity)]
-    congr 1
-    ring
-  simp_rw [collectionWeight,hn,hp,rotationLower,hs,hj]
-  rw [←Real.exp_add,←Real.exp_add,←Real.exp_nat_mul,←Real.exp_sum,←Real.exp_add]
-  congr 1
-  rw [←Finset.sum_div,←Finset.sum_div]
-  have hl : Real.log 2 ≠ 0 := ne_of_gt (Real.log_pos (by norm_num))
-  field_simp
-  ring
-
-lemma butterfly_cycleSlots_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) ≤
-        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
-  have h := cycleSlots_moment S r hr
-  rw [collectionWeight_eq_sqrt,div_mul_eq_div_mul_one_div] at h
-  rw [div_pow]
-  simpa only [one_div,Finset.prod_inv_distrib] using h
-
-end Collections
-
-variable {d A : ℕ}
-
-abbrev SelectedCycle (p : Equiv.Perm (Card d)) (S : Finset (Card d)) :=
-  {C : Finset (Card d) // (∃ x, C = orbitSet p x) ∧ C ⊆ S}
-
-noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) :
-    Fintype (SelectedCycle p S) := Fintype.ofFinite _
-
-lemma SelectedCycle.nonempty {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
-    (C : SelectedCycle p S) : C.val.Nonempty := by
-  obtain ⟨x,hx⟩ := C.property.1
-  rw [hx]
-  exact orbitSet_nonempty p x
-
-lemma SelectedCycle.orbit_eq {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
-    (C : SelectedCycle p S) {x : Card d} (hx : x ∈ C.val) : orbitSet p x = C.val := by
-  obtain ⟨y,hy⟩ := C.property.1
-  rw [hy] at hx ⊢
-  exact (orbitSet_eq p ((mem_orbitSet _ _ _).mp hx)).symm
-
-lemma SelectedCycle.eq_of_mem {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
-    (C D : SelectedCycle p S) {x : Card d} (hC : x ∈ C.val) (hD : x ∈ D.val) : C = D := by
-  apply Subtype.ext
-  exact (C.orbit_eq hC).symm.trans (D.orbit_eq hD)
-
-abbrev SlotAssignment (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
-  {f : Fin A → SelectedCycle p S // Function.Injective f ∧ ∀ i, (f i).val.card = r i+1}
-
-noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
-    Fintype (SlotAssignment p S r) := Fintype.ofFinite _
-
-noncomputable def slotsToAssignment {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (c : CycleSlots p S r) : SlotAssignment p S r := by
-  classical
-  let f (i : Fin A) : SelectedCycle p S := ⟨cycleFiber c.val i,by
-    obtain ⟨x,hx⟩ := c.terminal_nonempty i
-    exact ⟨⟨x,(c.property.2.2 i x hx).symm⟩,c.property.2.1 i⟩⟩
-  refine ⟨f,?_,fun i => c.property.1 i⟩
-  intro i j hij
-  have he : cycleFiber c.val i = cycleFiber c.val j := congrArg Subtype.val hij
-  obtain ⟨x,hx⟩ := c.terminal_nonempty i
-  have hxi : x ∈ cycleFiber c.val i := (mem_cycleFiber _ _ _).mpr hx
-  rw [he] at hxi
-  exact Option.some.inj (hx.symm.trans ((mem_cycleFiber _ _ _).mp hxi))
-
-noncomputable def assignmentOwner {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) (x : Card d) : Option (Fin A) := by
-  classical
-  exact if h : ∃ i, x ∈ (f.val i).val then some (Classical.choose h) else none
-
-lemma assignmentOwner_eq_some {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) (x : Card d) (i : Fin A) :
-    assignmentOwner f x = some i ↔ x ∈ (f.val i).val := by
-  classical
-  unfold assignmentOwner
-  split_ifs with h
-  · constructor
-    · intro hi
-      have hx := Classical.choose_spec h
-      rw [Option.some.inj hi] at hx
-      exact hx
-    · intro hx
-      have he := (f.val (Classical.choose h)).eq_of_mem (f.val i) (Classical.choose_spec h) hx
-      exact congrArg some (f.property.1 he)
-  · constructor
-    · intro hi; contradiction
-    · intro hx; exact (h ⟨i,hx⟩).elim
-
-end Thorp
-
-end ThorpNine.Dense
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Dense.ExposedCoins
+
+namespace OAI
+
+namespace ThorpNine.Dense
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+section Rotation
+variable {d A : ℕ}
+
+lemma card_cycleFiber (o : Card d → Option (Fin A)) (i : Fin A) :
+    Fintype.card {x : Card d // o x = some i} = (cycleFiber o i).card := by
+  classical
+  simp [cycleFiber,Fintype.card_subtype]
+
+lemma terminal_mean {o : Card d → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
+    (i : Fin A) (x : Card d) (hx : o x = some i) :
+    finiteMean (fun z : Terminal o => if (z i).val = x then (1:ℝ) else 0) = slotWeight o i := by
+  classical
+  have hev := finiteMean_pi_eval (A := fun i : Fin A => {x : Card d // o x = some i})
+    i (fun z => if z.val = x then (1:ℝ) else 0)
+  rw [hev]
+  have he : (fun z : {x : Card d // o x = some i} => if z.val = x then (1:ℝ) else 0) =
+      (fun z => if z = ⟨x,hx⟩ then 1 else 0) := by
+    funext z
+    simp only [Subtype.ext_iff]
+  rw [he,finiteMean_eq_indicator,card_cycleFiber,slotWeight]
+
+lemma slotWeight_antitone (o : Card d → Option (Fin A))
+    (hn : ∀ i, (cycleFiber o i).Nonempty)
+    (hm : Monotone (fun i => (cycleFiber o i).card)) : Antitone (slotWeight o) := by
+  intro i j hij
+  unfold slotWeight
+  apply one_div_le_one_div_of_le
+  · exact_mod_cast Finset.card_pos.mpr (hn i)
+  · exact_mod_cast hm hij
+
+lemma rotationMeeting_mean {o : Card d → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
+    (hm : Monotone (fun i => (cycleFiber o i).card)) (u v : Card d) :
+    finiteMean (fun z : Terminal o => rotationMeeting o z u v) =
+      min (pathWeight o u) (pathWeight o v) + withinWeight o u v := by
+  classical
+  have hn : ∀ i, (cycleFiber o i).Nonempty := by
+    intro i
+    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : Card d // o x = some i})
+    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
+  have hw := slotWeight_antitone o hn hm
+  cases hu : o u with
+  | none =>
+    cases hv : o v with
+    | none => simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean]
+    | some j =>
+      have hj : 0 ≤ slotWeight o j := by unfold slotWeight; positivity
+      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_left hj]
+  | some i =>
+    cases hv : o v with
+    | none =>
+      have hi : 0 ≤ slotWeight o i := by unfold slotWeight; positivity
+      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_right hi]
+    | some j =>
+      simp only [rotationMeeting,pathWeight,withinWeight,hu,hv]
+      rcases lt_trichotomy i j with hij | rfl | hji
+      · simp only [hij,↓reduceIte,ne_of_lt hij]
+        rw [terminal_mean j v hv,min_eq_right (hw (le_of_lt hij)),add_zero]
+      · simp only [lt_self_iff_false,↓reduceIte,min_self]
+        rw [finiteMean_add,terminal_mean i u hu,terminal_mean i v hv]
+      · simp only [not_lt_of_ge (le_of_lt hji),hji,↓reduceIte,ne_of_gt hji]
+        rw [terminal_mean i u hu,min_eq_left (hw (le_of_lt hji)),add_zero]
+
+lemma pathWeight_sum (o : Card d → Option (Fin A)) (hn : ∀ i, (cycleFiber o i).Nonempty) :
+    ∑ x : Card d, pathWeight o x = (A : ℝ) := by
+  classical
+  have he (x : Card d) : pathWeight o x =
+      ∑ i : Fin A, if o x = some i then slotWeight o i else 0 := by
+    cases hx : o x with
+    | none => simp [pathWeight,hx]
+    | some j => simp [pathWeight,hx]
+  simp only [he]
+  rw [Finset.sum_comm]
+  have hf (i : Fin A) : (∑ x : Card d, if o x = some i then slotWeight o i else 0) = 1 := by
+    rw [← Finset.sum_filter]
+    change (∑ _x ∈ cycleFiber o i, slotWeight o i) = 1
+    simp only [Finset.sum_const,nsmul_eq_mul,slotWeight]
+    exact mul_one_div_cancel (by exact_mod_cast (Finset.card_pos.mpr (hn i)).ne')
+  simp only [hf,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,mul_one]
+
+lemma withinWeight_sum_bound (o : Card d → Option (Fin A))
+    (hn : ∀ i, (cycleFiber o i).Nonempty) (b : Butterfly d) :
+    (∑ s : SwitchIndex d, withinWeight o (switchUsers d b s).1 (switchUsers d b s).2) ≤
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
+  classical
+  have he (u v : Card d) : withinWeight o u v =
+      ∑ i : Fin A, (if u ∈ cycleFiber o i ∧ v ∈ cycleFiber o i then (1:ℝ) else 0) * slotWeight o i := by
+    simp only [mem_cycleFiber]
+    cases hu : o u <;> cases hv : o v <;> simp [withinWeight,hu,hv,ite_and,eq_comm]
+  simp only [he]
+  rw [Finset.sum_comm]
+  apply Finset.sum_le_sum
+  intro i _
+  rw [← Finset.sum_mul]
+  have hi := marked_switch_entropy_bound d b (cycleFiber o i)
+  have hpos : (0:ℝ) < (cycleFiber o i).card := by exact_mod_cast Finset.card_pos.mpr (hn i)
+  have hmul := mul_le_mul_of_nonneg_right hi (show 0 ≤ slotWeight o i by unfold slotWeight; positivity)
+  calc
+    _ ≤ ((cycleFiber o i).card : ℝ) * Real.log (cycleFiber o i).card /
+        (2 * Real.log 2) * slotWeight o i := hmul
+    _ = _ := by unfold slotWeight; field_simp
+
+noncomputable def rotationCharge (o : Card d → Option (Fin A)) (b : Butterfly d)
+    (z : Terminal o) : ℝ :=
+  ∑ s : SwitchIndex d, rotationMeeting o z (switchUsers d b s).1 (switchUsers d b s).2
+
+lemma rotationCharge_mean_bound {o : Card d → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
+    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : Butterfly d) :
+    finiteMean (rotationCharge o b) ≤ (A:ℝ) * d / 2 +
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
+  classical
+  have hn : ∀ i, (cycleFiber o i).Nonempty := by
+    intro i
+    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : Card d // o x = some i})
+    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
+  unfold rotationCharge
+  rw [finiteMean_sum]
+  simp only [rotationMeeting_mean hm,Finset.sum_add_distrib]
+  have h₁ := weighted_switch_min_bound d b (pathWeight o)
+  rw [pathWeight_sum o hn] at h₁
+  have h₂ := withinWeight_sum_bound o hn b
+  linarith
+
+lemma rotation_exp_bound {o : Card d → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
+    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : Butterfly d) :
+    Real.exp (-Real.log 2 * ((A:ℝ)*d/2 +
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2))) ≤
+      finiteMean (fun z : Terminal o => Real.exp (-Real.log 2 * rotationCharge o b z)) := by
+  have hb := rotationCharge_mean_bound hm b
+  have hlog : 0 ≤ Real.log 2 := Real.log_nonneg (by norm_num)
+  have hm' : -Real.log 2 * ((A:ℝ)*d/2 +
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2)) ≤
+      finiteMean (fun z : Terminal o => -Real.log 2 * rotationCharge o b z) := by
+    have he := finiteMean_mul_const (rotationCharge o b) (-Real.log 2)
+    simp only [mul_comm] at he
+    rw [he]
+    exact mul_le_mul_of_nonpos_left hb (neg_nonpos.mpr hlog)
+  exact (Real.exp_le_exp.mpr hm').trans (exp_finiteMean_le _)
+
+end Rotation
+
+section OrbitCore
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+@[simp] lemma mem_orbitSet (p : Equiv.Perm α) (x y : α) :
+    y ∈ orbitSet p x ↔ p.SameCycle x y := by
+  classical
+  simp [orbitSet]
+
+lemma orbitSet_nonempty (p : Equiv.Perm α) (x : α) : (orbitSet p x).Nonempty :=
+  ⟨x,(mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.refl _ _)⟩
+
+lemma orbitSet_eq (p : Equiv.Perm α) {x y : α} (h : p.SameCycle x y) :
+    orbitSet p x = orbitSet p y := by
+  ext z
+  simp only [mem_orbitSet]
+  exact ⟨fun hz => h.symm.trans hz,fun hz => h.trans hz⟩
+
+@[simp] lemma orbitSet_apply (p : Equiv.Perm α) (x : α) : orbitSet p (p x) = orbitSet p x := by
+  ext y
+  simp only [mem_orbitSet,Equiv.Perm.sameCycle_apply_left]
+
+lemma orbitSet_eq_or_disjoint (p : Equiv.Perm α) (x y : α) :
+    orbitSet p x = orbitSet p y ∨ Disjoint (orbitSet p x) (orbitSet p y) := by
+  classical
+  by_cases h : p.SameCycle x y
+  · exact Or.inl (orbitSet_eq p h)
+  · right
+    apply Finset.disjoint_left.mpr
+    intro z hz hz'
+    exact h (((mem_orbitSet _ _ _).mp hz).trans ((mem_orbitSet _ _ _).mp hz').symm)
+
+omit [DecidableEq α] in
+lemma perm_minimalPeriod_pos (p : Equiv.Perm α) (x : α) :
+    0 < Function.minimalPeriod p x :=
+  Function.minimalPeriod_pos_of_mem_periodicPts (p.injective.mem_periodicPts x)
+
+lemma orbitSet_card (p : Equiv.Perm α) (x : α) :
+    (orbitSet p x).card = Function.minimalPeriod p x := by
+  classical
+  let f : Fin (Function.minimalPeriod p x) → {y // p.SameCycle x y} := fun n =>
+    ⟨p^[n.val] x,by rw [Equiv.Perm.iterate_eq_pow]; exact ⟨(n.val:ℤ),by simp⟩⟩
+  have hf : Function.Bijective f := by
+    constructor
+    · intro i j hij
+      apply Fin.ext
+      exact Function.iterate_injOn_Iio_minimalPeriod i.isLt j.isLt (congrArg Subtype.val hij)
+    · intro y
+      obtain ⟨n,hn⟩ := y.property.exists_nat_pow_eq
+      refine ⟨⟨n % Function.minimalPeriod p x, Nat.mod_lt _ (perm_minimalPeriod_pos p x)⟩,?_⟩
+      apply Subtype.ext
+      change p^[n % Function.minimalPeriod p x] x = y.val
+      rw [Function.iterate_mod_minimalPeriod_eq,Equiv.Perm.iterate_eq_pow,hn]
+  have hc := Fintype.card_congr (Equiv.ofBijective f hf)
+  simpa [orbitSet,Fintype.card_subtype] using hc.symm
+
+def walkSet (p : Equiv.Perm α) (r : ℕ) (x : α) : Finset α :=
+  (Finset.range r).image (fun n => p^[n] x)
+
+omit [Fintype α] in
+@[simp] lemma mem_walkSet (p : Equiv.Perm α) (r : ℕ) (x y : α) :
+    y ∈ walkSet p r x ↔ ∃ n < r, p^[n] x = y := by
+  simp [walkSet]
+
+omit [Fintype α] in
+@[simp] lemma walkSet_zero (p : Equiv.Perm α) (x : α) : walkSet p 0 x = ∅ := by
+  simp [walkSet]
+
+omit [Fintype α] in
+lemma walkSet_succ_last (p : Equiv.Perm α) (r : ℕ) (x : α) :
+    walkSet p (r+1) x = insert (p^[r] x) (walkSet p r x) := by
+  simp [walkSet,Finset.range_add_one]
+
+omit [Fintype α] in
+lemma walkSet_succ_first (p : Equiv.Perm α) (r : ℕ) (x : α) :
+    walkSet p (r+1) x = insert x (walkSet p r (p x)) := by
+  ext y
+  simp only [mem_walkSet,Finset.mem_insert]
+  constructor
+  · rintro ⟨n,hn,hny⟩
+    cases n with
+    | zero => exact Or.inl (by simpa using hny.symm)
+    | succ n => exact Or.inr ⟨n,by omega,by simpa only [Function.iterate_succ_apply] using hny⟩
+  · rintro (rfl | ⟨n,hn,hny⟩)
+    · exact ⟨0,by omega,rfl⟩
+    · exact ⟨n+1,by omega,by simpa only [Function.iterate_succ_apply] using hny⟩
+
+omit [Fintype α] in
+lemma not_mem_short_walk (p : Equiv.Perm α) (x : α) {r : ℕ}
+    (hr : r < Function.minimalPeriod p x) : x ∉ walkSet p r (p x) := by
+  rintro hx
+  obtain ⟨n,hn,hnx⟩ := (mem_walkSet _ _ _ _).mp hx
+  have hperiod : Function.IsPeriodicPt p (n+1) x := by
+    simpa only [Function.IsPeriodicPt,Function.IsFixedPt,Function.iterate_succ_apply] using hnx
+  have hp := hperiod.minimalPeriod_le (by omega : 0 < n+1)
+  omega
+
+lemma walkSet_period (p : Equiv.Perm α) (x : α) :
+    walkSet p (Function.minimalPeriod p x) x = orbitSet p x := by
+  ext y
+  simp only [mem_walkSet,mem_orbitSet]
+  constructor
+  · rintro ⟨n,_,rfl⟩
+    rw [Equiv.Perm.iterate_eq_pow]
+    exact ⟨(n:ℤ),by simp⟩
+  · intro h
+    obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
+    refine ⟨n % Function.minimalPeriod p x,Nat.mod_lt _ (perm_minimalPeriod_pos p x),?_⟩
+    rw [Function.iterate_mod_minimalPeriod_eq,Equiv.Perm.iterate_eq_pow,hn]
+
+omit [Fintype α] in
+lemma walkSet_card (p : Equiv.Perm α) (x : α) {r : ℕ}
+    (hr : r ≤ Function.minimalPeriod p x) : (walkSet p r x).card = r := by
+  rw [walkSet,Finset.card_image_of_injOn]
+  · exact Finset.card_range _
+  · intro i hi j hj hij
+    exact Function.iterate_injOn_Iio_minimalPeriod
+      (lt_of_lt_of_le (Finset.mem_range.mp hi) hr)
+      (lt_of_lt_of_le (Finset.mem_range.mp hj) hr) hij
+
+lemma walkSet_subset_orbit (p : Equiv.Perm α) (r : ℕ) (x : α) :
+    walkSet p r x ⊆ orbitSet p x := by
+  intro y hy
+  obtain ⟨n,_,rfl⟩ := (mem_walkSet _ _ _ _).mp hy
+  rw [mem_orbitSet,Equiv.Perm.iterate_eq_pow]
+  exact ⟨(n:ℤ),by simp⟩
+
+def cycleTrace (p : Equiv.Perm α) (S : Finset α) (a : α) :
+    ℕ → α → Finset α → Option (Finset α × α)
+  | 0, x, V => if x ∈ S ∧ x ∉ V ∧ p x = a then some (insert x V,x) else none
+  | r+1, x, V => if x ∈ S ∧ x ∉ V then cycleTrace p S a r (p x) (insert x V) else none
+
+lemma cycleTrace_walk (p : Equiv.Perm α) (S : Finset α) (r : ℕ) (x : α) (V : Finset α)
+    (hr : r < Function.minimalPeriod p x) (hS : walkSet p (r+1) x ⊆ S)
+    (hV : Disjoint V (walkSet p (r+1) x)) :
+    cycleTrace p S (p^[r+1] x) r x V = some (V ∪ walkSet p (r+1) x,p^[r] x) := by
+  induction r generalizing x V with
+  | zero =>
+    have hxS : x ∈ S := hS (by simp [walkSet_succ_first])
+    have hxV : x ∉ V := fun h => Finset.disjoint_left.mp hV h (by simp [walkSet_succ_first])
+    simp [cycleTrace,hxS,hxV,walkSet_succ_first]
+  | succ r ih =>
+    have hxS : x ∈ S := hS (by rw [walkSet_succ_first]; exact Finset.mem_insert_self _ _)
+    have hxV : x ∉ V := fun h => Finset.disjoint_left.mp hV h
+      (by rw [walkSet_succ_first]; exact Finset.mem_insert_self _ _)
+    have hr' : r < Function.minimalPeriod p (p x) := by
+      rw [Function.minimalPeriod_apply (p.injective.mem_periodicPts x)]
+      omega
+    have hS' : walkSet p (r+1) (p x) ⊆ S := by
+      intro y hy
+      apply hS
+      rw [walkSet_succ_first]
+      exact Finset.mem_insert_of_mem hy
+    have hV' : Disjoint (insert x V) (walkSet p (r+1) (p x)) := by
+      apply Finset.disjoint_left.mpr
+      intro y hy hy'
+      rcases Finset.mem_insert.mp hy with hy | hy
+      · subst y
+        exact not_mem_short_walk p x hr hy'
+      · apply Finset.disjoint_left.mp hV hy
+        rw [walkSet_succ_first]
+        exact Finset.mem_insert_of_mem hy'
+    rw [cycleTrace,ite_eq_left ⟨hxS,hxV⟩,Function.iterate_succ_apply p (r+1) x,
+      ih _ _ hr' hS' hV',walkSet_succ_first p (r+1) x,Function.iterate_succ_apply]
+    simp only [Finset.insert_union,Finset.union_insert]
+
+lemma cycleTrace_orbit (p : Equiv.Perm α) (S V : Finset α) (z : α)
+    (hS : orbitSet p z ⊆ S) (hV : Disjoint V (orbitSet p z)) :
+    cycleTrace p S (p z) (Function.minimalPeriod p z - 1) (p z) V =
+      some (V ∪ orbitSet p z,z) := by
+  have hpos := perm_minimalPeriod_pos p z
+  have he : Function.minimalPeriod p (p z) = Function.minimalPeriod p z :=
+    Function.minimalPeriod_apply (p.injective.mem_periodicPts z)
+  have hp : Function.minimalPeriod p z - 1 + 1 = Function.minimalPeriod p z := by omega
+  have hw : walkSet p (Function.minimalPeriod p z - 1 + 1) (p z) = orbitSet p z := by
+    rw [hp,←he,walkSet_period,orbitSet_apply]
+  have hr : Function.minimalPeriod p z - 1 < Function.minimalPeriod p (p z) := by rw [he]; omega
+  have h := cycleTrace_walk p S (Function.minimalPeriod p z-1) (p z) V hr
+    (by simpa only [hw] using hS) (by simpa only [hw] using hV)
+  have ht : p^[Function.minimalPeriod p z-1] (p z) = z := by
+    rw [← Function.iterate_succ_apply,show (Function.minimalPeriod p z-1).succ =
+      Function.minimalPeriod p z by omega,Function.iterate_minimalPeriod]
+  have ha : p^[Function.minimalPeriod p z-1+1] (p z) = p z := by
+    rw [hp,←he,Function.iterate_minimalPeriod]
+  simpa only [ha,ht,hw] using h
+
+end OrbitCore
+
+noncomputable def exposedPaths (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) :
+    Finset (SwitchIndex d) :=
+  V.biUnion (fun x => routeDomain d x (butterflyPerm d b x))
+
+@[simp] lemma exposedPaths_empty (d : ℕ) (b : Butterfly d) : exposedPaths d b ∅ = ∅ := by
+  classical
+  simp [exposedPaths]
+
+lemma exposedPaths_insert (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) (x : Card d) :
+    exposedPaths d b (insert x V) = exposedPaths d b V ∪ routeDomain d x (butterflyPerm d b x) := by
+  classical
+  simp [exposedPaths,Finset.union_comm]
+
+noncomputable def terminalCharge (d : ℕ) (b : Butterfly d) (z : Card d) (V : Finset (Card d)) : ℕ :=
+  (routeDomain d z (butterflyPerm d b z) ∩ exposedPaths d b V).card
+
+namespace QueryTree
+
+lemma followPath_stale_charge {ι : Type*} [DecidableEq ι]
+    (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
+    (k : Card d → QueryTree ι) (R : Finset ι) (ω : ι → Bool) :
+    staleCharge (followPath d e x k) R ω =
+      staleCharge (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x))
+        (R ∪ (routeDomain d x (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).image e) ω := by
+  have h₀ := fresh_add_stale (followPath d e x k) R ω
+  have h₁ := fresh_add_stale (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x))
+    (R ∪ (routeDomain d x (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).image e) ω
+  rw [followPath_charge,followPath_raw_charge] at h₀
+  omega
+
+lemma seekCycle_semantics (d : ℕ) (S : Finset (Card d)) (a : Card d) (r : ℕ)
+    (x : Card d) (V W : Finset (Card d)) (z : Card d)
+    (k : Finset (Card d) → QueryTree (SwitchIndex d)) (ω : SwitchIndex d → Bool)
+    (h : cycleTrace (butterflyPerm d (decodeButterfly d ω)) S a r x V = some (W,z)) :
+    (seekCycle d S a r x V k).succeeds ω = (k W).succeeds ω ∧
+      (seekCycle d S a r x V k).staleCharge (exposedPaths d (decodeButterfly d ω) V) ω =
+        (k W).staleCharge (exposedPaths d (decodeButterfly d ω) W) ω +
+          terminalCharge d (decodeButterfly d ω) z (W.erase z) := by
+  classical
+  induction r generalizing x V with
+  | zero =>
+    simp only [cycleTrace] at h
+    split_ifs at h with hv
+    obtain ⟨hxS,hxV,hxa⟩ := hv
+    obtain ⟨rfl,rfl⟩ := Prod.mk.inj (Option.some.inj h)
+    simp only [seekCycle,ite_eq_left (show x ∈ S ∧ x ∉ V from ⟨hxS,hxV⟩),closePath_succeeds,ite_eq_left hxa]
+    constructor
+    · trivial
+    · rw [closePath_stale_charge _ _ _ _ _ _ hxa,exposedPaths_insert,
+        Finset.erase_insert hxV]
+      simp only [terminalCharge,hxa]
+  | succ r ih =>
+    simp only [cycleTrace] at h
+    split_ifs at h with hv
+    obtain ⟨hxS,hxV⟩ := hv
+    rw [seekCycle,ite_eq_left ⟨hxS,hxV⟩,followPath_succeeds,followPath_stale_charge]
+    simp only [Function.comp_id,Finset.image_id]
+    rw [← exposedPaths_insert]
+    exact ih _ _ h
+
+lemma seekCycle_orbit (d : ℕ) (S V : Finset (Card d)) (z : Card d)
+    (k : Finset (Card d) → QueryTree (SwitchIndex d)) (ω : SwitchIndex d → Bool)
+    (hS : orbitSet (butterflyPerm d (decodeButterfly d ω)) z ⊆ S)
+    (hV : Disjoint V (orbitSet (butterflyPerm d (decodeButterfly d ω)) z)) :
+    let p := butterflyPerm d (decodeButterfly d ω)
+    let W := V ∪ orbitSet p z
+    (seekCycle d S (p z) (Function.minimalPeriod p z-1) (p z) V k).succeeds ω = (k W).succeeds ω ∧
+      (seekCycle d S (p z) (Function.minimalPeriod p z-1) (p z) V k).staleCharge
+          (exposedPaths d (decodeButterfly d ω) V) ω =
+        (k W).staleCharge (exposedPaths d (decodeButterfly d ω) W) ω +
+          terminalCharge d (decodeButterfly d ω) z (W.erase z) := by
+  exact seekCycle_semantics _ _ _ _ _ _ _ _ _ _ (cycleTrace_orbit _ _ _ _ hS hV)
+
+end QueryTree
+
+lemma mem_exposedPaths (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) (s : SwitchIndex d) :
+    s ∈ exposedPaths d b V ↔ (switchUsers d b s).1 ∈ V ∨ (switchUsers d b s).2 ∈ V := by
+  classical
+  simp only [exposedPaths,Finset.mem_biUnion,mem_routeDomain_iff_user]
+  constructor
+  · rintro ⟨x,hx,h | h⟩
+    · exact Or.inl (h ▸ hx)
+    · exact Or.inr (h ▸ hx)
+  · rintro (h | h)
+    · exact ⟨_,h,Or.inl rfl⟩
+    · exact ⟨_,h,Or.inr rfl⟩
+
+lemma terminalCharge_sum (d : ℕ) (b : Butterfly d) (z : Card d) (V : Finset (Card d))
+    (hz : z ∉ V) :
+    (terminalCharge d b z V : ℝ) = ∑ s : SwitchIndex d,
+      ((if (switchUsers d b s).1 = z ∧ (switchUsers d b s).2 ∈ V then (1:ℝ) else 0) +
+      (if (switchUsers d b s).2 = z ∧ (switchUsers d b s).1 ∈ V then (1:ℝ) else 0)) := by
+  classical
+  have he : routeDomain d z (butterflyPerm d b z) ∩ exposedPaths d b V =
+      Finset.univ.filter (fun s =>
+        ((switchUsers d b s).1 = z ∧ (switchUsers d b s).2 ∈ V) ∨
+        ((switchUsers d b s).2 = z ∧ (switchUsers d b s).1 ∈ V)) := by
+    ext s
+    simp only [Finset.mem_inter,mem_routeDomain_iff_user,mem_exposedPaths,
+      Finset.mem_filter,Finset.mem_univ,true_and]
+    by_cases h₁ : (switchUsers d b s).1 = z <;>
+      by_cases h₂ : (switchUsers d b s).2 = z <;> simp [h₁,h₂,hz]
+  rw [terminalCharge,he]
+  simp only [← Finset.sum_boole]
+  apply Finset.sum_congr rfl
+  intro s _
+  by_cases h₁ : (switchUsers d b s).1 = z <;>
+    by_cases h₂ : (switchUsers d b s).2 = z <;> simp [h₁,h₂,hz]
+
+noncomputable def priorPaths {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) : Finset (Card d) :=
+  Finset.univ.filter (fun x => ∃ i : Fin A, o x = some i ∧ i.val < t)
+
+@[simp] lemma mem_priorPaths {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) (x : Card d) :
+    x ∈ priorPaths o t ↔ ∃ i : Fin A, o x = some i ∧ i.val < t := by
+  classical
+  simp [priorPaths]
+
+lemma terminalCharge_prior_sum {d A : ℕ} (o : Card d → Option (Fin A))
+    (b : Butterfly d) (z : Terminal o) (i : Fin A) :
+    (terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) =
+      ∑ s : SwitchIndex d,
+        ((if (z i).val = (switchUsers d b s).1 ∧ (switchUsers d b s).2 ∈ priorPaths o (i.val+1)
+          then (1:ℝ) else 0) +
+        (if (z i).val = (switchUsers d b s).2 ∧ (switchUsers d b s).1 ∈ priorPaths o (i.val+1)
+          then (1:ℝ) else 0)) := by
+  classical
+  rw [terminalCharge_sum _ _ _ _ (Finset.notMem_erase _ _)]
+  apply Finset.sum_congr rfl
+  intro s _
+  have hn := switchUsers_ne d b s
+  simp only [Finset.mem_erase]
+  by_cases h₁ : (z i).val = (switchUsers d b s).1 <;>
+    by_cases h₂ : (z i).val = (switchUsers d b s).2 <;> simp_all [eq_comm]
+
+lemma rotationMeeting_eq_slot_sum {d A : ℕ} (o : Card d → Option (Fin A))
+    (z : Terminal o) (u v : Card d) :
+    rotationMeeting o z u v = ∑ i : Fin A,
+      ((if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) +
+      (if (z i).val = v ∧ u ∈ priorPaths o (i.val+1) then (1:ℝ) else 0)) := by
+  classical
+  have only_owner (u : Card d) (a : Fin A) (hou : o u = some a)
+      (i : Fin A) (hia : i ≠ a) : (z i).val ≠ u := by
+    intro h
+    have hi := (z i).property
+    rw [h,hou] at hi
+    exact hia (Option.some.inj hi).symm
+  have sum_one (u v : Card d) (a : Fin A) (hou : o u = some a) :
+      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) =
+      if (z a).val = u ∧ v ∈ priorPaths o (a.val+1) then 1 else 0 := by
+    apply Finset.sum_eq_single a
+    · intro i _ hia
+      simp [only_owner u a hou i hia]
+    · simp
+  have sum_none (u v : Card d) (hou : o u = none) :
+      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) = 0 := by
+    apply Finset.sum_eq_zero
+    intro i _
+    have hn : (z i).val ≠ u := by
+      intro h
+      have hi := (z i).property
+      rw [h,hou] at hi
+      contradiction
+    simp [hn]
+  rw [Finset.sum_add_distrib]
+  cases hou : o u with
+  | none =>
+    rw [sum_none u v hou]
+    simp [rotationMeeting,hou,mem_priorPaths]
+  | some a =>
+    rw [sum_one u v a hou]
+    cases hov : o v with
+    | none =>
+      rw [sum_none v u hov]
+      simp [rotationMeeting,hou,hov,mem_priorPaths]
+    | some b =>
+      rw [sum_one v u b hov]
+      simp only [rotationMeeting,hou,hov,mem_priorPaths,Option.some.injEq]
+      rcases lt_trichotomy a b with hab | rfl | hba
+      · have hn : ¬ b.val < a.val+1 := by omega
+        have hy : a.val < b.val+1 := by omega
+        simp [hab,hn,hy]
+      · simp
+      · have hn : ¬ a.val < b.val+1 := by omega
+        have hy : b.val < a.val+1 := by omega
+        simp [hba,not_lt_of_ge (le_of_lt hba),hn,hy]
+
+lemma rotationCharge_eq_terminal_sum {d A : ℕ} (o : Card d → Option (Fin A))
+    (b : Butterfly d) (z : Terminal o) :
+    rotationCharge o b z = ∑ i : Fin A,
+      (terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) := by
+  simp only [rotationCharge,rotationMeeting_eq_slot_sum,terminalCharge_prior_sum]
+  exact Finset.sum_comm
+
+@[simp] lemma priorPaths_zero {d A : ℕ} (o : Card d → Option (Fin A)) : priorPaths o 0 = ∅ := by
+  classical
+  ext x
+  simp
+
+lemma priorPaths_succ {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) (ht : t < A) :
+    priorPaths o (t+1) = priorPaths o t ∪ cycleFiber o ⟨t,ht⟩ := by
+  classical
+  ext x
+  simp only [mem_priorPaths,Finset.mem_union,mem_cycleFiber]
+  constructor
+  · rintro ⟨i,hi,hit⟩
+    by_cases h : i.val < t
+    · exact Or.inl ⟨i,hi,h⟩
+    · have he : i = ⟨t,ht⟩ := Fin.ext (by change i.val = t; omega)
+      exact Or.inr (he ▸ hi)
+  · rintro (⟨i,hi,hit⟩ | hi)
+    · exact ⟨i,hi,by omega⟩
+    · exact ⟨⟨t,ht⟩,hi,by change t < t+1; omega⟩
+
+lemma priorPaths_disjoint {d A : ℕ} (o : Card d → Option (Fin A)) (i : Fin A) :
+    Disjoint (priorPaths o i.val) (cycleFiber o i) := by
+  classical
+  apply Finset.disjoint_left.mpr
+  intro x hx hx'
+  obtain ⟨j,hj,hji⟩ := (mem_priorPaths _ _ _).mp hx
+  have he : j = i := Option.some.inj (hj.symm.trans ((mem_cycleFiber _ _ _).mp hx'))
+  subst j
+  exact (lt_irrefl _ hji)
+
+namespace QueryTree
+
+lemma seekCycles_owned {d A : ℕ} (S : Finset (Card d)) (r : Fin A → ℕ)
+    (o : Card d → Option (Fin A)) (z : Terminal o) (ω : SwitchIndex d → Bool)
+    (hS : ∀ i, cycleFiber o i ⊆ S)
+    (hO : ∀ i, orbitSet (butterflyPerm d (decodeButterfly d ω)) (z i).val = cycleFiber o i)
+    (hC : ∀ i, (cycleFiber o i).card = r i+1) :
+    let p := butterflyPerm d (decodeButterfly d ω)
+    let L := List.ofFn (fun i : Fin A => (p (z i).val,r i))
+    (seekCycles d S L ∅).succeeds ω = true ∧
+      ((seekCycles d S L ∅).staleCharge ∅ ω : ℝ) = rotationCharge o (decodeButterfly d ω) z := by
+  classical
+  let b := decodeButterfly d ω
+  let p := butterflyPerm d b
+  let f (i : Fin A) := (p (z i).val,r i)
+  let c (i : Fin A) : ℝ := terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val)
+  have hc (i : Fin A) : Function.minimalPeriod p (z i).val - 1 = r i := by
+    have he := orbitSet_card p (z i).val
+    rw [hO,hC] at he
+    omega
+  have hs (t : ℕ) (ht : t ≤ A) :
+      (seekCycles d S ((List.ofFn f).drop t) (priorPaths o t)).succeeds ω = true ∧
+        ((seekCycles d S ((List.ofFn f).drop t) (priorPaths o t)).staleCharge (exposedPaths d b (priorPaths o t)) ω : ℝ) =
+          ((List.ofFn c).drop t).sum := by
+    induction ht using Nat.decreasingInduction with
+    | self => simp only [List.length_ofFn,le_refl,List.drop_eq_nil_of_le,List.sum_nil,seekCycles,succeeds,staleCharge,Nat.cast_zero,and_self]
+    | of_succ t ht ih =>
+      have htf : t < (List.ofFn f).length := by simpa using ht
+      have htc : t < (List.ofFn c).length := by simpa using ht
+      rw [List.drop_eq_getElem_cons htf,List.drop_eq_getElem_cons htc]
+      simp only [List.getElem_ofFn,List.sum_cons,seekCycles]
+      have h := seekCycle_orbit d S (priorPaths o t) (z ⟨t,ht⟩).val
+        (fun V => seekCycles d S ((List.ofFn f).drop (t+1)) V) ω
+        (by rw [hO]; exact hS _) (by rw [hO]; exact priorPaths_disjoint o ⟨t,ht⟩)
+      dsimp only at h
+      rw [hc,hO,←priorPaths_succ o t ht] at h
+      dsimp only [f]
+      constructor
+      · exact h.1.trans ih.1
+      · rw [h.2,Nat.cast_add,ih.2]
+        exact add_comm _ _
+  have h := hs 0 (Nat.zero_le _)
+  simp only [List.drop_zero,priorPaths_zero,exposedPaths_empty,List.sum_ofFn] at h
+  dsimp only
+  exact ⟨h.1,h.2.trans (rotationCharge_eq_terminal_sum o b z).symm⟩
+
+end QueryTree
+
+section Collections
+variable {d A : ℕ}
+
+def ValidCycleSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ)
+    (o : Card d → Option (Fin A)) : Prop :=
+  (∀ i, (cycleFiber o i).card = r i + 1) ∧
+  (∀ i, cycleFiber o i ⊆ S) ∧
+  (∀ i x, o x = some i → orbitSet p x = cycleFiber o i)
+
+abbrev CycleSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
+  {o : Card d → Option (Fin A) // ValidCycleSlots p S r o}
+
+noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
+    Fintype (CycleSlots p S r) := Fintype.ofFinite _
+
+lemma CycleSlots.terminal_nonempty {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (c : CycleSlots p S r) (i : Fin A) : Nonempty {x : Card d // c.val x = some i} := by
+  have hn : (cycleFiber c.val i).Nonempty := Finset.card_pos.mp (by rw [c.property.1]; omega)
+  obtain ⟨x,hx⟩ := hn
+  exact ⟨x,(mem_cycleFiber _ _ _).mp hx⟩
+
+lemma owner_ext {o o' : Card d → Option (Fin A)}
+    (h : ∀ i, cycleFiber o i = cycleFiber o' i) : o = o' := by
+  funext x
+  have he (i : Fin A) : o x = some i ↔ o' x = some i := by
+    simpa only [mem_cycleFiber] using (Finset.ext_iff.mp (h i) x)
+  cases ho : o x with
+  | some i => exact ((he i).mp ho).symm
+  | none =>
+    cases ho' : o' x with
+    | none => rfl
+    | some i => have hn := (he i).mpr ho'; rw [ho] at hn; contradiction
+
+abbrev RootedSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
+  Σ c : CycleSlots p S r, Terminal c.val
+
+noncomputable def rootedStart {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (c : RootedSlots p S r) : Fin A → S := fun i =>
+  ⟨p (c.2 i).val,c.1.property.2.1 i (by
+    rw [←c.1.property.2.2 i (c.2 i).val (c.2 i).property]
+    exact (mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.apply_right (Equiv.Perm.SameCycle.refl p _)))⟩
+
+lemma rootedStart_injective {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ} :
+    Function.Injective (rootedStart (p:=p) (S:=S) (r:=r)) := by
+  rintro ⟨c,z⟩ ⟨c',z'⟩ h
+  have hz (i : Fin A) : (z i).val = (z' i).val :=
+    p.injective (congrArg Subtype.val (congrFun h i))
+  have hc : c = c' := Subtype.ext (owner_ext (fun i => by
+    rw [←c.property.2.2 i (z i).val (z i).property,
+      ←c'.property.2.2 i (z' i).val (z' i).property,hz]))
+  subst c'
+  have hzz : z = z' := funext (fun i => Subtype.ext (hz i))
+  subst z'
+  rfl
+
+lemma sum_comp_injective_le {α β : Type*} [Fintype α] [Fintype β]
+    (e : α → β) (he : Function.Injective e) (f : β → ℝ) (hf : ∀ b, 0 ≤ f b) :
+    ∑ a, f (e a) ≤ ∑ b, f b := by
+  classical
+  calc
+    _ = ∑ b ∈ Finset.univ.image e, f b := (Finset.sum_image (fun _ _ _ _ h => he h)).symm
+    _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _) (fun b _ _ => hf b)
+
+lemma rooted_weight_sum_le (S : Finset (Card d)) (r : Fin A → ℕ) (ω : SwitchIndex d → Bool) :
+    (∑ c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r,
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2)) ≤
+    ∑ a : Fin A → S,
+      if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+        ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+      else 0 := by
+  classical
+  let p := butterflyPerm d (decodeButterfly d ω)
+  let g (a : Fin A → S) : ℝ :=
+    if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+    else 0
+  have he (c : RootedSlots p S r) :
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2) =
+      g (rootedStart c) := by
+    have hs := QueryTree.seekCycles_owned S r c.1.val c.2 ω c.1.property.2.1
+      (fun i => c.1.property.2.2 i (c.2 i).val (c.2 i).property) c.1.property.1
+    dsimp only at hs
+    unfold g
+    rw [ite_eq_left (show (QueryTree.seekCycles d S (List.ofFn (fun i => ((rootedStart c i).val,r i))) ∅).succeeds ω = true from hs.1)]
+    rw [← hs.2,neg_mul,Real.exp_neg,mul_comm (Real.log 2),Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
+    rfl
+  simp_rw [he]
+  exact sum_comp_injective_le rootedStart rootedStart_injective g (fun a => by dsimp [g]; split_ifs <;> positivity)
+
+noncomputable def rotationLower (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : ℝ :=
+  Real.exp (-Real.log 2 * ((A:ℝ)*d/2 + ∑ i : Fin A, Real.log (r i+1) / (2 * Real.log 2)))
+
+noncomputable def collectionWeight (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : ℝ :=
+  ((2:ℝ)^d)^A * (∏ i : Fin A, (r i+1 : ℝ)) * rotationLower d r
+
+lemma collectionWeight_pos (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : 0 < collectionWeight d r := by
+  unfold collectionWeight rotationLower
+  positivity
+
+lemma collection_rotation_lower (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r)
+    (ω : SwitchIndex d → Bool) (c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) :
+    collectionWeight d r ≤ ∑ z : Terminal c.val,
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z) := by
+  classical
+  let (i : Fin A) : Nonempty {x : Card d // c.val x = some i} := c.terminal_nonempty i
+  have hm : Monotone (fun i => (cycleFiber c.val i).card) := by
+    intro i j hij
+    dsimp only
+    rw [c.property.1,c.property.1]
+    exact Nat.add_le_add_right (hr hij) 1
+  have hb := rotation_exp_bound hm (decodeButterfly d ω)
+  have hcard : (Fintype.card (Terminal c.val) : ℝ) = ∏ i : Fin A, (r i+1 : ℝ) := by
+    rw [Fintype.card_pi,Nat.cast_prod]
+    apply Finset.prod_congr rfl
+    intro i _
+    rw [card_cycleFiber,c.property.1,Nat.cast_add,Nat.cast_one]
+  have hp : (0:ℝ) < Fintype.card (Terminal c.val) := Nat.cast_pos.mpr Fintype.card_pos
+  have hl : rotationLower d r ≤ finiteMean (fun z : Terminal c.val => Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z)) := by
+    simpa only [rotationLower,c.property.1,Nat.cast_add,Nat.cast_one] using hb
+  have hh := (le_div_iff₀ hp).mp hl
+  rw [hcard] at hh
+  unfold collectionWeight
+  rw [← Finset.mul_sum]
+  calc
+    _ = ((2:ℝ)^d)^A * (rotationLower d r * ∏ i : Fin A, (r i+1 : ℝ)) := by ring
+    _ ≤ _ := mul_le_mul_of_nonneg_left hh (by positivity)
+
+lemma cycleSlots_weighted_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) *
+      collectionWeight d r ≤ (S.card : ℝ)^A := by
+  classical
+  let g (a : Fin A → S) (ω : SwitchIndex d → Bool) : ℝ :=
+    if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+    else 0
+  have hω (ω : SwitchIndex d → Bool) :
+      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ) * collectionWeight d r ≤
+        ∑ a : Fin A → S, g a ω := by
+    calc
+      _ = ∑ _c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r, collectionWeight d r := by simp
+      _ ≤ ∑ c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r,
+          ∑ z : Terminal c.val, ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z) :=
+        Finset.sum_le_sum (fun c _ => collection_rotation_lower S r hr ω c)
+      _ = ∑ c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r,
+          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2) :=
+        (Fintype.sum_sigma (fun c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r =>
+          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2))).symm
+      _ ≤ _ := rooted_weight_sum_le S r ω
+  have hb := finiteMean_mono hω
+  rw [finiteMean_mul_const,finiteMean_sum] at hb
+  calc
+    _ ≤ ∑ a : Fin A → S, finiteMean (g a) := hb
+    _ ≤ ∑ _a : Fin A → S, (1:ℝ) := by
+      apply Finset.sum_le_sum
+      intro a _
+      have hh := QueryTree.cycles_weighted_success d S (List.ofFn (fun i : Fin A => ((a i).val,r i)))
+      simpa only [List.length_ofFn] using hh
+    _ = _ := by simp
+
+lemma cycleSlots_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) ≤
+        (S.card : ℝ)^A / collectionWeight d r := by
+  exact (le_div_iff₀ (collectionWeight_pos d r)).mpr (cycleSlots_weighted_moment S r hr)
+
+lemma collectionWeight_eq_sqrt (d : ℕ) {A : ℕ} (r : Fin A → ℕ) :
+    collectionWeight d r = (Real.sqrt ((2:ℝ)^d))^A * ∏ i : Fin A, Real.sqrt (r i+1) := by
+  have hn : ((2:ℝ)^d)^A = Real.exp ((A:ℝ)*d*Real.log 2) := by
+    rw [mul_assoc,Real.exp_nat_mul,Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
+  have hp : (∏ i : Fin A, (r i+1 : ℝ)) = Real.exp (∑ i : Fin A, Real.log (r i+1)) := by
+    rw [Real.exp_sum]
+    apply Finset.prod_congr rfl
+    intro i _
+    exact (Real.exp_log (by positivity)).symm
+  have hs : Real.sqrt ((2:ℝ)^d) = Real.exp ((d:ℝ)*Real.log 2/2) := by
+    rw [Real.sqrt_eq_rpow,Real.rpow_def_of_pos (by positivity),Real.log_pow]
+    congr 1
+    ring
+  have hj (i : Fin A) : Real.sqrt (r i+1) = Real.exp (Real.log (r i+1)/2) := by
+    rw [Real.sqrt_eq_rpow,Real.rpow_def_of_pos (by positivity)]
+    congr 1
+    ring
+  simp_rw [collectionWeight,hn,hp,rotationLower,hs,hj]
+  rw [←Real.exp_add,←Real.exp_add,←Real.exp_nat_mul,←Real.exp_sum,←Real.exp_add]
+  congr 1
+  rw [←Finset.sum_div,←Finset.sum_div]
+  have hl : Real.log 2 ≠ 0 := ne_of_gt (Real.log_pos (by norm_num))
+  field_simp
+  ring
+
+lemma butterfly_cycleSlots_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) ≤
+        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
+  have h := cycleSlots_moment S r hr
+  rw [collectionWeight_eq_sqrt,div_mul_eq_div_mul_one_div] at h
+  rw [div_pow]
+  simpa only [one_div,Finset.prod_inv_distrib] using h
+
+end Collections
+
+variable {d A : ℕ}
+
+abbrev SelectedCycle (p : Equiv.Perm (Card d)) (S : Finset (Card d)) :=
+  {C : Finset (Card d) // (∃ x, C = orbitSet p x) ∧ C ⊆ S}
+
+noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) :
+    Fintype (SelectedCycle p S) := Fintype.ofFinite _
+
+lemma SelectedCycle.nonempty {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
+    (C : SelectedCycle p S) : C.val.Nonempty := by
+  obtain ⟨x,hx⟩ := C.property.1
+  rw [hx]
+  exact orbitSet_nonempty p x
+
+lemma SelectedCycle.orbit_eq {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
+    (C : SelectedCycle p S) {x : Card d} (hx : x ∈ C.val) : orbitSet p x = C.val := by
+  obtain ⟨y,hy⟩ := C.property.1
+  rw [hy] at hx ⊢
+  exact (orbitSet_eq p ((mem_orbitSet _ _ _).mp hx)).symm
+
+lemma SelectedCycle.eq_of_mem {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
+    (C D : SelectedCycle p S) {x : Card d} (hC : x ∈ C.val) (hD : x ∈ D.val) : C = D := by
+  apply Subtype.ext
+  exact (C.orbit_eq hC).symm.trans (D.orbit_eq hD)
+
+abbrev SlotAssignment (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
+  {f : Fin A → SelectedCycle p S // Function.Injective f ∧ ∀ i, (f i).val.card = r i+1}
+
+noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
+    Fintype (SlotAssignment p S r) := Fintype.ofFinite _
+
+noncomputable def slotsToAssignment {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (c : CycleSlots p S r) : SlotAssignment p S r := by
+  classical
+  let f (i : Fin A) : SelectedCycle p S := ⟨cycleFiber c.val i,by
+    obtain ⟨x,hx⟩ := c.terminal_nonempty i
+    exact ⟨⟨x,(c.property.2.2 i x hx).symm⟩,c.property.2.1 i⟩⟩
+  refine ⟨f,?_,fun i => c.property.1 i⟩
+  intro i j hij
+  have he : cycleFiber c.val i = cycleFiber c.val j := congrArg Subtype.val hij
+  obtain ⟨x,hx⟩ := c.terminal_nonempty i
+  have hxi : x ∈ cycleFiber c.val i := (mem_cycleFiber _ _ _).mpr hx
+  rw [he] at hxi
+  exact Option.some.inj (hx.symm.trans ((mem_cycleFiber _ _ _).mp hxi))
+
+noncomputable def assignmentOwner {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) (x : Card d) : Option (Fin A) := by
+  classical
+  exact if h : ∃ i, x ∈ (f.val i).val then some (Classical.choose h) else none
+
+lemma assignmentOwner_eq_some {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) (x : Card d) (i : Fin A) :
+    assignmentOwner f x = some i ↔ x ∈ (f.val i).val := by
+  classical
+  unfold assignmentOwner
+  split_ifs with h
+  · constructor
+    · intro hi
+      have hx := Classical.choose_spec h
+      rw [Option.some.inj hi] at hx
+      exact hx
+    · intro hx
+      have he := (f.val (Classical.choose h)).eq_of_mem (f.val i) (Classical.choose_spec h) hx
+      exact congrArg some (f.property.1 he)
+  · constructor
+    · intro hi; contradiction
+    · intro hx; exact (h ⟨i,hx⟩).elim
+
+end Thorp
+
+end ThorpNine.Dense
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Dense/SingleSlotsEquiv.lean b/lean/OAI/Probability/ThorpRouting/Dense/SingleSlotsEquiv.lean
index f5e6a932a2f432871716e2fbac7a569778343188..a2d60665bebd05e50d1e57154abfe8c0f1eead49 100644
--- a/lean/OAI/Probability/ThorpRouting/Dense/SingleSlotsEquiv.lean
+++ b/lean/OAI/Probability/ThorpRouting/Dense/SingleSlotsEquiv.lean
@@ -1,916 +1,916 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Dense.PalindromeFamilyCost
-
-namespace OAI
-
-namespace ThorpNine.Dense
-
-open scoped BigOperators
-namespace Thorp.RoutingNetwork
-open scoped BigOperators
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-noncomputable def singleSlotsEquiv (p : Equiv.Perm α) (S : Finset α) (r : ℕ) :
-    CycleSlots p S (fun _ : Fin 1 => r) ≃
-      {C : SelectedCycle p S // C.val.card = r+1} :=
-  (cycleSlotsEquivAssignments p S (fun _ : Fin 1 => r)).trans
-  { toFun := fun a => ⟨a.val 0, a.property.2 0⟩
-    invFun := fun C => ⟨fun _ => C.val, Function.injective_of_subsingleton _, fun _ => C.property⟩
-    left_inv := fun a => by
-      apply Subtype.ext
-      funext i
-      have hi : i=0 := Subsingleton.elim _ _
-      subst i
-      rfl
-    right_inv := fun C => rfl }
-
-noncomputable def cycleCount (p : Equiv.Perm α) (S : Finset α) (r : ℕ) : ℝ :=
-  Fintype.card {C : SelectedCycle p S // C.val.card = r+1}
-
-lemma cycleCount_eq_sum (p : Equiv.Perm α) (S : Finset α) (r : ℕ) :
-    cycleCount p S r = ∑ C : SelectedCycle p S, if C.val.card=r+1 then (1:ℝ) else 0 := by
-  classical
-  simp only [cycleCount, Fintype.card_subtype, Finset.sum_boole]
-
-lemma hasSum_cycleCount_weight (p : Equiv.Perm α) (S : Finset α) (f : ℕ → ℝ) :
-    HasSum (fun l => cycleCount p S l * f (l+1))
-      (∑ C : SelectedCycle p S, f C.val.card) := by
-  classical
-  have h (C : SelectedCycle p S) :
-      HasSum (fun l : ℕ => if C.val.card=l+1 then f C.val.card else 0) (f C.val.card) := by
-    have hc : 1 ≤ C.val.card := C.nonempty.card_pos
-    have hs := hasSum_single (C.val.card-1)
-      (f := fun l : ℕ => if C.val.card=l+1 then f C.val.card else 0)
-      (fun l hl => ite_eq_right (by omega))
-    simpa only [Nat.sub_add_cancel hc, ↓reduceIte] using hs
-  convert hasSum_sum (s := Finset.univ) (fun C _ => h C) using 1
-  funext l
-  rw [cycleCount_eq_sum, Finset.sum_mul]
-  apply Finset.sum_congr rfl
-  intro C _
-  split_ifs with hc
-  · simp only [one_mul, hc]
-  · simp
-
-end Thorp.RoutingNetwork
-
-namespace Thorp
-open scoped BigOperators
-lemma hasSum_finiteMean {Ω : Type*} [Fintype Ω] {f : Ω → ℕ → ℝ} {a : Ω → ℝ}
-    (hf : ∀ ω, HasSum (f ω) (a ω)) :
-    HasSum (fun l => finiteMean (fun ω => f ω l)) (finiteMean a) :=
-  (hasSum_sum (s := Finset.univ) (fun ω _ => hf ω)).div_const (Fintype.card Ω : ℝ)
-end Thorp
-
-namespace Thorp.RoutingNetwork
-open scoped BigOperators
-variable {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι] [DecidableEq ι]
-  {d : ℕ} (R : RoutingNetwork α ι d)
-
-lemma selected_cycle_mean_le_h (S : Finset α) :
-    finiteMean (fun ω : ι → Bool => (Fintype.card (SelectedCycle (R.perm ω) S) : ℝ)) ≤
-      (S.card : ℝ) * DenseTruncation.h (d+1) := by
-  classical
-  by_cases hS : S.Nonempty
-  · have hpos : (0:ℝ) < S.card := Nat.cast_pos.mpr hS.card_pos
-    let w : ℕ → ℝ := fun l => finiteMean (fun ω : ι → Bool => cycleCount (R.perm ω) S l)
-    have hw : ∀ l, 0 ≤ w l := fun _ => finiteMean_nonneg (fun _ => Nat.cast_nonneg _)
-    have hc (l : ℕ) : w l ≤ (S.card : ℝ) / (2:ℝ)^d := by
-      apply (le_div_iff₀ (by positivity : 0 < (2:ℝ)^d)).mpr
-      have he (ω : ι → Bool) : cycleCount (R.perm ω) S l =
-          (Fintype.card (CycleSlots (R.perm ω) S (fun _ : Fin 1 => l)) : ℝ) := by
-        rw [cycleCount, Fintype.card_congr (singleSlotsEquiv (R.perm ω) S l)]
-      simpa only [w,he] using single_cycleSlots_weighted_moment R S l
-    have hmass := hasSum_finiteMean (fun ω : ι → Bool =>
-      hasSum_cycleCount_weight (R.perm ω) S (fun n => (n:ℝ)))
-    have hm : HasSum (fun l => ((l+1 : ℕ) : ℝ) * w l)
-        (finiteMean (fun ω : ι → Bool => (cycleMass (R.perm ω) S : ℝ))) := by
-      simp only [finiteMean_mul_const] at hmass
-      simpa only [w, mul_comm, cycleMass,Nat.cast_sum] using hmass
-    have hmt : finiteMean (fun ω : ι → Bool => (cycleMass (R.perm ω) S : ℝ)) ≤ S.card := by
-      calc
-        _ ≤ finiteMean (fun _ : ι → Bool => (S.card : ℝ)) :=
-          finiteMean_mono (fun ω => by exact_mod_cast cycleMass_le (R.perm ω) S)
-        _ = _ := finiteMean_const _
-    let q : ℕ → ℝ := fun l => ((l+1 : ℕ) : ℝ) * w l / S.card
-    have hq : ∀ l, 0 ≤ q l := fun l => div_nonneg (mul_nonneg (by positivity) (hw l)) hpos.le
-    have hqs : Summable q := (hm.div_const (S.card : ℝ)).summable
-    have hqt : (∑' l, q l) ≤ 1 := by
-      rw [(hm.div_const (S.card : ℝ)).tsum_eq]
-      exact (div_le_one hpos).mpr hmt
-    have hqc (l : ℕ) : q l ≤ ((l+1 : ℕ) : ℝ) / (2:ℝ)^((d+1)-1) := by
-      simp only [Nat.add_sub_cancel]
-      dsimp [q]
-      calc
-        _ ≤ (((l+1 : ℕ) : ℝ) * ((S.card : ℝ) / (2:ℝ)^d)) / S.card := by
-          gcongr
-          exact hc l
-        _ = _ := by field_simp
-    have hh := DenseTruncation.subprob_reciprocal_le_h hq hqs hqt hqc
-    have hcount := hasSum_finiteMean (fun ω : ι → Bool =>
-      hasSum_cycleCount_weight (R.perm ω) S (fun _ => (1:ℝ)))
-    have hwc : HasSum w (finiteMean (fun ω : ι → Bool =>
-        (Fintype.card (SelectedCycle (R.perm ω) S) : ℝ))) := by
-      simpa only [mul_one,Finset.sum_const,Finset.card_univ,nsmul_eq_mul,mul_one] using hcount
-    have he (l : ℕ) : q l / ((l+1 : ℕ) : ℝ) = w l / S.card := by
-      have hl : ((l+1 : ℕ) : ℝ) ≠ 0 := by positivity
-      dsimp [q]
-      field_simp
-    simp only [he, (hwc.div_const (S.card : ℝ)).tsum_eq] at hh
-    simpa only [mul_comm] using (div_le_iff₀ hpos).mp hh
-  · have he : S = ∅ := Finset.not_nonempty_iff_eq_empty.mp hS
-    subst S
-    have hz (ω : ι → Bool) : Fintype.card (SelectedCycle (R.perm ω) ∅) = 0 := by
-      let : IsEmpty (SelectedCycle (R.perm ω) ∅) :=
-        ⟨fun C => C.nonempty.ne_empty (Finset.subset_empty.mp C.property.2)⟩
-      exact Fintype.card_eq_zero
-    simp only [hz,Nat.cast_zero,finiteMean_const,Finset.card_empty,zero_mul,le_refl]
-
-end Thorp.RoutingNetwork
-
-namespace Thorp.DenseTruncation
-open scoped BigOperators
-
-lemma reciprocal_values_nonempty (j : ℕ) :
-    {z : ℝ | ∃ p : ℕ → ℝ, AdmissibleCycleLaw j p ∧
-      z = ∑' l : ℕ, p l / ((l+1 : ℕ) : ℝ)}.Nonempty := by
-  let t := 2^(j-1)-1
-  have ht : t+1 = 2^(j-1) := Nat.sub_add_cancel (Nat.one_le_pow _ _ (by norm_num))
-  let p : ℕ → ℝ := fun l => if l=t then 1 else 0
-  have hp : AdmissibleCycleLaw j p := by
-    refine ⟨fun l => by dsimp [p]; split_ifs <;> norm_num, hasSum_ite_eq t 1, ?_⟩
-    intro l
-    by_cases hl : l=t
-    · subst l
-      have he : ((t+1 : ℕ) : ℝ) = (2:ℝ)^(j-1) := by rw [ht]; norm_cast
-      simp only [p,↓reduceIte,he,div_self (by positivity : (2:ℝ)^(j-1) ≠ 0),le_refl]
-    · simp only [p,ite_eq_right hl]
-      positivity
-  exact ⟨_,p,hp,rfl⟩
-
-lemma h_nonneg (j : ℕ) : 0 ≤ h j := by
-  obtain ⟨z,p,hp,rfl⟩ := reciprocal_values_nonempty j
-  exact (law_reciprocal_nonneg hp).trans (reciprocal_le_h hp)
-
-lemma h_le_one (j : ℕ) : h j ≤ 1 :=
-  csSup_le (reciprocal_values_nonempty j) (fun _ ⟨_,hp,hz⟩ => hz ▸ law_reciprocal_le_one hp)
-
-lemma law_reciprocal_cutoff {j : ℕ} {p : ℕ → ℝ} (hp : AdmissibleCycleLaw j p)
-    (L : ℕ) (hL : 0 < L) :
-    (∑' l, p l / ((l+1 : ℕ) : ℝ)) ≤ (L : ℝ)/(2:ℝ)^(j-1) + 1/L := by
-  let a : ℕ → ℝ := fun l => if l<L then 1/(2:ℝ)^(j-1) else 0
-  have ha : HasSum a ((L : ℝ)/(2:ℝ)^(j-1)) := by
-    have hs : HasSum a (∑ l ∈ Finset.range L, a l) :=
-      hasSum_sum_of_ne_finset_zero (s := Finset.range L) (f := a)
-      (fun l hl => by simpa only [a,Finset.mem_range] using ite_eq_right (show ¬l<L from fun hh => hl (Finset.mem_range.mpr hh)))
-    simpa only [a,Finset.sum_ite_of_true (fun l hl => Finset.mem_range.mp hl),
-      Finset.sum_const,Finset.card_range,nsmul_eq_mul,mul_one_div] using hs
-  have hb := hp.2.1.div_const (L:ℝ)
-  have he (l : ℕ) : p l / ((l+1 : ℕ) : ℝ) ≤ a l + p l/L := by
-    by_cases hl : l<L
-    · have hc : p l / ((l+1 : ℕ) : ℝ) ≤ 1/(2:ℝ)^(j-1) := by
-        have hp1 := hp.2.2 l
-        have hlpos : (0:ℝ) < (l+1 : ℕ) := by positivity
-        apply (div_le_iff₀ hlpos).mpr
-        simpa only [one_div, div_eq_mul_inv,mul_comm,one_mul] using hp1
-      dsimp [a]
-      rw [ite_eq_left hl]
-      exact hc.trans (le_add_of_nonneg_right (div_nonneg (hp.1 l) (by positivity)))
-    · dsimp [a]
-      rw [ite_eq_right hl,zero_add]
-      exact div_le_div_of_nonneg_left (hp.1 l) (by exact_mod_cast hL)
-        (by exact_mod_cast (show L ≤ l+1 by omega))
-  have hh := Summable.tsum_le_tsum he (law_reciprocal_summable hp) (ha.add hb).summable
-  rwa [(ha.add hb).tsum_eq] at hh
-
-lemma h_cutoff (j L : ℕ) (hL : 0 < L) :
-    h j ≤ (L : ℝ)/(2:ℝ)^(j-1) + 1/L :=
-  csSup_le (reciprocal_values_nonempty j)
-    (fun _ ⟨_,hp,hz⟩ => hz ▸ law_reciprocal_cutoff hp L hL)
-
-lemma h_dyadic_bound (j : ℕ) : h (j+1) ≤ 2/(2:ℝ)^(j/2) := by
-  have h := h_cutoff (j+1) (2^(j/2)) (by positivity)
-  simp only [Nat.add_sub_cancel,Nat.cast_pow,Nat.cast_ofNat] at h
-  have hpow : ((2:ℝ)^(j/2))^2 ≤ (2:ℝ)^j := by
-    rw [←pow_mul]
-    exact pow_le_pow_right₀ (by norm_num) (by omega)
-  have hp : (0:ℝ) < (2:ℝ)^(j/2) := by positivity
-  have hq : (0:ℝ) < (2:ℝ)^j := by positivity
-  have hr : (2:ℝ)^(j/2)/(2:ℝ)^j ≤ 1/(2:ℝ)^(j/2) := by
-    apply (div_le_div_iff₀ hq hp).mpr
-    nlinarith
-  calc
-    _ ≤ 1/(2:ℝ)^(j/2) + 1/(2:ℝ)^(j/2) := h.trans (by linarith)
-    _ = _ := by ring
-
-lemma summable_half_dyadic : Summable (fun j : ℕ => 2/(2:ℝ)^(j/2)) := by
-  apply ((Nat.divModEquiv 2).symm.summable_iff).mp
-  have he (p : ℕ × Fin 2) : 2/(2:ℝ)^(((Nat.divModEquiv 2).symm p)/2) =
-      2*(1/2:ℝ)^p.1 := by
-    rw [Nat.divModEquiv_symm_apply]
-    have hi : (p.1*2+p.2.val)/2 = p.1 := by omega
-    rw [hi,div_pow,one_pow]
-    ring
-  simp only [Function.comp_def,he]
-  apply (summable_prod_of_nonneg (fun p => by positivity)).mpr
-  constructor
-  · intro n
-    exact (hasSum_fintype _).summable
-  · simpa only [tsum_fintype,Finset.sum_const,Finset.card_univ,Fintype.card_fin,
-      nsmul_eq_mul,Nat.cast_ofNat,←mul_assoc,show (2:ℝ)*2=4 from by norm_num] using summable_geometric_two.mul_left 4
-
-lemma h_summable : Summable (fun j => h (j+1)) :=
-  Summable.of_nonneg_of_le (fun j => h_nonneg (j+1)) h_dyadic_bound summable_half_dyadic
-
-lemma allocation_value_summable {ρ : ℝ} {x : ℕ → ℝ} (hx : AdmissibleAllocation ρ x) :
-    Summable (fun j => h (j+1)*x j) :=
-  Summable.of_nonneg_of_le (fun j => mul_nonneg (h_nonneg _) (hx.1 j).1)
-    (fun j => mul_le_of_le_one_right (h_nonneg _) (hx.1 j).2) h_summable
-
-lemma H_values_bdd (ρ : ℝ) : BddAbove {z : ℝ | ∃ x : ℕ → ℝ,
-    AdmissibleAllocation ρ x ∧ z = (1/2:ℝ)*∑' j, h (j+1)*x j} := by
-  refine ⟨(1/2:ℝ)*∑' j, h (j+1), ?_⟩
-  rintro z ⟨x,hx,rfl⟩
-  apply mul_le_mul_of_nonneg_left _ (by norm_num)
-  exact Summable.tsum_le_tsum (fun j => mul_le_of_le_one_right (h_nonneg _) (hx.1 j).2)
-    (allocation_value_summable hx) h_summable
-
-lemma allocation_le_H {ρ : ℝ} {x : ℕ → ℝ} (hx : AdmissibleAllocation ρ x) :
-    (1/2:ℝ)*∑' j, h (j+1)*x j ≤ H ρ :=
-  le_csSup (H_values_bdd ρ) ⟨x,hx,rfl⟩
-
-end Thorp.DenseTruncation
-
-namespace Thorp
-open scoped BigOperators
-
-def unitCardEquiv (d : ℕ) : Card d ≃ Unit × Card d where
-  toFun x := ((),x)
-  invFun x := x.2
-  left_inv _ := rfl
-  right_inv x := by cases x with | mk u x => cases u; rfl
-
-def unitBitsEquiv (d : ℕ) : (Unit × SwitchIndex d → Bool) ≃ (SwitchIndex d → Bool) where
-  toFun ω i := ω ((),i)
-  invFun ω i := ω i.2
-  left_inv ω := by funext i; rcases i with ⟨u,i⟩; cases u; rfl
-  right_inv _ := rfl
-
-lemma relabeled_single_cycle_mean (d : ℕ) (pre post : Equiv.Perm (Card d))
-    (S : Finset (Card d)) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      (Fintype.card (RoutingNetwork.SelectedCycle
-        (post * butterflyPerm d (decodeButterfly d ω) * pre) S) : ℝ)) ≤
-      (S.card : ℝ)*DenseTruncation.h (d+1) := by
-  have hh := RoutingNetwork.selected_cycle_mean_le_h
-    (parallelNetwork (pre.trans (unitCardEquiv d)) ((unitCardEquiv d).symm.trans post)) S
-  rw [finiteMean_equiv (unitBitsEquiv d)] at hh
-  exact hh
-
-namespace CycleColoring
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-noncomputable def projectSharedCycle (p : Equiv.Perm α) (A B : Finset α)
-    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) :
-    RoutingNetwork.SelectedCycle p (A ∩ B) := by
-  refine ⟨(projectCycle p A B C).val, (projectCycle p A B C).property.1, ?_⟩
-  intro x hx
-  apply Finset.mem_inter.mpr
-  refine ⟨(projectCycle p A B C).property.2 hx, ?_⟩
-  have he := selected_alternating_eq p A B C
-  have hc : (true,x) ∈ C.val := by
-    rw [he]
-    exact (mem_childJoin _ _ true x).mpr hx
-  exact (mem_childJoin A B true x).mp (C.property.2 hc)
-
-lemma projectSharedCycle_injective (p : Equiv.Perm α) (A B : Finset α) :
-    Function.Injective (projectSharedCycle p A B) := by
-  intro C D h
-  apply projectCycle_injective p A B
-  apply Subtype.ext
-  change (projectSharedCycle p A B C).val = (projectSharedCycle p A B D).val
-  exact congrArg Subtype.val h
-
-lemma card_alternating_cycles_le_shared (p : Equiv.Perm α) (A B : Finset α) :
-    Fintype.card (RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) ≤
-      Fintype.card (RoutingNetwork.SelectedCycle p (A ∩ B)) :=
-  Fintype.card_le_of_injective _ (projectSharedCycle_injective p A B)
-
-end CycleColoring
-
-namespace PairRouting
-variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
-
-noncomputable def sharedPairs (e : ι ↪ Bool × α) : Finset α :=
-  boolFiber (labelSet e) false ∩ boolFiber (labelSet e) true
-
-lemma alternatingCycles_le_sharedSelected (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    alternatingCycles e p ≤
-      Fintype.card (RoutingNetwork.SelectedCycle (alternatingProjection p) (sharedPairs e)) := by
-  unfold alternatingCycles sharedPairs
-  conv_lhs => rw [←childJoin_boolFiber (labelSet e)]
-  exact CycleColoring.card_alternating_cycles_le_shared _ _ _
-
-omit [Fintype α] in
-lemma sharedPairs_card_twice_le (e : ι ↪ Bool × α) :
-    2*(sharedPairs e).card ≤ Fintype.card ι := by
-  have he := card_childJoin (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
-  rw [childJoin_boolFiber,card_labelSet] at he
-  have h₀ := Finset.card_le_card (Finset.inter_subset_left : sharedPairs e ⊆ boolFiber (labelSet e) false)
-  have h₁ := Finset.card_le_card (Finset.inter_subset_right : sharedPairs e ⊆ boolFiber (labelSet e) true)
-  simp only [sharedPairs] at *
-  omega
-
-omit [Fintype α] in
-lemma labelSet_switched (e : ι ↪ Bool × α) (ξ : α → Bool) :
-    labelSet (switchedEmbedding e ξ) = (labelSet e).image (pairLayer ξ) := by
-  classical
-  simp only [labelSet,Finset.image_image,switchedEmbedding]
-  rfl
-
-omit [Fintype α] in
-lemma sharedPairs_switched (e : ι ↪ Bool × α) (ξ : α → Bool) :
-    sharedPairs (switchedEmbedding e ξ) = sharedPairs e := by
-  classical
-  ext x
-  simp only [sharedPairs,Finset.mem_inter,mem_boolFiber,labelSet_switched,mem_image_perm_iff]
-  change (Bool.xor false (ξ x),x) ∈ labelSet e ∧
-      (Bool.xor true (ξ x),x) ∈ labelSet e ↔
-      (false,x) ∈ labelSet e ∧ (true,x) ∈ labelSet e
-  cases ξ x <;> simp only [Bool.xor_false,Bool.xor_true,Bool.not_false,Bool.not_true,and_comm]
-
-lemma node_cycle_mean (d : ℕ) (e : ι ↪ Bool × Card d) :
-    finiteMean (fun c : BenesCoins d × BenesCoins d =>
-      (alternatingCycles e (fun b => palindromePerm d (if b then c.2 else c.1)) : ℝ)) ≤
-        (sharedPairs e).card * DenseTruncation.h (d+1) := by
-  classical
-  rw [finiteMean_prod]
-  have hone (c₀ : BenesCoins d) :
-      finiteMean (fun c₁ : BenesCoins d =>
-        (alternatingCycles e (fun b => palindromePerm d (if b then c₁ else c₀)) : ℝ)) ≤
-          (sharedPairs e).card * DenseTruncation.h (d+1) := by
-    rw [finiteMean_prod,finiteMean_comm]
-    have hf (X : SwitchIndex d → Bool) :
-        finiteMean (fun Y : SwitchIndex d → Bool =>
-          (alternatingCycles e (fun b => palindromePerm d (if b then (Y,X) else c₀)) : ℝ)) ≤
-          (sharedPairs e).card * DenseTruncation.h (d+1) := by
-      calc
-        _ ≤ finiteMean (fun Y : SwitchIndex d → Bool =>
-            (Fintype.card (RoutingNetwork.SelectedCycle
-              ((palindromePerm d c₀).symm * butterflyPerm d (decodeButterfly d Y) *
-                (butterflyPerm d (decodeButterfly d X)).symm) (sharedPairs e)) : ℝ)) := by
-          apply finiteMean_mono
-          intro Y
-          have hh := alternatingCycles_le_sharedSelected e
-            (fun b => palindromePerm d (if b then (Y,X) else c₀))
-          simpa only [alternatingProjection,Bool.false_eq_true,↓reduceIte,palindromePerm,mul_assoc] using
-            (Nat.cast_le.mpr hh : (alternatingCycles e _ : ℝ) ≤ _)
-        _ ≤ _ := relabeled_single_cycle_mean d _ _ _
-    calc
-      _ ≤ finiteMean (fun _ : SwitchIndex d → Bool =>
-          (sharedPairs e).card * DenseTruncation.h (d+1)) := finiteMean_mono hf
-      _ = _ := finiteMean_const _
-  calc
-    _ ≤ finiteMean (fun _ : BenesCoins d => (sharedPairs e).card * DenseTruncation.h (d+1)) :=
-      finiteMean_mono hone
-    _ = _ := finiteMean_const _
-
-end PairRouting
-end Thorp
-
-namespace Thorp
-
-lemma exp_secant {M x t : ℝ} (hM : 0 < M) (hx : 0 ≤ x) (hxM : x ≤ M) :
-    Real.exp (t*x) ≤ 1 + x*((Real.exp (t*M)-1)/M) := by
-  have hb : 0 ≤ x/M := div_nonneg hx hM.le
-  have ha : 0 ≤ 1-x/M := sub_nonneg.mpr ((div_le_one hM).mpr hxM)
-  have hh := convexOn_exp.2 (Set.mem_univ (0:ℝ)) (Set.mem_univ (t*M)) ha hb
-    (show (1-x/M)+(x/M)=1 by ring)
-  simp only [smul_eq_mul, mul_zero, zero_add, Real.exp_zero, mul_one] at hh
-  have he : x/M*(t*M) = t*x := by field_simp
-  rw [he] at hh
-  calc
-    _ ≤ 1-x/M+(x/M)*Real.exp (t*M) := hh
-    _ = _ := by ring
-
-lemma finiteMean_exp_bounded {Ω : Type*} [Fintype Ω] [Nonempty Ω]
-    (Z : Ω → ℝ) (M t μ : ℝ) (hM : 0 < M) (ht : 0 ≤ t)
-    (hZ : ∀ ω, 0 ≤ Z ω ∧ Z ω ≤ M) (hmean : finiteMean Z ≤ μ) :
-    finiteMean (fun ω => Real.exp (t*Z ω)) ≤
-      Real.exp (μ*((Real.exp (t*M)-1)/M)) := by
-  have hz : 0 ≤ (Real.exp (t*M)-1)/M :=
-    div_nonneg (sub_nonneg.mpr (Real.one_le_exp_iff.mpr (mul_nonneg ht hM.le))) hM.le
-  calc
-    _ ≤ finiteMean (fun ω => 1 + Z ω*((Real.exp (t*M)-1)/M)) :=
-      finiteMean_mono (fun ω => exp_secant hM (hZ ω).1 (hZ ω).2)
-    _ = 1 + finiteMean Z*((Real.exp (t*M)-1)/M) := by
-      rw [finiteMean_add,finiteMean_const,finiteMean_mul_const]
-    _ ≤ 1 + μ*((Real.exp (t*M)-1)/M) := by gcongr
-    _ ≤ _ := by simpa only [add_comm] using Real.add_one_le_exp (μ*((Real.exp (t*M)-1)/M))
-
-namespace PairRouting
-variable {ι : Type*} [Fintype ι]
-
-lemma node_cycle_mgf (d : ℕ) (e : ι ↪ Bool × Card d) (t : ℝ) (ht : 0 ≤ t) :
-    finiteMean (fun c : BenesCoins d × BenesCoins d =>
-      Real.exp (t * alternatingCycles e
-        (fun b => palindromePerm d (if b then c.2 else c.1)))) ≤
-      Real.exp ((sharedPairs e).card * DenseTruncation.h (d+1) *
-        ((Real.exp (t*(2:ℝ)^d)-1)/(2:ℝ)^d)) := by
-  apply finiteMean_exp_bounded _ _ _ _ (by positivity) ht _ (node_cycle_mean d e)
-  intro c
-  constructor
-  · exact Nat.cast_nonneg _
-  · have h₀ := alternatingCycles_double e
-      (fun b => palindromePerm d (if b then c.2 else c.1))
-    have h₁ := Fintype.card_le_of_injective e e.injective
-    simp only [Fintype.card_prod,Fintype.card_bool,card_positions] at h₁
-    have h₂ : alternatingCycles e (fun b => palindromePerm d (if b then c.2 else c.1)) ≤ 2^d := by omega
-    exact_mod_cast h₂
-
-end PairRouting
-
-open scoped BigOperators
-
-lemma shared_pair_mgf {β Ω : Type*} [Fintype β] [DecidableEq β]
-    [Fintype Ω] (F : Ω → Finset (Bool × β)) (p : Bool × β → ℝ)
-    (hp : ∀ x, 0 ≤ p x) (hc : CylinderBound (inclusionMoment F) p)
-    (z : ℝ) (hz : 0 ≤ z) :
-    finiteMean (fun ω => Real.exp (z * (boolFiber (F ω) false ∩ boolFiber (F ω) true).card)) ≤
-      Real.exp ((Real.exp z - 1) * ∑ b, p (false,b)*p (true,b)) := by
-  classical
-  let T : β → Bool → Finset (Bool × β) := fun b q =>
-    if q then {(false,b),(true,b)} else ∅
-  let c : β → Bool → ℝ := fun _ q => if q then Real.exp z - 1 else 1
-  have hcz : 0 ≤ Real.exp z - 1 := sub_nonneg.mpr (Real.one_le_exp_iff.mpr hz)
-  have hdis : ∀ b b', b ≠ b' → ∀ q q', Disjoint (T b q) (T b' q') := by
-    intro b b' h q q'
-    cases q <;> cases q' <;> simp [T,Finset.disjoint_left,h]
-  have hh := cylinder_block_polynomial (fun _ : β => Bool) F p hc T c
-    (fun b q => by cases q <;> simp only [c,↓reduceIte] <;> positivity) hdis
-  have hsum (A : Finset (Bool × β)) (b : β) :
-      (∑ q, c b q * if T b q ⊆ A then 1 else 0) =
-        Real.exp (z * if (false,b) ∈ A ∧ (true,b) ∈ A then 1 else 0) := by
-    simp only [Fintype.sum_bool,c,T,Bool.false_eq_true,↓reduceIte,Finset.empty_subset,
-      mul_one,Finset.insert_subset_iff,Finset.singleton_subset_iff]
-    split_ifs <;> simp
-  have hcard (A : Finset (Bool × β)) :
-      (∑ b : β, if (false,b) ∈ A ∧ (true,b) ∈ A then (1:ℝ) else 0) =
-        (boolFiber A false ∩ boolFiber A true).card := by
-    rw [Finset.sum_boole]
-    congr 2
-    ext b
-    simp [mem_boolFiber]
-  have hleft (ω : Ω) :
-      (∏ b, ∑ q, c b q * if T b q ⊆ F ω then 1 else 0) =
-        Real.exp (z*(boolFiber (F ω) false ∩ boolFiber (F ω) true).card) := by
-    simp_rw [hsum]
-    rw [←Real.exp_sum,←Finset.mul_sum,hcard]
-  have hright (b : β) :
-      (∑ q, c b q * ∏ x ∈ T b q, p x) =
-        1+(Real.exp z-1)*(p (false,b)*p (true,b)) := by
-    simp [c,T,add_comm]
-  simp_rw [hleft,hright] at hh
-  refine hh.trans ?_
-  rw [Finset.mul_sum,Real.exp_sum]
-  apply Finset.prod_le_prod₀
-  · intro b _
-    exact add_nonneg (by norm_num) (mul_nonneg hcz (mul_nonneg (hp _) (hp _)))
-  · intro b _
-    simpa only [add_comm] using Real.add_one_le_exp ((Real.exp z-1)*(p (false,b)*p (true,b)))
-
-end Thorp
-
-namespace Thorp
-variable {B : Type*} [Fintype B] [DecidableEq B]
-
-lemma prod_blockMarks (d : ℕ) (S : Finset (B × Card d)) (f : B × Card d → ℝ) :
-    (∏ x ∈ S, f x) = ∏ b, ∏ x ∈ blockMarks (Equiv.refl _) S b, f (b,x) := by
-  classical
-  have hb (b : B) : (∏ x ∈ blockMarks (Equiv.refl _) S b, f (b,x)) =
-      ∏ x, if (b,x) ∈ S then f (b,x) else 1 := by
-    exact Finset.prod_filter (fun x => (b,x) ∈ S) (fun x => f (b,x))
-  simp_rw [hb]
-  rw [←Fintype.prod_prod_type (fun x : B × Card d => if x ∈ S then f x else 1)]
-  symm
-  calc
-    _ = ∏ x ∈ Finset.univ.filter (fun x : B × Card d => x ∈ S), f x :=
-      (Finset.prod_filter _ _).symm
-    _ = _ := by simp
-
-omit [Fintype B] in
-lemma subset_parallel_inverse (d : ℕ) (A S : Finset (B × Card d))
-    (ω : B × SwitchIndex d → Bool) :
-    S ⊆ A.image (parallelPerm ω).symm ↔
-      ∀ b, blockMarks (Equiv.refl _) S b ⊆
-        inverseButterflyMarks d (blockMarks (Equiv.refl _) A b) (fun i => ω (b,i)) := by
-  classical
-  constructor
-  · intro h b x hx
-    have hh := h ((mem_blockMarks (Equiv.refl _) S b x).mp hx)
-    rw [mem_image_perm_iff] at hh
-    rw [inverseButterflyMarks,mem_image_perm_iff]
-    exact (mem_blockMarks (Equiv.refl _) A b _).mpr hh
-  · intro h ⟨b,x⟩ hx
-    have hh := h b ((mem_blockMarks (Equiv.refl _) S b x).mpr hx)
-    rw [inverseButterflyMarks,mem_image_perm_iff] at hh
-    rw [mem_image_perm_iff]
-    exact (mem_blockMarks (Equiv.refl _) A b _).mp hh
-
-lemma parallel_inverse_cylinder (d : ℕ) (A : Finset (B × Card d)) :
-    CylinderBound (inclusionMoment (fun ω : B × SwitchIndex d → Bool =>
-      A.image (parallelPerm ω).symm))
-      (fun x => (blockMarks (Equiv.refl _) A x.1).card / (2:ℝ)^d) := by
-  classical
-  intro S
-  rw [inclusionMoment_eq,finiteMean_equiv (Equiv.curry B (SwitchIndex d) Bool)]
-  have hi (ω : B → SwitchIndex d → Bool) :
-      (if S ⊆ A.image (parallelPerm ((Equiv.curry B (SwitchIndex d) Bool).symm ω)).symm
-        then (1:ℝ) else 0) =
-      ∏ b, if blockMarks (Equiv.refl _) S b ⊆ inverseButterflyMarks d
-        (blockMarks (Equiv.refl _) A b) (ω b) then 1 else 0 := by
-    simp only [subset_parallel_inverse]
-    change (if ∀ b, blockMarks (Equiv.refl _) S b ⊆ inverseButterflyMarks d
-      (blockMarks (Equiv.refl _) A b) (ω b) then (1:ℝ) else 0) = _
-    by_cases h : ∀ b, blockMarks (Equiv.refl _) S b ⊆
-        inverseButterflyMarks d (blockMarks (Equiv.refl _) A b) (ω b)
-    · simp [h]
-    · rw [ite_eq_right h]
-      obtain ⟨b,hb⟩ := not_forall.mp h
-      symm
-      apply Finset.prod_eq_zero (Finset.mem_univ b)
-      exact ite_eq_right hb
-  simp_rw [hi]
-  rw [finiteMean_pi_prod (fun b ω => if blockMarks (Equiv.refl _) S b ⊆
-    inverseButterflyMarks d (blockMarks (Equiv.refl _) A b) ω then 1 else 0),prod_blockMarks]
-  apply Finset.prod_le_prod₀
-  · intro b _
-    exact finiteMean_nonneg (fun ω => by split_ifs <;> norm_num)
-  · intro b _
-    simpa only [←inclusionMoment_eq,Finset.prod_const] using
-      inverseButterflyMarks_inclusion d (blockMarks (Equiv.refl _) A b)
-        (blockMarks (Equiv.refl _) S b)
-
-end Thorp
-
-namespace Thorp
-namespace PairRouting
-variable {ι : Type*} [Fintype ι]
-
-lemma node_cycle_mean_one_fresh (d : ℕ) (e : ι ↪ Bool × Card d)
-    (c₀ : BenesCoins d) (X : SwitchIndex d → Bool) :
-    finiteMean (fun Y : SwitchIndex d → Bool =>
-      (alternatingCycles e (fun b => palindromePerm d (if b then (Y,X) else c₀)) : ℝ)) ≤
-        (sharedPairs e).card * DenseTruncation.h (d+1) := by
-  classical
-  calc
-    _ ≤ finiteMean (fun Y : SwitchIndex d → Bool =>
-        (Fintype.card (RoutingNetwork.SelectedCycle
-          ((palindromePerm d c₀).symm * butterflyPerm d (decodeButterfly d Y) *
-            (butterflyPerm d (decodeButterfly d X)).symm) (sharedPairs e)) : ℝ)) := by
-      apply finiteMean_mono
-      intro Y
-      have hh := alternatingCycles_le_sharedSelected e
-        (fun b => palindromePerm d (if b then (Y,X) else c₀))
-      simpa only [alternatingProjection,Bool.false_eq_true,↓reduceIte,palindromePerm,mul_assoc] using
-        (Nat.cast_le.mpr hh : (alternatingCycles e _ : ℝ) ≤ _)
-    _ ≤ _ := relabeled_single_cycle_mean d _ _ _
-
-lemma node_cycle_mgf_fixed_input (d : ℕ) (e : ι ↪ Bool × Card d)
-    (X : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (t : ℝ) (ht : 0 ≤ t) :
-    finiteMean (fun Y : (SwitchIndex d → Bool) × (SwitchIndex d → Bool) =>
-      Real.exp (t * alternatingCycles e
-        (fun b => palindromePerm d (if b then (Y.2,X.2) else (Y.1,X.1))))) ≤
-      Real.exp ((sharedPairs e).card * DenseTruncation.h (d+1) *
-        ((Real.exp (t*(2:ℝ)^d)-1)/(2:ℝ)^d)) := by
-  apply finiteMean_exp_bounded _ _ _ _ (by positivity) ht
-  · intro Y
-    constructor
-    · exact Nat.cast_nonneg _
-    · have h₀ := alternatingCycles_double e
-        (fun b => palindromePerm d (if b then (Y.2,X.2) else (Y.1,X.1)))
-      have h₁ := Fintype.card_le_of_injective e e.injective
-      simp only [Fintype.card_prod,Fintype.card_bool,card_positions] at h₁
-      exact_mod_cast (show alternatingCycles e _ ≤ 2^d by omega)
-  · rw [finiteMean_prod]
-    calc
-      _ ≤ finiteMean (fun _ : SwitchIndex d → Bool =>
-          (sharedPairs e).card * DenseTruncation.h (d+1)) :=
-        finiteMean_mono (fun Y => node_cycle_mean_one_fresh d e (Y,X.1) X.2)
-      _ = _ := finiteMean_const _
-
-end PairRouting
-
-noncomputable def sharedLevel (t : ℕ) : (d : ℕ) → {ι : Type*} → [Fintype ι] →
-    (ι ↪ Card d) → (SwitchIndex d → Bool) → ℕ
-  | 0, _, _, _, _ => 0
-  | d+1, _, _, e, X =>
-      let χ := coinStepEquiv d X
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x χ.2
-      let hc := PairRouting.colors_compatible x χ.2
-      if t = d+1 then (PairRouting.sharedPairs x).card else
-        sharedLevel t d (PairRouting.childEmbedding x c hc false) χ.1.1 +
-          sharedLevel t d (PairRouting.childEmbedding x c hc true) χ.1.2
-
-noncomputable def denseLevelCoefficient (j : ℕ) (t : ℝ) : ℝ :=
-  DenseTruncation.h (j+1) * ((Real.exp (t*(2:ℝ)^j)-1)/(2:ℝ)^j)
-
-lemma denseLevelCoefficient_nonneg (j : ℕ) {t : ℝ} (ht : 0 ≤ t) :
-    0 ≤ denseLevelCoefficient j t := by
-  apply mul_nonneg (DenseTruncation.h_nonneg _)
-  apply div_nonneg _ (by positivity)
-  exact sub_nonneg.mpr (Real.one_le_exp_iff.mpr (mul_nonneg ht (by positivity)))
-
-lemma palindrome_level_dense_conditional (j d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (t : ℝ) (ht : 0 ≤ t) :
-    finiteMean (fun Y : SwitchIndex d → Bool =>
-      Real.exp (t * palindromeLevelCost (j+1) d e (Y,X))) ≤
-        Real.exp (denseLevelCoefficient j t * sharedLevel (j+1) d e X) := by
-  classical
-  induction d generalizing ι with
-  | zero => simp only [palindromeLevelCost,sharedLevel,Nat.cast_zero,mul_zero,
-      Real.exp_zero,finiteMean_const,le_refl]
-  | succ d ih =>
-    let χ := coinStepEquiv d X
-    let x := e.trans (headTailEquiv d).toEmbedding
-    let c := PairRouting.colors x χ.2
-    let hc := PairRouting.colors_compatible x χ.2
-    let e₀ := PairRouting.childEmbedding x c hc false
-    let e₁ := PairRouting.childEmbedding x c hc true
-    have hs (Y : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (η : Card d → Bool) :
-        palindromeLevelCost (j+1) (d+1) e ((coinStepEquiv d).symm (Y,η),X) =
-          if j+1 = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x χ.2)
-            (fun b => palindromePerm d (if b then (Y.2,χ.1.2) else (Y.1,χ.1.1))) else
-            palindromeLevelCost (j+1) d e₀ (Y.1,χ.1.1) +
-              palindromeLevelCost (j+1) d e₁ (Y.2,χ.1.2) := by
-      conv_lhs => rw [←(coinStepEquiv d).symm_apply_apply X]
-      rw [palindromeLevelCost_step]
-    rw [finiteMean_equiv (coinStepEquiv d),finiteMean_prod]
-    simp_rw [hs,finiteMean_const]
-    change _ ≤ Real.exp (denseLevelCoefficient j t *
-      (if j+1 = d+1 then (PairRouting.sharedPairs x).card else
-        sharedLevel (j+1) d e₀ χ.1.1 + sharedLevel (j+1) d e₁ χ.1.2))
-    by_cases hj : j = d
-    · subst j
-      simp only [ite_true]
-      have hh := PairRouting.node_cycle_mgf_fixed_input d
-        (PairRouting.switchedEmbedding x χ.2) χ.1 t ht
-      rw [PairRouting.sharedPairs_switched] at hh
-      simpa only [denseLevelCoefficient,mul_left_comm,mul_assoc,mul_comm] using hh
-    · have hne : j+1 ≠ d+1 := by omega
-      simp only [ite_eq_right hne,Nat.cast_add,mul_add,Real.exp_add]
-      rw [finiteMean_prod_mul
-        (fun Y : SwitchIndex d → Bool => Real.exp (t * palindromeLevelCost (j+1) d e₀ (Y,χ.1.1)))
-        (fun Y : SwitchIndex d → Bool => Real.exp (t * palindromeLevelCost (j+1) d e₁ (Y,χ.1.2)))]
-      exact mul_le_mul (ih e₀ χ.1.1) (ih e₁ χ.1.2)
-        (finiteMean_nonneg (fun _ => (Real.exp_pos _).le)) (Real.exp_pos _).le
-
-
-def pairCoords (r s : ℕ) : Card ((s+1)+r) ≃ Bool × (Card r × Card s) :=
-  (cardSplit r (s+1)).trans
-    { toFun := fun x => ((x.2) 0,(x.1,Fin.tail x.2))
-      invFun := fun x => (x.2.1,Fin.cons x.1 x.2.2)
-      left_inv := by intro x; simp only [Fin.cons_self_tail]
-      right_inv := by intro x; simp only [Fin.cons_zero,Fin.tail_cons] }
-
-noncomputable def gridPairs (r s : ℕ) (A : Finset (Card ((s+1)+r))) : Finset (Card r × Card s) :=
-  Finset.univ.filter (fun x => (pairCoords r s).symm (false,x) ∈ A ∧
-    (pairCoords r s).symm (true,x) ∈ A)
-
-noncomputable def gridShared (r s : ℕ) (A : Finset (Card ((s+1)+r))) : ℕ :=
-  (gridPairs r s A).card
-
-lemma gridPairs_eq (r s : ℕ) (A : Finset (Card ((s+1)+r))) :
-    gridPairs r s A = boolFiber (A.image (pairCoords r s)) false ∩
-      boolFiber (A.image (pairCoords r s)) true := by
-  classical
-  ext x
-  have hm (b : Bool) : (b,x) ∈ A.image (pairCoords r s) ↔
-      (pairCoords r s).symm (b,x) ∈ A := by
-    constructor
-    · rintro h
-      obtain ⟨a,ha,he⟩ := Finset.mem_image.mp h
-      rw [←he,Equiv.symm_apply_apply]
-      exact ha
-    · intro h
-      exact Finset.mem_image.mpr ⟨_,h,(pairCoords r s).apply_symm_apply _⟩
-  simp only [gridPairs,Finset.mem_filter,Finset.mem_univ,true_and,Finset.mem_inter,
-    mem_boolFiber,hm]
-
-lemma gridShared_zero (s : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+1)) :
-    gridShared 0 s (Finset.univ.image e) =
-      (PairRouting.sharedPairs (e.trans (headTailEquiv s).toEmbedding)).card := by
-  classical
-  let z : Card 0 := fun i => Fin.elim0 i
-  have he : gridPairs 0 s (Finset.univ.image e) =
-      (PairRouting.sharedPairs (e.trans (headTailEquiv s).toEmbedding)).image (fun y => (z,y)) := by
-    ext ⟨b,y⟩
-    have hb : b = z := Subsingleton.elim _ _
-    subst b
-    simp only [gridPairs,Finset.mem_filter,Finset.mem_univ,true_and,
-      PairRouting.sharedPairs,Finset.mem_inter,mem_boolFiber,PairRouting.labelSet,
-      Finset.mem_image,Prod.mk.injEq,true_and]
-    change ((∃ i, e i = Fin.cons false y) ∧ ∃ i, e i = Fin.cons true y) ↔ _
-    simp only [Function.Embedding.trans_apply,Equiv.toEmbedding_apply,
-      headTailEquiv]
-    constructor
-    · rintro ⟨⟨i,hi⟩,⟨i',hi'⟩⟩
-      refine ⟨y,⟨?_,?_⟩,rfl⟩
-      · exact ⟨i,by change (e i 0,Fin.tail (e i)) = (false,y); rw [hi]; rfl⟩
-      · exact ⟨i',by change (e i' 0,Fin.tail (e i')) = (true,y); rw [hi']; rfl⟩
-    · rintro ⟨y',⟨⟨i,hi⟩,⟨i',hi'⟩⟩,rfl⟩
-      constructor
-      · exact ⟨i,(headTailEquiv s).injective hi⟩
-      · exact ⟨i',(headTailEquiv s).injective hi'⟩
-  rw [gridShared,he,Finset.card_image_of_injective]
-  intro y y' h
-  exact congrArg Prod.snd h
-
-lemma gridShared_step (r s : ℕ) (A : Finset (Card ((s+1)+(r+1)))) :
-    gridShared (r+1) s A = gridShared r s (childMarks A false) +
-      gridShared r s (childMarks A true) := by
-  classical
-  have hsum (r : ℕ) (A : Finset (Card ((s+1)+r))) : gridShared r s A =
-      ∑ b : Card r, ∑ y : Card s,
-        if (pairCoords r s).symm (false,(b,y)) ∈ A ∧
-          (pairCoords r s).symm (true,(b,y)) ∈ A then (1:ℕ) else 0 := by
-    rw [gridShared,gridPairs,Finset.card_filter,Fintype.sum_prod_type]
-  rw [hsum,hsum,hsum,←Equiv.sum_comp (headTailEquiv r).symm,Fintype.sum_prod_type]
-  simp only [Fintype.sum_bool]
-  have hc (b : Bool) (x : Card r) (y : Card s) (ε : Bool) :
-      (pairCoords (r+1) s).symm (ε,((headTailEquiv r).symm (b,x),y)) =
-        Fin.cons b ((pairCoords r s).symm (ε,(x,y))) := rfl
-  simp only [hc,mem_childMarks]
-  exact add_comm _ _
-
-lemma butterfly_false (d : ℕ) :
-    butterflyPerm d (decodeButterfly d (fun _ => false)) = 1 := by
-  induction d with
-  | zero => exact Subsingleton.elim _ _
-  | succ d ih =>
-    have he : (fun _ : SwitchIndex (d+1) => false) = (coinStepEquiv d).symm
-        (((fun _ => false),(fun _ => false)),fun _ => false) := by
-      funext i
-      rcases i with y | ⟨b,i⟩
-      · rfl
-      · cases b <;> rfl
-    rw [he,butterflyPerm_step]
-    ext x i
-    simp [ih,Equiv.Perm.mul_apply,childLift]
-    change Fin.cons (Bool.xor (x 0) false) (Fin.tail x) i = x i
-    simp
-
-lemma inverseButterflyMarks_false (d : ℕ) (A : Finset (Card d)) :
-    inverseButterflyMarks d A (fun _ => false) = A := by
-  rw [inverseButterflyMarks,butterfly_false]
-  change Finset.image id A = A
-  exact Finset.image_id
-
-
-lemma sharedLevel_partial_marks (r s : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card ((s+1)+r))
-    (lo : Card r × SwitchIndex (s+1) → Bool)
-    (hi : Card (s+1) × SwitchIndex r → Bool) :
-    sharedLevel (s+1) ((s+1)+r) e (assembleBits r (s+1) lo hi) =
-      gridShared r s (inverseButterflyMarks ((s+1)+r) (Finset.univ.image e)
-        (assembleBits r (s+1) (fun _ => false) hi)) := by
-  classical
-  induction r generalizing ι with
-  | zero =>
-    rw [show assembleBits 0 (s+1) (fun _ => false) hi = (fun _ => false) from rfl,
-      inverseButterflyMarks_false,gridShared_zero]
-    simp only [sharedLevel,ite_true]
-  | succ r ih =>
-    obtain ⟨c,rfl⟩ := (highStepEquiv r (s+1)).symm.surjective hi
-    simp only [Nat.add_succ] at *
-    rw [assembleBits_step,sharedLevel]
-    simp only [Equiv.apply_symm_apply]
-    rw [ite_eq_right (by omega : s+1 ≠ s+1+r+1)]
-    rw [ih,ih,assembleBits_step,gridShared_step,
-      inverseButterflyMarks_child,inverseButterflyMarks_child]
-    rfl
-
-lemma partial_input_parallel (r s : ℕ) (hi : Card s × SwitchIndex r → Bool)
-    (x : Card (s+r)) :
-    ((cardSplit r s).trans (Equiv.prodComm _ _))
-      ((butterflyPerm (s+r) (decodeButterfly (s+r)
-        (assembleBits r s (fun _ => false) hi))).symm x) =
-      (parallelPerm hi).symm (((cardSplit r s).trans (Equiv.prodComm _ _)) x) := by
-  let y := (butterflyPerm (s+r) (decodeButterfly (s+r)
-    (assembleBits r s (fun _ => false) hi))).symm x
-  have hh := butterfly_split r s (fun _ => false) hi y
-  dsimp only [y] at hh
-  simp only [butterfly_false,Equiv.Perm.one_apply,Equiv.apply_symm_apply] at hh
-  change cardSplit r s x =
-    (butterflyPerm r (decodeButterfly r (fun i => hi ((cardSplit r s y).2,i)))
-      (cardSplit r s y).1,(cardSplit r s y).2) at hh
-  have hs := congrArg Prod.snd hh
-  have hf := congrArg Prod.fst hh
-  dsimp only at hs hf
-  change ((cardSplit r s y).2,(cardSplit r s y).1) =
-    ((cardSplit r s x).2,(butterflyPerm r
-      (decodeButterfly r (fun i => hi ((cardSplit r s x).2,i)))).symm (cardSplit r s x).1)
-  rw [hs,hf,Equiv.symm_apply_apply]
-
-lemma partial_marks_parallel (r s : ℕ) (A : Finset (Card (s+r)))
-    (hi : Card s × SwitchIndex r → Bool) :
-    (inverseButterflyMarks (s+r) A (assembleBits r s (fun _ => false) hi)).image
-      ((cardSplit r s).trans (Equiv.prodComm _ _)) =
-        (A.image ((cardSplit r s).trans (Equiv.prodComm _ _))).image (parallelPerm hi).symm := by
-  classical
-  simp only [inverseButterflyMarks,Finset.image_image]
-  congr 1
-  funext x
-  exact partial_input_parallel r s hi x
-
-lemma cylinder_image_equiv_bound {α β Ω : Type*} [DecidableEq α] [DecidableEq β]
-    [Fintype Ω] (e : α ≃ β) (F : Ω → Finset α) (p : α → ℝ)
-    (h : CylinderBound (inclusionMoment F) p) :
-    CylinderBound (inclusionMoment (fun ω => (F ω).image e)) (fun x => p (e.symm x)) := by
-  intro S
-  rw [inclusionMoment_image_equiv]
-  have hh := h (S.image e.symm)
-  simpa only [Finset.prod_image e.symm.injective.injOn] using hh
-
-
-noncomputable def suffixMarks (r s : ℕ) (A : Finset (Card (s+r))) (y : Card s) : Finset (Card r) :=
-  blockMarks ((cardSplit r s).trans (Equiv.prodComm _ _)) A y
-
-noncomputable def sharedPartnerMean (r s : ℕ) (A : Finset (Card ((s+1)+r))) : ℝ :=
-  (∑ y : Card s, ((suffixMarks r (s+1) A (Fin.cons false y)).card : ℝ) *
-    (suffixMarks r (s+1) A (Fin.cons true y)).card) / (2:ℝ)^r
-
-lemma blockMarks_image_equiv {α B : Type*} [Fintype α] [DecidableEq α]
-    [Fintype B] [DecidableEq B] (r : ℕ) (E : α ≃ B × Card r)
-    (A : Finset α) (b : B) :
-    blockMarks (Equiv.refl _) (A.image E) b = blockMarks E A b := by
-  classical
-  ext y
-  simp only [mem_blockMarks,Equiv.refl_symm,Equiv.refl_apply,Finset.mem_image]
-  constructor
-  · rintro ⟨x,hx,he⟩
-    rw [←he,Equiv.symm_apply_apply]
-    exact hx
-  · intro h
-    exact ⟨_,h,E.apply_symm_apply _⟩
-
-def pairReorder (r s : ℕ) : Card (s+1) × Card r ≃ Bool × (Card r × Card s) where
-  toFun x := (x.1 0,(x.2,Fin.tail x.1))
-  invFun x := (Fin.cons x.1 x.2.2,x.2.1)
-  left_inv x := by simp only [Fin.cons_self_tail]
-  right_inv x := by simp only [Fin.cons_zero,Fin.tail_cons]
-
-lemma pairCoords_reorder (r s : ℕ) :
-    pairCoords r s = ((cardSplit r (s+1)).trans (Equiv.prodComm _ _)).trans (pairReorder r s) := rfl
-
-lemma partial_grid_mgf (r s : ℕ) (A : Finset (Card ((s+1)+r)))
-    (z : ℝ) (hz : 0 ≤ z) :
-    finiteMean (fun hi : Card (s+1) × SwitchIndex r → Bool =>
-      Real.exp (z * gridShared r s (inverseButterflyMarks ((s+1)+r) A
-        (assembleBits r (s+1) (fun _ => false) hi)))) ≤
-      Real.exp ((Real.exp z-1)*sharedPartnerMean r s A) := by
-  classical
-  let E := (cardSplit r (s+1)).trans (Equiv.prodComm _ _)
-  let F (hi : Card (s+1) × SwitchIndex r → Bool) := (A.image E).image (parallelPerm hi).symm
-  let p (x : Card (s+1) × Card r) : ℝ :=
-    (suffixMarks r (s+1) A x.1).card / (2:ℝ)^r
-  have hc : CylinderBound (inclusionMoment F) p := by
-    have hh := parallel_inverse_cylinder r (A.image E)
-    simp only [blockMarks_image_equiv] at hh
-    exact hh
-  have hp : ∀ x, 0 ≤ p x := by intro x; exact div_nonneg (Nat.cast_nonneg _) (by positivity)
-  have hh := shared_pair_mgf (fun hi => (F hi).image (pairReorder r s))
-    (fun x => p ((pairReorder r s).symm x)) (fun x => hp _)
-    (cylinder_image_equiv_bound (pairReorder r s) F p hc) z hz
-  have he (hi : Card (s+1) × SwitchIndex r → Bool) :
-      (F hi).image (pairReorder r s) =
-        (inverseButterflyMarks ((s+1)+r) A
-          (assembleBits r (s+1) (fun _ => false) hi)).image (pairCoords r s) := by
-    rw [pairCoords_reorder]
-    change (F hi).image (pairReorder r s) =
-      (inverseButterflyMarks ((s+1)+r) A
-        (assembleBits r (s+1) (fun _ => false) hi)).image ((pairReorder r s) ∘ E)
-    rw [←Finset.image_image,partial_marks_parallel]
-  have hm : (∑ b : Card r × Card s,
-      p ((pairReorder r s).symm (false,b))*p ((pairReorder r s).symm (true,b))) =
-        sharedPartnerMean r s A := by
-    simp only [p,pairReorder,Equiv.coe_fn_symm_mk,Fintype.sum_prod_type]
-    rw [Finset.sum_const,Finset.card_univ,card_positions,nsmul_eq_mul]
-    simp only [sharedPartnerMean, div_mul_div_comm, ←Finset.sum_div]
-    rw [show (↑(2^r) : ℝ) = (2:ℝ)^r by rfl]
-    have hpow : (2:ℝ)^r ≠ 0 := by positivity
-    generalize (∑ y : Card s, (suffixMarks r (s+1) A (Fin.cons false y)).card *
-      (suffixMarks r (s+1) A (Fin.cons true y)).card : ℝ) = v
-    field_simp
-    ; push_cast
-    ; ring
-  simp_rw [he,←gridPairs_eq] at hh
-  rw [hm] at hh
-  exact hh
-
-end Thorp
-
-end ThorpNine.Dense
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Dense.PalindromeFamilyCost
+
+namespace OAI
+
+namespace ThorpNine.Dense
+
+open scoped BigOperators
+namespace Thorp.RoutingNetwork
+open scoped BigOperators
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+noncomputable def singleSlotsEquiv (p : Equiv.Perm α) (S : Finset α) (r : ℕ) :
+    CycleSlots p S (fun _ : Fin 1 => r) ≃
+      {C : SelectedCycle p S // C.val.card = r+1} :=
+  (cycleSlotsEquivAssignments p S (fun _ : Fin 1 => r)).trans
+  { toFun := fun a => ⟨a.val 0, a.property.2 0⟩
+    invFun := fun C => ⟨fun _ => C.val, Function.injective_of_subsingleton _, fun _ => C.property⟩
+    left_inv := fun a => by
+      apply Subtype.ext
+      funext i
+      have hi : i=0 := Subsingleton.elim _ _
+      subst i
+      rfl
+    right_inv := fun C => rfl }
+
+noncomputable def cycleCount (p : Equiv.Perm α) (S : Finset α) (r : ℕ) : ℝ :=
+  Fintype.card {C : SelectedCycle p S // C.val.card = r+1}
+
+lemma cycleCount_eq_sum (p : Equiv.Perm α) (S : Finset α) (r : ℕ) :
+    cycleCount p S r = ∑ C : SelectedCycle p S, if C.val.card=r+1 then (1:ℝ) else 0 := by
+  classical
+  simp only [cycleCount, Fintype.card_subtype, Finset.sum_boole]
+
+lemma hasSum_cycleCount_weight (p : Equiv.Perm α) (S : Finset α) (f : ℕ → ℝ) :
+    HasSum (fun l => cycleCount p S l * f (l+1))
+      (∑ C : SelectedCycle p S, f C.val.card) := by
+  classical
+  have h (C : SelectedCycle p S) :
+      HasSum (fun l : ℕ => if C.val.card=l+1 then f C.val.card else 0) (f C.val.card) := by
+    have hc : 1 ≤ C.val.card := C.nonempty.card_pos
+    have hs := hasSum_single (C.val.card-1)
+      (f := fun l : ℕ => if C.val.card=l+1 then f C.val.card else 0)
+      (fun l hl => ite_eq_right (by omega))
+    simpa only [Nat.sub_add_cancel hc, ↓reduceIte] using hs
+  convert hasSum_sum (s := Finset.univ) (fun C _ => h C) using 1
+  funext l
+  rw [cycleCount_eq_sum, Finset.sum_mul]
+  apply Finset.sum_congr rfl
+  intro C _
+  split_ifs with hc
+  · simp only [one_mul, hc]
+  · simp
+
+end Thorp.RoutingNetwork
+
+namespace Thorp
+open scoped BigOperators
+lemma hasSum_finiteMean {Ω : Type*} [Fintype Ω] {f : Ω → ℕ → ℝ} {a : Ω → ℝ}
+    (hf : ∀ ω, HasSum (f ω) (a ω)) :
+    HasSum (fun l => finiteMean (fun ω => f ω l)) (finiteMean a) :=
+  (hasSum_sum (s := Finset.univ) (fun ω _ => hf ω)).div_const (Fintype.card Ω : ℝ)
+end Thorp
+
+namespace Thorp.RoutingNetwork
+open scoped BigOperators
+variable {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι] [DecidableEq ι]
+  {d : ℕ} (R : RoutingNetwork α ι d)
+
+lemma selected_cycle_mean_le_h (S : Finset α) :
+    finiteMean (fun ω : ι → Bool => (Fintype.card (SelectedCycle (R.perm ω) S) : ℝ)) ≤
+      (S.card : ℝ) * DenseTruncation.h (d+1) := by
+  classical
+  by_cases hS : S.Nonempty
+  · have hpos : (0:ℝ) < S.card := Nat.cast_pos.mpr hS.card_pos
+    let w : ℕ → ℝ := fun l => finiteMean (fun ω : ι → Bool => cycleCount (R.perm ω) S l)
+    have hw : ∀ l, 0 ≤ w l := fun _ => finiteMean_nonneg (fun _ => Nat.cast_nonneg _)
+    have hc (l : ℕ) : w l ≤ (S.card : ℝ) / (2:ℝ)^d := by
+      apply (le_div_iff₀ (by positivity : 0 < (2:ℝ)^d)).mpr
+      have he (ω : ι → Bool) : cycleCount (R.perm ω) S l =
+          (Fintype.card (CycleSlots (R.perm ω) S (fun _ : Fin 1 => l)) : ℝ) := by
+        rw [cycleCount, Fintype.card_congr (singleSlotsEquiv (R.perm ω) S l)]
+      simpa only [w,he] using single_cycleSlots_weighted_moment R S l
+    have hmass := hasSum_finiteMean (fun ω : ι → Bool =>
+      hasSum_cycleCount_weight (R.perm ω) S (fun n => (n:ℝ)))
+    have hm : HasSum (fun l => ((l+1 : ℕ) : ℝ) * w l)
+        (finiteMean (fun ω : ι → Bool => (cycleMass (R.perm ω) S : ℝ))) := by
+      simp only [finiteMean_mul_const] at hmass
+      simpa only [w, mul_comm, cycleMass,Nat.cast_sum] using hmass
+    have hmt : finiteMean (fun ω : ι → Bool => (cycleMass (R.perm ω) S : ℝ)) ≤ S.card := by
+      calc
+        _ ≤ finiteMean (fun _ : ι → Bool => (S.card : ℝ)) :=
+          finiteMean_mono (fun ω => by exact_mod_cast cycleMass_le (R.perm ω) S)
+        _ = _ := finiteMean_const _
+    let q : ℕ → ℝ := fun l => ((l+1 : ℕ) : ℝ) * w l / S.card
+    have hq : ∀ l, 0 ≤ q l := fun l => div_nonneg (mul_nonneg (by positivity) (hw l)) hpos.le
+    have hqs : Summable q := (hm.div_const (S.card : ℝ)).summable
+    have hqt : (∑' l, q l) ≤ 1 := by
+      rw [(hm.div_const (S.card : ℝ)).tsum_eq]
+      exact (div_le_one hpos).mpr hmt
+    have hqc (l : ℕ) : q l ≤ ((l+1 : ℕ) : ℝ) / (2:ℝ)^((d+1)-1) := by
+      simp only [Nat.add_sub_cancel]
+      dsimp [q]
+      calc
+        _ ≤ (((l+1 : ℕ) : ℝ) * ((S.card : ℝ) / (2:ℝ)^d)) / S.card := by
+          gcongr
+          exact hc l
+        _ = _ := by field_simp
+    have hh := DenseTruncation.subprob_reciprocal_le_h hq hqs hqt hqc
+    have hcount := hasSum_finiteMean (fun ω : ι → Bool =>
+      hasSum_cycleCount_weight (R.perm ω) S (fun _ => (1:ℝ)))
+    have hwc : HasSum w (finiteMean (fun ω : ι → Bool =>
+        (Fintype.card (SelectedCycle (R.perm ω) S) : ℝ))) := by
+      simpa only [mul_one,Finset.sum_const,Finset.card_univ,nsmul_eq_mul,mul_one] using hcount
+    have he (l : ℕ) : q l / ((l+1 : ℕ) : ℝ) = w l / S.card := by
+      have hl : ((l+1 : ℕ) : ℝ) ≠ 0 := by positivity
+      dsimp [q]
+      field_simp
+    simp only [he, (hwc.div_const (S.card : ℝ)).tsum_eq] at hh
+    simpa only [mul_comm] using (div_le_iff₀ hpos).mp hh
+  · have he : S = ∅ := Finset.not_nonempty_iff_eq_empty.mp hS
+    subst S
+    have hz (ω : ι → Bool) : Fintype.card (SelectedCycle (R.perm ω) ∅) = 0 := by
+      let : IsEmpty (SelectedCycle (R.perm ω) ∅) :=
+        ⟨fun C => C.nonempty.ne_empty (Finset.subset_empty.mp C.property.2)⟩
+      exact Fintype.card_eq_zero
+    simp only [hz,Nat.cast_zero,finiteMean_const,Finset.card_empty,zero_mul,le_refl]
+
+end Thorp.RoutingNetwork
+
+namespace Thorp.DenseTruncation
+open scoped BigOperators
+
+lemma reciprocal_values_nonempty (j : ℕ) :
+    {z : ℝ | ∃ p : ℕ → ℝ, AdmissibleCycleLaw j p ∧
+      z = ∑' l : ℕ, p l / ((l+1 : ℕ) : ℝ)}.Nonempty := by
+  let t := 2^(j-1)-1
+  have ht : t+1 = 2^(j-1) := Nat.sub_add_cancel (Nat.one_le_pow _ _ (by norm_num))
+  let p : ℕ → ℝ := fun l => if l=t then 1 else 0
+  have hp : AdmissibleCycleLaw j p := by
+    refine ⟨fun l => by dsimp [p]; split_ifs <;> norm_num, hasSum_ite_eq t 1, ?_⟩
+    intro l
+    by_cases hl : l=t
+    · subst l
+      have he : ((t+1 : ℕ) : ℝ) = (2:ℝ)^(j-1) := by rw [ht]; norm_cast
+      simp only [p,↓reduceIte,he,div_self (by positivity : (2:ℝ)^(j-1) ≠ 0),le_refl]
+    · simp only [p,ite_eq_right hl]
+      positivity
+  exact ⟨_,p,hp,rfl⟩
+
+lemma h_nonneg (j : ℕ) : 0 ≤ h j := by
+  obtain ⟨z,p,hp,rfl⟩ := reciprocal_values_nonempty j
+  exact (law_reciprocal_nonneg hp).trans (reciprocal_le_h hp)
+
+lemma h_le_one (j : ℕ) : h j ≤ 1 :=
+  csSup_le (reciprocal_values_nonempty j) (fun _ ⟨_,hp,hz⟩ => hz ▸ law_reciprocal_le_one hp)
+
+lemma law_reciprocal_cutoff {j : ℕ} {p : ℕ → ℝ} (hp : AdmissibleCycleLaw j p)
+    (L : ℕ) (hL : 0 < L) :
+    (∑' l, p l / ((l+1 : ℕ) : ℝ)) ≤ (L : ℝ)/(2:ℝ)^(j-1) + 1/L := by
+  let a : ℕ → ℝ := fun l => if l<L then 1/(2:ℝ)^(j-1) else 0
+  have ha : HasSum a ((L : ℝ)/(2:ℝ)^(j-1)) := by
+    have hs : HasSum a (∑ l ∈ Finset.range L, a l) :=
+      hasSum_sum_of_ne_finset_zero (s := Finset.range L) (f := a)
+      (fun l hl => by simpa only [a,Finset.mem_range] using ite_eq_right (show ¬l<L from fun hh => hl (Finset.mem_range.mpr hh)))
+    simpa only [a,Finset.sum_ite_of_true (fun l hl => Finset.mem_range.mp hl),
+      Finset.sum_const,Finset.card_range,nsmul_eq_mul,mul_one_div] using hs
+  have hb := hp.2.1.div_const (L:ℝ)
+  have he (l : ℕ) : p l / ((l+1 : ℕ) : ℝ) ≤ a l + p l/L := by
+    by_cases hl : l<L
+    · have hc : p l / ((l+1 : ℕ) : ℝ) ≤ 1/(2:ℝ)^(j-1) := by
+        have hp1 := hp.2.2 l
+        have hlpos : (0:ℝ) < (l+1 : ℕ) := by positivity
+        apply (div_le_iff₀ hlpos).mpr
+        simpa only [one_div, div_eq_mul_inv,mul_comm,one_mul] using hp1
+      dsimp [a]
+      rw [ite_eq_left hl]
+      exact hc.trans (le_add_of_nonneg_right (div_nonneg (hp.1 l) (by positivity)))
+    · dsimp [a]
+      rw [ite_eq_right hl,zero_add]
+      exact div_le_div_of_nonneg_left (hp.1 l) (by exact_mod_cast hL)
+        (by exact_mod_cast (show L ≤ l+1 by omega))
+  have hh := Summable.tsum_le_tsum he (law_reciprocal_summable hp) (ha.add hb).summable
+  rwa [(ha.add hb).tsum_eq] at hh
+
+lemma h_cutoff (j L : ℕ) (hL : 0 < L) :
+    h j ≤ (L : ℝ)/(2:ℝ)^(j-1) + 1/L :=
+  csSup_le (reciprocal_values_nonempty j)
+    (fun _ ⟨_,hp,hz⟩ => hz ▸ law_reciprocal_cutoff hp L hL)
+
+lemma h_dyadic_bound (j : ℕ) : h (j+1) ≤ 2/(2:ℝ)^(j/2) := by
+  have h := h_cutoff (j+1) (2^(j/2)) (by positivity)
+  simp only [Nat.add_sub_cancel,Nat.cast_pow,Nat.cast_ofNat] at h
+  have hpow : ((2:ℝ)^(j/2))^2 ≤ (2:ℝ)^j := by
+    rw [←pow_mul]
+    exact pow_le_pow_right₀ (by norm_num) (by omega)
+  have hp : (0:ℝ) < (2:ℝ)^(j/2) := by positivity
+  have hq : (0:ℝ) < (2:ℝ)^j := by positivity
+  have hr : (2:ℝ)^(j/2)/(2:ℝ)^j ≤ 1/(2:ℝ)^(j/2) := by
+    apply (div_le_div_iff₀ hq hp).mpr
+    nlinarith
+  calc
+    _ ≤ 1/(2:ℝ)^(j/2) + 1/(2:ℝ)^(j/2) := h.trans (by linarith)
+    _ = _ := by ring
+
+lemma summable_half_dyadic : Summable (fun j : ℕ => 2/(2:ℝ)^(j/2)) := by
+  apply ((Nat.divModEquiv 2).symm.summable_iff).mp
+  have he (p : ℕ × Fin 2) : 2/(2:ℝ)^(((Nat.divModEquiv 2).symm p)/2) =
+      2*(1/2:ℝ)^p.1 := by
+    rw [Nat.divModEquiv_symm_apply]
+    have hi : (p.1*2+p.2.val)/2 = p.1 := by omega
+    rw [hi,div_pow,one_pow]
+    ring
+  simp only [Function.comp_def,he]
+  apply (summable_prod_of_nonneg (fun p => by positivity)).mpr
+  constructor
+  · intro n
+    exact (hasSum_fintype _).summable
+  · simpa only [tsum_fintype,Finset.sum_const,Finset.card_univ,Fintype.card_fin,
+      nsmul_eq_mul,Nat.cast_ofNat,←mul_assoc,show (2:ℝ)*2=4 from by norm_num] using summable_geometric_two.mul_left 4
+
+lemma h_summable : Summable (fun j => h (j+1)) :=
+  Summable.of_nonneg_of_le (fun j => h_nonneg (j+1)) h_dyadic_bound summable_half_dyadic
+
+lemma allocation_value_summable {ρ : ℝ} {x : ℕ → ℝ} (hx : AdmissibleAllocation ρ x) :
+    Summable (fun j => h (j+1)*x j) :=
+  Summable.of_nonneg_of_le (fun j => mul_nonneg (h_nonneg _) (hx.1 j).1)
+    (fun j => mul_le_of_le_one_right (h_nonneg _) (hx.1 j).2) h_summable
+
+lemma H_values_bdd (ρ : ℝ) : BddAbove {z : ℝ | ∃ x : ℕ → ℝ,
+    AdmissibleAllocation ρ x ∧ z = (1/2:ℝ)*∑' j, h (j+1)*x j} := by
+  refine ⟨(1/2:ℝ)*∑' j, h (j+1), ?_⟩
+  rintro z ⟨x,hx,rfl⟩
+  apply mul_le_mul_of_nonneg_left _ (by norm_num)
+  exact Summable.tsum_le_tsum (fun j => mul_le_of_le_one_right (h_nonneg _) (hx.1 j).2)
+    (allocation_value_summable hx) h_summable
+
+lemma allocation_le_H {ρ : ℝ} {x : ℕ → ℝ} (hx : AdmissibleAllocation ρ x) :
+    (1/2:ℝ)*∑' j, h (j+1)*x j ≤ H ρ :=
+  le_csSup (H_values_bdd ρ) ⟨x,hx,rfl⟩
+
+end Thorp.DenseTruncation
+
+namespace Thorp
+open scoped BigOperators
+
+def unitCardEquiv (d : ℕ) : Card d ≃ Unit × Card d where
+  toFun x := ((),x)
+  invFun x := x.2
+  left_inv _ := rfl
+  right_inv x := by cases x with | mk u x => cases u; rfl
+
+def unitBitsEquiv (d : ℕ) : (Unit × SwitchIndex d → Bool) ≃ (SwitchIndex d → Bool) where
+  toFun ω i := ω ((),i)
+  invFun ω i := ω i.2
+  left_inv ω := by funext i; rcases i with ⟨u,i⟩; cases u; rfl
+  right_inv _ := rfl
+
+lemma relabeled_single_cycle_mean (d : ℕ) (pre post : Equiv.Perm (Card d))
+    (S : Finset (Card d)) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      (Fintype.card (RoutingNetwork.SelectedCycle
+        (post * butterflyPerm d (decodeButterfly d ω) * pre) S) : ℝ)) ≤
+      (S.card : ℝ)*DenseTruncation.h (d+1) := by
+  have hh := RoutingNetwork.selected_cycle_mean_le_h
+    (parallelNetwork (pre.trans (unitCardEquiv d)) ((unitCardEquiv d).symm.trans post)) S
+  rw [finiteMean_equiv (unitBitsEquiv d)] at hh
+  exact hh
+
+namespace CycleColoring
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+noncomputable def projectSharedCycle (p : Equiv.Perm α) (A B : Finset α)
+    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) :
+    RoutingNetwork.SelectedCycle p (A ∩ B) := by
+  refine ⟨(projectCycle p A B C).val, (projectCycle p A B C).property.1, ?_⟩
+  intro x hx
+  apply Finset.mem_inter.mpr
+  refine ⟨(projectCycle p A B C).property.2 hx, ?_⟩
+  have he := selected_alternating_eq p A B C
+  have hc : (true,x) ∈ C.val := by
+    rw [he]
+    exact (mem_childJoin _ _ true x).mpr hx
+  exact (mem_childJoin A B true x).mp (C.property.2 hc)
+
+lemma projectSharedCycle_injective (p : Equiv.Perm α) (A B : Finset α) :
+    Function.Injective (projectSharedCycle p A B) := by
+  intro C D h
+  apply projectCycle_injective p A B
+  apply Subtype.ext
+  change (projectSharedCycle p A B C).val = (projectSharedCycle p A B D).val
+  exact congrArg Subtype.val h
+
+lemma card_alternating_cycles_le_shared (p : Equiv.Perm α) (A B : Finset α) :
+    Fintype.card (RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) ≤
+      Fintype.card (RoutingNetwork.SelectedCycle p (A ∩ B)) :=
+  Fintype.card_le_of_injective _ (projectSharedCycle_injective p A B)
+
+end CycleColoring
+
+namespace PairRouting
+variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
+
+noncomputable def sharedPairs (e : ι ↪ Bool × α) : Finset α :=
+  boolFiber (labelSet e) false ∩ boolFiber (labelSet e) true
+
+lemma alternatingCycles_le_sharedSelected (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    alternatingCycles e p ≤
+      Fintype.card (RoutingNetwork.SelectedCycle (alternatingProjection p) (sharedPairs e)) := by
+  unfold alternatingCycles sharedPairs
+  conv_lhs => rw [←childJoin_boolFiber (labelSet e)]
+  exact CycleColoring.card_alternating_cycles_le_shared _ _ _
+
+omit [Fintype α] in
+lemma sharedPairs_card_twice_le (e : ι ↪ Bool × α) :
+    2*(sharedPairs e).card ≤ Fintype.card ι := by
+  have he := card_childJoin (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
+  rw [childJoin_boolFiber,card_labelSet] at he
+  have h₀ := Finset.card_le_card (Finset.inter_subset_left : sharedPairs e ⊆ boolFiber (labelSet e) false)
+  have h₁ := Finset.card_le_card (Finset.inter_subset_right : sharedPairs e ⊆ boolFiber (labelSet e) true)
+  simp only [sharedPairs] at *
+  omega
+
+omit [Fintype α] in
+lemma labelSet_switched (e : ι ↪ Bool × α) (ξ : α → Bool) :
+    labelSet (switchedEmbedding e ξ) = (labelSet e).image (pairLayer ξ) := by
+  classical
+  simp only [labelSet,Finset.image_image,switchedEmbedding]
+  rfl
+
+omit [Fintype α] in
+lemma sharedPairs_switched (e : ι ↪ Bool × α) (ξ : α → Bool) :
+    sharedPairs (switchedEmbedding e ξ) = sharedPairs e := by
+  classical
+  ext x
+  simp only [sharedPairs,Finset.mem_inter,mem_boolFiber,labelSet_switched,mem_image_perm_iff]
+  change (Bool.xor false (ξ x),x) ∈ labelSet e ∧
+      (Bool.xor true (ξ x),x) ∈ labelSet e ↔
+      (false,x) ∈ labelSet e ∧ (true,x) ∈ labelSet e
+  cases ξ x <;> simp only [Bool.xor_false,Bool.xor_true,Bool.not_false,Bool.not_true,and_comm]
+
+lemma node_cycle_mean (d : ℕ) (e : ι ↪ Bool × Card d) :
+    finiteMean (fun c : BenesCoins d × BenesCoins d =>
+      (alternatingCycles e (fun b => palindromePerm d (if b then c.2 else c.1)) : ℝ)) ≤
+        (sharedPairs e).card * DenseTruncation.h (d+1) := by
+  classical
+  rw [finiteMean_prod]
+  have hone (c₀ : BenesCoins d) :
+      finiteMean (fun c₁ : BenesCoins d =>
+        (alternatingCycles e (fun b => palindromePerm d (if b then c₁ else c₀)) : ℝ)) ≤
+          (sharedPairs e).card * DenseTruncation.h (d+1) := by
+    rw [finiteMean_prod,finiteMean_comm]
+    have hf (X : SwitchIndex d → Bool) :
+        finiteMean (fun Y : SwitchIndex d → Bool =>
+          (alternatingCycles e (fun b => palindromePerm d (if b then (Y,X) else c₀)) : ℝ)) ≤
+          (sharedPairs e).card * DenseTruncation.h (d+1) := by
+      calc
+        _ ≤ finiteMean (fun Y : SwitchIndex d → Bool =>
+            (Fintype.card (RoutingNetwork.SelectedCycle
+              ((palindromePerm d c₀).symm * butterflyPerm d (decodeButterfly d Y) *
+                (butterflyPerm d (decodeButterfly d X)).symm) (sharedPairs e)) : ℝ)) := by
+          apply finiteMean_mono
+          intro Y
+          have hh := alternatingCycles_le_sharedSelected e
+            (fun b => palindromePerm d (if b then (Y,X) else c₀))
+          simpa only [alternatingProjection,Bool.false_eq_true,↓reduceIte,palindromePerm,mul_assoc] using
+            (Nat.cast_le.mpr hh : (alternatingCycles e _ : ℝ) ≤ _)
+        _ ≤ _ := relabeled_single_cycle_mean d _ _ _
+    calc
+      _ ≤ finiteMean (fun _ : SwitchIndex d → Bool =>
+          (sharedPairs e).card * DenseTruncation.h (d+1)) := finiteMean_mono hf
+      _ = _ := finiteMean_const _
+  calc
+    _ ≤ finiteMean (fun _ : BenesCoins d => (sharedPairs e).card * DenseTruncation.h (d+1)) :=
+      finiteMean_mono hone
+    _ = _ := finiteMean_const _
+
+end PairRouting
+end Thorp
+
+namespace Thorp
+
+lemma exp_secant {M x t : ℝ} (hM : 0 < M) (hx : 0 ≤ x) (hxM : x ≤ M) :
+    Real.exp (t*x) ≤ 1 + x*((Real.exp (t*M)-1)/M) := by
+  have hb : 0 ≤ x/M := div_nonneg hx hM.le
+  have ha : 0 ≤ 1-x/M := sub_nonneg.mpr ((div_le_one hM).mpr hxM)
+  have hh := convexOn_exp.2 (Set.mem_univ (0:ℝ)) (Set.mem_univ (t*M)) ha hb
+    (show (1-x/M)+(x/M)=1 by ring)
+  simp only [smul_eq_mul, mul_zero, zero_add, Real.exp_zero, mul_one] at hh
+  have he : x/M*(t*M) = t*x := by field_simp
+  rw [he] at hh
+  calc
+    _ ≤ 1-x/M+(x/M)*Real.exp (t*M) := hh
+    _ = _ := by ring
+
+lemma finiteMean_exp_bounded {Ω : Type*} [Fintype Ω] [Nonempty Ω]
+    (Z : Ω → ℝ) (M t μ : ℝ) (hM : 0 < M) (ht : 0 ≤ t)
+    (hZ : ∀ ω, 0 ≤ Z ω ∧ Z ω ≤ M) (hmean : finiteMean Z ≤ μ) :
+    finiteMean (fun ω => Real.exp (t*Z ω)) ≤
+      Real.exp (μ*((Real.exp (t*M)-1)/M)) := by
+  have hz : 0 ≤ (Real.exp (t*M)-1)/M :=
+    div_nonneg (sub_nonneg.mpr (Real.one_le_exp_iff.mpr (mul_nonneg ht hM.le))) hM.le
+  calc
+    _ ≤ finiteMean (fun ω => 1 + Z ω*((Real.exp (t*M)-1)/M)) :=
+      finiteMean_mono (fun ω => exp_secant hM (hZ ω).1 (hZ ω).2)
+    _ = 1 + finiteMean Z*((Real.exp (t*M)-1)/M) := by
+      rw [finiteMean_add,finiteMean_const,finiteMean_mul_const]
+    _ ≤ 1 + μ*((Real.exp (t*M)-1)/M) := by gcongr
+    _ ≤ _ := by simpa only [add_comm] using Real.add_one_le_exp (μ*((Real.exp (t*M)-1)/M))
+
+namespace PairRouting
+variable {ι : Type*} [Fintype ι]
+
+lemma node_cycle_mgf (d : ℕ) (e : ι ↪ Bool × Card d) (t : ℝ) (ht : 0 ≤ t) :
+    finiteMean (fun c : BenesCoins d × BenesCoins d =>
+      Real.exp (t * alternatingCycles e
+        (fun b => palindromePerm d (if b then c.2 else c.1)))) ≤
+      Real.exp ((sharedPairs e).card * DenseTruncation.h (d+1) *
+        ((Real.exp (t*(2:ℝ)^d)-1)/(2:ℝ)^d)) := by
+  apply finiteMean_exp_bounded _ _ _ _ (by positivity) ht _ (node_cycle_mean d e)
+  intro c
+  constructor
+  · exact Nat.cast_nonneg _
+  · have h₀ := alternatingCycles_double e
+      (fun b => palindromePerm d (if b then c.2 else c.1))
+    have h₁ := Fintype.card_le_of_injective e e.injective
+    simp only [Fintype.card_prod,Fintype.card_bool,card_positions] at h₁
+    have h₂ : alternatingCycles e (fun b => palindromePerm d (if b then c.2 else c.1)) ≤ 2^d := by omega
+    exact_mod_cast h₂
+
+end PairRouting
+
+open scoped BigOperators
+
+lemma shared_pair_mgf {β Ω : Type*} [Fintype β] [DecidableEq β]
+    [Fintype Ω] (F : Ω → Finset (Bool × β)) (p : Bool × β → ℝ)
+    (hp : ∀ x, 0 ≤ p x) (hc : CylinderBound (inclusionMoment F) p)
+    (z : ℝ) (hz : 0 ≤ z) :
+    finiteMean (fun ω => Real.exp (z * (boolFiber (F ω) false ∩ boolFiber (F ω) true).card)) ≤
+      Real.exp ((Real.exp z - 1) * ∑ b, p (false,b)*p (true,b)) := by
+  classical
+  let T : β → Bool → Finset (Bool × β) := fun b q =>
+    if q then {(false,b),(true,b)} else ∅
+  let c : β → Bool → ℝ := fun _ q => if q then Real.exp z - 1 else 1
+  have hcz : 0 ≤ Real.exp z - 1 := sub_nonneg.mpr (Real.one_le_exp_iff.mpr hz)
+  have hdis : ∀ b b', b ≠ b' → ∀ q q', Disjoint (T b q) (T b' q') := by
+    intro b b' h q q'
+    cases q <;> cases q' <;> simp [T,Finset.disjoint_left,h]
+  have hh := cylinder_block_polynomial (fun _ : β => Bool) F p hc T c
+    (fun b q => by cases q <;> simp only [c,↓reduceIte] <;> positivity) hdis
+  have hsum (A : Finset (Bool × β)) (b : β) :
+      (∑ q, c b q * if T b q ⊆ A then 1 else 0) =
+        Real.exp (z * if (false,b) ∈ A ∧ (true,b) ∈ A then 1 else 0) := by
+    simp only [Fintype.sum_bool,c,T,Bool.false_eq_true,↓reduceIte,Finset.empty_subset,
+      mul_one,Finset.insert_subset_iff,Finset.singleton_subset_iff]
+    split_ifs <;> simp
+  have hcard (A : Finset (Bool × β)) :
+      (∑ b : β, if (false,b) ∈ A ∧ (true,b) ∈ A then (1:ℝ) else 0) =
+        (boolFiber A false ∩ boolFiber A true).card := by
+    rw [Finset.sum_boole]
+    congr 2
+    ext b
+    simp [mem_boolFiber]
+  have hleft (ω : Ω) :
+      (∏ b, ∑ q, c b q * if T b q ⊆ F ω then 1 else 0) =
+        Real.exp (z*(boolFiber (F ω) false ∩ boolFiber (F ω) true).card) := by
+    simp_rw [hsum]
+    rw [←Real.exp_sum,←Finset.mul_sum,hcard]
+  have hright (b : β) :
+      (∑ q, c b q * ∏ x ∈ T b q, p x) =
+        1+(Real.exp z-1)*(p (false,b)*p (true,b)) := by
+    simp [c,T,add_comm]
+  simp_rw [hleft,hright] at hh
+  refine hh.trans ?_
+  rw [Finset.mul_sum,Real.exp_sum]
+  apply Finset.prod_le_prod₀
+  · intro b _
+    exact add_nonneg (by norm_num) (mul_nonneg hcz (mul_nonneg (hp _) (hp _)))
+  · intro b _
+    simpa only [add_comm] using Real.add_one_le_exp ((Real.exp z-1)*(p (false,b)*p (true,b)))
+
+end Thorp
+
+namespace Thorp
+variable {B : Type*} [Fintype B] [DecidableEq B]
+
+lemma prod_blockMarks (d : ℕ) (S : Finset (B × Card d)) (f : B × Card d → ℝ) :
+    (∏ x ∈ S, f x) = ∏ b, ∏ x ∈ blockMarks (Equiv.refl _) S b, f (b,x) := by
+  classical
+  have hb (b : B) : (∏ x ∈ blockMarks (Equiv.refl _) S b, f (b,x)) =
+      ∏ x, if (b,x) ∈ S then f (b,x) else 1 := by
+    exact Finset.prod_filter (fun x => (b,x) ∈ S) (fun x => f (b,x))
+  simp_rw [hb]
+  rw [←Fintype.prod_prod_type (fun x : B × Card d => if x ∈ S then f x else 1)]
+  symm
+  calc
+    _ = ∏ x ∈ Finset.univ.filter (fun x : B × Card d => x ∈ S), f x :=
+      (Finset.prod_filter _ _).symm
+    _ = _ := by simp
+
+omit [Fintype B] in
+lemma subset_parallel_inverse (d : ℕ) (A S : Finset (B × Card d))
+    (ω : B × SwitchIndex d → Bool) :
+    S ⊆ A.image (parallelPerm ω).symm ↔
+      ∀ b, blockMarks (Equiv.refl _) S b ⊆
+        inverseButterflyMarks d (blockMarks (Equiv.refl _) A b) (fun i => ω (b,i)) := by
+  classical
+  constructor
+  · intro h b x hx
+    have hh := h ((mem_blockMarks (Equiv.refl _) S b x).mp hx)
+    rw [mem_image_perm_iff] at hh
+    rw [inverseButterflyMarks,mem_image_perm_iff]
+    exact (mem_blockMarks (Equiv.refl _) A b _).mpr hh
+  · intro h ⟨b,x⟩ hx
+    have hh := h b ((mem_blockMarks (Equiv.refl _) S b x).mpr hx)
+    rw [inverseButterflyMarks,mem_image_perm_iff] at hh
+    rw [mem_image_perm_iff]
+    exact (mem_blockMarks (Equiv.refl _) A b _).mp hh
+
+lemma parallel_inverse_cylinder (d : ℕ) (A : Finset (B × Card d)) :
+    CylinderBound (inclusionMoment (fun ω : B × SwitchIndex d → Bool =>
+      A.image (parallelPerm ω).symm))
+      (fun x => (blockMarks (Equiv.refl _) A x.1).card / (2:ℝ)^d) := by
+  classical
+  intro S
+  rw [inclusionMoment_eq,finiteMean_equiv (Equiv.curry B (SwitchIndex d) Bool)]
+  have hi (ω : B → SwitchIndex d → Bool) :
+      (if S ⊆ A.image (parallelPerm ((Equiv.curry B (SwitchIndex d) Bool).symm ω)).symm
+        then (1:ℝ) else 0) =
+      ∏ b, if blockMarks (Equiv.refl _) S b ⊆ inverseButterflyMarks d
+        (blockMarks (Equiv.refl _) A b) (ω b) then 1 else 0 := by
+    simp only [subset_parallel_inverse]
+    change (if ∀ b, blockMarks (Equiv.refl _) S b ⊆ inverseButterflyMarks d
+      (blockMarks (Equiv.refl _) A b) (ω b) then (1:ℝ) else 0) = _
+    by_cases h : ∀ b, blockMarks (Equiv.refl _) S b ⊆
+        inverseButterflyMarks d (blockMarks (Equiv.refl _) A b) (ω b)
+    · simp [h]
+    · rw [ite_eq_right h]
+      obtain ⟨b,hb⟩ := not_forall.mp h
+      symm
+      apply Finset.prod_eq_zero (Finset.mem_univ b)
+      exact ite_eq_right hb
+  simp_rw [hi]
+  rw [finiteMean_pi_prod (fun b ω => if blockMarks (Equiv.refl _) S b ⊆
+    inverseButterflyMarks d (blockMarks (Equiv.refl _) A b) ω then 1 else 0),prod_blockMarks]
+  apply Finset.prod_le_prod₀
+  · intro b _
+    exact finiteMean_nonneg (fun ω => by split_ifs <;> norm_num)
+  · intro b _
+    simpa only [←inclusionMoment_eq,Finset.prod_const] using
+      inverseButterflyMarks_inclusion d (blockMarks (Equiv.refl _) A b)
+        (blockMarks (Equiv.refl _) S b)
+
+end Thorp
+
+namespace Thorp
+namespace PairRouting
+variable {ι : Type*} [Fintype ι]
+
+lemma node_cycle_mean_one_fresh (d : ℕ) (e : ι ↪ Bool × Card d)
+    (c₀ : BenesCoins d) (X : SwitchIndex d → Bool) :
+    finiteMean (fun Y : SwitchIndex d → Bool =>
+      (alternatingCycles e (fun b => palindromePerm d (if b then (Y,X) else c₀)) : ℝ)) ≤
+        (sharedPairs e).card * DenseTruncation.h (d+1) := by
+  classical
+  calc
+    _ ≤ finiteMean (fun Y : SwitchIndex d → Bool =>
+        (Fintype.card (RoutingNetwork.SelectedCycle
+          ((palindromePerm d c₀).symm * butterflyPerm d (decodeButterfly d Y) *
+            (butterflyPerm d (decodeButterfly d X)).symm) (sharedPairs e)) : ℝ)) := by
+      apply finiteMean_mono
+      intro Y
+      have hh := alternatingCycles_le_sharedSelected e
+        (fun b => palindromePerm d (if b then (Y,X) else c₀))
+      simpa only [alternatingProjection,Bool.false_eq_true,↓reduceIte,palindromePerm,mul_assoc] using
+        (Nat.cast_le.mpr hh : (alternatingCycles e _ : ℝ) ≤ _)
+    _ ≤ _ := relabeled_single_cycle_mean d _ _ _
+
+lemma node_cycle_mgf_fixed_input (d : ℕ) (e : ι ↪ Bool × Card d)
+    (X : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (t : ℝ) (ht : 0 ≤ t) :
+    finiteMean (fun Y : (SwitchIndex d → Bool) × (SwitchIndex d → Bool) =>
+      Real.exp (t * alternatingCycles e
+        (fun b => palindromePerm d (if b then (Y.2,X.2) else (Y.1,X.1))))) ≤
+      Real.exp ((sharedPairs e).card * DenseTruncation.h (d+1) *
+        ((Real.exp (t*(2:ℝ)^d)-1)/(2:ℝ)^d)) := by
+  apply finiteMean_exp_bounded _ _ _ _ (by positivity) ht
+  · intro Y
+    constructor
+    · exact Nat.cast_nonneg _
+    · have h₀ := alternatingCycles_double e
+        (fun b => palindromePerm d (if b then (Y.2,X.2) else (Y.1,X.1)))
+      have h₁ := Fintype.card_le_of_injective e e.injective
+      simp only [Fintype.card_prod,Fintype.card_bool,card_positions] at h₁
+      exact_mod_cast (show alternatingCycles e _ ≤ 2^d by omega)
+  · rw [finiteMean_prod]
+    calc
+      _ ≤ finiteMean (fun _ : SwitchIndex d → Bool =>
+          (sharedPairs e).card * DenseTruncation.h (d+1)) :=
+        finiteMean_mono (fun Y => node_cycle_mean_one_fresh d e (Y,X.1) X.2)
+      _ = _ := finiteMean_const _
+
+end PairRouting
+
+noncomputable def sharedLevel (t : ℕ) : (d : ℕ) → {ι : Type*} → [Fintype ι] →
+    (ι ↪ Card d) → (SwitchIndex d → Bool) → ℕ
+  | 0, _, _, _, _ => 0
+  | d+1, _, _, e, X =>
+      let χ := coinStepEquiv d X
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x χ.2
+      let hc := PairRouting.colors_compatible x χ.2
+      if t = d+1 then (PairRouting.sharedPairs x).card else
+        sharedLevel t d (PairRouting.childEmbedding x c hc false) χ.1.1 +
+          sharedLevel t d (PairRouting.childEmbedding x c hc true) χ.1.2
+
+noncomputable def denseLevelCoefficient (j : ℕ) (t : ℝ) : ℝ :=
+  DenseTruncation.h (j+1) * ((Real.exp (t*(2:ℝ)^j)-1)/(2:ℝ)^j)
+
+lemma denseLevelCoefficient_nonneg (j : ℕ) {t : ℝ} (ht : 0 ≤ t) :
+    0 ≤ denseLevelCoefficient j t := by
+  apply mul_nonneg (DenseTruncation.h_nonneg _)
+  apply div_nonneg _ (by positivity)
+  exact sub_nonneg.mpr (Real.one_le_exp_iff.mpr (mul_nonneg ht (by positivity)))
+
+lemma palindrome_level_dense_conditional (j d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (t : ℝ) (ht : 0 ≤ t) :
+    finiteMean (fun Y : SwitchIndex d → Bool =>
+      Real.exp (t * palindromeLevelCost (j+1) d e (Y,X))) ≤
+        Real.exp (denseLevelCoefficient j t * sharedLevel (j+1) d e X) := by
+  classical
+  induction d generalizing ι with
+  | zero => simp only [palindromeLevelCost,sharedLevel,Nat.cast_zero,mul_zero,
+      Real.exp_zero,finiteMean_const,le_refl]
+  | succ d ih =>
+    let χ := coinStepEquiv d X
+    let x := e.trans (headTailEquiv d).toEmbedding
+    let c := PairRouting.colors x χ.2
+    let hc := PairRouting.colors_compatible x χ.2
+    let e₀ := PairRouting.childEmbedding x c hc false
+    let e₁ := PairRouting.childEmbedding x c hc true
+    have hs (Y : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (η : Card d → Bool) :
+        palindromeLevelCost (j+1) (d+1) e ((coinStepEquiv d).symm (Y,η),X) =
+          if j+1 = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x χ.2)
+            (fun b => palindromePerm d (if b then (Y.2,χ.1.2) else (Y.1,χ.1.1))) else
+            palindromeLevelCost (j+1) d e₀ (Y.1,χ.1.1) +
+              palindromeLevelCost (j+1) d e₁ (Y.2,χ.1.2) := by
+      conv_lhs => rw [←(coinStepEquiv d).symm_apply_apply X]
+      rw [palindromeLevelCost_step]
+    rw [finiteMean_equiv (coinStepEquiv d),finiteMean_prod]
+    simp_rw [hs,finiteMean_const]
+    change _ ≤ Real.exp (denseLevelCoefficient j t *
+      (if j+1 = d+1 then (PairRouting.sharedPairs x).card else
+        sharedLevel (j+1) d e₀ χ.1.1 + sharedLevel (j+1) d e₁ χ.1.2))
+    by_cases hj : j = d
+    · subst j
+      simp only [ite_true]
+      have hh := PairRouting.node_cycle_mgf_fixed_input d
+        (PairRouting.switchedEmbedding x χ.2) χ.1 t ht
+      rw [PairRouting.sharedPairs_switched] at hh
+      simpa only [denseLevelCoefficient,mul_left_comm,mul_assoc,mul_comm] using hh
+    · have hne : j+1 ≠ d+1 := by omega
+      simp only [ite_eq_right hne,Nat.cast_add,mul_add,Real.exp_add]
+      rw [finiteMean_prod_mul
+        (fun Y : SwitchIndex d → Bool => Real.exp (t * palindromeLevelCost (j+1) d e₀ (Y,χ.1.1)))
+        (fun Y : SwitchIndex d → Bool => Real.exp (t * palindromeLevelCost (j+1) d e₁ (Y,χ.1.2)))]
+      exact mul_le_mul (ih e₀ χ.1.1) (ih e₁ χ.1.2)
+        (finiteMean_nonneg (fun _ => (Real.exp_pos _).le)) (Real.exp_pos _).le
+
+
+def pairCoords (r s : ℕ) : Card ((s+1)+r) ≃ Bool × (Card r × Card s) :=
+  (cardSplit r (s+1)).trans
+    { toFun := fun x => ((x.2) 0,(x.1,Fin.tail x.2))
+      invFun := fun x => (x.2.1,Fin.cons x.1 x.2.2)
+      left_inv := by intro x; simp only [Fin.cons_self_tail]
+      right_inv := by intro x; simp only [Fin.cons_zero,Fin.tail_cons] }
+
+noncomputable def gridPairs (r s : ℕ) (A : Finset (Card ((s+1)+r))) : Finset (Card r × Card s) :=
+  Finset.univ.filter (fun x => (pairCoords r s).symm (false,x) ∈ A ∧
+    (pairCoords r s).symm (true,x) ∈ A)
+
+noncomputable def gridShared (r s : ℕ) (A : Finset (Card ((s+1)+r))) : ℕ :=
+  (gridPairs r s A).card
+
+lemma gridPairs_eq (r s : ℕ) (A : Finset (Card ((s+1)+r))) :
+    gridPairs r s A = boolFiber (A.image (pairCoords r s)) false ∩
+      boolFiber (A.image (pairCoords r s)) true := by
+  classical
+  ext x
+  have hm (b : Bool) : (b,x) ∈ A.image (pairCoords r s) ↔
+      (pairCoords r s).symm (b,x) ∈ A := by
+    constructor
+    · rintro h
+      obtain ⟨a,ha,he⟩ := Finset.mem_image.mp h
+      rw [←he,Equiv.symm_apply_apply]
+      exact ha
+    · intro h
+      exact Finset.mem_image.mpr ⟨_,h,(pairCoords r s).apply_symm_apply _⟩
+  simp only [gridPairs,Finset.mem_filter,Finset.mem_univ,true_and,Finset.mem_inter,
+    mem_boolFiber,hm]
+
+lemma gridShared_zero (s : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+1)) :
+    gridShared 0 s (Finset.univ.image e) =
+      (PairRouting.sharedPairs (e.trans (headTailEquiv s).toEmbedding)).card := by
+  classical
+  let z : Card 0 := fun i => Fin.elim0 i
+  have he : gridPairs 0 s (Finset.univ.image e) =
+      (PairRouting.sharedPairs (e.trans (headTailEquiv s).toEmbedding)).image (fun y => (z,y)) := by
+    ext ⟨b,y⟩
+    have hb : b = z := Subsingleton.elim _ _
+    subst b
+    simp only [gridPairs,Finset.mem_filter,Finset.mem_univ,true_and,
+      PairRouting.sharedPairs,Finset.mem_inter,mem_boolFiber,PairRouting.labelSet,
+      Finset.mem_image,Prod.mk.injEq,true_and]
+    change ((∃ i, e i = Fin.cons false y) ∧ ∃ i, e i = Fin.cons true y) ↔ _
+    simp only [Function.Embedding.trans_apply,Equiv.toEmbedding_apply,
+      headTailEquiv]
+    constructor
+    · rintro ⟨⟨i,hi⟩,⟨i',hi'⟩⟩
+      refine ⟨y,⟨?_,?_⟩,rfl⟩
+      · exact ⟨i,by change (e i 0,Fin.tail (e i)) = (false,y); rw [hi]; rfl⟩
+      · exact ⟨i',by change (e i' 0,Fin.tail (e i')) = (true,y); rw [hi']; rfl⟩
+    · rintro ⟨y',⟨⟨i,hi⟩,⟨i',hi'⟩⟩,rfl⟩
+      constructor
+      · exact ⟨i,(headTailEquiv s).injective hi⟩
+      · exact ⟨i',(headTailEquiv s).injective hi'⟩
+  rw [gridShared,he,Finset.card_image_of_injective]
+  intro y y' h
+  exact congrArg Prod.snd h
+
+lemma gridShared_step (r s : ℕ) (A : Finset (Card ((s+1)+(r+1)))) :
+    gridShared (r+1) s A = gridShared r s (childMarks A false) +
+      gridShared r s (childMarks A true) := by
+  classical
+  have hsum (r : ℕ) (A : Finset (Card ((s+1)+r))) : gridShared r s A =
+      ∑ b : Card r, ∑ y : Card s,
+        if (pairCoords r s).symm (false,(b,y)) ∈ A ∧
+          (pairCoords r s).symm (true,(b,y)) ∈ A then (1:ℕ) else 0 := by
+    rw [gridShared,gridPairs,Finset.card_filter,Fintype.sum_prod_type]
+  rw [hsum,hsum,hsum,←Equiv.sum_comp (headTailEquiv r).symm,Fintype.sum_prod_type]
+  simp only [Fintype.sum_bool]
+  have hc (b : Bool) (x : Card r) (y : Card s) (ε : Bool) :
+      (pairCoords (r+1) s).symm (ε,((headTailEquiv r).symm (b,x),y)) =
+        Fin.cons b ((pairCoords r s).symm (ε,(x,y))) := rfl
+  simp only [hc,mem_childMarks]
+  exact add_comm _ _
+
+lemma butterfly_false (d : ℕ) :
+    butterflyPerm d (decodeButterfly d (fun _ => false)) = 1 := by
+  induction d with
+  | zero => exact Subsingleton.elim _ _
+  | succ d ih =>
+    have he : (fun _ : SwitchIndex (d+1) => false) = (coinStepEquiv d).symm
+        (((fun _ => false),(fun _ => false)),fun _ => false) := by
+      funext i
+      rcases i with y | ⟨b,i⟩
+      · rfl
+      · cases b <;> rfl
+    rw [he,butterflyPerm_step]
+    ext x i
+    simp [ih,Equiv.Perm.mul_apply,childLift]
+    change Fin.cons (Bool.xor (x 0) false) (Fin.tail x) i = x i
+    simp
+
+lemma inverseButterflyMarks_false (d : ℕ) (A : Finset (Card d)) :
+    inverseButterflyMarks d A (fun _ => false) = A := by
+  rw [inverseButterflyMarks,butterfly_false]
+  change Finset.image id A = A
+  exact Finset.image_id
+
+
+lemma sharedLevel_partial_marks (r s : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card ((s+1)+r))
+    (lo : Card r × SwitchIndex (s+1) → Bool)
+    (hi : Card (s+1) × SwitchIndex r → Bool) :
+    sharedLevel (s+1) ((s+1)+r) e (assembleBits r (s+1) lo hi) =
+      gridShared r s (inverseButterflyMarks ((s+1)+r) (Finset.univ.image e)
+        (assembleBits r (s+1) (fun _ => false) hi)) := by
+  classical
+  induction r generalizing ι with
+  | zero =>
+    rw [show assembleBits 0 (s+1) (fun _ => false) hi = (fun _ => false) from rfl,
+      inverseButterflyMarks_false,gridShared_zero]
+    simp only [sharedLevel,ite_true]
+  | succ r ih =>
+    obtain ⟨c,rfl⟩ := (highStepEquiv r (s+1)).symm.surjective hi
+    simp only [Nat.add_succ] at *
+    rw [assembleBits_step,sharedLevel]
+    simp only [Equiv.apply_symm_apply]
+    rw [ite_eq_right (by omega : s+1 ≠ s+1+r+1)]
+    rw [ih,ih,assembleBits_step,gridShared_step,
+      inverseButterflyMarks_child,inverseButterflyMarks_child]
+    rfl
+
+lemma partial_input_parallel (r s : ℕ) (hi : Card s × SwitchIndex r → Bool)
+    (x : Card (s+r)) :
+    ((cardSplit r s).trans (Equiv.prodComm _ _))
+      ((butterflyPerm (s+r) (decodeButterfly (s+r)
+        (assembleBits r s (fun _ => false) hi))).symm x) =
+      (parallelPerm hi).symm (((cardSplit r s).trans (Equiv.prodComm _ _)) x) := by
+  let y := (butterflyPerm (s+r) (decodeButterfly (s+r)
+    (assembleBits r s (fun _ => false) hi))).symm x
+  have hh := butterfly_split r s (fun _ => false) hi y
+  dsimp only [y] at hh
+  simp only [butterfly_false,Equiv.Perm.one_apply,Equiv.apply_symm_apply] at hh
+  change cardSplit r s x =
+    (butterflyPerm r (decodeButterfly r (fun i => hi ((cardSplit r s y).2,i)))
+      (cardSplit r s y).1,(cardSplit r s y).2) at hh
+  have hs := congrArg Prod.snd hh
+  have hf := congrArg Prod.fst hh
+  dsimp only at hs hf
+  change ((cardSplit r s y).2,(cardSplit r s y).1) =
+    ((cardSplit r s x).2,(butterflyPerm r
+      (decodeButterfly r (fun i => hi ((cardSplit r s x).2,i)))).symm (cardSplit r s x).1)
+  rw [hs,hf,Equiv.symm_apply_apply]
+
+lemma partial_marks_parallel (r s : ℕ) (A : Finset (Card (s+r)))
+    (hi : Card s × SwitchIndex r → Bool) :
+    (inverseButterflyMarks (s+r) A (assembleBits r s (fun _ => false) hi)).image
+      ((cardSplit r s).trans (Equiv.prodComm _ _)) =
+        (A.image ((cardSplit r s).trans (Equiv.prodComm _ _))).image (parallelPerm hi).symm := by
+  classical
+  simp only [inverseButterflyMarks,Finset.image_image]
+  congr 1
+  funext x
+  exact partial_input_parallel r s hi x
+
+lemma cylinder_image_equiv_bound {α β Ω : Type*} [DecidableEq α] [DecidableEq β]
+    [Fintype Ω] (e : α ≃ β) (F : Ω → Finset α) (p : α → ℝ)
+    (h : CylinderBound (inclusionMoment F) p) :
+    CylinderBound (inclusionMoment (fun ω => (F ω).image e)) (fun x => p (e.symm x)) := by
+  intro S
+  rw [inclusionMoment_image_equiv]
+  have hh := h (S.image e.symm)
+  simpa only [Finset.prod_image e.symm.injective.injOn] using hh
+
+
+noncomputable def suffixMarks (r s : ℕ) (A : Finset (Card (s+r))) (y : Card s) : Finset (Card r) :=
+  blockMarks ((cardSplit r s).trans (Equiv.prodComm _ _)) A y
+
+noncomputable def sharedPartnerMean (r s : ℕ) (A : Finset (Card ((s+1)+r))) : ℝ :=
+  (∑ y : Card s, ((suffixMarks r (s+1) A (Fin.cons false y)).card : ℝ) *
+    (suffixMarks r (s+1) A (Fin.cons true y)).card) / (2:ℝ)^r
+
+lemma blockMarks_image_equiv {α B : Type*} [Fintype α] [DecidableEq α]
+    [Fintype B] [DecidableEq B] (r : ℕ) (E : α ≃ B × Card r)
+    (A : Finset α) (b : B) :
+    blockMarks (Equiv.refl _) (A.image E) b = blockMarks E A b := by
+  classical
+  ext y
+  simp only [mem_blockMarks,Equiv.refl_symm,Equiv.refl_apply,Finset.mem_image]
+  constructor
+  · rintro ⟨x,hx,he⟩
+    rw [←he,Equiv.symm_apply_apply]
+    exact hx
+  · intro h
+    exact ⟨_,h,E.apply_symm_apply _⟩
+
+def pairReorder (r s : ℕ) : Card (s+1) × Card r ≃ Bool × (Card r × Card s) where
+  toFun x := (x.1 0,(x.2,Fin.tail x.1))
+  invFun x := (Fin.cons x.1 x.2.2,x.2.1)
+  left_inv x := by simp only [Fin.cons_self_tail]
+  right_inv x := by simp only [Fin.cons_zero,Fin.tail_cons]
+
+lemma pairCoords_reorder (r s : ℕ) :
+    pairCoords r s = ((cardSplit r (s+1)).trans (Equiv.prodComm _ _)).trans (pairReorder r s) := rfl
+
+lemma partial_grid_mgf (r s : ℕ) (A : Finset (Card ((s+1)+r)))
+    (z : ℝ) (hz : 0 ≤ z) :
+    finiteMean (fun hi : Card (s+1) × SwitchIndex r → Bool =>
+      Real.exp (z * gridShared r s (inverseButterflyMarks ((s+1)+r) A
+        (assembleBits r (s+1) (fun _ => false) hi)))) ≤
+      Real.exp ((Real.exp z-1)*sharedPartnerMean r s A) := by
+  classical
+  let E := (cardSplit r (s+1)).trans (Equiv.prodComm _ _)
+  let F (hi : Card (s+1) × SwitchIndex r → Bool) := (A.image E).image (parallelPerm hi).symm
+  let p (x : Card (s+1) × Card r) : ℝ :=
+    (suffixMarks r (s+1) A x.1).card / (2:ℝ)^r
+  have hc : CylinderBound (inclusionMoment F) p := by
+    have hh := parallel_inverse_cylinder r (A.image E)
+    simp only [blockMarks_image_equiv] at hh
+    exact hh
+  have hp : ∀ x, 0 ≤ p x := by intro x; exact div_nonneg (Nat.cast_nonneg _) (by positivity)
+  have hh := shared_pair_mgf (fun hi => (F hi).image (pairReorder r s))
+    (fun x => p ((pairReorder r s).symm x)) (fun x => hp _)
+    (cylinder_image_equiv_bound (pairReorder r s) F p hc) z hz
+  have he (hi : Card (s+1) × SwitchIndex r → Bool) :
+      (F hi).image (pairReorder r s) =
+        (inverseButterflyMarks ((s+1)+r) A
+          (assembleBits r (s+1) (fun _ => false) hi)).image (pairCoords r s) := by
+    rw [pairCoords_reorder]
+    change (F hi).image (pairReorder r s) =
+      (inverseButterflyMarks ((s+1)+r) A
+        (assembleBits r (s+1) (fun _ => false) hi)).image ((pairReorder r s) ∘ E)
+    rw [←Finset.image_image,partial_marks_parallel]
+  have hm : (∑ b : Card r × Card s,
+      p ((pairReorder r s).symm (false,b))*p ((pairReorder r s).symm (true,b))) =
+        sharedPartnerMean r s A := by
+    simp only [p,pairReorder,Equiv.coe_fn_symm_mk,Fintype.sum_prod_type]
+    rw [Finset.sum_const,Finset.card_univ,card_positions,nsmul_eq_mul]
+    simp only [sharedPartnerMean, div_mul_div_comm, ←Finset.sum_div]
+    rw [show (↑(2^r) : ℝ) = (2:ℝ)^r by rfl]
+    have hpow : (2:ℝ)^r ≠ 0 := by positivity
+    generalize (∑ y : Card s, (suffixMarks r (s+1) A (Fin.cons false y)).card *
+      (suffixMarks r (s+1) A (Fin.cons true y)).card : ℝ) = v
+    field_simp
+    ; push_cast
+    ; ring
+  simp_rw [he,←gridPairs_eq] at hh
+  rw [hm] at hh
+  exact hh
+
+end Thorp
+
+end ThorpNine.Dense
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Harmonic/AlternatingPerm.lean b/lean/OAI/Probability/ThorpRouting/Harmonic/AlternatingPerm.lean
index cd62ec1fe9f9e2de549d51a8fed06e163d16f06f..d1f0fbd30cae82bb5c1c7d73b07cd66771cc281a 100644
--- a/lean/OAI/Probability/ThorpRouting/Harmonic/AlternatingPerm.lean
+++ b/lean/OAI/Probability/ThorpRouting/Harmonic/AlternatingPerm.lean
@@ -1,857 +1,857 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Harmonic.ParallelFollow
-
-namespace OAI
-
-namespace ThorpNine.Harmonic
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-namespace CycleColoring
-variable {α : Type*} [Fintype α] [DecidableEq α]
-omit [Fintype α] in
-lemma bipartitionFlip_injective (p : Equiv.Perm α) (S : Finset α)
-    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x)) :
-    Function.Injective (bipartitionFlip p S χ hχ) := by
-  classical
-  intro f g he
-  apply Subtype.ext
-  funext x
-  by_cases hx : x ∈ S
-  · have hh := congrFun (congrArg Subtype.val he) x
-    change (if x ∈ S then Bool.xor (f.val x) (χ x) else false) =
-      (if x ∈ S then Bool.xor (g.val x) (χ x) else false) at hh
-    simp only [ite_eq_left hx] at hh
-    cases hfx : f.val x <;> cases hgx : g.val x <;> cases hcx : χ x <;> simp_all
-  · exact (f.property.1 x hx).trans (g.property.1 x hx).symm
-
-lemma card_opposite_le (p : Equiv.Perm α) (S : Finset α)
-    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x)) :
-    Fintype.card (OppositeColoring p S) ≤
-      2 ^ ((boundary p S).card + Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
-  exact (Fintype.card_le_of_injective _ (bipartitionFlip_injective p S χ hχ)).trans
-    (card_coloring_le p S)
-
-def alternatingPerm (p : Equiv.Perm α) : Equiv.Perm (Bool × α) where
-  toFun x := if x.1 then (false,p x.2) else (true,x.2)
-  invFun x := if x.1 then (false,x.2) else (true,p.symm x.2)
-  left_inv x := by rcases x with ⟨b,x⟩; cases b <;> simp
-  right_inv x := by rcases x with ⟨b,x⟩; cases b <;> simp
-
-omit [Fintype α] [DecidableEq α] in
-lemma alternatingPerm_flip (p : Equiv.Perm α) (x : Bool × α) :
-    (alternatingPerm p x).1 = !x.1 := by
-  rcases x with ⟨b,x⟩
-  cases b <;> rfl
-
-omit [Fintype α] [DecidableEq α] in
-lemma alternating_step_sameCycle (p : Equiv.Perm α) (x : Bool × α) :
-    p.SameCycle x.2 (alternatingPerm p x).2 := by
-  rcases x with ⟨b,x⟩
-  cases b
-  · exact Equiv.Perm.SameCycle.refl p x
-  · exact (Equiv.Perm.SameCycle.refl p x).apply_right
-
-omit [DecidableEq α] in
-lemma alternating_sameCycle_snd (p : Equiv.Perm α) {x y : Bool × α}
-    (h : (alternatingPerm p).SameCycle x y) : p.SameCycle x.2 y.2 := by
-  obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
-  have hp (k : ℕ) : p.SameCycle x.2 (((alternatingPerm p)^k) x).2 := by
-    induction k with
-    | zero => exact Equiv.Perm.SameCycle.refl p x.2
-    | succ k ih =>
-      rw [pow_succ',Equiv.Perm.mul_apply]
-      exact ih.trans (alternating_step_sameCycle p _)
-  exact hn ▸ hp n
-
-omit [DecidableEq α] in
-lemma alternating_zero_sameCycle (p : Equiv.Perm α) {x y : α}
-    (h : p.SameCycle x y) : (alternatingPerm p).SameCycle (false,x) (false,y) := by
-  obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
-  have hp (k : ℕ) : (alternatingPerm p).SameCycle (false,x) (false,(p^k) x) := by
-    induction k with
-    | zero => exact Equiv.Perm.SameCycle.refl _ _
-    | succ k ih =>
-      have hs := ih.apply_right.apply_right
-      simpa only [pow_succ',Equiv.Perm.mul_apply,alternatingPerm,Equiv.coe_fn_mk,
-        Bool.false_eq_true,↓reduceIte] using hs
-  simpa only [hn] using hp n
-
-omit [Fintype α] [DecidableEq α] in
-lemma alternating_sameCycle_zero (p : Equiv.Perm α) (x : Bool × α) :
-    (alternatingPerm p).SameCycle x (false,x.2) := by
-  rcases x with ⟨b,x⟩
-  cases b
-  · exact Equiv.Perm.SameCycle.refl _ _
-  · exact ((Equiv.Perm.SameCycle.refl (alternatingPerm p) (false,x)).apply_right).symm
-
-omit [DecidableEq α] in
-lemma alternating_sameCycle_iff (p : Equiv.Perm α) (x y : Bool × α) :
-    (alternatingPerm p).SameCycle x y ↔ p.SameCycle x.2 y.2 := by
-  constructor
-  · exact alternating_sameCycle_snd p
-  · intro h
-    exact (alternating_sameCycle_zero p x).trans
-      ((alternating_zero_sameCycle p h).trans (alternating_sameCycle_zero p y).symm)
-
-lemma orbit_alternating (p : Equiv.Perm α) (x : Bool × α) :
-    orbitSet (alternatingPerm p) x = childJoin (orbitSet p x.2) (orbitSet p x.2) := by
-  ext ⟨b,y⟩
-  simp only [mem_orbitSet,alternating_sameCycle_iff,mem_childJoin,ite_self]
-
-omit [Fintype α] in
-lemma boolFiber_childJoin (A B : Finset α) (b : Bool) :
-    boolFiber (childJoin A B) b = if b then B else A := by
-  ext x
-  cases b <;> simp only [mem_boolFiber,mem_childJoin,Bool.false_eq_true,↓reduceIte]
-
-noncomputable def projectCycle (p : Equiv.Perm α) (A B : Finset α)
-    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) :
-    RoutingNetwork.SelectedCycle p A := by
-  refine ⟨boolFiber C.val false,?_,?_⟩
-  · obtain ⟨x,hx⟩ := C.property.1
-    refine ⟨x.2,?_⟩
-    rw [hx,orbit_alternating,boolFiber_childJoin]
-    rfl
-  · intro x hx
-    exact (mem_childJoin A B false x).mp (C.property.2 ((mem_boolFiber _ _ _).mp hx))
-
-lemma selected_alternating_eq (p : Equiv.Perm α) (A B : Finset α)
-    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) :
-    C.val = childJoin (projectCycle p A B C).val (projectCycle p A B C).val := by
-  change C.val = childJoin (boolFiber C.val false) (boolFiber C.val false)
-  obtain ⟨x,hx⟩ := C.property.1
-  rw [hx,orbit_alternating,boolFiber_childJoin]
-  rfl
-
-lemma projectCycle_injective (p : Equiv.Perm α) (A B : Finset α) :
-    Function.Injective (projectCycle p A B) := by
-  intro C D he
-  apply Subtype.ext
-  rw [selected_alternating_eq p A B C,selected_alternating_eq p A B D,he]
-
-lemma card_alternating_cycles_le (p : Equiv.Perm α) (A B : Finset α) :
-    Fintype.card (RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) ≤
-      Fintype.card (RoutingNetwork.SelectedCycle p A) :=
-  Fintype.card_le_of_injective _ (projectCycle_injective p A B)
-
-end CycleColoring
-
-section PairAssignmentCount
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-omit [Fintype α] in
-lemma mem_image_perm_iff (p : Equiv.Perm α) (A : Finset α) (y : α) :
-    y ∈ A.image p ↔ p.symm y ∈ A := by
-  constructor
-  · intro h
-    obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp h
-    simpa only [←hxy,Equiv.symm_apply_apply] using hx
-  · intro h
-    exact Finset.mem_image.mpr ⟨p.symm y,h,p.apply_symm_apply y⟩
-
-omit [Fintype α] in
-lemma card_childJoin (A B : Finset α) : (childJoin A B).card = A.card+B.card := by
-  have hd : Disjoint (A.image (fun x => (false,x))) (B.image (fun x => (true,x))) := by
-    apply Finset.disjoint_left.mpr
-    rintro x hx hy
-    obtain ⟨a,ha,rfl⟩ := Finset.mem_image.mp hx
-    obtain ⟨b,hb,hba⟩ := Finset.mem_image.mp hy
-    exact Bool.false_ne_true (congrArg Prod.fst hba).symm
-  have hi (b : Bool) : Function.Injective (fun x : α => (b,x)) :=
-    fun _ _ h => congrArg Prod.snd h
-  rw [childJoin,Finset.card_union_of_disjoint hd,
-    Finset.card_image_of_injective A (hi false),Finset.card_image_of_injective B (hi true)]
-
-omit [Fintype α] in
-lemma alternating_internal (p : Equiv.Perm α) (A B : Finset α) :
-    ((childJoin A B).filter (fun x => (CycleColoring.alternatingPerm p).symm x ∈ childJoin A B)) =
-      childJoin (A ∩ B.image p) (B ∩ A) := by
-  ext ⟨b,x⟩
-  rw [Finset.mem_filter]
-  cases b <;> simp only [mem_childJoin,CycleColoring.alternatingPerm,
-    Equiv.symm_mk,Equiv.coe_fn_mk,Bool.false_eq_true,↓reduceIte,Finset.mem_inter,mem_image_perm_iff]
-
-omit [Fintype α] in
-lemma alternating_boundary_card (p : Equiv.Perm α) (A B : Finset α) :
-    (CycleColoring.boundary (CycleColoring.alternatingPerm p) (childJoin A B)).card =
-      (A ∪ B).card + (A ∪ B.image p).card - (A.card+B.card) := by
-  rw [CycleColoring.boundary_card,alternating_internal,card_childJoin,card_childJoin]
-  have h₀ := Finset.card_union_add_card_inter A B
-  have h₁ := Finset.card_union_add_card_inter A (B.image p)
-  rw [Finset.card_image_of_injective _ p.injective] at h₁
-  have h₂ := Finset.inter_comm B A
-  rw [h₂]
-  omega
-
-lemma compatible_assignment_bound (p : Equiv.Perm α) (A B : Finset α) :
-    Fintype.card (CycleColoring.OppositeColoring (CycleColoring.alternatingPerm p) (childJoin A B)) ≤
-      2 ^ ((A ∪ B).card + (A ∪ B.image p).card - (A.card+B.card) +
-        Fintype.card (RoutingNetwork.SelectedCycle p A)) := by
-  have h := CycleColoring.card_opposite_le (CycleColoring.alternatingPerm p) (childJoin A B)
-    Prod.fst (CycleColoring.alternatingPerm_flip p)
-  rw [alternating_boundary_card] at h
-  exact h.trans (Nat.pow_le_pow_right (by norm_num)
-    (Nat.add_le_add_left (CycleColoring.card_alternating_cycles_le p A B) _))
-
-end PairAssignmentCount
-
-lemma finiteMean_nonneg {Ω : Type*} [Fintype Ω] {f : Ω → ℝ}
-    (h : ∀ ω, 0 ≤ f ω) : 0 ≤ finiteMean f := by
-  exact div_nonneg (Finset.sum_nonneg (fun ω _ => h ω)) (Nat.cast_nonneg _)
-
-lemma boolFiber_headTail (d : ℕ) (S : Finset (Card (d+1))) (b : Bool) :
-    boolFiber (S.image (headTailEquiv d)) b = childMarks S b := by
-  classical
-  ext x
-  rw [mem_boolFiber,mem_childMarks]
-  constructor
-  · intro h
-    obtain ⟨y,hy,he⟩ := Finset.mem_image.mp h
-    have hh := congrArg (headTailEquiv d).symm he
-    rw [Equiv.symm_apply_apply] at hh
-    change y = Fin.cons b x at hh
-    exact hh ▸ hy
-  · intro h
-    exact Finset.mem_image.mpr ⟨Fin.cons b x,h,by rfl⟩
-
-lemma prod_childMarks (d : ℕ) (S : Finset (Card (d+1))) (f : Card (d+1) → ℝ) :
-    (∏ x ∈ S, f x) =
-      (∏ x ∈ childMarks S false, f (Fin.cons false x)) *
-      ∏ x ∈ childMarks S true, f (Fin.cons true x) := by
-  classical
-  have h := prod_boolFibers (S.image (headTailEquiv d)) (fun x => f ((headTailEquiv d).symm x))
-  rw [Finset.prod_image (headTailEquiv d).injective.injOn,
-    boolFiber_headTail,boolFiber_headTail] at h
-  simp only [Equiv.symm_apply_apply] at h
-  exact h
-
-noncomputable def headAverage {d : ℕ} (p : Card (d+1) → ℝ) (y : Card d) : ℝ :=
-  (p (Fin.cons false y)+p (Fin.cons true y))/2
-
-lemma finiteMean_headAverage (d : ℕ) (p : Card (d+1) → ℝ) :
-    finiteMean (headAverage p) = finiteMean p := by
-  rw [finiteMean_equiv (headTailEquiv d),finiteMean_prod,finiteMean_comm]
-  apply finiteMean_congr
-  intro y
-  rw [finiteMean_bool]
-  rfl
-
-lemma pairSwitch_product_bound (d : ℕ) (S : Finset (Card (d+1)))
-    (p : Card (d+1) → ℝ) (hp : ∀ x, 0 ≤ p x) :
-    finiteMean (fun ξ : Card d → Bool => ∏ x ∈ S, p (pairSwitch ξ x)) ≤
-      ∏ x ∈ S, headAverage p (Fin.tail x) := by
-  classical
-  have h : CylinderBound (fun T => ∏ x ∈ T, p ((headTailEquiv d).symm x))
-      (fun x => p ((headTailEquiv d).symm x)) := fun _ => le_rfl
-  have hh := h.fair_parallel (fun x => hp _) (S.image (headTailEquiv d))
-  simp only [Finset.prod_image (pairLayer _).injective.injOn,
-    Finset.prod_image (headTailEquiv d).injective.injOn] at hh
-  convert hh using 1
-  · apply finiteMean_congr
-    intro ξ
-    apply Finset.prod_congr rfl
-    intro x _
-    rfl
-  · apply Finset.prod_congr rfl
-    intro x _
-    change (p (Fin.cons false (Fin.tail x))+p (Fin.cons true (Fin.tail x)))/2 =
-      (p (Fin.cons (x 0) (Fin.tail x))+p (Fin.cons (!(x 0)) (Fin.tail x)))/2
-    cases x 0
-    · rfl
-    · simp only [Bool.not_true]
-      ring
-
-lemma butterflyPerm_step (d : ℕ)
-    (c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (ξ : Card d → Bool) :
-    butterflyPerm (d+1) (decodeButterfly (d+1) ((coinStepEquiv d).symm (c,ξ))) =
-    pairSwitch ξ * childLift (fun b => butterflyPerm d (decodeButterfly d (if b then c.2 else c.1))) := by
-  change pairSwitch (fun y => (coinStepEquiv d).symm (c,ξ) (Sum.inl y)) *
-    childLift (fun b => butterflyPerm d (decodeButterfly d
-      (fun i => (coinStepEquiv d).symm (c,ξ) (Sum.inr (b,i))))) = _
-  apply congrArg₂ (· * ·)
-  · rfl
-  · apply congrArg childLift
-    funext b
-    apply congrArg (fun ω => butterflyPerm d (decodeButterfly d ω))
-    funext i
-    cases b <;> rfl
-
-lemma butterfly_product_bound (d : ℕ) (S : Finset (Card d))
-    (p : Card d → ℝ) (hp : ∀ x, 0 ≤ p x) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      ∏ x ∈ S, p (butterflyPerm d (decodeButterfly d ω) x)) ≤ (finiteMean p)^S.card := by
-  classical
-  induction d with
-  | zero =>
-    have he : p = fun _ => p (fun i => Fin.elim0 i) := by
-      funext x
-      exact congrArg p (Subsingleton.elim _ _)
-    rw [he]
-    simp only [Finset.prod_const,finiteMean_const,le_refl]
-  | succ d ih =>
-    let g : Card d → ℝ := headAverage p
-    have hg : ∀ x, 0 ≤ g x := fun x => div_nonneg (add_nonneg (hp _) (hp _)) (by norm_num)
-    rw [finiteMean_equiv (coinStepEquiv d),finiteMean_prod]
-    calc
-      _ ≤ finiteMean (fun c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool) =>
-        (∏ x ∈ childMarks S false, g (butterflyPerm d (decodeButterfly d c.1) x)) *
-        ∏ x ∈ childMarks S true, g (butterflyPerm d (decodeButterfly d c.2) x)) := by
-        apply finiteMean_mono
-        intro c
-        let f : Equiv.Perm (Card (d+1)) := childLift (fun b =>
-          butterflyPerm d (decodeButterfly d (if b then c.2 else c.1)))
-        have hh := pairSwitch_product_bound d (S.image f) p hp
-        simp only [Finset.prod_image f.injective.injOn] at hh
-        convert hh using 1
-        · apply finiteMean_congr
-          intro ξ
-          apply Finset.prod_congr rfl
-          intro x _
-          rw [butterflyPerm_step]
-          rfl
-        · rw [prod_childMarks]
-          simp only [f,childLift,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
-            Bool.false_eq_true,↓reduceIte]
-          rfl
-      _ = (finiteMean (fun c : SwitchIndex d → Bool =>
-        ∏ x ∈ childMarks S false, g (butterflyPerm d (decodeButterfly d c) x))) *
-        finiteMean (fun c : SwitchIndex d → Bool =>
-        ∏ x ∈ childMarks S true, g (butterflyPerm d (decodeButterfly d c) x)) :=
-          finiteMean_prod_mul
-            (fun c : SwitchIndex d → Bool => ∏ x ∈ childMarks S false,
-              g (butterflyPerm d (decodeButterfly d c) x))
-            (fun c : SwitchIndex d → Bool => ∏ x ∈ childMarks S true,
-              g (butterflyPerm d (decodeButterfly d c) x))
-      _ ≤ (finiteMean g)^(childMarks S false).card * (finiteMean g)^(childMarks S true).card :=
-        mul_le_mul (ih (childMarks S false) g hg) (ih (childMarks S true) g hg)
-          (finiteMean_nonneg fun c => Finset.prod_nonneg (fun x _ => hg _))
-          (pow_nonneg (finiteMean_nonneg hg) _)
-      _ = (finiteMean p)^S.card := by
-        rw [←pow_add,card_childMarks_add,finiteMean_headAverage]
-
-noncomputable def inverseButterflyMarks (d : ℕ) (A : Finset (Card d))
-    (ω : SwitchIndex d → Bool) : Finset (Card d) :=
-  A.image (butterflyPerm d (decodeButterfly d ω)).symm
-
-lemma inverseButterflyMarks_inclusion (d : ℕ) (A S : Finset (Card d)) :
-    inclusionMoment (inverseButterflyMarks d A) S ≤
-      ((A.card : ℝ)/(2:ℝ)^d)^S.card := by
-  classical
-  let p : Card d → ℝ := fun x => if x ∈ A then 1 else 0
-  have hm : finiteMean p = (A.card : ℝ)/(2:ℝ)^d := by
-    simp only [finiteMean,p,card_positions,Nat.cast_pow,Nat.cast_ofNat]
-    congr 1
-    simp
-  have he : inclusionMoment (inverseButterflyMarks d A) S =
-      finiteMean (fun ω : SwitchIndex d → Bool =>
-        ∏ x ∈ S, p (butterflyPerm d (decodeButterfly d ω) x)) := by
-    rw [inclusionMoment_eq]
-    apply finiteMean_congr
-    intro ω
-    simp only [inverseButterflyMarks,subset_image_equiv_iff,Equiv.symm_symm,indicator_subset_prod,
-      Finset.prod_image (butterflyPerm d (decodeButterfly d ω)).injective.injOn,p]
-  rw [he]
-  simpa only [hm] using butterfly_product_bound d S p (fun x => by dsimp [p]; split_ifs <;> norm_num)
-
-namespace PairRouting
-
-section
-variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
-
-noncomputable def occupied (x : ι → Bool × α) : Finset α :=
-  Finset.univ.image (fun i => (x i).2)
-
-omit [Fintype α] in
-lemma mem_occupied (x : ι → Bool × α) (y : α) : y ∈ occupied x ↔ ∃ i, (x i).2 = y := by
-  simp [occupied]
-
-def required (x : ι → Bool × α) (c : ι → Bool) (i : ι) : Bool := Bool.xor (x i).1 (c i)
-
-def Compatible (x : ι → Bool × α) (c : ι → Bool) : Prop :=
-  ∀ i j, (x i).2 = (x j).2 → required x c i = required x c j
-
-def colors (x : ι → Bool × α) (ξ : α → Bool) (i : ι) : Bool :=
-  Bool.xor (x i).1 (ξ (x i).2)
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma required_colors (x : ι → Bool × α) (ξ : α → Bool) (i : ι) :
-    required x (colors x ξ) i = ξ (x i).2 := by
-  simp only [required,colors]
-  cases (x i).1 <;> cases ξ (x i).2 <;> rfl
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma colors_compatible (x : ι → Bool × α) (ξ : α → Bool) : Compatible x (colors x ξ) := by
-  intro i j he
-  simp only [required_colors,he]
-
-noncomputable def chosenCoin (x : ι → Bool × α) (c : ι → Bool) (y : α) : Bool := by
-  classical
-  exact if h : ∃ i, (x i).2 = y then required x c (Classical.choose h) else false
-
-omit [Fintype α] in
-lemma chosenCoin_apply (x : ι → Bool × α) (c : ι → Bool) (hc : Compatible x c) (i : ι) :
-    chosenCoin x c (x i).2 = required x c i := by
-  classical
-  simp only [chosenCoin,dite_eq_left (show ∃ j, (x j).2 = (x i).2 from ⟨i,rfl⟩)]
-  exact hc _ i (Classical.choose_spec (show ∃ j, (x j).2 = (x i).2 from ⟨i,rfl⟩))
-
-omit [Fintype α] in
-lemma colors_eq_iff (x : ι → Bool × α) (c : ι → Bool) (hc : Compatible x c) (ξ : α → Bool) :
-    colors x ξ = c ↔ ∀ y ∈ occupied x, ξ y = chosenCoin x c y := by
-  constructor
-  · intro he y hy
-    obtain ⟨i,rfl⟩ := (mem_occupied x y).mp hy
-    rw [chosenCoin_apply x c hc i]
-    rw [←he,required_colors]
-  · intro he
-    funext i
-    apply (xor_eq_iff _ _ _).mpr
-    exact (he _ ((mem_occupied x _).mpr ⟨i,rfl⟩)).trans (chosenCoin_apply x c hc i)
-
-noncomputable instance (x : ι → Bool × α) (c : ι → Bool) : Decidable (Compatible x c) :=
-  Classical.propDecidable _
-
-lemma colors_probability (x : ι → Bool × α) (c : ι → Bool) :
-    finiteMean (fun ξ : α → Bool => if colors x ξ = c then (1 : ℝ) else 0) =
-      if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0 := by
-  classical
-  by_cases hc : Compatible x c
-  · rw [ite_eq_left hc]
-    have he := fun ξ => colors_eq_iff x c hc ξ
-    simp only [he]
-    exact mean_coinCylinder (occupied x) (chosenCoin x c)
-  · rw [ite_eq_right hc]
-    have he : ∀ ξ, colors x ξ ≠ c := fun ξ hh => hc (hh ▸ colors_compatible x ξ)
-    simp only [he,ite_false,finiteMean_const]
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma Compatible.tail_injective {x : ι ↪ Bool × α} {c : ι → Bool} (hc : Compatible x c) (b : Bool) :
-    Function.Injective (fun i : {i : ι // c i = b} => (x i.val).2) := by
-  intro i j he
-  apply Subtype.ext
-  apply x.injective
-  apply Prod.ext
-  · have hh := hc i j he
-    simp only [required,i.property,j.property] at hh
-    cases h₀ : (x i.val).1 <;> cases h₁ : (x j.val).1 <;> cases b <;> simp_all
-  · exact he
-
-def childEmbedding (x : ι ↪ Bool × α) (c : ι → Bool) (hc : Compatible x c) (b : Bool) :
-    {i : ι // c i = b} ↪ α := ⟨fun i => (x i.val).2,hc.tail_injective b⟩
-
-
-def lift (p : Bool → Equiv.Perm α) : Equiv.Perm (Bool × α) where
-  toFun x := (x.1,p x.1 x.2)
-  invFun x := (x.1,(p x.1).symm x.2)
-  left_inv x := by simp
-  right_inv x := by simp
-
-def sandwich (ξ η : α → Bool) (p : Bool → Equiv.Perm α) : Equiv.Perm (Bool × α) :=
-  pairLayer η * lift p * pairLayer ξ
-
-def ChildMatches (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) : Prop :=
-  ∀ i, p (c i) (x i).2 = (y i).2
-
-noncomputable instance (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
-    Decidable (ChildMatches p x y c) := Classical.propDecidable _
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma sandwich_matches_iff (ξ η : α → Bool) (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) :
-    (∀ i, sandwich ξ η p (x i) = y i) ↔
-      colors x ξ = colors y η ∧ ChildMatches p x y (colors x ξ) := by
-  have he (i : ι) : sandwich ξ η p (x i) = y i ↔
-      colors x ξ i = colors y η i ∧ p (colors x ξ i) (x i).2 = (y i).2 := by
-    change pairLayer η (lift p (pairLayer ξ (x i))) = y i ↔ _
-    rw [←Equiv.eq_symm_apply]
-    change (colors x ξ i,p (colors x ξ i) (x i).2) = (colors y η i,(y i).2) ↔ _
-    exact ⟨Prod.mk.inj,fun h => Prod.ext h.1 h.2⟩
-  simp only [he,forall_and,←funext_iff,ChildMatches]
-
-lemma sandwich_probability [DecidableEq ι] (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) :
-    finiteMean (fun coins : (α → Bool) × (α → Bool) =>
-      if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then (1:ℝ) else 0) =
-    ∑ c : ι → Bool,
-      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
-      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) *
-      (if ChildMatches p x y c then 1 else 0) := by
-  classical
-  have he (coins : (α → Bool) × (α → Bool)) :
-      (if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then (1:ℝ) else 0) =
-      ∑ c : ι → Bool, (if colors x coins.1 = c then (1:ℝ) else 0) *
-        (if colors y coins.2 = c then 1 else 0) * (if ChildMatches p x y c then 1 else 0) := by
-    rw [Finset.sum_eq_single (colors x coins.1)]
-    · by_cases hc : colors x coins.1 = colors y coins.2
-      · simp only [sandwich_matches_iff,hc,true_and,↓reduceIte,one_mul]
-      · simp only [sandwich_matches_iff,hc,false_and,↓reduceIte,one_mul,Ne.symm hc,zero_mul]
-    · intro c _ hc
-      simp only [Ne.symm hc,ite_false,zero_mul]
-    · simp
-  simp_rw [he]
-  rw [finiteMean_sum]
-  apply Finset.sum_congr rfl
-  intro c _
-  rw [finiteMean_mul_const]
-  rw [finiteMean_prod_mul
-    (fun ξ : α → Bool => if colors x ξ = c then (1:ℝ) else 0)
-    (fun η : α → Bool => if colors y η = c then (1:ℝ) else 0),
-    colors_probability,colors_probability]
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma childMatches_split (p₀ p₁ : Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
-    ChildMatches (fun b => if b then p₁ else p₀) x y c ↔
-      (∀ i : {i // c i = false}, p₀ (x i.val).2 = (y i.val).2) ∧
-      (∀ i : {i // c i = true}, p₁ (x i.val).2 = (y i.val).2) := by
-  constructor
-  · intro h
-    constructor
-    · intro i
-      simpa only [i.property,Bool.false_eq_true,↓reduceIte] using h i.val
-    · intro i
-      simpa only [i.property,↓reduceIte] using h i.val
-  · rintro ⟨h₀,h₁⟩ i
-    cases hi : c i
-    · simpa only [hi,Bool.false_eq_true,↓reduceIte] using h₀ ⟨i,hi⟩
-    · simpa only [hi,↓reduceIte] using h₁ ⟨i,hi⟩
-
-omit [Fintype α] in
-lemma childMatches_mean {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
-    (P : Ω → Equiv.Perm α) (Q : Ω' → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
-    finiteMean (fun ω : Ω × Ω' =>
-      if ChildMatches (fun b => if b then Q ω.2 else P ω.1) x y c then (1:ℝ) else 0) =
-      tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
-        (fun i : {i // c i = false} => (y i.val).2) *
-      tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
-        (fun i : {i // c i = true} => (y i.val).2) := by
-  classical
-  let f : Ω → ℝ := fun ω => if ∀ i : {i // c i = false}, P ω (x i.val).2 = (y i.val).2 then 1 else 0
-  let g : Ω' → ℝ := fun ω => if ∀ i : {i // c i = true}, Q ω (x i.val).2 = (y i.val).2 then 1 else 0
-  have he (ω : Ω × Ω') :
-      (if ChildMatches (fun b => if b then Q ω.2 else P ω.1) x y c then (1:ℝ) else 0) = f ω.1*g ω.2 := by
-    simp only [childMatches_split,f,g]
-    split_ifs <;> simp_all
-  simp_rw [he]
-  exact finiteMean_prod_mul f g
-
-lemma sandwich_mean [DecidableEq ι] {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
-    (P : Ω → Equiv.Perm α) (Q : Ω' → Equiv.Perm α) (x y : ι → Bool × α) :
-    finiteMean (fun ω : Ω × Ω' => finiteMean (fun coins : (α → Bool) × (α → Bool) =>
-      if ∀ i, sandwich coins.1 coins.2 (fun b => if b then Q ω.2 else P ω.1) (x i) = y i
-      then (1:ℝ) else 0)) =
-    ∑ c : ι → Bool,
-      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
-      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) *
-      (tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
-        (fun i : {i // c i = false} => (y i.val).2) *
-      tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
-        (fun i : {i // c i = true} => (y i.val).2)) := by
-  classical
-  simp_rw [sandwich_probability]
-  rw [finiteMean_sum]
-  apply Finset.sum_congr rfl
-  intro c _
-  simp only [mul_comm ((if Compatible x c then _ else _) * _),finiteMean_mul_const,
-    childMatches_mean]
-
-
-abbrev BothAssignments (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :=
-  {c : ι → Bool // Compatible e c ∧ Compatible (fun i => lift p (e i)) c}
-
-noncomputable instance (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    Fintype (BothAssignments e p) := Fintype.ofFinite _
-
-noncomputable def labelSet (e : ι ↪ Bool × α) : Finset (Bool × α) := Finset.univ.image e
-
-def alternatingProjection (p : Bool → Equiv.Perm α) : Equiv.Perm α := (p false).symm * p true
-
-noncomputable def assignmentColoring (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α)
-    (c : BothAssignments e p) : CycleColoring.OppositeColoring
-      (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e) := by
-  classical
-  refine ⟨Function.extend e c.val (fun _ => false),?_,?_⟩
-  · intro z hz
-    apply Function.extend_apply'
-    rintro ⟨i,rfl⟩
-    exact hz (Finset.mem_image.mpr ⟨i,Finset.mem_univ _,rfl⟩)
-  · intro z hz hnext
-    obtain ⟨i,_,hi⟩ := Finset.mem_image.mp hz
-    obtain ⟨j,_,hj⟩ := Finset.mem_image.mp hnext
-    have hfi := e.injective.extend_apply c.val (fun _ => false) i
-    have hfj := e.injective.extend_apply c.val (fun _ => false) j
-    rw [hi] at hfi
-    rw [hj] at hfj
-    rw [hfi,hfj]
-    rcases z with ⟨b,a⟩
-    cases b
-    · have hj' : e j = (true,a) := hj
-      have ht : (e i).2 = (e j).2 := by rw [hi,hj']
-      have hh := c.property.1 i j ht
-      simp only [required,hi,hj'] at hh
-      revert hh
-      generalize c.val i = bi
-      generalize c.val j = bj
-      cases bi <;> cases bj <;> decide
-    · have hj' : e j = (false,alternatingProjection p a) := hj
-      have ht : (lift p (e i)).2 = (lift p (e j)).2 := by
-        simp only [hi,hj',lift,Equiv.coe_fn_mk,alternatingProjection,Equiv.Perm.coe_mul,
-          Function.comp_apply,Equiv.apply_symm_apply]
-      have hh := c.property.2 i j ht
-      simp only [required,hi,hj',lift,Equiv.coe_fn_mk] at hh
-      revert hh
-      generalize c.val i = bi
-      generalize c.val j = bj
-      cases bi <;> cases bj <;> decide
-
-omit [Fintype α] in
-lemma assignmentColoring_injective (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    Function.Injective (assignmentColoring e p) := by
-  intro c c' he
-  apply Subtype.ext
-  funext i
-  have hh := congrFun (congrArg Subtype.val he) (e i)
-  change Function.extend e c.val (fun _ => false) (e i) =
-    Function.extend e c'.val (fun _ => false) (e i) at hh
-  simpa only [e.injective.extend_apply] using hh
-
-lemma bothAssignments_card (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    Fintype.card (BothAssignments e p) ≤
-      2 ^ ((boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true).card +
-        (boolFiber (labelSet e) false ∪ (boolFiber (labelSet e) true).image
-          (alternatingProjection p)).card -
-        ((boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card) +
-        Fintype.card (RoutingNetwork.SelectedCycle (alternatingProjection p) (boolFiber (labelSet e) false))) := by
-  have h := compatible_assignment_bound (alternatingProjection p)
-    (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
-  rw [childJoin_boolFiber] at h
-  exact (Fintype.card_le_of_injective _ (assignmentColoring_injective e p)).trans h
-
-omit [Fintype α] in
-lemma occupied_labelSet (e : ι ↪ Bool × α) : occupied e = (labelSet e).image Prod.snd := by
-  classical
-  simp only [occupied,labelSet,Finset.image_image,Function.comp_def]
-
-omit [Fintype ι] [Fintype α] in
-lemma image_snd_childJoin (A B : Finset α) : (childJoin A B).image Prod.snd = A ∪ B := by
-  classical
-  simp only [childJoin,Finset.image_union,Finset.image_image,Function.comp_def,Finset.image_id']
-
-omit [Fintype ι] [Fintype α] in
-lemma image_lift_childJoin (p : Bool → Equiv.Perm α) (A B : Finset α) :
-    (childJoin A B).image (fun x => (lift p x).2) = A.image (p false) ∪ B.image (p true) := by
-  classical
-  simp only [childJoin,Finset.image_union,Finset.image_image,Function.comp_def,lift,Equiv.coe_fn_mk]
-
-omit [Fintype α] in
-lemma occupied_lift_card (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    (occupied (fun i => lift p (e i))).card =
-      (boolFiber (labelSet e) false ∪ (boolFiber (labelSet e) true).image (alternatingProjection p)).card := by
-  classical
-  have he : occupied (fun i => lift p (e i)) = (labelSet e).image (fun x => (lift p x).2) := by
-    simp only [occupied,labelSet,Finset.image_image,Function.comp_def]
-  rw [he,←childJoin_boolFiber (labelSet e),image_lift_childJoin]
-  simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
-  let A := boolFiber (labelSet e) false
-  let B := boolFiber (labelSet e) true
-  change (A.image (p false) ∪ B.image (p true)).card = (A ∪ B.image (alternatingProjection p)).card
-  have hh : A.image (p false) ∪ B.image (p true) =
-      (A ∪ B.image (alternatingProjection p)).image (p false) := by
-    simp only [Finset.image_union,Finset.image_image,Function.comp_def,
-      alternatingProjection,Equiv.Perm.coe_mul,Equiv.apply_symm_apply]
-  rw [hh,Finset.card_image_of_injective _ (p false).injective]
-
-lemma bothAssignments_occupied_bound (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    Fintype.card (BothAssignments e p) ≤
-      2 ^ ((occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι +
-        Fintype.card (RoutingNetwork.SelectedCycle (alternatingProjection p) (boolFiber (labelSet e) false))) := by
-  have he : (occupied e).card =
-      (boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true).card := by
-    rw [occupied_labelSet,←childJoin_boolFiber (labelSet e),image_snd_childJoin]
-    simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
-  have hc : (boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card = Fintype.card ι := by
-    rw [←card_childJoin,childJoin_boolFiber,labelSet,Finset.card_image_of_injective _ e.injective,
-      Finset.card_univ]
-  simpa only [he,occupied_lift_card,←hc] using bothAssignments_card e p
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma compatible_pairLayer_iff (x : ι → Bool × α) (c : ι → Bool) (ξ : α → Bool) :
-    Compatible (fun i => pairLayer ξ (x i)) c ↔ Compatible x c := by
-  have he (i j : ι) (ht : (x i).2 = (x j).2) :
-      required (fun i => pairLayer ξ (x i)) c i = required (fun i => pairLayer ξ (x i)) c j ↔
-      required x c i = required x c j := by
-    simp only [required,pairLayer,Equiv.coe_fn_mk,ht]
-    cases (x i).1 <;> cases (x j).1 <;> cases c i <;> cases c j <;> cases ξ (x j).2 <;> decide
-  constructor
-  · intro h i j ht
-    exact (he i j ht).mp (h i j ht)
-  · intro h i j ht
-    exact (he i j ht).mpr (h i j ht)
-
-
-noncomputable def alternatingCycles (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) : ℕ :=
-  Fintype.card (RoutingNetwork.SelectedCycle
-    (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e))
-
-omit [Fintype α] in
-lemma occupied_eq_fibers (e : ι ↪ Bool × α) :
-    occupied e = boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true := by
-  rw [occupied_labelSet,←childJoin_boolFiber (labelSet e),image_snd_childJoin]
-  simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
-
-omit [Fintype α] in
-lemma card_labelSet (e : ι ↪ Bool × α) : (labelSet e).card = Fintype.card ι := by
-  classical
-  exact (Finset.card_image_of_injective _ e.injective).trans (Finset.card_univ)
-
-omit [Fintype α] in
-lemma occupied_card_double (e : ι ↪ Bool × α) : Fintype.card ι ≤ 2*(occupied e).card := by
-  have h₀ : (boolFiber (labelSet e) false).card ≤ (occupied e).card := by
-    rw [occupied_eq_fibers]
-    exact Finset.card_le_card Finset.subset_union_left
-  have h₁ : (boolFiber (labelSet e) true).card ≤ (occupied e).card := by
-    rw [occupied_eq_fibers]
-    exact Finset.card_le_card Finset.subset_union_right
-  have he : (boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card = Fintype.card ι := by
-    rw [←card_childJoin,childJoin_boolFiber,card_labelSet]
-  omega
-
-lemma bothAssignments_exact_bound (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    Fintype.card (BothAssignments e p) ≤
-      2 ^ ((occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι +
-        alternatingCycles e p) := by
-  have h := CycleColoring.card_opposite_le
-    (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e)
-    Prod.fst (CycleColoring.alternatingPerm_flip (alternatingProjection p))
-  have hb : (CycleColoring.boundary (CycleColoring.alternatingPerm (alternatingProjection p))
-      (labelSet e)).card =
-      (occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι := by
-    have hh := alternating_boundary_card (alternatingProjection p)
-      (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
-    rw [childJoin_boolFiber] at hh
-    rw [occupied_eq_fibers,occupied_lift_card]
-    have he := card_childJoin (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
-    rw [childJoin_boolFiber,card_labelSet] at he
-    simpa only [he] using hh
-  rw [hb] at h
-  exact (Fintype.card_le_of_injective _ (assignmentColoring_injective e p)).trans h
-
-abbrev Assignments (x y : ι → Bool × α) :=
-  {c : ι → Bool // Compatible x c ∧ Compatible y c}
-
-noncomputable instance (x y : ι → Bool × α) : Fintype (Assignments x y) := Fintype.ofFinite _
-
-def switchedEmbedding (e : ι ↪ Bool × α) (ξ : α → Bool) : ι ↪ Bool × α :=
-  e.trans (pairLayer ξ).toEmbedding
-
-def routedEmbedding (e : ι ↪ Bool × α) (ξ η : α → Bool) (p : Bool → Equiv.Perm α) :
-    ι ↪ Bool × α := e.trans (sandwich ξ η p).toEmbedding
-
-omit [Fintype α] in
-lemma occupied_pairLayer (x : ι → Bool × α) (ξ : α → Bool) :
-    occupied (fun i => pairLayer ξ (x i)) = occupied x := rfl
-
-noncomputable def assignmentsEquiv (e : ι ↪ Bool × α) (ξ η : α → Bool)
-    (p : Bool → Equiv.Perm α) :
-    Assignments e (routedEmbedding e ξ η p) ≃ BothAssignments (switchedEmbedding e ξ) p := by
-  apply Equiv.subtypeEquivRight
-  intro c
-  change Compatible e c ∧ Compatible (fun i => pairLayer η (lift p (pairLayer ξ (e i)))) c ↔ _
-  rw [compatible_pairLayer_iff]
-  exact and_congr (compatible_pairLayer_iff e c ξ).symm Iff.rfl
-
-lemma assignments_actual_bound (e : ι ↪ Bool × α) (ξ η : α → Bool) (p : Bool → Equiv.Perm α) :
-    Fintype.card (Assignments e (routedEmbedding e ξ η p)) ≤
-      2 ^ ((occupied e).card + (occupied (routedEmbedding e ξ η p)).card - Fintype.card ι +
-        alternatingCycles (switchedEmbedding e ξ) p) := by
-  rw [Fintype.card_congr (assignmentsEquiv e ξ η p)]
-  exact bothAssignments_exact_bound (switchedEmbedding e ξ) p
-
-omit [Fintype α] in
-lemma assignments_actual_exponent_nonneg (e : ι ↪ Bool × α) (ξ η : α → Bool)
-    (p : Bool → Equiv.Perm α) :
-    Fintype.card ι ≤ (occupied e).card + (occupied (routedEmbedding e ξ η p)).card := by
-  have h₀ := occupied_card_double e
-  have h₁ := occupied_card_double (routedEmbedding e ξ η p)
-  omega
-
-end
-variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α] [DecidableEq ι]
-
-noncomputable def compatibleSet (x y : ι → Bool × α) : Finset (ι → Bool) := by
-  classical
-  exact Finset.univ.filter (fun c => Compatible x c ∧ Compatible y c)
-
-omit [Fintype α] [DecidableEq α] in
-lemma mem_compatibleSet (x y : ι → Bool × α) (c : ι → Bool) :
-    c ∈ compatibleSet x y ↔ Compatible x c ∧ Compatible y c := by
-  classical
-  simp only [compatibleSet,Finset.mem_filter,Finset.mem_univ,true_and]
-
-omit [Fintype α] [DecidableEq α] in
-lemma card_compatibleSet (x y : ι → Bool × α) :
-    (compatibleSet x y).card = Fintype.card (Assignments x y) := by
-  classical
-  exact (Fintype.card_subtype _).symm
-
-noncomputable def outerWeight (x y : ι → Bool × α) : ℝ :=
-  1/(2:ℝ)^((occupied x).card+(occupied y).card)
-
-omit [Fintype α] [DecidableEq ι] in
-lemma outerWeight_pos (x y : ι → Bool × α) : 0 < outerWeight x y := by
-  unfold outerWeight
-  positivity
-
-omit [Fintype α] in
-lemma sum_colors_factor (x y : ι → Bool × α) (f : (ι → Bool) → ℝ) :
-    (∑ c : ι → Bool,
-      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
-      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) * f c) =
-      outerWeight x y * ∑ c ∈ compatibleSet x y, f c := by
-  classical
-  rw [compatibleSet,Finset.sum_filter,Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro c _
-  by_cases hx : Compatible x c <;> by_cases hy : Compatible y c
-  · simp only [hx,hy,true_and,↓reduceIte,outerWeight,pow_add,one_div_mul_one_div]
-  · simp only [hx,hy,and_false,↓reduceIte,mul_zero,zero_mul]
-  · simp only [hx,hy,false_and,↓reduceIte,mul_zero,zero_mul]
-  · simp only [hx,hy,false_and,↓reduceIte,mul_zero,zero_mul]
-
-lemma sandwich_weighted_probability (p : Bool → Equiv.Perm α) (x y : ι → Bool × α)
-    (W : (ι → Bool) → ℝ) :
-    finiteMean (fun coins : (α → Bool) × (α → Bool) =>
-      if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then W (colors x coins.1) else 0) =
-    outerWeight x y * ∑ c ∈ compatibleSet x y, if ChildMatches p x y c then W c else 0 := by
-  classical
-  have he (coins : (α → Bool) × (α → Bool)) :
-      (if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then W (colors x coins.1) else 0) =
-      ∑ c : ι → Bool, (if colors x coins.1 = c then (1:ℝ) else 0) *
-        (if colors y coins.2 = c then 1 else 0) * (if ChildMatches p x y c then W c else 0) := by
-    rw [Finset.sum_eq_single (colors x coins.1)]
-    · by_cases hc : colors x coins.1 = colors y coins.2
-      · simp only [sandwich_matches_iff,hc,true_and,↓reduceIte,one_mul]
-      · simp only [sandwich_matches_iff,hc,false_and,↓reduceIte,one_mul,Ne.symm hc,zero_mul]
-    · intro c _ hc
-      simp only [Ne.symm hc,ite_false,zero_mul]
-    · simp
-  simp_rw [he]
-  rw [finiteMean_sum,←sum_colors_factor]
-  apply Finset.sum_congr rfl
-  intro c _
-  rw [finiteMean_mul_const,finiteMean_prod_mul
-    (fun ξ : α → Bool => if colors x ξ = c then (1:ℝ) else 0)
-    (fun η : α → Bool => if colors y η = c then (1:ℝ) else 0),
-    colors_probability,colors_probability]
-
-omit [Fintype α] [DecidableEq ι] in
-lemma tupleProbability_nonneg {Ω β : Type*} [Fintype Ω] [DecidableEq β]
-    (P : Ω → Equiv.Perm β) (x y : ι → β) : 0 ≤ tupleProbability P x y := by
-  classical
-  exact finiteMean_nonneg (fun ω => by split_ifs <;> norm_num)
-
-end PairRouting
-end Thorp
-
-end ThorpNine.Harmonic
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Harmonic.ParallelFollow
+
+namespace OAI
+
+namespace ThorpNine.Harmonic
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+namespace CycleColoring
+variable {α : Type*} [Fintype α] [DecidableEq α]
+omit [Fintype α] in
+lemma bipartitionFlip_injective (p : Equiv.Perm α) (S : Finset α)
+    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x)) :
+    Function.Injective (bipartitionFlip p S χ hχ) := by
+  classical
+  intro f g he
+  apply Subtype.ext
+  funext x
+  by_cases hx : x ∈ S
+  · have hh := congrFun (congrArg Subtype.val he) x
+    change (if x ∈ S then Bool.xor (f.val x) (χ x) else false) =
+      (if x ∈ S then Bool.xor (g.val x) (χ x) else false) at hh
+    simp only [ite_eq_left hx] at hh
+    cases hfx : f.val x <;> cases hgx : g.val x <;> cases hcx : χ x <;> simp_all
+  · exact (f.property.1 x hx).trans (g.property.1 x hx).symm
+
+lemma card_opposite_le (p : Equiv.Perm α) (S : Finset α)
+    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x)) :
+    Fintype.card (OppositeColoring p S) ≤
+      2 ^ ((boundary p S).card + Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
+  exact (Fintype.card_le_of_injective _ (bipartitionFlip_injective p S χ hχ)).trans
+    (card_coloring_le p S)
+
+def alternatingPerm (p : Equiv.Perm α) : Equiv.Perm (Bool × α) where
+  toFun x := if x.1 then (false,p x.2) else (true,x.2)
+  invFun x := if x.1 then (false,x.2) else (true,p.symm x.2)
+  left_inv x := by rcases x with ⟨b,x⟩; cases b <;> simp
+  right_inv x := by rcases x with ⟨b,x⟩; cases b <;> simp
+
+omit [Fintype α] [DecidableEq α] in
+lemma alternatingPerm_flip (p : Equiv.Perm α) (x : Bool × α) :
+    (alternatingPerm p x).1 = !x.1 := by
+  rcases x with ⟨b,x⟩
+  cases b <;> rfl
+
+omit [Fintype α] [DecidableEq α] in
+lemma alternating_step_sameCycle (p : Equiv.Perm α) (x : Bool × α) :
+    p.SameCycle x.2 (alternatingPerm p x).2 := by
+  rcases x with ⟨b,x⟩
+  cases b
+  · exact Equiv.Perm.SameCycle.refl p x
+  · exact (Equiv.Perm.SameCycle.refl p x).apply_right
+
+omit [DecidableEq α] in
+lemma alternating_sameCycle_snd (p : Equiv.Perm α) {x y : Bool × α}
+    (h : (alternatingPerm p).SameCycle x y) : p.SameCycle x.2 y.2 := by
+  obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
+  have hp (k : ℕ) : p.SameCycle x.2 (((alternatingPerm p)^k) x).2 := by
+    induction k with
+    | zero => exact Equiv.Perm.SameCycle.refl p x.2
+    | succ k ih =>
+      rw [pow_succ',Equiv.Perm.mul_apply]
+      exact ih.trans (alternating_step_sameCycle p _)
+  exact hn ▸ hp n
+
+omit [DecidableEq α] in
+lemma alternating_zero_sameCycle (p : Equiv.Perm α) {x y : α}
+    (h : p.SameCycle x y) : (alternatingPerm p).SameCycle (false,x) (false,y) := by
+  obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
+  have hp (k : ℕ) : (alternatingPerm p).SameCycle (false,x) (false,(p^k) x) := by
+    induction k with
+    | zero => exact Equiv.Perm.SameCycle.refl _ _
+    | succ k ih =>
+      have hs := ih.apply_right.apply_right
+      simpa only [pow_succ',Equiv.Perm.mul_apply,alternatingPerm,Equiv.coe_fn_mk,
+        Bool.false_eq_true,↓reduceIte] using hs
+  simpa only [hn] using hp n
+
+omit [Fintype α] [DecidableEq α] in
+lemma alternating_sameCycle_zero (p : Equiv.Perm α) (x : Bool × α) :
+    (alternatingPerm p).SameCycle x (false,x.2) := by
+  rcases x with ⟨b,x⟩
+  cases b
+  · exact Equiv.Perm.SameCycle.refl _ _
+  · exact ((Equiv.Perm.SameCycle.refl (alternatingPerm p) (false,x)).apply_right).symm
+
+omit [DecidableEq α] in
+lemma alternating_sameCycle_iff (p : Equiv.Perm α) (x y : Bool × α) :
+    (alternatingPerm p).SameCycle x y ↔ p.SameCycle x.2 y.2 := by
+  constructor
+  · exact alternating_sameCycle_snd p
+  · intro h
+    exact (alternating_sameCycle_zero p x).trans
+      ((alternating_zero_sameCycle p h).trans (alternating_sameCycle_zero p y).symm)
+
+lemma orbit_alternating (p : Equiv.Perm α) (x : Bool × α) :
+    orbitSet (alternatingPerm p) x = childJoin (orbitSet p x.2) (orbitSet p x.2) := by
+  ext ⟨b,y⟩
+  simp only [mem_orbitSet,alternating_sameCycle_iff,mem_childJoin,ite_self]
+
+omit [Fintype α] in
+lemma boolFiber_childJoin (A B : Finset α) (b : Bool) :
+    boolFiber (childJoin A B) b = if b then B else A := by
+  ext x
+  cases b <;> simp only [mem_boolFiber,mem_childJoin,Bool.false_eq_true,↓reduceIte]
+
+noncomputable def projectCycle (p : Equiv.Perm α) (A B : Finset α)
+    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) :
+    RoutingNetwork.SelectedCycle p A := by
+  refine ⟨boolFiber C.val false,?_,?_⟩
+  · obtain ⟨x,hx⟩ := C.property.1
+    refine ⟨x.2,?_⟩
+    rw [hx,orbit_alternating,boolFiber_childJoin]
+    rfl
+  · intro x hx
+    exact (mem_childJoin A B false x).mp (C.property.2 ((mem_boolFiber _ _ _).mp hx))
+
+lemma selected_alternating_eq (p : Equiv.Perm α) (A B : Finset α)
+    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) :
+    C.val = childJoin (projectCycle p A B C).val (projectCycle p A B C).val := by
+  change C.val = childJoin (boolFiber C.val false) (boolFiber C.val false)
+  obtain ⟨x,hx⟩ := C.property.1
+  rw [hx,orbit_alternating,boolFiber_childJoin]
+  rfl
+
+lemma projectCycle_injective (p : Equiv.Perm α) (A B : Finset α) :
+    Function.Injective (projectCycle p A B) := by
+  intro C D he
+  apply Subtype.ext
+  rw [selected_alternating_eq p A B C,selected_alternating_eq p A B D,he]
+
+lemma card_alternating_cycles_le (p : Equiv.Perm α) (A B : Finset α) :
+    Fintype.card (RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) ≤
+      Fintype.card (RoutingNetwork.SelectedCycle p A) :=
+  Fintype.card_le_of_injective _ (projectCycle_injective p A B)
+
+end CycleColoring
+
+section PairAssignmentCount
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+omit [Fintype α] in
+lemma mem_image_perm_iff (p : Equiv.Perm α) (A : Finset α) (y : α) :
+    y ∈ A.image p ↔ p.symm y ∈ A := by
+  constructor
+  · intro h
+    obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp h
+    simpa only [←hxy,Equiv.symm_apply_apply] using hx
+  · intro h
+    exact Finset.mem_image.mpr ⟨p.symm y,h,p.apply_symm_apply y⟩
+
+omit [Fintype α] in
+lemma card_childJoin (A B : Finset α) : (childJoin A B).card = A.card+B.card := by
+  have hd : Disjoint (A.image (fun x => (false,x))) (B.image (fun x => (true,x))) := by
+    apply Finset.disjoint_left.mpr
+    rintro x hx hy
+    obtain ⟨a,ha,rfl⟩ := Finset.mem_image.mp hx
+    obtain ⟨b,hb,hba⟩ := Finset.mem_image.mp hy
+    exact Bool.false_ne_true (congrArg Prod.fst hba).symm
+  have hi (b : Bool) : Function.Injective (fun x : α => (b,x)) :=
+    fun _ _ h => congrArg Prod.snd h
+  rw [childJoin,Finset.card_union_of_disjoint hd,
+    Finset.card_image_of_injective A (hi false),Finset.card_image_of_injective B (hi true)]
+
+omit [Fintype α] in
+lemma alternating_internal (p : Equiv.Perm α) (A B : Finset α) :
+    ((childJoin A B).filter (fun x => (CycleColoring.alternatingPerm p).symm x ∈ childJoin A B)) =
+      childJoin (A ∩ B.image p) (B ∩ A) := by
+  ext ⟨b,x⟩
+  rw [Finset.mem_filter]
+  cases b <;> simp only [mem_childJoin,CycleColoring.alternatingPerm,
+    Equiv.symm_mk,Equiv.coe_fn_mk,Bool.false_eq_true,↓reduceIte,Finset.mem_inter,mem_image_perm_iff]
+
+omit [Fintype α] in
+lemma alternating_boundary_card (p : Equiv.Perm α) (A B : Finset α) :
+    (CycleColoring.boundary (CycleColoring.alternatingPerm p) (childJoin A B)).card =
+      (A ∪ B).card + (A ∪ B.image p).card - (A.card+B.card) := by
+  rw [CycleColoring.boundary_card,alternating_internal,card_childJoin,card_childJoin]
+  have h₀ := Finset.card_union_add_card_inter A B
+  have h₁ := Finset.card_union_add_card_inter A (B.image p)
+  rw [Finset.card_image_of_injective _ p.injective] at h₁
+  have h₂ := Finset.inter_comm B A
+  rw [h₂]
+  omega
+
+lemma compatible_assignment_bound (p : Equiv.Perm α) (A B : Finset α) :
+    Fintype.card (CycleColoring.OppositeColoring (CycleColoring.alternatingPerm p) (childJoin A B)) ≤
+      2 ^ ((A ∪ B).card + (A ∪ B.image p).card - (A.card+B.card) +
+        Fintype.card (RoutingNetwork.SelectedCycle p A)) := by
+  have h := CycleColoring.card_opposite_le (CycleColoring.alternatingPerm p) (childJoin A B)
+    Prod.fst (CycleColoring.alternatingPerm_flip p)
+  rw [alternating_boundary_card] at h
+  exact h.trans (Nat.pow_le_pow_right (by norm_num)
+    (Nat.add_le_add_left (CycleColoring.card_alternating_cycles_le p A B) _))
+
+end PairAssignmentCount
+
+lemma finiteMean_nonneg {Ω : Type*} [Fintype Ω] {f : Ω → ℝ}
+    (h : ∀ ω, 0 ≤ f ω) : 0 ≤ finiteMean f := by
+  exact div_nonneg (Finset.sum_nonneg (fun ω _ => h ω)) (Nat.cast_nonneg _)
+
+lemma boolFiber_headTail (d : ℕ) (S : Finset (Card (d+1))) (b : Bool) :
+    boolFiber (S.image (headTailEquiv d)) b = childMarks S b := by
+  classical
+  ext x
+  rw [mem_boolFiber,mem_childMarks]
+  constructor
+  · intro h
+    obtain ⟨y,hy,he⟩ := Finset.mem_image.mp h
+    have hh := congrArg (headTailEquiv d).symm he
+    rw [Equiv.symm_apply_apply] at hh
+    change y = Fin.cons b x at hh
+    exact hh ▸ hy
+  · intro h
+    exact Finset.mem_image.mpr ⟨Fin.cons b x,h,by rfl⟩
+
+lemma prod_childMarks (d : ℕ) (S : Finset (Card (d+1))) (f : Card (d+1) → ℝ) :
+    (∏ x ∈ S, f x) =
+      (∏ x ∈ childMarks S false, f (Fin.cons false x)) *
+      ∏ x ∈ childMarks S true, f (Fin.cons true x) := by
+  classical
+  have h := prod_boolFibers (S.image (headTailEquiv d)) (fun x => f ((headTailEquiv d).symm x))
+  rw [Finset.prod_image (headTailEquiv d).injective.injOn,
+    boolFiber_headTail,boolFiber_headTail] at h
+  simp only [Equiv.symm_apply_apply] at h
+  exact h
+
+noncomputable def headAverage {d : ℕ} (p : Card (d+1) → ℝ) (y : Card d) : ℝ :=
+  (p (Fin.cons false y)+p (Fin.cons true y))/2
+
+lemma finiteMean_headAverage (d : ℕ) (p : Card (d+1) → ℝ) :
+    finiteMean (headAverage p) = finiteMean p := by
+  rw [finiteMean_equiv (headTailEquiv d),finiteMean_prod,finiteMean_comm]
+  apply finiteMean_congr
+  intro y
+  rw [finiteMean_bool]
+  rfl
+
+lemma pairSwitch_product_bound (d : ℕ) (S : Finset (Card (d+1)))
+    (p : Card (d+1) → ℝ) (hp : ∀ x, 0 ≤ p x) :
+    finiteMean (fun ξ : Card d → Bool => ∏ x ∈ S, p (pairSwitch ξ x)) ≤
+      ∏ x ∈ S, headAverage p (Fin.tail x) := by
+  classical
+  have h : CylinderBound (fun T => ∏ x ∈ T, p ((headTailEquiv d).symm x))
+      (fun x => p ((headTailEquiv d).symm x)) := fun _ => le_rfl
+  have hh := h.fair_parallel (fun x => hp _) (S.image (headTailEquiv d))
+  simp only [Finset.prod_image (pairLayer _).injective.injOn,
+    Finset.prod_image (headTailEquiv d).injective.injOn] at hh
+  convert hh using 1
+  · apply finiteMean_congr
+    intro ξ
+    apply Finset.prod_congr rfl
+    intro x _
+    rfl
+  · apply Finset.prod_congr rfl
+    intro x _
+    change (p (Fin.cons false (Fin.tail x))+p (Fin.cons true (Fin.tail x)))/2 =
+      (p (Fin.cons (x 0) (Fin.tail x))+p (Fin.cons (!(x 0)) (Fin.tail x)))/2
+    cases x 0
+    · rfl
+    · simp only [Bool.not_true]
+      ring
+
+lemma butterflyPerm_step (d : ℕ)
+    (c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (ξ : Card d → Bool) :
+    butterflyPerm (d+1) (decodeButterfly (d+1) ((coinStepEquiv d).symm (c,ξ))) =
+    pairSwitch ξ * childLift (fun b => butterflyPerm d (decodeButterfly d (if b then c.2 else c.1))) := by
+  change pairSwitch (fun y => (coinStepEquiv d).symm (c,ξ) (Sum.inl y)) *
+    childLift (fun b => butterflyPerm d (decodeButterfly d
+      (fun i => (coinStepEquiv d).symm (c,ξ) (Sum.inr (b,i))))) = _
+  apply congrArg₂ (· * ·)
+  · rfl
+  · apply congrArg childLift
+    funext b
+    apply congrArg (fun ω => butterflyPerm d (decodeButterfly d ω))
+    funext i
+    cases b <;> rfl
+
+lemma butterfly_product_bound (d : ℕ) (S : Finset (Card d))
+    (p : Card d → ℝ) (hp : ∀ x, 0 ≤ p x) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      ∏ x ∈ S, p (butterflyPerm d (decodeButterfly d ω) x)) ≤ (finiteMean p)^S.card := by
+  classical
+  induction d with
+  | zero =>
+    have he : p = fun _ => p (fun i => Fin.elim0 i) := by
+      funext x
+      exact congrArg p (Subsingleton.elim _ _)
+    rw [he]
+    simp only [Finset.prod_const,finiteMean_const,le_refl]
+  | succ d ih =>
+    let g : Card d → ℝ := headAverage p
+    have hg : ∀ x, 0 ≤ g x := fun x => div_nonneg (add_nonneg (hp _) (hp _)) (by norm_num)
+    rw [finiteMean_equiv (coinStepEquiv d),finiteMean_prod]
+    calc
+      _ ≤ finiteMean (fun c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool) =>
+        (∏ x ∈ childMarks S false, g (butterflyPerm d (decodeButterfly d c.1) x)) *
+        ∏ x ∈ childMarks S true, g (butterflyPerm d (decodeButterfly d c.2) x)) := by
+        apply finiteMean_mono
+        intro c
+        let f : Equiv.Perm (Card (d+1)) := childLift (fun b =>
+          butterflyPerm d (decodeButterfly d (if b then c.2 else c.1)))
+        have hh := pairSwitch_product_bound d (S.image f) p hp
+        simp only [Finset.prod_image f.injective.injOn] at hh
+        convert hh using 1
+        · apply finiteMean_congr
+          intro ξ
+          apply Finset.prod_congr rfl
+          intro x _
+          rw [butterflyPerm_step]
+          rfl
+        · rw [prod_childMarks]
+          simp only [f,childLift,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
+            Bool.false_eq_true,↓reduceIte]
+          rfl
+      _ = (finiteMean (fun c : SwitchIndex d → Bool =>
+        ∏ x ∈ childMarks S false, g (butterflyPerm d (decodeButterfly d c) x))) *
+        finiteMean (fun c : SwitchIndex d → Bool =>
+        ∏ x ∈ childMarks S true, g (butterflyPerm d (decodeButterfly d c) x)) :=
+          finiteMean_prod_mul
+            (fun c : SwitchIndex d → Bool => ∏ x ∈ childMarks S false,
+              g (butterflyPerm d (decodeButterfly d c) x))
+            (fun c : SwitchIndex d → Bool => ∏ x ∈ childMarks S true,
+              g (butterflyPerm d (decodeButterfly d c) x))
+      _ ≤ (finiteMean g)^(childMarks S false).card * (finiteMean g)^(childMarks S true).card :=
+        mul_le_mul (ih (childMarks S false) g hg) (ih (childMarks S true) g hg)
+          (finiteMean_nonneg fun c => Finset.prod_nonneg (fun x _ => hg _))
+          (pow_nonneg (finiteMean_nonneg hg) _)
+      _ = (finiteMean p)^S.card := by
+        rw [←pow_add,card_childMarks_add,finiteMean_headAverage]
+
+noncomputable def inverseButterflyMarks (d : ℕ) (A : Finset (Card d))
+    (ω : SwitchIndex d → Bool) : Finset (Card d) :=
+  A.image (butterflyPerm d (decodeButterfly d ω)).symm
+
+lemma inverseButterflyMarks_inclusion (d : ℕ) (A S : Finset (Card d)) :
+    inclusionMoment (inverseButterflyMarks d A) S ≤
+      ((A.card : ℝ)/(2:ℝ)^d)^S.card := by
+  classical
+  let p : Card d → ℝ := fun x => if x ∈ A then 1 else 0
+  have hm : finiteMean p = (A.card : ℝ)/(2:ℝ)^d := by
+    simp only [finiteMean,p,card_positions,Nat.cast_pow,Nat.cast_ofNat]
+    congr 1
+    simp
+  have he : inclusionMoment (inverseButterflyMarks d A) S =
+      finiteMean (fun ω : SwitchIndex d → Bool =>
+        ∏ x ∈ S, p (butterflyPerm d (decodeButterfly d ω) x)) := by
+    rw [inclusionMoment_eq]
+    apply finiteMean_congr
+    intro ω
+    simp only [inverseButterflyMarks,subset_image_equiv_iff,Equiv.symm_symm,indicator_subset_prod,
+      Finset.prod_image (butterflyPerm d (decodeButterfly d ω)).injective.injOn,p]
+  rw [he]
+  simpa only [hm] using butterfly_product_bound d S p (fun x => by dsimp [p]; split_ifs <;> norm_num)
+
+namespace PairRouting
+
+section
+variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
+
+noncomputable def occupied (x : ι → Bool × α) : Finset α :=
+  Finset.univ.image (fun i => (x i).2)
+
+omit [Fintype α] in
+lemma mem_occupied (x : ι → Bool × α) (y : α) : y ∈ occupied x ↔ ∃ i, (x i).2 = y := by
+  simp [occupied]
+
+def required (x : ι → Bool × α) (c : ι → Bool) (i : ι) : Bool := Bool.xor (x i).1 (c i)
+
+def Compatible (x : ι → Bool × α) (c : ι → Bool) : Prop :=
+  ∀ i j, (x i).2 = (x j).2 → required x c i = required x c j
+
+def colors (x : ι → Bool × α) (ξ : α → Bool) (i : ι) : Bool :=
+  Bool.xor (x i).1 (ξ (x i).2)
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma required_colors (x : ι → Bool × α) (ξ : α → Bool) (i : ι) :
+    required x (colors x ξ) i = ξ (x i).2 := by
+  simp only [required,colors]
+  cases (x i).1 <;> cases ξ (x i).2 <;> rfl
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma colors_compatible (x : ι → Bool × α) (ξ : α → Bool) : Compatible x (colors x ξ) := by
+  intro i j he
+  simp only [required_colors,he]
+
+noncomputable def chosenCoin (x : ι → Bool × α) (c : ι → Bool) (y : α) : Bool := by
+  classical
+  exact if h : ∃ i, (x i).2 = y then required x c (Classical.choose h) else false
+
+omit [Fintype α] in
+lemma chosenCoin_apply (x : ι → Bool × α) (c : ι → Bool) (hc : Compatible x c) (i : ι) :
+    chosenCoin x c (x i).2 = required x c i := by
+  classical
+  simp only [chosenCoin,dite_eq_left (show ∃ j, (x j).2 = (x i).2 from ⟨i,rfl⟩)]
+  exact hc _ i (Classical.choose_spec (show ∃ j, (x j).2 = (x i).2 from ⟨i,rfl⟩))
+
+omit [Fintype α] in
+lemma colors_eq_iff (x : ι → Bool × α) (c : ι → Bool) (hc : Compatible x c) (ξ : α → Bool) :
+    colors x ξ = c ↔ ∀ y ∈ occupied x, ξ y = chosenCoin x c y := by
+  constructor
+  · intro he y hy
+    obtain ⟨i,rfl⟩ := (mem_occupied x y).mp hy
+    rw [chosenCoin_apply x c hc i]
+    rw [←he,required_colors]
+  · intro he
+    funext i
+    apply (xor_eq_iff _ _ _).mpr
+    exact (he _ ((mem_occupied x _).mpr ⟨i,rfl⟩)).trans (chosenCoin_apply x c hc i)
+
+noncomputable instance (x : ι → Bool × α) (c : ι → Bool) : Decidable (Compatible x c) :=
+  Classical.propDecidable _
+
+lemma colors_probability (x : ι → Bool × α) (c : ι → Bool) :
+    finiteMean (fun ξ : α → Bool => if colors x ξ = c then (1 : ℝ) else 0) =
+      if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0 := by
+  classical
+  by_cases hc : Compatible x c
+  · rw [ite_eq_left hc]
+    have he := fun ξ => colors_eq_iff x c hc ξ
+    simp only [he]
+    exact mean_coinCylinder (occupied x) (chosenCoin x c)
+  · rw [ite_eq_right hc]
+    have he : ∀ ξ, colors x ξ ≠ c := fun ξ hh => hc (hh ▸ colors_compatible x ξ)
+    simp only [he,ite_false,finiteMean_const]
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma Compatible.tail_injective {x : ι ↪ Bool × α} {c : ι → Bool} (hc : Compatible x c) (b : Bool) :
+    Function.Injective (fun i : {i : ι // c i = b} => (x i.val).2) := by
+  intro i j he
+  apply Subtype.ext
+  apply x.injective
+  apply Prod.ext
+  · have hh := hc i j he
+    simp only [required,i.property,j.property] at hh
+    cases h₀ : (x i.val).1 <;> cases h₁ : (x j.val).1 <;> cases b <;> simp_all
+  · exact he
+
+def childEmbedding (x : ι ↪ Bool × α) (c : ι → Bool) (hc : Compatible x c) (b : Bool) :
+    {i : ι // c i = b} ↪ α := ⟨fun i => (x i.val).2,hc.tail_injective b⟩
+
+
+def lift (p : Bool → Equiv.Perm α) : Equiv.Perm (Bool × α) where
+  toFun x := (x.1,p x.1 x.2)
+  invFun x := (x.1,(p x.1).symm x.2)
+  left_inv x := by simp
+  right_inv x := by simp
+
+def sandwich (ξ η : α → Bool) (p : Bool → Equiv.Perm α) : Equiv.Perm (Bool × α) :=
+  pairLayer η * lift p * pairLayer ξ
+
+def ChildMatches (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) : Prop :=
+  ∀ i, p (c i) (x i).2 = (y i).2
+
+noncomputable instance (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
+    Decidable (ChildMatches p x y c) := Classical.propDecidable _
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma sandwich_matches_iff (ξ η : α → Bool) (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) :
+    (∀ i, sandwich ξ η p (x i) = y i) ↔
+      colors x ξ = colors y η ∧ ChildMatches p x y (colors x ξ) := by
+  have he (i : ι) : sandwich ξ η p (x i) = y i ↔
+      colors x ξ i = colors y η i ∧ p (colors x ξ i) (x i).2 = (y i).2 := by
+    change pairLayer η (lift p (pairLayer ξ (x i))) = y i ↔ _
+    rw [←Equiv.eq_symm_apply]
+    change (colors x ξ i,p (colors x ξ i) (x i).2) = (colors y η i,(y i).2) ↔ _
+    exact ⟨Prod.mk.inj,fun h => Prod.ext h.1 h.2⟩
+  simp only [he,forall_and,←funext_iff,ChildMatches]
+
+lemma sandwich_probability [DecidableEq ι] (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) :
+    finiteMean (fun coins : (α → Bool) × (α → Bool) =>
+      if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then (1:ℝ) else 0) =
+    ∑ c : ι → Bool,
+      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
+      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) *
+      (if ChildMatches p x y c then 1 else 0) := by
+  classical
+  have he (coins : (α → Bool) × (α → Bool)) :
+      (if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then (1:ℝ) else 0) =
+      ∑ c : ι → Bool, (if colors x coins.1 = c then (1:ℝ) else 0) *
+        (if colors y coins.2 = c then 1 else 0) * (if ChildMatches p x y c then 1 else 0) := by
+    rw [Finset.sum_eq_single (colors x coins.1)]
+    · by_cases hc : colors x coins.1 = colors y coins.2
+      · simp only [sandwich_matches_iff,hc,true_and,↓reduceIte,one_mul]
+      · simp only [sandwich_matches_iff,hc,false_and,↓reduceIte,one_mul,Ne.symm hc,zero_mul]
+    · intro c _ hc
+      simp only [Ne.symm hc,ite_false,zero_mul]
+    · simp
+  simp_rw [he]
+  rw [finiteMean_sum]
+  apply Finset.sum_congr rfl
+  intro c _
+  rw [finiteMean_mul_const]
+  rw [finiteMean_prod_mul
+    (fun ξ : α → Bool => if colors x ξ = c then (1:ℝ) else 0)
+    (fun η : α → Bool => if colors y η = c then (1:ℝ) else 0),
+    colors_probability,colors_probability]
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma childMatches_split (p₀ p₁ : Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
+    ChildMatches (fun b => if b then p₁ else p₀) x y c ↔
+      (∀ i : {i // c i = false}, p₀ (x i.val).2 = (y i.val).2) ∧
+      (∀ i : {i // c i = true}, p₁ (x i.val).2 = (y i.val).2) := by
+  constructor
+  · intro h
+    constructor
+    · intro i
+      simpa only [i.property,Bool.false_eq_true,↓reduceIte] using h i.val
+    · intro i
+      simpa only [i.property,↓reduceIte] using h i.val
+  · rintro ⟨h₀,h₁⟩ i
+    cases hi : c i
+    · simpa only [hi,Bool.false_eq_true,↓reduceIte] using h₀ ⟨i,hi⟩
+    · simpa only [hi,↓reduceIte] using h₁ ⟨i,hi⟩
+
+omit [Fintype α] in
+lemma childMatches_mean {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
+    (P : Ω → Equiv.Perm α) (Q : Ω' → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
+    finiteMean (fun ω : Ω × Ω' =>
+      if ChildMatches (fun b => if b then Q ω.2 else P ω.1) x y c then (1:ℝ) else 0) =
+      tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
+        (fun i : {i // c i = false} => (y i.val).2) *
+      tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
+        (fun i : {i // c i = true} => (y i.val).2) := by
+  classical
+  let f : Ω → ℝ := fun ω => if ∀ i : {i // c i = false}, P ω (x i.val).2 = (y i.val).2 then 1 else 0
+  let g : Ω' → ℝ := fun ω => if ∀ i : {i // c i = true}, Q ω (x i.val).2 = (y i.val).2 then 1 else 0
+  have he (ω : Ω × Ω') :
+      (if ChildMatches (fun b => if b then Q ω.2 else P ω.1) x y c then (1:ℝ) else 0) = f ω.1*g ω.2 := by
+    simp only [childMatches_split,f,g]
+    split_ifs <;> simp_all
+  simp_rw [he]
+  exact finiteMean_prod_mul f g
+
+lemma sandwich_mean [DecidableEq ι] {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
+    (P : Ω → Equiv.Perm α) (Q : Ω' → Equiv.Perm α) (x y : ι → Bool × α) :
+    finiteMean (fun ω : Ω × Ω' => finiteMean (fun coins : (α → Bool) × (α → Bool) =>
+      if ∀ i, sandwich coins.1 coins.2 (fun b => if b then Q ω.2 else P ω.1) (x i) = y i
+      then (1:ℝ) else 0)) =
+    ∑ c : ι → Bool,
+      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
+      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) *
+      (tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
+        (fun i : {i // c i = false} => (y i.val).2) *
+      tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
+        (fun i : {i // c i = true} => (y i.val).2)) := by
+  classical
+  simp_rw [sandwich_probability]
+  rw [finiteMean_sum]
+  apply Finset.sum_congr rfl
+  intro c _
+  simp only [mul_comm ((if Compatible x c then _ else _) * _),finiteMean_mul_const,
+    childMatches_mean]
+
+
+abbrev BothAssignments (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :=
+  {c : ι → Bool // Compatible e c ∧ Compatible (fun i => lift p (e i)) c}
+
+noncomputable instance (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    Fintype (BothAssignments e p) := Fintype.ofFinite _
+
+noncomputable def labelSet (e : ι ↪ Bool × α) : Finset (Bool × α) := Finset.univ.image e
+
+def alternatingProjection (p : Bool → Equiv.Perm α) : Equiv.Perm α := (p false).symm * p true
+
+noncomputable def assignmentColoring (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α)
+    (c : BothAssignments e p) : CycleColoring.OppositeColoring
+      (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e) := by
+  classical
+  refine ⟨Function.extend e c.val (fun _ => false),?_,?_⟩
+  · intro z hz
+    apply Function.extend_apply'
+    rintro ⟨i,rfl⟩
+    exact hz (Finset.mem_image.mpr ⟨i,Finset.mem_univ _,rfl⟩)
+  · intro z hz hnext
+    obtain ⟨i,_,hi⟩ := Finset.mem_image.mp hz
+    obtain ⟨j,_,hj⟩ := Finset.mem_image.mp hnext
+    have hfi := e.injective.extend_apply c.val (fun _ => false) i
+    have hfj := e.injective.extend_apply c.val (fun _ => false) j
+    rw [hi] at hfi
+    rw [hj] at hfj
+    rw [hfi,hfj]
+    rcases z with ⟨b,a⟩
+    cases b
+    · have hj' : e j = (true,a) := hj
+      have ht : (e i).2 = (e j).2 := by rw [hi,hj']
+      have hh := c.property.1 i j ht
+      simp only [required,hi,hj'] at hh
+      revert hh
+      generalize c.val i = bi
+      generalize c.val j = bj
+      cases bi <;> cases bj <;> decide
+    · have hj' : e j = (false,alternatingProjection p a) := hj
+      have ht : (lift p (e i)).2 = (lift p (e j)).2 := by
+        simp only [hi,hj',lift,Equiv.coe_fn_mk,alternatingProjection,Equiv.Perm.coe_mul,
+          Function.comp_apply,Equiv.apply_symm_apply]
+      have hh := c.property.2 i j ht
+      simp only [required,hi,hj',lift,Equiv.coe_fn_mk] at hh
+      revert hh
+      generalize c.val i = bi
+      generalize c.val j = bj
+      cases bi <;> cases bj <;> decide
+
+omit [Fintype α] in
+lemma assignmentColoring_injective (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    Function.Injective (assignmentColoring e p) := by
+  intro c c' he
+  apply Subtype.ext
+  funext i
+  have hh := congrFun (congrArg Subtype.val he) (e i)
+  change Function.extend e c.val (fun _ => false) (e i) =
+    Function.extend e c'.val (fun _ => false) (e i) at hh
+  simpa only [e.injective.extend_apply] using hh
+
+lemma bothAssignments_card (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    Fintype.card (BothAssignments e p) ≤
+      2 ^ ((boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true).card +
+        (boolFiber (labelSet e) false ∪ (boolFiber (labelSet e) true).image
+          (alternatingProjection p)).card -
+        ((boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card) +
+        Fintype.card (RoutingNetwork.SelectedCycle (alternatingProjection p) (boolFiber (labelSet e) false))) := by
+  have h := compatible_assignment_bound (alternatingProjection p)
+    (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
+  rw [childJoin_boolFiber] at h
+  exact (Fintype.card_le_of_injective _ (assignmentColoring_injective e p)).trans h
+
+omit [Fintype α] in
+lemma occupied_labelSet (e : ι ↪ Bool × α) : occupied e = (labelSet e).image Prod.snd := by
+  classical
+  simp only [occupied,labelSet,Finset.image_image,Function.comp_def]
+
+omit [Fintype ι] [Fintype α] in
+lemma image_snd_childJoin (A B : Finset α) : (childJoin A B).image Prod.snd = A ∪ B := by
+  classical
+  simp only [childJoin,Finset.image_union,Finset.image_image,Function.comp_def,Finset.image_id']
+
+omit [Fintype ι] [Fintype α] in
+lemma image_lift_childJoin (p : Bool → Equiv.Perm α) (A B : Finset α) :
+    (childJoin A B).image (fun x => (lift p x).2) = A.image (p false) ∪ B.image (p true) := by
+  classical
+  simp only [childJoin,Finset.image_union,Finset.image_image,Function.comp_def,lift,Equiv.coe_fn_mk]
+
+omit [Fintype α] in
+lemma occupied_lift_card (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    (occupied (fun i => lift p (e i))).card =
+      (boolFiber (labelSet e) false ∪ (boolFiber (labelSet e) true).image (alternatingProjection p)).card := by
+  classical
+  have he : occupied (fun i => lift p (e i)) = (labelSet e).image (fun x => (lift p x).2) := by
+    simp only [occupied,labelSet,Finset.image_image,Function.comp_def]
+  rw [he,←childJoin_boolFiber (labelSet e),image_lift_childJoin]
+  simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
+  let A := boolFiber (labelSet e) false
+  let B := boolFiber (labelSet e) true
+  change (A.image (p false) ∪ B.image (p true)).card = (A ∪ B.image (alternatingProjection p)).card
+  have hh : A.image (p false) ∪ B.image (p true) =
+      (A ∪ B.image (alternatingProjection p)).image (p false) := by
+    simp only [Finset.image_union,Finset.image_image,Function.comp_def,
+      alternatingProjection,Equiv.Perm.coe_mul,Equiv.apply_symm_apply]
+  rw [hh,Finset.card_image_of_injective _ (p false).injective]
+
+lemma bothAssignments_occupied_bound (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    Fintype.card (BothAssignments e p) ≤
+      2 ^ ((occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι +
+        Fintype.card (RoutingNetwork.SelectedCycle (alternatingProjection p) (boolFiber (labelSet e) false))) := by
+  have he : (occupied e).card =
+      (boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true).card := by
+    rw [occupied_labelSet,←childJoin_boolFiber (labelSet e),image_snd_childJoin]
+    simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
+  have hc : (boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card = Fintype.card ι := by
+    rw [←card_childJoin,childJoin_boolFiber,labelSet,Finset.card_image_of_injective _ e.injective,
+      Finset.card_univ]
+  simpa only [he,occupied_lift_card,←hc] using bothAssignments_card e p
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma compatible_pairLayer_iff (x : ι → Bool × α) (c : ι → Bool) (ξ : α → Bool) :
+    Compatible (fun i => pairLayer ξ (x i)) c ↔ Compatible x c := by
+  have he (i j : ι) (ht : (x i).2 = (x j).2) :
+      required (fun i => pairLayer ξ (x i)) c i = required (fun i => pairLayer ξ (x i)) c j ↔
+      required x c i = required x c j := by
+    simp only [required,pairLayer,Equiv.coe_fn_mk,ht]
+    cases (x i).1 <;> cases (x j).1 <;> cases c i <;> cases c j <;> cases ξ (x j).2 <;> decide
+  constructor
+  · intro h i j ht
+    exact (he i j ht).mp (h i j ht)
+  · intro h i j ht
+    exact (he i j ht).mpr (h i j ht)
+
+
+noncomputable def alternatingCycles (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) : ℕ :=
+  Fintype.card (RoutingNetwork.SelectedCycle
+    (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e))
+
+omit [Fintype α] in
+lemma occupied_eq_fibers (e : ι ↪ Bool × α) :
+    occupied e = boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true := by
+  rw [occupied_labelSet,←childJoin_boolFiber (labelSet e),image_snd_childJoin]
+  simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
+
+omit [Fintype α] in
+lemma card_labelSet (e : ι ↪ Bool × α) : (labelSet e).card = Fintype.card ι := by
+  classical
+  exact (Finset.card_image_of_injective _ e.injective).trans (Finset.card_univ)
+
+omit [Fintype α] in
+lemma occupied_card_double (e : ι ↪ Bool × α) : Fintype.card ι ≤ 2*(occupied e).card := by
+  have h₀ : (boolFiber (labelSet e) false).card ≤ (occupied e).card := by
+    rw [occupied_eq_fibers]
+    exact Finset.card_le_card Finset.subset_union_left
+  have h₁ : (boolFiber (labelSet e) true).card ≤ (occupied e).card := by
+    rw [occupied_eq_fibers]
+    exact Finset.card_le_card Finset.subset_union_right
+  have he : (boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card = Fintype.card ι := by
+    rw [←card_childJoin,childJoin_boolFiber,card_labelSet]
+  omega
+
+lemma bothAssignments_exact_bound (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    Fintype.card (BothAssignments e p) ≤
+      2 ^ ((occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι +
+        alternatingCycles e p) := by
+  have h := CycleColoring.card_opposite_le
+    (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e)
+    Prod.fst (CycleColoring.alternatingPerm_flip (alternatingProjection p))
+  have hb : (CycleColoring.boundary (CycleColoring.alternatingPerm (alternatingProjection p))
+      (labelSet e)).card =
+      (occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι := by
+    have hh := alternating_boundary_card (alternatingProjection p)
+      (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
+    rw [childJoin_boolFiber] at hh
+    rw [occupied_eq_fibers,occupied_lift_card]
+    have he := card_childJoin (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
+    rw [childJoin_boolFiber,card_labelSet] at he
+    simpa only [he] using hh
+  rw [hb] at h
+  exact (Fintype.card_le_of_injective _ (assignmentColoring_injective e p)).trans h
+
+abbrev Assignments (x y : ι → Bool × α) :=
+  {c : ι → Bool // Compatible x c ∧ Compatible y c}
+
+noncomputable instance (x y : ι → Bool × α) : Fintype (Assignments x y) := Fintype.ofFinite _
+
+def switchedEmbedding (e : ι ↪ Bool × α) (ξ : α → Bool) : ι ↪ Bool × α :=
+  e.trans (pairLayer ξ).toEmbedding
+
+def routedEmbedding (e : ι ↪ Bool × α) (ξ η : α → Bool) (p : Bool → Equiv.Perm α) :
+    ι ↪ Bool × α := e.trans (sandwich ξ η p).toEmbedding
+
+omit [Fintype α] in
+lemma occupied_pairLayer (x : ι → Bool × α) (ξ : α → Bool) :
+    occupied (fun i => pairLayer ξ (x i)) = occupied x := rfl
+
+noncomputable def assignmentsEquiv (e : ι ↪ Bool × α) (ξ η : α → Bool)
+    (p : Bool → Equiv.Perm α) :
+    Assignments e (routedEmbedding e ξ η p) ≃ BothAssignments (switchedEmbedding e ξ) p := by
+  apply Equiv.subtypeEquivRight
+  intro c
+  change Compatible e c ∧ Compatible (fun i => pairLayer η (lift p (pairLayer ξ (e i)))) c ↔ _
+  rw [compatible_pairLayer_iff]
+  exact and_congr (compatible_pairLayer_iff e c ξ).symm Iff.rfl
+
+lemma assignments_actual_bound (e : ι ↪ Bool × α) (ξ η : α → Bool) (p : Bool → Equiv.Perm α) :
+    Fintype.card (Assignments e (routedEmbedding e ξ η p)) ≤
+      2 ^ ((occupied e).card + (occupied (routedEmbedding e ξ η p)).card - Fintype.card ι +
+        alternatingCycles (switchedEmbedding e ξ) p) := by
+  rw [Fintype.card_congr (assignmentsEquiv e ξ η p)]
+  exact bothAssignments_exact_bound (switchedEmbedding e ξ) p
+
+omit [Fintype α] in
+lemma assignments_actual_exponent_nonneg (e : ι ↪ Bool × α) (ξ η : α → Bool)
+    (p : Bool → Equiv.Perm α) :
+    Fintype.card ι ≤ (occupied e).card + (occupied (routedEmbedding e ξ η p)).card := by
+  have h₀ := occupied_card_double e
+  have h₁ := occupied_card_double (routedEmbedding e ξ η p)
+  omega
+
+end
+variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α] [DecidableEq ι]
+
+noncomputable def compatibleSet (x y : ι → Bool × α) : Finset (ι → Bool) := by
+  classical
+  exact Finset.univ.filter (fun c => Compatible x c ∧ Compatible y c)
+
+omit [Fintype α] [DecidableEq α] in
+lemma mem_compatibleSet (x y : ι → Bool × α) (c : ι → Bool) :
+    c ∈ compatibleSet x y ↔ Compatible x c ∧ Compatible y c := by
+  classical
+  simp only [compatibleSet,Finset.mem_filter,Finset.mem_univ,true_and]
+
+omit [Fintype α] [DecidableEq α] in
+lemma card_compatibleSet (x y : ι → Bool × α) :
+    (compatibleSet x y).card = Fintype.card (Assignments x y) := by
+  classical
+  exact (Fintype.card_subtype _).symm
+
+noncomputable def outerWeight (x y : ι → Bool × α) : ℝ :=
+  1/(2:ℝ)^((occupied x).card+(occupied y).card)
+
+omit [Fintype α] [DecidableEq ι] in
+lemma outerWeight_pos (x y : ι → Bool × α) : 0 < outerWeight x y := by
+  unfold outerWeight
+  positivity
+
+omit [Fintype α] in
+lemma sum_colors_factor (x y : ι → Bool × α) (f : (ι → Bool) → ℝ) :
+    (∑ c : ι → Bool,
+      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
+      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) * f c) =
+      outerWeight x y * ∑ c ∈ compatibleSet x y, f c := by
+  classical
+  rw [compatibleSet,Finset.sum_filter,Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro c _
+  by_cases hx : Compatible x c <;> by_cases hy : Compatible y c
+  · simp only [hx,hy,true_and,↓reduceIte,outerWeight,pow_add,one_div_mul_one_div]
+  · simp only [hx,hy,and_false,↓reduceIte,mul_zero,zero_mul]
+  · simp only [hx,hy,false_and,↓reduceIte,mul_zero,zero_mul]
+  · simp only [hx,hy,false_and,↓reduceIte,mul_zero,zero_mul]
+
+lemma sandwich_weighted_probability (p : Bool → Equiv.Perm α) (x y : ι → Bool × α)
+    (W : (ι → Bool) → ℝ) :
+    finiteMean (fun coins : (α → Bool) × (α → Bool) =>
+      if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then W (colors x coins.1) else 0) =
+    outerWeight x y * ∑ c ∈ compatibleSet x y, if ChildMatches p x y c then W c else 0 := by
+  classical
+  have he (coins : (α → Bool) × (α → Bool)) :
+      (if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then W (colors x coins.1) else 0) =
+      ∑ c : ι → Bool, (if colors x coins.1 = c then (1:ℝ) else 0) *
+        (if colors y coins.2 = c then 1 else 0) * (if ChildMatches p x y c then W c else 0) := by
+    rw [Finset.sum_eq_single (colors x coins.1)]
+    · by_cases hc : colors x coins.1 = colors y coins.2
+      · simp only [sandwich_matches_iff,hc,true_and,↓reduceIte,one_mul]
+      · simp only [sandwich_matches_iff,hc,false_and,↓reduceIte,one_mul,Ne.symm hc,zero_mul]
+    · intro c _ hc
+      simp only [Ne.symm hc,ite_false,zero_mul]
+    · simp
+  simp_rw [he]
+  rw [finiteMean_sum,←sum_colors_factor]
+  apply Finset.sum_congr rfl
+  intro c _
+  rw [finiteMean_mul_const,finiteMean_prod_mul
+    (fun ξ : α → Bool => if colors x ξ = c then (1:ℝ) else 0)
+    (fun η : α → Bool => if colors y η = c then (1:ℝ) else 0),
+    colors_probability,colors_probability]
+
+omit [Fintype α] [DecidableEq ι] in
+lemma tupleProbability_nonneg {Ω β : Type*} [Fintype Ω] [DecidableEq β]
+    (P : Ω → Equiv.Perm β) (x y : ι → β) : 0 ≤ tupleProbability P x y := by
+  classical
+  exact finiteMean_nonneg (fun ω => by split_ifs <;> norm_num)
+
+end PairRouting
+end Thorp
+
+end ThorpNine.Harmonic
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Harmonic/AssignmentSlots.lean b/lean/OAI/Probability/ThorpRouting/Harmonic/AssignmentSlots.lean
index b4027d64e362965fb2ca104c077ac0c3d26009e0..4ef7ae5c8b22e70018fcdc018c3267c985d7cba7 100644
--- a/lean/OAI/Probability/ThorpRouting/Harmonic/AssignmentSlots.lean
+++ b/lean/OAI/Probability/ThorpRouting/Harmonic/AssignmentSlots.lean
@@ -1,834 +1,834 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Harmonic.RotationMeeting
-
-namespace OAI
-
-namespace ThorpNine.Harmonic
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-section CycleCounting
-variable {d A : ℕ}
-
-lemma assignmentOwner_eq_some {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) (x : Card d) (i : Fin A) :
-    assignmentOwner f x = some i ↔ x ∈ (f.val i).val := by
-  classical
-  unfold assignmentOwner
-  split_ifs with h
-  · constructor
-    · intro hi
-      have hx := Classical.choose_spec h
-      rw [Option.some.inj hi] at hx
-      exact hx
-    · intro hx
-      have he := (f.val (Classical.choose h)).eq_of_mem (f.val i) (Classical.choose_spec h) hx
-      exact congrArg some (f.property.1 he)
-  · constructor
-    · intro hi; contradiction
-    · intro hx; exact (h ⟨i,hx⟩).elim
-
-lemma assignmentOwner_fiber {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) (i : Fin A) : cycleFiber (assignmentOwner f) i = (f.val i).val := by
-  ext x
-  simp only [mem_cycleFiber,assignmentOwner_eq_some]
-
-noncomputable def assignmentToSlots {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) : CycleSlots p S r :=
-  ⟨assignmentOwner f,by
-    refine ⟨?_,?_,?_⟩
-    · intro i
-      rw [assignmentOwner_fiber]
-      exact f.property.2 i
-    · intro i
-      rw [assignmentOwner_fiber]
-      exact (f.val i).property.2
-    · intro i x hx
-      rw [assignmentOwner_fiber]
-      exact (f.val i).orbit_eq ((assignmentOwner_eq_some f x i).mp hx)⟩
-
-noncomputable def cycleSlotsEquivAssignments (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
-    CycleSlots p S r ≃ SlotAssignment p S r where
-  toFun := slotsToAssignment
-  invFun := assignmentToSlots
-  left_inv c := by
-    apply Subtype.ext
-    apply owner_ext
-    intro i
-    rw [assignmentToSlots,assignmentOwner_fiber]
-    rfl
-  right_inv f := by
-    apply Subtype.ext
-    funext i
-    apply Subtype.ext
-    exact assignmentOwner_fiber f i
-
-end CycleCounting
-
-section GroupedCounting
-variable {d A : ℕ}
-
-abbrev LengthClass (r : Fin A → ℕ) := {j : ℕ // j ∈ Finset.univ.image (fun i => r i + 1)}
-
-def slotClass (r : Fin A → ℕ) (i : Fin A) : LengthClass r :=
-  ⟨r i+1, Finset.mem_image.mpr ⟨i, Finset.mem_univ _, rfl⟩⟩
-
-abbrev SlotsOfLength (r : Fin A → ℕ) (j : LengthClass r) := {i : Fin A // slotClass r i = j}
-abbrev CyclesOfLength (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (j : ℕ) :=
-  {C : SelectedCycle p S // C.val.card = j}
-
-noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (j : ℕ) :
-    Fintype (CyclesOfLength p S j) := Fintype.ofFinite _
-
-abbrev GroupedAssignments (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
-  ∀ j : LengthClass r, SlotsOfLength r j ↪ CyclesOfLength p S j.val
-
-noncomputable def assignmentToGroups {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) : GroupedAssignments p S r := fun _j =>
-  { toFun := fun i => ⟨f.val i.val, (f.property.2 i.val).trans (congrArg Subtype.val i.property)⟩
-    inj' := fun _i _k h => Subtype.ext (f.property.1 (congrArg Subtype.val h)) }
-
-noncomputable def groupsEmbedding {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (F : GroupedAssignments p S r) : Fin A ↪ SelectedCycle p S :=
-  ((Equiv.sigmaFiberEquiv (slotClass r)).symm.toEmbedding.trans
-    (Function.Embedding.sigmaMap (Function.Embedding.refl _) F)).trans
-    ((Function.Embedding.sigmaMap (Function.Embedding.subtype _) (fun _ => Function.Embedding.refl _)).trans
-      (Equiv.sigmaFiberEquiv (fun C : SelectedCycle p S => C.val.card)).toEmbedding)
-
-noncomputable def groupsToAssignment {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (F : GroupedAssignments p S r) : SlotAssignment p S r :=
-  ⟨groupsEmbedding F, (groupsEmbedding F).injective,
-    fun i => (F (slotClass r i) ⟨i,rfl⟩).property⟩
-
-noncomputable def assignmentEquivGroups (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
-    SlotAssignment p S r ≃ GroupedAssignments p S r where
-  toFun := assignmentToGroups
-  invFun := groupsToAssignment
-  left_inv f := by
-    apply Subtype.ext
-    rfl
-  right_inv F := by
-    funext j
-    apply Function.Embedding.ext
-    intro i
-    rcases i with ⟨i,hi⟩
-    subst j
-    rfl
-
-lemma card_cycleSlots_factorial (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
-    Fintype.card (CycleSlots p S r) =
-      ∏ j : LengthClass r, (Fintype.card (CyclesOfLength p S j.val)).descFactorial
-        (Fintype.card (SlotsOfLength r j)) := by
-  classical
-  rw [Fintype.card_congr (cycleSlotsEquivAssignments p S r),
-    Fintype.card_congr (assignmentEquivGroups p S r), Fintype.card_pi]
-  simp only [Fintype.card_embedding_eq]
-
-lemma butterfly_factorial_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-       (∏ j : LengthClass r,
-         (Fintype.card (CyclesOfLength (butterflyPerm d (decodeButterfly d ω)) S j.val)).descFactorial
-           (Fintype.card (SlotsOfLength r j)) : ℕ)) ≤
-      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-        ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
-  simpa only [←card_cycleSlots_factorial] using butterfly_cycleSlots_moment S r hr
-
-end GroupedCounting
-
-section AllLengths
-variable {d A : ℕ}
-
-noncomputable def assignmentRelabel (p : Equiv.Perm (Card d)) (S : Finset (Card d))
-    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
-    SlotAssignment p S r ≃ SlotAssignment p S (r ∘ e) where
-  toFun f := ⟨f.val ∘ e, f.property.1.comp e.injective, fun i => f.property.2 (e i)⟩
-  invFun f := ⟨f.val ∘ e.symm, f.property.1.comp e.symm.injective, fun i => by
-    simpa only [Function.comp_apply,Equiv.apply_symm_apply] using f.property.2 (e.symm i)⟩
-  left_inv f := by apply Subtype.ext; funext i; simp
-  right_inv f := by apply Subtype.ext; funext i; simp
-
-lemma card_cycleSlots_relabel (p : Equiv.Perm (Card d)) (S : Finset (Card d))
-    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
-    Fintype.card (CycleSlots p S r) = Fintype.card (CycleSlots p S (r ∘ e)) := by
-  exact Fintype.card_congr ((cycleSlotsEquivAssignments p S r).trans
-    ((assignmentRelabel p S r e).trans (cycleSlotsEquivAssignments p S (r ∘ e)).symm))
-
-lemma butterfly_cycleSlots_moment_all (S : Finset (Card d)) (r : Fin A → ℕ) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) ≤
-        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
-  have h := butterfly_cycleSlots_moment S (r ∘ Tuple.sort r) (Tuple.monotone_sort r)
-  simp_rw [←card_cycleSlots_relabel] at h
-  have hp := Equiv.prod_comp (Tuple.sort r) (fun i : Fin A => (1 / Real.sqrt (r i+1) : ℝ))
-  simpa only [Function.comp_apply,hp] using h
-
-abbrev ShortCycle (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :=
-  {C : SelectedCycle p S // C.val.card ≤ J}
-
-noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
-    Fintype (ShortCycle p S J) := Fintype.ofFinite _
-
-noncomputable def shortCycleLength {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {J : ℕ}
-    (C : ShortCycle p S J) : Fin J :=
-  ⟨C.val.val.card-1, by have := C.val.nonempty.card_pos; have := C.property; omega⟩
-
-lemma shortCycleLength_add_one {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {J : ℕ}
-    (C : ShortCycle p S J) : (shortCycleLength C : ℕ)+1 = C.val.val.card := by
-  have := C.val.nonempty.card_pos
-  simp only [shortCycleLength]
-  omega
-
-noncomputable def shortEmbeddingEquiv (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
-    (Fin A ↪ ShortCycle p S J) ≃
-      Σ r : Fin A → Fin J, SlotAssignment p S (fun i => (r i : ℕ)) where
-  toFun f := ⟨fun i => shortCycleLength (f i),
-    ⟨fun i => (f i).val, fun i j h => f.injective (Subtype.ext h),
-      fun i => (shortCycleLength_add_one (f i)).symm⟩⟩
-  invFun c :=
-    { toFun := fun i => ⟨c.2.val i, by rw [c.2.property.2]; exact (c.1 i).isLt⟩
-      inj' := fun _i _j h => c.2.property.1 (congrArg Subtype.val h) }
-  left_inv f := by
-    apply Function.Embedding.ext
-    intro i
-    rfl
-  right_inv c := by
-    rcases c with ⟨r,f⟩
-    have he : (fun i => shortCycleLength (⟨f.val i, by rw [f.property.2]; exact (r i).isLt⟩ : ShortCycle p S J)) = r := by
-      funext i
-      apply Fin.ext
-      simp only [shortCycleLength,f.property.2,Nat.add_sub_cancel]
-    apply Sigma.ext he
-    apply (Subtype.heq_iff_coe_eq (by
-      intro x
-      change (Function.Injective x ∧ ∀ i, (x i).val.card = (f.val i).val.card - 1 + 1) ↔ _
-      simp only [f.property.2,Nat.add_sub_cancel])).mpr
-    rfl
-
-lemma shortCycle_factorial_eq (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
-    (Fintype.card (ShortCycle p S J)).descFactorial A =
-      ∑ r : Fin A → Fin J, Fintype.card (CycleSlots p S (fun i => (r i : ℕ))) := by
-  classical
-  have hc := Fintype.card_embedding_eq (α := Fin A) (β := ShortCycle p S J)
-  simp only [Fintype.card_fin] at hc
-  rw [←hc, Fintype.card_congr (shortEmbeddingEquiv p S J),Fintype.card_sigma]
-  apply Finset.sum_congr rfl
-  intro r _
-  exact (Fintype.card_congr (cycleSlotsEquivAssignments p S (fun i => (r i : ℕ)))).symm
-
-end AllLengths
-
-lemma factorial_mgf {Ω : Type*} [Fintype Ω] (F : Ω → ℕ) (k : ℕ) (Λ q : ℝ)
-    (hF : ∀ ω, F ω ≤ k) (hΛ : 0 ≤ Λ) (hq : 1 ≤ q)
-    (hm : ∀ a, finiteMean (fun ω => ((F ω).descFactorial a : ℝ)) ≤ Λ^a) :
-    finiteMean (fun ω => q^(F ω)) ≤ Real.exp ((q-1)*Λ) := by
-  have hc (n a : ℕ) : (n.choose a : ℝ) = (n.descFactorial a : ℝ) / (a.factorial : ℝ) := by
-    rw [Nat.descFactorial_eq_factorial_mul_choose,Nat.cast_mul]
-    exact (mul_div_cancel_left₀ _ (by exact_mod_cast Nat.factorial_ne_zero a)).symm
-  have hex (ω : Ω) : q^(F ω) = ∑ a ∈ Finset.range (k+1),
-      ((F ω).descFactorial a : ℝ) * ((q-1)^a / (a.factorial : ℝ)) := by
-    calc
-      _ = ((q-1)+1)^(F ω) := by congr 1; ring
-      _ = ∑ a ∈ Finset.range (F ω+1), (q-1)^a * (F ω).choose a := by
-        rw [add_pow]
-        simp only [one_pow,mul_one]
-      _ = ∑ a ∈ Finset.range (k+1), (q-1)^a * (F ω).choose a := by
-        apply Finset.sum_subset (Finset.range_mono (Nat.add_le_add_right (hF ω) _))
-        intro a _ ha
-        have hn : F ω < a := by have := (Finset.mem_range.not.mp ha); omega
-        rw [Nat.choose_eq_zero_of_lt hn,Nat.cast_zero,mul_zero]
-      _ = _ := by
-        apply Finset.sum_congr rfl
-        intro a _
-        rw [hc]
-        ring
-  simp_rw [hex,finiteMean_sum,finiteMean_mul_const]
-  calc
-    _ ≤ ∑ a ∈ Finset.range (k+1), Λ^a * ((q-1)^a / (a.factorial : ℝ)) := by
-      apply Finset.sum_le_sum
-      intro a _
-      exact mul_le_mul_of_nonneg_right (hm a) (by positivity)
-    _ = ∑ a ∈ Finset.range (k+1), ((q-1)*Λ)^a / (a.factorial : ℝ) := by
-      apply Finset.sum_congr rfl
-      intro a _
-      rw [mul_pow]
-      ring
-    _ ≤ _ := Real.sum_le_exp_of_nonneg (mul_nonneg (sub_nonneg.mpr hq) hΛ) _
-
-lemma inv_sqrt_step (j : ℕ) :
-    1 / Real.sqrt (j+1) ≤ 2*(Real.sqrt (j+1)-Real.sqrt j) := by
-  have hp : 0 < Real.sqrt (j+1) := Real.sqrt_pos.mpr (by positivity)
-  apply (div_le_iff₀ hp).mpr
-  have h1 := Real.sq_sqrt (show (0:ℝ) ≤ j+1 by positivity)
-  have h0 := Real.sq_sqrt (show (0:ℝ) ≤ j by positivity)
-  nlinarith [sq_nonneg (Real.sqrt (j+1)-Real.sqrt j)]
-
-lemma sum_inv_sqrt (J : ℕ) :
-    (∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1)) ≤ 2*Real.sqrt J := by
-  have h (n : ℕ) : (∑ j ∈ Finset.range n, 1 / Real.sqrt (j+1)) ≤ 2*Real.sqrt n := by
-    induction n with
-    | zero => simp
-    | succ n ih =>
-      rw [Finset.sum_range_succ]
-      have hs := inv_sqrt_step n
-      push_cast
-      linarith
-  rw [Fin.sum_univ_eq_sum_range (fun j : ℕ => (1 / Real.sqrt (j+1) : ℝ)) J]
-  exact h J
-section CycleMGF
-variable {d : ℕ}
-
-lemma butterfly_shortCycle_moment (S : Finset (Card d)) (J A : ℕ) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      ((Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J)).descFactorial A : ℝ)) ≤
-      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))^A := by
-  classical
-  simp_rw [shortCycle_factorial_eq,Nat.cast_sum,finiteMean_sum]
-  calc
-    _ ≤ ∑ r : Fin A → Fin J,
-        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-          ∏ i : Fin A, (1 / Real.sqrt ((r i : ℕ)+1)) := by
-      apply Finset.sum_le_sum
-      intro r _
-      exact butterfly_cycleSlots_moment_all S (fun i => (r i : ℕ))
-    _ = _ := by
-      rw [←Finset.mul_sum,mul_pow]
-      congr 1
-      rw [←Fintype.prod_sum (fun (_ : Fin A) (j : Fin J) => (1 / Real.sqrt ((j:ℕ)+1) : ℝ))]
-      simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fin]
-
-noncomputable def cycleMass (p : Equiv.Perm (Card d)) (S : Finset (Card d)) : ℕ :=
-  ∑ C : SelectedCycle p S, C.val.card
-
-lemma cycleMass_le (p : Equiv.Perm (Card d)) (S : Finset (Card d)) : cycleMass p S ≤ S.card := by
-  classical
-  let f : (Σ C : SelectedCycle p S, {x : Card d // x ∈ C.val}) → S :=
-    fun c => ⟨c.2.val,c.1.property.2 c.2.property⟩
-  have hi : Function.Injective f := by
-    intro c e h
-    have hx : c.2.val = e.2.val := congrArg Subtype.val h
-    have hC : c.1 = e.1 := c.1.eq_of_mem e.1 c.2.property (hx ▸ e.2.property)
-    rcases c with ⟨c,x⟩
-    rcases e with ⟨e,y⟩
-    dsimp only at hC
-    subst e
-    have he : x=y := Subtype.ext hx
-    cases he
-    rfl
-  have hh := Fintype.card_le_of_injective f hi
-  simpa only [Fintype.card_sigma,Fintype.card_coe,cycleMass] using hh
-
-lemma shortCycle_card_le (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
-    Fintype.card (ShortCycle p S J) ≤ S.card := by
-  classical
-  calc
-    _ ≤ Fintype.card (SelectedCycle p S) := Fintype.card_subtype_le _
-    _ = ∑ _ : SelectedCycle p S, 1 := by simp
-    _ ≤ cycleMass p S := Finset.sum_le_sum (fun C _ => C.nonempty.card_pos)
-    _ ≤ _ := cycleMass_le p S
-
-lemma butterfly_shortCycle_mgf (S : Finset (Card d)) (J : ℕ) (q : ℝ) (hq : 1 ≤ q) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      q^(Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J))) ≤
-      Real.exp ((q-1)*((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))) := by
-  apply factorial_mgf _ S.card _ q (fun ω => shortCycle_card_le _ _ _)
-  · positivity
-  · exact hq
-  · exact butterfly_shortCycle_moment S J
-
-end CycleMGF
-
-section FullCycleMGF
-variable {d : ℕ}
-
-abbrev LongCycle (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :=
-  {C : SelectedCycle p S // ¬ C.val.card ≤ J}
-noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
-    Fintype (LongCycle p S J) := Fintype.ofFinite _
-
-lemma selectedCycle_card_split (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
-    Fintype.card (SelectedCycle p S) =
-      Fintype.card (ShortCycle p S J) + Fintype.card (LongCycle p S J) := by
-  classical
-  rw [←Fintype.card_congr (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J)),Fintype.card_sum]
-
-lemma longCycle_card_bound (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
-    J * Fintype.card (LongCycle p S J) ≤ S.card := by
-  classical
-  have hs := Equiv.sum_comp (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J))
-    (fun C : SelectedCycle p S => C.val.card)
-  rw [Fintype.sum_sum_type] at hs
-  have h0 : J * Fintype.card (LongCycle p S J) ≤
-      ∑ C : LongCycle p S J, C.val.val.card := by
-    calc
-      _ = ∑ _C : LongCycle p S J, J := by simp [mul_comm]
-      _ ≤ _ := Finset.sum_le_sum (fun C _ => Nat.le_of_lt (Nat.lt_of_not_ge C.property))
-  have hh := cycleMass_le p S
-  unfold cycleMass at hh
-  rw [←hs] at hh
-  exact h0.trans ((Nat.le_add_left _ _).trans hh)
-
-lemma finiteMean_pos {Ω : Type*} [Fintype Ω] [Nonempty Ω] {f : Ω → ℝ}
-    (h : ∀ ω, 0 < f ω) : 0 < finiteMean f := by
-  apply div_pos
-  · exact Finset.sum_pos (fun ω _ => h ω) Finset.univ_nonempty
-  · exact Nat.cast_pos.mpr Fintype.card_pos
-
-lemma longCycle_pow_bound (p : Equiv.Perm (Card d)) (S : Finset (Card d))
-    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    q^(Fintype.card (LongCycle p S J)) ≤ Real.exp ((S.card : ℝ)/J * Real.log q) := by
-  have hq0 : 0 < q := lt_of_lt_of_le zero_lt_one hq
-  have hc : (Fintype.card (LongCycle p S J) : ℝ) ≤ (S.card : ℝ)/J := by
-    apply (le_div_iff₀ (by exact_mod_cast hJ)).mpr
-    have hh := longCycle_card_bound p S J
-    exact_mod_cast (by simpa only [mul_comm] using hh)
-  calc
-    _ = Real.exp ((Fintype.card (LongCycle p S J) : ℝ)*Real.log q) := by
-      rw [Real.exp_nat_mul,Real.exp_log hq0]
-    _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right hc (Real.log_nonneg hq))
-
-lemma butterfly_cycle_log_mgf (S : Finset (Card d)) (J : ℕ) (hJ : 0 < J)
-    (q : ℝ) (hq : 1 ≤ q) :
-    Real.log (finiteMean (fun ω : SwitchIndex d → Bool =>
-      q^(Fintype.card (SelectedCycle (butterflyPerm d (decodeButterfly d ω)) S)))) ≤
-      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
-  let lam : ℝ := (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1)
-  let c : ℝ := (S.card : ℝ)/J * Real.log q
-  have hnum : (q-1)*lam ≤ 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
-    have hsum := sum_inv_sqrt J
-    have hp : 0 ≤ (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) := by positivity
-    have h0 : 0 ≤ 2*Real.sqrt J := by positivity
-    have hh := mul_le_mul_of_nonneg_left hsum hp
-    have hq1 : 0 ≤ q-1 := sub_nonneg.mpr hq
-    calc
-      _ ≤ (q-1)*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
-        mul_le_mul_of_nonneg_left hh hq1
-      _ ≤ q*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
-        mul_le_mul_of_nonneg_right (by linarith) (mul_nonneg hp h0)
-      _ = _ := by ring
-  have hm : finiteMean (fun ω : SwitchIndex d → Bool =>
-      q^(Fintype.card (SelectedCycle (butterflyPerm d (decodeButterfly d ω)) S))) ≤
-      Real.exp (c+(q-1)*lam) := by
-    calc
-      _ ≤ finiteMean (fun ω : SwitchIndex d → Bool =>
-          q^(Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J))*Real.exp c) := by
-        apply finiteMean_mono
-        intro ω
-        rw [selectedCycle_card_split _ _ J,pow_add]
-        exact mul_le_mul_of_nonneg_left (longCycle_pow_bound _ S J hJ q hq) (by positivity)
-      _ = finiteMean (fun ω : SwitchIndex d → Bool =>
-          q^(Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J))) * Real.exp c :=
-        finiteMean_mul_const _ _
-      _ ≤ Real.exp ((q-1)*lam) * Real.exp c :=
-        mul_le_mul_of_nonneg_right (butterfly_shortCycle_mgf S J q hq) (Real.exp_nonneg _)
-      _ = _ := by rw [←Real.exp_add,add_comm]
-  have hpos : 0 < finiteMean (fun ω : SwitchIndex d → Bool =>
-      q^(Fintype.card (SelectedCycle (butterflyPerm d (decodeButterfly d ω)) S))) := by
-    apply finiteMean_pos
-    intro ω
-    positivity
-  have hlog := Real.log_le_log hpos hm
-  rw [Real.log_exp] at hlog
-  dsimp [c] at hlog
-  linarith
-
-end FullCycleMGF
-
-structure RoutingNetwork (α ι : Type*) [Fintype α] [DecidableEq α]
-    [Fintype ι] [DecidableEq ι] (d : ℕ) where
-  perm : (ι → Bool) → Equiv.Perm α
-  path : (ι → Bool) → α → Finset ι
-  users : (ι → Bool) → ι → α × α
-  users_ne : ∀ ω i, (users ω i).1 ≠ (users ω i).2
-  mem_path : ∀ ω x i, i ∈ path ω x ↔ (users ω i).1 = x ∨ (users ω i).2 = x
-  card_path : ∀ ω x, (path ω x).card = d
-  entropy : ∀ ω (S : Finset α),
-    (∑ i : ι, if (users ω i).1 ∈ S ∧ (users ω i).2 ∈ S then (1:ℝ) else 0) ≤
-      (S.card : ℝ) * Real.log S.card / (2 * Real.log 2)
-  follow : α → (α → QueryTree ι) → QueryTree ι
-  close : α → α → QueryTree ι → QueryTree ι
-  follow_succeeds : ∀ x k ω, (follow x k).succeeds ω = (k (perm ω x)).succeeds ω
-  follow_raw_charge : ∀ x k ω, (follow x k).charge ω = (k (perm ω x)).charge ω
-  follow_stale_charge : ∀ x k R ω,
-    (follow x k).staleCharge R ω = (k (perm ω x)).staleCharge (R ∪ path ω x) ω
-  close_succeeds : ∀ x y k ω,
-    (close x y k).succeeds ω = if perm ω x = y then k.succeeds ω else false
-  close_raw_charge : ∀ x y k ω, perm ω x = y → (close x y k).charge ω = k.charge ω + d
-  close_stale_charge : ∀ x y k R ω, perm ω x = y →
-    (close x y k).staleCharge R ω = k.staleCharge (R ∪ path ω x) ω + (path ω x ∩ R).card
-
-namespace RoutingNetwork
-variable {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι] [DecidableEq ι]
-  {d : ℕ} (R : RoutingNetwork α ι d)
-
-lemma card_switches_using (ω : ι → Bool) (x : α) :
-    (Finset.univ.filter (fun i : ι =>
-      (R.users ω i).1 = x ∨ (R.users ω i).2 = x)).card = d := by
-  classical
-  have he : Finset.univ.filter (fun i : ι =>
-      (R.users ω i).1 = x ∨ (R.users ω i).2 = x) = R.path ω x := by
-    ext i
-    simp only [Finset.mem_filter,Finset.mem_univ,true_and,R.mem_path]
-  rw [he,R.card_path]
-lemma switch_incidence_sum (b : ι → Bool) (w : α → ℝ) :
-    ∑ i : ι, (w (R.users b i).1 + w (R.users b i).2) =
-      (d : ℝ) * ∑ x : α, w x := by
-  classical
-  calc
-    _ = ∑ i : ι, ∑ x : α,
-        if (R.users b i).1 = x ∨ (R.users b i).2 = x then w x else 0 := by
-      apply Finset.sum_congr rfl
-      intro i _
-      rw [Finset.sum_ite]
-      simp only [Finset.sum_const_zero, add_zero]
-      have he : Finset.univ.filter (fun x : α =>
-          (R.users b i).1 = x ∨ (R.users b i).2 = x) =
-          {(R.users b i).1, (R.users b i).2} := by
-        ext x; simp [eq_comm]
-      rw [he, Finset.sum_pair (R.users_ne b i)]
-    _ = ∑ x : α, ∑ i : ι,
-        if (R.users b i).1 = x ∨ (R.users b i).2 = x then w x else 0 :=
-      Finset.sum_comm
-    _ = ∑ x : α, (d : ℝ) * w x := by
-      apply Finset.sum_congr rfl
-      intro x _
-      rw [Finset.sum_ite]
-      simp only [Finset.sum_const_zero, add_zero, Finset.sum_const, nsmul_eq_mul,
-        card_switches_using]
-    _ = _ := by rw [Finset.mul_sum]
-
-lemma weighted_switch_min_bound (b : ι → Bool) (w : α → ℝ) :
-    ∑ i : ι, min (w (R.users b i).1) (w (R.users b i).2) ≤
-      (d : ℝ) / 2 * ∑ x : α, w x := by
-  have h : 2 * (∑ i : ι,
-      min (w (R.users b i).1) (w (R.users b i).2)) ≤
-      ∑ i : ι, (w (R.users b i).1 + w (R.users b i).2) := by
-    rw [Finset.mul_sum]
-    apply Finset.sum_le_sum
-    intro i _
-    linarith [min_le_left (w (R.users b i).1) (w (R.users b i).2),
-      min_le_right (w (R.users b i).1) (w (R.users b i).2)]
-  rw [switch_incidence_sum] at h
-  linarith
-section Rotation
-variable {A : ℕ}
-
-def cycleFiber (o : α → Option (Fin A)) (i : Fin A) : Finset (α) := by
-  classical
-  exact Finset.univ.filter (fun x => o x = some i)
-
-omit [DecidableEq α] in
-@[simp] lemma mem_cycleFiber (o : α → Option (Fin A)) (i : Fin A) (x : α) :
-    x ∈ cycleFiber o i ↔ o x = some i := by
-  classical
-  simp [cycleFiber]
-
-abbrev Terminal (o : α → Option (Fin A)) := ∀ i : Fin A, {x : α // o x = some i}
-
-noncomputable def slotWeight (o : α → Option (Fin A)) (i : Fin A) : ℝ :=
-  1 / (cycleFiber o i).card
-
-noncomputable def pathWeight (o : α → Option (Fin A)) (x : α) : ℝ :=
-  match o x with
-  | none => 0
-  | some i => slotWeight o i
-
-noncomputable def withinWeight (o : α → Option (Fin A)) (u v : α) : ℝ :=
-  match o u, o v with
-  | some i, some j => if i = j then slotWeight o i else 0
-  | _, _ => 0
-
-noncomputable def rotationMeeting (o : α → Option (Fin A)) (z : Terminal o)
-    (u v : α) : ℝ :=
-  match o u, o v with
-  | some i, some j => if i < j then (if (z j).val = v then 1 else 0)
-      else if j < i then (if (z i).val = u then 1 else 0)
-      else (if (z i).val = u then 1 else 0) + (if (z i).val = v then 1 else 0)
-  | _, _ => 0
-
-omit [DecidableEq α] in
-lemma card_cycleFiber (o : α → Option (Fin A)) (i : Fin A) :
-    Fintype.card {x : α // o x = some i} = (cycleFiber o i).card := by
-  classical
-  simp [cycleFiber,Fintype.card_subtype]
-
-lemma terminal_mean {o : α → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
-    (i : Fin A) (x : α) (hx : o x = some i) :
-    finiteMean (fun z : Terminal o => if (z i).val = x then (1:ℝ) else 0) = slotWeight o i := by
-  classical
-  have hev := finiteMean_pi_eval (A := fun i : Fin A => {x : α // o x = some i})
-    i (fun z => if z.val = x then (1:ℝ) else 0)
-  rw [hev]
-  have he : (fun z : {x : α // o x = some i} => if z.val = x then (1:ℝ) else 0) =
-      (fun z => if z = ⟨x,hx⟩ then 1 else 0) := by
-    funext z
-    simp only [Subtype.ext_iff]
-  rw [he,finiteMean_eq_indicator,card_cycleFiber,slotWeight]
-
-omit [DecidableEq α] in
-lemma slotWeight_antitone (o : α → Option (Fin A))
-    (hn : ∀ i, (cycleFiber o i).Nonempty)
-    (hm : Monotone (fun i => (cycleFiber o i).card)) : Antitone (slotWeight o) := by
-  intro i j hij
-  unfold slotWeight
-  apply one_div_le_one_div_of_le
-  · exact_mod_cast Finset.card_pos.mpr (hn i)
-  · exact_mod_cast hm hij
-
-lemma rotationMeeting_mean {o : α → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
-    (hm : Monotone (fun i => (cycleFiber o i).card)) (u v : α) :
-    finiteMean (fun z : Terminal o => rotationMeeting o z u v) =
-      min (pathWeight o u) (pathWeight o v) + withinWeight o u v := by
-  classical
-  have hn : ∀ i, (cycleFiber o i).Nonempty := by
-    intro i
-    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : α // o x = some i})
-    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
-  have hw := slotWeight_antitone o hn hm
-  cases hu : o u with
-  | none =>
-    cases hv : o v with
-    | none => simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean]
-    | some j =>
-      have hj : 0 ≤ slotWeight o j := by unfold slotWeight; positivity
-      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_left hj]
-  | some i =>
-    cases hv : o v with
-    | none =>
-      have hi : 0 ≤ slotWeight o i := by unfold slotWeight; positivity
-      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_right hi]
-    | some j =>
-      simp only [rotationMeeting,pathWeight,withinWeight,hu,hv]
-      rcases lt_trichotomy i j with hij | rfl | hji
-      · simp only [hij,↓reduceIte,ne_of_lt hij]
-        rw [terminal_mean j v hv,min_eq_right (hw (le_of_lt hij)),add_zero]
-      · simp only [lt_self_iff_false,↓reduceIte,min_self]
-        rw [finiteMean_add,terminal_mean i u hu,terminal_mean i v hv]
-      · simp only [not_lt_of_ge (le_of_lt hji),hji,↓reduceIte,ne_of_gt hji]
-        rw [terminal_mean i u hu,min_eq_left (hw (le_of_lt hji)),add_zero]
-
-omit [DecidableEq α] in
-lemma pathWeight_sum (o : α → Option (Fin A)) (hn : ∀ i, (cycleFiber o i).Nonempty) :
-    ∑ x : α, pathWeight o x = (A : ℝ) := by
-  classical
-  have he (x : α) : pathWeight o x =
-      ∑ i : Fin A, if o x = some i then slotWeight o i else 0 := by
-    cases hx : o x with
-    | none => simp [pathWeight,hx]
-    | some j => simp [pathWeight,hx]
-  simp only [he]
-  rw [Finset.sum_comm]
-  have hf (i : Fin A) : (∑ x : α, if o x = some i then slotWeight o i else 0) = 1 := by
-    rw [← Finset.sum_filter]
-    change (∑ _x ∈ cycleFiber o i, slotWeight o i) = 1
-    simp only [Finset.sum_const,nsmul_eq_mul,slotWeight]
-    exact mul_one_div_cancel (by exact_mod_cast (Finset.card_pos.mpr (hn i)).ne')
-  simp only [hf,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,mul_one]
-
-lemma withinWeight_sum_bound (o : α → Option (Fin A))
-    (hn : ∀ i, (cycleFiber o i).Nonempty) (b : ι → Bool) :
-    (∑ s : ι, withinWeight o (R.users b s).1 (R.users b s).2) ≤
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
-  classical
-  have he (u v : α) : withinWeight o u v =
-      ∑ i : Fin A, (if u ∈ cycleFiber o i ∧ v ∈ cycleFiber o i then (1:ℝ) else 0) * slotWeight o i := by
-    simp only [mem_cycleFiber]
-    cases hu : o u <;> cases hv : o v <;> simp [withinWeight,hu,hv,ite_and,eq_comm]
-  simp only [he]
-  rw [Finset.sum_comm]
-  apply Finset.sum_le_sum
-  intro i _
-  rw [← Finset.sum_mul]
-  have hi := R.entropy b (cycleFiber o i)
-  have hpos : (0:ℝ) < (cycleFiber o i).card := by exact_mod_cast Finset.card_pos.mpr (hn i)
-  have hmul := mul_le_mul_of_nonneg_right hi (show 0 ≤ slotWeight o i by unfold slotWeight; positivity)
-  calc
-    _ ≤ ((cycleFiber o i).card : ℝ) * Real.log (cycleFiber o i).card /
-        (2 * Real.log 2) * slotWeight o i := hmul
-    _ = _ := by unfold slotWeight; field_simp
-
-noncomputable def rotationCharge (o : α → Option (Fin A)) (b : ι → Bool)
-    (z : Terminal o) : ℝ :=
-  ∑ s : ι, rotationMeeting o z (R.users b s).1 (R.users b s).2
-
-lemma rotationCharge_mean_bound {o : α → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
-    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : ι → Bool) :
-    finiteMean (rotationCharge R o b) ≤ (A:ℝ) * d / 2 +
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
-  classical
-  have hn : ∀ i, (cycleFiber o i).Nonempty := by
-    intro i
-    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : α // o x = some i})
-    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
-  unfold rotationCharge
-  rw [finiteMean_sum]
-  simp only [rotationMeeting_mean hm,Finset.sum_add_distrib]
-  have h₁ := weighted_switch_min_bound R b (pathWeight o)
-  rw [pathWeight_sum o hn] at h₁
-  have h₂ := withinWeight_sum_bound R o hn b
-  linarith
-
-lemma rotation_exp_bound {o : α → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
-    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : ι → Bool) :
-    Real.exp (-Real.log 2 * ((A:ℝ)*d/2 +
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2))) ≤
-      finiteMean (fun z : Terminal o => Real.exp (-Real.log 2 * rotationCharge R o b z)) := by
-  have hb := rotationCharge_mean_bound R hm b
-  have hlog : 0 ≤ Real.log 2 := Real.log_nonneg (by norm_num)
-  have hm' : -Real.log 2 * ((A:ℝ)*d/2 +
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2)) ≤
-      finiteMean (fun z : Terminal o => -Real.log 2 * rotationCharge R o b z) := by
-    have he := finiteMean_mul_const (rotationCharge R o b) (-Real.log 2)
-    simp only [mul_comm] at he
-    rw [he]
-    exact mul_le_mul_of_nonpos_left hb (neg_nonpos.mpr hlog)
-  exact (Real.exp_le_exp.mpr hm').trans (exp_finiteMean_le _)
-
-end Rotation
-
-namespace QueryTree
-open Thorp.QueryTree
-
-noncomputable def seekCycle (R : RoutingNetwork α ι d) (S : Finset (α)) (a : α) :
-    ℕ → α → Finset (α) → (Finset (α) → QueryTree (ι)) →
-      QueryTree (ι)
-  | 0, x, V, k => if x ∈ S ∧ x ∉ V then R.close x a (k (insert x V)) else reject
-  | r + 1, x, V, k => if x ∈ S ∧ x ∉ V then R.follow x
-      (fun y => seekCycle R S a r y (insert x V) k) else reject
-
-lemma seekCycle_raw_charge (S : Finset (α)) (a : α) (r : ℕ)
-    (x : α) (V : Finset (α)) (k : Finset (α) → QueryTree (ι))
-    (ω : ι → Bool) (K : ℕ)
-    (hK : ∀ V, (k V).succeeds ω = true → (k V).charge ω = K)
-    (hs : (seekCycle R S a r x V k).succeeds ω = true) :
-    (seekCycle R S a r x V k).charge ω = K + d := by
-  induction r generalizing x V with
-  | zero =>
-    by_cases hv : x ∈ S ∧ x ∉ V
-    · rw [seekCycle,ite_eq_left hv,R.close_succeeds] at hs
-      split_ifs at hs with he
-      · rw [seekCycle,ite_eq_left hv,R.close_raw_charge _ _ _ _ he,hK _ hs]
-    · simp [seekCycle,hv,Thorp.QueryTree.succeeds] at hs
-  | succ r ih =>
-    by_cases hv : x ∈ S ∧ x ∉ V
-    · rw [seekCycle,ite_eq_left hv,R.follow_succeeds] at hs
-      rw [seekCycle,ite_eq_left hv,R.follow_raw_charge]
-      exact ih _ _ hs
-    · simp [seekCycle,hv,Thorp.QueryTree.succeeds] at hs
-
-noncomputable def seekCycles (R : RoutingNetwork α ι d) (S : Finset (α)) :
-    List (α × ℕ) → Finset (α) → QueryTree (ι)
-  | [], _ => accept
-  | (a,r) :: L, V => seekCycle R S a r a V (fun V => seekCycles R S L V)
-
-lemma seekCycles_raw_charge (S : Finset (α)) (L : List (α × ℕ))
-    (V : Finset (α)) (ω : ι → Bool)
-    (hs : (seekCycles R S L V).succeeds ω = true) :
-    (seekCycles R S L V).charge ω = L.length * d := by
-  induction L generalizing V with
-  | nil => simp [seekCycles,charge]
-  | cons ar L ih =>
-    obtain ⟨a,r⟩ := ar
-    rw [seekCycles]
-    rw [seekCycle_raw_charge R S a r a V (fun V => seekCycles R S L V) ω (L.length*d)
-      (fun V hs => ih V hs) hs]
-    simp [Nat.add_mul]
-
-lemma cycles_weighted_success (S : Finset (α)) (L : List (α × ℕ)) :
-    finiteMean (fun ω : ι → Bool =>
-      if (seekCycles R S L ∅).succeeds ω then
-        ((2:ℝ)^d)^L.length / (2:ℝ)^(seekCycles R S L ∅).staleCharge ∅ ω else 0) ≤ 1 := by
-  let T := seekCycles R S L ∅
-  have hh := mean_freshCharge_le_one T
-  have he : finiteMean (fun ω : ι → Bool =>
-      if T.succeeds ω then (2:ℝ)^T.freshCharge ∅ ω else 0) =
-      finiteMean (fun ω : ι → Bool =>
-        if T.succeeds ω then ((2:ℝ)^d)^L.length / (2:ℝ)^T.staleCharge ∅ ω else 0) := by
-    apply finiteMean_congr
-    intro ω
-    cases hs : T.succeeds ω
-    · simp
-    · simp only [↓reduceIte]
-      have hc := fresh_add_stale T ∅ ω
-      rw [seekCycles_raw_charge R S L ∅ ω hs] at hc
-      have hp : ((2:ℝ)^d)^L.length = (2:ℝ)^T.freshCharge ∅ ω * (2:ℝ)^T.staleCharge ∅ ω := by
-        rw [←pow_add,hc,←pow_mul,Nat.mul_comm]
-      rw [hp,mul_div_cancel_right₀ _ (pow_ne_zero _ (by norm_num))]
-  rw [← he]
-  exact hh
-
-end QueryTree
-noncomputable def exposedPaths (b : ι → Bool) (V : Finset (α)) :
-    Finset (ι) :=
-  V.biUnion (fun x => R.path b x)
-
-@[simp] lemma exposedPaths_empty (b : ι → Bool) : exposedPaths R b ∅ = ∅ := by
-  classical
-  simp [exposedPaths]
-
-lemma exposedPaths_insert (b : ι → Bool) (V : Finset (α)) (x : α) :
-    exposedPaths R b (insert x V) = exposedPaths R b V ∪ R.path b x := by
-  classical
-  simp [exposedPaths,Finset.union_comm]
-
-noncomputable def terminalCharge (b : ι → Bool) (z : α) (V : Finset (α)) : ℕ :=
-  (R.path b z ∩ exposedPaths R b V).card
-
-namespace QueryTree
-open Thorp.QueryTree
-
-lemma seekCycle_semantics (S : Finset (α)) (a : α) (r : ℕ)
-    (x : α) (V W : Finset (α)) (z : α)
-    (k : Finset (α) → QueryTree (ι)) (ω : ι → Bool)
-    (h : cycleTrace (R.perm ω) S a r x V = some (W,z)) :
-    (seekCycle R S a r x V k).succeeds ω = (k W).succeeds ω ∧
-      (seekCycle R S a r x V k).staleCharge (exposedPaths R (ω) V) ω =
-        (k W).staleCharge (exposedPaths R (ω) W) ω +
-          terminalCharge R (ω) z (W.erase z) := by
-  classical
-  induction r generalizing x V with
-  | zero =>
-    simp only [cycleTrace] at h
-    split_ifs at h with hv
-    obtain ⟨hxS,hxV,hxa⟩ := hv
-    obtain ⟨rfl,rfl⟩ := Prod.mk.inj (Option.some.inj h)
-    simp only [seekCycle,ite_eq_left (show x ∈ S ∧ x ∉ V from ⟨hxS,hxV⟩),R.close_succeeds,ite_eq_left hxa]
-    constructor
-    · trivial
-    · rw [R.close_stale_charge _ _ _ _ _ hxa,exposedPaths_insert,
-        Finset.erase_insert hxV]
-      simp only [terminalCharge]
-  | succ r ih =>
-    simp only [cycleTrace] at h
-    split_ifs at h with hv
-    obtain ⟨hxS,hxV⟩ := hv
-    rw [seekCycle,ite_eq_left ⟨hxS,hxV⟩,R.follow_succeeds,R.follow_stale_charge]
-    rw [← exposedPaths_insert]
-    exact ih _ _ h
-
-lemma seekCycle_orbit (S V : Finset (α)) (z : α)
-    (k : Finset (α) → QueryTree (ι)) (ω : ι → Bool)
-    (hS : orbitSet (R.perm ω) z ⊆ S)
-    (hV : Disjoint V (orbitSet (R.perm ω) z)) :
-    let p := R.perm ω
-    let W := V ∪ orbitSet p z
-    (seekCycle R S (p z) (Function.minimalPeriod p z-1) (p z) V k).succeeds ω = (k W).succeeds ω ∧
-      (seekCycle R S (p z) (Function.minimalPeriod p z-1) (p z) V k).staleCharge
-          (exposedPaths R (ω) V) ω =
-        (k W).staleCharge (exposedPaths R (ω) W) ω +
-          terminalCharge R (ω) z (W.erase z) := by
-  exact seekCycle_semantics _ _ _ _ _ _ _ _ _ _ (cycleTrace_orbit _ _ _ _ hS hV)
-
-end QueryTree
-
-end RoutingNetwork
-end Thorp
-
-end ThorpNine.Harmonic
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Harmonic.RotationMeeting
+
+namespace OAI
+
+namespace ThorpNine.Harmonic
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+section CycleCounting
+variable {d A : ℕ}
+
+lemma assignmentOwner_eq_some {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) (x : Card d) (i : Fin A) :
+    assignmentOwner f x = some i ↔ x ∈ (f.val i).val := by
+  classical
+  unfold assignmentOwner
+  split_ifs with h
+  · constructor
+    · intro hi
+      have hx := Classical.choose_spec h
+      rw [Option.some.inj hi] at hx
+      exact hx
+    · intro hx
+      have he := (f.val (Classical.choose h)).eq_of_mem (f.val i) (Classical.choose_spec h) hx
+      exact congrArg some (f.property.1 he)
+  · constructor
+    · intro hi; contradiction
+    · intro hx; exact (h ⟨i,hx⟩).elim
+
+lemma assignmentOwner_fiber {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) (i : Fin A) : cycleFiber (assignmentOwner f) i = (f.val i).val := by
+  ext x
+  simp only [mem_cycleFiber,assignmentOwner_eq_some]
+
+noncomputable def assignmentToSlots {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) : CycleSlots p S r :=
+  ⟨assignmentOwner f,by
+    refine ⟨?_,?_,?_⟩
+    · intro i
+      rw [assignmentOwner_fiber]
+      exact f.property.2 i
+    · intro i
+      rw [assignmentOwner_fiber]
+      exact (f.val i).property.2
+    · intro i x hx
+      rw [assignmentOwner_fiber]
+      exact (f.val i).orbit_eq ((assignmentOwner_eq_some f x i).mp hx)⟩
+
+noncomputable def cycleSlotsEquivAssignments (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
+    CycleSlots p S r ≃ SlotAssignment p S r where
+  toFun := slotsToAssignment
+  invFun := assignmentToSlots
+  left_inv c := by
+    apply Subtype.ext
+    apply owner_ext
+    intro i
+    rw [assignmentToSlots,assignmentOwner_fiber]
+    rfl
+  right_inv f := by
+    apply Subtype.ext
+    funext i
+    apply Subtype.ext
+    exact assignmentOwner_fiber f i
+
+end CycleCounting
+
+section GroupedCounting
+variable {d A : ℕ}
+
+abbrev LengthClass (r : Fin A → ℕ) := {j : ℕ // j ∈ Finset.univ.image (fun i => r i + 1)}
+
+def slotClass (r : Fin A → ℕ) (i : Fin A) : LengthClass r :=
+  ⟨r i+1, Finset.mem_image.mpr ⟨i, Finset.mem_univ _, rfl⟩⟩
+
+abbrev SlotsOfLength (r : Fin A → ℕ) (j : LengthClass r) := {i : Fin A // slotClass r i = j}
+abbrev CyclesOfLength (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (j : ℕ) :=
+  {C : SelectedCycle p S // C.val.card = j}
+
+noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (j : ℕ) :
+    Fintype (CyclesOfLength p S j) := Fintype.ofFinite _
+
+abbrev GroupedAssignments (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
+  ∀ j : LengthClass r, SlotsOfLength r j ↪ CyclesOfLength p S j.val
+
+noncomputable def assignmentToGroups {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) : GroupedAssignments p S r := fun _j =>
+  { toFun := fun i => ⟨f.val i.val, (f.property.2 i.val).trans (congrArg Subtype.val i.property)⟩
+    inj' := fun _i _k h => Subtype.ext (f.property.1 (congrArg Subtype.val h)) }
+
+noncomputable def groupsEmbedding {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (F : GroupedAssignments p S r) : Fin A ↪ SelectedCycle p S :=
+  ((Equiv.sigmaFiberEquiv (slotClass r)).symm.toEmbedding.trans
+    (Function.Embedding.sigmaMap (Function.Embedding.refl _) F)).trans
+    ((Function.Embedding.sigmaMap (Function.Embedding.subtype _) (fun _ => Function.Embedding.refl _)).trans
+      (Equiv.sigmaFiberEquiv (fun C : SelectedCycle p S => C.val.card)).toEmbedding)
+
+noncomputable def groupsToAssignment {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (F : GroupedAssignments p S r) : SlotAssignment p S r :=
+  ⟨groupsEmbedding F, (groupsEmbedding F).injective,
+    fun i => (F (slotClass r i) ⟨i,rfl⟩).property⟩
+
+noncomputable def assignmentEquivGroups (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
+    SlotAssignment p S r ≃ GroupedAssignments p S r where
+  toFun := assignmentToGroups
+  invFun := groupsToAssignment
+  left_inv f := by
+    apply Subtype.ext
+    rfl
+  right_inv F := by
+    funext j
+    apply Function.Embedding.ext
+    intro i
+    rcases i with ⟨i,hi⟩
+    subst j
+    rfl
+
+lemma card_cycleSlots_factorial (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
+    Fintype.card (CycleSlots p S r) =
+      ∏ j : LengthClass r, (Fintype.card (CyclesOfLength p S j.val)).descFactorial
+        (Fintype.card (SlotsOfLength r j)) := by
+  classical
+  rw [Fintype.card_congr (cycleSlotsEquivAssignments p S r),
+    Fintype.card_congr (assignmentEquivGroups p S r), Fintype.card_pi]
+  simp only [Fintype.card_embedding_eq]
+
+lemma butterfly_factorial_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+       (∏ j : LengthClass r,
+         (Fintype.card (CyclesOfLength (butterflyPerm d (decodeButterfly d ω)) S j.val)).descFactorial
+           (Fintype.card (SlotsOfLength r j)) : ℕ)) ≤
+      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+        ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
+  simpa only [←card_cycleSlots_factorial] using butterfly_cycleSlots_moment S r hr
+
+end GroupedCounting
+
+section AllLengths
+variable {d A : ℕ}
+
+noncomputable def assignmentRelabel (p : Equiv.Perm (Card d)) (S : Finset (Card d))
+    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
+    SlotAssignment p S r ≃ SlotAssignment p S (r ∘ e) where
+  toFun f := ⟨f.val ∘ e, f.property.1.comp e.injective, fun i => f.property.2 (e i)⟩
+  invFun f := ⟨f.val ∘ e.symm, f.property.1.comp e.symm.injective, fun i => by
+    simpa only [Function.comp_apply,Equiv.apply_symm_apply] using f.property.2 (e.symm i)⟩
+  left_inv f := by apply Subtype.ext; funext i; simp
+  right_inv f := by apply Subtype.ext; funext i; simp
+
+lemma card_cycleSlots_relabel (p : Equiv.Perm (Card d)) (S : Finset (Card d))
+    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
+    Fintype.card (CycleSlots p S r) = Fintype.card (CycleSlots p S (r ∘ e)) := by
+  exact Fintype.card_congr ((cycleSlotsEquivAssignments p S r).trans
+    ((assignmentRelabel p S r e).trans (cycleSlotsEquivAssignments p S (r ∘ e)).symm))
+
+lemma butterfly_cycleSlots_moment_all (S : Finset (Card d)) (r : Fin A → ℕ) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) ≤
+        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
+  have h := butterfly_cycleSlots_moment S (r ∘ Tuple.sort r) (Tuple.monotone_sort r)
+  simp_rw [←card_cycleSlots_relabel] at h
+  have hp := Equiv.prod_comp (Tuple.sort r) (fun i : Fin A => (1 / Real.sqrt (r i+1) : ℝ))
+  simpa only [Function.comp_apply,hp] using h
+
+abbrev ShortCycle (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :=
+  {C : SelectedCycle p S // C.val.card ≤ J}
+
+noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
+    Fintype (ShortCycle p S J) := Fintype.ofFinite _
+
+noncomputable def shortCycleLength {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {J : ℕ}
+    (C : ShortCycle p S J) : Fin J :=
+  ⟨C.val.val.card-1, by have := C.val.nonempty.card_pos; have := C.property; omega⟩
+
+lemma shortCycleLength_add_one {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {J : ℕ}
+    (C : ShortCycle p S J) : (shortCycleLength C : ℕ)+1 = C.val.val.card := by
+  have := C.val.nonempty.card_pos
+  simp only [shortCycleLength]
+  omega
+
+noncomputable def shortEmbeddingEquiv (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
+    (Fin A ↪ ShortCycle p S J) ≃
+      Σ r : Fin A → Fin J, SlotAssignment p S (fun i => (r i : ℕ)) where
+  toFun f := ⟨fun i => shortCycleLength (f i),
+    ⟨fun i => (f i).val, fun i j h => f.injective (Subtype.ext h),
+      fun i => (shortCycleLength_add_one (f i)).symm⟩⟩
+  invFun c :=
+    { toFun := fun i => ⟨c.2.val i, by rw [c.2.property.2]; exact (c.1 i).isLt⟩
+      inj' := fun _i _j h => c.2.property.1 (congrArg Subtype.val h) }
+  left_inv f := by
+    apply Function.Embedding.ext
+    intro i
+    rfl
+  right_inv c := by
+    rcases c with ⟨r,f⟩
+    have he : (fun i => shortCycleLength (⟨f.val i, by rw [f.property.2]; exact (r i).isLt⟩ : ShortCycle p S J)) = r := by
+      funext i
+      apply Fin.ext
+      simp only [shortCycleLength,f.property.2,Nat.add_sub_cancel]
+    apply Sigma.ext he
+    apply (Subtype.heq_iff_coe_eq (by
+      intro x
+      change (Function.Injective x ∧ ∀ i, (x i).val.card = (f.val i).val.card - 1 + 1) ↔ _
+      simp only [f.property.2,Nat.add_sub_cancel])).mpr
+    rfl
+
+lemma shortCycle_factorial_eq (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
+    (Fintype.card (ShortCycle p S J)).descFactorial A =
+      ∑ r : Fin A → Fin J, Fintype.card (CycleSlots p S (fun i => (r i : ℕ))) := by
+  classical
+  have hc := Fintype.card_embedding_eq (α := Fin A) (β := ShortCycle p S J)
+  simp only [Fintype.card_fin] at hc
+  rw [←hc, Fintype.card_congr (shortEmbeddingEquiv p S J),Fintype.card_sigma]
+  apply Finset.sum_congr rfl
+  intro r _
+  exact (Fintype.card_congr (cycleSlotsEquivAssignments p S (fun i => (r i : ℕ)))).symm
+
+end AllLengths
+
+lemma factorial_mgf {Ω : Type*} [Fintype Ω] (F : Ω → ℕ) (k : ℕ) (Λ q : ℝ)
+    (hF : ∀ ω, F ω ≤ k) (hΛ : 0 ≤ Λ) (hq : 1 ≤ q)
+    (hm : ∀ a, finiteMean (fun ω => ((F ω).descFactorial a : ℝ)) ≤ Λ^a) :
+    finiteMean (fun ω => q^(F ω)) ≤ Real.exp ((q-1)*Λ) := by
+  have hc (n a : ℕ) : (n.choose a : ℝ) = (n.descFactorial a : ℝ) / (a.factorial : ℝ) := by
+    rw [Nat.descFactorial_eq_factorial_mul_choose,Nat.cast_mul]
+    exact (mul_div_cancel_left₀ _ (by exact_mod_cast Nat.factorial_ne_zero a)).symm
+  have hex (ω : Ω) : q^(F ω) = ∑ a ∈ Finset.range (k+1),
+      ((F ω).descFactorial a : ℝ) * ((q-1)^a / (a.factorial : ℝ)) := by
+    calc
+      _ = ((q-1)+1)^(F ω) := by congr 1; ring
+      _ = ∑ a ∈ Finset.range (F ω+1), (q-1)^a * (F ω).choose a := by
+        rw [add_pow]
+        simp only [one_pow,mul_one]
+      _ = ∑ a ∈ Finset.range (k+1), (q-1)^a * (F ω).choose a := by
+        apply Finset.sum_subset (Finset.range_mono (Nat.add_le_add_right (hF ω) _))
+        intro a _ ha
+        have hn : F ω < a := by have := (Finset.mem_range.not.mp ha); omega
+        rw [Nat.choose_eq_zero_of_lt hn,Nat.cast_zero,mul_zero]
+      _ = _ := by
+        apply Finset.sum_congr rfl
+        intro a _
+        rw [hc]
+        ring
+  simp_rw [hex,finiteMean_sum,finiteMean_mul_const]
+  calc
+    _ ≤ ∑ a ∈ Finset.range (k+1), Λ^a * ((q-1)^a / (a.factorial : ℝ)) := by
+      apply Finset.sum_le_sum
+      intro a _
+      exact mul_le_mul_of_nonneg_right (hm a) (by positivity)
+    _ = ∑ a ∈ Finset.range (k+1), ((q-1)*Λ)^a / (a.factorial : ℝ) := by
+      apply Finset.sum_congr rfl
+      intro a _
+      rw [mul_pow]
+      ring
+    _ ≤ _ := Real.sum_le_exp_of_nonneg (mul_nonneg (sub_nonneg.mpr hq) hΛ) _
+
+lemma inv_sqrt_step (j : ℕ) :
+    1 / Real.sqrt (j+1) ≤ 2*(Real.sqrt (j+1)-Real.sqrt j) := by
+  have hp : 0 < Real.sqrt (j+1) := Real.sqrt_pos.mpr (by positivity)
+  apply (div_le_iff₀ hp).mpr
+  have h1 := Real.sq_sqrt (show (0:ℝ) ≤ j+1 by positivity)
+  have h0 := Real.sq_sqrt (show (0:ℝ) ≤ j by positivity)
+  nlinarith [sq_nonneg (Real.sqrt (j+1)-Real.sqrt j)]
+
+lemma sum_inv_sqrt (J : ℕ) :
+    (∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1)) ≤ 2*Real.sqrt J := by
+  have h (n : ℕ) : (∑ j ∈ Finset.range n, 1 / Real.sqrt (j+1)) ≤ 2*Real.sqrt n := by
+    induction n with
+    | zero => simp
+    | succ n ih =>
+      rw [Finset.sum_range_succ]
+      have hs := inv_sqrt_step n
+      push_cast
+      linarith
+  rw [Fin.sum_univ_eq_sum_range (fun j : ℕ => (1 / Real.sqrt (j+1) : ℝ)) J]
+  exact h J
+section CycleMGF
+variable {d : ℕ}
+
+lemma butterfly_shortCycle_moment (S : Finset (Card d)) (J A : ℕ) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      ((Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J)).descFactorial A : ℝ)) ≤
+      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))^A := by
+  classical
+  simp_rw [shortCycle_factorial_eq,Nat.cast_sum,finiteMean_sum]
+  calc
+    _ ≤ ∑ r : Fin A → Fin J,
+        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+          ∏ i : Fin A, (1 / Real.sqrt ((r i : ℕ)+1)) := by
+      apply Finset.sum_le_sum
+      intro r _
+      exact butterfly_cycleSlots_moment_all S (fun i => (r i : ℕ))
+    _ = _ := by
+      rw [←Finset.mul_sum,mul_pow]
+      congr 1
+      rw [←Fintype.prod_sum (fun (_ : Fin A) (j : Fin J) => (1 / Real.sqrt ((j:ℕ)+1) : ℝ))]
+      simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fin]
+
+noncomputable def cycleMass (p : Equiv.Perm (Card d)) (S : Finset (Card d)) : ℕ :=
+  ∑ C : SelectedCycle p S, C.val.card
+
+lemma cycleMass_le (p : Equiv.Perm (Card d)) (S : Finset (Card d)) : cycleMass p S ≤ S.card := by
+  classical
+  let f : (Σ C : SelectedCycle p S, {x : Card d // x ∈ C.val}) → S :=
+    fun c => ⟨c.2.val,c.1.property.2 c.2.property⟩
+  have hi : Function.Injective f := by
+    intro c e h
+    have hx : c.2.val = e.2.val := congrArg Subtype.val h
+    have hC : c.1 = e.1 := c.1.eq_of_mem e.1 c.2.property (hx ▸ e.2.property)
+    rcases c with ⟨c,x⟩
+    rcases e with ⟨e,y⟩
+    dsimp only at hC
+    subst e
+    have he : x=y := Subtype.ext hx
+    cases he
+    rfl
+  have hh := Fintype.card_le_of_injective f hi
+  simpa only [Fintype.card_sigma,Fintype.card_coe,cycleMass] using hh
+
+lemma shortCycle_card_le (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
+    Fintype.card (ShortCycle p S J) ≤ S.card := by
+  classical
+  calc
+    _ ≤ Fintype.card (SelectedCycle p S) := Fintype.card_subtype_le _
+    _ = ∑ _ : SelectedCycle p S, 1 := by simp
+    _ ≤ cycleMass p S := Finset.sum_le_sum (fun C _ => C.nonempty.card_pos)
+    _ ≤ _ := cycleMass_le p S
+
+lemma butterfly_shortCycle_mgf (S : Finset (Card d)) (J : ℕ) (q : ℝ) (hq : 1 ≤ q) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      q^(Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J))) ≤
+      Real.exp ((q-1)*((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))) := by
+  apply factorial_mgf _ S.card _ q (fun ω => shortCycle_card_le _ _ _)
+  · positivity
+  · exact hq
+  · exact butterfly_shortCycle_moment S J
+
+end CycleMGF
+
+section FullCycleMGF
+variable {d : ℕ}
+
+abbrev LongCycle (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :=
+  {C : SelectedCycle p S // ¬ C.val.card ≤ J}
+noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
+    Fintype (LongCycle p S J) := Fintype.ofFinite _
+
+lemma selectedCycle_card_split (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
+    Fintype.card (SelectedCycle p S) =
+      Fintype.card (ShortCycle p S J) + Fintype.card (LongCycle p S J) := by
+  classical
+  rw [←Fintype.card_congr (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J)),Fintype.card_sum]
+
+lemma longCycle_card_bound (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
+    J * Fintype.card (LongCycle p S J) ≤ S.card := by
+  classical
+  have hs := Equiv.sum_comp (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J))
+    (fun C : SelectedCycle p S => C.val.card)
+  rw [Fintype.sum_sum_type] at hs
+  have h0 : J * Fintype.card (LongCycle p S J) ≤
+      ∑ C : LongCycle p S J, C.val.val.card := by
+    calc
+      _ = ∑ _C : LongCycle p S J, J := by simp [mul_comm]
+      _ ≤ _ := Finset.sum_le_sum (fun C _ => Nat.le_of_lt (Nat.lt_of_not_ge C.property))
+  have hh := cycleMass_le p S
+  unfold cycleMass at hh
+  rw [←hs] at hh
+  exact h0.trans ((Nat.le_add_left _ _).trans hh)
+
+lemma finiteMean_pos {Ω : Type*} [Fintype Ω] [Nonempty Ω] {f : Ω → ℝ}
+    (h : ∀ ω, 0 < f ω) : 0 < finiteMean f := by
+  apply div_pos
+  · exact Finset.sum_pos (fun ω _ => h ω) Finset.univ_nonempty
+  · exact Nat.cast_pos.mpr Fintype.card_pos
+
+lemma longCycle_pow_bound (p : Equiv.Perm (Card d)) (S : Finset (Card d))
+    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    q^(Fintype.card (LongCycle p S J)) ≤ Real.exp ((S.card : ℝ)/J * Real.log q) := by
+  have hq0 : 0 < q := lt_of_lt_of_le zero_lt_one hq
+  have hc : (Fintype.card (LongCycle p S J) : ℝ) ≤ (S.card : ℝ)/J := by
+    apply (le_div_iff₀ (by exact_mod_cast hJ)).mpr
+    have hh := longCycle_card_bound p S J
+    exact_mod_cast (by simpa only [mul_comm] using hh)
+  calc
+    _ = Real.exp ((Fintype.card (LongCycle p S J) : ℝ)*Real.log q) := by
+      rw [Real.exp_nat_mul,Real.exp_log hq0]
+    _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right hc (Real.log_nonneg hq))
+
+lemma butterfly_cycle_log_mgf (S : Finset (Card d)) (J : ℕ) (hJ : 0 < J)
+    (q : ℝ) (hq : 1 ≤ q) :
+    Real.log (finiteMean (fun ω : SwitchIndex d → Bool =>
+      q^(Fintype.card (SelectedCycle (butterflyPerm d (decodeButterfly d ω)) S)))) ≤
+      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
+  let lam : ℝ := (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1)
+  let c : ℝ := (S.card : ℝ)/J * Real.log q
+  have hnum : (q-1)*lam ≤ 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
+    have hsum := sum_inv_sqrt J
+    have hp : 0 ≤ (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) := by positivity
+    have h0 : 0 ≤ 2*Real.sqrt J := by positivity
+    have hh := mul_le_mul_of_nonneg_left hsum hp
+    have hq1 : 0 ≤ q-1 := sub_nonneg.mpr hq
+    calc
+      _ ≤ (q-1)*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
+        mul_le_mul_of_nonneg_left hh hq1
+      _ ≤ q*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
+        mul_le_mul_of_nonneg_right (by linarith) (mul_nonneg hp h0)
+      _ = _ := by ring
+  have hm : finiteMean (fun ω : SwitchIndex d → Bool =>
+      q^(Fintype.card (SelectedCycle (butterflyPerm d (decodeButterfly d ω)) S))) ≤
+      Real.exp (c+(q-1)*lam) := by
+    calc
+      _ ≤ finiteMean (fun ω : SwitchIndex d → Bool =>
+          q^(Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J))*Real.exp c) := by
+        apply finiteMean_mono
+        intro ω
+        rw [selectedCycle_card_split _ _ J,pow_add]
+        exact mul_le_mul_of_nonneg_left (longCycle_pow_bound _ S J hJ q hq) (by positivity)
+      _ = finiteMean (fun ω : SwitchIndex d → Bool =>
+          q^(Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J))) * Real.exp c :=
+        finiteMean_mul_const _ _
+      _ ≤ Real.exp ((q-1)*lam) * Real.exp c :=
+        mul_le_mul_of_nonneg_right (butterfly_shortCycle_mgf S J q hq) (Real.exp_nonneg _)
+      _ = _ := by rw [←Real.exp_add,add_comm]
+  have hpos : 0 < finiteMean (fun ω : SwitchIndex d → Bool =>
+      q^(Fintype.card (SelectedCycle (butterflyPerm d (decodeButterfly d ω)) S))) := by
+    apply finiteMean_pos
+    intro ω
+    positivity
+  have hlog := Real.log_le_log hpos hm
+  rw [Real.log_exp] at hlog
+  dsimp [c] at hlog
+  linarith
+
+end FullCycleMGF
+
+structure RoutingNetwork (α ι : Type*) [Fintype α] [DecidableEq α]
+    [Fintype ι] [DecidableEq ι] (d : ℕ) where
+  perm : (ι → Bool) → Equiv.Perm α
+  path : (ι → Bool) → α → Finset ι
+  users : (ι → Bool) → ι → α × α
+  users_ne : ∀ ω i, (users ω i).1 ≠ (users ω i).2
+  mem_path : ∀ ω x i, i ∈ path ω x ↔ (users ω i).1 = x ∨ (users ω i).2 = x
+  card_path : ∀ ω x, (path ω x).card = d
+  entropy : ∀ ω (S : Finset α),
+    (∑ i : ι, if (users ω i).1 ∈ S ∧ (users ω i).2 ∈ S then (1:ℝ) else 0) ≤
+      (S.card : ℝ) * Real.log S.card / (2 * Real.log 2)
+  follow : α → (α → QueryTree ι) → QueryTree ι
+  close : α → α → QueryTree ι → QueryTree ι
+  follow_succeeds : ∀ x k ω, (follow x k).succeeds ω = (k (perm ω x)).succeeds ω
+  follow_raw_charge : ∀ x k ω, (follow x k).charge ω = (k (perm ω x)).charge ω
+  follow_stale_charge : ∀ x k R ω,
+    (follow x k).staleCharge R ω = (k (perm ω x)).staleCharge (R ∪ path ω x) ω
+  close_succeeds : ∀ x y k ω,
+    (close x y k).succeeds ω = if perm ω x = y then k.succeeds ω else false
+  close_raw_charge : ∀ x y k ω, perm ω x = y → (close x y k).charge ω = k.charge ω + d
+  close_stale_charge : ∀ x y k R ω, perm ω x = y →
+    (close x y k).staleCharge R ω = k.staleCharge (R ∪ path ω x) ω + (path ω x ∩ R).card
+
+namespace RoutingNetwork
+variable {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι] [DecidableEq ι]
+  {d : ℕ} (R : RoutingNetwork α ι d)
+
+lemma card_switches_using (ω : ι → Bool) (x : α) :
+    (Finset.univ.filter (fun i : ι =>
+      (R.users ω i).1 = x ∨ (R.users ω i).2 = x)).card = d := by
+  classical
+  have he : Finset.univ.filter (fun i : ι =>
+      (R.users ω i).1 = x ∨ (R.users ω i).2 = x) = R.path ω x := by
+    ext i
+    simp only [Finset.mem_filter,Finset.mem_univ,true_and,R.mem_path]
+  rw [he,R.card_path]
+lemma switch_incidence_sum (b : ι → Bool) (w : α → ℝ) :
+    ∑ i : ι, (w (R.users b i).1 + w (R.users b i).2) =
+      (d : ℝ) * ∑ x : α, w x := by
+  classical
+  calc
+    _ = ∑ i : ι, ∑ x : α,
+        if (R.users b i).1 = x ∨ (R.users b i).2 = x then w x else 0 := by
+      apply Finset.sum_congr rfl
+      intro i _
+      rw [Finset.sum_ite]
+      simp only [Finset.sum_const_zero, add_zero]
+      have he : Finset.univ.filter (fun x : α =>
+          (R.users b i).1 = x ∨ (R.users b i).2 = x) =
+          {(R.users b i).1, (R.users b i).2} := by
+        ext x; simp [eq_comm]
+      rw [he, Finset.sum_pair (R.users_ne b i)]
+    _ = ∑ x : α, ∑ i : ι,
+        if (R.users b i).1 = x ∨ (R.users b i).2 = x then w x else 0 :=
+      Finset.sum_comm
+    _ = ∑ x : α, (d : ℝ) * w x := by
+      apply Finset.sum_congr rfl
+      intro x _
+      rw [Finset.sum_ite]
+      simp only [Finset.sum_const_zero, add_zero, Finset.sum_const, nsmul_eq_mul,
+        card_switches_using]
+    _ = _ := by rw [Finset.mul_sum]
+
+lemma weighted_switch_min_bound (b : ι → Bool) (w : α → ℝ) :
+    ∑ i : ι, min (w (R.users b i).1) (w (R.users b i).2) ≤
+      (d : ℝ) / 2 * ∑ x : α, w x := by
+  have h : 2 * (∑ i : ι,
+      min (w (R.users b i).1) (w (R.users b i).2)) ≤
+      ∑ i : ι, (w (R.users b i).1 + w (R.users b i).2) := by
+    rw [Finset.mul_sum]
+    apply Finset.sum_le_sum
+    intro i _
+    linarith [min_le_left (w (R.users b i).1) (w (R.users b i).2),
+      min_le_right (w (R.users b i).1) (w (R.users b i).2)]
+  rw [switch_incidence_sum] at h
+  linarith
+section Rotation
+variable {A : ℕ}
+
+def cycleFiber (o : α → Option (Fin A)) (i : Fin A) : Finset (α) := by
+  classical
+  exact Finset.univ.filter (fun x => o x = some i)
+
+omit [DecidableEq α] in
+@[simp] lemma mem_cycleFiber (o : α → Option (Fin A)) (i : Fin A) (x : α) :
+    x ∈ cycleFiber o i ↔ o x = some i := by
+  classical
+  simp [cycleFiber]
+
+abbrev Terminal (o : α → Option (Fin A)) := ∀ i : Fin A, {x : α // o x = some i}
+
+noncomputable def slotWeight (o : α → Option (Fin A)) (i : Fin A) : ℝ :=
+  1 / (cycleFiber o i).card
+
+noncomputable def pathWeight (o : α → Option (Fin A)) (x : α) : ℝ :=
+  match o x with
+  | none => 0
+  | some i => slotWeight o i
+
+noncomputable def withinWeight (o : α → Option (Fin A)) (u v : α) : ℝ :=
+  match o u, o v with
+  | some i, some j => if i = j then slotWeight o i else 0
+  | _, _ => 0
+
+noncomputable def rotationMeeting (o : α → Option (Fin A)) (z : Terminal o)
+    (u v : α) : ℝ :=
+  match o u, o v with
+  | some i, some j => if i < j then (if (z j).val = v then 1 else 0)
+      else if j < i then (if (z i).val = u then 1 else 0)
+      else (if (z i).val = u then 1 else 0) + (if (z i).val = v then 1 else 0)
+  | _, _ => 0
+
+omit [DecidableEq α] in
+lemma card_cycleFiber (o : α → Option (Fin A)) (i : Fin A) :
+    Fintype.card {x : α // o x = some i} = (cycleFiber o i).card := by
+  classical
+  simp [cycleFiber,Fintype.card_subtype]
+
+lemma terminal_mean {o : α → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
+    (i : Fin A) (x : α) (hx : o x = some i) :
+    finiteMean (fun z : Terminal o => if (z i).val = x then (1:ℝ) else 0) = slotWeight o i := by
+  classical
+  have hev := finiteMean_pi_eval (A := fun i : Fin A => {x : α // o x = some i})
+    i (fun z => if z.val = x then (1:ℝ) else 0)
+  rw [hev]
+  have he : (fun z : {x : α // o x = some i} => if z.val = x then (1:ℝ) else 0) =
+      (fun z => if z = ⟨x,hx⟩ then 1 else 0) := by
+    funext z
+    simp only [Subtype.ext_iff]
+  rw [he,finiteMean_eq_indicator,card_cycleFiber,slotWeight]
+
+omit [DecidableEq α] in
+lemma slotWeight_antitone (o : α → Option (Fin A))
+    (hn : ∀ i, (cycleFiber o i).Nonempty)
+    (hm : Monotone (fun i => (cycleFiber o i).card)) : Antitone (slotWeight o) := by
+  intro i j hij
+  unfold slotWeight
+  apply one_div_le_one_div_of_le
+  · exact_mod_cast Finset.card_pos.mpr (hn i)
+  · exact_mod_cast hm hij
+
+lemma rotationMeeting_mean {o : α → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
+    (hm : Monotone (fun i => (cycleFiber o i).card)) (u v : α) :
+    finiteMean (fun z : Terminal o => rotationMeeting o z u v) =
+      min (pathWeight o u) (pathWeight o v) + withinWeight o u v := by
+  classical
+  have hn : ∀ i, (cycleFiber o i).Nonempty := by
+    intro i
+    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : α // o x = some i})
+    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
+  have hw := slotWeight_antitone o hn hm
+  cases hu : o u with
+  | none =>
+    cases hv : o v with
+    | none => simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean]
+    | some j =>
+      have hj : 0 ≤ slotWeight o j := by unfold slotWeight; positivity
+      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_left hj]
+  | some i =>
+    cases hv : o v with
+    | none =>
+      have hi : 0 ≤ slotWeight o i := by unfold slotWeight; positivity
+      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_right hi]
+    | some j =>
+      simp only [rotationMeeting,pathWeight,withinWeight,hu,hv]
+      rcases lt_trichotomy i j with hij | rfl | hji
+      · simp only [hij,↓reduceIte,ne_of_lt hij]
+        rw [terminal_mean j v hv,min_eq_right (hw (le_of_lt hij)),add_zero]
+      · simp only [lt_self_iff_false,↓reduceIte,min_self]
+        rw [finiteMean_add,terminal_mean i u hu,terminal_mean i v hv]
+      · simp only [not_lt_of_ge (le_of_lt hji),hji,↓reduceIte,ne_of_gt hji]
+        rw [terminal_mean i u hu,min_eq_left (hw (le_of_lt hji)),add_zero]
+
+omit [DecidableEq α] in
+lemma pathWeight_sum (o : α → Option (Fin A)) (hn : ∀ i, (cycleFiber o i).Nonempty) :
+    ∑ x : α, pathWeight o x = (A : ℝ) := by
+  classical
+  have he (x : α) : pathWeight o x =
+      ∑ i : Fin A, if o x = some i then slotWeight o i else 0 := by
+    cases hx : o x with
+    | none => simp [pathWeight,hx]
+    | some j => simp [pathWeight,hx]
+  simp only [he]
+  rw [Finset.sum_comm]
+  have hf (i : Fin A) : (∑ x : α, if o x = some i then slotWeight o i else 0) = 1 := by
+    rw [← Finset.sum_filter]
+    change (∑ _x ∈ cycleFiber o i, slotWeight o i) = 1
+    simp only [Finset.sum_const,nsmul_eq_mul,slotWeight]
+    exact mul_one_div_cancel (by exact_mod_cast (Finset.card_pos.mpr (hn i)).ne')
+  simp only [hf,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,mul_one]
+
+lemma withinWeight_sum_bound (o : α → Option (Fin A))
+    (hn : ∀ i, (cycleFiber o i).Nonempty) (b : ι → Bool) :
+    (∑ s : ι, withinWeight o (R.users b s).1 (R.users b s).2) ≤
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
+  classical
+  have he (u v : α) : withinWeight o u v =
+      ∑ i : Fin A, (if u ∈ cycleFiber o i ∧ v ∈ cycleFiber o i then (1:ℝ) else 0) * slotWeight o i := by
+    simp only [mem_cycleFiber]
+    cases hu : o u <;> cases hv : o v <;> simp [withinWeight,hu,hv,ite_and,eq_comm]
+  simp only [he]
+  rw [Finset.sum_comm]
+  apply Finset.sum_le_sum
+  intro i _
+  rw [← Finset.sum_mul]
+  have hi := R.entropy b (cycleFiber o i)
+  have hpos : (0:ℝ) < (cycleFiber o i).card := by exact_mod_cast Finset.card_pos.mpr (hn i)
+  have hmul := mul_le_mul_of_nonneg_right hi (show 0 ≤ slotWeight o i by unfold slotWeight; positivity)
+  calc
+    _ ≤ ((cycleFiber o i).card : ℝ) * Real.log (cycleFiber o i).card /
+        (2 * Real.log 2) * slotWeight o i := hmul
+    _ = _ := by unfold slotWeight; field_simp
+
+noncomputable def rotationCharge (o : α → Option (Fin A)) (b : ι → Bool)
+    (z : Terminal o) : ℝ :=
+  ∑ s : ι, rotationMeeting o z (R.users b s).1 (R.users b s).2
+
+lemma rotationCharge_mean_bound {o : α → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
+    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : ι → Bool) :
+    finiteMean (rotationCharge R o b) ≤ (A:ℝ) * d / 2 +
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
+  classical
+  have hn : ∀ i, (cycleFiber o i).Nonempty := by
+    intro i
+    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : α // o x = some i})
+    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
+  unfold rotationCharge
+  rw [finiteMean_sum]
+  simp only [rotationMeeting_mean hm,Finset.sum_add_distrib]
+  have h₁ := weighted_switch_min_bound R b (pathWeight o)
+  rw [pathWeight_sum o hn] at h₁
+  have h₂ := withinWeight_sum_bound R o hn b
+  linarith
+
+lemma rotation_exp_bound {o : α → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
+    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : ι → Bool) :
+    Real.exp (-Real.log 2 * ((A:ℝ)*d/2 +
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2))) ≤
+      finiteMean (fun z : Terminal o => Real.exp (-Real.log 2 * rotationCharge R o b z)) := by
+  have hb := rotationCharge_mean_bound R hm b
+  have hlog : 0 ≤ Real.log 2 := Real.log_nonneg (by norm_num)
+  have hm' : -Real.log 2 * ((A:ℝ)*d/2 +
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2)) ≤
+      finiteMean (fun z : Terminal o => -Real.log 2 * rotationCharge R o b z) := by
+    have he := finiteMean_mul_const (rotationCharge R o b) (-Real.log 2)
+    simp only [mul_comm] at he
+    rw [he]
+    exact mul_le_mul_of_nonpos_left hb (neg_nonpos.mpr hlog)
+  exact (Real.exp_le_exp.mpr hm').trans (exp_finiteMean_le _)
+
+end Rotation
+
+namespace QueryTree
+open Thorp.QueryTree
+
+noncomputable def seekCycle (R : RoutingNetwork α ι d) (S : Finset (α)) (a : α) :
+    ℕ → α → Finset (α) → (Finset (α) → QueryTree (ι)) →
+      QueryTree (ι)
+  | 0, x, V, k => if x ∈ S ∧ x ∉ V then R.close x a (k (insert x V)) else reject
+  | r + 1, x, V, k => if x ∈ S ∧ x ∉ V then R.follow x
+      (fun y => seekCycle R S a r y (insert x V) k) else reject
+
+lemma seekCycle_raw_charge (S : Finset (α)) (a : α) (r : ℕ)
+    (x : α) (V : Finset (α)) (k : Finset (α) → QueryTree (ι))
+    (ω : ι → Bool) (K : ℕ)
+    (hK : ∀ V, (k V).succeeds ω = true → (k V).charge ω = K)
+    (hs : (seekCycle R S a r x V k).succeeds ω = true) :
+    (seekCycle R S a r x V k).charge ω = K + d := by
+  induction r generalizing x V with
+  | zero =>
+    by_cases hv : x ∈ S ∧ x ∉ V
+    · rw [seekCycle,ite_eq_left hv,R.close_succeeds] at hs
+      split_ifs at hs with he
+      · rw [seekCycle,ite_eq_left hv,R.close_raw_charge _ _ _ _ he,hK _ hs]
+    · simp [seekCycle,hv,Thorp.QueryTree.succeeds] at hs
+  | succ r ih =>
+    by_cases hv : x ∈ S ∧ x ∉ V
+    · rw [seekCycle,ite_eq_left hv,R.follow_succeeds] at hs
+      rw [seekCycle,ite_eq_left hv,R.follow_raw_charge]
+      exact ih _ _ hs
+    · simp [seekCycle,hv,Thorp.QueryTree.succeeds] at hs
+
+noncomputable def seekCycles (R : RoutingNetwork α ι d) (S : Finset (α)) :
+    List (α × ℕ) → Finset (α) → QueryTree (ι)
+  | [], _ => accept
+  | (a,r) :: L, V => seekCycle R S a r a V (fun V => seekCycles R S L V)
+
+lemma seekCycles_raw_charge (S : Finset (α)) (L : List (α × ℕ))
+    (V : Finset (α)) (ω : ι → Bool)
+    (hs : (seekCycles R S L V).succeeds ω = true) :
+    (seekCycles R S L V).charge ω = L.length * d := by
+  induction L generalizing V with
+  | nil => simp [seekCycles,charge]
+  | cons ar L ih =>
+    obtain ⟨a,r⟩ := ar
+    rw [seekCycles]
+    rw [seekCycle_raw_charge R S a r a V (fun V => seekCycles R S L V) ω (L.length*d)
+      (fun V hs => ih V hs) hs]
+    simp [Nat.add_mul]
+
+lemma cycles_weighted_success (S : Finset (α)) (L : List (α × ℕ)) :
+    finiteMean (fun ω : ι → Bool =>
+      if (seekCycles R S L ∅).succeeds ω then
+        ((2:ℝ)^d)^L.length / (2:ℝ)^(seekCycles R S L ∅).staleCharge ∅ ω else 0) ≤ 1 := by
+  let T := seekCycles R S L ∅
+  have hh := mean_freshCharge_le_one T
+  have he : finiteMean (fun ω : ι → Bool =>
+      if T.succeeds ω then (2:ℝ)^T.freshCharge ∅ ω else 0) =
+      finiteMean (fun ω : ι → Bool =>
+        if T.succeeds ω then ((2:ℝ)^d)^L.length / (2:ℝ)^T.staleCharge ∅ ω else 0) := by
+    apply finiteMean_congr
+    intro ω
+    cases hs : T.succeeds ω
+    · simp
+    · simp only [↓reduceIte]
+      have hc := fresh_add_stale T ∅ ω
+      rw [seekCycles_raw_charge R S L ∅ ω hs] at hc
+      have hp : ((2:ℝ)^d)^L.length = (2:ℝ)^T.freshCharge ∅ ω * (2:ℝ)^T.staleCharge ∅ ω := by
+        rw [←pow_add,hc,←pow_mul,Nat.mul_comm]
+      rw [hp,mul_div_cancel_right₀ _ (pow_ne_zero _ (by norm_num))]
+  rw [← he]
+  exact hh
+
+end QueryTree
+noncomputable def exposedPaths (b : ι → Bool) (V : Finset (α)) :
+    Finset (ι) :=
+  V.biUnion (fun x => R.path b x)
+
+@[simp] lemma exposedPaths_empty (b : ι → Bool) : exposedPaths R b ∅ = ∅ := by
+  classical
+  simp [exposedPaths]
+
+lemma exposedPaths_insert (b : ι → Bool) (V : Finset (α)) (x : α) :
+    exposedPaths R b (insert x V) = exposedPaths R b V ∪ R.path b x := by
+  classical
+  simp [exposedPaths,Finset.union_comm]
+
+noncomputable def terminalCharge (b : ι → Bool) (z : α) (V : Finset (α)) : ℕ :=
+  (R.path b z ∩ exposedPaths R b V).card
+
+namespace QueryTree
+open Thorp.QueryTree
+
+lemma seekCycle_semantics (S : Finset (α)) (a : α) (r : ℕ)
+    (x : α) (V W : Finset (α)) (z : α)
+    (k : Finset (α) → QueryTree (ι)) (ω : ι → Bool)
+    (h : cycleTrace (R.perm ω) S a r x V = some (W,z)) :
+    (seekCycle R S a r x V k).succeeds ω = (k W).succeeds ω ∧
+      (seekCycle R S a r x V k).staleCharge (exposedPaths R (ω) V) ω =
+        (k W).staleCharge (exposedPaths R (ω) W) ω +
+          terminalCharge R (ω) z (W.erase z) := by
+  classical
+  induction r generalizing x V with
+  | zero =>
+    simp only [cycleTrace] at h
+    split_ifs at h with hv
+    obtain ⟨hxS,hxV,hxa⟩ := hv
+    obtain ⟨rfl,rfl⟩ := Prod.mk.inj (Option.some.inj h)
+    simp only [seekCycle,ite_eq_left (show x ∈ S ∧ x ∉ V from ⟨hxS,hxV⟩),R.close_succeeds,ite_eq_left hxa]
+    constructor
+    · trivial
+    · rw [R.close_stale_charge _ _ _ _ _ hxa,exposedPaths_insert,
+        Finset.erase_insert hxV]
+      simp only [terminalCharge]
+  | succ r ih =>
+    simp only [cycleTrace] at h
+    split_ifs at h with hv
+    obtain ⟨hxS,hxV⟩ := hv
+    rw [seekCycle,ite_eq_left ⟨hxS,hxV⟩,R.follow_succeeds,R.follow_stale_charge]
+    rw [← exposedPaths_insert]
+    exact ih _ _ h
+
+lemma seekCycle_orbit (S V : Finset (α)) (z : α)
+    (k : Finset (α) → QueryTree (ι)) (ω : ι → Bool)
+    (hS : orbitSet (R.perm ω) z ⊆ S)
+    (hV : Disjoint V (orbitSet (R.perm ω) z)) :
+    let p := R.perm ω
+    let W := V ∪ orbitSet p z
+    (seekCycle R S (p z) (Function.minimalPeriod p z-1) (p z) V k).succeeds ω = (k W).succeeds ω ∧
+      (seekCycle R S (p z) (Function.minimalPeriod p z-1) (p z) V k).staleCharge
+          (exposedPaths R (ω) V) ω =
+        (k W).staleCharge (exposedPaths R (ω) W) ω +
+          terminalCharge R (ω) z (W.erase z) := by
+  exact seekCycle_semantics _ _ _ _ _ _ _ _ _ _ (cycleTrace_orbit _ _ _ _ hS hV)
+
+end QueryTree
+
+end RoutingNetwork
+end Thorp
+
+end ThorpNine.Harmonic
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Harmonic/AverageMap.lean b/lean/OAI/Probability/ThorpRouting/Harmonic/AverageMap.lean
index 438753c4b4c3509f361cd2ea981781acef426f4b..2ec82cbd4239a7bbb61c1e4476a2ae95f52bc927 100644
--- a/lean/OAI/Probability/ThorpRouting/Harmonic/AverageMap.lean
+++ b/lean/OAI/Probability/ThorpRouting/Harmonic/AverageMap.lean
@@ -1,883 +1,883 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Harmonic.Shapes
-
-namespace OAI
-
-namespace ThorpNine.Harmonic
-
-namespace Thorp.UnitaryFinite
-open scoped BigOperators ComplexConjugate Classical
-variable {G : Type*} [Group G] [Fintype G]
-variable {V W : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-  [NormedAddCommGroup W] [InnerProductSpace ℂ W] [FiniteDimensional ℂ W]
-
-noncomputable def averageMap (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
-    (A : V →ₗ[ℂ] W) : Representation.IntertwiningMap ρ σ where
-  toLinearMap := ∑ g, (σ g).comp (A.comp (ρ g⁻¹))
-  isIntertwining' h := by
-    ext v
-    simp only [LinearMap.comp_apply,LinearMap.sum_apply,map_sum]
-    apply Fintype.sum_equiv (Equiv.mulLeft h⁻¹)
-    intro g
-    simp only [Equiv.coe_mulLeft,mul_inv_rev,inv_inv,map_mul,Module.End.mul_apply]
-    have hh (w : W) : σ h (σ h⁻¹ w)=w := by
-      change (σ h*σ h⁻¹) w=w
-      rw [←map_mul,mul_inv_cancel,map_one]
-      rfl
-    rw [hh]
-
-omit [FiniteDimensional ℂ V] in
-lemma averageMap_self_trace (ρ : Representation ℂ G V) (A : Module.End ℂ V) :
-    LinearMap.trace ℂ V (averageMap ρ ρ A).toLinearMap =
-      (Fintype.card G:ℂ) * LinearMap.trace ℂ V A := by
-  change LinearMap.trace ℂ V (∑ g, ρ g * A * ρ g⁻¹) = _
-  rw [map_sum]
-  have hj (g : G) : LinearMap.trace ℂ V (ρ g*A*ρ g⁻¹) = LinearMap.trace ℂ V A := by
-    rw [LinearMap.trace_mul_cycle,←map_mul,inv_mul_cancel,map_one,one_mul]
-  simp only [hj,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
-
-lemma averageMap_scalar (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
-    (A : Module.End ℂ V) :
-    ∃ c : ℂ, (averageMap ρ ρ A).toLinearMap = c • LinearMap.id ∧
-      c * (Module.finrank ℂ V:ℂ) = (Fintype.card G:ℂ)*LinearMap.trace ℂ V A := by
-  obtain ⟨c,hc⟩ := (Representation.IsIrreducible.algebraMap_intertwiningMap_bijective_of_isAlgClosed (ρ := ρ)).surjective (averageMap ρ ρ A)
-  have he : (averageMap ρ ρ A).toLinearMap = c • LinearMap.id := by
-    rw [←hc]
-    rfl
-  refine ⟨c,he,?_⟩
-  rw [←averageMap_self_trace ρ A,he,map_smul,LinearMap.trace_id,smul_eq_mul]
-
-theorem coefficient_orthogonality (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
-    (hρ : IsUnitary ρ) (u v w z : V) :
-    (Module.finrank ℂ V:ℂ) * ∑ g, inner ℂ u (ρ g v) * inner ℂ (ρ g w) z =
-      (Fintype.card G:ℂ) * inner ℂ w v * inner ℂ u z := by
-  obtain ⟨c,hc,htrace⟩ := averageMap_scalar ρ (InnerProductSpace.rankOne ℂ v w).toLinearMap
-  rw [InnerProductSpace.trace_rankOne] at htrace
-  have happ := congrArg (fun A : Module.End ℂ V => inner ℂ u (A z)) hc
-  change inner ℂ u ((∑ g, ρ g * (InnerProductSpace.rankOne ℂ v w).toLinearMap * ρ g⁻¹) z) = _ at happ
-  simp only [LinearMap.sum_apply,Module.End.mul_apply,ContinuousLinearMap.coe_coe,
-    InnerProductSpace.rankOne_apply,map_smul,unitary_inner_inv ρ hρ,inner_sum,
-    inner_smul_right,LinearMap.smul_apply,LinearMap.id_apply] at happ
-  have hsum : ∑ g, inner ℂ u (ρ g v) * inner ℂ (ρ g w) z = c * inner ℂ u z := by
-    simpa only [mul_comm] using happ
-  rw [hsum]
-  calc
-    _ = (c*(Module.finrank ℂ V:ℂ))*inner ℂ u z := by ring
-    _ = _ := by rw [htrace]
-
-end Thorp.UnitaryFinite
-namespace Thorp.UnitaryFinite
-open scoped BigOperators ComplexConjugate Classical
-
-variable {G X : Type*} [Group G] [Fintype G] [Fintype X]
-  (a : G →* Equiv.Perm X) (x₀ : X) (ha : ∀ x, ∃ g, a g x₀=x)
-
-noncomputable def representative (x : X) : G := Classical.choose (ha x)
-
-omit [Fintype G] [Fintype X] in
-lemma representative_apply (x : X) : a (representative a x₀ ha x) x₀=x :=
-  Classical.choose_spec (ha x)
-
-include ha in
-lemma orbit_sum (F : X → ℂ) :
-    (Fintype.card X:ℂ)*∑ g, F (a g x₀) = (Fintype.card G:ℂ)*∑ x, F x := by
-  have ht (x : X) : ∑ g, F (a g x)=∑ g, F (a g x₀) := by
-    obtain ⟨t,rfl⟩ := ha x
-    symm
-    apply Fintype.sum_equiv (Equiv.mulRight t⁻¹)
-    intro g
-    simp
-  calc
-    _ = ∑ x : X, ∑ g : G, F (a g x) := by simp only [ht,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
-    _ = ∑ g : G, ∑ x : X, F (a g x) := Finset.sum_comm
-    _ = ∑ g : G, ∑ x : X, F x := by
-      apply Finset.sum_congr rfl
-      intro g _
-      exact Equiv.sum_comp (a g) F
-    _ = _ := by simp
-
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V)
-
-def IsFixed (w : V) : Prop := ∀ g, a g x₀=x₀ → ρ g w=w
-
-noncomputable def coefficient (w v : V) : EuclideanSpace ℂ X :=
-  WithLp.toLp 2 (fun x => inner ℂ (ρ (representative a x₀ ha x) w) v)
-
-omit [Fintype G] [Fintype X] [FiniteDimensional ℂ V] in
-lemma coefficient_at (w : V) (hw : IsFixed a x₀ ρ w) (v : V) (g : G) :
-    coefficient a x₀ ha ρ w v (a g x₀) = inner ℂ (ρ g w) v := by
-  let r := representative a x₀ ha (a g x₀)
-  have hr : a r x₀=a g x₀ := representative_apply a x₀ ha _
-  have hs : a (g⁻¹*r) x₀=x₀ := by
-    simp only [map_mul,Equiv.Perm.mul_apply]
-    rw [hr,←Equiv.Perm.mul_apply,←map_mul,inv_mul_cancel,map_one]
-    rfl
-  have he := congrArg (ρ g) (hw (g⁻¹*r) hs)
-  have hh : ρ r w=ρ g w := by
-    simpa only [map_mul,Module.End.mul_apply,←Module.End.mul_apply,←map_mul,
-      mul_inv_cancel_left] using he
-  change inner ℂ (ρ r w) v= _
-  rw [hh]
-
-lemma coefficient_inner [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
-    (w z : V) (hw : IsFixed a x₀ ρ w) (hz : IsFixed a x₀ ρ z) (v t : V) :
-    (Module.finrank ℂ V:ℂ)*inner ℂ (coefficient a x₀ ha ρ w v) (coefficient a x₀ ha ρ z t) =
-      (Fintype.card X:ℂ)*inner ℂ z w*inner ℂ v t := by
-  have hs := orbit_sum a x₀ ha (fun x => star (coefficient a x₀ ha ρ w v x)*coefficient a x₀ ha ρ z t x)
-  simp only [coefficient_at a x₀ ha ρ w hw,coefficient_at a x₀ ha ρ z hz] at hs
-  simp only [←starRingEnd_apply,inner_conj_symm] at hs
-  simp only [starRingEnd_apply] at hs
-  have ho := coefficient_orthogonality ρ hρ v w z t
-  have hcard : (Fintype.card G:ℂ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
-  apply mul_left_cancel₀ hcard
-  rw [PiLp.inner_apply]
-  simp only [RCLike.inner_apply,starRingEnd_apply]
-  have hsum : ∑ x, coefficient a x₀ ha ρ z t x * star (coefficient a x₀ ha ρ w v x) =
-      ∑ x, star (coefficient a x₀ ha ρ w v x)*coefficient a x₀ ha ρ z t x := by
-    apply Finset.sum_congr rfl
-    intro x _
-    ring
-  rw [hsum]
-  calc
-    _ = (Module.finrank ℂ V:ℂ)*((Fintype.card G:ℂ)*∑ x,
-        star (coefficient a x₀ ha ρ w v x)*coefficient a x₀ ha ρ z t x) := by ring
-    _ = (Fintype.card X:ℂ)*((Module.finrank ℂ V:ℂ)*∑ g,
-        inner ℂ v (ρ g w)*inner ℂ (ρ g z) t) := by rw [←hs]; ring
-    _ = _ := by rw [ho]; ring
-
-omit [Fintype G] [Fintype X] [FiniteDimensional ℂ V] in
-lemma coefficient_action (hρ : IsUnitary ρ) (w : V) (hw : IsFixed a x₀ ρ w)
-    (v : V) (g : G) (x : X) :
-    coefficient a x₀ ha ρ w v (a g x) = coefficient a x₀ ha ρ w (ρ g⁻¹ v) x := by
-  obtain ⟨r,rfl⟩ := ha x
-  rw [←Equiv.Perm.mul_apply,←map_mul,coefficient_at a x₀ ha ρ w hw,
-    coefficient_at a x₀ ha ρ w hw]
-  simp only [map_mul,Module.End.mul_apply,unitary_inner_inv ρ hρ]
-
-noncomputable def normalizedCoefficient (w v : V) : EuclideanSpace ℂ X :=
-  (Real.sqrt ((Module.finrank ℂ V:ℝ)/(Fintype.card X:ℝ)) : ℂ) • coefficient a x₀ ha ρ w v
-
-lemma normalizedCoefficient_inner [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
-    (w z : V) (hw : IsFixed a x₀ ρ w) (hz : IsFixed a x₀ ρ z) (v t : V) :
-    inner ℂ (normalizedCoefficient a x₀ ha ρ w v) (normalizedCoefficient a x₀ ha ρ z t) =
-      inner ℂ z w*inner ℂ v t := by
-  let : Nonempty X := ⟨x₀⟩
-  have hX : (Fintype.card X:ℂ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
-  have hs : (Real.sqrt ((Module.finrank ℂ V:ℝ)/(Fintype.card X:ℝ)) : ℂ)^2 =
-      (Module.finrank ℂ V:ℂ)/(Fintype.card X:ℂ) := by
-    rw [←Complex.ofReal_pow,Real.sq_sqrt (by positivity)]
-    simp
-  simp only [normalizedCoefficient,inner_smul_left,inner_smul_right]
-  simp only [Complex.conj_ofReal]
-  rw [←mul_assoc,←pow_two,hs]
-  have hi := coefficient_inner a x₀ ha ρ hρ w z hw hz v t
-  field_simp
-  simpa only [mul_assoc] using hi
-
-lemma normalizedCoefficient_orthonormal [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
-    {I : Type*} (w : I → V) (hw : Orthonormal ℂ w)
-    (hfix : ∀ i, IsFixed a x₀ ρ (w i)) (v : V) (hv : ‖v‖=1) :
-    Orthonormal ℂ (fun i => normalizedCoefficient a x₀ ha ρ (w i) v) := by
-  rw [orthonormal_iff_ite] at hw ⊢
-  intro i j
-  rw [normalizedCoefficient_inner a x₀ ha ρ hρ _ _ (hfix i) (hfix j),hw,
-    inner_self_eq_norm_sq_to_K,hv]
-  simp [eq_comm]
-
-lemma normalizedCoefficient_norm [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
-    (w : V) (hw : IsFixed a x₀ ρ w) (hn : ‖w‖=1) (v : V) :
-    ‖normalizedCoefficient a x₀ ha ρ w v‖=‖v‖ := by
-  have he := normalizedCoefficient_inner a x₀ ha ρ hρ w w hw hw v v
-  rw [inner_self_eq_norm_sq_to_K,inner_self_eq_norm_sq_to_K,hn] at he
-  simp only [inner_self_eq_norm_sq_to_K] at he
-  norm_num at he
-  have hs : ‖normalizedCoefficient a x₀ ha ρ w v‖^2=‖v‖^2 := by exact_mod_cast he
-  nlinarith [norm_nonneg (normalizedCoefficient a x₀ ha ρ w v),norm_nonneg v]
-
-end Thorp.UnitaryFinite
-
-namespace Thorp.FiniteConvolution
-open scoped BigOperators Classical
-variable {G : Type*} [Group G] [Fintype G]
-
-noncomputable def conv (p q : G → ℝ) (g : G) : ℝ := ∑ h, p h*q (h⁻¹*g)
-noncomputable def power (p : G → ℝ) : ℕ → G → ℝ
-  | 0 => fun g => if g=1 then 1 else 0
-  | n+1 => conv p (power p n)
-
-lemma conv_nonneg {p q : G → ℝ} (hp : ∀ g, 0 ≤ p g) (hq : ∀ g, 0 ≤ q g) :
-    ∀ g, 0 ≤ conv p q g := fun _ => Finset.sum_nonneg (fun h _ => mul_nonneg (hp h) (hq _))
-
-lemma conv_pos_of {p q : G → ℝ} (hp : ∀ g, 0 ≤ p g) (hq : ∀ g, 0 ≤ q g)
-    {a b : G} (ha : 0 < p a) (hb : 0 < q b) : 0 < conv p q (a*b) := by
-  apply lt_of_lt_of_le (mul_pos ha hb)
-  have h := Finset.single_le_sum (fun g _ => mul_nonneg (hp g) (hq (g⁻¹*(a*b))))
-    (Finset.mem_univ a)
-  simpa [conv] using h
-
-lemma sum_conv (p q : G → ℝ) : ∑ g, conv p q g=(∑ g,p g)*(∑ g,q g) := by
-  unfold conv
-  rw [Finset.sum_comm,Finset.sum_mul]
-  apply Finset.sum_congr rfl
-  intro h _
-  rw [←Finset.mul_sum]
-  congr 1
-  exact Fintype.sum_equiv (Equiv.mulLeft h⁻¹) _ _ (fun _ => rfl)
-
-lemma power_nonneg {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) : ∀ n g, 0 ≤ power p n g := by
-  intro n
-  induction n with
-  | zero => intro g; simp only [power]; split_ifs  <;> norm_num
-  | succ n ih => exact conv_nonneg hp ih
-
-lemma sum_power {p : G → ℝ} (hp : ∑ g,p g=1) : ∀ n, ∑ g,power p n g=1 := by
-  intro n
-  induction n with
-  | zero => simp [power]
-  | succ n ih => rw [power,sum_conv,hp,ih,one_mul]
-
-lemma power_list_pos {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) (l : List G)
-    (hl : ∀ g ∈ l, 0 < p g) : 0 < power p l.length l.prod := by
-  induction l with
-  | nil => simp [power]
-  | cons a l ih =>
-    simpa only [List.length_cons,List.prod_cons,power] using
-      conv_pos_of hp (power_nonneg hp l.length) (hl a (by simp))
-        (ih (fun g hg => hl g (by simp [hg])))
-
-lemma power_pos_mono {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) (h1 : 0 < p 1)
-    {n m : ℕ} (hnm : n ≤ m) {g : G} (hg : 0 < power p n g) : 0 < power p m g := by
-  induction m, hnm using Nat.le_induction with
-  | base => exact hg
-  | succ m _ ih =>
-    simpa only [one_mul,power] using conv_pos_of hp (power_nonneg hp m) h1 ih
-
-theorem exists_positive_power {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) (h1 : 0 < p 1)
-    (hgen : Subgroup.closure {g | 0 < p g}=⊤) :
-    ∃ n : ℕ, ∀ g, 0 < power p (2^n) g := by
-  have hx (g : G) : g ∈ Submonoid.closure {g | 0 < p g} := by
-    rw [←Subgroup.closure_toSubmonoid_of_finite,hgen]
-    trivial
-  choose l hl hprod using fun g => Submonoid.exists_list_of_mem_closure (hx g)
-  let n := Finset.univ.sup (fun g => (l g).length)
-  refine ⟨n,fun g => ?_⟩
-  have hlen : (l g).length ≤ n := Finset.le_sup (f := fun g => (l g).length) (Finset.mem_univ g)
-  have hpow : n ≤ 2^n := Nat.le_of_lt (Nat.lt_two_pow_self)
-  have hpos := power_list_pos hp (l g) (hl g)
-  rw [hprod g] at hpos
-  exact power_pos_mono hp h1 (hlen.trans hpow) hpos
-
-structure Minorant (p : G → ℝ) where
-  exponent : ℕ
-  mass : ℝ
-  mass_pos : 0 < mass
-  mass_le_one : mass ≤ 1
-  lower : ∀ g, mass/(Fintype.card G:ℝ) ≤ power p (2^exponent) g
-
-noncomputable def minorant {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) (hsum : ∑ g,p g=1)
-    (h1 : 0 < p 1) (hgen : Subgroup.closure {g | 0 < p g}=⊤) : Minorant p := by
-  let n := Classical.choose (exists_positive_power hp h1 hgen)
-  have hn := Classical.choose_spec (exists_positive_power hp h1 hgen)
-  let a := Finset.univ.inf' Finset.univ_nonempty (power p (2^n))
-  have ha : 0 < a := (Finset.lt_inf'_iff _).mpr (fun g _ => hn g)
-  have hag (g : G) : a ≤ power p (2^n) g := Finset.inf'_le _ (Finset.mem_univ g)
-  have hN : (0:ℝ) < Fintype.card G := Nat.cast_pos.mpr Fintype.card_pos
-  have hsum' := Finset.sum_le_sum (s := Finset.univ) (fun g _ => hag g)
-  simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,sum_power hsum] at hsum'
-  refine ⟨n,(Fintype.card G:ℝ)*a,mul_pos hN ha,hsum',fun g => ?_⟩
-  have he : (Fintype.card G:ℝ)*a/(Fintype.card G:ℝ)=a := by field_simp
-  rw [he]
-  exact hag g
-
-end Thorp.FiniteConvolution
-namespace Thorp.UnitaryFinite
-
-section
-open scoped BigOperators ComplexConjugate Classical
-
-variable {G : Type*} [Group G] [Fintype G]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-omit [Fintype G] in
-@[simp] lemma continuousRepresentation_apply (ρ : Representation ℂ G V) (g : G) (v : V) :
-    continuousRepresentation ρ g v=ρ g v := rfl
-
-omit [Fintype G] [FiniteDimensional ℂ V] in
-lemma unitary_norm (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (g : G) (v : V) :
-    ‖ρ g v‖=‖v‖ := by
-  rw [norm_eq_sqrt_re_inner (𝕜 := ℂ),hρ,norm_eq_sqrt_re_inner (𝕜 := ℂ)]
-
-omit [Fintype G] in
-lemma continuousRepresentation_norm_le (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (g : G) :
-    ‖continuousRepresentation ρ g‖≤1 := by
-  apply ContinuousLinearMap.opNorm_le_bound _ zero_le_one
-  intro v
-  simp only [continuousRepresentation_apply,unitary_norm ρ hρ,one_mul,le_refl]
-
-omit [Fintype G] in
-lemma continuousRepresentation_adjoint (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (g : G) :
-    (continuousRepresentation ρ g).adjoint=continuousRepresentation ρ g⁻¹ := by
-  ext w
-  apply ext_inner_left ℂ
-  intro v
-  rw [ContinuousLinearMap.adjoint_inner_right]
-  exact (unitary_inner_inv ρ hρ g v w).symm
-
-noncomputable def weightedOperator (ρ : Representation ℂ G V) (p : G → ℝ) : V →L[ℂ] V :=
-  ∑ g, (p g : ℂ) • continuousRepresentation ρ g
-
-lemma weightedOperator_norm_le (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
-    (p : G → ℝ) (hp : ∀ g, 0≤p g) : ‖weightedOperator ρ p‖ ≤ ∑ g, p g := by
-  apply (norm_sum_le _ _).trans
-  apply Finset.sum_le_sum
-  intro g _
-  rw [norm_smul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (hp g)]
-  exact mul_le_of_le_one_right (hp g) (continuousRepresentation_norm_le ρ hρ g)
-
-lemma weightedOperator_symmetric (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
-    (p : G → ℝ) (hp : ∀ g, p g⁻¹=p g) : IsSelfAdjoint (weightedOperator ρ p) := by
-  change (weightedOperator ρ p).adjoint=weightedOperator ρ p
-  unfold weightedOperator
-  rw [map_sum]
-  simp only [map_smulₛₗ,Complex.conj_ofReal,continuousRepresentation_adjoint ρ hρ]
-  apply Fintype.sum_equiv (Equiv.inv G)
-  intro g
-  simp [hp]
-
-noncomputable def uniformOperator (ρ : Representation ℂ G V) : V →L[ℂ] V :=
-  weightedOperator ρ (fun _ => (Fintype.card G:ℝ)⁻¹)
-
-lemma uniformOperator_symmetric (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) :
-    IsSelfAdjoint (uniformOperator ρ) := weightedOperator_symmetric ρ hρ _ (fun _ => rfl)
-
-lemma uniformOperator_left (ρ : Representation ℂ G V) (g : G) :
-    continuousRepresentation ρ g * uniformOperator ρ=uniformOperator ρ := by
-  unfold uniformOperator weightedOperator
-  rw [Finset.mul_sum]
-  simp only [mul_smul_comm,←map_mul]
-  exact Fintype.sum_equiv (Equiv.mulLeft g) _ _ (fun _ => rfl)
-
-lemma uniformOperator_right (ρ : Representation ℂ G V) (g : G) :
-    uniformOperator ρ * continuousRepresentation ρ g=uniformOperator ρ := by
-  unfold uniformOperator weightedOperator
-  rw [Finset.sum_mul]
-  simp only [smul_mul_assoc,←map_mul]
-  exact Fintype.sum_equiv (Equiv.mulRight g) _ _ (fun _ => rfl)
-
-lemma uniformOperator_idempotent (ρ : Representation ℂ G V) :
-    IsIdempotentElem (uniformOperator ρ) := by
-  change uniformOperator ρ * uniformOperator ρ=uniformOperator ρ
-  nth_rw 1 [uniformOperator,weightedOperator]
-  rw [Finset.sum_mul]
-  simp only [smul_mul_assoc,uniformOperator_left,Finset.sum_const,Finset.card_univ,
-    ←Nat.cast_smul_eq_nsmul ℂ,smul_smul,Complex.ofReal_inv,Complex.ofReal_natCast]
-  rw [mul_inv_cancel₀ (by exact_mod_cast Fintype.card_ne_zero),one_smul]
-
-lemma uniformOperator_projection (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) :
-    IsStarProjection (uniformOperator ρ) :=
-  ⟨uniformOperator_idempotent ρ,uniformOperator_symmetric ρ hρ⟩
-
-lemma weighted_uniform_left (ρ : Representation ℂ G V) (p : G → ℝ) :
-    weightedOperator ρ p * uniformOperator ρ=(∑ g,p g : ℝ) • uniformOperator ρ := by
-  unfold weightedOperator
-  rw [Finset.sum_mul]
-  simp only [smul_mul_assoc,uniformOperator_left,←Finset.sum_smul,←Complex.ofReal_sum]
-  rfl
-
-lemma weighted_uniform_right (ρ : Representation ℂ G V) (p : G → ℝ) :
-    uniformOperator ρ * weightedOperator ρ p=(∑ g,p g : ℝ) • uniformOperator ρ := by
-  unfold weightedOperator
-  rw [Finset.mul_sum]
-  simp only [mul_smul_comm,uniformOperator_right,←Finset.sum_smul,←Complex.ofReal_sum]
-  rfl
-
-theorem doeblin_off_invariants (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
-    (p : G → ℝ) (hp : ∑ g,p g=1) (ε : ℝ)
-    (hminor : ∀ g, ε/(Fintype.card G:ℝ)≤p g) :
-    ‖weightedOperator ρ p * (1-uniformOperator ρ)‖≤1-ε := by
-  let q : G → ℝ := fun g => p g-ε/(Fintype.card G:ℝ)
-  have hq : ∀ g, 0≤q g := fun g => sub_nonneg.mpr (hminor g)
-  have hcard : (Fintype.card G:ℝ)≠0 := by exact_mod_cast Fintype.card_ne_zero
-  have hsum : ∑ g,q g=1-ε := by
-    simp only [q,Finset.sum_sub_distrib,hp,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
-    field_simp
-  have he : weightedOperator ρ p=weightedOperator ρ q+(ε:ℂ) • uniformOperator ρ := by
-    simp only [uniformOperator,weightedOperator]
-    rw [Finset.smul_sum,←Finset.sum_add_distrib]
-    apply Finset.sum_congr rfl
-    intro g _
-    simp only [q,Complex.ofReal_sub,Complex.ofReal_div,smul_smul,←add_smul,
-      Complex.ofReal_inv,Complex.ofReal_natCast]
-    congr 1
-    field_simp
-    ring
-  have hz : uniformOperator ρ*(1-uniformOperator ρ)=0 := by
-    rw [mul_sub,mul_one,(uniformOperator_idempotent ρ).eq,sub_self]
-  rw [he,add_mul,smul_mul_assoc,hz,smul_zero,add_zero]
-  calc
-    _ ≤ ‖weightedOperator ρ q‖*‖1-uniformOperator ρ‖ := norm_mul_le _ _
-    _ ≤ (1-ε)*1 := mul_le_mul (by rw [←hsum]; exact weightedOperator_norm_le ρ hρ q hq)
-      ((uniformOperator_projection ρ hρ).one_sub.norm_le _) (norm_nonneg _)
-      (by rw [←hsum]; exact Finset.sum_nonneg (fun g _ => hq g))
-    _ = _ := mul_one _
-
-end
-open scoped BigOperators Classical
-variable {G : Type*} [Group G] [Fintype G]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-lemma weightedOperator_conv (ρ : Representation ℂ G V) (p q : G → ℝ) :
-    weightedOperator ρ (FiniteConvolution.conv p q)=weightedOperator ρ p*weightedOperator ρ q := by
-  unfold weightedOperator FiniteConvolution.conv
-  simp only [Complex.ofReal_sum,Finset.sum_smul]
-  rw [Finset.sum_comm,Finset.sum_mul]
-  apply Finset.sum_congr rfl
-  intro h _
-  rw [Finset.mul_sum]
-  simp only [smul_mul_assoc,mul_smul_comm,smul_smul,←map_mul]
-  apply Fintype.sum_equiv (Equiv.mulLeft h⁻¹)
-  intro g
-  simp [Complex.ofReal_mul,mul_comm]
-
-lemma weightedOperator_power (ρ : Representation ℂ G V) (p : G → ℝ) (n : ℕ) :
-    weightedOperator ρ (FiniteConvolution.power p n)=(weightedOperator ρ p)^n := by
-  induction n with
-  | zero => simp [FiniteConvolution.power,weightedOperator]
-  | succ n ih => rw [FiniteConvolution.power,weightedOperator_conv,ih,pow_succ']
-
-lemma off_invariants_pow (ρ : Representation ℂ G V) (p : G → ℝ)
-    (hp : ∑ g,p g=1) (n : ℕ) (hn : n≠0) :
-    (weightedOperator ρ p-uniformOperator ρ)^n=
-      (weightedOperator ρ p)^n*(1-uniformOperator ρ) := by
-  have hAP := weighted_uniform_left ρ p
-  have hPA := weighted_uniform_right ρ p
-  rw [hp,one_smul] at hAP hPA
-  have hc : Commute (weightedOperator ρ p) (1-uniformOperator ρ) := by
-    show _*_= _*_
-    rw [mul_sub,sub_mul,mul_one,one_mul,hAP,hPA]
-  have he : weightedOperator ρ p-uniformOperator ρ=
-      weightedOperator ρ p*(1-uniformOperator ρ) := by rw [mul_sub,mul_one,hAP]
-  rw [he,hc.mul_pow,((uniformOperator_idempotent ρ).one_sub).pow_eq hn]
-
-noncomputable def minorantGap {p : G → ℝ} (c : FiniteConvolution.Minorant p) : ℝ :=
-  1-c.mass/(2*(2:ℝ)^c.exponent)
-
-omit [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V] in
-lemma minorantGap_pos {p : G → ℝ} (c : FiniteConvolution.Minorant p) : 0 < minorantGap c := by
-  have hn : (1:ℝ) ≤ 2^c.exponent := one_le_pow₀ (by norm_num)
-  have hq : c.mass/(2*(2:ℝ)^c.exponent) ≤ 1/2 := by
-    apply (div_le_iff₀ (by positivity)).mpr
-    nlinarith [c.mass_le_one]
-  unfold minorantGap
-  linarith
-
-omit [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V] in
-lemma minorantGap_lt_one {p : G → ℝ} (c : FiniteConvolution.Minorant p) : minorantGap c < 1 := by
-  unfold minorantGap
-  exact sub_lt_self _ (div_pos c.mass_pos (by positivity))
-
-theorem norm_off_invariants_le_gap (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
-    (p : G → ℝ) (hp : ∑ g,p g=1) (hsym : ∀ g,p g⁻¹=p g)
-    (c : FiniteConvolution.Minorant p) :
-    ‖weightedOperator ρ p-uniformOperator ρ‖ ≤ minorantGap c := by
-  have hself := (weightedOperator_symmetric ρ hρ p hsym).sub (uniformOperator_symmetric ρ hρ)
-  have hbound := doeblin_off_invariants ρ hρ (FiniteConvolution.power p (2^c.exponent))
-    (FiniteConvolution.sum_power hp _) c.mass c.lower
-  rw [weightedOperator_power,←off_invariants_pow ρ p hp _ (by positivity),
-    hself.norm_pow_two_pow] at hbound
-  have hn : (1:ℝ) ≤ 2^c.exponent := one_le_pow₀ (by norm_num)
-  have hq : c.mass/(2*(2:ℝ)^c.exponent) ≤ 1/2 := by
-    apply (div_le_iff₀ (by positivity)).mpr
-    nlinarith [c.mass_le_one]
-  have hbern := one_add_mul_le_pow
-    (a := -(c.mass/(2*(2:ℝ)^c.exponent))) (by linarith : (-2:ℝ) ≤ -(c.mass/(2*(2:ℝ)^c.exponent)))
-    (2^c.exponent)
-  have he : 1+(2^c.exponent:ℕ)*(-(c.mass/(2*(2:ℝ)^c.exponent)))=1-c.mass/2 := by
-    push_cast
-    field_simp
-    ring
-  rw [he,show 1+ -(c.mass/(2*(2:ℝ)^c.exponent))=minorantGap c by rfl] at hbern
-  by_contra h
-  have hlt : minorantGap c < ‖weightedOperator ρ p-uniformOperator ρ‖ := lt_of_not_ge h
-  have hpows := pow_lt_pow_left₀ hlt (minorantGap_pos c).le (by positivity : (2^c.exponent:ℕ)≠0)
-  linarith [c.mass_pos]
-
-end Thorp.UnitaryFinite
-namespace Thorp
-open scoped BigOperators Classical
-
-def butterflyIdentity : (d : ℕ) → Butterfly d
-  | 0 => ()
-  | d+1 => (fun _ => false,fun _ => butterflyIdentity d)
-
-lemma pairSwitch_false (d : ℕ) : pairSwitch (d := d) (fun _ => false)=1 := by
-  apply Equiv.ext
-  intro x
-  change switchFun (fun _ => false) x=x
-  simp [switchFun]
-
-lemma childLift_one (d : ℕ) : childLift (d := d) (fun _ => 1)=1 := by
-  ext x i
-  simp [childLift]
-
-lemma butterflyIdentity_perm (d : ℕ) : butterflyPerm d (butterflyIdentity d)=1 := by
-  induction d with
-  | zero => rfl
-  | succ d ih => simp only [butterflyIdentity,butterflyPerm,ih,pairSwitch_false,childLift_one,mul_one]
-
-noncomputable def childInjection (d : ℕ) (b : Bool) : Equiv.Perm (Card d) →* Equiv.Perm (Card (d+1)) where
-  toFun p := childLift (fun c => if c=b then p else 1)
-  map_one' := by simpa only [ite_self] using childLift_one d
-  map_mul' p q := by
-    rw [childLift_mul]
-    congr 1
-    funext c
-    split_ifs <;> simp
-
-lemma childInjection_swap (d : ℕ) (b : Bool) (u v : Card d) :
-    childInjection d b (Equiv.swap u v)=Equiv.swap (Fin.cons b u) (Fin.cons b v) := by
-  apply Equiv.ext
-  intro x
-  obtain ⟨c,w,rfl⟩ : ∃ c w, x=Fin.cons c w := ⟨x 0,Fin.tail x,(Fin.cons_self_tail x).symm⟩
-  change (Fin.cons c ((if c=b then Equiv.swap u v else 1) w) : Card (d+1))=_
-  by_cases hc : c=b
-  · subst c
-    simp only [↓reduceIte]
-    rw [Equiv.swap_apply_def]
-    split_ifs with h₁ h₂
-    · subst w; simp
-    · subst w; simp
-    · rw [Equiv.swap_apply_of_ne_of_ne]
-      · exact fun h => h₁ (by simpa using congrArg Fin.tail h)
-      · exact fun h => h₂ (by simpa using congrArg Fin.tail h)
-  · rw [ite_eq_right hc,Equiv.Perm.one_apply,Equiv.swap_apply_of_ne_of_ne]
-    · intro h; exact hc (by simpa using congrArg (fun z => z 0) h)
-    · intro h; exact hc (by simpa using congrArg (fun z => z 0) h)
-
-lemma pairSwitch_single (d : ℕ) (u : Card d) :
-    pairSwitch (fun v => decide (v=u))=Equiv.swap (Fin.cons false u) (Fin.cons true u) := by
-  apply Equiv.ext
-  intro x
-  obtain ⟨c,w,rfl⟩ : ∃ c w, x=Fin.cons c w := ⟨x 0,Fin.tail x,(Fin.cons_self_tail x).symm⟩
-  change switchFun (fun v => decide (v=u)) (Fin.cons c w)=_
-  by_cases h : w=u
-  · subst w
-    cases c <;> simp [switchFun]
-  · have ha : (Fin.cons c w : Card (d+1))≠Fin.cons false u := fun he => h (by simpa using congrArg Fin.tail he)
-    have hb : (Fin.cons c w : Card (d+1))≠Fin.cons true u := fun he => h (by simpa using congrArg Fin.tail he)
-    simp [switchFun,h,Equiv.swap_apply_of_ne_of_ne ha hb]
-
-theorem butterfly_generates (d : ℕ) (H : Subgroup (Equiv.Perm (Card d)))
-    (hH : ∀ ω : Butterfly d, butterflyPerm d ω∈H) : H=⊤ := by
-  induction d with
-  | zero =>
-    apply top_unique
-    intro p _
-    have hp : p=1 := Subsingleton.elim _ _
-    rw [hp]
-    exact H.one_mem
-  | succ d ih =>
-    have hchild (b : Bool) (p : Equiv.Perm (Card d)) : childInjection d b p∈H := by
-      have hc : H.comap (childInjection d b)=⊤ := by
-        apply ih
-        intro ω
-        change childLift (fun c => if c=b then butterflyPerm d ω else 1)∈H
-        have h := hH ((fun _ => false),fun c => if c=b then ω else butterflyIdentity d)
-        simpa only [butterflyPerm,pairSwitch_false,one_mul,apply_ite,butterflyIdentity_perm] using h
-      have hp : p∈H.comap (childInjection d b) := by rw [hc]; trivial
-      exact hp
-    have hpair (u : Card d) : Equiv.swap (Fin.cons false u) (Fin.cons true u)∈H := by
-      rw [←pairSwitch_single]
-      have h := hH ((fun v => decide (v=u)),fun _ => butterflyIdentity d)
-      simpa only [butterflyPerm,butterflyIdentity_perm,childLift_one,mul_one] using h
-    have hswap (x y : Card (d+1)) : Equiv.swap x y∈H := by
-      have ht (a b : Bool) (u : Card d) : Equiv.swap (Fin.cons a u) (Fin.cons b u)∈H := by
-        cases a <;> cases b
-        · rw [Equiv.swap_self]; exact H.one_mem
-        · exact hpair _
-        · rw [Equiv.swap_comm]; exact hpair _
-        · rw [Equiv.swap_self]; exact H.one_mem
-      have h₁ : Equiv.swap x (Fin.cons (y 0) (Fin.tail x))∈H := by
-        convert ht (x 0) (y 0) (Fin.tail x) using 1
-        rw [Fin.cons_self_tail]
-      have h₂ := hchild (y 0) (Equiv.swap (Fin.tail x) (Fin.tail y))
-      rw [childInjection_swap,Fin.cons_self_tail] at h₂
-      exact SubmonoidClass.swap_mem_trans H h₁ h₂
-    rw [eq_top_iff,←Equiv.Perm.closure_isSwap,Subgroup.closure_le]
-    rintro _ ⟨x,y,_,rfl⟩
-    exact hswap x y
-
-lemma butterfly_closure (d : ℕ) : Subgroup.closure (Set.range (butterflyPerm d))=⊤ :=
-  butterfly_generates d _ (fun ω => Subgroup.subset_closure ⟨ω,rfl⟩)
-
-variable {Ω G : Type*} [Fintype Ω] [Fintype G]
-
-noncomputable def sampleLaw (P : Ω → G) (g : G) : ℝ :=
-  finiteMean (fun ω => if P ω=g then 1 else 0)
-
-omit [Fintype G] in
-lemma sampleLaw_nonneg (P : Ω → G) (g : G) : 0 ≤ sampleLaw P g :=
-  finiteMean_nonneg (fun _ => by split_ifs <;> norm_num)
-
-lemma sampleLaw_sum [Nonempty Ω] (P : Ω → G) : ∑ g,sampleLaw P g=1 := by
-  unfold sampleLaw
-  rw [←finiteMean_sum]
-  simp only [Finset.sum_ite_eq,Finset.mem_univ,↓reduceIte,finiteMean_const]
-
-omit [Fintype G] in
-lemma sampleLaw_pos_iff [Nonempty Ω] (P : Ω → G) (g : G) :
-    0<sampleLaw P g ↔ ∃ ω,P ω=g := by
-  unfold sampleLaw finiteMean
-  rw [div_pos_iff_of_pos_right (Nat.cast_pos.mpr Fintype.card_pos)]
-  rw [Finset.sum_pos_iff_of_nonneg (fun _ _ => by split_ifs <;> norm_num)]
-  simp only [Finset.mem_univ,true_and]
-  apply exists_congr
-  intro ω
-  split_ifs <;> simp_all
-
-omit [Fintype G] in
-lemma sampleLaw_equiv {Ω' : Type*} [Fintype Ω'] (P : Ω → G) (e : Ω' ≃ Ω) :
-    sampleLaw (P ∘ e)=sampleLaw P := by
-  funext g
-  simpa only [sampleLaw,Function.comp_apply,Equiv.apply_symm_apply] using
-    finiteMean_equiv e (fun ω => if P (e ω)=g then (1:ℝ) else 0)
-
-variable [Group G]
-omit [Fintype G] in
-lemma sampleLaw_inverse (P : Ω → G) (g : G) :
-    sampleLaw (fun ω => (P ω)⁻¹) g=sampleLaw P g⁻¹ := by
-  apply finiteMean_congr
-  intro ω
-  rw [inv_eq_iff_eq_inv]
-
-omit [Fintype G] in
-lemma sampleLaw_symmetric (P : Ω → G) (e : Equiv.Perm Ω)
-    (he : ∀ ω,P (e ω)=(P ω)⁻¹) (g : G) : sampleLaw P g⁻¹=sampleLaw P g := by
-  rw [←sampleLaw_inverse]
-  have hh : (fun ω => (P ω)⁻¹)=P ∘ e := funext (fun ω => (he ω).symm)
-  rw [hh,sampleLaw_equiv]
-
-end Thorp
-
-namespace Thorp
-open scoped BigOperators Classical
-
-lemma palindrome_identity (d : ℕ) : ∃ ω,palindromePerm d ω=1 := by
-  refine ⟨((butterflyCoinEquiv d).symm (butterflyIdentity d),
-    (butterflyCoinEquiv d).symm (butterflyIdentity d)),?_⟩
-  exact mul_inv_cancel _
-
-lemma palindrome_range_contains_butterfly (d : ℕ) (ω : Butterfly d) :
-    ∃ ξ,palindromePerm d ξ=butterflyPerm d ω := by
-  refine ⟨((butterflyCoinEquiv d).symm ω,(butterflyCoinEquiv d).symm (butterflyIdentity d)),?_⟩
-  unfold palindromePerm
-  have h (β : Butterfly d) : decodeButterfly d ((butterflyCoinEquiv d).symm β)=β :=
-    (butterflyCoinEquiv d).apply_symm_apply β
-  rw [h,h,butterflyIdentity_perm,show (1 : Equiv.Perm (Card d)).symm=1 from rfl,mul_one]
-
-lemma palindrome_support_generates (d : ℕ) :
-    Subgroup.closure {g | 0<sampleLaw (palindromePerm d) g}=⊤ := by
-  apply butterfly_generates
-  intro ω
-  apply Subgroup.subset_closure
-  exact (sampleLaw_pos_iff _ _).mpr (palindrome_range_contains_butterfly d ω)
-
-noncomputable def palindromeMinorant (d : ℕ) :
-    FiniteConvolution.Minorant (sampleLaw (palindromePerm d)) :=
-  FiniteConvolution.minorant (sampleLaw_nonneg _) (sampleLaw_sum _)
-    ((sampleLaw_pos_iff _ _).mpr (palindrome_identity d)) (palindrome_support_generates d)
-
-noncomputable def blockGap (d : ℕ) : ℝ := UnitaryFinite.minorantGap (palindromeMinorant d)
-
-lemma blockGap_pos (d : ℕ) : 0<blockGap d := UnitaryFinite.minorantGap_pos _
-lemma blockGap_lt_one (d : ℕ) : blockGap d<1 := UnitaryFinite.minorantGap_lt_one _
-
-theorem palindrome_off_invariants_gap (d : ℕ) {V : Type*}
-    [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-    (ρ : Representation ℂ (Equiv.Perm (Card d)) V) (hρ : UnitaryFinite.IsUnitary ρ) :
-    ‖UnitaryFinite.weightedOperator ρ (sampleLaw (palindromePerm d))-
-      UnitaryFinite.uniformOperator ρ‖≤blockGap d := by
-  apply UnitaryFinite.norm_off_invariants_le_gap ρ hρ _ (sampleLaw_sum _)
-  exact sampleLaw_symmetric _ (Equiv.prodComm _ _) (palindromePerm_reverse d)
-
-end Thorp
-
-namespace Thorp.CommutingBlocks
-open scoped BigOperators Classical
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
-variable {ι : Type*} [Fintype ι]
-
-omit [CompleteSpace V] in
-lemma orthogonal_sum_norm_sq (v : ι → V) (hv : ∀ i j, i≠j → inner ℂ (v i) (v j)=0) :
-    ‖∑ i,v i‖^2=∑ i,‖v i‖^2 := by
-  rw [norm_sq_eq_re_inner (𝕜 := ℂ),sum_inner]
-  simp only [inner_sum,map_sum]
-  apply Finset.sum_congr rfl
-  intro i _
-  rw [Finset.sum_eq_single i]
-  · exact (norm_sq_eq_re_inner (𝕜 := ℂ) (v i)).symm
-  · intro j _ hji
-    rw [hv i j hji.symm]
-    rfl
-  · simp
-
-lemma projection_inner_zero (P Q : V →L[ℂ] V) (hP : IsStarProjection P)
-    (hPQ : P*Q=0) (v w : V) : inner ℂ (P v) (Q w)=0 := by
-  rw [←ContinuousLinearMap.adjoint_inner_right]
-  change inner ℂ v ((star P*Q) w)=0
-  rw [hP.isSelfAdjoint.star_eq,hPQ]
-  simp
-
-lemma projection_sandwich_norm (P B : V →L[ℂ] V) (hP : IsStarProjection P) :
-    ‖B.adjoint*P*B‖=‖P*B‖^2 := by
-  have he : star (P*B)*(P*B)=B.adjoint*P*B := by
-    rw [star_mul,hP.isSelfAdjoint.star_eq]
-    change B.adjoint*P*(P*B)=_
-    rw [←mul_assoc,mul_assoc B.adjoint P P,hP.isIdempotentElem.eq]
-  rw [←he,CStarRing.norm_star_mul_self]
-  ring
-
-theorem resolution_norm_bound (P : ι → V →L[ℂ] V)
-    (hP : ∀ i, IsStarProjection (P i))
-    (horth : ∀ i j, i≠j → P i*P j=0) (hsum : ∑ i,P i=1)
-    (A : V →L[ℂ] V) (hcomm : ∀ i, Commute A (P i))
-    (c : ℝ) (hc : 0≤c) (hbound : ∀ i, ‖A*P i‖≤c) : ‖A‖≤c := by
-  apply ContinuousLinearMap.opNorm_le_bound _ hc
-  intro v
-  have hsplit (w : V) : ∑ i,P i w=w := by
-    rw [←sum_apply,hsum]
-    rfl
-  have hp := orthogonal_sum_norm_sq (fun i => P i v)
-    (fun i j hij => projection_inner_zero _ _ (hP i) (horth i j hij) v v)
-  rw [hsplit] at hp
-  have ha := orthogonal_sum_norm_sq (fun i => P i (A v))
-    (fun i j hij => projection_inner_zero _ _ (hP i) (horth i j hij) (A v) (A v))
-  rw [hsplit] at ha
-  have he (i : ι) : P i (A v)=(A*P i) (P i v) := by
-    change (P i*A) v=(A*P i*P i) v
-    rw [mul_assoc,(hP i).isIdempotentElem.eq,(hcomm i).eq]
-  have hh (i : ι) : ‖P i (A v)‖≤c*‖P i v‖ := by
-    rw [he]
-    exact ((A*P i).le_opNorm _).trans (mul_le_mul_of_nonneg_right (hbound i) (norm_nonneg _))
-  have hsq := Finset.sum_le_sum (s := Finset.univ) (fun i _ =>
-    pow_le_pow_left₀ (norm_nonneg _) (hh i) 2)
-  simp_rw [mul_pow] at hsq
-  rw [←Finset.mul_sum,←hp,←ha,←mul_pow] at hsq
-  exact (sq_le_sq₀ (norm_nonneg _) (mul_nonneg hc (norm_nonneg _))).mp hsq
-
-theorem sector_sandwich_le (T P A B : V →L[ℂ] V)
-    (hT : IsStarProjection T) (hP : IsStarProjection P)
-    (hTP : T*P=T) (hcomm : Commute A T) (hA : ‖A‖≤1) :
-    ‖B.adjoint*A*T*B‖≤‖B.adjoint*P*B‖ := by
-  have he : B.adjoint*A*T*B=(T*B).adjoint*A*(T*B) := by
-    change _=star (T*B)*A*(T*B)
-    rw [star_mul,hT.isSelfAdjoint.star_eq]
-    change _=B.adjoint*T*A*(T*B)
-    rw [mul_assoc B.adjoint T A,←hcomm.eq]
-    simp only [mul_assoc]
-    rw [←mul_assoc T T B,hT.isIdempotentElem.eq]
-  have hTB : ‖T*B‖≤‖P*B‖ := by
-    have he : T*B=T*(P*B) := by rw [←mul_assoc,hTP]
-    rw [he]
-    exact (norm_mul_le _ _).trans ((mul_le_mul_of_nonneg_right (hT.norm_le _) (norm_nonneg _)).trans_eq (one_mul _))
-  rw [he,projection_sandwich_norm P B hP]
-  calc
-    _ ≤ ‖(T*B).adjoint‖*‖A‖*‖T*B‖ := (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right (norm_mul_le _ _) (norm_nonneg _))
-    _ ≤ ‖T*B‖^2 := by rw [ContinuousLinearMap.adjoint.norm_map]; nlinarith [norm_nonneg (T*B)]
-    _ ≤ _ := pow_le_pow_left₀ (norm_nonneg _) hTB 2
-
-omit [CompleteSpace V] [Fintype ι] in
-lemma resolution_commute (P : ι → V →L[ℂ] V)
-    (horth : ∀ i j, i≠j → P i*P j=0) (i j : ι) : Commute (P i) (P j) := by
-  by_cases hij : i=j
-  · subst j; exact Commute.refl _
-  · exact (horth i j hij).trans (horth j i (Ne.symm hij)).symm
-
-omit [Fintype ι] in
-lemma resolution_sum_mul (P : ι → V →L[ℂ] V) (hP : ∀ i, IsStarProjection (P i))
-    (horth : ∀ i j, i≠j → P i*P j=0) (S : Finset ι) (j : ι) :
-    (∑ i∈S,P i)*P j=if j∈S then P j else 0 := by
-  rw [Finset.sum_mul]
-  have he (i : ι) : P i*P j=if i=j then P j else 0 := by
-    split_ifs with h
-    · subst i; exact (hP j).isIdempotentElem.eq
-    · exact horth i j h
-  simp only [he,Finset.sum_ite_eq']
-
-lemma resolution_truncated_norm (P : ι → V →L[ℂ] V)
-    (hP : ∀ i, IsStarProjection (P i))
-    (horth : ∀ i j, i≠j → P i*P j=0) (hsum : ∑ i,P i=1)
-    (A : V →L[ℂ] V) (hcomm : ∀ i, Commute A (P i))
-    (S : Finset ι) (c : ℝ) (hc : 0≤c) (hbound : ∀ i∈S, ‖A*P i‖≤c) :
-    ‖A*(∑ i∈S,P i)‖≤c := by
-  apply resolution_norm_bound P hP horth hsum _ _ c hc
-  · intro i
-    rw [mul_assoc,resolution_sum_mul P hP horth]
-    split_ifs with hi
-    · exact hbound i hi
-    · simp [hc]
-  · intro i
-    exact (hcomm i).mul_left (Commute.sum_left S P (P i) fun j _ => resolution_commute P horth j i)
-
-theorem resolution_exception_bound (T Q : ι → V →L[ℂ] V)
-    (hT : ∀ i, IsStarProjection (T i))
-    (horth : ∀ i j, i≠j → T i*T j=0) (hsum : ∑ i,T i=1)
-    (hQ : ∀ i, IsStarProjection (Q i)) (hTQ : ∀ i,T i*Q i=T i)
-    (A B : V →L[ℂ] V) (hcomm : ∀ i, Commute A (T i))
-    (hA : ‖A‖≤1) (hB : ‖B‖≤1)
-    (S : Finset ι) (c : ℝ) (hc : 0≤c) (hbound : ∀ i∉S, ‖A*T i‖≤c) :
-    ‖B.adjoint*A*B‖≤c+∑ i∈S,‖B.adjoint*Q i*B‖ := by
-  have hhigh : ‖A*(∑ i∈Sᶜ,T i)‖≤c := resolution_truncated_norm T hT horth hsum A hcomm
-    Sᶜ c hc (fun i hi => hbound i (Finset.mem_compl.mp hi))
-  have hsplit : (∑ i∈Sᶜ,T i)+(∑ i∈S,T i)=1 := by
-    rw [Finset.sum_compl_add_sum,hsum]
-  have he : B.adjoint*A*B=B.adjoint*(A*(∑ i∈Sᶜ,T i))*B+
-      ∑ i∈S,B.adjoint*A*T i*B := by
-    rw [←Finset.sum_mul,←Finset.mul_sum]
-    rw [←add_mul]
-    simp only [mul_assoc]
-    rw [←mul_add,←mul_add,hsplit]
-    simp only [mul_one,mul_assoc]
-  rw [he]
-  apply (norm_add_le _ _).trans
-  apply add_le_add
-  · calc
-      _ ≤ ‖B.adjoint‖*‖A*(∑ i∈Sᶜ,T i)‖*‖B‖ :=
-        (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right (norm_mul_le _ _) (norm_nonneg _))
-      _ ≤ 1*c*1 := mul_le_mul (mul_le_mul (by simpa only [ContinuousLinearMap.adjoint.norm_map] using hB)
-        hhigh (norm_nonneg _) zero_le_one) hB (norm_nonneg _) (by simpa using hc)
-      _ = c := by ring
-  · apply (norm_sum_le _ _).trans
-    exact Finset.sum_le_sum fun i _ => sector_sandwich_le _ _ _ _ (hT i) (hQ i) (hTQ i) (hcomm i) hA
-
-variable [DecidableEq ι]
-variable (P : ι → V →L[ℂ] V) (hcomm : ∀ i j, Commute (P i) (P j))
-
-noncomputable def choice (E : Finset ι) (i : ι) : V →L[ℂ] V :=
-  if i∈E then 1-P i else P i
-
-include hcomm
-omit [CompleteSpace V] [Fintype ι] in
-lemma choice_commute (E F : Finset ι) (i j : ι) :
-    Commute (choice P E i) (choice P F j) := by
-  unfold choice
-  split_ifs
-  · exact (Commute.one_left _).sub_left ((Commute.one_right _).sub_right (hcomm i j))
-  · exact (Commute.one_left _).sub_left (hcomm i j)
-  · exact (Commute.one_right _).sub_right (hcomm i j)
-  · exact hcomm i j
-
-noncomputable def sector (s E : Finset ι) : V →L[ℂ] V :=
-  s.noncommProd (choice P E) (fun i _ j _ _ => choice_commute P hcomm E E i j)
-
-omit [CompleteSpace V] [Fintype ι] in
-lemma sector_insert (s E : Finset ι) (a : ι) (ha : a∉s) :
-    sector P hcomm (insert a s) E=choice P E a*sector P hcomm s E := by
-  exact Finset.noncommProd_insert_of_notMem _ _ _ _ ha
-
-end Thorp.CommutingBlocks
-
-end ThorpNine.Harmonic
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Harmonic.Shapes
+
+namespace OAI
+
+namespace ThorpNine.Harmonic
+
+namespace Thorp.UnitaryFinite
+open scoped BigOperators ComplexConjugate Classical
+variable {G : Type*} [Group G] [Fintype G]
+variable {V W : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+  [NormedAddCommGroup W] [InnerProductSpace ℂ W] [FiniteDimensional ℂ W]
+
+noncomputable def averageMap (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
+    (A : V →ₗ[ℂ] W) : Representation.IntertwiningMap ρ σ where
+  toLinearMap := ∑ g, (σ g).comp (A.comp (ρ g⁻¹))
+  isIntertwining' h := by
+    ext v
+    simp only [LinearMap.comp_apply,LinearMap.sum_apply,map_sum]
+    apply Fintype.sum_equiv (Equiv.mulLeft h⁻¹)
+    intro g
+    simp only [Equiv.coe_mulLeft,mul_inv_rev,inv_inv,map_mul,Module.End.mul_apply]
+    have hh (w : W) : σ h (σ h⁻¹ w)=w := by
+      change (σ h*σ h⁻¹) w=w
+      rw [←map_mul,mul_inv_cancel,map_one]
+      rfl
+    rw [hh]
+
+omit [FiniteDimensional ℂ V] in
+lemma averageMap_self_trace (ρ : Representation ℂ G V) (A : Module.End ℂ V) :
+    LinearMap.trace ℂ V (averageMap ρ ρ A).toLinearMap =
+      (Fintype.card G:ℂ) * LinearMap.trace ℂ V A := by
+  change LinearMap.trace ℂ V (∑ g, ρ g * A * ρ g⁻¹) = _
+  rw [map_sum]
+  have hj (g : G) : LinearMap.trace ℂ V (ρ g*A*ρ g⁻¹) = LinearMap.trace ℂ V A := by
+    rw [LinearMap.trace_mul_cycle,←map_mul,inv_mul_cancel,map_one,one_mul]
+  simp only [hj,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
+
+lemma averageMap_scalar (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
+    (A : Module.End ℂ V) :
+    ∃ c : ℂ, (averageMap ρ ρ A).toLinearMap = c • LinearMap.id ∧
+      c * (Module.finrank ℂ V:ℂ) = (Fintype.card G:ℂ)*LinearMap.trace ℂ V A := by
+  obtain ⟨c,hc⟩ := (Representation.IsIrreducible.algebraMap_intertwiningMap_bijective_of_isAlgClosed (ρ := ρ)).surjective (averageMap ρ ρ A)
+  have he : (averageMap ρ ρ A).toLinearMap = c • LinearMap.id := by
+    rw [←hc]
+    rfl
+  refine ⟨c,he,?_⟩
+  rw [←averageMap_self_trace ρ A,he,map_smul,LinearMap.trace_id,smul_eq_mul]
+
+theorem coefficient_orthogonality (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
+    (hρ : IsUnitary ρ) (u v w z : V) :
+    (Module.finrank ℂ V:ℂ) * ∑ g, inner ℂ u (ρ g v) * inner ℂ (ρ g w) z =
+      (Fintype.card G:ℂ) * inner ℂ w v * inner ℂ u z := by
+  obtain ⟨c,hc,htrace⟩ := averageMap_scalar ρ (InnerProductSpace.rankOne ℂ v w).toLinearMap
+  rw [InnerProductSpace.trace_rankOne] at htrace
+  have happ := congrArg (fun A : Module.End ℂ V => inner ℂ u (A z)) hc
+  change inner ℂ u ((∑ g, ρ g * (InnerProductSpace.rankOne ℂ v w).toLinearMap * ρ g⁻¹) z) = _ at happ
+  simp only [LinearMap.sum_apply,Module.End.mul_apply,ContinuousLinearMap.coe_coe,
+    InnerProductSpace.rankOne_apply,map_smul,unitary_inner_inv ρ hρ,inner_sum,
+    inner_smul_right,LinearMap.smul_apply,LinearMap.id_apply] at happ
+  have hsum : ∑ g, inner ℂ u (ρ g v) * inner ℂ (ρ g w) z = c * inner ℂ u z := by
+    simpa only [mul_comm] using happ
+  rw [hsum]
+  calc
+    _ = (c*(Module.finrank ℂ V:ℂ))*inner ℂ u z := by ring
+    _ = _ := by rw [htrace]
+
+end Thorp.UnitaryFinite
+namespace Thorp.UnitaryFinite
+open scoped BigOperators ComplexConjugate Classical
+
+variable {G X : Type*} [Group G] [Fintype G] [Fintype X]
+  (a : G →* Equiv.Perm X) (x₀ : X) (ha : ∀ x, ∃ g, a g x₀=x)
+
+noncomputable def representative (x : X) : G := Classical.choose (ha x)
+
+omit [Fintype G] [Fintype X] in
+lemma representative_apply (x : X) : a (representative a x₀ ha x) x₀=x :=
+  Classical.choose_spec (ha x)
+
+include ha in
+lemma orbit_sum (F : X → ℂ) :
+    (Fintype.card X:ℂ)*∑ g, F (a g x₀) = (Fintype.card G:ℂ)*∑ x, F x := by
+  have ht (x : X) : ∑ g, F (a g x)=∑ g, F (a g x₀) := by
+    obtain ⟨t,rfl⟩ := ha x
+    symm
+    apply Fintype.sum_equiv (Equiv.mulRight t⁻¹)
+    intro g
+    simp
+  calc
+    _ = ∑ x : X, ∑ g : G, F (a g x) := by simp only [ht,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
+    _ = ∑ g : G, ∑ x : X, F (a g x) := Finset.sum_comm
+    _ = ∑ g : G, ∑ x : X, F x := by
+      apply Finset.sum_congr rfl
+      intro g _
+      exact Equiv.sum_comp (a g) F
+    _ = _ := by simp
+
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V)
+
+def IsFixed (w : V) : Prop := ∀ g, a g x₀=x₀ → ρ g w=w
+
+noncomputable def coefficient (w v : V) : EuclideanSpace ℂ X :=
+  WithLp.toLp 2 (fun x => inner ℂ (ρ (representative a x₀ ha x) w) v)
+
+omit [Fintype G] [Fintype X] [FiniteDimensional ℂ V] in
+lemma coefficient_at (w : V) (hw : IsFixed a x₀ ρ w) (v : V) (g : G) :
+    coefficient a x₀ ha ρ w v (a g x₀) = inner ℂ (ρ g w) v := by
+  let r := representative a x₀ ha (a g x₀)
+  have hr : a r x₀=a g x₀ := representative_apply a x₀ ha _
+  have hs : a (g⁻¹*r) x₀=x₀ := by
+    simp only [map_mul,Equiv.Perm.mul_apply]
+    rw [hr,←Equiv.Perm.mul_apply,←map_mul,inv_mul_cancel,map_one]
+    rfl
+  have he := congrArg (ρ g) (hw (g⁻¹*r) hs)
+  have hh : ρ r w=ρ g w := by
+    simpa only [map_mul,Module.End.mul_apply,←Module.End.mul_apply,←map_mul,
+      mul_inv_cancel_left] using he
+  change inner ℂ (ρ r w) v= _
+  rw [hh]
+
+lemma coefficient_inner [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
+    (w z : V) (hw : IsFixed a x₀ ρ w) (hz : IsFixed a x₀ ρ z) (v t : V) :
+    (Module.finrank ℂ V:ℂ)*inner ℂ (coefficient a x₀ ha ρ w v) (coefficient a x₀ ha ρ z t) =
+      (Fintype.card X:ℂ)*inner ℂ z w*inner ℂ v t := by
+  have hs := orbit_sum a x₀ ha (fun x => star (coefficient a x₀ ha ρ w v x)*coefficient a x₀ ha ρ z t x)
+  simp only [coefficient_at a x₀ ha ρ w hw,coefficient_at a x₀ ha ρ z hz] at hs
+  simp only [←starRingEnd_apply,inner_conj_symm] at hs
+  simp only [starRingEnd_apply] at hs
+  have ho := coefficient_orthogonality ρ hρ v w z t
+  have hcard : (Fintype.card G:ℂ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
+  apply mul_left_cancel₀ hcard
+  rw [PiLp.inner_apply]
+  simp only [RCLike.inner_apply,starRingEnd_apply]
+  have hsum : ∑ x, coefficient a x₀ ha ρ z t x * star (coefficient a x₀ ha ρ w v x) =
+      ∑ x, star (coefficient a x₀ ha ρ w v x)*coefficient a x₀ ha ρ z t x := by
+    apply Finset.sum_congr rfl
+    intro x _
+    ring
+  rw [hsum]
+  calc
+    _ = (Module.finrank ℂ V:ℂ)*((Fintype.card G:ℂ)*∑ x,
+        star (coefficient a x₀ ha ρ w v x)*coefficient a x₀ ha ρ z t x) := by ring
+    _ = (Fintype.card X:ℂ)*((Module.finrank ℂ V:ℂ)*∑ g,
+        inner ℂ v (ρ g w)*inner ℂ (ρ g z) t) := by rw [←hs]; ring
+    _ = _ := by rw [ho]; ring
+
+omit [Fintype G] [Fintype X] [FiniteDimensional ℂ V] in
+lemma coefficient_action (hρ : IsUnitary ρ) (w : V) (hw : IsFixed a x₀ ρ w)
+    (v : V) (g : G) (x : X) :
+    coefficient a x₀ ha ρ w v (a g x) = coefficient a x₀ ha ρ w (ρ g⁻¹ v) x := by
+  obtain ⟨r,rfl⟩ := ha x
+  rw [←Equiv.Perm.mul_apply,←map_mul,coefficient_at a x₀ ha ρ w hw,
+    coefficient_at a x₀ ha ρ w hw]
+  simp only [map_mul,Module.End.mul_apply,unitary_inner_inv ρ hρ]
+
+noncomputable def normalizedCoefficient (w v : V) : EuclideanSpace ℂ X :=
+  (Real.sqrt ((Module.finrank ℂ V:ℝ)/(Fintype.card X:ℝ)) : ℂ) • coefficient a x₀ ha ρ w v
+
+lemma normalizedCoefficient_inner [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
+    (w z : V) (hw : IsFixed a x₀ ρ w) (hz : IsFixed a x₀ ρ z) (v t : V) :
+    inner ℂ (normalizedCoefficient a x₀ ha ρ w v) (normalizedCoefficient a x₀ ha ρ z t) =
+      inner ℂ z w*inner ℂ v t := by
+  let : Nonempty X := ⟨x₀⟩
+  have hX : (Fintype.card X:ℂ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
+  have hs : (Real.sqrt ((Module.finrank ℂ V:ℝ)/(Fintype.card X:ℝ)) : ℂ)^2 =
+      (Module.finrank ℂ V:ℂ)/(Fintype.card X:ℂ) := by
+    rw [←Complex.ofReal_pow,Real.sq_sqrt (by positivity)]
+    simp
+  simp only [normalizedCoefficient,inner_smul_left,inner_smul_right]
+  simp only [Complex.conj_ofReal]
+  rw [←mul_assoc,←pow_two,hs]
+  have hi := coefficient_inner a x₀ ha ρ hρ w z hw hz v t
+  field_simp
+  simpa only [mul_assoc] using hi
+
+lemma normalizedCoefficient_orthonormal [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
+    {I : Type*} (w : I → V) (hw : Orthonormal ℂ w)
+    (hfix : ∀ i, IsFixed a x₀ ρ (w i)) (v : V) (hv : ‖v‖=1) :
+    Orthonormal ℂ (fun i => normalizedCoefficient a x₀ ha ρ (w i) v) := by
+  rw [orthonormal_iff_ite] at hw ⊢
+  intro i j
+  rw [normalizedCoefficient_inner a x₀ ha ρ hρ _ _ (hfix i) (hfix j),hw,
+    inner_self_eq_norm_sq_to_K,hv]
+  simp [eq_comm]
+
+lemma normalizedCoefficient_norm [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
+    (w : V) (hw : IsFixed a x₀ ρ w) (hn : ‖w‖=1) (v : V) :
+    ‖normalizedCoefficient a x₀ ha ρ w v‖=‖v‖ := by
+  have he := normalizedCoefficient_inner a x₀ ha ρ hρ w w hw hw v v
+  rw [inner_self_eq_norm_sq_to_K,inner_self_eq_norm_sq_to_K,hn] at he
+  simp only [inner_self_eq_norm_sq_to_K] at he
+  norm_num at he
+  have hs : ‖normalizedCoefficient a x₀ ha ρ w v‖^2=‖v‖^2 := by exact_mod_cast he
+  nlinarith [norm_nonneg (normalizedCoefficient a x₀ ha ρ w v),norm_nonneg v]
+
+end Thorp.UnitaryFinite
+
+namespace Thorp.FiniteConvolution
+open scoped BigOperators Classical
+variable {G : Type*} [Group G] [Fintype G]
+
+noncomputable def conv (p q : G → ℝ) (g : G) : ℝ := ∑ h, p h*q (h⁻¹*g)
+noncomputable def power (p : G → ℝ) : ℕ → G → ℝ
+  | 0 => fun g => if g=1 then 1 else 0
+  | n+1 => conv p (power p n)
+
+lemma conv_nonneg {p q : G → ℝ} (hp : ∀ g, 0 ≤ p g) (hq : ∀ g, 0 ≤ q g) :
+    ∀ g, 0 ≤ conv p q g := fun _ => Finset.sum_nonneg (fun h _ => mul_nonneg (hp h) (hq _))
+
+lemma conv_pos_of {p q : G → ℝ} (hp : ∀ g, 0 ≤ p g) (hq : ∀ g, 0 ≤ q g)
+    {a b : G} (ha : 0 < p a) (hb : 0 < q b) : 0 < conv p q (a*b) := by
+  apply lt_of_lt_of_le (mul_pos ha hb)
+  have h := Finset.single_le_sum (fun g _ => mul_nonneg (hp g) (hq (g⁻¹*(a*b))))
+    (Finset.mem_univ a)
+  simpa [conv] using h
+
+lemma sum_conv (p q : G → ℝ) : ∑ g, conv p q g=(∑ g,p g)*(∑ g,q g) := by
+  unfold conv
+  rw [Finset.sum_comm,Finset.sum_mul]
+  apply Finset.sum_congr rfl
+  intro h _
+  rw [←Finset.mul_sum]
+  congr 1
+  exact Fintype.sum_equiv (Equiv.mulLeft h⁻¹) _ _ (fun _ => rfl)
+
+lemma power_nonneg {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) : ∀ n g, 0 ≤ power p n g := by
+  intro n
+  induction n with
+  | zero => intro g; simp only [power]; split_ifs  <;> norm_num
+  | succ n ih => exact conv_nonneg hp ih
+
+lemma sum_power {p : G → ℝ} (hp : ∑ g,p g=1) : ∀ n, ∑ g,power p n g=1 := by
+  intro n
+  induction n with
+  | zero => simp [power]
+  | succ n ih => rw [power,sum_conv,hp,ih,one_mul]
+
+lemma power_list_pos {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) (l : List G)
+    (hl : ∀ g ∈ l, 0 < p g) : 0 < power p l.length l.prod := by
+  induction l with
+  | nil => simp [power]
+  | cons a l ih =>
+    simpa only [List.length_cons,List.prod_cons,power] using
+      conv_pos_of hp (power_nonneg hp l.length) (hl a (by simp))
+        (ih (fun g hg => hl g (by simp [hg])))
+
+lemma power_pos_mono {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) (h1 : 0 < p 1)
+    {n m : ℕ} (hnm : n ≤ m) {g : G} (hg : 0 < power p n g) : 0 < power p m g := by
+  induction m, hnm using Nat.le_induction with
+  | base => exact hg
+  | succ m _ ih =>
+    simpa only [one_mul,power] using conv_pos_of hp (power_nonneg hp m) h1 ih
+
+theorem exists_positive_power {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) (h1 : 0 < p 1)
+    (hgen : Subgroup.closure {g | 0 < p g}=⊤) :
+    ∃ n : ℕ, ∀ g, 0 < power p (2^n) g := by
+  have hx (g : G) : g ∈ Submonoid.closure {g | 0 < p g} := by
+    rw [←Subgroup.closure_toSubmonoid_of_finite,hgen]
+    trivial
+  choose l hl hprod using fun g => Submonoid.exists_list_of_mem_closure (hx g)
+  let n := Finset.univ.sup (fun g => (l g).length)
+  refine ⟨n,fun g => ?_⟩
+  have hlen : (l g).length ≤ n := Finset.le_sup (f := fun g => (l g).length) (Finset.mem_univ g)
+  have hpow : n ≤ 2^n := Nat.le_of_lt (Nat.lt_two_pow_self)
+  have hpos := power_list_pos hp (l g) (hl g)
+  rw [hprod g] at hpos
+  exact power_pos_mono hp h1 (hlen.trans hpow) hpos
+
+structure Minorant (p : G → ℝ) where
+  exponent : ℕ
+  mass : ℝ
+  mass_pos : 0 < mass
+  mass_le_one : mass ≤ 1
+  lower : ∀ g, mass/(Fintype.card G:ℝ) ≤ power p (2^exponent) g
+
+noncomputable def minorant {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) (hsum : ∑ g,p g=1)
+    (h1 : 0 < p 1) (hgen : Subgroup.closure {g | 0 < p g}=⊤) : Minorant p := by
+  let n := Classical.choose (exists_positive_power hp h1 hgen)
+  have hn := Classical.choose_spec (exists_positive_power hp h1 hgen)
+  let a := Finset.univ.inf' Finset.univ_nonempty (power p (2^n))
+  have ha : 0 < a := (Finset.lt_inf'_iff _).mpr (fun g _ => hn g)
+  have hag (g : G) : a ≤ power p (2^n) g := Finset.inf'_le _ (Finset.mem_univ g)
+  have hN : (0:ℝ) < Fintype.card G := Nat.cast_pos.mpr Fintype.card_pos
+  have hsum' := Finset.sum_le_sum (s := Finset.univ) (fun g _ => hag g)
+  simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,sum_power hsum] at hsum'
+  refine ⟨n,(Fintype.card G:ℝ)*a,mul_pos hN ha,hsum',fun g => ?_⟩
+  have he : (Fintype.card G:ℝ)*a/(Fintype.card G:ℝ)=a := by field_simp
+  rw [he]
+  exact hag g
+
+end Thorp.FiniteConvolution
+namespace Thorp.UnitaryFinite
+
+section
+open scoped BigOperators ComplexConjugate Classical
+
+variable {G : Type*} [Group G] [Fintype G]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+omit [Fintype G] in
+@[simp] lemma continuousRepresentation_apply (ρ : Representation ℂ G V) (g : G) (v : V) :
+    continuousRepresentation ρ g v=ρ g v := rfl
+
+omit [Fintype G] [FiniteDimensional ℂ V] in
+lemma unitary_norm (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (g : G) (v : V) :
+    ‖ρ g v‖=‖v‖ := by
+  rw [norm_eq_sqrt_re_inner (𝕜 := ℂ),hρ,norm_eq_sqrt_re_inner (𝕜 := ℂ)]
+
+omit [Fintype G] in
+lemma continuousRepresentation_norm_le (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (g : G) :
+    ‖continuousRepresentation ρ g‖≤1 := by
+  apply ContinuousLinearMap.opNorm_le_bound _ zero_le_one
+  intro v
+  simp only [continuousRepresentation_apply,unitary_norm ρ hρ,one_mul,le_refl]
+
+omit [Fintype G] in
+lemma continuousRepresentation_adjoint (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (g : G) :
+    (continuousRepresentation ρ g).adjoint=continuousRepresentation ρ g⁻¹ := by
+  ext w
+  apply ext_inner_left ℂ
+  intro v
+  rw [ContinuousLinearMap.adjoint_inner_right]
+  exact (unitary_inner_inv ρ hρ g v w).symm
+
+noncomputable def weightedOperator (ρ : Representation ℂ G V) (p : G → ℝ) : V →L[ℂ] V :=
+  ∑ g, (p g : ℂ) • continuousRepresentation ρ g
+
+lemma weightedOperator_norm_le (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
+    (p : G → ℝ) (hp : ∀ g, 0≤p g) : ‖weightedOperator ρ p‖ ≤ ∑ g, p g := by
+  apply (norm_sum_le _ _).trans
+  apply Finset.sum_le_sum
+  intro g _
+  rw [norm_smul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (hp g)]
+  exact mul_le_of_le_one_right (hp g) (continuousRepresentation_norm_le ρ hρ g)
+
+lemma weightedOperator_symmetric (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
+    (p : G → ℝ) (hp : ∀ g, p g⁻¹=p g) : IsSelfAdjoint (weightedOperator ρ p) := by
+  change (weightedOperator ρ p).adjoint=weightedOperator ρ p
+  unfold weightedOperator
+  rw [map_sum]
+  simp only [map_smulₛₗ,Complex.conj_ofReal,continuousRepresentation_adjoint ρ hρ]
+  apply Fintype.sum_equiv (Equiv.inv G)
+  intro g
+  simp [hp]
+
+noncomputable def uniformOperator (ρ : Representation ℂ G V) : V →L[ℂ] V :=
+  weightedOperator ρ (fun _ => (Fintype.card G:ℝ)⁻¹)
+
+lemma uniformOperator_symmetric (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) :
+    IsSelfAdjoint (uniformOperator ρ) := weightedOperator_symmetric ρ hρ _ (fun _ => rfl)
+
+lemma uniformOperator_left (ρ : Representation ℂ G V) (g : G) :
+    continuousRepresentation ρ g * uniformOperator ρ=uniformOperator ρ := by
+  unfold uniformOperator weightedOperator
+  rw [Finset.mul_sum]
+  simp only [mul_smul_comm,←map_mul]
+  exact Fintype.sum_equiv (Equiv.mulLeft g) _ _ (fun _ => rfl)
+
+lemma uniformOperator_right (ρ : Representation ℂ G V) (g : G) :
+    uniformOperator ρ * continuousRepresentation ρ g=uniformOperator ρ := by
+  unfold uniformOperator weightedOperator
+  rw [Finset.sum_mul]
+  simp only [smul_mul_assoc,←map_mul]
+  exact Fintype.sum_equiv (Equiv.mulRight g) _ _ (fun _ => rfl)
+
+lemma uniformOperator_idempotent (ρ : Representation ℂ G V) :
+    IsIdempotentElem (uniformOperator ρ) := by
+  change uniformOperator ρ * uniformOperator ρ=uniformOperator ρ
+  nth_rw 1 [uniformOperator,weightedOperator]
+  rw [Finset.sum_mul]
+  simp only [smul_mul_assoc,uniformOperator_left,Finset.sum_const,Finset.card_univ,
+    ←Nat.cast_smul_eq_nsmul ℂ,smul_smul,Complex.ofReal_inv,Complex.ofReal_natCast]
+  rw [mul_inv_cancel₀ (by exact_mod_cast Fintype.card_ne_zero),one_smul]
+
+lemma uniformOperator_projection (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) :
+    IsStarProjection (uniformOperator ρ) :=
+  ⟨uniformOperator_idempotent ρ,uniformOperator_symmetric ρ hρ⟩
+
+lemma weighted_uniform_left (ρ : Representation ℂ G V) (p : G → ℝ) :
+    weightedOperator ρ p * uniformOperator ρ=(∑ g,p g : ℝ) • uniformOperator ρ := by
+  unfold weightedOperator
+  rw [Finset.sum_mul]
+  simp only [smul_mul_assoc,uniformOperator_left,←Finset.sum_smul,←Complex.ofReal_sum]
+  rfl
+
+lemma weighted_uniform_right (ρ : Representation ℂ G V) (p : G → ℝ) :
+    uniformOperator ρ * weightedOperator ρ p=(∑ g,p g : ℝ) • uniformOperator ρ := by
+  unfold weightedOperator
+  rw [Finset.mul_sum]
+  simp only [mul_smul_comm,uniformOperator_right,←Finset.sum_smul,←Complex.ofReal_sum]
+  rfl
+
+theorem doeblin_off_invariants (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
+    (p : G → ℝ) (hp : ∑ g,p g=1) (ε : ℝ)
+    (hminor : ∀ g, ε/(Fintype.card G:ℝ)≤p g) :
+    ‖weightedOperator ρ p * (1-uniformOperator ρ)‖≤1-ε := by
+  let q : G → ℝ := fun g => p g-ε/(Fintype.card G:ℝ)
+  have hq : ∀ g, 0≤q g := fun g => sub_nonneg.mpr (hminor g)
+  have hcard : (Fintype.card G:ℝ)≠0 := by exact_mod_cast Fintype.card_ne_zero
+  have hsum : ∑ g,q g=1-ε := by
+    simp only [q,Finset.sum_sub_distrib,hp,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
+    field_simp
+  have he : weightedOperator ρ p=weightedOperator ρ q+(ε:ℂ) • uniformOperator ρ := by
+    simp only [uniformOperator,weightedOperator]
+    rw [Finset.smul_sum,←Finset.sum_add_distrib]
+    apply Finset.sum_congr rfl
+    intro g _
+    simp only [q,Complex.ofReal_sub,Complex.ofReal_div,smul_smul,←add_smul,
+      Complex.ofReal_inv,Complex.ofReal_natCast]
+    congr 1
+    field_simp
+    ring
+  have hz : uniformOperator ρ*(1-uniformOperator ρ)=0 := by
+    rw [mul_sub,mul_one,(uniformOperator_idempotent ρ).eq,sub_self]
+  rw [he,add_mul,smul_mul_assoc,hz,smul_zero,add_zero]
+  calc
+    _ ≤ ‖weightedOperator ρ q‖*‖1-uniformOperator ρ‖ := norm_mul_le _ _
+    _ ≤ (1-ε)*1 := mul_le_mul (by rw [←hsum]; exact weightedOperator_norm_le ρ hρ q hq)
+      ((uniformOperator_projection ρ hρ).one_sub.norm_le _) (norm_nonneg _)
+      (by rw [←hsum]; exact Finset.sum_nonneg (fun g _ => hq g))
+    _ = _ := mul_one _
+
+end
+open scoped BigOperators Classical
+variable {G : Type*} [Group G] [Fintype G]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+lemma weightedOperator_conv (ρ : Representation ℂ G V) (p q : G → ℝ) :
+    weightedOperator ρ (FiniteConvolution.conv p q)=weightedOperator ρ p*weightedOperator ρ q := by
+  unfold weightedOperator FiniteConvolution.conv
+  simp only [Complex.ofReal_sum,Finset.sum_smul]
+  rw [Finset.sum_comm,Finset.sum_mul]
+  apply Finset.sum_congr rfl
+  intro h _
+  rw [Finset.mul_sum]
+  simp only [smul_mul_assoc,mul_smul_comm,smul_smul,←map_mul]
+  apply Fintype.sum_equiv (Equiv.mulLeft h⁻¹)
+  intro g
+  simp [Complex.ofReal_mul,mul_comm]
+
+lemma weightedOperator_power (ρ : Representation ℂ G V) (p : G → ℝ) (n : ℕ) :
+    weightedOperator ρ (FiniteConvolution.power p n)=(weightedOperator ρ p)^n := by
+  induction n with
+  | zero => simp [FiniteConvolution.power,weightedOperator]
+  | succ n ih => rw [FiniteConvolution.power,weightedOperator_conv,ih,pow_succ']
+
+lemma off_invariants_pow (ρ : Representation ℂ G V) (p : G → ℝ)
+    (hp : ∑ g,p g=1) (n : ℕ) (hn : n≠0) :
+    (weightedOperator ρ p-uniformOperator ρ)^n=
+      (weightedOperator ρ p)^n*(1-uniformOperator ρ) := by
+  have hAP := weighted_uniform_left ρ p
+  have hPA := weighted_uniform_right ρ p
+  rw [hp,one_smul] at hAP hPA
+  have hc : Commute (weightedOperator ρ p) (1-uniformOperator ρ) := by
+    show _*_= _*_
+    rw [mul_sub,sub_mul,mul_one,one_mul,hAP,hPA]
+  have he : weightedOperator ρ p-uniformOperator ρ=
+      weightedOperator ρ p*(1-uniformOperator ρ) := by rw [mul_sub,mul_one,hAP]
+  rw [he,hc.mul_pow,((uniformOperator_idempotent ρ).one_sub).pow_eq hn]
+
+noncomputable def minorantGap {p : G → ℝ} (c : FiniteConvolution.Minorant p) : ℝ :=
+  1-c.mass/(2*(2:ℝ)^c.exponent)
+
+omit [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V] in
+lemma minorantGap_pos {p : G → ℝ} (c : FiniteConvolution.Minorant p) : 0 < minorantGap c := by
+  have hn : (1:ℝ) ≤ 2^c.exponent := one_le_pow₀ (by norm_num)
+  have hq : c.mass/(2*(2:ℝ)^c.exponent) ≤ 1/2 := by
+    apply (div_le_iff₀ (by positivity)).mpr
+    nlinarith [c.mass_le_one]
+  unfold minorantGap
+  linarith
+
+omit [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V] in
+lemma minorantGap_lt_one {p : G → ℝ} (c : FiniteConvolution.Minorant p) : minorantGap c < 1 := by
+  unfold minorantGap
+  exact sub_lt_self _ (div_pos c.mass_pos (by positivity))
+
+theorem norm_off_invariants_le_gap (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
+    (p : G → ℝ) (hp : ∑ g,p g=1) (hsym : ∀ g,p g⁻¹=p g)
+    (c : FiniteConvolution.Minorant p) :
+    ‖weightedOperator ρ p-uniformOperator ρ‖ ≤ minorantGap c := by
+  have hself := (weightedOperator_symmetric ρ hρ p hsym).sub (uniformOperator_symmetric ρ hρ)
+  have hbound := doeblin_off_invariants ρ hρ (FiniteConvolution.power p (2^c.exponent))
+    (FiniteConvolution.sum_power hp _) c.mass c.lower
+  rw [weightedOperator_power,←off_invariants_pow ρ p hp _ (by positivity),
+    hself.norm_pow_two_pow] at hbound
+  have hn : (1:ℝ) ≤ 2^c.exponent := one_le_pow₀ (by norm_num)
+  have hq : c.mass/(2*(2:ℝ)^c.exponent) ≤ 1/2 := by
+    apply (div_le_iff₀ (by positivity)).mpr
+    nlinarith [c.mass_le_one]
+  have hbern := one_add_mul_le_pow
+    (a := -(c.mass/(2*(2:ℝ)^c.exponent))) (by linarith : (-2:ℝ) ≤ -(c.mass/(2*(2:ℝ)^c.exponent)))
+    (2^c.exponent)
+  have he : 1+(2^c.exponent:ℕ)*(-(c.mass/(2*(2:ℝ)^c.exponent)))=1-c.mass/2 := by
+    push_cast
+    field_simp
+    ring
+  rw [he,show 1+ -(c.mass/(2*(2:ℝ)^c.exponent))=minorantGap c by rfl] at hbern
+  by_contra h
+  have hlt : minorantGap c < ‖weightedOperator ρ p-uniformOperator ρ‖ := lt_of_not_ge h
+  have hpows := pow_lt_pow_left₀ hlt (minorantGap_pos c).le (by positivity : (2^c.exponent:ℕ)≠0)
+  linarith [c.mass_pos]
+
+end Thorp.UnitaryFinite
+namespace Thorp
+open scoped BigOperators Classical
+
+def butterflyIdentity : (d : ℕ) → Butterfly d
+  | 0 => ()
+  | d+1 => (fun _ => false,fun _ => butterflyIdentity d)
+
+lemma pairSwitch_false (d : ℕ) : pairSwitch (d := d) (fun _ => false)=1 := by
+  apply Equiv.ext
+  intro x
+  change switchFun (fun _ => false) x=x
+  simp [switchFun]
+
+lemma childLift_one (d : ℕ) : childLift (d := d) (fun _ => 1)=1 := by
+  ext x i
+  simp [childLift]
+
+lemma butterflyIdentity_perm (d : ℕ) : butterflyPerm d (butterflyIdentity d)=1 := by
+  induction d with
+  | zero => rfl
+  | succ d ih => simp only [butterflyIdentity,butterflyPerm,ih,pairSwitch_false,childLift_one,mul_one]
+
+noncomputable def childInjection (d : ℕ) (b : Bool) : Equiv.Perm (Card d) →* Equiv.Perm (Card (d+1)) where
+  toFun p := childLift (fun c => if c=b then p else 1)
+  map_one' := by simpa only [ite_self] using childLift_one d
+  map_mul' p q := by
+    rw [childLift_mul]
+    congr 1
+    funext c
+    split_ifs <;> simp
+
+lemma childInjection_swap (d : ℕ) (b : Bool) (u v : Card d) :
+    childInjection d b (Equiv.swap u v)=Equiv.swap (Fin.cons b u) (Fin.cons b v) := by
+  apply Equiv.ext
+  intro x
+  obtain ⟨c,w,rfl⟩ : ∃ c w, x=Fin.cons c w := ⟨x 0,Fin.tail x,(Fin.cons_self_tail x).symm⟩
+  change (Fin.cons c ((if c=b then Equiv.swap u v else 1) w) : Card (d+1))=_
+  by_cases hc : c=b
+  · subst c
+    simp only [↓reduceIte]
+    rw [Equiv.swap_apply_def]
+    split_ifs with h₁ h₂
+    · subst w; simp
+    · subst w; simp
+    · rw [Equiv.swap_apply_of_ne_of_ne]
+      · exact fun h => h₁ (by simpa using congrArg Fin.tail h)
+      · exact fun h => h₂ (by simpa using congrArg Fin.tail h)
+  · rw [ite_eq_right hc,Equiv.Perm.one_apply,Equiv.swap_apply_of_ne_of_ne]
+    · intro h; exact hc (by simpa using congrArg (fun z => z 0) h)
+    · intro h; exact hc (by simpa using congrArg (fun z => z 0) h)
+
+lemma pairSwitch_single (d : ℕ) (u : Card d) :
+    pairSwitch (fun v => decide (v=u))=Equiv.swap (Fin.cons false u) (Fin.cons true u) := by
+  apply Equiv.ext
+  intro x
+  obtain ⟨c,w,rfl⟩ : ∃ c w, x=Fin.cons c w := ⟨x 0,Fin.tail x,(Fin.cons_self_tail x).symm⟩
+  change switchFun (fun v => decide (v=u)) (Fin.cons c w)=_
+  by_cases h : w=u
+  · subst w
+    cases c <;> simp [switchFun]
+  · have ha : (Fin.cons c w : Card (d+1))≠Fin.cons false u := fun he => h (by simpa using congrArg Fin.tail he)
+    have hb : (Fin.cons c w : Card (d+1))≠Fin.cons true u := fun he => h (by simpa using congrArg Fin.tail he)
+    simp [switchFun,h,Equiv.swap_apply_of_ne_of_ne ha hb]
+
+theorem butterfly_generates (d : ℕ) (H : Subgroup (Equiv.Perm (Card d)))
+    (hH : ∀ ω : Butterfly d, butterflyPerm d ω∈H) : H=⊤ := by
+  induction d with
+  | zero =>
+    apply top_unique
+    intro p _
+    have hp : p=1 := Subsingleton.elim _ _
+    rw [hp]
+    exact H.one_mem
+  | succ d ih =>
+    have hchild (b : Bool) (p : Equiv.Perm (Card d)) : childInjection d b p∈H := by
+      have hc : H.comap (childInjection d b)=⊤ := by
+        apply ih
+        intro ω
+        change childLift (fun c => if c=b then butterflyPerm d ω else 1)∈H
+        have h := hH ((fun _ => false),fun c => if c=b then ω else butterflyIdentity d)
+        simpa only [butterflyPerm,pairSwitch_false,one_mul,apply_ite,butterflyIdentity_perm] using h
+      have hp : p∈H.comap (childInjection d b) := by rw [hc]; trivial
+      exact hp
+    have hpair (u : Card d) : Equiv.swap (Fin.cons false u) (Fin.cons true u)∈H := by
+      rw [←pairSwitch_single]
+      have h := hH ((fun v => decide (v=u)),fun _ => butterflyIdentity d)
+      simpa only [butterflyPerm,butterflyIdentity_perm,childLift_one,mul_one] using h
+    have hswap (x y : Card (d+1)) : Equiv.swap x y∈H := by
+      have ht (a b : Bool) (u : Card d) : Equiv.swap (Fin.cons a u) (Fin.cons b u)∈H := by
+        cases a <;> cases b
+        · rw [Equiv.swap_self]; exact H.one_mem
+        · exact hpair _
+        · rw [Equiv.swap_comm]; exact hpair _
+        · rw [Equiv.swap_self]; exact H.one_mem
+      have h₁ : Equiv.swap x (Fin.cons (y 0) (Fin.tail x))∈H := by
+        convert ht (x 0) (y 0) (Fin.tail x) using 1
+        rw [Fin.cons_self_tail]
+      have h₂ := hchild (y 0) (Equiv.swap (Fin.tail x) (Fin.tail y))
+      rw [childInjection_swap,Fin.cons_self_tail] at h₂
+      exact SubmonoidClass.swap_mem_trans H h₁ h₂
+    rw [eq_top_iff,←Equiv.Perm.closure_isSwap,Subgroup.closure_le]
+    rintro _ ⟨x,y,_,rfl⟩
+    exact hswap x y
+
+lemma butterfly_closure (d : ℕ) : Subgroup.closure (Set.range (butterflyPerm d))=⊤ :=
+  butterfly_generates d _ (fun ω => Subgroup.subset_closure ⟨ω,rfl⟩)
+
+variable {Ω G : Type*} [Fintype Ω] [Fintype G]
+
+noncomputable def sampleLaw (P : Ω → G) (g : G) : ℝ :=
+  finiteMean (fun ω => if P ω=g then 1 else 0)
+
+omit [Fintype G] in
+lemma sampleLaw_nonneg (P : Ω → G) (g : G) : 0 ≤ sampleLaw P g :=
+  finiteMean_nonneg (fun _ => by split_ifs <;> norm_num)
+
+lemma sampleLaw_sum [Nonempty Ω] (P : Ω → G) : ∑ g,sampleLaw P g=1 := by
+  unfold sampleLaw
+  rw [←finiteMean_sum]
+  simp only [Finset.sum_ite_eq,Finset.mem_univ,↓reduceIte,finiteMean_const]
+
+omit [Fintype G] in
+lemma sampleLaw_pos_iff [Nonempty Ω] (P : Ω → G) (g : G) :
+    0<sampleLaw P g ↔ ∃ ω,P ω=g := by
+  unfold sampleLaw finiteMean
+  rw [div_pos_iff_of_pos_right (Nat.cast_pos.mpr Fintype.card_pos)]
+  rw [Finset.sum_pos_iff_of_nonneg (fun _ _ => by split_ifs <;> norm_num)]
+  simp only [Finset.mem_univ,true_and]
+  apply exists_congr
+  intro ω
+  split_ifs <;> simp_all
+
+omit [Fintype G] in
+lemma sampleLaw_equiv {Ω' : Type*} [Fintype Ω'] (P : Ω → G) (e : Ω' ≃ Ω) :
+    sampleLaw (P ∘ e)=sampleLaw P := by
+  funext g
+  simpa only [sampleLaw,Function.comp_apply,Equiv.apply_symm_apply] using
+    finiteMean_equiv e (fun ω => if P (e ω)=g then (1:ℝ) else 0)
+
+variable [Group G]
+omit [Fintype G] in
+lemma sampleLaw_inverse (P : Ω → G) (g : G) :
+    sampleLaw (fun ω => (P ω)⁻¹) g=sampleLaw P g⁻¹ := by
+  apply finiteMean_congr
+  intro ω
+  rw [inv_eq_iff_eq_inv]
+
+omit [Fintype G] in
+lemma sampleLaw_symmetric (P : Ω → G) (e : Equiv.Perm Ω)
+    (he : ∀ ω,P (e ω)=(P ω)⁻¹) (g : G) : sampleLaw P g⁻¹=sampleLaw P g := by
+  rw [←sampleLaw_inverse]
+  have hh : (fun ω => (P ω)⁻¹)=P ∘ e := funext (fun ω => (he ω).symm)
+  rw [hh,sampleLaw_equiv]
+
+end Thorp
+
+namespace Thorp
+open scoped BigOperators Classical
+
+lemma palindrome_identity (d : ℕ) : ∃ ω,palindromePerm d ω=1 := by
+  refine ⟨((butterflyCoinEquiv d).symm (butterflyIdentity d),
+    (butterflyCoinEquiv d).symm (butterflyIdentity d)),?_⟩
+  exact mul_inv_cancel _
+
+lemma palindrome_range_contains_butterfly (d : ℕ) (ω : Butterfly d) :
+    ∃ ξ,palindromePerm d ξ=butterflyPerm d ω := by
+  refine ⟨((butterflyCoinEquiv d).symm ω,(butterflyCoinEquiv d).symm (butterflyIdentity d)),?_⟩
+  unfold palindromePerm
+  have h (β : Butterfly d) : decodeButterfly d ((butterflyCoinEquiv d).symm β)=β :=
+    (butterflyCoinEquiv d).apply_symm_apply β
+  rw [h,h,butterflyIdentity_perm,show (1 : Equiv.Perm (Card d)).symm=1 from rfl,mul_one]
+
+lemma palindrome_support_generates (d : ℕ) :
+    Subgroup.closure {g | 0<sampleLaw (palindromePerm d) g}=⊤ := by
+  apply butterfly_generates
+  intro ω
+  apply Subgroup.subset_closure
+  exact (sampleLaw_pos_iff _ _).mpr (palindrome_range_contains_butterfly d ω)
+
+noncomputable def palindromeMinorant (d : ℕ) :
+    FiniteConvolution.Minorant (sampleLaw (palindromePerm d)) :=
+  FiniteConvolution.minorant (sampleLaw_nonneg _) (sampleLaw_sum _)
+    ((sampleLaw_pos_iff _ _).mpr (palindrome_identity d)) (palindrome_support_generates d)
+
+noncomputable def blockGap (d : ℕ) : ℝ := UnitaryFinite.minorantGap (palindromeMinorant d)
+
+lemma blockGap_pos (d : ℕ) : 0<blockGap d := UnitaryFinite.minorantGap_pos _
+lemma blockGap_lt_one (d : ℕ) : blockGap d<1 := UnitaryFinite.minorantGap_lt_one _
+
+theorem palindrome_off_invariants_gap (d : ℕ) {V : Type*}
+    [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+    (ρ : Representation ℂ (Equiv.Perm (Card d)) V) (hρ : UnitaryFinite.IsUnitary ρ) :
+    ‖UnitaryFinite.weightedOperator ρ (sampleLaw (palindromePerm d))-
+      UnitaryFinite.uniformOperator ρ‖≤blockGap d := by
+  apply UnitaryFinite.norm_off_invariants_le_gap ρ hρ _ (sampleLaw_sum _)
+  exact sampleLaw_symmetric _ (Equiv.prodComm _ _) (palindromePerm_reverse d)
+
+end Thorp
+
+namespace Thorp.CommutingBlocks
+open scoped BigOperators Classical
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
+variable {ι : Type*} [Fintype ι]
+
+omit [CompleteSpace V] in
+lemma orthogonal_sum_norm_sq (v : ι → V) (hv : ∀ i j, i≠j → inner ℂ (v i) (v j)=0) :
+    ‖∑ i,v i‖^2=∑ i,‖v i‖^2 := by
+  rw [norm_sq_eq_re_inner (𝕜 := ℂ),sum_inner]
+  simp only [inner_sum,map_sum]
+  apply Finset.sum_congr rfl
+  intro i _
+  rw [Finset.sum_eq_single i]
+  · exact (norm_sq_eq_re_inner (𝕜 := ℂ) (v i)).symm
+  · intro j _ hji
+    rw [hv i j hji.symm]
+    rfl
+  · simp
+
+lemma projection_inner_zero (P Q : V →L[ℂ] V) (hP : IsStarProjection P)
+    (hPQ : P*Q=0) (v w : V) : inner ℂ (P v) (Q w)=0 := by
+  rw [←ContinuousLinearMap.adjoint_inner_right]
+  change inner ℂ v ((star P*Q) w)=0
+  rw [hP.isSelfAdjoint.star_eq,hPQ]
+  simp
+
+lemma projection_sandwich_norm (P B : V →L[ℂ] V) (hP : IsStarProjection P) :
+    ‖B.adjoint*P*B‖=‖P*B‖^2 := by
+  have he : star (P*B)*(P*B)=B.adjoint*P*B := by
+    rw [star_mul,hP.isSelfAdjoint.star_eq]
+    change B.adjoint*P*(P*B)=_
+    rw [←mul_assoc,mul_assoc B.adjoint P P,hP.isIdempotentElem.eq]
+  rw [←he,CStarRing.norm_star_mul_self]
+  ring
+
+theorem resolution_norm_bound (P : ι → V →L[ℂ] V)
+    (hP : ∀ i, IsStarProjection (P i))
+    (horth : ∀ i j, i≠j → P i*P j=0) (hsum : ∑ i,P i=1)
+    (A : V →L[ℂ] V) (hcomm : ∀ i, Commute A (P i))
+    (c : ℝ) (hc : 0≤c) (hbound : ∀ i, ‖A*P i‖≤c) : ‖A‖≤c := by
+  apply ContinuousLinearMap.opNorm_le_bound _ hc
+  intro v
+  have hsplit (w : V) : ∑ i,P i w=w := by
+    rw [←sum_apply,hsum]
+    rfl
+  have hp := orthogonal_sum_norm_sq (fun i => P i v)
+    (fun i j hij => projection_inner_zero _ _ (hP i) (horth i j hij) v v)
+  rw [hsplit] at hp
+  have ha := orthogonal_sum_norm_sq (fun i => P i (A v))
+    (fun i j hij => projection_inner_zero _ _ (hP i) (horth i j hij) (A v) (A v))
+  rw [hsplit] at ha
+  have he (i : ι) : P i (A v)=(A*P i) (P i v) := by
+    change (P i*A) v=(A*P i*P i) v
+    rw [mul_assoc,(hP i).isIdempotentElem.eq,(hcomm i).eq]
+  have hh (i : ι) : ‖P i (A v)‖≤c*‖P i v‖ := by
+    rw [he]
+    exact ((A*P i).le_opNorm _).trans (mul_le_mul_of_nonneg_right (hbound i) (norm_nonneg _))
+  have hsq := Finset.sum_le_sum (s := Finset.univ) (fun i _ =>
+    pow_le_pow_left₀ (norm_nonneg _) (hh i) 2)
+  simp_rw [mul_pow] at hsq
+  rw [←Finset.mul_sum,←hp,←ha,←mul_pow] at hsq
+  exact (sq_le_sq₀ (norm_nonneg _) (mul_nonneg hc (norm_nonneg _))).mp hsq
+
+theorem sector_sandwich_le (T P A B : V →L[ℂ] V)
+    (hT : IsStarProjection T) (hP : IsStarProjection P)
+    (hTP : T*P=T) (hcomm : Commute A T) (hA : ‖A‖≤1) :
+    ‖B.adjoint*A*T*B‖≤‖B.adjoint*P*B‖ := by
+  have he : B.adjoint*A*T*B=(T*B).adjoint*A*(T*B) := by
+    change _=star (T*B)*A*(T*B)
+    rw [star_mul,hT.isSelfAdjoint.star_eq]
+    change _=B.adjoint*T*A*(T*B)
+    rw [mul_assoc B.adjoint T A,←hcomm.eq]
+    simp only [mul_assoc]
+    rw [←mul_assoc T T B,hT.isIdempotentElem.eq]
+  have hTB : ‖T*B‖≤‖P*B‖ := by
+    have he : T*B=T*(P*B) := by rw [←mul_assoc,hTP]
+    rw [he]
+    exact (norm_mul_le _ _).trans ((mul_le_mul_of_nonneg_right (hT.norm_le _) (norm_nonneg _)).trans_eq (one_mul _))
+  rw [he,projection_sandwich_norm P B hP]
+  calc
+    _ ≤ ‖(T*B).adjoint‖*‖A‖*‖T*B‖ := (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right (norm_mul_le _ _) (norm_nonneg _))
+    _ ≤ ‖T*B‖^2 := by rw [ContinuousLinearMap.adjoint.norm_map]; nlinarith [norm_nonneg (T*B)]
+    _ ≤ _ := pow_le_pow_left₀ (norm_nonneg _) hTB 2
+
+omit [CompleteSpace V] [Fintype ι] in
+lemma resolution_commute (P : ι → V →L[ℂ] V)
+    (horth : ∀ i j, i≠j → P i*P j=0) (i j : ι) : Commute (P i) (P j) := by
+  by_cases hij : i=j
+  · subst j; exact Commute.refl _
+  · exact (horth i j hij).trans (horth j i (Ne.symm hij)).symm
+
+omit [Fintype ι] in
+lemma resolution_sum_mul (P : ι → V →L[ℂ] V) (hP : ∀ i, IsStarProjection (P i))
+    (horth : ∀ i j, i≠j → P i*P j=0) (S : Finset ι) (j : ι) :
+    (∑ i∈S,P i)*P j=if j∈S then P j else 0 := by
+  rw [Finset.sum_mul]
+  have he (i : ι) : P i*P j=if i=j then P j else 0 := by
+    split_ifs with h
+    · subst i; exact (hP j).isIdempotentElem.eq
+    · exact horth i j h
+  simp only [he,Finset.sum_ite_eq']
+
+lemma resolution_truncated_norm (P : ι → V →L[ℂ] V)
+    (hP : ∀ i, IsStarProjection (P i))
+    (horth : ∀ i j, i≠j → P i*P j=0) (hsum : ∑ i,P i=1)
+    (A : V →L[ℂ] V) (hcomm : ∀ i, Commute A (P i))
+    (S : Finset ι) (c : ℝ) (hc : 0≤c) (hbound : ∀ i∈S, ‖A*P i‖≤c) :
+    ‖A*(∑ i∈S,P i)‖≤c := by
+  apply resolution_norm_bound P hP horth hsum _ _ c hc
+  · intro i
+    rw [mul_assoc,resolution_sum_mul P hP horth]
+    split_ifs with hi
+    · exact hbound i hi
+    · simp [hc]
+  · intro i
+    exact (hcomm i).mul_left (Commute.sum_left S P (P i) fun j _ => resolution_commute P horth j i)
+
+theorem resolution_exception_bound (T Q : ι → V →L[ℂ] V)
+    (hT : ∀ i, IsStarProjection (T i))
+    (horth : ∀ i j, i≠j → T i*T j=0) (hsum : ∑ i,T i=1)
+    (hQ : ∀ i, IsStarProjection (Q i)) (hTQ : ∀ i,T i*Q i=T i)
+    (A B : V →L[ℂ] V) (hcomm : ∀ i, Commute A (T i))
+    (hA : ‖A‖≤1) (hB : ‖B‖≤1)
+    (S : Finset ι) (c : ℝ) (hc : 0≤c) (hbound : ∀ i∉S, ‖A*T i‖≤c) :
+    ‖B.adjoint*A*B‖≤c+∑ i∈S,‖B.adjoint*Q i*B‖ := by
+  have hhigh : ‖A*(∑ i∈Sᶜ,T i)‖≤c := resolution_truncated_norm T hT horth hsum A hcomm
+    Sᶜ c hc (fun i hi => hbound i (Finset.mem_compl.mp hi))
+  have hsplit : (∑ i∈Sᶜ,T i)+(∑ i∈S,T i)=1 := by
+    rw [Finset.sum_compl_add_sum,hsum]
+  have he : B.adjoint*A*B=B.adjoint*(A*(∑ i∈Sᶜ,T i))*B+
+      ∑ i∈S,B.adjoint*A*T i*B := by
+    rw [←Finset.sum_mul,←Finset.mul_sum]
+    rw [←add_mul]
+    simp only [mul_assoc]
+    rw [←mul_add,←mul_add,hsplit]
+    simp only [mul_one,mul_assoc]
+  rw [he]
+  apply (norm_add_le _ _).trans
+  apply add_le_add
+  · calc
+      _ ≤ ‖B.adjoint‖*‖A*(∑ i∈Sᶜ,T i)‖*‖B‖ :=
+        (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right (norm_mul_le _ _) (norm_nonneg _))
+      _ ≤ 1*c*1 := mul_le_mul (mul_le_mul (by simpa only [ContinuousLinearMap.adjoint.norm_map] using hB)
+        hhigh (norm_nonneg _) zero_le_one) hB (norm_nonneg _) (by simpa using hc)
+      _ = c := by ring
+  · apply (norm_sum_le _ _).trans
+    exact Finset.sum_le_sum fun i _ => sector_sandwich_le _ _ _ _ (hT i) (hQ i) (hTQ i) (hcomm i) hA
+
+variable [DecidableEq ι]
+variable (P : ι → V →L[ℂ] V) (hcomm : ∀ i j, Commute (P i) (P j))
+
+noncomputable def choice (E : Finset ι) (i : ι) : V →L[ℂ] V :=
+  if i∈E then 1-P i else P i
+
+include hcomm
+omit [CompleteSpace V] [Fintype ι] in
+lemma choice_commute (E F : Finset ι) (i j : ι) :
+    Commute (choice P E i) (choice P F j) := by
+  unfold choice
+  split_ifs
+  · exact (Commute.one_left _).sub_left ((Commute.one_right _).sub_right (hcomm i j))
+  · exact (Commute.one_left _).sub_left (hcomm i j)
+  · exact (Commute.one_right _).sub_right (hcomm i j)
+  · exact hcomm i j
+
+noncomputable def sector (s E : Finset ι) : V →L[ℂ] V :=
+  s.noncommProd (choice P E) (fun i _ j _ _ => choice_commute P hcomm E E i j)
+
+omit [CompleteSpace V] [Fintype ι] in
+lemma sector_insert (s E : Finset ι) (a : ι) (ha : a∉s) :
+    sector P hcomm (insert a s) E=choice P E a*sector P hcomm s E := by
+  exact Finset.noncommProd_insert_of_notMem _ _ _ _ ha
+
+end Thorp.CommutingBlocks
+
+end ThorpNine.Harmonic
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Harmonic/BlockProduct.lean b/lean/OAI/Probability/ThorpRouting/Harmonic/BlockProduct.lean
index 80161aeba7bfdc24dc3df6c8119377f886d7ba27..c812ca4936a8a9cb09c373cb912f3b1f03da59b1 100644
--- a/lean/OAI/Probability/ThorpRouting/Harmonic/BlockProduct.lean
+++ b/lean/OAI/Probability/ThorpRouting/Harmonic/BlockProduct.lean
@@ -1,892 +1,892 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Harmonic.AverageMap
-
-namespace OAI
-
-namespace ThorpNine.Harmonic
-
-namespace Thorp.CommutingBlocks
-open scoped BigOperators Classical
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
-variable {ι : Type*} [Fintype ι]
-variable [DecidableEq ι]
-variable (P : ι → V →L[ℂ] V) (hcomm : ∀ i j, Commute (P i) (P j))
-include hcomm
-omit [CompleteSpace V] [Fintype ι] in
-lemma sector_congr (s E F : Finset ι) (h : ∀ i∈s, (i∈E ↔ i∈F)) :
-    sector P hcomm s E=sector P hcomm s F := by
-  apply Finset.noncommProd_congr rfl
-  intro i hi
-  simp only [choice,h i hi]
-
-omit [CompleteSpace V] [Fintype ι] in
-lemma sector_factor_commute (s E F : Finset ι) (i : ι) :
-    Commute (choice P E i) (sector P hcomm s F) :=
-  Finset.noncommProd_commute _ _ _ _ (fun j _ => choice_commute P hcomm E F i j)
-
-omit [Fintype ι] in
-lemma sector_projection (hP : ∀ i, IsStarProjection (P i)) (s E : Finset ι) :
-    IsStarProjection (sector P hcomm s E) := by
-  induction s using Finset.induction_on with
-  | empty => exact IsStarProjection.one _
-  | @insert a s ha ih =>
-    rw [sector_insert P hcomm s E a ha]
-    have hc : IsStarProjection (choice P E a) := by
-      unfold choice
-      split_ifs
-      · exact (hP a).one_sub
-      · exact hP a
-    exact hc.mul ih (sector_factor_commute P hcomm s E E a)
-
-omit [CompleteSpace V] [Fintype ι] in
-lemma sector_sum (s : Finset ι) : ∑ E∈s.powerset, sector P hcomm s E=1 := by
-  induction s using Finset.induction_on with
-  | empty =>
-    simp only [Finset.powerset_empty,Finset.sum_singleton]
-    exact Finset.noncommProd_empty _ _
-  | @insert a s ha ih =>
-    rw [Finset.sum_powerset_insert ha]
-    have h₁ (E : Finset ι) (hE : E∈s.powerset) :
-        sector P hcomm (insert a s) E=P a*sector P hcomm s E := by
-      rw [sector_insert P hcomm s E a ha]
-      congr 1
-      exact ite_eq_right (fun h => ha ((Finset.mem_powerset.mp hE) h))
-    have h₂ (E : Finset ι) (_hE : E∈s.powerset) :
-        sector P hcomm (insert a s) (insert a E)=(1-P a)*sector P hcomm s E := by
-      rw [sector_insert P hcomm s (insert a E) a ha]
-      simp only [choice,Finset.mem_insert_self,↓reduceIte]
-      congr 1
-      apply sector_congr
-      intro i hi
-      simp [ne_of_mem_of_not_mem hi ha]
-    rw [Finset.sum_congr rfl h₁,Finset.sum_congr rfl h₂,
-      ←Finset.mul_sum,←Finset.mul_sum,ih]
-    simp
-
-omit [Fintype ι] in
-lemma sector_absorb_factor (s E : Finset ι) (i : ι) (hi : i∈s)
-    (hP : ∀ i, IsStarProjection (P i)) :
-    choice P E i*sector P hcomm s E=sector P hcomm s E := by
-  have hc : IsIdempotentElem (choice P E i) := by
-    unfold choice
-    split_ifs
-    · exact (hP i).one_sub.isIdempotentElem
-    · exact (hP i).isIdempotentElem
-  have hsplit := Finset.mul_noncommProd_erase s hi (choice P E)
-    (fun i _ j _ _ => choice_commute P hcomm E E i j)
-  change choice P E i*sector P hcomm s E=sector P hcomm s E
-  change _=sector P hcomm s E at hsplit
-  nth_rw 1 [←hsplit]
-  rw [←mul_assoc,hc.eq,hsplit]
-
-omit [Fintype ι] in
-lemma sector_orthogonal (s E F : Finset ι) (hE : E⊆s) (hF : F⊆s) (hEF : E≠F)
-    (hP : ∀ i, IsStarProjection (P i)) :
-    sector P hcomm s E*sector P hcomm s F=0 := by
-  have hdiff : ∃ i∈s, ¬(i∈E ↔ i∈F) := by
-    by_contra h
-    push Not at h
-    exact hEF (Finset.ext fun i => by
-      by_cases hi : i∈s
-      · exact h i hi
-      · exact iff_of_false (fun h => hi (hE h)) (fun h => hi (hF h)))
-  obtain ⟨i,hi,hdiff⟩ := hdiff
-  have hz : choice P E i*choice P F i=0 := by
-    by_cases he : i∈E <;> by_cases hf : i∈F
-    · exact (hdiff (iff_of_true he hf)).elim
-    · simpa only [choice,ite_eq_left he,ite_eq_right hf] using (hP i).one_sub_mul_self
-    · simpa only [choice,ite_eq_right he,ite_eq_left hf] using (hP i).mul_one_sub_self
-    · exact (hdiff (iff_of_false he hf)).elim
-  have hEi := sector_absorb_factor P hcomm s E i hi hP
-  have hFi := sector_absorb_factor P hcomm s F i hi hP
-  calc
-    _ = sector P hcomm s E*(choice P E i*choice P F i)*sector P hcomm s F := by
-      simp only [mul_assoc]
-      rw [hFi]
-      rw [←mul_assoc,(sector_factor_commute P hcomm s E E i).eq.symm,hEi]
-    _ = 0 := by rw [hz,mul_zero,zero_mul]
-
-noncomputable def blockProduct (A : ι → V →L[ℂ] V)
-    (hA : ∀ i j, Commute (A i) (A j)) (s : Finset ι) : V →L[ℂ] V :=
-  s.noncommProd A (fun i _ j _ _ => hA i j)
-
-omit hcomm [Fintype ι] in
-lemma norm_noncommProd_le (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
-    (s : Finset ι) : ‖blockProduct A hA s‖≤∏ i∈s,‖A i‖ := by
-  induction s using Finset.induction_on with
-  | empty => exact ContinuousLinearMap.norm_id_le
-  | @insert a s ha ih =>
-    have he : blockProduct A hA (insert a s)=A a*blockProduct A hA s :=
-      Finset.noncommProd_insert_of_notMem _ _ _ _ ha
-    rw [he,Finset.prod_insert ha]
-    exact (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_left ih (norm_nonneg _))
-
-noncomputable def blocks (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j)) : V →L[ℂ] V :=
-  blockProduct A hA Finset.univ
-
-omit hcomm in
-lemma blocks_norm_le_one (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
-    (hn : ∀ i,‖A i‖≤1) : ‖blocks A hA‖≤1 :=
-  (norm_noncommProd_le A hA Finset.univ).trans (Finset.prod_le_one₀ (fun _ _ => norm_nonneg _) (fun i _ => hn i))
-
-omit [CompleteSpace V] in
-lemma blocks_sector_commute (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
-    (hAP : ∀ i j, Commute (A i) (P j)) (E : Finset ι) :
-    Commute (blocks A hA) (sector P hcomm Finset.univ E) := by
-  unfold blocks blockProduct
-  apply (Finset.noncommProd_commute _ _ (fun i _ j _ _ => hA i j) _ _).symm
-  intro i _
-  unfold sector
-  apply (Finset.noncommProd_commute _ _ (fun i _ j _ _ => choice_commute P hcomm E E i j) _ _).symm
-  intro j _
-  unfold choice
-  split_ifs
-  · exact (Commute.one_right _).sub_right (hAP i j)
-  · exact hAP i j
-
-lemma blocks_sector_norm (hP : ∀ i, IsStarProjection (P i))
-    (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
-    (hAP : ∀ i j, Commute (A i) (P j)) (hn : ∀ i,‖A i‖≤1)
-    (θ : ℝ) (_hθ : 0≤θ) (hgap : ∀ i,‖A i*(1-P i)‖≤θ) (E : Finset ι) :
-    ‖blocks A hA*sector P hcomm Finset.univ E‖≤θ^E.card := by
-  have hAC (i j : ι) : Commute (A i) (choice P E j) := by
-    unfold choice
-    split_ifs
-    · exact (Commute.one_right _).sub_right (hAP i j)
-    · exact hAP i j
-  have hprod := Finset.noncommProd_mul_distrib (s := Finset.univ) A (choice P E)
-    (fun i _ j _ _ => hA i j) (fun i _ j _ _ => choice_commute P hcomm E E i j)
-    (fun i _ j _ _ => (hAC j i).symm)
-  change _=blocks A hA*sector P hcomm Finset.univ E at hprod
-  rw [←hprod]
-  have hc (i j : ι) : Commute ((A*choice P E) i) ((A*choice P E) j) :=
-    ((hA i j).mul_right (hAC i j)).mul_left ((hAC j i).symm.mul_right (choice_commute P hcomm E E i j))
-  apply (norm_noncommProd_le _ hc Finset.univ).trans
-  calc
-    _ ≤ ∏ i : ι, if i∈E then θ else 1 := by
-      apply Finset.prod_le_prod₀ (fun i _ => norm_nonneg _)
-      intro i _
-      dsimp only [Pi.mul_apply]
-      unfold choice
-      split_ifs with hi
-      · exact hgap i
-      · exact (norm_mul_le _ _).trans (by nlinarith [hn i,(hP i).norm_le (P i),norm_nonneg (A i),norm_nonneg (P i)])
-    _ = _ := by rw [Fintype.prod_ite_mem,Finset.prod_const]
-
-noncomputable def coarse (E : Finset ι) : V →L[ℂ] V :=
-  Eᶜ.noncommProd P (fun i _ j _ _ => hcomm i j)
-
-lemma coarse_projection (hP : ∀ i, IsStarProjection (P i)) (E : Finset ι) :
-    IsStarProjection (coarse P hcomm E) := by
-  have he : coarse P hcomm E=sector P hcomm Eᶜ ∅ := by
-    apply Finset.noncommProd_congr rfl
-    intro i _
-    simp [choice]
-  rw [he]
-  exact sector_projection P hcomm hP _ _
-
-lemma sector_coarse (hP : ∀ i, IsStarProjection (P i)) (E : Finset ι) :
-    sector P hcomm Finset.univ E*coarse P hcomm E=sector P hcomm Finset.univ E := by
-  have hpi (i : ι) (hi : i∈Eᶜ) : sector P hcomm Finset.univ E*P i=sector P hcomm Finset.univ E := by
-    have h := sector_absorb_factor P hcomm Finset.univ E i (Finset.mem_univ i) hP
-    rw [(sector_factor_commute P hcomm Finset.univ E E i).eq] at h
-    simpa only [choice,ite_eq_right (Finset.mem_compl.mp hi)] using h
-  have hp (s : Finset ι) (hs : ∀ i∈s,
-      sector P hcomm Finset.univ E*P i=sector P hcomm Finset.univ E) :
-      sector P hcomm Finset.univ E*blockProduct P hcomm s=sector P hcomm Finset.univ E := by
-    induction s using Finset.induction_on with
-    | empty => exact mul_one _
-    | @insert a s ha ih =>
-      have he : blockProduct P hcomm (insert a s)=P a*blockProduct P hcomm s :=
-        Finset.noncommProd_insert_of_notMem _ _ _ _ ha
-      rw [he,←mul_assoc,hs a (Finset.mem_insert_self _ _)]
-      exact ih (fun i hi => hs i (Finset.mem_insert_of_mem hi))
-  exact hp Eᶜ hpi
-
-theorem blocks_exception_bound (hP : ∀ i, IsStarProjection (P i))
-    (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
-    (hAP : ∀ i j, Commute (A i) (P j)) (hn : ∀ i,‖A i‖≤1)
-    (θ : ℝ) (hθ : 0≤θ) (hθ1 : θ≤1) (hgap : ∀ i,‖A i*(1-P i)‖≤θ)
-    (B : V →L[ℂ] V) (hB : ‖B‖≤1) (n : ℕ) :
-    ‖B.adjoint*blocks A hA*B‖≤θ^n+
-      ∑ E : Finset ι, if E.card<n then ‖B.adjoint*coarse P hcomm E*B‖ else 0 := by
-  let T : Finset ι → V →L[ℂ] V := sector P hcomm Finset.univ
-  have hsum : ∑ E,T E=1 := by
-    simpa only [Finset.powerset_univ] using sector_sum P hcomm Finset.univ
-  have hbound := resolution_exception_bound T (coarse P hcomm)
-    (fun E => sector_projection P hcomm hP _ E)
-    (fun E F hEF => sector_orthogonal P hcomm _ E F (Finset.subset_univ _) (Finset.subset_univ _) hEF hP)
-    hsum (coarse_projection P hcomm hP) (sector_coarse P hcomm hP)
-    (blocks A hA) B (blocks_sector_commute P hcomm A hA hAP)
-    (blocks_norm_le_one A hA hn) hB (Finset.univ.filter fun E : Finset ι => E.card<n)
-    (θ^n) (pow_nonneg hθ _) (by
-      intro E hE
-      have hEn : n≤E.card := by simpa using hE
-      exact (blocks_sector_norm P hcomm hP A hA hAP hn θ hθ hgap E).trans
-        (pow_le_pow_of_le_one hθ hθ1 hEn))
-  simpa only [Finset.sum_filter] using hbound
-
-
-end Thorp.CommutingBlocks
-
-namespace Thorp.UnitaryFinite
-open scoped BigOperators Classical
-
-variable {G : Type*} [Group G] [Fintype G]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-variable {Ω Ξ : Type*} [Fintype Ω] [Fintype Ξ]
-
-lemma sampleOperator_weighted (ρ : Representation ℂ G V) (P : Ω → G) :
-    sampleOperator ρ P=weightedOperator ρ (sampleLaw P) := by
-  unfold sampleOperator weightedOperator sampleLaw finiteMean
-  simp only [Complex.ofReal_sum,div_eq_mul_inv,Finset.sum_mul,Finset.sum_smul]
-  rw [Finset.sum_comm]
-  simp only [Finset.smul_sum]
-  apply Finset.sum_congr rfl
-  intro ω _
-  calc
-    _ = ∑ g, if P ω=g then (Fintype.card Ω:ℂ)⁻¹ • continuousRepresentation ρ g else 0 := by
-      symm
-      simp only [Finset.sum_ite_eq,Finset.mem_univ,↓reduceIte]
-    _ = _ := by
-      apply Finset.sum_congr rfl
-      intro g _
-      split_ifs <;> simp_all
-
-lemma sampleOperator_norm_le [Nonempty Ω] (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (P : Ω → G) :
-    ‖sampleOperator ρ P‖≤1 := by
-  rw [sampleOperator_weighted]
-  exact (weightedOperator_norm_le ρ hρ _ (sampleLaw_nonneg _)).trans_eq (sampleLaw_sum P)
-
-omit [Fintype G] in
-lemma sampleOperator_equiv (ρ : Representation ℂ G V) (P : Ω → G) (e : Ξ ≃ Ω) :
-    sampleOperator ρ (P ∘ e)=sampleOperator ρ P := by
-  simp only [sampleOperator,Fintype.card_congr e,Function.comp_apply]
-  rw [Equiv.sum_comp e (fun ω => continuousRepresentation ρ (P ω))]
-
-omit [Fintype G] in
-lemma sampleOperator_prod (ρ : Representation ℂ G V) (P : Ω → G) (Q : Ξ → G) :
-    sampleOperator ρ (fun ω : Ω × Ξ => P ω.1*Q ω.2)=sampleOperator ρ P*sampleOperator ρ Q := by
-  simp only [sampleOperator,Fintype.card_prod,Nat.cast_mul,mul_inv_rev,map_mul,Fintype.sum_prod_type,
-    Finset.sum_mul,Finset.mul_sum,smul_mul_smul,Finset.smul_sum]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro ω _
-  apply Finset.sum_congr rfl
-  intro ξ _
-  rw [mul_comm]
-
-omit [Fintype G] in
-lemma sampleOperator_inv (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (P : Ω → G) :
-    sampleOperator ρ (fun ω => (P ω)⁻¹)=(sampleOperator ρ P).adjoint := by
-  simp only [sampleOperator,map_smulₛₗ,map_sum,map_inv₀,map_natCast,
-    continuousRepresentation_adjoint ρ hρ]
-
-lemma sampleOperator_palindrome (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V)
-    (hρ : IsUnitary ρ) :
-    sampleOperator ρ (palindromePerm d)=
-      sampleOperator ρ (fun ω => butterflyPerm d (decodeButterfly d ω))*
-      (sampleOperator ρ (fun ω => butterflyPerm d (decodeButterfly d ω))).adjoint := by
-  rw [←sampleOperator_inv ρ hρ,←sampleOperator_prod]
-  rfl
-
-omit [Fintype G] in
-lemma sampleOperator_one [Nonempty Ω] (ρ : Representation ℂ G V) :
-    sampleOperator ρ (fun _ : Ω => 1)=1 := by
-  simp only [sampleOperator,map_one,Finset.sum_const,Finset.card_univ,←Nat.cast_smul_eq_nsmul ℂ,
-    smul_smul]
-  rw [inv_mul_cancel₀ (by exact_mod_cast Fintype.card_ne_zero),one_smul]
-
-lemma sampleOperator_uniform (ρ : Representation ℂ G V) : sampleOperator ρ (id : G → G)=uniformOperator ρ := by
-  simp only [sampleOperator,uniformOperator,weightedOperator,Complex.ofReal_inv,Complex.ofReal_natCast,
-    Finset.smul_sum,id_eq]
-
-lemma sampleOperator_uniform_mul [Nonempty Ω] (ρ : Representation ℂ G V) (P : Ω → G) :
-    sampleOperator ρ (fun ω : Ω × G => P ω.1*ω.2⁻¹)=uniformOperator ρ := by
-  rw [sampleOperator_prod]
-  have he : sampleOperator ρ (fun g : G => g⁻¹)=uniformOperator ρ := by
-    rw [←sampleOperator_uniform]
-    exact sampleOperator_equiv ρ id (Equiv.inv G)
-  rw [he,sampleOperator_weighted,weighted_uniform_left,sampleLaw_sum,one_smul]
-
-end Thorp.UnitaryFinite
-
-namespace Thorp.FiniteAverage
-open scoped BigOperators Classical
-variable {I Ω A : Type*} [Fintype I] [DecidableEq I] [Fintype Ω]
-    [Ring A] [Algebra ℂ A]
-
-noncomputable def orderedProduct : MultilinearMap ℂ (fun _ : I => A) A :=
-  MultilinearMap.domDomCongr (Fintype.equivFin I).symm
-    (MultilinearMap.mkPiAlgebraFin ℂ (Fintype.card I) A)
-
-lemma orderedProduct_eq (f : I → A) (hc : Pairwise fun i j => Commute (f i) (f j)) :
-    orderedProduct f=Finset.univ.noncommProd f (fun _ _ _ _ hij => hc hij) := by
-  unfold orderedProduct
-  rw [MultilinearMap.domDomCongr_apply,MultilinearMap.mkPiAlgebraFin_apply]
-  let l := List.ofFn (Fintype.equivFin I).symm
-  have hl : l.Nodup := List.nodup_ofFn.mpr (Fintype.equivFin I).symm.injective
-  have hu : l.toFinset=Finset.univ := by
-    ext i
-    simp only [List.mem_toFinset,List.mem_ofFn,Finset.mem_univ,iff_true,l]
-    exact (Fintype.equivFin I).symm.surjective i
-  have he := Finset.noncommProd_toFinset l f (by intro i hi j hj hij; exact hc hij) hl
-  rw [hu] at he
-  rw [he]
-  congr 1
-  exact List.map_ofFn.symm
-
-lemma multilinear_average (F : MultilinearMap ℂ (fun _ : I => A) A) (x : I → Ω → A) :
-    F (fun i => (Fintype.card Ω:ℂ)⁻¹ • ∑ ω,x i ω)=
-      (Fintype.card (I → Ω):ℂ)⁻¹ • ∑ ω : I → Ω,F (fun i => x i (ω i)) := by
-  rw [F.map_smul_univ,F.map_sum]
-  simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fun,Nat.cast_pow,inv_pow]
-
-lemma ordered_average (x : I → Ω → A)
-    (hc : ∀ ω, Pairwise fun i j => Commute (x i (ω i)) (x j (ω j)))
-    (havg : Pairwise fun i j => Commute ((Fintype.card Ω:ℂ)⁻¹ • ∑ ω,x i ω)
-      ((Fintype.card Ω:ℂ)⁻¹ • ∑ ω,x j ω)) :
-    (Finset.univ.noncommProd (fun i => (Fintype.card Ω:ℂ)⁻¹ • ∑ ω,x i ω)
-      (fun _ _ _ _ hij => havg hij))=
-    (Fintype.card (I → Ω):ℂ)⁻¹ • ∑ ω : I → Ω,
-      Finset.univ.noncommProd (fun i => x i (ω i)) (fun _ _ _ _ hij => hc ω hij) := by
-  rw [←orderedProduct_eq _ havg,multilinear_average]
-  simp only [orderedProduct_eq _ (hc _)]
-
-end Thorp.FiniteAverage
-namespace Thorp.FiniteCharacters
-open Module
-open scoped BigOperators Classical
-
-variable {G ι : Type*} [Group G] [Fintype G] [Fintype ι]
-variable (V : ι → Type*) [∀ i, AddCommGroup (V i)] [∀ i, Module ℂ (V i)]
-  [∀ i, FiniteDimensional ℂ (V i)]
-variable (ρ : ∀ i, Representation ℂ G (V i)) [∀ i, Representation.IsIrreducible (ρ i)]
-variable (hneq : ∀ i j, i≠j → ¬Nonempty (Representation.Equiv (ρ i) (ρ j)))
-  (hcard : Nat.card (ConjClasses G)≤Fintype.card ι)
-include hneq hcard
-
-lemma character_expansion (f : ConjClasses G → ℂ) :
-    f=∑ i, characterDual (ρ i) f • characterClass (ρ i) := by
-  let : Fintype (ConjClasses G) := Fintype.ofFinite _
-  have horth (i j : ι) : characterDual (ρ i) (characterClass (ρ j))=if i=j then 1 else 0 := by
-    rw [characterDual_characterClass]
-    by_cases h : i=j
-    · subst j; rw [ite_eq_left ⟨Representation.Equiv.refl _⟩,ite_eq_left rfl]
-    · rw [ite_eq_right h,ite_eq_right (hneq i j h)]
-  have hLI : LinearIndependent ℂ (fun i => characterClass (ρ i)) := by
-    apply LinearIndependent.of_pairwise_dual_eq_zero_one _ (fun i => characterDual (ρ i))
-    · intro i j hij; exact (horth i j).trans (ite_eq_right hij)
-    · intro i; simpa using horth i i
-  have heq : Fintype.card ι=Module.finrank ℂ (ConjClasses G → ℂ) := by
-    apply Nat.le_antisymm hLI.fintype_card_le_finrank
-    simpa only [Module.finrank_pi,Module.finrank_self,Finset.sum_const,Finset.card_univ,
-      smul_eq_mul,mul_one,Nat.card_eq_fintype_card] using hcard
-  let b := Basis.mk hLI (by rw [hLI.span_eq_top_of_card_eq_finrank' heq])
-  have hb (i : ι) : b i=characterClass (ρ i) := Basis.mk_apply _ _ _
-  have hd (i : ι) : b.coord i=characterDual (ρ i) := by
-    apply b.ext
-    intro j
-    rw [hb,horth]
-    simp only [←hb,Basis.coord_apply,Basis.repr_self,Finsupp.single_apply]
-    split_ifs <;> simp_all
-  have h := b.sum_repr f
-  simpa only [hb,show ∀ i,b.repr f i=characterDual (ρ i) f from fun i => congrArg (fun l => l f) (hd i)] using h.symm
-
-theorem sum_dimension_sq : ∑ i, (Module.finrank ℂ (V i))^2=Fintype.card G := by
-  let f : ConjClasses G → ℂ := fun c => if c=ConjClasses.mk (1:G) then 1 else 0
-  have hf (g : G) : f (ConjClasses.mk g)=if g=1 then 1 else 0 := by
-    simp only [f,ConjClasses.mk_eq_mk_iff_isConj,isConj_one_left]
-  have hd (i : ι) : characterDual (ρ i) f=(Fintype.card G:ℂ)⁻¹*(Module.finrank ℂ (V i):ℂ) := by
-    change (Nat.card G:ℂ)⁻¹ * ∑ g, f (ConjClasses.mk g)*(ρ i).character g⁻¹=_
-    simp only [hf,ite_mul,one_mul,zero_mul,Finset.sum_ite_eq',Finset.mem_univ,↓reduceIte,inv_one,
-      Representation.char_one,Nat.card_eq_fintype_card]
-  have he := congrFun (character_expansion V ρ hneq hcard f) (ConjClasses.mk (1:G))
-  rw [hf,ite_eq_left rfl] at he
-  simp only [Finset.sum_apply,Pi.smul_apply,smul_eq_mul,hd,characterClass_mk,Representation.char_one] at he
-  have hN : (Fintype.card G:ℂ)≠0 := by exact_mod_cast Fintype.card_ne_zero
-  have hs : (∑ i,(Module.finrank ℂ (V i):ℂ)^2)=(Fintype.card G:ℂ) := by
-    calc
-      _ = (Fintype.card G:ℂ) * ∑ i,(Fintype.card G:ℂ)⁻¹ *
-          (Module.finrank ℂ (V i):ℂ)*(Module.finrank ℂ (V i):ℂ) := by
-        rw [Finset.mul_sum]
-        apply Finset.sum_congr rfl
-        intro i _
-        field_simp
-      _ = _ := by rw [←he,mul_one]
-  exact_mod_cast hs
-
-end Thorp.FiniteCharacters
-
-namespace Thorp.UnitaryFinite
-open scoped BigOperators ComplexConjugate Classical
-
-variable {G : Type*} [Group G] [Fintype G]
-variable {V W : Type*} [NormedAddCommGroup V] [NormedAddCommGroup W]
-  [InnerProductSpace ℂ V] [InnerProductSpace ℂ W]
-  [FiniteDimensional ℂ V] [FiniteDimensional ℂ W]
-
-omit [FiniteDimensional ℂ V] [FiniteDimensional ℂ W] in
-lemma averageMap_zero (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
-    [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ]
-    (hne : ¬Nonempty (Representation.Equiv ρ σ)) (A : V →ₗ[ℂ] W) :
-    averageMap ρ σ A=0 := by
-  rcases Representation.IsIrreducible.bijective_or_eq_zero (averageMap ρ σ A) with h|h
-  · exact (hne ⟨Representation.IntertwiningMap.ofBijective _ h⟩).elim
-  · exact h
-
-omit [FiniteDimensional ℂ V] [FiniteDimensional ℂ W] in
-lemma coefficient_cross_orthogonality (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
-    [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ]
-    (hρ : IsUnitary ρ) (hne : ¬Nonempty (Representation.Equiv ρ σ))
-    (u z : W) (v w : V) :
-    ∑ g, inner ℂ u (σ g z)*inner ℂ (ρ g w) v=0 := by
-  have he := averageMap_zero ρ σ hne (InnerProductSpace.rankOne ℂ z w).toLinearMap
-  have hv := congrArg (fun T : Representation.IntertwiningMap ρ σ => inner ℂ u (T v)) he
-  change inner ℂ u ((∑ g, (σ g).comp ((InnerProductSpace.rankOne ℂ z w).toLinearMap.comp (ρ g⁻¹))) v)=_ at hv
-  simpa only [LinearMap.sum_apply,LinearMap.comp_apply,ContinuousLinearMap.coe_coe,
-    InnerProductSpace.rankOne_apply,map_smul,unitary_inner_inv ρ hρ,inner_sum,
-    inner_smul_right,Representation.IntertwiningMap.coe_zero,Pi.zero_apply,inner_zero_right,mul_comm] using hv
-
-end Thorp.UnitaryFinite
-namespace Thorp.UnitaryFinite
-open scoped BigOperators ComplexConjugate Classical
-
-variable {G : Type*} [Group G] [Fintype G]
-
-def regularAction : G →* Equiv.Perm G where
-  toFun := Equiv.mulLeft
-  map_one' := by ext; simp
-  map_mul' _ _ := by ext; simp
-
-omit [Fintype G] in
-lemma regular_transitive (x : G) : ∃ g,regularAction g (1:G)=x := ⟨x,mul_one x⟩
-
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-omit [Fintype G] [FiniteDimensional ℂ V] in
-lemma regular_fixed (ρ : Representation ℂ G V) (w : V) : IsFixed regularAction (1:G) ρ w := by
-  intro g hg
-  have he : g=1 := by simpa only [regularAction,MonoidHom.coe_mk,OneHom.coe_mk,Equiv.coe_mulLeft,mul_one] using hg
-  subst g
-  simp
-
-noncomputable def regularCoefficient (ρ : Representation ℂ G V) (w v : V) : EuclideanSpace ℂ G :=
-  normalizedCoefficient regularAction 1 regular_transitive ρ w v
-
-omit [FiniteDimensional ℂ V] in
-lemma regularCoefficient_apply (ρ : Representation ℂ G V) (w v : V) (g : G) :
-    regularCoefficient ρ w v g = (Real.sqrt ((Module.finrank ℂ V:ℝ)/(Fintype.card G:ℝ)):ℂ)*inner ℂ (ρ g w) v := by
-  unfold regularCoefficient normalizedCoefficient
-  simp only [PiLp.smul_apply,smul_eq_mul]
-  congr 1
-  have he := coefficient_at regularAction 1 regular_transitive ρ w (regular_fixed ρ w) v g
-  simpa only [regularAction,MonoidHom.coe_mk,OneHom.coe_mk,Equiv.coe_mulLeft,mul_one] using he
-
-lemma regularCoefficient_inner (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
-    (hρ : IsUnitary ρ) (w z v t : V) :
-    inner ℂ (regularCoefficient ρ w v) (regularCoefficient ρ z t)=inner ℂ z w*inner ℂ v t :=
-  normalizedCoefficient_inner regularAction 1 regular_transitive ρ hρ w z
-    (regular_fixed ρ w) (regular_fixed ρ z) v t
-
-variable {W : Type*} [NormedAddCommGroup W] [InnerProductSpace ℂ W] [FiniteDimensional ℂ W]
-
-omit [FiniteDimensional ℂ V] [FiniteDimensional ℂ W] in
-lemma regularCoefficient_cross (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
-    [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ]
-    (hσ : IsUnitary σ) (hne : ¬Nonempty (Representation.Equiv σ ρ))
-    (w v : V) (z t : W) : inner ℂ (regularCoefficient ρ w v) (regularCoefficient σ z t)=0 := by
-  rw [PiLp.inner_apply]
-  simp only [RCLike.inner_apply,regularCoefficient_apply,map_mul,Complex.conj_ofReal,
-    inner_conj_symm]
-  have he := coefficient_cross_orthogonality σ ρ hσ hne v w t z
-  calc
-    _ = ((Real.sqrt ((Module.finrank ℂ V:ℝ)/(Fintype.card G:ℝ)):ℂ)*
-        (Real.sqrt ((Module.finrank ℂ W:ℝ)/(Fintype.card G:ℝ)):ℂ)) *
-        ∑ g,inner ℂ v (ρ g w)*inner ℂ (σ g z) t := by
-      rw [Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro g _
-      ring
-    _ = 0 := by rw [he,mul_zero]
-
-end Thorp.UnitaryFinite
-
-namespace Thorp.FiniteFourier
-open scoped BigOperators ComplexConjugate Classical
-open UnitaryFinite
-variable {G ι : Type*} [Group G] [Fintype G] [Fintype ι]
-variable (V : ι → Type*) [∀ i, NormedAddCommGroup (V i)] [∀ i, InnerProductSpace ℂ (V i)]
-  [∀ i, FiniteDimensional ℂ (V i)]
-variable (ρ : ∀ i, Representation ℂ G (V i)) [∀ i, Representation.IsIrreducible (ρ i)]
-variable (hρ : ∀ i, IsUnitary (ρ i))
-variable (hneq : ∀ i j, i≠j → ¬Nonempty (Representation.Equiv (ρ i) (ρ j)))
-  (hcard : Nat.card (ConjClasses G)≤Fintype.card ι)
-
-
-abbrev CoefficientIndex := Σ i,Fin (Module.finrank ℂ (V i)) × Fin (Module.finrank ℂ (V i))
-
-noncomputable def coefficientFamily (a : CoefficientIndex V) : EuclideanSpace ℂ G :=
-  regularCoefficient (ρ a.1) (stdOrthonormalBasis ℂ (V a.1) a.2.1) (stdOrthonormalBasis ℂ (V a.1) a.2.2)
-
-omit [Fintype ι] in
-include hρ hneq in
-lemma coefficientFamily_orthonormal : Orthonormal ℂ (coefficientFamily V ρ) := by
-  rw [orthonormal_iff_ite]
-  rintro ⟨i,a,b⟩ ⟨j,c,d⟩
-  dsimp only [coefficientFamily]
-  by_cases hij : i=j
-  · subst j
-    rw [regularCoefficient_inner _ (hρ i)]
-    simp only [(stdOrthonormalBasis ℂ (V i)).inner_eq_ite]
-    by_cases hac : a=c <;> by_cases hbd : b=d <;> simp [hac,hbd,eq_comm]
-  · rw [regularCoefficient_cross _ _ (hρ j) (hneq j i (Ne.symm hij))]
-    rw [ite_eq_right (by intro h; exact hij (congrArg Sigma.fst h))]
-
-include ρ hneq hcard in
-lemma coefficientIndex_card : Fintype.card (CoefficientIndex V)=Fintype.card G := by
-  simp only [CoefficientIndex,Fintype.card_sigma,Fintype.card_prod,Fintype.card_fin,←pow_two]
-  exact FiniteCharacters.sum_dimension_sq V ρ hneq hcard
-
-noncomputable def coefficientBasis : OrthonormalBasis (CoefficientIndex V) ℂ (EuclideanSpace ℂ G) :=
-  OrthonormalBasis.mk (coefficientFamily_orthonormal V ρ hρ hneq)
-    (by rw [(coefficientFamily_orthonormal V ρ hρ hneq).linearIndependent.span_eq_top_of_card_eq_finrank'
-          (by rw [coefficientIndex_card V ρ hneq hcard,finrank_euclideanSpace])])
-
-lemma coefficientBasis_apply (a : CoefficientIndex V) :
-    coefficientBasis V ρ hρ hneq hcard a=coefficientFamily V ρ a := congrFun (OrthonormalBasis.coe_mk _ _) a
-
-include hρ hneq hcard in
-
-theorem parseval (f : EuclideanSpace ℂ G) :
-    ∑ a : CoefficientIndex V, ‖inner ℂ (coefficientFamily V ρ a) f‖^2=‖f‖^2 := by
-  simpa only [coefficientBasis_apply] using
-    (coefficientBasis V ρ hρ hneq hcard).sum_sq_norm_inner_right f
-
-
-noncomputable def realVector (p : G → ℝ) : EuclideanSpace ℂ G := WithLp.toLp 2 (fun g => (p g:ℂ))
-
-omit [Fintype ι] [∀ i, Representation.IsIrreducible (ρ i)] in
-lemma coefficient_realVector (p : G → ℝ) (a : CoefficientIndex V) :
-    inner ℂ (coefficientFamily V ρ a) (realVector p)=
-      (Real.sqrt ((Module.finrank ℂ (V a.1):ℝ)/(Fintype.card G:ℝ)):ℂ)*
-      inner ℂ (stdOrthonormalBasis ℂ (V a.1) a.2.2)
-        (weightedOperator (ρ a.1) p (stdOrthonormalBasis ℂ (V a.1) a.2.1)) := by
-  simp only [PiLp.inner_apply,coefficientFamily,regularCoefficient_apply,realVector,
-    RCLike.inner_apply,map_mul,Complex.conj_ofReal,inner_conj_symm,
-    weightedOperator,sum_apply,smul_apply,
-    continuousRepresentation_apply,inner_sum,inner_smul_right,Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro g _
-  ring
-
-omit [Fintype ι] [∀ i, Representation.IsIrreducible (ρ i)] in
-lemma coefficient_realVector_sq (p : G → ℝ) (a : CoefficientIndex V) :
-    ‖inner ℂ (coefficientFamily V ρ a) (realVector p)‖^2=
-      ((Module.finrank ℂ (V a.1):ℝ)/(Fintype.card G:ℝ))*
-      ‖inner ℂ (stdOrthonormalBasis ℂ (V a.1) a.2.2)
-        (weightedOperator (ρ a.1) p (stdOrthonormalBasis ℂ (V a.1) a.2.1))‖^2 := by
-  rw [coefficient_realVector,norm_mul,Complex.norm_real,Real.norm_eq_abs,
-    abs_of_nonneg (Real.sqrt_nonneg _),mul_pow,Real.sq_sqrt (div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _))]
-
-include hρ hneq hcard in
-lemma parseval_operator (p : G → ℝ) :
-    (Fintype.card G:ℝ)*‖realVector p‖^2=
-      ∑ i,(Module.finrank ℂ (V i):ℝ)*∑ a : Fin (Module.finrank ℂ (V i)),
-        ‖weightedOperator (ρ i) p (stdOrthonormalBasis ℂ (V i) a)‖^2 := by
-  rw [←parseval V ρ hρ hneq hcard (realVector p)]
-  simp only [Fintype.sum_sigma,Fintype.sum_prod_type,coefficient_realVector_sq]
-  have hN : (Fintype.card G:ℝ)≠0 := by exact_mod_cast Fintype.card_ne_zero
-  rw [Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro i _
-  have he (a : Fin (Module.finrank ℂ (V i))) :
-      (∑ b : Fin (Module.finrank ℂ (V i)),
-        (Module.finrank ℂ (V i):ℝ)/(Fintype.card G:ℝ)*
-          ‖inner ℂ (stdOrthonormalBasis ℂ (V i) b)
-            (weightedOperator (ρ i) p (stdOrthonormalBasis ℂ (V i) a))‖^2)=
-        (Module.finrank ℂ (V i):ℝ)/(Fintype.card G:ℝ)*
-          ‖weightedOperator (ρ i) p (stdOrthonormalBasis ℂ (V i) a)‖^2 := by
-    rw [←Finset.mul_sum,(stdOrthonormalBasis ℂ (V i)).sum_sq_norm_inner_right]
-  simp_rw [he]
-  rw [←Finset.mul_sum]
-  field_simp
-
-include hρ hneq hcard in
-lemma realVector_bound (p : G → ℝ) :
-    (Fintype.card G:ℝ)*‖realVector p‖^2≤
-      ∑ i,(Module.finrank ℂ (V i):ℝ)^2*‖weightedOperator (ρ i) p‖^2 := by
-  rw [parseval_operator V ρ hρ hneq hcard]
-  apply Finset.sum_le_sum
-  intro i _
-  calc
-    _ ≤ (Module.finrank ℂ (V i):ℝ)*∑ _a : Fin (Module.finrank ℂ (V i)),‖weightedOperator (ρ i) p‖^2 := by
-      apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
-      apply Finset.sum_le_sum
-      intro a _
-      apply pow_le_pow_left₀ (norm_nonneg _)
-      simpa only [(stdOrthonormalBasis ℂ (V i)).norm_eq_one,mul_one] using
-        (weightedOperator (ρ i) p).le_opNorm (stdOrthonormalBasis ℂ (V i) a)
-    _ = _ := by simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]; ring
-
-include hρ hneq hcard in
-
-theorem l1_fourier_bound (p : G → ℝ) :
-    (∑ g,|p g|)^2≤∑ i,(Module.finrank ℂ (V i):ℝ)^2*‖weightedOperator (ρ i) p‖^2 := by
-  apply le_trans _ (realVector_bound V ρ hρ hneq hcard p)
-  have he : ‖realVector p‖^2=∑ g,|p g|^2 := by
-    rw [EuclideanSpace.norm_sq_eq]
-    simp only [realVector,PiLp.toLp_apply,Complex.norm_real,Real.norm_eq_abs]
-  rw [he]
-  simpa only [Finset.card_univ] using sq_sum_le_card_mul_sum_sq (s := Finset.univ) (f := fun g => |p g|)
-
-end Thorp.FiniteFourier
-
-namespace Thorp.ParallelBlocks
-open scoped BigOperators Classical
-variable {B X : Type*}
-
-def lift : (B → Equiv.Perm X) →* Equiv.Perm (B × X) where
-  toFun p :=
-    { toFun := fun x => (x.1,p x.1 x.2)
-      invFun := fun x => (x.1,(p x.1).symm x.2)
-      left_inv := fun x => by simp
-      right_inv := fun x => by simp }
-  map_one' := rfl
-  map_mul' _ _ := rfl
-
-@[simp] lemma lift_apply (p : B → Equiv.Perm X) (x : B × X) : lift p x=(x.1,p x.1 x.2) := rfl
-
-noncomputable def injection (b : B) : Equiv.Perm X →* Equiv.Perm (B × X) :=
-  lift.comp
-    { toFun := fun p c => if c=b then p else 1
-      map_one' := by funext c; simp
-      map_mul' := by intro p q; funext c; by_cases h : c=b <;> simp [h] }
-
-@[simp] lemma injection_apply (b : B) (p : Equiv.Perm X) (x : B × X) :
-    injection b p x=(x.1,if x.1=b then p x.2 else x.2) := by
-  simp only [injection,MonoidHom.comp_apply,lift_apply]
-  by_cases h : x.1=b <;> simp [h]
-
-lemma injection_commute {b c : B} (h : b≠c) (p q : Equiv.Perm X) :
-    Commute (injection b p) (injection c q) := by
-  apply Equiv.ext
-  intro x
-  simp only [Equiv.Perm.mul_apply,injection_apply]
-  by_cases hb : x.1=b <;> by_cases hc : x.1=c <;> simp_all
-
-variable [Fintype B]
-
-lemma lift_product (p : B → Equiv.Perm X) :
-    lift p=Finset.univ.noncommProd (fun b => injection b (p b))
-      (fun _ _ _ _ h => injection_commute h _ _) := by
-  have he (s : Finset B) (x : B × X) :
-      s.noncommProd (fun b => injection b (p b))
-        (fun _ _ _ _ h => injection_commute h _ _) x=
-          (x.1,if x.1∈s then p x.1 x.2 else x.2) := by
-    induction s using Finset.induction_on with
-    | empty =>
-      have hz : Finset.noncommProd (∅ : Finset B) (fun b => injection b (p b))
-          (fun _ _ _ _ h => injection_commute h _ _)=1 := Finset.noncommProd_empty _ _
-      rw [hz]; simp
-    | @insert a s ha ih =>
-      have hz : (insert a s).noncommProd (fun b => injection b (p b))
-          (fun _ _ _ _ h => injection_commute h _ _)=injection a (p a)*
-          s.noncommProd (fun b => injection b (p b))
-            (fun _ _ _ _ h => injection_commute h _ _) :=
-        Finset.noncommProd_insert_of_notMem _ _ _ _ ha
-      rw [hz,Equiv.Perm.mul_apply,ih,injection_apply]
-      by_cases h : x.1=a
-      · subst a; simp [ha]
-      · simp [h]
-  apply Equiv.ext
-  intro x
-  rw [he]
-  simp
-
-end Thorp.ParallelBlocks
-namespace Thorp.UnitaryFinite
-open scoped BigOperators Classical
-
-variable {G H : Type*} [Group G] [Group H]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-variable {Ω Ξ : Type*} [Fintype Ω] [Fintype Ξ]
-
-lemma continuousRepresentation_comp (ρ : Representation ℂ G V) (φ : H →* G) (h : H) :
-    continuousRepresentation (ρ.comp φ) h=continuousRepresentation ρ (φ h) := rfl
-
-lemma sampleOperator_comp (ρ : Representation ℂ G V) (φ : H →* G) (P : Ω → H) :
-    sampleOperator (ρ.comp φ) P=sampleOperator ρ (φ ∘ P) := rfl
-
-omit [FiniteDimensional ℂ V] in
-lemma unitary_comp (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (φ : H →* G) :
-    IsUnitary (ρ.comp φ) := fun h v w => hρ (φ h) v w
-
-lemma sampleOperator_commute (ρ : Representation ℂ G V) (P : Ω → G) (Q : Ξ → G)
-    (h : ∀ ω ξ, Commute (P ω) (Q ξ)) :
-    Commute (sampleOperator ρ P) (sampleOperator ρ Q) := by
-  unfold sampleOperator
-  apply Commute.smul_left
-  apply Commute.smul_right
-  apply Commute.sum_left
-  intro ω _
-  apply Commute.sum_right
-  intro ξ _
-  exact (h ω ξ).map (continuousRepresentation ρ)
-
-lemma sampleOperator_uniform_absorb [Fintype G] [Nonempty Ω]
-    (ρ : Representation ℂ G V) (P : Ω → G) :
-    sampleOperator ρ P*uniformOperator ρ=uniformOperator ρ := by
-  rw [sampleOperator_weighted,weighted_uniform_left,sampleLaw_sum,one_smul]
-
-end Thorp.UnitaryFinite
-namespace Thorp.ParallelBlocks
-open scoped BigOperators Classical
-open UnitaryFinite
-variable {B X Ω : Type*} [Fintype B] [DecidableEq B] [Fintype Ω]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-omit [Fintype B] [DecidableEq B] in
-lemma local_average_commute (ρ : Representation ℂ (Equiv.Perm (B × X)) V)
-    (P : B → Ω → Equiv.Perm X) (b c : B) :
-    Commute (sampleOperator (ρ.comp (injection b)) (P b))
-      (sampleOperator (ρ.comp (injection c)) (P c)) := by
-  by_cases h : b=c
-  · subst c; exact Commute.refl _
-  · rw [sampleOperator_comp,sampleOperator_comp]
-    exact sampleOperator_commute ρ _ _ (fun _ _ => injection_commute h _ _)
-
-lemma sampleOperator_parallel (ρ : Representation ℂ (Equiv.Perm (B × X)) V)
-    (P : B → Ω → Equiv.Perm X) :
-    sampleOperator ρ (fun ω : B → Ω => lift (fun b => P b (ω b)))=
-      CommutingBlocks.blocks (fun b => sampleOperator (ρ.comp (injection b)) (P b))
-        (local_average_commute ρ P) := by
-  let A (b : B) (w : Ω) := continuousRepresentation ρ (injection b (P b w))
-  have hc : ∀ ω : B → Ω, Pairwise fun b c => Commute (A b (ω b)) (A c (ω c)) := by
-    intro ω b c h
-    exact (injection_commute h _ _).map (continuousRepresentation ρ)
-  have havg : Pairwise fun b c => Commute ((Fintype.card Ω:ℂ)⁻¹ • ∑ w,A b w)
-      ((Fintype.card Ω:ℂ)⁻¹ • ∑ w,A c w) := by
-    intro b c _
-    exact local_average_commute ρ P b c
-  have he (ω : B → Ω) : continuousRepresentation ρ (lift (fun b => P b (ω b)))=
-      Finset.univ.noncommProd (fun b => A b (ω b)) (fun _ _ _ _ h => hc ω h) := by
-    rw [lift_product]
-    exact Finset.map_noncommProd _ _ _ (continuousRepresentation ρ)
-  unfold sampleOperator
-  simp_rw [he]
-  exact (FiniteAverage.ordered_average A hc havg).symm
-
-end Thorp.ParallelBlocks
-namespace Thorp
-open scoped BigOperators Classical
-
-noncomputable def splitPerm (r L : ℕ) : Equiv.Perm (Card (L+r)) ≃* Equiv.Perm (Card r × Card L) :=
-  (cardSplit r L).permCongrHom
-
-noncomputable def outerPerm (r L : ℕ) (hi : Card L × SwitchIndex r → Bool) :
-    Equiv.Perm (Card r × Card L) :=
-  (Equiv.prodComm (Card L) (Card r)).permCongrHom
-    (ParallelBlocks.lift (fun x => butterflyPerm r (decodeButterfly r (fun i => hi (x,i)))))
-
-lemma outerPerm_apply (r L : ℕ) (hi : Card L × SwitchIndex r → Bool) (x : Card r × Card L) :
-    outerPerm r L hi x=(butterflyPerm r (decodeButterfly r (fun i => hi (x.2,i))) x.1,x.2) := rfl
-
-lemma splitPerm_butterfly (r L : ℕ) (lo : Card r × SwitchIndex L → Bool)
-    (hi : Card L × SwitchIndex r → Bool) :
-    splitPerm r L (butterflyPerm (L+r) (decodeButterfly (L+r) (assembleBits r L lo hi)))=
-      outerPerm r L hi*ParallelBlocks.lift
-        (fun b => butterflyPerm L (decodeButterfly L (fun i => lo (b,i)))) := by
-  apply Equiv.ext
-  intro x
-  change cardSplit r L (butterflyPerm (L+r) (decodeButterfly (L+r) (assembleBits r L lo hi))
-    ((cardSplit r L).symm x))=_
-  rw [butterfly_split,Equiv.apply_symm_apply]
-  rfl
-
-lemma boundaryTree_node_raw (L r : ℕ) (U : Card (r+1) → Equiv.Perm (Card L))
-    (X : SwitchIndex (L+r+1) → Bool) :
-    boundaryTree L (r+1) U X=InputTree.node (fun y => X (Sum.inl y))
-      (fun b => boundaryTree L r (fun u => U (Fin.cons b u)) (fun i => X (Sum.inr (b,i)))) := by
-  conv_lhs => unfold boundaryTree
-  congr 1
-  funext b
-  cases b <;> simp only [Bool.false_eq_true,ite_false,ite_true] <;> rfl
-
-lemma boundaryOutput_node_raw (L r : ℕ) (Y : SwitchIndex (L+r+1) → Bool) (b : Card (r+1)) :
-    boundaryOutput L (r+1) Y b=
-      boundaryOutput L r (fun i => Y (Sum.inr (b 0,i))) (Fin.tail b) := by
-  conv_lhs => unfold boundaryOutput
-  cases b 0 <;> simp only [Bool.false_eq_true,ite_false,ite_true] <;> rfl
-
-lemma boundaryTree_split (L r : ℕ) (U : Card r → Equiv.Perm (Card L))
-    (lo : Card r × SwitchIndex L → Bool) (hi : Card L × SwitchIndex r → Bool) (x : Card (L+r)) :
-    cardSplit r L ((boundaryTree L r U (assembleBits r L lo hi)).perm x)=
-      let y := cardSplit r L x
-      let z := U y.1 y.2
-      (butterflyPerm r (decodeButterfly r (fun i => hi (z,i))) y.1,z) := by
-  induction r with
-  | zero =>
-    apply Prod.ext
-    · exact Subsingleton.elim _ _
-    · rfl
-  | succ r ih =>
-    let lo' : Card r × SwitchIndex L → Bool := fun j => lo (Fin.cons (x 0) j.1,j.2)
-    let hi' : Card L × SwitchIndex r → Bool := fun j => hi (j.1,Sum.inr (x 0,j.2))
-    let U' : Card r → Equiv.Perm (Card L) := fun b => U (Fin.cons (x 0) b)
-    have hh := ih U' lo' hi' (Fin.tail x)
-    let u := (boundaryTree L r U' (assembleBits r L lo' hi')).perm (Fin.tail x)
-    let y := cardSplit r L (Fin.tail x)
-    let z := U' y.1 y.2
-    let w := butterflyPerm r (decodeButterfly r (fun i => hi' (z,i))) y.1
-    change cardSplit r L u=(w,z) at hh
-    rw [boundaryTree_node_raw]
-    change ((Fin.cons (Bool.xor (x 0) (hi ((cardSplit r L u).2,Sum.inl (cardSplit r L u).1)))
-      ((cardSplit r L u).1) : Card (r+1)),(cardSplit r L u).2)=
-        ((Fin.cons (Bool.xor (x 0) (hi (z,Sum.inl w))) w : Card (r+1)),z)
-    rw [hh]
-
-lemma splitPerm_boundaryTree (L r : ℕ) (U : Card r → Equiv.Perm (Card L))
-    (lo : Card r × SwitchIndex L → Bool) (hi : Card L × SwitchIndex r → Bool) :
-    splitPerm r L (boundaryTree L r U (assembleBits r L lo hi)).perm=
-      outerPerm r L hi*ParallelBlocks.lift U := by
-  apply Equiv.ext
-  intro x
-  change cardSplit r L ((boundaryTree L r U (assembleBits r L lo hi)).perm
-    ((cardSplit r L).symm x))=_
-  rw [boundaryTree_split,Equiv.apply_symm_apply]
-  rfl
-
-lemma boundaryOutput_split (L r : ℕ) (lo : Card r × SwitchIndex L → Bool)
-    (hi : Card L × SwitchIndex r → Bool) (b : Card r) :
-    boundaryOutput L r (assembleBits r L lo hi) b=
-      butterflyPerm L (decodeButterfly L (fun i => lo (b,i))) := by
-  induction r with
-  | zero =>
-    rw [Subsingleton.elim b (fun j => Fin.elim0 j)]
-    rfl
-  | succ r ih =>
-    rw [boundaryOutput_node_raw]
-    change boundaryOutput L r (assembleBits r L (fun j => lo (Fin.cons (b 0) j.1,j.2))
-      (fun j => hi (j.1,Sum.inr (b 0,j.2)))) (Fin.tail b)=_
-    rw [ih]
-    simp only [Fin.cons_self_tail]
-
-
-abbrev OuterCoins (r L : ℕ) := Card L × SwitchIndex r → Bool
-abbrev LocalCoins (L : ℕ) := SwitchIndex L → Bool
-
-def parallelSplitCoinEquiv (r L : ℕ) :
-    ((Card r → LocalCoins L) × OuterCoins r L) ≃ LocalCoins (L+r) :=
-  (Equiv.prodCongr (Equiv.curry (Card r) (SwitchIndex L) Bool).symm (Equiv.refl _)).trans
-    (splitCoinEquiv r L)
-
-lemma parallelSplitCoinEquiv_apply (r L : ℕ) (c : Card r → LocalCoins L) (h : OuterCoins r L) :
-    parallelSplitCoinEquiv r L (c,h)=assembleBits r L (fun x => c x.1 x.2) h := by
-  exact splitCoinEquiv_apply _ _ _ _
-
-end Thorp
-
-end ThorpNine.Harmonic
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Harmonic.AverageMap
+
+namespace OAI
+
+namespace ThorpNine.Harmonic
+
+namespace Thorp.CommutingBlocks
+open scoped BigOperators Classical
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
+variable {ι : Type*} [Fintype ι]
+variable [DecidableEq ι]
+variable (P : ι → V →L[ℂ] V) (hcomm : ∀ i j, Commute (P i) (P j))
+include hcomm
+omit [CompleteSpace V] [Fintype ι] in
+lemma sector_congr (s E F : Finset ι) (h : ∀ i∈s, (i∈E ↔ i∈F)) :
+    sector P hcomm s E=sector P hcomm s F := by
+  apply Finset.noncommProd_congr rfl
+  intro i hi
+  simp only [choice,h i hi]
+
+omit [CompleteSpace V] [Fintype ι] in
+lemma sector_factor_commute (s E F : Finset ι) (i : ι) :
+    Commute (choice P E i) (sector P hcomm s F) :=
+  Finset.noncommProd_commute _ _ _ _ (fun j _ => choice_commute P hcomm E F i j)
+
+omit [Fintype ι] in
+lemma sector_projection (hP : ∀ i, IsStarProjection (P i)) (s E : Finset ι) :
+    IsStarProjection (sector P hcomm s E) := by
+  induction s using Finset.induction_on with
+  | empty => exact IsStarProjection.one _
+  | @insert a s ha ih =>
+    rw [sector_insert P hcomm s E a ha]
+    have hc : IsStarProjection (choice P E a) := by
+      unfold choice
+      split_ifs
+      · exact (hP a).one_sub
+      · exact hP a
+    exact hc.mul ih (sector_factor_commute P hcomm s E E a)
+
+omit [CompleteSpace V] [Fintype ι] in
+lemma sector_sum (s : Finset ι) : ∑ E∈s.powerset, sector P hcomm s E=1 := by
+  induction s using Finset.induction_on with
+  | empty =>
+    simp only [Finset.powerset_empty,Finset.sum_singleton]
+    exact Finset.noncommProd_empty _ _
+  | @insert a s ha ih =>
+    rw [Finset.sum_powerset_insert ha]
+    have h₁ (E : Finset ι) (hE : E∈s.powerset) :
+        sector P hcomm (insert a s) E=P a*sector P hcomm s E := by
+      rw [sector_insert P hcomm s E a ha]
+      congr 1
+      exact ite_eq_right (fun h => ha ((Finset.mem_powerset.mp hE) h))
+    have h₂ (E : Finset ι) (_hE : E∈s.powerset) :
+        sector P hcomm (insert a s) (insert a E)=(1-P a)*sector P hcomm s E := by
+      rw [sector_insert P hcomm s (insert a E) a ha]
+      simp only [choice,Finset.mem_insert_self,↓reduceIte]
+      congr 1
+      apply sector_congr
+      intro i hi
+      simp [ne_of_mem_of_not_mem hi ha]
+    rw [Finset.sum_congr rfl h₁,Finset.sum_congr rfl h₂,
+      ←Finset.mul_sum,←Finset.mul_sum,ih]
+    simp
+
+omit [Fintype ι] in
+lemma sector_absorb_factor (s E : Finset ι) (i : ι) (hi : i∈s)
+    (hP : ∀ i, IsStarProjection (P i)) :
+    choice P E i*sector P hcomm s E=sector P hcomm s E := by
+  have hc : IsIdempotentElem (choice P E i) := by
+    unfold choice
+    split_ifs
+    · exact (hP i).one_sub.isIdempotentElem
+    · exact (hP i).isIdempotentElem
+  have hsplit := Finset.mul_noncommProd_erase s hi (choice P E)
+    (fun i _ j _ _ => choice_commute P hcomm E E i j)
+  change choice P E i*sector P hcomm s E=sector P hcomm s E
+  change _=sector P hcomm s E at hsplit
+  nth_rw 1 [←hsplit]
+  rw [←mul_assoc,hc.eq,hsplit]
+
+omit [Fintype ι] in
+lemma sector_orthogonal (s E F : Finset ι) (hE : E⊆s) (hF : F⊆s) (hEF : E≠F)
+    (hP : ∀ i, IsStarProjection (P i)) :
+    sector P hcomm s E*sector P hcomm s F=0 := by
+  have hdiff : ∃ i∈s, ¬(i∈E ↔ i∈F) := by
+    by_contra h
+    push Not at h
+    exact hEF (Finset.ext fun i => by
+      by_cases hi : i∈s
+      · exact h i hi
+      · exact iff_of_false (fun h => hi (hE h)) (fun h => hi (hF h)))
+  obtain ⟨i,hi,hdiff⟩ := hdiff
+  have hz : choice P E i*choice P F i=0 := by
+    by_cases he : i∈E <;> by_cases hf : i∈F
+    · exact (hdiff (iff_of_true he hf)).elim
+    · simpa only [choice,ite_eq_left he,ite_eq_right hf] using (hP i).one_sub_mul_self
+    · simpa only [choice,ite_eq_right he,ite_eq_left hf] using (hP i).mul_one_sub_self
+    · exact (hdiff (iff_of_false he hf)).elim
+  have hEi := sector_absorb_factor P hcomm s E i hi hP
+  have hFi := sector_absorb_factor P hcomm s F i hi hP
+  calc
+    _ = sector P hcomm s E*(choice P E i*choice P F i)*sector P hcomm s F := by
+      simp only [mul_assoc]
+      rw [hFi]
+      rw [←mul_assoc,(sector_factor_commute P hcomm s E E i).eq.symm,hEi]
+    _ = 0 := by rw [hz,mul_zero,zero_mul]
+
+noncomputable def blockProduct (A : ι → V →L[ℂ] V)
+    (hA : ∀ i j, Commute (A i) (A j)) (s : Finset ι) : V →L[ℂ] V :=
+  s.noncommProd A (fun i _ j _ _ => hA i j)
+
+omit hcomm [Fintype ι] in
+lemma norm_noncommProd_le (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
+    (s : Finset ι) : ‖blockProduct A hA s‖≤∏ i∈s,‖A i‖ := by
+  induction s using Finset.induction_on with
+  | empty => exact ContinuousLinearMap.norm_id_le
+  | @insert a s ha ih =>
+    have he : blockProduct A hA (insert a s)=A a*blockProduct A hA s :=
+      Finset.noncommProd_insert_of_notMem _ _ _ _ ha
+    rw [he,Finset.prod_insert ha]
+    exact (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_left ih (norm_nonneg _))
+
+noncomputable def blocks (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j)) : V →L[ℂ] V :=
+  blockProduct A hA Finset.univ
+
+omit hcomm in
+lemma blocks_norm_le_one (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
+    (hn : ∀ i,‖A i‖≤1) : ‖blocks A hA‖≤1 :=
+  (norm_noncommProd_le A hA Finset.univ).trans (Finset.prod_le_one₀ (fun _ _ => norm_nonneg _) (fun i _ => hn i))
+
+omit [CompleteSpace V] in
+lemma blocks_sector_commute (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
+    (hAP : ∀ i j, Commute (A i) (P j)) (E : Finset ι) :
+    Commute (blocks A hA) (sector P hcomm Finset.univ E) := by
+  unfold blocks blockProduct
+  apply (Finset.noncommProd_commute _ _ (fun i _ j _ _ => hA i j) _ _).symm
+  intro i _
+  unfold sector
+  apply (Finset.noncommProd_commute _ _ (fun i _ j _ _ => choice_commute P hcomm E E i j) _ _).symm
+  intro j _
+  unfold choice
+  split_ifs
+  · exact (Commute.one_right _).sub_right (hAP i j)
+  · exact hAP i j
+
+lemma blocks_sector_norm (hP : ∀ i, IsStarProjection (P i))
+    (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
+    (hAP : ∀ i j, Commute (A i) (P j)) (hn : ∀ i,‖A i‖≤1)
+    (θ : ℝ) (_hθ : 0≤θ) (hgap : ∀ i,‖A i*(1-P i)‖≤θ) (E : Finset ι) :
+    ‖blocks A hA*sector P hcomm Finset.univ E‖≤θ^E.card := by
+  have hAC (i j : ι) : Commute (A i) (choice P E j) := by
+    unfold choice
+    split_ifs
+    · exact (Commute.one_right _).sub_right (hAP i j)
+    · exact hAP i j
+  have hprod := Finset.noncommProd_mul_distrib (s := Finset.univ) A (choice P E)
+    (fun i _ j _ _ => hA i j) (fun i _ j _ _ => choice_commute P hcomm E E i j)
+    (fun i _ j _ _ => (hAC j i).symm)
+  change _=blocks A hA*sector P hcomm Finset.univ E at hprod
+  rw [←hprod]
+  have hc (i j : ι) : Commute ((A*choice P E) i) ((A*choice P E) j) :=
+    ((hA i j).mul_right (hAC i j)).mul_left ((hAC j i).symm.mul_right (choice_commute P hcomm E E i j))
+  apply (norm_noncommProd_le _ hc Finset.univ).trans
+  calc
+    _ ≤ ∏ i : ι, if i∈E then θ else 1 := by
+      apply Finset.prod_le_prod₀ (fun i _ => norm_nonneg _)
+      intro i _
+      dsimp only [Pi.mul_apply]
+      unfold choice
+      split_ifs with hi
+      · exact hgap i
+      · exact (norm_mul_le _ _).trans (by nlinarith [hn i,(hP i).norm_le (P i),norm_nonneg (A i),norm_nonneg (P i)])
+    _ = _ := by rw [Fintype.prod_ite_mem,Finset.prod_const]
+
+noncomputable def coarse (E : Finset ι) : V →L[ℂ] V :=
+  Eᶜ.noncommProd P (fun i _ j _ _ => hcomm i j)
+
+lemma coarse_projection (hP : ∀ i, IsStarProjection (P i)) (E : Finset ι) :
+    IsStarProjection (coarse P hcomm E) := by
+  have he : coarse P hcomm E=sector P hcomm Eᶜ ∅ := by
+    apply Finset.noncommProd_congr rfl
+    intro i _
+    simp [choice]
+  rw [he]
+  exact sector_projection P hcomm hP _ _
+
+lemma sector_coarse (hP : ∀ i, IsStarProjection (P i)) (E : Finset ι) :
+    sector P hcomm Finset.univ E*coarse P hcomm E=sector P hcomm Finset.univ E := by
+  have hpi (i : ι) (hi : i∈Eᶜ) : sector P hcomm Finset.univ E*P i=sector P hcomm Finset.univ E := by
+    have h := sector_absorb_factor P hcomm Finset.univ E i (Finset.mem_univ i) hP
+    rw [(sector_factor_commute P hcomm Finset.univ E E i).eq] at h
+    simpa only [choice,ite_eq_right (Finset.mem_compl.mp hi)] using h
+  have hp (s : Finset ι) (hs : ∀ i∈s,
+      sector P hcomm Finset.univ E*P i=sector P hcomm Finset.univ E) :
+      sector P hcomm Finset.univ E*blockProduct P hcomm s=sector P hcomm Finset.univ E := by
+    induction s using Finset.induction_on with
+    | empty => exact mul_one _
+    | @insert a s ha ih =>
+      have he : blockProduct P hcomm (insert a s)=P a*blockProduct P hcomm s :=
+        Finset.noncommProd_insert_of_notMem _ _ _ _ ha
+      rw [he,←mul_assoc,hs a (Finset.mem_insert_self _ _)]
+      exact ih (fun i hi => hs i (Finset.mem_insert_of_mem hi))
+  exact hp Eᶜ hpi
+
+theorem blocks_exception_bound (hP : ∀ i, IsStarProjection (P i))
+    (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
+    (hAP : ∀ i j, Commute (A i) (P j)) (hn : ∀ i,‖A i‖≤1)
+    (θ : ℝ) (hθ : 0≤θ) (hθ1 : θ≤1) (hgap : ∀ i,‖A i*(1-P i)‖≤θ)
+    (B : V →L[ℂ] V) (hB : ‖B‖≤1) (n : ℕ) :
+    ‖B.adjoint*blocks A hA*B‖≤θ^n+
+      ∑ E : Finset ι, if E.card<n then ‖B.adjoint*coarse P hcomm E*B‖ else 0 := by
+  let T : Finset ι → V →L[ℂ] V := sector P hcomm Finset.univ
+  have hsum : ∑ E,T E=1 := by
+    simpa only [Finset.powerset_univ] using sector_sum P hcomm Finset.univ
+  have hbound := resolution_exception_bound T (coarse P hcomm)
+    (fun E => sector_projection P hcomm hP _ E)
+    (fun E F hEF => sector_orthogonal P hcomm _ E F (Finset.subset_univ _) (Finset.subset_univ _) hEF hP)
+    hsum (coarse_projection P hcomm hP) (sector_coarse P hcomm hP)
+    (blocks A hA) B (blocks_sector_commute P hcomm A hA hAP)
+    (blocks_norm_le_one A hA hn) hB (Finset.univ.filter fun E : Finset ι => E.card<n)
+    (θ^n) (pow_nonneg hθ _) (by
+      intro E hE
+      have hEn : n≤E.card := by simpa using hE
+      exact (blocks_sector_norm P hcomm hP A hA hAP hn θ hθ hgap E).trans
+        (pow_le_pow_of_le_one hθ hθ1 hEn))
+  simpa only [Finset.sum_filter] using hbound
+
+
+end Thorp.CommutingBlocks
+
+namespace Thorp.UnitaryFinite
+open scoped BigOperators Classical
+
+variable {G : Type*} [Group G] [Fintype G]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+variable {Ω Ξ : Type*} [Fintype Ω] [Fintype Ξ]
+
+lemma sampleOperator_weighted (ρ : Representation ℂ G V) (P : Ω → G) :
+    sampleOperator ρ P=weightedOperator ρ (sampleLaw P) := by
+  unfold sampleOperator weightedOperator sampleLaw finiteMean
+  simp only [Complex.ofReal_sum,div_eq_mul_inv,Finset.sum_mul,Finset.sum_smul]
+  rw [Finset.sum_comm]
+  simp only [Finset.smul_sum]
+  apply Finset.sum_congr rfl
+  intro ω _
+  calc
+    _ = ∑ g, if P ω=g then (Fintype.card Ω:ℂ)⁻¹ • continuousRepresentation ρ g else 0 := by
+      symm
+      simp only [Finset.sum_ite_eq,Finset.mem_univ,↓reduceIte]
+    _ = _ := by
+      apply Finset.sum_congr rfl
+      intro g _
+      split_ifs <;> simp_all
+
+lemma sampleOperator_norm_le [Nonempty Ω] (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (P : Ω → G) :
+    ‖sampleOperator ρ P‖≤1 := by
+  rw [sampleOperator_weighted]
+  exact (weightedOperator_norm_le ρ hρ _ (sampleLaw_nonneg _)).trans_eq (sampleLaw_sum P)
+
+omit [Fintype G] in
+lemma sampleOperator_equiv (ρ : Representation ℂ G V) (P : Ω → G) (e : Ξ ≃ Ω) :
+    sampleOperator ρ (P ∘ e)=sampleOperator ρ P := by
+  simp only [sampleOperator,Fintype.card_congr e,Function.comp_apply]
+  rw [Equiv.sum_comp e (fun ω => continuousRepresentation ρ (P ω))]
+
+omit [Fintype G] in
+lemma sampleOperator_prod (ρ : Representation ℂ G V) (P : Ω → G) (Q : Ξ → G) :
+    sampleOperator ρ (fun ω : Ω × Ξ => P ω.1*Q ω.2)=sampleOperator ρ P*sampleOperator ρ Q := by
+  simp only [sampleOperator,Fintype.card_prod,Nat.cast_mul,mul_inv_rev,map_mul,Fintype.sum_prod_type,
+    Finset.sum_mul,Finset.mul_sum,smul_mul_smul,Finset.smul_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro ω _
+  apply Finset.sum_congr rfl
+  intro ξ _
+  rw [mul_comm]
+
+omit [Fintype G] in
+lemma sampleOperator_inv (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (P : Ω → G) :
+    sampleOperator ρ (fun ω => (P ω)⁻¹)=(sampleOperator ρ P).adjoint := by
+  simp only [sampleOperator,map_smulₛₗ,map_sum,map_inv₀,map_natCast,
+    continuousRepresentation_adjoint ρ hρ]
+
+lemma sampleOperator_palindrome (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V)
+    (hρ : IsUnitary ρ) :
+    sampleOperator ρ (palindromePerm d)=
+      sampleOperator ρ (fun ω => butterflyPerm d (decodeButterfly d ω))*
+      (sampleOperator ρ (fun ω => butterflyPerm d (decodeButterfly d ω))).adjoint := by
+  rw [←sampleOperator_inv ρ hρ,←sampleOperator_prod]
+  rfl
+
+omit [Fintype G] in
+lemma sampleOperator_one [Nonempty Ω] (ρ : Representation ℂ G V) :
+    sampleOperator ρ (fun _ : Ω => 1)=1 := by
+  simp only [sampleOperator,map_one,Finset.sum_const,Finset.card_univ,←Nat.cast_smul_eq_nsmul ℂ,
+    smul_smul]
+  rw [inv_mul_cancel₀ (by exact_mod_cast Fintype.card_ne_zero),one_smul]
+
+lemma sampleOperator_uniform (ρ : Representation ℂ G V) : sampleOperator ρ (id : G → G)=uniformOperator ρ := by
+  simp only [sampleOperator,uniformOperator,weightedOperator,Complex.ofReal_inv,Complex.ofReal_natCast,
+    Finset.smul_sum,id_eq]
+
+lemma sampleOperator_uniform_mul [Nonempty Ω] (ρ : Representation ℂ G V) (P : Ω → G) :
+    sampleOperator ρ (fun ω : Ω × G => P ω.1*ω.2⁻¹)=uniformOperator ρ := by
+  rw [sampleOperator_prod]
+  have he : sampleOperator ρ (fun g : G => g⁻¹)=uniformOperator ρ := by
+    rw [←sampleOperator_uniform]
+    exact sampleOperator_equiv ρ id (Equiv.inv G)
+  rw [he,sampleOperator_weighted,weighted_uniform_left,sampleLaw_sum,one_smul]
+
+end Thorp.UnitaryFinite
+
+namespace Thorp.FiniteAverage
+open scoped BigOperators Classical
+variable {I Ω A : Type*} [Fintype I] [DecidableEq I] [Fintype Ω]
+    [Ring A] [Algebra ℂ A]
+
+noncomputable def orderedProduct : MultilinearMap ℂ (fun _ : I => A) A :=
+  MultilinearMap.domDomCongr (Fintype.equivFin I).symm
+    (MultilinearMap.mkPiAlgebraFin ℂ (Fintype.card I) A)
+
+lemma orderedProduct_eq (f : I → A) (hc : Pairwise fun i j => Commute (f i) (f j)) :
+    orderedProduct f=Finset.univ.noncommProd f (fun _ _ _ _ hij => hc hij) := by
+  unfold orderedProduct
+  rw [MultilinearMap.domDomCongr_apply,MultilinearMap.mkPiAlgebraFin_apply]
+  let l := List.ofFn (Fintype.equivFin I).symm
+  have hl : l.Nodup := List.nodup_ofFn.mpr (Fintype.equivFin I).symm.injective
+  have hu : l.toFinset=Finset.univ := by
+    ext i
+    simp only [List.mem_toFinset,List.mem_ofFn,Finset.mem_univ,iff_true,l]
+    exact (Fintype.equivFin I).symm.surjective i
+  have he := Finset.noncommProd_toFinset l f (by intro i hi j hj hij; exact hc hij) hl
+  rw [hu] at he
+  rw [he]
+  congr 1
+  exact List.map_ofFn.symm
+
+lemma multilinear_average (F : MultilinearMap ℂ (fun _ : I => A) A) (x : I → Ω → A) :
+    F (fun i => (Fintype.card Ω:ℂ)⁻¹ • ∑ ω,x i ω)=
+      (Fintype.card (I → Ω):ℂ)⁻¹ • ∑ ω : I → Ω,F (fun i => x i (ω i)) := by
+  rw [F.map_smul_univ,F.map_sum]
+  simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fun,Nat.cast_pow,inv_pow]
+
+lemma ordered_average (x : I → Ω → A)
+    (hc : ∀ ω, Pairwise fun i j => Commute (x i (ω i)) (x j (ω j)))
+    (havg : Pairwise fun i j => Commute ((Fintype.card Ω:ℂ)⁻¹ • ∑ ω,x i ω)
+      ((Fintype.card Ω:ℂ)⁻¹ • ∑ ω,x j ω)) :
+    (Finset.univ.noncommProd (fun i => (Fintype.card Ω:ℂ)⁻¹ • ∑ ω,x i ω)
+      (fun _ _ _ _ hij => havg hij))=
+    (Fintype.card (I → Ω):ℂ)⁻¹ • ∑ ω : I → Ω,
+      Finset.univ.noncommProd (fun i => x i (ω i)) (fun _ _ _ _ hij => hc ω hij) := by
+  rw [←orderedProduct_eq _ havg,multilinear_average]
+  simp only [orderedProduct_eq _ (hc _)]
+
+end Thorp.FiniteAverage
+namespace Thorp.FiniteCharacters
+open Module
+open scoped BigOperators Classical
+
+variable {G ι : Type*} [Group G] [Fintype G] [Fintype ι]
+variable (V : ι → Type*) [∀ i, AddCommGroup (V i)] [∀ i, Module ℂ (V i)]
+  [∀ i, FiniteDimensional ℂ (V i)]
+variable (ρ : ∀ i, Representation ℂ G (V i)) [∀ i, Representation.IsIrreducible (ρ i)]
+variable (hneq : ∀ i j, i≠j → ¬Nonempty (Representation.Equiv (ρ i) (ρ j)))
+  (hcard : Nat.card (ConjClasses G)≤Fintype.card ι)
+include hneq hcard
+
+lemma character_expansion (f : ConjClasses G → ℂ) :
+    f=∑ i, characterDual (ρ i) f • characterClass (ρ i) := by
+  let : Fintype (ConjClasses G) := Fintype.ofFinite _
+  have horth (i j : ι) : characterDual (ρ i) (characterClass (ρ j))=if i=j then 1 else 0 := by
+    rw [characterDual_characterClass]
+    by_cases h : i=j
+    · subst j; rw [ite_eq_left ⟨Representation.Equiv.refl _⟩,ite_eq_left rfl]
+    · rw [ite_eq_right h,ite_eq_right (hneq i j h)]
+  have hLI : LinearIndependent ℂ (fun i => characterClass (ρ i)) := by
+    apply LinearIndependent.of_pairwise_dual_eq_zero_one _ (fun i => characterDual (ρ i))
+    · intro i j hij; exact (horth i j).trans (ite_eq_right hij)
+    · intro i; simpa using horth i i
+  have heq : Fintype.card ι=Module.finrank ℂ (ConjClasses G → ℂ) := by
+    apply Nat.le_antisymm hLI.fintype_card_le_finrank
+    simpa only [Module.finrank_pi,Module.finrank_self,Finset.sum_const,Finset.card_univ,
+      smul_eq_mul,mul_one,Nat.card_eq_fintype_card] using hcard
+  let b := Basis.mk hLI (by rw [hLI.span_eq_top_of_card_eq_finrank' heq])
+  have hb (i : ι) : b i=characterClass (ρ i) := Basis.mk_apply _ _ _
+  have hd (i : ι) : b.coord i=characterDual (ρ i) := by
+    apply b.ext
+    intro j
+    rw [hb,horth]
+    simp only [←hb,Basis.coord_apply,Basis.repr_self,Finsupp.single_apply]
+    split_ifs <;> simp_all
+  have h := b.sum_repr f
+  simpa only [hb,show ∀ i,b.repr f i=characterDual (ρ i) f from fun i => congrArg (fun l => l f) (hd i)] using h.symm
+
+theorem sum_dimension_sq : ∑ i, (Module.finrank ℂ (V i))^2=Fintype.card G := by
+  let f : ConjClasses G → ℂ := fun c => if c=ConjClasses.mk (1:G) then 1 else 0
+  have hf (g : G) : f (ConjClasses.mk g)=if g=1 then 1 else 0 := by
+    simp only [f,ConjClasses.mk_eq_mk_iff_isConj,isConj_one_left]
+  have hd (i : ι) : characterDual (ρ i) f=(Fintype.card G:ℂ)⁻¹*(Module.finrank ℂ (V i):ℂ) := by
+    change (Nat.card G:ℂ)⁻¹ * ∑ g, f (ConjClasses.mk g)*(ρ i).character g⁻¹=_
+    simp only [hf,ite_mul,one_mul,zero_mul,Finset.sum_ite_eq',Finset.mem_univ,↓reduceIte,inv_one,
+      Representation.char_one,Nat.card_eq_fintype_card]
+  have he := congrFun (character_expansion V ρ hneq hcard f) (ConjClasses.mk (1:G))
+  rw [hf,ite_eq_left rfl] at he
+  simp only [Finset.sum_apply,Pi.smul_apply,smul_eq_mul,hd,characterClass_mk,Representation.char_one] at he
+  have hN : (Fintype.card G:ℂ)≠0 := by exact_mod_cast Fintype.card_ne_zero
+  have hs : (∑ i,(Module.finrank ℂ (V i):ℂ)^2)=(Fintype.card G:ℂ) := by
+    calc
+      _ = (Fintype.card G:ℂ) * ∑ i,(Fintype.card G:ℂ)⁻¹ *
+          (Module.finrank ℂ (V i):ℂ)*(Module.finrank ℂ (V i):ℂ) := by
+        rw [Finset.mul_sum]
+        apply Finset.sum_congr rfl
+        intro i _
+        field_simp
+      _ = _ := by rw [←he,mul_one]
+  exact_mod_cast hs
+
+end Thorp.FiniteCharacters
+
+namespace Thorp.UnitaryFinite
+open scoped BigOperators ComplexConjugate Classical
+
+variable {G : Type*} [Group G] [Fintype G]
+variable {V W : Type*} [NormedAddCommGroup V] [NormedAddCommGroup W]
+  [InnerProductSpace ℂ V] [InnerProductSpace ℂ W]
+  [FiniteDimensional ℂ V] [FiniteDimensional ℂ W]
+
+omit [FiniteDimensional ℂ V] [FiniteDimensional ℂ W] in
+lemma averageMap_zero (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
+    [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ]
+    (hne : ¬Nonempty (Representation.Equiv ρ σ)) (A : V →ₗ[ℂ] W) :
+    averageMap ρ σ A=0 := by
+  rcases Representation.IsIrreducible.bijective_or_eq_zero (averageMap ρ σ A) with h|h
+  · exact (hne ⟨Representation.IntertwiningMap.ofBijective _ h⟩).elim
+  · exact h
+
+omit [FiniteDimensional ℂ V] [FiniteDimensional ℂ W] in
+lemma coefficient_cross_orthogonality (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
+    [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ]
+    (hρ : IsUnitary ρ) (hne : ¬Nonempty (Representation.Equiv ρ σ))
+    (u z : W) (v w : V) :
+    ∑ g, inner ℂ u (σ g z)*inner ℂ (ρ g w) v=0 := by
+  have he := averageMap_zero ρ σ hne (InnerProductSpace.rankOne ℂ z w).toLinearMap
+  have hv := congrArg (fun T : Representation.IntertwiningMap ρ σ => inner ℂ u (T v)) he
+  change inner ℂ u ((∑ g, (σ g).comp ((InnerProductSpace.rankOne ℂ z w).toLinearMap.comp (ρ g⁻¹))) v)=_ at hv
+  simpa only [LinearMap.sum_apply,LinearMap.comp_apply,ContinuousLinearMap.coe_coe,
+    InnerProductSpace.rankOne_apply,map_smul,unitary_inner_inv ρ hρ,inner_sum,
+    inner_smul_right,Representation.IntertwiningMap.coe_zero,Pi.zero_apply,inner_zero_right,mul_comm] using hv
+
+end Thorp.UnitaryFinite
+namespace Thorp.UnitaryFinite
+open scoped BigOperators ComplexConjugate Classical
+
+variable {G : Type*} [Group G] [Fintype G]
+
+def regularAction : G →* Equiv.Perm G where
+  toFun := Equiv.mulLeft
+  map_one' := by ext; simp
+  map_mul' _ _ := by ext; simp
+
+omit [Fintype G] in
+lemma regular_transitive (x : G) : ∃ g,regularAction g (1:G)=x := ⟨x,mul_one x⟩
+
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+omit [Fintype G] [FiniteDimensional ℂ V] in
+lemma regular_fixed (ρ : Representation ℂ G V) (w : V) : IsFixed regularAction (1:G) ρ w := by
+  intro g hg
+  have he : g=1 := by simpa only [regularAction,MonoidHom.coe_mk,OneHom.coe_mk,Equiv.coe_mulLeft,mul_one] using hg
+  subst g
+  simp
+
+noncomputable def regularCoefficient (ρ : Representation ℂ G V) (w v : V) : EuclideanSpace ℂ G :=
+  normalizedCoefficient regularAction 1 regular_transitive ρ w v
+
+omit [FiniteDimensional ℂ V] in
+lemma regularCoefficient_apply (ρ : Representation ℂ G V) (w v : V) (g : G) :
+    regularCoefficient ρ w v g = (Real.sqrt ((Module.finrank ℂ V:ℝ)/(Fintype.card G:ℝ)):ℂ)*inner ℂ (ρ g w) v := by
+  unfold regularCoefficient normalizedCoefficient
+  simp only [PiLp.smul_apply,smul_eq_mul]
+  congr 1
+  have he := coefficient_at regularAction 1 regular_transitive ρ w (regular_fixed ρ w) v g
+  simpa only [regularAction,MonoidHom.coe_mk,OneHom.coe_mk,Equiv.coe_mulLeft,mul_one] using he
+
+lemma regularCoefficient_inner (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
+    (hρ : IsUnitary ρ) (w z v t : V) :
+    inner ℂ (regularCoefficient ρ w v) (regularCoefficient ρ z t)=inner ℂ z w*inner ℂ v t :=
+  normalizedCoefficient_inner regularAction 1 regular_transitive ρ hρ w z
+    (regular_fixed ρ w) (regular_fixed ρ z) v t
+
+variable {W : Type*} [NormedAddCommGroup W] [InnerProductSpace ℂ W] [FiniteDimensional ℂ W]
+
+omit [FiniteDimensional ℂ V] [FiniteDimensional ℂ W] in
+lemma regularCoefficient_cross (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
+    [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ]
+    (hσ : IsUnitary σ) (hne : ¬Nonempty (Representation.Equiv σ ρ))
+    (w v : V) (z t : W) : inner ℂ (regularCoefficient ρ w v) (regularCoefficient σ z t)=0 := by
+  rw [PiLp.inner_apply]
+  simp only [RCLike.inner_apply,regularCoefficient_apply,map_mul,Complex.conj_ofReal,
+    inner_conj_symm]
+  have he := coefficient_cross_orthogonality σ ρ hσ hne v w t z
+  calc
+    _ = ((Real.sqrt ((Module.finrank ℂ V:ℝ)/(Fintype.card G:ℝ)):ℂ)*
+        (Real.sqrt ((Module.finrank ℂ W:ℝ)/(Fintype.card G:ℝ)):ℂ)) *
+        ∑ g,inner ℂ v (ρ g w)*inner ℂ (σ g z) t := by
+      rw [Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro g _
+      ring
+    _ = 0 := by rw [he,mul_zero]
+
+end Thorp.UnitaryFinite
+
+namespace Thorp.FiniteFourier
+open scoped BigOperators ComplexConjugate Classical
+open UnitaryFinite
+variable {G ι : Type*} [Group G] [Fintype G] [Fintype ι]
+variable (V : ι → Type*) [∀ i, NormedAddCommGroup (V i)] [∀ i, InnerProductSpace ℂ (V i)]
+  [∀ i, FiniteDimensional ℂ (V i)]
+variable (ρ : ∀ i, Representation ℂ G (V i)) [∀ i, Representation.IsIrreducible (ρ i)]
+variable (hρ : ∀ i, IsUnitary (ρ i))
+variable (hneq : ∀ i j, i≠j → ¬Nonempty (Representation.Equiv (ρ i) (ρ j)))
+  (hcard : Nat.card (ConjClasses G)≤Fintype.card ι)
+
+
+abbrev CoefficientIndex := Σ i,Fin (Module.finrank ℂ (V i)) × Fin (Module.finrank ℂ (V i))
+
+noncomputable def coefficientFamily (a : CoefficientIndex V) : EuclideanSpace ℂ G :=
+  regularCoefficient (ρ a.1) (stdOrthonormalBasis ℂ (V a.1) a.2.1) (stdOrthonormalBasis ℂ (V a.1) a.2.2)
+
+omit [Fintype ι] in
+include hρ hneq in
+lemma coefficientFamily_orthonormal : Orthonormal ℂ (coefficientFamily V ρ) := by
+  rw [orthonormal_iff_ite]
+  rintro ⟨i,a,b⟩ ⟨j,c,d⟩
+  dsimp only [coefficientFamily]
+  by_cases hij : i=j
+  · subst j
+    rw [regularCoefficient_inner _ (hρ i)]
+    simp only [(stdOrthonormalBasis ℂ (V i)).inner_eq_ite]
+    by_cases hac : a=c <;> by_cases hbd : b=d <;> simp [hac,hbd,eq_comm]
+  · rw [regularCoefficient_cross _ _ (hρ j) (hneq j i (Ne.symm hij))]
+    rw [ite_eq_right (by intro h; exact hij (congrArg Sigma.fst h))]
+
+include ρ hneq hcard in
+lemma coefficientIndex_card : Fintype.card (CoefficientIndex V)=Fintype.card G := by
+  simp only [CoefficientIndex,Fintype.card_sigma,Fintype.card_prod,Fintype.card_fin,←pow_two]
+  exact FiniteCharacters.sum_dimension_sq V ρ hneq hcard
+
+noncomputable def coefficientBasis : OrthonormalBasis (CoefficientIndex V) ℂ (EuclideanSpace ℂ G) :=
+  OrthonormalBasis.mk (coefficientFamily_orthonormal V ρ hρ hneq)
+    (by rw [(coefficientFamily_orthonormal V ρ hρ hneq).linearIndependent.span_eq_top_of_card_eq_finrank'
+          (by rw [coefficientIndex_card V ρ hneq hcard,finrank_euclideanSpace])])
+
+lemma coefficientBasis_apply (a : CoefficientIndex V) :
+    coefficientBasis V ρ hρ hneq hcard a=coefficientFamily V ρ a := congrFun (OrthonormalBasis.coe_mk _ _) a
+
+include hρ hneq hcard in
+
+theorem parseval (f : EuclideanSpace ℂ G) :
+    ∑ a : CoefficientIndex V, ‖inner ℂ (coefficientFamily V ρ a) f‖^2=‖f‖^2 := by
+  simpa only [coefficientBasis_apply] using
+    (coefficientBasis V ρ hρ hneq hcard).sum_sq_norm_inner_right f
+
+
+noncomputable def realVector (p : G → ℝ) : EuclideanSpace ℂ G := WithLp.toLp 2 (fun g => (p g:ℂ))
+
+omit [Fintype ι] [∀ i, Representation.IsIrreducible (ρ i)] in
+lemma coefficient_realVector (p : G → ℝ) (a : CoefficientIndex V) :
+    inner ℂ (coefficientFamily V ρ a) (realVector p)=
+      (Real.sqrt ((Module.finrank ℂ (V a.1):ℝ)/(Fintype.card G:ℝ)):ℂ)*
+      inner ℂ (stdOrthonormalBasis ℂ (V a.1) a.2.2)
+        (weightedOperator (ρ a.1) p (stdOrthonormalBasis ℂ (V a.1) a.2.1)) := by
+  simp only [PiLp.inner_apply,coefficientFamily,regularCoefficient_apply,realVector,
+    RCLike.inner_apply,map_mul,Complex.conj_ofReal,inner_conj_symm,
+    weightedOperator,sum_apply,smul_apply,
+    continuousRepresentation_apply,inner_sum,inner_smul_right,Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro g _
+  ring
+
+omit [Fintype ι] [∀ i, Representation.IsIrreducible (ρ i)] in
+lemma coefficient_realVector_sq (p : G → ℝ) (a : CoefficientIndex V) :
+    ‖inner ℂ (coefficientFamily V ρ a) (realVector p)‖^2=
+      ((Module.finrank ℂ (V a.1):ℝ)/(Fintype.card G:ℝ))*
+      ‖inner ℂ (stdOrthonormalBasis ℂ (V a.1) a.2.2)
+        (weightedOperator (ρ a.1) p (stdOrthonormalBasis ℂ (V a.1) a.2.1))‖^2 := by
+  rw [coefficient_realVector,norm_mul,Complex.norm_real,Real.norm_eq_abs,
+    abs_of_nonneg (Real.sqrt_nonneg _),mul_pow,Real.sq_sqrt (div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _))]
+
+include hρ hneq hcard in
+lemma parseval_operator (p : G → ℝ) :
+    (Fintype.card G:ℝ)*‖realVector p‖^2=
+      ∑ i,(Module.finrank ℂ (V i):ℝ)*∑ a : Fin (Module.finrank ℂ (V i)),
+        ‖weightedOperator (ρ i) p (stdOrthonormalBasis ℂ (V i) a)‖^2 := by
+  rw [←parseval V ρ hρ hneq hcard (realVector p)]
+  simp only [Fintype.sum_sigma,Fintype.sum_prod_type,coefficient_realVector_sq]
+  have hN : (Fintype.card G:ℝ)≠0 := by exact_mod_cast Fintype.card_ne_zero
+  rw [Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro i _
+  have he (a : Fin (Module.finrank ℂ (V i))) :
+      (∑ b : Fin (Module.finrank ℂ (V i)),
+        (Module.finrank ℂ (V i):ℝ)/(Fintype.card G:ℝ)*
+          ‖inner ℂ (stdOrthonormalBasis ℂ (V i) b)
+            (weightedOperator (ρ i) p (stdOrthonormalBasis ℂ (V i) a))‖^2)=
+        (Module.finrank ℂ (V i):ℝ)/(Fintype.card G:ℝ)*
+          ‖weightedOperator (ρ i) p (stdOrthonormalBasis ℂ (V i) a)‖^2 := by
+    rw [←Finset.mul_sum,(stdOrthonormalBasis ℂ (V i)).sum_sq_norm_inner_right]
+  simp_rw [he]
+  rw [←Finset.mul_sum]
+  field_simp
+
+include hρ hneq hcard in
+lemma realVector_bound (p : G → ℝ) :
+    (Fintype.card G:ℝ)*‖realVector p‖^2≤
+      ∑ i,(Module.finrank ℂ (V i):ℝ)^2*‖weightedOperator (ρ i) p‖^2 := by
+  rw [parseval_operator V ρ hρ hneq hcard]
+  apply Finset.sum_le_sum
+  intro i _
+  calc
+    _ ≤ (Module.finrank ℂ (V i):ℝ)*∑ _a : Fin (Module.finrank ℂ (V i)),‖weightedOperator (ρ i) p‖^2 := by
+      apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
+      apply Finset.sum_le_sum
+      intro a _
+      apply pow_le_pow_left₀ (norm_nonneg _)
+      simpa only [(stdOrthonormalBasis ℂ (V i)).norm_eq_one,mul_one] using
+        (weightedOperator (ρ i) p).le_opNorm (stdOrthonormalBasis ℂ (V i) a)
+    _ = _ := by simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]; ring
+
+include hρ hneq hcard in
+
+theorem l1_fourier_bound (p : G → ℝ) :
+    (∑ g,|p g|)^2≤∑ i,(Module.finrank ℂ (V i):ℝ)^2*‖weightedOperator (ρ i) p‖^2 := by
+  apply le_trans _ (realVector_bound V ρ hρ hneq hcard p)
+  have he : ‖realVector p‖^2=∑ g,|p g|^2 := by
+    rw [EuclideanSpace.norm_sq_eq]
+    simp only [realVector,PiLp.toLp_apply,Complex.norm_real,Real.norm_eq_abs]
+  rw [he]
+  simpa only [Finset.card_univ] using sq_sum_le_card_mul_sum_sq (s := Finset.univ) (f := fun g => |p g|)
+
+end Thorp.FiniteFourier
+
+namespace Thorp.ParallelBlocks
+open scoped BigOperators Classical
+variable {B X : Type*}
+
+def lift : (B → Equiv.Perm X) →* Equiv.Perm (B × X) where
+  toFun p :=
+    { toFun := fun x => (x.1,p x.1 x.2)
+      invFun := fun x => (x.1,(p x.1).symm x.2)
+      left_inv := fun x => by simp
+      right_inv := fun x => by simp }
+  map_one' := rfl
+  map_mul' _ _ := rfl
+
+@[simp] lemma lift_apply (p : B → Equiv.Perm X) (x : B × X) : lift p x=(x.1,p x.1 x.2) := rfl
+
+noncomputable def injection (b : B) : Equiv.Perm X →* Equiv.Perm (B × X) :=
+  lift.comp
+    { toFun := fun p c => if c=b then p else 1
+      map_one' := by funext c; simp
+      map_mul' := by intro p q; funext c; by_cases h : c=b <;> simp [h] }
+
+@[simp] lemma injection_apply (b : B) (p : Equiv.Perm X) (x : B × X) :
+    injection b p x=(x.1,if x.1=b then p x.2 else x.2) := by
+  simp only [injection,MonoidHom.comp_apply,lift_apply]
+  by_cases h : x.1=b <;> simp [h]
+
+lemma injection_commute {b c : B} (h : b≠c) (p q : Equiv.Perm X) :
+    Commute (injection b p) (injection c q) := by
+  apply Equiv.ext
+  intro x
+  simp only [Equiv.Perm.mul_apply,injection_apply]
+  by_cases hb : x.1=b <;> by_cases hc : x.1=c <;> simp_all
+
+variable [Fintype B]
+
+lemma lift_product (p : B → Equiv.Perm X) :
+    lift p=Finset.univ.noncommProd (fun b => injection b (p b))
+      (fun _ _ _ _ h => injection_commute h _ _) := by
+  have he (s : Finset B) (x : B × X) :
+      s.noncommProd (fun b => injection b (p b))
+        (fun _ _ _ _ h => injection_commute h _ _) x=
+          (x.1,if x.1∈s then p x.1 x.2 else x.2) := by
+    induction s using Finset.induction_on with
+    | empty =>
+      have hz : Finset.noncommProd (∅ : Finset B) (fun b => injection b (p b))
+          (fun _ _ _ _ h => injection_commute h _ _)=1 := Finset.noncommProd_empty _ _
+      rw [hz]; simp
+    | @insert a s ha ih =>
+      have hz : (insert a s).noncommProd (fun b => injection b (p b))
+          (fun _ _ _ _ h => injection_commute h _ _)=injection a (p a)*
+          s.noncommProd (fun b => injection b (p b))
+            (fun _ _ _ _ h => injection_commute h _ _) :=
+        Finset.noncommProd_insert_of_notMem _ _ _ _ ha
+      rw [hz,Equiv.Perm.mul_apply,ih,injection_apply]
+      by_cases h : x.1=a
+      · subst a; simp [ha]
+      · simp [h]
+  apply Equiv.ext
+  intro x
+  rw [he]
+  simp
+
+end Thorp.ParallelBlocks
+namespace Thorp.UnitaryFinite
+open scoped BigOperators Classical
+
+variable {G H : Type*} [Group G] [Group H]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+variable {Ω Ξ : Type*} [Fintype Ω] [Fintype Ξ]
+
+lemma continuousRepresentation_comp (ρ : Representation ℂ G V) (φ : H →* G) (h : H) :
+    continuousRepresentation (ρ.comp φ) h=continuousRepresentation ρ (φ h) := rfl
+
+lemma sampleOperator_comp (ρ : Representation ℂ G V) (φ : H →* G) (P : Ω → H) :
+    sampleOperator (ρ.comp φ) P=sampleOperator ρ (φ ∘ P) := rfl
+
+omit [FiniteDimensional ℂ V] in
+lemma unitary_comp (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (φ : H →* G) :
+    IsUnitary (ρ.comp φ) := fun h v w => hρ (φ h) v w
+
+lemma sampleOperator_commute (ρ : Representation ℂ G V) (P : Ω → G) (Q : Ξ → G)
+    (h : ∀ ω ξ, Commute (P ω) (Q ξ)) :
+    Commute (sampleOperator ρ P) (sampleOperator ρ Q) := by
+  unfold sampleOperator
+  apply Commute.smul_left
+  apply Commute.smul_right
+  apply Commute.sum_left
+  intro ω _
+  apply Commute.sum_right
+  intro ξ _
+  exact (h ω ξ).map (continuousRepresentation ρ)
+
+lemma sampleOperator_uniform_absorb [Fintype G] [Nonempty Ω]
+    (ρ : Representation ℂ G V) (P : Ω → G) :
+    sampleOperator ρ P*uniformOperator ρ=uniformOperator ρ := by
+  rw [sampleOperator_weighted,weighted_uniform_left,sampleLaw_sum,one_smul]
+
+end Thorp.UnitaryFinite
+namespace Thorp.ParallelBlocks
+open scoped BigOperators Classical
+open UnitaryFinite
+variable {B X Ω : Type*} [Fintype B] [DecidableEq B] [Fintype Ω]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+omit [Fintype B] [DecidableEq B] in
+lemma local_average_commute (ρ : Representation ℂ (Equiv.Perm (B × X)) V)
+    (P : B → Ω → Equiv.Perm X) (b c : B) :
+    Commute (sampleOperator (ρ.comp (injection b)) (P b))
+      (sampleOperator (ρ.comp (injection c)) (P c)) := by
+  by_cases h : b=c
+  · subst c; exact Commute.refl _
+  · rw [sampleOperator_comp,sampleOperator_comp]
+    exact sampleOperator_commute ρ _ _ (fun _ _ => injection_commute h _ _)
+
+lemma sampleOperator_parallel (ρ : Representation ℂ (Equiv.Perm (B × X)) V)
+    (P : B → Ω → Equiv.Perm X) :
+    sampleOperator ρ (fun ω : B → Ω => lift (fun b => P b (ω b)))=
+      CommutingBlocks.blocks (fun b => sampleOperator (ρ.comp (injection b)) (P b))
+        (local_average_commute ρ P) := by
+  let A (b : B) (w : Ω) := continuousRepresentation ρ (injection b (P b w))
+  have hc : ∀ ω : B → Ω, Pairwise fun b c => Commute (A b (ω b)) (A c (ω c)) := by
+    intro ω b c h
+    exact (injection_commute h _ _).map (continuousRepresentation ρ)
+  have havg : Pairwise fun b c => Commute ((Fintype.card Ω:ℂ)⁻¹ • ∑ w,A b w)
+      ((Fintype.card Ω:ℂ)⁻¹ • ∑ w,A c w) := by
+    intro b c _
+    exact local_average_commute ρ P b c
+  have he (ω : B → Ω) : continuousRepresentation ρ (lift (fun b => P b (ω b)))=
+      Finset.univ.noncommProd (fun b => A b (ω b)) (fun _ _ _ _ h => hc ω h) := by
+    rw [lift_product]
+    exact Finset.map_noncommProd _ _ _ (continuousRepresentation ρ)
+  unfold sampleOperator
+  simp_rw [he]
+  exact (FiniteAverage.ordered_average A hc havg).symm
+
+end Thorp.ParallelBlocks
+namespace Thorp
+open scoped BigOperators Classical
+
+noncomputable def splitPerm (r L : ℕ) : Equiv.Perm (Card (L+r)) ≃* Equiv.Perm (Card r × Card L) :=
+  (cardSplit r L).permCongrHom
+
+noncomputable def outerPerm (r L : ℕ) (hi : Card L × SwitchIndex r → Bool) :
+    Equiv.Perm (Card r × Card L) :=
+  (Equiv.prodComm (Card L) (Card r)).permCongrHom
+    (ParallelBlocks.lift (fun x => butterflyPerm r (decodeButterfly r (fun i => hi (x,i)))))
+
+lemma outerPerm_apply (r L : ℕ) (hi : Card L × SwitchIndex r → Bool) (x : Card r × Card L) :
+    outerPerm r L hi x=(butterflyPerm r (decodeButterfly r (fun i => hi (x.2,i))) x.1,x.2) := rfl
+
+lemma splitPerm_butterfly (r L : ℕ) (lo : Card r × SwitchIndex L → Bool)
+    (hi : Card L × SwitchIndex r → Bool) :
+    splitPerm r L (butterflyPerm (L+r) (decodeButterfly (L+r) (assembleBits r L lo hi)))=
+      outerPerm r L hi*ParallelBlocks.lift
+        (fun b => butterflyPerm L (decodeButterfly L (fun i => lo (b,i)))) := by
+  apply Equiv.ext
+  intro x
+  change cardSplit r L (butterflyPerm (L+r) (decodeButterfly (L+r) (assembleBits r L lo hi))
+    ((cardSplit r L).symm x))=_
+  rw [butterfly_split,Equiv.apply_symm_apply]
+  rfl
+
+lemma boundaryTree_node_raw (L r : ℕ) (U : Card (r+1) → Equiv.Perm (Card L))
+    (X : SwitchIndex (L+r+1) → Bool) :
+    boundaryTree L (r+1) U X=InputTree.node (fun y => X (Sum.inl y))
+      (fun b => boundaryTree L r (fun u => U (Fin.cons b u)) (fun i => X (Sum.inr (b,i)))) := by
+  conv_lhs => unfold boundaryTree
+  congr 1
+  funext b
+  cases b <;> simp only [Bool.false_eq_true,ite_false,ite_true] <;> rfl
+
+lemma boundaryOutput_node_raw (L r : ℕ) (Y : SwitchIndex (L+r+1) → Bool) (b : Card (r+1)) :
+    boundaryOutput L (r+1) Y b=
+      boundaryOutput L r (fun i => Y (Sum.inr (b 0,i))) (Fin.tail b) := by
+  conv_lhs => unfold boundaryOutput
+  cases b 0 <;> simp only [Bool.false_eq_true,ite_false,ite_true] <;> rfl
+
+lemma boundaryTree_split (L r : ℕ) (U : Card r → Equiv.Perm (Card L))
+    (lo : Card r × SwitchIndex L → Bool) (hi : Card L × SwitchIndex r → Bool) (x : Card (L+r)) :
+    cardSplit r L ((boundaryTree L r U (assembleBits r L lo hi)).perm x)=
+      let y := cardSplit r L x
+      let z := U y.1 y.2
+      (butterflyPerm r (decodeButterfly r (fun i => hi (z,i))) y.1,z) := by
+  induction r with
+  | zero =>
+    apply Prod.ext
+    · exact Subsingleton.elim _ _
+    · rfl
+  | succ r ih =>
+    let lo' : Card r × SwitchIndex L → Bool := fun j => lo (Fin.cons (x 0) j.1,j.2)
+    let hi' : Card L × SwitchIndex r → Bool := fun j => hi (j.1,Sum.inr (x 0,j.2))
+    let U' : Card r → Equiv.Perm (Card L) := fun b => U (Fin.cons (x 0) b)
+    have hh := ih U' lo' hi' (Fin.tail x)
+    let u := (boundaryTree L r U' (assembleBits r L lo' hi')).perm (Fin.tail x)
+    let y := cardSplit r L (Fin.tail x)
+    let z := U' y.1 y.2
+    let w := butterflyPerm r (decodeButterfly r (fun i => hi' (z,i))) y.1
+    change cardSplit r L u=(w,z) at hh
+    rw [boundaryTree_node_raw]
+    change ((Fin.cons (Bool.xor (x 0) (hi ((cardSplit r L u).2,Sum.inl (cardSplit r L u).1)))
+      ((cardSplit r L u).1) : Card (r+1)),(cardSplit r L u).2)=
+        ((Fin.cons (Bool.xor (x 0) (hi (z,Sum.inl w))) w : Card (r+1)),z)
+    rw [hh]
+
+lemma splitPerm_boundaryTree (L r : ℕ) (U : Card r → Equiv.Perm (Card L))
+    (lo : Card r × SwitchIndex L → Bool) (hi : Card L × SwitchIndex r → Bool) :
+    splitPerm r L (boundaryTree L r U (assembleBits r L lo hi)).perm=
+      outerPerm r L hi*ParallelBlocks.lift U := by
+  apply Equiv.ext
+  intro x
+  change cardSplit r L ((boundaryTree L r U (assembleBits r L lo hi)).perm
+    ((cardSplit r L).symm x))=_
+  rw [boundaryTree_split,Equiv.apply_symm_apply]
+  rfl
+
+lemma boundaryOutput_split (L r : ℕ) (lo : Card r × SwitchIndex L → Bool)
+    (hi : Card L × SwitchIndex r → Bool) (b : Card r) :
+    boundaryOutput L r (assembleBits r L lo hi) b=
+      butterflyPerm L (decodeButterfly L (fun i => lo (b,i))) := by
+  induction r with
+  | zero =>
+    rw [Subsingleton.elim b (fun j => Fin.elim0 j)]
+    rfl
+  | succ r ih =>
+    rw [boundaryOutput_node_raw]
+    change boundaryOutput L r (assembleBits r L (fun j => lo (Fin.cons (b 0) j.1,j.2))
+      (fun j => hi (j.1,Sum.inr (b 0,j.2)))) (Fin.tail b)=_
+    rw [ih]
+    simp only [Fin.cons_self_tail]
+
+
+abbrev OuterCoins (r L : ℕ) := Card L × SwitchIndex r → Bool
+abbrev LocalCoins (L : ℕ) := SwitchIndex L → Bool
+
+def parallelSplitCoinEquiv (r L : ℕ) :
+    ((Card r → LocalCoins L) × OuterCoins r L) ≃ LocalCoins (L+r) :=
+  (Equiv.prodCongr (Equiv.curry (Card r) (SwitchIndex L) Bool).symm (Equiv.refl _)).trans
+    (splitCoinEquiv r L)
+
+lemma parallelSplitCoinEquiv_apply (r L : ℕ) (c : Card r → LocalCoins L) (h : OuterCoins r L) :
+    parallelSplitCoinEquiv r L (c,h)=assembleBits r L (fun x => c x.1 x.2) h := by
+  exact splitCoinEquiv_apply _ _ _ _
+
+end Thorp
+
+end ThorpNine.Harmonic
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Harmonic/Coins.lean b/lean/OAI/Probability/ThorpRouting/Harmonic/Coins.lean
index ac37b363f1ff06ecf2f078915f725275e466a3a4..8f54a3cb1b4dcce22deb0132d78c458673bc0e4d 100644
--- a/lean/OAI/Probability/ThorpRouting/Harmonic/Coins.lean
+++ b/lean/OAI/Probability/ThorpRouting/Harmonic/Coins.lean
@@ -1,764 +1,764 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Harmonic.Model
-
-namespace OAI
-
-namespace ThorpNine.Harmonic
-
-namespace Thorp
-
-open scoped BigOperators
-open Filter
-
-def Coins : ℕ → Type
-  | 0 => Unit
-  | d + 1 => Card d → Bool
-
-instance (d : ℕ) : Fintype (Coins d) := by
-  cases d <;> dsimp [Coins] <;> infer_instance
-
-instance (d : ℕ) : Nonempty (Coins d) := by
-  cases d <;> dsimp [Coins] <;> infer_instance
-
-def rotate (d : ℕ) : Equiv.Perm (Card d) where
-  toFun x i := x (finRotate d i)
-  invFun x i := x ((finRotate d).symm i)
-  left_inv x := by funext i; simp only [Equiv.apply_symm_apply]
-  right_inv x := by funext i; simp only [Equiv.symm_apply_apply]
-
-def step : (d : ℕ) → Coins d → Equiv.Perm (Card d)
-  | 0, _ => 1
-  | d + 1, ξ => rotate (d + 1) * pairSwitch ξ
-
-def run (d : ℕ) : (t : ℕ) → (Fin t → Coins d) → Equiv.Perm (Card d)
-  | 0, _ => 1
-  | t + 1, ω => step d (ω (Fin.last t)) * run d t (fun i => ω i.castSucc)
-
-noncomputable def law (d t : ℕ) (g : Equiv.Perm (Card d)) : ℝ :=
-  (Fintype.card {ω : Fin t → Coins d // run d t ω = g} : ℝ) /
-    (Fintype.card (Fin t → Coins d) : ℝ)
-
-noncomputable def distance (d t : ℕ) : ℝ :=
-  (1 / 2 : ℝ) * ∑ g : Equiv.Perm (Card d),
-    |law d t g - 1 / (Nat.factorial (2 ^ d) : ℝ)|
-
-noncomputable def mixingTime (d : ℕ) : ℕ :=
-  sInf {t : ℕ | distance d t ≤ (1 / 4 : ℝ)}
-
-noncomputable def supportTimeBound (d : ℕ) : ℤ :=
-  ⌈(2 / (2 : ℝ) ^ d) *
-    (Real.log ((3 / 4 : ℝ) * (Nat.factorial (2 ^ d) : ℝ)) / Real.log 2)⌉
-
-def OptimalOrder : Prop :=
-  ∃ C : ℕ, 0 < C ∧
-    Tendsto (fun d : ℕ => distance d (C * d)) atTop (nhds 0) ∧
-    (∀ d t : ℕ, 1 - (2 : ℝ) ^ ((t : ℝ) * (2 : ℝ) ^ d / 2) /
-      (Nat.factorial (2 ^ d) : ℝ) ≤ distance d t) ∧
-    (∀ᶠ d : ℕ in atTop,
-      supportTimeBound d ≤ (mixingTime d : ℤ) ∧ mixingTime d ≤ C * d) ∧
-    (∃ K : ℝ, ∀ᶠ d : ℕ in atTop, 2 * (d : ℝ) - K ≤ (mixingTime d : ℝ)) ∧
-    Asymptotics.IsTheta atTop (fun d : ℕ => (mixingTime d : ℝ)) (fun d : ℕ => (d : ℝ))
-
-noncomputable def finiteLaw {α β : Type*} [Fintype α] (f : α → β) (b : β) : ℝ := by
-  classical
-  exact (Fintype.card {a : α // f a = b} : ℝ) / (Fintype.card α : ℝ)
-
-lemma sum_card_fibers {α β : Type*} [Fintype α] [Fintype β] [DecidableEq β] (f : α → β) :
-    ∑ b, Fintype.card {a : α // f a = b} = Fintype.card α := by
-  classical
-  rw [← Fintype.card_sigma]
-  exact Fintype.card_congr (Equiv.sigmaFiberEquiv f)
-
-lemma finiteLaw_nonneg {α β : Type*} [Fintype α] (f : α → β) (b : β) :
-    0 ≤ finiteLaw f b := by
-  exact div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _)
-
-lemma finiteLaw_sum {α β : Type*} [Fintype α] [Nonempty α] [Fintype β] (f : α → β) :
-    ∑ b, finiteLaw f b = 1 := by
-  classical
-  simp only [finiteLaw, ← Finset.sum_div]
-  have hc : (∑ b, (Fintype.card {a : α // f a = b} : ℝ)) = (Fintype.card α : ℝ) := by
-    exact_mod_cast sum_card_fibers f
-  rw [hc]
-  exact div_self (Nat.cast_ne_zero.mpr Fintype.card_ne_zero)
-
-lemma finiteLaw_eq_zero_of_not_mem {α β : Type*} [Fintype α] [DecidableEq β]
-    (f : α → β) {b : β} (hb : b ∉ Finset.univ.image f) : finiteLaw f b = 0 := by
-  classical
-  have : IsEmpty {a : α // f a = b} := ⟨fun a => hb (Finset.mem_image.mpr
-    ⟨a.val, Finset.mem_univ _, a.property⟩)⟩
-  simp [finiteLaw]
-
-lemma event_le_totalVariation {β : Type*} [Fintype β] [DecidableEq β]
-    (p q : β → ℝ) (hpq : ∑ b, (p b - q b) = 0) (s : Finset β) :
-    (∑ b ∈ s, p b) - (∑ b ∈ s, q b) ≤ (1 / 2 : ℝ) * ∑ b, |p b - q b| := by
-  have h₁ : (∑ b ∈ s, (p b - q b)) ≤ ∑ b ∈ s, |p b - q b| :=
-    Finset.sum_le_sum (fun b _ => le_abs_self (p b - q b))
-  have h₂ : -(∑ b ∈ sᶜ, (p b - q b)) ≤ ∑ b ∈ sᶜ, |p b - q b| := by
-    rw [← Finset.sum_neg_distrib]
-    exact Finset.sum_le_sum (fun b _ => neg_le_abs (p b - q b))
-  have hz := Finset.sum_add_sum_compl s (fun b => p b - q b)
-  have ha := Finset.sum_add_sum_compl s (fun b => |p b - q b|)
-  rw [hpq] at hz
-  rw [← Finset.sum_sub_distrib]
-  linarith
-
-lemma finiteLaw_support_bound {α β : Type*} [Fintype α] [Nonempty α]
-    [Fintype β] [Nonempty β] (f : α → β) :
-    1 - (Fintype.card α : ℝ) / (Fintype.card β : ℝ) ≤
-      (1 / 2 : ℝ) * ∑ b, |finiteLaw f b - 1 / (Fintype.card β : ℝ)| := by
-  classical
-  let s := Finset.univ.image f
-  have hs : ∑ b ∈ s, finiteLaw f b = 1 := by
-    have h := Finset.sum_subset (Finset.subset_univ s)
-      (fun b _ hb => finiteLaw_eq_zero_of_not_mem f hb)
-    simpa only [finiteLaw_sum] using h
-  have hβ : (Fintype.card β : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
-  have hq : (∑ _b : β, 1 / (Fintype.card β : ℝ)) = 1 := by simp [hβ]
-  have hd : (∑ b, (finiteLaw f b - 1 / (Fintype.card β : ℝ))) = 0 := by
-    rw [Finset.sum_sub_distrib, finiteLaw_sum, hq, sub_self]
-  have he := event_le_totalVariation (finiteLaw f) (fun _ => 1 / (Fintype.card β : ℝ)) hd s
-  rw [hs] at he
-  have hsq : (∑ _b ∈ s, 1 / (Fintype.card β : ℝ)) = (s.card : ℝ) / Fintype.card β := by
-    simp [div_eq_mul_inv]
-  rw [hsq] at he
-  have hcard : (s.card : ℝ) ≤ Fintype.card α := by
-    exact_mod_cast (Finset.card_image_le (s := (Finset.univ : Finset α)) (f := f))
-  have hdiv := div_le_div_of_nonneg_right hcard (Nat.cast_nonneg (Fintype.card β) : (0 : ℝ) ≤ _)
-  linarith
-
-lemma card_positions (d : ℕ) : Fintype.card (Card d) = 2 ^ d := by
-  simp [Card]
-
-lemma card_permutations (d : ℕ) : Fintype.card (Equiv.Perm (Card d)) = Nat.factorial (2 ^ d) := by
-  rw [Fintype.card_perm, card_positions]
-
-lemma card_coins_succ (d : ℕ) : Fintype.card (Coins (d + 1)) = 2 ^ (2 ^ d) := by
-  simp [Coins]
-
-lemma card_tapes_succ (d t : ℕ) : Fintype.card (Fin t → Coins (d + 1)) = 2 ^ (t * 2 ^ d) := by
-  rw [Fintype.card_fun, card_coins_succ, Fintype.card_fin, ← pow_mul, Nat.mul_comm]
-
-lemma law_eq_finiteLaw (d t : ℕ) (g : Equiv.Perm (Card d)) :
-    law d t g = finiteLaw (run d t) g := by
-  unfold law finiteLaw
-  apply congrArg (fun n : ℕ => (n : ℝ) / (Fintype.card (Fin t → Coins d) : ℝ))
-  exact congrArg (@Fintype.card _) (Subsingleton.elim _ _)
-
-lemma law_nonneg (d t : ℕ) (g : Equiv.Perm (Card d)) : 0 ≤ law d t g := by
-  rw [law_eq_finiteLaw]
-  exact finiteLaw_nonneg (run d t) g
-
-lemma law_sum (d t : ℕ) : ∑ g, law d t g = 1 := by
-  simp only [law_eq_finiteLaw]
-  exact finiteLaw_sum (run d t)
-
-lemma physical_support_bound_succ (d t : ℕ) :
-    1 - (2 : ℝ) ^ (t * 2 ^ d) / (Nat.factorial (2 ^ (d + 1)) : ℝ) ≤ distance (d + 1) t := by
-  have : Nonempty (Equiv.Perm (Card (d + 1))) := ⟨1⟩
-  have h := finiteLaw_support_bound (run (d + 1) t)
-  rw [card_tapes_succ, card_permutations] at h
-  simpa only [Nat.cast_pow, Nat.cast_ofNat, distance, law_eq_finiteLaw] using h
-
-lemma step_succ_formula (d : ℕ) (ξ : Coins (d + 1)) (x : Card (d + 1)) :
-    step (d + 1) ξ x = Fin.snoc (Fin.tail x) (Bool.xor (x 0) (ξ (Fin.tail x))) := by
-  change (fun i => switchFun ξ x (finRotate (d + 1) i)) = _
-  rw [Fin.snoc_eq_cons_rotate]
-  rfl
-
-lemma distance_nonneg (d t : ℕ) : 0 ≤ distance d t := by
-  unfold distance
-  exact mul_nonneg (by norm_num) (Finset.sum_nonneg (fun _ _ => abs_nonneg _))
-
-lemma physical_support_bound (d t : ℕ) :
-    1 - (2 : ℝ) ^ ((t : ℝ) * (2 : ℝ) ^ d / 2) /
-      (Nat.factorial (2 ^ d) : ℝ) ≤ distance d t := by
-  cases d with
-  | zero =>
-    have h : 1 ≤ (2 : ℝ) ^ ((t : ℝ) / 2) := Real.one_le_rpow (by norm_num) (by positivity)
-    simpa only [pow_zero, Nat.factorial_one, Nat.cast_one, div_one, mul_one] using
-      le_trans (sub_nonpos.mpr h) (distance_nonneg 0 t)
-  | succ d =>
-    have he : (t : ℝ) * (2 : ℝ) ^ (d + 1) / 2 = (t * 2 ^ d : ℕ) := by
-      push_cast
-      rw [pow_succ]
-      ring
-    rw [he, Real.rpow_natCast]
-    exact physical_support_bound_succ d t
-
-lemma supportTimeBound_le_of_distance_le (d t : ℕ) (ht : distance d t ≤ (1 / 4 : ℝ)) :
-    supportTimeBound d ≤ (t : ℤ) := by
-  have hF : (0 : ℝ) < Nat.factorial (2 ^ d) := by positivity
-  have hN : (0 : ℝ) < (2 : ℝ) ^ d := by positivity
-  have hL : (0 : ℝ) < Real.log 2 := Real.log_pos (by norm_num)
-  have hb := physical_support_bound d t
-  have hp : (3 / 4 : ℝ) ≤
-      (2 : ℝ) ^ ((t : ℝ) * (2 : ℝ) ^ d / 2) / (Nat.factorial (2 ^ d) : ℝ) := by
-    linarith
-  have hr := (le_div_iff₀ hF).mp hp
-  have hlog := Real.log_le_log (show (0 : ℝ) < (3 / 4 : ℝ) * Nat.factorial (2 ^ d) by positivity) hr
-  rw [Real.log_rpow (by norm_num : (0 : ℝ) < 2)] at hlog
-  unfold supportTimeBound
-  apply Int.ceil_le.mpr
-  simp only [Int.cast_natCast]
-  calc
-    2 / (2 : ℝ) ^ d * (Real.log ((3 / 4 : ℝ) * Nat.factorial (2 ^ d)) / Real.log 2) =
-        Real.log ((3 / 4 : ℝ) * Nat.factorial (2 ^ d)) / ((2 : ℝ) ^ d / 2 * Real.log 2) := by
-      field_simp
-    _ ≤ (t : ℝ) := (div_le_iff₀ (mul_pos (div_pos hN (by norm_num)) hL)).mpr (by
-      nlinarith [hlog])
-
-lemma supportTimeBound_le_mixingTime {d : ℕ}
-    (h : ({t : ℕ | distance d t ≤ (1 / 4 : ℝ)} : Set ℕ).Nonempty) :
-    supportTimeBound d ≤ (mixingTime d : ℤ) := by
-  exact supportTimeBound_le_of_distance_le d (mixingTime d) (Nat.sInf_mem h)
-
-lemma log_factorial_lower {n : ℕ} (hn : 1 ≤ n) :
-    (n : ℝ) * Real.log n - n ≤ Real.log (Nat.factorial n) := by
-  have h := Stirling.le_log_factorial_stirling (show n ≠ 0 by omega)
-  have hln : 0 ≤ Real.log (n : ℝ) := Real.log_nonneg (by exact_mod_cast hn)
-  have hlp : 0 ≤ Real.log (2 * Real.pi) := Real.log_nonneg (by
-    have := Real.pi_gt_three
-    linarith)
-  linarith
-
-lemma log_three_quarters_lower : -(1 : ℝ) ≤ Real.log (3 / 4 : ℝ) := by
-  have h := Real.log_le_sub_one_of_pos (show (0 : ℝ) < 4 / 3 by norm_num)
-  have he : Real.log (4 / 3 : ℝ) = - Real.log (3 / 4 : ℝ) := by
-    rw [← Real.log_inv]
-    congr 1
-    norm_num
-  rw [he] at h
-  linarith
-
-lemma linear_le_supportTimeBound (d : ℕ) :
-    2 * (d : ℝ) - 4 / Real.log 2 ≤ (supportTimeBound d : ℝ) := by
-  have hN : 1 ≤ (2 : ℝ) ^ d := one_le_pow₀ (by norm_num)
-  have hNp : 0 < (2 : ℝ) ^ d := by positivity
-  have hL : 0 < Real.log 2 := Real.log_pos (by norm_num)
-  have hf := log_factorial_lower (show 1 ≤ 2 ^ d by exact one_le_pow₀ (by norm_num))
-  have hlog : (2 : ℝ) ^ d * ((d : ℝ) * Real.log 2) - (2 : ℝ) ^ d - 1 ≤
-      Real.log ((3 / 4 : ℝ) * (Nat.factorial (2 ^ d) : ℝ)) := by
-    rw [Real.log_mul (by norm_num) (by positivity)]
-    push_cast at hf
-    rw [Real.log_pow] at hf
-    linarith [log_three_quarters_lower]
-  apply le_trans _ (Int.le_ceil _)
-  have hmul : (2 * (d : ℝ) - 4 / Real.log 2) *
-      ((2 : ℝ) ^ d / 2 * Real.log 2) ≤
-      Real.log ((3 / 4 : ℝ) * (Nat.factorial (2 ^ d) : ℝ)) := by
-    have hcancel : (2 * (d : ℝ) - 4 / Real.log 2) *
-        ((2 : ℝ) ^ d / 2 * Real.log 2) =
-        (2 : ℝ) ^ d * ((d : ℝ) * Real.log 2) - 2 * (2 : ℝ) ^ d := by
-      field_simp
-      ring
-    rw [hcancel]
-    linarith
-  convert (le_div_iff₀ (mul_pos (div_pos hNp (by norm_num)) hL)).mpr hmul using 1
-  field_simp
-
-lemma linear_le_time_of_distance_le (d t : ℕ)
-    (ht : distance d t ≤ (1 / 4 : ℝ)) :
-    2 * (d : ℝ) - 4 / Real.log 2 ≤ (t : ℝ) := by
-  have h := supportTimeBound_le_of_distance_le d t ht
-  have hr : (supportTimeBound d : ℝ) ≤ (t : ℝ) := by exact_mod_cast h
-  exact (linear_le_supportTimeBound d).trans hr
-
-lemma entropy_chord (p : ℝ) (hp : 0 ≤ p) (hq : p ≤ 1 / 2) :
-    2 * p * Real.log 2 ≤ Real.binEntropy p := by
-  have hc := Real.strictConcave_binEntropy.concaveOn.2
-    (show (0 : ℝ) ∈ Set.Icc (0 : ℝ) 1 by norm_num)
-    (show (2 : ℝ)⁻¹ ∈ Set.Icc (0 : ℝ) 1 by norm_num)
-    (show 0 ≤ 1 - 2 * p by linarith)
-    (show 0 ≤ 2 * p by positivity)
-    (show (1 - 2 * p) + 2 * p = 1 by ring)
-  simpa only [Real.binEntropy_zero, Real.binEntropy_two_inv,
-    smul_eq_mul, mul_zero, zero_add, show 2 * p * (2 : ℝ)⁻¹ = p by ring] using hc
-
-lemma split_entropy_lower (a b : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) :
-    2 * min a b * Real.log 2 ≤
-      (a + b) * Real.log (a + b) - a * Real.log a - b * Real.log b := by
-  wlog hab : a ≤ b generalizing a b
-  · have hba := this b a hb ha (le_of_not_ge hab)
-    simpa only [min_comm, add_comm b a] using hba.trans_eq (show (b + a) * Real.log (b + a) - b * Real.log b - a * Real.log a =
-      (a + b) * Real.log (a + b) - a * Real.log a - b * Real.log b by rw [add_comm b a]; ring)
-  rw [min_eq_left hab]
-  by_cases hz : a = 0
-  · subst a
-    simp
-  have hap : 0 < a := lt_of_le_of_ne ha (Ne.symm hz)
-  have hbp : 0 < b := hap.trans_le hab
-  have hs : 0 < a + b := add_pos hap hbp
-  have hp : 0 ≤ a / (a + b) := div_nonneg ha hs.le
-  have hq : a / (a + b) ≤ 1 / 2 := (div_le_iff₀ hs).mpr (by linarith)
-  have hc := mul_le_mul_of_nonneg_left (entropy_chord (a / (a + b)) hp hq) hs.le
-  have he : (a + b) * Real.binEntropy (a / (a + b)) =
-      (a + b) * Real.log (a + b) - a * Real.log a - b * Real.log b := by
-    unfold Real.binEntropy
-    rw [show 1 - a / (a + b) = b / (a + b) by field_simp; ring]
-    simp only [Real.log_inv]
-    rw [Real.log_div (ne_of_gt hap) (ne_of_gt hs),
-      Real.log_div (ne_of_gt hbp) (ne_of_gt hs)]
-    field_simp
-    ring
-  rw [he] at hc
-  calc
-    2 * a * Real.log 2 = (a + b) * (2 * (a / (a + b)) * Real.log 2) := by
-      field_simp
-    _ ≤ _ := hc
-
-noncomputable def childMarks {d : ℕ} (s : Finset (Card (d + 1))) (ε : Bool) :
-    Finset (Card d) := (s.filter (fun x => x 0 = ε)).image Fin.tail
-
-lemma card_childMarks {d : ℕ} (s : Finset (Card (d + 1))) (ε : Bool) :
-    (childMarks s ε).card = (s.filter (fun x => x 0 = ε)).card := by
-  classical
-  unfold childMarks
-  apply Finset.card_image_iff.mpr
-  intro x hx y hy hxy
-  have hx0 := (Finset.mem_filter.mp hx).2
-  have hy0 := (Finset.mem_filter.mp hy).2
-  rw [← Fin.cons_self_tail x, ← Fin.cons_self_tail y, hx0, hy0, hxy]
-
-lemma card_childMarks_add {d : ℕ} (s : Finset (Card (d + 1))) :
-    (childMarks s false).card + (childMarks s true).card = s.card := by
-  classical
-  rw [card_childMarks, card_childMarks]
-  simpa only [Bool.not_eq_false] using
-    (Finset.card_filter_add_card_filter_not (s := s) (fun x => x 0 = false))
-
-noncomputable def topMeetings {d : ℕ} (b : Butterfly (d + 1))
-    (s : Finset (Card (d + 1))) : Finset (Card d) :=
-  ((childMarks s false).image (butterflyPerm d (b.2 false))) ∩
-    ((childMarks s true).image (butterflyPerm d (b.2 true)))
-
-lemma card_topMeetings_le {d : ℕ} (b : Butterfly (d + 1)) (s : Finset (Card (d + 1))) :
-    (topMeetings b s).card ≤ min (childMarks s false).card (childMarks s true).card := by
-  classical
-  apply le_min
-  · exact (Finset.card_le_card Finset.inter_subset_left).trans_eq
-      (Finset.card_image_of_injective _ (butterflyPerm d (b.2 false)).injective)
-  · exact (Finset.card_le_card Finset.inter_subset_right).trans_eq
-      (Finset.card_image_of_injective _ (butterflyPerm d (b.2 true)).injective)
-
-noncomputable def encounters : (d : ℕ) → Butterfly d → Finset (Card d) → ℕ
-  | 0, _, _ => 0
-  | d + 1, b, s => encounters d (b.2 false) (childMarks s false) +
-      encounters d (b.2 true) (childMarks s true) + (topMeetings b s).card
-
-lemma butterfly_encounters (d : ℕ) (b : Butterfly d) (s : Finset (Card d)) :
-    (encounters d b s : ℝ) ≤ (s.card : ℝ) * Real.log s.card / (2 * Real.log 2) := by
-  induction d with
-  | zero =>
-    simp only [encounters, Nat.cast_zero]
-    have hs : s.card ≤ 1 := by
-      simpa [Card] using Finset.card_le_univ s
-    interval_cases s.card <;> simp
-  | succ d ih =>
-    have h₀ := ih (b.2 false) (childMarks s false)
-    have h₁ := ih (b.2 true) (childMarks s true)
-    have ht : ((topMeetings b s).card : ℝ) ≤
-        min ((childMarks s false).card : ℝ) ((childMarks s true).card : ℝ) := by
-      exact_mod_cast card_topMeetings_le b s
-    have he := split_entropy_lower ((childMarks s false).card : ℝ)
-      ((childMarks s true).card : ℝ) (by positivity) (by positivity)
-    have hs : ((childMarks s false).card : ℝ) + (childMarks s true).card = s.card := by
-      exact_mod_cast card_childMarks_add s
-    rw [hs] at he
-    have hL : 0 < 2 * Real.log 2 := by positivity
-    rw [le_div_iff₀ hL] at h₀ h₁ ⊢
-    simp only [encounters, Nat.cast_add]
-    nlinarith [mul_le_mul_of_nonneg_right ht hL.le]
-
-lemma prod_split_at {α : Type*} [DecidableEq α] (s : Finset α) (f : α → ℝ) (a : α) :
-    ∏ i ∈ s, f i = (if a ∈ s then f a else 1) * ∏ i ∈ s.erase a, f i := by
-  by_cases ha : a ∈ s
-  · rw [ite_eq_left ha]
-    exact (Finset.mul_prod_erase s f ha).symm
-  · simp [ha]
-
-lemma prod_split_two {α : Type*} [DecidableEq α] (s : Finset α) (f : α → ℝ)
-    (a b : α) (hab : a ≠ b) :
-    ∏ i ∈ s, f i = (if a ∈ s then f a else 1) * (if b ∈ s then f b else 1) *
-      ∏ i ∈ (s.erase a).erase b, f i := by
-  calc
-    ∏ i ∈ s, f i = (if a ∈ s then f a else 1) * ∏ i ∈ s.erase a, f i :=
-      prod_split_at s f a
-    _ = _ := by
-      rw [prod_split_at (s.erase a) f b]
-      have hm : (b ∈ s.erase a) ↔ b ∈ s :=
-        ⟨fun h => (Finset.mem_erase.mp h).2, fun h => Finset.mem_erase.mpr ⟨Ne.symm hab, h⟩⟩
-      simp only [hm, mul_assoc]
-
-lemma swap_average_product {α : Type*} [DecidableEq α] (s : Finset α) (p : α → ℝ)
-    (hp : ∀ i, 0 ≤ p i) (a b : α) :
-    ((∏ i ∈ s, p i) + ∏ i ∈ s, p (Equiv.swap a b i)) / 2 ≤
-      ∏ i ∈ s, (p i + p (Equiv.swap a b i)) / 2 := by
-  by_cases hab : a = b
-  · subst b
-    simp only [Equiv.swap_self, Equiv.refl_apply]
-    have h : (fun i => (p i + p i) / 2) = p := by funext i; ring
-    rw [h]
-    linarith
-  let t := (s.erase a).erase b
-  have hfix : ∀ i ∈ t, Equiv.swap a b i = i := by
-    intro i hi
-    exact Equiv.swap_apply_of_ne_of_ne (Finset.mem_erase.mp (Finset.mem_erase.mp hi).2).1
-      (Finset.mem_erase.mp hi).1
-  have hprod : (∏ i ∈ t, p (Equiv.swap a b i)) = ∏ i ∈ t, p i :=
-    Finset.prod_congr rfl (fun i hi => by rw [hfix i hi])
-  have havg : (∏ i ∈ t, (p i + p (Equiv.swap a b i)) / 2) = ∏ i ∈ t, p i :=
-    Finset.prod_congr rfl (fun i hi => by rw [hfix i hi]; ring)
-  rw [prod_split_two s p a b hab,
-    prod_split_two s (fun i => p (Equiv.swap a b i)) a b hab,
-    prod_split_two s (fun i => (p i + p (Equiv.swap a b i)) / 2) a b hab]
-  change ((if a ∈ s then p a else 1) * (if b ∈ s then p b else 1) * (∏ i ∈ t, p i) +
-    (if a ∈ s then p (Equiv.swap a b a) else 1) *
-      (if b ∈ s then p (Equiv.swap a b b) else 1) * (∏ i ∈ t, p (Equiv.swap a b i))) / 2 ≤ _
-  rw [hprod]
-  change _ ≤ (if a ∈ s then (p a + p (Equiv.swap a b a)) / 2 else 1) *
-    (if b ∈ s then (p b + p (Equiv.swap a b b)) / 2 else 1) *
-      (∏ i ∈ t, (p i + p (Equiv.swap a b i)) / 2)
-  rw [havg, Equiv.swap_apply_left, Equiv.swap_apply_right]
-  have ht : 0 ≤ ∏ i ∈ t, p i := Finset.prod_nonneg (fun i _ => hp i)
-  by_cases ha : a ∈ s <;> by_cases hb : b ∈ s <;> simp only [ha, hb, ↓reduceIte]
-  · have hsq := mul_nonneg (sq_nonneg (p a - p b)) ht
-    nlinarith
-  all_goals ring_nf; rfl
-
-def CylinderBound {α : Type*} (m : Finset α → ℝ) (p : α → ℝ) : Prop :=
-  ∀ s, m s ≤ ∏ i ∈ s, p i
-
-lemma CylinderBound.fair_swap {α : Type*} [DecidableEq α]
-    {m : Finset α → ℝ} {p : α → ℝ} (h : CylinderBound m p)
-    (hp : ∀ i, 0 ≤ p i) (a b : α) :
-    CylinderBound (fun s => (m s + m (s.image (Equiv.swap a b))) / 2)
-      (fun i => (p i + p (Equiv.swap a b i)) / 2) := by
-  intro s
-  have h₀ := h s
-  have h₁ := h (s.image (Equiv.swap a b))
-  rw [Finset.prod_image (Equiv.swap a b).injective.injOn] at h₁
-  exact (show (m s + m (s.image (Equiv.swap a b))) / 2 ≤
-      ((∏ i ∈ s, p i) + ∏ i ∈ s, p (Equiv.swap a b i)) / 2 by linarith).trans
-    (swap_average_product s p hp a b)
-
-noncomputable def inclusionMoment {α Ω : Type*} [Fintype Ω]
-    (F : Ω → Finset α) (s : Finset α) : ℝ := by
-  classical
-  exact finiteMean (fun ω => if s ⊆ F ω then 1 else 0)
-
-lemma inclusionMoment_nonneg {α Ω : Type*} [Fintype Ω]
-    (F : Ω → Finset α) (s : Finset α) : 0 ≤ inclusionMoment F s := by
-  classical
-  unfold inclusionMoment finiteMean
-  positivity
-
-lemma indicator_subset_prod {α : Type*} [DecidableEq α] (A s : Finset α) :
-    (if s ⊆ A then (1 : ℝ) else 0) = ∏ i ∈ s, (if i ∈ A then 1 else 0) := by
-  by_cases h : s ⊆ A
-  · rw [ite_eq_left h]
-    exact (Finset.prod_eq_one (fun i hi => ite_eq_left (h hi))).symm
-  · rw [ite_eq_right h]
-    obtain ⟨i, hi, hiA⟩ := Finset.not_subset.mp h
-    exact (Finset.prod_eq_zero hi (ite_eq_right hiA)).symm
-
-lemma inclusionMoment_const {α Ω : Type*} [DecidableEq α] [Fintype Ω] [Nonempty Ω]
-    (A s : Finset α) : inclusionMoment (fun _ : Ω => A) s =
-      if s ⊆ A then 1 else 0 := by
-  classical
-  have hn : (Fintype.card Ω : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
-  by_cases h : s ⊆ A <;> simp [inclusionMoment, finiteMean, h, hn]
-
-lemma cylinder_const {α Ω : Type*} [Fintype Ω] [Nonempty Ω] (A : Finset α) :
-    CylinderBound (inclusionMoment (fun _ : Ω => A))
-      (fun i => inclusionMoment (fun _ : Ω => A) {i}) := by
-  classical
-  intro s
-  simp only [inclusionMoment_const, Finset.singleton_subset_iff]
-  exact (indicator_subset_prod A s).le
-
-noncomputable def fairSetSwap {α Ω : Type*} [DecidableEq α]
-    (F : Ω → Finset α) (a b : α) : Ω × Bool → Finset α :=
-  fun ω => if ω.2 then (F ω.1).image (Equiv.swap a b) else F ω.1
-
-lemma subset_swap_image_iff {α : Type*} [DecidableEq α]
-    (A s : Finset α) (a b : α) :
-    s ⊆ A.image (Equiv.swap a b) ↔ s.image (Equiv.swap a b) ⊆ A := by
-  constructor
-  · intro h
-    have h' := Finset.image_subset_image h (f := Equiv.swap a b)
-    simpa only [Finset.image_image, Function.comp_def, Equiv.swap_apply_self, Finset.image_id'] using h'
-  · intro h
-    have h' := Finset.image_subset_image h (f := Equiv.swap a b)
-    simpa only [Finset.image_image, Function.comp_def, Equiv.swap_apply_self, Finset.image_id'] using h'
-
-lemma inclusionMoment_fairSetSwap {α Ω : Type*} [DecidableEq α] [Fintype Ω]
-    (F : Ω → Finset α) (a b : α) (s : Finset α) :
-    inclusionMoment (fairSetSwap F a b) s =
-      (inclusionMoment F s + inclusionMoment F (s.image (Equiv.swap a b))) / 2 := by
-  classical
-  simp only [inclusionMoment, finiteMean, Fintype.sum_prod_type, Fintype.sum_bool,
-    fairSetSwap, Bool.false_eq_true, ↓reduceIte, Fintype.card_prod, Fintype.card_bool,
-    Nat.cast_mul, Nat.cast_ofNat, subset_swap_image_iff, Finset.sum_add_distrib]
-  rw [div_mul_eq_div_div, add_div]
-  ring_nf
-
-lemma cylinder_fairSetSwap {α Ω : Type*} [DecidableEq α] [Fintype Ω]
-    (F : Ω → Finset α) (a b : α)
-    (h : CylinderBound (inclusionMoment F) (fun i => inclusionMoment F {i})) :
-    CylinderBound (inclusionMoment (fairSetSwap F a b))
-      (fun i => inclusionMoment (fairSetSwap F a b) {i}) := by
-  classical
-  have h' := h.fair_swap (fun i => inclusionMoment_nonneg F {i}) a b
-  intro s
-  simpa only [inclusionMoment_fairSetSwap, Finset.image_singleton] using h' s
-
-def SwitchSamples : ℕ → Type
-  | 0 => Unit
-  | n + 1 => SwitchSamples n × Bool
-
-instance (n : ℕ) : Fintype (SwitchSamples n) := by
-  induction n with
-  | zero => exact inferInstanceAs (Fintype Unit)
-  | succ n ih => exact inferInstanceAs (Fintype (SwitchSamples n × Bool))
-
-instance (n : ℕ) : Nonempty (SwitchSamples n) := by
-  induction n with
-  | zero => exact inferInstanceAs (Nonempty Unit)
-  | succ n ih => exact inferInstanceAs (Nonempty (SwitchSamples n × Bool))
-
-noncomputable def switchedSet {α : Type*} [DecidableEq α] (A : Finset α) :
-    (L : List (α × α)) → SwitchSamples L.length → Finset α
-  | [], _ => A
-  | (a,b) :: L, ω => fairSetSwap (switchedSet A L) a b ω
-
-lemma switchedSet_cylinder {α : Type*} [DecidableEq α] (A : Finset α)
-    (L : List (α × α)) :
-    CylinderBound (inclusionMoment (switchedSet A L))
-      (fun i => inclusionMoment (switchedSet A L) {i}) := by
-  induction L with
-  | nil => exact cylinder_const A
-  | cons e L ih => exact cylinder_fairSetSwap (switchedSet A L) e.1 e.2 ih
-
-inductive QueryTree (ι : Type*)
-  | reject : QueryTree ι
-  | accept : QueryTree ι
-  | branch (i : ι) (zero one : QueryTree ι) : QueryTree ι
-  | force (i : ι) (b : Bool) (next : QueryTree ι) : QueryTree ι
-
-namespace QueryTree
-
-variable {ι : Type*} [DecidableEq ι]
-
-def queries : QueryTree ι → Finset ι
-  | reject => ∅
-  | accept => ∅
-  | branch i zero one => insert i (queries zero ∪ queries one)
-  | force i _ next => insert i (queries next)
-
-def Fresh : QueryTree ι → Prop
-  | reject => True
-  | accept => True
-  | branch i zero one => i ∉ queries zero ∧ i ∉ queries one ∧ Fresh zero ∧ Fresh one
-  | force i _ next => i ∉ queries next ∧ Fresh next
-
-def weighted : QueryTree ι → (ι → Bool) → ℝ
-  | reject, _ => 0
-  | accept, _ => 1
-  | branch i zero one, ω => if ω i then weighted one ω else weighted zero ω
-  | force i b next, ω => if ω i = b then 2 * weighted next ω else 0
-
-omit [DecidableEq ι] in
-lemma weighted_nonneg (T : QueryTree ι) (ω : ι → Bool) : 0 ≤ T.weighted ω := by
-  induction T with
-  | reject => exact le_rfl
-  | accept => exact zero_le_one
-  | branch i zero one ih₀ ih₁ => simpa only [weighted] using ite_nonneg ih₁ ih₀
-  | force i b next ih => simpa only [weighted] using ite_nonneg (mul_nonneg (by norm_num) ih) le_rfl
-
-lemma weighted_congr (T : QueryTree ι) {ω η : ι → Bool}
-    (h : ∀ i ∈ T.queries, ω i = η i) : T.weighted ω = T.weighted η := by
-  induction T with
-  | reject => rfl
-  | accept => rfl
-  | branch i zero one ih₀ ih₁ =>
-    have hi := h i (Finset.mem_insert_self _ _)
-    simp only [weighted, hi]
-    rw [ih₀ (fun j hj => h j (Finset.mem_insert_of_mem (Finset.mem_union_left _ hj))),
-      ih₁ (fun j hj => h j (Finset.mem_insert_of_mem (Finset.mem_union_right _ hj)))]
-  | force i b next ih =>
-    have hi := h i (Finset.mem_insert_self _ _)
-    simp only [weighted, hi]
-    rw [ih (fun j hj => h j (Finset.mem_insert_of_mem hj))]
-
-def flipCoin (i : ι) : Equiv.Perm (ι → Bool) :=
-  (show Function.Involutive (fun ω : ι → Bool => Function.update ω i (!(ω i))) from by
-    intro ω
-    funext j
-    by_cases h : j = i
-    · subst j; simp
-    · simp [Function.update_of_ne h]).toPerm
-
-lemma flipCoin_apply_eq (i : ι) (ω : ι → Bool) : flipCoin i ω i = !(ω i) := by
-  change Function.update ω i (!(ω i)) i = _
-  exact Function.update_self _ _ _
-
-lemma flipCoin_apply_ne (i j : ι) (h : j ≠ i) (ω : ι → Bool) :
-    flipCoin i ω j = ω j := by
-  change Function.update ω i (!(ω i)) j = _
-  exact Function.update_of_ne h _ _
-
-lemma weighted_flip_of_not_mem (T : QueryTree ι) (i : ι) (hi : i ∉ T.queries)
-    (ω : ι → Bool) : T.weighted (flipCoin i ω) = T.weighted ω := by
-  apply weighted_congr
-  intro j hj
-  exact flipCoin_apply_ne i j (fun h => hi (h ▸ hj)) ω
-
-lemma mean_indicator_bit [Fintype ι] (f : (ι → Bool) → ℝ) (i : ι) (b : Bool)
-    (hf : ∀ ω, f (flipCoin i ω) = f ω) :
-    finiteMean (fun ω => if ω i = b then f ω else 0) = finiteMean f / 2 := by
-  let g := fun ω : ι → Bool => if ω i = b then f ω else 0
-  have hp (ω : ι → Bool) : g ω + g (flipCoin i ω) = f ω := by
-    dsimp [g]
-    rw [flipCoin_apply_eq, hf]
-    cases hω : ω i <;> cases b <;> simp
-  have hs := Equiv.sum_comp (flipCoin i) g
-  have hadd : (∑ ω, g ω) + (∑ ω, g (flipCoin i ω)) = ∑ ω, f ω := by
-    rw [← Finset.sum_add_distrib]
-    exact Finset.sum_congr rfl (fun ω _ => hp ω)
-  unfold finiteMean
-  change (∑ ω, g ω) / _ = _
-  rw [hs] at hadd
-  have hh : (∑ ω, g ω) = (∑ ω, f ω) / 2 := by linarith
-  rw [hh]
-  ring
-
-lemma mean_weighted_le_one [Fintype ι] (T : QueryTree ι) (hT : T.Fresh) :
-    finiteMean T.weighted ≤ 1 := by
-  induction T with
-  | reject => simp [weighted, finiteMean]
-  | accept => simp [weighted, finiteMean]
-  | branch i zero one ih₀ ih₁ =>
-    obtain ⟨hz, ho, hfz, hfo⟩ := hT
-    have h₀ := mean_indicator_bit zero.weighted i false (zero.weighted_flip_of_not_mem i hz)
-    have h₁ := mean_indicator_bit one.weighted i true (one.weighted_flip_of_not_mem i ho)
-    have he : finiteMean (branch i zero one).weighted =
-        finiteMean (fun ω => if ω i = false then zero.weighted ω else 0) +
-        finiteMean (fun ω => if ω i = true then one.weighted ω else 0) := by
-      unfold finiteMean
-      rw [← add_div, ← Finset.sum_add_distrib]
-      congr 1
-      apply Finset.sum_congr rfl
-      intro ω _
-      cases hω : ω i <;> simp [weighted, hω]
-    rw [he, h₀, h₁]
-    linarith [ih₀ hfz, ih₁ hfo]
-  | force i b next ih =>
-    obtain ⟨hi, hn⟩ := hT
-    have hm := mean_indicator_bit next.weighted i b (next.weighted_flip_of_not_mem i hi)
-    have he : finiteMean (force i b next).weighted =
-        2 * finiteMean (fun ω => if ω i = b then next.weighted ω else 0) := by
-      unfold finiteMean
-      rw [← mul_div_assoc, Finset.mul_sum]
-      congr 1
-      apply Finset.sum_congr rfl
-      intro ω _
-      by_cases h : ω i = b <;> simp [weighted, h]
-    rw [he, hm]
-    linarith [ih hn]
-
-def succeeds : QueryTree ι → (ι → Bool) → Bool
-  | reject, _ => false
-  | accept, _ => true
-  | branch i zero one, ω => if ω i then succeeds one ω else succeeds zero ω
-  | force i b next, ω => if ω i = b then succeeds next ω else false
-
-def charge : QueryTree ι → (ι → Bool) → ℕ
-  | reject, _ => 0
-  | accept, _ => 0
-  | branch i zero one, ω => if ω i then charge one ω else charge zero ω
-  | force i b next, ω => if ω i = b then charge next ω + 1 else 0
-
-omit [DecidableEq ι] in
-lemma weighted_eq (T : QueryTree ι) (ω : ι → Bool) :
-    T.weighted ω = if T.succeeds ω then (2 : ℝ) ^ T.charge ω else 0 := by
-  induction T with
-  | reject => rfl
-  | accept => norm_num [weighted, succeeds, charge]
-  | branch i zero one ih₀ ih₁ =>
-    cases h : ω i <;> simp [weighted, succeeds, charge, h, ih₀, ih₁]
-  | force i b next ih =>
-    by_cases h : ω i = b
-    · simp only [weighted, succeeds, charge, h, ↓reduceIte, ih]
-      cases hs : next.succeeds ω <;> simp [pow_succ, mul_comm]
-    · simp [weighted, succeeds, h]
-
-lemma success_probability_le [Fintype ι] (T : QueryTree ι) (hT : T.Fresh)
-    (K : ℕ) (hK : ∀ ω, T.succeeds ω = true → K ≤ T.charge ω) :
-    finiteMean (fun ω => if T.succeeds ω then (1 : ℝ) else 0) ≤ 1 / (2 : ℝ) ^ K := by
-  have hw (ω : ι → Bool) :
-      (2 : ℝ) ^ K * (if T.succeeds ω then (1 : ℝ) else 0) ≤ T.weighted ω := by
-    rw [weighted_eq]
-    cases hs : T.succeeds ω
-    · simp
-    · simpa using pow_le_pow_right₀ (show (1 : ℝ) ≤ 2 by norm_num) (hK ω hs)
-  have hn : 0 ≤ (Fintype.card (ι → Bool) : ℝ) := Nat.cast_nonneg _
-  have hm := div_le_div_of_nonneg_right
-    (Finset.sum_le_sum (fun ω (_ : ω ∈ Finset.univ) => hw ω)) hn
-  rw [← Finset.mul_sum, mul_div_assoc] at hm
-  change (2 : ℝ) ^ K * finiteMean (fun ω => if T.succeeds ω then 1 else 0) ≤
-    finiteMean T.weighted at hm
-  exact (le_div_iff₀ (pow_pos (by norm_num) K)).mpr (by
-    rw [mul_comm]
-    exact hm.trans (mean_weighted_le_one T hT))
-
-end QueryTree
-
-def encodeButterfly : (d : ℕ) → Butterfly d → SwitchIndex d → Bool
-  | 0, _, i => nomatch i
-  | _d + 1, b, Sum.inl y => b.1 y
-  | d + 1, b, Sum.inr (ε,i) => encodeButterfly d (b.2 ε) i
-
-lemma decode_encodeButterfly (d : ℕ) (b : Butterfly d) :
-    decodeButterfly d (encodeButterfly d b) = b := by
-  induction d with
-  | zero => exact Unit.ext _ _
-  | succ d ih =>
-    apply Prod.ext
-    · rfl
-    · funext ε
-      exact ih (b.2 ε)
-
-lemma encode_decodeButterfly (d : ℕ) (ω : SwitchIndex d → Bool) :
-    encodeButterfly d (decodeButterfly d ω) = ω := by
-  induction d with
-  | zero => funext i; exact Empty.elim i
-  | succ d ih =>
-    funext i
-    cases i with
-    | inl y => rfl
-    | inr z => exact congrFun (ih (fun i => ω (Sum.inr (z.1,i)))) z.2
-
-def butterflyCoinEquiv (d : ℕ) : (SwitchIndex d → Bool) ≃ Butterfly d where
-  toFun := decodeButterfly d
-  invFun := encodeButterfly d
-  left_inv := encode_decodeButterfly d
-  right_inv := decode_encodeButterfly d
-
-noncomputable def routeDomain : (d : ℕ) → Card d → Card d → Finset (SwitchIndex d)
-  | 0, _, _ => ∅
-  | d + 1, x, y => by
-      classical
-      exact insert (Sum.inl (Fin.tail y))
-        ((routeDomain d (Fin.tail x) (Fin.tail y)).image (fun i => Sum.inr (x 0,i)))
-
-def routeValue : (d : ℕ) → Card d → Card d → SwitchIndex d → Bool
-  | 0, _, _, i => nomatch i
-  | _d + 1, x, y, Sum.inl _ => Bool.xor (x 0) (y 0)
-  | d + 1, x, y, Sum.inr (_,i) => routeValue d (Fin.tail x) (Fin.tail y) i
-
-end Thorp
-
-end ThorpNine.Harmonic
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Harmonic.Model
+
+namespace OAI
+
+namespace ThorpNine.Harmonic
+
+namespace Thorp
+
+open scoped BigOperators
+open Filter
+
+def Coins : ℕ → Type
+  | 0 => Unit
+  | d + 1 => Card d → Bool
+
+instance (d : ℕ) : Fintype (Coins d) := by
+  cases d <;> dsimp [Coins] <;> infer_instance
+
+instance (d : ℕ) : Nonempty (Coins d) := by
+  cases d <;> dsimp [Coins] <;> infer_instance
+
+def rotate (d : ℕ) : Equiv.Perm (Card d) where
+  toFun x i := x (finRotate d i)
+  invFun x i := x ((finRotate d).symm i)
+  left_inv x := by funext i; simp only [Equiv.apply_symm_apply]
+  right_inv x := by funext i; simp only [Equiv.symm_apply_apply]
+
+def step : (d : ℕ) → Coins d → Equiv.Perm (Card d)
+  | 0, _ => 1
+  | d + 1, ξ => rotate (d + 1) * pairSwitch ξ
+
+def run (d : ℕ) : (t : ℕ) → (Fin t → Coins d) → Equiv.Perm (Card d)
+  | 0, _ => 1
+  | t + 1, ω => step d (ω (Fin.last t)) * run d t (fun i => ω i.castSucc)
+
+noncomputable def law (d t : ℕ) (g : Equiv.Perm (Card d)) : ℝ :=
+  (Fintype.card {ω : Fin t → Coins d // run d t ω = g} : ℝ) /
+    (Fintype.card (Fin t → Coins d) : ℝ)
+
+noncomputable def distance (d t : ℕ) : ℝ :=
+  (1 / 2 : ℝ) * ∑ g : Equiv.Perm (Card d),
+    |law d t g - 1 / (Nat.factorial (2 ^ d) : ℝ)|
+
+noncomputable def mixingTime (d : ℕ) : ℕ :=
+  sInf {t : ℕ | distance d t ≤ (1 / 4 : ℝ)}
+
+noncomputable def supportTimeBound (d : ℕ) : ℤ :=
+  ⌈(2 / (2 : ℝ) ^ d) *
+    (Real.log ((3 / 4 : ℝ) * (Nat.factorial (2 ^ d) : ℝ)) / Real.log 2)⌉
+
+def OptimalOrder : Prop :=
+  ∃ C : ℕ, 0 < C ∧
+    Tendsto (fun d : ℕ => distance d (C * d)) atTop (nhds 0) ∧
+    (∀ d t : ℕ, 1 - (2 : ℝ) ^ ((t : ℝ) * (2 : ℝ) ^ d / 2) /
+      (Nat.factorial (2 ^ d) : ℝ) ≤ distance d t) ∧
+    (∀ᶠ d : ℕ in atTop,
+      supportTimeBound d ≤ (mixingTime d : ℤ) ∧ mixingTime d ≤ C * d) ∧
+    (∃ K : ℝ, ∀ᶠ d : ℕ in atTop, 2 * (d : ℝ) - K ≤ (mixingTime d : ℝ)) ∧
+    Asymptotics.IsTheta atTop (fun d : ℕ => (mixingTime d : ℝ)) (fun d : ℕ => (d : ℝ))
+
+noncomputable def finiteLaw {α β : Type*} [Fintype α] (f : α → β) (b : β) : ℝ := by
+  classical
+  exact (Fintype.card {a : α // f a = b} : ℝ) / (Fintype.card α : ℝ)
+
+lemma sum_card_fibers {α β : Type*} [Fintype α] [Fintype β] [DecidableEq β] (f : α → β) :
+    ∑ b, Fintype.card {a : α // f a = b} = Fintype.card α := by
+  classical
+  rw [← Fintype.card_sigma]
+  exact Fintype.card_congr (Equiv.sigmaFiberEquiv f)
+
+lemma finiteLaw_nonneg {α β : Type*} [Fintype α] (f : α → β) (b : β) :
+    0 ≤ finiteLaw f b := by
+  exact div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _)
+
+lemma finiteLaw_sum {α β : Type*} [Fintype α] [Nonempty α] [Fintype β] (f : α → β) :
+    ∑ b, finiteLaw f b = 1 := by
+  classical
+  simp only [finiteLaw, ← Finset.sum_div]
+  have hc : (∑ b, (Fintype.card {a : α // f a = b} : ℝ)) = (Fintype.card α : ℝ) := by
+    exact_mod_cast sum_card_fibers f
+  rw [hc]
+  exact div_self (Nat.cast_ne_zero.mpr Fintype.card_ne_zero)
+
+lemma finiteLaw_eq_zero_of_not_mem {α β : Type*} [Fintype α] [DecidableEq β]
+    (f : α → β) {b : β} (hb : b ∉ Finset.univ.image f) : finiteLaw f b = 0 := by
+  classical
+  have : IsEmpty {a : α // f a = b} := ⟨fun a => hb (Finset.mem_image.mpr
+    ⟨a.val, Finset.mem_univ _, a.property⟩)⟩
+  simp [finiteLaw]
+
+lemma event_le_totalVariation {β : Type*} [Fintype β] [DecidableEq β]
+    (p q : β → ℝ) (hpq : ∑ b, (p b - q b) = 0) (s : Finset β) :
+    (∑ b ∈ s, p b) - (∑ b ∈ s, q b) ≤ (1 / 2 : ℝ) * ∑ b, |p b - q b| := by
+  have h₁ : (∑ b ∈ s, (p b - q b)) ≤ ∑ b ∈ s, |p b - q b| :=
+    Finset.sum_le_sum (fun b _ => le_abs_self (p b - q b))
+  have h₂ : -(∑ b ∈ sᶜ, (p b - q b)) ≤ ∑ b ∈ sᶜ, |p b - q b| := by
+    rw [← Finset.sum_neg_distrib]
+    exact Finset.sum_le_sum (fun b _ => neg_le_abs (p b - q b))
+  have hz := Finset.sum_add_sum_compl s (fun b => p b - q b)
+  have ha := Finset.sum_add_sum_compl s (fun b => |p b - q b|)
+  rw [hpq] at hz
+  rw [← Finset.sum_sub_distrib]
+  linarith
+
+lemma finiteLaw_support_bound {α β : Type*} [Fintype α] [Nonempty α]
+    [Fintype β] [Nonempty β] (f : α → β) :
+    1 - (Fintype.card α : ℝ) / (Fintype.card β : ℝ) ≤
+      (1 / 2 : ℝ) * ∑ b, |finiteLaw f b - 1 / (Fintype.card β : ℝ)| := by
+  classical
+  let s := Finset.univ.image f
+  have hs : ∑ b ∈ s, finiteLaw f b = 1 := by
+    have h := Finset.sum_subset (Finset.subset_univ s)
+      (fun b _ hb => finiteLaw_eq_zero_of_not_mem f hb)
+    simpa only [finiteLaw_sum] using h
+  have hβ : (Fintype.card β : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
+  have hq : (∑ _b : β, 1 / (Fintype.card β : ℝ)) = 1 := by simp [hβ]
+  have hd : (∑ b, (finiteLaw f b - 1 / (Fintype.card β : ℝ))) = 0 := by
+    rw [Finset.sum_sub_distrib, finiteLaw_sum, hq, sub_self]
+  have he := event_le_totalVariation (finiteLaw f) (fun _ => 1 / (Fintype.card β : ℝ)) hd s
+  rw [hs] at he
+  have hsq : (∑ _b ∈ s, 1 / (Fintype.card β : ℝ)) = (s.card : ℝ) / Fintype.card β := by
+    simp [div_eq_mul_inv]
+  rw [hsq] at he
+  have hcard : (s.card : ℝ) ≤ Fintype.card α := by
+    exact_mod_cast (Finset.card_image_le (s := (Finset.univ : Finset α)) (f := f))
+  have hdiv := div_le_div_of_nonneg_right hcard (Nat.cast_nonneg (Fintype.card β) : (0 : ℝ) ≤ _)
+  linarith
+
+lemma card_positions (d : ℕ) : Fintype.card (Card d) = 2 ^ d := by
+  simp [Card]
+
+lemma card_permutations (d : ℕ) : Fintype.card (Equiv.Perm (Card d)) = Nat.factorial (2 ^ d) := by
+  rw [Fintype.card_perm, card_positions]
+
+lemma card_coins_succ (d : ℕ) : Fintype.card (Coins (d + 1)) = 2 ^ (2 ^ d) := by
+  simp [Coins]
+
+lemma card_tapes_succ (d t : ℕ) : Fintype.card (Fin t → Coins (d + 1)) = 2 ^ (t * 2 ^ d) := by
+  rw [Fintype.card_fun, card_coins_succ, Fintype.card_fin, ← pow_mul, Nat.mul_comm]
+
+lemma law_eq_finiteLaw (d t : ℕ) (g : Equiv.Perm (Card d)) :
+    law d t g = finiteLaw (run d t) g := by
+  unfold law finiteLaw
+  apply congrArg (fun n : ℕ => (n : ℝ) / (Fintype.card (Fin t → Coins d) : ℝ))
+  exact congrArg (@Fintype.card _) (Subsingleton.elim _ _)
+
+lemma law_nonneg (d t : ℕ) (g : Equiv.Perm (Card d)) : 0 ≤ law d t g := by
+  rw [law_eq_finiteLaw]
+  exact finiteLaw_nonneg (run d t) g
+
+lemma law_sum (d t : ℕ) : ∑ g, law d t g = 1 := by
+  simp only [law_eq_finiteLaw]
+  exact finiteLaw_sum (run d t)
+
+lemma physical_support_bound_succ (d t : ℕ) :
+    1 - (2 : ℝ) ^ (t * 2 ^ d) / (Nat.factorial (2 ^ (d + 1)) : ℝ) ≤ distance (d + 1) t := by
+  have : Nonempty (Equiv.Perm (Card (d + 1))) := ⟨1⟩
+  have h := finiteLaw_support_bound (run (d + 1) t)
+  rw [card_tapes_succ, card_permutations] at h
+  simpa only [Nat.cast_pow, Nat.cast_ofNat, distance, law_eq_finiteLaw] using h
+
+lemma step_succ_formula (d : ℕ) (ξ : Coins (d + 1)) (x : Card (d + 1)) :
+    step (d + 1) ξ x = Fin.snoc (Fin.tail x) (Bool.xor (x 0) (ξ (Fin.tail x))) := by
+  change (fun i => switchFun ξ x (finRotate (d + 1) i)) = _
+  rw [Fin.snoc_eq_cons_rotate]
+  rfl
+
+lemma distance_nonneg (d t : ℕ) : 0 ≤ distance d t := by
+  unfold distance
+  exact mul_nonneg (by norm_num) (Finset.sum_nonneg (fun _ _ => abs_nonneg _))
+
+lemma physical_support_bound (d t : ℕ) :
+    1 - (2 : ℝ) ^ ((t : ℝ) * (2 : ℝ) ^ d / 2) /
+      (Nat.factorial (2 ^ d) : ℝ) ≤ distance d t := by
+  cases d with
+  | zero =>
+    have h : 1 ≤ (2 : ℝ) ^ ((t : ℝ) / 2) := Real.one_le_rpow (by norm_num) (by positivity)
+    simpa only [pow_zero, Nat.factorial_one, Nat.cast_one, div_one, mul_one] using
+      le_trans (sub_nonpos.mpr h) (distance_nonneg 0 t)
+  | succ d =>
+    have he : (t : ℝ) * (2 : ℝ) ^ (d + 1) / 2 = (t * 2 ^ d : ℕ) := by
+      push_cast
+      rw [pow_succ]
+      ring
+    rw [he, Real.rpow_natCast]
+    exact physical_support_bound_succ d t
+
+lemma supportTimeBound_le_of_distance_le (d t : ℕ) (ht : distance d t ≤ (1 / 4 : ℝ)) :
+    supportTimeBound d ≤ (t : ℤ) := by
+  have hF : (0 : ℝ) < Nat.factorial (2 ^ d) := by positivity
+  have hN : (0 : ℝ) < (2 : ℝ) ^ d := by positivity
+  have hL : (0 : ℝ) < Real.log 2 := Real.log_pos (by norm_num)
+  have hb := physical_support_bound d t
+  have hp : (3 / 4 : ℝ) ≤
+      (2 : ℝ) ^ ((t : ℝ) * (2 : ℝ) ^ d / 2) / (Nat.factorial (2 ^ d) : ℝ) := by
+    linarith
+  have hr := (le_div_iff₀ hF).mp hp
+  have hlog := Real.log_le_log (show (0 : ℝ) < (3 / 4 : ℝ) * Nat.factorial (2 ^ d) by positivity) hr
+  rw [Real.log_rpow (by norm_num : (0 : ℝ) < 2)] at hlog
+  unfold supportTimeBound
+  apply Int.ceil_le.mpr
+  simp only [Int.cast_natCast]
+  calc
+    2 / (2 : ℝ) ^ d * (Real.log ((3 / 4 : ℝ) * Nat.factorial (2 ^ d)) / Real.log 2) =
+        Real.log ((3 / 4 : ℝ) * Nat.factorial (2 ^ d)) / ((2 : ℝ) ^ d / 2 * Real.log 2) := by
+      field_simp
+    _ ≤ (t : ℝ) := (div_le_iff₀ (mul_pos (div_pos hN (by norm_num)) hL)).mpr (by
+      nlinarith [hlog])
+
+lemma supportTimeBound_le_mixingTime {d : ℕ}
+    (h : ({t : ℕ | distance d t ≤ (1 / 4 : ℝ)} : Set ℕ).Nonempty) :
+    supportTimeBound d ≤ (mixingTime d : ℤ) := by
+  exact supportTimeBound_le_of_distance_le d (mixingTime d) (Nat.sInf_mem h)
+
+lemma log_factorial_lower {n : ℕ} (hn : 1 ≤ n) :
+    (n : ℝ) * Real.log n - n ≤ Real.log (Nat.factorial n) := by
+  have h := Stirling.le_log_factorial_stirling (show n ≠ 0 by omega)
+  have hln : 0 ≤ Real.log (n : ℝ) := Real.log_nonneg (by exact_mod_cast hn)
+  have hlp : 0 ≤ Real.log (2 * Real.pi) := Real.log_nonneg (by
+    have := Real.pi_gt_three
+    linarith)
+  linarith
+
+lemma log_three_quarters_lower : -(1 : ℝ) ≤ Real.log (3 / 4 : ℝ) := by
+  have h := Real.log_le_sub_one_of_pos (show (0 : ℝ) < 4 / 3 by norm_num)
+  have he : Real.log (4 / 3 : ℝ) = - Real.log (3 / 4 : ℝ) := by
+    rw [← Real.log_inv]
+    congr 1
+    norm_num
+  rw [he] at h
+  linarith
+
+lemma linear_le_supportTimeBound (d : ℕ) :
+    2 * (d : ℝ) - 4 / Real.log 2 ≤ (supportTimeBound d : ℝ) := by
+  have hN : 1 ≤ (2 : ℝ) ^ d := one_le_pow₀ (by norm_num)
+  have hNp : 0 < (2 : ℝ) ^ d := by positivity
+  have hL : 0 < Real.log 2 := Real.log_pos (by norm_num)
+  have hf := log_factorial_lower (show 1 ≤ 2 ^ d by exact one_le_pow₀ (by norm_num))
+  have hlog : (2 : ℝ) ^ d * ((d : ℝ) * Real.log 2) - (2 : ℝ) ^ d - 1 ≤
+      Real.log ((3 / 4 : ℝ) * (Nat.factorial (2 ^ d) : ℝ)) := by
+    rw [Real.log_mul (by norm_num) (by positivity)]
+    push_cast at hf
+    rw [Real.log_pow] at hf
+    linarith [log_three_quarters_lower]
+  apply le_trans _ (Int.le_ceil _)
+  have hmul : (2 * (d : ℝ) - 4 / Real.log 2) *
+      ((2 : ℝ) ^ d / 2 * Real.log 2) ≤
+      Real.log ((3 / 4 : ℝ) * (Nat.factorial (2 ^ d) : ℝ)) := by
+    have hcancel : (2 * (d : ℝ) - 4 / Real.log 2) *
+        ((2 : ℝ) ^ d / 2 * Real.log 2) =
+        (2 : ℝ) ^ d * ((d : ℝ) * Real.log 2) - 2 * (2 : ℝ) ^ d := by
+      field_simp
+      ring
+    rw [hcancel]
+    linarith
+  convert (le_div_iff₀ (mul_pos (div_pos hNp (by norm_num)) hL)).mpr hmul using 1
+  field_simp
+
+lemma linear_le_time_of_distance_le (d t : ℕ)
+    (ht : distance d t ≤ (1 / 4 : ℝ)) :
+    2 * (d : ℝ) - 4 / Real.log 2 ≤ (t : ℝ) := by
+  have h := supportTimeBound_le_of_distance_le d t ht
+  have hr : (supportTimeBound d : ℝ) ≤ (t : ℝ) := by exact_mod_cast h
+  exact (linear_le_supportTimeBound d).trans hr
+
+lemma entropy_chord (p : ℝ) (hp : 0 ≤ p) (hq : p ≤ 1 / 2) :
+    2 * p * Real.log 2 ≤ Real.binEntropy p := by
+  have hc := Real.strictConcave_binEntropy.concaveOn.2
+    (show (0 : ℝ) ∈ Set.Icc (0 : ℝ) 1 by norm_num)
+    (show (2 : ℝ)⁻¹ ∈ Set.Icc (0 : ℝ) 1 by norm_num)
+    (show 0 ≤ 1 - 2 * p by linarith)
+    (show 0 ≤ 2 * p by positivity)
+    (show (1 - 2 * p) + 2 * p = 1 by ring)
+  simpa only [Real.binEntropy_zero, Real.binEntropy_two_inv,
+    smul_eq_mul, mul_zero, zero_add, show 2 * p * (2 : ℝ)⁻¹ = p by ring] using hc
+
+lemma split_entropy_lower (a b : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) :
+    2 * min a b * Real.log 2 ≤
+      (a + b) * Real.log (a + b) - a * Real.log a - b * Real.log b := by
+  wlog hab : a ≤ b generalizing a b
+  · have hba := this b a hb ha (le_of_not_ge hab)
+    simpa only [min_comm, add_comm b a] using hba.trans_eq (show (b + a) * Real.log (b + a) - b * Real.log b - a * Real.log a =
+      (a + b) * Real.log (a + b) - a * Real.log a - b * Real.log b by rw [add_comm b a]; ring)
+  rw [min_eq_left hab]
+  by_cases hz : a = 0
+  · subst a
+    simp
+  have hap : 0 < a := lt_of_le_of_ne ha (Ne.symm hz)
+  have hbp : 0 < b := hap.trans_le hab
+  have hs : 0 < a + b := add_pos hap hbp
+  have hp : 0 ≤ a / (a + b) := div_nonneg ha hs.le
+  have hq : a / (a + b) ≤ 1 / 2 := (div_le_iff₀ hs).mpr (by linarith)
+  have hc := mul_le_mul_of_nonneg_left (entropy_chord (a / (a + b)) hp hq) hs.le
+  have he : (a + b) * Real.binEntropy (a / (a + b)) =
+      (a + b) * Real.log (a + b) - a * Real.log a - b * Real.log b := by
+    unfold Real.binEntropy
+    rw [show 1 - a / (a + b) = b / (a + b) by field_simp; ring]
+    simp only [Real.log_inv]
+    rw [Real.log_div (ne_of_gt hap) (ne_of_gt hs),
+      Real.log_div (ne_of_gt hbp) (ne_of_gt hs)]
+    field_simp
+    ring
+  rw [he] at hc
+  calc
+    2 * a * Real.log 2 = (a + b) * (2 * (a / (a + b)) * Real.log 2) := by
+      field_simp
+    _ ≤ _ := hc
+
+noncomputable def childMarks {d : ℕ} (s : Finset (Card (d + 1))) (ε : Bool) :
+    Finset (Card d) := (s.filter (fun x => x 0 = ε)).image Fin.tail
+
+lemma card_childMarks {d : ℕ} (s : Finset (Card (d + 1))) (ε : Bool) :
+    (childMarks s ε).card = (s.filter (fun x => x 0 = ε)).card := by
+  classical
+  unfold childMarks
+  apply Finset.card_image_iff.mpr
+  intro x hx y hy hxy
+  have hx0 := (Finset.mem_filter.mp hx).2
+  have hy0 := (Finset.mem_filter.mp hy).2
+  rw [← Fin.cons_self_tail x, ← Fin.cons_self_tail y, hx0, hy0, hxy]
+
+lemma card_childMarks_add {d : ℕ} (s : Finset (Card (d + 1))) :
+    (childMarks s false).card + (childMarks s true).card = s.card := by
+  classical
+  rw [card_childMarks, card_childMarks]
+  simpa only [Bool.not_eq_false] using
+    (Finset.card_filter_add_card_filter_not (s := s) (fun x => x 0 = false))
+
+noncomputable def topMeetings {d : ℕ} (b : Butterfly (d + 1))
+    (s : Finset (Card (d + 1))) : Finset (Card d) :=
+  ((childMarks s false).image (butterflyPerm d (b.2 false))) ∩
+    ((childMarks s true).image (butterflyPerm d (b.2 true)))
+
+lemma card_topMeetings_le {d : ℕ} (b : Butterfly (d + 1)) (s : Finset (Card (d + 1))) :
+    (topMeetings b s).card ≤ min (childMarks s false).card (childMarks s true).card := by
+  classical
+  apply le_min
+  · exact (Finset.card_le_card Finset.inter_subset_left).trans_eq
+      (Finset.card_image_of_injective _ (butterflyPerm d (b.2 false)).injective)
+  · exact (Finset.card_le_card Finset.inter_subset_right).trans_eq
+      (Finset.card_image_of_injective _ (butterflyPerm d (b.2 true)).injective)
+
+noncomputable def encounters : (d : ℕ) → Butterfly d → Finset (Card d) → ℕ
+  | 0, _, _ => 0
+  | d + 1, b, s => encounters d (b.2 false) (childMarks s false) +
+      encounters d (b.2 true) (childMarks s true) + (topMeetings b s).card
+
+lemma butterfly_encounters (d : ℕ) (b : Butterfly d) (s : Finset (Card d)) :
+    (encounters d b s : ℝ) ≤ (s.card : ℝ) * Real.log s.card / (2 * Real.log 2) := by
+  induction d with
+  | zero =>
+    simp only [encounters, Nat.cast_zero]
+    have hs : s.card ≤ 1 := by
+      simpa [Card] using Finset.card_le_univ s
+    interval_cases s.card <;> simp
+  | succ d ih =>
+    have h₀ := ih (b.2 false) (childMarks s false)
+    have h₁ := ih (b.2 true) (childMarks s true)
+    have ht : ((topMeetings b s).card : ℝ) ≤
+        min ((childMarks s false).card : ℝ) ((childMarks s true).card : ℝ) := by
+      exact_mod_cast card_topMeetings_le b s
+    have he := split_entropy_lower ((childMarks s false).card : ℝ)
+      ((childMarks s true).card : ℝ) (by positivity) (by positivity)
+    have hs : ((childMarks s false).card : ℝ) + (childMarks s true).card = s.card := by
+      exact_mod_cast card_childMarks_add s
+    rw [hs] at he
+    have hL : 0 < 2 * Real.log 2 := by positivity
+    rw [le_div_iff₀ hL] at h₀ h₁ ⊢
+    simp only [encounters, Nat.cast_add]
+    nlinarith [mul_le_mul_of_nonneg_right ht hL.le]
+
+lemma prod_split_at {α : Type*} [DecidableEq α] (s : Finset α) (f : α → ℝ) (a : α) :
+    ∏ i ∈ s, f i = (if a ∈ s then f a else 1) * ∏ i ∈ s.erase a, f i := by
+  by_cases ha : a ∈ s
+  · rw [ite_eq_left ha]
+    exact (Finset.mul_prod_erase s f ha).symm
+  · simp [ha]
+
+lemma prod_split_two {α : Type*} [DecidableEq α] (s : Finset α) (f : α → ℝ)
+    (a b : α) (hab : a ≠ b) :
+    ∏ i ∈ s, f i = (if a ∈ s then f a else 1) * (if b ∈ s then f b else 1) *
+      ∏ i ∈ (s.erase a).erase b, f i := by
+  calc
+    ∏ i ∈ s, f i = (if a ∈ s then f a else 1) * ∏ i ∈ s.erase a, f i :=
+      prod_split_at s f a
+    _ = _ := by
+      rw [prod_split_at (s.erase a) f b]
+      have hm : (b ∈ s.erase a) ↔ b ∈ s :=
+        ⟨fun h => (Finset.mem_erase.mp h).2, fun h => Finset.mem_erase.mpr ⟨Ne.symm hab, h⟩⟩
+      simp only [hm, mul_assoc]
+
+lemma swap_average_product {α : Type*} [DecidableEq α] (s : Finset α) (p : α → ℝ)
+    (hp : ∀ i, 0 ≤ p i) (a b : α) :
+    ((∏ i ∈ s, p i) + ∏ i ∈ s, p (Equiv.swap a b i)) / 2 ≤
+      ∏ i ∈ s, (p i + p (Equiv.swap a b i)) / 2 := by
+  by_cases hab : a = b
+  · subst b
+    simp only [Equiv.swap_self, Equiv.refl_apply]
+    have h : (fun i => (p i + p i) / 2) = p := by funext i; ring
+    rw [h]
+    linarith
+  let t := (s.erase a).erase b
+  have hfix : ∀ i ∈ t, Equiv.swap a b i = i := by
+    intro i hi
+    exact Equiv.swap_apply_of_ne_of_ne (Finset.mem_erase.mp (Finset.mem_erase.mp hi).2).1
+      (Finset.mem_erase.mp hi).1
+  have hprod : (∏ i ∈ t, p (Equiv.swap a b i)) = ∏ i ∈ t, p i :=
+    Finset.prod_congr rfl (fun i hi => by rw [hfix i hi])
+  have havg : (∏ i ∈ t, (p i + p (Equiv.swap a b i)) / 2) = ∏ i ∈ t, p i :=
+    Finset.prod_congr rfl (fun i hi => by rw [hfix i hi]; ring)
+  rw [prod_split_two s p a b hab,
+    prod_split_two s (fun i => p (Equiv.swap a b i)) a b hab,
+    prod_split_two s (fun i => (p i + p (Equiv.swap a b i)) / 2) a b hab]
+  change ((if a ∈ s then p a else 1) * (if b ∈ s then p b else 1) * (∏ i ∈ t, p i) +
+    (if a ∈ s then p (Equiv.swap a b a) else 1) *
+      (if b ∈ s then p (Equiv.swap a b b) else 1) * (∏ i ∈ t, p (Equiv.swap a b i))) / 2 ≤ _
+  rw [hprod]
+  change _ ≤ (if a ∈ s then (p a + p (Equiv.swap a b a)) / 2 else 1) *
+    (if b ∈ s then (p b + p (Equiv.swap a b b)) / 2 else 1) *
+      (∏ i ∈ t, (p i + p (Equiv.swap a b i)) / 2)
+  rw [havg, Equiv.swap_apply_left, Equiv.swap_apply_right]
+  have ht : 0 ≤ ∏ i ∈ t, p i := Finset.prod_nonneg (fun i _ => hp i)
+  by_cases ha : a ∈ s <;> by_cases hb : b ∈ s <;> simp only [ha, hb, ↓reduceIte]
+  · have hsq := mul_nonneg (sq_nonneg (p a - p b)) ht
+    nlinarith
+  all_goals ring_nf; rfl
+
+def CylinderBound {α : Type*} (m : Finset α → ℝ) (p : α → ℝ) : Prop :=
+  ∀ s, m s ≤ ∏ i ∈ s, p i
+
+lemma CylinderBound.fair_swap {α : Type*} [DecidableEq α]
+    {m : Finset α → ℝ} {p : α → ℝ} (h : CylinderBound m p)
+    (hp : ∀ i, 0 ≤ p i) (a b : α) :
+    CylinderBound (fun s => (m s + m (s.image (Equiv.swap a b))) / 2)
+      (fun i => (p i + p (Equiv.swap a b i)) / 2) := by
+  intro s
+  have h₀ := h s
+  have h₁ := h (s.image (Equiv.swap a b))
+  rw [Finset.prod_image (Equiv.swap a b).injective.injOn] at h₁
+  exact (show (m s + m (s.image (Equiv.swap a b))) / 2 ≤
+      ((∏ i ∈ s, p i) + ∏ i ∈ s, p (Equiv.swap a b i)) / 2 by linarith).trans
+    (swap_average_product s p hp a b)
+
+noncomputable def inclusionMoment {α Ω : Type*} [Fintype Ω]
+    (F : Ω → Finset α) (s : Finset α) : ℝ := by
+  classical
+  exact finiteMean (fun ω => if s ⊆ F ω then 1 else 0)
+
+lemma inclusionMoment_nonneg {α Ω : Type*} [Fintype Ω]
+    (F : Ω → Finset α) (s : Finset α) : 0 ≤ inclusionMoment F s := by
+  classical
+  unfold inclusionMoment finiteMean
+  positivity
+
+lemma indicator_subset_prod {α : Type*} [DecidableEq α] (A s : Finset α) :
+    (if s ⊆ A then (1 : ℝ) else 0) = ∏ i ∈ s, (if i ∈ A then 1 else 0) := by
+  by_cases h : s ⊆ A
+  · rw [ite_eq_left h]
+    exact (Finset.prod_eq_one (fun i hi => ite_eq_left (h hi))).symm
+  · rw [ite_eq_right h]
+    obtain ⟨i, hi, hiA⟩ := Finset.not_subset.mp h
+    exact (Finset.prod_eq_zero hi (ite_eq_right hiA)).symm
+
+lemma inclusionMoment_const {α Ω : Type*} [DecidableEq α] [Fintype Ω] [Nonempty Ω]
+    (A s : Finset α) : inclusionMoment (fun _ : Ω => A) s =
+      if s ⊆ A then 1 else 0 := by
+  classical
+  have hn : (Fintype.card Ω : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
+  by_cases h : s ⊆ A <;> simp [inclusionMoment, finiteMean, h, hn]
+
+lemma cylinder_const {α Ω : Type*} [Fintype Ω] [Nonempty Ω] (A : Finset α) :
+    CylinderBound (inclusionMoment (fun _ : Ω => A))
+      (fun i => inclusionMoment (fun _ : Ω => A) {i}) := by
+  classical
+  intro s
+  simp only [inclusionMoment_const, Finset.singleton_subset_iff]
+  exact (indicator_subset_prod A s).le
+
+noncomputable def fairSetSwap {α Ω : Type*} [DecidableEq α]
+    (F : Ω → Finset α) (a b : α) : Ω × Bool → Finset α :=
+  fun ω => if ω.2 then (F ω.1).image (Equiv.swap a b) else F ω.1
+
+lemma subset_swap_image_iff {α : Type*} [DecidableEq α]
+    (A s : Finset α) (a b : α) :
+    s ⊆ A.image (Equiv.swap a b) ↔ s.image (Equiv.swap a b) ⊆ A := by
+  constructor
+  · intro h
+    have h' := Finset.image_subset_image h (f := Equiv.swap a b)
+    simpa only [Finset.image_image, Function.comp_def, Equiv.swap_apply_self, Finset.image_id'] using h'
+  · intro h
+    have h' := Finset.image_subset_image h (f := Equiv.swap a b)
+    simpa only [Finset.image_image, Function.comp_def, Equiv.swap_apply_self, Finset.image_id'] using h'
+
+lemma inclusionMoment_fairSetSwap {α Ω : Type*} [DecidableEq α] [Fintype Ω]
+    (F : Ω → Finset α) (a b : α) (s : Finset α) :
+    inclusionMoment (fairSetSwap F a b) s =
+      (inclusionMoment F s + inclusionMoment F (s.image (Equiv.swap a b))) / 2 := by
+  classical
+  simp only [inclusionMoment, finiteMean, Fintype.sum_prod_type, Fintype.sum_bool,
+    fairSetSwap, Bool.false_eq_true, ↓reduceIte, Fintype.card_prod, Fintype.card_bool,
+    Nat.cast_mul, Nat.cast_ofNat, subset_swap_image_iff, Finset.sum_add_distrib]
+  rw [div_mul_eq_div_div, add_div]
+  ring_nf
+
+lemma cylinder_fairSetSwap {α Ω : Type*} [DecidableEq α] [Fintype Ω]
+    (F : Ω → Finset α) (a b : α)
+    (h : CylinderBound (inclusionMoment F) (fun i => inclusionMoment F {i})) :
+    CylinderBound (inclusionMoment (fairSetSwap F a b))
+      (fun i => inclusionMoment (fairSetSwap F a b) {i}) := by
+  classical
+  have h' := h.fair_swap (fun i => inclusionMoment_nonneg F {i}) a b
+  intro s
+  simpa only [inclusionMoment_fairSetSwap, Finset.image_singleton] using h' s
+
+def SwitchSamples : ℕ → Type
+  | 0 => Unit
+  | n + 1 => SwitchSamples n × Bool
+
+instance (n : ℕ) : Fintype (SwitchSamples n) := by
+  induction n with
+  | zero => exact inferInstanceAs (Fintype Unit)
+  | succ n ih => exact inferInstanceAs (Fintype (SwitchSamples n × Bool))
+
+instance (n : ℕ) : Nonempty (SwitchSamples n) := by
+  induction n with
+  | zero => exact inferInstanceAs (Nonempty Unit)
+  | succ n ih => exact inferInstanceAs (Nonempty (SwitchSamples n × Bool))
+
+noncomputable def switchedSet {α : Type*} [DecidableEq α] (A : Finset α) :
+    (L : List (α × α)) → SwitchSamples L.length → Finset α
+  | [], _ => A
+  | (a,b) :: L, ω => fairSetSwap (switchedSet A L) a b ω
+
+lemma switchedSet_cylinder {α : Type*} [DecidableEq α] (A : Finset α)
+    (L : List (α × α)) :
+    CylinderBound (inclusionMoment (switchedSet A L))
+      (fun i => inclusionMoment (switchedSet A L) {i}) := by
+  induction L with
+  | nil => exact cylinder_const A
+  | cons e L ih => exact cylinder_fairSetSwap (switchedSet A L) e.1 e.2 ih
+
+inductive QueryTree (ι : Type*)
+  | reject : QueryTree ι
+  | accept : QueryTree ι
+  | branch (i : ι) (zero one : QueryTree ι) : QueryTree ι
+  | force (i : ι) (b : Bool) (next : QueryTree ι) : QueryTree ι
+
+namespace QueryTree
+
+variable {ι : Type*} [DecidableEq ι]
+
+def queries : QueryTree ι → Finset ι
+  | reject => ∅
+  | accept => ∅
+  | branch i zero one => insert i (queries zero ∪ queries one)
+  | force i _ next => insert i (queries next)
+
+def Fresh : QueryTree ι → Prop
+  | reject => True
+  | accept => True
+  | branch i zero one => i ∉ queries zero ∧ i ∉ queries one ∧ Fresh zero ∧ Fresh one
+  | force i _ next => i ∉ queries next ∧ Fresh next
+
+def weighted : QueryTree ι → (ι → Bool) → ℝ
+  | reject, _ => 0
+  | accept, _ => 1
+  | branch i zero one, ω => if ω i then weighted one ω else weighted zero ω
+  | force i b next, ω => if ω i = b then 2 * weighted next ω else 0
+
+omit [DecidableEq ι] in
+lemma weighted_nonneg (T : QueryTree ι) (ω : ι → Bool) : 0 ≤ T.weighted ω := by
+  induction T with
+  | reject => exact le_rfl
+  | accept => exact zero_le_one
+  | branch i zero one ih₀ ih₁ => simpa only [weighted] using ite_nonneg ih₁ ih₀
+  | force i b next ih => simpa only [weighted] using ite_nonneg (mul_nonneg (by norm_num) ih) le_rfl
+
+lemma weighted_congr (T : QueryTree ι) {ω η : ι → Bool}
+    (h : ∀ i ∈ T.queries, ω i = η i) : T.weighted ω = T.weighted η := by
+  induction T with
+  | reject => rfl
+  | accept => rfl
+  | branch i zero one ih₀ ih₁ =>
+    have hi := h i (Finset.mem_insert_self _ _)
+    simp only [weighted, hi]
+    rw [ih₀ (fun j hj => h j (Finset.mem_insert_of_mem (Finset.mem_union_left _ hj))),
+      ih₁ (fun j hj => h j (Finset.mem_insert_of_mem (Finset.mem_union_right _ hj)))]
+  | force i b next ih =>
+    have hi := h i (Finset.mem_insert_self _ _)
+    simp only [weighted, hi]
+    rw [ih (fun j hj => h j (Finset.mem_insert_of_mem hj))]
+
+def flipCoin (i : ι) : Equiv.Perm (ι → Bool) :=
+  (show Function.Involutive (fun ω : ι → Bool => Function.update ω i (!(ω i))) from by
+    intro ω
+    funext j
+    by_cases h : j = i
+    · subst j; simp
+    · simp [Function.update_of_ne h]).toPerm
+
+lemma flipCoin_apply_eq (i : ι) (ω : ι → Bool) : flipCoin i ω i = !(ω i) := by
+  change Function.update ω i (!(ω i)) i = _
+  exact Function.update_self _ _ _
+
+lemma flipCoin_apply_ne (i j : ι) (h : j ≠ i) (ω : ι → Bool) :
+    flipCoin i ω j = ω j := by
+  change Function.update ω i (!(ω i)) j = _
+  exact Function.update_of_ne h _ _
+
+lemma weighted_flip_of_not_mem (T : QueryTree ι) (i : ι) (hi : i ∉ T.queries)
+    (ω : ι → Bool) : T.weighted (flipCoin i ω) = T.weighted ω := by
+  apply weighted_congr
+  intro j hj
+  exact flipCoin_apply_ne i j (fun h => hi (h ▸ hj)) ω
+
+lemma mean_indicator_bit [Fintype ι] (f : (ι → Bool) → ℝ) (i : ι) (b : Bool)
+    (hf : ∀ ω, f (flipCoin i ω) = f ω) :
+    finiteMean (fun ω => if ω i = b then f ω else 0) = finiteMean f / 2 := by
+  let g := fun ω : ι → Bool => if ω i = b then f ω else 0
+  have hp (ω : ι → Bool) : g ω + g (flipCoin i ω) = f ω := by
+    dsimp [g]
+    rw [flipCoin_apply_eq, hf]
+    cases hω : ω i <;> cases b <;> simp
+  have hs := Equiv.sum_comp (flipCoin i) g
+  have hadd : (∑ ω, g ω) + (∑ ω, g (flipCoin i ω)) = ∑ ω, f ω := by
+    rw [← Finset.sum_add_distrib]
+    exact Finset.sum_congr rfl (fun ω _ => hp ω)
+  unfold finiteMean
+  change (∑ ω, g ω) / _ = _
+  rw [hs] at hadd
+  have hh : (∑ ω, g ω) = (∑ ω, f ω) / 2 := by linarith
+  rw [hh]
+  ring
+
+lemma mean_weighted_le_one [Fintype ι] (T : QueryTree ι) (hT : T.Fresh) :
+    finiteMean T.weighted ≤ 1 := by
+  induction T with
+  | reject => simp [weighted, finiteMean]
+  | accept => simp [weighted, finiteMean]
+  | branch i zero one ih₀ ih₁ =>
+    obtain ⟨hz, ho, hfz, hfo⟩ := hT
+    have h₀ := mean_indicator_bit zero.weighted i false (zero.weighted_flip_of_not_mem i hz)
+    have h₁ := mean_indicator_bit one.weighted i true (one.weighted_flip_of_not_mem i ho)
+    have he : finiteMean (branch i zero one).weighted =
+        finiteMean (fun ω => if ω i = false then zero.weighted ω else 0) +
+        finiteMean (fun ω => if ω i = true then one.weighted ω else 0) := by
+      unfold finiteMean
+      rw [← add_div, ← Finset.sum_add_distrib]
+      congr 1
+      apply Finset.sum_congr rfl
+      intro ω _
+      cases hω : ω i <;> simp [weighted, hω]
+    rw [he, h₀, h₁]
+    linarith [ih₀ hfz, ih₁ hfo]
+  | force i b next ih =>
+    obtain ⟨hi, hn⟩ := hT
+    have hm := mean_indicator_bit next.weighted i b (next.weighted_flip_of_not_mem i hi)
+    have he : finiteMean (force i b next).weighted =
+        2 * finiteMean (fun ω => if ω i = b then next.weighted ω else 0) := by
+      unfold finiteMean
+      rw [← mul_div_assoc, Finset.mul_sum]
+      congr 1
+      apply Finset.sum_congr rfl
+      intro ω _
+      by_cases h : ω i = b <;> simp [weighted, h]
+    rw [he, hm]
+    linarith [ih hn]
+
+def succeeds : QueryTree ι → (ι → Bool) → Bool
+  | reject, _ => false
+  | accept, _ => true
+  | branch i zero one, ω => if ω i then succeeds one ω else succeeds zero ω
+  | force i b next, ω => if ω i = b then succeeds next ω else false
+
+def charge : QueryTree ι → (ι → Bool) → ℕ
+  | reject, _ => 0
+  | accept, _ => 0
+  | branch i zero one, ω => if ω i then charge one ω else charge zero ω
+  | force i b next, ω => if ω i = b then charge next ω + 1 else 0
+
+omit [DecidableEq ι] in
+lemma weighted_eq (T : QueryTree ι) (ω : ι → Bool) :
+    T.weighted ω = if T.succeeds ω then (2 : ℝ) ^ T.charge ω else 0 := by
+  induction T with
+  | reject => rfl
+  | accept => norm_num [weighted, succeeds, charge]
+  | branch i zero one ih₀ ih₁ =>
+    cases h : ω i <;> simp [weighted, succeeds, charge, h, ih₀, ih₁]
+  | force i b next ih =>
+    by_cases h : ω i = b
+    · simp only [weighted, succeeds, charge, h, ↓reduceIte, ih]
+      cases hs : next.succeeds ω <;> simp [pow_succ, mul_comm]
+    · simp [weighted, succeeds, h]
+
+lemma success_probability_le [Fintype ι] (T : QueryTree ι) (hT : T.Fresh)
+    (K : ℕ) (hK : ∀ ω, T.succeeds ω = true → K ≤ T.charge ω) :
+    finiteMean (fun ω => if T.succeeds ω then (1 : ℝ) else 0) ≤ 1 / (2 : ℝ) ^ K := by
+  have hw (ω : ι → Bool) :
+      (2 : ℝ) ^ K * (if T.succeeds ω then (1 : ℝ) else 0) ≤ T.weighted ω := by
+    rw [weighted_eq]
+    cases hs : T.succeeds ω
+    · simp
+    · simpa using pow_le_pow_right₀ (show (1 : ℝ) ≤ 2 by norm_num) (hK ω hs)
+  have hn : 0 ≤ (Fintype.card (ι → Bool) : ℝ) := Nat.cast_nonneg _
+  have hm := div_le_div_of_nonneg_right
+    (Finset.sum_le_sum (fun ω (_ : ω ∈ Finset.univ) => hw ω)) hn
+  rw [← Finset.mul_sum, mul_div_assoc] at hm
+  change (2 : ℝ) ^ K * finiteMean (fun ω => if T.succeeds ω then 1 else 0) ≤
+    finiteMean T.weighted at hm
+  exact (le_div_iff₀ (pow_pos (by norm_num) K)).mpr (by
+    rw [mul_comm]
+    exact hm.trans (mean_weighted_le_one T hT))
+
+end QueryTree
+
+def encodeButterfly : (d : ℕ) → Butterfly d → SwitchIndex d → Bool
+  | 0, _, i => nomatch i
+  | _d + 1, b, Sum.inl y => b.1 y
+  | d + 1, b, Sum.inr (ε,i) => encodeButterfly d (b.2 ε) i
+
+lemma decode_encodeButterfly (d : ℕ) (b : Butterfly d) :
+    decodeButterfly d (encodeButterfly d b) = b := by
+  induction d with
+  | zero => exact Unit.ext _ _
+  | succ d ih =>
+    apply Prod.ext
+    · rfl
+    · funext ε
+      exact ih (b.2 ε)
+
+lemma encode_decodeButterfly (d : ℕ) (ω : SwitchIndex d → Bool) :
+    encodeButterfly d (decodeButterfly d ω) = ω := by
+  induction d with
+  | zero => funext i; exact Empty.elim i
+  | succ d ih =>
+    funext i
+    cases i with
+    | inl y => rfl
+    | inr z => exact congrFun (ih (fun i => ω (Sum.inr (z.1,i)))) z.2
+
+def butterflyCoinEquiv (d : ℕ) : (SwitchIndex d → Bool) ≃ Butterfly d where
+  toFun := decodeButterfly d
+  invFun := encodeButterfly d
+  left_inv := encode_decodeButterfly d
+  right_inv := decode_encodeButterfly d
+
+noncomputable def routeDomain : (d : ℕ) → Card d → Card d → Finset (SwitchIndex d)
+  | 0, _, _ => ∅
+  | d + 1, x, y => by
+      classical
+      exact insert (Sum.inl (Fin.tail y))
+        ((routeDomain d (Fin.tail x) (Fin.tail y)).image (fun i => Sum.inr (x 0,i)))
+
+def routeValue : (d : ℕ) → Card d → Card d → SwitchIndex d → Bool
+  | 0, _, _, i => nomatch i
+  | _d + 1, x, y, Sum.inl _ => Bool.xor (x 0) (y 0)
+  | d + 1, x, y, Sum.inr (_,i) => routeValue d (Fin.tail x) (Fin.tail y) i
+
+end Thorp
+
+end ThorpNine.Harmonic
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Harmonic/CutoffMass.lean b/lean/OAI/Probability/ThorpRouting/Harmonic/CutoffMass.lean
index 04ac1081b7e7b5e598082f585652a117c7a5cdaf..3aa2b96f09bcdf8898f2cc4cf82716c7fc872c00 100644
--- a/lean/OAI/Probability/ThorpRouting/Harmonic/CutoffMass.lean
+++ b/lean/OAI/Probability/ThorpRouting/Harmonic/CutoffMass.lean
@@ -1,862 +1,862 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Harmonic.SandwichShuffle
-
-namespace OAI
-
-namespace ThorpNine.Harmonic
-
-namespace Thorp
-
-noncomputable def cutoffMass (H : ℕ) : (d : ℕ) → {ι : Type*} → [Fintype ι] →
-    (ι ↪ Card d) → BenesCoins d → ℕ
-  | 0, _, _, _, _ => 0
-  | d+1, ι, _, e, ω =>
-      if d+1 ≤ H then crowdedMass (Fintype.card ι) else
-      let σ := benesStepEquiv d ω
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x σ.2.1
-      let hc := PairRouting.colors_compatible x σ.2.1
-      cutoffMass H d (PairRouting.childEmbedding x c hc false) σ.1.1 +
-        cutoffMass H d (PairRouting.childEmbedding x c hc true) σ.1.2
-
-lemma palindromeLowCost_eq (H d : ℕ) (h : d ≤ H) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) :
-    palindromeLowCost H d e ω = palindromeCost d e ω := by
-  classical
-  induction d generalizing ι with
-  | zero => rfl
-  | succ d ih =>
-    rw [palindromeLowCost,palindromeCost,ite_eq_left h]
-    congr 2 <;> exact ih (by omega) _ _
-
-lemma palindromeLowCost_le_cutoff (H d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) :
-    palindromeLowCost H d e ω ≤ H*cutoffMass H d e ω := by
-  classical
-  induction d generalizing ι with
-  | zero => simp [palindromeLowCost,cutoffMass]
-  | succ d ih =>
-    by_cases h : d+1 ≤ H
-    · rw [palindromeLowCost_eq H (d+1) h,cutoffMass,ite_eq_left h]
-      exact (palindromeCost_le_crowded (d+1) e ω).trans (Nat.mul_le_mul_right _ h)
-    · rw [palindromeLowCost,cutoffMass,ite_eq_right h,ite_eq_right h,zero_add,Nat.mul_add]
-      exact Nat.add_le_add (ih _ _) (ih _ _)
-
-
-noncomputable def centralBlock (r H : ℕ) (b : Card r) : Finset (Card (H+r)) :=
-  Finset.univ.image (fun z : Card H => (cardSplit r H).symm (b,z))
-
-lemma mem_centralBlock (r H : ℕ) (b : Card r) (x : Card (H+r)) :
-    x ∈ centralBlock r H b ↔ (cardSplit r H x).1 = b := by
-  classical
-  constructor
-  · intro h
-    obtain ⟨z,_,rfl⟩ := Finset.mem_image.mp h
-    simp only [Equiv.apply_symm_apply]
-  · intro h
-    apply Finset.mem_image.mpr
-    refine ⟨(cardSplit r H x).2,Finset.mem_univ _,?_⟩
-    rw [←h,Prod.mk.eta,Equiv.symm_apply_apply]
-
-lemma centralBlock_card (r H : ℕ) (b : Card r) : (centralBlock r H b).card = 2^H := by
-  classical
-  rw [centralBlock,Finset.card_image_of_injective]
-  · exact card_positions H
-  · intro x y h
-    exact congrArg Prod.snd ((cardSplit r H).symm.injective h)
-
-lemma centralBlock_disjoint (r H : ℕ) :
-    Pairwise (fun b b' => Disjoint (centralBlock r H b) (centralBlock r H b')) := by
-  classical
-  intro b b' h
-  apply Finset.disjoint_left.mpr
-  intro x hx hy
-  exact h (((mem_centralBlock r H b x).mp hx).symm.trans
-    ((mem_centralBlock r H b' x).mp hy))
-
-lemma centralBlock_zero (H : ℕ) (b : Card 0) : centralBlock 0 H b = Finset.univ := by
-  classical
-  ext x
-  simp only [mem_centralBlock,Finset.mem_univ,iff_true]
-  exact Subsingleton.elim _ _
-
-lemma centralBlock_step (r H : ℕ) (b : Bool) (c : Card r) :
-    centralBlock (r+1) H (Fin.cons b c) = (centralBlock r H c).image (Fin.cons b) := by
-  classical
-  simp only [centralBlock,Finset.image_image,cardSplit,Equiv.coe_fn_symm_mk,
-    Fin.cons_zero,Fin.tail_cons,Function.comp_def]
-
-lemma block_inter_step (r H : ℕ) (b : Bool) (c : Card r) (A : Finset (Card (H+(r+1)))) :
-    centralBlock (r+1) H (Fin.cons b c) ∩ A =
-      (centralBlock r H c ∩ childMarks A b).image (Fin.cons b) := by
-  classical
-  rw [centralBlock_step]
-  ext x
-  constructor
-  · intro hx
-    obtain ⟨hxB,hxA⟩ := Finset.mem_inter.mp hx
-    obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hxB
-    exact Finset.mem_image.mpr ⟨y,Finset.mem_inter.mpr ⟨hy,(mem_childMarks A b y).mpr hxA⟩,rfl⟩
-  · intro hx
-    obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
-    exact Finset.mem_inter.mpr ⟨Finset.mem_image.mpr ⟨y,(Finset.mem_inter.mp hy).1,rfl⟩,
-      (mem_childMarks A b y).mp (Finset.mem_inter.mp hy).2⟩
-
-lemma sparseMass_central_zero (H : ℕ) (A : Finset (Card H)) :
-    sparseMass (centralBlock 0 H) A = crowdedMass A.card := by
-  classical
-  simp only [sparseMass,centralBlock_zero,Finset.univ_inter,crowdedMass,
-    Finset.sum_const,Finset.card_univ,card_positions,pow_zero,one_nsmul]
-
-lemma sparseMass_central_step (r H : ℕ) (A : Finset (Card (H+(r+1)))) :
-    sparseMass (centralBlock (r+1) H) A =
-      sparseMass (centralBlock r H) (childMarks A false) +
-        sparseMass (centralBlock r H) (childMarks A true) := by
-  classical
-  unfold sparseMass
-  rw [←Equiv.sum_comp (headTailEquiv r).symm,Fintype.sum_prod_type]
-  have ht (b : Bool) (c : Card r) : (headTailEquiv r).symm (b,c) = Fin.cons b c := rfl
-  simp only [Fintype.sum_bool,ht]
-  have hc (b : Bool) (c : Card r) :
-      (centralBlock (r+1) H (Fin.cons b c) ∩ A).card =
-      (centralBlock r H c ∩ childMarks A b).card := by
-    rw [block_inter_step]
-    apply Finset.card_image_of_injective
-    intro x y h
-    simpa only [Fin.tail_cons] using congrArg Fin.tail h
-  simp only [hc]
-  exact add_comm _ _
-
-
-lemma inverseButterfly_coordinates_step (d : ℕ)
-    (c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (ξ : Card d → Bool)
-    (x : Card (d+1)) :
-    headTailEquiv d ((butterflyPerm (d+1)
-      (decodeButterfly (d+1) ((coinStepEquiv d).symm (c,ξ)))).symm x) =
-      (Bool.xor (x 0) (ξ (Fin.tail x)),
-        (butterflyPerm d (decodeButterfly d
-          (if Bool.xor (x 0) (ξ (Fin.tail x)) then c.2 else c.1))).symm (Fin.tail x)) := by
-  rw [butterflyPerm_step]
-  rfl
-
-lemma inverseButterflyMarks_child (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (d+1))
-    (c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (ξ : Card d → Bool) (b : Bool) :
-    childMarks (inverseButterflyMarks (d+1) (Finset.univ.image e)
-      ((coinStepEquiv d).symm (c,ξ))) b =
-      inverseButterflyMarks d
-        (Finset.univ.image (PairRouting.childEmbedding (e.trans (headTailEquiv d).toEmbedding)
-          (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) ξ)
-            (PairRouting.colors_compatible _ _) b)) (if b then c.2 else c.1) := by
-  classical
-  let x := e.trans (headTailEquiv d).toEmbedding
-  let col := PairRouting.colors x ξ
-  let child := PairRouting.childEmbedding x col (PairRouting.colors_compatible x ξ) b
-  have hp (i : ι) : headTailEquiv d ((butterflyPerm (d+1)
-      (decodeButterfly (d+1) ((coinStepEquiv d).symm (c,ξ)))).symm (e i)) =
-      (col i,(butterflyPerm d (decodeButterfly d (if col i then c.2 else c.1))).symm (x i).2) :=
-    inverseButterfly_coordinates_step d c ξ (e i)
-  ext y
-  simp only [mem_childMarks,inverseButterflyMarks,Finset.image_image,
-    Finset.mem_image,Finset.mem_univ,true_and,Function.comp_def]
-  constructor
-  · rintro ⟨i,hi⟩
-    have he := congrArg (headTailEquiv d) hi
-    rw [hp i] at he
-    change (col i,_) = (b,y) at he
-    have hb := congrArg Prod.fst he
-    have hy := congrArg Prod.snd he
-    dsimp only at hb hy
-    refine ⟨⟨i,hb⟩,?_⟩
-    change (butterflyPerm d (decodeButterfly d (if b then c.2 else c.1))).symm (x i).2 = y
-    simpa only [hb] using hy
-  · rintro ⟨i,hi⟩
-    refine ⟨i.val,?_⟩
-    apply (headTailEquiv d).injective
-    rw [hp]
-    change (col i.val,_) = (b,y)
-    have hb : col i.val = b := i.property
-    rw [hb]
-    exact Prod.ext rfl hi
-
-lemma inverseButterflyMarks_card (d : ℕ) (A : Finset (Card d)) (ω : SwitchIndex d → Bool) :
-    (inverseButterflyMarks d A ω).card = A.card := by
-  classical
-  exact Finset.card_image_of_injective _ (butterflyPerm d (decodeButterfly d ω)).symm.injective
-
-lemma benes_input_coordinates (d : ℕ) (ω : BenesCoins (d+1)) :
-    ω.2 = (coinStepEquiv d).symm
-      (((benesStepEquiv d ω).1.1.2,(benesStepEquiv d ω).1.2.2),(benesStepEquiv d ω).2.1) := by
-  change ω.2 = (coinStepEquiv d).symm (coinStepEquiv d ω.2)
-  exact ((coinStepEquiv d).symm_apply_apply ω.2).symm
-
-lemma cutoffMass_central (r H : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (H+r)) (ω : BenesCoins (H+r)) :
-    cutoffMass H (H+r) e ω =
-      sparseMass (centralBlock r H) (inverseButterflyMarks (H+r) (Finset.univ.image e) ω.2) := by
-  classical
-  induction r generalizing ι with
-  | zero =>
-    rw [sparseMass_central_zero,inverseButterflyMarks_card,
-      Finset.card_image_of_injective _ e.injective,Finset.card_univ]
-    cases H with
-    | zero =>
-      have hcard : Fintype.card ι ≤ 1 := by
-        have hh := Fintype.card_le_of_injective e e.injective
-        simpa only [Nat.add_zero,card_positions,pow_zero] using hh
-      simp [cutoffMass,crowdedMass,show ¬2≤Fintype.card ι by omega]
-    | succ H => simp only [cutoffMass,le_refl,ite_true]
-  | succ r ih =>
-    simp only [Nat.add_succ] at *
-    rw [cutoffMass,ite_eq_right (by omega : ¬H+r+1≤H)]
-    let σ := benesStepEquiv (H+r) ω
-    let x := e.trans (headTailEquiv (H+r)).toEmbedding
-    let col := PairRouting.colors x σ.2.1
-    let hc := PairRouting.colors_compatible x σ.2.1
-    change cutoffMass H (H+r) (PairRouting.childEmbedding x col hc false) σ.1.1 +
-      cutoffMass H (H+r) (PairRouting.childEmbedding x col hc true) σ.1.2 = _
-    rw [ih,ih,sparseMass_central_step,benes_input_coordinates (H+r) ω]
-    rw [inverseButterflyMarks_child,inverseButterflyMarks_child]
-    rfl
-
-
-theorem palindrome_low_mgf (r H : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (H+r)) (q : ℝ) (hq : 1 ≤ q)
-    (hr : (2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*q^H) ≤ 1/2) :
-    finiteMean (fun ω : BenesCoins (H+r) => q^(palindromeLowCost H (H+r) e ω)) ≤
-      Real.exp ((2:ℝ)^r * (2 * ((2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*q^H))^2)) := by
-  classical
-  have hec : (Finset.univ.image e).card = Fintype.card ι := by
-    rw [Finset.card_image_of_injective _ e.injective,Finset.card_univ]
-  have hm := inverseButterfly_sparse_mgf (H+r) (centralBlock r H)
-    (centralBlock_disjoint r H) (Finset.univ.image e) (q^H)
-    (pow_nonneg (by linarith) _) (fun b => by
-      simpa only [centralBlock_card,hec,Nat.cast_pow,Nat.cast_ofNat] using hr)
-  calc
-    _ ≤ finiteMean (fun ω : BenesCoins (H+r) => (q^H)^(cutoffMass H (H+r) e ω)) := by
-      apply finiteMean_mono
-      intro ω
-      rw [←pow_mul]
-      exact pow_le_pow_right₀ hq (palindromeLowCost_le_cutoff H (H+r) e ω)
-    _ = finiteMean (fun ω : SwitchIndex (H+r) → Bool =>
-        (q^H)^(sparseMass (centralBlock r H)
-          (inverseButterflyMarks (H+r) (Finset.univ.image e) ω))) := by
-      simp_rw [cutoffMass_central]
-      simp only [finiteMean_prod,finiteMean_const]
-    _ ≤ _ := by
-      simpa only [centralBlock_card,hec,Nat.cast_pow,Nat.cast_ofNat,Finset.sum_const,
-        Finset.card_univ,card_positions,nsmul_eq_mul] using hm
-
-variable {α B : Type*} {r s : ℕ}
-
-def parallelFreshPre (pre : α ≃ B × Card (s+r))
-    (lo : B → Card r × SwitchIndex s → Bool) : α ≃ (B × Card s) × Card r :=
-  ((pre.trans (Equiv.prodCongr (Equiv.refl B) (cardSplit r s))).trans
-    (Equiv.prodCongrRight (fun b =>
-      (parallelPerm (lo b)).trans (Equiv.prodComm _ _)))).trans
-    (Equiv.prodAssoc _ _ _).symm
-
-def parallelFreshPost (post : B × Card (s+r) ≃ α) : (B × Card s) × Card r ≃ α :=
-  (((Equiv.prodAssoc _ _ _).trans (Equiv.prodCongr (Equiv.refl B)
-    (Equiv.prodComm _ _))).trans (Equiv.prodCongr (Equiv.refl B) (cardSplit r s).symm)).trans post
-
-def parallelAssembled (lo : B → Card r × SwitchIndex s → Bool)
-    (hi : (B × Card s) × SwitchIndex r → Bool) : B × SwitchIndex (s+r) → Bool :=
-  fun i => assembleBits r s (lo i.1) (fun j => hi ((i.1,j.1),j.2)) i.2
-
-lemma parallel_exposed_perm (pre : α ≃ B × Card (s+r)) (post : B × Card (s+r) ≃ α)
-    (lo : B → Card r × SwitchIndex s → Bool)
-    (hi : (B × Card s) × SwitchIndex r → Bool) :
-    relabeledPerm pre post (parallelAssembled lo hi) =
-      relabeledPerm (parallelFreshPre pre lo) (parallelFreshPost post) hi := by
-  ext x
-  have h := butterfly_split r s (lo (pre x).1)
-    (fun j => hi (((pre x).1,j.1),j.2)) (pre x).2
-  have hh := congrArg (fun y => post ((pre x).1,(cardSplit r s).symm y)) h
-  simpa only [relabeledPerm,parallelAssembled,parallelPerm,parallelFreshPre,parallelFreshPost,
-    Equiv.trans_apply,Equiv.prodCongr_apply,Equiv.prodCongrRight,Equiv.coe_fn_mk,
-    Equiv.refl_apply,Equiv.prodComm_apply,Prod.swap,Equiv.prodAssoc_apply,
-    Equiv.prodAssoc_symm_apply,Prod.map,Equiv.symm_apply_apply] using hh
-
-theorem parallel_partially_exposed_log_mgf [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B]
-    (pre : α ≃ B × Card (s+r)) (post : B × Card (s+r) ≃ α)
-    (lo : B → Card r × SwitchIndex s → Bool) (S : Finset α)
-    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    Real.log (finiteMean (fun hi : (B × Card s) × SwitchIndex r → Bool =>
-      q^(Fintype.card (RoutingNetwork.SelectedCycle
-        (relabeledPerm pre post (parallelAssembled lo hi)) S)))) ≤
-      (S.card:ℝ)/J*Real.log q + 2*q*S.card*Real.sqrt J/Real.sqrt ((2:ℝ)^r) := by
-  simp_rw [parallel_exposed_perm]
-  exact parallel_cycle_log_mgf (parallelFreshPre pre lo) (parallelFreshPost post) S J hJ q hq
-
-end Thorp
-namespace Thorp
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-lemma orbitSet_inverse (p : Equiv.Perm α) (x : α) : orbitSet p⁻¹ x = orbitSet p x := by
-  ext y
-  simp only [mem_orbitSet,Equiv.Perm.sameCycle_inv]
-
-lemma selectedCycle_inverse (p : Equiv.Perm α) (S : Finset α) :
-    Fintype.card (RoutingNetwork.SelectedCycle p⁻¹ S) =
-      Fintype.card (RoutingNetwork.SelectedCycle p S) := by
-  apply Fintype.card_congr
-  refine { toFun := fun C => ⟨C.val,?_,C.property.2⟩
-           invFun := fun C => ⟨C.val,?_,C.property.2⟩
-           left_inv := fun _ => rfl
-           right_inv := fun _ => rfl }
-  · obtain ⟨x,hx⟩ := C.property.1
-    exact ⟨x,hx.trans (orbitSet_inverse p x)⟩
-  · obtain ⟨x,hx⟩ := C.property.1
-    exact ⟨x,hx.trans (orbitSet_inverse p x).symm⟩
-
-lemma orbitSet_conjugate (p g : Equiv.Perm α) (x : α) :
-    orbitSet (g*p*g⁻¹) (g x) = (orbitSet p x).image g := by
-  ext y
-  simp only [mem_orbitSet,Finset.mem_image]
-  rw [Equiv.Perm.sameCycle_conj]
-  change p.SameCycle (g.symm (g x)) (g.symm y) ↔ ∃ z, p.SameCycle x z ∧ g z = y
-  rw [Equiv.symm_apply_apply]
-  constructor
-  · intro h
-    exact ⟨g.symm y,h,g.apply_symm_apply y⟩
-  · rintro ⟨z,hz,rfl⟩
-    simpa only [Equiv.symm_apply_apply] using hz
-
-noncomputable def conjugateCycle (p g : Equiv.Perm α) (S : Finset α)
-    (C : RoutingNetwork.SelectedCycle p S) :
-    RoutingNetwork.SelectedCycle (g*p*g⁻¹) (S.image g) := by
-  refine ⟨C.val.image g,?_,Finset.image_subset_image C.property.2⟩
-  obtain ⟨x,hx⟩ := C.property.1
-  exact ⟨g x,by rw [hx,orbitSet_conjugate]⟩
-
-lemma conjugateCycle_injective (p g : Equiv.Perm α) (S : Finset α) :
-    Function.Injective (conjugateCycle p g S) := by
-  intro C D he
-  apply Subtype.ext
-  exact (Finset.image_injective g.injective) (congrArg Subtype.val he)
-
-lemma selectedCycle_conjugate (p g : Equiv.Perm α) (S : Finset α) :
-    Fintype.card (RoutingNetwork.SelectedCycle (g*p*g⁻¹) (S.image g)) =
-      Fintype.card (RoutingNetwork.SelectedCycle p S) := by
-  apply Nat.le_antisymm
-  · have h := Fintype.card_le_of_injective _
-      (conjugateCycle_injective (g*p*g⁻¹) g⁻¹ (S.image g))
-    have hp : g⁻¹*(g*p*g⁻¹)*g = p := by group
-    rw [inv_inv,hp] at h
-    have hs : (S.image g).image g.symm = S := by
-      ext x
-      simp only [Finset.mem_image]
-      constructor
-      · rintro ⟨y,⟨z,hz,rfl⟩,he⟩
-        have hz' : z = x := by simpa only [Equiv.symm_apply_apply] using he
-        exact hz' ▸ hz
-      · intro hx
-        exact ⟨g x,⟨x,hx,rfl⟩,g.symm_apply_apply x⟩
-    change _ ≤ Fintype.card (RoutingNetwork.SelectedCycle p ((S.image g).image g.symm)) at h
-    rw [hs] at h
-    exact h
-  · exact Fintype.card_le_of_injective _ (conjugateCycle_injective p g S)
-
-lemma alternatingCycles_le_projection {ι : Type*} [Fintype ι]
-    (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    PairRouting.alternatingCycles e p ≤
-      Fintype.card (RoutingNetwork.SelectedCycle (PairRouting.alternatingProjection p)
-        (boolFiber (PairRouting.labelSet e) false)) := by
-  have h := CycleColoring.card_alternating_cycles_le (PairRouting.alternatingProjection p)
-    (boolFiber (PairRouting.labelSet e) false) (boolFiber (PairRouting.labelSet e) true)
-  simpa only [PairRouting.alternatingCycles,childJoin_boolFiber] using h
-
-lemma projectionCycles_forward (X₀ X₁ Y₀ Y₁ : Equiv.Perm α) (S : Finset α) :
-    Fintype.card (RoutingNetwork.SelectedCycle
-      ((Y₀*X₀⁻¹)⁻¹*(Y₁*X₁⁻¹)) S) =
-    Fintype.card (RoutingNetwork.SelectedCycle
-      (X₀⁻¹*X₁*Y₁⁻¹*Y₀) (S.image X₀.symm)) := by
-  let p := (Y₀*X₀⁻¹)⁻¹*(Y₁*X₁⁻¹)
-  calc
-    _ = Fintype.card (RoutingNetwork.SelectedCycle (X₀⁻¹*p*X₀)
-        (S.image X₀.symm)) := by
-      change Fintype.card (RoutingNetwork.SelectedCycle p S) = _
-      have h := (selectedCycle_conjugate p X₀⁻¹ S).symm
-      rw [inv_inv] at h
-      exact h
-    _ = Fintype.card (RoutingNetwork.SelectedCycle (X₀⁻¹*p*X₀)⁻¹
-        (S.image X₀.symm)) := (selectedCycle_inverse _ _).symm
-    _ = _ := by
-      have he : (X₀⁻¹*p*X₀)⁻¹ = X₀⁻¹*X₁*Y₁⁻¹*Y₀ := by dsimp only [p]; group
-      rw [he]
-
-end Thorp
-namespace Thorp
-
-noncomputable def cycleCoefficient (r J : ℕ) (q : ℝ) : ℝ :=
-  Real.log q / J + 2*q*Real.sqrt J/Real.sqrt ((2:ℝ)^r)
-
-lemma cycleCoefficient_nonneg (r J : ℕ) (q : ℝ) (hq : 1 ≤ q) :
-    0 ≤ cycleCoefficient r J q := by
-  unfold cycleCoefficient
-  exact add_nonneg (div_nonneg (Real.log_nonneg hq) (Nat.cast_nonneg _))
-    (div_nonneg (mul_nonneg (by linarith) (Real.sqrt_nonneg _)) (Real.sqrt_nonneg _))
-
-lemma projectionSide_card_le {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι]
-    (e : ι ↪ Bool × α) : (boolFiber (PairRouting.labelSet e) false).card ≤ Fintype.card ι := by
-  have h := card_childJoin (boolFiber (PairRouting.labelSet e) false)
-    (boolFiber (PairRouting.labelSet e) true)
-  rw [childJoin_boolFiber,PairRouting.card_labelSet] at h
-  omega
-
-theorem node_partial_mgf (r s : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Bool × Card (s+r)) (X₀ X₁ Y₁ : Equiv.Perm (Card (s+r)))
-    (lo : Card r × SwitchIndex s → Bool) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(PairRouting.alternatingCycles e (fun b => if b then Y₁*X₁⁻¹ else
-        butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi))*X₀⁻¹))) ≤
-      Real.exp (Fintype.card ι * cycleCoefficient r J q) := by
-  classical
-  let S := boolFiber (PairRouting.labelSet e) false
-  let S' := S.image X₀.symm
-  let post := X₀⁻¹*X₁*Y₁⁻¹
-  let count := fun hi : Card s × SwitchIndex r → Bool =>
-    Fintype.card (RoutingNetwork.SelectedCycle
-      ((Equiv.refl _).trans (butterflyPerm (s+r)
-        (decodeButterfly (s+r) (assembleBits r s lo hi))) |>.trans post) S')
-  have hpoint (hi : Card s × SwitchIndex r → Bool) :
-      PairRouting.alternatingCycles e (fun b => if b then Y₁*X₁⁻¹ else
-        butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi))*X₀⁻¹) ≤ count hi := by
-    refine (alternatingCycles_le_projection e _).trans_eq ?_
-    simp only [PairRouting.alternatingProjection,Bool.false_eq_true,↓reduceIte]
-    change Fintype.card (RoutingNetwork.SelectedCycle
-      ((butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi))*X₀⁻¹)⁻¹ * (Y₁*X₁⁻¹)) S) = count hi
-    exact projectionCycles_forward X₀ X₁ _ Y₁ S
-  have hm := partially_exposed_cycle_log_mgf (Equiv.refl (Card (s+r))) post lo S' J hJ q hq
-  have hp : 0 < finiteMean (fun hi : Card s × SwitchIndex r → Bool => q^(count hi)) :=
-    finiteMean_pos (fun _ => pow_pos (by linarith) _)
-  have hb : finiteMean (fun hi : Card s × SwitchIndex r → Bool => q^(count hi)) ≤
-      Real.exp (S'.card * cycleCoefficient r J q) := by
-    rw [←Real.exp_log hp]
-    apply Real.exp_le_exp.mpr
-    calc
-      _ ≤ _ := hm
-      _ = _ := by unfold cycleCoefficient; ring
-  have hcard : S'.card ≤ Fintype.card ι := by
-    rw [show S'.card = S.card from Finset.card_image_of_injective _ X₀.symm.injective]
-    exact projectionSide_card_le e
-  calc
-    _ ≤ finiteMean (fun hi : Card s × SwitchIndex r → Bool => q^(count hi)) :=
-      finiteMean_mono (fun hi => pow_le_pow_right₀ hq (hpoint hi))
-    _ ≤ Real.exp (S'.card * cycleCoefficient r J q) := hb
-    _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right
-      (by exact_mod_cast hcard) (cycleCoefficient_nonneg r J q hq))
-
-
-def highStepEquiv (r s : ℕ) : (Card s × SwitchIndex (r+1) → Bool) ≃
-    (((Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool)) ×
-      (Card s × Card r → Bool)) where
-  toFun hi := ((fun j => hi (j.1,Sum.inr (false,j.2)),
-    fun j => hi (j.1,Sum.inr (true,j.2))),fun j => hi (j.1,Sum.inl j.2))
-  invFun c := fun j => match j.2 with
-    | Sum.inl y => c.2 (j.1,y)
-    | Sum.inr (b,i) => if b then c.1.2 (j.1,i) else c.1.1 (j.1,i)
-  left_inv hi := by
-    funext j
-    rcases j with ⟨z,y|⟨b,i⟩⟩
-    · rfl
-    · cases b <;> rfl
-  right_inv c := by
-    rcases c with ⟨⟨h₀,h₁⟩,η⟩
-    rfl
-
-lemma assembleBits_step (r s : ℕ) (lo : Card (r+1) × SwitchIndex s → Bool)
-    (c : ((Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool)) ×
-      (Card s × Card r → Bool)) :
-    assembleBits (r+1) s lo ((highStepEquiv r s).symm c) =
-      (coinStepEquiv (s+r)).symm
-        ((assembleBits r s (fun j => lo (Fin.cons false j.1,j.2)) c.1.1,
-          assembleBits r s (fun j => lo (Fin.cons true j.1,j.2)) c.1.2),
-        fun y => c.2 ((cardSplit r s y).2,(cardSplit r s y).1)) := by
-  funext i
-  rcases i with y | ⟨b,i⟩
-  · rfl
-  · cases b <;> rfl
-
-noncomputable def palindromeLevelCost (t : ℕ) : (d : ℕ) → {ι : Type*} → [Fintype ι] →
-    (ι ↪ Card d) → BenesCoins d → ℕ
-  | 0, _, _, _, _ => 0
-  | d+1, _, _, e, ω =>
-      let σ := benesStepEquiv d ω
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x σ.2.1
-      let hc := PairRouting.colors_compatible x σ.2.1
-      if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
-          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) else
-        palindromeLevelCost t d (PairRouting.childEmbedding x c hc false) σ.1.1 +
-          palindromeLevelCost t d (PairRouting.childEmbedding x c hc true) σ.1.2
-
-lemma palindromeLevelCost_step (t d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (d+1)) (X Y : (SwitchIndex d → Bool) × (SwitchIndex d → Bool))
-    (ξ η : Card d → Bool) :
-    palindromeLevelCost t (d+1) e
-      ((coinStepEquiv d).symm (Y,η),(coinStepEquiv d).symm (X,ξ)) =
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x ξ
-      let hc := PairRouting.colors_compatible x ξ
-      if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x ξ)
-        (fun b => palindromePerm d (if b then (Y.2,X.2) else (Y.1,X.1))) else
-        palindromeLevelCost t d (PairRouting.childEmbedding x c hc false) (Y.1,X.1) +
-        palindromeLevelCost t d (PairRouting.childEmbedding x c hc true) (Y.2,X.2) := by
-  rw [palindromeLevelCost]
-  rfl
-
-lemma palindromeLevelCost_gt (t d : ℕ) (h : d < t) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) : palindromeLevelCost t d e ω = 0 := by
-  classical
-  induction d generalizing ι with
-  | zero => rfl
-  | succ d ih =>
-    rw [palindromeLevelCost,ite_eq_right (by omega : t ≠ d+1),ih (by omega),ih (by omega),Nat.zero_add]
-
-
-theorem palindrome_level_partial_mgf (r s u : ℕ) (hu : u < r)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+r)) (X : SwitchIndex (s+r) → Bool)
-    (lo : Card r × SwitchIndex s → Bool) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(palindromeLevelCost (s+u+1) (s+r) e (assembleBits r s lo hi,X))) ≤
-      Real.exp (Fintype.card ι * cycleCoefficient u J q) := by
-  classical
-  induction r generalizing u ι with
-  | zero => omega
-  | succ r ih =>
-    simp only [Nat.add_succ] at *
-    let χ := coinStepEquiv (s+r) X
-    let x := e.trans (headTailEquiv (s+r)).toEmbedding
-    let c := PairRouting.colors x χ.2
-    let hc := PairRouting.colors_compatible x χ.2
-    let e₀ := PairRouting.childEmbedding x c hc false
-    let e₁ := PairRouting.childEmbedding x c hc true
-    let e' := PairRouting.switchedEmbedding x χ.2
-    let lo₀ : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons false j.1,j.2)
-    let lo₁ : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons true j.1,j.2)
-    let Y₀ := fun hi : Card s × SwitchIndex r → Bool => assembleBits r s lo₀ hi
-    let Y₁ := fun hi : Card s × SwitchIndex r → Bool => assembleBits r s lo₁ hi
-    let N := fun hi : (Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool) =>
-      PairRouting.alternatingCycles e' (fun b => if b then palindromePerm (s+r) (Y₁ hi.2,χ.1.2)
-        else palindromePerm (s+r) (Y₀ hi.1,χ.1.1))
-    let L₀ := fun hi : Card s × SwitchIndex r → Bool =>
-      palindromeLevelCost (s+u+1) (s+r) e₀ (Y₀ hi,χ.1.1)
-    let L₁ := fun hi : Card s × SwitchIndex r → Bool =>
-      palindromeLevelCost (s+u+1) (s+r) e₁ (Y₁ hi,χ.1.2)
-    have hs (hi : (Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool))
-        (η : Card s × Card r → Bool) :
-        palindromeLevelCost (s+u+1) (s+r+1) e
-          (assembleBits (r+1) s lo ((highStepEquiv r s).symm (hi,η)),X) =
-          if u = r then N hi else L₀ hi.1 + L₁ hi.2 := by
-      rw [assembleBits_step]
-      conv_lhs => rw [←(coinStepEquiv (s+r)).symm_apply_apply X]
-      rw [palindromeLevelCost_step]
-      simp only [Nat.add_right_cancel_iff,Nat.add_left_cancel_iff]
-      dsimp only [χ,x,c,hc,e₀,e₁,e',lo₀,lo₁,Y₀,Y₁,N,L₀,L₁]
-      congr 2
-      funext b
-      cases b <;> rfl
-    rw [finiteMean_equiv (highStepEquiv r s),finiteMean_prod]
-    simp_rw [hs,finiteMean_const]
-    by_cases hur : u = r
-    · subst u
-      simp only [ite_true]
-      rw [finiteMean_prod,finiteMean_comm]
-      calc
-        _ ≤ finiteMean (fun _hi : Card s × SwitchIndex r → Bool =>
-          Real.exp (Fintype.card ι * cycleCoefficient r J q)) := by
-          apply finiteMean_mono
-          intro hi
-          have hn := node_partial_mgf r s e'
-            (butterflyPerm (s+r) (decodeButterfly (s+r) χ.1.1))
-            (butterflyPerm (s+r) (decodeButterfly (s+r) χ.1.2))
-            (butterflyPerm (s+r) (decodeButterfly (s+r) (Y₁ hi))) lo₀ J hJ q hq
-          exact hn
-        _ = _ := finiteMean_const _
-    · simp only [ite_eq_right hur,pow_add]
-      rw [finiteMean_prod_mul (fun hi => q ^ L₀ hi) (fun hi => q ^ L₁ hi)]
-      have hu' : u < r := by omega
-      have h₀ := ih u hu' e₀ χ.1.1 lo₀
-      have h₁ := ih u hu' e₁ χ.1.2 lo₁
-      have hp := mul_le_mul h₀ h₁ (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
-        (Real.exp_nonneg _)
-      refine hp.trans_eq ?_
-      rw [←Real.exp_add]
-      congr 1
-      have hcount := PairRouting.color_card_add c
-      have hcount' : (Fintype.card {i : ι // c i = false} : ℝ) +
-          Fintype.card {i : ι // c i = true} = Fintype.card ι := by exact_mod_cast hcount
-      change (Fintype.card {i : ι // c i = false} : ℝ) * cycleCoefficient u J q +
-          Fintype.card {i : ι // c i = true} * cycleCoefficient u J q = _
-      rw [←add_mul,hcount']
-
-
-def switchHeight : (d : ℕ) → SwitchIndex d → ℕ
-  | 0, i => nomatch i
-  | d+1, Sum.inl _ => d+1
-  | d+1, Sum.inr (_,i) => switchHeight d i
-
-lemma switchHeight_le (d : ℕ) (i : SwitchIndex d) : switchHeight d i ≤ d := by
-  induction d with
-  | zero => exact Empty.elim i
-  | succ d ih =>
-    rcases i with y | ⟨b,i⟩
-    · exact le_refl _
-    · exact (ih i).trans (Nat.le_succ d)
-
-def AgreeThrough (d s : ℕ) (Y Y' : SwitchIndex d → Bool) : Prop :=
-  ∀ i, switchHeight d i ≤ s → Y i = Y' i
-
-def LayerAdapted (d s : ℕ) {β : Type*} (f : (SwitchIndex d → Bool) → β) : Prop :=
-  ∀ Y Y', AgreeThrough d s Y Y' → f Y = f Y'
-
-lemma agreeThrough_mono {d s t : ℕ} (h : s ≤ t) {Y Y' : SwitchIndex d → Bool}
-    (hh : AgreeThrough d t Y Y') : AgreeThrough d s Y Y' :=
-  fun i hi => hh i (hi.trans h)
-
-lemma layerAdapted_mono {d s t : ℕ} {β : Type*} {f : (SwitchIndex d → Bool) → β}
-    (hf : LayerAdapted d s f) (h : s ≤ t) : LayerAdapted d t f :=
-  fun Y Y' hh => hf Y Y' (agreeThrough_mono h hh)
-
-lemma layerAdapted_comp {d s : ℕ} {α β : Type*} {f : (SwitchIndex d → Bool) → α}
-    (hf : LayerAdapted d s f) (g : α → β) : LayerAdapted d s (g ∘ f) :=
-  fun Y Y' h => congrArg g (hf Y Y' h)
-
-lemma layerAdapted_sum {d s : ℕ} {ι : Type*} (A : Finset ι)
-    (f : ι → (SwitchIndex d → Bool) → ℕ)
-    (hf : ∀ i ∈ A, LayerAdapted d s (f i)) :
-    LayerAdapted d s (fun Y => ∑ i ∈ A, f i Y) := by
-  intro Y Y' h
-  exact Finset.sum_congr rfl (fun i hi => hf i hi Y Y' h)
-
-lemma assembleBits_agree (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
-    (hi hi' : Card s × SwitchIndex r → Bool) :
-    AgreeThrough (s+r) s (assembleBits r s lo hi) (assembleBits r s lo hi') := by
-  induction r with
-  | zero => intro i _; rfl
-  | succ r ih =>
-    intro i h
-    rcases i with y | ⟨b,i⟩
-    · have : s+r+1 ≤ s := h
-      omega
-    · exact ih (fun j => lo (Fin.cons b j.1,j.2))
-        (fun j => hi (j.1,Sum.inr (b,j.2))) (fun j => hi' (j.1,Sum.inr (b,j.2))) i h
-
-lemma palindromeLevelCost_adapted (t d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) :
-    LayerAdapted d (t-1) (fun Y => palindromeLevelCost t d e (Y,X)) := by
-  classical
-  induction d generalizing ι with
-  | zero => intro Y Y' _; rfl
-  | succ d ih =>
-    intro Y Y' h
-    change palindromeLevelCost t (d+1) e (Y,X) = palindromeLevelCost t (d+1) e (Y',X)
-    conv_lhs => rw [←(coinStepEquiv d).symm_apply_apply Y,←(coinStepEquiv d).symm_apply_apply X]
-    conv_rhs => rw [←(coinStepEquiv d).symm_apply_apply Y',←(coinStepEquiv d).symm_apply_apply X]
-    rw [palindromeLevelCost_step,palindromeLevelCost_step]
-    by_cases ht : t = d+1
-    · subst t
-      rw [ite_eq_left rfl,ite_eq_left rfl]
-      have he₀ : (coinStepEquiv d Y).1.1 = (coinStepEquiv d Y').1.1 := by
-        funext i
-        exact h (Sum.inr (false,i)) (by simpa only [switchHeight,Nat.add_sub_cancel] using switchHeight_le d i)
-      have he₁ : (coinStepEquiv d Y).1.2 = (coinStepEquiv d Y').1.2 := by
-        funext i
-        exact h (Sum.inr (true,i)) (by simpa only [switchHeight,Nat.add_sub_cancel] using switchHeight_le d i)
-      rw [he₀,he₁]
-    · rw [ite_eq_right ht,ite_eq_right ht]
-      congr 1
-      · apply ih
-        intro i hi
-        exact h (Sum.inr (false,i)) hi
-      · apply ih
-        intro i hi
-        exact h (Sum.inr (true,i)) hi
-
-lemma adapted_mul_mean_le (r s : ℕ) (F G : (SwitchIndex (s+r) → Bool) → ℝ)
-    (hF : ∀ Y, 0 ≤ F Y) (hA : LayerAdapted (s+r) s F) (B : ℝ)
-    (hG : ∀ lo : Card r × SwitchIndex s → Bool,
-      finiteMean (fun hi : Card s × SwitchIndex r → Bool => G (assembleBits r s lo hi)) ≤ B) :
-    finiteMean (fun Y => F Y*G Y) ≤ finiteMean F * B := by
-  rw [finiteMean_split r s,finiteMean_split r s F,←finiteMean_mul_const]
-  apply finiteMean_mono
-  intro lo
-  let H : Card s × SwitchIndex r → Bool := fun _ => false
-  have hh (hi : Card s × SwitchIndex r → Bool) :
-      F (assembleBits r s lo hi) = F (assembleBits r s lo H) :=
-    hA _ _ (assembleBits_agree r s lo hi H)
-  simp_rw [hh,finiteMean_const]
-  rw [show (fun hi => F (assembleBits r s lo H) * G (assembleBits r s lo hi)) =
-      (fun hi => G (assembleBits r s lo hi) * F (assembleBits r s lo H)) by
-        funext hi; exact mul_comm _ _,finiteMean_mul_const]
-  simpa only [mul_comm] using mul_le_mul_of_nonneg_left (hG lo) (hF _)
-
-
-theorem cycleCoefficient_decay (l r : ℕ) (hr : (l:ℝ)/2 ≤ r)
-    (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ (l:ℝ)/250) :
-    cycleCoefficient r ⌈Real.exp ((l:ℝ)/32)⌉₊ q ≤
-      68 * Real.exp (-(l:ℝ)/64) := by
-  let E := Real.exp ((l:ℝ)/32)
-  let J := ⌈E⌉₊
-  have hl : (0:ℝ) ≤ l := Nat.cast_nonneg _
-  have hE : 1 ≤ E := Real.one_le_exp_iff.mpr (by positivity)
-  have hEpos : 0 < E := Real.exp_pos _
-  have hEJ : E ≤ (J:ℝ) := Nat.le_ceil E
-  have hJ : (1:ℝ) ≤ J := hE.trans hEJ
-  have hJpos : (0:ℝ) < J := lt_of_lt_of_le (by norm_num) hJ
-  have hJE : (J:ℝ) ≤ 2*E := by
-    have h := Nat.ceil_lt_add_one (le_of_lt hEpos)
-    change (J:ℝ) < E+1 at h
-    linarith
-  have hsqJ : Real.sqrt J ≤ (J:ℝ) := by
-    apply Real.sqrt_le_iff.mpr
-    constructor
-    · positivity
-    · nlinarith
-  have hqp : 0 < q := lt_of_lt_of_le (by norm_num) hq
-  have hqE : q ≤ Real.exp ((l:ℝ)/250) := by
-    rw [←Real.exp_log hqp]
-    exact Real.exp_le_exp.mpr hlog
-  have hln : (1/2:ℝ) ≤ Real.log 2 := by
-    have hh := Real.one_sub_inv_le_log_of_pos (by norm_num : (0:ℝ)<2)
-    norm_num at hh ⊢
-    exact hh
-  have hden : Real.exp ((l:ℝ)/8) ≤ Real.sqrt ((2:ℝ)^r) := by
-    have he : (2:ℝ)^r = Real.exp ((r:ℝ)*Real.log 2) := by
-      rw [Real.exp_nat_mul,Real.exp_log (by norm_num)]
-    rw [he,Real.sqrt_eq_rpow,Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp]
-    apply Real.exp_le_exp.mpr
-    have hrn : (0:ℝ) ≤ r := Nat.cast_nonneg _
-    nlinarith
-  have hlog' : Real.log q ≤ (l:ℝ) := by linarith
-  have h₁ : Real.log q / (J:ℝ) ≤ 64*Real.exp (-(l:ℝ)/64) := by
-    calc
-      _ ≤ (l:ℝ)/E := div_le_div₀ (by positivity) hlog' hEpos hEJ
-      _ ≤ (64*Real.exp ((l:ℝ)/64))/E := by
-        apply div_le_div_of_nonneg_right _ (le_of_lt hEpos)
-        have hh := Real.add_one_le_exp ((l:ℝ)/64)
-        linarith
-      _ = _ := by
-        dsimp [E]
-        rw [mul_div_assoc,←Real.exp_sub]
-        congr 2
-        ring
-  have h₂ : 2*q*Real.sqrt J / Real.sqrt ((2:ℝ)^r) ≤
-      4*Real.exp (-(l:ℝ)/64) := by
-    calc
-      _ ≤ (4*Real.exp ((l:ℝ)/250)*E)/Real.exp ((l:ℝ)/8) := by
-        apply div_le_div₀ (by positivity) _ (Real.exp_pos _) hden
-        have hs := hsqJ.trans hJE
-        have hm := mul_le_mul hqE hs (Real.sqrt_nonneg _) (Real.exp_nonneg _)
-        nlinarith
-      _ = 4*Real.exp ((l:ℝ)/250+(l:ℝ)/32-(l:ℝ)/8) := by
-        dsimp [E]
-        rw [mul_assoc,←Real.exp_add,mul_div_assoc,←Real.exp_sub]
-      _ ≤ _ := by
-        apply mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr _) (by norm_num)
-        linarith
-  change Real.log q / (J:ℝ) + 2*q*Real.sqrt J / Real.sqrt ((2:ℝ)^r) ≤ _
-  linarith
-
-lemma cycleCutoff_pos (l : ℕ) : 0 < ⌈Real.exp ((l:ℝ)/32)⌉₊ := by
-  have h := Nat.le_ceil (Real.exp ((l:ℝ)/32))
-  have he := Real.exp_pos ((l:ℝ)/32)
-  have : (0:ℝ) < ⌈Real.exp ((l:ℝ)/32)⌉₊ := lt_of_lt_of_le he h
-  exact_mod_cast this
-
-
-lemma pow_sum_le_mean_powers {ι : Type*} [Fintype ι] [Nonempty ι]
-    (z : ι → ℕ) (q : ℝ) (hq : 0 < q) :
-    q^(∑ i, z i) ≤ finiteMean (fun i => q^(Fintype.card ι*z i)) := by
-  have hn : (Fintype.card ι:ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
-  have hm : finiteMean (fun i => (Fintype.card ι*z i:ℕ)*Real.log q) =
-      (∑ i, z i:ℕ)*Real.log q := by
-    simp only [finiteMean,Nat.cast_mul,Nat.cast_sum,←Finset.mul_sum,←Finset.sum_mul]
-    field_simp
-  have he := exp_finiteMean_le (fun i => (Fintype.card ι*z i:ℕ)*Real.log q)
-  rw [hm,Real.exp_nat_mul,Real.exp_log hq] at he
-  simpa only [Real.exp_nat_mul,Real.exp_log hq] using he
-
-noncomputable def palindromeSlabCost (l d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) : ℕ :=
-  ∑ i : Fin l, palindromeLevelCost (l+i.1+1) d e ω
-
-lemma palindromeSlabCost_adapted (l d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) :
-    LayerAdapted d (2*l) (fun Y => palindromeSlabCost l d e (Y,X)) := by
-  apply layerAdapted_sum
-  intro i _
-  exact layerAdapted_mono (palindromeLevelCost_adapted (l+i.1+1) d e X) (by omega)
-
-theorem palindrome_slab_partial_mgf (r s l : ℕ) (hl : 0 < l) (hs : 2*s ≤ l)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+r)) (X : SwitchIndex (s+r) → Bool)
-    (lo : Card r × SwitchIndex s → Bool) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(palindromeSlabCost l (s+r) e (assembleBits r s lo hi,X))) ≤
-      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
-  classical
-  let : NeZero l := ⟨ne_of_gt hl⟩
-  have hqp : 0 < q := lt_of_lt_of_le (by norm_num) hq
-  calc
-    _ ≤ finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      finiteMean (fun i : Fin l => q^(l*palindromeLevelCost (l+i.1+1) (s+r) e
-        (assembleBits r s lo hi,X)))) := by
-      apply finiteMean_mono
-      intro hi
-      simpa only [palindromeSlabCost,Fintype.card_fin] using
-        pow_sum_le_mean_powers (fun i : Fin l => palindromeLevelCost (l+i.1+1) (s+r) e
-          (assembleBits r s lo hi,X)) q hqp
-    _ = finiteMean (fun i : Fin l => finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(l*palindromeLevelCost (l+i.1+1) (s+r) e (assembleBits r s lo hi,X)))) := finiteMean_comm _
-    _ ≤ finiteMean (fun _i : Fin l =>
-      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64)))) := by
-      apply finiteMean_mono
-      intro i
-      by_cases ht : l+i.1+1 ≤ s+r
-      · have hs' : s ≤ l+i.1 := by omega
-        have he : l+i.1+1 = s+(l+i.1-s)+1 := by omega
-        have hu : l+i.1-s < r := by omega
-        have hm := palindrome_level_partial_mgf r s (l+i.1-s) hu e X lo
-          ⌈Real.exp ((l:ℝ)/32)⌉₊ (cycleCutoff_pos l) (q^l) (one_le_pow₀ hq)
-        have hc := cycleCoefficient_decay l (l+i.1-s) (by
-          rw [Nat.cast_sub hs',Nat.cast_add]
-          have hsr : (2:ℝ)*s ≤ l := by exact_mod_cast hs
-          have hi : (0:ℝ) ≤ i.1 := Nat.cast_nonneg _
-          linarith) (q^l) (one_le_pow₀ hq) (by
-            rw [Real.log_pow]
-            nlinarith [mul_le_mul_of_nonneg_left hlog (Nat.cast_nonneg l)])
-        simp_rw [he,pow_mul]
-        exact hm.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left hc (Nat.cast_nonneg _)))
-      · have hz := palindromeLevelCost_gt (l+i.1+1) (s+r) (by omega) e
-        simp only [hz,Nat.mul_zero,pow_zero,finiteMean_const]
-        exact Real.one_le_exp_iff.mpr (by positivity)
-    _ = _ := finiteMean_const _
-
-
-lemma palindromeSlabCost_gt (l d : ℕ) (h : d ≤ l) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) : palindromeSlabCost l d e ω = 0 := by
-  apply Finset.sum_eq_zero
-  intro i _
-  exact palindromeLevelCost_gt _ _ (by omega) _ _
-
-lemma palindrome_slab_adapted_mul_le (d s l : ℕ) (hsd : s ≤ d) (hl : 0 < l)
-    (hs : 2*s ≤ l) {ι : Type*} [Fintype ι] (e : ι ↪ Card d)
-    (X : SwitchIndex d → Bool) (F : (SwitchIndex d → Bool) → ℝ)
-    (hF : ∀ Y, 0 ≤ F Y) (hA : LayerAdapted d s F) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun Y => F Y*q^(palindromeSlabCost l d e (Y,X))) ≤
-      finiteMean F * Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
-  obtain ⟨r,rfl⟩ := Nat.exists_eq_add_of_le hsd
-  exact adapted_mul_mean_le r s F _ hF hA _
-    (fun lo => palindrome_slab_partial_mgf r s l hl hs e X lo q hq hlog)
-
-def familyEnd (L n : ℕ) : ℕ := if n = 0 then 0 else 2*(L*4^(n-1))
-
-end Thorp
-
-end ThorpNine.Harmonic
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Harmonic.SandwichShuffle
+
+namespace OAI
+
+namespace ThorpNine.Harmonic
+
+namespace Thorp
+
+noncomputable def cutoffMass (H : ℕ) : (d : ℕ) → {ι : Type*} → [Fintype ι] →
+    (ι ↪ Card d) → BenesCoins d → ℕ
+  | 0, _, _, _, _ => 0
+  | d+1, ι, _, e, ω =>
+      if d+1 ≤ H then crowdedMass (Fintype.card ι) else
+      let σ := benesStepEquiv d ω
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x σ.2.1
+      let hc := PairRouting.colors_compatible x σ.2.1
+      cutoffMass H d (PairRouting.childEmbedding x c hc false) σ.1.1 +
+        cutoffMass H d (PairRouting.childEmbedding x c hc true) σ.1.2
+
+lemma palindromeLowCost_eq (H d : ℕ) (h : d ≤ H) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) :
+    palindromeLowCost H d e ω = palindromeCost d e ω := by
+  classical
+  induction d generalizing ι with
+  | zero => rfl
+  | succ d ih =>
+    rw [palindromeLowCost,palindromeCost,ite_eq_left h]
+    congr 2 <;> exact ih (by omega) _ _
+
+lemma palindromeLowCost_le_cutoff (H d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) :
+    palindromeLowCost H d e ω ≤ H*cutoffMass H d e ω := by
+  classical
+  induction d generalizing ι with
+  | zero => simp [palindromeLowCost,cutoffMass]
+  | succ d ih =>
+    by_cases h : d+1 ≤ H
+    · rw [palindromeLowCost_eq H (d+1) h,cutoffMass,ite_eq_left h]
+      exact (palindromeCost_le_crowded (d+1) e ω).trans (Nat.mul_le_mul_right _ h)
+    · rw [palindromeLowCost,cutoffMass,ite_eq_right h,ite_eq_right h,zero_add,Nat.mul_add]
+      exact Nat.add_le_add (ih _ _) (ih _ _)
+
+
+noncomputable def centralBlock (r H : ℕ) (b : Card r) : Finset (Card (H+r)) :=
+  Finset.univ.image (fun z : Card H => (cardSplit r H).symm (b,z))
+
+lemma mem_centralBlock (r H : ℕ) (b : Card r) (x : Card (H+r)) :
+    x ∈ centralBlock r H b ↔ (cardSplit r H x).1 = b := by
+  classical
+  constructor
+  · intro h
+    obtain ⟨z,_,rfl⟩ := Finset.mem_image.mp h
+    simp only [Equiv.apply_symm_apply]
+  · intro h
+    apply Finset.mem_image.mpr
+    refine ⟨(cardSplit r H x).2,Finset.mem_univ _,?_⟩
+    rw [←h,Prod.mk.eta,Equiv.symm_apply_apply]
+
+lemma centralBlock_card (r H : ℕ) (b : Card r) : (centralBlock r H b).card = 2^H := by
+  classical
+  rw [centralBlock,Finset.card_image_of_injective]
+  · exact card_positions H
+  · intro x y h
+    exact congrArg Prod.snd ((cardSplit r H).symm.injective h)
+
+lemma centralBlock_disjoint (r H : ℕ) :
+    Pairwise (fun b b' => Disjoint (centralBlock r H b) (centralBlock r H b')) := by
+  classical
+  intro b b' h
+  apply Finset.disjoint_left.mpr
+  intro x hx hy
+  exact h (((mem_centralBlock r H b x).mp hx).symm.trans
+    ((mem_centralBlock r H b' x).mp hy))
+
+lemma centralBlock_zero (H : ℕ) (b : Card 0) : centralBlock 0 H b = Finset.univ := by
+  classical
+  ext x
+  simp only [mem_centralBlock,Finset.mem_univ,iff_true]
+  exact Subsingleton.elim _ _
+
+lemma centralBlock_step (r H : ℕ) (b : Bool) (c : Card r) :
+    centralBlock (r+1) H (Fin.cons b c) = (centralBlock r H c).image (Fin.cons b) := by
+  classical
+  simp only [centralBlock,Finset.image_image,cardSplit,Equiv.coe_fn_symm_mk,
+    Fin.cons_zero,Fin.tail_cons,Function.comp_def]
+
+lemma block_inter_step (r H : ℕ) (b : Bool) (c : Card r) (A : Finset (Card (H+(r+1)))) :
+    centralBlock (r+1) H (Fin.cons b c) ∩ A =
+      (centralBlock r H c ∩ childMarks A b).image (Fin.cons b) := by
+  classical
+  rw [centralBlock_step]
+  ext x
+  constructor
+  · intro hx
+    obtain ⟨hxB,hxA⟩ := Finset.mem_inter.mp hx
+    obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hxB
+    exact Finset.mem_image.mpr ⟨y,Finset.mem_inter.mpr ⟨hy,(mem_childMarks A b y).mpr hxA⟩,rfl⟩
+  · intro hx
+    obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
+    exact Finset.mem_inter.mpr ⟨Finset.mem_image.mpr ⟨y,(Finset.mem_inter.mp hy).1,rfl⟩,
+      (mem_childMarks A b y).mp (Finset.mem_inter.mp hy).2⟩
+
+lemma sparseMass_central_zero (H : ℕ) (A : Finset (Card H)) :
+    sparseMass (centralBlock 0 H) A = crowdedMass A.card := by
+  classical
+  simp only [sparseMass,centralBlock_zero,Finset.univ_inter,crowdedMass,
+    Finset.sum_const,Finset.card_univ,card_positions,pow_zero,one_nsmul]
+
+lemma sparseMass_central_step (r H : ℕ) (A : Finset (Card (H+(r+1)))) :
+    sparseMass (centralBlock (r+1) H) A =
+      sparseMass (centralBlock r H) (childMarks A false) +
+        sparseMass (centralBlock r H) (childMarks A true) := by
+  classical
+  unfold sparseMass
+  rw [←Equiv.sum_comp (headTailEquiv r).symm,Fintype.sum_prod_type]
+  have ht (b : Bool) (c : Card r) : (headTailEquiv r).symm (b,c) = Fin.cons b c := rfl
+  simp only [Fintype.sum_bool,ht]
+  have hc (b : Bool) (c : Card r) :
+      (centralBlock (r+1) H (Fin.cons b c) ∩ A).card =
+      (centralBlock r H c ∩ childMarks A b).card := by
+    rw [block_inter_step]
+    apply Finset.card_image_of_injective
+    intro x y h
+    simpa only [Fin.tail_cons] using congrArg Fin.tail h
+  simp only [hc]
+  exact add_comm _ _
+
+
+lemma inverseButterfly_coordinates_step (d : ℕ)
+    (c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (ξ : Card d → Bool)
+    (x : Card (d+1)) :
+    headTailEquiv d ((butterflyPerm (d+1)
+      (decodeButterfly (d+1) ((coinStepEquiv d).symm (c,ξ)))).symm x) =
+      (Bool.xor (x 0) (ξ (Fin.tail x)),
+        (butterflyPerm d (decodeButterfly d
+          (if Bool.xor (x 0) (ξ (Fin.tail x)) then c.2 else c.1))).symm (Fin.tail x)) := by
+  rw [butterflyPerm_step]
+  rfl
+
+lemma inverseButterflyMarks_child (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (d+1))
+    (c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (ξ : Card d → Bool) (b : Bool) :
+    childMarks (inverseButterflyMarks (d+1) (Finset.univ.image e)
+      ((coinStepEquiv d).symm (c,ξ))) b =
+      inverseButterflyMarks d
+        (Finset.univ.image (PairRouting.childEmbedding (e.trans (headTailEquiv d).toEmbedding)
+          (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) ξ)
+            (PairRouting.colors_compatible _ _) b)) (if b then c.2 else c.1) := by
+  classical
+  let x := e.trans (headTailEquiv d).toEmbedding
+  let col := PairRouting.colors x ξ
+  let child := PairRouting.childEmbedding x col (PairRouting.colors_compatible x ξ) b
+  have hp (i : ι) : headTailEquiv d ((butterflyPerm (d+1)
+      (decodeButterfly (d+1) ((coinStepEquiv d).symm (c,ξ)))).symm (e i)) =
+      (col i,(butterflyPerm d (decodeButterfly d (if col i then c.2 else c.1))).symm (x i).2) :=
+    inverseButterfly_coordinates_step d c ξ (e i)
+  ext y
+  simp only [mem_childMarks,inverseButterflyMarks,Finset.image_image,
+    Finset.mem_image,Finset.mem_univ,true_and,Function.comp_def]
+  constructor
+  · rintro ⟨i,hi⟩
+    have he := congrArg (headTailEquiv d) hi
+    rw [hp i] at he
+    change (col i,_) = (b,y) at he
+    have hb := congrArg Prod.fst he
+    have hy := congrArg Prod.snd he
+    dsimp only at hb hy
+    refine ⟨⟨i,hb⟩,?_⟩
+    change (butterflyPerm d (decodeButterfly d (if b then c.2 else c.1))).symm (x i).2 = y
+    simpa only [hb] using hy
+  · rintro ⟨i,hi⟩
+    refine ⟨i.val,?_⟩
+    apply (headTailEquiv d).injective
+    rw [hp]
+    change (col i.val,_) = (b,y)
+    have hb : col i.val = b := i.property
+    rw [hb]
+    exact Prod.ext rfl hi
+
+lemma inverseButterflyMarks_card (d : ℕ) (A : Finset (Card d)) (ω : SwitchIndex d → Bool) :
+    (inverseButterflyMarks d A ω).card = A.card := by
+  classical
+  exact Finset.card_image_of_injective _ (butterflyPerm d (decodeButterfly d ω)).symm.injective
+
+lemma benes_input_coordinates (d : ℕ) (ω : BenesCoins (d+1)) :
+    ω.2 = (coinStepEquiv d).symm
+      (((benesStepEquiv d ω).1.1.2,(benesStepEquiv d ω).1.2.2),(benesStepEquiv d ω).2.1) := by
+  change ω.2 = (coinStepEquiv d).symm (coinStepEquiv d ω.2)
+  exact ((coinStepEquiv d).symm_apply_apply ω.2).symm
+
+lemma cutoffMass_central (r H : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (H+r)) (ω : BenesCoins (H+r)) :
+    cutoffMass H (H+r) e ω =
+      sparseMass (centralBlock r H) (inverseButterflyMarks (H+r) (Finset.univ.image e) ω.2) := by
+  classical
+  induction r generalizing ι with
+  | zero =>
+    rw [sparseMass_central_zero,inverseButterflyMarks_card,
+      Finset.card_image_of_injective _ e.injective,Finset.card_univ]
+    cases H with
+    | zero =>
+      have hcard : Fintype.card ι ≤ 1 := by
+        have hh := Fintype.card_le_of_injective e e.injective
+        simpa only [Nat.add_zero,card_positions,pow_zero] using hh
+      simp [cutoffMass,crowdedMass,show ¬2≤Fintype.card ι by omega]
+    | succ H => simp only [cutoffMass,le_refl,ite_true]
+  | succ r ih =>
+    simp only [Nat.add_succ] at *
+    rw [cutoffMass,ite_eq_right (by omega : ¬H+r+1≤H)]
+    let σ := benesStepEquiv (H+r) ω
+    let x := e.trans (headTailEquiv (H+r)).toEmbedding
+    let col := PairRouting.colors x σ.2.1
+    let hc := PairRouting.colors_compatible x σ.2.1
+    change cutoffMass H (H+r) (PairRouting.childEmbedding x col hc false) σ.1.1 +
+      cutoffMass H (H+r) (PairRouting.childEmbedding x col hc true) σ.1.2 = _
+    rw [ih,ih,sparseMass_central_step,benes_input_coordinates (H+r) ω]
+    rw [inverseButterflyMarks_child,inverseButterflyMarks_child]
+    rfl
+
+
+theorem palindrome_low_mgf (r H : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (H+r)) (q : ℝ) (hq : 1 ≤ q)
+    (hr : (2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*q^H) ≤ 1/2) :
+    finiteMean (fun ω : BenesCoins (H+r) => q^(palindromeLowCost H (H+r) e ω)) ≤
+      Real.exp ((2:ℝ)^r * (2 * ((2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*q^H))^2)) := by
+  classical
+  have hec : (Finset.univ.image e).card = Fintype.card ι := by
+    rw [Finset.card_image_of_injective _ e.injective,Finset.card_univ]
+  have hm := inverseButterfly_sparse_mgf (H+r) (centralBlock r H)
+    (centralBlock_disjoint r H) (Finset.univ.image e) (q^H)
+    (pow_nonneg (by linarith) _) (fun b => by
+      simpa only [centralBlock_card,hec,Nat.cast_pow,Nat.cast_ofNat] using hr)
+  calc
+    _ ≤ finiteMean (fun ω : BenesCoins (H+r) => (q^H)^(cutoffMass H (H+r) e ω)) := by
+      apply finiteMean_mono
+      intro ω
+      rw [←pow_mul]
+      exact pow_le_pow_right₀ hq (palindromeLowCost_le_cutoff H (H+r) e ω)
+    _ = finiteMean (fun ω : SwitchIndex (H+r) → Bool =>
+        (q^H)^(sparseMass (centralBlock r H)
+          (inverseButterflyMarks (H+r) (Finset.univ.image e) ω))) := by
+      simp_rw [cutoffMass_central]
+      simp only [finiteMean_prod,finiteMean_const]
+    _ ≤ _ := by
+      simpa only [centralBlock_card,hec,Nat.cast_pow,Nat.cast_ofNat,Finset.sum_const,
+        Finset.card_univ,card_positions,nsmul_eq_mul] using hm
+
+variable {α B : Type*} {r s : ℕ}
+
+def parallelFreshPre (pre : α ≃ B × Card (s+r))
+    (lo : B → Card r × SwitchIndex s → Bool) : α ≃ (B × Card s) × Card r :=
+  ((pre.trans (Equiv.prodCongr (Equiv.refl B) (cardSplit r s))).trans
+    (Equiv.prodCongrRight (fun b =>
+      (parallelPerm (lo b)).trans (Equiv.prodComm _ _)))).trans
+    (Equiv.prodAssoc _ _ _).symm
+
+def parallelFreshPost (post : B × Card (s+r) ≃ α) : (B × Card s) × Card r ≃ α :=
+  (((Equiv.prodAssoc _ _ _).trans (Equiv.prodCongr (Equiv.refl B)
+    (Equiv.prodComm _ _))).trans (Equiv.prodCongr (Equiv.refl B) (cardSplit r s).symm)).trans post
+
+def parallelAssembled (lo : B → Card r × SwitchIndex s → Bool)
+    (hi : (B × Card s) × SwitchIndex r → Bool) : B × SwitchIndex (s+r) → Bool :=
+  fun i => assembleBits r s (lo i.1) (fun j => hi ((i.1,j.1),j.2)) i.2
+
+lemma parallel_exposed_perm (pre : α ≃ B × Card (s+r)) (post : B × Card (s+r) ≃ α)
+    (lo : B → Card r × SwitchIndex s → Bool)
+    (hi : (B × Card s) × SwitchIndex r → Bool) :
+    relabeledPerm pre post (parallelAssembled lo hi) =
+      relabeledPerm (parallelFreshPre pre lo) (parallelFreshPost post) hi := by
+  ext x
+  have h := butterfly_split r s (lo (pre x).1)
+    (fun j => hi (((pre x).1,j.1),j.2)) (pre x).2
+  have hh := congrArg (fun y => post ((pre x).1,(cardSplit r s).symm y)) h
+  simpa only [relabeledPerm,parallelAssembled,parallelPerm,parallelFreshPre,parallelFreshPost,
+    Equiv.trans_apply,Equiv.prodCongr_apply,Equiv.prodCongrRight,Equiv.coe_fn_mk,
+    Equiv.refl_apply,Equiv.prodComm_apply,Prod.swap,Equiv.prodAssoc_apply,
+    Equiv.prodAssoc_symm_apply,Prod.map,Equiv.symm_apply_apply] using hh
+
+theorem parallel_partially_exposed_log_mgf [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B]
+    (pre : α ≃ B × Card (s+r)) (post : B × Card (s+r) ≃ α)
+    (lo : B → Card r × SwitchIndex s → Bool) (S : Finset α)
+    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    Real.log (finiteMean (fun hi : (B × Card s) × SwitchIndex r → Bool =>
+      q^(Fintype.card (RoutingNetwork.SelectedCycle
+        (relabeledPerm pre post (parallelAssembled lo hi)) S)))) ≤
+      (S.card:ℝ)/J*Real.log q + 2*q*S.card*Real.sqrt J/Real.sqrt ((2:ℝ)^r) := by
+  simp_rw [parallel_exposed_perm]
+  exact parallel_cycle_log_mgf (parallelFreshPre pre lo) (parallelFreshPost post) S J hJ q hq
+
+end Thorp
+namespace Thorp
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+lemma orbitSet_inverse (p : Equiv.Perm α) (x : α) : orbitSet p⁻¹ x = orbitSet p x := by
+  ext y
+  simp only [mem_orbitSet,Equiv.Perm.sameCycle_inv]
+
+lemma selectedCycle_inverse (p : Equiv.Perm α) (S : Finset α) :
+    Fintype.card (RoutingNetwork.SelectedCycle p⁻¹ S) =
+      Fintype.card (RoutingNetwork.SelectedCycle p S) := by
+  apply Fintype.card_congr
+  refine { toFun := fun C => ⟨C.val,?_,C.property.2⟩
+           invFun := fun C => ⟨C.val,?_,C.property.2⟩
+           left_inv := fun _ => rfl
+           right_inv := fun _ => rfl }
+  · obtain ⟨x,hx⟩ := C.property.1
+    exact ⟨x,hx.trans (orbitSet_inverse p x)⟩
+  · obtain ⟨x,hx⟩ := C.property.1
+    exact ⟨x,hx.trans (orbitSet_inverse p x).symm⟩
+
+lemma orbitSet_conjugate (p g : Equiv.Perm α) (x : α) :
+    orbitSet (g*p*g⁻¹) (g x) = (orbitSet p x).image g := by
+  ext y
+  simp only [mem_orbitSet,Finset.mem_image]
+  rw [Equiv.Perm.sameCycle_conj]
+  change p.SameCycle (g.symm (g x)) (g.symm y) ↔ ∃ z, p.SameCycle x z ∧ g z = y
+  rw [Equiv.symm_apply_apply]
+  constructor
+  · intro h
+    exact ⟨g.symm y,h,g.apply_symm_apply y⟩
+  · rintro ⟨z,hz,rfl⟩
+    simpa only [Equiv.symm_apply_apply] using hz
+
+noncomputable def conjugateCycle (p g : Equiv.Perm α) (S : Finset α)
+    (C : RoutingNetwork.SelectedCycle p S) :
+    RoutingNetwork.SelectedCycle (g*p*g⁻¹) (S.image g) := by
+  refine ⟨C.val.image g,?_,Finset.image_subset_image C.property.2⟩
+  obtain ⟨x,hx⟩ := C.property.1
+  exact ⟨g x,by rw [hx,orbitSet_conjugate]⟩
+
+lemma conjugateCycle_injective (p g : Equiv.Perm α) (S : Finset α) :
+    Function.Injective (conjugateCycle p g S) := by
+  intro C D he
+  apply Subtype.ext
+  exact (Finset.image_injective g.injective) (congrArg Subtype.val he)
+
+lemma selectedCycle_conjugate (p g : Equiv.Perm α) (S : Finset α) :
+    Fintype.card (RoutingNetwork.SelectedCycle (g*p*g⁻¹) (S.image g)) =
+      Fintype.card (RoutingNetwork.SelectedCycle p S) := by
+  apply Nat.le_antisymm
+  · have h := Fintype.card_le_of_injective _
+      (conjugateCycle_injective (g*p*g⁻¹) g⁻¹ (S.image g))
+    have hp : g⁻¹*(g*p*g⁻¹)*g = p := by group
+    rw [inv_inv,hp] at h
+    have hs : (S.image g).image g.symm = S := by
+      ext x
+      simp only [Finset.mem_image]
+      constructor
+      · rintro ⟨y,⟨z,hz,rfl⟩,he⟩
+        have hz' : z = x := by simpa only [Equiv.symm_apply_apply] using he
+        exact hz' ▸ hz
+      · intro hx
+        exact ⟨g x,⟨x,hx,rfl⟩,g.symm_apply_apply x⟩
+    change _ ≤ Fintype.card (RoutingNetwork.SelectedCycle p ((S.image g).image g.symm)) at h
+    rw [hs] at h
+    exact h
+  · exact Fintype.card_le_of_injective _ (conjugateCycle_injective p g S)
+
+lemma alternatingCycles_le_projection {ι : Type*} [Fintype ι]
+    (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    PairRouting.alternatingCycles e p ≤
+      Fintype.card (RoutingNetwork.SelectedCycle (PairRouting.alternatingProjection p)
+        (boolFiber (PairRouting.labelSet e) false)) := by
+  have h := CycleColoring.card_alternating_cycles_le (PairRouting.alternatingProjection p)
+    (boolFiber (PairRouting.labelSet e) false) (boolFiber (PairRouting.labelSet e) true)
+  simpa only [PairRouting.alternatingCycles,childJoin_boolFiber] using h
+
+lemma projectionCycles_forward (X₀ X₁ Y₀ Y₁ : Equiv.Perm α) (S : Finset α) :
+    Fintype.card (RoutingNetwork.SelectedCycle
+      ((Y₀*X₀⁻¹)⁻¹*(Y₁*X₁⁻¹)) S) =
+    Fintype.card (RoutingNetwork.SelectedCycle
+      (X₀⁻¹*X₁*Y₁⁻¹*Y₀) (S.image X₀.symm)) := by
+  let p := (Y₀*X₀⁻¹)⁻¹*(Y₁*X₁⁻¹)
+  calc
+    _ = Fintype.card (RoutingNetwork.SelectedCycle (X₀⁻¹*p*X₀)
+        (S.image X₀.symm)) := by
+      change Fintype.card (RoutingNetwork.SelectedCycle p S) = _
+      have h := (selectedCycle_conjugate p X₀⁻¹ S).symm
+      rw [inv_inv] at h
+      exact h
+    _ = Fintype.card (RoutingNetwork.SelectedCycle (X₀⁻¹*p*X₀)⁻¹
+        (S.image X₀.symm)) := (selectedCycle_inverse _ _).symm
+    _ = _ := by
+      have he : (X₀⁻¹*p*X₀)⁻¹ = X₀⁻¹*X₁*Y₁⁻¹*Y₀ := by dsimp only [p]; group
+      rw [he]
+
+end Thorp
+namespace Thorp
+
+noncomputable def cycleCoefficient (r J : ℕ) (q : ℝ) : ℝ :=
+  Real.log q / J + 2*q*Real.sqrt J/Real.sqrt ((2:ℝ)^r)
+
+lemma cycleCoefficient_nonneg (r J : ℕ) (q : ℝ) (hq : 1 ≤ q) :
+    0 ≤ cycleCoefficient r J q := by
+  unfold cycleCoefficient
+  exact add_nonneg (div_nonneg (Real.log_nonneg hq) (Nat.cast_nonneg _))
+    (div_nonneg (mul_nonneg (by linarith) (Real.sqrt_nonneg _)) (Real.sqrt_nonneg _))
+
+lemma projectionSide_card_le {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι]
+    (e : ι ↪ Bool × α) : (boolFiber (PairRouting.labelSet e) false).card ≤ Fintype.card ι := by
+  have h := card_childJoin (boolFiber (PairRouting.labelSet e) false)
+    (boolFiber (PairRouting.labelSet e) true)
+  rw [childJoin_boolFiber,PairRouting.card_labelSet] at h
+  omega
+
+theorem node_partial_mgf (r s : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Bool × Card (s+r)) (X₀ X₁ Y₁ : Equiv.Perm (Card (s+r)))
+    (lo : Card r × SwitchIndex s → Bool) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(PairRouting.alternatingCycles e (fun b => if b then Y₁*X₁⁻¹ else
+        butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi))*X₀⁻¹))) ≤
+      Real.exp (Fintype.card ι * cycleCoefficient r J q) := by
+  classical
+  let S := boolFiber (PairRouting.labelSet e) false
+  let S' := S.image X₀.symm
+  let post := X₀⁻¹*X₁*Y₁⁻¹
+  let count := fun hi : Card s × SwitchIndex r → Bool =>
+    Fintype.card (RoutingNetwork.SelectedCycle
+      ((Equiv.refl _).trans (butterflyPerm (s+r)
+        (decodeButterfly (s+r) (assembleBits r s lo hi))) |>.trans post) S')
+  have hpoint (hi : Card s × SwitchIndex r → Bool) :
+      PairRouting.alternatingCycles e (fun b => if b then Y₁*X₁⁻¹ else
+        butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi))*X₀⁻¹) ≤ count hi := by
+    refine (alternatingCycles_le_projection e _).trans_eq ?_
+    simp only [PairRouting.alternatingProjection,Bool.false_eq_true,↓reduceIte]
+    change Fintype.card (RoutingNetwork.SelectedCycle
+      ((butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi))*X₀⁻¹)⁻¹ * (Y₁*X₁⁻¹)) S) = count hi
+    exact projectionCycles_forward X₀ X₁ _ Y₁ S
+  have hm := partially_exposed_cycle_log_mgf (Equiv.refl (Card (s+r))) post lo S' J hJ q hq
+  have hp : 0 < finiteMean (fun hi : Card s × SwitchIndex r → Bool => q^(count hi)) :=
+    finiteMean_pos (fun _ => pow_pos (by linarith) _)
+  have hb : finiteMean (fun hi : Card s × SwitchIndex r → Bool => q^(count hi)) ≤
+      Real.exp (S'.card * cycleCoefficient r J q) := by
+    rw [←Real.exp_log hp]
+    apply Real.exp_le_exp.mpr
+    calc
+      _ ≤ _ := hm
+      _ = _ := by unfold cycleCoefficient; ring
+  have hcard : S'.card ≤ Fintype.card ι := by
+    rw [show S'.card = S.card from Finset.card_image_of_injective _ X₀.symm.injective]
+    exact projectionSide_card_le e
+  calc
+    _ ≤ finiteMean (fun hi : Card s × SwitchIndex r → Bool => q^(count hi)) :=
+      finiteMean_mono (fun hi => pow_le_pow_right₀ hq (hpoint hi))
+    _ ≤ Real.exp (S'.card * cycleCoefficient r J q) := hb
+    _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right
+      (by exact_mod_cast hcard) (cycleCoefficient_nonneg r J q hq))
+
+
+def highStepEquiv (r s : ℕ) : (Card s × SwitchIndex (r+1) → Bool) ≃
+    (((Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool)) ×
+      (Card s × Card r → Bool)) where
+  toFun hi := ((fun j => hi (j.1,Sum.inr (false,j.2)),
+    fun j => hi (j.1,Sum.inr (true,j.2))),fun j => hi (j.1,Sum.inl j.2))
+  invFun c := fun j => match j.2 with
+    | Sum.inl y => c.2 (j.1,y)
+    | Sum.inr (b,i) => if b then c.1.2 (j.1,i) else c.1.1 (j.1,i)
+  left_inv hi := by
+    funext j
+    rcases j with ⟨z,y|⟨b,i⟩⟩
+    · rfl
+    · cases b <;> rfl
+  right_inv c := by
+    rcases c with ⟨⟨h₀,h₁⟩,η⟩
+    rfl
+
+lemma assembleBits_step (r s : ℕ) (lo : Card (r+1) × SwitchIndex s → Bool)
+    (c : ((Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool)) ×
+      (Card s × Card r → Bool)) :
+    assembleBits (r+1) s lo ((highStepEquiv r s).symm c) =
+      (coinStepEquiv (s+r)).symm
+        ((assembleBits r s (fun j => lo (Fin.cons false j.1,j.2)) c.1.1,
+          assembleBits r s (fun j => lo (Fin.cons true j.1,j.2)) c.1.2),
+        fun y => c.2 ((cardSplit r s y).2,(cardSplit r s y).1)) := by
+  funext i
+  rcases i with y | ⟨b,i⟩
+  · rfl
+  · cases b <;> rfl
+
+noncomputable def palindromeLevelCost (t : ℕ) : (d : ℕ) → {ι : Type*} → [Fintype ι] →
+    (ι ↪ Card d) → BenesCoins d → ℕ
+  | 0, _, _, _, _ => 0
+  | d+1, _, _, e, ω =>
+      let σ := benesStepEquiv d ω
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x σ.2.1
+      let hc := PairRouting.colors_compatible x σ.2.1
+      if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
+          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) else
+        palindromeLevelCost t d (PairRouting.childEmbedding x c hc false) σ.1.1 +
+          palindromeLevelCost t d (PairRouting.childEmbedding x c hc true) σ.1.2
+
+lemma palindromeLevelCost_step (t d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (d+1)) (X Y : (SwitchIndex d → Bool) × (SwitchIndex d → Bool))
+    (ξ η : Card d → Bool) :
+    palindromeLevelCost t (d+1) e
+      ((coinStepEquiv d).symm (Y,η),(coinStepEquiv d).symm (X,ξ)) =
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x ξ
+      let hc := PairRouting.colors_compatible x ξ
+      if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x ξ)
+        (fun b => palindromePerm d (if b then (Y.2,X.2) else (Y.1,X.1))) else
+        palindromeLevelCost t d (PairRouting.childEmbedding x c hc false) (Y.1,X.1) +
+        palindromeLevelCost t d (PairRouting.childEmbedding x c hc true) (Y.2,X.2) := by
+  rw [palindromeLevelCost]
+  rfl
+
+lemma palindromeLevelCost_gt (t d : ℕ) (h : d < t) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) : palindromeLevelCost t d e ω = 0 := by
+  classical
+  induction d generalizing ι with
+  | zero => rfl
+  | succ d ih =>
+    rw [palindromeLevelCost,ite_eq_right (by omega : t ≠ d+1),ih (by omega),ih (by omega),Nat.zero_add]
+
+
+theorem palindrome_level_partial_mgf (r s u : ℕ) (hu : u < r)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+r)) (X : SwitchIndex (s+r) → Bool)
+    (lo : Card r × SwitchIndex s → Bool) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(palindromeLevelCost (s+u+1) (s+r) e (assembleBits r s lo hi,X))) ≤
+      Real.exp (Fintype.card ι * cycleCoefficient u J q) := by
+  classical
+  induction r generalizing u ι with
+  | zero => omega
+  | succ r ih =>
+    simp only [Nat.add_succ] at *
+    let χ := coinStepEquiv (s+r) X
+    let x := e.trans (headTailEquiv (s+r)).toEmbedding
+    let c := PairRouting.colors x χ.2
+    let hc := PairRouting.colors_compatible x χ.2
+    let e₀ := PairRouting.childEmbedding x c hc false
+    let e₁ := PairRouting.childEmbedding x c hc true
+    let e' := PairRouting.switchedEmbedding x χ.2
+    let lo₀ : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons false j.1,j.2)
+    let lo₁ : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons true j.1,j.2)
+    let Y₀ := fun hi : Card s × SwitchIndex r → Bool => assembleBits r s lo₀ hi
+    let Y₁ := fun hi : Card s × SwitchIndex r → Bool => assembleBits r s lo₁ hi
+    let N := fun hi : (Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool) =>
+      PairRouting.alternatingCycles e' (fun b => if b then palindromePerm (s+r) (Y₁ hi.2,χ.1.2)
+        else palindromePerm (s+r) (Y₀ hi.1,χ.1.1))
+    let L₀ := fun hi : Card s × SwitchIndex r → Bool =>
+      palindromeLevelCost (s+u+1) (s+r) e₀ (Y₀ hi,χ.1.1)
+    let L₁ := fun hi : Card s × SwitchIndex r → Bool =>
+      palindromeLevelCost (s+u+1) (s+r) e₁ (Y₁ hi,χ.1.2)
+    have hs (hi : (Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool))
+        (η : Card s × Card r → Bool) :
+        palindromeLevelCost (s+u+1) (s+r+1) e
+          (assembleBits (r+1) s lo ((highStepEquiv r s).symm (hi,η)),X) =
+          if u = r then N hi else L₀ hi.1 + L₁ hi.2 := by
+      rw [assembleBits_step]
+      conv_lhs => rw [←(coinStepEquiv (s+r)).symm_apply_apply X]
+      rw [palindromeLevelCost_step]
+      simp only [Nat.add_right_cancel_iff,Nat.add_left_cancel_iff]
+      dsimp only [χ,x,c,hc,e₀,e₁,e',lo₀,lo₁,Y₀,Y₁,N,L₀,L₁]
+      congr 2
+      funext b
+      cases b <;> rfl
+    rw [finiteMean_equiv (highStepEquiv r s),finiteMean_prod]
+    simp_rw [hs,finiteMean_const]
+    by_cases hur : u = r
+    · subst u
+      simp only [ite_true]
+      rw [finiteMean_prod,finiteMean_comm]
+      calc
+        _ ≤ finiteMean (fun _hi : Card s × SwitchIndex r → Bool =>
+          Real.exp (Fintype.card ι * cycleCoefficient r J q)) := by
+          apply finiteMean_mono
+          intro hi
+          have hn := node_partial_mgf r s e'
+            (butterflyPerm (s+r) (decodeButterfly (s+r) χ.1.1))
+            (butterflyPerm (s+r) (decodeButterfly (s+r) χ.1.2))
+            (butterflyPerm (s+r) (decodeButterfly (s+r) (Y₁ hi))) lo₀ J hJ q hq
+          exact hn
+        _ = _ := finiteMean_const _
+    · simp only [ite_eq_right hur,pow_add]
+      rw [finiteMean_prod_mul (fun hi => q ^ L₀ hi) (fun hi => q ^ L₁ hi)]
+      have hu' : u < r := by omega
+      have h₀ := ih u hu' e₀ χ.1.1 lo₀
+      have h₁ := ih u hu' e₁ χ.1.2 lo₁
+      have hp := mul_le_mul h₀ h₁ (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
+        (Real.exp_nonneg _)
+      refine hp.trans_eq ?_
+      rw [←Real.exp_add]
+      congr 1
+      have hcount := PairRouting.color_card_add c
+      have hcount' : (Fintype.card {i : ι // c i = false} : ℝ) +
+          Fintype.card {i : ι // c i = true} = Fintype.card ι := by exact_mod_cast hcount
+      change (Fintype.card {i : ι // c i = false} : ℝ) * cycleCoefficient u J q +
+          Fintype.card {i : ι // c i = true} * cycleCoefficient u J q = _
+      rw [←add_mul,hcount']
+
+
+def switchHeight : (d : ℕ) → SwitchIndex d → ℕ
+  | 0, i => nomatch i
+  | d+1, Sum.inl _ => d+1
+  | d+1, Sum.inr (_,i) => switchHeight d i
+
+lemma switchHeight_le (d : ℕ) (i : SwitchIndex d) : switchHeight d i ≤ d := by
+  induction d with
+  | zero => exact Empty.elim i
+  | succ d ih =>
+    rcases i with y | ⟨b,i⟩
+    · exact le_refl _
+    · exact (ih i).trans (Nat.le_succ d)
+
+def AgreeThrough (d s : ℕ) (Y Y' : SwitchIndex d → Bool) : Prop :=
+  ∀ i, switchHeight d i ≤ s → Y i = Y' i
+
+def LayerAdapted (d s : ℕ) {β : Type*} (f : (SwitchIndex d → Bool) → β) : Prop :=
+  ∀ Y Y', AgreeThrough d s Y Y' → f Y = f Y'
+
+lemma agreeThrough_mono {d s t : ℕ} (h : s ≤ t) {Y Y' : SwitchIndex d → Bool}
+    (hh : AgreeThrough d t Y Y') : AgreeThrough d s Y Y' :=
+  fun i hi => hh i (hi.trans h)
+
+lemma layerAdapted_mono {d s t : ℕ} {β : Type*} {f : (SwitchIndex d → Bool) → β}
+    (hf : LayerAdapted d s f) (h : s ≤ t) : LayerAdapted d t f :=
+  fun Y Y' hh => hf Y Y' (agreeThrough_mono h hh)
+
+lemma layerAdapted_comp {d s : ℕ} {α β : Type*} {f : (SwitchIndex d → Bool) → α}
+    (hf : LayerAdapted d s f) (g : α → β) : LayerAdapted d s (g ∘ f) :=
+  fun Y Y' h => congrArg g (hf Y Y' h)
+
+lemma layerAdapted_sum {d s : ℕ} {ι : Type*} (A : Finset ι)
+    (f : ι → (SwitchIndex d → Bool) → ℕ)
+    (hf : ∀ i ∈ A, LayerAdapted d s (f i)) :
+    LayerAdapted d s (fun Y => ∑ i ∈ A, f i Y) := by
+  intro Y Y' h
+  exact Finset.sum_congr rfl (fun i hi => hf i hi Y Y' h)
+
+lemma assembleBits_agree (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
+    (hi hi' : Card s × SwitchIndex r → Bool) :
+    AgreeThrough (s+r) s (assembleBits r s lo hi) (assembleBits r s lo hi') := by
+  induction r with
+  | zero => intro i _; rfl
+  | succ r ih =>
+    intro i h
+    rcases i with y | ⟨b,i⟩
+    · have : s+r+1 ≤ s := h
+      omega
+    · exact ih (fun j => lo (Fin.cons b j.1,j.2))
+        (fun j => hi (j.1,Sum.inr (b,j.2))) (fun j => hi' (j.1,Sum.inr (b,j.2))) i h
+
+lemma palindromeLevelCost_adapted (t d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) :
+    LayerAdapted d (t-1) (fun Y => palindromeLevelCost t d e (Y,X)) := by
+  classical
+  induction d generalizing ι with
+  | zero => intro Y Y' _; rfl
+  | succ d ih =>
+    intro Y Y' h
+    change palindromeLevelCost t (d+1) e (Y,X) = palindromeLevelCost t (d+1) e (Y',X)
+    conv_lhs => rw [←(coinStepEquiv d).symm_apply_apply Y,←(coinStepEquiv d).symm_apply_apply X]
+    conv_rhs => rw [←(coinStepEquiv d).symm_apply_apply Y',←(coinStepEquiv d).symm_apply_apply X]
+    rw [palindromeLevelCost_step,palindromeLevelCost_step]
+    by_cases ht : t = d+1
+    · subst t
+      rw [ite_eq_left rfl,ite_eq_left rfl]
+      have he₀ : (coinStepEquiv d Y).1.1 = (coinStepEquiv d Y').1.1 := by
+        funext i
+        exact h (Sum.inr (false,i)) (by simpa only [switchHeight,Nat.add_sub_cancel] using switchHeight_le d i)
+      have he₁ : (coinStepEquiv d Y).1.2 = (coinStepEquiv d Y').1.2 := by
+        funext i
+        exact h (Sum.inr (true,i)) (by simpa only [switchHeight,Nat.add_sub_cancel] using switchHeight_le d i)
+      rw [he₀,he₁]
+    · rw [ite_eq_right ht,ite_eq_right ht]
+      congr 1
+      · apply ih
+        intro i hi
+        exact h (Sum.inr (false,i)) hi
+      · apply ih
+        intro i hi
+        exact h (Sum.inr (true,i)) hi
+
+lemma adapted_mul_mean_le (r s : ℕ) (F G : (SwitchIndex (s+r) → Bool) → ℝ)
+    (hF : ∀ Y, 0 ≤ F Y) (hA : LayerAdapted (s+r) s F) (B : ℝ)
+    (hG : ∀ lo : Card r × SwitchIndex s → Bool,
+      finiteMean (fun hi : Card s × SwitchIndex r → Bool => G (assembleBits r s lo hi)) ≤ B) :
+    finiteMean (fun Y => F Y*G Y) ≤ finiteMean F * B := by
+  rw [finiteMean_split r s,finiteMean_split r s F,←finiteMean_mul_const]
+  apply finiteMean_mono
+  intro lo
+  let H : Card s × SwitchIndex r → Bool := fun _ => false
+  have hh (hi : Card s × SwitchIndex r → Bool) :
+      F (assembleBits r s lo hi) = F (assembleBits r s lo H) :=
+    hA _ _ (assembleBits_agree r s lo hi H)
+  simp_rw [hh,finiteMean_const]
+  rw [show (fun hi => F (assembleBits r s lo H) * G (assembleBits r s lo hi)) =
+      (fun hi => G (assembleBits r s lo hi) * F (assembleBits r s lo H)) by
+        funext hi; exact mul_comm _ _,finiteMean_mul_const]
+  simpa only [mul_comm] using mul_le_mul_of_nonneg_left (hG lo) (hF _)
+
+
+theorem cycleCoefficient_decay (l r : ℕ) (hr : (l:ℝ)/2 ≤ r)
+    (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ (l:ℝ)/250) :
+    cycleCoefficient r ⌈Real.exp ((l:ℝ)/32)⌉₊ q ≤
+      68 * Real.exp (-(l:ℝ)/64) := by
+  let E := Real.exp ((l:ℝ)/32)
+  let J := ⌈E⌉₊
+  have hl : (0:ℝ) ≤ l := Nat.cast_nonneg _
+  have hE : 1 ≤ E := Real.one_le_exp_iff.mpr (by positivity)
+  have hEpos : 0 < E := Real.exp_pos _
+  have hEJ : E ≤ (J:ℝ) := Nat.le_ceil E
+  have hJ : (1:ℝ) ≤ J := hE.trans hEJ
+  have hJpos : (0:ℝ) < J := lt_of_lt_of_le (by norm_num) hJ
+  have hJE : (J:ℝ) ≤ 2*E := by
+    have h := Nat.ceil_lt_add_one (le_of_lt hEpos)
+    change (J:ℝ) < E+1 at h
+    linarith
+  have hsqJ : Real.sqrt J ≤ (J:ℝ) := by
+    apply Real.sqrt_le_iff.mpr
+    constructor
+    · positivity
+    · nlinarith
+  have hqp : 0 < q := lt_of_lt_of_le (by norm_num) hq
+  have hqE : q ≤ Real.exp ((l:ℝ)/250) := by
+    rw [←Real.exp_log hqp]
+    exact Real.exp_le_exp.mpr hlog
+  have hln : (1/2:ℝ) ≤ Real.log 2 := by
+    have hh := Real.one_sub_inv_le_log_of_pos (by norm_num : (0:ℝ)<2)
+    norm_num at hh ⊢
+    exact hh
+  have hden : Real.exp ((l:ℝ)/8) ≤ Real.sqrt ((2:ℝ)^r) := by
+    have he : (2:ℝ)^r = Real.exp ((r:ℝ)*Real.log 2) := by
+      rw [Real.exp_nat_mul,Real.exp_log (by norm_num)]
+    rw [he,Real.sqrt_eq_rpow,Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp]
+    apply Real.exp_le_exp.mpr
+    have hrn : (0:ℝ) ≤ r := Nat.cast_nonneg _
+    nlinarith
+  have hlog' : Real.log q ≤ (l:ℝ) := by linarith
+  have h₁ : Real.log q / (J:ℝ) ≤ 64*Real.exp (-(l:ℝ)/64) := by
+    calc
+      _ ≤ (l:ℝ)/E := div_le_div₀ (by positivity) hlog' hEpos hEJ
+      _ ≤ (64*Real.exp ((l:ℝ)/64))/E := by
+        apply div_le_div_of_nonneg_right _ (le_of_lt hEpos)
+        have hh := Real.add_one_le_exp ((l:ℝ)/64)
+        linarith
+      _ = _ := by
+        dsimp [E]
+        rw [mul_div_assoc,←Real.exp_sub]
+        congr 2
+        ring
+  have h₂ : 2*q*Real.sqrt J / Real.sqrt ((2:ℝ)^r) ≤
+      4*Real.exp (-(l:ℝ)/64) := by
+    calc
+      _ ≤ (4*Real.exp ((l:ℝ)/250)*E)/Real.exp ((l:ℝ)/8) := by
+        apply div_le_div₀ (by positivity) _ (Real.exp_pos _) hden
+        have hs := hsqJ.trans hJE
+        have hm := mul_le_mul hqE hs (Real.sqrt_nonneg _) (Real.exp_nonneg _)
+        nlinarith
+      _ = 4*Real.exp ((l:ℝ)/250+(l:ℝ)/32-(l:ℝ)/8) := by
+        dsimp [E]
+        rw [mul_assoc,←Real.exp_add,mul_div_assoc,←Real.exp_sub]
+      _ ≤ _ := by
+        apply mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr _) (by norm_num)
+        linarith
+  change Real.log q / (J:ℝ) + 2*q*Real.sqrt J / Real.sqrt ((2:ℝ)^r) ≤ _
+  linarith
+
+lemma cycleCutoff_pos (l : ℕ) : 0 < ⌈Real.exp ((l:ℝ)/32)⌉₊ := by
+  have h := Nat.le_ceil (Real.exp ((l:ℝ)/32))
+  have he := Real.exp_pos ((l:ℝ)/32)
+  have : (0:ℝ) < ⌈Real.exp ((l:ℝ)/32)⌉₊ := lt_of_lt_of_le he h
+  exact_mod_cast this
+
+
+lemma pow_sum_le_mean_powers {ι : Type*} [Fintype ι] [Nonempty ι]
+    (z : ι → ℕ) (q : ℝ) (hq : 0 < q) :
+    q^(∑ i, z i) ≤ finiteMean (fun i => q^(Fintype.card ι*z i)) := by
+  have hn : (Fintype.card ι:ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
+  have hm : finiteMean (fun i => (Fintype.card ι*z i:ℕ)*Real.log q) =
+      (∑ i, z i:ℕ)*Real.log q := by
+    simp only [finiteMean,Nat.cast_mul,Nat.cast_sum,←Finset.mul_sum,←Finset.sum_mul]
+    field_simp
+  have he := exp_finiteMean_le (fun i => (Fintype.card ι*z i:ℕ)*Real.log q)
+  rw [hm,Real.exp_nat_mul,Real.exp_log hq] at he
+  simpa only [Real.exp_nat_mul,Real.exp_log hq] using he
+
+noncomputable def palindromeSlabCost (l d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) : ℕ :=
+  ∑ i : Fin l, palindromeLevelCost (l+i.1+1) d e ω
+
+lemma palindromeSlabCost_adapted (l d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) :
+    LayerAdapted d (2*l) (fun Y => palindromeSlabCost l d e (Y,X)) := by
+  apply layerAdapted_sum
+  intro i _
+  exact layerAdapted_mono (palindromeLevelCost_adapted (l+i.1+1) d e X) (by omega)
+
+theorem palindrome_slab_partial_mgf (r s l : ℕ) (hl : 0 < l) (hs : 2*s ≤ l)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+r)) (X : SwitchIndex (s+r) → Bool)
+    (lo : Card r × SwitchIndex s → Bool) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(palindromeSlabCost l (s+r) e (assembleBits r s lo hi,X))) ≤
+      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
+  classical
+  let : NeZero l := ⟨ne_of_gt hl⟩
+  have hqp : 0 < q := lt_of_lt_of_le (by norm_num) hq
+  calc
+    _ ≤ finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      finiteMean (fun i : Fin l => q^(l*palindromeLevelCost (l+i.1+1) (s+r) e
+        (assembleBits r s lo hi,X)))) := by
+      apply finiteMean_mono
+      intro hi
+      simpa only [palindromeSlabCost,Fintype.card_fin] using
+        pow_sum_le_mean_powers (fun i : Fin l => palindromeLevelCost (l+i.1+1) (s+r) e
+          (assembleBits r s lo hi,X)) q hqp
+    _ = finiteMean (fun i : Fin l => finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(l*palindromeLevelCost (l+i.1+1) (s+r) e (assembleBits r s lo hi,X)))) := finiteMean_comm _
+    _ ≤ finiteMean (fun _i : Fin l =>
+      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64)))) := by
+      apply finiteMean_mono
+      intro i
+      by_cases ht : l+i.1+1 ≤ s+r
+      · have hs' : s ≤ l+i.1 := by omega
+        have he : l+i.1+1 = s+(l+i.1-s)+1 := by omega
+        have hu : l+i.1-s < r := by omega
+        have hm := palindrome_level_partial_mgf r s (l+i.1-s) hu e X lo
+          ⌈Real.exp ((l:ℝ)/32)⌉₊ (cycleCutoff_pos l) (q^l) (one_le_pow₀ hq)
+        have hc := cycleCoefficient_decay l (l+i.1-s) (by
+          rw [Nat.cast_sub hs',Nat.cast_add]
+          have hsr : (2:ℝ)*s ≤ l := by exact_mod_cast hs
+          have hi : (0:ℝ) ≤ i.1 := Nat.cast_nonneg _
+          linarith) (q^l) (one_le_pow₀ hq) (by
+            rw [Real.log_pow]
+            nlinarith [mul_le_mul_of_nonneg_left hlog (Nat.cast_nonneg l)])
+        simp_rw [he,pow_mul]
+        exact hm.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left hc (Nat.cast_nonneg _)))
+      · have hz := palindromeLevelCost_gt (l+i.1+1) (s+r) (by omega) e
+        simp only [hz,Nat.mul_zero,pow_zero,finiteMean_const]
+        exact Real.one_le_exp_iff.mpr (by positivity)
+    _ = _ := finiteMean_const _
+
+
+lemma palindromeSlabCost_gt (l d : ℕ) (h : d ≤ l) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) : palindromeSlabCost l d e ω = 0 := by
+  apply Finset.sum_eq_zero
+  intro i _
+  exact palindromeLevelCost_gt _ _ (by omega) _ _
+
+lemma palindrome_slab_adapted_mul_le (d s l : ℕ) (hsd : s ≤ d) (hl : 0 < l)
+    (hs : 2*s ≤ l) {ι : Type*} [Fintype ι] (e : ι ↪ Card d)
+    (X : SwitchIndex d → Bool) (F : (SwitchIndex d → Bool) → ℝ)
+    (hF : ∀ Y, 0 ≤ F Y) (hA : LayerAdapted d s F) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun Y => F Y*q^(palindromeSlabCost l d e (Y,X))) ≤
+      finiteMean F * Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
+  obtain ⟨r,rfl⟩ := Nat.exists_eq_add_of_le hsd
+  exact adapted_mul_mean_le r s F _ hF hA _
+    (fun lo => palindrome_slab_partial_mgf r s l hl hs e X lo q hq hlog)
+
+def familyEnd (L n : ℕ) : ℕ := if n = 0 then 0 else 2*(L*4^(n-1))
+
+end Thorp
+
+end ThorpNine.Harmonic
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Harmonic/ExposedCoins.lean b/lean/OAI/Probability/ThorpRouting/Harmonic/ExposedCoins.lean
index 0e6119cbee9094f78bf87034309919e42e58ccc2..5e828b5a5ec55777a656ccb21d325be4aa71ddc0 100644
--- a/lean/OAI/Probability/ThorpRouting/Harmonic/ExposedCoins.lean
+++ b/lean/OAI/Probability/ThorpRouting/Harmonic/ExposedCoins.lean
@@ -1,860 +1,860 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Harmonic.Coins
-
-namespace OAI
-
-namespace ThorpNine.Harmonic
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-
-lemma card_routeDomain (d : ℕ) (x y : Card d) : (routeDomain d x y).card = d := by
-  classical
-  induction d with
-  | zero => rfl
-  | succ d ih =>
-    change (insert (Sum.inl (Fin.tail y))
-      ((routeDomain d (Fin.tail x) (Fin.tail y)).image
-        (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) :
-      Finset (Sum (Card d) (Bool × SwitchIndex d))).card = d + 1
-    rw [Finset.card_insert_of_notMem (by simp)]
-    rw [Finset.card_image_of_injective _ (by intro i j h; cases h; rfl), ih]
-
-lemma xor_eq_iff (a b c : Bool) : Bool.xor a b = c ↔ b = Bool.xor a c := by
-  cases a <;> cases b <;> cases c <;> decide
-
-lemma butterfly_endpoint_iff (d : ℕ) (ω : SwitchIndex d → Bool) (x y : Card d) :
-    butterflyPerm d (decodeButterfly d ω) x = y ↔
-      ∀ i ∈ routeDomain d x y, ω i = routeValue d x y i := by
-  classical
-  induction d with
-  | zero =>
-    have he : butterflyPerm 0 (decodeButterfly 0 ω) x = y := Subsingleton.elim _ _
-    simp [he, routeDomain]
-  | succ d ih =>
-    let v := butterflyPerm d (decodeButterfly d (fun i => ω (Sum.inr (x 0,i)))) (Fin.tail x)
-    change Fin.cons (Bool.xor (x 0) (ω (Sum.inl v))) v = y ↔ _
-    have hv : v = Fin.tail y ↔
-        ∀ i ∈ routeDomain d (Fin.tail x) (Fin.tail y),
-          ω (Sum.inr (x 0,i)) = routeValue d (Fin.tail x) (Fin.tail y) i :=
-      ih _ _ _
-    constructor
-    · intro h i hi
-      have ht : v = Fin.tail y := by simpa using congrArg Fin.tail h
-      have hh : ω (Sum.inl (Fin.tail y)) = Bool.xor (x 0) (y 0) := by
-        have he := congrFun h 0
-        simp only [Fin.cons_zero, ht] at he
-        exact (xor_eq_iff _ _ _).mp he
-      rcases Finset.mem_insert.mp hi with htop | hchild
-      · subst i
-        exact hh
-      · obtain ⟨j, hj, rfl⟩ := Finset.mem_image.mp hchild
-        exact hv.mp ht j hj
-    · intro h
-      have hh := h (Sum.inl (Fin.tail y)) (Finset.mem_insert_self _ _)
-      have ht : v = Fin.tail y := hv.mpr (fun i hi =>
-        h (Sum.inr (x 0,i)) (Finset.mem_insert_of_mem (Finset.mem_image_of_mem _ hi)))
-      have hh' : Bool.xor (x 0) (ω (Sum.inl (Fin.tail y))) = y 0 :=
-        (xor_eq_iff _ _ _).mpr hh
-      rw [ht, hh', Fin.cons_self_tail]
-
-lemma finiteMean_congr {Ω : Type*} [Fintype Ω] {f g : Ω → ℝ}
-    (h : ∀ ω, f ω = g ω) : finiteMean f = finiteMean g := by
-  unfold finiteMean
-  congr 1
-  exact Finset.sum_congr rfl (fun ω _ => h ω)
-
-lemma mean_coinCylinder {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (s : Finset ι) (v : ι → Bool) :
-    finiteMean (fun ω : ι → Bool => if ∀ i ∈ s, ω i = v i then (1 : ℝ) else 0) =
-      1 / (2 : ℝ) ^ s.card := by
-  induction s using Finset.induction_on with
-  | empty => simp [finiteMean]
-  | @insert i s hi ih =>
-    let f := fun ω : ι → Bool => if ∀ j ∈ s, ω j = v j then (1 : ℝ) else 0
-    have hf : ∀ ω, f (QueryTree.flipCoin i ω) = f ω := by
-      intro ω
-      have he : (∀ j ∈ s, QueryTree.flipCoin i ω j = v j) ↔ ∀ j ∈ s, ω j = v j := by
-        apply forall₂_congr
-        intro j hj
-        rw [QueryTree.flipCoin_apply_ne i j (fun h => hi (h ▸ hj))]
-      dsimp [f]
-      simp only [he]
-    have hm := QueryTree.mean_indicator_bit f i (v i) hf
-    have he : (fun ω : ι → Bool => if ∀ j ∈ insert i s, ω j = v j then (1 : ℝ) else 0) =
-        (fun ω => if ω i = v i then f ω else 0) := by
-      funext ω
-      by_cases h : ω i = v i <;> simp [f, h]
-    rw [he, hm]
-    change finiteMean (fun ω : ι → Bool => if ∀ j ∈ s, ω j = v j then (1 : ℝ) else 0) / 2 = _
-    rw [ih, Finset.card_insert_of_notMem hi, pow_succ]
-    ring
-
-def exposedCoins {ι : Type*} [DecidableEq ι] (R : Finset ι) (η ω : ι → Bool) : ι → Bool :=
-  fun i => if i ∈ R then η i else ω i
-
-lemma mean_coinCylinder_exposed {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (s R : Finset ι) (v η : ι → Bool) :
-    finiteMean (fun ω : ι → Bool =>
-      if ∀ i ∈ s, exposedCoins R η ω i = v i then (1 : ℝ) else 0) =
-    if (∀ i ∈ s ∩ R, η i = v i) then 1 / (2 : ℝ) ^ (s \ R).card else 0 := by
-  have he (ω : ι → Bool) : (∀ i ∈ s, exposedCoins R η ω i = v i) ↔
-      (∀ i ∈ s ∩ R, η i = v i) ∧ (∀ i ∈ s \ R, ω i = v i) := by
-    constructor
-    · intro h
-      constructor
-      · intro i hi
-        simpa [exposedCoins, (Finset.mem_inter.mp hi).2] using h i (Finset.mem_inter.mp hi).1
-      · intro i hi
-        simpa [exposedCoins, (Finset.mem_sdiff.mp hi).2] using h i (Finset.mem_sdiff.mp hi).1
-    · rintro ⟨ho, hf⟩ i hi
-      by_cases hR : i ∈ R
-      · simpa [exposedCoins, hR] using ho i (Finset.mem_inter.mpr ⟨hi,hR⟩)
-      · simpa [exposedCoins, hR] using hf i (Finset.mem_sdiff.mpr ⟨hi,hR⟩)
-  have hm : finiteMean (fun ω : ι → Bool =>
-      if ∀ i ∈ s, exposedCoins R η ω i = v i then (1 : ℝ) else 0) =
-      finiteMean (fun ω : ι → Bool =>
-        if (∀ i ∈ s ∩ R, η i = v i) ∧ (∀ i ∈ s \ R, ω i = v i) then (1 : ℝ) else 0) :=
-    finiteMean_congr (fun ω => by simp only [he ω])
-  rw [hm]
-  by_cases h : ∀ i ∈ s ∩ R, η i = v i
-  · have hp : (∀ i ∈ s ∩ R, η i = v i) ↔ True := iff_true_intro h
-    simp only [hp, true_and, ↓reduceIte]
-    exact mean_coinCylinder (s \ R) v
-  · have hp : (∀ i ∈ s ∩ R, η i = v i) ↔ False := iff_false_intro h
-    simp only [hp, false_and, ↓reduceIte]
-    simp [finiteMean]
-
-lemma butterfly_closure_probability (d : ℕ) (x y : Card d)
-    (R : Finset (SwitchIndex d)) (η : SwitchIndex d → Bool) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      if butterflyPerm d (decodeButterfly d (exposedCoins R η ω)) x = y then (1 : ℝ) else 0) =
-      if (∀ i ∈ routeDomain d x y ∩ R, η i = routeValue d x y i) then
-        (2 : ℝ) ^ (routeDomain d x y ∩ R).card / (2 : ℝ) ^ d else 0 := by
-  have he : finiteMean (fun ω : SwitchIndex d → Bool =>
-      if butterflyPerm d (decodeButterfly d (exposedCoins R η ω)) x = y then (1 : ℝ) else 0) =
-      finiteMean (fun ω : SwitchIndex d → Bool =>
-        if ∀ i ∈ routeDomain d x y, exposedCoins R η ω i = routeValue d x y i then (1 : ℝ) else 0) :=
-    finiteMean_congr (fun ω => by simp only [butterfly_endpoint_iff])
-  rw [he, mean_coinCylinder_exposed]
-  congr 1
-  have hc : (routeDomain d x y \ R).card + (routeDomain d x y ∩ R).card = d := by
-    rw [Finset.card_sdiff_add_card_inter, card_routeDomain]
-  have hp : (2 : ℝ) ^ d = (2 : ℝ) ^ (routeDomain d x y \ R).card *
-      (2 : ℝ) ^ (routeDomain d x y ∩ R).card := by
-    rw [← pow_add]
-    exact congrArg (fun n => (2 : ℝ) ^ n) hc.symm
-  rw [hp]
-  have hn : (2 : ℝ) ^ (routeDomain d x y ∩ R).card ≠ 0 := pow_ne_zero _ (by norm_num)
-  field_simp
-
-lemma butterfly_single_card_uniform (d : ℕ) (x y : Card d) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      if butterflyPerm d (decodeButterfly d ω) x = y then (1 : ℝ) else 0) = 1 / (2 : ℝ) ^ d := by
-  have he : finiteMean (fun ω : SwitchIndex d → Bool =>
-      if butterflyPerm d (decodeButterfly d ω) x = y then (1 : ℝ) else 0) =
-      finiteMean (fun ω : SwitchIndex d → Bool =>
-        if ∀ i ∈ routeDomain d x y, ω i = routeValue d x y i then (1 : ℝ) else 0) :=
-    finiteMean_congr (fun ω => by simp only [butterfly_endpoint_iff])
-  rw [he, mean_coinCylinder, card_routeDomain]
-
-def switchUsers : (d : ℕ) → Butterfly d → SwitchIndex d → Card d × Card d
-  | 0, _, i => nomatch i
-  | d + 1, b, Sum.inl y =>
-      (Fin.cons false ((butterflyPerm d (b.2 false)).symm y),
-       Fin.cons true ((butterflyPerm d (b.2 true)).symm y))
-  | d + 1, b, Sum.inr (ε,i) =>
-      (Fin.cons ε (switchUsers d (b.2 ε) i).1,
-       Fin.cons ε (switchUsers d (b.2 ε) i).2)
-
-lemma switchUsers_ne (d : ℕ) (b : Butterfly d) (i : SwitchIndex d) :
-    (switchUsers d b i).1 ≠ (switchUsers d b i).2 := by
-  induction d with
-  | zero => exact Empty.elim i
-  | succ d ih =>
-    cases i with
-    | inl y =>
-      intro h
-      have := congrFun h 0
-      simp only [switchUsers, Fin.cons_zero] at this
-      exact Bool.false_ne_true this
-    | inr z =>
-      intro h
-      exact ih (b.2 z.1) z.2 (by simpa only [switchUsers, Fin.tail_cons] using congrArg Fin.tail h)
-
-lemma cons_eq_iff {d : ℕ} (ε : Bool) (a : Card d) (x : Card (d + 1)) :
-    Fin.cons ε a = x ↔ ε = x 0 ∧ a = Fin.tail x := by
-  constructor
-  · intro h
-    exact ⟨by simpa only [Fin.cons_zero] using congrFun h 0,
-      by simpa only [Fin.tail_cons] using congrArg Fin.tail h⟩
-  · rintro ⟨rfl,rfl⟩
-    exact Fin.cons_self_tail x
-
-lemma mem_routeDomain_iff_user (d : ℕ) (b : Butterfly d) (x : Card d) (i : SwitchIndex d) :
-    i ∈ routeDomain d x (butterflyPerm d b x) ↔
-      (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x := by
-  classical
-  induction d with
-  | zero => exact Empty.elim i
-  | succ d ih =>
-    cases i with
-    | inl y =>
-      change Sum.inl y ∈ insert (Sum.inl (Fin.tail (butterflyPerm (d+1) b x)))
-        ((routeDomain d (Fin.tail x) (Fin.tail (butterflyPerm (d+1) b x))).image
-          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) ↔ _
-      simp only [Finset.mem_insert, Sum.inl.injEq, Finset.mem_image,
-        Sum.inr_ne_inl, and_false, exists_false, or_false]
-      change y = butterflyPerm d (b.2 (x 0)) (Fin.tail x) ↔ _
-      simp only [switchUsers, cons_eq_iff]
-      cases x 0 <;> simp [Equiv.symm_apply_eq]
-    | inr z =>
-      obtain ⟨ε,i⟩ := z
-      change Sum.inr (ε,i) ∈ insert (Sum.inl (Fin.tail (butterflyPerm (d+1) b x)))
-        ((routeDomain d (Fin.tail x) (Fin.tail (butterflyPerm (d+1) b x))).image
-          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) ↔ _
-      simp only [Finset.mem_insert, Sum.inr_ne_inl, false_or, Finset.mem_image,
-        Sum.inr.injEq, Prod.mk.injEq]
-      change (∃ j, j ∈ routeDomain d (Fin.tail x) (butterflyPerm d (b.2 (x 0)) (Fin.tail x)) ∧
-        x 0 = ε ∧ j = i) ↔ _
-      simp only [switchUsers, cons_eq_iff]
-      constructor
-      · rintro ⟨j,hj,hε,rfl⟩
-        subst ε
-        rcases (ih (b.2 (x 0)) (Fin.tail x) j).mp hj with h | h
-        · exact Or.inl ⟨rfl,h⟩
-        · exact Or.inr ⟨rfl,h⟩
-      · rintro (⟨hε,h⟩ | ⟨hε,h⟩)
-        · subst ε
-          exact ⟨i,(ih (b.2 (x 0)) (Fin.tail x) i).mpr (Or.inl h),rfl,rfl⟩
-        · subst ε
-          exact ⟨i,(ih (b.2 (x 0)) (Fin.tail x) i).mpr (Or.inr h),rfl,rfl⟩
-
-lemma card_switches_using (d : ℕ) (b : Butterfly d) (x : Card d) :
-    (Finset.univ.filter (fun i : SwitchIndex d =>
-      (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x)).card = d := by
-  classical
-  have he : Finset.univ.filter (fun i : SwitchIndex d =>
-      (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x) =
-      routeDomain d x (butterflyPerm d b x) := by
-    ext i
-    simp only [Finset.mem_filter, Finset.mem_univ, true_and, mem_routeDomain_iff_user]
-  rw [he,card_routeDomain]
-
-lemma switch_incidence_sum (d : ℕ) (b : Butterfly d) (w : Card d → ℝ) :
-    ∑ i : SwitchIndex d, (w (switchUsers d b i).1 + w (switchUsers d b i).2) =
-      (d : ℝ) * ∑ x : Card d, w x := by
-  classical
-  calc
-    _ = ∑ i : SwitchIndex d, ∑ x : Card d,
-        if (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x then w x else 0 := by
-      apply Finset.sum_congr rfl
-      intro i _
-      rw [Finset.sum_ite]
-      simp only [Finset.sum_const_zero, add_zero]
-      have he : Finset.univ.filter (fun x : Card d =>
-          (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x) =
-          {(switchUsers d b i).1, (switchUsers d b i).2} := by
-        ext x; simp [eq_comm]
-      rw [he, Finset.sum_pair (switchUsers_ne d b i)]
-    _ = ∑ x : Card d, ∑ i : SwitchIndex d,
-        if (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x then w x else 0 :=
-      Finset.sum_comm
-    _ = ∑ x : Card d, (d : ℝ) * w x := by
-      apply Finset.sum_congr rfl
-      intro x _
-      rw [Finset.sum_ite]
-      simp only [Finset.sum_const_zero, add_zero, Finset.sum_const, nsmul_eq_mul,
-        card_switches_using]
-    _ = _ := by rw [Finset.mul_sum]
-
-lemma weighted_switch_min_bound (d : ℕ) (b : Butterfly d) (w : Card d → ℝ) :
-    ∑ i : SwitchIndex d, min (w (switchUsers d b i).1) (w (switchUsers d b i).2) ≤
-      (d : ℝ) / 2 * ∑ x : Card d, w x := by
-  have h : 2 * (∑ i : SwitchIndex d,
-      min (w (switchUsers d b i).1) (w (switchUsers d b i).2)) ≤
-      ∑ i : SwitchIndex d, (w (switchUsers d b i).1 + w (switchUsers d b i).2) := by
-    rw [Finset.mul_sum]
-    apply Finset.sum_le_sum
-    intro i _
-    linarith [min_le_left (w (switchUsers d b i).1) (w (switchUsers d b i).2),
-      min_le_right (w (switchUsers d b i).1) (w (switchUsers d b i).2)]
-  rw [switch_incidence_sum] at h
-  linarith
-
-lemma mem_childMarks {d : ℕ} (s : Finset (Card (d + 1))) (ε : Bool) (x : Card d) :
-    x ∈ childMarks s ε ↔ Fin.cons ε x ∈ s := by
-  classical
-  constructor
-  · intro h
-    obtain ⟨y,hy,hxy⟩ := Finset.mem_image.mp h
-    obtain ⟨hys,hyε⟩ := Finset.mem_filter.mp hy
-    rw [← hxy, ← hyε, Fin.cons_self_tail]
-    exact hys
-  · intro h
-    exact Finset.mem_image.mpr ⟨Fin.cons ε x,
-      Finset.mem_filter.mpr ⟨h,by simp⟩,by simp⟩
-
-lemma mem_topMeetings {d : ℕ} (b : Butterfly (d + 1))
-    (s : Finset (Card (d + 1))) (y : Card d) :
-    y ∈ topMeetings b s ↔
-      (switchUsers (d+1) b (Sum.inl y)).1 ∈ s ∧
-      (switchUsers (d+1) b (Sum.inl y)).2 ∈ s := by
-  classical
-  have he (ε : Bool) : y ∈ (childMarks s ε).image (butterflyPerm d (b.2 ε)) ↔
-      (butterflyPerm d (b.2 ε)).symm y ∈ childMarks s ε := by
-    constructor
-    · intro h
-      obtain ⟨x,hx,rfl⟩ := Finset.mem_image.mp h
-      simpa only [Equiv.symm_apply_apply] using hx
-    · intro h
-      exact Finset.mem_image.mpr ⟨_,h,Equiv.apply_symm_apply _ _⟩
-  simp only [topMeetings, Finset.mem_inter, he, mem_childMarks, switchUsers]
-
-lemma encounters_eq_switch_sum (d : ℕ) (b : Butterfly d) (s : Finset (Card d)) :
-    encounters d b s = ∑ i : SwitchIndex d,
-      if (switchUsers d b i).1 ∈ s ∧ (switchUsers d b i).2 ∈ s then 1 else 0 := by
-  classical
-  induction d with
-  | zero =>
-    simp only [encounters]
-    symm
-    apply Finset.sum_eq_zero
-    intro i _
-    exact Empty.elim i
-  | succ d ih =>
-    change encounters d (b.2 false) (childMarks s false) +
-      encounters d (b.2 true) (childMarks s true) + (topMeetings b s).card =
-        ∑ i : Sum (Card d) (Bool × SwitchIndex d), _
-    rw [Fintype.sum_sum_type, Fintype.sum_prod_type]
-    simp only [Fintype.sum_bool]
-    rw [ih,ih]
-    have ht : (∑ y : Card d,
-        if (switchUsers (d+1) b (Sum.inl y)).1 ∈ s ∧
-          (switchUsers (d+1) b (Sum.inl y)).2 ∈ s then 1 else 0) =
-        (topMeetings b s).card := by
-      simp only [← mem_topMeetings]
-      simp
-    rw [ht]
-    simp only [mem_childMarks, switchUsers]
-    ac_rfl
-
-lemma marked_switch_entropy_bound (d : ℕ) (b : Butterfly d) (s : Finset (Card d)) :
-    (∑ i : SwitchIndex d,
-      if (switchUsers d b i).1 ∈ s ∧ (switchUsers d b i).2 ∈ s then (1:ℝ) else 0) ≤
-      (s.card : ℝ) * Real.log s.card / (2 * Real.log 2) := by
-  classical
-  have he : (encounters d b s : ℝ) =
-      ∑ i : SwitchIndex d,
-        if (switchUsers d b i).1 ∈ s ∧ (switchUsers d b i).2 ∈ s then (1:ℝ) else 0 := by
-    rw [encounters_eq_switch_sum, Nat.cast_sum]
-    apply Finset.sum_congr rfl
-    intro i _
-    split_ifs <;> simp
-  rw [← he]
-  exact butterfly_encounters d b s
-
-namespace QueryTree
-variable {ι : Type*} [DecidableEq ι]
-
-def memo : QueryTree ι → Finset ι → (ι → Bool) → QueryTree ι
-  | reject, _, _ => reject
-  | accept, _, _ => accept
-  | branch i zero one, R, η =>
-      if i ∈ R then (if η i then memo one R η else memo zero R η) else
-      branch i (memo zero (insert i R) (Function.update η i false))
-        (memo one (insert i R) (Function.update η i true))
-  | force i b next, R, η =>
-      if i ∈ R then (if η i = b then memo next R η else reject) else
-      force i b (memo next (insert i R) (Function.update η i b))
-
-lemma memo_queries_disjoint (T : QueryTree ι) (R : Finset ι) (η : ι → Bool) :
-    Disjoint (memo T R η).queries R := by
-  induction T generalizing R η with
-  | reject => simp [memo,queries]
-  | accept => simp [memo,queries]
-  | branch i zero one ih₀ ih₁ =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte]
-      cases η i <;> simp only [Bool.false_eq_true,↓reduceIte]
-      · exact ih₀ R η
-      · exact ih₁ R η
-    · simp only [memo,hi,↓reduceIte,queries,Finset.disjoint_insert_left,Finset.disjoint_union_left]
-      exact ⟨by simp, (ih₀ (insert i R) _).mono_right (Finset.subset_insert _ _),
-        (ih₁ (insert i R) _).mono_right (Finset.subset_insert _ _)⟩
-  | force i b next ih =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte]
-      by_cases hb : η i = b
-      · simpa only [hb,↓reduceIte] using ih R η
-      · simp [hb,queries]
-    · simp only [memo,hi,↓reduceIte,queries,Finset.disjoint_insert_left]
-      exact ⟨by simp,(ih (insert i R) _).mono_right (Finset.subset_insert _ _)⟩
-
-lemma memo_fresh (T : QueryTree ι) (R : Finset ι) (η : ι → Bool) :
-    (memo T R η).Fresh := by
-  induction T generalizing R η with
-  | reject => trivial
-  | accept => trivial
-  | branch i zero one ih₀ ih₁ =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte]
-      cases η i <;> simp only [Bool.false_eq_true,↓reduceIte]
-      · exact ih₀ R η
-      · exact ih₁ R η
-    · simp only [memo,hi,↓reduceIte,Fresh]
-      exact ⟨fun hm => Finset.disjoint_left.mp (memo_queries_disjoint zero (insert i R) _) hm
-          (Finset.mem_insert_self _ _),
-        fun hm => Finset.disjoint_left.mp (memo_queries_disjoint one (insert i R) _) hm
-          (Finset.mem_insert_self _ _), ih₀ _ _,ih₁ _ _⟩
-  | force i b next ih =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte]
-      by_cases hb : η i = b
-      · simpa only [hb,↓reduceIte] using ih R η
-      · simp [hb,Fresh]
-    · simp only [memo,hi,↓reduceIte,Fresh]
-      exact ⟨fun hm => Finset.disjoint_left.mp (memo_queries_disjoint next (insert i R) _) hm
-        (Finset.mem_insert_self _ _),ih _ _⟩
-
-lemma update_agrees {R : Finset ι} {η ω : ι → Bool}
-    (h : ∀ i ∈ R, η i = ω i) (i : ι) :
-    ∀ j ∈ insert i R, Function.update η i (ω i) j = ω j := by
-  intro j hj
-  rcases Finset.mem_insert.mp hj with rfl | hj
-  · simp
-  · by_cases hji : j = i
-    · subst j; simp
-    · rw [Function.update_of_ne hji]
-      exact h j hj
-
-lemma memo_succeeds (T : QueryTree ι) (R : Finset ι) (η ω : ι → Bool)
-    (h : ∀ i ∈ R, η i = ω i) : (memo T R η).succeeds ω = T.succeeds ω := by
-  induction T generalizing R η with
-  | reject => rfl
-  | accept => rfl
-  | branch i zero one ih₀ ih₁ =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte,succeeds,h i hi]
-      cases ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
-      · exact ih₀ R η h
-      · exact ih₁ R η h
-    · simp only [memo,hi,↓reduceIte,succeeds]
-      cases hb : ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
-      · exact ih₀ _ _ (by simpa only [hb] using update_agrees h i)
-      · exact ih₁ _ _ (by simpa only [hb] using update_agrees h i)
-  | force i b next ih =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte,h i hi]
-      by_cases hb : ω i = b
-      · simp only [hb,↓reduceIte,succeeds]
-        exact ih R η h
-      · simp [hb,succeeds]
-    · simp only [memo,hi,↓reduceIte,succeeds]
-      by_cases hb : ω i = b
-      · simp only [hb,↓reduceIte]
-        exact ih _ _ (by simpa only [hb] using update_agrees h i)
-      · simp [hb]
-
-def freshCharge : QueryTree ι → Finset ι → (ι → Bool) → ℕ
-  | reject, _, _ => 0
-  | accept, _, _ => 0
-  | branch i zero one, R, ω => if ω i then freshCharge one (insert i R) ω else freshCharge zero (insert i R) ω
-  | force i b next, R, ω => if ω i = b then
-      freshCharge next (insert i R) ω + (if i ∈ R then 0 else 1) else 0
-
-lemma memo_charge (T : QueryTree ι) (R : Finset ι) (η ω : ι → Bool)
-    (h : ∀ i ∈ R, η i = ω i) : (memo T R η).charge ω = freshCharge T R ω := by
-  induction T generalizing R η with
-  | reject => rfl
-  | accept => rfl
-  | branch i zero one ih₀ ih₁ =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte,freshCharge,h i hi,Finset.insert_eq_of_mem hi]
-      cases ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
-      · exact ih₀ R η h
-      · exact ih₁ R η h
-    · simp only [memo,hi,↓reduceIte,charge,freshCharge]
-      cases hb : ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
-      · exact ih₀ _ _ (by simpa only [hb] using update_agrees h i)
-      · exact ih₁ _ _ (by simpa only [hb] using update_agrees h i)
-  | force i b next ih =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte,h i hi,freshCharge,Finset.insert_eq_of_mem hi,Nat.add_zero]
-      by_cases hb : ω i = b
-      · simp only [hb,↓reduceIte]
-        exact ih R η h
-      · simp [hb,charge]
-    · simp only [memo,hi,↓reduceIte,charge,freshCharge]
-      by_cases hb : ω i = b
-      · simp only [hb,↓reduceIte]
-        rw [ih _ _ (by simpa only [hb] using update_agrees h i)]
-      · simp [hb]
-
-lemma mean_freshCharge_le_one [Fintype ι] (T : QueryTree ι) :
-    finiteMean (fun ω : ι → Bool =>
-      if T.succeeds ω then (2:ℝ) ^ T.freshCharge ∅ ω else 0) ≤ 1 := by
-  let η : ι → Bool := fun _ => false
-  have hm := mean_weighted_le_one (memo T ∅ η) (memo_fresh T ∅ η)
-  have he : finiteMean (memo T ∅ η).weighted =
-      finiteMean (fun ω : ι → Bool =>
-        if T.succeeds ω then (2:ℝ) ^ T.freshCharge ∅ ω else 0) := by
-    apply finiteMean_congr
-    intro ω
-    rw [weighted_eq, memo_succeeds _ _ _ _ (by simp), memo_charge _ _ _ _ (by simp)]
-  rw [he] at hm
-  exact hm
-
-
-def followPath : (d : ℕ) → (SwitchIndex d → ι) → Card d →
-    (Card d → QueryTree ι) → QueryTree ι
-  | 0, _, x, k => k x
-  | d + 1, e, x, k => followPath d (fun i => e (Sum.inr (x 0,i))) (Fin.tail x)
-      (fun y => branch (e (Sum.inl y))
-        (k (Fin.cons (Bool.xor (x 0) false) y))
-        (k (Fin.cons (Bool.xor (x 0) true) y)))
-
-omit [DecidableEq ι] in
-lemma followPath_succeeds (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
-    (k : Card d → QueryTree ι) (ω : ι → Bool) :
-    (followPath d e x k).succeeds ω =
-      (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).succeeds ω := by
-  induction d with
-  | zero => rfl
-  | succ d ih =>
-    rw [followPath, ih]
-    let y := butterflyPerm d
-      (decodeButterfly d (fun i => ω (e (Sum.inr (x 0,i))))) (Fin.tail x)
-    change (if ω (e (Sum.inl y)) then
-      (k (Fin.cons (Bool.xor (x 0) true) y)).succeeds ω else
-      (k (Fin.cons (Bool.xor (x 0) false) y)).succeeds ω) =
-      (k (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).succeeds ω
-    cases ω (e (Sum.inl y)) <;> rfl
-
-lemma followPath_charge (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
-    (k : Card d → QueryTree ι) (R : Finset ι) (ω : ι → Bool) :
-    freshCharge (followPath d e x k) R ω =
-      freshCharge (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x))
-        (R ∪ (routeDomain d x (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).image e) ω := by
-  classical
-  induction d generalizing R with
-  | zero => simp only [followPath, routeDomain, Finset.image_empty, Finset.union_empty]; rfl
-  | succ d ih =>
-    rw [followPath, ih]
-    let y := butterflyPerm d
-      (decodeButterfly d (fun i => ω (e (Sum.inr (x 0,i))))) (Fin.tail x)
-    let S := (routeDomain d (Fin.tail x) y).image (fun i => e (Sum.inr (x 0,i)))
-    change (if ω (e (Sum.inl y)) then
-      freshCharge (k (Fin.cons (Bool.xor (x 0) true) y)) (insert (e (Sum.inl y)) (R ∪ S)) ω else
-      freshCharge (k (Fin.cons (Bool.xor (x 0) false) y)) (insert (e (Sum.inl y)) (R ∪ S)) ω) =
-      freshCharge (k (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y))
-        (R ∪ (routeDomain (d+1) x
-          (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).image e) ω
-    have hs : (routeDomain (d+1) x
-        (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).image e =
-        insert (e (Sum.inl y)) S := by
-      let e' : Sum (Card d) (Bool × SwitchIndex d) → ι := e
-      change (insert (Sum.inl y)
-        ((routeDomain d (Fin.tail x) y).image
-          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) :
-          Finset (Sum (Card d) (Bool × SwitchIndex d))).image e' = _
-      rw [Finset.image_insert, Finset.image_image]
-      rfl
-    rw [hs,Finset.union_insert]
-    cases ω (e (Sum.inl y)) <;> rfl
-
-def forceList : List ι → (ι → Bool) → QueryTree ι → QueryTree ι
-  | [], _, k => k
-  | i :: L, v, k => force i (v i) (forceList L v k)
-
-omit [DecidableEq ι] in
-lemma forceList_succeeds (L : List ι) (v : ι → Bool) (k : QueryTree ι) (ω : ι → Bool) :
-    (forceList L v k).succeeds ω =
-      if ∀ i ∈ L, ω i = v i then k.succeeds ω else false := by
-  induction L with
-  | nil => simp [forceList]
-  | cons i L ih =>
-    simp only [forceList,succeeds,ih,List.mem_cons,forall_eq_or_imp]
-    by_cases h : ω i = v i <;> simp [h]
-
-lemma forceList_charge (L : List ι) (v : ι → Bool) (k : QueryTree ι)
-    (R : Finset ι) (ω : ι → Bool) (h : ∀ i ∈ L, ω i = v i) :
-    freshCharge (forceList L v k) R ω =
-      freshCharge k (R ∪ L.toFinset) ω + (L.toFinset \ R).card := by
-  induction L generalizing R with
-  | nil => simp [forceList]
-  | cons i L ih =>
-    rw [forceList,freshCharge,ite_eq_left (h i List.mem_cons_self),
-      ih (insert i R) (fun j hj => h j (List.mem_cons_of_mem _ hj))]
-    simp only [List.toFinset_cons]
-    have hs : insert i R ∪ L.toFinset = R ∪ insert i L.toFinset := by ext j; simp
-    rw [hs]
-    by_cases hi : i ∈ R
-    · rw [ite_eq_left hi]
-      have he : L.toFinset \ insert i R = insert i L.toFinset \ R := by
-        ext j
-        by_cases hji : j = i
-        · subst j; simp [hi]
-        · simp [hji]
-      rw [he,Nat.add_zero]
-    · rw [ite_eq_right hi]
-      have he : insert i L.toFinset \ R = insert i (L.toFinset \ insert i R) := by
-        ext j
-        by_cases hji : j = i
-        · subst j; simp [hi]
-        · simp [hji]
-      rw [he, Finset.card_insert_of_notMem (by simp)]
-      omega
-
-
-def staleCharge : QueryTree ι → Finset ι → (ι → Bool) → ℕ
-  | reject, _, _ => 0
-  | accept, _, _ => 0
-  | branch i zero one, R, ω => if ω i then staleCharge one (insert i R) ω
-      else staleCharge zero (insert i R) ω
-  | force i b next, R, ω => if ω i = b then
-      staleCharge next (insert i R) ω + (if i ∈ R then 1 else 0) else 0
-
-lemma fresh_add_stale (T : QueryTree ι) (R : Finset ι) (ω : ι → Bool) :
-    T.freshCharge R ω + T.staleCharge R ω = T.charge ω := by
-  induction T generalizing R with
-  | reject => rfl
-  | accept => rfl
-  | branch i zero one ih₀ ih₁ =>
-    cases hb : ω i <;> simp [freshCharge,staleCharge,charge,hb,ih₀,ih₁]
-  | force i b next ih =>
-    by_cases hb : ω i = b
-    · simp only [freshCharge,staleCharge,charge,hb,↓reduceIte]
-      have he := ih (insert i R)
-      by_cases hi : i ∈ R <;> simp only [hi,↓reduceIte] <;> omega
-    · simp [freshCharge,staleCharge,charge,hb]
-
-omit [DecidableEq ι] in
-lemma followPath_raw_charge (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
-    (k : Card d → QueryTree ι) (ω : ι → Bool) :
-    (followPath d e x k).charge ω =
-      (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).charge ω := by
-  induction d with
-  | zero => rfl
-  | succ d ih =>
-    rw [followPath, ih]
-    let y := butterflyPerm d
-      (decodeButterfly d (fun i => ω (e (Sum.inr (x 0,i))))) (Fin.tail x)
-    change (if ω (e (Sum.inl y)) then
-      (k (Fin.cons (Bool.xor (x 0) true) y)).charge ω else
-      (k (Fin.cons (Bool.xor (x 0) false) y)).charge ω) =
-      (k (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).charge ω
-    cases ω (e (Sum.inl y)) <;> rfl
-
-omit [DecidableEq ι] in
-lemma forceList_raw_charge (L : List ι) (v : ι → Bool) (k : QueryTree ι) (ω : ι → Bool)
-    (h : ∀ i ∈ L, ω i = v i) :
-    (forceList L v k).charge ω = k.charge ω + L.length := by
-  induction L with
-  | nil => simp [forceList]
-  | cons i L ih =>
-    rw [forceList,charge,ite_eq_left (h i List.mem_cons_self),
-      ih (fun j hj => h j (List.mem_cons_of_mem _ hj))]
-    simp only [List.length_cons,Nat.add_assoc]
-
-noncomputable def closePath (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d)) :
-    QueryTree (SwitchIndex d) :=
-  forceList (routeDomain d x y).toList (routeValue d x y) k
-
-lemma closePath_succeeds (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
-    (ω : SwitchIndex d → Bool) :
-    (closePath d x y k).succeeds ω =
-      if butterflyPerm d (decodeButterfly d ω) x = y then k.succeeds ω else false := by
-  unfold closePath
-  rw [forceList_succeeds]
-  simp only [Finset.mem_toList, ← butterfly_endpoint_iff]
-
-lemma closePath_raw_charge (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
-    (ω : SwitchIndex d → Bool) (h : butterflyPerm d (decodeButterfly d ω) x = y) :
-    (closePath d x y k).charge ω = k.charge ω + d := by
-  unfold closePath
-  rw [forceList_raw_charge _ _ _ _ (by simpa only [Finset.mem_toList] using
-    (butterfly_endpoint_iff d ω x y).mp h), Finset.length_toList, card_routeDomain]
-
-lemma closePath_fresh_charge (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
-    (R : Finset (SwitchIndex d)) (ω : SwitchIndex d → Bool)
-    (h : butterflyPerm d (decodeButterfly d ω) x = y) :
-    (closePath d x y k).freshCharge R ω =
-      k.freshCharge (R ∪ routeDomain d x y) ω + (routeDomain d x y \ R).card := by
-  unfold closePath
-  rw [forceList_charge _ _ _ _ _ (by simpa only [Finset.mem_toList] using
-    (butterfly_endpoint_iff d ω x y).mp h), Finset.toList_toFinset]
-
-lemma closePath_stale_charge (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
-    (R : Finset (SwitchIndex d)) (ω : SwitchIndex d → Bool)
-    (h : butterflyPerm d (decodeButterfly d ω) x = y) :
-    (closePath d x y k).staleCharge R ω =
-      k.staleCharge (R ∪ routeDomain d x y) ω + (routeDomain d x y ∩ R).card := by
-  have h₀ := fresh_add_stale (closePath d x y k) R ω
-  have h₁ := fresh_add_stale k (R ∪ routeDomain d x y) ω
-  rw [closePath_raw_charge _ _ _ _ _ h,closePath_fresh_charge _ _ _ _ _ _ h] at h₀
-  have h₂ : (routeDomain d x y \ R).card + (routeDomain d x y ∩ R).card = d := by
-    rw [Finset.card_sdiff_add_card_inter,card_routeDomain]
-  omega
-
-noncomputable def seekCycle (d : ℕ) (S : Finset (Card d)) (a : Card d) :
-    ℕ → Card d → Finset (Card d) → (Finset (Card d) → QueryTree (SwitchIndex d)) →
-      QueryTree (SwitchIndex d)
-  | 0, x, V, k => if x ∈ S ∧ x ∉ V then closePath d x a (k (insert x V)) else reject
-  | r + 1, x, V, k => if x ∈ S ∧ x ∉ V then followPath d id x
-      (fun y => seekCycle d S a r y (insert x V) k) else reject
-
-lemma seekCycle_raw_charge (d : ℕ) (S : Finset (Card d)) (a : Card d) (r : ℕ)
-    (x : Card d) (V : Finset (Card d)) (k : Finset (Card d) → QueryTree (SwitchIndex d))
-    (ω : SwitchIndex d → Bool) (K : ℕ)
-    (hK : ∀ V, (k V).succeeds ω = true → (k V).charge ω = K)
-    (hs : (seekCycle d S a r x V k).succeeds ω = true) :
-    (seekCycle d S a r x V k).charge ω = K + d := by
-  induction r generalizing x V with
-  | zero =>
-    by_cases hv : x ∈ S ∧ x ∉ V
-    · rw [seekCycle,ite_eq_left hv,closePath_succeeds] at hs
-      split_ifs at hs with he
-      · rw [seekCycle,ite_eq_left hv,closePath_raw_charge _ _ _ _ _ he,hK _ hs]
-    · simp [seekCycle,hv,succeeds] at hs
-  | succ r ih =>
-    by_cases hv : x ∈ S ∧ x ∉ V
-    · rw [seekCycle,ite_eq_left hv,followPath_succeeds] at hs
-      rw [seekCycle,ite_eq_left hv,followPath_raw_charge]
-      exact ih _ _ hs
-    · simp [seekCycle,hv,succeeds] at hs
-
-noncomputable def seekCycles (d : ℕ) (S : Finset (Card d)) :
-    List (Card d × ℕ) → Finset (Card d) → QueryTree (SwitchIndex d)
-  | [], _ => accept
-  | (a,r) :: L, V => seekCycle d S a r a V (fun V => seekCycles d S L V)
-
-lemma seekCycles_raw_charge (d : ℕ) (S : Finset (Card d)) (L : List (Card d × ℕ))
-    (V : Finset (Card d)) (ω : SwitchIndex d → Bool)
-    (hs : (seekCycles d S L V).succeeds ω = true) :
-    (seekCycles d S L V).charge ω = L.length * d := by
-  induction L generalizing V with
-  | nil => simp [seekCycles,charge]
-  | cons ar L ih =>
-    obtain ⟨a,r⟩ := ar
-    rw [seekCycles]
-    rw [seekCycle_raw_charge d S a r a V (fun V => seekCycles d S L V) ω (L.length*d)
-      (fun V hs => ih V hs) hs]
-    simp [Nat.add_mul]
-
-lemma cycles_weighted_success (d : ℕ) (S : Finset (Card d)) (L : List (Card d × ℕ)) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      if (seekCycles d S L ∅).succeeds ω then
-        ((2:ℝ)^d)^L.length / (2:ℝ)^(seekCycles d S L ∅).staleCharge ∅ ω else 0) ≤ 1 := by
-  let T := seekCycles d S L ∅
-  have hh := mean_freshCharge_le_one T
-  have he : finiteMean (fun ω : SwitchIndex d → Bool =>
-      if T.succeeds ω then (2:ℝ)^T.freshCharge ∅ ω else 0) =
-      finiteMean (fun ω : SwitchIndex d → Bool =>
-        if T.succeeds ω then ((2:ℝ)^d)^L.length / (2:ℝ)^T.staleCharge ∅ ω else 0) := by
-    apply finiteMean_congr
-    intro ω
-    cases hs : T.succeeds ω
-    · simp
-    · simp only [↓reduceIte]
-      have hc := fresh_add_stale T ∅ ω
-      rw [seekCycles_raw_charge d S L ∅ ω hs] at hc
-      have hp : ((2:ℝ)^d)^L.length = (2:ℝ)^T.freshCharge ∅ ω * (2:ℝ)^T.staleCharge ∅ ω := by
-        rw [←pow_add,hc,←pow_mul,Nat.mul_comm]
-      rw [hp,mul_div_cancel_right₀ _ (pow_ne_zero _ (by norm_num))]
-  rw [← he]
-  exact hh
-
-end QueryTree
-
-lemma finiteMean_add {Ω : Type*} [Fintype Ω] (f g : Ω → ℝ) :
-    finiteMean (fun ω => f ω + g ω) = finiteMean f + finiteMean g := by
-  simp only [finiteMean,Finset.sum_add_distrib,add_div]
-
-lemma finiteMean_mul_const {Ω : Type*} [Fintype Ω] (f : Ω → ℝ) (c : ℝ) :
-    finiteMean (fun ω => f ω * c) = finiteMean f * c := by
-  unfold finiteMean
-  rw [← Finset.sum_mul]
-  ring
-
-lemma finiteMean_sum {Ω α : Type*} [Fintype Ω] (s : Finset α) (f : α → Ω → ℝ) :
-    finiteMean (fun ω => ∑ i ∈ s, f i ω) = ∑ i ∈ s, finiteMean (f i) := by
-  simp only [finiteMean,Finset.sum_div]
-  exact Finset.sum_comm
-
-lemma finiteMean_mono {Ω : Type*} [Fintype Ω] {f g : Ω → ℝ}
-    (h : ∀ ω, f ω ≤ g ω) : finiteMean f ≤ finiteMean g := by
-  exact div_le_div_of_nonneg_right (Finset.sum_le_sum (fun ω _ => h ω)) (Nat.cast_nonneg _)
-
-lemma finiteMean_equiv {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
-    (e : Ω ≃ Ω') (f : Ω → ℝ) :
-    finiteMean f = finiteMean (fun ω => f (e.symm ω)) := by
-  unfold finiteMean
-  rw [Fintype.card_congr e]
-  congr 1
-  exact Fintype.sum_equiv e _ _ (fun ω => by simp)
-
-lemma finiteMean_prod_left {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] [Nonempty Ω']
-    (f : Ω → ℝ) : finiteMean (fun ω : Ω × Ω' => f ω.1) = finiteMean f := by
-  simp only [finiteMean,Fintype.sum_prod_type,Finset.sum_const,nsmul_eq_mul,
-    ← Finset.mul_sum,Fintype.card_prod,Nat.cast_mul,Finset.card_univ]
-  have hn : (Fintype.card Ω' : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
-  field_simp
-
-lemma finiteMean_pi_eval {C : Type*} [Fintype C] [DecidableEq C]
-    {A : C → Type*} [∀ c, Fintype (A c)] [∀ c, Nonempty (A c)]
-    (c : C) (g : A c → ℝ) :
-    finiteMean (fun z : ∀ c, A c => g (z c)) = finiteMean g := by
-  rw [finiteMean_equiv (Equiv.piSplitAt c A)]
-  have he : (fun z : A c × ((j : {j // j ≠ c}) → A j) =>
-      g ((Equiv.piSplitAt c A).symm z c)) = (fun z => g z.1) := by
-    funext z
-    simp [Equiv.piSplitAt]
-  rw [he,finiteMean_prod_left]
-
-lemma finiteMean_eq_indicator {Ω : Type*} [Fintype Ω] [DecidableEq Ω] (a : Ω) :
-    finiteMean (fun ω : Ω => if ω = a then (1:ℝ) else 0) = 1 / Fintype.card Ω := by
-  simp [finiteMean]
-
-lemma exp_finiteMean_le {Ω : Type*} [Fintype Ω] [Nonempty Ω] (f : Ω → ℝ) :
-    Real.exp (finiteMean f) ≤ finiteMean (fun ω => Real.exp (f ω)) := by
-  have hn : (Fintype.card Ω : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
-  have hw : ∑ ω : Ω, (1 / (Fintype.card Ω : ℝ)) = 1 := by simp [hn]
-  have hj := convexOn_exp.map_sum_le (t := Finset.univ)
-    (w := fun _ : Ω => 1 / (Fintype.card Ω : ℝ)) (p := f)
-    (fun _ _ => by positivity) hw (fun _ _ => Set.mem_univ _)
-  simp only [smul_eq_mul, ← Finset.mul_sum] at hj
-  simpa only [finiteMean, div_eq_mul_inv, one_div, mul_comm, mul_one, one_mul] using hj
-
-variable {d A : ℕ}
-
-def cycleFiber (o : Card d → Option (Fin A)) (i : Fin A) : Finset (Card d) := by
-  classical
-  exact Finset.univ.filter (fun x => o x = some i)
-
-@[simp] lemma mem_cycleFiber (o : Card d → Option (Fin A)) (i : Fin A) (x : Card d) :
-    x ∈ cycleFiber o i ↔ o x = some i := by
-  classical
-  simp [cycleFiber]
-
-abbrev Terminal (o : Card d → Option (Fin A)) := ∀ i : Fin A, {x : Card d // o x = some i}
-
-noncomputable def slotWeight (o : Card d → Option (Fin A)) (i : Fin A) : ℝ :=
-  1 / (cycleFiber o i).card
-
-noncomputable def pathWeight (o : Card d → Option (Fin A)) (x : Card d) : ℝ :=
-  match o x with
-  | none => 0
-  | some i => slotWeight o i
-
-noncomputable def withinWeight (o : Card d → Option (Fin A)) (u v : Card d) : ℝ :=
-  match o u, o v with
-  | some i, some j => if i = j then slotWeight o i else 0
-  | _, _ => 0
-
-end Thorp
-
-end ThorpNine.Harmonic
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Harmonic.Coins
+
+namespace OAI
+
+namespace ThorpNine.Harmonic
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+
+lemma card_routeDomain (d : ℕ) (x y : Card d) : (routeDomain d x y).card = d := by
+  classical
+  induction d with
+  | zero => rfl
+  | succ d ih =>
+    change (insert (Sum.inl (Fin.tail y))
+      ((routeDomain d (Fin.tail x) (Fin.tail y)).image
+        (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) :
+      Finset (Sum (Card d) (Bool × SwitchIndex d))).card = d + 1
+    rw [Finset.card_insert_of_notMem (by simp)]
+    rw [Finset.card_image_of_injective _ (by intro i j h; cases h; rfl), ih]
+
+lemma xor_eq_iff (a b c : Bool) : Bool.xor a b = c ↔ b = Bool.xor a c := by
+  cases a <;> cases b <;> cases c <;> decide
+
+lemma butterfly_endpoint_iff (d : ℕ) (ω : SwitchIndex d → Bool) (x y : Card d) :
+    butterflyPerm d (decodeButterfly d ω) x = y ↔
+      ∀ i ∈ routeDomain d x y, ω i = routeValue d x y i := by
+  classical
+  induction d with
+  | zero =>
+    have he : butterflyPerm 0 (decodeButterfly 0 ω) x = y := Subsingleton.elim _ _
+    simp [he, routeDomain]
+  | succ d ih =>
+    let v := butterflyPerm d (decodeButterfly d (fun i => ω (Sum.inr (x 0,i)))) (Fin.tail x)
+    change Fin.cons (Bool.xor (x 0) (ω (Sum.inl v))) v = y ↔ _
+    have hv : v = Fin.tail y ↔
+        ∀ i ∈ routeDomain d (Fin.tail x) (Fin.tail y),
+          ω (Sum.inr (x 0,i)) = routeValue d (Fin.tail x) (Fin.tail y) i :=
+      ih _ _ _
+    constructor
+    · intro h i hi
+      have ht : v = Fin.tail y := by simpa using congrArg Fin.tail h
+      have hh : ω (Sum.inl (Fin.tail y)) = Bool.xor (x 0) (y 0) := by
+        have he := congrFun h 0
+        simp only [Fin.cons_zero, ht] at he
+        exact (xor_eq_iff _ _ _).mp he
+      rcases Finset.mem_insert.mp hi with htop | hchild
+      · subst i
+        exact hh
+      · obtain ⟨j, hj, rfl⟩ := Finset.mem_image.mp hchild
+        exact hv.mp ht j hj
+    · intro h
+      have hh := h (Sum.inl (Fin.tail y)) (Finset.mem_insert_self _ _)
+      have ht : v = Fin.tail y := hv.mpr (fun i hi =>
+        h (Sum.inr (x 0,i)) (Finset.mem_insert_of_mem (Finset.mem_image_of_mem _ hi)))
+      have hh' : Bool.xor (x 0) (ω (Sum.inl (Fin.tail y))) = y 0 :=
+        (xor_eq_iff _ _ _).mpr hh
+      rw [ht, hh', Fin.cons_self_tail]
+
+lemma finiteMean_congr {Ω : Type*} [Fintype Ω] {f g : Ω → ℝ}
+    (h : ∀ ω, f ω = g ω) : finiteMean f = finiteMean g := by
+  unfold finiteMean
+  congr 1
+  exact Finset.sum_congr rfl (fun ω _ => h ω)
+
+lemma mean_coinCylinder {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (s : Finset ι) (v : ι → Bool) :
+    finiteMean (fun ω : ι → Bool => if ∀ i ∈ s, ω i = v i then (1 : ℝ) else 0) =
+      1 / (2 : ℝ) ^ s.card := by
+  induction s using Finset.induction_on with
+  | empty => simp [finiteMean]
+  | @insert i s hi ih =>
+    let f := fun ω : ι → Bool => if ∀ j ∈ s, ω j = v j then (1 : ℝ) else 0
+    have hf : ∀ ω, f (QueryTree.flipCoin i ω) = f ω := by
+      intro ω
+      have he : (∀ j ∈ s, QueryTree.flipCoin i ω j = v j) ↔ ∀ j ∈ s, ω j = v j := by
+        apply forall₂_congr
+        intro j hj
+        rw [QueryTree.flipCoin_apply_ne i j (fun h => hi (h ▸ hj))]
+      dsimp [f]
+      simp only [he]
+    have hm := QueryTree.mean_indicator_bit f i (v i) hf
+    have he : (fun ω : ι → Bool => if ∀ j ∈ insert i s, ω j = v j then (1 : ℝ) else 0) =
+        (fun ω => if ω i = v i then f ω else 0) := by
+      funext ω
+      by_cases h : ω i = v i <;> simp [f, h]
+    rw [he, hm]
+    change finiteMean (fun ω : ι → Bool => if ∀ j ∈ s, ω j = v j then (1 : ℝ) else 0) / 2 = _
+    rw [ih, Finset.card_insert_of_notMem hi, pow_succ]
+    ring
+
+def exposedCoins {ι : Type*} [DecidableEq ι] (R : Finset ι) (η ω : ι → Bool) : ι → Bool :=
+  fun i => if i ∈ R then η i else ω i
+
+lemma mean_coinCylinder_exposed {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (s R : Finset ι) (v η : ι → Bool) :
+    finiteMean (fun ω : ι → Bool =>
+      if ∀ i ∈ s, exposedCoins R η ω i = v i then (1 : ℝ) else 0) =
+    if (∀ i ∈ s ∩ R, η i = v i) then 1 / (2 : ℝ) ^ (s \ R).card else 0 := by
+  have he (ω : ι → Bool) : (∀ i ∈ s, exposedCoins R η ω i = v i) ↔
+      (∀ i ∈ s ∩ R, η i = v i) ∧ (∀ i ∈ s \ R, ω i = v i) := by
+    constructor
+    · intro h
+      constructor
+      · intro i hi
+        simpa [exposedCoins, (Finset.mem_inter.mp hi).2] using h i (Finset.mem_inter.mp hi).1
+      · intro i hi
+        simpa [exposedCoins, (Finset.mem_sdiff.mp hi).2] using h i (Finset.mem_sdiff.mp hi).1
+    · rintro ⟨ho, hf⟩ i hi
+      by_cases hR : i ∈ R
+      · simpa [exposedCoins, hR] using ho i (Finset.mem_inter.mpr ⟨hi,hR⟩)
+      · simpa [exposedCoins, hR] using hf i (Finset.mem_sdiff.mpr ⟨hi,hR⟩)
+  have hm : finiteMean (fun ω : ι → Bool =>
+      if ∀ i ∈ s, exposedCoins R η ω i = v i then (1 : ℝ) else 0) =
+      finiteMean (fun ω : ι → Bool =>
+        if (∀ i ∈ s ∩ R, η i = v i) ∧ (∀ i ∈ s \ R, ω i = v i) then (1 : ℝ) else 0) :=
+    finiteMean_congr (fun ω => by simp only [he ω])
+  rw [hm]
+  by_cases h : ∀ i ∈ s ∩ R, η i = v i
+  · have hp : (∀ i ∈ s ∩ R, η i = v i) ↔ True := iff_true_intro h
+    simp only [hp, true_and, ↓reduceIte]
+    exact mean_coinCylinder (s \ R) v
+  · have hp : (∀ i ∈ s ∩ R, η i = v i) ↔ False := iff_false_intro h
+    simp only [hp, false_and, ↓reduceIte]
+    simp [finiteMean]
+
+lemma butterfly_closure_probability (d : ℕ) (x y : Card d)
+    (R : Finset (SwitchIndex d)) (η : SwitchIndex d → Bool) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      if butterflyPerm d (decodeButterfly d (exposedCoins R η ω)) x = y then (1 : ℝ) else 0) =
+      if (∀ i ∈ routeDomain d x y ∩ R, η i = routeValue d x y i) then
+        (2 : ℝ) ^ (routeDomain d x y ∩ R).card / (2 : ℝ) ^ d else 0 := by
+  have he : finiteMean (fun ω : SwitchIndex d → Bool =>
+      if butterflyPerm d (decodeButterfly d (exposedCoins R η ω)) x = y then (1 : ℝ) else 0) =
+      finiteMean (fun ω : SwitchIndex d → Bool =>
+        if ∀ i ∈ routeDomain d x y, exposedCoins R η ω i = routeValue d x y i then (1 : ℝ) else 0) :=
+    finiteMean_congr (fun ω => by simp only [butterfly_endpoint_iff])
+  rw [he, mean_coinCylinder_exposed]
+  congr 1
+  have hc : (routeDomain d x y \ R).card + (routeDomain d x y ∩ R).card = d := by
+    rw [Finset.card_sdiff_add_card_inter, card_routeDomain]
+  have hp : (2 : ℝ) ^ d = (2 : ℝ) ^ (routeDomain d x y \ R).card *
+      (2 : ℝ) ^ (routeDomain d x y ∩ R).card := by
+    rw [← pow_add]
+    exact congrArg (fun n => (2 : ℝ) ^ n) hc.symm
+  rw [hp]
+  have hn : (2 : ℝ) ^ (routeDomain d x y ∩ R).card ≠ 0 := pow_ne_zero _ (by norm_num)
+  field_simp
+
+lemma butterfly_single_card_uniform (d : ℕ) (x y : Card d) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      if butterflyPerm d (decodeButterfly d ω) x = y then (1 : ℝ) else 0) = 1 / (2 : ℝ) ^ d := by
+  have he : finiteMean (fun ω : SwitchIndex d → Bool =>
+      if butterflyPerm d (decodeButterfly d ω) x = y then (1 : ℝ) else 0) =
+      finiteMean (fun ω : SwitchIndex d → Bool =>
+        if ∀ i ∈ routeDomain d x y, ω i = routeValue d x y i then (1 : ℝ) else 0) :=
+    finiteMean_congr (fun ω => by simp only [butterfly_endpoint_iff])
+  rw [he, mean_coinCylinder, card_routeDomain]
+
+def switchUsers : (d : ℕ) → Butterfly d → SwitchIndex d → Card d × Card d
+  | 0, _, i => nomatch i
+  | d + 1, b, Sum.inl y =>
+      (Fin.cons false ((butterflyPerm d (b.2 false)).symm y),
+       Fin.cons true ((butterflyPerm d (b.2 true)).symm y))
+  | d + 1, b, Sum.inr (ε,i) =>
+      (Fin.cons ε (switchUsers d (b.2 ε) i).1,
+       Fin.cons ε (switchUsers d (b.2 ε) i).2)
+
+lemma switchUsers_ne (d : ℕ) (b : Butterfly d) (i : SwitchIndex d) :
+    (switchUsers d b i).1 ≠ (switchUsers d b i).2 := by
+  induction d with
+  | zero => exact Empty.elim i
+  | succ d ih =>
+    cases i with
+    | inl y =>
+      intro h
+      have := congrFun h 0
+      simp only [switchUsers, Fin.cons_zero] at this
+      exact Bool.false_ne_true this
+    | inr z =>
+      intro h
+      exact ih (b.2 z.1) z.2 (by simpa only [switchUsers, Fin.tail_cons] using congrArg Fin.tail h)
+
+lemma cons_eq_iff {d : ℕ} (ε : Bool) (a : Card d) (x : Card (d + 1)) :
+    Fin.cons ε a = x ↔ ε = x 0 ∧ a = Fin.tail x := by
+  constructor
+  · intro h
+    exact ⟨by simpa only [Fin.cons_zero] using congrFun h 0,
+      by simpa only [Fin.tail_cons] using congrArg Fin.tail h⟩
+  · rintro ⟨rfl,rfl⟩
+    exact Fin.cons_self_tail x
+
+lemma mem_routeDomain_iff_user (d : ℕ) (b : Butterfly d) (x : Card d) (i : SwitchIndex d) :
+    i ∈ routeDomain d x (butterflyPerm d b x) ↔
+      (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x := by
+  classical
+  induction d with
+  | zero => exact Empty.elim i
+  | succ d ih =>
+    cases i with
+    | inl y =>
+      change Sum.inl y ∈ insert (Sum.inl (Fin.tail (butterflyPerm (d+1) b x)))
+        ((routeDomain d (Fin.tail x) (Fin.tail (butterflyPerm (d+1) b x))).image
+          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) ↔ _
+      simp only [Finset.mem_insert, Sum.inl.injEq, Finset.mem_image,
+        Sum.inr_ne_inl, and_false, exists_false, or_false]
+      change y = butterflyPerm d (b.2 (x 0)) (Fin.tail x) ↔ _
+      simp only [switchUsers, cons_eq_iff]
+      cases x 0 <;> simp [Equiv.symm_apply_eq]
+    | inr z =>
+      obtain ⟨ε,i⟩ := z
+      change Sum.inr (ε,i) ∈ insert (Sum.inl (Fin.tail (butterflyPerm (d+1) b x)))
+        ((routeDomain d (Fin.tail x) (Fin.tail (butterflyPerm (d+1) b x))).image
+          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) ↔ _
+      simp only [Finset.mem_insert, Sum.inr_ne_inl, false_or, Finset.mem_image,
+        Sum.inr.injEq, Prod.mk.injEq]
+      change (∃ j, j ∈ routeDomain d (Fin.tail x) (butterflyPerm d (b.2 (x 0)) (Fin.tail x)) ∧
+        x 0 = ε ∧ j = i) ↔ _
+      simp only [switchUsers, cons_eq_iff]
+      constructor
+      · rintro ⟨j,hj,hε,rfl⟩
+        subst ε
+        rcases (ih (b.2 (x 0)) (Fin.tail x) j).mp hj with h | h
+        · exact Or.inl ⟨rfl,h⟩
+        · exact Or.inr ⟨rfl,h⟩
+      · rintro (⟨hε,h⟩ | ⟨hε,h⟩)
+        · subst ε
+          exact ⟨i,(ih (b.2 (x 0)) (Fin.tail x) i).mpr (Or.inl h),rfl,rfl⟩
+        · subst ε
+          exact ⟨i,(ih (b.2 (x 0)) (Fin.tail x) i).mpr (Or.inr h),rfl,rfl⟩
+
+lemma card_switches_using (d : ℕ) (b : Butterfly d) (x : Card d) :
+    (Finset.univ.filter (fun i : SwitchIndex d =>
+      (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x)).card = d := by
+  classical
+  have he : Finset.univ.filter (fun i : SwitchIndex d =>
+      (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x) =
+      routeDomain d x (butterflyPerm d b x) := by
+    ext i
+    simp only [Finset.mem_filter, Finset.mem_univ, true_and, mem_routeDomain_iff_user]
+  rw [he,card_routeDomain]
+
+lemma switch_incidence_sum (d : ℕ) (b : Butterfly d) (w : Card d → ℝ) :
+    ∑ i : SwitchIndex d, (w (switchUsers d b i).1 + w (switchUsers d b i).2) =
+      (d : ℝ) * ∑ x : Card d, w x := by
+  classical
+  calc
+    _ = ∑ i : SwitchIndex d, ∑ x : Card d,
+        if (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x then w x else 0 := by
+      apply Finset.sum_congr rfl
+      intro i _
+      rw [Finset.sum_ite]
+      simp only [Finset.sum_const_zero, add_zero]
+      have he : Finset.univ.filter (fun x : Card d =>
+          (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x) =
+          {(switchUsers d b i).1, (switchUsers d b i).2} := by
+        ext x; simp [eq_comm]
+      rw [he, Finset.sum_pair (switchUsers_ne d b i)]
+    _ = ∑ x : Card d, ∑ i : SwitchIndex d,
+        if (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x then w x else 0 :=
+      Finset.sum_comm
+    _ = ∑ x : Card d, (d : ℝ) * w x := by
+      apply Finset.sum_congr rfl
+      intro x _
+      rw [Finset.sum_ite]
+      simp only [Finset.sum_const_zero, add_zero, Finset.sum_const, nsmul_eq_mul,
+        card_switches_using]
+    _ = _ := by rw [Finset.mul_sum]
+
+lemma weighted_switch_min_bound (d : ℕ) (b : Butterfly d) (w : Card d → ℝ) :
+    ∑ i : SwitchIndex d, min (w (switchUsers d b i).1) (w (switchUsers d b i).2) ≤
+      (d : ℝ) / 2 * ∑ x : Card d, w x := by
+  have h : 2 * (∑ i : SwitchIndex d,
+      min (w (switchUsers d b i).1) (w (switchUsers d b i).2)) ≤
+      ∑ i : SwitchIndex d, (w (switchUsers d b i).1 + w (switchUsers d b i).2) := by
+    rw [Finset.mul_sum]
+    apply Finset.sum_le_sum
+    intro i _
+    linarith [min_le_left (w (switchUsers d b i).1) (w (switchUsers d b i).2),
+      min_le_right (w (switchUsers d b i).1) (w (switchUsers d b i).2)]
+  rw [switch_incidence_sum] at h
+  linarith
+
+lemma mem_childMarks {d : ℕ} (s : Finset (Card (d + 1))) (ε : Bool) (x : Card d) :
+    x ∈ childMarks s ε ↔ Fin.cons ε x ∈ s := by
+  classical
+  constructor
+  · intro h
+    obtain ⟨y,hy,hxy⟩ := Finset.mem_image.mp h
+    obtain ⟨hys,hyε⟩ := Finset.mem_filter.mp hy
+    rw [← hxy, ← hyε, Fin.cons_self_tail]
+    exact hys
+  · intro h
+    exact Finset.mem_image.mpr ⟨Fin.cons ε x,
+      Finset.mem_filter.mpr ⟨h,by simp⟩,by simp⟩
+
+lemma mem_topMeetings {d : ℕ} (b : Butterfly (d + 1))
+    (s : Finset (Card (d + 1))) (y : Card d) :
+    y ∈ topMeetings b s ↔
+      (switchUsers (d+1) b (Sum.inl y)).1 ∈ s ∧
+      (switchUsers (d+1) b (Sum.inl y)).2 ∈ s := by
+  classical
+  have he (ε : Bool) : y ∈ (childMarks s ε).image (butterflyPerm d (b.2 ε)) ↔
+      (butterflyPerm d (b.2 ε)).symm y ∈ childMarks s ε := by
+    constructor
+    · intro h
+      obtain ⟨x,hx,rfl⟩ := Finset.mem_image.mp h
+      simpa only [Equiv.symm_apply_apply] using hx
+    · intro h
+      exact Finset.mem_image.mpr ⟨_,h,Equiv.apply_symm_apply _ _⟩
+  simp only [topMeetings, Finset.mem_inter, he, mem_childMarks, switchUsers]
+
+lemma encounters_eq_switch_sum (d : ℕ) (b : Butterfly d) (s : Finset (Card d)) :
+    encounters d b s = ∑ i : SwitchIndex d,
+      if (switchUsers d b i).1 ∈ s ∧ (switchUsers d b i).2 ∈ s then 1 else 0 := by
+  classical
+  induction d with
+  | zero =>
+    simp only [encounters]
+    symm
+    apply Finset.sum_eq_zero
+    intro i _
+    exact Empty.elim i
+  | succ d ih =>
+    change encounters d (b.2 false) (childMarks s false) +
+      encounters d (b.2 true) (childMarks s true) + (topMeetings b s).card =
+        ∑ i : Sum (Card d) (Bool × SwitchIndex d), _
+    rw [Fintype.sum_sum_type, Fintype.sum_prod_type]
+    simp only [Fintype.sum_bool]
+    rw [ih,ih]
+    have ht : (∑ y : Card d,
+        if (switchUsers (d+1) b (Sum.inl y)).1 ∈ s ∧
+          (switchUsers (d+1) b (Sum.inl y)).2 ∈ s then 1 else 0) =
+        (topMeetings b s).card := by
+      simp only [← mem_topMeetings]
+      simp
+    rw [ht]
+    simp only [mem_childMarks, switchUsers]
+    ac_rfl
+
+lemma marked_switch_entropy_bound (d : ℕ) (b : Butterfly d) (s : Finset (Card d)) :
+    (∑ i : SwitchIndex d,
+      if (switchUsers d b i).1 ∈ s ∧ (switchUsers d b i).2 ∈ s then (1:ℝ) else 0) ≤
+      (s.card : ℝ) * Real.log s.card / (2 * Real.log 2) := by
+  classical
+  have he : (encounters d b s : ℝ) =
+      ∑ i : SwitchIndex d,
+        if (switchUsers d b i).1 ∈ s ∧ (switchUsers d b i).2 ∈ s then (1:ℝ) else 0 := by
+    rw [encounters_eq_switch_sum, Nat.cast_sum]
+    apply Finset.sum_congr rfl
+    intro i _
+    split_ifs <;> simp
+  rw [← he]
+  exact butterfly_encounters d b s
+
+namespace QueryTree
+variable {ι : Type*} [DecidableEq ι]
+
+def memo : QueryTree ι → Finset ι → (ι → Bool) → QueryTree ι
+  | reject, _, _ => reject
+  | accept, _, _ => accept
+  | branch i zero one, R, η =>
+      if i ∈ R then (if η i then memo one R η else memo zero R η) else
+      branch i (memo zero (insert i R) (Function.update η i false))
+        (memo one (insert i R) (Function.update η i true))
+  | force i b next, R, η =>
+      if i ∈ R then (if η i = b then memo next R η else reject) else
+      force i b (memo next (insert i R) (Function.update η i b))
+
+lemma memo_queries_disjoint (T : QueryTree ι) (R : Finset ι) (η : ι → Bool) :
+    Disjoint (memo T R η).queries R := by
+  induction T generalizing R η with
+  | reject => simp [memo,queries]
+  | accept => simp [memo,queries]
+  | branch i zero one ih₀ ih₁ =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte]
+      cases η i <;> simp only [Bool.false_eq_true,↓reduceIte]
+      · exact ih₀ R η
+      · exact ih₁ R η
+    · simp only [memo,hi,↓reduceIte,queries,Finset.disjoint_insert_left,Finset.disjoint_union_left]
+      exact ⟨by simp, (ih₀ (insert i R) _).mono_right (Finset.subset_insert _ _),
+        (ih₁ (insert i R) _).mono_right (Finset.subset_insert _ _)⟩
+  | force i b next ih =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte]
+      by_cases hb : η i = b
+      · simpa only [hb,↓reduceIte] using ih R η
+      · simp [hb,queries]
+    · simp only [memo,hi,↓reduceIte,queries,Finset.disjoint_insert_left]
+      exact ⟨by simp,(ih (insert i R) _).mono_right (Finset.subset_insert _ _)⟩
+
+lemma memo_fresh (T : QueryTree ι) (R : Finset ι) (η : ι → Bool) :
+    (memo T R η).Fresh := by
+  induction T generalizing R η with
+  | reject => trivial
+  | accept => trivial
+  | branch i zero one ih₀ ih₁ =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte]
+      cases η i <;> simp only [Bool.false_eq_true,↓reduceIte]
+      · exact ih₀ R η
+      · exact ih₁ R η
+    · simp only [memo,hi,↓reduceIte,Fresh]
+      exact ⟨fun hm => Finset.disjoint_left.mp (memo_queries_disjoint zero (insert i R) _) hm
+          (Finset.mem_insert_self _ _),
+        fun hm => Finset.disjoint_left.mp (memo_queries_disjoint one (insert i R) _) hm
+          (Finset.mem_insert_self _ _), ih₀ _ _,ih₁ _ _⟩
+  | force i b next ih =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte]
+      by_cases hb : η i = b
+      · simpa only [hb,↓reduceIte] using ih R η
+      · simp [hb,Fresh]
+    · simp only [memo,hi,↓reduceIte,Fresh]
+      exact ⟨fun hm => Finset.disjoint_left.mp (memo_queries_disjoint next (insert i R) _) hm
+        (Finset.mem_insert_self _ _),ih _ _⟩
+
+lemma update_agrees {R : Finset ι} {η ω : ι → Bool}
+    (h : ∀ i ∈ R, η i = ω i) (i : ι) :
+    ∀ j ∈ insert i R, Function.update η i (ω i) j = ω j := by
+  intro j hj
+  rcases Finset.mem_insert.mp hj with rfl | hj
+  · simp
+  · by_cases hji : j = i
+    · subst j; simp
+    · rw [Function.update_of_ne hji]
+      exact h j hj
+
+lemma memo_succeeds (T : QueryTree ι) (R : Finset ι) (η ω : ι → Bool)
+    (h : ∀ i ∈ R, η i = ω i) : (memo T R η).succeeds ω = T.succeeds ω := by
+  induction T generalizing R η with
+  | reject => rfl
+  | accept => rfl
+  | branch i zero one ih₀ ih₁ =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte,succeeds,h i hi]
+      cases ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
+      · exact ih₀ R η h
+      · exact ih₁ R η h
+    · simp only [memo,hi,↓reduceIte,succeeds]
+      cases hb : ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
+      · exact ih₀ _ _ (by simpa only [hb] using update_agrees h i)
+      · exact ih₁ _ _ (by simpa only [hb] using update_agrees h i)
+  | force i b next ih =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte,h i hi]
+      by_cases hb : ω i = b
+      · simp only [hb,↓reduceIte,succeeds]
+        exact ih R η h
+      · simp [hb,succeeds]
+    · simp only [memo,hi,↓reduceIte,succeeds]
+      by_cases hb : ω i = b
+      · simp only [hb,↓reduceIte]
+        exact ih _ _ (by simpa only [hb] using update_agrees h i)
+      · simp [hb]
+
+def freshCharge : QueryTree ι → Finset ι → (ι → Bool) → ℕ
+  | reject, _, _ => 0
+  | accept, _, _ => 0
+  | branch i zero one, R, ω => if ω i then freshCharge one (insert i R) ω else freshCharge zero (insert i R) ω
+  | force i b next, R, ω => if ω i = b then
+      freshCharge next (insert i R) ω + (if i ∈ R then 0 else 1) else 0
+
+lemma memo_charge (T : QueryTree ι) (R : Finset ι) (η ω : ι → Bool)
+    (h : ∀ i ∈ R, η i = ω i) : (memo T R η).charge ω = freshCharge T R ω := by
+  induction T generalizing R η with
+  | reject => rfl
+  | accept => rfl
+  | branch i zero one ih₀ ih₁ =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte,freshCharge,h i hi,Finset.insert_eq_of_mem hi]
+      cases ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
+      · exact ih₀ R η h
+      · exact ih₁ R η h
+    · simp only [memo,hi,↓reduceIte,charge,freshCharge]
+      cases hb : ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
+      · exact ih₀ _ _ (by simpa only [hb] using update_agrees h i)
+      · exact ih₁ _ _ (by simpa only [hb] using update_agrees h i)
+  | force i b next ih =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte,h i hi,freshCharge,Finset.insert_eq_of_mem hi,Nat.add_zero]
+      by_cases hb : ω i = b
+      · simp only [hb,↓reduceIte]
+        exact ih R η h
+      · simp [hb,charge]
+    · simp only [memo,hi,↓reduceIte,charge,freshCharge]
+      by_cases hb : ω i = b
+      · simp only [hb,↓reduceIte]
+        rw [ih _ _ (by simpa only [hb] using update_agrees h i)]
+      · simp [hb]
+
+lemma mean_freshCharge_le_one [Fintype ι] (T : QueryTree ι) :
+    finiteMean (fun ω : ι → Bool =>
+      if T.succeeds ω then (2:ℝ) ^ T.freshCharge ∅ ω else 0) ≤ 1 := by
+  let η : ι → Bool := fun _ => false
+  have hm := mean_weighted_le_one (memo T ∅ η) (memo_fresh T ∅ η)
+  have he : finiteMean (memo T ∅ η).weighted =
+      finiteMean (fun ω : ι → Bool =>
+        if T.succeeds ω then (2:ℝ) ^ T.freshCharge ∅ ω else 0) := by
+    apply finiteMean_congr
+    intro ω
+    rw [weighted_eq, memo_succeeds _ _ _ _ (by simp), memo_charge _ _ _ _ (by simp)]
+  rw [he] at hm
+  exact hm
+
+
+def followPath : (d : ℕ) → (SwitchIndex d → ι) → Card d →
+    (Card d → QueryTree ι) → QueryTree ι
+  | 0, _, x, k => k x
+  | d + 1, e, x, k => followPath d (fun i => e (Sum.inr (x 0,i))) (Fin.tail x)
+      (fun y => branch (e (Sum.inl y))
+        (k (Fin.cons (Bool.xor (x 0) false) y))
+        (k (Fin.cons (Bool.xor (x 0) true) y)))
+
+omit [DecidableEq ι] in
+lemma followPath_succeeds (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
+    (k : Card d → QueryTree ι) (ω : ι → Bool) :
+    (followPath d e x k).succeeds ω =
+      (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).succeeds ω := by
+  induction d with
+  | zero => rfl
+  | succ d ih =>
+    rw [followPath, ih]
+    let y := butterflyPerm d
+      (decodeButterfly d (fun i => ω (e (Sum.inr (x 0,i))))) (Fin.tail x)
+    change (if ω (e (Sum.inl y)) then
+      (k (Fin.cons (Bool.xor (x 0) true) y)).succeeds ω else
+      (k (Fin.cons (Bool.xor (x 0) false) y)).succeeds ω) =
+      (k (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).succeeds ω
+    cases ω (e (Sum.inl y)) <;> rfl
+
+lemma followPath_charge (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
+    (k : Card d → QueryTree ι) (R : Finset ι) (ω : ι → Bool) :
+    freshCharge (followPath d e x k) R ω =
+      freshCharge (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x))
+        (R ∪ (routeDomain d x (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).image e) ω := by
+  classical
+  induction d generalizing R with
+  | zero => simp only [followPath, routeDomain, Finset.image_empty, Finset.union_empty]; rfl
+  | succ d ih =>
+    rw [followPath, ih]
+    let y := butterflyPerm d
+      (decodeButterfly d (fun i => ω (e (Sum.inr (x 0,i))))) (Fin.tail x)
+    let S := (routeDomain d (Fin.tail x) y).image (fun i => e (Sum.inr (x 0,i)))
+    change (if ω (e (Sum.inl y)) then
+      freshCharge (k (Fin.cons (Bool.xor (x 0) true) y)) (insert (e (Sum.inl y)) (R ∪ S)) ω else
+      freshCharge (k (Fin.cons (Bool.xor (x 0) false) y)) (insert (e (Sum.inl y)) (R ∪ S)) ω) =
+      freshCharge (k (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y))
+        (R ∪ (routeDomain (d+1) x
+          (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).image e) ω
+    have hs : (routeDomain (d+1) x
+        (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).image e =
+        insert (e (Sum.inl y)) S := by
+      let e' : Sum (Card d) (Bool × SwitchIndex d) → ι := e
+      change (insert (Sum.inl y)
+        ((routeDomain d (Fin.tail x) y).image
+          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) :
+          Finset (Sum (Card d) (Bool × SwitchIndex d))).image e' = _
+      rw [Finset.image_insert, Finset.image_image]
+      rfl
+    rw [hs,Finset.union_insert]
+    cases ω (e (Sum.inl y)) <;> rfl
+
+def forceList : List ι → (ι → Bool) → QueryTree ι → QueryTree ι
+  | [], _, k => k
+  | i :: L, v, k => force i (v i) (forceList L v k)
+
+omit [DecidableEq ι] in
+lemma forceList_succeeds (L : List ι) (v : ι → Bool) (k : QueryTree ι) (ω : ι → Bool) :
+    (forceList L v k).succeeds ω =
+      if ∀ i ∈ L, ω i = v i then k.succeeds ω else false := by
+  induction L with
+  | nil => simp [forceList]
+  | cons i L ih =>
+    simp only [forceList,succeeds,ih,List.mem_cons,forall_eq_or_imp]
+    by_cases h : ω i = v i <;> simp [h]
+
+lemma forceList_charge (L : List ι) (v : ι → Bool) (k : QueryTree ι)
+    (R : Finset ι) (ω : ι → Bool) (h : ∀ i ∈ L, ω i = v i) :
+    freshCharge (forceList L v k) R ω =
+      freshCharge k (R ∪ L.toFinset) ω + (L.toFinset \ R).card := by
+  induction L generalizing R with
+  | nil => simp [forceList]
+  | cons i L ih =>
+    rw [forceList,freshCharge,ite_eq_left (h i List.mem_cons_self),
+      ih (insert i R) (fun j hj => h j (List.mem_cons_of_mem _ hj))]
+    simp only [List.toFinset_cons]
+    have hs : insert i R ∪ L.toFinset = R ∪ insert i L.toFinset := by ext j; simp
+    rw [hs]
+    by_cases hi : i ∈ R
+    · rw [ite_eq_left hi]
+      have he : L.toFinset \ insert i R = insert i L.toFinset \ R := by
+        ext j
+        by_cases hji : j = i
+        · subst j; simp [hi]
+        · simp [hji]
+      rw [he,Nat.add_zero]
+    · rw [ite_eq_right hi]
+      have he : insert i L.toFinset \ R = insert i (L.toFinset \ insert i R) := by
+        ext j
+        by_cases hji : j = i
+        · subst j; simp [hi]
+        · simp [hji]
+      rw [he, Finset.card_insert_of_notMem (by simp)]
+      omega
+
+
+def staleCharge : QueryTree ι → Finset ι → (ι → Bool) → ℕ
+  | reject, _, _ => 0
+  | accept, _, _ => 0
+  | branch i zero one, R, ω => if ω i then staleCharge one (insert i R) ω
+      else staleCharge zero (insert i R) ω
+  | force i b next, R, ω => if ω i = b then
+      staleCharge next (insert i R) ω + (if i ∈ R then 1 else 0) else 0
+
+lemma fresh_add_stale (T : QueryTree ι) (R : Finset ι) (ω : ι → Bool) :
+    T.freshCharge R ω + T.staleCharge R ω = T.charge ω := by
+  induction T generalizing R with
+  | reject => rfl
+  | accept => rfl
+  | branch i zero one ih₀ ih₁ =>
+    cases hb : ω i <;> simp [freshCharge,staleCharge,charge,hb,ih₀,ih₁]
+  | force i b next ih =>
+    by_cases hb : ω i = b
+    · simp only [freshCharge,staleCharge,charge,hb,↓reduceIte]
+      have he := ih (insert i R)
+      by_cases hi : i ∈ R <;> simp only [hi,↓reduceIte] <;> omega
+    · simp [freshCharge,staleCharge,charge,hb]
+
+omit [DecidableEq ι] in
+lemma followPath_raw_charge (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
+    (k : Card d → QueryTree ι) (ω : ι → Bool) :
+    (followPath d e x k).charge ω =
+      (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).charge ω := by
+  induction d with
+  | zero => rfl
+  | succ d ih =>
+    rw [followPath, ih]
+    let y := butterflyPerm d
+      (decodeButterfly d (fun i => ω (e (Sum.inr (x 0,i))))) (Fin.tail x)
+    change (if ω (e (Sum.inl y)) then
+      (k (Fin.cons (Bool.xor (x 0) true) y)).charge ω else
+      (k (Fin.cons (Bool.xor (x 0) false) y)).charge ω) =
+      (k (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).charge ω
+    cases ω (e (Sum.inl y)) <;> rfl
+
+omit [DecidableEq ι] in
+lemma forceList_raw_charge (L : List ι) (v : ι → Bool) (k : QueryTree ι) (ω : ι → Bool)
+    (h : ∀ i ∈ L, ω i = v i) :
+    (forceList L v k).charge ω = k.charge ω + L.length := by
+  induction L with
+  | nil => simp [forceList]
+  | cons i L ih =>
+    rw [forceList,charge,ite_eq_left (h i List.mem_cons_self),
+      ih (fun j hj => h j (List.mem_cons_of_mem _ hj))]
+    simp only [List.length_cons,Nat.add_assoc]
+
+noncomputable def closePath (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d)) :
+    QueryTree (SwitchIndex d) :=
+  forceList (routeDomain d x y).toList (routeValue d x y) k
+
+lemma closePath_succeeds (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
+    (ω : SwitchIndex d → Bool) :
+    (closePath d x y k).succeeds ω =
+      if butterflyPerm d (decodeButterfly d ω) x = y then k.succeeds ω else false := by
+  unfold closePath
+  rw [forceList_succeeds]
+  simp only [Finset.mem_toList, ← butterfly_endpoint_iff]
+
+lemma closePath_raw_charge (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
+    (ω : SwitchIndex d → Bool) (h : butterflyPerm d (decodeButterfly d ω) x = y) :
+    (closePath d x y k).charge ω = k.charge ω + d := by
+  unfold closePath
+  rw [forceList_raw_charge _ _ _ _ (by simpa only [Finset.mem_toList] using
+    (butterfly_endpoint_iff d ω x y).mp h), Finset.length_toList, card_routeDomain]
+
+lemma closePath_fresh_charge (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
+    (R : Finset (SwitchIndex d)) (ω : SwitchIndex d → Bool)
+    (h : butterflyPerm d (decodeButterfly d ω) x = y) :
+    (closePath d x y k).freshCharge R ω =
+      k.freshCharge (R ∪ routeDomain d x y) ω + (routeDomain d x y \ R).card := by
+  unfold closePath
+  rw [forceList_charge _ _ _ _ _ (by simpa only [Finset.mem_toList] using
+    (butterfly_endpoint_iff d ω x y).mp h), Finset.toList_toFinset]
+
+lemma closePath_stale_charge (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
+    (R : Finset (SwitchIndex d)) (ω : SwitchIndex d → Bool)
+    (h : butterflyPerm d (decodeButterfly d ω) x = y) :
+    (closePath d x y k).staleCharge R ω =
+      k.staleCharge (R ∪ routeDomain d x y) ω + (routeDomain d x y ∩ R).card := by
+  have h₀ := fresh_add_stale (closePath d x y k) R ω
+  have h₁ := fresh_add_stale k (R ∪ routeDomain d x y) ω
+  rw [closePath_raw_charge _ _ _ _ _ h,closePath_fresh_charge _ _ _ _ _ _ h] at h₀
+  have h₂ : (routeDomain d x y \ R).card + (routeDomain d x y ∩ R).card = d := by
+    rw [Finset.card_sdiff_add_card_inter,card_routeDomain]
+  omega
+
+noncomputable def seekCycle (d : ℕ) (S : Finset (Card d)) (a : Card d) :
+    ℕ → Card d → Finset (Card d) → (Finset (Card d) → QueryTree (SwitchIndex d)) →
+      QueryTree (SwitchIndex d)
+  | 0, x, V, k => if x ∈ S ∧ x ∉ V then closePath d x a (k (insert x V)) else reject
+  | r + 1, x, V, k => if x ∈ S ∧ x ∉ V then followPath d id x
+      (fun y => seekCycle d S a r y (insert x V) k) else reject
+
+lemma seekCycle_raw_charge (d : ℕ) (S : Finset (Card d)) (a : Card d) (r : ℕ)
+    (x : Card d) (V : Finset (Card d)) (k : Finset (Card d) → QueryTree (SwitchIndex d))
+    (ω : SwitchIndex d → Bool) (K : ℕ)
+    (hK : ∀ V, (k V).succeeds ω = true → (k V).charge ω = K)
+    (hs : (seekCycle d S a r x V k).succeeds ω = true) :
+    (seekCycle d S a r x V k).charge ω = K + d := by
+  induction r generalizing x V with
+  | zero =>
+    by_cases hv : x ∈ S ∧ x ∉ V
+    · rw [seekCycle,ite_eq_left hv,closePath_succeeds] at hs
+      split_ifs at hs with he
+      · rw [seekCycle,ite_eq_left hv,closePath_raw_charge _ _ _ _ _ he,hK _ hs]
+    · simp [seekCycle,hv,succeeds] at hs
+  | succ r ih =>
+    by_cases hv : x ∈ S ∧ x ∉ V
+    · rw [seekCycle,ite_eq_left hv,followPath_succeeds] at hs
+      rw [seekCycle,ite_eq_left hv,followPath_raw_charge]
+      exact ih _ _ hs
+    · simp [seekCycle,hv,succeeds] at hs
+
+noncomputable def seekCycles (d : ℕ) (S : Finset (Card d)) :
+    List (Card d × ℕ) → Finset (Card d) → QueryTree (SwitchIndex d)
+  | [], _ => accept
+  | (a,r) :: L, V => seekCycle d S a r a V (fun V => seekCycles d S L V)
+
+lemma seekCycles_raw_charge (d : ℕ) (S : Finset (Card d)) (L : List (Card d × ℕ))
+    (V : Finset (Card d)) (ω : SwitchIndex d → Bool)
+    (hs : (seekCycles d S L V).succeeds ω = true) :
+    (seekCycles d S L V).charge ω = L.length * d := by
+  induction L generalizing V with
+  | nil => simp [seekCycles,charge]
+  | cons ar L ih =>
+    obtain ⟨a,r⟩ := ar
+    rw [seekCycles]
+    rw [seekCycle_raw_charge d S a r a V (fun V => seekCycles d S L V) ω (L.length*d)
+      (fun V hs => ih V hs) hs]
+    simp [Nat.add_mul]
+
+lemma cycles_weighted_success (d : ℕ) (S : Finset (Card d)) (L : List (Card d × ℕ)) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      if (seekCycles d S L ∅).succeeds ω then
+        ((2:ℝ)^d)^L.length / (2:ℝ)^(seekCycles d S L ∅).staleCharge ∅ ω else 0) ≤ 1 := by
+  let T := seekCycles d S L ∅
+  have hh := mean_freshCharge_le_one T
+  have he : finiteMean (fun ω : SwitchIndex d → Bool =>
+      if T.succeeds ω then (2:ℝ)^T.freshCharge ∅ ω else 0) =
+      finiteMean (fun ω : SwitchIndex d → Bool =>
+        if T.succeeds ω then ((2:ℝ)^d)^L.length / (2:ℝ)^T.staleCharge ∅ ω else 0) := by
+    apply finiteMean_congr
+    intro ω
+    cases hs : T.succeeds ω
+    · simp
+    · simp only [↓reduceIte]
+      have hc := fresh_add_stale T ∅ ω
+      rw [seekCycles_raw_charge d S L ∅ ω hs] at hc
+      have hp : ((2:ℝ)^d)^L.length = (2:ℝ)^T.freshCharge ∅ ω * (2:ℝ)^T.staleCharge ∅ ω := by
+        rw [←pow_add,hc,←pow_mul,Nat.mul_comm]
+      rw [hp,mul_div_cancel_right₀ _ (pow_ne_zero _ (by norm_num))]
+  rw [← he]
+  exact hh
+
+end QueryTree
+
+lemma finiteMean_add {Ω : Type*} [Fintype Ω] (f g : Ω → ℝ) :
+    finiteMean (fun ω => f ω + g ω) = finiteMean f + finiteMean g := by
+  simp only [finiteMean,Finset.sum_add_distrib,add_div]
+
+lemma finiteMean_mul_const {Ω : Type*} [Fintype Ω] (f : Ω → ℝ) (c : ℝ) :
+    finiteMean (fun ω => f ω * c) = finiteMean f * c := by
+  unfold finiteMean
+  rw [← Finset.sum_mul]
+  ring
+
+lemma finiteMean_sum {Ω α : Type*} [Fintype Ω] (s : Finset α) (f : α → Ω → ℝ) :
+    finiteMean (fun ω => ∑ i ∈ s, f i ω) = ∑ i ∈ s, finiteMean (f i) := by
+  simp only [finiteMean,Finset.sum_div]
+  exact Finset.sum_comm
+
+lemma finiteMean_mono {Ω : Type*} [Fintype Ω] {f g : Ω → ℝ}
+    (h : ∀ ω, f ω ≤ g ω) : finiteMean f ≤ finiteMean g := by
+  exact div_le_div_of_nonneg_right (Finset.sum_le_sum (fun ω _ => h ω)) (Nat.cast_nonneg _)
+
+lemma finiteMean_equiv {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
+    (e : Ω ≃ Ω') (f : Ω → ℝ) :
+    finiteMean f = finiteMean (fun ω => f (e.symm ω)) := by
+  unfold finiteMean
+  rw [Fintype.card_congr e]
+  congr 1
+  exact Fintype.sum_equiv e _ _ (fun ω => by simp)
+
+lemma finiteMean_prod_left {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] [Nonempty Ω']
+    (f : Ω → ℝ) : finiteMean (fun ω : Ω × Ω' => f ω.1) = finiteMean f := by
+  simp only [finiteMean,Fintype.sum_prod_type,Finset.sum_const,nsmul_eq_mul,
+    ← Finset.mul_sum,Fintype.card_prod,Nat.cast_mul,Finset.card_univ]
+  have hn : (Fintype.card Ω' : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
+  field_simp
+
+lemma finiteMean_pi_eval {C : Type*} [Fintype C] [DecidableEq C]
+    {A : C → Type*} [∀ c, Fintype (A c)] [∀ c, Nonempty (A c)]
+    (c : C) (g : A c → ℝ) :
+    finiteMean (fun z : ∀ c, A c => g (z c)) = finiteMean g := by
+  rw [finiteMean_equiv (Equiv.piSplitAt c A)]
+  have he : (fun z : A c × ((j : {j // j ≠ c}) → A j) =>
+      g ((Equiv.piSplitAt c A).symm z c)) = (fun z => g z.1) := by
+    funext z
+    simp [Equiv.piSplitAt]
+  rw [he,finiteMean_prod_left]
+
+lemma finiteMean_eq_indicator {Ω : Type*} [Fintype Ω] [DecidableEq Ω] (a : Ω) :
+    finiteMean (fun ω : Ω => if ω = a then (1:ℝ) else 0) = 1 / Fintype.card Ω := by
+  simp [finiteMean]
+
+lemma exp_finiteMean_le {Ω : Type*} [Fintype Ω] [Nonempty Ω] (f : Ω → ℝ) :
+    Real.exp (finiteMean f) ≤ finiteMean (fun ω => Real.exp (f ω)) := by
+  have hn : (Fintype.card Ω : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
+  have hw : ∑ ω : Ω, (1 / (Fintype.card Ω : ℝ)) = 1 := by simp [hn]
+  have hj := convexOn_exp.map_sum_le (t := Finset.univ)
+    (w := fun _ : Ω => 1 / (Fintype.card Ω : ℝ)) (p := f)
+    (fun _ _ => by positivity) hw (fun _ _ => Set.mem_univ _)
+  simp only [smul_eq_mul, ← Finset.mul_sum] at hj
+  simpa only [finiteMean, div_eq_mul_inv, one_div, mul_comm, mul_one, one_mul] using hj
+
+variable {d A : ℕ}
+
+def cycleFiber (o : Card d → Option (Fin A)) (i : Fin A) : Finset (Card d) := by
+  classical
+  exact Finset.univ.filter (fun x => o x = some i)
+
+@[simp] lemma mem_cycleFiber (o : Card d → Option (Fin A)) (i : Fin A) (x : Card d) :
+    x ∈ cycleFiber o i ↔ o x = some i := by
+  classical
+  simp [cycleFiber]
+
+abbrev Terminal (o : Card d → Option (Fin A)) := ∀ i : Fin A, {x : Card d // o x = some i}
+
+noncomputable def slotWeight (o : Card d → Option (Fin A)) (i : Fin A) : ℝ :=
+  1 / (cycleFiber o i).card
+
+noncomputable def pathWeight (o : Card d → Option (Fin A)) (x : Card d) : ℝ :=
+  match o x with
+  | none => 0
+  | some i => slotWeight o i
+
+noncomputable def withinWeight (o : Card d → Option (Fin A)) (u v : Card d) : ℝ :=
+  match o u, o v with
+  | some i, some j => if i = j then slotWeight o i else 0
+  | _, _ => 0
+
+end Thorp
+
+end ThorpNine.Harmonic
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Harmonic/InputTree.lean b/lean/OAI/Probability/ThorpRouting/Harmonic/InputTree.lean
index 376ba4ed80bacfa7377fe34235f4d989b18f7e8f..9d0536b69a9b583a2542e2c10c2fb39df021b1cd 100644
--- a/lean/OAI/Probability/ThorpRouting/Harmonic/InputTree.lean
+++ b/lean/OAI/Probability/ThorpRouting/Harmonic/InputTree.lean
@@ -1,869 +1,869 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Harmonic.PalindromeFamilyCost
-
-namespace OAI
-
-namespace ThorpNine.Harmonic
-
-namespace Thorp
-
-lemma centralSlabCost_perturb (l L r : ℕ) (C D : Card r → Equiv.Perm (Card L))
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (ω : BenesCoins (L+r)) :
-    centralSlabCost l L r C e ω ≤ centralSlabCost l L r D e ω+l*centralDistance L r C D := by
-  have hh := Finset.sum_le_sum (fun (i : Fin l) (_ : i ∈ Finset.univ) =>
-    centralLevelCost_perturb (l+i.val+1) L r C D e ω)
-  simpa only [Finset.sum_add_distrib,Finset.sum_const,Finset.card_univ,Fintype.card_fin,
-    nsmul_eq_mul,Nat.cast_id,centralSlabCost] using hh
-
-lemma centralDistance_exceptional (L r : ℕ) (C D : Card r → Equiv.Perm (Card L))
-    (E : Finset (Card r)) (h : ∀ u ∉ E, C u = D u) :
-    centralDistance L r C D ≤ E.card*2^L := by
-  classical
-  calc
-    _ ≤ ∑ u : Card r, if u ∈ E then 2^L else 0 := by
-      apply Finset.sum_le_sum
-      intro u _
-      split_ifs with hu
-      · simpa only [card_positions] using permDistance_le (C u) (D u)
-      · rw [h u hu,permDistance_refl]
-    _ = _ := by simp
-
-
-inductive InputTree : ℕ → Type
-  | tip {d : ℕ} (p : Equiv.Perm (Card d)) : InputTree d
-  | node {d : ℕ} (ξ : Card d → Bool) (T : Bool → InputTree d) : InputTree (d+1)
-
-def InputTree.perm : {d : ℕ} → InputTree d → Equiv.Perm (Card d)
-  | _, .tip p => p
-  | _, .node ξ T => pairSwitch ξ * childLift (fun b => (T b).perm)
-
-noncomputable def InputTree.levelCost (t : ℕ) : {d : ℕ} → InputTree d →
-    {ι : Type*} → [Fintype ι] → (ι ↪ Card d) → (SwitchIndex d → Bool) → ℕ
-  | _, .tip _, _, _, _, _ => 0
-  | d+1, .node ξ T, _, _, e, Y =>
-    let x := e.trans (headTailEquiv d).toEmbedding
-    let c := PairRouting.colors x ξ
-    let hc := PairRouting.colors_compatible x ξ
-    let σ := coinStepEquiv d Y
-    if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x ξ)
-      (fun b => if b then butterflyPerm d (decodeButterfly d σ.1.2) * (T true).perm⁻¹
-        else butterflyPerm d (decodeButterfly d σ.1.1) * (T false).perm⁻¹)
-    else (T false).levelCost t (PairRouting.childEmbedding x c hc false) σ.1.1 +
-      (T true).levelCost t (PairRouting.childEmbedding x c hc true) σ.1.2
-
-lemma InputTree.levelCost_node (t d : ℕ) (ξ : Card d → Bool) (T : Bool → InputTree d)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (d+1))
-    (Y : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (η : Card d → Bool) :
-    (InputTree.node ξ T).levelCost t e ((coinStepEquiv d).symm (Y,η)) =
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x ξ
-      let hc := PairRouting.colors_compatible x ξ
-      if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x ξ)
-        (fun b => if b then butterflyPerm d (decodeButterfly d Y.2) * (T true).perm⁻¹
-          else butterflyPerm d (decodeButterfly d Y.1) * (T false).perm⁻¹)
-      else (T false).levelCost t (PairRouting.childEmbedding x c hc false) Y.1 +
-        (T true).levelCost t (PairRouting.childEmbedding x c hc true) Y.2 := by
-  rw [InputTree.levelCost]
-  rfl
-
-lemma InputTree.levelCost_gt (t d : ℕ) (T : InputTree d) (ht : d < t)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) :
-    T.levelCost t e Y = 0 := by
-  induction T generalizing ι with
-  | tip p => rfl
-  | @node d ξ T ih =>
-    rw [InputTree.levelCost,ite_eq_right (by omega : t ≠ d+1),ih false (by omega),ih true (by omega)]
-
-lemma InputTree.levelCost_adapted (t d : ℕ) (T : InputTree d)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) :
-    LayerAdapted d (t-1) (fun Y => T.levelCost t e Y) := by
-  induction T generalizing ι with
-  | tip p => intro Y Y' _; rfl
-  | @node d ξ T ih =>
-    intro Y Y' h
-    dsimp only
-    conv_lhs => rw [←(coinStepEquiv d).symm_apply_apply Y]
-    conv_rhs => rw [←(coinStepEquiv d).symm_apply_apply Y']
-    rw [InputTree.levelCost_node t d ξ T e ((coinStepEquiv d Y).1) ((coinStepEquiv d Y).2),
-      InputTree.levelCost_node t d ξ T e ((coinStepEquiv d Y').1) ((coinStepEquiv d Y').2)]
-    by_cases ht : t = d+1
-    · subst t
-      rw [ite_eq_left rfl,ite_eq_left rfl]
-      have he₀ : (coinStepEquiv d Y).1.1 = (coinStepEquiv d Y').1.1 := by
-        funext i
-        exact h (Sum.inr (false,i)) (by simpa only [switchHeight,Nat.add_sub_cancel] using switchHeight_le d i)
-      have he₁ : (coinStepEquiv d Y).1.2 = (coinStepEquiv d Y').1.2 := by
-        funext i
-        exact h (Sum.inr (true,i)) (by simpa only [switchHeight,Nat.add_sub_cancel] using switchHeight_le d i)
-      rw [he₀,he₁]
-    · rw [ite_eq_right ht,ite_eq_right ht]
-      congr 1
-      · apply ih false
-        intro i hi
-        exact h (Sum.inr (false,i)) hi
-      · apply ih true
-        intro i hi
-        exact h (Sum.inr (true,i)) hi
-
-theorem InputTree.level_partial_mgf (r s u : ℕ) (hu : u < r)
-    (T : InputTree (s+r)) {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+r))
-    (lo : Card r × SwitchIndex s → Bool) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(T.levelCost (s+u+1) e (assembleBits r s lo hi))) ≤
-      Real.exp (Fintype.card ι * cycleCoefficient u J q) := by
-  classical
-  induction r generalizing u ι with
-  | zero => omega
-  | succ r ih =>
-    simp only [Nat.add_succ] at *
-    cases T with
-    | tip p =>
-      simp only [InputTree.levelCost,pow_zero,finiteMean_const]
-      exact Real.one_le_exp (mul_nonneg (Nat.cast_nonneg _) (cycleCoefficient_nonneg u J q hq))
-    | node ξ T =>
-      let x := e.trans (headTailEquiv (s+r)).toEmbedding
-      let c := PairRouting.colors x ξ
-      let hc := PairRouting.colors_compatible x ξ
-      let e₀ := PairRouting.childEmbedding x c hc false
-      let e₁ := PairRouting.childEmbedding x c hc true
-      let e' := PairRouting.switchedEmbedding x ξ
-      let lo₀ : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons false j.1,j.2)
-      let lo₁ : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons true j.1,j.2)
-      let Y₀ := fun hi : Card s × SwitchIndex r → Bool => assembleBits r s lo₀ hi
-      let Y₁ := fun hi : Card s × SwitchIndex r → Bool => assembleBits r s lo₁ hi
-      let N := fun hi : (Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool) =>
-        PairRouting.alternatingCycles e' (fun b =>
-          if b then butterflyPerm (s+r) (decodeButterfly (s+r) (Y₁ hi.2)) * (T true).perm⁻¹
-          else butterflyPerm (s+r) (decodeButterfly (s+r) (Y₀ hi.1)) * (T false).perm⁻¹)
-      let L₀ := fun hi : Card s × SwitchIndex r → Bool =>
-        (T false).levelCost (s+u+1) e₀ (Y₀ hi)
-      let L₁ := fun hi : Card s × SwitchIndex r → Bool =>
-        (T true).levelCost (s+u+1) e₁ (Y₁ hi)
-      have hs (hi : (Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool))
-          (η : Card s × Card r → Bool) :
-          (InputTree.node ξ T).levelCost (s+u+1) e
-            (assembleBits (r+1) s lo ((highStepEquiv r s).symm (hi,η))) =
-            if u = r then N hi else L₀ hi.1 + L₁ hi.2 := by
-        rw [assembleBits_step]
-        rw [InputTree.levelCost_node (s+u+1) (s+r) ξ T e (Y₀ hi.1,Y₁ hi.2) (fun y => η (((cardSplit r s) y).2,((cardSplit r s) y).1))]
-        simp only [Nat.add_right_cancel_iff,Nat.add_left_cancel_iff]
-        rfl
-      rw [finiteMean_equiv (highStepEquiv r s),finiteMean_prod]
-      simp_rw [hs,finiteMean_const]
-      by_cases hur : u = r
-      · subst u
-        simp only [ite_true]
-        rw [finiteMean_prod,finiteMean_comm]
-        calc
-          _ ≤ finiteMean (fun _hi : Card s × SwitchIndex r → Bool =>
-            Real.exp (Fintype.card ι * cycleCoefficient r J q)) := by
-            apply finiteMean_mono
-            intro hi
-            have hn := node_partial_mgf r s e' (T false).perm (T true).perm
-              (butterflyPerm (s+r) (decodeButterfly (s+r) (Y₁ hi))) lo₀ J hJ q hq
-            exact hn
-          _ = _ := finiteMean_const _
-      · simp only [ite_eq_right hur,pow_add]
-        rw [finiteMean_prod_mul (fun hi => q ^ L₀ hi) (fun hi => q ^ L₁ hi)]
-        have hu' : u < r := by omega
-        have h₀ := ih u hu' (T false) e₀ lo₀
-        have h₁ := ih u hu' (T true) e₁ lo₁
-        have hp := mul_le_mul h₀ h₁ (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
-          (Real.exp_nonneg _)
-        refine hp.trans_eq ?_
-        rw [←Real.exp_add]
-        congr 1
-        have hcount := PairRouting.color_card_add c
-        have hcount' : (Fintype.card {i : ι // c i = false} : ℝ) +
-            Fintype.card {i : ι // c i = true} = Fintype.card ι := by exact_mod_cast hcount
-        change (Fintype.card {i : ι // c i = false} : ℝ) * cycleCoefficient u J q +
-            Fintype.card {i : ι // c i = true} * cycleCoefficient u J q = _
-        rw [←add_mul,hcount']
-
-
-noncomputable def InputTree.slabCost (l d : ℕ) (T : InputTree d) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) : ℕ :=
-  ∑ i : Fin l, T.levelCost (l+i.1+1) e Y
-
-lemma InputTree.slabCost_adapted (l d : ℕ) (T : InputTree d) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) :
-    LayerAdapted d (2*l) (fun Y => InputTree.slabCost l d T e Y) := by
-  apply layerAdapted_sum
-  intro i _
-  exact layerAdapted_mono (InputTree.levelCost_adapted (l+i.1+1) d T e) (by omega)
-
-theorem InputTree.slab_partial_mgf (r s l : ℕ) (hl : 0 < l) (hs : 2*s ≤ l) (T : InputTree (s+r))
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+r))
-    (lo : Card r × SwitchIndex s → Bool) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(InputTree.slabCost l (s+r) T e (assembleBits r s lo hi))) ≤
-      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
-  classical
-  let : NeZero l := ⟨ne_of_gt hl⟩
-  have hqp : 0 < q := lt_of_lt_of_le (by norm_num) hq
-  calc
-    _ ≤ finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      finiteMean (fun i : Fin l => q^(l*T.levelCost (l+i.1+1) e
-        (assembleBits r s lo hi)))) := by
-      apply finiteMean_mono
-      intro hi
-      simpa only [InputTree.slabCost,Fintype.card_fin] using
-        pow_sum_le_mean_powers (fun i : Fin l => T.levelCost (l+i.1+1) e
-          (assembleBits r s lo hi)) q hqp
-    _ = finiteMean (fun i : Fin l => finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(l*T.levelCost (l+i.1+1) e (assembleBits r s lo hi)))) := finiteMean_comm _
-    _ ≤ finiteMean (fun _i : Fin l =>
-      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64)))) := by
-      apply finiteMean_mono
-      intro i
-      by_cases ht : l+i.1+1 ≤ s+r
-      · have hs' : s ≤ l+i.1 := by omega
-        have he : l+i.1+1 = s+(l+i.1-s)+1 := by omega
-        have hu : l+i.1-s < r := by omega
-        have hm := InputTree.level_partial_mgf r s (l+i.1-s) hu T e lo
-          ⌈Real.exp ((l:ℝ)/32)⌉₊ (cycleCutoff_pos l) (q^l) (one_le_pow₀ hq)
-        have hc := cycleCoefficient_decay l (l+i.1-s) (by
-          rw [Nat.cast_sub hs',Nat.cast_add]
-          have hsr : (2:ℝ)*s ≤ l := by exact_mod_cast hs
-          have hi : (0:ℝ) ≤ i.1 := Nat.cast_nonneg _
-          linarith) (q^l) (one_le_pow₀ hq) (by
-            rw [Real.log_pow]
-            nlinarith [mul_le_mul_of_nonneg_left hlog (Nat.cast_nonneg l)])
-        simp_rw [he,pow_mul]
-        exact hm.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left hc (Nat.cast_nonneg _)))
-      · have hz := InputTree.levelCost_gt (l+i.1+1) (s+r) T (by omega) e
-        simp only [hz,Nat.mul_zero,pow_zero,finiteMean_const]
-        exact Real.one_le_exp_iff.mpr (by positivity)
-    _ = _ := finiteMean_const _
-
-
-def boundaryTree (L : ℕ) : (r : ℕ) → (Card r → Equiv.Perm (Card L)) →
-    (SwitchIndex (L+r) → Bool) → InputTree (L+r)
-  | 0, U, _ => .tip (U (fun j => Fin.elim0 j))
-  | r+1, U, X => .node (coinStepEquiv (L+r) X).2 (fun b =>
-      boundaryTree L r (fun u => U (Fin.cons b u))
-        (if b then (coinStepEquiv (L+r) X).1.2 else (coinStepEquiv (L+r) X).1.1))
-
-def boundaryOutput (L : ℕ) : (r : ℕ) → (SwitchIndex (L+r) → Bool) →
-    Card r → Equiv.Perm (Card L)
-  | 0, Y, _ => butterflyPerm L (decodeButterfly L Y)
-  | r+1, Y, u => boundaryOutput L r
-      (if u 0 then (coinStepEquiv (L+r) Y).1.2 else (coinStepEquiv (L+r) Y).1.1) (Fin.tail u)
-
-lemma boundaryOutput_adapted (L r : ℕ) :
-    LayerAdapted (L+r) L (boundaryOutput L r) := by
-  induction r with
-  | zero =>
-    intro Y Y' h
-    have he : Y = Y' := funext (fun i => h i (by simpa only [Nat.add_zero] using switchHeight_le L i))
-    rw [he]
-  | succ r ih =>
-    intro Y Y' h
-    funext u
-    dsimp only [boundaryOutput]
-    cases hb : u 0 <;> simp only [Bool.false_eq_true,↓reduceIte]
-    · exact congrFun (ih _ _ (fun i hi => h (Sum.inr (false,i)) hi)) (Fin.tail u)
-    · exact congrFun (ih _ _ (fun i hi => h (Sum.inr (true,i)) hi)) (Fin.tail u)
-
-noncomputable def modifiedBoundary (L r : ℕ) (E : Finset (Card r))
-    (U : Card r → Equiv.Perm (Card L)) (Y : SwitchIndex (L+r) → Bool) :
-    Card r → Equiv.Perm (Card L) := fun u => if u ∈ E then boundaryOutput L r Y u else U u
-
-lemma modifiedBoundary_adapted (L r : ℕ) (E : Finset (Card r))
-    (U : Card r → Equiv.Perm (Card L)) :
-    LayerAdapted (L+r) L (modifiedBoundary L r E U) := by
-  intro Y Y' h
-  unfold modifiedBoundary
-  rw [boundaryOutput_adapted L r Y Y' h]
-
-lemma permDistance_boundaryTree (L r : ℕ) (U V : Card r → Equiv.Perm (Card L))
-    (X : SwitchIndex (L+r) → Bool) :
-    permDistance (boundaryTree L r U X).perm (boundaryTree L r V X).perm =
-      centralDistance L r U V := by
-  induction r with
-  | zero =>
-    simp only [boundaryTree,InputTree.perm,centralDistance,Fintype.sum_unique]
-    congr 1
-  | succ r ih =>
-    simp only [boundaryTree,InputTree.perm,permDistance_mul_left]
-    rw [permDistance_childLift,centralDistance_step,ih,ih]
-
-lemma boundaryTree_levelCost_perturb (t L r : ℕ) (U V : Card r → Equiv.Perm (Card L))
-    (X : SwitchIndex (L+r) → Bool) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (L+r)) (Y : SwitchIndex (L+r) → Bool) :
-    (boundaryTree L r U X).levelCost t e Y ≤
-      (boundaryTree L r V X).levelCost t e Y + centralDistance L r U V := by
-  induction r generalizing ι with
-  | zero => simp only [boundaryTree,InputTree.levelCost,Nat.zero_add,Nat.zero_le]
-  | succ r ih =>
-    rw [boundaryTree,boundaryTree,InputTree.levelCost,InputTree.levelCost]
-    by_cases ht : t = L+r+1
-    · simp only [ite_eq_left ht]
-      have hh := alternatingCycles_distance_le
-        (PairRouting.switchedEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
-          (coinStepEquiv (L+r) X).2)
-        (fun b => if b then butterflyPerm (L+r) (decodeButterfly (L+r) (coinStepEquiv (L+r) Y).1.2) *
-            (boundaryTree L r (fun u => U (Fin.cons true u)) (coinStepEquiv (L+r) X).1.2).perm⁻¹
-          else butterflyPerm (L+r) (decodeButterfly (L+r) (coinStepEquiv (L+r) Y).1.1) *
-            (boundaryTree L r (fun u => U (Fin.cons false u)) (coinStepEquiv (L+r) X).1.1).perm⁻¹)
-        (fun b => if b then butterflyPerm (L+r) (decodeButterfly (L+r) (coinStepEquiv (L+r) Y).1.2) *
-            (boundaryTree L r (fun u => V (Fin.cons true u)) (coinStepEquiv (L+r) X).1.2).perm⁻¹
-          else butterflyPerm (L+r) (decodeButterfly (L+r) (coinStepEquiv (L+r) Y).1.1) *
-            (boundaryTree L r (fun u => V (Fin.cons false u)) (coinStepEquiv (L+r) X).1.1).perm⁻¹)
-      simpa only [Bool.false_eq_true,↓reduceIte,permDistance_mul_left,permDistance_inv,
-        permDistance_boundaryTree,←centralDistance_step] using hh
-    · simp only [ite_eq_right ht,Bool.false_eq_true,↓reduceIte]
-      rw [centralDistance_step]
-      have h₀ := ih (fun u => U (Fin.cons false u)) (fun u => V (Fin.cons false u))
-        (coinStepEquiv (L+r) X).1.1
-        (PairRouting.childEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
-          (PairRouting.colors (e.trans (headTailEquiv (L+r)).toEmbedding) (coinStepEquiv (L+r) X).2)
-          (PairRouting.colors_compatible _ _) false) (coinStepEquiv (L+r) Y).1.1
-      have h₁ := ih (fun u => U (Fin.cons true u)) (fun u => V (Fin.cons true u))
-        (coinStepEquiv (L+r) X).1.2
-        (PairRouting.childEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
-          (PairRouting.colors (e.trans (headTailEquiv (L+r)).toEmbedding) (coinStepEquiv (L+r) X).2)
-          (PairRouting.colors_compatible _ _) true) (coinStepEquiv (L+r) Y).1.2
-      omega
-
-lemma boundaryTree_slabCost_perturb (l L r : ℕ) (U V : Card r → Equiv.Perm (Card L))
-    (X : SwitchIndex (L+r) → Bool) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (L+r)) (Y : SwitchIndex (L+r) → Bool) :
-    InputTree.slabCost l (L+r) (boundaryTree L r U X) e Y ≤
-      InputTree.slabCost l (L+r) (boundaryTree L r V X) e Y + l*centralDistance L r U V := by
-  have hh := Finset.sum_le_sum (fun (i : Fin l) (_ : i ∈ Finset.univ) =>
-    boundaryTree_levelCost_perturb (l+i.val+1) L r U V X e Y)
-  simpa only [Finset.sum_add_distrib,Finset.sum_const,Finset.card_univ,Fintype.card_fin,
-    nsmul_eq_mul,Nat.cast_id,InputTree.slabCost] using hh
-
-lemma modifiedBoundary_distance_le (L r : ℕ) (E : Finset (Card r))
-    (U : Card r → Equiv.Perm (Card L)) (Y : SwitchIndex (L+r) → Bool) :
-    centralDistance L r (modifiedBoundary L r E U Y) U ≤ E.card*2^L := by
-  apply centralDistance_exceptional
-  intro u hu
-  exact ite_eq_right hu
-
-
-lemma InputTree.slabCost_gt (l d : ℕ) (T : InputTree d) (hd : d ≤ l)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) :
-    InputTree.slabCost l d T e Y = 0 := by
-  apply Finset.sum_eq_zero
-  intro i _
-  exact InputTree.levelCost_gt _ _ T (by omega) _ _
-
-lemma InputTree.random_slab_adapted_mul_le (d s l : ℕ) (hsd : s ≤ d) (hl : 0 < l)
-    (hs : 2*s ≤ l) (T : (SwitchIndex d → Bool) → InputTree d)
-    (hT : LayerAdapted d s T) {ι : Type*} [Fintype ι] (e : ι ↪ Card d)
-    (F : (SwitchIndex d → Bool) → ℝ) (hF : ∀ Y, 0 ≤ F Y) (hA : LayerAdapted d s F)
-    (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun Y => F Y*q^(InputTree.slabCost l d (T Y) e Y)) ≤
-      finiteMean F * Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
-  obtain ⟨r,rfl⟩ := Nat.exists_eq_add_of_le hsd
-  apply adapted_mul_mean_le r s F _ hF hA
-  intro lo
-  let hi₀ : Card s × SwitchIndex r → Bool := fun _ => false
-  have he (hi : Card s × SwitchIndex r → Bool) :
-      T (assembleBits r s lo hi) = T (assembleBits r s lo hi₀) :=
-    hT _ _ (assembleBits_agree r s lo hi hi₀)
-  simp_rw [he]
-  exact InputTree.slab_partial_mgf r s l hl hs _ e lo q hq hlog
-
-lemma InputTree.slab_mgf (l d : ℕ) (hl : 0 < l) (T : InputTree d)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun Y => q^(InputTree.slabCost l d T e Y)) ≤
-      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
-  have hh := InputTree.random_slab_adapted_mul_le d 0 l (Nat.zero_le _) hl (Nat.zero_le _)
-    (fun _ => T) (fun _ _ _ => rfl) e (fun _ => 1) (fun _ => by norm_num)
-    (fun _ _ _ => rfl) q hq hlog
-  simpa only [one_mul,finiteMean_const] using hh
-
-noncomputable def InputTree.familyCost (L n d : ℕ)
-    (T : (SwitchIndex d → Bool) → InputTree d) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) : ℕ :=
-  ∑ j ∈ Finset.range n, InputTree.slabCost (L*4^j) d (T Y) e Y
-
-lemma InputTree.random_slabCost_adapted (l d s : ℕ) (T : (SwitchIndex d → Bool) → InputTree d)
-    (hT : LayerAdapted d s T) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) :
-    LayerAdapted d (max s (2*l)) (fun Y => InputTree.slabCost l d (T Y) e Y) := by
-  intro Y Y' h
-  dsimp only
-  rw [hT _ _ (agreeThrough_mono (le_max_left _ _) h)]
-  exact InputTree.slabCost_adapted l d (T Y') e Y Y'
-    (agreeThrough_mono (le_max_right _ _) h)
-
-lemma InputTree.familyCost_adapted (L n d s : ℕ) (T : (SwitchIndex d → Bool) → InputTree d)
-    (hT : LayerAdapted d s T) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) :
-    LayerAdapted d (max s (familyEnd L n)) (fun Y => InputTree.familyCost L n d T e Y) := by
-  apply layerAdapted_sum
-  intro j hj
-  exact layerAdapted_mono (InputTree.random_slabCost_adapted (L*4^j) d s T hT e)
-    (max_le_max_left s (familyEnd_ge L n j (Finset.mem_range.mp hj)))
-
-theorem InputTree.family_mgf_first (L n d s : ℕ) (hL : 0 < L) (hs : s ≤ L)
-    (T : (SwitchIndex d → Bool) → InputTree d) (hT : LayerAdapted d s T)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) (p : ℝ) (hp : 0 ≤ p)
-    (hfirst : finiteMean (fun Y => q^(InputTree.slabCost L d (T Y) e Y)) ≤
-      Real.exp (p+Fintype.card ι*(68*Real.exp (-(L:ℝ)/64)))) :
-    finiteMean (fun Y => q^(InputTree.familyCost L n d T e Y)) ≤
-      Real.exp (p+Fintype.card ι*(68*∑ j ∈ Finset.range n,
-        Real.exp (-((L*4^j:ℕ):ℝ)/64))) := by
-  induction n with
-  | zero =>
-    simp only [InputTree.familyCost,Finset.range_zero,Finset.sum_empty,pow_zero,
-      finiteMean_const,mul_zero,add_zero]
-    exact Real.one_le_exp hp
-  | succ n ih =>
-    by_cases hn : n = 0
-    · subst n
-      simpa only [InputTree.familyCost,Nat.zero_add,Finset.sum_range_one,pow_zero,mul_one] using hfirst
-    · let l := L*4^n
-      have hl : 0 < l := Nat.mul_pos hL (pow_pos (by decide) _)
-      have hsl : 2*familyEnd L n ≤ l := familyEnd_two L n
-      have hs' : s ≤ familyEnd L n := by
-        have hh := familyEnd_ge L n 0 (Nat.pos_of_ne_zero hn)
-        simp only [pow_zero,mul_one] at hh
-        omega
-      have hc : ∀ Y, InputTree.familyCost L (n+1) d T e Y =
-          InputTree.familyCost L n d T e Y+InputTree.slabCost l d (T Y) e Y := by
-        intro Y
-        exact Finset.sum_range_succ _ _
-      have hm : finiteMean (fun Y => q^(InputTree.familyCost L (n+1) d T e Y)) ≤
-          finiteMean (fun Y => q^(InputTree.familyCost L n d T e Y)) *
-            Real.exp (Fintype.card ι*(68*Real.exp (-(l:ℝ)/64))) := by
-        by_cases hld : l ≤ d
-        · simp_rw [hc,pow_add]
-          apply InputTree.random_slab_adapted_mul_le d (familyEnd L n) l (by omega) hl hsl T
-            (layerAdapted_mono hT hs') e _ (fun _ => pow_nonneg (by linarith) _)
-          · apply layerAdapted_comp
-            exact layerAdapted_mono (InputTree.familyCost_adapted L n d s T hT e) (max_le hs' le_rfl)
-          · exact hq
-          · exact hlog
-        · simp_rw [hc,InputTree.slabCost_gt l d _ (by omega),Nat.add_zero]
-          exact le_mul_of_one_le_right
-            (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
-            (Real.one_le_exp_iff.mpr (by positivity))
-      refine (hm.trans (mul_le_mul_of_nonneg_right ih (Real.exp_nonneg _))).trans_eq ?_
-      rw [←Real.exp_add,Finset.sum_range_succ]
-      congr 1
-      dsimp only [l]
-      ring
-
-lemma InputTree.family_mgf_first_decay (L n d s : ℕ) (hL : 0 < L) (hs : s ≤ L)
-    (T : (SwitchIndex d → Bool) → InputTree d) (hT : LayerAdapted d s T)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) (p : ℝ) (hp : 0 ≤ p)
-    (hfirst : finiteMean (fun Y => q^(InputTree.slabCost L d (T Y) e Y)) ≤
-      Real.exp (p+Fintype.card ι*(68*Real.exp (-(L:ℝ)/64)))) :
-    finiteMean (fun Y => q^(InputTree.familyCost L n d T e Y)) ≤
-      Real.exp (p+Fintype.card ι*heightDecay L) := by
-  apply (InputTree.family_mgf_first L n d s hL hs T hT e q hq hlog p hp hfirst).trans
-  apply Real.exp_le_exp.mpr
-  apply add_le_add_right
-  apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
-  have hh := mul_le_mul_of_nonneg_left (sum_family_decay L n hL) (by norm_num : (0:ℝ)≤68)
-  simpa only [heightDecay,mul_div_assoc] using hh
-
-
-noncomputable def modifiedInputTree (L r : ℕ) (E : Finset (Card r))
-    (U : Card r → Equiv.Perm (Card L)) (X Y : SwitchIndex (L+r) → Bool) : InputTree (L+r) :=
-  boundaryTree L r (modifiedBoundary L r E U Y) X
-
-lemma modifiedInputTree_adapted (L r : ℕ) (E : Finset (Card r))
-    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool) :
-    LayerAdapted (L+r) L (modifiedInputTree L r E U X) := by
-  intro Y Y' h
-  unfold modifiedInputTree
-  rw [modifiedBoundary_adapted L r E U Y Y' h]
-
-lemma modified_slab_mgf (l L r : ℕ) (hl : 0 < l) (E : Finset (Card r))
-    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun Y => q^(InputTree.slabCost l (L+r) (modifiedInputTree L r E U X Y) e Y)) ≤
-      Real.exp (((l*E.card*2^L:ℕ):ℝ)*Real.log q+
-        Fintype.card ι*(68*Real.exp (-(l:ℝ)/64))) := by
-  have hqp : 0 < q := by linarith
-  have hpoint (Y : SwitchIndex (L+r) → Bool) :
-      InputTree.slabCost l (L+r) (modifiedInputTree L r E U X Y) e Y ≤
-        InputTree.slabCost l (L+r) (boundaryTree L r U X) e Y+l*E.card*2^L := by
-    apply (boundaryTree_slabCost_perturb l L r (modifiedBoundary L r E U Y) U X e Y).trans
-    apply Nat.add_le_add_left
-    simpa only [Nat.mul_assoc] using Nat.mul_le_mul_left l (modifiedBoundary_distance_le L r E U Y)
-  calc
-    _ ≤ finiteMean (fun Y => q^(InputTree.slabCost l (L+r) (boundaryTree L r U X) e Y+l*E.card*2^L)) :=
-      finiteMean_mono (fun Y => pow_le_pow_right₀ hq (hpoint Y))
-    _ = finiteMean (fun Y => q^(InputTree.slabCost l (L+r) (boundaryTree L r U X) e Y)) *
-        q^(l*E.card*2^L) := by simp_rw [pow_add]; exact finiteMean_mul_const _ _
-    _ ≤ Real.exp (Fintype.card ι*(68*Real.exp (-(l:ℝ)/64))) * q^(l*E.card*2^L) :=
-      mul_le_mul_of_nonneg_right (InputTree.slab_mgf l (L+r) hl _ e q hq hlog) (pow_nonneg (by linarith) _)
-    _ = _ := by
-      have he : Real.exp (((l*E.card*2^L:ℕ):ℝ)*Real.log q) = q^(l*E.card*2^L) := by
-        rw [Real.exp_nat_mul,Real.exp_log hqp]
-      rw [Real.exp_add,he,mul_comm]
-
-lemma modified_family_mgf (l n L r : ℕ) (hl : 0 < l) (hLl : L ≤ l) (E : Finset (Card r))
-    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun Y => q^(InputTree.familyCost l n (L+r) (modifiedInputTree L r E U X) e Y)) ≤
-      Real.exp (((l*E.card*2^L:ℕ):ℝ)*Real.log q+Fintype.card ι*heightDecay l) := by
-  exact InputTree.family_mgf_first_decay l n (L+r) L hl hLl _
-    (modifiedInputTree_adapted L r E U X) e q hq hlog _
-    (mul_nonneg (Nat.cast_nonneg _) (Real.log_nonneg hq))
-    (modified_slab_mgf l L r hl E U X e q hq hlog)
-
-theorem modified_high_mgf (L n r : ℕ) (hL : 0 < L) (E : Finset (Card r))
-    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/500) :
-    finiteMean (fun Y => q^(InputTree.familyCost L n (L+r) (modifiedInputTree L r E U X) e Y+
-      InputTree.familyCost (2*L) n (L+r) (modifiedInputTree L r E U X) e Y)) ≤
-      Real.exp (Fintype.card ι*heightDecay L +
-        3*((L*E.card*2^L:ℕ):ℝ)*Real.log q) := by
-  have hq2 := one_le_pow₀ (n:=2) hq
-  have hlog2 : Real.log (q^2) ≤ 1/250 := by rw [Real.log_pow]; norm_num; linarith
-  have h₀ := modified_family_mgf L n L r hL le_rfl E U X e (q^2) hq2 hlog2
-  have h₁ := modified_family_mgf (2*L) n L r (by omega) (by omega) E U X e (q^2) hq2 hlog2
-  have h₁' : finiteMean (fun Y => (q^2)^(InputTree.familyCost (2*L) n (L+r)
-      (modifiedInputTree L r E U X) e Y)) ≤
-      Real.exp ((((2*L)*E.card*2^L:ℕ):ℝ)*Real.log (q^2)+Fintype.card ι*heightDecay L) := by
-    apply h₁.trans
-    apply Real.exp_le_exp.mpr
-    exact add_le_add_right (mul_le_mul_of_nonneg_left
-      (heightDecay_antitone (show L ≤ 2*L by omega)) (Nat.cast_nonneg _)) _
-  have hh := finiteMean_mul_exp_le
-    (fun Y => q^(InputTree.familyCost L n (L+r) (modifiedInputTree L r E U X) e Y))
-    (fun Y => q^(InputTree.familyCost (2*L) n (L+r) (modifiedInputTree L r E U X) e Y))
-    _ _ (by simpa only [pow_right_comm] using h₀) (by simpa only [pow_right_comm] using h₁')
-  simp only [←pow_add] at hh
-  refine hh.trans_eq ?_
-  congr 1
-  simp only [Real.log_pow,Nat.cast_mul,Nat.cast_ofNat]
-  ring
-
-noncomputable def InputTree.lowCost (H d : ℕ) (T : InputTree d)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) : ℕ :=
-  ∑ i ∈ Finset.range H, T.levelCost (i+1) e Y
-
-lemma InputTree.lowCost_slab (L d : ℕ) (T : InputTree d)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) :
-    InputTree.lowCost L d T e Y+InputTree.slabCost L d T e Y = InputTree.lowCost (2*L) d T e Y := by
-  rw [InputTree.lowCost,InputTree.lowCost,show 2*L=L+L by omega,Finset.sum_range_add]
-  congr 1
-  exact Fin.sum_univ_eq_sum_range (fun i => T.levelCost (L+i+1) e Y) L
-
-lemma InputTree.cost_family (L n d : ℕ) (T : (SwitchIndex d → Bool) → InputTree d)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) :
-    InputTree.lowCost L d (T Y) e Y + InputTree.familyCost L n d T e Y +
-      InputTree.familyCost (2*L) n d T e Y = InputTree.lowCost (L*4^n) d (T Y) e Y := by
-  induction n with
-  | zero => simp only [InputTree.familyCost,Finset.range_zero,Finset.sum_empty,Nat.add_zero,
-      pow_zero,mul_one]
-  | succ n ih =>
-    have he : (2*L)*4^n = 2*(L*4^n) := by ring
-    have he' : L*4^(n+1) = 2*(2*(L*4^n)) := by rw [pow_succ]; ring
-    simp only [InputTree.familyCost,Finset.sum_range_succ] at ih ⊢
-    rw [he,he',←InputTree.lowCost_slab,←InputTree.lowCost_slab,←ih]
-    omega
-
-lemma InputTree.lowCost_ge (H d : ℕ) (hd : d ≤ H) (T : InputTree d)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) :
-    InputTree.lowCost H d T e Y = InputTree.lowCost d d T e Y := by
-  symm
-  apply Finset.sum_subset (Finset.range_mono hd)
-  intro i _ hi
-  have hi' : d ≤ i := by simpa only [Finset.mem_range,not_lt] using hi
-  exact InputTree.levelCost_gt (i+1) d T (by omega) e Y
-
-lemma boundaryTree_levelCost_low (t L r : ℕ) (ht : t ≤ L)
-    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (Y : SwitchIndex (L+r) → Bool) :
-    (boundaryTree L r U X).levelCost t e Y = 0 := by
-  induction r generalizing ι with
-  | zero => rfl
-  | succ r ih =>
-    rw [boundaryTree,InputTree.levelCost,ite_eq_right (by omega : t ≠ L+r+1),ih,ih]
-
-lemma boundaryTree_lowCost_zero (L r : ℕ) (U : Card r → Equiv.Perm (Card L))
-    (X : SwitchIndex (L+r) → Bool) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (L+r)) (Y : SwitchIndex (L+r) → Bool) :
-    InputTree.lowCost L (L+r) (boundaryTree L r U X) e Y = 0 := by
-  apply Finset.sum_eq_zero
-  intro i hi
-  exact boundaryTree_levelCost_low (i+1) L r (by simpa only [Finset.mem_range,Nat.lt_iff_add_one_le] using hi) U X e Y
-
-theorem modified_cost_mgf (L r : ℕ) (hL : 0 < L) (E : Finset (Card r))
-    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/500) :
-    finiteMean (fun Y => q^(InputTree.lowCost (L+r) (L+r) (modifiedInputTree L r E U X Y) e Y)) ≤
-      Real.exp (Fintype.card ι*heightDecay L +
-        3*((L*E.card*2^L:ℕ):ℝ)*Real.log q) := by
-  have hbound : L+r ≤ L*4^(L+r) := by
-    have h₀ := pow_four_ge (L+r)
-    have h₁ := Nat.mul_le_mul_right (4^(L+r)) hL
-    nlinarith
-  have he (Y : SwitchIndex (L+r) → Bool) :=
-    InputTree.cost_family L (L+r) (L+r) (modifiedInputTree L r E U X) e Y
-  have hz (Y : SwitchIndex (L+r) → Bool) :
-      InputTree.lowCost L (L+r) (modifiedInputTree L r E U X Y) e Y = 0 :=
-    boundaryTree_lowCost_zero L r _ X e Y
-  simp only [hz,Nat.zero_add,InputTree.lowCost_ge _ _ hbound] at he
-  simp_rw [←he]
-  exact modified_high_mgf L (L+r) r hL E U X e q hq hlog
-
-
-noncomputable def exceptionChild {r : ℕ} (E : Finset (Card (r+1))) (b : Bool) : Finset (Card r) :=
-  Finset.univ.filter (fun u => Fin.cons b u ∈ E)
-
-lemma mem_exceptionChild {r : ℕ} (E : Finset (Card (r+1))) (b : Bool) (u : Card r) :
-    u ∈ exceptionChild E b ↔ Fin.cons b u ∈ E := by
-  classical
-  simp only [exceptionChild,Finset.mem_filter,Finset.mem_univ,true_and]
-
-lemma exceptionChild_card_add {r : ℕ} (E : Finset (Card (r+1))) :
-    (exceptionChild E false).card+(exceptionChild E true).card = E.card := by
-  classical
-  have he : (∑ u : Card (r+1), if u ∈ E then 1 else 0) = E.card := by simp
-  rw [Fintype.sum_equiv (headTailEquiv r) _
-    (fun x : Bool × Card r => if Fin.cons x.1 x.2 ∈ E then 1 else 0)
-    (fun x => by change (if x ∈ E then 1 else 0) = (if Fin.cons (x 0) (Fin.tail x) ∈ E then 1 else 0); rw [Fin.cons_self_tail])] at he
-  simp only [Fintype.sum_prod_type,Fintype.sum_bool] at he
-  simpa only [exceptionChild,Finset.card_filter,add_comm] using he
-
-def seedSplitEquiv (r : ℕ) (A : Type*) : (Card (r+1) → A) ≃ (Card r → A) × (Card r → A) where
-  toFun U := (fun u => U (Fin.cons false u),fun u => U (Fin.cons true u))
-  invFun U := fun u => if u 0 then U.2 (Fin.tail u) else U.1 (Fin.tail u)
-  left_inv U := by
-    funext u
-    dsimp only
-    split_ifs with h
-    · rw [←h,Fin.cons_self_tail]
-    · have h' : u 0 = false := by cases hh : u 0 <;> simp_all
-      rw [←h',Fin.cons_self_tail]
-  right_inv U := by simp only [Fin.cons_zero,Fin.tail_cons,Bool.false_eq_true,↓reduceIte]
-
-def seedSandwichShuffle {A B C D E : Type*} :
-    (A × B) × ((C × D) × E) ≃ ((A × C) × (B × D)) × E where
-  toFun x := (((x.1.1,x.2.1.1),(x.1.2,x.2.1.2)),x.2.2)
-  invFun x := ((x.1.1.1,x.1.2.1),((x.1.1.2,x.1.2.2),x.2))
-  left_inv _ := rfl
-  right_inv _ := rfl
-
-abbrev ModifiedCoins (L r : ℕ) := (Card r → Equiv.Perm (Card L)) × BenesCoins (L+r)
-
-def modifiedStepEquiv (L r : ℕ) : ModifiedCoins L (r+1) ≃
-    (ModifiedCoins L r × ModifiedCoins L r) × ((Card (L+r) → Bool) × (Card (L+r) → Bool)) :=
-  (Equiv.prodCongr (seedSplitEquiv r (Equiv.Perm (Card L))) (benesStepEquiv (L+r))).trans
-    seedSandwichShuffle
-
-noncomputable def modifiedPerm (L r : ℕ) (E : Finset (Card r)) (ω : ModifiedCoins L r) :
-    Equiv.Perm (Card (L+r)) :=
-  butterflyPerm (L+r) (decodeButterfly (L+r) ω.2.1) *
-    (modifiedInputTree L r E ω.1 ω.2.2 ω.2.1).perm⁻¹
-
-noncomputable def modifiedCost (L r : ℕ) (E : Finset (Card r))
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (ω : ModifiedCoins L r) : ℕ :=
-  InputTree.lowCost (L+r) (L+r) (modifiedInputTree L r E ω.1 ω.2.2 ω.2.1) e ω.2.1
-
-lemma modifiedBoundary_cons (L r : ℕ) (E : Finset (Card (r+1)))
-    (U : Card (r+1) → Equiv.Perm (Card L)) (Y : SwitchIndex (L+r+1) → Bool) (b : Bool) :
-    (fun u => modifiedBoundary L (r+1) E U Y (Fin.cons b u)) =
-    modifiedBoundary L r (exceptionChild E b) (fun u => U (Fin.cons b u))
-      (if b then (coinStepEquiv (L+r) Y).1.2 else (coinStepEquiv (L+r) Y).1.1) := by
-  funext u
-  simp only [modifiedBoundary,mem_exceptionChild,boundaryOutput,Fin.cons_zero,Fin.tail_cons]
-
-lemma modifiedInputTree_node (L r : ℕ) (E : Finset (Card (r+1)))
-    (ω : ModifiedCoins L r × ModifiedCoins L r) (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
-    let z := (modifiedStepEquiv L r).symm (ω,coins)
-    modifiedInputTree L (r+1) E z.1 z.2.2 z.2.1 =
-      InputTree.node coins.1 (fun b =>
-        let w := if b then ω.2 else ω.1
-        modifiedInputTree L r (exceptionChild E b) w.1 w.2.2 w.2.1) := by
-  dsimp only
-  unfold modifiedInputTree
-  rw [boundaryTree]
-  change InputTree.node coins.1 (fun b => boundaryTree L r _ _) = _
-  congr 1
-  funext b
-  rw [modifiedBoundary_cons]
-  cases b <;> rfl
-
-lemma modifiedPerm_step (L r : ℕ) (E : Finset (Card (r+1)))
-    (ω : ModifiedCoins L r × ModifiedCoins L r) (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
-    modifiedPerm L (r+1) E ((modifiedStepEquiv L r).symm (ω,coins)) =
-      pairSwitch coins.2 * childLift (fun b => modifiedPerm L r (exceptionChild E b)
-        (if b then ω.2 else ω.1)) * pairSwitch coins.1 := by
-  rw [modifiedPerm,modifiedInputTree_node,InputTree.perm,mul_inv_rev,pairSwitch_inv,childLift_inv]
-  change (butterflyPerm (L+r+1) (decodeButterfly (L+r+1)
-    ((coinStepEquiv (L+r)).symm ((ω.1.2.1,ω.2.2.1),coins.2)))) * _ = _
-  rw [butterflyPerm_step,mul_assoc,←mul_assoc (childLift _),childLift_mul,←mul_assoc]
-  congr 2
-  apply congrArg childLift
-  funext b
-  cases b <;> rfl
-
-lemma modifiedPerm_coordinates (L r : ℕ) (E : Finset (Card (r+1)))
-    (ω : ModifiedCoins L r × ModifiedCoins L r) (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool))
-    (x : Card (L+r+1)) :
-    headTailEquiv (L+r) (modifiedPerm L (r+1) E ((modifiedStepEquiv L r).symm (ω,coins)) x) =
-      PairRouting.sandwich coins.1 coins.2 (fun b => modifiedPerm L r (exceptionChild E b)
-        (if b then ω.2 else ω.1)) (headTailEquiv (L+r) x) := by
-  rw [modifiedPerm_step]
-  simp only [Equiv.Perm.mul_apply,headTail_switch,headTail_child]
-  rfl
-
-lemma InputTree.lowCost_node (d : ℕ) (ξ : Card d → Bool) (T : Bool → InputTree d)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (d+1)) (Y : (SwitchIndex d → Bool) × (SwitchIndex d → Bool))
-    (η : Card d → Bool) :
-    InputTree.lowCost (d+1) (d+1) (.node ξ T) e ((coinStepEquiv d).symm (Y,η)) =
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x ξ
-      let hc := PairRouting.colors_compatible x ξ
-      PairRouting.alternatingCycles (PairRouting.switchedEmbedding x ξ)
-        (fun b => if b then butterflyPerm d (decodeButterfly d Y.2) * (T true).perm⁻¹
-          else butterflyPerm d (decodeButterfly d Y.1) * (T false).perm⁻¹) +
-        InputTree.lowCost d d (T false) (PairRouting.childEmbedding x c hc false) Y.1 +
-        InputTree.lowCost d d (T true) (PairRouting.childEmbedding x c hc true) Y.2 := by
-  dsimp only
-  rw [InputTree.lowCost,Finset.sum_range_succ,InputTree.levelCost_node,ite_eq_left rfl]
-  have he : (∑ i ∈ Finset.range d,
-      (.node ξ T : InputTree (d+1)).levelCost (i+1) e ((coinStepEquiv d).symm (Y,η))) =
-      InputTree.lowCost d d (T false)
-        (PairRouting.childEmbedding (e.trans (headTailEquiv d).toEmbedding)
-          (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) ξ) (PairRouting.colors_compatible _ _) false) Y.1 +
-      InputTree.lowCost d d (T true)
-        (PairRouting.childEmbedding (e.trans (headTailEquiv d).toEmbedding)
-          (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) ξ) (PairRouting.colors_compatible _ _) true) Y.2 := by
-    rw [InputTree.lowCost,InputTree.lowCost,←Finset.sum_add_distrib]
-    apply Finset.sum_congr rfl
-    intro i hi
-    rw [InputTree.levelCost_node,ite_eq_right (by simp only [Finset.mem_range] at hi; omega : i+1 ≠ d+1)]
-  rw [he]
-  omega
-
-noncomputable def modifiedAssignmentCost (L r : ℕ) (E : Finset (Card (r+1)))
-    {ι : Type*} [Fintype ι] (x : ι ↪ Bool × Card (L+r)) (c : ι → Bool)
-    (hc : PairRouting.Compatible x c) (ω : ModifiedCoins L r × ModifiedCoins L r) : ℕ :=
-  PairRouting.alternatingCycles (PairRouting.coloredEmbedding x c hc)
-    (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) +
-  modifiedCost L r (exceptionChild E false) (PairRouting.childEmbedding x c hc false) ω.1 +
-  modifiedCost L r (exceptionChild E true) (PairRouting.childEmbedding x c hc true) ω.2
-
-lemma modifiedCost_step (L r : ℕ) (E : Finset (Card (r+1)))
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r+1))
-    (ω : ModifiedCoins L r × ModifiedCoins L r) (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
-    modifiedCost L (r+1) E e ((modifiedStepEquiv L r).symm (ω,coins)) =
-      modifiedAssignmentCost L r E (e.trans (headTailEquiv (L+r)).toEmbedding)
-        (PairRouting.colors (e.trans (headTailEquiv (L+r)).toEmbedding) coins.1)
-        (PairRouting.colors_compatible _ _) ω := by
-  rw [modifiedCost,modifiedInputTree_node]
-  change InputTree.lowCost (L+r+1) (L+r+1) _ e
-    ((coinStepEquiv (L+r)).symm ((ω.1.2.1,ω.2.2.1),coins.2)) = _
-  rw [InputTree.lowCost_node]
-  unfold modifiedAssignmentCost
-  dsimp only
-  simp only [Bool.false_eq_true,↓reduceIte]
-  rw [PairRouting.switchedEmbedding_eq_colored _ _ (PairRouting.colors_compatible _ _) _ rfl]
-  congr 2
-
-
-namespace PairRouting
-variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α] [DecidableEq ι]
-
-omit [Fintype α] [DecidableEq ι] in
-lemma childEmbedding_surjective (x : ι ↪ Bool × α) (hx : Function.Surjective x)
-    (c : ι → Bool) (hc : Compatible x c) (b : Bool) :
-    Function.Surjective (childEmbedding x c hc b) := by
-  intro y
-  have hs : Function.Surjective (coloredEmbedding x c hc) :=
-    (pairLayer (chosenCoin x c)).surjective.comp hx
-  obtain ⟨i,hi⟩ := hs (b,y)
-  rw [coloredEmbedding_apply] at hi
-  exact ⟨⟨i,congrArg Prod.fst hi⟩,congrArg Prod.snd hi⟩
-
-omit [DecidableEq ι] in
-lemma uniformPerm_tuple_upper (e f : ι ↪ α) (he : Function.Surjective e) (p : Equiv.Perm α) :
-    finiteMean (fun u : Equiv.Perm α => if ∀ i, (p*u⁻¹) (e i) = f i then (1:ℝ) else 0) ≤
-      1/(Fintype.card (Equiv.Perm α):ℝ) := by
-  classical
-  let M : Equiv.Perm α → Prop := fun u => ∀ i, (p*u⁻¹) (e i) = f i
-  have huniq (u v : Equiv.Perm α) (hu : M u) (hv : M v) : u = v := by
-    have hh : p*u⁻¹ = p*v⁻¹ := by
-      ext x
-      obtain ⟨i,rfl⟩ := he x
-      exact (hu i).trans (hv i).symm
-    exact inv_injective (mul_left_cancel hh)
-  unfold finiteMean
-  apply div_le_div_of_nonneg_right _ (Nat.cast_nonneg _)
-  change (∑ u, if M u then (1:ℝ) else 0) ≤ 1
-  by_cases h : ∃ u, M u
-  · obtain ⟨u,hu⟩ := h
-    rw [Finset.sum_eq_single u]
-    · simp only [hu,↓reduceIte,le_refl]
-    · intro v _ hv
-      exact ite_eq_right (fun hh => hv (huniq v u hh hu))
-    · simp
-  · push Not at h
-    simp only [h,ite_false,Finset.sum_const_zero,zero_le_one]
-
-end PairRouting
-
-noncomputable def centralFactor (L r : ℕ) (E : Finset (Card r)) (a : ℝ) : ℝ :=
-  (Nat.factorial (2^L):ℝ)^(-((2:ℝ)^r-E.card)*a)
-
-lemma centralFactor_nonneg (L r : ℕ) (E : Finset (Card r)) (a : ℝ) :
-    0 ≤ centralFactor L r E a := Real.rpow_nonneg (Nat.cast_nonneg _) _
-
-lemma centralFactor_step (L r : ℕ) (E : Finset (Card (r+1))) (a : ℝ) :
-    centralFactor L (r+1) E a =
-      centralFactor L r (exceptionChild E false) a * centralFactor L r (exceptionChild E true) a := by
-  unfold centralFactor
-  rw [←Real.rpow_add (Nat.cast_pos.mpr (Nat.factorial_pos _))]
-  congr 1
-  have he : ((exceptionChild E false).card:ℝ)+(exceptionChild E true).card = E.card := by
-    exact_mod_cast exceptionChild_card_add E
-  rw [pow_succ,←he]
-  ring
-
-lemma modifiedPerm_zero (L : ℕ) (E : Finset (Card 0)) (ω : ModifiedCoins L 0) :
-    modifiedPerm L 0 E ω =
-      if (fun j => Fin.elim0 j : Card 0) ∈ E then 1
-      else butterflyPerm L (decodeButterfly L ω.2.1)*(ω.1 (fun j => Fin.elim0 j))⁻¹ := by
-  unfold modifiedPerm modifiedInputTree boundaryTree InputTree.perm modifiedBoundary boundaryOutput
-  split_ifs
-  · exact mul_inv_cancel _
-  · rfl
-
-lemma modifiedCost_zero (L : ℕ) (E : Finset (Card 0)) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card L) (ω : ModifiedCoins L 0) : modifiedCost L 0 E e ω = 0 := by
-  simp only [modifiedCost,modifiedInputTree,boundaryTree,InputTree.lowCost,
-    InputTree.levelCost,Finset.sum_const_zero]
-
-lemma centralFactor_zero (L : ℕ) (E : Finset (Card 0)) (a : ℝ) :
-    centralFactor L 0 E a =
-      if (fun j => Fin.elim0 j : Card 0) ∈ E then 1 else (Nat.factorial (2^L):ℝ)^(-a) := by
-  classical
-  let z : Card 0 := fun j => Fin.elim0 j
-  have he : (E.card:ℝ) = if z ∈ E then 1 else 0 := by
-    by_cases hz : z ∈ E
-    · have he : E = {z} := by
-        ext u
-        rw [Subsingleton.elim u z]
-        simp only [hz,Finset.mem_singleton]
-      rw [he,ite_eq_left (by simpa only [he] using hz),Finset.card_singleton,Nat.cast_one]
-    · have he : E = ∅ := Finset.eq_empty_iff_forall_notMem.mpr (fun u hu => hz (Subsingleton.elim u z ▸ hu))
-      rw [he,Finset.card_empty,Nat.cast_zero,ite_eq_right (by simp)]
-  unfold centralFactor
-  rw [pow_zero,he]
-  split_ifs <;> simp
-
-end Thorp
-
-end ThorpNine.Harmonic
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Harmonic.PalindromeFamilyCost
+
+namespace OAI
+
+namespace ThorpNine.Harmonic
+
+namespace Thorp
+
+lemma centralSlabCost_perturb (l L r : ℕ) (C D : Card r → Equiv.Perm (Card L))
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (ω : BenesCoins (L+r)) :
+    centralSlabCost l L r C e ω ≤ centralSlabCost l L r D e ω+l*centralDistance L r C D := by
+  have hh := Finset.sum_le_sum (fun (i : Fin l) (_ : i ∈ Finset.univ) =>
+    centralLevelCost_perturb (l+i.val+1) L r C D e ω)
+  simpa only [Finset.sum_add_distrib,Finset.sum_const,Finset.card_univ,Fintype.card_fin,
+    nsmul_eq_mul,Nat.cast_id,centralSlabCost] using hh
+
+lemma centralDistance_exceptional (L r : ℕ) (C D : Card r → Equiv.Perm (Card L))
+    (E : Finset (Card r)) (h : ∀ u ∉ E, C u = D u) :
+    centralDistance L r C D ≤ E.card*2^L := by
+  classical
+  calc
+    _ ≤ ∑ u : Card r, if u ∈ E then 2^L else 0 := by
+      apply Finset.sum_le_sum
+      intro u _
+      split_ifs with hu
+      · simpa only [card_positions] using permDistance_le (C u) (D u)
+      · rw [h u hu,permDistance_refl]
+    _ = _ := by simp
+
+
+inductive InputTree : ℕ → Type
+  | tip {d : ℕ} (p : Equiv.Perm (Card d)) : InputTree d
+  | node {d : ℕ} (ξ : Card d → Bool) (T : Bool → InputTree d) : InputTree (d+1)
+
+def InputTree.perm : {d : ℕ} → InputTree d → Equiv.Perm (Card d)
+  | _, .tip p => p
+  | _, .node ξ T => pairSwitch ξ * childLift (fun b => (T b).perm)
+
+noncomputable def InputTree.levelCost (t : ℕ) : {d : ℕ} → InputTree d →
+    {ι : Type*} → [Fintype ι] → (ι ↪ Card d) → (SwitchIndex d → Bool) → ℕ
+  | _, .tip _, _, _, _, _ => 0
+  | d+1, .node ξ T, _, _, e, Y =>
+    let x := e.trans (headTailEquiv d).toEmbedding
+    let c := PairRouting.colors x ξ
+    let hc := PairRouting.colors_compatible x ξ
+    let σ := coinStepEquiv d Y
+    if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x ξ)
+      (fun b => if b then butterflyPerm d (decodeButterfly d σ.1.2) * (T true).perm⁻¹
+        else butterflyPerm d (decodeButterfly d σ.1.1) * (T false).perm⁻¹)
+    else (T false).levelCost t (PairRouting.childEmbedding x c hc false) σ.1.1 +
+      (T true).levelCost t (PairRouting.childEmbedding x c hc true) σ.1.2
+
+lemma InputTree.levelCost_node (t d : ℕ) (ξ : Card d → Bool) (T : Bool → InputTree d)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (d+1))
+    (Y : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (η : Card d → Bool) :
+    (InputTree.node ξ T).levelCost t e ((coinStepEquiv d).symm (Y,η)) =
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x ξ
+      let hc := PairRouting.colors_compatible x ξ
+      if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x ξ)
+        (fun b => if b then butterflyPerm d (decodeButterfly d Y.2) * (T true).perm⁻¹
+          else butterflyPerm d (decodeButterfly d Y.1) * (T false).perm⁻¹)
+      else (T false).levelCost t (PairRouting.childEmbedding x c hc false) Y.1 +
+        (T true).levelCost t (PairRouting.childEmbedding x c hc true) Y.2 := by
+  rw [InputTree.levelCost]
+  rfl
+
+lemma InputTree.levelCost_gt (t d : ℕ) (T : InputTree d) (ht : d < t)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) :
+    T.levelCost t e Y = 0 := by
+  induction T generalizing ι with
+  | tip p => rfl
+  | @node d ξ T ih =>
+    rw [InputTree.levelCost,ite_eq_right (by omega : t ≠ d+1),ih false (by omega),ih true (by omega)]
+
+lemma InputTree.levelCost_adapted (t d : ℕ) (T : InputTree d)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) :
+    LayerAdapted d (t-1) (fun Y => T.levelCost t e Y) := by
+  induction T generalizing ι with
+  | tip p => intro Y Y' _; rfl
+  | @node d ξ T ih =>
+    intro Y Y' h
+    dsimp only
+    conv_lhs => rw [←(coinStepEquiv d).symm_apply_apply Y]
+    conv_rhs => rw [←(coinStepEquiv d).symm_apply_apply Y']
+    rw [InputTree.levelCost_node t d ξ T e ((coinStepEquiv d Y).1) ((coinStepEquiv d Y).2),
+      InputTree.levelCost_node t d ξ T e ((coinStepEquiv d Y').1) ((coinStepEquiv d Y').2)]
+    by_cases ht : t = d+1
+    · subst t
+      rw [ite_eq_left rfl,ite_eq_left rfl]
+      have he₀ : (coinStepEquiv d Y).1.1 = (coinStepEquiv d Y').1.1 := by
+        funext i
+        exact h (Sum.inr (false,i)) (by simpa only [switchHeight,Nat.add_sub_cancel] using switchHeight_le d i)
+      have he₁ : (coinStepEquiv d Y).1.2 = (coinStepEquiv d Y').1.2 := by
+        funext i
+        exact h (Sum.inr (true,i)) (by simpa only [switchHeight,Nat.add_sub_cancel] using switchHeight_le d i)
+      rw [he₀,he₁]
+    · rw [ite_eq_right ht,ite_eq_right ht]
+      congr 1
+      · apply ih false
+        intro i hi
+        exact h (Sum.inr (false,i)) hi
+      · apply ih true
+        intro i hi
+        exact h (Sum.inr (true,i)) hi
+
+theorem InputTree.level_partial_mgf (r s u : ℕ) (hu : u < r)
+    (T : InputTree (s+r)) {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+r))
+    (lo : Card r × SwitchIndex s → Bool) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(T.levelCost (s+u+1) e (assembleBits r s lo hi))) ≤
+      Real.exp (Fintype.card ι * cycleCoefficient u J q) := by
+  classical
+  induction r generalizing u ι with
+  | zero => omega
+  | succ r ih =>
+    simp only [Nat.add_succ] at *
+    cases T with
+    | tip p =>
+      simp only [InputTree.levelCost,pow_zero,finiteMean_const]
+      exact Real.one_le_exp (mul_nonneg (Nat.cast_nonneg _) (cycleCoefficient_nonneg u J q hq))
+    | node ξ T =>
+      let x := e.trans (headTailEquiv (s+r)).toEmbedding
+      let c := PairRouting.colors x ξ
+      let hc := PairRouting.colors_compatible x ξ
+      let e₀ := PairRouting.childEmbedding x c hc false
+      let e₁ := PairRouting.childEmbedding x c hc true
+      let e' := PairRouting.switchedEmbedding x ξ
+      let lo₀ : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons false j.1,j.2)
+      let lo₁ : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons true j.1,j.2)
+      let Y₀ := fun hi : Card s × SwitchIndex r → Bool => assembleBits r s lo₀ hi
+      let Y₁ := fun hi : Card s × SwitchIndex r → Bool => assembleBits r s lo₁ hi
+      let N := fun hi : (Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool) =>
+        PairRouting.alternatingCycles e' (fun b =>
+          if b then butterflyPerm (s+r) (decodeButterfly (s+r) (Y₁ hi.2)) * (T true).perm⁻¹
+          else butterflyPerm (s+r) (decodeButterfly (s+r) (Y₀ hi.1)) * (T false).perm⁻¹)
+      let L₀ := fun hi : Card s × SwitchIndex r → Bool =>
+        (T false).levelCost (s+u+1) e₀ (Y₀ hi)
+      let L₁ := fun hi : Card s × SwitchIndex r → Bool =>
+        (T true).levelCost (s+u+1) e₁ (Y₁ hi)
+      have hs (hi : (Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool))
+          (η : Card s × Card r → Bool) :
+          (InputTree.node ξ T).levelCost (s+u+1) e
+            (assembleBits (r+1) s lo ((highStepEquiv r s).symm (hi,η))) =
+            if u = r then N hi else L₀ hi.1 + L₁ hi.2 := by
+        rw [assembleBits_step]
+        rw [InputTree.levelCost_node (s+u+1) (s+r) ξ T e (Y₀ hi.1,Y₁ hi.2) (fun y => η (((cardSplit r s) y).2,((cardSplit r s) y).1))]
+        simp only [Nat.add_right_cancel_iff,Nat.add_left_cancel_iff]
+        rfl
+      rw [finiteMean_equiv (highStepEquiv r s),finiteMean_prod]
+      simp_rw [hs,finiteMean_const]
+      by_cases hur : u = r
+      · subst u
+        simp only [ite_true]
+        rw [finiteMean_prod,finiteMean_comm]
+        calc
+          _ ≤ finiteMean (fun _hi : Card s × SwitchIndex r → Bool =>
+            Real.exp (Fintype.card ι * cycleCoefficient r J q)) := by
+            apply finiteMean_mono
+            intro hi
+            have hn := node_partial_mgf r s e' (T false).perm (T true).perm
+              (butterflyPerm (s+r) (decodeButterfly (s+r) (Y₁ hi))) lo₀ J hJ q hq
+            exact hn
+          _ = _ := finiteMean_const _
+      · simp only [ite_eq_right hur,pow_add]
+        rw [finiteMean_prod_mul (fun hi => q ^ L₀ hi) (fun hi => q ^ L₁ hi)]
+        have hu' : u < r := by omega
+        have h₀ := ih u hu' (T false) e₀ lo₀
+        have h₁ := ih u hu' (T true) e₁ lo₁
+        have hp := mul_le_mul h₀ h₁ (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
+          (Real.exp_nonneg _)
+        refine hp.trans_eq ?_
+        rw [←Real.exp_add]
+        congr 1
+        have hcount := PairRouting.color_card_add c
+        have hcount' : (Fintype.card {i : ι // c i = false} : ℝ) +
+            Fintype.card {i : ι // c i = true} = Fintype.card ι := by exact_mod_cast hcount
+        change (Fintype.card {i : ι // c i = false} : ℝ) * cycleCoefficient u J q +
+            Fintype.card {i : ι // c i = true} * cycleCoefficient u J q = _
+        rw [←add_mul,hcount']
+
+
+noncomputable def InputTree.slabCost (l d : ℕ) (T : InputTree d) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) : ℕ :=
+  ∑ i : Fin l, T.levelCost (l+i.1+1) e Y
+
+lemma InputTree.slabCost_adapted (l d : ℕ) (T : InputTree d) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) :
+    LayerAdapted d (2*l) (fun Y => InputTree.slabCost l d T e Y) := by
+  apply layerAdapted_sum
+  intro i _
+  exact layerAdapted_mono (InputTree.levelCost_adapted (l+i.1+1) d T e) (by omega)
+
+theorem InputTree.slab_partial_mgf (r s l : ℕ) (hl : 0 < l) (hs : 2*s ≤ l) (T : InputTree (s+r))
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+r))
+    (lo : Card r × SwitchIndex s → Bool) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(InputTree.slabCost l (s+r) T e (assembleBits r s lo hi))) ≤
+      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
+  classical
+  let : NeZero l := ⟨ne_of_gt hl⟩
+  have hqp : 0 < q := lt_of_lt_of_le (by norm_num) hq
+  calc
+    _ ≤ finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      finiteMean (fun i : Fin l => q^(l*T.levelCost (l+i.1+1) e
+        (assembleBits r s lo hi)))) := by
+      apply finiteMean_mono
+      intro hi
+      simpa only [InputTree.slabCost,Fintype.card_fin] using
+        pow_sum_le_mean_powers (fun i : Fin l => T.levelCost (l+i.1+1) e
+          (assembleBits r s lo hi)) q hqp
+    _ = finiteMean (fun i : Fin l => finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(l*T.levelCost (l+i.1+1) e (assembleBits r s lo hi)))) := finiteMean_comm _
+    _ ≤ finiteMean (fun _i : Fin l =>
+      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64)))) := by
+      apply finiteMean_mono
+      intro i
+      by_cases ht : l+i.1+1 ≤ s+r
+      · have hs' : s ≤ l+i.1 := by omega
+        have he : l+i.1+1 = s+(l+i.1-s)+1 := by omega
+        have hu : l+i.1-s < r := by omega
+        have hm := InputTree.level_partial_mgf r s (l+i.1-s) hu T e lo
+          ⌈Real.exp ((l:ℝ)/32)⌉₊ (cycleCutoff_pos l) (q^l) (one_le_pow₀ hq)
+        have hc := cycleCoefficient_decay l (l+i.1-s) (by
+          rw [Nat.cast_sub hs',Nat.cast_add]
+          have hsr : (2:ℝ)*s ≤ l := by exact_mod_cast hs
+          have hi : (0:ℝ) ≤ i.1 := Nat.cast_nonneg _
+          linarith) (q^l) (one_le_pow₀ hq) (by
+            rw [Real.log_pow]
+            nlinarith [mul_le_mul_of_nonneg_left hlog (Nat.cast_nonneg l)])
+        simp_rw [he,pow_mul]
+        exact hm.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left hc (Nat.cast_nonneg _)))
+      · have hz := InputTree.levelCost_gt (l+i.1+1) (s+r) T (by omega) e
+        simp only [hz,Nat.mul_zero,pow_zero,finiteMean_const]
+        exact Real.one_le_exp_iff.mpr (by positivity)
+    _ = _ := finiteMean_const _
+
+
+def boundaryTree (L : ℕ) : (r : ℕ) → (Card r → Equiv.Perm (Card L)) →
+    (SwitchIndex (L+r) → Bool) → InputTree (L+r)
+  | 0, U, _ => .tip (U (fun j => Fin.elim0 j))
+  | r+1, U, X => .node (coinStepEquiv (L+r) X).2 (fun b =>
+      boundaryTree L r (fun u => U (Fin.cons b u))
+        (if b then (coinStepEquiv (L+r) X).1.2 else (coinStepEquiv (L+r) X).1.1))
+
+def boundaryOutput (L : ℕ) : (r : ℕ) → (SwitchIndex (L+r) → Bool) →
+    Card r → Equiv.Perm (Card L)
+  | 0, Y, _ => butterflyPerm L (decodeButterfly L Y)
+  | r+1, Y, u => boundaryOutput L r
+      (if u 0 then (coinStepEquiv (L+r) Y).1.2 else (coinStepEquiv (L+r) Y).1.1) (Fin.tail u)
+
+lemma boundaryOutput_adapted (L r : ℕ) :
+    LayerAdapted (L+r) L (boundaryOutput L r) := by
+  induction r with
+  | zero =>
+    intro Y Y' h
+    have he : Y = Y' := funext (fun i => h i (by simpa only [Nat.add_zero] using switchHeight_le L i))
+    rw [he]
+  | succ r ih =>
+    intro Y Y' h
+    funext u
+    dsimp only [boundaryOutput]
+    cases hb : u 0 <;> simp only [Bool.false_eq_true,↓reduceIte]
+    · exact congrFun (ih _ _ (fun i hi => h (Sum.inr (false,i)) hi)) (Fin.tail u)
+    · exact congrFun (ih _ _ (fun i hi => h (Sum.inr (true,i)) hi)) (Fin.tail u)
+
+noncomputable def modifiedBoundary (L r : ℕ) (E : Finset (Card r))
+    (U : Card r → Equiv.Perm (Card L)) (Y : SwitchIndex (L+r) → Bool) :
+    Card r → Equiv.Perm (Card L) := fun u => if u ∈ E then boundaryOutput L r Y u else U u
+
+lemma modifiedBoundary_adapted (L r : ℕ) (E : Finset (Card r))
+    (U : Card r → Equiv.Perm (Card L)) :
+    LayerAdapted (L+r) L (modifiedBoundary L r E U) := by
+  intro Y Y' h
+  unfold modifiedBoundary
+  rw [boundaryOutput_adapted L r Y Y' h]
+
+lemma permDistance_boundaryTree (L r : ℕ) (U V : Card r → Equiv.Perm (Card L))
+    (X : SwitchIndex (L+r) → Bool) :
+    permDistance (boundaryTree L r U X).perm (boundaryTree L r V X).perm =
+      centralDistance L r U V := by
+  induction r with
+  | zero =>
+    simp only [boundaryTree,InputTree.perm,centralDistance,Fintype.sum_unique]
+    congr 1
+  | succ r ih =>
+    simp only [boundaryTree,InputTree.perm,permDistance_mul_left]
+    rw [permDistance_childLift,centralDistance_step,ih,ih]
+
+lemma boundaryTree_levelCost_perturb (t L r : ℕ) (U V : Card r → Equiv.Perm (Card L))
+    (X : SwitchIndex (L+r) → Bool) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (L+r)) (Y : SwitchIndex (L+r) → Bool) :
+    (boundaryTree L r U X).levelCost t e Y ≤
+      (boundaryTree L r V X).levelCost t e Y + centralDistance L r U V := by
+  induction r generalizing ι with
+  | zero => simp only [boundaryTree,InputTree.levelCost,Nat.zero_add,Nat.zero_le]
+  | succ r ih =>
+    rw [boundaryTree,boundaryTree,InputTree.levelCost,InputTree.levelCost]
+    by_cases ht : t = L+r+1
+    · simp only [ite_eq_left ht]
+      have hh := alternatingCycles_distance_le
+        (PairRouting.switchedEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
+          (coinStepEquiv (L+r) X).2)
+        (fun b => if b then butterflyPerm (L+r) (decodeButterfly (L+r) (coinStepEquiv (L+r) Y).1.2) *
+            (boundaryTree L r (fun u => U (Fin.cons true u)) (coinStepEquiv (L+r) X).1.2).perm⁻¹
+          else butterflyPerm (L+r) (decodeButterfly (L+r) (coinStepEquiv (L+r) Y).1.1) *
+            (boundaryTree L r (fun u => U (Fin.cons false u)) (coinStepEquiv (L+r) X).1.1).perm⁻¹)
+        (fun b => if b then butterflyPerm (L+r) (decodeButterfly (L+r) (coinStepEquiv (L+r) Y).1.2) *
+            (boundaryTree L r (fun u => V (Fin.cons true u)) (coinStepEquiv (L+r) X).1.2).perm⁻¹
+          else butterflyPerm (L+r) (decodeButterfly (L+r) (coinStepEquiv (L+r) Y).1.1) *
+            (boundaryTree L r (fun u => V (Fin.cons false u)) (coinStepEquiv (L+r) X).1.1).perm⁻¹)
+      simpa only [Bool.false_eq_true,↓reduceIte,permDistance_mul_left,permDistance_inv,
+        permDistance_boundaryTree,←centralDistance_step] using hh
+    · simp only [ite_eq_right ht,Bool.false_eq_true,↓reduceIte]
+      rw [centralDistance_step]
+      have h₀ := ih (fun u => U (Fin.cons false u)) (fun u => V (Fin.cons false u))
+        (coinStepEquiv (L+r) X).1.1
+        (PairRouting.childEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
+          (PairRouting.colors (e.trans (headTailEquiv (L+r)).toEmbedding) (coinStepEquiv (L+r) X).2)
+          (PairRouting.colors_compatible _ _) false) (coinStepEquiv (L+r) Y).1.1
+      have h₁ := ih (fun u => U (Fin.cons true u)) (fun u => V (Fin.cons true u))
+        (coinStepEquiv (L+r) X).1.2
+        (PairRouting.childEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
+          (PairRouting.colors (e.trans (headTailEquiv (L+r)).toEmbedding) (coinStepEquiv (L+r) X).2)
+          (PairRouting.colors_compatible _ _) true) (coinStepEquiv (L+r) Y).1.2
+      omega
+
+lemma boundaryTree_slabCost_perturb (l L r : ℕ) (U V : Card r → Equiv.Perm (Card L))
+    (X : SwitchIndex (L+r) → Bool) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (L+r)) (Y : SwitchIndex (L+r) → Bool) :
+    InputTree.slabCost l (L+r) (boundaryTree L r U X) e Y ≤
+      InputTree.slabCost l (L+r) (boundaryTree L r V X) e Y + l*centralDistance L r U V := by
+  have hh := Finset.sum_le_sum (fun (i : Fin l) (_ : i ∈ Finset.univ) =>
+    boundaryTree_levelCost_perturb (l+i.val+1) L r U V X e Y)
+  simpa only [Finset.sum_add_distrib,Finset.sum_const,Finset.card_univ,Fintype.card_fin,
+    nsmul_eq_mul,Nat.cast_id,InputTree.slabCost] using hh
+
+lemma modifiedBoundary_distance_le (L r : ℕ) (E : Finset (Card r))
+    (U : Card r → Equiv.Perm (Card L)) (Y : SwitchIndex (L+r) → Bool) :
+    centralDistance L r (modifiedBoundary L r E U Y) U ≤ E.card*2^L := by
+  apply centralDistance_exceptional
+  intro u hu
+  exact ite_eq_right hu
+
+
+lemma InputTree.slabCost_gt (l d : ℕ) (T : InputTree d) (hd : d ≤ l)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) :
+    InputTree.slabCost l d T e Y = 0 := by
+  apply Finset.sum_eq_zero
+  intro i _
+  exact InputTree.levelCost_gt _ _ T (by omega) _ _
+
+lemma InputTree.random_slab_adapted_mul_le (d s l : ℕ) (hsd : s ≤ d) (hl : 0 < l)
+    (hs : 2*s ≤ l) (T : (SwitchIndex d → Bool) → InputTree d)
+    (hT : LayerAdapted d s T) {ι : Type*} [Fintype ι] (e : ι ↪ Card d)
+    (F : (SwitchIndex d → Bool) → ℝ) (hF : ∀ Y, 0 ≤ F Y) (hA : LayerAdapted d s F)
+    (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun Y => F Y*q^(InputTree.slabCost l d (T Y) e Y)) ≤
+      finiteMean F * Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
+  obtain ⟨r,rfl⟩ := Nat.exists_eq_add_of_le hsd
+  apply adapted_mul_mean_le r s F _ hF hA
+  intro lo
+  let hi₀ : Card s × SwitchIndex r → Bool := fun _ => false
+  have he (hi : Card s × SwitchIndex r → Bool) :
+      T (assembleBits r s lo hi) = T (assembleBits r s lo hi₀) :=
+    hT _ _ (assembleBits_agree r s lo hi hi₀)
+  simp_rw [he]
+  exact InputTree.slab_partial_mgf r s l hl hs _ e lo q hq hlog
+
+lemma InputTree.slab_mgf (l d : ℕ) (hl : 0 < l) (T : InputTree d)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun Y => q^(InputTree.slabCost l d T e Y)) ≤
+      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
+  have hh := InputTree.random_slab_adapted_mul_le d 0 l (Nat.zero_le _) hl (Nat.zero_le _)
+    (fun _ => T) (fun _ _ _ => rfl) e (fun _ => 1) (fun _ => by norm_num)
+    (fun _ _ _ => rfl) q hq hlog
+  simpa only [one_mul,finiteMean_const] using hh
+
+noncomputable def InputTree.familyCost (L n d : ℕ)
+    (T : (SwitchIndex d → Bool) → InputTree d) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) : ℕ :=
+  ∑ j ∈ Finset.range n, InputTree.slabCost (L*4^j) d (T Y) e Y
+
+lemma InputTree.random_slabCost_adapted (l d s : ℕ) (T : (SwitchIndex d → Bool) → InputTree d)
+    (hT : LayerAdapted d s T) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) :
+    LayerAdapted d (max s (2*l)) (fun Y => InputTree.slabCost l d (T Y) e Y) := by
+  intro Y Y' h
+  dsimp only
+  rw [hT _ _ (agreeThrough_mono (le_max_left _ _) h)]
+  exact InputTree.slabCost_adapted l d (T Y') e Y Y'
+    (agreeThrough_mono (le_max_right _ _) h)
+
+lemma InputTree.familyCost_adapted (L n d s : ℕ) (T : (SwitchIndex d → Bool) → InputTree d)
+    (hT : LayerAdapted d s T) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) :
+    LayerAdapted d (max s (familyEnd L n)) (fun Y => InputTree.familyCost L n d T e Y) := by
+  apply layerAdapted_sum
+  intro j hj
+  exact layerAdapted_mono (InputTree.random_slabCost_adapted (L*4^j) d s T hT e)
+    (max_le_max_left s (familyEnd_ge L n j (Finset.mem_range.mp hj)))
+
+theorem InputTree.family_mgf_first (L n d s : ℕ) (hL : 0 < L) (hs : s ≤ L)
+    (T : (SwitchIndex d → Bool) → InputTree d) (hT : LayerAdapted d s T)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) (p : ℝ) (hp : 0 ≤ p)
+    (hfirst : finiteMean (fun Y => q^(InputTree.slabCost L d (T Y) e Y)) ≤
+      Real.exp (p+Fintype.card ι*(68*Real.exp (-(L:ℝ)/64)))) :
+    finiteMean (fun Y => q^(InputTree.familyCost L n d T e Y)) ≤
+      Real.exp (p+Fintype.card ι*(68*∑ j ∈ Finset.range n,
+        Real.exp (-((L*4^j:ℕ):ℝ)/64))) := by
+  induction n with
+  | zero =>
+    simp only [InputTree.familyCost,Finset.range_zero,Finset.sum_empty,pow_zero,
+      finiteMean_const,mul_zero,add_zero]
+    exact Real.one_le_exp hp
+  | succ n ih =>
+    by_cases hn : n = 0
+    · subst n
+      simpa only [InputTree.familyCost,Nat.zero_add,Finset.sum_range_one,pow_zero,mul_one] using hfirst
+    · let l := L*4^n
+      have hl : 0 < l := Nat.mul_pos hL (pow_pos (by decide) _)
+      have hsl : 2*familyEnd L n ≤ l := familyEnd_two L n
+      have hs' : s ≤ familyEnd L n := by
+        have hh := familyEnd_ge L n 0 (Nat.pos_of_ne_zero hn)
+        simp only [pow_zero,mul_one] at hh
+        omega
+      have hc : ∀ Y, InputTree.familyCost L (n+1) d T e Y =
+          InputTree.familyCost L n d T e Y+InputTree.slabCost l d (T Y) e Y := by
+        intro Y
+        exact Finset.sum_range_succ _ _
+      have hm : finiteMean (fun Y => q^(InputTree.familyCost L (n+1) d T e Y)) ≤
+          finiteMean (fun Y => q^(InputTree.familyCost L n d T e Y)) *
+            Real.exp (Fintype.card ι*(68*Real.exp (-(l:ℝ)/64))) := by
+        by_cases hld : l ≤ d
+        · simp_rw [hc,pow_add]
+          apply InputTree.random_slab_adapted_mul_le d (familyEnd L n) l (by omega) hl hsl T
+            (layerAdapted_mono hT hs') e _ (fun _ => pow_nonneg (by linarith) _)
+          · apply layerAdapted_comp
+            exact layerAdapted_mono (InputTree.familyCost_adapted L n d s T hT e) (max_le hs' le_rfl)
+          · exact hq
+          · exact hlog
+        · simp_rw [hc,InputTree.slabCost_gt l d _ (by omega),Nat.add_zero]
+          exact le_mul_of_one_le_right
+            (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
+            (Real.one_le_exp_iff.mpr (by positivity))
+      refine (hm.trans (mul_le_mul_of_nonneg_right ih (Real.exp_nonneg _))).trans_eq ?_
+      rw [←Real.exp_add,Finset.sum_range_succ]
+      congr 1
+      dsimp only [l]
+      ring
+
+lemma InputTree.family_mgf_first_decay (L n d s : ℕ) (hL : 0 < L) (hs : s ≤ L)
+    (T : (SwitchIndex d → Bool) → InputTree d) (hT : LayerAdapted d s T)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) (p : ℝ) (hp : 0 ≤ p)
+    (hfirst : finiteMean (fun Y => q^(InputTree.slabCost L d (T Y) e Y)) ≤
+      Real.exp (p+Fintype.card ι*(68*Real.exp (-(L:ℝ)/64)))) :
+    finiteMean (fun Y => q^(InputTree.familyCost L n d T e Y)) ≤
+      Real.exp (p+Fintype.card ι*heightDecay L) := by
+  apply (InputTree.family_mgf_first L n d s hL hs T hT e q hq hlog p hp hfirst).trans
+  apply Real.exp_le_exp.mpr
+  apply add_le_add_right
+  apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
+  have hh := mul_le_mul_of_nonneg_left (sum_family_decay L n hL) (by norm_num : (0:ℝ)≤68)
+  simpa only [heightDecay,mul_div_assoc] using hh
+
+
+noncomputable def modifiedInputTree (L r : ℕ) (E : Finset (Card r))
+    (U : Card r → Equiv.Perm (Card L)) (X Y : SwitchIndex (L+r) → Bool) : InputTree (L+r) :=
+  boundaryTree L r (modifiedBoundary L r E U Y) X
+
+lemma modifiedInputTree_adapted (L r : ℕ) (E : Finset (Card r))
+    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool) :
+    LayerAdapted (L+r) L (modifiedInputTree L r E U X) := by
+  intro Y Y' h
+  unfold modifiedInputTree
+  rw [modifiedBoundary_adapted L r E U Y Y' h]
+
+lemma modified_slab_mgf (l L r : ℕ) (hl : 0 < l) (E : Finset (Card r))
+    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun Y => q^(InputTree.slabCost l (L+r) (modifiedInputTree L r E U X Y) e Y)) ≤
+      Real.exp (((l*E.card*2^L:ℕ):ℝ)*Real.log q+
+        Fintype.card ι*(68*Real.exp (-(l:ℝ)/64))) := by
+  have hqp : 0 < q := by linarith
+  have hpoint (Y : SwitchIndex (L+r) → Bool) :
+      InputTree.slabCost l (L+r) (modifiedInputTree L r E U X Y) e Y ≤
+        InputTree.slabCost l (L+r) (boundaryTree L r U X) e Y+l*E.card*2^L := by
+    apply (boundaryTree_slabCost_perturb l L r (modifiedBoundary L r E U Y) U X e Y).trans
+    apply Nat.add_le_add_left
+    simpa only [Nat.mul_assoc] using Nat.mul_le_mul_left l (modifiedBoundary_distance_le L r E U Y)
+  calc
+    _ ≤ finiteMean (fun Y => q^(InputTree.slabCost l (L+r) (boundaryTree L r U X) e Y+l*E.card*2^L)) :=
+      finiteMean_mono (fun Y => pow_le_pow_right₀ hq (hpoint Y))
+    _ = finiteMean (fun Y => q^(InputTree.slabCost l (L+r) (boundaryTree L r U X) e Y)) *
+        q^(l*E.card*2^L) := by simp_rw [pow_add]; exact finiteMean_mul_const _ _
+    _ ≤ Real.exp (Fintype.card ι*(68*Real.exp (-(l:ℝ)/64))) * q^(l*E.card*2^L) :=
+      mul_le_mul_of_nonneg_right (InputTree.slab_mgf l (L+r) hl _ e q hq hlog) (pow_nonneg (by linarith) _)
+    _ = _ := by
+      have he : Real.exp (((l*E.card*2^L:ℕ):ℝ)*Real.log q) = q^(l*E.card*2^L) := by
+        rw [Real.exp_nat_mul,Real.exp_log hqp]
+      rw [Real.exp_add,he,mul_comm]
+
+lemma modified_family_mgf (l n L r : ℕ) (hl : 0 < l) (hLl : L ≤ l) (E : Finset (Card r))
+    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun Y => q^(InputTree.familyCost l n (L+r) (modifiedInputTree L r E U X) e Y)) ≤
+      Real.exp (((l*E.card*2^L:ℕ):ℝ)*Real.log q+Fintype.card ι*heightDecay l) := by
+  exact InputTree.family_mgf_first_decay l n (L+r) L hl hLl _
+    (modifiedInputTree_adapted L r E U X) e q hq hlog _
+    (mul_nonneg (Nat.cast_nonneg _) (Real.log_nonneg hq))
+    (modified_slab_mgf l L r hl E U X e q hq hlog)
+
+theorem modified_high_mgf (L n r : ℕ) (hL : 0 < L) (E : Finset (Card r))
+    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/500) :
+    finiteMean (fun Y => q^(InputTree.familyCost L n (L+r) (modifiedInputTree L r E U X) e Y+
+      InputTree.familyCost (2*L) n (L+r) (modifiedInputTree L r E U X) e Y)) ≤
+      Real.exp (Fintype.card ι*heightDecay L +
+        3*((L*E.card*2^L:ℕ):ℝ)*Real.log q) := by
+  have hq2 := one_le_pow₀ (n:=2) hq
+  have hlog2 : Real.log (q^2) ≤ 1/250 := by rw [Real.log_pow]; norm_num; linarith
+  have h₀ := modified_family_mgf L n L r hL le_rfl E U X e (q^2) hq2 hlog2
+  have h₁ := modified_family_mgf (2*L) n L r (by omega) (by omega) E U X e (q^2) hq2 hlog2
+  have h₁' : finiteMean (fun Y => (q^2)^(InputTree.familyCost (2*L) n (L+r)
+      (modifiedInputTree L r E U X) e Y)) ≤
+      Real.exp ((((2*L)*E.card*2^L:ℕ):ℝ)*Real.log (q^2)+Fintype.card ι*heightDecay L) := by
+    apply h₁.trans
+    apply Real.exp_le_exp.mpr
+    exact add_le_add_right (mul_le_mul_of_nonneg_left
+      (heightDecay_antitone (show L ≤ 2*L by omega)) (Nat.cast_nonneg _)) _
+  have hh := finiteMean_mul_exp_le
+    (fun Y => q^(InputTree.familyCost L n (L+r) (modifiedInputTree L r E U X) e Y))
+    (fun Y => q^(InputTree.familyCost (2*L) n (L+r) (modifiedInputTree L r E U X) e Y))
+    _ _ (by simpa only [pow_right_comm] using h₀) (by simpa only [pow_right_comm] using h₁')
+  simp only [←pow_add] at hh
+  refine hh.trans_eq ?_
+  congr 1
+  simp only [Real.log_pow,Nat.cast_mul,Nat.cast_ofNat]
+  ring
+
+noncomputable def InputTree.lowCost (H d : ℕ) (T : InputTree d)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) : ℕ :=
+  ∑ i ∈ Finset.range H, T.levelCost (i+1) e Y
+
+lemma InputTree.lowCost_slab (L d : ℕ) (T : InputTree d)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) :
+    InputTree.lowCost L d T e Y+InputTree.slabCost L d T e Y = InputTree.lowCost (2*L) d T e Y := by
+  rw [InputTree.lowCost,InputTree.lowCost,show 2*L=L+L by omega,Finset.sum_range_add]
+  congr 1
+  exact Fin.sum_univ_eq_sum_range (fun i => T.levelCost (L+i+1) e Y) L
+
+lemma InputTree.cost_family (L n d : ℕ) (T : (SwitchIndex d → Bool) → InputTree d)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) :
+    InputTree.lowCost L d (T Y) e Y + InputTree.familyCost L n d T e Y +
+      InputTree.familyCost (2*L) n d T e Y = InputTree.lowCost (L*4^n) d (T Y) e Y := by
+  induction n with
+  | zero => simp only [InputTree.familyCost,Finset.range_zero,Finset.sum_empty,Nat.add_zero,
+      pow_zero,mul_one]
+  | succ n ih =>
+    have he : (2*L)*4^n = 2*(L*4^n) := by ring
+    have he' : L*4^(n+1) = 2*(2*(L*4^n)) := by rw [pow_succ]; ring
+    simp only [InputTree.familyCost,Finset.sum_range_succ] at ih ⊢
+    rw [he,he',←InputTree.lowCost_slab,←InputTree.lowCost_slab,←ih]
+    omega
+
+lemma InputTree.lowCost_ge (H d : ℕ) (hd : d ≤ H) (T : InputTree d)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) :
+    InputTree.lowCost H d T e Y = InputTree.lowCost d d T e Y := by
+  symm
+  apply Finset.sum_subset (Finset.range_mono hd)
+  intro i _ hi
+  have hi' : d ≤ i := by simpa only [Finset.mem_range,not_lt] using hi
+  exact InputTree.levelCost_gt (i+1) d T (by omega) e Y
+
+lemma boundaryTree_levelCost_low (t L r : ℕ) (ht : t ≤ L)
+    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (Y : SwitchIndex (L+r) → Bool) :
+    (boundaryTree L r U X).levelCost t e Y = 0 := by
+  induction r generalizing ι with
+  | zero => rfl
+  | succ r ih =>
+    rw [boundaryTree,InputTree.levelCost,ite_eq_right (by omega : t ≠ L+r+1),ih,ih]
+
+lemma boundaryTree_lowCost_zero (L r : ℕ) (U : Card r → Equiv.Perm (Card L))
+    (X : SwitchIndex (L+r) → Bool) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (L+r)) (Y : SwitchIndex (L+r) → Bool) :
+    InputTree.lowCost L (L+r) (boundaryTree L r U X) e Y = 0 := by
+  apply Finset.sum_eq_zero
+  intro i hi
+  exact boundaryTree_levelCost_low (i+1) L r (by simpa only [Finset.mem_range,Nat.lt_iff_add_one_le] using hi) U X e Y
+
+theorem modified_cost_mgf (L r : ℕ) (hL : 0 < L) (E : Finset (Card r))
+    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/500) :
+    finiteMean (fun Y => q^(InputTree.lowCost (L+r) (L+r) (modifiedInputTree L r E U X Y) e Y)) ≤
+      Real.exp (Fintype.card ι*heightDecay L +
+        3*((L*E.card*2^L:ℕ):ℝ)*Real.log q) := by
+  have hbound : L+r ≤ L*4^(L+r) := by
+    have h₀ := pow_four_ge (L+r)
+    have h₁ := Nat.mul_le_mul_right (4^(L+r)) hL
+    nlinarith
+  have he (Y : SwitchIndex (L+r) → Bool) :=
+    InputTree.cost_family L (L+r) (L+r) (modifiedInputTree L r E U X) e Y
+  have hz (Y : SwitchIndex (L+r) → Bool) :
+      InputTree.lowCost L (L+r) (modifiedInputTree L r E U X Y) e Y = 0 :=
+    boundaryTree_lowCost_zero L r _ X e Y
+  simp only [hz,Nat.zero_add,InputTree.lowCost_ge _ _ hbound] at he
+  simp_rw [←he]
+  exact modified_high_mgf L (L+r) r hL E U X e q hq hlog
+
+
+noncomputable def exceptionChild {r : ℕ} (E : Finset (Card (r+1))) (b : Bool) : Finset (Card r) :=
+  Finset.univ.filter (fun u => Fin.cons b u ∈ E)
+
+lemma mem_exceptionChild {r : ℕ} (E : Finset (Card (r+1))) (b : Bool) (u : Card r) :
+    u ∈ exceptionChild E b ↔ Fin.cons b u ∈ E := by
+  classical
+  simp only [exceptionChild,Finset.mem_filter,Finset.mem_univ,true_and]
+
+lemma exceptionChild_card_add {r : ℕ} (E : Finset (Card (r+1))) :
+    (exceptionChild E false).card+(exceptionChild E true).card = E.card := by
+  classical
+  have he : (∑ u : Card (r+1), if u ∈ E then 1 else 0) = E.card := by simp
+  rw [Fintype.sum_equiv (headTailEquiv r) _
+    (fun x : Bool × Card r => if Fin.cons x.1 x.2 ∈ E then 1 else 0)
+    (fun x => by change (if x ∈ E then 1 else 0) = (if Fin.cons (x 0) (Fin.tail x) ∈ E then 1 else 0); rw [Fin.cons_self_tail])] at he
+  simp only [Fintype.sum_prod_type,Fintype.sum_bool] at he
+  simpa only [exceptionChild,Finset.card_filter,add_comm] using he
+
+def seedSplitEquiv (r : ℕ) (A : Type*) : (Card (r+1) → A) ≃ (Card r → A) × (Card r → A) where
+  toFun U := (fun u => U (Fin.cons false u),fun u => U (Fin.cons true u))
+  invFun U := fun u => if u 0 then U.2 (Fin.tail u) else U.1 (Fin.tail u)
+  left_inv U := by
+    funext u
+    dsimp only
+    split_ifs with h
+    · rw [←h,Fin.cons_self_tail]
+    · have h' : u 0 = false := by cases hh : u 0 <;> simp_all
+      rw [←h',Fin.cons_self_tail]
+  right_inv U := by simp only [Fin.cons_zero,Fin.tail_cons,Bool.false_eq_true,↓reduceIte]
+
+def seedSandwichShuffle {A B C D E : Type*} :
+    (A × B) × ((C × D) × E) ≃ ((A × C) × (B × D)) × E where
+  toFun x := (((x.1.1,x.2.1.1),(x.1.2,x.2.1.2)),x.2.2)
+  invFun x := ((x.1.1.1,x.1.2.1),((x.1.1.2,x.1.2.2),x.2))
+  left_inv _ := rfl
+  right_inv _ := rfl
+
+abbrev ModifiedCoins (L r : ℕ) := (Card r → Equiv.Perm (Card L)) × BenesCoins (L+r)
+
+def modifiedStepEquiv (L r : ℕ) : ModifiedCoins L (r+1) ≃
+    (ModifiedCoins L r × ModifiedCoins L r) × ((Card (L+r) → Bool) × (Card (L+r) → Bool)) :=
+  (Equiv.prodCongr (seedSplitEquiv r (Equiv.Perm (Card L))) (benesStepEquiv (L+r))).trans
+    seedSandwichShuffle
+
+noncomputable def modifiedPerm (L r : ℕ) (E : Finset (Card r)) (ω : ModifiedCoins L r) :
+    Equiv.Perm (Card (L+r)) :=
+  butterflyPerm (L+r) (decodeButterfly (L+r) ω.2.1) *
+    (modifiedInputTree L r E ω.1 ω.2.2 ω.2.1).perm⁻¹
+
+noncomputable def modifiedCost (L r : ℕ) (E : Finset (Card r))
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (ω : ModifiedCoins L r) : ℕ :=
+  InputTree.lowCost (L+r) (L+r) (modifiedInputTree L r E ω.1 ω.2.2 ω.2.1) e ω.2.1
+
+lemma modifiedBoundary_cons (L r : ℕ) (E : Finset (Card (r+1)))
+    (U : Card (r+1) → Equiv.Perm (Card L)) (Y : SwitchIndex (L+r+1) → Bool) (b : Bool) :
+    (fun u => modifiedBoundary L (r+1) E U Y (Fin.cons b u)) =
+    modifiedBoundary L r (exceptionChild E b) (fun u => U (Fin.cons b u))
+      (if b then (coinStepEquiv (L+r) Y).1.2 else (coinStepEquiv (L+r) Y).1.1) := by
+  funext u
+  simp only [modifiedBoundary,mem_exceptionChild,boundaryOutput,Fin.cons_zero,Fin.tail_cons]
+
+lemma modifiedInputTree_node (L r : ℕ) (E : Finset (Card (r+1)))
+    (ω : ModifiedCoins L r × ModifiedCoins L r) (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
+    let z := (modifiedStepEquiv L r).symm (ω,coins)
+    modifiedInputTree L (r+1) E z.1 z.2.2 z.2.1 =
+      InputTree.node coins.1 (fun b =>
+        let w := if b then ω.2 else ω.1
+        modifiedInputTree L r (exceptionChild E b) w.1 w.2.2 w.2.1) := by
+  dsimp only
+  unfold modifiedInputTree
+  rw [boundaryTree]
+  change InputTree.node coins.1 (fun b => boundaryTree L r _ _) = _
+  congr 1
+  funext b
+  rw [modifiedBoundary_cons]
+  cases b <;> rfl
+
+lemma modifiedPerm_step (L r : ℕ) (E : Finset (Card (r+1)))
+    (ω : ModifiedCoins L r × ModifiedCoins L r) (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
+    modifiedPerm L (r+1) E ((modifiedStepEquiv L r).symm (ω,coins)) =
+      pairSwitch coins.2 * childLift (fun b => modifiedPerm L r (exceptionChild E b)
+        (if b then ω.2 else ω.1)) * pairSwitch coins.1 := by
+  rw [modifiedPerm,modifiedInputTree_node,InputTree.perm,mul_inv_rev,pairSwitch_inv,childLift_inv]
+  change (butterflyPerm (L+r+1) (decodeButterfly (L+r+1)
+    ((coinStepEquiv (L+r)).symm ((ω.1.2.1,ω.2.2.1),coins.2)))) * _ = _
+  rw [butterflyPerm_step,mul_assoc,←mul_assoc (childLift _),childLift_mul,←mul_assoc]
+  congr 2
+  apply congrArg childLift
+  funext b
+  cases b <;> rfl
+
+lemma modifiedPerm_coordinates (L r : ℕ) (E : Finset (Card (r+1)))
+    (ω : ModifiedCoins L r × ModifiedCoins L r) (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool))
+    (x : Card (L+r+1)) :
+    headTailEquiv (L+r) (modifiedPerm L (r+1) E ((modifiedStepEquiv L r).symm (ω,coins)) x) =
+      PairRouting.sandwich coins.1 coins.2 (fun b => modifiedPerm L r (exceptionChild E b)
+        (if b then ω.2 else ω.1)) (headTailEquiv (L+r) x) := by
+  rw [modifiedPerm_step]
+  simp only [Equiv.Perm.mul_apply,headTail_switch,headTail_child]
+  rfl
+
+lemma InputTree.lowCost_node (d : ℕ) (ξ : Card d → Bool) (T : Bool → InputTree d)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (d+1)) (Y : (SwitchIndex d → Bool) × (SwitchIndex d → Bool))
+    (η : Card d → Bool) :
+    InputTree.lowCost (d+1) (d+1) (.node ξ T) e ((coinStepEquiv d).symm (Y,η)) =
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x ξ
+      let hc := PairRouting.colors_compatible x ξ
+      PairRouting.alternatingCycles (PairRouting.switchedEmbedding x ξ)
+        (fun b => if b then butterflyPerm d (decodeButterfly d Y.2) * (T true).perm⁻¹
+          else butterflyPerm d (decodeButterfly d Y.1) * (T false).perm⁻¹) +
+        InputTree.lowCost d d (T false) (PairRouting.childEmbedding x c hc false) Y.1 +
+        InputTree.lowCost d d (T true) (PairRouting.childEmbedding x c hc true) Y.2 := by
+  dsimp only
+  rw [InputTree.lowCost,Finset.sum_range_succ,InputTree.levelCost_node,ite_eq_left rfl]
+  have he : (∑ i ∈ Finset.range d,
+      (.node ξ T : InputTree (d+1)).levelCost (i+1) e ((coinStepEquiv d).symm (Y,η))) =
+      InputTree.lowCost d d (T false)
+        (PairRouting.childEmbedding (e.trans (headTailEquiv d).toEmbedding)
+          (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) ξ) (PairRouting.colors_compatible _ _) false) Y.1 +
+      InputTree.lowCost d d (T true)
+        (PairRouting.childEmbedding (e.trans (headTailEquiv d).toEmbedding)
+          (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) ξ) (PairRouting.colors_compatible _ _) true) Y.2 := by
+    rw [InputTree.lowCost,InputTree.lowCost,←Finset.sum_add_distrib]
+    apply Finset.sum_congr rfl
+    intro i hi
+    rw [InputTree.levelCost_node,ite_eq_right (by simp only [Finset.mem_range] at hi; omega : i+1 ≠ d+1)]
+  rw [he]
+  omega
+
+noncomputable def modifiedAssignmentCost (L r : ℕ) (E : Finset (Card (r+1)))
+    {ι : Type*} [Fintype ι] (x : ι ↪ Bool × Card (L+r)) (c : ι → Bool)
+    (hc : PairRouting.Compatible x c) (ω : ModifiedCoins L r × ModifiedCoins L r) : ℕ :=
+  PairRouting.alternatingCycles (PairRouting.coloredEmbedding x c hc)
+    (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) +
+  modifiedCost L r (exceptionChild E false) (PairRouting.childEmbedding x c hc false) ω.1 +
+  modifiedCost L r (exceptionChild E true) (PairRouting.childEmbedding x c hc true) ω.2
+
+lemma modifiedCost_step (L r : ℕ) (E : Finset (Card (r+1)))
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r+1))
+    (ω : ModifiedCoins L r × ModifiedCoins L r) (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
+    modifiedCost L (r+1) E e ((modifiedStepEquiv L r).symm (ω,coins)) =
+      modifiedAssignmentCost L r E (e.trans (headTailEquiv (L+r)).toEmbedding)
+        (PairRouting.colors (e.trans (headTailEquiv (L+r)).toEmbedding) coins.1)
+        (PairRouting.colors_compatible _ _) ω := by
+  rw [modifiedCost,modifiedInputTree_node]
+  change InputTree.lowCost (L+r+1) (L+r+1) _ e
+    ((coinStepEquiv (L+r)).symm ((ω.1.2.1,ω.2.2.1),coins.2)) = _
+  rw [InputTree.lowCost_node]
+  unfold modifiedAssignmentCost
+  dsimp only
+  simp only [Bool.false_eq_true,↓reduceIte]
+  rw [PairRouting.switchedEmbedding_eq_colored _ _ (PairRouting.colors_compatible _ _) _ rfl]
+  congr 2
+
+
+namespace PairRouting
+variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α] [DecidableEq ι]
+
+omit [Fintype α] [DecidableEq ι] in
+lemma childEmbedding_surjective (x : ι ↪ Bool × α) (hx : Function.Surjective x)
+    (c : ι → Bool) (hc : Compatible x c) (b : Bool) :
+    Function.Surjective (childEmbedding x c hc b) := by
+  intro y
+  have hs : Function.Surjective (coloredEmbedding x c hc) :=
+    (pairLayer (chosenCoin x c)).surjective.comp hx
+  obtain ⟨i,hi⟩ := hs (b,y)
+  rw [coloredEmbedding_apply] at hi
+  exact ⟨⟨i,congrArg Prod.fst hi⟩,congrArg Prod.snd hi⟩
+
+omit [DecidableEq ι] in
+lemma uniformPerm_tuple_upper (e f : ι ↪ α) (he : Function.Surjective e) (p : Equiv.Perm α) :
+    finiteMean (fun u : Equiv.Perm α => if ∀ i, (p*u⁻¹) (e i) = f i then (1:ℝ) else 0) ≤
+      1/(Fintype.card (Equiv.Perm α):ℝ) := by
+  classical
+  let M : Equiv.Perm α → Prop := fun u => ∀ i, (p*u⁻¹) (e i) = f i
+  have huniq (u v : Equiv.Perm α) (hu : M u) (hv : M v) : u = v := by
+    have hh : p*u⁻¹ = p*v⁻¹ := by
+      ext x
+      obtain ⟨i,rfl⟩ := he x
+      exact (hu i).trans (hv i).symm
+    exact inv_injective (mul_left_cancel hh)
+  unfold finiteMean
+  apply div_le_div_of_nonneg_right _ (Nat.cast_nonneg _)
+  change (∑ u, if M u then (1:ℝ) else 0) ≤ 1
+  by_cases h : ∃ u, M u
+  · obtain ⟨u,hu⟩ := h
+    rw [Finset.sum_eq_single u]
+    · simp only [hu,↓reduceIte,le_refl]
+    · intro v _ hv
+      exact ite_eq_right (fun hh => hv (huniq v u hh hu))
+    · simp
+  · push Not at h
+    simp only [h,ite_false,Finset.sum_const_zero,zero_le_one]
+
+end PairRouting
+
+noncomputable def centralFactor (L r : ℕ) (E : Finset (Card r)) (a : ℝ) : ℝ :=
+  (Nat.factorial (2^L):ℝ)^(-((2:ℝ)^r-E.card)*a)
+
+lemma centralFactor_nonneg (L r : ℕ) (E : Finset (Card r)) (a : ℝ) :
+    0 ≤ centralFactor L r E a := Real.rpow_nonneg (Nat.cast_nonneg _) _
+
+lemma centralFactor_step (L r : ℕ) (E : Finset (Card (r+1))) (a : ℝ) :
+    centralFactor L (r+1) E a =
+      centralFactor L r (exceptionChild E false) a * centralFactor L r (exceptionChild E true) a := by
+  unfold centralFactor
+  rw [←Real.rpow_add (Nat.cast_pos.mpr (Nat.factorial_pos _))]
+  congr 1
+  have he : ((exceptionChild E false).card:ℝ)+(exceptionChild E true).card = E.card := by
+    exact_mod_cast exceptionChild_card_add E
+  rw [pow_succ,←he]
+  ring
+
+lemma modifiedPerm_zero (L : ℕ) (E : Finset (Card 0)) (ω : ModifiedCoins L 0) :
+    modifiedPerm L 0 E ω =
+      if (fun j => Fin.elim0 j : Card 0) ∈ E then 1
+      else butterflyPerm L (decodeButterfly L ω.2.1)*(ω.1 (fun j => Fin.elim0 j))⁻¹ := by
+  unfold modifiedPerm modifiedInputTree boundaryTree InputTree.perm modifiedBoundary boundaryOutput
+  split_ifs
+  · exact mul_inv_cancel _
+  · rfl
+
+lemma modifiedCost_zero (L : ℕ) (E : Finset (Card 0)) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card L) (ω : ModifiedCoins L 0) : modifiedCost L 0 E e ω = 0 := by
+  simp only [modifiedCost,modifiedInputTree,boundaryTree,InputTree.lowCost,
+    InputTree.levelCost,Finset.sum_const_zero]
+
+lemma centralFactor_zero (L : ℕ) (E : Finset (Card 0)) (a : ℝ) :
+    centralFactor L 0 E a =
+      if (fun j => Fin.elim0 j : Card 0) ∈ E then 1 else (Nat.factorial (2^L):ℝ)^(-a) := by
+  classical
+  let z : Card 0 := fun j => Fin.elim0 j
+  have he : (E.card:ℝ) = if z ∈ E then 1 else 0 := by
+    by_cases hz : z ∈ E
+    · have he : E = {z} := by
+        ext u
+        rw [Subsingleton.elim u z]
+        simp only [hz,Finset.mem_singleton]
+      rw [he,ite_eq_left (by simpa only [he] using hz),Finset.card_singleton,Nat.cast_one]
+    · have he : E = ∅ := Finset.eq_empty_iff_forall_notMem.mpr (fun u hu => hz (Subsingleton.elim u z ▸ hu))
+      rw [he,Finset.card_empty,Nat.cast_zero,ite_eq_right (by simp)]
+  unfold centralFactor
+  rw [pow_zero,he]
+  split_ifs <;> simp
+
+end Thorp
+
+end ThorpNine.Harmonic
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Harmonic/InsertionPerm.lean b/lean/OAI/Probability/ThorpRouting/Harmonic/InsertionPerm.lean
index f2cb5e0d78bcdd3c4e8df3138ac1212d5d690ff9..76ca2a98518d138f5b603af2da82ccfea47227d9 100644
--- a/lean/OAI/Probability/ThorpRouting/Harmonic/InsertionPerm.lean
+++ b/lean/OAI/Probability/ThorpRouting/Harmonic/InsertionPerm.lean
@@ -1,891 +1,891 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Harmonic.ZeroExtend
-
-namespace OAI
-
-namespace ThorpNine.Harmonic
-
-namespace Thorp.DimensionEstimates
-open scoped BigOperators Classical
-
-section
-open Filter
-
-lemma chosenTuple_density_bound (N j : ℕ) (hj : 0<j) (hx : 1≤(N:ℝ)/j)
-    (hhalf : ((N:ℝ)/j)^(nearPower-1)≤1/2) (C : ℝ) (hC : 0≤C) :
-    C*(chosenTuple N j:ℝ)*((chosenTuple N j:ℝ)/N)^(1/512:ℝ) ≤
-      C*2^(1+(1/512:ℝ))*(j:ℝ)*((N:ℝ)/j)^(-tailPower) := by
-  let x : ℝ := (N:ℝ)/j
-  have hjp : (0:ℝ)<j := by exact_mod_cast hj
-  have hxp : 0<x := lt_of_lt_of_le zero_lt_one hx
-  have hNp : (0:ℝ)<N := by
-    have : (0:ℝ)<(N:ℝ)/j := hxp
-    exact (div_pos_iff.mp this).resolve_right (by intro h; exact (not_lt_of_ge hjp.le) h.2) |>.1
-  have he : (j:ℝ)*x=N := by dsimp [x]; field_simp
-  have hk := (chosenTuple_bounds N j hj hx hhalf).2.2
-  change (chosenTuple N j:ℝ)≤2*(j:ℝ)*x^nearPower at hk
-  have hratio : (chosenTuple N j:ℝ)/N≤2*x^(nearPower-1) := by
-    rw [Real.rpow_sub hxp,Real.rpow_one]
-    apply (div_le_iff₀ hNp).mpr
-    rw [←he]
-    have : 2*(x^nearPower/x)*((j:ℝ)*x)=2*(j:ℝ)*x^nearPower := by field_simp
-    rw [this]
-    exact hk
-  have hpow := Real.rpow_le_rpow (by positivity : (0:ℝ)≤(chosenTuple N j:ℝ)/N) hratio
-    (by norm_num : (0:ℝ)≤1/512)
-  have hmain := mul_le_mul hk hpow (by positivity) (by positivity)
-  have hrpow : (2*x^(nearPower-1))^(1/512:ℝ)=
-      (2:ℝ)^(1/512:ℝ)*x^((nearPower-1)*(1/512)) := by
-    rw [Real.mul_rpow (by norm_num) (by positivity),←Real.rpow_mul hxp.le]
-  rw [hrpow] at hmain
-  have heq : 2*(j:ℝ)*x^nearPower*((2:ℝ)^(1/512:ℝ)*x^((nearPower-1)*(1/512)))=
-      (2:ℝ)^(1+(1/512:ℝ))*(j:ℝ)*x^(-tailPower) := by
-    rw [Real.rpow_add (by norm_num : (0:ℝ)<2),Real.rpow_one]
-    have hp : nearPower+(nearPower-1)*(1/512) = -tailPower := by norm_num [nearPower,tailPower]
-    rw [←hp,Real.rpow_add hxp]
-    ring
-  rw [heq] at hmain
-  have h := mul_le_mul_of_nonneg_left hmain hC
-  simpa only [mul_assoc] using h
-
-lemma near_conditions_eventually (C : ℝ) :
-    ∀ᶠ x : ℝ in atTop, 2≤x ∧ 2≤x^nearPower ∧ x^(nearPower-1)≤1/2 ∧
-      8*Real.log 2/nearPower≤Real.log x ∧ 4*Real.log 2+2≤Real.log x ∧
-      C*2^(1+(1/512:ℝ))*x^(-tailPower) ≤ nearFraction/4000 := by
-  have hpow := (tendsto_rpow_atTop (by norm_num [nearPower] : 0<nearPower)).eventually
-    (eventually_ge_atTop (2:ℝ))
-  have hhalf : ∀ᶠ x : ℝ in atTop, x^(nearPower-1)≤1/2 := by
-    have h := (tendsto_rpow_neg_atTop (by norm_num [nearPower] : 0<1-nearPower)).eventually
-      (gt_mem_nhds (by norm_num : (0:ℝ)<1/2))
-    filter_upwards [h] with x hx
-    simpa only [neg_sub] using hx.le
-  have hdec : Tendsto (fun x : ℝ => C*2^(1+(1/512:ℝ))*x^(-tailPower)) atTop (nhds 0) := by
-    simpa only [mul_zero] using
-      (tendsto_const_nhds.mul (tendsto_rpow_neg_atTop (by norm_num [tailPower] : 0<tailPower)))
-  have hd := hdec.eventually (gt_mem_nhds (by norm_num [nearFraction] : (0:ℝ)<nearFraction/4000))
-  filter_upwards [eventually_ge_atTop (2:ℝ),hpow,hhalf,
-    Real.tendsto_log_atTop.eventually (eventually_ge_atTop (8*Real.log 2/nearPower)),
-    Real.tendsto_log_atTop.eventually (eventually_ge_atTop (4*Real.log 2+2)),hd] with x hx hp hh hl hl' hd
-  exact ⟨hx,hp,hh,hl,hl',hd.le⟩
-
-theorem exists_near_cutoff (C : ℝ) (hC : 0≤C) :
-    ∃ X : ℝ, 2≤X ∧ ∀ (N j D m : ℕ), 0<j → X≤(N:ℝ)/j →
-      N.choose j≤D*2^j →
-      D * (chosenTuple N j).choose j ≤ m * N.choose j →
-      j≤chosenTuple N j ∧ chosenTuple N j≤N ∧
-      nearFraction*Real.log D≤Real.log m ∧
-      C*(chosenTuple N j:ℝ)*((chosenTuple N j:ℝ)/N)^(1/512:ℝ)≤Real.log m/4000 ∧
-      (1/2:ℝ)*Real.log N≤Real.log D ∧ (j:ℝ)≤Real.log D := by
-  obtain ⟨X,hX⟩ := eventually_atTop.mp (near_conditions_eventually C)
-  refine ⟨max 2 X,le_max_left _ _,?_⟩
-  intro N j D m hj hNx hD hfoot
-  have hc := hX ((N:ℝ)/j) ((le_max_right _ _).trans hNx)
-  have hjp : (0:ℝ)<j := by exact_mod_cast hj
-  have hN : 2*j≤N := by exact_mod_cast (le_div_iff₀ hjp).mp hc.1
-  have hk := chosenTuple_bounds N j hj (by linarith [hc.1]) hc.2.2.1
-  have hm := chosenTuple_log_footprint N j D m hj hN hD hc.2.1 hc.2.2.2.1 hfoot
-  have hd := near_row_log_uniform N j D hj hN hD hc.2.2.2.2.1
-  refine ⟨hk.1,hk.2.1,hm,?_,hd⟩
-  apply (chosenTuple_density_bound N j hj (by linarith [hc.1]) hc.2.2.1 C hC).trans
-  have hh := mul_le_mul_of_nonneg_right hc.2.2.2.2.2 hjp.le
-  dsimp [nearFraction] at hm ⊢
-  dsimp [nearFraction] at hh
-  nlinarith [hd.2]
-
-end
-
-lemma amplification_numerical (m u : ℝ) (hm : 1 ≤ m) (hu : u ≤ Real.log m/4000) :
-    Real.exp u*(Real.exp (Real.log m/2))^(-(1/1000:ℝ))+
-      Real.sqrt (Real.exp (Real.log m/2)/m) ≤ 2*Real.exp (-Real.log m/4000) := by
-  have hmp : 0 < m := lt_of_lt_of_le zero_lt_one hm
-  have hl : 0≤Real.log m := Real.log_nonneg hm
-  have he : Real.exp (Real.log m/2)/m=Real.exp (-Real.log m/2) := by
-    calc
-      _ = Real.exp (Real.log m/2)/Real.exp (Real.log m) := by rw [Real.exp_log hmp]
-      _ = _ := by rw [←Real.exp_sub]; congr 1; ring
-  have hs : Real.sqrt (Real.exp (-Real.log m/2))=Real.exp (-Real.log m/4) := by
-    rw [Real.sqrt_eq_rpow,Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp]
-    congr 1
-    ring
-  rw [Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp,←Real.exp_add,he,hs]
-  have h₁ : u+(Real.log m/2)*(-(1/1000:ℝ))≤ -Real.log m/4000 := by linarith
-  have h₂ : -Real.log m/4≤ -Real.log m/4000 := by linarith
-  linarith [Real.exp_le_exp.mpr h₁,Real.exp_le_exp.mpr h₂]
-
-lemma amplification_absorb (x : ℝ) (hx : 8000*Real.log 2≤x) :
-    2*Real.exp (-x/4000)≤Real.exp (-x/8000) := by
-  rw [show (2:ℝ)=Real.exp (Real.log 2) by rw [Real.exp_log (by norm_num)],←Real.exp_add]
-  exact Real.exp_le_exp.mpr (by linarith)
-
-lemma amplification_decay (m u : ℝ) (hm : 1 ≤ m) (hu : u ≤ Real.log m/4000)
-    (hl : 8000*Real.log 2≤Real.log m) :
-    Real.exp u*(Real.exp (Real.log m/2))^(-(1/1000:ℝ))+
-      Real.sqrt (Real.exp (Real.log m/2)/m) ≤ Real.exp (-Real.log m/8000) :=
-  (amplification_numerical m u hm hu).trans (amplification_absorb _ hl)
-
-end Thorp.DimensionEstimates
-
-namespace Thorp.UnitaryFinite
-open scoped BigOperators Classical
-
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-omit [FiniteDimensional ℂ V] in
-lemma full_tuple_fixed (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V) :
-    fixedSpace tupleAction (Function.Embedding.refl (Card d)) ρ=⊤ := by
-  apply Submodule.eq_top_iff'.mpr
-  intro v g hg
-  have he : g=1 := by
-    apply Equiv.ext
-    intro x
-    exact congrArg (fun f : Card d ↪ Card d => f x) hg
-  rw [he,map_one]
-  rfl
-
-omit [FiniteDimensional ℂ V] in
-lemma full_tuple_finrank (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V) :
-    Module.finrank ℂ (fixedSpace tupleAction (Function.Embedding.refl (Card d)) ρ)=Module.finrank ℂ V := by
-  rw [full_tuple_fixed,finrank_top]
-
-theorem palindrome_irrep_amplification (d : ℕ) {ι : Type*} [Fintype ι]
-    (f : ι ↪ Card d) (hk : 0<Fintype.card ι)
-    (ρ : Representation ℂ (Equiv.Perm (Card d)) V) [Representation.IsIrreducible ρ]
-    (hρ : IsUnitary ρ) (hm : 0<Module.finrank ℂ (fixedSpace tupleAction f ρ))
-    (z : ℝ) (hz : 0<z) :
-    ‖sampleOperator ρ (palindromePerm d)‖ ≤
-      Real.exp (densityConstant*Fintype.card ι*((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ))*
-        z^(-(1/1000:ℝ))+Real.sqrt (z/(Module.finrank ℂ (fixedSpace tupleAction f ρ):ℝ)) := by
-  apply sampleOperator_amplification tupleAction f (tupleAction_transitive f) ρ hρ
-    (palindromePerm d) _ _ (1/1000) z _ (by norm_num) hz (by positivity) _ hm
-  · intro e f
-    simp only [tupleAction_kernel]
-    exact palindrome_kernel_symmetric d e f
-  · intro e
-    simp only [tupleAction_kernel]
-    exact PairRouting.tupleProbability_row_sum _ e
-  · intro e
-    simp only [tupleAction_kernel]
-    exact palindrome_row_moment_bound d e hk
-
-theorem modified_irrep_amplification (L r : ℕ) (E : Finset (Card r))
-    (ρ : Representation ℂ (Equiv.Perm (Card (L+r))) V) [Representation.IsIrreducible ρ]
-    (hρ : IsUnitary ρ) (hm : 0<Module.finrank ℂ V)
-    (u z : ℝ) (hz : 0<z)
-    (hM : ∀ e : Card (L+r) ↪ Card (L+r), modifiedRowMoment L r E e (1/1000)≤Real.exp u) :
-    ‖sampleOperator ρ (modifiedPerm L r E)‖ ≤
-      Real.exp u*z^(-(1/1000:ℝ))+Real.sqrt (z/(Module.finrank ℂ V:ℝ)) := by
-  let f := Function.Embedding.refl (Card (L+r))
-  have h := sampleOperator_amplification tupleAction f (tupleAction_transitive f) ρ hρ
-    (modifiedPerm L r E) (fun e f => by simpa only [tupleAction_kernel] using modified_kernel_symmetric L r E e f)
-    (fun e => by simpa only [tupleAction_kernel] using PairRouting.tupleProbability_row_sum (modifiedPerm L r E) e)
-    (1/1000) z (Real.exp u) (by norm_num) hz (by positivity)
-    (fun e => by simpa only [tupleAction_kernel,modifiedRowMoment] using hM e)
-    (by simpa only [f,full_tuple_finrank] using hm)
-  simpa only [f,full_tuple_finrank] using h
-
-end Thorp.UnitaryFinite
-
-namespace Thorp.Specht
-open scoped BigOperators Classical
-
-theorem choose_tail_le_of_factorial (μ : YoungDiagram) (D : ℕ)
-    (hd : μ.card.factorial ≤ D * Fintype.card (rowGroup μ) * Fintype.card (colGroup μ)) :
-    μ.card.choose (μ.card-μ.rowLen 0) ≤ D * 2^(μ.card-μ.rowLen 0) := by
-  let j := μ.card-μ.rowLen 0
-  have htail := card_fiberGroups_mul_le
-    (fun x : Tail μ => rowIndex x.1) (fun x : Tail μ => colIndex x.1)
-    (fun x y hr hc => Subtype.ext (Subtype.ext (Prod.ext (congrArg Fin.val hr) (congrArg Fin.val hc))))
-  rw [tail_card] at htail
-  have hc := column_group_le_tail μ
-  have hr := tail_row_group μ
-  simp only [←Nat.card_eq_fintype_card] at htail hc hr hd
-  have hh : μ.card.factorial ≤ D * 2^j *
-      ((μ.rowLen 0).factorial * j.factorial) := by
-    calc
-      _ ≤ _ := hd
-      _ ≤ D * Nat.card (rowGroup μ) *
-          (2^j * Nat.card (fiberGroup (fun x : Tail μ => colIndex x.1))) := Nat.mul_le_mul_left _ hc
-      _ = D * 2^j * (μ.rowLen 0).factorial *
-          (Nat.card (fiberGroup (fun x : Tail μ => rowIndex x.1)) *
-           Nat.card (fiberGroup (fun x : Tail μ => colIndex x.1))) := by rw [←hr]; ring
-      _ ≤ _ := by
-        calc
-          _ ≤ D * 2^j * (μ.rowLen 0).factorial * j.factorial :=
-            Nat.mul_le_mul_left _ htail
-          _ = _ := by ring
-  have hj : j ≤ μ.card := Nat.sub_le _ _
-  have hs : μ.card-j = μ.rowLen 0 := by dsimp [j]; have := rowLen_zero_le_card μ; omega
-  have hf := Nat.choose_mul_factorial_mul_factorial hj
-  rw [hs] at hf
-  rw [←hf] at hh
-  have hpos : 0 < (μ.rowLen 0).factorial * j.factorial := Nat.mul_pos (Nat.factorial_pos _) (Nat.factorial_pos _)
-  nlinarith
-
-lemma card_rowGroup_transpose (μ : YoungDiagram) :
-    Fintype.card (rowGroup μ.transpose) = Fintype.card (colGroup μ) := by
-  rw [card_rowGroup,card_colGroup]
-  apply Fintype.prod_equiv (finCongr (μ.colLen_transpose 0))
-  intro i
-  change (μ.transpose.rowLen i).factorial = (μ.colLen i).factorial
-  rw [YoungDiagram.rowLen_transpose]
-
-lemma card_colGroup_transpose (μ : YoungDiagram) :
-    Fintype.card (colGroup μ.transpose) = Fintype.card (rowGroup μ) := by
-  rw [card_colGroup,card_rowGroup]
-  apply Fintype.prod_equiv (finCongr (μ.rowLen_transpose 0))
-  intro i
-  change (μ.transpose.colLen i).factorial = (μ.rowLen i).factorial
-  rw [YoungDiagram.colLen_transpose]
-
-lemma card_transpose (μ : YoungDiagram) : μ.transpose.card=μ.card := by
-  unfold YoungDiagram.card YoungDiagram.transpose
-  exact Finset.card_map _
-
-theorem choose_column_tail_le_dimension (μ : YoungDiagram) :
-    μ.card.choose (μ.card-μ.colLen 0) ≤ Module.finrank ℂ (space μ) * 2^(μ.card-μ.colLen 0) := by
-  have hd := factorial_le_dimension_row_column μ
-  have ht := choose_tail_le_of_factorial μ.transpose (Module.finrank ℂ (space μ))
-  simp only [card_transpose,card_rowGroup_transpose,card_colGroup_transpose,YoungDiagram.rowLen_transpose] at ht
-  apply ht
-  nlinarith
-
-open DimensionEstimates Filter
-
-lemma dimension_log_incidence (μ : YoungDiagram) (hμ : 0<μ.card) :
-    (2*Real.log 2-1)*(μ.card:ℝ)-Real.log 2*((μ.rowLen 0:ℝ)+μ.colLen 0)+1≤
-      Real.log (Module.finrank ℂ (space μ)) := by
-  let : Nonempty (Cell μ) := Fintype.card_pos_iff.mp (by simpa only [card_cell] using hμ)
-  have hd := factorial_le_dimension_row_column μ
-  rw [card_rowGroup,card_colGroup] at hd
-  have hr (i : Fin (μ.colLen 0)) : Nat.card {x : Cell μ // rowIndex x=i}=μ.rowLen i := by
-    let e : {x : Cell μ // rowIndex x=i} ≃ {x : Cell μ // row x=i} :=
-      Equiv.subtypeEquivRight (fun x => Fin.ext_iff)
-    rw [Nat.card_congr e,Nat.card_congr (rowEquiv μ i),Nat.card_fin]
-  have hc (i : Fin (μ.rowLen 0)) : Nat.card {x : Cell μ // colIndex x=i}=μ.colLen i := by
-    let e : {x : Cell μ // colIndex x=i} ≃ {x : Cell μ // col x=i} :=
-      Equiv.subtypeEquivRight (fun x => Fin.ext_iff)
-    rw [Nat.card_congr e,Nat.card_congr (columnEquiv μ i),Nat.card_fin]
-  have hcell : Nat.card (Cell μ)=μ.card := by rw [Nat.card_eq_fintype_card,card_cell]
-  have h := dimension_log_from_incidence (rowIndex (μ := μ)) (colIndex (μ := μ))
-    (fun x y h => Subtype.ext (Prod.ext (congrArg (fun z => (z.1:ℕ)) h)
-      (congrArg (fun z => (z.2:ℕ)) h))) (Module.finrank ℂ (space μ))
-    (by simpa only [←Nat.card_eq_fintype_card,hr,hc,hcell] using hd)
-  simpa only [card_cell,Fintype.card_fin,add_comm (μ.colLen 0:ℝ)] using h
-
-lemma medium_dimension_raw (μ : YoungDiagram) (hμ : 0<μ.card) (δ : ℝ) (hδ : 0<δ)
-    (hδ2 : δ≤1/2) (hrow : δ*(μ.card:ℝ)≤μ.card-μ.rowLen 0)
-    (hcol : (μ.colLen 0:ℝ)≤(μ.card:ℝ)/2) :
-    min ((3/2:ℝ)*Real.log 2-1) (entropyGap*δ)*(μ.card:ℝ)-2*Real.log μ.card-1≤
-      Real.log (Module.finrank ℂ (space μ)) := by
-  let c := (3/2:ℝ)*Real.log 2-1
-  have hN : (0:ℝ)<μ.card := by exact_mod_cast hμ
-  have hlog : 0≤Real.log μ.card := Real.log_nonneg (by exact_mod_cast hμ)
-  have hc : 0≤Real.log 2 := Real.log_nonneg (by norm_num)
-  have hr := rowLen_zero_le_card μ
-  have hcl : μ.colLen 0≤μ.card := by
-    simpa only [YoungDiagram.rowLen_transpose,card_transpose] using rowLen_zero_le_card μ.transpose
-  by_cases hsum : (μ.rowLen 0:ℝ)+μ.colLen 0≤(μ.card:ℝ)/2
-  · have hi := dimension_log_incidence μ hμ
-    have hm := mul_le_mul_of_nonneg_right (min_le_left c (entropyGap*δ)) hN.le
-    dsimp [c] at hm
-    nlinarith
-  · have hbig : (μ.card:ℝ)/4≤μ.rowLen 0 ∨ (μ.card:ℝ)/4≤μ.colLen 0 := by
-      by_contra h
-      push Not at h
-      linarith
-    rcases hbig with hrb | hcb
-    · have hj : 0<μ.card-μ.rowLen 0 := by
-        rw [←Nat.cast_pos (α := ℝ),Nat.cast_sub hr]
-        exact lt_of_lt_of_le (mul_pos hδ hN) hrow
-      have hjN : μ.card-μ.rowLen 0<μ.card := by
-        have : 0<μ.rowLen 0 := by exact_mod_cast (by linarith : (0:ℝ)<μ.rowLen 0)
-        omega
-      have hi := log_dimension_from_binomial μ.card (μ.card-μ.rowLen 0)
-        (Module.finrank ℂ (space μ)) hj hjN
-        (by rw [Nat.cast_sub hr]; linarith) (choose_tail_le_dimension μ)
-      rw [Nat.cast_sub hr] at hi
-      have hm := mul_le_mul_of_nonneg_right (min_le_right c (entropyGap*δ)) hN.le
-      nlinarith [entropyGap_pos]
-    · have hj : 0<μ.card-μ.colLen 0 := by
-        rw [←Nat.cast_pos (α := ℝ),Nat.cast_sub hcl]
-        linarith
-      have hjN : μ.card-μ.colLen 0<μ.card := by
-        have : 0<μ.colLen 0 := by exact_mod_cast (by linarith : (0:ℝ)<μ.colLen 0)
-        omega
-      have hi := log_dimension_from_binomial μ.card (μ.card-μ.colLen 0)
-        (Module.finrank ℂ (space μ)) hj hjN
-        (by rw [Nat.cast_sub hcl]; linarith) (choose_column_tail_le_dimension μ)
-      rw [Nat.cast_sub hcl] at hi
-      have hm := mul_le_mul_of_nonneg_right (min_le_right c (entropyGap*δ)) hN.le
-      have ht := mul_le_mul_of_nonneg_left (show δ*(μ.card:ℝ)≤(μ.card:ℝ)-μ.colLen 0 by nlinarith) entropyGap_pos.le
-      linarith
-
-end Thorp.Specht
-
-namespace Thorp.DimensionEstimates
-
-section
-open scoped BigOperators Classical
-open Filter
-
-lemma finset_power_sum {ι : Type*} [Fintype ι] (x : ℝ) :
-    ∑ E : Finset ι, x^E.card=(1+x)^Fintype.card ι := by
-  have h := Finset.prod_one_add (f := fun _ : ι => x) Finset.univ
-  simpa only [Finset.prod_const,Finset.card_univ,Finset.powerset_univ] using h.symm
-
-lemma small_sets_count {ι : Type*} [Fintype ι] (ε t : ℝ) (ht : 0≤t) :
-    (∑ E : Finset ι, if (E.card:ℝ)<ε*Fintype.card ι then (1:ℝ) else 0) ≤
-      Real.exp ((t*ε+Real.log (1+Real.exp (-t)))*Fintype.card ι) := by
-  have hpow (E : Finset ι) : (Real.exp (-t))^E.card=Real.exp (-t*(E.card:ℝ)) := by
-    rw [←Real.exp_nat_mul]
-    congr 1
-    ring
-  calc
-    _ ≤ ∑ E : Finset ι, Real.exp (t*ε*Fintype.card ι)*(Real.exp (-t))^E.card := by
-      apply Finset.sum_le_sum
-      intro E _
-      split_ifs with hE
-      · rw [hpow,←Real.exp_add,←Real.exp_zero]
-        apply Real.exp_le_exp.mpr
-        nlinarith [mul_le_mul_of_nonneg_left hE.le ht]
-      · positivity
-    _ = Real.exp (t*ε*Fintype.card ι)*(1+Real.exp (-t))^Fintype.card ι := by
-      rw [←Finset.mul_sum,finset_power_sum]
-    _ = _ := by
-      have he : (1+Real.exp (-t))^Fintype.card ι=
-          Real.exp (Real.log (1+Real.exp (-t))*(Fintype.card ι:ℝ)) := by
-        rw [mul_comm,Real.exp_nat_mul,Real.exp_log (by positivity)]
-      rw [he,←Real.exp_add]
-      congr 1
-      ring
-
-lemma small_sets_ceil_bound {ι : Type*} [Fintype ι] (ε t : ℝ) (ht : 0≤t) :
-    (∑ E : Finset ι, if E.card<⌈ε*Fintype.card ι⌉₊ then (1:ℝ) else 0) ≤
-      Real.exp ((t*ε+Real.log (1+Real.exp (-t)))*Fintype.card ι) := by
-  have he (E : Finset ι) : E.card<⌈ε*Fintype.card ι⌉₊ ↔ (E.card:ℝ)<ε*Fintype.card ι := Nat.lt_ceil
-  simp only [he]
-  exact small_sets_count ε t ht
-
-lemma exists_small_set_rate (κ ε₀ : ℝ) (hκ : 0<κ) (hε₀ : 0<ε₀) :
-    ∃ ε : ℝ, 0<ε ∧ ε≤ε₀ ∧ ε<1/2 ∧ ∃ t : ℝ, 0≤t ∧
-      t*ε+Real.log (1+Real.exp (-t))≤κ := by
-  have ht : Tendsto (fun t : ℝ => Real.log (1+Real.exp (-t))) atTop (nhds 0) := by
-    have he : Tendsto (fun t : ℝ => Real.exp (-t)) atTop (nhds 0) :=
-      Real.tendsto_exp_atBot.comp tendsto_neg_atTop_atBot
-    have hl := (tendsto_const_nhds.add he).log (by norm_num : (1:ℝ)+0≠0)
-    simpa only [add_zero,Real.log_one] using hl
-  obtain ⟨t,ht,hlog⟩ := ((eventually_ge_atTop (1:ℝ)).and
-    (ht.eventually (gt_mem_nhds (by linarith : (0:ℝ)<κ/2)))).exists
-  let ε := min ε₀ (min (1/4) (κ/(2*t)))
-  have htp : 0<t := by linarith
-  have hε : 0<ε := lt_min hε₀ (lt_min (by norm_num) (by positivity))
-  refine ⟨ε,hε,min_le_left _ _,(min_le_right _ _).trans_lt ((min_le_left _ _).trans_lt (by norm_num)),t,htp.le,?_⟩
-  have hsmall : ε≤κ/(2*t) := (min_le_right _ _).trans (min_le_right _ _)
-  have hm := (le_div_iff₀ (by positivity : (0:ℝ)<2*t)).mp hsmall
-  linarith
-
-end
-open Filter
-
-lemma log_error_eventually (a : ℝ) (ha : 0<a) :
-    ∀ᶠ N : ℕ in atTop, 2*Real.log N+1≤(a/2)*(N:ℝ) := by
-  have h := (tendsto_natCast_atTop_atTop (R := ℝ)).eventually
-    (Real.isLittleO_log_id_atTop.bound (by linarith : 0<a/8))
-  have hg := (tendsto_natCast_atTop_atTop (R := ℝ)).eventually (eventually_ge_atTop (4/a))
-  filter_upwards [h,hg,eventually_ge_atTop 1] with N h hg hN
-  have hn : (1:ℝ)≤N := by exact_mod_cast hN
-  simp only [Real.norm_eq_abs,abs_of_nonneg (Real.log_nonneg hn),id_eq,abs_of_nonneg (by positivity : (0:ℝ)≤N)] at h
-  have hh : 4≤(N:ℝ)*a := (div_le_iff₀ ha).mp hg
-  linarith
-
-end Thorp.DimensionEstimates
-
-namespace Thorp.Specht
-open scoped BigOperators Classical
-open DimensionEstimates Filter
-
-theorem medium_dimension (δ : ℝ) (hδ : 0<δ) (hδ2 : δ≤1/2) :
-    ∃ h : ℝ, 0<h ∧ ∀ᶠ N : ℕ in atTop, ∀ μ : YoungDiagram, μ.card=N →
-      δ*(N:ℝ)≤N-μ.rowLen 0 → (μ.colLen 0:ℝ)≤(N:ℝ)/2 →
-      h*(N:ℝ)≤Real.log (Module.finrank ℂ (space μ)) := by
-  let a := min ((3/2:ℝ)*Real.log 2-1) (entropyGap*δ)
-  have ha : 0<a := by
-    apply lt_min
-    · nlinarith [Real.log_two_gt_d9]
-    · exact mul_pos entropyGap_pos hδ
-  refine ⟨a/2,by positivity,?_⟩
-  filter_upwards [log_error_eventually a ha,eventually_ge_atTop 1] with N hN hn
-  intro μ hμ hr hc
-  have hp : 0<μ.card := by omega
-  have hh := medium_dimension_raw μ hp δ hδ hδ2 (by simpa only [hμ] using hr) (by simpa only [hμ] using hc)
-  rw [hμ] at hh
-  change a*(N:ℝ)-2*Real.log N-1≤_ at hh
-  linarith
-
-end Thorp.Specht
-
-namespace Thorp
-open scoped BigOperators Classical
-
-lemma insertNth_castSucc_snoc {α : Type*} {d : ℕ} (i : Fin (d+1)) (b a : α) (y : Fin d → α) :
-    Fin.insertNth i.castSucc b (Fin.snoc (α := fun _ => α) y a)=Fin.snoc (α := fun _ => α) (Fin.insertNth i b y) a := by
-  apply Fin.lastCases ?_ (fun j => ?_) |> fun h => funext h
-  · rw [←show i.castSucc.succAbove (Fin.last d)=Fin.last (d+1) from
-        Fin.succAbove_castSucc_of_le i (Fin.last d) (Fin.le_last i),Fin.insertNth_apply_succAbove]
-    simp
-  · cases j using Fin.succAboveCases i with
-    | x => simp
-    | p j =>
-      have he : i.castSucc.succAbove j.castSucc=(i.succAbove j).castSucc := by
-        simp only [Fin.succAbove,Fin.castSucc_lt_castSucc_iff]
-        split_ifs <;> rfl
-      rw [←he,Fin.insertNth_apply_succAbove]
-      simp
-
-lemma rotate_cons (d : ℕ) (b : Bool) (y : Card d) :
-    rotate (d+1) (Fin.cons b y)=Fin.snoc y b := (Fin.snoc_eq_cons_rotate y b).symm
-
-lemma rotate_insert (d : ℕ) (i : Fin (d+1)) (b : Bool) (y : Card (d+1)) :
-    rotate (d+2) (Fin.insertNth i.succ b y)=Fin.insertNth i.castSucc b (rotate (d+1) y) := by
-  rw [←Fin.cons_self_tail y,Fin.insertNth_succ_cons,rotate_cons,rotate_cons,insertNth_castSucc_snoc]
-
-def insertionPerm (d : ℕ) (i : Fin (d+1)) : Equiv.Perm (Card (d+1)) :=
-  (Fin.consEquiv (fun _ : Fin (d+1) => Bool)).symm.trans
-    (Fin.insertNthEquiv (fun _ : Fin (d+1) => Bool) i)
-
-lemma insertionPerm_apply (d : ℕ) (i : Fin (d+1)) (x : Card (d+1)) :
-    insertionPerm d i x=Fin.insertNth i (x 0) (Fin.tail x) := rfl
-
-lemma insertionPerm_zero (d : ℕ) : insertionPerm d 0=1 := by
-  ext x j
-  simp [insertionPerm_apply]
-
-lemma insertionPerm_last (d : ℕ) : insertionPerm d (Fin.last d)=rotate (d+1) := by
-  apply Equiv.ext
-  intro x
-  rw [insertionPerm_apply,Fin.insertNth_last',←rotate_cons,Fin.cons_self_tail]
-
-lemma physical_carry_step (d : ℕ) (i : Fin (d+1)) (ξ : Bool → Coins (d+1)) :
-    step (d+2) (fun y => ξ (y i) (Fin.removeNth i y))*insertionPerm (d+1) i.succ=
-      insertionPerm (d+1) i.castSucc*childLift (fun b => step (d+1) (ξ b)) := by
-  apply Equiv.ext
-  intro x
-  have he : Fin.cons (Bool.xor ((Fin.tail x) 0) (ξ (x 0) (Fin.tail (Fin.tail x))))
-      (Fin.insertNth i (x 0) (Fin.tail (Fin.tail x))) =
-      (Fin.insertNth i.succ (x 0) (switchFun (ξ (x 0)) (Fin.tail x)) : Card (d+2)) := by
-    simp only [switchFun,Fin.insertNth_succ_cons]
-  have hs : switchFun (fun y => ξ (y i) (Fin.removeNth i y))
-      (Fin.insertNth i.succ (x 0) (Fin.tail x)) =
-      (Fin.insertNth i.succ (x 0) (switchFun (ξ (x 0)) (Fin.tail x)) : Card (d+2)) := by
-    conv_lhs => rw [←Fin.cons_self_tail (Fin.tail x),Fin.insertNth_succ_cons]
-    simp only [switchFun,Fin.cons_zero,Fin.tail_cons,Fin.insertNth_apply_same,Fin.removeNth_insertNth]
-    exact he
-  change rotate (d+2) (switchFun (fun y => ξ (y i) (Fin.removeNth i y))
-      (Fin.insertNth i.succ (x 0) (Fin.tail x))) = _
-  rw [hs,rotate_insert]
-  rfl
-
-end Thorp
-namespace Thorp.UnitaryFinite
-open scoped BigOperators Classical
-variable {G : Type*} [Group G]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-variable {Ω : Type*} [Fintype Ω]
-
-lemma sampleOperator_mul_const (ρ : Representation ℂ G V) (P : Ω → G) (g : G) :
-    sampleOperator ρ (fun ω => P ω*g)=sampleOperator ρ P*continuousRepresentation ρ g := by
-  simp only [sampleOperator,map_mul,Finset.sum_mul,smul_mul_assoc]
-
-lemma sampleOperator_const_mul (ρ : Representation ℂ G V) (P : Ω → G) (g : G) :
-    sampleOperator ρ (fun ω => g*P ω)=continuousRepresentation ρ g*sampleOperator ρ P := by
-  simp only [sampleOperator,map_mul,Finset.mul_sum,mul_smul_comm]
-
-lemma sampleOperator_run (d t : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V) :
-    sampleOperator ρ (run d t)=(sampleOperator ρ (step d))^t := by
-  induction t with
-  | zero => exact sampleOperator_one ρ
-  | succ t ih =>
-    let e : Coins d × (Fin t → Coins d) ≃ (Fin (t+1) → Coins d) :=
-      Fin.snocEquiv (fun _ => Coins d)
-    have he : run d (t+1) ∘ e=fun w => step d w.1*run d t w.2 := by
-      funext w
-      simp only [Function.comp_apply,e,Fin.snocEquiv_apply,
-        run,Fin.snoc_last,Fin.snoc_castSucc]
-    rw [←sampleOperator_equiv ρ (run d (t+1)) e,he,sampleOperator_prod,ih,pow_succ']
-
-end Thorp.UnitaryFinite
-
-namespace Thorp
-open scoped BigOperators Classical
-open UnitaryFinite
-
-def carryCoins (d : ℕ) (i : Fin (d+1)) : (Bool → Coins (d+1)) ≃ Coins (d+2) where
-  toFun ξ y := ξ (y i) (Fin.removeNth i y)
-  invFun ξ b y := ξ (Fin.insertNth i b y)
-  left_inv ξ := by funext b y; simp
-  right_inv ξ := by funext y; simp
-
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-lemma physical_carry_average (d : ℕ) (i : Fin d)
-    (ρ : Representation ℂ (Equiv.Perm (Card (d+1))) V) :
-    sampleOperator ρ (step (d+1))*continuousRepresentation ρ (insertionPerm d i.succ)=
-      continuousRepresentation ρ (insertionPerm d i.castSucc)*
-        sampleOperator ρ (fun ξ : Bool → Coins d => childLift (fun b => step d (ξ b))) := by
-  cases d with
-  | zero => exact Fin.elim0 i
-  | succ d =>
-    have he : (fun ξ : Bool → Coins (d+1) =>
-        step (d+2) (carryCoins d i ξ)*insertionPerm (d+1) i.succ)=
-      (fun ξ => insertionPerm (d+1) i.castSucc*childLift (fun b => step (d+1) (ξ b))) := by
-      funext ξ
-      exact physical_carry_step d i ξ
-    have h := congrArg (sampleOperator ρ) he
-    rw [sampleOperator_mul_const,sampleOperator_const_mul] at h
-    change sampleOperator ρ (step (d+2) ∘ carryCoins d i)*_=_ at h
-    rw [sampleOperator_equiv] at h
-    exact h
-
-theorem physical_sweep_recursion (d : ℕ)
-    (ρ : Representation ℂ (Equiv.Perm (Card (d+1))) V) :
-    (sampleOperator ρ (step (d+1)))^(d+1)=
-      (sampleOperator ρ (fun ξ : Bool → Coins d => childLift (fun b => step d (ξ b))))^d*
-        sampleOperator ρ (pairSwitch (d := d)) := by
-  let A := sampleOperator ρ (step (d+1))
-  let K := sampleOperator ρ (fun ξ : Bool → Coins d => childLift (fun b => step d (ξ b)))
-  have h (t : ℕ) (ht : t≤d) : A^t*continuousRepresentation ρ (insertionPerm d ⟨t,by omega⟩)=K^t := by
-    induction t with
-    | zero =>
-      change 1*continuousRepresentation ρ (insertionPerm d 0)=1
-      rw [insertionPerm_zero,map_one,one_mul]
-    | succ t ih =>
-      have hc := physical_carry_average d (⟨t,by omega⟩ : Fin d) ρ
-      change A*continuousRepresentation ρ (insertionPerm d ⟨t+1,by omega⟩)=
-        continuousRepresentation ρ (insertionPerm d ⟨t,by omega⟩)*K at hc
-      rw [pow_succ,mul_assoc,hc,←mul_assoc,ih (by omega),←pow_succ]
-  have he : sampleOperator ρ (step (d+1))=
-      continuousRepresentation ρ (rotate (d+1))*sampleOperator ρ (pairSwitch (d := d)) :=
-    sampleOperator_const_mul ρ (pairSwitch (d := d)) (rotate (d+1))
-  calc
-    _ = A^d*A := pow_succ A d
-    _ = A^d*(continuousRepresentation ρ (insertionPerm d (Fin.last d))*
-        sampleOperator ρ (pairSwitch (d := d))) := by rw [insertionPerm_last]; rw [←he]
-    _ = _ := by
-      rw [←mul_assoc]
-      exact congrArg (fun z => z*sampleOperator ρ (pairSwitch (d := d))) (h d le_rfl)
-
-end Thorp
-namespace Thorp
-open scoped BigOperators Classical
-open UnitaryFinite
-
-lemma childInjection_product (d : ℕ) (p : Bool → Equiv.Perm (Card d)) :
-    childInjection d false (p false)*childInjection d true (p true)=childLift p := by
-  change childLift _*childLift _=_
-  rw [childLift_mul]
-  congr 1
-  funext b
-  cases b <;> simp
-
-lemma childInjection_commute (d : ℕ) (p q : Equiv.Perm (Card d)) :
-    Commute (childInjection d false p) (childInjection d true q) := by
-  change childLift _*childLift _=childLift _*childLift _
-  rw [childLift_mul,childLift_mul]
-  congr 1
-  funext b
-  cases b <;> simp
-
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-variable {Ω : Type*} [Fintype Ω]
-
-lemma sampleOperator_child (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card (d+1))) V)
-    (P : Ω → Equiv.Perm (Card d)) :
-    sampleOperator ρ (fun ω : Bool → Ω => childLift (fun b => P (ω b)))=
-      sampleOperator (ρ.comp (childInjection d false)) P*
-        sampleOperator (ρ.comp (childInjection d true)) P := by
-  let e := (Equiv.boolArrowEquivProd Ω).symm
-  have he : (fun ω : Bool → Ω => childLift (fun b => P (ω b))) ∘ e=
-      fun w : Ω × Ω => childInjection d false (P w.1)*childInjection d true (P w.2) := by
-    funext w
-    change childLift (fun b => P (e w b))=_
-    rw [←childInjection_product]
-    rfl
-  rw [←sampleOperator_equiv ρ _ e,he]
-  exact sampleOperator_prod ρ (fun w => childInjection d false (P w))
-    (fun w => childInjection d true (P w))
-
-lemma sampleOperator_child_commute (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card (d+1))) V)
-    (P : Ω → Equiv.Perm (Card d)) :
-    Commute (sampleOperator (ρ.comp (childInjection d false)) P)
-      (sampleOperator (ρ.comp (childInjection d true)) P) := by
-  rw [sampleOperator_comp,sampleOperator_comp]
-  exact sampleOperator_commute ρ _ _ (fun _ _ => childInjection_commute d _ _)
-
-lemma inverse_butterfly_average_step (d : ℕ)
-    (ρ : Representation ℂ (Equiv.Perm (Card (d+1))) V) :
-    sampleOperator ρ (fun ω => (butterflyPerm (d+1) (decodeButterfly (d+1) ω))⁻¹)=
-      sampleOperator ρ (fun ω : Bool → (SwitchIndex d → Bool) =>
-        childLift (fun b => (butterflyPerm d (decodeButterfly d (ω b)))⁻¹))*
-      sampleOperator ρ (pairSwitch (d := d)) := by
-  let e := ((Equiv.prodCongr (Equiv.boolArrowEquivProd (SwitchIndex d → Bool)) (Equiv.refl (Card d → Bool))).trans
-    (coinStepEquiv d).symm)
-  have he : (fun ω => (butterflyPerm (d+1) (decodeButterfly (d+1) ω))⁻¹) ∘ e=
-      fun w : (Bool → (SwitchIndex d → Bool)) × (Card d → Bool) =>
-        childLift (fun b => (butterflyPerm d (decodeButterfly d (w.1 b)))⁻¹)*pairSwitch w.2 := by
-    funext w
-    change (butterflyPerm (d+1) (decodeButterfly (d+1)
-      ((coinStepEquiv d).symm ((w.1 false,w.1 true),w.2))))⁻¹=_
-    rw [butterflyPerm_step,mul_inv_rev,childLift_inv,pairSwitch_inv]
-    congr 2
-    funext b
-    cases b <;> rfl
-  rw [←sampleOperator_equiv ρ _ e,he]
-  exact sampleOperator_prod ρ
-    (fun ω : Bool → (SwitchIndex d → Bool) => childLift (fun b => (butterflyPerm d (decodeButterfly d (ω b)))⁻¹))
-    (pairSwitch (d := d))
-
-theorem physical_sweep_inverse_butterfly (d : ℕ)
-    (ρ : Representation ℂ (Equiv.Perm (Card d)) V) :
-    (sampleOperator ρ (step d))^d=
-      sampleOperator ρ (fun ω => (butterflyPerm d (decodeButterfly d ω))⁻¹) := by
-  induction d with
-  | zero =>
-    symm
-    exact sampleOperator_one ρ
-  | succ d ih =>
-    rw [physical_sweep_recursion,sampleOperator_child,
-      (sampleOperator_child_commute d ρ (step d)).mul_pow,ih,ih,
-      inverse_butterfly_average_step]
-    rw [sampleOperator_child d ρ (fun ω => (butterflyPerm d (decodeButterfly d ω))⁻¹)]
-
-lemma physical_sweep_adjoint (d : ℕ)
-    (ρ : Representation ℂ (Equiv.Perm (Card d)) V) (hρ : IsUnitary ρ) :
-    sampleOperator ρ (run d d)=
-      (sampleOperator ρ (fun ω => butterflyPerm d (decodeButterfly d ω))).adjoint := by
-  rw [sampleOperator_run,physical_sweep_inverse_butterfly,sampleOperator_inv ρ hρ]
-
-end Thorp
-
-namespace Thorp.Specht
-open scoped BigOperators Classical
-open UnitaryFinite DimensionEstimates Filter
-
-lemma dimension_pos (μ : YoungDiagram) : 0 < Module.finrank ℂ (space μ) := by
-  have h := factorial_le_dimension_row_column μ
-  by_contra hn
-  have hz : Module.finrank ℂ (space μ)=0 := by omega
-  rw [hz,zero_mul,zero_mul] at h
-  exact (Nat.factorial_pos μ.card).not_ge h
-
-lemma near_numerical_cutoff :
-    ∃ X : ℝ, 2 ≤ X ∧ ∀ (N j D : ℕ), 0 < j → X ≤ (N:ℝ)/j → N.choose j ≤ D * 2^j →
-      ∃ k : ℕ, j ≤ k ∧ k ≤ N ∧ ∀ m : ℕ, D * k.choose j ≤ m * N.choose j →
-        nearFraction * Real.log D ≤ Real.log m ∧
-        densityConstant * (k:ℝ) * ((k:ℝ)/N)^(1/512:ℝ) ≤ Real.log m/4000 ∧
-        (1/2:ℝ) * Real.log N ≤ Real.log D ∧ (j:ℝ) ≤ Real.log D := by
-  obtain ⟨X,hX,h⟩ := exists_near_cutoff densityConstant densityConstant_pos.le
-  refine ⟨X,hX,?_⟩
-  intro N j D hj hx hd
-  have hjN : j ≤ N := by
-    have hjp : (0:ℝ) < j := by exact_mod_cast hj
-    have hh : (j:ℝ) ≤ N := by
-      have := (le_div_iff₀ hjp).mp (hX.trans hx)
-      linarith
-    exact_mod_cast hh
-  have hc : 1 ≤ N.choose j := Nat.choose_pos hjN
-  have hfoot : D * (chosenTuple N j).choose j ≤ (D * (chosenTuple N j).choose j) * N.choose j :=
-    Nat.le_mul_of_pos_right _ hc
-  have hh := h N j D _ hj hx hd hfoot
-  refine ⟨chosenTuple N j,hh.1,hh.2.1,?_⟩
-  intro m hm
-  exact (h N j D m hj hx hd hm).2.2
-
-lemma near_row_amplification (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Cell μ)
-    {k : ℕ} (f : Fin k ↪ Card d) (hk : 0<k)
-    (hm : 0<Module.finrank ℂ (fixedSpace (V := hilbertSpace μ) tupleAction f (relabelledUnitary μ e))) :
-    ‖sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e) (palindromePerm d)‖ ≤
-      Real.exp (densityConstant*(k:ℝ)*((k:ℝ)/(2:ℝ)^d)^(1/512:ℝ))*
-        (Real.exp (Real.log (Module.finrank ℂ (fixedSpace (V := hilbertSpace μ) tupleAction f (relabelledUnitary μ e)))/2))^(-(1/1000:ℝ))+
-      Real.sqrt ((Real.exp (Real.log (Module.finrank ℂ (fixedSpace (V := hilbertSpace μ) tupleAction f (relabelledUnitary μ e)))/2))/
-        (Module.finrank ℂ (fixedSpace (V := hilbertSpace μ) tupleAction f (relabelledUnitary μ e)):ℝ)) := by
-  let := relabelledUnitary_irreducible μ e
-  let : NormedAddCommGroup (hilbertSpace μ) := inferInstance
-  let : InnerProductSpace ℂ (hilbertSpace μ) := inferInstance
-  let : FiniteDimensional ℂ (hilbertSpace μ) := inferInstance
-  have aux := @palindrome_irrep_amplification (hilbertSpace μ) _ _ _ d (Fin k) _ f
-    (by simpa only [Fintype.card_fin] using hk) (relabelledUnitary μ e) inferInstance
-    (relabelledUnitary_unitary μ e) hm
-    (Real.exp (Real.log (Module.finrank ℂ (fixedSpace (V := hilbertSpace μ) tupleAction f (relabelledUnitary μ e)))/2))
-    (Real.exp_pos _)
-  simpa only [Fintype.card_fin] using aux
-
-theorem near_row_spectral :
-    ∃ X : ℝ, 2 ≤ X ∧ ∀ (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Cell μ),
-      0 < μ.card-μ.rowLen 0 → X ≤ (μ.card:ℝ)/(μ.card-μ.rowLen 0:ℕ) →
-      32768000 * Real.log 2 ≤ (1/2:ℝ) * Real.log μ.card →
-      ‖sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e) (palindromePerm d)‖ ≤
-        Real.exp (-(nearFraction/8000) * Real.log (Module.finrank ℂ (space μ))) ∧
-      (1/2:ℝ) * Real.log μ.card ≤ Real.log (Module.finrank ℂ (space μ)) ∧
-      (μ.card-μ.rowLen 0:ℕ) ≤ Real.log (Module.finrank ℂ (space μ)) := by
-  obtain ⟨X,hX,h⟩ := near_numerical_cutoff
-  refine ⟨X,hX,?_⟩
-  intro d μ e hj hx hlarge
-  let j := μ.card-μ.rowLen 0
-  let D := Module.finrank ℂ (space μ)
-  have hcard : μ.card=2^d := by rw [←card_cell,←Fintype.card_congr e,card_positions]
-  obtain ⟨k,hjk,hk,hnum⟩ := h μ.card j D hj hx (choose_tail_le_dimension μ)
-  obtain ⟨f⟩ := Function.Embedding.nonempty_of_card_le
-    (show Fintype.card (Fin k) ≤ Fintype.card (Card d) by simpa only [Fintype.card_fin,card_positions,←hcard] using hk)
-  let m := Module.finrank ℂ (fixedSpace (V := hilbertSpace μ) tupleAction f (relabelledUnitary μ e))
-  have hfoot : D * k.choose j ≤ m * μ.card.choose j := by
-    simpa only [Fintype.card_fin,finrank_hilbertSpace] using relabelled_tuple_dimension μ e f
-  have hm : 0 < m := by
-    have hc := Nat.choose_pos hjk
-    have hd := dimension_pos μ
-    by_contra hn
-    have hz : m=0 := by omega
-    rw [hz,zero_mul] at hfoot
-    exact (Nat.mul_pos hd hc).not_ge hfoot
-  obtain ⟨hlm,hden,hlog,hjlog⟩ := hnum m hfoot
-  have ha := near_row_amplification d μ e f (hj.trans_le hjk) hm
-  have hcardR : (μ.card:ℝ)=(2:ℝ)^d := by exact_mod_cast hcard
-  rw [←hcardR] at ha
-  have hl : 8000 * Real.log 2 ≤ Real.log m := by
-    dsimp [nearFraction] at hlm
-    linarith
-  have hn := amplification_decay (m:ℝ) (densityConstant * k * ((k:ℝ)/μ.card)^(1/512:ℝ))
-    (by exact_mod_cast hm) hden hl
-  refine ⟨ha.trans (hn.trans ?_),hlog,hjlog⟩
-  apply Real.exp_le_exp.mpr
-  dsimp [D] at hlm
-  linarith
-
-end Thorp.Specht
-
-namespace Thorp.UnitaryFinite
-open scoped BigOperators Classical
-open DimensionEstimates
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-theorem full_palindrome_decay (d : ℕ)
-    (ρ : Representation ℂ (Equiv.Perm (Card d)) V) [Representation.IsIrreducible ρ]
-    (hρ : IsUnitary ρ) (hm : 0<Module.finrank ℂ V)
-    (hden : densityConstant*(2:ℝ)^d≤Real.log (Module.finrank ℂ V)/4000)
-    (hl : 8000*Real.log 2≤Real.log (Module.finrank ℂ V)) :
-    ‖sampleOperator ρ (palindromePerm d)‖≤Real.exp (-Real.log (Module.finrank ℂ V)/8000) := by
-  have h := palindrome_irrep_amplification d (Function.Embedding.refl (Card d))
-    (Fintype.card_pos) ρ hρ (by simpa only [full_tuple_finrank] using hm)
-    (Real.exp (Real.log (Module.finrank ℂ V)/2)) (Real.exp_pos _)
-  simp only [full_tuple_finrank,card_positions,Nat.cast_pow,Nat.cast_ofNat,
-    div_self (by positivity : (2:ℝ)^d≠0),Real.one_rpow,mul_one] at h
-  exact h.trans (amplification_decay _ _ (by exact_mod_cast hm) hden hl)
-
-theorem full_modified_decay (L r : ℕ) (E : Finset (Card r))
-    (ρ : Representation ℂ (Equiv.Perm (Card (L+r))) V) [Representation.IsIrreducible ρ]
-    (hρ : IsUnitary ρ) (hm : 0<Module.finrank ℂ V) (u : ℝ)
-    (hM : ∀ e : Card (L+r) ↪ Card (L+r), modifiedRowMoment L r E e (1/1000)≤Real.exp u)
-    (hden : u≤Real.log (Module.finrank ℂ V)/4000)
-    (hl : 8000*Real.log 2≤Real.log (Module.finrank ℂ V)) :
-    ‖sampleOperator ρ (modifiedPerm L r E)‖≤Real.exp (-Real.log (Module.finrank ℂ V)/8000) := by
-  exact (modified_irrep_amplification L r E ρ hρ hm u
-    (Real.exp (Real.log (Module.finrank ℂ V)/2)) (Real.exp_pos _) hM).trans
-    (amplification_decay _ _ (by exact_mod_cast hm) hden hl)
-
-lemma palindrome_exception_cards (r L : ℕ)
-    (ρ : Representation ℂ (Equiv.Perm (Card (L+r))) V) (hρ : IsUnitary ρ) (n : ℕ) :
-    ‖sampleOperator ρ (palindromePerm (L+r))‖ ≤
-      (blockGap L)^n+∑ E : Finset (Card r), if E.card<n then
-        ‖sampleOperator ρ (modifiedPerm L r E)‖ else 0 := by
-  let σ := ρ.comp (splitPerm r L).symm.toMonoidHom
-  have he : σ.comp (splitPerm r L).toMonoidHom=ρ := by
-    ext g v
-    change ρ ((splitPerm r L).symm ((splitPerm r L) g)) v=ρ g v
-    rw [MulEquiv.symm_apply_apply]
-  have h := palindrome_exception_bound r L σ (unitary_comp ρ hρ _) n
-  simpa only [he] using h
-
-end Thorp.UnitaryFinite
-
-namespace Thorp.DimensionEstimates
-open scoped BigOperators Classical
-open Filter
-
-lemma medium_parameters (h : ℝ) (hh : 0<h) :
-    ∃ L : ℕ, 0<L ∧ ∃ ε t : ℝ, 0<ε ∧ ε<1/2 ∧ 0≤t ∧
-      t*ε+Real.log (1+Real.exp (-t))≤h*(2:ℝ)^L/16000 ∧
-      ∀ᶠ r : ℕ in atTop, ∀ (E : Finset (Card r)), (E.card:ℝ)<ε*(2:ℝ)^r →
-        ∀ e : Card (L+r) ↪ Card (L+r),
-          modifiedRowMoment L r E e (1/1000)≤Real.exp ((h/4000)*(2:ℝ)^(L+r)) := by
-  obtain ⟨L,hL,ε₀,hε₀,_,hM⟩ := modified_arbitrarily_small_rate (h/4000) (by positivity)
-  obtain ⟨ε,hε,hεle,hεhalf,t,ht,hcount⟩ := exists_small_set_rate
-    (h*(2:ℝ)^L/16000) ε₀ (by positivity) hε₀
-  refine ⟨L,hL,ε,t,hε,hεhalf,ht,hcount,?_⟩
-  filter_upwards [hM] with r hM
-  intro E hE e
-  exact hM E (hE.le.trans (mul_le_mul_of_nonneg_right hεle (by positivity))) e
-
-lemma geometric_ceil_bound (θ ε : ℝ) (hθ : 0<θ) (hθ1 : θ<1) (M : ℕ) :
-    θ^⌈ε*M⌉₊≤Real.exp (Real.log θ*ε*M) := by
-  conv_lhs => rw [←Real.exp_log hθ,←Real.exp_nat_mul]
-  apply Real.exp_le_exp.mpr
-  have hg := mul_le_mul_of_nonpos_left (Nat.le_ceil (ε*M)) (Real.log_nonpos hθ.le hθ1.le)
-  nlinarith
-
-end Thorp.DimensionEstimates
-
-namespace Thorp.UnitaryFinite
-open scoped BigOperators Classical
-open DimensionEstimates
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-theorem medium_exception_decay (L r : ℕ) (h ε t : ℝ) (ht : 0≤t)
-    (hcount : t*ε+Real.log (1+Real.exp (-t))≤h*(2:ℝ)^L/16000)
-    (ρ : Representation ℂ (Equiv.Perm (Card (L+r))) V) [Representation.IsIrreducible ρ]
-    (hρ : IsUnitary ρ) (hm : 0<Module.finrank ℂ V)
-    (hd : h*(2:ℝ)^(L+r)≤Real.log (Module.finrank ℂ V))
-    (hl : 8000*Real.log 2≤Real.log (Module.finrank ℂ V))
-    (hM : ∀ (E : Finset (Card r)), (E.card:ℝ)<ε*(2:ℝ)^r →
-      ∀ e : Card (L+r) ↪ Card (L+r),
-        modifiedRowMoment L r E e (1/1000)≤Real.exp ((h/4000)*(2:ℝ)^(L+r))) :
-    ‖sampleOperator ρ (palindromePerm (L+r))‖≤
-      Real.exp (Real.log (blockGap L)*ε*(2:ℝ)^r)+Real.exp (-h*(2:ℝ)^(L+r)/16000) := by
-  have hmE (E : Finset (Card r)) (hE : E.card<⌈ε*(2:ℝ)^r⌉₊) :
-      ‖sampleOperator ρ (modifiedPerm L r E)‖≤Real.exp (-h*(2:ℝ)^(L+r)/8000) := by
-    apply (full_modified_decay L r E ρ hρ hm ((h/4000)*(2:ℝ)^(L+r))
-      (hM E (Nat.lt_ceil.mp hE)) (by linarith) hl).trans
-    apply Real.exp_le_exp.mpr
-    linarith
-  have hs : (∑ E : Finset (Card r), if E.card<⌈ε*(2:ℝ)^r⌉₊ then
-      ‖sampleOperator ρ (modifiedPerm L r E)‖ else 0) ≤
-      Real.exp (-h*(2:ℝ)^(L+r)/16000) := by
-    calc
-      _ ≤ (∑ E : Finset (Card r), if E.card<⌈ε*(2:ℝ)^r⌉₊ then (1:ℝ) else 0)*
-          Real.exp (-h*(2:ℝ)^(L+r)/8000) := by
-        rw [Finset.sum_mul]
-        apply Finset.sum_le_sum
-        intro E _
-        split_ifs with hE
-        · simpa only [one_mul] using hmE E hE
-        · simp
-      _ ≤ Real.exp ((t*ε+Real.log (1+Real.exp (-t)))*(2:ℝ)^r)*
-          Real.exp (-h*(2:ℝ)^(L+r)/8000) := by
-        apply mul_le_mul_of_nonneg_right _ (Real.exp_pos _).le
-        simpa only [card_positions,Nat.cast_pow,Nat.cast_ofNat] using
-          small_sets_ceil_bound (ι := Card r) ε t ht
-      _ ≤ _ := by
-        rw [←Real.exp_add]
-        apply Real.exp_le_exp.mpr
-        have hh := mul_le_mul_of_nonneg_right hcount (by positivity : (0:ℝ)≤(2:ℝ)^r)
-        rw [pow_add]
-        nlinarith
-  apply (palindrome_exception_cards r L ρ hρ ⌈ε*(2:ℝ)^r⌉₊).trans
-  apply add_le_add _ hs
-  simpa only [Nat.cast_pow,Nat.cast_ofNat] using
-    geometric_ceil_bound (blockGap L) ε (blockGap_pos L) (blockGap_lt_one L) (2^r)
-
-end Thorp.UnitaryFinite
-
-
-end ThorpNine.Harmonic
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Harmonic.ZeroExtend
+
+namespace OAI
+
+namespace ThorpNine.Harmonic
+
+namespace Thorp.DimensionEstimates
+open scoped BigOperators Classical
+
+section
+open Filter
+
+lemma chosenTuple_density_bound (N j : ℕ) (hj : 0<j) (hx : 1≤(N:ℝ)/j)
+    (hhalf : ((N:ℝ)/j)^(nearPower-1)≤1/2) (C : ℝ) (hC : 0≤C) :
+    C*(chosenTuple N j:ℝ)*((chosenTuple N j:ℝ)/N)^(1/512:ℝ) ≤
+      C*2^(1+(1/512:ℝ))*(j:ℝ)*((N:ℝ)/j)^(-tailPower) := by
+  let x : ℝ := (N:ℝ)/j
+  have hjp : (0:ℝ)<j := by exact_mod_cast hj
+  have hxp : 0<x := lt_of_lt_of_le zero_lt_one hx
+  have hNp : (0:ℝ)<N := by
+    have : (0:ℝ)<(N:ℝ)/j := hxp
+    exact (div_pos_iff.mp this).resolve_right (by intro h; exact (not_lt_of_ge hjp.le) h.2) |>.1
+  have he : (j:ℝ)*x=N := by dsimp [x]; field_simp
+  have hk := (chosenTuple_bounds N j hj hx hhalf).2.2
+  change (chosenTuple N j:ℝ)≤2*(j:ℝ)*x^nearPower at hk
+  have hratio : (chosenTuple N j:ℝ)/N≤2*x^(nearPower-1) := by
+    rw [Real.rpow_sub hxp,Real.rpow_one]
+    apply (div_le_iff₀ hNp).mpr
+    rw [←he]
+    have : 2*(x^nearPower/x)*((j:ℝ)*x)=2*(j:ℝ)*x^nearPower := by field_simp
+    rw [this]
+    exact hk
+  have hpow := Real.rpow_le_rpow (by positivity : (0:ℝ)≤(chosenTuple N j:ℝ)/N) hratio
+    (by norm_num : (0:ℝ)≤1/512)
+  have hmain := mul_le_mul hk hpow (by positivity) (by positivity)
+  have hrpow : (2*x^(nearPower-1))^(1/512:ℝ)=
+      (2:ℝ)^(1/512:ℝ)*x^((nearPower-1)*(1/512)) := by
+    rw [Real.mul_rpow (by norm_num) (by positivity),←Real.rpow_mul hxp.le]
+  rw [hrpow] at hmain
+  have heq : 2*(j:ℝ)*x^nearPower*((2:ℝ)^(1/512:ℝ)*x^((nearPower-1)*(1/512)))=
+      (2:ℝ)^(1+(1/512:ℝ))*(j:ℝ)*x^(-tailPower) := by
+    rw [Real.rpow_add (by norm_num : (0:ℝ)<2),Real.rpow_one]
+    have hp : nearPower+(nearPower-1)*(1/512) = -tailPower := by norm_num [nearPower,tailPower]
+    rw [←hp,Real.rpow_add hxp]
+    ring
+  rw [heq] at hmain
+  have h := mul_le_mul_of_nonneg_left hmain hC
+  simpa only [mul_assoc] using h
+
+lemma near_conditions_eventually (C : ℝ) :
+    ∀ᶠ x : ℝ in atTop, 2≤x ∧ 2≤x^nearPower ∧ x^(nearPower-1)≤1/2 ∧
+      8*Real.log 2/nearPower≤Real.log x ∧ 4*Real.log 2+2≤Real.log x ∧
+      C*2^(1+(1/512:ℝ))*x^(-tailPower) ≤ nearFraction/4000 := by
+  have hpow := (tendsto_rpow_atTop (by norm_num [nearPower] : 0<nearPower)).eventually
+    (eventually_ge_atTop (2:ℝ))
+  have hhalf : ∀ᶠ x : ℝ in atTop, x^(nearPower-1)≤1/2 := by
+    have h := (tendsto_rpow_neg_atTop (by norm_num [nearPower] : 0<1-nearPower)).eventually
+      (gt_mem_nhds (by norm_num : (0:ℝ)<1/2))
+    filter_upwards [h] with x hx
+    simpa only [neg_sub] using hx.le
+  have hdec : Tendsto (fun x : ℝ => C*2^(1+(1/512:ℝ))*x^(-tailPower)) atTop (nhds 0) := by
+    simpa only [mul_zero] using
+      (tendsto_const_nhds.mul (tendsto_rpow_neg_atTop (by norm_num [tailPower] : 0<tailPower)))
+  have hd := hdec.eventually (gt_mem_nhds (by norm_num [nearFraction] : (0:ℝ)<nearFraction/4000))
+  filter_upwards [eventually_ge_atTop (2:ℝ),hpow,hhalf,
+    Real.tendsto_log_atTop.eventually (eventually_ge_atTop (8*Real.log 2/nearPower)),
+    Real.tendsto_log_atTop.eventually (eventually_ge_atTop (4*Real.log 2+2)),hd] with x hx hp hh hl hl' hd
+  exact ⟨hx,hp,hh,hl,hl',hd.le⟩
+
+theorem exists_near_cutoff (C : ℝ) (hC : 0≤C) :
+    ∃ X : ℝ, 2≤X ∧ ∀ (N j D m : ℕ), 0<j → X≤(N:ℝ)/j →
+      N.choose j≤D*2^j →
+      D * (chosenTuple N j).choose j ≤ m * N.choose j →
+      j≤chosenTuple N j ∧ chosenTuple N j≤N ∧
+      nearFraction*Real.log D≤Real.log m ∧
+      C*(chosenTuple N j:ℝ)*((chosenTuple N j:ℝ)/N)^(1/512:ℝ)≤Real.log m/4000 ∧
+      (1/2:ℝ)*Real.log N≤Real.log D ∧ (j:ℝ)≤Real.log D := by
+  obtain ⟨X,hX⟩ := eventually_atTop.mp (near_conditions_eventually C)
+  refine ⟨max 2 X,le_max_left _ _,?_⟩
+  intro N j D m hj hNx hD hfoot
+  have hc := hX ((N:ℝ)/j) ((le_max_right _ _).trans hNx)
+  have hjp : (0:ℝ)<j := by exact_mod_cast hj
+  have hN : 2*j≤N := by exact_mod_cast (le_div_iff₀ hjp).mp hc.1
+  have hk := chosenTuple_bounds N j hj (by linarith [hc.1]) hc.2.2.1
+  have hm := chosenTuple_log_footprint N j D m hj hN hD hc.2.1 hc.2.2.2.1 hfoot
+  have hd := near_row_log_uniform N j D hj hN hD hc.2.2.2.2.1
+  refine ⟨hk.1,hk.2.1,hm,?_,hd⟩
+  apply (chosenTuple_density_bound N j hj (by linarith [hc.1]) hc.2.2.1 C hC).trans
+  have hh := mul_le_mul_of_nonneg_right hc.2.2.2.2.2 hjp.le
+  dsimp [nearFraction] at hm ⊢
+  dsimp [nearFraction] at hh
+  nlinarith [hd.2]
+
+end
+
+lemma amplification_numerical (m u : ℝ) (hm : 1 ≤ m) (hu : u ≤ Real.log m/4000) :
+    Real.exp u*(Real.exp (Real.log m/2))^(-(1/1000:ℝ))+
+      Real.sqrt (Real.exp (Real.log m/2)/m) ≤ 2*Real.exp (-Real.log m/4000) := by
+  have hmp : 0 < m := lt_of_lt_of_le zero_lt_one hm
+  have hl : 0≤Real.log m := Real.log_nonneg hm
+  have he : Real.exp (Real.log m/2)/m=Real.exp (-Real.log m/2) := by
+    calc
+      _ = Real.exp (Real.log m/2)/Real.exp (Real.log m) := by rw [Real.exp_log hmp]
+      _ = _ := by rw [←Real.exp_sub]; congr 1; ring
+  have hs : Real.sqrt (Real.exp (-Real.log m/2))=Real.exp (-Real.log m/4) := by
+    rw [Real.sqrt_eq_rpow,Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp]
+    congr 1
+    ring
+  rw [Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp,←Real.exp_add,he,hs]
+  have h₁ : u+(Real.log m/2)*(-(1/1000:ℝ))≤ -Real.log m/4000 := by linarith
+  have h₂ : -Real.log m/4≤ -Real.log m/4000 := by linarith
+  linarith [Real.exp_le_exp.mpr h₁,Real.exp_le_exp.mpr h₂]
+
+lemma amplification_absorb (x : ℝ) (hx : 8000*Real.log 2≤x) :
+    2*Real.exp (-x/4000)≤Real.exp (-x/8000) := by
+  rw [show (2:ℝ)=Real.exp (Real.log 2) by rw [Real.exp_log (by norm_num)],←Real.exp_add]
+  exact Real.exp_le_exp.mpr (by linarith)
+
+lemma amplification_decay (m u : ℝ) (hm : 1 ≤ m) (hu : u ≤ Real.log m/4000)
+    (hl : 8000*Real.log 2≤Real.log m) :
+    Real.exp u*(Real.exp (Real.log m/2))^(-(1/1000:ℝ))+
+      Real.sqrt (Real.exp (Real.log m/2)/m) ≤ Real.exp (-Real.log m/8000) :=
+  (amplification_numerical m u hm hu).trans (amplification_absorb _ hl)
+
+end Thorp.DimensionEstimates
+
+namespace Thorp.UnitaryFinite
+open scoped BigOperators Classical
+
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+omit [FiniteDimensional ℂ V] in
+lemma full_tuple_fixed (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V) :
+    fixedSpace tupleAction (Function.Embedding.refl (Card d)) ρ=⊤ := by
+  apply Submodule.eq_top_iff'.mpr
+  intro v g hg
+  have he : g=1 := by
+    apply Equiv.ext
+    intro x
+    exact congrArg (fun f : Card d ↪ Card d => f x) hg
+  rw [he,map_one]
+  rfl
+
+omit [FiniteDimensional ℂ V] in
+lemma full_tuple_finrank (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V) :
+    Module.finrank ℂ (fixedSpace tupleAction (Function.Embedding.refl (Card d)) ρ)=Module.finrank ℂ V := by
+  rw [full_tuple_fixed,finrank_top]
+
+theorem palindrome_irrep_amplification (d : ℕ) {ι : Type*} [Fintype ι]
+    (f : ι ↪ Card d) (hk : 0<Fintype.card ι)
+    (ρ : Representation ℂ (Equiv.Perm (Card d)) V) [Representation.IsIrreducible ρ]
+    (hρ : IsUnitary ρ) (hm : 0<Module.finrank ℂ (fixedSpace tupleAction f ρ))
+    (z : ℝ) (hz : 0<z) :
+    ‖sampleOperator ρ (palindromePerm d)‖ ≤
+      Real.exp (densityConstant*Fintype.card ι*((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ))*
+        z^(-(1/1000:ℝ))+Real.sqrt (z/(Module.finrank ℂ (fixedSpace tupleAction f ρ):ℝ)) := by
+  apply sampleOperator_amplification tupleAction f (tupleAction_transitive f) ρ hρ
+    (palindromePerm d) _ _ (1/1000) z _ (by norm_num) hz (by positivity) _ hm
+  · intro e f
+    simp only [tupleAction_kernel]
+    exact palindrome_kernel_symmetric d e f
+  · intro e
+    simp only [tupleAction_kernel]
+    exact PairRouting.tupleProbability_row_sum _ e
+  · intro e
+    simp only [tupleAction_kernel]
+    exact palindrome_row_moment_bound d e hk
+
+theorem modified_irrep_amplification (L r : ℕ) (E : Finset (Card r))
+    (ρ : Representation ℂ (Equiv.Perm (Card (L+r))) V) [Representation.IsIrreducible ρ]
+    (hρ : IsUnitary ρ) (hm : 0<Module.finrank ℂ V)
+    (u z : ℝ) (hz : 0<z)
+    (hM : ∀ e : Card (L+r) ↪ Card (L+r), modifiedRowMoment L r E e (1/1000)≤Real.exp u) :
+    ‖sampleOperator ρ (modifiedPerm L r E)‖ ≤
+      Real.exp u*z^(-(1/1000:ℝ))+Real.sqrt (z/(Module.finrank ℂ V:ℝ)) := by
+  let f := Function.Embedding.refl (Card (L+r))
+  have h := sampleOperator_amplification tupleAction f (tupleAction_transitive f) ρ hρ
+    (modifiedPerm L r E) (fun e f => by simpa only [tupleAction_kernel] using modified_kernel_symmetric L r E e f)
+    (fun e => by simpa only [tupleAction_kernel] using PairRouting.tupleProbability_row_sum (modifiedPerm L r E) e)
+    (1/1000) z (Real.exp u) (by norm_num) hz (by positivity)
+    (fun e => by simpa only [tupleAction_kernel,modifiedRowMoment] using hM e)
+    (by simpa only [f,full_tuple_finrank] using hm)
+  simpa only [f,full_tuple_finrank] using h
+
+end Thorp.UnitaryFinite
+
+namespace Thorp.Specht
+open scoped BigOperators Classical
+
+theorem choose_tail_le_of_factorial (μ : YoungDiagram) (D : ℕ)
+    (hd : μ.card.factorial ≤ D * Fintype.card (rowGroup μ) * Fintype.card (colGroup μ)) :
+    μ.card.choose (μ.card-μ.rowLen 0) ≤ D * 2^(μ.card-μ.rowLen 0) := by
+  let j := μ.card-μ.rowLen 0
+  have htail := card_fiberGroups_mul_le
+    (fun x : Tail μ => rowIndex x.1) (fun x : Tail μ => colIndex x.1)
+    (fun x y hr hc => Subtype.ext (Subtype.ext (Prod.ext (congrArg Fin.val hr) (congrArg Fin.val hc))))
+  rw [tail_card] at htail
+  have hc := column_group_le_tail μ
+  have hr := tail_row_group μ
+  simp only [←Nat.card_eq_fintype_card] at htail hc hr hd
+  have hh : μ.card.factorial ≤ D * 2^j *
+      ((μ.rowLen 0).factorial * j.factorial) := by
+    calc
+      _ ≤ _ := hd
+      _ ≤ D * Nat.card (rowGroup μ) *
+          (2^j * Nat.card (fiberGroup (fun x : Tail μ => colIndex x.1))) := Nat.mul_le_mul_left _ hc
+      _ = D * 2^j * (μ.rowLen 0).factorial *
+          (Nat.card (fiberGroup (fun x : Tail μ => rowIndex x.1)) *
+           Nat.card (fiberGroup (fun x : Tail μ => colIndex x.1))) := by rw [←hr]; ring
+      _ ≤ _ := by
+        calc
+          _ ≤ D * 2^j * (μ.rowLen 0).factorial * j.factorial :=
+            Nat.mul_le_mul_left _ htail
+          _ = _ := by ring
+  have hj : j ≤ μ.card := Nat.sub_le _ _
+  have hs : μ.card-j = μ.rowLen 0 := by dsimp [j]; have := rowLen_zero_le_card μ; omega
+  have hf := Nat.choose_mul_factorial_mul_factorial hj
+  rw [hs] at hf
+  rw [←hf] at hh
+  have hpos : 0 < (μ.rowLen 0).factorial * j.factorial := Nat.mul_pos (Nat.factorial_pos _) (Nat.factorial_pos _)
+  nlinarith
+
+lemma card_rowGroup_transpose (μ : YoungDiagram) :
+    Fintype.card (rowGroup μ.transpose) = Fintype.card (colGroup μ) := by
+  rw [card_rowGroup,card_colGroup]
+  apply Fintype.prod_equiv (finCongr (μ.colLen_transpose 0))
+  intro i
+  change (μ.transpose.rowLen i).factorial = (μ.colLen i).factorial
+  rw [YoungDiagram.rowLen_transpose]
+
+lemma card_colGroup_transpose (μ : YoungDiagram) :
+    Fintype.card (colGroup μ.transpose) = Fintype.card (rowGroup μ) := by
+  rw [card_colGroup,card_rowGroup]
+  apply Fintype.prod_equiv (finCongr (μ.rowLen_transpose 0))
+  intro i
+  change (μ.transpose.colLen i).factorial = (μ.rowLen i).factorial
+  rw [YoungDiagram.colLen_transpose]
+
+lemma card_transpose (μ : YoungDiagram) : μ.transpose.card=μ.card := by
+  unfold YoungDiagram.card YoungDiagram.transpose
+  exact Finset.card_map _
+
+theorem choose_column_tail_le_dimension (μ : YoungDiagram) :
+    μ.card.choose (μ.card-μ.colLen 0) ≤ Module.finrank ℂ (space μ) * 2^(μ.card-μ.colLen 0) := by
+  have hd := factorial_le_dimension_row_column μ
+  have ht := choose_tail_le_of_factorial μ.transpose (Module.finrank ℂ (space μ))
+  simp only [card_transpose,card_rowGroup_transpose,card_colGroup_transpose,YoungDiagram.rowLen_transpose] at ht
+  apply ht
+  nlinarith
+
+open DimensionEstimates Filter
+
+lemma dimension_log_incidence (μ : YoungDiagram) (hμ : 0<μ.card) :
+    (2*Real.log 2-1)*(μ.card:ℝ)-Real.log 2*((μ.rowLen 0:ℝ)+μ.colLen 0)+1≤
+      Real.log (Module.finrank ℂ (space μ)) := by
+  let : Nonempty (Cell μ) := Fintype.card_pos_iff.mp (by simpa only [card_cell] using hμ)
+  have hd := factorial_le_dimension_row_column μ
+  rw [card_rowGroup,card_colGroup] at hd
+  have hr (i : Fin (μ.colLen 0)) : Nat.card {x : Cell μ // rowIndex x=i}=μ.rowLen i := by
+    let e : {x : Cell μ // rowIndex x=i} ≃ {x : Cell μ // row x=i} :=
+      Equiv.subtypeEquivRight (fun x => Fin.ext_iff)
+    rw [Nat.card_congr e,Nat.card_congr (rowEquiv μ i),Nat.card_fin]
+  have hc (i : Fin (μ.rowLen 0)) : Nat.card {x : Cell μ // colIndex x=i}=μ.colLen i := by
+    let e : {x : Cell μ // colIndex x=i} ≃ {x : Cell μ // col x=i} :=
+      Equiv.subtypeEquivRight (fun x => Fin.ext_iff)
+    rw [Nat.card_congr e,Nat.card_congr (columnEquiv μ i),Nat.card_fin]
+  have hcell : Nat.card (Cell μ)=μ.card := by rw [Nat.card_eq_fintype_card,card_cell]
+  have h := dimension_log_from_incidence (rowIndex (μ := μ)) (colIndex (μ := μ))
+    (fun x y h => Subtype.ext (Prod.ext (congrArg (fun z => (z.1:ℕ)) h)
+      (congrArg (fun z => (z.2:ℕ)) h))) (Module.finrank ℂ (space μ))
+    (by simpa only [←Nat.card_eq_fintype_card,hr,hc,hcell] using hd)
+  simpa only [card_cell,Fintype.card_fin,add_comm (μ.colLen 0:ℝ)] using h
+
+lemma medium_dimension_raw (μ : YoungDiagram) (hμ : 0<μ.card) (δ : ℝ) (hδ : 0<δ)
+    (hδ2 : δ≤1/2) (hrow : δ*(μ.card:ℝ)≤μ.card-μ.rowLen 0)
+    (hcol : (μ.colLen 0:ℝ)≤(μ.card:ℝ)/2) :
+    min ((3/2:ℝ)*Real.log 2-1) (entropyGap*δ)*(μ.card:ℝ)-2*Real.log μ.card-1≤
+      Real.log (Module.finrank ℂ (space μ)) := by
+  let c := (3/2:ℝ)*Real.log 2-1
+  have hN : (0:ℝ)<μ.card := by exact_mod_cast hμ
+  have hlog : 0≤Real.log μ.card := Real.log_nonneg (by exact_mod_cast hμ)
+  have hc : 0≤Real.log 2 := Real.log_nonneg (by norm_num)
+  have hr := rowLen_zero_le_card μ
+  have hcl : μ.colLen 0≤μ.card := by
+    simpa only [YoungDiagram.rowLen_transpose,card_transpose] using rowLen_zero_le_card μ.transpose
+  by_cases hsum : (μ.rowLen 0:ℝ)+μ.colLen 0≤(μ.card:ℝ)/2
+  · have hi := dimension_log_incidence μ hμ
+    have hm := mul_le_mul_of_nonneg_right (min_le_left c (entropyGap*δ)) hN.le
+    dsimp [c] at hm
+    nlinarith
+  · have hbig : (μ.card:ℝ)/4≤μ.rowLen 0 ∨ (μ.card:ℝ)/4≤μ.colLen 0 := by
+      by_contra h
+      push Not at h
+      linarith
+    rcases hbig with hrb | hcb
+    · have hj : 0<μ.card-μ.rowLen 0 := by
+        rw [←Nat.cast_pos (α := ℝ),Nat.cast_sub hr]
+        exact lt_of_lt_of_le (mul_pos hδ hN) hrow
+      have hjN : μ.card-μ.rowLen 0<μ.card := by
+        have : 0<μ.rowLen 0 := by exact_mod_cast (by linarith : (0:ℝ)<μ.rowLen 0)
+        omega
+      have hi := log_dimension_from_binomial μ.card (μ.card-μ.rowLen 0)
+        (Module.finrank ℂ (space μ)) hj hjN
+        (by rw [Nat.cast_sub hr]; linarith) (choose_tail_le_dimension μ)
+      rw [Nat.cast_sub hr] at hi
+      have hm := mul_le_mul_of_nonneg_right (min_le_right c (entropyGap*δ)) hN.le
+      nlinarith [entropyGap_pos]
+    · have hj : 0<μ.card-μ.colLen 0 := by
+        rw [←Nat.cast_pos (α := ℝ),Nat.cast_sub hcl]
+        linarith
+      have hjN : μ.card-μ.colLen 0<μ.card := by
+        have : 0<μ.colLen 0 := by exact_mod_cast (by linarith : (0:ℝ)<μ.colLen 0)
+        omega
+      have hi := log_dimension_from_binomial μ.card (μ.card-μ.colLen 0)
+        (Module.finrank ℂ (space μ)) hj hjN
+        (by rw [Nat.cast_sub hcl]; linarith) (choose_column_tail_le_dimension μ)
+      rw [Nat.cast_sub hcl] at hi
+      have hm := mul_le_mul_of_nonneg_right (min_le_right c (entropyGap*δ)) hN.le
+      have ht := mul_le_mul_of_nonneg_left (show δ*(μ.card:ℝ)≤(μ.card:ℝ)-μ.colLen 0 by nlinarith) entropyGap_pos.le
+      linarith
+
+end Thorp.Specht
+
+namespace Thorp.DimensionEstimates
+
+section
+open scoped BigOperators Classical
+open Filter
+
+lemma finset_power_sum {ι : Type*} [Fintype ι] (x : ℝ) :
+    ∑ E : Finset ι, x^E.card=(1+x)^Fintype.card ι := by
+  have h := Finset.prod_one_add (f := fun _ : ι => x) Finset.univ
+  simpa only [Finset.prod_const,Finset.card_univ,Finset.powerset_univ] using h.symm
+
+lemma small_sets_count {ι : Type*} [Fintype ι] (ε t : ℝ) (ht : 0≤t) :
+    (∑ E : Finset ι, if (E.card:ℝ)<ε*Fintype.card ι then (1:ℝ) else 0) ≤
+      Real.exp ((t*ε+Real.log (1+Real.exp (-t)))*Fintype.card ι) := by
+  have hpow (E : Finset ι) : (Real.exp (-t))^E.card=Real.exp (-t*(E.card:ℝ)) := by
+    rw [←Real.exp_nat_mul]
+    congr 1
+    ring
+  calc
+    _ ≤ ∑ E : Finset ι, Real.exp (t*ε*Fintype.card ι)*(Real.exp (-t))^E.card := by
+      apply Finset.sum_le_sum
+      intro E _
+      split_ifs with hE
+      · rw [hpow,←Real.exp_add,←Real.exp_zero]
+        apply Real.exp_le_exp.mpr
+        nlinarith [mul_le_mul_of_nonneg_left hE.le ht]
+      · positivity
+    _ = Real.exp (t*ε*Fintype.card ι)*(1+Real.exp (-t))^Fintype.card ι := by
+      rw [←Finset.mul_sum,finset_power_sum]
+    _ = _ := by
+      have he : (1+Real.exp (-t))^Fintype.card ι=
+          Real.exp (Real.log (1+Real.exp (-t))*(Fintype.card ι:ℝ)) := by
+        rw [mul_comm,Real.exp_nat_mul,Real.exp_log (by positivity)]
+      rw [he,←Real.exp_add]
+      congr 1
+      ring
+
+lemma small_sets_ceil_bound {ι : Type*} [Fintype ι] (ε t : ℝ) (ht : 0≤t) :
+    (∑ E : Finset ι, if E.card<⌈ε*Fintype.card ι⌉₊ then (1:ℝ) else 0) ≤
+      Real.exp ((t*ε+Real.log (1+Real.exp (-t)))*Fintype.card ι) := by
+  have he (E : Finset ι) : E.card<⌈ε*Fintype.card ι⌉₊ ↔ (E.card:ℝ)<ε*Fintype.card ι := Nat.lt_ceil
+  simp only [he]
+  exact small_sets_count ε t ht
+
+lemma exists_small_set_rate (κ ε₀ : ℝ) (hκ : 0<κ) (hε₀ : 0<ε₀) :
+    ∃ ε : ℝ, 0<ε ∧ ε≤ε₀ ∧ ε<1/2 ∧ ∃ t : ℝ, 0≤t ∧
+      t*ε+Real.log (1+Real.exp (-t))≤κ := by
+  have ht : Tendsto (fun t : ℝ => Real.log (1+Real.exp (-t))) atTop (nhds 0) := by
+    have he : Tendsto (fun t : ℝ => Real.exp (-t)) atTop (nhds 0) :=
+      Real.tendsto_exp_atBot.comp tendsto_neg_atTop_atBot
+    have hl := (tendsto_const_nhds.add he).log (by norm_num : (1:ℝ)+0≠0)
+    simpa only [add_zero,Real.log_one] using hl
+  obtain ⟨t,ht,hlog⟩ := ((eventually_ge_atTop (1:ℝ)).and
+    (ht.eventually (gt_mem_nhds (by linarith : (0:ℝ)<κ/2)))).exists
+  let ε := min ε₀ (min (1/4) (κ/(2*t)))
+  have htp : 0<t := by linarith
+  have hε : 0<ε := lt_min hε₀ (lt_min (by norm_num) (by positivity))
+  refine ⟨ε,hε,min_le_left _ _,(min_le_right _ _).trans_lt ((min_le_left _ _).trans_lt (by norm_num)),t,htp.le,?_⟩
+  have hsmall : ε≤κ/(2*t) := (min_le_right _ _).trans (min_le_right _ _)
+  have hm := (le_div_iff₀ (by positivity : (0:ℝ)<2*t)).mp hsmall
+  linarith
+
+end
+open Filter
+
+lemma log_error_eventually (a : ℝ) (ha : 0<a) :
+    ∀ᶠ N : ℕ in atTop, 2*Real.log N+1≤(a/2)*(N:ℝ) := by
+  have h := (tendsto_natCast_atTop_atTop (R := ℝ)).eventually
+    (Real.isLittleO_log_id_atTop.bound (by linarith : 0<a/8))
+  have hg := (tendsto_natCast_atTop_atTop (R := ℝ)).eventually (eventually_ge_atTop (4/a))
+  filter_upwards [h,hg,eventually_ge_atTop 1] with N h hg hN
+  have hn : (1:ℝ)≤N := by exact_mod_cast hN
+  simp only [Real.norm_eq_abs,abs_of_nonneg (Real.log_nonneg hn),id_eq,abs_of_nonneg (by positivity : (0:ℝ)≤N)] at h
+  have hh : 4≤(N:ℝ)*a := (div_le_iff₀ ha).mp hg
+  linarith
+
+end Thorp.DimensionEstimates
+
+namespace Thorp.Specht
+open scoped BigOperators Classical
+open DimensionEstimates Filter
+
+theorem medium_dimension (δ : ℝ) (hδ : 0<δ) (hδ2 : δ≤1/2) :
+    ∃ h : ℝ, 0<h ∧ ∀ᶠ N : ℕ in atTop, ∀ μ : YoungDiagram, μ.card=N →
+      δ*(N:ℝ)≤N-μ.rowLen 0 → (μ.colLen 0:ℝ)≤(N:ℝ)/2 →
+      h*(N:ℝ)≤Real.log (Module.finrank ℂ (space μ)) := by
+  let a := min ((3/2:ℝ)*Real.log 2-1) (entropyGap*δ)
+  have ha : 0<a := by
+    apply lt_min
+    · nlinarith [Real.log_two_gt_d9]
+    · exact mul_pos entropyGap_pos hδ
+  refine ⟨a/2,by positivity,?_⟩
+  filter_upwards [log_error_eventually a ha,eventually_ge_atTop 1] with N hN hn
+  intro μ hμ hr hc
+  have hp : 0<μ.card := by omega
+  have hh := medium_dimension_raw μ hp δ hδ hδ2 (by simpa only [hμ] using hr) (by simpa only [hμ] using hc)
+  rw [hμ] at hh
+  change a*(N:ℝ)-2*Real.log N-1≤_ at hh
+  linarith
+
+end Thorp.Specht
+
+namespace Thorp
+open scoped BigOperators Classical
+
+lemma insertNth_castSucc_snoc {α : Type*} {d : ℕ} (i : Fin (d+1)) (b a : α) (y : Fin d → α) :
+    Fin.insertNth i.castSucc b (Fin.snoc (α := fun _ => α) y a)=Fin.snoc (α := fun _ => α) (Fin.insertNth i b y) a := by
+  apply Fin.lastCases ?_ (fun j => ?_) |> fun h => funext h
+  · rw [←show i.castSucc.succAbove (Fin.last d)=Fin.last (d+1) from
+        Fin.succAbove_castSucc_of_le i (Fin.last d) (Fin.le_last i),Fin.insertNth_apply_succAbove]
+    simp
+  · cases j using Fin.succAboveCases i with
+    | x => simp
+    | p j =>
+      have he : i.castSucc.succAbove j.castSucc=(i.succAbove j).castSucc := by
+        simp only [Fin.succAbove,Fin.castSucc_lt_castSucc_iff]
+        split_ifs <;> rfl
+      rw [←he,Fin.insertNth_apply_succAbove]
+      simp
+
+lemma rotate_cons (d : ℕ) (b : Bool) (y : Card d) :
+    rotate (d+1) (Fin.cons b y)=Fin.snoc y b := (Fin.snoc_eq_cons_rotate y b).symm
+
+lemma rotate_insert (d : ℕ) (i : Fin (d+1)) (b : Bool) (y : Card (d+1)) :
+    rotate (d+2) (Fin.insertNth i.succ b y)=Fin.insertNth i.castSucc b (rotate (d+1) y) := by
+  rw [←Fin.cons_self_tail y,Fin.insertNth_succ_cons,rotate_cons,rotate_cons,insertNth_castSucc_snoc]
+
+def insertionPerm (d : ℕ) (i : Fin (d+1)) : Equiv.Perm (Card (d+1)) :=
+  (Fin.consEquiv (fun _ : Fin (d+1) => Bool)).symm.trans
+    (Fin.insertNthEquiv (fun _ : Fin (d+1) => Bool) i)
+
+lemma insertionPerm_apply (d : ℕ) (i : Fin (d+1)) (x : Card (d+1)) :
+    insertionPerm d i x=Fin.insertNth i (x 0) (Fin.tail x) := rfl
+
+lemma insertionPerm_zero (d : ℕ) : insertionPerm d 0=1 := by
+  ext x j
+  simp [insertionPerm_apply]
+
+lemma insertionPerm_last (d : ℕ) : insertionPerm d (Fin.last d)=rotate (d+1) := by
+  apply Equiv.ext
+  intro x
+  rw [insertionPerm_apply,Fin.insertNth_last',←rotate_cons,Fin.cons_self_tail]
+
+lemma physical_carry_step (d : ℕ) (i : Fin (d+1)) (ξ : Bool → Coins (d+1)) :
+    step (d+2) (fun y => ξ (y i) (Fin.removeNth i y))*insertionPerm (d+1) i.succ=
+      insertionPerm (d+1) i.castSucc*childLift (fun b => step (d+1) (ξ b)) := by
+  apply Equiv.ext
+  intro x
+  have he : Fin.cons (Bool.xor ((Fin.tail x) 0) (ξ (x 0) (Fin.tail (Fin.tail x))))
+      (Fin.insertNth i (x 0) (Fin.tail (Fin.tail x))) =
+      (Fin.insertNth i.succ (x 0) (switchFun (ξ (x 0)) (Fin.tail x)) : Card (d+2)) := by
+    simp only [switchFun,Fin.insertNth_succ_cons]
+  have hs : switchFun (fun y => ξ (y i) (Fin.removeNth i y))
+      (Fin.insertNth i.succ (x 0) (Fin.tail x)) =
+      (Fin.insertNth i.succ (x 0) (switchFun (ξ (x 0)) (Fin.tail x)) : Card (d+2)) := by
+    conv_lhs => rw [←Fin.cons_self_tail (Fin.tail x),Fin.insertNth_succ_cons]
+    simp only [switchFun,Fin.cons_zero,Fin.tail_cons,Fin.insertNth_apply_same,Fin.removeNth_insertNth]
+    exact he
+  change rotate (d+2) (switchFun (fun y => ξ (y i) (Fin.removeNth i y))
+      (Fin.insertNth i.succ (x 0) (Fin.tail x))) = _
+  rw [hs,rotate_insert]
+  rfl
+
+end Thorp
+namespace Thorp.UnitaryFinite
+open scoped BigOperators Classical
+variable {G : Type*} [Group G]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+variable {Ω : Type*} [Fintype Ω]
+
+lemma sampleOperator_mul_const (ρ : Representation ℂ G V) (P : Ω → G) (g : G) :
+    sampleOperator ρ (fun ω => P ω*g)=sampleOperator ρ P*continuousRepresentation ρ g := by
+  simp only [sampleOperator,map_mul,Finset.sum_mul,smul_mul_assoc]
+
+lemma sampleOperator_const_mul (ρ : Representation ℂ G V) (P : Ω → G) (g : G) :
+    sampleOperator ρ (fun ω => g*P ω)=continuousRepresentation ρ g*sampleOperator ρ P := by
+  simp only [sampleOperator,map_mul,Finset.mul_sum,mul_smul_comm]
+
+lemma sampleOperator_run (d t : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V) :
+    sampleOperator ρ (run d t)=(sampleOperator ρ (step d))^t := by
+  induction t with
+  | zero => exact sampleOperator_one ρ
+  | succ t ih =>
+    let e : Coins d × (Fin t → Coins d) ≃ (Fin (t+1) → Coins d) :=
+      Fin.snocEquiv (fun _ => Coins d)
+    have he : run d (t+1) ∘ e=fun w => step d w.1*run d t w.2 := by
+      funext w
+      simp only [Function.comp_apply,e,Fin.snocEquiv_apply,
+        run,Fin.snoc_last,Fin.snoc_castSucc]
+    rw [←sampleOperator_equiv ρ (run d (t+1)) e,he,sampleOperator_prod,ih,pow_succ']
+
+end Thorp.UnitaryFinite
+
+namespace Thorp
+open scoped BigOperators Classical
+open UnitaryFinite
+
+def carryCoins (d : ℕ) (i : Fin (d+1)) : (Bool → Coins (d+1)) ≃ Coins (d+2) where
+  toFun ξ y := ξ (y i) (Fin.removeNth i y)
+  invFun ξ b y := ξ (Fin.insertNth i b y)
+  left_inv ξ := by funext b y; simp
+  right_inv ξ := by funext y; simp
+
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+lemma physical_carry_average (d : ℕ) (i : Fin d)
+    (ρ : Representation ℂ (Equiv.Perm (Card (d+1))) V) :
+    sampleOperator ρ (step (d+1))*continuousRepresentation ρ (insertionPerm d i.succ)=
+      continuousRepresentation ρ (insertionPerm d i.castSucc)*
+        sampleOperator ρ (fun ξ : Bool → Coins d => childLift (fun b => step d (ξ b))) := by
+  cases d with
+  | zero => exact Fin.elim0 i
+  | succ d =>
+    have he : (fun ξ : Bool → Coins (d+1) =>
+        step (d+2) (carryCoins d i ξ)*insertionPerm (d+1) i.succ)=
+      (fun ξ => insertionPerm (d+1) i.castSucc*childLift (fun b => step (d+1) (ξ b))) := by
+      funext ξ
+      exact physical_carry_step d i ξ
+    have h := congrArg (sampleOperator ρ) he
+    rw [sampleOperator_mul_const,sampleOperator_const_mul] at h
+    change sampleOperator ρ (step (d+2) ∘ carryCoins d i)*_=_ at h
+    rw [sampleOperator_equiv] at h
+    exact h
+
+theorem physical_sweep_recursion (d : ℕ)
+    (ρ : Representation ℂ (Equiv.Perm (Card (d+1))) V) :
+    (sampleOperator ρ (step (d+1)))^(d+1)=
+      (sampleOperator ρ (fun ξ : Bool → Coins d => childLift (fun b => step d (ξ b))))^d*
+        sampleOperator ρ (pairSwitch (d := d)) := by
+  let A := sampleOperator ρ (step (d+1))
+  let K := sampleOperator ρ (fun ξ : Bool → Coins d => childLift (fun b => step d (ξ b)))
+  have h (t : ℕ) (ht : t≤d) : A^t*continuousRepresentation ρ (insertionPerm d ⟨t,by omega⟩)=K^t := by
+    induction t with
+    | zero =>
+      change 1*continuousRepresentation ρ (insertionPerm d 0)=1
+      rw [insertionPerm_zero,map_one,one_mul]
+    | succ t ih =>
+      have hc := physical_carry_average d (⟨t,by omega⟩ : Fin d) ρ
+      change A*continuousRepresentation ρ (insertionPerm d ⟨t+1,by omega⟩)=
+        continuousRepresentation ρ (insertionPerm d ⟨t,by omega⟩)*K at hc
+      rw [pow_succ,mul_assoc,hc,←mul_assoc,ih (by omega),←pow_succ]
+  have he : sampleOperator ρ (step (d+1))=
+      continuousRepresentation ρ (rotate (d+1))*sampleOperator ρ (pairSwitch (d := d)) :=
+    sampleOperator_const_mul ρ (pairSwitch (d := d)) (rotate (d+1))
+  calc
+    _ = A^d*A := pow_succ A d
+    _ = A^d*(continuousRepresentation ρ (insertionPerm d (Fin.last d))*
+        sampleOperator ρ (pairSwitch (d := d))) := by rw [insertionPerm_last]; rw [←he]
+    _ = _ := by
+      rw [←mul_assoc]
+      exact congrArg (fun z => z*sampleOperator ρ (pairSwitch (d := d))) (h d le_rfl)
+
+end Thorp
+namespace Thorp
+open scoped BigOperators Classical
+open UnitaryFinite
+
+lemma childInjection_product (d : ℕ) (p : Bool → Equiv.Perm (Card d)) :
+    childInjection d false (p false)*childInjection d true (p true)=childLift p := by
+  change childLift _*childLift _=_
+  rw [childLift_mul]
+  congr 1
+  funext b
+  cases b <;> simp
+
+lemma childInjection_commute (d : ℕ) (p q : Equiv.Perm (Card d)) :
+    Commute (childInjection d false p) (childInjection d true q) := by
+  change childLift _*childLift _=childLift _*childLift _
+  rw [childLift_mul,childLift_mul]
+  congr 1
+  funext b
+  cases b <;> simp
+
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+variable {Ω : Type*} [Fintype Ω]
+
+lemma sampleOperator_child (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card (d+1))) V)
+    (P : Ω → Equiv.Perm (Card d)) :
+    sampleOperator ρ (fun ω : Bool → Ω => childLift (fun b => P (ω b)))=
+      sampleOperator (ρ.comp (childInjection d false)) P*
+        sampleOperator (ρ.comp (childInjection d true)) P := by
+  let e := (Equiv.boolArrowEquivProd Ω).symm
+  have he : (fun ω : Bool → Ω => childLift (fun b => P (ω b))) ∘ e=
+      fun w : Ω × Ω => childInjection d false (P w.1)*childInjection d true (P w.2) := by
+    funext w
+    change childLift (fun b => P (e w b))=_
+    rw [←childInjection_product]
+    rfl
+  rw [←sampleOperator_equiv ρ _ e,he]
+  exact sampleOperator_prod ρ (fun w => childInjection d false (P w))
+    (fun w => childInjection d true (P w))
+
+lemma sampleOperator_child_commute (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card (d+1))) V)
+    (P : Ω → Equiv.Perm (Card d)) :
+    Commute (sampleOperator (ρ.comp (childInjection d false)) P)
+      (sampleOperator (ρ.comp (childInjection d true)) P) := by
+  rw [sampleOperator_comp,sampleOperator_comp]
+  exact sampleOperator_commute ρ _ _ (fun _ _ => childInjection_commute d _ _)
+
+lemma inverse_butterfly_average_step (d : ℕ)
+    (ρ : Representation ℂ (Equiv.Perm (Card (d+1))) V) :
+    sampleOperator ρ (fun ω => (butterflyPerm (d+1) (decodeButterfly (d+1) ω))⁻¹)=
+      sampleOperator ρ (fun ω : Bool → (SwitchIndex d → Bool) =>
+        childLift (fun b => (butterflyPerm d (decodeButterfly d (ω b)))⁻¹))*
+      sampleOperator ρ (pairSwitch (d := d)) := by
+  let e := ((Equiv.prodCongr (Equiv.boolArrowEquivProd (SwitchIndex d → Bool)) (Equiv.refl (Card d → Bool))).trans
+    (coinStepEquiv d).symm)
+  have he : (fun ω => (butterflyPerm (d+1) (decodeButterfly (d+1) ω))⁻¹) ∘ e=
+      fun w : (Bool → (SwitchIndex d → Bool)) × (Card d → Bool) =>
+        childLift (fun b => (butterflyPerm d (decodeButterfly d (w.1 b)))⁻¹)*pairSwitch w.2 := by
+    funext w
+    change (butterflyPerm (d+1) (decodeButterfly (d+1)
+      ((coinStepEquiv d).symm ((w.1 false,w.1 true),w.2))))⁻¹=_
+    rw [butterflyPerm_step,mul_inv_rev,childLift_inv,pairSwitch_inv]
+    congr 2
+    funext b
+    cases b <;> rfl
+  rw [←sampleOperator_equiv ρ _ e,he]
+  exact sampleOperator_prod ρ
+    (fun ω : Bool → (SwitchIndex d → Bool) => childLift (fun b => (butterflyPerm d (decodeButterfly d (ω b)))⁻¹))
+    (pairSwitch (d := d))
+
+theorem physical_sweep_inverse_butterfly (d : ℕ)
+    (ρ : Representation ℂ (Equiv.Perm (Card d)) V) :
+    (sampleOperator ρ (step d))^d=
+      sampleOperator ρ (fun ω => (butterflyPerm d (decodeButterfly d ω))⁻¹) := by
+  induction d with
+  | zero =>
+    symm
+    exact sampleOperator_one ρ
+  | succ d ih =>
+    rw [physical_sweep_recursion,sampleOperator_child,
+      (sampleOperator_child_commute d ρ (step d)).mul_pow,ih,ih,
+      inverse_butterfly_average_step]
+    rw [sampleOperator_child d ρ (fun ω => (butterflyPerm d (decodeButterfly d ω))⁻¹)]
+
+lemma physical_sweep_adjoint (d : ℕ)
+    (ρ : Representation ℂ (Equiv.Perm (Card d)) V) (hρ : IsUnitary ρ) :
+    sampleOperator ρ (run d d)=
+      (sampleOperator ρ (fun ω => butterflyPerm d (decodeButterfly d ω))).adjoint := by
+  rw [sampleOperator_run,physical_sweep_inverse_butterfly,sampleOperator_inv ρ hρ]
+
+end Thorp
+
+namespace Thorp.Specht
+open scoped BigOperators Classical
+open UnitaryFinite DimensionEstimates Filter
+
+lemma dimension_pos (μ : YoungDiagram) : 0 < Module.finrank ℂ (space μ) := by
+  have h := factorial_le_dimension_row_column μ
+  by_contra hn
+  have hz : Module.finrank ℂ (space μ)=0 := by omega
+  rw [hz,zero_mul,zero_mul] at h
+  exact (Nat.factorial_pos μ.card).not_ge h
+
+lemma near_numerical_cutoff :
+    ∃ X : ℝ, 2 ≤ X ∧ ∀ (N j D : ℕ), 0 < j → X ≤ (N:ℝ)/j → N.choose j ≤ D * 2^j →
+      ∃ k : ℕ, j ≤ k ∧ k ≤ N ∧ ∀ m : ℕ, D * k.choose j ≤ m * N.choose j →
+        nearFraction * Real.log D ≤ Real.log m ∧
+        densityConstant * (k:ℝ) * ((k:ℝ)/N)^(1/512:ℝ) ≤ Real.log m/4000 ∧
+        (1/2:ℝ) * Real.log N ≤ Real.log D ∧ (j:ℝ) ≤ Real.log D := by
+  obtain ⟨X,hX,h⟩ := exists_near_cutoff densityConstant densityConstant_pos.le
+  refine ⟨X,hX,?_⟩
+  intro N j D hj hx hd
+  have hjN : j ≤ N := by
+    have hjp : (0:ℝ) < j := by exact_mod_cast hj
+    have hh : (j:ℝ) ≤ N := by
+      have := (le_div_iff₀ hjp).mp (hX.trans hx)
+      linarith
+    exact_mod_cast hh
+  have hc : 1 ≤ N.choose j := Nat.choose_pos hjN
+  have hfoot : D * (chosenTuple N j).choose j ≤ (D * (chosenTuple N j).choose j) * N.choose j :=
+    Nat.le_mul_of_pos_right _ hc
+  have hh := h N j D _ hj hx hd hfoot
+  refine ⟨chosenTuple N j,hh.1,hh.2.1,?_⟩
+  intro m hm
+  exact (h N j D m hj hx hd hm).2.2
+
+lemma near_row_amplification (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Cell μ)
+    {k : ℕ} (f : Fin k ↪ Card d) (hk : 0<k)
+    (hm : 0<Module.finrank ℂ (fixedSpace (V := hilbertSpace μ) tupleAction f (relabelledUnitary μ e))) :
+    ‖sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e) (palindromePerm d)‖ ≤
+      Real.exp (densityConstant*(k:ℝ)*((k:ℝ)/(2:ℝ)^d)^(1/512:ℝ))*
+        (Real.exp (Real.log (Module.finrank ℂ (fixedSpace (V := hilbertSpace μ) tupleAction f (relabelledUnitary μ e)))/2))^(-(1/1000:ℝ))+
+      Real.sqrt ((Real.exp (Real.log (Module.finrank ℂ (fixedSpace (V := hilbertSpace μ) tupleAction f (relabelledUnitary μ e)))/2))/
+        (Module.finrank ℂ (fixedSpace (V := hilbertSpace μ) tupleAction f (relabelledUnitary μ e)):ℝ)) := by
+  let := relabelledUnitary_irreducible μ e
+  let : NormedAddCommGroup (hilbertSpace μ) := inferInstance
+  let : InnerProductSpace ℂ (hilbertSpace μ) := inferInstance
+  let : FiniteDimensional ℂ (hilbertSpace μ) := inferInstance
+  have aux := @palindrome_irrep_amplification (hilbertSpace μ) _ _ _ d (Fin k) _ f
+    (by simpa only [Fintype.card_fin] using hk) (relabelledUnitary μ e) inferInstance
+    (relabelledUnitary_unitary μ e) hm
+    (Real.exp (Real.log (Module.finrank ℂ (fixedSpace (V := hilbertSpace μ) tupleAction f (relabelledUnitary μ e)))/2))
+    (Real.exp_pos _)
+  simpa only [Fintype.card_fin] using aux
+
+theorem near_row_spectral :
+    ∃ X : ℝ, 2 ≤ X ∧ ∀ (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Cell μ),
+      0 < μ.card-μ.rowLen 0 → X ≤ (μ.card:ℝ)/(μ.card-μ.rowLen 0:ℕ) →
+      32768000 * Real.log 2 ≤ (1/2:ℝ) * Real.log μ.card →
+      ‖sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e) (palindromePerm d)‖ ≤
+        Real.exp (-(nearFraction/8000) * Real.log (Module.finrank ℂ (space μ))) ∧
+      (1/2:ℝ) * Real.log μ.card ≤ Real.log (Module.finrank ℂ (space μ)) ∧
+      (μ.card-μ.rowLen 0:ℕ) ≤ Real.log (Module.finrank ℂ (space μ)) := by
+  obtain ⟨X,hX,h⟩ := near_numerical_cutoff
+  refine ⟨X,hX,?_⟩
+  intro d μ e hj hx hlarge
+  let j := μ.card-μ.rowLen 0
+  let D := Module.finrank ℂ (space μ)
+  have hcard : μ.card=2^d := by rw [←card_cell,←Fintype.card_congr e,card_positions]
+  obtain ⟨k,hjk,hk,hnum⟩ := h μ.card j D hj hx (choose_tail_le_dimension μ)
+  obtain ⟨f⟩ := Function.Embedding.nonempty_of_card_le
+    (show Fintype.card (Fin k) ≤ Fintype.card (Card d) by simpa only [Fintype.card_fin,card_positions,←hcard] using hk)
+  let m := Module.finrank ℂ (fixedSpace (V := hilbertSpace μ) tupleAction f (relabelledUnitary μ e))
+  have hfoot : D * k.choose j ≤ m * μ.card.choose j := by
+    simpa only [Fintype.card_fin,finrank_hilbertSpace] using relabelled_tuple_dimension μ e f
+  have hm : 0 < m := by
+    have hc := Nat.choose_pos hjk
+    have hd := dimension_pos μ
+    by_contra hn
+    have hz : m=0 := by omega
+    rw [hz,zero_mul] at hfoot
+    exact (Nat.mul_pos hd hc).not_ge hfoot
+  obtain ⟨hlm,hden,hlog,hjlog⟩ := hnum m hfoot
+  have ha := near_row_amplification d μ e f (hj.trans_le hjk) hm
+  have hcardR : (μ.card:ℝ)=(2:ℝ)^d := by exact_mod_cast hcard
+  rw [←hcardR] at ha
+  have hl : 8000 * Real.log 2 ≤ Real.log m := by
+    dsimp [nearFraction] at hlm
+    linarith
+  have hn := amplification_decay (m:ℝ) (densityConstant * k * ((k:ℝ)/μ.card)^(1/512:ℝ))
+    (by exact_mod_cast hm) hden hl
+  refine ⟨ha.trans (hn.trans ?_),hlog,hjlog⟩
+  apply Real.exp_le_exp.mpr
+  dsimp [D] at hlm
+  linarith
+
+end Thorp.Specht
+
+namespace Thorp.UnitaryFinite
+open scoped BigOperators Classical
+open DimensionEstimates
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+theorem full_palindrome_decay (d : ℕ)
+    (ρ : Representation ℂ (Equiv.Perm (Card d)) V) [Representation.IsIrreducible ρ]
+    (hρ : IsUnitary ρ) (hm : 0<Module.finrank ℂ V)
+    (hden : densityConstant*(2:ℝ)^d≤Real.log (Module.finrank ℂ V)/4000)
+    (hl : 8000*Real.log 2≤Real.log (Module.finrank ℂ V)) :
+    ‖sampleOperator ρ (palindromePerm d)‖≤Real.exp (-Real.log (Module.finrank ℂ V)/8000) := by
+  have h := palindrome_irrep_amplification d (Function.Embedding.refl (Card d))
+    (Fintype.card_pos) ρ hρ (by simpa only [full_tuple_finrank] using hm)
+    (Real.exp (Real.log (Module.finrank ℂ V)/2)) (Real.exp_pos _)
+  simp only [full_tuple_finrank,card_positions,Nat.cast_pow,Nat.cast_ofNat,
+    div_self (by positivity : (2:ℝ)^d≠0),Real.one_rpow,mul_one] at h
+  exact h.trans (amplification_decay _ _ (by exact_mod_cast hm) hden hl)
+
+theorem full_modified_decay (L r : ℕ) (E : Finset (Card r))
+    (ρ : Representation ℂ (Equiv.Perm (Card (L+r))) V) [Representation.IsIrreducible ρ]
+    (hρ : IsUnitary ρ) (hm : 0<Module.finrank ℂ V) (u : ℝ)
+    (hM : ∀ e : Card (L+r) ↪ Card (L+r), modifiedRowMoment L r E e (1/1000)≤Real.exp u)
+    (hden : u≤Real.log (Module.finrank ℂ V)/4000)
+    (hl : 8000*Real.log 2≤Real.log (Module.finrank ℂ V)) :
+    ‖sampleOperator ρ (modifiedPerm L r E)‖≤Real.exp (-Real.log (Module.finrank ℂ V)/8000) := by
+  exact (modified_irrep_amplification L r E ρ hρ hm u
+    (Real.exp (Real.log (Module.finrank ℂ V)/2)) (Real.exp_pos _) hM).trans
+    (amplification_decay _ _ (by exact_mod_cast hm) hden hl)
+
+lemma palindrome_exception_cards (r L : ℕ)
+    (ρ : Representation ℂ (Equiv.Perm (Card (L+r))) V) (hρ : IsUnitary ρ) (n : ℕ) :
+    ‖sampleOperator ρ (palindromePerm (L+r))‖ ≤
+      (blockGap L)^n+∑ E : Finset (Card r), if E.card<n then
+        ‖sampleOperator ρ (modifiedPerm L r E)‖ else 0 := by
+  let σ := ρ.comp (splitPerm r L).symm.toMonoidHom
+  have he : σ.comp (splitPerm r L).toMonoidHom=ρ := by
+    ext g v
+    change ρ ((splitPerm r L).symm ((splitPerm r L) g)) v=ρ g v
+    rw [MulEquiv.symm_apply_apply]
+  have h := palindrome_exception_bound r L σ (unitary_comp ρ hρ _) n
+  simpa only [he] using h
+
+end Thorp.UnitaryFinite
+
+namespace Thorp.DimensionEstimates
+open scoped BigOperators Classical
+open Filter
+
+lemma medium_parameters (h : ℝ) (hh : 0<h) :
+    ∃ L : ℕ, 0<L ∧ ∃ ε t : ℝ, 0<ε ∧ ε<1/2 ∧ 0≤t ∧
+      t*ε+Real.log (1+Real.exp (-t))≤h*(2:ℝ)^L/16000 ∧
+      ∀ᶠ r : ℕ in atTop, ∀ (E : Finset (Card r)), (E.card:ℝ)<ε*(2:ℝ)^r →
+        ∀ e : Card (L+r) ↪ Card (L+r),
+          modifiedRowMoment L r E e (1/1000)≤Real.exp ((h/4000)*(2:ℝ)^(L+r)) := by
+  obtain ⟨L,hL,ε₀,hε₀,_,hM⟩ := modified_arbitrarily_small_rate (h/4000) (by positivity)
+  obtain ⟨ε,hε,hεle,hεhalf,t,ht,hcount⟩ := exists_small_set_rate
+    (h*(2:ℝ)^L/16000) ε₀ (by positivity) hε₀
+  refine ⟨L,hL,ε,t,hε,hεhalf,ht,hcount,?_⟩
+  filter_upwards [hM] with r hM
+  intro E hE e
+  exact hM E (hE.le.trans (mul_le_mul_of_nonneg_right hεle (by positivity))) e
+
+lemma geometric_ceil_bound (θ ε : ℝ) (hθ : 0<θ) (hθ1 : θ<1) (M : ℕ) :
+    θ^⌈ε*M⌉₊≤Real.exp (Real.log θ*ε*M) := by
+  conv_lhs => rw [←Real.exp_log hθ,←Real.exp_nat_mul]
+  apply Real.exp_le_exp.mpr
+  have hg := mul_le_mul_of_nonpos_left (Nat.le_ceil (ε*M)) (Real.log_nonpos hθ.le hθ1.le)
+  nlinarith
+
+end Thorp.DimensionEstimates
+
+namespace Thorp.UnitaryFinite
+open scoped BigOperators Classical
+open DimensionEstimates
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+theorem medium_exception_decay (L r : ℕ) (h ε t : ℝ) (ht : 0≤t)
+    (hcount : t*ε+Real.log (1+Real.exp (-t))≤h*(2:ℝ)^L/16000)
+    (ρ : Representation ℂ (Equiv.Perm (Card (L+r))) V) [Representation.IsIrreducible ρ]
+    (hρ : IsUnitary ρ) (hm : 0<Module.finrank ℂ V)
+    (hd : h*(2:ℝ)^(L+r)≤Real.log (Module.finrank ℂ V))
+    (hl : 8000*Real.log 2≤Real.log (Module.finrank ℂ V))
+    (hM : ∀ (E : Finset (Card r)), (E.card:ℝ)<ε*(2:ℝ)^r →
+      ∀ e : Card (L+r) ↪ Card (L+r),
+        modifiedRowMoment L r E e (1/1000)≤Real.exp ((h/4000)*(2:ℝ)^(L+r))) :
+    ‖sampleOperator ρ (palindromePerm (L+r))‖≤
+      Real.exp (Real.log (blockGap L)*ε*(2:ℝ)^r)+Real.exp (-h*(2:ℝ)^(L+r)/16000) := by
+  have hmE (E : Finset (Card r)) (hE : E.card<⌈ε*(2:ℝ)^r⌉₊) :
+      ‖sampleOperator ρ (modifiedPerm L r E)‖≤Real.exp (-h*(2:ℝ)^(L+r)/8000) := by
+    apply (full_modified_decay L r E ρ hρ hm ((h/4000)*(2:ℝ)^(L+r))
+      (hM E (Nat.lt_ceil.mp hE)) (by linarith) hl).trans
+    apply Real.exp_le_exp.mpr
+    linarith
+  have hs : (∑ E : Finset (Card r), if E.card<⌈ε*(2:ℝ)^r⌉₊ then
+      ‖sampleOperator ρ (modifiedPerm L r E)‖ else 0) ≤
+      Real.exp (-h*(2:ℝ)^(L+r)/16000) := by
+    calc
+      _ ≤ (∑ E : Finset (Card r), if E.card<⌈ε*(2:ℝ)^r⌉₊ then (1:ℝ) else 0)*
+          Real.exp (-h*(2:ℝ)^(L+r)/8000) := by
+        rw [Finset.sum_mul]
+        apply Finset.sum_le_sum
+        intro E _
+        split_ifs with hE
+        · simpa only [one_mul] using hmE E hE
+        · simp
+      _ ≤ Real.exp ((t*ε+Real.log (1+Real.exp (-t)))*(2:ℝ)^r)*
+          Real.exp (-h*(2:ℝ)^(L+r)/8000) := by
+        apply mul_le_mul_of_nonneg_right _ (Real.exp_pos _).le
+        simpa only [card_positions,Nat.cast_pow,Nat.cast_ofNat] using
+          small_sets_ceil_bound (ι := Card r) ε t ht
+      _ ≤ _ := by
+        rw [←Real.exp_add]
+        apply Real.exp_le_exp.mpr
+        have hh := mul_le_mul_of_nonneg_right hcount (by positivity : (0:ℝ)≤(2:ℝ)^r)
+        rw [pow_add]
+        nlinarith
+  apply (palindrome_exception_cards r L ρ hρ ⌈ε*(2:ℝ)^r⌉₊).trans
+  apply add_le_add _ hs
+  simpa only [Nat.cast_pow,Nat.cast_ofNat] using
+    geometric_ceil_bound (blockGap L) ε (blockGap_pos L) (blockGap_lt_one L) (2^r)
+
+end Thorp.UnitaryFinite
+
+
+end ThorpNine.Harmonic
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Harmonic/Main.lean b/lean/OAI/Probability/ThorpRouting/Harmonic/Main.lean
index 771f71a9c8877f1bae0389fb5890108de17adc00..3b772f2ef286c97fb1970de1a98f43e5f8171d5f 100644
--- a/lean/OAI/Probability/ThorpRouting/Harmonic/Main.lean
+++ b/lean/OAI/Probability/ThorpRouting/Harmonic/Main.lean
@@ -1,1226 +1,1226 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Harmonic.Site
-
-namespace OAI
-
-namespace ThorpNine.Harmonic
-
-namespace Thorp.Casimir
-open scoped BigOperators Classical
-
-theorem palindrome_cost_mgf_coarse (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/32) :
-    finiteMean (fun ω : BenesCoins d => q^(palindromeCost d e ω)) ≤
-      Real.exp (Fintype.card ι*(1+heightDecay 1)) := by
-  have hd : d ≤ 1*4^d := by have hh := pow_four_ge d; omega
-  have hcost (ω : BenesCoins d) : palindromeCost d e ω = palindromeLowCost 1 d e ω+
-      (palindromeFamilyCost 1 d d e ω+palindromeFamilyCost 2 d d e ω) := by
-    have hh := palindromeCost_family 1 d d e ω
-    rw [palindromeLowCost_eq _ _ hd] at hh
-    simpa only [Nat.add_assoc] using hh.symm
-  have hqp : 0 < q := lt_of_lt_of_le (by norm_num) hq
-  have hlow (ω : BenesCoins d) : q^(palindromeLowCost 1 d e ω) ≤ Real.exp (Fintype.card ι) := by
-    calc
-      _ ≤ q^(Fintype.card ι) := pow_le_pow_right₀ hq (by simpa using palindromeLowCost_le 1 d e ω)
-      _ = Real.exp (Fintype.card ι*Real.log q) := by rw [Real.exp_nat_mul,Real.exp_log hqp]
-      _ ≤ _ := Real.exp_le_exp.mpr (by nlinarith [Nat.cast_nonneg (α:=ℝ) (Fintype.card ι)])
-  calc
-    _ ≤ finiteMean (fun ω : BenesCoins d => Real.exp (Fintype.card ι)*
-        q^(palindromeFamilyCost 1 d d e ω+palindromeFamilyCost 2 d d e ω)) := by
-      apply finiteMean_mono
-      intro ω
-      rw [hcost,pow_add]
-      exact mul_le_mul_of_nonneg_right (hlow ω) (pow_nonneg hqp.le _)
-    _ = Real.exp (Fintype.card ι)*finiteMean (fun ω : BenesCoins d =>
-        q^(palindromeFamilyCost 1 d d e ω+palindromeFamilyCost 2 d d e ω)) := by simpa only [mul_comm] using finiteMean_mul_const (fun ω : BenesCoins d => q^(palindromeFamilyCost 1 d d e ω+palindromeFamilyCost 2 d d e ω)) (Real.exp (Fintype.card ι))
-    _ ≤ Real.exp (Fintype.card ι)*Real.exp (Fintype.card ι*heightDecay 1) := by
-      exact mul_le_mul_of_nonneg_left
-        (palindrome_high_full_mgf 1 d d (by decide) e q hq hlog) (Real.exp_nonneg _)
-    _ = _ := by rw [←Real.exp_add]; congr 1; ring
-
-lemma row_moment_coarse (d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d)
-    (a : ℝ) (ha : 0 < a) (ha32 : a ≤ 1/32) :
-    palindromeRowMoment d e a ≤ Real.exp (Fintype.card ι*(1+heightDecay 1)) := by
-  let q : ℝ := (2:ℝ)^a
-  have hq : 1 ≤ q := Real.one_le_rpow (by norm_num) ha.le
-  have hlog : Real.log q ≤ 1/32 := by
-    dsimp [q]
-    rw [Real.log_rpow (by norm_num)]
-    have hh := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
-    norm_num at hh
-    nlinarith
-  have hn : (0:ℝ) < ((2^d:ℕ):ℝ)^(Fintype.card ι) := by positivity
-  have hn0 : (0:ℝ) ≤ ((2^d).descFactorial (Fintype.card ι):ℝ) /
-      ((2^d:ℕ):ℝ)^(Fintype.card ι) := by positivity
-  have hn1 : ((2^d).descFactorial (Fintype.card ι):ℝ) /
-      ((2^d:ℕ):ℝ)^(Fintype.card ι) ≤ 1 := by
-    apply (div_le_one hn).mpr
-    exact_mod_cast Nat.descFactorial_le_pow (2^d) (Fintype.card ι)
-  have hpow := Real.rpow_le_one hn0 hn1 ha.le
-  have hm := palindrome_row_density_moment d e a ha.le
-  have he (c : ℕ) : q^c = (2:ℝ)^((c:ℝ)*a) := by
-    rw [mul_comm,Real.rpow_mul_natCast (by norm_num)]
-  simp_rw [←he] at hm
-  exact (hm.trans (mul_le_of_le_one_left (finiteMean_nonneg (fun _ =>
-    pow_nonneg (le_trans (by norm_num) hq) _)) hpow)).trans
-      (palindrome_cost_mgf_coarse d e q hq hlog)
-
-
-lemma applyKernel_mul {X : Type*} [Fintype X] (P Q : Matrix X X ℝ)
-    (v : EuclideanSpace ℂ X) :
-    DensityTransfer.applyKernel (fun x y => (P*Q) x y) v =
-      DensityTransfer.applyKernel (fun x y => P x y) (DensityTransfer.applyKernel (fun x y => Q x y) v) := by
-  apply PiLp.ext
-  intro x
-  change DensityTransfer.applyKernel (fun x y => (P*Q) x y) v x = DensityTransfer.applyKernel (fun x y => P x y) (DensityTransfer.applyKernel (fun x y => Q x y) v) x
-  simp only [DensityTransfer.applyKernel_apply,Matrix.mul_apply,Complex.ofReal_sum,
-    Complex.ofReal_mul,Finset.sum_mul,Finset.mul_sum]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro y _
-  apply Finset.sum_congr rfl
-  intro z _
-  ring
-
-lemma applyKernel_one {X : Type*} [Fintype X] [DecidableEq X]
-    (v : EuclideanSpace ℂ X) : DensityTransfer.applyKernel (fun x y => (1 : Matrix X X ℝ) x y) v = v := by
-  apply PiLp.ext
-  intro x
-  change DensityTransfer.applyKernel (fun x y => (1 : Matrix X X ℝ) x y) v x = v x
-  simp [DensityTransfer.applyKernel_apply,Matrix.one_apply,apply_ite,ite_mul]
-
-lemma symmetric_hs_power {X : Type*} [Fintype X] [DecidableEq X]
-    (P : Matrix X X ℝ) (hs : P.IsSymm) (r : ℕ) :
-    (∑ x, ∑ y, ((P^r) x y)^2) = ∑ x, (P^(2*r)) x x := by
-  rw [two_mul,pow_add]
-  apply Finset.sum_congr rfl
-  intro x _
-  simp only [Matrix.mul_apply]
-  apply Finset.sum_congr rfl
-  intro y _
-  rw [(hs.pow r).apply x y,pow_two]
-
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-  [FiniteDimensional ℂ V]
-
-lemma trace_power_le_hs (A : V →L[ℂ] V) (hs : IsSelfAdjoint A) (hn : ‖A‖ ≤ 1)
-    (r : ℕ) :
-    (LinearMap.trace ℂ V ((A^(2*r+1)).toLinearMap)).re ≤
-      ∑ j, ‖(A^r) (stdOrthonormalBasis ℂ V j)‖^2 := by
-  let B := stdOrthonormalBasis ℂ V
-  rw [LinearMap.trace_eq_matrix_trace ℂ B.toBasis]
-  simp only [Matrix.trace,Matrix.diag,Complex.re_sum,LinearMap.toMatrix_apply,
-    OrthonormalBasis.coe_toBasis_repr_apply,OrthonormalBasis.repr_apply_apply]
-  apply Finset.sum_le_sum
-  intro j _
-  have he : (A^(2*r+1)) (B j) = (A^r) (A ((A^r) (B j))) := by
-    have hnat : 2*r+1=r+(1+r) := by omega
-    rw [hnat,pow_add,pow_add,pow_one]
-    rfl
-  change (inner ℂ (B j) ((A^(2*r+1)) (B j))).re ≤ ‖(A^r) (B j)‖^2
-  rw [he]
-  conv_lhs => rw [←(hs.pow r).adjoint_eq,ContinuousLinearMap.adjoint_inner_right]
-  rw [(hs.pow r).adjoint_eq]
-  apply (Complex.re_le_norm _).trans
-  apply (norm_inner_le_norm _ _).trans
-  have hh := A.le_opNorm ((A^r) (B j))
-  have ha : ‖A ((A^r) (B j))‖ ≤ ‖(A^r) (B j)‖ := by
-    exact hh.trans (by simpa using mul_le_mul_of_nonneg_right hn (norm_nonneg _))
-  nlinarith [norm_nonneg ((A^r) (B j))]
-
-open UnitaryFinite
-lemma palindrome_selfadjoint (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V)
-    (hρ : IsUnitary ρ) : IsSelfAdjoint (sampleOperator ρ (palindromePerm d)) := by
-  rw [sampleOperator_palindrome _ _ hρ]
-  change IsSelfAdjoint (_ * star _)
-  simp only [isSelfAdjoint_iff,star_mul,star_star]
-
-variable {G X : Type*} [Group G] [Fintype G] [Fintype X] [DecidableEq X]
-
-lemma coefficient_trace_amplification (a : G →* Equiv.Perm X) (x₀ : X)
-    (ha : ∀ x, ∃ g, a g x₀=x) (ρ : Representation ℂ G V)
-    [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
-    (P : Matrix X X ℝ) (A : V →L[ℂ] V)
-    (hmap : ∀ w, IsFixed a x₀ ρ w → ∀ v,
-      DensityTransfer.applyKernel (fun x y => P x y) (normalizedCoefficient a x₀ ha ρ w v) =
-        normalizedCoefficient a x₀ ha ρ w (A v))
-    (hs : IsSelfAdjoint A) (hn : ‖A‖ ≤ 1) (r : ℕ) :
-    (Module.finrank ℂ (fixedSpace a x₀ ρ):ℝ) *
-      (LinearMap.trace ℂ V ((A^(2*r+1)).toLinearMap)).re ≤
-      ∑ x, ∑ y, ((P^r) x y)^2 := by
-  let W := fixedSpace a x₀ ρ
-  let B := stdOrthonormalBasis ℂ V
-  let C := stdOrthonormalBasis ℂ W
-  let w : Fin (Module.finrank ℂ W) → V := fun i => (C i : V)
-  have hfix (i) : IsFixed a x₀ ρ (w i) := (C i).2
-  have hw : Orthonormal ℂ w := C.orthonormal.comp_linearIsometry W.subtypeₗᵢ
-  let v : Fin (Module.finrank ℂ W) × Fin (Module.finrank ℂ V) → EuclideanSpace ℂ X :=
-    fun ij => normalizedCoefficient a x₀ ha ρ (w ij.1) (B ij.2)
-  have hv : Orthonormal ℂ v := by
-    rw [orthonormal_iff_ite]
-    intro i j
-    dsimp [v]
-    rw [normalizedCoefficient_inner a x₀ ha ρ hρ _ _ (hfix _) (hfix _)]
-    rw [(orthonormal_iff_ite.mp hw), (orthonormal_iff_ite.mp B.orthonormal)]
-    by_cases h1 : i.1=j.1 <;> by_cases h2 : i.2=j.2 <;>
-      simp [h1,h2,Prod.ext_iff,eq_comm]
-  have hp (s : ℕ) (i) (t : V) :
-      DensityTransfer.applyKernel (fun x y => (P^s) x y) (normalizedCoefficient a x₀ ha ρ (w i) t) =
-        normalizedCoefficient a x₀ ha ρ (w i) ((A^s) t) := by
-    induction s with
-    | zero => simp [applyKernel_one]
-    | succ s ih =>
-      rw [pow_succ',applyKernel_mul,ih,hmap _ (hfix _),pow_succ']
-      rfl
-  have hhs := DensityTransfer.hilbert_schmidt_family (fun x y => (P^r) x y) v hv
-  have he : (∑ ij, ‖DensityTransfer.applyKernel (fun x y => (P^r) x y) (v ij)‖^2) =
-      (Module.finrank ℂ W:ℝ) * ∑ j, ‖(A^r) (B j)‖^2 := by
-    simp only [v,hp,normalizedCoefficient_norm a x₀ ha ρ hρ _ (hfix _)
-      (hw.norm_eq_one _),Fintype.sum_prod_type]
-    simp
-  rw [he] at hhs
-  exact (mul_le_mul_of_nonneg_left (trace_power_le_hs A hs hn r) (by positivity)).trans hhs
-
-end Thorp.Casimir
-
-namespace Thorp.Casimir
-open scoped BigOperators Classical
-open UnitaryFinite
-
-section
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-noncomputable def tupleKernel (d k : ℕ) : Matrix (Fin k ↪ Card d) (Fin k ↪ Card d) ℝ :=
-  fun x y => PairRouting.tupleProbability (palindromePerm d) x y
-
-lemma palindrome_trace_amplification (d k : ℕ) (x₀ : Fin k ↪ Card d)
-    (ρ : Representation ℂ (Equiv.Perm (Card d)) V) [Representation.IsIrreducible ρ]
-    (hρ : IsUnitary ρ) (r : ℕ) :
-    (Module.finrank ℂ (fixedSpace tupleAction x₀ ρ):ℝ) *
-      (LinearMap.trace ℂ V (((sampleOperator ρ (palindromePerm d))^(2*r+1)).toLinearMap)).re ≤
-    ∑ x, ∑ y, ((tupleKernel d k)^r) x y ^ 2 := by
-  let ha := tupleAction_transitive x₀
-  have hmap (w : V) (hw : IsFixed tupleAction x₀ ρ w) (v : V) :
-      DensityTransfer.applyKernel (fun x y : Fin k ↪ Card d =>
-        PairRouting.tupleProbability (palindromePerm d) x y)
-        (normalizedCoefficient tupleAction x₀ ha ρ w v) =
-      normalizedCoefficient tupleAction x₀ ha ρ w (sampleOperator ρ (palindromePerm d) v) := by
-    have hh := coefficient_random tupleAction x₀ ha ρ hρ (palindromePerm d) w hw v
-    have he : actionKernel tupleAction (palindromePerm d) =
-        fun x y : Fin k ↪ Card d => PairRouting.tupleProbability (palindromePerm d) x y := by
-      funext x y
-      exact tupleAction_kernel _ x y
-    rw [he,←sampleOperator_inv_apply,sampleOperator_inv _ hρ,
-      (palindrome_selfadjoint d ρ hρ).adjoint_eq] at hh
-    exact hh
-  exact coefficient_trace_amplification tupleAction x₀ ha ρ hρ _ _ hmap
-    (palindrome_selfadjoint d ρ hρ) (sampleOperator_norm_le ρ hρ _) r
-
-end
-
-lemma tupleKernel_symm (d k : ℕ) : (tupleKernel d k).IsSymm := by
-  ext x y
-  exact palindrome_kernel_symmetric d y x
-
-lemma tupleKernel_power64 (d k : ℕ) (x y : Fin k ↪ Card d) :
-    ((tupleKernel d k)^64) x y ≤
-      Real.exp ((64*(1+heightDecay 1))*k)/(Fintype.card (Fin k ↪ Card d):ℝ) := by
-  let : Nonempty (Fin k ↪ Card d) := ⟨x⟩
-  have hr (x : Fin k ↪ Card d) : ∑ y, tupleKernel d k x y = 1 :=
-    PairRouting.tupleProbability_row_sum _ x
-  have hc (y : Fin k ↪ Card d) : ∑ x, tupleKernel d k x y = 1 := by
-    simp_rw [(tupleKernel_symm d k).apply y]
-    exact hr y
-  have hm (x : Fin k ↪ Card d) :
-      finiteMean (fun y => ((Fintype.card (Fin k ↪ Card d):ℝ)*tupleKernel d k x y)^
-        (1+(1/32:ℝ))) ≤ Real.exp ((k:ℝ)*(1+heightDecay 1)) := by
-    simpa only [palindromeRowMoment,Fintype.card_fin,tupleKernel] using
-      row_moment_coarse d x (1/32) (by norm_num) (by norm_num)
-  have hnum : 2 ≤ (1+(1/32:ℝ))^(32:ℕ) := by
-    have hh := one_add_mul_le_pow (a := (1/32:ℝ)) (by norm_num) 32
-    norm_num at hh ⊢
-  have hh := StrongSmoothing.symmetric_kernel_smoothing (tupleKernel d k)
-    (fun x y => PairRouting.tupleProbability_nonneg _ x y) hr hc (tupleKernel_symm d k)
-    (p := 1+(1/32:ℝ)) (A := k*(1+heightDecay 1)) (by norm_num)
-    (mul_nonneg (Nat.cast_nonneg _) (by linarith [heightDecay_nonneg 1])) hm 32 hnum x y
-  have he : 2*((k:ℝ)*(1+heightDecay 1))*(32:ℝ) = (64*(1+heightDecay 1))*k := by ring
-  simpa only [show 2*32=64 by norm_num,Nat.cast_ofNat,he] using hh
-
-lemma K_selfadjoint (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ) :
-    @IsSelfAdjoint _ (ContinuousLinearMap.instStarId (𝕜:=ℂ) (E:=Specht.hilbertSpace μ)) (K d μ e) := by
-  exact palindrome_selfadjoint (V:=Specht.hilbertSpace μ) d _ (Specht.relabelledUnitary_unitary μ e)
-
-lemma K_norm_le_one (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ) :
-    ‖K d μ e‖ ≤ 1 :=
-  sampleOperator_norm_le _ (Specht.relabelledUnitary_unitary μ e) _
-
-lemma fixed_trace_bound (d k : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ)
-    (x₀ : Fin k ↪ Card d) :
-    (Module.finrank ℂ (fixedSpace (V := Specht.hilbertSpace μ) tupleAction x₀
-      (Specht.relabelledUnitary μ e)):ℝ) * traceMoment d μ e 65 ≤
-        Real.exp ((64*(1+heightDecay 1))*k) := by
-  let ρ := Specht.relabelledUnitary μ e
-  let := Specht.relabelledUnitary_irreducible μ e
-  have hh := palindrome_trace_amplification (V:=Specht.hilbertSpace μ) d k x₀ ρ
-    (Specht.relabelledUnitary_unitary μ e) 32
-  change _ * traceMoment d μ e 65 ≤ _ at hh
-  rw [symmetric_hs_power _ (tupleKernel_symm d k)] at hh
-  apply hh.trans
-  have hX : (0:ℝ) < Fintype.card (Fin k ↪ Card d) := by
-    let : Nonempty (Fin k ↪ Card d) := ⟨x₀⟩
-    exact Nat.cast_pos.mpr Fintype.card_pos
-  calc
-    _ ≤ ∑ x : Fin k ↪ Card d,
-        Real.exp ((64*(1+heightDecay 1))*k)/(Fintype.card (Fin k ↪ Card d):ℝ) :=
-      Finset.sum_le_sum (fun x _ => tupleKernel_power64 d k x x)
-    _ = _ := by
-      simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
-      exact mul_div_cancel₀ _ (ne_of_gt hX)
-
-lemma levelScale_ge (n k : ℕ) (hk : k ≤ n) : (k:ℝ) ≤ levelScale n k := by
-  by_cases h : k=0
-  · simp [h,levelScale]
-  have hkp : (0:ℝ)<k := by exact_mod_cast Nat.pos_of_ne_zero h
-  have hlog : 0 ≤ Real.log ((n:ℝ)/k) := Real.log_nonneg ((one_le_div hkp).mpr (by exact_mod_cast hk))
-  dsimp [levelScale]
-  nlinarith
-
-lemma choose_le_exp_levelScale (n k : ℕ) (hn : 0<n) (hk : k≤n) :
-    (n.choose k:ℝ) ≤ Real.exp (levelScale n k) := by
-  by_cases h : k=0
-  · simp [h,levelScale]
-  have hkp : 0<k := Nat.pos_of_ne_zero h
-  have hh := DimensionEstimates.log_choose_upper n k hn hk
-  have hf := DimensionEstimates.log_factorial_lower k hkp
-  have he : Real.log ((n:ℝ)/k) = Real.log n-Real.log k :=
-    Real.log_div (by positivity) (by positivity)
-  have hl : Real.log (n.choose k:ℝ) ≤ levelScale n k := by
-    dsimp [levelScale]
-    rw [he]
-    linarith
-  rw [←Real.exp_log (by exact_mod_cast Nat.choose_pos hk : (0:ℝ)<n.choose k)]
-  exact Real.exp_le_exp.mpr hl
-
-lemma trace_bound (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ) :
-    (Module.finrank ℂ (Specht.space μ):ℝ)*traceMoment d μ e 65 ≤
-      Real.exp ((1+64*(1+heightDecay 1))*levelScale (2^d) (μ.card-μ.rowLen 0)) := by
-  let k := μ.card-μ.rowLen 0
-  have hn : μ.card=2^d := by
-    rw [←Specht.card_cell,←Fintype.card_congr e,card_positions]
-  have hk : k≤2^d := by dsimp [k]; omega
-  let x₀ : Fin k ↪ Card d := (Fin.castLEEmb (show k≤Fintype.card (Card d) by simpa [card_positions] using hk)).trans
-    (Fintype.equivFin (Card d)).symm.toEmbedding
-  let m := Module.finrank ℂ (fixedSpace (V := Specht.hilbertSpace μ) tupleAction x₀
-    (Specht.relabelledUnitary μ e))
-  have hd : (Module.finrank ℂ (Specht.space μ):ℝ) ≤ (m:ℝ)*((2^d).choose k:ℝ) := by
-    have hh := Specht.relabelled_tuple_dimension μ e x₀
-    simp only [Fintype.card_fin] at hh
-    change Module.finrank ℂ (Specht.hilbertSpace μ) * k.choose k ≤ m * μ.card.choose k at hh
-    rw [Nat.choose_self,mul_one,Specht.finrank_hilbertSpace,hn] at hh
-    exact_mod_cast hh
-  have hm := fixed_trace_bound d k μ e x₀
-  by_cases ht : 0≤traceMoment d μ e 65
-  · calc
-      _ ≤ ((m:ℝ)*((2^d).choose k:ℝ))*traceMoment d μ e 65 :=
-        mul_le_mul_of_nonneg_right hd ht
-      _ = ((2^d).choose k:ℝ)*((m:ℝ)*traceMoment d μ e 65) := by ring
-      _ ≤ ((2^d).choose k:ℝ)*Real.exp ((64*(1+heightDecay 1))*k) :=
-        mul_le_mul_of_nonneg_left hm (Nat.cast_nonneg _)
-      _ ≤ Real.exp (levelScale (2^d) k)*Real.exp ((64*(1+heightDecay 1))*levelScale (2^d) k) := by
-        apply mul_le_mul (choose_le_exp_levelScale _ _ (by positivity) hk) _ (by positivity) (Real.exp_nonneg _)
-        apply Real.exp_le_exp.mpr
-        exact mul_le_mul_of_nonneg_left (levelScale_ge _ _ hk) (by linarith [heightDecay_nonneg 1])
-      _ = _ := by rw [←Real.exp_add]; congr 1; ring
-  · exact (mul_nonpos_of_nonneg_of_nonpos (Nat.cast_nonneg _) (le_of_not_ge ht)).trans (Real.exp_nonneg _)
-
-end Thorp.Casimir
-
-namespace Thorp.Casimir
-open scoped BigOperators Classical
-open Filter UnitaryFinite DimensionEstimates
-
-lemma levelScale_le (n k : ℕ) (hn : 0<n) (_ : k≤n) : levelScale n k ≤ n := by
-  by_cases hz : k=0
-  · simp [hz,levelScale]
-  have hkp : (0:ℝ)<k := by exact_mod_cast Nat.pos_of_ne_zero hz
-  have hnp : (0:ℝ)<n := by exact_mod_cast hn
-  have hh := Real.log_le_sub_one_of_pos (div_pos hnp hkp)
-  have hm := mul_le_mul_of_nonneg_left hh hkp.le
-  have he : (k:ℝ)*((n:ℝ)/k)=n := mul_div_cancel₀ _ (ne_of_gt hkp)
-  dsimp [levelScale]
-  nlinarith
-
-lemma nontrivial_uniform_zero (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ)
-    (hk : 0<μ.card-μ.rowLen 0) :
-    uniformOperator (V:=Specht.hilbertSpace μ) (Specht.relabelledUnitary μ e)=0 := by
-  apply ContinuousLinearMap.ext
-  intro v
-  apply SparseContact.point_relabelled_fixed_zero μ e ∅ (by simpa using hk)
-  intro g _
-  have he := congrArg (fun A : Specht.hilbertSpace μ →L[ℂ] Specht.hilbertSpace μ => A v)
-    (uniformOperator_left (V:=Specht.hilbertSpace μ) (Specht.relabelledUnitary μ e) g)
-  exact he
-
-lemma gap_of_uniform_zero {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-    [FiniteDimensional ℂ V] (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V)
-    (hρ : IsUnitary ρ) (hz : uniformOperator ρ=0) :
-    ‖sampleOperator ρ (palindromePerm d)‖ ≤ blockGap d := by
-  have hh := palindrome_off_invariants_gap d ρ hρ
-  rw [hz,sub_zero,←sampleOperator_weighted] at hh
-  exact hh
-
-lemma nontrivial_gap (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ)
-    (hk : 0<μ.card-μ.rowLen 0) : ‖K d μ e‖ ≤ blockGap d := by
-  exact gap_of_uniform_zero (V:=Specht.hilbertSpace μ) d _
-    (Specht.relabelledUnitary_unitary μ e) (nontrivial_uniform_zero d μ e hk)
-
-lemma eventual_level_dimension :
-    ∃ η : ℝ, 0<η ∧ ∀ᶠ d : ℕ in atTop,
-      ∀ μ : YoungDiagram, μ.card=2^d → 0<μ.card-μ.rowLen 0 →
-        (μ.colLen 0:ℝ)≤(μ.card:ℝ)/2 →
-        η*levelScale (2^d) (μ.card-μ.rowLen 0) ≤
-          Real.log (Module.finrank ℂ (Specht.space μ)) := by
-  obtain ⟨h,hh,hm⟩ := Specht.medium_dimension (1/16) (by norm_num) (by norm_num)
-  refine ⟨min (1/3) h,lt_min (by norm_num) hh,?_⟩
-  filter_upwards [twopow_nat_tendsto.eventually hm] with d hd
-  intro μ hμ hk hc
-  let k := μ.card-μ.rowLen 0
-  have hkn : k≤2^d := by dsimp [k]; omega
-  have hΛ : 0≤levelScale (2^d) k := (Nat.cast_nonneg k).trans (levelScale_ge _ _ hkn)
-  have hr : ((μ.card-μ.rowLen 0:ℕ):ℝ)=(μ.card:ℝ)-μ.rowLen 0 :=
-    Nat.cast_sub (Specht.rowLen_zero_le_card μ)
-  by_cases hs : (k:ℝ)≤(2^d:ℕ)/16
-  · have hk2 : 2*k≤2^d := by exact_mod_cast (show (2:ℝ)*k≤(2^d:ℕ) by linarith)
-    have hdim := DimensionEstimates.near_row_log_dimension (2^d) k _ hk hk2
-      (by simpa only [←hμ] using Specht.choose_tail_le_dimension μ)
-    have hkp : (0:ℝ)<k := by exact_mod_cast hk
-    have hratio : (16:ℝ)≤(2^d:ℕ)/k := (le_div_iff₀ hkp).mpr (by linarith)
-    have hlog : 4*Real.log 2 ≤ Real.log ((2^d:ℕ)/k) := by
-      have ht := Real.log_le_log (by norm_num : (0:ℝ)<16) hratio
-      rw [show (16:ℝ)=2^(4:ℕ) by norm_num,Real.log_pow] at ht
-      simpa using ht
-    have hlog2 : (1/2:ℝ)≤Real.log 2 := by linarith [Real.log_two_gt_d9]
-    have hL : levelScale (2^d) k ≤ 3*Real.log (Module.finrank ℂ (Specht.space μ)) := by
-      dsimp [levelScale]
-      nlinarith
-    have hη := mul_le_mul_of_nonneg_right (min_le_left (1/3:ℝ) h) hΛ
-    nlinarith
-  · have hdim := hd μ hμ (by rw [←hμ,←hr]; change (1/16:ℝ)*(μ.card:ℝ)≤(k:ℝ); rw [hμ]; linarith)
-      (by simpa only [←hμ] using hc)
-    have hL := levelScale_le (2^d) k (by positivity) hkn
-    have ht := mul_le_mul_of_nonneg_left hL hh.le
-    have hη := mul_le_mul_of_nonneg_right (min_le_right (1/3:ℝ) h) hΛ
-    nlinarith
-
-lemma eventual_level_contraction :
-    ∃ a : ℝ, 0<a ∧ ∀ᶠ d : ℕ in atTop,
-      ∀ (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ), 0<μ.card-μ.rowLen 0 →
-        ‖K d μ e‖ ≤ Real.exp (-a*levelScale (2^d) (μ.card-μ.rowLen 0)) := by
-  obtain ⟨a,h,ha,hh,hsp⟩ := Specht.spectral_regimes
-  obtain ⟨η,hη,hdim⟩ := eventual_level_dimension
-  refine ⟨a*η,mul_pos ha hη,?_⟩
-  filter_upwards [hsp,hdim,eventually_ge_atTop 1] with d hs hd hd1
-  intro μ e hk
-  by_cases ht : (μ.card:ℝ)/2 < μ.colLen 0
-  · have hz := Specht.palindrome_tall_zero_general d (by omega) μ e ht
-    change K d μ e=0 at hz
-    rw [hz,@norm_zero (Specht.hilbertSpace μ →L[ℂ] Specht.hilbertSpace μ) _]
-    exact (Real.exp_pos _).le
-  · have hc := le_of_not_gt ht
-    have hμ : μ.card=2^d := by rw [←Specht.card_cell,←Fintype.card_congr e,card_positions]
-    apply (hs μ e hk hc).1.trans
-    apply Real.exp_le_exp.mpr
-    have hh := mul_le_mul_of_nonneg_left (hd μ hμ hk hc) ha.le
-    nlinarith
-
-lemma finite_positive_lower (f : ℕ → ℝ) (D : ℕ) (hf : ∀ d<D,0<f d) :
-    ∃ b : ℝ, 0<b ∧ ∀ d<D,b≤f d := by
-  induction D with
-  | zero => exact ⟨1,by norm_num,by omega⟩
-  | succ D ih =>
-    obtain ⟨b,hb,hh⟩ := ih (fun d hd => hf d (by omega))
-    refine ⟨min b (f D),lt_min hb (hf D (by omega)),?_⟩
-    intro d hd
-    by_cases he : d=D
-    · subst d; exact min_le_right _ _
-    · exact (min_le_left _ _).trans (hh d (by omega))
-
-theorem level_contraction :
-    ∃ a : ℝ, 0<a ∧ ∀ (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ),
-      0<μ.card-μ.rowLen 0 →
-        ‖K d μ e‖ ≤ Real.exp (-a*levelScale (2^d) (μ.card-μ.rowLen 0)) := by
-  obtain ⟨a,ha,he⟩ := eventual_level_contraction
-  obtain ⟨D,hD⟩ := eventually_atTop.mp he
-  let f (d : ℕ) : ℝ := -Real.log (blockGap d)/(2:ℝ)^d
-  have hf (d : ℕ) : 0<f d := by
-    have hl := Real.log_neg (blockGap_pos d) (blockGap_lt_one d)
-    exact div_pos (neg_pos.mpr hl) (by positivity)
-  obtain ⟨b,hb,hbD⟩ := finite_positive_lower f D (fun d _ => hf d)
-  refine ⟨min a b,lt_min ha hb,?_⟩
-  intro d μ e hk
-  have hμ : μ.card=2^d := by rw [←Specht.card_cell,←Fintype.card_congr e,card_positions]
-  have hkn : μ.card-μ.rowLen 0≤2^d := by omega
-  have hΛ : 0≤levelScale (2^d) (μ.card-μ.rowLen 0) :=
-    (Nat.cast_nonneg _).trans (levelScale_ge _ _ hkn)
-  by_cases hd : D≤d
-  · apply (hD d hd μ e hk).trans (Real.exp_le_exp.mpr _)
-    have hh := mul_le_mul_of_nonneg_right (min_le_left a b) hΛ
-    linarith
-  · have hab : min a b≤f d := (min_le_right _ _).trans (hbD d (by omega))
-    have hscale : levelScale (2^d) (μ.card-μ.rowLen 0) ≤ (2:ℝ)^d := by
-      simpa only [Nat.cast_pow,Nat.cast_ofNat] using levelScale_le (2^d) _ (by positivity) hkn
-    have hprod := mul_le_mul_of_nonneg_right hab (by positivity : (0:ℝ)≤(2:ℝ)^d)
-    have hfprod : f d*(2:ℝ)^d = -Real.log (blockGap d) := by dsimp [f]; field_simp
-    rw [hfprod] at hprod
-    have hs := mul_le_mul_of_nonneg_left hscale (le_of_lt (lt_min ha hb))
-    calc
-      _ ≤ blockGap d := nontrivial_gap d μ e hk
-      _ = Real.exp (Real.log (blockGap d)) := (Real.exp_log (blockGap_pos d)).symm
-      _ ≤ _ := Real.exp_le_exp.mpr (by linarith)
-
-end Thorp.Casimir
-
-namespace Thorp.Specht
-open scoped BigOperators Classical
-open UnitaryFinite
-
-noncomputable def cardLabels (d : ℕ) (μ : Shapes (2^d)) : Card d ≃ Cell μ.1 :=
-  ((Fintype.equivFin (Card d)).trans (finCongr (card_positions d))).trans (labels μ).symm
-
-noncomputable def cardRepresentation (d : ℕ) (μ : Shapes (2^d)) :
-    Representation ℂ (Equiv.Perm (Card d)) (hilbertSpace μ.1) :=
-  relabelledUnitary μ.1 (cardLabels d μ)
-
-lemma cardRepresentation_unitary (d : ℕ) (μ : Shapes (2^d)) :
-    IsUnitary (V := hilbertSpace μ.1) (cardRepresentation d μ) :=
-  relabelledUnitary_unitary _ _
-
-noncomputable instance cardRepresentation_irreducible (d : ℕ) (μ : Shapes (2^d)) :
-    Representation.IsIrreducible (cardRepresentation d μ) :=
-  relabelledUnitary_irreducible _ _
-
-lemma relabelledUnitary_neq {α : Type*} (μ ν : YoungDiagram)
-    (e : α ≃ Cell μ) (f : α ≃ Cell ν)
-    (T : Representation.Equiv (relabelledUnitary μ e) (relabelledUnitary ν f)) : μ=ν := by
-  let S := (spaceHilbertEquiv μ).trans (T.toLinearEquiv.trans (spaceHilbertEquiv ν).symm)
-  apply shapes_eq_of_equivariant_equiv μ ν (e.symm.trans f) S
-  intro p v
-  have hT := Representation.IntertwiningMap.isIntertwining _ _ T.toIntertwiningMap
-    (e.symm.permCongr p) (spaceHilbertEquiv μ v)
-  change T.toLinearEquiv (spaceHilbertEquiv μ
-    (representation μ (e.permCongr (e.symm.permCongr p))
-      ((spaceHilbertEquiv μ).symm (spaceHilbertEquiv μ v))))=
-    spaceHilbertEquiv ν (representation ν (f.permCongr (e.symm.permCongr p))
-      ((spaceHilbertEquiv ν).symm (T.toLinearEquiv (spaceHilbertEquiv μ v)))) at hT
-  have he : e.permCongr (e.symm.permCongr p)=p := by
-    apply Equiv.ext
-    intro x
-    simp only [Equiv.permCongr_apply,Equiv.symm_symm,e.apply_symm_apply]
-  have hf : f.permCongr (e.symm.permCongr p)=(e.symm.trans f).permCongr p := by
-    apply Equiv.ext
-    intro x
-    rfl
-  rw [he,hf,LinearEquiv.symm_apply_apply] at hT
-  have ht := congrArg (spaceHilbertEquiv ν).symm hT
-  simpa only [LinearEquiv.symm_apply_apply,S,LinearEquiv.trans_apply] using ht
-
-lemma cardRepresentation_distinct (d : ℕ) (μ ν : Shapes (2^d)) (hne : μ≠ν) :
-    ¬Nonempty (Representation.Equiv (cardRepresentation d μ) (cardRepresentation d ν)) := by
-  rintro ⟨T⟩
-  exact hne (Subtype.ext (relabelledUnitary_neq μ.1 ν.1 (cardLabels d μ) (cardLabels d ν) T))
-
-lemma card_conjugacy_shapes (d : ℕ) :
-    Nat.card (ConjClasses (Equiv.Perm (Card d)))≤Fintype.card (Shapes (2^d)) := by
-  have h := Nat.card_le_card_of_injective _ (conjugacyPartition_injective (α := Card d))
-  simpa only [card_positions,Nat.card_eq_fintype_card,Fintype.card_congr (shapePartitionEquiv (2^d))] using h
-
-lemma row_zero_of_full_row (μ : YoungDiagram) (hμ : μ.rowLen 0=μ.card) (x : Cell μ) : row x=0 := by
-  have hr : μ.row 0=μ.cells := Finset.eq_of_subset_of_card_le
-    (Finset.filter_subset _ _) (by rw [←μ.rowLen_eq_card]; exact hμ.ge)
-  have hx : x.1∈μ.row 0 := by rw [hr]; exact x.2
-  exact (YoungDiagram.mem_row_iff.mp hx).2
-
-lemma tabloid_subsingleton_of_full_row (μ : YoungDiagram) (hμ : μ.rowLen 0=μ.card) :
-    Subsingleton (Tabloid μ) := by
-  refine ⟨fun f g => ?_⟩
-  apply Subtype.ext
-  funext x
-  obtain ⟨p,hp⟩ := f.2
-  obtain ⟨q,hq⟩ := g.2
-  simp only [hp,hq,Function.comp_apply]
-  apply Fin.ext
-  exact (row_zero_of_full_row μ hμ (p x)).trans (row_zero_of_full_row μ hμ (q x)).symm
-
-lemma representation_full_row (μ : YoungDiagram) (hμ : μ.rowLen 0=μ.card)
-    (p : Equiv.Perm (Cell μ)) : representation μ p=1 := by
-  let := tabloid_subsingleton_of_full_row μ hμ
-  apply LinearMap.ext
-  intro v
-  apply Subtype.ext
-  funext f
-  change v.1 (tabloidAct p⁻¹ f)=v.1 f
-  rw [show tabloidAct p⁻¹ f=f from Subsingleton.elim _ _]
-
-lemma relabelled_full_row {α : Type*} (μ : YoungDiagram) (hμ : μ.rowLen 0=μ.card)
-    (e : α ≃ Cell μ) (p : Equiv.Perm α) : relabelledUnitary μ e p=1 := by
-  change ((spaceHilbertEquiv μ).conjAlgEquiv ℂ) (representation μ (e.permCongr p))=1
-  rw [representation_full_row μ hμ,map_one]
-
-end Thorp.Specht
-namespace Thorp.UnitaryFinite
-open scoped BigOperators Classical
-variable {G Ω V : Type*} [Group G] [Fintype G] [Fintype Ω] [Nonempty Ω]
-    [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-lemma sample_uniform_left (ρ : Representation ℂ G V) (P : Ω → G) :
-    sampleOperator ρ P*uniformOperator ρ=uniformOperator ρ := by
-  rw [sampleOperator_weighted,weighted_uniform_left,sampleLaw_sum,one_smul]
-
-lemma uniform_zero_of_sample_norm_lt_one (ρ : Representation ℂ G V) (P : Ω → G)
-    (h : ‖sampleOperator ρ P‖<1) : uniformOperator ρ=0 := by
-  have hn := norm_mul_le (sampleOperator ρ P) (uniformOperator ρ)
-  rw [sample_uniform_left] at hn
-  apply norm_eq_zero.mp
-  nlinarith [norm_nonneg (uniformOperator ρ)]
-
-omit [Fintype G] in
-lemma sampleOperator_trivial (ρ : Representation ℂ G V) (hρ : ∀ g,ρ g=1) (P : Ω → G) :
-    sampleOperator ρ P=1 := by
-  convert sampleOperator_one (Ω := Ω) ρ using 1
-  unfold sampleOperator
-  congr 1
-  apply Finset.sum_congr rfl
-  intro ω _
-  apply ContinuousLinearMap.ext
-  intro v
-  change ρ (P ω) v=ρ 1 v
-  rw [hρ,hρ]
-
-lemma uniformOperator_trivial (ρ : Representation ℂ G V) (hρ : ∀ g,ρ g=1) :
-    uniformOperator ρ=1 := by
-  rw [←sampleOperator_uniform]
-  exact sampleOperator_trivial ρ hρ id
-
-lemma weighted_sub (ρ : Representation ℂ G V) (p q : G → ℝ) :
-    weightedOperator ρ (fun g => p g-q g)=weightedOperator ρ p-weightedOperator ρ q := by
-  simp only [weightedOperator,Complex.ofReal_sub,sub_smul,Finset.sum_sub_distrib]
-
-end Thorp.UnitaryFinite
-
-namespace Thorp
-open scoped BigOperators Classical
-open UnitaryFinite
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-lemma physical_palindrome_norm (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V)
-    (hρ : IsUnitary ρ) :
-    ‖sampleOperator ρ (run d d)‖^2=‖sampleOperator ρ (palindromePerm d)‖ := by
-  rw [physical_sweep_adjoint d ρ hρ,sampleOperator_palindrome d ρ hρ]
-  have h := ContinuousLinearMap.norm_adjoint_comp_self
-    ((sampleOperator ρ (fun ω => butterflyPerm d (decodeButterfly d ω))).adjoint)
-  rw [ContinuousLinearMap.adjoint_adjoint] at h
-  simpa only [ContinuousLinearMap.mul_def,←sq] using h.symm
-
-lemma physical_power_bound (d C : ℕ) (hC : 0 < C) (ρ : Representation ℂ (Equiv.Perm (Card d)) V)
-    (hρ : IsUnitary ρ) :
-    ‖sampleOperator ρ (run d (C*d))‖^2≤‖sampleOperator ρ (palindromePerm d)‖^C := by
-  have he : sampleOperator ρ (run d (C*d))=(sampleOperator ρ (run d d))^C := by
-    simp only [sampleOperator_run,←pow_mul,Nat.mul_comm C d]
-  rw [he,←physical_palindrome_norm d ρ hρ]
-  calc
-    _ ≤ (‖sampleOperator ρ (run d d)‖^C)^2 :=
-      pow_le_pow_left₀ (norm_nonneg _) (norm_pow_le' _ hC) _
-    _ = _ := by rw [←pow_mul,←pow_mul,Nat.mul_comm C 2]
-
-end Thorp
-namespace Thorp.Specht
-open scoped BigOperators Classical
-
-section
-open UnitaryFinite
-
-lemma card_fourier_decay (d C : ℕ) (μ : Shapes (2^d)) (a s : ℝ)
-    (ha : 0 < a) (hs : 0 ≤ s) (hC : s+2 ≤ a*C)
-    (hD : 1≤Real.log (Module.finrank ℂ (space μ.1)))
-    (hK : ‖sampleOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) (palindromePerm d)‖≤
-      Real.exp (-a*Real.log (Module.finrank ℂ (space μ.1)))) :
-    (Module.finrank ℂ (hilbertSpace μ.1):ℝ)^2*
-      ‖sampleOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) (run d (C*d))-uniformOperator (V := hilbertSpace μ.1) (cardRepresentation d μ)‖^2≤
-        Real.exp (-s*Real.log (Module.finrank ℂ (space μ.1))) := by
-  let : NormedAddCommGroup (hilbertSpace μ.1) := inferInstance
-  let : InnerProductSpace ℂ (hilbertSpace μ.1) := inferInstance
-  let : FiniteDimensional ℂ (hilbertSpace μ.1) := inferInstance
-  have hu : uniformOperator (V := hilbertSpace μ.1) (cardRepresentation d μ)=0 :=
-    uniform_zero_of_sample_norm_lt_one _ (palindromePerm d) (hK.trans_lt (by
-      rw [Real.exp_lt_one_iff]; nlinarith))
-  rw [hu,sub_zero,finrank_hilbertSpace]
-  have hCp : 0 < C := by
-    by_contra hn
-    have hz : C=0 := by omega
-    rw [hz,Nat.cast_zero,mul_zero] at hC
-    linarith
-  have hp := physical_power_bound d C hCp (cardRepresentation d μ) (cardRepresentation_unitary d μ)
-  have hkpow := pow_le_pow_left₀ (norm_nonneg _) hK C
-  have hDp : (0:ℝ)<Module.finrank ℂ (space μ.1) := by exact_mod_cast dimension_pos μ.1
-  calc
-    _ ≤ (Module.finrank ℂ (space μ.1):ℝ)^2*
-        (Real.exp (-a*Real.log (Module.finrank ℂ (space μ.1))))^C :=
-      mul_le_mul_of_nonneg_left (hp.trans hkpow) (sq_nonneg _)
-    _ = Real.exp ((2-a*C)*Real.log (Module.finrank ℂ (space μ.1))) := by
-      rw [←Real.exp_log hDp,←Real.exp_nat_mul,←Real.exp_nat_mul,←Real.exp_add,Real.log_exp]
-      congr 1
-      ring
-    _ ≤ _ := Real.exp_le_exp.mpr (by nlinarith)
-
-lemma card_fourier_tall (d C : ℕ) (hd : 0<d) (hC : 0<C) (μ : Shapes (2^d))
-    (hc : (μ.1.card:ℝ)/2<(μ.1.colLen 0:ℝ)) :
-    sampleOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) (run d (C*d))-uniformOperator (V := hilbertSpace μ.1) (cardRepresentation d μ)=0 := by
-  let : NormedAddCommGroup (hilbertSpace μ.1) := inferInstance
-  let : InnerProductSpace ℂ (hilbertSpace μ.1) := inferInstance
-  let : FiniteDimensional ℂ (hilbertSpace μ.1) := inferInstance
-  have hk : sampleOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) (palindromePerm d)=0 :=
-    palindrome_tall_zero_general d hd μ.1 (cardLabels d μ) hc
-  have hu : uniformOperator (V := hilbertSpace μ.1) (cardRepresentation d μ)=0 :=
-    uniform_zero_of_sample_norm_lt_one _ (palindromePerm d) (by rw [hk,norm_zero]; norm_num)
-  rw [hu,sub_zero]
-  apply norm_eq_zero.mp
-  have hp := physical_power_bound d C hC (cardRepresentation d μ) (cardRepresentation_unitary d μ)
-  rw [hk,norm_zero,zero_pow (by omega : C≠0)] at hp
-  nlinarith [norm_nonneg (sampleOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) (run d (C*d)))]
-
-lemma card_fourier_full_row (d t : ℕ) (μ : Shapes (2^d)) (hj : μ.1.card-μ.1.rowLen 0=0) :
-    sampleOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) (run d t)-uniformOperator (V := hilbertSpace μ.1) (cardRepresentation d μ)=0 := by
-  have hr : μ.1.rowLen 0=μ.1.card := by have := rowLen_zero_le_card μ.1; omega
-  have ht (p : Equiv.Perm (Card d)) : cardRepresentation d μ p=1 :=
-    relabelled_full_row μ.1 hr (cardLabels d μ) p
-  rw [sampleOperator_trivial _ ht,uniformOperator_trivial _ ht,sub_self]
-
-end
-
-lemma half_geom_sum (n : ℕ) : (∑ j∈Finset.range n,(1/2:ℝ)^j)≤2 := by
-  have he : (∑ j∈Finset.range n,(1/2:ℝ)^j)=2*(1-(1/2:ℝ)^n) := by
-    induction n with
-    | zero => simp
-    | succ n ih => rw [Finset.sum_range_succ,ih,pow_succ]; ring
-  rw [he]
-  nlinarith [pow_nonneg (by norm_num : (0:ℝ)≤1/2) n]
-
-lemma shape_weight_sum (N : ℕ) :
-    (∑ μ : Shapes N,(1/4:ℝ)^(μ.1.card-μ.1.rowLen 0))≤2 := by
-  have he := Finset.sum_fiberwise_of_maps_to'
-    (s := (Finset.univ : Finset (Shapes N))) (t := Finset.range (N+1))
-    (g := fun μ : Shapes N => μ.1.card-μ.1.rowLen 0)
-    (by intro μ _; simp only [Finset.mem_range]; have := μ.2; omega)
-    (fun j => (1/4:ℝ)^j)
-  rw [←he]
-  apply le_trans _ (half_geom_sum (N+1))
-  apply Finset.sum_le_sum
-  intro j _
-  have hc : ((Finset.univ.filter (fun μ : Shapes N => μ.1.card-μ.1.rowLen 0=j)).card:ℝ)≤(2:ℝ)^j := by
-    have hh := tail_shapes_count N j
-    rw [Fintype.card_subtype] at hh
-    exact_mod_cast hh
-  simp only [Finset.sum_const,nsmul_eq_mul]
-  calc
-    _ ≤ (2:ℝ)^j*(1/4:ℝ)^j := mul_le_mul_of_nonneg_right hc (by positivity)
-    _ = (1/2:ℝ)^j := by rw [←mul_pow]; norm_num
-
-lemma near_shape_term (N j : ℕ) (L s : ℝ) (hN : 0<N) (hs : 4≤ s)
-    (hL : 0≤L) (hl : (1/2:ℝ)*Real.log N≤L) (hj : (j:ℝ)≤L) :
-    Real.exp (-s*L)≤(1/(N:ℝ))*(1/4:ℝ)^j := by
-  have hlog : Real.log (4:ℝ)≤2 := by
-    have h2 : Real.log (2:ℝ)<1 := by
-      have ht := Real.log_lt_sub_one_of_pos (by norm_num : (0:ℝ)<2) (by norm_num : (2:ℝ)≠1)
-      norm_num at ht
-      exact ht
-    have he : Real.log (4:ℝ)=2*Real.log 2 := by rw [show (4:ℝ)=2^2 by norm_num,Real.log_pow]; norm_num
-    linarith
-  have hnR : (0:ℝ)<N := by exact_mod_cast hN
-  calc
-    _ ≤ Real.exp (-(Real.log N+(j:ℝ)*Real.log 4)) := Real.exp_le_exp.mpr (by
-      have hb := mul_le_mul_of_nonneg_left hlog (Nat.cast_nonneg j : (0:ℝ)≤j)
-      nlinarith)
-    _ = _ := by
-      rw [neg_add,Real.exp_add,Real.exp_neg,Real.exp_log hnR,one_div]
-      congr 1
-      rw [show -((j:ℝ)*Real.log (4:ℝ))=(j:ℝ)*(-Real.log 4) by ring,
-        Real.exp_nat_mul,Real.exp_neg,Real.exp_log (by norm_num : (0:ℝ)<4)]
-      norm_num
-
-lemma shapes_medium_sum (N : ℕ) (h s : ℝ) (hsh : 2≤ s*h) :
-    (Fintype.card (Shapes N):ℝ)*Real.exp (-s*h*N)≤Real.exp (-(N:ℝ)) := by
-  have hc : (Fintype.card (Shapes N):ℝ)≤(2:ℝ)^N := by exact_mod_cast shapes_count N
-  have hl : Real.log (2:ℝ)≤1 := by
-    have := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
-    norm_num at this ⊢
-    exact this
-  calc
-    _ ≤ (2:ℝ)^N*Real.exp (-s*h*N) := mul_le_mul_of_nonneg_right hc (Real.exp_pos _).le
-    _ = Real.exp ((Real.log 2-s*h)*N) := by
-      rw [←Real.exp_log (by norm_num : (0:ℝ)<2),←Real.exp_nat_mul,←Real.exp_add,Real.log_exp]
-      congr 1
-      ring
-    _ ≤ _ := Real.exp_le_exp.mpr (by
-      have hb := mul_le_mul_of_nonneg_right (show Real.log 2-s*h≤-1 by linarith) (Nat.cast_nonneg N : (0:ℝ)≤N)
-      linarith)
-
-end Thorp.Specht
-
-namespace Thorp
-open scoped BigOperators Classical
-
-section
-open UnitaryFinite Specht DimensionEstimates Filter
-
-lemma sampleLaw_run (d t : ℕ) : sampleLaw (run d t)=law d t := by
-  classical
-  funext g
-  unfold sampleLaw finiteMean law
-  rw [Fintype.card_subtype]
-  apply congrArg (fun z : ℝ => z/(Fintype.card (Fin t → Coins d):ℝ))
-  convert Finset.sum_boole (R := ℝ) (fun x => run d t x = g) Finset.univ using 1
-  apply Finset.sum_congr rfl
-  intro ω _
-  split_ifs with h <;> simp only [h, ite_true, ite_false]
-
-lemma physical_fourier_bound (d t : ℕ) :
-    distance d t^2≤∑ μ : Shapes (2^d),(Module.finrank ℂ (hilbertSpace μ.1):ℝ)^2*
-      ‖sampleOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) (run d t)-
-        uniformOperator (V := hilbertSpace μ.1) (cardRepresentation d μ)‖^2 := by
-  let V : Shapes (2^d) → Type := fun μ => hilbertSpace μ.1
-  let : ∀ μ,NormedAddCommGroup (V μ) := fun _ => inferInstance
-  let : ∀ μ,InnerProductSpace ℂ (V μ) := fun _ => inferInstance
-  let : ∀ μ,FiniteDimensional ℂ (V μ) := fun _ => inferInstance
-  let p := fun g : Equiv.Perm (Card d) => law d t g-1/(Nat.factorial (2^d):ℝ)
-  have hb := FiniteFourier.l1_fourier_bound V (cardRepresentation d)
-    (cardRepresentation_unitary d) (cardRepresentation_distinct d) (card_conjugacy_shapes d) p
-  have he (μ : Shapes (2^d)) : weightedOperator (V := V μ) (cardRepresentation d μ) p=
-      sampleOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) (run d t)-
-        uniformOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) := by
-    rw [sampleOperator_weighted,sampleLaw_run]
-    dsimp only [p]
-    rw [weighted_sub]
-    congr 1
-    unfold uniformOperator
-    rw [card_permutations]
-    simp only [one_div]
-  simp_rw [he] at hb
-  apply le_trans _ hb
-  change ((1/2:ℝ)*(∑ g,|p g|))^2≤(∑ g,|p g|)^2
-  nlinarith [sq_nonneg (∑ g,|p g|)]
-
-theorem eventual_physical_bound :
-    ∃ C : ℕ, 0<C ∧ ∀ᶠ d : ℕ in atTop,
-      distance d (C*d)^2≤2/(2:ℝ)^d+Real.exp (-(2:ℝ)^d) := by
-  obtain ⟨a,h,ha,hh,hreg⟩ := spectral_regimes
-  let s := max 4 (2/h)
-  have hs4 : 4≤ s := le_max_left _ _
-  have hs : 0≤ s := by dsimp [s]; positivity
-  have hsh : 2≤ s*h := by
-    have hc := le_max_right (4:ℝ) (2/h)
-    exact (div_le_iff₀ hh).mp hc
-  obtain ⟨C,hC₀⟩ := exists_nat_ge ((s+2)/a)
-  have hC : s+2≤a*C := by
-    have hC := (div_le_iff₀ ha).mp hC₀
-    nlinarith
-  have hCp : 0<C := by
-    by_contra hn
-    have hz : C=0 := by omega
-    rw [hz,Nat.cast_zero,mul_zero] at hC
-    linarith
-  refine ⟨C,hCp,?_⟩
-  filter_upwards [hreg,eventually_gt_atTop 0] with d hd hdp
-  have hbound (μ : Shapes (2^d)) :
-      (Module.finrank ℂ (hilbertSpace μ.1):ℝ)^2*
-        ‖sampleOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) (run d (C*d))-
-          uniformOperator (V := hilbertSpace μ.1) (cardRepresentation d μ)‖^2≤
-        (1/(2:ℝ)^d)*(1/4:ℝ)^(μ.1.card-μ.1.rowLen 0)+Real.exp (-s*h*(2:ℝ)^d) := by
-    let : NormedAddCommGroup (hilbertSpace μ.1) := inferInstance
-    let : InnerProductSpace ℂ (hilbertSpace μ.1) := inferInstance
-    let : FiniteDimensional ℂ (hilbertSpace μ.1) := inferInstance
-    have hr : 0≤(1/(2:ℝ)^d)*(1/4:ℝ)^(μ.1.card-μ.1.rowLen 0) := by positivity
-    by_cases hj : μ.1.card-μ.1.rowLen 0=0
-    · rw [card_fourier_full_row d (C*d) μ hj,norm_zero,zero_pow (by norm_num : 2≠0),mul_zero]
-      positivity
-    by_cases hc : (μ.1.colLen 0:ℝ)≤(μ.1.card:ℝ)/2
-    · obtain ⟨hK,hD,halt⟩ := hd μ.1 (cardLabels d μ) (by omega) hc
-      have ht := card_fourier_decay d C μ a s ha hs hC hD hK
-      apply ht.trans
-      rcases halt with ⟨hlog,hjlog⟩ | hlog
-      · have hnear := near_shape_term μ.1.card (μ.1.card-μ.1.rowLen 0)
-          (Real.log (Module.finrank ℂ (space μ.1))) s
-          (by rw [μ.2]; positivity) hs4 (by linarith) hlog hjlog
-        rw [μ.2,Nat.cast_pow,Nat.cast_ofNat] at hnear
-        rw [μ.2]
-        exact hnear.trans (le_add_of_nonneg_right (Real.exp_pos _).le)
-      · apply le_trans (b := Real.exp (-s*h*(2:ℝ)^d)) _ (le_add_of_nonneg_left hr)
-        apply Real.exp_le_exp.mpr
-        nlinarith
-    · rw [card_fourier_tall d C hdp hCp μ (lt_of_not_ge hc),norm_zero,
-        zero_pow (by norm_num : 2≠0),mul_zero]
-      positivity
-  calc
-    _ ≤ ∑ μ : Shapes (2^d),(Module.finrank ℂ (hilbertSpace μ.1):ℝ)^2*
-        ‖sampleOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) (run d (C*d))-
-          uniformOperator (V := hilbertSpace μ.1) (cardRepresentation d μ)‖^2 := physical_fourier_bound d (C*d)
-    _ ≤ ∑ μ : Shapes (2^d),((1/(2:ℝ)^d)*(1/4:ℝ)^(μ.1.card-μ.1.rowLen 0)+
-        Real.exp (-s*h*(2:ℝ)^d)) := Finset.sum_le_sum (fun μ _ => hbound μ)
-    _ = (1/(2:ℝ)^d)*(∑ μ : Shapes (2^d),(1/4:ℝ)^(μ.1.card-μ.1.rowLen 0))+
-        (Fintype.card (Shapes (2^d)):ℝ)*Real.exp (-s*h*(2:ℝ)^d) := by
-      rw [Finset.sum_add_distrib,←Finset.mul_sum,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
-    _ ≤ (1/(2:ℝ)^d)*2+Real.exp (-(2:ℝ)^d) := by
-      apply add_le_add
-      · exact mul_le_mul_of_nonneg_left (shape_weight_sum (2^d)) (by positivity)
-      · simpa only [Nat.cast_pow,Nat.cast_ofNat] using shapes_medium_sum (2^d) h s hsh
-    _ = _ := by ring
-
-end
-open Filter DimensionEstimates
-
-lemma physical_convergence : ∃ C : ℕ, 0<C ∧
-    Tendsto (fun d : ℕ => distance d (C*d)) atTop (nhds 0) := by
-  obtain ⟨C,hC,hbound⟩ := eventual_physical_bound
-  refine ⟨C,hC,?_⟩
-  have hrec : Tendsto (fun d : ℕ => 2/(2:ℝ)^d) atTop (nhds 0) :=
-    tendsto_const_nhds.div_atTop twopow_tendsto
-  have hexp : Tendsto (fun d : ℕ => Real.exp (-(2:ℝ)^d)) atTop (nhds 0) :=
-    Real.tendsto_exp_neg_atTop_nhds_zero.comp twopow_tendsto
-  have hr : Tendsto (fun d : ℕ => 2/(2:ℝ)^d+Real.exp (-(2:ℝ)^d)) atTop (nhds 0) := by
-    simpa only [zero_add] using hrec.add hexp
-  have hsq : Tendsto (fun d : ℕ => distance d (C*d)^2) atTop (nhds 0) :=
-    squeeze_zero' (Eventually.of_forall (fun d => sq_nonneg (distance d (C*d)))) hbound hr
-  have hs := (Real.continuous_sqrt.tendsto (0:ℝ)).comp hsq
-  have he : (Real.sqrt ∘ fun d : ℕ => distance d (C*d)^2)=(fun d => distance d (C*d)) := by
-    funext d
-    exact Real.sqrt_sq (distance_nonneg _ _)
-  rw [he,Real.sqrt_zero] at hs
-  exact hs
-
-end Thorp
-
-namespace Thorp.Casimir
-open scoped BigOperators Classical
-open Filter UnitaryFinite
-
-lemma K_source_square (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ) :
-    K d μ e = (ContinuousLinearMap.adjoint (𝕜:=ℂ) (E:=Specht.hilbertSpace μ) (F:=Specht.hilbertSpace μ)
-      (SparseContact.sweepOperator d μ e true)) *
-      SparseContact.sweepOperator d μ e true := by
-  simp only [SparseContact.sweepOperator,ite_true]
-  rw [sampleOperator_inv _ (Specht.relabelledUnitary_unitary μ e),
-    ContinuousLinearMap.adjoint_adjoint]
-  exact sampleOperator_palindrome _ _ (Specht.relabelledUnitary_unitary μ e)
-
-lemma density_bound (d k : ℕ) (hk : k≤2^d) (x : Fin k ↪ Card d) :
-    (palindromeRowMoment d x (1/64))^(64/65:ℝ) ≤
-      Real.exp ((1+64*(1+heightDecay 1))*levelScale (2^d) k) := by
-  have hm := row_moment_coarse d x (1/64) (by norm_num) (by norm_num)
-  have hh := Real.rpow_le_rpow (palindromeRowMoment_nonneg d x (1/64)) hm
-    (by norm_num : (0:ℝ)≤64/65)
-  simp only [Fintype.card_fin] at hh
-  apply hh.trans
-  rw [Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp]
-  apply Real.exp_le_exp.mpr
-  have hL := levelScale_ge (2^d) k hk
-  have hA := heightDecay_nonneg 1
-  have hk0 : (0:ℝ)≤k := by positivity
-  nlinarith
-
-lemma fromDistance_eq (d t : ℕ) (σ : Equiv.Perm (Card d)) : fromDistance d t σ=distance d t := by
-  unfold fromDistance distance
-  congr 1
-  exact Fintype.sum_equiv (Equiv.mulRight σ⁻¹) _ _ (fun _ => rfl)
-
-lemma trace_real {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-    [FiniteDimensional ℂ V] (A : V →L[ℂ] V) (hA : IsSelfAdjoint A) :
-    (((LinearMap.trace ℂ V A.toLinearMap).re : ℝ):ℂ) = LinearMap.trace ℂ V A.toLinearMap := by
-  let B := stdOrthonormalBasis ℂ V
-  rw [LinearMap.trace_eq_matrix_trace ℂ B.toBasis]
-  simp only [Matrix.trace,Matrix.diag,Complex.re_sum,Complex.ofReal_sum,
-    LinearMap.toMatrix_apply,OrthonormalBasis.coe_toBasis_repr_apply,
-    OrthonormalBasis.repr_apply_apply]
-  apply Finset.sum_congr rfl
-  intro j _
-  exact hA.isSymmetric.coe_re_inner_self_apply (B j)
-
-lemma trace_power_real {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-    [FiniteDimensional ℂ V] (A : V →L[ℂ] V) (hA : IsSelfAdjoint A) (r : ℕ) :
-    (((LinearMap.trace ℂ V (A^r).toLinearMap).re : ℝ):ℂ) =
-      LinearMap.trace ℂ V (A^r).toLinearMap := trace_real _ (hA.pow r)
-
-lemma traceMoment_real (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ) (r : ℕ) :
-    (traceMoment d μ e r:ℂ) =
-      LinearMap.trace ℂ (Specht.hilbertSpace μ) ((K d μ e)^r).toLinearMap := by
-  exact trace_power_real (V:=Specht.hilbertSpace μ) _ (K_selfadjoint d μ e) r
-
-theorem casimir_main : MainStatement := by
-  obtain ⟨a,ha,hnorm⟩ := level_contraction
-  refine ⟨a,1+64*(1+heightDecay 1),ha,by linarith [heightDecay_nonneg 1],?_,?_,?_⟩
-  · intro d k _ hk x
-    exact density_bound d k hk x
-  · intro d μ e hk
-    exact ⟨hnorm d μ e hk,trace_bound d μ e⟩
-  · obtain ⟨l,hl,hm⟩ := physical_convergence
-    refine ⟨l,hl,fun starts => ?_⟩
-    simpa only [fromDistance_eq] using hm
-
-end Thorp.Casimir
-
-namespace Thorp.StrongTail
-open scoped BigOperators Classical
-open Specht UnitaryFinite
-
-def MainStatement : Prop :=
-  ∃ p Cdensity C : ℝ, 1 < p ∧ p ≤ 2 ∧ DensityAt p Cdensity ∧
-    ∀ (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ) (rev : Bool),
-      ‖SparseContact.sweepOperator d μ e rev‖^2 ≤
-       Real.exp (C*(μ.card-μ.rowLen 0 : ℕ)) *
-       (Module.finrank ℂ (Specht.space (tailDiagram μ)):ℝ)^(-((p-1)/p))
-
-lemma dimension_le_factorial (μ : YoungDiagram) :
-    Module.finrank ℂ (space μ) ≤ μ.card.factorial := by
-  have hdim : Module.finrank ℂ (space μ) ≤ Fintype.card (Tabloid μ) := by
-    simpa using Submodule.finrank_le (space μ)
-  have hrow : 1 ≤ Fintype.card (rowGroup μ) := Fintype.card_pos
-  have hc := card_tabloid_mul_row μ
-  nlinarith
-
-lemma tail_card_exact (μ : YoungDiagram) :
-    (tailDiagram μ).card = μ.card - μ.rowLen 0 := by
-  rw [←card_cell,Fintype.card_congr (tailCellEquiv μ),tail_card]
-
-lemma tail_dimension_le_factorial (μ : YoungDiagram) :
-    Module.finrank ℂ (space (tailDiagram μ)) ≤ (μ.card - μ.rowLen 0).factorial := by
-  simpa only [tail_card_exact] using dimension_le_factorial (tailDiagram μ)
-
-lemma sweep_sq_palindrome (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Cell μ) (rev : Bool) :
-    ‖SparseContact.sweepOperator d μ e rev‖^2 =
-      ‖sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e) (palindromePerm d)‖ := by
-  rw [SparseContact.sweep_forward_norm, sampleOperator_palindrome _ _ (relabelledUnitary_unitary μ e)]
-  change ‖sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e)
-    (fun ω => butterflyPerm d (decodeButterfly d ω))‖^2 = _
-  rw [show (sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e)
-    (fun ω => butterflyPerm d (decodeButterfly d ω))).adjoint =
-    star (sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e)
-    (fun ω => butterflyPerm d (decodeButterfly d ω))) from rfl]
-  simpa only [pow_two] using (CStarRing.norm_self_mul_star (x := sampleOperator (V := hilbertSpace μ)
-    (relabelledUnitary μ e) (fun ω => butterflyPerm d (decodeButterfly d ω)))).symm
-
-lemma sweep_sq_le_one (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Cell μ) (rev : Bool) :
-    ‖SparseContact.sweepOperator d μ e rev‖^2 ≤ 1 := by
-  have h := SparseContact.sweep_norm_le_one d μ e rev
-  nlinarith [norm_nonneg (SparseContact.sweepOperator d μ e rev)]
-
-theorem strong_tail : MainStatement := by
-  obtain ⟨a,h,ha,hh,dE⟩ := Specht.spectral_regimes
-  obtain ⟨D0,hD0⟩ := Filter.eventually_atTop.mp dE
-  let d0 := max D0 1
-  let t : ℝ := min a (1/1000)
-  have ht : 0 < t := lt_min ha (by norm_num)
-  have hta : t ≤ a := min_le_left _ _
-  have htq : t ≤ 1/1000 := min_le_right _ _
-  let b := t/(1+t)
-  have hden : 0 < 1+t := by linarith
-  have hb : 0 < b := div_pos ht hden
-  have hbt : b ≤ t := (div_le_iff₀ hden).mpr (by nlinarith [sq_nonneg t])
-  have hba : b ≤ a := hbt.trans hta
-  have hb1 : b ≤ 1 := (div_le_one hden).mpr (by linarith)
-  let C := Real.log (((2^d0).factorial:ℕ):ℝ)
-  have hC : 0 ≤ C := Real.log_nonneg (by exact_mod_cast Nat.factorial_pos (2^d0))
-  refine ⟨1+t,3+heightDecay 1,C,by linarith,by linarith,
-    density_at_small_exponent (by linarith) (by linarith),?_⟩
-  intro d μ e rev
-  let k := μ.card - μ.rowLen 0
-  let D : ℝ := Module.finrank ℂ (space (tailDiagram μ))
-  have hD : 1 ≤ D := by dsimp [D]; exact_mod_cast dimension_pos (tailDiagram μ)
-  have hDp : 0 < D := by linarith
-  have hlog : 0 ≤ Real.log D := Real.log_nonneg hD
-  have hμ : μ.card=2^d := by rw [←card_cell,←Fintype.card_congr e,card_positions]
-  have hform : -(((1+t)-1)/(1+t)) = -b := by dsimp [b]; congr 1; ring
-  rw [hform]
-  change ‖SparseContact.sweepOperator d μ e rev‖^2 ≤ Real.exp (C*k)*D^(-b)
-  rw [Real.rpow_def_of_pos hDp,←Real.exp_add]
-  by_cases hk : k=0
-  · have hDone : D=1 := by
-      have hn := tail_dimension_le_factorial μ
-      change Module.finrank ℂ (space (tailDiagram μ)) ≤ k.factorial at hn
-      rw [hk,Nat.factorial_zero] at hn
-      have hu : D ≤ 1 := by dsimp [D]; exact_mod_cast hn
-      linarith
-    rw [hk,Nat.cast_zero,mul_zero,hDone,Real.log_one,zero_mul,add_zero,Real.exp_zero]
-    exact sweep_sq_le_one d μ e rev
-  · have hkpos : 0<k := by omega
-    by_cases hd : d0 ≤ d
-    · have hdpos : 0<d := by have := le_max_right D0 1; omega
-      by_cases htall : (μ.card:ℝ)/2 < μ.colLen 0
-      · rw [sweep_sq_palindrome,Specht.palindrome_tall_zero_general d hdpos μ e htall,
-          @norm_zero (hilbertSpace μ →L[ℂ] hilbertSpace μ) _]
-        exact (Real.exp_pos _).le
-      · have hg := (hD0 d ((le_max_left _ _).trans hd) μ e hkpos (le_of_not_gt htall)).1
-        have hrootpos : (0:ℝ)<Module.finrank ℂ (space μ) := by exact_mod_cast dimension_pos μ
-        have hlogle : Real.log D ≤ Real.log (Module.finrank ℂ (space μ):ℝ) :=
-          Real.log_le_log hDp (by dsimp [D]; exact_mod_cast tail_dimension_le μ)
-        rw [sweep_sq_palindrome]
-        apply hg.trans
-        apply Real.exp_le_exp.mpr
-        have hbound := mul_le_mul hba hlogle hlog ha.le
-        have hck : 0 ≤ C*(k:ℝ) := mul_nonneg hC (Nat.cast_nonneg _)
-        nlinarith
-    · have hpow : 2^d ≤ 2^d0 := Nat.pow_le_pow_right (by decide) (by omega)
-      have hklarge : k ≤ 2^d0 := (by simp [k,hμ] : k ≤ 2^d).trans hpow
-      have hdim : D ≤ ((2^d0).factorial:ℝ) := by
-        dsimp [D]
-        exact_mod_cast (tail_dimension_le_factorial μ).trans (Nat.factorial_le hklarge)
-      have hlC : Real.log D ≤ C := Real.log_le_log hDp hdim
-      have hk1 : (1:ℝ) ≤ k := by exact_mod_cast hkpos
-      have hprod : b*Real.log D ≤ C*(k:ℝ) := by nlinarith
-      apply (sweep_sq_le_one d μ e rev).trans
-      exact Real.one_le_exp_iff.mpr (by nlinarith)
-
-end Thorp.StrongTail
-
-
-namespace Thorp.HarmonicBounds
-open scoped BigOperators Classical
-open Filter Casimir StrongSmoothing
-
-lemma dimension_contraction :
-    ∃ b : ℝ, 0<b ∧ ∀ (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ),
-      0<μ.card-μ.rowLen 0 →
-      W d μ e ≤ Real.exp (-b*Real.log (Module.finrank ℂ (Specht.space μ):ℝ)) := by
-  obtain ⟨a,h,ha,hh,he⟩ := Specht.spectral_regimes
-  obtain ⟨D,hD⟩ := eventually_atTop.mp (he.and (eventually_ge_atTop 1))
-  let B (d : ℕ) : ℝ := 1+Real.log ((2^d).factorial:ℝ)
-  have hB (d : ℕ) : 0<B d := by
-    have hh : 0≤Real.log ((2^d).factorial:ℝ) := Real.log_nonneg (by exact_mod_cast Nat.factorial_pos (2^d))
-    dsimp [B]; linarith
-  let f (d : ℕ) : ℝ := -Real.log (blockGap d)/B d
-  have hf (d : ℕ) : 0<f d :=
-    div_pos (neg_pos.mpr (Real.log_neg (blockGap_pos d) (blockGap_lt_one d))) (hB d)
-  obtain ⟨b,hb,hbD⟩ := finite_positive_lower f D (fun d _ => hf d)
-  refine ⟨min a b,lt_min ha hb,?_⟩
-  intro d μ e hk
-  have hμ : μ.card=2^d := by rw [←Specht.card_cell,←Fintype.card_congr e,card_positions]
-  have hDpos : (0:ℝ)<Module.finrank ℂ (Specht.space μ) := by exact_mod_cast Specht.dimension_pos μ
-  have hL : 0≤Real.log (Module.finrank ℂ (Specht.space μ):ℝ) :=
-    Real.log_nonneg (by exact_mod_cast Specht.dimension_pos μ)
-  by_cases hd : D≤d
-  · obtain ⟨hs,hd1⟩ := hD d hd
-    by_cases ht : (μ.card:ℝ)/2<μ.colLen 0
-    · have hz := Specht.palindrome_tall_zero_general d (by omega) μ e ht
-      change K d μ e=0 at hz
-      change ‖K d μ e‖≤_
-      rw [hz,@norm_zero (Specht.hilbertSpace μ →L[ℂ] Specht.hilbertSpace μ) _]
-      exact (Real.exp_pos _).le
-    · apply (hs μ e hk (le_of_not_gt ht)).1.trans
-      apply Real.exp_le_exp.mpr
-      have hh := mul_le_mul_of_nonneg_right (min_le_left a b) hL
-      linarith
-  · have hdim : Real.log (Module.finrank ℂ (Specht.space μ):ℝ) ≤ B d := by
-      have hh := StrongTail.dimension_le_factorial μ
-      rw [hμ] at hh
-      have hl := Real.log_le_log hDpos (show (Module.finrank ℂ (Specht.space μ):ℝ)≤(2^d).factorial by exact_mod_cast hh)
-      dsimp [B]; linarith
-    have hab : min a b≤f d := (min_le_right _ _).trans (hbD d (by omega))
-    have hp := mul_le_mul_of_nonneg_right hab (hB d).le
-    have heq : f d*B d = -Real.log (blockGap d) := by dsimp [f]; field_simp [ne_of_gt (hB d)]
-    rw [heq] at hp
-    have hscale := mul_le_mul_of_nonneg_left hdim (le_of_lt (lt_min ha hb))
-    calc
-      W d μ e ≤ blockGap d := nontrivial_gap d μ e hk
-      _ = Real.exp (Real.log (blockGap d)) := (Real.exp_log (blockGap_pos d)).symm
-      _ ≤ _ := Real.exp_le_exp.mpr (by linarith)
-
-lemma exp_intersection {x A B : ℝ} (hx : 0≤x) (hA : x≤Real.exp A) (hB : x≤Real.exp B) :
-    x≤Real.exp ((A+B)/2) := by
-  have hh := mul_le_mul hA hB hx (Real.exp_pos A).le
-  rw [←Real.exp_add] at hh
-  have he : Real.exp ((A+B)/2)^2=Real.exp (A+B) := by
-    rw [pow_two,←Real.exp_add]
-    congr 1; ring
-  nlinarith [Real.exp_pos ((A+B)/2)]
-
-lemma flat_tuple_trace (d k : ℕ) (hk : k≤2^d) (order : Equiv.Perm (Fin d)) (rev : Bool) :
-    tupleTrace d k order rev 2001 ≤ Real.exp ((2000*(1+heightDecay 1))*k) := by
-  have hpos : 0<Fintype.card (Tuples d k) := by
-    rw [tuples_card]
-    exact Nat.descFactorial_pos.mpr hk
-  have hne : (Fintype.card (Tuples d k):ℝ)≠0 := by exact_mod_cast ne_of_gt hpos
-  calc
-    _ ≤ ∑ x : Tuples d k, Real.exp ((2000*(1+heightDecay 1))*k)/
-        ((2^d).descFactorial k:ℝ) := by
-      apply Finset.sum_le_sum
-      intro x _
-      exact both_pointwise d k order rev x x
-    _ = _ := by
-      simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,←tuples_card]
-      field_simp
-
-theorem harmonic_cycle_main : MainStatement := by
-  obtain ⟨a,ha,hlevel⟩ := level_contraction
-  obtain ⟨b,hb,hdim⟩ := dimension_contraction
-  let C : ℝ := 2000*(1+heightDecay 1)
-  have hA : 0≤heightDecay 1 := heightDecay_nonneg 1
-  have hC : 0<C := by dsimp [C]; positivity
-  refine ⟨a,C,b,1/1000,min a b/4,a/2,b/2,2001,
-    ha,hC,hb,by norm_num,by norm_num,by positivity,by positivity,by positivity,by norm_num,?_,?_⟩
-  · intro d μ e hk
-    dsimp only
-    let L := levelScale (2^d) (μ.card-μ.rowLen 0)
-    let f : ℝ := Module.finrank ℂ (Specht.space (tail μ))
-    let D : ℝ := Module.finrank ℂ (Specht.space μ)
-    have hμ : μ.card=2^d := by rw [←Specht.card_cell,←Fintype.card_congr e,card_positions]
-    have hkn : μ.card-μ.rowLen 0≤2^d := by omega
-    have hL : 0≤L := (Nat.cast_nonneg _).trans (levelScale_ge _ _ hkn)
-    have hfp : 0<f := by dsimp [f,tail]; exact_mod_cast Specht.dimension_pos (StrongTail.tailDiagram μ)
-    have hfl : 0≤Real.log f := Real.log_nonneg (by dsimp [f,tail]; exact_mod_cast Specht.dimension_pos (StrongTail.tailDiagram μ))
-    have hDp : 0<D := by dsimp [D]; exact_mod_cast Specht.dimension_pos μ
-    have hle : Real.log f≤Real.log D := Real.log_le_log hfp (by
-      dsimp [f,D,tail]; exact_mod_cast StrongTail.tail_dimension_le μ)
-    have hw := hlevel d μ e hk
-    have hd := hdim d μ e hk
-    change W d μ e≤Real.exp (-a*L) at hw
-    change W d μ e≤Real.exp (-b*Real.log D) at hd
-    have ht : W d μ e≤Real.exp (-b*Real.log f) := hd.trans (Real.exp_le_exp.mpr (by nlinarith))
-    refine ⟨hw,?_,?_,?_⟩
-    · change W d μ e≤Real.exp (C*L)*f^(-b)
-      rw [Real.rpow_def_of_pos hfp,←Real.exp_add]
-      exact ht.trans (Real.exp_le_exp.mpr (by nlinarith))
-    · intro rev
-      have hg := exp_intersection (norm_nonneg (K d μ e)) hw ht
-      have hcoef : (-a*L+-b*Real.log f)/2 ≤ -2*(min a b/4)*(L+Real.log f) := by
-        have h₁ := mul_le_mul_of_nonneg_right (min_le_left a b) hL
-        have h₂ := mul_le_mul_of_nonneg_right (min_le_right a b) hfl
-        nlinarith
-      have hg' := hg.trans (Real.exp_le_exp.mpr hcoef)
-      have hs := StrongTail.sweep_sq_palindrome d μ e rev
-      change ‖SparseContact.sweepOperator d μ e rev‖^2=W d μ e at hs
-      change W d μ e≤_ at hg'
-      rw [←hs] at hg'
-      have he : Real.exp (-(min a b/4)*(L+Real.log f))^2 =
-          Real.exp (-2*(min a b/4)*(L+Real.log f)) := by
-        rw [pow_two,←Real.exp_add]
-        congr 1; ring
-      change ‖SparseContact.sweepOperator d μ e rev‖≤Real.exp (-(min a b/4)*(L+Real.log f))
-      nlinarith [Real.exp_pos (-(min a b/4)*(L+Real.log f))]
-    · change W d μ e≤Real.exp (-(a/2)*L)*D^(-(b/2))
-      rw [Real.rpow_def_of_pos hDp,←Real.exp_add]
-      have hg := exp_intersection (norm_nonneg (K d μ e)) hw hd
-      have he : -(a/2)*L+Real.log D*(-(b/2))=(-a*L+-b*Real.log D)/2 := by ring
-      rw [he]
-      exact hg
-  · intro d k _ hk order rev
-    have hL := levelScale_ge (2^d) k hk
-    have hcoef : k*(1+heightDecay 1) ≤ C*levelScale (2^d) k := by
-      have hk0 : (0:ℝ)≤k := by positivity
-      dsimp [C]; nlinarith
-    refine ⟨?_,?_⟩
-    · intro x
-      exact (StrongTail.reflected_moment_both d k order rev x).trans (Real.exp_le_exp.mpr hcoef)
-    · exact (flat_tuple_trace d k hk order rev).trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left hL hC.le))
-
-end Thorp.HarmonicBounds
-
-end ThorpNine.Harmonic
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Harmonic.Site
+
+namespace OAI
+
+namespace ThorpNine.Harmonic
+
+namespace Thorp.Casimir
+open scoped BigOperators Classical
+
+theorem palindrome_cost_mgf_coarse (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/32) :
+    finiteMean (fun ω : BenesCoins d => q^(palindromeCost d e ω)) ≤
+      Real.exp (Fintype.card ι*(1+heightDecay 1)) := by
+  have hd : d ≤ 1*4^d := by have hh := pow_four_ge d; omega
+  have hcost (ω : BenesCoins d) : palindromeCost d e ω = palindromeLowCost 1 d e ω+
+      (palindromeFamilyCost 1 d d e ω+palindromeFamilyCost 2 d d e ω) := by
+    have hh := palindromeCost_family 1 d d e ω
+    rw [palindromeLowCost_eq _ _ hd] at hh
+    simpa only [Nat.add_assoc] using hh.symm
+  have hqp : 0 < q := lt_of_lt_of_le (by norm_num) hq
+  have hlow (ω : BenesCoins d) : q^(palindromeLowCost 1 d e ω) ≤ Real.exp (Fintype.card ι) := by
+    calc
+      _ ≤ q^(Fintype.card ι) := pow_le_pow_right₀ hq (by simpa using palindromeLowCost_le 1 d e ω)
+      _ = Real.exp (Fintype.card ι*Real.log q) := by rw [Real.exp_nat_mul,Real.exp_log hqp]
+      _ ≤ _ := Real.exp_le_exp.mpr (by nlinarith [Nat.cast_nonneg (α:=ℝ) (Fintype.card ι)])
+  calc
+    _ ≤ finiteMean (fun ω : BenesCoins d => Real.exp (Fintype.card ι)*
+        q^(palindromeFamilyCost 1 d d e ω+palindromeFamilyCost 2 d d e ω)) := by
+      apply finiteMean_mono
+      intro ω
+      rw [hcost,pow_add]
+      exact mul_le_mul_of_nonneg_right (hlow ω) (pow_nonneg hqp.le _)
+    _ = Real.exp (Fintype.card ι)*finiteMean (fun ω : BenesCoins d =>
+        q^(palindromeFamilyCost 1 d d e ω+palindromeFamilyCost 2 d d e ω)) := by simpa only [mul_comm] using finiteMean_mul_const (fun ω : BenesCoins d => q^(palindromeFamilyCost 1 d d e ω+palindromeFamilyCost 2 d d e ω)) (Real.exp (Fintype.card ι))
+    _ ≤ Real.exp (Fintype.card ι)*Real.exp (Fintype.card ι*heightDecay 1) := by
+      exact mul_le_mul_of_nonneg_left
+        (palindrome_high_full_mgf 1 d d (by decide) e q hq hlog) (Real.exp_nonneg _)
+    _ = _ := by rw [←Real.exp_add]; congr 1; ring
+
+lemma row_moment_coarse (d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d)
+    (a : ℝ) (ha : 0 < a) (ha32 : a ≤ 1/32) :
+    palindromeRowMoment d e a ≤ Real.exp (Fintype.card ι*(1+heightDecay 1)) := by
+  let q : ℝ := (2:ℝ)^a
+  have hq : 1 ≤ q := Real.one_le_rpow (by norm_num) ha.le
+  have hlog : Real.log q ≤ 1/32 := by
+    dsimp [q]
+    rw [Real.log_rpow (by norm_num)]
+    have hh := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
+    norm_num at hh
+    nlinarith
+  have hn : (0:ℝ) < ((2^d:ℕ):ℝ)^(Fintype.card ι) := by positivity
+  have hn0 : (0:ℝ) ≤ ((2^d).descFactorial (Fintype.card ι):ℝ) /
+      ((2^d:ℕ):ℝ)^(Fintype.card ι) := by positivity
+  have hn1 : ((2^d).descFactorial (Fintype.card ι):ℝ) /
+      ((2^d:ℕ):ℝ)^(Fintype.card ι) ≤ 1 := by
+    apply (div_le_one hn).mpr
+    exact_mod_cast Nat.descFactorial_le_pow (2^d) (Fintype.card ι)
+  have hpow := Real.rpow_le_one hn0 hn1 ha.le
+  have hm := palindrome_row_density_moment d e a ha.le
+  have he (c : ℕ) : q^c = (2:ℝ)^((c:ℝ)*a) := by
+    rw [mul_comm,Real.rpow_mul_natCast (by norm_num)]
+  simp_rw [←he] at hm
+  exact (hm.trans (mul_le_of_le_one_left (finiteMean_nonneg (fun _ =>
+    pow_nonneg (le_trans (by norm_num) hq) _)) hpow)).trans
+      (palindrome_cost_mgf_coarse d e q hq hlog)
+
+
+lemma applyKernel_mul {X : Type*} [Fintype X] (P Q : Matrix X X ℝ)
+    (v : EuclideanSpace ℂ X) :
+    DensityTransfer.applyKernel (fun x y => (P*Q) x y) v =
+      DensityTransfer.applyKernel (fun x y => P x y) (DensityTransfer.applyKernel (fun x y => Q x y) v) := by
+  apply PiLp.ext
+  intro x
+  change DensityTransfer.applyKernel (fun x y => (P*Q) x y) v x = DensityTransfer.applyKernel (fun x y => P x y) (DensityTransfer.applyKernel (fun x y => Q x y) v) x
+  simp only [DensityTransfer.applyKernel_apply,Matrix.mul_apply,Complex.ofReal_sum,
+    Complex.ofReal_mul,Finset.sum_mul,Finset.mul_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro y _
+  apply Finset.sum_congr rfl
+  intro z _
+  ring
+
+lemma applyKernel_one {X : Type*} [Fintype X] [DecidableEq X]
+    (v : EuclideanSpace ℂ X) : DensityTransfer.applyKernel (fun x y => (1 : Matrix X X ℝ) x y) v = v := by
+  apply PiLp.ext
+  intro x
+  change DensityTransfer.applyKernel (fun x y => (1 : Matrix X X ℝ) x y) v x = v x
+  simp [DensityTransfer.applyKernel_apply,Matrix.one_apply,apply_ite,ite_mul]
+
+lemma symmetric_hs_power {X : Type*} [Fintype X] [DecidableEq X]
+    (P : Matrix X X ℝ) (hs : P.IsSymm) (r : ℕ) :
+    (∑ x, ∑ y, ((P^r) x y)^2) = ∑ x, (P^(2*r)) x x := by
+  rw [two_mul,pow_add]
+  apply Finset.sum_congr rfl
+  intro x _
+  simp only [Matrix.mul_apply]
+  apply Finset.sum_congr rfl
+  intro y _
+  rw [(hs.pow r).apply x y,pow_two]
+
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+  [FiniteDimensional ℂ V]
+
+lemma trace_power_le_hs (A : V →L[ℂ] V) (hs : IsSelfAdjoint A) (hn : ‖A‖ ≤ 1)
+    (r : ℕ) :
+    (LinearMap.trace ℂ V ((A^(2*r+1)).toLinearMap)).re ≤
+      ∑ j, ‖(A^r) (stdOrthonormalBasis ℂ V j)‖^2 := by
+  let B := stdOrthonormalBasis ℂ V
+  rw [LinearMap.trace_eq_matrix_trace ℂ B.toBasis]
+  simp only [Matrix.trace,Matrix.diag,Complex.re_sum,LinearMap.toMatrix_apply,
+    OrthonormalBasis.coe_toBasis_repr_apply,OrthonormalBasis.repr_apply_apply]
+  apply Finset.sum_le_sum
+  intro j _
+  have he : (A^(2*r+1)) (B j) = (A^r) (A ((A^r) (B j))) := by
+    have hnat : 2*r+1=r+(1+r) := by omega
+    rw [hnat,pow_add,pow_add,pow_one]
+    rfl
+  change (inner ℂ (B j) ((A^(2*r+1)) (B j))).re ≤ ‖(A^r) (B j)‖^2
+  rw [he]
+  conv_lhs => rw [←(hs.pow r).adjoint_eq,ContinuousLinearMap.adjoint_inner_right]
+  rw [(hs.pow r).adjoint_eq]
+  apply (Complex.re_le_norm _).trans
+  apply (norm_inner_le_norm _ _).trans
+  have hh := A.le_opNorm ((A^r) (B j))
+  have ha : ‖A ((A^r) (B j))‖ ≤ ‖(A^r) (B j)‖ := by
+    exact hh.trans (by simpa using mul_le_mul_of_nonneg_right hn (norm_nonneg _))
+  nlinarith [norm_nonneg ((A^r) (B j))]
+
+open UnitaryFinite
+lemma palindrome_selfadjoint (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V)
+    (hρ : IsUnitary ρ) : IsSelfAdjoint (sampleOperator ρ (palindromePerm d)) := by
+  rw [sampleOperator_palindrome _ _ hρ]
+  change IsSelfAdjoint (_ * star _)
+  simp only [isSelfAdjoint_iff,star_mul,star_star]
+
+variable {G X : Type*} [Group G] [Fintype G] [Fintype X] [DecidableEq X]
+
+lemma coefficient_trace_amplification (a : G →* Equiv.Perm X) (x₀ : X)
+    (ha : ∀ x, ∃ g, a g x₀=x) (ρ : Representation ℂ G V)
+    [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
+    (P : Matrix X X ℝ) (A : V →L[ℂ] V)
+    (hmap : ∀ w, IsFixed a x₀ ρ w → ∀ v,
+      DensityTransfer.applyKernel (fun x y => P x y) (normalizedCoefficient a x₀ ha ρ w v) =
+        normalizedCoefficient a x₀ ha ρ w (A v))
+    (hs : IsSelfAdjoint A) (hn : ‖A‖ ≤ 1) (r : ℕ) :
+    (Module.finrank ℂ (fixedSpace a x₀ ρ):ℝ) *
+      (LinearMap.trace ℂ V ((A^(2*r+1)).toLinearMap)).re ≤
+      ∑ x, ∑ y, ((P^r) x y)^2 := by
+  let W := fixedSpace a x₀ ρ
+  let B := stdOrthonormalBasis ℂ V
+  let C := stdOrthonormalBasis ℂ W
+  let w : Fin (Module.finrank ℂ W) → V := fun i => (C i : V)
+  have hfix (i) : IsFixed a x₀ ρ (w i) := (C i).2
+  have hw : Orthonormal ℂ w := C.orthonormal.comp_linearIsometry W.subtypeₗᵢ
+  let v : Fin (Module.finrank ℂ W) × Fin (Module.finrank ℂ V) → EuclideanSpace ℂ X :=
+    fun ij => normalizedCoefficient a x₀ ha ρ (w ij.1) (B ij.2)
+  have hv : Orthonormal ℂ v := by
+    rw [orthonormal_iff_ite]
+    intro i j
+    dsimp [v]
+    rw [normalizedCoefficient_inner a x₀ ha ρ hρ _ _ (hfix _) (hfix _)]
+    rw [(orthonormal_iff_ite.mp hw), (orthonormal_iff_ite.mp B.orthonormal)]
+    by_cases h1 : i.1=j.1 <;> by_cases h2 : i.2=j.2 <;>
+      simp [h1,h2,Prod.ext_iff,eq_comm]
+  have hp (s : ℕ) (i) (t : V) :
+      DensityTransfer.applyKernel (fun x y => (P^s) x y) (normalizedCoefficient a x₀ ha ρ (w i) t) =
+        normalizedCoefficient a x₀ ha ρ (w i) ((A^s) t) := by
+    induction s with
+    | zero => simp [applyKernel_one]
+    | succ s ih =>
+      rw [pow_succ',applyKernel_mul,ih,hmap _ (hfix _),pow_succ']
+      rfl
+  have hhs := DensityTransfer.hilbert_schmidt_family (fun x y => (P^r) x y) v hv
+  have he : (∑ ij, ‖DensityTransfer.applyKernel (fun x y => (P^r) x y) (v ij)‖^2) =
+      (Module.finrank ℂ W:ℝ) * ∑ j, ‖(A^r) (B j)‖^2 := by
+    simp only [v,hp,normalizedCoefficient_norm a x₀ ha ρ hρ _ (hfix _)
+      (hw.norm_eq_one _),Fintype.sum_prod_type]
+    simp
+  rw [he] at hhs
+  exact (mul_le_mul_of_nonneg_left (trace_power_le_hs A hs hn r) (by positivity)).trans hhs
+
+end Thorp.Casimir
+
+namespace Thorp.Casimir
+open scoped BigOperators Classical
+open UnitaryFinite
+
+section
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+noncomputable def tupleKernel (d k : ℕ) : Matrix (Fin k ↪ Card d) (Fin k ↪ Card d) ℝ :=
+  fun x y => PairRouting.tupleProbability (palindromePerm d) x y
+
+lemma palindrome_trace_amplification (d k : ℕ) (x₀ : Fin k ↪ Card d)
+    (ρ : Representation ℂ (Equiv.Perm (Card d)) V) [Representation.IsIrreducible ρ]
+    (hρ : IsUnitary ρ) (r : ℕ) :
+    (Module.finrank ℂ (fixedSpace tupleAction x₀ ρ):ℝ) *
+      (LinearMap.trace ℂ V (((sampleOperator ρ (palindromePerm d))^(2*r+1)).toLinearMap)).re ≤
+    ∑ x, ∑ y, ((tupleKernel d k)^r) x y ^ 2 := by
+  let ha := tupleAction_transitive x₀
+  have hmap (w : V) (hw : IsFixed tupleAction x₀ ρ w) (v : V) :
+      DensityTransfer.applyKernel (fun x y : Fin k ↪ Card d =>
+        PairRouting.tupleProbability (palindromePerm d) x y)
+        (normalizedCoefficient tupleAction x₀ ha ρ w v) =
+      normalizedCoefficient tupleAction x₀ ha ρ w (sampleOperator ρ (palindromePerm d) v) := by
+    have hh := coefficient_random tupleAction x₀ ha ρ hρ (palindromePerm d) w hw v
+    have he : actionKernel tupleAction (palindromePerm d) =
+        fun x y : Fin k ↪ Card d => PairRouting.tupleProbability (palindromePerm d) x y := by
+      funext x y
+      exact tupleAction_kernel _ x y
+    rw [he,←sampleOperator_inv_apply,sampleOperator_inv _ hρ,
+      (palindrome_selfadjoint d ρ hρ).adjoint_eq] at hh
+    exact hh
+  exact coefficient_trace_amplification tupleAction x₀ ha ρ hρ _ _ hmap
+    (palindrome_selfadjoint d ρ hρ) (sampleOperator_norm_le ρ hρ _) r
+
+end
+
+lemma tupleKernel_symm (d k : ℕ) : (tupleKernel d k).IsSymm := by
+  ext x y
+  exact palindrome_kernel_symmetric d y x
+
+lemma tupleKernel_power64 (d k : ℕ) (x y : Fin k ↪ Card d) :
+    ((tupleKernel d k)^64) x y ≤
+      Real.exp ((64*(1+heightDecay 1))*k)/(Fintype.card (Fin k ↪ Card d):ℝ) := by
+  let : Nonempty (Fin k ↪ Card d) := ⟨x⟩
+  have hr (x : Fin k ↪ Card d) : ∑ y, tupleKernel d k x y = 1 :=
+    PairRouting.tupleProbability_row_sum _ x
+  have hc (y : Fin k ↪ Card d) : ∑ x, tupleKernel d k x y = 1 := by
+    simp_rw [(tupleKernel_symm d k).apply y]
+    exact hr y
+  have hm (x : Fin k ↪ Card d) :
+      finiteMean (fun y => ((Fintype.card (Fin k ↪ Card d):ℝ)*tupleKernel d k x y)^
+        (1+(1/32:ℝ))) ≤ Real.exp ((k:ℝ)*(1+heightDecay 1)) := by
+    simpa only [palindromeRowMoment,Fintype.card_fin,tupleKernel] using
+      row_moment_coarse d x (1/32) (by norm_num) (by norm_num)
+  have hnum : 2 ≤ (1+(1/32:ℝ))^(32:ℕ) := by
+    have hh := one_add_mul_le_pow (a := (1/32:ℝ)) (by norm_num) 32
+    norm_num at hh ⊢
+  have hh := StrongSmoothing.symmetric_kernel_smoothing (tupleKernel d k)
+    (fun x y => PairRouting.tupleProbability_nonneg _ x y) hr hc (tupleKernel_symm d k)
+    (p := 1+(1/32:ℝ)) (A := k*(1+heightDecay 1)) (by norm_num)
+    (mul_nonneg (Nat.cast_nonneg _) (by linarith [heightDecay_nonneg 1])) hm 32 hnum x y
+  have he : 2*((k:ℝ)*(1+heightDecay 1))*(32:ℝ) = (64*(1+heightDecay 1))*k := by ring
+  simpa only [show 2*32=64 by norm_num,Nat.cast_ofNat,he] using hh
+
+lemma K_selfadjoint (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ) :
+    @IsSelfAdjoint _ (ContinuousLinearMap.instStarId (𝕜:=ℂ) (E:=Specht.hilbertSpace μ)) (K d μ e) := by
+  exact palindrome_selfadjoint (V:=Specht.hilbertSpace μ) d _ (Specht.relabelledUnitary_unitary μ e)
+
+lemma K_norm_le_one (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ) :
+    ‖K d μ e‖ ≤ 1 :=
+  sampleOperator_norm_le _ (Specht.relabelledUnitary_unitary μ e) _
+
+lemma fixed_trace_bound (d k : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ)
+    (x₀ : Fin k ↪ Card d) :
+    (Module.finrank ℂ (fixedSpace (V := Specht.hilbertSpace μ) tupleAction x₀
+      (Specht.relabelledUnitary μ e)):ℝ) * traceMoment d μ e 65 ≤
+        Real.exp ((64*(1+heightDecay 1))*k) := by
+  let ρ := Specht.relabelledUnitary μ e
+  let := Specht.relabelledUnitary_irreducible μ e
+  have hh := palindrome_trace_amplification (V:=Specht.hilbertSpace μ) d k x₀ ρ
+    (Specht.relabelledUnitary_unitary μ e) 32
+  change _ * traceMoment d μ e 65 ≤ _ at hh
+  rw [symmetric_hs_power _ (tupleKernel_symm d k)] at hh
+  apply hh.trans
+  have hX : (0:ℝ) < Fintype.card (Fin k ↪ Card d) := by
+    let : Nonempty (Fin k ↪ Card d) := ⟨x₀⟩
+    exact Nat.cast_pos.mpr Fintype.card_pos
+  calc
+    _ ≤ ∑ x : Fin k ↪ Card d,
+        Real.exp ((64*(1+heightDecay 1))*k)/(Fintype.card (Fin k ↪ Card d):ℝ) :=
+      Finset.sum_le_sum (fun x _ => tupleKernel_power64 d k x x)
+    _ = _ := by
+      simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
+      exact mul_div_cancel₀ _ (ne_of_gt hX)
+
+lemma levelScale_ge (n k : ℕ) (hk : k ≤ n) : (k:ℝ) ≤ levelScale n k := by
+  by_cases h : k=0
+  · simp [h,levelScale]
+  have hkp : (0:ℝ)<k := by exact_mod_cast Nat.pos_of_ne_zero h
+  have hlog : 0 ≤ Real.log ((n:ℝ)/k) := Real.log_nonneg ((one_le_div hkp).mpr (by exact_mod_cast hk))
+  dsimp [levelScale]
+  nlinarith
+
+lemma choose_le_exp_levelScale (n k : ℕ) (hn : 0<n) (hk : k≤n) :
+    (n.choose k:ℝ) ≤ Real.exp (levelScale n k) := by
+  by_cases h : k=0
+  · simp [h,levelScale]
+  have hkp : 0<k := Nat.pos_of_ne_zero h
+  have hh := DimensionEstimates.log_choose_upper n k hn hk
+  have hf := DimensionEstimates.log_factorial_lower k hkp
+  have he : Real.log ((n:ℝ)/k) = Real.log n-Real.log k :=
+    Real.log_div (by positivity) (by positivity)
+  have hl : Real.log (n.choose k:ℝ) ≤ levelScale n k := by
+    dsimp [levelScale]
+    rw [he]
+    linarith
+  rw [←Real.exp_log (by exact_mod_cast Nat.choose_pos hk : (0:ℝ)<n.choose k)]
+  exact Real.exp_le_exp.mpr hl
+
+lemma trace_bound (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ) :
+    (Module.finrank ℂ (Specht.space μ):ℝ)*traceMoment d μ e 65 ≤
+      Real.exp ((1+64*(1+heightDecay 1))*levelScale (2^d) (μ.card-μ.rowLen 0)) := by
+  let k := μ.card-μ.rowLen 0
+  have hn : μ.card=2^d := by
+    rw [←Specht.card_cell,←Fintype.card_congr e,card_positions]
+  have hk : k≤2^d := by dsimp [k]; omega
+  let x₀ : Fin k ↪ Card d := (Fin.castLEEmb (show k≤Fintype.card (Card d) by simpa [card_positions] using hk)).trans
+    (Fintype.equivFin (Card d)).symm.toEmbedding
+  let m := Module.finrank ℂ (fixedSpace (V := Specht.hilbertSpace μ) tupleAction x₀
+    (Specht.relabelledUnitary μ e))
+  have hd : (Module.finrank ℂ (Specht.space μ):ℝ) ≤ (m:ℝ)*((2^d).choose k:ℝ) := by
+    have hh := Specht.relabelled_tuple_dimension μ e x₀
+    simp only [Fintype.card_fin] at hh
+    change Module.finrank ℂ (Specht.hilbertSpace μ) * k.choose k ≤ m * μ.card.choose k at hh
+    rw [Nat.choose_self,mul_one,Specht.finrank_hilbertSpace,hn] at hh
+    exact_mod_cast hh
+  have hm := fixed_trace_bound d k μ e x₀
+  by_cases ht : 0≤traceMoment d μ e 65
+  · calc
+      _ ≤ ((m:ℝ)*((2^d).choose k:ℝ))*traceMoment d μ e 65 :=
+        mul_le_mul_of_nonneg_right hd ht
+      _ = ((2^d).choose k:ℝ)*((m:ℝ)*traceMoment d μ e 65) := by ring
+      _ ≤ ((2^d).choose k:ℝ)*Real.exp ((64*(1+heightDecay 1))*k) :=
+        mul_le_mul_of_nonneg_left hm (Nat.cast_nonneg _)
+      _ ≤ Real.exp (levelScale (2^d) k)*Real.exp ((64*(1+heightDecay 1))*levelScale (2^d) k) := by
+        apply mul_le_mul (choose_le_exp_levelScale _ _ (by positivity) hk) _ (by positivity) (Real.exp_nonneg _)
+        apply Real.exp_le_exp.mpr
+        exact mul_le_mul_of_nonneg_left (levelScale_ge _ _ hk) (by linarith [heightDecay_nonneg 1])
+      _ = _ := by rw [←Real.exp_add]; congr 1; ring
+  · exact (mul_nonpos_of_nonneg_of_nonpos (Nat.cast_nonneg _) (le_of_not_ge ht)).trans (Real.exp_nonneg _)
+
+end Thorp.Casimir
+
+namespace Thorp.Casimir
+open scoped BigOperators Classical
+open Filter UnitaryFinite DimensionEstimates
+
+lemma levelScale_le (n k : ℕ) (hn : 0<n) (_ : k≤n) : levelScale n k ≤ n := by
+  by_cases hz : k=0
+  · simp [hz,levelScale]
+  have hkp : (0:ℝ)<k := by exact_mod_cast Nat.pos_of_ne_zero hz
+  have hnp : (0:ℝ)<n := by exact_mod_cast hn
+  have hh := Real.log_le_sub_one_of_pos (div_pos hnp hkp)
+  have hm := mul_le_mul_of_nonneg_left hh hkp.le
+  have he : (k:ℝ)*((n:ℝ)/k)=n := mul_div_cancel₀ _ (ne_of_gt hkp)
+  dsimp [levelScale]
+  nlinarith
+
+lemma nontrivial_uniform_zero (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ)
+    (hk : 0<μ.card-μ.rowLen 0) :
+    uniformOperator (V:=Specht.hilbertSpace μ) (Specht.relabelledUnitary μ e)=0 := by
+  apply ContinuousLinearMap.ext
+  intro v
+  apply SparseContact.point_relabelled_fixed_zero μ e ∅ (by simpa using hk)
+  intro g _
+  have he := congrArg (fun A : Specht.hilbertSpace μ →L[ℂ] Specht.hilbertSpace μ => A v)
+    (uniformOperator_left (V:=Specht.hilbertSpace μ) (Specht.relabelledUnitary μ e) g)
+  exact he
+
+lemma gap_of_uniform_zero {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+    [FiniteDimensional ℂ V] (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V)
+    (hρ : IsUnitary ρ) (hz : uniformOperator ρ=0) :
+    ‖sampleOperator ρ (palindromePerm d)‖ ≤ blockGap d := by
+  have hh := palindrome_off_invariants_gap d ρ hρ
+  rw [hz,sub_zero,←sampleOperator_weighted] at hh
+  exact hh
+
+lemma nontrivial_gap (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ)
+    (hk : 0<μ.card-μ.rowLen 0) : ‖K d μ e‖ ≤ blockGap d := by
+  exact gap_of_uniform_zero (V:=Specht.hilbertSpace μ) d _
+    (Specht.relabelledUnitary_unitary μ e) (nontrivial_uniform_zero d μ e hk)
+
+lemma eventual_level_dimension :
+    ∃ η : ℝ, 0<η ∧ ∀ᶠ d : ℕ in atTop,
+      ∀ μ : YoungDiagram, μ.card=2^d → 0<μ.card-μ.rowLen 0 →
+        (μ.colLen 0:ℝ)≤(μ.card:ℝ)/2 →
+        η*levelScale (2^d) (μ.card-μ.rowLen 0) ≤
+          Real.log (Module.finrank ℂ (Specht.space μ)) := by
+  obtain ⟨h,hh,hm⟩ := Specht.medium_dimension (1/16) (by norm_num) (by norm_num)
+  refine ⟨min (1/3) h,lt_min (by norm_num) hh,?_⟩
+  filter_upwards [twopow_nat_tendsto.eventually hm] with d hd
+  intro μ hμ hk hc
+  let k := μ.card-μ.rowLen 0
+  have hkn : k≤2^d := by dsimp [k]; omega
+  have hΛ : 0≤levelScale (2^d) k := (Nat.cast_nonneg k).trans (levelScale_ge _ _ hkn)
+  have hr : ((μ.card-μ.rowLen 0:ℕ):ℝ)=(μ.card:ℝ)-μ.rowLen 0 :=
+    Nat.cast_sub (Specht.rowLen_zero_le_card μ)
+  by_cases hs : (k:ℝ)≤(2^d:ℕ)/16
+  · have hk2 : 2*k≤2^d := by exact_mod_cast (show (2:ℝ)*k≤(2^d:ℕ) by linarith)
+    have hdim := DimensionEstimates.near_row_log_dimension (2^d) k _ hk hk2
+      (by simpa only [←hμ] using Specht.choose_tail_le_dimension μ)
+    have hkp : (0:ℝ)<k := by exact_mod_cast hk
+    have hratio : (16:ℝ)≤(2^d:ℕ)/k := (le_div_iff₀ hkp).mpr (by linarith)
+    have hlog : 4*Real.log 2 ≤ Real.log ((2^d:ℕ)/k) := by
+      have ht := Real.log_le_log (by norm_num : (0:ℝ)<16) hratio
+      rw [show (16:ℝ)=2^(4:ℕ) by norm_num,Real.log_pow] at ht
+      simpa using ht
+    have hlog2 : (1/2:ℝ)≤Real.log 2 := by linarith [Real.log_two_gt_d9]
+    have hL : levelScale (2^d) k ≤ 3*Real.log (Module.finrank ℂ (Specht.space μ)) := by
+      dsimp [levelScale]
+      nlinarith
+    have hη := mul_le_mul_of_nonneg_right (min_le_left (1/3:ℝ) h) hΛ
+    nlinarith
+  · have hdim := hd μ hμ (by rw [←hμ,←hr]; change (1/16:ℝ)*(μ.card:ℝ)≤(k:ℝ); rw [hμ]; linarith)
+      (by simpa only [←hμ] using hc)
+    have hL := levelScale_le (2^d) k (by positivity) hkn
+    have ht := mul_le_mul_of_nonneg_left hL hh.le
+    have hη := mul_le_mul_of_nonneg_right (min_le_right (1/3:ℝ) h) hΛ
+    nlinarith
+
+lemma eventual_level_contraction :
+    ∃ a : ℝ, 0<a ∧ ∀ᶠ d : ℕ in atTop,
+      ∀ (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ), 0<μ.card-μ.rowLen 0 →
+        ‖K d μ e‖ ≤ Real.exp (-a*levelScale (2^d) (μ.card-μ.rowLen 0)) := by
+  obtain ⟨a,h,ha,hh,hsp⟩ := Specht.spectral_regimes
+  obtain ⟨η,hη,hdim⟩ := eventual_level_dimension
+  refine ⟨a*η,mul_pos ha hη,?_⟩
+  filter_upwards [hsp,hdim,eventually_ge_atTop 1] with d hs hd hd1
+  intro μ e hk
+  by_cases ht : (μ.card:ℝ)/2 < μ.colLen 0
+  · have hz := Specht.palindrome_tall_zero_general d (by omega) μ e ht
+    change K d μ e=0 at hz
+    rw [hz,@norm_zero (Specht.hilbertSpace μ →L[ℂ] Specht.hilbertSpace μ) _]
+    exact (Real.exp_pos _).le
+  · have hc := le_of_not_gt ht
+    have hμ : μ.card=2^d := by rw [←Specht.card_cell,←Fintype.card_congr e,card_positions]
+    apply (hs μ e hk hc).1.trans
+    apply Real.exp_le_exp.mpr
+    have hh := mul_le_mul_of_nonneg_left (hd μ hμ hk hc) ha.le
+    nlinarith
+
+lemma finite_positive_lower (f : ℕ → ℝ) (D : ℕ) (hf : ∀ d<D,0<f d) :
+    ∃ b : ℝ, 0<b ∧ ∀ d<D,b≤f d := by
+  induction D with
+  | zero => exact ⟨1,by norm_num,by omega⟩
+  | succ D ih =>
+    obtain ⟨b,hb,hh⟩ := ih (fun d hd => hf d (by omega))
+    refine ⟨min b (f D),lt_min hb (hf D (by omega)),?_⟩
+    intro d hd
+    by_cases he : d=D
+    · subst d; exact min_le_right _ _
+    · exact (min_le_left _ _).trans (hh d (by omega))
+
+theorem level_contraction :
+    ∃ a : ℝ, 0<a ∧ ∀ (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ),
+      0<μ.card-μ.rowLen 0 →
+        ‖K d μ e‖ ≤ Real.exp (-a*levelScale (2^d) (μ.card-μ.rowLen 0)) := by
+  obtain ⟨a,ha,he⟩ := eventual_level_contraction
+  obtain ⟨D,hD⟩ := eventually_atTop.mp he
+  let f (d : ℕ) : ℝ := -Real.log (blockGap d)/(2:ℝ)^d
+  have hf (d : ℕ) : 0<f d := by
+    have hl := Real.log_neg (blockGap_pos d) (blockGap_lt_one d)
+    exact div_pos (neg_pos.mpr hl) (by positivity)
+  obtain ⟨b,hb,hbD⟩ := finite_positive_lower f D (fun d _ => hf d)
+  refine ⟨min a b,lt_min ha hb,?_⟩
+  intro d μ e hk
+  have hμ : μ.card=2^d := by rw [←Specht.card_cell,←Fintype.card_congr e,card_positions]
+  have hkn : μ.card-μ.rowLen 0≤2^d := by omega
+  have hΛ : 0≤levelScale (2^d) (μ.card-μ.rowLen 0) :=
+    (Nat.cast_nonneg _).trans (levelScale_ge _ _ hkn)
+  by_cases hd : D≤d
+  · apply (hD d hd μ e hk).trans (Real.exp_le_exp.mpr _)
+    have hh := mul_le_mul_of_nonneg_right (min_le_left a b) hΛ
+    linarith
+  · have hab : min a b≤f d := (min_le_right _ _).trans (hbD d (by omega))
+    have hscale : levelScale (2^d) (μ.card-μ.rowLen 0) ≤ (2:ℝ)^d := by
+      simpa only [Nat.cast_pow,Nat.cast_ofNat] using levelScale_le (2^d) _ (by positivity) hkn
+    have hprod := mul_le_mul_of_nonneg_right hab (by positivity : (0:ℝ)≤(2:ℝ)^d)
+    have hfprod : f d*(2:ℝ)^d = -Real.log (blockGap d) := by dsimp [f]; field_simp
+    rw [hfprod] at hprod
+    have hs := mul_le_mul_of_nonneg_left hscale (le_of_lt (lt_min ha hb))
+    calc
+      _ ≤ blockGap d := nontrivial_gap d μ e hk
+      _ = Real.exp (Real.log (blockGap d)) := (Real.exp_log (blockGap_pos d)).symm
+      _ ≤ _ := Real.exp_le_exp.mpr (by linarith)
+
+end Thorp.Casimir
+
+namespace Thorp.Specht
+open scoped BigOperators Classical
+open UnitaryFinite
+
+noncomputable def cardLabels (d : ℕ) (μ : Shapes (2^d)) : Card d ≃ Cell μ.1 :=
+  ((Fintype.equivFin (Card d)).trans (finCongr (card_positions d))).trans (labels μ).symm
+
+noncomputable def cardRepresentation (d : ℕ) (μ : Shapes (2^d)) :
+    Representation ℂ (Equiv.Perm (Card d)) (hilbertSpace μ.1) :=
+  relabelledUnitary μ.1 (cardLabels d μ)
+
+lemma cardRepresentation_unitary (d : ℕ) (μ : Shapes (2^d)) :
+    IsUnitary (V := hilbertSpace μ.1) (cardRepresentation d μ) :=
+  relabelledUnitary_unitary _ _
+
+noncomputable instance cardRepresentation_irreducible (d : ℕ) (μ : Shapes (2^d)) :
+    Representation.IsIrreducible (cardRepresentation d μ) :=
+  relabelledUnitary_irreducible _ _
+
+lemma relabelledUnitary_neq {α : Type*} (μ ν : YoungDiagram)
+    (e : α ≃ Cell μ) (f : α ≃ Cell ν)
+    (T : Representation.Equiv (relabelledUnitary μ e) (relabelledUnitary ν f)) : μ=ν := by
+  let S := (spaceHilbertEquiv μ).trans (T.toLinearEquiv.trans (spaceHilbertEquiv ν).symm)
+  apply shapes_eq_of_equivariant_equiv μ ν (e.symm.trans f) S
+  intro p v
+  have hT := Representation.IntertwiningMap.isIntertwining _ _ T.toIntertwiningMap
+    (e.symm.permCongr p) (spaceHilbertEquiv μ v)
+  change T.toLinearEquiv (spaceHilbertEquiv μ
+    (representation μ (e.permCongr (e.symm.permCongr p))
+      ((spaceHilbertEquiv μ).symm (spaceHilbertEquiv μ v))))=
+    spaceHilbertEquiv ν (representation ν (f.permCongr (e.symm.permCongr p))
+      ((spaceHilbertEquiv ν).symm (T.toLinearEquiv (spaceHilbertEquiv μ v)))) at hT
+  have he : e.permCongr (e.symm.permCongr p)=p := by
+    apply Equiv.ext
+    intro x
+    simp only [Equiv.permCongr_apply,Equiv.symm_symm,e.apply_symm_apply]
+  have hf : f.permCongr (e.symm.permCongr p)=(e.symm.trans f).permCongr p := by
+    apply Equiv.ext
+    intro x
+    rfl
+  rw [he,hf,LinearEquiv.symm_apply_apply] at hT
+  have ht := congrArg (spaceHilbertEquiv ν).symm hT
+  simpa only [LinearEquiv.symm_apply_apply,S,LinearEquiv.trans_apply] using ht
+
+lemma cardRepresentation_distinct (d : ℕ) (μ ν : Shapes (2^d)) (hne : μ≠ν) :
+    ¬Nonempty (Representation.Equiv (cardRepresentation d μ) (cardRepresentation d ν)) := by
+  rintro ⟨T⟩
+  exact hne (Subtype.ext (relabelledUnitary_neq μ.1 ν.1 (cardLabels d μ) (cardLabels d ν) T))
+
+lemma card_conjugacy_shapes (d : ℕ) :
+    Nat.card (ConjClasses (Equiv.Perm (Card d)))≤Fintype.card (Shapes (2^d)) := by
+  have h := Nat.card_le_card_of_injective _ (conjugacyPartition_injective (α := Card d))
+  simpa only [card_positions,Nat.card_eq_fintype_card,Fintype.card_congr (shapePartitionEquiv (2^d))] using h
+
+lemma row_zero_of_full_row (μ : YoungDiagram) (hμ : μ.rowLen 0=μ.card) (x : Cell μ) : row x=0 := by
+  have hr : μ.row 0=μ.cells := Finset.eq_of_subset_of_card_le
+    (Finset.filter_subset _ _) (by rw [←μ.rowLen_eq_card]; exact hμ.ge)
+  have hx : x.1∈μ.row 0 := by rw [hr]; exact x.2
+  exact (YoungDiagram.mem_row_iff.mp hx).2
+
+lemma tabloid_subsingleton_of_full_row (μ : YoungDiagram) (hμ : μ.rowLen 0=μ.card) :
+    Subsingleton (Tabloid μ) := by
+  refine ⟨fun f g => ?_⟩
+  apply Subtype.ext
+  funext x
+  obtain ⟨p,hp⟩ := f.2
+  obtain ⟨q,hq⟩ := g.2
+  simp only [hp,hq,Function.comp_apply]
+  apply Fin.ext
+  exact (row_zero_of_full_row μ hμ (p x)).trans (row_zero_of_full_row μ hμ (q x)).symm
+
+lemma representation_full_row (μ : YoungDiagram) (hμ : μ.rowLen 0=μ.card)
+    (p : Equiv.Perm (Cell μ)) : representation μ p=1 := by
+  let := tabloid_subsingleton_of_full_row μ hμ
+  apply LinearMap.ext
+  intro v
+  apply Subtype.ext
+  funext f
+  change v.1 (tabloidAct p⁻¹ f)=v.1 f
+  rw [show tabloidAct p⁻¹ f=f from Subsingleton.elim _ _]
+
+lemma relabelled_full_row {α : Type*} (μ : YoungDiagram) (hμ : μ.rowLen 0=μ.card)
+    (e : α ≃ Cell μ) (p : Equiv.Perm α) : relabelledUnitary μ e p=1 := by
+  change ((spaceHilbertEquiv μ).conjAlgEquiv ℂ) (representation μ (e.permCongr p))=1
+  rw [representation_full_row μ hμ,map_one]
+
+end Thorp.Specht
+namespace Thorp.UnitaryFinite
+open scoped BigOperators Classical
+variable {G Ω V : Type*} [Group G] [Fintype G] [Fintype Ω] [Nonempty Ω]
+    [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+lemma sample_uniform_left (ρ : Representation ℂ G V) (P : Ω → G) :
+    sampleOperator ρ P*uniformOperator ρ=uniformOperator ρ := by
+  rw [sampleOperator_weighted,weighted_uniform_left,sampleLaw_sum,one_smul]
+
+lemma uniform_zero_of_sample_norm_lt_one (ρ : Representation ℂ G V) (P : Ω → G)
+    (h : ‖sampleOperator ρ P‖<1) : uniformOperator ρ=0 := by
+  have hn := norm_mul_le (sampleOperator ρ P) (uniformOperator ρ)
+  rw [sample_uniform_left] at hn
+  apply norm_eq_zero.mp
+  nlinarith [norm_nonneg (uniformOperator ρ)]
+
+omit [Fintype G] in
+lemma sampleOperator_trivial (ρ : Representation ℂ G V) (hρ : ∀ g,ρ g=1) (P : Ω → G) :
+    sampleOperator ρ P=1 := by
+  convert sampleOperator_one (Ω := Ω) ρ using 1
+  unfold sampleOperator
+  congr 1
+  apply Finset.sum_congr rfl
+  intro ω _
+  apply ContinuousLinearMap.ext
+  intro v
+  change ρ (P ω) v=ρ 1 v
+  rw [hρ,hρ]
+
+lemma uniformOperator_trivial (ρ : Representation ℂ G V) (hρ : ∀ g,ρ g=1) :
+    uniformOperator ρ=1 := by
+  rw [←sampleOperator_uniform]
+  exact sampleOperator_trivial ρ hρ id
+
+lemma weighted_sub (ρ : Representation ℂ G V) (p q : G → ℝ) :
+    weightedOperator ρ (fun g => p g-q g)=weightedOperator ρ p-weightedOperator ρ q := by
+  simp only [weightedOperator,Complex.ofReal_sub,sub_smul,Finset.sum_sub_distrib]
+
+end Thorp.UnitaryFinite
+
+namespace Thorp
+open scoped BigOperators Classical
+open UnitaryFinite
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+lemma physical_palindrome_norm (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V)
+    (hρ : IsUnitary ρ) :
+    ‖sampleOperator ρ (run d d)‖^2=‖sampleOperator ρ (palindromePerm d)‖ := by
+  rw [physical_sweep_adjoint d ρ hρ,sampleOperator_palindrome d ρ hρ]
+  have h := ContinuousLinearMap.norm_adjoint_comp_self
+    ((sampleOperator ρ (fun ω => butterflyPerm d (decodeButterfly d ω))).adjoint)
+  rw [ContinuousLinearMap.adjoint_adjoint] at h
+  simpa only [ContinuousLinearMap.mul_def,←sq] using h.symm
+
+lemma physical_power_bound (d C : ℕ) (hC : 0 < C) (ρ : Representation ℂ (Equiv.Perm (Card d)) V)
+    (hρ : IsUnitary ρ) :
+    ‖sampleOperator ρ (run d (C*d))‖^2≤‖sampleOperator ρ (palindromePerm d)‖^C := by
+  have he : sampleOperator ρ (run d (C*d))=(sampleOperator ρ (run d d))^C := by
+    simp only [sampleOperator_run,←pow_mul,Nat.mul_comm C d]
+  rw [he,←physical_palindrome_norm d ρ hρ]
+  calc
+    _ ≤ (‖sampleOperator ρ (run d d)‖^C)^2 :=
+      pow_le_pow_left₀ (norm_nonneg _) (norm_pow_le' _ hC) _
+    _ = _ := by rw [←pow_mul,←pow_mul,Nat.mul_comm C 2]
+
+end Thorp
+namespace Thorp.Specht
+open scoped BigOperators Classical
+
+section
+open UnitaryFinite
+
+lemma card_fourier_decay (d C : ℕ) (μ : Shapes (2^d)) (a s : ℝ)
+    (ha : 0 < a) (hs : 0 ≤ s) (hC : s+2 ≤ a*C)
+    (hD : 1≤Real.log (Module.finrank ℂ (space μ.1)))
+    (hK : ‖sampleOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) (palindromePerm d)‖≤
+      Real.exp (-a*Real.log (Module.finrank ℂ (space μ.1)))) :
+    (Module.finrank ℂ (hilbertSpace μ.1):ℝ)^2*
+      ‖sampleOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) (run d (C*d))-uniformOperator (V := hilbertSpace μ.1) (cardRepresentation d μ)‖^2≤
+        Real.exp (-s*Real.log (Module.finrank ℂ (space μ.1))) := by
+  let : NormedAddCommGroup (hilbertSpace μ.1) := inferInstance
+  let : InnerProductSpace ℂ (hilbertSpace μ.1) := inferInstance
+  let : FiniteDimensional ℂ (hilbertSpace μ.1) := inferInstance
+  have hu : uniformOperator (V := hilbertSpace μ.1) (cardRepresentation d μ)=0 :=
+    uniform_zero_of_sample_norm_lt_one _ (palindromePerm d) (hK.trans_lt (by
+      rw [Real.exp_lt_one_iff]; nlinarith))
+  rw [hu,sub_zero,finrank_hilbertSpace]
+  have hCp : 0 < C := by
+    by_contra hn
+    have hz : C=0 := by omega
+    rw [hz,Nat.cast_zero,mul_zero] at hC
+    linarith
+  have hp := physical_power_bound d C hCp (cardRepresentation d μ) (cardRepresentation_unitary d μ)
+  have hkpow := pow_le_pow_left₀ (norm_nonneg _) hK C
+  have hDp : (0:ℝ)<Module.finrank ℂ (space μ.1) := by exact_mod_cast dimension_pos μ.1
+  calc
+    _ ≤ (Module.finrank ℂ (space μ.1):ℝ)^2*
+        (Real.exp (-a*Real.log (Module.finrank ℂ (space μ.1))))^C :=
+      mul_le_mul_of_nonneg_left (hp.trans hkpow) (sq_nonneg _)
+    _ = Real.exp ((2-a*C)*Real.log (Module.finrank ℂ (space μ.1))) := by
+      rw [←Real.exp_log hDp,←Real.exp_nat_mul,←Real.exp_nat_mul,←Real.exp_add,Real.log_exp]
+      congr 1
+      ring
+    _ ≤ _ := Real.exp_le_exp.mpr (by nlinarith)
+
+lemma card_fourier_tall (d C : ℕ) (hd : 0<d) (hC : 0<C) (μ : Shapes (2^d))
+    (hc : (μ.1.card:ℝ)/2<(μ.1.colLen 0:ℝ)) :
+    sampleOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) (run d (C*d))-uniformOperator (V := hilbertSpace μ.1) (cardRepresentation d μ)=0 := by
+  let : NormedAddCommGroup (hilbertSpace μ.1) := inferInstance
+  let : InnerProductSpace ℂ (hilbertSpace μ.1) := inferInstance
+  let : FiniteDimensional ℂ (hilbertSpace μ.1) := inferInstance
+  have hk : sampleOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) (palindromePerm d)=0 :=
+    palindrome_tall_zero_general d hd μ.1 (cardLabels d μ) hc
+  have hu : uniformOperator (V := hilbertSpace μ.1) (cardRepresentation d μ)=0 :=
+    uniform_zero_of_sample_norm_lt_one _ (palindromePerm d) (by rw [hk,norm_zero]; norm_num)
+  rw [hu,sub_zero]
+  apply norm_eq_zero.mp
+  have hp := physical_power_bound d C hC (cardRepresentation d μ) (cardRepresentation_unitary d μ)
+  rw [hk,norm_zero,zero_pow (by omega : C≠0)] at hp
+  nlinarith [norm_nonneg (sampleOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) (run d (C*d)))]
+
+lemma card_fourier_full_row (d t : ℕ) (μ : Shapes (2^d)) (hj : μ.1.card-μ.1.rowLen 0=0) :
+    sampleOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) (run d t)-uniformOperator (V := hilbertSpace μ.1) (cardRepresentation d μ)=0 := by
+  have hr : μ.1.rowLen 0=μ.1.card := by have := rowLen_zero_le_card μ.1; omega
+  have ht (p : Equiv.Perm (Card d)) : cardRepresentation d μ p=1 :=
+    relabelled_full_row μ.1 hr (cardLabels d μ) p
+  rw [sampleOperator_trivial _ ht,uniformOperator_trivial _ ht,sub_self]
+
+end
+
+lemma half_geom_sum (n : ℕ) : (∑ j∈Finset.range n,(1/2:ℝ)^j)≤2 := by
+  have he : (∑ j∈Finset.range n,(1/2:ℝ)^j)=2*(1-(1/2:ℝ)^n) := by
+    induction n with
+    | zero => simp
+    | succ n ih => rw [Finset.sum_range_succ,ih,pow_succ]; ring
+  rw [he]
+  nlinarith [pow_nonneg (by norm_num : (0:ℝ)≤1/2) n]
+
+lemma shape_weight_sum (N : ℕ) :
+    (∑ μ : Shapes N,(1/4:ℝ)^(μ.1.card-μ.1.rowLen 0))≤2 := by
+  have he := Finset.sum_fiberwise_of_maps_to'
+    (s := (Finset.univ : Finset (Shapes N))) (t := Finset.range (N+1))
+    (g := fun μ : Shapes N => μ.1.card-μ.1.rowLen 0)
+    (by intro μ _; simp only [Finset.mem_range]; have := μ.2; omega)
+    (fun j => (1/4:ℝ)^j)
+  rw [←he]
+  apply le_trans _ (half_geom_sum (N+1))
+  apply Finset.sum_le_sum
+  intro j _
+  have hc : ((Finset.univ.filter (fun μ : Shapes N => μ.1.card-μ.1.rowLen 0=j)).card:ℝ)≤(2:ℝ)^j := by
+    have hh := tail_shapes_count N j
+    rw [Fintype.card_subtype] at hh
+    exact_mod_cast hh
+  simp only [Finset.sum_const,nsmul_eq_mul]
+  calc
+    _ ≤ (2:ℝ)^j*(1/4:ℝ)^j := mul_le_mul_of_nonneg_right hc (by positivity)
+    _ = (1/2:ℝ)^j := by rw [←mul_pow]; norm_num
+
+lemma near_shape_term (N j : ℕ) (L s : ℝ) (hN : 0<N) (hs : 4≤ s)
+    (hL : 0≤L) (hl : (1/2:ℝ)*Real.log N≤L) (hj : (j:ℝ)≤L) :
+    Real.exp (-s*L)≤(1/(N:ℝ))*(1/4:ℝ)^j := by
+  have hlog : Real.log (4:ℝ)≤2 := by
+    have h2 : Real.log (2:ℝ)<1 := by
+      have ht := Real.log_lt_sub_one_of_pos (by norm_num : (0:ℝ)<2) (by norm_num : (2:ℝ)≠1)
+      norm_num at ht
+      exact ht
+    have he : Real.log (4:ℝ)=2*Real.log 2 := by rw [show (4:ℝ)=2^2 by norm_num,Real.log_pow]; norm_num
+    linarith
+  have hnR : (0:ℝ)<N := by exact_mod_cast hN
+  calc
+    _ ≤ Real.exp (-(Real.log N+(j:ℝ)*Real.log 4)) := Real.exp_le_exp.mpr (by
+      have hb := mul_le_mul_of_nonneg_left hlog (Nat.cast_nonneg j : (0:ℝ)≤j)
+      nlinarith)
+    _ = _ := by
+      rw [neg_add,Real.exp_add,Real.exp_neg,Real.exp_log hnR,one_div]
+      congr 1
+      rw [show -((j:ℝ)*Real.log (4:ℝ))=(j:ℝ)*(-Real.log 4) by ring,
+        Real.exp_nat_mul,Real.exp_neg,Real.exp_log (by norm_num : (0:ℝ)<4)]
+      norm_num
+
+lemma shapes_medium_sum (N : ℕ) (h s : ℝ) (hsh : 2≤ s*h) :
+    (Fintype.card (Shapes N):ℝ)*Real.exp (-s*h*N)≤Real.exp (-(N:ℝ)) := by
+  have hc : (Fintype.card (Shapes N):ℝ)≤(2:ℝ)^N := by exact_mod_cast shapes_count N
+  have hl : Real.log (2:ℝ)≤1 := by
+    have := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
+    norm_num at this ⊢
+    exact this
+  calc
+    _ ≤ (2:ℝ)^N*Real.exp (-s*h*N) := mul_le_mul_of_nonneg_right hc (Real.exp_pos _).le
+    _ = Real.exp ((Real.log 2-s*h)*N) := by
+      rw [←Real.exp_log (by norm_num : (0:ℝ)<2),←Real.exp_nat_mul,←Real.exp_add,Real.log_exp]
+      congr 1
+      ring
+    _ ≤ _ := Real.exp_le_exp.mpr (by
+      have hb := mul_le_mul_of_nonneg_right (show Real.log 2-s*h≤-1 by linarith) (Nat.cast_nonneg N : (0:ℝ)≤N)
+      linarith)
+
+end Thorp.Specht
+
+namespace Thorp
+open scoped BigOperators Classical
+
+section
+open UnitaryFinite Specht DimensionEstimates Filter
+
+lemma sampleLaw_run (d t : ℕ) : sampleLaw (run d t)=law d t := by
+  classical
+  funext g
+  unfold sampleLaw finiteMean law
+  rw [Fintype.card_subtype]
+  apply congrArg (fun z : ℝ => z/(Fintype.card (Fin t → Coins d):ℝ))
+  convert Finset.sum_boole (R := ℝ) (fun x => run d t x = g) Finset.univ using 1
+  apply Finset.sum_congr rfl
+  intro ω _
+  split_ifs with h <;> simp only [h, ite_true, ite_false]
+
+lemma physical_fourier_bound (d t : ℕ) :
+    distance d t^2≤∑ μ : Shapes (2^d),(Module.finrank ℂ (hilbertSpace μ.1):ℝ)^2*
+      ‖sampleOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) (run d t)-
+        uniformOperator (V := hilbertSpace μ.1) (cardRepresentation d μ)‖^2 := by
+  let V : Shapes (2^d) → Type := fun μ => hilbertSpace μ.1
+  let : ∀ μ,NormedAddCommGroup (V μ) := fun _ => inferInstance
+  let : ∀ μ,InnerProductSpace ℂ (V μ) := fun _ => inferInstance
+  let : ∀ μ,FiniteDimensional ℂ (V μ) := fun _ => inferInstance
+  let p := fun g : Equiv.Perm (Card d) => law d t g-1/(Nat.factorial (2^d):ℝ)
+  have hb := FiniteFourier.l1_fourier_bound V (cardRepresentation d)
+    (cardRepresentation_unitary d) (cardRepresentation_distinct d) (card_conjugacy_shapes d) p
+  have he (μ : Shapes (2^d)) : weightedOperator (V := V μ) (cardRepresentation d μ) p=
+      sampleOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) (run d t)-
+        uniformOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) := by
+    rw [sampleOperator_weighted,sampleLaw_run]
+    dsimp only [p]
+    rw [weighted_sub]
+    congr 1
+    unfold uniformOperator
+    rw [card_permutations]
+    simp only [one_div]
+  simp_rw [he] at hb
+  apply le_trans _ hb
+  change ((1/2:ℝ)*(∑ g,|p g|))^2≤(∑ g,|p g|)^2
+  nlinarith [sq_nonneg (∑ g,|p g|)]
+
+theorem eventual_physical_bound :
+    ∃ C : ℕ, 0<C ∧ ∀ᶠ d : ℕ in atTop,
+      distance d (C*d)^2≤2/(2:ℝ)^d+Real.exp (-(2:ℝ)^d) := by
+  obtain ⟨a,h,ha,hh,hreg⟩ := spectral_regimes
+  let s := max 4 (2/h)
+  have hs4 : 4≤ s := le_max_left _ _
+  have hs : 0≤ s := by dsimp [s]; positivity
+  have hsh : 2≤ s*h := by
+    have hc := le_max_right (4:ℝ) (2/h)
+    exact (div_le_iff₀ hh).mp hc
+  obtain ⟨C,hC₀⟩ := exists_nat_ge ((s+2)/a)
+  have hC : s+2≤a*C := by
+    have hC := (div_le_iff₀ ha).mp hC₀
+    nlinarith
+  have hCp : 0<C := by
+    by_contra hn
+    have hz : C=0 := by omega
+    rw [hz,Nat.cast_zero,mul_zero] at hC
+    linarith
+  refine ⟨C,hCp,?_⟩
+  filter_upwards [hreg,eventually_gt_atTop 0] with d hd hdp
+  have hbound (μ : Shapes (2^d)) :
+      (Module.finrank ℂ (hilbertSpace μ.1):ℝ)^2*
+        ‖sampleOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) (run d (C*d))-
+          uniformOperator (V := hilbertSpace μ.1) (cardRepresentation d μ)‖^2≤
+        (1/(2:ℝ)^d)*(1/4:ℝ)^(μ.1.card-μ.1.rowLen 0)+Real.exp (-s*h*(2:ℝ)^d) := by
+    let : NormedAddCommGroup (hilbertSpace μ.1) := inferInstance
+    let : InnerProductSpace ℂ (hilbertSpace μ.1) := inferInstance
+    let : FiniteDimensional ℂ (hilbertSpace μ.1) := inferInstance
+    have hr : 0≤(1/(2:ℝ)^d)*(1/4:ℝ)^(μ.1.card-μ.1.rowLen 0) := by positivity
+    by_cases hj : μ.1.card-μ.1.rowLen 0=0
+    · rw [card_fourier_full_row d (C*d) μ hj,norm_zero,zero_pow (by norm_num : 2≠0),mul_zero]
+      positivity
+    by_cases hc : (μ.1.colLen 0:ℝ)≤(μ.1.card:ℝ)/2
+    · obtain ⟨hK,hD,halt⟩ := hd μ.1 (cardLabels d μ) (by omega) hc
+      have ht := card_fourier_decay d C μ a s ha hs hC hD hK
+      apply ht.trans
+      rcases halt with ⟨hlog,hjlog⟩ | hlog
+      · have hnear := near_shape_term μ.1.card (μ.1.card-μ.1.rowLen 0)
+          (Real.log (Module.finrank ℂ (space μ.1))) s
+          (by rw [μ.2]; positivity) hs4 (by linarith) hlog hjlog
+        rw [μ.2,Nat.cast_pow,Nat.cast_ofNat] at hnear
+        rw [μ.2]
+        exact hnear.trans (le_add_of_nonneg_right (Real.exp_pos _).le)
+      · apply le_trans (b := Real.exp (-s*h*(2:ℝ)^d)) _ (le_add_of_nonneg_left hr)
+        apply Real.exp_le_exp.mpr
+        nlinarith
+    · rw [card_fourier_tall d C hdp hCp μ (lt_of_not_ge hc),norm_zero,
+        zero_pow (by norm_num : 2≠0),mul_zero]
+      positivity
+  calc
+    _ ≤ ∑ μ : Shapes (2^d),(Module.finrank ℂ (hilbertSpace μ.1):ℝ)^2*
+        ‖sampleOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) (run d (C*d))-
+          uniformOperator (V := hilbertSpace μ.1) (cardRepresentation d μ)‖^2 := physical_fourier_bound d (C*d)
+    _ ≤ ∑ μ : Shapes (2^d),((1/(2:ℝ)^d)*(1/4:ℝ)^(μ.1.card-μ.1.rowLen 0)+
+        Real.exp (-s*h*(2:ℝ)^d)) := Finset.sum_le_sum (fun μ _ => hbound μ)
+    _ = (1/(2:ℝ)^d)*(∑ μ : Shapes (2^d),(1/4:ℝ)^(μ.1.card-μ.1.rowLen 0))+
+        (Fintype.card (Shapes (2^d)):ℝ)*Real.exp (-s*h*(2:ℝ)^d) := by
+      rw [Finset.sum_add_distrib,←Finset.mul_sum,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
+    _ ≤ (1/(2:ℝ)^d)*2+Real.exp (-(2:ℝ)^d) := by
+      apply add_le_add
+      · exact mul_le_mul_of_nonneg_left (shape_weight_sum (2^d)) (by positivity)
+      · simpa only [Nat.cast_pow,Nat.cast_ofNat] using shapes_medium_sum (2^d) h s hsh
+    _ = _ := by ring
+
+end
+open Filter DimensionEstimates
+
+lemma physical_convergence : ∃ C : ℕ, 0<C ∧
+    Tendsto (fun d : ℕ => distance d (C*d)) atTop (nhds 0) := by
+  obtain ⟨C,hC,hbound⟩ := eventual_physical_bound
+  refine ⟨C,hC,?_⟩
+  have hrec : Tendsto (fun d : ℕ => 2/(2:ℝ)^d) atTop (nhds 0) :=
+    tendsto_const_nhds.div_atTop twopow_tendsto
+  have hexp : Tendsto (fun d : ℕ => Real.exp (-(2:ℝ)^d)) atTop (nhds 0) :=
+    Real.tendsto_exp_neg_atTop_nhds_zero.comp twopow_tendsto
+  have hr : Tendsto (fun d : ℕ => 2/(2:ℝ)^d+Real.exp (-(2:ℝ)^d)) atTop (nhds 0) := by
+    simpa only [zero_add] using hrec.add hexp
+  have hsq : Tendsto (fun d : ℕ => distance d (C*d)^2) atTop (nhds 0) :=
+    squeeze_zero' (Eventually.of_forall (fun d => sq_nonneg (distance d (C*d)))) hbound hr
+  have hs := (Real.continuous_sqrt.tendsto (0:ℝ)).comp hsq
+  have he : (Real.sqrt ∘ fun d : ℕ => distance d (C*d)^2)=(fun d => distance d (C*d)) := by
+    funext d
+    exact Real.sqrt_sq (distance_nonneg _ _)
+  rw [he,Real.sqrt_zero] at hs
+  exact hs
+
+end Thorp
+
+namespace Thorp.Casimir
+open scoped BigOperators Classical
+open Filter UnitaryFinite
+
+lemma K_source_square (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ) :
+    K d μ e = (ContinuousLinearMap.adjoint (𝕜:=ℂ) (E:=Specht.hilbertSpace μ) (F:=Specht.hilbertSpace μ)
+      (SparseContact.sweepOperator d μ e true)) *
+      SparseContact.sweepOperator d μ e true := by
+  simp only [SparseContact.sweepOperator,ite_true]
+  rw [sampleOperator_inv _ (Specht.relabelledUnitary_unitary μ e),
+    ContinuousLinearMap.adjoint_adjoint]
+  exact sampleOperator_palindrome _ _ (Specht.relabelledUnitary_unitary μ e)
+
+lemma density_bound (d k : ℕ) (hk : k≤2^d) (x : Fin k ↪ Card d) :
+    (palindromeRowMoment d x (1/64))^(64/65:ℝ) ≤
+      Real.exp ((1+64*(1+heightDecay 1))*levelScale (2^d) k) := by
+  have hm := row_moment_coarse d x (1/64) (by norm_num) (by norm_num)
+  have hh := Real.rpow_le_rpow (palindromeRowMoment_nonneg d x (1/64)) hm
+    (by norm_num : (0:ℝ)≤64/65)
+  simp only [Fintype.card_fin] at hh
+  apply hh.trans
+  rw [Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp]
+  apply Real.exp_le_exp.mpr
+  have hL := levelScale_ge (2^d) k hk
+  have hA := heightDecay_nonneg 1
+  have hk0 : (0:ℝ)≤k := by positivity
+  nlinarith
+
+lemma fromDistance_eq (d t : ℕ) (σ : Equiv.Perm (Card d)) : fromDistance d t σ=distance d t := by
+  unfold fromDistance distance
+  congr 1
+  exact Fintype.sum_equiv (Equiv.mulRight σ⁻¹) _ _ (fun _ => rfl)
+
+lemma trace_real {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+    [FiniteDimensional ℂ V] (A : V →L[ℂ] V) (hA : IsSelfAdjoint A) :
+    (((LinearMap.trace ℂ V A.toLinearMap).re : ℝ):ℂ) = LinearMap.trace ℂ V A.toLinearMap := by
+  let B := stdOrthonormalBasis ℂ V
+  rw [LinearMap.trace_eq_matrix_trace ℂ B.toBasis]
+  simp only [Matrix.trace,Matrix.diag,Complex.re_sum,Complex.ofReal_sum,
+    LinearMap.toMatrix_apply,OrthonormalBasis.coe_toBasis_repr_apply,
+    OrthonormalBasis.repr_apply_apply]
+  apply Finset.sum_congr rfl
+  intro j _
+  exact hA.isSymmetric.coe_re_inner_self_apply (B j)
+
+lemma trace_power_real {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+    [FiniteDimensional ℂ V] (A : V →L[ℂ] V) (hA : IsSelfAdjoint A) (r : ℕ) :
+    (((LinearMap.trace ℂ V (A^r).toLinearMap).re : ℝ):ℂ) =
+      LinearMap.trace ℂ V (A^r).toLinearMap := trace_real _ (hA.pow r)
+
+lemma traceMoment_real (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ) (r : ℕ) :
+    (traceMoment d μ e r:ℂ) =
+      LinearMap.trace ℂ (Specht.hilbertSpace μ) ((K d μ e)^r).toLinearMap := by
+  exact trace_power_real (V:=Specht.hilbertSpace μ) _ (K_selfadjoint d μ e) r
+
+theorem casimir_main : MainStatement := by
+  obtain ⟨a,ha,hnorm⟩ := level_contraction
+  refine ⟨a,1+64*(1+heightDecay 1),ha,by linarith [heightDecay_nonneg 1],?_,?_,?_⟩
+  · intro d k _ hk x
+    exact density_bound d k hk x
+  · intro d μ e hk
+    exact ⟨hnorm d μ e hk,trace_bound d μ e⟩
+  · obtain ⟨l,hl,hm⟩ := physical_convergence
+    refine ⟨l,hl,fun starts => ?_⟩
+    simpa only [fromDistance_eq] using hm
+
+end Thorp.Casimir
+
+namespace Thorp.StrongTail
+open scoped BigOperators Classical
+open Specht UnitaryFinite
+
+def MainStatement : Prop :=
+  ∃ p Cdensity C : ℝ, 1 < p ∧ p ≤ 2 ∧ DensityAt p Cdensity ∧
+    ∀ (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ) (rev : Bool),
+      ‖SparseContact.sweepOperator d μ e rev‖^2 ≤
+       Real.exp (C*(μ.card-μ.rowLen 0 : ℕ)) *
+       (Module.finrank ℂ (Specht.space (tailDiagram μ)):ℝ)^(-((p-1)/p))
+
+lemma dimension_le_factorial (μ : YoungDiagram) :
+    Module.finrank ℂ (space μ) ≤ μ.card.factorial := by
+  have hdim : Module.finrank ℂ (space μ) ≤ Fintype.card (Tabloid μ) := by
+    simpa using Submodule.finrank_le (space μ)
+  have hrow : 1 ≤ Fintype.card (rowGroup μ) := Fintype.card_pos
+  have hc := card_tabloid_mul_row μ
+  nlinarith
+
+lemma tail_card_exact (μ : YoungDiagram) :
+    (tailDiagram μ).card = μ.card - μ.rowLen 0 := by
+  rw [←card_cell,Fintype.card_congr (tailCellEquiv μ),tail_card]
+
+lemma tail_dimension_le_factorial (μ : YoungDiagram) :
+    Module.finrank ℂ (space (tailDiagram μ)) ≤ (μ.card - μ.rowLen 0).factorial := by
+  simpa only [tail_card_exact] using dimension_le_factorial (tailDiagram μ)
+
+lemma sweep_sq_palindrome (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Cell μ) (rev : Bool) :
+    ‖SparseContact.sweepOperator d μ e rev‖^2 =
+      ‖sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e) (palindromePerm d)‖ := by
+  rw [SparseContact.sweep_forward_norm, sampleOperator_palindrome _ _ (relabelledUnitary_unitary μ e)]
+  change ‖sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e)
+    (fun ω => butterflyPerm d (decodeButterfly d ω))‖^2 = _
+  rw [show (sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e)
+    (fun ω => butterflyPerm d (decodeButterfly d ω))).adjoint =
+    star (sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e)
+    (fun ω => butterflyPerm d (decodeButterfly d ω))) from rfl]
+  simpa only [pow_two] using (CStarRing.norm_self_mul_star (x := sampleOperator (V := hilbertSpace μ)
+    (relabelledUnitary μ e) (fun ω => butterflyPerm d (decodeButterfly d ω)))).symm
+
+lemma sweep_sq_le_one (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Cell μ) (rev : Bool) :
+    ‖SparseContact.sweepOperator d μ e rev‖^2 ≤ 1 := by
+  have h := SparseContact.sweep_norm_le_one d μ e rev
+  nlinarith [norm_nonneg (SparseContact.sweepOperator d μ e rev)]
+
+theorem strong_tail : MainStatement := by
+  obtain ⟨a,h,ha,hh,dE⟩ := Specht.spectral_regimes
+  obtain ⟨D0,hD0⟩ := Filter.eventually_atTop.mp dE
+  let d0 := max D0 1
+  let t : ℝ := min a (1/1000)
+  have ht : 0 < t := lt_min ha (by norm_num)
+  have hta : t ≤ a := min_le_left _ _
+  have htq : t ≤ 1/1000 := min_le_right _ _
+  let b := t/(1+t)
+  have hden : 0 < 1+t := by linarith
+  have hb : 0 < b := div_pos ht hden
+  have hbt : b ≤ t := (div_le_iff₀ hden).mpr (by nlinarith [sq_nonneg t])
+  have hba : b ≤ a := hbt.trans hta
+  have hb1 : b ≤ 1 := (div_le_one hden).mpr (by linarith)
+  let C := Real.log (((2^d0).factorial:ℕ):ℝ)
+  have hC : 0 ≤ C := Real.log_nonneg (by exact_mod_cast Nat.factorial_pos (2^d0))
+  refine ⟨1+t,3+heightDecay 1,C,by linarith,by linarith,
+    density_at_small_exponent (by linarith) (by linarith),?_⟩
+  intro d μ e rev
+  let k := μ.card - μ.rowLen 0
+  let D : ℝ := Module.finrank ℂ (space (tailDiagram μ))
+  have hD : 1 ≤ D := by dsimp [D]; exact_mod_cast dimension_pos (tailDiagram μ)
+  have hDp : 0 < D := by linarith
+  have hlog : 0 ≤ Real.log D := Real.log_nonneg hD
+  have hμ : μ.card=2^d := by rw [←card_cell,←Fintype.card_congr e,card_positions]
+  have hform : -(((1+t)-1)/(1+t)) = -b := by dsimp [b]; congr 1; ring
+  rw [hform]
+  change ‖SparseContact.sweepOperator d μ e rev‖^2 ≤ Real.exp (C*k)*D^(-b)
+  rw [Real.rpow_def_of_pos hDp,←Real.exp_add]
+  by_cases hk : k=0
+  · have hDone : D=1 := by
+      have hn := tail_dimension_le_factorial μ
+      change Module.finrank ℂ (space (tailDiagram μ)) ≤ k.factorial at hn
+      rw [hk,Nat.factorial_zero] at hn
+      have hu : D ≤ 1 := by dsimp [D]; exact_mod_cast hn
+      linarith
+    rw [hk,Nat.cast_zero,mul_zero,hDone,Real.log_one,zero_mul,add_zero,Real.exp_zero]
+    exact sweep_sq_le_one d μ e rev
+  · have hkpos : 0<k := by omega
+    by_cases hd : d0 ≤ d
+    · have hdpos : 0<d := by have := le_max_right D0 1; omega
+      by_cases htall : (μ.card:ℝ)/2 < μ.colLen 0
+      · rw [sweep_sq_palindrome,Specht.palindrome_tall_zero_general d hdpos μ e htall,
+          @norm_zero (hilbertSpace μ →L[ℂ] hilbertSpace μ) _]
+        exact (Real.exp_pos _).le
+      · have hg := (hD0 d ((le_max_left _ _).trans hd) μ e hkpos (le_of_not_gt htall)).1
+        have hrootpos : (0:ℝ)<Module.finrank ℂ (space μ) := by exact_mod_cast dimension_pos μ
+        have hlogle : Real.log D ≤ Real.log (Module.finrank ℂ (space μ):ℝ) :=
+          Real.log_le_log hDp (by dsimp [D]; exact_mod_cast tail_dimension_le μ)
+        rw [sweep_sq_palindrome]
+        apply hg.trans
+        apply Real.exp_le_exp.mpr
+        have hbound := mul_le_mul hba hlogle hlog ha.le
+        have hck : 0 ≤ C*(k:ℝ) := mul_nonneg hC (Nat.cast_nonneg _)
+        nlinarith
+    · have hpow : 2^d ≤ 2^d0 := Nat.pow_le_pow_right (by decide) (by omega)
+      have hklarge : k ≤ 2^d0 := (by simp [k,hμ] : k ≤ 2^d).trans hpow
+      have hdim : D ≤ ((2^d0).factorial:ℝ) := by
+        dsimp [D]
+        exact_mod_cast (tail_dimension_le_factorial μ).trans (Nat.factorial_le hklarge)
+      have hlC : Real.log D ≤ C := Real.log_le_log hDp hdim
+      have hk1 : (1:ℝ) ≤ k := by exact_mod_cast hkpos
+      have hprod : b*Real.log D ≤ C*(k:ℝ) := by nlinarith
+      apply (sweep_sq_le_one d μ e rev).trans
+      exact Real.one_le_exp_iff.mpr (by nlinarith)
+
+end Thorp.StrongTail
+
+
+namespace Thorp.HarmonicBounds
+open scoped BigOperators Classical
+open Filter Casimir StrongSmoothing
+
+lemma dimension_contraction :
+    ∃ b : ℝ, 0<b ∧ ∀ (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ),
+      0<μ.card-μ.rowLen 0 →
+      W d μ e ≤ Real.exp (-b*Real.log (Module.finrank ℂ (Specht.space μ):ℝ)) := by
+  obtain ⟨a,h,ha,hh,he⟩ := Specht.spectral_regimes
+  obtain ⟨D,hD⟩ := eventually_atTop.mp (he.and (eventually_ge_atTop 1))
+  let B (d : ℕ) : ℝ := 1+Real.log ((2^d).factorial:ℝ)
+  have hB (d : ℕ) : 0<B d := by
+    have hh : 0≤Real.log ((2^d).factorial:ℝ) := Real.log_nonneg (by exact_mod_cast Nat.factorial_pos (2^d))
+    dsimp [B]; linarith
+  let f (d : ℕ) : ℝ := -Real.log (blockGap d)/B d
+  have hf (d : ℕ) : 0<f d :=
+    div_pos (neg_pos.mpr (Real.log_neg (blockGap_pos d) (blockGap_lt_one d))) (hB d)
+  obtain ⟨b,hb,hbD⟩ := finite_positive_lower f D (fun d _ => hf d)
+  refine ⟨min a b,lt_min ha hb,?_⟩
+  intro d μ e hk
+  have hμ : μ.card=2^d := by rw [←Specht.card_cell,←Fintype.card_congr e,card_positions]
+  have hDpos : (0:ℝ)<Module.finrank ℂ (Specht.space μ) := by exact_mod_cast Specht.dimension_pos μ
+  have hL : 0≤Real.log (Module.finrank ℂ (Specht.space μ):ℝ) :=
+    Real.log_nonneg (by exact_mod_cast Specht.dimension_pos μ)
+  by_cases hd : D≤d
+  · obtain ⟨hs,hd1⟩ := hD d hd
+    by_cases ht : (μ.card:ℝ)/2<μ.colLen 0
+    · have hz := Specht.palindrome_tall_zero_general d (by omega) μ e ht
+      change K d μ e=0 at hz
+      change ‖K d μ e‖≤_
+      rw [hz,@norm_zero (Specht.hilbertSpace μ →L[ℂ] Specht.hilbertSpace μ) _]
+      exact (Real.exp_pos _).le
+    · apply (hs μ e hk (le_of_not_gt ht)).1.trans
+      apply Real.exp_le_exp.mpr
+      have hh := mul_le_mul_of_nonneg_right (min_le_left a b) hL
+      linarith
+  · have hdim : Real.log (Module.finrank ℂ (Specht.space μ):ℝ) ≤ B d := by
+      have hh := StrongTail.dimension_le_factorial μ
+      rw [hμ] at hh
+      have hl := Real.log_le_log hDpos (show (Module.finrank ℂ (Specht.space μ):ℝ)≤(2^d).factorial by exact_mod_cast hh)
+      dsimp [B]; linarith
+    have hab : min a b≤f d := (min_le_right _ _).trans (hbD d (by omega))
+    have hp := mul_le_mul_of_nonneg_right hab (hB d).le
+    have heq : f d*B d = -Real.log (blockGap d) := by dsimp [f]; field_simp [ne_of_gt (hB d)]
+    rw [heq] at hp
+    have hscale := mul_le_mul_of_nonneg_left hdim (le_of_lt (lt_min ha hb))
+    calc
+      W d μ e ≤ blockGap d := nontrivial_gap d μ e hk
+      _ = Real.exp (Real.log (blockGap d)) := (Real.exp_log (blockGap_pos d)).symm
+      _ ≤ _ := Real.exp_le_exp.mpr (by linarith)
+
+lemma exp_intersection {x A B : ℝ} (hx : 0≤x) (hA : x≤Real.exp A) (hB : x≤Real.exp B) :
+    x≤Real.exp ((A+B)/2) := by
+  have hh := mul_le_mul hA hB hx (Real.exp_pos A).le
+  rw [←Real.exp_add] at hh
+  have he : Real.exp ((A+B)/2)^2=Real.exp (A+B) := by
+    rw [pow_two,←Real.exp_add]
+    congr 1; ring
+  nlinarith [Real.exp_pos ((A+B)/2)]
+
+lemma flat_tuple_trace (d k : ℕ) (hk : k≤2^d) (order : Equiv.Perm (Fin d)) (rev : Bool) :
+    tupleTrace d k order rev 2001 ≤ Real.exp ((2000*(1+heightDecay 1))*k) := by
+  have hpos : 0<Fintype.card (Tuples d k) := by
+    rw [tuples_card]
+    exact Nat.descFactorial_pos.mpr hk
+  have hne : (Fintype.card (Tuples d k):ℝ)≠0 := by exact_mod_cast ne_of_gt hpos
+  calc
+    _ ≤ ∑ x : Tuples d k, Real.exp ((2000*(1+heightDecay 1))*k)/
+        ((2^d).descFactorial k:ℝ) := by
+      apply Finset.sum_le_sum
+      intro x _
+      exact both_pointwise d k order rev x x
+    _ = _ := by
+      simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,←tuples_card]
+      field_simp
+
+theorem harmonic_cycle_main : MainStatement := by
+  obtain ⟨a,ha,hlevel⟩ := level_contraction
+  obtain ⟨b,hb,hdim⟩ := dimension_contraction
+  let C : ℝ := 2000*(1+heightDecay 1)
+  have hA : 0≤heightDecay 1 := heightDecay_nonneg 1
+  have hC : 0<C := by dsimp [C]; positivity
+  refine ⟨a,C,b,1/1000,min a b/4,a/2,b/2,2001,
+    ha,hC,hb,by norm_num,by norm_num,by positivity,by positivity,by positivity,by norm_num,?_,?_⟩
+  · intro d μ e hk
+    dsimp only
+    let L := levelScale (2^d) (μ.card-μ.rowLen 0)
+    let f : ℝ := Module.finrank ℂ (Specht.space (tail μ))
+    let D : ℝ := Module.finrank ℂ (Specht.space μ)
+    have hμ : μ.card=2^d := by rw [←Specht.card_cell,←Fintype.card_congr e,card_positions]
+    have hkn : μ.card-μ.rowLen 0≤2^d := by omega
+    have hL : 0≤L := (Nat.cast_nonneg _).trans (levelScale_ge _ _ hkn)
+    have hfp : 0<f := by dsimp [f,tail]; exact_mod_cast Specht.dimension_pos (StrongTail.tailDiagram μ)
+    have hfl : 0≤Real.log f := Real.log_nonneg (by dsimp [f,tail]; exact_mod_cast Specht.dimension_pos (StrongTail.tailDiagram μ))
+    have hDp : 0<D := by dsimp [D]; exact_mod_cast Specht.dimension_pos μ
+    have hle : Real.log f≤Real.log D := Real.log_le_log hfp (by
+      dsimp [f,D,tail]; exact_mod_cast StrongTail.tail_dimension_le μ)
+    have hw := hlevel d μ e hk
+    have hd := hdim d μ e hk
+    change W d μ e≤Real.exp (-a*L) at hw
+    change W d μ e≤Real.exp (-b*Real.log D) at hd
+    have ht : W d μ e≤Real.exp (-b*Real.log f) := hd.trans (Real.exp_le_exp.mpr (by nlinarith))
+    refine ⟨hw,?_,?_,?_⟩
+    · change W d μ e≤Real.exp (C*L)*f^(-b)
+      rw [Real.rpow_def_of_pos hfp,←Real.exp_add]
+      exact ht.trans (Real.exp_le_exp.mpr (by nlinarith))
+    · intro rev
+      have hg := exp_intersection (norm_nonneg (K d μ e)) hw ht
+      have hcoef : (-a*L+-b*Real.log f)/2 ≤ -2*(min a b/4)*(L+Real.log f) := by
+        have h₁ := mul_le_mul_of_nonneg_right (min_le_left a b) hL
+        have h₂ := mul_le_mul_of_nonneg_right (min_le_right a b) hfl
+        nlinarith
+      have hg' := hg.trans (Real.exp_le_exp.mpr hcoef)
+      have hs := StrongTail.sweep_sq_palindrome d μ e rev
+      change ‖SparseContact.sweepOperator d μ e rev‖^2=W d μ e at hs
+      change W d μ e≤_ at hg'
+      rw [←hs] at hg'
+      have he : Real.exp (-(min a b/4)*(L+Real.log f))^2 =
+          Real.exp (-2*(min a b/4)*(L+Real.log f)) := by
+        rw [pow_two,←Real.exp_add]
+        congr 1; ring
+      change ‖SparseContact.sweepOperator d μ e rev‖≤Real.exp (-(min a b/4)*(L+Real.log f))
+      nlinarith [Real.exp_pos (-(min a b/4)*(L+Real.log f))]
+    · change W d μ e≤Real.exp (-(a/2)*L)*D^(-(b/2))
+      rw [Real.rpow_def_of_pos hDp,←Real.exp_add]
+      have hg := exp_intersection (norm_nonneg (K d μ e)) hw hd
+      have he : -(a/2)*L+Real.log D*(-(b/2))=(-a*L+-b*Real.log D)/2 := by ring
+      rw [he]
+      exact hg
+  · intro d k _ hk order rev
+    have hL := levelScale_ge (2^d) k hk
+    have hcoef : k*(1+heightDecay 1) ≤ C*levelScale (2^d) k := by
+      have hk0 : (0:ℝ)≤k := by positivity
+      dsimp [C]; nlinarith
+    refine ⟨?_,?_⟩
+    · intro x
+      exact (StrongTail.reflected_moment_both d k order rev x).trans (Real.exp_le_exp.mpr hcoef)
+    · exact (flat_tuple_trace d k hk order rev).trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left hL hC.le))
+
+end Thorp.HarmonicBounds
+
+end ThorpNine.Harmonic
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Harmonic/Model.lean b/lean/OAI/Probability/ThorpRouting/Harmonic/Model.lean
index bbcc5111f939b3fca02acea8763c2aec0bda6462..25c65fa63ca3a4b419268742b9ab3c31146b68f5 100644
--- a/lean/OAI/Probability/ThorpRouting/Harmonic/Model.lean
+++ b/lean/OAI/Probability/ThorpRouting/Harmonic/Model.lean
@@ -1,364 +1,364 @@
-import Mathlib
-
-namespace OAI
-
-namespace ThorpNine.Harmonic
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-
-abbrev Card (d : ℕ) := Fin d → Bool
-
-def switchFun {d : ℕ} (ξ : Card d → Bool) (x : Card (d + 1)) : Card (d + 1) :=
-  Fin.cons (Bool.xor (x 0) (ξ (Fin.tail x))) (Fin.tail x)
-
-theorem switchFun_involutive {d : ℕ} (ξ : Card d → Bool) :
-    Function.Involutive (switchFun ξ) := by
-  intro x
-  funext i
-  refine Fin.cases ?_ (fun j => ?_) i
-  · simp only [switchFun, Fin.cons_zero, Fin.tail_cons]
-    cases x 0 <;> cases ξ (Fin.tail x) <;> rfl
-  · simp [switchFun, Fin.tail]
-
-def pairSwitch {d : ℕ} (ξ : Card d → Bool) : Equiv.Perm (Card (d + 1)) :=
-  { toFun := switchFun ξ
-    invFun := switchFun ξ
-    left_inv := switchFun_involutive ξ
-    right_inv := switchFun_involutive ξ }
-
-def Butterfly : ℕ → Type
-  | 0 => Unit
-  | d + 1 => (Card d → Bool) × (Bool → Butterfly d)
-
-def childLift {d : ℕ} (p : Bool → Equiv.Perm (Card d)) : Equiv.Perm (Card (d + 1)) where
-  toFun x := Fin.cons (x 0) (p (x 0) (Fin.tail x))
-  invFun x := Fin.cons (x 0) ((p (x 0)).symm (Fin.tail x))
-  left_inv x := by simp only [Fin.cons_zero, Fin.tail_cons, Equiv.symm_apply_apply, Fin.cons_self_tail]
-  right_inv x := by simp only [Fin.cons_zero, Fin.tail_cons, Equiv.apply_symm_apply, Fin.cons_self_tail]
-
-def butterflyPerm : (d : ℕ) → Butterfly d → Equiv.Perm (Card d)
-  | 0, _ => 1
-  | d + 1, b => pairSwitch b.1 * childLift (fun ε => butterflyPerm d (b.2 ε))
-
-noncomputable def finiteMean {Ω : Type*} [Fintype Ω] (f : Ω → ℝ) : ℝ :=
-  (∑ ω, f ω) / Fintype.card Ω
-
-def SwitchIndex : ℕ → Type
-  | 0 => Empty
-  | d + 1 => Sum (Card d) (Bool × SwitchIndex d)
-
-instance (d : ℕ) : Fintype (SwitchIndex d) := by
-  induction d with
-  | zero => exact inferInstanceAs (Fintype Empty)
-  | succ d ih => exact inferInstanceAs (Fintype (Sum (Card d) (Bool × SwitchIndex d)))
-
-instance (d : ℕ) : DecidableEq (SwitchIndex d) := by
-  induction d with
-  | zero => exact inferInstanceAs (DecidableEq Empty)
-  | succ d ih => exact inferInstanceAs (DecidableEq (Sum (Card d) (Bool × SwitchIndex d)))
-
-def decodeButterfly : (d : ℕ) → (SwitchIndex d → Bool) → Butterfly d
-  | 0, _ => ()
-  | d + 1, ω => (fun y => ω (Sum.inl y), fun ε =>
-      decodeButterfly d (fun i => ω (Sum.inr (ε,i))))
-
-end Thorp
-
-namespace Thorp.PairRouting
-open scoped BigOperators
-open Filter
-variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
-
-noncomputable def tupleProbability {Ω β : Type*} [Fintype Ω] [DecidableEq β]
-    (P : Ω → Equiv.Perm β) (x y : ι → β) : ℝ := by
-  classical
-  exact finiteMean (fun ω => if ∀ i, P ω (x i) = y i then 1 else 0)
-
-end Thorp.PairRouting
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-
-abbrev BenesCoins (d : ℕ) := (SwitchIndex d → Bool) × (SwitchIndex d → Bool)
-
-def palindromePerm (d : ℕ) (ω : BenesCoins d) : Equiv.Perm (Card d) :=
-  butterflyPerm d (decodeButterfly d ω.1) * (butterflyPerm d (decodeButterfly d ω.2)).symm
-
-end Thorp
-
-namespace Thorp.Specht
-open scoped BigOperators Classical
-
-abbrev Cell (μ : YoungDiagram) := {x : ℕ × ℕ // x ∈ μ.cells}
-
-def row {μ : YoungDiagram} (x : Cell μ) : ℕ := x.1.1
-
-def col {μ : YoungDiagram} (x : Cell μ) : ℕ := x.1.2
-
-lemma row_lt_colLen {μ : YoungDiagram} (x : Cell μ) : row x < μ.colLen (col x) :=
-  YoungDiagram.mem_iff_lt_colLen.mp x.2
-
-noncomputable def rowIndex {μ : YoungDiagram} (x : Cell μ) : Fin (μ.colLen 0) :=
-  ⟨row x, (row_lt_colLen x).trans_le (μ.colLen_anti 0 (col x) (Nat.zero_le _))⟩
-
-abbrev Tabloid (μ : YoungDiagram) :=
-  {f : Cell μ → Fin (μ.colLen 0) // ∃ p : Equiv.Perm (Cell μ), f = rowIndex ∘ p}
-
-noncomputable instance (μ : YoungDiagram) : Fintype (Tabloid μ) := Fintype.ofFinite _
-
-noncomputable def baseTabloid (μ : YoungDiagram) : Tabloid μ := ⟨rowIndex, 1, rfl⟩
-
-noncomputable def tabloidAct {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) :
-    Equiv.Perm (Tabloid μ) where
-  toFun f := ⟨fun x => f.1 (p⁻¹ x), by
-    obtain ⟨q,hq⟩ := f.2
-    refine ⟨q*p⁻¹, ?_⟩
-    funext x
-    simp only [hq,Function.comp_apply,Equiv.Perm.mul_apply]⟩
-  invFun f := ⟨fun x => f.1 (p x), by
-    obtain ⟨q,hq⟩ := f.2
-    refine ⟨q*p, ?_⟩
-    funext x
-    simp only [hq,Function.comp_apply,Equiv.Perm.mul_apply]⟩
-  left_inv f := by apply Subtype.ext; funext x; simp
-  right_inv f := by apply Subtype.ext; funext x; simp
-
-@[simp] lemma tabloidAct_apply {μ : YoungDiagram} (p : Equiv.Perm (Cell μ))
-    (f : Tabloid μ) (x : Cell μ) : (tabloidAct p f).1 x = f.1 (p⁻¹ x) := rfl
-
-@[simp] lemma tabloidAct_one (μ : YoungDiagram) : tabloidAct (1 : Equiv.Perm (Cell μ)) = 1 := by
-  ext f x
-  rfl
-
-@[simp] lemma tabloidAct_mul {μ : YoungDiagram} (p q : Equiv.Perm (Cell μ)) :
-    tabloidAct (p*q) = tabloidAct p * tabloidAct q := by
-  ext f x
-  simp only [tabloidAct_apply, mul_inv_rev,Equiv.Perm.mul_apply]
-
-noncomputable def tabloidRep (μ : YoungDiagram) :
-    Representation ℂ (Equiv.Perm (Cell μ)) (Tabloid μ → ℂ) where
-  toFun p :=
-    { toFun := fun v f => v (tabloidAct p⁻¹ f)
-      map_add' := by intros; rfl
-      map_smul' := by intros; rfl }
-  map_one' := by ext v f; simp
-  map_mul' p q := by
-    ext v f
-    simp only [mul_inv_rev,tabloidAct_mul,Equiv.Perm.mul_apply]
-    rfl
-
-noncomputable def delta {μ : YoungDiagram} (f : Tabloid μ) : Tabloid μ → ℂ :=
-  fun g => if g=f then 1 else 0
-
-noncomputable def colGroup (μ : YoungDiagram) : Subgroup (Equiv.Perm (Cell μ)) where
-  carrier := {p | ∀ x, col (p x) = col x}
-  one_mem' := by intro x; rfl
-  mul_mem' := by intro p q hp hq x; exact (hp (q x)).trans (hq x)
-  inv_mem' := by intro p hp x; simpa using (hp (p⁻¹ x)).symm
-
-noncomputable def permSign {α : Type*} [Fintype α] [DecidableEq α] (p : Equiv.Perm α) : ℂ :=
-  ((Equiv.Perm.sign p : ℤ) : ℂ)
-
-
-section
-variable {α V : Type*} [Fintype α] [DecidableEq α] [AddCommGroup V] [Module ℂ V]
-
-noncomputable def alternator (C : Subgroup (Equiv.Perm α))
-    (ρ : Representation ℂ (Equiv.Perm α) V) : Module.End ℂ V := by
-  classical
-  exact ∑ c : C, permSign (c : Equiv.Perm α) • ρ c
-
-end
-
-noncomputable def polytabloid (μ : YoungDiagram) : Tabloid μ → ℂ :=
-  alternator (colGroup μ) (tabloidRep μ) (delta (baseTabloid μ))
-
-noncomputable def space (μ : YoungDiagram) : Submodule ℂ (Tabloid μ → ℂ) :=
-  Submodule.span ℂ (Set.range (fun p : Equiv.Perm (Cell μ) => tabloidRep μ p (polytabloid μ)))
-
-lemma orbit_mem_space (μ : YoungDiagram) (p : Equiv.Perm (Cell μ)) :
-    tabloidRep μ p (polytabloid μ) ∈ space μ := Submodule.subset_span ⟨p,rfl⟩
-
-lemma action_mem_space (μ : YoungDiagram) (p : Equiv.Perm (Cell μ))
-    (v : Tabloid μ → ℂ) (hv : v ∈ space μ) : tabloidRep μ p v ∈ space μ := by
-  induction hv using Submodule.span_induction with
-  | mem v hv =>
-    obtain ⟨q,rfl⟩ := hv
-    change ((tabloidRep μ p)*(tabloidRep μ q)) (polytabloid μ) ∈ space μ
-    rw [←map_mul]
-    exact orbit_mem_space μ (p*q)
-  | zero => simp
-  | add v w hv hw ihv ihw => simpa using (space μ).add_mem ihv ihw
-  | smul a v hv ih => simpa using (space μ).smul_mem a ih
-
-noncomputable def subrepresentation (μ : YoungDiagram) : Subrepresentation (tabloidRep μ) :=
-  ⟨space μ, fun p v hv => action_mem_space μ p v hv⟩
-
-noncomputable def representation (μ : YoungDiagram) :
-    Representation ℂ (Equiv.Perm (Cell μ)) (space μ) :=
-  (subrepresentation μ).toRepresentation
-
-end Thorp.Specht
-
-namespace Thorp.PermutationHilbert
-open scoped BigOperators ComplexConjugate Classical
-open Complex
-variable {X : Type*} [Fintype X]
-
-abbrev H (X : Type*) [Fintype X] := EuclideanSpace ℂ X
-
-end Thorp.PermutationHilbert
-
-namespace Thorp.Specht
-open scoped BigOperators Classical
-
-noncomputable def hilbertEquiv (μ : YoungDiagram) :
-    (Tabloid μ → ℂ) ≃ₗ[ℂ] PermutationHilbert.H (Tabloid μ) :=
-  (WithLp.linearEquiv 2 ℂ (Tabloid μ → ℂ)).symm
-
-noncomputable def hilbertSpace (μ : YoungDiagram) :
-    Submodule ℂ (PermutationHilbert.H (Tabloid μ)) := (space μ).map (hilbertEquiv μ).toLinearMap
-
-end Thorp.Specht
-
-namespace Thorp.UnitaryFinite
-
-section
-open scoped BigOperators ComplexConjugate Classical
-variable {G : Type*} [Group G] [Fintype G]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-noncomputable def continuousRepresentation (ρ : Representation ℂ G V) : G →* (V →L[ℂ] V) where
-  toFun g := (ρ g).toContinuousLinearMap
-  map_one' := by ext v; simp
-  map_mul' g h := by ext v; simp
-
-end
-open scoped BigOperators Classical
-variable {G : Type*} [Group G] [Fintype G]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-variable {Ω Ξ : Type*} [Fintype Ω] [Fintype Ξ]
-
-noncomputable def sampleOperator (ρ : Representation ℂ G V) (P : Ω → G) : V →L[ℂ] V :=
-  (Fintype.card Ω:ℂ)⁻¹ • ∑ ω,continuousRepresentation ρ (P ω)
-
-end Thorp.UnitaryFinite
-
-namespace Thorp.Specht
-open scoped BigOperators Classical
-
-noncomputable def spaceHilbertEquiv (μ : YoungDiagram) : space μ ≃ₗ[ℂ] hilbertSpace μ :=
-  (hilbertEquiv μ).submoduleMap (space μ)
-
-noncomputable def unitaryRepresentation (μ : YoungDiagram) :
-    Representation ℂ (Equiv.Perm (Cell μ)) (hilbertSpace μ) :=
-  ((spaceHilbertEquiv μ).conjAlgEquiv ℂ).toMonoidHom.comp (representation μ)
-
-variable {α ι : Type*} [Fintype α] [Fintype ι]
-
-noncomputable def relabelledUnitary (μ : YoungDiagram) (e : α ≃ Cell μ) :
-    Representation ℂ (Equiv.Perm α) (hilbertSpace μ) :=
-  (unitaryRepresentation μ).comp e.permCongrHom.toMonoidHom
-
-end Thorp.Specht
-
-namespace Thorp.SparseContact
-open scoped BigOperators
-
-noncomputable def sweepOperator (d : ℕ) (μ : YoungDiagram)
-    (e : Card d ≃ Specht.Cell μ) (reverse : Bool) :
-    Specht.hilbertSpace μ →L[ℂ] Specht.hilbertSpace μ :=
-  UnitaryFinite.sampleOperator (V := Specht.hilbertSpace μ) (Specht.relabelledUnitary μ e)
-    (fun ω : SwitchIndex d → Bool => if reverse then
-      (butterflyPerm d (decodeButterfly d ω))⁻¹ else butterflyPerm d (decodeButterfly d ω))
-
-end Thorp.SparseContact
-
-namespace Thorp.StrongSmoothing
-
-def coordinateRelabel (d : ℕ) (order : Equiv.Perm (Fin d)) : Equiv.Perm (Card d) where
-  toFun x := fun i => x (order.symm i)
-  invFun x := fun i => x (order i)
-  left_inv x := by funext i; simp
-  right_inv x := by funext i; simp
-
-def sweep (d : ℕ) (order : Equiv.Perm (Fin d)) (ω : SwitchIndex d → Bool) :
-    Equiv.Perm (Card d) :=
-  coordinateRelabel d order * butterflyPerm d (decodeButterfly d ω) *
-    (coordinateRelabel d order)⁻¹
-
-abbrev Tuples (d l : ℕ) := Fin l ↪ Card d
-
-noncomputable def sweepKernel (d l : ℕ) (order : Equiv.Perm (Fin d)) :
-    Matrix (Tuples d l) (Tuples d l) ℝ :=
-  fun x y => PairRouting.tupleProbability (sweep d order) x y
-
-noncomputable def reflectedKernel (d l : ℕ) (order : Equiv.Perm (Fin d)) (orientation : Bool) :
-    Matrix (Tuples d l) (Tuples d l) ℝ :=
-  let A := sweepKernel d l order
-  if orientation then A * A.transpose else A.transpose * A
-
-end Thorp.StrongSmoothing
-
-namespace Thorp.StrongTail
-open scoped BigOperators Classical
-open Specht
-
-def tailDiagram (μ : YoungDiagram) : YoungDiagram :=
-  YoungDiagram.ofRowLens μ.rowLens.tail μ.rowLens_sorted.pairwise.tail.sortedGE
-
-end Thorp.StrongTail
-
-namespace Thorp.Casimir
-open scoped BigOperators Classical
-open Filter
-
-noncomputable def levelScale (n k : ℕ) : ℝ :=
-  (k:ℝ) * (1 + Real.log ((n:ℝ)/(k:ℝ)))
-
-noncomputable def K (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ) :
-    Specht.hilbertSpace μ →L[ℂ] Specht.hilbertSpace μ :=
-  UnitaryFinite.sampleOperator (V := Specht.hilbertSpace μ)
-    (Specht.relabelledUnitary μ e) (palindromePerm d)
-
-end Thorp.Casimir
-
-namespace Thorp.HarmonicBounds
-open scoped BigOperators Classical
-open Casimir StrongSmoothing
-
-abbrev tail (μ : YoungDiagram) := StrongTail.tailDiagram μ
-
-noncomputable def W (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ) : ℝ := ‖K d μ e‖
-
-noncomputable def tupleTrace (d k : ℕ) (order : Equiv.Perm (Fin d)) (rev : Bool) (p : ℕ) : ℝ :=
-  Matrix.trace ((reflectedKernel d k order rev)^p)
-
-def MainStatement : Prop :=
-  ∃ c C ζ δ cTail c₁ c₂ : ℝ, ∃ p₀ : ℕ,
-    0<c ∧ 0<C ∧ 0<ζ ∧ 0<δ ∧ δ<1 ∧ 0<cTail ∧ 0<c₁ ∧ 0<c₂ ∧ 0<p₀ ∧
-    (∀ (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ),
-      0<μ.card-μ.rowLen 0 →
-      let L := levelScale (2^d) (μ.card-μ.rowLen 0)
-      let f : ℝ := Module.finrank ℂ (Specht.space (tail μ))
-      let D : ℝ := Module.finrank ℂ (Specht.space μ)
-      W d μ e ≤ Real.exp (-c*L) ∧
-      W d μ e ≤ Real.exp (C*L)*f^(-ζ) ∧
-      (∀ rev : Bool, ‖SparseContact.sweepOperator d μ e rev‖ ≤
-        Real.exp (-cTail*(L+Real.log f))) ∧
-      W d μ e ≤ Real.exp (-c₁*L)*D^(-c₂)) ∧
-    (∀ d k : ℕ, 1≤k → k≤2^d → ∀ (order : Equiv.Perm (Fin d)) (rev : Bool),
-      (∀ x : Tuples d k,
-        finiteMean (fun y => ((Fintype.card (Tuples d k):ℝ)*
-          reflectedKernel d k order rev x y)^(1+δ)) ≤
-        Real.exp (C*levelScale (2^d) k)) ∧
-      tupleTrace d k order rev p₀ ≤ Real.exp (C*levelScale (2^d) k))
-
-end Thorp.HarmonicBounds
-
-end ThorpNine.Harmonic
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+namespace ThorpNine.Harmonic
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+
+abbrev Card (d : ℕ) := Fin d → Bool
+
+def switchFun {d : ℕ} (ξ : Card d → Bool) (x : Card (d + 1)) : Card (d + 1) :=
+  Fin.cons (Bool.xor (x 0) (ξ (Fin.tail x))) (Fin.tail x)
+
+theorem switchFun_involutive {d : ℕ} (ξ : Card d → Bool) :
+    Function.Involutive (switchFun ξ) := by
+  intro x
+  funext i
+  refine Fin.cases ?_ (fun j => ?_) i
+  · simp only [switchFun, Fin.cons_zero, Fin.tail_cons]
+    cases x 0 <;> cases ξ (Fin.tail x) <;> rfl
+  · simp [switchFun, Fin.tail]
+
+def pairSwitch {d : ℕ} (ξ : Card d → Bool) : Equiv.Perm (Card (d + 1)) :=
+  { toFun := switchFun ξ
+    invFun := switchFun ξ
+    left_inv := switchFun_involutive ξ
+    right_inv := switchFun_involutive ξ }
+
+def Butterfly : ℕ → Type
+  | 0 => Unit
+  | d + 1 => (Card d → Bool) × (Bool → Butterfly d)
+
+def childLift {d : ℕ} (p : Bool → Equiv.Perm (Card d)) : Equiv.Perm (Card (d + 1)) where
+  toFun x := Fin.cons (x 0) (p (x 0) (Fin.tail x))
+  invFun x := Fin.cons (x 0) ((p (x 0)).symm (Fin.tail x))
+  left_inv x := by simp only [Fin.cons_zero, Fin.tail_cons, Equiv.symm_apply_apply, Fin.cons_self_tail]
+  right_inv x := by simp only [Fin.cons_zero, Fin.tail_cons, Equiv.apply_symm_apply, Fin.cons_self_tail]
+
+def butterflyPerm : (d : ℕ) → Butterfly d → Equiv.Perm (Card d)
+  | 0, _ => 1
+  | d + 1, b => pairSwitch b.1 * childLift (fun ε => butterflyPerm d (b.2 ε))
+
+noncomputable def finiteMean {Ω : Type*} [Fintype Ω] (f : Ω → ℝ) : ℝ :=
+  (∑ ω, f ω) / Fintype.card Ω
+
+def SwitchIndex : ℕ → Type
+  | 0 => Empty
+  | d + 1 => Sum (Card d) (Bool × SwitchIndex d)
+
+instance (d : ℕ) : Fintype (SwitchIndex d) := by
+  induction d with
+  | zero => exact inferInstanceAs (Fintype Empty)
+  | succ d ih => exact inferInstanceAs (Fintype (Sum (Card d) (Bool × SwitchIndex d)))
+
+instance (d : ℕ) : DecidableEq (SwitchIndex d) := by
+  induction d with
+  | zero => exact inferInstanceAs (DecidableEq Empty)
+  | succ d ih => exact inferInstanceAs (DecidableEq (Sum (Card d) (Bool × SwitchIndex d)))
+
+def decodeButterfly : (d : ℕ) → (SwitchIndex d → Bool) → Butterfly d
+  | 0, _ => ()
+  | d + 1, ω => (fun y => ω (Sum.inl y), fun ε =>
+      decodeButterfly d (fun i => ω (Sum.inr (ε,i))))
+
+end Thorp
+
+namespace Thorp.PairRouting
+open scoped BigOperators
+open Filter
+variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
+
+noncomputable def tupleProbability {Ω β : Type*} [Fintype Ω] [DecidableEq β]
+    (P : Ω → Equiv.Perm β) (x y : ι → β) : ℝ := by
+  classical
+  exact finiteMean (fun ω => if ∀ i, P ω (x i) = y i then 1 else 0)
+
+end Thorp.PairRouting
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+
+abbrev BenesCoins (d : ℕ) := (SwitchIndex d → Bool) × (SwitchIndex d → Bool)
+
+def palindromePerm (d : ℕ) (ω : BenesCoins d) : Equiv.Perm (Card d) :=
+  butterflyPerm d (decodeButterfly d ω.1) * (butterflyPerm d (decodeButterfly d ω.2)).symm
+
+end Thorp
+
+namespace Thorp.Specht
+open scoped BigOperators Classical
+
+abbrev Cell (μ : YoungDiagram) := {x : ℕ × ℕ // x ∈ μ.cells}
+
+def row {μ : YoungDiagram} (x : Cell μ) : ℕ := x.1.1
+
+def col {μ : YoungDiagram} (x : Cell μ) : ℕ := x.1.2
+
+lemma row_lt_colLen {μ : YoungDiagram} (x : Cell μ) : row x < μ.colLen (col x) :=
+  YoungDiagram.mem_iff_lt_colLen.mp x.2
+
+noncomputable def rowIndex {μ : YoungDiagram} (x : Cell μ) : Fin (μ.colLen 0) :=
+  ⟨row x, (row_lt_colLen x).trans_le (μ.colLen_anti 0 (col x) (Nat.zero_le _))⟩
+
+abbrev Tabloid (μ : YoungDiagram) :=
+  {f : Cell μ → Fin (μ.colLen 0) // ∃ p : Equiv.Perm (Cell μ), f = rowIndex ∘ p}
+
+noncomputable instance (μ : YoungDiagram) : Fintype (Tabloid μ) := Fintype.ofFinite _
+
+noncomputable def baseTabloid (μ : YoungDiagram) : Tabloid μ := ⟨rowIndex, 1, rfl⟩
+
+noncomputable def tabloidAct {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) :
+    Equiv.Perm (Tabloid μ) where
+  toFun f := ⟨fun x => f.1 (p⁻¹ x), by
+    obtain ⟨q,hq⟩ := f.2
+    refine ⟨q*p⁻¹, ?_⟩
+    funext x
+    simp only [hq,Function.comp_apply,Equiv.Perm.mul_apply]⟩
+  invFun f := ⟨fun x => f.1 (p x), by
+    obtain ⟨q,hq⟩ := f.2
+    refine ⟨q*p, ?_⟩
+    funext x
+    simp only [hq,Function.comp_apply,Equiv.Perm.mul_apply]⟩
+  left_inv f := by apply Subtype.ext; funext x; simp
+  right_inv f := by apply Subtype.ext; funext x; simp
+
+@[simp] lemma tabloidAct_apply {μ : YoungDiagram} (p : Equiv.Perm (Cell μ))
+    (f : Tabloid μ) (x : Cell μ) : (tabloidAct p f).1 x = f.1 (p⁻¹ x) := rfl
+
+@[simp] lemma tabloidAct_one (μ : YoungDiagram) : tabloidAct (1 : Equiv.Perm (Cell μ)) = 1 := by
+  ext f x
+  rfl
+
+@[simp] lemma tabloidAct_mul {μ : YoungDiagram} (p q : Equiv.Perm (Cell μ)) :
+    tabloidAct (p*q) = tabloidAct p * tabloidAct q := by
+  ext f x
+  simp only [tabloidAct_apply, mul_inv_rev,Equiv.Perm.mul_apply]
+
+noncomputable def tabloidRep (μ : YoungDiagram) :
+    Representation ℂ (Equiv.Perm (Cell μ)) (Tabloid μ → ℂ) where
+  toFun p :=
+    { toFun := fun v f => v (tabloidAct p⁻¹ f)
+      map_add' := by intros; rfl
+      map_smul' := by intros; rfl }
+  map_one' := by ext v f; simp
+  map_mul' p q := by
+    ext v f
+    simp only [mul_inv_rev,tabloidAct_mul,Equiv.Perm.mul_apply]
+    rfl
+
+noncomputable def delta {μ : YoungDiagram} (f : Tabloid μ) : Tabloid μ → ℂ :=
+  fun g => if g=f then 1 else 0
+
+noncomputable def colGroup (μ : YoungDiagram) : Subgroup (Equiv.Perm (Cell μ)) where
+  carrier := {p | ∀ x, col (p x) = col x}
+  one_mem' := by intro x; rfl
+  mul_mem' := by intro p q hp hq x; exact (hp (q x)).trans (hq x)
+  inv_mem' := by intro p hp x; simpa using (hp (p⁻¹ x)).symm
+
+noncomputable def permSign {α : Type*} [Fintype α] [DecidableEq α] (p : Equiv.Perm α) : ℂ :=
+  ((Equiv.Perm.sign p : ℤ) : ℂ)
+
+
+section
+variable {α V : Type*} [Fintype α] [DecidableEq α] [AddCommGroup V] [Module ℂ V]
+
+noncomputable def alternator (C : Subgroup (Equiv.Perm α))
+    (ρ : Representation ℂ (Equiv.Perm α) V) : Module.End ℂ V := by
+  classical
+  exact ∑ c : C, permSign (c : Equiv.Perm α) • ρ c
+
+end
+
+noncomputable def polytabloid (μ : YoungDiagram) : Tabloid μ → ℂ :=
+  alternator (colGroup μ) (tabloidRep μ) (delta (baseTabloid μ))
+
+noncomputable def space (μ : YoungDiagram) : Submodule ℂ (Tabloid μ → ℂ) :=
+  Submodule.span ℂ (Set.range (fun p : Equiv.Perm (Cell μ) => tabloidRep μ p (polytabloid μ)))
+
+lemma orbit_mem_space (μ : YoungDiagram) (p : Equiv.Perm (Cell μ)) :
+    tabloidRep μ p (polytabloid μ) ∈ space μ := Submodule.subset_span ⟨p,rfl⟩
+
+lemma action_mem_space (μ : YoungDiagram) (p : Equiv.Perm (Cell μ))
+    (v : Tabloid μ → ℂ) (hv : v ∈ space μ) : tabloidRep μ p v ∈ space μ := by
+  induction hv using Submodule.span_induction with
+  | mem v hv =>
+    obtain ⟨q,rfl⟩ := hv
+    change ((tabloidRep μ p)*(tabloidRep μ q)) (polytabloid μ) ∈ space μ
+    rw [←map_mul]
+    exact orbit_mem_space μ (p*q)
+  | zero => simp
+  | add v w hv hw ihv ihw => simpa using (space μ).add_mem ihv ihw
+  | smul a v hv ih => simpa using (space μ).smul_mem a ih
+
+noncomputable def subrepresentation (μ : YoungDiagram) : Subrepresentation (tabloidRep μ) :=
+  ⟨space μ, fun p v hv => action_mem_space μ p v hv⟩
+
+noncomputable def representation (μ : YoungDiagram) :
+    Representation ℂ (Equiv.Perm (Cell μ)) (space μ) :=
+  (subrepresentation μ).toRepresentation
+
+end Thorp.Specht
+
+namespace Thorp.PermutationHilbert
+open scoped BigOperators ComplexConjugate Classical
+open Complex
+variable {X : Type*} [Fintype X]
+
+abbrev H (X : Type*) [Fintype X] := EuclideanSpace ℂ X
+
+end Thorp.PermutationHilbert
+
+namespace Thorp.Specht
+open scoped BigOperators Classical
+
+noncomputable def hilbertEquiv (μ : YoungDiagram) :
+    (Tabloid μ → ℂ) ≃ₗ[ℂ] PermutationHilbert.H (Tabloid μ) :=
+  (WithLp.linearEquiv 2 ℂ (Tabloid μ → ℂ)).symm
+
+noncomputable def hilbertSpace (μ : YoungDiagram) :
+    Submodule ℂ (PermutationHilbert.H (Tabloid μ)) := (space μ).map (hilbertEquiv μ).toLinearMap
+
+end Thorp.Specht
+
+namespace Thorp.UnitaryFinite
+
+section
+open scoped BigOperators ComplexConjugate Classical
+variable {G : Type*} [Group G] [Fintype G]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+noncomputable def continuousRepresentation (ρ : Representation ℂ G V) : G →* (V →L[ℂ] V) where
+  toFun g := (ρ g).toContinuousLinearMap
+  map_one' := by ext v; simp
+  map_mul' g h := by ext v; simp
+
+end
+open scoped BigOperators Classical
+variable {G : Type*} [Group G] [Fintype G]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+variable {Ω Ξ : Type*} [Fintype Ω] [Fintype Ξ]
+
+noncomputable def sampleOperator (ρ : Representation ℂ G V) (P : Ω → G) : V →L[ℂ] V :=
+  (Fintype.card Ω:ℂ)⁻¹ • ∑ ω,continuousRepresentation ρ (P ω)
+
+end Thorp.UnitaryFinite
+
+namespace Thorp.Specht
+open scoped BigOperators Classical
+
+noncomputable def spaceHilbertEquiv (μ : YoungDiagram) : space μ ≃ₗ[ℂ] hilbertSpace μ :=
+  (hilbertEquiv μ).submoduleMap (space μ)
+
+noncomputable def unitaryRepresentation (μ : YoungDiagram) :
+    Representation ℂ (Equiv.Perm (Cell μ)) (hilbertSpace μ) :=
+  ((spaceHilbertEquiv μ).conjAlgEquiv ℂ).toMonoidHom.comp (representation μ)
+
+variable {α ι : Type*} [Fintype α] [Fintype ι]
+
+noncomputable def relabelledUnitary (μ : YoungDiagram) (e : α ≃ Cell μ) :
+    Representation ℂ (Equiv.Perm α) (hilbertSpace μ) :=
+  (unitaryRepresentation μ).comp e.permCongrHom.toMonoidHom
+
+end Thorp.Specht
+
+namespace Thorp.SparseContact
+open scoped BigOperators
+
+noncomputable def sweepOperator (d : ℕ) (μ : YoungDiagram)
+    (e : Card d ≃ Specht.Cell μ) (reverse : Bool) :
+    Specht.hilbertSpace μ →L[ℂ] Specht.hilbertSpace μ :=
+  UnitaryFinite.sampleOperator (V := Specht.hilbertSpace μ) (Specht.relabelledUnitary μ e)
+    (fun ω : SwitchIndex d → Bool => if reverse then
+      (butterflyPerm d (decodeButterfly d ω))⁻¹ else butterflyPerm d (decodeButterfly d ω))
+
+end Thorp.SparseContact
+
+namespace Thorp.StrongSmoothing
+
+def coordinateRelabel (d : ℕ) (order : Equiv.Perm (Fin d)) : Equiv.Perm (Card d) where
+  toFun x := fun i => x (order.symm i)
+  invFun x := fun i => x (order i)
+  left_inv x := by funext i; simp
+  right_inv x := by funext i; simp
+
+def sweep (d : ℕ) (order : Equiv.Perm (Fin d)) (ω : SwitchIndex d → Bool) :
+    Equiv.Perm (Card d) :=
+  coordinateRelabel d order * butterflyPerm d (decodeButterfly d ω) *
+    (coordinateRelabel d order)⁻¹
+
+abbrev Tuples (d l : ℕ) := Fin l ↪ Card d
+
+noncomputable def sweepKernel (d l : ℕ) (order : Equiv.Perm (Fin d)) :
+    Matrix (Tuples d l) (Tuples d l) ℝ :=
+  fun x y => PairRouting.tupleProbability (sweep d order) x y
+
+noncomputable def reflectedKernel (d l : ℕ) (order : Equiv.Perm (Fin d)) (orientation : Bool) :
+    Matrix (Tuples d l) (Tuples d l) ℝ :=
+  let A := sweepKernel d l order
+  if orientation then A * A.transpose else A.transpose * A
+
+end Thorp.StrongSmoothing
+
+namespace Thorp.StrongTail
+open scoped BigOperators Classical
+open Specht
+
+def tailDiagram (μ : YoungDiagram) : YoungDiagram :=
+  YoungDiagram.ofRowLens μ.rowLens.tail μ.rowLens_sorted.pairwise.tail.sortedGE
+
+end Thorp.StrongTail
+
+namespace Thorp.Casimir
+open scoped BigOperators Classical
+open Filter
+
+noncomputable def levelScale (n k : ℕ) : ℝ :=
+  (k:ℝ) * (1 + Real.log ((n:ℝ)/(k:ℝ)))
+
+noncomputable def K (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ) :
+    Specht.hilbertSpace μ →L[ℂ] Specht.hilbertSpace μ :=
+  UnitaryFinite.sampleOperator (V := Specht.hilbertSpace μ)
+    (Specht.relabelledUnitary μ e) (palindromePerm d)
+
+end Thorp.Casimir
+
+namespace Thorp.HarmonicBounds
+open scoped BigOperators Classical
+open Casimir StrongSmoothing
+
+abbrev tail (μ : YoungDiagram) := StrongTail.tailDiagram μ
+
+noncomputable def W (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ) : ℝ := ‖K d μ e‖
+
+noncomputable def tupleTrace (d k : ℕ) (order : Equiv.Perm (Fin d)) (rev : Bool) (p : ℕ) : ℝ :=
+  Matrix.trace ((reflectedKernel d k order rev)^p)
+
+def MainStatement : Prop :=
+  ∃ c C ζ δ cTail c₁ c₂ : ℝ, ∃ p₀ : ℕ,
+    0<c ∧ 0<C ∧ 0<ζ ∧ 0<δ ∧ δ<1 ∧ 0<cTail ∧ 0<c₁ ∧ 0<c₂ ∧ 0<p₀ ∧
+    (∀ (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ),
+      0<μ.card-μ.rowLen 0 →
+      let L := levelScale (2^d) (μ.card-μ.rowLen 0)
+      let f : ℝ := Module.finrank ℂ (Specht.space (tail μ))
+      let D : ℝ := Module.finrank ℂ (Specht.space μ)
+      W d μ e ≤ Real.exp (-c*L) ∧
+      W d μ e ≤ Real.exp (C*L)*f^(-ζ) ∧
+      (∀ rev : Bool, ‖SparseContact.sweepOperator d μ e rev‖ ≤
+        Real.exp (-cTail*(L+Real.log f))) ∧
+      W d μ e ≤ Real.exp (-c₁*L)*D^(-c₂)) ∧
+    (∀ d k : ℕ, 1≤k → k≤2^d → ∀ (order : Equiv.Perm (Fin d)) (rev : Bool),
+      (∀ x : Tuples d k,
+        finiteMean (fun y => ((Fintype.card (Tuples d k):ℝ)*
+          reflectedKernel d k order rev x y)^(1+δ)) ≤
+        Real.exp (C*levelScale (2^d) k)) ∧
+      tupleTrace d k order rev p₀ ≤ Real.exp (C*levelScale (2^d) k))
+
+end Thorp.HarmonicBounds
+
+end ThorpNine.Harmonic
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Harmonic/ModifiedReverseCoins.lean b/lean/OAI/Probability/ThorpRouting/Harmonic/ModifiedReverseCoins.lean
index 9108027a38e464cd7ca276e9407fddd5c25d4c8d..505dbfe909a6f97fe110fe481791b291125dc6db 100644
--- a/lean/OAI/Probability/ThorpRouting/Harmonic/ModifiedReverseCoins.lean
+++ b/lean/OAI/Probability/ThorpRouting/Harmonic/ModifiedReverseCoins.lean
@@ -1,761 +1,761 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Harmonic.ModifiedRoutingMoment
-
-namespace OAI
-
-namespace ThorpNine.Harmonic
-
-namespace Thorp
-
-noncomputable def modifiedReverseCoins (L : ℕ) : (r : ℕ) → ModifiedCoins L r → ModifiedCoins L r
-  | 0, ω => ((fun u => butterflyPerm L (decodeButterfly L ω.2.1) * (ω.1 u)⁻¹ *
-      butterflyPerm L (decodeButterfly L ω.2.1)),ω.2)
-  | r+1, ω =>
-      let z := modifiedStepEquiv L r ω
-      (modifiedStepEquiv L r).symm
-        ((modifiedReverseCoins L r z.1.1,modifiedReverseCoins L r z.1.2),(z.2.2,z.2.1))
-
-lemma modifiedReverseCoins_involutive (L r : ℕ) :
-    Function.Involutive (modifiedReverseCoins L r) := by
-  induction r with
-  | zero =>
-      intro ω
-      apply Prod.ext
-      · funext u
-        simp only [modifiedReverseCoins,mul_inv_rev,inv_inv]
-        simp [mul_assoc]
-      · rfl
-  | succ r ih =>
-      change ∀ ω, modifiedReverseCoins L r (modifiedReverseCoins L r ω) = ω at ih
-      intro ω
-      simp only [modifiedReverseCoins,Equiv.apply_symm_apply,ih]
-      exact (modifiedStepEquiv L r).symm_apply_apply ω
-
-noncomputable def modifiedReverseEquiv (L r : ℕ) : Equiv.Perm (ModifiedCoins L r) where
-  toFun := modifiedReverseCoins L r
-  invFun := modifiedReverseCoins L r
-  left_inv := modifiedReverseCoins_involutive L r
-  right_inv := modifiedReverseCoins_involutive L r
-
-lemma modifiedPerm_reverse (L r : ℕ) (E : Finset (Card r)) (ω : ModifiedCoins L r) :
-    modifiedPerm L r E (modifiedReverseCoins L r ω) = (modifiedPerm L r E ω)⁻¹ := by
-  induction r with
-  | zero =>
-      rw [modifiedPerm_zero,modifiedPerm_zero]
-      split_ifs
-      · simp only [inv_one]
-      · simp only [modifiedReverseCoins,mul_inv_rev,inv_inv]
-        simp
-  | succ r ih =>
-      obtain ⟨z,rfl⟩ := (modifiedStepEquiv L r).symm.surjective ω
-      rcases z with ⟨⟨ω₀,ω₁⟩,⟨ξ,η⟩⟩
-      simp only [modifiedReverseCoins,Equiv.apply_symm_apply,modifiedPerm_step,
-        mul_inv_rev,pairSwitch_inv,childLift_inv]
-      rw [mul_assoc]
-      congr 1
-      congr 1
-      apply congrArg childLift
-      funext b
-      cases b <;> simp only [Bool.false_eq_true,↓reduceIte,ih]
-
-lemma modified_kernel_symmetric (L r : ℕ) (E : Finset (Card r))
-    {ι : Type*} [Fintype ι] (e f : ι ↪ Card (L+r)) :
-    PairRouting.tupleProbability (modifiedPerm L r E) e f =
-      PairRouting.tupleProbability (modifiedPerm L r E) f e :=
-  PairRouting.tupleProbability_symmetric _ (modifiedReverseEquiv L r)
-    (modifiedPerm_reverse L r E) e f
-
-end Thorp
-
-namespace Thorp.Specht
-open scoped BigOperators Classical
-
-@[simp] lemma card_cell (μ : YoungDiagram) : Fintype.card (Cell μ) = μ.card := by
-  simp [Cell, YoungDiagram.card]
-
-noncomputable def columnEquiv (μ : YoungDiagram) (j : ℕ) :
-    {x : Cell μ // col x = j} ≃ Fin (μ.colLen j) where
-  toFun x := ⟨row x.1, by simpa [x.2] using row_lt_colLen x.1⟩
-  invFun i := ⟨⟨(i,j), YoungDiagram.mem_iff_lt_colLen.mpr i.2⟩, rfl⟩
-  left_inv x := by apply Subtype.ext; apply Subtype.ext; exact Prod.ext rfl x.2.symm
-  right_inv i := by rfl
-
-noncomputable def columnValues {μ : YoungDiagram} (f : Cell μ → ℕ) (j : ℕ) : Finset ℕ :=
-  (Finset.univ.filter (fun x => col x = j)).image f
-
-lemma mem_columnValues {μ : YoungDiagram} (f : Cell μ → ℕ) (x : Cell μ) :
-    f x ∈ columnValues f (col x) := by
-  classical
-  exact Finset.mem_image.mpr ⟨x, by simp, rfl⟩
-
-lemma card_columnValues {μ : YoungDiagram} (f : Cell μ → ℕ)
-    (hf : ∀ x y, col x = col y → f x = f y → x = y) (j : ℕ) :
-    (columnValues f j).card = μ.colLen j := by
-  classical
-  rw [columnValues, Finset.card_image_iff.mpr]
-  · rw [← Fintype.card_subtype]
-    exact Fintype.card_congr (columnEquiv μ j) |>.trans (Fintype.card_fin _)
-  intro x hx y hy he
-  exact hf x y ((Finset.mem_filter.mp hx).2.trans (Finset.mem_filter.mp hy).2.symm) he
-
-lemma strictMono_fin_nat_ge {n : ℕ} (f : Fin n → ℕ) (hf : StrictMono f) :
-    ∀ i, i.val ≤ f i := by
-  intro i
-  induction h : i.val using Nat.strong_induction_on generalizing i with
-  | h n ih =>
-    by_cases hn : n = 0
-    · omega
-    · have hn' : n-1 < i.val := by omega
-      let j : Fin _ := ⟨n-1, hn'.trans i.isLt⟩
-      have hj := ih (n-1) (by omega) j rfl
-      have hij := hf (show j < i from hn')
-      dsimp [j] at hj hij
-      omega
-
-noncomputable def rankCell {μ : YoungDiagram} (f : Cell μ → ℕ)
-    (hf : ∀ x y, col x = col y → f x = f y → x = y) (x : Cell μ) : Cell μ :=
-  let k := ((columnValues f (col x)).orderIsoOfFin (card_columnValues f hf (col x))).symm
-    ⟨f x, mem_columnValues f x⟩
-  ⟨(k.val,col x), YoungDiagram.mem_iff_lt_colLen.mpr k.isLt⟩
-
-@[simp] lemma rankCell_col {μ : YoungDiagram} (f : Cell μ → ℕ)
-    (hf : ∀ x y, col x = col y → f x = f y → x = y) (x : Cell μ) :
-    col (rankCell f hf x) = col x := rfl
-
-lemma rankCell_orderEmb {μ : YoungDiagram} (f : Cell μ → ℕ)
-    (hf : ∀ x y, col x = col y → f x = f y → x = y) (x : Cell μ) :
-    (columnValues f (col x)).orderEmbOfFin (card_columnValues f hf (col x))
-      ⟨row (rankCell f hf x), row_lt_colLen (rankCell f hf x)⟩ = f x := by
-  change ((columnValues f (col x)).orderIsoOfFin (card_columnValues f hf (col x))
-    (((columnValues f (col x)).orderIsoOfFin (card_columnValues f hf (col x))).symm
-      ⟨f x, mem_columnValues f x⟩)).val = f x
-  simp
-
-lemma rankCell_injective {μ : YoungDiagram} (f : Cell μ → ℕ)
-    (hf : ∀ x y, col x = col y → f x = f y → x = y) :
-    Function.Injective (rankCell f hf) := by
-  intro x y h
-  have hc : col x = col y := by simpa only [rankCell_col] using congrArg (fun z : Cell μ => col z) h
-  apply hf x y hc
-  have hx := rankCell_orderEmb f hf x
-  have hy := rankCell_orderEmb f hf y
-  change (columnValues f (col (rankCell f hf x))).orderEmbOfFin
-    (card_columnValues f hf (col (rankCell f hf x)))
-      ⟨row (rankCell f hf x), row_lt_colLen (rankCell f hf x)⟩ = f x at hx
-  rw [h] at hx
-  exact hx.symm.trans hy
-
-noncomputable def rankPerm {μ : YoungDiagram} (f : Cell μ → ℕ)
-    (hf : ∀ x y, col x = col y → f x = f y → x = y) : Equiv.Perm (Cell μ) :=
-  Equiv.ofBijective (rankCell f hf) ((Finite.injective_iff_bijective).mp (rankCell_injective f hf))
-
-lemma rankCell_row_le {μ : YoungDiagram} (f : Cell μ → ℕ)
-    (hf : ∀ x y, col x = col y → f x = f y → x = y) (x : Cell μ) :
-    row (rankCell f hf x) ≤ f x := by
-  rw [←rankCell_orderEmb f hf x]
-  exact strictMono_fin_nat_ge _ (OrderEmbedding.strictMono _) _
-
-theorem column_transporter (μ : YoungDiagram) (p : Equiv.Perm (Cell μ))
-    (hp : ∀ x y, col x = col y → row (p x) = row (p y) → x = y) :
-    ∃ c : Equiv.Perm (Cell μ), (∀ x, col (c x) = col x) ∧
-      ∀ x, row (c x) = row (p x) := by
-  classical
-  let c := rankPerm (fun x => row (p x)) hp
-  refine ⟨c, fun x => rfl, ?_⟩
-  have hle (x : Cell μ) : row (c x) ≤ row (p x) := rankCell_row_le _ hp x
-  have hsum : ∑ x, row (c x) = ∑ x, row (p x) := by
-    rw [Equiv.sum_comp, Equiv.sum_comp]
-  exact fun x => (Finset.sum_eq_sum_iff_of_le (fun x _ => hle x)).mp hsum x (Finset.mem_univ _)
-
-
-@[simp] lemma tabloidAct_inv {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) :
-    tabloidAct p⁻¹ = (tabloidAct p)⁻¹ := by
-  ext f x
-  simp [tabloidAct,Equiv.Perm.inv_def]
-
-@[simp] lemma tabloidRep_apply {μ : YoungDiagram} (p : Equiv.Perm (Cell μ))
-    (v : Tabloid μ → ℂ) (f : Tabloid μ) : tabloidRep μ p v f = v (tabloidAct p⁻¹ f) := rfl
-
-lemma tabloidRep_delta {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) (f : Tabloid μ) :
-    tabloidRep μ p (delta f) = delta (tabloidAct p f) := by
-  classical
-  ext g
-  simp only [tabloidRep_apply,delta,Equiv.Perm.inv_def]
-  congr 1
-  exact propext (Equiv.symm_apply_eq (tabloidAct p))
-
-@[simp] lemma mem_colGroup {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) :
-    p ∈ colGroup μ ↔ ∀ x, col (p x) = col x := Iff.rfl
-
-lemma colGroup_row_stabilizer {μ : YoungDiagram} (c : colGroup μ)
-    (hc : tabloidAct c (baseTabloid μ) = baseTabloid μ) : (c : Equiv.Perm (Cell μ)) = 1 := by
-  apply Equiv.ext
-  intro x
-  have hr := congrArg (fun f : Tabloid μ => ((f.1 ((c : Equiv.Perm _) x)) : ℕ)) hc
-  simp only [tabloidAct_apply,Equiv.Perm.inv_def,Equiv.symm_apply_apply,baseTabloid,rowIndex] at hr
-  apply Subtype.ext
-  exact Prod.ext hr.symm (c.2 x)
-
-lemma tabloid_column_transporter {μ : YoungDiagram} (f : Tabloid μ)
-    (hf : ∀ x y, col x = col y → f.1 x = f.1 y → x = y) :
-    ∃ c : colGroup μ, f = tabloidAct c (baseTabloid μ) := by
-  obtain ⟨p,hp⟩ := f.2
-  have hp' : ∀ x y, col x = col y → row (p x) = row (p y) → x = y := by
-    intro x y hcol hrow
-    apply hf x y hcol
-    rw [hp]
-    exact Fin.ext hrow
-  obtain ⟨c,hcol,hrow⟩ := column_transporter μ p hp'
-  refine ⟨⟨c⁻¹, (colGroup μ).inv_mem hcol⟩, ?_⟩
-  apply Subtype.ext
-  funext x
-  apply Fin.ext
-  simp only [tabloidAct_apply,baseTabloid,rowIndex,inv_inv,hp,Function.comp_apply]
-  exact (hrow x).symm
-
-
-@[simp] lemma permSign_one {α : Type*} [Fintype α] [DecidableEq α] :
-    permSign (1 : Equiv.Perm α) = 1 := by simp [permSign]
-@[simp] lemma permSign_mul {α : Type*} [Fintype α] [DecidableEq α] (p q : Equiv.Perm α) :
-    permSign (p*q) = permSign p * permSign q := by simp [permSign]
-@[simp] lemma permSign_inv {α : Type*} [Fintype α] [DecidableEq α] (p : Equiv.Perm α) :
-    permSign p⁻¹ = permSign p := by simp [permSign,Equiv.Perm.inv_def]
-lemma permSign_sq {α : Type*} [Fintype α] [DecidableEq α] (p : Equiv.Perm α) :
-    permSign p * permSign p = 1 := by
-  calc
-    permSign p * permSign p = permSign (p*p⁻¹) := by rw [permSign_mul,permSign_inv]
-    _ = 1 := by simp
-@[simp] lemma permSign_swap {α : Type*} [Fintype α] [DecidableEq α] {x y : α} (h : x ≠ y) :
-    permSign (Equiv.swap x y) = -1 := by simp [permSign, h]
-@[simp] lemma conj_permSign {α : Type*} [Fintype α] [DecidableEq α] (p : Equiv.Perm α) :
-    starRingEnd ℂ (permSign p) = permSign p := by simp [permSign]
-
-variable {α V : Type*} [Fintype α] [DecidableEq α] [AddCommGroup V] [Module ℂ V]
-
-lemma alternator_mul (C : Subgroup (Equiv.Perm α))
-    (ρ : Representation ℂ (Equiv.Perm α) V) (c : C) :
-    alternator C ρ * ρ c = permSign (c : Equiv.Perm α) • alternator C ρ := by
-  classical
-  rw [alternator,Finset.sum_mul,Finset.smul_sum]
-  apply Fintype.sum_equiv (Equiv.mulRight c)
-  intro a
-  simp only [smul_mul_assoc,←map_mul,Equiv.coe_mulRight,Subgroup.coe_mul,permSign_mul,smul_smul]
-  congr 1
-  have h := permSign_sq (c : Equiv.Perm α)
-  calc
-    permSign (a : Equiv.Perm α) = permSign (a : Equiv.Perm α)*1 := by ring
-    _ = permSign (a : Equiv.Perm α)*(permSign (c : Equiv.Perm α)*permSign (c : Equiv.Perm α)) := by rw [h]
-    _ = _ := by ring
-
-lemma mul_alternator (C : Subgroup (Equiv.Perm α))
-    (ρ : Representation ℂ (Equiv.Perm α) V) (c : C) :
-    ρ c * alternator C ρ = permSign (c : Equiv.Perm α) • alternator C ρ := by
-  classical
-  rw [alternator,Finset.mul_sum,Finset.smul_sum]
-  apply Fintype.sum_equiv (Equiv.mulLeft c)
-  intro a
-  simp only [mul_smul_comm,←map_mul,Equiv.coe_mulLeft,Subgroup.coe_mul,permSign_mul,smul_smul]
-  congr 1
-  have h := permSign_sq (c : Equiv.Perm α)
-  calc
-    permSign (a : Equiv.Perm α) = 1*permSign (a : Equiv.Perm α) := by ring
-    _ = (permSign (c : Equiv.Perm α)*permSign (c : Equiv.Perm α))*permSign (a : Equiv.Perm α) := by rw [h]
-    _ = _ := by ring
-
-lemma alternator_apply_action (C : Subgroup (Equiv.Perm α))
-    (ρ : Representation ℂ (Equiv.Perm α) V) (c : C) (v : V) :
-    alternator C ρ (ρ c v) = permSign (c : Equiv.Perm α) • alternator C ρ v := by
-  exact congrArg (fun T : Module.End ℂ V => T v) (alternator_mul C ρ c)
-
-lemma action_alternator_apply (C : Subgroup (Equiv.Perm α))
-    (ρ : Representation ℂ (Equiv.Perm α) V) (c : C) (v : V) :
-    ρ c (alternator C ρ v) = permSign (c : Equiv.Perm α) • alternator C ρ v := by
-  exact congrArg (fun T : Module.End ℂ V => T v) (mul_alternator C ρ c)
-
-lemma alternator_eq_zero_of_odd_stabilizer (C : Subgroup (Equiv.Perm α))
-    (ρ : Representation ℂ (Equiv.Perm α) V) (c : C) (v : V)
-    (hc : permSign (c : Equiv.Perm α) = -1) (hv : ρ c v = v) :
-    alternator C ρ v = 0 := by
-  have h := alternator_apply_action C ρ c v
-  rw [hv,hc,neg_one_smul] at h
-  have h2 : (2:ℂ) • alternator C ρ v = 0 := by
-    rw [two_smul]
-    nth_rw 1 [h]
-    exact neg_add_cancel _
-  exact (smul_eq_zero.mp h2).resolve_left (by norm_num)
-
-end Thorp.Specht
-namespace Thorp.Specht
-open scoped BigOperators Classical
-
-lemma swap_mem_colGroup {μ : YoungDiagram} (x y : Cell μ) (hc : col x = col y) :
-    Equiv.swap x y ∈ colGroup μ := by
-  classical
-  intro z
-  by_cases hx : z=x
-  · subst z; simpa using hc.symm
-  by_cases hy : z=y
-  · subst z; simpa using hc
-  simp [Equiv.swap_apply_of_ne_of_ne hx hy]
-
-lemma swap_fixes_tabloid {μ : YoungDiagram} (f : Tabloid μ) (x y : Cell μ)
-    (hxy : f.1 x = f.1 y) : tabloidAct (Equiv.swap x y) f = f := by
-  classical
-  apply Subtype.ext
-  funext z
-  simp only [tabloidAct_apply, Equiv.swap_inv]
-  by_cases hx : z=x
-  · subst z; simpa using hxy.symm
-  by_cases hy : z=y
-  · subst z; simpa using hxy
-  simp [Equiv.swap_apply_of_ne_of_ne hx hy]
-
-lemma alternator_delta_rank_one {μ : YoungDiagram} (f : Tabloid μ) :
-    ∃ z : ℂ, alternator (colGroup μ) (tabloidRep μ) (delta f) = z • polytabloid μ := by
-  classical
-  by_cases hf : ∀ x y, col x = col y → f.1 x = f.1 y → x = y
-  · obtain ⟨c,rfl⟩ := tabloid_column_transporter f hf
-    refine ⟨permSign (c:Equiv.Perm (Cell μ)), ?_⟩
-    rw [←tabloidRep_delta]
-    exact alternator_apply_action _ _ c _
-  · push Not at hf
-    obtain ⟨x,y,hcol,hrow,hne⟩ := hf
-    refine ⟨0, ?_⟩
-    rw [zero_smul]
-    apply alternator_eq_zero_of_odd_stabilizer _ _ ⟨Equiv.swap x y,swap_mem_colGroup x y hcol⟩
-    · exact permSign_swap hne
-    · rw [tabloidRep_delta,swap_fixes_tabloid f x y hrow]
-
-lemma polytabloid_formula (μ : YoungDiagram) :
-    polytabloid μ = ∑ c : colGroup μ, permSign (c:Equiv.Perm (Cell μ)) •
-      delta (tabloidAct c (baseTabloid μ)) := by
-  classical
-  simp only [polytabloid,alternator,LinearMap.sum_apply,LinearMap.smul_apply,tabloidRep_delta]
-
-lemma polytabloid_base (μ : YoungDiagram) : polytabloid μ (baseTabloid μ) = 1 := by
-  classical
-  rw [polytabloid_formula,Finset.sum_apply]
-  rw [Finset.sum_eq_single (1 : colGroup μ)]
-  · simp [delta]
-  · intro c hc hne
-    have hn : baseTabloid μ ≠ tabloidAct c (baseTabloid μ) := by
-      intro h
-      apply hne
-      exact Subtype.ext (colGroup_row_stabilizer c h.symm)
-    simp [Pi.smul_apply,delta,hn]
-  · simp
-
-lemma polytabloid_ne_zero (μ : YoungDiagram) : polytabloid μ ≠ 0 := by
-  intro h
-  have hv := congrArg (fun v : Tabloid μ → ℂ => v (baseTabloid μ)) h
-  simp [polytabloid_base] at hv
-
-lemma delta_decomposition {μ : YoungDiagram} (v : Tabloid μ → ℂ) :
-    v = ∑ f : Tabloid μ, v f • delta f := by
-  classical
-  funext g
-  simp only [Finset.sum_apply,Pi.smul_apply,delta,smul_eq_mul,mul_ite,mul_one,mul_zero]
-  simp
-
-theorem alternator_rank_one (μ : YoungDiagram) (v : Tabloid μ → ℂ) :
-    ∃ z : ℂ, alternator (colGroup μ) (tabloidRep μ) v = z • polytabloid μ := by
-  classical
-  choose z hz using fun f : Tabloid μ => alternator_delta_rank_one f
-  refine ⟨∑ f, v f * z f, ?_⟩
-  conv_lhs => rw [delta_decomposition v]
-  simp only [map_sum,map_smul,hz,smul_smul,Finset.sum_smul]
-
-lemma alternator_rank_one_exact (μ : YoungDiagram) (v : Tabloid μ → ℂ) :
-    alternator (colGroup μ) (tabloidRep μ) v =
-      (alternator (colGroup μ) (tabloidRep μ) v (baseTabloid μ)) • polytabloid μ := by
-  obtain ⟨z,hz⟩ := alternator_rank_one μ v
-  rw [hz]
-  simp only [Pi.smul_apply,polytabloid_base,smul_eq_mul,mul_one]
-
-
-noncomputable def pairing {μ : YoungDiagram} (v w : Tabloid μ → ℂ) : ℂ :=
-  ∑ f, w f * starRingEnd ℂ (v f)
-
-lemma pairing_delta {μ : YoungDiagram} (f : Tabloid μ) (v : Tabloid μ → ℂ) :
-    pairing (delta f) v = v f := by
-  classical
-  simp [pairing,delta]
-
-lemma pairing_sum_left {μ : YoungDiagram} {ι : Type*} [Fintype ι]
-    (v : ι → Tabloid μ → ℂ) (w : Tabloid μ → ℂ) :
-    pairing (∑ i, v i) w = ∑ i, pairing (v i) w := by
-  simp only [pairing,Finset.sum_apply,map_sum,Finset.mul_sum]
-  exact Finset.sum_comm
-
-lemma pairing_smul_left {μ : YoungDiagram} (a : ℂ) (v w : Tabloid μ → ℂ) :
-    pairing (a • v) w = starRingEnd ℂ a * pairing v w := by
-  simp only [pairing,Pi.smul_apply,smul_eq_mul,map_mul,Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro f hf
-  ring
-
-lemma pairing_zero_left {μ : YoungDiagram} (v : Tabloid μ → ℂ) : pairing 0 v = 0 := by
-  simp [pairing]
-
-lemma pairing_add_left {μ : YoungDiagram} (u v w : Tabloid μ → ℂ) :
-    pairing (u+v) w = pairing u w + pairing v w := by
-  simp [pairing,mul_add,Finset.sum_add_distrib]
-
-lemma pairing_self_eq_zero {μ : YoungDiagram} (v : Tabloid μ → ℂ) :
-    pairing v v = 0 ↔ v = 0 := by
-  change inner ℂ (WithLp.toLp 2 v : EuclideanSpace ℂ (Tabloid μ)) (WithLp.toLp 2 v) = 0 ↔ v = 0
-  rw [inner_self_eq_zero]
-  constructor
-  · intro h; exact congrArg WithLp.ofLp h
-  · intro h; subst v; rfl
-
-lemma pairing_action {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) (v w : Tabloid μ → ℂ) :
-    pairing (tabloidRep μ p v) w = pairing v (tabloidRep μ p⁻¹ w) := by
-  classical
-  simp only [pairing,tabloidRep_apply,inv_inv]
-  apply Fintype.sum_equiv (tabloidAct p⁻¹)
-  intro f
-  have hcancel : tabloidAct p (tabloidAct p⁻¹ f) = f := by
-    rw [tabloidAct_inv]
-    exact (tabloidAct p).apply_symm_apply f
-  rw [hcancel]
-
-lemma alternator_base_pairing (μ : YoungDiagram) (v : Tabloid μ → ℂ) :
-    alternator (colGroup μ) (tabloidRep μ) v (baseTabloid μ) = pairing (polytabloid μ) v := by
-  classical
-  simp only [polytabloid_formula,pairing_sum_left,pairing_smul_left,conj_permSign,pairing_delta]
-  simp only [alternator,LinearMap.sum_apply,LinearMap.smul_apply,Finset.sum_apply,Pi.smul_apply,
-    smul_eq_mul,tabloidRep_apply]
-  apply Fintype.sum_equiv (Equiv.inv (colGroup μ))
-  intro c
-  simp only [Equiv.inv_apply,Subgroup.coe_inv,permSign_inv]
-
-theorem alternator_pairing (μ : YoungDiagram) (v : Tabloid μ → ℂ) :
-    alternator (colGroup μ) (tabloidRep μ) v = pairing (polytabloid μ) v • polytabloid μ := by
-  rw [alternator_rank_one_exact,alternator_base_pairing]
-
-
-lemma polytabloid_mem_space (μ : YoungDiagram) : polytabloid μ ∈ space μ := by
-  simpa using orbit_mem_space μ 1
-
-lemma alternator_mem_invariant (μ : YoungDiagram) (W : Submodule ℂ (Tabloid μ → ℂ))
-    (hW : ∀ p v, v ∈ W → tabloidRep μ p v ∈ W) (v : Tabloid μ → ℂ) (hv : v ∈ W) :
-    alternator (colGroup μ) (tabloidRep μ) v ∈ W := by
-  classical
-  simp only [alternator,LinearMap.sum_apply,LinearMap.smul_apply]
-  exact W.sum_mem (fun c _ => W.smul_mem _ (hW c v hv))
-
-theorem submodule_theorem (μ : YoungDiagram) (W : Submodule ℂ (Tabloid μ → ℂ))
-    (hW : ∀ p v, v ∈ W → tabloidRep μ p v ∈ W) :
-    space μ ≤ W ∨ ∀ v ∈ W, ∀ w ∈ space μ, pairing w v = 0 := by
-  classical
-  by_cases hSW : space μ ≤ W
-  · exact Or.inl hSW
-  right
-  have he : polytabloid μ ∉ W := by
-    intro he
-    apply hSW
-    apply Submodule.span_le.mpr
-    rintro _ ⟨p,rfl⟩
-    exact hW p _ he
-  have hpair : ∀ v ∈ W, pairing (polytabloid μ) v = 0 := by
-    intro v hv
-    by_contra hne
-    apply he
-    have hal := alternator_mem_invariant μ W hW v hv
-    rw [alternator_pairing] at hal
-    have hh := W.smul_mem (pairing (polytabloid μ) v)⁻¹ hal
-    simpa only [smul_smul,inv_mul_cancel₀ hne,one_smul] using hh
-  intro v hv w hw
-  induction hw using Submodule.span_induction with
-  | mem w hw =>
-    obtain ⟨p,rfl⟩ := hw
-    rw [pairing_action]
-    exact hpair _ (hW p⁻¹ v hv)
-  | zero => exact pairing_zero_left v
-  | add u w hu hw ihu ihw => rw [pairing_add_left,ihu,ihw,add_zero]
-  | smul a w hw ih => rw [pairing_smul_left,ih,mul_zero]
-
-theorem space_simple (μ : YoungDiagram) (W : Submodule ℂ (Tabloid μ → ℂ))
-    (hW : ∀ p v, v ∈ W → tabloidRep μ p v ∈ W) (hWS : W ≤ space μ) :
-    W = ⊥ ∨ W = space μ := by
-  obtain hSW | horth := submodule_theorem μ W hW
-  · exact Or.inr (le_antisymm hWS hSW)
-  · left
-    apply le_antisymm _ bot_le
-    intro v hv
-    exact (pairing_self_eq_zero v).mp (horth v hv v (hWS hv))
-
-
-lemma space_ne_bot (μ : YoungDiagram) : space μ ≠ ⊥ := by
-  intro h
-  have he := polytabloid_mem_space μ
-  rw [h,Submodule.mem_bot] at he
-  exact polytabloid_ne_zero μ he
-
-theorem representation_irreducible (μ : YoungDiagram) :
-    Representation.IsIrreducible (representation μ) := by
-  have hnt : (⊥ : Subrepresentation (representation μ)) ≠ ⊤ := by
-    intro h
-    have he : (⟨polytabloid μ,polytabloid_mem_space μ⟩ : space μ) ∈
-        (⊥ : Subrepresentation (representation μ)) := by rw [h]; trivial
-    have he' : (⟨polytabloid μ,polytabloid_mem_space μ⟩ : space μ) = 0 := he
-    exact polytabloid_ne_zero μ (congrArg Subtype.val he')
-  refine { exists_pair_ne := ⟨⊥,⊤,hnt⟩, eq_bot_or_eq_top := ?_ }
-  intro U
-  let W : Submodule ℂ (Tabloid μ → ℂ) := U.toSubmodule.map (space μ).subtype
-  have hW : ∀ p v, v ∈ W → tabloidRep μ p v ∈ W := by
-    intro p v hv
-    obtain ⟨u,hu,rfl⟩ := hv
-    exact ⟨representation μ p u,U.apply_mem_toSubmodule p hu,rfl⟩
-  have hWS : W ≤ space μ := by
-    intro v hv
-    obtain ⟨u,hu,rfl⟩ := hv
-    exact u.2
-  obtain hbot | htop := space_simple μ W hW hWS
-  · left
-    apply le_antisymm _ bot_le
-    intro u hu
-    have hmem : (u : Tabloid μ → ℂ) ∈ W := ⟨u,hu,rfl⟩
-    rw [hbot,Submodule.mem_bot] at hmem
-    change u = 0
-    exact Subtype.ext hmem
-  · right
-    apply le_antisymm le_top
-    intro u hu
-    have hmem : (u : Tabloid μ → ℂ) ∈ W := by rw [htop]; exact u.2
-    obtain ⟨v,hv,hvu⟩ := hmem
-    have heq : v = u := Subtype.ext hvu
-    exact heq ▸ hv
-
-
-section Symmetrizer
-variable {α V : Type*} [Fintype α] [DecidableEq α] [AddCommGroup V] [Module ℂ V]
-
-noncomputable def symmetrizer (H : Subgroup (Equiv.Perm α))
-    (ρ : Representation ℂ (Equiv.Perm α) V) : Module.End ℂ V :=
-  ∑ h : H, ρ h
-
-lemma symmetrizer_mul (H : Subgroup (Equiv.Perm α))
-    (ρ : Representation ℂ (Equiv.Perm α) V) (h : H) :
-    symmetrizer H ρ * ρ h = symmetrizer H ρ := by
-  rw [symmetrizer,Finset.sum_mul]
-  exact Fintype.sum_equiv (Equiv.mulRight h) _ _ (fun g => by simp [←map_mul])
-
-lemma symmetrizer_eq_zero_of_neg (H : Subgroup (Equiv.Perm α))
-    (ρ : Representation ℂ (Equiv.Perm α) V) (h : H) (v : V) (hv : ρ h v = -v) :
-    symmetrizer H ρ v = 0 := by
-  have hh := congrArg (fun T : Module.End ℂ V => T v) (symmetrizer_mul H ρ h)
-  change symmetrizer H ρ (ρ h v) = symmetrizer H ρ v at hh
-  rw [hv,map_neg] at hh
-  have h2 : (2:ℂ) • symmetrizer H ρ v = 0 := by
-    rw [two_smul]
-    nth_rw 1 [←hh]
-    exact neg_add_cancel _
-  exact (smul_eq_zero.mp h2).resolve_left (by norm_num)
-
-noncomputable def fiberGroup {β : Type*} (f : α → β) : Subgroup (Equiv.Perm α) where
-  carrier := {p | ∀ x, f (p x) = f x}
-  one_mem' := by intro x; rfl
-  mul_mem' := by intro p q hp hq x; exact (hp (q x)).trans (hq x)
-  inv_mem' := by intro p hp x; simpa using (hp (p⁻¹ x)).symm
-
-omit [Fintype α] in
-lemma swap_mem_fiberGroup {β : Type*} (f : α → β) (x y : α) (hf : f x = f y) :
-    Equiv.swap x y ∈ fiberGroup f := by
-  intro z
-  by_cases hx : z=x
-  · subst z; simpa using hf.symm
-  by_cases hy : z=y
-  · subst z; simpa using hf
-  simp [Equiv.swap_apply_of_ne_of_ne hx hy]
-
-end Symmetrizer
-
-lemma swap_polytabloid {μ : YoungDiagram} (x y : Cell μ) (hne : x ≠ y)
-    (hcol : col x = col y) :
-    tabloidRep μ (Equiv.swap x y) (polytabloid μ) = -polytabloid μ := by
-  have h := action_alternator_apply (colGroup μ) (tabloidRep μ)
-    ⟨Equiv.swap x y,swap_mem_colGroup x y hcol⟩ (delta (baseTabloid μ))
-  simpa only [polytabloid,permSign_swap hne,neg_one_smul] using h
-
-theorem tall_column_average_zero (μ : YoungDiagram) {β : Type*} [Fintype β]
-    (f : Cell μ → β) (j : ℕ) (hj : Fintype.card β < μ.colLen j)
-    (v : Tabloid μ → ℂ) (hv : v ∈ space μ) :
-    symmetrizer (fiberGroup f) (tabloidRep μ) v = 0 := by
-  induction hv using Submodule.span_induction with
-  | mem v hv =>
-    obtain ⟨p,rfl⟩ := hv
-    have hcard : Fintype.card β < Fintype.card {x : Cell μ // col x = j} := by
-      rw [Fintype.card_congr (columnEquiv μ j),Fintype.card_fin]
-      exact hj
-    obtain ⟨x,y,hne,hf⟩ := Fintype.exists_ne_map_eq_of_card_lt
-      (fun x : {x : Cell μ // col x = j} => f (p x.1)) hcard
-    have hxy : x.1 ≠ y.1 := fun h => hne (Subtype.ext h)
-    let h : fiberGroup f := ⟨Equiv.swap (p x.1) (p y.1),swap_mem_fiberGroup f _ _ hf⟩
-    apply symmetrizer_eq_zero_of_neg _ _ h
-    change tabloidRep μ (Equiv.swap (p x.1) (p y.1))
-      (tabloidRep μ p (polytabloid μ)) = -tabloidRep μ p (polytabloid μ)
-    rw [←Module.End.mul_apply,←map_mul,←Equiv.mul_swap_eq_swap_mul]
-    rw [map_mul,Module.End.mul_apply,swap_polytabloid x.1 y.1 hxy (x.2.trans y.2.symm),map_neg]
-  | zero => exact map_zero _
-  | add v w hv hw ihv ihw => simp only [map_add,ihv,ihw,add_zero]
-  | smul a v hv ih => simp only [map_smul,ih,smul_zero]
-
-
-noncomputable def rowEquiv (μ : YoungDiagram) (i : ℕ) :
-    {x : Cell μ // row x = i} ≃ Fin (μ.rowLen i) where
-  toFun x := ⟨col x.1, by
-    have h := YoungDiagram.mem_iff_lt_rowLen.mp x.1.2
-    change col x.1 < μ.rowLen (row x.1) at h
-    simpa only [x.2] using h⟩
-  invFun j := ⟨⟨(i,j),YoungDiagram.mem_iff_lt_rowLen.mpr j.2⟩,rfl⟩
-  left_inv x := by apply Subtype.ext; apply Subtype.ext; exact Prod.ext x.2.symm rfl
-  right_inv j := rfl
-
-def rowCost (μ : YoungDiagram) : ℕ := ∑ x : Cell μ, row x
-
-lemma eq_of_row_preserving (μ ν : YoungDiagram) (e : Cell μ ≃ Cell ν)
-    (he : ∀ x, row (e x) = row x) : μ = ν := by
-  have hlen (i : ℕ) : μ.rowLen i = ν.rowLen i := by
-    let ei : {x : Cell μ // row x = i} ≃ {x : Cell ν // row x = i} :=
-      { toFun := fun x => ⟨e x.1,(he x.1).trans x.2⟩
-        invFun := fun x => ⟨e.symm x.1,by
-          have h := he (e.symm x.1)
-          simp only [Equiv.apply_symm_apply] at h
-          exact h.symm.trans x.2⟩
-        left_inv := fun x => by apply Subtype.ext; exact e.symm_apply_apply _
-        right_inv := fun x => by apply Subtype.ext; exact e.apply_symm_apply _ }
-    simpa only [Fintype.card_fin] using Fintype.card_congr
-      ((rowEquiv μ i).symm.trans (ei.trans (rowEquiv ν i)))
-  apply YoungDiagram.ext
-  ext ⟨i,j⟩
-  simp only [YoungDiagram.mem_cells,YoungDiagram.mem_iff_lt_rowLen,hlen]
-
-lemma column_injective_rowCost (μ ν : YoungDiagram) (e : Cell μ ≃ Cell ν)
-    (h : ∀ x y : Cell μ, col x = col y → row (e x) = row (e y) → x = y) :
-    rowCost μ ≤ rowCost ν ∧ (rowCost μ = rowCost ν → μ = ν) := by
-  let c := rankPerm (fun x => row (e x)) h
-  have hle (x : Cell μ) : row (c x) ≤ row (e x) := rankCell_row_le _ h x
-  have hsumc : ∑ x, row (c x) = rowCost μ := Equiv.sum_comp c _
-  have hsume : ∑ x, row (e x) = rowCost ν := Equiv.sum_comp e _
-  constructor
-  · rw [←hsumc,←hsume]
-    exact Finset.sum_le_sum (fun x _ => hle x)
-  · intro heq
-    have he : ∀ x, row (c x) = row (e x) := by
-      apply fun x => (Finset.sum_eq_sum_iff_of_le (fun x _ => hle x)).mp
-        (hsumc.trans (heq.trans hsume.symm)) x (Finset.mem_univ _)
-    apply eq_of_row_preserving μ ν (c.symm.trans e)
-    intro x
-    simpa using (he (c.symm x)).symm
-
-lemma swap_relabel {α β : Type*} [DecidableEq α] [DecidableEq β]
-    (e : α ≃ β) (x y : α) : e.permCongr (Equiv.swap x y) = Equiv.swap (e x) (e y) := by
-  ext z
-  simp only [Equiv.permCongr_apply]
-  exact e.injective.map_swap x y (e.symm z) |>.trans (by simp)
-
-lemma sign_vector_rowCost (μ ν : YoungDiagram) (e : Cell μ ≃ Cell ν)
-    (v : Tabloid ν → ℂ) (hv : v ≠ 0)
-    (hs : ∀ c : colGroup μ,
-      tabloidRep ν (e.permCongr c) v = permSign (c : Equiv.Perm (Cell μ)) • v) :
-    rowCost μ ≤ rowCost ν ∧ (rowCost μ = rowCost ν → μ = ν) := by
-  obtain ⟨f,hf⟩ : ∃ f, v f ≠ 0 := by
-    by_contra h
-    push Not at h
-    exact hv (funext h)
-  obtain ⟨p,hp⟩ := f.2
-  apply column_injective_rowCost μ ν (e.trans p)
-  intro x y hcol hrow
-  by_contra hne
-  have hfix : tabloidAct (Equiv.swap (e x) (e y)) f = f := by
-    apply swap_fixes_tabloid
-    rw [hp]
-    exact Fin.ext hrow
-  have heq := congrArg (fun w : Tabloid ν → ℂ => w f)
-    (hs ⟨Equiv.swap x y,swap_mem_colGroup x y hcol⟩)
-  rw [swap_relabel,permSign_swap hne] at heq
-  simp only [tabloidRep_apply,Equiv.swap_inv,hfix,Pi.smul_apply,neg_one_smul] at heq
-  apply hf
-  have h2 : (2:ℂ)*(v f) = 0 := by linear_combination heq
-  exact (mul_eq_zero.mp h2).resolve_left (by norm_num)
-
-lemma polytabloid_col_action (μ : YoungDiagram) (c : colGroup μ) :
-    tabloidRep μ c (polytabloid μ) = permSign (c:Equiv.Perm (Cell μ)) • polytabloid μ :=
-  action_alternator_apply _ _ c _
-
-theorem shapes_eq_of_equivariant_equiv (μ ν : YoungDiagram) (e : Cell μ ≃ Cell ν)
-    (T : space μ ≃ₗ[ℂ] space ν)
-    (hT : ∀ p v, T (representation μ p v) = representation ν (e.permCongr p) (T v)) :
-    μ = ν := by
-  let u : space μ := ⟨polytabloid μ,polytabloid_mem_space μ⟩
-  have hu : u ≠ 0 := fun h => polytabloid_ne_zero μ (congrArg Subtype.val h)
-  have hTcol (c : colGroup μ) :
-      tabloidRep ν (e.permCongr c) (T u) = permSign (c:Equiv.Perm (Cell μ)) • (T u : Tabloid ν → ℂ) := by
-    change (representation ν (e.permCongr c) (T u) : Tabloid ν → ℂ) = _
-    rw [←hT]
-    have hh : representation μ c u = permSign (c:Equiv.Perm (Cell μ)) • u :=
-      Subtype.ext (polytabloid_col_action μ c)
-    rw [hh,map_smul]
-    rfl
-  have hfwd := sign_vector_rowCost μ ν e (T u)
-    (fun h => hu (T.injective (by apply Subtype.ext; simpa using h))) hTcol
-  let w : space ν := ⟨polytabloid ν,polytabloid_mem_space ν⟩
-  have hw : w ≠ 0 := fun h => polytabloid_ne_zero ν (congrArg Subtype.val h)
-  have hTi (p : Equiv.Perm (Cell ν)) (v : space ν) :
-      T.symm (representation ν p v) = representation μ (e.symm.permCongr p) (T.symm v) := by
-    apply T.injective
-    rw [T.apply_symm_apply,hT,T.apply_symm_apply]
-    have hp : e.permCongr (e.symm.permCongr p) = p := by
-      apply Equiv.ext
-      intro x
-      simp [Equiv.permCongr_apply]
-    rw [hp]
-  have hTicol (c : colGroup ν) :
-      tabloidRep μ (e.symm.permCongr c) (T.symm w) =
-        permSign (c:Equiv.Perm (Cell ν)) • (T.symm w : Tabloid μ → ℂ) := by
-    change (representation μ (e.symm.permCongr c) (T.symm w) : Tabloid μ → ℂ) = _
-    rw [←hTi]
-    have hh : representation ν c w = permSign (c:Equiv.Perm (Cell ν)) • w :=
-      Subtype.ext (polytabloid_col_action ν c)
-    rw [hh,map_smul]
-    rfl
-  have hbwd := sign_vector_rowCost ν μ e.symm (T.symm w)
-    (fun h => hw (T.symm.injective (by apply Subtype.ext; simpa using h))) hTicol
-  exact hfwd.2 (Nat.le_antisymm hfwd.1 hbwd.1)
-
-
-noncomputable def cellRowsEquiv (μ : YoungDiagram) :
-    Cell μ ≃ (i : Fin (μ.colLen 0)) × Fin (μ.rowLen i) where
-  toFun x := ⟨rowIndex x,⟨col x,YoungDiagram.mem_iff_lt_rowLen.mp x.2⟩⟩
-  invFun p := ⟨(p.1,p.2),YoungDiagram.mem_iff_lt_rowLen.mpr p.2.2⟩
-  left_inv x := by apply Subtype.ext; rfl
-  right_inv p := by rcases p with ⟨⟨i,hi⟩,⟨j,hj⟩⟩; rfl
-
-lemma card_eq_sum_rowLen (μ : YoungDiagram) : μ.card = ∑ i : Fin (μ.colLen 0), μ.rowLen i := by
-  rw [←card_cell,Fintype.card_congr (cellRowsEquiv μ),Fintype.card_sigma]
-  simp only [Fintype.card_fin]
-
-lemma sum_rowLens (μ : YoungDiagram) : μ.rowLens.sum = μ.card := by
-  rw [card_eq_sum_rowLen,YoungDiagram.rowLens]
-  rw [Fin.sum_univ_eq_sum_range]
-  have hh (n : ℕ) : (List.map μ.rowLen (List.range n)).sum = ∑ i ∈ Finset.range n, μ.rowLen i := by
-    induction n with
-    | zero => simp
-    | succ n ih => simp [List.range_succ,List.map_append,Finset.sum_range_succ,ih]
-  exact hh _
-
-noncomputable def diagramPartition (μ : YoungDiagram) : Nat.Partition μ.card where
-  parts := μ.rowLens
-  parts_pos := fun {i} hi => μ.pos_of_mem_rowLens i (by simpa using hi)
-  parts_sum := by simp
-
-noncomputable def partitionDiagram {N : ℕ} (p : Nat.Partition N) : YoungDiagram :=
-  YoungDiagram.ofRowLens (p.parts.sort (· ≥ ·)) (Multiset.pairwise_sort _ _).sortedGE
-
-end Thorp.Specht
-
-end ThorpNine.Harmonic
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Harmonic.ModifiedRoutingMoment
+
+namespace OAI
+
+namespace ThorpNine.Harmonic
+
+namespace Thorp
+
+noncomputable def modifiedReverseCoins (L : ℕ) : (r : ℕ) → ModifiedCoins L r → ModifiedCoins L r
+  | 0, ω => ((fun u => butterflyPerm L (decodeButterfly L ω.2.1) * (ω.1 u)⁻¹ *
+      butterflyPerm L (decodeButterfly L ω.2.1)),ω.2)
+  | r+1, ω =>
+      let z := modifiedStepEquiv L r ω
+      (modifiedStepEquiv L r).symm
+        ((modifiedReverseCoins L r z.1.1,modifiedReverseCoins L r z.1.2),(z.2.2,z.2.1))
+
+lemma modifiedReverseCoins_involutive (L r : ℕ) :
+    Function.Involutive (modifiedReverseCoins L r) := by
+  induction r with
+  | zero =>
+      intro ω
+      apply Prod.ext
+      · funext u
+        simp only [modifiedReverseCoins,mul_inv_rev,inv_inv]
+        simp [mul_assoc]
+      · rfl
+  | succ r ih =>
+      change ∀ ω, modifiedReverseCoins L r (modifiedReverseCoins L r ω) = ω at ih
+      intro ω
+      simp only [modifiedReverseCoins,Equiv.apply_symm_apply,ih]
+      exact (modifiedStepEquiv L r).symm_apply_apply ω
+
+noncomputable def modifiedReverseEquiv (L r : ℕ) : Equiv.Perm (ModifiedCoins L r) where
+  toFun := modifiedReverseCoins L r
+  invFun := modifiedReverseCoins L r
+  left_inv := modifiedReverseCoins_involutive L r
+  right_inv := modifiedReverseCoins_involutive L r
+
+lemma modifiedPerm_reverse (L r : ℕ) (E : Finset (Card r)) (ω : ModifiedCoins L r) :
+    modifiedPerm L r E (modifiedReverseCoins L r ω) = (modifiedPerm L r E ω)⁻¹ := by
+  induction r with
+  | zero =>
+      rw [modifiedPerm_zero,modifiedPerm_zero]
+      split_ifs
+      · simp only [inv_one]
+      · simp only [modifiedReverseCoins,mul_inv_rev,inv_inv]
+        simp
+  | succ r ih =>
+      obtain ⟨z,rfl⟩ := (modifiedStepEquiv L r).symm.surjective ω
+      rcases z with ⟨⟨ω₀,ω₁⟩,⟨ξ,η⟩⟩
+      simp only [modifiedReverseCoins,Equiv.apply_symm_apply,modifiedPerm_step,
+        mul_inv_rev,pairSwitch_inv,childLift_inv]
+      rw [mul_assoc]
+      congr 1
+      congr 1
+      apply congrArg childLift
+      funext b
+      cases b <;> simp only [Bool.false_eq_true,↓reduceIte,ih]
+
+lemma modified_kernel_symmetric (L r : ℕ) (E : Finset (Card r))
+    {ι : Type*} [Fintype ι] (e f : ι ↪ Card (L+r)) :
+    PairRouting.tupleProbability (modifiedPerm L r E) e f =
+      PairRouting.tupleProbability (modifiedPerm L r E) f e :=
+  PairRouting.tupleProbability_symmetric _ (modifiedReverseEquiv L r)
+    (modifiedPerm_reverse L r E) e f
+
+end Thorp
+
+namespace Thorp.Specht
+open scoped BigOperators Classical
+
+@[simp] lemma card_cell (μ : YoungDiagram) : Fintype.card (Cell μ) = μ.card := by
+  simp [Cell, YoungDiagram.card]
+
+noncomputable def columnEquiv (μ : YoungDiagram) (j : ℕ) :
+    {x : Cell μ // col x = j} ≃ Fin (μ.colLen j) where
+  toFun x := ⟨row x.1, by simpa [x.2] using row_lt_colLen x.1⟩
+  invFun i := ⟨⟨(i,j), YoungDiagram.mem_iff_lt_colLen.mpr i.2⟩, rfl⟩
+  left_inv x := by apply Subtype.ext; apply Subtype.ext; exact Prod.ext rfl x.2.symm
+  right_inv i := by rfl
+
+noncomputable def columnValues {μ : YoungDiagram} (f : Cell μ → ℕ) (j : ℕ) : Finset ℕ :=
+  (Finset.univ.filter (fun x => col x = j)).image f
+
+lemma mem_columnValues {μ : YoungDiagram} (f : Cell μ → ℕ) (x : Cell μ) :
+    f x ∈ columnValues f (col x) := by
+  classical
+  exact Finset.mem_image.mpr ⟨x, by simp, rfl⟩
+
+lemma card_columnValues {μ : YoungDiagram} (f : Cell μ → ℕ)
+    (hf : ∀ x y, col x = col y → f x = f y → x = y) (j : ℕ) :
+    (columnValues f j).card = μ.colLen j := by
+  classical
+  rw [columnValues, Finset.card_image_iff.mpr]
+  · rw [← Fintype.card_subtype]
+    exact Fintype.card_congr (columnEquiv μ j) |>.trans (Fintype.card_fin _)
+  intro x hx y hy he
+  exact hf x y ((Finset.mem_filter.mp hx).2.trans (Finset.mem_filter.mp hy).2.symm) he
+
+lemma strictMono_fin_nat_ge {n : ℕ} (f : Fin n → ℕ) (hf : StrictMono f) :
+    ∀ i, i.val ≤ f i := by
+  intro i
+  induction h : i.val using Nat.strong_induction_on generalizing i with
+  | h n ih =>
+    by_cases hn : n = 0
+    · omega
+    · have hn' : n-1 < i.val := by omega
+      let j : Fin _ := ⟨n-1, hn'.trans i.isLt⟩
+      have hj := ih (n-1) (by omega) j rfl
+      have hij := hf (show j < i from hn')
+      dsimp [j] at hj hij
+      omega
+
+noncomputable def rankCell {μ : YoungDiagram} (f : Cell μ → ℕ)
+    (hf : ∀ x y, col x = col y → f x = f y → x = y) (x : Cell μ) : Cell μ :=
+  let k := ((columnValues f (col x)).orderIsoOfFin (card_columnValues f hf (col x))).symm
+    ⟨f x, mem_columnValues f x⟩
+  ⟨(k.val,col x), YoungDiagram.mem_iff_lt_colLen.mpr k.isLt⟩
+
+@[simp] lemma rankCell_col {μ : YoungDiagram} (f : Cell μ → ℕ)
+    (hf : ∀ x y, col x = col y → f x = f y → x = y) (x : Cell μ) :
+    col (rankCell f hf x) = col x := rfl
+
+lemma rankCell_orderEmb {μ : YoungDiagram} (f : Cell μ → ℕ)
+    (hf : ∀ x y, col x = col y → f x = f y → x = y) (x : Cell μ) :
+    (columnValues f (col x)).orderEmbOfFin (card_columnValues f hf (col x))
+      ⟨row (rankCell f hf x), row_lt_colLen (rankCell f hf x)⟩ = f x := by
+  change ((columnValues f (col x)).orderIsoOfFin (card_columnValues f hf (col x))
+    (((columnValues f (col x)).orderIsoOfFin (card_columnValues f hf (col x))).symm
+      ⟨f x, mem_columnValues f x⟩)).val = f x
+  simp
+
+lemma rankCell_injective {μ : YoungDiagram} (f : Cell μ → ℕ)
+    (hf : ∀ x y, col x = col y → f x = f y → x = y) :
+    Function.Injective (rankCell f hf) := by
+  intro x y h
+  have hc : col x = col y := by simpa only [rankCell_col] using congrArg (fun z : Cell μ => col z) h
+  apply hf x y hc
+  have hx := rankCell_orderEmb f hf x
+  have hy := rankCell_orderEmb f hf y
+  change (columnValues f (col (rankCell f hf x))).orderEmbOfFin
+    (card_columnValues f hf (col (rankCell f hf x)))
+      ⟨row (rankCell f hf x), row_lt_colLen (rankCell f hf x)⟩ = f x at hx
+  rw [h] at hx
+  exact hx.symm.trans hy
+
+noncomputable def rankPerm {μ : YoungDiagram} (f : Cell μ → ℕ)
+    (hf : ∀ x y, col x = col y → f x = f y → x = y) : Equiv.Perm (Cell μ) :=
+  Equiv.ofBijective (rankCell f hf) ((Finite.injective_iff_bijective).mp (rankCell_injective f hf))
+
+lemma rankCell_row_le {μ : YoungDiagram} (f : Cell μ → ℕ)
+    (hf : ∀ x y, col x = col y → f x = f y → x = y) (x : Cell μ) :
+    row (rankCell f hf x) ≤ f x := by
+  rw [←rankCell_orderEmb f hf x]
+  exact strictMono_fin_nat_ge _ (OrderEmbedding.strictMono _) _
+
+theorem column_transporter (μ : YoungDiagram) (p : Equiv.Perm (Cell μ))
+    (hp : ∀ x y, col x = col y → row (p x) = row (p y) → x = y) :
+    ∃ c : Equiv.Perm (Cell μ), (∀ x, col (c x) = col x) ∧
+      ∀ x, row (c x) = row (p x) := by
+  classical
+  let c := rankPerm (fun x => row (p x)) hp
+  refine ⟨c, fun x => rfl, ?_⟩
+  have hle (x : Cell μ) : row (c x) ≤ row (p x) := rankCell_row_le _ hp x
+  have hsum : ∑ x, row (c x) = ∑ x, row (p x) := by
+    rw [Equiv.sum_comp, Equiv.sum_comp]
+  exact fun x => (Finset.sum_eq_sum_iff_of_le (fun x _ => hle x)).mp hsum x (Finset.mem_univ _)
+
+
+@[simp] lemma tabloidAct_inv {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) :
+    tabloidAct p⁻¹ = (tabloidAct p)⁻¹ := by
+  ext f x
+  simp [tabloidAct,Equiv.Perm.inv_def]
+
+@[simp] lemma tabloidRep_apply {μ : YoungDiagram} (p : Equiv.Perm (Cell μ))
+    (v : Tabloid μ → ℂ) (f : Tabloid μ) : tabloidRep μ p v f = v (tabloidAct p⁻¹ f) := rfl
+
+lemma tabloidRep_delta {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) (f : Tabloid μ) :
+    tabloidRep μ p (delta f) = delta (tabloidAct p f) := by
+  classical
+  ext g
+  simp only [tabloidRep_apply,delta,Equiv.Perm.inv_def]
+  congr 1
+  exact propext (Equiv.symm_apply_eq (tabloidAct p))
+
+@[simp] lemma mem_colGroup {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) :
+    p ∈ colGroup μ ↔ ∀ x, col (p x) = col x := Iff.rfl
+
+lemma colGroup_row_stabilizer {μ : YoungDiagram} (c : colGroup μ)
+    (hc : tabloidAct c (baseTabloid μ) = baseTabloid μ) : (c : Equiv.Perm (Cell μ)) = 1 := by
+  apply Equiv.ext
+  intro x
+  have hr := congrArg (fun f : Tabloid μ => ((f.1 ((c : Equiv.Perm _) x)) : ℕ)) hc
+  simp only [tabloidAct_apply,Equiv.Perm.inv_def,Equiv.symm_apply_apply,baseTabloid,rowIndex] at hr
+  apply Subtype.ext
+  exact Prod.ext hr.symm (c.2 x)
+
+lemma tabloid_column_transporter {μ : YoungDiagram} (f : Tabloid μ)
+    (hf : ∀ x y, col x = col y → f.1 x = f.1 y → x = y) :
+    ∃ c : colGroup μ, f = tabloidAct c (baseTabloid μ) := by
+  obtain ⟨p,hp⟩ := f.2
+  have hp' : ∀ x y, col x = col y → row (p x) = row (p y) → x = y := by
+    intro x y hcol hrow
+    apply hf x y hcol
+    rw [hp]
+    exact Fin.ext hrow
+  obtain ⟨c,hcol,hrow⟩ := column_transporter μ p hp'
+  refine ⟨⟨c⁻¹, (colGroup μ).inv_mem hcol⟩, ?_⟩
+  apply Subtype.ext
+  funext x
+  apply Fin.ext
+  simp only [tabloidAct_apply,baseTabloid,rowIndex,inv_inv,hp,Function.comp_apply]
+  exact (hrow x).symm
+
+
+@[simp] lemma permSign_one {α : Type*} [Fintype α] [DecidableEq α] :
+    permSign (1 : Equiv.Perm α) = 1 := by simp [permSign]
+@[simp] lemma permSign_mul {α : Type*} [Fintype α] [DecidableEq α] (p q : Equiv.Perm α) :
+    permSign (p*q) = permSign p * permSign q := by simp [permSign]
+@[simp] lemma permSign_inv {α : Type*} [Fintype α] [DecidableEq α] (p : Equiv.Perm α) :
+    permSign p⁻¹ = permSign p := by simp [permSign,Equiv.Perm.inv_def]
+lemma permSign_sq {α : Type*} [Fintype α] [DecidableEq α] (p : Equiv.Perm α) :
+    permSign p * permSign p = 1 := by
+  calc
+    permSign p * permSign p = permSign (p*p⁻¹) := by rw [permSign_mul,permSign_inv]
+    _ = 1 := by simp
+@[simp] lemma permSign_swap {α : Type*} [Fintype α] [DecidableEq α] {x y : α} (h : x ≠ y) :
+    permSign (Equiv.swap x y) = -1 := by simp [permSign, h]
+@[simp] lemma conj_permSign {α : Type*} [Fintype α] [DecidableEq α] (p : Equiv.Perm α) :
+    starRingEnd ℂ (permSign p) = permSign p := by simp [permSign]
+
+variable {α V : Type*} [Fintype α] [DecidableEq α] [AddCommGroup V] [Module ℂ V]
+
+lemma alternator_mul (C : Subgroup (Equiv.Perm α))
+    (ρ : Representation ℂ (Equiv.Perm α) V) (c : C) :
+    alternator C ρ * ρ c = permSign (c : Equiv.Perm α) • alternator C ρ := by
+  classical
+  rw [alternator,Finset.sum_mul,Finset.smul_sum]
+  apply Fintype.sum_equiv (Equiv.mulRight c)
+  intro a
+  simp only [smul_mul_assoc,←map_mul,Equiv.coe_mulRight,Subgroup.coe_mul,permSign_mul,smul_smul]
+  congr 1
+  have h := permSign_sq (c : Equiv.Perm α)
+  calc
+    permSign (a : Equiv.Perm α) = permSign (a : Equiv.Perm α)*1 := by ring
+    _ = permSign (a : Equiv.Perm α)*(permSign (c : Equiv.Perm α)*permSign (c : Equiv.Perm α)) := by rw [h]
+    _ = _ := by ring
+
+lemma mul_alternator (C : Subgroup (Equiv.Perm α))
+    (ρ : Representation ℂ (Equiv.Perm α) V) (c : C) :
+    ρ c * alternator C ρ = permSign (c : Equiv.Perm α) • alternator C ρ := by
+  classical
+  rw [alternator,Finset.mul_sum,Finset.smul_sum]
+  apply Fintype.sum_equiv (Equiv.mulLeft c)
+  intro a
+  simp only [mul_smul_comm,←map_mul,Equiv.coe_mulLeft,Subgroup.coe_mul,permSign_mul,smul_smul]
+  congr 1
+  have h := permSign_sq (c : Equiv.Perm α)
+  calc
+    permSign (a : Equiv.Perm α) = 1*permSign (a : Equiv.Perm α) := by ring
+    _ = (permSign (c : Equiv.Perm α)*permSign (c : Equiv.Perm α))*permSign (a : Equiv.Perm α) := by rw [h]
+    _ = _ := by ring
+
+lemma alternator_apply_action (C : Subgroup (Equiv.Perm α))
+    (ρ : Representation ℂ (Equiv.Perm α) V) (c : C) (v : V) :
+    alternator C ρ (ρ c v) = permSign (c : Equiv.Perm α) • alternator C ρ v := by
+  exact congrArg (fun T : Module.End ℂ V => T v) (alternator_mul C ρ c)
+
+lemma action_alternator_apply (C : Subgroup (Equiv.Perm α))
+    (ρ : Representation ℂ (Equiv.Perm α) V) (c : C) (v : V) :
+    ρ c (alternator C ρ v) = permSign (c : Equiv.Perm α) • alternator C ρ v := by
+  exact congrArg (fun T : Module.End ℂ V => T v) (mul_alternator C ρ c)
+
+lemma alternator_eq_zero_of_odd_stabilizer (C : Subgroup (Equiv.Perm α))
+    (ρ : Representation ℂ (Equiv.Perm α) V) (c : C) (v : V)
+    (hc : permSign (c : Equiv.Perm α) = -1) (hv : ρ c v = v) :
+    alternator C ρ v = 0 := by
+  have h := alternator_apply_action C ρ c v
+  rw [hv,hc,neg_one_smul] at h
+  have h2 : (2:ℂ) • alternator C ρ v = 0 := by
+    rw [two_smul]
+    nth_rw 1 [h]
+    exact neg_add_cancel _
+  exact (smul_eq_zero.mp h2).resolve_left (by norm_num)
+
+end Thorp.Specht
+namespace Thorp.Specht
+open scoped BigOperators Classical
+
+lemma swap_mem_colGroup {μ : YoungDiagram} (x y : Cell μ) (hc : col x = col y) :
+    Equiv.swap x y ∈ colGroup μ := by
+  classical
+  intro z
+  by_cases hx : z=x
+  · subst z; simpa using hc.symm
+  by_cases hy : z=y
+  · subst z; simpa using hc
+  simp [Equiv.swap_apply_of_ne_of_ne hx hy]
+
+lemma swap_fixes_tabloid {μ : YoungDiagram} (f : Tabloid μ) (x y : Cell μ)
+    (hxy : f.1 x = f.1 y) : tabloidAct (Equiv.swap x y) f = f := by
+  classical
+  apply Subtype.ext
+  funext z
+  simp only [tabloidAct_apply, Equiv.swap_inv]
+  by_cases hx : z=x
+  · subst z; simpa using hxy.symm
+  by_cases hy : z=y
+  · subst z; simpa using hxy
+  simp [Equiv.swap_apply_of_ne_of_ne hx hy]
+
+lemma alternator_delta_rank_one {μ : YoungDiagram} (f : Tabloid μ) :
+    ∃ z : ℂ, alternator (colGroup μ) (tabloidRep μ) (delta f) = z • polytabloid μ := by
+  classical
+  by_cases hf : ∀ x y, col x = col y → f.1 x = f.1 y → x = y
+  · obtain ⟨c,rfl⟩ := tabloid_column_transporter f hf
+    refine ⟨permSign (c:Equiv.Perm (Cell μ)), ?_⟩
+    rw [←tabloidRep_delta]
+    exact alternator_apply_action _ _ c _
+  · push Not at hf
+    obtain ⟨x,y,hcol,hrow,hne⟩ := hf
+    refine ⟨0, ?_⟩
+    rw [zero_smul]
+    apply alternator_eq_zero_of_odd_stabilizer _ _ ⟨Equiv.swap x y,swap_mem_colGroup x y hcol⟩
+    · exact permSign_swap hne
+    · rw [tabloidRep_delta,swap_fixes_tabloid f x y hrow]
+
+lemma polytabloid_formula (μ : YoungDiagram) :
+    polytabloid μ = ∑ c : colGroup μ, permSign (c:Equiv.Perm (Cell μ)) •
+      delta (tabloidAct c (baseTabloid μ)) := by
+  classical
+  simp only [polytabloid,alternator,LinearMap.sum_apply,LinearMap.smul_apply,tabloidRep_delta]
+
+lemma polytabloid_base (μ : YoungDiagram) : polytabloid μ (baseTabloid μ) = 1 := by
+  classical
+  rw [polytabloid_formula,Finset.sum_apply]
+  rw [Finset.sum_eq_single (1 : colGroup μ)]
+  · simp [delta]
+  · intro c hc hne
+    have hn : baseTabloid μ ≠ tabloidAct c (baseTabloid μ) := by
+      intro h
+      apply hne
+      exact Subtype.ext (colGroup_row_stabilizer c h.symm)
+    simp [Pi.smul_apply,delta,hn]
+  · simp
+
+lemma polytabloid_ne_zero (μ : YoungDiagram) : polytabloid μ ≠ 0 := by
+  intro h
+  have hv := congrArg (fun v : Tabloid μ → ℂ => v (baseTabloid μ)) h
+  simp [polytabloid_base] at hv
+
+lemma delta_decomposition {μ : YoungDiagram} (v : Tabloid μ → ℂ) :
+    v = ∑ f : Tabloid μ, v f • delta f := by
+  classical
+  funext g
+  simp only [Finset.sum_apply,Pi.smul_apply,delta,smul_eq_mul,mul_ite,mul_one,mul_zero]
+  simp
+
+theorem alternator_rank_one (μ : YoungDiagram) (v : Tabloid μ → ℂ) :
+    ∃ z : ℂ, alternator (colGroup μ) (tabloidRep μ) v = z • polytabloid μ := by
+  classical
+  choose z hz using fun f : Tabloid μ => alternator_delta_rank_one f
+  refine ⟨∑ f, v f * z f, ?_⟩
+  conv_lhs => rw [delta_decomposition v]
+  simp only [map_sum,map_smul,hz,smul_smul,Finset.sum_smul]
+
+lemma alternator_rank_one_exact (μ : YoungDiagram) (v : Tabloid μ → ℂ) :
+    alternator (colGroup μ) (tabloidRep μ) v =
+      (alternator (colGroup μ) (tabloidRep μ) v (baseTabloid μ)) • polytabloid μ := by
+  obtain ⟨z,hz⟩ := alternator_rank_one μ v
+  rw [hz]
+  simp only [Pi.smul_apply,polytabloid_base,smul_eq_mul,mul_one]
+
+
+noncomputable def pairing {μ : YoungDiagram} (v w : Tabloid μ → ℂ) : ℂ :=
+  ∑ f, w f * starRingEnd ℂ (v f)
+
+lemma pairing_delta {μ : YoungDiagram} (f : Tabloid μ) (v : Tabloid μ → ℂ) :
+    pairing (delta f) v = v f := by
+  classical
+  simp [pairing,delta]
+
+lemma pairing_sum_left {μ : YoungDiagram} {ι : Type*} [Fintype ι]
+    (v : ι → Tabloid μ → ℂ) (w : Tabloid μ → ℂ) :
+    pairing (∑ i, v i) w = ∑ i, pairing (v i) w := by
+  simp only [pairing,Finset.sum_apply,map_sum,Finset.mul_sum]
+  exact Finset.sum_comm
+
+lemma pairing_smul_left {μ : YoungDiagram} (a : ℂ) (v w : Tabloid μ → ℂ) :
+    pairing (a • v) w = starRingEnd ℂ a * pairing v w := by
+  simp only [pairing,Pi.smul_apply,smul_eq_mul,map_mul,Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro f hf
+  ring
+
+lemma pairing_zero_left {μ : YoungDiagram} (v : Tabloid μ → ℂ) : pairing 0 v = 0 := by
+  simp [pairing]
+
+lemma pairing_add_left {μ : YoungDiagram} (u v w : Tabloid μ → ℂ) :
+    pairing (u+v) w = pairing u w + pairing v w := by
+  simp [pairing,mul_add,Finset.sum_add_distrib]
+
+lemma pairing_self_eq_zero {μ : YoungDiagram} (v : Tabloid μ → ℂ) :
+    pairing v v = 0 ↔ v = 0 := by
+  change inner ℂ (WithLp.toLp 2 v : EuclideanSpace ℂ (Tabloid μ)) (WithLp.toLp 2 v) = 0 ↔ v = 0
+  rw [inner_self_eq_zero]
+  constructor
+  · intro h; exact congrArg WithLp.ofLp h
+  · intro h; subst v; rfl
+
+lemma pairing_action {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) (v w : Tabloid μ → ℂ) :
+    pairing (tabloidRep μ p v) w = pairing v (tabloidRep μ p⁻¹ w) := by
+  classical
+  simp only [pairing,tabloidRep_apply,inv_inv]
+  apply Fintype.sum_equiv (tabloidAct p⁻¹)
+  intro f
+  have hcancel : tabloidAct p (tabloidAct p⁻¹ f) = f := by
+    rw [tabloidAct_inv]
+    exact (tabloidAct p).apply_symm_apply f
+  rw [hcancel]
+
+lemma alternator_base_pairing (μ : YoungDiagram) (v : Tabloid μ → ℂ) :
+    alternator (colGroup μ) (tabloidRep μ) v (baseTabloid μ) = pairing (polytabloid μ) v := by
+  classical
+  simp only [polytabloid_formula,pairing_sum_left,pairing_smul_left,conj_permSign,pairing_delta]
+  simp only [alternator,LinearMap.sum_apply,LinearMap.smul_apply,Finset.sum_apply,Pi.smul_apply,
+    smul_eq_mul,tabloidRep_apply]
+  apply Fintype.sum_equiv (Equiv.inv (colGroup μ))
+  intro c
+  simp only [Equiv.inv_apply,Subgroup.coe_inv,permSign_inv]
+
+theorem alternator_pairing (μ : YoungDiagram) (v : Tabloid μ → ℂ) :
+    alternator (colGroup μ) (tabloidRep μ) v = pairing (polytabloid μ) v • polytabloid μ := by
+  rw [alternator_rank_one_exact,alternator_base_pairing]
+
+
+lemma polytabloid_mem_space (μ : YoungDiagram) : polytabloid μ ∈ space μ := by
+  simpa using orbit_mem_space μ 1
+
+lemma alternator_mem_invariant (μ : YoungDiagram) (W : Submodule ℂ (Tabloid μ → ℂ))
+    (hW : ∀ p v, v ∈ W → tabloidRep μ p v ∈ W) (v : Tabloid μ → ℂ) (hv : v ∈ W) :
+    alternator (colGroup μ) (tabloidRep μ) v ∈ W := by
+  classical
+  simp only [alternator,LinearMap.sum_apply,LinearMap.smul_apply]
+  exact W.sum_mem (fun c _ => W.smul_mem _ (hW c v hv))
+
+theorem submodule_theorem (μ : YoungDiagram) (W : Submodule ℂ (Tabloid μ → ℂ))
+    (hW : ∀ p v, v ∈ W → tabloidRep μ p v ∈ W) :
+    space μ ≤ W ∨ ∀ v ∈ W, ∀ w ∈ space μ, pairing w v = 0 := by
+  classical
+  by_cases hSW : space μ ≤ W
+  · exact Or.inl hSW
+  right
+  have he : polytabloid μ ∉ W := by
+    intro he
+    apply hSW
+    apply Submodule.span_le.mpr
+    rintro _ ⟨p,rfl⟩
+    exact hW p _ he
+  have hpair : ∀ v ∈ W, pairing (polytabloid μ) v = 0 := by
+    intro v hv
+    by_contra hne
+    apply he
+    have hal := alternator_mem_invariant μ W hW v hv
+    rw [alternator_pairing] at hal
+    have hh := W.smul_mem (pairing (polytabloid μ) v)⁻¹ hal
+    simpa only [smul_smul,inv_mul_cancel₀ hne,one_smul] using hh
+  intro v hv w hw
+  induction hw using Submodule.span_induction with
+  | mem w hw =>
+    obtain ⟨p,rfl⟩ := hw
+    rw [pairing_action]
+    exact hpair _ (hW p⁻¹ v hv)
+  | zero => exact pairing_zero_left v
+  | add u w hu hw ihu ihw => rw [pairing_add_left,ihu,ihw,add_zero]
+  | smul a w hw ih => rw [pairing_smul_left,ih,mul_zero]
+
+theorem space_simple (μ : YoungDiagram) (W : Submodule ℂ (Tabloid μ → ℂ))
+    (hW : ∀ p v, v ∈ W → tabloidRep μ p v ∈ W) (hWS : W ≤ space μ) :
+    W = ⊥ ∨ W = space μ := by
+  obtain hSW | horth := submodule_theorem μ W hW
+  · exact Or.inr (le_antisymm hWS hSW)
+  · left
+    apply le_antisymm _ bot_le
+    intro v hv
+    exact (pairing_self_eq_zero v).mp (horth v hv v (hWS hv))
+
+
+lemma space_ne_bot (μ : YoungDiagram) : space μ ≠ ⊥ := by
+  intro h
+  have he := polytabloid_mem_space μ
+  rw [h,Submodule.mem_bot] at he
+  exact polytabloid_ne_zero μ he
+
+theorem representation_irreducible (μ : YoungDiagram) :
+    Representation.IsIrreducible (representation μ) := by
+  have hnt : (⊥ : Subrepresentation (representation μ)) ≠ ⊤ := by
+    intro h
+    have he : (⟨polytabloid μ,polytabloid_mem_space μ⟩ : space μ) ∈
+        (⊥ : Subrepresentation (representation μ)) := by rw [h]; trivial
+    have he' : (⟨polytabloid μ,polytabloid_mem_space μ⟩ : space μ) = 0 := he
+    exact polytabloid_ne_zero μ (congrArg Subtype.val he')
+  refine { exists_pair_ne := ⟨⊥,⊤,hnt⟩, eq_bot_or_eq_top := ?_ }
+  intro U
+  let W : Submodule ℂ (Tabloid μ → ℂ) := U.toSubmodule.map (space μ).subtype
+  have hW : ∀ p v, v ∈ W → tabloidRep μ p v ∈ W := by
+    intro p v hv
+    obtain ⟨u,hu,rfl⟩ := hv
+    exact ⟨representation μ p u,U.apply_mem_toSubmodule p hu,rfl⟩
+  have hWS : W ≤ space μ := by
+    intro v hv
+    obtain ⟨u,hu,rfl⟩ := hv
+    exact u.2
+  obtain hbot | htop := space_simple μ W hW hWS
+  · left
+    apply le_antisymm _ bot_le
+    intro u hu
+    have hmem : (u : Tabloid μ → ℂ) ∈ W := ⟨u,hu,rfl⟩
+    rw [hbot,Submodule.mem_bot] at hmem
+    change u = 0
+    exact Subtype.ext hmem
+  · right
+    apply le_antisymm le_top
+    intro u hu
+    have hmem : (u : Tabloid μ → ℂ) ∈ W := by rw [htop]; exact u.2
+    obtain ⟨v,hv,hvu⟩ := hmem
+    have heq : v = u := Subtype.ext hvu
+    exact heq ▸ hv
+
+
+section Symmetrizer
+variable {α V : Type*} [Fintype α] [DecidableEq α] [AddCommGroup V] [Module ℂ V]
+
+noncomputable def symmetrizer (H : Subgroup (Equiv.Perm α))
+    (ρ : Representation ℂ (Equiv.Perm α) V) : Module.End ℂ V :=
+  ∑ h : H, ρ h
+
+lemma symmetrizer_mul (H : Subgroup (Equiv.Perm α))
+    (ρ : Representation ℂ (Equiv.Perm α) V) (h : H) :
+    symmetrizer H ρ * ρ h = symmetrizer H ρ := by
+  rw [symmetrizer,Finset.sum_mul]
+  exact Fintype.sum_equiv (Equiv.mulRight h) _ _ (fun g => by simp [←map_mul])
+
+lemma symmetrizer_eq_zero_of_neg (H : Subgroup (Equiv.Perm α))
+    (ρ : Representation ℂ (Equiv.Perm α) V) (h : H) (v : V) (hv : ρ h v = -v) :
+    symmetrizer H ρ v = 0 := by
+  have hh := congrArg (fun T : Module.End ℂ V => T v) (symmetrizer_mul H ρ h)
+  change symmetrizer H ρ (ρ h v) = symmetrizer H ρ v at hh
+  rw [hv,map_neg] at hh
+  have h2 : (2:ℂ) • symmetrizer H ρ v = 0 := by
+    rw [two_smul]
+    nth_rw 1 [←hh]
+    exact neg_add_cancel _
+  exact (smul_eq_zero.mp h2).resolve_left (by norm_num)
+
+noncomputable def fiberGroup {β : Type*} (f : α → β) : Subgroup (Equiv.Perm α) where
+  carrier := {p | ∀ x, f (p x) = f x}
+  one_mem' := by intro x; rfl
+  mul_mem' := by intro p q hp hq x; exact (hp (q x)).trans (hq x)
+  inv_mem' := by intro p hp x; simpa using (hp (p⁻¹ x)).symm
+
+omit [Fintype α] in
+lemma swap_mem_fiberGroup {β : Type*} (f : α → β) (x y : α) (hf : f x = f y) :
+    Equiv.swap x y ∈ fiberGroup f := by
+  intro z
+  by_cases hx : z=x
+  · subst z; simpa using hf.symm
+  by_cases hy : z=y
+  · subst z; simpa using hf
+  simp [Equiv.swap_apply_of_ne_of_ne hx hy]
+
+end Symmetrizer
+
+lemma swap_polytabloid {μ : YoungDiagram} (x y : Cell μ) (hne : x ≠ y)
+    (hcol : col x = col y) :
+    tabloidRep μ (Equiv.swap x y) (polytabloid μ) = -polytabloid μ := by
+  have h := action_alternator_apply (colGroup μ) (tabloidRep μ)
+    ⟨Equiv.swap x y,swap_mem_colGroup x y hcol⟩ (delta (baseTabloid μ))
+  simpa only [polytabloid,permSign_swap hne,neg_one_smul] using h
+
+theorem tall_column_average_zero (μ : YoungDiagram) {β : Type*} [Fintype β]
+    (f : Cell μ → β) (j : ℕ) (hj : Fintype.card β < μ.colLen j)
+    (v : Tabloid μ → ℂ) (hv : v ∈ space μ) :
+    symmetrizer (fiberGroup f) (tabloidRep μ) v = 0 := by
+  induction hv using Submodule.span_induction with
+  | mem v hv =>
+    obtain ⟨p,rfl⟩ := hv
+    have hcard : Fintype.card β < Fintype.card {x : Cell μ // col x = j} := by
+      rw [Fintype.card_congr (columnEquiv μ j),Fintype.card_fin]
+      exact hj
+    obtain ⟨x,y,hne,hf⟩ := Fintype.exists_ne_map_eq_of_card_lt
+      (fun x : {x : Cell μ // col x = j} => f (p x.1)) hcard
+    have hxy : x.1 ≠ y.1 := fun h => hne (Subtype.ext h)
+    let h : fiberGroup f := ⟨Equiv.swap (p x.1) (p y.1),swap_mem_fiberGroup f _ _ hf⟩
+    apply symmetrizer_eq_zero_of_neg _ _ h
+    change tabloidRep μ (Equiv.swap (p x.1) (p y.1))
+      (tabloidRep μ p (polytabloid μ)) = -tabloidRep μ p (polytabloid μ)
+    rw [←Module.End.mul_apply,←map_mul,←Equiv.mul_swap_eq_swap_mul]
+    rw [map_mul,Module.End.mul_apply,swap_polytabloid x.1 y.1 hxy (x.2.trans y.2.symm),map_neg]
+  | zero => exact map_zero _
+  | add v w hv hw ihv ihw => simp only [map_add,ihv,ihw,add_zero]
+  | smul a v hv ih => simp only [map_smul,ih,smul_zero]
+
+
+noncomputable def rowEquiv (μ : YoungDiagram) (i : ℕ) :
+    {x : Cell μ // row x = i} ≃ Fin (μ.rowLen i) where
+  toFun x := ⟨col x.1, by
+    have h := YoungDiagram.mem_iff_lt_rowLen.mp x.1.2
+    change col x.1 < μ.rowLen (row x.1) at h
+    simpa only [x.2] using h⟩
+  invFun j := ⟨⟨(i,j),YoungDiagram.mem_iff_lt_rowLen.mpr j.2⟩,rfl⟩
+  left_inv x := by apply Subtype.ext; apply Subtype.ext; exact Prod.ext x.2.symm rfl
+  right_inv j := rfl
+
+def rowCost (μ : YoungDiagram) : ℕ := ∑ x : Cell μ, row x
+
+lemma eq_of_row_preserving (μ ν : YoungDiagram) (e : Cell μ ≃ Cell ν)
+    (he : ∀ x, row (e x) = row x) : μ = ν := by
+  have hlen (i : ℕ) : μ.rowLen i = ν.rowLen i := by
+    let ei : {x : Cell μ // row x = i} ≃ {x : Cell ν // row x = i} :=
+      { toFun := fun x => ⟨e x.1,(he x.1).trans x.2⟩
+        invFun := fun x => ⟨e.symm x.1,by
+          have h := he (e.symm x.1)
+          simp only [Equiv.apply_symm_apply] at h
+          exact h.symm.trans x.2⟩
+        left_inv := fun x => by apply Subtype.ext; exact e.symm_apply_apply _
+        right_inv := fun x => by apply Subtype.ext; exact e.apply_symm_apply _ }
+    simpa only [Fintype.card_fin] using Fintype.card_congr
+      ((rowEquiv μ i).symm.trans (ei.trans (rowEquiv ν i)))
+  apply YoungDiagram.ext
+  ext ⟨i,j⟩
+  simp only [YoungDiagram.mem_cells,YoungDiagram.mem_iff_lt_rowLen,hlen]
+
+lemma column_injective_rowCost (μ ν : YoungDiagram) (e : Cell μ ≃ Cell ν)
+    (h : ∀ x y : Cell μ, col x = col y → row (e x) = row (e y) → x = y) :
+    rowCost μ ≤ rowCost ν ∧ (rowCost μ = rowCost ν → μ = ν) := by
+  let c := rankPerm (fun x => row (e x)) h
+  have hle (x : Cell μ) : row (c x) ≤ row (e x) := rankCell_row_le _ h x
+  have hsumc : ∑ x, row (c x) = rowCost μ := Equiv.sum_comp c _
+  have hsume : ∑ x, row (e x) = rowCost ν := Equiv.sum_comp e _
+  constructor
+  · rw [←hsumc,←hsume]
+    exact Finset.sum_le_sum (fun x _ => hle x)
+  · intro heq
+    have he : ∀ x, row (c x) = row (e x) := by
+      apply fun x => (Finset.sum_eq_sum_iff_of_le (fun x _ => hle x)).mp
+        (hsumc.trans (heq.trans hsume.symm)) x (Finset.mem_univ _)
+    apply eq_of_row_preserving μ ν (c.symm.trans e)
+    intro x
+    simpa using (he (c.symm x)).symm
+
+lemma swap_relabel {α β : Type*} [DecidableEq α] [DecidableEq β]
+    (e : α ≃ β) (x y : α) : e.permCongr (Equiv.swap x y) = Equiv.swap (e x) (e y) := by
+  ext z
+  simp only [Equiv.permCongr_apply]
+  exact e.injective.map_swap x y (e.symm z) |>.trans (by simp)
+
+lemma sign_vector_rowCost (μ ν : YoungDiagram) (e : Cell μ ≃ Cell ν)
+    (v : Tabloid ν → ℂ) (hv : v ≠ 0)
+    (hs : ∀ c : colGroup μ,
+      tabloidRep ν (e.permCongr c) v = permSign (c : Equiv.Perm (Cell μ)) • v) :
+    rowCost μ ≤ rowCost ν ∧ (rowCost μ = rowCost ν → μ = ν) := by
+  obtain ⟨f,hf⟩ : ∃ f, v f ≠ 0 := by
+    by_contra h
+    push Not at h
+    exact hv (funext h)
+  obtain ⟨p,hp⟩ := f.2
+  apply column_injective_rowCost μ ν (e.trans p)
+  intro x y hcol hrow
+  by_contra hne
+  have hfix : tabloidAct (Equiv.swap (e x) (e y)) f = f := by
+    apply swap_fixes_tabloid
+    rw [hp]
+    exact Fin.ext hrow
+  have heq := congrArg (fun w : Tabloid ν → ℂ => w f)
+    (hs ⟨Equiv.swap x y,swap_mem_colGroup x y hcol⟩)
+  rw [swap_relabel,permSign_swap hne] at heq
+  simp only [tabloidRep_apply,Equiv.swap_inv,hfix,Pi.smul_apply,neg_one_smul] at heq
+  apply hf
+  have h2 : (2:ℂ)*(v f) = 0 := by linear_combination heq
+  exact (mul_eq_zero.mp h2).resolve_left (by norm_num)
+
+lemma polytabloid_col_action (μ : YoungDiagram) (c : colGroup μ) :
+    tabloidRep μ c (polytabloid μ) = permSign (c:Equiv.Perm (Cell μ)) • polytabloid μ :=
+  action_alternator_apply _ _ c _
+
+theorem shapes_eq_of_equivariant_equiv (μ ν : YoungDiagram) (e : Cell μ ≃ Cell ν)
+    (T : space μ ≃ₗ[ℂ] space ν)
+    (hT : ∀ p v, T (representation μ p v) = representation ν (e.permCongr p) (T v)) :
+    μ = ν := by
+  let u : space μ := ⟨polytabloid μ,polytabloid_mem_space μ⟩
+  have hu : u ≠ 0 := fun h => polytabloid_ne_zero μ (congrArg Subtype.val h)
+  have hTcol (c : colGroup μ) :
+      tabloidRep ν (e.permCongr c) (T u) = permSign (c:Equiv.Perm (Cell μ)) • (T u : Tabloid ν → ℂ) := by
+    change (representation ν (e.permCongr c) (T u) : Tabloid ν → ℂ) = _
+    rw [←hT]
+    have hh : representation μ c u = permSign (c:Equiv.Perm (Cell μ)) • u :=
+      Subtype.ext (polytabloid_col_action μ c)
+    rw [hh,map_smul]
+    rfl
+  have hfwd := sign_vector_rowCost μ ν e (T u)
+    (fun h => hu (T.injective (by apply Subtype.ext; simpa using h))) hTcol
+  let w : space ν := ⟨polytabloid ν,polytabloid_mem_space ν⟩
+  have hw : w ≠ 0 := fun h => polytabloid_ne_zero ν (congrArg Subtype.val h)
+  have hTi (p : Equiv.Perm (Cell ν)) (v : space ν) :
+      T.symm (representation ν p v) = representation μ (e.symm.permCongr p) (T.symm v) := by
+    apply T.injective
+    rw [T.apply_symm_apply,hT,T.apply_symm_apply]
+    have hp : e.permCongr (e.symm.permCongr p) = p := by
+      apply Equiv.ext
+      intro x
+      simp [Equiv.permCongr_apply]
+    rw [hp]
+  have hTicol (c : colGroup ν) :
+      tabloidRep μ (e.symm.permCongr c) (T.symm w) =
+        permSign (c:Equiv.Perm (Cell ν)) • (T.symm w : Tabloid μ → ℂ) := by
+    change (representation μ (e.symm.permCongr c) (T.symm w) : Tabloid μ → ℂ) = _
+    rw [←hTi]
+    have hh : representation ν c w = permSign (c:Equiv.Perm (Cell ν)) • w :=
+      Subtype.ext (polytabloid_col_action ν c)
+    rw [hh,map_smul]
+    rfl
+  have hbwd := sign_vector_rowCost ν μ e.symm (T.symm w)
+    (fun h => hw (T.symm.injective (by apply Subtype.ext; simpa using h))) hTicol
+  exact hfwd.2 (Nat.le_antisymm hfwd.1 hbwd.1)
+
+
+noncomputable def cellRowsEquiv (μ : YoungDiagram) :
+    Cell μ ≃ (i : Fin (μ.colLen 0)) × Fin (μ.rowLen i) where
+  toFun x := ⟨rowIndex x,⟨col x,YoungDiagram.mem_iff_lt_rowLen.mp x.2⟩⟩
+  invFun p := ⟨(p.1,p.2),YoungDiagram.mem_iff_lt_rowLen.mpr p.2.2⟩
+  left_inv x := by apply Subtype.ext; rfl
+  right_inv p := by rcases p with ⟨⟨i,hi⟩,⟨j,hj⟩⟩; rfl
+
+lemma card_eq_sum_rowLen (μ : YoungDiagram) : μ.card = ∑ i : Fin (μ.colLen 0), μ.rowLen i := by
+  rw [←card_cell,Fintype.card_congr (cellRowsEquiv μ),Fintype.card_sigma]
+  simp only [Fintype.card_fin]
+
+lemma sum_rowLens (μ : YoungDiagram) : μ.rowLens.sum = μ.card := by
+  rw [card_eq_sum_rowLen,YoungDiagram.rowLens]
+  rw [Fin.sum_univ_eq_sum_range]
+  have hh (n : ℕ) : (List.map μ.rowLen (List.range n)).sum = ∑ i ∈ Finset.range n, μ.rowLen i := by
+    induction n with
+    | zero => simp
+    | succ n ih => simp [List.range_succ,List.map_append,Finset.sum_range_succ,ih]
+  exact hh _
+
+noncomputable def diagramPartition (μ : YoungDiagram) : Nat.Partition μ.card where
+  parts := μ.rowLens
+  parts_pos := fun {i} hi => μ.pos_of_mem_rowLens i (by simpa using hi)
+  parts_sum := by simp
+
+noncomputable def partitionDiagram {N : ℕ} (p : Nat.Partition N) : YoungDiagram :=
+  YoungDiagram.ofRowLens (p.parts.sort (· ≥ ·)) (Multiset.pairwise_sort _ _).sortedGE
+
+end Thorp.Specht
+
+end ThorpNine.Harmonic
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Harmonic/ModifiedRoutingMoment.lean b/lean/OAI/Probability/ThorpRouting/Harmonic/ModifiedRoutingMoment.lean
index b1f2c299ee7f571ccaebfd2f1c934fa3c727c239..d0cd688cb82eff840479a59cd12abf0fa1754696 100644
--- a/lean/OAI/Probability/ThorpRouting/Harmonic/ModifiedRoutingMoment.lean
+++ b/lean/OAI/Probability/ThorpRouting/Harmonic/ModifiedRoutingMoment.lean
@@ -1,856 +1,856 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Harmonic.InputTree
-
-namespace OAI
-
-namespace ThorpNine.Harmonic
-
-namespace Thorp
-
-lemma modified_probability_zero_le (L : ℕ) (E : Finset (Card 0))
-    {ι : Type*} [Fintype ι] (e f : ι ↪ Card L) (he : Function.Surjective e) :
-    PairRouting.tupleProbability (modifiedPerm L 0 E) e f ≤
-      if (fun j => Fin.elim0 j : Card 0) ∈ E then 1 else 1/(Nat.factorial (2^L):ℝ) := by
-  classical
-  rw [PairRouting.tupleProbability,finiteMean_prod,finiteMean_comm]
-  let B : ℝ := if (fun j => Fin.elim0 j : Card 0) ∈ E then 1 else 1/(Nat.factorial (2^L):ℝ)
-  change _ ≤ B
-  calc
-    _ ≤ finiteMean (fun _ω : BenesCoins L => B) := by
-      apply finiteMean_mono
-      intro ω
-      simp only [modifiedPerm_zero]
-      by_cases h : (fun j => Fin.elim0 j : Card 0) ∈ E
-      · simp only [h,↓reduceIte,Equiv.Perm.one_apply,finiteMean_const,B]
-        split_ifs <;> norm_num
-      · simp only [h,↓reduceIte,B]
-        rw [finiteMean_pi_eval (A := fun _ : Card 0 => Equiv.Perm (Card L)) (fun j => Fin.elim0 j) (fun u : Equiv.Perm (Card L) =>
-          if ∀ i, (butterflyPerm L (decodeButterfly L ω.1)*u⁻¹) (e i) = f i then (1:ℝ) else 0)]
-        convert PairRouting.uniformPerm_tuple_upper e f he (butterflyPerm L (decodeButterfly L ω.1)) using 1
-        rw [Fintype.card_perm,card_positions]
-    _ = _ := finiteMean_const B
-
-noncomputable def modifiedRoutingMoment (L r : ℕ) (E : Finset (Card r))
-    {ι : Type*} [Fintype ι] (e f : ι ↪ Card (L+r)) (a : ℝ) : ℝ := by
-  classical
-  exact finiteMean (fun ω : ModifiedCoins L r =>
-    if ∀ i, modifiedPerm L r E ω (e i) = f i then (2:ℝ)^((modifiedCost L r E e ω:ℝ)*a) else 0)
-
-lemma modifiedRoutingMoment_nonneg (L r : ℕ) (E : Finset (Card r))
-    {ι : Type*} [Fintype ι] (e f : ι ↪ Card (L+r)) (a : ℝ) :
-    0 ≤ modifiedRoutingMoment L r E e f a := by
-  classical
-  exact finiteMean_nonneg (fun _ => by split_ifs <;> positivity)
-
-lemma modifiedRoutingMoment_zero (L : ℕ) (E : Finset (Card 0))
-    {ι : Type*} [Fintype ι] (e f : ι ↪ Card L) (a : ℝ) :
-    modifiedRoutingMoment L 0 E e f a = PairRouting.tupleProbability (modifiedPerm L 0 E) e f := by
-  simp only [modifiedRoutingMoment,modifiedCost_zero,Nat.cast_zero,zero_mul,Real.rpow_zero,
-    PairRouting.tupleProbability]
-
-lemma modified_pointwise_zero (L : ℕ) (E : Finset (Card 0))
-    {ι : Type*} [Fintype ι] (e f : ι ↪ Card L) (he : Function.Surjective e) (a : ℝ) (ha : 0 ≤ a) :
-    (PairRouting.tupleProbability (modifiedPerm L 0 E) e f)^(1+a) ≤
-      centralFactor L 0 E a * modifiedRoutingMoment L 0 E e f a := by
-  have hp := PairRouting.tupleProbability_nonneg (modifiedPerm L 0 E) e f
-  rw [modifiedRoutingMoment_zero]
-  have hbound := modified_probability_zero_le L E e f he
-  have hpow := Real.rpow_le_rpow hp hbound ha
-  have hfpos : (0:ℝ) < Nat.factorial (2^L) := Nat.cast_pos.mpr (Nat.factorial_pos _)
-  have heq : (if (fun j => Fin.elim0 j : Card 0) ∈ E then (1:ℝ) else 1/(Nat.factorial (2^L):ℝ))^a =
-      centralFactor L 0 E a := by
-    rw [centralFactor_zero]
-    split_ifs
-    · exact Real.one_rpow _
-    · rw [one_div,Real.inv_rpow (le_of_lt hfpos),Real.rpow_neg (le_of_lt hfpos)]
-  rw [heq] at hpow
-  rw [Real.rpow_add' hp (by linarith : (1+a:ℝ) ≠ 0),Real.rpow_one,mul_comm]
-  exact mul_le_mul_of_nonneg_right hpow hp
-
-
-lemma modified_matches_step (L r : ℕ) (E : Finset (Card (r+1))) {ι : Type*}
-    (e f : ι ↪ Card (L+r+1)) (ω : ModifiedCoins L r × ModifiedCoins L r)
-    (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
-    (∀ i, modifiedPerm L (r+1) E ((modifiedStepEquiv L r).symm (ω,coins)) (e i) = f i) ↔
-    (∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1))
-      ((e.trans (headTailEquiv (L+r)).toEmbedding) i) = ((f.trans (headTailEquiv (L+r)).toEmbedding) i)) := by
-  apply forall_congr'
-  intro i
-  rw [←(headTailEquiv (L+r)).injective.eq_iff,modifiedPerm_coordinates]
-  rfl
-
-lemma modifiedRoutingMoment_step (L r : ℕ) (E : Finset (Card (r+1))) {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (e f : ι ↪ Card (L+r+1)) (a : ℝ) :
-    let x := e.trans (headTailEquiv (L+r)).toEmbedding
-    let y := f.trans (headTailEquiv (L+r)).toEmbedding
-    modifiedRoutingMoment L (r+1) E e f a = PairRouting.outerWeight x y *
-      ∑ c : {c // c ∈ PairRouting.compatibleSet x y},
-        finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
-          if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c.val
-          then (2:ℝ)^((modifiedAssignmentCost L r E x c.val
-            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0) := by
-  classical
-  intro x y
-  let : Fintype (ModifiedCoins L r) := inferInstance
-  rw [modifiedRoutingMoment,finiteMean_equiv (modifiedStepEquiv L r),finiteMean_prod]
-  have he (ω : ModifiedCoins L r × ModifiedCoins L r) :
-      finiteMean (fun coins : (Card (L+r) → Bool) × (Card (L+r) → Bool) =>
-        if ∀ i, modifiedPerm L (r+1) E ((modifiedStepEquiv L r).symm (ω,coins)) (e i) = f i
-        then (2:ℝ)^((modifiedCost L (r+1) E e ((modifiedStepEquiv L r).symm (ω,coins)):ℝ)*a) else 0) =
-      PairRouting.outerWeight x y *
-        ∑ c : {c // c ∈ PairRouting.compatibleSet x y},
-          if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c.val
-          then (2:ℝ)^((modifiedAssignmentCost L r E x c.val
-            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0 := by
-    let W : (ι → Bool) → ℝ := fun c =>
-      if hc : PairRouting.Compatible x c then (2:ℝ)^((modifiedAssignmentCost L r E x c hc ω:ℝ)*a) else 0
-    calc
-      _ = finiteMean (fun coins : (Card (L+r) → Bool) × (Card (L+r) → Bool) =>
-        if ∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1))
-          (x i) = y i then W (PairRouting.colors x coins.1) else 0) := by
-        apply finiteMean_congr
-        intro coins
-        simp only [modified_matches_step,modifiedCost_step]
-        simp only [W,PairRouting.colors_compatible,↓reduceDIte]
-        rfl
-      _ = _ := by
-        rw [PairRouting.sandwich_weighted_probability_subtype]
-        congr 1
-        apply Finset.sum_congr rfl
-        intro c _
-        simp only [W,((PairRouting.mem_compatibleSet x y c.val).mp c.property).1,↓reduceDIte]
-  simp_rw [he]
-  simp_rw [mul_comm (PairRouting.outerWeight x y)]
-  rw [finiteMean_mul_const,finiteMean_sum,mul_comm]
-
-lemma modified_child_moment_product (L r : ℕ) (E : Finset (Card (r+1))) {ι : Type*} [Fintype ι]
-    (x y : ι ↪ Bool × Card (L+r)) (c : ι → Bool)
-    (hcx : PairRouting.Compatible x c) (hcy : PairRouting.Compatible y c) (a : ℝ) :
-    modifiedRoutingMoment L r (exceptionChild E false) (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a *
-      modifiedRoutingMoment L r (exceptionChild E true) (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a =
-    finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
-      if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c
-      then (2:ℝ)^(((modifiedCost L r (exceptionChild E false) (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
-        modifiedCost L r (exceptionChild E true) (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0) := by
-  classical
-  unfold modifiedRoutingMoment
-  rw [←finiteMean_prod_mul]
-  apply finiteMean_congr
-  intro ω
-  let M₀ := ∀ i : {i // c i = false}, modifiedPerm L r (exceptionChild E false) ω.1 (x i.val).2 = (y i.val).2
-  let M₁ := ∀ i : {i // c i = true}, modifiedPerm L r (exceptionChild E true) ω.2 (x i.val).2 = (y i.val).2
-  have hp : (fun b : Bool => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) =
-      (fun b => if b then modifiedPerm L r (exceptionChild E true) ω.2 else modifiedPerm L r (exceptionChild E false) ω.1) := by
-    funext b
-    cases b <;> rfl
-  have hs : PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c ↔
-      M₀ ∧ M₁ := by rw [hp]; exact PairRouting.childMatches_split _ _ _ _ _
-  change (if M₀ then _ else 0) * (if M₁ then _ else 0) = _
-  by_cases h₀ : M₀ <;> by_cases h₁ : M₁
-  · simp only [hs,h₀,h₁,true_and,↓reduceIte]
-    rw [←Real.rpow_add (by norm_num : (0:ℝ)<2)]
-    congr 1
-    ring
-  · simp only [hs,h₀,h₁,and_false,↓reduceIte,mul_zero]
-  · simp only [hs,h₀,h₁,false_and,↓reduceIte,zero_mul]
-  · simp only [hs,h₀,h₁,false_and,↓reduceIte,mul_zero]
-
-lemma modified_probability_step (L r : ℕ) (E : Finset (Card (r+1))) {ι : Type*} [Fintype ι]
-    (e f : ι ↪ Card (L+r+1)) :
-    PairRouting.tupleProbability (modifiedPerm L (r+1) E) e f =
-    finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
-      finiteMean (fun coins : (Card (L+r) → Bool) × (Card (L+r) → Bool) =>
-        if ∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => if b then modifiedPerm L r (exceptionChild E true) ω.2 else modifiedPerm L r (exceptionChild E false) ω.1)
-          ((e.trans (headTailEquiv (L+r)).toEmbedding) i) = ((f.trans (headTailEquiv (L+r)).toEmbedding) i)
-        then (1:ℝ) else 0)) := by
-  classical
-  let : Fintype (ModifiedCoins L r) := inferInstance
-  rw [PairRouting.tupleProbability,finiteMean_equiv (modifiedStepEquiv L r),finiteMean_prod]
-  have hp (b : Bool) (p q : ModifiedCoins L r) :
-      modifiedPerm L r (exceptionChild E b) (if b then p else q) =
-      (if b then modifiedPerm L r (exceptionChild E true) p else modifiedPerm L r (exceptionChild E false) q) := by
-    cases b <;> rfl
-  simp only [modified_matches_step,hp]
-
-
-lemma modified_assignment_moment_bound (L r : ℕ) (E : Finset (Card (r+1))) {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (x y : ι ↪ Bool × Card (L+r)) (c : ι → Bool) (hc : c ∈ PairRouting.compatibleSet x y)
-    (a : ℝ) (ha : 0 ≤ a) :
-    let hcx := ((PairRouting.mem_compatibleSet x y c).mp hc).1
-    PairRouting.outerWeight x y ^ a * ((PairRouting.compatibleSet x y).card:ℝ)^a *
-      ((2:ℝ)^(-(r:ℝ)*Fintype.card ι*a) *
-        finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
-          if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c
-          then (2:ℝ)^(((modifiedCost L r (exceptionChild E false) (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
-            modifiedCost L r (exceptionChild E true) (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0)) ≤
-    (2:ℝ)^(-((r:ℝ)+1)*Fintype.card ι*a) *
-      finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
-        if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c
-        then (2:ℝ)^((modifiedAssignmentCost L r E x c hcx ω:ℝ)*a) else 0) := by
-  classical
-  intro hcx
-  have hscale (q : ℝ) (F : ModifiedCoins L r × ModifiedCoins L r → ℝ) :
-      q * finiteMean F = finiteMean (fun ω => q * F ω) := by
-    simp_rw [mul_comm q,finiteMean_mul_const]
-  rw [←mul_assoc,hscale,hscale]
-  apply finiteMean_mono
-  intro ω
-  split_ifs with hm
-  · have hh := PairRouting.assignment_factor_bound x y c hc
-      (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) hm a ha
-    calc
-      _ ≤ ((2:ℝ)^(((PairRouting.alternatingCycles (PairRouting.coloredEmbedding x c hcx)
-          (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)):ℝ)-Fintype.card ι)*a) *
-          (2:ℝ)^(-(r:ℝ)*Fintype.card ι*a)) *
-          (2:ℝ)^(((modifiedCost L r (exceptionChild E false) (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
-            modifiedCost L r (exceptionChild E true) (PairRouting.childEmbedding x c hcx true) ω.2)*a) :=
-        mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hh (by positivity)) (by positivity)
-      _ = _ := by
-        simp only [←Real.rpow_add (by norm_num : (0:ℝ)<2),modifiedAssignmentCost,Nat.cast_add]
-        congr 1
-        ring
-  · simp only [mul_zero]
-    exact le_rfl
-
-
-lemma modified_child_power_product_bound (L r : ℕ) (E : Finset (Card (r+1)))
-    {ι : Type*} [Fintype ι] (x y : ι ↪ Bool × Card (L+r)) (c : ι → Bool)
-    (hcx : PairRouting.Compatible x c) (hcy : PairRouting.Compatible y c) (a : ℝ)
-    (hfalse : (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E false))
-        (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false))^(1+a) ≤
-      ((2:ℝ)^(-(r:ℝ)*Fintype.card {i // c i = false}*a)*centralFactor L r (exceptionChild E false) a) *
-        modifiedRoutingMoment L r (exceptionChild E false)
-          (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a)
-    (htrue : (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E true))
-        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true))^(1+a) ≤
-      ((2:ℝ)^(-(r:ℝ)*Fintype.card {i // c i = true}*a)*centralFactor L r (exceptionChild E true) a) *
-        modifiedRoutingMoment L r (exceptionChild E true)
-          (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a) :
-    (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E false))
-      (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false))^(1+a) *
-      (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E true))
-        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true))^(1+a) ≤
-    ((2:ℝ)^(-(r:ℝ)*Fintype.card ι*a)*centralFactor L (r+1) E a) *
-      finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
-        if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c
-        then (2:ℝ)^(((modifiedCost L r (exceptionChild E false) (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
-          modifiedCost L r (exceptionChild E true) (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0) := by
-  have hh := mul_le_mul hfalse htrue
-    (Real.rpow_nonneg (PairRouting.tupleProbability_nonneg _ _ _) _)
-    (mul_nonneg (mul_nonneg (by positivity) (centralFactor_nonneg _ _ _ _))
-      (modifiedRoutingMoment_nonneg _ _ _ _ _ _))
-  have he : -(r:ℝ)*Fintype.card {i // c i = false}*a + -(r:ℝ)*Fintype.card {i // c i = true}*a =
-      -(r:ℝ)*Fintype.card ι*a := by
-    rw [←add_mul,←mul_add,←Nat.cast_add,PairRouting.color_card_add]
-  calc
-    _ ≤ _ := hh
-    _ = ((2:ℝ)^(-(r:ℝ)*Fintype.card {i // c i = false}*a)*
-        (2:ℝ)^(-(r:ℝ)*Fintype.card {i // c i = true}*a)) *
-      (centralFactor L r (exceptionChild E false) a * centralFactor L r (exceptionChild E true) a) *
-      (modifiedRoutingMoment L r (exceptionChild E false)
-        (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a *
-      modifiedRoutingMoment L r (exceptionChild E true)
-        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a) := by ring
-    _ = _ := by rw [←Real.rpow_add (by norm_num : (0:ℝ)<2),he,←centralFactor_step,modified_child_moment_product]
-
-
-theorem modified_pointwise_moment (L r : ℕ) (E : Finset (Card r)) {ι : Type*} [Fintype ι]
-    (e f : ι ↪ Card (L+r)) (he : Function.Surjective e) (a : ℝ) (ha : 0 ≤ a) :
-    (PairRouting.tupleProbability (modifiedPerm L r E) e f)^(1+a) ≤
-      ((2:ℝ)^(-(r:ℝ)*Fintype.card ι*a)*centralFactor L r E a) * modifiedRoutingMoment L r E e f a := by
-  classical
-  induction r generalizing ι with
-  | zero =>
-      simpa only [Nat.cast_zero,neg_zero,zero_mul,Real.rpow_zero,one_mul] using
-        modified_pointwise_zero L E e f he a ha
-  | succ r ih =>
-      let x := e.trans (headTailEquiv (L+r)).toEmbedding
-      let y := f.trans (headTailEquiv (L+r)).toEmbedding
-      let W := PairRouting.outerWeight x y
-      let C := PairRouting.compatibleSet x y
-      let p : (ι → Bool) → ℝ := fun c =>
-        (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E false))
-          (fun i : {i // c i = false} => (x i.val).2)
-          (fun i : {i // c i = false} => (y i.val).2))^(1+a) *
-        (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E true))
-          (fun i : {i // c i = true} => (x i.val).2)
-          (fun i : {i // c i = true} => (y i.val).2))^(1+a)
-      let m : {c // c ∈ C} → ℝ := fun c =>
-        finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
-          if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c.val
-          then (2:ℝ)^((modifiedAssignmentCost L r E x c.val
-            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0)
-      have hp (c : {c // c ∈ C}) : W^a * (C.card:ℝ)^a * p c.val ≤
-          ((2:ℝ)^(-((r:ℝ)+1)*Fintype.card ι*a)*centralFactor L (r+1) E a) * m c := by
-        obtain ⟨hcx,hcy⟩ := (PairRouting.mem_compatibleSet x y c.val).mp c.property
-        have h₀ := ih (exceptionChild E false) (PairRouting.childEmbedding x c.val hcx false)
-          (PairRouting.childEmbedding y c.val hcy false)
-          (PairRouting.childEmbedding_surjective x ((headTailEquiv (L+r)).surjective.comp he) c.val hcx false)
-        have h₁ := ih (exceptionChild E true) (PairRouting.childEmbedding x c.val hcx true)
-          (PairRouting.childEmbedding y c.val hcy true)
-          (PairRouting.childEmbedding_surjective x ((headTailEquiv (L+r)).surjective.comp he) c.val hcx true)
-        have hh := modified_child_power_product_bound L r E x y c.val hcx hcy a h₀ h₁
-        have hh' := mul_le_mul_of_nonneg_left hh
-          (mul_nonneg (Real.rpow_nonneg (le_of_lt (PairRouting.outerWeight_pos x y)) a)
-            (Real.rpow_nonneg (Nat.cast_nonneg C.card) a))
-        apply hh'.trans
-        have hm := modified_assignment_moment_bound L r E x y c.val c.property a ha
-        convert mul_le_mul_of_nonneg_left hm (centralFactor_nonneg L (r+1) E a) using 1 <;> ring
-      have hnode := PairRouting.sandwich_power_bound (modifiedPerm L r (exceptionChild E false)) (modifiedPerm L r (exceptionChild E true)) x y a ha
-      rw [←modified_probability_step L r E e f] at hnode
-      change (PairRouting.tupleProbability (modifiedPerm L (r+1) E) e f)^(1+a) ≤
-        W^(1+a) * (C.card:ℝ)^a * ∑ c ∈ C, p c at hnode
-      have hsum : ∑ c ∈ C, p c = ∑ c : {c // c ∈ C}, p c.val := by
-        rw [Finset.sum_subtype C (fun _ => Iff.rfl)]
-      have hW : 0 < W := PairRouting.outerWeight_pos x y
-      calc
-        _ ≤ W^(1+a) * (C.card:ℝ)^a * ∑ c ∈ C, p c := hnode
-        _ = W * ∑ c : {c // c ∈ C}, W^a * (C.card:ℝ)^a * p c.val := by
-          rw [Real.rpow_add hW,Real.rpow_one,hsum,←Finset.mul_sum]
-          ring
-        _ ≤ W * ∑ c : {c // c ∈ C}, ((2:ℝ)^(-((r:ℝ)+1)*Fintype.card ι*a)*centralFactor L (r+1) E a) * m c :=
-          mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun c _ => hp c)) (le_of_lt hW)
-        _ = _ := by
-          rw [←Finset.mul_sum,modifiedRoutingMoment_step]
-          simp only [Nat.cast_add,Nat.cast_one]
-          change W * (((2:ℝ)^(-((r:ℝ)+1)*Fintype.card ι*a)*centralFactor L (r+1) E a) * ∑ c, m c) =
-            ((2:ℝ)^(-((r:ℝ)+1)*Fintype.card ι*a)*centralFactor L (r+1) E a) * (W * ∑ c, m c)
-          ring
-
-
-lemma modifiedRoutingMoment_sum (L r : ℕ) (E : Finset (Card r)) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (L+r)) (a : ℝ) :
-    (∑ f : ι ↪ Card (L+r), modifiedRoutingMoment L r E e f a) =
-      finiteMean (fun ω : ModifiedCoins L r => (2:ℝ)^((modifiedCost L r E e ω:ℝ)*a)) := by
-  classical
-  unfold modifiedRoutingMoment
-  rw [←finiteMean_sum]
-  apply finiteMean_congr
-  intro ω
-  let f₀ : ι ↪ Card (L+r) := e.trans (modifiedPerm L r E ω).toEmbedding
-  rw [Finset.sum_eq_single f₀]
-  · have hm : ∀ i, modifiedPerm L r E ω (e i) = f₀ i := fun _ => rfl
-    simp only [hm,implies_true,↓reduceIte]
-  · intro f _ hf
-    have hm : ¬ ∀ i, modifiedPerm L r E ω (e i) = f i := by
-      intro hh
-      apply hf
-      ext i : 1
-      exact (hh i).symm
-    exact ite_eq_right hm
-  · simp
-
-noncomputable def modifiedRowMoment (L r : ℕ) (E : Finset (Card r)) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (L+r)) (a : ℝ) : ℝ :=
-  finiteMean (fun f : ι ↪ Card (L+r) =>
-    ((Fintype.card (ι ↪ Card (L+r)):ℝ)*PairRouting.tupleProbability (modifiedPerm L r E) e f)^(1+a))
-
-lemma full_tuple_card (d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d)
-    (he : Function.Surjective e) : Fintype.card ι = 2^d := by
-  rw [←card_positions d]
-  exact Fintype.card_congr (Equiv.ofBijective e ⟨e.injective,he⟩)
-
-lemma exception_card_le (r : ℕ) (E : Finset (Card r)) : E.card ≤ 2^r := by
-  simpa only [card_positions] using E.card_le_univ
-
-noncomputable def modifiedNormalization (L r : ℕ) (E : Finset (Card r)) : ℝ :=
-  (Nat.factorial (2^(L+r)):ℝ) /
-    (((2:ℝ)^r)^(2^(L+r)) * (Nat.factorial (2^L):ℝ)^(2^r-E.card))
-
-lemma modifiedNormalization_pos (L r : ℕ) (E : Finset (Card r)) :
-    0 < modifiedNormalization L r E := by unfold modifiedNormalization; positivity
-
-lemma modified_normalization_identity (L r : ℕ) (E : Finset (Card r)) (a : ℝ) :
-    (Nat.factorial (2^(L+r)):ℝ)^a *
-      ((2:ℝ)^(-(r:ℝ)*(2^(L+r):ℕ)*a)*centralFactor L r E a) =
-    modifiedNormalization L r E ^ a := by
-  rw [←mul_assoc,density_normalization,centralFactor,modifiedNormalization,
-    Real.div_rpow (by positivity) (by positivity),Real.div_rpow (by positivity) (by positivity),
-    Real.mul_rpow (by positivity) (by positivity)]
-  have hc : ((2^r-E.card:ℕ):ℝ) = (2:ℝ)^r-E.card := by
-    rw [Nat.cast_sub (exception_card_le r E),Nat.cast_pow,Nat.cast_ofNat]
-  rw [←Real.rpow_natCast (Nat.factorial (2^L):ℝ),←Real.rpow_mul (by positivity),hc,
-    show -((2:ℝ)^r-E.card)*a = -(((2:ℝ)^r-E.card)*a) by ring,
-    Real.rpow_neg (by positivity)]
-  ring
-
-theorem modified_row_density_moment (L r : ℕ) (E : Finset (Card r)) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (L+r)) (he : Function.Surjective e) (a : ℝ) (ha : 0 ≤ a) :
-    modifiedRowMoment L r E e a ≤ modifiedNormalization L r E ^ a *
-      finiteMean (fun ω : ModifiedCoins L r => (2:ℝ)^((modifiedCost L r E e ω:ℝ)*a)) := by
-  classical
-  have : Nonempty (ι ↪ Card (L+r)) := ⟨e⟩
-  rw [modifiedRowMoment,finite_row_power _ (fun f => PairRouting.tupleProbability_nonneg _ _ _)]
-  calc
-    _ ≤ (Fintype.card (ι ↪ Card (L+r)):ℝ)^a *
-      ∑ f : ι ↪ Card (L+r), ((2:ℝ)^(-(r:ℝ)*Fintype.card ι*a)*centralFactor L r E a) *
-        modifiedRoutingMoment L r E e f a :=
-      mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun f _ => modified_pointwise_moment L r E e f he a ha))
-        (Real.rpow_nonneg (Nat.cast_nonneg _) _)
-    _ = _ := by
-      rw [←Finset.mul_sum,modifiedRoutingMoment_sum,←mul_assoc,Fintype.card_embedding_eq,
-        card_positions,full_tuple_card (L+r) e he,Nat.descFactorial_self,modified_normalization_identity]
-
-lemma modified_total_cost_mgf (L r : ℕ) (hL : 0 < L) (E : Finset (Card r))
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) :
-    finiteMean (fun ω : ModifiedCoins L r => momentBase^(modifiedCost L r E e ω)) ≤
-      Real.exp (Fintype.card ι*heightDecay L +
-        3*((L*E.card*2^L:ℕ):ℝ)*Real.log momentBase) := by
-  let B := Real.exp (Fintype.card ι*heightDecay L +
-    3*((L*E.card*2^L:ℕ):ℝ)*Real.log momentBase)
-  rw [finiteMean_prod]
-  apply le_trans (finiteMean_mono (fun U => ?_)) (finiteMean_const B).le
-  rw [finiteMean_prod,finiteMean_comm]
-  apply le_trans (finiteMean_mono (fun X => ?_)) (finiteMean_const B).le
-  exact modified_cost_mgf L r hL E U X e momentBase momentBase_one_le (momentBase_log.trans (by norm_num))
-
-lemma modifiedRowMoment_bound (L r : ℕ) (hL : 0 < L) (E : Finset (Card r))
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (he : Function.Surjective e) :
-    modifiedRowMoment L r E e (1/1000) ≤
-      modifiedNormalization L r E ^ (1/1000:ℝ) *
-        Real.exp ((2^(L+r):ℕ)*heightDecay L +
-          3*((L*E.card*2^L:ℕ):ℝ)*Real.log momentBase) := by
-  have hm := modified_row_density_moment L r E e he (1/1000) (by norm_num)
-  simp_rw [←momentBase_pow] at hm
-  apply hm.trans
-  apply mul_le_mul_of_nonneg_left _ (Real.rpow_nonneg (modifiedNormalization_pos L r E).le _)
-  simpa only [full_tuple_card (L+r) e he] using modified_total_cost_mgf L r hL E e
-
-
-lemma log_factorial_upper {n : ℕ} (hn : 1 ≤ n) :
-    Real.log (Nat.factorial n) ≤ (n:ℝ)*Real.log n-n+Real.log n+1 := by
-  induction n with
-  | zero => omega
-  | succ n ih =>
-      by_cases hz : n = 0
-      · subst n; norm_num
-      have hn0 : (0:ℝ) < n := Nat.cast_pos.mpr (Nat.pos_of_ne_zero hz)
-      have hn1 : (0:ℝ) < n+1 := by positivity
-      have h := Real.one_sub_inv_le_log_of_pos (div_pos hn1 hn0)
-      rw [Real.log_div (ne_of_gt hn1) (ne_of_gt hn0),inv_div] at h
-      have hh := mul_le_mul_of_nonneg_left h hn1.le
-      have hcancel : ((n:ℝ)+1)*(1-(n:ℝ)/(n+1)) = 1 := by field_simp; ring
-      rw [hcancel] at hh
-      have hi := ih (by omega)
-      rw [Nat.factorial_succ,Nat.cast_mul,Real.log_mul (by positivity) (by positivity),Nat.cast_add,Nat.cast_one]
-      nlinarith
-
-lemma log_modifiedNormalization (L r : ℕ) (E : Finset (Card r)) :
-    Real.log (modifiedNormalization L r E) =
-      Real.log (Nat.factorial (2^(L+r))) -
-        ((2^(L+r):ℕ)*((r:ℝ)*Real.log 2)+((2:ℝ)^r-E.card)*Real.log (Nat.factorial (2^L))) := by
-  unfold modifiedNormalization
-  rw [Real.log_div (by positivity) (by positivity),Real.log_mul (by positivity) (by positivity),
-    Real.log_pow,Real.log_pow,Real.log_pow,
-    Nat.cast_sub (exception_card_le r E),Nat.cast_pow,Nat.cast_ofNat]
-  simp only [Nat.cast_pow,Nat.cast_ofNat]
-
-lemma log_modifiedNormalization_le (L r : ℕ) (E : Finset (Card r)) :
-    Real.log (modifiedNormalization L r E) ≤
-      ((L+r:ℕ):ℝ)*Real.log 2+1+E.card*Real.log (Nat.factorial (2^L)) := by
-  have hN : 1 ≤ 2^(L+r) := one_le_pow₀ (by norm_num)
-  have hS : 1 ≤ 2^L := one_le_pow₀ (by norm_num)
-  have hu := log_factorial_upper hN
-  have hl := mul_le_mul_of_nonneg_left (log_factorial_lower hS) (by positivity : (0:ℝ) ≤ (2:ℝ)^r)
-  simp only [Nat.cast_pow,Nat.cast_ofNat,Real.log_pow,Nat.cast_add] at hu hl ⊢
-  rw [log_modifiedNormalization]
-  simp only [pow_add,Nat.cast_mul,Nat.cast_pow,Nat.cast_ofNat] at hu hl ⊢
-  nlinarith
-
-lemma modified_moment_exponential (L r : ℕ) (hL : 0 < L) (E : Finset (Card r))
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (he : Function.Surjective e) :
-    modifiedRowMoment L r E e (1/1000) ≤
-      Real.exp (((L+r:ℕ)*Real.log 2+1+E.card*Real.log (Nat.factorial (2^L)))/1000 +
-        (2^(L+r):ℕ)*heightDecay L+3*((L*E.card*2^L:ℕ):ℝ)*Real.log momentBase) := by
-  apply (modifiedRowMoment_bound L r hL E e he).trans
-  rw [Real.rpow_def_of_pos (modifiedNormalization_pos L r E),←Real.exp_add]
-  apply Real.exp_le_exp.mpr
-  have h := log_modifiedNormalization_le L r E
-  linarith
-
-lemma log_power_factorial_le (L : ℕ) :
-    Real.log (Nat.factorial (2^L)) ≤ (2:ℝ)^L*L := by
-  have hf : (Nat.factorial (2^L):ℝ) ≤ ((2^L:ℕ):ℝ)^(2^L) := by
-    exact_mod_cast Nat.factorial_le_pow (2^L)
-  have hh := Real.log_le_log (by positivity) hf
-  rw [Real.log_pow,Nat.cast_pow,Nat.cast_ofNat,Real.log_pow] at hh
-  apply hh.trans
-  have hl := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
-  have hn : 0 ≤ (2:ℝ)^L*(L:ℝ) := by positivity
-  nlinarith
-
-lemma modified_moment_exception_bound (L r : ℕ) (hL : 0 < L) (E : Finset (Card r))
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (he : Function.Surjective e)
-    (ε : ℝ) (hE : (E.card:ℝ) ≤ ε*(2:ℝ)^r) :
-    modifiedRowMoment L r E e (1/1000) ≤
-      Real.exp (((L+r:ℕ)*Real.log 2+1)/1000 +
-        (2:ℝ)^(L+r)*(heightDecay L+ε*L/250)) := by
-  apply (modified_moment_exponential L r hL E e he).trans
-  apply Real.exp_le_exp.mpr
-  have hs := mul_le_mul_of_nonneg_left (log_power_factorial_le L) (Nat.cast_nonneg E.card)
-  have hb := mul_le_mul_of_nonneg_left momentBase_log
-    (by positivity : (0:ℝ) ≤ 3*((L*E.card*2^L:ℕ):ℝ))
-  have hexc := mul_le_mul_of_nonneg_left hE (by positivity : (0:ℝ) ≤ (2:ℝ)^L*L/250)
-  simp only [Nat.cast_mul,Nat.cast_pow,Nat.cast_ofNat,pow_add] at hs hb hexc ⊢
-  nlinarith
-
-
-lemma heightDecay_tendsto : Filter.Tendsto heightDecay Filter.atTop (nhds 0) := by
-  have h : Filter.Tendsto (fun L : ℕ => (-(1:ℝ)/64)*(L:ℝ)) Filter.atTop Filter.atBot :=
-    (Filter.tendsto_const_mul_atBot_of_neg (by norm_num : -(1:ℝ)/64 < 0)).mpr
-      tendsto_natCast_atTop_atTop
-  have ht := ((Real.tendsto_exp_atBot.comp h).const_mul 68).div_const (1-Real.exp (-(1:ℝ)/64))
-  convert ht using 1
-  · funext L
-    unfold heightDecay
-    dsimp only [Function.comp_apply]
-    rw [show -(L:ℝ)/64 = (-(1:ℝ)/64)*(L:ℝ) by ring]
-  · simp
-
-noncomputable def modifiedOverhead (L r : ℕ) : ℝ :=
-  (((L+r:ℕ):ℝ)*Real.log 2+1)/((2:ℝ)^(L+r)*1000)
-
-lemma modifiedOverhead_tendsto (L : ℕ) :
-    Filter.Tendsto (modifiedOverhead L) Filter.atTop (nhds 0) := by
-  have h₁ := tendsto_pow_const_div_const_pow_of_one_lt 1 (by norm_num : (1:ℝ)<2)
-  have h₀ := tendsto_pow_const_div_const_pow_of_one_lt 0 (by norm_num : (1:ℝ)<2)
-  simp only [pow_one] at h₁
-  simp only [pow_zero] at h₀
-  have ht := ((h₁.const_mul (Real.log 2)).add (h₀.const_mul ((L:ℝ)*Real.log 2+1))).div_const
-    ((2:ℝ)^L*1000)
-  convert ht using 1
-  · funext r
-    simp only [modifiedOverhead,Nat.cast_add,pow_add]
-    field_simp
-    ring
-  · simp
-
-theorem modified_arbitrarily_small_rate (η : ℝ) (hη : 0 < η) :
-    ∃ L : ℕ, 0 < L ∧ ∃ ε : ℝ, 0 < ε ∧ ε < 1/2 ∧
-      ∀ᶠ r : ℕ in Filter.atTop, ∀ (E : Finset (Card r)), (E.card:ℝ) ≤ ε*(2:ℝ)^r →
-        ∀ (e : Card (L+r) ↪ Card (L+r)),
-          modifiedRowMoment L r E e (1/1000) ≤ Real.exp (η*(2:ℝ)^(L+r)) := by
-  have hsmall : ∀ᶠ L : ℕ in Filter.atTop, heightDecay L < η/3 :=
-    heightDecay_tendsto.eventually (gt_mem_nhds (by linarith : (0:ℝ)<η/3))
-  obtain ⟨L,hdec,hL⟩ := (hsmall.and (Filter.eventually_ge_atTop 1)).exists
-  have hL0 : (0:ℝ) < L := Nat.cast_pos.mpr hL
-  let ε : ℝ := min (1/4) (η*250/(3*L))
-  have hε : 0 < ε := lt_min (by norm_num) (by positivity)
-  have hε1 : ε < 1/2 := (min_le_left _ _).trans_lt (by norm_num)
-  have hεL : ε*L/250 ≤ η/3 := by
-    have h := min_le_right (1/4:ℝ) (η*250/(3*L))
-    change ε ≤ η*250/(3*L) at h
-    have hh := mul_le_mul_of_nonneg_right h hL0.le
-    have he : (η*250/(3*(L:ℝ)))*(L:ℝ)/250 = η/3 := by field_simp
-    exact (div_le_div_of_nonneg_right hh (by norm_num)).trans he.le
-  refine ⟨L,hL,ε,hε,hε1,?_⟩
-  have hr : ∀ᶠ r : ℕ in Filter.atTop, modifiedOverhead L r < η/3 :=
-    (modifiedOverhead_tendsto L).eventually (gt_mem_nhds (by linarith : (0:ℝ)<η/3))
-  filter_upwards [hr] with r hr
-  intro E hE e
-  have he := Finite.surjective_of_injective e.injective
-  apply (modified_moment_exception_bound L r hL E e he ε hE).trans
-  apply Real.exp_le_exp.mpr
-  have hn : (0:ℝ) < (2:ℝ)^(L+r) := by positivity
-  have hd : modifiedOverhead L r * (2:ℝ)^(L+r) =
-      (((L+r:ℕ):ℝ)*Real.log 2+1)/1000 := by
-    unfold modifiedOverhead
-    field_simp
-  have hh := mul_le_mul_of_nonneg_right hr.le hn.le
-  rw [hd] at hh
-  have ht := mul_le_mul_of_nonneg_left (show heightDecay L+ε*L/250 ≤ 2*η/3 by linarith) hn.le
-  linarith
-
-
-namespace DensityTransfer
-open scoped BigOperators
-
-noncomputable def applyKernel {X : Type*} [Fintype X] (P : X → X → ℝ)
-    (v : EuclideanSpace ℂ X) : EuclideanSpace ℂ X :=
-  WithLp.toLp 2 (fun x => ∑ y, (P x y : ℂ)*v y)
-
-lemma applyKernel_apply {X : Type*} [Fintype X] (P : X → X → ℝ)
-    (v : EuclideanSpace ℂ X) (x : X) :
-    applyKernel P v x = ∑ y, (P x y : ℂ)*v y := rfl
-
-lemma applyKernel_add {X : Type*} [Fintype X] (P Q : X → X → ℝ)
-    (v : EuclideanSpace ℂ X) :
-    applyKernel (fun x y => P x y+Q x y) v = applyKernel P v+applyKernel Q v := by
-  ext x
-  simp only [applyKernel_apply,Complex.ofReal_add,add_mul,Finset.sum_add_distrib]
-  rfl
-
-lemma kernel_component_sq {X : Type*} [Fintype X] (P : X → X → ℝ)
-    (hP : ∀ x y, 0 ≤ P x y) (v : EuclideanSpace ℂ X) (x : X) :
-    ‖applyKernel P v x‖^2 ≤ (∑ y, P x y) * ∑ y, P x y*‖v y‖^2 := by
-  have hs : ‖applyKernel P v x‖ ≤ ∑ y, P x y*‖v y‖ := by
-    rw [applyKernel_apply]
-    apply (norm_sum_le _ _).trans
-    apply Finset.sum_le_sum
-    intro y _
-    rw [norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (hP x y)]
-  apply (pow_le_pow_left₀ (norm_nonneg _) hs 2).trans
-  exact Finset.sum_sq_le_sum_mul_sum_of_sq_le_mul Finset.univ
-    (fun y _ => hP x y) (fun y _ => mul_nonneg (hP x y) (sq_nonneg _))
-    (fun y _ => by ring_nf; exact le_rfl)
-
-lemma schur_bound {X : Type*} [Fintype X] (P : X → X → ℝ)
-    (hP : ∀ x y, 0 ≤ P x y) (δ : ℝ) (hδ : 0 ≤ δ)
-    (hrow : ∀ x, ∑ y, P x y ≤ δ) (hcol : ∀ y, ∑ x, P x y ≤ δ)
-    (v : EuclideanSpace ℂ X) : ‖applyKernel P v‖ ≤ δ*‖v‖ := by
-  have hsq : ‖applyKernel P v‖^2 ≤ δ^2*‖v‖^2 := by
-    rw [EuclideanSpace.norm_sq_eq,EuclideanSpace.norm_sq_eq]
-    calc
-      _ ≤ ∑ x, δ * ∑ y, P x y*‖v y‖^2 := by
-        apply Finset.sum_le_sum
-        intro x _
-        apply (kernel_component_sq P hP v x).trans
-        exact mul_le_mul_of_nonneg_right (hrow x)
-          (Finset.sum_nonneg (fun y _ => mul_nonneg (hP x y) (sq_nonneg _)))
-      _ = δ * ∑ y, (∑ x, P x y)*‖v y‖^2 := by
-        rw [←Finset.mul_sum,Finset.sum_comm]
-        congr 1
-        apply Finset.sum_congr rfl
-        intro y _
-        exact (Finset.sum_mul Finset.univ _ _).symm
-      _ ≤ δ * ∑ y, δ*‖v y‖^2 := by
-        apply mul_le_mul_of_nonneg_left _ hδ
-        exact Finset.sum_le_sum (fun y _ => mul_le_mul_of_nonneg_right (hcol y) (sq_nonneg _))
-      _ = _ := by rw [←Finset.mul_sum]; ring
-  have hn := norm_nonneg (applyKernel P v)
-  have hr := mul_nonneg hδ (norm_nonneg v)
-  nlinarith
-
-lemma hilbert_schmidt_family {X I : Type*} [Fintype X] [Fintype I]
-    (P : X → X → ℝ) (v : I → EuclideanSpace ℂ X) (hv : Orthonormal ℂ v) :
-    (∑ i, ‖applyKernel P (v i)‖^2) ≤ ∑ x, ∑ y, (P x y)^2 := by
-  simp_rw [EuclideanSpace.norm_sq_eq]
-  rw [Finset.sum_comm]
-  apply Finset.sum_le_sum
-  intro x _
-  let w : EuclideanSpace ℂ X := WithLp.toLp 2 (fun y => (P x y : ℂ))
-  have he (i : I) : ‖applyKernel P (v i) x‖ = ‖inner ℂ (v i) w‖ := by
-    rw [norm_inner_symm]
-    congr 1
-    simp only [applyKernel_apply,PiLp.inner_apply]
-    apply Finset.sum_congr rfl
-    intro y _
-    simp [w,RCLike.inner_apply,mul_comm]
-  simp_rw [he]
-  have h := hv.sum_inner_products_le (s := Finset.univ) w
-  apply h.trans_eq
-  rw [EuclideanSpace.norm_sq_eq]
-  apply Finset.sum_congr rfl
-  intro y _
-  simp [w,Complex.norm_real,Real.norm_eq_abs,sq_abs]
-
-
-lemma tail_density_pointwise (t a z : ℝ) (ht : 0 ≤ t) (ha : 0 ≤ a) (hz : 0 < z) :
-    (if z < t then t else 0) ≤ t^(1+a)*z^(-a) := by
-  split_ifs with h
-  · rw [Real.rpow_neg hz.le,←div_eq_mul_inv]
-    apply (le_div_iff₀ (Real.rpow_pos_of_pos hz a)).mpr
-    rw [Real.rpow_add (lt_trans hz h),Real.rpow_one]
-    exact mul_le_mul_of_nonneg_left (Real.rpow_le_rpow hz.le h.le ha) ht
-  · positivity
-
-lemma tail_density_mean {X : Type*} [Fintype X] (f : X → ℝ) (hf : ∀ x, 0 ≤ f x)
-    (a z : ℝ) (ha : 0 ≤ a) (hz : 0 < z) :
-    finiteMean (fun x => if z < f x then f x else 0) ≤ finiteMean (fun x => f x^(1+a))*z^(-a) := by
-  rw [←finiteMean_mul_const]
-  exact finiteMean_mono (fun x => tail_density_pointwise (f x) a z (hf x) ha hz)
-
-noncomputable def tailKernel {X : Type*} [Fintype X] (P : X → X → ℝ) (z : ℝ) : X → X → ℝ :=
-  fun x y => if z < (Fintype.card X:ℝ)*P x y then P x y else 0
-
-noncomputable def headKernel {X : Type*} [Fintype X] (P : X → X → ℝ) (z : ℝ) : X → X → ℝ :=
-  fun x y => if z < (Fintype.card X:ℝ)*P x y then 0 else P x y
-
-lemma tail_add_head {X : Type*} [Fintype X] (P : X → X → ℝ) (z : ℝ) :
-    (fun x y => tailKernel P z x y+headKernel P z x y) = P := by
-  funext x y
-  unfold tailKernel headKernel
-  split_ifs <;> simp
-
-lemma tailKernel_nonneg {X : Type*} [Fintype X] (P : X → X → ℝ) (z : ℝ)
-    (hP : ∀ x y, 0 ≤ P x y) (x y : X) : 0 ≤ tailKernel P z x y := by
-  unfold tailKernel
-  split_ifs <;> simp only [hP,le_refl]
-
-lemma tailKernel_row {X : Type*} [Fintype X] [Nonempty X] (P : X → X → ℝ)
-    (hP : ∀ x y, 0 ≤ P x y) (a z M : ℝ) (ha : 0 ≤ a) (hz : 0 < z)
-    (hM : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*P x y)^(1+a)) ≤ M) (x : X) :
-    ∑ y, tailKernel P z x y ≤ M*z^(-a) := by
-  have hn : (Fintype.card X:ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
-  have he : finiteMean (fun y => if z < (Fintype.card X:ℝ)*P x y then
-      (Fintype.card X:ℝ)*P x y else 0) = ∑ y, tailKernel P z x y := by
-    unfold finiteMean
-    calc
-      _ = (∑ y, (Fintype.card X:ℝ)*tailKernel P z x y)/(Fintype.card X:ℝ) := by
-        congr 1
-        apply Finset.sum_congr rfl
-        intro y _
-        by_cases hy : z < (Fintype.card X:ℝ)*P x y
-        · simp only [tailKernel,hy,↓reduceIte]
-        · simp only [tailKernel,hy,↓reduceIte,mul_zero]
-      _ = _ := by rw [←Finset.mul_sum]; field_simp
-  rw [←he]
-  exact (tail_density_mean _ (fun y => mul_nonneg (Nat.cast_nonneg _) (hP x y)) a z ha hz).trans
-    (mul_le_mul_of_nonneg_right (hM x) (Real.rpow_nonneg hz.le _))
-
-lemma tailKernel_schur {X : Type*} [Fintype X] [Nonempty X] (P : X → X → ℝ)
-    (hP : ∀ x y, 0 ≤ P x y) (hsym : ∀ x y, P x y = P y x)
-    (a z M : ℝ) (ha : 0 ≤ a) (hz : 0 < z) (hM0 : 0 ≤ M)
-    (hM : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*P x y)^(1+a)) ≤ M)
-    (v : EuclideanSpace ℂ X) :
-    ‖applyKernel (tailKernel P z) v‖ ≤ (M*z^(-a))*‖v‖ := by
-  apply schur_bound _ (tailKernel_nonneg P z hP) _ (by positivity)
-    (tailKernel_row P hP a z M ha hz hM)
-  intro y
-  have hs : (fun x => tailKernel P z x y) = (fun x => tailKernel P z y x) := by
-    funext x
-    simp only [tailKernel,hsym x y]
-  rw [hs]
-  exact tailKernel_row P hP a z M ha hz hM y
-
-lemma headKernel_hs {X : Type*} [Fintype X] [Nonempty X] (P : X → X → ℝ)
-    (hP : ∀ x y, 0 ≤ P x y) (hrow : ∀ x, ∑ y, P x y = 1) (z : ℝ) (hz : 0 ≤ z) :
-    (∑ x, ∑ y, (headKernel P z x y)^2) ≤ z := by
-  have hn : (0:ℝ) < Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
-  have he (x y : X) : (headKernel P z x y)^2 ≤ (z/(Fintype.card X:ℝ))*P x y := by
-    unfold headKernel
-    split_ifs with h
-    · simp only [zero_pow (by omega : (2:ℕ) ≠ 0)]
-      exact mul_nonneg (div_nonneg hz hn.le) (hP x y)
-    · have h' : (Fintype.card X:ℝ)*P x y ≤ z := le_of_not_gt h
-      have hh := mul_le_mul_of_nonneg_right h' (hP x y)
-      rw [show (z/(Fintype.card X:ℝ))*P x y = (z*P x y)/(Fintype.card X:ℝ) by ring]
-      apply (le_div_iff₀ hn).mpr
-      nlinarith
-  calc
-    _ ≤ ∑ x, ∑ y, (z/(Fintype.card X:ℝ))*P x y :=
-      Finset.sum_le_sum (fun x _ => Finset.sum_le_sum (fun y _ => he x y))
-    _ = z := by
-      simp only [←Finset.mul_sum,hrow,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
-      field_simp
-
-
-theorem multiplicity_amplification {X I : Type*} [Fintype X] [Nonempty X]
-    [Fintype I] [Nonempty I] (P : X → X → ℝ)
-    (hP : ∀ x y, 0 ≤ P x y) (hsym : ∀ x y, P x y = P y x)
-    (hrow : ∀ x, ∑ y, P x y = 1) (a z M : ℝ) (ha : 0 ≤ a) (hz : 0 < z) (hM0 : 0 ≤ M)
-    (hM : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*P x y)^(1+a)) ≤ M)
-    (v : I → EuclideanSpace ℂ X) (hv : Orthonormal ℂ v)
-    (μ : ℝ) (hvμ : ∀ i, μ ≤ ‖applyKernel P (v i)‖) :
-    μ ≤ M*z^(-a)+Real.sqrt (z/(Fintype.card I:ℝ)) := by
-  let δ : ℝ := M*z^(-a)
-  have hδ : 0 ≤ δ := by dsimp [δ]; positivity
-  have hm : (0:ℝ) < Fintype.card I := Nat.cast_pos.mpr Fintype.card_pos
-  change μ ≤ δ+_
-  by_cases hμ : μ ≤ δ
-  · exact hμ.trans (le_add_of_nonneg_right (Real.sqrt_nonneg _))
-  have hmd : 0 ≤ μ-δ := by linarith
-  have hsmall (i : I) : μ-δ ≤ ‖applyKernel (headKernel P z) (v i)‖ := by
-    have hr := tailKernel_schur P hP hsym a z M ha hz hM0 hM (v i)
-    rw [hv.norm_eq_one i,mul_one] at hr
-    have he := applyKernel_add (tailKernel P z) (headKernel P z) (v i)
-    rw [tail_add_head] at he
-    have hn := norm_add_le (applyKernel (tailKernel P z) (v i)) (applyKernel (headKernel P z) (v i))
-    rw [←he] at hn
-    have hh := hvμ i
-    change ‖applyKernel (tailKernel P z) (v i)‖ ≤ δ at hr
-    linarith
-  have hsum : (Fintype.card I:ℝ)*(μ-δ)^2 ≤ z := by
-    calc
-      _ = ∑ i : I, (μ-δ)^2 := by simp
-      _ ≤ ∑ i : I, ‖applyKernel (headKernel P z) (v i)‖^2 :=
-        Finset.sum_le_sum (fun i _ => pow_le_pow_left₀ hmd (hsmall i) 2)
-      _ ≤ ∑ x, ∑ y, (headKernel P z x y)^2 := hilbert_schmidt_family _ v hv
-      _ ≤ z := headKernel_hs P hP hrow z hz.le
-  have hsq : (μ-δ)^2 ≤ z/(Fintype.card I:ℝ) := (le_div_iff₀ hm).mpr (by nlinarith)
-  have hs := (Real.le_sqrt hmd (div_nonneg hz.le hm.le)).mpr hsq
-  linarith
-
-end DensityTransfer
-namespace PairRouting
-open scoped BigOperators
-
-lemma tupleProbability_row_sum {ι β Ω : Type*} [Fintype ι] [Fintype β]
-    [DecidableEq β] [Fintype Ω] [Nonempty Ω]
-    (P : Ω → Equiv.Perm β) (e : ι ↪ β) :
-    (∑ f : ι ↪ β, tupleProbability P e f) = 1 := by
-  classical
-  simp only [tupleProbability]
-  rw [←finiteMean_sum]
-  have h (ω : Ω) (f : ι ↪ β) : (∀ i, P ω (e i) = f i) ↔ e.trans (P ω).toEmbedding = f := by
-    constructor
-    · intro h; ext i; exact h i
-    · intro h i; exact congrArg (fun g : ι ↪ β => g i) h
-  simp_rw [h]
-  simp only [Finset.sum_ite_eq,Finset.mem_univ,↓reduceIte,finiteMean_const]
-
-lemma tupleProbability_inverse {ι β Ω : Type*} [Fintype ι] [DecidableEq β]
-    [Fintype Ω] (P : Ω → Equiv.Perm β) (e f : ι → β) :
-    tupleProbability (fun ω => (P ω)⁻¹) e f = tupleProbability P f e := by
-  classical
-  apply finiteMean_congr
-  intro ω
-  congr 1
-  apply propext
-  constructor
-  · intro h i
-    have hh := congrArg (P ω) (h i)
-    simpa only [Equiv.Perm.inv_def,Equiv.apply_symm_apply] using hh.symm
-  · intro h i
-    have hh := congrArg (fun x => (P ω).symm x) (h i)
-    simpa only [Equiv.Perm.inv_def,Equiv.symm_apply_apply] using hh.symm
-
-lemma tupleProbability_equiv {ι β Ω Ω' : Type*} [Fintype ι] [DecidableEq β]
-    [Fintype Ω] [Fintype Ω'] (P : Ω → Equiv.Perm β) (τ : Ω ≃ Ω') (e f : ι → β) :
-    tupleProbability P e f = tupleProbability (fun ω => P (τ.symm ω)) e f :=
-  finiteMean_equiv τ _
-
-lemma tupleProbability_symmetric {ι β Ω : Type*} [Fintype ι] [DecidableEq β]
-    [Fintype Ω] (P : Ω → Equiv.Perm β) (τ : Equiv.Perm Ω)
-    (hτ : ∀ ω, P (τ ω) = (P ω)⁻¹) (e f : ι → β) :
-    tupleProbability P e f = tupleProbability P f e := by
-  rw [tupleProbability_equiv P τ.symm]
-  simp only [Equiv.symm_symm,hτ]
-  exact tupleProbability_inverse P e f
-
-end PairRouting
-
-lemma palindromePerm_reverse (d : ℕ) (ω : BenesCoins d) :
-    palindromePerm d (ω.2,ω.1) = (palindromePerm d ω)⁻¹ := by
-  simp only [palindromePerm,mul_inv_rev]
-  rfl
-
-lemma palindrome_kernel_symmetric (d : ℕ) {ι : Type*} [Fintype ι]
-    (e f : ι ↪ Card d) :
-    PairRouting.tupleProbability (palindromePerm d) e f =
-      PairRouting.tupleProbability (palindromePerm d) f e :=
-  PairRouting.tupleProbability_symmetric _ (Equiv.prodComm _ _) (palindromePerm_reverse d) e f
-
-theorem palindrome_density_transfer (d : ℕ) {ι I : Type*} [Fintype ι] [Fintype I]
-    [Nonempty I] (e₀ : ι ↪ Card d) (hk : 0 < Fintype.card ι)
-    (v : I → EuclideanSpace ℂ (ι ↪ Card d)) (hv : Orthonormal ℂ v)
-    (μ : ℝ) (hvμ : ∀ i, μ ≤ ‖DensityTransfer.applyKernel
-      (fun e f : ι ↪ Card d => PairRouting.tupleProbability (palindromePerm d) e f) (v i)‖)
-    (z : ℝ) (hz : 0 < z) :
-    μ ≤ Real.exp (densityConstant*Fintype.card ι*
-      ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ))*z^(-(1/1000:ℝ))+
-      Real.sqrt (z/(Fintype.card I:ℝ)) := by
-  let : Nonempty (ι ↪ Card d) := ⟨e₀⟩
-  apply DensityTransfer.multiplicity_amplification
-    (fun e f : ι ↪ Card d => PairRouting.tupleProbability (palindromePerm d) e f)
-    (fun e f => PairRouting.tupleProbability_nonneg _ e f)
-    (palindrome_kernel_symmetric d) (PairRouting.tupleProbability_row_sum _)
-    (1/1000) z _ (by norm_num) hz (Real.exp_nonneg _) _ v hv μ hvμ
-  intro e
-  exact palindrome_row_moment_bound d e hk
-
-end Thorp
-
-end ThorpNine.Harmonic
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Harmonic.InputTree
+
+namespace OAI
+
+namespace ThorpNine.Harmonic
+
+namespace Thorp
+
+lemma modified_probability_zero_le (L : ℕ) (E : Finset (Card 0))
+    {ι : Type*} [Fintype ι] (e f : ι ↪ Card L) (he : Function.Surjective e) :
+    PairRouting.tupleProbability (modifiedPerm L 0 E) e f ≤
+      if (fun j => Fin.elim0 j : Card 0) ∈ E then 1 else 1/(Nat.factorial (2^L):ℝ) := by
+  classical
+  rw [PairRouting.tupleProbability,finiteMean_prod,finiteMean_comm]
+  let B : ℝ := if (fun j => Fin.elim0 j : Card 0) ∈ E then 1 else 1/(Nat.factorial (2^L):ℝ)
+  change _ ≤ B
+  calc
+    _ ≤ finiteMean (fun _ω : BenesCoins L => B) := by
+      apply finiteMean_mono
+      intro ω
+      simp only [modifiedPerm_zero]
+      by_cases h : (fun j => Fin.elim0 j : Card 0) ∈ E
+      · simp only [h,↓reduceIte,Equiv.Perm.one_apply,finiteMean_const,B]
+        split_ifs <;> norm_num
+      · simp only [h,↓reduceIte,B]
+        rw [finiteMean_pi_eval (A := fun _ : Card 0 => Equiv.Perm (Card L)) (fun j => Fin.elim0 j) (fun u : Equiv.Perm (Card L) =>
+          if ∀ i, (butterflyPerm L (decodeButterfly L ω.1)*u⁻¹) (e i) = f i then (1:ℝ) else 0)]
+        convert PairRouting.uniformPerm_tuple_upper e f he (butterflyPerm L (decodeButterfly L ω.1)) using 1
+        rw [Fintype.card_perm,card_positions]
+    _ = _ := finiteMean_const B
+
+noncomputable def modifiedRoutingMoment (L r : ℕ) (E : Finset (Card r))
+    {ι : Type*} [Fintype ι] (e f : ι ↪ Card (L+r)) (a : ℝ) : ℝ := by
+  classical
+  exact finiteMean (fun ω : ModifiedCoins L r =>
+    if ∀ i, modifiedPerm L r E ω (e i) = f i then (2:ℝ)^((modifiedCost L r E e ω:ℝ)*a) else 0)
+
+lemma modifiedRoutingMoment_nonneg (L r : ℕ) (E : Finset (Card r))
+    {ι : Type*} [Fintype ι] (e f : ι ↪ Card (L+r)) (a : ℝ) :
+    0 ≤ modifiedRoutingMoment L r E e f a := by
+  classical
+  exact finiteMean_nonneg (fun _ => by split_ifs <;> positivity)
+
+lemma modifiedRoutingMoment_zero (L : ℕ) (E : Finset (Card 0))
+    {ι : Type*} [Fintype ι] (e f : ι ↪ Card L) (a : ℝ) :
+    modifiedRoutingMoment L 0 E e f a = PairRouting.tupleProbability (modifiedPerm L 0 E) e f := by
+  simp only [modifiedRoutingMoment,modifiedCost_zero,Nat.cast_zero,zero_mul,Real.rpow_zero,
+    PairRouting.tupleProbability]
+
+lemma modified_pointwise_zero (L : ℕ) (E : Finset (Card 0))
+    {ι : Type*} [Fintype ι] (e f : ι ↪ Card L) (he : Function.Surjective e) (a : ℝ) (ha : 0 ≤ a) :
+    (PairRouting.tupleProbability (modifiedPerm L 0 E) e f)^(1+a) ≤
+      centralFactor L 0 E a * modifiedRoutingMoment L 0 E e f a := by
+  have hp := PairRouting.tupleProbability_nonneg (modifiedPerm L 0 E) e f
+  rw [modifiedRoutingMoment_zero]
+  have hbound := modified_probability_zero_le L E e f he
+  have hpow := Real.rpow_le_rpow hp hbound ha
+  have hfpos : (0:ℝ) < Nat.factorial (2^L) := Nat.cast_pos.mpr (Nat.factorial_pos _)
+  have heq : (if (fun j => Fin.elim0 j : Card 0) ∈ E then (1:ℝ) else 1/(Nat.factorial (2^L):ℝ))^a =
+      centralFactor L 0 E a := by
+    rw [centralFactor_zero]
+    split_ifs
+    · exact Real.one_rpow _
+    · rw [one_div,Real.inv_rpow (le_of_lt hfpos),Real.rpow_neg (le_of_lt hfpos)]
+  rw [heq] at hpow
+  rw [Real.rpow_add' hp (by linarith : (1+a:ℝ) ≠ 0),Real.rpow_one,mul_comm]
+  exact mul_le_mul_of_nonneg_right hpow hp
+
+
+lemma modified_matches_step (L r : ℕ) (E : Finset (Card (r+1))) {ι : Type*}
+    (e f : ι ↪ Card (L+r+1)) (ω : ModifiedCoins L r × ModifiedCoins L r)
+    (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
+    (∀ i, modifiedPerm L (r+1) E ((modifiedStepEquiv L r).symm (ω,coins)) (e i) = f i) ↔
+    (∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1))
+      ((e.trans (headTailEquiv (L+r)).toEmbedding) i) = ((f.trans (headTailEquiv (L+r)).toEmbedding) i)) := by
+  apply forall_congr'
+  intro i
+  rw [←(headTailEquiv (L+r)).injective.eq_iff,modifiedPerm_coordinates]
+  rfl
+
+lemma modifiedRoutingMoment_step (L r : ℕ) (E : Finset (Card (r+1))) {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (e f : ι ↪ Card (L+r+1)) (a : ℝ) :
+    let x := e.trans (headTailEquiv (L+r)).toEmbedding
+    let y := f.trans (headTailEquiv (L+r)).toEmbedding
+    modifiedRoutingMoment L (r+1) E e f a = PairRouting.outerWeight x y *
+      ∑ c : {c // c ∈ PairRouting.compatibleSet x y},
+        finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
+          if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c.val
+          then (2:ℝ)^((modifiedAssignmentCost L r E x c.val
+            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0) := by
+  classical
+  intro x y
+  let : Fintype (ModifiedCoins L r) := inferInstance
+  rw [modifiedRoutingMoment,finiteMean_equiv (modifiedStepEquiv L r),finiteMean_prod]
+  have he (ω : ModifiedCoins L r × ModifiedCoins L r) :
+      finiteMean (fun coins : (Card (L+r) → Bool) × (Card (L+r) → Bool) =>
+        if ∀ i, modifiedPerm L (r+1) E ((modifiedStepEquiv L r).symm (ω,coins)) (e i) = f i
+        then (2:ℝ)^((modifiedCost L (r+1) E e ((modifiedStepEquiv L r).symm (ω,coins)):ℝ)*a) else 0) =
+      PairRouting.outerWeight x y *
+        ∑ c : {c // c ∈ PairRouting.compatibleSet x y},
+          if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c.val
+          then (2:ℝ)^((modifiedAssignmentCost L r E x c.val
+            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0 := by
+    let W : (ι → Bool) → ℝ := fun c =>
+      if hc : PairRouting.Compatible x c then (2:ℝ)^((modifiedAssignmentCost L r E x c hc ω:ℝ)*a) else 0
+    calc
+      _ = finiteMean (fun coins : (Card (L+r) → Bool) × (Card (L+r) → Bool) =>
+        if ∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1))
+          (x i) = y i then W (PairRouting.colors x coins.1) else 0) := by
+        apply finiteMean_congr
+        intro coins
+        simp only [modified_matches_step,modifiedCost_step]
+        simp only [W,PairRouting.colors_compatible,↓reduceDIte]
+        rfl
+      _ = _ := by
+        rw [PairRouting.sandwich_weighted_probability_subtype]
+        congr 1
+        apply Finset.sum_congr rfl
+        intro c _
+        simp only [W,((PairRouting.mem_compatibleSet x y c.val).mp c.property).1,↓reduceDIte]
+  simp_rw [he]
+  simp_rw [mul_comm (PairRouting.outerWeight x y)]
+  rw [finiteMean_mul_const,finiteMean_sum,mul_comm]
+
+lemma modified_child_moment_product (L r : ℕ) (E : Finset (Card (r+1))) {ι : Type*} [Fintype ι]
+    (x y : ι ↪ Bool × Card (L+r)) (c : ι → Bool)
+    (hcx : PairRouting.Compatible x c) (hcy : PairRouting.Compatible y c) (a : ℝ) :
+    modifiedRoutingMoment L r (exceptionChild E false) (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a *
+      modifiedRoutingMoment L r (exceptionChild E true) (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a =
+    finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
+      if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c
+      then (2:ℝ)^(((modifiedCost L r (exceptionChild E false) (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
+        modifiedCost L r (exceptionChild E true) (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0) := by
+  classical
+  unfold modifiedRoutingMoment
+  rw [←finiteMean_prod_mul]
+  apply finiteMean_congr
+  intro ω
+  let M₀ := ∀ i : {i // c i = false}, modifiedPerm L r (exceptionChild E false) ω.1 (x i.val).2 = (y i.val).2
+  let M₁ := ∀ i : {i // c i = true}, modifiedPerm L r (exceptionChild E true) ω.2 (x i.val).2 = (y i.val).2
+  have hp : (fun b : Bool => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) =
+      (fun b => if b then modifiedPerm L r (exceptionChild E true) ω.2 else modifiedPerm L r (exceptionChild E false) ω.1) := by
+    funext b
+    cases b <;> rfl
+  have hs : PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c ↔
+      M₀ ∧ M₁ := by rw [hp]; exact PairRouting.childMatches_split _ _ _ _ _
+  change (if M₀ then _ else 0) * (if M₁ then _ else 0) = _
+  by_cases h₀ : M₀ <;> by_cases h₁ : M₁
+  · simp only [hs,h₀,h₁,true_and,↓reduceIte]
+    rw [←Real.rpow_add (by norm_num : (0:ℝ)<2)]
+    congr 1
+    ring
+  · simp only [hs,h₀,h₁,and_false,↓reduceIte,mul_zero]
+  · simp only [hs,h₀,h₁,false_and,↓reduceIte,zero_mul]
+  · simp only [hs,h₀,h₁,false_and,↓reduceIte,mul_zero]
+
+lemma modified_probability_step (L r : ℕ) (E : Finset (Card (r+1))) {ι : Type*} [Fintype ι]
+    (e f : ι ↪ Card (L+r+1)) :
+    PairRouting.tupleProbability (modifiedPerm L (r+1) E) e f =
+    finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
+      finiteMean (fun coins : (Card (L+r) → Bool) × (Card (L+r) → Bool) =>
+        if ∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => if b then modifiedPerm L r (exceptionChild E true) ω.2 else modifiedPerm L r (exceptionChild E false) ω.1)
+          ((e.trans (headTailEquiv (L+r)).toEmbedding) i) = ((f.trans (headTailEquiv (L+r)).toEmbedding) i)
+        then (1:ℝ) else 0)) := by
+  classical
+  let : Fintype (ModifiedCoins L r) := inferInstance
+  rw [PairRouting.tupleProbability,finiteMean_equiv (modifiedStepEquiv L r),finiteMean_prod]
+  have hp (b : Bool) (p q : ModifiedCoins L r) :
+      modifiedPerm L r (exceptionChild E b) (if b then p else q) =
+      (if b then modifiedPerm L r (exceptionChild E true) p else modifiedPerm L r (exceptionChild E false) q) := by
+    cases b <;> rfl
+  simp only [modified_matches_step,hp]
+
+
+lemma modified_assignment_moment_bound (L r : ℕ) (E : Finset (Card (r+1))) {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (x y : ι ↪ Bool × Card (L+r)) (c : ι → Bool) (hc : c ∈ PairRouting.compatibleSet x y)
+    (a : ℝ) (ha : 0 ≤ a) :
+    let hcx := ((PairRouting.mem_compatibleSet x y c).mp hc).1
+    PairRouting.outerWeight x y ^ a * ((PairRouting.compatibleSet x y).card:ℝ)^a *
+      ((2:ℝ)^(-(r:ℝ)*Fintype.card ι*a) *
+        finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
+          if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c
+          then (2:ℝ)^(((modifiedCost L r (exceptionChild E false) (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
+            modifiedCost L r (exceptionChild E true) (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0)) ≤
+    (2:ℝ)^(-((r:ℝ)+1)*Fintype.card ι*a) *
+      finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
+        if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c
+        then (2:ℝ)^((modifiedAssignmentCost L r E x c hcx ω:ℝ)*a) else 0) := by
+  classical
+  intro hcx
+  have hscale (q : ℝ) (F : ModifiedCoins L r × ModifiedCoins L r → ℝ) :
+      q * finiteMean F = finiteMean (fun ω => q * F ω) := by
+    simp_rw [mul_comm q,finiteMean_mul_const]
+  rw [←mul_assoc,hscale,hscale]
+  apply finiteMean_mono
+  intro ω
+  split_ifs with hm
+  · have hh := PairRouting.assignment_factor_bound x y c hc
+      (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) hm a ha
+    calc
+      _ ≤ ((2:ℝ)^(((PairRouting.alternatingCycles (PairRouting.coloredEmbedding x c hcx)
+          (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)):ℝ)-Fintype.card ι)*a) *
+          (2:ℝ)^(-(r:ℝ)*Fintype.card ι*a)) *
+          (2:ℝ)^(((modifiedCost L r (exceptionChild E false) (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
+            modifiedCost L r (exceptionChild E true) (PairRouting.childEmbedding x c hcx true) ω.2)*a) :=
+        mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hh (by positivity)) (by positivity)
+      _ = _ := by
+        simp only [←Real.rpow_add (by norm_num : (0:ℝ)<2),modifiedAssignmentCost,Nat.cast_add]
+        congr 1
+        ring
+  · simp only [mul_zero]
+    exact le_rfl
+
+
+lemma modified_child_power_product_bound (L r : ℕ) (E : Finset (Card (r+1)))
+    {ι : Type*} [Fintype ι] (x y : ι ↪ Bool × Card (L+r)) (c : ι → Bool)
+    (hcx : PairRouting.Compatible x c) (hcy : PairRouting.Compatible y c) (a : ℝ)
+    (hfalse : (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E false))
+        (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false))^(1+a) ≤
+      ((2:ℝ)^(-(r:ℝ)*Fintype.card {i // c i = false}*a)*centralFactor L r (exceptionChild E false) a) *
+        modifiedRoutingMoment L r (exceptionChild E false)
+          (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a)
+    (htrue : (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E true))
+        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true))^(1+a) ≤
+      ((2:ℝ)^(-(r:ℝ)*Fintype.card {i // c i = true}*a)*centralFactor L r (exceptionChild E true) a) *
+        modifiedRoutingMoment L r (exceptionChild E true)
+          (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a) :
+    (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E false))
+      (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false))^(1+a) *
+      (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E true))
+        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true))^(1+a) ≤
+    ((2:ℝ)^(-(r:ℝ)*Fintype.card ι*a)*centralFactor L (r+1) E a) *
+      finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
+        if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c
+        then (2:ℝ)^(((modifiedCost L r (exceptionChild E false) (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
+          modifiedCost L r (exceptionChild E true) (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0) := by
+  have hh := mul_le_mul hfalse htrue
+    (Real.rpow_nonneg (PairRouting.tupleProbability_nonneg _ _ _) _)
+    (mul_nonneg (mul_nonneg (by positivity) (centralFactor_nonneg _ _ _ _))
+      (modifiedRoutingMoment_nonneg _ _ _ _ _ _))
+  have he : -(r:ℝ)*Fintype.card {i // c i = false}*a + -(r:ℝ)*Fintype.card {i // c i = true}*a =
+      -(r:ℝ)*Fintype.card ι*a := by
+    rw [←add_mul,←mul_add,←Nat.cast_add,PairRouting.color_card_add]
+  calc
+    _ ≤ _ := hh
+    _ = ((2:ℝ)^(-(r:ℝ)*Fintype.card {i // c i = false}*a)*
+        (2:ℝ)^(-(r:ℝ)*Fintype.card {i // c i = true}*a)) *
+      (centralFactor L r (exceptionChild E false) a * centralFactor L r (exceptionChild E true) a) *
+      (modifiedRoutingMoment L r (exceptionChild E false)
+        (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a *
+      modifiedRoutingMoment L r (exceptionChild E true)
+        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a) := by ring
+    _ = _ := by rw [←Real.rpow_add (by norm_num : (0:ℝ)<2),he,←centralFactor_step,modified_child_moment_product]
+
+
+theorem modified_pointwise_moment (L r : ℕ) (E : Finset (Card r)) {ι : Type*} [Fintype ι]
+    (e f : ι ↪ Card (L+r)) (he : Function.Surjective e) (a : ℝ) (ha : 0 ≤ a) :
+    (PairRouting.tupleProbability (modifiedPerm L r E) e f)^(1+a) ≤
+      ((2:ℝ)^(-(r:ℝ)*Fintype.card ι*a)*centralFactor L r E a) * modifiedRoutingMoment L r E e f a := by
+  classical
+  induction r generalizing ι with
+  | zero =>
+      simpa only [Nat.cast_zero,neg_zero,zero_mul,Real.rpow_zero,one_mul] using
+        modified_pointwise_zero L E e f he a ha
+  | succ r ih =>
+      let x := e.trans (headTailEquiv (L+r)).toEmbedding
+      let y := f.trans (headTailEquiv (L+r)).toEmbedding
+      let W := PairRouting.outerWeight x y
+      let C := PairRouting.compatibleSet x y
+      let p : (ι → Bool) → ℝ := fun c =>
+        (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E false))
+          (fun i : {i // c i = false} => (x i.val).2)
+          (fun i : {i // c i = false} => (y i.val).2))^(1+a) *
+        (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E true))
+          (fun i : {i // c i = true} => (x i.val).2)
+          (fun i : {i // c i = true} => (y i.val).2))^(1+a)
+      let m : {c // c ∈ C} → ℝ := fun c =>
+        finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
+          if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c.val
+          then (2:ℝ)^((modifiedAssignmentCost L r E x c.val
+            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0)
+      have hp (c : {c // c ∈ C}) : W^a * (C.card:ℝ)^a * p c.val ≤
+          ((2:ℝ)^(-((r:ℝ)+1)*Fintype.card ι*a)*centralFactor L (r+1) E a) * m c := by
+        obtain ⟨hcx,hcy⟩ := (PairRouting.mem_compatibleSet x y c.val).mp c.property
+        have h₀ := ih (exceptionChild E false) (PairRouting.childEmbedding x c.val hcx false)
+          (PairRouting.childEmbedding y c.val hcy false)
+          (PairRouting.childEmbedding_surjective x ((headTailEquiv (L+r)).surjective.comp he) c.val hcx false)
+        have h₁ := ih (exceptionChild E true) (PairRouting.childEmbedding x c.val hcx true)
+          (PairRouting.childEmbedding y c.val hcy true)
+          (PairRouting.childEmbedding_surjective x ((headTailEquiv (L+r)).surjective.comp he) c.val hcx true)
+        have hh := modified_child_power_product_bound L r E x y c.val hcx hcy a h₀ h₁
+        have hh' := mul_le_mul_of_nonneg_left hh
+          (mul_nonneg (Real.rpow_nonneg (le_of_lt (PairRouting.outerWeight_pos x y)) a)
+            (Real.rpow_nonneg (Nat.cast_nonneg C.card) a))
+        apply hh'.trans
+        have hm := modified_assignment_moment_bound L r E x y c.val c.property a ha
+        convert mul_le_mul_of_nonneg_left hm (centralFactor_nonneg L (r+1) E a) using 1 <;> ring
+      have hnode := PairRouting.sandwich_power_bound (modifiedPerm L r (exceptionChild E false)) (modifiedPerm L r (exceptionChild E true)) x y a ha
+      rw [←modified_probability_step L r E e f] at hnode
+      change (PairRouting.tupleProbability (modifiedPerm L (r+1) E) e f)^(1+a) ≤
+        W^(1+a) * (C.card:ℝ)^a * ∑ c ∈ C, p c at hnode
+      have hsum : ∑ c ∈ C, p c = ∑ c : {c // c ∈ C}, p c.val := by
+        rw [Finset.sum_subtype C (fun _ => Iff.rfl)]
+      have hW : 0 < W := PairRouting.outerWeight_pos x y
+      calc
+        _ ≤ W^(1+a) * (C.card:ℝ)^a * ∑ c ∈ C, p c := hnode
+        _ = W * ∑ c : {c // c ∈ C}, W^a * (C.card:ℝ)^a * p c.val := by
+          rw [Real.rpow_add hW,Real.rpow_one,hsum,←Finset.mul_sum]
+          ring
+        _ ≤ W * ∑ c : {c // c ∈ C}, ((2:ℝ)^(-((r:ℝ)+1)*Fintype.card ι*a)*centralFactor L (r+1) E a) * m c :=
+          mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun c _ => hp c)) (le_of_lt hW)
+        _ = _ := by
+          rw [←Finset.mul_sum,modifiedRoutingMoment_step]
+          simp only [Nat.cast_add,Nat.cast_one]
+          change W * (((2:ℝ)^(-((r:ℝ)+1)*Fintype.card ι*a)*centralFactor L (r+1) E a) * ∑ c, m c) =
+            ((2:ℝ)^(-((r:ℝ)+1)*Fintype.card ι*a)*centralFactor L (r+1) E a) * (W * ∑ c, m c)
+          ring
+
+
+lemma modifiedRoutingMoment_sum (L r : ℕ) (E : Finset (Card r)) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (L+r)) (a : ℝ) :
+    (∑ f : ι ↪ Card (L+r), modifiedRoutingMoment L r E e f a) =
+      finiteMean (fun ω : ModifiedCoins L r => (2:ℝ)^((modifiedCost L r E e ω:ℝ)*a)) := by
+  classical
+  unfold modifiedRoutingMoment
+  rw [←finiteMean_sum]
+  apply finiteMean_congr
+  intro ω
+  let f₀ : ι ↪ Card (L+r) := e.trans (modifiedPerm L r E ω).toEmbedding
+  rw [Finset.sum_eq_single f₀]
+  · have hm : ∀ i, modifiedPerm L r E ω (e i) = f₀ i := fun _ => rfl
+    simp only [hm,implies_true,↓reduceIte]
+  · intro f _ hf
+    have hm : ¬ ∀ i, modifiedPerm L r E ω (e i) = f i := by
+      intro hh
+      apply hf
+      ext i : 1
+      exact (hh i).symm
+    exact ite_eq_right hm
+  · simp
+
+noncomputable def modifiedRowMoment (L r : ℕ) (E : Finset (Card r)) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (L+r)) (a : ℝ) : ℝ :=
+  finiteMean (fun f : ι ↪ Card (L+r) =>
+    ((Fintype.card (ι ↪ Card (L+r)):ℝ)*PairRouting.tupleProbability (modifiedPerm L r E) e f)^(1+a))
+
+lemma full_tuple_card (d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d)
+    (he : Function.Surjective e) : Fintype.card ι = 2^d := by
+  rw [←card_positions d]
+  exact Fintype.card_congr (Equiv.ofBijective e ⟨e.injective,he⟩)
+
+lemma exception_card_le (r : ℕ) (E : Finset (Card r)) : E.card ≤ 2^r := by
+  simpa only [card_positions] using E.card_le_univ
+
+noncomputable def modifiedNormalization (L r : ℕ) (E : Finset (Card r)) : ℝ :=
+  (Nat.factorial (2^(L+r)):ℝ) /
+    (((2:ℝ)^r)^(2^(L+r)) * (Nat.factorial (2^L):ℝ)^(2^r-E.card))
+
+lemma modifiedNormalization_pos (L r : ℕ) (E : Finset (Card r)) :
+    0 < modifiedNormalization L r E := by unfold modifiedNormalization; positivity
+
+lemma modified_normalization_identity (L r : ℕ) (E : Finset (Card r)) (a : ℝ) :
+    (Nat.factorial (2^(L+r)):ℝ)^a *
+      ((2:ℝ)^(-(r:ℝ)*(2^(L+r):ℕ)*a)*centralFactor L r E a) =
+    modifiedNormalization L r E ^ a := by
+  rw [←mul_assoc,density_normalization,centralFactor,modifiedNormalization,
+    Real.div_rpow (by positivity) (by positivity),Real.div_rpow (by positivity) (by positivity),
+    Real.mul_rpow (by positivity) (by positivity)]
+  have hc : ((2^r-E.card:ℕ):ℝ) = (2:ℝ)^r-E.card := by
+    rw [Nat.cast_sub (exception_card_le r E),Nat.cast_pow,Nat.cast_ofNat]
+  rw [←Real.rpow_natCast (Nat.factorial (2^L):ℝ),←Real.rpow_mul (by positivity),hc,
+    show -((2:ℝ)^r-E.card)*a = -(((2:ℝ)^r-E.card)*a) by ring,
+    Real.rpow_neg (by positivity)]
+  ring
+
+theorem modified_row_density_moment (L r : ℕ) (E : Finset (Card r)) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (L+r)) (he : Function.Surjective e) (a : ℝ) (ha : 0 ≤ a) :
+    modifiedRowMoment L r E e a ≤ modifiedNormalization L r E ^ a *
+      finiteMean (fun ω : ModifiedCoins L r => (2:ℝ)^((modifiedCost L r E e ω:ℝ)*a)) := by
+  classical
+  have : Nonempty (ι ↪ Card (L+r)) := ⟨e⟩
+  rw [modifiedRowMoment,finite_row_power _ (fun f => PairRouting.tupleProbability_nonneg _ _ _)]
+  calc
+    _ ≤ (Fintype.card (ι ↪ Card (L+r)):ℝ)^a *
+      ∑ f : ι ↪ Card (L+r), ((2:ℝ)^(-(r:ℝ)*Fintype.card ι*a)*centralFactor L r E a) *
+        modifiedRoutingMoment L r E e f a :=
+      mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun f _ => modified_pointwise_moment L r E e f he a ha))
+        (Real.rpow_nonneg (Nat.cast_nonneg _) _)
+    _ = _ := by
+      rw [←Finset.mul_sum,modifiedRoutingMoment_sum,←mul_assoc,Fintype.card_embedding_eq,
+        card_positions,full_tuple_card (L+r) e he,Nat.descFactorial_self,modified_normalization_identity]
+
+lemma modified_total_cost_mgf (L r : ℕ) (hL : 0 < L) (E : Finset (Card r))
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) :
+    finiteMean (fun ω : ModifiedCoins L r => momentBase^(modifiedCost L r E e ω)) ≤
+      Real.exp (Fintype.card ι*heightDecay L +
+        3*((L*E.card*2^L:ℕ):ℝ)*Real.log momentBase) := by
+  let B := Real.exp (Fintype.card ι*heightDecay L +
+    3*((L*E.card*2^L:ℕ):ℝ)*Real.log momentBase)
+  rw [finiteMean_prod]
+  apply le_trans (finiteMean_mono (fun U => ?_)) (finiteMean_const B).le
+  rw [finiteMean_prod,finiteMean_comm]
+  apply le_trans (finiteMean_mono (fun X => ?_)) (finiteMean_const B).le
+  exact modified_cost_mgf L r hL E U X e momentBase momentBase_one_le (momentBase_log.trans (by norm_num))
+
+lemma modifiedRowMoment_bound (L r : ℕ) (hL : 0 < L) (E : Finset (Card r))
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (he : Function.Surjective e) :
+    modifiedRowMoment L r E e (1/1000) ≤
+      modifiedNormalization L r E ^ (1/1000:ℝ) *
+        Real.exp ((2^(L+r):ℕ)*heightDecay L +
+          3*((L*E.card*2^L:ℕ):ℝ)*Real.log momentBase) := by
+  have hm := modified_row_density_moment L r E e he (1/1000) (by norm_num)
+  simp_rw [←momentBase_pow] at hm
+  apply hm.trans
+  apply mul_le_mul_of_nonneg_left _ (Real.rpow_nonneg (modifiedNormalization_pos L r E).le _)
+  simpa only [full_tuple_card (L+r) e he] using modified_total_cost_mgf L r hL E e
+
+
+lemma log_factorial_upper {n : ℕ} (hn : 1 ≤ n) :
+    Real.log (Nat.factorial n) ≤ (n:ℝ)*Real.log n-n+Real.log n+1 := by
+  induction n with
+  | zero => omega
+  | succ n ih =>
+      by_cases hz : n = 0
+      · subst n; norm_num
+      have hn0 : (0:ℝ) < n := Nat.cast_pos.mpr (Nat.pos_of_ne_zero hz)
+      have hn1 : (0:ℝ) < n+1 := by positivity
+      have h := Real.one_sub_inv_le_log_of_pos (div_pos hn1 hn0)
+      rw [Real.log_div (ne_of_gt hn1) (ne_of_gt hn0),inv_div] at h
+      have hh := mul_le_mul_of_nonneg_left h hn1.le
+      have hcancel : ((n:ℝ)+1)*(1-(n:ℝ)/(n+1)) = 1 := by field_simp; ring
+      rw [hcancel] at hh
+      have hi := ih (by omega)
+      rw [Nat.factorial_succ,Nat.cast_mul,Real.log_mul (by positivity) (by positivity),Nat.cast_add,Nat.cast_one]
+      nlinarith
+
+lemma log_modifiedNormalization (L r : ℕ) (E : Finset (Card r)) :
+    Real.log (modifiedNormalization L r E) =
+      Real.log (Nat.factorial (2^(L+r))) -
+        ((2^(L+r):ℕ)*((r:ℝ)*Real.log 2)+((2:ℝ)^r-E.card)*Real.log (Nat.factorial (2^L))) := by
+  unfold modifiedNormalization
+  rw [Real.log_div (by positivity) (by positivity),Real.log_mul (by positivity) (by positivity),
+    Real.log_pow,Real.log_pow,Real.log_pow,
+    Nat.cast_sub (exception_card_le r E),Nat.cast_pow,Nat.cast_ofNat]
+  simp only [Nat.cast_pow,Nat.cast_ofNat]
+
+lemma log_modifiedNormalization_le (L r : ℕ) (E : Finset (Card r)) :
+    Real.log (modifiedNormalization L r E) ≤
+      ((L+r:ℕ):ℝ)*Real.log 2+1+E.card*Real.log (Nat.factorial (2^L)) := by
+  have hN : 1 ≤ 2^(L+r) := one_le_pow₀ (by norm_num)
+  have hS : 1 ≤ 2^L := one_le_pow₀ (by norm_num)
+  have hu := log_factorial_upper hN
+  have hl := mul_le_mul_of_nonneg_left (log_factorial_lower hS) (by positivity : (0:ℝ) ≤ (2:ℝ)^r)
+  simp only [Nat.cast_pow,Nat.cast_ofNat,Real.log_pow,Nat.cast_add] at hu hl ⊢
+  rw [log_modifiedNormalization]
+  simp only [pow_add,Nat.cast_mul,Nat.cast_pow,Nat.cast_ofNat] at hu hl ⊢
+  nlinarith
+
+lemma modified_moment_exponential (L r : ℕ) (hL : 0 < L) (E : Finset (Card r))
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (he : Function.Surjective e) :
+    modifiedRowMoment L r E e (1/1000) ≤
+      Real.exp (((L+r:ℕ)*Real.log 2+1+E.card*Real.log (Nat.factorial (2^L)))/1000 +
+        (2^(L+r):ℕ)*heightDecay L+3*((L*E.card*2^L:ℕ):ℝ)*Real.log momentBase) := by
+  apply (modifiedRowMoment_bound L r hL E e he).trans
+  rw [Real.rpow_def_of_pos (modifiedNormalization_pos L r E),←Real.exp_add]
+  apply Real.exp_le_exp.mpr
+  have h := log_modifiedNormalization_le L r E
+  linarith
+
+lemma log_power_factorial_le (L : ℕ) :
+    Real.log (Nat.factorial (2^L)) ≤ (2:ℝ)^L*L := by
+  have hf : (Nat.factorial (2^L):ℝ) ≤ ((2^L:ℕ):ℝ)^(2^L) := by
+    exact_mod_cast Nat.factorial_le_pow (2^L)
+  have hh := Real.log_le_log (by positivity) hf
+  rw [Real.log_pow,Nat.cast_pow,Nat.cast_ofNat,Real.log_pow] at hh
+  apply hh.trans
+  have hl := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
+  have hn : 0 ≤ (2:ℝ)^L*(L:ℝ) := by positivity
+  nlinarith
+
+lemma modified_moment_exception_bound (L r : ℕ) (hL : 0 < L) (E : Finset (Card r))
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (he : Function.Surjective e)
+    (ε : ℝ) (hE : (E.card:ℝ) ≤ ε*(2:ℝ)^r) :
+    modifiedRowMoment L r E e (1/1000) ≤
+      Real.exp (((L+r:ℕ)*Real.log 2+1)/1000 +
+        (2:ℝ)^(L+r)*(heightDecay L+ε*L/250)) := by
+  apply (modified_moment_exponential L r hL E e he).trans
+  apply Real.exp_le_exp.mpr
+  have hs := mul_le_mul_of_nonneg_left (log_power_factorial_le L) (Nat.cast_nonneg E.card)
+  have hb := mul_le_mul_of_nonneg_left momentBase_log
+    (by positivity : (0:ℝ) ≤ 3*((L*E.card*2^L:ℕ):ℝ))
+  have hexc := mul_le_mul_of_nonneg_left hE (by positivity : (0:ℝ) ≤ (2:ℝ)^L*L/250)
+  simp only [Nat.cast_mul,Nat.cast_pow,Nat.cast_ofNat,pow_add] at hs hb hexc ⊢
+  nlinarith
+
+
+lemma heightDecay_tendsto : Filter.Tendsto heightDecay Filter.atTop (nhds 0) := by
+  have h : Filter.Tendsto (fun L : ℕ => (-(1:ℝ)/64)*(L:ℝ)) Filter.atTop Filter.atBot :=
+    (Filter.tendsto_const_mul_atBot_of_neg (by norm_num : -(1:ℝ)/64 < 0)).mpr
+      tendsto_natCast_atTop_atTop
+  have ht := ((Real.tendsto_exp_atBot.comp h).const_mul 68).div_const (1-Real.exp (-(1:ℝ)/64))
+  convert ht using 1
+  · funext L
+    unfold heightDecay
+    dsimp only [Function.comp_apply]
+    rw [show -(L:ℝ)/64 = (-(1:ℝ)/64)*(L:ℝ) by ring]
+  · simp
+
+noncomputable def modifiedOverhead (L r : ℕ) : ℝ :=
+  (((L+r:ℕ):ℝ)*Real.log 2+1)/((2:ℝ)^(L+r)*1000)
+
+lemma modifiedOverhead_tendsto (L : ℕ) :
+    Filter.Tendsto (modifiedOverhead L) Filter.atTop (nhds 0) := by
+  have h₁ := tendsto_pow_const_div_const_pow_of_one_lt 1 (by norm_num : (1:ℝ)<2)
+  have h₀ := tendsto_pow_const_div_const_pow_of_one_lt 0 (by norm_num : (1:ℝ)<2)
+  simp only [pow_one] at h₁
+  simp only [pow_zero] at h₀
+  have ht := ((h₁.const_mul (Real.log 2)).add (h₀.const_mul ((L:ℝ)*Real.log 2+1))).div_const
+    ((2:ℝ)^L*1000)
+  convert ht using 1
+  · funext r
+    simp only [modifiedOverhead,Nat.cast_add,pow_add]
+    field_simp
+    ring
+  · simp
+
+theorem modified_arbitrarily_small_rate (η : ℝ) (hη : 0 < η) :
+    ∃ L : ℕ, 0 < L ∧ ∃ ε : ℝ, 0 < ε ∧ ε < 1/2 ∧
+      ∀ᶠ r : ℕ in Filter.atTop, ∀ (E : Finset (Card r)), (E.card:ℝ) ≤ ε*(2:ℝ)^r →
+        ∀ (e : Card (L+r) ↪ Card (L+r)),
+          modifiedRowMoment L r E e (1/1000) ≤ Real.exp (η*(2:ℝ)^(L+r)) := by
+  have hsmall : ∀ᶠ L : ℕ in Filter.atTop, heightDecay L < η/3 :=
+    heightDecay_tendsto.eventually (gt_mem_nhds (by linarith : (0:ℝ)<η/3))
+  obtain ⟨L,hdec,hL⟩ := (hsmall.and (Filter.eventually_ge_atTop 1)).exists
+  have hL0 : (0:ℝ) < L := Nat.cast_pos.mpr hL
+  let ε : ℝ := min (1/4) (η*250/(3*L))
+  have hε : 0 < ε := lt_min (by norm_num) (by positivity)
+  have hε1 : ε < 1/2 := (min_le_left _ _).trans_lt (by norm_num)
+  have hεL : ε*L/250 ≤ η/3 := by
+    have h := min_le_right (1/4:ℝ) (η*250/(3*L))
+    change ε ≤ η*250/(3*L) at h
+    have hh := mul_le_mul_of_nonneg_right h hL0.le
+    have he : (η*250/(3*(L:ℝ)))*(L:ℝ)/250 = η/3 := by field_simp
+    exact (div_le_div_of_nonneg_right hh (by norm_num)).trans he.le
+  refine ⟨L,hL,ε,hε,hε1,?_⟩
+  have hr : ∀ᶠ r : ℕ in Filter.atTop, modifiedOverhead L r < η/3 :=
+    (modifiedOverhead_tendsto L).eventually (gt_mem_nhds (by linarith : (0:ℝ)<η/3))
+  filter_upwards [hr] with r hr
+  intro E hE e
+  have he := Finite.surjective_of_injective e.injective
+  apply (modified_moment_exception_bound L r hL E e he ε hE).trans
+  apply Real.exp_le_exp.mpr
+  have hn : (0:ℝ) < (2:ℝ)^(L+r) := by positivity
+  have hd : modifiedOverhead L r * (2:ℝ)^(L+r) =
+      (((L+r:ℕ):ℝ)*Real.log 2+1)/1000 := by
+    unfold modifiedOverhead
+    field_simp
+  have hh := mul_le_mul_of_nonneg_right hr.le hn.le
+  rw [hd] at hh
+  have ht := mul_le_mul_of_nonneg_left (show heightDecay L+ε*L/250 ≤ 2*η/3 by linarith) hn.le
+  linarith
+
+
+namespace DensityTransfer
+open scoped BigOperators
+
+noncomputable def applyKernel {X : Type*} [Fintype X] (P : X → X → ℝ)
+    (v : EuclideanSpace ℂ X) : EuclideanSpace ℂ X :=
+  WithLp.toLp 2 (fun x => ∑ y, (P x y : ℂ)*v y)
+
+lemma applyKernel_apply {X : Type*} [Fintype X] (P : X → X → ℝ)
+    (v : EuclideanSpace ℂ X) (x : X) :
+    applyKernel P v x = ∑ y, (P x y : ℂ)*v y := rfl
+
+lemma applyKernel_add {X : Type*} [Fintype X] (P Q : X → X → ℝ)
+    (v : EuclideanSpace ℂ X) :
+    applyKernel (fun x y => P x y+Q x y) v = applyKernel P v+applyKernel Q v := by
+  ext x
+  simp only [applyKernel_apply,Complex.ofReal_add,add_mul,Finset.sum_add_distrib]
+  rfl
+
+lemma kernel_component_sq {X : Type*} [Fintype X] (P : X → X → ℝ)
+    (hP : ∀ x y, 0 ≤ P x y) (v : EuclideanSpace ℂ X) (x : X) :
+    ‖applyKernel P v x‖^2 ≤ (∑ y, P x y) * ∑ y, P x y*‖v y‖^2 := by
+  have hs : ‖applyKernel P v x‖ ≤ ∑ y, P x y*‖v y‖ := by
+    rw [applyKernel_apply]
+    apply (norm_sum_le _ _).trans
+    apply Finset.sum_le_sum
+    intro y _
+    rw [norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (hP x y)]
+  apply (pow_le_pow_left₀ (norm_nonneg _) hs 2).trans
+  exact Finset.sum_sq_le_sum_mul_sum_of_sq_le_mul Finset.univ
+    (fun y _ => hP x y) (fun y _ => mul_nonneg (hP x y) (sq_nonneg _))
+    (fun y _ => by ring_nf; exact le_rfl)
+
+lemma schur_bound {X : Type*} [Fintype X] (P : X → X → ℝ)
+    (hP : ∀ x y, 0 ≤ P x y) (δ : ℝ) (hδ : 0 ≤ δ)
+    (hrow : ∀ x, ∑ y, P x y ≤ δ) (hcol : ∀ y, ∑ x, P x y ≤ δ)
+    (v : EuclideanSpace ℂ X) : ‖applyKernel P v‖ ≤ δ*‖v‖ := by
+  have hsq : ‖applyKernel P v‖^2 ≤ δ^2*‖v‖^2 := by
+    rw [EuclideanSpace.norm_sq_eq,EuclideanSpace.norm_sq_eq]
+    calc
+      _ ≤ ∑ x, δ * ∑ y, P x y*‖v y‖^2 := by
+        apply Finset.sum_le_sum
+        intro x _
+        apply (kernel_component_sq P hP v x).trans
+        exact mul_le_mul_of_nonneg_right (hrow x)
+          (Finset.sum_nonneg (fun y _ => mul_nonneg (hP x y) (sq_nonneg _)))
+      _ = δ * ∑ y, (∑ x, P x y)*‖v y‖^2 := by
+        rw [←Finset.mul_sum,Finset.sum_comm]
+        congr 1
+        apply Finset.sum_congr rfl
+        intro y _
+        exact (Finset.sum_mul Finset.univ _ _).symm
+      _ ≤ δ * ∑ y, δ*‖v y‖^2 := by
+        apply mul_le_mul_of_nonneg_left _ hδ
+        exact Finset.sum_le_sum (fun y _ => mul_le_mul_of_nonneg_right (hcol y) (sq_nonneg _))
+      _ = _ := by rw [←Finset.mul_sum]; ring
+  have hn := norm_nonneg (applyKernel P v)
+  have hr := mul_nonneg hδ (norm_nonneg v)
+  nlinarith
+
+lemma hilbert_schmidt_family {X I : Type*} [Fintype X] [Fintype I]
+    (P : X → X → ℝ) (v : I → EuclideanSpace ℂ X) (hv : Orthonormal ℂ v) :
+    (∑ i, ‖applyKernel P (v i)‖^2) ≤ ∑ x, ∑ y, (P x y)^2 := by
+  simp_rw [EuclideanSpace.norm_sq_eq]
+  rw [Finset.sum_comm]
+  apply Finset.sum_le_sum
+  intro x _
+  let w : EuclideanSpace ℂ X := WithLp.toLp 2 (fun y => (P x y : ℂ))
+  have he (i : I) : ‖applyKernel P (v i) x‖ = ‖inner ℂ (v i) w‖ := by
+    rw [norm_inner_symm]
+    congr 1
+    simp only [applyKernel_apply,PiLp.inner_apply]
+    apply Finset.sum_congr rfl
+    intro y _
+    simp [w,RCLike.inner_apply,mul_comm]
+  simp_rw [he]
+  have h := hv.sum_inner_products_le (s := Finset.univ) w
+  apply h.trans_eq
+  rw [EuclideanSpace.norm_sq_eq]
+  apply Finset.sum_congr rfl
+  intro y _
+  simp [w,Complex.norm_real,Real.norm_eq_abs,sq_abs]
+
+
+lemma tail_density_pointwise (t a z : ℝ) (ht : 0 ≤ t) (ha : 0 ≤ a) (hz : 0 < z) :
+    (if z < t then t else 0) ≤ t^(1+a)*z^(-a) := by
+  split_ifs with h
+  · rw [Real.rpow_neg hz.le,←div_eq_mul_inv]
+    apply (le_div_iff₀ (Real.rpow_pos_of_pos hz a)).mpr
+    rw [Real.rpow_add (lt_trans hz h),Real.rpow_one]
+    exact mul_le_mul_of_nonneg_left (Real.rpow_le_rpow hz.le h.le ha) ht
+  · positivity
+
+lemma tail_density_mean {X : Type*} [Fintype X] (f : X → ℝ) (hf : ∀ x, 0 ≤ f x)
+    (a z : ℝ) (ha : 0 ≤ a) (hz : 0 < z) :
+    finiteMean (fun x => if z < f x then f x else 0) ≤ finiteMean (fun x => f x^(1+a))*z^(-a) := by
+  rw [←finiteMean_mul_const]
+  exact finiteMean_mono (fun x => tail_density_pointwise (f x) a z (hf x) ha hz)
+
+noncomputable def tailKernel {X : Type*} [Fintype X] (P : X → X → ℝ) (z : ℝ) : X → X → ℝ :=
+  fun x y => if z < (Fintype.card X:ℝ)*P x y then P x y else 0
+
+noncomputable def headKernel {X : Type*} [Fintype X] (P : X → X → ℝ) (z : ℝ) : X → X → ℝ :=
+  fun x y => if z < (Fintype.card X:ℝ)*P x y then 0 else P x y
+
+lemma tail_add_head {X : Type*} [Fintype X] (P : X → X → ℝ) (z : ℝ) :
+    (fun x y => tailKernel P z x y+headKernel P z x y) = P := by
+  funext x y
+  unfold tailKernel headKernel
+  split_ifs <;> simp
+
+lemma tailKernel_nonneg {X : Type*} [Fintype X] (P : X → X → ℝ) (z : ℝ)
+    (hP : ∀ x y, 0 ≤ P x y) (x y : X) : 0 ≤ tailKernel P z x y := by
+  unfold tailKernel
+  split_ifs <;> simp only [hP,le_refl]
+
+lemma tailKernel_row {X : Type*} [Fintype X] [Nonempty X] (P : X → X → ℝ)
+    (hP : ∀ x y, 0 ≤ P x y) (a z M : ℝ) (ha : 0 ≤ a) (hz : 0 < z)
+    (hM : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*P x y)^(1+a)) ≤ M) (x : X) :
+    ∑ y, tailKernel P z x y ≤ M*z^(-a) := by
+  have hn : (Fintype.card X:ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
+  have he : finiteMean (fun y => if z < (Fintype.card X:ℝ)*P x y then
+      (Fintype.card X:ℝ)*P x y else 0) = ∑ y, tailKernel P z x y := by
+    unfold finiteMean
+    calc
+      _ = (∑ y, (Fintype.card X:ℝ)*tailKernel P z x y)/(Fintype.card X:ℝ) := by
+        congr 1
+        apply Finset.sum_congr rfl
+        intro y _
+        by_cases hy : z < (Fintype.card X:ℝ)*P x y
+        · simp only [tailKernel,hy,↓reduceIte]
+        · simp only [tailKernel,hy,↓reduceIte,mul_zero]
+      _ = _ := by rw [←Finset.mul_sum]; field_simp
+  rw [←he]
+  exact (tail_density_mean _ (fun y => mul_nonneg (Nat.cast_nonneg _) (hP x y)) a z ha hz).trans
+    (mul_le_mul_of_nonneg_right (hM x) (Real.rpow_nonneg hz.le _))
+
+lemma tailKernel_schur {X : Type*} [Fintype X] [Nonempty X] (P : X → X → ℝ)
+    (hP : ∀ x y, 0 ≤ P x y) (hsym : ∀ x y, P x y = P y x)
+    (a z M : ℝ) (ha : 0 ≤ a) (hz : 0 < z) (hM0 : 0 ≤ M)
+    (hM : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*P x y)^(1+a)) ≤ M)
+    (v : EuclideanSpace ℂ X) :
+    ‖applyKernel (tailKernel P z) v‖ ≤ (M*z^(-a))*‖v‖ := by
+  apply schur_bound _ (tailKernel_nonneg P z hP) _ (by positivity)
+    (tailKernel_row P hP a z M ha hz hM)
+  intro y
+  have hs : (fun x => tailKernel P z x y) = (fun x => tailKernel P z y x) := by
+    funext x
+    simp only [tailKernel,hsym x y]
+  rw [hs]
+  exact tailKernel_row P hP a z M ha hz hM y
+
+lemma headKernel_hs {X : Type*} [Fintype X] [Nonempty X] (P : X → X → ℝ)
+    (hP : ∀ x y, 0 ≤ P x y) (hrow : ∀ x, ∑ y, P x y = 1) (z : ℝ) (hz : 0 ≤ z) :
+    (∑ x, ∑ y, (headKernel P z x y)^2) ≤ z := by
+  have hn : (0:ℝ) < Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
+  have he (x y : X) : (headKernel P z x y)^2 ≤ (z/(Fintype.card X:ℝ))*P x y := by
+    unfold headKernel
+    split_ifs with h
+    · simp only [zero_pow (by omega : (2:ℕ) ≠ 0)]
+      exact mul_nonneg (div_nonneg hz hn.le) (hP x y)
+    · have h' : (Fintype.card X:ℝ)*P x y ≤ z := le_of_not_gt h
+      have hh := mul_le_mul_of_nonneg_right h' (hP x y)
+      rw [show (z/(Fintype.card X:ℝ))*P x y = (z*P x y)/(Fintype.card X:ℝ) by ring]
+      apply (le_div_iff₀ hn).mpr
+      nlinarith
+  calc
+    _ ≤ ∑ x, ∑ y, (z/(Fintype.card X:ℝ))*P x y :=
+      Finset.sum_le_sum (fun x _ => Finset.sum_le_sum (fun y _ => he x y))
+    _ = z := by
+      simp only [←Finset.mul_sum,hrow,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
+      field_simp
+
+
+theorem multiplicity_amplification {X I : Type*} [Fintype X] [Nonempty X]
+    [Fintype I] [Nonempty I] (P : X → X → ℝ)
+    (hP : ∀ x y, 0 ≤ P x y) (hsym : ∀ x y, P x y = P y x)
+    (hrow : ∀ x, ∑ y, P x y = 1) (a z M : ℝ) (ha : 0 ≤ a) (hz : 0 < z) (hM0 : 0 ≤ M)
+    (hM : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*P x y)^(1+a)) ≤ M)
+    (v : I → EuclideanSpace ℂ X) (hv : Orthonormal ℂ v)
+    (μ : ℝ) (hvμ : ∀ i, μ ≤ ‖applyKernel P (v i)‖) :
+    μ ≤ M*z^(-a)+Real.sqrt (z/(Fintype.card I:ℝ)) := by
+  let δ : ℝ := M*z^(-a)
+  have hδ : 0 ≤ δ := by dsimp [δ]; positivity
+  have hm : (0:ℝ) < Fintype.card I := Nat.cast_pos.mpr Fintype.card_pos
+  change μ ≤ δ+_
+  by_cases hμ : μ ≤ δ
+  · exact hμ.trans (le_add_of_nonneg_right (Real.sqrt_nonneg _))
+  have hmd : 0 ≤ μ-δ := by linarith
+  have hsmall (i : I) : μ-δ ≤ ‖applyKernel (headKernel P z) (v i)‖ := by
+    have hr := tailKernel_schur P hP hsym a z M ha hz hM0 hM (v i)
+    rw [hv.norm_eq_one i,mul_one] at hr
+    have he := applyKernel_add (tailKernel P z) (headKernel P z) (v i)
+    rw [tail_add_head] at he
+    have hn := norm_add_le (applyKernel (tailKernel P z) (v i)) (applyKernel (headKernel P z) (v i))
+    rw [←he] at hn
+    have hh := hvμ i
+    change ‖applyKernel (tailKernel P z) (v i)‖ ≤ δ at hr
+    linarith
+  have hsum : (Fintype.card I:ℝ)*(μ-δ)^2 ≤ z := by
+    calc
+      _ = ∑ i : I, (μ-δ)^2 := by simp
+      _ ≤ ∑ i : I, ‖applyKernel (headKernel P z) (v i)‖^2 :=
+        Finset.sum_le_sum (fun i _ => pow_le_pow_left₀ hmd (hsmall i) 2)
+      _ ≤ ∑ x, ∑ y, (headKernel P z x y)^2 := hilbert_schmidt_family _ v hv
+      _ ≤ z := headKernel_hs P hP hrow z hz.le
+  have hsq : (μ-δ)^2 ≤ z/(Fintype.card I:ℝ) := (le_div_iff₀ hm).mpr (by nlinarith)
+  have hs := (Real.le_sqrt hmd (div_nonneg hz.le hm.le)).mpr hsq
+  linarith
+
+end DensityTransfer
+namespace PairRouting
+open scoped BigOperators
+
+lemma tupleProbability_row_sum {ι β Ω : Type*} [Fintype ι] [Fintype β]
+    [DecidableEq β] [Fintype Ω] [Nonempty Ω]
+    (P : Ω → Equiv.Perm β) (e : ι ↪ β) :
+    (∑ f : ι ↪ β, tupleProbability P e f) = 1 := by
+  classical
+  simp only [tupleProbability]
+  rw [←finiteMean_sum]
+  have h (ω : Ω) (f : ι ↪ β) : (∀ i, P ω (e i) = f i) ↔ e.trans (P ω).toEmbedding = f := by
+    constructor
+    · intro h; ext i; exact h i
+    · intro h i; exact congrArg (fun g : ι ↪ β => g i) h
+  simp_rw [h]
+  simp only [Finset.sum_ite_eq,Finset.mem_univ,↓reduceIte,finiteMean_const]
+
+lemma tupleProbability_inverse {ι β Ω : Type*} [Fintype ι] [DecidableEq β]
+    [Fintype Ω] (P : Ω → Equiv.Perm β) (e f : ι → β) :
+    tupleProbability (fun ω => (P ω)⁻¹) e f = tupleProbability P f e := by
+  classical
+  apply finiteMean_congr
+  intro ω
+  congr 1
+  apply propext
+  constructor
+  · intro h i
+    have hh := congrArg (P ω) (h i)
+    simpa only [Equiv.Perm.inv_def,Equiv.apply_symm_apply] using hh.symm
+  · intro h i
+    have hh := congrArg (fun x => (P ω).symm x) (h i)
+    simpa only [Equiv.Perm.inv_def,Equiv.symm_apply_apply] using hh.symm
+
+lemma tupleProbability_equiv {ι β Ω Ω' : Type*} [Fintype ι] [DecidableEq β]
+    [Fintype Ω] [Fintype Ω'] (P : Ω → Equiv.Perm β) (τ : Ω ≃ Ω') (e f : ι → β) :
+    tupleProbability P e f = tupleProbability (fun ω => P (τ.symm ω)) e f :=
+  finiteMean_equiv τ _
+
+lemma tupleProbability_symmetric {ι β Ω : Type*} [Fintype ι] [DecidableEq β]
+    [Fintype Ω] (P : Ω → Equiv.Perm β) (τ : Equiv.Perm Ω)
+    (hτ : ∀ ω, P (τ ω) = (P ω)⁻¹) (e f : ι → β) :
+    tupleProbability P e f = tupleProbability P f e := by
+  rw [tupleProbability_equiv P τ.symm]
+  simp only [Equiv.symm_symm,hτ]
+  exact tupleProbability_inverse P e f
+
+end PairRouting
+
+lemma palindromePerm_reverse (d : ℕ) (ω : BenesCoins d) :
+    palindromePerm d (ω.2,ω.1) = (palindromePerm d ω)⁻¹ := by
+  simp only [palindromePerm,mul_inv_rev]
+  rfl
+
+lemma palindrome_kernel_symmetric (d : ℕ) {ι : Type*} [Fintype ι]
+    (e f : ι ↪ Card d) :
+    PairRouting.tupleProbability (palindromePerm d) e f =
+      PairRouting.tupleProbability (palindromePerm d) f e :=
+  PairRouting.tupleProbability_symmetric _ (Equiv.prodComm _ _) (palindromePerm_reverse d) e f
+
+theorem palindrome_density_transfer (d : ℕ) {ι I : Type*} [Fintype ι] [Fintype I]
+    [Nonempty I] (e₀ : ι ↪ Card d) (hk : 0 < Fintype.card ι)
+    (v : I → EuclideanSpace ℂ (ι ↪ Card d)) (hv : Orthonormal ℂ v)
+    (μ : ℝ) (hvμ : ∀ i, μ ≤ ‖DensityTransfer.applyKernel
+      (fun e f : ι ↪ Card d => PairRouting.tupleProbability (palindromePerm d) e f) (v i)‖)
+    (z : ℝ) (hz : 0 < z) :
+    μ ≤ Real.exp (densityConstant*Fintype.card ι*
+      ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ))*z^(-(1/1000:ℝ))+
+      Real.sqrt (z/(Fintype.card I:ℝ)) := by
+  let : Nonempty (ι ↪ Card d) := ⟨e₀⟩
+  apply DensityTransfer.multiplicity_amplification
+    (fun e f : ι ↪ Card d => PairRouting.tupleProbability (palindromePerm d) e f)
+    (fun e f => PairRouting.tupleProbability_nonneg _ e f)
+    (palindrome_kernel_symmetric d) (PairRouting.tupleProbability_row_sum _)
+    (1/1000) z _ (by norm_num) hz (Real.exp_nonneg _) _ v hv μ hvμ
+  intro e
+  exact palindrome_row_moment_bound d e hk
+
+end Thorp
+
+end ThorpNine.Harmonic
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Harmonic/PairAssembleEquiv.lean b/lean/OAI/Probability/ThorpRouting/Harmonic/PairAssembleEquiv.lean
index 9e7104213e9ec904b85fa1e09e170cd1d52cea43..fa34095949efdefaff728903add8ffbdb6f2d130 100644
--- a/lean/OAI/Probability/ThorpRouting/Harmonic/PairAssembleEquiv.lean
+++ b/lean/OAI/Probability/ThorpRouting/Harmonic/PairAssembleEquiv.lean
@@ -1,835 +1,835 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Harmonic.BlockProduct
-
-namespace OAI
-
-namespace ThorpNine.Harmonic
-
-namespace Thorp
-open scoped BigOperators Classical
-
-def pairAssembleEquiv (B C H : Type*) :
-    ((H × (B → C × C)) × H) ≃ (((B → C) × H) × ((B → C) × H)) where
-  toFun w := (((fun b => (w.1.2 b).1),w.1.1),((fun b => (w.1.2 b).2),w.2))
-  invFun w := ((w.1.2,fun b => (w.1.1 b,w.2.1 b)),w.2.2)
-  left_inv _w := rfl
-  right_inv _w := rfl
-
-def palindromeStageEquiv (r L : ℕ) :
-    ((OuterCoins r L × (Card r → BenesCoins L)) × OuterCoins r L) ≃ BenesCoins (L+r) :=
-  (pairAssembleEquiv _ _ _).trans
-    (Equiv.prodCongr (parallelSplitCoinEquiv r L) (parallelSplitCoinEquiv r L))
-
-lemma palindromeStageEquiv_apply (r L : ℕ)
-    (w : (OuterCoins r L × (Card r → BenesCoins L)) × OuterCoins r L) :
-    palindromeStageEquiv r L w=
-      (assembleBits r L (fun x => (w.1.2 x.1).1 x.2) w.1.1,
-       assembleBits r L (fun x => (w.1.2 x.1).2 x.2) w.2) := by
-  simp only [palindromeStageEquiv,Equiv.trans_apply,Equiv.prodCongr_apply,pairAssembleEquiv,
-    Equiv.coe_fn_mk,Prod.map_apply,parallelSplitCoinEquiv_apply]
-
-def modifiedAssembleEquiv (B C H U : Type*) :
-    ((H × (B → U × (C × C))) × H) ≃ (B → U) × (((B → C) × H) × ((B → C) × H)) where
-  toFun w := ((fun b => (w.1.2 b).1),
-    (((fun b => (w.1.2 b).2.1),w.1.1),((fun b => (w.1.2 b).2.2),w.2)))
-  invFun w := ((w.2.1.2,fun b => (w.1 b,(w.2.1.1 b,w.2.2.1 b))),w.2.2.2)
-  left_inv _w := rfl
-  right_inv _w := rfl
-
-def modifiedStageEquiv (r L : ℕ) :
-    ((OuterCoins r L × (Card r → Equiv.Perm (Card L) × BenesCoins L)) × OuterCoins r L) ≃
-      ModifiedCoins L r :=
-  (modifiedAssembleEquiv _ _ _ _).trans
-    (Equiv.prodCongr (Equiv.refl _) (Equiv.prodCongr (parallelSplitCoinEquiv r L) (parallelSplitCoinEquiv r L)))
-
-lemma modifiedStageEquiv_apply (r L : ℕ)
-    (w : (OuterCoins r L × (Card r → Equiv.Perm (Card L) × BenesCoins L)) × OuterCoins r L) :
-    modifiedStageEquiv r L w=
-      ((fun b => (w.1.2 b).1),
-       (assembleBits r L (fun x => (w.1.2 x.1).2.1 x.2) w.1.1,
-        assembleBits r L (fun x => (w.1.2 x.1).2.2 x.2) w.2)) := by
-  simp only [modifiedStageEquiv,Equiv.trans_apply,Equiv.prodCongr_apply,modifiedAssembleEquiv,
-    Equiv.coe_fn_mk,Prod.map_apply,Equiv.refl_apply,parallelSplitCoinEquiv_apply]
-
-lemma palindrome_stage (r L : ℕ)
-    (w : (OuterCoins r L × (Card r → BenesCoins L)) × OuterCoins r L) :
-    splitPerm r L (palindromePerm (L+r) (palindromeStageEquiv r L w))=
-      outerPerm r L w.1.1*ParallelBlocks.lift (fun b => palindromePerm L (w.1.2 b))*
-        (outerPerm r L w.2)⁻¹ := by
-  rw [palindromeStageEquiv_apply]
-  change splitPerm r L (_*_⁻¹)=_
-  rw [map_mul,map_inv,splitPerm_butterfly,splitPerm_butterfly,mul_inv_rev]
-  rw [mul_assoc,←mul_assoc (ParallelBlocks.lift _),←map_inv,←map_mul,←mul_assoc]
-  rfl
-
-noncomputable def localModifiedPerm (L : ℕ) (e : Bool)
-    (w : Equiv.Perm (Card L) × BenesCoins L) : Equiv.Perm (Card L) :=
-  if e then 1 else butterflyPerm L (decodeButterfly L w.2.1)*w.1⁻¹
-
-lemma modified_stage (r L : ℕ) (E : Finset (Card r))
-    (w : (OuterCoins r L × (Card r → Equiv.Perm (Card L) × BenesCoins L)) × OuterCoins r L) :
-    splitPerm r L (modifiedPerm L r E (modifiedStageEquiv r L w))=
-      outerPerm r L w.1.1*ParallelBlocks.lift (fun b => localModifiedPerm L (decide (b∈E)) (w.1.2 b))*
-        (outerPerm r L w.2)⁻¹ := by
-  rw [modifiedStageEquiv_apply]
-  unfold modifiedPerm modifiedInputTree
-  rw [map_mul,map_inv,splitPerm_butterfly,splitPerm_boundaryTree,mul_inv_rev]
-  rw [mul_assoc,←mul_assoc (ParallelBlocks.lift _),←map_inv,←map_mul,←mul_assoc]
-  congr 2
-  apply congrArg ParallelBlocks.lift
-  funext b
-  simp only [Pi.mul_apply,Pi.inv_apply,modifiedBoundary,boundaryOutput_split,localModifiedPerm]
-  by_cases h : b∈E <;> simp [h]
-
-end Thorp
-namespace Thorp.UnitaryFinite
-open scoped BigOperators Classical
-variable {G : Type*} [Group G]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-variable {Ω Ξ : Type*} [Fintype Ω] [Fintype Ξ]
-
-lemma sampleOperator_sandwich (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
-    (P : Ω → G) (Q : Ξ → G) :
-    sampleOperator ρ (fun w : (Ω × Ξ) × Ω => P w.1.1*Q w.1.2*(P w.2)⁻¹)=
-      sampleOperator ρ P*sampleOperator ρ Q*(sampleOperator ρ P).adjoint := by
-  rw [sampleOperator_prod ρ (fun w : Ω × Ξ => P w.1*Q w.2) (fun w : Ω => (P w)⁻¹),
-    sampleOperator_prod ρ P Q,sampleOperator_inv ρ hρ]
-
-lemma localModified_average (L : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card L)) V) (e : Bool) :
-    sampleOperator ρ (localModifiedPerm L e)=if e then 1 else uniformOperator ρ := by
-  cases e
-  · simp only [Bool.false_eq_true,ite_false]
-    rw [←sampleOperator_equiv ρ _ (Equiv.prodComm _ _)]
-    exact sampleOperator_uniform_mul ρ (fun w : BenesCoins L => butterflyPerm L (decodeButterfly L w.1))
-  · simp only [ite_true]
-    exact sampleOperator_one ρ
-
-end Thorp.UnitaryFinite
-
-namespace Thorp
-open scoped BigOperators Classical
-open UnitaryFinite
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-lemma sampleOperator_palindrome_stage (r L : ℕ)
-    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) (hρ : IsUnitary ρ) :
-    sampleOperator (ρ.comp (splitPerm r L).toMonoidHom) (palindromePerm (L+r))=
-      sampleOperator ρ (outerPerm r L)*
-        sampleOperator ρ (fun w : Card r → BenesCoins L => ParallelBlocks.lift (fun b => palindromePerm L (w b)))*
-          (sampleOperator ρ (outerPerm r L)).adjoint := by
-  have he : ((splitPerm r L).toMonoidHom ∘ palindromePerm (L+r)) ∘ palindromeStageEquiv r L=
-      fun w => outerPerm r L w.1.1*ParallelBlocks.lift (fun b => palindromePerm L (w.1.2 b))*
-        (outerPerm r L w.2)⁻¹ := funext (palindrome_stage r L)
-  calc
-    _ = sampleOperator ρ (((splitPerm r L).toMonoidHom ∘ palindromePerm (L+r)) ∘ palindromeStageEquiv r L) :=
-      (sampleOperator_equiv ρ ((splitPerm r L).toMonoidHom ∘ palindromePerm (L+r)) (palindromeStageEquiv r L)).symm
-    _ = _ := (congrArg (sampleOperator ρ) he).trans
-      (sampleOperator_sandwich ρ hρ (outerPerm r L)
-        (fun w : Card r → BenesCoins L => ParallelBlocks.lift (fun b => palindromePerm L (w b))))
-
-lemma sampleOperator_modified_stage (r L : ℕ) (E : Finset (Card r))
-    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) (hρ : IsUnitary ρ) :
-    sampleOperator (ρ.comp (splitPerm r L).toMonoidHom) (modifiedPerm L r E)=
-      sampleOperator ρ (outerPerm r L)*
-        sampleOperator ρ (fun w : Card r → Equiv.Perm (Card L) × BenesCoins L =>
-          ParallelBlocks.lift (fun b => localModifiedPerm L (decide (b∈E)) (w b)))*
-          (sampleOperator ρ (outerPerm r L)).adjoint := by
-  have he : ((splitPerm r L).toMonoidHom ∘ modifiedPerm L r E) ∘ modifiedStageEquiv r L=
-      fun w => outerPerm r L w.1.1*ParallelBlocks.lift
-        (fun b => localModifiedPerm L (decide (b∈E)) (w.1.2 b))*(outerPerm r L w.2)⁻¹ :=
-    funext (modified_stage r L E)
-  calc
-    _ = sampleOperator ρ (((splitPerm r L).toMonoidHom ∘ modifiedPerm L r E) ∘ modifiedStageEquiv r L) :=
-      (sampleOperator_equiv ρ ((splitPerm r L).toMonoidHom ∘ modifiedPerm L r E) (modifiedStageEquiv r L)).symm
-    _ = _ := (congrArg (sampleOperator ρ) he).trans
-      (sampleOperator_sandwich ρ hρ (outerPerm r L)
-        (fun w : Card r → Equiv.Perm (Card L) × BenesCoins L =>
-          ParallelBlocks.lift (fun b => localModifiedPerm L (decide (b∈E)) (w b))))
-
-end Thorp
-namespace Thorp.CommutingBlocks
-open scoped BigOperators Classical
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
-variable {ι : Type*} [Fintype ι] [DecidableEq ι]
-
-noncomputable def retain (P : ι → V →L[ℂ] V) (E : Finset ι) (i : ι) : V →L[ℂ] V :=
-  if i∈E then 1 else P i
-
-omit [CompleteSpace V] [Fintype ι] in
-lemma retain_commute (P : ι → V →L[ℂ] V) (hP : ∀ i j, Commute (P i) (P j))
-    (E : Finset ι) (i j : ι) : Commute (retain P E i) (retain P E j) := by
-  unfold retain
-  split_ifs <;> first | exact Commute.one_left _ | exact Commute.one_right _ | exact hP _ _
-
-omit [CompleteSpace V] in
-lemma blocks_retain (P : ι → V →L[ℂ] V) (hP : ∀ i j, Commute (P i) (P j)) (E : Finset ι) :
-    blocks (retain P E) (retain_commute P hP E)=coarse P hP E := by
-  have hs : E∪Eᶜ=(Finset.univ : Finset ι) := Finset.union_compl E
-  have hh := Finset.noncommProd_union_of_disjoint (s := E) (t := Eᶜ) disjoint_compl_right
-    (retain P E) (fun _ _ _ _ _ => retain_commute P hP E _ _)
-  simp only [hs] at hh
-  change blocks (retain P E) (retain_commute P hP E)=_ at hh
-  rw [hh]
-  have he : E.noncommProd (retain P E) (fun _ _ _ _ _ => retain_commute P hP E _ _)=1 := by
-    exact (Finset.noncommProd_eq_pow_card E (retain P E)
-      (fun i _ j _ _ => retain_commute P hP E i j) 1 (by intro i hi; exact ite_eq_left hi)).trans (one_pow _)
-  rw [he,one_mul]
-  apply Finset.noncommProd_congr rfl
-  intro i hi
-  exact ite_eq_right (Finset.mem_compl.mp hi)
-
-end Thorp.CommutingBlocks
-
-namespace Thorp.ParallelBlocks
-open scoped BigOperators Classical
-open UnitaryFinite
-variable {B X : Type*} [Fintype B] [Fintype X] [DecidableEq X]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-noncomputable def centralProjection (ρ : Representation ℂ (Equiv.Perm (B × X)) V) (b : B) : V →L[ℂ] V :=
-  uniformOperator (ρ.comp (injection b))
-
-omit [Fintype B] in
-lemma centralProjection_commute (ρ : Representation ℂ (Equiv.Perm (B × X)) V) (b c : B) :
-    Commute (centralProjection ρ b) (centralProjection ρ c) := by
-  simpa only [centralProjection,sampleOperator_uniform] using
-    local_average_commute ρ (fun _ => id : B → Equiv.Perm X → Equiv.Perm X) b c
-
-omit [Fintype B] in
-lemma local_central_commute {Ω : Type*} [Fintype Ω] [Nonempty Ω]
-    (ρ : Representation ℂ (Equiv.Perm (B × X)) V) (P : B → Ω → Equiv.Perm X) (b c : B) :
-    Commute (sampleOperator (ρ.comp (injection b)) (P b)) (centralProjection ρ c) := by
-  by_cases h : b=c
-  · subst c
-    change _*uniformOperator _=uniformOperator _*_
-    rw [sampleOperator_uniform_absorb,sampleOperator_weighted,weighted_uniform_right,sampleLaw_sum,one_smul]
-  · unfold centralProjection
-    rw [←sampleOperator_uniform,sampleOperator_comp,sampleOperator_comp]
-    exact sampleOperator_commute ρ _ _ (fun _ _ => injection_commute h _ _)
-
-end Thorp.ParallelBlocks
-namespace Thorp
-open scoped BigOperators Classical
-open UnitaryFinite ParallelBlocks CommutingBlocks
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-lemma modified_central_average (r L : ℕ) (E : Finset (Card r))
-    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) :
-    sampleOperator ρ (fun w : Card r → Equiv.Perm (Card L) × BenesCoins L =>
-      lift (fun b => localModifiedPerm L (decide (b∈E)) (w b)))=
-      coarse (centralProjection ρ) (centralProjection_commute ρ) E := by
-  rw [sampleOperator_parallel ρ (fun b => localModifiedPerm L (decide (b∈E))),←blocks_retain]
-  apply Finset.noncommProd_congr rfl
-  · intro b _
-    rw [localModified_average]
-    simp only [retain,centralProjection,decide_eq_true_eq]
-  · intro b _ c _ _
-    exact local_average_commute ρ (fun b => localModifiedPerm L (decide (b∈E))) b c
-
-lemma central_local_gap (r L : ℕ)
-    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) (hρ : IsUnitary ρ) (b : Card r) :
-    ‖sampleOperator (ρ.comp (injection b)) (palindromePerm L)*(1-centralProjection ρ b)‖≤blockGap L := by
-  change ‖sampleOperator _ _*(1-uniformOperator _)‖≤_
-  rw [mul_sub,mul_one,sampleOperator_uniform_absorb,sampleOperator_weighted]
-  exact palindrome_off_invariants_gap L _ (unitary_comp ρ hρ _)
-
-lemma central_blocks_exception_bound (r L : ℕ)
-    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) (hρ : IsUnitary ρ)
-    (B : V →L[ℂ] V) (hB0 : ‖B‖≤1) (n : ℕ) :
-    ‖B*blocks (fun b : Card r => sampleOperator (ρ.comp (injection b)) (palindromePerm L))
-      (local_average_commute ρ (fun _ => palindromePerm L))*B.adjoint‖ ≤
-      (blockGap L)^n + ∑ E : Finset (Card r), if E.card<n then
-        ‖B*coarse (centralProjection ρ) (centralProjection_commute ρ) E*B.adjoint‖ else 0 := by
-  have hB : ‖B.adjoint‖≤1 := by
-    rw [ContinuousLinearMap.adjoint.norm_map]
-    exact hB0
-  have h := blocks_exception_bound (centralProjection ρ) (centralProjection_commute ρ)
-    (fun b => uniformOperator_projection (ρ.comp (injection b)) (unitary_comp ρ hρ _))
-    (fun b : Card r => sampleOperator (ρ.comp (injection b)) (palindromePerm L))
-    (local_average_commute ρ (fun _ => palindromePerm L))
-    (local_central_commute ρ (fun _ => palindromePerm L))
-    (fun b => sampleOperator_norm_le (ρ.comp (injection b)) (unitary_comp ρ hρ _) (palindromePerm L))
-    (blockGap L) (blockGap_pos L).le (blockGap_lt_one L).le (central_local_gap r L ρ hρ)
-    B.adjoint hB n
-  simpa only [ContinuousLinearMap.adjoint_adjoint] using h
-
-theorem palindrome_exception_bound (r L : ℕ)
-    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) (hρ : IsUnitary ρ) (n : ℕ) :
-    ‖sampleOperator (ρ.comp (splitPerm r L).toMonoidHom) (palindromePerm (L+r))‖ ≤
-      (blockGap L)^n + ∑ E : Finset (Card r), if E.card<n then
-        ‖sampleOperator (ρ.comp (splitPerm r L).toMonoidHom) (modifiedPerm L r E)‖ else 0 := by
-  rw [sampleOperator_palindrome_stage r L ρ hρ]
-  rw [sampleOperator_parallel ρ (fun _ => palindromePerm L)]
-  apply (central_blocks_exception_bound r L ρ hρ (sampleOperator ρ (outerPerm r L))
-    (sampleOperator_norm_le ρ hρ _) n).trans_eq
-  congr 1
-  apply Finset.sum_congr rfl
-  intro E _
-  split_ifs
-  · rw [sampleOperator_modified_stage r L E ρ hρ,modified_central_average]
-  · rfl
-
-end Thorp
-
-namespace Thorp.UnitaryFinite
-open scoped BigOperators ComplexConjugate Classical
-
-variable {G X Ω : Type*} [Group G] [Fintype G] [Fintype X] [Fintype Ω]
-
-noncomputable def actionKernel (a : G →* Equiv.Perm X) (P : Ω → G) (x y : X) : ℝ :=
-  finiteMean (fun ω => if a (P ω) x=y then 1 else 0)
-
-omit [Fintype G] in
-lemma actionKernel_apply (a : G →* Equiv.Perm X) (P : Ω → G)
-    (v : EuclideanSpace ℂ X) (x : X) :
-    DensityTransfer.applyKernel (actionKernel a P) v x =
-      (Fintype.card Ω:ℂ)⁻¹ * ∑ ω, v (a (P ω) x) := by
-  simp only [DensityTransfer.applyKernel_apply,actionKernel,finiteMean,
-    Complex.ofReal_sum,div_eq_mul_inv,Finset.sum_mul]
-  rw [Finset.sum_comm]
-  calc
-    _ = ∑ ω, (Fintype.card Ω:ℂ)⁻¹ * ∑ y : X, if a (P ω) x=y then v y else 0 := by
-      apply Finset.sum_congr rfl
-      intro ω _
-      rw [Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro y _
-      split_ifs <;> simp_all
-    _ = _ := by simp [Finset.mul_sum]
-
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-noncomputable def randomOperator (ρ : Representation ℂ G V) (P : Ω → G) : Module.End ℂ V :=
-  (Fintype.card Ω:ℂ)⁻¹ • ∑ ω, ρ (P ω)⁻¹
-
-omit [Fintype G] [FiniteDimensional ℂ V] in
-lemma coefficient_random (a : G →* Equiv.Perm X) (x₀ : X)
-    (ha : ∀ x, ∃ g, a g x₀=x) (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
-    (P : Ω → G) (w : V) (hw : IsFixed a x₀ ρ w) (v : V) :
-    DensityTransfer.applyKernel (actionKernel a P) (normalizedCoefficient a x₀ ha ρ w v) =
-      normalizedCoefficient a x₀ ha ρ w (randomOperator ρ P v) := by
-  ext x
-  rw [actionKernel_apply]
-  simp only [normalizedCoefficient,PiLp.smul_apply,smul_eq_mul,Finset.mul_sum,
-    coefficient,randomOperator,LinearMap.smul_apply,LinearMap.sum_apply,
-    inner_smul_right,inner_sum]
-  apply Finset.sum_congr rfl
-  intro ω _
-  have h := coefficient_action a x₀ ha ρ hρ w hw v (P ω) x
-  change inner ℂ (ρ (representative a x₀ ha (a (P ω) x)) w) v =
-    inner ℂ (ρ (representative a x₀ ha x) w) (ρ (P ω)⁻¹ v) at h
-  rw [h]
-  ring
-
-noncomputable def fixedSpace (a : G →* Equiv.Perm X) (x₀ : X) (ρ : Representation ℂ G V) :
-    Submodule ℂ V where
-  carrier := {w | IsFixed a x₀ ρ w}
-  zero_mem' := fun g hg => map_zero _
-  add_mem' := fun hw hz g hg => by rw [map_add,hw g hg,hz g hg]
-  smul_mem' := fun c w hw g hg => by rw [map_smul,hw g hg]
-
-theorem operator_amplification (a : G →* Equiv.Perm X) (x₀ : X)
-    (ha : ∀ x, ∃ g, a g x₀=x) (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
-    (hρ : IsUnitary ρ) (P : Ω → G)
-    (hsym : ∀ x y, actionKernel a P x y=actionKernel a P y x)
-    (hrow : ∀ x, ∑ y, actionKernel a P x y=1)
-    (b z M : ℝ) (hb : 0≤b) (hz : 0<z) (hM0 : 0≤M)
-    (hM : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*actionKernel a P x y)^(1+b)) ≤ M)
-    (hm : 0 < Module.finrank ℂ (fixedSpace a x₀ ρ)) (v : V) (hv : ‖v‖=1) :
-    ‖randomOperator ρ P v‖ ≤ M*z^(-b)+Real.sqrt (z/(Module.finrank ℂ (fixedSpace a x₀ ρ):ℝ)) := by
-  let K := fixedSpace a x₀ ρ
-  let m := Module.finrank ℂ K
-  let : Nonempty X := ⟨x₀⟩
-  let : Nonempty (Fin m) := ⟨⟨0,hm⟩⟩
-  let B := stdOrthonormalBasis ℂ K
-  let w : Fin m → V := fun i => (B i : V)
-  have hfix (i : Fin m) : IsFixed a x₀ ρ (w i) := (B i).2
-  have hw : Orthonormal ℂ w := B.orthonormal.comp_linearIsometry K.subtypeₗᵢ
-  have hc := normalizedCoefficient_orthonormal a x₀ ha ρ hρ w hw hfix v hv
-  have hn (i : Fin m) : ‖DensityTransfer.applyKernel (actionKernel a P)
-      (normalizedCoefficient a x₀ ha ρ (w i) v)‖ = ‖randomOperator ρ P v‖ := by
-    rw [coefficient_random a x₀ ha ρ hρ P (w i) (hfix i),
-      normalizedCoefficient_norm a x₀ ha ρ hρ (w i) (hfix i) (hw.norm_eq_one i)]
-  have h := DensityTransfer.multiplicity_amplification (actionKernel a P)
-    (fun x y => finiteMean_nonneg (fun _ => by split_ifs <;> norm_num)) hsym hrow
-    b z M hb hz hM0 hM _ hc (‖randomOperator ρ P v‖) (fun i => (hn i).ge)
-  simpa only [Fintype.card_fin] using h
-
-end Thorp.UnitaryFinite
-namespace Thorp.UnitaryFinite
-open scoped BigOperators Classical
-
-variable {α ι : Type*} [Fintype α] [Fintype ι]
-
-def tupleAction : Equiv.Perm α →* Equiv.Perm (ι ↪ α) where
-  toFun p :=
-    { toFun := fun e => e.trans p.toEmbedding
-      invFun := fun e => e.trans p.symm.toEmbedding
-      left_inv := fun e => by ext i; exact p.symm_apply_apply (e i)
-      right_inv := fun e => by ext i; exact p.apply_symm_apply (e i) }
-  map_one' := by ext e i; rfl
-  map_mul' p q := by ext e i; rfl
-
-omit [Fintype α] [Fintype ι] in
-@[simp] lemma tupleAction_apply (p : Equiv.Perm α) (e : ι ↪ α) (i : ι) :
-    tupleAction p e i=p (e i) := rfl
-
-omit [Fintype α] in
-lemma tupleAction_transitive (e f : ι ↪ α) : ∃ p, tupleAction p e=f := by
-  obtain ⟨p,hp⟩ := Equiv.Perm.exists_extending_pair e f e.injective f.injective
-  exact ⟨p,Function.Embedding.ext hp⟩
-
-omit [Fintype α] in
-lemma tupleAction_fixed_iff [DecidableEq α] (e : ι ↪ α) (p : Equiv.Perm α) :
-    tupleAction p e=e ↔ ∀ x ∈ Finset.univ.image e, p x=x := by
-  constructor
-  · intro h x hx
-    obtain ⟨i,_,rfl⟩ := Finset.mem_image.mp hx
-    exact congrArg (fun f : ι ↪ α => f i) h
-  · intro h
-    ext i
-    exact h (e i) (Finset.mem_image.mpr ⟨i,Finset.mem_univ _,rfl⟩)
-
-omit [Fintype α] in
-lemma tupleAction_kernel [DecidableEq α] {Ω : Type*} [Fintype Ω] (P : Ω → Equiv.Perm α)
-    (e f : ι ↪ α) : actionKernel tupleAction P e f=PairRouting.tupleProbability P e f := by
-  unfold actionKernel PairRouting.tupleProbability
-  congr 1
-  funext ω
-  congr 1
-  exact propext Function.Embedding.ext_iff
-
-end Thorp.UnitaryFinite
-namespace Thorp.UnitaryFinite
-open scoped BigOperators Classical
-
-lemma irreducible_of_equiv {G V W : Type*} [Group G]
-    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
-    (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
-    (e : Representation.Equiv ρ σ) [Representation.IsIrreducible ρ] :
-    Representation.IsIrreducible σ := by
-  rw [Representation.irreducible_iff_isSimpleModule_asModule]
-  have hi := (Representation.irreducible_iff_isSimpleModule_asModule ρ).mp inferInstance
-  let f : ρ.asModule →ₗ[MonoidAlgebra ℂ G] σ.asModule :=
-    Representation.IntertwiningMap.equivLinearMapAsModule ρ σ e.toIntertwiningMap
-  have hf : Function.Bijective f := e.toLinearEquiv.bijective
-  exact (f.isSimpleModule_iff_of_bijective hf).mp hi
-
-end Thorp.UnitaryFinite
-
-namespace Thorp.Specht
-open scoped BigOperators Classical
-
-lemma unitaryRepresentation_coe (μ : YoungDiagram) (p : Equiv.Perm (Cell μ))
-    (v : hilbertSpace μ) :
-    (unitaryRepresentation μ p v : PermutationHilbert.H (Tabloid μ)) =
-      PermutationHilbert.act (tabloidAct p) v := by
-  obtain ⟨w,rfl⟩ := (spaceHilbertEquiv μ).surjective v
-  change hilbertEquiv μ (tabloidRep μ p w) = _
-  exact hilbert_action μ p w
-
-noncomputable def unitaryEquiv (μ : YoungDiagram) :
-    Representation.Equiv (representation μ) (unitaryRepresentation μ) :=
-  Representation.Equiv.mk (spaceHilbertEquiv μ) (by
-    intro p
-    apply LinearMap.ext
-    intro v
-    change spaceHilbertEquiv μ (representation μ p v) =
-      spaceHilbertEquiv μ (representation μ p ((spaceHilbertEquiv μ).symm (spaceHilbertEquiv μ v)))
-    rw [LinearEquiv.symm_apply_apply])
-
-lemma unitaryRepresentation_irreducible (μ : YoungDiagram) :
-    Representation.IsIrreducible (unitaryRepresentation μ) := by
-  let := representation_irreducible μ
-  exact UnitaryFinite.irreducible_of_equiv _ _ (unitaryEquiv μ)
-
-lemma unitaryRepresentation_unitary (μ : YoungDiagram) :
-    UnitaryFinite.IsUnitary (G := Equiv.Perm (Cell μ)) (V := hilbertSpace μ) (unitaryRepresentation μ) := by
-  intro p v w
-  change inner ℂ (unitaryRepresentation μ p v : PermutationHilbert.H (Tabloid μ))
-      (unitaryRepresentation μ p w : PermutationHilbert.H (Tabloid μ)) = _
-  rw [unitaryRepresentation_coe,unitaryRepresentation_coe]
-  exact (PermutationHilbert.act (tabloidAct p)).inner_map_map v w
-
-
-noncomputable def tupleFixedEquiv (μ : YoungDiagram) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Cell μ) :
-    UnitaryFinite.fixedSpace (V := hilbertSpace μ) UnitaryFinite.tupleAction e (unitaryRepresentation μ) ≃ₗ[ℂ]
-      fixedHilbert μ (Finset.univ.image e) where
-  toFun v := ⟨v.1.1,v.1.2,by
-    intro p hp
-    have h := v.2 p ((UnitaryFinite.tupleAction_fixed_iff e p).mpr hp)
-    exact (unitaryRepresentation_coe μ p v.1).symm.trans (congrArg Subtype.val h)⟩
-  invFun v := ⟨⟨v.1,v.2.1⟩,by
-    intro p hp
-    apply Subtype.ext
-    exact (unitaryRepresentation_coe μ p _).trans
-      (v.2.2 p ((UnitaryFinite.tupleAction_fixed_iff e p).mp hp))⟩
-  left_inv _ := rfl
-  right_inv _ := rfl
-  map_add' _ _ := rfl
-  map_smul' _ _ := rfl
-
-theorem tupleFixed_dimension (μ : YoungDiagram) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Cell μ) :
-    Module.finrank ℂ (hilbertSpace μ) * (Fintype.card ι).choose (μ.card-μ.rowLen 0) ≤
-      Module.finrank ℂ (UnitaryFinite.fixedSpace (V := hilbertSpace μ) UnitaryFinite.tupleAction e (unitaryRepresentation μ)) *
-        μ.card.choose (μ.card-μ.rowLen 0) := by
-  rw [(tupleFixedEquiv μ e).finrank_eq,finrank_hilbertSpace]
-  have h := fixedHilbert_dimension μ (Finset.univ.image e)
-  simpa only [Finset.card_image_of_injective _ e.injective,Finset.card_univ] using h
-
-end Thorp.Specht
-namespace Thorp.UnitaryFinite
-open scoped BigOperators Classical
-
-variable {G H X Ω : Type*} [Group G] [Group H] [Fintype G] [Fintype X] [Fintype Ω]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-omit [Fintype G] in
-lemma sampleOperator_inv_apply (ρ : Representation ℂ G V) (P : Ω → G) (v : V) :
-    sampleOperator ρ (fun ω => (P ω)⁻¹) v=randomOperator ρ P v := by
-  simp only [sampleOperator,randomOperator,smul_apply,
-    sum_apply,continuousRepresentation_apply,LinearMap.smul_apply,LinearMap.sum_apply]
-
-theorem sampleOperator_amplification (a : G →* Equiv.Perm X) (x₀ : X)
-    (ha : ∀ x, ∃ g, a g x₀=x) (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
-    (hρ : IsUnitary ρ) (P : Ω → G)
-    (hsym : ∀ x y, actionKernel a P x y=actionKernel a P y x)
-    (hrow : ∀ x, ∑ y, actionKernel a P x y=1)
-    (b z M : ℝ) (hb : 0≤b) (hz : 0<z) (hM0 : 0≤M)
-    (hM : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*actionKernel a P x y)^(1+b)) ≤ M)
-    (hm : 0 < Module.finrank ℂ (fixedSpace a x₀ ρ)) :
-    ‖sampleOperator ρ P‖ ≤ M*z^(-b)+Real.sqrt (z/(Module.finrank ℂ (fixedSpace a x₀ ρ):ℝ)) := by
-  rw [←ContinuousLinearMap.adjoint.norm_map (sampleOperator ρ P),←sampleOperator_inv ρ hρ]
-  apply ContinuousLinearMap.opNorm_le_of_unit_norm (by positivity)
-  intro v hv
-  rw [sampleOperator_inv_apply]
-  exact operator_amplification a x₀ ha ρ hρ P hsym hrow b z M hb hz hM0 hM hm v hv
-
-omit [Fintype G] [Fintype X] [Fintype Ω] [FiniteDimensional ℂ V] in
-noncomputable def subrepresentationEquiv (ρ : Representation ℂ G V) (e : H ≃* G) :
-    Subrepresentation (ρ.comp e.toMonoidHom) ≃o Subrepresentation ρ where
-  toFun U :=
-    { toSubmodule := U.toSubmodule
-      apply_mem_toSubmodule := fun g v hv => by
-        have h := U.apply_mem_toSubmodule (e.symm g) hv
-        change ρ (e (e.symm g)) v∈U.toSubmodule at h
-        simpa only [e.apply_symm_apply] using h }
-  invFun U :=
-    { toSubmodule := U.toSubmodule
-      apply_mem_toSubmodule := fun h _ hv => U.apply_mem_toSubmodule (e h) hv }
-  left_inv U := by apply Subrepresentation.toSubmodule_injective; rfl
-  right_inv U := by apply Subrepresentation.toSubmodule_injective; rfl
-  map_rel_iff' := Iff.rfl
-
-omit [Fintype G] [Fintype X] [Fintype Ω] [FiniteDimensional ℂ V] in
-lemma irreducible_comp_equiv (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
-    (e : H ≃* G) : Representation.IsIrreducible (ρ.comp e.toMonoidHom) := by
-  exact (subrepresentationEquiv ρ e).isSimpleOrder_iff.mpr inferInstance
-
-end Thorp.UnitaryFinite
-
-namespace Thorp.Specht
-open scoped BigOperators Classical
-
-variable {α ι : Type*} [Fintype α] [Fintype ι]
-
-omit [Fintype α] in
-lemma relabelledUnitary_unitary (μ : YoungDiagram) (e : α ≃ Cell μ) :
-    UnitaryFinite.IsUnitary (V := hilbertSpace μ) (relabelledUnitary μ e) := by
-  intro g v w
-  exact unitaryRepresentation_unitary μ (e.permCongr g) v w
-
-omit [Fintype α] in
-lemma relabelledUnitary_irreducible (μ : YoungDiagram) (e : α ≃ Cell μ) :
-    Representation.IsIrreducible (relabelledUnitary μ e) := by
-  let := unitaryRepresentation_irreducible μ
-  exact UnitaryFinite.irreducible_comp_equiv (V := hilbertSpace μ) (unitaryRepresentation μ) e.permCongrHom
-
-noncomputable def relabelledTupleFixed (μ : YoungDiagram) (e : α ≃ Cell μ) (f : ι ↪ α) :
-    UnitaryFinite.fixedSpace (V := hilbertSpace μ) UnitaryFinite.tupleAction f (relabelledUnitary μ e) ≃ₗ[ℂ]
-    UnitaryFinite.fixedSpace (V := hilbertSpace μ) UnitaryFinite.tupleAction (f.trans e.toEmbedding) (unitaryRepresentation μ) where
-  toFun v := ⟨v.1,by
-    intro g hg
-    have hp : UnitaryFinite.tupleAction (e.symm.permCongr g) f=f := by
-      apply Function.Embedding.ext
-      intro i
-      have hh := congrArg (fun f : ι ↪ Cell μ => f i) hg
-      change g (e (f i))=e (f i) at hh
-      simp only [UnitaryFinite.tupleAction_apply,Equiv.permCongr_apply,Equiv.symm_symm,e.symm_apply_apply,hh]
-    have hh := v.2 (e.symm.permCongr g) hp
-    change unitaryRepresentation μ (e.permCongr (e.symm.permCongr g)) v=v at hh
-    have he : e.permCongr (e.symm.permCongr g)=g := by
-      apply Equiv.ext
-      intro x
-      simp only [Equiv.permCongr_apply,Equiv.symm_symm,e.apply_symm_apply]
-    simpa only [he] using hh⟩
-  invFun v := ⟨v.1,by
-    intro g hg
-    have hp : UnitaryFinite.tupleAction (e.permCongr g) (f.trans e.toEmbedding)=f.trans e.toEmbedding := by
-      apply Function.Embedding.ext
-      intro i
-      have hh := congrArg (fun f : ι ↪ α => f i) hg
-      change g (f i)=f i at hh
-      simp only [UnitaryFinite.tupleAction_apply,Function.Embedding.trans_apply,
-        Equiv.toEmbedding_apply,Equiv.permCongr_apply,e.symm_apply_apply,hh]
-    exact v.2 (e.permCongr g) hp⟩
-  left_inv _ := rfl
-  right_inv _ := rfl
-  map_add' _ _ := rfl
-  map_smul' _ _ := rfl
-
-omit [Fintype α] in
-lemma relabelled_tuple_dimension (μ : YoungDiagram) (e : α ≃ Cell μ) (f : ι ↪ α) :
-    Module.finrank ℂ (hilbertSpace μ) * (Fintype.card ι).choose (μ.card-μ.rowLen 0) ≤
-      Module.finrank ℂ (UnitaryFinite.fixedSpace (V := hilbertSpace μ) UnitaryFinite.tupleAction f (relabelledUnitary μ e)) *
-        μ.card.choose (μ.card-μ.rowLen 0) := by
-  rw [(relabelledTupleFixed μ e f).finrank_eq]
-  exact tupleFixed_dimension μ (f.trans e.toEmbedding)
-
-end Thorp.Specht
-
-namespace Thorp.SparseContact
-open scoped BigOperators Classical
-
-noncomputable def pointStabilizer {α : Type*} (A : Finset α) : Subgroup (Equiv.Perm α) where
-  carrier := {p | ∀ x ∈ A, p x = x}
-  one_mem' := by intro x _; rfl
-  mul_mem' := by intro p q hp hq x hx; change p (q x) = x; rw [hq x hx, hp x hx]
-  inv_mem' := by intro p hp x hx; exact (Equiv.Perm.inv_eq_iff_eq).mpr (hp x hx).symm
-
-lemma swap_mem_pointStabilizer {α : Type*} [DecidableEq α] (A : Finset α)
-    (x y : α) (hx : x ∉ A) (hy : y ∉ A) : Equiv.swap x y ∈ pointStabilizer A := by
-  intro z hz
-  exact Equiv.swap_apply_of_ne_of_ne (fun h => hx (h ▸ hz)) (fun h => hy (h ▸ hz))
-
-lemma point_symmetrizer_zero (μ : YoungDiagram) (A : Finset (Specht.Cell μ))
-    (hA : A.card < μ.card - μ.rowLen 0) (v : Specht.Tabloid μ → ℂ) (hv : v ∈ Specht.space μ) :
-    Specht.symmetrizer (pointStabilizer A) (Specht.tabloidRep μ) v = 0 := by
-  classical
-  induction hv using Submodule.span_induction with
-  | mem v hv =>
-    obtain ⟨p,rfl⟩ := hv
-    let e : {x : Specht.Cell μ // p x ∉ A} ≃ {x : Specht.Cell μ // x ∉ A} :=
-      { toFun := fun x => ⟨p x.1,x.2⟩
-        invFun := fun x => ⟨p.symm x.1, by simpa only [p.apply_symm_apply] using x.2⟩
-        left_inv := fun x => by apply Subtype.ext; exact p.symm_apply_apply _
-        right_inv := fun x => by apply Subtype.ext; exact p.apply_symm_apply _ }
-    have hc : μ.rowLen 0 < Fintype.card {x : Specht.Cell μ // p x ∉ A} := by
-      rw [Fintype.card_congr e, Fintype.card_subtype_compl, Specht.card_cell, Fintype.card_coe]
-      have hr := Specht.rowLen_zero_le_card μ
-      omega
-    obtain ⟨x,y,hne,hcol⟩ := Fintype.exists_ne_map_eq_of_card_lt
-      (fun x : {x : Specht.Cell μ // p x ∉ A} => Specht.colIndex x.1)
-      (by simpa only [Fintype.card_fin] using hc)
-    have hxy : x.1 ≠ y.1 := fun h => hne (Subtype.ext h)
-    let h : pointStabilizer A := ⟨Equiv.swap (p x.1) (p y.1),
-      swap_mem_pointStabilizer A _ _ x.2 y.2⟩
-    apply Specht.symmetrizer_eq_zero_of_neg _ _ h
-    change Specht.tabloidRep μ (Equiv.swap (p x.1) (p y.1))
-      (Specht.tabloidRep μ p (Specht.polytabloid μ)) = -Specht.tabloidRep μ p (Specht.polytabloid μ)
-    rw [←Module.End.mul_apply, ←map_mul, ←Equiv.mul_swap_eq_swap_mul]
-    rw [map_mul, Module.End.mul_apply,
-      Specht.swap_polytabloid x.1 y.1 hxy (congrArg Fin.val hcol), map_neg]
-  | zero => exact map_zero _
-  | add v w hv hw ihv ihw => simp only [map_add,ihv,ihw,add_zero]
-  | smul a v hv ih => simp only [map_smul,ih,smul_zero]
-
-lemma point_fixed_zero (μ : YoungDiagram) (A : Finset (Specht.Cell μ))
-    (hA : A.card < μ.card - μ.rowLen 0) (v : Specht.Tabloid μ → ℂ) (hv : v ∈ Specht.space μ)
-    (hfix : ∀ p, p ∈ pointStabilizer A → Specht.tabloidRep μ p v = v) : v = 0 := by
-  classical
-  have hz := point_symmetrizer_zero μ A hA v hv
-  have he : Specht.symmetrizer (pointStabilizer A) (Specht.tabloidRep μ) v =
-      (Fintype.card (pointStabilizer A) : ℂ) • v := by
-    simp only [Specht.symmetrizer, LinearMap.sum_apply, hfix _ (Subtype.prop _),
-      Finset.sum_const, Finset.card_univ, nsmul_eq_mul, Nat.cast_smul_eq_nsmul]
-  rw [he] at hz
-  exact (smul_eq_zero.mp hz).resolve_left (by exact_mod_cast Fintype.card_ne_zero)
-
-lemma point_unitary_fixed_zero (μ : YoungDiagram) (A : Finset (Specht.Cell μ))
-    (hA : A.card < μ.card - μ.rowLen 0) (v : Specht.hilbertSpace μ)
-    (hfix : ∀ p, p ∈ pointStabilizer A → Specht.unitaryRepresentation μ p v = v) : v = 0 := by
-  let w := (Specht.spaceHilbertEquiv μ).symm v
-  have hw : ∀ p, p ∈ pointStabilizer A → Specht.tabloidRep μ p w.val = w.val := by
-    intro p hp
-    have he := congrArg ((Specht.spaceHilbertEquiv μ).symm) (hfix p hp)
-    change Specht.representation μ p w = w at he
-    exact congrArg Subtype.val he
-  have hz := point_fixed_zero μ A hA w.val w.prop hw
-  have hw0 : w = 0 := Subtype.ext hz
-  have he := congrArg (Specht.spaceHilbertEquiv μ) hw0
-  simpa only [w, LinearEquiv.apply_symm_apply, map_zero] using he
-
-lemma point_relabelled_fixed_zero {α : Type*} [Fintype α] (μ : YoungDiagram)
-    (e : α ≃ Specht.Cell μ) (A : Finset α) (hA : A.card < μ.card - μ.rowLen 0)
-    (v : Specht.hilbertSpace μ)
-    (hfix : ∀ p, p ∈ pointStabilizer A → Specht.relabelledUnitary μ e p v = v) : v = 0 := by
-  classical
-  apply point_unitary_fixed_zero μ (A.image e) (by simpa only [Finset.card_image_of_injective _ e.injective] using hA) v
-  intro p hp
-  have hf : e.symm.permCongr p ∈ pointStabilizer A := by
-    intro x hx
-    have hh := hp (e x) (Finset.mem_image.mpr ⟨x,hx,rfl⟩)
-    simp only [Equiv.permCongr_apply, Equiv.symm_symm, hh, e.symm_apply_apply]
-  have hh := hfix _ hf
-  change Specht.unitaryRepresentation μ (e.permCongr (e.symm.permCongr p)) v = v at hh
-  have he : e.permCongr (e.symm.permCongr p) = p := by
-    apply Equiv.ext
-    intro x
-    simp only [Equiv.permCongr_apply, Equiv.symm_symm, e.apply_symm_apply]
-  simpa only [he] using hh
-
-
-lemma small_dominating_set {V : Type*} [Fintype V] (G : SimpleGraph V)
-    (hG : ∀ v, ∃ w, G.Adj v w) :
-    ∃ R : Finset V, 2 * R.card ≤ Fintype.card V ∧
-      ∀ v, v ∉ R → ∃ w ∈ R, G.Adj v w := by
-  classical
-  obtain ⟨F, hFG, hF, hreach⟩ := G.exists_isAcyclic_reachable_eq_le
-  let c := hF.coloringTwo
-  have hneigh (v : V) : ∃ w, F.Adj v w := by
-    obtain ⟨w, hvw⟩ := hG v
-    have hr : F.Reachable v w := by rw [hreach]; exact hvw.reachable
-    exact hr.nonempty_neighborSet_left hvw.ne
-  let A : Finset V := Finset.univ.filter (fun v => c v = 0)
-  let B : Finset V := Finset.univ.filter (fun v => c v ≠ 0)
-  have hab : A.card + B.card = Fintype.card V := by
-    exact Finset.card_filter_add_card_filter_not _
-  have hcolor {v w : V} (hvw : F.Adj v w) : c v ≠ c w := c.valid hvw
-  by_cases h : A.card ≤ B.card
-  · refine ⟨A, by omega, ?_⟩
-    intro v hv
-    obtain ⟨w, hvw⟩ := hneigh v
-    have hv0 : c v ≠ 0 := by simpa [A] using hv
-    have hw0 : c w = 0 := by
-      have hv1 : c v = 1 := by omega
-      have hn := hcolor hvw
-      omega
-    exact ⟨w, by simp [A, hw0], hFG hvw⟩
-  · refine ⟨B, by omega, ?_⟩
-    intro v hv
-    obtain ⟨w, hvw⟩ := hneigh v
-    have hv0 : c v = 0 := by simpa [B] using hv
-    have hw0 : c w ≠ 0 := by simpa [hv0] using (hcolor hvw).symm
-    exact ⟨w, by simp [B, hw0], hFG hvw⟩
-
-lemma powerset_weight_le_exp {V : Type*} [DecidableEq V] (S : Finset V)
-    (p : V → ℝ) (hp : ∀ v, 0 ≤ p v) :
-    (∑ L ∈ S.powerset, ∏ v ∈ L, p v) ≤ Real.exp (∑ v ∈ S, p v) := by
-  rw [← Finset.prod_one_add, Real.exp_sum]
-  apply Finset.prod_le_prod₀
-  · intro v _; exact add_nonneg (by norm_num) (hp v)
-  · intro v _
-    have h := Real.add_one_le_exp (p v)
-    linarith
-
-lemma fixed_size_weight_le {V : Type*} [DecidableEq V] (S : Finset V)
-    (p : V → ℝ) (hp : ∀ v, 0 ≤ p v) (m : ℕ) (δ : ℝ) (hδ : 0 < δ) :
-    (∑ L ∈ S.powersetCard m, ∏ v ∈ L, p v) ≤
-      δ^m * Real.exp ((∑ v ∈ S, p v)/δ) := by
-  have hsplit (L : Finset V) (hL : L ∈ S.powersetCard m) :
-      (∏ v ∈ L, p v) = δ^m * ∏ v ∈ L, (p v / δ) := by
-    rw [Finset.prod_div_distrib, Finset.prod_const, (Finset.mem_powersetCard.mp hL).2]
-    field_simp
-  calc
-    _ = δ^m * ∑ L ∈ S.powersetCard m, ∏ v ∈ L, (p v/δ) := by
-      rw [Finset.mul_sum]
-      exact Finset.sum_congr rfl hsplit
-    _ ≤ δ^m * ∑ L ∈ S.powerset, ∏ v ∈ L, (p v/δ) := by
-      apply mul_le_mul_of_nonneg_left _ (pow_nonneg hδ.le _)
-      apply Finset.sum_le_sum_of_subset_of_nonneg
-      · intro L hL
-        exact Finset.mem_powerset.mpr (Finset.mem_powersetCard.mp hL).1
-      · intro L _ _; exact Finset.prod_nonneg (fun v _ => div_nonneg (hp v) hδ.le)
-    _ ≤ δ^m * Real.exp (∑ v ∈ S, p v/δ) :=
-      mul_le_mul_of_nonneg_left
-        (powerset_weight_le_exp S (fun v => p v/δ) (fun v => div_nonneg (hp v) hδ.le))
-        (pow_nonneg hδ.le _)
-    _ = _ := by rw [Finset.sum_div]
-
-noncomputable def neighborhood {V : Type*} [Fintype V] (G : SimpleGraph V)
-    (R : Finset V) : Finset V := by
-  classical
-  exact Finset.univ.filter (fun w => ∃ v ∈ R, G.Adj v w)
-
-lemma neighborhood_weight_le {V : Type*} [Fintype V] (G : SimpleGraph V)
-    (p : V → ℝ) (hp : ∀ v, 0 ≤ p v) (δ : ℝ)
-    (hdeg : ∀ v, (∑ w, if G.Adj v w then p w else 0) ≤ δ) (R : Finset V) :
-    (∑ w ∈ neighborhood G R, p w) ≤ R.card * δ := by
-  classical
-  calc
-    _ ≤ ∑ w ∈ neighborhood G R, ∑ v ∈ R, if G.Adj v w then p w else 0 := by
-      apply Finset.sum_le_sum
-      intro w hw
-      obtain ⟨v, hv, hvw⟩ := (Finset.mem_filter.mp hw).2
-      have h : (if G.Adj v w then p w else 0) ≤
-          ∑ u ∈ R, if G.Adj u w then p w else 0 :=
-        Finset.single_le_sum (f := fun u => if G.Adj u w then p w else 0) (fun u _ => by split_ifs; exact hp w; exact le_rfl) hv
-      simpa only [ite_eq_left hvw] using h
-    _ ≤ ∑ w, ∑ v ∈ R, if G.Adj v w then p w else 0 := by
-      apply Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _)
-      intro w _ _
-      exact Finset.sum_nonneg (fun v _ => by split_ifs; exact hp w; exact le_rfl)
-    _ = ∑ v ∈ R, ∑ w, if G.Adj v w then p w else 0 := Finset.sum_comm
-    _ ≤ ∑ v ∈ R, δ := Finset.sum_le_sum (fun v _ => hdeg v)
-    _ = _ := by simp
-
-lemma root_leaves_weight_le {V : Type*} [Fintype V] (G : SimpleGraph V)
-    (p : V → ℝ) (hp : ∀ v, 0 ≤ p v) (k : ℕ) (δ : ℝ)
-    (hδ : 0 < δ) (hδ1 : δ ≤ 1)
-    (hdeg : ∀ v, (∑ w, if G.Adj v w then p w else 0) ≤ δ)
-    (R : Finset V) (hR : 2 * R.card ≤ k) :
-    (∑ L ∈ (neighborhood G R).powersetCard (k-R.card), ∏ v ∈ L, p v) ≤
-      δ^((k:ℝ)/2) * Real.exp R.card := by
-  classical
-  have hmass := neighborhood_weight_le G p hp δ hdeg R
-  have hpow : δ^(k-R.card) ≤ δ^((k:ℝ)/2) := by
-    rw [← Real.rpow_natCast]
-    apply Real.rpow_le_rpow_of_exponent_ge hδ hδ1
-    have hr : R.card ≤ k := by omega
-    rw [Nat.cast_sub hr]
-    have hh : (2:ℝ)*R.card ≤ k := by exact_mod_cast hR
-    linarith
-  apply (fixed_size_weight_le (neighborhood G R) p hp _ δ hδ).trans
-  apply mul_le_mul hpow
-  · apply Real.exp_le_exp.mpr
-    exact (div_le_iff₀ hδ).mpr hmass
-  · positivity
-  · positivity
-
-def NoIsolated {V : Type*} (G : SimpleGraph V) (S : Finset V) : Prop :=
-  ∀ v ∈ S, ∃ w ∈ S, G.Adj v w
-
-lemma small_dominating_subset {V : Type*} [Fintype V] (G : SimpleGraph V)
-    (S : Finset V) (hS : NoIsolated G S) :
-    ∃ R : Finset V, R ⊆ S ∧ 2 * R.card ≤ S.card ∧ S \ R ⊆ neighborhood G R := by
-  classical
-  let H := G.induce (S : Set V)
-  have hH : ∀ v : S, ∃ w : S, H.Adj v w := by
-    intro v
-    obtain ⟨w, hw, hvw⟩ := hS v v.2
-    exact ⟨⟨w, hw⟩, hvw⟩
-  obtain ⟨R, hR, hdom⟩ := small_dominating_set H hH
-  let T := R.image (fun v : S => (v : V))
-  have hc : T.card = R.card := Finset.card_image_of_injective _ Subtype.val_injective
-  refine ⟨T, ?_, ?_, ?_⟩
-  · intro v hv
-    obtain ⟨w, _, rfl⟩ := Finset.mem_image.mp hv
-    exact w.2
-  · simpa [hc, H] using hR
-  · intro v hv
-    have hvS := (Finset.mem_sdiff.mp hv).1
-    have hvR : (⟨v, hvS⟩ : S) ∉ R := by
-      intro hh
-      exact (Finset.mem_sdiff.mp hv).2 (Finset.mem_image.mpr ⟨⟨v,hvS⟩, hh, rfl⟩)
-    obtain ⟨w, hw, hvw⟩ := hdom ⟨v,hvS⟩ hvR
-    exact Finset.mem_filter.mpr ⟨Finset.mem_univ _,
-      ⟨w, Finset.mem_image.mpr ⟨w,hw,rfl⟩, hvw.symm⟩⟩
-
-end Thorp.SparseContact
-
-end ThorpNine.Harmonic
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Harmonic.BlockProduct
+
+namespace OAI
+
+namespace ThorpNine.Harmonic
+
+namespace Thorp
+open scoped BigOperators Classical
+
+def pairAssembleEquiv (B C H : Type*) :
+    ((H × (B → C × C)) × H) ≃ (((B → C) × H) × ((B → C) × H)) where
+  toFun w := (((fun b => (w.1.2 b).1),w.1.1),((fun b => (w.1.2 b).2),w.2))
+  invFun w := ((w.1.2,fun b => (w.1.1 b,w.2.1 b)),w.2.2)
+  left_inv _w := rfl
+  right_inv _w := rfl
+
+def palindromeStageEquiv (r L : ℕ) :
+    ((OuterCoins r L × (Card r → BenesCoins L)) × OuterCoins r L) ≃ BenesCoins (L+r) :=
+  (pairAssembleEquiv _ _ _).trans
+    (Equiv.prodCongr (parallelSplitCoinEquiv r L) (parallelSplitCoinEquiv r L))
+
+lemma palindromeStageEquiv_apply (r L : ℕ)
+    (w : (OuterCoins r L × (Card r → BenesCoins L)) × OuterCoins r L) :
+    palindromeStageEquiv r L w=
+      (assembleBits r L (fun x => (w.1.2 x.1).1 x.2) w.1.1,
+       assembleBits r L (fun x => (w.1.2 x.1).2 x.2) w.2) := by
+  simp only [palindromeStageEquiv,Equiv.trans_apply,Equiv.prodCongr_apply,pairAssembleEquiv,
+    Equiv.coe_fn_mk,Prod.map_apply,parallelSplitCoinEquiv_apply]
+
+def modifiedAssembleEquiv (B C H U : Type*) :
+    ((H × (B → U × (C × C))) × H) ≃ (B → U) × (((B → C) × H) × ((B → C) × H)) where
+  toFun w := ((fun b => (w.1.2 b).1),
+    (((fun b => (w.1.2 b).2.1),w.1.1),((fun b => (w.1.2 b).2.2),w.2)))
+  invFun w := ((w.2.1.2,fun b => (w.1 b,(w.2.1.1 b,w.2.2.1 b))),w.2.2.2)
+  left_inv _w := rfl
+  right_inv _w := rfl
+
+def modifiedStageEquiv (r L : ℕ) :
+    ((OuterCoins r L × (Card r → Equiv.Perm (Card L) × BenesCoins L)) × OuterCoins r L) ≃
+      ModifiedCoins L r :=
+  (modifiedAssembleEquiv _ _ _ _).trans
+    (Equiv.prodCongr (Equiv.refl _) (Equiv.prodCongr (parallelSplitCoinEquiv r L) (parallelSplitCoinEquiv r L)))
+
+lemma modifiedStageEquiv_apply (r L : ℕ)
+    (w : (OuterCoins r L × (Card r → Equiv.Perm (Card L) × BenesCoins L)) × OuterCoins r L) :
+    modifiedStageEquiv r L w=
+      ((fun b => (w.1.2 b).1),
+       (assembleBits r L (fun x => (w.1.2 x.1).2.1 x.2) w.1.1,
+        assembleBits r L (fun x => (w.1.2 x.1).2.2 x.2) w.2)) := by
+  simp only [modifiedStageEquiv,Equiv.trans_apply,Equiv.prodCongr_apply,modifiedAssembleEquiv,
+    Equiv.coe_fn_mk,Prod.map_apply,Equiv.refl_apply,parallelSplitCoinEquiv_apply]
+
+lemma palindrome_stage (r L : ℕ)
+    (w : (OuterCoins r L × (Card r → BenesCoins L)) × OuterCoins r L) :
+    splitPerm r L (palindromePerm (L+r) (palindromeStageEquiv r L w))=
+      outerPerm r L w.1.1*ParallelBlocks.lift (fun b => palindromePerm L (w.1.2 b))*
+        (outerPerm r L w.2)⁻¹ := by
+  rw [palindromeStageEquiv_apply]
+  change splitPerm r L (_*_⁻¹)=_
+  rw [map_mul,map_inv,splitPerm_butterfly,splitPerm_butterfly,mul_inv_rev]
+  rw [mul_assoc,←mul_assoc (ParallelBlocks.lift _),←map_inv,←map_mul,←mul_assoc]
+  rfl
+
+noncomputable def localModifiedPerm (L : ℕ) (e : Bool)
+    (w : Equiv.Perm (Card L) × BenesCoins L) : Equiv.Perm (Card L) :=
+  if e then 1 else butterflyPerm L (decodeButterfly L w.2.1)*w.1⁻¹
+
+lemma modified_stage (r L : ℕ) (E : Finset (Card r))
+    (w : (OuterCoins r L × (Card r → Equiv.Perm (Card L) × BenesCoins L)) × OuterCoins r L) :
+    splitPerm r L (modifiedPerm L r E (modifiedStageEquiv r L w))=
+      outerPerm r L w.1.1*ParallelBlocks.lift (fun b => localModifiedPerm L (decide (b∈E)) (w.1.2 b))*
+        (outerPerm r L w.2)⁻¹ := by
+  rw [modifiedStageEquiv_apply]
+  unfold modifiedPerm modifiedInputTree
+  rw [map_mul,map_inv,splitPerm_butterfly,splitPerm_boundaryTree,mul_inv_rev]
+  rw [mul_assoc,←mul_assoc (ParallelBlocks.lift _),←map_inv,←map_mul,←mul_assoc]
+  congr 2
+  apply congrArg ParallelBlocks.lift
+  funext b
+  simp only [Pi.mul_apply,Pi.inv_apply,modifiedBoundary,boundaryOutput_split,localModifiedPerm]
+  by_cases h : b∈E <;> simp [h]
+
+end Thorp
+namespace Thorp.UnitaryFinite
+open scoped BigOperators Classical
+variable {G : Type*} [Group G]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+variable {Ω Ξ : Type*} [Fintype Ω] [Fintype Ξ]
+
+lemma sampleOperator_sandwich (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
+    (P : Ω → G) (Q : Ξ → G) :
+    sampleOperator ρ (fun w : (Ω × Ξ) × Ω => P w.1.1*Q w.1.2*(P w.2)⁻¹)=
+      sampleOperator ρ P*sampleOperator ρ Q*(sampleOperator ρ P).adjoint := by
+  rw [sampleOperator_prod ρ (fun w : Ω × Ξ => P w.1*Q w.2) (fun w : Ω => (P w)⁻¹),
+    sampleOperator_prod ρ P Q,sampleOperator_inv ρ hρ]
+
+lemma localModified_average (L : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card L)) V) (e : Bool) :
+    sampleOperator ρ (localModifiedPerm L e)=if e then 1 else uniformOperator ρ := by
+  cases e
+  · simp only [Bool.false_eq_true,ite_false]
+    rw [←sampleOperator_equiv ρ _ (Equiv.prodComm _ _)]
+    exact sampleOperator_uniform_mul ρ (fun w : BenesCoins L => butterflyPerm L (decodeButterfly L w.1))
+  · simp only [ite_true]
+    exact sampleOperator_one ρ
+
+end Thorp.UnitaryFinite
+
+namespace Thorp
+open scoped BigOperators Classical
+open UnitaryFinite
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+lemma sampleOperator_palindrome_stage (r L : ℕ)
+    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) (hρ : IsUnitary ρ) :
+    sampleOperator (ρ.comp (splitPerm r L).toMonoidHom) (palindromePerm (L+r))=
+      sampleOperator ρ (outerPerm r L)*
+        sampleOperator ρ (fun w : Card r → BenesCoins L => ParallelBlocks.lift (fun b => palindromePerm L (w b)))*
+          (sampleOperator ρ (outerPerm r L)).adjoint := by
+  have he : ((splitPerm r L).toMonoidHom ∘ palindromePerm (L+r)) ∘ palindromeStageEquiv r L=
+      fun w => outerPerm r L w.1.1*ParallelBlocks.lift (fun b => palindromePerm L (w.1.2 b))*
+        (outerPerm r L w.2)⁻¹ := funext (palindrome_stage r L)
+  calc
+    _ = sampleOperator ρ (((splitPerm r L).toMonoidHom ∘ palindromePerm (L+r)) ∘ palindromeStageEquiv r L) :=
+      (sampleOperator_equiv ρ ((splitPerm r L).toMonoidHom ∘ palindromePerm (L+r)) (palindromeStageEquiv r L)).symm
+    _ = _ := (congrArg (sampleOperator ρ) he).trans
+      (sampleOperator_sandwich ρ hρ (outerPerm r L)
+        (fun w : Card r → BenesCoins L => ParallelBlocks.lift (fun b => palindromePerm L (w b))))
+
+lemma sampleOperator_modified_stage (r L : ℕ) (E : Finset (Card r))
+    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) (hρ : IsUnitary ρ) :
+    sampleOperator (ρ.comp (splitPerm r L).toMonoidHom) (modifiedPerm L r E)=
+      sampleOperator ρ (outerPerm r L)*
+        sampleOperator ρ (fun w : Card r → Equiv.Perm (Card L) × BenesCoins L =>
+          ParallelBlocks.lift (fun b => localModifiedPerm L (decide (b∈E)) (w b)))*
+          (sampleOperator ρ (outerPerm r L)).adjoint := by
+  have he : ((splitPerm r L).toMonoidHom ∘ modifiedPerm L r E) ∘ modifiedStageEquiv r L=
+      fun w => outerPerm r L w.1.1*ParallelBlocks.lift
+        (fun b => localModifiedPerm L (decide (b∈E)) (w.1.2 b))*(outerPerm r L w.2)⁻¹ :=
+    funext (modified_stage r L E)
+  calc
+    _ = sampleOperator ρ (((splitPerm r L).toMonoidHom ∘ modifiedPerm L r E) ∘ modifiedStageEquiv r L) :=
+      (sampleOperator_equiv ρ ((splitPerm r L).toMonoidHom ∘ modifiedPerm L r E) (modifiedStageEquiv r L)).symm
+    _ = _ := (congrArg (sampleOperator ρ) he).trans
+      (sampleOperator_sandwich ρ hρ (outerPerm r L)
+        (fun w : Card r → Equiv.Perm (Card L) × BenesCoins L =>
+          ParallelBlocks.lift (fun b => localModifiedPerm L (decide (b∈E)) (w b))))
+
+end Thorp
+namespace Thorp.CommutingBlocks
+open scoped BigOperators Classical
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
+variable {ι : Type*} [Fintype ι] [DecidableEq ι]
+
+noncomputable def retain (P : ι → V →L[ℂ] V) (E : Finset ι) (i : ι) : V →L[ℂ] V :=
+  if i∈E then 1 else P i
+
+omit [CompleteSpace V] [Fintype ι] in
+lemma retain_commute (P : ι → V →L[ℂ] V) (hP : ∀ i j, Commute (P i) (P j))
+    (E : Finset ι) (i j : ι) : Commute (retain P E i) (retain P E j) := by
+  unfold retain
+  split_ifs <;> first | exact Commute.one_left _ | exact Commute.one_right _ | exact hP _ _
+
+omit [CompleteSpace V] in
+lemma blocks_retain (P : ι → V →L[ℂ] V) (hP : ∀ i j, Commute (P i) (P j)) (E : Finset ι) :
+    blocks (retain P E) (retain_commute P hP E)=coarse P hP E := by
+  have hs : E∪Eᶜ=(Finset.univ : Finset ι) := Finset.union_compl E
+  have hh := Finset.noncommProd_union_of_disjoint (s := E) (t := Eᶜ) disjoint_compl_right
+    (retain P E) (fun _ _ _ _ _ => retain_commute P hP E _ _)
+  simp only [hs] at hh
+  change blocks (retain P E) (retain_commute P hP E)=_ at hh
+  rw [hh]
+  have he : E.noncommProd (retain P E) (fun _ _ _ _ _ => retain_commute P hP E _ _)=1 := by
+    exact (Finset.noncommProd_eq_pow_card E (retain P E)
+      (fun i _ j _ _ => retain_commute P hP E i j) 1 (by intro i hi; exact ite_eq_left hi)).trans (one_pow _)
+  rw [he,one_mul]
+  apply Finset.noncommProd_congr rfl
+  intro i hi
+  exact ite_eq_right (Finset.mem_compl.mp hi)
+
+end Thorp.CommutingBlocks
+
+namespace Thorp.ParallelBlocks
+open scoped BigOperators Classical
+open UnitaryFinite
+variable {B X : Type*} [Fintype B] [Fintype X] [DecidableEq X]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+noncomputable def centralProjection (ρ : Representation ℂ (Equiv.Perm (B × X)) V) (b : B) : V →L[ℂ] V :=
+  uniformOperator (ρ.comp (injection b))
+
+omit [Fintype B] in
+lemma centralProjection_commute (ρ : Representation ℂ (Equiv.Perm (B × X)) V) (b c : B) :
+    Commute (centralProjection ρ b) (centralProjection ρ c) := by
+  simpa only [centralProjection,sampleOperator_uniform] using
+    local_average_commute ρ (fun _ => id : B → Equiv.Perm X → Equiv.Perm X) b c
+
+omit [Fintype B] in
+lemma local_central_commute {Ω : Type*} [Fintype Ω] [Nonempty Ω]
+    (ρ : Representation ℂ (Equiv.Perm (B × X)) V) (P : B → Ω → Equiv.Perm X) (b c : B) :
+    Commute (sampleOperator (ρ.comp (injection b)) (P b)) (centralProjection ρ c) := by
+  by_cases h : b=c
+  · subst c
+    change _*uniformOperator _=uniformOperator _*_
+    rw [sampleOperator_uniform_absorb,sampleOperator_weighted,weighted_uniform_right,sampleLaw_sum,one_smul]
+  · unfold centralProjection
+    rw [←sampleOperator_uniform,sampleOperator_comp,sampleOperator_comp]
+    exact sampleOperator_commute ρ _ _ (fun _ _ => injection_commute h _ _)
+
+end Thorp.ParallelBlocks
+namespace Thorp
+open scoped BigOperators Classical
+open UnitaryFinite ParallelBlocks CommutingBlocks
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+lemma modified_central_average (r L : ℕ) (E : Finset (Card r))
+    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) :
+    sampleOperator ρ (fun w : Card r → Equiv.Perm (Card L) × BenesCoins L =>
+      lift (fun b => localModifiedPerm L (decide (b∈E)) (w b)))=
+      coarse (centralProjection ρ) (centralProjection_commute ρ) E := by
+  rw [sampleOperator_parallel ρ (fun b => localModifiedPerm L (decide (b∈E))),←blocks_retain]
+  apply Finset.noncommProd_congr rfl
+  · intro b _
+    rw [localModified_average]
+    simp only [retain,centralProjection,decide_eq_true_eq]
+  · intro b _ c _ _
+    exact local_average_commute ρ (fun b => localModifiedPerm L (decide (b∈E))) b c
+
+lemma central_local_gap (r L : ℕ)
+    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) (hρ : IsUnitary ρ) (b : Card r) :
+    ‖sampleOperator (ρ.comp (injection b)) (palindromePerm L)*(1-centralProjection ρ b)‖≤blockGap L := by
+  change ‖sampleOperator _ _*(1-uniformOperator _)‖≤_
+  rw [mul_sub,mul_one,sampleOperator_uniform_absorb,sampleOperator_weighted]
+  exact palindrome_off_invariants_gap L _ (unitary_comp ρ hρ _)
+
+lemma central_blocks_exception_bound (r L : ℕ)
+    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) (hρ : IsUnitary ρ)
+    (B : V →L[ℂ] V) (hB0 : ‖B‖≤1) (n : ℕ) :
+    ‖B*blocks (fun b : Card r => sampleOperator (ρ.comp (injection b)) (palindromePerm L))
+      (local_average_commute ρ (fun _ => palindromePerm L))*B.adjoint‖ ≤
+      (blockGap L)^n + ∑ E : Finset (Card r), if E.card<n then
+        ‖B*coarse (centralProjection ρ) (centralProjection_commute ρ) E*B.adjoint‖ else 0 := by
+  have hB : ‖B.adjoint‖≤1 := by
+    rw [ContinuousLinearMap.adjoint.norm_map]
+    exact hB0
+  have h := blocks_exception_bound (centralProjection ρ) (centralProjection_commute ρ)
+    (fun b => uniformOperator_projection (ρ.comp (injection b)) (unitary_comp ρ hρ _))
+    (fun b : Card r => sampleOperator (ρ.comp (injection b)) (palindromePerm L))
+    (local_average_commute ρ (fun _ => palindromePerm L))
+    (local_central_commute ρ (fun _ => palindromePerm L))
+    (fun b => sampleOperator_norm_le (ρ.comp (injection b)) (unitary_comp ρ hρ _) (palindromePerm L))
+    (blockGap L) (blockGap_pos L).le (blockGap_lt_one L).le (central_local_gap r L ρ hρ)
+    B.adjoint hB n
+  simpa only [ContinuousLinearMap.adjoint_adjoint] using h
+
+theorem palindrome_exception_bound (r L : ℕ)
+    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) (hρ : IsUnitary ρ) (n : ℕ) :
+    ‖sampleOperator (ρ.comp (splitPerm r L).toMonoidHom) (palindromePerm (L+r))‖ ≤
+      (blockGap L)^n + ∑ E : Finset (Card r), if E.card<n then
+        ‖sampleOperator (ρ.comp (splitPerm r L).toMonoidHom) (modifiedPerm L r E)‖ else 0 := by
+  rw [sampleOperator_palindrome_stage r L ρ hρ]
+  rw [sampleOperator_parallel ρ (fun _ => palindromePerm L)]
+  apply (central_blocks_exception_bound r L ρ hρ (sampleOperator ρ (outerPerm r L))
+    (sampleOperator_norm_le ρ hρ _) n).trans_eq
+  congr 1
+  apply Finset.sum_congr rfl
+  intro E _
+  split_ifs
+  · rw [sampleOperator_modified_stage r L E ρ hρ,modified_central_average]
+  · rfl
+
+end Thorp
+
+namespace Thorp.UnitaryFinite
+open scoped BigOperators ComplexConjugate Classical
+
+variable {G X Ω : Type*} [Group G] [Fintype G] [Fintype X] [Fintype Ω]
+
+noncomputable def actionKernel (a : G →* Equiv.Perm X) (P : Ω → G) (x y : X) : ℝ :=
+  finiteMean (fun ω => if a (P ω) x=y then 1 else 0)
+
+omit [Fintype G] in
+lemma actionKernel_apply (a : G →* Equiv.Perm X) (P : Ω → G)
+    (v : EuclideanSpace ℂ X) (x : X) :
+    DensityTransfer.applyKernel (actionKernel a P) v x =
+      (Fintype.card Ω:ℂ)⁻¹ * ∑ ω, v (a (P ω) x) := by
+  simp only [DensityTransfer.applyKernel_apply,actionKernel,finiteMean,
+    Complex.ofReal_sum,div_eq_mul_inv,Finset.sum_mul]
+  rw [Finset.sum_comm]
+  calc
+    _ = ∑ ω, (Fintype.card Ω:ℂ)⁻¹ * ∑ y : X, if a (P ω) x=y then v y else 0 := by
+      apply Finset.sum_congr rfl
+      intro ω _
+      rw [Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro y _
+      split_ifs <;> simp_all
+    _ = _ := by simp [Finset.mul_sum]
+
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+noncomputable def randomOperator (ρ : Representation ℂ G V) (P : Ω → G) : Module.End ℂ V :=
+  (Fintype.card Ω:ℂ)⁻¹ • ∑ ω, ρ (P ω)⁻¹
+
+omit [Fintype G] [FiniteDimensional ℂ V] in
+lemma coefficient_random (a : G →* Equiv.Perm X) (x₀ : X)
+    (ha : ∀ x, ∃ g, a g x₀=x) (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
+    (P : Ω → G) (w : V) (hw : IsFixed a x₀ ρ w) (v : V) :
+    DensityTransfer.applyKernel (actionKernel a P) (normalizedCoefficient a x₀ ha ρ w v) =
+      normalizedCoefficient a x₀ ha ρ w (randomOperator ρ P v) := by
+  ext x
+  rw [actionKernel_apply]
+  simp only [normalizedCoefficient,PiLp.smul_apply,smul_eq_mul,Finset.mul_sum,
+    coefficient,randomOperator,LinearMap.smul_apply,LinearMap.sum_apply,
+    inner_smul_right,inner_sum]
+  apply Finset.sum_congr rfl
+  intro ω _
+  have h := coefficient_action a x₀ ha ρ hρ w hw v (P ω) x
+  change inner ℂ (ρ (representative a x₀ ha (a (P ω) x)) w) v =
+    inner ℂ (ρ (representative a x₀ ha x) w) (ρ (P ω)⁻¹ v) at h
+  rw [h]
+  ring
+
+noncomputable def fixedSpace (a : G →* Equiv.Perm X) (x₀ : X) (ρ : Representation ℂ G V) :
+    Submodule ℂ V where
+  carrier := {w | IsFixed a x₀ ρ w}
+  zero_mem' := fun g hg => map_zero _
+  add_mem' := fun hw hz g hg => by rw [map_add,hw g hg,hz g hg]
+  smul_mem' := fun c w hw g hg => by rw [map_smul,hw g hg]
+
+theorem operator_amplification (a : G →* Equiv.Perm X) (x₀ : X)
+    (ha : ∀ x, ∃ g, a g x₀=x) (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
+    (hρ : IsUnitary ρ) (P : Ω → G)
+    (hsym : ∀ x y, actionKernel a P x y=actionKernel a P y x)
+    (hrow : ∀ x, ∑ y, actionKernel a P x y=1)
+    (b z M : ℝ) (hb : 0≤b) (hz : 0<z) (hM0 : 0≤M)
+    (hM : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*actionKernel a P x y)^(1+b)) ≤ M)
+    (hm : 0 < Module.finrank ℂ (fixedSpace a x₀ ρ)) (v : V) (hv : ‖v‖=1) :
+    ‖randomOperator ρ P v‖ ≤ M*z^(-b)+Real.sqrt (z/(Module.finrank ℂ (fixedSpace a x₀ ρ):ℝ)) := by
+  let K := fixedSpace a x₀ ρ
+  let m := Module.finrank ℂ K
+  let : Nonempty X := ⟨x₀⟩
+  let : Nonempty (Fin m) := ⟨⟨0,hm⟩⟩
+  let B := stdOrthonormalBasis ℂ K
+  let w : Fin m → V := fun i => (B i : V)
+  have hfix (i : Fin m) : IsFixed a x₀ ρ (w i) := (B i).2
+  have hw : Orthonormal ℂ w := B.orthonormal.comp_linearIsometry K.subtypeₗᵢ
+  have hc := normalizedCoefficient_orthonormal a x₀ ha ρ hρ w hw hfix v hv
+  have hn (i : Fin m) : ‖DensityTransfer.applyKernel (actionKernel a P)
+      (normalizedCoefficient a x₀ ha ρ (w i) v)‖ = ‖randomOperator ρ P v‖ := by
+    rw [coefficient_random a x₀ ha ρ hρ P (w i) (hfix i),
+      normalizedCoefficient_norm a x₀ ha ρ hρ (w i) (hfix i) (hw.norm_eq_one i)]
+  have h := DensityTransfer.multiplicity_amplification (actionKernel a P)
+    (fun x y => finiteMean_nonneg (fun _ => by split_ifs <;> norm_num)) hsym hrow
+    b z M hb hz hM0 hM _ hc (‖randomOperator ρ P v‖) (fun i => (hn i).ge)
+  simpa only [Fintype.card_fin] using h
+
+end Thorp.UnitaryFinite
+namespace Thorp.UnitaryFinite
+open scoped BigOperators Classical
+
+variable {α ι : Type*} [Fintype α] [Fintype ι]
+
+def tupleAction : Equiv.Perm α →* Equiv.Perm (ι ↪ α) where
+  toFun p :=
+    { toFun := fun e => e.trans p.toEmbedding
+      invFun := fun e => e.trans p.symm.toEmbedding
+      left_inv := fun e => by ext i; exact p.symm_apply_apply (e i)
+      right_inv := fun e => by ext i; exact p.apply_symm_apply (e i) }
+  map_one' := by ext e i; rfl
+  map_mul' p q := by ext e i; rfl
+
+omit [Fintype α] [Fintype ι] in
+@[simp] lemma tupleAction_apply (p : Equiv.Perm α) (e : ι ↪ α) (i : ι) :
+    tupleAction p e i=p (e i) := rfl
+
+omit [Fintype α] in
+lemma tupleAction_transitive (e f : ι ↪ α) : ∃ p, tupleAction p e=f := by
+  obtain ⟨p,hp⟩ := Equiv.Perm.exists_extending_pair e f e.injective f.injective
+  exact ⟨p,Function.Embedding.ext hp⟩
+
+omit [Fintype α] in
+lemma tupleAction_fixed_iff [DecidableEq α] (e : ι ↪ α) (p : Equiv.Perm α) :
+    tupleAction p e=e ↔ ∀ x ∈ Finset.univ.image e, p x=x := by
+  constructor
+  · intro h x hx
+    obtain ⟨i,_,rfl⟩ := Finset.mem_image.mp hx
+    exact congrArg (fun f : ι ↪ α => f i) h
+  · intro h
+    ext i
+    exact h (e i) (Finset.mem_image.mpr ⟨i,Finset.mem_univ _,rfl⟩)
+
+omit [Fintype α] in
+lemma tupleAction_kernel [DecidableEq α] {Ω : Type*} [Fintype Ω] (P : Ω → Equiv.Perm α)
+    (e f : ι ↪ α) : actionKernel tupleAction P e f=PairRouting.tupleProbability P e f := by
+  unfold actionKernel PairRouting.tupleProbability
+  congr 1
+  funext ω
+  congr 1
+  exact propext Function.Embedding.ext_iff
+
+end Thorp.UnitaryFinite
+namespace Thorp.UnitaryFinite
+open scoped BigOperators Classical
+
+lemma irreducible_of_equiv {G V W : Type*} [Group G]
+    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
+    (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
+    (e : Representation.Equiv ρ σ) [Representation.IsIrreducible ρ] :
+    Representation.IsIrreducible σ := by
+  rw [Representation.irreducible_iff_isSimpleModule_asModule]
+  have hi := (Representation.irreducible_iff_isSimpleModule_asModule ρ).mp inferInstance
+  let f : ρ.asModule →ₗ[MonoidAlgebra ℂ G] σ.asModule :=
+    Representation.IntertwiningMap.equivLinearMapAsModule ρ σ e.toIntertwiningMap
+  have hf : Function.Bijective f := e.toLinearEquiv.bijective
+  exact (f.isSimpleModule_iff_of_bijective hf).mp hi
+
+end Thorp.UnitaryFinite
+
+namespace Thorp.Specht
+open scoped BigOperators Classical
+
+lemma unitaryRepresentation_coe (μ : YoungDiagram) (p : Equiv.Perm (Cell μ))
+    (v : hilbertSpace μ) :
+    (unitaryRepresentation μ p v : PermutationHilbert.H (Tabloid μ)) =
+      PermutationHilbert.act (tabloidAct p) v := by
+  obtain ⟨w,rfl⟩ := (spaceHilbertEquiv μ).surjective v
+  change hilbertEquiv μ (tabloidRep μ p w) = _
+  exact hilbert_action μ p w
+
+noncomputable def unitaryEquiv (μ : YoungDiagram) :
+    Representation.Equiv (representation μ) (unitaryRepresentation μ) :=
+  Representation.Equiv.mk (spaceHilbertEquiv μ) (by
+    intro p
+    apply LinearMap.ext
+    intro v
+    change spaceHilbertEquiv μ (representation μ p v) =
+      spaceHilbertEquiv μ (representation μ p ((spaceHilbertEquiv μ).symm (spaceHilbertEquiv μ v)))
+    rw [LinearEquiv.symm_apply_apply])
+
+lemma unitaryRepresentation_irreducible (μ : YoungDiagram) :
+    Representation.IsIrreducible (unitaryRepresentation μ) := by
+  let := representation_irreducible μ
+  exact UnitaryFinite.irreducible_of_equiv _ _ (unitaryEquiv μ)
+
+lemma unitaryRepresentation_unitary (μ : YoungDiagram) :
+    UnitaryFinite.IsUnitary (G := Equiv.Perm (Cell μ)) (V := hilbertSpace μ) (unitaryRepresentation μ) := by
+  intro p v w
+  change inner ℂ (unitaryRepresentation μ p v : PermutationHilbert.H (Tabloid μ))
+      (unitaryRepresentation μ p w : PermutationHilbert.H (Tabloid μ)) = _
+  rw [unitaryRepresentation_coe,unitaryRepresentation_coe]
+  exact (PermutationHilbert.act (tabloidAct p)).inner_map_map v w
+
+
+noncomputable def tupleFixedEquiv (μ : YoungDiagram) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Cell μ) :
+    UnitaryFinite.fixedSpace (V := hilbertSpace μ) UnitaryFinite.tupleAction e (unitaryRepresentation μ) ≃ₗ[ℂ]
+      fixedHilbert μ (Finset.univ.image e) where
+  toFun v := ⟨v.1.1,v.1.2,by
+    intro p hp
+    have h := v.2 p ((UnitaryFinite.tupleAction_fixed_iff e p).mpr hp)
+    exact (unitaryRepresentation_coe μ p v.1).symm.trans (congrArg Subtype.val h)⟩
+  invFun v := ⟨⟨v.1,v.2.1⟩,by
+    intro p hp
+    apply Subtype.ext
+    exact (unitaryRepresentation_coe μ p _).trans
+      (v.2.2 p ((UnitaryFinite.tupleAction_fixed_iff e p).mp hp))⟩
+  left_inv _ := rfl
+  right_inv _ := rfl
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+
+theorem tupleFixed_dimension (μ : YoungDiagram) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Cell μ) :
+    Module.finrank ℂ (hilbertSpace μ) * (Fintype.card ι).choose (μ.card-μ.rowLen 0) ≤
+      Module.finrank ℂ (UnitaryFinite.fixedSpace (V := hilbertSpace μ) UnitaryFinite.tupleAction e (unitaryRepresentation μ)) *
+        μ.card.choose (μ.card-μ.rowLen 0) := by
+  rw [(tupleFixedEquiv μ e).finrank_eq,finrank_hilbertSpace]
+  have h := fixedHilbert_dimension μ (Finset.univ.image e)
+  simpa only [Finset.card_image_of_injective _ e.injective,Finset.card_univ] using h
+
+end Thorp.Specht
+namespace Thorp.UnitaryFinite
+open scoped BigOperators Classical
+
+variable {G H X Ω : Type*} [Group G] [Group H] [Fintype G] [Fintype X] [Fintype Ω]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+omit [Fintype G] in
+lemma sampleOperator_inv_apply (ρ : Representation ℂ G V) (P : Ω → G) (v : V) :
+    sampleOperator ρ (fun ω => (P ω)⁻¹) v=randomOperator ρ P v := by
+  simp only [sampleOperator,randomOperator,smul_apply,
+    sum_apply,continuousRepresentation_apply,LinearMap.smul_apply,LinearMap.sum_apply]
+
+theorem sampleOperator_amplification (a : G →* Equiv.Perm X) (x₀ : X)
+    (ha : ∀ x, ∃ g, a g x₀=x) (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
+    (hρ : IsUnitary ρ) (P : Ω → G)
+    (hsym : ∀ x y, actionKernel a P x y=actionKernel a P y x)
+    (hrow : ∀ x, ∑ y, actionKernel a P x y=1)
+    (b z M : ℝ) (hb : 0≤b) (hz : 0<z) (hM0 : 0≤M)
+    (hM : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*actionKernel a P x y)^(1+b)) ≤ M)
+    (hm : 0 < Module.finrank ℂ (fixedSpace a x₀ ρ)) :
+    ‖sampleOperator ρ P‖ ≤ M*z^(-b)+Real.sqrt (z/(Module.finrank ℂ (fixedSpace a x₀ ρ):ℝ)) := by
+  rw [←ContinuousLinearMap.adjoint.norm_map (sampleOperator ρ P),←sampleOperator_inv ρ hρ]
+  apply ContinuousLinearMap.opNorm_le_of_unit_norm (by positivity)
+  intro v hv
+  rw [sampleOperator_inv_apply]
+  exact operator_amplification a x₀ ha ρ hρ P hsym hrow b z M hb hz hM0 hM hm v hv
+
+omit [Fintype G] [Fintype X] [Fintype Ω] [FiniteDimensional ℂ V] in
+noncomputable def subrepresentationEquiv (ρ : Representation ℂ G V) (e : H ≃* G) :
+    Subrepresentation (ρ.comp e.toMonoidHom) ≃o Subrepresentation ρ where
+  toFun U :=
+    { toSubmodule := U.toSubmodule
+      apply_mem_toSubmodule := fun g v hv => by
+        have h := U.apply_mem_toSubmodule (e.symm g) hv
+        change ρ (e (e.symm g)) v∈U.toSubmodule at h
+        simpa only [e.apply_symm_apply] using h }
+  invFun U :=
+    { toSubmodule := U.toSubmodule
+      apply_mem_toSubmodule := fun h _ hv => U.apply_mem_toSubmodule (e h) hv }
+  left_inv U := by apply Subrepresentation.toSubmodule_injective; rfl
+  right_inv U := by apply Subrepresentation.toSubmodule_injective; rfl
+  map_rel_iff' := Iff.rfl
+
+omit [Fintype G] [Fintype X] [Fintype Ω] [FiniteDimensional ℂ V] in
+lemma irreducible_comp_equiv (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
+    (e : H ≃* G) : Representation.IsIrreducible (ρ.comp e.toMonoidHom) := by
+  exact (subrepresentationEquiv ρ e).isSimpleOrder_iff.mpr inferInstance
+
+end Thorp.UnitaryFinite
+
+namespace Thorp.Specht
+open scoped BigOperators Classical
+
+variable {α ι : Type*} [Fintype α] [Fintype ι]
+
+omit [Fintype α] in
+lemma relabelledUnitary_unitary (μ : YoungDiagram) (e : α ≃ Cell μ) :
+    UnitaryFinite.IsUnitary (V := hilbertSpace μ) (relabelledUnitary μ e) := by
+  intro g v w
+  exact unitaryRepresentation_unitary μ (e.permCongr g) v w
+
+omit [Fintype α] in
+lemma relabelledUnitary_irreducible (μ : YoungDiagram) (e : α ≃ Cell μ) :
+    Representation.IsIrreducible (relabelledUnitary μ e) := by
+  let := unitaryRepresentation_irreducible μ
+  exact UnitaryFinite.irreducible_comp_equiv (V := hilbertSpace μ) (unitaryRepresentation μ) e.permCongrHom
+
+noncomputable def relabelledTupleFixed (μ : YoungDiagram) (e : α ≃ Cell μ) (f : ι ↪ α) :
+    UnitaryFinite.fixedSpace (V := hilbertSpace μ) UnitaryFinite.tupleAction f (relabelledUnitary μ e) ≃ₗ[ℂ]
+    UnitaryFinite.fixedSpace (V := hilbertSpace μ) UnitaryFinite.tupleAction (f.trans e.toEmbedding) (unitaryRepresentation μ) where
+  toFun v := ⟨v.1,by
+    intro g hg
+    have hp : UnitaryFinite.tupleAction (e.symm.permCongr g) f=f := by
+      apply Function.Embedding.ext
+      intro i
+      have hh := congrArg (fun f : ι ↪ Cell μ => f i) hg
+      change g (e (f i))=e (f i) at hh
+      simp only [UnitaryFinite.tupleAction_apply,Equiv.permCongr_apply,Equiv.symm_symm,e.symm_apply_apply,hh]
+    have hh := v.2 (e.symm.permCongr g) hp
+    change unitaryRepresentation μ (e.permCongr (e.symm.permCongr g)) v=v at hh
+    have he : e.permCongr (e.symm.permCongr g)=g := by
+      apply Equiv.ext
+      intro x
+      simp only [Equiv.permCongr_apply,Equiv.symm_symm,e.apply_symm_apply]
+    simpa only [he] using hh⟩
+  invFun v := ⟨v.1,by
+    intro g hg
+    have hp : UnitaryFinite.tupleAction (e.permCongr g) (f.trans e.toEmbedding)=f.trans e.toEmbedding := by
+      apply Function.Embedding.ext
+      intro i
+      have hh := congrArg (fun f : ι ↪ α => f i) hg
+      change g (f i)=f i at hh
+      simp only [UnitaryFinite.tupleAction_apply,Function.Embedding.trans_apply,
+        Equiv.toEmbedding_apply,Equiv.permCongr_apply,e.symm_apply_apply,hh]
+    exact v.2 (e.permCongr g) hp⟩
+  left_inv _ := rfl
+  right_inv _ := rfl
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+
+omit [Fintype α] in
+lemma relabelled_tuple_dimension (μ : YoungDiagram) (e : α ≃ Cell μ) (f : ι ↪ α) :
+    Module.finrank ℂ (hilbertSpace μ) * (Fintype.card ι).choose (μ.card-μ.rowLen 0) ≤
+      Module.finrank ℂ (UnitaryFinite.fixedSpace (V := hilbertSpace μ) UnitaryFinite.tupleAction f (relabelledUnitary μ e)) *
+        μ.card.choose (μ.card-μ.rowLen 0) := by
+  rw [(relabelledTupleFixed μ e f).finrank_eq]
+  exact tupleFixed_dimension μ (f.trans e.toEmbedding)
+
+end Thorp.Specht
+
+namespace Thorp.SparseContact
+open scoped BigOperators Classical
+
+noncomputable def pointStabilizer {α : Type*} (A : Finset α) : Subgroup (Equiv.Perm α) where
+  carrier := {p | ∀ x ∈ A, p x = x}
+  one_mem' := by intro x _; rfl
+  mul_mem' := by intro p q hp hq x hx; change p (q x) = x; rw [hq x hx, hp x hx]
+  inv_mem' := by intro p hp x hx; exact (Equiv.Perm.inv_eq_iff_eq).mpr (hp x hx).symm
+
+lemma swap_mem_pointStabilizer {α : Type*} [DecidableEq α] (A : Finset α)
+    (x y : α) (hx : x ∉ A) (hy : y ∉ A) : Equiv.swap x y ∈ pointStabilizer A := by
+  intro z hz
+  exact Equiv.swap_apply_of_ne_of_ne (fun h => hx (h ▸ hz)) (fun h => hy (h ▸ hz))
+
+lemma point_symmetrizer_zero (μ : YoungDiagram) (A : Finset (Specht.Cell μ))
+    (hA : A.card < μ.card - μ.rowLen 0) (v : Specht.Tabloid μ → ℂ) (hv : v ∈ Specht.space μ) :
+    Specht.symmetrizer (pointStabilizer A) (Specht.tabloidRep μ) v = 0 := by
+  classical
+  induction hv using Submodule.span_induction with
+  | mem v hv =>
+    obtain ⟨p,rfl⟩ := hv
+    let e : {x : Specht.Cell μ // p x ∉ A} ≃ {x : Specht.Cell μ // x ∉ A} :=
+      { toFun := fun x => ⟨p x.1,x.2⟩
+        invFun := fun x => ⟨p.symm x.1, by simpa only [p.apply_symm_apply] using x.2⟩
+        left_inv := fun x => by apply Subtype.ext; exact p.symm_apply_apply _
+        right_inv := fun x => by apply Subtype.ext; exact p.apply_symm_apply _ }
+    have hc : μ.rowLen 0 < Fintype.card {x : Specht.Cell μ // p x ∉ A} := by
+      rw [Fintype.card_congr e, Fintype.card_subtype_compl, Specht.card_cell, Fintype.card_coe]
+      have hr := Specht.rowLen_zero_le_card μ
+      omega
+    obtain ⟨x,y,hne,hcol⟩ := Fintype.exists_ne_map_eq_of_card_lt
+      (fun x : {x : Specht.Cell μ // p x ∉ A} => Specht.colIndex x.1)
+      (by simpa only [Fintype.card_fin] using hc)
+    have hxy : x.1 ≠ y.1 := fun h => hne (Subtype.ext h)
+    let h : pointStabilizer A := ⟨Equiv.swap (p x.1) (p y.1),
+      swap_mem_pointStabilizer A _ _ x.2 y.2⟩
+    apply Specht.symmetrizer_eq_zero_of_neg _ _ h
+    change Specht.tabloidRep μ (Equiv.swap (p x.1) (p y.1))
+      (Specht.tabloidRep μ p (Specht.polytabloid μ)) = -Specht.tabloidRep μ p (Specht.polytabloid μ)
+    rw [←Module.End.mul_apply, ←map_mul, ←Equiv.mul_swap_eq_swap_mul]
+    rw [map_mul, Module.End.mul_apply,
+      Specht.swap_polytabloid x.1 y.1 hxy (congrArg Fin.val hcol), map_neg]
+  | zero => exact map_zero _
+  | add v w hv hw ihv ihw => simp only [map_add,ihv,ihw,add_zero]
+  | smul a v hv ih => simp only [map_smul,ih,smul_zero]
+
+lemma point_fixed_zero (μ : YoungDiagram) (A : Finset (Specht.Cell μ))
+    (hA : A.card < μ.card - μ.rowLen 0) (v : Specht.Tabloid μ → ℂ) (hv : v ∈ Specht.space μ)
+    (hfix : ∀ p, p ∈ pointStabilizer A → Specht.tabloidRep μ p v = v) : v = 0 := by
+  classical
+  have hz := point_symmetrizer_zero μ A hA v hv
+  have he : Specht.symmetrizer (pointStabilizer A) (Specht.tabloidRep μ) v =
+      (Fintype.card (pointStabilizer A) : ℂ) • v := by
+    simp only [Specht.symmetrizer, LinearMap.sum_apply, hfix _ (Subtype.prop _),
+      Finset.sum_const, Finset.card_univ, nsmul_eq_mul, Nat.cast_smul_eq_nsmul]
+  rw [he] at hz
+  exact (smul_eq_zero.mp hz).resolve_left (by exact_mod_cast Fintype.card_ne_zero)
+
+lemma point_unitary_fixed_zero (μ : YoungDiagram) (A : Finset (Specht.Cell μ))
+    (hA : A.card < μ.card - μ.rowLen 0) (v : Specht.hilbertSpace μ)
+    (hfix : ∀ p, p ∈ pointStabilizer A → Specht.unitaryRepresentation μ p v = v) : v = 0 := by
+  let w := (Specht.spaceHilbertEquiv μ).symm v
+  have hw : ∀ p, p ∈ pointStabilizer A → Specht.tabloidRep μ p w.val = w.val := by
+    intro p hp
+    have he := congrArg ((Specht.spaceHilbertEquiv μ).symm) (hfix p hp)
+    change Specht.representation μ p w = w at he
+    exact congrArg Subtype.val he
+  have hz := point_fixed_zero μ A hA w.val w.prop hw
+  have hw0 : w = 0 := Subtype.ext hz
+  have he := congrArg (Specht.spaceHilbertEquiv μ) hw0
+  simpa only [w, LinearEquiv.apply_symm_apply, map_zero] using he
+
+lemma point_relabelled_fixed_zero {α : Type*} [Fintype α] (μ : YoungDiagram)
+    (e : α ≃ Specht.Cell μ) (A : Finset α) (hA : A.card < μ.card - μ.rowLen 0)
+    (v : Specht.hilbertSpace μ)
+    (hfix : ∀ p, p ∈ pointStabilizer A → Specht.relabelledUnitary μ e p v = v) : v = 0 := by
+  classical
+  apply point_unitary_fixed_zero μ (A.image e) (by simpa only [Finset.card_image_of_injective _ e.injective] using hA) v
+  intro p hp
+  have hf : e.symm.permCongr p ∈ pointStabilizer A := by
+    intro x hx
+    have hh := hp (e x) (Finset.mem_image.mpr ⟨x,hx,rfl⟩)
+    simp only [Equiv.permCongr_apply, Equiv.symm_symm, hh, e.symm_apply_apply]
+  have hh := hfix _ hf
+  change Specht.unitaryRepresentation μ (e.permCongr (e.symm.permCongr p)) v = v at hh
+  have he : e.permCongr (e.symm.permCongr p) = p := by
+    apply Equiv.ext
+    intro x
+    simp only [Equiv.permCongr_apply, Equiv.symm_symm, e.apply_symm_apply]
+  simpa only [he] using hh
+
+
+lemma small_dominating_set {V : Type*} [Fintype V] (G : SimpleGraph V)
+    (hG : ∀ v, ∃ w, G.Adj v w) :
+    ∃ R : Finset V, 2 * R.card ≤ Fintype.card V ∧
+      ∀ v, v ∉ R → ∃ w ∈ R, G.Adj v w := by
+  classical
+  obtain ⟨F, hFG, hF, hreach⟩ := G.exists_isAcyclic_reachable_eq_le
+  let c := hF.coloringTwo
+  have hneigh (v : V) : ∃ w, F.Adj v w := by
+    obtain ⟨w, hvw⟩ := hG v
+    have hr : F.Reachable v w := by rw [hreach]; exact hvw.reachable
+    exact hr.nonempty_neighborSet_left hvw.ne
+  let A : Finset V := Finset.univ.filter (fun v => c v = 0)
+  let B : Finset V := Finset.univ.filter (fun v => c v ≠ 0)
+  have hab : A.card + B.card = Fintype.card V := by
+    exact Finset.card_filter_add_card_filter_not _
+  have hcolor {v w : V} (hvw : F.Adj v w) : c v ≠ c w := c.valid hvw
+  by_cases h : A.card ≤ B.card
+  · refine ⟨A, by omega, ?_⟩
+    intro v hv
+    obtain ⟨w, hvw⟩ := hneigh v
+    have hv0 : c v ≠ 0 := by simpa [A] using hv
+    have hw0 : c w = 0 := by
+      have hv1 : c v = 1 := by omega
+      have hn := hcolor hvw
+      omega
+    exact ⟨w, by simp [A, hw0], hFG hvw⟩
+  · refine ⟨B, by omega, ?_⟩
+    intro v hv
+    obtain ⟨w, hvw⟩ := hneigh v
+    have hv0 : c v = 0 := by simpa [B] using hv
+    have hw0 : c w ≠ 0 := by simpa [hv0] using (hcolor hvw).symm
+    exact ⟨w, by simp [B, hw0], hFG hvw⟩
+
+lemma powerset_weight_le_exp {V : Type*} [DecidableEq V] (S : Finset V)
+    (p : V → ℝ) (hp : ∀ v, 0 ≤ p v) :
+    (∑ L ∈ S.powerset, ∏ v ∈ L, p v) ≤ Real.exp (∑ v ∈ S, p v) := by
+  rw [← Finset.prod_one_add, Real.exp_sum]
+  apply Finset.prod_le_prod₀
+  · intro v _; exact add_nonneg (by norm_num) (hp v)
+  · intro v _
+    have h := Real.add_one_le_exp (p v)
+    linarith
+
+lemma fixed_size_weight_le {V : Type*} [DecidableEq V] (S : Finset V)
+    (p : V → ℝ) (hp : ∀ v, 0 ≤ p v) (m : ℕ) (δ : ℝ) (hδ : 0 < δ) :
+    (∑ L ∈ S.powersetCard m, ∏ v ∈ L, p v) ≤
+      δ^m * Real.exp ((∑ v ∈ S, p v)/δ) := by
+  have hsplit (L : Finset V) (hL : L ∈ S.powersetCard m) :
+      (∏ v ∈ L, p v) = δ^m * ∏ v ∈ L, (p v / δ) := by
+    rw [Finset.prod_div_distrib, Finset.prod_const, (Finset.mem_powersetCard.mp hL).2]
+    field_simp
+  calc
+    _ = δ^m * ∑ L ∈ S.powersetCard m, ∏ v ∈ L, (p v/δ) := by
+      rw [Finset.mul_sum]
+      exact Finset.sum_congr rfl hsplit
+    _ ≤ δ^m * ∑ L ∈ S.powerset, ∏ v ∈ L, (p v/δ) := by
+      apply mul_le_mul_of_nonneg_left _ (pow_nonneg hδ.le _)
+      apply Finset.sum_le_sum_of_subset_of_nonneg
+      · intro L hL
+        exact Finset.mem_powerset.mpr (Finset.mem_powersetCard.mp hL).1
+      · intro L _ _; exact Finset.prod_nonneg (fun v _ => div_nonneg (hp v) hδ.le)
+    _ ≤ δ^m * Real.exp (∑ v ∈ S, p v/δ) :=
+      mul_le_mul_of_nonneg_left
+        (powerset_weight_le_exp S (fun v => p v/δ) (fun v => div_nonneg (hp v) hδ.le))
+        (pow_nonneg hδ.le _)
+    _ = _ := by rw [Finset.sum_div]
+
+noncomputable def neighborhood {V : Type*} [Fintype V] (G : SimpleGraph V)
+    (R : Finset V) : Finset V := by
+  classical
+  exact Finset.univ.filter (fun w => ∃ v ∈ R, G.Adj v w)
+
+lemma neighborhood_weight_le {V : Type*} [Fintype V] (G : SimpleGraph V)
+    (p : V → ℝ) (hp : ∀ v, 0 ≤ p v) (δ : ℝ)
+    (hdeg : ∀ v, (∑ w, if G.Adj v w then p w else 0) ≤ δ) (R : Finset V) :
+    (∑ w ∈ neighborhood G R, p w) ≤ R.card * δ := by
+  classical
+  calc
+    _ ≤ ∑ w ∈ neighborhood G R, ∑ v ∈ R, if G.Adj v w then p w else 0 := by
+      apply Finset.sum_le_sum
+      intro w hw
+      obtain ⟨v, hv, hvw⟩ := (Finset.mem_filter.mp hw).2
+      have h : (if G.Adj v w then p w else 0) ≤
+          ∑ u ∈ R, if G.Adj u w then p w else 0 :=
+        Finset.single_le_sum (f := fun u => if G.Adj u w then p w else 0) (fun u _ => by split_ifs; exact hp w; exact le_rfl) hv
+      simpa only [ite_eq_left hvw] using h
+    _ ≤ ∑ w, ∑ v ∈ R, if G.Adj v w then p w else 0 := by
+      apply Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _)
+      intro w _ _
+      exact Finset.sum_nonneg (fun v _ => by split_ifs; exact hp w; exact le_rfl)
+    _ = ∑ v ∈ R, ∑ w, if G.Adj v w then p w else 0 := Finset.sum_comm
+    _ ≤ ∑ v ∈ R, δ := Finset.sum_le_sum (fun v _ => hdeg v)
+    _ = _ := by simp
+
+lemma root_leaves_weight_le {V : Type*} [Fintype V] (G : SimpleGraph V)
+    (p : V → ℝ) (hp : ∀ v, 0 ≤ p v) (k : ℕ) (δ : ℝ)
+    (hδ : 0 < δ) (hδ1 : δ ≤ 1)
+    (hdeg : ∀ v, (∑ w, if G.Adj v w then p w else 0) ≤ δ)
+    (R : Finset V) (hR : 2 * R.card ≤ k) :
+    (∑ L ∈ (neighborhood G R).powersetCard (k-R.card), ∏ v ∈ L, p v) ≤
+      δ^((k:ℝ)/2) * Real.exp R.card := by
+  classical
+  have hmass := neighborhood_weight_le G p hp δ hdeg R
+  have hpow : δ^(k-R.card) ≤ δ^((k:ℝ)/2) := by
+    rw [← Real.rpow_natCast]
+    apply Real.rpow_le_rpow_of_exponent_ge hδ hδ1
+    have hr : R.card ≤ k := by omega
+    rw [Nat.cast_sub hr]
+    have hh : (2:ℝ)*R.card ≤ k := by exact_mod_cast hR
+    linarith
+  apply (fixed_size_weight_le (neighborhood G R) p hp _ δ hδ).trans
+  apply mul_le_mul hpow
+  · apply Real.exp_le_exp.mpr
+    exact (div_le_iff₀ hδ).mpr hmass
+  · positivity
+  · positivity
+
+def NoIsolated {V : Type*} (G : SimpleGraph V) (S : Finset V) : Prop :=
+  ∀ v ∈ S, ∃ w ∈ S, G.Adj v w
+
+lemma small_dominating_subset {V : Type*} [Fintype V] (G : SimpleGraph V)
+    (S : Finset V) (hS : NoIsolated G S) :
+    ∃ R : Finset V, R ⊆ S ∧ 2 * R.card ≤ S.card ∧ S \ R ⊆ neighborhood G R := by
+  classical
+  let H := G.induce (S : Set V)
+  have hH : ∀ v : S, ∃ w : S, H.Adj v w := by
+    intro v
+    obtain ⟨w, hw, hvw⟩ := hS v v.2
+    exact ⟨⟨w, hw⟩, hvw⟩
+  obtain ⟨R, hR, hdom⟩ := small_dominating_set H hH
+  let T := R.image (fun v : S => (v : V))
+  have hc : T.card = R.card := Finset.card_image_of_injective _ Subtype.val_injective
+  refine ⟨T, ?_, ?_, ?_⟩
+  · intro v hv
+    obtain ⟨w, _, rfl⟩ := Finset.mem_image.mp hv
+    exact w.2
+  · simpa [hc, H] using hR
+  · intro v hv
+    have hvS := (Finset.mem_sdiff.mp hv).1
+    have hvR : (⟨v, hvS⟩ : S) ∉ R := by
+      intro hh
+      exact (Finset.mem_sdiff.mp hv).2 (Finset.mem_image.mpr ⟨⟨v,hvS⟩, hh, rfl⟩)
+    obtain ⟨w, hw, hvw⟩ := hdom ⟨v,hvS⟩ hvR
+    exact Finset.mem_filter.mpr ⟨Finset.mem_univ _,
+      ⟨w, Finset.mem_image.mpr ⟨w,hw,rfl⟩, hvw.symm⟩⟩
+
+end Thorp.SparseContact
+
+end ThorpNine.Harmonic
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Harmonic/PalindromeFamilyCost.lean b/lean/OAI/Probability/ThorpRouting/Harmonic/PalindromeFamilyCost.lean
index b2cf9a579ef63cc544292658384ad7ec5d31bcd8..3756d494f6a3312442c1862d8cefdfc52aed306e 100644
--- a/lean/OAI/Probability/ThorpRouting/Harmonic/PalindromeFamilyCost.lean
+++ b/lean/OAI/Probability/ThorpRouting/Harmonic/PalindromeFamilyCost.lean
@@ -1,880 +1,880 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Harmonic.CutoffMass
-
-namespace OAI
-
-namespace ThorpNine.Harmonic
-
-namespace Thorp
-
-lemma familyEnd_two (L n : ℕ) : 2*familyEnd L n ≤ L*4^n := by
-  cases n with
-  | zero => simp [familyEnd]
-  | succ n => simp only [familyEnd,Nat.succ_ne_zero,ite_false,Nat.add_one_sub_one,pow_succ]; nlinarith
-
-lemma familyEnd_ge (L n j : ℕ) (hj : j < n) : 2*(L*4^j) ≤ familyEnd L n := by
-  cases n with
-  | zero => omega
-  | succ n =>
-    simp only [familyEnd,Nat.succ_ne_zero,ite_false,Nat.add_one_sub_one]
-    apply Nat.mul_le_mul_left
-    exact Nat.mul_le_mul_left L (Nat.pow_le_pow_right (by decide) (by omega : j ≤ n))
-
-noncomputable def palindromeFamilyCost (L n d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) : ℕ :=
-  ∑ j ∈ Finset.range n, palindromeSlabCost (L*4^j) d e ω
-
-lemma palindromeFamilyCost_adapted (L n d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) :
-    LayerAdapted d (familyEnd L n) (fun Y => palindromeFamilyCost L n d e (Y,X)) := by
-  apply layerAdapted_sum
-  intro j hj
-  exact layerAdapted_mono (palindromeSlabCost_adapted (L*4^j) d e X)
-    (familyEnd_ge L n j (Finset.mem_range.mp hj))
-
-theorem palindrome_family_mgf (L n d : ℕ) (hL : 0 < L) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) ≤
-      Real.exp (Fintype.card ι * (68*∑ j ∈ Finset.range n,
-        Real.exp (-((L*4^j:ℕ):ℝ)/64))) := by
-  induction n with
-  | zero => simp only [palindromeFamilyCost,Finset.range_zero,Finset.sum_empty,pow_zero,
-      finiteMean_const,mul_zero,Real.exp_zero,le_refl]
-  | succ n ih =>
-    let l := L*4^n
-    have hl : 0 < l := Nat.mul_pos hL (pow_pos (by decide) _)
-    have hs : 2*familyEnd L n ≤ l := familyEnd_two L n
-    have hc : ∀ Y, palindromeFamilyCost L (n+1) d e (Y,X) =
-        palindromeFamilyCost L n d e (Y,X)+palindromeSlabCost l d e (Y,X) := by
-      intro Y
-      exact Finset.sum_range_succ _ _
-    have hm : finiteMean (fun Y => q^(palindromeFamilyCost L (n+1) d e (Y,X))) ≤
-        finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) *
-          Real.exp (Fintype.card ι*(68*Real.exp (-(l:ℝ)/64))) := by
-      by_cases hld : l ≤ d
-      · simp_rw [hc,pow_add]
-        exact palindrome_slab_adapted_mul_le d (familyEnd L n) l (by omega) hl hs e X _
-          (fun _ => pow_nonneg (by linarith) _)
-          (layerAdapted_comp (palindromeFamilyCost_adapted L n d e X) (fun c => q^c))
-          q hq hlog
-      · simp_rw [hc,palindromeSlabCost_gt l d (by omega) e,Nat.add_zero]
-        exact le_mul_of_one_le_right
-          (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
-          (Real.one_le_exp_iff.mpr (by positivity))
-    refine (hm.trans (mul_le_mul_of_nonneg_right ih (Real.exp_nonneg _))).trans_eq ?_
-    rw [←Real.exp_add,Finset.sum_range_succ]
-    congr 1
-    dsimp [l]
-    ring
-
-lemma pow_four_ge (j : ℕ) : j+1 ≤ 4^j := by
-  induction j with
-  | zero => norm_num
-  | succ j ih => rw [pow_succ]; omega
-
-lemma sum_family_decay (L n : ℕ) (hL : 0 < L) :
-    (∑ j ∈ Finset.range n, Real.exp (-((L*4^j:ℕ):ℝ)/64)) ≤
-      Real.exp (-(L:ℝ)/64)/(1-Real.exp (-(1:ℝ)/64)) := by
-  let r := Real.exp (-(1:ℝ)/64)
-  have hr : 0 < r := Real.exp_pos _
-  have hr1 : r < 1 := Real.exp_lt_one_iff.mpr (by norm_num)
-  have hg : (∑ j ∈ Finset.range n, r^j) ≤ 1/(1-r) := by
-    apply (le_div_iff₀ (by linarith : (0:ℝ)<1-r)).mpr
-    have hh := geom_sum_mul r n
-    nlinarith [pow_nonneg (le_of_lt hr) n]
-  calc
-    _ ≤ ∑ j ∈ Finset.range n, Real.exp (-(L:ℝ)/64)*r^j := by
-      apply Finset.sum_le_sum
-      intro j _
-      have hj := pow_four_ge j
-      have hlj : L+j ≤ L*4^j := by nlinarith
-      have hlj' : (L:ℝ)+j ≤ (L*4^j:ℕ) := by exact_mod_cast hlj
-      rw [←Real.exp_nat_mul,←Real.exp_add]
-      exact Real.exp_le_exp.mpr (by linarith)
-    _ = Real.exp (-(L:ℝ)/64)*(∑ j ∈ Finset.range n, r^j) := by rw [Finset.mul_sum]
-    _ ≤ Real.exp (-(L:ℝ)/64)*(1/(1-r)) :=
-      mul_le_mul_of_nonneg_left hg (Real.exp_nonneg _)
-    _ = _ := by ring
-
-
-lemma palindromeLevelCost_step_add (t d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (d+1)) (ω : BenesCoins (d+1)) :
-    palindromeLevelCost t (d+1) e ω =
-      let σ := benesStepEquiv d ω
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x σ.2.1
-      let hc := PairRouting.colors_compatible x σ.2.1
-      (if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
-          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) else 0) +
-        palindromeLevelCost t d (PairRouting.childEmbedding x c hc false) σ.1.1 +
-        palindromeLevelCost t d (PairRouting.childEmbedding x c hc true) σ.1.2 := by
-  classical
-  rw [palindromeLevelCost]
-  dsimp only
-  split_ifs with h
-  · simp only [palindromeLevelCost_gt t d (by omega),Nat.add_zero]
-  · simp only [Nat.zero_add]
-
-lemma palindromeLowCost_sum (H d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) :
-    palindromeLowCost H d e ω = ∑ i ∈ Finset.range H, palindromeLevelCost (i+1) d e ω := by
-  classical
-  induction d generalizing ι with
-  | zero => simp only [palindromeLowCost,palindromeLevelCost,Finset.sum_const_zero]
-  | succ d ih =>
-    rw [palindromeLowCost]
-    simp only [palindromeLevelCost_step_add,Finset.sum_add_distrib]
-    rw [←ih,←ih]
-    congr 2
-    simp only [Nat.add_right_cancel_iff,Finset.sum_ite_eq',Finset.mem_range,Nat.succ_le_iff]
-
-lemma palindromeLowCost_slab (L d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) :
-    palindromeLowCost L d e ω + palindromeSlabCost L d e ω = palindromeLowCost (2*L) d e ω := by
-  rw [palindromeLowCost_sum,palindromeLowCost_sum,show 2*L=L+L by omega,Finset.sum_range_add]
-  congr 1
-  exact Fin.sum_univ_eq_sum_range (fun i => palindromeLevelCost (L+i+1) d e ω) L
-
-lemma palindromeCost_family (L n d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) :
-    palindromeLowCost L d e ω + palindromeFamilyCost L n d e ω +
-      palindromeFamilyCost (2*L) n d e ω = palindromeLowCost (L*4^n) d e ω := by
-  induction n with
-  | zero => simp only [palindromeFamilyCost,Finset.range_zero,Finset.sum_empty,Nat.add_zero,
-      pow_zero,mul_one]
-  | succ n ih =>
-    have he : (2*L)*4^n = 2*(L*4^n) := by ring
-    have he' : L*4^(n+1) = 2*(2*(L*4^n)) := by rw [pow_succ]; ring
-    simp only [palindromeFamilyCost,Finset.sum_range_succ] at ih ⊢
-    rw [he,he',←palindromeLowCost_slab,←palindromeLowCost_slab,←ih]
-    omega
-
-
-lemma finiteMean_mul_sq_le {Ω : Type*} [Fintype Ω] (f g : Ω → ℝ) :
-    (finiteMean (fun ω => f ω*g ω))^2 ≤ finiteMean (fun ω => (f ω)^2)*finiteMean (fun ω => (g ω)^2) := by
-  have h := Finset.sum_mul_sq_le_sq_mul_sq Finset.univ f g
-  simp only [finiteMean,div_pow,div_mul_div_comm,←pow_two]
-  exact div_le_div_of_nonneg_right h (sq_nonneg _)
-
-lemma finiteMean_mul_exp_le {Ω : Type*} [Fintype Ω] (f g : Ω → ℝ) (A B : ℝ)
-    (hf : finiteMean (fun ω => (f ω)^2) ≤ Real.exp A)
-    (hg : finiteMean (fun ω => (g ω)^2) ≤ Real.exp B) :
-    finiteMean (fun ω => f ω*g ω) ≤ Real.exp ((A+B)/2) := by
-  have hc := finiteMean_mul_sq_le f g
-  have hn : 0 ≤ finiteMean (fun ω => (g ω)^2) := finiteMean_nonneg (fun _ => sq_nonneg _)
-  have hu := mul_le_mul hf hg hn (Real.exp_nonneg _)
-  have he : Real.exp A*Real.exp B = (Real.exp ((A+B)/2))^2 := by
-    rw [←Real.exp_add,←Real.exp_nat_mul]
-    congr 1
-    push_cast
-    ring
-  rw [he] at hu
-  have hh := hc.trans hu
-  have hep := Real.exp_pos ((A+B)/2)
-  nlinarith
-
-
-noncomputable def heightDecay (H : ℕ) : ℝ :=
-  68*Real.exp (-(H:ℝ)/64)/(1-Real.exp (-(1:ℝ)/64))
-
-lemma heightDecay_nonneg (H : ℕ) : 0 ≤ heightDecay H := by
-  have h : Real.exp (-(1:ℝ)/64) < 1 := Real.exp_lt_one_iff.mpr (by norm_num)
-  apply div_nonneg (by positivity) (by linarith)
-
-lemma heightDecay_antitone : Antitone heightDecay := by
-  intro L H h
-  have hden : 0 ≤ 1-Real.exp (-(1:ℝ)/64) := by
-    have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
-    linarith
-  apply div_le_div_of_nonneg_right _ hden
-  apply mul_le_mul_of_nonneg_left _ (by norm_num)
-  apply Real.exp_le_exp.mpr
-  have h' : (L:ℝ) ≤ H := by exact_mod_cast h
-  linarith
-
-lemma palindrome_family_mgf_decay (L n d : ℕ) (hL : 0 < L) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) ≤
-      Real.exp (Fintype.card ι*heightDecay L) := by
-  apply (palindrome_family_mgf L n d hL e X q hq hlog).trans
-  apply Real.exp_le_exp.mpr
-  apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
-  have h := mul_le_mul_of_nonneg_left (sum_family_decay L n hL) (by norm_num : (0:ℝ)≤68)
-  simpa only [heightDecay,mul_div_assoc] using h
-
-theorem palindrome_high_mgf (H n d : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/500) :
-    finiteMean (fun Y => q^(palindromeFamilyCost H n d e (Y,X)+
-      palindromeFamilyCost (2*H) n d e (Y,X))) ≤ Real.exp (Fintype.card ι*heightDecay H) := by
-  have hq2 := one_le_pow₀ (n:=2) hq
-  have hlog2 : Real.log (q^2) ≤ 1/250 := by rw [Real.log_pow]; norm_num; linarith
-  have h₀ := palindrome_family_mgf_decay H n d hH e X (q^2) hq2 hlog2
-  have h₁ := palindrome_family_mgf_decay (2*H) n d (by omega) e X (q^2) hq2 hlog2
-  have h₁' := h₁.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left
-    (heightDecay_antitone (show H ≤ 2*H by omega)) (Nat.cast_nonneg _)))
-  have hh := finiteMean_mul_exp_le (fun Y => q^(palindromeFamilyCost H n d e (Y,X)))
-    (fun Y => q^(palindromeFamilyCost (2*H) n d e (Y,X)))
-    (Fintype.card ι*heightDecay H) (Fintype.card ι*heightDecay H)
-    (by simpa only [pow_right_comm] using h₀) (by simpa only [pow_right_comm] using h₁')
-  simpa only [←pow_add,add_self_div_two] using hh
-
-lemma palindrome_high_full_mgf (H n d : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/500) :
-    finiteMean (fun ω : BenesCoins d => q^(palindromeFamilyCost H n d e ω+
-      palindromeFamilyCost (2*H) n d e ω)) ≤ Real.exp (Fintype.card ι*heightDecay H) := by
-  rw [finiteMean_prod,finiteMean_comm]
-  calc
-    _ ≤ finiteMean (fun _X : SwitchIndex d → Bool => Real.exp (Fintype.card ι*heightDecay H)) := by
-      apply finiteMean_mono
-      intro X
-      exact palindrome_high_mgf H n d hH e X q hq hlog
-    _ = _ := finiteMean_const _
-
-theorem palindrome_cost_mgf_cutoff (r H : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (H+r)) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/1000)
-    (hr : (2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*(q^2)^H) ≤ 1/2) :
-    finiteMean (fun ω : BenesCoins (H+r) => q^(palindromeCost (H+r) e ω)) ≤
-      Real.exp (((2:ℝ)^r*(2*((2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*(q^2)^H))^2)+
-        Fintype.card ι*heightDecay H)/2) := by
-  let d := H+r
-  have hd : d ≤ H*4^d := by
-    have hh := pow_four_ge d
-    nlinarith
-  have hcost (ω : BenesCoins d) : palindromeCost d e ω = palindromeLowCost H d e ω+
-      (palindromeFamilyCost H d d e ω+palindromeFamilyCost (2*H) d d e ω) := by
-    have hh := palindromeCost_family H d d e ω
-    rw [palindromeLowCost_eq _ _ hd] at hh
-    simpa only [Nat.add_assoc] using hh.symm
-  have hq2 := one_le_pow₀ (n:=2) hq
-  have hlog2 : Real.log (q^2) ≤ 1/500 := by rw [Real.log_pow]; norm_num; linarith
-  have hlow := palindrome_low_mgf r H e (q^2) hq2 hr
-  have hhigh := palindrome_high_full_mgf H d d hH e (q^2) hq2 hlog2
-  have hh := finiteMean_mul_exp_le (fun ω : BenesCoins d => q^(palindromeLowCost H d e ω))
-    (fun ω : BenesCoins d => q^(palindromeFamilyCost H d d e ω+palindromeFamilyCost (2*H) d d e ω))
-    _ _ (by simpa only [pow_right_comm] using hlow) (by simpa only [pow_right_comm] using hhigh)
-  change finiteMean (fun ω : BenesCoins d => q^(palindromeCost d e ω)) ≤ _
-  simpa only [hcost,pow_add,pow_right_comm] using hh
-
-
-lemma cutoffMass_le (H d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (ω : BenesCoins d) :
-    cutoffMass H d e ω ≤ Fintype.card ι := by
-  classical
-  induction d generalizing ι with
-  | zero => exact Nat.zero_le _
-  | succ d ih =>
-    rw [cutoffMass]
-    split_ifs
-    · exact crowdedMass_le _
-    · exact (Nat.add_le_add (ih _ _) (ih _ _)).trans_eq (PairRouting.color_card_add _)
-
-lemma palindromeLowCost_le (H d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) : palindromeLowCost H d e ω ≤ H*Fintype.card ι :=
-  (palindromeLowCost_le_cutoff H d e ω).trans (Nat.mul_le_mul_left H (cutoffMass_le H d e ω))
-
-theorem palindrome_cost_mgf_coarse (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/1000) :
-    finiteMean (fun ω : BenesCoins d => q^(palindromeCost d e ω)) ≤
-      Real.exp (Fintype.card ι*(1+heightDecay 1)) := by
-  have hd : d ≤ 1*4^d := by have hh := pow_four_ge d; omega
-  have hcost (ω : BenesCoins d) : palindromeCost d e ω = palindromeLowCost 1 d e ω+
-      (palindromeFamilyCost 1 d d e ω+palindromeFamilyCost 2 d d e ω) := by
-    have hh := palindromeCost_family 1 d d e ω
-    rw [palindromeLowCost_eq _ _ hd] at hh
-    simpa only [Nat.add_assoc] using hh.symm
-  have hq2 := one_le_pow₀ (n:=2) hq
-  have hlog2 : Real.log (q^2) ≤ 1/500 := by rw [Real.log_pow]; norm_num; linarith
-  have hlow : finiteMean (fun ω : BenesCoins d => (q^2)^(palindromeLowCost 1 d e ω)) ≤
-      Real.exp (Fintype.card ι) := by
-    calc
-      _ ≤ finiteMean (fun _ω : BenesCoins d => (q^2)^(Fintype.card ι)) := by
-        apply finiteMean_mono
-        intro ω
-        apply pow_le_pow_right₀ hq2
-        simpa only [one_mul] using palindromeLowCost_le 1 d e ω
-      _ = (q^2)^(Fintype.card ι) := finiteMean_const _
-      _ = Real.exp (Fintype.card ι * Real.log (q^2)) := by
-        rw [Real.exp_nat_mul,Real.exp_log (by positivity : 0<q^2)]
-      _ ≤ _ := Real.exp_le_exp.mpr (by
-        have hh := mul_le_mul_of_nonneg_left hlog2 (Nat.cast_nonneg (Fintype.card ι) : (0:ℝ)≤_)
-        nlinarith [Nat.cast_nonneg (α:=ℝ) (Fintype.card ι)])
-  have hhigh := palindrome_high_full_mgf 1 d d (by decide) e (q^2) hq2 hlog2
-  have hh := finiteMean_mul_exp_le (fun ω : BenesCoins d => q^(palindromeLowCost 1 d e ω))
-    (fun ω : BenesCoins d => q^(palindromeFamilyCost 1 d d e ω+palindromeFamilyCost 2 d d e ω))
-    (Fintype.card ι) (Fintype.card ι*heightDecay 1)
-    (by simpa only [pow_right_comm] using hlow) (by simpa only [pow_right_comm] using hhigh)
-  simp only [←pow_add,←hcost] at hh
-  apply hh.trans (Real.exp_le_exp.mpr _)
-  have hk : (0:ℝ) ≤ Fintype.card ι := Nat.cast_nonneg _
-  have hD := heightDecay_nonneg 1
-  nlinarith
-
-noncomputable def momentBase : ℝ := (2:ℝ)^(1/1000:ℝ)
-
-lemma momentBase_one_le : 1 ≤ momentBase := by
-  exact Real.one_le_rpow (by norm_num) (by norm_num)
-
-lemma momentBase_log : Real.log momentBase ≤ 1/1000 := by
-  rw [momentBase,Real.log_rpow (by norm_num)]
-  have hh := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
-  norm_num at hh ⊢
-  linarith
-
-lemma momentBase_pow (c : ℕ) : momentBase^c = (2:ℝ)^((c:ℝ)*(1/1000:ℝ)) := by
-  rw [mul_comm,Real.rpow_mul_natCast (by norm_num)]
-  rfl
-
-lemma palindromeRowMoment_le_cost (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) :
-    palindromeRowMoment d e (1/1000) ≤
-      finiteMean (fun ω : BenesCoins d => momentBase^(palindromeCost d e ω)) := by
-  have hn : (0:ℝ) < ((2^d:ℕ):ℝ)^(Fintype.card ι) := by positivity
-  have hn0 : (0:ℝ) ≤ ((2^d).descFactorial (Fintype.card ι):ℝ) /
-      ((2^d:ℕ):ℝ)^(Fintype.card ι) := by positivity
-  have hn1 : ((2^d).descFactorial (Fintype.card ι):ℝ) /
-      ((2^d:ℕ):ℝ)^(Fintype.card ι) ≤ 1 := by
-    apply (div_le_one hn).mpr
-    exact_mod_cast Nat.descFactorial_le_pow (2^d) (Fintype.card ι)
-  have hpow := Real.rpow_le_one hn0 hn1 (by norm_num : (0:ℝ)≤1/1000)
-  have hm := palindrome_row_density_moment d e (1/1000) (by norm_num)
-  simp_rw [←momentBase_pow] at hm
-  exact hm.trans (mul_le_of_le_one_left (finiteMean_nonneg (fun _ =>
-    pow_nonneg (le_trans (by norm_num) momentBase_one_le) _)) hpow)
-
-theorem palindrome_row_moment_coarse (d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) :
-    palindromeRowMoment d e (1/1000) ≤ Real.exp (Fintype.card ι*(1+heightDecay 1)) :=
-  (palindromeRowMoment_le_cost d e).trans
-    (palindrome_cost_mgf_coarse d e momentBase momentBase_one_le momentBase_log)
-
-
-noncomputable def densityCutoff (p : ℝ) : ℕ := ⌊-Real.log p/(4*Real.log 2)⌋₊
-
-lemma log_small_density {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
-    Real.log p ≤ -8*Real.log 2 := by
-  have he : Real.log (1/256:ℝ) = -8*Real.log 2 := by
-    rw [show (1/256:ℝ) = 1/2^8 by norm_num,
-      Real.log_div (by norm_num) (by positivity),Real.log_one,Real.log_pow]
-    ring
-  exact (Real.log_le_log hp hs).trans_eq he
-
-lemma densityCutoff_bounds {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
-    0 < densityCutoff p ∧
-    (densityCutoff p:ℝ)*(4*Real.log 2) ≤ -Real.log p ∧
-    -Real.log p < ((densityCutoff p:ℝ)+1)*(4*Real.log 2) := by
-  have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
-  have hd : 0 < 4*Real.log 2 := by positivity
-  have hh := log_small_density hp hs
-  have hx : 1 ≤ -Real.log p/(4*Real.log 2) := by
-    apply (le_div_iff₀ hd).mpr
-    nlinarith
-  refine ⟨Nat.floor_pos.mpr hx,?_,?_⟩
-  · exact (le_div_iff₀ hd).mp (Nat.floor_le (by linarith : 0 ≤ -Real.log p/(4*Real.log 2)))
-  · exact (div_lt_iff₀ hd).mp (Nat.lt_floor_add_one (-Real.log p/(4*Real.log 2)))
-
-lemma densityCutoff_decay {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
-    Real.exp (-(densityCutoff p:ℝ)/64) ≤ Real.exp (1/64) * p^(1/512:ℝ) := by
-  have hh := (densityCutoff_bounds hp hs).2.2
-  have hb : Real.log 2 ≤ 2 := by
-    have := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
-    linarith
-  have hH : 0 ≤ (densityCutoff p:ℝ)+1 := by positivity
-  have hu := mul_le_mul_of_nonneg_left hb hH
-  rw [Real.rpow_def_of_pos hp,←Real.exp_add]
-  apply Real.exp_le_exp.mpr
-  nlinarith
-
-lemma densityCutoff_power {p q : ℝ} (hp : 0 < p) (hs : p ≤ 1/256)
-    (hq : 0 < q) (hlog : Real.log q ≤ Real.log 2/4) (n : ℕ) (hn : n ≤ 4)
-    (hq1 : 1 ≤ q) :
-    p*(2:ℝ)^(densityCutoff p)*q^(n*densityCutoff p) ≤ p^(1/2:ℝ) := by
-  have hcut := (densityCutoff_bounds hp hs).2.1
-  have hH : 0 ≤ (densityCutoff p:ℝ) := Nat.cast_nonneg _
-  have hn' : (n:ℝ) ≤ 4 := by exact_mod_cast hn
-  have hln := Real.log_nonneg hq1
-  have hl := mul_le_mul_of_nonneg_right hn' hln
-  have hnl : (n:ℝ)*Real.log q ≤ Real.log 2 := by linarith
-  have hm := mul_le_mul_of_nonneg_left hnl hH
-  have heq : p*(2:ℝ)^(densityCutoff p)*q^(n*densityCutoff p) =
-      Real.exp (Real.log p+(densityCutoff p:ℝ)*Real.log 2+
-        (n*densityCutoff p:ℕ)*Real.log q) := by
-    rw [Real.exp_add,Real.exp_add,Real.exp_nat_mul,Real.exp_nat_mul,
-      Real.exp_log hp,Real.exp_log hq,Real.exp_log (by norm_num : (0:ℝ)<2)]
-  rw [heq,Real.rpow_def_of_pos hp]
-  apply Real.exp_le_exp.mpr
-  push_cast
-  nlinarith
-
-lemma densityCutoff_sparse {p q : ℝ} (hp : 0 < p) (hs : p ≤ 1/256)
-    (hq : 0 < q) (hlog : Real.log q ≤ Real.log 2/4) (hq1 : 1 ≤ q) :
-    (2:ℝ)^(densityCutoff p)*(p*(q^2)^(densityCutoff p)) ≤ 1/2 := by
-  have hh := densityCutoff_power hp hs hq hlog 2 (by decide) hq1
-  rw [←pow_mul] at ⊢
-  have hle : p^(1/2:ℝ) ≤ 1/2 := by
-    have hlogp := log_small_density hp hs
-    have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
-    rw [Real.rpow_def_of_pos hp]
-    calc
-      _ ≤ Real.exp (-Real.log 2) := by apply Real.exp_le_exp.mpr; nlinarith
-      _ = 1/2 := by rw [Real.exp_neg,Real.exp_log (by norm_num : (0:ℝ)<2)]; norm_num
-  nlinarith [hh.trans hle]
-
-
-lemma momentBase_log_exact : Real.log momentBase = Real.log 2/1000 := by
-  rw [momentBase,Real.log_rpow (by norm_num)]
-  ring
-
-lemma densityCutoff_le_height (d k : ℕ) (hk : 0 < k) :
-    densityCutoff ((k:ℝ)/(2:ℝ)^d) ≤ d := by
-  have hk' : (1:ℝ) ≤ k := by exact_mod_cast hk
-  have hp : 0 < (k:ℝ)/(2:ℝ)^d := by positivity
-  have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
-  have hlogk := Real.log_nonneg hk'
-  by_cases hx : 0 ≤ -Real.log ((k:ℝ)/(2:ℝ)^d)/(4*Real.log 2)
-  · have hh := (le_div_iff₀ (by positivity : 0 < 4*Real.log 2)).mp (Nat.floor_le hx)
-    have hd : (0:ℝ) ≤ d := Nat.cast_nonneg _
-    have hH : (0:ℝ) ≤ densityCutoff ((k:ℝ)/(2:ℝ)^d) := Nat.cast_nonneg _
-    change (densityCutoff ((k:ℝ)/(2:ℝ)^d):ℝ)*(4*Real.log 2) ≤ -Real.log ((k:ℝ)/(2:ℝ)^d) at hh
-    rw [Real.log_div (by positivity) (by positivity),Real.log_pow] at hh
-    have hle : (densityCutoff ((k:ℝ)/(2:ℝ)^d):ℝ) ≤ d := by nlinarith
-    exact_mod_cast hle
-  · have hz := Nat.floor_of_nonpos (le_of_not_ge hx)
-    simp only [densityCutoff,hz,Nat.zero_le]
-
-noncomputable def densityTailConstant : ℝ :=
-  68*Real.exp (1/64)/(1-Real.exp (-(1:ℝ)/64))
-
-lemma densityTailConstant_nonneg : 0 ≤ densityTailConstant := by
-  have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
-  apply div_nonneg (by positivity) (by linarith)
-
-lemma heightDecay_cutoff {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
-    heightDecay (densityCutoff p) ≤ densityTailConstant*p^(1/512:ℝ) := by
-  have hden : 0 ≤ 1-Real.exp (-(1:ℝ)/64) := by
-    have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
-    linarith
-  have hh := div_le_div_of_nonneg_right
-    (mul_le_mul_of_nonneg_left (densityCutoff_decay hp hs) (by norm_num : (0:ℝ)≤68)) hden
-  simpa only [heightDecay,densityTailConstant,mul_assoc,div_mul_eq_mul_div] using hh
-
-lemma sparse_low_algebra (r H k : ℕ) (q : ℝ) :
-    (2:ℝ)^r*(2*((2:ℝ)^H*(((k:ℝ)/(2:ℝ)^(H+r))*(q^2)^H))^2) =
-    2*k*(((k:ℝ)/(2:ℝ)^(H+r))*(2:ℝ)^H*q^(4*H)) := by
-  rw [pow_add,←pow_mul,show 4*H = (2*H)*2 by omega,pow_mul]
-  have hH : (2:ℝ)^H ≠ 0 := by positivity
-  have hr : (2:ℝ)^r ≠ 0 := by positivity
-  field_simp
-  ring
-
-lemma palindrome_row_moment_small (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι)
-    (hs : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1/256) :
-    palindromeRowMoment d e (1/1000) ≤
-      Real.exp (Fintype.card ι*(1+densityTailConstant)*
-        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ)) := by
-  generalize hpdef : ((Fintype.card ι:ℝ)/(2:ℝ)^d) = p at hs ⊢
-  have hp : 0 < p := by rw [←hpdef]; positivity
-  have hle := densityCutoff_le_height d (Fintype.card ι) hk
-  rw [hpdef] at hle
-  obtain ⟨r,hr⟩ := Nat.exists_eq_add_of_le hle
-  subst d
-  have hH := (densityCutoff_bounds hp hs).1
-  have hq := momentBase_one_le
-  have hqp : 0 < momentBase := by linarith
-  have hlog : Real.log momentBase ≤ Real.log 2/4 := by
-    rw [momentBase_log_exact]
-    have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
-    linarith
-  have hsp := densityCutoff_sparse hp hs hqp hlog hq
-  rw [←hpdef] at hsp
-  have hh := palindrome_cost_mgf_cutoff r (densityCutoff p) hH e momentBase hq momentBase_log
-    (by simpa only [hpdef] using hsp)
-  apply (palindromeRowMoment_le_cost _ e).trans (hh.trans (Real.exp_le_exp.mpr _))
-  rw [sparse_low_algebra,hpdef]
-  have hlow := densityCutoff_power hp hs hqp hlog 4 (by decide) hq
-  have hp1 : p ≤ 1 := by linarith
-  have he := Real.rpow_le_rpow_of_exponent_ge hp hp1 (by norm_num : (1:ℝ)/512 ≤ 1/2)
-  have hlow' := mul_le_mul_of_nonneg_left (hlow.trans he)
-    (by positivity : (0:ℝ)≤2*Fintype.card ι)
-  have hhigh := mul_le_mul_of_nonneg_left (heightDecay_cutoff hp hs)
-    (Nat.cast_nonneg (Fintype.card ι): (0:ℝ)≤_)
-  have hnon : 0 ≤ Fintype.card ι*densityTailConstant*p^(1/512:ℝ) :=
-    mul_nonneg (mul_nonneg (Nat.cast_nonneg _) densityTailConstant_nonneg)
-      (Real.rpow_nonneg (le_of_lt hp) _)
-  nlinarith
-
-noncomputable def densityConstant : ℝ :=
-  256*(1+heightDecay 1)+1+densityTailConstant
-
-lemma densityConstant_pos : 0 < densityConstant := by
-  dsimp [densityConstant]
-  have hh := heightDecay_nonneg 1
-  have ht := densityTailConstant_nonneg
-  linarith
-
-theorem palindrome_row_moment_bound (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι) :
-    palindromeRowMoment d e (1/1000) ≤ Real.exp (densityConstant*Fintype.card ι*
-        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ)) := by
-  have hp : 0 < (Fintype.card ι:ℝ)/(2:ℝ)^d := by positivity
-  have hp1 : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1 := by
-    apply (div_le_one (by positivity)).mpr
-    have h := Fintype.card_le_of_injective e e.injective
-    rw [card_positions] at h
-    exact_mod_cast h
-  have hk0 : (0:ℝ) ≤ Fintype.card ι := Nat.cast_nonneg _
-  have hpow := Real.rpow_nonneg (le_of_lt hp) (1/512:ℝ)
-  by_cases hs : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1/256
-  · apply (palindrome_row_moment_small d e hk hs).trans (Real.exp_le_exp.mpr _)
-    have hD : 1+densityTailConstant ≤ densityConstant := by
-      dsimp [densityConstant]
-      have := heightDecay_nonneg 1
-      linarith
-    have hh := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hD hk0) hpow
-    nlinarith
-  · have hlow : (1:ℝ)/256 ≤ ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ) := by
-      have he := Real.rpow_le_rpow_of_exponent_ge hp hp1 (by norm_num : (1:ℝ)/512 ≤ 1)
-      rw [Real.rpow_one] at he
-      exact (le_of_not_ge hs).trans he
-    have hD : 256*(1+heightDecay 1) ≤ densityConstant := by
-      dsimp [densityConstant]
-      have := densityTailConstant_nonneg
-      linarith
-    apply (palindrome_row_moment_coarse d e).trans (Real.exp_le_exp.mpr _)
-    have hD0 : 0 ≤ 1+heightDecay 1 := by have := heightDecay_nonneg 1; linarith
-    have ha := mul_le_mul_of_nonneg_left hlow (mul_nonneg (by norm_num : (0:ℝ)≤256) hD0)
-    have hb := mul_le_mul_of_nonneg_right hD hpow
-    have hc := mul_le_mul_of_nonneg_right (ha.trans hb) hk0
-    nlinarith
-
-lemma palindromeRowMoment_nonneg (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (a : ℝ) : 0 ≤ palindromeRowMoment d e a := by
-  apply finiteMean_nonneg
-  intro f
-  exact Real.rpow_nonneg (mul_nonneg (Nat.cast_nonneg _)
-    (PairRouting.tupleProbability_nonneg _ _ _)) _
-
-theorem palindrome_log_row_moment (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι) :
-    Real.log (palindromeRowMoment d e (1/1000)) ≤ densityConstant*Fintype.card ι*
-        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ) := by
-  by_cases hz : palindromeRowMoment d e (1/1000) = 0
-  · rw [hz,Real.log_zero]
-    exact mul_nonneg (mul_nonneg (le_of_lt densityConstant_pos) (Nat.cast_nonneg _))
-      (Real.rpow_nonneg (by positivity) _)
-  · exact (Real.log_le_iff_le_exp (lt_of_le_of_ne (palindromeRowMoment_nonneg _ _ _) (Ne.symm hz))).mpr
-      (palindrome_row_moment_bound d e hk)
-
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-lemma orbitSet_eq_of_agree (p q : Equiv.Perm α) (x : α)
-    (h : ∀ z ∈ orbitSet p x, p z = q z) : orbitSet p x = orbitSet q x := by
-  have hp (n : ℕ) : (p^n) x ∈ orbitSet p x := by
-    apply (mem_orbitSet _ _ _).mpr
-    exact Equiv.Perm.sameCycle_pow_right.mpr (Equiv.Perm.SameCycle.refl _ _)
-  have he (n : ℕ) : (p^n) x = (q^n) x := by
-    induction n with
-    | zero => rfl
-    | succ n ih =>
-      rw [pow_succ',pow_succ',Equiv.Perm.mul_apply,Equiv.Perm.mul_apply,←ih]
-      exact h _ (hp n)
-  ext y
-  simp only [mem_orbitSet]
-  constructor
-  · intro hy
-    obtain ⟨n,hn⟩ := hy.exists_nat_pow_eq
-    rw [←hn,he]
-    exact Equiv.Perm.sameCycle_pow_right.mpr (Equiv.Perm.SameCycle.refl _ _)
-  · intro hy
-    obtain ⟨n,hn⟩ := hy.exists_nat_pow_eq
-    rw [←hn,←he]
-    exact Equiv.Perm.sameCycle_pow_right.mpr (Equiv.Perm.SameCycle.refl _ _)
-
-noncomputable def survivingCycle (p q : Equiv.Perm α) (S B : Finset α)
-    (h : ∀ x ∉ B, p x = q x) (C : RoutingNetwork.SelectedCycle p S)
-    (hC : ∀ x ∈ C.val, x ∉ B) : RoutingNetwork.SelectedCycle q S := by
-  refine ⟨C.val,?_,C.property.2⟩
-  obtain ⟨x,hx⟩ := C.property.1
-  refine ⟨x,hx.trans (orbitSet_eq_of_agree p q x ?_)⟩
-  intro z hz
-  exact h z (hC z (hx ▸ hz))
-
-noncomputable def perturbCycleMap (p q : Equiv.Perm α) (S B : Finset α)
-    (h : ∀ x ∉ B, p x = q x) (C : RoutingNetwork.SelectedCycle p S) :
-    RoutingNetwork.SelectedCycle q S ⊕ {x // x ∈ B} := by
-  classical
-  by_cases hb : ∃ x ∈ C.val, x ∈ B
-  · exact Sum.inr ⟨hb.choose,hb.choose_spec.2⟩
-  · exact Sum.inl (survivingCycle p q S B h C (by simpa only [not_exists,not_and] using hb))
-
-lemma perturbCycleMap_injective (p q : Equiv.Perm α) (S B : Finset α)
-    (h : ∀ x ∉ B, p x = q x) : Function.Injective (perturbCycleMap p q S B h) := by
-  classical
-  intro C D he
-  by_cases hC : ∃ x ∈ C.val, x ∈ B <;>
-    by_cases hD : ∃ x ∈ D.val, x ∈ B
-  · simp only [perturbCycleMap,dite_eq_left hC,dite_eq_left hD,Sum.inr.injEq] at he
-    have hx : hC.choose = hD.choose := congrArg Subtype.val he
-    exact RoutingNetwork.SelectedCycle.eq_of_mem C D hC.choose_spec.1
-      (hx.symm ▸ hD.choose_spec.1)
-  · simp only [perturbCycleMap,dite_eq_left hC,dite_eq_right hD] at he
-    cases he
-  · simp only [perturbCycleMap,dite_eq_right hC,dite_eq_left hD] at he
-    cases he
-  · simp only [perturbCycleMap,dite_eq_right hC,dite_eq_right hD,Sum.inl.injEq] at he
-    have hval : C.val = D.val := congrArg (fun Z : RoutingNetwork.SelectedCycle q S => Z.val) he
-    exact Subtype.ext hval
-
-theorem selectedCycle_perturb_le (p q : Equiv.Perm α) (S B : Finset α)
-    (h : ∀ x ∉ B, p x = q x) :
-    Fintype.card (RoutingNetwork.SelectedCycle p S) ≤
-      Fintype.card (RoutingNetwork.SelectedCycle q S)+B.card := by
-  have hh := Fintype.card_le_of_injective _ (perturbCycleMap_injective p q S B h)
-  simpa only [Fintype.card_sum,Fintype.card_coe] using hh
-
-
-noncomputable def permDistance (p q : Equiv.Perm α) : ℕ :=
-  (Finset.univ.filter (fun x => p x ≠ q x)).card
-
-lemma permDistance_sum (p q : Equiv.Perm α) :
-    permDistance p q = ∑ x : α, if p x = q x then 0 else 1 := by
-  classical
-  simp [permDistance,Finset.card_filter]
-
-lemma permDistance_refl (p : Equiv.Perm α) : permDistance p p = 0 := by
-  simp [permDistance]
-
-lemma permDistance_le (p q : Equiv.Perm α) : permDistance p q ≤ Fintype.card α :=
-  (Finset.card_filter_le _ _).trans_eq (Finset.card_univ)
-
-lemma permDistance_symm (p q : Equiv.Perm α) : permDistance p q = permDistance q p := by
-  simp only [permDistance,ne_comm]
-
-lemma permDistance_triangle (p q r : Equiv.Perm α) :
-    permDistance p r ≤ permDistance p q+permDistance q r := by
-  classical
-  simp only [permDistance_sum,←Finset.sum_add_distrib]
-  apply Finset.sum_le_sum
-  intro x _
-  by_cases hpq : p x = q x <;> by_cases hqr : q x = r x <;>
-    by_cases hpr : p x = r x <;> simp only [hpq,hqr,hpr,↓reduceIte] <;>
-    omega
-
-lemma permDistance_mul_left (p q g : Equiv.Perm α) :
-    permDistance (g*p) (g*q) = permDistance p q := by
-  classical
-  unfold permDistance
-  congr 1
-  ext x
-  simp only [Finset.mem_filter,Finset.mem_univ,true_and]
-  exact not_congr g.injective.eq_iff
-
-lemma permDistance_mul_right (p q g : Equiv.Perm α) :
-    permDistance (p*g) (q*g) = permDistance p q := by
-  classical
-  simp only [permDistance_sum,Equiv.Perm.mul_apply]
-  exact Fintype.sum_equiv g _ _ (fun _ => rfl)
-
-lemma permDistance_inv (p q : Equiv.Perm α) : permDistance p⁻¹ q⁻¹ = permDistance p q := by
-  have hh := permDistance_mul_left p⁻¹ q⁻¹ q
-  have hhh := permDistance_mul_right (q*p⁻¹) 1 p
-  simp only [mul_inv_cancel,one_mul,inv_mul_cancel_right] at hh hhh
-  rw [←hhh] at hh
-  rw [←hh,permDistance_symm]
-
-lemma permDistance_mul_le (p q g h : Equiv.Perm α) :
-    permDistance (p*g) (q*h) ≤ permDistance p q+permDistance g h := by
-  have hh := permDistance_triangle (p*g) (q*g) (q*h)
-  simpa only [permDistance_mul_left,permDistance_mul_right] using hh
-
-lemma selectedCycle_distance_le (p q : Equiv.Perm α) (S : Finset α) :
-    Fintype.card (RoutingNetwork.SelectedCycle p S) ≤
-      Fintype.card (RoutingNetwork.SelectedCycle q S)+permDistance p q := by
-  apply selectedCycle_perturb_le
-  intro x hx
-  simpa only [Finset.mem_filter,Finset.mem_univ,true_and,not_not] using hx
-
-lemma permDistance_alternating (p q : Equiv.Perm α) :
-    permDistance (CycleColoring.alternatingPerm p) (CycleColoring.alternatingPerm q) = permDistance p q := by
-  classical
-  simp only [permDistance_sum,Fintype.sum_prod_type,Fintype.sum_bool]
-  simp [CycleColoring.alternatingPerm]
-
-lemma alternatingCycles_distance_le {ι : Type*} [Fintype ι]
-    (e : ι ↪ Bool × α) (p q : Bool → Equiv.Perm α) :
-    PairRouting.alternatingCycles e p ≤ PairRouting.alternatingCycles e q+
-      (permDistance (p false) (q false)+permDistance (p true) (q true)) := by
-  have hh := selectedCycle_distance_le
-    (CycleColoring.alternatingPerm (PairRouting.alternatingProjection p))
-    (CycleColoring.alternatingPerm (PairRouting.alternatingProjection q)) (PairRouting.labelSet e)
-  rw [permDistance_alternating] at hh
-  apply hh.trans
-  apply Nat.add_le_add_left
-  dsimp only [PairRouting.alternatingProjection]
-  exact (permDistance_mul_le _ _ _ _).trans_eq (by
-    change permDistance (p false)⁻¹ (q false)⁻¹ + _ = _
-    rw [permDistance_inv])
-
-end Thorp
-namespace Thorp
-
-def centralBlockPerm (L : ℕ) : (r : ℕ) → (Card r → Equiv.Perm (Card L)) →
-    Equiv.Perm (Card (L+r))
-  | 0, C => C (fun j => Fin.elim0 j)
-  | r+1, C => childLift (fun b => centralBlockPerm L r (fun u => C (Fin.cons b u)))
-
-def centralPalindromePerm (L r : ℕ) (C : Card r → Equiv.Perm (Card L))
-    (ω : BenesCoins (L+r)) : Equiv.Perm (Card (L+r)) :=
-  butterflyPerm (L+r) (decodeButterfly (L+r) ω.1) * centralBlockPerm L r C *
-    (butterflyPerm (L+r) (decodeButterfly (L+r) ω.2))⁻¹
-
-lemma centralPalindromePerm_step (L r : ℕ) (C : Card (r+1) → Equiv.Perm (Card L))
-    (c : BenesCoins (L+r) × BenesCoins (L+r))
-    (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
-    centralPalindromePerm L (r+1) C ((benesStepEquiv (L+r)).symm (c,coins)) =
-      pairSwitch coins.2 * childLift (fun b => centralPalindromePerm L r
-        (fun u => C (Fin.cons b u)) (if b then c.2 else c.1)) * pairSwitch coins.1 := by
-  change ((butterflyPerm (L+r+1) (decodeButterfly (L+r+1)
-      ((coinStepEquiv (L+r)).symm ((c.1.1,c.2.1),coins.2))))) *
-    childLift (fun b => centralBlockPerm L r (fun u => C (Fin.cons b u))) *
-    (butterflyPerm (L+r+1) (decodeButterfly (L+r+1)
-      ((coinStepEquiv (L+r)).symm ((c.1.2,c.2.2),coins.1))))⁻¹ = _
-  rw [butterflyPerm_step,butterflyPerm_step,mul_inv_rev,childLift_inv,pairSwitch_inv]
-  simp only [mul_assoc]
-  rw [←mul_assoc (childLift _),childLift_mul,←mul_assoc (childLift _),childLift_mul]
-  congr 2
-  apply congrArg childLift
-  funext b
-  cases b <;> rfl
-
-noncomputable def centralLevelCost (t L : ℕ) : (r : ℕ) →
-    (Card r → Equiv.Perm (Card L)) → {ι : Type*} → [Fintype ι] →
-    (ι ↪ Card (L+r)) → BenesCoins (L+r) → ℕ
-  | 0, _, _, _, _, _ => 0
-  | r+1, C, _, _, e, ω =>
-    let σ := benesStepEquiv (L+r) ω
-    let x := e.trans (headTailEquiv (L+r)).toEmbedding
-    let c := PairRouting.colors x σ.2.1
-    let hc := PairRouting.colors_compatible x σ.2.1
-    if t = L+r+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
-      (fun b => centralPalindromePerm L r (fun u => C (Fin.cons b u)) (if b then σ.1.2 else σ.1.1))
-    else centralLevelCost t L r (fun u => C (Fin.cons false u))
-      (PairRouting.childEmbedding x c hc false) σ.1.1 +
-      centralLevelCost t L r (fun u => C (Fin.cons true u))
-      (PairRouting.childEmbedding x c hc true) σ.1.2
-
-noncomputable def centralSlabCost (l L r : ℕ) (C : Card r → Equiv.Perm (Card L))
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (ω : BenesCoins (L+r)) : ℕ :=
-  ∑ i : Fin l, centralLevelCost (l+i.val+1) L r C e ω
-
-lemma permDistance_childLift (d : ℕ) (p q : Bool → Equiv.Perm (Card d)) :
-    permDistance (childLift p) (childLift q) =
-      permDistance (p false) (q false)+permDistance (p true) (q true) := by
-  classical
-  rw [permDistance_sum]
-  have hh : (∑ x : Card (d+1), if childLift p x = childLift q x then 0 else 1) =
-      ∑ x : Bool × Card d, if p x.1 x.2 = q x.1 x.2 then 0 else 1 := by
-    apply Fintype.sum_equiv (headTailEquiv d)
-    intro x
-    have hx : (headTailEquiv d) x = (x 0,Fin.tail x) := rfl
-    rw [hx]
-    congr 1
-    apply propext
-    constructor
-    · intro h
-      have ht := congrArg Fin.tail h
-      simpa only [childLift,Equiv.coe_fn_mk,Fin.tail_cons] using ht
-    · intro h
-      dsimp only [childLift,Equiv.coe_fn_mk]
-      exact congrArg (fun z : Card d => (Fin.cons (x 0) z : Card (d+1))) h
-  rw [hh]
-  simp only [Fintype.sum_prod_type,Fintype.sum_bool,permDistance_sum]
-  omega
-
-noncomputable def centralDistance (L r : ℕ) (C D : Card r → Equiv.Perm (Card L)) : ℕ :=
-  ∑ u : Card r, permDistance (C u) (D u)
-
-lemma centralDistance_step (L r : ℕ) (C D : Card (r+1) → Equiv.Perm (Card L)) :
-    centralDistance L (r+1) C D =
-      centralDistance L r (fun u => C (Fin.cons false u)) (fun u => D (Fin.cons false u))+
-      centralDistance L r (fun u => C (Fin.cons true u)) (fun u => D (Fin.cons true u)) := by
-  classical
-  unfold centralDistance
-  rw [Fintype.sum_equiv (headTailEquiv r) _
-    (fun x : Bool × Card r => permDistance (C (Fin.cons x.1 x.2)) (D (Fin.cons x.1 x.2)))
-      (fun x => by
-        change permDistance (C x) (D x) =
-          permDistance (C (Fin.cons (x 0) (Fin.tail x))) (D (Fin.cons (x 0) (Fin.tail x)))
-        rw [Fin.cons_self_tail])]
-  simp only [Fintype.sum_prod_type,Fintype.sum_bool]
-  omega
-
-lemma permDistance_centralBlock (L r : ℕ) (C D : Card r → Equiv.Perm (Card L)) :
-    permDistance (centralBlockPerm L r C) (centralBlockPerm L r D) = centralDistance L r C D := by
-  induction r with
-  | zero =>
-    simp only [centralBlockPerm,centralDistance,Fintype.sum_unique]
-    congr 1
-  | succ r ih =>
-    rw [centralBlockPerm,centralBlockPerm,permDistance_childLift,centralDistance_step,ih,ih]
-
-lemma permDistance_centralPalindrome (L r : ℕ) (C D : Card r → Equiv.Perm (Card L))
-    (ω : BenesCoins (L+r)) :
-    permDistance (centralPalindromePerm L r C ω) (centralPalindromePerm L r D ω) =
-      centralDistance L r C D := by
-  rw [centralPalindromePerm,centralPalindromePerm,permDistance_mul_right,
-    permDistance_mul_left,permDistance_centralBlock]
-
-lemma centralLevelCost_step (t L r : ℕ) (C : Card (r+1) → Equiv.Perm (Card L))
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r+1))
-    (ω : BenesCoins (L+r) × BenesCoins (L+r))
-    (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
-    centralLevelCost t L (r+1) C e ((benesStepEquiv (L+r)).symm (ω,coins)) =
-      let x := e.trans (headTailEquiv (L+r)).toEmbedding
-      let c := PairRouting.colors x coins.1
-      let hc := PairRouting.colors_compatible x coins.1
-      if t = L+r+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x coins.1)
-        (fun b => centralPalindromePerm L r (fun u => C (Fin.cons b u)) (if b then ω.2 else ω.1))
-      else centralLevelCost t L r (fun u => C (Fin.cons false u))
-        (PairRouting.childEmbedding x c hc false) ω.1 +
-        centralLevelCost t L r (fun u => C (Fin.cons true u))
-        (PairRouting.childEmbedding x c hc true) ω.2 := by
-  rw [centralLevelCost]
-  rfl
-
-lemma centralLevelCost_perturb (t L r : ℕ) (C D : Card r → Equiv.Perm (Card L))
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (ω : BenesCoins (L+r)) :
-    centralLevelCost t L r C e ω ≤ centralLevelCost t L r D e ω+centralDistance L r C D := by
-  induction r generalizing ι with
-  | zero => simp only [centralLevelCost,Nat.zero_add,Nat.zero_le]
-  | succ r ih =>
-    rw [←(benesStepEquiv (L+r)).symm_apply_apply ω,centralLevelCost_step,centralLevelCost_step]
-    dsimp only
-    split_ifs with ht
-    · have hh := alternatingCycles_distance_le
-        (PairRouting.switchedEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
-          (benesStepEquiv (L+r) ω).2.1)
-        (fun b => centralPalindromePerm L r (fun u => C (Fin.cons b u))
-          (if b then (benesStepEquiv (L+r) ω).1.2 else (benesStepEquiv (L+r) ω).1.1))
-        (fun b => centralPalindromePerm L r (fun u => D (Fin.cons b u))
-          (if b then (benesStepEquiv (L+r) ω).1.2 else (benesStepEquiv (L+r) ω).1.1))
-      simpa only [Bool.false_eq_true,↓reduceIte,permDistance_centralPalindrome,←centralDistance_step] using hh
-    · rw [centralDistance_step]
-      have h₀ := ih (fun u => C (Fin.cons false u)) (fun u => D (Fin.cons false u))
-        (PairRouting.childEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
-          (PairRouting.colors (e.trans (headTailEquiv (L+r)).toEmbedding) (benesStepEquiv (L+r) ω).2.1)
-          (PairRouting.colors_compatible _ _) false)
-        (benesStepEquiv (L+r) ω).1.1
-      have h₁ := ih (fun u => C (Fin.cons true u)) (fun u => D (Fin.cons true u))
-        (PairRouting.childEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
-          (PairRouting.colors (e.trans (headTailEquiv (L+r)).toEmbedding) (benesStepEquiv (L+r) ω).2.1)
-          (PairRouting.colors_compatible _ _) true)
-        (benesStepEquiv (L+r) ω).1.2
-      omega
-
-end Thorp
-
-end ThorpNine.Harmonic
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Harmonic.CutoffMass
+
+namespace OAI
+
+namespace ThorpNine.Harmonic
+
+namespace Thorp
+
+lemma familyEnd_two (L n : ℕ) : 2*familyEnd L n ≤ L*4^n := by
+  cases n with
+  | zero => simp [familyEnd]
+  | succ n => simp only [familyEnd,Nat.succ_ne_zero,ite_false,Nat.add_one_sub_one,pow_succ]; nlinarith
+
+lemma familyEnd_ge (L n j : ℕ) (hj : j < n) : 2*(L*4^j) ≤ familyEnd L n := by
+  cases n with
+  | zero => omega
+  | succ n =>
+    simp only [familyEnd,Nat.succ_ne_zero,ite_false,Nat.add_one_sub_one]
+    apply Nat.mul_le_mul_left
+    exact Nat.mul_le_mul_left L (Nat.pow_le_pow_right (by decide) (by omega : j ≤ n))
+
+noncomputable def palindromeFamilyCost (L n d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) : ℕ :=
+  ∑ j ∈ Finset.range n, palindromeSlabCost (L*4^j) d e ω
+
+lemma palindromeFamilyCost_adapted (L n d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) :
+    LayerAdapted d (familyEnd L n) (fun Y => palindromeFamilyCost L n d e (Y,X)) := by
+  apply layerAdapted_sum
+  intro j hj
+  exact layerAdapted_mono (palindromeSlabCost_adapted (L*4^j) d e X)
+    (familyEnd_ge L n j (Finset.mem_range.mp hj))
+
+theorem palindrome_family_mgf (L n d : ℕ) (hL : 0 < L) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) ≤
+      Real.exp (Fintype.card ι * (68*∑ j ∈ Finset.range n,
+        Real.exp (-((L*4^j:ℕ):ℝ)/64))) := by
+  induction n with
+  | zero => simp only [palindromeFamilyCost,Finset.range_zero,Finset.sum_empty,pow_zero,
+      finiteMean_const,mul_zero,Real.exp_zero,le_refl]
+  | succ n ih =>
+    let l := L*4^n
+    have hl : 0 < l := Nat.mul_pos hL (pow_pos (by decide) _)
+    have hs : 2*familyEnd L n ≤ l := familyEnd_two L n
+    have hc : ∀ Y, palindromeFamilyCost L (n+1) d e (Y,X) =
+        palindromeFamilyCost L n d e (Y,X)+palindromeSlabCost l d e (Y,X) := by
+      intro Y
+      exact Finset.sum_range_succ _ _
+    have hm : finiteMean (fun Y => q^(palindromeFamilyCost L (n+1) d e (Y,X))) ≤
+        finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) *
+          Real.exp (Fintype.card ι*(68*Real.exp (-(l:ℝ)/64))) := by
+      by_cases hld : l ≤ d
+      · simp_rw [hc,pow_add]
+        exact palindrome_slab_adapted_mul_le d (familyEnd L n) l (by omega) hl hs e X _
+          (fun _ => pow_nonneg (by linarith) _)
+          (layerAdapted_comp (palindromeFamilyCost_adapted L n d e X) (fun c => q^c))
+          q hq hlog
+      · simp_rw [hc,palindromeSlabCost_gt l d (by omega) e,Nat.add_zero]
+        exact le_mul_of_one_le_right
+          (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
+          (Real.one_le_exp_iff.mpr (by positivity))
+    refine (hm.trans (mul_le_mul_of_nonneg_right ih (Real.exp_nonneg _))).trans_eq ?_
+    rw [←Real.exp_add,Finset.sum_range_succ]
+    congr 1
+    dsimp [l]
+    ring
+
+lemma pow_four_ge (j : ℕ) : j+1 ≤ 4^j := by
+  induction j with
+  | zero => norm_num
+  | succ j ih => rw [pow_succ]; omega
+
+lemma sum_family_decay (L n : ℕ) (hL : 0 < L) :
+    (∑ j ∈ Finset.range n, Real.exp (-((L*4^j:ℕ):ℝ)/64)) ≤
+      Real.exp (-(L:ℝ)/64)/(1-Real.exp (-(1:ℝ)/64)) := by
+  let r := Real.exp (-(1:ℝ)/64)
+  have hr : 0 < r := Real.exp_pos _
+  have hr1 : r < 1 := Real.exp_lt_one_iff.mpr (by norm_num)
+  have hg : (∑ j ∈ Finset.range n, r^j) ≤ 1/(1-r) := by
+    apply (le_div_iff₀ (by linarith : (0:ℝ)<1-r)).mpr
+    have hh := geom_sum_mul r n
+    nlinarith [pow_nonneg (le_of_lt hr) n]
+  calc
+    _ ≤ ∑ j ∈ Finset.range n, Real.exp (-(L:ℝ)/64)*r^j := by
+      apply Finset.sum_le_sum
+      intro j _
+      have hj := pow_four_ge j
+      have hlj : L+j ≤ L*4^j := by nlinarith
+      have hlj' : (L:ℝ)+j ≤ (L*4^j:ℕ) := by exact_mod_cast hlj
+      rw [←Real.exp_nat_mul,←Real.exp_add]
+      exact Real.exp_le_exp.mpr (by linarith)
+    _ = Real.exp (-(L:ℝ)/64)*(∑ j ∈ Finset.range n, r^j) := by rw [Finset.mul_sum]
+    _ ≤ Real.exp (-(L:ℝ)/64)*(1/(1-r)) :=
+      mul_le_mul_of_nonneg_left hg (Real.exp_nonneg _)
+    _ = _ := by ring
+
+
+lemma palindromeLevelCost_step_add (t d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (d+1)) (ω : BenesCoins (d+1)) :
+    palindromeLevelCost t (d+1) e ω =
+      let σ := benesStepEquiv d ω
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x σ.2.1
+      let hc := PairRouting.colors_compatible x σ.2.1
+      (if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
+          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) else 0) +
+        palindromeLevelCost t d (PairRouting.childEmbedding x c hc false) σ.1.1 +
+        palindromeLevelCost t d (PairRouting.childEmbedding x c hc true) σ.1.2 := by
+  classical
+  rw [palindromeLevelCost]
+  dsimp only
+  split_ifs with h
+  · simp only [palindromeLevelCost_gt t d (by omega),Nat.add_zero]
+  · simp only [Nat.zero_add]
+
+lemma palindromeLowCost_sum (H d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) :
+    palindromeLowCost H d e ω = ∑ i ∈ Finset.range H, palindromeLevelCost (i+1) d e ω := by
+  classical
+  induction d generalizing ι with
+  | zero => simp only [palindromeLowCost,palindromeLevelCost,Finset.sum_const_zero]
+  | succ d ih =>
+    rw [palindromeLowCost]
+    simp only [palindromeLevelCost_step_add,Finset.sum_add_distrib]
+    rw [←ih,←ih]
+    congr 2
+    simp only [Nat.add_right_cancel_iff,Finset.sum_ite_eq',Finset.mem_range,Nat.succ_le_iff]
+
+lemma palindromeLowCost_slab (L d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) :
+    palindromeLowCost L d e ω + palindromeSlabCost L d e ω = palindromeLowCost (2*L) d e ω := by
+  rw [palindromeLowCost_sum,palindromeLowCost_sum,show 2*L=L+L by omega,Finset.sum_range_add]
+  congr 1
+  exact Fin.sum_univ_eq_sum_range (fun i => palindromeLevelCost (L+i+1) d e ω) L
+
+lemma palindromeCost_family (L n d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) :
+    palindromeLowCost L d e ω + palindromeFamilyCost L n d e ω +
+      palindromeFamilyCost (2*L) n d e ω = palindromeLowCost (L*4^n) d e ω := by
+  induction n with
+  | zero => simp only [palindromeFamilyCost,Finset.range_zero,Finset.sum_empty,Nat.add_zero,
+      pow_zero,mul_one]
+  | succ n ih =>
+    have he : (2*L)*4^n = 2*(L*4^n) := by ring
+    have he' : L*4^(n+1) = 2*(2*(L*4^n)) := by rw [pow_succ]; ring
+    simp only [palindromeFamilyCost,Finset.sum_range_succ] at ih ⊢
+    rw [he,he',←palindromeLowCost_slab,←palindromeLowCost_slab,←ih]
+    omega
+
+
+lemma finiteMean_mul_sq_le {Ω : Type*} [Fintype Ω] (f g : Ω → ℝ) :
+    (finiteMean (fun ω => f ω*g ω))^2 ≤ finiteMean (fun ω => (f ω)^2)*finiteMean (fun ω => (g ω)^2) := by
+  have h := Finset.sum_mul_sq_le_sq_mul_sq Finset.univ f g
+  simp only [finiteMean,div_pow,div_mul_div_comm,←pow_two]
+  exact div_le_div_of_nonneg_right h (sq_nonneg _)
+
+lemma finiteMean_mul_exp_le {Ω : Type*} [Fintype Ω] (f g : Ω → ℝ) (A B : ℝ)
+    (hf : finiteMean (fun ω => (f ω)^2) ≤ Real.exp A)
+    (hg : finiteMean (fun ω => (g ω)^2) ≤ Real.exp B) :
+    finiteMean (fun ω => f ω*g ω) ≤ Real.exp ((A+B)/2) := by
+  have hc := finiteMean_mul_sq_le f g
+  have hn : 0 ≤ finiteMean (fun ω => (g ω)^2) := finiteMean_nonneg (fun _ => sq_nonneg _)
+  have hu := mul_le_mul hf hg hn (Real.exp_nonneg _)
+  have he : Real.exp A*Real.exp B = (Real.exp ((A+B)/2))^2 := by
+    rw [←Real.exp_add,←Real.exp_nat_mul]
+    congr 1
+    push_cast
+    ring
+  rw [he] at hu
+  have hh := hc.trans hu
+  have hep := Real.exp_pos ((A+B)/2)
+  nlinarith
+
+
+noncomputable def heightDecay (H : ℕ) : ℝ :=
+  68*Real.exp (-(H:ℝ)/64)/(1-Real.exp (-(1:ℝ)/64))
+
+lemma heightDecay_nonneg (H : ℕ) : 0 ≤ heightDecay H := by
+  have h : Real.exp (-(1:ℝ)/64) < 1 := Real.exp_lt_one_iff.mpr (by norm_num)
+  apply div_nonneg (by positivity) (by linarith)
+
+lemma heightDecay_antitone : Antitone heightDecay := by
+  intro L H h
+  have hden : 0 ≤ 1-Real.exp (-(1:ℝ)/64) := by
+    have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
+    linarith
+  apply div_le_div_of_nonneg_right _ hden
+  apply mul_le_mul_of_nonneg_left _ (by norm_num)
+  apply Real.exp_le_exp.mpr
+  have h' : (L:ℝ) ≤ H := by exact_mod_cast h
+  linarith
+
+lemma palindrome_family_mgf_decay (L n d : ℕ) (hL : 0 < L) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) ≤
+      Real.exp (Fintype.card ι*heightDecay L) := by
+  apply (palindrome_family_mgf L n d hL e X q hq hlog).trans
+  apply Real.exp_le_exp.mpr
+  apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
+  have h := mul_le_mul_of_nonneg_left (sum_family_decay L n hL) (by norm_num : (0:ℝ)≤68)
+  simpa only [heightDecay,mul_div_assoc] using h
+
+theorem palindrome_high_mgf (H n d : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/500) :
+    finiteMean (fun Y => q^(palindromeFamilyCost H n d e (Y,X)+
+      palindromeFamilyCost (2*H) n d e (Y,X))) ≤ Real.exp (Fintype.card ι*heightDecay H) := by
+  have hq2 := one_le_pow₀ (n:=2) hq
+  have hlog2 : Real.log (q^2) ≤ 1/250 := by rw [Real.log_pow]; norm_num; linarith
+  have h₀ := palindrome_family_mgf_decay H n d hH e X (q^2) hq2 hlog2
+  have h₁ := palindrome_family_mgf_decay (2*H) n d (by omega) e X (q^2) hq2 hlog2
+  have h₁' := h₁.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left
+    (heightDecay_antitone (show H ≤ 2*H by omega)) (Nat.cast_nonneg _)))
+  have hh := finiteMean_mul_exp_le (fun Y => q^(palindromeFamilyCost H n d e (Y,X)))
+    (fun Y => q^(palindromeFamilyCost (2*H) n d e (Y,X)))
+    (Fintype.card ι*heightDecay H) (Fintype.card ι*heightDecay H)
+    (by simpa only [pow_right_comm] using h₀) (by simpa only [pow_right_comm] using h₁')
+  simpa only [←pow_add,add_self_div_two] using hh
+
+lemma palindrome_high_full_mgf (H n d : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/500) :
+    finiteMean (fun ω : BenesCoins d => q^(palindromeFamilyCost H n d e ω+
+      palindromeFamilyCost (2*H) n d e ω)) ≤ Real.exp (Fintype.card ι*heightDecay H) := by
+  rw [finiteMean_prod,finiteMean_comm]
+  calc
+    _ ≤ finiteMean (fun _X : SwitchIndex d → Bool => Real.exp (Fintype.card ι*heightDecay H)) := by
+      apply finiteMean_mono
+      intro X
+      exact palindrome_high_mgf H n d hH e X q hq hlog
+    _ = _ := finiteMean_const _
+
+theorem palindrome_cost_mgf_cutoff (r H : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (H+r)) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/1000)
+    (hr : (2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*(q^2)^H) ≤ 1/2) :
+    finiteMean (fun ω : BenesCoins (H+r) => q^(palindromeCost (H+r) e ω)) ≤
+      Real.exp (((2:ℝ)^r*(2*((2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*(q^2)^H))^2)+
+        Fintype.card ι*heightDecay H)/2) := by
+  let d := H+r
+  have hd : d ≤ H*4^d := by
+    have hh := pow_four_ge d
+    nlinarith
+  have hcost (ω : BenesCoins d) : palindromeCost d e ω = palindromeLowCost H d e ω+
+      (palindromeFamilyCost H d d e ω+palindromeFamilyCost (2*H) d d e ω) := by
+    have hh := palindromeCost_family H d d e ω
+    rw [palindromeLowCost_eq _ _ hd] at hh
+    simpa only [Nat.add_assoc] using hh.symm
+  have hq2 := one_le_pow₀ (n:=2) hq
+  have hlog2 : Real.log (q^2) ≤ 1/500 := by rw [Real.log_pow]; norm_num; linarith
+  have hlow := palindrome_low_mgf r H e (q^2) hq2 hr
+  have hhigh := palindrome_high_full_mgf H d d hH e (q^2) hq2 hlog2
+  have hh := finiteMean_mul_exp_le (fun ω : BenesCoins d => q^(palindromeLowCost H d e ω))
+    (fun ω : BenesCoins d => q^(palindromeFamilyCost H d d e ω+palindromeFamilyCost (2*H) d d e ω))
+    _ _ (by simpa only [pow_right_comm] using hlow) (by simpa only [pow_right_comm] using hhigh)
+  change finiteMean (fun ω : BenesCoins d => q^(palindromeCost d e ω)) ≤ _
+  simpa only [hcost,pow_add,pow_right_comm] using hh
+
+
+lemma cutoffMass_le (H d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (ω : BenesCoins d) :
+    cutoffMass H d e ω ≤ Fintype.card ι := by
+  classical
+  induction d generalizing ι with
+  | zero => exact Nat.zero_le _
+  | succ d ih =>
+    rw [cutoffMass]
+    split_ifs
+    · exact crowdedMass_le _
+    · exact (Nat.add_le_add (ih _ _) (ih _ _)).trans_eq (PairRouting.color_card_add _)
+
+lemma palindromeLowCost_le (H d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) : palindromeLowCost H d e ω ≤ H*Fintype.card ι :=
+  (palindromeLowCost_le_cutoff H d e ω).trans (Nat.mul_le_mul_left H (cutoffMass_le H d e ω))
+
+theorem palindrome_cost_mgf_coarse (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/1000) :
+    finiteMean (fun ω : BenesCoins d => q^(palindromeCost d e ω)) ≤
+      Real.exp (Fintype.card ι*(1+heightDecay 1)) := by
+  have hd : d ≤ 1*4^d := by have hh := pow_four_ge d; omega
+  have hcost (ω : BenesCoins d) : palindromeCost d e ω = palindromeLowCost 1 d e ω+
+      (palindromeFamilyCost 1 d d e ω+palindromeFamilyCost 2 d d e ω) := by
+    have hh := palindromeCost_family 1 d d e ω
+    rw [palindromeLowCost_eq _ _ hd] at hh
+    simpa only [Nat.add_assoc] using hh.symm
+  have hq2 := one_le_pow₀ (n:=2) hq
+  have hlog2 : Real.log (q^2) ≤ 1/500 := by rw [Real.log_pow]; norm_num; linarith
+  have hlow : finiteMean (fun ω : BenesCoins d => (q^2)^(palindromeLowCost 1 d e ω)) ≤
+      Real.exp (Fintype.card ι) := by
+    calc
+      _ ≤ finiteMean (fun _ω : BenesCoins d => (q^2)^(Fintype.card ι)) := by
+        apply finiteMean_mono
+        intro ω
+        apply pow_le_pow_right₀ hq2
+        simpa only [one_mul] using palindromeLowCost_le 1 d e ω
+      _ = (q^2)^(Fintype.card ι) := finiteMean_const _
+      _ = Real.exp (Fintype.card ι * Real.log (q^2)) := by
+        rw [Real.exp_nat_mul,Real.exp_log (by positivity : 0<q^2)]
+      _ ≤ _ := Real.exp_le_exp.mpr (by
+        have hh := mul_le_mul_of_nonneg_left hlog2 (Nat.cast_nonneg (Fintype.card ι) : (0:ℝ)≤_)
+        nlinarith [Nat.cast_nonneg (α:=ℝ) (Fintype.card ι)])
+  have hhigh := palindrome_high_full_mgf 1 d d (by decide) e (q^2) hq2 hlog2
+  have hh := finiteMean_mul_exp_le (fun ω : BenesCoins d => q^(palindromeLowCost 1 d e ω))
+    (fun ω : BenesCoins d => q^(palindromeFamilyCost 1 d d e ω+palindromeFamilyCost 2 d d e ω))
+    (Fintype.card ι) (Fintype.card ι*heightDecay 1)
+    (by simpa only [pow_right_comm] using hlow) (by simpa only [pow_right_comm] using hhigh)
+  simp only [←pow_add,←hcost] at hh
+  apply hh.trans (Real.exp_le_exp.mpr _)
+  have hk : (0:ℝ) ≤ Fintype.card ι := Nat.cast_nonneg _
+  have hD := heightDecay_nonneg 1
+  nlinarith
+
+noncomputable def momentBase : ℝ := (2:ℝ)^(1/1000:ℝ)
+
+lemma momentBase_one_le : 1 ≤ momentBase := by
+  exact Real.one_le_rpow (by norm_num) (by norm_num)
+
+lemma momentBase_log : Real.log momentBase ≤ 1/1000 := by
+  rw [momentBase,Real.log_rpow (by norm_num)]
+  have hh := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
+  norm_num at hh ⊢
+  linarith
+
+lemma momentBase_pow (c : ℕ) : momentBase^c = (2:ℝ)^((c:ℝ)*(1/1000:ℝ)) := by
+  rw [mul_comm,Real.rpow_mul_natCast (by norm_num)]
+  rfl
+
+lemma palindromeRowMoment_le_cost (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) :
+    palindromeRowMoment d e (1/1000) ≤
+      finiteMean (fun ω : BenesCoins d => momentBase^(palindromeCost d e ω)) := by
+  have hn : (0:ℝ) < ((2^d:ℕ):ℝ)^(Fintype.card ι) := by positivity
+  have hn0 : (0:ℝ) ≤ ((2^d).descFactorial (Fintype.card ι):ℝ) /
+      ((2^d:ℕ):ℝ)^(Fintype.card ι) := by positivity
+  have hn1 : ((2^d).descFactorial (Fintype.card ι):ℝ) /
+      ((2^d:ℕ):ℝ)^(Fintype.card ι) ≤ 1 := by
+    apply (div_le_one hn).mpr
+    exact_mod_cast Nat.descFactorial_le_pow (2^d) (Fintype.card ι)
+  have hpow := Real.rpow_le_one hn0 hn1 (by norm_num : (0:ℝ)≤1/1000)
+  have hm := palindrome_row_density_moment d e (1/1000) (by norm_num)
+  simp_rw [←momentBase_pow] at hm
+  exact hm.trans (mul_le_of_le_one_left (finiteMean_nonneg (fun _ =>
+    pow_nonneg (le_trans (by norm_num) momentBase_one_le) _)) hpow)
+
+theorem palindrome_row_moment_coarse (d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) :
+    palindromeRowMoment d e (1/1000) ≤ Real.exp (Fintype.card ι*(1+heightDecay 1)) :=
+  (palindromeRowMoment_le_cost d e).trans
+    (palindrome_cost_mgf_coarse d e momentBase momentBase_one_le momentBase_log)
+
+
+noncomputable def densityCutoff (p : ℝ) : ℕ := ⌊-Real.log p/(4*Real.log 2)⌋₊
+
+lemma log_small_density {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
+    Real.log p ≤ -8*Real.log 2 := by
+  have he : Real.log (1/256:ℝ) = -8*Real.log 2 := by
+    rw [show (1/256:ℝ) = 1/2^8 by norm_num,
+      Real.log_div (by norm_num) (by positivity),Real.log_one,Real.log_pow]
+    ring
+  exact (Real.log_le_log hp hs).trans_eq he
+
+lemma densityCutoff_bounds {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
+    0 < densityCutoff p ∧
+    (densityCutoff p:ℝ)*(4*Real.log 2) ≤ -Real.log p ∧
+    -Real.log p < ((densityCutoff p:ℝ)+1)*(4*Real.log 2) := by
+  have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
+  have hd : 0 < 4*Real.log 2 := by positivity
+  have hh := log_small_density hp hs
+  have hx : 1 ≤ -Real.log p/(4*Real.log 2) := by
+    apply (le_div_iff₀ hd).mpr
+    nlinarith
+  refine ⟨Nat.floor_pos.mpr hx,?_,?_⟩
+  · exact (le_div_iff₀ hd).mp (Nat.floor_le (by linarith : 0 ≤ -Real.log p/(4*Real.log 2)))
+  · exact (div_lt_iff₀ hd).mp (Nat.lt_floor_add_one (-Real.log p/(4*Real.log 2)))
+
+lemma densityCutoff_decay {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
+    Real.exp (-(densityCutoff p:ℝ)/64) ≤ Real.exp (1/64) * p^(1/512:ℝ) := by
+  have hh := (densityCutoff_bounds hp hs).2.2
+  have hb : Real.log 2 ≤ 2 := by
+    have := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
+    linarith
+  have hH : 0 ≤ (densityCutoff p:ℝ)+1 := by positivity
+  have hu := mul_le_mul_of_nonneg_left hb hH
+  rw [Real.rpow_def_of_pos hp,←Real.exp_add]
+  apply Real.exp_le_exp.mpr
+  nlinarith
+
+lemma densityCutoff_power {p q : ℝ} (hp : 0 < p) (hs : p ≤ 1/256)
+    (hq : 0 < q) (hlog : Real.log q ≤ Real.log 2/4) (n : ℕ) (hn : n ≤ 4)
+    (hq1 : 1 ≤ q) :
+    p*(2:ℝ)^(densityCutoff p)*q^(n*densityCutoff p) ≤ p^(1/2:ℝ) := by
+  have hcut := (densityCutoff_bounds hp hs).2.1
+  have hH : 0 ≤ (densityCutoff p:ℝ) := Nat.cast_nonneg _
+  have hn' : (n:ℝ) ≤ 4 := by exact_mod_cast hn
+  have hln := Real.log_nonneg hq1
+  have hl := mul_le_mul_of_nonneg_right hn' hln
+  have hnl : (n:ℝ)*Real.log q ≤ Real.log 2 := by linarith
+  have hm := mul_le_mul_of_nonneg_left hnl hH
+  have heq : p*(2:ℝ)^(densityCutoff p)*q^(n*densityCutoff p) =
+      Real.exp (Real.log p+(densityCutoff p:ℝ)*Real.log 2+
+        (n*densityCutoff p:ℕ)*Real.log q) := by
+    rw [Real.exp_add,Real.exp_add,Real.exp_nat_mul,Real.exp_nat_mul,
+      Real.exp_log hp,Real.exp_log hq,Real.exp_log (by norm_num : (0:ℝ)<2)]
+  rw [heq,Real.rpow_def_of_pos hp]
+  apply Real.exp_le_exp.mpr
+  push_cast
+  nlinarith
+
+lemma densityCutoff_sparse {p q : ℝ} (hp : 0 < p) (hs : p ≤ 1/256)
+    (hq : 0 < q) (hlog : Real.log q ≤ Real.log 2/4) (hq1 : 1 ≤ q) :
+    (2:ℝ)^(densityCutoff p)*(p*(q^2)^(densityCutoff p)) ≤ 1/2 := by
+  have hh := densityCutoff_power hp hs hq hlog 2 (by decide) hq1
+  rw [←pow_mul] at ⊢
+  have hle : p^(1/2:ℝ) ≤ 1/2 := by
+    have hlogp := log_small_density hp hs
+    have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
+    rw [Real.rpow_def_of_pos hp]
+    calc
+      _ ≤ Real.exp (-Real.log 2) := by apply Real.exp_le_exp.mpr; nlinarith
+      _ = 1/2 := by rw [Real.exp_neg,Real.exp_log (by norm_num : (0:ℝ)<2)]; norm_num
+  nlinarith [hh.trans hle]
+
+
+lemma momentBase_log_exact : Real.log momentBase = Real.log 2/1000 := by
+  rw [momentBase,Real.log_rpow (by norm_num)]
+  ring
+
+lemma densityCutoff_le_height (d k : ℕ) (hk : 0 < k) :
+    densityCutoff ((k:ℝ)/(2:ℝ)^d) ≤ d := by
+  have hk' : (1:ℝ) ≤ k := by exact_mod_cast hk
+  have hp : 0 < (k:ℝ)/(2:ℝ)^d := by positivity
+  have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
+  have hlogk := Real.log_nonneg hk'
+  by_cases hx : 0 ≤ -Real.log ((k:ℝ)/(2:ℝ)^d)/(4*Real.log 2)
+  · have hh := (le_div_iff₀ (by positivity : 0 < 4*Real.log 2)).mp (Nat.floor_le hx)
+    have hd : (0:ℝ) ≤ d := Nat.cast_nonneg _
+    have hH : (0:ℝ) ≤ densityCutoff ((k:ℝ)/(2:ℝ)^d) := Nat.cast_nonneg _
+    change (densityCutoff ((k:ℝ)/(2:ℝ)^d):ℝ)*(4*Real.log 2) ≤ -Real.log ((k:ℝ)/(2:ℝ)^d) at hh
+    rw [Real.log_div (by positivity) (by positivity),Real.log_pow] at hh
+    have hle : (densityCutoff ((k:ℝ)/(2:ℝ)^d):ℝ) ≤ d := by nlinarith
+    exact_mod_cast hle
+  · have hz := Nat.floor_of_nonpos (le_of_not_ge hx)
+    simp only [densityCutoff,hz,Nat.zero_le]
+
+noncomputable def densityTailConstant : ℝ :=
+  68*Real.exp (1/64)/(1-Real.exp (-(1:ℝ)/64))
+
+lemma densityTailConstant_nonneg : 0 ≤ densityTailConstant := by
+  have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
+  apply div_nonneg (by positivity) (by linarith)
+
+lemma heightDecay_cutoff {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
+    heightDecay (densityCutoff p) ≤ densityTailConstant*p^(1/512:ℝ) := by
+  have hden : 0 ≤ 1-Real.exp (-(1:ℝ)/64) := by
+    have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
+    linarith
+  have hh := div_le_div_of_nonneg_right
+    (mul_le_mul_of_nonneg_left (densityCutoff_decay hp hs) (by norm_num : (0:ℝ)≤68)) hden
+  simpa only [heightDecay,densityTailConstant,mul_assoc,div_mul_eq_mul_div] using hh
+
+lemma sparse_low_algebra (r H k : ℕ) (q : ℝ) :
+    (2:ℝ)^r*(2*((2:ℝ)^H*(((k:ℝ)/(2:ℝ)^(H+r))*(q^2)^H))^2) =
+    2*k*(((k:ℝ)/(2:ℝ)^(H+r))*(2:ℝ)^H*q^(4*H)) := by
+  rw [pow_add,←pow_mul,show 4*H = (2*H)*2 by omega,pow_mul]
+  have hH : (2:ℝ)^H ≠ 0 := by positivity
+  have hr : (2:ℝ)^r ≠ 0 := by positivity
+  field_simp
+  ring
+
+lemma palindrome_row_moment_small (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι)
+    (hs : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1/256) :
+    palindromeRowMoment d e (1/1000) ≤
+      Real.exp (Fintype.card ι*(1+densityTailConstant)*
+        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ)) := by
+  generalize hpdef : ((Fintype.card ι:ℝ)/(2:ℝ)^d) = p at hs ⊢
+  have hp : 0 < p := by rw [←hpdef]; positivity
+  have hle := densityCutoff_le_height d (Fintype.card ι) hk
+  rw [hpdef] at hle
+  obtain ⟨r,hr⟩ := Nat.exists_eq_add_of_le hle
+  subst d
+  have hH := (densityCutoff_bounds hp hs).1
+  have hq := momentBase_one_le
+  have hqp : 0 < momentBase := by linarith
+  have hlog : Real.log momentBase ≤ Real.log 2/4 := by
+    rw [momentBase_log_exact]
+    have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
+    linarith
+  have hsp := densityCutoff_sparse hp hs hqp hlog hq
+  rw [←hpdef] at hsp
+  have hh := palindrome_cost_mgf_cutoff r (densityCutoff p) hH e momentBase hq momentBase_log
+    (by simpa only [hpdef] using hsp)
+  apply (palindromeRowMoment_le_cost _ e).trans (hh.trans (Real.exp_le_exp.mpr _))
+  rw [sparse_low_algebra,hpdef]
+  have hlow := densityCutoff_power hp hs hqp hlog 4 (by decide) hq
+  have hp1 : p ≤ 1 := by linarith
+  have he := Real.rpow_le_rpow_of_exponent_ge hp hp1 (by norm_num : (1:ℝ)/512 ≤ 1/2)
+  have hlow' := mul_le_mul_of_nonneg_left (hlow.trans he)
+    (by positivity : (0:ℝ)≤2*Fintype.card ι)
+  have hhigh := mul_le_mul_of_nonneg_left (heightDecay_cutoff hp hs)
+    (Nat.cast_nonneg (Fintype.card ι): (0:ℝ)≤_)
+  have hnon : 0 ≤ Fintype.card ι*densityTailConstant*p^(1/512:ℝ) :=
+    mul_nonneg (mul_nonneg (Nat.cast_nonneg _) densityTailConstant_nonneg)
+      (Real.rpow_nonneg (le_of_lt hp) _)
+  nlinarith
+
+noncomputable def densityConstant : ℝ :=
+  256*(1+heightDecay 1)+1+densityTailConstant
+
+lemma densityConstant_pos : 0 < densityConstant := by
+  dsimp [densityConstant]
+  have hh := heightDecay_nonneg 1
+  have ht := densityTailConstant_nonneg
+  linarith
+
+theorem palindrome_row_moment_bound (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι) :
+    palindromeRowMoment d e (1/1000) ≤ Real.exp (densityConstant*Fintype.card ι*
+        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ)) := by
+  have hp : 0 < (Fintype.card ι:ℝ)/(2:ℝ)^d := by positivity
+  have hp1 : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1 := by
+    apply (div_le_one (by positivity)).mpr
+    have h := Fintype.card_le_of_injective e e.injective
+    rw [card_positions] at h
+    exact_mod_cast h
+  have hk0 : (0:ℝ) ≤ Fintype.card ι := Nat.cast_nonneg _
+  have hpow := Real.rpow_nonneg (le_of_lt hp) (1/512:ℝ)
+  by_cases hs : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1/256
+  · apply (palindrome_row_moment_small d e hk hs).trans (Real.exp_le_exp.mpr _)
+    have hD : 1+densityTailConstant ≤ densityConstant := by
+      dsimp [densityConstant]
+      have := heightDecay_nonneg 1
+      linarith
+    have hh := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hD hk0) hpow
+    nlinarith
+  · have hlow : (1:ℝ)/256 ≤ ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ) := by
+      have he := Real.rpow_le_rpow_of_exponent_ge hp hp1 (by norm_num : (1:ℝ)/512 ≤ 1)
+      rw [Real.rpow_one] at he
+      exact (le_of_not_ge hs).trans he
+    have hD : 256*(1+heightDecay 1) ≤ densityConstant := by
+      dsimp [densityConstant]
+      have := densityTailConstant_nonneg
+      linarith
+    apply (palindrome_row_moment_coarse d e).trans (Real.exp_le_exp.mpr _)
+    have hD0 : 0 ≤ 1+heightDecay 1 := by have := heightDecay_nonneg 1; linarith
+    have ha := mul_le_mul_of_nonneg_left hlow (mul_nonneg (by norm_num : (0:ℝ)≤256) hD0)
+    have hb := mul_le_mul_of_nonneg_right hD hpow
+    have hc := mul_le_mul_of_nonneg_right (ha.trans hb) hk0
+    nlinarith
+
+lemma palindromeRowMoment_nonneg (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (a : ℝ) : 0 ≤ palindromeRowMoment d e a := by
+  apply finiteMean_nonneg
+  intro f
+  exact Real.rpow_nonneg (mul_nonneg (Nat.cast_nonneg _)
+    (PairRouting.tupleProbability_nonneg _ _ _)) _
+
+theorem palindrome_log_row_moment (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι) :
+    Real.log (palindromeRowMoment d e (1/1000)) ≤ densityConstant*Fintype.card ι*
+        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ) := by
+  by_cases hz : palindromeRowMoment d e (1/1000) = 0
+  · rw [hz,Real.log_zero]
+    exact mul_nonneg (mul_nonneg (le_of_lt densityConstant_pos) (Nat.cast_nonneg _))
+      (Real.rpow_nonneg (by positivity) _)
+  · exact (Real.log_le_iff_le_exp (lt_of_le_of_ne (palindromeRowMoment_nonneg _ _ _) (Ne.symm hz))).mpr
+      (palindrome_row_moment_bound d e hk)
+
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+lemma orbitSet_eq_of_agree (p q : Equiv.Perm α) (x : α)
+    (h : ∀ z ∈ orbitSet p x, p z = q z) : orbitSet p x = orbitSet q x := by
+  have hp (n : ℕ) : (p^n) x ∈ orbitSet p x := by
+    apply (mem_orbitSet _ _ _).mpr
+    exact Equiv.Perm.sameCycle_pow_right.mpr (Equiv.Perm.SameCycle.refl _ _)
+  have he (n : ℕ) : (p^n) x = (q^n) x := by
+    induction n with
+    | zero => rfl
+    | succ n ih =>
+      rw [pow_succ',pow_succ',Equiv.Perm.mul_apply,Equiv.Perm.mul_apply,←ih]
+      exact h _ (hp n)
+  ext y
+  simp only [mem_orbitSet]
+  constructor
+  · intro hy
+    obtain ⟨n,hn⟩ := hy.exists_nat_pow_eq
+    rw [←hn,he]
+    exact Equiv.Perm.sameCycle_pow_right.mpr (Equiv.Perm.SameCycle.refl _ _)
+  · intro hy
+    obtain ⟨n,hn⟩ := hy.exists_nat_pow_eq
+    rw [←hn,←he]
+    exact Equiv.Perm.sameCycle_pow_right.mpr (Equiv.Perm.SameCycle.refl _ _)
+
+noncomputable def survivingCycle (p q : Equiv.Perm α) (S B : Finset α)
+    (h : ∀ x ∉ B, p x = q x) (C : RoutingNetwork.SelectedCycle p S)
+    (hC : ∀ x ∈ C.val, x ∉ B) : RoutingNetwork.SelectedCycle q S := by
+  refine ⟨C.val,?_,C.property.2⟩
+  obtain ⟨x,hx⟩ := C.property.1
+  refine ⟨x,hx.trans (orbitSet_eq_of_agree p q x ?_)⟩
+  intro z hz
+  exact h z (hC z (hx ▸ hz))
+
+noncomputable def perturbCycleMap (p q : Equiv.Perm α) (S B : Finset α)
+    (h : ∀ x ∉ B, p x = q x) (C : RoutingNetwork.SelectedCycle p S) :
+    RoutingNetwork.SelectedCycle q S ⊕ {x // x ∈ B} := by
+  classical
+  by_cases hb : ∃ x ∈ C.val, x ∈ B
+  · exact Sum.inr ⟨hb.choose,hb.choose_spec.2⟩
+  · exact Sum.inl (survivingCycle p q S B h C (by simpa only [not_exists,not_and] using hb))
+
+lemma perturbCycleMap_injective (p q : Equiv.Perm α) (S B : Finset α)
+    (h : ∀ x ∉ B, p x = q x) : Function.Injective (perturbCycleMap p q S B h) := by
+  classical
+  intro C D he
+  by_cases hC : ∃ x ∈ C.val, x ∈ B <;>
+    by_cases hD : ∃ x ∈ D.val, x ∈ B
+  · simp only [perturbCycleMap,dite_eq_left hC,dite_eq_left hD,Sum.inr.injEq] at he
+    have hx : hC.choose = hD.choose := congrArg Subtype.val he
+    exact RoutingNetwork.SelectedCycle.eq_of_mem C D hC.choose_spec.1
+      (hx.symm ▸ hD.choose_spec.1)
+  · simp only [perturbCycleMap,dite_eq_left hC,dite_eq_right hD] at he
+    cases he
+  · simp only [perturbCycleMap,dite_eq_right hC,dite_eq_left hD] at he
+    cases he
+  · simp only [perturbCycleMap,dite_eq_right hC,dite_eq_right hD,Sum.inl.injEq] at he
+    have hval : C.val = D.val := congrArg (fun Z : RoutingNetwork.SelectedCycle q S => Z.val) he
+    exact Subtype.ext hval
+
+theorem selectedCycle_perturb_le (p q : Equiv.Perm α) (S B : Finset α)
+    (h : ∀ x ∉ B, p x = q x) :
+    Fintype.card (RoutingNetwork.SelectedCycle p S) ≤
+      Fintype.card (RoutingNetwork.SelectedCycle q S)+B.card := by
+  have hh := Fintype.card_le_of_injective _ (perturbCycleMap_injective p q S B h)
+  simpa only [Fintype.card_sum,Fintype.card_coe] using hh
+
+
+noncomputable def permDistance (p q : Equiv.Perm α) : ℕ :=
+  (Finset.univ.filter (fun x => p x ≠ q x)).card
+
+lemma permDistance_sum (p q : Equiv.Perm α) :
+    permDistance p q = ∑ x : α, if p x = q x then 0 else 1 := by
+  classical
+  simp [permDistance,Finset.card_filter]
+
+lemma permDistance_refl (p : Equiv.Perm α) : permDistance p p = 0 := by
+  simp [permDistance]
+
+lemma permDistance_le (p q : Equiv.Perm α) : permDistance p q ≤ Fintype.card α :=
+  (Finset.card_filter_le _ _).trans_eq (Finset.card_univ)
+
+lemma permDistance_symm (p q : Equiv.Perm α) : permDistance p q = permDistance q p := by
+  simp only [permDistance,ne_comm]
+
+lemma permDistance_triangle (p q r : Equiv.Perm α) :
+    permDistance p r ≤ permDistance p q+permDistance q r := by
+  classical
+  simp only [permDistance_sum,←Finset.sum_add_distrib]
+  apply Finset.sum_le_sum
+  intro x _
+  by_cases hpq : p x = q x <;> by_cases hqr : q x = r x <;>
+    by_cases hpr : p x = r x <;> simp only [hpq,hqr,hpr,↓reduceIte] <;>
+    omega
+
+lemma permDistance_mul_left (p q g : Equiv.Perm α) :
+    permDistance (g*p) (g*q) = permDistance p q := by
+  classical
+  unfold permDistance
+  congr 1
+  ext x
+  simp only [Finset.mem_filter,Finset.mem_univ,true_and]
+  exact not_congr g.injective.eq_iff
+
+lemma permDistance_mul_right (p q g : Equiv.Perm α) :
+    permDistance (p*g) (q*g) = permDistance p q := by
+  classical
+  simp only [permDistance_sum,Equiv.Perm.mul_apply]
+  exact Fintype.sum_equiv g _ _ (fun _ => rfl)
+
+lemma permDistance_inv (p q : Equiv.Perm α) : permDistance p⁻¹ q⁻¹ = permDistance p q := by
+  have hh := permDistance_mul_left p⁻¹ q⁻¹ q
+  have hhh := permDistance_mul_right (q*p⁻¹) 1 p
+  simp only [mul_inv_cancel,one_mul,inv_mul_cancel_right] at hh hhh
+  rw [←hhh] at hh
+  rw [←hh,permDistance_symm]
+
+lemma permDistance_mul_le (p q g h : Equiv.Perm α) :
+    permDistance (p*g) (q*h) ≤ permDistance p q+permDistance g h := by
+  have hh := permDistance_triangle (p*g) (q*g) (q*h)
+  simpa only [permDistance_mul_left,permDistance_mul_right] using hh
+
+lemma selectedCycle_distance_le (p q : Equiv.Perm α) (S : Finset α) :
+    Fintype.card (RoutingNetwork.SelectedCycle p S) ≤
+      Fintype.card (RoutingNetwork.SelectedCycle q S)+permDistance p q := by
+  apply selectedCycle_perturb_le
+  intro x hx
+  simpa only [Finset.mem_filter,Finset.mem_univ,true_and,not_not] using hx
+
+lemma permDistance_alternating (p q : Equiv.Perm α) :
+    permDistance (CycleColoring.alternatingPerm p) (CycleColoring.alternatingPerm q) = permDistance p q := by
+  classical
+  simp only [permDistance_sum,Fintype.sum_prod_type,Fintype.sum_bool]
+  simp [CycleColoring.alternatingPerm]
+
+lemma alternatingCycles_distance_le {ι : Type*} [Fintype ι]
+    (e : ι ↪ Bool × α) (p q : Bool → Equiv.Perm α) :
+    PairRouting.alternatingCycles e p ≤ PairRouting.alternatingCycles e q+
+      (permDistance (p false) (q false)+permDistance (p true) (q true)) := by
+  have hh := selectedCycle_distance_le
+    (CycleColoring.alternatingPerm (PairRouting.alternatingProjection p))
+    (CycleColoring.alternatingPerm (PairRouting.alternatingProjection q)) (PairRouting.labelSet e)
+  rw [permDistance_alternating] at hh
+  apply hh.trans
+  apply Nat.add_le_add_left
+  dsimp only [PairRouting.alternatingProjection]
+  exact (permDistance_mul_le _ _ _ _).trans_eq (by
+    change permDistance (p false)⁻¹ (q false)⁻¹ + _ = _
+    rw [permDistance_inv])
+
+end Thorp
+namespace Thorp
+
+def centralBlockPerm (L : ℕ) : (r : ℕ) → (Card r → Equiv.Perm (Card L)) →
+    Equiv.Perm (Card (L+r))
+  | 0, C => C (fun j => Fin.elim0 j)
+  | r+1, C => childLift (fun b => centralBlockPerm L r (fun u => C (Fin.cons b u)))
+
+def centralPalindromePerm (L r : ℕ) (C : Card r → Equiv.Perm (Card L))
+    (ω : BenesCoins (L+r)) : Equiv.Perm (Card (L+r)) :=
+  butterflyPerm (L+r) (decodeButterfly (L+r) ω.1) * centralBlockPerm L r C *
+    (butterflyPerm (L+r) (decodeButterfly (L+r) ω.2))⁻¹
+
+lemma centralPalindromePerm_step (L r : ℕ) (C : Card (r+1) → Equiv.Perm (Card L))
+    (c : BenesCoins (L+r) × BenesCoins (L+r))
+    (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
+    centralPalindromePerm L (r+1) C ((benesStepEquiv (L+r)).symm (c,coins)) =
+      pairSwitch coins.2 * childLift (fun b => centralPalindromePerm L r
+        (fun u => C (Fin.cons b u)) (if b then c.2 else c.1)) * pairSwitch coins.1 := by
+  change ((butterflyPerm (L+r+1) (decodeButterfly (L+r+1)
+      ((coinStepEquiv (L+r)).symm ((c.1.1,c.2.1),coins.2))))) *
+    childLift (fun b => centralBlockPerm L r (fun u => C (Fin.cons b u))) *
+    (butterflyPerm (L+r+1) (decodeButterfly (L+r+1)
+      ((coinStepEquiv (L+r)).symm ((c.1.2,c.2.2),coins.1))))⁻¹ = _
+  rw [butterflyPerm_step,butterflyPerm_step,mul_inv_rev,childLift_inv,pairSwitch_inv]
+  simp only [mul_assoc]
+  rw [←mul_assoc (childLift _),childLift_mul,←mul_assoc (childLift _),childLift_mul]
+  congr 2
+  apply congrArg childLift
+  funext b
+  cases b <;> rfl
+
+noncomputable def centralLevelCost (t L : ℕ) : (r : ℕ) →
+    (Card r → Equiv.Perm (Card L)) → {ι : Type*} → [Fintype ι] →
+    (ι ↪ Card (L+r)) → BenesCoins (L+r) → ℕ
+  | 0, _, _, _, _, _ => 0
+  | r+1, C, _, _, e, ω =>
+    let σ := benesStepEquiv (L+r) ω
+    let x := e.trans (headTailEquiv (L+r)).toEmbedding
+    let c := PairRouting.colors x σ.2.1
+    let hc := PairRouting.colors_compatible x σ.2.1
+    if t = L+r+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
+      (fun b => centralPalindromePerm L r (fun u => C (Fin.cons b u)) (if b then σ.1.2 else σ.1.1))
+    else centralLevelCost t L r (fun u => C (Fin.cons false u))
+      (PairRouting.childEmbedding x c hc false) σ.1.1 +
+      centralLevelCost t L r (fun u => C (Fin.cons true u))
+      (PairRouting.childEmbedding x c hc true) σ.1.2
+
+noncomputable def centralSlabCost (l L r : ℕ) (C : Card r → Equiv.Perm (Card L))
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (ω : BenesCoins (L+r)) : ℕ :=
+  ∑ i : Fin l, centralLevelCost (l+i.val+1) L r C e ω
+
+lemma permDistance_childLift (d : ℕ) (p q : Bool → Equiv.Perm (Card d)) :
+    permDistance (childLift p) (childLift q) =
+      permDistance (p false) (q false)+permDistance (p true) (q true) := by
+  classical
+  rw [permDistance_sum]
+  have hh : (∑ x : Card (d+1), if childLift p x = childLift q x then 0 else 1) =
+      ∑ x : Bool × Card d, if p x.1 x.2 = q x.1 x.2 then 0 else 1 := by
+    apply Fintype.sum_equiv (headTailEquiv d)
+    intro x
+    have hx : (headTailEquiv d) x = (x 0,Fin.tail x) := rfl
+    rw [hx]
+    congr 1
+    apply propext
+    constructor
+    · intro h
+      have ht := congrArg Fin.tail h
+      simpa only [childLift,Equiv.coe_fn_mk,Fin.tail_cons] using ht
+    · intro h
+      dsimp only [childLift,Equiv.coe_fn_mk]
+      exact congrArg (fun z : Card d => (Fin.cons (x 0) z : Card (d+1))) h
+  rw [hh]
+  simp only [Fintype.sum_prod_type,Fintype.sum_bool,permDistance_sum]
+  omega
+
+noncomputable def centralDistance (L r : ℕ) (C D : Card r → Equiv.Perm (Card L)) : ℕ :=
+  ∑ u : Card r, permDistance (C u) (D u)
+
+lemma centralDistance_step (L r : ℕ) (C D : Card (r+1) → Equiv.Perm (Card L)) :
+    centralDistance L (r+1) C D =
+      centralDistance L r (fun u => C (Fin.cons false u)) (fun u => D (Fin.cons false u))+
+      centralDistance L r (fun u => C (Fin.cons true u)) (fun u => D (Fin.cons true u)) := by
+  classical
+  unfold centralDistance
+  rw [Fintype.sum_equiv (headTailEquiv r) _
+    (fun x : Bool × Card r => permDistance (C (Fin.cons x.1 x.2)) (D (Fin.cons x.1 x.2)))
+      (fun x => by
+        change permDistance (C x) (D x) =
+          permDistance (C (Fin.cons (x 0) (Fin.tail x))) (D (Fin.cons (x 0) (Fin.tail x)))
+        rw [Fin.cons_self_tail])]
+  simp only [Fintype.sum_prod_type,Fintype.sum_bool]
+  omega
+
+lemma permDistance_centralBlock (L r : ℕ) (C D : Card r → Equiv.Perm (Card L)) :
+    permDistance (centralBlockPerm L r C) (centralBlockPerm L r D) = centralDistance L r C D := by
+  induction r with
+  | zero =>
+    simp only [centralBlockPerm,centralDistance,Fintype.sum_unique]
+    congr 1
+  | succ r ih =>
+    rw [centralBlockPerm,centralBlockPerm,permDistance_childLift,centralDistance_step,ih,ih]
+
+lemma permDistance_centralPalindrome (L r : ℕ) (C D : Card r → Equiv.Perm (Card L))
+    (ω : BenesCoins (L+r)) :
+    permDistance (centralPalindromePerm L r C ω) (centralPalindromePerm L r D ω) =
+      centralDistance L r C D := by
+  rw [centralPalindromePerm,centralPalindromePerm,permDistance_mul_right,
+    permDistance_mul_left,permDistance_centralBlock]
+
+lemma centralLevelCost_step (t L r : ℕ) (C : Card (r+1) → Equiv.Perm (Card L))
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r+1))
+    (ω : BenesCoins (L+r) × BenesCoins (L+r))
+    (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
+    centralLevelCost t L (r+1) C e ((benesStepEquiv (L+r)).symm (ω,coins)) =
+      let x := e.trans (headTailEquiv (L+r)).toEmbedding
+      let c := PairRouting.colors x coins.1
+      let hc := PairRouting.colors_compatible x coins.1
+      if t = L+r+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x coins.1)
+        (fun b => centralPalindromePerm L r (fun u => C (Fin.cons b u)) (if b then ω.2 else ω.1))
+      else centralLevelCost t L r (fun u => C (Fin.cons false u))
+        (PairRouting.childEmbedding x c hc false) ω.1 +
+        centralLevelCost t L r (fun u => C (Fin.cons true u))
+        (PairRouting.childEmbedding x c hc true) ω.2 := by
+  rw [centralLevelCost]
+  rfl
+
+lemma centralLevelCost_perturb (t L r : ℕ) (C D : Card r → Equiv.Perm (Card L))
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (ω : BenesCoins (L+r)) :
+    centralLevelCost t L r C e ω ≤ centralLevelCost t L r D e ω+centralDistance L r C D := by
+  induction r generalizing ι with
+  | zero => simp only [centralLevelCost,Nat.zero_add,Nat.zero_le]
+  | succ r ih =>
+    rw [←(benesStepEquiv (L+r)).symm_apply_apply ω,centralLevelCost_step,centralLevelCost_step]
+    dsimp only
+    split_ifs with ht
+    · have hh := alternatingCycles_distance_le
+        (PairRouting.switchedEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
+          (benesStepEquiv (L+r) ω).2.1)
+        (fun b => centralPalindromePerm L r (fun u => C (Fin.cons b u))
+          (if b then (benesStepEquiv (L+r) ω).1.2 else (benesStepEquiv (L+r) ω).1.1))
+        (fun b => centralPalindromePerm L r (fun u => D (Fin.cons b u))
+          (if b then (benesStepEquiv (L+r) ω).1.2 else (benesStepEquiv (L+r) ω).1.1))
+      simpa only [Bool.false_eq_true,↓reduceIte,permDistance_centralPalindrome,←centralDistance_step] using hh
+    · rw [centralDistance_step]
+      have h₀ := ih (fun u => C (Fin.cons false u)) (fun u => D (Fin.cons false u))
+        (PairRouting.childEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
+          (PairRouting.colors (e.trans (headTailEquiv (L+r)).toEmbedding) (benesStepEquiv (L+r) ω).2.1)
+          (PairRouting.colors_compatible _ _) false)
+        (benesStepEquiv (L+r) ω).1.1
+      have h₁ := ih (fun u => C (Fin.cons true u)) (fun u => D (Fin.cons true u))
+        (PairRouting.childEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
+          (PairRouting.colors (e.trans (headTailEquiv (L+r)).toEmbedding) (benesStepEquiv (L+r) ω).2.1)
+          (PairRouting.colors_compatible _ _) true)
+        (benesStepEquiv (L+r) ω).1.2
+      omega
+
+end Thorp
+
+end ThorpNine.Harmonic
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Harmonic/ParallelFollow.lean b/lean/OAI/Probability/ThorpRouting/Harmonic/ParallelFollow.lean
index 6cdd396a0d457832461494fb8761a25eba765077..8ff10d317e6106a290320bc2c426b3aae27ae52d 100644
--- a/lean/OAI/Probability/ThorpRouting/Harmonic/ParallelFollow.lean
+++ b/lean/OAI/Probability/ThorpRouting/Harmonic/ParallelFollow.lean
@@ -1,883 +1,883 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Harmonic.PriorPaths
-
-namespace OAI
-
-namespace ThorpNine.Harmonic
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-section ParallelButterfly
-variable {α B : Type*} [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] {d : ℕ}
-omit [DecidableEq B] in
-lemma parallel_entropy (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool) (S : Finset α) :
-    (∑ i : B × SwitchIndex d,
-      if (parallelUsers pre ω i).1 ∈ S ∧ (parallelUsers pre ω i).2 ∈ S then (1:ℝ) else 0) ≤
-      (S.card : ℝ) * Real.log S.card / (2 * Real.log 2) := by
-  classical
-  rw [Fintype.sum_prod_type]
-  calc
-    _ ≤ ∑ b : B, (blockMarks pre S b).card * Real.log (blockMarks pre S b).card / (2*Real.log 2) := by
-      apply Finset.sum_le_sum
-      intro b _
-      simpa only [←mem_blockMarks,parallelUsers] using
-        marked_switch_entropy_bound d (decodeButterfly d (fun i => ω (b,i))) (blockMarks pre S b)
-    _ ≤ ∑ b : B, (blockMarks pre S b).card * Real.log S.card / (2*Real.log 2) := by
-      apply Finset.sum_le_sum
-      intro b _
-      by_cases hz : (blockMarks pre S b).card = 0
-      · simp [hz]
-      · apply div_le_div_of_nonneg_right _ (by positivity)
-        apply mul_le_mul_of_nonneg_left _ (by positivity)
-        exact Real.log_le_log (by exact_mod_cast Nat.pos_of_ne_zero hz)
-          (by exact_mod_cast card_blockMarks_le pre S b)
-    _ = _ := by rw [←Finset.sum_div,←Finset.sum_mul,←Nat.cast_sum,sum_card_blockMarks]
-
-noncomputable def parallelFollow (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x : α) (k : α → QueryTree (B × SwitchIndex d)) : QueryTree (B × SwitchIndex d) :=
-  QueryTree.followPath d (fun i => ((pre x).1,i)) (pre x).2 (fun y => k (post ((pre x).1,y)))
-
-omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
-lemma parallelFollow_succeeds (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
-    (parallelFollow pre post x k).succeeds ω = (k (relabeledPerm pre post ω x)).succeeds ω := by
-  exact QueryTree.followPath_succeeds _ _ _ _ _
-
-omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
-lemma parallelFollow_raw_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
-    (parallelFollow pre post x k).charge ω = (k (relabeledPerm pre post ω x)).charge ω := by
-  exact QueryTree.followPath_raw_charge _ _ _ _ _
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma parallelFollow_stale_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
-    (ω : B × SwitchIndex d → Bool) :
-    (parallelFollow pre post x k).staleCharge R ω =
-      (k (relabeledPerm pre post ω x)).staleCharge (R ∪ parallelPath pre ω x) ω := by
-  exact QueryTree.followPath_stale_charge _ _ _ _ _ _
-
-noncomputable def closingDomain (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) : Finset (B × SwitchIndex d) :=
-  (routeDomain d (pre x).2 (post.symm y).2).image (fun i => ((pre x).1,i))
-
-def closingValue (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (i : B × SwitchIndex d) : Bool := routeValue d (pre x).2 (post.symm y).2 i.2
-
-omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
-lemma relabeled_endpoint_pair (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (ω : B × SwitchIndex d → Bool) (x y : α) :
-    relabeledPerm pre post ω x = y ↔
-      (pre x).1 = (post.symm y).1 ∧
-      butterflyPerm d (decodeButterfly d (fun i => ω ((pre x).1,i))) (pre x).2 = (post.symm y).2 := by
-  change post ((pre x).1,_) = y ↔ _
-  rw [←post.eq_symm_apply,Prod.mk.injEq]
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma relabeled_endpoint_iff (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (ω : B × SwitchIndex d → Bool) (x y : α) :
-    relabeledPerm pre post ω x = y ↔
-      (pre x).1 = (post.symm y).1 ∧
-        ∀ i ∈ closingDomain pre post x y, ω i = closingValue pre post x y i := by
-  classical
-  rw [relabeled_endpoint_pair,butterfly_endpoint_iff]
-  simp only [closingDomain,Finset.forall_mem_image,closingValue]
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma closingDomain_eq (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (ω : B × SwitchIndex d → Bool) (x y : α) (h : relabeledPerm pre post ω x = y) :
-    closingDomain pre post x y = parallelPath pre ω x := by
-  rw [closingDomain,parallelPath,(relabeled_endpoint_pair _ _ _ _ _).mp h |>.2]
-
-noncomputable def parallelClose (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (k : QueryTree (B × SwitchIndex d)) : QueryTree (B × SwitchIndex d) :=
-  if (pre x).1 = (post.symm y).1 then
-    QueryTree.forceList (closingDomain pre post x y).toList (closingValue pre post x y) k
-  else QueryTree.reject
-
-omit [Fintype α] in
-lemma parallelClose_succeeds (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (k : QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
-    (parallelClose pre post x y k).succeeds ω =
-      if relabeledPerm pre post ω x = y then k.succeeds ω else false := by
-  classical
-  simp only [relabeled_endpoint_iff,parallelClose]
-  by_cases hb : (pre x).1 = (post.symm y).1
-  · rw [ite_eq_left hb,QueryTree.forceList_succeeds]
-    simp only [Finset.mem_toList,hb,true_and]
-  · simp [hb,QueryTree.succeeds]
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma parallelClose_raw_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (k : QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool)
-    (h : relabeledPerm pre post ω x = y) :
-    (parallelClose pre post x y k).charge ω = k.charge ω + d := by
-  have hh := (relabeled_endpoint_iff _ _ _ _ _).mp h
-  rw [parallelClose,ite_eq_left hh.1,QueryTree.forceList_raw_charge _ _ _ _
-    (by simpa only [Finset.mem_toList] using hh.2),Finset.length_toList,
-    closingDomain_eq _ _ _ _ _ h,card_parallelPath]
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma parallelClose_fresh_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (k : QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
-    (ω : B × SwitchIndex d → Bool) (h : relabeledPerm pre post ω x = y) :
-    (parallelClose pre post x y k).freshCharge R ω =
-      k.freshCharge (R ∪ parallelPath pre ω x) ω + (parallelPath pre ω x \ R).card := by
-  have hh := (relabeled_endpoint_iff _ _ _ _ _).mp h
-  rw [parallelClose,ite_eq_left hh.1,QueryTree.forceList_charge _ _ _ _ _
-    (by simpa only [Finset.mem_toList] using hh.2),Finset.toList_toFinset,
-    closingDomain_eq _ _ _ _ _ h]
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma parallelClose_stale_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (k : QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
-    (ω : B × SwitchIndex d → Bool) (h : relabeledPerm pre post ω x = y) :
-    (parallelClose pre post x y k).staleCharge R ω =
-      k.staleCharge (R ∪ parallelPath pre ω x) ω + (parallelPath pre ω x ∩ R).card := by
-  have h₀ := QueryTree.fresh_add_stale (parallelClose pre post x y k) R ω
-  have h₁ := QueryTree.fresh_add_stale k (R ∪ parallelPath pre ω x) ω
-  rw [parallelClose_raw_charge _ _ _ _ _ _ h,parallelClose_fresh_charge _ _ _ _ _ _ _ h] at h₀
-  have h₂ : (parallelPath pre ω x \ R).card + (parallelPath pre ω x ∩ R).card = d := by
-    rw [Finset.card_sdiff_add_card_inter,card_parallelPath]
-  omega
-
-noncomputable def parallelNetwork (pre : α ≃ B × Card d) (post : B × Card d ≃ α) :
-    RoutingNetwork α (B × SwitchIndex d) d where
-  perm := relabeledPerm pre post
-  path := parallelPath pre
-  users := parallelUsers pre
-  users_ne := parallelUsers_ne pre
-  mem_path := mem_parallelPath pre
-  card_path := card_parallelPath pre
-  entropy := parallel_entropy pre
-  follow := parallelFollow pre post
-  close := parallelClose pre post
-  follow_succeeds := parallelFollow_succeeds pre post
-  follow_raw_charge := parallelFollow_raw_charge pre post
-  follow_stale_charge := parallelFollow_stale_charge pre post
-  close_succeeds := parallelClose_succeeds pre post
-  close_raw_charge := parallelClose_raw_charge pre post
-  close_stale_charge := parallelClose_stale_charge pre post
-
-lemma parallel_cycle_log_mgf (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (S : Finset α) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    Real.log (finiteMean (fun ω : B × SwitchIndex d → Bool =>
-      q^(Fintype.card (RoutingNetwork.SelectedCycle (relabeledPerm pre post ω) S)))) ≤
-      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
-  exact (parallelNetwork pre post).butterfly_cycle_log_mgf S J hJ q hq
-
-end ParallelButterfly
-
-def cardSplit : (r s : ℕ) → Card (s+r) ≃ Card r × Card s
-  | 0, s =>
-    { toFun := fun x => ((fun i => Fin.elim0 i), x)
-      invFun := fun x => x.2
-      left_inv := fun _ => rfl
-      right_inv := fun x => by
-        apply Prod.ext
-        · exact Subsingleton.elim _ _
-        · rfl }
-  | r+1, s =>
-    { toFun := fun x => (Fin.cons (x 0) ((cardSplit r s (Fin.tail x)).1),
-        (cardSplit r s (Fin.tail x)).2)
-      invFun := fun x => Fin.cons (x.1 0) ((cardSplit r s).symm (Fin.tail x.1,x.2))
-      left_inv := by
-        intro x
-        simp only [Fin.cons_zero, Fin.tail_cons, Prod.mk.eta, Equiv.symm_apply_apply, Fin.cons_self_tail]
-      right_inv := by
-        intro x
-        simp only [Fin.cons_zero, Fin.tail_cons, Equiv.apply_symm_apply, Fin.cons_self_tail] }
-
-def assembleBits : (r s : ℕ) → (Card r × SwitchIndex s → Bool) →
-    (Card s × SwitchIndex r → Bool) → SwitchIndex (s+r) → Bool
-  | 0, _, lo, _, i => lo ((fun j => Fin.elim0 j), i)
-  | r+1, s, _, hi, Sum.inl y => hi ((cardSplit r s y).2,Sum.inl (cardSplit r s y).1)
-  | r+1, s, lo, hi, Sum.inr (b,i) => assembleBits r s
-      (fun j => lo (Fin.cons b j.1,j.2)) (fun j => hi (j.1,Sum.inr (b,j.2))) i
-
-lemma butterfly_split (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
-    (hi : Card s × SwitchIndex r → Bool) (x : Card (s+r)) :
-    cardSplit r s (butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi)) x) =
-      let y := cardSplit r s x
-      let z := butterflyPerm s (decodeButterfly s (fun i => lo (y.1,i))) y.2
-      (butterflyPerm r (decodeButterfly r (fun i => hi (z,i))) y.1,z) := by
-  induction r with
-  | zero =>
-    apply Prod.ext
-    · funext i
-      exact Fin.elim0 i
-    · rfl
-  | succ r ih =>
-    let lo' : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons (x 0) j.1,j.2)
-    let hi' : Card s × SwitchIndex r → Bool := fun j => hi (j.1,Sum.inr (x 0,j.2))
-    have hh := ih lo' hi' (Fin.tail x)
-    let u := butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo' hi')) (Fin.tail x)
-    let y := cardSplit r s (Fin.tail x)
-    let z := butterflyPerm s (decodeButterfly s (fun i => lo' (y.1,i))) y.2
-    let w := butterflyPerm r (decodeButterfly r (fun i => hi' (z,i))) y.1
-    change cardSplit r s u = (w,z) at hh
-    change ((Fin.cons (Bool.xor (x 0) (hi ((cardSplit r s u).2,Sum.inl (cardSplit r s u).1)))
-        ((cardSplit r s u).1) : Card (r+1)), (cardSplit r s u).2) =
-      ((Fin.cons (Bool.xor (x 0) (hi (z,Sum.inl w))) w : Card (r+1)),z)
-    rw [hh]
-
-def splitSwitch : (r s : ℕ) → SwitchIndex (s+r) ≃
-    (Card r × SwitchIndex s) ⊕ (Card s × SwitchIndex r)
-  | 0, _ =>
-    { toFun := fun i => Sum.inl ((fun j => Fin.elim0 j),i)
-      invFun := fun i => match i with
-        | Sum.inl (_,j) => j
-        | Sum.inr (_,j) => nomatch j
-      left_inv := fun _ => rfl
-      right_inv := by
-        intro i
-        rcases i with ⟨x,j⟩ | ⟨x,j⟩
-        · exact congrArg Sum.inl (Prod.ext (Subsingleton.elim _ _) rfl)
-        · exact Empty.elim j }
-  | r+1, s =>
-    { toFun := fun i => match i with
-        | Sum.inl y => Sum.inr ((cardSplit r s y).2,Sum.inl (cardSplit r s y).1)
-        | Sum.inr (b,j) => match splitSwitch r s j with
-          | Sum.inl (p,j) => Sum.inl (Fin.cons b p,j)
-          | Sum.inr (q,j) => Sum.inr (q,Sum.inr (b,j))
-      invFun := fun i => match i with
-        | Sum.inl (p,j) => Sum.inr (p 0,(splitSwitch r s).symm (Sum.inl (Fin.tail p,j)))
-        | Sum.inr (q,Sum.inl p) => Sum.inl ((cardSplit r s).symm (p,q))
-        | Sum.inr (q,Sum.inr (b,j)) => Sum.inr (b,(splitSwitch r s).symm (Sum.inr (q,j)))
-      left_inv := by
-        intro i
-        rcases i with y | ⟨b,j⟩
-        · dsimp only
-          rw [Prod.mk.eta, Equiv.symm_apply_apply]
-        · dsimp only
-          cases hz : splitSwitch r s j with
-          | inl z =>
-            rcases z with ⟨p,k⟩
-            simp only [Fin.cons_zero, Fin.tail_cons]
-            rw [←hz,Equiv.symm_apply_apply]
-          | inr z =>
-            rcases z with ⟨q,k⟩
-            dsimp only
-            rw [←hz,Equiv.symm_apply_apply]
-      right_inv := by
-        intro i
-        rcases i with ⟨p,j⟩ | ⟨q,j⟩
-        · simp only [Equiv.apply_symm_apply, Fin.cons_self_tail]
-        · rcases j with p | ⟨b,j⟩ <;> simp only [Equiv.apply_symm_apply] }
-
-lemma assembleBits_eq (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
-    (hi : Card s × SwitchIndex r → Bool) (i : SwitchIndex (s+r)) :
-    assembleBits r s lo hi i = Sum.elim lo hi (splitSwitch r s i) := by
-  induction r with
-  | zero => rfl
-  | succ r ih =>
-    rcases i with y | ⟨b,j⟩
-    · rfl
-    · simp only [assembleBits,splitSwitch,Equiv.coe_fn_mk]
-      rw [ih]
-      cases splitSwitch r s j <;> rfl
-
-def splitCoinEquiv (r s : ℕ) :
-    ((Card r × SwitchIndex s → Bool) × (Card s × SwitchIndex r → Bool)) ≃
-      (SwitchIndex (s+r) → Bool) :=
-  (Equiv.sumArrowEquivProdArrow _ _ Bool).symm.trans
-    (Equiv.arrowCongr (splitSwitch r s).symm (Equiv.refl Bool))
-
-lemma splitCoinEquiv_apply (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
-    (hi : Card s × SwitchIndex r → Bool) :
-    splitCoinEquiv r s (lo,hi) = assembleBits r s lo hi := by
-  funext i
-  exact (assembleBits_eq r s lo hi i).symm
-
-lemma finiteMean_prod {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] (f : Ω × Ω' → ℝ) :
-    finiteMean f = finiteMean (fun x => finiteMean (fun y => f (x,y))) := by
-  simp only [finiteMean, Fintype.sum_prod_type, Fintype.card_prod, Nat.cast_mul,
-    ←Finset.sum_div, div_div, mul_comm]
-
-lemma finiteMean_split (r s : ℕ) (f : (SwitchIndex (s+r) → Bool) → ℝ) :
-    finiteMean f = finiteMean (fun lo : Card r × SwitchIndex s → Bool =>
-      finiteMean (fun hi : Card s × SwitchIndex r → Bool => f (assembleBits r s lo hi))) := by
-  rw [finiteMean_equiv (splitCoinEquiv r s).symm]
-  simp only [Equiv.symm_symm]
-  rw [finiteMean_prod]
-  apply finiteMean_congr
-  intro lo
-  apply finiteMean_congr
-  intro hi
-  rw [splitCoinEquiv_apply]
-
-section PartiallyExposed
-variable {α : Type*} {r s : ℕ}
-
-def freshPre (pre : α ≃ Card (s+r)) (lo : Card r × SwitchIndex s → Bool) :
-    α ≃ Card s × Card r :=
-  ((pre.trans (cardSplit r s)).trans (parallelPerm lo)).trans (Equiv.prodComm _ _)
-
-def freshPost (post : Card (s+r) ≃ α) : Card s × Card r ≃ α :=
-  ((Equiv.prodComm _ _).trans (cardSplit r s).symm).trans post
-
-lemma exposed_perm_eq (pre : α ≃ Card (s+r)) (post : Card (s+r) ≃ α)
-    (lo : Card r × SwitchIndex s → Bool) (hi : Card s × SwitchIndex r → Bool) :
-    (pre.trans (butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi)))).trans post =
-      relabeledPerm (freshPre pre lo) (freshPost post) hi := by
-  ext x
-  simpa only [relabeledPerm, freshPre, freshPost, parallelPerm, Equiv.trans_apply,
-    Equiv.coe_fn_mk, Equiv.prodComm_apply, Prod.swap, Equiv.symm_apply_apply] using
-    congrArg (fun y => post ((cardSplit r s).symm y)) (butterfly_split r s lo hi (pre x))
-
-lemma partially_exposed_cycle_log_mgf [Fintype α] [DecidableEq α]
-    (pre : α ≃ Card (s+r)) (post : Card (s+r) ≃ α)
-    (lo : Card r × SwitchIndex s → Bool) (S : Finset α)
-    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    Real.log (finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(Fintype.card (RoutingNetwork.SelectedCycle
-        ((pre.trans (butterflyPerm (s+r)
-          (decodeButterfly (s+r) (assembleBits r s lo hi)))).trans post) S)))) ≤
-      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^r) := by
-  simp only [exposed_perm_eq]
-  exact parallel_cycle_log_mgf (freshPre pre lo) (freshPost post) S J hJ q hq
-
-end PartiallyExposed
-
-lemma inclusionMoment_eq {α Ω : Type*} [DecidableEq α] [Fintype Ω]
-    (F : Ω → Finset α) (S : Finset α) :
-    inclusionMoment F S = finiteMean (fun ω => if S ⊆ F ω then (1:ℝ) else 0) := by
-  unfold inclusionMoment
-  apply finiteMean_congr
-  intro ω
-  split_ifs <;> rfl
-
-lemma finiteMean_bool (f : Bool → ℝ) : finiteMean f = (f false+f true)/2 := by
-  simp only [finiteMean, Fintype.sum_bool, Fintype.card_bool, Nat.cast_ofNat]
-  ring
-
-lemma finiteMean_const {Ω : Type*} [Fintype Ω] [Nonempty Ω] (c : ℝ) :
-    finiteMean (fun _ : Ω => c) = c := by
-  simp only [finiteMean,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
-  exact mul_div_cancel_left₀ c (Nat.cast_ne_zero.mpr Fintype.card_ne_zero)
-
-lemma finiteMean_pi_prod {ι : Type*} [Fintype ι] [DecidableEq ι] {Ω : ι → Type*}
-    [∀ i, Fintype (Ω i)] (f : (i : ι) → Ω i → ℝ) :
-    finiteMean (fun ω : (i : ι) → Ω i => ∏ i, f i (ω i)) = ∏ i, finiteMean (f i) := by
-  simp only [finiteMean,←Fintype.prod_sum,Fintype.card_pi,Nat.cast_prod,Finset.prod_div_distrib]
-
-section ParallelFairLayer
-variable {β : Type*} [Fintype β] [DecidableEq β]
-
-def pairLayer (ξ : β → Bool) : Equiv.Perm (Bool × β) where
-  toFun x := (Bool.xor x.1 (ξ x.2),x.2)
-  invFun x := (Bool.xor x.1 (ξ x.2),x.2)
-  left_inv x := by
-    rcases x with ⟨b,x⟩
-    change (Bool.xor (Bool.xor b (ξ x)) (ξ x),x) = (b,x)
-    cases b <;> cases ξ x <;> rfl
-  right_inv x := by
-    rcases x with ⟨b,x⟩
-    change (Bool.xor (Bool.xor b (ξ x)) (ξ x),x) = (b,x)
-    cases b <;> cases ξ x <;> rfl
-
-noncomputable def pairFiber (S : Finset (Bool × β)) (b : β) : Finset Bool :=
-  Finset.univ.filter (fun ε => (ε,b) ∈ S)
-
-lemma prod_pairFiber (S : Finset (Bool × β)) (f : Bool × β → ℝ) :
-    (∏ x ∈ S, f x) = ∏ b : β, ∏ ε ∈ pairFiber S b, f (ε,b) := by
-  classical
-  simp only [pairFiber,Finset.prod_filter]
-  rw [Finset.prod_comm,←Fintype.prod_prod_type (fun x : Bool × β => if x ∈ S then f x else 1)]
-  rw [←Finset.prod_filter]
-  congr 1
-  ext x
-  simp
-
-lemma fiber_average_bound (S : Finset Bool) (p : Bool → ℝ) (hp : ∀ b, 0 ≤ p b) :
-    finiteMean (fun c : Bool => ∏ ε ∈ S, p (Bool.xor ε c)) ≤
-      ∏ ε ∈ S, (p ε + p (!ε))/2 := by
-  rw [finiteMean_bool]
-  have hxor : ∀ ε : Bool, Bool.xor ε true = Equiv.swap false true ε := by decide
-  simp only [Bool.xor_false,hxor]
-  calc
-    _ ≤ ∏ ε ∈ S, (p ε+p (Equiv.swap false true ε))/2 :=
-      swap_average_product S p hp false true
-    _ = _ := Finset.prod_congr rfl (fun ε _ => by cases ε <;> rfl)
-
-lemma CylinderBound.fair_parallel {m : Finset (Bool × β) → ℝ} {p : Bool × β → ℝ}
-    (h : CylinderBound m p) (hp : ∀ x, 0 ≤ p x) :
-    CylinderBound (fun S => finiteMean (fun ξ : β → Bool => m (S.image (pairLayer ξ))))
-      (fun x => (p x+p (!x.1,x.2))/2) := by
-  intro S
-  calc
-    _ ≤ finiteMean (fun ξ : β → Bool => ∏ x ∈ S, p (pairLayer ξ x)) := by
-      apply finiteMean_mono
-      intro ξ
-      simpa only [Finset.prod_image (pairLayer ξ).injective.injOn] using h (S.image (pairLayer ξ))
-    _ = ∏ b : β, finiteMean (fun c : Bool => ∏ ε ∈ pairFiber S b, p (Bool.xor ε c,b)) := by
-      calc
-        _ = finiteMean (fun ξ : β → Bool => ∏ b : β,
-            ∏ ε ∈ pairFiber S b, p (Bool.xor ε (ξ b),b)) :=
-          finiteMean_congr (fun ξ => prod_pairFiber S (fun x => p (pairLayer ξ x)))
-        _ = _ := finiteMean_pi_prod (fun b c => ∏ ε ∈ pairFiber S b, p (Bool.xor ε c,b))
-    _ ≤ ∏ b : β, ∏ ε ∈ pairFiber S b, (p (ε,b)+p (!ε,b))/2 := by
-      apply Finset.prod_le_prod₀
-      · intro b _
-        exact div_nonneg (Finset.sum_nonneg (fun c _ =>
-          Finset.prod_nonneg (fun ε _ => hp (Bool.xor ε c,b)))) (Nat.cast_nonneg _)
-      · intro b _
-        exact fiber_average_bound (pairFiber S b) (fun ε => p (ε,b)) (fun ε => hp (ε,b))
-    _ = _ := (prod_pairFiber S (fun x => (p x+p (!x.1,x.2))/2)).symm
-
-omit [Fintype β] in
-lemma subset_pairLayer_image_iff (S A : Finset (Bool × β)) (ξ : β → Bool) :
-    S ⊆ A.image (pairLayer ξ) ↔ S.image (pairLayer ξ) ⊆ A := by
-  have hi (x : Bool × β) : pairLayer ξ (pairLayer ξ x) = x := (pairLayer ξ).left_inv x
-  constructor
-  · intro h
-    have h' := Finset.image_subset_image h (f := pairLayer ξ)
-    simpa only [Finset.image_image,Function.comp_def,hi,Finset.image_id'] using h'
-  · intro h
-    have h' := Finset.image_subset_image h (f := pairLayer ξ)
-    simpa only [Finset.image_image,Function.comp_def,hi,Finset.image_id'] using h'
-
-noncomputable def fairSetLayer {Ω : Type*} (F : Ω → Finset (Bool × β)) :
-    Ω × (β → Bool) → Finset (Bool × β) := fun ω => (F ω.1).image (pairLayer ω.2)
-
-lemma finiteMean_comm {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] (f : Ω → Ω' → ℝ) :
-    finiteMean (fun x => finiteMean (f x)) = finiteMean (fun y => finiteMean (fun x => f x y)) := by
-  simp only [finiteMean,←Finset.sum_div,div_div]
-  rw [Finset.sum_comm,mul_comm]
-
-lemma inclusionMoment_fairSetLayer {Ω : Type*} [Fintype Ω]
-    (F : Ω → Finset (Bool × β)) (S : Finset (Bool × β)) :
-    inclusionMoment (fairSetLayer F) S =
-      finiteMean (fun ξ : β → Bool => inclusionMoment F (S.image (pairLayer ξ))) := by
-  classical
-  simp only [inclusionMoment_eq]
-  have he : (fun ω : Ω × (β → Bool) => if S ⊆ fairSetLayer F ω then (1:ℝ) else 0) =
-      (fun ω : Ω × (β → Bool) => if S.image (pairLayer ω.2) ⊆ F ω.1 then (1:ℝ) else 0) := by
-    funext ω
-    simp only [fairSetLayer,subset_pairLayer_image_iff]
-  apply (congrArg finiteMean he).trans
-  exact (finiteMean_prod (fun ω : Ω × (β → Bool) =>
-    if S.image (pairLayer ω.2) ⊆ F ω.1 then (1:ℝ) else 0)).trans
-      (finiteMean_comm (fun ω ξ => if S.image (pairLayer ξ) ⊆ F ω then (1:ℝ) else 0))
-
-lemma inclusionMoment_fairSetLayer_singleton {Ω : Type*} [Fintype Ω]
-    (F : Ω → Finset (Bool × β)) (x : Bool × β) :
-    inclusionMoment (fairSetLayer F) {x} =
-      (inclusionMoment F {x}+inclusionMoment F {(!x.1,x.2)})/2 := by
-  rw [inclusionMoment_fairSetLayer]
-  simp only [Finset.image_singleton,pairLayer,Equiv.coe_fn_mk]
-  rw [finiteMean_pi_eval (A := fun _ : β => Bool) x.2 (fun c : Bool => inclusionMoment F {(Bool.xor x.1 c,x.2)})]
-  rw [finiteMean_bool]
-  rcases x with ⟨b,x⟩
-  cases b <;> simp only [Bool.xor_false,Bool.false_xor,Bool.true_xor, Bool.not_false,Bool.not_true]
-
-lemma cylinder_fairSetLayer {Ω : Type*} [Fintype Ω]
-    (F : Ω → Finset (Bool × β))
-    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
-    CylinderBound (inclusionMoment (fairSetLayer F))
-      (fun x => inclusionMoment (fairSetLayer F) {x}) := by
-  have hh := h.fair_parallel (fun x => inclusionMoment_nonneg F {x})
-  intro S
-  rw [inclusionMoment_fairSetLayer]
-  simp only [inclusionMoment_fairSetLayer_singleton]
-  exact hh S
-
-end ParallelFairLayer
-
-section RelabelMarks
-variable {α β Ω Ω' : Type*} [DecidableEq α] [DecidableEq β] [Fintype Ω] [Fintype Ω']
-
-omit [Fintype Ω] [Fintype Ω'] in
-lemma subset_image_equiv_iff (e : α ≃ β) (A : Finset α) (S : Finset β) :
-    S ⊆ A.image e ↔ S.image e.symm ⊆ A := by
-  constructor
-  · intro h x hx
-    obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
-    obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp (h hy)
-    simpa only [←hxy,Equiv.symm_apply_apply] using hx
-  · intro h y hy
-    exact Finset.mem_image.mpr ⟨e.symm y,h (Finset.mem_image_of_mem _ hy),e.apply_symm_apply y⟩
-
-omit [Fintype Ω'] in
-lemma inclusionMoment_image_equiv (e : α ≃ β) (F : Ω → Finset α) (S : Finset β) :
-    inclusionMoment (fun ω => (F ω).image e) S = inclusionMoment F (S.image e.symm) := by
-  classical
-  simp only [inclusionMoment_eq]
-  apply finiteMean_congr
-  intro ω
-  simp only [subset_image_equiv_iff]
-
-omit [Fintype Ω'] in
-lemma cylinder_image_equiv (e : α ≃ β) (F : Ω → Finset α)
-    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
-    CylinderBound (inclusionMoment (fun ω => (F ω).image e))
-      (fun x => inclusionMoment (fun ω => (F ω).image e) {x}) := by
-  intro S
-  simp only [inclusionMoment_image_equiv,Finset.image_singleton]
-  have hh := h (S.image e.symm)
-  simpa only [Finset.prod_image e.symm.injective.injOn] using hh
-
-omit [DecidableEq β] in
-lemma inclusionMoment_comp_equiv (e : Ω ≃ Ω') (F : Ω' → Finset α) (S : Finset α) :
-    inclusionMoment (fun ω => F (e ω)) S = inclusionMoment F S := by
-  classical
-  simp only [inclusionMoment_eq]
-  simpa only [Equiv.apply_symm_apply] using
-    finiteMean_equiv e (fun ω => if S ⊆ F (e ω) then (1 : ℝ) else 0)
-
-omit [DecidableEq β] in
-lemma cylinder_comp_equiv (e : Ω ≃ Ω') (F : Ω' → Finset α)
-    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
-    CylinderBound (inclusionMoment (fun ω => F (e ω)))
-      (fun x => inclusionMoment (fun ω => F (e ω)) {x}) := by
-  intro S
-  simpa only [inclusionMoment_comp_equiv] using h S
-end RelabelMarks
-
-lemma finiteMean_prod_mul {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
-    (f : Ω → ℝ) (g : Ω' → ℝ) :
-    finiteMean (fun ω : Ω × Ω' => f ω.1*g ω.2) = finiteMean f * finiteMean g := by
-  simp only [finiteMean,Fintype.sum_prod_type,Fintype.card_prod,Nat.cast_mul,
-    ←Finset.mul_sum,←Finset.sum_mul]
-  ring
-
-section JoinChildren
-variable {β : Type*} [DecidableEq β]
-
-noncomputable def childJoin (A B : Finset β) : Finset (Bool × β) :=
-  A.image (fun x => (false,x)) ∪ B.image (fun x => (true,x))
-
-lemma mem_childJoin (A B : Finset β) (b : Bool) (x : β) :
-    (b,x) ∈ childJoin A B ↔ (if b then x ∈ B else x ∈ A) := by
-  cases b <;> simp [childJoin]
-
-noncomputable def boolFiber (S : Finset (Bool × β)) (b : Bool) : Finset β :=
-  (S.filter (fun x => x.1 = b)).image Prod.snd
-
-lemma mem_boolFiber (S : Finset (Bool × β)) (b : Bool) (x : β) :
-    x ∈ boolFiber S b ↔ (b,x) ∈ S := by
-  constructor
-  · intro h
-    obtain ⟨⟨c,y⟩,hy,hxy⟩ := Finset.mem_image.mp h
-    obtain ⟨hy,hc⟩ := Finset.mem_filter.mp hy
-    change c = b at hc
-    change y = x at hxy
-    simpa only [hc,hxy] using hy
-  · intro h
-    exact Finset.mem_image.mpr ⟨(b,x),Finset.mem_filter.mpr ⟨h,rfl⟩,rfl⟩
-
-lemma childJoin_boolFiber (S : Finset (Bool × β)) :
-    childJoin (boolFiber S false) (boolFiber S true) = S := by
-  ext ⟨b,x⟩
-  rw [mem_childJoin]
-  cases b <;> simp only [Bool.false_eq_true,↓reduceIte,mem_boolFiber]
-
-lemma subset_childJoin_iff (S : Finset (Bool × β)) (A B : Finset β) :
-    S ⊆ childJoin A B ↔ boolFiber S false ⊆ A ∧ boolFiber S true ⊆ B := by
-  constructor
-  · intro h
-    constructor
-    · intro x hx
-      exact (mem_childJoin A B false x).mp (h ((mem_boolFiber S false x).mp hx))
-    · intro x hx
-      exact (mem_childJoin A B true x).mp (h ((mem_boolFiber S true x).mp hx))
-  · rintro ⟨h₀,h₁⟩ ⟨b,x⟩ hx
-    rw [mem_childJoin]
-    cases b
-    · exact h₀ ((mem_boolFiber S false x).mpr hx)
-    · exact h₁ ((mem_boolFiber S true x).mpr hx)
-
-lemma prod_childJoin (A B : Finset β) (f : Bool × β → ℝ) :
-    (∏ x ∈ childJoin A B, f x) = (∏ x ∈ A, f (false,x)) * ∏ x ∈ B, f (true,x) := by
-  classical
-  have hd : Disjoint (A.image (fun x => (false,x))) (B.image (fun x => (true,x))) := by
-    apply Finset.disjoint_left.mpr
-    rintro x hx hy
-    obtain ⟨a,ha,rfl⟩ := Finset.mem_image.mp hx
-    obtain ⟨b,hb,hba⟩ := Finset.mem_image.mp hy
-    exact Bool.false_ne_true (congrArg Prod.fst hba).symm
-  rw [childJoin,Finset.prod_union hd,
-    Finset.prod_image (fun _ _ _ _ h => congrArg Prod.snd h),
-    Finset.prod_image (fun _ _ _ _ h => congrArg Prod.snd h)]
-
-lemma prod_boolFibers (S : Finset (Bool × β)) (f : Bool × β → ℝ) :
-    (∏ x ∈ S, f x) = (∏ x ∈ boolFiber S false, f (false,x)) *
-      ∏ x ∈ boolFiber S true, f (true,x) := by
-  rw [←prod_childJoin,childJoin_boolFiber]
-
-lemma inclusionMoment_childJoin {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
-    (F : Ω → Finset β) (G : Ω' → Finset β) (S : Finset (Bool × β)) :
-    inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) S =
-      inclusionMoment F (boolFiber S false) * inclusionMoment G (boolFiber S true) := by
-  classical
-  simp only [inclusionMoment_eq]
-  rw [←finiteMean_prod_mul]
-  apply finiteMean_congr
-  intro ω
-  simp only [subset_childJoin_iff]
-  split_ifs <;> simp_all
-
-lemma cylinder_childJoin {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] [Nonempty Ω] [Nonempty Ω']
-    (F : Ω → Finset β) (G : Ω' → Finset β)
-    (hF : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x}))
-    (hG : CylinderBound (inclusionMoment G) (fun x => inclusionMoment G {x})) :
-    CylinderBound (inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)))
-      (fun x => inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {x}) := by
-  intro S
-  rw [inclusionMoment_childJoin,prod_boolFibers]
-  have hf (b : Bool) (x : β) : boolFiber {(b,x)} b = {x} := by
-    ext y
-    simp only [mem_boolFiber,Finset.mem_singleton,Prod.mk.injEq,true_and]
-  have hn (b : Bool) (x : β) : boolFiber {(!b,x)} b = ∅ := by
-    ext y
-    cases b <;> simp [mem_boolFiber]
-  have h₀ (x : β) : inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {(false,x)} =
-      inclusionMoment F {x} * inclusionMoment G ∅ := by
-    rw [inclusionMoment_childJoin,hf]
-    have hh := hn true x
-    exact congrArg (fun T => inclusionMoment F {x} * inclusionMoment G T) hh
-  have h₁ (x : β) : inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {(true,x)} =
-      inclusionMoment F ∅ * inclusionMoment G {x} := by
-    rw [inclusionMoment_childJoin,hf]
-    have hh := hn false x
-    exact congrArg (fun T => inclusionMoment F T * inclusionMoment G {x}) hh
-  have emF : inclusionMoment F ∅ = 1 := by simp [inclusionMoment,finiteMean_const]
-  have emG : inclusionMoment G ∅ = 1 := by simp [inclusionMoment,finiteMean_const]
-  simp only [h₀,h₁,emF,emG,mul_one,one_mul]
-  exact mul_le_mul (hF _) (hG _) (inclusionMoment_nonneg _ _)
-    (Finset.prod_nonneg (fun x _ => inclusionMoment_nonneg F {x}))
-end JoinChildren
-
-def coinStepEquiv (d : ℕ) : (SwitchIndex (d+1) → Bool) ≃
-    ((SwitchIndex d → Bool) × (SwitchIndex d → Bool)) × (Card d → Bool) where
-  toFun ω := ((fun i => ω (Sum.inr (false,i)),fun i => ω (Sum.inr (true,i))),
-    fun x => ω (Sum.inl x))
-  invFun ω := Sum.elim ω.2 (fun bi => if bi.1 then ω.1.2 bi.2 else ω.1.1 bi.2)
-  left_inv ω := by
-    funext i
-    cases i with
-    | inl x => rfl
-    | inr bi => cases bi with | mk b i => cases b <;> rfl
-  right_inv ω := by
-    rcases ω with ⟨⟨ω₀,ω₁⟩,ξ⟩
-    rfl
-
-def headTailEquiv (d : ℕ) : Card (d+1) ≃ Bool × Card d :=
-  (Fin.consEquiv (fun _ : Fin (d+1) => Bool)).symm
-
-lemma childMarks_join (d : ℕ) (A : Finset (Card (d+1))) :
-    A = (childMarks A false).image (Fin.cons false) ∪
-      (childMarks A true).image (Fin.cons true) := by
-  classical
-  ext x
-  constructor
-  · intro hx
-    cases hb : x 0
-    · exact Finset.mem_union_left _ (Finset.mem_image.mpr
-        ⟨Fin.tail x,(mem_childMarks _ _ _).mpr (by simpa only [←hb,Fin.cons_self_tail] using hx),
-          by simp only [←hb,Fin.cons_self_tail]⟩)
-    · exact Finset.mem_union_right _ (Finset.mem_image.mpr
-        ⟨Fin.tail x,(mem_childMarks _ _ _).mpr (by simpa only [←hb,Fin.cons_self_tail] using hx),
-          by simp only [←hb,Fin.cons_self_tail]⟩)
-  · intro hx
-    rcases Finset.mem_union.mp hx with hx | hx
-    · obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
-      exact (mem_childMarks _ _ _).mp hy
-    · obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
-      exact (mem_childMarks _ _ _).mp hy
-
-noncomputable def butterflyMarks (d : ℕ) (A : Finset (Card d))
-    (ω : SwitchIndex d → Bool) : Finset (Card d) :=
-  A.image (butterflyPerm d (decodeButterfly d ω))
-
-lemma butterflyMarks_succ (d : ℕ) (A : Finset (Card (d+1))) (ω : SwitchIndex (d+1) → Bool) :
-    butterflyMarks (d+1) A ω =
-      (fairSetLayer (fun p : (SwitchIndex d → Bool) × (SwitchIndex d → Bool) =>
-        childJoin (butterflyMarks d (childMarks A false) p.1)
-          (butterflyMarks d (childMarks A true) p.2)) (coinStepEquiv d ω)).image
-            (headTailEquiv d).symm := by
-  classical
-  unfold butterflyMarks fairSetLayer childJoin
-  rw [Finset.image_union,Finset.image_union]
-  simp only [Finset.image_image]
-  conv_lhs => rw [childMarks_join d A]
-  rw [Finset.image_union]
-  simp only [Finset.image_image]
-  apply congrArg₂ (· ∪ ·)
-  · apply Finset.image_congr
-    intro x _
-    simp only [Function.comp_def,butterflyPerm,decodeButterfly,Equiv.Perm.coe_mul,
-      childLift,pairSwitch,switchFun,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
-      coinStepEquiv,headTailEquiv,Equiv.symm_symm,pairLayer]
-    rfl
-  · apply Finset.image_congr
-    intro x _
-    simp only [Function.comp_def,butterflyPerm,decodeButterfly,Equiv.Perm.coe_mul,
-      childLift,pairSwitch,switchFun,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
-      coinStepEquiv,headTailEquiv,Equiv.symm_symm,pairLayer]
-    rfl
-
-lemma butterflyMarks_cylinder (d : ℕ) (A : Finset (Card d)) :
-    CylinderBound (inclusionMoment (butterflyMarks d A))
-      (fun x => inclusionMoment (butterflyMarks d A) {x}) := by
-  induction d with
-  | zero =>
-    have he : butterflyMarks 0 A = fun _ : SwitchIndex 0 → Bool => A := by
-      funext ω
-      simp [butterflyMarks,butterflyPerm]
-    rw [he]
-    exact cylinder_const A
-  | succ d ih =>
-    have h := cylinder_childJoin (butterflyMarks d (childMarks A false))
-      (butterflyMarks d (childMarks A true)) (ih _) (ih _)
-    have hh := cylinder_image_equiv (headTailEquiv d).symm _ (cylinder_fairSetLayer _ h)
-    have hs := cylinder_comp_equiv (coinStepEquiv d) _ hh
-    simpa only [←butterflyMarks_succ] using hs
-
-lemma indicator_mem_image_perm {α : Type*} [DecidableEq α]
-    (A : Finset α) (p : Equiv.Perm α) (y : α) :
-    (if y ∈ A.image p then (1 : ℝ) else 0) = ∑ x ∈ A, if p x = y then 1 else 0 := by
-  classical
-  by_cases h : y ∈ A.image p
-  · obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp h
-    rw [ite_eq_left (Finset.mem_image.mpr ⟨x,hx,hxy⟩)]
-    symm
-    calc
-      _ = (if p x = y then (1:ℝ) else 0) := Finset.sum_eq_single_of_mem x hx (by
-        intro z _ hzx
-        exact ite_eq_right (fun hz => hzx (p.injective (hz.trans hxy.symm))))
-      _ = 1 := ite_eq_left hxy
-  · rw [ite_eq_right h]
-    symm
-    apply Finset.sum_eq_zero
-    intro x hx
-    exact ite_eq_right (fun hxy => h (Finset.mem_image.mpr ⟨x,hx,hxy⟩))
-
-lemma butterflyMarks_singleton (d : ℕ) (A : Finset (Card d)) (y : Card d) :
-    inclusionMoment (butterflyMarks d A) {y} = (A.card : ℝ)/(2:ℝ)^d := by
-  classical
-  unfold inclusionMoment
-  simp only [Finset.singleton_subset_iff,butterflyMarks,indicator_mem_image_perm]
-  rw [finiteMean_sum]
-  simp only [butterfly_single_card_uniform,Finset.sum_const,nsmul_eq_mul]
-  ring
-
-lemma butterflyMarks_inclusion (d : ℕ) (A S : Finset (Card d)) :
-    inclusionMoment (butterflyMarks d A) S ≤ ((A.card : ℝ)/(2:ℝ)^d)^S.card := by
-  have h := butterflyMarks_cylinder d A S
-  simpa only [butterflyMarks_singleton,Finset.prod_const] using h
-
-namespace CycleColoring
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-abbrev Coloring (p : Equiv.Perm α) (S : Finset α) :=
-  {f : α → Bool // (∀ x, x ∉ S → f x = false) ∧
-    ∀ x ∈ S, p x ∈ S → f (p x) = f x}
-
-noncomputable instance (p : Equiv.Perm α) (S : Finset α) : Fintype (Coloring p S) :=
-  Fintype.ofFinite _
-
-noncomputable def boundary (p : Equiv.Perm α) (S : Finset α) : Finset α :=
-  S.filter (fun x => p.symm x ∉ S)
-
-noncomputable def root {p : Equiv.Perm α} {S : Finset α}
-    (C : RoutingNetwork.SelectedCycle p S) : α := C.nonempty.choose
-
-lemma root_mem {p : Equiv.Perm α} {S : Finset α} (C : RoutingNetwork.SelectedCycle p S) :
-    root C ∈ C.val := C.nonempty.choose_spec
-
-abbrev Seed (p : Equiv.Perm α) (S : Finset α) :=
-  {x // x ∈ boundary p S} ⊕ RoutingNetwork.SelectedCycle p S
-
-noncomputable def record {p : Equiv.Perm α} {S : Finset α}
-    (f : Coloring p S) : Seed p S → Bool :=
-  Sum.elim (fun x => f.val x.val) (fun C => f.val (root C))
-
-lemma orbit_subset_of_predecessor_closed (p : Equiv.Perm α) (D : Finset α)
-    (hD : ∀ x ∈ D, p.symm x ∈ D) {x : α} (hx : x ∈ D) : orbitSet p x ⊆ D := by
-  intro y hy
-  have hh := ((mem_orbitSet p x y).mp hy).inv
-  obtain ⟨n,hn⟩ := hh.exists_nat_pow_eq
-  have hp (k : ℕ) : (p⁻¹ ^ k) x ∈ D := by
-    induction k with
-    | zero => simpa using hx
-    | succ k ih => simpa only [pow_succ',Equiv.Perm.mul_apply,Equiv.Perm.coe_inv] using hD _ ih
-  exact hn ▸ hp n
-
-lemma record_injective (p : Equiv.Perm α) (S : Finset α) :
-    Function.Injective (record (p := p) (S := S)) := by
-  classical
-  intro f g hfg
-  apply Subtype.ext
-  funext x
-  by_contra hne
-  have hx : x ∈ S := by
-    by_contra h
-    exact hne ((f.property.1 x h).trans (g.property.1 x h).symm)
-  let D := S.filter (fun y => f.val y ≠ g.val y)
-  have hxD : x ∈ D := Finset.mem_filter.mpr ⟨hx,hne⟩
-  have hprev (y : α) (hy : y ∈ D) : p.symm y ∈ D := by
-    obtain ⟨hyS,hyne⟩ := Finset.mem_filter.mp hy
-    have hpS : p.symm y ∈ S := by
-      by_contra hn
-      have hseed : y ∈ boundary p S := Finset.mem_filter.mpr ⟨hyS,hn⟩
-      exact hyne (congrFun hfg (Sum.inl ⟨y,hseed⟩))
-    refine Finset.mem_filter.mpr ⟨hpS,?_⟩
-    have hf := f.property.2 (p.symm y) hpS (by simpa only [Equiv.apply_symm_apply] using hyS)
-    have hg := g.property.2 (p.symm y) hpS (by simpa only [Equiv.apply_symm_apply] using hyS)
-    simp only [Equiv.apply_symm_apply] at hf hg
-    intro he
-    exact hyne (hf.trans (he.trans hg.symm))
-  have horb := orbit_subset_of_predecessor_closed p D hprev hxD
-  let C : RoutingNetwork.SelectedCycle p S :=
-    ⟨orbitSet p x,⟨x,rfl⟩,fun y hy => (Finset.mem_filter.mp (horb hy)).1⟩
-  have hrD : root C ∈ D := horb (root_mem C)
-  have hrne := (Finset.mem_filter.mp hrD).2
-  exact hrne (congrFun hfg (Sum.inr C))
-
-lemma card_coloring_le (p : Equiv.Perm α) (S : Finset α) :
-    Fintype.card (Coloring p S) ≤
-      2 ^ ((boundary p S).card + Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
-  have h := Fintype.card_le_of_injective (record (p := p) (S := S)) (record_injective p S)
-  simpa only [Fintype.card_fun,Fintype.card_sum,Fintype.card_coe,Fintype.card_bool] using h
-
-omit [Fintype α] in
-lemma boundary_card (p : Equiv.Perm α) (S : Finset α) :
-    (boundary p S).card = S.card - (S.filter (fun x => p.symm x ∈ S)).card := by
-  classical
-  have h := Finset.card_filter_add_card_filter_not (s := S) (fun x => p.symm x ∈ S)
-  unfold boundary
-  omega
-
-lemma card_coloring_le_edges (p : Equiv.Perm α) (S : Finset α) :
-    Fintype.card (Coloring p S) ≤
-      2 ^ (S.card - (S.filter (fun x => p.symm x ∈ S)).card +
-        Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
-  simpa only [boundary_card] using card_coloring_le p S
-
-
-abbrev OppositeColoring (p : Equiv.Perm α) (S : Finset α) :=
-  {f : α → Bool // (∀ x, x ∉ S → f x = false) ∧
-    ∀ x ∈ S, p x ∈ S → f (p x) = !(f x)}
-
-noncomputable instance (p : Equiv.Perm α) (S : Finset α) : Fintype (OppositeColoring p S) :=
-  Fintype.ofFinite _
-
-noncomputable def bipartitionFlip (p : Equiv.Perm α) (S : Finset α)
-    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x))
-    (f : OppositeColoring p S) : Coloring p S := by
-  classical
-  refine ⟨(fun x => if x ∈ S then Bool.xor (f.val x) (χ x) else false),?_,?_⟩
-  · intro x hx
-    exact ite_eq_right hx
-  · intro x hx hpx
-    simp only [ite_eq_left hx,ite_eq_left hpx,f.property.2 x hx hpx,hχ x]
-    cases f.val x <;> cases χ x <;> rfl
-
-end CycleColoring
-end Thorp
-
-end ThorpNine.Harmonic
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Harmonic.PriorPaths
+
+namespace OAI
+
+namespace ThorpNine.Harmonic
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+section ParallelButterfly
+variable {α B : Type*} [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] {d : ℕ}
+omit [DecidableEq B] in
+lemma parallel_entropy (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool) (S : Finset α) :
+    (∑ i : B × SwitchIndex d,
+      if (parallelUsers pre ω i).1 ∈ S ∧ (parallelUsers pre ω i).2 ∈ S then (1:ℝ) else 0) ≤
+      (S.card : ℝ) * Real.log S.card / (2 * Real.log 2) := by
+  classical
+  rw [Fintype.sum_prod_type]
+  calc
+    _ ≤ ∑ b : B, (blockMarks pre S b).card * Real.log (blockMarks pre S b).card / (2*Real.log 2) := by
+      apply Finset.sum_le_sum
+      intro b _
+      simpa only [←mem_blockMarks,parallelUsers] using
+        marked_switch_entropy_bound d (decodeButterfly d (fun i => ω (b,i))) (blockMarks pre S b)
+    _ ≤ ∑ b : B, (blockMarks pre S b).card * Real.log S.card / (2*Real.log 2) := by
+      apply Finset.sum_le_sum
+      intro b _
+      by_cases hz : (blockMarks pre S b).card = 0
+      · simp [hz]
+      · apply div_le_div_of_nonneg_right _ (by positivity)
+        apply mul_le_mul_of_nonneg_left _ (by positivity)
+        exact Real.log_le_log (by exact_mod_cast Nat.pos_of_ne_zero hz)
+          (by exact_mod_cast card_blockMarks_le pre S b)
+    _ = _ := by rw [←Finset.sum_div,←Finset.sum_mul,←Nat.cast_sum,sum_card_blockMarks]
+
+noncomputable def parallelFollow (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x : α) (k : α → QueryTree (B × SwitchIndex d)) : QueryTree (B × SwitchIndex d) :=
+  QueryTree.followPath d (fun i => ((pre x).1,i)) (pre x).2 (fun y => k (post ((pre x).1,y)))
+
+omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
+lemma parallelFollow_succeeds (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
+    (parallelFollow pre post x k).succeeds ω = (k (relabeledPerm pre post ω x)).succeeds ω := by
+  exact QueryTree.followPath_succeeds _ _ _ _ _
+
+omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
+lemma parallelFollow_raw_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
+    (parallelFollow pre post x k).charge ω = (k (relabeledPerm pre post ω x)).charge ω := by
+  exact QueryTree.followPath_raw_charge _ _ _ _ _
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma parallelFollow_stale_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
+    (ω : B × SwitchIndex d → Bool) :
+    (parallelFollow pre post x k).staleCharge R ω =
+      (k (relabeledPerm pre post ω x)).staleCharge (R ∪ parallelPath pre ω x) ω := by
+  exact QueryTree.followPath_stale_charge _ _ _ _ _ _
+
+noncomputable def closingDomain (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) : Finset (B × SwitchIndex d) :=
+  (routeDomain d (pre x).2 (post.symm y).2).image (fun i => ((pre x).1,i))
+
+def closingValue (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (i : B × SwitchIndex d) : Bool := routeValue d (pre x).2 (post.symm y).2 i.2
+
+omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
+lemma relabeled_endpoint_pair (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (ω : B × SwitchIndex d → Bool) (x y : α) :
+    relabeledPerm pre post ω x = y ↔
+      (pre x).1 = (post.symm y).1 ∧
+      butterflyPerm d (decodeButterfly d (fun i => ω ((pre x).1,i))) (pre x).2 = (post.symm y).2 := by
+  change post ((pre x).1,_) = y ↔ _
+  rw [←post.eq_symm_apply,Prod.mk.injEq]
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma relabeled_endpoint_iff (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (ω : B × SwitchIndex d → Bool) (x y : α) :
+    relabeledPerm pre post ω x = y ↔
+      (pre x).1 = (post.symm y).1 ∧
+        ∀ i ∈ closingDomain pre post x y, ω i = closingValue pre post x y i := by
+  classical
+  rw [relabeled_endpoint_pair,butterfly_endpoint_iff]
+  simp only [closingDomain,Finset.forall_mem_image,closingValue]
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma closingDomain_eq (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (ω : B × SwitchIndex d → Bool) (x y : α) (h : relabeledPerm pre post ω x = y) :
+    closingDomain pre post x y = parallelPath pre ω x := by
+  rw [closingDomain,parallelPath,(relabeled_endpoint_pair _ _ _ _ _).mp h |>.2]
+
+noncomputable def parallelClose (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (k : QueryTree (B × SwitchIndex d)) : QueryTree (B × SwitchIndex d) :=
+  if (pre x).1 = (post.symm y).1 then
+    QueryTree.forceList (closingDomain pre post x y).toList (closingValue pre post x y) k
+  else QueryTree.reject
+
+omit [Fintype α] in
+lemma parallelClose_succeeds (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (k : QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
+    (parallelClose pre post x y k).succeeds ω =
+      if relabeledPerm pre post ω x = y then k.succeeds ω else false := by
+  classical
+  simp only [relabeled_endpoint_iff,parallelClose]
+  by_cases hb : (pre x).1 = (post.symm y).1
+  · rw [ite_eq_left hb,QueryTree.forceList_succeeds]
+    simp only [Finset.mem_toList,hb,true_and]
+  · simp [hb,QueryTree.succeeds]
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma parallelClose_raw_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (k : QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool)
+    (h : relabeledPerm pre post ω x = y) :
+    (parallelClose pre post x y k).charge ω = k.charge ω + d := by
+  have hh := (relabeled_endpoint_iff _ _ _ _ _).mp h
+  rw [parallelClose,ite_eq_left hh.1,QueryTree.forceList_raw_charge _ _ _ _
+    (by simpa only [Finset.mem_toList] using hh.2),Finset.length_toList,
+    closingDomain_eq _ _ _ _ _ h,card_parallelPath]
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma parallelClose_fresh_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (k : QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
+    (ω : B × SwitchIndex d → Bool) (h : relabeledPerm pre post ω x = y) :
+    (parallelClose pre post x y k).freshCharge R ω =
+      k.freshCharge (R ∪ parallelPath pre ω x) ω + (parallelPath pre ω x \ R).card := by
+  have hh := (relabeled_endpoint_iff _ _ _ _ _).mp h
+  rw [parallelClose,ite_eq_left hh.1,QueryTree.forceList_charge _ _ _ _ _
+    (by simpa only [Finset.mem_toList] using hh.2),Finset.toList_toFinset,
+    closingDomain_eq _ _ _ _ _ h]
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma parallelClose_stale_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (k : QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
+    (ω : B × SwitchIndex d → Bool) (h : relabeledPerm pre post ω x = y) :
+    (parallelClose pre post x y k).staleCharge R ω =
+      k.staleCharge (R ∪ parallelPath pre ω x) ω + (parallelPath pre ω x ∩ R).card := by
+  have h₀ := QueryTree.fresh_add_stale (parallelClose pre post x y k) R ω
+  have h₁ := QueryTree.fresh_add_stale k (R ∪ parallelPath pre ω x) ω
+  rw [parallelClose_raw_charge _ _ _ _ _ _ h,parallelClose_fresh_charge _ _ _ _ _ _ _ h] at h₀
+  have h₂ : (parallelPath pre ω x \ R).card + (parallelPath pre ω x ∩ R).card = d := by
+    rw [Finset.card_sdiff_add_card_inter,card_parallelPath]
+  omega
+
+noncomputable def parallelNetwork (pre : α ≃ B × Card d) (post : B × Card d ≃ α) :
+    RoutingNetwork α (B × SwitchIndex d) d where
+  perm := relabeledPerm pre post
+  path := parallelPath pre
+  users := parallelUsers pre
+  users_ne := parallelUsers_ne pre
+  mem_path := mem_parallelPath pre
+  card_path := card_parallelPath pre
+  entropy := parallel_entropy pre
+  follow := parallelFollow pre post
+  close := parallelClose pre post
+  follow_succeeds := parallelFollow_succeeds pre post
+  follow_raw_charge := parallelFollow_raw_charge pre post
+  follow_stale_charge := parallelFollow_stale_charge pre post
+  close_succeeds := parallelClose_succeeds pre post
+  close_raw_charge := parallelClose_raw_charge pre post
+  close_stale_charge := parallelClose_stale_charge pre post
+
+lemma parallel_cycle_log_mgf (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (S : Finset α) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    Real.log (finiteMean (fun ω : B × SwitchIndex d → Bool =>
+      q^(Fintype.card (RoutingNetwork.SelectedCycle (relabeledPerm pre post ω) S)))) ≤
+      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
+  exact (parallelNetwork pre post).butterfly_cycle_log_mgf S J hJ q hq
+
+end ParallelButterfly
+
+def cardSplit : (r s : ℕ) → Card (s+r) ≃ Card r × Card s
+  | 0, s =>
+    { toFun := fun x => ((fun i => Fin.elim0 i), x)
+      invFun := fun x => x.2
+      left_inv := fun _ => rfl
+      right_inv := fun x => by
+        apply Prod.ext
+        · exact Subsingleton.elim _ _
+        · rfl }
+  | r+1, s =>
+    { toFun := fun x => (Fin.cons (x 0) ((cardSplit r s (Fin.tail x)).1),
+        (cardSplit r s (Fin.tail x)).2)
+      invFun := fun x => Fin.cons (x.1 0) ((cardSplit r s).symm (Fin.tail x.1,x.2))
+      left_inv := by
+        intro x
+        simp only [Fin.cons_zero, Fin.tail_cons, Prod.mk.eta, Equiv.symm_apply_apply, Fin.cons_self_tail]
+      right_inv := by
+        intro x
+        simp only [Fin.cons_zero, Fin.tail_cons, Equiv.apply_symm_apply, Fin.cons_self_tail] }
+
+def assembleBits : (r s : ℕ) → (Card r × SwitchIndex s → Bool) →
+    (Card s × SwitchIndex r → Bool) → SwitchIndex (s+r) → Bool
+  | 0, _, lo, _, i => lo ((fun j => Fin.elim0 j), i)
+  | r+1, s, _, hi, Sum.inl y => hi ((cardSplit r s y).2,Sum.inl (cardSplit r s y).1)
+  | r+1, s, lo, hi, Sum.inr (b,i) => assembleBits r s
+      (fun j => lo (Fin.cons b j.1,j.2)) (fun j => hi (j.1,Sum.inr (b,j.2))) i
+
+lemma butterfly_split (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
+    (hi : Card s × SwitchIndex r → Bool) (x : Card (s+r)) :
+    cardSplit r s (butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi)) x) =
+      let y := cardSplit r s x
+      let z := butterflyPerm s (decodeButterfly s (fun i => lo (y.1,i))) y.2
+      (butterflyPerm r (decodeButterfly r (fun i => hi (z,i))) y.1,z) := by
+  induction r with
+  | zero =>
+    apply Prod.ext
+    · funext i
+      exact Fin.elim0 i
+    · rfl
+  | succ r ih =>
+    let lo' : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons (x 0) j.1,j.2)
+    let hi' : Card s × SwitchIndex r → Bool := fun j => hi (j.1,Sum.inr (x 0,j.2))
+    have hh := ih lo' hi' (Fin.tail x)
+    let u := butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo' hi')) (Fin.tail x)
+    let y := cardSplit r s (Fin.tail x)
+    let z := butterflyPerm s (decodeButterfly s (fun i => lo' (y.1,i))) y.2
+    let w := butterflyPerm r (decodeButterfly r (fun i => hi' (z,i))) y.1
+    change cardSplit r s u = (w,z) at hh
+    change ((Fin.cons (Bool.xor (x 0) (hi ((cardSplit r s u).2,Sum.inl (cardSplit r s u).1)))
+        ((cardSplit r s u).1) : Card (r+1)), (cardSplit r s u).2) =
+      ((Fin.cons (Bool.xor (x 0) (hi (z,Sum.inl w))) w : Card (r+1)),z)
+    rw [hh]
+
+def splitSwitch : (r s : ℕ) → SwitchIndex (s+r) ≃
+    (Card r × SwitchIndex s) ⊕ (Card s × SwitchIndex r)
+  | 0, _ =>
+    { toFun := fun i => Sum.inl ((fun j => Fin.elim0 j),i)
+      invFun := fun i => match i with
+        | Sum.inl (_,j) => j
+        | Sum.inr (_,j) => nomatch j
+      left_inv := fun _ => rfl
+      right_inv := by
+        intro i
+        rcases i with ⟨x,j⟩ | ⟨x,j⟩
+        · exact congrArg Sum.inl (Prod.ext (Subsingleton.elim _ _) rfl)
+        · exact Empty.elim j }
+  | r+1, s =>
+    { toFun := fun i => match i with
+        | Sum.inl y => Sum.inr ((cardSplit r s y).2,Sum.inl (cardSplit r s y).1)
+        | Sum.inr (b,j) => match splitSwitch r s j with
+          | Sum.inl (p,j) => Sum.inl (Fin.cons b p,j)
+          | Sum.inr (q,j) => Sum.inr (q,Sum.inr (b,j))
+      invFun := fun i => match i with
+        | Sum.inl (p,j) => Sum.inr (p 0,(splitSwitch r s).symm (Sum.inl (Fin.tail p,j)))
+        | Sum.inr (q,Sum.inl p) => Sum.inl ((cardSplit r s).symm (p,q))
+        | Sum.inr (q,Sum.inr (b,j)) => Sum.inr (b,(splitSwitch r s).symm (Sum.inr (q,j)))
+      left_inv := by
+        intro i
+        rcases i with y | ⟨b,j⟩
+        · dsimp only
+          rw [Prod.mk.eta, Equiv.symm_apply_apply]
+        · dsimp only
+          cases hz : splitSwitch r s j with
+          | inl z =>
+            rcases z with ⟨p,k⟩
+            simp only [Fin.cons_zero, Fin.tail_cons]
+            rw [←hz,Equiv.symm_apply_apply]
+          | inr z =>
+            rcases z with ⟨q,k⟩
+            dsimp only
+            rw [←hz,Equiv.symm_apply_apply]
+      right_inv := by
+        intro i
+        rcases i with ⟨p,j⟩ | ⟨q,j⟩
+        · simp only [Equiv.apply_symm_apply, Fin.cons_self_tail]
+        · rcases j with p | ⟨b,j⟩ <;> simp only [Equiv.apply_symm_apply] }
+
+lemma assembleBits_eq (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
+    (hi : Card s × SwitchIndex r → Bool) (i : SwitchIndex (s+r)) :
+    assembleBits r s lo hi i = Sum.elim lo hi (splitSwitch r s i) := by
+  induction r with
+  | zero => rfl
+  | succ r ih =>
+    rcases i with y | ⟨b,j⟩
+    · rfl
+    · simp only [assembleBits,splitSwitch,Equiv.coe_fn_mk]
+      rw [ih]
+      cases splitSwitch r s j <;> rfl
+
+def splitCoinEquiv (r s : ℕ) :
+    ((Card r × SwitchIndex s → Bool) × (Card s × SwitchIndex r → Bool)) ≃
+      (SwitchIndex (s+r) → Bool) :=
+  (Equiv.sumArrowEquivProdArrow _ _ Bool).symm.trans
+    (Equiv.arrowCongr (splitSwitch r s).symm (Equiv.refl Bool))
+
+lemma splitCoinEquiv_apply (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
+    (hi : Card s × SwitchIndex r → Bool) :
+    splitCoinEquiv r s (lo,hi) = assembleBits r s lo hi := by
+  funext i
+  exact (assembleBits_eq r s lo hi i).symm
+
+lemma finiteMean_prod {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] (f : Ω × Ω' → ℝ) :
+    finiteMean f = finiteMean (fun x => finiteMean (fun y => f (x,y))) := by
+  simp only [finiteMean, Fintype.sum_prod_type, Fintype.card_prod, Nat.cast_mul,
+    ←Finset.sum_div, div_div, mul_comm]
+
+lemma finiteMean_split (r s : ℕ) (f : (SwitchIndex (s+r) → Bool) → ℝ) :
+    finiteMean f = finiteMean (fun lo : Card r × SwitchIndex s → Bool =>
+      finiteMean (fun hi : Card s × SwitchIndex r → Bool => f (assembleBits r s lo hi))) := by
+  rw [finiteMean_equiv (splitCoinEquiv r s).symm]
+  simp only [Equiv.symm_symm]
+  rw [finiteMean_prod]
+  apply finiteMean_congr
+  intro lo
+  apply finiteMean_congr
+  intro hi
+  rw [splitCoinEquiv_apply]
+
+section PartiallyExposed
+variable {α : Type*} {r s : ℕ}
+
+def freshPre (pre : α ≃ Card (s+r)) (lo : Card r × SwitchIndex s → Bool) :
+    α ≃ Card s × Card r :=
+  ((pre.trans (cardSplit r s)).trans (parallelPerm lo)).trans (Equiv.prodComm _ _)
+
+def freshPost (post : Card (s+r) ≃ α) : Card s × Card r ≃ α :=
+  ((Equiv.prodComm _ _).trans (cardSplit r s).symm).trans post
+
+lemma exposed_perm_eq (pre : α ≃ Card (s+r)) (post : Card (s+r) ≃ α)
+    (lo : Card r × SwitchIndex s → Bool) (hi : Card s × SwitchIndex r → Bool) :
+    (pre.trans (butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi)))).trans post =
+      relabeledPerm (freshPre pre lo) (freshPost post) hi := by
+  ext x
+  simpa only [relabeledPerm, freshPre, freshPost, parallelPerm, Equiv.trans_apply,
+    Equiv.coe_fn_mk, Equiv.prodComm_apply, Prod.swap, Equiv.symm_apply_apply] using
+    congrArg (fun y => post ((cardSplit r s).symm y)) (butterfly_split r s lo hi (pre x))
+
+lemma partially_exposed_cycle_log_mgf [Fintype α] [DecidableEq α]
+    (pre : α ≃ Card (s+r)) (post : Card (s+r) ≃ α)
+    (lo : Card r × SwitchIndex s → Bool) (S : Finset α)
+    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    Real.log (finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(Fintype.card (RoutingNetwork.SelectedCycle
+        ((pre.trans (butterflyPerm (s+r)
+          (decodeButterfly (s+r) (assembleBits r s lo hi)))).trans post) S)))) ≤
+      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^r) := by
+  simp only [exposed_perm_eq]
+  exact parallel_cycle_log_mgf (freshPre pre lo) (freshPost post) S J hJ q hq
+
+end PartiallyExposed
+
+lemma inclusionMoment_eq {α Ω : Type*} [DecidableEq α] [Fintype Ω]
+    (F : Ω → Finset α) (S : Finset α) :
+    inclusionMoment F S = finiteMean (fun ω => if S ⊆ F ω then (1:ℝ) else 0) := by
+  unfold inclusionMoment
+  apply finiteMean_congr
+  intro ω
+  split_ifs <;> rfl
+
+lemma finiteMean_bool (f : Bool → ℝ) : finiteMean f = (f false+f true)/2 := by
+  simp only [finiteMean, Fintype.sum_bool, Fintype.card_bool, Nat.cast_ofNat]
+  ring
+
+lemma finiteMean_const {Ω : Type*} [Fintype Ω] [Nonempty Ω] (c : ℝ) :
+    finiteMean (fun _ : Ω => c) = c := by
+  simp only [finiteMean,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
+  exact mul_div_cancel_left₀ c (Nat.cast_ne_zero.mpr Fintype.card_ne_zero)
+
+lemma finiteMean_pi_prod {ι : Type*} [Fintype ι] [DecidableEq ι] {Ω : ι → Type*}
+    [∀ i, Fintype (Ω i)] (f : (i : ι) → Ω i → ℝ) :
+    finiteMean (fun ω : (i : ι) → Ω i => ∏ i, f i (ω i)) = ∏ i, finiteMean (f i) := by
+  simp only [finiteMean,←Fintype.prod_sum,Fintype.card_pi,Nat.cast_prod,Finset.prod_div_distrib]
+
+section ParallelFairLayer
+variable {β : Type*} [Fintype β] [DecidableEq β]
+
+def pairLayer (ξ : β → Bool) : Equiv.Perm (Bool × β) where
+  toFun x := (Bool.xor x.1 (ξ x.2),x.2)
+  invFun x := (Bool.xor x.1 (ξ x.2),x.2)
+  left_inv x := by
+    rcases x with ⟨b,x⟩
+    change (Bool.xor (Bool.xor b (ξ x)) (ξ x),x) = (b,x)
+    cases b <;> cases ξ x <;> rfl
+  right_inv x := by
+    rcases x with ⟨b,x⟩
+    change (Bool.xor (Bool.xor b (ξ x)) (ξ x),x) = (b,x)
+    cases b <;> cases ξ x <;> rfl
+
+noncomputable def pairFiber (S : Finset (Bool × β)) (b : β) : Finset Bool :=
+  Finset.univ.filter (fun ε => (ε,b) ∈ S)
+
+lemma prod_pairFiber (S : Finset (Bool × β)) (f : Bool × β → ℝ) :
+    (∏ x ∈ S, f x) = ∏ b : β, ∏ ε ∈ pairFiber S b, f (ε,b) := by
+  classical
+  simp only [pairFiber,Finset.prod_filter]
+  rw [Finset.prod_comm,←Fintype.prod_prod_type (fun x : Bool × β => if x ∈ S then f x else 1)]
+  rw [←Finset.prod_filter]
+  congr 1
+  ext x
+  simp
+
+lemma fiber_average_bound (S : Finset Bool) (p : Bool → ℝ) (hp : ∀ b, 0 ≤ p b) :
+    finiteMean (fun c : Bool => ∏ ε ∈ S, p (Bool.xor ε c)) ≤
+      ∏ ε ∈ S, (p ε + p (!ε))/2 := by
+  rw [finiteMean_bool]
+  have hxor : ∀ ε : Bool, Bool.xor ε true = Equiv.swap false true ε := by decide
+  simp only [Bool.xor_false,hxor]
+  calc
+    _ ≤ ∏ ε ∈ S, (p ε+p (Equiv.swap false true ε))/2 :=
+      swap_average_product S p hp false true
+    _ = _ := Finset.prod_congr rfl (fun ε _ => by cases ε <;> rfl)
+
+lemma CylinderBound.fair_parallel {m : Finset (Bool × β) → ℝ} {p : Bool × β → ℝ}
+    (h : CylinderBound m p) (hp : ∀ x, 0 ≤ p x) :
+    CylinderBound (fun S => finiteMean (fun ξ : β → Bool => m (S.image (pairLayer ξ))))
+      (fun x => (p x+p (!x.1,x.2))/2) := by
+  intro S
+  calc
+    _ ≤ finiteMean (fun ξ : β → Bool => ∏ x ∈ S, p (pairLayer ξ x)) := by
+      apply finiteMean_mono
+      intro ξ
+      simpa only [Finset.prod_image (pairLayer ξ).injective.injOn] using h (S.image (pairLayer ξ))
+    _ = ∏ b : β, finiteMean (fun c : Bool => ∏ ε ∈ pairFiber S b, p (Bool.xor ε c,b)) := by
+      calc
+        _ = finiteMean (fun ξ : β → Bool => ∏ b : β,
+            ∏ ε ∈ pairFiber S b, p (Bool.xor ε (ξ b),b)) :=
+          finiteMean_congr (fun ξ => prod_pairFiber S (fun x => p (pairLayer ξ x)))
+        _ = _ := finiteMean_pi_prod (fun b c => ∏ ε ∈ pairFiber S b, p (Bool.xor ε c,b))
+    _ ≤ ∏ b : β, ∏ ε ∈ pairFiber S b, (p (ε,b)+p (!ε,b))/2 := by
+      apply Finset.prod_le_prod₀
+      · intro b _
+        exact div_nonneg (Finset.sum_nonneg (fun c _ =>
+          Finset.prod_nonneg (fun ε _ => hp (Bool.xor ε c,b)))) (Nat.cast_nonneg _)
+      · intro b _
+        exact fiber_average_bound (pairFiber S b) (fun ε => p (ε,b)) (fun ε => hp (ε,b))
+    _ = _ := (prod_pairFiber S (fun x => (p x+p (!x.1,x.2))/2)).symm
+
+omit [Fintype β] in
+lemma subset_pairLayer_image_iff (S A : Finset (Bool × β)) (ξ : β → Bool) :
+    S ⊆ A.image (pairLayer ξ) ↔ S.image (pairLayer ξ) ⊆ A := by
+  have hi (x : Bool × β) : pairLayer ξ (pairLayer ξ x) = x := (pairLayer ξ).left_inv x
+  constructor
+  · intro h
+    have h' := Finset.image_subset_image h (f := pairLayer ξ)
+    simpa only [Finset.image_image,Function.comp_def,hi,Finset.image_id'] using h'
+  · intro h
+    have h' := Finset.image_subset_image h (f := pairLayer ξ)
+    simpa only [Finset.image_image,Function.comp_def,hi,Finset.image_id'] using h'
+
+noncomputable def fairSetLayer {Ω : Type*} (F : Ω → Finset (Bool × β)) :
+    Ω × (β → Bool) → Finset (Bool × β) := fun ω => (F ω.1).image (pairLayer ω.2)
+
+lemma finiteMean_comm {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] (f : Ω → Ω' → ℝ) :
+    finiteMean (fun x => finiteMean (f x)) = finiteMean (fun y => finiteMean (fun x => f x y)) := by
+  simp only [finiteMean,←Finset.sum_div,div_div]
+  rw [Finset.sum_comm,mul_comm]
+
+lemma inclusionMoment_fairSetLayer {Ω : Type*} [Fintype Ω]
+    (F : Ω → Finset (Bool × β)) (S : Finset (Bool × β)) :
+    inclusionMoment (fairSetLayer F) S =
+      finiteMean (fun ξ : β → Bool => inclusionMoment F (S.image (pairLayer ξ))) := by
+  classical
+  simp only [inclusionMoment_eq]
+  have he : (fun ω : Ω × (β → Bool) => if S ⊆ fairSetLayer F ω then (1:ℝ) else 0) =
+      (fun ω : Ω × (β → Bool) => if S.image (pairLayer ω.2) ⊆ F ω.1 then (1:ℝ) else 0) := by
+    funext ω
+    simp only [fairSetLayer,subset_pairLayer_image_iff]
+  apply (congrArg finiteMean he).trans
+  exact (finiteMean_prod (fun ω : Ω × (β → Bool) =>
+    if S.image (pairLayer ω.2) ⊆ F ω.1 then (1:ℝ) else 0)).trans
+      (finiteMean_comm (fun ω ξ => if S.image (pairLayer ξ) ⊆ F ω then (1:ℝ) else 0))
+
+lemma inclusionMoment_fairSetLayer_singleton {Ω : Type*} [Fintype Ω]
+    (F : Ω → Finset (Bool × β)) (x : Bool × β) :
+    inclusionMoment (fairSetLayer F) {x} =
+      (inclusionMoment F {x}+inclusionMoment F {(!x.1,x.2)})/2 := by
+  rw [inclusionMoment_fairSetLayer]
+  simp only [Finset.image_singleton,pairLayer,Equiv.coe_fn_mk]
+  rw [finiteMean_pi_eval (A := fun _ : β => Bool) x.2 (fun c : Bool => inclusionMoment F {(Bool.xor x.1 c,x.2)})]
+  rw [finiteMean_bool]
+  rcases x with ⟨b,x⟩
+  cases b <;> simp only [Bool.xor_false,Bool.false_xor,Bool.true_xor, Bool.not_false,Bool.not_true]
+
+lemma cylinder_fairSetLayer {Ω : Type*} [Fintype Ω]
+    (F : Ω → Finset (Bool × β))
+    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
+    CylinderBound (inclusionMoment (fairSetLayer F))
+      (fun x => inclusionMoment (fairSetLayer F) {x}) := by
+  have hh := h.fair_parallel (fun x => inclusionMoment_nonneg F {x})
+  intro S
+  rw [inclusionMoment_fairSetLayer]
+  simp only [inclusionMoment_fairSetLayer_singleton]
+  exact hh S
+
+end ParallelFairLayer
+
+section RelabelMarks
+variable {α β Ω Ω' : Type*} [DecidableEq α] [DecidableEq β] [Fintype Ω] [Fintype Ω']
+
+omit [Fintype Ω] [Fintype Ω'] in
+lemma subset_image_equiv_iff (e : α ≃ β) (A : Finset α) (S : Finset β) :
+    S ⊆ A.image e ↔ S.image e.symm ⊆ A := by
+  constructor
+  · intro h x hx
+    obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
+    obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp (h hy)
+    simpa only [←hxy,Equiv.symm_apply_apply] using hx
+  · intro h y hy
+    exact Finset.mem_image.mpr ⟨e.symm y,h (Finset.mem_image_of_mem _ hy),e.apply_symm_apply y⟩
+
+omit [Fintype Ω'] in
+lemma inclusionMoment_image_equiv (e : α ≃ β) (F : Ω → Finset α) (S : Finset β) :
+    inclusionMoment (fun ω => (F ω).image e) S = inclusionMoment F (S.image e.symm) := by
+  classical
+  simp only [inclusionMoment_eq]
+  apply finiteMean_congr
+  intro ω
+  simp only [subset_image_equiv_iff]
+
+omit [Fintype Ω'] in
+lemma cylinder_image_equiv (e : α ≃ β) (F : Ω → Finset α)
+    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
+    CylinderBound (inclusionMoment (fun ω => (F ω).image e))
+      (fun x => inclusionMoment (fun ω => (F ω).image e) {x}) := by
+  intro S
+  simp only [inclusionMoment_image_equiv,Finset.image_singleton]
+  have hh := h (S.image e.symm)
+  simpa only [Finset.prod_image e.symm.injective.injOn] using hh
+
+omit [DecidableEq β] in
+lemma inclusionMoment_comp_equiv (e : Ω ≃ Ω') (F : Ω' → Finset α) (S : Finset α) :
+    inclusionMoment (fun ω => F (e ω)) S = inclusionMoment F S := by
+  classical
+  simp only [inclusionMoment_eq]
+  simpa only [Equiv.apply_symm_apply] using
+    finiteMean_equiv e (fun ω => if S ⊆ F (e ω) then (1 : ℝ) else 0)
+
+omit [DecidableEq β] in
+lemma cylinder_comp_equiv (e : Ω ≃ Ω') (F : Ω' → Finset α)
+    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
+    CylinderBound (inclusionMoment (fun ω => F (e ω)))
+      (fun x => inclusionMoment (fun ω => F (e ω)) {x}) := by
+  intro S
+  simpa only [inclusionMoment_comp_equiv] using h S
+end RelabelMarks
+
+lemma finiteMean_prod_mul {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
+    (f : Ω → ℝ) (g : Ω' → ℝ) :
+    finiteMean (fun ω : Ω × Ω' => f ω.1*g ω.2) = finiteMean f * finiteMean g := by
+  simp only [finiteMean,Fintype.sum_prod_type,Fintype.card_prod,Nat.cast_mul,
+    ←Finset.mul_sum,←Finset.sum_mul]
+  ring
+
+section JoinChildren
+variable {β : Type*} [DecidableEq β]
+
+noncomputable def childJoin (A B : Finset β) : Finset (Bool × β) :=
+  A.image (fun x => (false,x)) ∪ B.image (fun x => (true,x))
+
+lemma mem_childJoin (A B : Finset β) (b : Bool) (x : β) :
+    (b,x) ∈ childJoin A B ↔ (if b then x ∈ B else x ∈ A) := by
+  cases b <;> simp [childJoin]
+
+noncomputable def boolFiber (S : Finset (Bool × β)) (b : Bool) : Finset β :=
+  (S.filter (fun x => x.1 = b)).image Prod.snd
+
+lemma mem_boolFiber (S : Finset (Bool × β)) (b : Bool) (x : β) :
+    x ∈ boolFiber S b ↔ (b,x) ∈ S := by
+  constructor
+  · intro h
+    obtain ⟨⟨c,y⟩,hy,hxy⟩ := Finset.mem_image.mp h
+    obtain ⟨hy,hc⟩ := Finset.mem_filter.mp hy
+    change c = b at hc
+    change y = x at hxy
+    simpa only [hc,hxy] using hy
+  · intro h
+    exact Finset.mem_image.mpr ⟨(b,x),Finset.mem_filter.mpr ⟨h,rfl⟩,rfl⟩
+
+lemma childJoin_boolFiber (S : Finset (Bool × β)) :
+    childJoin (boolFiber S false) (boolFiber S true) = S := by
+  ext ⟨b,x⟩
+  rw [mem_childJoin]
+  cases b <;> simp only [Bool.false_eq_true,↓reduceIte,mem_boolFiber]
+
+lemma subset_childJoin_iff (S : Finset (Bool × β)) (A B : Finset β) :
+    S ⊆ childJoin A B ↔ boolFiber S false ⊆ A ∧ boolFiber S true ⊆ B := by
+  constructor
+  · intro h
+    constructor
+    · intro x hx
+      exact (mem_childJoin A B false x).mp (h ((mem_boolFiber S false x).mp hx))
+    · intro x hx
+      exact (mem_childJoin A B true x).mp (h ((mem_boolFiber S true x).mp hx))
+  · rintro ⟨h₀,h₁⟩ ⟨b,x⟩ hx
+    rw [mem_childJoin]
+    cases b
+    · exact h₀ ((mem_boolFiber S false x).mpr hx)
+    · exact h₁ ((mem_boolFiber S true x).mpr hx)
+
+lemma prod_childJoin (A B : Finset β) (f : Bool × β → ℝ) :
+    (∏ x ∈ childJoin A B, f x) = (∏ x ∈ A, f (false,x)) * ∏ x ∈ B, f (true,x) := by
+  classical
+  have hd : Disjoint (A.image (fun x => (false,x))) (B.image (fun x => (true,x))) := by
+    apply Finset.disjoint_left.mpr
+    rintro x hx hy
+    obtain ⟨a,ha,rfl⟩ := Finset.mem_image.mp hx
+    obtain ⟨b,hb,hba⟩ := Finset.mem_image.mp hy
+    exact Bool.false_ne_true (congrArg Prod.fst hba).symm
+  rw [childJoin,Finset.prod_union hd,
+    Finset.prod_image (fun _ _ _ _ h => congrArg Prod.snd h),
+    Finset.prod_image (fun _ _ _ _ h => congrArg Prod.snd h)]
+
+lemma prod_boolFibers (S : Finset (Bool × β)) (f : Bool × β → ℝ) :
+    (∏ x ∈ S, f x) = (∏ x ∈ boolFiber S false, f (false,x)) *
+      ∏ x ∈ boolFiber S true, f (true,x) := by
+  rw [←prod_childJoin,childJoin_boolFiber]
+
+lemma inclusionMoment_childJoin {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
+    (F : Ω → Finset β) (G : Ω' → Finset β) (S : Finset (Bool × β)) :
+    inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) S =
+      inclusionMoment F (boolFiber S false) * inclusionMoment G (boolFiber S true) := by
+  classical
+  simp only [inclusionMoment_eq]
+  rw [←finiteMean_prod_mul]
+  apply finiteMean_congr
+  intro ω
+  simp only [subset_childJoin_iff]
+  split_ifs <;> simp_all
+
+lemma cylinder_childJoin {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] [Nonempty Ω] [Nonempty Ω']
+    (F : Ω → Finset β) (G : Ω' → Finset β)
+    (hF : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x}))
+    (hG : CylinderBound (inclusionMoment G) (fun x => inclusionMoment G {x})) :
+    CylinderBound (inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)))
+      (fun x => inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {x}) := by
+  intro S
+  rw [inclusionMoment_childJoin,prod_boolFibers]
+  have hf (b : Bool) (x : β) : boolFiber {(b,x)} b = {x} := by
+    ext y
+    simp only [mem_boolFiber,Finset.mem_singleton,Prod.mk.injEq,true_and]
+  have hn (b : Bool) (x : β) : boolFiber {(!b,x)} b = ∅ := by
+    ext y
+    cases b <;> simp [mem_boolFiber]
+  have h₀ (x : β) : inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {(false,x)} =
+      inclusionMoment F {x} * inclusionMoment G ∅ := by
+    rw [inclusionMoment_childJoin,hf]
+    have hh := hn true x
+    exact congrArg (fun T => inclusionMoment F {x} * inclusionMoment G T) hh
+  have h₁ (x : β) : inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {(true,x)} =
+      inclusionMoment F ∅ * inclusionMoment G {x} := by
+    rw [inclusionMoment_childJoin,hf]
+    have hh := hn false x
+    exact congrArg (fun T => inclusionMoment F T * inclusionMoment G {x}) hh
+  have emF : inclusionMoment F ∅ = 1 := by simp [inclusionMoment,finiteMean_const]
+  have emG : inclusionMoment G ∅ = 1 := by simp [inclusionMoment,finiteMean_const]
+  simp only [h₀,h₁,emF,emG,mul_one,one_mul]
+  exact mul_le_mul (hF _) (hG _) (inclusionMoment_nonneg _ _)
+    (Finset.prod_nonneg (fun x _ => inclusionMoment_nonneg F {x}))
+end JoinChildren
+
+def coinStepEquiv (d : ℕ) : (SwitchIndex (d+1) → Bool) ≃
+    ((SwitchIndex d → Bool) × (SwitchIndex d → Bool)) × (Card d → Bool) where
+  toFun ω := ((fun i => ω (Sum.inr (false,i)),fun i => ω (Sum.inr (true,i))),
+    fun x => ω (Sum.inl x))
+  invFun ω := Sum.elim ω.2 (fun bi => if bi.1 then ω.1.2 bi.2 else ω.1.1 bi.2)
+  left_inv ω := by
+    funext i
+    cases i with
+    | inl x => rfl
+    | inr bi => cases bi with | mk b i => cases b <;> rfl
+  right_inv ω := by
+    rcases ω with ⟨⟨ω₀,ω₁⟩,ξ⟩
+    rfl
+
+def headTailEquiv (d : ℕ) : Card (d+1) ≃ Bool × Card d :=
+  (Fin.consEquiv (fun _ : Fin (d+1) => Bool)).symm
+
+lemma childMarks_join (d : ℕ) (A : Finset (Card (d+1))) :
+    A = (childMarks A false).image (Fin.cons false) ∪
+      (childMarks A true).image (Fin.cons true) := by
+  classical
+  ext x
+  constructor
+  · intro hx
+    cases hb : x 0
+    · exact Finset.mem_union_left _ (Finset.mem_image.mpr
+        ⟨Fin.tail x,(mem_childMarks _ _ _).mpr (by simpa only [←hb,Fin.cons_self_tail] using hx),
+          by simp only [←hb,Fin.cons_self_tail]⟩)
+    · exact Finset.mem_union_right _ (Finset.mem_image.mpr
+        ⟨Fin.tail x,(mem_childMarks _ _ _).mpr (by simpa only [←hb,Fin.cons_self_tail] using hx),
+          by simp only [←hb,Fin.cons_self_tail]⟩)
+  · intro hx
+    rcases Finset.mem_union.mp hx with hx | hx
+    · obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
+      exact (mem_childMarks _ _ _).mp hy
+    · obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
+      exact (mem_childMarks _ _ _).mp hy
+
+noncomputable def butterflyMarks (d : ℕ) (A : Finset (Card d))
+    (ω : SwitchIndex d → Bool) : Finset (Card d) :=
+  A.image (butterflyPerm d (decodeButterfly d ω))
+
+lemma butterflyMarks_succ (d : ℕ) (A : Finset (Card (d+1))) (ω : SwitchIndex (d+1) → Bool) :
+    butterflyMarks (d+1) A ω =
+      (fairSetLayer (fun p : (SwitchIndex d → Bool) × (SwitchIndex d → Bool) =>
+        childJoin (butterflyMarks d (childMarks A false) p.1)
+          (butterflyMarks d (childMarks A true) p.2)) (coinStepEquiv d ω)).image
+            (headTailEquiv d).symm := by
+  classical
+  unfold butterflyMarks fairSetLayer childJoin
+  rw [Finset.image_union,Finset.image_union]
+  simp only [Finset.image_image]
+  conv_lhs => rw [childMarks_join d A]
+  rw [Finset.image_union]
+  simp only [Finset.image_image]
+  apply congrArg₂ (· ∪ ·)
+  · apply Finset.image_congr
+    intro x _
+    simp only [Function.comp_def,butterflyPerm,decodeButterfly,Equiv.Perm.coe_mul,
+      childLift,pairSwitch,switchFun,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
+      coinStepEquiv,headTailEquiv,Equiv.symm_symm,pairLayer]
+    rfl
+  · apply Finset.image_congr
+    intro x _
+    simp only [Function.comp_def,butterflyPerm,decodeButterfly,Equiv.Perm.coe_mul,
+      childLift,pairSwitch,switchFun,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
+      coinStepEquiv,headTailEquiv,Equiv.symm_symm,pairLayer]
+    rfl
+
+lemma butterflyMarks_cylinder (d : ℕ) (A : Finset (Card d)) :
+    CylinderBound (inclusionMoment (butterflyMarks d A))
+      (fun x => inclusionMoment (butterflyMarks d A) {x}) := by
+  induction d with
+  | zero =>
+    have he : butterflyMarks 0 A = fun _ : SwitchIndex 0 → Bool => A := by
+      funext ω
+      simp [butterflyMarks,butterflyPerm]
+    rw [he]
+    exact cylinder_const A
+  | succ d ih =>
+    have h := cylinder_childJoin (butterflyMarks d (childMarks A false))
+      (butterflyMarks d (childMarks A true)) (ih _) (ih _)
+    have hh := cylinder_image_equiv (headTailEquiv d).symm _ (cylinder_fairSetLayer _ h)
+    have hs := cylinder_comp_equiv (coinStepEquiv d) _ hh
+    simpa only [←butterflyMarks_succ] using hs
+
+lemma indicator_mem_image_perm {α : Type*} [DecidableEq α]
+    (A : Finset α) (p : Equiv.Perm α) (y : α) :
+    (if y ∈ A.image p then (1 : ℝ) else 0) = ∑ x ∈ A, if p x = y then 1 else 0 := by
+  classical
+  by_cases h : y ∈ A.image p
+  · obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp h
+    rw [ite_eq_left (Finset.mem_image.mpr ⟨x,hx,hxy⟩)]
+    symm
+    calc
+      _ = (if p x = y then (1:ℝ) else 0) := Finset.sum_eq_single_of_mem x hx (by
+        intro z _ hzx
+        exact ite_eq_right (fun hz => hzx (p.injective (hz.trans hxy.symm))))
+      _ = 1 := ite_eq_left hxy
+  · rw [ite_eq_right h]
+    symm
+    apply Finset.sum_eq_zero
+    intro x hx
+    exact ite_eq_right (fun hxy => h (Finset.mem_image.mpr ⟨x,hx,hxy⟩))
+
+lemma butterflyMarks_singleton (d : ℕ) (A : Finset (Card d)) (y : Card d) :
+    inclusionMoment (butterflyMarks d A) {y} = (A.card : ℝ)/(2:ℝ)^d := by
+  classical
+  unfold inclusionMoment
+  simp only [Finset.singleton_subset_iff,butterflyMarks,indicator_mem_image_perm]
+  rw [finiteMean_sum]
+  simp only [butterfly_single_card_uniform,Finset.sum_const,nsmul_eq_mul]
+  ring
+
+lemma butterflyMarks_inclusion (d : ℕ) (A S : Finset (Card d)) :
+    inclusionMoment (butterflyMarks d A) S ≤ ((A.card : ℝ)/(2:ℝ)^d)^S.card := by
+  have h := butterflyMarks_cylinder d A S
+  simpa only [butterflyMarks_singleton,Finset.prod_const] using h
+
+namespace CycleColoring
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+abbrev Coloring (p : Equiv.Perm α) (S : Finset α) :=
+  {f : α → Bool // (∀ x, x ∉ S → f x = false) ∧
+    ∀ x ∈ S, p x ∈ S → f (p x) = f x}
+
+noncomputable instance (p : Equiv.Perm α) (S : Finset α) : Fintype (Coloring p S) :=
+  Fintype.ofFinite _
+
+noncomputable def boundary (p : Equiv.Perm α) (S : Finset α) : Finset α :=
+  S.filter (fun x => p.symm x ∉ S)
+
+noncomputable def root {p : Equiv.Perm α} {S : Finset α}
+    (C : RoutingNetwork.SelectedCycle p S) : α := C.nonempty.choose
+
+lemma root_mem {p : Equiv.Perm α} {S : Finset α} (C : RoutingNetwork.SelectedCycle p S) :
+    root C ∈ C.val := C.nonempty.choose_spec
+
+abbrev Seed (p : Equiv.Perm α) (S : Finset α) :=
+  {x // x ∈ boundary p S} ⊕ RoutingNetwork.SelectedCycle p S
+
+noncomputable def record {p : Equiv.Perm α} {S : Finset α}
+    (f : Coloring p S) : Seed p S → Bool :=
+  Sum.elim (fun x => f.val x.val) (fun C => f.val (root C))
+
+lemma orbit_subset_of_predecessor_closed (p : Equiv.Perm α) (D : Finset α)
+    (hD : ∀ x ∈ D, p.symm x ∈ D) {x : α} (hx : x ∈ D) : orbitSet p x ⊆ D := by
+  intro y hy
+  have hh := ((mem_orbitSet p x y).mp hy).inv
+  obtain ⟨n,hn⟩ := hh.exists_nat_pow_eq
+  have hp (k : ℕ) : (p⁻¹ ^ k) x ∈ D := by
+    induction k with
+    | zero => simpa using hx
+    | succ k ih => simpa only [pow_succ',Equiv.Perm.mul_apply,Equiv.Perm.coe_inv] using hD _ ih
+  exact hn ▸ hp n
+
+lemma record_injective (p : Equiv.Perm α) (S : Finset α) :
+    Function.Injective (record (p := p) (S := S)) := by
+  classical
+  intro f g hfg
+  apply Subtype.ext
+  funext x
+  by_contra hne
+  have hx : x ∈ S := by
+    by_contra h
+    exact hne ((f.property.1 x h).trans (g.property.1 x h).symm)
+  let D := S.filter (fun y => f.val y ≠ g.val y)
+  have hxD : x ∈ D := Finset.mem_filter.mpr ⟨hx,hne⟩
+  have hprev (y : α) (hy : y ∈ D) : p.symm y ∈ D := by
+    obtain ⟨hyS,hyne⟩ := Finset.mem_filter.mp hy
+    have hpS : p.symm y ∈ S := by
+      by_contra hn
+      have hseed : y ∈ boundary p S := Finset.mem_filter.mpr ⟨hyS,hn⟩
+      exact hyne (congrFun hfg (Sum.inl ⟨y,hseed⟩))
+    refine Finset.mem_filter.mpr ⟨hpS,?_⟩
+    have hf := f.property.2 (p.symm y) hpS (by simpa only [Equiv.apply_symm_apply] using hyS)
+    have hg := g.property.2 (p.symm y) hpS (by simpa only [Equiv.apply_symm_apply] using hyS)
+    simp only [Equiv.apply_symm_apply] at hf hg
+    intro he
+    exact hyne (hf.trans (he.trans hg.symm))
+  have horb := orbit_subset_of_predecessor_closed p D hprev hxD
+  let C : RoutingNetwork.SelectedCycle p S :=
+    ⟨orbitSet p x,⟨x,rfl⟩,fun y hy => (Finset.mem_filter.mp (horb hy)).1⟩
+  have hrD : root C ∈ D := horb (root_mem C)
+  have hrne := (Finset.mem_filter.mp hrD).2
+  exact hrne (congrFun hfg (Sum.inr C))
+
+lemma card_coloring_le (p : Equiv.Perm α) (S : Finset α) :
+    Fintype.card (Coloring p S) ≤
+      2 ^ ((boundary p S).card + Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
+  have h := Fintype.card_le_of_injective (record (p := p) (S := S)) (record_injective p S)
+  simpa only [Fintype.card_fun,Fintype.card_sum,Fintype.card_coe,Fintype.card_bool] using h
+
+omit [Fintype α] in
+lemma boundary_card (p : Equiv.Perm α) (S : Finset α) :
+    (boundary p S).card = S.card - (S.filter (fun x => p.symm x ∈ S)).card := by
+  classical
+  have h := Finset.card_filter_add_card_filter_not (s := S) (fun x => p.symm x ∈ S)
+  unfold boundary
+  omega
+
+lemma card_coloring_le_edges (p : Equiv.Perm α) (S : Finset α) :
+    Fintype.card (Coloring p S) ≤
+      2 ^ (S.card - (S.filter (fun x => p.symm x ∈ S)).card +
+        Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
+  simpa only [boundary_card] using card_coloring_le p S
+
+
+abbrev OppositeColoring (p : Equiv.Perm α) (S : Finset α) :=
+  {f : α → Bool // (∀ x, x ∉ S → f x = false) ∧
+    ∀ x ∈ S, p x ∈ S → f (p x) = !(f x)}
+
+noncomputable instance (p : Equiv.Perm α) (S : Finset α) : Fintype (OppositeColoring p S) :=
+  Fintype.ofFinite _
+
+noncomputable def bipartitionFlip (p : Equiv.Perm α) (S : Finset α)
+    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x))
+    (f : OppositeColoring p S) : Coloring p S := by
+  classical
+  refine ⟨(fun x => if x ∈ S then Bool.xor (f.val x) (χ x) else false),?_,?_⟩
+  · intro x hx
+    exact ite_eq_right hx
+  · intro x hx hpx
+    simp only [ite_eq_left hx,ite_eq_left hpx,f.property.2 x hx hpx,hχ x]
+    cases f.val x <;> cases χ x <;> rfl
+
+end CycleColoring
+end Thorp
+
+end ThorpNine.Harmonic
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Harmonic/PriorPaths.lean b/lean/OAI/Probability/ThorpRouting/Harmonic/PriorPaths.lean
index c5ec921fff698cb8568f7a9f765d68b4d78f186b..e683839901f5f6dcc138faa3aee7cc1015061675 100644
--- a/lean/OAI/Probability/ThorpRouting/Harmonic/PriorPaths.lean
+++ b/lean/OAI/Probability/ThorpRouting/Harmonic/PriorPaths.lean
@@ -1,882 +1,882 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Harmonic.AssignmentSlots
-
-namespace OAI
-
-namespace ThorpNine.Harmonic
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-namespace RoutingNetwork
-variable {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι] [DecidableEq ι]
-  {d : ℕ} (R : RoutingNetwork α ι d)
-lemma mem_exposedPaths (b : ι → Bool) (V : Finset (α)) (s : ι) :
-    s ∈ exposedPaths R b V ↔ (R.users b s).1 ∈ V ∨ (R.users b s).2 ∈ V := by
-  classical
-  simp only [exposedPaths,Finset.mem_biUnion,R.mem_path]
-  constructor
-  · rintro ⟨x,hx,h | h⟩
-    · exact Or.inl (h ▸ hx)
-    · exact Or.inr (h ▸ hx)
-  · rintro (h | h)
-    · exact ⟨_,h,Or.inl rfl⟩
-    · exact ⟨_,h,Or.inr rfl⟩
-
-lemma terminalCharge_sum (b : ι → Bool) (z : α) (V : Finset (α))
-    (hz : z ∉ V) :
-    (terminalCharge R b z V : ℝ) = ∑ s : ι,
-      ((if (R.users b s).1 = z ∧ (R.users b s).2 ∈ V then (1:ℝ) else 0) +
-      (if (R.users b s).2 = z ∧ (R.users b s).1 ∈ V then (1:ℝ) else 0)) := by
-  classical
-  have he : R.path b z ∩ exposedPaths R b V =
-      Finset.univ.filter (fun s =>
-        ((R.users b s).1 = z ∧ (R.users b s).2 ∈ V) ∨
-        ((R.users b s).2 = z ∧ (R.users b s).1 ∈ V)) := by
-    ext s
-    simp only [Finset.mem_inter,R.mem_path,mem_exposedPaths,
-      Finset.mem_filter,Finset.mem_univ,true_and]
-    by_cases h₁ : (R.users b s).1 = z <;>
-      by_cases h₂ : (R.users b s).2 = z <;> simp [h₁,h₂,hz]
-  rw [terminalCharge,he]
-  simp only [← Finset.sum_boole]
-  apply Finset.sum_congr rfl
-  intro s _
-  by_cases h₁ : (R.users b s).1 = z <;>
-    by_cases h₂ : (R.users b s).2 = z <;> simp [h₁,h₂,hz]
-
-noncomputable def priorPaths {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) : Finset (α) :=
-  Finset.univ.filter (fun x => ∃ i : Fin A, o x = some i ∧ i.val < t)
-
-omit [DecidableEq α] in
-@[simp] lemma mem_priorPaths {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) (x : α) :
-    x ∈ priorPaths o t ↔ ∃ i : Fin A, o x = some i ∧ i.val < t := by
-  classical
-  simp [priorPaths]
-
-lemma terminalCharge_prior_sum {A : ℕ} (o : α → Option (Fin A))
-    (b : ι → Bool) (z : Terminal o) (i : Fin A) :
-    (terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) =
-      ∑ s : ι,
-        ((if (z i).val = (R.users b s).1 ∧ (R.users b s).2 ∈ priorPaths o (i.val+1)
-          then (1:ℝ) else 0) +
-        (if (z i).val = (R.users b s).2 ∧ (R.users b s).1 ∈ priorPaths o (i.val+1)
-          then (1:ℝ) else 0)) := by
-  classical
-  rw [terminalCharge_sum _ _ _ _ (Finset.notMem_erase _ _)]
-  apply Finset.sum_congr rfl
-  intro s _
-  have hn := R.users_ne b s
-  simp only [Finset.mem_erase]
-  by_cases h₁ : (z i).val = (R.users b s).1 <;>
-    by_cases h₂ : (z i).val = (R.users b s).2 <;> simp_all [eq_comm]
-
-lemma rotationMeeting_eq_slot_sum {A : ℕ} (o : α → Option (Fin A))
-    (z : Terminal o) (u v : α) :
-    rotationMeeting o z u v = ∑ i : Fin A,
-      ((if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) +
-      (if (z i).val = v ∧ u ∈ priorPaths o (i.val+1) then (1:ℝ) else 0)) := by
-  classical
-  have only_owner (u : α) (a : Fin A) (hou : o u = some a)
-      (i : Fin A) (hia : i ≠ a) : (z i).val ≠ u := by
-    intro h
-    have hi := (z i).property
-    rw [h,hou] at hi
-    exact hia (Option.some.inj hi).symm
-  have sum_one (u v : α) (a : Fin A) (hou : o u = some a) :
-      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) =
-      if (z a).val = u ∧ v ∈ priorPaths o (a.val+1) then 1 else 0 := by
-    apply Finset.sum_eq_single a
-    · intro i _ hia
-      simp [only_owner u a hou i hia]
-    · simp
-  have sum_none (u v : α) (hou : o u = none) :
-      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) = 0 := by
-    apply Finset.sum_eq_zero
-    intro i _
-    have hn : (z i).val ≠ u := by
-      intro h
-      have hi := (z i).property
-      rw [h,hou] at hi
-      contradiction
-    simp [hn]
-  rw [Finset.sum_add_distrib]
-  cases hou : o u with
-  | none =>
-    rw [sum_none u v hou]
-    simp [rotationMeeting,hou,mem_priorPaths]
-  | some a =>
-    rw [sum_one u v a hou]
-    cases hov : o v with
-    | none =>
-      rw [sum_none v u hov]
-      simp [rotationMeeting,hou,hov,mem_priorPaths]
-    | some b =>
-      rw [sum_one v u b hov]
-      simp only [rotationMeeting,hou,hov,mem_priorPaths,Option.some.injEq]
-      rcases lt_trichotomy a b with hab | rfl | hba
-      · have hn : ¬ b.val < a.val+1 := by omega
-        have hy : a.val < b.val+1 := by omega
-        simp [hab,hn,hy]
-      · simp
-      · have hn : ¬ a.val < b.val+1 := by omega
-        have hy : b.val < a.val+1 := by omega
-        simp [hba,not_lt_of_ge (le_of_lt hba),hn,hy]
-
-lemma rotationCharge_eq_terminal_sum {A : ℕ} (o : α → Option (Fin A))
-    (b : ι → Bool) (z : Terminal o) :
-    rotationCharge R o b z = ∑ i : Fin A,
-      (terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) := by
-  simp only [rotationCharge,rotationMeeting_eq_slot_sum,terminalCharge_prior_sum]
-  exact Finset.sum_comm
-
-omit [DecidableEq α] in
-@[simp] lemma priorPaths_zero {A : ℕ} (o : α → Option (Fin A)) : priorPaths o 0 = ∅ := by
-  classical
-  ext x
-  simp
-
-lemma priorPaths_succ {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) (ht : t < A) :
-    priorPaths o (t+1) = priorPaths o t ∪ cycleFiber o ⟨t,ht⟩ := by
-  classical
-  ext x
-  simp only [mem_priorPaths,Finset.mem_union,mem_cycleFiber]
-  constructor
-  · rintro ⟨i,hi,hit⟩
-    by_cases h : i.val < t
-    · exact Or.inl ⟨i,hi,h⟩
-    · have he : i = ⟨t,ht⟩ := Fin.ext (by change i.val = t; omega)
-      exact Or.inr (he ▸ hi)
-  · rintro (⟨i,hi,hit⟩ | hi)
-    · exact ⟨i,hi,by omega⟩
-    · exact ⟨⟨t,ht⟩,hi,by change t < t+1; omega⟩
-
-omit [DecidableEq α] in
-lemma priorPaths_disjoint {A : ℕ} (o : α → Option (Fin A)) (i : Fin A) :
-    Disjoint (priorPaths o i.val) (cycleFiber o i) := by
-  classical
-  apply Finset.disjoint_left.mpr
-  intro x hx hx'
-  obtain ⟨j,hj,hji⟩ := (mem_priorPaths _ _ _).mp hx
-  have he : j = i := Option.some.inj (hj.symm.trans ((mem_cycleFiber _ _ _).mp hx'))
-  subst j
-  exact (lt_irrefl _ hji)
-
-namespace QueryTree
-open Thorp.QueryTree
-
-lemma seekCycles_owned {A : ℕ} (S : Finset (α)) (r : Fin A → ℕ)
-    (o : α → Option (Fin A)) (z : Terminal o) (ω : ι → Bool)
-    (hS : ∀ i, cycleFiber o i ⊆ S)
-    (hO : ∀ i, orbitSet (R.perm ω) (z i).val = cycleFiber o i)
-    (hC : ∀ i, (cycleFiber o i).card = r i+1) :
-    let p := R.perm ω
-    let L := List.ofFn (fun i : Fin A => (p (z i).val,r i))
-    (seekCycles R S L ∅).succeeds ω = true ∧
-      ((seekCycles R S L ∅).staleCharge ∅ ω : ℝ) = rotationCharge R o (ω) z := by
-  classical
-  let b := ω
-  let p := R.perm b
-  let f (i : Fin A) := (p (z i).val,r i)
-  let c (i : Fin A) : ℝ := terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val)
-  have hc (i : Fin A) : Function.minimalPeriod p (z i).val - 1 = r i := by
-    have he := orbitSet_card p (z i).val
-    rw [hO,hC] at he
-    omega
-  have hs (t : ℕ) (ht : t ≤ A) :
-      (seekCycles R S ((List.ofFn f).drop t) (priorPaths o t)).succeeds ω = true ∧
-        ((seekCycles R S ((List.ofFn f).drop t) (priorPaths o t)).staleCharge (exposedPaths R b (priorPaths o t)) ω : ℝ) =
-          ((List.ofFn c).drop t).sum := by
-    induction ht using Nat.decreasingInduction with
-    | self => simp only [List.length_ofFn,le_refl,List.drop_eq_nil_of_le,List.sum_nil,seekCycles,Thorp.QueryTree.succeeds,staleCharge,Nat.cast_zero,and_self]
-    | of_succ t ht ih =>
-      have htf : t < (List.ofFn f).length := by simpa using ht
-      have htc : t < (List.ofFn c).length := by simpa using ht
-      rw [List.drop_eq_getElem_cons htf,List.drop_eq_getElem_cons htc]
-      simp only [List.getElem_ofFn,List.sum_cons,seekCycles]
-      have h := seekCycle_orbit R S (priorPaths o t) (z ⟨t,ht⟩).val
-        (fun V => seekCycles R S ((List.ofFn f).drop (t+1)) V) ω
-        (by rw [hO]; exact hS _) (by rw [hO]; exact priorPaths_disjoint o ⟨t,ht⟩)
-      dsimp only at h
-      rw [hc,hO,←priorPaths_succ o t ht] at h
-      dsimp only [f]
-      constructor
-      · exact h.1.trans ih.1
-      · rw [h.2,Nat.cast_add,ih.2]
-        exact add_comm _ _
-  have h := hs 0 (Nat.zero_le _)
-  simp only [List.drop_zero,priorPaths_zero,exposedPaths_empty,List.sum_ofFn] at h
-  dsimp only
-  exact ⟨h.1,h.2.trans (rotationCharge_eq_terminal_sum R o b z).symm⟩
-
-end QueryTree
-section Collections
-variable {A : ℕ}
-
-def ValidCycleSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ)
-    (o : α → Option (Fin A)) : Prop :=
-  (∀ i, (cycleFiber o i).card = r i + 1) ∧
-  (∀ i, cycleFiber o i ⊆ S) ∧
-  (∀ i x, o x = some i → orbitSet p x = cycleFiber o i)
-
-abbrev CycleSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
-  {o : α → Option (Fin A) // ValidCycleSlots p S r o}
-
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
-    Fintype (CycleSlots p S r) := Fintype.ofFinite _
-
-lemma CycleSlots.terminal_nonempty {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (c : CycleSlots p S r) (i : Fin A) : Nonempty {x : α // c.val x = some i} := by
-  have hn : (cycleFiber c.val i).Nonempty := Finset.card_pos.mp (by rw [c.property.1]; omega)
-  obtain ⟨x,hx⟩ := hn
-  exact ⟨x,(mem_cycleFiber _ _ _).mp hx⟩
-
-omit [DecidableEq α] in
-lemma owner_ext {o o' : α → Option (Fin A)}
-    (h : ∀ i, cycleFiber o i = cycleFiber o' i) : o = o' := by
-  funext x
-  have he (i : Fin A) : o x = some i ↔ o' x = some i := by
-    simpa only [mem_cycleFiber] using (Finset.ext_iff.mp (h i) x)
-  cases ho : o x with
-  | some i => exact ((he i).mp ho).symm
-  | none =>
-    cases ho' : o' x with
-    | none => rfl
-    | some i => have hn := (he i).mpr ho'; rw [ho] at hn; contradiction
-
-abbrev RootedSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
-  Σ c : CycleSlots p S r, Terminal c.val
-
-noncomputable def rootedStart {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (c : RootedSlots p S r) : Fin A → S := fun i =>
-  ⟨p (c.2 i).val,c.1.property.2.1 i (by
-    rw [←c.1.property.2.2 i (c.2 i).val (c.2 i).property]
-    exact (mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.apply_right (Equiv.Perm.SameCycle.refl p _)))⟩
-
-lemma rootedStart_injective {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ} :
-    Function.Injective (rootedStart (p:=p) (S:=S) (r:=r)) := by
-  rintro ⟨c,z⟩ ⟨c',z'⟩ h
-  have hz (i : Fin A) : (z i).val = (z' i).val :=
-    p.injective (congrArg Subtype.val (congrFun h i))
-  have hc : c = c' := Subtype.ext (owner_ext (fun i => by
-    rw [←c.property.2.2 i (z i).val (z i).property,
-      ←c'.property.2.2 i (z' i).val (z' i).property,hz]))
-  subst c'
-  have hzz : z = z' := funext (fun i => Subtype.ext (hz i))
-  subst z'
-  rfl
-
-lemma rooted_weight_sum_le (S : Finset (α)) (r : Fin A → ℕ) (ω : ι → Bool) :
-    (∑ c : RootedSlots (R.perm ω) S r,
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2)) ≤
-    ∑ a : Fin A → S,
-      if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-        ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-      else 0 := by
-  classical
-  let p := R.perm ω
-  let g (a : Fin A → S) : ℝ :=
-    if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-    else 0
-  have he (c : RootedSlots p S r) :
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2) =
-      g (rootedStart c) := by
-    have hs := QueryTree.seekCycles_owned R S r c.1.val c.2 ω c.1.property.2.1
-      (fun i => c.1.property.2.2 i (c.2 i).val (c.2 i).property) c.1.property.1
-    dsimp only at hs
-    unfold g
-    rw [ite_eq_left (show (QueryTree.seekCycles R S (List.ofFn (fun i => ((rootedStart c i).val,r i))) ∅).succeeds ω = true from hs.1)]
-    rw [← hs.2,neg_mul,Real.exp_neg,mul_comm (Real.log 2),Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
-    rfl
-  simp_rw [he]
-  exact sum_comp_injective_le rootedStart rootedStart_injective g (fun a => by dsimp [g]; split_ifs <;> positivity)
-
-lemma collection_rotation_lower (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r)
-    (ω : ι → Bool) (c : CycleSlots (R.perm ω) S r) :
-    collectionWeight d r ≤ ∑ z : Terminal c.val,
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z) := by
-  classical
-  let (i : Fin A) : Nonempty {x : α // c.val x = some i} := c.terminal_nonempty i
-  have hm : Monotone (fun i => (cycleFiber c.val i).card) := by
-    intro i j hij
-    dsimp only
-    rw [c.property.1,c.property.1]
-    exact Nat.add_le_add_right (hr hij) 1
-  have hb := rotation_exp_bound R hm (ω)
-  have hcard : (Fintype.card (Terminal c.val) : ℝ) = ∏ i : Fin A, (r i+1 : ℝ) := by
-    rw [Fintype.card_pi,Nat.cast_prod]
-    apply Finset.prod_congr rfl
-    intro i _
-    rw [card_cycleFiber,c.property.1,Nat.cast_add,Nat.cast_one]
-  have hp : (0:ℝ) < Fintype.card (Terminal c.val) := Nat.cast_pos.mpr Fintype.card_pos
-  have hl : rotationLower d r ≤ finiteMean (fun z : Terminal c.val => Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z)) := by
-    simpa only [rotationLower,c.property.1,Nat.cast_add,Nat.cast_one] using hb
-  have hh := (le_div_iff₀ hp).mp hl
-  rw [hcard] at hh
-  unfold collectionWeight
-  rw [← Finset.mul_sum]
-  calc
-    _ = ((2:ℝ)^d)^A * (rotationLower d r * ∏ i : Fin A, (r i+1 : ℝ)) := by ring
-    _ ≤ _ := mul_le_mul_of_nonneg_left hh (by positivity)
-
-lemma cycleSlots_weighted_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : ι → Bool =>
-      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) *
-      collectionWeight d r ≤ (S.card : ℝ)^A := by
-  classical
-  let g (a : Fin A → S) (ω : ι → Bool) : ℝ :=
-    if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-    else 0
-  have hω (ω : ι → Bool) :
-      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ) * collectionWeight d r ≤
-        ∑ a : Fin A → S, g a ω := by
-    calc
-      _ = ∑ _c : CycleSlots (R.perm ω) S r, collectionWeight d r := by simp
-      _ ≤ ∑ c : CycleSlots (R.perm ω) S r,
-          ∑ z : Terminal c.val, ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z) :=
-        Finset.sum_le_sum (fun c _ => collection_rotation_lower R S r hr ω c)
-      _ = ∑ c : RootedSlots (R.perm ω) S r,
-          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2) :=
-        (Fintype.sum_sigma (fun c : RootedSlots (R.perm ω) S r =>
-          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2))).symm
-      _ ≤ _ := rooted_weight_sum_le R S r ω
-  have hb := finiteMean_mono hω
-  rw [finiteMean_mul_const,finiteMean_sum] at hb
-  calc
-    _ ≤ ∑ a : Fin A → S, finiteMean (g a) := hb
-    _ ≤ ∑ _a : Fin A → S, (1:ℝ) := by
-      apply Finset.sum_le_sum
-      intro a _
-      have hh := QueryTree.cycles_weighted_success R S (List.ofFn (fun i : Fin A => ((a i).val,r i)))
-      simpa only [List.length_ofFn] using hh
-    _ = _ := by simp
-
-lemma cycleSlots_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : ι → Bool =>
-      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
-        (S.card : ℝ)^A / collectionWeight d r := by
-  exact (le_div_iff₀ (collectionWeight_pos d r)).mpr (cycleSlots_weighted_moment R S r hr)
-
-lemma butterfly_cycleSlots_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : ι → Bool =>
-      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
-        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
-  have h := cycleSlots_moment R S r hr
-  rw [collectionWeight_eq_sqrt,div_mul_eq_div_mul_one_div] at h
-  rw [div_pow]
-  simpa only [one_div,Finset.prod_inv_distrib] using h
-
-end Collections
-
-section CycleCounting
-variable {A : ℕ}
-
-abbrev SelectedCycle (p : Equiv.Perm (α)) (S : Finset (α)) :=
-  {C : Finset (α) // (∃ x, C = orbitSet p x) ∧ C ⊆ S}
-
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) :
-    Fintype (SelectedCycle p S) := Fintype.ofFinite _
-
-lemma SelectedCycle.nonempty {p : Equiv.Perm (α)} {S : Finset (α)}
-    (C : SelectedCycle p S) : C.val.Nonempty := by
-  obtain ⟨x,hx⟩ := C.property.1
-  rw [hx]
-  exact orbitSet_nonempty p x
-
-lemma SelectedCycle.orbit_eq {p : Equiv.Perm (α)} {S : Finset (α)}
-    (C : SelectedCycle p S) {x : α} (hx : x ∈ C.val) : orbitSet p x = C.val := by
-  obtain ⟨y,hy⟩ := C.property.1
-  rw [hy] at hx ⊢
-  exact (orbitSet_eq p ((mem_orbitSet _ _ _).mp hx)).symm
-
-lemma SelectedCycle.eq_of_mem {p : Equiv.Perm (α)} {S : Finset (α)}
-    (C D : SelectedCycle p S) {x : α} (hC : x ∈ C.val) (hD : x ∈ D.val) : C = D := by
-  apply Subtype.ext
-  exact (C.orbit_eq hC).symm.trans (D.orbit_eq hD)
-
-abbrev SlotAssignment (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
-  {f : Fin A → SelectedCycle p S // Function.Injective f ∧ ∀ i, (f i).val.card = r i+1}
-
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
-    Fintype (SlotAssignment p S r) := Fintype.ofFinite _
-
-noncomputable def slotsToAssignment {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (c : CycleSlots p S r) : SlotAssignment p S r := by
-  classical
-  let f (i : Fin A) : SelectedCycle p S := ⟨cycleFiber c.val i,by
-    obtain ⟨x,hx⟩ := c.terminal_nonempty i
-    exact ⟨⟨x,(c.property.2.2 i x hx).symm⟩,c.property.2.1 i⟩⟩
-  refine ⟨f,?_,fun i => c.property.1 i⟩
-  intro i j hij
-  have he : cycleFiber c.val i = cycleFiber c.val j := congrArg Subtype.val hij
-  obtain ⟨x,hx⟩ := c.terminal_nonempty i
-  have hxi : x ∈ cycleFiber c.val i := (mem_cycleFiber _ _ _).mpr hx
-  rw [he] at hxi
-  exact Option.some.inj (hx.symm.trans ((mem_cycleFiber _ _ _).mp hxi))
-
-noncomputable def assignmentOwner {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) (x : α) : Option (Fin A) := by
-  classical
-  exact if h : ∃ i, x ∈ (f.val i).val then some (Classical.choose h) else none
-
-lemma assignmentOwner_eq_some {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) (x : α) (i : Fin A) :
-    assignmentOwner f x = some i ↔ x ∈ (f.val i).val := by
-  classical
-  unfold assignmentOwner
-  split_ifs with h
-  · constructor
-    · intro hi
-      have hx := Classical.choose_spec h
-      rw [Option.some.inj hi] at hx
-      exact hx
-    · intro hx
-      have he := (f.val (Classical.choose h)).eq_of_mem (f.val i) (Classical.choose_spec h) hx
-      exact congrArg some (f.property.1 he)
-  · constructor
-    · intro hi; contradiction
-    · intro hx; exact (h ⟨i,hx⟩).elim
-
-lemma assignmentOwner_fiber {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) (i : Fin A) : cycleFiber (assignmentOwner f) i = (f.val i).val := by
-  ext x
-  simp only [mem_cycleFiber,assignmentOwner_eq_some]
-
-noncomputable def assignmentToSlots {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) : CycleSlots p S r :=
-  ⟨assignmentOwner f,by
-    refine ⟨?_,?_,?_⟩
-    · intro i
-      rw [assignmentOwner_fiber]
-      exact f.property.2 i
-    · intro i
-      rw [assignmentOwner_fiber]
-      exact (f.val i).property.2
-    · intro i x hx
-      rw [assignmentOwner_fiber]
-      exact (f.val i).orbit_eq ((assignmentOwner_eq_some f x i).mp hx)⟩
-
-noncomputable def cycleSlotsEquivAssignments (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
-    CycleSlots p S r ≃ SlotAssignment p S r where
-  toFun := slotsToAssignment
-  invFun := assignmentToSlots
-  left_inv c := by
-    apply Subtype.ext
-    apply owner_ext
-    intro i
-    rw [assignmentToSlots,assignmentOwner_fiber]
-    rfl
-  right_inv f := by
-    apply Subtype.ext
-    funext i
-    apply Subtype.ext
-    exact assignmentOwner_fiber f i
-
-end CycleCounting
-
-section GroupedCounting
-variable {A : ℕ}
-
-abbrev LengthClass (r : Fin A → ℕ) := {j : ℕ // j ∈ Finset.univ.image (fun i => r i + 1)}
-
-def slotClass (r : Fin A → ℕ) (i : Fin A) : LengthClass r :=
-  ⟨r i+1, Finset.mem_image.mpr ⟨i, Finset.mem_univ _, rfl⟩⟩
-
-abbrev SlotsOfLength (r : Fin A → ℕ) (j : LengthClass r) := {i : Fin A // slotClass r i = j}
-abbrev CyclesOfLength (p : Equiv.Perm (α)) (S : Finset (α)) (j : ℕ) :=
-  {C : SelectedCycle p S // C.val.card = j}
-
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (j : ℕ) :
-    Fintype (CyclesOfLength p S j) := Fintype.ofFinite _
-
-abbrev GroupedAssignments (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
-  ∀ j : LengthClass r, SlotsOfLength r j ↪ CyclesOfLength p S j.val
-
-noncomputable def assignmentToGroups {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) : GroupedAssignments p S r := fun _j =>
-  { toFun := fun i => ⟨f.val i.val, (f.property.2 i.val).trans (congrArg Subtype.val i.property)⟩
-    inj' := fun _i _k h => Subtype.ext (f.property.1 (congrArg Subtype.val h)) }
-
-noncomputable def groupsEmbedding {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (F : GroupedAssignments p S r) : Fin A ↪ SelectedCycle p S :=
-  ((Equiv.sigmaFiberEquiv (slotClass r)).symm.toEmbedding.trans
-    (Function.Embedding.sigmaMap (Function.Embedding.refl _) F)).trans
-    ((Function.Embedding.sigmaMap (Function.Embedding.subtype _) (fun _ => Function.Embedding.refl _)).trans
-      (Equiv.sigmaFiberEquiv (fun C : SelectedCycle p S => C.val.card)).toEmbedding)
-
-noncomputable def groupsToAssignment {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (F : GroupedAssignments p S r) : SlotAssignment p S r :=
-  ⟨groupsEmbedding F, (groupsEmbedding F).injective,
-    fun i => (F (slotClass r i) ⟨i,rfl⟩).property⟩
-
-noncomputable def assignmentEquivGroups (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
-    SlotAssignment p S r ≃ GroupedAssignments p S r where
-  toFun := assignmentToGroups
-  invFun := groupsToAssignment
-  left_inv f := by
-    apply Subtype.ext
-    rfl
-  right_inv F := by
-    funext j
-    apply Function.Embedding.ext
-    intro i
-    rcases i with ⟨i,hi⟩
-    subst j
-    rfl
-
-lemma card_cycleSlots_factorial (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
-    Fintype.card (CycleSlots p S r) =
-      ∏ j : LengthClass r, (Fintype.card (CyclesOfLength p S j.val)).descFactorial
-        (Fintype.card (SlotsOfLength r j)) := by
-  classical
-  rw [Fintype.card_congr (cycleSlotsEquivAssignments p S r),
-    Fintype.card_congr (assignmentEquivGroups p S r), Fintype.card_pi]
-  simp only [Fintype.card_embedding_eq]
-
-lemma butterfly_factorial_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : ι → Bool =>
-       (∏ j : LengthClass r,
-         (Fintype.card (CyclesOfLength (R.perm ω) S j.val)).descFactorial
-           (Fintype.card (SlotsOfLength r j)) : ℕ)) ≤
-      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-        ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
-  simpa only [←card_cycleSlots_factorial] using butterfly_cycleSlots_moment R S r hr
-
-end GroupedCounting
-
-section AllLengths
-variable {A : ℕ}
-
-noncomputable def assignmentRelabel (p : Equiv.Perm (α)) (S : Finset (α))
-    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
-    SlotAssignment p S r ≃ SlotAssignment p S (r ∘ e) where
-  toFun f := ⟨f.val ∘ e, f.property.1.comp e.injective, fun i => f.property.2 (e i)⟩
-  invFun f := ⟨f.val ∘ e.symm, f.property.1.comp e.symm.injective, fun i => by
-    simpa only [Function.comp_apply,Equiv.apply_symm_apply] using f.property.2 (e.symm i)⟩
-  left_inv f := by apply Subtype.ext; funext i; simp
-  right_inv f := by apply Subtype.ext; funext i; simp
-
-lemma card_cycleSlots_relabel (p : Equiv.Perm (α)) (S : Finset (α))
-    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
-    Fintype.card (CycleSlots p S r) = Fintype.card (CycleSlots p S (r ∘ e)) := by
-  exact Fintype.card_congr ((cycleSlotsEquivAssignments p S r).trans
-    ((assignmentRelabel p S r e).trans (cycleSlotsEquivAssignments p S (r ∘ e)).symm))
-
-lemma butterfly_cycleSlots_moment_all (S : Finset (α)) (r : Fin A → ℕ) :
-    finiteMean (fun ω : ι → Bool =>
-      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
-        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
-  have h := butterfly_cycleSlots_moment R S (r ∘ Tuple.sort r) (Tuple.monotone_sort r)
-  simp_rw [←card_cycleSlots_relabel] at h
-  have hp := Equiv.prod_comp (Tuple.sort r) (fun i : Fin A => (1 / Real.sqrt (r i+1) : ℝ))
-  simpa only [Function.comp_apply,hp] using h
-
-abbrev ShortCycle (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :=
-  {C : SelectedCycle p S // C.val.card ≤ J}
-
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    Fintype (ShortCycle p S J) := Fintype.ofFinite _
-
-noncomputable def shortCycleLength {p : Equiv.Perm (α)} {S : Finset (α)} {J : ℕ}
-    (C : ShortCycle p S J) : Fin J :=
-  ⟨C.val.val.card-1, by have := C.val.nonempty.card_pos; have := C.property; omega⟩
-
-lemma shortCycleLength_add_one {p : Equiv.Perm (α)} {S : Finset (α)} {J : ℕ}
-    (C : ShortCycle p S J) : (shortCycleLength C : ℕ)+1 = C.val.val.card := by
-  have := C.val.nonempty.card_pos
-  simp only [shortCycleLength]
-  omega
-
-noncomputable def shortEmbeddingEquiv (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    (Fin A ↪ ShortCycle p S J) ≃
-      Σ r : Fin A → Fin J, SlotAssignment p S (fun i => (r i : ℕ)) where
-  toFun f := ⟨fun i => shortCycleLength (f i),
-    ⟨fun i => (f i).val, fun i j h => f.injective (Subtype.ext h),
-      fun i => (shortCycleLength_add_one (f i)).symm⟩⟩
-  invFun c :=
-    { toFun := fun i => ⟨c.2.val i, by rw [c.2.property.2]; exact (c.1 i).isLt⟩
-      inj' := fun _i _j h => c.2.property.1 (congrArg Subtype.val h) }
-  left_inv f := by
-    apply Function.Embedding.ext
-    intro i
-    rfl
-  right_inv c := by
-    rcases c with ⟨r,f⟩
-    have he : (fun i => shortCycleLength (⟨f.val i, by rw [f.property.2]; exact (r i).isLt⟩ : ShortCycle p S J)) = r := by
-      funext i
-      apply Fin.ext
-      simp only [shortCycleLength,f.property.2,Nat.add_sub_cancel]
-    apply Sigma.ext he
-    apply (Subtype.heq_iff_coe_eq (by
-      intro x
-      change (Function.Injective x ∧ ∀ i, (x i).val.card = (f.val i).val.card - 1 + 1) ↔ _
-      simp only [f.property.2,Nat.add_sub_cancel])).mpr
-    rfl
-
-lemma shortCycle_factorial_eq (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    (Fintype.card (ShortCycle p S J)).descFactorial A =
-      ∑ r : Fin A → Fin J, Fintype.card (CycleSlots p S (fun i => (r i : ℕ))) := by
-  classical
-  have hc := Fintype.card_embedding_eq (α := Fin A) (β := ShortCycle p S J)
-  simp only [Fintype.card_fin] at hc
-  rw [←hc, Fintype.card_congr (shortEmbeddingEquiv p S J),Fintype.card_sigma]
-  apply Finset.sum_congr rfl
-  intro r _
-  exact (Fintype.card_congr (cycleSlotsEquivAssignments p S (fun i => (r i : ℕ)))).symm
-
-end AllLengths
-
-section CycleMGF
-
-lemma butterfly_shortCycle_moment (S : Finset (α)) (J A : ℕ) :
-    finiteMean (fun ω : ι → Bool =>
-      ((Fintype.card (ShortCycle (R.perm ω) S J)).descFactorial A : ℝ)) ≤
-      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))^A := by
-  classical
-  simp_rw [shortCycle_factorial_eq,Nat.cast_sum,finiteMean_sum]
-  calc
-    _ ≤ ∑ r : Fin A → Fin J,
-        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-          ∏ i : Fin A, (1 / Real.sqrt ((r i : ℕ)+1)) := by
-      apply Finset.sum_le_sum
-      intro r _
-      exact butterfly_cycleSlots_moment_all R S (fun i => (r i : ℕ))
-    _ = _ := by
-      rw [←Finset.mul_sum,mul_pow]
-      congr 1
-      rw [←Fintype.prod_sum (fun (_ : Fin A) (j : Fin J) => (1 / Real.sqrt ((j:ℕ)+1) : ℝ))]
-      simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fin]
-
-noncomputable def cycleMass (p : Equiv.Perm (α)) (S : Finset (α)) : ℕ :=
-  ∑ C : SelectedCycle p S, C.val.card
-
-lemma cycleMass_le (p : Equiv.Perm (α)) (S : Finset (α)) : cycleMass p S ≤ S.card := by
-  classical
-  let f : (Σ C : SelectedCycle p S, {x : α // x ∈ C.val}) → S :=
-    fun c => ⟨c.2.val,c.1.property.2 c.2.property⟩
-  have hi : Function.Injective f := by
-    intro c e h
-    have hx : c.2.val = e.2.val := congrArg Subtype.val h
-    have hC : c.1 = e.1 := c.1.eq_of_mem e.1 c.2.property (hx ▸ e.2.property)
-    rcases c with ⟨c,x⟩
-    rcases e with ⟨e,y⟩
-    dsimp only at hC
-    subst e
-    have he : x=y := Subtype.ext hx
-    cases he
-    rfl
-  have hh := Fintype.card_le_of_injective f hi
-  simpa only [Fintype.card_sigma,Fintype.card_coe,cycleMass] using hh
-
-lemma shortCycle_card_le (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    Fintype.card (ShortCycle p S J) ≤ S.card := by
-  classical
-  calc
-    _ ≤ Fintype.card (SelectedCycle p S) := Fintype.card_subtype_le _
-    _ = ∑ _ : SelectedCycle p S, 1 := by simp
-    _ ≤ cycleMass p S := Finset.sum_le_sum (fun C _ => C.nonempty.card_pos)
-    _ ≤ _ := cycleMass_le p S
-
-lemma butterfly_shortCycle_mgf (S : Finset (α)) (J : ℕ) (q : ℝ) (hq : 1 ≤ q) :
-    finiteMean (fun ω : ι → Bool =>
-      q^(Fintype.card (ShortCycle (R.perm ω) S J))) ≤
-      Real.exp ((q-1)*((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))) := by
-  apply factorial_mgf _ S.card _ q (fun ω => shortCycle_card_le _ _ _)
-  · positivity
-  · exact hq
-  · exact butterfly_shortCycle_moment R S J
-
-end CycleMGF
-
-
-abbrev LongCycle (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :=
-  {C : SelectedCycle p S // ¬ C.val.card ≤ J}
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    Fintype (LongCycle p S J) := Fintype.ofFinite _
-
-lemma selectedCycle_card_split (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    Fintype.card (SelectedCycle p S) =
-      Fintype.card (ShortCycle p S J) + Fintype.card (LongCycle p S J) := by
-  classical
-  rw [←Fintype.card_congr (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J)),Fintype.card_sum]
-
-lemma longCycle_card_bound (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    J * Fintype.card (LongCycle p S J) ≤ S.card := by
-  classical
-  have hs := Equiv.sum_comp (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J))
-    (fun C : SelectedCycle p S => C.val.card)
-  rw [Fintype.sum_sum_type] at hs
-  have h0 : J * Fintype.card (LongCycle p S J) ≤
-      ∑ C : LongCycle p S J, C.val.val.card := by
-    calc
-      _ = ∑ _C : LongCycle p S J, J := by simp [mul_comm]
-      _ ≤ _ := Finset.sum_le_sum (fun C _ => Nat.le_of_lt (Nat.lt_of_not_ge C.property))
-  have hh := cycleMass_le p S
-  unfold cycleMass at hh
-  rw [←hs] at hh
-  exact h0.trans ((Nat.le_add_left _ _).trans hh)
-
-lemma longCycle_pow_bound (p : Equiv.Perm (α)) (S : Finset (α))
-    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    q^(Fintype.card (LongCycle p S J)) ≤ Real.exp ((S.card : ℝ)/J * Real.log q) := by
-  have hq0 : 0 < q := lt_of_lt_of_le zero_lt_one hq
-  have hc : (Fintype.card (LongCycle p S J) : ℝ) ≤ (S.card : ℝ)/J := by
-    apply (le_div_iff₀ (by exact_mod_cast hJ)).mpr
-    have hh := longCycle_card_bound p S J
-    exact_mod_cast (by simpa only [mul_comm] using hh)
-  calc
-    _ = Real.exp ((Fintype.card (LongCycle p S J) : ℝ)*Real.log q) := by
-      rw [Real.exp_nat_mul,Real.exp_log hq0]
-    _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right hc (Real.log_nonneg hq))
-
-lemma butterfly_cycle_log_mgf (S : Finset (α)) (J : ℕ) (hJ : 0 < J)
-    (q : ℝ) (hq : 1 ≤ q) :
-    Real.log (finiteMean (fun ω : ι → Bool =>
-      q^(Fintype.card (SelectedCycle (R.perm ω) S)))) ≤
-      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
-  let lam : ℝ := (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1)
-  let c : ℝ := (S.card : ℝ)/J * Real.log q
-  have hnum : (q-1)*lam ≤ 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
-    have hsum := sum_inv_sqrt J
-    have hp : 0 ≤ (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) := by positivity
-    have h0 : 0 ≤ 2*Real.sqrt J := by positivity
-    have hh := mul_le_mul_of_nonneg_left hsum hp
-    have hq1 : 0 ≤ q-1 := sub_nonneg.mpr hq
-    calc
-      _ ≤ (q-1)*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
-        mul_le_mul_of_nonneg_left hh hq1
-      _ ≤ q*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
-        mul_le_mul_of_nonneg_right (by linarith) (mul_nonneg hp h0)
-      _ = _ := by ring
-  have hm : finiteMean (fun ω : ι → Bool =>
-      q^(Fintype.card (SelectedCycle (R.perm ω) S))) ≤
-      Real.exp (c+(q-1)*lam) := by
-    calc
-      _ ≤ finiteMean (fun ω : ι → Bool =>
-          q^(Fintype.card (ShortCycle (R.perm ω) S J))*Real.exp c) := by
-        apply finiteMean_mono
-        intro ω
-        rw [selectedCycle_card_split _ _ J,pow_add]
-        exact mul_le_mul_of_nonneg_left (longCycle_pow_bound _ S J hJ q hq) (by positivity)
-      _ = finiteMean (fun ω : ι → Bool =>
-          q^(Fintype.card (ShortCycle (R.perm ω) S J))) * Real.exp c :=
-        finiteMean_mul_const _ _
-      _ ≤ Real.exp ((q-1)*lam) * Real.exp c :=
-        mul_le_mul_of_nonneg_right (butterfly_shortCycle_mgf R S J q hq) (Real.exp_nonneg _)
-      _ = _ := by rw [←Real.exp_add,add_comm]
-  have hpos : 0 < finiteMean (fun ω : ι → Bool =>
-      q^(Fintype.card (SelectedCycle (R.perm ω) S))) := by
-    apply finiteMean_pos
-    intro ω
-    positivity
-  have hlog := Real.log_le_log hpos hm
-  rw [Real.log_exp] at hlog
-  dsimp [c] at hlog
-  linarith
-
-
-end RoutingNetwork
-
-variable {α B : Type*} [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] {d : ℕ}
-
-def parallelPerm (ω : B × SwitchIndex d → Bool) : Equiv.Perm (B × Card d) where
-  toFun x := (x.1,butterflyPerm d (decodeButterfly d (fun i => ω (x.1,i))) x.2)
-  invFun y := (y.1,(butterflyPerm d (decodeButterfly d (fun i => ω (y.1,i)))).symm y.2)
-  left_inv x := by simp
-  right_inv y := by simp
-
-def relabeledPerm (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (ω : B × SwitchIndex d → Bool) : Equiv.Perm α :=
-  (pre.trans (parallelPerm ω)).trans post
-
-noncomputable def parallelPath (pre : α ≃ B × Card d)
-    (ω : B × SwitchIndex d → Bool) (x : α) : Finset (B × SwitchIndex d) :=
-  (routeDomain d (pre x).2
-    (butterflyPerm d (decodeButterfly d (fun i => ω ((pre x).1,i))) (pre x).2)).image
-      (fun i => ((pre x).1,i))
-
-def parallelUsers (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
-    (i : B × SwitchIndex d) : α × α :=
-  let u := switchUsers d (decodeButterfly d (fun j => ω (i.1,j))) i.2
-  (pre.symm (i.1,u.1),pre.symm (i.1,u.2))
-
-omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
-lemma parallelUsers_ne (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
-    (i : B × SwitchIndex d) : (parallelUsers pre ω i).1 ≠ (parallelUsers pre ω i).2 := by
-  intro h
-  exact switchUsers_ne d _ _ (congrArg Prod.snd (pre.symm.injective h))
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma mem_parallelPath (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
-    (x : α) (i : B × SwitchIndex d) :
-    i ∈ parallelPath pre ω x ↔ (parallelUsers pre ω i).1 = x ∨ (parallelUsers pre ω i).2 = x := by
-  classical
-  obtain ⟨b,j⟩ := i
-  simp only [parallelPath,Finset.mem_image,Prod.mk.injEq]
-  constructor
-  · rintro ⟨k,hk,hb,rfl⟩
-    subst b
-    rcases (mem_routeDomain_iff_user d _ _ _).mp hk with h | h
-    · left
-      apply pre.injective
-      change pre (pre.symm ((pre x).1,_)) = pre x
-      rw [Equiv.apply_symm_apply]
-      exact Prod.ext rfl h
-    · right
-      apply pre.injective
-      change pre (pre.symm ((pre x).1,_)) = pre x
-      rw [Equiv.apply_symm_apply]
-      exact Prod.ext rfl h
-  · intro h
-    have hh : (b,(switchUsers d (decodeButterfly d (fun j => ω (b,j))) j).1) = pre x ∨
-        (b,(switchUsers d (decodeButterfly d (fun j => ω (b,j))) j).2) = pre x := by
-      rcases h with h | h
-      · exact Or.inl (by simpa only [parallelUsers,Equiv.apply_symm_apply] using congrArg pre h)
-      · exact Or.inr (by simpa only [parallelUsers,Equiv.apply_symm_apply] using congrArg pre h)
-    have hb : b = (pre x).1 := hh.elim (congrArg Prod.fst) (congrArg Prod.fst)
-    subst b
-    refine ⟨j,?_,rfl,rfl⟩
-    apply (mem_routeDomain_iff_user d _ _ _).mpr
-    exact hh.elim (fun h => Or.inl (congrArg Prod.snd h)) (fun h => Or.inr (congrArg Prod.snd h))
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma card_parallelPath (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool) (x : α) :
-    (parallelPath pre ω x).card = d := by
-  classical
-  rw [parallelPath,Finset.card_image_of_injective _ (fun _ _ h => Prod.mk.inj h |>.2),card_routeDomain]
-
-noncomputable def blockMarks (pre : α ≃ B × Card d) (S : Finset α) (b : B) : Finset (Card d) :=
-  Finset.univ.filter (fun x => pre.symm (b,x) ∈ S)
-
-omit [Fintype α] [Fintype B] [DecidableEq B] in
-@[simp] lemma mem_blockMarks (pre : α ≃ B × Card d) (S : Finset α) (b : B) (x : Card d) :
-    x ∈ blockMarks pre S b ↔ pre.symm (b,x) ∈ S := by
-  classical
-  simp [blockMarks]
-
-omit [DecidableEq B] in
-lemma sum_card_blockMarks (pre : α ≃ B × Card d) (S : Finset α) :
-    ∑ b : B, (blockMarks pre S b).card = S.card := by
-  classical
-  calc
-    _ = ∑ b : B, ∑ x : Card d, if pre.symm (b,x) ∈ S then 1 else 0 := by
-      simp only [blockMarks,Finset.card_filter]
-    _ = ∑ u : B × Card d, if pre.symm u ∈ S then 1 else 0 := (Fintype.sum_prod_type (fun u : B × Card d => if pre.symm u ∈ S then (1:ℕ) else 0)).symm
-    _ = ∑ x : α, if x ∈ S then 1 else 0 := Equiv.sum_comp pre.symm (fun x : α => if x ∈ S then (1:ℕ) else 0)
-    _ = _ := by simp
-
-omit [DecidableEq B] in
-lemma card_blockMarks_le (pre : α ≃ B × Card d) (S : Finset α) (b : B) :
-    (blockMarks pre S b).card ≤ S.card := by
-  rw [←sum_card_blockMarks pre S]
-  exact Finset.single_le_sum (f := fun b : B => (blockMarks pre S b).card) (fun _ _ => Nat.zero_le _) (Finset.mem_univ b)
-
-end Thorp
-
-end ThorpNine.Harmonic
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Harmonic.AssignmentSlots
+
+namespace OAI
+
+namespace ThorpNine.Harmonic
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+namespace RoutingNetwork
+variable {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι] [DecidableEq ι]
+  {d : ℕ} (R : RoutingNetwork α ι d)
+lemma mem_exposedPaths (b : ι → Bool) (V : Finset (α)) (s : ι) :
+    s ∈ exposedPaths R b V ↔ (R.users b s).1 ∈ V ∨ (R.users b s).2 ∈ V := by
+  classical
+  simp only [exposedPaths,Finset.mem_biUnion,R.mem_path]
+  constructor
+  · rintro ⟨x,hx,h | h⟩
+    · exact Or.inl (h ▸ hx)
+    · exact Or.inr (h ▸ hx)
+  · rintro (h | h)
+    · exact ⟨_,h,Or.inl rfl⟩
+    · exact ⟨_,h,Or.inr rfl⟩
+
+lemma terminalCharge_sum (b : ι → Bool) (z : α) (V : Finset (α))
+    (hz : z ∉ V) :
+    (terminalCharge R b z V : ℝ) = ∑ s : ι,
+      ((if (R.users b s).1 = z ∧ (R.users b s).2 ∈ V then (1:ℝ) else 0) +
+      (if (R.users b s).2 = z ∧ (R.users b s).1 ∈ V then (1:ℝ) else 0)) := by
+  classical
+  have he : R.path b z ∩ exposedPaths R b V =
+      Finset.univ.filter (fun s =>
+        ((R.users b s).1 = z ∧ (R.users b s).2 ∈ V) ∨
+        ((R.users b s).2 = z ∧ (R.users b s).1 ∈ V)) := by
+    ext s
+    simp only [Finset.mem_inter,R.mem_path,mem_exposedPaths,
+      Finset.mem_filter,Finset.mem_univ,true_and]
+    by_cases h₁ : (R.users b s).1 = z <;>
+      by_cases h₂ : (R.users b s).2 = z <;> simp [h₁,h₂,hz]
+  rw [terminalCharge,he]
+  simp only [← Finset.sum_boole]
+  apply Finset.sum_congr rfl
+  intro s _
+  by_cases h₁ : (R.users b s).1 = z <;>
+    by_cases h₂ : (R.users b s).2 = z <;> simp [h₁,h₂,hz]
+
+noncomputable def priorPaths {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) : Finset (α) :=
+  Finset.univ.filter (fun x => ∃ i : Fin A, o x = some i ∧ i.val < t)
+
+omit [DecidableEq α] in
+@[simp] lemma mem_priorPaths {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) (x : α) :
+    x ∈ priorPaths o t ↔ ∃ i : Fin A, o x = some i ∧ i.val < t := by
+  classical
+  simp [priorPaths]
+
+lemma terminalCharge_prior_sum {A : ℕ} (o : α → Option (Fin A))
+    (b : ι → Bool) (z : Terminal o) (i : Fin A) :
+    (terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) =
+      ∑ s : ι,
+        ((if (z i).val = (R.users b s).1 ∧ (R.users b s).2 ∈ priorPaths o (i.val+1)
+          then (1:ℝ) else 0) +
+        (if (z i).val = (R.users b s).2 ∧ (R.users b s).1 ∈ priorPaths o (i.val+1)
+          then (1:ℝ) else 0)) := by
+  classical
+  rw [terminalCharge_sum _ _ _ _ (Finset.notMem_erase _ _)]
+  apply Finset.sum_congr rfl
+  intro s _
+  have hn := R.users_ne b s
+  simp only [Finset.mem_erase]
+  by_cases h₁ : (z i).val = (R.users b s).1 <;>
+    by_cases h₂ : (z i).val = (R.users b s).2 <;> simp_all [eq_comm]
+
+lemma rotationMeeting_eq_slot_sum {A : ℕ} (o : α → Option (Fin A))
+    (z : Terminal o) (u v : α) :
+    rotationMeeting o z u v = ∑ i : Fin A,
+      ((if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) +
+      (if (z i).val = v ∧ u ∈ priorPaths o (i.val+1) then (1:ℝ) else 0)) := by
+  classical
+  have only_owner (u : α) (a : Fin A) (hou : o u = some a)
+      (i : Fin A) (hia : i ≠ a) : (z i).val ≠ u := by
+    intro h
+    have hi := (z i).property
+    rw [h,hou] at hi
+    exact hia (Option.some.inj hi).symm
+  have sum_one (u v : α) (a : Fin A) (hou : o u = some a) :
+      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) =
+      if (z a).val = u ∧ v ∈ priorPaths o (a.val+1) then 1 else 0 := by
+    apply Finset.sum_eq_single a
+    · intro i _ hia
+      simp [only_owner u a hou i hia]
+    · simp
+  have sum_none (u v : α) (hou : o u = none) :
+      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) = 0 := by
+    apply Finset.sum_eq_zero
+    intro i _
+    have hn : (z i).val ≠ u := by
+      intro h
+      have hi := (z i).property
+      rw [h,hou] at hi
+      contradiction
+    simp [hn]
+  rw [Finset.sum_add_distrib]
+  cases hou : o u with
+  | none =>
+    rw [sum_none u v hou]
+    simp [rotationMeeting,hou,mem_priorPaths]
+  | some a =>
+    rw [sum_one u v a hou]
+    cases hov : o v with
+    | none =>
+      rw [sum_none v u hov]
+      simp [rotationMeeting,hou,hov,mem_priorPaths]
+    | some b =>
+      rw [sum_one v u b hov]
+      simp only [rotationMeeting,hou,hov,mem_priorPaths,Option.some.injEq]
+      rcases lt_trichotomy a b with hab | rfl | hba
+      · have hn : ¬ b.val < a.val+1 := by omega
+        have hy : a.val < b.val+1 := by omega
+        simp [hab,hn,hy]
+      · simp
+      · have hn : ¬ a.val < b.val+1 := by omega
+        have hy : b.val < a.val+1 := by omega
+        simp [hba,not_lt_of_ge (le_of_lt hba),hn,hy]
+
+lemma rotationCharge_eq_terminal_sum {A : ℕ} (o : α → Option (Fin A))
+    (b : ι → Bool) (z : Terminal o) :
+    rotationCharge R o b z = ∑ i : Fin A,
+      (terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) := by
+  simp only [rotationCharge,rotationMeeting_eq_slot_sum,terminalCharge_prior_sum]
+  exact Finset.sum_comm
+
+omit [DecidableEq α] in
+@[simp] lemma priorPaths_zero {A : ℕ} (o : α → Option (Fin A)) : priorPaths o 0 = ∅ := by
+  classical
+  ext x
+  simp
+
+lemma priorPaths_succ {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) (ht : t < A) :
+    priorPaths o (t+1) = priorPaths o t ∪ cycleFiber o ⟨t,ht⟩ := by
+  classical
+  ext x
+  simp only [mem_priorPaths,Finset.mem_union,mem_cycleFiber]
+  constructor
+  · rintro ⟨i,hi,hit⟩
+    by_cases h : i.val < t
+    · exact Or.inl ⟨i,hi,h⟩
+    · have he : i = ⟨t,ht⟩ := Fin.ext (by change i.val = t; omega)
+      exact Or.inr (he ▸ hi)
+  · rintro (⟨i,hi,hit⟩ | hi)
+    · exact ⟨i,hi,by omega⟩
+    · exact ⟨⟨t,ht⟩,hi,by change t < t+1; omega⟩
+
+omit [DecidableEq α] in
+lemma priorPaths_disjoint {A : ℕ} (o : α → Option (Fin A)) (i : Fin A) :
+    Disjoint (priorPaths o i.val) (cycleFiber o i) := by
+  classical
+  apply Finset.disjoint_left.mpr
+  intro x hx hx'
+  obtain ⟨j,hj,hji⟩ := (mem_priorPaths _ _ _).mp hx
+  have he : j = i := Option.some.inj (hj.symm.trans ((mem_cycleFiber _ _ _).mp hx'))
+  subst j
+  exact (lt_irrefl _ hji)
+
+namespace QueryTree
+open Thorp.QueryTree
+
+lemma seekCycles_owned {A : ℕ} (S : Finset (α)) (r : Fin A → ℕ)
+    (o : α → Option (Fin A)) (z : Terminal o) (ω : ι → Bool)
+    (hS : ∀ i, cycleFiber o i ⊆ S)
+    (hO : ∀ i, orbitSet (R.perm ω) (z i).val = cycleFiber o i)
+    (hC : ∀ i, (cycleFiber o i).card = r i+1) :
+    let p := R.perm ω
+    let L := List.ofFn (fun i : Fin A => (p (z i).val,r i))
+    (seekCycles R S L ∅).succeeds ω = true ∧
+      ((seekCycles R S L ∅).staleCharge ∅ ω : ℝ) = rotationCharge R o (ω) z := by
+  classical
+  let b := ω
+  let p := R.perm b
+  let f (i : Fin A) := (p (z i).val,r i)
+  let c (i : Fin A) : ℝ := terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val)
+  have hc (i : Fin A) : Function.minimalPeriod p (z i).val - 1 = r i := by
+    have he := orbitSet_card p (z i).val
+    rw [hO,hC] at he
+    omega
+  have hs (t : ℕ) (ht : t ≤ A) :
+      (seekCycles R S ((List.ofFn f).drop t) (priorPaths o t)).succeeds ω = true ∧
+        ((seekCycles R S ((List.ofFn f).drop t) (priorPaths o t)).staleCharge (exposedPaths R b (priorPaths o t)) ω : ℝ) =
+          ((List.ofFn c).drop t).sum := by
+    induction ht using Nat.decreasingInduction with
+    | self => simp only [List.length_ofFn,le_refl,List.drop_eq_nil_of_le,List.sum_nil,seekCycles,Thorp.QueryTree.succeeds,staleCharge,Nat.cast_zero,and_self]
+    | of_succ t ht ih =>
+      have htf : t < (List.ofFn f).length := by simpa using ht
+      have htc : t < (List.ofFn c).length := by simpa using ht
+      rw [List.drop_eq_getElem_cons htf,List.drop_eq_getElem_cons htc]
+      simp only [List.getElem_ofFn,List.sum_cons,seekCycles]
+      have h := seekCycle_orbit R S (priorPaths o t) (z ⟨t,ht⟩).val
+        (fun V => seekCycles R S ((List.ofFn f).drop (t+1)) V) ω
+        (by rw [hO]; exact hS _) (by rw [hO]; exact priorPaths_disjoint o ⟨t,ht⟩)
+      dsimp only at h
+      rw [hc,hO,←priorPaths_succ o t ht] at h
+      dsimp only [f]
+      constructor
+      · exact h.1.trans ih.1
+      · rw [h.2,Nat.cast_add,ih.2]
+        exact add_comm _ _
+  have h := hs 0 (Nat.zero_le _)
+  simp only [List.drop_zero,priorPaths_zero,exposedPaths_empty,List.sum_ofFn] at h
+  dsimp only
+  exact ⟨h.1,h.2.trans (rotationCharge_eq_terminal_sum R o b z).symm⟩
+
+end QueryTree
+section Collections
+variable {A : ℕ}
+
+def ValidCycleSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ)
+    (o : α → Option (Fin A)) : Prop :=
+  (∀ i, (cycleFiber o i).card = r i + 1) ∧
+  (∀ i, cycleFiber o i ⊆ S) ∧
+  (∀ i x, o x = some i → orbitSet p x = cycleFiber o i)
+
+abbrev CycleSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
+  {o : α → Option (Fin A) // ValidCycleSlots p S r o}
+
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
+    Fintype (CycleSlots p S r) := Fintype.ofFinite _
+
+lemma CycleSlots.terminal_nonempty {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (c : CycleSlots p S r) (i : Fin A) : Nonempty {x : α // c.val x = some i} := by
+  have hn : (cycleFiber c.val i).Nonempty := Finset.card_pos.mp (by rw [c.property.1]; omega)
+  obtain ⟨x,hx⟩ := hn
+  exact ⟨x,(mem_cycleFiber _ _ _).mp hx⟩
+
+omit [DecidableEq α] in
+lemma owner_ext {o o' : α → Option (Fin A)}
+    (h : ∀ i, cycleFiber o i = cycleFiber o' i) : o = o' := by
+  funext x
+  have he (i : Fin A) : o x = some i ↔ o' x = some i := by
+    simpa only [mem_cycleFiber] using (Finset.ext_iff.mp (h i) x)
+  cases ho : o x with
+  | some i => exact ((he i).mp ho).symm
+  | none =>
+    cases ho' : o' x with
+    | none => rfl
+    | some i => have hn := (he i).mpr ho'; rw [ho] at hn; contradiction
+
+abbrev RootedSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
+  Σ c : CycleSlots p S r, Terminal c.val
+
+noncomputable def rootedStart {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (c : RootedSlots p S r) : Fin A → S := fun i =>
+  ⟨p (c.2 i).val,c.1.property.2.1 i (by
+    rw [←c.1.property.2.2 i (c.2 i).val (c.2 i).property]
+    exact (mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.apply_right (Equiv.Perm.SameCycle.refl p _)))⟩
+
+lemma rootedStart_injective {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ} :
+    Function.Injective (rootedStart (p:=p) (S:=S) (r:=r)) := by
+  rintro ⟨c,z⟩ ⟨c',z'⟩ h
+  have hz (i : Fin A) : (z i).val = (z' i).val :=
+    p.injective (congrArg Subtype.val (congrFun h i))
+  have hc : c = c' := Subtype.ext (owner_ext (fun i => by
+    rw [←c.property.2.2 i (z i).val (z i).property,
+      ←c'.property.2.2 i (z' i).val (z' i).property,hz]))
+  subst c'
+  have hzz : z = z' := funext (fun i => Subtype.ext (hz i))
+  subst z'
+  rfl
+
+lemma rooted_weight_sum_le (S : Finset (α)) (r : Fin A → ℕ) (ω : ι → Bool) :
+    (∑ c : RootedSlots (R.perm ω) S r,
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2)) ≤
+    ∑ a : Fin A → S,
+      if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+        ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+      else 0 := by
+  classical
+  let p := R.perm ω
+  let g (a : Fin A → S) : ℝ :=
+    if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+    else 0
+  have he (c : RootedSlots p S r) :
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2) =
+      g (rootedStart c) := by
+    have hs := QueryTree.seekCycles_owned R S r c.1.val c.2 ω c.1.property.2.1
+      (fun i => c.1.property.2.2 i (c.2 i).val (c.2 i).property) c.1.property.1
+    dsimp only at hs
+    unfold g
+    rw [ite_eq_left (show (QueryTree.seekCycles R S (List.ofFn (fun i => ((rootedStart c i).val,r i))) ∅).succeeds ω = true from hs.1)]
+    rw [← hs.2,neg_mul,Real.exp_neg,mul_comm (Real.log 2),Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
+    rfl
+  simp_rw [he]
+  exact sum_comp_injective_le rootedStart rootedStart_injective g (fun a => by dsimp [g]; split_ifs <;> positivity)
+
+lemma collection_rotation_lower (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r)
+    (ω : ι → Bool) (c : CycleSlots (R.perm ω) S r) :
+    collectionWeight d r ≤ ∑ z : Terminal c.val,
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z) := by
+  classical
+  let (i : Fin A) : Nonempty {x : α // c.val x = some i} := c.terminal_nonempty i
+  have hm : Monotone (fun i => (cycleFiber c.val i).card) := by
+    intro i j hij
+    dsimp only
+    rw [c.property.1,c.property.1]
+    exact Nat.add_le_add_right (hr hij) 1
+  have hb := rotation_exp_bound R hm (ω)
+  have hcard : (Fintype.card (Terminal c.val) : ℝ) = ∏ i : Fin A, (r i+1 : ℝ) := by
+    rw [Fintype.card_pi,Nat.cast_prod]
+    apply Finset.prod_congr rfl
+    intro i _
+    rw [card_cycleFiber,c.property.1,Nat.cast_add,Nat.cast_one]
+  have hp : (0:ℝ) < Fintype.card (Terminal c.val) := Nat.cast_pos.mpr Fintype.card_pos
+  have hl : rotationLower d r ≤ finiteMean (fun z : Terminal c.val => Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z)) := by
+    simpa only [rotationLower,c.property.1,Nat.cast_add,Nat.cast_one] using hb
+  have hh := (le_div_iff₀ hp).mp hl
+  rw [hcard] at hh
+  unfold collectionWeight
+  rw [← Finset.mul_sum]
+  calc
+    _ = ((2:ℝ)^d)^A * (rotationLower d r * ∏ i : Fin A, (r i+1 : ℝ)) := by ring
+    _ ≤ _ := mul_le_mul_of_nonneg_left hh (by positivity)
+
+lemma cycleSlots_weighted_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : ι → Bool =>
+      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) *
+      collectionWeight d r ≤ (S.card : ℝ)^A := by
+  classical
+  let g (a : Fin A → S) (ω : ι → Bool) : ℝ :=
+    if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+    else 0
+  have hω (ω : ι → Bool) :
+      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ) * collectionWeight d r ≤
+        ∑ a : Fin A → S, g a ω := by
+    calc
+      _ = ∑ _c : CycleSlots (R.perm ω) S r, collectionWeight d r := by simp
+      _ ≤ ∑ c : CycleSlots (R.perm ω) S r,
+          ∑ z : Terminal c.val, ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z) :=
+        Finset.sum_le_sum (fun c _ => collection_rotation_lower R S r hr ω c)
+      _ = ∑ c : RootedSlots (R.perm ω) S r,
+          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2) :=
+        (Fintype.sum_sigma (fun c : RootedSlots (R.perm ω) S r =>
+          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2))).symm
+      _ ≤ _ := rooted_weight_sum_le R S r ω
+  have hb := finiteMean_mono hω
+  rw [finiteMean_mul_const,finiteMean_sum] at hb
+  calc
+    _ ≤ ∑ a : Fin A → S, finiteMean (g a) := hb
+    _ ≤ ∑ _a : Fin A → S, (1:ℝ) := by
+      apply Finset.sum_le_sum
+      intro a _
+      have hh := QueryTree.cycles_weighted_success R S (List.ofFn (fun i : Fin A => ((a i).val,r i)))
+      simpa only [List.length_ofFn] using hh
+    _ = _ := by simp
+
+lemma cycleSlots_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : ι → Bool =>
+      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
+        (S.card : ℝ)^A / collectionWeight d r := by
+  exact (le_div_iff₀ (collectionWeight_pos d r)).mpr (cycleSlots_weighted_moment R S r hr)
+
+lemma butterfly_cycleSlots_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : ι → Bool =>
+      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
+        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
+  have h := cycleSlots_moment R S r hr
+  rw [collectionWeight_eq_sqrt,div_mul_eq_div_mul_one_div] at h
+  rw [div_pow]
+  simpa only [one_div,Finset.prod_inv_distrib] using h
+
+end Collections
+
+section CycleCounting
+variable {A : ℕ}
+
+abbrev SelectedCycle (p : Equiv.Perm (α)) (S : Finset (α)) :=
+  {C : Finset (α) // (∃ x, C = orbitSet p x) ∧ C ⊆ S}
+
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) :
+    Fintype (SelectedCycle p S) := Fintype.ofFinite _
+
+lemma SelectedCycle.nonempty {p : Equiv.Perm (α)} {S : Finset (α)}
+    (C : SelectedCycle p S) : C.val.Nonempty := by
+  obtain ⟨x,hx⟩ := C.property.1
+  rw [hx]
+  exact orbitSet_nonempty p x
+
+lemma SelectedCycle.orbit_eq {p : Equiv.Perm (α)} {S : Finset (α)}
+    (C : SelectedCycle p S) {x : α} (hx : x ∈ C.val) : orbitSet p x = C.val := by
+  obtain ⟨y,hy⟩ := C.property.1
+  rw [hy] at hx ⊢
+  exact (orbitSet_eq p ((mem_orbitSet _ _ _).mp hx)).symm
+
+lemma SelectedCycle.eq_of_mem {p : Equiv.Perm (α)} {S : Finset (α)}
+    (C D : SelectedCycle p S) {x : α} (hC : x ∈ C.val) (hD : x ∈ D.val) : C = D := by
+  apply Subtype.ext
+  exact (C.orbit_eq hC).symm.trans (D.orbit_eq hD)
+
+abbrev SlotAssignment (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
+  {f : Fin A → SelectedCycle p S // Function.Injective f ∧ ∀ i, (f i).val.card = r i+1}
+
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
+    Fintype (SlotAssignment p S r) := Fintype.ofFinite _
+
+noncomputable def slotsToAssignment {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (c : CycleSlots p S r) : SlotAssignment p S r := by
+  classical
+  let f (i : Fin A) : SelectedCycle p S := ⟨cycleFiber c.val i,by
+    obtain ⟨x,hx⟩ := c.terminal_nonempty i
+    exact ⟨⟨x,(c.property.2.2 i x hx).symm⟩,c.property.2.1 i⟩⟩
+  refine ⟨f,?_,fun i => c.property.1 i⟩
+  intro i j hij
+  have he : cycleFiber c.val i = cycleFiber c.val j := congrArg Subtype.val hij
+  obtain ⟨x,hx⟩ := c.terminal_nonempty i
+  have hxi : x ∈ cycleFiber c.val i := (mem_cycleFiber _ _ _).mpr hx
+  rw [he] at hxi
+  exact Option.some.inj (hx.symm.trans ((mem_cycleFiber _ _ _).mp hxi))
+
+noncomputable def assignmentOwner {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) (x : α) : Option (Fin A) := by
+  classical
+  exact if h : ∃ i, x ∈ (f.val i).val then some (Classical.choose h) else none
+
+lemma assignmentOwner_eq_some {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) (x : α) (i : Fin A) :
+    assignmentOwner f x = some i ↔ x ∈ (f.val i).val := by
+  classical
+  unfold assignmentOwner
+  split_ifs with h
+  · constructor
+    · intro hi
+      have hx := Classical.choose_spec h
+      rw [Option.some.inj hi] at hx
+      exact hx
+    · intro hx
+      have he := (f.val (Classical.choose h)).eq_of_mem (f.val i) (Classical.choose_spec h) hx
+      exact congrArg some (f.property.1 he)
+  · constructor
+    · intro hi; contradiction
+    · intro hx; exact (h ⟨i,hx⟩).elim
+
+lemma assignmentOwner_fiber {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) (i : Fin A) : cycleFiber (assignmentOwner f) i = (f.val i).val := by
+  ext x
+  simp only [mem_cycleFiber,assignmentOwner_eq_some]
+
+noncomputable def assignmentToSlots {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) : CycleSlots p S r :=
+  ⟨assignmentOwner f,by
+    refine ⟨?_,?_,?_⟩
+    · intro i
+      rw [assignmentOwner_fiber]
+      exact f.property.2 i
+    · intro i
+      rw [assignmentOwner_fiber]
+      exact (f.val i).property.2
+    · intro i x hx
+      rw [assignmentOwner_fiber]
+      exact (f.val i).orbit_eq ((assignmentOwner_eq_some f x i).mp hx)⟩
+
+noncomputable def cycleSlotsEquivAssignments (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
+    CycleSlots p S r ≃ SlotAssignment p S r where
+  toFun := slotsToAssignment
+  invFun := assignmentToSlots
+  left_inv c := by
+    apply Subtype.ext
+    apply owner_ext
+    intro i
+    rw [assignmentToSlots,assignmentOwner_fiber]
+    rfl
+  right_inv f := by
+    apply Subtype.ext
+    funext i
+    apply Subtype.ext
+    exact assignmentOwner_fiber f i
+
+end CycleCounting
+
+section GroupedCounting
+variable {A : ℕ}
+
+abbrev LengthClass (r : Fin A → ℕ) := {j : ℕ // j ∈ Finset.univ.image (fun i => r i + 1)}
+
+def slotClass (r : Fin A → ℕ) (i : Fin A) : LengthClass r :=
+  ⟨r i+1, Finset.mem_image.mpr ⟨i, Finset.mem_univ _, rfl⟩⟩
+
+abbrev SlotsOfLength (r : Fin A → ℕ) (j : LengthClass r) := {i : Fin A // slotClass r i = j}
+abbrev CyclesOfLength (p : Equiv.Perm (α)) (S : Finset (α)) (j : ℕ) :=
+  {C : SelectedCycle p S // C.val.card = j}
+
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (j : ℕ) :
+    Fintype (CyclesOfLength p S j) := Fintype.ofFinite _
+
+abbrev GroupedAssignments (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
+  ∀ j : LengthClass r, SlotsOfLength r j ↪ CyclesOfLength p S j.val
+
+noncomputable def assignmentToGroups {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) : GroupedAssignments p S r := fun _j =>
+  { toFun := fun i => ⟨f.val i.val, (f.property.2 i.val).trans (congrArg Subtype.val i.property)⟩
+    inj' := fun _i _k h => Subtype.ext (f.property.1 (congrArg Subtype.val h)) }
+
+noncomputable def groupsEmbedding {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (F : GroupedAssignments p S r) : Fin A ↪ SelectedCycle p S :=
+  ((Equiv.sigmaFiberEquiv (slotClass r)).symm.toEmbedding.trans
+    (Function.Embedding.sigmaMap (Function.Embedding.refl _) F)).trans
+    ((Function.Embedding.sigmaMap (Function.Embedding.subtype _) (fun _ => Function.Embedding.refl _)).trans
+      (Equiv.sigmaFiberEquiv (fun C : SelectedCycle p S => C.val.card)).toEmbedding)
+
+noncomputable def groupsToAssignment {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (F : GroupedAssignments p S r) : SlotAssignment p S r :=
+  ⟨groupsEmbedding F, (groupsEmbedding F).injective,
+    fun i => (F (slotClass r i) ⟨i,rfl⟩).property⟩
+
+noncomputable def assignmentEquivGroups (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
+    SlotAssignment p S r ≃ GroupedAssignments p S r where
+  toFun := assignmentToGroups
+  invFun := groupsToAssignment
+  left_inv f := by
+    apply Subtype.ext
+    rfl
+  right_inv F := by
+    funext j
+    apply Function.Embedding.ext
+    intro i
+    rcases i with ⟨i,hi⟩
+    subst j
+    rfl
+
+lemma card_cycleSlots_factorial (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
+    Fintype.card (CycleSlots p S r) =
+      ∏ j : LengthClass r, (Fintype.card (CyclesOfLength p S j.val)).descFactorial
+        (Fintype.card (SlotsOfLength r j)) := by
+  classical
+  rw [Fintype.card_congr (cycleSlotsEquivAssignments p S r),
+    Fintype.card_congr (assignmentEquivGroups p S r), Fintype.card_pi]
+  simp only [Fintype.card_embedding_eq]
+
+lemma butterfly_factorial_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : ι → Bool =>
+       (∏ j : LengthClass r,
+         (Fintype.card (CyclesOfLength (R.perm ω) S j.val)).descFactorial
+           (Fintype.card (SlotsOfLength r j)) : ℕ)) ≤
+      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+        ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
+  simpa only [←card_cycleSlots_factorial] using butterfly_cycleSlots_moment R S r hr
+
+end GroupedCounting
+
+section AllLengths
+variable {A : ℕ}
+
+noncomputable def assignmentRelabel (p : Equiv.Perm (α)) (S : Finset (α))
+    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
+    SlotAssignment p S r ≃ SlotAssignment p S (r ∘ e) where
+  toFun f := ⟨f.val ∘ e, f.property.1.comp e.injective, fun i => f.property.2 (e i)⟩
+  invFun f := ⟨f.val ∘ e.symm, f.property.1.comp e.symm.injective, fun i => by
+    simpa only [Function.comp_apply,Equiv.apply_symm_apply] using f.property.2 (e.symm i)⟩
+  left_inv f := by apply Subtype.ext; funext i; simp
+  right_inv f := by apply Subtype.ext; funext i; simp
+
+lemma card_cycleSlots_relabel (p : Equiv.Perm (α)) (S : Finset (α))
+    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
+    Fintype.card (CycleSlots p S r) = Fintype.card (CycleSlots p S (r ∘ e)) := by
+  exact Fintype.card_congr ((cycleSlotsEquivAssignments p S r).trans
+    ((assignmentRelabel p S r e).trans (cycleSlotsEquivAssignments p S (r ∘ e)).symm))
+
+lemma butterfly_cycleSlots_moment_all (S : Finset (α)) (r : Fin A → ℕ) :
+    finiteMean (fun ω : ι → Bool =>
+      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
+        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
+  have h := butterfly_cycleSlots_moment R S (r ∘ Tuple.sort r) (Tuple.monotone_sort r)
+  simp_rw [←card_cycleSlots_relabel] at h
+  have hp := Equiv.prod_comp (Tuple.sort r) (fun i : Fin A => (1 / Real.sqrt (r i+1) : ℝ))
+  simpa only [Function.comp_apply,hp] using h
+
+abbrev ShortCycle (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :=
+  {C : SelectedCycle p S // C.val.card ≤ J}
+
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    Fintype (ShortCycle p S J) := Fintype.ofFinite _
+
+noncomputable def shortCycleLength {p : Equiv.Perm (α)} {S : Finset (α)} {J : ℕ}
+    (C : ShortCycle p S J) : Fin J :=
+  ⟨C.val.val.card-1, by have := C.val.nonempty.card_pos; have := C.property; omega⟩
+
+lemma shortCycleLength_add_one {p : Equiv.Perm (α)} {S : Finset (α)} {J : ℕ}
+    (C : ShortCycle p S J) : (shortCycleLength C : ℕ)+1 = C.val.val.card := by
+  have := C.val.nonempty.card_pos
+  simp only [shortCycleLength]
+  omega
+
+noncomputable def shortEmbeddingEquiv (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    (Fin A ↪ ShortCycle p S J) ≃
+      Σ r : Fin A → Fin J, SlotAssignment p S (fun i => (r i : ℕ)) where
+  toFun f := ⟨fun i => shortCycleLength (f i),
+    ⟨fun i => (f i).val, fun i j h => f.injective (Subtype.ext h),
+      fun i => (shortCycleLength_add_one (f i)).symm⟩⟩
+  invFun c :=
+    { toFun := fun i => ⟨c.2.val i, by rw [c.2.property.2]; exact (c.1 i).isLt⟩
+      inj' := fun _i _j h => c.2.property.1 (congrArg Subtype.val h) }
+  left_inv f := by
+    apply Function.Embedding.ext
+    intro i
+    rfl
+  right_inv c := by
+    rcases c with ⟨r,f⟩
+    have he : (fun i => shortCycleLength (⟨f.val i, by rw [f.property.2]; exact (r i).isLt⟩ : ShortCycle p S J)) = r := by
+      funext i
+      apply Fin.ext
+      simp only [shortCycleLength,f.property.2,Nat.add_sub_cancel]
+    apply Sigma.ext he
+    apply (Subtype.heq_iff_coe_eq (by
+      intro x
+      change (Function.Injective x ∧ ∀ i, (x i).val.card = (f.val i).val.card - 1 + 1) ↔ _
+      simp only [f.property.2,Nat.add_sub_cancel])).mpr
+    rfl
+
+lemma shortCycle_factorial_eq (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    (Fintype.card (ShortCycle p S J)).descFactorial A =
+      ∑ r : Fin A → Fin J, Fintype.card (CycleSlots p S (fun i => (r i : ℕ))) := by
+  classical
+  have hc := Fintype.card_embedding_eq (α := Fin A) (β := ShortCycle p S J)
+  simp only [Fintype.card_fin] at hc
+  rw [←hc, Fintype.card_congr (shortEmbeddingEquiv p S J),Fintype.card_sigma]
+  apply Finset.sum_congr rfl
+  intro r _
+  exact (Fintype.card_congr (cycleSlotsEquivAssignments p S (fun i => (r i : ℕ)))).symm
+
+end AllLengths
+
+section CycleMGF
+
+lemma butterfly_shortCycle_moment (S : Finset (α)) (J A : ℕ) :
+    finiteMean (fun ω : ι → Bool =>
+      ((Fintype.card (ShortCycle (R.perm ω) S J)).descFactorial A : ℝ)) ≤
+      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))^A := by
+  classical
+  simp_rw [shortCycle_factorial_eq,Nat.cast_sum,finiteMean_sum]
+  calc
+    _ ≤ ∑ r : Fin A → Fin J,
+        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+          ∏ i : Fin A, (1 / Real.sqrt ((r i : ℕ)+1)) := by
+      apply Finset.sum_le_sum
+      intro r _
+      exact butterfly_cycleSlots_moment_all R S (fun i => (r i : ℕ))
+    _ = _ := by
+      rw [←Finset.mul_sum,mul_pow]
+      congr 1
+      rw [←Fintype.prod_sum (fun (_ : Fin A) (j : Fin J) => (1 / Real.sqrt ((j:ℕ)+1) : ℝ))]
+      simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fin]
+
+noncomputable def cycleMass (p : Equiv.Perm (α)) (S : Finset (α)) : ℕ :=
+  ∑ C : SelectedCycle p S, C.val.card
+
+lemma cycleMass_le (p : Equiv.Perm (α)) (S : Finset (α)) : cycleMass p S ≤ S.card := by
+  classical
+  let f : (Σ C : SelectedCycle p S, {x : α // x ∈ C.val}) → S :=
+    fun c => ⟨c.2.val,c.1.property.2 c.2.property⟩
+  have hi : Function.Injective f := by
+    intro c e h
+    have hx : c.2.val = e.2.val := congrArg Subtype.val h
+    have hC : c.1 = e.1 := c.1.eq_of_mem e.1 c.2.property (hx ▸ e.2.property)
+    rcases c with ⟨c,x⟩
+    rcases e with ⟨e,y⟩
+    dsimp only at hC
+    subst e
+    have he : x=y := Subtype.ext hx
+    cases he
+    rfl
+  have hh := Fintype.card_le_of_injective f hi
+  simpa only [Fintype.card_sigma,Fintype.card_coe,cycleMass] using hh
+
+lemma shortCycle_card_le (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    Fintype.card (ShortCycle p S J) ≤ S.card := by
+  classical
+  calc
+    _ ≤ Fintype.card (SelectedCycle p S) := Fintype.card_subtype_le _
+    _ = ∑ _ : SelectedCycle p S, 1 := by simp
+    _ ≤ cycleMass p S := Finset.sum_le_sum (fun C _ => C.nonempty.card_pos)
+    _ ≤ _ := cycleMass_le p S
+
+lemma butterfly_shortCycle_mgf (S : Finset (α)) (J : ℕ) (q : ℝ) (hq : 1 ≤ q) :
+    finiteMean (fun ω : ι → Bool =>
+      q^(Fintype.card (ShortCycle (R.perm ω) S J))) ≤
+      Real.exp ((q-1)*((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))) := by
+  apply factorial_mgf _ S.card _ q (fun ω => shortCycle_card_le _ _ _)
+  · positivity
+  · exact hq
+  · exact butterfly_shortCycle_moment R S J
+
+end CycleMGF
+
+
+abbrev LongCycle (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :=
+  {C : SelectedCycle p S // ¬ C.val.card ≤ J}
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    Fintype (LongCycle p S J) := Fintype.ofFinite _
+
+lemma selectedCycle_card_split (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    Fintype.card (SelectedCycle p S) =
+      Fintype.card (ShortCycle p S J) + Fintype.card (LongCycle p S J) := by
+  classical
+  rw [←Fintype.card_congr (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J)),Fintype.card_sum]
+
+lemma longCycle_card_bound (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    J * Fintype.card (LongCycle p S J) ≤ S.card := by
+  classical
+  have hs := Equiv.sum_comp (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J))
+    (fun C : SelectedCycle p S => C.val.card)
+  rw [Fintype.sum_sum_type] at hs
+  have h0 : J * Fintype.card (LongCycle p S J) ≤
+      ∑ C : LongCycle p S J, C.val.val.card := by
+    calc
+      _ = ∑ _C : LongCycle p S J, J := by simp [mul_comm]
+      _ ≤ _ := Finset.sum_le_sum (fun C _ => Nat.le_of_lt (Nat.lt_of_not_ge C.property))
+  have hh := cycleMass_le p S
+  unfold cycleMass at hh
+  rw [←hs] at hh
+  exact h0.trans ((Nat.le_add_left _ _).trans hh)
+
+lemma longCycle_pow_bound (p : Equiv.Perm (α)) (S : Finset (α))
+    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    q^(Fintype.card (LongCycle p S J)) ≤ Real.exp ((S.card : ℝ)/J * Real.log q) := by
+  have hq0 : 0 < q := lt_of_lt_of_le zero_lt_one hq
+  have hc : (Fintype.card (LongCycle p S J) : ℝ) ≤ (S.card : ℝ)/J := by
+    apply (le_div_iff₀ (by exact_mod_cast hJ)).mpr
+    have hh := longCycle_card_bound p S J
+    exact_mod_cast (by simpa only [mul_comm] using hh)
+  calc
+    _ = Real.exp ((Fintype.card (LongCycle p S J) : ℝ)*Real.log q) := by
+      rw [Real.exp_nat_mul,Real.exp_log hq0]
+    _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right hc (Real.log_nonneg hq))
+
+lemma butterfly_cycle_log_mgf (S : Finset (α)) (J : ℕ) (hJ : 0 < J)
+    (q : ℝ) (hq : 1 ≤ q) :
+    Real.log (finiteMean (fun ω : ι → Bool =>
+      q^(Fintype.card (SelectedCycle (R.perm ω) S)))) ≤
+      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
+  let lam : ℝ := (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1)
+  let c : ℝ := (S.card : ℝ)/J * Real.log q
+  have hnum : (q-1)*lam ≤ 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
+    have hsum := sum_inv_sqrt J
+    have hp : 0 ≤ (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) := by positivity
+    have h0 : 0 ≤ 2*Real.sqrt J := by positivity
+    have hh := mul_le_mul_of_nonneg_left hsum hp
+    have hq1 : 0 ≤ q-1 := sub_nonneg.mpr hq
+    calc
+      _ ≤ (q-1)*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
+        mul_le_mul_of_nonneg_left hh hq1
+      _ ≤ q*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
+        mul_le_mul_of_nonneg_right (by linarith) (mul_nonneg hp h0)
+      _ = _ := by ring
+  have hm : finiteMean (fun ω : ι → Bool =>
+      q^(Fintype.card (SelectedCycle (R.perm ω) S))) ≤
+      Real.exp (c+(q-1)*lam) := by
+    calc
+      _ ≤ finiteMean (fun ω : ι → Bool =>
+          q^(Fintype.card (ShortCycle (R.perm ω) S J))*Real.exp c) := by
+        apply finiteMean_mono
+        intro ω
+        rw [selectedCycle_card_split _ _ J,pow_add]
+        exact mul_le_mul_of_nonneg_left (longCycle_pow_bound _ S J hJ q hq) (by positivity)
+      _ = finiteMean (fun ω : ι → Bool =>
+          q^(Fintype.card (ShortCycle (R.perm ω) S J))) * Real.exp c :=
+        finiteMean_mul_const _ _
+      _ ≤ Real.exp ((q-1)*lam) * Real.exp c :=
+        mul_le_mul_of_nonneg_right (butterfly_shortCycle_mgf R S J q hq) (Real.exp_nonneg _)
+      _ = _ := by rw [←Real.exp_add,add_comm]
+  have hpos : 0 < finiteMean (fun ω : ι → Bool =>
+      q^(Fintype.card (SelectedCycle (R.perm ω) S))) := by
+    apply finiteMean_pos
+    intro ω
+    positivity
+  have hlog := Real.log_le_log hpos hm
+  rw [Real.log_exp] at hlog
+  dsimp [c] at hlog
+  linarith
+
+
+end RoutingNetwork
+
+variable {α B : Type*} [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] {d : ℕ}
+
+def parallelPerm (ω : B × SwitchIndex d → Bool) : Equiv.Perm (B × Card d) where
+  toFun x := (x.1,butterflyPerm d (decodeButterfly d (fun i => ω (x.1,i))) x.2)
+  invFun y := (y.1,(butterflyPerm d (decodeButterfly d (fun i => ω (y.1,i)))).symm y.2)
+  left_inv x := by simp
+  right_inv y := by simp
+
+def relabeledPerm (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (ω : B × SwitchIndex d → Bool) : Equiv.Perm α :=
+  (pre.trans (parallelPerm ω)).trans post
+
+noncomputable def parallelPath (pre : α ≃ B × Card d)
+    (ω : B × SwitchIndex d → Bool) (x : α) : Finset (B × SwitchIndex d) :=
+  (routeDomain d (pre x).2
+    (butterflyPerm d (decodeButterfly d (fun i => ω ((pre x).1,i))) (pre x).2)).image
+      (fun i => ((pre x).1,i))
+
+def parallelUsers (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
+    (i : B × SwitchIndex d) : α × α :=
+  let u := switchUsers d (decodeButterfly d (fun j => ω (i.1,j))) i.2
+  (pre.symm (i.1,u.1),pre.symm (i.1,u.2))
+
+omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
+lemma parallelUsers_ne (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
+    (i : B × SwitchIndex d) : (parallelUsers pre ω i).1 ≠ (parallelUsers pre ω i).2 := by
+  intro h
+  exact switchUsers_ne d _ _ (congrArg Prod.snd (pre.symm.injective h))
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma mem_parallelPath (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
+    (x : α) (i : B × SwitchIndex d) :
+    i ∈ parallelPath pre ω x ↔ (parallelUsers pre ω i).1 = x ∨ (parallelUsers pre ω i).2 = x := by
+  classical
+  obtain ⟨b,j⟩ := i
+  simp only [parallelPath,Finset.mem_image,Prod.mk.injEq]
+  constructor
+  · rintro ⟨k,hk,hb,rfl⟩
+    subst b
+    rcases (mem_routeDomain_iff_user d _ _ _).mp hk with h | h
+    · left
+      apply pre.injective
+      change pre (pre.symm ((pre x).1,_)) = pre x
+      rw [Equiv.apply_symm_apply]
+      exact Prod.ext rfl h
+    · right
+      apply pre.injective
+      change pre (pre.symm ((pre x).1,_)) = pre x
+      rw [Equiv.apply_symm_apply]
+      exact Prod.ext rfl h
+  · intro h
+    have hh : (b,(switchUsers d (decodeButterfly d (fun j => ω (b,j))) j).1) = pre x ∨
+        (b,(switchUsers d (decodeButterfly d (fun j => ω (b,j))) j).2) = pre x := by
+      rcases h with h | h
+      · exact Or.inl (by simpa only [parallelUsers,Equiv.apply_symm_apply] using congrArg pre h)
+      · exact Or.inr (by simpa only [parallelUsers,Equiv.apply_symm_apply] using congrArg pre h)
+    have hb : b = (pre x).1 := hh.elim (congrArg Prod.fst) (congrArg Prod.fst)
+    subst b
+    refine ⟨j,?_,rfl,rfl⟩
+    apply (mem_routeDomain_iff_user d _ _ _).mpr
+    exact hh.elim (fun h => Or.inl (congrArg Prod.snd h)) (fun h => Or.inr (congrArg Prod.snd h))
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma card_parallelPath (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool) (x : α) :
+    (parallelPath pre ω x).card = d := by
+  classical
+  rw [parallelPath,Finset.card_image_of_injective _ (fun _ _ h => Prod.mk.inj h |>.2),card_routeDomain]
+
+noncomputable def blockMarks (pre : α ≃ B × Card d) (S : Finset α) (b : B) : Finset (Card d) :=
+  Finset.univ.filter (fun x => pre.symm (b,x) ∈ S)
+
+omit [Fintype α] [Fintype B] [DecidableEq B] in
+@[simp] lemma mem_blockMarks (pre : α ≃ B × Card d) (S : Finset α) (b : B) (x : Card d) :
+    x ∈ blockMarks pre S b ↔ pre.symm (b,x) ∈ S := by
+  classical
+  simp [blockMarks]
+
+omit [DecidableEq B] in
+lemma sum_card_blockMarks (pre : α ≃ B × Card d) (S : Finset α) :
+    ∑ b : B, (blockMarks pre S b).card = S.card := by
+  classical
+  calc
+    _ = ∑ b : B, ∑ x : Card d, if pre.symm (b,x) ∈ S then 1 else 0 := by
+      simp only [blockMarks,Finset.card_filter]
+    _ = ∑ u : B × Card d, if pre.symm u ∈ S then 1 else 0 := (Fintype.sum_prod_type (fun u : B × Card d => if pre.symm u ∈ S then (1:ℕ) else 0)).symm
+    _ = ∑ x : α, if x ∈ S then 1 else 0 := Equiv.sum_comp pre.symm (fun x : α => if x ∈ S then (1:ℕ) else 0)
+    _ = _ := by simp
+
+omit [DecidableEq B] in
+lemma card_blockMarks_le (pre : α ≃ B × Card d) (S : Finset α) (b : B) :
+    (blockMarks pre S b).card ≤ S.card := by
+  rw [←sum_card_blockMarks pre S]
+  exact Finset.single_le_sum (f := fun b : B => (blockMarks pre S b).card) (fun _ _ => Nat.zero_le _) (Finset.mem_univ b)
+
+end Thorp
+
+end ThorpNine.Harmonic
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Harmonic/RotationMeeting.lean b/lean/OAI/Probability/ThorpRouting/Harmonic/RotationMeeting.lean
index abbae9a3da050019c607c0c3aa3d5fd2f1435fcd..d2b484610273d17dda5d260f05fbbb4862669ad8 100644
--- a/lean/OAI/Probability/ThorpRouting/Harmonic/RotationMeeting.lean
+++ b/lean/OAI/Probability/ThorpRouting/Harmonic/RotationMeeting.lean
@@ -1,881 +1,881 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Harmonic.ExposedCoins
-
-namespace OAI
-
-namespace ThorpNine.Harmonic
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-section Rotation
-variable {d A : ℕ}
-
-noncomputable def rotationMeeting (o : Card d → Option (Fin A)) (z : Terminal o)
-    (u v : Card d) : ℝ :=
-  match o u, o v with
-  | some i, some j => if i < j then (if (z j).val = v then 1 else 0)
-      else if j < i then (if (z i).val = u then 1 else 0)
-      else (if (z i).val = u then 1 else 0) + (if (z i).val = v then 1 else 0)
-  | _, _ => 0
-
-lemma card_cycleFiber (o : Card d → Option (Fin A)) (i : Fin A) :
-    Fintype.card {x : Card d // o x = some i} = (cycleFiber o i).card := by
-  classical
-  simp [cycleFiber,Fintype.card_subtype]
-
-lemma terminal_mean {o : Card d → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
-    (i : Fin A) (x : Card d) (hx : o x = some i) :
-    finiteMean (fun z : Terminal o => if (z i).val = x then (1:ℝ) else 0) = slotWeight o i := by
-  classical
-  have hev := finiteMean_pi_eval (A := fun i : Fin A => {x : Card d // o x = some i})
-    i (fun z => if z.val = x then (1:ℝ) else 0)
-  rw [hev]
-  have he : (fun z : {x : Card d // o x = some i} => if z.val = x then (1:ℝ) else 0) =
-      (fun z => if z = ⟨x,hx⟩ then 1 else 0) := by
-    funext z
-    simp only [Subtype.ext_iff]
-  rw [he,finiteMean_eq_indicator,card_cycleFiber,slotWeight]
-
-lemma slotWeight_antitone (o : Card d → Option (Fin A))
-    (hn : ∀ i, (cycleFiber o i).Nonempty)
-    (hm : Monotone (fun i => (cycleFiber o i).card)) : Antitone (slotWeight o) := by
-  intro i j hij
-  unfold slotWeight
-  apply one_div_le_one_div_of_le
-  · exact_mod_cast Finset.card_pos.mpr (hn i)
-  · exact_mod_cast hm hij
-
-lemma rotationMeeting_mean {o : Card d → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
-    (hm : Monotone (fun i => (cycleFiber o i).card)) (u v : Card d) :
-    finiteMean (fun z : Terminal o => rotationMeeting o z u v) =
-      min (pathWeight o u) (pathWeight o v) + withinWeight o u v := by
-  classical
-  have hn : ∀ i, (cycleFiber o i).Nonempty := by
-    intro i
-    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : Card d // o x = some i})
-    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
-  have hw := slotWeight_antitone o hn hm
-  cases hu : o u with
-  | none =>
-    cases hv : o v with
-    | none => simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean]
-    | some j =>
-      have hj : 0 ≤ slotWeight o j := by unfold slotWeight; positivity
-      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_left hj]
-  | some i =>
-    cases hv : o v with
-    | none =>
-      have hi : 0 ≤ slotWeight o i := by unfold slotWeight; positivity
-      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_right hi]
-    | some j =>
-      simp only [rotationMeeting,pathWeight,withinWeight,hu,hv]
-      rcases lt_trichotomy i j with hij | rfl | hji
-      · simp only [hij,↓reduceIte,ne_of_lt hij]
-        rw [terminal_mean j v hv,min_eq_right (hw (le_of_lt hij)),add_zero]
-      · simp only [lt_self_iff_false,↓reduceIte,min_self]
-        rw [finiteMean_add,terminal_mean i u hu,terminal_mean i v hv]
-      · simp only [not_lt_of_ge (le_of_lt hji),hji,↓reduceIte,ne_of_gt hji]
-        rw [terminal_mean i u hu,min_eq_left (hw (le_of_lt hji)),add_zero]
-
-lemma pathWeight_sum (o : Card d → Option (Fin A)) (hn : ∀ i, (cycleFiber o i).Nonempty) :
-    ∑ x : Card d, pathWeight o x = (A : ℝ) := by
-  classical
-  have he (x : Card d) : pathWeight o x =
-      ∑ i : Fin A, if o x = some i then slotWeight o i else 0 := by
-    cases hx : o x with
-    | none => simp [pathWeight,hx]
-    | some j => simp [pathWeight,hx]
-  simp only [he]
-  rw [Finset.sum_comm]
-  have hf (i : Fin A) : (∑ x : Card d, if o x = some i then slotWeight o i else 0) = 1 := by
-    rw [← Finset.sum_filter]
-    change (∑ _x ∈ cycleFiber o i, slotWeight o i) = 1
-    simp only [Finset.sum_const,nsmul_eq_mul,slotWeight]
-    exact mul_one_div_cancel (by exact_mod_cast (Finset.card_pos.mpr (hn i)).ne')
-  simp only [hf,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,mul_one]
-
-lemma withinWeight_sum_bound (o : Card d → Option (Fin A))
-    (hn : ∀ i, (cycleFiber o i).Nonempty) (b : Butterfly d) :
-    (∑ s : SwitchIndex d, withinWeight o (switchUsers d b s).1 (switchUsers d b s).2) ≤
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
-  classical
-  have he (u v : Card d) : withinWeight o u v =
-      ∑ i : Fin A, (if u ∈ cycleFiber o i ∧ v ∈ cycleFiber o i then (1:ℝ) else 0) * slotWeight o i := by
-    simp only [mem_cycleFiber]
-    cases hu : o u <;> cases hv : o v <;> simp [withinWeight,hu,hv,ite_and,eq_comm]
-  simp only [he]
-  rw [Finset.sum_comm]
-  apply Finset.sum_le_sum
-  intro i _
-  rw [← Finset.sum_mul]
-  have hi := marked_switch_entropy_bound d b (cycleFiber o i)
-  have hpos : (0:ℝ) < (cycleFiber o i).card := by exact_mod_cast Finset.card_pos.mpr (hn i)
-  have hmul := mul_le_mul_of_nonneg_right hi (show 0 ≤ slotWeight o i by unfold slotWeight; positivity)
-  calc
-    _ ≤ ((cycleFiber o i).card : ℝ) * Real.log (cycleFiber o i).card /
-        (2 * Real.log 2) * slotWeight o i := hmul
-    _ = _ := by unfold slotWeight; field_simp
-
-noncomputable def rotationCharge (o : Card d → Option (Fin A)) (b : Butterfly d)
-    (z : Terminal o) : ℝ :=
-  ∑ s : SwitchIndex d, rotationMeeting o z (switchUsers d b s).1 (switchUsers d b s).2
-
-lemma rotationCharge_mean_bound {o : Card d → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
-    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : Butterfly d) :
-    finiteMean (rotationCharge o b) ≤ (A:ℝ) * d / 2 +
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
-  classical
-  have hn : ∀ i, (cycleFiber o i).Nonempty := by
-    intro i
-    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : Card d // o x = some i})
-    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
-  unfold rotationCharge
-  rw [finiteMean_sum]
-  simp only [rotationMeeting_mean hm,Finset.sum_add_distrib]
-  have h₁ := weighted_switch_min_bound d b (pathWeight o)
-  rw [pathWeight_sum o hn] at h₁
-  have h₂ := withinWeight_sum_bound o hn b
-  linarith
-
-lemma rotation_exp_bound {o : Card d → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
-    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : Butterfly d) :
-    Real.exp (-Real.log 2 * ((A:ℝ)*d/2 +
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2))) ≤
-      finiteMean (fun z : Terminal o => Real.exp (-Real.log 2 * rotationCharge o b z)) := by
-  have hb := rotationCharge_mean_bound hm b
-  have hlog : 0 ≤ Real.log 2 := Real.log_nonneg (by norm_num)
-  have hm' : -Real.log 2 * ((A:ℝ)*d/2 +
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2)) ≤
-      finiteMean (fun z : Terminal o => -Real.log 2 * rotationCharge o b z) := by
-    have he := finiteMean_mul_const (rotationCharge o b) (-Real.log 2)
-    simp only [mul_comm] at he
-    rw [he]
-    exact mul_le_mul_of_nonpos_left hb (neg_nonpos.mpr hlog)
-  exact (Real.exp_le_exp.mpr hm').trans (exp_finiteMean_le _)
-
-end Rotation
-
-section OrbitCore
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-noncomputable def orbitSet (p : Equiv.Perm α) (x : α) : Finset α :=
-  Finset.univ.filter (p.SameCycle x)
-
-@[simp] lemma mem_orbitSet (p : Equiv.Perm α) (x y : α) :
-    y ∈ orbitSet p x ↔ p.SameCycle x y := by
-  classical
-  simp [orbitSet]
-
-lemma orbitSet_nonempty (p : Equiv.Perm α) (x : α) : (orbitSet p x).Nonempty :=
-  ⟨x,(mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.refl _ _)⟩
-
-lemma orbitSet_eq (p : Equiv.Perm α) {x y : α} (h : p.SameCycle x y) :
-    orbitSet p x = orbitSet p y := by
-  ext z
-  simp only [mem_orbitSet]
-  exact ⟨fun hz => h.symm.trans hz,fun hz => h.trans hz⟩
-
-@[simp] lemma orbitSet_apply (p : Equiv.Perm α) (x : α) : orbitSet p (p x) = orbitSet p x := by
-  ext y
-  simp only [mem_orbitSet,Equiv.Perm.sameCycle_apply_left]
-
-lemma orbitSet_eq_or_disjoint (p : Equiv.Perm α) (x y : α) :
-    orbitSet p x = orbitSet p y ∨ Disjoint (orbitSet p x) (orbitSet p y) := by
-  classical
-  by_cases h : p.SameCycle x y
-  · exact Or.inl (orbitSet_eq p h)
-  · right
-    apply Finset.disjoint_left.mpr
-    intro z hz hz'
-    exact h (((mem_orbitSet _ _ _).mp hz).trans ((mem_orbitSet _ _ _).mp hz').symm)
-
-omit [DecidableEq α] in
-lemma perm_minimalPeriod_pos (p : Equiv.Perm α) (x : α) :
-    0 < Function.minimalPeriod p x :=
-  Function.minimalPeriod_pos_of_mem_periodicPts (p.injective.mem_periodicPts x)
-
-lemma orbitSet_card (p : Equiv.Perm α) (x : α) :
-    (orbitSet p x).card = Function.minimalPeriod p x := by
-  classical
-  let f : Fin (Function.minimalPeriod p x) → {y // p.SameCycle x y} := fun n =>
-    ⟨p^[n.val] x,by rw [Equiv.Perm.iterate_eq_pow]; exact ⟨(n.val:ℤ),by simp⟩⟩
-  have hf : Function.Bijective f := by
-    constructor
-    · intro i j hij
-      apply Fin.ext
-      exact Function.iterate_injOn_Iio_minimalPeriod i.isLt j.isLt (congrArg Subtype.val hij)
-    · intro y
-      obtain ⟨n,hn⟩ := y.property.exists_nat_pow_eq
-      refine ⟨⟨n % Function.minimalPeriod p x, Nat.mod_lt _ (perm_minimalPeriod_pos p x)⟩,?_⟩
-      apply Subtype.ext
-      change p^[n % Function.minimalPeriod p x] x = y.val
-      rw [Function.iterate_mod_minimalPeriod_eq,Equiv.Perm.iterate_eq_pow,hn]
-  have hc := Fintype.card_congr (Equiv.ofBijective f hf)
-  simpa [orbitSet,Fintype.card_subtype] using hc.symm
-
-def walkSet (p : Equiv.Perm α) (r : ℕ) (x : α) : Finset α :=
-  (Finset.range r).image (fun n => p^[n] x)
-
-omit [Fintype α] in
-@[simp] lemma mem_walkSet (p : Equiv.Perm α) (r : ℕ) (x y : α) :
-    y ∈ walkSet p r x ↔ ∃ n < r, p^[n] x = y := by
-  simp [walkSet]
-
-omit [Fintype α] in
-@[simp] lemma walkSet_zero (p : Equiv.Perm α) (x : α) : walkSet p 0 x = ∅ := by
-  simp [walkSet]
-
-omit [Fintype α] in
-lemma walkSet_succ_last (p : Equiv.Perm α) (r : ℕ) (x : α) :
-    walkSet p (r+1) x = insert (p^[r] x) (walkSet p r x) := by
-  simp [walkSet,Finset.range_add_one]
-
-omit [Fintype α] in
-lemma walkSet_succ_first (p : Equiv.Perm α) (r : ℕ) (x : α) :
-    walkSet p (r+1) x = insert x (walkSet p r (p x)) := by
-  ext y
-  simp only [mem_walkSet,Finset.mem_insert]
-  constructor
-  · rintro ⟨n,hn,hny⟩
-    cases n with
-    | zero => exact Or.inl (by simpa using hny.symm)
-    | succ n => exact Or.inr ⟨n,by omega,by simpa only [Function.iterate_succ_apply] using hny⟩
-  · rintro (rfl | ⟨n,hn,hny⟩)
-    · exact ⟨0,by omega,rfl⟩
-    · exact ⟨n+1,by omega,by simpa only [Function.iterate_succ_apply] using hny⟩
-
-omit [Fintype α] in
-lemma not_mem_short_walk (p : Equiv.Perm α) (x : α) {r : ℕ}
-    (hr : r < Function.minimalPeriod p x) : x ∉ walkSet p r (p x) := by
-  rintro hx
-  obtain ⟨n,hn,hnx⟩ := (mem_walkSet _ _ _ _).mp hx
-  have hperiod : Function.IsPeriodicPt p (n+1) x := by
-    simpa only [Function.IsPeriodicPt,Function.IsFixedPt,Function.iterate_succ_apply] using hnx
-  have hp := hperiod.minimalPeriod_le (by omega : 0 < n+1)
-  omega
-
-lemma walkSet_period (p : Equiv.Perm α) (x : α) :
-    walkSet p (Function.minimalPeriod p x) x = orbitSet p x := by
-  ext y
-  simp only [mem_walkSet,mem_orbitSet]
-  constructor
-  · rintro ⟨n,_,rfl⟩
-    rw [Equiv.Perm.iterate_eq_pow]
-    exact ⟨(n:ℤ),by simp⟩
-  · intro h
-    obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
-    refine ⟨n % Function.minimalPeriod p x,Nat.mod_lt _ (perm_minimalPeriod_pos p x),?_⟩
-    rw [Function.iterate_mod_minimalPeriod_eq,Equiv.Perm.iterate_eq_pow,hn]
-
-omit [Fintype α] in
-lemma walkSet_card (p : Equiv.Perm α) (x : α) {r : ℕ}
-    (hr : r ≤ Function.minimalPeriod p x) : (walkSet p r x).card = r := by
-  rw [walkSet,Finset.card_image_of_injOn]
-  · exact Finset.card_range _
-  · intro i hi j hj hij
-    exact Function.iterate_injOn_Iio_minimalPeriod
-      (lt_of_lt_of_le (Finset.mem_range.mp hi) hr)
-      (lt_of_lt_of_le (Finset.mem_range.mp hj) hr) hij
-
-lemma walkSet_subset_orbit (p : Equiv.Perm α) (r : ℕ) (x : α) :
-    walkSet p r x ⊆ orbitSet p x := by
-  intro y hy
-  obtain ⟨n,_,rfl⟩ := (mem_walkSet _ _ _ _).mp hy
-  rw [mem_orbitSet,Equiv.Perm.iterate_eq_pow]
-  exact ⟨(n:ℤ),by simp⟩
-
-def cycleTrace (p : Equiv.Perm α) (S : Finset α) (a : α) :
-    ℕ → α → Finset α → Option (Finset α × α)
-  | 0, x, V => if x ∈ S ∧ x ∉ V ∧ p x = a then some (insert x V,x) else none
-  | r+1, x, V => if x ∈ S ∧ x ∉ V then cycleTrace p S a r (p x) (insert x V) else none
-
-lemma cycleTrace_walk (p : Equiv.Perm α) (S : Finset α) (r : ℕ) (x : α) (V : Finset α)
-    (hr : r < Function.minimalPeriod p x) (hS : walkSet p (r+1) x ⊆ S)
-    (hV : Disjoint V (walkSet p (r+1) x)) :
-    cycleTrace p S (p^[r+1] x) r x V = some (V ∪ walkSet p (r+1) x,p^[r] x) := by
-  induction r generalizing x V with
-  | zero =>
-    have hxS : x ∈ S := hS (by simp [walkSet_succ_first])
-    have hxV : x ∉ V := fun h => Finset.disjoint_left.mp hV h (by simp [walkSet_succ_first])
-    simp [cycleTrace,hxS,hxV,walkSet_succ_first]
-  | succ r ih =>
-    have hxS : x ∈ S := hS (by rw [walkSet_succ_first]; exact Finset.mem_insert_self _ _)
-    have hxV : x ∉ V := fun h => Finset.disjoint_left.mp hV h
-      (by rw [walkSet_succ_first]; exact Finset.mem_insert_self _ _)
-    have hr' : r < Function.minimalPeriod p (p x) := by
-      rw [Function.minimalPeriod_apply (p.injective.mem_periodicPts x)]
-      omega
-    have hS' : walkSet p (r+1) (p x) ⊆ S := by
-      intro y hy
-      apply hS
-      rw [walkSet_succ_first]
-      exact Finset.mem_insert_of_mem hy
-    have hV' : Disjoint (insert x V) (walkSet p (r+1) (p x)) := by
-      apply Finset.disjoint_left.mpr
-      intro y hy hy'
-      rcases Finset.mem_insert.mp hy with hy | hy
-      · subst y
-        exact not_mem_short_walk p x hr hy'
-      · apply Finset.disjoint_left.mp hV hy
-        rw [walkSet_succ_first]
-        exact Finset.mem_insert_of_mem hy'
-    rw [cycleTrace,ite_eq_left ⟨hxS,hxV⟩,Function.iterate_succ_apply p (r+1) x,
-      ih _ _ hr' hS' hV',walkSet_succ_first p (r+1) x,Function.iterate_succ_apply]
-    simp only [Finset.insert_union,Finset.union_insert]
-
-lemma cycleTrace_orbit (p : Equiv.Perm α) (S V : Finset α) (z : α)
-    (hS : orbitSet p z ⊆ S) (hV : Disjoint V (orbitSet p z)) :
-    cycleTrace p S (p z) (Function.minimalPeriod p z - 1) (p z) V =
-      some (V ∪ orbitSet p z,z) := by
-  have hpos := perm_minimalPeriod_pos p z
-  have he : Function.minimalPeriod p (p z) = Function.minimalPeriod p z :=
-    Function.minimalPeriod_apply (p.injective.mem_periodicPts z)
-  have hp : Function.minimalPeriod p z - 1 + 1 = Function.minimalPeriod p z := by omega
-  have hw : walkSet p (Function.minimalPeriod p z - 1 + 1) (p z) = orbitSet p z := by
-    rw [hp,←he,walkSet_period,orbitSet_apply]
-  have hr : Function.minimalPeriod p z - 1 < Function.minimalPeriod p (p z) := by rw [he]; omega
-  have h := cycleTrace_walk p S (Function.minimalPeriod p z-1) (p z) V hr
-    (by simpa only [hw] using hS) (by simpa only [hw] using hV)
-  have ht : p^[Function.minimalPeriod p z-1] (p z) = z := by
-    rw [← Function.iterate_succ_apply,show (Function.minimalPeriod p z-1).succ =
-      Function.minimalPeriod p z by omega,Function.iterate_minimalPeriod]
-  have ha : p^[Function.minimalPeriod p z-1+1] (p z) = p z := by
-    rw [hp,←he,Function.iterate_minimalPeriod]
-  simpa only [ha,ht,hw] using h
-
-end OrbitCore
-
-noncomputable def exposedPaths (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) :
-    Finset (SwitchIndex d) :=
-  V.biUnion (fun x => routeDomain d x (butterflyPerm d b x))
-
-@[simp] lemma exposedPaths_empty (d : ℕ) (b : Butterfly d) : exposedPaths d b ∅ = ∅ := by
-  classical
-  simp [exposedPaths]
-
-lemma exposedPaths_insert (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) (x : Card d) :
-    exposedPaths d b (insert x V) = exposedPaths d b V ∪ routeDomain d x (butterflyPerm d b x) := by
-  classical
-  simp [exposedPaths,Finset.union_comm]
-
-noncomputable def terminalCharge (d : ℕ) (b : Butterfly d) (z : Card d) (V : Finset (Card d)) : ℕ :=
-  (routeDomain d z (butterflyPerm d b z) ∩ exposedPaths d b V).card
-
-namespace QueryTree
-
-lemma followPath_stale_charge {ι : Type*} [DecidableEq ι]
-    (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
-    (k : Card d → QueryTree ι) (R : Finset ι) (ω : ι → Bool) :
-    staleCharge (followPath d e x k) R ω =
-      staleCharge (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x))
-        (R ∪ (routeDomain d x (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).image e) ω := by
-  have h₀ := fresh_add_stale (followPath d e x k) R ω
-  have h₁ := fresh_add_stale (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x))
-    (R ∪ (routeDomain d x (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).image e) ω
-  rw [followPath_charge,followPath_raw_charge] at h₀
-  omega
-
-lemma seekCycle_semantics (d : ℕ) (S : Finset (Card d)) (a : Card d) (r : ℕ)
-    (x : Card d) (V W : Finset (Card d)) (z : Card d)
-    (k : Finset (Card d) → QueryTree (SwitchIndex d)) (ω : SwitchIndex d → Bool)
-    (h : cycleTrace (butterflyPerm d (decodeButterfly d ω)) S a r x V = some (W,z)) :
-    (seekCycle d S a r x V k).succeeds ω = (k W).succeeds ω ∧
-      (seekCycle d S a r x V k).staleCharge (exposedPaths d (decodeButterfly d ω) V) ω =
-        (k W).staleCharge (exposedPaths d (decodeButterfly d ω) W) ω +
-          terminalCharge d (decodeButterfly d ω) z (W.erase z) := by
-  classical
-  induction r generalizing x V with
-  | zero =>
-    simp only [cycleTrace] at h
-    split_ifs at h with hv
-    obtain ⟨hxS,hxV,hxa⟩ := hv
-    obtain ⟨rfl,rfl⟩ := Prod.mk.inj (Option.some.inj h)
-    simp only [seekCycle,ite_eq_left (show x ∈ S ∧ x ∉ V from ⟨hxS,hxV⟩),closePath_succeeds,ite_eq_left hxa]
-    constructor
-    · trivial
-    · rw [closePath_stale_charge _ _ _ _ _ _ hxa,exposedPaths_insert,
-        Finset.erase_insert hxV]
-      simp only [terminalCharge,hxa]
-  | succ r ih =>
-    simp only [cycleTrace] at h
-    split_ifs at h with hv
-    obtain ⟨hxS,hxV⟩ := hv
-    rw [seekCycle,ite_eq_left ⟨hxS,hxV⟩,followPath_succeeds,followPath_stale_charge]
-    simp only [Function.comp_id,Finset.image_id]
-    rw [← exposedPaths_insert]
-    exact ih _ _ h
-
-lemma seekCycle_orbit (d : ℕ) (S V : Finset (Card d)) (z : Card d)
-    (k : Finset (Card d) → QueryTree (SwitchIndex d)) (ω : SwitchIndex d → Bool)
-    (hS : orbitSet (butterflyPerm d (decodeButterfly d ω)) z ⊆ S)
-    (hV : Disjoint V (orbitSet (butterflyPerm d (decodeButterfly d ω)) z)) :
-    let p := butterflyPerm d (decodeButterfly d ω)
-    let W := V ∪ orbitSet p z
-    (seekCycle d S (p z) (Function.minimalPeriod p z-1) (p z) V k).succeeds ω = (k W).succeeds ω ∧
-      (seekCycle d S (p z) (Function.minimalPeriod p z-1) (p z) V k).staleCharge
-          (exposedPaths d (decodeButterfly d ω) V) ω =
-        (k W).staleCharge (exposedPaths d (decodeButterfly d ω) W) ω +
-          terminalCharge d (decodeButterfly d ω) z (W.erase z) := by
-  exact seekCycle_semantics _ _ _ _ _ _ _ _ _ _ (cycleTrace_orbit _ _ _ _ hS hV)
-
-end QueryTree
-
-lemma mem_exposedPaths (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) (s : SwitchIndex d) :
-    s ∈ exposedPaths d b V ↔ (switchUsers d b s).1 ∈ V ∨ (switchUsers d b s).2 ∈ V := by
-  classical
-  simp only [exposedPaths,Finset.mem_biUnion,mem_routeDomain_iff_user]
-  constructor
-  · rintro ⟨x,hx,h | h⟩
-    · exact Or.inl (h ▸ hx)
-    · exact Or.inr (h ▸ hx)
-  · rintro (h | h)
-    · exact ⟨_,h,Or.inl rfl⟩
-    · exact ⟨_,h,Or.inr rfl⟩
-
-lemma terminalCharge_sum (d : ℕ) (b : Butterfly d) (z : Card d) (V : Finset (Card d))
-    (hz : z ∉ V) :
-    (terminalCharge d b z V : ℝ) = ∑ s : SwitchIndex d,
-      ((if (switchUsers d b s).1 = z ∧ (switchUsers d b s).2 ∈ V then (1:ℝ) else 0) +
-      (if (switchUsers d b s).2 = z ∧ (switchUsers d b s).1 ∈ V then (1:ℝ) else 0)) := by
-  classical
-  have he : routeDomain d z (butterflyPerm d b z) ∩ exposedPaths d b V =
-      Finset.univ.filter (fun s =>
-        ((switchUsers d b s).1 = z ∧ (switchUsers d b s).2 ∈ V) ∨
-        ((switchUsers d b s).2 = z ∧ (switchUsers d b s).1 ∈ V)) := by
-    ext s
-    simp only [Finset.mem_inter,mem_routeDomain_iff_user,mem_exposedPaths,
-      Finset.mem_filter,Finset.mem_univ,true_and]
-    by_cases h₁ : (switchUsers d b s).1 = z <;>
-      by_cases h₂ : (switchUsers d b s).2 = z <;> simp [h₁,h₂,hz]
-  rw [terminalCharge,he]
-  simp only [← Finset.sum_boole]
-  apply Finset.sum_congr rfl
-  intro s _
-  by_cases h₁ : (switchUsers d b s).1 = z <;>
-    by_cases h₂ : (switchUsers d b s).2 = z <;> simp [h₁,h₂,hz]
-
-noncomputable def priorPaths {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) : Finset (Card d) :=
-  Finset.univ.filter (fun x => ∃ i : Fin A, o x = some i ∧ i.val < t)
-
-@[simp] lemma mem_priorPaths {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) (x : Card d) :
-    x ∈ priorPaths o t ↔ ∃ i : Fin A, o x = some i ∧ i.val < t := by
-  classical
-  simp [priorPaths]
-
-lemma terminalCharge_prior_sum {d A : ℕ} (o : Card d → Option (Fin A))
-    (b : Butterfly d) (z : Terminal o) (i : Fin A) :
-    (terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) =
-      ∑ s : SwitchIndex d,
-        ((if (z i).val = (switchUsers d b s).1 ∧ (switchUsers d b s).2 ∈ priorPaths o (i.val+1)
-          then (1:ℝ) else 0) +
-        (if (z i).val = (switchUsers d b s).2 ∧ (switchUsers d b s).1 ∈ priorPaths o (i.val+1)
-          then (1:ℝ) else 0)) := by
-  classical
-  rw [terminalCharge_sum _ _ _ _ (Finset.notMem_erase _ _)]
-  apply Finset.sum_congr rfl
-  intro s _
-  have hn := switchUsers_ne d b s
-  simp only [Finset.mem_erase]
-  by_cases h₁ : (z i).val = (switchUsers d b s).1 <;>
-    by_cases h₂ : (z i).val = (switchUsers d b s).2 <;> simp_all [eq_comm]
-
-lemma rotationMeeting_eq_slot_sum {d A : ℕ} (o : Card d → Option (Fin A))
-    (z : Terminal o) (u v : Card d) :
-    rotationMeeting o z u v = ∑ i : Fin A,
-      ((if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) +
-      (if (z i).val = v ∧ u ∈ priorPaths o (i.val+1) then (1:ℝ) else 0)) := by
-  classical
-  have only_owner (u : Card d) (a : Fin A) (hou : o u = some a)
-      (i : Fin A) (hia : i ≠ a) : (z i).val ≠ u := by
-    intro h
-    have hi := (z i).property
-    rw [h,hou] at hi
-    exact hia (Option.some.inj hi).symm
-  have sum_one (u v : Card d) (a : Fin A) (hou : o u = some a) :
-      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) =
-      if (z a).val = u ∧ v ∈ priorPaths o (a.val+1) then 1 else 0 := by
-    apply Finset.sum_eq_single a
-    · intro i _ hia
-      simp [only_owner u a hou i hia]
-    · simp
-  have sum_none (u v : Card d) (hou : o u = none) :
-      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) = 0 := by
-    apply Finset.sum_eq_zero
-    intro i _
-    have hn : (z i).val ≠ u := by
-      intro h
-      have hi := (z i).property
-      rw [h,hou] at hi
-      contradiction
-    simp [hn]
-  rw [Finset.sum_add_distrib]
-  cases hou : o u with
-  | none =>
-    rw [sum_none u v hou]
-    simp [rotationMeeting,hou,mem_priorPaths]
-  | some a =>
-    rw [sum_one u v a hou]
-    cases hov : o v with
-    | none =>
-      rw [sum_none v u hov]
-      simp [rotationMeeting,hou,hov,mem_priorPaths]
-    | some b =>
-      rw [sum_one v u b hov]
-      simp only [rotationMeeting,hou,hov,mem_priorPaths,Option.some.injEq]
-      rcases lt_trichotomy a b with hab | rfl | hba
-      · have hn : ¬ b.val < a.val+1 := by omega
-        have hy : a.val < b.val+1 := by omega
-        simp [hab,hn,hy]
-      · simp
-      · have hn : ¬ a.val < b.val+1 := by omega
-        have hy : b.val < a.val+1 := by omega
-        simp [hba,not_lt_of_ge (le_of_lt hba),hn,hy]
-
-lemma rotationCharge_eq_terminal_sum {d A : ℕ} (o : Card d → Option (Fin A))
-    (b : Butterfly d) (z : Terminal o) :
-    rotationCharge o b z = ∑ i : Fin A,
-      (terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) := by
-  simp only [rotationCharge,rotationMeeting_eq_slot_sum,terminalCharge_prior_sum]
-  exact Finset.sum_comm
-
-@[simp] lemma priorPaths_zero {d A : ℕ} (o : Card d → Option (Fin A)) : priorPaths o 0 = ∅ := by
-  classical
-  ext x
-  simp
-
-lemma priorPaths_succ {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) (ht : t < A) :
-    priorPaths o (t+1) = priorPaths o t ∪ cycleFiber o ⟨t,ht⟩ := by
-  classical
-  ext x
-  simp only [mem_priorPaths,Finset.mem_union,mem_cycleFiber]
-  constructor
-  · rintro ⟨i,hi,hit⟩
-    by_cases h : i.val < t
-    · exact Or.inl ⟨i,hi,h⟩
-    · have he : i = ⟨t,ht⟩ := Fin.ext (by change i.val = t; omega)
-      exact Or.inr (he ▸ hi)
-  · rintro (⟨i,hi,hit⟩ | hi)
-    · exact ⟨i,hi,by omega⟩
-    · exact ⟨⟨t,ht⟩,hi,by change t < t+1; omega⟩
-
-lemma priorPaths_disjoint {d A : ℕ} (o : Card d → Option (Fin A)) (i : Fin A) :
-    Disjoint (priorPaths o i.val) (cycleFiber o i) := by
-  classical
-  apply Finset.disjoint_left.mpr
-  intro x hx hx'
-  obtain ⟨j,hj,hji⟩ := (mem_priorPaths _ _ _).mp hx
-  have he : j = i := Option.some.inj (hj.symm.trans ((mem_cycleFiber _ _ _).mp hx'))
-  subst j
-  exact (lt_irrefl _ hji)
-
-namespace QueryTree
-
-lemma seekCycles_owned {d A : ℕ} (S : Finset (Card d)) (r : Fin A → ℕ)
-    (o : Card d → Option (Fin A)) (z : Terminal o) (ω : SwitchIndex d → Bool)
-    (hS : ∀ i, cycleFiber o i ⊆ S)
-    (hO : ∀ i, orbitSet (butterflyPerm d (decodeButterfly d ω)) (z i).val = cycleFiber o i)
-    (hC : ∀ i, (cycleFiber o i).card = r i+1) :
-    let p := butterflyPerm d (decodeButterfly d ω)
-    let L := List.ofFn (fun i : Fin A => (p (z i).val,r i))
-    (seekCycles d S L ∅).succeeds ω = true ∧
-      ((seekCycles d S L ∅).staleCharge ∅ ω : ℝ) = rotationCharge o (decodeButterfly d ω) z := by
-  classical
-  let b := decodeButterfly d ω
-  let p := butterflyPerm d b
-  let f (i : Fin A) := (p (z i).val,r i)
-  let c (i : Fin A) : ℝ := terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val)
-  have hc (i : Fin A) : Function.minimalPeriod p (z i).val - 1 = r i := by
-    have he := orbitSet_card p (z i).val
-    rw [hO,hC] at he
-    omega
-  have hs (t : ℕ) (ht : t ≤ A) :
-      (seekCycles d S ((List.ofFn f).drop t) (priorPaths o t)).succeeds ω = true ∧
-        ((seekCycles d S ((List.ofFn f).drop t) (priorPaths o t)).staleCharge (exposedPaths d b (priorPaths o t)) ω : ℝ) =
-          ((List.ofFn c).drop t).sum := by
-    induction ht using Nat.decreasingInduction with
-    | self => simp only [List.length_ofFn,le_refl,List.drop_eq_nil_of_le,List.sum_nil,seekCycles,succeeds,staleCharge,Nat.cast_zero,and_self]
-    | of_succ t ht ih =>
-      have htf : t < (List.ofFn f).length := by simpa using ht
-      have htc : t < (List.ofFn c).length := by simpa using ht
-      rw [List.drop_eq_getElem_cons htf,List.drop_eq_getElem_cons htc]
-      simp only [List.getElem_ofFn,List.sum_cons,seekCycles]
-      have h := seekCycle_orbit d S (priorPaths o t) (z ⟨t,ht⟩).val
-        (fun V => seekCycles d S ((List.ofFn f).drop (t+1)) V) ω
-        (by rw [hO]; exact hS _) (by rw [hO]; exact priorPaths_disjoint o ⟨t,ht⟩)
-      dsimp only at h
-      rw [hc,hO,←priorPaths_succ o t ht] at h
-      dsimp only [f]
-      constructor
-      · exact h.1.trans ih.1
-      · rw [h.2,Nat.cast_add,ih.2]
-        exact add_comm _ _
-  have h := hs 0 (Nat.zero_le _)
-  simp only [List.drop_zero,priorPaths_zero,exposedPaths_empty,List.sum_ofFn] at h
-  dsimp only
-  exact ⟨h.1,h.2.trans (rotationCharge_eq_terminal_sum o b z).symm⟩
-
-end QueryTree
-
-section Collections
-variable {d A : ℕ}
-
-def ValidCycleSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ)
-    (o : Card d → Option (Fin A)) : Prop :=
-  (∀ i, (cycleFiber o i).card = r i + 1) ∧
-  (∀ i, cycleFiber o i ⊆ S) ∧
-  (∀ i x, o x = some i → orbitSet p x = cycleFiber o i)
-
-abbrev CycleSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
-  {o : Card d → Option (Fin A) // ValidCycleSlots p S r o}
-
-noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
-    Fintype (CycleSlots p S r) := Fintype.ofFinite _
-
-lemma CycleSlots.terminal_nonempty {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (c : CycleSlots p S r) (i : Fin A) : Nonempty {x : Card d // c.val x = some i} := by
-  have hn : (cycleFiber c.val i).Nonempty := Finset.card_pos.mp (by rw [c.property.1]; omega)
-  obtain ⟨x,hx⟩ := hn
-  exact ⟨x,(mem_cycleFiber _ _ _).mp hx⟩
-
-lemma owner_ext {o o' : Card d → Option (Fin A)}
-    (h : ∀ i, cycleFiber o i = cycleFiber o' i) : o = o' := by
-  funext x
-  have he (i : Fin A) : o x = some i ↔ o' x = some i := by
-    simpa only [mem_cycleFiber] using (Finset.ext_iff.mp (h i) x)
-  cases ho : o x with
-  | some i => exact ((he i).mp ho).symm
-  | none =>
-    cases ho' : o' x with
-    | none => rfl
-    | some i => have hn := (he i).mpr ho'; rw [ho] at hn; contradiction
-
-abbrev RootedSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
-  Σ c : CycleSlots p S r, Terminal c.val
-
-noncomputable def rootedStart {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (c : RootedSlots p S r) : Fin A → S := fun i =>
-  ⟨p (c.2 i).val,c.1.property.2.1 i (by
-    rw [←c.1.property.2.2 i (c.2 i).val (c.2 i).property]
-    exact (mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.apply_right (Equiv.Perm.SameCycle.refl p _)))⟩
-
-lemma rootedStart_injective {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ} :
-    Function.Injective (rootedStart (p:=p) (S:=S) (r:=r)) := by
-  rintro ⟨c,z⟩ ⟨c',z'⟩ h
-  have hz (i : Fin A) : (z i).val = (z' i).val :=
-    p.injective (congrArg Subtype.val (congrFun h i))
-  have hc : c = c' := Subtype.ext (owner_ext (fun i => by
-    rw [←c.property.2.2 i (z i).val (z i).property,
-      ←c'.property.2.2 i (z' i).val (z' i).property,hz]))
-  subst c'
-  have hzz : z = z' := funext (fun i => Subtype.ext (hz i))
-  subst z'
-  rfl
-
-lemma sum_comp_injective_le {α β : Type*} [Fintype α] [Fintype β]
-    (e : α → β) (he : Function.Injective e) (f : β → ℝ) (hf : ∀ b, 0 ≤ f b) :
-    ∑ a, f (e a) ≤ ∑ b, f b := by
-  classical
-  calc
-    _ = ∑ b ∈ Finset.univ.image e, f b := (Finset.sum_image (fun _ _ _ _ h => he h)).symm
-    _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _) (fun b _ _ => hf b)
-
-lemma rooted_weight_sum_le (S : Finset (Card d)) (r : Fin A → ℕ) (ω : SwitchIndex d → Bool) :
-    (∑ c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r,
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2)) ≤
-    ∑ a : Fin A → S,
-      if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-        ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-      else 0 := by
-  classical
-  let p := butterflyPerm d (decodeButterfly d ω)
-  let g (a : Fin A → S) : ℝ :=
-    if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-    else 0
-  have he (c : RootedSlots p S r) :
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2) =
-      g (rootedStart c) := by
-    have hs := QueryTree.seekCycles_owned S r c.1.val c.2 ω c.1.property.2.1
-      (fun i => c.1.property.2.2 i (c.2 i).val (c.2 i).property) c.1.property.1
-    dsimp only at hs
-    unfold g
-    rw [ite_eq_left (show (QueryTree.seekCycles d S (List.ofFn (fun i => ((rootedStart c i).val,r i))) ∅).succeeds ω = true from hs.1)]
-    rw [← hs.2,neg_mul,Real.exp_neg,mul_comm (Real.log 2),Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
-    rfl
-  simp_rw [he]
-  exact sum_comp_injective_le rootedStart rootedStart_injective g (fun a => by dsimp [g]; split_ifs <;> positivity)
-
-noncomputable def rotationLower (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : ℝ :=
-  Real.exp (-Real.log 2 * ((A:ℝ)*d/2 + ∑ i : Fin A, Real.log (r i+1) / (2 * Real.log 2)))
-
-noncomputable def collectionWeight (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : ℝ :=
-  ((2:ℝ)^d)^A * (∏ i : Fin A, (r i+1 : ℝ)) * rotationLower d r
-
-lemma collectionWeight_pos (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : 0 < collectionWeight d r := by
-  unfold collectionWeight rotationLower
-  positivity
-
-lemma collection_rotation_lower (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r)
-    (ω : SwitchIndex d → Bool) (c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) :
-    collectionWeight d r ≤ ∑ z : Terminal c.val,
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z) := by
-  classical
-  let (i : Fin A) : Nonempty {x : Card d // c.val x = some i} := c.terminal_nonempty i
-  have hm : Monotone (fun i => (cycleFiber c.val i).card) := by
-    intro i j hij
-    dsimp only
-    rw [c.property.1,c.property.1]
-    exact Nat.add_le_add_right (hr hij) 1
-  have hb := rotation_exp_bound hm (decodeButterfly d ω)
-  have hcard : (Fintype.card (Terminal c.val) : ℝ) = ∏ i : Fin A, (r i+1 : ℝ) := by
-    rw [Fintype.card_pi,Nat.cast_prod]
-    apply Finset.prod_congr rfl
-    intro i _
-    rw [card_cycleFiber,c.property.1,Nat.cast_add,Nat.cast_one]
-  have hp : (0:ℝ) < Fintype.card (Terminal c.val) := Nat.cast_pos.mpr Fintype.card_pos
-  have hl : rotationLower d r ≤ finiteMean (fun z : Terminal c.val => Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z)) := by
-    simpa only [rotationLower,c.property.1,Nat.cast_add,Nat.cast_one] using hb
-  have hh := (le_div_iff₀ hp).mp hl
-  rw [hcard] at hh
-  unfold collectionWeight
-  rw [← Finset.mul_sum]
-  calc
-    _ = ((2:ℝ)^d)^A * (rotationLower d r * ∏ i : Fin A, (r i+1 : ℝ)) := by ring
-    _ ≤ _ := mul_le_mul_of_nonneg_left hh (by positivity)
-
-lemma cycleSlots_weighted_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) *
-      collectionWeight d r ≤ (S.card : ℝ)^A := by
-  classical
-  let g (a : Fin A → S) (ω : SwitchIndex d → Bool) : ℝ :=
-    if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-    else 0
-  have hω (ω : SwitchIndex d → Bool) :
-      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ) * collectionWeight d r ≤
-        ∑ a : Fin A → S, g a ω := by
-    calc
-      _ = ∑ _c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r, collectionWeight d r := by simp
-      _ ≤ ∑ c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r,
-          ∑ z : Terminal c.val, ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z) :=
-        Finset.sum_le_sum (fun c _ => collection_rotation_lower S r hr ω c)
-      _ = ∑ c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r,
-          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2) :=
-        (Fintype.sum_sigma (fun c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r =>
-          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2))).symm
-      _ ≤ _ := rooted_weight_sum_le S r ω
-  have hb := finiteMean_mono hω
-  rw [finiteMean_mul_const,finiteMean_sum] at hb
-  calc
-    _ ≤ ∑ a : Fin A → S, finiteMean (g a) := hb
-    _ ≤ ∑ _a : Fin A → S, (1:ℝ) := by
-      apply Finset.sum_le_sum
-      intro a _
-      have hh := QueryTree.cycles_weighted_success d S (List.ofFn (fun i : Fin A => ((a i).val,r i)))
-      simpa only [List.length_ofFn] using hh
-    _ = _ := by simp
-
-lemma cycleSlots_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) ≤
-        (S.card : ℝ)^A / collectionWeight d r := by
-  exact (le_div_iff₀ (collectionWeight_pos d r)).mpr (cycleSlots_weighted_moment S r hr)
-
-lemma collectionWeight_eq_sqrt (d : ℕ) {A : ℕ} (r : Fin A → ℕ) :
-    collectionWeight d r = (Real.sqrt ((2:ℝ)^d))^A * ∏ i : Fin A, Real.sqrt (r i+1) := by
-  have hn : ((2:ℝ)^d)^A = Real.exp ((A:ℝ)*d*Real.log 2) := by
-    rw [mul_assoc,Real.exp_nat_mul,Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
-  have hp : (∏ i : Fin A, (r i+1 : ℝ)) = Real.exp (∑ i : Fin A, Real.log (r i+1)) := by
-    rw [Real.exp_sum]
-    apply Finset.prod_congr rfl
-    intro i _
-    exact (Real.exp_log (by positivity)).symm
-  have hs : Real.sqrt ((2:ℝ)^d) = Real.exp ((d:ℝ)*Real.log 2/2) := by
-    rw [Real.sqrt_eq_rpow,Real.rpow_def_of_pos (by positivity),Real.log_pow]
-    congr 1
-    ring
-  have hj (i : Fin A) : Real.sqrt (r i+1) = Real.exp (Real.log (r i+1)/2) := by
-    rw [Real.sqrt_eq_rpow,Real.rpow_def_of_pos (by positivity)]
-    congr 1
-    ring
-  simp_rw [collectionWeight,hn,hp,rotationLower,hs,hj]
-  rw [←Real.exp_add,←Real.exp_add,←Real.exp_nat_mul,←Real.exp_sum,←Real.exp_add]
-  congr 1
-  rw [←Finset.sum_div,←Finset.sum_div]
-  have hl : Real.log 2 ≠ 0 := ne_of_gt (Real.log_pos (by norm_num))
-  field_simp
-  ring
-
-lemma butterfly_cycleSlots_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) ≤
-        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
-  have h := cycleSlots_moment S r hr
-  rw [collectionWeight_eq_sqrt,div_mul_eq_div_mul_one_div] at h
-  rw [div_pow]
-  simpa only [one_div,Finset.prod_inv_distrib] using h
-
-end Collections
-
-variable {d A : ℕ}
-
-abbrev SelectedCycle (p : Equiv.Perm (Card d)) (S : Finset (Card d)) :=
-  {C : Finset (Card d) // (∃ x, C = orbitSet p x) ∧ C ⊆ S}
-
-noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) :
-    Fintype (SelectedCycle p S) := Fintype.ofFinite _
-
-lemma SelectedCycle.nonempty {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
-    (C : SelectedCycle p S) : C.val.Nonempty := by
-  obtain ⟨x,hx⟩ := C.property.1
-  rw [hx]
-  exact orbitSet_nonempty p x
-
-lemma SelectedCycle.orbit_eq {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
-    (C : SelectedCycle p S) {x : Card d} (hx : x ∈ C.val) : orbitSet p x = C.val := by
-  obtain ⟨y,hy⟩ := C.property.1
-  rw [hy] at hx ⊢
-  exact (orbitSet_eq p ((mem_orbitSet _ _ _).mp hx)).symm
-
-lemma SelectedCycle.eq_of_mem {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
-    (C D : SelectedCycle p S) {x : Card d} (hC : x ∈ C.val) (hD : x ∈ D.val) : C = D := by
-  apply Subtype.ext
-  exact (C.orbit_eq hC).symm.trans (D.orbit_eq hD)
-
-abbrev SlotAssignment (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
-  {f : Fin A → SelectedCycle p S // Function.Injective f ∧ ∀ i, (f i).val.card = r i+1}
-
-noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
-    Fintype (SlotAssignment p S r) := Fintype.ofFinite _
-
-noncomputable def slotsToAssignment {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (c : CycleSlots p S r) : SlotAssignment p S r := by
-  classical
-  let f (i : Fin A) : SelectedCycle p S := ⟨cycleFiber c.val i,by
-    obtain ⟨x,hx⟩ := c.terminal_nonempty i
-    exact ⟨⟨x,(c.property.2.2 i x hx).symm⟩,c.property.2.1 i⟩⟩
-  refine ⟨f,?_,fun i => c.property.1 i⟩
-  intro i j hij
-  have he : cycleFiber c.val i = cycleFiber c.val j := congrArg Subtype.val hij
-  obtain ⟨x,hx⟩ := c.terminal_nonempty i
-  have hxi : x ∈ cycleFiber c.val i := (mem_cycleFiber _ _ _).mpr hx
-  rw [he] at hxi
-  exact Option.some.inj (hx.symm.trans ((mem_cycleFiber _ _ _).mp hxi))
-
-noncomputable def assignmentOwner {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) (x : Card d) : Option (Fin A) := by
-  classical
-  exact if h : ∃ i, x ∈ (f.val i).val then some (Classical.choose h) else none
-
-end Thorp
-
-end ThorpNine.Harmonic
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Harmonic.ExposedCoins
+
+namespace OAI
+
+namespace ThorpNine.Harmonic
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+section Rotation
+variable {d A : ℕ}
+
+noncomputable def rotationMeeting (o : Card d → Option (Fin A)) (z : Terminal o)
+    (u v : Card d) : ℝ :=
+  match o u, o v with
+  | some i, some j => if i < j then (if (z j).val = v then 1 else 0)
+      else if j < i then (if (z i).val = u then 1 else 0)
+      else (if (z i).val = u then 1 else 0) + (if (z i).val = v then 1 else 0)
+  | _, _ => 0
+
+lemma card_cycleFiber (o : Card d → Option (Fin A)) (i : Fin A) :
+    Fintype.card {x : Card d // o x = some i} = (cycleFiber o i).card := by
+  classical
+  simp [cycleFiber,Fintype.card_subtype]
+
+lemma terminal_mean {o : Card d → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
+    (i : Fin A) (x : Card d) (hx : o x = some i) :
+    finiteMean (fun z : Terminal o => if (z i).val = x then (1:ℝ) else 0) = slotWeight o i := by
+  classical
+  have hev := finiteMean_pi_eval (A := fun i : Fin A => {x : Card d // o x = some i})
+    i (fun z => if z.val = x then (1:ℝ) else 0)
+  rw [hev]
+  have he : (fun z : {x : Card d // o x = some i} => if z.val = x then (1:ℝ) else 0) =
+      (fun z => if z = ⟨x,hx⟩ then 1 else 0) := by
+    funext z
+    simp only [Subtype.ext_iff]
+  rw [he,finiteMean_eq_indicator,card_cycleFiber,slotWeight]
+
+lemma slotWeight_antitone (o : Card d → Option (Fin A))
+    (hn : ∀ i, (cycleFiber o i).Nonempty)
+    (hm : Monotone (fun i => (cycleFiber o i).card)) : Antitone (slotWeight o) := by
+  intro i j hij
+  unfold slotWeight
+  apply one_div_le_one_div_of_le
+  · exact_mod_cast Finset.card_pos.mpr (hn i)
+  · exact_mod_cast hm hij
+
+lemma rotationMeeting_mean {o : Card d → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
+    (hm : Monotone (fun i => (cycleFiber o i).card)) (u v : Card d) :
+    finiteMean (fun z : Terminal o => rotationMeeting o z u v) =
+      min (pathWeight o u) (pathWeight o v) + withinWeight o u v := by
+  classical
+  have hn : ∀ i, (cycleFiber o i).Nonempty := by
+    intro i
+    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : Card d // o x = some i})
+    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
+  have hw := slotWeight_antitone o hn hm
+  cases hu : o u with
+  | none =>
+    cases hv : o v with
+    | none => simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean]
+    | some j =>
+      have hj : 0 ≤ slotWeight o j := by unfold slotWeight; positivity
+      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_left hj]
+  | some i =>
+    cases hv : o v with
+    | none =>
+      have hi : 0 ≤ slotWeight o i := by unfold slotWeight; positivity
+      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_right hi]
+    | some j =>
+      simp only [rotationMeeting,pathWeight,withinWeight,hu,hv]
+      rcases lt_trichotomy i j with hij | rfl | hji
+      · simp only [hij,↓reduceIte,ne_of_lt hij]
+        rw [terminal_mean j v hv,min_eq_right (hw (le_of_lt hij)),add_zero]
+      · simp only [lt_self_iff_false,↓reduceIte,min_self]
+        rw [finiteMean_add,terminal_mean i u hu,terminal_mean i v hv]
+      · simp only [not_lt_of_ge (le_of_lt hji),hji,↓reduceIte,ne_of_gt hji]
+        rw [terminal_mean i u hu,min_eq_left (hw (le_of_lt hji)),add_zero]
+
+lemma pathWeight_sum (o : Card d → Option (Fin A)) (hn : ∀ i, (cycleFiber o i).Nonempty) :
+    ∑ x : Card d, pathWeight o x = (A : ℝ) := by
+  classical
+  have he (x : Card d) : pathWeight o x =
+      ∑ i : Fin A, if o x = some i then slotWeight o i else 0 := by
+    cases hx : o x with
+    | none => simp [pathWeight,hx]
+    | some j => simp [pathWeight,hx]
+  simp only [he]
+  rw [Finset.sum_comm]
+  have hf (i : Fin A) : (∑ x : Card d, if o x = some i then slotWeight o i else 0) = 1 := by
+    rw [← Finset.sum_filter]
+    change (∑ _x ∈ cycleFiber o i, slotWeight o i) = 1
+    simp only [Finset.sum_const,nsmul_eq_mul,slotWeight]
+    exact mul_one_div_cancel (by exact_mod_cast (Finset.card_pos.mpr (hn i)).ne')
+  simp only [hf,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,mul_one]
+
+lemma withinWeight_sum_bound (o : Card d → Option (Fin A))
+    (hn : ∀ i, (cycleFiber o i).Nonempty) (b : Butterfly d) :
+    (∑ s : SwitchIndex d, withinWeight o (switchUsers d b s).1 (switchUsers d b s).2) ≤
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
+  classical
+  have he (u v : Card d) : withinWeight o u v =
+      ∑ i : Fin A, (if u ∈ cycleFiber o i ∧ v ∈ cycleFiber o i then (1:ℝ) else 0) * slotWeight o i := by
+    simp only [mem_cycleFiber]
+    cases hu : o u <;> cases hv : o v <;> simp [withinWeight,hu,hv,ite_and,eq_comm]
+  simp only [he]
+  rw [Finset.sum_comm]
+  apply Finset.sum_le_sum
+  intro i _
+  rw [← Finset.sum_mul]
+  have hi := marked_switch_entropy_bound d b (cycleFiber o i)
+  have hpos : (0:ℝ) < (cycleFiber o i).card := by exact_mod_cast Finset.card_pos.mpr (hn i)
+  have hmul := mul_le_mul_of_nonneg_right hi (show 0 ≤ slotWeight o i by unfold slotWeight; positivity)
+  calc
+    _ ≤ ((cycleFiber o i).card : ℝ) * Real.log (cycleFiber o i).card /
+        (2 * Real.log 2) * slotWeight o i := hmul
+    _ = _ := by unfold slotWeight; field_simp
+
+noncomputable def rotationCharge (o : Card d → Option (Fin A)) (b : Butterfly d)
+    (z : Terminal o) : ℝ :=
+  ∑ s : SwitchIndex d, rotationMeeting o z (switchUsers d b s).1 (switchUsers d b s).2
+
+lemma rotationCharge_mean_bound {o : Card d → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
+    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : Butterfly d) :
+    finiteMean (rotationCharge o b) ≤ (A:ℝ) * d / 2 +
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
+  classical
+  have hn : ∀ i, (cycleFiber o i).Nonempty := by
+    intro i
+    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : Card d // o x = some i})
+    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
+  unfold rotationCharge
+  rw [finiteMean_sum]
+  simp only [rotationMeeting_mean hm,Finset.sum_add_distrib]
+  have h₁ := weighted_switch_min_bound d b (pathWeight o)
+  rw [pathWeight_sum o hn] at h₁
+  have h₂ := withinWeight_sum_bound o hn b
+  linarith
+
+lemma rotation_exp_bound {o : Card d → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
+    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : Butterfly d) :
+    Real.exp (-Real.log 2 * ((A:ℝ)*d/2 +
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2))) ≤
+      finiteMean (fun z : Terminal o => Real.exp (-Real.log 2 * rotationCharge o b z)) := by
+  have hb := rotationCharge_mean_bound hm b
+  have hlog : 0 ≤ Real.log 2 := Real.log_nonneg (by norm_num)
+  have hm' : -Real.log 2 * ((A:ℝ)*d/2 +
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2)) ≤
+      finiteMean (fun z : Terminal o => -Real.log 2 * rotationCharge o b z) := by
+    have he := finiteMean_mul_const (rotationCharge o b) (-Real.log 2)
+    simp only [mul_comm] at he
+    rw [he]
+    exact mul_le_mul_of_nonpos_left hb (neg_nonpos.mpr hlog)
+  exact (Real.exp_le_exp.mpr hm').trans (exp_finiteMean_le _)
+
+end Rotation
+
+section OrbitCore
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+noncomputable def orbitSet (p : Equiv.Perm α) (x : α) : Finset α :=
+  Finset.univ.filter (p.SameCycle x)
+
+@[simp] lemma mem_orbitSet (p : Equiv.Perm α) (x y : α) :
+    y ∈ orbitSet p x ↔ p.SameCycle x y := by
+  classical
+  simp [orbitSet]
+
+lemma orbitSet_nonempty (p : Equiv.Perm α) (x : α) : (orbitSet p x).Nonempty :=
+  ⟨x,(mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.refl _ _)⟩
+
+lemma orbitSet_eq (p : Equiv.Perm α) {x y : α} (h : p.SameCycle x y) :
+    orbitSet p x = orbitSet p y := by
+  ext z
+  simp only [mem_orbitSet]
+  exact ⟨fun hz => h.symm.trans hz,fun hz => h.trans hz⟩
+
+@[simp] lemma orbitSet_apply (p : Equiv.Perm α) (x : α) : orbitSet p (p x) = orbitSet p x := by
+  ext y
+  simp only [mem_orbitSet,Equiv.Perm.sameCycle_apply_left]
+
+lemma orbitSet_eq_or_disjoint (p : Equiv.Perm α) (x y : α) :
+    orbitSet p x = orbitSet p y ∨ Disjoint (orbitSet p x) (orbitSet p y) := by
+  classical
+  by_cases h : p.SameCycle x y
+  · exact Or.inl (orbitSet_eq p h)
+  · right
+    apply Finset.disjoint_left.mpr
+    intro z hz hz'
+    exact h (((mem_orbitSet _ _ _).mp hz).trans ((mem_orbitSet _ _ _).mp hz').symm)
+
+omit [DecidableEq α] in
+lemma perm_minimalPeriod_pos (p : Equiv.Perm α) (x : α) :
+    0 < Function.minimalPeriod p x :=
+  Function.minimalPeriod_pos_of_mem_periodicPts (p.injective.mem_periodicPts x)
+
+lemma orbitSet_card (p : Equiv.Perm α) (x : α) :
+    (orbitSet p x).card = Function.minimalPeriod p x := by
+  classical
+  let f : Fin (Function.minimalPeriod p x) → {y // p.SameCycle x y} := fun n =>
+    ⟨p^[n.val] x,by rw [Equiv.Perm.iterate_eq_pow]; exact ⟨(n.val:ℤ),by simp⟩⟩
+  have hf : Function.Bijective f := by
+    constructor
+    · intro i j hij
+      apply Fin.ext
+      exact Function.iterate_injOn_Iio_minimalPeriod i.isLt j.isLt (congrArg Subtype.val hij)
+    · intro y
+      obtain ⟨n,hn⟩ := y.property.exists_nat_pow_eq
+      refine ⟨⟨n % Function.minimalPeriod p x, Nat.mod_lt _ (perm_minimalPeriod_pos p x)⟩,?_⟩
+      apply Subtype.ext
+      change p^[n % Function.minimalPeriod p x] x = y.val
+      rw [Function.iterate_mod_minimalPeriod_eq,Equiv.Perm.iterate_eq_pow,hn]
+  have hc := Fintype.card_congr (Equiv.ofBijective f hf)
+  simpa [orbitSet,Fintype.card_subtype] using hc.symm
+
+def walkSet (p : Equiv.Perm α) (r : ℕ) (x : α) : Finset α :=
+  (Finset.range r).image (fun n => p^[n] x)
+
+omit [Fintype α] in
+@[simp] lemma mem_walkSet (p : Equiv.Perm α) (r : ℕ) (x y : α) :
+    y ∈ walkSet p r x ↔ ∃ n < r, p^[n] x = y := by
+  simp [walkSet]
+
+omit [Fintype α] in
+@[simp] lemma walkSet_zero (p : Equiv.Perm α) (x : α) : walkSet p 0 x = ∅ := by
+  simp [walkSet]
+
+omit [Fintype α] in
+lemma walkSet_succ_last (p : Equiv.Perm α) (r : ℕ) (x : α) :
+    walkSet p (r+1) x = insert (p^[r] x) (walkSet p r x) := by
+  simp [walkSet,Finset.range_add_one]
+
+omit [Fintype α] in
+lemma walkSet_succ_first (p : Equiv.Perm α) (r : ℕ) (x : α) :
+    walkSet p (r+1) x = insert x (walkSet p r (p x)) := by
+  ext y
+  simp only [mem_walkSet,Finset.mem_insert]
+  constructor
+  · rintro ⟨n,hn,hny⟩
+    cases n with
+    | zero => exact Or.inl (by simpa using hny.symm)
+    | succ n => exact Or.inr ⟨n,by omega,by simpa only [Function.iterate_succ_apply] using hny⟩
+  · rintro (rfl | ⟨n,hn,hny⟩)
+    · exact ⟨0,by omega,rfl⟩
+    · exact ⟨n+1,by omega,by simpa only [Function.iterate_succ_apply] using hny⟩
+
+omit [Fintype α] in
+lemma not_mem_short_walk (p : Equiv.Perm α) (x : α) {r : ℕ}
+    (hr : r < Function.minimalPeriod p x) : x ∉ walkSet p r (p x) := by
+  rintro hx
+  obtain ⟨n,hn,hnx⟩ := (mem_walkSet _ _ _ _).mp hx
+  have hperiod : Function.IsPeriodicPt p (n+1) x := by
+    simpa only [Function.IsPeriodicPt,Function.IsFixedPt,Function.iterate_succ_apply] using hnx
+  have hp := hperiod.minimalPeriod_le (by omega : 0 < n+1)
+  omega
+
+lemma walkSet_period (p : Equiv.Perm α) (x : α) :
+    walkSet p (Function.minimalPeriod p x) x = orbitSet p x := by
+  ext y
+  simp only [mem_walkSet,mem_orbitSet]
+  constructor
+  · rintro ⟨n,_,rfl⟩
+    rw [Equiv.Perm.iterate_eq_pow]
+    exact ⟨(n:ℤ),by simp⟩
+  · intro h
+    obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
+    refine ⟨n % Function.minimalPeriod p x,Nat.mod_lt _ (perm_minimalPeriod_pos p x),?_⟩
+    rw [Function.iterate_mod_minimalPeriod_eq,Equiv.Perm.iterate_eq_pow,hn]
+
+omit [Fintype α] in
+lemma walkSet_card (p : Equiv.Perm α) (x : α) {r : ℕ}
+    (hr : r ≤ Function.minimalPeriod p x) : (walkSet p r x).card = r := by
+  rw [walkSet,Finset.card_image_of_injOn]
+  · exact Finset.card_range _
+  · intro i hi j hj hij
+    exact Function.iterate_injOn_Iio_minimalPeriod
+      (lt_of_lt_of_le (Finset.mem_range.mp hi) hr)
+      (lt_of_lt_of_le (Finset.mem_range.mp hj) hr) hij
+
+lemma walkSet_subset_orbit (p : Equiv.Perm α) (r : ℕ) (x : α) :
+    walkSet p r x ⊆ orbitSet p x := by
+  intro y hy
+  obtain ⟨n,_,rfl⟩ := (mem_walkSet _ _ _ _).mp hy
+  rw [mem_orbitSet,Equiv.Perm.iterate_eq_pow]
+  exact ⟨(n:ℤ),by simp⟩
+
+def cycleTrace (p : Equiv.Perm α) (S : Finset α) (a : α) :
+    ℕ → α → Finset α → Option (Finset α × α)
+  | 0, x, V => if x ∈ S ∧ x ∉ V ∧ p x = a then some (insert x V,x) else none
+  | r+1, x, V => if x ∈ S ∧ x ∉ V then cycleTrace p S a r (p x) (insert x V) else none
+
+lemma cycleTrace_walk (p : Equiv.Perm α) (S : Finset α) (r : ℕ) (x : α) (V : Finset α)
+    (hr : r < Function.minimalPeriod p x) (hS : walkSet p (r+1) x ⊆ S)
+    (hV : Disjoint V (walkSet p (r+1) x)) :
+    cycleTrace p S (p^[r+1] x) r x V = some (V ∪ walkSet p (r+1) x,p^[r] x) := by
+  induction r generalizing x V with
+  | zero =>
+    have hxS : x ∈ S := hS (by simp [walkSet_succ_first])
+    have hxV : x ∉ V := fun h => Finset.disjoint_left.mp hV h (by simp [walkSet_succ_first])
+    simp [cycleTrace,hxS,hxV,walkSet_succ_first]
+  | succ r ih =>
+    have hxS : x ∈ S := hS (by rw [walkSet_succ_first]; exact Finset.mem_insert_self _ _)
+    have hxV : x ∉ V := fun h => Finset.disjoint_left.mp hV h
+      (by rw [walkSet_succ_first]; exact Finset.mem_insert_self _ _)
+    have hr' : r < Function.minimalPeriod p (p x) := by
+      rw [Function.minimalPeriod_apply (p.injective.mem_periodicPts x)]
+      omega
+    have hS' : walkSet p (r+1) (p x) ⊆ S := by
+      intro y hy
+      apply hS
+      rw [walkSet_succ_first]
+      exact Finset.mem_insert_of_mem hy
+    have hV' : Disjoint (insert x V) (walkSet p (r+1) (p x)) := by
+      apply Finset.disjoint_left.mpr
+      intro y hy hy'
+      rcases Finset.mem_insert.mp hy with hy | hy
+      · subst y
+        exact not_mem_short_walk p x hr hy'
+      · apply Finset.disjoint_left.mp hV hy
+        rw [walkSet_succ_first]
+        exact Finset.mem_insert_of_mem hy'
+    rw [cycleTrace,ite_eq_left ⟨hxS,hxV⟩,Function.iterate_succ_apply p (r+1) x,
+      ih _ _ hr' hS' hV',walkSet_succ_first p (r+1) x,Function.iterate_succ_apply]
+    simp only [Finset.insert_union,Finset.union_insert]
+
+lemma cycleTrace_orbit (p : Equiv.Perm α) (S V : Finset α) (z : α)
+    (hS : orbitSet p z ⊆ S) (hV : Disjoint V (orbitSet p z)) :
+    cycleTrace p S (p z) (Function.minimalPeriod p z - 1) (p z) V =
+      some (V ∪ orbitSet p z,z) := by
+  have hpos := perm_minimalPeriod_pos p z
+  have he : Function.minimalPeriod p (p z) = Function.minimalPeriod p z :=
+    Function.minimalPeriod_apply (p.injective.mem_periodicPts z)
+  have hp : Function.minimalPeriod p z - 1 + 1 = Function.minimalPeriod p z := by omega
+  have hw : walkSet p (Function.minimalPeriod p z - 1 + 1) (p z) = orbitSet p z := by
+    rw [hp,←he,walkSet_period,orbitSet_apply]
+  have hr : Function.minimalPeriod p z - 1 < Function.minimalPeriod p (p z) := by rw [he]; omega
+  have h := cycleTrace_walk p S (Function.minimalPeriod p z-1) (p z) V hr
+    (by simpa only [hw] using hS) (by simpa only [hw] using hV)
+  have ht : p^[Function.minimalPeriod p z-1] (p z) = z := by
+    rw [← Function.iterate_succ_apply,show (Function.minimalPeriod p z-1).succ =
+      Function.minimalPeriod p z by omega,Function.iterate_minimalPeriod]
+  have ha : p^[Function.minimalPeriod p z-1+1] (p z) = p z := by
+    rw [hp,←he,Function.iterate_minimalPeriod]
+  simpa only [ha,ht,hw] using h
+
+end OrbitCore
+
+noncomputable def exposedPaths (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) :
+    Finset (SwitchIndex d) :=
+  V.biUnion (fun x => routeDomain d x (butterflyPerm d b x))
+
+@[simp] lemma exposedPaths_empty (d : ℕ) (b : Butterfly d) : exposedPaths d b ∅ = ∅ := by
+  classical
+  simp [exposedPaths]
+
+lemma exposedPaths_insert (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) (x : Card d) :
+    exposedPaths d b (insert x V) = exposedPaths d b V ∪ routeDomain d x (butterflyPerm d b x) := by
+  classical
+  simp [exposedPaths,Finset.union_comm]
+
+noncomputable def terminalCharge (d : ℕ) (b : Butterfly d) (z : Card d) (V : Finset (Card d)) : ℕ :=
+  (routeDomain d z (butterflyPerm d b z) ∩ exposedPaths d b V).card
+
+namespace QueryTree
+
+lemma followPath_stale_charge {ι : Type*} [DecidableEq ι]
+    (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
+    (k : Card d → QueryTree ι) (R : Finset ι) (ω : ι → Bool) :
+    staleCharge (followPath d e x k) R ω =
+      staleCharge (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x))
+        (R ∪ (routeDomain d x (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).image e) ω := by
+  have h₀ := fresh_add_stale (followPath d e x k) R ω
+  have h₁ := fresh_add_stale (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x))
+    (R ∪ (routeDomain d x (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).image e) ω
+  rw [followPath_charge,followPath_raw_charge] at h₀
+  omega
+
+lemma seekCycle_semantics (d : ℕ) (S : Finset (Card d)) (a : Card d) (r : ℕ)
+    (x : Card d) (V W : Finset (Card d)) (z : Card d)
+    (k : Finset (Card d) → QueryTree (SwitchIndex d)) (ω : SwitchIndex d → Bool)
+    (h : cycleTrace (butterflyPerm d (decodeButterfly d ω)) S a r x V = some (W,z)) :
+    (seekCycle d S a r x V k).succeeds ω = (k W).succeeds ω ∧
+      (seekCycle d S a r x V k).staleCharge (exposedPaths d (decodeButterfly d ω) V) ω =
+        (k W).staleCharge (exposedPaths d (decodeButterfly d ω) W) ω +
+          terminalCharge d (decodeButterfly d ω) z (W.erase z) := by
+  classical
+  induction r generalizing x V with
+  | zero =>
+    simp only [cycleTrace] at h
+    split_ifs at h with hv
+    obtain ⟨hxS,hxV,hxa⟩ := hv
+    obtain ⟨rfl,rfl⟩ := Prod.mk.inj (Option.some.inj h)
+    simp only [seekCycle,ite_eq_left (show x ∈ S ∧ x ∉ V from ⟨hxS,hxV⟩),closePath_succeeds,ite_eq_left hxa]
+    constructor
+    · trivial
+    · rw [closePath_stale_charge _ _ _ _ _ _ hxa,exposedPaths_insert,
+        Finset.erase_insert hxV]
+      simp only [terminalCharge,hxa]
+  | succ r ih =>
+    simp only [cycleTrace] at h
+    split_ifs at h with hv
+    obtain ⟨hxS,hxV⟩ := hv
+    rw [seekCycle,ite_eq_left ⟨hxS,hxV⟩,followPath_succeeds,followPath_stale_charge]
+    simp only [Function.comp_id,Finset.image_id]
+    rw [← exposedPaths_insert]
+    exact ih _ _ h
+
+lemma seekCycle_orbit (d : ℕ) (S V : Finset (Card d)) (z : Card d)
+    (k : Finset (Card d) → QueryTree (SwitchIndex d)) (ω : SwitchIndex d → Bool)
+    (hS : orbitSet (butterflyPerm d (decodeButterfly d ω)) z ⊆ S)
+    (hV : Disjoint V (orbitSet (butterflyPerm d (decodeButterfly d ω)) z)) :
+    let p := butterflyPerm d (decodeButterfly d ω)
+    let W := V ∪ orbitSet p z
+    (seekCycle d S (p z) (Function.minimalPeriod p z-1) (p z) V k).succeeds ω = (k W).succeeds ω ∧
+      (seekCycle d S (p z) (Function.minimalPeriod p z-1) (p z) V k).staleCharge
+          (exposedPaths d (decodeButterfly d ω) V) ω =
+        (k W).staleCharge (exposedPaths d (decodeButterfly d ω) W) ω +
+          terminalCharge d (decodeButterfly d ω) z (W.erase z) := by
+  exact seekCycle_semantics _ _ _ _ _ _ _ _ _ _ (cycleTrace_orbit _ _ _ _ hS hV)
+
+end QueryTree
+
+lemma mem_exposedPaths (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) (s : SwitchIndex d) :
+    s ∈ exposedPaths d b V ↔ (switchUsers d b s).1 ∈ V ∨ (switchUsers d b s).2 ∈ V := by
+  classical
+  simp only [exposedPaths,Finset.mem_biUnion,mem_routeDomain_iff_user]
+  constructor
+  · rintro ⟨x,hx,h | h⟩
+    · exact Or.inl (h ▸ hx)
+    · exact Or.inr (h ▸ hx)
+  · rintro (h | h)
+    · exact ⟨_,h,Or.inl rfl⟩
+    · exact ⟨_,h,Or.inr rfl⟩
+
+lemma terminalCharge_sum (d : ℕ) (b : Butterfly d) (z : Card d) (V : Finset (Card d))
+    (hz : z ∉ V) :
+    (terminalCharge d b z V : ℝ) = ∑ s : SwitchIndex d,
+      ((if (switchUsers d b s).1 = z ∧ (switchUsers d b s).2 ∈ V then (1:ℝ) else 0) +
+      (if (switchUsers d b s).2 = z ∧ (switchUsers d b s).1 ∈ V then (1:ℝ) else 0)) := by
+  classical
+  have he : routeDomain d z (butterflyPerm d b z) ∩ exposedPaths d b V =
+      Finset.univ.filter (fun s =>
+        ((switchUsers d b s).1 = z ∧ (switchUsers d b s).2 ∈ V) ∨
+        ((switchUsers d b s).2 = z ∧ (switchUsers d b s).1 ∈ V)) := by
+    ext s
+    simp only [Finset.mem_inter,mem_routeDomain_iff_user,mem_exposedPaths,
+      Finset.mem_filter,Finset.mem_univ,true_and]
+    by_cases h₁ : (switchUsers d b s).1 = z <;>
+      by_cases h₂ : (switchUsers d b s).2 = z <;> simp [h₁,h₂,hz]
+  rw [terminalCharge,he]
+  simp only [← Finset.sum_boole]
+  apply Finset.sum_congr rfl
+  intro s _
+  by_cases h₁ : (switchUsers d b s).1 = z <;>
+    by_cases h₂ : (switchUsers d b s).2 = z <;> simp [h₁,h₂,hz]
+
+noncomputable def priorPaths {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) : Finset (Card d) :=
+  Finset.univ.filter (fun x => ∃ i : Fin A, o x = some i ∧ i.val < t)
+
+@[simp] lemma mem_priorPaths {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) (x : Card d) :
+    x ∈ priorPaths o t ↔ ∃ i : Fin A, o x = some i ∧ i.val < t := by
+  classical
+  simp [priorPaths]
+
+lemma terminalCharge_prior_sum {d A : ℕ} (o : Card d → Option (Fin A))
+    (b : Butterfly d) (z : Terminal o) (i : Fin A) :
+    (terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) =
+      ∑ s : SwitchIndex d,
+        ((if (z i).val = (switchUsers d b s).1 ∧ (switchUsers d b s).2 ∈ priorPaths o (i.val+1)
+          then (1:ℝ) else 0) +
+        (if (z i).val = (switchUsers d b s).2 ∧ (switchUsers d b s).1 ∈ priorPaths o (i.val+1)
+          then (1:ℝ) else 0)) := by
+  classical
+  rw [terminalCharge_sum _ _ _ _ (Finset.notMem_erase _ _)]
+  apply Finset.sum_congr rfl
+  intro s _
+  have hn := switchUsers_ne d b s
+  simp only [Finset.mem_erase]
+  by_cases h₁ : (z i).val = (switchUsers d b s).1 <;>
+    by_cases h₂ : (z i).val = (switchUsers d b s).2 <;> simp_all [eq_comm]
+
+lemma rotationMeeting_eq_slot_sum {d A : ℕ} (o : Card d → Option (Fin A))
+    (z : Terminal o) (u v : Card d) :
+    rotationMeeting o z u v = ∑ i : Fin A,
+      ((if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) +
+      (if (z i).val = v ∧ u ∈ priorPaths o (i.val+1) then (1:ℝ) else 0)) := by
+  classical
+  have only_owner (u : Card d) (a : Fin A) (hou : o u = some a)
+      (i : Fin A) (hia : i ≠ a) : (z i).val ≠ u := by
+    intro h
+    have hi := (z i).property
+    rw [h,hou] at hi
+    exact hia (Option.some.inj hi).symm
+  have sum_one (u v : Card d) (a : Fin A) (hou : o u = some a) :
+      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) =
+      if (z a).val = u ∧ v ∈ priorPaths o (a.val+1) then 1 else 0 := by
+    apply Finset.sum_eq_single a
+    · intro i _ hia
+      simp [only_owner u a hou i hia]
+    · simp
+  have sum_none (u v : Card d) (hou : o u = none) :
+      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) = 0 := by
+    apply Finset.sum_eq_zero
+    intro i _
+    have hn : (z i).val ≠ u := by
+      intro h
+      have hi := (z i).property
+      rw [h,hou] at hi
+      contradiction
+    simp [hn]
+  rw [Finset.sum_add_distrib]
+  cases hou : o u with
+  | none =>
+    rw [sum_none u v hou]
+    simp [rotationMeeting,hou,mem_priorPaths]
+  | some a =>
+    rw [sum_one u v a hou]
+    cases hov : o v with
+    | none =>
+      rw [sum_none v u hov]
+      simp [rotationMeeting,hou,hov,mem_priorPaths]
+    | some b =>
+      rw [sum_one v u b hov]
+      simp only [rotationMeeting,hou,hov,mem_priorPaths,Option.some.injEq]
+      rcases lt_trichotomy a b with hab | rfl | hba
+      · have hn : ¬ b.val < a.val+1 := by omega
+        have hy : a.val < b.val+1 := by omega
+        simp [hab,hn,hy]
+      · simp
+      · have hn : ¬ a.val < b.val+1 := by omega
+        have hy : b.val < a.val+1 := by omega
+        simp [hba,not_lt_of_ge (le_of_lt hba),hn,hy]
+
+lemma rotationCharge_eq_terminal_sum {d A : ℕ} (o : Card d → Option (Fin A))
+    (b : Butterfly d) (z : Terminal o) :
+    rotationCharge o b z = ∑ i : Fin A,
+      (terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) := by
+  simp only [rotationCharge,rotationMeeting_eq_slot_sum,terminalCharge_prior_sum]
+  exact Finset.sum_comm
+
+@[simp] lemma priorPaths_zero {d A : ℕ} (o : Card d → Option (Fin A)) : priorPaths o 0 = ∅ := by
+  classical
+  ext x
+  simp
+
+lemma priorPaths_succ {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) (ht : t < A) :
+    priorPaths o (t+1) = priorPaths o t ∪ cycleFiber o ⟨t,ht⟩ := by
+  classical
+  ext x
+  simp only [mem_priorPaths,Finset.mem_union,mem_cycleFiber]
+  constructor
+  · rintro ⟨i,hi,hit⟩
+    by_cases h : i.val < t
+    · exact Or.inl ⟨i,hi,h⟩
+    · have he : i = ⟨t,ht⟩ := Fin.ext (by change i.val = t; omega)
+      exact Or.inr (he ▸ hi)
+  · rintro (⟨i,hi,hit⟩ | hi)
+    · exact ⟨i,hi,by omega⟩
+    · exact ⟨⟨t,ht⟩,hi,by change t < t+1; omega⟩
+
+lemma priorPaths_disjoint {d A : ℕ} (o : Card d → Option (Fin A)) (i : Fin A) :
+    Disjoint (priorPaths o i.val) (cycleFiber o i) := by
+  classical
+  apply Finset.disjoint_left.mpr
+  intro x hx hx'
+  obtain ⟨j,hj,hji⟩ := (mem_priorPaths _ _ _).mp hx
+  have he : j = i := Option.some.inj (hj.symm.trans ((mem_cycleFiber _ _ _).mp hx'))
+  subst j
+  exact (lt_irrefl _ hji)
+
+namespace QueryTree
+
+lemma seekCycles_owned {d A : ℕ} (S : Finset (Card d)) (r : Fin A → ℕ)
+    (o : Card d → Option (Fin A)) (z : Terminal o) (ω : SwitchIndex d → Bool)
+    (hS : ∀ i, cycleFiber o i ⊆ S)
+    (hO : ∀ i, orbitSet (butterflyPerm d (decodeButterfly d ω)) (z i).val = cycleFiber o i)
+    (hC : ∀ i, (cycleFiber o i).card = r i+1) :
+    let p := butterflyPerm d (decodeButterfly d ω)
+    let L := List.ofFn (fun i : Fin A => (p (z i).val,r i))
+    (seekCycles d S L ∅).succeeds ω = true ∧
+      ((seekCycles d S L ∅).staleCharge ∅ ω : ℝ) = rotationCharge o (decodeButterfly d ω) z := by
+  classical
+  let b := decodeButterfly d ω
+  let p := butterflyPerm d b
+  let f (i : Fin A) := (p (z i).val,r i)
+  let c (i : Fin A) : ℝ := terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val)
+  have hc (i : Fin A) : Function.minimalPeriod p (z i).val - 1 = r i := by
+    have he := orbitSet_card p (z i).val
+    rw [hO,hC] at he
+    omega
+  have hs (t : ℕ) (ht : t ≤ A) :
+      (seekCycles d S ((List.ofFn f).drop t) (priorPaths o t)).succeeds ω = true ∧
+        ((seekCycles d S ((List.ofFn f).drop t) (priorPaths o t)).staleCharge (exposedPaths d b (priorPaths o t)) ω : ℝ) =
+          ((List.ofFn c).drop t).sum := by
+    induction ht using Nat.decreasingInduction with
+    | self => simp only [List.length_ofFn,le_refl,List.drop_eq_nil_of_le,List.sum_nil,seekCycles,succeeds,staleCharge,Nat.cast_zero,and_self]
+    | of_succ t ht ih =>
+      have htf : t < (List.ofFn f).length := by simpa using ht
+      have htc : t < (List.ofFn c).length := by simpa using ht
+      rw [List.drop_eq_getElem_cons htf,List.drop_eq_getElem_cons htc]
+      simp only [List.getElem_ofFn,List.sum_cons,seekCycles]
+      have h := seekCycle_orbit d S (priorPaths o t) (z ⟨t,ht⟩).val
+        (fun V => seekCycles d S ((List.ofFn f).drop (t+1)) V) ω
+        (by rw [hO]; exact hS _) (by rw [hO]; exact priorPaths_disjoint o ⟨t,ht⟩)
+      dsimp only at h
+      rw [hc,hO,←priorPaths_succ o t ht] at h
+      dsimp only [f]
+      constructor
+      · exact h.1.trans ih.1
+      · rw [h.2,Nat.cast_add,ih.2]
+        exact add_comm _ _
+  have h := hs 0 (Nat.zero_le _)
+  simp only [List.drop_zero,priorPaths_zero,exposedPaths_empty,List.sum_ofFn] at h
+  dsimp only
+  exact ⟨h.1,h.2.trans (rotationCharge_eq_terminal_sum o b z).symm⟩
+
+end QueryTree
+
+section Collections
+variable {d A : ℕ}
+
+def ValidCycleSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ)
+    (o : Card d → Option (Fin A)) : Prop :=
+  (∀ i, (cycleFiber o i).card = r i + 1) ∧
+  (∀ i, cycleFiber o i ⊆ S) ∧
+  (∀ i x, o x = some i → orbitSet p x = cycleFiber o i)
+
+abbrev CycleSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
+  {o : Card d → Option (Fin A) // ValidCycleSlots p S r o}
+
+noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
+    Fintype (CycleSlots p S r) := Fintype.ofFinite _
+
+lemma CycleSlots.terminal_nonempty {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (c : CycleSlots p S r) (i : Fin A) : Nonempty {x : Card d // c.val x = some i} := by
+  have hn : (cycleFiber c.val i).Nonempty := Finset.card_pos.mp (by rw [c.property.1]; omega)
+  obtain ⟨x,hx⟩ := hn
+  exact ⟨x,(mem_cycleFiber _ _ _).mp hx⟩
+
+lemma owner_ext {o o' : Card d → Option (Fin A)}
+    (h : ∀ i, cycleFiber o i = cycleFiber o' i) : o = o' := by
+  funext x
+  have he (i : Fin A) : o x = some i ↔ o' x = some i := by
+    simpa only [mem_cycleFiber] using (Finset.ext_iff.mp (h i) x)
+  cases ho : o x with
+  | some i => exact ((he i).mp ho).symm
+  | none =>
+    cases ho' : o' x with
+    | none => rfl
+    | some i => have hn := (he i).mpr ho'; rw [ho] at hn; contradiction
+
+abbrev RootedSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
+  Σ c : CycleSlots p S r, Terminal c.val
+
+noncomputable def rootedStart {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (c : RootedSlots p S r) : Fin A → S := fun i =>
+  ⟨p (c.2 i).val,c.1.property.2.1 i (by
+    rw [←c.1.property.2.2 i (c.2 i).val (c.2 i).property]
+    exact (mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.apply_right (Equiv.Perm.SameCycle.refl p _)))⟩
+
+lemma rootedStart_injective {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ} :
+    Function.Injective (rootedStart (p:=p) (S:=S) (r:=r)) := by
+  rintro ⟨c,z⟩ ⟨c',z'⟩ h
+  have hz (i : Fin A) : (z i).val = (z' i).val :=
+    p.injective (congrArg Subtype.val (congrFun h i))
+  have hc : c = c' := Subtype.ext (owner_ext (fun i => by
+    rw [←c.property.2.2 i (z i).val (z i).property,
+      ←c'.property.2.2 i (z' i).val (z' i).property,hz]))
+  subst c'
+  have hzz : z = z' := funext (fun i => Subtype.ext (hz i))
+  subst z'
+  rfl
+
+lemma sum_comp_injective_le {α β : Type*} [Fintype α] [Fintype β]
+    (e : α → β) (he : Function.Injective e) (f : β → ℝ) (hf : ∀ b, 0 ≤ f b) :
+    ∑ a, f (e a) ≤ ∑ b, f b := by
+  classical
+  calc
+    _ = ∑ b ∈ Finset.univ.image e, f b := (Finset.sum_image (fun _ _ _ _ h => he h)).symm
+    _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _) (fun b _ _ => hf b)
+
+lemma rooted_weight_sum_le (S : Finset (Card d)) (r : Fin A → ℕ) (ω : SwitchIndex d → Bool) :
+    (∑ c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r,
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2)) ≤
+    ∑ a : Fin A → S,
+      if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+        ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+      else 0 := by
+  classical
+  let p := butterflyPerm d (decodeButterfly d ω)
+  let g (a : Fin A → S) : ℝ :=
+    if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+    else 0
+  have he (c : RootedSlots p S r) :
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2) =
+      g (rootedStart c) := by
+    have hs := QueryTree.seekCycles_owned S r c.1.val c.2 ω c.1.property.2.1
+      (fun i => c.1.property.2.2 i (c.2 i).val (c.2 i).property) c.1.property.1
+    dsimp only at hs
+    unfold g
+    rw [ite_eq_left (show (QueryTree.seekCycles d S (List.ofFn (fun i => ((rootedStart c i).val,r i))) ∅).succeeds ω = true from hs.1)]
+    rw [← hs.2,neg_mul,Real.exp_neg,mul_comm (Real.log 2),Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
+    rfl
+  simp_rw [he]
+  exact sum_comp_injective_le rootedStart rootedStart_injective g (fun a => by dsimp [g]; split_ifs <;> positivity)
+
+noncomputable def rotationLower (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : ℝ :=
+  Real.exp (-Real.log 2 * ((A:ℝ)*d/2 + ∑ i : Fin A, Real.log (r i+1) / (2 * Real.log 2)))
+
+noncomputable def collectionWeight (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : ℝ :=
+  ((2:ℝ)^d)^A * (∏ i : Fin A, (r i+1 : ℝ)) * rotationLower d r
+
+lemma collectionWeight_pos (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : 0 < collectionWeight d r := by
+  unfold collectionWeight rotationLower
+  positivity
+
+lemma collection_rotation_lower (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r)
+    (ω : SwitchIndex d → Bool) (c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) :
+    collectionWeight d r ≤ ∑ z : Terminal c.val,
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z) := by
+  classical
+  let (i : Fin A) : Nonempty {x : Card d // c.val x = some i} := c.terminal_nonempty i
+  have hm : Monotone (fun i => (cycleFiber c.val i).card) := by
+    intro i j hij
+    dsimp only
+    rw [c.property.1,c.property.1]
+    exact Nat.add_le_add_right (hr hij) 1
+  have hb := rotation_exp_bound hm (decodeButterfly d ω)
+  have hcard : (Fintype.card (Terminal c.val) : ℝ) = ∏ i : Fin A, (r i+1 : ℝ) := by
+    rw [Fintype.card_pi,Nat.cast_prod]
+    apply Finset.prod_congr rfl
+    intro i _
+    rw [card_cycleFiber,c.property.1,Nat.cast_add,Nat.cast_one]
+  have hp : (0:ℝ) < Fintype.card (Terminal c.val) := Nat.cast_pos.mpr Fintype.card_pos
+  have hl : rotationLower d r ≤ finiteMean (fun z : Terminal c.val => Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z)) := by
+    simpa only [rotationLower,c.property.1,Nat.cast_add,Nat.cast_one] using hb
+  have hh := (le_div_iff₀ hp).mp hl
+  rw [hcard] at hh
+  unfold collectionWeight
+  rw [← Finset.mul_sum]
+  calc
+    _ = ((2:ℝ)^d)^A * (rotationLower d r * ∏ i : Fin A, (r i+1 : ℝ)) := by ring
+    _ ≤ _ := mul_le_mul_of_nonneg_left hh (by positivity)
+
+lemma cycleSlots_weighted_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) *
+      collectionWeight d r ≤ (S.card : ℝ)^A := by
+  classical
+  let g (a : Fin A → S) (ω : SwitchIndex d → Bool) : ℝ :=
+    if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+    else 0
+  have hω (ω : SwitchIndex d → Bool) :
+      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ) * collectionWeight d r ≤
+        ∑ a : Fin A → S, g a ω := by
+    calc
+      _ = ∑ _c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r, collectionWeight d r := by simp
+      _ ≤ ∑ c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r,
+          ∑ z : Terminal c.val, ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z) :=
+        Finset.sum_le_sum (fun c _ => collection_rotation_lower S r hr ω c)
+      _ = ∑ c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r,
+          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2) :=
+        (Fintype.sum_sigma (fun c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r =>
+          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2))).symm
+      _ ≤ _ := rooted_weight_sum_le S r ω
+  have hb := finiteMean_mono hω
+  rw [finiteMean_mul_const,finiteMean_sum] at hb
+  calc
+    _ ≤ ∑ a : Fin A → S, finiteMean (g a) := hb
+    _ ≤ ∑ _a : Fin A → S, (1:ℝ) := by
+      apply Finset.sum_le_sum
+      intro a _
+      have hh := QueryTree.cycles_weighted_success d S (List.ofFn (fun i : Fin A => ((a i).val,r i)))
+      simpa only [List.length_ofFn] using hh
+    _ = _ := by simp
+
+lemma cycleSlots_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) ≤
+        (S.card : ℝ)^A / collectionWeight d r := by
+  exact (le_div_iff₀ (collectionWeight_pos d r)).mpr (cycleSlots_weighted_moment S r hr)
+
+lemma collectionWeight_eq_sqrt (d : ℕ) {A : ℕ} (r : Fin A → ℕ) :
+    collectionWeight d r = (Real.sqrt ((2:ℝ)^d))^A * ∏ i : Fin A, Real.sqrt (r i+1) := by
+  have hn : ((2:ℝ)^d)^A = Real.exp ((A:ℝ)*d*Real.log 2) := by
+    rw [mul_assoc,Real.exp_nat_mul,Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
+  have hp : (∏ i : Fin A, (r i+1 : ℝ)) = Real.exp (∑ i : Fin A, Real.log (r i+1)) := by
+    rw [Real.exp_sum]
+    apply Finset.prod_congr rfl
+    intro i _
+    exact (Real.exp_log (by positivity)).symm
+  have hs : Real.sqrt ((2:ℝ)^d) = Real.exp ((d:ℝ)*Real.log 2/2) := by
+    rw [Real.sqrt_eq_rpow,Real.rpow_def_of_pos (by positivity),Real.log_pow]
+    congr 1
+    ring
+  have hj (i : Fin A) : Real.sqrt (r i+1) = Real.exp (Real.log (r i+1)/2) := by
+    rw [Real.sqrt_eq_rpow,Real.rpow_def_of_pos (by positivity)]
+    congr 1
+    ring
+  simp_rw [collectionWeight,hn,hp,rotationLower,hs,hj]
+  rw [←Real.exp_add,←Real.exp_add,←Real.exp_nat_mul,←Real.exp_sum,←Real.exp_add]
+  congr 1
+  rw [←Finset.sum_div,←Finset.sum_div]
+  have hl : Real.log 2 ≠ 0 := ne_of_gt (Real.log_pos (by norm_num))
+  field_simp
+  ring
+
+lemma butterfly_cycleSlots_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) ≤
+        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
+  have h := cycleSlots_moment S r hr
+  rw [collectionWeight_eq_sqrt,div_mul_eq_div_mul_one_div] at h
+  rw [div_pow]
+  simpa only [one_div,Finset.prod_inv_distrib] using h
+
+end Collections
+
+variable {d A : ℕ}
+
+abbrev SelectedCycle (p : Equiv.Perm (Card d)) (S : Finset (Card d)) :=
+  {C : Finset (Card d) // (∃ x, C = orbitSet p x) ∧ C ⊆ S}
+
+noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) :
+    Fintype (SelectedCycle p S) := Fintype.ofFinite _
+
+lemma SelectedCycle.nonempty {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
+    (C : SelectedCycle p S) : C.val.Nonempty := by
+  obtain ⟨x,hx⟩ := C.property.1
+  rw [hx]
+  exact orbitSet_nonempty p x
+
+lemma SelectedCycle.orbit_eq {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
+    (C : SelectedCycle p S) {x : Card d} (hx : x ∈ C.val) : orbitSet p x = C.val := by
+  obtain ⟨y,hy⟩ := C.property.1
+  rw [hy] at hx ⊢
+  exact (orbitSet_eq p ((mem_orbitSet _ _ _).mp hx)).symm
+
+lemma SelectedCycle.eq_of_mem {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
+    (C D : SelectedCycle p S) {x : Card d} (hC : x ∈ C.val) (hD : x ∈ D.val) : C = D := by
+  apply Subtype.ext
+  exact (C.orbit_eq hC).symm.trans (D.orbit_eq hD)
+
+abbrev SlotAssignment (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
+  {f : Fin A → SelectedCycle p S // Function.Injective f ∧ ∀ i, (f i).val.card = r i+1}
+
+noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
+    Fintype (SlotAssignment p S r) := Fintype.ofFinite _
+
+noncomputable def slotsToAssignment {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (c : CycleSlots p S r) : SlotAssignment p S r := by
+  classical
+  let f (i : Fin A) : SelectedCycle p S := ⟨cycleFiber c.val i,by
+    obtain ⟨x,hx⟩ := c.terminal_nonempty i
+    exact ⟨⟨x,(c.property.2.2 i x hx).symm⟩,c.property.2.1 i⟩⟩
+  refine ⟨f,?_,fun i => c.property.1 i⟩
+  intro i j hij
+  have he : cycleFiber c.val i = cycleFiber c.val j := congrArg Subtype.val hij
+  obtain ⟨x,hx⟩ := c.terminal_nonempty i
+  have hxi : x ∈ cycleFiber c.val i := (mem_cycleFiber _ _ _).mpr hx
+  rw [he] at hxi
+  exact Option.some.inj (hx.symm.trans ((mem_cycleFiber _ _ _).mp hxi))
+
+noncomputable def assignmentOwner {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) (x : Card d) : Option (Fin A) := by
+  classical
+  exact if h : ∃ i, x ∈ (f.val i).val then some (Classical.choose h) else none
+
+end Thorp
+
+end ThorpNine.Harmonic
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Harmonic/SandwichShuffle.lean b/lean/OAI/Probability/ThorpRouting/Harmonic/SandwichShuffle.lean
index f8a86b4ce2efdcb9e4f36e39e6c8883dc9fa09ca..e7f52ea033face800c2748cd4f140c8ca440a826 100644
--- a/lean/OAI/Probability/ThorpRouting/Harmonic/SandwichShuffle.lean
+++ b/lean/OAI/Probability/ThorpRouting/Harmonic/SandwichShuffle.lean
@@ -1,879 +1,879 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Harmonic.AlternatingPerm
-
-namespace OAI
-
-namespace ThorpNine.Harmonic
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-namespace PairRouting
-variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α] [DecidableEq ι]
-
-lemma sandwich_power_bound {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
-    (P : Ω → Equiv.Perm α) (Q : Ω' → Equiv.Perm α) (x y : ι → Bool × α)
-    (a : ℝ) (ha : 0 ≤ a) :
-    (finiteMean (fun ω : Ω × Ω' => finiteMean (fun coins : (α → Bool) × (α → Bool) =>
-      if ∀ i, sandwich coins.1 coins.2 (fun b => if b then Q ω.2 else P ω.1) (x i) = y i
-      then (1:ℝ) else 0)))^(1+a) ≤
-    (outerWeight x y)^(1+a) * ((compatibleSet x y).card:ℝ)^a *
-      ∑ c ∈ compatibleSet x y,
-      (tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
-        (fun i : {i // c i = false} => (y i.val).2))^(1+a) *
-      (tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
-        (fun i : {i // c i = true} => (y i.val).2))^(1+a) := by
-  classical
-  rw [sandwich_mean,sum_colors_factor]
-  let f (c : ι → Bool) : ℝ :=
-      tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
-        (fun i : {i // c i = false} => (y i.val).2) *
-      tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
-        (fun i : {i // c i = true} => (y i.val).2)
-  have hf (c : ι → Bool) : 0 ≤ f c := mul_nonneg (tupleProbability_nonneg _ _ _)
-    (tupleProbability_nonneg _ _ _)
-  rw [Real.mul_rpow (le_of_lt (outerWeight_pos _ _)) (Finset.sum_nonneg (fun c _ => hf c))]
-  have hh := Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg (compatibleSet x y)
-    (show 1 ≤ 1+a by linarith) (fun c _ => hf c)
-  have he : 1+a-1=a := by ring
-  rw [he] at hh
-  calc
-    _ ≤ (outerWeight x y)^(1+a) *
-        (((compatibleSet x y).card:ℝ)^a * ∑ c ∈ compatibleSet x y, (f c)^(1+a)) :=
-      mul_le_mul_of_nonneg_left hh (Real.rpow_nonneg (le_of_lt (outerWeight_pos _ _)) _)
-    _ = _ := by
-      rw [←mul_assoc]
-      congr 1
-      apply Finset.sum_congr rfl
-      intro c _
-      exact Real.mul_rpow (tupleProbability_nonneg _ _ _) (tupleProbability_nonneg _ _ _)
-
-end PairRouting
-
-def sandwichShuffle {A B C D E F : Type*} :
-    ((A × B) × C) × ((D × E) × F) ≃ ((A × D) × (B × E)) × (F × C) where
-  toFun x := (((x.1.1.1,x.2.1.1),(x.1.1.2,x.2.1.2)),(x.2.2,x.1.2))
-  invFun x := (((x.1.1.1,x.1.2.1),x.2.2),((x.1.1.2,x.1.2.2),x.2.1))
-  left_inv _ := rfl
-  right_inv _ := rfl
-
-def benesStepEquiv (d : ℕ) : BenesCoins (d+1) ≃
-    (BenesCoins d × BenesCoins d) × ((Card d → Bool) × (Card d → Bool)) :=
-  (Equiv.prodCongr (coinStepEquiv d) (coinStepEquiv d)).trans sandwichShuffle
-
-lemma childLift_mul (d : ℕ) (p q : Bool → Equiv.Perm (Card d)) :
-    childLift p * childLift q = childLift (fun b => p b * q b) := by
-  ext x i
-  simp only [Equiv.Perm.mul_apply,childLift,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons]
-
-lemma childLift_inv (d : ℕ) (p : Bool → Equiv.Perm (Card d)) :
-    (childLift p)⁻¹ = childLift (fun b => (p b)⁻¹) := rfl
-
-lemma pairSwitch_inv (d : ℕ) (ξ : Card d → Bool) : (pairSwitch ξ)⁻¹ = pairSwitch ξ := rfl
-
-lemma palindromePerm_step (d : ℕ) (c : BenesCoins d × BenesCoins d)
-    (coins : (Card d → Bool) × (Card d → Bool)) :
-    palindromePerm (d+1) ((benesStepEquiv d).symm (c,coins)) =
-      pairSwitch coins.2 * childLift (fun b => palindromePerm d (if b then c.2 else c.1)) *
-        pairSwitch coins.1 := by
-  change (butterflyPerm (d+1) (decodeButterfly (d+1)
-      ((coinStepEquiv d).symm ((c.1.1,c.2.1),coins.2)))) *
-    (butterflyPerm (d+1) (decodeButterfly (d+1)
-      ((coinStepEquiv d).symm ((c.1.2,c.2.2),coins.1))))⁻¹ = _
-  rw [butterflyPerm_step,butterflyPerm_step,mul_inv_rev,childLift_inv,pairSwitch_inv]
-  rw [mul_assoc (pairSwitch coins.2),←mul_assoc (childLift _),childLift_mul,←mul_assoc]
-  congr 2
-  apply congrArg childLift
-  funext b
-  cases b <;> rfl
-
-lemma headTail_switch (d : ℕ) (ξ : Card d → Bool) (x : Card (d+1)) :
-    headTailEquiv d (pairSwitch ξ x) = pairLayer ξ (headTailEquiv d x) := rfl
-
-lemma headTail_child (d : ℕ) (p : Bool → Equiv.Perm (Card d)) (x : Card (d+1)) :
-    headTailEquiv d (childLift p x) = PairRouting.lift p (headTailEquiv d x) := rfl
-
-lemma palindromePerm_coordinates (d : ℕ) (c : BenesCoins d × BenesCoins d)
-    (coins : (Card d → Bool) × (Card d → Bool)) (x : Card (d+1)) :
-    headTailEquiv d (palindromePerm (d+1) ((benesStepEquiv d).symm (c,coins)) x) =
-      PairRouting.sandwich coins.1 coins.2 (fun b => palindromePerm d (if b then c.2 else c.1))
-        (headTailEquiv d x) := by
-  rw [palindromePerm_step]
-  simp only [Equiv.Perm.mul_apply,headTail_switch,headTail_child]
-  rfl
-
-noncomputable def palindromeCost : (d : ℕ) → {ι : Type*} → [Fintype ι] →
-    (ι ↪ Card d) → BenesCoins d → ℕ
-  | 0, _, _, _, _ => 0
-  | d+1, _, _, e, ω =>
-      let σ := benesStepEquiv d ω
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x σ.2.1
-      let hc := PairRouting.colors_compatible x σ.2.1
-      PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
-          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) +
-        palindromeCost d (PairRouting.childEmbedding x c hc false) σ.1.1 +
-        palindromeCost d (PairRouting.childEmbedding x c hc true) σ.1.2
-
-namespace CycleColoring
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-lemma alternating_cycle_two (p : Equiv.Perm α) (S : Finset (Bool × α))
-    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) S) : 2 ≤ C.val.card := by
-  obtain ⟨x,hx⟩ := C.nonempty
-  have hpx : alternatingPerm p x ∈ C.val := by
-    rw [←C.orbit_eq hx,mem_orbitSet]
-    exact (Equiv.Perm.SameCycle.refl _ _).apply_right
-  have hne : x ≠ alternatingPerm p x := by
-    intro he
-    have hh := congrArg Prod.fst he
-    rw [alternatingPerm_flip] at hh
-    cases hb : x.1 <;> simp only [hb, Bool.not_false, Bool.not_true] at hh <;> cases hh
-  have hs : {x,alternatingPerm p x} ⊆ C.val := by
-    intro y hy
-    simp only [Finset.mem_insert,Finset.mem_singleton] at hy
-    rcases hy with rfl | rfl <;> assumption
-  have hh := Finset.card_le_card hs
-  simpa only [Finset.card_pair hne] using hh
-
-lemma alternating_cycle_count_double (p : Equiv.Perm α) (S : Finset (Bool × α)) :
-    2 * Fintype.card (RoutingNetwork.SelectedCycle (alternatingPerm p) S) ≤ S.card := by
-  classical
-  calc
-    _ = ∑ _C : RoutingNetwork.SelectedCycle (alternatingPerm p) S, 2 := by simp [Nat.mul_comm]
-    _ ≤ RoutingNetwork.cycleMass (alternatingPerm p) S :=
-      Finset.sum_le_sum (fun C _ => alternating_cycle_two p S C)
-    _ ≤ _ := RoutingNetwork.cycleMass_le _ _
-
-lemma alternating_cycle_count_zero (p : Equiv.Perm α) (S : Finset (Bool × α))
-    (hS : S.card ≤ 1) : Fintype.card (RoutingNetwork.SelectedCycle (alternatingPerm p) S) = 0 := by
-  have h := alternating_cycle_count_double p S
-  omega
-
-end CycleColoring
-
-namespace PairRouting
-
-section
-variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
-
-lemma alternatingCycles_double (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    2 * alternatingCycles e p ≤ Fintype.card ι := by
-  have hh := CycleColoring.alternating_cycle_count_double (alternatingProjection p) (labelSet e)
-  simpa only [alternatingCycles, card_labelSet] using hh
-
-lemma alternatingCycles_zero (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α)
-    (hι : Fintype.card ι ≤ 1) : alternatingCycles e p = 0 := by
-  have hh := alternatingCycles_double e p
-  omega
-
-end
-variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α] [DecidableEq ι]
-
-omit [Fintype α] [DecidableEq ι] in
-lemma colors_chosenCoin (x : ι → Bool × α) (c : ι → Bool) (hc : Compatible x c) :
-    colors x (chosenCoin x c) = c := by
-  funext i
-  simp only [colors,chosenCoin_apply x c hc i,required]
-  cases (x i).1 <;> cases c i <;> rfl
-
-noncomputable def coloredEmbedding (x : ι ↪ Bool × α) (c : ι → Bool) (_hc : Compatible x c) :
-    ι ↪ Bool × α := switchedEmbedding x (chosenCoin x c)
-
-omit [Fintype α] [DecidableEq ι] in
-lemma coloredEmbedding_apply (x : ι ↪ Bool × α) (c : ι → Bool) (hc : Compatible x c) (i : ι) :
-    coloredEmbedding x c hc i = (c i,(x i).2) := by
-  apply Prod.ext
-  · exact congrFun (colors_chosenCoin x c hc) i
-  · rfl
-
-omit [Fintype α] [DecidableEq ι] in
-lemma switchedEmbedding_eq_colored (x : ι ↪ Bool × α) (c : ι → Bool) (hc : Compatible x c)
-    (ξ : α → Bool) (he : colors x ξ = c) :
-    switchedEmbedding x ξ = coloredEmbedding x c hc := by
-  ext i : 1
-  rw [coloredEmbedding_apply]
-  apply Prod.ext
-  · exact congrFun he i
-  · rfl
-
-lemma assignments_match_bound (x y : ι ↪ Bool × α) (c : ι → Bool)
-    (hc : c ∈ compatibleSet x y) (p : Bool → Equiv.Perm α) (hm : ChildMatches p x y c) :
-    (compatibleSet x y).card ≤
-      2 ^ ((occupied x).card + (occupied y).card - Fintype.card ι +
-        alternatingCycles (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p) := by
-  obtain ⟨hcx,hcy⟩ := (mem_compatibleSet x y c).mp hc
-  let ξ := chosenCoin x c
-  let η := chosenCoin y c
-  have he : routedEmbedding x ξ η p = y := by
-    ext i : 1
-    have h := (sandwich_matches_iff ξ η p x y).mpr
-      ⟨(colors_chosenCoin x c hcx).trans (colors_chosenCoin y c hcy).symm,by
-        simpa only [ξ,colors_chosenCoin x c hcx] using hm⟩
-    exact h i
-  have hh := assignments_actual_bound x ξ η p
-  rw [he] at hh
-  rw [card_compatibleSet]
-  exact hh
-
-omit [Fintype α] [DecidableEq ι] in
-lemma occupied_sum_ge (x y : ι ↪ Bool × α) :
-    Fintype.card ι ≤ (occupied x).card + (occupied y).card := by
-  have h₀ := occupied_card_double x
-  have h₁ := occupied_card_double y
-  omega
-
-lemma assignment_weight_bound (x y : ι ↪ Bool × α) (c : ι → Bool)
-    (hc : c ∈ compatibleSet x y) (p : Bool → Equiv.Perm α) (hm : ChildMatches p x y c) :
-    ((compatibleSet x y).card:ℝ) * outerWeight x y ≤
-      (2:ℝ)^((alternatingCycles
-        (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p : ℝ) - Fintype.card ι) := by
-  let z := alternatingCycles (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p
-  let b := (occupied x).card+(occupied y).card
-  let h := Fintype.card ι
-  have hb : h ≤ b := occupied_sum_ge x y
-  have hn : ((compatibleSet x y).card:ℝ) ≤ (2:ℝ)^(b-h+z) := by
-    exact_mod_cast assignments_match_bound x y c hc p hm
-  change ((compatibleSet x y).card:ℝ) * (1/(2:ℝ)^b) ≤ (2:ℝ)^((z:ℝ)-h)
-  rw [mul_one_div]
-  calc
-    _ ≤ (2:ℝ)^(b-h+z)/(2:ℝ)^b := div_le_div_of_nonneg_right hn (by positivity)
-    _ = _ := by
-      rw [←Real.rpow_natCast,←Real.rpow_natCast,←Real.rpow_sub (by norm_num : (0:ℝ)<2)]
-      congr 1
-      rw [Nat.cast_add,Nat.cast_sub hb]
-      ring
-
-lemma assignment_factor_bound (x y : ι ↪ Bool × α) (c : ι → Bool)
-    (hc : c ∈ compatibleSet x y) (p : Bool → Equiv.Perm α) (hm : ChildMatches p x y c)
-    (a : ℝ) (ha : 0 ≤ a) :
-    (outerWeight x y)^a * ((compatibleSet x y).card:ℝ)^a ≤
-      (2:ℝ)^(((alternatingCycles
-        (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p : ℝ) - Fintype.card ι)*a) := by
-  have h := Real.rpow_le_rpow (mul_nonneg (Nat.cast_nonneg _) (le_of_lt (outerWeight_pos x y)))
-    (assignment_weight_bound x y c hc p hm) ha
-  rw [Real.mul_rpow (Nat.cast_nonneg _) (le_of_lt (outerWeight_pos x y)),
-    ←Real.rpow_mul (by norm_num : (0:ℝ)≤2)] at h
-  simpa only [mul_comm] using h
-
-
-lemma sandwich_weighted_probability_subtype (p : Bool → Equiv.Perm α) (x y : ι → Bool × α)
-    (W : (ι → Bool) → ℝ) :
-    finiteMean (fun coins : (α → Bool) × (α → Bool) =>
-      if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then W (colors x coins.1) else 0) =
-    outerWeight x y * ∑ c : {c // c ∈ compatibleSet x y},
-      if ChildMatches p x y c.val then W c.val else 0 := by
-  rw [sandwich_weighted_probability]
-  congr 1
-  rw [Finset.sum_subtype (compatibleSet x y) (fun _ => Iff.rfl)]
-
-omit [Fintype α] [DecidableEq α] [DecidableEq ι] in
-lemma color_card_add (c : ι → Bool) :
-    Fintype.card {i // c i = false} + Fintype.card {i // c i = true} = Fintype.card ι := by
-  have he : {i // c i = true} ≃ {i // ¬c i = false} := Equiv.subtypeEquivRight fun i => by
-    cases c i <;> simp
-  rw [Fintype.card_congr he]
-  rw [Fintype.card_subtype_compl]
-  exact Nat.add_sub_of_le (Fintype.card_subtype_le _)
-
-end PairRouting
-
-noncomputable def assignmentCost (d : ℕ) {ι : Type*} [Fintype ι]
-    (x : ι ↪ Bool × Card d) (c : ι → Bool) (hc : PairRouting.Compatible x c)
-    (ω : BenesCoins d × BenesCoins d) : ℕ :=
-  PairRouting.alternatingCycles (PairRouting.coloredEmbedding x c hc)
-      (fun b => palindromePerm d (if b then ω.2 else ω.1)) +
-    palindromeCost d (PairRouting.childEmbedding x c hc false) ω.1 +
-    palindromeCost d (PairRouting.childEmbedding x c hc true) ω.2
-
-lemma palindromeCost_eq (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (d+1)) (σ : BenesCoins (d+1)) :
-    palindromeCost (d+1) e σ =
-      assignmentCost d (e.trans (headTailEquiv d).toEmbedding)
-        (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) (benesStepEquiv d σ).2.1)
-        (PairRouting.colors_compatible _ _) (benesStepEquiv d σ).1 := by
-  unfold palindromeCost assignmentCost
-  dsimp only
-  rw [PairRouting.switchedEmbedding_eq_colored _ _ (PairRouting.colors_compatible _ _) _ rfl]
-
-lemma palindromeCost_step (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (d+1)) (ω : BenesCoins d × BenesCoins d)
-    (coins : (Card d → Bool) × (Card d → Bool)) :
-    palindromeCost (d+1) e ((benesStepEquiv d).symm (ω,coins)) =
-      assignmentCost d (e.trans (headTailEquiv d).toEmbedding)
-        (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) coins.1)
-        (PairRouting.colors_compatible _ _) ω := by
-  rw [palindromeCost_eq]
-  simp only [Equiv.apply_symm_apply]
-
-noncomputable def routingMoment (d : ℕ) {ι : Type*} [Fintype ι]
-    (e f : ι ↪ Card d) (a : ℝ) : ℝ := by
-  classical
-  exact finiteMean (fun ω : BenesCoins d =>
-    if ∀ i, palindromePerm d ω (e i) = f i then (2:ℝ)^((palindromeCost d e ω:ℝ)*a) else 0)
-
-lemma routingMoment_nonneg (d : ℕ) {ι : Type*} [Fintype ι]
-    (e f : ι ↪ Card d) (a : ℝ) : 0 ≤ routingMoment d e f a := by
-  classical
-  exact finiteMean_nonneg (fun _ => by split_ifs <;> positivity)
-
-lemma palindrome_matches_step (d : ℕ) {ι : Type*}
-    (e f : ι ↪ Card (d+1)) (ω : BenesCoins d × BenesCoins d)
-    (coins : (Card d → Bool) × (Card d → Bool)) :
-    (∀ i, palindromePerm (d+1) ((benesStepEquiv d).symm (ω,coins)) (e i) = f i) ↔
-    (∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => palindromePerm d (if b then ω.2 else ω.1))
-      ((e.trans (headTailEquiv d).toEmbedding) i) = ((f.trans (headTailEquiv d).toEmbedding) i)) := by
-  apply forall_congr'
-  intro i
-  rw [←(headTailEquiv d).injective.eq_iff,palindromePerm_coordinates]
-  rfl
-
-lemma routingMoment_step (d : ℕ) {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (e f : ι ↪ Card (d+1)) (a : ℝ) :
-    let x := e.trans (headTailEquiv d).toEmbedding
-    let y := f.trans (headTailEquiv d).toEmbedding
-    routingMoment (d+1) e f a = PairRouting.outerWeight x y *
-      ∑ c : {c // c ∈ PairRouting.compatibleSet x y},
-        finiteMean (fun ω : BenesCoins d × BenesCoins d =>
-          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c.val
-          then (2:ℝ)^((assignmentCost d x c.val
-            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0) := by
-  classical
-  intro x y
-  rw [routingMoment,finiteMean_equiv (benesStepEquiv d),finiteMean_prod]
-  have he (ω : BenesCoins d × BenesCoins d) :
-      finiteMean (fun coins : (Card d → Bool) × (Card d → Bool) =>
-        if ∀ i, palindromePerm (d+1) ((benesStepEquiv d).symm (ω,coins)) (e i) = f i
-        then (2:ℝ)^((palindromeCost (d+1) e ((benesStepEquiv d).symm (ω,coins)):ℝ)*a) else 0) =
-      PairRouting.outerWeight x y *
-        ∑ c : {c // c ∈ PairRouting.compatibleSet x y},
-          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c.val
-          then (2:ℝ)^((assignmentCost d x c.val
-            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0 := by
-    let W : (ι → Bool) → ℝ := fun c =>
-      if hc : PairRouting.Compatible x c then (2:ℝ)^((assignmentCost d x c hc ω:ℝ)*a) else 0
-    calc
-      _ = finiteMean (fun coins : (Card d → Bool) × (Card d → Bool) =>
-        if ∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => palindromePerm d (if b then ω.2 else ω.1))
-          (x i) = y i then W (PairRouting.colors x coins.1) else 0) := by
-        apply finiteMean_congr
-        intro coins
-        simp only [palindrome_matches_step,palindromeCost_step]
-        simp only [W,PairRouting.colors_compatible,↓reduceDIte]
-        rfl
-      _ = _ := by
-        rw [PairRouting.sandwich_weighted_probability_subtype]
-        congr 1
-        apply Finset.sum_congr rfl
-        intro c _
-        simp only [W,((PairRouting.mem_compatibleSet x y c.val).mp c.property).1,↓reduceDIte]
-  simp_rw [he]
-  simp_rw [mul_comm (PairRouting.outerWeight x y)]
-  rw [finiteMean_mul_const,finiteMean_sum,mul_comm]
-
-lemma child_moment_product (d : ℕ) {ι : Type*} [Fintype ι]
-    (x y : ι ↪ Bool × Card d) (c : ι → Bool)
-    (hcx : PairRouting.Compatible x c) (hcy : PairRouting.Compatible y c) (a : ℝ) :
-    routingMoment d (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a *
-      routingMoment d (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a =
-    finiteMean (fun ω : BenesCoins d × BenesCoins d =>
-      if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
-      then (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
-        palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0) := by
-  classical
-  unfold routingMoment
-  rw [←finiteMean_prod_mul]
-  apply finiteMean_congr
-  intro ω
-  let M₀ := ∀ i : {i // c i = false}, palindromePerm d ω.1 (x i.val).2 = (y i.val).2
-  let M₁ := ∀ i : {i // c i = true}, palindromePerm d ω.2 (x i.val).2 = (y i.val).2
-  have hp : (fun b : Bool => palindromePerm d (if b then ω.2 else ω.1)) =
-      (fun b => if b then palindromePerm d ω.2 else palindromePerm d ω.1) := by
-    funext b
-    cases b <;> rfl
-  have hs : PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c ↔
-      M₀ ∧ M₁ := by rw [hp]; exact PairRouting.childMatches_split _ _ _ _ _
-  change (if M₀ then _ else 0) * (if M₁ then _ else 0) = _
-  by_cases h₀ : M₀ <;> by_cases h₁ : M₁
-  · simp only [hs,h₀,h₁,true_and,↓reduceIte]
-    rw [←Real.rpow_add (by norm_num : (0:ℝ)<2)]
-    congr 1
-    ring
-  · simp only [hs,h₀,h₁,and_false,↓reduceIte,mul_zero]
-  · simp only [hs,h₀,h₁,false_and,↓reduceIte,zero_mul]
-  · simp only [hs,h₀,h₁,false_and,↓reduceIte,mul_zero]
-
-lemma palindrome_probability_step (d : ℕ) {ι : Type*} [Fintype ι]
-    (e f : ι ↪ Card (d+1)) :
-    PairRouting.tupleProbability (palindromePerm (d+1)) e f =
-    finiteMean (fun ω : BenesCoins d × BenesCoins d =>
-      finiteMean (fun coins : (Card d → Bool) × (Card d → Bool) =>
-        if ∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => if b then palindromePerm d ω.2 else palindromePerm d ω.1)
-          ((e.trans (headTailEquiv d).toEmbedding) i) = ((f.trans (headTailEquiv d).toEmbedding) i)
-        then (1:ℝ) else 0)) := by
-  classical
-  rw [PairRouting.tupleProbability,finiteMean_equiv (benesStepEquiv d),finiteMean_prod]
-  have hp (b : Bool) (p q : BenesCoins d) :
-      palindromePerm d (if b then p else q) = (if b then palindromePerm d p else palindromePerm d q) := by
-    cases b <;> rfl
-  simp only [palindrome_matches_step,hp]
-
-lemma child_power_product_bound (d : ℕ) {ι : Type*} [Fintype ι]
-    (x y : ι ↪ Bool × Card d) (c : ι → Bool)
-    (hcx : PairRouting.Compatible x c) (hcy : PairRouting.Compatible y c) (a : ℝ)
-    (hfalse : (PairRouting.tupleProbability (palindromePerm d)
-        (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false))^(1+a) ≤
-      (2:ℝ)^(-(d:ℝ)*Fintype.card {i // c i = false}*a) *
-        routingMoment d (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a)
-    (htrue : (PairRouting.tupleProbability (palindromePerm d)
-        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true))^(1+a) ≤
-      (2:ℝ)^(-(d:ℝ)*Fintype.card {i // c i = true}*a) *
-        routingMoment d (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a) :
-    (PairRouting.tupleProbability (palindromePerm d)
-      (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false))^(1+a) *
-      (PairRouting.tupleProbability (palindromePerm d)
-        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true))^(1+a) ≤
-    (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) *
-      finiteMean (fun ω : BenesCoins d × BenesCoins d =>
-        if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
-        then (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
-          palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0) := by
-  have hh := mul_le_mul hfalse htrue
-    (Real.rpow_nonneg (PairRouting.tupleProbability_nonneg _ _ _) _)
-    (mul_nonneg (by positivity) (routingMoment_nonneg _ _ _ _))
-  rw [mul_mul_mul_comm,child_moment_product,←Real.rpow_add (by norm_num : (0:ℝ)<2)] at hh
-  have he : -(d:ℝ)*Fintype.card {i // c i = false}*a + -(d:ℝ)*Fintype.card {i // c i = true}*a =
-      -(d:ℝ)*Fintype.card ι*a := by
-    rw [←add_mul,←mul_add,←Nat.cast_add,PairRouting.color_card_add]
-  rwa [he] at hh
-
-lemma assignment_moment_bound (d : ℕ) {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (x y : ι ↪ Bool × Card d) (c : ι → Bool) (hc : c ∈ PairRouting.compatibleSet x y)
-    (a : ℝ) (ha : 0 ≤ a) :
-    let hcx := ((PairRouting.mem_compatibleSet x y c).mp hc).1
-    PairRouting.outerWeight x y ^ a * ((PairRouting.compatibleSet x y).card:ℝ)^a *
-      ((2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) *
-        finiteMean (fun ω : BenesCoins d × BenesCoins d =>
-          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
-          then (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
-            palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0)) ≤
-    (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) *
-      finiteMean (fun ω : BenesCoins d × BenesCoins d =>
-        if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
-        then (2:ℝ)^((assignmentCost d x c hcx ω:ℝ)*a) else 0) := by
-  classical
-  intro hcx
-  have hscale (q : ℝ) (F : BenesCoins d × BenesCoins d → ℝ) :
-      q * finiteMean F = finiteMean (fun ω => q * F ω) := by
-    simp_rw [mul_comm q,finiteMean_mul_const]
-  rw [←mul_assoc,hscale,hscale]
-  apply finiteMean_mono
-  intro ω
-  split_ifs with hm
-  · have hh := PairRouting.assignment_factor_bound x y c hc
-      (fun b => palindromePerm d (if b then ω.2 else ω.1)) hm a ha
-    calc
-      _ ≤ ((2:ℝ)^(((PairRouting.alternatingCycles (PairRouting.coloredEmbedding x c hcx)
-          (fun b => palindromePerm d (if b then ω.2 else ω.1)):ℝ)-Fintype.card ι)*a) *
-          (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a)) *
-          (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
-            palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) :=
-        mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hh (by positivity)) (by positivity)
-      _ = _ := by
-        simp only [←Real.rpow_add (by norm_num : (0:ℝ)<2),assignmentCost,Nat.cast_add]
-        congr 1
-        ring
-  · simp only [mul_zero]
-    exact le_rfl
-
-theorem palindrome_pointwise_moment (d : ℕ) {ι : Type*} [Fintype ι]
-    (e f : ι ↪ Card d) (a : ℝ) (ha : 0 ≤ a) :
-    (PairRouting.tupleProbability (palindromePerm d) e f)^(1+a) ≤
-      (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) * routingMoment d e f a := by
-  classical
-  induction d generalizing ι with
-  | zero =>
-      have he : ∀ (ω : BenesCoins 0) (i : ι), palindromePerm 0 ω (e i) = f i :=
-        fun _ _ => Subsingleton.elim _ _
-      simp [PairRouting.tupleProbability,routingMoment,he,palindromeCost,finiteMean_const]
-  | succ d ih =>
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let y := f.trans (headTailEquiv d).toEmbedding
-      let W := PairRouting.outerWeight x y
-      let C := PairRouting.compatibleSet x y
-      let p : (ι → Bool) → ℝ := fun c =>
-        (PairRouting.tupleProbability (palindromePerm d)
-          (fun i : {i // c i = false} => (x i.val).2)
-          (fun i : {i // c i = false} => (y i.val).2))^(1+a) *
-        (PairRouting.tupleProbability (palindromePerm d)
-          (fun i : {i // c i = true} => (x i.val).2)
-          (fun i : {i // c i = true} => (y i.val).2))^(1+a)
-      let m : {c // c ∈ C} → ℝ := fun c =>
-        finiteMean (fun ω : BenesCoins d × BenesCoins d =>
-          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c.val
-          then (2:ℝ)^((assignmentCost d x c.val
-            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0)
-      have hp (c : {c // c ∈ C}) : W^a * (C.card:ℝ)^a * p c.val ≤
-          (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * m c := by
-        obtain ⟨hcx,hcy⟩ := (PairRouting.mem_compatibleSet x y c.val).mp c.property
-        have h₀ := ih (PairRouting.childEmbedding x c.val hcx false)
-          (PairRouting.childEmbedding y c.val hcy false)
-        have h₁ := ih (PairRouting.childEmbedding x c.val hcx true)
-          (PairRouting.childEmbedding y c.val hcy true)
-        have hh := child_power_product_bound d x y c.val hcx hcy a h₀ h₁
-        have hh' := mul_le_mul_of_nonneg_left hh
-          (mul_nonneg (Real.rpow_nonneg (le_of_lt (PairRouting.outerWeight_pos x y)) a)
-            (Real.rpow_nonneg (Nat.cast_nonneg C.card) a))
-        exact hh'.trans (assignment_moment_bound d x y c.val c.property a ha)
-      have hnode := PairRouting.sandwich_power_bound (palindromePerm d) (palindromePerm d) x y a ha
-      rw [←palindrome_probability_step d e f] at hnode
-      change (PairRouting.tupleProbability (palindromePerm (d+1)) e f)^(1+a) ≤
-        W^(1+a) * (C.card:ℝ)^a * ∑ c ∈ C, p c at hnode
-      have hsum : ∑ c ∈ C, p c = ∑ c : {c // c ∈ C}, p c.val := by
-        rw [Finset.sum_subtype C (fun _ => Iff.rfl)]
-      have hW : 0 < W := PairRouting.outerWeight_pos x y
-      calc
-        _ ≤ W^(1+a) * (C.card:ℝ)^a * ∑ c ∈ C, p c := hnode
-        _ = W * ∑ c : {c // c ∈ C}, W^a * (C.card:ℝ)^a * p c.val := by
-          rw [Real.rpow_add hW,Real.rpow_one,hsum,←Finset.mul_sum]
-          ring
-        _ ≤ W * ∑ c : {c // c ∈ C}, (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * m c :=
-          mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun c _ => hp c)) (le_of_lt hW)
-        _ = _ := by
-          rw [←Finset.mul_sum,routingMoment_step]
-          simp only [Nat.cast_add,Nat.cast_one]
-          change W * ((2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * ∑ c, m c) =
-            (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * (W * ∑ c, m c)
-          ring
-
-lemma routingMoment_sum (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (a : ℝ) :
-    (∑ f : ι ↪ Card d, routingMoment d e f a) =
-      finiteMean (fun ω : BenesCoins d => (2:ℝ)^((palindromeCost d e ω:ℝ)*a)) := by
-  classical
-  unfold routingMoment
-  rw [←finiteMean_sum]
-  apply finiteMean_congr
-  intro ω
-  let f₀ : ι ↪ Card d := e.trans (palindromePerm d ω).toEmbedding
-  rw [Finset.sum_eq_single f₀]
-  · have hm : ∀ i, palindromePerm d ω (e i) = f₀ i := fun _ => rfl
-    simp only [hm,implies_true,↓reduceIte]
-  · intro f _ hf
-    have hm : ¬ ∀ i, palindromePerm d ω (e i) = f i := by
-      intro hh
-      apply hf
-      ext i : 1
-      exact (hh i).symm
-    exact ite_eq_right hm
-  · simp
-
-lemma finite_row_power {X : Type*} [Fintype X] [Nonempty X]
-    (p : X → ℝ) (hp : ∀ x, 0 ≤ p x) (a : ℝ) :
-    finiteMean (fun x => ((Fintype.card X:ℝ)*p x)^(1+a)) =
-      (Fintype.card X:ℝ)^a * ∑ x, (p x)^(1+a) := by
-  have hn : (0:ℝ) < Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
-  simp only [finiteMean,Real.mul_rpow (le_of_lt hn) (hp _),←Finset.mul_sum]
-  rw [Real.rpow_add hn,Real.rpow_one]
-  field_simp
-
-noncomputable def palindromeRowMoment (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (a : ℝ) : ℝ :=
-  finiteMean (fun f : ι ↪ Card d =>
-    ((Fintype.card (ι ↪ Card d):ℝ)*PairRouting.tupleProbability (palindromePerm d) e f)^(1+a))
-
-lemma density_normalization (d h n : ℕ) (a : ℝ) :
-    (n:ℝ)^a * (2:ℝ)^(-(d:ℝ)*h*a) = ((n:ℝ)/((2:ℝ)^d)^h)^a := by
-  rw [Real.div_rpow (Nat.cast_nonneg _) (by positivity),←pow_mul,
-    ←Real.rpow_natCast,←Real.rpow_mul (by norm_num : (0:ℝ) ≤ 2),Nat.cast_mul]
-  rw [show -(d:ℝ)*h*a = -((d:ℝ)*h*a) by ring,
-    Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2),div_eq_mul_inv]
-
-theorem palindrome_row_density_moment (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (a : ℝ) (ha : 0 ≤ a) :
-    palindromeRowMoment d e a ≤
-      (((2^d).descFactorial (Fintype.card ι):ℝ)/((2^d:ℕ):ℝ)^(Fintype.card ι))^a *
-        finiteMean (fun ω : BenesCoins d => (2:ℝ)^((palindromeCost d e ω:ℝ)*a)) := by
-  classical
-  have : Nonempty (ι ↪ Card d) := ⟨e⟩
-  rw [palindromeRowMoment,finite_row_power _ (fun f => PairRouting.tupleProbability_nonneg _ _ _)]
-  calc
-    _ ≤ (Fintype.card (ι ↪ Card d):ℝ)^a *
-      ∑ f : ι ↪ Card d, (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) * routingMoment d e f a :=
-      mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun f _ => palindrome_pointwise_moment d e f a ha))
-        (Real.rpow_nonneg (Nat.cast_nonneg _) _)
-    _ = _ := by
-      rw [←Finset.mul_sum,routingMoment_sum,←mul_assoc,density_normalization,
-        Fintype.card_embedding_eq,card_positions]
-      norm_cast
-
-end Thorp
-namespace Thorp
-
-lemma subset_biUnion_indicator {α β : Type*} [DecidableEq α] [Fintype β]
-    (A : Finset α) (T : β → Finset α) :
-    (if Finset.univ.biUnion T ⊆ A then (1:ℝ) else 0) =
-      ∏ b, if T b ⊆ A then (1:ℝ) else 0 := by
-  classical
-  by_cases h : Finset.univ.biUnion T ⊆ A
-  · rw [ite_eq_left h]
-    symm
-    apply Finset.prod_eq_one
-    intro b _
-    rw [ite_eq_left]
-    exact (Finset.subset_biUnion_of_mem T (Finset.mem_univ b)).trans h
-  · rw [ite_eq_right h]
-    obtain ⟨x,hx,hxA⟩ := Finset.not_subset.mp h
-    obtain ⟨b,_,hxb⟩ := Finset.mem_biUnion.mp hx
-    symm
-    apply Finset.prod_eq_zero (Finset.mem_univ b)
-    rw [ite_eq_right]
-    exact fun hb => hxA (hb hxb)
-
-lemma cylinder_block_polynomial {α β Ω : Type*} [DecidableEq α] [Fintype β]
-    [Fintype Ω] (τ : β → Type*) [∀ b, Fintype (τ b)]
-    (F : Ω → Finset α) (p : α → ℝ)
-    (hF : CylinderBound (inclusionMoment F) p)
-    (T : ∀ b, τ b → Finset α) (c : ∀ b, τ b → ℝ)
-    (hc : ∀ b s, 0 ≤ c b s)
-    (hd : ∀ b b', b ≠ b' → ∀ s s', Disjoint (T b s) (T b' s')) :
-    finiteMean (fun ω => ∏ b, ∑ s, c b s * if T b s ⊆ F ω then 1 else 0) ≤
-      ∏ b, ∑ s, c b s * ∏ x ∈ T b s, p x := by
-  classical
-  simp_rw [Fintype.prod_sum]
-  rw [finiteMean_sum]
-  apply Finset.sum_le_sum
-  intro s _
-  let U := Finset.univ.biUnion (fun b => T b (s b))
-  have he : (fun ω => ∏ b, c b (s b) * if T b (s b) ⊆ F ω then 1 else 0) =
-      fun ω => (if U ⊆ F ω then 1 else 0) * ∏ b, c b (s b) := by
-    funext ω
-    rw [Finset.prod_mul_distrib,←subset_biUnion_indicator]
-    exact mul_comm _ _
-  rw [he,finiteMean_mul_const]
-  have hdis : ((Finset.univ : Finset β) : Set β).PairwiseDisjoint (fun b => T b (s b)) := by
-    intro b _ b' _ hbb
-    exact hd b b' hbb (s b) (s b')
-  have hmean : finiteMean (fun ω => if U ⊆ F ω then 1 else 0) = inclusionMoment F U := by
-    unfold inclusionMoment
-    apply finiteMean_congr
-    intro ω
-    by_cases hh : U ⊆ F ω <;> simp [hh]
-  calc
-    _ ≤ (∏ x ∈ U, p x) * ∏ b, c b (s b) :=
-      mul_le_mul_of_nonneg_right (hmean.le.trans (hF U)) (Finset.prod_nonneg (fun b _ => hc b (s b)))
-    _ = _ := by
-      dsimp only [U]
-      rw [Finset.prod_biUnion hdis,←Finset.prod_mul_distrib]
-      apply Finset.prod_congr rfl
-      intro b _
-      exact mul_comm _ _
-
-
-noncomputable def sparseCoefficient {α : Type*} (w : ℝ) (T : Finset α) : ℝ :=
-  if T.card = 1 then 0 else w^T.card
-
-lemma sparseCoefficient_nonneg {α : Type*} (w : ℝ) (hw : 0 ≤ w) (T : Finset α) :
-    0 ≤ sparseCoefficient w T := by
-  unfold sparseCoefficient
-  split_ifs <;> positivity
-
-lemma sparse_block_pointwise {α : Type*} [DecidableEq α] (A D : Finset α)
-    (w : ℝ) (hw : 0 ≤ w) :
-    w^(if 2 ≤ (D ∩ A).card then (D ∩ A).card else 0) ≤
-      ∑ T : ↑D.powerset, sparseCoefficient w T.val * if T.val ⊆ A then 1 else 0 := by
-  classical
-  let V := D ∩ A
-  let T : ↑D.powerset := if 2 ≤ V.card then
-    ⟨V,Finset.mem_powerset.mpr Finset.inter_subset_left⟩ else
-    ⟨∅,Finset.mem_powerset.mpr (Finset.empty_subset D)⟩
-  calc
-    _ = sparseCoefficient w T.val * (if T.val ⊆ A then 1 else 0) := by
-      by_cases h : 2 ≤ V.card
-      · have hn : V.card ≠ 1 := by omega
-        simp [T,ite_eq_left h,sparseCoefficient,hn,show V ⊆ A from Finset.inter_subset_right,V]
-      · simp [T,ite_eq_right h,sparseCoefficient,V] at *
-    _ ≤ _ := Finset.single_le_sum (f := fun S : ↑D.powerset =>
-      sparseCoefficient w S.val * (if S.val ⊆ A then 1 else 0))
-      (fun S _ => mul_nonneg (sparseCoefficient_nonneg w hw S.val)
-        (by split_ifs <;> norm_num)) (Finset.mem_univ T)
-
-lemma geometric_sum_half (n : ℕ) {x : ℝ} (hx : 0 ≤ x) (hhalf : x ≤ 1/2) :
-    ∑ i ∈ Finset.range n, x^i ≤ 2 := by
-  induction n with
-  | zero => simp
-  | succ n ih =>
-    rw [Finset.sum_range_succ',pow_zero]
-    simp only [pow_succ',←Finset.mul_sum]
-    nlinarith
-
-lemma sparse_binomial_bound (R : ℕ) (z : ℝ) (hz : 0 ≤ z) (hr : (R:ℝ)*z ≤ 1/2) :
-    ∑ i ∈ Finset.range (R+1), (R.choose i:ℝ) * (if i = 1 then 0 else z^i) ≤
-      1 + 2*((R:ℝ)*z)^2 := by
-  classical
-  let x := (R:ℝ)*z
-  have hx : 0 ≤ x := mul_nonneg (Nat.cast_nonneg _) hz
-  have hbound (i : ℕ) : (R.choose i:ℝ)*(if i=1 then 0 else z^i) ≤
-      if i=1 then 0 else x^i := by
-    by_cases h : i=1
-    · simp [h]
-    · simp only [ite_eq_right h]
-      dsimp [x]
-      rw [mul_pow]
-      apply mul_le_mul_of_nonneg_right _ (pow_nonneg hz i)
-      exact_mod_cast Nat.choose_le_pow R i
-  calc
-    _ ≤ ∑ i ∈ Finset.range (R+1), if i=1 then 0 else x^i :=
-      Finset.sum_le_sum (fun i _ => hbound i)
-    _ ≤ 1+2*x^2 := by
-      rcases R with _ | R
-      · simp [x]
-      rcases R with _ | R
-      · norm_num [Finset.sum_range_succ]
-        positivity
-      have he : ∑ i ∈ Finset.range (R+2+1), (if i=1 then 0 else x^i) =
-          1 + x^2*∑ i ∈ Finset.range (R+1), x^i := by
-        rw [show R+2+1=2+(R+1) by omega,Finset.sum_range_add]
-        have hn (i : ℕ) : 2+i ≠ 1 := by omega
-        simp only [hn,ite_false,pow_add,←Finset.mul_sum]
-        norm_num [Finset.sum_range_succ]
-      rw [he]
-      have hg := geometric_sum_half (R+1) hx hr
-      nlinarith [sq_nonneg x]
-
-lemma sparse_block_product_bound {α : Type*} [DecidableEq α]
-    (D : Finset α) (p w : ℝ) (hp : 0 ≤ p) (hw : 0 ≤ w)
-    (hr : (D.card:ℝ)*(p*w) ≤ 1/2) :
-    ∑ T : ↑D.powerset, sparseCoefficient w T.val * p^T.val.card ≤
-      Real.exp (2*((D.card:ℝ)*(p*w))^2) := by
-  classical
-  have he : ∑ T : ↑D.powerset, sparseCoefficient w T.val * p^T.val.card =
-      ∑ i ∈ Finset.range (D.card+1), (D.card.choose i:ℝ)*(if i=1 then 0 else (p*w)^i) := by
-    rw [Finset.sum_coe_sort D.powerset (fun T => sparseCoefficient w T*p^T.card)]
-    have hh : (fun T : Finset α => sparseCoefficient w T*p^T.card) =
-        fun T => if T.card=1 then 0 else (p*w)^T.card := by
-      funext T
-      unfold sparseCoefficient
-      split_ifs <;> simp [mul_pow,mul_comm]
-    change (∑ T ∈ D.powerset, (fun T : Finset α => sparseCoefficient w T*p^T.card) T) = _
-    rw [hh,Finset.sum_powerset_apply_card (fun i => if i=1 then (0:ℝ) else (p*w)^i)]
-    simp only [nsmul_eq_mul]
-  rw [he]
-  exact (sparse_binomial_bound D.card (p*w) (mul_nonneg hp hw) hr).trans
-    (by simpa only [add_comm] using Real.add_one_le_exp (2*((D.card:ℝ)*(p*w))^2))
-
-
-noncomputable def sparseMass {α β : Type*} [DecidableEq α] [Fintype β]
-    (D : β → Finset α) (A : Finset α) : ℕ :=
-  ∑ b, if 2 ≤ (D b ∩ A).card then (D b ∩ A).card else 0
-
-theorem sparse_block_mgf {α β Ω : Type*} [DecidableEq α] [Fintype β] [Fintype Ω]
-    (D : β → Finset α) (hd : Pairwise (fun b b' => Disjoint (D b) (D b')))
-    (F : Ω → Finset α) (p w : ℝ) (hp : 0 ≤ p) (hw : 0 ≤ w)
-    (hF : ∀ T, inclusionMoment F T ≤ p^T.card)
-    (hr : ∀ b, ((D b).card:ℝ)*(p*w) ≤ 1/2) :
-    finiteMean (fun ω => w^(sparseMass D (F ω))) ≤
-      Real.exp (∑ b, 2*(((D b).card:ℝ)*(p*w))^2) := by
-  classical
-  have hc : CylinderBound (inclusionMoment F) (fun _ => p) := by
-    intro T
-    simpa only [Finset.prod_const] using hF T
-  have hb := cylinder_block_polynomial (fun b => ↑(D b).powerset) F (fun _ => p) hc
-    (fun _ S => S.val) (fun _ S => sparseCoefficient w S.val)
-    (fun _ S => sparseCoefficient_nonneg w hw S.val)
-    (fun b b' h S S' => (hd h).mono
-      (Finset.mem_powerset.mp S.property) (Finset.mem_powerset.mp S'.property))
-  calc
-    _ ≤ finiteMean (fun ω => ∏ b, ∑ S : ↑(D b).powerset,
-        sparseCoefficient w S.val * if S.val ⊆ F ω then 1 else 0) := by
-      apply finiteMean_mono
-      intro ω
-      rw [sparseMass,←Finset.prod_pow_eq_pow_sum]
-      apply Finset.prod_le_prod₀ (fun _ _ => pow_nonneg hw _)
-      intro b _
-      exact sparse_block_pointwise (F ω) (D b) w hw
-    _ ≤ ∏ b, ∑ S : ↑(D b).powerset, sparseCoefficient w S.val * p^S.val.card := by
-      simpa only [Finset.prod_const] using hb
-    _ ≤ _ := by
-      rw [Real.exp_sum]
-      apply Finset.prod_le_prod₀
-      · intro b _
-        exact Finset.sum_nonneg (fun S _ => mul_nonneg
-          (sparseCoefficient_nonneg w hw S.val) (pow_nonneg hp _))
-      · intro b _
-        exact sparse_block_product_bound (D b) p w hp hw (hr b)
-
-theorem inverseButterfly_sparse_mgf (d : ℕ) {β : Type*} [Fintype β]
-    (D : β → Finset (Card d)) (hd : Pairwise (fun b b' => Disjoint (D b) (D b')))
-    (A : Finset (Card d)) (w : ℝ) (hw : 0 ≤ w)
-    (hr : ∀ b, ((D b).card:ℝ)*((A.card/(2:ℝ)^d)*w) ≤ 1/2) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      w^(sparseMass D (inverseButterflyMarks d A ω))) ≤
-      Real.exp (∑ b, 2*(((D b).card:ℝ)*((A.card/(2:ℝ)^d)*w))^2) := by
-  apply sparse_block_mgf D hd (inverseButterflyMarks d A) (A.card/(2:ℝ)^d) w
-    (by positivity) hw
-  · exact inverseButterflyMarks_inclusion d A
-  · exact hr
-
-
-def crowdedMass (h : ℕ) : ℕ := if 2 ≤ h then h else 0
-
-lemma crowdedMass_le (h : ℕ) : crowdedMass h ≤ h := by
-  unfold crowdedMass
-  split_ifs <;> omega
-
-lemma crowdedMass_add (u v : ℕ) : crowdedMass u + crowdedMass v ≤ crowdedMass (u+v) := by
-  unfold crowdedMass
-  split_ifs <;> omega
-
-lemma alternatingCycles_le_crowded {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
-    (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    PairRouting.alternatingCycles e p ≤ crowdedMass (Fintype.card ι) := by
-  unfold crowdedMass
-  split_ifs with h
-  · have hh := PairRouting.alternatingCycles_double e p
-    omega
-  · rw [PairRouting.alternatingCycles_zero e p (by omega)]
-
-lemma palindromeCost_le_crowded (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) :
-    palindromeCost d e ω ≤ d*crowdedMass (Fintype.card ι) := by
-  classical
-  induction d generalizing ι with
-  | zero => simp [palindromeCost]
-  | succ d ih =>
-    let σ := benesStepEquiv d ω
-    let x := e.trans (headTailEquiv d).toEmbedding
-    let c := PairRouting.colors x σ.2.1
-    let hc := PairRouting.colors_compatible x σ.2.1
-    have h₀ := ih (PairRouting.childEmbedding x c hc false) σ.1.1
-    have h₁ := ih (PairRouting.childEmbedding x c hc true) σ.1.2
-    have hz := alternatingCycles_le_crowded (PairRouting.switchedEmbedding x σ.2.1)
-      (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1))
-    have hs := crowdedMass_add (Fintype.card {i // c i = false})
-      (Fintype.card {i // c i = true})
-    rw [PairRouting.color_card_add] at hs
-    change PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
-        (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) +
-        palindromeCost d (PairRouting.childEmbedding x c hc false) σ.1.1 +
-        palindromeCost d (PairRouting.childEmbedding x c hc true) σ.1.2 ≤ _
-    nlinarith
-
-noncomputable def palindromeLowCost (H : ℕ) : (d : ℕ) → {ι : Type*} → [Fintype ι] →
-    (ι ↪ Card d) → BenesCoins d → ℕ
-  | 0, _, _, _, _ => 0
-  | d+1, _, _, e, ω =>
-      let σ := benesStepEquiv d ω
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x σ.2.1
-      let hc := PairRouting.colors_compatible x σ.2.1
-      (if d+1 ≤ H then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
-          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) else 0) +
-        palindromeLowCost H d (PairRouting.childEmbedding x c hc false) σ.1.1 +
-        palindromeLowCost H d (PairRouting.childEmbedding x c hc true) σ.1.2
-
-end Thorp
-
-end ThorpNine.Harmonic
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Harmonic.AlternatingPerm
+
+namespace OAI
+
+namespace ThorpNine.Harmonic
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+namespace PairRouting
+variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α] [DecidableEq ι]
+
+lemma sandwich_power_bound {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
+    (P : Ω → Equiv.Perm α) (Q : Ω' → Equiv.Perm α) (x y : ι → Bool × α)
+    (a : ℝ) (ha : 0 ≤ a) :
+    (finiteMean (fun ω : Ω × Ω' => finiteMean (fun coins : (α → Bool) × (α → Bool) =>
+      if ∀ i, sandwich coins.1 coins.2 (fun b => if b then Q ω.2 else P ω.1) (x i) = y i
+      then (1:ℝ) else 0)))^(1+a) ≤
+    (outerWeight x y)^(1+a) * ((compatibleSet x y).card:ℝ)^a *
+      ∑ c ∈ compatibleSet x y,
+      (tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
+        (fun i : {i // c i = false} => (y i.val).2))^(1+a) *
+      (tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
+        (fun i : {i // c i = true} => (y i.val).2))^(1+a) := by
+  classical
+  rw [sandwich_mean,sum_colors_factor]
+  let f (c : ι → Bool) : ℝ :=
+      tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
+        (fun i : {i // c i = false} => (y i.val).2) *
+      tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
+        (fun i : {i // c i = true} => (y i.val).2)
+  have hf (c : ι → Bool) : 0 ≤ f c := mul_nonneg (tupleProbability_nonneg _ _ _)
+    (tupleProbability_nonneg _ _ _)
+  rw [Real.mul_rpow (le_of_lt (outerWeight_pos _ _)) (Finset.sum_nonneg (fun c _ => hf c))]
+  have hh := Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg (compatibleSet x y)
+    (show 1 ≤ 1+a by linarith) (fun c _ => hf c)
+  have he : 1+a-1=a := by ring
+  rw [he] at hh
+  calc
+    _ ≤ (outerWeight x y)^(1+a) *
+        (((compatibleSet x y).card:ℝ)^a * ∑ c ∈ compatibleSet x y, (f c)^(1+a)) :=
+      mul_le_mul_of_nonneg_left hh (Real.rpow_nonneg (le_of_lt (outerWeight_pos _ _)) _)
+    _ = _ := by
+      rw [←mul_assoc]
+      congr 1
+      apply Finset.sum_congr rfl
+      intro c _
+      exact Real.mul_rpow (tupleProbability_nonneg _ _ _) (tupleProbability_nonneg _ _ _)
+
+end PairRouting
+
+def sandwichShuffle {A B C D E F : Type*} :
+    ((A × B) × C) × ((D × E) × F) ≃ ((A × D) × (B × E)) × (F × C) where
+  toFun x := (((x.1.1.1,x.2.1.1),(x.1.1.2,x.2.1.2)),(x.2.2,x.1.2))
+  invFun x := (((x.1.1.1,x.1.2.1),x.2.2),((x.1.1.2,x.1.2.2),x.2.1))
+  left_inv _ := rfl
+  right_inv _ := rfl
+
+def benesStepEquiv (d : ℕ) : BenesCoins (d+1) ≃
+    (BenesCoins d × BenesCoins d) × ((Card d → Bool) × (Card d → Bool)) :=
+  (Equiv.prodCongr (coinStepEquiv d) (coinStepEquiv d)).trans sandwichShuffle
+
+lemma childLift_mul (d : ℕ) (p q : Bool → Equiv.Perm (Card d)) :
+    childLift p * childLift q = childLift (fun b => p b * q b) := by
+  ext x i
+  simp only [Equiv.Perm.mul_apply,childLift,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons]
+
+lemma childLift_inv (d : ℕ) (p : Bool → Equiv.Perm (Card d)) :
+    (childLift p)⁻¹ = childLift (fun b => (p b)⁻¹) := rfl
+
+lemma pairSwitch_inv (d : ℕ) (ξ : Card d → Bool) : (pairSwitch ξ)⁻¹ = pairSwitch ξ := rfl
+
+lemma palindromePerm_step (d : ℕ) (c : BenesCoins d × BenesCoins d)
+    (coins : (Card d → Bool) × (Card d → Bool)) :
+    palindromePerm (d+1) ((benesStepEquiv d).symm (c,coins)) =
+      pairSwitch coins.2 * childLift (fun b => palindromePerm d (if b then c.2 else c.1)) *
+        pairSwitch coins.1 := by
+  change (butterflyPerm (d+1) (decodeButterfly (d+1)
+      ((coinStepEquiv d).symm ((c.1.1,c.2.1),coins.2)))) *
+    (butterflyPerm (d+1) (decodeButterfly (d+1)
+      ((coinStepEquiv d).symm ((c.1.2,c.2.2),coins.1))))⁻¹ = _
+  rw [butterflyPerm_step,butterflyPerm_step,mul_inv_rev,childLift_inv,pairSwitch_inv]
+  rw [mul_assoc (pairSwitch coins.2),←mul_assoc (childLift _),childLift_mul,←mul_assoc]
+  congr 2
+  apply congrArg childLift
+  funext b
+  cases b <;> rfl
+
+lemma headTail_switch (d : ℕ) (ξ : Card d → Bool) (x : Card (d+1)) :
+    headTailEquiv d (pairSwitch ξ x) = pairLayer ξ (headTailEquiv d x) := rfl
+
+lemma headTail_child (d : ℕ) (p : Bool → Equiv.Perm (Card d)) (x : Card (d+1)) :
+    headTailEquiv d (childLift p x) = PairRouting.lift p (headTailEquiv d x) := rfl
+
+lemma palindromePerm_coordinates (d : ℕ) (c : BenesCoins d × BenesCoins d)
+    (coins : (Card d → Bool) × (Card d → Bool)) (x : Card (d+1)) :
+    headTailEquiv d (palindromePerm (d+1) ((benesStepEquiv d).symm (c,coins)) x) =
+      PairRouting.sandwich coins.1 coins.2 (fun b => palindromePerm d (if b then c.2 else c.1))
+        (headTailEquiv d x) := by
+  rw [palindromePerm_step]
+  simp only [Equiv.Perm.mul_apply,headTail_switch,headTail_child]
+  rfl
+
+noncomputable def palindromeCost : (d : ℕ) → {ι : Type*} → [Fintype ι] →
+    (ι ↪ Card d) → BenesCoins d → ℕ
+  | 0, _, _, _, _ => 0
+  | d+1, _, _, e, ω =>
+      let σ := benesStepEquiv d ω
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x σ.2.1
+      let hc := PairRouting.colors_compatible x σ.2.1
+      PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
+          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) +
+        palindromeCost d (PairRouting.childEmbedding x c hc false) σ.1.1 +
+        palindromeCost d (PairRouting.childEmbedding x c hc true) σ.1.2
+
+namespace CycleColoring
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+lemma alternating_cycle_two (p : Equiv.Perm α) (S : Finset (Bool × α))
+    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) S) : 2 ≤ C.val.card := by
+  obtain ⟨x,hx⟩ := C.nonempty
+  have hpx : alternatingPerm p x ∈ C.val := by
+    rw [←C.orbit_eq hx,mem_orbitSet]
+    exact (Equiv.Perm.SameCycle.refl _ _).apply_right
+  have hne : x ≠ alternatingPerm p x := by
+    intro he
+    have hh := congrArg Prod.fst he
+    rw [alternatingPerm_flip] at hh
+    cases hb : x.1 <;> simp only [hb, Bool.not_false, Bool.not_true] at hh <;> cases hh
+  have hs : {x,alternatingPerm p x} ⊆ C.val := by
+    intro y hy
+    simp only [Finset.mem_insert,Finset.mem_singleton] at hy
+    rcases hy with rfl | rfl <;> assumption
+  have hh := Finset.card_le_card hs
+  simpa only [Finset.card_pair hne] using hh
+
+lemma alternating_cycle_count_double (p : Equiv.Perm α) (S : Finset (Bool × α)) :
+    2 * Fintype.card (RoutingNetwork.SelectedCycle (alternatingPerm p) S) ≤ S.card := by
+  classical
+  calc
+    _ = ∑ _C : RoutingNetwork.SelectedCycle (alternatingPerm p) S, 2 := by simp [Nat.mul_comm]
+    _ ≤ RoutingNetwork.cycleMass (alternatingPerm p) S :=
+      Finset.sum_le_sum (fun C _ => alternating_cycle_two p S C)
+    _ ≤ _ := RoutingNetwork.cycleMass_le _ _
+
+lemma alternating_cycle_count_zero (p : Equiv.Perm α) (S : Finset (Bool × α))
+    (hS : S.card ≤ 1) : Fintype.card (RoutingNetwork.SelectedCycle (alternatingPerm p) S) = 0 := by
+  have h := alternating_cycle_count_double p S
+  omega
+
+end CycleColoring
+
+namespace PairRouting
+
+section
+variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
+
+lemma alternatingCycles_double (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    2 * alternatingCycles e p ≤ Fintype.card ι := by
+  have hh := CycleColoring.alternating_cycle_count_double (alternatingProjection p) (labelSet e)
+  simpa only [alternatingCycles, card_labelSet] using hh
+
+lemma alternatingCycles_zero (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α)
+    (hι : Fintype.card ι ≤ 1) : alternatingCycles e p = 0 := by
+  have hh := alternatingCycles_double e p
+  omega
+
+end
+variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α] [DecidableEq ι]
+
+omit [Fintype α] [DecidableEq ι] in
+lemma colors_chosenCoin (x : ι → Bool × α) (c : ι → Bool) (hc : Compatible x c) :
+    colors x (chosenCoin x c) = c := by
+  funext i
+  simp only [colors,chosenCoin_apply x c hc i,required]
+  cases (x i).1 <;> cases c i <;> rfl
+
+noncomputable def coloredEmbedding (x : ι ↪ Bool × α) (c : ι → Bool) (_hc : Compatible x c) :
+    ι ↪ Bool × α := switchedEmbedding x (chosenCoin x c)
+
+omit [Fintype α] [DecidableEq ι] in
+lemma coloredEmbedding_apply (x : ι ↪ Bool × α) (c : ι → Bool) (hc : Compatible x c) (i : ι) :
+    coloredEmbedding x c hc i = (c i,(x i).2) := by
+  apply Prod.ext
+  · exact congrFun (colors_chosenCoin x c hc) i
+  · rfl
+
+omit [Fintype α] [DecidableEq ι] in
+lemma switchedEmbedding_eq_colored (x : ι ↪ Bool × α) (c : ι → Bool) (hc : Compatible x c)
+    (ξ : α → Bool) (he : colors x ξ = c) :
+    switchedEmbedding x ξ = coloredEmbedding x c hc := by
+  ext i : 1
+  rw [coloredEmbedding_apply]
+  apply Prod.ext
+  · exact congrFun he i
+  · rfl
+
+lemma assignments_match_bound (x y : ι ↪ Bool × α) (c : ι → Bool)
+    (hc : c ∈ compatibleSet x y) (p : Bool → Equiv.Perm α) (hm : ChildMatches p x y c) :
+    (compatibleSet x y).card ≤
+      2 ^ ((occupied x).card + (occupied y).card - Fintype.card ι +
+        alternatingCycles (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p) := by
+  obtain ⟨hcx,hcy⟩ := (mem_compatibleSet x y c).mp hc
+  let ξ := chosenCoin x c
+  let η := chosenCoin y c
+  have he : routedEmbedding x ξ η p = y := by
+    ext i : 1
+    have h := (sandwich_matches_iff ξ η p x y).mpr
+      ⟨(colors_chosenCoin x c hcx).trans (colors_chosenCoin y c hcy).symm,by
+        simpa only [ξ,colors_chosenCoin x c hcx] using hm⟩
+    exact h i
+  have hh := assignments_actual_bound x ξ η p
+  rw [he] at hh
+  rw [card_compatibleSet]
+  exact hh
+
+omit [Fintype α] [DecidableEq ι] in
+lemma occupied_sum_ge (x y : ι ↪ Bool × α) :
+    Fintype.card ι ≤ (occupied x).card + (occupied y).card := by
+  have h₀ := occupied_card_double x
+  have h₁ := occupied_card_double y
+  omega
+
+lemma assignment_weight_bound (x y : ι ↪ Bool × α) (c : ι → Bool)
+    (hc : c ∈ compatibleSet x y) (p : Bool → Equiv.Perm α) (hm : ChildMatches p x y c) :
+    ((compatibleSet x y).card:ℝ) * outerWeight x y ≤
+      (2:ℝ)^((alternatingCycles
+        (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p : ℝ) - Fintype.card ι) := by
+  let z := alternatingCycles (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p
+  let b := (occupied x).card+(occupied y).card
+  let h := Fintype.card ι
+  have hb : h ≤ b := occupied_sum_ge x y
+  have hn : ((compatibleSet x y).card:ℝ) ≤ (2:ℝ)^(b-h+z) := by
+    exact_mod_cast assignments_match_bound x y c hc p hm
+  change ((compatibleSet x y).card:ℝ) * (1/(2:ℝ)^b) ≤ (2:ℝ)^((z:ℝ)-h)
+  rw [mul_one_div]
+  calc
+    _ ≤ (2:ℝ)^(b-h+z)/(2:ℝ)^b := div_le_div_of_nonneg_right hn (by positivity)
+    _ = _ := by
+      rw [←Real.rpow_natCast,←Real.rpow_natCast,←Real.rpow_sub (by norm_num : (0:ℝ)<2)]
+      congr 1
+      rw [Nat.cast_add,Nat.cast_sub hb]
+      ring
+
+lemma assignment_factor_bound (x y : ι ↪ Bool × α) (c : ι → Bool)
+    (hc : c ∈ compatibleSet x y) (p : Bool → Equiv.Perm α) (hm : ChildMatches p x y c)
+    (a : ℝ) (ha : 0 ≤ a) :
+    (outerWeight x y)^a * ((compatibleSet x y).card:ℝ)^a ≤
+      (2:ℝ)^(((alternatingCycles
+        (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p : ℝ) - Fintype.card ι)*a) := by
+  have h := Real.rpow_le_rpow (mul_nonneg (Nat.cast_nonneg _) (le_of_lt (outerWeight_pos x y)))
+    (assignment_weight_bound x y c hc p hm) ha
+  rw [Real.mul_rpow (Nat.cast_nonneg _) (le_of_lt (outerWeight_pos x y)),
+    ←Real.rpow_mul (by norm_num : (0:ℝ)≤2)] at h
+  simpa only [mul_comm] using h
+
+
+lemma sandwich_weighted_probability_subtype (p : Bool → Equiv.Perm α) (x y : ι → Bool × α)
+    (W : (ι → Bool) → ℝ) :
+    finiteMean (fun coins : (α → Bool) × (α → Bool) =>
+      if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then W (colors x coins.1) else 0) =
+    outerWeight x y * ∑ c : {c // c ∈ compatibleSet x y},
+      if ChildMatches p x y c.val then W c.val else 0 := by
+  rw [sandwich_weighted_probability]
+  congr 1
+  rw [Finset.sum_subtype (compatibleSet x y) (fun _ => Iff.rfl)]
+
+omit [Fintype α] [DecidableEq α] [DecidableEq ι] in
+lemma color_card_add (c : ι → Bool) :
+    Fintype.card {i // c i = false} + Fintype.card {i // c i = true} = Fintype.card ι := by
+  have he : {i // c i = true} ≃ {i // ¬c i = false} := Equiv.subtypeEquivRight fun i => by
+    cases c i <;> simp
+  rw [Fintype.card_congr he]
+  rw [Fintype.card_subtype_compl]
+  exact Nat.add_sub_of_le (Fintype.card_subtype_le _)
+
+end PairRouting
+
+noncomputable def assignmentCost (d : ℕ) {ι : Type*} [Fintype ι]
+    (x : ι ↪ Bool × Card d) (c : ι → Bool) (hc : PairRouting.Compatible x c)
+    (ω : BenesCoins d × BenesCoins d) : ℕ :=
+  PairRouting.alternatingCycles (PairRouting.coloredEmbedding x c hc)
+      (fun b => palindromePerm d (if b then ω.2 else ω.1)) +
+    palindromeCost d (PairRouting.childEmbedding x c hc false) ω.1 +
+    palindromeCost d (PairRouting.childEmbedding x c hc true) ω.2
+
+lemma palindromeCost_eq (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (d+1)) (σ : BenesCoins (d+1)) :
+    palindromeCost (d+1) e σ =
+      assignmentCost d (e.trans (headTailEquiv d).toEmbedding)
+        (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) (benesStepEquiv d σ).2.1)
+        (PairRouting.colors_compatible _ _) (benesStepEquiv d σ).1 := by
+  unfold palindromeCost assignmentCost
+  dsimp only
+  rw [PairRouting.switchedEmbedding_eq_colored _ _ (PairRouting.colors_compatible _ _) _ rfl]
+
+lemma palindromeCost_step (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (d+1)) (ω : BenesCoins d × BenesCoins d)
+    (coins : (Card d → Bool) × (Card d → Bool)) :
+    palindromeCost (d+1) e ((benesStepEquiv d).symm (ω,coins)) =
+      assignmentCost d (e.trans (headTailEquiv d).toEmbedding)
+        (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) coins.1)
+        (PairRouting.colors_compatible _ _) ω := by
+  rw [palindromeCost_eq]
+  simp only [Equiv.apply_symm_apply]
+
+noncomputable def routingMoment (d : ℕ) {ι : Type*} [Fintype ι]
+    (e f : ι ↪ Card d) (a : ℝ) : ℝ := by
+  classical
+  exact finiteMean (fun ω : BenesCoins d =>
+    if ∀ i, palindromePerm d ω (e i) = f i then (2:ℝ)^((palindromeCost d e ω:ℝ)*a) else 0)
+
+lemma routingMoment_nonneg (d : ℕ) {ι : Type*} [Fintype ι]
+    (e f : ι ↪ Card d) (a : ℝ) : 0 ≤ routingMoment d e f a := by
+  classical
+  exact finiteMean_nonneg (fun _ => by split_ifs <;> positivity)
+
+lemma palindrome_matches_step (d : ℕ) {ι : Type*}
+    (e f : ι ↪ Card (d+1)) (ω : BenesCoins d × BenesCoins d)
+    (coins : (Card d → Bool) × (Card d → Bool)) :
+    (∀ i, palindromePerm (d+1) ((benesStepEquiv d).symm (ω,coins)) (e i) = f i) ↔
+    (∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => palindromePerm d (if b then ω.2 else ω.1))
+      ((e.trans (headTailEquiv d).toEmbedding) i) = ((f.trans (headTailEquiv d).toEmbedding) i)) := by
+  apply forall_congr'
+  intro i
+  rw [←(headTailEquiv d).injective.eq_iff,palindromePerm_coordinates]
+  rfl
+
+lemma routingMoment_step (d : ℕ) {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (e f : ι ↪ Card (d+1)) (a : ℝ) :
+    let x := e.trans (headTailEquiv d).toEmbedding
+    let y := f.trans (headTailEquiv d).toEmbedding
+    routingMoment (d+1) e f a = PairRouting.outerWeight x y *
+      ∑ c : {c // c ∈ PairRouting.compatibleSet x y},
+        finiteMean (fun ω : BenesCoins d × BenesCoins d =>
+          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c.val
+          then (2:ℝ)^((assignmentCost d x c.val
+            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0) := by
+  classical
+  intro x y
+  rw [routingMoment,finiteMean_equiv (benesStepEquiv d),finiteMean_prod]
+  have he (ω : BenesCoins d × BenesCoins d) :
+      finiteMean (fun coins : (Card d → Bool) × (Card d → Bool) =>
+        if ∀ i, palindromePerm (d+1) ((benesStepEquiv d).symm (ω,coins)) (e i) = f i
+        then (2:ℝ)^((palindromeCost (d+1) e ((benesStepEquiv d).symm (ω,coins)):ℝ)*a) else 0) =
+      PairRouting.outerWeight x y *
+        ∑ c : {c // c ∈ PairRouting.compatibleSet x y},
+          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c.val
+          then (2:ℝ)^((assignmentCost d x c.val
+            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0 := by
+    let W : (ι → Bool) → ℝ := fun c =>
+      if hc : PairRouting.Compatible x c then (2:ℝ)^((assignmentCost d x c hc ω:ℝ)*a) else 0
+    calc
+      _ = finiteMean (fun coins : (Card d → Bool) × (Card d → Bool) =>
+        if ∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => palindromePerm d (if b then ω.2 else ω.1))
+          (x i) = y i then W (PairRouting.colors x coins.1) else 0) := by
+        apply finiteMean_congr
+        intro coins
+        simp only [palindrome_matches_step,palindromeCost_step]
+        simp only [W,PairRouting.colors_compatible,↓reduceDIte]
+        rfl
+      _ = _ := by
+        rw [PairRouting.sandwich_weighted_probability_subtype]
+        congr 1
+        apply Finset.sum_congr rfl
+        intro c _
+        simp only [W,((PairRouting.mem_compatibleSet x y c.val).mp c.property).1,↓reduceDIte]
+  simp_rw [he]
+  simp_rw [mul_comm (PairRouting.outerWeight x y)]
+  rw [finiteMean_mul_const,finiteMean_sum,mul_comm]
+
+lemma child_moment_product (d : ℕ) {ι : Type*} [Fintype ι]
+    (x y : ι ↪ Bool × Card d) (c : ι → Bool)
+    (hcx : PairRouting.Compatible x c) (hcy : PairRouting.Compatible y c) (a : ℝ) :
+    routingMoment d (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a *
+      routingMoment d (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a =
+    finiteMean (fun ω : BenesCoins d × BenesCoins d =>
+      if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
+      then (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
+        palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0) := by
+  classical
+  unfold routingMoment
+  rw [←finiteMean_prod_mul]
+  apply finiteMean_congr
+  intro ω
+  let M₀ := ∀ i : {i // c i = false}, palindromePerm d ω.1 (x i.val).2 = (y i.val).2
+  let M₁ := ∀ i : {i // c i = true}, palindromePerm d ω.2 (x i.val).2 = (y i.val).2
+  have hp : (fun b : Bool => palindromePerm d (if b then ω.2 else ω.1)) =
+      (fun b => if b then palindromePerm d ω.2 else palindromePerm d ω.1) := by
+    funext b
+    cases b <;> rfl
+  have hs : PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c ↔
+      M₀ ∧ M₁ := by rw [hp]; exact PairRouting.childMatches_split _ _ _ _ _
+  change (if M₀ then _ else 0) * (if M₁ then _ else 0) = _
+  by_cases h₀ : M₀ <;> by_cases h₁ : M₁
+  · simp only [hs,h₀,h₁,true_and,↓reduceIte]
+    rw [←Real.rpow_add (by norm_num : (0:ℝ)<2)]
+    congr 1
+    ring
+  · simp only [hs,h₀,h₁,and_false,↓reduceIte,mul_zero]
+  · simp only [hs,h₀,h₁,false_and,↓reduceIte,zero_mul]
+  · simp only [hs,h₀,h₁,false_and,↓reduceIte,mul_zero]
+
+lemma palindrome_probability_step (d : ℕ) {ι : Type*} [Fintype ι]
+    (e f : ι ↪ Card (d+1)) :
+    PairRouting.tupleProbability (palindromePerm (d+1)) e f =
+    finiteMean (fun ω : BenesCoins d × BenesCoins d =>
+      finiteMean (fun coins : (Card d → Bool) × (Card d → Bool) =>
+        if ∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => if b then palindromePerm d ω.2 else palindromePerm d ω.1)
+          ((e.trans (headTailEquiv d).toEmbedding) i) = ((f.trans (headTailEquiv d).toEmbedding) i)
+        then (1:ℝ) else 0)) := by
+  classical
+  rw [PairRouting.tupleProbability,finiteMean_equiv (benesStepEquiv d),finiteMean_prod]
+  have hp (b : Bool) (p q : BenesCoins d) :
+      palindromePerm d (if b then p else q) = (if b then palindromePerm d p else palindromePerm d q) := by
+    cases b <;> rfl
+  simp only [palindrome_matches_step,hp]
+
+lemma child_power_product_bound (d : ℕ) {ι : Type*} [Fintype ι]
+    (x y : ι ↪ Bool × Card d) (c : ι → Bool)
+    (hcx : PairRouting.Compatible x c) (hcy : PairRouting.Compatible y c) (a : ℝ)
+    (hfalse : (PairRouting.tupleProbability (palindromePerm d)
+        (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false))^(1+a) ≤
+      (2:ℝ)^(-(d:ℝ)*Fintype.card {i // c i = false}*a) *
+        routingMoment d (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a)
+    (htrue : (PairRouting.tupleProbability (palindromePerm d)
+        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true))^(1+a) ≤
+      (2:ℝ)^(-(d:ℝ)*Fintype.card {i // c i = true}*a) *
+        routingMoment d (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a) :
+    (PairRouting.tupleProbability (palindromePerm d)
+      (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false))^(1+a) *
+      (PairRouting.tupleProbability (palindromePerm d)
+        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true))^(1+a) ≤
+    (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) *
+      finiteMean (fun ω : BenesCoins d × BenesCoins d =>
+        if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
+        then (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
+          palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0) := by
+  have hh := mul_le_mul hfalse htrue
+    (Real.rpow_nonneg (PairRouting.tupleProbability_nonneg _ _ _) _)
+    (mul_nonneg (by positivity) (routingMoment_nonneg _ _ _ _))
+  rw [mul_mul_mul_comm,child_moment_product,←Real.rpow_add (by norm_num : (0:ℝ)<2)] at hh
+  have he : -(d:ℝ)*Fintype.card {i // c i = false}*a + -(d:ℝ)*Fintype.card {i // c i = true}*a =
+      -(d:ℝ)*Fintype.card ι*a := by
+    rw [←add_mul,←mul_add,←Nat.cast_add,PairRouting.color_card_add]
+  rwa [he] at hh
+
+lemma assignment_moment_bound (d : ℕ) {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (x y : ι ↪ Bool × Card d) (c : ι → Bool) (hc : c ∈ PairRouting.compatibleSet x y)
+    (a : ℝ) (ha : 0 ≤ a) :
+    let hcx := ((PairRouting.mem_compatibleSet x y c).mp hc).1
+    PairRouting.outerWeight x y ^ a * ((PairRouting.compatibleSet x y).card:ℝ)^a *
+      ((2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) *
+        finiteMean (fun ω : BenesCoins d × BenesCoins d =>
+          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
+          then (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
+            palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0)) ≤
+    (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) *
+      finiteMean (fun ω : BenesCoins d × BenesCoins d =>
+        if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
+        then (2:ℝ)^((assignmentCost d x c hcx ω:ℝ)*a) else 0) := by
+  classical
+  intro hcx
+  have hscale (q : ℝ) (F : BenesCoins d × BenesCoins d → ℝ) :
+      q * finiteMean F = finiteMean (fun ω => q * F ω) := by
+    simp_rw [mul_comm q,finiteMean_mul_const]
+  rw [←mul_assoc,hscale,hscale]
+  apply finiteMean_mono
+  intro ω
+  split_ifs with hm
+  · have hh := PairRouting.assignment_factor_bound x y c hc
+      (fun b => palindromePerm d (if b then ω.2 else ω.1)) hm a ha
+    calc
+      _ ≤ ((2:ℝ)^(((PairRouting.alternatingCycles (PairRouting.coloredEmbedding x c hcx)
+          (fun b => palindromePerm d (if b then ω.2 else ω.1)):ℝ)-Fintype.card ι)*a) *
+          (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a)) *
+          (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
+            palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) :=
+        mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hh (by positivity)) (by positivity)
+      _ = _ := by
+        simp only [←Real.rpow_add (by norm_num : (0:ℝ)<2),assignmentCost,Nat.cast_add]
+        congr 1
+        ring
+  · simp only [mul_zero]
+    exact le_rfl
+
+theorem palindrome_pointwise_moment (d : ℕ) {ι : Type*} [Fintype ι]
+    (e f : ι ↪ Card d) (a : ℝ) (ha : 0 ≤ a) :
+    (PairRouting.tupleProbability (palindromePerm d) e f)^(1+a) ≤
+      (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) * routingMoment d e f a := by
+  classical
+  induction d generalizing ι with
+  | zero =>
+      have he : ∀ (ω : BenesCoins 0) (i : ι), palindromePerm 0 ω (e i) = f i :=
+        fun _ _ => Subsingleton.elim _ _
+      simp [PairRouting.tupleProbability,routingMoment,he,palindromeCost,finiteMean_const]
+  | succ d ih =>
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let y := f.trans (headTailEquiv d).toEmbedding
+      let W := PairRouting.outerWeight x y
+      let C := PairRouting.compatibleSet x y
+      let p : (ι → Bool) → ℝ := fun c =>
+        (PairRouting.tupleProbability (palindromePerm d)
+          (fun i : {i // c i = false} => (x i.val).2)
+          (fun i : {i // c i = false} => (y i.val).2))^(1+a) *
+        (PairRouting.tupleProbability (palindromePerm d)
+          (fun i : {i // c i = true} => (x i.val).2)
+          (fun i : {i // c i = true} => (y i.val).2))^(1+a)
+      let m : {c // c ∈ C} → ℝ := fun c =>
+        finiteMean (fun ω : BenesCoins d × BenesCoins d =>
+          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c.val
+          then (2:ℝ)^((assignmentCost d x c.val
+            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0)
+      have hp (c : {c // c ∈ C}) : W^a * (C.card:ℝ)^a * p c.val ≤
+          (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * m c := by
+        obtain ⟨hcx,hcy⟩ := (PairRouting.mem_compatibleSet x y c.val).mp c.property
+        have h₀ := ih (PairRouting.childEmbedding x c.val hcx false)
+          (PairRouting.childEmbedding y c.val hcy false)
+        have h₁ := ih (PairRouting.childEmbedding x c.val hcx true)
+          (PairRouting.childEmbedding y c.val hcy true)
+        have hh := child_power_product_bound d x y c.val hcx hcy a h₀ h₁
+        have hh' := mul_le_mul_of_nonneg_left hh
+          (mul_nonneg (Real.rpow_nonneg (le_of_lt (PairRouting.outerWeight_pos x y)) a)
+            (Real.rpow_nonneg (Nat.cast_nonneg C.card) a))
+        exact hh'.trans (assignment_moment_bound d x y c.val c.property a ha)
+      have hnode := PairRouting.sandwich_power_bound (palindromePerm d) (palindromePerm d) x y a ha
+      rw [←palindrome_probability_step d e f] at hnode
+      change (PairRouting.tupleProbability (palindromePerm (d+1)) e f)^(1+a) ≤
+        W^(1+a) * (C.card:ℝ)^a * ∑ c ∈ C, p c at hnode
+      have hsum : ∑ c ∈ C, p c = ∑ c : {c // c ∈ C}, p c.val := by
+        rw [Finset.sum_subtype C (fun _ => Iff.rfl)]
+      have hW : 0 < W := PairRouting.outerWeight_pos x y
+      calc
+        _ ≤ W^(1+a) * (C.card:ℝ)^a * ∑ c ∈ C, p c := hnode
+        _ = W * ∑ c : {c // c ∈ C}, W^a * (C.card:ℝ)^a * p c.val := by
+          rw [Real.rpow_add hW,Real.rpow_one,hsum,←Finset.mul_sum]
+          ring
+        _ ≤ W * ∑ c : {c // c ∈ C}, (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * m c :=
+          mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun c _ => hp c)) (le_of_lt hW)
+        _ = _ := by
+          rw [←Finset.mul_sum,routingMoment_step]
+          simp only [Nat.cast_add,Nat.cast_one]
+          change W * ((2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * ∑ c, m c) =
+            (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * (W * ∑ c, m c)
+          ring
+
+lemma routingMoment_sum (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (a : ℝ) :
+    (∑ f : ι ↪ Card d, routingMoment d e f a) =
+      finiteMean (fun ω : BenesCoins d => (2:ℝ)^((palindromeCost d e ω:ℝ)*a)) := by
+  classical
+  unfold routingMoment
+  rw [←finiteMean_sum]
+  apply finiteMean_congr
+  intro ω
+  let f₀ : ι ↪ Card d := e.trans (palindromePerm d ω).toEmbedding
+  rw [Finset.sum_eq_single f₀]
+  · have hm : ∀ i, palindromePerm d ω (e i) = f₀ i := fun _ => rfl
+    simp only [hm,implies_true,↓reduceIte]
+  · intro f _ hf
+    have hm : ¬ ∀ i, palindromePerm d ω (e i) = f i := by
+      intro hh
+      apply hf
+      ext i : 1
+      exact (hh i).symm
+    exact ite_eq_right hm
+  · simp
+
+lemma finite_row_power {X : Type*} [Fintype X] [Nonempty X]
+    (p : X → ℝ) (hp : ∀ x, 0 ≤ p x) (a : ℝ) :
+    finiteMean (fun x => ((Fintype.card X:ℝ)*p x)^(1+a)) =
+      (Fintype.card X:ℝ)^a * ∑ x, (p x)^(1+a) := by
+  have hn : (0:ℝ) < Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
+  simp only [finiteMean,Real.mul_rpow (le_of_lt hn) (hp _),←Finset.mul_sum]
+  rw [Real.rpow_add hn,Real.rpow_one]
+  field_simp
+
+noncomputable def palindromeRowMoment (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (a : ℝ) : ℝ :=
+  finiteMean (fun f : ι ↪ Card d =>
+    ((Fintype.card (ι ↪ Card d):ℝ)*PairRouting.tupleProbability (palindromePerm d) e f)^(1+a))
+
+lemma density_normalization (d h n : ℕ) (a : ℝ) :
+    (n:ℝ)^a * (2:ℝ)^(-(d:ℝ)*h*a) = ((n:ℝ)/((2:ℝ)^d)^h)^a := by
+  rw [Real.div_rpow (Nat.cast_nonneg _) (by positivity),←pow_mul,
+    ←Real.rpow_natCast,←Real.rpow_mul (by norm_num : (0:ℝ) ≤ 2),Nat.cast_mul]
+  rw [show -(d:ℝ)*h*a = -((d:ℝ)*h*a) by ring,
+    Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2),div_eq_mul_inv]
+
+theorem palindrome_row_density_moment (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (a : ℝ) (ha : 0 ≤ a) :
+    palindromeRowMoment d e a ≤
+      (((2^d).descFactorial (Fintype.card ι):ℝ)/((2^d:ℕ):ℝ)^(Fintype.card ι))^a *
+        finiteMean (fun ω : BenesCoins d => (2:ℝ)^((palindromeCost d e ω:ℝ)*a)) := by
+  classical
+  have : Nonempty (ι ↪ Card d) := ⟨e⟩
+  rw [palindromeRowMoment,finite_row_power _ (fun f => PairRouting.tupleProbability_nonneg _ _ _)]
+  calc
+    _ ≤ (Fintype.card (ι ↪ Card d):ℝ)^a *
+      ∑ f : ι ↪ Card d, (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) * routingMoment d e f a :=
+      mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun f _ => palindrome_pointwise_moment d e f a ha))
+        (Real.rpow_nonneg (Nat.cast_nonneg _) _)
+    _ = _ := by
+      rw [←Finset.mul_sum,routingMoment_sum,←mul_assoc,density_normalization,
+        Fintype.card_embedding_eq,card_positions]
+      norm_cast
+
+end Thorp
+namespace Thorp
+
+lemma subset_biUnion_indicator {α β : Type*} [DecidableEq α] [Fintype β]
+    (A : Finset α) (T : β → Finset α) :
+    (if Finset.univ.biUnion T ⊆ A then (1:ℝ) else 0) =
+      ∏ b, if T b ⊆ A then (1:ℝ) else 0 := by
+  classical
+  by_cases h : Finset.univ.biUnion T ⊆ A
+  · rw [ite_eq_left h]
+    symm
+    apply Finset.prod_eq_one
+    intro b _
+    rw [ite_eq_left]
+    exact (Finset.subset_biUnion_of_mem T (Finset.mem_univ b)).trans h
+  · rw [ite_eq_right h]
+    obtain ⟨x,hx,hxA⟩ := Finset.not_subset.mp h
+    obtain ⟨b,_,hxb⟩ := Finset.mem_biUnion.mp hx
+    symm
+    apply Finset.prod_eq_zero (Finset.mem_univ b)
+    rw [ite_eq_right]
+    exact fun hb => hxA (hb hxb)
+
+lemma cylinder_block_polynomial {α β Ω : Type*} [DecidableEq α] [Fintype β]
+    [Fintype Ω] (τ : β → Type*) [∀ b, Fintype (τ b)]
+    (F : Ω → Finset α) (p : α → ℝ)
+    (hF : CylinderBound (inclusionMoment F) p)
+    (T : ∀ b, τ b → Finset α) (c : ∀ b, τ b → ℝ)
+    (hc : ∀ b s, 0 ≤ c b s)
+    (hd : ∀ b b', b ≠ b' → ∀ s s', Disjoint (T b s) (T b' s')) :
+    finiteMean (fun ω => ∏ b, ∑ s, c b s * if T b s ⊆ F ω then 1 else 0) ≤
+      ∏ b, ∑ s, c b s * ∏ x ∈ T b s, p x := by
+  classical
+  simp_rw [Fintype.prod_sum]
+  rw [finiteMean_sum]
+  apply Finset.sum_le_sum
+  intro s _
+  let U := Finset.univ.biUnion (fun b => T b (s b))
+  have he : (fun ω => ∏ b, c b (s b) * if T b (s b) ⊆ F ω then 1 else 0) =
+      fun ω => (if U ⊆ F ω then 1 else 0) * ∏ b, c b (s b) := by
+    funext ω
+    rw [Finset.prod_mul_distrib,←subset_biUnion_indicator]
+    exact mul_comm _ _
+  rw [he,finiteMean_mul_const]
+  have hdis : ((Finset.univ : Finset β) : Set β).PairwiseDisjoint (fun b => T b (s b)) := by
+    intro b _ b' _ hbb
+    exact hd b b' hbb (s b) (s b')
+  have hmean : finiteMean (fun ω => if U ⊆ F ω then 1 else 0) = inclusionMoment F U := by
+    unfold inclusionMoment
+    apply finiteMean_congr
+    intro ω
+    by_cases hh : U ⊆ F ω <;> simp [hh]
+  calc
+    _ ≤ (∏ x ∈ U, p x) * ∏ b, c b (s b) :=
+      mul_le_mul_of_nonneg_right (hmean.le.trans (hF U)) (Finset.prod_nonneg (fun b _ => hc b (s b)))
+    _ = _ := by
+      dsimp only [U]
+      rw [Finset.prod_biUnion hdis,←Finset.prod_mul_distrib]
+      apply Finset.prod_congr rfl
+      intro b _
+      exact mul_comm _ _
+
+
+noncomputable def sparseCoefficient {α : Type*} (w : ℝ) (T : Finset α) : ℝ :=
+  if T.card = 1 then 0 else w^T.card
+
+lemma sparseCoefficient_nonneg {α : Type*} (w : ℝ) (hw : 0 ≤ w) (T : Finset α) :
+    0 ≤ sparseCoefficient w T := by
+  unfold sparseCoefficient
+  split_ifs <;> positivity
+
+lemma sparse_block_pointwise {α : Type*} [DecidableEq α] (A D : Finset α)
+    (w : ℝ) (hw : 0 ≤ w) :
+    w^(if 2 ≤ (D ∩ A).card then (D ∩ A).card else 0) ≤
+      ∑ T : ↑D.powerset, sparseCoefficient w T.val * if T.val ⊆ A then 1 else 0 := by
+  classical
+  let V := D ∩ A
+  let T : ↑D.powerset := if 2 ≤ V.card then
+    ⟨V,Finset.mem_powerset.mpr Finset.inter_subset_left⟩ else
+    ⟨∅,Finset.mem_powerset.mpr (Finset.empty_subset D)⟩
+  calc
+    _ = sparseCoefficient w T.val * (if T.val ⊆ A then 1 else 0) := by
+      by_cases h : 2 ≤ V.card
+      · have hn : V.card ≠ 1 := by omega
+        simp [T,ite_eq_left h,sparseCoefficient,hn,show V ⊆ A from Finset.inter_subset_right,V]
+      · simp [T,ite_eq_right h,sparseCoefficient,V] at *
+    _ ≤ _ := Finset.single_le_sum (f := fun S : ↑D.powerset =>
+      sparseCoefficient w S.val * (if S.val ⊆ A then 1 else 0))
+      (fun S _ => mul_nonneg (sparseCoefficient_nonneg w hw S.val)
+        (by split_ifs <;> norm_num)) (Finset.mem_univ T)
+
+lemma geometric_sum_half (n : ℕ) {x : ℝ} (hx : 0 ≤ x) (hhalf : x ≤ 1/2) :
+    ∑ i ∈ Finset.range n, x^i ≤ 2 := by
+  induction n with
+  | zero => simp
+  | succ n ih =>
+    rw [Finset.sum_range_succ',pow_zero]
+    simp only [pow_succ',←Finset.mul_sum]
+    nlinarith
+
+lemma sparse_binomial_bound (R : ℕ) (z : ℝ) (hz : 0 ≤ z) (hr : (R:ℝ)*z ≤ 1/2) :
+    ∑ i ∈ Finset.range (R+1), (R.choose i:ℝ) * (if i = 1 then 0 else z^i) ≤
+      1 + 2*((R:ℝ)*z)^2 := by
+  classical
+  let x := (R:ℝ)*z
+  have hx : 0 ≤ x := mul_nonneg (Nat.cast_nonneg _) hz
+  have hbound (i : ℕ) : (R.choose i:ℝ)*(if i=1 then 0 else z^i) ≤
+      if i=1 then 0 else x^i := by
+    by_cases h : i=1
+    · simp [h]
+    · simp only [ite_eq_right h]
+      dsimp [x]
+      rw [mul_pow]
+      apply mul_le_mul_of_nonneg_right _ (pow_nonneg hz i)
+      exact_mod_cast Nat.choose_le_pow R i
+  calc
+    _ ≤ ∑ i ∈ Finset.range (R+1), if i=1 then 0 else x^i :=
+      Finset.sum_le_sum (fun i _ => hbound i)
+    _ ≤ 1+2*x^2 := by
+      rcases R with _ | R
+      · simp [x]
+      rcases R with _ | R
+      · norm_num [Finset.sum_range_succ]
+        positivity
+      have he : ∑ i ∈ Finset.range (R+2+1), (if i=1 then 0 else x^i) =
+          1 + x^2*∑ i ∈ Finset.range (R+1), x^i := by
+        rw [show R+2+1=2+(R+1) by omega,Finset.sum_range_add]
+        have hn (i : ℕ) : 2+i ≠ 1 := by omega
+        simp only [hn,ite_false,pow_add,←Finset.mul_sum]
+        norm_num [Finset.sum_range_succ]
+      rw [he]
+      have hg := geometric_sum_half (R+1) hx hr
+      nlinarith [sq_nonneg x]
+
+lemma sparse_block_product_bound {α : Type*} [DecidableEq α]
+    (D : Finset α) (p w : ℝ) (hp : 0 ≤ p) (hw : 0 ≤ w)
+    (hr : (D.card:ℝ)*(p*w) ≤ 1/2) :
+    ∑ T : ↑D.powerset, sparseCoefficient w T.val * p^T.val.card ≤
+      Real.exp (2*((D.card:ℝ)*(p*w))^2) := by
+  classical
+  have he : ∑ T : ↑D.powerset, sparseCoefficient w T.val * p^T.val.card =
+      ∑ i ∈ Finset.range (D.card+1), (D.card.choose i:ℝ)*(if i=1 then 0 else (p*w)^i) := by
+    rw [Finset.sum_coe_sort D.powerset (fun T => sparseCoefficient w T*p^T.card)]
+    have hh : (fun T : Finset α => sparseCoefficient w T*p^T.card) =
+        fun T => if T.card=1 then 0 else (p*w)^T.card := by
+      funext T
+      unfold sparseCoefficient
+      split_ifs <;> simp [mul_pow,mul_comm]
+    change (∑ T ∈ D.powerset, (fun T : Finset α => sparseCoefficient w T*p^T.card) T) = _
+    rw [hh,Finset.sum_powerset_apply_card (fun i => if i=1 then (0:ℝ) else (p*w)^i)]
+    simp only [nsmul_eq_mul]
+  rw [he]
+  exact (sparse_binomial_bound D.card (p*w) (mul_nonneg hp hw) hr).trans
+    (by simpa only [add_comm] using Real.add_one_le_exp (2*((D.card:ℝ)*(p*w))^2))
+
+
+noncomputable def sparseMass {α β : Type*} [DecidableEq α] [Fintype β]
+    (D : β → Finset α) (A : Finset α) : ℕ :=
+  ∑ b, if 2 ≤ (D b ∩ A).card then (D b ∩ A).card else 0
+
+theorem sparse_block_mgf {α β Ω : Type*} [DecidableEq α] [Fintype β] [Fintype Ω]
+    (D : β → Finset α) (hd : Pairwise (fun b b' => Disjoint (D b) (D b')))
+    (F : Ω → Finset α) (p w : ℝ) (hp : 0 ≤ p) (hw : 0 ≤ w)
+    (hF : ∀ T, inclusionMoment F T ≤ p^T.card)
+    (hr : ∀ b, ((D b).card:ℝ)*(p*w) ≤ 1/2) :
+    finiteMean (fun ω => w^(sparseMass D (F ω))) ≤
+      Real.exp (∑ b, 2*(((D b).card:ℝ)*(p*w))^2) := by
+  classical
+  have hc : CylinderBound (inclusionMoment F) (fun _ => p) := by
+    intro T
+    simpa only [Finset.prod_const] using hF T
+  have hb := cylinder_block_polynomial (fun b => ↑(D b).powerset) F (fun _ => p) hc
+    (fun _ S => S.val) (fun _ S => sparseCoefficient w S.val)
+    (fun _ S => sparseCoefficient_nonneg w hw S.val)
+    (fun b b' h S S' => (hd h).mono
+      (Finset.mem_powerset.mp S.property) (Finset.mem_powerset.mp S'.property))
+  calc
+    _ ≤ finiteMean (fun ω => ∏ b, ∑ S : ↑(D b).powerset,
+        sparseCoefficient w S.val * if S.val ⊆ F ω then 1 else 0) := by
+      apply finiteMean_mono
+      intro ω
+      rw [sparseMass,←Finset.prod_pow_eq_pow_sum]
+      apply Finset.prod_le_prod₀ (fun _ _ => pow_nonneg hw _)
+      intro b _
+      exact sparse_block_pointwise (F ω) (D b) w hw
+    _ ≤ ∏ b, ∑ S : ↑(D b).powerset, sparseCoefficient w S.val * p^S.val.card := by
+      simpa only [Finset.prod_const] using hb
+    _ ≤ _ := by
+      rw [Real.exp_sum]
+      apply Finset.prod_le_prod₀
+      · intro b _
+        exact Finset.sum_nonneg (fun S _ => mul_nonneg
+          (sparseCoefficient_nonneg w hw S.val) (pow_nonneg hp _))
+      · intro b _
+        exact sparse_block_product_bound (D b) p w hp hw (hr b)
+
+theorem inverseButterfly_sparse_mgf (d : ℕ) {β : Type*} [Fintype β]
+    (D : β → Finset (Card d)) (hd : Pairwise (fun b b' => Disjoint (D b) (D b')))
+    (A : Finset (Card d)) (w : ℝ) (hw : 0 ≤ w)
+    (hr : ∀ b, ((D b).card:ℝ)*((A.card/(2:ℝ)^d)*w) ≤ 1/2) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      w^(sparseMass D (inverseButterflyMarks d A ω))) ≤
+      Real.exp (∑ b, 2*(((D b).card:ℝ)*((A.card/(2:ℝ)^d)*w))^2) := by
+  apply sparse_block_mgf D hd (inverseButterflyMarks d A) (A.card/(2:ℝ)^d) w
+    (by positivity) hw
+  · exact inverseButterflyMarks_inclusion d A
+  · exact hr
+
+
+def crowdedMass (h : ℕ) : ℕ := if 2 ≤ h then h else 0
+
+lemma crowdedMass_le (h : ℕ) : crowdedMass h ≤ h := by
+  unfold crowdedMass
+  split_ifs <;> omega
+
+lemma crowdedMass_add (u v : ℕ) : crowdedMass u + crowdedMass v ≤ crowdedMass (u+v) := by
+  unfold crowdedMass
+  split_ifs <;> omega
+
+lemma alternatingCycles_le_crowded {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
+    (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    PairRouting.alternatingCycles e p ≤ crowdedMass (Fintype.card ι) := by
+  unfold crowdedMass
+  split_ifs with h
+  · have hh := PairRouting.alternatingCycles_double e p
+    omega
+  · rw [PairRouting.alternatingCycles_zero e p (by omega)]
+
+lemma palindromeCost_le_crowded (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) :
+    palindromeCost d e ω ≤ d*crowdedMass (Fintype.card ι) := by
+  classical
+  induction d generalizing ι with
+  | zero => simp [palindromeCost]
+  | succ d ih =>
+    let σ := benesStepEquiv d ω
+    let x := e.trans (headTailEquiv d).toEmbedding
+    let c := PairRouting.colors x σ.2.1
+    let hc := PairRouting.colors_compatible x σ.2.1
+    have h₀ := ih (PairRouting.childEmbedding x c hc false) σ.1.1
+    have h₁ := ih (PairRouting.childEmbedding x c hc true) σ.1.2
+    have hz := alternatingCycles_le_crowded (PairRouting.switchedEmbedding x σ.2.1)
+      (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1))
+    have hs := crowdedMass_add (Fintype.card {i // c i = false})
+      (Fintype.card {i // c i = true})
+    rw [PairRouting.color_card_add] at hs
+    change PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
+        (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) +
+        palindromeCost d (PairRouting.childEmbedding x c hc false) σ.1.1 +
+        palindromeCost d (PairRouting.childEmbedding x c hc true) σ.1.2 ≤ _
+    nlinarith
+
+noncomputable def palindromeLowCost (H : ℕ) : (d : ℕ) → {ι : Type*} → [Fintype ι] →
+    (ι ↪ Card d) → BenesCoins d → ℕ
+  | 0, _, _, _, _ => 0
+  | d+1, _, _, e, ω =>
+      let σ := benesStepEquiv d ω
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x σ.2.1
+      let hc := PairRouting.colors_compatible x σ.2.1
+      (if d+1 ≤ H then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
+          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) else 0) +
+        palindromeLowCost H d (PairRouting.childEmbedding x c hc false) σ.1.1 +
+        palindromeLowCost H d (PairRouting.childEmbedding x c hc true) σ.1.2
+
+end Thorp
+
+end ThorpNine.Harmonic
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Harmonic/Shapes.lean b/lean/OAI/Probability/ThorpRouting/Harmonic/Shapes.lean
index e68d8dd984b908d5b968bdedcb78580be2f011e9..9c3f91689ae4343735d3cc2c2e15c735502171f1 100644
--- a/lean/OAI/Probability/ThorpRouting/Harmonic/Shapes.lean
+++ b/lean/OAI/Probability/ThorpRouting/Harmonic/Shapes.lean
@@ -1,882 +1,882 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Harmonic.ModifiedReverseCoins
-
-namespace OAI
-
-namespace ThorpNine.Harmonic
-
-namespace Thorp.Specht
-open scoped BigOperators Classical
-
-lemma partitionDiagram_rowLens {N : ℕ} (p : Nat.Partition N) :
-    (partitionDiagram p).rowLens = p.parts.sort (· ≥ ·) := by
-  apply YoungDiagram.rowLens_ofRowLens_eq_self
-  intro x hx
-  exact p.parts_pos ((Multiset.mem_sort _).mp hx)
-
-lemma partitionDiagram_card {N : ℕ} (p : Nat.Partition N) : (partitionDiagram p).card = N := by
-  rw [←sum_rowLens,partitionDiagram_rowLens]
-  change (↑(p.parts.sort (· ≥ ·)) : Multiset ℕ).sum = N
-  rw [Multiset.sort_eq,p.parts_sum]
-
-lemma partitionDiagram_diagramPartition (μ : YoungDiagram) :
-    partitionDiagram (diagramPartition μ) = μ := by
-  apply YoungDiagram.equivListRowLens.injective
-  apply Subtype.ext
-  change (partitionDiagram (diagramPartition μ)).rowLens = μ.rowLens
-  rw [partitionDiagram_rowLens]
-  change (↑μ.rowLens : Multiset ℕ).sort (· ≥ ·) = μ.rowLens
-  rw [Multiset.coe_sort,List.mergeSort_eq_self _ μ.rowLens_sorted.pairwise]
-
-abbrev Shapes (N : ℕ) := {μ : YoungDiagram // μ.card = N}
-
-noncomputable def shapePartitionEquiv (N : ℕ) : Shapes N ≃ Nat.Partition N where
-  toFun μ := { parts := μ.1.rowLens
-               parts_pos := fun {i} hi => μ.1.pos_of_mem_rowLens i (by simpa using hi)
-               parts_sum := by simpa using (sum_rowLens μ.1).trans μ.2 }
-  invFun p := ⟨partitionDiagram p,partitionDiagram_card p⟩
-  left_inv μ := by
-    apply Subtype.ext
-    obtain ⟨μ,rfl⟩ := μ
-    exact partitionDiagram_diagramPartition μ
-  right_inv p := by
-    apply Nat.Partition.ext
-    change ((partitionDiagram p).rowLens : Multiset ℕ) = p.parts
-    rw [partitionDiagram_rowLens,Multiset.sort_eq]
-
-noncomputable instance (N : ℕ) : Fintype (Shapes N) := Fintype.ofEquiv _ (shapePartitionEquiv N).symm
-
-noncomputable def conjugacyPartition {α : Type*} [Fintype α] [DecidableEq α] :
-    ConjClasses (Equiv.Perm α) → Nat.Partition (Fintype.card α) :=
-  Quotient.lift Equiv.Perm.partition (fun _ _ hpq => (Equiv.Perm.partition_eq_of_isConj).mp hpq)
-
-lemma conjugacyPartition_injective {α : Type*} [Fintype α] [DecidableEq α] :
-    Function.Injective (conjugacyPartition (α := α)) := by
-  intro p q
-  induction p using Quotient.inductionOn with | _ p =>
-    induction q using Quotient.inductionOn with | _ q =>
-      intro h
-      exact Quotient.sound ((Equiv.Perm.partition_eq_of_isConj).mpr h)
-
-lemma card_conjugacy_le_shapes (N : ℕ) :
-    Nat.card (ConjClasses (Equiv.Perm (Fin N))) ≤ Fintype.card (Shapes N) := by
-  have h := Nat.card_le_card_of_injective _ (conjugacyPartition_injective (α := Fin N))
-  simpa only [Fintype.card_fin, Nat.card_eq_fintype_card, Fintype.card_congr (shapePartitionEquiv N)] using h
-
-noncomputable def labels {N : ℕ} (μ : Shapes N) : Cell μ.1 ≃ Fin N :=
-  (Fintype.equivFin (Cell μ.1)).trans (finCongr ((card_cell μ.1).trans μ.2))
-
-noncomputable def labelledRepresentation {N : ℕ} (μ : Shapes N) :
-    Representation ℂ (Equiv.Perm (Fin N)) (space μ.1) :=
-  (representation μ.1).comp (labels μ).symm.permCongrHom.toMonoidHom
-
-lemma labelledRepresentation_irreducible {N : ℕ} (μ : Shapes N) :
-    Representation.IsIrreducible (labelledRepresentation μ) := by
-  let e := (labels μ).symm.permCongrHom
-  let ρ := representation μ.1
-  have hn : (⊥ : Subrepresentation (labelledRepresentation μ)) ≠ ⊤ := by
-    intro h
-    have he : (⟨polytabloid μ.1,polytabloid_mem_space μ.1⟩ : space μ.1) ∈
-        (⊥ : Subrepresentation (labelledRepresentation μ)) := by rw [h]; trivial
-    exact polytabloid_ne_zero μ.1 (congrArg Subtype.val he)
-  refine { exists_pair_ne := ⟨⊥,⊤,hn⟩,eq_bot_or_eq_top := ?_ }
-  intro U
-  let U' : Subrepresentation ρ :=
-    { toSubmodule := U.toSubmodule
-      apply_mem_toSubmodule := fun p v hv => by
-        have hh := U.apply_mem_toSubmodule (e.symm p) hv
-        change ρ (e (e.symm p)) v ∈ U.toSubmodule at hh
-        simpa using hh }
-  have := representation_irreducible μ.1
-  obtain h | h := IsSimpleOrder.eq_bot_or_eq_top U'
-  · left
-    apply Subrepresentation.toSubmodule_injective
-    exact congrArg (fun R : Subrepresentation ρ => R.toSubmodule) h
-  · right
-    apply Subrepresentation.toSubmodule_injective
-    exact congrArg (fun R : Subrepresentation ρ => R.toSubmodule) h
-
-lemma labelledRepresentation_equiv_shapes {N : ℕ} (μ ν : Shapes N)
-    (T : Representation.Equiv (labelledRepresentation μ) (labelledRepresentation ν)) : μ = ν := by
-  apply Subtype.ext
-  apply shapes_eq_of_equivariant_equiv μ.1 ν.1 ((labels μ).trans (labels ν).symm) T.toLinearEquiv
-  intro p v
-  have hh := Representation.IntertwiningMap.isIntertwining _ _ T.toIntertwiningMap ((labels μ).permCongr p) v
-  change T.toLinearEquiv
-    (representation μ.1 ((labels μ).symm.permCongr ((labels μ).permCongr p)) v) =
-      representation ν.1 ((labels ν).symm.permCongr ((labels μ).permCongr p)) (T.toLinearEquiv v) at hh
-  have hp : (labels μ).symm.permCongr ((labels μ).permCongr p) = p := by
-    apply Equiv.ext
-    intro x
-    simp [Equiv.permCongr_apply]
-  rw [hp] at hh
-  convert hh using 2
-  congr 1
-
-end Thorp.Specht
-
-namespace Thorp.FiniteCharacters
-open scoped BigOperators Classical
-
-section Group
-variable {G : Type*} [Group G] [Fintype G]
-
-noncomputable def classRep (c : ConjClasses G) : G := (ConjClasses.exists_rep c).choose
-omit [Fintype G] in
-lemma classRep_spec (c : ConjClasses G) : ConjClasses.mk (classRep c) = c :=
-  (ConjClasses.exists_rep c).choose_spec
-
-variable {V : Type*} [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
-
-noncomputable def characterClass (ρ : Representation ℂ G V) : ConjClasses G → ℂ :=
-  fun c => ρ.character (classRep c)
-
-omit [Fintype G] [FiniteDimensional ℂ V] in
-lemma characterClass_mk (ρ : Representation ℂ G V) (g : G) :
-    characterClass ρ (ConjClasses.mk g) = ρ.character g := by
-  have h := ConjClasses.mk_eq_mk_iff_isConj.mp (classRep_spec (ConjClasses.mk g))
-  obtain ⟨p,hp⟩ := isConj_iff.mp h
-  change ρ.character (classRep (ConjClasses.mk g)) = ρ.character g
-  exact (Representation.char_conj ρ _ p).symm.trans (congrArg ρ.character hp)
-
-noncomputable def characterDual (ρ : Representation ℂ G V) :
-    (ConjClasses G → ℂ) →ₗ[ℂ] ℂ where
-  toFun f := (Nat.card G : ℂ)⁻¹ * ∑ g, f (ConjClasses.mk g) * ρ.character g⁻¹
-  map_add' f h := by simp [Finset.sum_add_distrib,add_mul,mul_add]
-  map_smul' a f := by
-    simp only [Pi.smul_apply,smul_eq_mul,RingHom.id_apply]
-    simp_rw [mul_assoc]
-    rw [←Finset.mul_sum]
-    ring
-
-lemma characterDual_characterClass (ρ : Representation ℂ G V)
-    {W : Type*} [AddCommGroup W] [Module ℂ W] [FiniteDimensional ℂ W]
-    (σ : Representation ℂ G W) [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ] :
-    characterDual ρ (characterClass σ) = if Nonempty (Representation.Equiv ρ σ) then 1 else 0 := by
-  let : Invertible (Nat.card G : ℂ) := invertibleOfNonzero (by
-    exact_mod_cast (Nat.card_pos (α := G)).ne')
-  change (Nat.card G : ℂ)⁻¹ * ∑ g, characterClass σ (ConjClasses.mk g) * ρ.character g⁻¹ = _
-  simp only [characterClass_mk]
-  exact Representation.char_orthonormal σ ρ
-
-end Group
-
-theorem complete_of_card_conjugacy_le
-    {G ι : Type*} [Group G] [Fintype G] [Fintype ι]
-    (V : ι → Type*) [∀ i, AddCommGroup (V i)] [∀ i, Module ℂ (V i)]
-    [∀ i, FiniteDimensional ℂ (V i)]
-    (ρ : ∀ i, Representation ℂ G (V i)) [∀ i, Representation.IsIrreducible (ρ i)]
-    (hneq : ∀ i j, i ≠ j → ¬Nonempty (Representation.Equiv (ρ i) (ρ j)))
-    (hcard : Nat.card (ConjClasses G) ≤ Fintype.card ι)
-    {W : Type*} [AddCommGroup W] [Module ℂ W] [FiniteDimensional ℂ W]
-    (σ : Representation ℂ G W) [Representation.IsIrreducible σ] :
-    ∃ i, Nonempty (Representation.Equiv (ρ i) σ) := by
-  let : Fintype (ConjClasses G) := Fintype.ofFinite _
-  have horth (i j : ι) : characterDual (ρ i) (characterClass (ρ j)) = if i=j then 1 else 0 := by
-    rw [characterDual_characterClass]
-    by_cases hij : i=j
-    · subst j
-      simp only [ite_true]
-      exact ite_eq_left ⟨Representation.Equiv.refl _⟩
-    · rw [ite_eq_right hij,ite_eq_right (hneq i j hij)]
-  have hLI : LinearIndependent ℂ (fun i => characterClass (ρ i)) := by
-    apply LinearIndependent.of_pairwise_dual_eq_zero_one _ (fun i => characterDual (ρ i))
-    · intro i j hij
-      exact (horth i j).trans (ite_eq_right hij)
-    · intro i
-      simpa using horth i i
-  have heq : Fintype.card ι = Module.finrank ℂ (ConjClasses G → ℂ) := by
-    apply Nat.le_antisymm hLI.fintype_card_le_finrank
-    simpa only [Module.finrank_pi,Module.finrank_self,Finset.sum_const,Finset.card_univ,
-      smul_eq_mul,mul_one,Nat.card_eq_fintype_card] using hcard
-  have hspan := hLI.span_eq_top_of_card_eq_finrank' heq
-  by_contra h
-  push Not at h
-  have hker : Submodule.span ℂ (Set.range (fun i => characterClass (ρ i))) ≤
-      (characterDual σ).ker := by
-    apply Submodule.span_le.mpr
-    rintro f ⟨i,rfl⟩
-    change characterDual σ (characterClass (ρ i)) = 0
-    rw [characterDual_characterClass,ite_eq_right]
-    rintro ⟨T⟩
-    exact (h i).false T.symm
-  rw [hspan] at hker
-  have hz : characterDual σ (characterClass σ) = 0 := hker (show characterClass σ ∈ ⊤ from trivial)
-  rw [characterDual_characterClass,ite_eq_left ⟨Representation.Equiv.refl σ⟩] at hz
-  exact one_ne_zero hz
-
-end Thorp.FiniteCharacters
-
-namespace Thorp.Specht
-open scoped Classical
-
-theorem labelled_complete {N : ℕ}
-    {V : Type*} [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
-    (ρ : Representation ℂ (Equiv.Perm (Fin N)) V) [Representation.IsIrreducible ρ] :
-    ∃ μ : Shapes N, Nonempty (Representation.Equiv (labelledRepresentation μ) ρ) := by
-  let (μ : Shapes N) : Representation.IsIrreducible (labelledRepresentation μ) :=
-    labelledRepresentation_irreducible μ
-  apply FiniteCharacters.complete_of_card_conjugacy_le
-    (fun μ : Shapes N => space μ.1) labelledRepresentation _ (card_conjugacy_le_shapes N) ρ
-  intro μ ν h ⟨T⟩
-  exact h (labelledRepresentation_equiv_shapes μ ν T)
-
-end Thorp.Specht
-
-namespace Thorp.PermutationHilbert
-
-section
-open scoped BigOperators ComplexConjugate Classical
-open Complex
-
-variable {X : Type*} [Fintype X]
-
-noncomputable def act (p : Equiv.Perm X) : H X ≃ₗᵢ[ℂ] H X :=
-  LinearIsometryEquiv.piLpCongrLeft 2 ℂ ℂ p
-
-@[simp] lemma act_apply (p : Equiv.Perm X) (v : H X) (x : X) :
-    act p v x = v (p⁻¹ x) := rfl
-
-lemma act_single [DecidableEq X] (p : Equiv.Perm X) (x : X) :
-    act p (EuclideanSpace.single x (1:ℂ)) = EuclideanSpace.single (p x) 1 := by
-  exact EuclideanSpace.piLpCongrLeft_single p x 1
-
-lemma trace_projection (W : Submodule ℂ (H X)) :
-    ∑ x, (W.starProjection (EuclideanSpace.single x (1:ℂ))) x = Module.finrank ℂ W := by
-  classical
-  have hp : LinearMap.IsProj W W.starProjection.toLinearMap := by
-    constructor
-    · intro x; exact (W.orthogonalProjectionOnto x).2
-    · intro x hx; exact W.starProjection_eq_self_iff.mpr hx
-  have ht := hp.trace
-  rw [LinearMap.trace_eq_matrix_trace ℂ (EuclideanSpace.basisFun X ℂ).toBasis] at ht
-  simpa only [Matrix.trace,Matrix.diag_apply,LinearMap.toMatrix_apply,EuclideanSpace.basisFun_repr,
-    EuclideanSpace.basisFun_apply,OrthonormalBasis.coe_toBasis,
-    OrthonormalBasis.coe_toBasis_repr_apply,ContinuousLinearMap.coe_coe] using ht
-
-lemma projection_diagonal (W : Submodule ℂ (H X)) (x : X) :
-    (W.starProjection (EuclideanSpace.single x (1:ℂ))) x =
-      (‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 : ℝ) := by
-  classical
-  have hh := W.inner_starProjection_left_eq_right
-    (EuclideanSpace.single x (1:ℂ)) (W.starProjection (EuclideanSpace.single x (1:ℂ)))
-  have hf : W.starProjection (W.starProjection (EuclideanSpace.single x (1:ℂ))) =
-      W.starProjection (EuclideanSpace.single x (1:ℂ)) :=
-    W.starProjection_eq_self_iff.mpr (W.orthogonalProjectionOnto _).2
-  rw [hf] at hh
-  rw [inner_self_eq_norm_sq_to_K,EuclideanSpace.inner_single_left] at hh
-  simpa using hh.symm
-
-lemma projection_comm (W : Submodule ℂ (H X)) (p : Equiv.Perm X)
-    (hW : W.map (act p).toLinearEquiv.toLinearMap = W) (v : H X) :
-    act p (W.starProjection v) = W.starProjection (act p v) := by
-  have hh := (act p).toLinearIsometry.map_starProjection W v
-  change act p (W.starProjection v) =
-    (W.map (act p).toLinearEquiv.toLinearMap).starProjection (act p v) at hh
-  simpa only [hW] using hh
-
-lemma uniform_diagonal (W : Submodule ℂ (H X))
-    (htrans : ∀ x y : X, ∃ p : Equiv.Perm X, p x = y ∧
-      W.map (act p).toLinearEquiv.toLinearMap = W) (x : X) :
-    (Fintype.card X:ℝ) * ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 =
-      Module.finrank ℂ W := by
-  classical
-  have hh (y : X) : (W.starProjection (EuclideanSpace.single y (1:ℂ))) y =
-      (‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 : ℝ) := by
-    obtain ⟨p,hp,hW⟩ := htrans x y
-    subst y
-    rw [projection_diagonal,←act_single p x,←projection_comm W p hW,(act p).norm_map]
-  have ht := trace_projection W
-  simp_rw [hh] at ht
-  have hhreal : (∑ _y : X, ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2) =
-      (Module.finrank ℂ W:ℝ) := by exact_mod_cast ht
-  simpa using hhreal
-
-theorem uncertainty (W : Submodule ℂ (H X))
-    (htrans : ∀ x y : X, ∃ p : Equiv.Perm X, p x = y ∧
-      W.map (act p).toLinearEquiv.toLinearMap = W)
-    (v : H X) (hv : v ∈ W) (x : X) :
-    (Fintype.card X:ℝ) * ‖v x‖^2 ≤ (Module.finrank ℂ W:ℝ) * ‖v‖^2 := by
-  classical
-  have hh : inner ℂ (W.starProjection (EuclideanSpace.single x (1:ℂ))) v = v x := by
-    rw [W.inner_starProjection_left_eq_right,W.starProjection_eq_self_iff.mpr hv,
-      EuclideanSpace.inner_single_left]
-    simp
-  have hc := norm_inner_le_norm (𝕜 := ℂ) (W.starProjection (EuclideanSpace.single x (1:ℂ))) v
-  rw [hh] at hc
-  have hs : ‖v x‖^2 ≤ ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 * ‖v‖^2 := by
-    nlinarith [norm_nonneg (v x),norm_nonneg v,
-      norm_nonneg (W.starProjection (EuclideanSpace.single x (1:ℂ)))]
-  calc
-    (Fintype.card X:ℝ) * ‖v x‖^2 ≤
-      (Fintype.card X:ℝ) * (‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 * ‖v‖^2) :=
-      mul_le_mul_of_nonneg_left hs (Nat.cast_nonneg _)
-    _ = _ := by rw [←mul_assoc,uniform_diagonal W htrans x]
-
-end
-open scoped BigOperators Classical
-
-variable {X : Type*} [Fintype X]
-
-lemma trace_projection_real (W : Submodule ℂ (H X)) :
-    ∑ x, ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 = (Module.finrank ℂ W:ℝ) := by
-  have hh := trace_projection W
-  simp_rw [projection_diagonal] at hh
-  exact_mod_cast hh
-
-theorem footprint_dimension (W K : Submodule ℂ (H X)) (hKW : K ≤ W)
-    (htrans : ∀ x y : X, ∃ p : Equiv.Perm X, p x = y ∧
-      W.map (act p).toLinearEquiv.toLinearMap = W)
-    (S : Finset X) (hS : ∀ x ∈ S, W.starProjection (EuclideanSpace.single x (1:ℂ)) ∈ K) :
-    (S.card:ℝ) * (Module.finrank ℂ W:ℝ) ≤
-      (Fintype.card X:ℝ) * (Module.finrank ℂ K:ℝ) := by
-  have he (x : X) (hx : x ∈ S) :
-      K.starProjection (EuclideanSpace.single x (1:ℂ)) =
-        W.starProjection (EuclideanSpace.single x (1:ℂ)) := by
-    apply K.eq_starProjection_of_mem_of_inner_eq_zero (hS x hx)
-    intro v hv
-    exact W.starProjection_inner_eq_zero _ v (hKW hv)
-  have hs : ∑ x ∈ S, ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 ≤
-      (Module.finrank ℂ K:ℝ) := by
-    rw [←trace_projection_real K]
-    calc
-      _ = ∑ x ∈ S, ‖K.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 := by
-        apply Finset.sum_congr rfl
-        intro x hx
-        rw [he x hx]
-      _ ≤ _ := Finset.sum_le_univ_sum_of_nonneg (fun x => sq_nonneg _)
-  have hmul := mul_le_mul_of_nonneg_left hs (Nat.cast_nonneg (Fintype.card X): (0:ℝ) ≤ _)
-  rw [Finset.mul_sum] at hmul
-  simp_rw [uniform_diagonal W htrans] at hmul
-  simpa using hmul
-
-end Thorp.PermutationHilbert
-namespace Thorp.Specht
-open scoped BigOperators Classical
-
-lemma finrank_hilbertSpace (μ : YoungDiagram) :
-    Module.finrank ℂ (hilbertSpace μ) = Module.finrank ℂ (space μ) :=
-  (hilbertEquiv μ).finrank_map_eq (space μ)
-
-lemma hilbert_action (μ : YoungDiagram) (p : Equiv.Perm (Cell μ)) (v : Tabloid μ → ℂ) :
-    hilbertEquiv μ (tabloidRep μ p v) =
-      PermutationHilbert.act (tabloidAct p) (hilbertEquiv μ v) := by
-  ext f
-  simp [hilbertEquiv,PermutationHilbert.act_apply]
-
-lemma hilbertSpace_invariant (μ : YoungDiagram) (p : Equiv.Perm (Cell μ)) :
-    (hilbertSpace μ).map (PermutationHilbert.act (tabloidAct p)).toLinearEquiv.toLinearMap =
-      hilbertSpace μ := by
-  apply le_antisymm
-  · rintro _ ⟨v,⟨w,hw,rfl⟩,rfl⟩
-    exact ⟨tabloidRep μ p w,action_mem_space μ p w hw,hilbert_action μ p w⟩
-  · rintro _ ⟨w,hw,rfl⟩
-    refine ⟨hilbertEquiv μ (tabloidRep μ p⁻¹ w),
-      ⟨tabloidRep μ p⁻¹ w,action_mem_space μ p⁻¹ w hw,rfl⟩, ?_⟩
-    change PermutationHilbert.act (tabloidAct p) (hilbertEquiv μ (tabloidRep μ p⁻¹ w)) = hilbertEquiv μ w
-    rw [←hilbert_action]
-    change hilbertEquiv μ ((tabloidRep μ p * tabloidRep μ p⁻¹) w) = _
-    rw [←map_mul,mul_inv_cancel,map_one]
-    rfl
-
-lemma tabloidAct_transitive (μ : YoungDiagram) (f g : Tabloid μ) :
-    ∃ p : Equiv.Perm (Cell μ), tabloidAct p f = g := by
-  obtain ⟨q,hq⟩ := f.2
-  obtain ⟨r,hr⟩ := g.2
-  refine ⟨r⁻¹*q, ?_⟩
-  apply Subtype.ext
-  funext x
-  simp only [tabloidAct_apply,mul_inv_rev,Equiv.Perm.mul_apply,hq,hr,
-    Function.comp_apply,Equiv.Perm.inv_def,Equiv.apply_symm_apply,Equiv.symm_symm]
-
-lemma polytabloid_pairing (μ : YoungDiagram) :
-    pairing (polytabloid μ) (polytabloid μ) = Fintype.card (colGroup μ) := by
-  have h := alternator_pairing μ (polytabloid μ)
-  have he (c : colGroup μ) : tabloidRep μ c (polytabloid μ) =
-      permSign (c:Equiv.Perm (Cell μ)) • polytabloid μ :=
-    action_alternator_apply (colGroup μ) (tabloidRep μ) c _
-  have hb := congrArg (fun v : Tabloid μ → ℂ => v (baseTabloid μ)) h
-  simp only [alternator,LinearMap.sum_apply,LinearMap.smul_apply,he,
-    Pi.smul_apply,smul_eq_mul,Finset.sum_apply,polytabloid_base,mul_one] at hb
-  simp only [permSign_sq,Finset.sum_const,Finset.card_univ,nsmul_eq_mul,mul_one] at hb
-  exact hb.symm
-
-lemma polytabloid_norm_sq (μ : YoungDiagram) :
-    ‖hilbertEquiv μ (polytabloid μ)‖^2 = (Fintype.card (colGroup μ):ℝ) := by
-  have hp := polytabloid_pairing μ
-  change inner ℂ (hilbertEquiv μ (polytabloid μ)) (hilbertEquiv μ (polytabloid μ)) = _ at hp
-  rw [inner_self_eq_norm_sq_to_K] at hp
-  have hre := congrArg Complex.re hp
-  simpa [pow_two] using hre
-
-theorem dimension_times_column_ge_tabloids (μ : YoungDiagram) :
-    Fintype.card (Tabloid μ) ≤ Module.finrank ℂ (space μ) * Fintype.card (colGroup μ) := by
-  have ht (f g : Tabloid μ) : ∃ p : Equiv.Perm (Tabloid μ), p f = g ∧
-      (hilbertSpace μ).map (PermutationHilbert.act p).toLinearEquiv.toLinearMap = hilbertSpace μ := by
-    obtain ⟨p,hp⟩ := tabloidAct_transitive μ f g
-    exact ⟨tabloidAct p,hp,hilbertSpace_invariant μ p⟩
-  have hu := PermutationHilbert.uncertainty (hilbertSpace μ) ht
-    (hilbertEquiv μ (polytabloid μ)) ⟨polytabloid μ,polytabloid_mem_space μ,rfl⟩ (baseTabloid μ)
-  have he : hilbertEquiv μ (polytabloid μ) (baseTabloid μ) = 1 := polytabloid_base μ
-  rw [he,norm_one,one_pow,mul_one,finrank_hilbertSpace,polytabloid_norm_sq] at hu
-  exact_mod_cast hu
-
-
-section FiberGroup
-variable {α β γ : Type*} [Fintype α] [Fintype β] [Fintype γ]
-
-noncomputable def fiberGroupEquiv (f : α → β) :
-    fiberGroup f ≃ (∀ b, Equiv.Perm {x // f x = b}) where
-  toFun p b :=
-    { toFun x := ⟨p.1 x.1,(p.2 x.1).trans x.2⟩
-      invFun x := ⟨p.1⁻¹ x.1, by
-        have hh := p.2 (p.1⁻¹ x.1)
-        simp only [Equiv.Perm.inv_def,Equiv.apply_symm_apply] at hh
-        exact hh.symm.trans x.2⟩
-      left_inv x := by apply Subtype.ext; simp
-      right_inv x := by apply Subtype.ext; simp }
-  invFun q :=
-    ⟨(Equiv.sigmaFiberEquiv f).permCongr (Equiv.Perm.sigmaCongrRight q),
-      fun x => (q (f x) ⟨x,rfl⟩).2⟩
-  left_inv p := by
-    apply Subtype.ext
-    apply Equiv.ext
-    intro x
-    rfl
-  right_inv q := by
-    funext b
-    apply Equiv.ext
-    intro x
-    apply Subtype.ext
-    change (q (f x.1) ⟨x.1,rfl⟩).1 = (q b x).1
-    cases x with
-    | mk x hx => dsimp only at *; subst b; rfl
-
-lemma nat_card_fiberGroup (f : α → β) :
-    Nat.card (fiberGroup f) = ∏ b, (Nat.card {x // f x=b}).factorial := by
-  rw [Nat.card_congr (fiberGroupEquiv f),Nat.card_pi]
-  simp only [Nat.card_eq_fintype_card,Fintype.card_perm]
-
-omit [Fintype β] [Fintype γ] in
-
-lemma card_fiberGroups_mul_le (f : α → β) (g : α → γ)
-    (hinj : ∀ x y, f x = f y → g x = g y → x=y) :
-    Fintype.card (fiberGroup f) * Fintype.card (fiberGroup g) ≤ (Fintype.card α).factorial := by
-  have hi : Function.Injective (fun p : fiberGroup f × fiberGroup g => p.1.1*p.2.1) := by
-    intro p q hpq
-    dsimp only at hpq
-    have he : p.1.1⁻¹*q.1.1 = p.2.1*q.2.1⁻¹ := by
-      apply inv_mul_eq_iff_eq_mul.mpr
-      rw [←mul_assoc,hpq,mul_assoc,mul_inv_cancel,mul_one]
-    have hf : p.1.1⁻¹*q.1.1 ∈ fiberGroup f := (fiberGroup f).mul_mem ((fiberGroup f).inv_mem p.1.2) q.1.2
-    have hg : p.1.1⁻¹*q.1.1 ∈ fiberGroup g := he ▸ (fiberGroup g).mul_mem p.2.2 ((fiberGroup g).inv_mem q.2.2)
-    have h1 : p.1.1⁻¹*q.1.1 = 1 := by
-      apply Equiv.ext
-      intro x
-      exact hinj _ x (hf x) (hg x)
-    have hpf : p.1=q.1 := Subtype.ext (inv_mul_eq_one.mp h1)
-    have hpg : p.2=q.2 := by
-      apply Subtype.ext
-      have hh := hpq
-      rw [show p.1.1=q.1.1 from congrArg Subtype.val hpf] at hh
-      exact mul_left_cancel hh
-    exact Prod.ext hpf hpg
-  have hc := Fintype.card_le_of_injective _ hi
-  simpa only [Fintype.card_prod,Fintype.card_perm] using hc
-
-end FiberGroup
-
-noncomputable def rowGroup (μ : YoungDiagram) : Subgroup (Equiv.Perm (Cell μ)) :=
-  fiberGroup (rowIndex (μ := μ))
-
-noncomputable def colIndex {μ : YoungDiagram} (x : Cell μ) : Fin (μ.rowLen 0) :=
-  ⟨col x, (YoungDiagram.mem_iff_lt_rowLen.mp x.2).trans_le (μ.rowLen_anti 0 (row x) (Nat.zero_le _))⟩
-
-lemma colGroup_eq (μ : YoungDiagram) : colGroup μ = fiberGroup (colIndex (μ := μ)) := by
-  ext p
-  constructor
-  · intro hp x; exact Fin.ext (hp x)
-  · intro hp x; exact congrArg Fin.val (hp x)
-
-noncomputable instance tabloidMulAction (μ : YoungDiagram) : MulAction (Equiv.Perm (Cell μ)) (Tabloid μ) where
-  smul p f := tabloidAct p f
-  one_smul f := by change tabloidAct 1 f = f; rw [tabloidAct_one]; rfl
-  mul_smul p q f := by change tabloidAct (p*q) f = tabloidAct p (tabloidAct q f); rw [tabloidAct_mul]; rfl
-
-lemma tabloid_orbit_all (μ : YoungDiagram) :
-    MulAction.orbit (Equiv.Perm (Cell μ)) (baseTabloid μ) = Set.univ := by
-  ext f
-  simp only [Set.mem_univ,iff_true]
-  obtain ⟨p,hp⟩ := tabloidAct_transitive μ (baseTabloid μ) f
-  exact ⟨p,hp⟩
-
-lemma tabloid_stabilizer (μ : YoungDiagram) :
-    MulAction.stabilizer (Equiv.Perm (Cell μ)) (baseTabloid μ) = rowGroup μ := by
-  ext p
-  change tabloidAct p (baseTabloid μ) = baseTabloid μ ↔ ∀ x, rowIndex (p x) = rowIndex x
-  constructor
-  · intro h x
-    have hh := congrArg (fun f : Tabloid μ => f.1 (p x)) h
-    simpa only [tabloidAct_apply,baseTabloid,Equiv.Perm.inv_def,Equiv.symm_apply_apply] using hh.symm
-  · intro h
-    apply Subtype.ext
-    funext x
-    exact (h (p⁻¹ x)).symm.trans (by simp [baseTabloid])
-
-lemma card_tabloid_mul_row (μ : YoungDiagram) :
-    Fintype.card (Tabloid μ)*Fintype.card (rowGroup μ) = μ.card.factorial := by
-  have hh := MulAction.card_orbit_mul_card_stabilizer_eq_card_group
-    (Equiv.Perm (Cell μ)) (baseTabloid μ)
-  have hc : Fintype.card (MulAction.orbit (Equiv.Perm (Cell μ)) (baseTabloid μ)) =
-      Fintype.card (Tabloid μ) := by
-    rw [tabloid_orbit_all,Fintype.card_setUniv]
-  simpa only [hc,tabloid_stabilizer,Fintype.card_perm,card_cell] using hh
-
-theorem factorial_le_dimension_row_column (μ : YoungDiagram) :
-    μ.card.factorial ≤ Module.finrank ℂ (space μ)*Fintype.card (rowGroup μ)*Fintype.card (colGroup μ) := by
-  rw [←card_tabloid_mul_row]
-  nlinarith [dimension_times_column_ge_tabloids μ]
-
-
-lemma card_row_fiber (μ : YoungDiagram) (i : Fin (μ.colLen 0)) :
-    Fintype.card {x : Cell μ // rowIndex x = i} = μ.rowLen i := by
-  let e : {x : Cell μ // rowIndex x = i} ≃ {x : Cell μ // row x = i} :=
-    Equiv.subtypeEquivRight (fun x => Fin.ext_iff)
-  rw [Fintype.card_congr e,Fintype.card_congr (rowEquiv μ i),Fintype.card_fin]
-
-lemma card_col_fiber (μ : YoungDiagram) (j : Fin (μ.rowLen 0)) :
-    Fintype.card {x : Cell μ // colIndex x = j} = μ.colLen j := by
-  let e : {x : Cell μ // colIndex x = j} ≃ {x : Cell μ // col x = j} :=
-    Equiv.subtypeEquivRight (fun x => Fin.ext_iff)
-  rw [Fintype.card_congr e,Fintype.card_congr (columnEquiv μ j),Fintype.card_fin]
-
-lemma card_rowGroup (μ : YoungDiagram) :
-    Fintype.card (rowGroup μ) = ∏ i : Fin (μ.colLen 0), (μ.rowLen i).factorial := by
-  rw [←Nat.card_eq_fintype_card,rowGroup,nat_card_fiberGroup]
-  simp only [Nat.card_eq_fintype_card]
-  simp only [card_row_fiber]
-
-lemma card_colGroup (μ : YoungDiagram) :
-    Fintype.card (colGroup μ) = ∏ j : Fin (μ.rowLen 0), (μ.colLen j).factorial := by
-  rw [←Nat.card_eq_fintype_card,colGroup_eq,nat_card_fiberGroup]
-  simp only [Nat.card_eq_fintype_card,card_col_fiber]
-
-abbrev Tail (μ : YoungDiagram) := {x : Cell μ // row x ≠ 0}
-
-lemma tail_card (μ : YoungDiagram) : Fintype.card (Tail μ) = μ.card - μ.rowLen 0 := by
-  have hh := Fintype.card_subtype_compl (p := fun x : Cell μ => row x=0)
-  rw [Fintype.card_congr (rowEquiv μ 0),Fintype.card_fin,card_cell] at hh
-  exact hh
-
-lemma rowLen_zero_le_card (μ : YoungDiagram) : μ.rowLen 0 ≤ μ.card := by
-  have h := Fintype.card_subtype_le (fun x : Cell μ => row x=0)
-  rwa [Fintype.card_congr (rowEquiv μ 0),Fintype.card_fin,card_cell] at h
-
-noncomputable def tailColumnEquiv (μ : YoungDiagram) (j : Fin (μ.rowLen 0)) :
-    {x : Tail μ // colIndex x.1 = j} ≃ Fin (μ.colLen j-1) where
-  toFun x := ⟨row x.1.1-1, by
-    have hx := row_lt_colLen x.1.1
-    have hc : col x.1.1 = j := congrArg Fin.val x.2
-    rw [hc] at hx
-    have hz := x.1.2
-    omega⟩
-  invFun i := ⟨⟨⟨(i+1,j),YoungDiagram.mem_iff_lt_colLen.mpr (by omega)⟩,by simp [row]⟩,rfl⟩
-  left_inv x := by
-    apply Subtype.ext
-    apply Subtype.ext
-    apply Subtype.ext
-    apply Prod.ext
-    · change row x.1.1 - 1 + 1 = row x.1.1
-      have := x.1.2
-      omega
-    · exact (congrArg Fin.val x.2).symm
-  right_inv i := by apply Fin.ext; simp [row]
-
-lemma card_tail_col (μ : YoungDiagram) (j : Fin (μ.rowLen 0)) :
-    Fintype.card {x : Tail μ // colIndex x.1 = j} = μ.colLen j-1 := by
-  rw [Fintype.card_congr (tailColumnEquiv μ j),Fintype.card_fin]
-
-lemma card_tail_row (μ : YoungDiagram) (i : Fin (μ.colLen 0)) :
-    Fintype.card {x : Tail μ // rowIndex x.1 = i} = if i.val=0 then 0 else μ.rowLen i := by
-  by_cases hi : i.val=0
-  · rw [ite_eq_left hi]
-    apply Fintype.card_eq_zero_iff.mpr
-    refine ⟨fun x => x.1.2 ?_⟩
-    exact (congrArg Fin.val x.2).trans hi
-  · rw [ite_eq_right hi]
-    let e : {x : Tail μ // rowIndex x.1 = i} ≃ {x : Cell μ // rowIndex x=i} :=
-      { toFun := fun x => ⟨x.1.1,x.2⟩
-        invFun := fun x => ⟨⟨x.1,fun hz => hi ((congrArg Fin.val x.2).symm.trans hz)⟩,x.2⟩
-        left_inv := fun x => rfl
-        right_inv := fun x => rfl }
-    rw [Fintype.card_congr e,card_row_fiber]
-
-lemma tail_row_group (μ : YoungDiagram) :
-    (μ.rowLen 0).factorial * Fintype.card (fiberGroup (fun x : Tail μ => rowIndex x.1)) =
-      Fintype.card (rowGroup μ) := by
-  rw [←Nat.card_eq_fintype_card (α := fiberGroup (fun x : Tail μ => rowIndex x.1)),nat_card_fiberGroup,card_rowGroup]
-  simp only [Nat.card_eq_fintype_card]
-  simp only [card_tail_row]
-  by_cases hn : μ.colLen 0=0
-  · have hr : μ.rowLen 0=0 := by
-      by_contra h
-      have hmem : (0,0) ∈ μ := YoungDiagram.mem_iff_lt_rowLen.mpr (Nat.pos_of_ne_zero h)
-      have hh := YoungDiagram.mem_iff_lt_colLen.mp hmem
-      omega
-    let : IsEmpty (Fin (μ.colLen 0)) := ⟨fun i => by have := i.isLt; omega⟩
-    simp [hr]
-  · let z : Fin (μ.colLen 0) := ⟨0,Nat.pos_of_ne_zero hn⟩
-    have hz : ∀ i : Fin (μ.colLen 0), i.val=0 ↔ i=z := fun i => by simp [z,Fin.ext_iff]
-    simp only [hz]
-    have hprod : (∏ x : Fin (μ.colLen 0), (if x=z then 0 else μ.rowLen x).factorial) =
-        ∏ x ∈ Finset.univ.erase z, (μ.rowLen x).factorial := by
-      rw [←Finset.prod_erase_mul Finset.univ (fun x : Fin (μ.colLen 0) => (if x=z then 0 else μ.rowLen x).factorial) (Finset.mem_univ z)]
-      simp only [ite_true,Nat.factorial_zero,mul_one]
-      apply Finset.prod_congr rfl
-      intro x hx
-      rw [ite_eq_right (Finset.mem_erase.mp hx).1]
-    rw [hprod]
-    exact Finset.mul_prod_erase Finset.univ (fun i : Fin (μ.colLen 0) => (μ.rowLen i).factorial) (Finset.mem_univ z)
-
-lemma sum_tail_columns (μ : YoungDiagram) :
-    ∑ j : Fin (μ.rowLen 0), (μ.colLen j-1) = μ.card-μ.rowLen 0 := by
-  simp_rw [←card_tail_col]
-  rw [←Fintype.card_sigma,Fintype.card_congr (Equiv.sigmaFiberEquiv (fun x : Tail μ => colIndex x.1)),tail_card]
-
-lemma column_pos (μ : YoungDiagram) (j : Fin (μ.rowLen 0)) : 0 < μ.colLen j :=
-  YoungDiagram.mem_iff_lt_colLen.mp (YoungDiagram.mem_iff_lt_rowLen.mpr j.isLt)
-
-lemma nat_le_two_pow_pred (n : ℕ) (hn : 0<n) : n ≤ 2^(n-1) := by
-  obtain ⟨n,rfl⟩ := Nat.exists_eq_succ_of_ne_zero hn.ne'
-  simp only [Nat.add_one_sub_one]
-  induction n with
-  | zero => norm_num
-  | succ n ih => rw [pow_succ]; omega
-
-lemma column_group_le_tail (μ : YoungDiagram) :
-    Fintype.card (colGroup μ) ≤ 2^(μ.card-μ.rowLen 0) *
-      Fintype.card (fiberGroup (fun x : Tail μ => colIndex x.1)) := by
-  rw [card_colGroup,←Nat.card_eq_fintype_card (α := fiberGroup (fun x : Tail μ => colIndex x.1)),nat_card_fiberGroup]
-  simp only [Nat.card_eq_fintype_card]
-  simp only [card_tail_col]
-  calc
-    _ = ∏ j : Fin (μ.rowLen 0), μ.colLen j * (μ.colLen j-1).factorial := by
-      apply Finset.prod_congr rfl
-      intro j hj
-      exact (Nat.mul_factorial_pred (column_pos μ j).ne').symm
-    _ ≤ ∏ j : Fin (μ.rowLen 0), 2^(μ.colLen j-1) * (μ.colLen j-1).factorial := by
-      apply Finset.prod_le_prod
-      intro j hj
-      exact Nat.mul_le_mul_right _ (nat_le_two_pow_pred _ (column_pos μ j))
-    _ = _ := by rw [Finset.prod_mul_distrib,Finset.prod_pow_eq_pow_sum,sum_tail_columns]
-
-theorem choose_tail_le_dimension (μ : YoungDiagram) :
-    μ.card.choose (μ.card-μ.rowLen 0) ≤ Module.finrank ℂ (space μ) * 2^(μ.card-μ.rowLen 0) := by
-  let j := μ.card-μ.rowLen 0
-  have htail := card_fiberGroups_mul_le
-    (fun x : Tail μ => rowIndex x.1) (fun x : Tail μ => colIndex x.1)
-    (fun x y hr hc => Subtype.ext (Subtype.ext (Prod.ext (congrArg Fin.val hr) (congrArg Fin.val hc))))
-  rw [tail_card] at htail
-  have hc := column_group_le_tail μ
-  have hr := tail_row_group μ
-  have hd := factorial_le_dimension_row_column μ
-  simp only [←Nat.card_eq_fintype_card] at htail hc hr hd
-  have hh : μ.card.factorial ≤ Module.finrank ℂ (space μ) * 2^j *
-      ((μ.rowLen 0).factorial * j.factorial) := by
-    calc
-      _ ≤ _ := hd
-      _ ≤ Module.finrank ℂ (space μ) * Nat.card (rowGroup μ) *
-          (2^j * Nat.card (fiberGroup (fun x : Tail μ => colIndex x.1))) := Nat.mul_le_mul_left _ hc
-      _ = Module.finrank ℂ (space μ) * 2^j * (μ.rowLen 0).factorial *
-          (Nat.card (fiberGroup (fun x : Tail μ => rowIndex x.1)) *
-           Nat.card (fiberGroup (fun x : Tail μ => colIndex x.1))) := by rw [←hr]; ring
-      _ ≤ _ := by
-        calc
-          _ ≤ Module.finrank ℂ (space μ) * 2^j * (μ.rowLen 0).factorial * j.factorial :=
-            Nat.mul_le_mul_left _ htail
-          _ = _ := by ring
-  have hj : j ≤ μ.card := Nat.sub_le _ _
-  have hs : μ.card-j = μ.rowLen 0 := by dsimp [j]; have := rowLen_zero_le_card μ; omega
-  have hf := Nat.choose_mul_factorial_mul_factorial hj
-  rw [hs] at hf
-  rw [←hf] at hh
-  have hpos : 0 < (μ.rowLen 0).factorial * j.factorial := Nat.mul_pos (Nat.factorial_pos _) (Nat.factorial_pos _)
-  nlinarith
-
-
-lemma exists_perm_mem_iff {α : Type*} [Fintype α] (s t : Finset α) (hc : s.card=t.card) :
-    ∃ p : Equiv.Perm α, ∀ x, p x ∈ t ↔ x ∈ s := by
-  let e : s ≃ t := Finset.equivOfCardEq hc
-  let ec : {x : α // x ∉ s} ≃ {x : α // x ∉ t} := Fintype.equivOfCardEq (by
-    simp only [Fintype.card_subtype_compl,Fintype.card_coe,hc])
-  let p := (Equiv.sumCompl (· ∈ s)).symm.trans ((Equiv.sumCongr e ec).trans (Equiv.sumCompl (· ∈ t)))
-  refine ⟨p,fun x => ?_⟩
-  by_cases hx : x ∈ s
-  · have hp : p x ∈ t := by
-      change ((Equiv.sumCompl (· ∈ t)) ((Equiv.sumCongr e ec) ((Equiv.sumCompl (· ∈ s)).symm x))) ∈ t
-      rw [Equiv.sumCompl_symm_apply_of_pos hx]
-      exact (e ⟨x,hx⟩).2
-    exact iff_of_true hp hx
-  · have hp : p x ∉ t := by
-      change ((Equiv.sumCompl (· ∈ t)) ((Equiv.sumCongr e ec) ((Equiv.sumCompl (· ∈ s)).symm x))) ∉ t
-      rw [Equiv.sumCompl_symm_apply_of_neg hx]
-      exact (ec ⟨x,hx⟩).2
-    exact iff_of_false hp hx
-
-noncomputable def tailSet {μ : YoungDiagram} (f : Tabloid μ) : Finset (Cell μ) :=
-  Finset.univ.filter (fun x => (f.1 x).val ≠ 0)
-
-@[simp] lemma mem_tailSet {μ : YoungDiagram} (f : Tabloid μ) (x : Cell μ) :
-    x ∈ tailSet f ↔ (f.1 x).val ≠ 0 := by simp [tailSet]
-
-lemma tailSet_act {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) (f : Tabloid μ) :
-    tailSet (tabloidAct p f) = (tailSet f).image p := by
-  ext x
-  simp only [mem_tailSet,tabloidAct_apply,Finset.mem_image]
-  constructor
-  · intro h; exact ⟨p⁻¹ x,h,by simp⟩
-  · rintro ⟨y,hy,rfl⟩; simpa using hy
-
-lemma card_tailSet {μ : YoungDiagram} (f : Tabloid μ) :
-    (tailSet f).card = μ.card-μ.rowLen 0 := by
-  obtain ⟨p,rfl⟩ := tabloidAct_transitive μ (baseTabloid μ) f
-  rw [tailSet_act,Finset.card_image_of_injective _ p.injective]
-  change (Finset.univ.filter (fun x : Cell μ => row x ≠ 0)).card = _
-  rw [←Fintype.card_subtype]
-  exact tail_card μ
-
-noncomputable def footprint (μ : YoungDiagram) (A : Finset (Cell μ)) : Finset (Tabloid μ) :=
-  Finset.univ.filter (fun f => tailSet f ⊆ A)
-
-lemma tail_fiber_card_eq (μ : YoungDiagram) (s t : Finset (Cell μ)) (hst : s.card=t.card) :
-    (Finset.univ.filter (fun f : Tabloid μ => tailSet f=s)).card =
-      (Finset.univ.filter (fun f : Tabloid μ => tailSet f=t)).card := by
-  obtain ⟨p,hp⟩ := exists_perm_mem_iff s t hst
-  have he : s.image p=t := by
-    ext x
-    constructor
-    · intro hx
-      obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
-      exact (hp y).mpr hy
-    · intro hx
-      exact Finset.mem_image.mpr ⟨p⁻¹ x,(hp (p⁻¹ x)).mp (by simpa using hx),by simp⟩
-  apply Finset.card_equiv (tabloidAct p)
-  intro f
-  simp only [Finset.mem_filter,Finset.mem_univ,true_and,tailSet_act,←he]
-  exact (Finset.image_inj p.injective).symm
-
-lemma footprint_card_product (μ : YoungDiagram) (A : Finset (Cell μ)) :
-    (footprint μ A).card * μ.card.choose (μ.card-μ.rowLen 0) =
-      Fintype.card (Tabloid μ) * A.card.choose (μ.card-μ.rowLen 0) := by
-  let j := μ.card-μ.rowLen 0
-  let s := tailSet (baseTabloid μ)
-  let c := (Finset.univ.filter (fun f : Tabloid μ => tailSet f=s)).card
-  have hconstant (t : Finset (Cell μ)) (ht : t.card=j) :
-      (Finset.univ.filter (fun f : Tabloid μ => tailSet f=t)).card = c :=
-    tail_fiber_card_eq μ t s (ht.trans (card_tailSet _).symm)
-  have hcount (A : Finset (Cell μ)) : (footprint μ A).card = A.card.choose j*c := by
-    have h := Finset.sum_card_fiberwise_eq_card_filter
-      (Finset.univ : Finset (Tabloid μ)) (A.powersetCard j) tailSet
-    have he : (Finset.univ.filter (fun f : Tabloid μ => tailSet f ∈ A.powersetCard j)) = footprint μ A := by
-      ext f
-      simp only [Finset.mem_filter,Finset.mem_univ,true_and,Finset.mem_powersetCard,
-        card_tailSet,footprint]
-      simp [j]
-    rw [he] at h
-    rw [←h]
-    calc
-      _ = ∑ _t ∈ A.powersetCard j, c := by
-        apply Finset.sum_congr rfl
-        intro t ht
-        exact hconstant t (Finset.mem_powersetCard.mp ht).2
-      _ = _ := by simp [Finset.card_powersetCard]
-  have hfull : (footprint μ Finset.univ).card = Fintype.card (Tabloid μ) := by
-    simp only [footprint,Finset.subset_univ,Finset.filter_true,Finset.card_univ]
-  have hN := hcount Finset.univ
-  rw [hfull,Finset.card_univ,card_cell] at hN
-  rw [hcount,hN]
-  ring
-
-lemma footprint_fixed {μ : YoungDiagram} (A : Finset (Cell μ))
-    (f : Tabloid μ) (hf : f ∈ footprint μ A) (p : Equiv.Perm (Cell μ))
-    (hp : ∀ x ∈ A, p x=x) : tabloidAct p f=f := by
-  have htail : tailSet f ⊆ A := (Finset.mem_filter.mp hf).2
-  apply Subtype.ext
-  funext x
-  change f.1 (p⁻¹ x) = f.1 x
-  by_cases hx : x ∈ A
-  · rw [show p⁻¹ x=x from p.injective (by simpa using (hp x hx).symm)]
-  · have hy : p⁻¹ x ∉ A := by
-      intro h
-      have hh := hp (p⁻¹ x) h
-      have he : x=p⁻¹ x := by simpa using hh
-      exact hx (he ▸ h)
-    apply Fin.ext
-    have hx0 : (f.1 x).val=0 := by by_contra h; exact hx (htail (mem_tailSet f x |>.mpr h))
-    have hy0 : (f.1 (p⁻¹ x)).val=0 := by by_contra h; exact hy (htail (mem_tailSet f _ |>.mpr h))
-    exact hy0.trans hx0.symm
-
-noncomputable def fixedHilbert (μ : YoungDiagram) (A : Finset (Cell μ)) :
-    Submodule ℂ (PermutationHilbert.H (Tabloid μ)) where
-  carrier := {v | v ∈ hilbertSpace μ ∧ ∀ p : Equiv.Perm (Cell μ),
-    (∀ x ∈ A, p x=x) → PermutationHilbert.act (tabloidAct p) v=v}
-  zero_mem' := ⟨Submodule.zero_mem _,fun p hp => map_zero _⟩
-  add_mem' := fun hv hw => ⟨Submodule.add_mem _ hv.1 hw.1,fun p hp => by
-    rw [map_add,hv.2 p hp,hw.2 p hp]⟩
-  smul_mem' := fun c v hv => ⟨Submodule.smul_mem _ c hv.1,fun p hp => by
-    rw [map_smul,hv.2 p hp]⟩
-
-lemma fixedHilbert_le (μ : YoungDiagram) (A : Finset (Cell μ)) : fixedHilbert μ A ≤ hilbertSpace μ :=
-  fun _ hv => hv.1
-
-theorem fixedHilbert_dimension (μ : YoungDiagram) (A : Finset (Cell μ)) :
-    Module.finrank ℂ (space μ) * A.card.choose (μ.card-μ.rowLen 0) ≤
-      Module.finrank ℂ (fixedHilbert μ A) * μ.card.choose (μ.card-μ.rowLen 0) := by
-  let : DecidableEq (Tabloid μ) := Classical.decEq _
-  have ht (f g : Tabloid μ) : ∃ p : Equiv.Perm (Tabloid μ), p f=g ∧
-      (hilbertSpace μ).map (PermutationHilbert.act p).toLinearEquiv.toLinearMap = hilbertSpace μ := by
-    obtain ⟨p,hp⟩ := tabloidAct_transitive μ f g
-    exact ⟨tabloidAct p,hp,hilbertSpace_invariant μ p⟩
-  have hproj (f : Tabloid μ) (hf : f ∈ footprint μ A) :
-      (hilbertSpace μ).starProjection (EuclideanSpace.single f (1:ℂ)) ∈ fixedHilbert μ A := by
-    refine ⟨((hilbertSpace μ).orthogonalProjectionOnto _).2,fun p hp => ?_⟩
-    rw [PermutationHilbert.projection_comm _ _ (hilbertSpace_invariant μ p),
-      PermutationHilbert.act_single,footprint_fixed A f hf p hp]
-  have h := PermutationHilbert.footprint_dimension (hilbertSpace μ) (fixedHilbert μ A)
-    (fixedHilbert_le μ A) ht (footprint μ A) hproj
-  rw [finrank_hilbertSpace] at h
-  have hn : (footprint μ A).card * Module.finrank ℂ (space μ) ≤
-      Fintype.card (Tabloid μ)*Module.finrank ℂ (fixedHilbert μ A) := by exact_mod_cast h
-  have hc := footprint_card_product μ A
-  have hpos : 0 < Fintype.card (Tabloid μ) := Fintype.card_pos_iff.mpr ⟨baseTabloid μ⟩
-  have hh := Nat.mul_le_mul_right (μ.card.choose (μ.card-μ.rowLen 0)) hn
-  nlinarith
-
-end Thorp.Specht
-
-namespace Thorp.UnitaryFinite
-open scoped BigOperators ComplexConjugate Classical
-
-variable {G : Type*} [Group G] [Fintype G]
-variable {V W : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-  [NormedAddCommGroup W] [InnerProductSpace ℂ W] [FiniteDimensional ℂ W]
-
-def IsUnitary (ρ : Representation ℂ G V) : Prop :=
-  ∀ g v w, inner ℂ (ρ g v) (ρ g w) = inner ℂ v w
-
-omit [Fintype G] [FiniteDimensional ℂ V] in
-lemma unitary_inner_inv (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (g : G) (v w : V) :
-    inner ℂ v (ρ g⁻¹ w) = inner ℂ (ρ g v) w := by
-  have h := hρ g v (ρ g⁻¹ w)
-  have he : ρ g (ρ g⁻¹ w) = w := by
-    change (ρ g * ρ g⁻¹) w = w
-    rw [←map_mul,mul_inv_cancel,map_one]
-    rfl
-  rw [he] at h
-  exact h.symm
-
-end Thorp.UnitaryFinite
-
-end ThorpNine.Harmonic
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Harmonic.ModifiedReverseCoins
+
+namespace OAI
+
+namespace ThorpNine.Harmonic
+
+namespace Thorp.Specht
+open scoped BigOperators Classical
+
+lemma partitionDiagram_rowLens {N : ℕ} (p : Nat.Partition N) :
+    (partitionDiagram p).rowLens = p.parts.sort (· ≥ ·) := by
+  apply YoungDiagram.rowLens_ofRowLens_eq_self
+  intro x hx
+  exact p.parts_pos ((Multiset.mem_sort _).mp hx)
+
+lemma partitionDiagram_card {N : ℕ} (p : Nat.Partition N) : (partitionDiagram p).card = N := by
+  rw [←sum_rowLens,partitionDiagram_rowLens]
+  change (↑(p.parts.sort (· ≥ ·)) : Multiset ℕ).sum = N
+  rw [Multiset.sort_eq,p.parts_sum]
+
+lemma partitionDiagram_diagramPartition (μ : YoungDiagram) :
+    partitionDiagram (diagramPartition μ) = μ := by
+  apply YoungDiagram.equivListRowLens.injective
+  apply Subtype.ext
+  change (partitionDiagram (diagramPartition μ)).rowLens = μ.rowLens
+  rw [partitionDiagram_rowLens]
+  change (↑μ.rowLens : Multiset ℕ).sort (· ≥ ·) = μ.rowLens
+  rw [Multiset.coe_sort,List.mergeSort_eq_self _ μ.rowLens_sorted.pairwise]
+
+abbrev Shapes (N : ℕ) := {μ : YoungDiagram // μ.card = N}
+
+noncomputable def shapePartitionEquiv (N : ℕ) : Shapes N ≃ Nat.Partition N where
+  toFun μ := { parts := μ.1.rowLens
+               parts_pos := fun {i} hi => μ.1.pos_of_mem_rowLens i (by simpa using hi)
+               parts_sum := by simpa using (sum_rowLens μ.1).trans μ.2 }
+  invFun p := ⟨partitionDiagram p,partitionDiagram_card p⟩
+  left_inv μ := by
+    apply Subtype.ext
+    obtain ⟨μ,rfl⟩ := μ
+    exact partitionDiagram_diagramPartition μ
+  right_inv p := by
+    apply Nat.Partition.ext
+    change ((partitionDiagram p).rowLens : Multiset ℕ) = p.parts
+    rw [partitionDiagram_rowLens,Multiset.sort_eq]
+
+noncomputable instance (N : ℕ) : Fintype (Shapes N) := Fintype.ofEquiv _ (shapePartitionEquiv N).symm
+
+noncomputable def conjugacyPartition {α : Type*} [Fintype α] [DecidableEq α] :
+    ConjClasses (Equiv.Perm α) → Nat.Partition (Fintype.card α) :=
+  Quotient.lift Equiv.Perm.partition (fun _ _ hpq => (Equiv.Perm.partition_eq_of_isConj).mp hpq)
+
+lemma conjugacyPartition_injective {α : Type*} [Fintype α] [DecidableEq α] :
+    Function.Injective (conjugacyPartition (α := α)) := by
+  intro p q
+  induction p using Quotient.inductionOn with | _ p =>
+    induction q using Quotient.inductionOn with | _ q =>
+      intro h
+      exact Quotient.sound ((Equiv.Perm.partition_eq_of_isConj).mpr h)
+
+lemma card_conjugacy_le_shapes (N : ℕ) :
+    Nat.card (ConjClasses (Equiv.Perm (Fin N))) ≤ Fintype.card (Shapes N) := by
+  have h := Nat.card_le_card_of_injective _ (conjugacyPartition_injective (α := Fin N))
+  simpa only [Fintype.card_fin, Nat.card_eq_fintype_card, Fintype.card_congr (shapePartitionEquiv N)] using h
+
+noncomputable def labels {N : ℕ} (μ : Shapes N) : Cell μ.1 ≃ Fin N :=
+  (Fintype.equivFin (Cell μ.1)).trans (finCongr ((card_cell μ.1).trans μ.2))
+
+noncomputable def labelledRepresentation {N : ℕ} (μ : Shapes N) :
+    Representation ℂ (Equiv.Perm (Fin N)) (space μ.1) :=
+  (representation μ.1).comp (labels μ).symm.permCongrHom.toMonoidHom
+
+lemma labelledRepresentation_irreducible {N : ℕ} (μ : Shapes N) :
+    Representation.IsIrreducible (labelledRepresentation μ) := by
+  let e := (labels μ).symm.permCongrHom
+  let ρ := representation μ.1
+  have hn : (⊥ : Subrepresentation (labelledRepresentation μ)) ≠ ⊤ := by
+    intro h
+    have he : (⟨polytabloid μ.1,polytabloid_mem_space μ.1⟩ : space μ.1) ∈
+        (⊥ : Subrepresentation (labelledRepresentation μ)) := by rw [h]; trivial
+    exact polytabloid_ne_zero μ.1 (congrArg Subtype.val he)
+  refine { exists_pair_ne := ⟨⊥,⊤,hn⟩,eq_bot_or_eq_top := ?_ }
+  intro U
+  let U' : Subrepresentation ρ :=
+    { toSubmodule := U.toSubmodule
+      apply_mem_toSubmodule := fun p v hv => by
+        have hh := U.apply_mem_toSubmodule (e.symm p) hv
+        change ρ (e (e.symm p)) v ∈ U.toSubmodule at hh
+        simpa using hh }
+  have := representation_irreducible μ.1
+  obtain h | h := IsSimpleOrder.eq_bot_or_eq_top U'
+  · left
+    apply Subrepresentation.toSubmodule_injective
+    exact congrArg (fun R : Subrepresentation ρ => R.toSubmodule) h
+  · right
+    apply Subrepresentation.toSubmodule_injective
+    exact congrArg (fun R : Subrepresentation ρ => R.toSubmodule) h
+
+lemma labelledRepresentation_equiv_shapes {N : ℕ} (μ ν : Shapes N)
+    (T : Representation.Equiv (labelledRepresentation μ) (labelledRepresentation ν)) : μ = ν := by
+  apply Subtype.ext
+  apply shapes_eq_of_equivariant_equiv μ.1 ν.1 ((labels μ).trans (labels ν).symm) T.toLinearEquiv
+  intro p v
+  have hh := Representation.IntertwiningMap.isIntertwining _ _ T.toIntertwiningMap ((labels μ).permCongr p) v
+  change T.toLinearEquiv
+    (representation μ.1 ((labels μ).symm.permCongr ((labels μ).permCongr p)) v) =
+      representation ν.1 ((labels ν).symm.permCongr ((labels μ).permCongr p)) (T.toLinearEquiv v) at hh
+  have hp : (labels μ).symm.permCongr ((labels μ).permCongr p) = p := by
+    apply Equiv.ext
+    intro x
+    simp [Equiv.permCongr_apply]
+  rw [hp] at hh
+  convert hh using 2
+  congr 1
+
+end Thorp.Specht
+
+namespace Thorp.FiniteCharacters
+open scoped BigOperators Classical
+
+section Group
+variable {G : Type*} [Group G] [Fintype G]
+
+noncomputable def classRep (c : ConjClasses G) : G := (ConjClasses.exists_rep c).choose
+omit [Fintype G] in
+lemma classRep_spec (c : ConjClasses G) : ConjClasses.mk (classRep c) = c :=
+  (ConjClasses.exists_rep c).choose_spec
+
+variable {V : Type*} [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
+
+noncomputable def characterClass (ρ : Representation ℂ G V) : ConjClasses G → ℂ :=
+  fun c => ρ.character (classRep c)
+
+omit [Fintype G] [FiniteDimensional ℂ V] in
+lemma characterClass_mk (ρ : Representation ℂ G V) (g : G) :
+    characterClass ρ (ConjClasses.mk g) = ρ.character g := by
+  have h := ConjClasses.mk_eq_mk_iff_isConj.mp (classRep_spec (ConjClasses.mk g))
+  obtain ⟨p,hp⟩ := isConj_iff.mp h
+  change ρ.character (classRep (ConjClasses.mk g)) = ρ.character g
+  exact (Representation.char_conj ρ _ p).symm.trans (congrArg ρ.character hp)
+
+noncomputable def characterDual (ρ : Representation ℂ G V) :
+    (ConjClasses G → ℂ) →ₗ[ℂ] ℂ where
+  toFun f := (Nat.card G : ℂ)⁻¹ * ∑ g, f (ConjClasses.mk g) * ρ.character g⁻¹
+  map_add' f h := by simp [Finset.sum_add_distrib,add_mul,mul_add]
+  map_smul' a f := by
+    simp only [Pi.smul_apply,smul_eq_mul,RingHom.id_apply]
+    simp_rw [mul_assoc]
+    rw [←Finset.mul_sum]
+    ring
+
+lemma characterDual_characterClass (ρ : Representation ℂ G V)
+    {W : Type*} [AddCommGroup W] [Module ℂ W] [FiniteDimensional ℂ W]
+    (σ : Representation ℂ G W) [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ] :
+    characterDual ρ (characterClass σ) = if Nonempty (Representation.Equiv ρ σ) then 1 else 0 := by
+  let : Invertible (Nat.card G : ℂ) := invertibleOfNonzero (by
+    exact_mod_cast (Nat.card_pos (α := G)).ne')
+  change (Nat.card G : ℂ)⁻¹ * ∑ g, characterClass σ (ConjClasses.mk g) * ρ.character g⁻¹ = _
+  simp only [characterClass_mk]
+  exact Representation.char_orthonormal σ ρ
+
+end Group
+
+theorem complete_of_card_conjugacy_le
+    {G ι : Type*} [Group G] [Fintype G] [Fintype ι]
+    (V : ι → Type*) [∀ i, AddCommGroup (V i)] [∀ i, Module ℂ (V i)]
+    [∀ i, FiniteDimensional ℂ (V i)]
+    (ρ : ∀ i, Representation ℂ G (V i)) [∀ i, Representation.IsIrreducible (ρ i)]
+    (hneq : ∀ i j, i ≠ j → ¬Nonempty (Representation.Equiv (ρ i) (ρ j)))
+    (hcard : Nat.card (ConjClasses G) ≤ Fintype.card ι)
+    {W : Type*} [AddCommGroup W] [Module ℂ W] [FiniteDimensional ℂ W]
+    (σ : Representation ℂ G W) [Representation.IsIrreducible σ] :
+    ∃ i, Nonempty (Representation.Equiv (ρ i) σ) := by
+  let : Fintype (ConjClasses G) := Fintype.ofFinite _
+  have horth (i j : ι) : characterDual (ρ i) (characterClass (ρ j)) = if i=j then 1 else 0 := by
+    rw [characterDual_characterClass]
+    by_cases hij : i=j
+    · subst j
+      simp only [ite_true]
+      exact ite_eq_left ⟨Representation.Equiv.refl _⟩
+    · rw [ite_eq_right hij,ite_eq_right (hneq i j hij)]
+  have hLI : LinearIndependent ℂ (fun i => characterClass (ρ i)) := by
+    apply LinearIndependent.of_pairwise_dual_eq_zero_one _ (fun i => characterDual (ρ i))
+    · intro i j hij
+      exact (horth i j).trans (ite_eq_right hij)
+    · intro i
+      simpa using horth i i
+  have heq : Fintype.card ι = Module.finrank ℂ (ConjClasses G → ℂ) := by
+    apply Nat.le_antisymm hLI.fintype_card_le_finrank
+    simpa only [Module.finrank_pi,Module.finrank_self,Finset.sum_const,Finset.card_univ,
+      smul_eq_mul,mul_one,Nat.card_eq_fintype_card] using hcard
+  have hspan := hLI.span_eq_top_of_card_eq_finrank' heq
+  by_contra h
+  push Not at h
+  have hker : Submodule.span ℂ (Set.range (fun i => characterClass (ρ i))) ≤
+      (characterDual σ).ker := by
+    apply Submodule.span_le.mpr
+    rintro f ⟨i,rfl⟩
+    change characterDual σ (characterClass (ρ i)) = 0
+    rw [characterDual_characterClass,ite_eq_right]
+    rintro ⟨T⟩
+    exact (h i).false T.symm
+  rw [hspan] at hker
+  have hz : characterDual σ (characterClass σ) = 0 := hker (show characterClass σ ∈ ⊤ from trivial)
+  rw [characterDual_characterClass,ite_eq_left ⟨Representation.Equiv.refl σ⟩] at hz
+  exact one_ne_zero hz
+
+end Thorp.FiniteCharacters
+
+namespace Thorp.Specht
+open scoped Classical
+
+theorem labelled_complete {N : ℕ}
+    {V : Type*} [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
+    (ρ : Representation ℂ (Equiv.Perm (Fin N)) V) [Representation.IsIrreducible ρ] :
+    ∃ μ : Shapes N, Nonempty (Representation.Equiv (labelledRepresentation μ) ρ) := by
+  let (μ : Shapes N) : Representation.IsIrreducible (labelledRepresentation μ) :=
+    labelledRepresentation_irreducible μ
+  apply FiniteCharacters.complete_of_card_conjugacy_le
+    (fun μ : Shapes N => space μ.1) labelledRepresentation _ (card_conjugacy_le_shapes N) ρ
+  intro μ ν h ⟨T⟩
+  exact h (labelledRepresentation_equiv_shapes μ ν T)
+
+end Thorp.Specht
+
+namespace Thorp.PermutationHilbert
+
+section
+open scoped BigOperators ComplexConjugate Classical
+open Complex
+
+variable {X : Type*} [Fintype X]
+
+noncomputable def act (p : Equiv.Perm X) : H X ≃ₗᵢ[ℂ] H X :=
+  LinearIsometryEquiv.piLpCongrLeft 2 ℂ ℂ p
+
+@[simp] lemma act_apply (p : Equiv.Perm X) (v : H X) (x : X) :
+    act p v x = v (p⁻¹ x) := rfl
+
+lemma act_single [DecidableEq X] (p : Equiv.Perm X) (x : X) :
+    act p (EuclideanSpace.single x (1:ℂ)) = EuclideanSpace.single (p x) 1 := by
+  exact EuclideanSpace.piLpCongrLeft_single p x 1
+
+lemma trace_projection (W : Submodule ℂ (H X)) :
+    ∑ x, (W.starProjection (EuclideanSpace.single x (1:ℂ))) x = Module.finrank ℂ W := by
+  classical
+  have hp : LinearMap.IsProj W W.starProjection.toLinearMap := by
+    constructor
+    · intro x; exact (W.orthogonalProjectionOnto x).2
+    · intro x hx; exact W.starProjection_eq_self_iff.mpr hx
+  have ht := hp.trace
+  rw [LinearMap.trace_eq_matrix_trace ℂ (EuclideanSpace.basisFun X ℂ).toBasis] at ht
+  simpa only [Matrix.trace,Matrix.diag_apply,LinearMap.toMatrix_apply,EuclideanSpace.basisFun_repr,
+    EuclideanSpace.basisFun_apply,OrthonormalBasis.coe_toBasis,
+    OrthonormalBasis.coe_toBasis_repr_apply,ContinuousLinearMap.coe_coe] using ht
+
+lemma projection_diagonal (W : Submodule ℂ (H X)) (x : X) :
+    (W.starProjection (EuclideanSpace.single x (1:ℂ))) x =
+      (‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 : ℝ) := by
+  classical
+  have hh := W.inner_starProjection_left_eq_right
+    (EuclideanSpace.single x (1:ℂ)) (W.starProjection (EuclideanSpace.single x (1:ℂ)))
+  have hf : W.starProjection (W.starProjection (EuclideanSpace.single x (1:ℂ))) =
+      W.starProjection (EuclideanSpace.single x (1:ℂ)) :=
+    W.starProjection_eq_self_iff.mpr (W.orthogonalProjectionOnto _).2
+  rw [hf] at hh
+  rw [inner_self_eq_norm_sq_to_K,EuclideanSpace.inner_single_left] at hh
+  simpa using hh.symm
+
+lemma projection_comm (W : Submodule ℂ (H X)) (p : Equiv.Perm X)
+    (hW : W.map (act p).toLinearEquiv.toLinearMap = W) (v : H X) :
+    act p (W.starProjection v) = W.starProjection (act p v) := by
+  have hh := (act p).toLinearIsometry.map_starProjection W v
+  change act p (W.starProjection v) =
+    (W.map (act p).toLinearEquiv.toLinearMap).starProjection (act p v) at hh
+  simpa only [hW] using hh
+
+lemma uniform_diagonal (W : Submodule ℂ (H X))
+    (htrans : ∀ x y : X, ∃ p : Equiv.Perm X, p x = y ∧
+      W.map (act p).toLinearEquiv.toLinearMap = W) (x : X) :
+    (Fintype.card X:ℝ) * ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 =
+      Module.finrank ℂ W := by
+  classical
+  have hh (y : X) : (W.starProjection (EuclideanSpace.single y (1:ℂ))) y =
+      (‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 : ℝ) := by
+    obtain ⟨p,hp,hW⟩ := htrans x y
+    subst y
+    rw [projection_diagonal,←act_single p x,←projection_comm W p hW,(act p).norm_map]
+  have ht := trace_projection W
+  simp_rw [hh] at ht
+  have hhreal : (∑ _y : X, ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2) =
+      (Module.finrank ℂ W:ℝ) := by exact_mod_cast ht
+  simpa using hhreal
+
+theorem uncertainty (W : Submodule ℂ (H X))
+    (htrans : ∀ x y : X, ∃ p : Equiv.Perm X, p x = y ∧
+      W.map (act p).toLinearEquiv.toLinearMap = W)
+    (v : H X) (hv : v ∈ W) (x : X) :
+    (Fintype.card X:ℝ) * ‖v x‖^2 ≤ (Module.finrank ℂ W:ℝ) * ‖v‖^2 := by
+  classical
+  have hh : inner ℂ (W.starProjection (EuclideanSpace.single x (1:ℂ))) v = v x := by
+    rw [W.inner_starProjection_left_eq_right,W.starProjection_eq_self_iff.mpr hv,
+      EuclideanSpace.inner_single_left]
+    simp
+  have hc := norm_inner_le_norm (𝕜 := ℂ) (W.starProjection (EuclideanSpace.single x (1:ℂ))) v
+  rw [hh] at hc
+  have hs : ‖v x‖^2 ≤ ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 * ‖v‖^2 := by
+    nlinarith [norm_nonneg (v x),norm_nonneg v,
+      norm_nonneg (W.starProjection (EuclideanSpace.single x (1:ℂ)))]
+  calc
+    (Fintype.card X:ℝ) * ‖v x‖^2 ≤
+      (Fintype.card X:ℝ) * (‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 * ‖v‖^2) :=
+      mul_le_mul_of_nonneg_left hs (Nat.cast_nonneg _)
+    _ = _ := by rw [←mul_assoc,uniform_diagonal W htrans x]
+
+end
+open scoped BigOperators Classical
+
+variable {X : Type*} [Fintype X]
+
+lemma trace_projection_real (W : Submodule ℂ (H X)) :
+    ∑ x, ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 = (Module.finrank ℂ W:ℝ) := by
+  have hh := trace_projection W
+  simp_rw [projection_diagonal] at hh
+  exact_mod_cast hh
+
+theorem footprint_dimension (W K : Submodule ℂ (H X)) (hKW : K ≤ W)
+    (htrans : ∀ x y : X, ∃ p : Equiv.Perm X, p x = y ∧
+      W.map (act p).toLinearEquiv.toLinearMap = W)
+    (S : Finset X) (hS : ∀ x ∈ S, W.starProjection (EuclideanSpace.single x (1:ℂ)) ∈ K) :
+    (S.card:ℝ) * (Module.finrank ℂ W:ℝ) ≤
+      (Fintype.card X:ℝ) * (Module.finrank ℂ K:ℝ) := by
+  have he (x : X) (hx : x ∈ S) :
+      K.starProjection (EuclideanSpace.single x (1:ℂ)) =
+        W.starProjection (EuclideanSpace.single x (1:ℂ)) := by
+    apply K.eq_starProjection_of_mem_of_inner_eq_zero (hS x hx)
+    intro v hv
+    exact W.starProjection_inner_eq_zero _ v (hKW hv)
+  have hs : ∑ x ∈ S, ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 ≤
+      (Module.finrank ℂ K:ℝ) := by
+    rw [←trace_projection_real K]
+    calc
+      _ = ∑ x ∈ S, ‖K.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 := by
+        apply Finset.sum_congr rfl
+        intro x hx
+        rw [he x hx]
+      _ ≤ _ := Finset.sum_le_univ_sum_of_nonneg (fun x => sq_nonneg _)
+  have hmul := mul_le_mul_of_nonneg_left hs (Nat.cast_nonneg (Fintype.card X): (0:ℝ) ≤ _)
+  rw [Finset.mul_sum] at hmul
+  simp_rw [uniform_diagonal W htrans] at hmul
+  simpa using hmul
+
+end Thorp.PermutationHilbert
+namespace Thorp.Specht
+open scoped BigOperators Classical
+
+lemma finrank_hilbertSpace (μ : YoungDiagram) :
+    Module.finrank ℂ (hilbertSpace μ) = Module.finrank ℂ (space μ) :=
+  (hilbertEquiv μ).finrank_map_eq (space μ)
+
+lemma hilbert_action (μ : YoungDiagram) (p : Equiv.Perm (Cell μ)) (v : Tabloid μ → ℂ) :
+    hilbertEquiv μ (tabloidRep μ p v) =
+      PermutationHilbert.act (tabloidAct p) (hilbertEquiv μ v) := by
+  ext f
+  simp [hilbertEquiv,PermutationHilbert.act_apply]
+
+lemma hilbertSpace_invariant (μ : YoungDiagram) (p : Equiv.Perm (Cell μ)) :
+    (hilbertSpace μ).map (PermutationHilbert.act (tabloidAct p)).toLinearEquiv.toLinearMap =
+      hilbertSpace μ := by
+  apply le_antisymm
+  · rintro _ ⟨v,⟨w,hw,rfl⟩,rfl⟩
+    exact ⟨tabloidRep μ p w,action_mem_space μ p w hw,hilbert_action μ p w⟩
+  · rintro _ ⟨w,hw,rfl⟩
+    refine ⟨hilbertEquiv μ (tabloidRep μ p⁻¹ w),
+      ⟨tabloidRep μ p⁻¹ w,action_mem_space μ p⁻¹ w hw,rfl⟩, ?_⟩
+    change PermutationHilbert.act (tabloidAct p) (hilbertEquiv μ (tabloidRep μ p⁻¹ w)) = hilbertEquiv μ w
+    rw [←hilbert_action]
+    change hilbertEquiv μ ((tabloidRep μ p * tabloidRep μ p⁻¹) w) = _
+    rw [←map_mul,mul_inv_cancel,map_one]
+    rfl
+
+lemma tabloidAct_transitive (μ : YoungDiagram) (f g : Tabloid μ) :
+    ∃ p : Equiv.Perm (Cell μ), tabloidAct p f = g := by
+  obtain ⟨q,hq⟩ := f.2
+  obtain ⟨r,hr⟩ := g.2
+  refine ⟨r⁻¹*q, ?_⟩
+  apply Subtype.ext
+  funext x
+  simp only [tabloidAct_apply,mul_inv_rev,Equiv.Perm.mul_apply,hq,hr,
+    Function.comp_apply,Equiv.Perm.inv_def,Equiv.apply_symm_apply,Equiv.symm_symm]
+
+lemma polytabloid_pairing (μ : YoungDiagram) :
+    pairing (polytabloid μ) (polytabloid μ) = Fintype.card (colGroup μ) := by
+  have h := alternator_pairing μ (polytabloid μ)
+  have he (c : colGroup μ) : tabloidRep μ c (polytabloid μ) =
+      permSign (c:Equiv.Perm (Cell μ)) • polytabloid μ :=
+    action_alternator_apply (colGroup μ) (tabloidRep μ) c _
+  have hb := congrArg (fun v : Tabloid μ → ℂ => v (baseTabloid μ)) h
+  simp only [alternator,LinearMap.sum_apply,LinearMap.smul_apply,he,
+    Pi.smul_apply,smul_eq_mul,Finset.sum_apply,polytabloid_base,mul_one] at hb
+  simp only [permSign_sq,Finset.sum_const,Finset.card_univ,nsmul_eq_mul,mul_one] at hb
+  exact hb.symm
+
+lemma polytabloid_norm_sq (μ : YoungDiagram) :
+    ‖hilbertEquiv μ (polytabloid μ)‖^2 = (Fintype.card (colGroup μ):ℝ) := by
+  have hp := polytabloid_pairing μ
+  change inner ℂ (hilbertEquiv μ (polytabloid μ)) (hilbertEquiv μ (polytabloid μ)) = _ at hp
+  rw [inner_self_eq_norm_sq_to_K] at hp
+  have hre := congrArg Complex.re hp
+  simpa [pow_two] using hre
+
+theorem dimension_times_column_ge_tabloids (μ : YoungDiagram) :
+    Fintype.card (Tabloid μ) ≤ Module.finrank ℂ (space μ) * Fintype.card (colGroup μ) := by
+  have ht (f g : Tabloid μ) : ∃ p : Equiv.Perm (Tabloid μ), p f = g ∧
+      (hilbertSpace μ).map (PermutationHilbert.act p).toLinearEquiv.toLinearMap = hilbertSpace μ := by
+    obtain ⟨p,hp⟩ := tabloidAct_transitive μ f g
+    exact ⟨tabloidAct p,hp,hilbertSpace_invariant μ p⟩
+  have hu := PermutationHilbert.uncertainty (hilbertSpace μ) ht
+    (hilbertEquiv μ (polytabloid μ)) ⟨polytabloid μ,polytabloid_mem_space μ,rfl⟩ (baseTabloid μ)
+  have he : hilbertEquiv μ (polytabloid μ) (baseTabloid μ) = 1 := polytabloid_base μ
+  rw [he,norm_one,one_pow,mul_one,finrank_hilbertSpace,polytabloid_norm_sq] at hu
+  exact_mod_cast hu
+
+
+section FiberGroup
+variable {α β γ : Type*} [Fintype α] [Fintype β] [Fintype γ]
+
+noncomputable def fiberGroupEquiv (f : α → β) :
+    fiberGroup f ≃ (∀ b, Equiv.Perm {x // f x = b}) where
+  toFun p b :=
+    { toFun x := ⟨p.1 x.1,(p.2 x.1).trans x.2⟩
+      invFun x := ⟨p.1⁻¹ x.1, by
+        have hh := p.2 (p.1⁻¹ x.1)
+        simp only [Equiv.Perm.inv_def,Equiv.apply_symm_apply] at hh
+        exact hh.symm.trans x.2⟩
+      left_inv x := by apply Subtype.ext; simp
+      right_inv x := by apply Subtype.ext; simp }
+  invFun q :=
+    ⟨(Equiv.sigmaFiberEquiv f).permCongr (Equiv.Perm.sigmaCongrRight q),
+      fun x => (q (f x) ⟨x,rfl⟩).2⟩
+  left_inv p := by
+    apply Subtype.ext
+    apply Equiv.ext
+    intro x
+    rfl
+  right_inv q := by
+    funext b
+    apply Equiv.ext
+    intro x
+    apply Subtype.ext
+    change (q (f x.1) ⟨x.1,rfl⟩).1 = (q b x).1
+    cases x with
+    | mk x hx => dsimp only at *; subst b; rfl
+
+lemma nat_card_fiberGroup (f : α → β) :
+    Nat.card (fiberGroup f) = ∏ b, (Nat.card {x // f x=b}).factorial := by
+  rw [Nat.card_congr (fiberGroupEquiv f),Nat.card_pi]
+  simp only [Nat.card_eq_fintype_card,Fintype.card_perm]
+
+omit [Fintype β] [Fintype γ] in
+
+lemma card_fiberGroups_mul_le (f : α → β) (g : α → γ)
+    (hinj : ∀ x y, f x = f y → g x = g y → x=y) :
+    Fintype.card (fiberGroup f) * Fintype.card (fiberGroup g) ≤ (Fintype.card α).factorial := by
+  have hi : Function.Injective (fun p : fiberGroup f × fiberGroup g => p.1.1*p.2.1) := by
+    intro p q hpq
+    dsimp only at hpq
+    have he : p.1.1⁻¹*q.1.1 = p.2.1*q.2.1⁻¹ := by
+      apply inv_mul_eq_iff_eq_mul.mpr
+      rw [←mul_assoc,hpq,mul_assoc,mul_inv_cancel,mul_one]
+    have hf : p.1.1⁻¹*q.1.1 ∈ fiberGroup f := (fiberGroup f).mul_mem ((fiberGroup f).inv_mem p.1.2) q.1.2
+    have hg : p.1.1⁻¹*q.1.1 ∈ fiberGroup g := he ▸ (fiberGroup g).mul_mem p.2.2 ((fiberGroup g).inv_mem q.2.2)
+    have h1 : p.1.1⁻¹*q.1.1 = 1 := by
+      apply Equiv.ext
+      intro x
+      exact hinj _ x (hf x) (hg x)
+    have hpf : p.1=q.1 := Subtype.ext (inv_mul_eq_one.mp h1)
+    have hpg : p.2=q.2 := by
+      apply Subtype.ext
+      have hh := hpq
+      rw [show p.1.1=q.1.1 from congrArg Subtype.val hpf] at hh
+      exact mul_left_cancel hh
+    exact Prod.ext hpf hpg
+  have hc := Fintype.card_le_of_injective _ hi
+  simpa only [Fintype.card_prod,Fintype.card_perm] using hc
+
+end FiberGroup
+
+noncomputable def rowGroup (μ : YoungDiagram) : Subgroup (Equiv.Perm (Cell μ)) :=
+  fiberGroup (rowIndex (μ := μ))
+
+noncomputable def colIndex {μ : YoungDiagram} (x : Cell μ) : Fin (μ.rowLen 0) :=
+  ⟨col x, (YoungDiagram.mem_iff_lt_rowLen.mp x.2).trans_le (μ.rowLen_anti 0 (row x) (Nat.zero_le _))⟩
+
+lemma colGroup_eq (μ : YoungDiagram) : colGroup μ = fiberGroup (colIndex (μ := μ)) := by
+  ext p
+  constructor
+  · intro hp x; exact Fin.ext (hp x)
+  · intro hp x; exact congrArg Fin.val (hp x)
+
+noncomputable instance tabloidMulAction (μ : YoungDiagram) : MulAction (Equiv.Perm (Cell μ)) (Tabloid μ) where
+  smul p f := tabloidAct p f
+  one_smul f := by change tabloidAct 1 f = f; rw [tabloidAct_one]; rfl
+  mul_smul p q f := by change tabloidAct (p*q) f = tabloidAct p (tabloidAct q f); rw [tabloidAct_mul]; rfl
+
+lemma tabloid_orbit_all (μ : YoungDiagram) :
+    MulAction.orbit (Equiv.Perm (Cell μ)) (baseTabloid μ) = Set.univ := by
+  ext f
+  simp only [Set.mem_univ,iff_true]
+  obtain ⟨p,hp⟩ := tabloidAct_transitive μ (baseTabloid μ) f
+  exact ⟨p,hp⟩
+
+lemma tabloid_stabilizer (μ : YoungDiagram) :
+    MulAction.stabilizer (Equiv.Perm (Cell μ)) (baseTabloid μ) = rowGroup μ := by
+  ext p
+  change tabloidAct p (baseTabloid μ) = baseTabloid μ ↔ ∀ x, rowIndex (p x) = rowIndex x
+  constructor
+  · intro h x
+    have hh := congrArg (fun f : Tabloid μ => f.1 (p x)) h
+    simpa only [tabloidAct_apply,baseTabloid,Equiv.Perm.inv_def,Equiv.symm_apply_apply] using hh.symm
+  · intro h
+    apply Subtype.ext
+    funext x
+    exact (h (p⁻¹ x)).symm.trans (by simp [baseTabloid])
+
+lemma card_tabloid_mul_row (μ : YoungDiagram) :
+    Fintype.card (Tabloid μ)*Fintype.card (rowGroup μ) = μ.card.factorial := by
+  have hh := MulAction.card_orbit_mul_card_stabilizer_eq_card_group
+    (Equiv.Perm (Cell μ)) (baseTabloid μ)
+  have hc : Fintype.card (MulAction.orbit (Equiv.Perm (Cell μ)) (baseTabloid μ)) =
+      Fintype.card (Tabloid μ) := by
+    rw [tabloid_orbit_all,Fintype.card_setUniv]
+  simpa only [hc,tabloid_stabilizer,Fintype.card_perm,card_cell] using hh
+
+theorem factorial_le_dimension_row_column (μ : YoungDiagram) :
+    μ.card.factorial ≤ Module.finrank ℂ (space μ)*Fintype.card (rowGroup μ)*Fintype.card (colGroup μ) := by
+  rw [←card_tabloid_mul_row]
+  nlinarith [dimension_times_column_ge_tabloids μ]
+
+
+lemma card_row_fiber (μ : YoungDiagram) (i : Fin (μ.colLen 0)) :
+    Fintype.card {x : Cell μ // rowIndex x = i} = μ.rowLen i := by
+  let e : {x : Cell μ // rowIndex x = i} ≃ {x : Cell μ // row x = i} :=
+    Equiv.subtypeEquivRight (fun x => Fin.ext_iff)
+  rw [Fintype.card_congr e,Fintype.card_congr (rowEquiv μ i),Fintype.card_fin]
+
+lemma card_col_fiber (μ : YoungDiagram) (j : Fin (μ.rowLen 0)) :
+    Fintype.card {x : Cell μ // colIndex x = j} = μ.colLen j := by
+  let e : {x : Cell μ // colIndex x = j} ≃ {x : Cell μ // col x = j} :=
+    Equiv.subtypeEquivRight (fun x => Fin.ext_iff)
+  rw [Fintype.card_congr e,Fintype.card_congr (columnEquiv μ j),Fintype.card_fin]
+
+lemma card_rowGroup (μ : YoungDiagram) :
+    Fintype.card (rowGroup μ) = ∏ i : Fin (μ.colLen 0), (μ.rowLen i).factorial := by
+  rw [←Nat.card_eq_fintype_card,rowGroup,nat_card_fiberGroup]
+  simp only [Nat.card_eq_fintype_card]
+  simp only [card_row_fiber]
+
+lemma card_colGroup (μ : YoungDiagram) :
+    Fintype.card (colGroup μ) = ∏ j : Fin (μ.rowLen 0), (μ.colLen j).factorial := by
+  rw [←Nat.card_eq_fintype_card,colGroup_eq,nat_card_fiberGroup]
+  simp only [Nat.card_eq_fintype_card,card_col_fiber]
+
+abbrev Tail (μ : YoungDiagram) := {x : Cell μ // row x ≠ 0}
+
+lemma tail_card (μ : YoungDiagram) : Fintype.card (Tail μ) = μ.card - μ.rowLen 0 := by
+  have hh := Fintype.card_subtype_compl (p := fun x : Cell μ => row x=0)
+  rw [Fintype.card_congr (rowEquiv μ 0),Fintype.card_fin,card_cell] at hh
+  exact hh
+
+lemma rowLen_zero_le_card (μ : YoungDiagram) : μ.rowLen 0 ≤ μ.card := by
+  have h := Fintype.card_subtype_le (fun x : Cell μ => row x=0)
+  rwa [Fintype.card_congr (rowEquiv μ 0),Fintype.card_fin,card_cell] at h
+
+noncomputable def tailColumnEquiv (μ : YoungDiagram) (j : Fin (μ.rowLen 0)) :
+    {x : Tail μ // colIndex x.1 = j} ≃ Fin (μ.colLen j-1) where
+  toFun x := ⟨row x.1.1-1, by
+    have hx := row_lt_colLen x.1.1
+    have hc : col x.1.1 = j := congrArg Fin.val x.2
+    rw [hc] at hx
+    have hz := x.1.2
+    omega⟩
+  invFun i := ⟨⟨⟨(i+1,j),YoungDiagram.mem_iff_lt_colLen.mpr (by omega)⟩,by simp [row]⟩,rfl⟩
+  left_inv x := by
+    apply Subtype.ext
+    apply Subtype.ext
+    apply Subtype.ext
+    apply Prod.ext
+    · change row x.1.1 - 1 + 1 = row x.1.1
+      have := x.1.2
+      omega
+    · exact (congrArg Fin.val x.2).symm
+  right_inv i := by apply Fin.ext; simp [row]
+
+lemma card_tail_col (μ : YoungDiagram) (j : Fin (μ.rowLen 0)) :
+    Fintype.card {x : Tail μ // colIndex x.1 = j} = μ.colLen j-1 := by
+  rw [Fintype.card_congr (tailColumnEquiv μ j),Fintype.card_fin]
+
+lemma card_tail_row (μ : YoungDiagram) (i : Fin (μ.colLen 0)) :
+    Fintype.card {x : Tail μ // rowIndex x.1 = i} = if i.val=0 then 0 else μ.rowLen i := by
+  by_cases hi : i.val=0
+  · rw [ite_eq_left hi]
+    apply Fintype.card_eq_zero_iff.mpr
+    refine ⟨fun x => x.1.2 ?_⟩
+    exact (congrArg Fin.val x.2).trans hi
+  · rw [ite_eq_right hi]
+    let e : {x : Tail μ // rowIndex x.1 = i} ≃ {x : Cell μ // rowIndex x=i} :=
+      { toFun := fun x => ⟨x.1.1,x.2⟩
+        invFun := fun x => ⟨⟨x.1,fun hz => hi ((congrArg Fin.val x.2).symm.trans hz)⟩,x.2⟩
+        left_inv := fun x => rfl
+        right_inv := fun x => rfl }
+    rw [Fintype.card_congr e,card_row_fiber]
+
+lemma tail_row_group (μ : YoungDiagram) :
+    (μ.rowLen 0).factorial * Fintype.card (fiberGroup (fun x : Tail μ => rowIndex x.1)) =
+      Fintype.card (rowGroup μ) := by
+  rw [←Nat.card_eq_fintype_card (α := fiberGroup (fun x : Tail μ => rowIndex x.1)),nat_card_fiberGroup,card_rowGroup]
+  simp only [Nat.card_eq_fintype_card]
+  simp only [card_tail_row]
+  by_cases hn : μ.colLen 0=0
+  · have hr : μ.rowLen 0=0 := by
+      by_contra h
+      have hmem : (0,0) ∈ μ := YoungDiagram.mem_iff_lt_rowLen.mpr (Nat.pos_of_ne_zero h)
+      have hh := YoungDiagram.mem_iff_lt_colLen.mp hmem
+      omega
+    let : IsEmpty (Fin (μ.colLen 0)) := ⟨fun i => by have := i.isLt; omega⟩
+    simp [hr]
+  · let z : Fin (μ.colLen 0) := ⟨0,Nat.pos_of_ne_zero hn⟩
+    have hz : ∀ i : Fin (μ.colLen 0), i.val=0 ↔ i=z := fun i => by simp [z,Fin.ext_iff]
+    simp only [hz]
+    have hprod : (∏ x : Fin (μ.colLen 0), (if x=z then 0 else μ.rowLen x).factorial) =
+        ∏ x ∈ Finset.univ.erase z, (μ.rowLen x).factorial := by
+      rw [←Finset.prod_erase_mul Finset.univ (fun x : Fin (μ.colLen 0) => (if x=z then 0 else μ.rowLen x).factorial) (Finset.mem_univ z)]
+      simp only [ite_true,Nat.factorial_zero,mul_one]
+      apply Finset.prod_congr rfl
+      intro x hx
+      rw [ite_eq_right (Finset.mem_erase.mp hx).1]
+    rw [hprod]
+    exact Finset.mul_prod_erase Finset.univ (fun i : Fin (μ.colLen 0) => (μ.rowLen i).factorial) (Finset.mem_univ z)
+
+lemma sum_tail_columns (μ : YoungDiagram) :
+    ∑ j : Fin (μ.rowLen 0), (μ.colLen j-1) = μ.card-μ.rowLen 0 := by
+  simp_rw [←card_tail_col]
+  rw [←Fintype.card_sigma,Fintype.card_congr (Equiv.sigmaFiberEquiv (fun x : Tail μ => colIndex x.1)),tail_card]
+
+lemma column_pos (μ : YoungDiagram) (j : Fin (μ.rowLen 0)) : 0 < μ.colLen j :=
+  YoungDiagram.mem_iff_lt_colLen.mp (YoungDiagram.mem_iff_lt_rowLen.mpr j.isLt)
+
+lemma nat_le_two_pow_pred (n : ℕ) (hn : 0<n) : n ≤ 2^(n-1) := by
+  obtain ⟨n,rfl⟩ := Nat.exists_eq_succ_of_ne_zero hn.ne'
+  simp only [Nat.add_one_sub_one]
+  induction n with
+  | zero => norm_num
+  | succ n ih => rw [pow_succ]; omega
+
+lemma column_group_le_tail (μ : YoungDiagram) :
+    Fintype.card (colGroup μ) ≤ 2^(μ.card-μ.rowLen 0) *
+      Fintype.card (fiberGroup (fun x : Tail μ => colIndex x.1)) := by
+  rw [card_colGroup,←Nat.card_eq_fintype_card (α := fiberGroup (fun x : Tail μ => colIndex x.1)),nat_card_fiberGroup]
+  simp only [Nat.card_eq_fintype_card]
+  simp only [card_tail_col]
+  calc
+    _ = ∏ j : Fin (μ.rowLen 0), μ.colLen j * (μ.colLen j-1).factorial := by
+      apply Finset.prod_congr rfl
+      intro j hj
+      exact (Nat.mul_factorial_pred (column_pos μ j).ne').symm
+    _ ≤ ∏ j : Fin (μ.rowLen 0), 2^(μ.colLen j-1) * (μ.colLen j-1).factorial := by
+      apply Finset.prod_le_prod
+      intro j hj
+      exact Nat.mul_le_mul_right _ (nat_le_two_pow_pred _ (column_pos μ j))
+    _ = _ := by rw [Finset.prod_mul_distrib,Finset.prod_pow_eq_pow_sum,sum_tail_columns]
+
+theorem choose_tail_le_dimension (μ : YoungDiagram) :
+    μ.card.choose (μ.card-μ.rowLen 0) ≤ Module.finrank ℂ (space μ) * 2^(μ.card-μ.rowLen 0) := by
+  let j := μ.card-μ.rowLen 0
+  have htail := card_fiberGroups_mul_le
+    (fun x : Tail μ => rowIndex x.1) (fun x : Tail μ => colIndex x.1)
+    (fun x y hr hc => Subtype.ext (Subtype.ext (Prod.ext (congrArg Fin.val hr) (congrArg Fin.val hc))))
+  rw [tail_card] at htail
+  have hc := column_group_le_tail μ
+  have hr := tail_row_group μ
+  have hd := factorial_le_dimension_row_column μ
+  simp only [←Nat.card_eq_fintype_card] at htail hc hr hd
+  have hh : μ.card.factorial ≤ Module.finrank ℂ (space μ) * 2^j *
+      ((μ.rowLen 0).factorial * j.factorial) := by
+    calc
+      _ ≤ _ := hd
+      _ ≤ Module.finrank ℂ (space μ) * Nat.card (rowGroup μ) *
+          (2^j * Nat.card (fiberGroup (fun x : Tail μ => colIndex x.1))) := Nat.mul_le_mul_left _ hc
+      _ = Module.finrank ℂ (space μ) * 2^j * (μ.rowLen 0).factorial *
+          (Nat.card (fiberGroup (fun x : Tail μ => rowIndex x.1)) *
+           Nat.card (fiberGroup (fun x : Tail μ => colIndex x.1))) := by rw [←hr]; ring
+      _ ≤ _ := by
+        calc
+          _ ≤ Module.finrank ℂ (space μ) * 2^j * (μ.rowLen 0).factorial * j.factorial :=
+            Nat.mul_le_mul_left _ htail
+          _ = _ := by ring
+  have hj : j ≤ μ.card := Nat.sub_le _ _
+  have hs : μ.card-j = μ.rowLen 0 := by dsimp [j]; have := rowLen_zero_le_card μ; omega
+  have hf := Nat.choose_mul_factorial_mul_factorial hj
+  rw [hs] at hf
+  rw [←hf] at hh
+  have hpos : 0 < (μ.rowLen 0).factorial * j.factorial := Nat.mul_pos (Nat.factorial_pos _) (Nat.factorial_pos _)
+  nlinarith
+
+
+lemma exists_perm_mem_iff {α : Type*} [Fintype α] (s t : Finset α) (hc : s.card=t.card) :
+    ∃ p : Equiv.Perm α, ∀ x, p x ∈ t ↔ x ∈ s := by
+  let e : s ≃ t := Finset.equivOfCardEq hc
+  let ec : {x : α // x ∉ s} ≃ {x : α // x ∉ t} := Fintype.equivOfCardEq (by
+    simp only [Fintype.card_subtype_compl,Fintype.card_coe,hc])
+  let p := (Equiv.sumCompl (· ∈ s)).symm.trans ((Equiv.sumCongr e ec).trans (Equiv.sumCompl (· ∈ t)))
+  refine ⟨p,fun x => ?_⟩
+  by_cases hx : x ∈ s
+  · have hp : p x ∈ t := by
+      change ((Equiv.sumCompl (· ∈ t)) ((Equiv.sumCongr e ec) ((Equiv.sumCompl (· ∈ s)).symm x))) ∈ t
+      rw [Equiv.sumCompl_symm_apply_of_pos hx]
+      exact (e ⟨x,hx⟩).2
+    exact iff_of_true hp hx
+  · have hp : p x ∉ t := by
+      change ((Equiv.sumCompl (· ∈ t)) ((Equiv.sumCongr e ec) ((Equiv.sumCompl (· ∈ s)).symm x))) ∉ t
+      rw [Equiv.sumCompl_symm_apply_of_neg hx]
+      exact (ec ⟨x,hx⟩).2
+    exact iff_of_false hp hx
+
+noncomputable def tailSet {μ : YoungDiagram} (f : Tabloid μ) : Finset (Cell μ) :=
+  Finset.univ.filter (fun x => (f.1 x).val ≠ 0)
+
+@[simp] lemma mem_tailSet {μ : YoungDiagram} (f : Tabloid μ) (x : Cell μ) :
+    x ∈ tailSet f ↔ (f.1 x).val ≠ 0 := by simp [tailSet]
+
+lemma tailSet_act {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) (f : Tabloid μ) :
+    tailSet (tabloidAct p f) = (tailSet f).image p := by
+  ext x
+  simp only [mem_tailSet,tabloidAct_apply,Finset.mem_image]
+  constructor
+  · intro h; exact ⟨p⁻¹ x,h,by simp⟩
+  · rintro ⟨y,hy,rfl⟩; simpa using hy
+
+lemma card_tailSet {μ : YoungDiagram} (f : Tabloid μ) :
+    (tailSet f).card = μ.card-μ.rowLen 0 := by
+  obtain ⟨p,rfl⟩ := tabloidAct_transitive μ (baseTabloid μ) f
+  rw [tailSet_act,Finset.card_image_of_injective _ p.injective]
+  change (Finset.univ.filter (fun x : Cell μ => row x ≠ 0)).card = _
+  rw [←Fintype.card_subtype]
+  exact tail_card μ
+
+noncomputable def footprint (μ : YoungDiagram) (A : Finset (Cell μ)) : Finset (Tabloid μ) :=
+  Finset.univ.filter (fun f => tailSet f ⊆ A)
+
+lemma tail_fiber_card_eq (μ : YoungDiagram) (s t : Finset (Cell μ)) (hst : s.card=t.card) :
+    (Finset.univ.filter (fun f : Tabloid μ => tailSet f=s)).card =
+      (Finset.univ.filter (fun f : Tabloid μ => tailSet f=t)).card := by
+  obtain ⟨p,hp⟩ := exists_perm_mem_iff s t hst
+  have he : s.image p=t := by
+    ext x
+    constructor
+    · intro hx
+      obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
+      exact (hp y).mpr hy
+    · intro hx
+      exact Finset.mem_image.mpr ⟨p⁻¹ x,(hp (p⁻¹ x)).mp (by simpa using hx),by simp⟩
+  apply Finset.card_equiv (tabloidAct p)
+  intro f
+  simp only [Finset.mem_filter,Finset.mem_univ,true_and,tailSet_act,←he]
+  exact (Finset.image_inj p.injective).symm
+
+lemma footprint_card_product (μ : YoungDiagram) (A : Finset (Cell μ)) :
+    (footprint μ A).card * μ.card.choose (μ.card-μ.rowLen 0) =
+      Fintype.card (Tabloid μ) * A.card.choose (μ.card-μ.rowLen 0) := by
+  let j := μ.card-μ.rowLen 0
+  let s := tailSet (baseTabloid μ)
+  let c := (Finset.univ.filter (fun f : Tabloid μ => tailSet f=s)).card
+  have hconstant (t : Finset (Cell μ)) (ht : t.card=j) :
+      (Finset.univ.filter (fun f : Tabloid μ => tailSet f=t)).card = c :=
+    tail_fiber_card_eq μ t s (ht.trans (card_tailSet _).symm)
+  have hcount (A : Finset (Cell μ)) : (footprint μ A).card = A.card.choose j*c := by
+    have h := Finset.sum_card_fiberwise_eq_card_filter
+      (Finset.univ : Finset (Tabloid μ)) (A.powersetCard j) tailSet
+    have he : (Finset.univ.filter (fun f : Tabloid μ => tailSet f ∈ A.powersetCard j)) = footprint μ A := by
+      ext f
+      simp only [Finset.mem_filter,Finset.mem_univ,true_and,Finset.mem_powersetCard,
+        card_tailSet,footprint]
+      simp [j]
+    rw [he] at h
+    rw [←h]
+    calc
+      _ = ∑ _t ∈ A.powersetCard j, c := by
+        apply Finset.sum_congr rfl
+        intro t ht
+        exact hconstant t (Finset.mem_powersetCard.mp ht).2
+      _ = _ := by simp [Finset.card_powersetCard]
+  have hfull : (footprint μ Finset.univ).card = Fintype.card (Tabloid μ) := by
+    simp only [footprint,Finset.subset_univ,Finset.filter_true,Finset.card_univ]
+  have hN := hcount Finset.univ
+  rw [hfull,Finset.card_univ,card_cell] at hN
+  rw [hcount,hN]
+  ring
+
+lemma footprint_fixed {μ : YoungDiagram} (A : Finset (Cell μ))
+    (f : Tabloid μ) (hf : f ∈ footprint μ A) (p : Equiv.Perm (Cell μ))
+    (hp : ∀ x ∈ A, p x=x) : tabloidAct p f=f := by
+  have htail : tailSet f ⊆ A := (Finset.mem_filter.mp hf).2
+  apply Subtype.ext
+  funext x
+  change f.1 (p⁻¹ x) = f.1 x
+  by_cases hx : x ∈ A
+  · rw [show p⁻¹ x=x from p.injective (by simpa using (hp x hx).symm)]
+  · have hy : p⁻¹ x ∉ A := by
+      intro h
+      have hh := hp (p⁻¹ x) h
+      have he : x=p⁻¹ x := by simpa using hh
+      exact hx (he ▸ h)
+    apply Fin.ext
+    have hx0 : (f.1 x).val=0 := by by_contra h; exact hx (htail (mem_tailSet f x |>.mpr h))
+    have hy0 : (f.1 (p⁻¹ x)).val=0 := by by_contra h; exact hy (htail (mem_tailSet f _ |>.mpr h))
+    exact hy0.trans hx0.symm
+
+noncomputable def fixedHilbert (μ : YoungDiagram) (A : Finset (Cell μ)) :
+    Submodule ℂ (PermutationHilbert.H (Tabloid μ)) where
+  carrier := {v | v ∈ hilbertSpace μ ∧ ∀ p : Equiv.Perm (Cell μ),
+    (∀ x ∈ A, p x=x) → PermutationHilbert.act (tabloidAct p) v=v}
+  zero_mem' := ⟨Submodule.zero_mem _,fun p hp => map_zero _⟩
+  add_mem' := fun hv hw => ⟨Submodule.add_mem _ hv.1 hw.1,fun p hp => by
+    rw [map_add,hv.2 p hp,hw.2 p hp]⟩
+  smul_mem' := fun c v hv => ⟨Submodule.smul_mem _ c hv.1,fun p hp => by
+    rw [map_smul,hv.2 p hp]⟩
+
+lemma fixedHilbert_le (μ : YoungDiagram) (A : Finset (Cell μ)) : fixedHilbert μ A ≤ hilbertSpace μ :=
+  fun _ hv => hv.1
+
+theorem fixedHilbert_dimension (μ : YoungDiagram) (A : Finset (Cell μ)) :
+    Module.finrank ℂ (space μ) * A.card.choose (μ.card-μ.rowLen 0) ≤
+      Module.finrank ℂ (fixedHilbert μ A) * μ.card.choose (μ.card-μ.rowLen 0) := by
+  let : DecidableEq (Tabloid μ) := Classical.decEq _
+  have ht (f g : Tabloid μ) : ∃ p : Equiv.Perm (Tabloid μ), p f=g ∧
+      (hilbertSpace μ).map (PermutationHilbert.act p).toLinearEquiv.toLinearMap = hilbertSpace μ := by
+    obtain ⟨p,hp⟩ := tabloidAct_transitive μ f g
+    exact ⟨tabloidAct p,hp,hilbertSpace_invariant μ p⟩
+  have hproj (f : Tabloid μ) (hf : f ∈ footprint μ A) :
+      (hilbertSpace μ).starProjection (EuclideanSpace.single f (1:ℂ)) ∈ fixedHilbert μ A := by
+    refine ⟨((hilbertSpace μ).orthogonalProjectionOnto _).2,fun p hp => ?_⟩
+    rw [PermutationHilbert.projection_comm _ _ (hilbertSpace_invariant μ p),
+      PermutationHilbert.act_single,footprint_fixed A f hf p hp]
+  have h := PermutationHilbert.footprint_dimension (hilbertSpace μ) (fixedHilbert μ A)
+    (fixedHilbert_le μ A) ht (footprint μ A) hproj
+  rw [finrank_hilbertSpace] at h
+  have hn : (footprint μ A).card * Module.finrank ℂ (space μ) ≤
+      Fintype.card (Tabloid μ)*Module.finrank ℂ (fixedHilbert μ A) := by exact_mod_cast h
+  have hc := footprint_card_product μ A
+  have hpos : 0 < Fintype.card (Tabloid μ) := Fintype.card_pos_iff.mpr ⟨baseTabloid μ⟩
+  have hh := Nat.mul_le_mul_right (μ.card.choose (μ.card-μ.rowLen 0)) hn
+  nlinarith
+
+end Thorp.Specht
+
+namespace Thorp.UnitaryFinite
+open scoped BigOperators ComplexConjugate Classical
+
+variable {G : Type*} [Group G] [Fintype G]
+variable {V W : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+  [NormedAddCommGroup W] [InnerProductSpace ℂ W] [FiniteDimensional ℂ W]
+
+def IsUnitary (ρ : Representation ℂ G V) : Prop :=
+  ∀ g v w, inner ℂ (ρ g v) (ρ g w) = inner ℂ v w
+
+omit [Fintype G] [FiniteDimensional ℂ V] in
+lemma unitary_inner_inv (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (g : G) (v w : V) :
+    inner ℂ v (ρ g⁻¹ w) = inner ℂ (ρ g v) w := by
+  have h := hρ g v (ρ g⁻¹ w)
+  have he : ρ g (ρ g⁻¹ w) = w := by
+    change (ρ g * ρ g⁻¹) w = w
+    rw [←map_mul,mul_inv_cancel,map_one]
+    rfl
+  rw [he] at h
+  exact h.symm
+
+end Thorp.UnitaryFinite
+
+end ThorpNine.Harmonic
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Harmonic/Site.lean b/lean/OAI/Probability/ThorpRouting/Harmonic/Site.lean
index 40807f32b8f0f86b3d89e8a2c608315fb8c30d0b..4c9c1d0fe1edf4af52ef3f768f9b0477de06d426 100644
--- a/lean/OAI/Probability/ThorpRouting/Harmonic/Site.lean
+++ b/lean/OAI/Probability/ThorpRouting/Harmonic/Site.lean
@@ -1,873 +1,873 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Harmonic.TailShapes
-
-namespace OAI
-
-namespace ThorpNine.Harmonic
-
-namespace Thorp.StrongSmoothing
-open scoped BigOperators
-
-lemma regular_trace_bound (d : ℕ) (order : Equiv.Perm (Fin d)) :
-    regularAbsoluteEvenTrace d order 2001 ≤ Real.exp ((2000*(1+heightDecay 1))*(2:ℝ)^d) := by
-  have hn : 0 < Fintype.card (Tuples d (2^d)) := by
-    rw [tuples_card,Nat.descFactorial_self]
-    exact Nat.factorial_pos _
-  have hn' : (Fintype.card (Tuples d (2^d)):ℝ) ≠ 0 := Nat.cast_ne_zero.mpr (ne_of_gt hn)
-  calc
-    _ ≤ ∑ x : Tuples d (2^d), Real.exp ((2000*(1+heightDecay 1))*(2^d:ℕ)) /
-        ((2^d).descFactorial (2^d):ℝ) := by
-      apply Finset.sum_le_sum
-      intro x _
-      exact both_pointwise d (2^d) order false x x
-    _ = _ := by
-      simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,←tuples_card]
-      push_cast
-      field_simp
-
-theorem strong_smoothing : MainStatement := by
-  refine ⟨2001,by norm_num,2000*(1+heightDecay 1),?_,regular_trace_bound⟩
-  intro d l _ order orientation x y
-  exact both_pointwise d l order orientation x y
-
-end Thorp.StrongSmoothing
-
-namespace Thorp.ButterflyReverse
-open scoped BigOperators
-
-abbrev Site (d : ℕ) := {t : Fin d × Card d // t.2 t.1 = false}
-
-def site : (d : ℕ) → SwitchIndex d → Site d
-  | 0, t => nomatch t
-  | _+1, Sum.inl y => ⟨(0, Fin.cons false y), rfl⟩
-  | d+1, Sum.inr (b,t) =>
-      ⟨((site d t).1.1.succ, Fin.cons b (site d t).1.2), (site d t).2⟩
-
-lemma site_injective (d : ℕ) : Function.Injective (site d) := by
-  induction d with
-  | zero => intro t; exact Empty.elim t
-  | succ d ih =>
-    intro s t h
-    cases s with
-    | inl y =>
-      cases t with
-      | inl z =>
-        have he := congrArg (fun a : Site (d+1) => Fin.tail a.1.2) h
-        exact congrArg Sum.inl he
-      | inr t =>
-        have he := congrArg (fun a : Site (d+1) => a.1.1.val) h
-        simp [site] at he
-    | inr s =>
-      cases t with
-      | inl y =>
-        have he := congrArg (fun a : Site (d+1) => a.1.1.val) h
-        simp [site] at he
-      | inr t =>
-        obtain ⟨b,s⟩ := s
-        obtain ⟨c,t⟩ := t
-        have hbc := congrArg (fun a : Site (d+1) => a.1.2 0) h
-        simp only [site, Fin.cons_zero] at hbc
-        have hst : site d s = site d t := by
-          apply Subtype.ext
-          apply Prod.ext
-          · exact Fin.succ_inj.mp (congrArg (fun a : Site (d+1) => a.1.1) h)
-          · exact congrArg (fun a : Site (d+1) => Fin.tail a.1.2) h
-        cases hbc
-        cases ih hst
-        rfl
-
-lemma site_surjective (d : ℕ) : Function.Surjective (site d) := by
-  induction d with
-  | zero => intro t; exact Fin.elim0 t.1.1
-  | succ d ih =>
-    rintro ⟨⟨i,x⟩,hx⟩
-    refine Fin.cases ?_ (fun j => ?_) i hx
-    · intro hx0
-      refine ⟨Sum.inl (Fin.tail x), ?_⟩
-      apply Subtype.ext
-      simp only [site, Prod.mk.injEq, true_and]
-      rw [←hx0,Fin.cons_self_tail]
-    · intro hxj
-      obtain ⟨t,ht⟩ := ih ⟨(j,Fin.tail x),hxj⟩
-      refine ⟨Sum.inr (x 0,t), ?_⟩
-      apply Subtype.ext
-      simp only [site,ht,Fin.cons_self_tail]
-
-noncomputable def siteEquiv (d : ℕ) : SwitchIndex d ≃ Site d :=
-  Equiv.ofBijective (site d) ⟨site_injective d,site_surjective d⟩
-
-def reverseCard (d : ℕ) : Equiv.Perm (Card d) where
-  toFun x := fun i => x i.rev
-  invFun x := fun i => x i.rev
-  left_inv x := by funext i; simp
-  right_inv x := by funext i; simp
-
-@[simp] lemma reverseCard_apply (d : ℕ) (x : Card d) (i : Fin d) :
-    reverseCard d x i = x i.rev := rfl
-
-@[simp] lemma reverseCard_involutive (d : ℕ) (x : Card d) :
-    reverseCard d (reverseCard d x) = x := by funext i; simp
-
-def mirrorSite (d : ℕ) : Equiv.Perm (Site d) where
-  toFun s := ⟨(s.1.1.rev,reverseCard d s.1.2),by simpa using s.2⟩
-  invFun s := ⟨(s.1.1.rev,reverseCard d s.1.2),by simpa using s.2⟩
-  left_inv s := by apply Subtype.ext; simp
-  right_inv s := by apply Subtype.ext; simp
-
-noncomputable def mirrorSwitch (d : ℕ) : Equiv.Perm (SwitchIndex d) :=
-  (siteEquiv d).trans ((mirrorSite d).trans (siteEquiv d).symm)
-
-lemma site_mirrorSwitch (d : ℕ) (t : SwitchIndex d) :
-    site d (mirrorSwitch d t) = mirrorSite d (site d t) := by
-  exact (siteEquiv d).apply_symm_apply _
-
-lemma routeDomain_site (d : ℕ) (x y : Card d) (t : SwitchIndex d) :
-    t ∈ routeDomain d x y ↔
-      ∀ j : Fin d, j ≠ (site d t).1.1 →
-      (site d t).1.2 j = if j < (site d t).1.1 then x j else y j := by
-  classical
-  induction d with
-  | zero => exact Empty.elim t
-  | succ d ih =>
-    cases t with
-    | inl z =>
-      change Sum.inl z ∈ (insert (Sum.inl (Fin.tail y))
-        ((routeDomain d (Fin.tail x) (Fin.tail y)).image
-          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) :
-        Finset (Sum (Card d) (Bool × SwitchIndex d))) ↔ _
-      simp only [Finset.mem_insert, Sum.inl.injEq,Finset.mem_image,
-        reduceCtorEq,and_false,exists_false,or_false]
-      constructor
-      · intro hz j hj
-        subst z
-        have hj0 : j ≠ 0 := hj
-        obtain ⟨i,rfl⟩ := Fin.eq_succ_of_ne_zero hj0
-        simp [site,Fin.tail]
-      · intro h
-        funext j
-        have he := h j.succ (by simp [site])
-        simpa [site,Fin.tail] using he
-    | inr z =>
-      obtain ⟨b,t⟩ := z
-      change Sum.inr (b,t) ∈ (insert (Sum.inl (Fin.tail y))
-        ((routeDomain d (Fin.tail x) (Fin.tail y)).image
-          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) :
-        Finset (Sum (Card d) (Bool × SwitchIndex d))) ↔ _
-      simp only [Finset.mem_insert,reduceCtorEq,false_or,Finset.mem_image,Sum.inr.injEq,Prod.mk.injEq]
-      constructor
-      · rintro ⟨s,hs,hb,ht⟩
-        subst s
-        have hst := (ih (Fin.tail x) (Fin.tail y) t).mp hs
-        intro j hj
-        refine Fin.cases ?_ (fun i => ?_) j hj
-        · intro _
-          simp [site,hb]
-        · intro hi
-          have hi' : i ≠ (site d t).1.1 := by simpa [site] using hi
-          simpa [site,Fin.tail] using hst i hi'
-      · intro h
-        have hb := h 0 (by simpa only [site] using (Fin.succ_ne_zero (site d t).1.1).symm)
-        have hb' : x 0 = b := by simpa [site] using hb.symm
-        refine ⟨t,?_,hb',rfl⟩
-        apply (ih (Fin.tail x) (Fin.tail y) t).mpr
-        intro j hj
-        have he := h j.succ (by simpa [site] using hj)
-        simpa [site,Fin.tail] using he
-
-lemma routeValue_site (d : ℕ) (x y : Card d) (t : SwitchIndex d) :
-    routeValue d x y t = Bool.xor (x (site d t).1.1) (y (site d t).1.1) := by
-  induction d with
-  | zero => exact Empty.elim t
-  | succ d ih =>
-    cases t with
-    | inl z => rfl
-    | inr z => exact ih (Fin.tail x) (Fin.tail y) z.2
-
-lemma mirrorSwitch_involutive (d : ℕ) (t : SwitchIndex d) :
-    mirrorSwitch d (mirrorSwitch d t) = t := by
-  apply site_injective d
-  rw [site_mirrorSwitch,site_mirrorSwitch]
-  apply Subtype.ext
-  simp [mirrorSite]
-
-lemma mirror_routeDomain (d : ℕ) (x y : Card d) (t : SwitchIndex d)
-    (ht : t ∈ routeDomain d x y) :
-    mirrorSwitch d t ∈ routeDomain d (reverseCard d y) (reverseCard d x) := by
-  rw [routeDomain_site,site_mirrorSwitch]
-  intro j hj
-  have hj' : j.rev ≠ (site d t).1.1 := by
-    intro h
-    apply hj
-    change j = (site d t).1.1.rev
-    rw [←h,Fin.rev_rev]
-  have he := (routeDomain_site d x y t).mp ht j.rev hj'
-  change (site d t).1.2 j.rev =
-    if j < (site d t).1.1.rev then y j.rev else x j.rev
-  rw [he]
-  have hord : j < (site d t).1.1.rev ↔ ¬j.rev < (site d t).1.1 := by
-    rw [Fin.lt_rev_iff]
-    constructor
-    · intro h
-      exact not_lt_of_ge h.le
-    · intro h
-      exact lt_of_le_of_ne (le_of_not_gt h) (Ne.symm hj')
-  by_cases h : j.rev < (site d t).1.1 <;> simp [h,hord]
-
-lemma mirror_routeValue (d : ℕ) (x y : Card d) (t : SwitchIndex d) :
-    routeValue d (reverseCard d y) (reverseCard d x) (mirrorSwitch d t) =
-      routeValue d x y t := by
-  rw [routeValue_site,routeValue_site,site_mirrorSwitch]
-  simp only [mirrorSite,Equiv.coe_fn_mk,reverseCard_apply,Fin.rev_rev]
-  exact Bool.xor_comm _ _
-
-noncomputable def mirrorCoins (d : ℕ) : Equiv.Perm (SwitchIndex d → Bool) where
-  toFun ω := fun t => ω (mirrorSwitch d t)
-  invFun ω := fun t => ω (mirrorSwitch d t)
-  left_inv ω := by funext t; change ω (mirrorSwitch d (mirrorSwitch d t)) = ω t; rw [mirrorSwitch_involutive]
-  right_inv ω := by funext t; change ω (mirrorSwitch d (mirrorSwitch d t)) = ω t; rw [mirrorSwitch_involutive]
-
-lemma mirror_endpoint (d : ℕ) (ω : SwitchIndex d → Bool) (x y : Card d)
-    (hxy : butterflyPerm d (decodeButterfly d ω) x = y) :
-    butterflyPerm d (decodeButterfly d (mirrorCoins d ω)) (reverseCard d y) =
-      reverseCard d x := by
-  apply (butterfly_endpoint_iff d _ _ _).mpr
-  intro t ht
-  have ht' := mirror_routeDomain d (reverseCard d y) (reverseCard d x) t ht
-  simp only [reverseCard_involutive] at ht'
-  have he := (butterfly_endpoint_iff d ω x y).mp hxy (mirrorSwitch d t) ht'
-  change ω (mirrorSwitch d t) = _
-  rw [he]
-  have hv := mirror_routeValue d x y (mirrorSwitch d t)
-  rw [mirrorSwitch_involutive] at hv
-  exact hv.symm
-
-lemma butterfly_mirror (d : ℕ) (ω : SwitchIndex d → Bool) :
-    butterflyPerm d (decodeButterfly d (mirrorCoins d ω)) =
-      reverseCard d * (butterflyPerm d (decodeButterfly d ω))⁻¹ * reverseCard d := by
-  apply Equiv.ext
-  intro x
-  have he := mirror_endpoint d ω
-    ((butterflyPerm d (decodeButterfly d ω))⁻¹ (reverseCard d x)) (reverseCard d x)
-    (Equiv.apply_symm_apply _ _)
-  simpa only [reverseCard_involutive,Equiv.Perm.mul_apply] using he
-
-def reverseOrder (d : ℕ) : Equiv.Perm (Fin d) where
-  toFun := Fin.rev
-  invFun := Fin.rev
-  left_inv := Fin.rev_rev
-  right_inv := Fin.rev_rev
-
-lemma relabel_mul (d : ℕ) (a b : Equiv.Perm (Fin d)) :
-    StrongSmoothing.coordinateRelabel d (a*b) =
-      StrongSmoothing.coordinateRelabel d a * StrongSmoothing.coordinateRelabel d b := rfl
-
-lemma relabel_reverse (d : ℕ) :
-    StrongSmoothing.coordinateRelabel d (reverseOrder d) = reverseCard d := rfl
-
-lemma reverseCard_sq (d : ℕ) : reverseCard d * reverseCard d = 1 := by
-  ext x i
-  simp
-
-lemma sweep_mirror (d : ℕ) (order : Equiv.Perm (Fin d)) (ω : SwitchIndex d → Bool) :
-    StrongSmoothing.sweep d (order*reverseOrder d) (mirrorCoins d ω) =
-      (StrongSmoothing.sweep d order ω)⁻¹ := by
-  simp only [StrongSmoothing.sweep,relabel_mul,relabel_reverse,butterfly_mirror,mul_inv_rev]
-  have hR : (reverseCard d)⁻¹ = reverseCard d := by
-    exact inv_eq_of_mul_eq_one_right (reverseCard_sq d)
-  rw [hR]
-  calc
-    _ = StrongSmoothing.coordinateRelabel d order *
-        (reverseCard d*reverseCard d) *
-        (butterflyPerm d (decodeButterfly d ω))⁻¹ *
-        (reverseCard d*reverseCard d) * (StrongSmoothing.coordinateRelabel d order)⁻¹ := by group
-    _ = _ := by rw [reverseCard_sq]; group
-
-lemma sweepKernel_reverse (d l : ℕ) (order : Equiv.Perm (Fin d)) :
-    StrongSmoothing.sweepKernel d l (order*reverseOrder d) =
-      (StrongSmoothing.sweepKernel d l order).transpose := by
-  classical
-  rw [StrongSmoothing.sweepKernel_law,StrongSmoothing.sweepKernel_law,
-    ←StrongSmoothing.PermLaw.inverse]
-  ext x y
-  unfold StrongSmoothing.PermLaw.kernel
-  rw [finiteMean_equiv (mirrorCoins d)]
-  apply finiteMean_congr
-  intro ω
-  change (if StrongSmoothing.tuplePerm
-      (StrongSmoothing.sweep d (order*reverseOrder d) (mirrorCoins d ω)) x = y then (1:ℝ) else 0) = _
-  rw [sweep_mirror,StrongSmoothing.tuplePerm_inverse]
-
-lemma reflectedKernel_reverse (d l : ℕ) (order : Equiv.Perm (Fin d)) :
-    StrongSmoothing.reflectedKernel d l order true =
-      StrongSmoothing.reflectedKernel d l (order*reverseOrder d) false := by
-  simp only [StrongSmoothing.reflectedKernel,Bool.false_eq_true,
-    ↓reduceIte,sweepKernel_reverse,Matrix.transpose_transpose]
-
-end Thorp.ButterflyReverse
-
-namespace Thorp.StrongTail
-open scoped BigOperators Classical
-
-section
-open Specht
-
-lemma mem_tailDiagram (μ : YoungDiagram) (i j : ℕ) :
-    (i,j) ∈ tailDiagram μ ↔ (i+1,j) ∈ μ := by
-  rw [tailDiagram, YoungDiagram.mem_ofRowLens]
-  constructor
-  · rintro ⟨hi,hj⟩
-    rw [List.getElem_tail,YoungDiagram.get_rowLens] at hj
-    exact YoungDiagram.mem_iff_lt_rowLen.mpr hj
-  · intro h
-    have hi : i+1 < μ.colLen 0 :=
-      (YoungDiagram.mem_iff_lt_colLen.mp h).trans_le (μ.colLen_anti 0 j (Nat.zero_le _))
-    have hi' : i < μ.rowLens.tail.length := by simp only [List.length_tail,YoungDiagram.length_rowLens]; omega
-    refine ⟨hi', ?_⟩
-    simpa only [List.getElem_tail,YoungDiagram.get_rowLens] using YoungDiagram.mem_iff_lt_rowLen.mp h
-
-lemma tail_height (μ : YoungDiagram) : (tailDiagram μ).colLen 0 = μ.colLen 0 - 1 := by
-  rw [←YoungDiagram.length_rowLens]
-  exact (YoungDiagram.rowLens_length_ofRowLens (fun x hx => μ.pos_of_mem_rowLens x (List.mem_of_mem_tail hx))).trans
-    (by simp only [List.length_tail,YoungDiagram.length_rowLens])
-
-noncomputable def tailCellEquiv (μ : YoungDiagram) : Cell (tailDiagram μ) ≃ Tail μ where
-  toFun x := ⟨⟨(x.1.1+1,x.1.2),(mem_tailDiagram μ _ _).mp x.2⟩,by simp [row]⟩
-  invFun x := ⟨(row x.1-1,col x.1),(mem_tailDiagram μ _ _).mpr (by
-    have he : row x.1-1+1 = row x.1 := by have := x.2; omega
-    change (row x.1-1+1,col x.1) ∈ μ
-    rw [he]
-    exact x.1.2)⟩
-  left_inv x := by apply Subtype.ext; simp [row,col]
-  right_inv x := by
-    apply Subtype.ext; apply Subtype.ext; apply Prod.ext
-    · change row x.1-1+1 = row x.1
-      have := x.2; omega
-    · rfl
-
-noncomputable abbrev upCell (μ : YoungDiagram) (x : Cell (tailDiagram μ)) : Cell μ := (tailCellEquiv μ x).1
-
-@[simp] lemma upCell_row (μ : YoungDiagram) (x : Cell (tailDiagram μ)) : row (upCell μ x) = row x+1 := rfl
-@[simp] lemma upCell_col (μ : YoungDiagram) (x : Cell (tailDiagram μ)) : col (upCell μ x) = col x := rfl
-
-lemma tail_card_eq (μ : YoungDiagram) : (tailDiagram μ).card = μ.card-μ.rowLen 0 := by
-  rw [←card_cell, Fintype.card_congr (tailCellEquiv μ), tail_card]
-
-noncomputable def extendPerm (μ : YoungDiagram) (p : Equiv.Perm (Cell (tailDiagram μ))) :
-    Equiv.Perm (Cell μ) := p.extendDomain (tailCellEquiv μ)
-
-@[simp] lemma extendPerm_up (μ : YoungDiagram) (p : Equiv.Perm (Cell (tailDiagram μ)))
-    (x : Cell (tailDiagram μ)) : extendPerm μ p (upCell μ x) = upCell μ (p x) :=
-  Equiv.Perm.extendDomain_apply_image p (tailCellEquiv μ) x
-
-lemma extendPerm_top (μ : YoungDiagram) (p : Equiv.Perm (Cell (tailDiagram μ)))
-    (x : Cell μ) (hx : row x=0) : extendPerm μ p x = x :=
-  Equiv.Perm.extendDomain_apply_not_subtype p (tailCellEquiv μ) (not_not.mpr hx)
-
-@[simp] lemma extendPerm_inv (μ : YoungDiagram) (p : Equiv.Perm (Cell (tailDiagram μ))) :
-    (extendPerm μ p)⁻¹ = extendPerm μ p⁻¹ := Equiv.Perm.extendDomain_symm _ _
-
-@[simp] lemma extendPerm_sign (μ : YoungDiagram) (p : Equiv.Perm (Cell (tailDiagram μ))) :
-    permSign (extendPerm μ p) = permSign p := by
-  simp [extendPerm,permSign,Equiv.Perm.sign_extendDomain]
-
-noncomputable def liftRows (μ : YoungDiagram) (f : Tabloid (tailDiagram μ)) (x : Cell μ) : Fin (μ.colLen 0) :=
-  if hx : row x ≠ 0 then
-    ⟨(f.1 ((tailCellEquiv μ).symm ⟨x,hx⟩)).val+1,by
-      have h := (f.1 ((tailCellEquiv μ).symm ⟨x,hx⟩)).isLt
-      have hh := tail_height μ
-      omega⟩
-  else rowIndex x
-
-@[simp] lemma liftRows_up (μ : YoungDiagram) (f : Tabloid (tailDiagram μ))
-    (x : Cell (tailDiagram μ)) : (liftRows μ f (upCell μ x)).val = (f.1 x).val+1 := by
-  simp [liftRows,upCell]
-
-lemma liftRows_top (μ : YoungDiagram) (f : Tabloid (tailDiagram μ)) (x : Cell μ) (hx : row x=0) :
-    liftRows μ f x = rowIndex x := by simp [liftRows,hx]
-
-lemma liftRows_perm (μ : YoungDiagram) (f : Tabloid (tailDiagram μ))
-    (p : Equiv.Perm (Cell (tailDiagram μ))) (hf : f.1 = rowIndex ∘ p) :
-    liftRows μ f = rowIndex ∘ extendPerm μ p := by
-  funext x
-  by_cases hx : row x = 0
-  · simp [liftRows_top μ f x hx,extendPerm_top μ p x hx]
-  · let y := (tailCellEquiv μ).symm ⟨x,hx⟩
-    have he : upCell μ y = x := congrArg Subtype.val ((tailCellEquiv μ).apply_symm_apply ⟨x,hx⟩)
-    rw [←he]
-    apply Fin.ext
-    simp [hf,Function.comp_apply,rowIndex]
-
-noncomputable def liftTabloid (μ : YoungDiagram) (f : Tabloid (tailDiagram μ)) : Tabloid μ :=
-  ⟨liftRows μ f, by obtain ⟨p,hp⟩ := f.2; exact ⟨extendPerm μ p,liftRows_perm μ f p hp⟩⟩
-
-@[simp] lemma liftTabloid_base (μ : YoungDiagram) :
-    liftTabloid μ (baseTabloid (tailDiagram μ)) = baseTabloid μ := by
-  apply Subtype.ext
-  funext x
-  by_cases hx : row x=0
-  · exact liftRows_top μ _ x hx
-  · let y := (tailCellEquiv μ).symm ⟨x,hx⟩
-    have he : upCell μ y = x := congrArg Subtype.val ((tailCellEquiv μ).apply_symm_apply ⟨x,hx⟩)
-    rw [←he]
-    apply Fin.ext
-    exact liftRows_up μ _ y
-
-lemma liftTabloid_action (μ : YoungDiagram) (p : Equiv.Perm (Cell (tailDiagram μ)))
-    (f : Tabloid (tailDiagram μ)) :
-    liftTabloid μ (tabloidAct p f) = tabloidAct (extendPerm μ p) (liftTabloid μ f) := by
-  apply Subtype.ext
-  funext x
-  change liftRows μ (tabloidAct p f) x = liftRows μ f ((extendPerm μ p)⁻¹ x)
-  rw [extendPerm_inv]
-  by_cases hx : row x=0
-  · rw [extendPerm_top μ p⁻¹ x hx,liftRows_top μ _ x hx,liftRows_top μ _ x hx]
-  · let y := (tailCellEquiv μ).symm ⟨x,hx⟩
-    have he : upCell μ y = x := congrArg Subtype.val ((tailCellEquiv μ).apply_symm_apply ⟨x,hx⟩)
-    rw [←he,extendPerm_up]
-    apply Fin.ext
-    simp only [liftRows_up,tabloidAct_apply]
-
-noncomputable def restrictTabloids (μ : YoungDiagram) :
-    (Tabloid μ → ℂ) →ₗ[ℂ] (Tabloid (tailDiagram μ) → ℂ) where
-  toFun v := fun f => v (liftTabloid μ f)
-  map_add' := by intros; rfl
-  map_smul' := by intros; rfl
-
-lemma restrict_action (μ : YoungDiagram) (p : Equiv.Perm (Cell (tailDiagram μ)))
-    (v : Tabloid μ → ℂ) :
-    restrictTabloids μ (tabloidRep μ (extendPerm μ p) v) =
-      tabloidRep (tailDiagram μ) p (restrictTabloids μ v) := by
-  ext f
-  change v (tabloidAct (extendPerm μ p)⁻¹ (liftTabloid μ f)) = v (liftTabloid μ (tabloidAct p⁻¹ f))
-  rw [extendPerm_inv,liftTabloid_action]
-
-lemma extend_colGroup (μ : YoungDiagram) (c : colGroup (tailDiagram μ)) :
-    extendPerm μ c ∈ colGroup μ := by
-  intro x
-  by_cases hx : row x=0
-  · rw [extendPerm_top μ c x hx]
-  · let y := (tailCellEquiv μ).symm ⟨x,hx⟩
-    have he : upCell μ y = x := congrArg Subtype.val ((tailCellEquiv μ).apply_symm_apply ⟨x,hx⟩)
-    rw [←he,extendPerm_up,upCell_col,upCell_col]
-    exact c.2 y
-
-lemma restrict_polytabloid (μ : YoungDiagram) :
-    restrictTabloids μ (polytabloid μ) = polytabloid (tailDiagram μ) := by
-  let v := restrictTabloids μ (polytabloid μ)
-  have hv (c : colGroup (tailDiagram μ)) :
-      tabloidRep (tailDiagram μ) c v = permSign (c:Equiv.Perm (Cell (tailDiagram μ))) • v := by
-    rw [←restrict_action]
-    rw [polytabloid_col_action μ ⟨extendPerm μ c,extend_colGroup μ c⟩]
-    simp only [map_smul,extendPerm_sign]
-    rfl
-  have ha : alternator (colGroup (tailDiagram μ)) (tabloidRep (tailDiagram μ)) v =
-      (Fintype.card (colGroup (tailDiagram μ)):ℂ) • v := by
-    simp only [alternator,LinearMap.sum_apply,LinearMap.smul_apply,hv,smul_smul,permSign_sq,one_smul]
-    ext f
-    simp
-  have hb : v (baseTabloid (tailDiagram μ)) = 1 := by
-    change polytabloid μ (liftTabloid μ (baseTabloid (tailDiagram μ))) = 1
-    rw [liftTabloid_base, polytabloid_base]
-  have he := alternator_rank_one_exact (tailDiagram μ) v
-  rw [ha] at he
-  simp only [Pi.smul_apply,hb,smul_eq_mul,mul_one] at he
-  have hc : (Fintype.card (colGroup (tailDiagram μ)):ℂ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
-  exact (smul_right_injective _ hc) he
-
-lemma tail_space_le_image (μ : YoungDiagram) :
-    space (tailDiagram μ) ≤ (space μ).map (restrictTabloids μ) := by
-  apply Submodule.span_le.mpr
-  rintro _ ⟨p,rfl⟩
-  refine ⟨tabloidRep μ (extendPerm μ p) (polytabloid μ), orbit_mem_space μ _, ?_⟩
-  rw [restrict_action,restrict_polytabloid]
-
-theorem tail_dimension_le (μ : YoungDiagram) :
-    Module.finrank ℂ (space (tailDiagram μ)) ≤ Module.finrank ℂ (space μ) :=
-  (Submodule.finrank_mono (tail_space_le_image μ)).trans (Submodule.finrank_map_le _ _)
-
-end
-open StrongSmoothing
-
-def DensityAt (p C : ℝ) : Prop :=
-  ∀ (d l : ℕ), l ≤ 2^d → ∀ (order : Equiv.Perm (Fin d)) (rev : Bool)
-    (x : StrongSmoothing.Tuples d l),
-    let H := StrongSmoothing.reflectedKernel d l order rev
-    (Thorp.finiteMean (fun y => ((Fintype.card (StrongSmoothing.Tuples d l):ℝ)*H x y)^p)
-      ≤ Real.exp (C*l)) ∧
-    (Thorp.finiteMean (fun y => ((Fintype.card (StrongSmoothing.Tuples d l):ℝ)*H y x)^p)
-      ≤ Real.exp (C*l)) ∧
-    (((2:ℝ)^d)^l)⁻¹ * (∑ y, ((((2:ℝ)^d)^l)*H x y)^p) ≤ Real.exp (C*l) ∧
-    (((2:ℝ)^d)^l)⁻¹ * (∑ y, ((((2:ℝ)^d)^l)*H y x)^p) ≤ Real.exp (C*l)
-
-lemma reflected_nonneg_both (d l : ℕ) (order : Equiv.Perm (Fin d)) (rev : Bool)
-    (x y : Tuples d l) : 0 ≤ reflectedKernel d l order rev x y := by
-  cases rev
-  · exact reflected_nonneg _ _ _ _ _
-  · rw [ButterflyReverse.reflectedKernel_reverse]
-    exact reflected_nonneg _ _ _ _ _
-
-lemma reflected_symm_both (d l : ℕ) (order : Equiv.Perm (Fin d)) (rev : Bool) :
-    (reflectedKernel d l order rev).IsSymm := by
-  cases rev
-  · exact reflected_symmetric _ _ _
-  · rw [ButterflyReverse.reflectedKernel_reverse]
-    exact reflected_symmetric _ _ _
-
-lemma reflected_moment_both (d l : ℕ) (order : Equiv.Perm (Fin d)) (rev : Bool)
-    (x : Tuples d l) :
-    finiteMean (fun y : Tuples d l =>
-      ((Fintype.card (Tuples d l):ℝ)*reflectedKernel d l order rev x y)^(1+(1/1000:ℝ))) ≤
-      Real.exp (l*(1+heightDecay 1)) := by
-  cases rev
-  · exact reflected_moment _ _ _ _
-  · rw [ButterflyReverse.reflectedKernel_reverse]
-    exact reflected_moment _ _ _ _
-
-lemma reflected_moment_small (d l : ℕ) (order : Equiv.Perm (Fin d)) (rev : Bool)
-    (x : Tuples d l) {p : ℝ} (hp : 0 < p) (hpq : p ≤ 1+(1/1000:ℝ)) :
-    finiteMean (fun y : Tuples d l =>
-      ((Fintype.card (Tuples d l):ℝ)*reflectedKernel d l order rev x y)^p) ≤
-      Real.exp (l*(1+heightDecay 1)) := by
-  let : Nonempty (Tuples d l) := ⟨x⟩
-  let q : ℝ := 1+1/1000
-  let f := fun y : Tuples d l => (Fintype.card (Tuples d l):ℝ)*reflectedKernel d l order rev x y
-  have hf : ∀ y, 0 ≤ f y := fun y => mul_nonneg (Nat.cast_nonneg _) (reflected_nonneg_both _ _ _ _ _ _)
-  have hq : 0 < q := by norm_num [q]
-  have hr : 0 ≤ p/q := (div_pos hp hq).le
-  have hr1 : p/q ≤ 1 := (div_le_one hq).mpr hpq
-  have he (y) : (f y)^p = ((f y)^q)^(p/q) := by
-    rw [←Real.rpow_mul (hf y)]
-    congr 1
-    field_simp
-  simp_rw [show (fun y : Tuples d l => ((Fintype.card (Tuples d l):ℝ)*reflectedKernel d l order rev x y)^p)
-    = fun y => ((f y)^q)^(p/q) from funext he]
-  apply (mean_rpow_concave _ (fun y => Real.rpow_nonneg (hf y) _) hr hr1).trans
-  apply (Real.rpow_le_rpow (finiteMean_nonneg (fun y => Real.rpow_nonneg (hf y) _))
-    (reflected_moment_both d l order rev x) hr).trans
-  rw [Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp]
-  apply Real.exp_le_exp.mpr
-  have hA : 0 ≤ (l:ℝ)*(1+heightDecay 1) := by
-    have hh := heightDecay_nonneg 1
-    positivity
-  nlinarith
-
-lemma log_factorial_le_power (n N : ℕ) (hn : n ≤ N) (_ : 0 < N) :
-    Real.log (n.factorial:ℝ) ≤ (n:ℝ)*Real.log N := by
-  have hb : (n.factorial:ℝ) ≤ (N:ℝ)^n := by
-    exact_mod_cast (Nat.factorial_le_pow n).trans (Nat.pow_le_pow_left hn n)
-  simpa only [Real.log_pow] using Real.log_le_log (by positivity : (0:ℝ)<n.factorial) hb
-
-lemma log_normalization_le (N l : ℕ) (hN : 0 < N) (hl : l ≤ N) :
-    Real.log ((N:ℝ)^l/(N.descFactorial l:ℝ)) ≤ 2*l := by
-  have hNp : (0:ℝ)<N := by exact_mod_cast hN
-  have hMp : (0:ℝ)<N.descFactorial l := by exact_mod_cast Nat.descFactorial_pos.mpr hl
-  rw [Real.log_div (by positivity) hMp.ne',Real.log_pow]
-  by_cases hlow : 2*l ≤ N
-  · have hhalf : (N:ℝ)/2 ≤ (N+1-l:ℕ) := by
-      have hNat : N ≤ 2*(N+1-l) := by omega
-      exact (div_le_iff₀ (by norm_num : (0:ℝ)<2)).mpr (by simpa only [mul_comm] using (show (N:ℝ) ≤ 2*(N+1-l:ℕ) by exact_mod_cast hNat))
-    have hb : ((N:ℝ)/2)^l ≤ (N.descFactorial l:ℝ) := by
-      apply (pow_le_pow_left₀ (by positivity) hhalf l).trans
-      exact_mod_cast Nat.pow_sub_le_descFactorial N l
-    have hh := Real.log_le_log (pow_pos (by positivity : (0:ℝ)<(N:ℝ)/2) l) hb
-    rw [Real.log_pow,Real.log_div hNp.ne' (by norm_num)] at hh
-    have ht : Real.log (2:ℝ) ≤ 1 := by simpa only [show (2:ℝ)-1=1 by norm_num] using Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
-    nlinarith [(show (0:ℝ) ≤ l by positivity)]
-  · have he : (N.descFactorial l:ℝ)*(Nat.factorial (N-l):ℝ) = N.factorial := by
-      have he := Nat.factorial_mul_descFactorial hl
-      exact_mod_cast (by simpa only [mul_comm] using he : N.descFactorial l * (N-l).factorial = N.factorial)
-    have hh := congrArg Real.log he
-    rw [Real.log_mul hMp.ne' (by positivity)] at hh
-    have hfull := log_factorial_lower (show 1 ≤ N by omega)
-    have hrem := log_factorial_le_power (N-l) N (Nat.sub_le _ _) hN
-    rw [Nat.cast_sub hl] at hrem
-    have hNle : (N:ℝ) ≤ 2*l := by exact_mod_cast (by omega : N ≤ 2*l)
-    linarith
-
-lemma normalized_moment_identity {X : Type*} [Fintype X] [Nonempty X]
-    (f : X → ℝ) (hf : ∀ x, 0 ≤ f x) {N : ℝ} (hN : 0 < N) (p : ℝ) :
-    N⁻¹*(∑ x, (N*f x)^p) =
-     (N/(Fintype.card X:ℝ))^(p-1) * finiteMean (fun x => ((Fintype.card X:ℝ)*f x)^p) := by
-  let M : ℝ := Fintype.card X
-  have hM : 0 < M := Nat.cast_pos.mpr Fintype.card_pos
-  have he (x) : (N*f x)^p = (N/M)^p*(M*f x)^p := by
-    rw [←Real.mul_rpow (div_nonneg hN.le hM.le) (mul_nonneg hM.le (hf x))]
-    congr 1
-    field_simp
-  simp_rw [he]
-  rw [←Finset.mul_sum,Real.rpow_sub (div_pos hN hM),Real.rpow_one]
-  unfold finiteMean
-  dsimp [M] at hM ⊢
-  field_simp
-
-lemma density_at_small_exponent {p : ℝ} (hp : 1 < p) (hpq : p ≤ 1+(1/1000:ℝ)) :
-    DensityAt p (3+heightDecay 1) := by
-  intro d l hl order rev x
-  dsimp only
-  let : Nonempty (Tuples d l) := ⟨x⟩
-  let H := reflectedKernel d l order rev
-  have hm := reflected_moment_small d l order rev x (by linarith : 0<p) hpq
-  have hb : Real.exp ((l:ℝ)*(1+heightDecay 1)) ≤ Real.exp ((3+heightDecay 1)*l) := by
-    apply Real.exp_le_exp.mpr
-    nlinarith [(show (0:ℝ) ≤ l by positivity)]
-  have hs (y) : reflectedKernel d l order rev y x = reflectedKernel d l order rev x y := (reflected_symm_both d l order rev).apply _ _
-  have hpow : (((2:ℝ)^d)^l)⁻¹ * (∑ y, ((((2:ℝ)^d)^l)*H x y)^p) ≤
-      Real.exp ((3+heightDecay 1)*l) := by
-    rw [normalized_moment_identity (fun y => H x y) (reflected_nonneg_both d l order rev x)
-      (by positivity) p]
-    have hrat : ((2:ℝ)^d)^l/(Fintype.card (Tuples d l):ℝ) ≤ Real.exp (2*l) := by
-      apply (Real.log_le_iff_le_exp (by positivity)).mp
-      rw [tuples_card]
-      exact_mod_cast log_normalization_le (2^d) l (by positivity) hl
-    have hr : 0 ≤ ((2:ℝ)^d)^l/(Fintype.card (Tuples d l):ℝ) := by positivity
-    have hrp : (((2:ℝ)^d)^l/(Fintype.card (Tuples d l):ℝ))^(p-1) ≤ Real.exp (2*l) := by
-      apply (Real.rpow_le_rpow hr hrat (by linarith)).trans
-      rw [Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp]
-      apply Real.exp_le_exp.mpr
-      have hpl : p-1 ≤ 1 := by linarith
-      nlinarith [(show (0:ℝ) ≤ l by positivity)]
-    apply (mul_le_mul hrp hm (finiteMean_nonneg (fun y => Real.rpow_nonneg
-      (mul_nonneg (Nat.cast_nonneg _) (reflected_nonneg_both d l order rev x y)) _)) (Real.exp_pos _).le).trans
-    rw [←Real.exp_add]
-    apply Real.exp_le_exp.mpr
-    ring_nf
-    rfl
-  dsimp [H] at hpow
-  refine ⟨hm.trans hb,?_,hpow,?_⟩
-  · simpa only [hs] using hm.trans hb
-  · simpa only [hs] using hpow
-
-end Thorp.StrongTail
-
-namespace Thorp.Casimir
-open scoped BigOperators Classical
-
-section
-open Filter
-
-noncomputable def traceMoment (d : ℕ) (μ : YoungDiagram)
-    (e : Card d ≃ Specht.Cell μ) (r : ℕ) : ℝ :=
-  (LinearMap.trace ℂ (Specht.hilbertSpace μ) ((K d μ e)^r).toLinearMap).re
-
-noncomputable def fromDistance (d t : ℕ) (σ : Equiv.Perm (Card d)) : ℝ :=
-  (1/2:ℝ) * ∑ g : Equiv.Perm (Card d),
-    |law d t (g*σ⁻¹) - 1/(Nat.factorial (2^d):ℝ)|
-
-def MainStatement : Prop :=
-  ∃ a C : ℝ, 0 < a ∧ 0 < C ∧
-    (∀ d k : ℕ, 1 ≤ k → k ≤ 2^d → ∀ x : Fin k ↪ Card d,
-      (palindromeRowMoment d x (1/64))^(64/65:ℝ) ≤
-        Real.exp (C*levelScale (2^d) k)) ∧
-    (∀ (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ),
-      0 < μ.card-μ.rowLen 0 →
-        ‖K d μ e‖ ≤ Real.exp (-a*levelScale (2^d) (μ.card-μ.rowLen 0)) ∧
-        (Module.finrank ℂ (Specht.space μ):ℝ) * traceMoment d μ e 65 ≤
-          Real.exp (C*levelScale (2^d) (μ.card-μ.rowLen 0))) ∧
-    (∃ l : ℕ, 0 < l ∧ ∀ starts : (d : ℕ) → Equiv.Perm (Card d),
-      Tendsto (fun d => fromDistance d (l*d) (starts d)) atTop (nhds 0))
-
-end
-
-theorem cycleCoefficient_decay (l r : ℕ) (hr : (l:ℝ)/2 ≤ r)
-    (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ (l:ℝ)/16) :
-    cycleCoefficient r ⌈Real.exp ((l:ℝ)/32)⌉₊ q ≤
-      68 * Real.exp (-(l:ℝ)/64) := by
-  let E := Real.exp ((l:ℝ)/32)
-  let J := ⌈E⌉₊
-  have hl : (0:ℝ) ≤ l := Nat.cast_nonneg _
-  have hE : 1 ≤ E := Real.one_le_exp_iff.mpr (by positivity)
-  have hEpos : 0 < E := Real.exp_pos _
-  have hEJ : E ≤ (J:ℝ) := Nat.le_ceil E
-  have hJ : (1:ℝ) ≤ J := hE.trans hEJ
-  have hJpos : (0:ℝ) < J := lt_of_lt_of_le (by norm_num) hJ
-  have hJE : (J:ℝ) ≤ 2*E := by
-    have h := Nat.ceil_lt_add_one (le_of_lt hEpos)
-    change (J:ℝ) < E+1 at h
-    linarith
-  have hsqJ : Real.sqrt J ≤ (J:ℝ) := by
-    apply Real.sqrt_le_iff.mpr
-    constructor
-    · positivity
-    · nlinarith
-  have hqp : 0 < q := lt_of_lt_of_le (by norm_num) hq
-  have hqE : q ≤ Real.exp ((l:ℝ)/16) := by
-    rw [←Real.exp_log hqp]
-    exact Real.exp_le_exp.mpr hlog
-  have hln : (1/2:ℝ) ≤ Real.log 2 := by
-    have hh := Real.one_sub_inv_le_log_of_pos (by norm_num : (0:ℝ)<2)
-    norm_num at hh ⊢
-    exact hh
-  have hden : Real.exp ((l:ℝ)/8) ≤ Real.sqrt ((2:ℝ)^r) := by
-    have he : (2:ℝ)^r = Real.exp ((r:ℝ)*Real.log 2) := by
-      rw [Real.exp_nat_mul,Real.exp_log (by norm_num)]
-    rw [he,Real.sqrt_eq_rpow,Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp]
-    apply Real.exp_le_exp.mpr
-    have hrn : (0:ℝ) ≤ r := Nat.cast_nonneg _
-    nlinarith
-  have hlog' : Real.log q ≤ (l:ℝ) := by linarith
-  have h₁ : Real.log q / (J:ℝ) ≤ 64*Real.exp (-(l:ℝ)/64) := by
-    calc
-      _ ≤ (l:ℝ)/E := div_le_div₀ (by positivity) hlog' hEpos hEJ
-      _ ≤ (64*Real.exp ((l:ℝ)/64))/E := by
-        apply div_le_div_of_nonneg_right _ (le_of_lt hEpos)
-        have hh := Real.add_one_le_exp ((l:ℝ)/64)
-        linarith
-      _ = _ := by
-        dsimp [E]
-        rw [mul_div_assoc,←Real.exp_sub]
-        congr 2
-        ring
-  have h₂ : 2*q*Real.sqrt J / Real.sqrt ((2:ℝ)^r) ≤
-      4*Real.exp (-(l:ℝ)/64) := by
-    calc
-      _ ≤ (4*Real.exp ((l:ℝ)/16)*E)/Real.exp ((l:ℝ)/8) := by
-        apply div_le_div₀ (by positivity) _ (Real.exp_pos _) hden
-        have hs := hsqJ.trans hJE
-        have hm := mul_le_mul hqE hs (Real.sqrt_nonneg _) (Real.exp_nonneg _)
-        nlinarith
-      _ = 4*Real.exp ((l:ℝ)/16+(l:ℝ)/32-(l:ℝ)/8) := by
-        dsimp [E]
-        rw [mul_assoc,←Real.exp_add,mul_div_assoc,←Real.exp_sub]
-      _ ≤ _ := by
-        apply mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr _) (by norm_num)
-        linarith
-  change Real.log q / (J:ℝ) + 2*q*Real.sqrt J / Real.sqrt ((2:ℝ)^r) ≤ _
-  linarith
-
-theorem palindrome_slab_partial_mgf (r s l : ℕ) (hl : 0 < l) (hs : 2*s ≤ l)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+r)) (X : SwitchIndex (s+r) → Bool)
-    (lo : Card r × SwitchIndex s → Bool) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/16) :
-    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(palindromeSlabCost l (s+r) e (assembleBits r s lo hi,X))) ≤
-      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
-  classical
-  let : NeZero l := ⟨ne_of_gt hl⟩
-  have hqp : 0 < q := lt_of_lt_of_le (by norm_num) hq
-  calc
-    _ ≤ finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      finiteMean (fun i : Fin l => q^(l*palindromeLevelCost (l+i.1+1) (s+r) e
-        (assembleBits r s lo hi,X)))) := by
-      apply finiteMean_mono
-      intro hi
-      simpa only [palindromeSlabCost,Fintype.card_fin] using
-        pow_sum_le_mean_powers (fun i : Fin l => palindromeLevelCost (l+i.1+1) (s+r) e
-          (assembleBits r s lo hi,X)) q hqp
-    _ = finiteMean (fun i : Fin l => finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(l*palindromeLevelCost (l+i.1+1) (s+r) e (assembleBits r s lo hi,X)))) := finiteMean_comm _
-    _ ≤ finiteMean (fun _i : Fin l =>
-      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64)))) := by
-      apply finiteMean_mono
-      intro i
-      by_cases ht : l+i.1+1 ≤ s+r
-      · have hs' : s ≤ l+i.1 := by omega
-        have he : l+i.1+1 = s+(l+i.1-s)+1 := by omega
-        have hu : l+i.1-s < r := by omega
-        have hm := palindrome_level_partial_mgf r s (l+i.1-s) hu e X lo
-          ⌈Real.exp ((l:ℝ)/32)⌉₊ (cycleCutoff_pos l) (q^l) (one_le_pow₀ hq)
-        have hc := cycleCoefficient_decay l (l+i.1-s) (by
-          rw [Nat.cast_sub hs',Nat.cast_add]
-          have hsr : (2:ℝ)*s ≤ l := by exact_mod_cast hs
-          have hi : (0:ℝ) ≤ i.1 := Nat.cast_nonneg _
-          linarith) (q^l) (one_le_pow₀ hq) (by
-            rw [Real.log_pow]
-            nlinarith [mul_le_mul_of_nonneg_left hlog (Nat.cast_nonneg l)])
-        simp_rw [he,pow_mul]
-        exact hm.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left hc (Nat.cast_nonneg _)))
-      · have hz := palindromeLevelCost_gt (l+i.1+1) (s+r) (by omega) e
-        simp only [hz,Nat.mul_zero,pow_zero,finiteMean_const]
-        exact Real.one_le_exp_iff.mpr (by positivity)
-    _ = _ := finiteMean_const _
-
-lemma palindrome_slab_adapted_mul_le (d s l : ℕ) (hsd : s ≤ d) (hl : 0 < l)
-    (hs : 2*s ≤ l) {ι : Type*} [Fintype ι] (e : ι ↪ Card d)
-    (X : SwitchIndex d → Bool) (F : (SwitchIndex d → Bool) → ℝ)
-    (hF : ∀ Y, 0 ≤ F Y) (hA : LayerAdapted d s F) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/16) :
-    finiteMean (fun Y => F Y*q^(palindromeSlabCost l d e (Y,X))) ≤
-      finiteMean F * Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
-  obtain ⟨r,rfl⟩ := Nat.exists_eq_add_of_le hsd
-  exact adapted_mul_mean_le r s F _ hF hA _
-    (fun lo => palindrome_slab_partial_mgf r s l hl hs e X lo q hq hlog)
-
-theorem palindrome_family_mgf (L n d : ℕ) (hL : 0 < L) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/16) :
-    finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) ≤
-      Real.exp (Fintype.card ι * (68*∑ j ∈ Finset.range n,
-        Real.exp (-((L*4^j:ℕ):ℝ)/64))) := by
-  induction n with
-  | zero => simp only [palindromeFamilyCost,Finset.range_zero,Finset.sum_empty,pow_zero,
-      finiteMean_const,mul_zero,Real.exp_zero,le_refl]
-  | succ n ih =>
-    let l := L*4^n
-    have hl : 0 < l := Nat.mul_pos hL (pow_pos (by decide) _)
-    have hs : 2*familyEnd L n ≤ l := familyEnd_two L n
-    have hc : ∀ Y, palindromeFamilyCost L (n+1) d e (Y,X) =
-        palindromeFamilyCost L n d e (Y,X)+palindromeSlabCost l d e (Y,X) := by
-      intro Y
-      exact Finset.sum_range_succ _ _
-    have hm : finiteMean (fun Y => q^(palindromeFamilyCost L (n+1) d e (Y,X))) ≤
-        finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) *
-          Real.exp (Fintype.card ι*(68*Real.exp (-(l:ℝ)/64))) := by
-      by_cases hld : l ≤ d
-      · simp_rw [hc,pow_add]
-        exact palindrome_slab_adapted_mul_le d (familyEnd L n) l (by omega) hl hs e X _
-          (fun _ => pow_nonneg (by linarith) _)
-          (layerAdapted_comp (palindromeFamilyCost_adapted L n d e X) (fun c => q^c))
-          q hq hlog
-      · simp_rw [hc,palindromeSlabCost_gt l d (by omega) e,Nat.add_zero]
-        exact le_mul_of_one_le_right
-          (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
-          (Real.one_le_exp_iff.mpr (by positivity))
-    refine (hm.trans (mul_le_mul_of_nonneg_right ih (Real.exp_nonneg _))).trans_eq ?_
-    rw [←Real.exp_add,Finset.sum_range_succ]
-    congr 1
-    dsimp [l]
-    ring
-
-lemma palindrome_family_mgf_decay (L n d : ℕ) (hL : 0 < L) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/16) :
-    finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) ≤
-      Real.exp (Fintype.card ι*heightDecay L) := by
-  apply (palindrome_family_mgf L n d hL e X q hq hlog).trans
-  apply Real.exp_le_exp.mpr
-  apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
-  have h := mul_le_mul_of_nonneg_left (sum_family_decay L n hL) (by norm_num : (0:ℝ)≤68)
-  simpa only [heightDecay,mul_div_assoc] using h
-
-theorem palindrome_high_mgf (H n d : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/32) :
-    finiteMean (fun Y => q^(palindromeFamilyCost H n d e (Y,X)+
-      palindromeFamilyCost (2*H) n d e (Y,X))) ≤ Real.exp (Fintype.card ι*heightDecay H) := by
-  have hq2 := one_le_pow₀ (n:=2) hq
-  have hlog2 : Real.log (q^2) ≤ 1/16 := by rw [Real.log_pow]; norm_num; linarith
-  have h₀ := palindrome_family_mgf_decay H n d hH e X (q^2) hq2 hlog2
-  have h₁ := palindrome_family_mgf_decay (2*H) n d (by omega) e X (q^2) hq2 hlog2
-  have h₁' := h₁.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left
-    (heightDecay_antitone (show H ≤ 2*H by omega)) (Nat.cast_nonneg _)))
-  have hh := finiteMean_mul_exp_le (fun Y => q^(palindromeFamilyCost H n d e (Y,X)))
-    (fun Y => q^(palindromeFamilyCost (2*H) n d e (Y,X)))
-    (Fintype.card ι*heightDecay H) (Fintype.card ι*heightDecay H)
-    (by simpa only [pow_right_comm] using h₀) (by simpa only [pow_right_comm] using h₁')
-  simpa only [←pow_add,add_self_div_two] using hh
-
-lemma palindrome_high_full_mgf (H n d : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/32) :
-    finiteMean (fun ω : BenesCoins d => q^(palindromeFamilyCost H n d e ω+
-      palindromeFamilyCost (2*H) n d e ω)) ≤ Real.exp (Fintype.card ι*heightDecay H) := by
-  rw [finiteMean_prod,finiteMean_comm]
-  calc
-    _ ≤ finiteMean (fun _X : SwitchIndex d → Bool => Real.exp (Fintype.card ι*heightDecay H)) := by
-      apply finiteMean_mono
-      intro X
-      exact palindrome_high_mgf H n d hH e X q hq hlog
-    _ = _ := finiteMean_const _
-
-end Thorp.Casimir
-
-end ThorpNine.Harmonic
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Harmonic.TailShapes
+
+namespace OAI
+
+namespace ThorpNine.Harmonic
+
+namespace Thorp.StrongSmoothing
+open scoped BigOperators
+
+lemma regular_trace_bound (d : ℕ) (order : Equiv.Perm (Fin d)) :
+    regularAbsoluteEvenTrace d order 2001 ≤ Real.exp ((2000*(1+heightDecay 1))*(2:ℝ)^d) := by
+  have hn : 0 < Fintype.card (Tuples d (2^d)) := by
+    rw [tuples_card,Nat.descFactorial_self]
+    exact Nat.factorial_pos _
+  have hn' : (Fintype.card (Tuples d (2^d)):ℝ) ≠ 0 := Nat.cast_ne_zero.mpr (ne_of_gt hn)
+  calc
+    _ ≤ ∑ x : Tuples d (2^d), Real.exp ((2000*(1+heightDecay 1))*(2^d:ℕ)) /
+        ((2^d).descFactorial (2^d):ℝ) := by
+      apply Finset.sum_le_sum
+      intro x _
+      exact both_pointwise d (2^d) order false x x
+    _ = _ := by
+      simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,←tuples_card]
+      push_cast
+      field_simp
+
+theorem strong_smoothing : MainStatement := by
+  refine ⟨2001,by norm_num,2000*(1+heightDecay 1),?_,regular_trace_bound⟩
+  intro d l _ order orientation x y
+  exact both_pointwise d l order orientation x y
+
+end Thorp.StrongSmoothing
+
+namespace Thorp.ButterflyReverse
+open scoped BigOperators
+
+abbrev Site (d : ℕ) := {t : Fin d × Card d // t.2 t.1 = false}
+
+def site : (d : ℕ) → SwitchIndex d → Site d
+  | 0, t => nomatch t
+  | _+1, Sum.inl y => ⟨(0, Fin.cons false y), rfl⟩
+  | d+1, Sum.inr (b,t) =>
+      ⟨((site d t).1.1.succ, Fin.cons b (site d t).1.2), (site d t).2⟩
+
+lemma site_injective (d : ℕ) : Function.Injective (site d) := by
+  induction d with
+  | zero => intro t; exact Empty.elim t
+  | succ d ih =>
+    intro s t h
+    cases s with
+    | inl y =>
+      cases t with
+      | inl z =>
+        have he := congrArg (fun a : Site (d+1) => Fin.tail a.1.2) h
+        exact congrArg Sum.inl he
+      | inr t =>
+        have he := congrArg (fun a : Site (d+1) => a.1.1.val) h
+        simp [site] at he
+    | inr s =>
+      cases t with
+      | inl y =>
+        have he := congrArg (fun a : Site (d+1) => a.1.1.val) h
+        simp [site] at he
+      | inr t =>
+        obtain ⟨b,s⟩ := s
+        obtain ⟨c,t⟩ := t
+        have hbc := congrArg (fun a : Site (d+1) => a.1.2 0) h
+        simp only [site, Fin.cons_zero] at hbc
+        have hst : site d s = site d t := by
+          apply Subtype.ext
+          apply Prod.ext
+          · exact Fin.succ_inj.mp (congrArg (fun a : Site (d+1) => a.1.1) h)
+          · exact congrArg (fun a : Site (d+1) => Fin.tail a.1.2) h
+        cases hbc
+        cases ih hst
+        rfl
+
+lemma site_surjective (d : ℕ) : Function.Surjective (site d) := by
+  induction d with
+  | zero => intro t; exact Fin.elim0 t.1.1
+  | succ d ih =>
+    rintro ⟨⟨i,x⟩,hx⟩
+    refine Fin.cases ?_ (fun j => ?_) i hx
+    · intro hx0
+      refine ⟨Sum.inl (Fin.tail x), ?_⟩
+      apply Subtype.ext
+      simp only [site, Prod.mk.injEq, true_and]
+      rw [←hx0,Fin.cons_self_tail]
+    · intro hxj
+      obtain ⟨t,ht⟩ := ih ⟨(j,Fin.tail x),hxj⟩
+      refine ⟨Sum.inr (x 0,t), ?_⟩
+      apply Subtype.ext
+      simp only [site,ht,Fin.cons_self_tail]
+
+noncomputable def siteEquiv (d : ℕ) : SwitchIndex d ≃ Site d :=
+  Equiv.ofBijective (site d) ⟨site_injective d,site_surjective d⟩
+
+def reverseCard (d : ℕ) : Equiv.Perm (Card d) where
+  toFun x := fun i => x i.rev
+  invFun x := fun i => x i.rev
+  left_inv x := by funext i; simp
+  right_inv x := by funext i; simp
+
+@[simp] lemma reverseCard_apply (d : ℕ) (x : Card d) (i : Fin d) :
+    reverseCard d x i = x i.rev := rfl
+
+@[simp] lemma reverseCard_involutive (d : ℕ) (x : Card d) :
+    reverseCard d (reverseCard d x) = x := by funext i; simp
+
+def mirrorSite (d : ℕ) : Equiv.Perm (Site d) where
+  toFun s := ⟨(s.1.1.rev,reverseCard d s.1.2),by simpa using s.2⟩
+  invFun s := ⟨(s.1.1.rev,reverseCard d s.1.2),by simpa using s.2⟩
+  left_inv s := by apply Subtype.ext; simp
+  right_inv s := by apply Subtype.ext; simp
+
+noncomputable def mirrorSwitch (d : ℕ) : Equiv.Perm (SwitchIndex d) :=
+  (siteEquiv d).trans ((mirrorSite d).trans (siteEquiv d).symm)
+
+lemma site_mirrorSwitch (d : ℕ) (t : SwitchIndex d) :
+    site d (mirrorSwitch d t) = mirrorSite d (site d t) := by
+  exact (siteEquiv d).apply_symm_apply _
+
+lemma routeDomain_site (d : ℕ) (x y : Card d) (t : SwitchIndex d) :
+    t ∈ routeDomain d x y ↔
+      ∀ j : Fin d, j ≠ (site d t).1.1 →
+      (site d t).1.2 j = if j < (site d t).1.1 then x j else y j := by
+  classical
+  induction d with
+  | zero => exact Empty.elim t
+  | succ d ih =>
+    cases t with
+    | inl z =>
+      change Sum.inl z ∈ (insert (Sum.inl (Fin.tail y))
+        ((routeDomain d (Fin.tail x) (Fin.tail y)).image
+          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) :
+        Finset (Sum (Card d) (Bool × SwitchIndex d))) ↔ _
+      simp only [Finset.mem_insert, Sum.inl.injEq,Finset.mem_image,
+        reduceCtorEq,and_false,exists_false,or_false]
+      constructor
+      · intro hz j hj
+        subst z
+        have hj0 : j ≠ 0 := hj
+        obtain ⟨i,rfl⟩ := Fin.eq_succ_of_ne_zero hj0
+        simp [site,Fin.tail]
+      · intro h
+        funext j
+        have he := h j.succ (by simp [site])
+        simpa [site,Fin.tail] using he
+    | inr z =>
+      obtain ⟨b,t⟩ := z
+      change Sum.inr (b,t) ∈ (insert (Sum.inl (Fin.tail y))
+        ((routeDomain d (Fin.tail x) (Fin.tail y)).image
+          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) :
+        Finset (Sum (Card d) (Bool × SwitchIndex d))) ↔ _
+      simp only [Finset.mem_insert,reduceCtorEq,false_or,Finset.mem_image,Sum.inr.injEq,Prod.mk.injEq]
+      constructor
+      · rintro ⟨s,hs,hb,ht⟩
+        subst s
+        have hst := (ih (Fin.tail x) (Fin.tail y) t).mp hs
+        intro j hj
+        refine Fin.cases ?_ (fun i => ?_) j hj
+        · intro _
+          simp [site,hb]
+        · intro hi
+          have hi' : i ≠ (site d t).1.1 := by simpa [site] using hi
+          simpa [site,Fin.tail] using hst i hi'
+      · intro h
+        have hb := h 0 (by simpa only [site] using (Fin.succ_ne_zero (site d t).1.1).symm)
+        have hb' : x 0 = b := by simpa [site] using hb.symm
+        refine ⟨t,?_,hb',rfl⟩
+        apply (ih (Fin.tail x) (Fin.tail y) t).mpr
+        intro j hj
+        have he := h j.succ (by simpa [site] using hj)
+        simpa [site,Fin.tail] using he
+
+lemma routeValue_site (d : ℕ) (x y : Card d) (t : SwitchIndex d) :
+    routeValue d x y t = Bool.xor (x (site d t).1.1) (y (site d t).1.1) := by
+  induction d with
+  | zero => exact Empty.elim t
+  | succ d ih =>
+    cases t with
+    | inl z => rfl
+    | inr z => exact ih (Fin.tail x) (Fin.tail y) z.2
+
+lemma mirrorSwitch_involutive (d : ℕ) (t : SwitchIndex d) :
+    mirrorSwitch d (mirrorSwitch d t) = t := by
+  apply site_injective d
+  rw [site_mirrorSwitch,site_mirrorSwitch]
+  apply Subtype.ext
+  simp [mirrorSite]
+
+lemma mirror_routeDomain (d : ℕ) (x y : Card d) (t : SwitchIndex d)
+    (ht : t ∈ routeDomain d x y) :
+    mirrorSwitch d t ∈ routeDomain d (reverseCard d y) (reverseCard d x) := by
+  rw [routeDomain_site,site_mirrorSwitch]
+  intro j hj
+  have hj' : j.rev ≠ (site d t).1.1 := by
+    intro h
+    apply hj
+    change j = (site d t).1.1.rev
+    rw [←h,Fin.rev_rev]
+  have he := (routeDomain_site d x y t).mp ht j.rev hj'
+  change (site d t).1.2 j.rev =
+    if j < (site d t).1.1.rev then y j.rev else x j.rev
+  rw [he]
+  have hord : j < (site d t).1.1.rev ↔ ¬j.rev < (site d t).1.1 := by
+    rw [Fin.lt_rev_iff]
+    constructor
+    · intro h
+      exact not_lt_of_ge h.le
+    · intro h
+      exact lt_of_le_of_ne (le_of_not_gt h) (Ne.symm hj')
+  by_cases h : j.rev < (site d t).1.1 <;> simp [h,hord]
+
+lemma mirror_routeValue (d : ℕ) (x y : Card d) (t : SwitchIndex d) :
+    routeValue d (reverseCard d y) (reverseCard d x) (mirrorSwitch d t) =
+      routeValue d x y t := by
+  rw [routeValue_site,routeValue_site,site_mirrorSwitch]
+  simp only [mirrorSite,Equiv.coe_fn_mk,reverseCard_apply,Fin.rev_rev]
+  exact Bool.xor_comm _ _
+
+noncomputable def mirrorCoins (d : ℕ) : Equiv.Perm (SwitchIndex d → Bool) where
+  toFun ω := fun t => ω (mirrorSwitch d t)
+  invFun ω := fun t => ω (mirrorSwitch d t)
+  left_inv ω := by funext t; change ω (mirrorSwitch d (mirrorSwitch d t)) = ω t; rw [mirrorSwitch_involutive]
+  right_inv ω := by funext t; change ω (mirrorSwitch d (mirrorSwitch d t)) = ω t; rw [mirrorSwitch_involutive]
+
+lemma mirror_endpoint (d : ℕ) (ω : SwitchIndex d → Bool) (x y : Card d)
+    (hxy : butterflyPerm d (decodeButterfly d ω) x = y) :
+    butterflyPerm d (decodeButterfly d (mirrorCoins d ω)) (reverseCard d y) =
+      reverseCard d x := by
+  apply (butterfly_endpoint_iff d _ _ _).mpr
+  intro t ht
+  have ht' := mirror_routeDomain d (reverseCard d y) (reverseCard d x) t ht
+  simp only [reverseCard_involutive] at ht'
+  have he := (butterfly_endpoint_iff d ω x y).mp hxy (mirrorSwitch d t) ht'
+  change ω (mirrorSwitch d t) = _
+  rw [he]
+  have hv := mirror_routeValue d x y (mirrorSwitch d t)
+  rw [mirrorSwitch_involutive] at hv
+  exact hv.symm
+
+lemma butterfly_mirror (d : ℕ) (ω : SwitchIndex d → Bool) :
+    butterflyPerm d (decodeButterfly d (mirrorCoins d ω)) =
+      reverseCard d * (butterflyPerm d (decodeButterfly d ω))⁻¹ * reverseCard d := by
+  apply Equiv.ext
+  intro x
+  have he := mirror_endpoint d ω
+    ((butterflyPerm d (decodeButterfly d ω))⁻¹ (reverseCard d x)) (reverseCard d x)
+    (Equiv.apply_symm_apply _ _)
+  simpa only [reverseCard_involutive,Equiv.Perm.mul_apply] using he
+
+def reverseOrder (d : ℕ) : Equiv.Perm (Fin d) where
+  toFun := Fin.rev
+  invFun := Fin.rev
+  left_inv := Fin.rev_rev
+  right_inv := Fin.rev_rev
+
+lemma relabel_mul (d : ℕ) (a b : Equiv.Perm (Fin d)) :
+    StrongSmoothing.coordinateRelabel d (a*b) =
+      StrongSmoothing.coordinateRelabel d a * StrongSmoothing.coordinateRelabel d b := rfl
+
+lemma relabel_reverse (d : ℕ) :
+    StrongSmoothing.coordinateRelabel d (reverseOrder d) = reverseCard d := rfl
+
+lemma reverseCard_sq (d : ℕ) : reverseCard d * reverseCard d = 1 := by
+  ext x i
+  simp
+
+lemma sweep_mirror (d : ℕ) (order : Equiv.Perm (Fin d)) (ω : SwitchIndex d → Bool) :
+    StrongSmoothing.sweep d (order*reverseOrder d) (mirrorCoins d ω) =
+      (StrongSmoothing.sweep d order ω)⁻¹ := by
+  simp only [StrongSmoothing.sweep,relabel_mul,relabel_reverse,butterfly_mirror,mul_inv_rev]
+  have hR : (reverseCard d)⁻¹ = reverseCard d := by
+    exact inv_eq_of_mul_eq_one_right (reverseCard_sq d)
+  rw [hR]
+  calc
+    _ = StrongSmoothing.coordinateRelabel d order *
+        (reverseCard d*reverseCard d) *
+        (butterflyPerm d (decodeButterfly d ω))⁻¹ *
+        (reverseCard d*reverseCard d) * (StrongSmoothing.coordinateRelabel d order)⁻¹ := by group
+    _ = _ := by rw [reverseCard_sq]; group
+
+lemma sweepKernel_reverse (d l : ℕ) (order : Equiv.Perm (Fin d)) :
+    StrongSmoothing.sweepKernel d l (order*reverseOrder d) =
+      (StrongSmoothing.sweepKernel d l order).transpose := by
+  classical
+  rw [StrongSmoothing.sweepKernel_law,StrongSmoothing.sweepKernel_law,
+    ←StrongSmoothing.PermLaw.inverse]
+  ext x y
+  unfold StrongSmoothing.PermLaw.kernel
+  rw [finiteMean_equiv (mirrorCoins d)]
+  apply finiteMean_congr
+  intro ω
+  change (if StrongSmoothing.tuplePerm
+      (StrongSmoothing.sweep d (order*reverseOrder d) (mirrorCoins d ω)) x = y then (1:ℝ) else 0) = _
+  rw [sweep_mirror,StrongSmoothing.tuplePerm_inverse]
+
+lemma reflectedKernel_reverse (d l : ℕ) (order : Equiv.Perm (Fin d)) :
+    StrongSmoothing.reflectedKernel d l order true =
+      StrongSmoothing.reflectedKernel d l (order*reverseOrder d) false := by
+  simp only [StrongSmoothing.reflectedKernel,Bool.false_eq_true,
+    ↓reduceIte,sweepKernel_reverse,Matrix.transpose_transpose]
+
+end Thorp.ButterflyReverse
+
+namespace Thorp.StrongTail
+open scoped BigOperators Classical
+
+section
+open Specht
+
+lemma mem_tailDiagram (μ : YoungDiagram) (i j : ℕ) :
+    (i,j) ∈ tailDiagram μ ↔ (i+1,j) ∈ μ := by
+  rw [tailDiagram, YoungDiagram.mem_ofRowLens]
+  constructor
+  · rintro ⟨hi,hj⟩
+    rw [List.getElem_tail,YoungDiagram.get_rowLens] at hj
+    exact YoungDiagram.mem_iff_lt_rowLen.mpr hj
+  · intro h
+    have hi : i+1 < μ.colLen 0 :=
+      (YoungDiagram.mem_iff_lt_colLen.mp h).trans_le (μ.colLen_anti 0 j (Nat.zero_le _))
+    have hi' : i < μ.rowLens.tail.length := by simp only [List.length_tail,YoungDiagram.length_rowLens]; omega
+    refine ⟨hi', ?_⟩
+    simpa only [List.getElem_tail,YoungDiagram.get_rowLens] using YoungDiagram.mem_iff_lt_rowLen.mp h
+
+lemma tail_height (μ : YoungDiagram) : (tailDiagram μ).colLen 0 = μ.colLen 0 - 1 := by
+  rw [←YoungDiagram.length_rowLens]
+  exact (YoungDiagram.rowLens_length_ofRowLens (fun x hx => μ.pos_of_mem_rowLens x (List.mem_of_mem_tail hx))).trans
+    (by simp only [List.length_tail,YoungDiagram.length_rowLens])
+
+noncomputable def tailCellEquiv (μ : YoungDiagram) : Cell (tailDiagram μ) ≃ Tail μ where
+  toFun x := ⟨⟨(x.1.1+1,x.1.2),(mem_tailDiagram μ _ _).mp x.2⟩,by simp [row]⟩
+  invFun x := ⟨(row x.1-1,col x.1),(mem_tailDiagram μ _ _).mpr (by
+    have he : row x.1-1+1 = row x.1 := by have := x.2; omega
+    change (row x.1-1+1,col x.1) ∈ μ
+    rw [he]
+    exact x.1.2)⟩
+  left_inv x := by apply Subtype.ext; simp [row,col]
+  right_inv x := by
+    apply Subtype.ext; apply Subtype.ext; apply Prod.ext
+    · change row x.1-1+1 = row x.1
+      have := x.2; omega
+    · rfl
+
+noncomputable abbrev upCell (μ : YoungDiagram) (x : Cell (tailDiagram μ)) : Cell μ := (tailCellEquiv μ x).1
+
+@[simp] lemma upCell_row (μ : YoungDiagram) (x : Cell (tailDiagram μ)) : row (upCell μ x) = row x+1 := rfl
+@[simp] lemma upCell_col (μ : YoungDiagram) (x : Cell (tailDiagram μ)) : col (upCell μ x) = col x := rfl
+
+lemma tail_card_eq (μ : YoungDiagram) : (tailDiagram μ).card = μ.card-μ.rowLen 0 := by
+  rw [←card_cell, Fintype.card_congr (tailCellEquiv μ), tail_card]
+
+noncomputable def extendPerm (μ : YoungDiagram) (p : Equiv.Perm (Cell (tailDiagram μ))) :
+    Equiv.Perm (Cell μ) := p.extendDomain (tailCellEquiv μ)
+
+@[simp] lemma extendPerm_up (μ : YoungDiagram) (p : Equiv.Perm (Cell (tailDiagram μ)))
+    (x : Cell (tailDiagram μ)) : extendPerm μ p (upCell μ x) = upCell μ (p x) :=
+  Equiv.Perm.extendDomain_apply_image p (tailCellEquiv μ) x
+
+lemma extendPerm_top (μ : YoungDiagram) (p : Equiv.Perm (Cell (tailDiagram μ)))
+    (x : Cell μ) (hx : row x=0) : extendPerm μ p x = x :=
+  Equiv.Perm.extendDomain_apply_not_subtype p (tailCellEquiv μ) (not_not.mpr hx)
+
+@[simp] lemma extendPerm_inv (μ : YoungDiagram) (p : Equiv.Perm (Cell (tailDiagram μ))) :
+    (extendPerm μ p)⁻¹ = extendPerm μ p⁻¹ := Equiv.Perm.extendDomain_symm _ _
+
+@[simp] lemma extendPerm_sign (μ : YoungDiagram) (p : Equiv.Perm (Cell (tailDiagram μ))) :
+    permSign (extendPerm μ p) = permSign p := by
+  simp [extendPerm,permSign,Equiv.Perm.sign_extendDomain]
+
+noncomputable def liftRows (μ : YoungDiagram) (f : Tabloid (tailDiagram μ)) (x : Cell μ) : Fin (μ.colLen 0) :=
+  if hx : row x ≠ 0 then
+    ⟨(f.1 ((tailCellEquiv μ).symm ⟨x,hx⟩)).val+1,by
+      have h := (f.1 ((tailCellEquiv μ).symm ⟨x,hx⟩)).isLt
+      have hh := tail_height μ
+      omega⟩
+  else rowIndex x
+
+@[simp] lemma liftRows_up (μ : YoungDiagram) (f : Tabloid (tailDiagram μ))
+    (x : Cell (tailDiagram μ)) : (liftRows μ f (upCell μ x)).val = (f.1 x).val+1 := by
+  simp [liftRows,upCell]
+
+lemma liftRows_top (μ : YoungDiagram) (f : Tabloid (tailDiagram μ)) (x : Cell μ) (hx : row x=0) :
+    liftRows μ f x = rowIndex x := by simp [liftRows,hx]
+
+lemma liftRows_perm (μ : YoungDiagram) (f : Tabloid (tailDiagram μ))
+    (p : Equiv.Perm (Cell (tailDiagram μ))) (hf : f.1 = rowIndex ∘ p) :
+    liftRows μ f = rowIndex ∘ extendPerm μ p := by
+  funext x
+  by_cases hx : row x = 0
+  · simp [liftRows_top μ f x hx,extendPerm_top μ p x hx]
+  · let y := (tailCellEquiv μ).symm ⟨x,hx⟩
+    have he : upCell μ y = x := congrArg Subtype.val ((tailCellEquiv μ).apply_symm_apply ⟨x,hx⟩)
+    rw [←he]
+    apply Fin.ext
+    simp [hf,Function.comp_apply,rowIndex]
+
+noncomputable def liftTabloid (μ : YoungDiagram) (f : Tabloid (tailDiagram μ)) : Tabloid μ :=
+  ⟨liftRows μ f, by obtain ⟨p,hp⟩ := f.2; exact ⟨extendPerm μ p,liftRows_perm μ f p hp⟩⟩
+
+@[simp] lemma liftTabloid_base (μ : YoungDiagram) :
+    liftTabloid μ (baseTabloid (tailDiagram μ)) = baseTabloid μ := by
+  apply Subtype.ext
+  funext x
+  by_cases hx : row x=0
+  · exact liftRows_top μ _ x hx
+  · let y := (tailCellEquiv μ).symm ⟨x,hx⟩
+    have he : upCell μ y = x := congrArg Subtype.val ((tailCellEquiv μ).apply_symm_apply ⟨x,hx⟩)
+    rw [←he]
+    apply Fin.ext
+    exact liftRows_up μ _ y
+
+lemma liftTabloid_action (μ : YoungDiagram) (p : Equiv.Perm (Cell (tailDiagram μ)))
+    (f : Tabloid (tailDiagram μ)) :
+    liftTabloid μ (tabloidAct p f) = tabloidAct (extendPerm μ p) (liftTabloid μ f) := by
+  apply Subtype.ext
+  funext x
+  change liftRows μ (tabloidAct p f) x = liftRows μ f ((extendPerm μ p)⁻¹ x)
+  rw [extendPerm_inv]
+  by_cases hx : row x=0
+  · rw [extendPerm_top μ p⁻¹ x hx,liftRows_top μ _ x hx,liftRows_top μ _ x hx]
+  · let y := (tailCellEquiv μ).symm ⟨x,hx⟩
+    have he : upCell μ y = x := congrArg Subtype.val ((tailCellEquiv μ).apply_symm_apply ⟨x,hx⟩)
+    rw [←he,extendPerm_up]
+    apply Fin.ext
+    simp only [liftRows_up,tabloidAct_apply]
+
+noncomputable def restrictTabloids (μ : YoungDiagram) :
+    (Tabloid μ → ℂ) →ₗ[ℂ] (Tabloid (tailDiagram μ) → ℂ) where
+  toFun v := fun f => v (liftTabloid μ f)
+  map_add' := by intros; rfl
+  map_smul' := by intros; rfl
+
+lemma restrict_action (μ : YoungDiagram) (p : Equiv.Perm (Cell (tailDiagram μ)))
+    (v : Tabloid μ → ℂ) :
+    restrictTabloids μ (tabloidRep μ (extendPerm μ p) v) =
+      tabloidRep (tailDiagram μ) p (restrictTabloids μ v) := by
+  ext f
+  change v (tabloidAct (extendPerm μ p)⁻¹ (liftTabloid μ f)) = v (liftTabloid μ (tabloidAct p⁻¹ f))
+  rw [extendPerm_inv,liftTabloid_action]
+
+lemma extend_colGroup (μ : YoungDiagram) (c : colGroup (tailDiagram μ)) :
+    extendPerm μ c ∈ colGroup μ := by
+  intro x
+  by_cases hx : row x=0
+  · rw [extendPerm_top μ c x hx]
+  · let y := (tailCellEquiv μ).symm ⟨x,hx⟩
+    have he : upCell μ y = x := congrArg Subtype.val ((tailCellEquiv μ).apply_symm_apply ⟨x,hx⟩)
+    rw [←he,extendPerm_up,upCell_col,upCell_col]
+    exact c.2 y
+
+lemma restrict_polytabloid (μ : YoungDiagram) :
+    restrictTabloids μ (polytabloid μ) = polytabloid (tailDiagram μ) := by
+  let v := restrictTabloids μ (polytabloid μ)
+  have hv (c : colGroup (tailDiagram μ)) :
+      tabloidRep (tailDiagram μ) c v = permSign (c:Equiv.Perm (Cell (tailDiagram μ))) • v := by
+    rw [←restrict_action]
+    rw [polytabloid_col_action μ ⟨extendPerm μ c,extend_colGroup μ c⟩]
+    simp only [map_smul,extendPerm_sign]
+    rfl
+  have ha : alternator (colGroup (tailDiagram μ)) (tabloidRep (tailDiagram μ)) v =
+      (Fintype.card (colGroup (tailDiagram μ)):ℂ) • v := by
+    simp only [alternator,LinearMap.sum_apply,LinearMap.smul_apply,hv,smul_smul,permSign_sq,one_smul]
+    ext f
+    simp
+  have hb : v (baseTabloid (tailDiagram μ)) = 1 := by
+    change polytabloid μ (liftTabloid μ (baseTabloid (tailDiagram μ))) = 1
+    rw [liftTabloid_base, polytabloid_base]
+  have he := alternator_rank_one_exact (tailDiagram μ) v
+  rw [ha] at he
+  simp only [Pi.smul_apply,hb,smul_eq_mul,mul_one] at he
+  have hc : (Fintype.card (colGroup (tailDiagram μ)):ℂ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
+  exact (smul_right_injective _ hc) he
+
+lemma tail_space_le_image (μ : YoungDiagram) :
+    space (tailDiagram μ) ≤ (space μ).map (restrictTabloids μ) := by
+  apply Submodule.span_le.mpr
+  rintro _ ⟨p,rfl⟩
+  refine ⟨tabloidRep μ (extendPerm μ p) (polytabloid μ), orbit_mem_space μ _, ?_⟩
+  rw [restrict_action,restrict_polytabloid]
+
+theorem tail_dimension_le (μ : YoungDiagram) :
+    Module.finrank ℂ (space (tailDiagram μ)) ≤ Module.finrank ℂ (space μ) :=
+  (Submodule.finrank_mono (tail_space_le_image μ)).trans (Submodule.finrank_map_le _ _)
+
+end
+open StrongSmoothing
+
+def DensityAt (p C : ℝ) : Prop :=
+  ∀ (d l : ℕ), l ≤ 2^d → ∀ (order : Equiv.Perm (Fin d)) (rev : Bool)
+    (x : StrongSmoothing.Tuples d l),
+    let H := StrongSmoothing.reflectedKernel d l order rev
+    (Thorp.finiteMean (fun y => ((Fintype.card (StrongSmoothing.Tuples d l):ℝ)*H x y)^p)
+      ≤ Real.exp (C*l)) ∧
+    (Thorp.finiteMean (fun y => ((Fintype.card (StrongSmoothing.Tuples d l):ℝ)*H y x)^p)
+      ≤ Real.exp (C*l)) ∧
+    (((2:ℝ)^d)^l)⁻¹ * (∑ y, ((((2:ℝ)^d)^l)*H x y)^p) ≤ Real.exp (C*l) ∧
+    (((2:ℝ)^d)^l)⁻¹ * (∑ y, ((((2:ℝ)^d)^l)*H y x)^p) ≤ Real.exp (C*l)
+
+lemma reflected_nonneg_both (d l : ℕ) (order : Equiv.Perm (Fin d)) (rev : Bool)
+    (x y : Tuples d l) : 0 ≤ reflectedKernel d l order rev x y := by
+  cases rev
+  · exact reflected_nonneg _ _ _ _ _
+  · rw [ButterflyReverse.reflectedKernel_reverse]
+    exact reflected_nonneg _ _ _ _ _
+
+lemma reflected_symm_both (d l : ℕ) (order : Equiv.Perm (Fin d)) (rev : Bool) :
+    (reflectedKernel d l order rev).IsSymm := by
+  cases rev
+  · exact reflected_symmetric _ _ _
+  · rw [ButterflyReverse.reflectedKernel_reverse]
+    exact reflected_symmetric _ _ _
+
+lemma reflected_moment_both (d l : ℕ) (order : Equiv.Perm (Fin d)) (rev : Bool)
+    (x : Tuples d l) :
+    finiteMean (fun y : Tuples d l =>
+      ((Fintype.card (Tuples d l):ℝ)*reflectedKernel d l order rev x y)^(1+(1/1000:ℝ))) ≤
+      Real.exp (l*(1+heightDecay 1)) := by
+  cases rev
+  · exact reflected_moment _ _ _ _
+  · rw [ButterflyReverse.reflectedKernel_reverse]
+    exact reflected_moment _ _ _ _
+
+lemma reflected_moment_small (d l : ℕ) (order : Equiv.Perm (Fin d)) (rev : Bool)
+    (x : Tuples d l) {p : ℝ} (hp : 0 < p) (hpq : p ≤ 1+(1/1000:ℝ)) :
+    finiteMean (fun y : Tuples d l =>
+      ((Fintype.card (Tuples d l):ℝ)*reflectedKernel d l order rev x y)^p) ≤
+      Real.exp (l*(1+heightDecay 1)) := by
+  let : Nonempty (Tuples d l) := ⟨x⟩
+  let q : ℝ := 1+1/1000
+  let f := fun y : Tuples d l => (Fintype.card (Tuples d l):ℝ)*reflectedKernel d l order rev x y
+  have hf : ∀ y, 0 ≤ f y := fun y => mul_nonneg (Nat.cast_nonneg _) (reflected_nonneg_both _ _ _ _ _ _)
+  have hq : 0 < q := by norm_num [q]
+  have hr : 0 ≤ p/q := (div_pos hp hq).le
+  have hr1 : p/q ≤ 1 := (div_le_one hq).mpr hpq
+  have he (y) : (f y)^p = ((f y)^q)^(p/q) := by
+    rw [←Real.rpow_mul (hf y)]
+    congr 1
+    field_simp
+  simp_rw [show (fun y : Tuples d l => ((Fintype.card (Tuples d l):ℝ)*reflectedKernel d l order rev x y)^p)
+    = fun y => ((f y)^q)^(p/q) from funext he]
+  apply (mean_rpow_concave _ (fun y => Real.rpow_nonneg (hf y) _) hr hr1).trans
+  apply (Real.rpow_le_rpow (finiteMean_nonneg (fun y => Real.rpow_nonneg (hf y) _))
+    (reflected_moment_both d l order rev x) hr).trans
+  rw [Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp]
+  apply Real.exp_le_exp.mpr
+  have hA : 0 ≤ (l:ℝ)*(1+heightDecay 1) := by
+    have hh := heightDecay_nonneg 1
+    positivity
+  nlinarith
+
+lemma log_factorial_le_power (n N : ℕ) (hn : n ≤ N) (_ : 0 < N) :
+    Real.log (n.factorial:ℝ) ≤ (n:ℝ)*Real.log N := by
+  have hb : (n.factorial:ℝ) ≤ (N:ℝ)^n := by
+    exact_mod_cast (Nat.factorial_le_pow n).trans (Nat.pow_le_pow_left hn n)
+  simpa only [Real.log_pow] using Real.log_le_log (by positivity : (0:ℝ)<n.factorial) hb
+
+lemma log_normalization_le (N l : ℕ) (hN : 0 < N) (hl : l ≤ N) :
+    Real.log ((N:ℝ)^l/(N.descFactorial l:ℝ)) ≤ 2*l := by
+  have hNp : (0:ℝ)<N := by exact_mod_cast hN
+  have hMp : (0:ℝ)<N.descFactorial l := by exact_mod_cast Nat.descFactorial_pos.mpr hl
+  rw [Real.log_div (by positivity) hMp.ne',Real.log_pow]
+  by_cases hlow : 2*l ≤ N
+  · have hhalf : (N:ℝ)/2 ≤ (N+1-l:ℕ) := by
+      have hNat : N ≤ 2*(N+1-l) := by omega
+      exact (div_le_iff₀ (by norm_num : (0:ℝ)<2)).mpr (by simpa only [mul_comm] using (show (N:ℝ) ≤ 2*(N+1-l:ℕ) by exact_mod_cast hNat))
+    have hb : ((N:ℝ)/2)^l ≤ (N.descFactorial l:ℝ) := by
+      apply (pow_le_pow_left₀ (by positivity) hhalf l).trans
+      exact_mod_cast Nat.pow_sub_le_descFactorial N l
+    have hh := Real.log_le_log (pow_pos (by positivity : (0:ℝ)<(N:ℝ)/2) l) hb
+    rw [Real.log_pow,Real.log_div hNp.ne' (by norm_num)] at hh
+    have ht : Real.log (2:ℝ) ≤ 1 := by simpa only [show (2:ℝ)-1=1 by norm_num] using Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
+    nlinarith [(show (0:ℝ) ≤ l by positivity)]
+  · have he : (N.descFactorial l:ℝ)*(Nat.factorial (N-l):ℝ) = N.factorial := by
+      have he := Nat.factorial_mul_descFactorial hl
+      exact_mod_cast (by simpa only [mul_comm] using he : N.descFactorial l * (N-l).factorial = N.factorial)
+    have hh := congrArg Real.log he
+    rw [Real.log_mul hMp.ne' (by positivity)] at hh
+    have hfull := log_factorial_lower (show 1 ≤ N by omega)
+    have hrem := log_factorial_le_power (N-l) N (Nat.sub_le _ _) hN
+    rw [Nat.cast_sub hl] at hrem
+    have hNle : (N:ℝ) ≤ 2*l := by exact_mod_cast (by omega : N ≤ 2*l)
+    linarith
+
+lemma normalized_moment_identity {X : Type*} [Fintype X] [Nonempty X]
+    (f : X → ℝ) (hf : ∀ x, 0 ≤ f x) {N : ℝ} (hN : 0 < N) (p : ℝ) :
+    N⁻¹*(∑ x, (N*f x)^p) =
+     (N/(Fintype.card X:ℝ))^(p-1) * finiteMean (fun x => ((Fintype.card X:ℝ)*f x)^p) := by
+  let M : ℝ := Fintype.card X
+  have hM : 0 < M := Nat.cast_pos.mpr Fintype.card_pos
+  have he (x) : (N*f x)^p = (N/M)^p*(M*f x)^p := by
+    rw [←Real.mul_rpow (div_nonneg hN.le hM.le) (mul_nonneg hM.le (hf x))]
+    congr 1
+    field_simp
+  simp_rw [he]
+  rw [←Finset.mul_sum,Real.rpow_sub (div_pos hN hM),Real.rpow_one]
+  unfold finiteMean
+  dsimp [M] at hM ⊢
+  field_simp
+
+lemma density_at_small_exponent {p : ℝ} (hp : 1 < p) (hpq : p ≤ 1+(1/1000:ℝ)) :
+    DensityAt p (3+heightDecay 1) := by
+  intro d l hl order rev x
+  dsimp only
+  let : Nonempty (Tuples d l) := ⟨x⟩
+  let H := reflectedKernel d l order rev
+  have hm := reflected_moment_small d l order rev x (by linarith : 0<p) hpq
+  have hb : Real.exp ((l:ℝ)*(1+heightDecay 1)) ≤ Real.exp ((3+heightDecay 1)*l) := by
+    apply Real.exp_le_exp.mpr
+    nlinarith [(show (0:ℝ) ≤ l by positivity)]
+  have hs (y) : reflectedKernel d l order rev y x = reflectedKernel d l order rev x y := (reflected_symm_both d l order rev).apply _ _
+  have hpow : (((2:ℝ)^d)^l)⁻¹ * (∑ y, ((((2:ℝ)^d)^l)*H x y)^p) ≤
+      Real.exp ((3+heightDecay 1)*l) := by
+    rw [normalized_moment_identity (fun y => H x y) (reflected_nonneg_both d l order rev x)
+      (by positivity) p]
+    have hrat : ((2:ℝ)^d)^l/(Fintype.card (Tuples d l):ℝ) ≤ Real.exp (2*l) := by
+      apply (Real.log_le_iff_le_exp (by positivity)).mp
+      rw [tuples_card]
+      exact_mod_cast log_normalization_le (2^d) l (by positivity) hl
+    have hr : 0 ≤ ((2:ℝ)^d)^l/(Fintype.card (Tuples d l):ℝ) := by positivity
+    have hrp : (((2:ℝ)^d)^l/(Fintype.card (Tuples d l):ℝ))^(p-1) ≤ Real.exp (2*l) := by
+      apply (Real.rpow_le_rpow hr hrat (by linarith)).trans
+      rw [Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp]
+      apply Real.exp_le_exp.mpr
+      have hpl : p-1 ≤ 1 := by linarith
+      nlinarith [(show (0:ℝ) ≤ l by positivity)]
+    apply (mul_le_mul hrp hm (finiteMean_nonneg (fun y => Real.rpow_nonneg
+      (mul_nonneg (Nat.cast_nonneg _) (reflected_nonneg_both d l order rev x y)) _)) (Real.exp_pos _).le).trans
+    rw [←Real.exp_add]
+    apply Real.exp_le_exp.mpr
+    ring_nf
+    rfl
+  dsimp [H] at hpow
+  refine ⟨hm.trans hb,?_,hpow,?_⟩
+  · simpa only [hs] using hm.trans hb
+  · simpa only [hs] using hpow
+
+end Thorp.StrongTail
+
+namespace Thorp.Casimir
+open scoped BigOperators Classical
+
+section
+open Filter
+
+noncomputable def traceMoment (d : ℕ) (μ : YoungDiagram)
+    (e : Card d ≃ Specht.Cell μ) (r : ℕ) : ℝ :=
+  (LinearMap.trace ℂ (Specht.hilbertSpace μ) ((K d μ e)^r).toLinearMap).re
+
+noncomputable def fromDistance (d t : ℕ) (σ : Equiv.Perm (Card d)) : ℝ :=
+  (1/2:ℝ) * ∑ g : Equiv.Perm (Card d),
+    |law d t (g*σ⁻¹) - 1/(Nat.factorial (2^d):ℝ)|
+
+def MainStatement : Prop :=
+  ∃ a C : ℝ, 0 < a ∧ 0 < C ∧
+    (∀ d k : ℕ, 1 ≤ k → k ≤ 2^d → ∀ x : Fin k ↪ Card d,
+      (palindromeRowMoment d x (1/64))^(64/65:ℝ) ≤
+        Real.exp (C*levelScale (2^d) k)) ∧
+    (∀ (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ),
+      0 < μ.card-μ.rowLen 0 →
+        ‖K d μ e‖ ≤ Real.exp (-a*levelScale (2^d) (μ.card-μ.rowLen 0)) ∧
+        (Module.finrank ℂ (Specht.space μ):ℝ) * traceMoment d μ e 65 ≤
+          Real.exp (C*levelScale (2^d) (μ.card-μ.rowLen 0))) ∧
+    (∃ l : ℕ, 0 < l ∧ ∀ starts : (d : ℕ) → Equiv.Perm (Card d),
+      Tendsto (fun d => fromDistance d (l*d) (starts d)) atTop (nhds 0))
+
+end
+
+theorem cycleCoefficient_decay (l r : ℕ) (hr : (l:ℝ)/2 ≤ r)
+    (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ (l:ℝ)/16) :
+    cycleCoefficient r ⌈Real.exp ((l:ℝ)/32)⌉₊ q ≤
+      68 * Real.exp (-(l:ℝ)/64) := by
+  let E := Real.exp ((l:ℝ)/32)
+  let J := ⌈E⌉₊
+  have hl : (0:ℝ) ≤ l := Nat.cast_nonneg _
+  have hE : 1 ≤ E := Real.one_le_exp_iff.mpr (by positivity)
+  have hEpos : 0 < E := Real.exp_pos _
+  have hEJ : E ≤ (J:ℝ) := Nat.le_ceil E
+  have hJ : (1:ℝ) ≤ J := hE.trans hEJ
+  have hJpos : (0:ℝ) < J := lt_of_lt_of_le (by norm_num) hJ
+  have hJE : (J:ℝ) ≤ 2*E := by
+    have h := Nat.ceil_lt_add_one (le_of_lt hEpos)
+    change (J:ℝ) < E+1 at h
+    linarith
+  have hsqJ : Real.sqrt J ≤ (J:ℝ) := by
+    apply Real.sqrt_le_iff.mpr
+    constructor
+    · positivity
+    · nlinarith
+  have hqp : 0 < q := lt_of_lt_of_le (by norm_num) hq
+  have hqE : q ≤ Real.exp ((l:ℝ)/16) := by
+    rw [←Real.exp_log hqp]
+    exact Real.exp_le_exp.mpr hlog
+  have hln : (1/2:ℝ) ≤ Real.log 2 := by
+    have hh := Real.one_sub_inv_le_log_of_pos (by norm_num : (0:ℝ)<2)
+    norm_num at hh ⊢
+    exact hh
+  have hden : Real.exp ((l:ℝ)/8) ≤ Real.sqrt ((2:ℝ)^r) := by
+    have he : (2:ℝ)^r = Real.exp ((r:ℝ)*Real.log 2) := by
+      rw [Real.exp_nat_mul,Real.exp_log (by norm_num)]
+    rw [he,Real.sqrt_eq_rpow,Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp]
+    apply Real.exp_le_exp.mpr
+    have hrn : (0:ℝ) ≤ r := Nat.cast_nonneg _
+    nlinarith
+  have hlog' : Real.log q ≤ (l:ℝ) := by linarith
+  have h₁ : Real.log q / (J:ℝ) ≤ 64*Real.exp (-(l:ℝ)/64) := by
+    calc
+      _ ≤ (l:ℝ)/E := div_le_div₀ (by positivity) hlog' hEpos hEJ
+      _ ≤ (64*Real.exp ((l:ℝ)/64))/E := by
+        apply div_le_div_of_nonneg_right _ (le_of_lt hEpos)
+        have hh := Real.add_one_le_exp ((l:ℝ)/64)
+        linarith
+      _ = _ := by
+        dsimp [E]
+        rw [mul_div_assoc,←Real.exp_sub]
+        congr 2
+        ring
+  have h₂ : 2*q*Real.sqrt J / Real.sqrt ((2:ℝ)^r) ≤
+      4*Real.exp (-(l:ℝ)/64) := by
+    calc
+      _ ≤ (4*Real.exp ((l:ℝ)/16)*E)/Real.exp ((l:ℝ)/8) := by
+        apply div_le_div₀ (by positivity) _ (Real.exp_pos _) hden
+        have hs := hsqJ.trans hJE
+        have hm := mul_le_mul hqE hs (Real.sqrt_nonneg _) (Real.exp_nonneg _)
+        nlinarith
+      _ = 4*Real.exp ((l:ℝ)/16+(l:ℝ)/32-(l:ℝ)/8) := by
+        dsimp [E]
+        rw [mul_assoc,←Real.exp_add,mul_div_assoc,←Real.exp_sub]
+      _ ≤ _ := by
+        apply mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr _) (by norm_num)
+        linarith
+  change Real.log q / (J:ℝ) + 2*q*Real.sqrt J / Real.sqrt ((2:ℝ)^r) ≤ _
+  linarith
+
+theorem palindrome_slab_partial_mgf (r s l : ℕ) (hl : 0 < l) (hs : 2*s ≤ l)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+r)) (X : SwitchIndex (s+r) → Bool)
+    (lo : Card r × SwitchIndex s → Bool) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/16) :
+    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(palindromeSlabCost l (s+r) e (assembleBits r s lo hi,X))) ≤
+      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
+  classical
+  let : NeZero l := ⟨ne_of_gt hl⟩
+  have hqp : 0 < q := lt_of_lt_of_le (by norm_num) hq
+  calc
+    _ ≤ finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      finiteMean (fun i : Fin l => q^(l*palindromeLevelCost (l+i.1+1) (s+r) e
+        (assembleBits r s lo hi,X)))) := by
+      apply finiteMean_mono
+      intro hi
+      simpa only [palindromeSlabCost,Fintype.card_fin] using
+        pow_sum_le_mean_powers (fun i : Fin l => palindromeLevelCost (l+i.1+1) (s+r) e
+          (assembleBits r s lo hi,X)) q hqp
+    _ = finiteMean (fun i : Fin l => finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(l*palindromeLevelCost (l+i.1+1) (s+r) e (assembleBits r s lo hi,X)))) := finiteMean_comm _
+    _ ≤ finiteMean (fun _i : Fin l =>
+      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64)))) := by
+      apply finiteMean_mono
+      intro i
+      by_cases ht : l+i.1+1 ≤ s+r
+      · have hs' : s ≤ l+i.1 := by omega
+        have he : l+i.1+1 = s+(l+i.1-s)+1 := by omega
+        have hu : l+i.1-s < r := by omega
+        have hm := palindrome_level_partial_mgf r s (l+i.1-s) hu e X lo
+          ⌈Real.exp ((l:ℝ)/32)⌉₊ (cycleCutoff_pos l) (q^l) (one_le_pow₀ hq)
+        have hc := cycleCoefficient_decay l (l+i.1-s) (by
+          rw [Nat.cast_sub hs',Nat.cast_add]
+          have hsr : (2:ℝ)*s ≤ l := by exact_mod_cast hs
+          have hi : (0:ℝ) ≤ i.1 := Nat.cast_nonneg _
+          linarith) (q^l) (one_le_pow₀ hq) (by
+            rw [Real.log_pow]
+            nlinarith [mul_le_mul_of_nonneg_left hlog (Nat.cast_nonneg l)])
+        simp_rw [he,pow_mul]
+        exact hm.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left hc (Nat.cast_nonneg _)))
+      · have hz := palindromeLevelCost_gt (l+i.1+1) (s+r) (by omega) e
+        simp only [hz,Nat.mul_zero,pow_zero,finiteMean_const]
+        exact Real.one_le_exp_iff.mpr (by positivity)
+    _ = _ := finiteMean_const _
+
+lemma palindrome_slab_adapted_mul_le (d s l : ℕ) (hsd : s ≤ d) (hl : 0 < l)
+    (hs : 2*s ≤ l) {ι : Type*} [Fintype ι] (e : ι ↪ Card d)
+    (X : SwitchIndex d → Bool) (F : (SwitchIndex d → Bool) → ℝ)
+    (hF : ∀ Y, 0 ≤ F Y) (hA : LayerAdapted d s F) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/16) :
+    finiteMean (fun Y => F Y*q^(palindromeSlabCost l d e (Y,X))) ≤
+      finiteMean F * Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
+  obtain ⟨r,rfl⟩ := Nat.exists_eq_add_of_le hsd
+  exact adapted_mul_mean_le r s F _ hF hA _
+    (fun lo => palindrome_slab_partial_mgf r s l hl hs e X lo q hq hlog)
+
+theorem palindrome_family_mgf (L n d : ℕ) (hL : 0 < L) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/16) :
+    finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) ≤
+      Real.exp (Fintype.card ι * (68*∑ j ∈ Finset.range n,
+        Real.exp (-((L*4^j:ℕ):ℝ)/64))) := by
+  induction n with
+  | zero => simp only [palindromeFamilyCost,Finset.range_zero,Finset.sum_empty,pow_zero,
+      finiteMean_const,mul_zero,Real.exp_zero,le_refl]
+  | succ n ih =>
+    let l := L*4^n
+    have hl : 0 < l := Nat.mul_pos hL (pow_pos (by decide) _)
+    have hs : 2*familyEnd L n ≤ l := familyEnd_two L n
+    have hc : ∀ Y, palindromeFamilyCost L (n+1) d e (Y,X) =
+        palindromeFamilyCost L n d e (Y,X)+palindromeSlabCost l d e (Y,X) := by
+      intro Y
+      exact Finset.sum_range_succ _ _
+    have hm : finiteMean (fun Y => q^(palindromeFamilyCost L (n+1) d e (Y,X))) ≤
+        finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) *
+          Real.exp (Fintype.card ι*(68*Real.exp (-(l:ℝ)/64))) := by
+      by_cases hld : l ≤ d
+      · simp_rw [hc,pow_add]
+        exact palindrome_slab_adapted_mul_le d (familyEnd L n) l (by omega) hl hs e X _
+          (fun _ => pow_nonneg (by linarith) _)
+          (layerAdapted_comp (palindromeFamilyCost_adapted L n d e X) (fun c => q^c))
+          q hq hlog
+      · simp_rw [hc,palindromeSlabCost_gt l d (by omega) e,Nat.add_zero]
+        exact le_mul_of_one_le_right
+          (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
+          (Real.one_le_exp_iff.mpr (by positivity))
+    refine (hm.trans (mul_le_mul_of_nonneg_right ih (Real.exp_nonneg _))).trans_eq ?_
+    rw [←Real.exp_add,Finset.sum_range_succ]
+    congr 1
+    dsimp [l]
+    ring
+
+lemma palindrome_family_mgf_decay (L n d : ℕ) (hL : 0 < L) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/16) :
+    finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) ≤
+      Real.exp (Fintype.card ι*heightDecay L) := by
+  apply (palindrome_family_mgf L n d hL e X q hq hlog).trans
+  apply Real.exp_le_exp.mpr
+  apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
+  have h := mul_le_mul_of_nonneg_left (sum_family_decay L n hL) (by norm_num : (0:ℝ)≤68)
+  simpa only [heightDecay,mul_div_assoc] using h
+
+theorem palindrome_high_mgf (H n d : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/32) :
+    finiteMean (fun Y => q^(palindromeFamilyCost H n d e (Y,X)+
+      palindromeFamilyCost (2*H) n d e (Y,X))) ≤ Real.exp (Fintype.card ι*heightDecay H) := by
+  have hq2 := one_le_pow₀ (n:=2) hq
+  have hlog2 : Real.log (q^2) ≤ 1/16 := by rw [Real.log_pow]; norm_num; linarith
+  have h₀ := palindrome_family_mgf_decay H n d hH e X (q^2) hq2 hlog2
+  have h₁ := palindrome_family_mgf_decay (2*H) n d (by omega) e X (q^2) hq2 hlog2
+  have h₁' := h₁.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left
+    (heightDecay_antitone (show H ≤ 2*H by omega)) (Nat.cast_nonneg _)))
+  have hh := finiteMean_mul_exp_le (fun Y => q^(palindromeFamilyCost H n d e (Y,X)))
+    (fun Y => q^(palindromeFamilyCost (2*H) n d e (Y,X)))
+    (Fintype.card ι*heightDecay H) (Fintype.card ι*heightDecay H)
+    (by simpa only [pow_right_comm] using h₀) (by simpa only [pow_right_comm] using h₁')
+  simpa only [←pow_add,add_self_div_two] using hh
+
+lemma palindrome_high_full_mgf (H n d : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/32) :
+    finiteMean (fun ω : BenesCoins d => q^(palindromeFamilyCost H n d e ω+
+      palindromeFamilyCost (2*H) n d e ω)) ≤ Real.exp (Fintype.card ι*heightDecay H) := by
+  rw [finiteMean_prod,finiteMean_comm]
+  calc
+    _ ≤ finiteMean (fun _X : SwitchIndex d → Bool => Real.exp (Fintype.card ι*heightDecay H)) := by
+      apply finiteMean_mono
+      intro X
+      exact palindrome_high_mgf H n d hH e X q hq hlog
+    _ = _ := finiteMean_const _
+
+end Thorp.Casimir
+
+end ThorpNine.Harmonic
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Harmonic/TailShapes.lean b/lean/OAI/Probability/ThorpRouting/Harmonic/TailShapes.lean
index 3188ed1556e7de8e65ed20dac157f8f875e6ab36..8fd8c83b725d6eb2a3dc2ce6cc668d51366d3fec 100644
--- a/lean/OAI/Probability/ThorpRouting/Harmonic/TailShapes.lean
+++ b/lean/OAI/Probability/ThorpRouting/Harmonic/TailShapes.lean
@@ -1,870 +1,870 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Harmonic.InsertionPerm
-
-namespace OAI
-
-namespace ThorpNine.Harmonic
-
-namespace Thorp.Specht
-open scoped BigOperators Classical
-
-lemma partition_count (N : ℕ) : Fintype.card (Nat.Partition N)≤2^N := by
-  have h := Fintype.card_le_of_surjective (Nat.Partition.ofComposition N) Nat.Partition.ofComposition_surj
-  rw [composition_card] at h
-  exact h.trans (Nat.pow_le_pow_right (by norm_num) (Nat.sub_le N 1))
-
-lemma shapes_count (N : ℕ) : Fintype.card (Shapes N)≤2^N := by
-  rw [Fintype.card_congr (shapePartitionEquiv N)]
-  exact partition_count N
-
-lemma rowLens_headD (μ : YoungDiagram) : μ.rowLens.headD 0=μ.rowLen 0 := by
-  cases he : μ.rowLens with
-  | nil =>
-    have hc := sum_rowLens μ
-    rw [he,List.sum_nil] at hc
-    have hr := rowLen_zero_le_card μ
-    simp only [List.headD_nil]
-    omega
-  | cons a l =>
-    have hh := YoungDiagram.get_rowLens (μ := μ) (i := 0) (h := by simp [he])
-    simpa only [he,List.getElem_cons_zero,List.headD_cons] using hh
-
-lemma rowLens_tail_sum (μ : YoungDiagram) : μ.rowLens.tail.sum=μ.card-μ.rowLen 0 := by
-  have hh := rowLens_headD μ
-  have hc := sum_rowLens μ
-  cases he : μ.rowLens with
-  | nil => simp only [he,List.headD_nil] at hh; simp only [he,List.sum_nil] at hc; simp [←hc]
-  | cons a l =>
-    simp only [he,List.headD_cons] at hh
-    simp only [he,List.sum_cons] at hc
-    simp only [List.tail_cons]
-    omega
-
-lemma positive_list_eq_of_tail_sum {a b : List ℕ}
-    (ha : ∀ x∈a, 0<x) (hb : ∀ x∈b, 0<x) (ht : a.tail=b.tail) (hs : a.sum=b.sum) : a=b := by
-  cases a with
-  | nil =>
-    cases b with
-    | nil => rfl
-    | cons x l =>
-      simp only [List.tail_nil,List.tail_cons] at ht
-      subst l
-      have hx := hb x (by simp)
-      simp only [List.sum_nil,List.sum_cons,add_zero] at hs
-      omega
-  | cons x l =>
-    cases b with
-    | nil =>
-      simp only [List.tail_nil,List.tail_cons] at ht
-      subst l
-      have hx := ha x (by simp)
-      simp only [List.sum_nil,List.sum_cons,add_zero] at hs
-      omega
-    | cons y m =>
-      simp only [List.tail_cons] at ht
-      subst m
-      simp only [List.sum_cons] at hs
-      congr 1
-      omega
-
-abbrev TailShapes (N j : ℕ) := {μ : Shapes N // μ.1.card-μ.1.rowLen 0=j}
-
-noncomputable def tailShapePartition (N j : ℕ) (μ : TailShapes N j) : Nat.Partition j where
-  parts := μ.1.1.rowLens.tail
-  parts_pos := fun {x} hx => μ.1.1.pos_of_mem_rowLens x (List.mem_of_mem_tail (by simpa using hx))
-  parts_sum := by simpa using (rowLens_tail_sum μ.1.1).trans μ.2
-
-lemma tailShapePartition_injective (N j : ℕ) : Function.Injective (tailShapePartition N j) := by
-  intro μ ν he
-  have hh : (μ.1.1.rowLens.tail : Multiset ℕ)=(ν.1.1.rowLens.tail : Multiset ℕ) :=
-    congrArg Nat.Partition.parts he
-  have ht : μ.1.1.rowLens.tail=ν.1.1.rowLens.tail := by
-    have hm := congrArg (fun s : Multiset ℕ => s.sort (· ≥ ·)) hh
-    simpa only [Multiset.coe_sort,
-      List.mergeSort_eq_self _ μ.1.1.rowLens_sorted.pairwise.tail,
-      List.mergeSort_eq_self _ ν.1.1.rowLens_sorted.pairwise.tail] using hm
-  have hl := positive_list_eq_of_tail_sum (μ.1.1.pos_of_mem_rowLens) (ν.1.1.pos_of_mem_rowLens)
-    ht (by rw [sum_rowLens,sum_rowLens,μ.1.2,ν.1.2])
-  apply Subtype.ext
-  apply Subtype.ext
-  apply YoungDiagram.equivListRowLens.injective
-  exact Subtype.ext hl
-
-lemma tail_shapes_count (N j : ℕ) : Fintype.card (TailShapes N j)≤2^j := by
-  exact (Fintype.card_le_of_injective _ (tailShapePartition_injective N j)).trans (partition_count j)
-
-end Thorp.Specht
-
-namespace Thorp
-open scoped BigOperators Classical
-
-lemma tail_preserving_pairSwitch (d : ℕ) (p : Equiv.Perm (Card (d+1)))
-    (hp : ∀ x, Fin.tail (p x)=Fin.tail x) :
-    pairSwitch (fun u => p (Fin.cons false u) 0)=p := by
-  apply Equiv.ext
-  intro x
-  rw [←Fin.cons_self_tail x]
-  generalize x 0=b
-  generalize Fin.tail x=u
-  apply funext
-  intro i
-  refine Fin.cases ?_ (fun j => ?_) i
-  · change Bool.xor b (p (Fin.cons false u) 0)=p (Fin.cons b u) 0
-    cases b with
-    | false => simp
-    | true =>
-      have hn : p (Fin.cons false u) 0 ≠ p (Fin.cons true u) 0 := by
-        intro he
-        have hh : p (Fin.cons false u)=p (Fin.cons true u) := by
-          rw [←Fin.cons_self_tail (p (Fin.cons false u)),←Fin.cons_self_tail (p (Fin.cons true u)),
-            hp,hp,Fin.tail_cons,Fin.tail_cons,he]
-        have hh := congrFun (p.injective hh) 0
-        simp only [Fin.cons_zero] at hh
-        contradiction
-      cases h0 : p (Fin.cons false u) 0 <;> cases h1 : p (Fin.cons true u) 0 <;> simp_all
-  · change u j=p (Fin.cons b u) j.succ
-    have h := congrFun (hp (Fin.cons b u)) j
-    simpa only [Fin.tail_cons,Fin.tail,Fin.cons_succ] using h.symm
-
-lemma pairSwitch_injective (d : ℕ) : Function.Injective (pairSwitch (d := d)) := by
-  intro ξ η h
-  funext u
-  have hh := congrArg (fun p : Equiv.Perm (Card (d+1)) => p (Fin.cons false u) 0) h
-  change Bool.xor false (ξ u)=Bool.xor false (η u) at hh
-  simpa only [Bool.false_xor] using hh
-
-end Thorp
-namespace Thorp.Specht
-open scoped BigOperators Classical
-
-noncomputable def pairFiberEquiv (d : ℕ) (μ : YoungDiagram) (e : Card (d+1) ≃ Cell μ) :
-    (Card d → Bool) ≃ fiberGroup (fun x => Fin.tail (e.symm x)) :=
-  Equiv.ofBijective (fun ξ => ⟨e.permCongr (pairSwitch ξ),by
-    intro x
-    simp only [Equiv.permCongr_apply,e.symm_apply_apply]
-    exact Fin.tail_cons _ _⟩)
-    ⟨fun ξ η h => pairSwitch_injective d (e.permCongr.injective (congrArg Subtype.val h)),by
-      intro p
-      let q := e.symm.permCongr p.1
-      have hq : ∀ x, Fin.tail (q x)=Fin.tail x := by
-        intro x
-        have h := p.2 (e x)
-        simpa only [q,Equiv.permCongr_apply,Equiv.symm_symm,e.symm_apply_apply] using h
-      refine ⟨fun u => q (Fin.cons false u) 0,?_⟩
-      apply Subtype.ext
-      change e.permCongr (pairSwitch (fun u => q (Fin.cons false u) 0))=p.1
-      rw [tail_preserving_pairSwitch d q hq]
-      apply Equiv.ext
-      intro x
-      simp only [q,Equiv.permCongr_apply,Equiv.symm_symm,e.apply_symm_apply]⟩
-
-lemma pair_sum_taboid_zero (d : ℕ) (μ : YoungDiagram) (e : Card (d+1) ≃ Cell μ)
-    (hj : 2^d < μ.colLen 0) (v : space μ) :
-    (∑ ξ : Card d → Bool, tabloidRep μ (e.permCongr (pairSwitch ξ))) v.1=0 := by
-  have he : (∑ ξ : Card d → Bool, tabloidRep μ (e.permCongr (pairSwitch ξ)))=
-      symmetrizer (fiberGroup (fun x => Fin.tail (e.symm x))) (tabloidRep μ) :=
-    Fintype.sum_equiv (pairFiberEquiv d μ e) _ _ (fun _ => rfl)
-  rw [he]
-  exact tall_column_average_zero μ _ 0 (by simpa only [card_positions] using hj) v.1 v.2
-
-lemma pair_operator_tall_zero (d : ℕ) (μ : YoungDiagram) (e : Card (d+1) ≃ Cell μ)
-    (hj : 2^d < μ.colLen 0) :
-    UnitaryFinite.sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e) pairSwitch=0 := by
-  apply ContinuousLinearMap.ext
-  intro v
-  obtain ⟨w,rfl⟩ := (spaceHilbertEquiv μ).surjective v
-  have hh : (∑ ξ : Card d → Bool, representation μ (e.permCongr (pairSwitch ξ))) w=0 := by
-    apply Subtype.ext
-    simp only [LinearMap.sum_apply,Submodule.coe_sum,Submodule.coe_zero]
-    change (∑ ξ : Card d → Bool, tabloidRep μ (e.permCongr (pairSwitch ξ)) w.1)=0
-    simpa only [LinearMap.sum_apply] using pair_sum_taboid_zero d μ e hj w
-  have hh' := congrArg (spaceHilbertEquiv μ) hh
-  simp only [LinearMap.sum_apply,map_sum,map_zero] at hh'
-  change ((Fintype.card (Card d → Bool):ℂ)⁻¹ •
-    ∑ ξ : Card d → Bool, UnitaryFinite.continuousRepresentation (relabelledUnitary μ e) (pairSwitch ξ))
-      (spaceHilbertEquiv μ w)=0
-  rw [smul_apply,sum_apply]
-  change (Fintype.card (Card d → Bool):ℂ)⁻¹ •
-    (∑ ξ : Card d → Bool, spaceHilbertEquiv μ (representation μ (e.permCongr (pairSwitch ξ))
-      ((spaceHilbertEquiv μ).symm (spaceHilbertEquiv μ w))))=0
-  simp only [LinearEquiv.symm_apply_apply,hh',smul_zero]
-
-open UnitaryFinite
-
-lemma palindrome_tall_zero (d : ℕ) (μ : YoungDiagram) (e : Card (d+1) ≃ Cell μ)
-    (hj : 2^d<μ.colLen 0) :
-    sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e) (palindromePerm (d+1))=0 := by
-  let ρ := relabelledUnitary μ e
-  let : NormedAddCommGroup (hilbertSpace μ) := inferInstance
-  let : InnerProductSpace ℂ (hilbertSpace μ) := inferInstance
-  let : FiniteDimensional ℂ (hilbertSpace μ) := inferInstance
-  have hpair : sampleOperator (V := hilbertSpace μ) ρ (pairSwitch (d := d))=0 := pair_operator_tall_zero d μ e hj
-  have hi : sampleOperator ρ (fun ω => (butterflyPerm (d+1) (decodeButterfly (d+1) ω))⁻¹)=0 := by
-    rw [inverse_butterfly_average_step,hpair,mul_zero]
-  rw [sampleOperator_inv ρ (relabelledUnitary_unitary μ e)] at hi
-  rw [sampleOperator_palindrome _ _ (relabelledUnitary_unitary μ e),hi,mul_zero]
-
-lemma palindrome_tall_zero_general (d : ℕ) (hd : 0<d) (μ : YoungDiagram) (e : Card d ≃ Cell μ)
-    (hj : (μ.card:ℝ)/2<(μ.colLen 0:ℝ)) :
-    sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e) (palindromePerm d)=0 := by
-  obtain ⟨r,rfl⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : d≠0)
-  apply palindrome_tall_zero r μ e
-  have hc : μ.card=2^(r+1) := by rw [←card_cell,←Fintype.card_congr e,card_positions]
-  rw [hc,Nat.cast_pow,Nat.cast_ofNat,pow_succ] at hj
-  have hh : (2:ℝ)^r<(μ.colLen 0:ℝ) := by linarith
-  exact_mod_cast hh
-
-end Thorp.Specht
-
-namespace Thorp.DimensionEstimates
-open Filter
-
-lemma twopow_tendsto : Tendsto (fun d : ℕ => (2:ℝ)^d) atTop atTop :=
-  tendsto_pow_atTop_atTop_of_one_lt (by norm_num)
-
-lemma twopow_nat_tendsto : Tendsto (fun d : ℕ => (2:ℕ)^d) atTop atTop :=
-  tendsto_pow_atTop_atTop_of_one_lt (by norm_num)
-
-lemma twopow_mul_eventually (c : ℝ) (hc : 0<c) (B : ℝ) :
-    ∀ᶠ d : ℕ in atTop, B≤c*(2:ℝ)^d := by
-  filter_upwards [twopow_tendsto.eventually (eventually_ge_atTop (B/c))] with d hd
-  exact (div_le_iff₀ hc).mp hd |>.trans_eq (mul_comm _ _)
-
-lemma exp_two_absorb (β n : ℝ) (hx : Real.log 2≤β*n/2) :
-    Real.exp (-β*n)+Real.exp (-β*n)≤Real.exp (-β*n/2) := by
-  calc
-    _ = Real.exp (Real.log 2-β*n) := by rw [Real.exp_sub,Real.exp_log (by norm_num)]; rw [neg_mul,Real.exp_neg]; ring
-    _ ≤ _ := Real.exp_le_exp.mpr (by linarith)
-
-end Thorp.DimensionEstimates
-namespace Thorp.Specht
-open scoped BigOperators Classical
-open Filter UnitaryFinite DimensionEstimates
-
-theorem exponential_dimension_spectral (h : ℝ) (hh : 0<h) :
-    ∃ a : ℝ, 0<a ∧ ∀ᶠ d : ℕ in atTop, ∀ (μ : YoungDiagram) (e : Card d ≃ Cell μ),
-      h*(2:ℝ)^d≤Real.log (Module.finrank ℂ (space μ)) →
-      ‖sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e) (palindromePerm d)‖ ≤
-        Real.exp (-a*Real.log (Module.finrank ℂ (space μ))) := by
-  obtain ⟨L,hL,ε,t,hε,_,ht,hcount,hM⟩ := medium_parameters h hh
-  obtain ⟨R,hR⟩ := eventually_atTop.mp hM
-  let A := 4000*densityConstant+1
-  have hA : 0<A := by dsimp [A]; have := densityConstant_pos; positivity
-  let β := min (-Real.log (blockGap L)*ε/(2:ℝ)^L) (h/16000)
-  have hβ : 0<β := by
-    apply lt_min
-    · exact div_pos (mul_pos (neg_pos.mpr (Real.log_neg (blockGap_pos L) (blockGap_lt_one L))) hε) (by positivity)
-    · positivity
-  let a := min (1/8000:ℝ) (β/(2*A))
-  have ha : 0<a := lt_min (by norm_num) (div_pos hβ (by positivity))
-  refine ⟨a,ha,?_⟩
-  filter_upwards [eventually_ge_atTop (L+R),twopow_mul_eventually h hh (8000*Real.log 2),
-    twopow_mul_eventually (β/2) (by positivity) (Real.log 2)] with d hd hl hb
-  intro μ e hdim
-  let : NormedAddCommGroup (hilbertSpace μ) := inferInstance
-  let : InnerProductSpace ℂ (hilbertSpace μ) := inferInstance
-  let : FiniteDimensional ℂ (hilbertSpace μ) := inferInstance
-  let := relabelledUnitary_irreducible μ e
-  have hpos : 0<Module.finrank ℂ (hilbertSpace μ) := by rw [finrank_hilbertSpace]; exact dimension_pos μ
-  have hlarge : 8000*Real.log 2≤Real.log (Module.finrank ℂ (hilbertSpace μ)) := by rw [finrank_hilbertSpace]; exact hl.trans hdim
-  have hlog : 0≤Real.log (Module.finrank ℂ (space μ)) := (by positivity : (0:ℝ)≤h*(2:ℝ)^d).trans hdim
-  by_cases hhigh : A*(2:ℝ)^d≤Real.log (Module.finrank ℂ (space μ))
-  · have hden : densityConstant*(2:ℝ)^d≤Real.log (Module.finrank ℂ (hilbertSpace μ))/4000 := by
-      rw [finrank_hilbertSpace]
-      dsimp [A] at hhigh
-      nlinarith [pow_pos (by norm_num : (0:ℝ)<2) d]
-    have hspec := @full_palindrome_decay (hilbertSpace μ) _ _ _ d (relabelledUnitary μ e)
-      inferInstance (relabelledUnitary_unitary μ e) hpos hden hlarge
-    rw [finrank_hilbertSpace] at hspec
-    apply hspec.trans (Real.exp_le_exp.mpr ?_)
-    have := min_le_left (1/8000:ℝ) (β/(2*A))
-    nlinarith
-  · obtain ⟨r,rfl⟩ := Nat.exists_eq_add_of_le (show L≤d by omega)
-    have hr : R≤r := by omega
-    have hdim' : h*(2:ℝ)^(L+r)≤Real.log (Module.finrank ℂ (hilbertSpace μ)) := by rwa [finrank_hilbertSpace]
-    have hs := @medium_exception_decay (hilbertSpace μ) _ _ _ L r h ε t ht hcount
-      (relabelledUnitary μ e) inferInstance (relabelledUnitary_unitary μ e) hpos hdim' hlarge (hR r hr)
-    have hb1 : Real.log (blockGap L)*ε*(2:ℝ)^r≤-β*(2:ℝ)^(L+r) := by
-      have hb0 := min_le_left (-Real.log (blockGap L)*ε/(2:ℝ)^L) (h/16000)
-      have hp := (le_div_iff₀ (by positivity : (0:ℝ)<(2:ℝ)^L)).mp hb0
-      have hp := mul_le_mul_of_nonneg_right hp (by positivity : (0:ℝ)≤(2:ℝ)^r)
-      rw [pow_add]
-      dsimp [β]
-      nlinarith
-    have hb2 : -h*(2:ℝ)^(L+r)/16000≤-β*(2:ℝ)^(L+r) := by
-      have hp := mul_le_mul_of_nonneg_right (min_le_right (-Real.log (blockGap L)*ε/(2:ℝ)^L) (h/16000))
-        (by positivity : (0:ℝ)≤(2:ℝ)^(L+r))
-      dsimp [β]
-      linarith
-    apply hs.trans ((add_le_add (Real.exp_le_exp.mpr hb1) (Real.exp_le_exp.mpr hb2)).trans ?_)
-    apply (exp_two_absorb β ((2:ℝ)^(L+r)) (by nlinarith)).trans
-    apply Real.exp_le_exp.mpr
-    have hdmax := le_of_not_ge hhigh
-    have hp := mul_le_mul_of_nonneg_left hdmax (by positivity : 0≤β/(2*A))
-    have he : β/(2*A)*(A*(2:ℝ)^(L+r))=β*(2:ℝ)^(L+r)/2 := by field_simp
-    rw [he] at hp
-    have hba := mul_le_mul_of_nonneg_right (min_le_right (1/8000:ℝ) (β/(2*A))) hlog
-    dsimp [a] at *
-    linarith
-
-
-theorem spectral_regimes :
-    ∃ a h : ℝ, 0<a ∧ 0<h ∧ ∀ᶠ d : ℕ in atTop,
-      ∀ (μ : YoungDiagram) (e : Card d ≃ Cell μ), 0<μ.card-μ.rowLen 0 →
-        (μ.colLen 0:ℝ)≤(μ.card:ℝ)/2 →
-        ‖sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e) (palindromePerm d)‖≤
-          Real.exp (-a*Real.log (Module.finrank ℂ (space μ))) ∧
-        1≤Real.log (Module.finrank ℂ (space μ)) ∧
-        (((1/2:ℝ)*Real.log μ.card≤Real.log (Module.finrank ℂ (space μ)) ∧
-          (μ.card-μ.rowLen 0:ℕ)≤Real.log (Module.finrank ℂ (space μ))) ∨
-          h*(2:ℝ)^d≤Real.log (Module.finrank ℂ (space μ))) := by
-  obtain ⟨X,hX,hnear⟩ := near_row_spectral
-  have hXpos : 0<X := by linarith
-  have hδ : 0<(1:ℝ)/X := by positivity
-  have hδ2 : (1:ℝ)/X≤1/2 := (div_le_iff₀ hXpos).mpr (by linarith)
-  obtain ⟨h,hh,hmedium⟩ := medium_dimension (1/X) hδ hδ2
-  obtain ⟨a₀,ha₀,hlarge⟩ := exponential_dimension_spectral h hh
-  let a := min (nearFraction/8000) a₀
-  have ha : 0<a := lt_min (by norm_num [nearFraction]) ha₀
-  refine ⟨a,h,ha,hh,?_⟩
-  have hl := (Real.tendsto_log_atTop.comp twopow_tendsto).eventually
-    (eventually_ge_atTop (2*max (32768000*Real.log 2) 1))
-  filter_upwards [hlarge,twopow_nat_tendsto.eventually hmedium,hl,
-    twopow_mul_eventually h hh 1] with d hspec hdim hlog h1
-  intro μ e hj hc
-  simp only [Function.comp_apply] at hlog
-  have hcard : μ.card=2^d := by rw [←card_cell,←Fintype.card_congr e,card_positions]
-  have hcardR : (μ.card:ℝ)=(2:ℝ)^d := by exact_mod_cast hcard
-  have hμlog : max (32768000*Real.log 2) 1≤(1/2:ℝ)*Real.log μ.card := by rw [hcardR]; linarith
-  by_cases hx : X≤(μ.card:ℝ)/(μ.card-μ.rowLen 0:ℕ)
-  · obtain ⟨hs,hl,hjD⟩ := hnear d μ e hj hx ((le_max_left _ _).trans hμlog)
-    have h1D : 1≤Real.log (Module.finrank ℂ (space μ)) := ((le_max_right _ _).trans hμlog).trans hl
-    refine ⟨hs.trans (Real.exp_le_exp.mpr ?_),h1D,Or.inl ⟨hl,hjD⟩⟩
-    have hp := mul_le_mul_of_nonneg_right (min_le_left (nearFraction/8000) a₀) (by linarith : 0≤Real.log (Module.finrank ℂ (space μ)))
-    dsimp [a]
-    linarith
-  · have htail : (1/X)*(μ.card:ℝ)≤(μ.card:ℝ)-μ.rowLen 0 := by
-      have hjR : (0:ℝ)<(μ.card-μ.rowLen 0:ℕ) := by exact_mod_cast hj
-      have hh := (div_lt_iff₀ hjR).mp (lt_of_not_ge hx)
-      have hs : ((μ.card-μ.rowLen 0:ℕ):ℝ)=(μ.card:ℝ)-μ.rowLen 0 := by
-        rw [Nat.cast_sub (rowLen_zero_le_card μ)]
-      rw [hs] at hh
-      have hd : (μ.card:ℝ)/X≤(μ.card:ℝ)-μ.rowLen 0 :=
-        (div_le_iff₀ hXpos).mpr (by nlinarith)
-      convert hd using 1; ring
-    have hdim' : h*(2:ℝ)^d≤Real.log (Module.finrank ℂ (space μ)) := by
-      simpa only [Nat.cast_pow,Nat.cast_ofNat] using
-        hdim μ hcard (by simpa only [hcard] using htail) (by simpa only [hcard] using hc)
-    have h1D := h1.trans hdim'
-    refine ⟨(hspec μ e hdim').trans (Real.exp_le_exp.mpr ?_),h1D,Or.inr hdim'⟩
-    have hp := mul_le_mul_of_nonneg_right (min_le_right (nearFraction/8000) a₀) (by linarith : 0≤Real.log (Module.finrank ℂ (space μ)))
-    dsimp [a]
-    linarith
-
-end Thorp.Specht
-
-namespace Thorp.StrongSmoothing
-
-noncomputable def regularAbsoluteEvenTrace (d : ℕ) (order : Equiv.Perm (Fin d)) (u : ℕ) : ℝ :=
-  Matrix.trace ((reflectedKernel d (2^d) order false)^u)
-
-def MainStatement : Prop :=
-  ∃ u : ℕ, 1 ≤ u ∧ ∃ C : ℝ,
-    (∀ d l : ℕ, l ≤ 2^d → ∀ order : Equiv.Perm (Fin d), ∀ orientation : Bool,
-      ∀ x y : Tuples d l,
-        ((reflectedKernel d l order orientation)^u) x y ≤
-          Real.exp (C*l)/((2^d).descFactorial l:ℝ)) ∧
-    (∀ d : ℕ, ∀ order : Equiv.Perm (Fin d),
-      regularAbsoluteEvenTrace d order u ≤ Real.exp (C*(2:ℝ)^d))
-
-
-open scoped BigOperators
-
-variable {X : Type*} [Fintype X] [Nonempty X]
-
-omit [Nonempty X] in
-lemma mean_nonneg (f : X → ℝ) (hf : ∀ x, 0 ≤ f x) : 0 ≤ finiteMean f :=
-  finiteMean_nonneg hf
-
-lemma mean_rpow_concave (f : X → ℝ) (hf : ∀ x, 0 ≤ f x)
-    {a : ℝ} (ha : 0 ≤ a) (ha1 : a ≤ 1) :
-    finiteMean (fun x => (f x)^a) ≤ (finiteMean f)^a := by
-  have hN : (0:ℝ) < Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
-  have hh := (Real.concaveOn_rpow ha ha1).le_map_sum
-    (t := Finset.univ) (w := fun _ : X => 1/(Fintype.card X:ℝ))
-    (p := f) (fun _ _ => by positivity)
-    (by simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]; field_simp)
-    (fun x _ => hf x)
-  simpa only [smul_eq_mul,one_div,←Finset.mul_sum,mul_comm _ (∑ _ : X,_),
-    ←div_eq_mul_inv,finiteMean] using hh
-
-noncomputable def act (K : Matrix X X ℝ) (f : X → ℝ) (y : X) : ℝ := ∑ x, K x y*f x
-
-omit [Nonempty X] in
-lemma act_nonneg (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
-    (f : X → ℝ) (hf : ∀ x, 0 ≤ f x) (y : X) : 0 ≤ act K f y :=
-  Finset.sum_nonneg (fun x _ => mul_nonneg (hK x y) (hf x))
-
-omit [Nonempty X] in
-lemma act_rpow_le (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
-    (hcol : ∀ y, ∑ x, K x y = 1) (f : X → ℝ) (hf : ∀ x, 0 ≤ f x)
-    {b : ℝ} (hb : 1 ≤ b) (y : X) :
-    (act K f y)^b ≤ act K (fun x => (f x)^b) y := by
-  simpa only [smul_eq_mul,act] using
-    (convexOn_rpow hb).map_sum_le (t := Finset.univ) (w := fun x => K x y)
-      (p := f) (fun x _ => hK x y) (hcol y) (fun x _ => hf x)
-
-lemma act_one_to_p (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
-    {p A : ℝ} (hp : 1 ≤ p)
-    (hm : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) ≤ Real.exp A)
-    (f : X → ℝ) (hf : ∀ x, 0 ≤ f x) :
-    finiteMean (fun y => (act K f y)^p) ≤ Real.exp A*(finiteMean f)^p := by
-  classical
-  let s : ℝ := ∑ x, f x
-  have hs : 0 ≤ s := Finset.sum_nonneg (fun x _ => hf x)
-  have hN : (0:ℝ) < Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
-  by_cases hz : s=0
-  · have hfz : ∀ x, f x=0 := by
-      intro x
-      exact (Finset.sum_eq_zero_iff_of_nonneg (fun x _ => hf x)).mp hz x (Finset.mem_univ _)
-    simp only [act,hfz,mul_zero,Finset.sum_const_zero,finiteMean,zero_div,
-      Real.zero_rpow (show p ≠ 0 by linarith),mul_zero,le_refl]
-  · have hsp : 0 < s := lt_of_le_of_ne hs (Ne.symm hz)
-    let w : X → ℝ := fun x => f x/s
-    have hw : ∀ x, 0 ≤ w x := fun x => div_nonneg (hf x) hs
-    have hws : ∑ x, w x = 1 := by simp only [w,←Finset.sum_div]; exact div_self hz
-    have hact (y : X) : act K f y = finiteMean f * ∑ x, w x*((Fintype.card X:ℝ)*K x y) := by
-      simp only [act,w,finiteMean,Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro x _
-      have hz' : (∑ x, f x) ≠ 0 := hz
-      dsimp [s]
-      field_simp [hz',ne_of_gt hN]
-    have hj (y : X) : (act K f y)^p ≤ (finiteMean f)^p*
-        ∑ x, w x*((Fintype.card X:ℝ)*K x y)^p := by
-      rw [hact,Real.mul_rpow (mean_nonneg f hf)
-        (Finset.sum_nonneg (fun x _ => mul_nonneg (hw x) (mul_nonneg hN.le (hK x y))))]
-      apply mul_le_mul_of_nonneg_left _ (Real.rpow_nonneg (mean_nonneg f hf) _)
-      exact (convexOn_rpow hp).map_sum_le
-        (fun x _ => hw x) hws (fun x _ => mul_nonneg hN.le (hK x y))
-    have hh := finiteMean_mono hj
-    have he : finiteMean (fun y => (finiteMean f)^p*
-        ∑ x, w x*((Fintype.card X:ℝ)*K x y)^p) =
-        (finiteMean f)^p * ∑ x, w x*finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) := by
-      simp_rw [mul_comm ((finiteMean f)^p), finiteMean_mul_const,
-        finiteMean_sum, mul_comm (w _), finiteMean_mul_const]
-    rw [he] at hh
-    apply hh.trans
-    have hsum : (∑ x, w x*finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p)) ≤ Real.exp A := by
-      calc
-        _ ≤ ∑ x, w x*Real.exp A := Finset.sum_le_sum
-          (fun x _ => mul_le_mul_of_nonneg_left (hm x) (hw x))
-        _ = Real.exp A := by rw [←Finset.sum_mul,hws,one_mul]
-    simpa only [mul_comm (Real.exp A)] using
-      mul_le_mul_of_nonneg_left hsum (Real.rpow_nonneg (mean_nonneg f hf) _)
-
-lemma act_b_to_pb (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
-    (hcol : ∀ y, ∑ x, K x y = 1) {p A b : ℝ} (hp : 1 ≤ p) (hb : 1 ≤ b)
-    (hm : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) ≤ Real.exp A)
-    (f : X → ℝ) (hf : ∀ x, 0 ≤ f x) :
-    finiteMean (fun y => (act K f y)^(b*p)) ≤
-      Real.exp A*(finiteMean (fun x => (f x)^b))^p := by
-  calc
-    _ ≤ finiteMean (fun y => (act K (fun x => (f x)^b) y)^p) := by
-      apply finiteMean_mono
-      intro y
-      rw [Real.rpow_mul (act_nonneg K hK f hf y)]
-      exact Real.rpow_le_rpow (Real.rpow_nonneg (act_nonneg K hK f hf y) _) (act_rpow_le K hK hcol f hf hb y) (by linarith)
-    _ ≤ _ := act_one_to_p K hK hp hm _ (fun x => Real.rpow_nonneg (hf x) _)
-
-variable [DecidableEq X]
-
-omit [Nonempty X] in
-lemma kernel_pow_nonneg (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y) (r : ℕ) :
-    ∀ x y, 0 ≤ (K^r) x y := by
-  classical
-  induction r with
-  | zero => intro x y; simp only [pow_zero,Matrix.one_apply]; split_ifs <;> norm_num
-  | succ r ih =>
-    intro x y
-    rw [pow_succ,Matrix.mul_apply]
-    exact Finset.sum_nonneg (fun z _ => mul_nonneg (ih x z) (hK z y))
-
-omit [Nonempty X] in
-lemma kernel_pow_row (K : Matrix X X ℝ) (hrow : ∀ x, ∑ y, K x y = 1) (r : ℕ) :
-    ∀ x, ∑ y, (K^r) x y = 1 := by
-  classical
-  induction r with
-  | zero => intro x; simp [Matrix.one_apply]
-  | succ r ih =>
-    intro x
-    simp only [pow_succ,Matrix.mul_apply]
-    rw [Finset.sum_comm]
-    simp only [←Finset.mul_sum,hrow,mul_one,ih]
-
-noncomputable def rowDensity (K : Matrix X X ℝ) (r : ℕ) (x y : X) : ℝ :=
-  (Fintype.card X:ℝ)*(K^r) x y
-
-omit [Nonempty X] in
-lemma rowDensity_nonneg (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y) (r : ℕ) (x y : X) :
-    0 ≤ rowDensity K r x y := mul_nonneg (Nat.cast_nonneg _) (kernel_pow_nonneg K hK r x y)
-
-omit [Nonempty X] in
-lemma rowDensity_succ (K : Matrix X X ℝ) (r : ℕ) (x y : X) :
-    rowDensity K (r+1) x y = act K (rowDensity K r x) y := by
-  simp only [rowDensity,pow_succ,Matrix.mul_apply,act,Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro z _
-  ring
-
-lemma rowDensity_mean (K : Matrix X X ℝ) (hrow : ∀ x, ∑ y, K x y = 1) (r : ℕ) (x : X) :
-    finiteMean (rowDensity K r x) = 1 := by
-  have hN : (Fintype.card X:ℝ) ≠ 0 := ne_of_gt (Nat.cast_pos.mpr Fintype.card_pos)
-  simp only [finiteMean,rowDensity,←Finset.mul_sum,kernel_pow_row K hrow,mul_one,div_self hN]
-
-lemma rowDensity_moment_iter (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
-    (hrow : ∀ x, ∑ y, K x y = 1) (hcol : ∀ y, ∑ x, K x y = 1)
-    {p A : ℝ} (hp : 1 ≤ p) (hA : 0 ≤ A)
-    (hm : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) ≤ Real.exp A)
-    (r : ℕ) (x : X) :
-    finiteMean (fun y => (rowDensity K r x y)^(p^r)) ≤ Real.exp (A*r*p^r) := by
-  induction r with
-  | zero => simp [rowDensity_mean K hrow]
-  | succ r ih =>
-    have hpr : 1 ≤ p^r := one_le_pow₀ hp
-    have hps : 1 ≤ p^(r+1) := one_le_pow₀ hp
-    simp_rw [rowDensity_succ]
-    calc
-      _ ≤ Real.exp A*(finiteMean (fun y => (rowDensity K r x y)^(p^r)))^p := by
-        rw [pow_succ]
-        exact act_b_to_pb K hK hcol hp hpr hm _ (rowDensity_nonneg K hK r x)
-      _ ≤ Real.exp A*(Real.exp (A*r*p^r))^p := by
-        apply mul_le_mul_of_nonneg_left _ (Real.exp_pos _).le
-        exact Real.rpow_le_rpow (mean_nonneg _ (fun y => Real.rpow_nonneg
-          (rowDensity_nonneg K hK r x y) _)) ih (le_trans zero_le_one hp)
-      _ = Real.exp (A+A*r*p^(r+1)) := by rw [←Real.exp_mul,←Real.exp_add,pow_succ]; congr 1; ring
-      _ ≤ _ := by
-        apply Real.exp_le_exp.mpr
-        have hh := mul_le_mul_of_nonneg_left hps hA
-        push_cast
-        nlinarith
-
-lemma rowDensity_second_moment (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
-    (hrow : ∀ x, ∑ y, K x y = 1) (hcol : ∀ y, ∑ x, K x y = 1)
-    {p A : ℝ} (hp : 1 ≤ p) (hA : 0 ≤ A)
-    (hm : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) ≤ Real.exp A)
-    (r : ℕ) (hr : 2 ≤ p^r) (x : X) :
-    finiteMean (fun y => (rowDensity K r x y)^2) ≤ Real.exp (2*A*r) := by
-  have hpos : 0 < p^r := by linarith
-  have hj := mean_rpow_concave (fun y => (rowDensity K r x y)^(p^r))
-    (fun y => Real.rpow_nonneg (rowDensity_nonneg K hK r x y) _)
-    (a := 2/(p^r)) (by positivity) ((div_le_one hpos).mpr hr)
-  have he (y : X) : ((rowDensity K r x y)^(p^r))^(2/(p^r)) = (rowDensity K r x y)^2 := by
-    rw [←Real.rpow_mul (rowDensity_nonneg K hK r x y)]
-    rw [mul_div_cancel₀ _ (ne_of_gt hpos),Real.rpow_two]
-  simp_rw [he] at hj
-  apply hj.trans
-  calc
-    _ ≤ (Real.exp (A*r*p^r))^(2/(p^r)) :=
-      Real.rpow_le_rpow (mean_nonneg _ (fun y => Real.rpow_nonneg
-        (rowDensity_nonneg K hK r x y) _)) (rowDensity_moment_iter K hK hrow hcol hp hA hm r x)
-        (by positivity)
-    _ = _ := by rw [←Real.exp_mul]; congr 1; field_simp [ne_of_gt hpos]
-
-lemma symmetric_kernel_smoothing (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
-    (hrow : ∀ x, ∑ y, K x y = 1) (hcol : ∀ y, ∑ x, K x y = 1)
-    (hsym : K.IsSymm) {p A : ℝ} (hp : 1 ≤ p) (hA : 0 ≤ A)
-    (hm : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) ≤ Real.exp A)
-    (r : ℕ) (hr : 2 ≤ p^r) (x y : X) :
-    (K^(2*r)) x y ≤ Real.exp (2*A*r)/(Fintype.card X:ℝ) := by
-  have hN : (0:ℝ) < Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
-  have hs : ∀ z, (K^r) z y = (K^r) y z := fun z => (hsym.pow r).apply y z
-  have he : (Fintype.card X:ℝ)*(K^(2*r)) x y =
-      finiteMean (fun z => rowDensity K r x z*rowDensity K r y z) := by
-    simp only [finiteMean,rowDensity,two_mul,pow_add,Matrix.mul_apply,hs,Finset.mul_sum]
-    rw [Finset.sum_div]
-    apply Finset.sum_congr rfl
-    intro z _
-    field_simp
-  have hh : finiteMean (fun z => rowDensity K r x z*rowDensity K r y z) ≤
-      Real.exp (2*A*r) := by
-    calc
-      _ ≤ finiteMean (fun z => ((rowDensity K r x z)^2+(rowDensity K r y z)^2)/2) := by
-        apply finiteMean_mono
-        intro z
-        nlinarith [sq_nonneg (rowDensity K r x z-rowDensity K r y z)]
-      _ = (finiteMean (fun z => (rowDensity K r x z)^2)+
-          finiteMean (fun z => (rowDensity K r y z)^2))/2 := by
-        simp only [div_eq_mul_inv,finiteMean_mul_const,finiteMean_add]
-      _ ≤ _ := by
-        have hx := rowDensity_second_moment K hK hrow hcol hp hA hm r hr x
-        have hy := rowDensity_second_moment K hK hrow hcol hp hA hm r hr y
-        linarith
-  apply (le_div_iff₀ hN).mpr
-  simpa only [←he,mul_comm (Fintype.card X:ℝ)] using hh
-
-lemma density_exponent_numeric : 2 ≤ (1+(1/1000:ℝ))^(1000:ℕ) := by
-  calc
-    (2:ℝ) = 1+(1000:ℝ)*(1/1000) := by norm_num
-    _ ≤ _ := by simpa only [Nat.cast_ofNat] using
-      (one_add_mul_le_pow (a := (1/1000:ℝ)) (by norm_num) 1000)
-
-end Thorp.StrongSmoothing
-
-namespace Thorp.StrongSmoothing.PermLaw
-open scoped BigOperators
-variable {X Ω Ω' : Type*} [Fintype X] [DecidableEq X] [Fintype Ω] [Fintype Ω']
-
-noncomputable def kernel (P : Ω → Equiv.Perm X) : Matrix X X ℝ :=
-  fun x y => finiteMean (fun ω => if P ω x = y then (1:ℝ) else 0)
-
-omit [Fintype X] in
-lemma nonneg (P : Ω → Equiv.Perm X) (x y : X) : 0 ≤ kernel P x y := by
-  exact finiteMean_nonneg (fun ω => by split_ifs <;> norm_num)
-
-lemma row [Nonempty Ω] (P : Ω → Equiv.Perm X) (x : X) : ∑ y, kernel P x y = 1 := by
-  simp only [kernel,←finiteMean_sum]
-  simp only [Finset.sum_ite_eq,Finset.mem_univ,↓reduceIte,finiteMean_const]
-
-omit [Fintype X] in
-lemma inverse (P : Ω → Equiv.Perm X) : kernel (fun ω => (P ω)⁻¹) = (kernel P).transpose := by
-  ext x y
-  apply finiteMean_congr
-  intro ω
-  have he : (P ω)⁻¹ x=y ↔ P ω y=x := by
-    change (P ω).symm x=y ↔ P ω y=x
-    exact (Equiv.symm_apply_eq (P ω)).trans eq_comm
-  simp only [he]
-
-lemma col [Nonempty Ω] (P : Ω → Equiv.Perm X) (y : X) : ∑ x, kernel P x y = 1 := by
-  have hh := row (fun ω => (P ω)⁻¹) y
-  rw [inverse] at hh
-  exact hh
-
-lemma mul (P : Ω → Equiv.Perm X) (Q : Ω' → Equiv.Perm X) :
-    kernel (fun ω : Ω × Ω' => P ω.1*Q ω.2) = kernel Q*kernel P := by
-  ext x y
-  simp only [Matrix.mul_apply,kernel]
-  have he (ω : Ω × Ω') :
-      (if (P ω.1*Q ω.2) x=y then (1:ℝ) else 0) =
-        ∑ z : X, (if P ω.1 z=y then (1:ℝ) else 0)*(if Q ω.2 x=z then (1:ℝ) else 0) := by
-    simp only [mul_ite,mul_one,mul_zero,Finset.sum_ite_eq,Finset.mem_univ,↓reduceIte]
-    rfl
-  change finiteMean (fun ω : Ω × Ω' => if (P ω.1*Q ω.2) x=y then (1:ℝ) else 0) = _
-  simp_rw [he]
-  rw [finiteMean_sum]
-  apply Finset.sum_congr rfl
-  intro z _
-  exact (finiteMean_prod_mul (fun ω : Ω => if P ω z=y then (1:ℝ) else 0)
-    (fun ω : Ω' => if Q ω x=z then (1:ℝ) else 0)).trans (mul_comm _ _)
-
-omit [Fintype X] in
-lemma conjugate (P : Ω → Equiv.Perm X) (e : Equiv.Perm X) (x y : X) :
-    kernel (fun ω => e*P ω*e⁻¹) x y = kernel P (e.symm x) (e.symm y) := by
-  apply finiteMean_congr
-  intro ω
-  have he : (e*P ω*e⁻¹) x=y ↔ P ω (e.symm x)=e.symm y := by
-    change e (P ω (e.symm x))=y ↔ _
-    exact (Equiv.eq_symm_apply e).symm
-  simp only [he]
-
-end Thorp.StrongSmoothing.PermLaw
-
-namespace Thorp.StrongSmoothing
-open scoped BigOperators
-
-section MatrixHelpers
-variable {X : Type*} [Fintype X]
-
-lemma mul_nonneg_entries (A B : Matrix X X ℝ) (hA : ∀ x y, 0 ≤ A x y)
-    (hB : ∀ x y, 0 ≤ B x y) (x y : X) : 0 ≤ (A*B) x y :=
-  Finset.sum_nonneg (fun z _ => mul_nonneg (hA x z) (hB z y))
-
-lemma mul_row (A B : Matrix X X ℝ) (hA : ∀ x, ∑ y, A x y = 1)
-    (hB : ∀ x, ∑ y, B x y = 1) (x : X) : ∑ y, (A*B) x y = 1 := by
-  simp only [Matrix.mul_apply]
-  rw [Finset.sum_comm]
-  simp only [←Finset.mul_sum,hB,mul_one,hA]
-
-lemma mul_col (A B : Matrix X X ℝ) (hA : ∀ y, ∑ x, A x y = 1)
-    (hB : ∀ y, ∑ x, B x y = 1) (y : X) : ∑ x, (A*B) x y = 1 := by
-  simp only [Matrix.mul_apply]
-  rw [Finset.sum_comm]
-  simp only [←Finset.sum_mul,hA,one_mul,hB]
-
-lemma bound_mul_left (A B : Matrix X X ℝ) (hA : ∀ x y, 0 ≤ A x y)
-    (hrow : ∀ x, ∑ y, A x y = 1) {D : ℝ} (hB : ∀ x y, B x y ≤ D) :
-    ∀ x y, (A*B) x y ≤ D := by
-  intro x y
-  calc
-    _ ≤ ∑ z, A x z*D := Finset.sum_le_sum (fun z _ => mul_le_mul_of_nonneg_left (hB z y) (hA x z))
-    _ = D := by rw [←Finset.sum_mul,hrow,one_mul]
-
-lemma bound_mul_right (A B : Matrix X X ℝ) (hB : ∀ x y, 0 ≤ B x y)
-    (hcol : ∀ y, ∑ x, B x y = 1) {D : ℝ} (hA : ∀ x y, A x y ≤ D) :
-    ∀ x y, (A*B) x y ≤ D := by
-  intro x y
-  calc
-    _ ≤ ∑ z, D*B z y := Finset.sum_le_sum (fun z _ => mul_le_mul_of_nonneg_right (hA x z) (hB z y))
-    _ = D := by rw [←Finset.mul_sum,hcol,mul_one]
-
-variable [DecidableEq X]
-
-lemma sandwich_power (A B : Matrix X X ℝ) (r : ℕ) : A*(B*A)^r*B = (A*B)^(r+1) := by
-  induction r with
-  | zero => simp
-  | succ r ih =>
-    calc
-      _ = (A*(B*A)^r*B)*(A*B) := by rw [pow_succ]; noncomm_ring
-      _ = (A*B)^(r+1)*(A*B) := by rw [ih]
-      _ = _ := (pow_succ (A*B) (r+1)).symm
-
-end MatrixHelpers
-
-section TupleAction
-variable {ι β Ω : Type*} [Fintype Ω]
-
-def tuplePerm (P : Equiv.Perm β) : Equiv.Perm (ι ↪ β) where
-  toFun x := x.trans P.toEmbedding
-  invFun x := x.trans P.symm.toEmbedding
-  left_inv x := by ext i; simp
-  right_inv x := by ext i; simp
-
-@[simp] lemma tuplePerm_apply (P : Equiv.Perm β) (x : ι ↪ β) (i : ι) :
-    tuplePerm P x i = P (x i) := rfl
-
-lemma tuplePerm_mul (P Q : Equiv.Perm β) :
-    tuplePerm (ι := ι) (P*Q) = tuplePerm P*tuplePerm Q := by
-  ext x i
-  rfl
-
-lemma tuplePerm_inverse (P : Equiv.Perm β) :
-    tuplePerm (ι := ι) P⁻¹ = (tuplePerm P)⁻¹ := by
-  ext x i
-  rfl
-
-variable [Fintype ι] [DecidableEq β]
-
-lemma tupleProbability_kernel (P : Ω → Equiv.Perm β) (x y : ι ↪ β) :
-    PairRouting.tupleProbability P x y = PermLaw.kernel (fun ω => tuplePerm (P ω)) x y := by
-  classical
-  apply finiteMean_congr
-  intro ω
-  have he : (∀ i, P ω (x i) = y i) ↔ tuplePerm (P ω) x = y := by
-    constructor
-    · intro h; ext i; exact h i
-    · intro h i; exact congrArg (fun z : ι ↪ β => z i) h
-  simp only [he]
-end TupleAction
-
-lemma sweepKernel_law (d l : ℕ) (order : Equiv.Perm (Fin d)) :
-    sweepKernel d l order = PermLaw.kernel (fun ω => tuplePerm (ι := Fin l) (sweep d order ω)) := by
-  ext x y
-  exact tupleProbability_kernel _ x y
-
-lemma reflectedKernel_conjugate (d l : ℕ) (order : Equiv.Perm (Fin d)) (x y : Tuples d l) :
-    reflectedKernel d l order false x y =
-      PairRouting.tupleProbability (palindromePerm d)
-        ((tuplePerm (coordinateRelabel d order)).symm x)
-        ((tuplePerm (coordinateRelabel d order)).symm y) := by
-  classical
-  let e : Equiv.Perm (Tuples d l) := tuplePerm (coordinateRelabel d order)
-  let P := fun ω => tuplePerm (ι := Fin l) (butterflyPerm d (decodeButterfly d ω))
-  have hA : sweepKernel d l order = PermLaw.kernel (fun ω => e*P ω*e⁻¹) := by
-    rw [sweepKernel_law]
-    simp only [sweep,tuplePerm_mul,tuplePerm_inverse]
-    rfl
-  rw [reflectedKernel,ite_eq_right Bool.false_ne_true,hA]
-  rw [←PermLaw.inverse,←PermLaw.mul]
-  have hprod : (fun ω : BenesCoins d => (e*P ω.1*e⁻¹)*((e*P ω.2*e⁻¹)⁻¹)) =
-      fun ω : BenesCoins d => e*tuplePerm (ι := Fin l) (palindromePerm d ω)*e⁻¹ := by
-    funext ω
-    simp only [mul_inv_rev,inv_inv,palindromePerm,tuplePerm_mul]
-    change (e*P ω.1*e⁻¹)*(e*(P ω.2)⁻¹*e⁻¹) = e*(P ω.1*(P ω.2)⁻¹)*e⁻¹
-    group
-  rw [hprod,PermLaw.conjugate,tupleProbability_kernel]
-
-
-lemma sweepKernel_nonneg (d l : ℕ) (order : Equiv.Perm (Fin d)) (x y : Tuples d l) :
-    0 ≤ sweepKernel d l order x y := PairRouting.tupleProbability_nonneg _ _ _
-
-lemma sweepKernel_row (d l : ℕ) (order : Equiv.Perm (Fin d)) (x : Tuples d l) :
-    ∑ y, sweepKernel d l order x y = 1 := by
-  rw [sweepKernel_law]
-  exact PermLaw.row _ _
-
-lemma sweepKernel_col (d l : ℕ) (order : Equiv.Perm (Fin d)) (y : Tuples d l) :
-    ∑ x, sweepKernel d l order x y = 1 := by
-  rw [sweepKernel_law]
-  exact PermLaw.col _ _
-
-lemma reflected_nonneg (d l : ℕ) (order : Equiv.Perm (Fin d)) (x y : Tuples d l) :
-    0 ≤ reflectedKernel d l order false x y := by
-  exact mul_nonneg_entries _ _ (fun x y => sweepKernel_nonneg d l order y x)
-    (sweepKernel_nonneg d l order) x y
-
-lemma reflected_row (d l : ℕ) (order : Equiv.Perm (Fin d)) (x : Tuples d l) :
-    ∑ y, reflectedKernel d l order false x y = 1 :=
-  mul_row _ _ (sweepKernel_col d l order) (sweepKernel_row d l order) x
-
-lemma reflected_col (d l : ℕ) (order : Equiv.Perm (Fin d)) (y : Tuples d l) :
-    ∑ x, reflectedKernel d l order false x y = 1 :=
-  mul_col _ _ (sweepKernel_row d l order) (sweepKernel_col d l order) y
-
-lemma reflected_symmetric (d l : ℕ) (order : Equiv.Perm (Fin d)) :
-    (reflectedKernel d l order false).IsSymm := Matrix.isSymm_transpose_mul_self _
-
-lemma reflected_moment (d l : ℕ) (order : Equiv.Perm (Fin d)) (x : Tuples d l) :
-    finiteMean (fun y : Tuples d l =>
-      ((Fintype.card (Tuples d l):ℝ)*reflectedKernel d l order false x y)^(1+(1/1000:ℝ))) ≤
-      Real.exp (l*(1+heightDecay 1)) := by
-  classical
-  let e : Equiv.Perm (Tuples d l) := tuplePerm (coordinateRelabel d order)
-  simp_rw [reflectedKernel_conjugate]
-  have he := finiteMean_equiv e (fun y : Tuples d l =>
-    ((Fintype.card (Tuples d l):ℝ)*PairRouting.tupleProbability (palindromePerm d) (e.symm x) y)^
-      (1+(1/1000:ℝ)))
-  change finiteMean (fun y =>
-    ((Fintype.card (Tuples d l):ℝ)*PairRouting.tupleProbability (palindromePerm d) (e.symm x) (e.symm y))^
-      (1+(1/1000:ℝ))) ≤ _
-  rw [←he]
-  simpa only [palindromeRowMoment,Fintype.card_fin] using palindrome_row_moment_coarse d (e.symm x)
-
-lemma tuples_card (d l : ℕ) : Fintype.card (Tuples d l) = (2^d).descFactorial l := by
-  simp only [Tuples,Fintype.card_embedding_eq,Fintype.card_fin,card_positions]
-
-lemma canonical_pointwise (d l : ℕ) (order : Equiv.Perm (Fin d)) (x y : Tuples d l) :
-    ((reflectedKernel d l order false)^2000) x y ≤
-      Real.exp ((2000*(1+heightDecay 1))*l)/((2^d).descFactorial l:ℝ) := by
-  let : Nonempty (Tuples d l) := ⟨x⟩
-  have hh := symmetric_kernel_smoothing (reflectedKernel d l order false)
-    (reflected_nonneg d l order) (reflected_row d l order) (reflected_col d l order)
-    (reflected_symmetric d l order) (p := 1+(1/1000:ℝ)) (A := l*(1+heightDecay 1))
-    (by norm_num) (mul_nonneg (Nat.cast_nonneg _) (by linarith [heightDecay_nonneg 1]))
-    (reflected_moment d l order) 1000 density_exponent_numeric x y
-  have he : 2*((l:ℝ)*(1+heightDecay 1))*(1000:ℝ) = (2000*(1+heightDecay 1))*l := by ring
-  norm_num only [show 2*1000=2000 by norm_num,Nat.cast_ofNat] at hh
-  simpa only [he,tuples_card] using hh
-
-lemma both_pointwise (d l : ℕ) (order : Equiv.Perm (Fin d)) (orientation : Bool)
-    (x y : Tuples d l) :
-    ((reflectedKernel d l order orientation)^2001) x y ≤
-      Real.exp ((2000*(1+heightDecay 1))*l)/((2^d).descFactorial l:ℝ) := by
-  cases orientation
-  · change ((reflectedKernel d l order false)^(2000+1)) x y ≤ _
-    rw [pow_succ]
-    exact bound_mul_right _ _ (reflected_nonneg d l order) (reflected_col d l order)
-      (canonical_pointwise d l order) x y
-  · let A := sweepKernel d l order
-    change ((A*A.transpose)^(2000+1)) x y ≤ _
-    rw [←sandwich_power]
-    apply bound_mul_right _ _ (fun x y => sweepKernel_nonneg d l order y x)
-      (sweepKernel_row d l order)
-    exact bound_mul_left _ _ (sweepKernel_nonneg d l order) (sweepKernel_row d l order)
-      (canonical_pointwise d l order)
-
-end Thorp.StrongSmoothing
-
-end ThorpNine.Harmonic
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Harmonic.InsertionPerm
+
+namespace OAI
+
+namespace ThorpNine.Harmonic
+
+namespace Thorp.Specht
+open scoped BigOperators Classical
+
+lemma partition_count (N : ℕ) : Fintype.card (Nat.Partition N)≤2^N := by
+  have h := Fintype.card_le_of_surjective (Nat.Partition.ofComposition N) Nat.Partition.ofComposition_surj
+  rw [composition_card] at h
+  exact h.trans (Nat.pow_le_pow_right (by norm_num) (Nat.sub_le N 1))
+
+lemma shapes_count (N : ℕ) : Fintype.card (Shapes N)≤2^N := by
+  rw [Fintype.card_congr (shapePartitionEquiv N)]
+  exact partition_count N
+
+lemma rowLens_headD (μ : YoungDiagram) : μ.rowLens.headD 0=μ.rowLen 0 := by
+  cases he : μ.rowLens with
+  | nil =>
+    have hc := sum_rowLens μ
+    rw [he,List.sum_nil] at hc
+    have hr := rowLen_zero_le_card μ
+    simp only [List.headD_nil]
+    omega
+  | cons a l =>
+    have hh := YoungDiagram.get_rowLens (μ := μ) (i := 0) (h := by simp [he])
+    simpa only [he,List.getElem_cons_zero,List.headD_cons] using hh
+
+lemma rowLens_tail_sum (μ : YoungDiagram) : μ.rowLens.tail.sum=μ.card-μ.rowLen 0 := by
+  have hh := rowLens_headD μ
+  have hc := sum_rowLens μ
+  cases he : μ.rowLens with
+  | nil => simp only [he,List.headD_nil] at hh; simp only [he,List.sum_nil] at hc; simp [←hc]
+  | cons a l =>
+    simp only [he,List.headD_cons] at hh
+    simp only [he,List.sum_cons] at hc
+    simp only [List.tail_cons]
+    omega
+
+lemma positive_list_eq_of_tail_sum {a b : List ℕ}
+    (ha : ∀ x∈a, 0<x) (hb : ∀ x∈b, 0<x) (ht : a.tail=b.tail) (hs : a.sum=b.sum) : a=b := by
+  cases a with
+  | nil =>
+    cases b with
+    | nil => rfl
+    | cons x l =>
+      simp only [List.tail_nil,List.tail_cons] at ht
+      subst l
+      have hx := hb x (by simp)
+      simp only [List.sum_nil,List.sum_cons,add_zero] at hs
+      omega
+  | cons x l =>
+    cases b with
+    | nil =>
+      simp only [List.tail_nil,List.tail_cons] at ht
+      subst l
+      have hx := ha x (by simp)
+      simp only [List.sum_nil,List.sum_cons,add_zero] at hs
+      omega
+    | cons y m =>
+      simp only [List.tail_cons] at ht
+      subst m
+      simp only [List.sum_cons] at hs
+      congr 1
+      omega
+
+abbrev TailShapes (N j : ℕ) := {μ : Shapes N // μ.1.card-μ.1.rowLen 0=j}
+
+noncomputable def tailShapePartition (N j : ℕ) (μ : TailShapes N j) : Nat.Partition j where
+  parts := μ.1.1.rowLens.tail
+  parts_pos := fun {x} hx => μ.1.1.pos_of_mem_rowLens x (List.mem_of_mem_tail (by simpa using hx))
+  parts_sum := by simpa using (rowLens_tail_sum μ.1.1).trans μ.2
+
+lemma tailShapePartition_injective (N j : ℕ) : Function.Injective (tailShapePartition N j) := by
+  intro μ ν he
+  have hh : (μ.1.1.rowLens.tail : Multiset ℕ)=(ν.1.1.rowLens.tail : Multiset ℕ) :=
+    congrArg Nat.Partition.parts he
+  have ht : μ.1.1.rowLens.tail=ν.1.1.rowLens.tail := by
+    have hm := congrArg (fun s : Multiset ℕ => s.sort (· ≥ ·)) hh
+    simpa only [Multiset.coe_sort,
+      List.mergeSort_eq_self _ μ.1.1.rowLens_sorted.pairwise.tail,
+      List.mergeSort_eq_self _ ν.1.1.rowLens_sorted.pairwise.tail] using hm
+  have hl := positive_list_eq_of_tail_sum (μ.1.1.pos_of_mem_rowLens) (ν.1.1.pos_of_mem_rowLens)
+    ht (by rw [sum_rowLens,sum_rowLens,μ.1.2,ν.1.2])
+  apply Subtype.ext
+  apply Subtype.ext
+  apply YoungDiagram.equivListRowLens.injective
+  exact Subtype.ext hl
+
+lemma tail_shapes_count (N j : ℕ) : Fintype.card (TailShapes N j)≤2^j := by
+  exact (Fintype.card_le_of_injective _ (tailShapePartition_injective N j)).trans (partition_count j)
+
+end Thorp.Specht
+
+namespace Thorp
+open scoped BigOperators Classical
+
+lemma tail_preserving_pairSwitch (d : ℕ) (p : Equiv.Perm (Card (d+1)))
+    (hp : ∀ x, Fin.tail (p x)=Fin.tail x) :
+    pairSwitch (fun u => p (Fin.cons false u) 0)=p := by
+  apply Equiv.ext
+  intro x
+  rw [←Fin.cons_self_tail x]
+  generalize x 0=b
+  generalize Fin.tail x=u
+  apply funext
+  intro i
+  refine Fin.cases ?_ (fun j => ?_) i
+  · change Bool.xor b (p (Fin.cons false u) 0)=p (Fin.cons b u) 0
+    cases b with
+    | false => simp
+    | true =>
+      have hn : p (Fin.cons false u) 0 ≠ p (Fin.cons true u) 0 := by
+        intro he
+        have hh : p (Fin.cons false u)=p (Fin.cons true u) := by
+          rw [←Fin.cons_self_tail (p (Fin.cons false u)),←Fin.cons_self_tail (p (Fin.cons true u)),
+            hp,hp,Fin.tail_cons,Fin.tail_cons,he]
+        have hh := congrFun (p.injective hh) 0
+        simp only [Fin.cons_zero] at hh
+        contradiction
+      cases h0 : p (Fin.cons false u) 0 <;> cases h1 : p (Fin.cons true u) 0 <;> simp_all
+  · change u j=p (Fin.cons b u) j.succ
+    have h := congrFun (hp (Fin.cons b u)) j
+    simpa only [Fin.tail_cons,Fin.tail,Fin.cons_succ] using h.symm
+
+lemma pairSwitch_injective (d : ℕ) : Function.Injective (pairSwitch (d := d)) := by
+  intro ξ η h
+  funext u
+  have hh := congrArg (fun p : Equiv.Perm (Card (d+1)) => p (Fin.cons false u) 0) h
+  change Bool.xor false (ξ u)=Bool.xor false (η u) at hh
+  simpa only [Bool.false_xor] using hh
+
+end Thorp
+namespace Thorp.Specht
+open scoped BigOperators Classical
+
+noncomputable def pairFiberEquiv (d : ℕ) (μ : YoungDiagram) (e : Card (d+1) ≃ Cell μ) :
+    (Card d → Bool) ≃ fiberGroup (fun x => Fin.tail (e.symm x)) :=
+  Equiv.ofBijective (fun ξ => ⟨e.permCongr (pairSwitch ξ),by
+    intro x
+    simp only [Equiv.permCongr_apply,e.symm_apply_apply]
+    exact Fin.tail_cons _ _⟩)
+    ⟨fun ξ η h => pairSwitch_injective d (e.permCongr.injective (congrArg Subtype.val h)),by
+      intro p
+      let q := e.symm.permCongr p.1
+      have hq : ∀ x, Fin.tail (q x)=Fin.tail x := by
+        intro x
+        have h := p.2 (e x)
+        simpa only [q,Equiv.permCongr_apply,Equiv.symm_symm,e.symm_apply_apply] using h
+      refine ⟨fun u => q (Fin.cons false u) 0,?_⟩
+      apply Subtype.ext
+      change e.permCongr (pairSwitch (fun u => q (Fin.cons false u) 0))=p.1
+      rw [tail_preserving_pairSwitch d q hq]
+      apply Equiv.ext
+      intro x
+      simp only [q,Equiv.permCongr_apply,Equiv.symm_symm,e.apply_symm_apply]⟩
+
+lemma pair_sum_taboid_zero (d : ℕ) (μ : YoungDiagram) (e : Card (d+1) ≃ Cell μ)
+    (hj : 2^d < μ.colLen 0) (v : space μ) :
+    (∑ ξ : Card d → Bool, tabloidRep μ (e.permCongr (pairSwitch ξ))) v.1=0 := by
+  have he : (∑ ξ : Card d → Bool, tabloidRep μ (e.permCongr (pairSwitch ξ)))=
+      symmetrizer (fiberGroup (fun x => Fin.tail (e.symm x))) (tabloidRep μ) :=
+    Fintype.sum_equiv (pairFiberEquiv d μ e) _ _ (fun _ => rfl)
+  rw [he]
+  exact tall_column_average_zero μ _ 0 (by simpa only [card_positions] using hj) v.1 v.2
+
+lemma pair_operator_tall_zero (d : ℕ) (μ : YoungDiagram) (e : Card (d+1) ≃ Cell μ)
+    (hj : 2^d < μ.colLen 0) :
+    UnitaryFinite.sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e) pairSwitch=0 := by
+  apply ContinuousLinearMap.ext
+  intro v
+  obtain ⟨w,rfl⟩ := (spaceHilbertEquiv μ).surjective v
+  have hh : (∑ ξ : Card d → Bool, representation μ (e.permCongr (pairSwitch ξ))) w=0 := by
+    apply Subtype.ext
+    simp only [LinearMap.sum_apply,Submodule.coe_sum,Submodule.coe_zero]
+    change (∑ ξ : Card d → Bool, tabloidRep μ (e.permCongr (pairSwitch ξ)) w.1)=0
+    simpa only [LinearMap.sum_apply] using pair_sum_taboid_zero d μ e hj w
+  have hh' := congrArg (spaceHilbertEquiv μ) hh
+  simp only [LinearMap.sum_apply,map_sum,map_zero] at hh'
+  change ((Fintype.card (Card d → Bool):ℂ)⁻¹ •
+    ∑ ξ : Card d → Bool, UnitaryFinite.continuousRepresentation (relabelledUnitary μ e) (pairSwitch ξ))
+      (spaceHilbertEquiv μ w)=0
+  rw [smul_apply,sum_apply]
+  change (Fintype.card (Card d → Bool):ℂ)⁻¹ •
+    (∑ ξ : Card d → Bool, spaceHilbertEquiv μ (representation μ (e.permCongr (pairSwitch ξ))
+      ((spaceHilbertEquiv μ).symm (spaceHilbertEquiv μ w))))=0
+  simp only [LinearEquiv.symm_apply_apply,hh',smul_zero]
+
+open UnitaryFinite
+
+lemma palindrome_tall_zero (d : ℕ) (μ : YoungDiagram) (e : Card (d+1) ≃ Cell μ)
+    (hj : 2^d<μ.colLen 0) :
+    sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e) (palindromePerm (d+1))=0 := by
+  let ρ := relabelledUnitary μ e
+  let : NormedAddCommGroup (hilbertSpace μ) := inferInstance
+  let : InnerProductSpace ℂ (hilbertSpace μ) := inferInstance
+  let : FiniteDimensional ℂ (hilbertSpace μ) := inferInstance
+  have hpair : sampleOperator (V := hilbertSpace μ) ρ (pairSwitch (d := d))=0 := pair_operator_tall_zero d μ e hj
+  have hi : sampleOperator ρ (fun ω => (butterflyPerm (d+1) (decodeButterfly (d+1) ω))⁻¹)=0 := by
+    rw [inverse_butterfly_average_step,hpair,mul_zero]
+  rw [sampleOperator_inv ρ (relabelledUnitary_unitary μ e)] at hi
+  rw [sampleOperator_palindrome _ _ (relabelledUnitary_unitary μ e),hi,mul_zero]
+
+lemma palindrome_tall_zero_general (d : ℕ) (hd : 0<d) (μ : YoungDiagram) (e : Card d ≃ Cell μ)
+    (hj : (μ.card:ℝ)/2<(μ.colLen 0:ℝ)) :
+    sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e) (palindromePerm d)=0 := by
+  obtain ⟨r,rfl⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : d≠0)
+  apply palindrome_tall_zero r μ e
+  have hc : μ.card=2^(r+1) := by rw [←card_cell,←Fintype.card_congr e,card_positions]
+  rw [hc,Nat.cast_pow,Nat.cast_ofNat,pow_succ] at hj
+  have hh : (2:ℝ)^r<(μ.colLen 0:ℝ) := by linarith
+  exact_mod_cast hh
+
+end Thorp.Specht
+
+namespace Thorp.DimensionEstimates
+open Filter
+
+lemma twopow_tendsto : Tendsto (fun d : ℕ => (2:ℝ)^d) atTop atTop :=
+  tendsto_pow_atTop_atTop_of_one_lt (by norm_num)
+
+lemma twopow_nat_tendsto : Tendsto (fun d : ℕ => (2:ℕ)^d) atTop atTop :=
+  tendsto_pow_atTop_atTop_of_one_lt (by norm_num)
+
+lemma twopow_mul_eventually (c : ℝ) (hc : 0<c) (B : ℝ) :
+    ∀ᶠ d : ℕ in atTop, B≤c*(2:ℝ)^d := by
+  filter_upwards [twopow_tendsto.eventually (eventually_ge_atTop (B/c))] with d hd
+  exact (div_le_iff₀ hc).mp hd |>.trans_eq (mul_comm _ _)
+
+lemma exp_two_absorb (β n : ℝ) (hx : Real.log 2≤β*n/2) :
+    Real.exp (-β*n)+Real.exp (-β*n)≤Real.exp (-β*n/2) := by
+  calc
+    _ = Real.exp (Real.log 2-β*n) := by rw [Real.exp_sub,Real.exp_log (by norm_num)]; rw [neg_mul,Real.exp_neg]; ring
+    _ ≤ _ := Real.exp_le_exp.mpr (by linarith)
+
+end Thorp.DimensionEstimates
+namespace Thorp.Specht
+open scoped BigOperators Classical
+open Filter UnitaryFinite DimensionEstimates
+
+theorem exponential_dimension_spectral (h : ℝ) (hh : 0<h) :
+    ∃ a : ℝ, 0<a ∧ ∀ᶠ d : ℕ in atTop, ∀ (μ : YoungDiagram) (e : Card d ≃ Cell μ),
+      h*(2:ℝ)^d≤Real.log (Module.finrank ℂ (space μ)) →
+      ‖sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e) (palindromePerm d)‖ ≤
+        Real.exp (-a*Real.log (Module.finrank ℂ (space μ))) := by
+  obtain ⟨L,hL,ε,t,hε,_,ht,hcount,hM⟩ := medium_parameters h hh
+  obtain ⟨R,hR⟩ := eventually_atTop.mp hM
+  let A := 4000*densityConstant+1
+  have hA : 0<A := by dsimp [A]; have := densityConstant_pos; positivity
+  let β := min (-Real.log (blockGap L)*ε/(2:ℝ)^L) (h/16000)
+  have hβ : 0<β := by
+    apply lt_min
+    · exact div_pos (mul_pos (neg_pos.mpr (Real.log_neg (blockGap_pos L) (blockGap_lt_one L))) hε) (by positivity)
+    · positivity
+  let a := min (1/8000:ℝ) (β/(2*A))
+  have ha : 0<a := lt_min (by norm_num) (div_pos hβ (by positivity))
+  refine ⟨a,ha,?_⟩
+  filter_upwards [eventually_ge_atTop (L+R),twopow_mul_eventually h hh (8000*Real.log 2),
+    twopow_mul_eventually (β/2) (by positivity) (Real.log 2)] with d hd hl hb
+  intro μ e hdim
+  let : NormedAddCommGroup (hilbertSpace μ) := inferInstance
+  let : InnerProductSpace ℂ (hilbertSpace μ) := inferInstance
+  let : FiniteDimensional ℂ (hilbertSpace μ) := inferInstance
+  let := relabelledUnitary_irreducible μ e
+  have hpos : 0<Module.finrank ℂ (hilbertSpace μ) := by rw [finrank_hilbertSpace]; exact dimension_pos μ
+  have hlarge : 8000*Real.log 2≤Real.log (Module.finrank ℂ (hilbertSpace μ)) := by rw [finrank_hilbertSpace]; exact hl.trans hdim
+  have hlog : 0≤Real.log (Module.finrank ℂ (space μ)) := (by positivity : (0:ℝ)≤h*(2:ℝ)^d).trans hdim
+  by_cases hhigh : A*(2:ℝ)^d≤Real.log (Module.finrank ℂ (space μ))
+  · have hden : densityConstant*(2:ℝ)^d≤Real.log (Module.finrank ℂ (hilbertSpace μ))/4000 := by
+      rw [finrank_hilbertSpace]
+      dsimp [A] at hhigh
+      nlinarith [pow_pos (by norm_num : (0:ℝ)<2) d]
+    have hspec := @full_palindrome_decay (hilbertSpace μ) _ _ _ d (relabelledUnitary μ e)
+      inferInstance (relabelledUnitary_unitary μ e) hpos hden hlarge
+    rw [finrank_hilbertSpace] at hspec
+    apply hspec.trans (Real.exp_le_exp.mpr ?_)
+    have := min_le_left (1/8000:ℝ) (β/(2*A))
+    nlinarith
+  · obtain ⟨r,rfl⟩ := Nat.exists_eq_add_of_le (show L≤d by omega)
+    have hr : R≤r := by omega
+    have hdim' : h*(2:ℝ)^(L+r)≤Real.log (Module.finrank ℂ (hilbertSpace μ)) := by rwa [finrank_hilbertSpace]
+    have hs := @medium_exception_decay (hilbertSpace μ) _ _ _ L r h ε t ht hcount
+      (relabelledUnitary μ e) inferInstance (relabelledUnitary_unitary μ e) hpos hdim' hlarge (hR r hr)
+    have hb1 : Real.log (blockGap L)*ε*(2:ℝ)^r≤-β*(2:ℝ)^(L+r) := by
+      have hb0 := min_le_left (-Real.log (blockGap L)*ε/(2:ℝ)^L) (h/16000)
+      have hp := (le_div_iff₀ (by positivity : (0:ℝ)<(2:ℝ)^L)).mp hb0
+      have hp := mul_le_mul_of_nonneg_right hp (by positivity : (0:ℝ)≤(2:ℝ)^r)
+      rw [pow_add]
+      dsimp [β]
+      nlinarith
+    have hb2 : -h*(2:ℝ)^(L+r)/16000≤-β*(2:ℝ)^(L+r) := by
+      have hp := mul_le_mul_of_nonneg_right (min_le_right (-Real.log (blockGap L)*ε/(2:ℝ)^L) (h/16000))
+        (by positivity : (0:ℝ)≤(2:ℝ)^(L+r))
+      dsimp [β]
+      linarith
+    apply hs.trans ((add_le_add (Real.exp_le_exp.mpr hb1) (Real.exp_le_exp.mpr hb2)).trans ?_)
+    apply (exp_two_absorb β ((2:ℝ)^(L+r)) (by nlinarith)).trans
+    apply Real.exp_le_exp.mpr
+    have hdmax := le_of_not_ge hhigh
+    have hp := mul_le_mul_of_nonneg_left hdmax (by positivity : 0≤β/(2*A))
+    have he : β/(2*A)*(A*(2:ℝ)^(L+r))=β*(2:ℝ)^(L+r)/2 := by field_simp
+    rw [he] at hp
+    have hba := mul_le_mul_of_nonneg_right (min_le_right (1/8000:ℝ) (β/(2*A))) hlog
+    dsimp [a] at *
+    linarith
+
+
+theorem spectral_regimes :
+    ∃ a h : ℝ, 0<a ∧ 0<h ∧ ∀ᶠ d : ℕ in atTop,
+      ∀ (μ : YoungDiagram) (e : Card d ≃ Cell μ), 0<μ.card-μ.rowLen 0 →
+        (μ.colLen 0:ℝ)≤(μ.card:ℝ)/2 →
+        ‖sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e) (palindromePerm d)‖≤
+          Real.exp (-a*Real.log (Module.finrank ℂ (space μ))) ∧
+        1≤Real.log (Module.finrank ℂ (space μ)) ∧
+        (((1/2:ℝ)*Real.log μ.card≤Real.log (Module.finrank ℂ (space μ)) ∧
+          (μ.card-μ.rowLen 0:ℕ)≤Real.log (Module.finrank ℂ (space μ))) ∨
+          h*(2:ℝ)^d≤Real.log (Module.finrank ℂ (space μ))) := by
+  obtain ⟨X,hX,hnear⟩ := near_row_spectral
+  have hXpos : 0<X := by linarith
+  have hδ : 0<(1:ℝ)/X := by positivity
+  have hδ2 : (1:ℝ)/X≤1/2 := (div_le_iff₀ hXpos).mpr (by linarith)
+  obtain ⟨h,hh,hmedium⟩ := medium_dimension (1/X) hδ hδ2
+  obtain ⟨a₀,ha₀,hlarge⟩ := exponential_dimension_spectral h hh
+  let a := min (nearFraction/8000) a₀
+  have ha : 0<a := lt_min (by norm_num [nearFraction]) ha₀
+  refine ⟨a,h,ha,hh,?_⟩
+  have hl := (Real.tendsto_log_atTop.comp twopow_tendsto).eventually
+    (eventually_ge_atTop (2*max (32768000*Real.log 2) 1))
+  filter_upwards [hlarge,twopow_nat_tendsto.eventually hmedium,hl,
+    twopow_mul_eventually h hh 1] with d hspec hdim hlog h1
+  intro μ e hj hc
+  simp only [Function.comp_apply] at hlog
+  have hcard : μ.card=2^d := by rw [←card_cell,←Fintype.card_congr e,card_positions]
+  have hcardR : (μ.card:ℝ)=(2:ℝ)^d := by exact_mod_cast hcard
+  have hμlog : max (32768000*Real.log 2) 1≤(1/2:ℝ)*Real.log μ.card := by rw [hcardR]; linarith
+  by_cases hx : X≤(μ.card:ℝ)/(μ.card-μ.rowLen 0:ℕ)
+  · obtain ⟨hs,hl,hjD⟩ := hnear d μ e hj hx ((le_max_left _ _).trans hμlog)
+    have h1D : 1≤Real.log (Module.finrank ℂ (space μ)) := ((le_max_right _ _).trans hμlog).trans hl
+    refine ⟨hs.trans (Real.exp_le_exp.mpr ?_),h1D,Or.inl ⟨hl,hjD⟩⟩
+    have hp := mul_le_mul_of_nonneg_right (min_le_left (nearFraction/8000) a₀) (by linarith : 0≤Real.log (Module.finrank ℂ (space μ)))
+    dsimp [a]
+    linarith
+  · have htail : (1/X)*(μ.card:ℝ)≤(μ.card:ℝ)-μ.rowLen 0 := by
+      have hjR : (0:ℝ)<(μ.card-μ.rowLen 0:ℕ) := by exact_mod_cast hj
+      have hh := (div_lt_iff₀ hjR).mp (lt_of_not_ge hx)
+      have hs : ((μ.card-μ.rowLen 0:ℕ):ℝ)=(μ.card:ℝ)-μ.rowLen 0 := by
+        rw [Nat.cast_sub (rowLen_zero_le_card μ)]
+      rw [hs] at hh
+      have hd : (μ.card:ℝ)/X≤(μ.card:ℝ)-μ.rowLen 0 :=
+        (div_le_iff₀ hXpos).mpr (by nlinarith)
+      convert hd using 1; ring
+    have hdim' : h*(2:ℝ)^d≤Real.log (Module.finrank ℂ (space μ)) := by
+      simpa only [Nat.cast_pow,Nat.cast_ofNat] using
+        hdim μ hcard (by simpa only [hcard] using htail) (by simpa only [hcard] using hc)
+    have h1D := h1.trans hdim'
+    refine ⟨(hspec μ e hdim').trans (Real.exp_le_exp.mpr ?_),h1D,Or.inr hdim'⟩
+    have hp := mul_le_mul_of_nonneg_right (min_le_right (nearFraction/8000) a₀) (by linarith : 0≤Real.log (Module.finrank ℂ (space μ)))
+    dsimp [a]
+    linarith
+
+end Thorp.Specht
+
+namespace Thorp.StrongSmoothing
+
+noncomputable def regularAbsoluteEvenTrace (d : ℕ) (order : Equiv.Perm (Fin d)) (u : ℕ) : ℝ :=
+  Matrix.trace ((reflectedKernel d (2^d) order false)^u)
+
+def MainStatement : Prop :=
+  ∃ u : ℕ, 1 ≤ u ∧ ∃ C : ℝ,
+    (∀ d l : ℕ, l ≤ 2^d → ∀ order : Equiv.Perm (Fin d), ∀ orientation : Bool,
+      ∀ x y : Tuples d l,
+        ((reflectedKernel d l order orientation)^u) x y ≤
+          Real.exp (C*l)/((2^d).descFactorial l:ℝ)) ∧
+    (∀ d : ℕ, ∀ order : Equiv.Perm (Fin d),
+      regularAbsoluteEvenTrace d order u ≤ Real.exp (C*(2:ℝ)^d))
+
+
+open scoped BigOperators
+
+variable {X : Type*} [Fintype X] [Nonempty X]
+
+omit [Nonempty X] in
+lemma mean_nonneg (f : X → ℝ) (hf : ∀ x, 0 ≤ f x) : 0 ≤ finiteMean f :=
+  finiteMean_nonneg hf
+
+lemma mean_rpow_concave (f : X → ℝ) (hf : ∀ x, 0 ≤ f x)
+    {a : ℝ} (ha : 0 ≤ a) (ha1 : a ≤ 1) :
+    finiteMean (fun x => (f x)^a) ≤ (finiteMean f)^a := by
+  have hN : (0:ℝ) < Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
+  have hh := (Real.concaveOn_rpow ha ha1).le_map_sum
+    (t := Finset.univ) (w := fun _ : X => 1/(Fintype.card X:ℝ))
+    (p := f) (fun _ _ => by positivity)
+    (by simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]; field_simp)
+    (fun x _ => hf x)
+  simpa only [smul_eq_mul,one_div,←Finset.mul_sum,mul_comm _ (∑ _ : X,_),
+    ←div_eq_mul_inv,finiteMean] using hh
+
+noncomputable def act (K : Matrix X X ℝ) (f : X → ℝ) (y : X) : ℝ := ∑ x, K x y*f x
+
+omit [Nonempty X] in
+lemma act_nonneg (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
+    (f : X → ℝ) (hf : ∀ x, 0 ≤ f x) (y : X) : 0 ≤ act K f y :=
+  Finset.sum_nonneg (fun x _ => mul_nonneg (hK x y) (hf x))
+
+omit [Nonempty X] in
+lemma act_rpow_le (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
+    (hcol : ∀ y, ∑ x, K x y = 1) (f : X → ℝ) (hf : ∀ x, 0 ≤ f x)
+    {b : ℝ} (hb : 1 ≤ b) (y : X) :
+    (act K f y)^b ≤ act K (fun x => (f x)^b) y := by
+  simpa only [smul_eq_mul,act] using
+    (convexOn_rpow hb).map_sum_le (t := Finset.univ) (w := fun x => K x y)
+      (p := f) (fun x _ => hK x y) (hcol y) (fun x _ => hf x)
+
+lemma act_one_to_p (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
+    {p A : ℝ} (hp : 1 ≤ p)
+    (hm : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) ≤ Real.exp A)
+    (f : X → ℝ) (hf : ∀ x, 0 ≤ f x) :
+    finiteMean (fun y => (act K f y)^p) ≤ Real.exp A*(finiteMean f)^p := by
+  classical
+  let s : ℝ := ∑ x, f x
+  have hs : 0 ≤ s := Finset.sum_nonneg (fun x _ => hf x)
+  have hN : (0:ℝ) < Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
+  by_cases hz : s=0
+  · have hfz : ∀ x, f x=0 := by
+      intro x
+      exact (Finset.sum_eq_zero_iff_of_nonneg (fun x _ => hf x)).mp hz x (Finset.mem_univ _)
+    simp only [act,hfz,mul_zero,Finset.sum_const_zero,finiteMean,zero_div,
+      Real.zero_rpow (show p ≠ 0 by linarith),mul_zero,le_refl]
+  · have hsp : 0 < s := lt_of_le_of_ne hs (Ne.symm hz)
+    let w : X → ℝ := fun x => f x/s
+    have hw : ∀ x, 0 ≤ w x := fun x => div_nonneg (hf x) hs
+    have hws : ∑ x, w x = 1 := by simp only [w,←Finset.sum_div]; exact div_self hz
+    have hact (y : X) : act K f y = finiteMean f * ∑ x, w x*((Fintype.card X:ℝ)*K x y) := by
+      simp only [act,w,finiteMean,Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro x _
+      have hz' : (∑ x, f x) ≠ 0 := hz
+      dsimp [s]
+      field_simp [hz',ne_of_gt hN]
+    have hj (y : X) : (act K f y)^p ≤ (finiteMean f)^p*
+        ∑ x, w x*((Fintype.card X:ℝ)*K x y)^p := by
+      rw [hact,Real.mul_rpow (mean_nonneg f hf)
+        (Finset.sum_nonneg (fun x _ => mul_nonneg (hw x) (mul_nonneg hN.le (hK x y))))]
+      apply mul_le_mul_of_nonneg_left _ (Real.rpow_nonneg (mean_nonneg f hf) _)
+      exact (convexOn_rpow hp).map_sum_le
+        (fun x _ => hw x) hws (fun x _ => mul_nonneg hN.le (hK x y))
+    have hh := finiteMean_mono hj
+    have he : finiteMean (fun y => (finiteMean f)^p*
+        ∑ x, w x*((Fintype.card X:ℝ)*K x y)^p) =
+        (finiteMean f)^p * ∑ x, w x*finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) := by
+      simp_rw [mul_comm ((finiteMean f)^p), finiteMean_mul_const,
+        finiteMean_sum, mul_comm (w _), finiteMean_mul_const]
+    rw [he] at hh
+    apply hh.trans
+    have hsum : (∑ x, w x*finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p)) ≤ Real.exp A := by
+      calc
+        _ ≤ ∑ x, w x*Real.exp A := Finset.sum_le_sum
+          (fun x _ => mul_le_mul_of_nonneg_left (hm x) (hw x))
+        _ = Real.exp A := by rw [←Finset.sum_mul,hws,one_mul]
+    simpa only [mul_comm (Real.exp A)] using
+      mul_le_mul_of_nonneg_left hsum (Real.rpow_nonneg (mean_nonneg f hf) _)
+
+lemma act_b_to_pb (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
+    (hcol : ∀ y, ∑ x, K x y = 1) {p A b : ℝ} (hp : 1 ≤ p) (hb : 1 ≤ b)
+    (hm : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) ≤ Real.exp A)
+    (f : X → ℝ) (hf : ∀ x, 0 ≤ f x) :
+    finiteMean (fun y => (act K f y)^(b*p)) ≤
+      Real.exp A*(finiteMean (fun x => (f x)^b))^p := by
+  calc
+    _ ≤ finiteMean (fun y => (act K (fun x => (f x)^b) y)^p) := by
+      apply finiteMean_mono
+      intro y
+      rw [Real.rpow_mul (act_nonneg K hK f hf y)]
+      exact Real.rpow_le_rpow (Real.rpow_nonneg (act_nonneg K hK f hf y) _) (act_rpow_le K hK hcol f hf hb y) (by linarith)
+    _ ≤ _ := act_one_to_p K hK hp hm _ (fun x => Real.rpow_nonneg (hf x) _)
+
+variable [DecidableEq X]
+
+omit [Nonempty X] in
+lemma kernel_pow_nonneg (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y) (r : ℕ) :
+    ∀ x y, 0 ≤ (K^r) x y := by
+  classical
+  induction r with
+  | zero => intro x y; simp only [pow_zero,Matrix.one_apply]; split_ifs <;> norm_num
+  | succ r ih =>
+    intro x y
+    rw [pow_succ,Matrix.mul_apply]
+    exact Finset.sum_nonneg (fun z _ => mul_nonneg (ih x z) (hK z y))
+
+omit [Nonempty X] in
+lemma kernel_pow_row (K : Matrix X X ℝ) (hrow : ∀ x, ∑ y, K x y = 1) (r : ℕ) :
+    ∀ x, ∑ y, (K^r) x y = 1 := by
+  classical
+  induction r with
+  | zero => intro x; simp [Matrix.one_apply]
+  | succ r ih =>
+    intro x
+    simp only [pow_succ,Matrix.mul_apply]
+    rw [Finset.sum_comm]
+    simp only [←Finset.mul_sum,hrow,mul_one,ih]
+
+noncomputable def rowDensity (K : Matrix X X ℝ) (r : ℕ) (x y : X) : ℝ :=
+  (Fintype.card X:ℝ)*(K^r) x y
+
+omit [Nonempty X] in
+lemma rowDensity_nonneg (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y) (r : ℕ) (x y : X) :
+    0 ≤ rowDensity K r x y := mul_nonneg (Nat.cast_nonneg _) (kernel_pow_nonneg K hK r x y)
+
+omit [Nonempty X] in
+lemma rowDensity_succ (K : Matrix X X ℝ) (r : ℕ) (x y : X) :
+    rowDensity K (r+1) x y = act K (rowDensity K r x) y := by
+  simp only [rowDensity,pow_succ,Matrix.mul_apply,act,Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro z _
+  ring
+
+lemma rowDensity_mean (K : Matrix X X ℝ) (hrow : ∀ x, ∑ y, K x y = 1) (r : ℕ) (x : X) :
+    finiteMean (rowDensity K r x) = 1 := by
+  have hN : (Fintype.card X:ℝ) ≠ 0 := ne_of_gt (Nat.cast_pos.mpr Fintype.card_pos)
+  simp only [finiteMean,rowDensity,←Finset.mul_sum,kernel_pow_row K hrow,mul_one,div_self hN]
+
+lemma rowDensity_moment_iter (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
+    (hrow : ∀ x, ∑ y, K x y = 1) (hcol : ∀ y, ∑ x, K x y = 1)
+    {p A : ℝ} (hp : 1 ≤ p) (hA : 0 ≤ A)
+    (hm : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) ≤ Real.exp A)
+    (r : ℕ) (x : X) :
+    finiteMean (fun y => (rowDensity K r x y)^(p^r)) ≤ Real.exp (A*r*p^r) := by
+  induction r with
+  | zero => simp [rowDensity_mean K hrow]
+  | succ r ih =>
+    have hpr : 1 ≤ p^r := one_le_pow₀ hp
+    have hps : 1 ≤ p^(r+1) := one_le_pow₀ hp
+    simp_rw [rowDensity_succ]
+    calc
+      _ ≤ Real.exp A*(finiteMean (fun y => (rowDensity K r x y)^(p^r)))^p := by
+        rw [pow_succ]
+        exact act_b_to_pb K hK hcol hp hpr hm _ (rowDensity_nonneg K hK r x)
+      _ ≤ Real.exp A*(Real.exp (A*r*p^r))^p := by
+        apply mul_le_mul_of_nonneg_left _ (Real.exp_pos _).le
+        exact Real.rpow_le_rpow (mean_nonneg _ (fun y => Real.rpow_nonneg
+          (rowDensity_nonneg K hK r x y) _)) ih (le_trans zero_le_one hp)
+      _ = Real.exp (A+A*r*p^(r+1)) := by rw [←Real.exp_mul,←Real.exp_add,pow_succ]; congr 1; ring
+      _ ≤ _ := by
+        apply Real.exp_le_exp.mpr
+        have hh := mul_le_mul_of_nonneg_left hps hA
+        push_cast
+        nlinarith
+
+lemma rowDensity_second_moment (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
+    (hrow : ∀ x, ∑ y, K x y = 1) (hcol : ∀ y, ∑ x, K x y = 1)
+    {p A : ℝ} (hp : 1 ≤ p) (hA : 0 ≤ A)
+    (hm : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) ≤ Real.exp A)
+    (r : ℕ) (hr : 2 ≤ p^r) (x : X) :
+    finiteMean (fun y => (rowDensity K r x y)^2) ≤ Real.exp (2*A*r) := by
+  have hpos : 0 < p^r := by linarith
+  have hj := mean_rpow_concave (fun y => (rowDensity K r x y)^(p^r))
+    (fun y => Real.rpow_nonneg (rowDensity_nonneg K hK r x y) _)
+    (a := 2/(p^r)) (by positivity) ((div_le_one hpos).mpr hr)
+  have he (y : X) : ((rowDensity K r x y)^(p^r))^(2/(p^r)) = (rowDensity K r x y)^2 := by
+    rw [←Real.rpow_mul (rowDensity_nonneg K hK r x y)]
+    rw [mul_div_cancel₀ _ (ne_of_gt hpos),Real.rpow_two]
+  simp_rw [he] at hj
+  apply hj.trans
+  calc
+    _ ≤ (Real.exp (A*r*p^r))^(2/(p^r)) :=
+      Real.rpow_le_rpow (mean_nonneg _ (fun y => Real.rpow_nonneg
+        (rowDensity_nonneg K hK r x y) _)) (rowDensity_moment_iter K hK hrow hcol hp hA hm r x)
+        (by positivity)
+    _ = _ := by rw [←Real.exp_mul]; congr 1; field_simp [ne_of_gt hpos]
+
+lemma symmetric_kernel_smoothing (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
+    (hrow : ∀ x, ∑ y, K x y = 1) (hcol : ∀ y, ∑ x, K x y = 1)
+    (hsym : K.IsSymm) {p A : ℝ} (hp : 1 ≤ p) (hA : 0 ≤ A)
+    (hm : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) ≤ Real.exp A)
+    (r : ℕ) (hr : 2 ≤ p^r) (x y : X) :
+    (K^(2*r)) x y ≤ Real.exp (2*A*r)/(Fintype.card X:ℝ) := by
+  have hN : (0:ℝ) < Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
+  have hs : ∀ z, (K^r) z y = (K^r) y z := fun z => (hsym.pow r).apply y z
+  have he : (Fintype.card X:ℝ)*(K^(2*r)) x y =
+      finiteMean (fun z => rowDensity K r x z*rowDensity K r y z) := by
+    simp only [finiteMean,rowDensity,two_mul,pow_add,Matrix.mul_apply,hs,Finset.mul_sum]
+    rw [Finset.sum_div]
+    apply Finset.sum_congr rfl
+    intro z _
+    field_simp
+  have hh : finiteMean (fun z => rowDensity K r x z*rowDensity K r y z) ≤
+      Real.exp (2*A*r) := by
+    calc
+      _ ≤ finiteMean (fun z => ((rowDensity K r x z)^2+(rowDensity K r y z)^2)/2) := by
+        apply finiteMean_mono
+        intro z
+        nlinarith [sq_nonneg (rowDensity K r x z-rowDensity K r y z)]
+      _ = (finiteMean (fun z => (rowDensity K r x z)^2)+
+          finiteMean (fun z => (rowDensity K r y z)^2))/2 := by
+        simp only [div_eq_mul_inv,finiteMean_mul_const,finiteMean_add]
+      _ ≤ _ := by
+        have hx := rowDensity_second_moment K hK hrow hcol hp hA hm r hr x
+        have hy := rowDensity_second_moment K hK hrow hcol hp hA hm r hr y
+        linarith
+  apply (le_div_iff₀ hN).mpr
+  simpa only [←he,mul_comm (Fintype.card X:ℝ)] using hh
+
+lemma density_exponent_numeric : 2 ≤ (1+(1/1000:ℝ))^(1000:ℕ) := by
+  calc
+    (2:ℝ) = 1+(1000:ℝ)*(1/1000) := by norm_num
+    _ ≤ _ := by simpa only [Nat.cast_ofNat] using
+      (one_add_mul_le_pow (a := (1/1000:ℝ)) (by norm_num) 1000)
+
+end Thorp.StrongSmoothing
+
+namespace Thorp.StrongSmoothing.PermLaw
+open scoped BigOperators
+variable {X Ω Ω' : Type*} [Fintype X] [DecidableEq X] [Fintype Ω] [Fintype Ω']
+
+noncomputable def kernel (P : Ω → Equiv.Perm X) : Matrix X X ℝ :=
+  fun x y => finiteMean (fun ω => if P ω x = y then (1:ℝ) else 0)
+
+omit [Fintype X] in
+lemma nonneg (P : Ω → Equiv.Perm X) (x y : X) : 0 ≤ kernel P x y := by
+  exact finiteMean_nonneg (fun ω => by split_ifs <;> norm_num)
+
+lemma row [Nonempty Ω] (P : Ω → Equiv.Perm X) (x : X) : ∑ y, kernel P x y = 1 := by
+  simp only [kernel,←finiteMean_sum]
+  simp only [Finset.sum_ite_eq,Finset.mem_univ,↓reduceIte,finiteMean_const]
+
+omit [Fintype X] in
+lemma inverse (P : Ω → Equiv.Perm X) : kernel (fun ω => (P ω)⁻¹) = (kernel P).transpose := by
+  ext x y
+  apply finiteMean_congr
+  intro ω
+  have he : (P ω)⁻¹ x=y ↔ P ω y=x := by
+    change (P ω).symm x=y ↔ P ω y=x
+    exact (Equiv.symm_apply_eq (P ω)).trans eq_comm
+  simp only [he]
+
+lemma col [Nonempty Ω] (P : Ω → Equiv.Perm X) (y : X) : ∑ x, kernel P x y = 1 := by
+  have hh := row (fun ω => (P ω)⁻¹) y
+  rw [inverse] at hh
+  exact hh
+
+lemma mul (P : Ω → Equiv.Perm X) (Q : Ω' → Equiv.Perm X) :
+    kernel (fun ω : Ω × Ω' => P ω.1*Q ω.2) = kernel Q*kernel P := by
+  ext x y
+  simp only [Matrix.mul_apply,kernel]
+  have he (ω : Ω × Ω') :
+      (if (P ω.1*Q ω.2) x=y then (1:ℝ) else 0) =
+        ∑ z : X, (if P ω.1 z=y then (1:ℝ) else 0)*(if Q ω.2 x=z then (1:ℝ) else 0) := by
+    simp only [mul_ite,mul_one,mul_zero,Finset.sum_ite_eq,Finset.mem_univ,↓reduceIte]
+    rfl
+  change finiteMean (fun ω : Ω × Ω' => if (P ω.1*Q ω.2) x=y then (1:ℝ) else 0) = _
+  simp_rw [he]
+  rw [finiteMean_sum]
+  apply Finset.sum_congr rfl
+  intro z _
+  exact (finiteMean_prod_mul (fun ω : Ω => if P ω z=y then (1:ℝ) else 0)
+    (fun ω : Ω' => if Q ω x=z then (1:ℝ) else 0)).trans (mul_comm _ _)
+
+omit [Fintype X] in
+lemma conjugate (P : Ω → Equiv.Perm X) (e : Equiv.Perm X) (x y : X) :
+    kernel (fun ω => e*P ω*e⁻¹) x y = kernel P (e.symm x) (e.symm y) := by
+  apply finiteMean_congr
+  intro ω
+  have he : (e*P ω*e⁻¹) x=y ↔ P ω (e.symm x)=e.symm y := by
+    change e (P ω (e.symm x))=y ↔ _
+    exact (Equiv.eq_symm_apply e).symm
+  simp only [he]
+
+end Thorp.StrongSmoothing.PermLaw
+
+namespace Thorp.StrongSmoothing
+open scoped BigOperators
+
+section MatrixHelpers
+variable {X : Type*} [Fintype X]
+
+lemma mul_nonneg_entries (A B : Matrix X X ℝ) (hA : ∀ x y, 0 ≤ A x y)
+    (hB : ∀ x y, 0 ≤ B x y) (x y : X) : 0 ≤ (A*B) x y :=
+  Finset.sum_nonneg (fun z _ => mul_nonneg (hA x z) (hB z y))
+
+lemma mul_row (A B : Matrix X X ℝ) (hA : ∀ x, ∑ y, A x y = 1)
+    (hB : ∀ x, ∑ y, B x y = 1) (x : X) : ∑ y, (A*B) x y = 1 := by
+  simp only [Matrix.mul_apply]
+  rw [Finset.sum_comm]
+  simp only [←Finset.mul_sum,hB,mul_one,hA]
+
+lemma mul_col (A B : Matrix X X ℝ) (hA : ∀ y, ∑ x, A x y = 1)
+    (hB : ∀ y, ∑ x, B x y = 1) (y : X) : ∑ x, (A*B) x y = 1 := by
+  simp only [Matrix.mul_apply]
+  rw [Finset.sum_comm]
+  simp only [←Finset.sum_mul,hA,one_mul,hB]
+
+lemma bound_mul_left (A B : Matrix X X ℝ) (hA : ∀ x y, 0 ≤ A x y)
+    (hrow : ∀ x, ∑ y, A x y = 1) {D : ℝ} (hB : ∀ x y, B x y ≤ D) :
+    ∀ x y, (A*B) x y ≤ D := by
+  intro x y
+  calc
+    _ ≤ ∑ z, A x z*D := Finset.sum_le_sum (fun z _ => mul_le_mul_of_nonneg_left (hB z y) (hA x z))
+    _ = D := by rw [←Finset.sum_mul,hrow,one_mul]
+
+lemma bound_mul_right (A B : Matrix X X ℝ) (hB : ∀ x y, 0 ≤ B x y)
+    (hcol : ∀ y, ∑ x, B x y = 1) {D : ℝ} (hA : ∀ x y, A x y ≤ D) :
+    ∀ x y, (A*B) x y ≤ D := by
+  intro x y
+  calc
+    _ ≤ ∑ z, D*B z y := Finset.sum_le_sum (fun z _ => mul_le_mul_of_nonneg_right (hA x z) (hB z y))
+    _ = D := by rw [←Finset.mul_sum,hcol,mul_one]
+
+variable [DecidableEq X]
+
+lemma sandwich_power (A B : Matrix X X ℝ) (r : ℕ) : A*(B*A)^r*B = (A*B)^(r+1) := by
+  induction r with
+  | zero => simp
+  | succ r ih =>
+    calc
+      _ = (A*(B*A)^r*B)*(A*B) := by rw [pow_succ]; noncomm_ring
+      _ = (A*B)^(r+1)*(A*B) := by rw [ih]
+      _ = _ := (pow_succ (A*B) (r+1)).symm
+
+end MatrixHelpers
+
+section TupleAction
+variable {ι β Ω : Type*} [Fintype Ω]
+
+def tuplePerm (P : Equiv.Perm β) : Equiv.Perm (ι ↪ β) where
+  toFun x := x.trans P.toEmbedding
+  invFun x := x.trans P.symm.toEmbedding
+  left_inv x := by ext i; simp
+  right_inv x := by ext i; simp
+
+@[simp] lemma tuplePerm_apply (P : Equiv.Perm β) (x : ι ↪ β) (i : ι) :
+    tuplePerm P x i = P (x i) := rfl
+
+lemma tuplePerm_mul (P Q : Equiv.Perm β) :
+    tuplePerm (ι := ι) (P*Q) = tuplePerm P*tuplePerm Q := by
+  ext x i
+  rfl
+
+lemma tuplePerm_inverse (P : Equiv.Perm β) :
+    tuplePerm (ι := ι) P⁻¹ = (tuplePerm P)⁻¹ := by
+  ext x i
+  rfl
+
+variable [Fintype ι] [DecidableEq β]
+
+lemma tupleProbability_kernel (P : Ω → Equiv.Perm β) (x y : ι ↪ β) :
+    PairRouting.tupleProbability P x y = PermLaw.kernel (fun ω => tuplePerm (P ω)) x y := by
+  classical
+  apply finiteMean_congr
+  intro ω
+  have he : (∀ i, P ω (x i) = y i) ↔ tuplePerm (P ω) x = y := by
+    constructor
+    · intro h; ext i; exact h i
+    · intro h i; exact congrArg (fun z : ι ↪ β => z i) h
+  simp only [he]
+end TupleAction
+
+lemma sweepKernel_law (d l : ℕ) (order : Equiv.Perm (Fin d)) :
+    sweepKernel d l order = PermLaw.kernel (fun ω => tuplePerm (ι := Fin l) (sweep d order ω)) := by
+  ext x y
+  exact tupleProbability_kernel _ x y
+
+lemma reflectedKernel_conjugate (d l : ℕ) (order : Equiv.Perm (Fin d)) (x y : Tuples d l) :
+    reflectedKernel d l order false x y =
+      PairRouting.tupleProbability (palindromePerm d)
+        ((tuplePerm (coordinateRelabel d order)).symm x)
+        ((tuplePerm (coordinateRelabel d order)).symm y) := by
+  classical
+  let e : Equiv.Perm (Tuples d l) := tuplePerm (coordinateRelabel d order)
+  let P := fun ω => tuplePerm (ι := Fin l) (butterflyPerm d (decodeButterfly d ω))
+  have hA : sweepKernel d l order = PermLaw.kernel (fun ω => e*P ω*e⁻¹) := by
+    rw [sweepKernel_law]
+    simp only [sweep,tuplePerm_mul,tuplePerm_inverse]
+    rfl
+  rw [reflectedKernel,ite_eq_right Bool.false_ne_true,hA]
+  rw [←PermLaw.inverse,←PermLaw.mul]
+  have hprod : (fun ω : BenesCoins d => (e*P ω.1*e⁻¹)*((e*P ω.2*e⁻¹)⁻¹)) =
+      fun ω : BenesCoins d => e*tuplePerm (ι := Fin l) (palindromePerm d ω)*e⁻¹ := by
+    funext ω
+    simp only [mul_inv_rev,inv_inv,palindromePerm,tuplePerm_mul]
+    change (e*P ω.1*e⁻¹)*(e*(P ω.2)⁻¹*e⁻¹) = e*(P ω.1*(P ω.2)⁻¹)*e⁻¹
+    group
+  rw [hprod,PermLaw.conjugate,tupleProbability_kernel]
+
+
+lemma sweepKernel_nonneg (d l : ℕ) (order : Equiv.Perm (Fin d)) (x y : Tuples d l) :
+    0 ≤ sweepKernel d l order x y := PairRouting.tupleProbability_nonneg _ _ _
+
+lemma sweepKernel_row (d l : ℕ) (order : Equiv.Perm (Fin d)) (x : Tuples d l) :
+    ∑ y, sweepKernel d l order x y = 1 := by
+  rw [sweepKernel_law]
+  exact PermLaw.row _ _
+
+lemma sweepKernel_col (d l : ℕ) (order : Equiv.Perm (Fin d)) (y : Tuples d l) :
+    ∑ x, sweepKernel d l order x y = 1 := by
+  rw [sweepKernel_law]
+  exact PermLaw.col _ _
+
+lemma reflected_nonneg (d l : ℕ) (order : Equiv.Perm (Fin d)) (x y : Tuples d l) :
+    0 ≤ reflectedKernel d l order false x y := by
+  exact mul_nonneg_entries _ _ (fun x y => sweepKernel_nonneg d l order y x)
+    (sweepKernel_nonneg d l order) x y
+
+lemma reflected_row (d l : ℕ) (order : Equiv.Perm (Fin d)) (x : Tuples d l) :
+    ∑ y, reflectedKernel d l order false x y = 1 :=
+  mul_row _ _ (sweepKernel_col d l order) (sweepKernel_row d l order) x
+
+lemma reflected_col (d l : ℕ) (order : Equiv.Perm (Fin d)) (y : Tuples d l) :
+    ∑ x, reflectedKernel d l order false x y = 1 :=
+  mul_col _ _ (sweepKernel_row d l order) (sweepKernel_col d l order) y
+
+lemma reflected_symmetric (d l : ℕ) (order : Equiv.Perm (Fin d)) :
+    (reflectedKernel d l order false).IsSymm := Matrix.isSymm_transpose_mul_self _
+
+lemma reflected_moment (d l : ℕ) (order : Equiv.Perm (Fin d)) (x : Tuples d l) :
+    finiteMean (fun y : Tuples d l =>
+      ((Fintype.card (Tuples d l):ℝ)*reflectedKernel d l order false x y)^(1+(1/1000:ℝ))) ≤
+      Real.exp (l*(1+heightDecay 1)) := by
+  classical
+  let e : Equiv.Perm (Tuples d l) := tuplePerm (coordinateRelabel d order)
+  simp_rw [reflectedKernel_conjugate]
+  have he := finiteMean_equiv e (fun y : Tuples d l =>
+    ((Fintype.card (Tuples d l):ℝ)*PairRouting.tupleProbability (palindromePerm d) (e.symm x) y)^
+      (1+(1/1000:ℝ)))
+  change finiteMean (fun y =>
+    ((Fintype.card (Tuples d l):ℝ)*PairRouting.tupleProbability (palindromePerm d) (e.symm x) (e.symm y))^
+      (1+(1/1000:ℝ))) ≤ _
+  rw [←he]
+  simpa only [palindromeRowMoment,Fintype.card_fin] using palindrome_row_moment_coarse d (e.symm x)
+
+lemma tuples_card (d l : ℕ) : Fintype.card (Tuples d l) = (2^d).descFactorial l := by
+  simp only [Tuples,Fintype.card_embedding_eq,Fintype.card_fin,card_positions]
+
+lemma canonical_pointwise (d l : ℕ) (order : Equiv.Perm (Fin d)) (x y : Tuples d l) :
+    ((reflectedKernel d l order false)^2000) x y ≤
+      Real.exp ((2000*(1+heightDecay 1))*l)/((2^d).descFactorial l:ℝ) := by
+  let : Nonempty (Tuples d l) := ⟨x⟩
+  have hh := symmetric_kernel_smoothing (reflectedKernel d l order false)
+    (reflected_nonneg d l order) (reflected_row d l order) (reflected_col d l order)
+    (reflected_symmetric d l order) (p := 1+(1/1000:ℝ)) (A := l*(1+heightDecay 1))
+    (by norm_num) (mul_nonneg (Nat.cast_nonneg _) (by linarith [heightDecay_nonneg 1]))
+    (reflected_moment d l order) 1000 density_exponent_numeric x y
+  have he : 2*((l:ℝ)*(1+heightDecay 1))*(1000:ℝ) = (2000*(1+heightDecay 1))*l := by ring
+  norm_num only [show 2*1000=2000 by norm_num,Nat.cast_ofNat] at hh
+  simpa only [he,tuples_card] using hh
+
+lemma both_pointwise (d l : ℕ) (order : Equiv.Perm (Fin d)) (orientation : Bool)
+    (x y : Tuples d l) :
+    ((reflectedKernel d l order orientation)^2001) x y ≤
+      Real.exp ((2000*(1+heightDecay 1))*l)/((2^d).descFactorial l:ℝ) := by
+  cases orientation
+  · change ((reflectedKernel d l order false)^(2000+1)) x y ≤ _
+    rw [pow_succ]
+    exact bound_mul_right _ _ (reflected_nonneg d l order) (reflected_col d l order)
+      (canonical_pointwise d l order) x y
+  · let A := sweepKernel d l order
+    change ((A*A.transpose)^(2000+1)) x y ≤ _
+    rw [←sandwich_power]
+    apply bound_mul_right _ _ (fun x y => sweepKernel_nonneg d l order y x)
+      (sweepKernel_row d l order)
+    exact bound_mul_left _ _ (sweepKernel_nonneg d l order) (sweepKernel_row d l order)
+      (canonical_pointwise d l order)
+
+end Thorp.StrongSmoothing
+
+end ThorpNine.Harmonic
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Harmonic/TypedSet.lean b/lean/OAI/Probability/ThorpRouting/Harmonic/TypedSet.lean
index 2ba0f83aa32473dec93615b4eb94fc1e50034e31..a91fc6bfb6353e095747d39edd1bb72feaad7b7f 100644
--- a/lean/OAI/Probability/ThorpRouting/Harmonic/TypedSet.lean
+++ b/lean/OAI/Probability/ThorpRouting/Harmonic/TypedSet.lean
@@ -1,871 +1,871 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Harmonic.PairAssembleEquiv
-
-namespace OAI
-
-namespace ThorpNine.Harmonic
-
-namespace Thorp.SparseContact
-open scoped BigOperators Classical
-
-lemma noIsolated_weight_le {V : Type*} [Fintype V] (G : SimpleGraph V)
-    (p : V → ℝ) (hp : ∀ v, 0 ≤ p v) (k : ℕ) (δ : ℝ)
-    (hδ : 0 < δ) (hδ1 : δ ≤ 1)
-    (hdeg : ∀ v, (∑ w, if G.Adj v w then p w else 0) ≤ δ) :
-    (∑ S : {S : Finset V // S.card = k ∧ NoIsolated G S}, ∏ v ∈ S.val, p v) ≤
-      δ^((k:ℝ)/2) * Real.exp (Real.exp 1 * ∑ v, p v) := by
-  classical
-  let I := {S : Finset V // S.card = k ∧ NoIsolated G S}
-  let root (S : I) := (small_dominating_subset G S.val S.property.2).choose
-  have hroot (S : I) : root S ⊆ S.val ∧ 2*(root S).card ≤ k ∧
-      S.val \ root S ⊆ neighborhood G (root S) := by
-    simpa only [root, S.property.1] using
-      (small_dominating_subset G S.val S.property.2).choose_spec
-  let enc (S : I) : Finset V × Finset V := (root S, S.val \ root S)
-  have hrec (S : I) : (enc S).1 ∪ (enc S).2 = S.val :=
-    Finset.union_sdiff_of_subset (hroot S).1
-  have hinj : Function.Injective enc := by
-    intro S T h
-    apply Subtype.ext
-    rw [← hrec S, ← hrec T, h]
-  let W (z : Finset V × Finset V) : ℝ :=
-    if 2*z.1.card ≤ k ∧ z.2 ∈ (neighborhood G z.1).powersetCard (k-z.1.card)
-    then (∏ v ∈ z.1, p v) * ∏ v ∈ z.2, p v else 0
-  have hW (z : Finset V × Finset V) : 0 ≤ W z := by
-    dsimp [W]
-    split_ifs
-    · exact mul_nonneg (Finset.prod_nonneg (fun v _ => hp v))
-        (Finset.prod_nonneg (fun v _ => hp v))
-    · exact le_rfl
-  have hwenc (S : I) : W (enc S) = ∏ v ∈ S.val, p v := by
-    have hmem : (enc S).2 ∈ (neighborhood G (enc S).1).powersetCard (k-(enc S).1.card) := by
-      refine Finset.mem_powersetCard.mpr ⟨(hroot S).2.2, ?_⟩
-      change (S.val \ root S).card = k-(root S).card
-      rw [Finset.card_sdiff_of_subset (hroot S).1, S.property.1]
-    dsimp only [W]
-    rw [ite_eq_left ⟨(hroot S).2.1,hmem⟩]
-    exact (mul_comm _ _).trans (Finset.prod_sdiff (hroot S).1)
-  have hsum : (∑ S : I, ∏ v ∈ S.val, p v) ≤ ∑ z, W z := by
-    calc
-      _ = ∑ z ∈ Finset.univ.image enc, W z := by
-        rw [Finset.sum_image (fun S _ T _ h => hinj h)]
-        exact Finset.sum_congr rfl (fun S _ => (hwenc S).symm)
-      _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _)
-        (fun z _ _ => hW z)
-  apply hsum.trans
-  rw [Fintype.sum_prod_type]
-  calc
-    (∑ R : Finset V, ∑ L : Finset V, W (R,L)) ≤
-        ∑ R : Finset V, δ^((k:ℝ)/2) * Real.exp R.card * ∏ v ∈ R, p v := by
-      apply Finset.sum_le_sum
-      intro R _
-      by_cases hR : 2*R.card ≤ k
-      · have he : (∑ L : Finset V, W (R,L)) =
-            (∏ v ∈ R, p v) *
-              ∑ L ∈ (neighborhood G R).powersetCard (k-R.card), ∏ v ∈ L, p v := by
-          simp only [W, hR, true_and]
-          rw [← Finset.sum_filter]
-          simp only [Finset.filter_mem_eq_inter, Finset.univ_inter, Finset.mul_sum]
-        rw [he]
-        have h := mul_le_mul_of_nonneg_left (root_leaves_weight_le G p hp k δ hδ hδ1 hdeg R hR)
-          (Finset.prod_nonneg (s := R) (fun v _ => hp v))
-        simpa only [mul_comm, mul_left_comm, mul_assoc] using h
-      · simp only [W, hR, false_and, ite_false, Finset.sum_const_zero]
-        exact mul_nonneg (mul_nonneg (Real.rpow_nonneg hδ.le _) (Real.exp_pos _).le)
-          (Finset.prod_nonneg (fun v _ => hp v))
-    _ = δ^((k:ℝ)/2) * ∑ R : Finset V, ∏ v ∈ R, (Real.exp 1 * p v) := by
-      rw [Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro R _
-      rw [Finset.prod_mul_distrib, Finset.prod_const, ← Real.exp_nat_mul]
-      simp only [mul_one, mul_assoc]
-    _ ≤ δ^((k:ℝ)/2) * Real.exp (∑ v, Real.exp 1 * p v) := by
-      apply mul_le_mul_of_nonneg_left _ (Real.rpow_nonneg hδ.le _)
-      simpa only [Finset.powerset_univ] using
-        powerset_weight_le_exp Finset.univ (fun v => Real.exp 1 * p v)
-          (fun v => mul_nonneg (Real.exp_pos _).le (hp v))
-    _ = _ := by rw [← Finset.mul_sum]
-
-
-lemma noIsolated_probability_le {V Ω : Type*} [Fintype V] [Fintype Ω]
-    (G : SimpleGraph V) (O : Ω → Finset V) (k : ℕ)
-    (hcard : ∀ ω, (O ω).card = k) (p : V → ℝ) (hp : ∀ v, 0 ≤ p v)
-    (hcyl : ∀ D, inclusionMoment O D ≤ ∏ v ∈ D, p v)
-    (δ : ℝ) (hδ : 0 < δ) (hδ1 : δ ≤ 1)
-    (hdeg : ∀ v, (∑ w, if G.Adj v w then p w else 0) ≤ δ) :
-    finiteMean (fun ω => if NoIsolated G (O ω) then (1:ℝ) else 0) ≤
-      δ^((k:ℝ)/2) * Real.exp (Real.exp 1 * ∑ v, p v) := by
-  classical
-  let I := {S : Finset V // S.card = k ∧ NoIsolated G S}
-  calc
-    _ ≤ finiteMean (fun ω => ∑ S : I, if S.val ⊆ O ω then (1:ℝ) else 0) := by
-      apply finiteMean_mono
-      intro ω
-      split_ifs with h
-      · exact (Finset.single_le_sum (f := fun S : I => if S.val ⊆ O ω then (1:ℝ) else 0)
-          (fun S _ => by positivity) (Finset.mem_univ (⟨O ω,hcard ω,h⟩ : I))) |>.trans' (by simp)
-      · positivity
-    _ = ∑ S : I, inclusionMoment O S.val := by
-      rw [finiteMean_sum]
-      rfl
-    _ ≤ ∑ S : I, ∏ v ∈ S.val, p v := Finset.sum_le_sum (fun S _ => hcyl S.val)
-    _ ≤ _ := noIsolated_weight_le G p hp k δ hδ hδ1 hdeg
-
-noncomputable def typedSet {N X : Type*} [Fintype N] [Fintype X]
-    (A : N → Finset X) : Finset (N × X) := by
-  classical
-  exact Finset.univ.filter (fun v => v.2 ∈ A v.1)
-
-@[simp] lemma mem_typedSet {N X : Type*} [Fintype N] [Fintype X]
-    (A : N → Finset X) (v : N × X) : v ∈ typedSet A ↔ v.2 ∈ A v.1 := by
-  classical
-  simp [typedSet]
-
-noncomputable def fiber {N X : Type*} [Fintype X]
-    (S : Finset (N × X)) (t : N) : Finset X := by
-  classical
-  exact Finset.univ.filter (fun x => (t,x) ∈ S)
-
-@[simp] lemma mem_fiber {N X : Type*} [Fintype X]
-    (S : Finset (N × X)) (t : N) (x : X) : x ∈ fiber S t ↔ (t,x) ∈ S := by
-  classical
-  simp [fiber]
-
-lemma typedSet_fiber {N X : Type*} [Fintype N] [Fintype X]
-    (S : Finset (N × X)) : typedSet (fiber S) = S := by
-  classical
-  ext v
-  simp
-
-lemma typedSet_prod {N X : Type*} [Fintype N] [Fintype X]
-    (A : N → Finset X) (p : N × X → ℝ) :
-    (∏ v ∈ typedSet A, p v) = ∏ t, ∏ x ∈ A t, p (t,x) := by
-  classical
-  simp only [typedSet, Finset.prod_filter, Fintype.prod_prod_type]
-  apply Finset.prod_congr rfl
-  intro t _
-  rw [← Finset.prod_filter]
-  simp only [Finset.filter_mem_eq_inter, Finset.univ_inter]
-
-lemma typedSet_card {N X : Type*} [Fintype N] [Fintype X]
-    (A : N → Finset X) : (typedSet A).card = ∑ t, (A t).card := by
-  classical
-  simp only [Finset.card_eq_sum_ones, typedSet, Finset.sum_filter, Fintype.sum_prod_type]
-  apply Finset.sum_congr rfl
-  intro t _
-  rw [← Finset.sum_filter]
-  simp only [Finset.filter_mem_eq_inter, Finset.univ_inter]
-
-lemma subset_typedSet {N X : Type*} [Fintype N] [Fintype X]
-    (A : N → Finset X) (D : Finset (N × X)) :
-    D ⊆ typedSet A ↔ ∀ t, fiber D t ⊆ A t := by
-  constructor
-  · intro h t x hx
-    exact mem_typedSet A (t,x) |>.mp (h (mem_fiber D t x |>.mp hx))
-  · intro h v hv
-    exact mem_typedSet A v |>.mpr (h v.1 (mem_fiber D v.1 v.2 |>.mpr hv))
-
-noncomputable def typedMarks {N : Type*} [Fintype N] (d : ℕ)
-    (A : N → Finset (Card d)) (ω : N → (SwitchIndex d → Bool)) : Finset (N × Card d) :=
-  typedSet (fun t => butterflyMarks d (A t) (ω t))
-
-lemma typedMarks_card {N : Type*} [Fintype N] (d : ℕ)
-    (A : N → Finset (Card d)) (ω : N → (SwitchIndex d → Bool)) :
-    (typedMarks d A ω).card = ∑ t, (A t).card := by
-  classical
-  rw [typedMarks, typedSet_card]
-  apply Finset.sum_congr rfl
-  intro t _
-  exact Finset.card_image_of_injective _ (butterflyPerm d (decodeButterfly d (ω t))).injective
-
-lemma typedMarks_inclusion {N : Type*} [Fintype N] (d : ℕ)
-    (A : N → Finset (Card d)) (D : Finset (N × Card d)) :
-    inclusionMoment (typedMarks d A) D ≤
-      ∏ v ∈ D, ((A v.1).card:ℝ)/(2:ℝ)^d := by
-  classical
-  have he : inclusionMoment (typedMarks d A) D =
-      ∏ t, inclusionMoment (butterflyMarks d (A t)) (fiber D t) := by
-    rw [inclusionMoment_eq]
-    have hh (ω : N → (SwitchIndex d → Bool)) :
-        (if D ⊆ typedMarks d A ω then (1:ℝ) else 0) =
-        ∏ t, if fiber D t ⊆ butterflyMarks d (A t) (ω t) then (1:ℝ) else 0 := by
-      rw [Fintype.prod_boole]
-      simp only [typedMarks, subset_typedSet]
-    rw [finiteMean_congr hh]
-    simp only [inclusionMoment_eq]
-    exact finiteMean_pi_prod (ι := N) (Ω := fun _ => SwitchIndex d → Bool)
-      (fun t (ω : SwitchIndex d → Bool) =>
-        if fiber D t ⊆ butterflyMarks d (A t) ω then (1:ℝ) else 0)
-  have hprod : (∏ v ∈ D, ((A v.1).card:ℝ)/(2:ℝ)^d) =
-      ∏ t, ∏ x ∈ fiber D t, ((A t).card:ℝ)/(2:ℝ)^d := by
-    conv_lhs => rw [← typedSet_fiber D]
-    exact typedSet_prod _ _
-  rw [he, hprod]
-  apply Finset.prod_le_prod₀
-  · intro t _; exact inclusionMoment_nonneg _ _
-  · intro t _
-    simpa only [Finset.prod_const] using butterflyMarks_inclusion d (A t) (fiber D t)
-
-noncomputable def returnRoute {N : Type*} (d : ℕ) (b : N → Butterfly d)
-    (v : N × Card d) : Finset (SwitchIndex d) :=
-  routeDomain d ((butterflyPerm d (b v.1)).symm v.2) v.2
-
-noncomputable def returnGraph {N : Type*} (d : ℕ) (b : N → Butterfly d) :
-    SimpleGraph (N × Card d) where
-  Adj v w := v ≠ w ∧ ¬Disjoint (returnRoute d b v) (returnRoute d b w)
-  symm := ⟨fun _ _ h => ⟨h.1.symm, fun hh => h.2 hh.symm⟩⟩
-  loopless := ⟨fun _ h => h.1 rfl⟩
-
-lemma returnRoute_card {N : Type*} (d : ℕ) (b : N → Butterfly d) (v : N × Card d) :
-    (returnRoute d b v).card = d := card_routeDomain _ _ _
-
-lemma mem_returnRoute {N : Type*} (d : ℕ) (b : N → Butterfly d)
-    (t : N) (y : Card d) (i : SwitchIndex d) :
-    i ∈ returnRoute d b (t,y) ↔
-      y = butterflyPerm d (b t) (switchUsers d (b t) i).1 ∨
-      y = butterflyPerm d (b t) (switchUsers d (b t) i).2 := by
-  have h := mem_routeDomain_iff_user d (b t) ((butterflyPerm d (b t)).symm y) i
-  simpa only [Equiv.apply_symm_apply, returnRoute, Equiv.eq_symm_apply, eq_comm] using h
-
-lemma returnRoute_incidence {N : Type*} [Fintype N] (d : ℕ) (b : N → Butterfly d)
-    (p : N → ℝ) (i : SwitchIndex d) :
-    (∑ v : N × Card d, if i ∈ returnRoute d b v then p v.1 else 0) =
-      2 * ∑ t, p t := by
-  classical
-  rw [Fintype.sum_prod_type, Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro t _
-  have hne : butterflyPerm d (b t) (switchUsers d (b t) i).1 ≠
-      butterflyPerm d (b t) (switchUsers d (b t) i).2 :=
-    (butterflyPerm d (b t)).injective.ne (switchUsers_ne d (b t) i)
-  simp only [mem_returnRoute]
-  have he (y : Card d) :
-      (if y = butterflyPerm d (b t) (switchUsers d (b t) i).1 ∨
-        y = butterflyPerm d (b t) (switchUsers d (b t) i).2 then p t else 0) =
-      (if y = butterflyPerm d (b t) (switchUsers d (b t) i).1 then p t else 0) +
-      (if y = butterflyPerm d (b t) (switchUsers d (b t) i).2 then p t else 0) := by
-    split_ifs <;> simp_all
-  simp only [he, Finset.sum_add_distrib, Finset.sum_ite_eq', Finset.mem_univ, ite_true]
-  ring
-
-lemma returnGraph_degree {N : Type*} [Fintype N] (d : ℕ) (b : N → Butterfly d)
-    (p : N → ℝ) (hp : ∀ t, 0 ≤ p t) (v : N × Card d) :
-    (∑ w, if (returnGraph d b).Adj v w then p w.1 else 0) ≤
-      2*d*∑ t, p t := by
-  classical
-  calc
-    _ ≤ ∑ w, ∑ i ∈ returnRoute d b v, if i ∈ returnRoute d b w then p w.1 else 0 := by
-      apply Finset.sum_le_sum
-      intro w _
-      split_ifs with h
-      · obtain ⟨i, hi, hw⟩ := Finset.not_disjoint_iff.mp h.2
-        have hh := Finset.single_le_sum
-          (f := fun i => if i ∈ returnRoute d b w then p w.1 else 0)
-          (fun i _ => by split_ifs; exact hp _; exact le_rfl) hi
-        simpa only [ite_eq_left hw] using hh
-      · exact Finset.sum_nonneg (fun i _ => by split_ifs; exact hp _; exact le_rfl)
-    _ = ∑ i ∈ returnRoute d b v, ∑ w, if i ∈ returnRoute d b w then p w.1 else 0 :=
-      Finset.sum_comm
-    _ = _ := by simp only [returnRoute_incidence, Finset.sum_const, returnRoute_card,
-        nsmul_eq_mul]; ring
-
-lemma typed_weight_sum {N : Type*} [Fintype N] (d : ℕ)
-    (A : N → Finset (Card d)) :
-    (∑ v : N × Card d, ((A v.1).card:ℝ)/(2:ℝ)^d) = ∑ t, ((A t).card:ℝ) := by
-  rw [Fintype.sum_prod_type]
-  apply Finset.sum_congr rfl
-  intro t _
-  simp only [Finset.sum_const, Finset.card_univ, card_positions, Nat.cast_pow,
-    Nat.cast_ofNat, nsmul_eq_mul]
-  field_simp
-
-lemma mixed_return_noIsolated {N : Type*} [Fintype N] (d : ℕ)
-    (A : N → Finset (Card d)) (b : N → Butterfly d) (k : ℕ)
-    (hk : ∑ t, (A t).card = k)
-    (hδ : 0 < 2*(d:ℝ)*k/(2:ℝ)^d) (hδ1 : 2*(d:ℝ)*k/(2:ℝ)^d ≤ 1) :
-    finiteMean (fun ω : N → (SwitchIndex d → Bool) =>
-      if NoIsolated (returnGraph d b) (typedMarks d A ω) then (1:ℝ) else 0) ≤
-      (2*(d:ℝ)*k/(2:ℝ)^d)^((k:ℝ)/2) * Real.exp (Real.exp 1*k) := by
-  have hs : (∑ t, ((A t).card:ℝ)) = k := by exact_mod_cast hk
-  have hm := noIsolated_probability_le (returnGraph d b) (typedMarks d A) k
-    (fun ω => (typedMarks_card d A ω).trans hk)
-    (fun v => ((A v.1).card:ℝ)/(2:ℝ)^d) (fun v => by positivity)
-    (typedMarks_inclusion d A) _ hδ hδ1
-    (fun v => (returnGraph_degree d b (fun t => ((A t).card:ℝ)/(2:ℝ)^d)
-      (fun t => by positivity) v).trans_eq (by rw [← Finset.sum_div, hs]; ring))
-  simpa only [typed_weight_sum, hs] using hm
-
-
-lemma mean_fresh_cylinder {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (s : Finset ι) (v : ι → Bool) (f : (ι → Bool) → ℝ)
-    (hf : ∀ i ∈ s, ∀ ω, f (QueryTree.flipCoin i ω) = f ω) :
-    finiteMean (fun ω => if ∀ i ∈ s, ω i = v i then f ω else 0) =
-      finiteMean f / (2:ℝ)^s.card := by
-  induction s using Finset.induction_on with
-  | empty => simp [finiteMean]
-  | @insert i s hi ih =>
-    let g := fun ω => if ∀ j ∈ s, ω j = v j then f ω else 0
-    have hg : ∀ ω, g (QueryTree.flipCoin i ω) = g ω := by
-      intro ω
-      have he : (∀ j ∈ s, QueryTree.flipCoin i ω j = v j) ↔ ∀ j ∈ s, ω j = v j := by
-        apply forall₂_congr
-        intro j hj
-        rw [QueryTree.flipCoin_apply_ne i j (fun h => hi (h ▸ hj))]
-      simp only [g, he, hf i (Finset.mem_insert_self _ _) ω]
-    have he (ω : ι → Bool) :
-        (if ∀ j ∈ insert i s, ω j = v j then f ω else 0) =
-        (if ω i = v i then g ω else 0) := by
-      by_cases h : ω i = v i <;> simp [h,g]
-    rw [finiteMean_congr he, QueryTree.mean_indicator_bit g i (v i) hg]
-    have hs := ih (fun j hj => hf j (Finset.mem_insert_of_mem hj))
-    change finiteMean (fun ω => if ∀ j ∈ s, ω j = v j then f ω else 0) / 2 = _
-    rw [hs, Finset.card_insert_of_notMem hi, pow_succ]
-    ring
-
-noncomputable def endpointProb {L : Type*} (d : ℕ) (A : Finset L)
-    (x y : L → Card d) : ℝ := by
-  classical
-  exact finiteMean (fun ω : SwitchIndex d → Bool =>
-    if ∀ i ∈ A, butterflyPerm d (decodeButterfly d ω) (x i) = y i then (1:ℝ) else 0)
-
-lemma endpointProb_nonneg {L : Type*} (d : ℕ) (A : Finset L)
-    (x y : L → Card d) : 0 ≤ endpointProb d A x y := by
-  classical
-  exact finiteMean_nonneg (fun _ => by positivity)
-
-lemma endpointProb_insert_isolated {L : Type*} (d : ℕ) (A : Finset L)
-    (x y : L → Card d) (i : L)
-    (hiso : ∀ j ∈ A, Disjoint (routeDomain d (x i) (y i)) (routeDomain d (x j) (y j))) :
-    endpointProb d (insert i A) x y = endpointProb d A x y / (2:ℝ)^d := by
-  classical
-  let f := fun ω : SwitchIndex d → Bool =>
-    if ∀ j ∈ A, butterflyPerm d (decodeButterfly d ω) (x j) = y j then (1:ℝ) else 0
-  have hf : ∀ t ∈ routeDomain d (x i) (y i), ∀ ω, f (QueryTree.flipCoin t ω) = f ω := by
-    intro t ht ω
-    have he : (∀ j ∈ A, butterflyPerm d (decodeButterfly d (QueryTree.flipCoin t ω)) (x j) = y j) ↔
-        ∀ j ∈ A, butterflyPerm d (decodeButterfly d ω) (x j) = y j := by
-      apply forall₂_congr
-      intro j hj
-      rw [butterfly_endpoint_iff, butterfly_endpoint_iff]
-      apply forall₂_congr
-      intro s hs
-      rw [QueryTree.flipCoin_apply_ne t s (fun h =>
-        Finset.disjoint_left.mp (hiso j hj) ht (h ▸ hs))]
-    simp only [f, he]
-  have he (ω : SwitchIndex d → Bool) :
-      (if ∀ j ∈ insert i A, butterflyPerm d (decodeButterfly d ω) (x j) = y j then (1:ℝ) else 0) =
-      (if ∀ t ∈ routeDomain d (x i) (y i), ω t = routeValue d (x i) (y i) t then f ω else 0) := by
-    by_cases h : butterflyPerm d (decodeButterfly d ω) (x i) = y i
-    · have hh := (butterfly_endpoint_iff d ω (x i) (y i)).mp h
-      rw [ite_eq_left hh]
-      simp only [Finset.forall_mem_insert, h, true_and, f]
-    · have hh := mt (butterfly_endpoint_iff d ω (x i) (y i)).mpr h
-      simp only [Finset.forall_mem_insert, h, false_and, hh, ite_false]
-  change finiteMean _ = _
-  rw [finiteMean_congr he, mean_fresh_cylinder _ _ f hf, card_routeDomain]
-  rfl
-
-noncomputable def partialKernel {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x y : L → Card d) : ℝ :=
-  ((2:ℝ)^d)^A.card / ((2:ℝ)^d)^(Fintype.card L) * endpointProb d A x y
-
-lemma partialKernel_insert_isolated {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x y : L → Card d) (i : L) (hi : i ∉ A)
-    (hiso : ∀ j ∈ A, Disjoint (routeDomain d (x i) (y i)) (routeDomain d (x j) (y j))) :
-    partialKernel d (insert i A) x y = partialKernel d A x y := by
-  classical
-  rw [partialKernel, endpointProb_insert_isolated d A x y i hiso,
-    Finset.card_insert_of_notMem hi, pow_succ, partialKernel]
-  field_simp
-
-def ContactEvent {L : Type*} (d : ℕ) (x y : L → Card d) : Prop :=
-  ∀ i, ∃ j, i ≠ j ∧ ¬Disjoint (routeDomain d (x i) (y i)) (routeDomain d (x j) (y j))
-
-lemma alternating_cube_cancel {L : Type*} [Fintype L] (f : Finset L → ℝ)
-    (i : L) (hf : ∀ A, i ∉ A → f (insert i A) = f A) :
-    (∑ A : Finset L, (-1:ℝ)^A.card * f A) = 0 := by
-  classical
-  have hi : i ∈ (Finset.univ : Finset L) := Finset.mem_univ _
-  rw [← Finset.powerset_univ, ← Finset.insert_erase hi, Finset.sum_powerset_insert]
-  · rw [← Finset.sum_add_distrib]
-    apply Finset.sum_eq_zero
-    intro A hA
-    have hiA : i ∉ A := fun hh => Finset.notMem_erase i Finset.univ
-      ((Finset.mem_powerset.mp hA) hh)
-    rw [hf A hiA, Finset.card_insert_of_notMem hiA, pow_succ]
-    ring
-  · exact Finset.notMem_erase _ _
-
-lemma alternating_kernel_cancel {L : Type*} [Fintype L] (d : ℕ)
-    (x y : L → Card d) (h : ¬ContactEvent d x y) :
-    (∑ A : Finset L, (-1:ℝ)^A.card * partialKernel d A x y) = 0 := by
-  classical
-  unfold ContactEvent at h
-  push Not at h
-  obtain ⟨i, hi⟩ := h
-  apply alternating_cube_cancel _ i
-  intro A hiA
-  apply partialKernel_insert_isolated d A x y i hiA
-  intro j hj
-  exact hi j (fun hij => hiA (hij ▸ hj))
-
-
-noncomputable def networkOf {L : Type*} (A : Finset L) (i : L) : Option L := by
-  classical
-  exact if i ∈ A then none else some i
-
-noncomputable def mixedTuple {L : Type*} (d : ℕ) (A : Finset L)
-    (ω : Option L → (SwitchIndex d → Bool)) (x : L → Card d) : L → Card d :=
-  fun i => butterflyPerm d (decodeButterfly d (ω (networkOf A i))) (x i)
-
-noncomputable def mixedReturn {L : Type*} (d : ℕ) (A : Finset L)
-    (ω : Option L → (SwitchIndex d → Bool)) (y : L → Card d) : L → Card d :=
-  fun i => (butterflyPerm d (decodeButterfly d (ω (networkOf A i)))).symm (y i)
-
-noncomputable def labelsOf {L : Type*} (A : Finset L) : Option L → Finset L := by
-  classical
-  exact fun t => match t with
-    | none => A
-    | some i => if i ∈ A then ∅ else {i}
-
-lemma mem_labelsOf {L : Type*} (A : Finset L) (i : L) (t : Option L) :
-    i ∈ labelsOf A t ↔ networkOf A i = t := by
-  classical
-  cases t with
-  | none => simp [labelsOf, networkOf]
-  | some j => by_cases hj : j ∈ A <;> by_cases hi : i ∈ A <;> simp_all [labelsOf, networkOf] <;> intro h <;> subst j <;> contradiction
-
-lemma mixedTuple_eq_iff {L : Type*} (d : ℕ) (A : Finset L)
-    (ω : Option L → (SwitchIndex d → Bool)) (x y : L → Card d) :
-    mixedTuple d A ω x = y ↔ ∀ t, ∀ i ∈ labelsOf A t,
-      butterflyPerm d (decodeButterfly d (ω t)) (x i) = y i := by
-  constructor
-  · intro h t i hi
-    have he := congrFun h i
-    simpa only [mixedTuple, mem_labelsOf A i t |>.mp hi] using he
-  · intro h
-    funext i
-    exact h (networkOf A i) i (mem_labelsOf A i _ |>.mpr rfl)
-
-lemma mixedReturn_eq_iff {L : Type*} (d : ℕ) (A : Finset L)
-    (ω : Option L → (SwitchIndex d → Bool)) (y z : L → Card d) :
-    mixedReturn d A ω y = z ↔ mixedTuple d A ω z = y := by
-  constructor
-  · intro h
-    funext i
-    exact (Equiv.symm_apply_eq _).mp (congrFun h i) |>.symm
-  · intro h
-    funext i
-    exact (Equiv.symm_apply_eq _).mpr (congrFun h i).symm
-
-lemma endpointProb_singleton {L : Type*} (d : ℕ) (i : L) (x y : L → Card d) :
-    endpointProb d {i} x y = 1/(2:ℝ)^d := by
-  classical
-  simpa [endpointProb] using butterfly_single_card_uniform d (x i) (y i)
-
-lemma mixedTuple_mass {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x y : L → Card d) :
-    finiteMean (fun ω : Option L → (SwitchIndex d → Bool) =>
-      if mixedTuple d A ω x = y then (1:ℝ) else 0) = partialKernel d A x y := by
-  classical
-  have he (ω : Option L → (SwitchIndex d → Bool)) :
-      (if mixedTuple d A ω x = y then (1:ℝ) else 0) =
-      ∏ t, if ∀ i ∈ labelsOf A t, butterflyPerm d (decodeButterfly d (ω t)) (x i) = y i
-        then (1:ℝ) else 0 := by
-    simp only [Fintype.prod_boole, mixedTuple_eq_iff]
-  rw [finiteMean_congr he]
-  rw [finiteMean_pi_prod (fun t (w : SwitchIndex d → Bool) =>
-    if ∀ i ∈ labelsOf A t, butterflyPerm d (decodeButterfly d w) (x i) = y i then (1:ℝ) else 0)]
-  have hfac (t : Option L) : (finiteMean (fun w : SwitchIndex d → Bool =>
-      if ∀ i ∈ labelsOf A t, butterflyPerm d (decodeButterfly d w) (x i) = y i then (1:ℝ) else 0)) =
-      endpointProb d (labelsOf A t) x y := by
-        unfold endpointProb
-        apply finiteMean_congr
-        intro w
-        split_ifs <;> rfl
-  simp only [hfac]
-  rw [Fintype.prod_option]
-  have hi (i : L) : endpointProb d (labelsOf A (some i)) x y =
-      if i ∈ A then 1 else 1/(2:ℝ)^d := by
-    by_cases h : i ∈ A
-    · simp [labelsOf, h, endpointProb, finiteMean_const]
-    · simp only [labelsOf, ite_eq_right h, endpointProb_singleton]
-  rw [show labelsOf A none = A from rfl]
-  simp only [hi]
-  have hp : (∏ i : L, if i ∈ A then (1:ℝ) else 1/(2:ℝ)^d) =
-      (1/(2:ℝ)^d)^((Fintype.card L) - A.card) := by
-    rw [Finset.prod_ite]
-    simp only [Finset.prod_const_one, one_mul]
-    simp only [Finset.filter_not, Finset.filter_mem_eq_inter, Finset.univ_inter,
-      Finset.prod_const, Finset.card_sdiff, Finset.inter_univ, Finset.card_univ]
-  rw [hp, partialKernel, one_div, inv_pow]
-  have hc : A.card ≤ Fintype.card L := Finset.card_le_univ _
-  have hn : (((2:ℝ)^d)^A.card) * (((2:ℝ)^d)^(Fintype.card L - A.card)) =
-      ((2:ℝ)^d)^(Fintype.card L) := by rw [← pow_add, Nat.add_sub_of_le hc]
-  have hnz : ((2:ℝ)^d)^(Fintype.card L) ≠ 0 := by positivity
-  have hmz : ((2:ℝ)^d)^(Fintype.card L - A.card) ≠ 0 := by positivity
-  field_simp
-  rw [← hn]
-  ring
-
-lemma mixedReturn_mass {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (y z : L → Card d) :
-    finiteMean (fun ω : Option L → (SwitchIndex d → Bool) =>
-      if mixedReturn d A ω y = z then (1:ℝ) else 0) = partialKernel d A z y := by
-  simpa only [mixedReturn_eq_iff] using mixedTuple_mass d A z y
-
-noncomputable def mixedInitial {L : Type*} (d : ℕ) (A : Finset L)
-    (x : L → Card d) (t : Option L) : Finset (Card d) := by
-  classical
-  exact (labelsOf A t).image x
-
-lemma typedMarks_mixedInitial {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x : L → Card d) (ω : Option L → (SwitchIndex d → Bool)) :
-    typedMarks d (mixedInitial d A x) ω = Finset.univ.image
-      (fun i => (networkOf A i, mixedTuple d A ω x i)) := by
-  classical
-  ext v
-  simp only [typedMarks, mem_typedSet, butterflyMarks, mixedInitial,
-    Finset.mem_image, Finset.mem_univ, true_and]
-  constructor
-  · rintro ⟨z, ⟨i,hi,rfl⟩, hv⟩
-    have ht := mem_labelsOf A i v.1 |>.mp hi
-    exact ⟨i, Prod.ext ht (by simpa only [mixedTuple, ht] using hv)⟩
-  · rintro ⟨i, hi⟩
-    have ht := congrArg Prod.fst hi
-    have hv := congrArg Prod.snd hi
-    change networkOf A i = v.1 at ht
-    change mixedTuple d A ω x i = v.2 at hv
-    exact ⟨x i, ⟨i, mem_labelsOf A i v.1 |>.mpr ht, rfl⟩,
-      by simpa only [mixedTuple, ht] using hv⟩
-
-lemma typedTuple_injective {L : Type*} (d : ℕ) (A : Finset L)
-    (x : L → Card d) (hx : Function.Injective x)
-    (ω : Option L → (SwitchIndex d → Bool)) :
-    Function.Injective (fun i => (networkOf A i, mixedTuple d A ω x i)) := by
-  intro i j h
-  have ht := congrArg Prod.fst h
-  have hv := congrArg Prod.snd h
-  change networkOf A i = networkOf A j at ht
-  dsimp only [mixedTuple] at hv
-  rw [ht] at hv
-  exact hx ((butterflyPerm d (decodeButterfly d (ω (networkOf A j)))).injective hv)
-
-lemma mixedInitial_card {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x : L → Card d) (hx : Function.Injective x) :
-    (∑ t, (mixedInitial d A x t).card) = Fintype.card L := by
-  classical
-  let ω : Option L → (SwitchIndex d → Bool) := fun _ _ => false
-  rw [← typedMarks_card d (mixedInitial d A x) ω, typedMarks_mixedInitial,
-    Finset.card_image_of_injective _ (typedTuple_injective d A x hx ω), Finset.card_univ]
-
-lemma contactEvent_mixedReturn_iff {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x : L → Card d) (hx : Function.Injective x)
-    (ω b : Option L → (SwitchIndex d → Bool)) :
-    ContactEvent d (mixedReturn d A b (mixedTuple d A ω x)) (mixedTuple d A ω x) ↔
-      NoIsolated (returnGraph d (fun t => decodeButterfly d (b t)))
-        (typedMarks d (mixedInitial d A x) ω) := by
-  classical
-  rw [typedMarks_mixedInitial]
-  let f := fun i => (networkOf A i, mixedTuple d A ω x i)
-  have hf : Function.Injective f := typedTuple_injective d A x hx ω
-  constructor
-  · intro h v hv
-    obtain ⟨i,_,rfl⟩ := Finset.mem_image.mp hv
-    obtain ⟨j,hij,hc⟩ := h i
-    exact ⟨f j, Finset.mem_image.mpr ⟨j, Finset.mem_univ _,rfl⟩,
-      hf.ne hij, hc⟩
-  · intro h i
-    obtain ⟨v,hv,hij,hc⟩ := h (f i) (Finset.mem_image.mpr ⟨i,Finset.mem_univ _,rfl⟩)
-    obtain ⟨j,_,rfl⟩ := Finset.mem_image.mp hv
-    exact ⟨j, fun he => hij (congrArg f he), hc⟩
-
-lemma probability_mixed_contact {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x : L → Card d) (hx : Function.Injective x)
-    (b : Option L → (SwitchIndex d → Bool))
-    (hδ : 0 < 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)
-    (hδ1 : 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d ≤ 1) :
-    finiteMean (fun ω : Option L → (SwitchIndex d → Bool) =>
-      if ContactEvent d (mixedReturn d A b (mixedTuple d A ω x)) (mixedTuple d A ω x)
-        then (1:ℝ) else 0) ≤
-      (2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
-        Real.exp (Real.exp 1*(Fintype.card L)) := by
-  classical
-  simp only [contactEvent_mixedReturn_iff d A x hx]
-  convert mixed_return_noIsolated d (mixedInitial d A x) (fun t => decodeButterfly d (b t))
-      (Fintype.card L) (mixedInitial_card d A x hx) hδ hδ1 using 1
-  congr 1
-  exact Subsingleton.elim _ _
-
-
-lemma finiteMean_pushforward {Ω X : Type*} [Fintype Ω] [Fintype X]
-    (g : Ω → X) (f : X → ℝ) :
-    finiteMean (fun ω => f (g ω)) =
-      ∑ y, finiteMean (fun ω => if g ω = y then (1:ℝ) else 0) * f y := by
-  classical
-  simp_rw [← finiteMean_mul_const, ← finiteMean_sum]
-  apply finiteMean_congr
-  intro ω
-  simp
-
-lemma mixedTuple_expectation {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x : L → Card d) (f : (L → Card d) → ℝ) :
-    finiteMean (fun ω : Option L → (SwitchIndex d → Bool) => f (mixedTuple d A ω x)) =
-      ∑ y, partialKernel d A x y * f y := by
-  classical
-  rw [finiteMean_pushforward (fun ω : Option L → (SwitchIndex d → Bool) => mixedTuple d A ω x) f]
-  apply Finset.sum_congr rfl
-  intro y _
-  congr 1
-  convert mixedTuple_mass d A x y using 1
-  congr 1
-  funext ω
-  split_ifs <;> rfl
-
-lemma mixedReturn_expectation {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (y : L → Card d) (f : (L → Card d) → ℝ) :
-    finiteMean (fun ω : Option L → (SwitchIndex d → Bool) => f (mixedReturn d A ω y)) =
-      ∑ z, partialKernel d A z y * f z := by
-  classical
-  rw [finiteMean_pushforward (fun ω : Option L → (SwitchIndex d → Bool) => mixedReturn d A ω y) f]
-  apply Finset.sum_congr rfl
-  intro z _
-  congr 1
-  convert mixedReturn_mass d A y z using 1
-  congr 1
-  funext ω
-  split_ifs <;> rfl
-
-lemma partialKernel_nonneg {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x y : L → Card d) : 0 ≤ partialKernel d A x y :=
-  mul_nonneg (by positivity) (endpointProb_nonneg d A x y)
-
-noncomputable def retainedKernel {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x : L ↪ Card d) (y : L → Card d) : ℝ := by
-  classical
-  exact if ContactEvent d x y then partialKernel d A x y else 0
-
-lemma retainedKernel_nonneg {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x : L ↪ Card d) (y : L → Card d) : 0 ≤ retainedKernel d A x y := by
-  classical
-  unfold retainedKernel
-  split_ifs
-  · exact partialKernel_nonneg d A x y
-  · exact le_rfl
-
-lemma retainedKernel_le {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x : L ↪ Card d) (y : L → Card d) : retainedKernel d A x y ≤ partialKernel d A x y := by
-  classical
-  unfold retainedKernel
-  split_ifs
-  · exact le_rfl
-  · exact partialKernel_nonneg d A x y
-
-lemma retained_square_row {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x : L ↪ Card d)
-    (hδ : 0 < 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)
-    (hδ1 : 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d ≤ 1) :
-    (∑ x' : L ↪ Card d, ∑ y : L → Card d, retainedKernel d A x y * retainedKernel d A x' y) ≤
-      (2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
-        Real.exp (Real.exp 1*(Fintype.card L)) := by
-  classical
-  let f := fun y z : L → Card d => if ContactEvent d z y then (1:ℝ) else 0
-  have hsub (y : L → Card d) : (∑ x' : L ↪ Card d, retainedKernel d A x' y) ≤
-      ∑ z : L → Card d, partialKernel d A z y * f y z := by
-    have he (x' : L ↪ Card d) : retainedKernel d A x' y = partialKernel d A x' y * f y x' := by
-      unfold retainedKernel f
-      split_ifs <;> ring
-    simp_rw [he]
-    rw [← Finset.sum_image (f := fun z => partialKernel d A z y * f y z)
-      (Function.Embedding.coe_injective.injOn)]
-    apply Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _)
-    intro z _ _
-    exact mul_nonneg (partialKernel_nonneg d A z y) (by dsimp [f]; positivity)
-  calc
-    _ = ∑ y, retainedKernel d A x y * ∑ x' : L ↪ Card d, retainedKernel d A x' y := by
-      rw [Finset.sum_comm]
-      simp only [Finset.mul_sum]
-    _ ≤ ∑ y, partialKernel d A x y * ∑ z : L → Card d, partialKernel d A z y * f y z := by
-      apply Finset.sum_le_sum
-      intro y _
-      exact mul_le_mul (retainedKernel_le d A x y) (hsub y)
-        (Finset.sum_nonneg (fun x' _ => retainedKernel_nonneg d A x' y)) (partialKernel_nonneg d A x y)
-    _ = finiteMean (fun ω : Option L → (SwitchIndex d → Bool) =>
-        finiteMean (fun b : Option L → (SwitchIndex d → Bool) =>
-          f (mixedTuple d A ω x) (mixedReturn d A b (mixedTuple d A ω x)))) := by
-      rw [mixedTuple_expectation d A x (fun y =>
-        finiteMean (fun b : Option L → (SwitchIndex d → Bool) => f y (mixedReturn d A b y)))]
-      simp only [mixedReturn_expectation]
-    _ = finiteMean (fun b : Option L → (SwitchIndex d → Bool) =>
-        finiteMean (fun ω : Option L → (SwitchIndex d → Bool) =>
-          f (mixedTuple d A ω x) (mixedReturn d A b (mixedTuple d A ω x)))) := finiteMean_comm _
-    _ ≤ _ := by
-      apply (finiteMean_mono (fun b => probability_mixed_contact d A x x.injective b hδ hδ1)).trans_eq
-      exact finiteMean_const _
-
-noncomputable def rectOperator {X Y : Type*} [Fintype X] [Fintype Y]
-    (P : X → Y → ℝ) : EuclideanSpace ℂ Y →L[ℂ] EuclideanSpace ℂ X :=
-  LinearMap.toContinuousLinearMap
-    { toFun := fun v => WithLp.toLp 2 (fun x => ∑ y, (P x y : ℂ)*v y)
-      map_add' := by
-        intro v w
-        ext x
-        simp only [PiLp.add_apply, mul_add, Finset.sum_add_distrib]
-      map_smul' := by
-        intro c v
-        ext x
-        simp only [PiLp.smul_apply, smul_eq_mul, RingHom.id_apply]
-        change (∑ y, (P x y : ℂ)*(c*v y)) = c*∑ y, (P x y : ℂ)*v y
-        rw [Finset.mul_sum]
-        apply Finset.sum_congr rfl
-        intro y _
-        ring }
-
-@[simp] lemma rectOperator_apply {X Y : Type*} [Fintype X] [Fintype Y]
-    (P : X → Y → ℝ) (v : EuclideanSpace ℂ Y) (x : X) :
-    rectOperator P v x = ∑ y, (P x y:ℂ)*v y := rfl
-
-lemma rectOperator_adjoint {X Y : Type*} [Fintype X] [Fintype Y]
-    (P : X → Y → ℝ) : ContinuousLinearMap.adjoint (rectOperator P) =
-      rectOperator (fun y x => P x y) := by
-  apply Eq.symm
-  rw [ContinuousLinearMap.eq_adjoint_iff]
-  intro v w
-  simp only [PiLp.inner_apply, RCLike.inner_apply, rectOperator_apply,
-    map_sum, map_mul, Complex.conj_ofReal, Finset.sum_mul, Finset.mul_sum]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro x _
-  apply Finset.sum_congr rfl
-  intro y _
-  ring
-
-lemma rectOperator_comp_transpose {X Y : Type*} [Fintype X] [Fintype Y]
-    (P : X → Y → ℝ) : (rectOperator P).comp (rectOperator (fun y x => P x y)) =
-      rectOperator (fun x x' => ∑ y, P x y * P x' y) := by
-  ext v x
-  simp only [ContinuousLinearMap.comp_apply, rectOperator_apply, Finset.mul_sum,
-    Complex.ofReal_sum, Complex.ofReal_mul, Finset.sum_mul]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro x' _
-  apply Finset.sum_congr rfl
-  intro y _
-  ring
-
-lemma rectOperator_norm_sq_le {X Y : Type*} [Fintype X] [Fintype Y]
-    (P : X → Y → ℝ) (hP : ∀ x y, 0 ≤ P x y) (B : ℝ) (hB : 0 ≤ B)
-    (hrow : ∀ x, (∑ x', ∑ y, P x y * P x' y) ≤ B) :
-    ‖rectOperator P‖^2 ≤ B := by
-  let S := fun x x' => ∑ y, P x y * P x' y
-  have hs : ∀ x x', S x x' = S x' x := by
-    intro x x'
-    unfold S
-    apply Finset.sum_congr rfl
-    intro y _
-    ring
-  have hbound : ‖rectOperator S‖ ≤ B := by
-    apply ContinuousLinearMap.opNorm_le_bound _ hB
-    intro v
-    exact DensityTransfer.schur_bound S
-      (fun x x' => Finset.sum_nonneg (fun y _ => mul_nonneg (hP x y) (hP x' y))) B hB
-      hrow (fun x' => by
-        calc (∑ x, S x x') = ∑ x, S x' x := Finset.sum_congr rfl (fun x _ => hs x x')
-             _ ≤ B := hrow x') v
-  have hn := ContinuousLinearMap.norm_adjoint_comp_self
-    (ContinuousLinearMap.adjoint (rectOperator P))
-  rw [ContinuousLinearMap.adjoint_adjoint, (ContinuousLinearMap.adjoint.norm_map _),
-    rectOperator_adjoint, rectOperator_comp_transpose] at hn
-  rw [pow_two, ← hn]
-  exact hbound
-
-lemma retained_norm_sq {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (hδ : 0 < 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)
-    (hδ1 : 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d ≤ 1) :
-    ‖rectOperator (retainedKernel d A)‖^2 ≤
-      (2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
-        Real.exp (Real.exp 1*(Fintype.card L)) :=
-  rectOperator_norm_sq_le _ (retainedKernel_nonneg d A) _ (by positivity)
-    (fun x => retained_square_row d A x hδ hδ1)
-
-
-section Orbit
-variable {G X V : Type*} [Group G] [Fintype G] [Fintype X]
-    [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-    (a : G →* Equiv.Perm X) (x₀ : X) (ha : ∀ x, ∃ g, a g x₀ = x)
-    (ρ : Representation ℂ G V) (w : V) (hw : UnitaryFinite.IsFixed a x₀ ρ w)
-
-noncomputable def orbitVector (x : X) : V := ρ (UnitaryFinite.representative a x₀ ha x) w
-
-include hw in
-omit [Fintype G] [Fintype X] [FiniteDimensional ℂ V] in
-lemma orbitVector_at (g : G) : orbitVector a x₀ ha ρ w (a g x₀) = ρ g w := by
-  apply ext_inner_right ℂ
-  intro v
-  exact UnitaryFinite.coefficient_at a x₀ ha ρ w hw v g
-
-include hw in
-omit [Fintype G] [Fintype X] [FiniteDimensional ℂ V] in
-lemma orbitVector_action (g : G) (x : X) :
-    orbitVector a x₀ ha ρ w (a g x) = ρ g (orbitVector a x₀ ha ρ w x) := by
-  obtain ⟨h,rfl⟩ := ha x
-  rw [←Equiv.Perm.mul_apply, ←map_mul, orbitVector_at a x₀ ha ρ w hw,
-    orbitVector_at a x₀ ha ρ w hw, map_mul, Module.End.mul_apply]
-end Orbit
-
-def IsHarmonic {L α : Type*} [Fintype L] [Fintype α] (f : EuclideanSpace ℂ (L ↪ α)) : Prop :=
-  ∀ (A : Finset L), A ≠ Finset.univ → ∀ z : L → α,
-    (∑ y : L ↪ α, if ∀ i ∈ A, y i = z i then f y else 0) = 0
-
-lemma normalizedCoefficient_harmonic {L α : Type*} [Fintype L] [Fintype α]
-    (μ : YoungDiagram) (e : α ≃ Specht.Cell μ)
-    (hk : Fintype.card L = μ.card - μ.rowLen 0) (x₀ : L ↪ α)
-    (w : Specht.hilbertSpace μ)
-    (hw : UnitaryFinite.IsFixed UnitaryFinite.tupleAction x₀ (Specht.relabelledUnitary μ e) w)
-    (v : Specht.hilbertSpace μ) :
-    IsHarmonic (UnitaryFinite.normalizedCoefficient UnitaryFinite.tupleAction x₀
-      (UnitaryFinite.tupleAction_transitive x₀) (Specht.relabelledUnitary μ e) w v) := by
-  classical
-  intro A hA z
-  let ρ := Specht.relabelledUnitary μ e
-  let F := orbitVector UnitaryFinite.tupleAction x₀ (UnitaryFinite.tupleAction_transitive x₀) ρ w
-  let S : Specht.hilbertSpace μ := ∑ y : L ↪ α, if ∀ i ∈ A, y i = z i then F y else 0
-  have hcard : (A.image z).card < μ.card - μ.rowLen 0 := by
-    apply (Finset.card_image_le).trans_lt
-    rw [← hk]
-    exact Finset.card_lt_card (Finset.ssubset_univ_iff.mpr hA)
-  have hz : S = 0 := by
-    apply point_relabelled_fixed_zero μ e (A.image z) hcard
-    intro p hp
-    change ρ p S = S
-    simp only [S, map_sum, apply_ite, map_zero]
-    apply Fintype.sum_equiv (UnitaryFinite.tupleAction p)
-    intro y
-    have he : (∀ i ∈ A, UnitaryFinite.tupleAction p y i = z i) ↔ ∀ i ∈ A, y i = z i := by
-      apply forall₂_congr
-      intro i hi
-      have hpi : p (z i) = z i := hp _ (Finset.mem_image.mpr ⟨i, hi,rfl⟩)
-      change p (y i) = z i ↔ y i = z i
-      exact (show (p (y i) = z i) ↔ p (y i) = p (z i) by rw [hpi]).trans p.injective.eq_iff
-    simp only [he]
-    split_ifs
-    · exact (orbitVector_action UnitaryFinite.tupleAction x₀
-        (UnitaryFinite.tupleAction_transitive x₀) ρ w hw p y).symm
-    · rfl
-  let c : ℂ := Real.sqrt ((Module.finrank ℂ (Specht.hilbertSpace μ):ℝ)/(Fintype.card (L ↪ α):ℝ))
-  calc
-    _ = c * inner ℂ S v := by
-      dsimp only [S]
-      rw [sum_inner (𝕜 := ℂ) Finset.univ _ v, Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro y _
-      split_ifs <;> simp_all only [UnitaryFinite.normalizedCoefficient, PiLp.smul_apply,
-        smul_eq_mul, UnitaryFinite.coefficient, F, orbitVector, inner_zero_left (𝕜 := ℂ), mul_zero, c, ρ]
-    _ = 0 := by rw [hz, inner_zero_left (𝕜 := ℂ) v, mul_zero]
-
-end Thorp.SparseContact
-
-
-end ThorpNine.Harmonic
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Harmonic.PairAssembleEquiv
+
+namespace OAI
+
+namespace ThorpNine.Harmonic
+
+namespace Thorp.SparseContact
+open scoped BigOperators Classical
+
+lemma noIsolated_weight_le {V : Type*} [Fintype V] (G : SimpleGraph V)
+    (p : V → ℝ) (hp : ∀ v, 0 ≤ p v) (k : ℕ) (δ : ℝ)
+    (hδ : 0 < δ) (hδ1 : δ ≤ 1)
+    (hdeg : ∀ v, (∑ w, if G.Adj v w then p w else 0) ≤ δ) :
+    (∑ S : {S : Finset V // S.card = k ∧ NoIsolated G S}, ∏ v ∈ S.val, p v) ≤
+      δ^((k:ℝ)/2) * Real.exp (Real.exp 1 * ∑ v, p v) := by
+  classical
+  let I := {S : Finset V // S.card = k ∧ NoIsolated G S}
+  let root (S : I) := (small_dominating_subset G S.val S.property.2).choose
+  have hroot (S : I) : root S ⊆ S.val ∧ 2*(root S).card ≤ k ∧
+      S.val \ root S ⊆ neighborhood G (root S) := by
+    simpa only [root, S.property.1] using
+      (small_dominating_subset G S.val S.property.2).choose_spec
+  let enc (S : I) : Finset V × Finset V := (root S, S.val \ root S)
+  have hrec (S : I) : (enc S).1 ∪ (enc S).2 = S.val :=
+    Finset.union_sdiff_of_subset (hroot S).1
+  have hinj : Function.Injective enc := by
+    intro S T h
+    apply Subtype.ext
+    rw [← hrec S, ← hrec T, h]
+  let W (z : Finset V × Finset V) : ℝ :=
+    if 2*z.1.card ≤ k ∧ z.2 ∈ (neighborhood G z.1).powersetCard (k-z.1.card)
+    then (∏ v ∈ z.1, p v) * ∏ v ∈ z.2, p v else 0
+  have hW (z : Finset V × Finset V) : 0 ≤ W z := by
+    dsimp [W]
+    split_ifs
+    · exact mul_nonneg (Finset.prod_nonneg (fun v _ => hp v))
+        (Finset.prod_nonneg (fun v _ => hp v))
+    · exact le_rfl
+  have hwenc (S : I) : W (enc S) = ∏ v ∈ S.val, p v := by
+    have hmem : (enc S).2 ∈ (neighborhood G (enc S).1).powersetCard (k-(enc S).1.card) := by
+      refine Finset.mem_powersetCard.mpr ⟨(hroot S).2.2, ?_⟩
+      change (S.val \ root S).card = k-(root S).card
+      rw [Finset.card_sdiff_of_subset (hroot S).1, S.property.1]
+    dsimp only [W]
+    rw [ite_eq_left ⟨(hroot S).2.1,hmem⟩]
+    exact (mul_comm _ _).trans (Finset.prod_sdiff (hroot S).1)
+  have hsum : (∑ S : I, ∏ v ∈ S.val, p v) ≤ ∑ z, W z := by
+    calc
+      _ = ∑ z ∈ Finset.univ.image enc, W z := by
+        rw [Finset.sum_image (fun S _ T _ h => hinj h)]
+        exact Finset.sum_congr rfl (fun S _ => (hwenc S).symm)
+      _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _)
+        (fun z _ _ => hW z)
+  apply hsum.trans
+  rw [Fintype.sum_prod_type]
+  calc
+    (∑ R : Finset V, ∑ L : Finset V, W (R,L)) ≤
+        ∑ R : Finset V, δ^((k:ℝ)/2) * Real.exp R.card * ∏ v ∈ R, p v := by
+      apply Finset.sum_le_sum
+      intro R _
+      by_cases hR : 2*R.card ≤ k
+      · have he : (∑ L : Finset V, W (R,L)) =
+            (∏ v ∈ R, p v) *
+              ∑ L ∈ (neighborhood G R).powersetCard (k-R.card), ∏ v ∈ L, p v := by
+          simp only [W, hR, true_and]
+          rw [← Finset.sum_filter]
+          simp only [Finset.filter_mem_eq_inter, Finset.univ_inter, Finset.mul_sum]
+        rw [he]
+        have h := mul_le_mul_of_nonneg_left (root_leaves_weight_le G p hp k δ hδ hδ1 hdeg R hR)
+          (Finset.prod_nonneg (s := R) (fun v _ => hp v))
+        simpa only [mul_comm, mul_left_comm, mul_assoc] using h
+      · simp only [W, hR, false_and, ite_false, Finset.sum_const_zero]
+        exact mul_nonneg (mul_nonneg (Real.rpow_nonneg hδ.le _) (Real.exp_pos _).le)
+          (Finset.prod_nonneg (fun v _ => hp v))
+    _ = δ^((k:ℝ)/2) * ∑ R : Finset V, ∏ v ∈ R, (Real.exp 1 * p v) := by
+      rw [Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro R _
+      rw [Finset.prod_mul_distrib, Finset.prod_const, ← Real.exp_nat_mul]
+      simp only [mul_one, mul_assoc]
+    _ ≤ δ^((k:ℝ)/2) * Real.exp (∑ v, Real.exp 1 * p v) := by
+      apply mul_le_mul_of_nonneg_left _ (Real.rpow_nonneg hδ.le _)
+      simpa only [Finset.powerset_univ] using
+        powerset_weight_le_exp Finset.univ (fun v => Real.exp 1 * p v)
+          (fun v => mul_nonneg (Real.exp_pos _).le (hp v))
+    _ = _ := by rw [← Finset.mul_sum]
+
+
+lemma noIsolated_probability_le {V Ω : Type*} [Fintype V] [Fintype Ω]
+    (G : SimpleGraph V) (O : Ω → Finset V) (k : ℕ)
+    (hcard : ∀ ω, (O ω).card = k) (p : V → ℝ) (hp : ∀ v, 0 ≤ p v)
+    (hcyl : ∀ D, inclusionMoment O D ≤ ∏ v ∈ D, p v)
+    (δ : ℝ) (hδ : 0 < δ) (hδ1 : δ ≤ 1)
+    (hdeg : ∀ v, (∑ w, if G.Adj v w then p w else 0) ≤ δ) :
+    finiteMean (fun ω => if NoIsolated G (O ω) then (1:ℝ) else 0) ≤
+      δ^((k:ℝ)/2) * Real.exp (Real.exp 1 * ∑ v, p v) := by
+  classical
+  let I := {S : Finset V // S.card = k ∧ NoIsolated G S}
+  calc
+    _ ≤ finiteMean (fun ω => ∑ S : I, if S.val ⊆ O ω then (1:ℝ) else 0) := by
+      apply finiteMean_mono
+      intro ω
+      split_ifs with h
+      · exact (Finset.single_le_sum (f := fun S : I => if S.val ⊆ O ω then (1:ℝ) else 0)
+          (fun S _ => by positivity) (Finset.mem_univ (⟨O ω,hcard ω,h⟩ : I))) |>.trans' (by simp)
+      · positivity
+    _ = ∑ S : I, inclusionMoment O S.val := by
+      rw [finiteMean_sum]
+      rfl
+    _ ≤ ∑ S : I, ∏ v ∈ S.val, p v := Finset.sum_le_sum (fun S _ => hcyl S.val)
+    _ ≤ _ := noIsolated_weight_le G p hp k δ hδ hδ1 hdeg
+
+noncomputable def typedSet {N X : Type*} [Fintype N] [Fintype X]
+    (A : N → Finset X) : Finset (N × X) := by
+  classical
+  exact Finset.univ.filter (fun v => v.2 ∈ A v.1)
+
+@[simp] lemma mem_typedSet {N X : Type*} [Fintype N] [Fintype X]
+    (A : N → Finset X) (v : N × X) : v ∈ typedSet A ↔ v.2 ∈ A v.1 := by
+  classical
+  simp [typedSet]
+
+noncomputable def fiber {N X : Type*} [Fintype X]
+    (S : Finset (N × X)) (t : N) : Finset X := by
+  classical
+  exact Finset.univ.filter (fun x => (t,x) ∈ S)
+
+@[simp] lemma mem_fiber {N X : Type*} [Fintype X]
+    (S : Finset (N × X)) (t : N) (x : X) : x ∈ fiber S t ↔ (t,x) ∈ S := by
+  classical
+  simp [fiber]
+
+lemma typedSet_fiber {N X : Type*} [Fintype N] [Fintype X]
+    (S : Finset (N × X)) : typedSet (fiber S) = S := by
+  classical
+  ext v
+  simp
+
+lemma typedSet_prod {N X : Type*} [Fintype N] [Fintype X]
+    (A : N → Finset X) (p : N × X → ℝ) :
+    (∏ v ∈ typedSet A, p v) = ∏ t, ∏ x ∈ A t, p (t,x) := by
+  classical
+  simp only [typedSet, Finset.prod_filter, Fintype.prod_prod_type]
+  apply Finset.prod_congr rfl
+  intro t _
+  rw [← Finset.prod_filter]
+  simp only [Finset.filter_mem_eq_inter, Finset.univ_inter]
+
+lemma typedSet_card {N X : Type*} [Fintype N] [Fintype X]
+    (A : N → Finset X) : (typedSet A).card = ∑ t, (A t).card := by
+  classical
+  simp only [Finset.card_eq_sum_ones, typedSet, Finset.sum_filter, Fintype.sum_prod_type]
+  apply Finset.sum_congr rfl
+  intro t _
+  rw [← Finset.sum_filter]
+  simp only [Finset.filter_mem_eq_inter, Finset.univ_inter]
+
+lemma subset_typedSet {N X : Type*} [Fintype N] [Fintype X]
+    (A : N → Finset X) (D : Finset (N × X)) :
+    D ⊆ typedSet A ↔ ∀ t, fiber D t ⊆ A t := by
+  constructor
+  · intro h t x hx
+    exact mem_typedSet A (t,x) |>.mp (h (mem_fiber D t x |>.mp hx))
+  · intro h v hv
+    exact mem_typedSet A v |>.mpr (h v.1 (mem_fiber D v.1 v.2 |>.mpr hv))
+
+noncomputable def typedMarks {N : Type*} [Fintype N] (d : ℕ)
+    (A : N → Finset (Card d)) (ω : N → (SwitchIndex d → Bool)) : Finset (N × Card d) :=
+  typedSet (fun t => butterflyMarks d (A t) (ω t))
+
+lemma typedMarks_card {N : Type*} [Fintype N] (d : ℕ)
+    (A : N → Finset (Card d)) (ω : N → (SwitchIndex d → Bool)) :
+    (typedMarks d A ω).card = ∑ t, (A t).card := by
+  classical
+  rw [typedMarks, typedSet_card]
+  apply Finset.sum_congr rfl
+  intro t _
+  exact Finset.card_image_of_injective _ (butterflyPerm d (decodeButterfly d (ω t))).injective
+
+lemma typedMarks_inclusion {N : Type*} [Fintype N] (d : ℕ)
+    (A : N → Finset (Card d)) (D : Finset (N × Card d)) :
+    inclusionMoment (typedMarks d A) D ≤
+      ∏ v ∈ D, ((A v.1).card:ℝ)/(2:ℝ)^d := by
+  classical
+  have he : inclusionMoment (typedMarks d A) D =
+      ∏ t, inclusionMoment (butterflyMarks d (A t)) (fiber D t) := by
+    rw [inclusionMoment_eq]
+    have hh (ω : N → (SwitchIndex d → Bool)) :
+        (if D ⊆ typedMarks d A ω then (1:ℝ) else 0) =
+        ∏ t, if fiber D t ⊆ butterflyMarks d (A t) (ω t) then (1:ℝ) else 0 := by
+      rw [Fintype.prod_boole]
+      simp only [typedMarks, subset_typedSet]
+    rw [finiteMean_congr hh]
+    simp only [inclusionMoment_eq]
+    exact finiteMean_pi_prod (ι := N) (Ω := fun _ => SwitchIndex d → Bool)
+      (fun t (ω : SwitchIndex d → Bool) =>
+        if fiber D t ⊆ butterflyMarks d (A t) ω then (1:ℝ) else 0)
+  have hprod : (∏ v ∈ D, ((A v.1).card:ℝ)/(2:ℝ)^d) =
+      ∏ t, ∏ x ∈ fiber D t, ((A t).card:ℝ)/(2:ℝ)^d := by
+    conv_lhs => rw [← typedSet_fiber D]
+    exact typedSet_prod _ _
+  rw [he, hprod]
+  apply Finset.prod_le_prod₀
+  · intro t _; exact inclusionMoment_nonneg _ _
+  · intro t _
+    simpa only [Finset.prod_const] using butterflyMarks_inclusion d (A t) (fiber D t)
+
+noncomputable def returnRoute {N : Type*} (d : ℕ) (b : N → Butterfly d)
+    (v : N × Card d) : Finset (SwitchIndex d) :=
+  routeDomain d ((butterflyPerm d (b v.1)).symm v.2) v.2
+
+noncomputable def returnGraph {N : Type*} (d : ℕ) (b : N → Butterfly d) :
+    SimpleGraph (N × Card d) where
+  Adj v w := v ≠ w ∧ ¬Disjoint (returnRoute d b v) (returnRoute d b w)
+  symm := ⟨fun _ _ h => ⟨h.1.symm, fun hh => h.2 hh.symm⟩⟩
+  loopless := ⟨fun _ h => h.1 rfl⟩
+
+lemma returnRoute_card {N : Type*} (d : ℕ) (b : N → Butterfly d) (v : N × Card d) :
+    (returnRoute d b v).card = d := card_routeDomain _ _ _
+
+lemma mem_returnRoute {N : Type*} (d : ℕ) (b : N → Butterfly d)
+    (t : N) (y : Card d) (i : SwitchIndex d) :
+    i ∈ returnRoute d b (t,y) ↔
+      y = butterflyPerm d (b t) (switchUsers d (b t) i).1 ∨
+      y = butterflyPerm d (b t) (switchUsers d (b t) i).2 := by
+  have h := mem_routeDomain_iff_user d (b t) ((butterflyPerm d (b t)).symm y) i
+  simpa only [Equiv.apply_symm_apply, returnRoute, Equiv.eq_symm_apply, eq_comm] using h
+
+lemma returnRoute_incidence {N : Type*} [Fintype N] (d : ℕ) (b : N → Butterfly d)
+    (p : N → ℝ) (i : SwitchIndex d) :
+    (∑ v : N × Card d, if i ∈ returnRoute d b v then p v.1 else 0) =
+      2 * ∑ t, p t := by
+  classical
+  rw [Fintype.sum_prod_type, Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro t _
+  have hne : butterflyPerm d (b t) (switchUsers d (b t) i).1 ≠
+      butterflyPerm d (b t) (switchUsers d (b t) i).2 :=
+    (butterflyPerm d (b t)).injective.ne (switchUsers_ne d (b t) i)
+  simp only [mem_returnRoute]
+  have he (y : Card d) :
+      (if y = butterflyPerm d (b t) (switchUsers d (b t) i).1 ∨
+        y = butterflyPerm d (b t) (switchUsers d (b t) i).2 then p t else 0) =
+      (if y = butterflyPerm d (b t) (switchUsers d (b t) i).1 then p t else 0) +
+      (if y = butterflyPerm d (b t) (switchUsers d (b t) i).2 then p t else 0) := by
+    split_ifs <;> simp_all
+  simp only [he, Finset.sum_add_distrib, Finset.sum_ite_eq', Finset.mem_univ, ite_true]
+  ring
+
+lemma returnGraph_degree {N : Type*} [Fintype N] (d : ℕ) (b : N → Butterfly d)
+    (p : N → ℝ) (hp : ∀ t, 0 ≤ p t) (v : N × Card d) :
+    (∑ w, if (returnGraph d b).Adj v w then p w.1 else 0) ≤
+      2*d*∑ t, p t := by
+  classical
+  calc
+    _ ≤ ∑ w, ∑ i ∈ returnRoute d b v, if i ∈ returnRoute d b w then p w.1 else 0 := by
+      apply Finset.sum_le_sum
+      intro w _
+      split_ifs with h
+      · obtain ⟨i, hi, hw⟩ := Finset.not_disjoint_iff.mp h.2
+        have hh := Finset.single_le_sum
+          (f := fun i => if i ∈ returnRoute d b w then p w.1 else 0)
+          (fun i _ => by split_ifs; exact hp _; exact le_rfl) hi
+        simpa only [ite_eq_left hw] using hh
+      · exact Finset.sum_nonneg (fun i _ => by split_ifs; exact hp _; exact le_rfl)
+    _ = ∑ i ∈ returnRoute d b v, ∑ w, if i ∈ returnRoute d b w then p w.1 else 0 :=
+      Finset.sum_comm
+    _ = _ := by simp only [returnRoute_incidence, Finset.sum_const, returnRoute_card,
+        nsmul_eq_mul]; ring
+
+lemma typed_weight_sum {N : Type*} [Fintype N] (d : ℕ)
+    (A : N → Finset (Card d)) :
+    (∑ v : N × Card d, ((A v.1).card:ℝ)/(2:ℝ)^d) = ∑ t, ((A t).card:ℝ) := by
+  rw [Fintype.sum_prod_type]
+  apply Finset.sum_congr rfl
+  intro t _
+  simp only [Finset.sum_const, Finset.card_univ, card_positions, Nat.cast_pow,
+    Nat.cast_ofNat, nsmul_eq_mul]
+  field_simp
+
+lemma mixed_return_noIsolated {N : Type*} [Fintype N] (d : ℕ)
+    (A : N → Finset (Card d)) (b : N → Butterfly d) (k : ℕ)
+    (hk : ∑ t, (A t).card = k)
+    (hδ : 0 < 2*(d:ℝ)*k/(2:ℝ)^d) (hδ1 : 2*(d:ℝ)*k/(2:ℝ)^d ≤ 1) :
+    finiteMean (fun ω : N → (SwitchIndex d → Bool) =>
+      if NoIsolated (returnGraph d b) (typedMarks d A ω) then (1:ℝ) else 0) ≤
+      (2*(d:ℝ)*k/(2:ℝ)^d)^((k:ℝ)/2) * Real.exp (Real.exp 1*k) := by
+  have hs : (∑ t, ((A t).card:ℝ)) = k := by exact_mod_cast hk
+  have hm := noIsolated_probability_le (returnGraph d b) (typedMarks d A) k
+    (fun ω => (typedMarks_card d A ω).trans hk)
+    (fun v => ((A v.1).card:ℝ)/(2:ℝ)^d) (fun v => by positivity)
+    (typedMarks_inclusion d A) _ hδ hδ1
+    (fun v => (returnGraph_degree d b (fun t => ((A t).card:ℝ)/(2:ℝ)^d)
+      (fun t => by positivity) v).trans_eq (by rw [← Finset.sum_div, hs]; ring))
+  simpa only [typed_weight_sum, hs] using hm
+
+
+lemma mean_fresh_cylinder {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (s : Finset ι) (v : ι → Bool) (f : (ι → Bool) → ℝ)
+    (hf : ∀ i ∈ s, ∀ ω, f (QueryTree.flipCoin i ω) = f ω) :
+    finiteMean (fun ω => if ∀ i ∈ s, ω i = v i then f ω else 0) =
+      finiteMean f / (2:ℝ)^s.card := by
+  induction s using Finset.induction_on with
+  | empty => simp [finiteMean]
+  | @insert i s hi ih =>
+    let g := fun ω => if ∀ j ∈ s, ω j = v j then f ω else 0
+    have hg : ∀ ω, g (QueryTree.flipCoin i ω) = g ω := by
+      intro ω
+      have he : (∀ j ∈ s, QueryTree.flipCoin i ω j = v j) ↔ ∀ j ∈ s, ω j = v j := by
+        apply forall₂_congr
+        intro j hj
+        rw [QueryTree.flipCoin_apply_ne i j (fun h => hi (h ▸ hj))]
+      simp only [g, he, hf i (Finset.mem_insert_self _ _) ω]
+    have he (ω : ι → Bool) :
+        (if ∀ j ∈ insert i s, ω j = v j then f ω else 0) =
+        (if ω i = v i then g ω else 0) := by
+      by_cases h : ω i = v i <;> simp [h,g]
+    rw [finiteMean_congr he, QueryTree.mean_indicator_bit g i (v i) hg]
+    have hs := ih (fun j hj => hf j (Finset.mem_insert_of_mem hj))
+    change finiteMean (fun ω => if ∀ j ∈ s, ω j = v j then f ω else 0) / 2 = _
+    rw [hs, Finset.card_insert_of_notMem hi, pow_succ]
+    ring
+
+noncomputable def endpointProb {L : Type*} (d : ℕ) (A : Finset L)
+    (x y : L → Card d) : ℝ := by
+  classical
+  exact finiteMean (fun ω : SwitchIndex d → Bool =>
+    if ∀ i ∈ A, butterflyPerm d (decodeButterfly d ω) (x i) = y i then (1:ℝ) else 0)
+
+lemma endpointProb_nonneg {L : Type*} (d : ℕ) (A : Finset L)
+    (x y : L → Card d) : 0 ≤ endpointProb d A x y := by
+  classical
+  exact finiteMean_nonneg (fun _ => by positivity)
+
+lemma endpointProb_insert_isolated {L : Type*} (d : ℕ) (A : Finset L)
+    (x y : L → Card d) (i : L)
+    (hiso : ∀ j ∈ A, Disjoint (routeDomain d (x i) (y i)) (routeDomain d (x j) (y j))) :
+    endpointProb d (insert i A) x y = endpointProb d A x y / (2:ℝ)^d := by
+  classical
+  let f := fun ω : SwitchIndex d → Bool =>
+    if ∀ j ∈ A, butterflyPerm d (decodeButterfly d ω) (x j) = y j then (1:ℝ) else 0
+  have hf : ∀ t ∈ routeDomain d (x i) (y i), ∀ ω, f (QueryTree.flipCoin t ω) = f ω := by
+    intro t ht ω
+    have he : (∀ j ∈ A, butterflyPerm d (decodeButterfly d (QueryTree.flipCoin t ω)) (x j) = y j) ↔
+        ∀ j ∈ A, butterflyPerm d (decodeButterfly d ω) (x j) = y j := by
+      apply forall₂_congr
+      intro j hj
+      rw [butterfly_endpoint_iff, butterfly_endpoint_iff]
+      apply forall₂_congr
+      intro s hs
+      rw [QueryTree.flipCoin_apply_ne t s (fun h =>
+        Finset.disjoint_left.mp (hiso j hj) ht (h ▸ hs))]
+    simp only [f, he]
+  have he (ω : SwitchIndex d → Bool) :
+      (if ∀ j ∈ insert i A, butterflyPerm d (decodeButterfly d ω) (x j) = y j then (1:ℝ) else 0) =
+      (if ∀ t ∈ routeDomain d (x i) (y i), ω t = routeValue d (x i) (y i) t then f ω else 0) := by
+    by_cases h : butterflyPerm d (decodeButterfly d ω) (x i) = y i
+    · have hh := (butterfly_endpoint_iff d ω (x i) (y i)).mp h
+      rw [ite_eq_left hh]
+      simp only [Finset.forall_mem_insert, h, true_and, f]
+    · have hh := mt (butterfly_endpoint_iff d ω (x i) (y i)).mpr h
+      simp only [Finset.forall_mem_insert, h, false_and, hh, ite_false]
+  change finiteMean _ = _
+  rw [finiteMean_congr he, mean_fresh_cylinder _ _ f hf, card_routeDomain]
+  rfl
+
+noncomputable def partialKernel {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x y : L → Card d) : ℝ :=
+  ((2:ℝ)^d)^A.card / ((2:ℝ)^d)^(Fintype.card L) * endpointProb d A x y
+
+lemma partialKernel_insert_isolated {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x y : L → Card d) (i : L) (hi : i ∉ A)
+    (hiso : ∀ j ∈ A, Disjoint (routeDomain d (x i) (y i)) (routeDomain d (x j) (y j))) :
+    partialKernel d (insert i A) x y = partialKernel d A x y := by
+  classical
+  rw [partialKernel, endpointProb_insert_isolated d A x y i hiso,
+    Finset.card_insert_of_notMem hi, pow_succ, partialKernel]
+  field_simp
+
+def ContactEvent {L : Type*} (d : ℕ) (x y : L → Card d) : Prop :=
+  ∀ i, ∃ j, i ≠ j ∧ ¬Disjoint (routeDomain d (x i) (y i)) (routeDomain d (x j) (y j))
+
+lemma alternating_cube_cancel {L : Type*} [Fintype L] (f : Finset L → ℝ)
+    (i : L) (hf : ∀ A, i ∉ A → f (insert i A) = f A) :
+    (∑ A : Finset L, (-1:ℝ)^A.card * f A) = 0 := by
+  classical
+  have hi : i ∈ (Finset.univ : Finset L) := Finset.mem_univ _
+  rw [← Finset.powerset_univ, ← Finset.insert_erase hi, Finset.sum_powerset_insert]
+  · rw [← Finset.sum_add_distrib]
+    apply Finset.sum_eq_zero
+    intro A hA
+    have hiA : i ∉ A := fun hh => Finset.notMem_erase i Finset.univ
+      ((Finset.mem_powerset.mp hA) hh)
+    rw [hf A hiA, Finset.card_insert_of_notMem hiA, pow_succ]
+    ring
+  · exact Finset.notMem_erase _ _
+
+lemma alternating_kernel_cancel {L : Type*} [Fintype L] (d : ℕ)
+    (x y : L → Card d) (h : ¬ContactEvent d x y) :
+    (∑ A : Finset L, (-1:ℝ)^A.card * partialKernel d A x y) = 0 := by
+  classical
+  unfold ContactEvent at h
+  push Not at h
+  obtain ⟨i, hi⟩ := h
+  apply alternating_cube_cancel _ i
+  intro A hiA
+  apply partialKernel_insert_isolated d A x y i hiA
+  intro j hj
+  exact hi j (fun hij => hiA (hij ▸ hj))
+
+
+noncomputable def networkOf {L : Type*} (A : Finset L) (i : L) : Option L := by
+  classical
+  exact if i ∈ A then none else some i
+
+noncomputable def mixedTuple {L : Type*} (d : ℕ) (A : Finset L)
+    (ω : Option L → (SwitchIndex d → Bool)) (x : L → Card d) : L → Card d :=
+  fun i => butterflyPerm d (decodeButterfly d (ω (networkOf A i))) (x i)
+
+noncomputable def mixedReturn {L : Type*} (d : ℕ) (A : Finset L)
+    (ω : Option L → (SwitchIndex d → Bool)) (y : L → Card d) : L → Card d :=
+  fun i => (butterflyPerm d (decodeButterfly d (ω (networkOf A i)))).symm (y i)
+
+noncomputable def labelsOf {L : Type*} (A : Finset L) : Option L → Finset L := by
+  classical
+  exact fun t => match t with
+    | none => A
+    | some i => if i ∈ A then ∅ else {i}
+
+lemma mem_labelsOf {L : Type*} (A : Finset L) (i : L) (t : Option L) :
+    i ∈ labelsOf A t ↔ networkOf A i = t := by
+  classical
+  cases t with
+  | none => simp [labelsOf, networkOf]
+  | some j => by_cases hj : j ∈ A <;> by_cases hi : i ∈ A <;> simp_all [labelsOf, networkOf] <;> intro h <;> subst j <;> contradiction
+
+lemma mixedTuple_eq_iff {L : Type*} (d : ℕ) (A : Finset L)
+    (ω : Option L → (SwitchIndex d → Bool)) (x y : L → Card d) :
+    mixedTuple d A ω x = y ↔ ∀ t, ∀ i ∈ labelsOf A t,
+      butterflyPerm d (decodeButterfly d (ω t)) (x i) = y i := by
+  constructor
+  · intro h t i hi
+    have he := congrFun h i
+    simpa only [mixedTuple, mem_labelsOf A i t |>.mp hi] using he
+  · intro h
+    funext i
+    exact h (networkOf A i) i (mem_labelsOf A i _ |>.mpr rfl)
+
+lemma mixedReturn_eq_iff {L : Type*} (d : ℕ) (A : Finset L)
+    (ω : Option L → (SwitchIndex d → Bool)) (y z : L → Card d) :
+    mixedReturn d A ω y = z ↔ mixedTuple d A ω z = y := by
+  constructor
+  · intro h
+    funext i
+    exact (Equiv.symm_apply_eq _).mp (congrFun h i) |>.symm
+  · intro h
+    funext i
+    exact (Equiv.symm_apply_eq _).mpr (congrFun h i).symm
+
+lemma endpointProb_singleton {L : Type*} (d : ℕ) (i : L) (x y : L → Card d) :
+    endpointProb d {i} x y = 1/(2:ℝ)^d := by
+  classical
+  simpa [endpointProb] using butterfly_single_card_uniform d (x i) (y i)
+
+lemma mixedTuple_mass {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x y : L → Card d) :
+    finiteMean (fun ω : Option L → (SwitchIndex d → Bool) =>
+      if mixedTuple d A ω x = y then (1:ℝ) else 0) = partialKernel d A x y := by
+  classical
+  have he (ω : Option L → (SwitchIndex d → Bool)) :
+      (if mixedTuple d A ω x = y then (1:ℝ) else 0) =
+      ∏ t, if ∀ i ∈ labelsOf A t, butterflyPerm d (decodeButterfly d (ω t)) (x i) = y i
+        then (1:ℝ) else 0 := by
+    simp only [Fintype.prod_boole, mixedTuple_eq_iff]
+  rw [finiteMean_congr he]
+  rw [finiteMean_pi_prod (fun t (w : SwitchIndex d → Bool) =>
+    if ∀ i ∈ labelsOf A t, butterflyPerm d (decodeButterfly d w) (x i) = y i then (1:ℝ) else 0)]
+  have hfac (t : Option L) : (finiteMean (fun w : SwitchIndex d → Bool =>
+      if ∀ i ∈ labelsOf A t, butterflyPerm d (decodeButterfly d w) (x i) = y i then (1:ℝ) else 0)) =
+      endpointProb d (labelsOf A t) x y := by
+        unfold endpointProb
+        apply finiteMean_congr
+        intro w
+        split_ifs <;> rfl
+  simp only [hfac]
+  rw [Fintype.prod_option]
+  have hi (i : L) : endpointProb d (labelsOf A (some i)) x y =
+      if i ∈ A then 1 else 1/(2:ℝ)^d := by
+    by_cases h : i ∈ A
+    · simp [labelsOf, h, endpointProb, finiteMean_const]
+    · simp only [labelsOf, ite_eq_right h, endpointProb_singleton]
+  rw [show labelsOf A none = A from rfl]
+  simp only [hi]
+  have hp : (∏ i : L, if i ∈ A then (1:ℝ) else 1/(2:ℝ)^d) =
+      (1/(2:ℝ)^d)^((Fintype.card L) - A.card) := by
+    rw [Finset.prod_ite]
+    simp only [Finset.prod_const_one, one_mul]
+    simp only [Finset.filter_not, Finset.filter_mem_eq_inter, Finset.univ_inter,
+      Finset.prod_const, Finset.card_sdiff, Finset.inter_univ, Finset.card_univ]
+  rw [hp, partialKernel, one_div, inv_pow]
+  have hc : A.card ≤ Fintype.card L := Finset.card_le_univ _
+  have hn : (((2:ℝ)^d)^A.card) * (((2:ℝ)^d)^(Fintype.card L - A.card)) =
+      ((2:ℝ)^d)^(Fintype.card L) := by rw [← pow_add, Nat.add_sub_of_le hc]
+  have hnz : ((2:ℝ)^d)^(Fintype.card L) ≠ 0 := by positivity
+  have hmz : ((2:ℝ)^d)^(Fintype.card L - A.card) ≠ 0 := by positivity
+  field_simp
+  rw [← hn]
+  ring
+
+lemma mixedReturn_mass {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (y z : L → Card d) :
+    finiteMean (fun ω : Option L → (SwitchIndex d → Bool) =>
+      if mixedReturn d A ω y = z then (1:ℝ) else 0) = partialKernel d A z y := by
+  simpa only [mixedReturn_eq_iff] using mixedTuple_mass d A z y
+
+noncomputable def mixedInitial {L : Type*} (d : ℕ) (A : Finset L)
+    (x : L → Card d) (t : Option L) : Finset (Card d) := by
+  classical
+  exact (labelsOf A t).image x
+
+lemma typedMarks_mixedInitial {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x : L → Card d) (ω : Option L → (SwitchIndex d → Bool)) :
+    typedMarks d (mixedInitial d A x) ω = Finset.univ.image
+      (fun i => (networkOf A i, mixedTuple d A ω x i)) := by
+  classical
+  ext v
+  simp only [typedMarks, mem_typedSet, butterflyMarks, mixedInitial,
+    Finset.mem_image, Finset.mem_univ, true_and]
+  constructor
+  · rintro ⟨z, ⟨i,hi,rfl⟩, hv⟩
+    have ht := mem_labelsOf A i v.1 |>.mp hi
+    exact ⟨i, Prod.ext ht (by simpa only [mixedTuple, ht] using hv)⟩
+  · rintro ⟨i, hi⟩
+    have ht := congrArg Prod.fst hi
+    have hv := congrArg Prod.snd hi
+    change networkOf A i = v.1 at ht
+    change mixedTuple d A ω x i = v.2 at hv
+    exact ⟨x i, ⟨i, mem_labelsOf A i v.1 |>.mpr ht, rfl⟩,
+      by simpa only [mixedTuple, ht] using hv⟩
+
+lemma typedTuple_injective {L : Type*} (d : ℕ) (A : Finset L)
+    (x : L → Card d) (hx : Function.Injective x)
+    (ω : Option L → (SwitchIndex d → Bool)) :
+    Function.Injective (fun i => (networkOf A i, mixedTuple d A ω x i)) := by
+  intro i j h
+  have ht := congrArg Prod.fst h
+  have hv := congrArg Prod.snd h
+  change networkOf A i = networkOf A j at ht
+  dsimp only [mixedTuple] at hv
+  rw [ht] at hv
+  exact hx ((butterflyPerm d (decodeButterfly d (ω (networkOf A j)))).injective hv)
+
+lemma mixedInitial_card {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x : L → Card d) (hx : Function.Injective x) :
+    (∑ t, (mixedInitial d A x t).card) = Fintype.card L := by
+  classical
+  let ω : Option L → (SwitchIndex d → Bool) := fun _ _ => false
+  rw [← typedMarks_card d (mixedInitial d A x) ω, typedMarks_mixedInitial,
+    Finset.card_image_of_injective _ (typedTuple_injective d A x hx ω), Finset.card_univ]
+
+lemma contactEvent_mixedReturn_iff {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x : L → Card d) (hx : Function.Injective x)
+    (ω b : Option L → (SwitchIndex d → Bool)) :
+    ContactEvent d (mixedReturn d A b (mixedTuple d A ω x)) (mixedTuple d A ω x) ↔
+      NoIsolated (returnGraph d (fun t => decodeButterfly d (b t)))
+        (typedMarks d (mixedInitial d A x) ω) := by
+  classical
+  rw [typedMarks_mixedInitial]
+  let f := fun i => (networkOf A i, mixedTuple d A ω x i)
+  have hf : Function.Injective f := typedTuple_injective d A x hx ω
+  constructor
+  · intro h v hv
+    obtain ⟨i,_,rfl⟩ := Finset.mem_image.mp hv
+    obtain ⟨j,hij,hc⟩ := h i
+    exact ⟨f j, Finset.mem_image.mpr ⟨j, Finset.mem_univ _,rfl⟩,
+      hf.ne hij, hc⟩
+  · intro h i
+    obtain ⟨v,hv,hij,hc⟩ := h (f i) (Finset.mem_image.mpr ⟨i,Finset.mem_univ _,rfl⟩)
+    obtain ⟨j,_,rfl⟩ := Finset.mem_image.mp hv
+    exact ⟨j, fun he => hij (congrArg f he), hc⟩
+
+lemma probability_mixed_contact {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x : L → Card d) (hx : Function.Injective x)
+    (b : Option L → (SwitchIndex d → Bool))
+    (hδ : 0 < 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)
+    (hδ1 : 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d ≤ 1) :
+    finiteMean (fun ω : Option L → (SwitchIndex d → Bool) =>
+      if ContactEvent d (mixedReturn d A b (mixedTuple d A ω x)) (mixedTuple d A ω x)
+        then (1:ℝ) else 0) ≤
+      (2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
+        Real.exp (Real.exp 1*(Fintype.card L)) := by
+  classical
+  simp only [contactEvent_mixedReturn_iff d A x hx]
+  convert mixed_return_noIsolated d (mixedInitial d A x) (fun t => decodeButterfly d (b t))
+      (Fintype.card L) (mixedInitial_card d A x hx) hδ hδ1 using 1
+  congr 1
+  exact Subsingleton.elim _ _
+
+
+lemma finiteMean_pushforward {Ω X : Type*} [Fintype Ω] [Fintype X]
+    (g : Ω → X) (f : X → ℝ) :
+    finiteMean (fun ω => f (g ω)) =
+      ∑ y, finiteMean (fun ω => if g ω = y then (1:ℝ) else 0) * f y := by
+  classical
+  simp_rw [← finiteMean_mul_const, ← finiteMean_sum]
+  apply finiteMean_congr
+  intro ω
+  simp
+
+lemma mixedTuple_expectation {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x : L → Card d) (f : (L → Card d) → ℝ) :
+    finiteMean (fun ω : Option L → (SwitchIndex d → Bool) => f (mixedTuple d A ω x)) =
+      ∑ y, partialKernel d A x y * f y := by
+  classical
+  rw [finiteMean_pushforward (fun ω : Option L → (SwitchIndex d → Bool) => mixedTuple d A ω x) f]
+  apply Finset.sum_congr rfl
+  intro y _
+  congr 1
+  convert mixedTuple_mass d A x y using 1
+  congr 1
+  funext ω
+  split_ifs <;> rfl
+
+lemma mixedReturn_expectation {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (y : L → Card d) (f : (L → Card d) → ℝ) :
+    finiteMean (fun ω : Option L → (SwitchIndex d → Bool) => f (mixedReturn d A ω y)) =
+      ∑ z, partialKernel d A z y * f z := by
+  classical
+  rw [finiteMean_pushforward (fun ω : Option L → (SwitchIndex d → Bool) => mixedReturn d A ω y) f]
+  apply Finset.sum_congr rfl
+  intro z _
+  congr 1
+  convert mixedReturn_mass d A y z using 1
+  congr 1
+  funext ω
+  split_ifs <;> rfl
+
+lemma partialKernel_nonneg {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x y : L → Card d) : 0 ≤ partialKernel d A x y :=
+  mul_nonneg (by positivity) (endpointProb_nonneg d A x y)
+
+noncomputable def retainedKernel {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x : L ↪ Card d) (y : L → Card d) : ℝ := by
+  classical
+  exact if ContactEvent d x y then partialKernel d A x y else 0
+
+lemma retainedKernel_nonneg {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x : L ↪ Card d) (y : L → Card d) : 0 ≤ retainedKernel d A x y := by
+  classical
+  unfold retainedKernel
+  split_ifs
+  · exact partialKernel_nonneg d A x y
+  · exact le_rfl
+
+lemma retainedKernel_le {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x : L ↪ Card d) (y : L → Card d) : retainedKernel d A x y ≤ partialKernel d A x y := by
+  classical
+  unfold retainedKernel
+  split_ifs
+  · exact le_rfl
+  · exact partialKernel_nonneg d A x y
+
+lemma retained_square_row {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x : L ↪ Card d)
+    (hδ : 0 < 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)
+    (hδ1 : 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d ≤ 1) :
+    (∑ x' : L ↪ Card d, ∑ y : L → Card d, retainedKernel d A x y * retainedKernel d A x' y) ≤
+      (2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
+        Real.exp (Real.exp 1*(Fintype.card L)) := by
+  classical
+  let f := fun y z : L → Card d => if ContactEvent d z y then (1:ℝ) else 0
+  have hsub (y : L → Card d) : (∑ x' : L ↪ Card d, retainedKernel d A x' y) ≤
+      ∑ z : L → Card d, partialKernel d A z y * f y z := by
+    have he (x' : L ↪ Card d) : retainedKernel d A x' y = partialKernel d A x' y * f y x' := by
+      unfold retainedKernel f
+      split_ifs <;> ring
+    simp_rw [he]
+    rw [← Finset.sum_image (f := fun z => partialKernel d A z y * f y z)
+      (Function.Embedding.coe_injective.injOn)]
+    apply Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _)
+    intro z _ _
+    exact mul_nonneg (partialKernel_nonneg d A z y) (by dsimp [f]; positivity)
+  calc
+    _ = ∑ y, retainedKernel d A x y * ∑ x' : L ↪ Card d, retainedKernel d A x' y := by
+      rw [Finset.sum_comm]
+      simp only [Finset.mul_sum]
+    _ ≤ ∑ y, partialKernel d A x y * ∑ z : L → Card d, partialKernel d A z y * f y z := by
+      apply Finset.sum_le_sum
+      intro y _
+      exact mul_le_mul (retainedKernel_le d A x y) (hsub y)
+        (Finset.sum_nonneg (fun x' _ => retainedKernel_nonneg d A x' y)) (partialKernel_nonneg d A x y)
+    _ = finiteMean (fun ω : Option L → (SwitchIndex d → Bool) =>
+        finiteMean (fun b : Option L → (SwitchIndex d → Bool) =>
+          f (mixedTuple d A ω x) (mixedReturn d A b (mixedTuple d A ω x)))) := by
+      rw [mixedTuple_expectation d A x (fun y =>
+        finiteMean (fun b : Option L → (SwitchIndex d → Bool) => f y (mixedReturn d A b y)))]
+      simp only [mixedReturn_expectation]
+    _ = finiteMean (fun b : Option L → (SwitchIndex d → Bool) =>
+        finiteMean (fun ω : Option L → (SwitchIndex d → Bool) =>
+          f (mixedTuple d A ω x) (mixedReturn d A b (mixedTuple d A ω x)))) := finiteMean_comm _
+    _ ≤ _ := by
+      apply (finiteMean_mono (fun b => probability_mixed_contact d A x x.injective b hδ hδ1)).trans_eq
+      exact finiteMean_const _
+
+noncomputable def rectOperator {X Y : Type*} [Fintype X] [Fintype Y]
+    (P : X → Y → ℝ) : EuclideanSpace ℂ Y →L[ℂ] EuclideanSpace ℂ X :=
+  LinearMap.toContinuousLinearMap
+    { toFun := fun v => WithLp.toLp 2 (fun x => ∑ y, (P x y : ℂ)*v y)
+      map_add' := by
+        intro v w
+        ext x
+        simp only [PiLp.add_apply, mul_add, Finset.sum_add_distrib]
+      map_smul' := by
+        intro c v
+        ext x
+        simp only [PiLp.smul_apply, smul_eq_mul, RingHom.id_apply]
+        change (∑ y, (P x y : ℂ)*(c*v y)) = c*∑ y, (P x y : ℂ)*v y
+        rw [Finset.mul_sum]
+        apply Finset.sum_congr rfl
+        intro y _
+        ring }
+
+@[simp] lemma rectOperator_apply {X Y : Type*} [Fintype X] [Fintype Y]
+    (P : X → Y → ℝ) (v : EuclideanSpace ℂ Y) (x : X) :
+    rectOperator P v x = ∑ y, (P x y:ℂ)*v y := rfl
+
+lemma rectOperator_adjoint {X Y : Type*} [Fintype X] [Fintype Y]
+    (P : X → Y → ℝ) : ContinuousLinearMap.adjoint (rectOperator P) =
+      rectOperator (fun y x => P x y) := by
+  apply Eq.symm
+  rw [ContinuousLinearMap.eq_adjoint_iff]
+  intro v w
+  simp only [PiLp.inner_apply, RCLike.inner_apply, rectOperator_apply,
+    map_sum, map_mul, Complex.conj_ofReal, Finset.sum_mul, Finset.mul_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro x _
+  apply Finset.sum_congr rfl
+  intro y _
+  ring
+
+lemma rectOperator_comp_transpose {X Y : Type*} [Fintype X] [Fintype Y]
+    (P : X → Y → ℝ) : (rectOperator P).comp (rectOperator (fun y x => P x y)) =
+      rectOperator (fun x x' => ∑ y, P x y * P x' y) := by
+  ext v x
+  simp only [ContinuousLinearMap.comp_apply, rectOperator_apply, Finset.mul_sum,
+    Complex.ofReal_sum, Complex.ofReal_mul, Finset.sum_mul]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro x' _
+  apply Finset.sum_congr rfl
+  intro y _
+  ring
+
+lemma rectOperator_norm_sq_le {X Y : Type*} [Fintype X] [Fintype Y]
+    (P : X → Y → ℝ) (hP : ∀ x y, 0 ≤ P x y) (B : ℝ) (hB : 0 ≤ B)
+    (hrow : ∀ x, (∑ x', ∑ y, P x y * P x' y) ≤ B) :
+    ‖rectOperator P‖^2 ≤ B := by
+  let S := fun x x' => ∑ y, P x y * P x' y
+  have hs : ∀ x x', S x x' = S x' x := by
+    intro x x'
+    unfold S
+    apply Finset.sum_congr rfl
+    intro y _
+    ring
+  have hbound : ‖rectOperator S‖ ≤ B := by
+    apply ContinuousLinearMap.opNorm_le_bound _ hB
+    intro v
+    exact DensityTransfer.schur_bound S
+      (fun x x' => Finset.sum_nonneg (fun y _ => mul_nonneg (hP x y) (hP x' y))) B hB
+      hrow (fun x' => by
+        calc (∑ x, S x x') = ∑ x, S x' x := Finset.sum_congr rfl (fun x _ => hs x x')
+             _ ≤ B := hrow x') v
+  have hn := ContinuousLinearMap.norm_adjoint_comp_self
+    (ContinuousLinearMap.adjoint (rectOperator P))
+  rw [ContinuousLinearMap.adjoint_adjoint, (ContinuousLinearMap.adjoint.norm_map _),
+    rectOperator_adjoint, rectOperator_comp_transpose] at hn
+  rw [pow_two, ← hn]
+  exact hbound
+
+lemma retained_norm_sq {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (hδ : 0 < 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)
+    (hδ1 : 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d ≤ 1) :
+    ‖rectOperator (retainedKernel d A)‖^2 ≤
+      (2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
+        Real.exp (Real.exp 1*(Fintype.card L)) :=
+  rectOperator_norm_sq_le _ (retainedKernel_nonneg d A) _ (by positivity)
+    (fun x => retained_square_row d A x hδ hδ1)
+
+
+section Orbit
+variable {G X V : Type*} [Group G] [Fintype G] [Fintype X]
+    [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+    (a : G →* Equiv.Perm X) (x₀ : X) (ha : ∀ x, ∃ g, a g x₀ = x)
+    (ρ : Representation ℂ G V) (w : V) (hw : UnitaryFinite.IsFixed a x₀ ρ w)
+
+noncomputable def orbitVector (x : X) : V := ρ (UnitaryFinite.representative a x₀ ha x) w
+
+include hw in
+omit [Fintype G] [Fintype X] [FiniteDimensional ℂ V] in
+lemma orbitVector_at (g : G) : orbitVector a x₀ ha ρ w (a g x₀) = ρ g w := by
+  apply ext_inner_right ℂ
+  intro v
+  exact UnitaryFinite.coefficient_at a x₀ ha ρ w hw v g
+
+include hw in
+omit [Fintype G] [Fintype X] [FiniteDimensional ℂ V] in
+lemma orbitVector_action (g : G) (x : X) :
+    orbitVector a x₀ ha ρ w (a g x) = ρ g (orbitVector a x₀ ha ρ w x) := by
+  obtain ⟨h,rfl⟩ := ha x
+  rw [←Equiv.Perm.mul_apply, ←map_mul, orbitVector_at a x₀ ha ρ w hw,
+    orbitVector_at a x₀ ha ρ w hw, map_mul, Module.End.mul_apply]
+end Orbit
+
+def IsHarmonic {L α : Type*} [Fintype L] [Fintype α] (f : EuclideanSpace ℂ (L ↪ α)) : Prop :=
+  ∀ (A : Finset L), A ≠ Finset.univ → ∀ z : L → α,
+    (∑ y : L ↪ α, if ∀ i ∈ A, y i = z i then f y else 0) = 0
+
+lemma normalizedCoefficient_harmonic {L α : Type*} [Fintype L] [Fintype α]
+    (μ : YoungDiagram) (e : α ≃ Specht.Cell μ)
+    (hk : Fintype.card L = μ.card - μ.rowLen 0) (x₀ : L ↪ α)
+    (w : Specht.hilbertSpace μ)
+    (hw : UnitaryFinite.IsFixed UnitaryFinite.tupleAction x₀ (Specht.relabelledUnitary μ e) w)
+    (v : Specht.hilbertSpace μ) :
+    IsHarmonic (UnitaryFinite.normalizedCoefficient UnitaryFinite.tupleAction x₀
+      (UnitaryFinite.tupleAction_transitive x₀) (Specht.relabelledUnitary μ e) w v) := by
+  classical
+  intro A hA z
+  let ρ := Specht.relabelledUnitary μ e
+  let F := orbitVector UnitaryFinite.tupleAction x₀ (UnitaryFinite.tupleAction_transitive x₀) ρ w
+  let S : Specht.hilbertSpace μ := ∑ y : L ↪ α, if ∀ i ∈ A, y i = z i then F y else 0
+  have hcard : (A.image z).card < μ.card - μ.rowLen 0 := by
+    apply (Finset.card_image_le).trans_lt
+    rw [← hk]
+    exact Finset.card_lt_card (Finset.ssubset_univ_iff.mpr hA)
+  have hz : S = 0 := by
+    apply point_relabelled_fixed_zero μ e (A.image z) hcard
+    intro p hp
+    change ρ p S = S
+    simp only [S, map_sum, apply_ite, map_zero]
+    apply Fintype.sum_equiv (UnitaryFinite.tupleAction p)
+    intro y
+    have he : (∀ i ∈ A, UnitaryFinite.tupleAction p y i = z i) ↔ ∀ i ∈ A, y i = z i := by
+      apply forall₂_congr
+      intro i hi
+      have hpi : p (z i) = z i := hp _ (Finset.mem_image.mpr ⟨i, hi,rfl⟩)
+      change p (y i) = z i ↔ y i = z i
+      exact (show (p (y i) = z i) ↔ p (y i) = p (z i) by rw [hpi]).trans p.injective.eq_iff
+    simp only [he]
+    split_ifs
+    · exact (orbitVector_action UnitaryFinite.tupleAction x₀
+        (UnitaryFinite.tupleAction_transitive x₀) ρ w hw p y).symm
+    · rfl
+  let c : ℂ := Real.sqrt ((Module.finrank ℂ (Specht.hilbertSpace μ):ℝ)/(Fintype.card (L ↪ α):ℝ))
+  calc
+    _ = c * inner ℂ S v := by
+      dsimp only [S]
+      rw [sum_inner (𝕜 := ℂ) Finset.univ _ v, Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro y _
+      split_ifs <;> simp_all only [UnitaryFinite.normalizedCoefficient, PiLp.smul_apply,
+        smul_eq_mul, UnitaryFinite.coefficient, F, orbitVector, inner_zero_left (𝕜 := ℂ), mul_zero, c, ρ]
+    _ = 0 := by rw [hz, inner_zero_left (𝕜 := ℂ) v, mul_zero]
+
+end Thorp.SparseContact
+
+
+end ThorpNine.Harmonic
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Harmonic/ZeroExtend.lean b/lean/OAI/Probability/ThorpRouting/Harmonic/ZeroExtend.lean
index 38b783ec87521a596f0de716906a9cfc6d9b3cef..b169d3050cb89541b97ce5a30198e61e8c91f15c 100644
--- a/lean/OAI/Probability/ThorpRouting/Harmonic/ZeroExtend.lean
+++ b/lean/OAI/Probability/ThorpRouting/Harmonic/ZeroExtend.lean
@@ -1,871 +1,871 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Harmonic.TypedSet
-
-namespace OAI
-
-namespace ThorpNine.Harmonic
-
-namespace Thorp.SparseContact
-
-section
-open scoped BigOperators Classical
-
-lemma sum_injections {L α M : Type*} [Fintype L] [Fintype α] [AddCommMonoid M]
-    (F : (L → α) → M) (hF : ∀ y, ¬Function.Injective y → F y = 0) :
-    (∑ y : L → α, F y) = ∑ y : L ↪ α, F y := by
-  classical
-  rw [← Finset.sum_image Function.Embedding.coe_injective.injOn]
-  apply Eq.symm
-  apply Finset.sum_subset (Finset.subset_univ _)
-  intro y _ hy
-  apply hF y
-  intro h
-  exact hy (Finset.mem_image.mpr ⟨⟨y,h⟩, Finset.mem_univ _, rfl⟩)
-
-noncomputable def zeroExtend {L α : Type*} [Fintype L] [Fintype α]
-    (f : EuclideanSpace ℂ (L ↪ α)) : EuclideanSpace ℂ (L → α) := by
-  classical
-  exact WithLp.toLp 2 (fun y => if h : Function.Injective y then f ⟨y,h⟩ else 0)
-
-@[simp] lemma zeroExtend_inj {L α : Type*} [Fintype L] [Fintype α]
-    (f : EuclideanSpace ℂ (L ↪ α)) (y : L ↪ α) : zeroExtend f y = f y := by
-  classical
-  simp only [zeroExtend, WithLp.ofLp_toLp, dite_eq_left y.injective]
-  congr 1
-
-lemma zeroExtend_noninj {L α : Type*} [Fintype L] [Fintype α]
-    (f : EuclideanSpace ℂ (L ↪ α)) (y : L → α) (hy : ¬Function.Injective y) : zeroExtend f y = 0 := by
-  classical
-  simp only [zeroExtend, WithLp.ofLp_toLp, dite_eq_right hy]
-
-lemma sum_zeroExtend_mul {L α : Type*} [Fintype L] [Fintype α]
-    (f : EuclideanSpace ℂ (L ↪ α)) (g : (L → α) → ℂ) :
-    (∑ y : L → α, g y * zeroExtend f y) = ∑ y : L ↪ α, g y * f y := by
-  rw [sum_injections _ (fun y hy => by rw [zeroExtend_noninj f y hy, mul_zero])]
-  simp only [zeroExtend_inj]
-
-lemma norm_zeroExtend {L α : Type*} [Fintype L] [Fintype α]
-    (f : EuclideanSpace ℂ (L ↪ α)) : ‖zeroExtend f‖ = ‖f‖ := by
-  have hs : ‖zeroExtend f‖^2 = ‖f‖^2 := by
-    rw [EuclideanSpace.norm_sq_eq, EuclideanSpace.norm_sq_eq,
-      sum_injections _ (fun y hy => by rw [zeroExtend_noninj f y hy]; simp)]
-    simp only [zeroExtend_inj]
-  nlinarith [norm_nonneg (zeroExtend f), norm_nonneg f]
-
-lemma partial_proper_zero {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (hA : A ≠ Finset.univ) (f : EuclideanSpace ℂ (L ↪ Card d)) (hf : IsHarmonic f) :
-    rectOperator (fun (x : L ↪ Card d) (y : L → Card d) => partialKernel d A x y) (zeroExtend f) = 0 := by
-  classical
-  ext x
-  rw [rectOperator_apply, sum_zeroExtend_mul]
-  change (∑ y : L ↪ Card d, (partialKernel d A x y : ℂ) * f y) = 0
-  simp only [partialKernel, endpointProb, finiteMean, Complex.ofReal_mul, Complex.ofReal_div,
-    Complex.ofReal_sum, Complex.ofReal_natCast]
-  simp_rw [div_eq_mul_inv, mul_assoc, ← Finset.mul_sum]
-  rw [mul_eq_zero]
-  right
-  simp only [Finset.sum_mul]
-  rw [Finset.sum_comm]
-  apply mul_eq_zero_of_right
-  apply Finset.sum_eq_zero
-  intro ω _
-  calc
-    _ = ∑ y : L ↪ Card d, (↑(Fintype.card (SwitchIndex d → Bool)) : ℂ)⁻¹ *
-        (if ∀ i ∈ A, y i = butterflyPerm d (decodeButterfly d ω) (x i) then f y else 0) :=
-      by
-        apply Finset.sum_congr rfl
-        intro y _
-        have hh : (∀ i ∈ A, butterflyPerm d (decodeButterfly d ω) (x i) = y i) ↔
-            (∀ i ∈ A, y i = butterflyPerm d (decodeButterfly d ω) (x i)) :=
-          forall₂_congr fun _ _ => eq_comm
-        by_cases h : ∀ i ∈ A, butterflyPerm d (decodeButterfly d ω) (x i) = y i
-        · simp only [ite_eq_left h, ite_eq_left (hh.mp h), Complex.ofReal_one, one_mul]
-        · simp only [ite_eq_right h, ite_eq_right (mt hh.mpr h), Complex.ofReal_zero, zero_mul, mul_zero]
-    _ = 0 := by
-      rw [← Finset.mul_sum]
-      apply mul_eq_zero_of_right
-      convert hf A hA (fun i => butterflyPerm d (decodeButterfly d ω) (x i)) using 1
-      apply Finset.sum_congr rfl
-      intro y _
-      split_ifs <;> rfl
-
-lemma partial_full_action {L : Type*} [Fintype L] (d : ℕ)
-    (f : EuclideanSpace ℂ (L ↪ Card d)) :
-    rectOperator (fun (x : L ↪ Card d) (y : L → Card d) => partialKernel d Finset.univ x y) (zeroExtend f) =
-      DensityTransfer.applyKernel (UnitaryFinite.actionKernel UnitaryFinite.tupleAction
-        (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))) f := by
-  classical
-  ext x
-  rw [rectOperator_apply, sum_zeroExtend_mul]
-  apply Finset.sum_congr rfl
-  intro y _
-  congr 2
-  simp only [partialKernel, Finset.card_univ, div_self (by positivity : ((2:ℝ)^d)^(Fintype.card L) ≠ 0),
-    one_mul, endpointProb, UnitaryFinite.actionKernel]
-  apply finiteMean_congr
-  intro ω
-  congr 1
-  simp only [Finset.mem_univ, forall_const, Function.Embedding.ext_iff]
-  rfl
-
-lemma alternating_retained_eq {L : Type*} [Fintype L] (d : ℕ)
-    (x : L ↪ Card d) (y : L → Card d) :
-    (∑ A : Finset L, (-1:ℝ)^A.card * retainedKernel d A x y) =
-      ∑ A : Finset L, (-1:ℝ)^A.card * partialKernel d A x y := by
-  classical
-  by_cases h : ContactEvent d x y
-  · simp only [retainedKernel, ite_eq_left h]
-  · simp only [retainedKernel, ite_eq_right h, mul_zero, Finset.sum_const_zero]
-    exact (alternating_kernel_cancel d x y h).symm
-
-lemma alternating_harmonic_action {L : Type*} [Fintype L] (d : ℕ)
-    (f : EuclideanSpace ℂ (L ↪ Card d)) (hf : IsHarmonic f) :
-    (∑ A : Finset L, ((-1:ℂ)^A.card) • rectOperator (retainedKernel d A) (zeroExtend f)) =
-      ((-1:ℂ)^(Fintype.card L)) • DensityTransfer.applyKernel
-        (UnitaryFinite.actionKernel UnitaryFinite.tupleAction
-          (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))) f := by
-  classical
-  calc
-    _ = ∑ A : Finset L, ((-1:ℂ)^A.card) •
-        rectOperator (fun (x : L ↪ Card d) y => partialKernel d A x y) (zeroExtend f) := by
-      ext x
-      simp only [WithLp.ofLp_sum, Finset.sum_apply, PiLp.smul_apply, rectOperator_apply, smul_eq_mul, Finset.mul_sum]
-      rw [Finset.sum_comm, Finset.sum_comm (s := (Finset.univ : Finset (Finset L)))]
-      apply Finset.sum_congr rfl
-      intro y _
-      simp_rw [← mul_assoc, ← Finset.sum_mul]
-      congr 1
-      have he := congrArg (fun t : ℝ => (t:ℂ)) (alternating_retained_eq d x y)
-      simpa only [Complex.ofReal_sum, Complex.ofReal_mul, Complex.ofReal_pow, Complex.ofReal_neg,
-        Complex.ofReal_one] using he
-    _ = ((-1:ℂ)^(Fintype.card L)) •
-        rectOperator (fun (x : L ↪ Card d) y => partialKernel d Finset.univ x y) (zeroExtend f) := by
-      rw [Finset.sum_eq_single (Finset.univ : Finset L)]
-      · simp only [Finset.card_univ]
-      · intro A _ hA
-        rw [partial_proper_zero d A hA f hf, smul_zero]
-      · simp
-    _ = _ := by rw [partial_full_action]
-
-lemma harmonic_action_bound {L : Type*} [Fintype L] (d : ℕ)
-    (f : EuclideanSpace ℂ (L ↪ Card d)) (hf : IsHarmonic f)
-    (hδ : 0 < 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)
-    (hδ1 : 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d ≤ 1) :
-    ‖DensityTransfer.applyKernel (UnitaryFinite.actionKernel UnitaryFinite.tupleAction
-      (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))) f‖ ≤
-      (2:ℝ)^(Fintype.card L) *
-        Real.sqrt ((2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
-          Real.exp (Real.exp 1*(Fintype.card L))) * ‖f‖ := by
-  classical
-  let B := (2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
-          Real.exp (Real.exp 1*(Fintype.card L))
-  have hB : 0 ≤ B := by dsimp [B]; positivity
-  have hnorm (A : Finset L) : ‖rectOperator (retainedKernel d A)‖ ≤ Real.sqrt B := by
-    have hh := retained_norm_sq d A hδ hδ1
-    have hs := Real.sq_sqrt hB
-    have hn := Real.sqrt_nonneg B
-    nlinarith
-  calc
-    _ = ‖((-1:ℂ)^(Fintype.card L)) • DensityTransfer.applyKernel
-        (UnitaryFinite.actionKernel UnitaryFinite.tupleAction
-          (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))) f‖ := by
-      simp only [norm_smul, norm_pow, norm_neg, norm_one, one_pow, one_mul]
-    _ = ‖∑ A : Finset L, ((-1:ℂ)^A.card) • rectOperator (retainedKernel d A) (zeroExtend f)‖ := by
-      rw [alternating_harmonic_action d f hf]
-    _ ≤ ∑ A : Finset L, ‖((-1:ℂ)^A.card) • rectOperator (retainedKernel d A) (zeroExtend f)‖ :=
-      norm_sum_le _ _
-    _ ≤ ∑ A : Finset L, Real.sqrt B * ‖f‖ := by
-      apply Finset.sum_le_sum
-      intro A _
-      simp only [norm_smul, norm_pow, norm_neg, norm_one, one_pow, one_mul]
-      calc _ ≤ ‖rectOperator (retainedKernel d A)‖ * ‖zeroExtend f‖ :=
-              ContinuousLinearMap.le_opNorm _ _
-           _ ≤ Real.sqrt B * ‖zeroExtend f‖ := mul_le_mul_of_nonneg_right (hnorm A) (norm_nonneg _)
-           _ = _ := by rw [norm_zeroExtend]
-    _ = _ := by simp only [Finset.sum_const, Finset.card_univ, Fintype.card_finset, nsmul_eq_mul,
-      Nat.cast_pow, Nat.cast_ofNat]; ring
-
-lemma harmonic_action_zero_one {L : Type*} [Fintype L] (d : ℕ)
-    (hL : Fintype.card L = 1) (f : EuclideanSpace ℂ (L ↪ Card d)) (hf : IsHarmonic f) :
-    DensityTransfer.applyKernel (UnitaryFinite.actionKernel UnitaryFinite.tupleAction
-      (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))) f = 0 := by
-  classical
-  obtain ⟨i,hi⟩ := Fintype.card_eq_one_iff.mp hL
-  have hz (A : Finset L) : rectOperator (retainedKernel d A) = 0 := by
-    ext v x
-    have hh (y : L → Card d) : ¬ContactEvent d x y := by
-      intro h
-      obtain ⟨j,hj,_⟩ := h i
-      exact hj (hi j).symm
-    simp only [rectOperator_apply, retainedKernel, ite_eq_right (hh _), Complex.ofReal_zero,
-      zero_mul, Finset.sum_const_zero, zero_apply, PiLp.zero_apply]
-  have he := alternating_harmonic_action d f hf
-  simp only [hz, zero_apply, smul_zero, Finset.sum_const_zero, hL, pow_one,
-    neg_smul, one_smul] at he
-  exact neg_eq_zero.mp he.symm
-
-end
-open scoped BigOperators
-
-def MainStatement : Prop :=
-  ∃ Cstar C : ℝ, 0 < Cstar ∧ 0 < C ∧
-    ∀ (d : ℕ), 1 ≤ d → ∀ (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ),
-      let k := μ.card - μ.rowLen 0
-      1 ≤ k → k < 2^d → ∀ reverse : Bool,
-        ‖sweepOperator d μ e reverse‖^2 ≤
-          min 1 ((Cstar * d * k / (2:ℝ)^d) ^ ((k:ℝ)/2)) ∧
-        (k = 1 → sweepOperator d μ e reverse = 0) ∧
-        ‖sweepOperator d μ e reverse‖^2 ≤
-          C^k * (1+(d:ℝ))^(C*k) * ((k:ℝ)/(2:ℝ)^d)^((k:ℝ)/2)
-
-end Thorp.SparseContact
-
-namespace Thorp.SparseContact
-open scoped BigOperators Classical
-
-lemma exists_exact_fixed_unit {α L : Type*} [Fintype α] [Fintype L]
-    (μ : YoungDiagram) (e : α ≃ Specht.Cell μ) (x₀ : L ↪ α)
-    (hL : Fintype.card L = μ.card - μ.rowLen 0) :
-    ∃ w : Specht.hilbertSpace μ,
-      UnitaryFinite.IsFixed UnitaryFinite.tupleAction x₀ (Specht.relabelledUnitary μ e) w ∧ ‖w‖ = 1 := by
-  let K := UnitaryFinite.fixedSpace (V := Specht.hilbertSpace μ) UnitaryFinite.tupleAction x₀ (Specht.relabelledUnitary μ e)
-  have hd : 0 < Module.finrank ℂ (Specht.hilbertSpace μ) := by
-    rw [Specht.finrank_hilbertSpace]
-    exact Module.finrank_pos_iff.mpr (Submodule.nontrivial_iff_ne_bot.mpr (Specht.space_ne_bot μ))
-  have hdim := Specht.relabelled_tuple_dimension μ e x₀
-  rw [hL, Nat.choose_self, mul_one] at hdim
-  have hm : 0 < Module.finrank ℂ K := by
-    by_contra h
-    have hz : Module.finrank ℂ K = 0 := by omega
-    change _ ≤ Module.finrank ℂ K * _ at hdim
-    rw [hz, zero_mul] at hdim
-    omega
-  let B := stdOrthonormalBasis ℂ K
-  let i : Fin (Module.finrank ℂ K) := ⟨0,hm⟩
-  exact ⟨(B i).val, (B i).prop, B.orthonormal.norm_eq_one i⟩
-
-lemma sweep_forward_norm {d : ℕ} (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ) (rev : Bool) :
-    ‖sweepOperator d μ e rev‖ = ‖sweepOperator d μ e false‖ := by
-  cases rev
-  · rfl
-  · simp only [sweepOperator, Bool.false_eq_true, ite_false, ite_true]
-    rw [UnitaryFinite.sampleOperator_inv _ (Specht.relabelledUnitary_unitary μ e),
-      ContinuousLinearMap.adjoint.norm_map]
-
-lemma sweep_norm_le_one (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ) (rev : Bool) :
-    ‖sweepOperator d μ e rev‖ ≤ 1 :=
-  UnitaryFinite.sampleOperator_norm_le _ (Specht.relabelledUnitary_unitary μ e) _
-
-section Generic
-variable {L V : Type*} [Fintype L] [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-  [FiniteDimensional ℂ V]
-
-lemma sample_sweep_bound (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V)
-    [Representation.IsIrreducible ρ] (hρ : UnitaryFinite.IsUnitary ρ)
-    (x₀ : L ↪ Card d) (w : V) (hw : UnitaryFinite.IsFixed UnitaryFinite.tupleAction x₀ ρ w)
-    (hn : ‖w‖ = 1)
-    (hh : ∀ v, IsHarmonic (UnitaryFinite.normalizedCoefficient UnitaryFinite.tupleAction x₀
-      (UnitaryFinite.tupleAction_transitive x₀) ρ w v))
-    (hδ : 0 < 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)
-    (hδ1 : 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d ≤ 1) :
-    ‖UnitaryFinite.sampleOperator ρ (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))‖ ≤
-      (2:ℝ)^(Fintype.card L) *
-        Real.sqrt ((2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
-          Real.exp (Real.exp 1*(Fintype.card L))) := by
-  rw [← ContinuousLinearMap.adjoint.norm_map (UnitaryFinite.sampleOperator ρ
-    (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))),
-    ← UnitaryFinite.sampleOperator_inv ρ hρ]
-  apply ContinuousLinearMap.opNorm_le_bound _ (by positivity)
-  intro v
-  rw [UnitaryFinite.sampleOperator_inv_apply]
-  have ha := harmonic_action_bound d _ (hh v) hδ hδ1
-  rw [UnitaryFinite.coefficient_random UnitaryFinite.tupleAction x₀
-    (UnitaryFinite.tupleAction_transitive x₀) ρ hρ _ w hw,
-    UnitaryFinite.normalizedCoefficient_norm _ _ _ ρ hρ w hw hn,
-    UnitaryFinite.normalizedCoefficient_norm _ _ _ ρ hρ w hw hn] at ha
-  exact ha
-
-lemma sample_sweep_zero_one (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V)
-    [Representation.IsIrreducible ρ] (hρ : UnitaryFinite.IsUnitary ρ)
-    (x₀ : L ↪ Card d) (w : V) (hw : UnitaryFinite.IsFixed UnitaryFinite.tupleAction x₀ ρ w)
-    (hn : ‖w‖ = 1)
-    (hh : ∀ v, IsHarmonic (UnitaryFinite.normalizedCoefficient UnitaryFinite.tupleAction x₀
-      (UnitaryFinite.tupleAction_transitive x₀) ρ w v))
-    (hL : Fintype.card L = 1) :
-    UnitaryFinite.sampleOperator ρ (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω)) = 0 := by
-  apply ContinuousLinearMap.adjoint.injective
-  rw [map_zero, ← UnitaryFinite.sampleOperator_inv ρ hρ]
-  apply ContinuousLinearMap.ext
-  intro v
-  rw [UnitaryFinite.sampleOperator_inv_apply]
-  have ha := harmonic_action_zero_one d hL _ (hh v)
-  have he := congrArg norm ha
-  rw [UnitaryFinite.coefficient_random UnitaryFinite.tupleAction x₀
-    (UnitaryFinite.tupleAction_transitive x₀) ρ hρ _ w hw,
-    UnitaryFinite.normalizedCoefficient_norm _ _ _ ρ hρ w hw hn, norm_zero] at he
-  exact norm_eq_zero.mp he
-end Generic
-
-lemma sweep_norm_bound (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ)
-    (hδ : 0 < 2*(d:ℝ)*(μ.card - μ.rowLen 0 : ℕ)/(2:ℝ)^d)
-    (hδ1 : 2*(d:ℝ)*(μ.card - μ.rowLen 0 : ℕ)/(2:ℝ)^d ≤ 1) (rev : Bool) :
-    ‖sweepOperator d μ e rev‖ ≤ (2:ℝ)^(μ.card - μ.rowLen 0) *
-      Real.sqrt ((2*(d:ℝ)*(μ.card - μ.rowLen 0 : ℕ)/(2:ℝ)^d)^(((μ.card - μ.rowLen 0 : ℕ):ℝ)/2) *
-        Real.exp (Real.exp 1*(μ.card - μ.rowLen 0 : ℕ))) := by
-  classical
-  let x₀ : Specht.Tail μ ↪ Card d := (Function.Embedding.subtype _).trans e.symm.toEmbedding
-  obtain ⟨w,hw,hn⟩ := exists_exact_fixed_unit μ e x₀ (Specht.tail_card μ)
-  let := Specht.relabelledUnitary_irreducible μ e
-  rw [sweep_forward_norm μ e rev]
-  have hh (v : Specht.hilbertSpace μ) : IsHarmonic (UnitaryFinite.normalizedCoefficient
-      (V := Specht.hilbertSpace μ) UnitaryFinite.tupleAction x₀
-      (UnitaryFinite.tupleAction_transitive x₀) (Specht.relabelledUnitary μ e) w v) :=
-    normalizedCoefficient_harmonic μ e (Specht.tail_card μ) x₀ w hw v
-  have he := sample_sweep_bound (V := Specht.hilbertSpace μ) (L := Specht.Tail μ) d (Specht.relabelledUnitary μ e)
-  have he := he (Specht.relabelledUnitary_unitary μ e) x₀ w hw hn hh
-  have he := he (by simpa only [Specht.tail_card] using hδ) (by simpa only [Specht.tail_card] using hδ1)
-  simpa only [sweepOperator, Bool.false_eq_true, ite_false, Specht.tail_card] using he
-
-lemma sweep_zero_one (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ)
-    (hk : μ.card - μ.rowLen 0 = 1) (rev : Bool) : sweepOperator d μ e rev = 0 := by
-  classical
-  let x₀ : Specht.Tail μ ↪ Card d := (Function.Embedding.subtype _).trans e.symm.toEmbedding
-  obtain ⟨w,hw,hn⟩ := exists_exact_fixed_unit μ e x₀ (Specht.tail_card μ)
-  let := Specht.relabelledUnitary_irreducible μ e
-  have hh (v : Specht.hilbertSpace μ) : IsHarmonic (UnitaryFinite.normalizedCoefficient
-      (V := Specht.hilbertSpace μ) UnitaryFinite.tupleAction x₀
-      (UnitaryFinite.tupleAction_transitive x₀) (Specht.relabelledUnitary μ e) w v) :=
-    normalizedCoefficient_harmonic μ e (Specht.tail_card μ) x₀ w hw v
-  have he := sample_sweep_zero_one (V := Specht.hilbertSpace μ) (L := Specht.Tail μ) d (Specht.relabelledUnitary μ e)
-  have he := he (Specht.relabelledUnitary_unitary μ e) x₀ w hw hn hh
-  have he := he ((Specht.tail_card μ).trans hk)
-  have hz : sweepOperator d μ e false = 0 := by
-    simpa only [sweepOperator, Bool.false_eq_true, ite_false] using he
-  apply (norm_eq_zero (a := sweepOperator d μ e rev)).mp
-  rw [sweep_forward_norm μ e rev, hz]
-  exact @norm_zero (Specht.hilbertSpace μ →L[ℂ] Specht.hilbertSpace μ) _
-
-end Thorp.SparseContact
-
-namespace Thorp.SparseContact
-
-noncomputable def contactConstant : ℝ := 32 * Real.exp (2 * Real.exp 1)
-
-lemma contactConstant_gt_one : 1 < contactConstant := by
-  have h : 1 ≤ Real.exp (2 * Real.exp 1) := Real.one_le_exp (by positivity)
-  unfold contactConstant
-  linarith
-
-lemma contactConstant_gt_two : 2 < contactConstant := by
-  have h : 1 ≤ Real.exp (2 * Real.exp 1) := Real.one_le_exp (by positivity)
-  unfold contactConstant
-  linarith
-
-lemma inflated_contact_bound {δ : ℝ} (hδ : 0 ≤ δ) (k : ℕ) :
-    ((2:ℝ)^k * Real.sqrt (δ^((k:ℝ)/2) * Real.exp (Real.exp 1*k)))^2 =
-      ((contactConstant/2)*δ)^((k:ℝ)/2) := by
-  have hC : 0 ≤ contactConstant/2 := by have := contactConstant_gt_one; positivity
-  rw [mul_pow, Real.sq_sqrt (by positivity), Real.mul_rpow hC hδ]
-  have hc : contactConstant/2 = Real.exp (2*(Real.log 4+Real.exp 1)) := by
-    unfold contactConstant
-    rw [show 2*(Real.log 4+Real.exp 1) = Real.log 4*2+2*Real.exp 1 by ring,
-      Real.exp_add, Real.exp_mul (Real.log 4) 2, Real.exp_log (by norm_num : (0:ℝ)<4)]
-    norm_num
-    ring
-  rw [hc, ←Real.exp_mul]
-  have hpow : ((2:ℝ)^k)^2 = Real.exp (Real.log 4 * k) := by
-    rw [← pow_mul, mul_comm k 2, pow_mul]
-    norm_num only [pow_two]
-    rw [Real.exp_mul (Real.log 4) k, Real.exp_log (by norm_num : (0:ℝ)<4), Real.rpow_natCast]
-  rw [hpow]
-  calc
-    _ = (Real.exp (Real.log 4*k)*Real.exp (Real.exp 1*k))*δ^((k:ℝ)/2) := by ring
-    _ = _ := by rw [←Real.exp_add]; congr 2; ring
-
-lemma contact_crude_arithmetic {C d q : ℝ} (hC : 1 ≤ C) (hd : 1 ≤ d) (hq : 0 ≤ q) (k : ℕ) :
-    (C*d*q)^((k:ℝ)/2) ≤ C^k*(1+d)^(C*k)*q^((k:ℝ)/2) := by
-  rw [Real.mul_rpow (by positivity) hq, Real.mul_rpow (by positivity) (by positivity)]
-  apply mul_le_mul_of_nonneg_right _ (by positivity)
-  apply mul_le_mul
-  · rw [← Real.rpow_natCast]
-    exact Real.rpow_le_rpow_of_exponent_le hC (by linarith [Nat.cast_nonneg (α := ℝ) k])
-  · calc
-      d^((k:ℝ)/2) ≤ (1+d)^((k:ℝ)/2) := Real.rpow_le_rpow (by positivity) (by linarith) (by positivity)
-      _ ≤ (1+d)^(C*k) := Real.rpow_le_rpow_of_exponent_le (by linarith) (by nlinarith [Nat.cast_nonneg (α := ℝ) k])
-  · positivity
-  · positivity
-
-
-lemma sweep_sharp_bound (d : ℕ) (hd : 1 ≤ d) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ)
-    (hk : 1 ≤ μ.card - μ.rowLen 0) (rev : Bool) :
-    ‖sweepOperator d μ e rev‖^2 ≤ min 1
-      ((contactConstant*d*(μ.card - μ.rowLen 0 : ℕ)/(2:ℝ)^d)^(((μ.card - μ.rowLen 0 : ℕ):ℝ)/2)) := by
-  let k := μ.card - μ.rowLen 0
-  have hk' : (0:ℝ) < k := by exact_mod_cast hk
-  have hd' : (0:ℝ) < d := by exact_mod_cast hd
-  have hδ : 0 < 2*(d:ℝ)*k/(2:ℝ)^d := by positivity
-  have hOne : ‖sweepOperator d μ e rev‖^2 ≤ 1 := by
-    have h := sweep_norm_le_one d μ e rev
-    have h0 := norm_nonneg (sweepOperator d μ e rev)
-    nlinarith
-  refine le_min hOne ?_
-  have he : contactConstant * d * (k:ℝ)/(2:ℝ)^d =
-      (contactConstant/2)*(2*(d:ℝ)*k/(2:ℝ)^d) := by ring
-  change _ ≤ (contactConstant * d * (k:ℝ)/(2:ℝ)^d)^((k:ℝ)/2)
-  rw [he]
-  by_cases hs : 2*(d:ℝ)*k/(2:ℝ)^d ≤ 1
-  · have h := sweep_norm_bound d μ e hδ hs rev
-    have hb : 0 ≤ (2:ℝ)^k * Real.sqrt
-        ((2*(d:ℝ)*k/(2:ℝ)^d)^((k:ℝ)/2)*Real.exp (Real.exp 1*k)) := by positivity
-    calc
-      _ ≤ ((2:ℝ)^k * Real.sqrt
-          ((2*(d:ℝ)*k/(2:ℝ)^d)^((k:ℝ)/2)*Real.exp (Real.exp 1*k)))^2 :=
-        (sq_le_sq₀ (norm_nonneg (sweepOperator d μ e rev)) hb).mpr h
-      _ = _ := inflated_contact_bound hδ.le k
-  · apply hOne.trans
-    apply Real.one_le_rpow
-    · have hc : 1 ≤ contactConstant/2 := by have := contactConstant_gt_two; linarith
-      exact one_le_mul_of_one_le_of_one_le hc (le_of_lt (lt_of_not_ge hs))
-    · positivity
-
-theorem uniform_sparse_contact : MainStatement := by
-  refine ⟨contactConstant, contactConstant, (lt_trans (by norm_num) contactConstant_gt_one),
-    (lt_trans (by norm_num) contactConstant_gt_one), ?_⟩
-  intro d hd μ e k hk _ rev
-  have hs := sweep_sharp_bound d hd μ e hk rev
-  refine ⟨hs, ?_, ?_⟩
-  · intro hk1
-    exact sweep_zero_one d μ e hk1 rev
-  · apply (hs.trans (min_le_right _ _)).trans
-    change (contactConstant * d * (k:ℝ)/(2:ℝ)^d)^((k:ℝ)/2) ≤ _
-    rw [mul_div_assoc]
-    exact contact_crude_arithmetic contactConstant_gt_one.le
-      (by exact_mod_cast hd) (by positivity) k
-
-end Thorp.SparseContact
-
-namespace Thorp.DimensionEstimates
-open scoped BigOperators
-
-lemma log_factorial_upper_binary (n : ℕ) (hn : 0<n) :
-    Real.log (n.factorial:ℝ) ≤ (n:ℝ)*Real.log n-Real.log 2*((n:ℝ)-1) := by
-  induction n, hn using Nat.le_induction with
-  | base => norm_num
-  | succ n hn ih =>
-    have hn0 : (0:ℝ)<n := by exact_mod_cast hn
-    have hb : (2:ℝ) ≤ (1+(n:ℝ)⁻¹)^n := by
-      have h := one_add_mul_le_pow (by have := inv_nonneg.mpr hn0.le; linarith : (-2:ℝ)≤(n:ℝ)⁻¹) n
-      norm_num [mul_inv_cancel₀ hn0.ne'] at h ⊢
-      exact h
-    have hl := Real.log_le_log (by norm_num : (0:ℝ)<2) hb
-    rw [Real.log_pow] at hl
-    have hs : Real.log (1+(n:ℝ)⁻¹)=Real.log ((n:ℝ)+1)-Real.log n := by
-      rw [←Real.log_div (by positivity) hn0.ne']
-      congr 1
-      field_simp
-    rw [hs] at hl
-    rw [Nat.factorial_succ,Nat.cast_mul,Real.log_mul (by positivity) (by positivity),Nat.cast_add,Nat.cast_one]
-    nlinarith
-
-lemma log_factorial_lower (n : ℕ) (hn : 0<n) :
-    (n:ℝ)*Real.log n-n+1 ≤ Real.log (n.factorial:ℝ) := by
-  induction n, hn using Nat.le_induction with
-  | base => norm_num
-  | succ n hn ih =>
-    have hn0 : (0:ℝ)<n := by exact_mod_cast hn
-    have hl := Real.log_le_sub_one_of_pos (show 0<((n:ℝ)+1)/n by positivity)
-    rw [Real.log_div (by positivity) hn0.ne'] at hl
-    have hh := mul_le_mul_of_nonneg_left hl hn0.le
-    have he : (n:ℝ)*(((n:ℝ)+1)/n-1)=1 := by field_simp; ring
-    rw [he] at hh
-    rw [Nat.factorial_succ,Nat.cast_mul,Real.log_mul (by positivity) (by positivity),Nat.cast_add,Nat.cast_one]
-    nlinarith
-
-lemma log_factorial_upper (n : ℕ) (hn : 0<n) :
-    Real.log (n.factorial:ℝ) ≤ (n:ℝ)*Real.log n-n+Real.log n+1 := by
-  induction n, hn using Nat.le_induction with
-  | base => norm_num
-  | succ n hn ih =>
-    have hn0 : (0:ℝ)<n := by exact_mod_cast hn
-    have hl := Real.one_sub_inv_le_log_of_pos (show 0<((n:ℝ)+1)/n by positivity)
-    rw [Real.log_div (by positivity) hn0.ne'] at hl
-    have hh := mul_le_mul_of_nonneg_left hl (show 0≤(n:ℝ)+1 by positivity)
-    have he : ((n:ℝ)+1)*(1-(((n:ℝ)+1)/n)⁻¹)=1 := by field_simp; ring
-    rw [he] at hh
-    rw [Nat.factorial_succ,Nat.cast_mul,Real.log_mul (by positivity) (by positivity),Nat.cast_add,Nat.cast_one]
-    nlinarith
-
-variable {X I J : Type*} [Fintype X] [Fintype I] [Fintype J]
-open scoped Classical
-
-noncomputable def fiberWeight (f : X → I) (i : I) : ℝ := Fintype.card {x // f x=i}
-
-omit [Fintype I] in
-lemma fiberWeight_nonneg (f : X → I) (i : I) : 0≤fiberWeight f i := Nat.cast_nonneg _
-
-omit [Fintype I] in
-lemma fiberWeight_pos (f : X → I) (x : X) : 0<fiberWeight f (f x) := by
-  let : Nonempty {y // f y=f x} := ⟨⟨x,rfl⟩⟩
-  exact Nat.cast_pos.mpr Fintype.card_pos
-
-lemma fiber_sum (f : X → I) (u : I → ℝ) :
-    ∑ x, u (f x)=∑ i, fiberWeight f i*u i := by
-  rw [←Equiv.sum_comp (Equiv.sigmaFiberEquiv f)]
-  simp only [Equiv.sigmaFiberEquiv_apply,Fintype.sum_sigma]
-  apply Finset.sum_congr rfl
-  intro i _
-  calc
-    (∑ x : {x // f x=i}, u (f x)) = ∑ _x : {x // f x=i}, u i :=
-      Finset.sum_congr rfl (fun x _ => congrArg u x.property)
-    _ = _ := by simp [fiberWeight]
-
-lemma sum_fiberWeight (f : X → I) : ∑ i, fiberWeight f i=Fintype.card X := by
-  have h := fiber_sum f (fun _ => 1)
-  simpa using h.symm
-
-theorem incidence_entropy (f : X → I) (g : X → J)
-    (hinj : Function.Injective (fun x => (f x,g x))) :
-    (∑ i, fiberWeight f i*Real.log (fiberWeight f i))+
-      (∑ j, fiberWeight g j*Real.log (fiberWeight g j)) ≤
-        (Fintype.card X:ℝ)*Real.log (Fintype.card X) := by
-  by_cases hX : Nonempty X
-  · let : Nonempty X := hX
-    have hN : (0:ℝ)<Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
-    have hprod : ∑ x, fiberWeight f (f x)*fiberWeight g (g x) ≤ (Fintype.card X:ℝ)^2 := by
-      calc
-        _ ≤ ∑ p : I × J, fiberWeight f p.1*fiberWeight g p.2 := by
-          calc
-            _ = ∑ p ∈ Finset.univ.image (fun x => (f x,g x)), fiberWeight f p.1*fiberWeight g p.2 := by
-              rw [Finset.sum_image (fun x _ y _ h => hinj h)]
-            _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _)
-              (fun p _ _ => mul_nonneg (fiberWeight_nonneg f p.1) (fiberWeight_nonneg g p.2))
-        _ = _ := by
-          rw [Fintype.sum_prod_type]
-          simp_rw [←Finset.mul_sum]
-          rw [←Finset.sum_mul,sum_fiberWeight,sum_fiberWeight,pow_two]
-    have hdiv : (∑ x, fiberWeight f (f x)*fiberWeight g (g x))/(Fintype.card X:ℝ) ≤ Fintype.card X := by
-      apply (div_le_iff₀ hN).mpr
-      simpa only [pow_two] using hprod
-    have hl (x : X) := Real.log_le_sub_one_of_pos
-      (show 0<fiberWeight f (f x)*fiberWeight g (g x)/(Fintype.card X:ℝ) from
-        div_pos (mul_pos (fiberWeight_pos f x) (fiberWeight_pos g x)) hN)
-    have hs := Finset.sum_le_sum (s := Finset.univ) (fun x _ => hl x)
-    simp_rw [Real.log_div (mul_pos (fiberWeight_pos f _) (fiberWeight_pos g _)).ne' hN.ne',
-      Real.log_mul (fiberWeight_pos f _).ne' (fiberWeight_pos g _).ne'] at hs
-    rw [Finset.sum_sub_distrib,Finset.sum_add_distrib,
-      fiber_sum f (fun i => Real.log (fiberWeight f i)),
-      fiber_sum g (fun j => Real.log (fiberWeight g j)),
-      Finset.sum_sub_distrib,←Finset.sum_div] at hs
-    simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,mul_one] at hs
-    linarith
-  · let : IsEmpty X := not_nonempty_iff.mp hX
-    have hf (i : I) : fiberWeight f i=0 := by simp [fiberWeight]
-    have hg (j : J) : fiberWeight g j=0 := by simp [fiberWeight]
-    simp [hf,hg]
-
-end Thorp.DimensionEstimates
-
-namespace Thorp.DimensionEstimates
-open scoped BigOperators Classical
-
-lemma log_factorial_upper_binary_all (n : ℕ) :
-    Real.log (n.factorial:ℝ) ≤ (n:ℝ)*Real.log n-Real.log 2*((n:ℝ)-1) := by
-  by_cases hn : n=0
-  · subst n
-    simpa using (Real.log_nonneg (by norm_num : (1:ℝ)≤2))
-  · exact log_factorial_upper_binary n (Nat.pos_of_ne_zero hn)
-
-lemma log_fiber_factorials {X I : Type*} [Fintype X] [Fintype I] (f : X → I) :
-    Real.log ((∏ i, (Fintype.card {x // f x=i}).factorial : ℕ):ℝ) ≤
-      ∑ i, fiberWeight f i*Real.log (fiberWeight f i) -
-        Real.log 2*((Fintype.card X:ℝ)-Fintype.card I) := by
-  rw [Nat.cast_prod,Real.log_prod (fun _ _ => by positivity)]
-  have h := Finset.sum_le_sum (s := Finset.univ) (fun i _ =>
-    log_factorial_upper_binary_all (Fintype.card {x // f x=i}))
-  change ∑ i, Real.log ((Fintype.card {x // f x=i}).factorial:ℝ) ≤
-    ∑ i, (fiberWeight f i*Real.log (fiberWeight f i)-Real.log 2*(fiberWeight f i-1)) at h
-  rw [Finset.sum_sub_distrib,←Finset.mul_sum,Finset.sum_sub_distrib,sum_fiberWeight] at h
-  simpa using h
-
-theorem dimension_log_from_incidence {X I J : Type*} [Fintype X] [Fintype I] [Fintype J]
-    [Nonempty X] (f : X → I) (g : X → J)
-    (hinj : Function.Injective (fun x => (f x,g x))) (D : ℕ)
-    (hD : (Fintype.card X).factorial ≤ D *
-      (∏ i, (Fintype.card {x // f x=i}).factorial) *
-      (∏ j, (Fintype.card {x // g x=j}).factorial)) :
-    ((2*Real.log 2-1)*(Fintype.card X:ℝ)-
-      Real.log 2*((Fintype.card I:ℝ)+Fintype.card J)+1) ≤ Real.log D := by
-  have hd : 0<D := by
-    by_contra h
-    have : D=0 := by omega
-    simp [this] at hD
-    exact (Nat.factorial_pos _).ne' hD
-  have hf := log_fiber_factorials f
-  have hg := log_fiber_factorials g
-  have hi := incidence_entropy f g hinj
-  have hlo := log_factorial_lower (Fintype.card X) Fintype.card_pos
-  have hlog := Real.log_le_log (by positivity : (0:ℝ)<(Fintype.card X).factorial)
-    (show ((Fintype.card X).factorial:ℝ) ≤ D *
-      (∏ i, (Fintype.card {x // f x=i}).factorial) *
-      (∏ j, (Fintype.card {x // g x=j}).factorial) by exact_mod_cast hD)
-  rw [Real.log_mul (by positivity) (by positivity),
-    Real.log_mul (by positivity) (by positivity)] at hlog
-  nlinarith
-
-
-lemma scaled_binEntropy (N j : ℝ) (hN : 0<N) (hj : 0<j) (hjN : j<N) :
-    N*Real.binEntropy (j/N)=N*Real.log N-j*Real.log j-(N-j)*Real.log (N-j) := by
-  have hsub : 0<N-j := sub_pos.mpr hjN
-  have he : 1-j/N=(N-j)/N := by field_simp
-  unfold Real.binEntropy
-  rw [he,Real.log_inv,Real.log_inv,Real.log_div hj.ne' hN.ne',
-    Real.log_div hsub.ne' hN.ne']
-  field_simp
-  ring
-
-lemma log_choose_lower_entropy (N j : ℕ) (hj : 0<j) (hjN : j<N) :
-    (N:ℝ)*Real.binEntropy ((j:ℝ)/N)-2*Real.log N-1 ≤ Real.log (N.choose j:ℝ) := by
-  have hN : 0<N := lt_trans hj hjN
-  have ht : 0<N-j := Nat.sub_pos_of_lt hjN
-  have hj0 : (0:ℝ)<j := by exact_mod_cast hj
-  have hN0 : (0:ℝ)<N := by exact_mod_cast hN
-  have hjN0 : (j:ℝ)<N := by exact_mod_cast hjN
-  have hchoose : (0:ℝ)<N.choose j := by exact_mod_cast Nat.choose_pos hjN.le
-  have hfac := congrArg (fun n : ℕ => Real.log (n:ℝ)) (Nat.choose_mul_factorial_mul_factorial hjN.le)
-  simp only [Nat.cast_mul] at hfac
-  rw [Real.log_mul (mul_pos hchoose (by positivity)).ne' (by positivity),
-    Real.log_mul hchoose.ne' (by positivity)] at hfac
-  have hlo := log_factorial_lower N hN
-  have hji := log_factorial_upper j hj
-  have hti := log_factorial_upper (N-j) ht
-  have hjlog := Real.log_le_log hj0 hjN0.le
-  have htlog := Real.log_le_log (by exact_mod_cast ht : (0:ℝ)<(N-j:ℕ))
-    (by exact_mod_cast Nat.sub_le N j : ((N-j:ℕ):ℝ)≤N)
-  rw [Nat.cast_sub hjN.le] at hti htlog
-  rw [scaled_binEntropy N j hN0 hj0 hjN0]
-  linarith
-
-lemma binEntropy_three_quarters :
-    Real.binEntropy (3/4:ℝ)=2*Real.log 2-(3/4:ℝ)*Real.log 3 := by
-  have h4 : Real.log (4:ℝ)=2*Real.log 2 := by
-    rw [show (4:ℝ)=2^2 by norm_num,Real.log_pow]; norm_num
-  unfold Real.binEntropy
-  norm_num only [show (1-(3/4:ℝ))=1/4 by norm_num]
-  rw [Real.log_div (by norm_num) (by norm_num),h4]
-  ring
-
-noncomputable def entropyGap : ℝ := Real.log (32/27:ℝ)/3
-
-lemma entropyGap_pos : 0<entropyGap := by
-  unfold entropyGap
-  exact div_pos (Real.log_pos (by norm_num)) (by norm_num)
-
-lemma entropyGap_eq : entropyGap=(5*Real.log 2-3*Real.log 3)/3 := by
-  unfold entropyGap
-  rw [Real.log_div (by norm_num) (by norm_num),show (32:ℝ)=2^5 by norm_num,
-    show (27:ℝ)=3^3 by norm_num,Real.log_pow,Real.log_pow]
-  norm_num
-
-lemma binEntropy_tail_lower (q : ℝ) (hq : 0≤q) (hq' : q≤3/4) :
-    q*Real.log 2+entropyGap*q ≤ Real.binEntropy q := by
-  have hc := Real.strictConcave_binEntropy.concaveOn.2
-    (show (0:ℝ)∈Set.Icc 0 1 by constructor <;> norm_num)
-    (show (3/4:ℝ)∈Set.Icc 0 1 by constructor <;> norm_num)
-    (show 0≤1-4*q/3 by linarith) (show 0≤4*q/3 by positivity)
-    (show (1-4*q/3)+(4*q/3)=1 by ring)
-  simp only [Real.binEntropy_zero,binEntropy_three_quarters,smul_eq_mul,mul_zero,zero_add] at hc
-  have he : 4*q/3*(3/4:ℝ)=q := by ring
-  rw [he] at hc
-  rw [entropyGap_eq]
-  linarith
-
-lemma log_dimension_from_binomial (N j D : ℕ) (hj : 0<j) (hjN : j<N)
-    (hfrac : (j:ℝ)≤(3/4:ℝ)*N) (hD : N.choose j≤D*2^j) :
-    entropyGap*(j:ℝ)-2*Real.log N-1 ≤ Real.log D := by
-  have hpos : 0<N.choose j := Nat.choose_pos hjN.le
-  have hd : 0<D := by nlinarith [Nat.zero_le (2^j)]
-  have hN : (0:ℝ)<N := by exact_mod_cast lt_trans hj hjN
-  have hm := Real.log_le_log (by exact_mod_cast hpos : (0:ℝ)<N.choose j)
-    (show (N.choose j:ℝ)≤D*2^j by exact_mod_cast hD)
-  rw [Real.log_mul (by positivity) (by positivity),Real.log_pow] at hm
-  have he := binEntropy_tail_lower ((j:ℝ)/N) (by positivity)
-    ((div_le_iff₀ hN).mpr hfrac)
-  have hel := mul_le_mul_of_nonneg_left he hN.le
-  have hbin := log_choose_lower_entropy N j hj hjN
-  have heq : (N:ℝ)*((j:ℝ)/N*Real.log 2+entropyGap*((j:ℝ)/N))=
-      (j:ℝ)*Real.log 2+entropyGap*j := by field_simp
-  rw [heq] at hel
-  linarith
-
-lemma balanced_dimension_coefficient_pos : 0<(3/2:ℝ)*Real.log 2-1 := by
-  linarith [Real.log_two_gt_d9]
-
-
-lemma log_choose_upper (N j : ℕ) (hN : 0<N) (hj : j≤N) :
-    Real.log (N.choose j:ℝ) ≤ (j:ℝ)*Real.log N-Real.log (j.factorial:ℝ) := by
-  have hh := Real.log_le_log (by exact_mod_cast Nat.choose_pos hj : (0:ℝ)<N.choose j)
-    (Nat.choose_le_pow_div j N : (N.choose j:ℝ)≤(N:ℝ)^j/(j.factorial:ℝ))
-  rw [Real.log_div (by positivity : (N:ℝ)^j≠0) (by positivity : (j.factorial:ℝ)≠0),Real.log_pow] at hh
-  exact hh
-
-lemma log_choose_lower (N j : ℕ) (hj : j≤N) :
-    (j:ℝ)*Real.log (N+1-j:ℕ)-Real.log (j.factorial:ℝ) ≤ Real.log (N.choose j:ℝ) := by
-  have hp : 0<N+1-j := by omega
-  have hh := Real.log_le_log (by positivity : (0:ℝ)<((N+1-j:ℕ):ℝ)^j/(j.factorial:ℝ))
-    (Nat.pow_le_choose j N : (((N+1-j:ℕ):ℝ)^j)/(j.factorial:ℝ)≤N.choose j)
-  rw [Real.log_div (by positivity : (((N+1-j:ℕ):ℝ)^j)≠0) (by positivity : (j.factorial:ℝ)≠0),Real.log_pow] at hh
-  exact hh
-
-lemma footprint_log_bound (N k j D m : ℕ) (hN : 0<N) (hk : j≤k) (hj : j≤N)
-    (hD : 0<D) (hfoot : D * k.choose j ≤ m * N.choose j) :
-    Real.log D-(j:ℝ)*(Real.log N-Real.log (k+1-j:ℕ)) ≤ Real.log m := by
-  have hkpos := Nat.choose_pos hk
-  have hprod : 0<D*k.choose j := Nat.mul_pos hD hkpos
-  have hm : 0 < m := by
-    by_contra h
-    have he : m=0 := by omega
-    simp only [he,zero_mul] at hfoot
-    omega
-  have hh := Real.log_le_log (by exact_mod_cast hprod : (0:ℝ)<D*k.choose j)
-    (show (D:ℝ)*k.choose j≤ m*N.choose j by exact_mod_cast hfoot)
-  rw [Real.log_mul (by positivity) (by exact_mod_cast hkpos.ne'),
-    Real.log_mul (by positivity) (by exact_mod_cast (Nat.choose_pos hj).ne')] at hh
-  have hlo := log_choose_lower k j hk
-  have hup := log_choose_upper N j hN hj
-  nlinarith
-
-lemma log_factorial_le_mul_log (j : ℕ) (_hj : 0<j) :
-    Real.log (j.factorial:ℝ)≤(j:ℝ)*Real.log j := by
-  have h := Real.log_le_log (by positivity : (0:ℝ)<j.factorial)
-    (show (j.factorial:ℝ)≤(j:ℝ)^j by exact_mod_cast Nat.factorial_le_pow j)
-  simpa only [Real.log_pow] using h
-
-lemma near_row_log_dimension (N j D : ℕ) (hj : 0<j) (hsmall : 2*j≤N)
-    (hD : N.choose j≤D*2^j) :
-    (j:ℝ)*(Real.log ((N:ℝ)/j)-2*Real.log 2) ≤ Real.log D := by
-  have hjN : j≤N := by omega
-  have hN : 0<N := by omega
-  have hNp : (0:ℝ)<N := by exact_mod_cast hN
-  have hjp : (0:ℝ)<j := by exact_mod_cast hj
-  have hn : 0<N+1-j := by omega
-  have hnreal : (N:ℝ)/2≤(N+1-j:ℕ) := by
-    rw [Nat.cast_sub (by omega : j≤N+1),Nat.cast_add,Nat.cast_one]
-    have hc : 2*(j:ℝ)≤N := by exact_mod_cast hsmall
-    linarith
-  have hDpos : 0<D := by
-    have hc := Nat.choose_pos hjN
-    by_contra h
-    have he : D=0 := by omega
-    simp only [he,zero_mul] at hD
-    omega
-  have hm := Real.log_le_log (by exact_mod_cast Nat.choose_pos hjN : (0:ℝ)<N.choose j)
-    (show (N.choose j:ℝ)≤D*2^j by exact_mod_cast hD)
-  rw [Real.log_mul (by positivity) (by positivity),Real.log_pow] at hm
-  have hl := Real.log_le_log (by positivity : (0:ℝ)<(N:ℝ)/2) hnreal
-  rw [Real.log_div hNp.ne' (by norm_num)] at hl
-  have hh := log_choose_lower N j hjN
-  have hf := log_factorial_le_mul_log j hj
-  rw [Real.log_div hNp.ne' hjp.ne']
-  nlinarith
-
-open Filter
-
-noncomputable def nearPower : ℝ := 1/2048
-noncomputable def nearFraction : ℝ := 1/4096
-noncomputable def tailPower : ℝ := 1535/1048576
-
-noncomputable def chosenTuple (N j : ℕ) : ℕ :=
-  ⌈(j:ℝ)*((N:ℝ)/j)^nearPower⌉₊
-
-lemma chosenTuple_bounds (N j : ℕ) (hj : 0<j) (hx : 1≤(N:ℝ)/j)
-    (hhalf : ((N:ℝ)/j)^(nearPower-1)≤1/2) :
-    j≤chosenTuple N j ∧ chosenTuple N j≤N ∧
-      (chosenTuple N j:ℝ)≤2*j*((N:ℝ)/j)^nearPower := by
-  let x : ℝ := (N:ℝ)/j
-  have hjp : (0:ℝ)<j := by exact_mod_cast hj
-  have hj1 : (1:ℝ)≤j := by exact_mod_cast hj
-  have hxp : 0<x := lt_of_lt_of_le zero_lt_one hx
-  have hpow : 1≤x^nearPower := Real.one_le_rpow hx (by norm_num [nearPower])
-  have hceil : (j:ℝ)*x^nearPower≤chosenTuple N j := Nat.le_ceil _
-  have hceil' : (chosenTuple N j:ℝ)<(j:ℝ)*x^nearPower+1 := Nat.ceil_lt_add_one (by positivity)
-  have he : (j:ℝ)*x=N := by dsimp [x]; field_simp
-  have hh : x^nearPower≤x/2 := by
-    rw [Real.rpow_sub hxp,Real.rpow_one] at hhalf
-    nlinarith [(div_le_iff₀ hxp).mp hhalf]
-  constructor
-  · exact_mod_cast (le_mul_of_one_le_right hjp.le hpow).trans hceil
-  constructor
-  · apply Nat.ceil_le.mpr
-    change (j:ℝ)*x^nearPower≤N
-    nlinarith
-  · change (chosenTuple N j:ℝ)≤2*(j:ℝ)*x^nearPower
-    nlinarith
-
-lemma chosenTuple_tail_lower (N j : ℕ) (hj : 0<j)
-    (hx : 2≤((N:ℝ)/j)^nearPower) :
-    (j:ℝ)*((N:ℝ)/j)^nearPower/2 ≤ (chosenTuple N j+1-j:ℕ) := by
-  have hjp : (0:ℝ)<j := by exact_mod_cast hj
-  have hc : (j:ℝ)*((N:ℝ)/j)^nearPower≤chosenTuple N j := Nat.le_ceil _
-  have hjk : j≤chosenTuple N j := by exact_mod_cast (by nlinarith : (j:ℝ)≤chosenTuple N j)
-  rw [Nat.cast_sub (by omega : j≤chosenTuple N j+1),Nat.cast_add,Nat.cast_one]
-  nlinarith
-
-lemma chosenTuple_log_footprint (N j D m : ℕ) (hj : 0<j)
-    (hN : 2*j≤N) (hD : N.choose j≤D*2^j)
-    (hx : 2≤((N:ℝ)/j)^nearPower)
-    (hlog : 8*Real.log 2/nearPower≤Real.log ((N:ℝ)/j))
-    (hfoot : D * (chosenTuple N j).choose j ≤ m * N.choose j) :
-    nearFraction*Real.log D≤Real.log m := by
-  let x : ℝ := (N:ℝ)/j
-  have hjp : (0:ℝ)<j := by exact_mod_cast hj
-  have hNp : (0:ℝ)<N := by exact_mod_cast (by omega : 0<N)
-  have hxp : 0<x := by dsimp [x]; positivity
-  have hdpos : 0<D := by
-    have hc := Nat.choose_pos (show j≤N by omega)
-    by_contra h
-    have hz : D=0 := by omega
-    simp only [hz,zero_mul] at hD
-    omega
-  have htail := chosenTuple_tail_lower N j hj hx
-  have hjk : j≤chosenTuple N j := by
-    have hc : (j:ℝ)*x^nearPower≤chosenTuple N j := Nat.le_ceil _
-    exact_mod_cast (by nlinarith : (j:ℝ)≤chosenTuple N j)
-  have hh := footprint_log_bound N (chosenTuple N j) j D m (by omega) hjk (by omega) hdpos hfoot
-  have htlog := Real.log_le_log (by positivity : (0:ℝ)<(j:ℝ)*x^nearPower/2) htail
-  rw [Real.log_div (by positivity) (by norm_num),Real.log_mul hjp.ne' (by positivity),Real.log_rpow hxp] at htlog
-  have hxlog : Real.log x=Real.log N-Real.log j := Real.log_div hNp.ne' hjp.ne'
-  have hdim := near_row_log_dimension N j D hj hN hD
-  change (j:ℝ)*(Real.log x-2*Real.log 2)≤Real.log D at hdim
-  have hh' : Real.log D-(j:ℝ)*((1-nearPower)*Real.log x+Real.log 2)≤Real.log m := by
-    have hm := mul_le_mul_of_nonneg_left htlog hjp.le
-    rw [hxlog] at hm
-    nlinarith
-  have hlg : (3-nearPower)*Real.log 2≤nearFraction*Real.log x := by
-    dsimp [nearPower,nearFraction] at hlog ⊢
-    have := Real.log_pos (by norm_num : (1:ℝ)<2)
-    linarith
-  have hlg' := mul_le_mul_of_nonneg_left hlg hjp.le
-  dsimp [nearPower,nearFraction] at hh' hlg' ⊢
-  nlinarith
-
-lemma near_row_log_uniform (N j D : ℕ) (hj : 0<j) (hN : 2*j≤N)
-    (hD : N.choose j≤D*2^j) (hlog : 4*Real.log 2+2≤Real.log ((N:ℝ)/j)) :
-    (1/2:ℝ)*Real.log N≤Real.log D ∧ (j:ℝ)≤Real.log D := by
-  have hjp : (0:ℝ)<j := by exact_mod_cast hj
-  have hj1 : (1:ℝ)≤j := by exact_mod_cast hj
-  have hNp : (0:ℝ)<N := by exact_mod_cast (by omega : 0<N)
-  have hd := near_row_log_dimension N j D hj hN hD
-  have hjlog := Real.log_le_sub_one_of_pos hjp
-  have hl2 : 0≤Real.log 2 := Real.log_nonneg (by norm_num)
-  have hi : 1≤Real.log ((N:ℝ)/j) := by linarith
-  have he : Real.log N=Real.log j+Real.log ((N:ℝ)/j) := by rw [Real.log_div hNp.ne' hjp.ne']; ring
-  constructor
-  · rw [he]
-    nlinarith
-  · nlinarith
-
-end Thorp.DimensionEstimates
-
-end ThorpNine.Harmonic
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Harmonic.TypedSet
+
+namespace OAI
+
+namespace ThorpNine.Harmonic
+
+namespace Thorp.SparseContact
+
+section
+open scoped BigOperators Classical
+
+lemma sum_injections {L α M : Type*} [Fintype L] [Fintype α] [AddCommMonoid M]
+    (F : (L → α) → M) (hF : ∀ y, ¬Function.Injective y → F y = 0) :
+    (∑ y : L → α, F y) = ∑ y : L ↪ α, F y := by
+  classical
+  rw [← Finset.sum_image Function.Embedding.coe_injective.injOn]
+  apply Eq.symm
+  apply Finset.sum_subset (Finset.subset_univ _)
+  intro y _ hy
+  apply hF y
+  intro h
+  exact hy (Finset.mem_image.mpr ⟨⟨y,h⟩, Finset.mem_univ _, rfl⟩)
+
+noncomputable def zeroExtend {L α : Type*} [Fintype L] [Fintype α]
+    (f : EuclideanSpace ℂ (L ↪ α)) : EuclideanSpace ℂ (L → α) := by
+  classical
+  exact WithLp.toLp 2 (fun y => if h : Function.Injective y then f ⟨y,h⟩ else 0)
+
+@[simp] lemma zeroExtend_inj {L α : Type*} [Fintype L] [Fintype α]
+    (f : EuclideanSpace ℂ (L ↪ α)) (y : L ↪ α) : zeroExtend f y = f y := by
+  classical
+  simp only [zeroExtend, WithLp.ofLp_toLp, dite_eq_left y.injective]
+  congr 1
+
+lemma zeroExtend_noninj {L α : Type*} [Fintype L] [Fintype α]
+    (f : EuclideanSpace ℂ (L ↪ α)) (y : L → α) (hy : ¬Function.Injective y) : zeroExtend f y = 0 := by
+  classical
+  simp only [zeroExtend, WithLp.ofLp_toLp, dite_eq_right hy]
+
+lemma sum_zeroExtend_mul {L α : Type*} [Fintype L] [Fintype α]
+    (f : EuclideanSpace ℂ (L ↪ α)) (g : (L → α) → ℂ) :
+    (∑ y : L → α, g y * zeroExtend f y) = ∑ y : L ↪ α, g y * f y := by
+  rw [sum_injections _ (fun y hy => by rw [zeroExtend_noninj f y hy, mul_zero])]
+  simp only [zeroExtend_inj]
+
+lemma norm_zeroExtend {L α : Type*} [Fintype L] [Fintype α]
+    (f : EuclideanSpace ℂ (L ↪ α)) : ‖zeroExtend f‖ = ‖f‖ := by
+  have hs : ‖zeroExtend f‖^2 = ‖f‖^2 := by
+    rw [EuclideanSpace.norm_sq_eq, EuclideanSpace.norm_sq_eq,
+      sum_injections _ (fun y hy => by rw [zeroExtend_noninj f y hy]; simp)]
+    simp only [zeroExtend_inj]
+  nlinarith [norm_nonneg (zeroExtend f), norm_nonneg f]
+
+lemma partial_proper_zero {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (hA : A ≠ Finset.univ) (f : EuclideanSpace ℂ (L ↪ Card d)) (hf : IsHarmonic f) :
+    rectOperator (fun (x : L ↪ Card d) (y : L → Card d) => partialKernel d A x y) (zeroExtend f) = 0 := by
+  classical
+  ext x
+  rw [rectOperator_apply, sum_zeroExtend_mul]
+  change (∑ y : L ↪ Card d, (partialKernel d A x y : ℂ) * f y) = 0
+  simp only [partialKernel, endpointProb, finiteMean, Complex.ofReal_mul, Complex.ofReal_div,
+    Complex.ofReal_sum, Complex.ofReal_natCast]
+  simp_rw [div_eq_mul_inv, mul_assoc, ← Finset.mul_sum]
+  rw [mul_eq_zero]
+  right
+  simp only [Finset.sum_mul]
+  rw [Finset.sum_comm]
+  apply mul_eq_zero_of_right
+  apply Finset.sum_eq_zero
+  intro ω _
+  calc
+    _ = ∑ y : L ↪ Card d, (↑(Fintype.card (SwitchIndex d → Bool)) : ℂ)⁻¹ *
+        (if ∀ i ∈ A, y i = butterflyPerm d (decodeButterfly d ω) (x i) then f y else 0) :=
+      by
+        apply Finset.sum_congr rfl
+        intro y _
+        have hh : (∀ i ∈ A, butterflyPerm d (decodeButterfly d ω) (x i) = y i) ↔
+            (∀ i ∈ A, y i = butterflyPerm d (decodeButterfly d ω) (x i)) :=
+          forall₂_congr fun _ _ => eq_comm
+        by_cases h : ∀ i ∈ A, butterflyPerm d (decodeButterfly d ω) (x i) = y i
+        · simp only [ite_eq_left h, ite_eq_left (hh.mp h), Complex.ofReal_one, one_mul]
+        · simp only [ite_eq_right h, ite_eq_right (mt hh.mpr h), Complex.ofReal_zero, zero_mul, mul_zero]
+    _ = 0 := by
+      rw [← Finset.mul_sum]
+      apply mul_eq_zero_of_right
+      convert hf A hA (fun i => butterflyPerm d (decodeButterfly d ω) (x i)) using 1
+      apply Finset.sum_congr rfl
+      intro y _
+      split_ifs <;> rfl
+
+lemma partial_full_action {L : Type*} [Fintype L] (d : ℕ)
+    (f : EuclideanSpace ℂ (L ↪ Card d)) :
+    rectOperator (fun (x : L ↪ Card d) (y : L → Card d) => partialKernel d Finset.univ x y) (zeroExtend f) =
+      DensityTransfer.applyKernel (UnitaryFinite.actionKernel UnitaryFinite.tupleAction
+        (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))) f := by
+  classical
+  ext x
+  rw [rectOperator_apply, sum_zeroExtend_mul]
+  apply Finset.sum_congr rfl
+  intro y _
+  congr 2
+  simp only [partialKernel, Finset.card_univ, div_self (by positivity : ((2:ℝ)^d)^(Fintype.card L) ≠ 0),
+    one_mul, endpointProb, UnitaryFinite.actionKernel]
+  apply finiteMean_congr
+  intro ω
+  congr 1
+  simp only [Finset.mem_univ, forall_const, Function.Embedding.ext_iff]
+  rfl
+
+lemma alternating_retained_eq {L : Type*} [Fintype L] (d : ℕ)
+    (x : L ↪ Card d) (y : L → Card d) :
+    (∑ A : Finset L, (-1:ℝ)^A.card * retainedKernel d A x y) =
+      ∑ A : Finset L, (-1:ℝ)^A.card * partialKernel d A x y := by
+  classical
+  by_cases h : ContactEvent d x y
+  · simp only [retainedKernel, ite_eq_left h]
+  · simp only [retainedKernel, ite_eq_right h, mul_zero, Finset.sum_const_zero]
+    exact (alternating_kernel_cancel d x y h).symm
+
+lemma alternating_harmonic_action {L : Type*} [Fintype L] (d : ℕ)
+    (f : EuclideanSpace ℂ (L ↪ Card d)) (hf : IsHarmonic f) :
+    (∑ A : Finset L, ((-1:ℂ)^A.card) • rectOperator (retainedKernel d A) (zeroExtend f)) =
+      ((-1:ℂ)^(Fintype.card L)) • DensityTransfer.applyKernel
+        (UnitaryFinite.actionKernel UnitaryFinite.tupleAction
+          (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))) f := by
+  classical
+  calc
+    _ = ∑ A : Finset L, ((-1:ℂ)^A.card) •
+        rectOperator (fun (x : L ↪ Card d) y => partialKernel d A x y) (zeroExtend f) := by
+      ext x
+      simp only [WithLp.ofLp_sum, Finset.sum_apply, PiLp.smul_apply, rectOperator_apply, smul_eq_mul, Finset.mul_sum]
+      rw [Finset.sum_comm, Finset.sum_comm (s := (Finset.univ : Finset (Finset L)))]
+      apply Finset.sum_congr rfl
+      intro y _
+      simp_rw [← mul_assoc, ← Finset.sum_mul]
+      congr 1
+      have he := congrArg (fun t : ℝ => (t:ℂ)) (alternating_retained_eq d x y)
+      simpa only [Complex.ofReal_sum, Complex.ofReal_mul, Complex.ofReal_pow, Complex.ofReal_neg,
+        Complex.ofReal_one] using he
+    _ = ((-1:ℂ)^(Fintype.card L)) •
+        rectOperator (fun (x : L ↪ Card d) y => partialKernel d Finset.univ x y) (zeroExtend f) := by
+      rw [Finset.sum_eq_single (Finset.univ : Finset L)]
+      · simp only [Finset.card_univ]
+      · intro A _ hA
+        rw [partial_proper_zero d A hA f hf, smul_zero]
+      · simp
+    _ = _ := by rw [partial_full_action]
+
+lemma harmonic_action_bound {L : Type*} [Fintype L] (d : ℕ)
+    (f : EuclideanSpace ℂ (L ↪ Card d)) (hf : IsHarmonic f)
+    (hδ : 0 < 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)
+    (hδ1 : 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d ≤ 1) :
+    ‖DensityTransfer.applyKernel (UnitaryFinite.actionKernel UnitaryFinite.tupleAction
+      (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))) f‖ ≤
+      (2:ℝ)^(Fintype.card L) *
+        Real.sqrt ((2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
+          Real.exp (Real.exp 1*(Fintype.card L))) * ‖f‖ := by
+  classical
+  let B := (2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
+          Real.exp (Real.exp 1*(Fintype.card L))
+  have hB : 0 ≤ B := by dsimp [B]; positivity
+  have hnorm (A : Finset L) : ‖rectOperator (retainedKernel d A)‖ ≤ Real.sqrt B := by
+    have hh := retained_norm_sq d A hδ hδ1
+    have hs := Real.sq_sqrt hB
+    have hn := Real.sqrt_nonneg B
+    nlinarith
+  calc
+    _ = ‖((-1:ℂ)^(Fintype.card L)) • DensityTransfer.applyKernel
+        (UnitaryFinite.actionKernel UnitaryFinite.tupleAction
+          (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))) f‖ := by
+      simp only [norm_smul, norm_pow, norm_neg, norm_one, one_pow, one_mul]
+    _ = ‖∑ A : Finset L, ((-1:ℂ)^A.card) • rectOperator (retainedKernel d A) (zeroExtend f)‖ := by
+      rw [alternating_harmonic_action d f hf]
+    _ ≤ ∑ A : Finset L, ‖((-1:ℂ)^A.card) • rectOperator (retainedKernel d A) (zeroExtend f)‖ :=
+      norm_sum_le _ _
+    _ ≤ ∑ A : Finset L, Real.sqrt B * ‖f‖ := by
+      apply Finset.sum_le_sum
+      intro A _
+      simp only [norm_smul, norm_pow, norm_neg, norm_one, one_pow, one_mul]
+      calc _ ≤ ‖rectOperator (retainedKernel d A)‖ * ‖zeroExtend f‖ :=
+              ContinuousLinearMap.le_opNorm _ _
+           _ ≤ Real.sqrt B * ‖zeroExtend f‖ := mul_le_mul_of_nonneg_right (hnorm A) (norm_nonneg _)
+           _ = _ := by rw [norm_zeroExtend]
+    _ = _ := by simp only [Finset.sum_const, Finset.card_univ, Fintype.card_finset, nsmul_eq_mul,
+      Nat.cast_pow, Nat.cast_ofNat]; ring
+
+lemma harmonic_action_zero_one {L : Type*} [Fintype L] (d : ℕ)
+    (hL : Fintype.card L = 1) (f : EuclideanSpace ℂ (L ↪ Card d)) (hf : IsHarmonic f) :
+    DensityTransfer.applyKernel (UnitaryFinite.actionKernel UnitaryFinite.tupleAction
+      (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))) f = 0 := by
+  classical
+  obtain ⟨i,hi⟩ := Fintype.card_eq_one_iff.mp hL
+  have hz (A : Finset L) : rectOperator (retainedKernel d A) = 0 := by
+    ext v x
+    have hh (y : L → Card d) : ¬ContactEvent d x y := by
+      intro h
+      obtain ⟨j,hj,_⟩ := h i
+      exact hj (hi j).symm
+    simp only [rectOperator_apply, retainedKernel, ite_eq_right (hh _), Complex.ofReal_zero,
+      zero_mul, Finset.sum_const_zero, zero_apply, PiLp.zero_apply]
+  have he := alternating_harmonic_action d f hf
+  simp only [hz, zero_apply, smul_zero, Finset.sum_const_zero, hL, pow_one,
+    neg_smul, one_smul] at he
+  exact neg_eq_zero.mp he.symm
+
+end
+open scoped BigOperators
+
+def MainStatement : Prop :=
+  ∃ Cstar C : ℝ, 0 < Cstar ∧ 0 < C ∧
+    ∀ (d : ℕ), 1 ≤ d → ∀ (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ),
+      let k := μ.card - μ.rowLen 0
+      1 ≤ k → k < 2^d → ∀ reverse : Bool,
+        ‖sweepOperator d μ e reverse‖^2 ≤
+          min 1 ((Cstar * d * k / (2:ℝ)^d) ^ ((k:ℝ)/2)) ∧
+        (k = 1 → sweepOperator d μ e reverse = 0) ∧
+        ‖sweepOperator d μ e reverse‖^2 ≤
+          C^k * (1+(d:ℝ))^(C*k) * ((k:ℝ)/(2:ℝ)^d)^((k:ℝ)/2)
+
+end Thorp.SparseContact
+
+namespace Thorp.SparseContact
+open scoped BigOperators Classical
+
+lemma exists_exact_fixed_unit {α L : Type*} [Fintype α] [Fintype L]
+    (μ : YoungDiagram) (e : α ≃ Specht.Cell μ) (x₀ : L ↪ α)
+    (hL : Fintype.card L = μ.card - μ.rowLen 0) :
+    ∃ w : Specht.hilbertSpace μ,
+      UnitaryFinite.IsFixed UnitaryFinite.tupleAction x₀ (Specht.relabelledUnitary μ e) w ∧ ‖w‖ = 1 := by
+  let K := UnitaryFinite.fixedSpace (V := Specht.hilbertSpace μ) UnitaryFinite.tupleAction x₀ (Specht.relabelledUnitary μ e)
+  have hd : 0 < Module.finrank ℂ (Specht.hilbertSpace μ) := by
+    rw [Specht.finrank_hilbertSpace]
+    exact Module.finrank_pos_iff.mpr (Submodule.nontrivial_iff_ne_bot.mpr (Specht.space_ne_bot μ))
+  have hdim := Specht.relabelled_tuple_dimension μ e x₀
+  rw [hL, Nat.choose_self, mul_one] at hdim
+  have hm : 0 < Module.finrank ℂ K := by
+    by_contra h
+    have hz : Module.finrank ℂ K = 0 := by omega
+    change _ ≤ Module.finrank ℂ K * _ at hdim
+    rw [hz, zero_mul] at hdim
+    omega
+  let B := stdOrthonormalBasis ℂ K
+  let i : Fin (Module.finrank ℂ K) := ⟨0,hm⟩
+  exact ⟨(B i).val, (B i).prop, B.orthonormal.norm_eq_one i⟩
+
+lemma sweep_forward_norm {d : ℕ} (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ) (rev : Bool) :
+    ‖sweepOperator d μ e rev‖ = ‖sweepOperator d μ e false‖ := by
+  cases rev
+  · rfl
+  · simp only [sweepOperator, Bool.false_eq_true, ite_false, ite_true]
+    rw [UnitaryFinite.sampleOperator_inv _ (Specht.relabelledUnitary_unitary μ e),
+      ContinuousLinearMap.adjoint.norm_map]
+
+lemma sweep_norm_le_one (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ) (rev : Bool) :
+    ‖sweepOperator d μ e rev‖ ≤ 1 :=
+  UnitaryFinite.sampleOperator_norm_le _ (Specht.relabelledUnitary_unitary μ e) _
+
+section Generic
+variable {L V : Type*} [Fintype L] [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+  [FiniteDimensional ℂ V]
+
+lemma sample_sweep_bound (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V)
+    [Representation.IsIrreducible ρ] (hρ : UnitaryFinite.IsUnitary ρ)
+    (x₀ : L ↪ Card d) (w : V) (hw : UnitaryFinite.IsFixed UnitaryFinite.tupleAction x₀ ρ w)
+    (hn : ‖w‖ = 1)
+    (hh : ∀ v, IsHarmonic (UnitaryFinite.normalizedCoefficient UnitaryFinite.tupleAction x₀
+      (UnitaryFinite.tupleAction_transitive x₀) ρ w v))
+    (hδ : 0 < 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)
+    (hδ1 : 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d ≤ 1) :
+    ‖UnitaryFinite.sampleOperator ρ (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))‖ ≤
+      (2:ℝ)^(Fintype.card L) *
+        Real.sqrt ((2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
+          Real.exp (Real.exp 1*(Fintype.card L))) := by
+  rw [← ContinuousLinearMap.adjoint.norm_map (UnitaryFinite.sampleOperator ρ
+    (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))),
+    ← UnitaryFinite.sampleOperator_inv ρ hρ]
+  apply ContinuousLinearMap.opNorm_le_bound _ (by positivity)
+  intro v
+  rw [UnitaryFinite.sampleOperator_inv_apply]
+  have ha := harmonic_action_bound d _ (hh v) hδ hδ1
+  rw [UnitaryFinite.coefficient_random UnitaryFinite.tupleAction x₀
+    (UnitaryFinite.tupleAction_transitive x₀) ρ hρ _ w hw,
+    UnitaryFinite.normalizedCoefficient_norm _ _ _ ρ hρ w hw hn,
+    UnitaryFinite.normalizedCoefficient_norm _ _ _ ρ hρ w hw hn] at ha
+  exact ha
+
+lemma sample_sweep_zero_one (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V)
+    [Representation.IsIrreducible ρ] (hρ : UnitaryFinite.IsUnitary ρ)
+    (x₀ : L ↪ Card d) (w : V) (hw : UnitaryFinite.IsFixed UnitaryFinite.tupleAction x₀ ρ w)
+    (hn : ‖w‖ = 1)
+    (hh : ∀ v, IsHarmonic (UnitaryFinite.normalizedCoefficient UnitaryFinite.tupleAction x₀
+      (UnitaryFinite.tupleAction_transitive x₀) ρ w v))
+    (hL : Fintype.card L = 1) :
+    UnitaryFinite.sampleOperator ρ (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω)) = 0 := by
+  apply ContinuousLinearMap.adjoint.injective
+  rw [map_zero, ← UnitaryFinite.sampleOperator_inv ρ hρ]
+  apply ContinuousLinearMap.ext
+  intro v
+  rw [UnitaryFinite.sampleOperator_inv_apply]
+  have ha := harmonic_action_zero_one d hL _ (hh v)
+  have he := congrArg norm ha
+  rw [UnitaryFinite.coefficient_random UnitaryFinite.tupleAction x₀
+    (UnitaryFinite.tupleAction_transitive x₀) ρ hρ _ w hw,
+    UnitaryFinite.normalizedCoefficient_norm _ _ _ ρ hρ w hw hn, norm_zero] at he
+  exact norm_eq_zero.mp he
+end Generic
+
+lemma sweep_norm_bound (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ)
+    (hδ : 0 < 2*(d:ℝ)*(μ.card - μ.rowLen 0 : ℕ)/(2:ℝ)^d)
+    (hδ1 : 2*(d:ℝ)*(μ.card - μ.rowLen 0 : ℕ)/(2:ℝ)^d ≤ 1) (rev : Bool) :
+    ‖sweepOperator d μ e rev‖ ≤ (2:ℝ)^(μ.card - μ.rowLen 0) *
+      Real.sqrt ((2*(d:ℝ)*(μ.card - μ.rowLen 0 : ℕ)/(2:ℝ)^d)^(((μ.card - μ.rowLen 0 : ℕ):ℝ)/2) *
+        Real.exp (Real.exp 1*(μ.card - μ.rowLen 0 : ℕ))) := by
+  classical
+  let x₀ : Specht.Tail μ ↪ Card d := (Function.Embedding.subtype _).trans e.symm.toEmbedding
+  obtain ⟨w,hw,hn⟩ := exists_exact_fixed_unit μ e x₀ (Specht.tail_card μ)
+  let := Specht.relabelledUnitary_irreducible μ e
+  rw [sweep_forward_norm μ e rev]
+  have hh (v : Specht.hilbertSpace μ) : IsHarmonic (UnitaryFinite.normalizedCoefficient
+      (V := Specht.hilbertSpace μ) UnitaryFinite.tupleAction x₀
+      (UnitaryFinite.tupleAction_transitive x₀) (Specht.relabelledUnitary μ e) w v) :=
+    normalizedCoefficient_harmonic μ e (Specht.tail_card μ) x₀ w hw v
+  have he := sample_sweep_bound (V := Specht.hilbertSpace μ) (L := Specht.Tail μ) d (Specht.relabelledUnitary μ e)
+  have he := he (Specht.relabelledUnitary_unitary μ e) x₀ w hw hn hh
+  have he := he (by simpa only [Specht.tail_card] using hδ) (by simpa only [Specht.tail_card] using hδ1)
+  simpa only [sweepOperator, Bool.false_eq_true, ite_false, Specht.tail_card] using he
+
+lemma sweep_zero_one (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ)
+    (hk : μ.card - μ.rowLen 0 = 1) (rev : Bool) : sweepOperator d μ e rev = 0 := by
+  classical
+  let x₀ : Specht.Tail μ ↪ Card d := (Function.Embedding.subtype _).trans e.symm.toEmbedding
+  obtain ⟨w,hw,hn⟩ := exists_exact_fixed_unit μ e x₀ (Specht.tail_card μ)
+  let := Specht.relabelledUnitary_irreducible μ e
+  have hh (v : Specht.hilbertSpace μ) : IsHarmonic (UnitaryFinite.normalizedCoefficient
+      (V := Specht.hilbertSpace μ) UnitaryFinite.tupleAction x₀
+      (UnitaryFinite.tupleAction_transitive x₀) (Specht.relabelledUnitary μ e) w v) :=
+    normalizedCoefficient_harmonic μ e (Specht.tail_card μ) x₀ w hw v
+  have he := sample_sweep_zero_one (V := Specht.hilbertSpace μ) (L := Specht.Tail μ) d (Specht.relabelledUnitary μ e)
+  have he := he (Specht.relabelledUnitary_unitary μ e) x₀ w hw hn hh
+  have he := he ((Specht.tail_card μ).trans hk)
+  have hz : sweepOperator d μ e false = 0 := by
+    simpa only [sweepOperator, Bool.false_eq_true, ite_false] using he
+  apply (norm_eq_zero (a := sweepOperator d μ e rev)).mp
+  rw [sweep_forward_norm μ e rev, hz]
+  exact @norm_zero (Specht.hilbertSpace μ →L[ℂ] Specht.hilbertSpace μ) _
+
+end Thorp.SparseContact
+
+namespace Thorp.SparseContact
+
+noncomputable def contactConstant : ℝ := 32 * Real.exp (2 * Real.exp 1)
+
+lemma contactConstant_gt_one : 1 < contactConstant := by
+  have h : 1 ≤ Real.exp (2 * Real.exp 1) := Real.one_le_exp (by positivity)
+  unfold contactConstant
+  linarith
+
+lemma contactConstant_gt_two : 2 < contactConstant := by
+  have h : 1 ≤ Real.exp (2 * Real.exp 1) := Real.one_le_exp (by positivity)
+  unfold contactConstant
+  linarith
+
+lemma inflated_contact_bound {δ : ℝ} (hδ : 0 ≤ δ) (k : ℕ) :
+    ((2:ℝ)^k * Real.sqrt (δ^((k:ℝ)/2) * Real.exp (Real.exp 1*k)))^2 =
+      ((contactConstant/2)*δ)^((k:ℝ)/2) := by
+  have hC : 0 ≤ contactConstant/2 := by have := contactConstant_gt_one; positivity
+  rw [mul_pow, Real.sq_sqrt (by positivity), Real.mul_rpow hC hδ]
+  have hc : contactConstant/2 = Real.exp (2*(Real.log 4+Real.exp 1)) := by
+    unfold contactConstant
+    rw [show 2*(Real.log 4+Real.exp 1) = Real.log 4*2+2*Real.exp 1 by ring,
+      Real.exp_add, Real.exp_mul (Real.log 4) 2, Real.exp_log (by norm_num : (0:ℝ)<4)]
+    norm_num
+    ring
+  rw [hc, ←Real.exp_mul]
+  have hpow : ((2:ℝ)^k)^2 = Real.exp (Real.log 4 * k) := by
+    rw [← pow_mul, mul_comm k 2, pow_mul]
+    norm_num only [pow_two]
+    rw [Real.exp_mul (Real.log 4) k, Real.exp_log (by norm_num : (0:ℝ)<4), Real.rpow_natCast]
+  rw [hpow]
+  calc
+    _ = (Real.exp (Real.log 4*k)*Real.exp (Real.exp 1*k))*δ^((k:ℝ)/2) := by ring
+    _ = _ := by rw [←Real.exp_add]; congr 2; ring
+
+lemma contact_crude_arithmetic {C d q : ℝ} (hC : 1 ≤ C) (hd : 1 ≤ d) (hq : 0 ≤ q) (k : ℕ) :
+    (C*d*q)^((k:ℝ)/2) ≤ C^k*(1+d)^(C*k)*q^((k:ℝ)/2) := by
+  rw [Real.mul_rpow (by positivity) hq, Real.mul_rpow (by positivity) (by positivity)]
+  apply mul_le_mul_of_nonneg_right _ (by positivity)
+  apply mul_le_mul
+  · rw [← Real.rpow_natCast]
+    exact Real.rpow_le_rpow_of_exponent_le hC (by linarith [Nat.cast_nonneg (α := ℝ) k])
+  · calc
+      d^((k:ℝ)/2) ≤ (1+d)^((k:ℝ)/2) := Real.rpow_le_rpow (by positivity) (by linarith) (by positivity)
+      _ ≤ (1+d)^(C*k) := Real.rpow_le_rpow_of_exponent_le (by linarith) (by nlinarith [Nat.cast_nonneg (α := ℝ) k])
+  · positivity
+  · positivity
+
+
+lemma sweep_sharp_bound (d : ℕ) (hd : 1 ≤ d) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ)
+    (hk : 1 ≤ μ.card - μ.rowLen 0) (rev : Bool) :
+    ‖sweepOperator d μ e rev‖^2 ≤ min 1
+      ((contactConstant*d*(μ.card - μ.rowLen 0 : ℕ)/(2:ℝ)^d)^(((μ.card - μ.rowLen 0 : ℕ):ℝ)/2)) := by
+  let k := μ.card - μ.rowLen 0
+  have hk' : (0:ℝ) < k := by exact_mod_cast hk
+  have hd' : (0:ℝ) < d := by exact_mod_cast hd
+  have hδ : 0 < 2*(d:ℝ)*k/(2:ℝ)^d := by positivity
+  have hOne : ‖sweepOperator d μ e rev‖^2 ≤ 1 := by
+    have h := sweep_norm_le_one d μ e rev
+    have h0 := norm_nonneg (sweepOperator d μ e rev)
+    nlinarith
+  refine le_min hOne ?_
+  have he : contactConstant * d * (k:ℝ)/(2:ℝ)^d =
+      (contactConstant/2)*(2*(d:ℝ)*k/(2:ℝ)^d) := by ring
+  change _ ≤ (contactConstant * d * (k:ℝ)/(2:ℝ)^d)^((k:ℝ)/2)
+  rw [he]
+  by_cases hs : 2*(d:ℝ)*k/(2:ℝ)^d ≤ 1
+  · have h := sweep_norm_bound d μ e hδ hs rev
+    have hb : 0 ≤ (2:ℝ)^k * Real.sqrt
+        ((2*(d:ℝ)*k/(2:ℝ)^d)^((k:ℝ)/2)*Real.exp (Real.exp 1*k)) := by positivity
+    calc
+      _ ≤ ((2:ℝ)^k * Real.sqrt
+          ((2*(d:ℝ)*k/(2:ℝ)^d)^((k:ℝ)/2)*Real.exp (Real.exp 1*k)))^2 :=
+        (sq_le_sq₀ (norm_nonneg (sweepOperator d μ e rev)) hb).mpr h
+      _ = _ := inflated_contact_bound hδ.le k
+  · apply hOne.trans
+    apply Real.one_le_rpow
+    · have hc : 1 ≤ contactConstant/2 := by have := contactConstant_gt_two; linarith
+      exact one_le_mul_of_one_le_of_one_le hc (le_of_lt (lt_of_not_ge hs))
+    · positivity
+
+theorem uniform_sparse_contact : MainStatement := by
+  refine ⟨contactConstant, contactConstant, (lt_trans (by norm_num) contactConstant_gt_one),
+    (lt_trans (by norm_num) contactConstant_gt_one), ?_⟩
+  intro d hd μ e k hk _ rev
+  have hs := sweep_sharp_bound d hd μ e hk rev
+  refine ⟨hs, ?_, ?_⟩
+  · intro hk1
+    exact sweep_zero_one d μ e hk1 rev
+  · apply (hs.trans (min_le_right _ _)).trans
+    change (contactConstant * d * (k:ℝ)/(2:ℝ)^d)^((k:ℝ)/2) ≤ _
+    rw [mul_div_assoc]
+    exact contact_crude_arithmetic contactConstant_gt_one.le
+      (by exact_mod_cast hd) (by positivity) k
+
+end Thorp.SparseContact
+
+namespace Thorp.DimensionEstimates
+open scoped BigOperators
+
+lemma log_factorial_upper_binary (n : ℕ) (hn : 0<n) :
+    Real.log (n.factorial:ℝ) ≤ (n:ℝ)*Real.log n-Real.log 2*((n:ℝ)-1) := by
+  induction n, hn using Nat.le_induction with
+  | base => norm_num
+  | succ n hn ih =>
+    have hn0 : (0:ℝ)<n := by exact_mod_cast hn
+    have hb : (2:ℝ) ≤ (1+(n:ℝ)⁻¹)^n := by
+      have h := one_add_mul_le_pow (by have := inv_nonneg.mpr hn0.le; linarith : (-2:ℝ)≤(n:ℝ)⁻¹) n
+      norm_num [mul_inv_cancel₀ hn0.ne'] at h ⊢
+      exact h
+    have hl := Real.log_le_log (by norm_num : (0:ℝ)<2) hb
+    rw [Real.log_pow] at hl
+    have hs : Real.log (1+(n:ℝ)⁻¹)=Real.log ((n:ℝ)+1)-Real.log n := by
+      rw [←Real.log_div (by positivity) hn0.ne']
+      congr 1
+      field_simp
+    rw [hs] at hl
+    rw [Nat.factorial_succ,Nat.cast_mul,Real.log_mul (by positivity) (by positivity),Nat.cast_add,Nat.cast_one]
+    nlinarith
+
+lemma log_factorial_lower (n : ℕ) (hn : 0<n) :
+    (n:ℝ)*Real.log n-n+1 ≤ Real.log (n.factorial:ℝ) := by
+  induction n, hn using Nat.le_induction with
+  | base => norm_num
+  | succ n hn ih =>
+    have hn0 : (0:ℝ)<n := by exact_mod_cast hn
+    have hl := Real.log_le_sub_one_of_pos (show 0<((n:ℝ)+1)/n by positivity)
+    rw [Real.log_div (by positivity) hn0.ne'] at hl
+    have hh := mul_le_mul_of_nonneg_left hl hn0.le
+    have he : (n:ℝ)*(((n:ℝ)+1)/n-1)=1 := by field_simp; ring
+    rw [he] at hh
+    rw [Nat.factorial_succ,Nat.cast_mul,Real.log_mul (by positivity) (by positivity),Nat.cast_add,Nat.cast_one]
+    nlinarith
+
+lemma log_factorial_upper (n : ℕ) (hn : 0<n) :
+    Real.log (n.factorial:ℝ) ≤ (n:ℝ)*Real.log n-n+Real.log n+1 := by
+  induction n, hn using Nat.le_induction with
+  | base => norm_num
+  | succ n hn ih =>
+    have hn0 : (0:ℝ)<n := by exact_mod_cast hn
+    have hl := Real.one_sub_inv_le_log_of_pos (show 0<((n:ℝ)+1)/n by positivity)
+    rw [Real.log_div (by positivity) hn0.ne'] at hl
+    have hh := mul_le_mul_of_nonneg_left hl (show 0≤(n:ℝ)+1 by positivity)
+    have he : ((n:ℝ)+1)*(1-(((n:ℝ)+1)/n)⁻¹)=1 := by field_simp; ring
+    rw [he] at hh
+    rw [Nat.factorial_succ,Nat.cast_mul,Real.log_mul (by positivity) (by positivity),Nat.cast_add,Nat.cast_one]
+    nlinarith
+
+variable {X I J : Type*} [Fintype X] [Fintype I] [Fintype J]
+open scoped Classical
+
+noncomputable def fiberWeight (f : X → I) (i : I) : ℝ := Fintype.card {x // f x=i}
+
+omit [Fintype I] in
+lemma fiberWeight_nonneg (f : X → I) (i : I) : 0≤fiberWeight f i := Nat.cast_nonneg _
+
+omit [Fintype I] in
+lemma fiberWeight_pos (f : X → I) (x : X) : 0<fiberWeight f (f x) := by
+  let : Nonempty {y // f y=f x} := ⟨⟨x,rfl⟩⟩
+  exact Nat.cast_pos.mpr Fintype.card_pos
+
+lemma fiber_sum (f : X → I) (u : I → ℝ) :
+    ∑ x, u (f x)=∑ i, fiberWeight f i*u i := by
+  rw [←Equiv.sum_comp (Equiv.sigmaFiberEquiv f)]
+  simp only [Equiv.sigmaFiberEquiv_apply,Fintype.sum_sigma]
+  apply Finset.sum_congr rfl
+  intro i _
+  calc
+    (∑ x : {x // f x=i}, u (f x)) = ∑ _x : {x // f x=i}, u i :=
+      Finset.sum_congr rfl (fun x _ => congrArg u x.property)
+    _ = _ := by simp [fiberWeight]
+
+lemma sum_fiberWeight (f : X → I) : ∑ i, fiberWeight f i=Fintype.card X := by
+  have h := fiber_sum f (fun _ => 1)
+  simpa using h.symm
+
+theorem incidence_entropy (f : X → I) (g : X → J)
+    (hinj : Function.Injective (fun x => (f x,g x))) :
+    (∑ i, fiberWeight f i*Real.log (fiberWeight f i))+
+      (∑ j, fiberWeight g j*Real.log (fiberWeight g j)) ≤
+        (Fintype.card X:ℝ)*Real.log (Fintype.card X) := by
+  by_cases hX : Nonempty X
+  · let : Nonempty X := hX
+    have hN : (0:ℝ)<Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
+    have hprod : ∑ x, fiberWeight f (f x)*fiberWeight g (g x) ≤ (Fintype.card X:ℝ)^2 := by
+      calc
+        _ ≤ ∑ p : I × J, fiberWeight f p.1*fiberWeight g p.2 := by
+          calc
+            _ = ∑ p ∈ Finset.univ.image (fun x => (f x,g x)), fiberWeight f p.1*fiberWeight g p.2 := by
+              rw [Finset.sum_image (fun x _ y _ h => hinj h)]
+            _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _)
+              (fun p _ _ => mul_nonneg (fiberWeight_nonneg f p.1) (fiberWeight_nonneg g p.2))
+        _ = _ := by
+          rw [Fintype.sum_prod_type]
+          simp_rw [←Finset.mul_sum]
+          rw [←Finset.sum_mul,sum_fiberWeight,sum_fiberWeight,pow_two]
+    have hdiv : (∑ x, fiberWeight f (f x)*fiberWeight g (g x))/(Fintype.card X:ℝ) ≤ Fintype.card X := by
+      apply (div_le_iff₀ hN).mpr
+      simpa only [pow_two] using hprod
+    have hl (x : X) := Real.log_le_sub_one_of_pos
+      (show 0<fiberWeight f (f x)*fiberWeight g (g x)/(Fintype.card X:ℝ) from
+        div_pos (mul_pos (fiberWeight_pos f x) (fiberWeight_pos g x)) hN)
+    have hs := Finset.sum_le_sum (s := Finset.univ) (fun x _ => hl x)
+    simp_rw [Real.log_div (mul_pos (fiberWeight_pos f _) (fiberWeight_pos g _)).ne' hN.ne',
+      Real.log_mul (fiberWeight_pos f _).ne' (fiberWeight_pos g _).ne'] at hs
+    rw [Finset.sum_sub_distrib,Finset.sum_add_distrib,
+      fiber_sum f (fun i => Real.log (fiberWeight f i)),
+      fiber_sum g (fun j => Real.log (fiberWeight g j)),
+      Finset.sum_sub_distrib,←Finset.sum_div] at hs
+    simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,mul_one] at hs
+    linarith
+  · let : IsEmpty X := not_nonempty_iff.mp hX
+    have hf (i : I) : fiberWeight f i=0 := by simp [fiberWeight]
+    have hg (j : J) : fiberWeight g j=0 := by simp [fiberWeight]
+    simp [hf,hg]
+
+end Thorp.DimensionEstimates
+
+namespace Thorp.DimensionEstimates
+open scoped BigOperators Classical
+
+lemma log_factorial_upper_binary_all (n : ℕ) :
+    Real.log (n.factorial:ℝ) ≤ (n:ℝ)*Real.log n-Real.log 2*((n:ℝ)-1) := by
+  by_cases hn : n=0
+  · subst n
+    simpa using (Real.log_nonneg (by norm_num : (1:ℝ)≤2))
+  · exact log_factorial_upper_binary n (Nat.pos_of_ne_zero hn)
+
+lemma log_fiber_factorials {X I : Type*} [Fintype X] [Fintype I] (f : X → I) :
+    Real.log ((∏ i, (Fintype.card {x // f x=i}).factorial : ℕ):ℝ) ≤
+      ∑ i, fiberWeight f i*Real.log (fiberWeight f i) -
+        Real.log 2*((Fintype.card X:ℝ)-Fintype.card I) := by
+  rw [Nat.cast_prod,Real.log_prod (fun _ _ => by positivity)]
+  have h := Finset.sum_le_sum (s := Finset.univ) (fun i _ =>
+    log_factorial_upper_binary_all (Fintype.card {x // f x=i}))
+  change ∑ i, Real.log ((Fintype.card {x // f x=i}).factorial:ℝ) ≤
+    ∑ i, (fiberWeight f i*Real.log (fiberWeight f i)-Real.log 2*(fiberWeight f i-1)) at h
+  rw [Finset.sum_sub_distrib,←Finset.mul_sum,Finset.sum_sub_distrib,sum_fiberWeight] at h
+  simpa using h
+
+theorem dimension_log_from_incidence {X I J : Type*} [Fintype X] [Fintype I] [Fintype J]
+    [Nonempty X] (f : X → I) (g : X → J)
+    (hinj : Function.Injective (fun x => (f x,g x))) (D : ℕ)
+    (hD : (Fintype.card X).factorial ≤ D *
+      (∏ i, (Fintype.card {x // f x=i}).factorial) *
+      (∏ j, (Fintype.card {x // g x=j}).factorial)) :
+    ((2*Real.log 2-1)*(Fintype.card X:ℝ)-
+      Real.log 2*((Fintype.card I:ℝ)+Fintype.card J)+1) ≤ Real.log D := by
+  have hd : 0<D := by
+    by_contra h
+    have : D=0 := by omega
+    simp [this] at hD
+    exact (Nat.factorial_pos _).ne' hD
+  have hf := log_fiber_factorials f
+  have hg := log_fiber_factorials g
+  have hi := incidence_entropy f g hinj
+  have hlo := log_factorial_lower (Fintype.card X) Fintype.card_pos
+  have hlog := Real.log_le_log (by positivity : (0:ℝ)<(Fintype.card X).factorial)
+    (show ((Fintype.card X).factorial:ℝ) ≤ D *
+      (∏ i, (Fintype.card {x // f x=i}).factorial) *
+      (∏ j, (Fintype.card {x // g x=j}).factorial) by exact_mod_cast hD)
+  rw [Real.log_mul (by positivity) (by positivity),
+    Real.log_mul (by positivity) (by positivity)] at hlog
+  nlinarith
+
+
+lemma scaled_binEntropy (N j : ℝ) (hN : 0<N) (hj : 0<j) (hjN : j<N) :
+    N*Real.binEntropy (j/N)=N*Real.log N-j*Real.log j-(N-j)*Real.log (N-j) := by
+  have hsub : 0<N-j := sub_pos.mpr hjN
+  have he : 1-j/N=(N-j)/N := by field_simp
+  unfold Real.binEntropy
+  rw [he,Real.log_inv,Real.log_inv,Real.log_div hj.ne' hN.ne',
+    Real.log_div hsub.ne' hN.ne']
+  field_simp
+  ring
+
+lemma log_choose_lower_entropy (N j : ℕ) (hj : 0<j) (hjN : j<N) :
+    (N:ℝ)*Real.binEntropy ((j:ℝ)/N)-2*Real.log N-1 ≤ Real.log (N.choose j:ℝ) := by
+  have hN : 0<N := lt_trans hj hjN
+  have ht : 0<N-j := Nat.sub_pos_of_lt hjN
+  have hj0 : (0:ℝ)<j := by exact_mod_cast hj
+  have hN0 : (0:ℝ)<N := by exact_mod_cast hN
+  have hjN0 : (j:ℝ)<N := by exact_mod_cast hjN
+  have hchoose : (0:ℝ)<N.choose j := by exact_mod_cast Nat.choose_pos hjN.le
+  have hfac := congrArg (fun n : ℕ => Real.log (n:ℝ)) (Nat.choose_mul_factorial_mul_factorial hjN.le)
+  simp only [Nat.cast_mul] at hfac
+  rw [Real.log_mul (mul_pos hchoose (by positivity)).ne' (by positivity),
+    Real.log_mul hchoose.ne' (by positivity)] at hfac
+  have hlo := log_factorial_lower N hN
+  have hji := log_factorial_upper j hj
+  have hti := log_factorial_upper (N-j) ht
+  have hjlog := Real.log_le_log hj0 hjN0.le
+  have htlog := Real.log_le_log (by exact_mod_cast ht : (0:ℝ)<(N-j:ℕ))
+    (by exact_mod_cast Nat.sub_le N j : ((N-j:ℕ):ℝ)≤N)
+  rw [Nat.cast_sub hjN.le] at hti htlog
+  rw [scaled_binEntropy N j hN0 hj0 hjN0]
+  linarith
+
+lemma binEntropy_three_quarters :
+    Real.binEntropy (3/4:ℝ)=2*Real.log 2-(3/4:ℝ)*Real.log 3 := by
+  have h4 : Real.log (4:ℝ)=2*Real.log 2 := by
+    rw [show (4:ℝ)=2^2 by norm_num,Real.log_pow]; norm_num
+  unfold Real.binEntropy
+  norm_num only [show (1-(3/4:ℝ))=1/4 by norm_num]
+  rw [Real.log_div (by norm_num) (by norm_num),h4]
+  ring
+
+noncomputable def entropyGap : ℝ := Real.log (32/27:ℝ)/3
+
+lemma entropyGap_pos : 0<entropyGap := by
+  unfold entropyGap
+  exact div_pos (Real.log_pos (by norm_num)) (by norm_num)
+
+lemma entropyGap_eq : entropyGap=(5*Real.log 2-3*Real.log 3)/3 := by
+  unfold entropyGap
+  rw [Real.log_div (by norm_num) (by norm_num),show (32:ℝ)=2^5 by norm_num,
+    show (27:ℝ)=3^3 by norm_num,Real.log_pow,Real.log_pow]
+  norm_num
+
+lemma binEntropy_tail_lower (q : ℝ) (hq : 0≤q) (hq' : q≤3/4) :
+    q*Real.log 2+entropyGap*q ≤ Real.binEntropy q := by
+  have hc := Real.strictConcave_binEntropy.concaveOn.2
+    (show (0:ℝ)∈Set.Icc 0 1 by constructor <;> norm_num)
+    (show (3/4:ℝ)∈Set.Icc 0 1 by constructor <;> norm_num)
+    (show 0≤1-4*q/3 by linarith) (show 0≤4*q/3 by positivity)
+    (show (1-4*q/3)+(4*q/3)=1 by ring)
+  simp only [Real.binEntropy_zero,binEntropy_three_quarters,smul_eq_mul,mul_zero,zero_add] at hc
+  have he : 4*q/3*(3/4:ℝ)=q := by ring
+  rw [he] at hc
+  rw [entropyGap_eq]
+  linarith
+
+lemma log_dimension_from_binomial (N j D : ℕ) (hj : 0<j) (hjN : j<N)
+    (hfrac : (j:ℝ)≤(3/4:ℝ)*N) (hD : N.choose j≤D*2^j) :
+    entropyGap*(j:ℝ)-2*Real.log N-1 ≤ Real.log D := by
+  have hpos : 0<N.choose j := Nat.choose_pos hjN.le
+  have hd : 0<D := by nlinarith [Nat.zero_le (2^j)]
+  have hN : (0:ℝ)<N := by exact_mod_cast lt_trans hj hjN
+  have hm := Real.log_le_log (by exact_mod_cast hpos : (0:ℝ)<N.choose j)
+    (show (N.choose j:ℝ)≤D*2^j by exact_mod_cast hD)
+  rw [Real.log_mul (by positivity) (by positivity),Real.log_pow] at hm
+  have he := binEntropy_tail_lower ((j:ℝ)/N) (by positivity)
+    ((div_le_iff₀ hN).mpr hfrac)
+  have hel := mul_le_mul_of_nonneg_left he hN.le
+  have hbin := log_choose_lower_entropy N j hj hjN
+  have heq : (N:ℝ)*((j:ℝ)/N*Real.log 2+entropyGap*((j:ℝ)/N))=
+      (j:ℝ)*Real.log 2+entropyGap*j := by field_simp
+  rw [heq] at hel
+  linarith
+
+lemma balanced_dimension_coefficient_pos : 0<(3/2:ℝ)*Real.log 2-1 := by
+  linarith [Real.log_two_gt_d9]
+
+
+lemma log_choose_upper (N j : ℕ) (hN : 0<N) (hj : j≤N) :
+    Real.log (N.choose j:ℝ) ≤ (j:ℝ)*Real.log N-Real.log (j.factorial:ℝ) := by
+  have hh := Real.log_le_log (by exact_mod_cast Nat.choose_pos hj : (0:ℝ)<N.choose j)
+    (Nat.choose_le_pow_div j N : (N.choose j:ℝ)≤(N:ℝ)^j/(j.factorial:ℝ))
+  rw [Real.log_div (by positivity : (N:ℝ)^j≠0) (by positivity : (j.factorial:ℝ)≠0),Real.log_pow] at hh
+  exact hh
+
+lemma log_choose_lower (N j : ℕ) (hj : j≤N) :
+    (j:ℝ)*Real.log (N+1-j:ℕ)-Real.log (j.factorial:ℝ) ≤ Real.log (N.choose j:ℝ) := by
+  have hp : 0<N+1-j := by omega
+  have hh := Real.log_le_log (by positivity : (0:ℝ)<((N+1-j:ℕ):ℝ)^j/(j.factorial:ℝ))
+    (Nat.pow_le_choose j N : (((N+1-j:ℕ):ℝ)^j)/(j.factorial:ℝ)≤N.choose j)
+  rw [Real.log_div (by positivity : (((N+1-j:ℕ):ℝ)^j)≠0) (by positivity : (j.factorial:ℝ)≠0),Real.log_pow] at hh
+  exact hh
+
+lemma footprint_log_bound (N k j D m : ℕ) (hN : 0<N) (hk : j≤k) (hj : j≤N)
+    (hD : 0<D) (hfoot : D * k.choose j ≤ m * N.choose j) :
+    Real.log D-(j:ℝ)*(Real.log N-Real.log (k+1-j:ℕ)) ≤ Real.log m := by
+  have hkpos := Nat.choose_pos hk
+  have hprod : 0<D*k.choose j := Nat.mul_pos hD hkpos
+  have hm : 0 < m := by
+    by_contra h
+    have he : m=0 := by omega
+    simp only [he,zero_mul] at hfoot
+    omega
+  have hh := Real.log_le_log (by exact_mod_cast hprod : (0:ℝ)<D*k.choose j)
+    (show (D:ℝ)*k.choose j≤ m*N.choose j by exact_mod_cast hfoot)
+  rw [Real.log_mul (by positivity) (by exact_mod_cast hkpos.ne'),
+    Real.log_mul (by positivity) (by exact_mod_cast (Nat.choose_pos hj).ne')] at hh
+  have hlo := log_choose_lower k j hk
+  have hup := log_choose_upper N j hN hj
+  nlinarith
+
+lemma log_factorial_le_mul_log (j : ℕ) (_hj : 0<j) :
+    Real.log (j.factorial:ℝ)≤(j:ℝ)*Real.log j := by
+  have h := Real.log_le_log (by positivity : (0:ℝ)<j.factorial)
+    (show (j.factorial:ℝ)≤(j:ℝ)^j by exact_mod_cast Nat.factorial_le_pow j)
+  simpa only [Real.log_pow] using h
+
+lemma near_row_log_dimension (N j D : ℕ) (hj : 0<j) (hsmall : 2*j≤N)
+    (hD : N.choose j≤D*2^j) :
+    (j:ℝ)*(Real.log ((N:ℝ)/j)-2*Real.log 2) ≤ Real.log D := by
+  have hjN : j≤N := by omega
+  have hN : 0<N := by omega
+  have hNp : (0:ℝ)<N := by exact_mod_cast hN
+  have hjp : (0:ℝ)<j := by exact_mod_cast hj
+  have hn : 0<N+1-j := by omega
+  have hnreal : (N:ℝ)/2≤(N+1-j:ℕ) := by
+    rw [Nat.cast_sub (by omega : j≤N+1),Nat.cast_add,Nat.cast_one]
+    have hc : 2*(j:ℝ)≤N := by exact_mod_cast hsmall
+    linarith
+  have hDpos : 0<D := by
+    have hc := Nat.choose_pos hjN
+    by_contra h
+    have he : D=0 := by omega
+    simp only [he,zero_mul] at hD
+    omega
+  have hm := Real.log_le_log (by exact_mod_cast Nat.choose_pos hjN : (0:ℝ)<N.choose j)
+    (show (N.choose j:ℝ)≤D*2^j by exact_mod_cast hD)
+  rw [Real.log_mul (by positivity) (by positivity),Real.log_pow] at hm
+  have hl := Real.log_le_log (by positivity : (0:ℝ)<(N:ℝ)/2) hnreal
+  rw [Real.log_div hNp.ne' (by norm_num)] at hl
+  have hh := log_choose_lower N j hjN
+  have hf := log_factorial_le_mul_log j hj
+  rw [Real.log_div hNp.ne' hjp.ne']
+  nlinarith
+
+open Filter
+
+noncomputable def nearPower : ℝ := 1/2048
+noncomputable def nearFraction : ℝ := 1/4096
+noncomputable def tailPower : ℝ := 1535/1048576
+
+noncomputable def chosenTuple (N j : ℕ) : ℕ :=
+  ⌈(j:ℝ)*((N:ℝ)/j)^nearPower⌉₊
+
+lemma chosenTuple_bounds (N j : ℕ) (hj : 0<j) (hx : 1≤(N:ℝ)/j)
+    (hhalf : ((N:ℝ)/j)^(nearPower-1)≤1/2) :
+    j≤chosenTuple N j ∧ chosenTuple N j≤N ∧
+      (chosenTuple N j:ℝ)≤2*j*((N:ℝ)/j)^nearPower := by
+  let x : ℝ := (N:ℝ)/j
+  have hjp : (0:ℝ)<j := by exact_mod_cast hj
+  have hj1 : (1:ℝ)≤j := by exact_mod_cast hj
+  have hxp : 0<x := lt_of_lt_of_le zero_lt_one hx
+  have hpow : 1≤x^nearPower := Real.one_le_rpow hx (by norm_num [nearPower])
+  have hceil : (j:ℝ)*x^nearPower≤chosenTuple N j := Nat.le_ceil _
+  have hceil' : (chosenTuple N j:ℝ)<(j:ℝ)*x^nearPower+1 := Nat.ceil_lt_add_one (by positivity)
+  have he : (j:ℝ)*x=N := by dsimp [x]; field_simp
+  have hh : x^nearPower≤x/2 := by
+    rw [Real.rpow_sub hxp,Real.rpow_one] at hhalf
+    nlinarith [(div_le_iff₀ hxp).mp hhalf]
+  constructor
+  · exact_mod_cast (le_mul_of_one_le_right hjp.le hpow).trans hceil
+  constructor
+  · apply Nat.ceil_le.mpr
+    change (j:ℝ)*x^nearPower≤N
+    nlinarith
+  · change (chosenTuple N j:ℝ)≤2*(j:ℝ)*x^nearPower
+    nlinarith
+
+lemma chosenTuple_tail_lower (N j : ℕ) (hj : 0<j)
+    (hx : 2≤((N:ℝ)/j)^nearPower) :
+    (j:ℝ)*((N:ℝ)/j)^nearPower/2 ≤ (chosenTuple N j+1-j:ℕ) := by
+  have hjp : (0:ℝ)<j := by exact_mod_cast hj
+  have hc : (j:ℝ)*((N:ℝ)/j)^nearPower≤chosenTuple N j := Nat.le_ceil _
+  have hjk : j≤chosenTuple N j := by exact_mod_cast (by nlinarith : (j:ℝ)≤chosenTuple N j)
+  rw [Nat.cast_sub (by omega : j≤chosenTuple N j+1),Nat.cast_add,Nat.cast_one]
+  nlinarith
+
+lemma chosenTuple_log_footprint (N j D m : ℕ) (hj : 0<j)
+    (hN : 2*j≤N) (hD : N.choose j≤D*2^j)
+    (hx : 2≤((N:ℝ)/j)^nearPower)
+    (hlog : 8*Real.log 2/nearPower≤Real.log ((N:ℝ)/j))
+    (hfoot : D * (chosenTuple N j).choose j ≤ m * N.choose j) :
+    nearFraction*Real.log D≤Real.log m := by
+  let x : ℝ := (N:ℝ)/j
+  have hjp : (0:ℝ)<j := by exact_mod_cast hj
+  have hNp : (0:ℝ)<N := by exact_mod_cast (by omega : 0<N)
+  have hxp : 0<x := by dsimp [x]; positivity
+  have hdpos : 0<D := by
+    have hc := Nat.choose_pos (show j≤N by omega)
+    by_contra h
+    have hz : D=0 := by omega
+    simp only [hz,zero_mul] at hD
+    omega
+  have htail := chosenTuple_tail_lower N j hj hx
+  have hjk : j≤chosenTuple N j := by
+    have hc : (j:ℝ)*x^nearPower≤chosenTuple N j := Nat.le_ceil _
+    exact_mod_cast (by nlinarith : (j:ℝ)≤chosenTuple N j)
+  have hh := footprint_log_bound N (chosenTuple N j) j D m (by omega) hjk (by omega) hdpos hfoot
+  have htlog := Real.log_le_log (by positivity : (0:ℝ)<(j:ℝ)*x^nearPower/2) htail
+  rw [Real.log_div (by positivity) (by norm_num),Real.log_mul hjp.ne' (by positivity),Real.log_rpow hxp] at htlog
+  have hxlog : Real.log x=Real.log N-Real.log j := Real.log_div hNp.ne' hjp.ne'
+  have hdim := near_row_log_dimension N j D hj hN hD
+  change (j:ℝ)*(Real.log x-2*Real.log 2)≤Real.log D at hdim
+  have hh' : Real.log D-(j:ℝ)*((1-nearPower)*Real.log x+Real.log 2)≤Real.log m := by
+    have hm := mul_le_mul_of_nonneg_left htlog hjp.le
+    rw [hxlog] at hm
+    nlinarith
+  have hlg : (3-nearPower)*Real.log 2≤nearFraction*Real.log x := by
+    dsimp [nearPower,nearFraction] at hlog ⊢
+    have := Real.log_pos (by norm_num : (1:ℝ)<2)
+    linarith
+  have hlg' := mul_le_mul_of_nonneg_left hlg hjp.le
+  dsimp [nearPower,nearFraction] at hh' hlg' ⊢
+  nlinarith
+
+lemma near_row_log_uniform (N j D : ℕ) (hj : 0<j) (hN : 2*j≤N)
+    (hD : N.choose j≤D*2^j) (hlog : 4*Real.log 2+2≤Real.log ((N:ℝ)/j)) :
+    (1/2:ℝ)*Real.log N≤Real.log D ∧ (j:ℝ)≤Real.log D := by
+  have hjp : (0:ℝ)<j := by exact_mod_cast hj
+  have hj1 : (1:ℝ)≤j := by exact_mod_cast hj
+  have hNp : (0:ℝ)<N := by exact_mod_cast (by omega : 0<N)
+  have hd := near_row_log_dimension N j D hj hN hD
+  have hjlog := Real.log_le_sub_one_of_pos hjp
+  have hl2 : 0≤Real.log 2 := Real.log_nonneg (by norm_num)
+  have hi : 1≤Real.log ((N:ℝ)/j) := by linarith
+  have he : Real.log N=Real.log j+Real.log ((N:ℝ)/j) := by rw [Real.log_div hNp.ne' hjp.ne']; ring
+  constructor
+  · rw [he]
+    nlinarith
+  · nlinarith
+
+end Thorp.DimensionEstimates
+
+end ThorpNine.Harmonic
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/HighTail/AssignmentSlots.lean b/lean/OAI/Probability/ThorpRouting/HighTail/AssignmentSlots.lean
index cacd09e88d3cf4898071a82ff8b693e592ac59bd..8d3603897028a4f9979a321ebcf6b727e2b399ae 100644
--- a/lean/OAI/Probability/ThorpRouting/HighTail/AssignmentSlots.lean
+++ b/lean/OAI/Probability/ThorpRouting/HighTail/AssignmentSlots.lean
@@ -1,834 +1,834 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.HighTail.RotationMeeting
-
-namespace OAI
-
-namespace ThorpNine.HighTail
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-section CycleCounting
-variable {d A : ℕ}
-
-lemma assignmentOwner_eq_some {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) (x : Card d) (i : Fin A) :
-    assignmentOwner f x = some i ↔ x ∈ (f.val i).val := by
-  classical
-  unfold assignmentOwner
-  split_ifs with h
-  · constructor
-    · intro hi
-      have hx := Classical.choose_spec h
-      rw [Option.some.inj hi] at hx
-      exact hx
-    · intro hx
-      have he := (f.val (Classical.choose h)).eq_of_mem (f.val i) (Classical.choose_spec h) hx
-      exact congrArg some (f.property.1 he)
-  · constructor
-    · intro hi; contradiction
-    · intro hx; exact (h ⟨i,hx⟩).elim
-
-lemma assignmentOwner_fiber {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) (i : Fin A) : cycleFiber (assignmentOwner f) i = (f.val i).val := by
-  ext x
-  simp only [mem_cycleFiber,assignmentOwner_eq_some]
-
-noncomputable def assignmentToSlots {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) : CycleSlots p S r :=
-  ⟨assignmentOwner f,by
-    refine ⟨?_,?_,?_⟩
-    · intro i
-      rw [assignmentOwner_fiber]
-      exact f.property.2 i
-    · intro i
-      rw [assignmentOwner_fiber]
-      exact (f.val i).property.2
-    · intro i x hx
-      rw [assignmentOwner_fiber]
-      exact (f.val i).orbit_eq ((assignmentOwner_eq_some f x i).mp hx)⟩
-
-noncomputable def cycleSlotsEquivAssignments (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
-    CycleSlots p S r ≃ SlotAssignment p S r where
-  toFun := slotsToAssignment
-  invFun := assignmentToSlots
-  left_inv c := by
-    apply Subtype.ext
-    apply owner_ext
-    intro i
-    rw [assignmentToSlots,assignmentOwner_fiber]
-    rfl
-  right_inv f := by
-    apply Subtype.ext
-    funext i
-    apply Subtype.ext
-    exact assignmentOwner_fiber f i
-
-end CycleCounting
-
-section GroupedCounting
-variable {d A : ℕ}
-
-abbrev LengthClass (r : Fin A → ℕ) := {j : ℕ // j ∈ Finset.univ.image (fun i => r i + 1)}
-
-def slotClass (r : Fin A → ℕ) (i : Fin A) : LengthClass r :=
-  ⟨r i+1, Finset.mem_image.mpr ⟨i, Finset.mem_univ _, rfl⟩⟩
-
-abbrev SlotsOfLength (r : Fin A → ℕ) (j : LengthClass r) := {i : Fin A // slotClass r i = j}
-abbrev CyclesOfLength (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (j : ℕ) :=
-  {C : SelectedCycle p S // C.val.card = j}
-
-noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (j : ℕ) :
-    Fintype (CyclesOfLength p S j) := Fintype.ofFinite _
-
-abbrev GroupedAssignments (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
-  ∀ j : LengthClass r, SlotsOfLength r j ↪ CyclesOfLength p S j.val
-
-noncomputable def assignmentToGroups {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) : GroupedAssignments p S r := fun _j =>
-  { toFun := fun i => ⟨f.val i.val, (f.property.2 i.val).trans (congrArg Subtype.val i.property)⟩
-    inj' := fun _i _k h => Subtype.ext (f.property.1 (congrArg Subtype.val h)) }
-
-noncomputable def groupsEmbedding {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (F : GroupedAssignments p S r) : Fin A ↪ SelectedCycle p S :=
-  ((Equiv.sigmaFiberEquiv (slotClass r)).symm.toEmbedding.trans
-    (Function.Embedding.sigmaMap (Function.Embedding.refl _) F)).trans
-    ((Function.Embedding.sigmaMap (Function.Embedding.subtype _) (fun _ => Function.Embedding.refl _)).trans
-      (Equiv.sigmaFiberEquiv (fun C : SelectedCycle p S => C.val.card)).toEmbedding)
-
-noncomputable def groupsToAssignment {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (F : GroupedAssignments p S r) : SlotAssignment p S r :=
-  ⟨groupsEmbedding F, (groupsEmbedding F).injective,
-    fun i => (F (slotClass r i) ⟨i,rfl⟩).property⟩
-
-noncomputable def assignmentEquivGroups (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
-    SlotAssignment p S r ≃ GroupedAssignments p S r where
-  toFun := assignmentToGroups
-  invFun := groupsToAssignment
-  left_inv f := by
-    apply Subtype.ext
-    rfl
-  right_inv F := by
-    funext j
-    apply Function.Embedding.ext
-    intro i
-    rcases i with ⟨i,hi⟩
-    subst j
-    rfl
-
-lemma card_cycleSlots_factorial (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
-    Fintype.card (CycleSlots p S r) =
-      ∏ j : LengthClass r, (Fintype.card (CyclesOfLength p S j.val)).descFactorial
-        (Fintype.card (SlotsOfLength r j)) := by
-  classical
-  rw [Fintype.card_congr (cycleSlotsEquivAssignments p S r),
-    Fintype.card_congr (assignmentEquivGroups p S r), Fintype.card_pi]
-  simp only [Fintype.card_embedding_eq]
-
-lemma butterfly_factorial_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-       (∏ j : LengthClass r,
-         (Fintype.card (CyclesOfLength (butterflyPerm d (decodeButterfly d ω)) S j.val)).descFactorial
-           (Fintype.card (SlotsOfLength r j)) : ℕ)) ≤
-      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-        ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
-  simpa only [←card_cycleSlots_factorial] using butterfly_cycleSlots_moment S r hr
-
-end GroupedCounting
-
-section AllLengths
-variable {d A : ℕ}
-
-noncomputable def assignmentRelabel (p : Equiv.Perm (Card d)) (S : Finset (Card d))
-    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
-    SlotAssignment p S r ≃ SlotAssignment p S (r ∘ e) where
-  toFun f := ⟨f.val ∘ e, f.property.1.comp e.injective, fun i => f.property.2 (e i)⟩
-  invFun f := ⟨f.val ∘ e.symm, f.property.1.comp e.symm.injective, fun i => by
-    simpa only [Function.comp_apply,Equiv.apply_symm_apply] using f.property.2 (e.symm i)⟩
-  left_inv f := by apply Subtype.ext; funext i; simp
-  right_inv f := by apply Subtype.ext; funext i; simp
-
-lemma card_cycleSlots_relabel (p : Equiv.Perm (Card d)) (S : Finset (Card d))
-    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
-    Fintype.card (CycleSlots p S r) = Fintype.card (CycleSlots p S (r ∘ e)) := by
-  exact Fintype.card_congr ((cycleSlotsEquivAssignments p S r).trans
-    ((assignmentRelabel p S r e).trans (cycleSlotsEquivAssignments p S (r ∘ e)).symm))
-
-lemma butterfly_cycleSlots_moment_all (S : Finset (Card d)) (r : Fin A → ℕ) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) ≤
-        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
-  have h := butterfly_cycleSlots_moment S (r ∘ Tuple.sort r) (Tuple.monotone_sort r)
-  simp_rw [←card_cycleSlots_relabel] at h
-  have hp := Equiv.prod_comp (Tuple.sort r) (fun i : Fin A => (1 / Real.sqrt (r i+1) : ℝ))
-  simpa only [Function.comp_apply,hp] using h
-
-abbrev ShortCycle (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :=
-  {C : SelectedCycle p S // C.val.card ≤ J}
-
-noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
-    Fintype (ShortCycle p S J) := Fintype.ofFinite _
-
-noncomputable def shortCycleLength {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {J : ℕ}
-    (C : ShortCycle p S J) : Fin J :=
-  ⟨C.val.val.card-1, by have := C.val.nonempty.card_pos; have := C.property; omega⟩
-
-lemma shortCycleLength_add_one {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {J : ℕ}
-    (C : ShortCycle p S J) : (shortCycleLength C : ℕ)+1 = C.val.val.card := by
-  have := C.val.nonempty.card_pos
-  simp only [shortCycleLength]
-  omega
-
-noncomputable def shortEmbeddingEquiv (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
-    (Fin A ↪ ShortCycle p S J) ≃
-      Σ r : Fin A → Fin J, SlotAssignment p S (fun i => (r i : ℕ)) where
-  toFun f := ⟨fun i => shortCycleLength (f i),
-    ⟨fun i => (f i).val, fun i j h => f.injective (Subtype.ext h),
-      fun i => (shortCycleLength_add_one (f i)).symm⟩⟩
-  invFun c :=
-    { toFun := fun i => ⟨c.2.val i, by rw [c.2.property.2]; exact (c.1 i).isLt⟩
-      inj' := fun _i _j h => c.2.property.1 (congrArg Subtype.val h) }
-  left_inv f := by
-    apply Function.Embedding.ext
-    intro i
-    rfl
-  right_inv c := by
-    rcases c with ⟨r,f⟩
-    have he : (fun i => shortCycleLength (⟨f.val i, by rw [f.property.2]; exact (r i).isLt⟩ : ShortCycle p S J)) = r := by
-      funext i
-      apply Fin.ext
-      simp only [shortCycleLength,f.property.2,Nat.add_sub_cancel]
-    apply Sigma.ext he
-    apply (Subtype.heq_iff_coe_eq (by
-      intro x
-      change (Function.Injective x ∧ ∀ i, (x i).val.card = (f.val i).val.card - 1 + 1) ↔ _
-      simp only [f.property.2,Nat.add_sub_cancel])).mpr
-    rfl
-
-lemma shortCycle_factorial_eq (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
-    (Fintype.card (ShortCycle p S J)).descFactorial A =
-      ∑ r : Fin A → Fin J, Fintype.card (CycleSlots p S (fun i => (r i : ℕ))) := by
-  classical
-  have hc := Fintype.card_embedding_eq (α := Fin A) (β := ShortCycle p S J)
-  simp only [Fintype.card_fin] at hc
-  rw [←hc, Fintype.card_congr (shortEmbeddingEquiv p S J),Fintype.card_sigma]
-  apply Finset.sum_congr rfl
-  intro r _
-  exact (Fintype.card_congr (cycleSlotsEquivAssignments p S (fun i => (r i : ℕ)))).symm
-
-end AllLengths
-
-lemma factorial_mgf {Ω : Type*} [Fintype Ω] (F : Ω → ℕ) (k : ℕ) (Λ q : ℝ)
-    (hF : ∀ ω, F ω ≤ k) (hΛ : 0 ≤ Λ) (hq : 1 ≤ q)
-    (hm : ∀ a, finiteMean (fun ω => ((F ω).descFactorial a : ℝ)) ≤ Λ^a) :
-    finiteMean (fun ω => q^(F ω)) ≤ Real.exp ((q-1)*Λ) := by
-  have hc (n a : ℕ) : (n.choose a : ℝ) = (n.descFactorial a : ℝ) / (a.factorial : ℝ) := by
-    rw [Nat.descFactorial_eq_factorial_mul_choose,Nat.cast_mul]
-    exact (mul_div_cancel_left₀ _ (by exact_mod_cast Nat.factorial_ne_zero a)).symm
-  have hex (ω : Ω) : q^(F ω) = ∑ a ∈ Finset.range (k+1),
-      ((F ω).descFactorial a : ℝ) * ((q-1)^a / (a.factorial : ℝ)) := by
-    calc
-      _ = ((q-1)+1)^(F ω) := by congr 1; ring
-      _ = ∑ a ∈ Finset.range (F ω+1), (q-1)^a * (F ω).choose a := by
-        rw [add_pow]
-        simp only [one_pow,mul_one]
-      _ = ∑ a ∈ Finset.range (k+1), (q-1)^a * (F ω).choose a := by
-        apply Finset.sum_subset (Finset.range_mono (Nat.add_le_add_right (hF ω) _))
-        intro a _ ha
-        have hn : F ω < a := by have := (Finset.mem_range.not.mp ha); omega
-        rw [Nat.choose_eq_zero_of_lt hn,Nat.cast_zero,mul_zero]
-      _ = _ := by
-        apply Finset.sum_congr rfl
-        intro a _
-        rw [hc]
-        ring
-  simp_rw [hex,finiteMean_sum,finiteMean_mul_const]
-  calc
-    _ ≤ ∑ a ∈ Finset.range (k+1), Λ^a * ((q-1)^a / (a.factorial : ℝ)) := by
-      apply Finset.sum_le_sum
-      intro a _
-      exact mul_le_mul_of_nonneg_right (hm a) (by positivity)
-    _ = ∑ a ∈ Finset.range (k+1), ((q-1)*Λ)^a / (a.factorial : ℝ) := by
-      apply Finset.sum_congr rfl
-      intro a _
-      rw [mul_pow]
-      ring
-    _ ≤ _ := Real.sum_le_exp_of_nonneg (mul_nonneg (sub_nonneg.mpr hq) hΛ) _
-
-lemma inv_sqrt_step (j : ℕ) :
-    1 / Real.sqrt (j+1) ≤ 2*(Real.sqrt (j+1)-Real.sqrt j) := by
-  have hp : 0 < Real.sqrt (j+1) := Real.sqrt_pos.mpr (by positivity)
-  apply (div_le_iff₀ hp).mpr
-  have h1 := Real.sq_sqrt (show (0:ℝ) ≤ j+1 by positivity)
-  have h0 := Real.sq_sqrt (show (0:ℝ) ≤ j by positivity)
-  nlinarith [sq_nonneg (Real.sqrt (j+1)-Real.sqrt j)]
-
-lemma sum_inv_sqrt (J : ℕ) :
-    (∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1)) ≤ 2*Real.sqrt J := by
-  have h (n : ℕ) : (∑ j ∈ Finset.range n, 1 / Real.sqrt (j+1)) ≤ 2*Real.sqrt n := by
-    induction n with
-    | zero => simp
-    | succ n ih =>
-      rw [Finset.sum_range_succ]
-      have hs := inv_sqrt_step n
-      push_cast
-      linarith
-  rw [Fin.sum_univ_eq_sum_range (fun j : ℕ => (1 / Real.sqrt (j+1) : ℝ)) J]
-  exact h J
-section CycleMGF
-variable {d : ℕ}
-
-lemma butterfly_shortCycle_moment (S : Finset (Card d)) (J A : ℕ) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      ((Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J)).descFactorial A : ℝ)) ≤
-      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))^A := by
-  classical
-  simp_rw [shortCycle_factorial_eq,Nat.cast_sum,finiteMean_sum]
-  calc
-    _ ≤ ∑ r : Fin A → Fin J,
-        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-          ∏ i : Fin A, (1 / Real.sqrt ((r i : ℕ)+1)) := by
-      apply Finset.sum_le_sum
-      intro r _
-      exact butterfly_cycleSlots_moment_all S (fun i => (r i : ℕ))
-    _ = _ := by
-      rw [←Finset.mul_sum,mul_pow]
-      congr 1
-      rw [←Fintype.prod_sum (fun (_ : Fin A) (j : Fin J) => (1 / Real.sqrt ((j:ℕ)+1) : ℝ))]
-      simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fin]
-
-noncomputable def cycleMass (p : Equiv.Perm (Card d)) (S : Finset (Card d)) : ℕ :=
-  ∑ C : SelectedCycle p S, C.val.card
-
-lemma cycleMass_le (p : Equiv.Perm (Card d)) (S : Finset (Card d)) : cycleMass p S ≤ S.card := by
-  classical
-  let f : (Σ C : SelectedCycle p S, {x : Card d // x ∈ C.val}) → S :=
-    fun c => ⟨c.2.val,c.1.property.2 c.2.property⟩
-  have hi : Function.Injective f := by
-    intro c e h
-    have hx : c.2.val = e.2.val := congrArg Subtype.val h
-    have hC : c.1 = e.1 := c.1.eq_of_mem e.1 c.2.property (hx ▸ e.2.property)
-    rcases c with ⟨c,x⟩
-    rcases e with ⟨e,y⟩
-    dsimp only at hC
-    subst e
-    have he : x=y := Subtype.ext hx
-    cases he
-    rfl
-  have hh := Fintype.card_le_of_injective f hi
-  simpa only [Fintype.card_sigma,Fintype.card_coe,cycleMass] using hh
-
-lemma shortCycle_card_le (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
-    Fintype.card (ShortCycle p S J) ≤ S.card := by
-  classical
-  calc
-    _ ≤ Fintype.card (SelectedCycle p S) := Fintype.card_subtype_le _
-    _ = ∑ _ : SelectedCycle p S, 1 := by simp
-    _ ≤ cycleMass p S := Finset.sum_le_sum (fun C _ => C.nonempty.card_pos)
-    _ ≤ _ := cycleMass_le p S
-
-lemma butterfly_shortCycle_mgf (S : Finset (Card d)) (J : ℕ) (q : ℝ) (hq : 1 ≤ q) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      q^(Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J))) ≤
-      Real.exp ((q-1)*((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))) := by
-  apply factorial_mgf _ S.card _ q (fun ω => shortCycle_card_le _ _ _)
-  · positivity
-  · exact hq
-  · exact butterfly_shortCycle_moment S J
-
-end CycleMGF
-
-section FullCycleMGF
-variable {d : ℕ}
-
-abbrev LongCycle (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :=
-  {C : SelectedCycle p S // ¬ C.val.card ≤ J}
-noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
-    Fintype (LongCycle p S J) := Fintype.ofFinite _
-
-lemma selectedCycle_card_split (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
-    Fintype.card (SelectedCycle p S) =
-      Fintype.card (ShortCycle p S J) + Fintype.card (LongCycle p S J) := by
-  classical
-  rw [←Fintype.card_congr (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J)),Fintype.card_sum]
-
-lemma longCycle_card_bound (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
-    J * Fintype.card (LongCycle p S J) ≤ S.card := by
-  classical
-  have hs := Equiv.sum_comp (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J))
-    (fun C : SelectedCycle p S => C.val.card)
-  rw [Fintype.sum_sum_type] at hs
-  have h0 : J * Fintype.card (LongCycle p S J) ≤
-      ∑ C : LongCycle p S J, C.val.val.card := by
-    calc
-      _ = ∑ _C : LongCycle p S J, J := by simp [mul_comm]
-      _ ≤ _ := Finset.sum_le_sum (fun C _ => Nat.le_of_lt (Nat.lt_of_not_ge C.property))
-  have hh := cycleMass_le p S
-  unfold cycleMass at hh
-  rw [←hs] at hh
-  exact h0.trans ((Nat.le_add_left _ _).trans hh)
-
-lemma finiteMean_pos {Ω : Type*} [Fintype Ω] [Nonempty Ω] {f : Ω → ℝ}
-    (h : ∀ ω, 0 < f ω) : 0 < finiteMean f := by
-  apply div_pos
-  · exact Finset.sum_pos (fun ω _ => h ω) Finset.univ_nonempty
-  · exact Nat.cast_pos.mpr Fintype.card_pos
-
-lemma longCycle_pow_bound (p : Equiv.Perm (Card d)) (S : Finset (Card d))
-    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    q^(Fintype.card (LongCycle p S J)) ≤ Real.exp ((S.card : ℝ)/J * Real.log q) := by
-  have hq0 : 0 < q := lt_of_lt_of_le zero_lt_one hq
-  have hc : (Fintype.card (LongCycle p S J) : ℝ) ≤ (S.card : ℝ)/J := by
-    apply (le_div_iff₀ (by exact_mod_cast hJ)).mpr
-    have hh := longCycle_card_bound p S J
-    exact_mod_cast (by simpa only [mul_comm] using hh)
-  calc
-    _ = Real.exp ((Fintype.card (LongCycle p S J) : ℝ)*Real.log q) := by
-      rw [Real.exp_nat_mul,Real.exp_log hq0]
-    _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right hc (Real.log_nonneg hq))
-
-lemma butterfly_cycle_log_mgf (S : Finset (Card d)) (J : ℕ) (hJ : 0 < J)
-    (q : ℝ) (hq : 1 ≤ q) :
-    Real.log (finiteMean (fun ω : SwitchIndex d → Bool =>
-      q^(Fintype.card (SelectedCycle (butterflyPerm d (decodeButterfly d ω)) S)))) ≤
-      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
-  let lam : ℝ := (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1)
-  let c : ℝ := (S.card : ℝ)/J * Real.log q
-  have hnum : (q-1)*lam ≤ 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
-    have hsum := sum_inv_sqrt J
-    have hp : 0 ≤ (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) := by positivity
-    have h0 : 0 ≤ 2*Real.sqrt J := by positivity
-    have hh := mul_le_mul_of_nonneg_left hsum hp
-    have hq1 : 0 ≤ q-1 := sub_nonneg.mpr hq
-    calc
-      _ ≤ (q-1)*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
-        mul_le_mul_of_nonneg_left hh hq1
-      _ ≤ q*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
-        mul_le_mul_of_nonneg_right (by linarith) (mul_nonneg hp h0)
-      _ = _ := by ring
-  have hm : finiteMean (fun ω : SwitchIndex d → Bool =>
-      q^(Fintype.card (SelectedCycle (butterflyPerm d (decodeButterfly d ω)) S))) ≤
-      Real.exp (c+(q-1)*lam) := by
-    calc
-      _ ≤ finiteMean (fun ω : SwitchIndex d → Bool =>
-          q^(Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J))*Real.exp c) := by
-        apply finiteMean_mono
-        intro ω
-        rw [selectedCycle_card_split _ _ J,pow_add]
-        exact mul_le_mul_of_nonneg_left (longCycle_pow_bound _ S J hJ q hq) (by positivity)
-      _ = finiteMean (fun ω : SwitchIndex d → Bool =>
-          q^(Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J))) * Real.exp c :=
-        finiteMean_mul_const _ _
-      _ ≤ Real.exp ((q-1)*lam) * Real.exp c :=
-        mul_le_mul_of_nonneg_right (butterfly_shortCycle_mgf S J q hq) (Real.exp_nonneg _)
-      _ = _ := by rw [←Real.exp_add,add_comm]
-  have hpos : 0 < finiteMean (fun ω : SwitchIndex d → Bool =>
-      q^(Fintype.card (SelectedCycle (butterflyPerm d (decodeButterfly d ω)) S))) := by
-    apply finiteMean_pos
-    intro ω
-    positivity
-  have hlog := Real.log_le_log hpos hm
-  rw [Real.log_exp] at hlog
-  dsimp [c] at hlog
-  linarith
-
-end FullCycleMGF
-
-structure RoutingNetwork (α ι : Type*) [Fintype α] [DecidableEq α]
-    [Fintype ι] [DecidableEq ι] (d : ℕ) where
-  perm : (ι → Bool) → Equiv.Perm α
-  path : (ι → Bool) → α → Finset ι
-  users : (ι → Bool) → ι → α × α
-  users_ne : ∀ ω i, (users ω i).1 ≠ (users ω i).2
-  mem_path : ∀ ω x i, i ∈ path ω x ↔ (users ω i).1 = x ∨ (users ω i).2 = x
-  card_path : ∀ ω x, (path ω x).card = d
-  entropy : ∀ ω (S : Finset α),
-    (∑ i : ι, if (users ω i).1 ∈ S ∧ (users ω i).2 ∈ S then (1:ℝ) else 0) ≤
-      (S.card : ℝ) * Real.log S.card / (2 * Real.log 2)
-  follow : α → (α → QueryTree ι) → QueryTree ι
-  close : α → α → QueryTree ι → QueryTree ι
-  follow_succeeds : ∀ x k ω, (follow x k).succeeds ω = (k (perm ω x)).succeeds ω
-  follow_raw_charge : ∀ x k ω, (follow x k).charge ω = (k (perm ω x)).charge ω
-  follow_stale_charge : ∀ x k R ω,
-    (follow x k).staleCharge R ω = (k (perm ω x)).staleCharge (R ∪ path ω x) ω
-  close_succeeds : ∀ x y k ω,
-    (close x y k).succeeds ω = if perm ω x = y then k.succeeds ω else false
-  close_raw_charge : ∀ x y k ω, perm ω x = y → (close x y k).charge ω = k.charge ω + d
-  close_stale_charge : ∀ x y k R ω, perm ω x = y →
-    (close x y k).staleCharge R ω = k.staleCharge (R ∪ path ω x) ω + (path ω x ∩ R).card
-
-namespace RoutingNetwork
-variable {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι] [DecidableEq ι]
-  {d : ℕ} (R : RoutingNetwork α ι d)
-
-lemma card_switches_using (ω : ι → Bool) (x : α) :
-    (Finset.univ.filter (fun i : ι =>
-      (R.users ω i).1 = x ∨ (R.users ω i).2 = x)).card = d := by
-  classical
-  have he : Finset.univ.filter (fun i : ι =>
-      (R.users ω i).1 = x ∨ (R.users ω i).2 = x) = R.path ω x := by
-    ext i
-    simp only [Finset.mem_filter,Finset.mem_univ,true_and,R.mem_path]
-  rw [he,R.card_path]
-lemma switch_incidence_sum (b : ι → Bool) (w : α → ℝ) :
-    ∑ i : ι, (w (R.users b i).1 + w (R.users b i).2) =
-      (d : ℝ) * ∑ x : α, w x := by
-  classical
-  calc
-    _ = ∑ i : ι, ∑ x : α,
-        if (R.users b i).1 = x ∨ (R.users b i).2 = x then w x else 0 := by
-      apply Finset.sum_congr rfl
-      intro i _
-      rw [Finset.sum_ite]
-      simp only [Finset.sum_const_zero, add_zero]
-      have he : Finset.univ.filter (fun x : α =>
-          (R.users b i).1 = x ∨ (R.users b i).2 = x) =
-          {(R.users b i).1, (R.users b i).2} := by
-        ext x; simp [eq_comm]
-      rw [he, Finset.sum_pair (R.users_ne b i)]
-    _ = ∑ x : α, ∑ i : ι,
-        if (R.users b i).1 = x ∨ (R.users b i).2 = x then w x else 0 :=
-      Finset.sum_comm
-    _ = ∑ x : α, (d : ℝ) * w x := by
-      apply Finset.sum_congr rfl
-      intro x _
-      rw [Finset.sum_ite]
-      simp only [Finset.sum_const_zero, add_zero, Finset.sum_const, nsmul_eq_mul,
-        card_switches_using]
-    _ = _ := by rw [Finset.mul_sum]
-
-lemma weighted_switch_min_bound (b : ι → Bool) (w : α → ℝ) :
-    ∑ i : ι, min (w (R.users b i).1) (w (R.users b i).2) ≤
-      (d : ℝ) / 2 * ∑ x : α, w x := by
-  have h : 2 * (∑ i : ι,
-      min (w (R.users b i).1) (w (R.users b i).2)) ≤
-      ∑ i : ι, (w (R.users b i).1 + w (R.users b i).2) := by
-    rw [Finset.mul_sum]
-    apply Finset.sum_le_sum
-    intro i _
-    linarith [min_le_left (w (R.users b i).1) (w (R.users b i).2),
-      min_le_right (w (R.users b i).1) (w (R.users b i).2)]
-  rw [switch_incidence_sum] at h
-  linarith
-section Rotation
-variable {A : ℕ}
-
-def cycleFiber (o : α → Option (Fin A)) (i : Fin A) : Finset (α) := by
-  classical
-  exact Finset.univ.filter (fun x => o x = some i)
-
-omit [DecidableEq α] in
-@[simp] lemma mem_cycleFiber (o : α → Option (Fin A)) (i : Fin A) (x : α) :
-    x ∈ cycleFiber o i ↔ o x = some i := by
-  classical
-  simp [cycleFiber]
-
-abbrev Terminal (o : α → Option (Fin A)) := ∀ i : Fin A, {x : α // o x = some i}
-
-noncomputable def slotWeight (o : α → Option (Fin A)) (i : Fin A) : ℝ :=
-  1 / (cycleFiber o i).card
-
-noncomputable def pathWeight (o : α → Option (Fin A)) (x : α) : ℝ :=
-  match o x with
-  | none => 0
-  | some i => slotWeight o i
-
-noncomputable def withinWeight (o : α → Option (Fin A)) (u v : α) : ℝ :=
-  match o u, o v with
-  | some i, some j => if i = j then slotWeight o i else 0
-  | _, _ => 0
-
-noncomputable def rotationMeeting (o : α → Option (Fin A)) (z : Terminal o)
-    (u v : α) : ℝ :=
-  match o u, o v with
-  | some i, some j => if i < j then (if (z j).val = v then 1 else 0)
-      else if j < i then (if (z i).val = u then 1 else 0)
-      else (if (z i).val = u then 1 else 0) + (if (z i).val = v then 1 else 0)
-  | _, _ => 0
-
-omit [DecidableEq α] in
-lemma card_cycleFiber (o : α → Option (Fin A)) (i : Fin A) :
-    Fintype.card {x : α // o x = some i} = (cycleFiber o i).card := by
-  classical
-  simp [cycleFiber,Fintype.card_subtype]
-
-lemma terminal_mean {o : α → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
-    (i : Fin A) (x : α) (hx : o x = some i) :
-    finiteMean (fun z : Terminal o => if (z i).val = x then (1:ℝ) else 0) = slotWeight o i := by
-  classical
-  have hev := finiteMean_pi_eval (A := fun i : Fin A => {x : α // o x = some i})
-    i (fun z => if z.val = x then (1:ℝ) else 0)
-  rw [hev]
-  have he : (fun z : {x : α // o x = some i} => if z.val = x then (1:ℝ) else 0) =
-      (fun z => if z = ⟨x,hx⟩ then 1 else 0) := by
-    funext z
-    simp only [Subtype.ext_iff]
-  rw [he,finiteMean_eq_indicator,card_cycleFiber,slotWeight]
-
-omit [DecidableEq α] in
-lemma slotWeight_antitone (o : α → Option (Fin A))
-    (hn : ∀ i, (cycleFiber o i).Nonempty)
-    (hm : Monotone (fun i => (cycleFiber o i).card)) : Antitone (slotWeight o) := by
-  intro i j hij
-  unfold slotWeight
-  apply one_div_le_one_div_of_le
-  · exact_mod_cast Finset.card_pos.mpr (hn i)
-  · exact_mod_cast hm hij
-
-lemma rotationMeeting_mean {o : α → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
-    (hm : Monotone (fun i => (cycleFiber o i).card)) (u v : α) :
-    finiteMean (fun z : Terminal o => rotationMeeting o z u v) =
-      min (pathWeight o u) (pathWeight o v) + withinWeight o u v := by
-  classical
-  have hn : ∀ i, (cycleFiber o i).Nonempty := by
-    intro i
-    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : α // o x = some i})
-    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
-  have hw := slotWeight_antitone o hn hm
-  cases hu : o u with
-  | none =>
-    cases hv : o v with
-    | none => simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean]
-    | some j =>
-      have hj : 0 ≤ slotWeight o j := by unfold slotWeight; positivity
-      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_left hj]
-  | some i =>
-    cases hv : o v with
-    | none =>
-      have hi : 0 ≤ slotWeight o i := by unfold slotWeight; positivity
-      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_right hi]
-    | some j =>
-      simp only [rotationMeeting,pathWeight,withinWeight,hu,hv]
-      rcases lt_trichotomy i j with hij | rfl | hji
-      · simp only [hij,↓reduceIte,ne_of_lt hij]
-        rw [terminal_mean j v hv,min_eq_right (hw (le_of_lt hij)),add_zero]
-      · simp only [lt_self_iff_false,↓reduceIte,min_self]
-        rw [finiteMean_add,terminal_mean i u hu,terminal_mean i v hv]
-      · simp only [not_lt_of_ge (le_of_lt hji),hji,↓reduceIte,ne_of_gt hji]
-        rw [terminal_mean i u hu,min_eq_left (hw (le_of_lt hji)),add_zero]
-
-omit [DecidableEq α] in
-lemma pathWeight_sum (o : α → Option (Fin A)) (hn : ∀ i, (cycleFiber o i).Nonempty) :
-    ∑ x : α, pathWeight o x = (A : ℝ) := by
-  classical
-  have he (x : α) : pathWeight o x =
-      ∑ i : Fin A, if o x = some i then slotWeight o i else 0 := by
-    cases hx : o x with
-    | none => simp [pathWeight,hx]
-    | some j => simp [pathWeight,hx]
-  simp only [he]
-  rw [Finset.sum_comm]
-  have hf (i : Fin A) : (∑ x : α, if o x = some i then slotWeight o i else 0) = 1 := by
-    rw [← Finset.sum_filter]
-    change (∑ _x ∈ cycleFiber o i, slotWeight o i) = 1
-    simp only [Finset.sum_const,nsmul_eq_mul,slotWeight]
-    exact mul_one_div_cancel (by exact_mod_cast (Finset.card_pos.mpr (hn i)).ne')
-  simp only [hf,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,mul_one]
-
-lemma withinWeight_sum_bound (o : α → Option (Fin A))
-    (hn : ∀ i, (cycleFiber o i).Nonempty) (b : ι → Bool) :
-    (∑ s : ι, withinWeight o (R.users b s).1 (R.users b s).2) ≤
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
-  classical
-  have he (u v : α) : withinWeight o u v =
-      ∑ i : Fin A, (if u ∈ cycleFiber o i ∧ v ∈ cycleFiber o i then (1:ℝ) else 0) * slotWeight o i := by
-    simp only [mem_cycleFiber]
-    cases hu : o u <;> cases hv : o v <;> simp [withinWeight,hu,hv,ite_and,eq_comm]
-  simp only [he]
-  rw [Finset.sum_comm]
-  apply Finset.sum_le_sum
-  intro i _
-  rw [← Finset.sum_mul]
-  have hi := R.entropy b (cycleFiber o i)
-  have hpos : (0:ℝ) < (cycleFiber o i).card := by exact_mod_cast Finset.card_pos.mpr (hn i)
-  have hmul := mul_le_mul_of_nonneg_right hi (show 0 ≤ slotWeight o i by unfold slotWeight; positivity)
-  calc
-    _ ≤ ((cycleFiber o i).card : ℝ) * Real.log (cycleFiber o i).card /
-        (2 * Real.log 2) * slotWeight o i := hmul
-    _ = _ := by unfold slotWeight; field_simp
-
-noncomputable def rotationCharge (o : α → Option (Fin A)) (b : ι → Bool)
-    (z : Terminal o) : ℝ :=
-  ∑ s : ι, rotationMeeting o z (R.users b s).1 (R.users b s).2
-
-lemma rotationCharge_mean_bound {o : α → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
-    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : ι → Bool) :
-    finiteMean (rotationCharge R o b) ≤ (A:ℝ) * d / 2 +
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
-  classical
-  have hn : ∀ i, (cycleFiber o i).Nonempty := by
-    intro i
-    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : α // o x = some i})
-    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
-  unfold rotationCharge
-  rw [finiteMean_sum]
-  simp only [rotationMeeting_mean hm,Finset.sum_add_distrib]
-  have h₁ := weighted_switch_min_bound R b (pathWeight o)
-  rw [pathWeight_sum o hn] at h₁
-  have h₂ := withinWeight_sum_bound R o hn b
-  linarith
-
-lemma rotation_exp_bound {o : α → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
-    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : ι → Bool) :
-    Real.exp (-Real.log 2 * ((A:ℝ)*d/2 +
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2))) ≤
-      finiteMean (fun z : Terminal o => Real.exp (-Real.log 2 * rotationCharge R o b z)) := by
-  have hb := rotationCharge_mean_bound R hm b
-  have hlog : 0 ≤ Real.log 2 := Real.log_nonneg (by norm_num)
-  have hm' : -Real.log 2 * ((A:ℝ)*d/2 +
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2)) ≤
-      finiteMean (fun z : Terminal o => -Real.log 2 * rotationCharge R o b z) := by
-    have he := finiteMean_mul_const (rotationCharge R o b) (-Real.log 2)
-    simp only [mul_comm] at he
-    rw [he]
-    exact mul_le_mul_of_nonpos_left hb (neg_nonpos.mpr hlog)
-  exact (Real.exp_le_exp.mpr hm').trans (exp_finiteMean_le _)
-
-end Rotation
-
-namespace QueryTree
-open Thorp.QueryTree
-
-noncomputable def seekCycle (R : RoutingNetwork α ι d) (S : Finset (α)) (a : α) :
-    ℕ → α → Finset (α) → (Finset (α) → QueryTree (ι)) →
-      QueryTree (ι)
-  | 0, x, V, k => if x ∈ S ∧ x ∉ V then R.close x a (k (insert x V)) else reject
-  | r + 1, x, V, k => if x ∈ S ∧ x ∉ V then R.follow x
-      (fun y => seekCycle R S a r y (insert x V) k) else reject
-
-lemma seekCycle_raw_charge (S : Finset (α)) (a : α) (r : ℕ)
-    (x : α) (V : Finset (α)) (k : Finset (α) → QueryTree (ι))
-    (ω : ι → Bool) (K : ℕ)
-    (hK : ∀ V, (k V).succeeds ω = true → (k V).charge ω = K)
-    (hs : (seekCycle R S a r x V k).succeeds ω = true) :
-    (seekCycle R S a r x V k).charge ω = K + d := by
-  induction r generalizing x V with
-  | zero =>
-    by_cases hv : x ∈ S ∧ x ∉ V
-    · rw [seekCycle,ite_eq_left hv,R.close_succeeds] at hs
-      split_ifs at hs with he
-      · rw [seekCycle,ite_eq_left hv,R.close_raw_charge _ _ _ _ he,hK _ hs]
-    · simp [seekCycle,hv,Thorp.QueryTree.succeeds] at hs
-  | succ r ih =>
-    by_cases hv : x ∈ S ∧ x ∉ V
-    · rw [seekCycle,ite_eq_left hv,R.follow_succeeds] at hs
-      rw [seekCycle,ite_eq_left hv,R.follow_raw_charge]
-      exact ih _ _ hs
-    · simp [seekCycle,hv,Thorp.QueryTree.succeeds] at hs
-
-noncomputable def seekCycles (R : RoutingNetwork α ι d) (S : Finset (α)) :
-    List (α × ℕ) → Finset (α) → QueryTree (ι)
-  | [], _ => accept
-  | (a,r) :: L, V => seekCycle R S a r a V (fun V => seekCycles R S L V)
-
-lemma seekCycles_raw_charge (S : Finset (α)) (L : List (α × ℕ))
-    (V : Finset (α)) (ω : ι → Bool)
-    (hs : (seekCycles R S L V).succeeds ω = true) :
-    (seekCycles R S L V).charge ω = L.length * d := by
-  induction L generalizing V with
-  | nil => simp [seekCycles,charge]
-  | cons ar L ih =>
-    obtain ⟨a,r⟩ := ar
-    rw [seekCycles]
-    rw [seekCycle_raw_charge R S a r a V (fun V => seekCycles R S L V) ω (L.length*d)
-      (fun V hs => ih V hs) hs]
-    simp [Nat.add_mul]
-
-lemma cycles_weighted_success (S : Finset (α)) (L : List (α × ℕ)) :
-    finiteMean (fun ω : ι → Bool =>
-      if (seekCycles R S L ∅).succeeds ω then
-        ((2:ℝ)^d)^L.length / (2:ℝ)^(seekCycles R S L ∅).staleCharge ∅ ω else 0) ≤ 1 := by
-  let T := seekCycles R S L ∅
-  have hh := mean_freshCharge_le_one T
-  have he : finiteMean (fun ω : ι → Bool =>
-      if T.succeeds ω then (2:ℝ)^T.freshCharge ∅ ω else 0) =
-      finiteMean (fun ω : ι → Bool =>
-        if T.succeeds ω then ((2:ℝ)^d)^L.length / (2:ℝ)^T.staleCharge ∅ ω else 0) := by
-    apply finiteMean_congr
-    intro ω
-    cases hs : T.succeeds ω
-    · simp
-    · simp only [↓reduceIte]
-      have hc := fresh_add_stale T ∅ ω
-      rw [seekCycles_raw_charge R S L ∅ ω hs] at hc
-      have hp : ((2:ℝ)^d)^L.length = (2:ℝ)^T.freshCharge ∅ ω * (2:ℝ)^T.staleCharge ∅ ω := by
-        rw [←pow_add,hc,←pow_mul,Nat.mul_comm]
-      rw [hp,mul_div_cancel_right₀ _ (pow_ne_zero _ (by norm_num))]
-  rw [← he]
-  exact hh
-
-end QueryTree
-noncomputable def exposedPaths (b : ι → Bool) (V : Finset (α)) :
-    Finset (ι) :=
-  V.biUnion (fun x => R.path b x)
-
-@[simp] lemma exposedPaths_empty (b : ι → Bool) : exposedPaths R b ∅ = ∅ := by
-  classical
-  simp [exposedPaths]
-
-lemma exposedPaths_insert (b : ι → Bool) (V : Finset (α)) (x : α) :
-    exposedPaths R b (insert x V) = exposedPaths R b V ∪ R.path b x := by
-  classical
-  simp [exposedPaths,Finset.union_comm]
-
-noncomputable def terminalCharge (b : ι → Bool) (z : α) (V : Finset (α)) : ℕ :=
-  (R.path b z ∩ exposedPaths R b V).card
-
-namespace QueryTree
-open Thorp.QueryTree
-
-lemma seekCycle_semantics (S : Finset (α)) (a : α) (r : ℕ)
-    (x : α) (V W : Finset (α)) (z : α)
-    (k : Finset (α) → QueryTree (ι)) (ω : ι → Bool)
-    (h : cycleTrace (R.perm ω) S a r x V = some (W,z)) :
-    (seekCycle R S a r x V k).succeeds ω = (k W).succeeds ω ∧
-      (seekCycle R S a r x V k).staleCharge (exposedPaths R (ω) V) ω =
-        (k W).staleCharge (exposedPaths R (ω) W) ω +
-          terminalCharge R (ω) z (W.erase z) := by
-  classical
-  induction r generalizing x V with
-  | zero =>
-    simp only [cycleTrace] at h
-    split_ifs at h with hv
-    obtain ⟨hxS,hxV,hxa⟩ := hv
-    obtain ⟨rfl,rfl⟩ := Prod.mk.inj (Option.some.inj h)
-    simp only [seekCycle,ite_eq_left (show x ∈ S ∧ x ∉ V from ⟨hxS,hxV⟩),R.close_succeeds,ite_eq_left hxa]
-    constructor
-    · trivial
-    · rw [R.close_stale_charge _ _ _ _ _ hxa,exposedPaths_insert,
-        Finset.erase_insert hxV]
-      simp only [terminalCharge]
-  | succ r ih =>
-    simp only [cycleTrace] at h
-    split_ifs at h with hv
-    obtain ⟨hxS,hxV⟩ := hv
-    rw [seekCycle,ite_eq_left ⟨hxS,hxV⟩,R.follow_succeeds,R.follow_stale_charge]
-    rw [← exposedPaths_insert]
-    exact ih _ _ h
-
-lemma seekCycle_orbit (S V : Finset (α)) (z : α)
-    (k : Finset (α) → QueryTree (ι)) (ω : ι → Bool)
-    (hS : orbitSet (R.perm ω) z ⊆ S)
-    (hV : Disjoint V (orbitSet (R.perm ω) z)) :
-    let p := R.perm ω
-    let W := V ∪ orbitSet p z
-    (seekCycle R S (p z) (Function.minimalPeriod p z-1) (p z) V k).succeeds ω = (k W).succeeds ω ∧
-      (seekCycle R S (p z) (Function.minimalPeriod p z-1) (p z) V k).staleCharge
-          (exposedPaths R (ω) V) ω =
-        (k W).staleCharge (exposedPaths R (ω) W) ω +
-          terminalCharge R (ω) z (W.erase z) := by
-  exact seekCycle_semantics _ _ _ _ _ _ _ _ _ _ (cycleTrace_orbit _ _ _ _ hS hV)
-
-end QueryTree
-
-end RoutingNetwork
-end Thorp
-
-end ThorpNine.HighTail
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.HighTail.RotationMeeting
+
+namespace OAI
+
+namespace ThorpNine.HighTail
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+section CycleCounting
+variable {d A : ℕ}
+
+lemma assignmentOwner_eq_some {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) (x : Card d) (i : Fin A) :
+    assignmentOwner f x = some i ↔ x ∈ (f.val i).val := by
+  classical
+  unfold assignmentOwner
+  split_ifs with h
+  · constructor
+    · intro hi
+      have hx := Classical.choose_spec h
+      rw [Option.some.inj hi] at hx
+      exact hx
+    · intro hx
+      have he := (f.val (Classical.choose h)).eq_of_mem (f.val i) (Classical.choose_spec h) hx
+      exact congrArg some (f.property.1 he)
+  · constructor
+    · intro hi; contradiction
+    · intro hx; exact (h ⟨i,hx⟩).elim
+
+lemma assignmentOwner_fiber {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) (i : Fin A) : cycleFiber (assignmentOwner f) i = (f.val i).val := by
+  ext x
+  simp only [mem_cycleFiber,assignmentOwner_eq_some]
+
+noncomputable def assignmentToSlots {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) : CycleSlots p S r :=
+  ⟨assignmentOwner f,by
+    refine ⟨?_,?_,?_⟩
+    · intro i
+      rw [assignmentOwner_fiber]
+      exact f.property.2 i
+    · intro i
+      rw [assignmentOwner_fiber]
+      exact (f.val i).property.2
+    · intro i x hx
+      rw [assignmentOwner_fiber]
+      exact (f.val i).orbit_eq ((assignmentOwner_eq_some f x i).mp hx)⟩
+
+noncomputable def cycleSlotsEquivAssignments (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
+    CycleSlots p S r ≃ SlotAssignment p S r where
+  toFun := slotsToAssignment
+  invFun := assignmentToSlots
+  left_inv c := by
+    apply Subtype.ext
+    apply owner_ext
+    intro i
+    rw [assignmentToSlots,assignmentOwner_fiber]
+    rfl
+  right_inv f := by
+    apply Subtype.ext
+    funext i
+    apply Subtype.ext
+    exact assignmentOwner_fiber f i
+
+end CycleCounting
+
+section GroupedCounting
+variable {d A : ℕ}
+
+abbrev LengthClass (r : Fin A → ℕ) := {j : ℕ // j ∈ Finset.univ.image (fun i => r i + 1)}
+
+def slotClass (r : Fin A → ℕ) (i : Fin A) : LengthClass r :=
+  ⟨r i+1, Finset.mem_image.mpr ⟨i, Finset.mem_univ _, rfl⟩⟩
+
+abbrev SlotsOfLength (r : Fin A → ℕ) (j : LengthClass r) := {i : Fin A // slotClass r i = j}
+abbrev CyclesOfLength (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (j : ℕ) :=
+  {C : SelectedCycle p S // C.val.card = j}
+
+noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (j : ℕ) :
+    Fintype (CyclesOfLength p S j) := Fintype.ofFinite _
+
+abbrev GroupedAssignments (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
+  ∀ j : LengthClass r, SlotsOfLength r j ↪ CyclesOfLength p S j.val
+
+noncomputable def assignmentToGroups {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) : GroupedAssignments p S r := fun _j =>
+  { toFun := fun i => ⟨f.val i.val, (f.property.2 i.val).trans (congrArg Subtype.val i.property)⟩
+    inj' := fun _i _k h => Subtype.ext (f.property.1 (congrArg Subtype.val h)) }
+
+noncomputable def groupsEmbedding {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (F : GroupedAssignments p S r) : Fin A ↪ SelectedCycle p S :=
+  ((Equiv.sigmaFiberEquiv (slotClass r)).symm.toEmbedding.trans
+    (Function.Embedding.sigmaMap (Function.Embedding.refl _) F)).trans
+    ((Function.Embedding.sigmaMap (Function.Embedding.subtype _) (fun _ => Function.Embedding.refl _)).trans
+      (Equiv.sigmaFiberEquiv (fun C : SelectedCycle p S => C.val.card)).toEmbedding)
+
+noncomputable def groupsToAssignment {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (F : GroupedAssignments p S r) : SlotAssignment p S r :=
+  ⟨groupsEmbedding F, (groupsEmbedding F).injective,
+    fun i => (F (slotClass r i) ⟨i,rfl⟩).property⟩
+
+noncomputable def assignmentEquivGroups (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
+    SlotAssignment p S r ≃ GroupedAssignments p S r where
+  toFun := assignmentToGroups
+  invFun := groupsToAssignment
+  left_inv f := by
+    apply Subtype.ext
+    rfl
+  right_inv F := by
+    funext j
+    apply Function.Embedding.ext
+    intro i
+    rcases i with ⟨i,hi⟩
+    subst j
+    rfl
+
+lemma card_cycleSlots_factorial (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
+    Fintype.card (CycleSlots p S r) =
+      ∏ j : LengthClass r, (Fintype.card (CyclesOfLength p S j.val)).descFactorial
+        (Fintype.card (SlotsOfLength r j)) := by
+  classical
+  rw [Fintype.card_congr (cycleSlotsEquivAssignments p S r),
+    Fintype.card_congr (assignmentEquivGroups p S r), Fintype.card_pi]
+  simp only [Fintype.card_embedding_eq]
+
+lemma butterfly_factorial_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+       (∏ j : LengthClass r,
+         (Fintype.card (CyclesOfLength (butterflyPerm d (decodeButterfly d ω)) S j.val)).descFactorial
+           (Fintype.card (SlotsOfLength r j)) : ℕ)) ≤
+      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+        ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
+  simpa only [←card_cycleSlots_factorial] using butterfly_cycleSlots_moment S r hr
+
+end GroupedCounting
+
+section AllLengths
+variable {d A : ℕ}
+
+noncomputable def assignmentRelabel (p : Equiv.Perm (Card d)) (S : Finset (Card d))
+    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
+    SlotAssignment p S r ≃ SlotAssignment p S (r ∘ e) where
+  toFun f := ⟨f.val ∘ e, f.property.1.comp e.injective, fun i => f.property.2 (e i)⟩
+  invFun f := ⟨f.val ∘ e.symm, f.property.1.comp e.symm.injective, fun i => by
+    simpa only [Function.comp_apply,Equiv.apply_symm_apply] using f.property.2 (e.symm i)⟩
+  left_inv f := by apply Subtype.ext; funext i; simp
+  right_inv f := by apply Subtype.ext; funext i; simp
+
+lemma card_cycleSlots_relabel (p : Equiv.Perm (Card d)) (S : Finset (Card d))
+    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
+    Fintype.card (CycleSlots p S r) = Fintype.card (CycleSlots p S (r ∘ e)) := by
+  exact Fintype.card_congr ((cycleSlotsEquivAssignments p S r).trans
+    ((assignmentRelabel p S r e).trans (cycleSlotsEquivAssignments p S (r ∘ e)).symm))
+
+lemma butterfly_cycleSlots_moment_all (S : Finset (Card d)) (r : Fin A → ℕ) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) ≤
+        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
+  have h := butterfly_cycleSlots_moment S (r ∘ Tuple.sort r) (Tuple.monotone_sort r)
+  simp_rw [←card_cycleSlots_relabel] at h
+  have hp := Equiv.prod_comp (Tuple.sort r) (fun i : Fin A => (1 / Real.sqrt (r i+1) : ℝ))
+  simpa only [Function.comp_apply,hp] using h
+
+abbrev ShortCycle (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :=
+  {C : SelectedCycle p S // C.val.card ≤ J}
+
+noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
+    Fintype (ShortCycle p S J) := Fintype.ofFinite _
+
+noncomputable def shortCycleLength {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {J : ℕ}
+    (C : ShortCycle p S J) : Fin J :=
+  ⟨C.val.val.card-1, by have := C.val.nonempty.card_pos; have := C.property; omega⟩
+
+lemma shortCycleLength_add_one {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {J : ℕ}
+    (C : ShortCycle p S J) : (shortCycleLength C : ℕ)+1 = C.val.val.card := by
+  have := C.val.nonempty.card_pos
+  simp only [shortCycleLength]
+  omega
+
+noncomputable def shortEmbeddingEquiv (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
+    (Fin A ↪ ShortCycle p S J) ≃
+      Σ r : Fin A → Fin J, SlotAssignment p S (fun i => (r i : ℕ)) where
+  toFun f := ⟨fun i => shortCycleLength (f i),
+    ⟨fun i => (f i).val, fun i j h => f.injective (Subtype.ext h),
+      fun i => (shortCycleLength_add_one (f i)).symm⟩⟩
+  invFun c :=
+    { toFun := fun i => ⟨c.2.val i, by rw [c.2.property.2]; exact (c.1 i).isLt⟩
+      inj' := fun _i _j h => c.2.property.1 (congrArg Subtype.val h) }
+  left_inv f := by
+    apply Function.Embedding.ext
+    intro i
+    rfl
+  right_inv c := by
+    rcases c with ⟨r,f⟩
+    have he : (fun i => shortCycleLength (⟨f.val i, by rw [f.property.2]; exact (r i).isLt⟩ : ShortCycle p S J)) = r := by
+      funext i
+      apply Fin.ext
+      simp only [shortCycleLength,f.property.2,Nat.add_sub_cancel]
+    apply Sigma.ext he
+    apply (Subtype.heq_iff_coe_eq (by
+      intro x
+      change (Function.Injective x ∧ ∀ i, (x i).val.card = (f.val i).val.card - 1 + 1) ↔ _
+      simp only [f.property.2,Nat.add_sub_cancel])).mpr
+    rfl
+
+lemma shortCycle_factorial_eq (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
+    (Fintype.card (ShortCycle p S J)).descFactorial A =
+      ∑ r : Fin A → Fin J, Fintype.card (CycleSlots p S (fun i => (r i : ℕ))) := by
+  classical
+  have hc := Fintype.card_embedding_eq (α := Fin A) (β := ShortCycle p S J)
+  simp only [Fintype.card_fin] at hc
+  rw [←hc, Fintype.card_congr (shortEmbeddingEquiv p S J),Fintype.card_sigma]
+  apply Finset.sum_congr rfl
+  intro r _
+  exact (Fintype.card_congr (cycleSlotsEquivAssignments p S (fun i => (r i : ℕ)))).symm
+
+end AllLengths
+
+lemma factorial_mgf {Ω : Type*} [Fintype Ω] (F : Ω → ℕ) (k : ℕ) (Λ q : ℝ)
+    (hF : ∀ ω, F ω ≤ k) (hΛ : 0 ≤ Λ) (hq : 1 ≤ q)
+    (hm : ∀ a, finiteMean (fun ω => ((F ω).descFactorial a : ℝ)) ≤ Λ^a) :
+    finiteMean (fun ω => q^(F ω)) ≤ Real.exp ((q-1)*Λ) := by
+  have hc (n a : ℕ) : (n.choose a : ℝ) = (n.descFactorial a : ℝ) / (a.factorial : ℝ) := by
+    rw [Nat.descFactorial_eq_factorial_mul_choose,Nat.cast_mul]
+    exact (mul_div_cancel_left₀ _ (by exact_mod_cast Nat.factorial_ne_zero a)).symm
+  have hex (ω : Ω) : q^(F ω) = ∑ a ∈ Finset.range (k+1),
+      ((F ω).descFactorial a : ℝ) * ((q-1)^a / (a.factorial : ℝ)) := by
+    calc
+      _ = ((q-1)+1)^(F ω) := by congr 1; ring
+      _ = ∑ a ∈ Finset.range (F ω+1), (q-1)^a * (F ω).choose a := by
+        rw [add_pow]
+        simp only [one_pow,mul_one]
+      _ = ∑ a ∈ Finset.range (k+1), (q-1)^a * (F ω).choose a := by
+        apply Finset.sum_subset (Finset.range_mono (Nat.add_le_add_right (hF ω) _))
+        intro a _ ha
+        have hn : F ω < a := by have := (Finset.mem_range.not.mp ha); omega
+        rw [Nat.choose_eq_zero_of_lt hn,Nat.cast_zero,mul_zero]
+      _ = _ := by
+        apply Finset.sum_congr rfl
+        intro a _
+        rw [hc]
+        ring
+  simp_rw [hex,finiteMean_sum,finiteMean_mul_const]
+  calc
+    _ ≤ ∑ a ∈ Finset.range (k+1), Λ^a * ((q-1)^a / (a.factorial : ℝ)) := by
+      apply Finset.sum_le_sum
+      intro a _
+      exact mul_le_mul_of_nonneg_right (hm a) (by positivity)
+    _ = ∑ a ∈ Finset.range (k+1), ((q-1)*Λ)^a / (a.factorial : ℝ) := by
+      apply Finset.sum_congr rfl
+      intro a _
+      rw [mul_pow]
+      ring
+    _ ≤ _ := Real.sum_le_exp_of_nonneg (mul_nonneg (sub_nonneg.mpr hq) hΛ) _
+
+lemma inv_sqrt_step (j : ℕ) :
+    1 / Real.sqrt (j+1) ≤ 2*(Real.sqrt (j+1)-Real.sqrt j) := by
+  have hp : 0 < Real.sqrt (j+1) := Real.sqrt_pos.mpr (by positivity)
+  apply (div_le_iff₀ hp).mpr
+  have h1 := Real.sq_sqrt (show (0:ℝ) ≤ j+1 by positivity)
+  have h0 := Real.sq_sqrt (show (0:ℝ) ≤ j by positivity)
+  nlinarith [sq_nonneg (Real.sqrt (j+1)-Real.sqrt j)]
+
+lemma sum_inv_sqrt (J : ℕ) :
+    (∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1)) ≤ 2*Real.sqrt J := by
+  have h (n : ℕ) : (∑ j ∈ Finset.range n, 1 / Real.sqrt (j+1)) ≤ 2*Real.sqrt n := by
+    induction n with
+    | zero => simp
+    | succ n ih =>
+      rw [Finset.sum_range_succ]
+      have hs := inv_sqrt_step n
+      push_cast
+      linarith
+  rw [Fin.sum_univ_eq_sum_range (fun j : ℕ => (1 / Real.sqrt (j+1) : ℝ)) J]
+  exact h J
+section CycleMGF
+variable {d : ℕ}
+
+lemma butterfly_shortCycle_moment (S : Finset (Card d)) (J A : ℕ) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      ((Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J)).descFactorial A : ℝ)) ≤
+      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))^A := by
+  classical
+  simp_rw [shortCycle_factorial_eq,Nat.cast_sum,finiteMean_sum]
+  calc
+    _ ≤ ∑ r : Fin A → Fin J,
+        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+          ∏ i : Fin A, (1 / Real.sqrt ((r i : ℕ)+1)) := by
+      apply Finset.sum_le_sum
+      intro r _
+      exact butterfly_cycleSlots_moment_all S (fun i => (r i : ℕ))
+    _ = _ := by
+      rw [←Finset.mul_sum,mul_pow]
+      congr 1
+      rw [←Fintype.prod_sum (fun (_ : Fin A) (j : Fin J) => (1 / Real.sqrt ((j:ℕ)+1) : ℝ))]
+      simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fin]
+
+noncomputable def cycleMass (p : Equiv.Perm (Card d)) (S : Finset (Card d)) : ℕ :=
+  ∑ C : SelectedCycle p S, C.val.card
+
+lemma cycleMass_le (p : Equiv.Perm (Card d)) (S : Finset (Card d)) : cycleMass p S ≤ S.card := by
+  classical
+  let f : (Σ C : SelectedCycle p S, {x : Card d // x ∈ C.val}) → S :=
+    fun c => ⟨c.2.val,c.1.property.2 c.2.property⟩
+  have hi : Function.Injective f := by
+    intro c e h
+    have hx : c.2.val = e.2.val := congrArg Subtype.val h
+    have hC : c.1 = e.1 := c.1.eq_of_mem e.1 c.2.property (hx ▸ e.2.property)
+    rcases c with ⟨c,x⟩
+    rcases e with ⟨e,y⟩
+    dsimp only at hC
+    subst e
+    have he : x=y := Subtype.ext hx
+    cases he
+    rfl
+  have hh := Fintype.card_le_of_injective f hi
+  simpa only [Fintype.card_sigma,Fintype.card_coe,cycleMass] using hh
+
+lemma shortCycle_card_le (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
+    Fintype.card (ShortCycle p S J) ≤ S.card := by
+  classical
+  calc
+    _ ≤ Fintype.card (SelectedCycle p S) := Fintype.card_subtype_le _
+    _ = ∑ _ : SelectedCycle p S, 1 := by simp
+    _ ≤ cycleMass p S := Finset.sum_le_sum (fun C _ => C.nonempty.card_pos)
+    _ ≤ _ := cycleMass_le p S
+
+lemma butterfly_shortCycle_mgf (S : Finset (Card d)) (J : ℕ) (q : ℝ) (hq : 1 ≤ q) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      q^(Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J))) ≤
+      Real.exp ((q-1)*((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))) := by
+  apply factorial_mgf _ S.card _ q (fun ω => shortCycle_card_le _ _ _)
+  · positivity
+  · exact hq
+  · exact butterfly_shortCycle_moment S J
+
+end CycleMGF
+
+section FullCycleMGF
+variable {d : ℕ}
+
+abbrev LongCycle (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :=
+  {C : SelectedCycle p S // ¬ C.val.card ≤ J}
+noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
+    Fintype (LongCycle p S J) := Fintype.ofFinite _
+
+lemma selectedCycle_card_split (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
+    Fintype.card (SelectedCycle p S) =
+      Fintype.card (ShortCycle p S J) + Fintype.card (LongCycle p S J) := by
+  classical
+  rw [←Fintype.card_congr (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J)),Fintype.card_sum]
+
+lemma longCycle_card_bound (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
+    J * Fintype.card (LongCycle p S J) ≤ S.card := by
+  classical
+  have hs := Equiv.sum_comp (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J))
+    (fun C : SelectedCycle p S => C.val.card)
+  rw [Fintype.sum_sum_type] at hs
+  have h0 : J * Fintype.card (LongCycle p S J) ≤
+      ∑ C : LongCycle p S J, C.val.val.card := by
+    calc
+      _ = ∑ _C : LongCycle p S J, J := by simp [mul_comm]
+      _ ≤ _ := Finset.sum_le_sum (fun C _ => Nat.le_of_lt (Nat.lt_of_not_ge C.property))
+  have hh := cycleMass_le p S
+  unfold cycleMass at hh
+  rw [←hs] at hh
+  exact h0.trans ((Nat.le_add_left _ _).trans hh)
+
+lemma finiteMean_pos {Ω : Type*} [Fintype Ω] [Nonempty Ω] {f : Ω → ℝ}
+    (h : ∀ ω, 0 < f ω) : 0 < finiteMean f := by
+  apply div_pos
+  · exact Finset.sum_pos (fun ω _ => h ω) Finset.univ_nonempty
+  · exact Nat.cast_pos.mpr Fintype.card_pos
+
+lemma longCycle_pow_bound (p : Equiv.Perm (Card d)) (S : Finset (Card d))
+    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    q^(Fintype.card (LongCycle p S J)) ≤ Real.exp ((S.card : ℝ)/J * Real.log q) := by
+  have hq0 : 0 < q := lt_of_lt_of_le zero_lt_one hq
+  have hc : (Fintype.card (LongCycle p S J) : ℝ) ≤ (S.card : ℝ)/J := by
+    apply (le_div_iff₀ (by exact_mod_cast hJ)).mpr
+    have hh := longCycle_card_bound p S J
+    exact_mod_cast (by simpa only [mul_comm] using hh)
+  calc
+    _ = Real.exp ((Fintype.card (LongCycle p S J) : ℝ)*Real.log q) := by
+      rw [Real.exp_nat_mul,Real.exp_log hq0]
+    _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right hc (Real.log_nonneg hq))
+
+lemma butterfly_cycle_log_mgf (S : Finset (Card d)) (J : ℕ) (hJ : 0 < J)
+    (q : ℝ) (hq : 1 ≤ q) :
+    Real.log (finiteMean (fun ω : SwitchIndex d → Bool =>
+      q^(Fintype.card (SelectedCycle (butterflyPerm d (decodeButterfly d ω)) S)))) ≤
+      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
+  let lam : ℝ := (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1)
+  let c : ℝ := (S.card : ℝ)/J * Real.log q
+  have hnum : (q-1)*lam ≤ 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
+    have hsum := sum_inv_sqrt J
+    have hp : 0 ≤ (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) := by positivity
+    have h0 : 0 ≤ 2*Real.sqrt J := by positivity
+    have hh := mul_le_mul_of_nonneg_left hsum hp
+    have hq1 : 0 ≤ q-1 := sub_nonneg.mpr hq
+    calc
+      _ ≤ (q-1)*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
+        mul_le_mul_of_nonneg_left hh hq1
+      _ ≤ q*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
+        mul_le_mul_of_nonneg_right (by linarith) (mul_nonneg hp h0)
+      _ = _ := by ring
+  have hm : finiteMean (fun ω : SwitchIndex d → Bool =>
+      q^(Fintype.card (SelectedCycle (butterflyPerm d (decodeButterfly d ω)) S))) ≤
+      Real.exp (c+(q-1)*lam) := by
+    calc
+      _ ≤ finiteMean (fun ω : SwitchIndex d → Bool =>
+          q^(Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J))*Real.exp c) := by
+        apply finiteMean_mono
+        intro ω
+        rw [selectedCycle_card_split _ _ J,pow_add]
+        exact mul_le_mul_of_nonneg_left (longCycle_pow_bound _ S J hJ q hq) (by positivity)
+      _ = finiteMean (fun ω : SwitchIndex d → Bool =>
+          q^(Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J))) * Real.exp c :=
+        finiteMean_mul_const _ _
+      _ ≤ Real.exp ((q-1)*lam) * Real.exp c :=
+        mul_le_mul_of_nonneg_right (butterfly_shortCycle_mgf S J q hq) (Real.exp_nonneg _)
+      _ = _ := by rw [←Real.exp_add,add_comm]
+  have hpos : 0 < finiteMean (fun ω : SwitchIndex d → Bool =>
+      q^(Fintype.card (SelectedCycle (butterflyPerm d (decodeButterfly d ω)) S))) := by
+    apply finiteMean_pos
+    intro ω
+    positivity
+  have hlog := Real.log_le_log hpos hm
+  rw [Real.log_exp] at hlog
+  dsimp [c] at hlog
+  linarith
+
+end FullCycleMGF
+
+structure RoutingNetwork (α ι : Type*) [Fintype α] [DecidableEq α]
+    [Fintype ι] [DecidableEq ι] (d : ℕ) where
+  perm : (ι → Bool) → Equiv.Perm α
+  path : (ι → Bool) → α → Finset ι
+  users : (ι → Bool) → ι → α × α
+  users_ne : ∀ ω i, (users ω i).1 ≠ (users ω i).2
+  mem_path : ∀ ω x i, i ∈ path ω x ↔ (users ω i).1 = x ∨ (users ω i).2 = x
+  card_path : ∀ ω x, (path ω x).card = d
+  entropy : ∀ ω (S : Finset α),
+    (∑ i : ι, if (users ω i).1 ∈ S ∧ (users ω i).2 ∈ S then (1:ℝ) else 0) ≤
+      (S.card : ℝ) * Real.log S.card / (2 * Real.log 2)
+  follow : α → (α → QueryTree ι) → QueryTree ι
+  close : α → α → QueryTree ι → QueryTree ι
+  follow_succeeds : ∀ x k ω, (follow x k).succeeds ω = (k (perm ω x)).succeeds ω
+  follow_raw_charge : ∀ x k ω, (follow x k).charge ω = (k (perm ω x)).charge ω
+  follow_stale_charge : ∀ x k R ω,
+    (follow x k).staleCharge R ω = (k (perm ω x)).staleCharge (R ∪ path ω x) ω
+  close_succeeds : ∀ x y k ω,
+    (close x y k).succeeds ω = if perm ω x = y then k.succeeds ω else false
+  close_raw_charge : ∀ x y k ω, perm ω x = y → (close x y k).charge ω = k.charge ω + d
+  close_stale_charge : ∀ x y k R ω, perm ω x = y →
+    (close x y k).staleCharge R ω = k.staleCharge (R ∪ path ω x) ω + (path ω x ∩ R).card
+
+namespace RoutingNetwork
+variable {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι] [DecidableEq ι]
+  {d : ℕ} (R : RoutingNetwork α ι d)
+
+lemma card_switches_using (ω : ι → Bool) (x : α) :
+    (Finset.univ.filter (fun i : ι =>
+      (R.users ω i).1 = x ∨ (R.users ω i).2 = x)).card = d := by
+  classical
+  have he : Finset.univ.filter (fun i : ι =>
+      (R.users ω i).1 = x ∨ (R.users ω i).2 = x) = R.path ω x := by
+    ext i
+    simp only [Finset.mem_filter,Finset.mem_univ,true_and,R.mem_path]
+  rw [he,R.card_path]
+lemma switch_incidence_sum (b : ι → Bool) (w : α → ℝ) :
+    ∑ i : ι, (w (R.users b i).1 + w (R.users b i).2) =
+      (d : ℝ) * ∑ x : α, w x := by
+  classical
+  calc
+    _ = ∑ i : ι, ∑ x : α,
+        if (R.users b i).1 = x ∨ (R.users b i).2 = x then w x else 0 := by
+      apply Finset.sum_congr rfl
+      intro i _
+      rw [Finset.sum_ite]
+      simp only [Finset.sum_const_zero, add_zero]
+      have he : Finset.univ.filter (fun x : α =>
+          (R.users b i).1 = x ∨ (R.users b i).2 = x) =
+          {(R.users b i).1, (R.users b i).2} := by
+        ext x; simp [eq_comm]
+      rw [he, Finset.sum_pair (R.users_ne b i)]
+    _ = ∑ x : α, ∑ i : ι,
+        if (R.users b i).1 = x ∨ (R.users b i).2 = x then w x else 0 :=
+      Finset.sum_comm
+    _ = ∑ x : α, (d : ℝ) * w x := by
+      apply Finset.sum_congr rfl
+      intro x _
+      rw [Finset.sum_ite]
+      simp only [Finset.sum_const_zero, add_zero, Finset.sum_const, nsmul_eq_mul,
+        card_switches_using]
+    _ = _ := by rw [Finset.mul_sum]
+
+lemma weighted_switch_min_bound (b : ι → Bool) (w : α → ℝ) :
+    ∑ i : ι, min (w (R.users b i).1) (w (R.users b i).2) ≤
+      (d : ℝ) / 2 * ∑ x : α, w x := by
+  have h : 2 * (∑ i : ι,
+      min (w (R.users b i).1) (w (R.users b i).2)) ≤
+      ∑ i : ι, (w (R.users b i).1 + w (R.users b i).2) := by
+    rw [Finset.mul_sum]
+    apply Finset.sum_le_sum
+    intro i _
+    linarith [min_le_left (w (R.users b i).1) (w (R.users b i).2),
+      min_le_right (w (R.users b i).1) (w (R.users b i).2)]
+  rw [switch_incidence_sum] at h
+  linarith
+section Rotation
+variable {A : ℕ}
+
+def cycleFiber (o : α → Option (Fin A)) (i : Fin A) : Finset (α) := by
+  classical
+  exact Finset.univ.filter (fun x => o x = some i)
+
+omit [DecidableEq α] in
+@[simp] lemma mem_cycleFiber (o : α → Option (Fin A)) (i : Fin A) (x : α) :
+    x ∈ cycleFiber o i ↔ o x = some i := by
+  classical
+  simp [cycleFiber]
+
+abbrev Terminal (o : α → Option (Fin A)) := ∀ i : Fin A, {x : α // o x = some i}
+
+noncomputable def slotWeight (o : α → Option (Fin A)) (i : Fin A) : ℝ :=
+  1 / (cycleFiber o i).card
+
+noncomputable def pathWeight (o : α → Option (Fin A)) (x : α) : ℝ :=
+  match o x with
+  | none => 0
+  | some i => slotWeight o i
+
+noncomputable def withinWeight (o : α → Option (Fin A)) (u v : α) : ℝ :=
+  match o u, o v with
+  | some i, some j => if i = j then slotWeight o i else 0
+  | _, _ => 0
+
+noncomputable def rotationMeeting (o : α → Option (Fin A)) (z : Terminal o)
+    (u v : α) : ℝ :=
+  match o u, o v with
+  | some i, some j => if i < j then (if (z j).val = v then 1 else 0)
+      else if j < i then (if (z i).val = u then 1 else 0)
+      else (if (z i).val = u then 1 else 0) + (if (z i).val = v then 1 else 0)
+  | _, _ => 0
+
+omit [DecidableEq α] in
+lemma card_cycleFiber (o : α → Option (Fin A)) (i : Fin A) :
+    Fintype.card {x : α // o x = some i} = (cycleFiber o i).card := by
+  classical
+  simp [cycleFiber,Fintype.card_subtype]
+
+lemma terminal_mean {o : α → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
+    (i : Fin A) (x : α) (hx : o x = some i) :
+    finiteMean (fun z : Terminal o => if (z i).val = x then (1:ℝ) else 0) = slotWeight o i := by
+  classical
+  have hev := finiteMean_pi_eval (A := fun i : Fin A => {x : α // o x = some i})
+    i (fun z => if z.val = x then (1:ℝ) else 0)
+  rw [hev]
+  have he : (fun z : {x : α // o x = some i} => if z.val = x then (1:ℝ) else 0) =
+      (fun z => if z = ⟨x,hx⟩ then 1 else 0) := by
+    funext z
+    simp only [Subtype.ext_iff]
+  rw [he,finiteMean_eq_indicator,card_cycleFiber,slotWeight]
+
+omit [DecidableEq α] in
+lemma slotWeight_antitone (o : α → Option (Fin A))
+    (hn : ∀ i, (cycleFiber o i).Nonempty)
+    (hm : Monotone (fun i => (cycleFiber o i).card)) : Antitone (slotWeight o) := by
+  intro i j hij
+  unfold slotWeight
+  apply one_div_le_one_div_of_le
+  · exact_mod_cast Finset.card_pos.mpr (hn i)
+  · exact_mod_cast hm hij
+
+lemma rotationMeeting_mean {o : α → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
+    (hm : Monotone (fun i => (cycleFiber o i).card)) (u v : α) :
+    finiteMean (fun z : Terminal o => rotationMeeting o z u v) =
+      min (pathWeight o u) (pathWeight o v) + withinWeight o u v := by
+  classical
+  have hn : ∀ i, (cycleFiber o i).Nonempty := by
+    intro i
+    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : α // o x = some i})
+    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
+  have hw := slotWeight_antitone o hn hm
+  cases hu : o u with
+  | none =>
+    cases hv : o v with
+    | none => simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean]
+    | some j =>
+      have hj : 0 ≤ slotWeight o j := by unfold slotWeight; positivity
+      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_left hj]
+  | some i =>
+    cases hv : o v with
+    | none =>
+      have hi : 0 ≤ slotWeight o i := by unfold slotWeight; positivity
+      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_right hi]
+    | some j =>
+      simp only [rotationMeeting,pathWeight,withinWeight,hu,hv]
+      rcases lt_trichotomy i j with hij | rfl | hji
+      · simp only [hij,↓reduceIte,ne_of_lt hij]
+        rw [terminal_mean j v hv,min_eq_right (hw (le_of_lt hij)),add_zero]
+      · simp only [lt_self_iff_false,↓reduceIte,min_self]
+        rw [finiteMean_add,terminal_mean i u hu,terminal_mean i v hv]
+      · simp only [not_lt_of_ge (le_of_lt hji),hji,↓reduceIte,ne_of_gt hji]
+        rw [terminal_mean i u hu,min_eq_left (hw (le_of_lt hji)),add_zero]
+
+omit [DecidableEq α] in
+lemma pathWeight_sum (o : α → Option (Fin A)) (hn : ∀ i, (cycleFiber o i).Nonempty) :
+    ∑ x : α, pathWeight o x = (A : ℝ) := by
+  classical
+  have he (x : α) : pathWeight o x =
+      ∑ i : Fin A, if o x = some i then slotWeight o i else 0 := by
+    cases hx : o x with
+    | none => simp [pathWeight,hx]
+    | some j => simp [pathWeight,hx]
+  simp only [he]
+  rw [Finset.sum_comm]
+  have hf (i : Fin A) : (∑ x : α, if o x = some i then slotWeight o i else 0) = 1 := by
+    rw [← Finset.sum_filter]
+    change (∑ _x ∈ cycleFiber o i, slotWeight o i) = 1
+    simp only [Finset.sum_const,nsmul_eq_mul,slotWeight]
+    exact mul_one_div_cancel (by exact_mod_cast (Finset.card_pos.mpr (hn i)).ne')
+  simp only [hf,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,mul_one]
+
+lemma withinWeight_sum_bound (o : α → Option (Fin A))
+    (hn : ∀ i, (cycleFiber o i).Nonempty) (b : ι → Bool) :
+    (∑ s : ι, withinWeight o (R.users b s).1 (R.users b s).2) ≤
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
+  classical
+  have he (u v : α) : withinWeight o u v =
+      ∑ i : Fin A, (if u ∈ cycleFiber o i ∧ v ∈ cycleFiber o i then (1:ℝ) else 0) * slotWeight o i := by
+    simp only [mem_cycleFiber]
+    cases hu : o u <;> cases hv : o v <;> simp [withinWeight,hu,hv,ite_and,eq_comm]
+  simp only [he]
+  rw [Finset.sum_comm]
+  apply Finset.sum_le_sum
+  intro i _
+  rw [← Finset.sum_mul]
+  have hi := R.entropy b (cycleFiber o i)
+  have hpos : (0:ℝ) < (cycleFiber o i).card := by exact_mod_cast Finset.card_pos.mpr (hn i)
+  have hmul := mul_le_mul_of_nonneg_right hi (show 0 ≤ slotWeight o i by unfold slotWeight; positivity)
+  calc
+    _ ≤ ((cycleFiber o i).card : ℝ) * Real.log (cycleFiber o i).card /
+        (2 * Real.log 2) * slotWeight o i := hmul
+    _ = _ := by unfold slotWeight; field_simp
+
+noncomputable def rotationCharge (o : α → Option (Fin A)) (b : ι → Bool)
+    (z : Terminal o) : ℝ :=
+  ∑ s : ι, rotationMeeting o z (R.users b s).1 (R.users b s).2
+
+lemma rotationCharge_mean_bound {o : α → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
+    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : ι → Bool) :
+    finiteMean (rotationCharge R o b) ≤ (A:ℝ) * d / 2 +
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
+  classical
+  have hn : ∀ i, (cycleFiber o i).Nonempty := by
+    intro i
+    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : α // o x = some i})
+    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
+  unfold rotationCharge
+  rw [finiteMean_sum]
+  simp only [rotationMeeting_mean hm,Finset.sum_add_distrib]
+  have h₁ := weighted_switch_min_bound R b (pathWeight o)
+  rw [pathWeight_sum o hn] at h₁
+  have h₂ := withinWeight_sum_bound R o hn b
+  linarith
+
+lemma rotation_exp_bound {o : α → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
+    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : ι → Bool) :
+    Real.exp (-Real.log 2 * ((A:ℝ)*d/2 +
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2))) ≤
+      finiteMean (fun z : Terminal o => Real.exp (-Real.log 2 * rotationCharge R o b z)) := by
+  have hb := rotationCharge_mean_bound R hm b
+  have hlog : 0 ≤ Real.log 2 := Real.log_nonneg (by norm_num)
+  have hm' : -Real.log 2 * ((A:ℝ)*d/2 +
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2)) ≤
+      finiteMean (fun z : Terminal o => -Real.log 2 * rotationCharge R o b z) := by
+    have he := finiteMean_mul_const (rotationCharge R o b) (-Real.log 2)
+    simp only [mul_comm] at he
+    rw [he]
+    exact mul_le_mul_of_nonpos_left hb (neg_nonpos.mpr hlog)
+  exact (Real.exp_le_exp.mpr hm').trans (exp_finiteMean_le _)
+
+end Rotation
+
+namespace QueryTree
+open Thorp.QueryTree
+
+noncomputable def seekCycle (R : RoutingNetwork α ι d) (S : Finset (α)) (a : α) :
+    ℕ → α → Finset (α) → (Finset (α) → QueryTree (ι)) →
+      QueryTree (ι)
+  | 0, x, V, k => if x ∈ S ∧ x ∉ V then R.close x a (k (insert x V)) else reject
+  | r + 1, x, V, k => if x ∈ S ∧ x ∉ V then R.follow x
+      (fun y => seekCycle R S a r y (insert x V) k) else reject
+
+lemma seekCycle_raw_charge (S : Finset (α)) (a : α) (r : ℕ)
+    (x : α) (V : Finset (α)) (k : Finset (α) → QueryTree (ι))
+    (ω : ι → Bool) (K : ℕ)
+    (hK : ∀ V, (k V).succeeds ω = true → (k V).charge ω = K)
+    (hs : (seekCycle R S a r x V k).succeeds ω = true) :
+    (seekCycle R S a r x V k).charge ω = K + d := by
+  induction r generalizing x V with
+  | zero =>
+    by_cases hv : x ∈ S ∧ x ∉ V
+    · rw [seekCycle,ite_eq_left hv,R.close_succeeds] at hs
+      split_ifs at hs with he
+      · rw [seekCycle,ite_eq_left hv,R.close_raw_charge _ _ _ _ he,hK _ hs]
+    · simp [seekCycle,hv,Thorp.QueryTree.succeeds] at hs
+  | succ r ih =>
+    by_cases hv : x ∈ S ∧ x ∉ V
+    · rw [seekCycle,ite_eq_left hv,R.follow_succeeds] at hs
+      rw [seekCycle,ite_eq_left hv,R.follow_raw_charge]
+      exact ih _ _ hs
+    · simp [seekCycle,hv,Thorp.QueryTree.succeeds] at hs
+
+noncomputable def seekCycles (R : RoutingNetwork α ι d) (S : Finset (α)) :
+    List (α × ℕ) → Finset (α) → QueryTree (ι)
+  | [], _ => accept
+  | (a,r) :: L, V => seekCycle R S a r a V (fun V => seekCycles R S L V)
+
+lemma seekCycles_raw_charge (S : Finset (α)) (L : List (α × ℕ))
+    (V : Finset (α)) (ω : ι → Bool)
+    (hs : (seekCycles R S L V).succeeds ω = true) :
+    (seekCycles R S L V).charge ω = L.length * d := by
+  induction L generalizing V with
+  | nil => simp [seekCycles,charge]
+  | cons ar L ih =>
+    obtain ⟨a,r⟩ := ar
+    rw [seekCycles]
+    rw [seekCycle_raw_charge R S a r a V (fun V => seekCycles R S L V) ω (L.length*d)
+      (fun V hs => ih V hs) hs]
+    simp [Nat.add_mul]
+
+lemma cycles_weighted_success (S : Finset (α)) (L : List (α × ℕ)) :
+    finiteMean (fun ω : ι → Bool =>
+      if (seekCycles R S L ∅).succeeds ω then
+        ((2:ℝ)^d)^L.length / (2:ℝ)^(seekCycles R S L ∅).staleCharge ∅ ω else 0) ≤ 1 := by
+  let T := seekCycles R S L ∅
+  have hh := mean_freshCharge_le_one T
+  have he : finiteMean (fun ω : ι → Bool =>
+      if T.succeeds ω then (2:ℝ)^T.freshCharge ∅ ω else 0) =
+      finiteMean (fun ω : ι → Bool =>
+        if T.succeeds ω then ((2:ℝ)^d)^L.length / (2:ℝ)^T.staleCharge ∅ ω else 0) := by
+    apply finiteMean_congr
+    intro ω
+    cases hs : T.succeeds ω
+    · simp
+    · simp only [↓reduceIte]
+      have hc := fresh_add_stale T ∅ ω
+      rw [seekCycles_raw_charge R S L ∅ ω hs] at hc
+      have hp : ((2:ℝ)^d)^L.length = (2:ℝ)^T.freshCharge ∅ ω * (2:ℝ)^T.staleCharge ∅ ω := by
+        rw [←pow_add,hc,←pow_mul,Nat.mul_comm]
+      rw [hp,mul_div_cancel_right₀ _ (pow_ne_zero _ (by norm_num))]
+  rw [← he]
+  exact hh
+
+end QueryTree
+noncomputable def exposedPaths (b : ι → Bool) (V : Finset (α)) :
+    Finset (ι) :=
+  V.biUnion (fun x => R.path b x)
+
+@[simp] lemma exposedPaths_empty (b : ι → Bool) : exposedPaths R b ∅ = ∅ := by
+  classical
+  simp [exposedPaths]
+
+lemma exposedPaths_insert (b : ι → Bool) (V : Finset (α)) (x : α) :
+    exposedPaths R b (insert x V) = exposedPaths R b V ∪ R.path b x := by
+  classical
+  simp [exposedPaths,Finset.union_comm]
+
+noncomputable def terminalCharge (b : ι → Bool) (z : α) (V : Finset (α)) : ℕ :=
+  (R.path b z ∩ exposedPaths R b V).card
+
+namespace QueryTree
+open Thorp.QueryTree
+
+lemma seekCycle_semantics (S : Finset (α)) (a : α) (r : ℕ)
+    (x : α) (V W : Finset (α)) (z : α)
+    (k : Finset (α) → QueryTree (ι)) (ω : ι → Bool)
+    (h : cycleTrace (R.perm ω) S a r x V = some (W,z)) :
+    (seekCycle R S a r x V k).succeeds ω = (k W).succeeds ω ∧
+      (seekCycle R S a r x V k).staleCharge (exposedPaths R (ω) V) ω =
+        (k W).staleCharge (exposedPaths R (ω) W) ω +
+          terminalCharge R (ω) z (W.erase z) := by
+  classical
+  induction r generalizing x V with
+  | zero =>
+    simp only [cycleTrace] at h
+    split_ifs at h with hv
+    obtain ⟨hxS,hxV,hxa⟩ := hv
+    obtain ⟨rfl,rfl⟩ := Prod.mk.inj (Option.some.inj h)
+    simp only [seekCycle,ite_eq_left (show x ∈ S ∧ x ∉ V from ⟨hxS,hxV⟩),R.close_succeeds,ite_eq_left hxa]
+    constructor
+    · trivial
+    · rw [R.close_stale_charge _ _ _ _ _ hxa,exposedPaths_insert,
+        Finset.erase_insert hxV]
+      simp only [terminalCharge]
+  | succ r ih =>
+    simp only [cycleTrace] at h
+    split_ifs at h with hv
+    obtain ⟨hxS,hxV⟩ := hv
+    rw [seekCycle,ite_eq_left ⟨hxS,hxV⟩,R.follow_succeeds,R.follow_stale_charge]
+    rw [← exposedPaths_insert]
+    exact ih _ _ h
+
+lemma seekCycle_orbit (S V : Finset (α)) (z : α)
+    (k : Finset (α) → QueryTree (ι)) (ω : ι → Bool)
+    (hS : orbitSet (R.perm ω) z ⊆ S)
+    (hV : Disjoint V (orbitSet (R.perm ω) z)) :
+    let p := R.perm ω
+    let W := V ∪ orbitSet p z
+    (seekCycle R S (p z) (Function.minimalPeriod p z-1) (p z) V k).succeeds ω = (k W).succeeds ω ∧
+      (seekCycle R S (p z) (Function.minimalPeriod p z-1) (p z) V k).staleCharge
+          (exposedPaths R (ω) V) ω =
+        (k W).staleCharge (exposedPaths R (ω) W) ω +
+          terminalCharge R (ω) z (W.erase z) := by
+  exact seekCycle_semantics _ _ _ _ _ _ _ _ _ _ (cycleTrace_orbit _ _ _ _ hS hV)
+
+end QueryTree
+
+end RoutingNetwork
+end Thorp
+
+end ThorpNine.HighTail
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/HighTail/Coins.lean b/lean/OAI/Probability/ThorpRouting/HighTail/Coins.lean
index f9ec989cec535346137960e0c21d2d0f931e72d9..9e727a42abda78acc5ce748276ee453b5cd12945 100644
--- a/lean/OAI/Probability/ThorpRouting/HighTail/Coins.lean
+++ b/lean/OAI/Probability/ThorpRouting/HighTail/Coins.lean
@@ -1,764 +1,764 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.HighTail.Model
-
-namespace OAI
-
-namespace ThorpNine.HighTail
-
-namespace Thorp
-
-open scoped BigOperators
-open Filter
-
-def Coins : ℕ → Type
-  | 0 => Unit
-  | d + 1 => Card d → Bool
-
-instance (d : ℕ) : Fintype (Coins d) := by
-  cases d <;> dsimp [Coins] <;> infer_instance
-
-instance (d : ℕ) : Nonempty (Coins d) := by
-  cases d <;> dsimp [Coins] <;> infer_instance
-
-def rotate (d : ℕ) : Equiv.Perm (Card d) where
-  toFun x i := x (finRotate d i)
-  invFun x i := x ((finRotate d).symm i)
-  left_inv x := by funext i; simp only [Equiv.apply_symm_apply]
-  right_inv x := by funext i; simp only [Equiv.symm_apply_apply]
-
-def step : (d : ℕ) → Coins d → Equiv.Perm (Card d)
-  | 0, _ => 1
-  | d + 1, ξ => rotate (d + 1) * pairSwitch ξ
-
-def run (d : ℕ) : (t : ℕ) → (Fin t → Coins d) → Equiv.Perm (Card d)
-  | 0, _ => 1
-  | t + 1, ω => step d (ω (Fin.last t)) * run d t (fun i => ω i.castSucc)
-
-noncomputable def law (d t : ℕ) (g : Equiv.Perm (Card d)) : ℝ :=
-  (Fintype.card {ω : Fin t → Coins d // run d t ω = g} : ℝ) /
-    (Fintype.card (Fin t → Coins d) : ℝ)
-
-noncomputable def distance (d t : ℕ) : ℝ :=
-  (1 / 2 : ℝ) * ∑ g : Equiv.Perm (Card d),
-    |law d t g - 1 / (Nat.factorial (2 ^ d) : ℝ)|
-
-noncomputable def mixingTime (d : ℕ) : ℕ :=
-  sInf {t : ℕ | distance d t ≤ (1 / 4 : ℝ)}
-
-noncomputable def supportTimeBound (d : ℕ) : ℤ :=
-  ⌈(2 / (2 : ℝ) ^ d) *
-    (Real.log ((3 / 4 : ℝ) * (Nat.factorial (2 ^ d) : ℝ)) / Real.log 2)⌉
-
-def OptimalOrder : Prop :=
-  ∃ C : ℕ, 0 < C ∧
-    Tendsto (fun d : ℕ => distance d (C * d)) atTop (nhds 0) ∧
-    (∀ d t : ℕ, 1 - (2 : ℝ) ^ ((t : ℝ) * (2 : ℝ) ^ d / 2) /
-      (Nat.factorial (2 ^ d) : ℝ) ≤ distance d t) ∧
-    (∀ᶠ d : ℕ in atTop,
-      supportTimeBound d ≤ (mixingTime d : ℤ) ∧ mixingTime d ≤ C * d) ∧
-    (∃ K : ℝ, ∀ᶠ d : ℕ in atTop, 2 * (d : ℝ) - K ≤ (mixingTime d : ℝ)) ∧
-    Asymptotics.IsTheta atTop (fun d : ℕ => (mixingTime d : ℝ)) (fun d : ℕ => (d : ℝ))
-
-noncomputable def finiteLaw {α β : Type*} [Fintype α] (f : α → β) (b : β) : ℝ := by
-  classical
-  exact (Fintype.card {a : α // f a = b} : ℝ) / (Fintype.card α : ℝ)
-
-lemma sum_card_fibers {α β : Type*} [Fintype α] [Fintype β] [DecidableEq β] (f : α → β) :
-    ∑ b, Fintype.card {a : α // f a = b} = Fintype.card α := by
-  classical
-  rw [← Fintype.card_sigma]
-  exact Fintype.card_congr (Equiv.sigmaFiberEquiv f)
-
-lemma finiteLaw_nonneg {α β : Type*} [Fintype α] (f : α → β) (b : β) :
-    0 ≤ finiteLaw f b := by
-  exact div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _)
-
-lemma finiteLaw_sum {α β : Type*} [Fintype α] [Nonempty α] [Fintype β] (f : α → β) :
-    ∑ b, finiteLaw f b = 1 := by
-  classical
-  simp only [finiteLaw, ← Finset.sum_div]
-  have hc : (∑ b, (Fintype.card {a : α // f a = b} : ℝ)) = (Fintype.card α : ℝ) := by
-    exact_mod_cast sum_card_fibers f
-  rw [hc]
-  exact div_self (Nat.cast_ne_zero.mpr Fintype.card_ne_zero)
-
-lemma finiteLaw_eq_zero_of_not_mem {α β : Type*} [Fintype α] [DecidableEq β]
-    (f : α → β) {b : β} (hb : b ∉ Finset.univ.image f) : finiteLaw f b = 0 := by
-  classical
-  have : IsEmpty {a : α // f a = b} := ⟨fun a => hb (Finset.mem_image.mpr
-    ⟨a.val, Finset.mem_univ _, a.property⟩)⟩
-  simp [finiteLaw]
-
-lemma event_le_totalVariation {β : Type*} [Fintype β] [DecidableEq β]
-    (p q : β → ℝ) (hpq : ∑ b, (p b - q b) = 0) (s : Finset β) :
-    (∑ b ∈ s, p b) - (∑ b ∈ s, q b) ≤ (1 / 2 : ℝ) * ∑ b, |p b - q b| := by
-  have h₁ : (∑ b ∈ s, (p b - q b)) ≤ ∑ b ∈ s, |p b - q b| :=
-    Finset.sum_le_sum (fun b _ => le_abs_self (p b - q b))
-  have h₂ : -(∑ b ∈ sᶜ, (p b - q b)) ≤ ∑ b ∈ sᶜ, |p b - q b| := by
-    rw [← Finset.sum_neg_distrib]
-    exact Finset.sum_le_sum (fun b _ => neg_le_abs (p b - q b))
-  have hz := Finset.sum_add_sum_compl s (fun b => p b - q b)
-  have ha := Finset.sum_add_sum_compl s (fun b => |p b - q b|)
-  rw [hpq] at hz
-  rw [← Finset.sum_sub_distrib]
-  linarith
-
-lemma finiteLaw_support_bound {α β : Type*} [Fintype α] [Nonempty α]
-    [Fintype β] [Nonempty β] (f : α → β) :
-    1 - (Fintype.card α : ℝ) / (Fintype.card β : ℝ) ≤
-      (1 / 2 : ℝ) * ∑ b, |finiteLaw f b - 1 / (Fintype.card β : ℝ)| := by
-  classical
-  let s := Finset.univ.image f
-  have hs : ∑ b ∈ s, finiteLaw f b = 1 := by
-    have h := Finset.sum_subset (Finset.subset_univ s)
-      (fun b _ hb => finiteLaw_eq_zero_of_not_mem f hb)
-    simpa only [finiteLaw_sum] using h
-  have hβ : (Fintype.card β : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
-  have hq : (∑ _b : β, 1 / (Fintype.card β : ℝ)) = 1 := by simp [hβ]
-  have hd : (∑ b, (finiteLaw f b - 1 / (Fintype.card β : ℝ))) = 0 := by
-    rw [Finset.sum_sub_distrib, finiteLaw_sum, hq, sub_self]
-  have he := event_le_totalVariation (finiteLaw f) (fun _ => 1 / (Fintype.card β : ℝ)) hd s
-  rw [hs] at he
-  have hsq : (∑ _b ∈ s, 1 / (Fintype.card β : ℝ)) = (s.card : ℝ) / Fintype.card β := by
-    simp [div_eq_mul_inv]
-  rw [hsq] at he
-  have hcard : (s.card : ℝ) ≤ Fintype.card α := by
-    exact_mod_cast (Finset.card_image_le (s := (Finset.univ : Finset α)) (f := f))
-  have hdiv := div_le_div_of_nonneg_right hcard (Nat.cast_nonneg (Fintype.card β) : (0 : ℝ) ≤ _)
-  linarith
-
-lemma card_positions (d : ℕ) : Fintype.card (Card d) = 2 ^ d := by
-  simp [Card]
-
-lemma card_permutations (d : ℕ) : Fintype.card (Equiv.Perm (Card d)) = Nat.factorial (2 ^ d) := by
-  rw [Fintype.card_perm, card_positions]
-
-lemma card_coins_succ (d : ℕ) : Fintype.card (Coins (d + 1)) = 2 ^ (2 ^ d) := by
-  simp [Coins]
-
-lemma card_tapes_succ (d t : ℕ) : Fintype.card (Fin t → Coins (d + 1)) = 2 ^ (t * 2 ^ d) := by
-  rw [Fintype.card_fun, card_coins_succ, Fintype.card_fin, ← pow_mul, Nat.mul_comm]
-
-lemma law_eq_finiteLaw (d t : ℕ) (g : Equiv.Perm (Card d)) :
-    law d t g = finiteLaw (run d t) g := by
-  unfold law finiteLaw
-  apply congrArg (fun n : ℕ => (n : ℝ) / (Fintype.card (Fin t → Coins d) : ℝ))
-  exact congrArg (@Fintype.card _) (Subsingleton.elim _ _)
-
-lemma law_nonneg (d t : ℕ) (g : Equiv.Perm (Card d)) : 0 ≤ law d t g := by
-  rw [law_eq_finiteLaw]
-  exact finiteLaw_nonneg (run d t) g
-
-lemma law_sum (d t : ℕ) : ∑ g, law d t g = 1 := by
-  simp only [law_eq_finiteLaw]
-  exact finiteLaw_sum (run d t)
-
-lemma physical_support_bound_succ (d t : ℕ) :
-    1 - (2 : ℝ) ^ (t * 2 ^ d) / (Nat.factorial (2 ^ (d + 1)) : ℝ) ≤ distance (d + 1) t := by
-  have : Nonempty (Equiv.Perm (Card (d + 1))) := ⟨1⟩
-  have h := finiteLaw_support_bound (run (d + 1) t)
-  rw [card_tapes_succ, card_permutations] at h
-  simpa only [Nat.cast_pow, Nat.cast_ofNat, distance, law_eq_finiteLaw] using h
-
-lemma step_succ_formula (d : ℕ) (ξ : Coins (d + 1)) (x : Card (d + 1)) :
-    step (d + 1) ξ x = Fin.snoc (Fin.tail x) (Bool.xor (x 0) (ξ (Fin.tail x))) := by
-  change (fun i => switchFun ξ x (finRotate (d + 1) i)) = _
-  rw [Fin.snoc_eq_cons_rotate]
-  rfl
-
-lemma distance_nonneg (d t : ℕ) : 0 ≤ distance d t := by
-  unfold distance
-  exact mul_nonneg (by norm_num) (Finset.sum_nonneg (fun _ _ => abs_nonneg _))
-
-lemma physical_support_bound (d t : ℕ) :
-    1 - (2 : ℝ) ^ ((t : ℝ) * (2 : ℝ) ^ d / 2) /
-      (Nat.factorial (2 ^ d) : ℝ) ≤ distance d t := by
-  cases d with
-  | zero =>
-    have h : 1 ≤ (2 : ℝ) ^ ((t : ℝ) / 2) := Real.one_le_rpow (by norm_num) (by positivity)
-    simpa only [pow_zero, Nat.factorial_one, Nat.cast_one, div_one, mul_one] using
-      le_trans (sub_nonpos.mpr h) (distance_nonneg 0 t)
-  | succ d =>
-    have he : (t : ℝ) * (2 : ℝ) ^ (d + 1) / 2 = (t * 2 ^ d : ℕ) := by
-      push_cast
-      rw [pow_succ]
-      ring
-    rw [he, Real.rpow_natCast]
-    exact physical_support_bound_succ d t
-
-lemma supportTimeBound_le_of_distance_le (d t : ℕ) (ht : distance d t ≤ (1 / 4 : ℝ)) :
-    supportTimeBound d ≤ (t : ℤ) := by
-  have hF : (0 : ℝ) < Nat.factorial (2 ^ d) := by positivity
-  have hN : (0 : ℝ) < (2 : ℝ) ^ d := by positivity
-  have hL : (0 : ℝ) < Real.log 2 := Real.log_pos (by norm_num)
-  have hb := physical_support_bound d t
-  have hp : (3 / 4 : ℝ) ≤
-      (2 : ℝ) ^ ((t : ℝ) * (2 : ℝ) ^ d / 2) / (Nat.factorial (2 ^ d) : ℝ) := by
-    linarith
-  have hr := (le_div_iff₀ hF).mp hp
-  have hlog := Real.log_le_log (show (0 : ℝ) < (3 / 4 : ℝ) * Nat.factorial (2 ^ d) by positivity) hr
-  rw [Real.log_rpow (by norm_num : (0 : ℝ) < 2)] at hlog
-  unfold supportTimeBound
-  apply Int.ceil_le.mpr
-  simp only [Int.cast_natCast]
-  calc
-    2 / (2 : ℝ) ^ d * (Real.log ((3 / 4 : ℝ) * Nat.factorial (2 ^ d)) / Real.log 2) =
-        Real.log ((3 / 4 : ℝ) * Nat.factorial (2 ^ d)) / ((2 : ℝ) ^ d / 2 * Real.log 2) := by
-      field_simp
-    _ ≤ (t : ℝ) := (div_le_iff₀ (mul_pos (div_pos hN (by norm_num)) hL)).mpr (by
-      nlinarith [hlog])
-
-lemma supportTimeBound_le_mixingTime {d : ℕ}
-    (h : ({t : ℕ | distance d t ≤ (1 / 4 : ℝ)} : Set ℕ).Nonempty) :
-    supportTimeBound d ≤ (mixingTime d : ℤ) := by
-  exact supportTimeBound_le_of_distance_le d (mixingTime d) (Nat.sInf_mem h)
-
-lemma log_factorial_lower {n : ℕ} (hn : 1 ≤ n) :
-    (n : ℝ) * Real.log n - n ≤ Real.log (Nat.factorial n) := by
-  have h := Stirling.le_log_factorial_stirling (show n ≠ 0 by omega)
-  have hln : 0 ≤ Real.log (n : ℝ) := Real.log_nonneg (by exact_mod_cast hn)
-  have hlp : 0 ≤ Real.log (2 * Real.pi) := Real.log_nonneg (by
-    have := Real.pi_gt_three
-    linarith)
-  linarith
-
-lemma log_three_quarters_lower : -(1 : ℝ) ≤ Real.log (3 / 4 : ℝ) := by
-  have h := Real.log_le_sub_one_of_pos (show (0 : ℝ) < 4 / 3 by norm_num)
-  have he : Real.log (4 / 3 : ℝ) = - Real.log (3 / 4 : ℝ) := by
-    rw [← Real.log_inv]
-    congr 1
-    norm_num
-  rw [he] at h
-  linarith
-
-lemma linear_le_supportTimeBound (d : ℕ) :
-    2 * (d : ℝ) - 4 / Real.log 2 ≤ (supportTimeBound d : ℝ) := by
-  have hN : 1 ≤ (2 : ℝ) ^ d := one_le_pow₀ (by norm_num)
-  have hNp : 0 < (2 : ℝ) ^ d := by positivity
-  have hL : 0 < Real.log 2 := Real.log_pos (by norm_num)
-  have hf := log_factorial_lower (show 1 ≤ 2 ^ d by exact one_le_pow₀ (by norm_num))
-  have hlog : (2 : ℝ) ^ d * ((d : ℝ) * Real.log 2) - (2 : ℝ) ^ d - 1 ≤
-      Real.log ((3 / 4 : ℝ) * (Nat.factorial (2 ^ d) : ℝ)) := by
-    rw [Real.log_mul (by norm_num) (by positivity)]
-    push_cast at hf
-    rw [Real.log_pow] at hf
-    linarith [log_three_quarters_lower]
-  apply le_trans _ (Int.le_ceil _)
-  have hmul : (2 * (d : ℝ) - 4 / Real.log 2) *
-      ((2 : ℝ) ^ d / 2 * Real.log 2) ≤
-      Real.log ((3 / 4 : ℝ) * (Nat.factorial (2 ^ d) : ℝ)) := by
-    have hcancel : (2 * (d : ℝ) - 4 / Real.log 2) *
-        ((2 : ℝ) ^ d / 2 * Real.log 2) =
-        (2 : ℝ) ^ d * ((d : ℝ) * Real.log 2) - 2 * (2 : ℝ) ^ d := by
-      field_simp
-      ring
-    rw [hcancel]
-    linarith
-  convert (le_div_iff₀ (mul_pos (div_pos hNp (by norm_num)) hL)).mpr hmul using 1
-  field_simp
-
-lemma linear_le_time_of_distance_le (d t : ℕ)
-    (ht : distance d t ≤ (1 / 4 : ℝ)) :
-    2 * (d : ℝ) - 4 / Real.log 2 ≤ (t : ℝ) := by
-  have h := supportTimeBound_le_of_distance_le d t ht
-  have hr : (supportTimeBound d : ℝ) ≤ (t : ℝ) := by exact_mod_cast h
-  exact (linear_le_supportTimeBound d).trans hr
-
-lemma entropy_chord (p : ℝ) (hp : 0 ≤ p) (hq : p ≤ 1 / 2) :
-    2 * p * Real.log 2 ≤ Real.binEntropy p := by
-  have hc := Real.strictConcave_binEntropy.concaveOn.2
-    (show (0 : ℝ) ∈ Set.Icc (0 : ℝ) 1 by norm_num)
-    (show (2 : ℝ)⁻¹ ∈ Set.Icc (0 : ℝ) 1 by norm_num)
-    (show 0 ≤ 1 - 2 * p by linarith)
-    (show 0 ≤ 2 * p by positivity)
-    (show (1 - 2 * p) + 2 * p = 1 by ring)
-  simpa only [Real.binEntropy_zero, Real.binEntropy_two_inv,
-    smul_eq_mul, mul_zero, zero_add, show 2 * p * (2 : ℝ)⁻¹ = p by ring] using hc
-
-lemma split_entropy_lower (a b : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) :
-    2 * min a b * Real.log 2 ≤
-      (a + b) * Real.log (a + b) - a * Real.log a - b * Real.log b := by
-  wlog hab : a ≤ b generalizing a b
-  · have hba := this b a hb ha (le_of_not_ge hab)
-    simpa only [min_comm, add_comm b a] using hba.trans_eq (show (b + a) * Real.log (b + a) - b * Real.log b - a * Real.log a =
-      (a + b) * Real.log (a + b) - a * Real.log a - b * Real.log b by rw [add_comm b a]; ring)
-  rw [min_eq_left hab]
-  by_cases hz : a = 0
-  · subst a
-    simp
-  have hap : 0 < a := lt_of_le_of_ne ha (Ne.symm hz)
-  have hbp : 0 < b := hap.trans_le hab
-  have hs : 0 < a + b := add_pos hap hbp
-  have hp : 0 ≤ a / (a + b) := div_nonneg ha hs.le
-  have hq : a / (a + b) ≤ 1 / 2 := (div_le_iff₀ hs).mpr (by linarith)
-  have hc := mul_le_mul_of_nonneg_left (entropy_chord (a / (a + b)) hp hq) hs.le
-  have he : (a + b) * Real.binEntropy (a / (a + b)) =
-      (a + b) * Real.log (a + b) - a * Real.log a - b * Real.log b := by
-    unfold Real.binEntropy
-    rw [show 1 - a / (a + b) = b / (a + b) by field_simp; ring]
-    simp only [Real.log_inv]
-    rw [Real.log_div (ne_of_gt hap) (ne_of_gt hs),
-      Real.log_div (ne_of_gt hbp) (ne_of_gt hs)]
-    field_simp
-    ring
-  rw [he] at hc
-  calc
-    2 * a * Real.log 2 = (a + b) * (2 * (a / (a + b)) * Real.log 2) := by
-      field_simp
-    _ ≤ _ := hc
-
-noncomputable def childMarks {d : ℕ} (s : Finset (Card (d + 1))) (ε : Bool) :
-    Finset (Card d) := (s.filter (fun x => x 0 = ε)).image Fin.tail
-
-lemma card_childMarks {d : ℕ} (s : Finset (Card (d + 1))) (ε : Bool) :
-    (childMarks s ε).card = (s.filter (fun x => x 0 = ε)).card := by
-  classical
-  unfold childMarks
-  apply Finset.card_image_iff.mpr
-  intro x hx y hy hxy
-  have hx0 := (Finset.mem_filter.mp hx).2
-  have hy0 := (Finset.mem_filter.mp hy).2
-  rw [← Fin.cons_self_tail x, ← Fin.cons_self_tail y, hx0, hy0, hxy]
-
-lemma card_childMarks_add {d : ℕ} (s : Finset (Card (d + 1))) :
-    (childMarks s false).card + (childMarks s true).card = s.card := by
-  classical
-  rw [card_childMarks, card_childMarks]
-  simpa only [Bool.not_eq_false] using
-    (Finset.card_filter_add_card_filter_not (s := s) (fun x => x 0 = false))
-
-noncomputable def topMeetings {d : ℕ} (b : Butterfly (d + 1))
-    (s : Finset (Card (d + 1))) : Finset (Card d) :=
-  ((childMarks s false).image (butterflyPerm d (b.2 false))) ∩
-    ((childMarks s true).image (butterflyPerm d (b.2 true)))
-
-lemma card_topMeetings_le {d : ℕ} (b : Butterfly (d + 1)) (s : Finset (Card (d + 1))) :
-    (topMeetings b s).card ≤ min (childMarks s false).card (childMarks s true).card := by
-  classical
-  apply le_min
-  · exact (Finset.card_le_card Finset.inter_subset_left).trans_eq
-      (Finset.card_image_of_injective _ (butterflyPerm d (b.2 false)).injective)
-  · exact (Finset.card_le_card Finset.inter_subset_right).trans_eq
-      (Finset.card_image_of_injective _ (butterflyPerm d (b.2 true)).injective)
-
-noncomputable def encounters : (d : ℕ) → Butterfly d → Finset (Card d) → ℕ
-  | 0, _, _ => 0
-  | d + 1, b, s => encounters d (b.2 false) (childMarks s false) +
-      encounters d (b.2 true) (childMarks s true) + (topMeetings b s).card
-
-lemma butterfly_encounters (d : ℕ) (b : Butterfly d) (s : Finset (Card d)) :
-    (encounters d b s : ℝ) ≤ (s.card : ℝ) * Real.log s.card / (2 * Real.log 2) := by
-  induction d with
-  | zero =>
-    simp only [encounters, Nat.cast_zero]
-    have hs : s.card ≤ 1 := by
-      simpa [Card] using Finset.card_le_univ s
-    interval_cases s.card <;> simp
-  | succ d ih =>
-    have h₀ := ih (b.2 false) (childMarks s false)
-    have h₁ := ih (b.2 true) (childMarks s true)
-    have ht : ((topMeetings b s).card : ℝ) ≤
-        min ((childMarks s false).card : ℝ) ((childMarks s true).card : ℝ) := by
-      exact_mod_cast card_topMeetings_le b s
-    have he := split_entropy_lower ((childMarks s false).card : ℝ)
-      ((childMarks s true).card : ℝ) (by positivity) (by positivity)
-    have hs : ((childMarks s false).card : ℝ) + (childMarks s true).card = s.card := by
-      exact_mod_cast card_childMarks_add s
-    rw [hs] at he
-    have hL : 0 < 2 * Real.log 2 := by positivity
-    rw [le_div_iff₀ hL] at h₀ h₁ ⊢
-    simp only [encounters, Nat.cast_add]
-    nlinarith [mul_le_mul_of_nonneg_right ht hL.le]
-
-lemma prod_split_at {α : Type*} [DecidableEq α] (s : Finset α) (f : α → ℝ) (a : α) :
-    ∏ i ∈ s, f i = (if a ∈ s then f a else 1) * ∏ i ∈ s.erase a, f i := by
-  by_cases ha : a ∈ s
-  · rw [ite_eq_left ha]
-    exact (Finset.mul_prod_erase s f ha).symm
-  · simp [ha]
-
-lemma prod_split_two {α : Type*} [DecidableEq α] (s : Finset α) (f : α → ℝ)
-    (a b : α) (hab : a ≠ b) :
-    ∏ i ∈ s, f i = (if a ∈ s then f a else 1) * (if b ∈ s then f b else 1) *
-      ∏ i ∈ (s.erase a).erase b, f i := by
-  calc
-    ∏ i ∈ s, f i = (if a ∈ s then f a else 1) * ∏ i ∈ s.erase a, f i :=
-      prod_split_at s f a
-    _ = _ := by
-      rw [prod_split_at (s.erase a) f b]
-      have hm : (b ∈ s.erase a) ↔ b ∈ s :=
-        ⟨fun h => (Finset.mem_erase.mp h).2, fun h => Finset.mem_erase.mpr ⟨Ne.symm hab, h⟩⟩
-      simp only [hm, mul_assoc]
-
-lemma swap_average_product {α : Type*} [DecidableEq α] (s : Finset α) (p : α → ℝ)
-    (hp : ∀ i, 0 ≤ p i) (a b : α) :
-    ((∏ i ∈ s, p i) + ∏ i ∈ s, p (Equiv.swap a b i)) / 2 ≤
-      ∏ i ∈ s, (p i + p (Equiv.swap a b i)) / 2 := by
-  by_cases hab : a = b
-  · subst b
-    simp only [Equiv.swap_self, Equiv.refl_apply]
-    have h : (fun i => (p i + p i) / 2) = p := by funext i; ring
-    rw [h]
-    linarith
-  let t := (s.erase a).erase b
-  have hfix : ∀ i ∈ t, Equiv.swap a b i = i := by
-    intro i hi
-    exact Equiv.swap_apply_of_ne_of_ne (Finset.mem_erase.mp (Finset.mem_erase.mp hi).2).1
-      (Finset.mem_erase.mp hi).1
-  have hprod : (∏ i ∈ t, p (Equiv.swap a b i)) = ∏ i ∈ t, p i :=
-    Finset.prod_congr rfl (fun i hi => by rw [hfix i hi])
-  have havg : (∏ i ∈ t, (p i + p (Equiv.swap a b i)) / 2) = ∏ i ∈ t, p i :=
-    Finset.prod_congr rfl (fun i hi => by rw [hfix i hi]; ring)
-  rw [prod_split_two s p a b hab,
-    prod_split_two s (fun i => p (Equiv.swap a b i)) a b hab,
-    prod_split_two s (fun i => (p i + p (Equiv.swap a b i)) / 2) a b hab]
-  change ((if a ∈ s then p a else 1) * (if b ∈ s then p b else 1) * (∏ i ∈ t, p i) +
-    (if a ∈ s then p (Equiv.swap a b a) else 1) *
-      (if b ∈ s then p (Equiv.swap a b b) else 1) * (∏ i ∈ t, p (Equiv.swap a b i))) / 2 ≤ _
-  rw [hprod]
-  change _ ≤ (if a ∈ s then (p a + p (Equiv.swap a b a)) / 2 else 1) *
-    (if b ∈ s then (p b + p (Equiv.swap a b b)) / 2 else 1) *
-      (∏ i ∈ t, (p i + p (Equiv.swap a b i)) / 2)
-  rw [havg, Equiv.swap_apply_left, Equiv.swap_apply_right]
-  have ht : 0 ≤ ∏ i ∈ t, p i := Finset.prod_nonneg (fun i _ => hp i)
-  by_cases ha : a ∈ s <;> by_cases hb : b ∈ s <;> simp only [ha, hb, ↓reduceIte]
-  · have hsq := mul_nonneg (sq_nonneg (p a - p b)) ht
-    nlinarith
-  all_goals ring_nf; rfl
-
-def CylinderBound {α : Type*} (m : Finset α → ℝ) (p : α → ℝ) : Prop :=
-  ∀ s, m s ≤ ∏ i ∈ s, p i
-
-lemma CylinderBound.fair_swap {α : Type*} [DecidableEq α]
-    {m : Finset α → ℝ} {p : α → ℝ} (h : CylinderBound m p)
-    (hp : ∀ i, 0 ≤ p i) (a b : α) :
-    CylinderBound (fun s => (m s + m (s.image (Equiv.swap a b))) / 2)
-      (fun i => (p i + p (Equiv.swap a b i)) / 2) := by
-  intro s
-  have h₀ := h s
-  have h₁ := h (s.image (Equiv.swap a b))
-  rw [Finset.prod_image (Equiv.swap a b).injective.injOn] at h₁
-  exact (show (m s + m (s.image (Equiv.swap a b))) / 2 ≤
-      ((∏ i ∈ s, p i) + ∏ i ∈ s, p (Equiv.swap a b i)) / 2 by linarith).trans
-    (swap_average_product s p hp a b)
-
-noncomputable def inclusionMoment {α Ω : Type*} [Fintype Ω]
-    (F : Ω → Finset α) (s : Finset α) : ℝ := by
-  classical
-  exact finiteMean (fun ω => if s ⊆ F ω then 1 else 0)
-
-lemma inclusionMoment_nonneg {α Ω : Type*} [Fintype Ω]
-    (F : Ω → Finset α) (s : Finset α) : 0 ≤ inclusionMoment F s := by
-  classical
-  unfold inclusionMoment finiteMean
-  positivity
-
-lemma indicator_subset_prod {α : Type*} [DecidableEq α] (A s : Finset α) :
-    (if s ⊆ A then (1 : ℝ) else 0) = ∏ i ∈ s, (if i ∈ A then 1 else 0) := by
-  by_cases h : s ⊆ A
-  · rw [ite_eq_left h]
-    exact (Finset.prod_eq_one (fun i hi => ite_eq_left (h hi))).symm
-  · rw [ite_eq_right h]
-    obtain ⟨i, hi, hiA⟩ := Finset.not_subset.mp h
-    exact (Finset.prod_eq_zero hi (ite_eq_right hiA)).symm
-
-lemma inclusionMoment_const {α Ω : Type*} [DecidableEq α] [Fintype Ω] [Nonempty Ω]
-    (A s : Finset α) : inclusionMoment (fun _ : Ω => A) s =
-      if s ⊆ A then 1 else 0 := by
-  classical
-  have hn : (Fintype.card Ω : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
-  by_cases h : s ⊆ A <;> simp [inclusionMoment, finiteMean, h, hn]
-
-lemma cylinder_const {α Ω : Type*} [Fintype Ω] [Nonempty Ω] (A : Finset α) :
-    CylinderBound (inclusionMoment (fun _ : Ω => A))
-      (fun i => inclusionMoment (fun _ : Ω => A) {i}) := by
-  classical
-  intro s
-  simp only [inclusionMoment_const, Finset.singleton_subset_iff]
-  exact (indicator_subset_prod A s).le
-
-noncomputable def fairSetSwap {α Ω : Type*} [DecidableEq α]
-    (F : Ω → Finset α) (a b : α) : Ω × Bool → Finset α :=
-  fun ω => if ω.2 then (F ω.1).image (Equiv.swap a b) else F ω.1
-
-lemma subset_swap_image_iff {α : Type*} [DecidableEq α]
-    (A s : Finset α) (a b : α) :
-    s ⊆ A.image (Equiv.swap a b) ↔ s.image (Equiv.swap a b) ⊆ A := by
-  constructor
-  · intro h
-    have h' := Finset.image_subset_image h (f := Equiv.swap a b)
-    simpa only [Finset.image_image, Function.comp_def, Equiv.swap_apply_self, Finset.image_id'] using h'
-  · intro h
-    have h' := Finset.image_subset_image h (f := Equiv.swap a b)
-    simpa only [Finset.image_image, Function.comp_def, Equiv.swap_apply_self, Finset.image_id'] using h'
-
-lemma inclusionMoment_fairSetSwap {α Ω : Type*} [DecidableEq α] [Fintype Ω]
-    (F : Ω → Finset α) (a b : α) (s : Finset α) :
-    inclusionMoment (fairSetSwap F a b) s =
-      (inclusionMoment F s + inclusionMoment F (s.image (Equiv.swap a b))) / 2 := by
-  classical
-  simp only [inclusionMoment, finiteMean, Fintype.sum_prod_type, Fintype.sum_bool,
-    fairSetSwap, Bool.false_eq_true, ↓reduceIte, Fintype.card_prod, Fintype.card_bool,
-    Nat.cast_mul, Nat.cast_ofNat, subset_swap_image_iff, Finset.sum_add_distrib]
-  rw [div_mul_eq_div_div, add_div]
-  ring_nf
-
-lemma cylinder_fairSetSwap {α Ω : Type*} [DecidableEq α] [Fintype Ω]
-    (F : Ω → Finset α) (a b : α)
-    (h : CylinderBound (inclusionMoment F) (fun i => inclusionMoment F {i})) :
-    CylinderBound (inclusionMoment (fairSetSwap F a b))
-      (fun i => inclusionMoment (fairSetSwap F a b) {i}) := by
-  classical
-  have h' := h.fair_swap (fun i => inclusionMoment_nonneg F {i}) a b
-  intro s
-  simpa only [inclusionMoment_fairSetSwap, Finset.image_singleton] using h' s
-
-def SwitchSamples : ℕ → Type
-  | 0 => Unit
-  | n + 1 => SwitchSamples n × Bool
-
-instance (n : ℕ) : Fintype (SwitchSamples n) := by
-  induction n with
-  | zero => exact inferInstanceAs (Fintype Unit)
-  | succ n ih => exact inferInstanceAs (Fintype (SwitchSamples n × Bool))
-
-instance (n : ℕ) : Nonempty (SwitchSamples n) := by
-  induction n with
-  | zero => exact inferInstanceAs (Nonempty Unit)
-  | succ n ih => exact inferInstanceAs (Nonempty (SwitchSamples n × Bool))
-
-noncomputable def switchedSet {α : Type*} [DecidableEq α] (A : Finset α) :
-    (L : List (α × α)) → SwitchSamples L.length → Finset α
-  | [], _ => A
-  | (a,b) :: L, ω => fairSetSwap (switchedSet A L) a b ω
-
-lemma switchedSet_cylinder {α : Type*} [DecidableEq α] (A : Finset α)
-    (L : List (α × α)) :
-    CylinderBound (inclusionMoment (switchedSet A L))
-      (fun i => inclusionMoment (switchedSet A L) {i}) := by
-  induction L with
-  | nil => exact cylinder_const A
-  | cons e L ih => exact cylinder_fairSetSwap (switchedSet A L) e.1 e.2 ih
-
-inductive QueryTree (ι : Type*)
-  | reject : QueryTree ι
-  | accept : QueryTree ι
-  | branch (i : ι) (zero one : QueryTree ι) : QueryTree ι
-  | force (i : ι) (b : Bool) (next : QueryTree ι) : QueryTree ι
-
-namespace QueryTree
-
-variable {ι : Type*} [DecidableEq ι]
-
-def queries : QueryTree ι → Finset ι
-  | reject => ∅
-  | accept => ∅
-  | branch i zero one => insert i (queries zero ∪ queries one)
-  | force i _ next => insert i (queries next)
-
-def Fresh : QueryTree ι → Prop
-  | reject => True
-  | accept => True
-  | branch i zero one => i ∉ queries zero ∧ i ∉ queries one ∧ Fresh zero ∧ Fresh one
-  | force i _ next => i ∉ queries next ∧ Fresh next
-
-def weighted : QueryTree ι → (ι → Bool) → ℝ
-  | reject, _ => 0
-  | accept, _ => 1
-  | branch i zero one, ω => if ω i then weighted one ω else weighted zero ω
-  | force i b next, ω => if ω i = b then 2 * weighted next ω else 0
-
-omit [DecidableEq ι] in
-lemma weighted_nonneg (T : QueryTree ι) (ω : ι → Bool) : 0 ≤ T.weighted ω := by
-  induction T with
-  | reject => exact le_rfl
-  | accept => exact zero_le_one
-  | branch i zero one ih₀ ih₁ => simpa only [weighted] using ite_nonneg ih₁ ih₀
-  | force i b next ih => simpa only [weighted] using ite_nonneg (mul_nonneg (by norm_num) ih) le_rfl
-
-lemma weighted_congr (T : QueryTree ι) {ω η : ι → Bool}
-    (h : ∀ i ∈ T.queries, ω i = η i) : T.weighted ω = T.weighted η := by
-  induction T with
-  | reject => rfl
-  | accept => rfl
-  | branch i zero one ih₀ ih₁ =>
-    have hi := h i (Finset.mem_insert_self _ _)
-    simp only [weighted, hi]
-    rw [ih₀ (fun j hj => h j (Finset.mem_insert_of_mem (Finset.mem_union_left _ hj))),
-      ih₁ (fun j hj => h j (Finset.mem_insert_of_mem (Finset.mem_union_right _ hj)))]
-  | force i b next ih =>
-    have hi := h i (Finset.mem_insert_self _ _)
-    simp only [weighted, hi]
-    rw [ih (fun j hj => h j (Finset.mem_insert_of_mem hj))]
-
-def flipCoin (i : ι) : Equiv.Perm (ι → Bool) :=
-  (show Function.Involutive (fun ω : ι → Bool => Function.update ω i (!(ω i))) from by
-    intro ω
-    funext j
-    by_cases h : j = i
-    · subst j; simp
-    · simp [Function.update_of_ne h]).toPerm
-
-lemma flipCoin_apply_eq (i : ι) (ω : ι → Bool) : flipCoin i ω i = !(ω i) := by
-  change Function.update ω i (!(ω i)) i = _
-  exact Function.update_self _ _ _
-
-lemma flipCoin_apply_ne (i j : ι) (h : j ≠ i) (ω : ι → Bool) :
-    flipCoin i ω j = ω j := by
-  change Function.update ω i (!(ω i)) j = _
-  exact Function.update_of_ne h _ _
-
-lemma weighted_flip_of_not_mem (T : QueryTree ι) (i : ι) (hi : i ∉ T.queries)
-    (ω : ι → Bool) : T.weighted (flipCoin i ω) = T.weighted ω := by
-  apply weighted_congr
-  intro j hj
-  exact flipCoin_apply_ne i j (fun h => hi (h ▸ hj)) ω
-
-lemma mean_indicator_bit [Fintype ι] (f : (ι → Bool) → ℝ) (i : ι) (b : Bool)
-    (hf : ∀ ω, f (flipCoin i ω) = f ω) :
-    finiteMean (fun ω => if ω i = b then f ω else 0) = finiteMean f / 2 := by
-  let g := fun ω : ι → Bool => if ω i = b then f ω else 0
-  have hp (ω : ι → Bool) : g ω + g (flipCoin i ω) = f ω := by
-    dsimp [g]
-    rw [flipCoin_apply_eq, hf]
-    cases hω : ω i <;> cases b <;> simp
-  have hs := Equiv.sum_comp (flipCoin i) g
-  have hadd : (∑ ω, g ω) + (∑ ω, g (flipCoin i ω)) = ∑ ω, f ω := by
-    rw [← Finset.sum_add_distrib]
-    exact Finset.sum_congr rfl (fun ω _ => hp ω)
-  unfold finiteMean
-  change (∑ ω, g ω) / _ = _
-  rw [hs] at hadd
-  have hh : (∑ ω, g ω) = (∑ ω, f ω) / 2 := by linarith
-  rw [hh]
-  ring
-
-lemma mean_weighted_le_one [Fintype ι] (T : QueryTree ι) (hT : T.Fresh) :
-    finiteMean T.weighted ≤ 1 := by
-  induction T with
-  | reject => simp [weighted, finiteMean]
-  | accept => simp [weighted, finiteMean]
-  | branch i zero one ih₀ ih₁ =>
-    obtain ⟨hz, ho, hfz, hfo⟩ := hT
-    have h₀ := mean_indicator_bit zero.weighted i false (zero.weighted_flip_of_not_mem i hz)
-    have h₁ := mean_indicator_bit one.weighted i true (one.weighted_flip_of_not_mem i ho)
-    have he : finiteMean (branch i zero one).weighted =
-        finiteMean (fun ω => if ω i = false then zero.weighted ω else 0) +
-        finiteMean (fun ω => if ω i = true then one.weighted ω else 0) := by
-      unfold finiteMean
-      rw [← add_div, ← Finset.sum_add_distrib]
-      congr 1
-      apply Finset.sum_congr rfl
-      intro ω _
-      cases hω : ω i <;> simp [weighted, hω]
-    rw [he, h₀, h₁]
-    linarith [ih₀ hfz, ih₁ hfo]
-  | force i b next ih =>
-    obtain ⟨hi, hn⟩ := hT
-    have hm := mean_indicator_bit next.weighted i b (next.weighted_flip_of_not_mem i hi)
-    have he : finiteMean (force i b next).weighted =
-        2 * finiteMean (fun ω => if ω i = b then next.weighted ω else 0) := by
-      unfold finiteMean
-      rw [← mul_div_assoc, Finset.mul_sum]
-      congr 1
-      apply Finset.sum_congr rfl
-      intro ω _
-      by_cases h : ω i = b <;> simp [weighted, h]
-    rw [he, hm]
-    linarith [ih hn]
-
-def succeeds : QueryTree ι → (ι → Bool) → Bool
-  | reject, _ => false
-  | accept, _ => true
-  | branch i zero one, ω => if ω i then succeeds one ω else succeeds zero ω
-  | force i b next, ω => if ω i = b then succeeds next ω else false
-
-def charge : QueryTree ι → (ι → Bool) → ℕ
-  | reject, _ => 0
-  | accept, _ => 0
-  | branch i zero one, ω => if ω i then charge one ω else charge zero ω
-  | force i b next, ω => if ω i = b then charge next ω + 1 else 0
-
-omit [DecidableEq ι] in
-lemma weighted_eq (T : QueryTree ι) (ω : ι → Bool) :
-    T.weighted ω = if T.succeeds ω then (2 : ℝ) ^ T.charge ω else 0 := by
-  induction T with
-  | reject => rfl
-  | accept => norm_num [weighted, succeeds, charge]
-  | branch i zero one ih₀ ih₁ =>
-    cases h : ω i <;> simp [weighted, succeeds, charge, h, ih₀, ih₁]
-  | force i b next ih =>
-    by_cases h : ω i = b
-    · simp only [weighted, succeeds, charge, h, ↓reduceIte, ih]
-      cases hs : next.succeeds ω <;> simp [pow_succ, mul_comm]
-    · simp [weighted, succeeds, h]
-
-lemma success_probability_le [Fintype ι] (T : QueryTree ι) (hT : T.Fresh)
-    (K : ℕ) (hK : ∀ ω, T.succeeds ω = true → K ≤ T.charge ω) :
-    finiteMean (fun ω => if T.succeeds ω then (1 : ℝ) else 0) ≤ 1 / (2 : ℝ) ^ K := by
-  have hw (ω : ι → Bool) :
-      (2 : ℝ) ^ K * (if T.succeeds ω then (1 : ℝ) else 0) ≤ T.weighted ω := by
-    rw [weighted_eq]
-    cases hs : T.succeeds ω
-    · simp
-    · simpa using pow_le_pow_right₀ (show (1 : ℝ) ≤ 2 by norm_num) (hK ω hs)
-  have hn : 0 ≤ (Fintype.card (ι → Bool) : ℝ) := Nat.cast_nonneg _
-  have hm := div_le_div_of_nonneg_right
-    (Finset.sum_le_sum (fun ω (_ : ω ∈ Finset.univ) => hw ω)) hn
-  rw [← Finset.mul_sum, mul_div_assoc] at hm
-  change (2 : ℝ) ^ K * finiteMean (fun ω => if T.succeeds ω then 1 else 0) ≤
-    finiteMean T.weighted at hm
-  exact (le_div_iff₀ (pow_pos (by norm_num) K)).mpr (by
-    rw [mul_comm]
-    exact hm.trans (mean_weighted_le_one T hT))
-
-end QueryTree
-
-def encodeButterfly : (d : ℕ) → Butterfly d → SwitchIndex d → Bool
-  | 0, _, i => nomatch i
-  | _d + 1, b, Sum.inl y => b.1 y
-  | d + 1, b, Sum.inr (ε,i) => encodeButterfly d (b.2 ε) i
-
-lemma decode_encodeButterfly (d : ℕ) (b : Butterfly d) :
-    decodeButterfly d (encodeButterfly d b) = b := by
-  induction d with
-  | zero => exact Unit.ext _ _
-  | succ d ih =>
-    apply Prod.ext
-    · rfl
-    · funext ε
-      exact ih (b.2 ε)
-
-lemma encode_decodeButterfly (d : ℕ) (ω : SwitchIndex d → Bool) :
-    encodeButterfly d (decodeButterfly d ω) = ω := by
-  induction d with
-  | zero => funext i; exact Empty.elim i
-  | succ d ih =>
-    funext i
-    cases i with
-    | inl y => rfl
-    | inr z => exact congrFun (ih (fun i => ω (Sum.inr (z.1,i)))) z.2
-
-def butterflyCoinEquiv (d : ℕ) : (SwitchIndex d → Bool) ≃ Butterfly d where
-  toFun := decodeButterfly d
-  invFun := encodeButterfly d
-  left_inv := encode_decodeButterfly d
-  right_inv := decode_encodeButterfly d
-
-noncomputable def routeDomain : (d : ℕ) → Card d → Card d → Finset (SwitchIndex d)
-  | 0, _, _ => ∅
-  | d + 1, x, y => by
-      classical
-      exact insert (Sum.inl (Fin.tail y))
-        ((routeDomain d (Fin.tail x) (Fin.tail y)).image (fun i => Sum.inr (x 0,i)))
-
-def routeValue : (d : ℕ) → Card d → Card d → SwitchIndex d → Bool
-  | 0, _, _, i => nomatch i
-  | _d + 1, x, y, Sum.inl _ => Bool.xor (x 0) (y 0)
-  | d + 1, x, y, Sum.inr (_,i) => routeValue d (Fin.tail x) (Fin.tail y) i
-
-end Thorp
-
-end ThorpNine.HighTail
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.HighTail.Model
+
+namespace OAI
+
+namespace ThorpNine.HighTail
+
+namespace Thorp
+
+open scoped BigOperators
+open Filter
+
+def Coins : ℕ → Type
+  | 0 => Unit
+  | d + 1 => Card d → Bool
+
+instance (d : ℕ) : Fintype (Coins d) := by
+  cases d <;> dsimp [Coins] <;> infer_instance
+
+instance (d : ℕ) : Nonempty (Coins d) := by
+  cases d <;> dsimp [Coins] <;> infer_instance
+
+def rotate (d : ℕ) : Equiv.Perm (Card d) where
+  toFun x i := x (finRotate d i)
+  invFun x i := x ((finRotate d).symm i)
+  left_inv x := by funext i; simp only [Equiv.apply_symm_apply]
+  right_inv x := by funext i; simp only [Equiv.symm_apply_apply]
+
+def step : (d : ℕ) → Coins d → Equiv.Perm (Card d)
+  | 0, _ => 1
+  | d + 1, ξ => rotate (d + 1) * pairSwitch ξ
+
+def run (d : ℕ) : (t : ℕ) → (Fin t → Coins d) → Equiv.Perm (Card d)
+  | 0, _ => 1
+  | t + 1, ω => step d (ω (Fin.last t)) * run d t (fun i => ω i.castSucc)
+
+noncomputable def law (d t : ℕ) (g : Equiv.Perm (Card d)) : ℝ :=
+  (Fintype.card {ω : Fin t → Coins d // run d t ω = g} : ℝ) /
+    (Fintype.card (Fin t → Coins d) : ℝ)
+
+noncomputable def distance (d t : ℕ) : ℝ :=
+  (1 / 2 : ℝ) * ∑ g : Equiv.Perm (Card d),
+    |law d t g - 1 / (Nat.factorial (2 ^ d) : ℝ)|
+
+noncomputable def mixingTime (d : ℕ) : ℕ :=
+  sInf {t : ℕ | distance d t ≤ (1 / 4 : ℝ)}
+
+noncomputable def supportTimeBound (d : ℕ) : ℤ :=
+  ⌈(2 / (2 : ℝ) ^ d) *
+    (Real.log ((3 / 4 : ℝ) * (Nat.factorial (2 ^ d) : ℝ)) / Real.log 2)⌉
+
+def OptimalOrder : Prop :=
+  ∃ C : ℕ, 0 < C ∧
+    Tendsto (fun d : ℕ => distance d (C * d)) atTop (nhds 0) ∧
+    (∀ d t : ℕ, 1 - (2 : ℝ) ^ ((t : ℝ) * (2 : ℝ) ^ d / 2) /
+      (Nat.factorial (2 ^ d) : ℝ) ≤ distance d t) ∧
+    (∀ᶠ d : ℕ in atTop,
+      supportTimeBound d ≤ (mixingTime d : ℤ) ∧ mixingTime d ≤ C * d) ∧
+    (∃ K : ℝ, ∀ᶠ d : ℕ in atTop, 2 * (d : ℝ) - K ≤ (mixingTime d : ℝ)) ∧
+    Asymptotics.IsTheta atTop (fun d : ℕ => (mixingTime d : ℝ)) (fun d : ℕ => (d : ℝ))
+
+noncomputable def finiteLaw {α β : Type*} [Fintype α] (f : α → β) (b : β) : ℝ := by
+  classical
+  exact (Fintype.card {a : α // f a = b} : ℝ) / (Fintype.card α : ℝ)
+
+lemma sum_card_fibers {α β : Type*} [Fintype α] [Fintype β] [DecidableEq β] (f : α → β) :
+    ∑ b, Fintype.card {a : α // f a = b} = Fintype.card α := by
+  classical
+  rw [← Fintype.card_sigma]
+  exact Fintype.card_congr (Equiv.sigmaFiberEquiv f)
+
+lemma finiteLaw_nonneg {α β : Type*} [Fintype α] (f : α → β) (b : β) :
+    0 ≤ finiteLaw f b := by
+  exact div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _)
+
+lemma finiteLaw_sum {α β : Type*} [Fintype α] [Nonempty α] [Fintype β] (f : α → β) :
+    ∑ b, finiteLaw f b = 1 := by
+  classical
+  simp only [finiteLaw, ← Finset.sum_div]
+  have hc : (∑ b, (Fintype.card {a : α // f a = b} : ℝ)) = (Fintype.card α : ℝ) := by
+    exact_mod_cast sum_card_fibers f
+  rw [hc]
+  exact div_self (Nat.cast_ne_zero.mpr Fintype.card_ne_zero)
+
+lemma finiteLaw_eq_zero_of_not_mem {α β : Type*} [Fintype α] [DecidableEq β]
+    (f : α → β) {b : β} (hb : b ∉ Finset.univ.image f) : finiteLaw f b = 0 := by
+  classical
+  have : IsEmpty {a : α // f a = b} := ⟨fun a => hb (Finset.mem_image.mpr
+    ⟨a.val, Finset.mem_univ _, a.property⟩)⟩
+  simp [finiteLaw]
+
+lemma event_le_totalVariation {β : Type*} [Fintype β] [DecidableEq β]
+    (p q : β → ℝ) (hpq : ∑ b, (p b - q b) = 0) (s : Finset β) :
+    (∑ b ∈ s, p b) - (∑ b ∈ s, q b) ≤ (1 / 2 : ℝ) * ∑ b, |p b - q b| := by
+  have h₁ : (∑ b ∈ s, (p b - q b)) ≤ ∑ b ∈ s, |p b - q b| :=
+    Finset.sum_le_sum (fun b _ => le_abs_self (p b - q b))
+  have h₂ : -(∑ b ∈ sᶜ, (p b - q b)) ≤ ∑ b ∈ sᶜ, |p b - q b| := by
+    rw [← Finset.sum_neg_distrib]
+    exact Finset.sum_le_sum (fun b _ => neg_le_abs (p b - q b))
+  have hz := Finset.sum_add_sum_compl s (fun b => p b - q b)
+  have ha := Finset.sum_add_sum_compl s (fun b => |p b - q b|)
+  rw [hpq] at hz
+  rw [← Finset.sum_sub_distrib]
+  linarith
+
+lemma finiteLaw_support_bound {α β : Type*} [Fintype α] [Nonempty α]
+    [Fintype β] [Nonempty β] (f : α → β) :
+    1 - (Fintype.card α : ℝ) / (Fintype.card β : ℝ) ≤
+      (1 / 2 : ℝ) * ∑ b, |finiteLaw f b - 1 / (Fintype.card β : ℝ)| := by
+  classical
+  let s := Finset.univ.image f
+  have hs : ∑ b ∈ s, finiteLaw f b = 1 := by
+    have h := Finset.sum_subset (Finset.subset_univ s)
+      (fun b _ hb => finiteLaw_eq_zero_of_not_mem f hb)
+    simpa only [finiteLaw_sum] using h
+  have hβ : (Fintype.card β : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
+  have hq : (∑ _b : β, 1 / (Fintype.card β : ℝ)) = 1 := by simp [hβ]
+  have hd : (∑ b, (finiteLaw f b - 1 / (Fintype.card β : ℝ))) = 0 := by
+    rw [Finset.sum_sub_distrib, finiteLaw_sum, hq, sub_self]
+  have he := event_le_totalVariation (finiteLaw f) (fun _ => 1 / (Fintype.card β : ℝ)) hd s
+  rw [hs] at he
+  have hsq : (∑ _b ∈ s, 1 / (Fintype.card β : ℝ)) = (s.card : ℝ) / Fintype.card β := by
+    simp [div_eq_mul_inv]
+  rw [hsq] at he
+  have hcard : (s.card : ℝ) ≤ Fintype.card α := by
+    exact_mod_cast (Finset.card_image_le (s := (Finset.univ : Finset α)) (f := f))
+  have hdiv := div_le_div_of_nonneg_right hcard (Nat.cast_nonneg (Fintype.card β) : (0 : ℝ) ≤ _)
+  linarith
+
+lemma card_positions (d : ℕ) : Fintype.card (Card d) = 2 ^ d := by
+  simp [Card]
+
+lemma card_permutations (d : ℕ) : Fintype.card (Equiv.Perm (Card d)) = Nat.factorial (2 ^ d) := by
+  rw [Fintype.card_perm, card_positions]
+
+lemma card_coins_succ (d : ℕ) : Fintype.card (Coins (d + 1)) = 2 ^ (2 ^ d) := by
+  simp [Coins]
+
+lemma card_tapes_succ (d t : ℕ) : Fintype.card (Fin t → Coins (d + 1)) = 2 ^ (t * 2 ^ d) := by
+  rw [Fintype.card_fun, card_coins_succ, Fintype.card_fin, ← pow_mul, Nat.mul_comm]
+
+lemma law_eq_finiteLaw (d t : ℕ) (g : Equiv.Perm (Card d)) :
+    law d t g = finiteLaw (run d t) g := by
+  unfold law finiteLaw
+  apply congrArg (fun n : ℕ => (n : ℝ) / (Fintype.card (Fin t → Coins d) : ℝ))
+  exact congrArg (@Fintype.card _) (Subsingleton.elim _ _)
+
+lemma law_nonneg (d t : ℕ) (g : Equiv.Perm (Card d)) : 0 ≤ law d t g := by
+  rw [law_eq_finiteLaw]
+  exact finiteLaw_nonneg (run d t) g
+
+lemma law_sum (d t : ℕ) : ∑ g, law d t g = 1 := by
+  simp only [law_eq_finiteLaw]
+  exact finiteLaw_sum (run d t)
+
+lemma physical_support_bound_succ (d t : ℕ) :
+    1 - (2 : ℝ) ^ (t * 2 ^ d) / (Nat.factorial (2 ^ (d + 1)) : ℝ) ≤ distance (d + 1) t := by
+  have : Nonempty (Equiv.Perm (Card (d + 1))) := ⟨1⟩
+  have h := finiteLaw_support_bound (run (d + 1) t)
+  rw [card_tapes_succ, card_permutations] at h
+  simpa only [Nat.cast_pow, Nat.cast_ofNat, distance, law_eq_finiteLaw] using h
+
+lemma step_succ_formula (d : ℕ) (ξ : Coins (d + 1)) (x : Card (d + 1)) :
+    step (d + 1) ξ x = Fin.snoc (Fin.tail x) (Bool.xor (x 0) (ξ (Fin.tail x))) := by
+  change (fun i => switchFun ξ x (finRotate (d + 1) i)) = _
+  rw [Fin.snoc_eq_cons_rotate]
+  rfl
+
+lemma distance_nonneg (d t : ℕ) : 0 ≤ distance d t := by
+  unfold distance
+  exact mul_nonneg (by norm_num) (Finset.sum_nonneg (fun _ _ => abs_nonneg _))
+
+lemma physical_support_bound (d t : ℕ) :
+    1 - (2 : ℝ) ^ ((t : ℝ) * (2 : ℝ) ^ d / 2) /
+      (Nat.factorial (2 ^ d) : ℝ) ≤ distance d t := by
+  cases d with
+  | zero =>
+    have h : 1 ≤ (2 : ℝ) ^ ((t : ℝ) / 2) := Real.one_le_rpow (by norm_num) (by positivity)
+    simpa only [pow_zero, Nat.factorial_one, Nat.cast_one, div_one, mul_one] using
+      le_trans (sub_nonpos.mpr h) (distance_nonneg 0 t)
+  | succ d =>
+    have he : (t : ℝ) * (2 : ℝ) ^ (d + 1) / 2 = (t * 2 ^ d : ℕ) := by
+      push_cast
+      rw [pow_succ]
+      ring
+    rw [he, Real.rpow_natCast]
+    exact physical_support_bound_succ d t
+
+lemma supportTimeBound_le_of_distance_le (d t : ℕ) (ht : distance d t ≤ (1 / 4 : ℝ)) :
+    supportTimeBound d ≤ (t : ℤ) := by
+  have hF : (0 : ℝ) < Nat.factorial (2 ^ d) := by positivity
+  have hN : (0 : ℝ) < (2 : ℝ) ^ d := by positivity
+  have hL : (0 : ℝ) < Real.log 2 := Real.log_pos (by norm_num)
+  have hb := physical_support_bound d t
+  have hp : (3 / 4 : ℝ) ≤
+      (2 : ℝ) ^ ((t : ℝ) * (2 : ℝ) ^ d / 2) / (Nat.factorial (2 ^ d) : ℝ) := by
+    linarith
+  have hr := (le_div_iff₀ hF).mp hp
+  have hlog := Real.log_le_log (show (0 : ℝ) < (3 / 4 : ℝ) * Nat.factorial (2 ^ d) by positivity) hr
+  rw [Real.log_rpow (by norm_num : (0 : ℝ) < 2)] at hlog
+  unfold supportTimeBound
+  apply Int.ceil_le.mpr
+  simp only [Int.cast_natCast]
+  calc
+    2 / (2 : ℝ) ^ d * (Real.log ((3 / 4 : ℝ) * Nat.factorial (2 ^ d)) / Real.log 2) =
+        Real.log ((3 / 4 : ℝ) * Nat.factorial (2 ^ d)) / ((2 : ℝ) ^ d / 2 * Real.log 2) := by
+      field_simp
+    _ ≤ (t : ℝ) := (div_le_iff₀ (mul_pos (div_pos hN (by norm_num)) hL)).mpr (by
+      nlinarith [hlog])
+
+lemma supportTimeBound_le_mixingTime {d : ℕ}
+    (h : ({t : ℕ | distance d t ≤ (1 / 4 : ℝ)} : Set ℕ).Nonempty) :
+    supportTimeBound d ≤ (mixingTime d : ℤ) := by
+  exact supportTimeBound_le_of_distance_le d (mixingTime d) (Nat.sInf_mem h)
+
+lemma log_factorial_lower {n : ℕ} (hn : 1 ≤ n) :
+    (n : ℝ) * Real.log n - n ≤ Real.log (Nat.factorial n) := by
+  have h := Stirling.le_log_factorial_stirling (show n ≠ 0 by omega)
+  have hln : 0 ≤ Real.log (n : ℝ) := Real.log_nonneg (by exact_mod_cast hn)
+  have hlp : 0 ≤ Real.log (2 * Real.pi) := Real.log_nonneg (by
+    have := Real.pi_gt_three
+    linarith)
+  linarith
+
+lemma log_three_quarters_lower : -(1 : ℝ) ≤ Real.log (3 / 4 : ℝ) := by
+  have h := Real.log_le_sub_one_of_pos (show (0 : ℝ) < 4 / 3 by norm_num)
+  have he : Real.log (4 / 3 : ℝ) = - Real.log (3 / 4 : ℝ) := by
+    rw [← Real.log_inv]
+    congr 1
+    norm_num
+  rw [he] at h
+  linarith
+
+lemma linear_le_supportTimeBound (d : ℕ) :
+    2 * (d : ℝ) - 4 / Real.log 2 ≤ (supportTimeBound d : ℝ) := by
+  have hN : 1 ≤ (2 : ℝ) ^ d := one_le_pow₀ (by norm_num)
+  have hNp : 0 < (2 : ℝ) ^ d := by positivity
+  have hL : 0 < Real.log 2 := Real.log_pos (by norm_num)
+  have hf := log_factorial_lower (show 1 ≤ 2 ^ d by exact one_le_pow₀ (by norm_num))
+  have hlog : (2 : ℝ) ^ d * ((d : ℝ) * Real.log 2) - (2 : ℝ) ^ d - 1 ≤
+      Real.log ((3 / 4 : ℝ) * (Nat.factorial (2 ^ d) : ℝ)) := by
+    rw [Real.log_mul (by norm_num) (by positivity)]
+    push_cast at hf
+    rw [Real.log_pow] at hf
+    linarith [log_three_quarters_lower]
+  apply le_trans _ (Int.le_ceil _)
+  have hmul : (2 * (d : ℝ) - 4 / Real.log 2) *
+      ((2 : ℝ) ^ d / 2 * Real.log 2) ≤
+      Real.log ((3 / 4 : ℝ) * (Nat.factorial (2 ^ d) : ℝ)) := by
+    have hcancel : (2 * (d : ℝ) - 4 / Real.log 2) *
+        ((2 : ℝ) ^ d / 2 * Real.log 2) =
+        (2 : ℝ) ^ d * ((d : ℝ) * Real.log 2) - 2 * (2 : ℝ) ^ d := by
+      field_simp
+      ring
+    rw [hcancel]
+    linarith
+  convert (le_div_iff₀ (mul_pos (div_pos hNp (by norm_num)) hL)).mpr hmul using 1
+  field_simp
+
+lemma linear_le_time_of_distance_le (d t : ℕ)
+    (ht : distance d t ≤ (1 / 4 : ℝ)) :
+    2 * (d : ℝ) - 4 / Real.log 2 ≤ (t : ℝ) := by
+  have h := supportTimeBound_le_of_distance_le d t ht
+  have hr : (supportTimeBound d : ℝ) ≤ (t : ℝ) := by exact_mod_cast h
+  exact (linear_le_supportTimeBound d).trans hr
+
+lemma entropy_chord (p : ℝ) (hp : 0 ≤ p) (hq : p ≤ 1 / 2) :
+    2 * p * Real.log 2 ≤ Real.binEntropy p := by
+  have hc := Real.strictConcave_binEntropy.concaveOn.2
+    (show (0 : ℝ) ∈ Set.Icc (0 : ℝ) 1 by norm_num)
+    (show (2 : ℝ)⁻¹ ∈ Set.Icc (0 : ℝ) 1 by norm_num)
+    (show 0 ≤ 1 - 2 * p by linarith)
+    (show 0 ≤ 2 * p by positivity)
+    (show (1 - 2 * p) + 2 * p = 1 by ring)
+  simpa only [Real.binEntropy_zero, Real.binEntropy_two_inv,
+    smul_eq_mul, mul_zero, zero_add, show 2 * p * (2 : ℝ)⁻¹ = p by ring] using hc
+
+lemma split_entropy_lower (a b : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) :
+    2 * min a b * Real.log 2 ≤
+      (a + b) * Real.log (a + b) - a * Real.log a - b * Real.log b := by
+  wlog hab : a ≤ b generalizing a b
+  · have hba := this b a hb ha (le_of_not_ge hab)
+    simpa only [min_comm, add_comm b a] using hba.trans_eq (show (b + a) * Real.log (b + a) - b * Real.log b - a * Real.log a =
+      (a + b) * Real.log (a + b) - a * Real.log a - b * Real.log b by rw [add_comm b a]; ring)
+  rw [min_eq_left hab]
+  by_cases hz : a = 0
+  · subst a
+    simp
+  have hap : 0 < a := lt_of_le_of_ne ha (Ne.symm hz)
+  have hbp : 0 < b := hap.trans_le hab
+  have hs : 0 < a + b := add_pos hap hbp
+  have hp : 0 ≤ a / (a + b) := div_nonneg ha hs.le
+  have hq : a / (a + b) ≤ 1 / 2 := (div_le_iff₀ hs).mpr (by linarith)
+  have hc := mul_le_mul_of_nonneg_left (entropy_chord (a / (a + b)) hp hq) hs.le
+  have he : (a + b) * Real.binEntropy (a / (a + b)) =
+      (a + b) * Real.log (a + b) - a * Real.log a - b * Real.log b := by
+    unfold Real.binEntropy
+    rw [show 1 - a / (a + b) = b / (a + b) by field_simp; ring]
+    simp only [Real.log_inv]
+    rw [Real.log_div (ne_of_gt hap) (ne_of_gt hs),
+      Real.log_div (ne_of_gt hbp) (ne_of_gt hs)]
+    field_simp
+    ring
+  rw [he] at hc
+  calc
+    2 * a * Real.log 2 = (a + b) * (2 * (a / (a + b)) * Real.log 2) := by
+      field_simp
+    _ ≤ _ := hc
+
+noncomputable def childMarks {d : ℕ} (s : Finset (Card (d + 1))) (ε : Bool) :
+    Finset (Card d) := (s.filter (fun x => x 0 = ε)).image Fin.tail
+
+lemma card_childMarks {d : ℕ} (s : Finset (Card (d + 1))) (ε : Bool) :
+    (childMarks s ε).card = (s.filter (fun x => x 0 = ε)).card := by
+  classical
+  unfold childMarks
+  apply Finset.card_image_iff.mpr
+  intro x hx y hy hxy
+  have hx0 := (Finset.mem_filter.mp hx).2
+  have hy0 := (Finset.mem_filter.mp hy).2
+  rw [← Fin.cons_self_tail x, ← Fin.cons_self_tail y, hx0, hy0, hxy]
+
+lemma card_childMarks_add {d : ℕ} (s : Finset (Card (d + 1))) :
+    (childMarks s false).card + (childMarks s true).card = s.card := by
+  classical
+  rw [card_childMarks, card_childMarks]
+  simpa only [Bool.not_eq_false] using
+    (Finset.card_filter_add_card_filter_not (s := s) (fun x => x 0 = false))
+
+noncomputable def topMeetings {d : ℕ} (b : Butterfly (d + 1))
+    (s : Finset (Card (d + 1))) : Finset (Card d) :=
+  ((childMarks s false).image (butterflyPerm d (b.2 false))) ∩
+    ((childMarks s true).image (butterflyPerm d (b.2 true)))
+
+lemma card_topMeetings_le {d : ℕ} (b : Butterfly (d + 1)) (s : Finset (Card (d + 1))) :
+    (topMeetings b s).card ≤ min (childMarks s false).card (childMarks s true).card := by
+  classical
+  apply le_min
+  · exact (Finset.card_le_card Finset.inter_subset_left).trans_eq
+      (Finset.card_image_of_injective _ (butterflyPerm d (b.2 false)).injective)
+  · exact (Finset.card_le_card Finset.inter_subset_right).trans_eq
+      (Finset.card_image_of_injective _ (butterflyPerm d (b.2 true)).injective)
+
+noncomputable def encounters : (d : ℕ) → Butterfly d → Finset (Card d) → ℕ
+  | 0, _, _ => 0
+  | d + 1, b, s => encounters d (b.2 false) (childMarks s false) +
+      encounters d (b.2 true) (childMarks s true) + (topMeetings b s).card
+
+lemma butterfly_encounters (d : ℕ) (b : Butterfly d) (s : Finset (Card d)) :
+    (encounters d b s : ℝ) ≤ (s.card : ℝ) * Real.log s.card / (2 * Real.log 2) := by
+  induction d with
+  | zero =>
+    simp only [encounters, Nat.cast_zero]
+    have hs : s.card ≤ 1 := by
+      simpa [Card] using Finset.card_le_univ s
+    interval_cases s.card <;> simp
+  | succ d ih =>
+    have h₀ := ih (b.2 false) (childMarks s false)
+    have h₁ := ih (b.2 true) (childMarks s true)
+    have ht : ((topMeetings b s).card : ℝ) ≤
+        min ((childMarks s false).card : ℝ) ((childMarks s true).card : ℝ) := by
+      exact_mod_cast card_topMeetings_le b s
+    have he := split_entropy_lower ((childMarks s false).card : ℝ)
+      ((childMarks s true).card : ℝ) (by positivity) (by positivity)
+    have hs : ((childMarks s false).card : ℝ) + (childMarks s true).card = s.card := by
+      exact_mod_cast card_childMarks_add s
+    rw [hs] at he
+    have hL : 0 < 2 * Real.log 2 := by positivity
+    rw [le_div_iff₀ hL] at h₀ h₁ ⊢
+    simp only [encounters, Nat.cast_add]
+    nlinarith [mul_le_mul_of_nonneg_right ht hL.le]
+
+lemma prod_split_at {α : Type*} [DecidableEq α] (s : Finset α) (f : α → ℝ) (a : α) :
+    ∏ i ∈ s, f i = (if a ∈ s then f a else 1) * ∏ i ∈ s.erase a, f i := by
+  by_cases ha : a ∈ s
+  · rw [ite_eq_left ha]
+    exact (Finset.mul_prod_erase s f ha).symm
+  · simp [ha]
+
+lemma prod_split_two {α : Type*} [DecidableEq α] (s : Finset α) (f : α → ℝ)
+    (a b : α) (hab : a ≠ b) :
+    ∏ i ∈ s, f i = (if a ∈ s then f a else 1) * (if b ∈ s then f b else 1) *
+      ∏ i ∈ (s.erase a).erase b, f i := by
+  calc
+    ∏ i ∈ s, f i = (if a ∈ s then f a else 1) * ∏ i ∈ s.erase a, f i :=
+      prod_split_at s f a
+    _ = _ := by
+      rw [prod_split_at (s.erase a) f b]
+      have hm : (b ∈ s.erase a) ↔ b ∈ s :=
+        ⟨fun h => (Finset.mem_erase.mp h).2, fun h => Finset.mem_erase.mpr ⟨Ne.symm hab, h⟩⟩
+      simp only [hm, mul_assoc]
+
+lemma swap_average_product {α : Type*} [DecidableEq α] (s : Finset α) (p : α → ℝ)
+    (hp : ∀ i, 0 ≤ p i) (a b : α) :
+    ((∏ i ∈ s, p i) + ∏ i ∈ s, p (Equiv.swap a b i)) / 2 ≤
+      ∏ i ∈ s, (p i + p (Equiv.swap a b i)) / 2 := by
+  by_cases hab : a = b
+  · subst b
+    simp only [Equiv.swap_self, Equiv.refl_apply]
+    have h : (fun i => (p i + p i) / 2) = p := by funext i; ring
+    rw [h]
+    linarith
+  let t := (s.erase a).erase b
+  have hfix : ∀ i ∈ t, Equiv.swap a b i = i := by
+    intro i hi
+    exact Equiv.swap_apply_of_ne_of_ne (Finset.mem_erase.mp (Finset.mem_erase.mp hi).2).1
+      (Finset.mem_erase.mp hi).1
+  have hprod : (∏ i ∈ t, p (Equiv.swap a b i)) = ∏ i ∈ t, p i :=
+    Finset.prod_congr rfl (fun i hi => by rw [hfix i hi])
+  have havg : (∏ i ∈ t, (p i + p (Equiv.swap a b i)) / 2) = ∏ i ∈ t, p i :=
+    Finset.prod_congr rfl (fun i hi => by rw [hfix i hi]; ring)
+  rw [prod_split_two s p a b hab,
+    prod_split_two s (fun i => p (Equiv.swap a b i)) a b hab,
+    prod_split_two s (fun i => (p i + p (Equiv.swap a b i)) / 2) a b hab]
+  change ((if a ∈ s then p a else 1) * (if b ∈ s then p b else 1) * (∏ i ∈ t, p i) +
+    (if a ∈ s then p (Equiv.swap a b a) else 1) *
+      (if b ∈ s then p (Equiv.swap a b b) else 1) * (∏ i ∈ t, p (Equiv.swap a b i))) / 2 ≤ _
+  rw [hprod]
+  change _ ≤ (if a ∈ s then (p a + p (Equiv.swap a b a)) / 2 else 1) *
+    (if b ∈ s then (p b + p (Equiv.swap a b b)) / 2 else 1) *
+      (∏ i ∈ t, (p i + p (Equiv.swap a b i)) / 2)
+  rw [havg, Equiv.swap_apply_left, Equiv.swap_apply_right]
+  have ht : 0 ≤ ∏ i ∈ t, p i := Finset.prod_nonneg (fun i _ => hp i)
+  by_cases ha : a ∈ s <;> by_cases hb : b ∈ s <;> simp only [ha, hb, ↓reduceIte]
+  · have hsq := mul_nonneg (sq_nonneg (p a - p b)) ht
+    nlinarith
+  all_goals ring_nf; rfl
+
+def CylinderBound {α : Type*} (m : Finset α → ℝ) (p : α → ℝ) : Prop :=
+  ∀ s, m s ≤ ∏ i ∈ s, p i
+
+lemma CylinderBound.fair_swap {α : Type*} [DecidableEq α]
+    {m : Finset α → ℝ} {p : α → ℝ} (h : CylinderBound m p)
+    (hp : ∀ i, 0 ≤ p i) (a b : α) :
+    CylinderBound (fun s => (m s + m (s.image (Equiv.swap a b))) / 2)
+      (fun i => (p i + p (Equiv.swap a b i)) / 2) := by
+  intro s
+  have h₀ := h s
+  have h₁ := h (s.image (Equiv.swap a b))
+  rw [Finset.prod_image (Equiv.swap a b).injective.injOn] at h₁
+  exact (show (m s + m (s.image (Equiv.swap a b))) / 2 ≤
+      ((∏ i ∈ s, p i) + ∏ i ∈ s, p (Equiv.swap a b i)) / 2 by linarith).trans
+    (swap_average_product s p hp a b)
+
+noncomputable def inclusionMoment {α Ω : Type*} [Fintype Ω]
+    (F : Ω → Finset α) (s : Finset α) : ℝ := by
+  classical
+  exact finiteMean (fun ω => if s ⊆ F ω then 1 else 0)
+
+lemma inclusionMoment_nonneg {α Ω : Type*} [Fintype Ω]
+    (F : Ω → Finset α) (s : Finset α) : 0 ≤ inclusionMoment F s := by
+  classical
+  unfold inclusionMoment finiteMean
+  positivity
+
+lemma indicator_subset_prod {α : Type*} [DecidableEq α] (A s : Finset α) :
+    (if s ⊆ A then (1 : ℝ) else 0) = ∏ i ∈ s, (if i ∈ A then 1 else 0) := by
+  by_cases h : s ⊆ A
+  · rw [ite_eq_left h]
+    exact (Finset.prod_eq_one (fun i hi => ite_eq_left (h hi))).symm
+  · rw [ite_eq_right h]
+    obtain ⟨i, hi, hiA⟩ := Finset.not_subset.mp h
+    exact (Finset.prod_eq_zero hi (ite_eq_right hiA)).symm
+
+lemma inclusionMoment_const {α Ω : Type*} [DecidableEq α] [Fintype Ω] [Nonempty Ω]
+    (A s : Finset α) : inclusionMoment (fun _ : Ω => A) s =
+      if s ⊆ A then 1 else 0 := by
+  classical
+  have hn : (Fintype.card Ω : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
+  by_cases h : s ⊆ A <;> simp [inclusionMoment, finiteMean, h, hn]
+
+lemma cylinder_const {α Ω : Type*} [Fintype Ω] [Nonempty Ω] (A : Finset α) :
+    CylinderBound (inclusionMoment (fun _ : Ω => A))
+      (fun i => inclusionMoment (fun _ : Ω => A) {i}) := by
+  classical
+  intro s
+  simp only [inclusionMoment_const, Finset.singleton_subset_iff]
+  exact (indicator_subset_prod A s).le
+
+noncomputable def fairSetSwap {α Ω : Type*} [DecidableEq α]
+    (F : Ω → Finset α) (a b : α) : Ω × Bool → Finset α :=
+  fun ω => if ω.2 then (F ω.1).image (Equiv.swap a b) else F ω.1
+
+lemma subset_swap_image_iff {α : Type*} [DecidableEq α]
+    (A s : Finset α) (a b : α) :
+    s ⊆ A.image (Equiv.swap a b) ↔ s.image (Equiv.swap a b) ⊆ A := by
+  constructor
+  · intro h
+    have h' := Finset.image_subset_image h (f := Equiv.swap a b)
+    simpa only [Finset.image_image, Function.comp_def, Equiv.swap_apply_self, Finset.image_id'] using h'
+  · intro h
+    have h' := Finset.image_subset_image h (f := Equiv.swap a b)
+    simpa only [Finset.image_image, Function.comp_def, Equiv.swap_apply_self, Finset.image_id'] using h'
+
+lemma inclusionMoment_fairSetSwap {α Ω : Type*} [DecidableEq α] [Fintype Ω]
+    (F : Ω → Finset α) (a b : α) (s : Finset α) :
+    inclusionMoment (fairSetSwap F a b) s =
+      (inclusionMoment F s + inclusionMoment F (s.image (Equiv.swap a b))) / 2 := by
+  classical
+  simp only [inclusionMoment, finiteMean, Fintype.sum_prod_type, Fintype.sum_bool,
+    fairSetSwap, Bool.false_eq_true, ↓reduceIte, Fintype.card_prod, Fintype.card_bool,
+    Nat.cast_mul, Nat.cast_ofNat, subset_swap_image_iff, Finset.sum_add_distrib]
+  rw [div_mul_eq_div_div, add_div]
+  ring_nf
+
+lemma cylinder_fairSetSwap {α Ω : Type*} [DecidableEq α] [Fintype Ω]
+    (F : Ω → Finset α) (a b : α)
+    (h : CylinderBound (inclusionMoment F) (fun i => inclusionMoment F {i})) :
+    CylinderBound (inclusionMoment (fairSetSwap F a b))
+      (fun i => inclusionMoment (fairSetSwap F a b) {i}) := by
+  classical
+  have h' := h.fair_swap (fun i => inclusionMoment_nonneg F {i}) a b
+  intro s
+  simpa only [inclusionMoment_fairSetSwap, Finset.image_singleton] using h' s
+
+def SwitchSamples : ℕ → Type
+  | 0 => Unit
+  | n + 1 => SwitchSamples n × Bool
+
+instance (n : ℕ) : Fintype (SwitchSamples n) := by
+  induction n with
+  | zero => exact inferInstanceAs (Fintype Unit)
+  | succ n ih => exact inferInstanceAs (Fintype (SwitchSamples n × Bool))
+
+instance (n : ℕ) : Nonempty (SwitchSamples n) := by
+  induction n with
+  | zero => exact inferInstanceAs (Nonempty Unit)
+  | succ n ih => exact inferInstanceAs (Nonempty (SwitchSamples n × Bool))
+
+noncomputable def switchedSet {α : Type*} [DecidableEq α] (A : Finset α) :
+    (L : List (α × α)) → SwitchSamples L.length → Finset α
+  | [], _ => A
+  | (a,b) :: L, ω => fairSetSwap (switchedSet A L) a b ω
+
+lemma switchedSet_cylinder {α : Type*} [DecidableEq α] (A : Finset α)
+    (L : List (α × α)) :
+    CylinderBound (inclusionMoment (switchedSet A L))
+      (fun i => inclusionMoment (switchedSet A L) {i}) := by
+  induction L with
+  | nil => exact cylinder_const A
+  | cons e L ih => exact cylinder_fairSetSwap (switchedSet A L) e.1 e.2 ih
+
+inductive QueryTree (ι : Type*)
+  | reject : QueryTree ι
+  | accept : QueryTree ι
+  | branch (i : ι) (zero one : QueryTree ι) : QueryTree ι
+  | force (i : ι) (b : Bool) (next : QueryTree ι) : QueryTree ι
+
+namespace QueryTree
+
+variable {ι : Type*} [DecidableEq ι]
+
+def queries : QueryTree ι → Finset ι
+  | reject => ∅
+  | accept => ∅
+  | branch i zero one => insert i (queries zero ∪ queries one)
+  | force i _ next => insert i (queries next)
+
+def Fresh : QueryTree ι → Prop
+  | reject => True
+  | accept => True
+  | branch i zero one => i ∉ queries zero ∧ i ∉ queries one ∧ Fresh zero ∧ Fresh one
+  | force i _ next => i ∉ queries next ∧ Fresh next
+
+def weighted : QueryTree ι → (ι → Bool) → ℝ
+  | reject, _ => 0
+  | accept, _ => 1
+  | branch i zero one, ω => if ω i then weighted one ω else weighted zero ω
+  | force i b next, ω => if ω i = b then 2 * weighted next ω else 0
+
+omit [DecidableEq ι] in
+lemma weighted_nonneg (T : QueryTree ι) (ω : ι → Bool) : 0 ≤ T.weighted ω := by
+  induction T with
+  | reject => exact le_rfl
+  | accept => exact zero_le_one
+  | branch i zero one ih₀ ih₁ => simpa only [weighted] using ite_nonneg ih₁ ih₀
+  | force i b next ih => simpa only [weighted] using ite_nonneg (mul_nonneg (by norm_num) ih) le_rfl
+
+lemma weighted_congr (T : QueryTree ι) {ω η : ι → Bool}
+    (h : ∀ i ∈ T.queries, ω i = η i) : T.weighted ω = T.weighted η := by
+  induction T with
+  | reject => rfl
+  | accept => rfl
+  | branch i zero one ih₀ ih₁ =>
+    have hi := h i (Finset.mem_insert_self _ _)
+    simp only [weighted, hi]
+    rw [ih₀ (fun j hj => h j (Finset.mem_insert_of_mem (Finset.mem_union_left _ hj))),
+      ih₁ (fun j hj => h j (Finset.mem_insert_of_mem (Finset.mem_union_right _ hj)))]
+  | force i b next ih =>
+    have hi := h i (Finset.mem_insert_self _ _)
+    simp only [weighted, hi]
+    rw [ih (fun j hj => h j (Finset.mem_insert_of_mem hj))]
+
+def flipCoin (i : ι) : Equiv.Perm (ι → Bool) :=
+  (show Function.Involutive (fun ω : ι → Bool => Function.update ω i (!(ω i))) from by
+    intro ω
+    funext j
+    by_cases h : j = i
+    · subst j; simp
+    · simp [Function.update_of_ne h]).toPerm
+
+lemma flipCoin_apply_eq (i : ι) (ω : ι → Bool) : flipCoin i ω i = !(ω i) := by
+  change Function.update ω i (!(ω i)) i = _
+  exact Function.update_self _ _ _
+
+lemma flipCoin_apply_ne (i j : ι) (h : j ≠ i) (ω : ι → Bool) :
+    flipCoin i ω j = ω j := by
+  change Function.update ω i (!(ω i)) j = _
+  exact Function.update_of_ne h _ _
+
+lemma weighted_flip_of_not_mem (T : QueryTree ι) (i : ι) (hi : i ∉ T.queries)
+    (ω : ι → Bool) : T.weighted (flipCoin i ω) = T.weighted ω := by
+  apply weighted_congr
+  intro j hj
+  exact flipCoin_apply_ne i j (fun h => hi (h ▸ hj)) ω
+
+lemma mean_indicator_bit [Fintype ι] (f : (ι → Bool) → ℝ) (i : ι) (b : Bool)
+    (hf : ∀ ω, f (flipCoin i ω) = f ω) :
+    finiteMean (fun ω => if ω i = b then f ω else 0) = finiteMean f / 2 := by
+  let g := fun ω : ι → Bool => if ω i = b then f ω else 0
+  have hp (ω : ι → Bool) : g ω + g (flipCoin i ω) = f ω := by
+    dsimp [g]
+    rw [flipCoin_apply_eq, hf]
+    cases hω : ω i <;> cases b <;> simp
+  have hs := Equiv.sum_comp (flipCoin i) g
+  have hadd : (∑ ω, g ω) + (∑ ω, g (flipCoin i ω)) = ∑ ω, f ω := by
+    rw [← Finset.sum_add_distrib]
+    exact Finset.sum_congr rfl (fun ω _ => hp ω)
+  unfold finiteMean
+  change (∑ ω, g ω) / _ = _
+  rw [hs] at hadd
+  have hh : (∑ ω, g ω) = (∑ ω, f ω) / 2 := by linarith
+  rw [hh]
+  ring
+
+lemma mean_weighted_le_one [Fintype ι] (T : QueryTree ι) (hT : T.Fresh) :
+    finiteMean T.weighted ≤ 1 := by
+  induction T with
+  | reject => simp [weighted, finiteMean]
+  | accept => simp [weighted, finiteMean]
+  | branch i zero one ih₀ ih₁ =>
+    obtain ⟨hz, ho, hfz, hfo⟩ := hT
+    have h₀ := mean_indicator_bit zero.weighted i false (zero.weighted_flip_of_not_mem i hz)
+    have h₁ := mean_indicator_bit one.weighted i true (one.weighted_flip_of_not_mem i ho)
+    have he : finiteMean (branch i zero one).weighted =
+        finiteMean (fun ω => if ω i = false then zero.weighted ω else 0) +
+        finiteMean (fun ω => if ω i = true then one.weighted ω else 0) := by
+      unfold finiteMean
+      rw [← add_div, ← Finset.sum_add_distrib]
+      congr 1
+      apply Finset.sum_congr rfl
+      intro ω _
+      cases hω : ω i <;> simp [weighted, hω]
+    rw [he, h₀, h₁]
+    linarith [ih₀ hfz, ih₁ hfo]
+  | force i b next ih =>
+    obtain ⟨hi, hn⟩ := hT
+    have hm := mean_indicator_bit next.weighted i b (next.weighted_flip_of_not_mem i hi)
+    have he : finiteMean (force i b next).weighted =
+        2 * finiteMean (fun ω => if ω i = b then next.weighted ω else 0) := by
+      unfold finiteMean
+      rw [← mul_div_assoc, Finset.mul_sum]
+      congr 1
+      apply Finset.sum_congr rfl
+      intro ω _
+      by_cases h : ω i = b <;> simp [weighted, h]
+    rw [he, hm]
+    linarith [ih hn]
+
+def succeeds : QueryTree ι → (ι → Bool) → Bool
+  | reject, _ => false
+  | accept, _ => true
+  | branch i zero one, ω => if ω i then succeeds one ω else succeeds zero ω
+  | force i b next, ω => if ω i = b then succeeds next ω else false
+
+def charge : QueryTree ι → (ι → Bool) → ℕ
+  | reject, _ => 0
+  | accept, _ => 0
+  | branch i zero one, ω => if ω i then charge one ω else charge zero ω
+  | force i b next, ω => if ω i = b then charge next ω + 1 else 0
+
+omit [DecidableEq ι] in
+lemma weighted_eq (T : QueryTree ι) (ω : ι → Bool) :
+    T.weighted ω = if T.succeeds ω then (2 : ℝ) ^ T.charge ω else 0 := by
+  induction T with
+  | reject => rfl
+  | accept => norm_num [weighted, succeeds, charge]
+  | branch i zero one ih₀ ih₁ =>
+    cases h : ω i <;> simp [weighted, succeeds, charge, h, ih₀, ih₁]
+  | force i b next ih =>
+    by_cases h : ω i = b
+    · simp only [weighted, succeeds, charge, h, ↓reduceIte, ih]
+      cases hs : next.succeeds ω <;> simp [pow_succ, mul_comm]
+    · simp [weighted, succeeds, h]
+
+lemma success_probability_le [Fintype ι] (T : QueryTree ι) (hT : T.Fresh)
+    (K : ℕ) (hK : ∀ ω, T.succeeds ω = true → K ≤ T.charge ω) :
+    finiteMean (fun ω => if T.succeeds ω then (1 : ℝ) else 0) ≤ 1 / (2 : ℝ) ^ K := by
+  have hw (ω : ι → Bool) :
+      (2 : ℝ) ^ K * (if T.succeeds ω then (1 : ℝ) else 0) ≤ T.weighted ω := by
+    rw [weighted_eq]
+    cases hs : T.succeeds ω
+    · simp
+    · simpa using pow_le_pow_right₀ (show (1 : ℝ) ≤ 2 by norm_num) (hK ω hs)
+  have hn : 0 ≤ (Fintype.card (ι → Bool) : ℝ) := Nat.cast_nonneg _
+  have hm := div_le_div_of_nonneg_right
+    (Finset.sum_le_sum (fun ω (_ : ω ∈ Finset.univ) => hw ω)) hn
+  rw [← Finset.mul_sum, mul_div_assoc] at hm
+  change (2 : ℝ) ^ K * finiteMean (fun ω => if T.succeeds ω then 1 else 0) ≤
+    finiteMean T.weighted at hm
+  exact (le_div_iff₀ (pow_pos (by norm_num) K)).mpr (by
+    rw [mul_comm]
+    exact hm.trans (mean_weighted_le_one T hT))
+
+end QueryTree
+
+def encodeButterfly : (d : ℕ) → Butterfly d → SwitchIndex d → Bool
+  | 0, _, i => nomatch i
+  | _d + 1, b, Sum.inl y => b.1 y
+  | d + 1, b, Sum.inr (ε,i) => encodeButterfly d (b.2 ε) i
+
+lemma decode_encodeButterfly (d : ℕ) (b : Butterfly d) :
+    decodeButterfly d (encodeButterfly d b) = b := by
+  induction d with
+  | zero => exact Unit.ext _ _
+  | succ d ih =>
+    apply Prod.ext
+    · rfl
+    · funext ε
+      exact ih (b.2 ε)
+
+lemma encode_decodeButterfly (d : ℕ) (ω : SwitchIndex d → Bool) :
+    encodeButterfly d (decodeButterfly d ω) = ω := by
+  induction d with
+  | zero => funext i; exact Empty.elim i
+  | succ d ih =>
+    funext i
+    cases i with
+    | inl y => rfl
+    | inr z => exact congrFun (ih (fun i => ω (Sum.inr (z.1,i)))) z.2
+
+def butterflyCoinEquiv (d : ℕ) : (SwitchIndex d → Bool) ≃ Butterfly d where
+  toFun := decodeButterfly d
+  invFun := encodeButterfly d
+  left_inv := encode_decodeButterfly d
+  right_inv := decode_encodeButterfly d
+
+noncomputable def routeDomain : (d : ℕ) → Card d → Card d → Finset (SwitchIndex d)
+  | 0, _, _ => ∅
+  | d + 1, x, y => by
+      classical
+      exact insert (Sum.inl (Fin.tail y))
+        ((routeDomain d (Fin.tail x) (Fin.tail y)).image (fun i => Sum.inr (x 0,i)))
+
+def routeValue : (d : ℕ) → Card d → Card d → SwitchIndex d → Bool
+  | 0, _, _, i => nomatch i
+  | _d + 1, x, y, Sum.inl _ => Bool.xor (x 0) (y 0)
+  | d + 1, x, y, Sum.inr (_,i) => routeValue d (Fin.tail x) (Fin.tail y) i
+
+end Thorp
+
+end ThorpNine.HighTail
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/HighTail/ColoredEmbedding.lean b/lean/OAI/Probability/ThorpRouting/HighTail/ColoredEmbedding.lean
index 64a9d939795da92fe762ea82ff0bdd776fdd3660..ec0e2aa35469e95de6bc1b27158e972082ec92a1 100644
--- a/lean/OAI/Probability/ThorpRouting/HighTail/ColoredEmbedding.lean
+++ b/lean/OAI/Probability/ThorpRouting/HighTail/ColoredEmbedding.lean
@@ -1,842 +1,842 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.HighTail.ProjectCycle
-
-namespace OAI
-
-namespace ThorpNine.HighTail
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-namespace PairRouting
-variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α] [DecidableEq ι]
-
-lemma sandwich_power_bound {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
-    (P : Ω → Equiv.Perm α) (Q : Ω' → Equiv.Perm α) (x y : ι → Bool × α)
-    (a : ℝ) (ha : 0 ≤ a) :
-    (finiteMean (fun ω : Ω × Ω' => finiteMean (fun coins : (α → Bool) × (α → Bool) =>
-      if ∀ i, sandwich coins.1 coins.2 (fun b => if b then Q ω.2 else P ω.1) (x i) = y i
-      then (1:ℝ) else 0)))^(1+a) ≤
-    (outerWeight x y)^(1+a) * ((compatibleSet x y).card:ℝ)^a *
-      ∑ c ∈ compatibleSet x y,
-      (tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
-        (fun i : {i // c i = false} => (y i.val).2))^(1+a) *
-      (tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
-        (fun i : {i // c i = true} => (y i.val).2))^(1+a) := by
-  classical
-  rw [sandwich_mean,sum_colors_factor]
-  let f (c : ι → Bool) : ℝ :=
-      tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
-        (fun i : {i // c i = false} => (y i.val).2) *
-      tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
-        (fun i : {i // c i = true} => (y i.val).2)
-  have hf (c : ι → Bool) : 0 ≤ f c := mul_nonneg (tupleProbability_nonneg _ _ _)
-    (tupleProbability_nonneg _ _ _)
-  rw [Real.mul_rpow (le_of_lt (outerWeight_pos _ _)) (Finset.sum_nonneg (fun c _ => hf c))]
-  have hh := Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg (compatibleSet x y)
-    (show 1 ≤ 1+a by linarith) (fun c _ => hf c)
-  have he : 1+a-1=a := by ring
-  rw [he] at hh
-  calc
-    _ ≤ (outerWeight x y)^(1+a) *
-        (((compatibleSet x y).card:ℝ)^a * ∑ c ∈ compatibleSet x y, (f c)^(1+a)) :=
-      mul_le_mul_of_nonneg_left hh (Real.rpow_nonneg (le_of_lt (outerWeight_pos _ _)) _)
-    _ = _ := by
-      rw [←mul_assoc]
-      congr 1
-      apply Finset.sum_congr rfl
-      intro c _
-      exact Real.mul_rpow (tupleProbability_nonneg _ _ _) (tupleProbability_nonneg _ _ _)
-
-end PairRouting
-
-lemma childLift_mul (d : ℕ) (p q : Bool → Equiv.Perm (Card d)) :
-    childLift p * childLift q = childLift (fun b => p b * q b) := by
-  ext x i
-  simp only [Equiv.Perm.mul_apply,childLift,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons]
-
-lemma childLift_inv (d : ℕ) (p : Bool → Equiv.Perm (Card d)) :
-    (childLift p)⁻¹ = childLift (fun b => (p b)⁻¹) := rfl
-
-lemma pairSwitch_inv (d : ℕ) (ξ : Card d → Bool) : (pairSwitch ξ)⁻¹ = pairSwitch ξ := rfl
-
-lemma palindromePerm_step (d : ℕ) (c : BenesCoins d × BenesCoins d)
-    (coins : (Card d → Bool) × (Card d → Bool)) :
-    palindromePerm (d+1) ((benesStepEquiv d).symm (c,coins)) =
-      pairSwitch coins.2 * childLift (fun b => palindromePerm d (if b then c.2 else c.1)) *
-        pairSwitch coins.1 := by
-  change (butterflyPerm (d+1) (decodeButterfly (d+1)
-      ((coinStepEquiv d).symm ((c.1.1,c.2.1),coins.2)))) *
-    (butterflyPerm (d+1) (decodeButterfly (d+1)
-      ((coinStepEquiv d).symm ((c.1.2,c.2.2),coins.1))))⁻¹ = _
-  rw [butterflyPerm_step,butterflyPerm_step,mul_inv_rev,childLift_inv,pairSwitch_inv]
-  rw [mul_assoc (pairSwitch coins.2),←mul_assoc (childLift _),childLift_mul,←mul_assoc]
-  congr 2
-  apply congrArg childLift
-  funext b
-  cases b <;> rfl
-
-lemma headTail_switch (d : ℕ) (ξ : Card d → Bool) (x : Card (d+1)) :
-    headTailEquiv d (pairSwitch ξ x) = pairLayer ξ (headTailEquiv d x) := rfl
-
-lemma headTail_child (d : ℕ) (p : Bool → Equiv.Perm (Card d)) (x : Card (d+1)) :
-    headTailEquiv d (childLift p x) = PairRouting.lift p (headTailEquiv d x) := rfl
-
-lemma palindromePerm_coordinates (d : ℕ) (c : BenesCoins d × BenesCoins d)
-    (coins : (Card d → Bool) × (Card d → Bool)) (x : Card (d+1)) :
-    headTailEquiv d (palindromePerm (d+1) ((benesStepEquiv d).symm (c,coins)) x) =
-      PairRouting.sandwich coins.1 coins.2 (fun b => palindromePerm d (if b then c.2 else c.1))
-        (headTailEquiv d x) := by
-  rw [palindromePerm_step]
-  simp only [Equiv.Perm.mul_apply,headTail_switch,headTail_child]
-  rfl
-
-namespace CycleColoring
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-lemma alternating_cycle_two (p : Equiv.Perm α) (S : Finset (Bool × α))
-    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) S) : 2 ≤ C.val.card := by
-  obtain ⟨x,hx⟩ := C.nonempty
-  have hpx : alternatingPerm p x ∈ C.val := by
-    rw [←C.orbit_eq hx,mem_orbitSet]
-    exact (Equiv.Perm.SameCycle.refl _ _).apply_right
-  have hne : x ≠ alternatingPerm p x := by
-    intro he
-    have hh := congrArg Prod.fst he
-    rw [alternatingPerm_flip] at hh
-    cases hb : x.1 <;> simp only [hb, Bool.not_false, Bool.not_true] at hh <;> cases hh
-  have hs : {x,alternatingPerm p x} ⊆ C.val := by
-    intro y hy
-    simp only [Finset.mem_insert,Finset.mem_singleton] at hy
-    rcases hy with rfl | rfl <;> assumption
-  have hh := Finset.card_le_card hs
-  simpa only [Finset.card_pair hne] using hh
-
-lemma alternating_cycle_count_double (p : Equiv.Perm α) (S : Finset (Bool × α)) :
-    2 * Fintype.card (RoutingNetwork.SelectedCycle (alternatingPerm p) S) ≤ S.card := by
-  classical
-  calc
-    _ = ∑ _C : RoutingNetwork.SelectedCycle (alternatingPerm p) S, 2 := by simp [Nat.mul_comm]
-    _ ≤ RoutingNetwork.cycleMass (alternatingPerm p) S :=
-      Finset.sum_le_sum (fun C _ => alternating_cycle_two p S C)
-    _ ≤ _ := RoutingNetwork.cycleMass_le _ _
-
-lemma alternating_cycle_count_zero (p : Equiv.Perm α) (S : Finset (Bool × α))
-    (hS : S.card ≤ 1) : Fintype.card (RoutingNetwork.SelectedCycle (alternatingPerm p) S) = 0 := by
-  have h := alternating_cycle_count_double p S
-  omega
-
-end CycleColoring
-
-namespace PairRouting
-
-section
-variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
-
-lemma alternatingCycles_double (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    2 * alternatingCycles e p ≤ Fintype.card ι := by
-  have hh := CycleColoring.alternating_cycle_count_double (alternatingProjection p) (labelSet e)
-  simpa only [alternatingCycles, card_labelSet] using hh
-
-lemma alternatingCycles_zero (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α)
-    (hι : Fintype.card ι ≤ 1) : alternatingCycles e p = 0 := by
-  have hh := alternatingCycles_double e p
-  omega
-
-end
-variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α] [DecidableEq ι]
-
-omit [Fintype α] [DecidableEq ι] in
-lemma colors_chosenCoin (x : ι → Bool × α) (c : ι → Bool) (hc : Compatible x c) :
-    colors x (chosenCoin x c) = c := by
-  funext i
-  simp only [colors,chosenCoin_apply x c hc i,required]
-  cases (x i).1 <;> cases c i <;> rfl
-
-noncomputable def coloredEmbedding (x : ι ↪ Bool × α) (c : ι → Bool) (_hc : Compatible x c) :
-    ι ↪ Bool × α := switchedEmbedding x (chosenCoin x c)
-
-omit [Fintype α] [DecidableEq ι] in
-lemma coloredEmbedding_apply (x : ι ↪ Bool × α) (c : ι → Bool) (hc : Compatible x c) (i : ι) :
-    coloredEmbedding x c hc i = (c i,(x i).2) := by
-  apply Prod.ext
-  · exact congrFun (colors_chosenCoin x c hc) i
-  · rfl
-
-omit [Fintype α] [DecidableEq ι] in
-lemma switchedEmbedding_eq_colored (x : ι ↪ Bool × α) (c : ι → Bool) (hc : Compatible x c)
-    (ξ : α → Bool) (he : colors x ξ = c) :
-    switchedEmbedding x ξ = coloredEmbedding x c hc := by
-  ext i : 1
-  rw [coloredEmbedding_apply]
-  apply Prod.ext
-  · exact congrFun he i
-  · rfl
-
-lemma assignments_match_bound (x y : ι ↪ Bool × α) (c : ι → Bool)
-    (hc : c ∈ compatibleSet x y) (p : Bool → Equiv.Perm α) (hm : ChildMatches p x y c) :
-    (compatibleSet x y).card ≤
-      2 ^ ((occupied x).card + (occupied y).card - Fintype.card ι +
-        alternatingCycles (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p) := by
-  obtain ⟨hcx,hcy⟩ := (mem_compatibleSet x y c).mp hc
-  let ξ := chosenCoin x c
-  let η := chosenCoin y c
-  have he : routedEmbedding x ξ η p = y := by
-    ext i : 1
-    have h := (sandwich_matches_iff ξ η p x y).mpr
-      ⟨(colors_chosenCoin x c hcx).trans (colors_chosenCoin y c hcy).symm,by
-        simpa only [ξ,colors_chosenCoin x c hcx] using hm⟩
-    exact h i
-  have hh := assignments_actual_bound x ξ η p
-  rw [he] at hh
-  rw [card_compatibleSet]
-  exact hh
-
-omit [Fintype α] [DecidableEq ι] in
-lemma occupied_sum_ge (x y : ι ↪ Bool × α) :
-    Fintype.card ι ≤ (occupied x).card + (occupied y).card := by
-  have h₀ := occupied_card_double x
-  have h₁ := occupied_card_double y
-  omega
-
-lemma assignment_weight_bound (x y : ι ↪ Bool × α) (c : ι → Bool)
-    (hc : c ∈ compatibleSet x y) (p : Bool → Equiv.Perm α) (hm : ChildMatches p x y c) :
-    ((compatibleSet x y).card:ℝ) * outerWeight x y ≤
-      (2:ℝ)^((alternatingCycles
-        (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p : ℝ) - Fintype.card ι) := by
-  let z := alternatingCycles (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p
-  let b := (occupied x).card+(occupied y).card
-  let h := Fintype.card ι
-  have hb : h ≤ b := occupied_sum_ge x y
-  have hn : ((compatibleSet x y).card:ℝ) ≤ (2:ℝ)^(b-h+z) := by
-    exact_mod_cast assignments_match_bound x y c hc p hm
-  change ((compatibleSet x y).card:ℝ) * (1/(2:ℝ)^b) ≤ (2:ℝ)^((z:ℝ)-h)
-  rw [mul_one_div]
-  calc
-    _ ≤ (2:ℝ)^(b-h+z)/(2:ℝ)^b := div_le_div_of_nonneg_right hn (by positivity)
-    _ = _ := by
-      rw [←Real.rpow_natCast,←Real.rpow_natCast,←Real.rpow_sub (by norm_num : (0:ℝ)<2)]
-      congr 1
-      rw [Nat.cast_add,Nat.cast_sub hb]
-      ring
-
-lemma assignment_factor_bound (x y : ι ↪ Bool × α) (c : ι → Bool)
-    (hc : c ∈ compatibleSet x y) (p : Bool → Equiv.Perm α) (hm : ChildMatches p x y c)
-    (a : ℝ) (ha : 0 ≤ a) :
-    (outerWeight x y)^a * ((compatibleSet x y).card:ℝ)^a ≤
-      (2:ℝ)^(((alternatingCycles
-        (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p : ℝ) - Fintype.card ι)*a) := by
-  have h := Real.rpow_le_rpow (mul_nonneg (Nat.cast_nonneg _) (le_of_lt (outerWeight_pos x y)))
-    (assignment_weight_bound x y c hc p hm) ha
-  rw [Real.mul_rpow (Nat.cast_nonneg _) (le_of_lt (outerWeight_pos x y)),
-    ←Real.rpow_mul (by norm_num : (0:ℝ)≤2)] at h
-  simpa only [mul_comm] using h
-
-
-lemma sandwich_weighted_probability_subtype (p : Bool → Equiv.Perm α) (x y : ι → Bool × α)
-    (W : (ι → Bool) → ℝ) :
-    finiteMean (fun coins : (α → Bool) × (α → Bool) =>
-      if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then W (colors x coins.1) else 0) =
-    outerWeight x y * ∑ c : {c // c ∈ compatibleSet x y},
-      if ChildMatches p x y c.val then W c.val else 0 := by
-  rw [sandwich_weighted_probability]
-  congr 1
-  rw [Finset.sum_subtype (compatibleSet x y) (fun _ => Iff.rfl)]
-
-omit [Fintype α] [DecidableEq α] [DecidableEq ι] in
-lemma color_card_add (c : ι → Bool) :
-    Fintype.card {i // c i = false} + Fintype.card {i // c i = true} = Fintype.card ι := by
-  have he : {i // c i = true} ≃ {i // ¬c i = false} := Equiv.subtypeEquivRight fun i => by
-    cases c i <;> simp
-  rw [Fintype.card_congr he]
-  rw [Fintype.card_subtype_compl]
-  exact Nat.add_sub_of_le (Fintype.card_subtype_le _)
-
-end PairRouting
-
-noncomputable def assignmentCost (d : ℕ) {ι : Type*} [Fintype ι]
-    (x : ι ↪ Bool × Card d) (c : ι → Bool) (hc : PairRouting.Compatible x c)
-    (ω : BenesCoins d × BenesCoins d) : ℕ :=
-  PairRouting.alternatingCycles (PairRouting.coloredEmbedding x c hc)
-      (fun b => palindromePerm d (if b then ω.2 else ω.1)) +
-    palindromeCost d (PairRouting.childEmbedding x c hc false) ω.1 +
-    palindromeCost d (PairRouting.childEmbedding x c hc true) ω.2
-
-lemma palindromeCost_eq (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (d+1)) (σ : BenesCoins (d+1)) :
-    palindromeCost (d+1) e σ =
-      assignmentCost d (e.trans (headTailEquiv d).toEmbedding)
-        (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) (benesStepEquiv d σ).2.1)
-        (PairRouting.colors_compatible _ _) (benesStepEquiv d σ).1 := by
-  unfold palindromeCost assignmentCost
-  dsimp only
-  rw [PairRouting.switchedEmbedding_eq_colored _ _ (PairRouting.colors_compatible _ _) _ rfl]
-
-lemma palindromeCost_step (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (d+1)) (ω : BenesCoins d × BenesCoins d)
-    (coins : (Card d → Bool) × (Card d → Bool)) :
-    palindromeCost (d+1) e ((benesStepEquiv d).symm (ω,coins)) =
-      assignmentCost d (e.trans (headTailEquiv d).toEmbedding)
-        (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) coins.1)
-        (PairRouting.colors_compatible _ _) ω := by
-  rw [palindromeCost_eq]
-  simp only [Equiv.apply_symm_apply]
-
-noncomputable def routingMoment (d : ℕ) {ι : Type*} [Fintype ι]
-    (e f : ι ↪ Card d) (a : ℝ) : ℝ := by
-  classical
-  exact finiteMean (fun ω : BenesCoins d =>
-    if ∀ i, palindromePerm d ω (e i) = f i then (2:ℝ)^((palindromeCost d e ω:ℝ)*a) else 0)
-
-lemma routingMoment_nonneg (d : ℕ) {ι : Type*} [Fintype ι]
-    (e f : ι ↪ Card d) (a : ℝ) : 0 ≤ routingMoment d e f a := by
-  classical
-  exact finiteMean_nonneg (fun _ => by split_ifs <;> positivity)
-
-lemma palindrome_matches_step (d : ℕ) {ι : Type*}
-    (e f : ι ↪ Card (d+1)) (ω : BenesCoins d × BenesCoins d)
-    (coins : (Card d → Bool) × (Card d → Bool)) :
-    (∀ i, palindromePerm (d+1) ((benesStepEquiv d).symm (ω,coins)) (e i) = f i) ↔
-    (∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => palindromePerm d (if b then ω.2 else ω.1))
-      ((e.trans (headTailEquiv d).toEmbedding) i) = ((f.trans (headTailEquiv d).toEmbedding) i)) := by
-  apply forall_congr'
-  intro i
-  rw [←(headTailEquiv d).injective.eq_iff,palindromePerm_coordinates]
-  rfl
-
-lemma routingMoment_step (d : ℕ) {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (e f : ι ↪ Card (d+1)) (a : ℝ) :
-    let x := e.trans (headTailEquiv d).toEmbedding
-    let y := f.trans (headTailEquiv d).toEmbedding
-    routingMoment (d+1) e f a = PairRouting.outerWeight x y *
-      ∑ c : {c // c ∈ PairRouting.compatibleSet x y},
-        finiteMean (fun ω : BenesCoins d × BenesCoins d =>
-          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c.val
-          then (2:ℝ)^((assignmentCost d x c.val
-            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0) := by
-  classical
-  intro x y
-  rw [routingMoment,finiteMean_equiv (benesStepEquiv d),finiteMean_prod]
-  have he (ω : BenesCoins d × BenesCoins d) :
-      finiteMean (fun coins : (Card d → Bool) × (Card d → Bool) =>
-        if ∀ i, palindromePerm (d+1) ((benesStepEquiv d).symm (ω,coins)) (e i) = f i
-        then (2:ℝ)^((palindromeCost (d+1) e ((benesStepEquiv d).symm (ω,coins)):ℝ)*a) else 0) =
-      PairRouting.outerWeight x y *
-        ∑ c : {c // c ∈ PairRouting.compatibleSet x y},
-          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c.val
-          then (2:ℝ)^((assignmentCost d x c.val
-            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0 := by
-    let W : (ι → Bool) → ℝ := fun c =>
-      if hc : PairRouting.Compatible x c then (2:ℝ)^((assignmentCost d x c hc ω:ℝ)*a) else 0
-    calc
-      _ = finiteMean (fun coins : (Card d → Bool) × (Card d → Bool) =>
-        if ∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => palindromePerm d (if b then ω.2 else ω.1))
-          (x i) = y i then W (PairRouting.colors x coins.1) else 0) := by
-        apply finiteMean_congr
-        intro coins
-        simp only [palindrome_matches_step,palindromeCost_step]
-        simp only [W,PairRouting.colors_compatible,↓reduceDIte]
-        rfl
-      _ = _ := by
-        rw [PairRouting.sandwich_weighted_probability_subtype]
-        congr 1
-        apply Finset.sum_congr rfl
-        intro c _
-        simp only [W,((PairRouting.mem_compatibleSet x y c.val).mp c.property).1,↓reduceDIte]
-  simp_rw [he]
-  simp_rw [mul_comm (PairRouting.outerWeight x y)]
-  rw [finiteMean_mul_const,finiteMean_sum,mul_comm]
-
-lemma child_moment_product (d : ℕ) {ι : Type*} [Fintype ι]
-    (x y : ι ↪ Bool × Card d) (c : ι → Bool)
-    (hcx : PairRouting.Compatible x c) (hcy : PairRouting.Compatible y c) (a : ℝ) :
-    routingMoment d (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a *
-      routingMoment d (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a =
-    finiteMean (fun ω : BenesCoins d × BenesCoins d =>
-      if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
-      then (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
-        palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0) := by
-  classical
-  unfold routingMoment
-  rw [←finiteMean_prod_mul]
-  apply finiteMean_congr
-  intro ω
-  let M₀ := ∀ i : {i // c i = false}, palindromePerm d ω.1 (x i.val).2 = (y i.val).2
-  let M₁ := ∀ i : {i // c i = true}, palindromePerm d ω.2 (x i.val).2 = (y i.val).2
-  have hp : (fun b : Bool => palindromePerm d (if b then ω.2 else ω.1)) =
-      (fun b => if b then palindromePerm d ω.2 else palindromePerm d ω.1) := by
-    funext b
-    cases b <;> rfl
-  have hs : PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c ↔
-      M₀ ∧ M₁ := by rw [hp]; exact PairRouting.childMatches_split _ _ _ _ _
-  change (if M₀ then _ else 0) * (if M₁ then _ else 0) = _
-  by_cases h₀ : M₀ <;> by_cases h₁ : M₁
-  · simp only [hs,h₀,h₁,true_and,↓reduceIte]
-    rw [←Real.rpow_add (by norm_num : (0:ℝ)<2)]
-    congr 1
-    ring
-  · simp only [hs,h₀,h₁,and_false,↓reduceIte,mul_zero]
-  · simp only [hs,h₀,h₁,false_and,↓reduceIte,zero_mul]
-  · simp only [hs,h₀,h₁,false_and,↓reduceIte,mul_zero]
-
-lemma palindrome_probability_step (d : ℕ) {ι : Type*} [Fintype ι]
-    (e f : ι ↪ Card (d+1)) :
-    PairRouting.tupleProbability (palindromePerm (d+1)) e f =
-    finiteMean (fun ω : BenesCoins d × BenesCoins d =>
-      finiteMean (fun coins : (Card d → Bool) × (Card d → Bool) =>
-        if ∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => if b then palindromePerm d ω.2 else palindromePerm d ω.1)
-          ((e.trans (headTailEquiv d).toEmbedding) i) = ((f.trans (headTailEquiv d).toEmbedding) i)
-        then (1:ℝ) else 0)) := by
-  classical
-  rw [PairRouting.tupleProbability,finiteMean_equiv (benesStepEquiv d),finiteMean_prod]
-  have hp (b : Bool) (p q : BenesCoins d) :
-      palindromePerm d (if b then p else q) = (if b then palindromePerm d p else palindromePerm d q) := by
-    cases b <;> rfl
-  simp only [palindrome_matches_step,hp]
-
-lemma child_power_product_bound (d : ℕ) {ι : Type*} [Fintype ι]
-    (x y : ι ↪ Bool × Card d) (c : ι → Bool)
-    (hcx : PairRouting.Compatible x c) (hcy : PairRouting.Compatible y c) (a : ℝ)
-    (hfalse : (PairRouting.tupleProbability (palindromePerm d)
-        (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false))^(1+a) ≤
-      (2:ℝ)^(-(d:ℝ)*Fintype.card {i // c i = false}*a) *
-        routingMoment d (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a)
-    (htrue : (PairRouting.tupleProbability (palindromePerm d)
-        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true))^(1+a) ≤
-      (2:ℝ)^(-(d:ℝ)*Fintype.card {i // c i = true}*a) *
-        routingMoment d (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a) :
-    (PairRouting.tupleProbability (palindromePerm d)
-      (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false))^(1+a) *
-      (PairRouting.tupleProbability (palindromePerm d)
-        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true))^(1+a) ≤
-    (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) *
-      finiteMean (fun ω : BenesCoins d × BenesCoins d =>
-        if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
-        then (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
-          palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0) := by
-  have hh := mul_le_mul hfalse htrue
-    (Real.rpow_nonneg (PairRouting.tupleProbability_nonneg _ _ _) _)
-    (mul_nonneg (by positivity) (routingMoment_nonneg _ _ _ _))
-  rw [mul_mul_mul_comm,child_moment_product,←Real.rpow_add (by norm_num : (0:ℝ)<2)] at hh
-  have he : -(d:ℝ)*Fintype.card {i // c i = false}*a + -(d:ℝ)*Fintype.card {i // c i = true}*a =
-      -(d:ℝ)*Fintype.card ι*a := by
-    rw [←add_mul,←mul_add,←Nat.cast_add,PairRouting.color_card_add]
-  rwa [he] at hh
-
-lemma assignment_moment_bound (d : ℕ) {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (x y : ι ↪ Bool × Card d) (c : ι → Bool) (hc : c ∈ PairRouting.compatibleSet x y)
-    (a : ℝ) (ha : 0 ≤ a) :
-    let hcx := ((PairRouting.mem_compatibleSet x y c).mp hc).1
-    PairRouting.outerWeight x y ^ a * ((PairRouting.compatibleSet x y).card:ℝ)^a *
-      ((2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) *
-        finiteMean (fun ω : BenesCoins d × BenesCoins d =>
-          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
-          then (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
-            palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0)) ≤
-    (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) *
-      finiteMean (fun ω : BenesCoins d × BenesCoins d =>
-        if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
-        then (2:ℝ)^((assignmentCost d x c hcx ω:ℝ)*a) else 0) := by
-  classical
-  intro hcx
-  have hscale (q : ℝ) (F : BenesCoins d × BenesCoins d → ℝ) :
-      q * finiteMean F = finiteMean (fun ω => q * F ω) := by
-    simp_rw [mul_comm q,finiteMean_mul_const]
-  rw [←mul_assoc,hscale,hscale]
-  apply finiteMean_mono
-  intro ω
-  split_ifs with hm
-  · have hh := PairRouting.assignment_factor_bound x y c hc
-      (fun b => palindromePerm d (if b then ω.2 else ω.1)) hm a ha
-    calc
-      _ ≤ ((2:ℝ)^(((PairRouting.alternatingCycles (PairRouting.coloredEmbedding x c hcx)
-          (fun b => palindromePerm d (if b then ω.2 else ω.1)):ℝ)-Fintype.card ι)*a) *
-          (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a)) *
-          (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
-            palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) :=
-        mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hh (by positivity)) (by positivity)
-      _ = _ := by
-        simp only [←Real.rpow_add (by norm_num : (0:ℝ)<2),assignmentCost,Nat.cast_add]
-        congr 1
-        ring
-  · simp only [mul_zero]
-    exact le_rfl
-
-theorem palindrome_pointwise_moment (d : ℕ) {ι : Type*} [Fintype ι]
-    (e f : ι ↪ Card d) (a : ℝ) (ha : 0 ≤ a) :
-    (PairRouting.tupleProbability (palindromePerm d) e f)^(1+a) ≤
-      (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) * routingMoment d e f a := by
-  classical
-  induction d generalizing ι with
-  | zero =>
-      have he : ∀ (ω : BenesCoins 0) (i : ι), palindromePerm 0 ω (e i) = f i :=
-        fun _ _ => Subsingleton.elim _ _
-      simp [PairRouting.tupleProbability,routingMoment,he,palindromeCost,finiteMean_const]
-  | succ d ih =>
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let y := f.trans (headTailEquiv d).toEmbedding
-      let W := PairRouting.outerWeight x y
-      let C := PairRouting.compatibleSet x y
-      let p : (ι → Bool) → ℝ := fun c =>
-        (PairRouting.tupleProbability (palindromePerm d)
-          (fun i : {i // c i = false} => (x i.val).2)
-          (fun i : {i // c i = false} => (y i.val).2))^(1+a) *
-        (PairRouting.tupleProbability (palindromePerm d)
-          (fun i : {i // c i = true} => (x i.val).2)
-          (fun i : {i // c i = true} => (y i.val).2))^(1+a)
-      let m : {c // c ∈ C} → ℝ := fun c =>
-        finiteMean (fun ω : BenesCoins d × BenesCoins d =>
-          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c.val
-          then (2:ℝ)^((assignmentCost d x c.val
-            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0)
-      have hp (c : {c // c ∈ C}) : W^a * (C.card:ℝ)^a * p c.val ≤
-          (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * m c := by
-        obtain ⟨hcx,hcy⟩ := (PairRouting.mem_compatibleSet x y c.val).mp c.property
-        have h₀ := ih (PairRouting.childEmbedding x c.val hcx false)
-          (PairRouting.childEmbedding y c.val hcy false)
-        have h₁ := ih (PairRouting.childEmbedding x c.val hcx true)
-          (PairRouting.childEmbedding y c.val hcy true)
-        have hh := child_power_product_bound d x y c.val hcx hcy a h₀ h₁
-        have hh' := mul_le_mul_of_nonneg_left hh
-          (mul_nonneg (Real.rpow_nonneg (le_of_lt (PairRouting.outerWeight_pos x y)) a)
-            (Real.rpow_nonneg (Nat.cast_nonneg C.card) a))
-        exact hh'.trans (assignment_moment_bound d x y c.val c.property a ha)
-      have hnode := PairRouting.sandwich_power_bound (palindromePerm d) (palindromePerm d) x y a ha
-      rw [←palindrome_probability_step d e f] at hnode
-      change (PairRouting.tupleProbability (palindromePerm (d+1)) e f)^(1+a) ≤
-        W^(1+a) * (C.card:ℝ)^a * ∑ c ∈ C, p c at hnode
-      have hsum : ∑ c ∈ C, p c = ∑ c : {c // c ∈ C}, p c.val := by
-        rw [Finset.sum_subtype C (fun _ => Iff.rfl)]
-      have hW : 0 < W := PairRouting.outerWeight_pos x y
-      calc
-        _ ≤ W^(1+a) * (C.card:ℝ)^a * ∑ c ∈ C, p c := hnode
-        _ = W * ∑ c : {c // c ∈ C}, W^a * (C.card:ℝ)^a * p c.val := by
-          rw [Real.rpow_add hW,Real.rpow_one,hsum,←Finset.mul_sum]
-          ring
-        _ ≤ W * ∑ c : {c // c ∈ C}, (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * m c :=
-          mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun c _ => hp c)) (le_of_lt hW)
-        _ = _ := by
-          rw [←Finset.mul_sum,routingMoment_step]
-          simp only [Nat.cast_add,Nat.cast_one]
-          change W * ((2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * ∑ c, m c) =
-            (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * (W * ∑ c, m c)
-          ring
-
-lemma routingMoment_sum (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (a : ℝ) :
-    (∑ f : ι ↪ Card d, routingMoment d e f a) =
-      finiteMean (fun ω : BenesCoins d => (2:ℝ)^((palindromeCost d e ω:ℝ)*a)) := by
-  classical
-  unfold routingMoment
-  rw [←finiteMean_sum]
-  apply finiteMean_congr
-  intro ω
-  let f₀ : ι ↪ Card d := e.trans (palindromePerm d ω).toEmbedding
-  rw [Finset.sum_eq_single f₀]
-  · have hm : ∀ i, palindromePerm d ω (e i) = f₀ i := fun _ => rfl
-    simp only [hm,implies_true,↓reduceIte]
-  · intro f _ hf
-    have hm : ¬ ∀ i, palindromePerm d ω (e i) = f i := by
-      intro hh
-      apply hf
-      ext i : 1
-      exact (hh i).symm
-    exact ite_eq_right hm
-  · simp
-
-lemma finite_row_power {X : Type*} [Fintype X] [Nonempty X]
-    (p : X → ℝ) (hp : ∀ x, 0 ≤ p x) (a : ℝ) :
-    finiteMean (fun x => ((Fintype.card X:ℝ)*p x)^(1+a)) =
-      (Fintype.card X:ℝ)^a * ∑ x, (p x)^(1+a) := by
-  have hn : (0:ℝ) < Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
-  simp only [finiteMean,Real.mul_rpow (le_of_lt hn) (hp _),←Finset.mul_sum]
-  rw [Real.rpow_add hn,Real.rpow_one]
-  field_simp
-
-noncomputable def palindromeRowMoment (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (a : ℝ) : ℝ :=
-  finiteMean (fun f : ι ↪ Card d =>
-    ((Fintype.card (ι ↪ Card d):ℝ)*PairRouting.tupleProbability (palindromePerm d) e f)^(1+a))
-
-lemma density_normalization (d h n : ℕ) (a : ℝ) :
-    (n:ℝ)^a * (2:ℝ)^(-(d:ℝ)*h*a) = ((n:ℝ)/((2:ℝ)^d)^h)^a := by
-  rw [Real.div_rpow (Nat.cast_nonneg _) (by positivity),←pow_mul,
-    ←Real.rpow_natCast,←Real.rpow_mul (by norm_num : (0:ℝ) ≤ 2),Nat.cast_mul]
-  rw [show -(d:ℝ)*h*a = -((d:ℝ)*h*a) by ring,
-    Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2),div_eq_mul_inv]
-
-theorem palindrome_row_density_moment (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (a : ℝ) (ha : 0 ≤ a) :
-    palindromeRowMoment d e a ≤
-      (((2^d).descFactorial (Fintype.card ι):ℝ)/((2^d:ℕ):ℝ)^(Fintype.card ι))^a *
-        finiteMean (fun ω : BenesCoins d => (2:ℝ)^((palindromeCost d e ω:ℝ)*a)) := by
-  classical
-  have : Nonempty (ι ↪ Card d) := ⟨e⟩
-  rw [palindromeRowMoment,finite_row_power _ (fun f => PairRouting.tupleProbability_nonneg _ _ _)]
-  calc
-    _ ≤ (Fintype.card (ι ↪ Card d):ℝ)^a *
-      ∑ f : ι ↪ Card d, (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) * routingMoment d e f a :=
-      mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun f _ => palindrome_pointwise_moment d e f a ha))
-        (Real.rpow_nonneg (Nat.cast_nonneg _) _)
-    _ = _ := by
-      rw [←Finset.mul_sum,routingMoment_sum,←mul_assoc,density_normalization,
-        Fintype.card_embedding_eq,card_positions]
-      norm_cast
-
-end Thorp
-namespace Thorp
-
-lemma subset_biUnion_indicator {α β : Type*} [DecidableEq α] [Fintype β]
-    (A : Finset α) (T : β → Finset α) :
-    (if Finset.univ.biUnion T ⊆ A then (1:ℝ) else 0) =
-      ∏ b, if T b ⊆ A then (1:ℝ) else 0 := by
-  classical
-  by_cases h : Finset.univ.biUnion T ⊆ A
-  · rw [ite_eq_left h]
-    symm
-    apply Finset.prod_eq_one
-    intro b _
-    rw [ite_eq_left]
-    exact (Finset.subset_biUnion_of_mem T (Finset.mem_univ b)).trans h
-  · rw [ite_eq_right h]
-    obtain ⟨x,hx,hxA⟩ := Finset.not_subset.mp h
-    obtain ⟨b,_,hxb⟩ := Finset.mem_biUnion.mp hx
-    symm
-    apply Finset.prod_eq_zero (Finset.mem_univ b)
-    rw [ite_eq_right]
-    exact fun hb => hxA (hb hxb)
-
-lemma cylinder_block_polynomial {α β Ω : Type*} [DecidableEq α] [Fintype β]
-    [Fintype Ω] (τ : β → Type*) [∀ b, Fintype (τ b)]
-    (F : Ω → Finset α) (p : α → ℝ)
-    (hF : CylinderBound (inclusionMoment F) p)
-    (T : ∀ b, τ b → Finset α) (c : ∀ b, τ b → ℝ)
-    (hc : ∀ b s, 0 ≤ c b s)
-    (hd : ∀ b b', b ≠ b' → ∀ s s', Disjoint (T b s) (T b' s')) :
-    finiteMean (fun ω => ∏ b, ∑ s, c b s * if T b s ⊆ F ω then 1 else 0) ≤
-      ∏ b, ∑ s, c b s * ∏ x ∈ T b s, p x := by
-  classical
-  simp_rw [Fintype.prod_sum]
-  rw [finiteMean_sum]
-  apply Finset.sum_le_sum
-  intro s _
-  let U := Finset.univ.biUnion (fun b => T b (s b))
-  have he : (fun ω => ∏ b, c b (s b) * if T b (s b) ⊆ F ω then 1 else 0) =
-      fun ω => (if U ⊆ F ω then 1 else 0) * ∏ b, c b (s b) := by
-    funext ω
-    rw [Finset.prod_mul_distrib,←subset_biUnion_indicator]
-    exact mul_comm _ _
-  rw [he,finiteMean_mul_const]
-  have hdis : ((Finset.univ : Finset β) : Set β).PairwiseDisjoint (fun b => T b (s b)) := by
-    intro b _ b' _ hbb
-    exact hd b b' hbb (s b) (s b')
-  have hmean : finiteMean (fun ω => if U ⊆ F ω then 1 else 0) = inclusionMoment F U := by
-    unfold inclusionMoment
-    apply finiteMean_congr
-    intro ω
-    by_cases hh : U ⊆ F ω <;> simp [hh]
-  calc
-    _ ≤ (∏ x ∈ U, p x) * ∏ b, c b (s b) :=
-      mul_le_mul_of_nonneg_right (hmean.le.trans (hF U)) (Finset.prod_nonneg (fun b _ => hc b (s b)))
-    _ = _ := by
-      dsimp only [U]
-      rw [Finset.prod_biUnion hdis,←Finset.prod_mul_distrib]
-      apply Finset.prod_congr rfl
-      intro b _
-      exact mul_comm _ _
-
-
-noncomputable def sparseCoefficient {α : Type*} (w : ℝ) (T : Finset α) : ℝ :=
-  if T.card = 1 then 0 else w^T.card
-
-lemma sparseCoefficient_nonneg {α : Type*} (w : ℝ) (hw : 0 ≤ w) (T : Finset α) :
-    0 ≤ sparseCoefficient w T := by
-  unfold sparseCoefficient
-  split_ifs <;> positivity
-
-lemma sparse_block_pointwise {α : Type*} [DecidableEq α] (A D : Finset α)
-    (w : ℝ) (hw : 0 ≤ w) :
-    w^(if 2 ≤ (D ∩ A).card then (D ∩ A).card else 0) ≤
-      ∑ T : ↑D.powerset, sparseCoefficient w T.val * if T.val ⊆ A then 1 else 0 := by
-  classical
-  let V := D ∩ A
-  let T : ↑D.powerset := if 2 ≤ V.card then
-    ⟨V,Finset.mem_powerset.mpr Finset.inter_subset_left⟩ else
-    ⟨∅,Finset.mem_powerset.mpr (Finset.empty_subset D)⟩
-  calc
-    _ = sparseCoefficient w T.val * (if T.val ⊆ A then 1 else 0) := by
-      by_cases h : 2 ≤ V.card
-      · have hn : V.card ≠ 1 := by omega
-        simp [T,ite_eq_left h,sparseCoefficient,hn,show V ⊆ A from Finset.inter_subset_right,V]
-      · simp [T,ite_eq_right h,sparseCoefficient,V] at *
-    _ ≤ _ := Finset.single_le_sum (f := fun S : ↑D.powerset =>
-      sparseCoefficient w S.val * (if S.val ⊆ A then 1 else 0))
-      (fun S _ => mul_nonneg (sparseCoefficient_nonneg w hw S.val)
-        (by split_ifs <;> norm_num)) (Finset.mem_univ T)
-
-lemma geometric_sum_half (n : ℕ) {x : ℝ} (hx : 0 ≤ x) (hhalf : x ≤ 1/2) :
-    ∑ i ∈ Finset.range n, x^i ≤ 2 := by
-  induction n with
-  | zero => simp
-  | succ n ih =>
-    rw [Finset.sum_range_succ',pow_zero]
-    simp only [pow_succ',←Finset.mul_sum]
-    nlinarith
-
-lemma sparse_binomial_bound (R : ℕ) (z : ℝ) (hz : 0 ≤ z) (hr : (R:ℝ)*z ≤ 1/2) :
-    ∑ i ∈ Finset.range (R+1), (R.choose i:ℝ) * (if i = 1 then 0 else z^i) ≤
-      1 + 2*((R:ℝ)*z)^2 := by
-  classical
-  let x := (R:ℝ)*z
-  have hx : 0 ≤ x := mul_nonneg (Nat.cast_nonneg _) hz
-  have hbound (i : ℕ) : (R.choose i:ℝ)*(if i=1 then 0 else z^i) ≤
-      if i=1 then 0 else x^i := by
-    by_cases h : i=1
-    · simp [h]
-    · simp only [ite_eq_right h]
-      dsimp [x]
-      rw [mul_pow]
-      apply mul_le_mul_of_nonneg_right _ (pow_nonneg hz i)
-      exact_mod_cast Nat.choose_le_pow R i
-  calc
-    _ ≤ ∑ i ∈ Finset.range (R+1), if i=1 then 0 else x^i :=
-      Finset.sum_le_sum (fun i _ => hbound i)
-    _ ≤ 1+2*x^2 := by
-      rcases R with _ | R
-      · simp [x]
-      rcases R with _ | R
-      · norm_num [Finset.sum_range_succ]
-        positivity
-      have he : ∑ i ∈ Finset.range (R+2+1), (if i=1 then 0 else x^i) =
-          1 + x^2*∑ i ∈ Finset.range (R+1), x^i := by
-        rw [show R+2+1=2+(R+1) by omega,Finset.sum_range_add]
-        have hn (i : ℕ) : 2+i ≠ 1 := by omega
-        simp only [hn,ite_false,pow_add,←Finset.mul_sum]
-        norm_num [Finset.sum_range_succ]
-      rw [he]
-      have hg := geometric_sum_half (R+1) hx hr
-      nlinarith [sq_nonneg x]
-
-lemma sparse_block_product_bound {α : Type*} [DecidableEq α]
-    (D : Finset α) (p w : ℝ) (hp : 0 ≤ p) (hw : 0 ≤ w)
-    (hr : (D.card:ℝ)*(p*w) ≤ 1/2) :
-    ∑ T : ↑D.powerset, sparseCoefficient w T.val * p^T.val.card ≤
-      Real.exp (2*((D.card:ℝ)*(p*w))^2) := by
-  classical
-  have he : ∑ T : ↑D.powerset, sparseCoefficient w T.val * p^T.val.card =
-      ∑ i ∈ Finset.range (D.card+1), (D.card.choose i:ℝ)*(if i=1 then 0 else (p*w)^i) := by
-    rw [Finset.sum_coe_sort D.powerset (fun T => sparseCoefficient w T*p^T.card)]
-    have hh : (fun T : Finset α => sparseCoefficient w T*p^T.card) =
-        fun T => if T.card=1 then 0 else (p*w)^T.card := by
-      funext T
-      unfold sparseCoefficient
-      split_ifs <;> simp [mul_pow,mul_comm]
-    change (∑ T ∈ D.powerset, (fun T : Finset α => sparseCoefficient w T*p^T.card) T) = _
-    rw [hh,Finset.sum_powerset_apply_card (fun i => if i=1 then (0:ℝ) else (p*w)^i)]
-    simp only [nsmul_eq_mul]
-  rw [he]
-  exact (sparse_binomial_bound D.card (p*w) (mul_nonneg hp hw) hr).trans
-    (by simpa only [add_comm] using Real.add_one_le_exp (2*((D.card:ℝ)*(p*w))^2))
-
-
-noncomputable def sparseMass {α β : Type*} [DecidableEq α] [Fintype β]
-    (D : β → Finset α) (A : Finset α) : ℕ :=
-  ∑ b, if 2 ≤ (D b ∩ A).card then (D b ∩ A).card else 0
-
-theorem sparse_block_mgf {α β Ω : Type*} [DecidableEq α] [Fintype β] [Fintype Ω]
-    (D : β → Finset α) (hd : Pairwise (fun b b' => Disjoint (D b) (D b')))
-    (F : Ω → Finset α) (p w : ℝ) (hp : 0 ≤ p) (hw : 0 ≤ w)
-    (hF : ∀ T, inclusionMoment F T ≤ p^T.card)
-    (hr : ∀ b, ((D b).card:ℝ)*(p*w) ≤ 1/2) :
-    finiteMean (fun ω => w^(sparseMass D (F ω))) ≤
-      Real.exp (∑ b, 2*(((D b).card:ℝ)*(p*w))^2) := by
-  classical
-  have hc : CylinderBound (inclusionMoment F) (fun _ => p) := by
-    intro T
-    simpa only [Finset.prod_const] using hF T
-  have hb := cylinder_block_polynomial (fun b => ↑(D b).powerset) F (fun _ => p) hc
-    (fun _ S => S.val) (fun _ S => sparseCoefficient w S.val)
-    (fun _ S => sparseCoefficient_nonneg w hw S.val)
-    (fun b b' h S S' => (hd h).mono
-      (Finset.mem_powerset.mp S.property) (Finset.mem_powerset.mp S'.property))
-  calc
-    _ ≤ finiteMean (fun ω => ∏ b, ∑ S : ↑(D b).powerset,
-        sparseCoefficient w S.val * if S.val ⊆ F ω then 1 else 0) := by
-      apply finiteMean_mono
-      intro ω
-      rw [sparseMass,←Finset.prod_pow_eq_pow_sum]
-      apply Finset.prod_le_prod₀ (fun _ _ => pow_nonneg hw _)
-      intro b _
-      exact sparse_block_pointwise (F ω) (D b) w hw
-    _ ≤ ∏ b, ∑ S : ↑(D b).powerset, sparseCoefficient w S.val * p^S.val.card := by
-      simpa only [Finset.prod_const] using hb
-    _ ≤ _ := by
-      rw [Real.exp_sum]
-      apply Finset.prod_le_prod₀
-      · intro b _
-        exact Finset.sum_nonneg (fun S _ => mul_nonneg
-          (sparseCoefficient_nonneg w hw S.val) (pow_nonneg hp _))
-      · intro b _
-        exact sparse_block_product_bound (D b) p w hp hw (hr b)
-
-theorem inverseButterfly_sparse_mgf (d : ℕ) {β : Type*} [Fintype β]
-    (D : β → Finset (Card d)) (hd : Pairwise (fun b b' => Disjoint (D b) (D b')))
-    (A : Finset (Card d)) (w : ℝ) (hw : 0 ≤ w)
-    (hr : ∀ b, ((D b).card:ℝ)*((A.card/(2:ℝ)^d)*w) ≤ 1/2) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      w^(sparseMass D (inverseButterflyMarks d A ω))) ≤
-      Real.exp (∑ b, 2*(((D b).card:ℝ)*((A.card/(2:ℝ)^d)*w))^2) := by
-  apply sparse_block_mgf D hd (inverseButterflyMarks d A) (A.card/(2:ℝ)^d) w
-    (by positivity) hw
-  · exact inverseButterflyMarks_inclusion d A
-  · exact hr
-
-
-def crowdedMass (h : ℕ) : ℕ := if 2 ≤ h then h else 0
-
-lemma crowdedMass_le (h : ℕ) : crowdedMass h ≤ h := by
-  unfold crowdedMass
-  split_ifs <;> omega
-
-lemma crowdedMass_add (u v : ℕ) : crowdedMass u + crowdedMass v ≤ crowdedMass (u+v) := by
-  unfold crowdedMass
-  split_ifs <;> omega
-
-lemma alternatingCycles_le_crowded {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
-    (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    PairRouting.alternatingCycles e p ≤ crowdedMass (Fintype.card ι) := by
-  unfold crowdedMass
-  split_ifs with h
-  · have hh := PairRouting.alternatingCycles_double e p
-    omega
-  · rw [PairRouting.alternatingCycles_zero e p (by omega)]
-
-lemma palindromeCost_le_crowded (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) :
-    palindromeCost d e ω ≤ d*crowdedMass (Fintype.card ι) := by
-  classical
-  induction d generalizing ι with
-  | zero => simp [palindromeCost]
-  | succ d ih =>
-    let σ := benesStepEquiv d ω
-    let x := e.trans (headTailEquiv d).toEmbedding
-    let c := PairRouting.colors x σ.2.1
-    let hc := PairRouting.colors_compatible x σ.2.1
-    have h₀ := ih (PairRouting.childEmbedding x c hc false) σ.1.1
-    have h₁ := ih (PairRouting.childEmbedding x c hc true) σ.1.2
-    have hz := alternatingCycles_le_crowded (PairRouting.switchedEmbedding x σ.2.1)
-      (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1))
-    have hs := crowdedMass_add (Fintype.card {i // c i = false})
-      (Fintype.card {i // c i = true})
-    rw [PairRouting.color_card_add] at hs
-    change PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
-        (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) +
-        palindromeCost d (PairRouting.childEmbedding x c hc false) σ.1.1 +
-        palindromeCost d (PairRouting.childEmbedding x c hc true) σ.1.2 ≤ _
-    nlinarith
-
-end Thorp
-
-end ThorpNine.HighTail
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.HighTail.ProjectCycle
+
+namespace OAI
+
+namespace ThorpNine.HighTail
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+namespace PairRouting
+variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α] [DecidableEq ι]
+
+lemma sandwich_power_bound {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
+    (P : Ω → Equiv.Perm α) (Q : Ω' → Equiv.Perm α) (x y : ι → Bool × α)
+    (a : ℝ) (ha : 0 ≤ a) :
+    (finiteMean (fun ω : Ω × Ω' => finiteMean (fun coins : (α → Bool) × (α → Bool) =>
+      if ∀ i, sandwich coins.1 coins.2 (fun b => if b then Q ω.2 else P ω.1) (x i) = y i
+      then (1:ℝ) else 0)))^(1+a) ≤
+    (outerWeight x y)^(1+a) * ((compatibleSet x y).card:ℝ)^a *
+      ∑ c ∈ compatibleSet x y,
+      (tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
+        (fun i : {i // c i = false} => (y i.val).2))^(1+a) *
+      (tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
+        (fun i : {i // c i = true} => (y i.val).2))^(1+a) := by
+  classical
+  rw [sandwich_mean,sum_colors_factor]
+  let f (c : ι → Bool) : ℝ :=
+      tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
+        (fun i : {i // c i = false} => (y i.val).2) *
+      tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
+        (fun i : {i // c i = true} => (y i.val).2)
+  have hf (c : ι → Bool) : 0 ≤ f c := mul_nonneg (tupleProbability_nonneg _ _ _)
+    (tupleProbability_nonneg _ _ _)
+  rw [Real.mul_rpow (le_of_lt (outerWeight_pos _ _)) (Finset.sum_nonneg (fun c _ => hf c))]
+  have hh := Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg (compatibleSet x y)
+    (show 1 ≤ 1+a by linarith) (fun c _ => hf c)
+  have he : 1+a-1=a := by ring
+  rw [he] at hh
+  calc
+    _ ≤ (outerWeight x y)^(1+a) *
+        (((compatibleSet x y).card:ℝ)^a * ∑ c ∈ compatibleSet x y, (f c)^(1+a)) :=
+      mul_le_mul_of_nonneg_left hh (Real.rpow_nonneg (le_of_lt (outerWeight_pos _ _)) _)
+    _ = _ := by
+      rw [←mul_assoc]
+      congr 1
+      apply Finset.sum_congr rfl
+      intro c _
+      exact Real.mul_rpow (tupleProbability_nonneg _ _ _) (tupleProbability_nonneg _ _ _)
+
+end PairRouting
+
+lemma childLift_mul (d : ℕ) (p q : Bool → Equiv.Perm (Card d)) :
+    childLift p * childLift q = childLift (fun b => p b * q b) := by
+  ext x i
+  simp only [Equiv.Perm.mul_apply,childLift,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons]
+
+lemma childLift_inv (d : ℕ) (p : Bool → Equiv.Perm (Card d)) :
+    (childLift p)⁻¹ = childLift (fun b => (p b)⁻¹) := rfl
+
+lemma pairSwitch_inv (d : ℕ) (ξ : Card d → Bool) : (pairSwitch ξ)⁻¹ = pairSwitch ξ := rfl
+
+lemma palindromePerm_step (d : ℕ) (c : BenesCoins d × BenesCoins d)
+    (coins : (Card d → Bool) × (Card d → Bool)) :
+    palindromePerm (d+1) ((benesStepEquiv d).symm (c,coins)) =
+      pairSwitch coins.2 * childLift (fun b => palindromePerm d (if b then c.2 else c.1)) *
+        pairSwitch coins.1 := by
+  change (butterflyPerm (d+1) (decodeButterfly (d+1)
+      ((coinStepEquiv d).symm ((c.1.1,c.2.1),coins.2)))) *
+    (butterflyPerm (d+1) (decodeButterfly (d+1)
+      ((coinStepEquiv d).symm ((c.1.2,c.2.2),coins.1))))⁻¹ = _
+  rw [butterflyPerm_step,butterflyPerm_step,mul_inv_rev,childLift_inv,pairSwitch_inv]
+  rw [mul_assoc (pairSwitch coins.2),←mul_assoc (childLift _),childLift_mul,←mul_assoc]
+  congr 2
+  apply congrArg childLift
+  funext b
+  cases b <;> rfl
+
+lemma headTail_switch (d : ℕ) (ξ : Card d → Bool) (x : Card (d+1)) :
+    headTailEquiv d (pairSwitch ξ x) = pairLayer ξ (headTailEquiv d x) := rfl
+
+lemma headTail_child (d : ℕ) (p : Bool → Equiv.Perm (Card d)) (x : Card (d+1)) :
+    headTailEquiv d (childLift p x) = PairRouting.lift p (headTailEquiv d x) := rfl
+
+lemma palindromePerm_coordinates (d : ℕ) (c : BenesCoins d × BenesCoins d)
+    (coins : (Card d → Bool) × (Card d → Bool)) (x : Card (d+1)) :
+    headTailEquiv d (palindromePerm (d+1) ((benesStepEquiv d).symm (c,coins)) x) =
+      PairRouting.sandwich coins.1 coins.2 (fun b => palindromePerm d (if b then c.2 else c.1))
+        (headTailEquiv d x) := by
+  rw [palindromePerm_step]
+  simp only [Equiv.Perm.mul_apply,headTail_switch,headTail_child]
+  rfl
+
+namespace CycleColoring
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+lemma alternating_cycle_two (p : Equiv.Perm α) (S : Finset (Bool × α))
+    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) S) : 2 ≤ C.val.card := by
+  obtain ⟨x,hx⟩ := C.nonempty
+  have hpx : alternatingPerm p x ∈ C.val := by
+    rw [←C.orbit_eq hx,mem_orbitSet]
+    exact (Equiv.Perm.SameCycle.refl _ _).apply_right
+  have hne : x ≠ alternatingPerm p x := by
+    intro he
+    have hh := congrArg Prod.fst he
+    rw [alternatingPerm_flip] at hh
+    cases hb : x.1 <;> simp only [hb, Bool.not_false, Bool.not_true] at hh <;> cases hh
+  have hs : {x,alternatingPerm p x} ⊆ C.val := by
+    intro y hy
+    simp only [Finset.mem_insert,Finset.mem_singleton] at hy
+    rcases hy with rfl | rfl <;> assumption
+  have hh := Finset.card_le_card hs
+  simpa only [Finset.card_pair hne] using hh
+
+lemma alternating_cycle_count_double (p : Equiv.Perm α) (S : Finset (Bool × α)) :
+    2 * Fintype.card (RoutingNetwork.SelectedCycle (alternatingPerm p) S) ≤ S.card := by
+  classical
+  calc
+    _ = ∑ _C : RoutingNetwork.SelectedCycle (alternatingPerm p) S, 2 := by simp [Nat.mul_comm]
+    _ ≤ RoutingNetwork.cycleMass (alternatingPerm p) S :=
+      Finset.sum_le_sum (fun C _ => alternating_cycle_two p S C)
+    _ ≤ _ := RoutingNetwork.cycleMass_le _ _
+
+lemma alternating_cycle_count_zero (p : Equiv.Perm α) (S : Finset (Bool × α))
+    (hS : S.card ≤ 1) : Fintype.card (RoutingNetwork.SelectedCycle (alternatingPerm p) S) = 0 := by
+  have h := alternating_cycle_count_double p S
+  omega
+
+end CycleColoring
+
+namespace PairRouting
+
+section
+variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
+
+lemma alternatingCycles_double (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    2 * alternatingCycles e p ≤ Fintype.card ι := by
+  have hh := CycleColoring.alternating_cycle_count_double (alternatingProjection p) (labelSet e)
+  simpa only [alternatingCycles, card_labelSet] using hh
+
+lemma alternatingCycles_zero (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α)
+    (hι : Fintype.card ι ≤ 1) : alternatingCycles e p = 0 := by
+  have hh := alternatingCycles_double e p
+  omega
+
+end
+variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α] [DecidableEq ι]
+
+omit [Fintype α] [DecidableEq ι] in
+lemma colors_chosenCoin (x : ι → Bool × α) (c : ι → Bool) (hc : Compatible x c) :
+    colors x (chosenCoin x c) = c := by
+  funext i
+  simp only [colors,chosenCoin_apply x c hc i,required]
+  cases (x i).1 <;> cases c i <;> rfl
+
+noncomputable def coloredEmbedding (x : ι ↪ Bool × α) (c : ι → Bool) (_hc : Compatible x c) :
+    ι ↪ Bool × α := switchedEmbedding x (chosenCoin x c)
+
+omit [Fintype α] [DecidableEq ι] in
+lemma coloredEmbedding_apply (x : ι ↪ Bool × α) (c : ι → Bool) (hc : Compatible x c) (i : ι) :
+    coloredEmbedding x c hc i = (c i,(x i).2) := by
+  apply Prod.ext
+  · exact congrFun (colors_chosenCoin x c hc) i
+  · rfl
+
+omit [Fintype α] [DecidableEq ι] in
+lemma switchedEmbedding_eq_colored (x : ι ↪ Bool × α) (c : ι → Bool) (hc : Compatible x c)
+    (ξ : α → Bool) (he : colors x ξ = c) :
+    switchedEmbedding x ξ = coloredEmbedding x c hc := by
+  ext i : 1
+  rw [coloredEmbedding_apply]
+  apply Prod.ext
+  · exact congrFun he i
+  · rfl
+
+lemma assignments_match_bound (x y : ι ↪ Bool × α) (c : ι → Bool)
+    (hc : c ∈ compatibleSet x y) (p : Bool → Equiv.Perm α) (hm : ChildMatches p x y c) :
+    (compatibleSet x y).card ≤
+      2 ^ ((occupied x).card + (occupied y).card - Fintype.card ι +
+        alternatingCycles (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p) := by
+  obtain ⟨hcx,hcy⟩ := (mem_compatibleSet x y c).mp hc
+  let ξ := chosenCoin x c
+  let η := chosenCoin y c
+  have he : routedEmbedding x ξ η p = y := by
+    ext i : 1
+    have h := (sandwich_matches_iff ξ η p x y).mpr
+      ⟨(colors_chosenCoin x c hcx).trans (colors_chosenCoin y c hcy).symm,by
+        simpa only [ξ,colors_chosenCoin x c hcx] using hm⟩
+    exact h i
+  have hh := assignments_actual_bound x ξ η p
+  rw [he] at hh
+  rw [card_compatibleSet]
+  exact hh
+
+omit [Fintype α] [DecidableEq ι] in
+lemma occupied_sum_ge (x y : ι ↪ Bool × α) :
+    Fintype.card ι ≤ (occupied x).card + (occupied y).card := by
+  have h₀ := occupied_card_double x
+  have h₁ := occupied_card_double y
+  omega
+
+lemma assignment_weight_bound (x y : ι ↪ Bool × α) (c : ι → Bool)
+    (hc : c ∈ compatibleSet x y) (p : Bool → Equiv.Perm α) (hm : ChildMatches p x y c) :
+    ((compatibleSet x y).card:ℝ) * outerWeight x y ≤
+      (2:ℝ)^((alternatingCycles
+        (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p : ℝ) - Fintype.card ι) := by
+  let z := alternatingCycles (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p
+  let b := (occupied x).card+(occupied y).card
+  let h := Fintype.card ι
+  have hb : h ≤ b := occupied_sum_ge x y
+  have hn : ((compatibleSet x y).card:ℝ) ≤ (2:ℝ)^(b-h+z) := by
+    exact_mod_cast assignments_match_bound x y c hc p hm
+  change ((compatibleSet x y).card:ℝ) * (1/(2:ℝ)^b) ≤ (2:ℝ)^((z:ℝ)-h)
+  rw [mul_one_div]
+  calc
+    _ ≤ (2:ℝ)^(b-h+z)/(2:ℝ)^b := div_le_div_of_nonneg_right hn (by positivity)
+    _ = _ := by
+      rw [←Real.rpow_natCast,←Real.rpow_natCast,←Real.rpow_sub (by norm_num : (0:ℝ)<2)]
+      congr 1
+      rw [Nat.cast_add,Nat.cast_sub hb]
+      ring
+
+lemma assignment_factor_bound (x y : ι ↪ Bool × α) (c : ι → Bool)
+    (hc : c ∈ compatibleSet x y) (p : Bool → Equiv.Perm α) (hm : ChildMatches p x y c)
+    (a : ℝ) (ha : 0 ≤ a) :
+    (outerWeight x y)^a * ((compatibleSet x y).card:ℝ)^a ≤
+      (2:ℝ)^(((alternatingCycles
+        (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p : ℝ) - Fintype.card ι)*a) := by
+  have h := Real.rpow_le_rpow (mul_nonneg (Nat.cast_nonneg _) (le_of_lt (outerWeight_pos x y)))
+    (assignment_weight_bound x y c hc p hm) ha
+  rw [Real.mul_rpow (Nat.cast_nonneg _) (le_of_lt (outerWeight_pos x y)),
+    ←Real.rpow_mul (by norm_num : (0:ℝ)≤2)] at h
+  simpa only [mul_comm] using h
+
+
+lemma sandwich_weighted_probability_subtype (p : Bool → Equiv.Perm α) (x y : ι → Bool × α)
+    (W : (ι → Bool) → ℝ) :
+    finiteMean (fun coins : (α → Bool) × (α → Bool) =>
+      if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then W (colors x coins.1) else 0) =
+    outerWeight x y * ∑ c : {c // c ∈ compatibleSet x y},
+      if ChildMatches p x y c.val then W c.val else 0 := by
+  rw [sandwich_weighted_probability]
+  congr 1
+  rw [Finset.sum_subtype (compatibleSet x y) (fun _ => Iff.rfl)]
+
+omit [Fintype α] [DecidableEq α] [DecidableEq ι] in
+lemma color_card_add (c : ι → Bool) :
+    Fintype.card {i // c i = false} + Fintype.card {i // c i = true} = Fintype.card ι := by
+  have he : {i // c i = true} ≃ {i // ¬c i = false} := Equiv.subtypeEquivRight fun i => by
+    cases c i <;> simp
+  rw [Fintype.card_congr he]
+  rw [Fintype.card_subtype_compl]
+  exact Nat.add_sub_of_le (Fintype.card_subtype_le _)
+
+end PairRouting
+
+noncomputable def assignmentCost (d : ℕ) {ι : Type*} [Fintype ι]
+    (x : ι ↪ Bool × Card d) (c : ι → Bool) (hc : PairRouting.Compatible x c)
+    (ω : BenesCoins d × BenesCoins d) : ℕ :=
+  PairRouting.alternatingCycles (PairRouting.coloredEmbedding x c hc)
+      (fun b => palindromePerm d (if b then ω.2 else ω.1)) +
+    palindromeCost d (PairRouting.childEmbedding x c hc false) ω.1 +
+    palindromeCost d (PairRouting.childEmbedding x c hc true) ω.2
+
+lemma palindromeCost_eq (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (d+1)) (σ : BenesCoins (d+1)) :
+    palindromeCost (d+1) e σ =
+      assignmentCost d (e.trans (headTailEquiv d).toEmbedding)
+        (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) (benesStepEquiv d σ).2.1)
+        (PairRouting.colors_compatible _ _) (benesStepEquiv d σ).1 := by
+  unfold palindromeCost assignmentCost
+  dsimp only
+  rw [PairRouting.switchedEmbedding_eq_colored _ _ (PairRouting.colors_compatible _ _) _ rfl]
+
+lemma palindromeCost_step (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (d+1)) (ω : BenesCoins d × BenesCoins d)
+    (coins : (Card d → Bool) × (Card d → Bool)) :
+    palindromeCost (d+1) e ((benesStepEquiv d).symm (ω,coins)) =
+      assignmentCost d (e.trans (headTailEquiv d).toEmbedding)
+        (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) coins.1)
+        (PairRouting.colors_compatible _ _) ω := by
+  rw [palindromeCost_eq]
+  simp only [Equiv.apply_symm_apply]
+
+noncomputable def routingMoment (d : ℕ) {ι : Type*} [Fintype ι]
+    (e f : ι ↪ Card d) (a : ℝ) : ℝ := by
+  classical
+  exact finiteMean (fun ω : BenesCoins d =>
+    if ∀ i, palindromePerm d ω (e i) = f i then (2:ℝ)^((palindromeCost d e ω:ℝ)*a) else 0)
+
+lemma routingMoment_nonneg (d : ℕ) {ι : Type*} [Fintype ι]
+    (e f : ι ↪ Card d) (a : ℝ) : 0 ≤ routingMoment d e f a := by
+  classical
+  exact finiteMean_nonneg (fun _ => by split_ifs <;> positivity)
+
+lemma palindrome_matches_step (d : ℕ) {ι : Type*}
+    (e f : ι ↪ Card (d+1)) (ω : BenesCoins d × BenesCoins d)
+    (coins : (Card d → Bool) × (Card d → Bool)) :
+    (∀ i, palindromePerm (d+1) ((benesStepEquiv d).symm (ω,coins)) (e i) = f i) ↔
+    (∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => palindromePerm d (if b then ω.2 else ω.1))
+      ((e.trans (headTailEquiv d).toEmbedding) i) = ((f.trans (headTailEquiv d).toEmbedding) i)) := by
+  apply forall_congr'
+  intro i
+  rw [←(headTailEquiv d).injective.eq_iff,palindromePerm_coordinates]
+  rfl
+
+lemma routingMoment_step (d : ℕ) {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (e f : ι ↪ Card (d+1)) (a : ℝ) :
+    let x := e.trans (headTailEquiv d).toEmbedding
+    let y := f.trans (headTailEquiv d).toEmbedding
+    routingMoment (d+1) e f a = PairRouting.outerWeight x y *
+      ∑ c : {c // c ∈ PairRouting.compatibleSet x y},
+        finiteMean (fun ω : BenesCoins d × BenesCoins d =>
+          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c.val
+          then (2:ℝ)^((assignmentCost d x c.val
+            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0) := by
+  classical
+  intro x y
+  rw [routingMoment,finiteMean_equiv (benesStepEquiv d),finiteMean_prod]
+  have he (ω : BenesCoins d × BenesCoins d) :
+      finiteMean (fun coins : (Card d → Bool) × (Card d → Bool) =>
+        if ∀ i, palindromePerm (d+1) ((benesStepEquiv d).symm (ω,coins)) (e i) = f i
+        then (2:ℝ)^((palindromeCost (d+1) e ((benesStepEquiv d).symm (ω,coins)):ℝ)*a) else 0) =
+      PairRouting.outerWeight x y *
+        ∑ c : {c // c ∈ PairRouting.compatibleSet x y},
+          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c.val
+          then (2:ℝ)^((assignmentCost d x c.val
+            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0 := by
+    let W : (ι → Bool) → ℝ := fun c =>
+      if hc : PairRouting.Compatible x c then (2:ℝ)^((assignmentCost d x c hc ω:ℝ)*a) else 0
+    calc
+      _ = finiteMean (fun coins : (Card d → Bool) × (Card d → Bool) =>
+        if ∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => palindromePerm d (if b then ω.2 else ω.1))
+          (x i) = y i then W (PairRouting.colors x coins.1) else 0) := by
+        apply finiteMean_congr
+        intro coins
+        simp only [palindrome_matches_step,palindromeCost_step]
+        simp only [W,PairRouting.colors_compatible,↓reduceDIte]
+        rfl
+      _ = _ := by
+        rw [PairRouting.sandwich_weighted_probability_subtype]
+        congr 1
+        apply Finset.sum_congr rfl
+        intro c _
+        simp only [W,((PairRouting.mem_compatibleSet x y c.val).mp c.property).1,↓reduceDIte]
+  simp_rw [he]
+  simp_rw [mul_comm (PairRouting.outerWeight x y)]
+  rw [finiteMean_mul_const,finiteMean_sum,mul_comm]
+
+lemma child_moment_product (d : ℕ) {ι : Type*} [Fintype ι]
+    (x y : ι ↪ Bool × Card d) (c : ι → Bool)
+    (hcx : PairRouting.Compatible x c) (hcy : PairRouting.Compatible y c) (a : ℝ) :
+    routingMoment d (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a *
+      routingMoment d (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a =
+    finiteMean (fun ω : BenesCoins d × BenesCoins d =>
+      if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
+      then (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
+        palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0) := by
+  classical
+  unfold routingMoment
+  rw [←finiteMean_prod_mul]
+  apply finiteMean_congr
+  intro ω
+  let M₀ := ∀ i : {i // c i = false}, palindromePerm d ω.1 (x i.val).2 = (y i.val).2
+  let M₁ := ∀ i : {i // c i = true}, palindromePerm d ω.2 (x i.val).2 = (y i.val).2
+  have hp : (fun b : Bool => palindromePerm d (if b then ω.2 else ω.1)) =
+      (fun b => if b then palindromePerm d ω.2 else palindromePerm d ω.1) := by
+    funext b
+    cases b <;> rfl
+  have hs : PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c ↔
+      M₀ ∧ M₁ := by rw [hp]; exact PairRouting.childMatches_split _ _ _ _ _
+  change (if M₀ then _ else 0) * (if M₁ then _ else 0) = _
+  by_cases h₀ : M₀ <;> by_cases h₁ : M₁
+  · simp only [hs,h₀,h₁,true_and,↓reduceIte]
+    rw [←Real.rpow_add (by norm_num : (0:ℝ)<2)]
+    congr 1
+    ring
+  · simp only [hs,h₀,h₁,and_false,↓reduceIte,mul_zero]
+  · simp only [hs,h₀,h₁,false_and,↓reduceIte,zero_mul]
+  · simp only [hs,h₀,h₁,false_and,↓reduceIte,mul_zero]
+
+lemma palindrome_probability_step (d : ℕ) {ι : Type*} [Fintype ι]
+    (e f : ι ↪ Card (d+1)) :
+    PairRouting.tupleProbability (palindromePerm (d+1)) e f =
+    finiteMean (fun ω : BenesCoins d × BenesCoins d =>
+      finiteMean (fun coins : (Card d → Bool) × (Card d → Bool) =>
+        if ∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => if b then palindromePerm d ω.2 else palindromePerm d ω.1)
+          ((e.trans (headTailEquiv d).toEmbedding) i) = ((f.trans (headTailEquiv d).toEmbedding) i)
+        then (1:ℝ) else 0)) := by
+  classical
+  rw [PairRouting.tupleProbability,finiteMean_equiv (benesStepEquiv d),finiteMean_prod]
+  have hp (b : Bool) (p q : BenesCoins d) :
+      palindromePerm d (if b then p else q) = (if b then palindromePerm d p else palindromePerm d q) := by
+    cases b <;> rfl
+  simp only [palindrome_matches_step,hp]
+
+lemma child_power_product_bound (d : ℕ) {ι : Type*} [Fintype ι]
+    (x y : ι ↪ Bool × Card d) (c : ι → Bool)
+    (hcx : PairRouting.Compatible x c) (hcy : PairRouting.Compatible y c) (a : ℝ)
+    (hfalse : (PairRouting.tupleProbability (palindromePerm d)
+        (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false))^(1+a) ≤
+      (2:ℝ)^(-(d:ℝ)*Fintype.card {i // c i = false}*a) *
+        routingMoment d (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a)
+    (htrue : (PairRouting.tupleProbability (palindromePerm d)
+        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true))^(1+a) ≤
+      (2:ℝ)^(-(d:ℝ)*Fintype.card {i // c i = true}*a) *
+        routingMoment d (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a) :
+    (PairRouting.tupleProbability (palindromePerm d)
+      (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false))^(1+a) *
+      (PairRouting.tupleProbability (palindromePerm d)
+        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true))^(1+a) ≤
+    (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) *
+      finiteMean (fun ω : BenesCoins d × BenesCoins d =>
+        if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
+        then (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
+          palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0) := by
+  have hh := mul_le_mul hfalse htrue
+    (Real.rpow_nonneg (PairRouting.tupleProbability_nonneg _ _ _) _)
+    (mul_nonneg (by positivity) (routingMoment_nonneg _ _ _ _))
+  rw [mul_mul_mul_comm,child_moment_product,←Real.rpow_add (by norm_num : (0:ℝ)<2)] at hh
+  have he : -(d:ℝ)*Fintype.card {i // c i = false}*a + -(d:ℝ)*Fintype.card {i // c i = true}*a =
+      -(d:ℝ)*Fintype.card ι*a := by
+    rw [←add_mul,←mul_add,←Nat.cast_add,PairRouting.color_card_add]
+  rwa [he] at hh
+
+lemma assignment_moment_bound (d : ℕ) {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (x y : ι ↪ Bool × Card d) (c : ι → Bool) (hc : c ∈ PairRouting.compatibleSet x y)
+    (a : ℝ) (ha : 0 ≤ a) :
+    let hcx := ((PairRouting.mem_compatibleSet x y c).mp hc).1
+    PairRouting.outerWeight x y ^ a * ((PairRouting.compatibleSet x y).card:ℝ)^a *
+      ((2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) *
+        finiteMean (fun ω : BenesCoins d × BenesCoins d =>
+          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
+          then (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
+            palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0)) ≤
+    (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) *
+      finiteMean (fun ω : BenesCoins d × BenesCoins d =>
+        if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
+        then (2:ℝ)^((assignmentCost d x c hcx ω:ℝ)*a) else 0) := by
+  classical
+  intro hcx
+  have hscale (q : ℝ) (F : BenesCoins d × BenesCoins d → ℝ) :
+      q * finiteMean F = finiteMean (fun ω => q * F ω) := by
+    simp_rw [mul_comm q,finiteMean_mul_const]
+  rw [←mul_assoc,hscale,hscale]
+  apply finiteMean_mono
+  intro ω
+  split_ifs with hm
+  · have hh := PairRouting.assignment_factor_bound x y c hc
+      (fun b => palindromePerm d (if b then ω.2 else ω.1)) hm a ha
+    calc
+      _ ≤ ((2:ℝ)^(((PairRouting.alternatingCycles (PairRouting.coloredEmbedding x c hcx)
+          (fun b => palindromePerm d (if b then ω.2 else ω.1)):ℝ)-Fintype.card ι)*a) *
+          (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a)) *
+          (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
+            palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) :=
+        mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hh (by positivity)) (by positivity)
+      _ = _ := by
+        simp only [←Real.rpow_add (by norm_num : (0:ℝ)<2),assignmentCost,Nat.cast_add]
+        congr 1
+        ring
+  · simp only [mul_zero]
+    exact le_rfl
+
+theorem palindrome_pointwise_moment (d : ℕ) {ι : Type*} [Fintype ι]
+    (e f : ι ↪ Card d) (a : ℝ) (ha : 0 ≤ a) :
+    (PairRouting.tupleProbability (palindromePerm d) e f)^(1+a) ≤
+      (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) * routingMoment d e f a := by
+  classical
+  induction d generalizing ι with
+  | zero =>
+      have he : ∀ (ω : BenesCoins 0) (i : ι), palindromePerm 0 ω (e i) = f i :=
+        fun _ _ => Subsingleton.elim _ _
+      simp [PairRouting.tupleProbability,routingMoment,he,palindromeCost,finiteMean_const]
+  | succ d ih =>
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let y := f.trans (headTailEquiv d).toEmbedding
+      let W := PairRouting.outerWeight x y
+      let C := PairRouting.compatibleSet x y
+      let p : (ι → Bool) → ℝ := fun c =>
+        (PairRouting.tupleProbability (palindromePerm d)
+          (fun i : {i // c i = false} => (x i.val).2)
+          (fun i : {i // c i = false} => (y i.val).2))^(1+a) *
+        (PairRouting.tupleProbability (palindromePerm d)
+          (fun i : {i // c i = true} => (x i.val).2)
+          (fun i : {i // c i = true} => (y i.val).2))^(1+a)
+      let m : {c // c ∈ C} → ℝ := fun c =>
+        finiteMean (fun ω : BenesCoins d × BenesCoins d =>
+          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c.val
+          then (2:ℝ)^((assignmentCost d x c.val
+            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0)
+      have hp (c : {c // c ∈ C}) : W^a * (C.card:ℝ)^a * p c.val ≤
+          (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * m c := by
+        obtain ⟨hcx,hcy⟩ := (PairRouting.mem_compatibleSet x y c.val).mp c.property
+        have h₀ := ih (PairRouting.childEmbedding x c.val hcx false)
+          (PairRouting.childEmbedding y c.val hcy false)
+        have h₁ := ih (PairRouting.childEmbedding x c.val hcx true)
+          (PairRouting.childEmbedding y c.val hcy true)
+        have hh := child_power_product_bound d x y c.val hcx hcy a h₀ h₁
+        have hh' := mul_le_mul_of_nonneg_left hh
+          (mul_nonneg (Real.rpow_nonneg (le_of_lt (PairRouting.outerWeight_pos x y)) a)
+            (Real.rpow_nonneg (Nat.cast_nonneg C.card) a))
+        exact hh'.trans (assignment_moment_bound d x y c.val c.property a ha)
+      have hnode := PairRouting.sandwich_power_bound (palindromePerm d) (palindromePerm d) x y a ha
+      rw [←palindrome_probability_step d e f] at hnode
+      change (PairRouting.tupleProbability (palindromePerm (d+1)) e f)^(1+a) ≤
+        W^(1+a) * (C.card:ℝ)^a * ∑ c ∈ C, p c at hnode
+      have hsum : ∑ c ∈ C, p c = ∑ c : {c // c ∈ C}, p c.val := by
+        rw [Finset.sum_subtype C (fun _ => Iff.rfl)]
+      have hW : 0 < W := PairRouting.outerWeight_pos x y
+      calc
+        _ ≤ W^(1+a) * (C.card:ℝ)^a * ∑ c ∈ C, p c := hnode
+        _ = W * ∑ c : {c // c ∈ C}, W^a * (C.card:ℝ)^a * p c.val := by
+          rw [Real.rpow_add hW,Real.rpow_one,hsum,←Finset.mul_sum]
+          ring
+        _ ≤ W * ∑ c : {c // c ∈ C}, (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * m c :=
+          mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun c _ => hp c)) (le_of_lt hW)
+        _ = _ := by
+          rw [←Finset.mul_sum,routingMoment_step]
+          simp only [Nat.cast_add,Nat.cast_one]
+          change W * ((2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * ∑ c, m c) =
+            (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * (W * ∑ c, m c)
+          ring
+
+lemma routingMoment_sum (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (a : ℝ) :
+    (∑ f : ι ↪ Card d, routingMoment d e f a) =
+      finiteMean (fun ω : BenesCoins d => (2:ℝ)^((palindromeCost d e ω:ℝ)*a)) := by
+  classical
+  unfold routingMoment
+  rw [←finiteMean_sum]
+  apply finiteMean_congr
+  intro ω
+  let f₀ : ι ↪ Card d := e.trans (palindromePerm d ω).toEmbedding
+  rw [Finset.sum_eq_single f₀]
+  · have hm : ∀ i, palindromePerm d ω (e i) = f₀ i := fun _ => rfl
+    simp only [hm,implies_true,↓reduceIte]
+  · intro f _ hf
+    have hm : ¬ ∀ i, palindromePerm d ω (e i) = f i := by
+      intro hh
+      apply hf
+      ext i : 1
+      exact (hh i).symm
+    exact ite_eq_right hm
+  · simp
+
+lemma finite_row_power {X : Type*} [Fintype X] [Nonempty X]
+    (p : X → ℝ) (hp : ∀ x, 0 ≤ p x) (a : ℝ) :
+    finiteMean (fun x => ((Fintype.card X:ℝ)*p x)^(1+a)) =
+      (Fintype.card X:ℝ)^a * ∑ x, (p x)^(1+a) := by
+  have hn : (0:ℝ) < Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
+  simp only [finiteMean,Real.mul_rpow (le_of_lt hn) (hp _),←Finset.mul_sum]
+  rw [Real.rpow_add hn,Real.rpow_one]
+  field_simp
+
+noncomputable def palindromeRowMoment (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (a : ℝ) : ℝ :=
+  finiteMean (fun f : ι ↪ Card d =>
+    ((Fintype.card (ι ↪ Card d):ℝ)*PairRouting.tupleProbability (palindromePerm d) e f)^(1+a))
+
+lemma density_normalization (d h n : ℕ) (a : ℝ) :
+    (n:ℝ)^a * (2:ℝ)^(-(d:ℝ)*h*a) = ((n:ℝ)/((2:ℝ)^d)^h)^a := by
+  rw [Real.div_rpow (Nat.cast_nonneg _) (by positivity),←pow_mul,
+    ←Real.rpow_natCast,←Real.rpow_mul (by norm_num : (0:ℝ) ≤ 2),Nat.cast_mul]
+  rw [show -(d:ℝ)*h*a = -((d:ℝ)*h*a) by ring,
+    Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2),div_eq_mul_inv]
+
+theorem palindrome_row_density_moment (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (a : ℝ) (ha : 0 ≤ a) :
+    palindromeRowMoment d e a ≤
+      (((2^d).descFactorial (Fintype.card ι):ℝ)/((2^d:ℕ):ℝ)^(Fintype.card ι))^a *
+        finiteMean (fun ω : BenesCoins d => (2:ℝ)^((palindromeCost d e ω:ℝ)*a)) := by
+  classical
+  have : Nonempty (ι ↪ Card d) := ⟨e⟩
+  rw [palindromeRowMoment,finite_row_power _ (fun f => PairRouting.tupleProbability_nonneg _ _ _)]
+  calc
+    _ ≤ (Fintype.card (ι ↪ Card d):ℝ)^a *
+      ∑ f : ι ↪ Card d, (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) * routingMoment d e f a :=
+      mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun f _ => palindrome_pointwise_moment d e f a ha))
+        (Real.rpow_nonneg (Nat.cast_nonneg _) _)
+    _ = _ := by
+      rw [←Finset.mul_sum,routingMoment_sum,←mul_assoc,density_normalization,
+        Fintype.card_embedding_eq,card_positions]
+      norm_cast
+
+end Thorp
+namespace Thorp
+
+lemma subset_biUnion_indicator {α β : Type*} [DecidableEq α] [Fintype β]
+    (A : Finset α) (T : β → Finset α) :
+    (if Finset.univ.biUnion T ⊆ A then (1:ℝ) else 0) =
+      ∏ b, if T b ⊆ A then (1:ℝ) else 0 := by
+  classical
+  by_cases h : Finset.univ.biUnion T ⊆ A
+  · rw [ite_eq_left h]
+    symm
+    apply Finset.prod_eq_one
+    intro b _
+    rw [ite_eq_left]
+    exact (Finset.subset_biUnion_of_mem T (Finset.mem_univ b)).trans h
+  · rw [ite_eq_right h]
+    obtain ⟨x,hx,hxA⟩ := Finset.not_subset.mp h
+    obtain ⟨b,_,hxb⟩ := Finset.mem_biUnion.mp hx
+    symm
+    apply Finset.prod_eq_zero (Finset.mem_univ b)
+    rw [ite_eq_right]
+    exact fun hb => hxA (hb hxb)
+
+lemma cylinder_block_polynomial {α β Ω : Type*} [DecidableEq α] [Fintype β]
+    [Fintype Ω] (τ : β → Type*) [∀ b, Fintype (τ b)]
+    (F : Ω → Finset α) (p : α → ℝ)
+    (hF : CylinderBound (inclusionMoment F) p)
+    (T : ∀ b, τ b → Finset α) (c : ∀ b, τ b → ℝ)
+    (hc : ∀ b s, 0 ≤ c b s)
+    (hd : ∀ b b', b ≠ b' → ∀ s s', Disjoint (T b s) (T b' s')) :
+    finiteMean (fun ω => ∏ b, ∑ s, c b s * if T b s ⊆ F ω then 1 else 0) ≤
+      ∏ b, ∑ s, c b s * ∏ x ∈ T b s, p x := by
+  classical
+  simp_rw [Fintype.prod_sum]
+  rw [finiteMean_sum]
+  apply Finset.sum_le_sum
+  intro s _
+  let U := Finset.univ.biUnion (fun b => T b (s b))
+  have he : (fun ω => ∏ b, c b (s b) * if T b (s b) ⊆ F ω then 1 else 0) =
+      fun ω => (if U ⊆ F ω then 1 else 0) * ∏ b, c b (s b) := by
+    funext ω
+    rw [Finset.prod_mul_distrib,←subset_biUnion_indicator]
+    exact mul_comm _ _
+  rw [he,finiteMean_mul_const]
+  have hdis : ((Finset.univ : Finset β) : Set β).PairwiseDisjoint (fun b => T b (s b)) := by
+    intro b _ b' _ hbb
+    exact hd b b' hbb (s b) (s b')
+  have hmean : finiteMean (fun ω => if U ⊆ F ω then 1 else 0) = inclusionMoment F U := by
+    unfold inclusionMoment
+    apply finiteMean_congr
+    intro ω
+    by_cases hh : U ⊆ F ω <;> simp [hh]
+  calc
+    _ ≤ (∏ x ∈ U, p x) * ∏ b, c b (s b) :=
+      mul_le_mul_of_nonneg_right (hmean.le.trans (hF U)) (Finset.prod_nonneg (fun b _ => hc b (s b)))
+    _ = _ := by
+      dsimp only [U]
+      rw [Finset.prod_biUnion hdis,←Finset.prod_mul_distrib]
+      apply Finset.prod_congr rfl
+      intro b _
+      exact mul_comm _ _
+
+
+noncomputable def sparseCoefficient {α : Type*} (w : ℝ) (T : Finset α) : ℝ :=
+  if T.card = 1 then 0 else w^T.card
+
+lemma sparseCoefficient_nonneg {α : Type*} (w : ℝ) (hw : 0 ≤ w) (T : Finset α) :
+    0 ≤ sparseCoefficient w T := by
+  unfold sparseCoefficient
+  split_ifs <;> positivity
+
+lemma sparse_block_pointwise {α : Type*} [DecidableEq α] (A D : Finset α)
+    (w : ℝ) (hw : 0 ≤ w) :
+    w^(if 2 ≤ (D ∩ A).card then (D ∩ A).card else 0) ≤
+      ∑ T : ↑D.powerset, sparseCoefficient w T.val * if T.val ⊆ A then 1 else 0 := by
+  classical
+  let V := D ∩ A
+  let T : ↑D.powerset := if 2 ≤ V.card then
+    ⟨V,Finset.mem_powerset.mpr Finset.inter_subset_left⟩ else
+    ⟨∅,Finset.mem_powerset.mpr (Finset.empty_subset D)⟩
+  calc
+    _ = sparseCoefficient w T.val * (if T.val ⊆ A then 1 else 0) := by
+      by_cases h : 2 ≤ V.card
+      · have hn : V.card ≠ 1 := by omega
+        simp [T,ite_eq_left h,sparseCoefficient,hn,show V ⊆ A from Finset.inter_subset_right,V]
+      · simp [T,ite_eq_right h,sparseCoefficient,V] at *
+    _ ≤ _ := Finset.single_le_sum (f := fun S : ↑D.powerset =>
+      sparseCoefficient w S.val * (if S.val ⊆ A then 1 else 0))
+      (fun S _ => mul_nonneg (sparseCoefficient_nonneg w hw S.val)
+        (by split_ifs <;> norm_num)) (Finset.mem_univ T)
+
+lemma geometric_sum_half (n : ℕ) {x : ℝ} (hx : 0 ≤ x) (hhalf : x ≤ 1/2) :
+    ∑ i ∈ Finset.range n, x^i ≤ 2 := by
+  induction n with
+  | zero => simp
+  | succ n ih =>
+    rw [Finset.sum_range_succ',pow_zero]
+    simp only [pow_succ',←Finset.mul_sum]
+    nlinarith
+
+lemma sparse_binomial_bound (R : ℕ) (z : ℝ) (hz : 0 ≤ z) (hr : (R:ℝ)*z ≤ 1/2) :
+    ∑ i ∈ Finset.range (R+1), (R.choose i:ℝ) * (if i = 1 then 0 else z^i) ≤
+      1 + 2*((R:ℝ)*z)^2 := by
+  classical
+  let x := (R:ℝ)*z
+  have hx : 0 ≤ x := mul_nonneg (Nat.cast_nonneg _) hz
+  have hbound (i : ℕ) : (R.choose i:ℝ)*(if i=1 then 0 else z^i) ≤
+      if i=1 then 0 else x^i := by
+    by_cases h : i=1
+    · simp [h]
+    · simp only [ite_eq_right h]
+      dsimp [x]
+      rw [mul_pow]
+      apply mul_le_mul_of_nonneg_right _ (pow_nonneg hz i)
+      exact_mod_cast Nat.choose_le_pow R i
+  calc
+    _ ≤ ∑ i ∈ Finset.range (R+1), if i=1 then 0 else x^i :=
+      Finset.sum_le_sum (fun i _ => hbound i)
+    _ ≤ 1+2*x^2 := by
+      rcases R with _ | R
+      · simp [x]
+      rcases R with _ | R
+      · norm_num [Finset.sum_range_succ]
+        positivity
+      have he : ∑ i ∈ Finset.range (R+2+1), (if i=1 then 0 else x^i) =
+          1 + x^2*∑ i ∈ Finset.range (R+1), x^i := by
+        rw [show R+2+1=2+(R+1) by omega,Finset.sum_range_add]
+        have hn (i : ℕ) : 2+i ≠ 1 := by omega
+        simp only [hn,ite_false,pow_add,←Finset.mul_sum]
+        norm_num [Finset.sum_range_succ]
+      rw [he]
+      have hg := geometric_sum_half (R+1) hx hr
+      nlinarith [sq_nonneg x]
+
+lemma sparse_block_product_bound {α : Type*} [DecidableEq α]
+    (D : Finset α) (p w : ℝ) (hp : 0 ≤ p) (hw : 0 ≤ w)
+    (hr : (D.card:ℝ)*(p*w) ≤ 1/2) :
+    ∑ T : ↑D.powerset, sparseCoefficient w T.val * p^T.val.card ≤
+      Real.exp (2*((D.card:ℝ)*(p*w))^2) := by
+  classical
+  have he : ∑ T : ↑D.powerset, sparseCoefficient w T.val * p^T.val.card =
+      ∑ i ∈ Finset.range (D.card+1), (D.card.choose i:ℝ)*(if i=1 then 0 else (p*w)^i) := by
+    rw [Finset.sum_coe_sort D.powerset (fun T => sparseCoefficient w T*p^T.card)]
+    have hh : (fun T : Finset α => sparseCoefficient w T*p^T.card) =
+        fun T => if T.card=1 then 0 else (p*w)^T.card := by
+      funext T
+      unfold sparseCoefficient
+      split_ifs <;> simp [mul_pow,mul_comm]
+    change (∑ T ∈ D.powerset, (fun T : Finset α => sparseCoefficient w T*p^T.card) T) = _
+    rw [hh,Finset.sum_powerset_apply_card (fun i => if i=1 then (0:ℝ) else (p*w)^i)]
+    simp only [nsmul_eq_mul]
+  rw [he]
+  exact (sparse_binomial_bound D.card (p*w) (mul_nonneg hp hw) hr).trans
+    (by simpa only [add_comm] using Real.add_one_le_exp (2*((D.card:ℝ)*(p*w))^2))
+
+
+noncomputable def sparseMass {α β : Type*} [DecidableEq α] [Fintype β]
+    (D : β → Finset α) (A : Finset α) : ℕ :=
+  ∑ b, if 2 ≤ (D b ∩ A).card then (D b ∩ A).card else 0
+
+theorem sparse_block_mgf {α β Ω : Type*} [DecidableEq α] [Fintype β] [Fintype Ω]
+    (D : β → Finset α) (hd : Pairwise (fun b b' => Disjoint (D b) (D b')))
+    (F : Ω → Finset α) (p w : ℝ) (hp : 0 ≤ p) (hw : 0 ≤ w)
+    (hF : ∀ T, inclusionMoment F T ≤ p^T.card)
+    (hr : ∀ b, ((D b).card:ℝ)*(p*w) ≤ 1/2) :
+    finiteMean (fun ω => w^(sparseMass D (F ω))) ≤
+      Real.exp (∑ b, 2*(((D b).card:ℝ)*(p*w))^2) := by
+  classical
+  have hc : CylinderBound (inclusionMoment F) (fun _ => p) := by
+    intro T
+    simpa only [Finset.prod_const] using hF T
+  have hb := cylinder_block_polynomial (fun b => ↑(D b).powerset) F (fun _ => p) hc
+    (fun _ S => S.val) (fun _ S => sparseCoefficient w S.val)
+    (fun _ S => sparseCoefficient_nonneg w hw S.val)
+    (fun b b' h S S' => (hd h).mono
+      (Finset.mem_powerset.mp S.property) (Finset.mem_powerset.mp S'.property))
+  calc
+    _ ≤ finiteMean (fun ω => ∏ b, ∑ S : ↑(D b).powerset,
+        sparseCoefficient w S.val * if S.val ⊆ F ω then 1 else 0) := by
+      apply finiteMean_mono
+      intro ω
+      rw [sparseMass,←Finset.prod_pow_eq_pow_sum]
+      apply Finset.prod_le_prod₀ (fun _ _ => pow_nonneg hw _)
+      intro b _
+      exact sparse_block_pointwise (F ω) (D b) w hw
+    _ ≤ ∏ b, ∑ S : ↑(D b).powerset, sparseCoefficient w S.val * p^S.val.card := by
+      simpa only [Finset.prod_const] using hb
+    _ ≤ _ := by
+      rw [Real.exp_sum]
+      apply Finset.prod_le_prod₀
+      · intro b _
+        exact Finset.sum_nonneg (fun S _ => mul_nonneg
+          (sparseCoefficient_nonneg w hw S.val) (pow_nonneg hp _))
+      · intro b _
+        exact sparse_block_product_bound (D b) p w hp hw (hr b)
+
+theorem inverseButterfly_sparse_mgf (d : ℕ) {β : Type*} [Fintype β]
+    (D : β → Finset (Card d)) (hd : Pairwise (fun b b' => Disjoint (D b) (D b')))
+    (A : Finset (Card d)) (w : ℝ) (hw : 0 ≤ w)
+    (hr : ∀ b, ((D b).card:ℝ)*((A.card/(2:ℝ)^d)*w) ≤ 1/2) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      w^(sparseMass D (inverseButterflyMarks d A ω))) ≤
+      Real.exp (∑ b, 2*(((D b).card:ℝ)*((A.card/(2:ℝ)^d)*w))^2) := by
+  apply sparse_block_mgf D hd (inverseButterflyMarks d A) (A.card/(2:ℝ)^d) w
+    (by positivity) hw
+  · exact inverseButterflyMarks_inclusion d A
+  · exact hr
+
+
+def crowdedMass (h : ℕ) : ℕ := if 2 ≤ h then h else 0
+
+lemma crowdedMass_le (h : ℕ) : crowdedMass h ≤ h := by
+  unfold crowdedMass
+  split_ifs <;> omega
+
+lemma crowdedMass_add (u v : ℕ) : crowdedMass u + crowdedMass v ≤ crowdedMass (u+v) := by
+  unfold crowdedMass
+  split_ifs <;> omega
+
+lemma alternatingCycles_le_crowded {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
+    (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    PairRouting.alternatingCycles e p ≤ crowdedMass (Fintype.card ι) := by
+  unfold crowdedMass
+  split_ifs with h
+  · have hh := PairRouting.alternatingCycles_double e p
+    omega
+  · rw [PairRouting.alternatingCycles_zero e p (by omega)]
+
+lemma palindromeCost_le_crowded (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) :
+    palindromeCost d e ω ≤ d*crowdedMass (Fintype.card ι) := by
+  classical
+  induction d generalizing ι with
+  | zero => simp [palindromeCost]
+  | succ d ih =>
+    let σ := benesStepEquiv d ω
+    let x := e.trans (headTailEquiv d).toEmbedding
+    let c := PairRouting.colors x σ.2.1
+    let hc := PairRouting.colors_compatible x σ.2.1
+    have h₀ := ih (PairRouting.childEmbedding x c hc false) σ.1.1
+    have h₁ := ih (PairRouting.childEmbedding x c hc true) σ.1.2
+    have hz := alternatingCycles_le_crowded (PairRouting.switchedEmbedding x σ.2.1)
+      (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1))
+    have hs := crowdedMass_add (Fintype.card {i // c i = false})
+      (Fintype.card {i // c i = true})
+    rw [PairRouting.color_card_add] at hs
+    change PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
+        (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) +
+        palindromeCost d (PairRouting.childEmbedding x c hc false) σ.1.1 +
+        palindromeCost d (PairRouting.childEmbedding x c hc true) σ.1.2 ≤ _
+    nlinarith
+
+end Thorp
+
+end ThorpNine.HighTail
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/HighTail/CutoffMass.lean b/lean/OAI/Probability/ThorpRouting/HighTail/CutoffMass.lean
index 78b5a4185cfab1add9a6adbc1d7a017ee6947065..645cc1f1c3d3ce879c2b53d94fbb6f9aad56125d 100644
--- a/lean/OAI/Probability/ThorpRouting/HighTail/CutoffMass.lean
+++ b/lean/OAI/Probability/ThorpRouting/HighTail/CutoffMass.lean
@@ -1,849 +1,849 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.HighTail.ColoredEmbedding
-
-namespace OAI
-
-namespace ThorpNine.HighTail
-
-namespace Thorp
-
-noncomputable def cutoffMass (H : ℕ) : (d : ℕ) → {ι : Type*} → [Fintype ι] →
-    (ι ↪ Card d) → BenesCoins d → ℕ
-  | 0, _, _, _, _ => 0
-  | d+1, ι, _, e, ω =>
-      if d+1 ≤ H then crowdedMass (Fintype.card ι) else
-      let σ := benesStepEquiv d ω
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x σ.2.1
-      let hc := PairRouting.colors_compatible x σ.2.1
-      cutoffMass H d (PairRouting.childEmbedding x c hc false) σ.1.1 +
-        cutoffMass H d (PairRouting.childEmbedding x c hc true) σ.1.2
-
-lemma palindromeLowCost_eq (H d : ℕ) (h : d ≤ H) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) :
-    palindromeLowCost H d e ω = palindromeCost d e ω := by
-  classical
-  induction d generalizing ι with
-  | zero => rfl
-  | succ d ih =>
-    rw [palindromeLowCost,palindromeCost,ite_eq_left h]
-    congr 2 <;> exact ih (by omega) _ _
-
-lemma palindromeLowCost_le_cutoff (H d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) :
-    palindromeLowCost H d e ω ≤ H*cutoffMass H d e ω := by
-  classical
-  induction d generalizing ι with
-  | zero => simp [palindromeLowCost,cutoffMass]
-  | succ d ih =>
-    by_cases h : d+1 ≤ H
-    · rw [palindromeLowCost_eq H (d+1) h,cutoffMass,ite_eq_left h]
-      exact (palindromeCost_le_crowded (d+1) e ω).trans (Nat.mul_le_mul_right _ h)
-    · rw [palindromeLowCost,cutoffMass,ite_eq_right h,ite_eq_right h,zero_add,Nat.mul_add]
-      exact Nat.add_le_add (ih _ _) (ih _ _)
-
-
-noncomputable def centralBlock (r H : ℕ) (b : Card r) : Finset (Card (H+r)) :=
-  Finset.univ.image (fun z : Card H => (cardSplit r H).symm (b,z))
-
-lemma mem_centralBlock (r H : ℕ) (b : Card r) (x : Card (H+r)) :
-    x ∈ centralBlock r H b ↔ (cardSplit r H x).1 = b := by
-  classical
-  constructor
-  · intro h
-    obtain ⟨z,_,rfl⟩ := Finset.mem_image.mp h
-    simp only [Equiv.apply_symm_apply]
-  · intro h
-    apply Finset.mem_image.mpr
-    refine ⟨(cardSplit r H x).2,Finset.mem_univ _,?_⟩
-    rw [←h,Prod.mk.eta,Equiv.symm_apply_apply]
-
-lemma centralBlock_card (r H : ℕ) (b : Card r) : (centralBlock r H b).card = 2^H := by
-  classical
-  rw [centralBlock,Finset.card_image_of_injective]
-  · exact card_positions H
-  · intro x y h
-    exact congrArg Prod.snd ((cardSplit r H).symm.injective h)
-
-lemma centralBlock_disjoint (r H : ℕ) :
-    Pairwise (fun b b' => Disjoint (centralBlock r H b) (centralBlock r H b')) := by
-  classical
-  intro b b' h
-  apply Finset.disjoint_left.mpr
-  intro x hx hy
-  exact h (((mem_centralBlock r H b x).mp hx).symm.trans
-    ((mem_centralBlock r H b' x).mp hy))
-
-lemma centralBlock_zero (H : ℕ) (b : Card 0) : centralBlock 0 H b = Finset.univ := by
-  classical
-  ext x
-  simp only [mem_centralBlock,Finset.mem_univ,iff_true]
-  exact Subsingleton.elim _ _
-
-lemma centralBlock_step (r H : ℕ) (b : Bool) (c : Card r) :
-    centralBlock (r+1) H (Fin.cons b c) = (centralBlock r H c).image (Fin.cons b) := by
-  classical
-  simp only [centralBlock,Finset.image_image,cardSplit,Equiv.coe_fn_symm_mk,
-    Fin.cons_zero,Fin.tail_cons,Function.comp_def]
-
-lemma block_inter_step (r H : ℕ) (b : Bool) (c : Card r) (A : Finset (Card (H+(r+1)))) :
-    centralBlock (r+1) H (Fin.cons b c) ∩ A =
-      (centralBlock r H c ∩ childMarks A b).image (Fin.cons b) := by
-  classical
-  rw [centralBlock_step]
-  ext x
-  constructor
-  · intro hx
-    obtain ⟨hxB,hxA⟩ := Finset.mem_inter.mp hx
-    obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hxB
-    exact Finset.mem_image.mpr ⟨y,Finset.mem_inter.mpr ⟨hy,(mem_childMarks A b y).mpr hxA⟩,rfl⟩
-  · intro hx
-    obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
-    exact Finset.mem_inter.mpr ⟨Finset.mem_image.mpr ⟨y,(Finset.mem_inter.mp hy).1,rfl⟩,
-      (mem_childMarks A b y).mp (Finset.mem_inter.mp hy).2⟩
-
-lemma sparseMass_central_zero (H : ℕ) (A : Finset (Card H)) :
-    sparseMass (centralBlock 0 H) A = crowdedMass A.card := by
-  classical
-  simp only [sparseMass,centralBlock_zero,Finset.univ_inter,crowdedMass,
-    Finset.sum_const,Finset.card_univ,card_positions,pow_zero,one_nsmul]
-
-lemma sparseMass_central_step (r H : ℕ) (A : Finset (Card (H+(r+1)))) :
-    sparseMass (centralBlock (r+1) H) A =
-      sparseMass (centralBlock r H) (childMarks A false) +
-        sparseMass (centralBlock r H) (childMarks A true) := by
-  classical
-  unfold sparseMass
-  rw [←Equiv.sum_comp (headTailEquiv r).symm,Fintype.sum_prod_type]
-  have ht (b : Bool) (c : Card r) : (headTailEquiv r).symm (b,c) = Fin.cons b c := rfl
-  simp only [Fintype.sum_bool,ht]
-  have hc (b : Bool) (c : Card r) :
-      (centralBlock (r+1) H (Fin.cons b c) ∩ A).card =
-      (centralBlock r H c ∩ childMarks A b).card := by
-    rw [block_inter_step]
-    apply Finset.card_image_of_injective
-    intro x y h
-    simpa only [Fin.tail_cons] using congrArg Fin.tail h
-  simp only [hc]
-  exact add_comm _ _
-
-
-lemma inverseButterfly_coordinates_step (d : ℕ)
-    (c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (ξ : Card d → Bool)
-    (x : Card (d+1)) :
-    headTailEquiv d ((butterflyPerm (d+1)
-      (decodeButterfly (d+1) ((coinStepEquiv d).symm (c,ξ)))).symm x) =
-      (Bool.xor (x 0) (ξ (Fin.tail x)),
-        (butterflyPerm d (decodeButterfly d
-          (if Bool.xor (x 0) (ξ (Fin.tail x)) then c.2 else c.1))).symm (Fin.tail x)) := by
-  rw [butterflyPerm_step]
-  rfl
-
-lemma inverseButterflyMarks_child (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (d+1))
-    (c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (ξ : Card d → Bool) (b : Bool) :
-    childMarks (inverseButterflyMarks (d+1) (Finset.univ.image e)
-      ((coinStepEquiv d).symm (c,ξ))) b =
-      inverseButterflyMarks d
-        (Finset.univ.image (PairRouting.childEmbedding (e.trans (headTailEquiv d).toEmbedding)
-          (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) ξ)
-            (PairRouting.colors_compatible _ _) b)) (if b then c.2 else c.1) := by
-  classical
-  let x := e.trans (headTailEquiv d).toEmbedding
-  let col := PairRouting.colors x ξ
-  let child := PairRouting.childEmbedding x col (PairRouting.colors_compatible x ξ) b
-  have hp (i : ι) : headTailEquiv d ((butterflyPerm (d+1)
-      (decodeButterfly (d+1) ((coinStepEquiv d).symm (c,ξ)))).symm (e i)) =
-      (col i,(butterflyPerm d (decodeButterfly d (if col i then c.2 else c.1))).symm (x i).2) :=
-    inverseButterfly_coordinates_step d c ξ (e i)
-  ext y
-  simp only [mem_childMarks,inverseButterflyMarks,Finset.image_image,
-    Finset.mem_image,Finset.mem_univ,true_and,Function.comp_def]
-  constructor
-  · rintro ⟨i,hi⟩
-    have he := congrArg (headTailEquiv d) hi
-    rw [hp i] at he
-    change (col i,_) = (b,y) at he
-    have hb := congrArg Prod.fst he
-    have hy := congrArg Prod.snd he
-    dsimp only at hb hy
-    refine ⟨⟨i,hb⟩,?_⟩
-    change (butterflyPerm d (decodeButterfly d (if b then c.2 else c.1))).symm (x i).2 = y
-    simpa only [hb] using hy
-  · rintro ⟨i,hi⟩
-    refine ⟨i.val,?_⟩
-    apply (headTailEquiv d).injective
-    rw [hp]
-    change (col i.val,_) = (b,y)
-    have hb : col i.val = b := i.property
-    rw [hb]
-    exact Prod.ext rfl hi
-
-lemma inverseButterflyMarks_card (d : ℕ) (A : Finset (Card d)) (ω : SwitchIndex d → Bool) :
-    (inverseButterflyMarks d A ω).card = A.card := by
-  classical
-  exact Finset.card_image_of_injective _ (butterflyPerm d (decodeButterfly d ω)).symm.injective
-
-lemma benes_input_coordinates (d : ℕ) (ω : BenesCoins (d+1)) :
-    ω.2 = (coinStepEquiv d).symm
-      (((benesStepEquiv d ω).1.1.2,(benesStepEquiv d ω).1.2.2),(benesStepEquiv d ω).2.1) := by
-  change ω.2 = (coinStepEquiv d).symm (coinStepEquiv d ω.2)
-  exact ((coinStepEquiv d).symm_apply_apply ω.2).symm
-
-lemma cutoffMass_central (r H : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (H+r)) (ω : BenesCoins (H+r)) :
-    cutoffMass H (H+r) e ω =
-      sparseMass (centralBlock r H) (inverseButterflyMarks (H+r) (Finset.univ.image e) ω.2) := by
-  classical
-  induction r generalizing ι with
-  | zero =>
-    rw [sparseMass_central_zero,inverseButterflyMarks_card,
-      Finset.card_image_of_injective _ e.injective,Finset.card_univ]
-    cases H with
-    | zero =>
-      have hcard : Fintype.card ι ≤ 1 := by
-        have hh := Fintype.card_le_of_injective e e.injective
-        simpa only [Nat.add_zero,card_positions,pow_zero] using hh
-      simp [cutoffMass,crowdedMass,show ¬2≤Fintype.card ι by omega]
-    | succ H => simp only [cutoffMass,le_refl,ite_true]
-  | succ r ih =>
-    simp only [Nat.add_succ] at *
-    rw [cutoffMass,ite_eq_right (by omega : ¬H+r+1≤H)]
-    let σ := benesStepEquiv (H+r) ω
-    let x := e.trans (headTailEquiv (H+r)).toEmbedding
-    let col := PairRouting.colors x σ.2.1
-    let hc := PairRouting.colors_compatible x σ.2.1
-    change cutoffMass H (H+r) (PairRouting.childEmbedding x col hc false) σ.1.1 +
-      cutoffMass H (H+r) (PairRouting.childEmbedding x col hc true) σ.1.2 = _
-    rw [ih,ih,sparseMass_central_step,benes_input_coordinates (H+r) ω]
-    rw [inverseButterflyMarks_child,inverseButterflyMarks_child]
-    rfl
-
-
-theorem palindrome_low_mgf (r H : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (H+r)) (q : ℝ) (hq : 1 ≤ q)
-    (hr : (2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*q^H) ≤ 1/2) :
-    finiteMean (fun ω : BenesCoins (H+r) => q^(palindromeLowCost H (H+r) e ω)) ≤
-      Real.exp ((2:ℝ)^r * (2 * ((2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*q^H))^2)) := by
-  classical
-  have hec : (Finset.univ.image e).card = Fintype.card ι := by
-    rw [Finset.card_image_of_injective _ e.injective,Finset.card_univ]
-  have hm := inverseButterfly_sparse_mgf (H+r) (centralBlock r H)
-    (centralBlock_disjoint r H) (Finset.univ.image e) (q^H)
-    (pow_nonneg (by linarith) _) (fun b => by
-      simpa only [centralBlock_card,hec,Nat.cast_pow,Nat.cast_ofNat] using hr)
-  calc
-    _ ≤ finiteMean (fun ω : BenesCoins (H+r) => (q^H)^(cutoffMass H (H+r) e ω)) := by
-      apply finiteMean_mono
-      intro ω
-      rw [←pow_mul]
-      exact pow_le_pow_right₀ hq (palindromeLowCost_le_cutoff H (H+r) e ω)
-    _ = finiteMean (fun ω : SwitchIndex (H+r) → Bool =>
-        (q^H)^(sparseMass (centralBlock r H)
-          (inverseButterflyMarks (H+r) (Finset.univ.image e) ω))) := by
-      simp_rw [cutoffMass_central]
-      simp only [finiteMean_prod,finiteMean_const]
-    _ ≤ _ := by
-      simpa only [centralBlock_card,hec,Nat.cast_pow,Nat.cast_ofNat,Finset.sum_const,
-        Finset.card_univ,card_positions,nsmul_eq_mul] using hm
-
-variable {α B : Type*} {r s : ℕ}
-
-def parallelFreshPre (pre : α ≃ B × Card (s+r))
-    (lo : B → Card r × SwitchIndex s → Bool) : α ≃ (B × Card s) × Card r :=
-  ((pre.trans (Equiv.prodCongr (Equiv.refl B) (cardSplit r s))).trans
-    (Equiv.prodCongrRight (fun b =>
-      (parallelPerm (lo b)).trans (Equiv.prodComm _ _)))).trans
-    (Equiv.prodAssoc _ _ _).symm
-
-def parallelFreshPost (post : B × Card (s+r) ≃ α) : (B × Card s) × Card r ≃ α :=
-  (((Equiv.prodAssoc _ _ _).trans (Equiv.prodCongr (Equiv.refl B)
-    (Equiv.prodComm _ _))).trans (Equiv.prodCongr (Equiv.refl B) (cardSplit r s).symm)).trans post
-
-def parallelAssembled (lo : B → Card r × SwitchIndex s → Bool)
-    (hi : (B × Card s) × SwitchIndex r → Bool) : B × SwitchIndex (s+r) → Bool :=
-  fun i => assembleBits r s (lo i.1) (fun j => hi ((i.1,j.1),j.2)) i.2
-
-lemma parallel_exposed_perm (pre : α ≃ B × Card (s+r)) (post : B × Card (s+r) ≃ α)
-    (lo : B → Card r × SwitchIndex s → Bool)
-    (hi : (B × Card s) × SwitchIndex r → Bool) :
-    relabeledPerm pre post (parallelAssembled lo hi) =
-      relabeledPerm (parallelFreshPre pre lo) (parallelFreshPost post) hi := by
-  ext x
-  have h := butterfly_split r s (lo (pre x).1)
-    (fun j => hi (((pre x).1,j.1),j.2)) (pre x).2
-  have hh := congrArg (fun y => post ((pre x).1,(cardSplit r s).symm y)) h
-  simpa only [relabeledPerm,parallelAssembled,parallelPerm,parallelFreshPre,parallelFreshPost,
-    Equiv.trans_apply,Equiv.prodCongr_apply,Equiv.prodCongrRight,Equiv.coe_fn_mk,
-    Equiv.refl_apply,Equiv.prodComm_apply,Prod.swap,Equiv.prodAssoc_apply,
-    Equiv.prodAssoc_symm_apply,Prod.map,Equiv.symm_apply_apply] using hh
-
-theorem parallel_partially_exposed_log_mgf [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B]
-    (pre : α ≃ B × Card (s+r)) (post : B × Card (s+r) ≃ α)
-    (lo : B → Card r × SwitchIndex s → Bool) (S : Finset α)
-    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    Real.log (finiteMean (fun hi : (B × Card s) × SwitchIndex r → Bool =>
-      q^(Fintype.card (RoutingNetwork.SelectedCycle
-        (relabeledPerm pre post (parallelAssembled lo hi)) S)))) ≤
-      (S.card:ℝ)/J*Real.log q + 2*q*S.card*Real.sqrt J/Real.sqrt ((2:ℝ)^r) := by
-  simp_rw [parallel_exposed_perm]
-  exact parallel_cycle_log_mgf (parallelFreshPre pre lo) (parallelFreshPost post) S J hJ q hq
-
-end Thorp
-namespace Thorp
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-lemma orbitSet_inverse (p : Equiv.Perm α) (x : α) : orbitSet p⁻¹ x = orbitSet p x := by
-  ext y
-  simp only [mem_orbitSet,Equiv.Perm.sameCycle_inv]
-
-lemma selectedCycle_inverse (p : Equiv.Perm α) (S : Finset α) :
-    Fintype.card (RoutingNetwork.SelectedCycle p⁻¹ S) =
-      Fintype.card (RoutingNetwork.SelectedCycle p S) := by
-  apply Fintype.card_congr
-  refine { toFun := fun C => ⟨C.val,?_,C.property.2⟩
-           invFun := fun C => ⟨C.val,?_,C.property.2⟩
-           left_inv := fun _ => rfl
-           right_inv := fun _ => rfl }
-  · obtain ⟨x,hx⟩ := C.property.1
-    exact ⟨x,hx.trans (orbitSet_inverse p x)⟩
-  · obtain ⟨x,hx⟩ := C.property.1
-    exact ⟨x,hx.trans (orbitSet_inverse p x).symm⟩
-
-lemma orbitSet_conjugate (p g : Equiv.Perm α) (x : α) :
-    orbitSet (g*p*g⁻¹) (g x) = (orbitSet p x).image g := by
-  ext y
-  simp only [mem_orbitSet,Finset.mem_image]
-  rw [Equiv.Perm.sameCycle_conj]
-  change p.SameCycle (g.symm (g x)) (g.symm y) ↔ ∃ z, p.SameCycle x z ∧ g z = y
-  rw [Equiv.symm_apply_apply]
-  constructor
-  · intro h
-    exact ⟨g.symm y,h,g.apply_symm_apply y⟩
-  · rintro ⟨z,hz,rfl⟩
-    simpa only [Equiv.symm_apply_apply] using hz
-
-noncomputable def conjugateCycle (p g : Equiv.Perm α) (S : Finset α)
-    (C : RoutingNetwork.SelectedCycle p S) :
-    RoutingNetwork.SelectedCycle (g*p*g⁻¹) (S.image g) := by
-  refine ⟨C.val.image g,?_,Finset.image_subset_image C.property.2⟩
-  obtain ⟨x,hx⟩ := C.property.1
-  exact ⟨g x,by rw [hx,orbitSet_conjugate]⟩
-
-lemma conjugateCycle_injective (p g : Equiv.Perm α) (S : Finset α) :
-    Function.Injective (conjugateCycle p g S) := by
-  intro C D he
-  apply Subtype.ext
-  exact (Finset.image_injective g.injective) (congrArg Subtype.val he)
-
-lemma selectedCycle_conjugate (p g : Equiv.Perm α) (S : Finset α) :
-    Fintype.card (RoutingNetwork.SelectedCycle (g*p*g⁻¹) (S.image g)) =
-      Fintype.card (RoutingNetwork.SelectedCycle p S) := by
-  apply Nat.le_antisymm
-  · have h := Fintype.card_le_of_injective _
-      (conjugateCycle_injective (g*p*g⁻¹) g⁻¹ (S.image g))
-    have hp : g⁻¹*(g*p*g⁻¹)*g = p := by group
-    rw [inv_inv,hp] at h
-    have hs : (S.image g).image g.symm = S := by
-      ext x
-      simp only [Finset.mem_image]
-      constructor
-      · rintro ⟨y,⟨z,hz,rfl⟩,he⟩
-        have hz' : z = x := by simpa only [Equiv.symm_apply_apply] using he
-        exact hz' ▸ hz
-      · intro hx
-        exact ⟨g x,⟨x,hx,rfl⟩,g.symm_apply_apply x⟩
-    change _ ≤ Fintype.card (RoutingNetwork.SelectedCycle p ((S.image g).image g.symm)) at h
-    rw [hs] at h
-    exact h
-  · exact Fintype.card_le_of_injective _ (conjugateCycle_injective p g S)
-
-lemma alternatingCycles_le_projection {ι : Type*} [Fintype ι]
-    (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    PairRouting.alternatingCycles e p ≤
-      Fintype.card (RoutingNetwork.SelectedCycle (PairRouting.alternatingProjection p)
-        (boolFiber (PairRouting.labelSet e) false)) := by
-  have h := CycleColoring.card_alternating_cycles_le (PairRouting.alternatingProjection p)
-    (boolFiber (PairRouting.labelSet e) false) (boolFiber (PairRouting.labelSet e) true)
-  simpa only [PairRouting.alternatingCycles,childJoin_boolFiber] using h
-
-lemma projectionCycles_forward (X₀ X₁ Y₀ Y₁ : Equiv.Perm α) (S : Finset α) :
-    Fintype.card (RoutingNetwork.SelectedCycle
-      ((Y₀*X₀⁻¹)⁻¹*(Y₁*X₁⁻¹)) S) =
-    Fintype.card (RoutingNetwork.SelectedCycle
-      (X₀⁻¹*X₁*Y₁⁻¹*Y₀) (S.image X₀.symm)) := by
-  let p := (Y₀*X₀⁻¹)⁻¹*(Y₁*X₁⁻¹)
-  calc
-    _ = Fintype.card (RoutingNetwork.SelectedCycle (X₀⁻¹*p*X₀)
-        (S.image X₀.symm)) := by
-      change Fintype.card (RoutingNetwork.SelectedCycle p S) = _
-      have h := (selectedCycle_conjugate p X₀⁻¹ S).symm
-      rw [inv_inv] at h
-      exact h
-    _ = Fintype.card (RoutingNetwork.SelectedCycle (X₀⁻¹*p*X₀)⁻¹
-        (S.image X₀.symm)) := (selectedCycle_inverse _ _).symm
-    _ = _ := by
-      have he : (X₀⁻¹*p*X₀)⁻¹ = X₀⁻¹*X₁*Y₁⁻¹*Y₀ := by dsimp only [p]; group
-      rw [he]
-
-end Thorp
-namespace Thorp
-
-noncomputable def cycleCoefficient (r J : ℕ) (q : ℝ) : ℝ :=
-  Real.log q / J + 2*q*Real.sqrt J/Real.sqrt ((2:ℝ)^r)
-
-lemma cycleCoefficient_nonneg (r J : ℕ) (q : ℝ) (hq : 1 ≤ q) :
-    0 ≤ cycleCoefficient r J q := by
-  unfold cycleCoefficient
-  exact add_nonneg (div_nonneg (Real.log_nonneg hq) (Nat.cast_nonneg _))
-    (div_nonneg (mul_nonneg (by linarith) (Real.sqrt_nonneg _)) (Real.sqrt_nonneg _))
-
-lemma projectionSide_card_le {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι]
-    (e : ι ↪ Bool × α) : (boolFiber (PairRouting.labelSet e) false).card ≤ Fintype.card ι := by
-  have h := card_childJoin (boolFiber (PairRouting.labelSet e) false)
-    (boolFiber (PairRouting.labelSet e) true)
-  rw [childJoin_boolFiber,PairRouting.card_labelSet] at h
-  omega
-
-theorem node_partial_mgf (r s : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Bool × Card (s+r)) (X₀ X₁ Y₁ : Equiv.Perm (Card (s+r)))
-    (lo : Card r × SwitchIndex s → Bool) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(PairRouting.alternatingCycles e (fun b => if b then Y₁*X₁⁻¹ else
-        butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi))*X₀⁻¹))) ≤
-      Real.exp (Fintype.card ι * cycleCoefficient r J q) := by
-  classical
-  let S := boolFiber (PairRouting.labelSet e) false
-  let S' := S.image X₀.symm
-  let post := X₀⁻¹*X₁*Y₁⁻¹
-  let count := fun hi : Card s × SwitchIndex r → Bool =>
-    Fintype.card (RoutingNetwork.SelectedCycle
-      ((Equiv.refl _).trans (butterflyPerm (s+r)
-        (decodeButterfly (s+r) (assembleBits r s lo hi))) |>.trans post) S')
-  have hpoint (hi : Card s × SwitchIndex r → Bool) :
-      PairRouting.alternatingCycles e (fun b => if b then Y₁*X₁⁻¹ else
-        butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi))*X₀⁻¹) ≤ count hi := by
-    refine (alternatingCycles_le_projection e _).trans_eq ?_
-    simp only [PairRouting.alternatingProjection,Bool.false_eq_true,↓reduceIte]
-    change Fintype.card (RoutingNetwork.SelectedCycle
-      ((butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi))*X₀⁻¹)⁻¹ * (Y₁*X₁⁻¹)) S) = count hi
-    exact projectionCycles_forward X₀ X₁ _ Y₁ S
-  have hm := partially_exposed_cycle_log_mgf (Equiv.refl (Card (s+r))) post lo S' J hJ q hq
-  have hp : 0 < finiteMean (fun hi : Card s × SwitchIndex r → Bool => q^(count hi)) :=
-    finiteMean_pos (fun _ => pow_pos (by linarith) _)
-  have hb : finiteMean (fun hi : Card s × SwitchIndex r → Bool => q^(count hi)) ≤
-      Real.exp (S'.card * cycleCoefficient r J q) := by
-    rw [←Real.exp_log hp]
-    apply Real.exp_le_exp.mpr
-    calc
-      _ ≤ _ := hm
-      _ = _ := by unfold cycleCoefficient; ring
-  have hcard : S'.card ≤ Fintype.card ι := by
-    rw [show S'.card = S.card from Finset.card_image_of_injective _ X₀.symm.injective]
-    exact projectionSide_card_le e
-  calc
-    _ ≤ finiteMean (fun hi : Card s × SwitchIndex r → Bool => q^(count hi)) :=
-      finiteMean_mono (fun hi => pow_le_pow_right₀ hq (hpoint hi))
-    _ ≤ Real.exp (S'.card * cycleCoefficient r J q) := hb
-    _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right
-      (by exact_mod_cast hcard) (cycleCoefficient_nonneg r J q hq))
-
-
-def highStepEquiv (r s : ℕ) : (Card s × SwitchIndex (r+1) → Bool) ≃
-    (((Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool)) ×
-      (Card s × Card r → Bool)) where
-  toFun hi := ((fun j => hi (j.1,Sum.inr (false,j.2)),
-    fun j => hi (j.1,Sum.inr (true,j.2))),fun j => hi (j.1,Sum.inl j.2))
-  invFun c := fun j => match j.2 with
-    | Sum.inl y => c.2 (j.1,y)
-    | Sum.inr (b,i) => if b then c.1.2 (j.1,i) else c.1.1 (j.1,i)
-  left_inv hi := by
-    funext j
-    rcases j with ⟨z,y|⟨b,i⟩⟩
-    · rfl
-    · cases b <;> rfl
-  right_inv c := by
-    rcases c with ⟨⟨h₀,h₁⟩,η⟩
-    rfl
-
-lemma assembleBits_step (r s : ℕ) (lo : Card (r+1) × SwitchIndex s → Bool)
-    (c : ((Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool)) ×
-      (Card s × Card r → Bool)) :
-    assembleBits (r+1) s lo ((highStepEquiv r s).symm c) =
-      (coinStepEquiv (s+r)).symm
-        ((assembleBits r s (fun j => lo (Fin.cons false j.1,j.2)) c.1.1,
-          assembleBits r s (fun j => lo (Fin.cons true j.1,j.2)) c.1.2),
-        fun y => c.2 ((cardSplit r s y).2,(cardSplit r s y).1)) := by
-  funext i
-  rcases i with y | ⟨b,i⟩
-  · rfl
-  · cases b <;> rfl
-
-noncomputable def palindromeLevelCost (t : ℕ) : (d : ℕ) → {ι : Type*} → [Fintype ι] →
-    (ι ↪ Card d) → BenesCoins d → ℕ
-  | 0, _, _, _, _ => 0
-  | d+1, _, _, e, ω =>
-      let σ := benesStepEquiv d ω
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x σ.2.1
-      let hc := PairRouting.colors_compatible x σ.2.1
-      if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
-          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) else
-        palindromeLevelCost t d (PairRouting.childEmbedding x c hc false) σ.1.1 +
-          palindromeLevelCost t d (PairRouting.childEmbedding x c hc true) σ.1.2
-
-lemma palindromeLevelCost_step (t d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (d+1)) (X Y : (SwitchIndex d → Bool) × (SwitchIndex d → Bool))
-    (ξ η : Card d → Bool) :
-    palindromeLevelCost t (d+1) e
-      ((coinStepEquiv d).symm (Y,η),(coinStepEquiv d).symm (X,ξ)) =
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x ξ
-      let hc := PairRouting.colors_compatible x ξ
-      if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x ξ)
-        (fun b => palindromePerm d (if b then (Y.2,X.2) else (Y.1,X.1))) else
-        palindromeLevelCost t d (PairRouting.childEmbedding x c hc false) (Y.1,X.1) +
-        palindromeLevelCost t d (PairRouting.childEmbedding x c hc true) (Y.2,X.2) := by
-  rw [palindromeLevelCost]
-  rfl
-
-lemma palindromeLevelCost_gt (t d : ℕ) (h : d < t) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) : palindromeLevelCost t d e ω = 0 := by
-  classical
-  induction d generalizing ι with
-  | zero => rfl
-  | succ d ih =>
-    rw [palindromeLevelCost,ite_eq_right (by omega : t ≠ d+1),ih (by omega),ih (by omega),Nat.zero_add]
-
-
-theorem palindrome_level_partial_mgf (r s u : ℕ) (hu : u < r)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+r)) (X : SwitchIndex (s+r) → Bool)
-    (lo : Card r × SwitchIndex s → Bool) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(palindromeLevelCost (s+u+1) (s+r) e (assembleBits r s lo hi,X))) ≤
-      Real.exp (Fintype.card ι * cycleCoefficient u J q) := by
-  classical
-  induction r generalizing u ι with
-  | zero => omega
-  | succ r ih =>
-    simp only [Nat.add_succ] at *
-    let χ := coinStepEquiv (s+r) X
-    let x := e.trans (headTailEquiv (s+r)).toEmbedding
-    let c := PairRouting.colors x χ.2
-    let hc := PairRouting.colors_compatible x χ.2
-    let e₀ := PairRouting.childEmbedding x c hc false
-    let e₁ := PairRouting.childEmbedding x c hc true
-    let e' := PairRouting.switchedEmbedding x χ.2
-    let lo₀ : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons false j.1,j.2)
-    let lo₁ : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons true j.1,j.2)
-    let Y₀ := fun hi : Card s × SwitchIndex r → Bool => assembleBits r s lo₀ hi
-    let Y₁ := fun hi : Card s × SwitchIndex r → Bool => assembleBits r s lo₁ hi
-    let N := fun hi : (Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool) =>
-      PairRouting.alternatingCycles e' (fun b => if b then palindromePerm (s+r) (Y₁ hi.2,χ.1.2)
-        else palindromePerm (s+r) (Y₀ hi.1,χ.1.1))
-    let L₀ := fun hi : Card s × SwitchIndex r → Bool =>
-      palindromeLevelCost (s+u+1) (s+r) e₀ (Y₀ hi,χ.1.1)
-    let L₁ := fun hi : Card s × SwitchIndex r → Bool =>
-      palindromeLevelCost (s+u+1) (s+r) e₁ (Y₁ hi,χ.1.2)
-    have hs (hi : (Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool))
-        (η : Card s × Card r → Bool) :
-        palindromeLevelCost (s+u+1) (s+r+1) e
-          (assembleBits (r+1) s lo ((highStepEquiv r s).symm (hi,η)),X) =
-          if u = r then N hi else L₀ hi.1 + L₁ hi.2 := by
-      rw [assembleBits_step]
-      conv_lhs => rw [←(coinStepEquiv (s+r)).symm_apply_apply X]
-      rw [palindromeLevelCost_step]
-      simp only [Nat.add_right_cancel_iff,Nat.add_left_cancel_iff]
-      dsimp only [χ,x,c,hc,e₀,e₁,e',lo₀,lo₁,Y₀,Y₁,N,L₀,L₁]
-      congr 2
-      funext b
-      cases b <;> rfl
-    rw [finiteMean_equiv (highStepEquiv r s),finiteMean_prod]
-    simp_rw [hs,finiteMean_const]
-    by_cases hur : u = r
-    · subst u
-      simp only [ite_true]
-      rw [finiteMean_prod,finiteMean_comm]
-      calc
-        _ ≤ finiteMean (fun _hi : Card s × SwitchIndex r → Bool =>
-          Real.exp (Fintype.card ι * cycleCoefficient r J q)) := by
-          apply finiteMean_mono
-          intro hi
-          have hn := node_partial_mgf r s e'
-            (butterflyPerm (s+r) (decodeButterfly (s+r) χ.1.1))
-            (butterflyPerm (s+r) (decodeButterfly (s+r) χ.1.2))
-            (butterflyPerm (s+r) (decodeButterfly (s+r) (Y₁ hi))) lo₀ J hJ q hq
-          exact hn
-        _ = _ := finiteMean_const _
-    · simp only [ite_eq_right hur,pow_add]
-      rw [finiteMean_prod_mul (fun hi => q ^ L₀ hi) (fun hi => q ^ L₁ hi)]
-      have hu' : u < r := by omega
-      have h₀ := ih u hu' e₀ χ.1.1 lo₀
-      have h₁ := ih u hu' e₁ χ.1.2 lo₁
-      have hp := mul_le_mul h₀ h₁ (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
-        (Real.exp_nonneg _)
-      refine hp.trans_eq ?_
-      rw [←Real.exp_add]
-      congr 1
-      have hcount := PairRouting.color_card_add c
-      have hcount' : (Fintype.card {i : ι // c i = false} : ℝ) +
-          Fintype.card {i : ι // c i = true} = Fintype.card ι := by exact_mod_cast hcount
-      change (Fintype.card {i : ι // c i = false} : ℝ) * cycleCoefficient u J q +
-          Fintype.card {i : ι // c i = true} * cycleCoefficient u J q = _
-      rw [←add_mul,hcount']
-
-
-def switchHeight : (d : ℕ) → SwitchIndex d → ℕ
-  | 0, i => nomatch i
-  | d+1, Sum.inl _ => d+1
-  | d+1, Sum.inr (_,i) => switchHeight d i
-
-lemma switchHeight_le (d : ℕ) (i : SwitchIndex d) : switchHeight d i ≤ d := by
-  induction d with
-  | zero => exact Empty.elim i
-  | succ d ih =>
-    rcases i with y | ⟨b,i⟩
-    · exact le_refl _
-    · exact (ih i).trans (Nat.le_succ d)
-
-def AgreeThrough (d s : ℕ) (Y Y' : SwitchIndex d → Bool) : Prop :=
-  ∀ i, switchHeight d i ≤ s → Y i = Y' i
-
-def LayerAdapted (d s : ℕ) {β : Type*} (f : (SwitchIndex d → Bool) → β) : Prop :=
-  ∀ Y Y', AgreeThrough d s Y Y' → f Y = f Y'
-
-lemma agreeThrough_mono {d s t : ℕ} (h : s ≤ t) {Y Y' : SwitchIndex d → Bool}
-    (hh : AgreeThrough d t Y Y') : AgreeThrough d s Y Y' :=
-  fun i hi => hh i (hi.trans h)
-
-lemma layerAdapted_mono {d s t : ℕ} {β : Type*} {f : (SwitchIndex d → Bool) → β}
-    (hf : LayerAdapted d s f) (h : s ≤ t) : LayerAdapted d t f :=
-  fun Y Y' hh => hf Y Y' (agreeThrough_mono h hh)
-
-lemma layerAdapted_comp {d s : ℕ} {α β : Type*} {f : (SwitchIndex d → Bool) → α}
-    (hf : LayerAdapted d s f) (g : α → β) : LayerAdapted d s (g ∘ f) :=
-  fun Y Y' h => congrArg g (hf Y Y' h)
-
-lemma layerAdapted_sum {d s : ℕ} {ι : Type*} (A : Finset ι)
-    (f : ι → (SwitchIndex d → Bool) → ℕ)
-    (hf : ∀ i ∈ A, LayerAdapted d s (f i)) :
-    LayerAdapted d s (fun Y => ∑ i ∈ A, f i Y) := by
-  intro Y Y' h
-  exact Finset.sum_congr rfl (fun i hi => hf i hi Y Y' h)
-
-lemma assembleBits_agree (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
-    (hi hi' : Card s × SwitchIndex r → Bool) :
-    AgreeThrough (s+r) s (assembleBits r s lo hi) (assembleBits r s lo hi') := by
-  induction r with
-  | zero => intro i _; rfl
-  | succ r ih =>
-    intro i h
-    rcases i with y | ⟨b,i⟩
-    · have : s+r+1 ≤ s := h
-      omega
-    · exact ih (fun j => lo (Fin.cons b j.1,j.2))
-        (fun j => hi (j.1,Sum.inr (b,j.2))) (fun j => hi' (j.1,Sum.inr (b,j.2))) i h
-
-lemma palindromeLevelCost_adapted (t d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) :
-    LayerAdapted d (t-1) (fun Y => palindromeLevelCost t d e (Y,X)) := by
-  classical
-  induction d generalizing ι with
-  | zero => intro Y Y' _; rfl
-  | succ d ih =>
-    intro Y Y' h
-    change palindromeLevelCost t (d+1) e (Y,X) = palindromeLevelCost t (d+1) e (Y',X)
-    conv_lhs => rw [←(coinStepEquiv d).symm_apply_apply Y,←(coinStepEquiv d).symm_apply_apply X]
-    conv_rhs => rw [←(coinStepEquiv d).symm_apply_apply Y',←(coinStepEquiv d).symm_apply_apply X]
-    rw [palindromeLevelCost_step,palindromeLevelCost_step]
-    by_cases ht : t = d+1
-    · subst t
-      rw [ite_eq_left rfl,ite_eq_left rfl]
-      have he₀ : (coinStepEquiv d Y).1.1 = (coinStepEquiv d Y').1.1 := by
-        funext i
-        exact h (Sum.inr (false,i)) (by simpa only [switchHeight,Nat.add_sub_cancel] using switchHeight_le d i)
-      have he₁ : (coinStepEquiv d Y).1.2 = (coinStepEquiv d Y').1.2 := by
-        funext i
-        exact h (Sum.inr (true,i)) (by simpa only [switchHeight,Nat.add_sub_cancel] using switchHeight_le d i)
-      rw [he₀,he₁]
-    · rw [ite_eq_right ht,ite_eq_right ht]
-      congr 1
-      · apply ih
-        intro i hi
-        exact h (Sum.inr (false,i)) hi
-      · apply ih
-        intro i hi
-        exact h (Sum.inr (true,i)) hi
-
-lemma adapted_mul_mean_le (r s : ℕ) (F G : (SwitchIndex (s+r) → Bool) → ℝ)
-    (hF : ∀ Y, 0 ≤ F Y) (hA : LayerAdapted (s+r) s F) (B : ℝ)
-    (hG : ∀ lo : Card r × SwitchIndex s → Bool,
-      finiteMean (fun hi : Card s × SwitchIndex r → Bool => G (assembleBits r s lo hi)) ≤ B) :
-    finiteMean (fun Y => F Y*G Y) ≤ finiteMean F * B := by
-  rw [finiteMean_split r s,finiteMean_split r s F,←finiteMean_mul_const]
-  apply finiteMean_mono
-  intro lo
-  let H : Card s × SwitchIndex r → Bool := fun _ => false
-  have hh (hi : Card s × SwitchIndex r → Bool) :
-      F (assembleBits r s lo hi) = F (assembleBits r s lo H) :=
-    hA _ _ (assembleBits_agree r s lo hi H)
-  simp_rw [hh,finiteMean_const]
-  rw [show (fun hi => F (assembleBits r s lo H) * G (assembleBits r s lo hi)) =
-      (fun hi => G (assembleBits r s lo hi) * F (assembleBits r s lo H)) by
-        funext hi; exact mul_comm _ _,finiteMean_mul_const]
-  simpa only [mul_comm] using mul_le_mul_of_nonneg_left (hG lo) (hF _)
-
-
-theorem cycleCoefficient_decay (l r : ℕ) (hr : (l:ℝ)/2 ≤ r)
-    (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ (l:ℝ)/250) :
-    cycleCoefficient r ⌈Real.exp ((l:ℝ)/32)⌉₊ q ≤
-      68 * Real.exp (-(l:ℝ)/64) := by
-  let E := Real.exp ((l:ℝ)/32)
-  let J := ⌈E⌉₊
-  have hl : (0:ℝ) ≤ l := Nat.cast_nonneg _
-  have hE : 1 ≤ E := Real.one_le_exp_iff.mpr (by positivity)
-  have hEpos : 0 < E := Real.exp_pos _
-  have hEJ : E ≤ (J:ℝ) := Nat.le_ceil E
-  have hJ : (1:ℝ) ≤ J := hE.trans hEJ
-  have hJpos : (0:ℝ) < J := lt_of_lt_of_le (by norm_num) hJ
-  have hJE : (J:ℝ) ≤ 2*E := by
-    have h := Nat.ceil_lt_add_one (le_of_lt hEpos)
-    change (J:ℝ) < E+1 at h
-    linarith
-  have hsqJ : Real.sqrt J ≤ (J:ℝ) := by
-    apply Real.sqrt_le_iff.mpr
-    constructor
-    · positivity
-    · nlinarith
-  have hqp : 0 < q := lt_of_lt_of_le (by norm_num) hq
-  have hqE : q ≤ Real.exp ((l:ℝ)/250) := by
-    rw [←Real.exp_log hqp]
-    exact Real.exp_le_exp.mpr hlog
-  have hln : (1/2:ℝ) ≤ Real.log 2 := by
-    have hh := Real.one_sub_inv_le_log_of_pos (by norm_num : (0:ℝ)<2)
-    norm_num at hh ⊢
-    exact hh
-  have hden : Real.exp ((l:ℝ)/8) ≤ Real.sqrt ((2:ℝ)^r) := by
-    have he : (2:ℝ)^r = Real.exp ((r:ℝ)*Real.log 2) := by
-      rw [Real.exp_nat_mul,Real.exp_log (by norm_num)]
-    rw [he,Real.sqrt_eq_rpow,Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp]
-    apply Real.exp_le_exp.mpr
-    have hrn : (0:ℝ) ≤ r := Nat.cast_nonneg _
-    nlinarith
-  have hlog' : Real.log q ≤ (l:ℝ) := by linarith
-  have h₁ : Real.log q / (J:ℝ) ≤ 64*Real.exp (-(l:ℝ)/64) := by
-    calc
-      _ ≤ (l:ℝ)/E := div_le_div₀ (by positivity) hlog' hEpos hEJ
-      _ ≤ (64*Real.exp ((l:ℝ)/64))/E := by
-        apply div_le_div_of_nonneg_right _ (le_of_lt hEpos)
-        have hh := Real.add_one_le_exp ((l:ℝ)/64)
-        linarith
-      _ = _ := by
-        dsimp [E]
-        rw [mul_div_assoc,←Real.exp_sub]
-        congr 2
-        ring
-  have h₂ : 2*q*Real.sqrt J / Real.sqrt ((2:ℝ)^r) ≤
-      4*Real.exp (-(l:ℝ)/64) := by
-    calc
-      _ ≤ (4*Real.exp ((l:ℝ)/250)*E)/Real.exp ((l:ℝ)/8) := by
-        apply div_le_div₀ (by positivity) _ (Real.exp_pos _) hden
-        have hs := hsqJ.trans hJE
-        have hm := mul_le_mul hqE hs (Real.sqrt_nonneg _) (Real.exp_nonneg _)
-        nlinarith
-      _ = 4*Real.exp ((l:ℝ)/250+(l:ℝ)/32-(l:ℝ)/8) := by
-        dsimp [E]
-        rw [mul_assoc,←Real.exp_add,mul_div_assoc,←Real.exp_sub]
-      _ ≤ _ := by
-        apply mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr _) (by norm_num)
-        linarith
-  change Real.log q / (J:ℝ) + 2*q*Real.sqrt J / Real.sqrt ((2:ℝ)^r) ≤ _
-  linarith
-
-lemma cycleCutoff_pos (l : ℕ) : 0 < ⌈Real.exp ((l:ℝ)/32)⌉₊ := by
-  have h := Nat.le_ceil (Real.exp ((l:ℝ)/32))
-  have he := Real.exp_pos ((l:ℝ)/32)
-  have : (0:ℝ) < ⌈Real.exp ((l:ℝ)/32)⌉₊ := lt_of_lt_of_le he h
-  exact_mod_cast this
-
-
-lemma pow_sum_le_mean_powers {ι : Type*} [Fintype ι] [Nonempty ι]
-    (z : ι → ℕ) (q : ℝ) (hq : 0 < q) :
-    q^(∑ i, z i) ≤ finiteMean (fun i => q^(Fintype.card ι*z i)) := by
-  have hn : (Fintype.card ι:ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
-  have hm : finiteMean (fun i => (Fintype.card ι*z i:ℕ)*Real.log q) =
-      (∑ i, z i:ℕ)*Real.log q := by
-    simp only [finiteMean,Nat.cast_mul,Nat.cast_sum,←Finset.mul_sum,←Finset.sum_mul]
-    field_simp
-  have he := exp_finiteMean_le (fun i => (Fintype.card ι*z i:ℕ)*Real.log q)
-  rw [hm,Real.exp_nat_mul,Real.exp_log hq] at he
-  simpa only [Real.exp_nat_mul,Real.exp_log hq] using he
-
-noncomputable def palindromeSlabCost (l d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) : ℕ :=
-  ∑ i : Fin l, palindromeLevelCost (l+i.1+1) d e ω
-
-lemma palindromeSlabCost_adapted (l d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) :
-    LayerAdapted d (2*l) (fun Y => palindromeSlabCost l d e (Y,X)) := by
-  apply layerAdapted_sum
-  intro i _
-  exact layerAdapted_mono (palindromeLevelCost_adapted (l+i.1+1) d e X) (by omega)
-
-theorem palindrome_slab_partial_mgf (r s l : ℕ) (hl : 0 < l) (hs : 2*s ≤ l)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+r)) (X : SwitchIndex (s+r) → Bool)
-    (lo : Card r × SwitchIndex s → Bool) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(palindromeSlabCost l (s+r) e (assembleBits r s lo hi,X))) ≤
-      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
-  classical
-  let : NeZero l := ⟨ne_of_gt hl⟩
-  have hqp : 0 < q := lt_of_lt_of_le (by norm_num) hq
-  calc
-    _ ≤ finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      finiteMean (fun i : Fin l => q^(l*palindromeLevelCost (l+i.1+1) (s+r) e
-        (assembleBits r s lo hi,X)))) := by
-      apply finiteMean_mono
-      intro hi
-      simpa only [palindromeSlabCost,Fintype.card_fin] using
-        pow_sum_le_mean_powers (fun i : Fin l => palindromeLevelCost (l+i.1+1) (s+r) e
-          (assembleBits r s lo hi,X)) q hqp
-    _ = finiteMean (fun i : Fin l => finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(l*palindromeLevelCost (l+i.1+1) (s+r) e (assembleBits r s lo hi,X)))) := finiteMean_comm _
-    _ ≤ finiteMean (fun _i : Fin l =>
-      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64)))) := by
-      apply finiteMean_mono
-      intro i
-      by_cases ht : l+i.1+1 ≤ s+r
-      · have hs' : s ≤ l+i.1 := by omega
-        have he : l+i.1+1 = s+(l+i.1-s)+1 := by omega
-        have hu : l+i.1-s < r := by omega
-        have hm := palindrome_level_partial_mgf r s (l+i.1-s) hu e X lo
-          ⌈Real.exp ((l:ℝ)/32)⌉₊ (cycleCutoff_pos l) (q^l) (one_le_pow₀ hq)
-        have hc := cycleCoefficient_decay l (l+i.1-s) (by
-          rw [Nat.cast_sub hs',Nat.cast_add]
-          have hsr : (2:ℝ)*s ≤ l := by exact_mod_cast hs
-          have hi : (0:ℝ) ≤ i.1 := Nat.cast_nonneg _
-          linarith) (q^l) (one_le_pow₀ hq) (by
-            rw [Real.log_pow]
-            nlinarith [mul_le_mul_of_nonneg_left hlog (Nat.cast_nonneg l)])
-        simp_rw [he,pow_mul]
-        exact hm.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left hc (Nat.cast_nonneg _)))
-      · have hz := palindromeLevelCost_gt (l+i.1+1) (s+r) (by omega) e
-        simp only [hz,Nat.mul_zero,pow_zero,finiteMean_const]
-        exact Real.one_le_exp_iff.mpr (by positivity)
-    _ = _ := finiteMean_const _
-
-
-lemma palindromeSlabCost_gt (l d : ℕ) (h : d ≤ l) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) : palindromeSlabCost l d e ω = 0 := by
-  apply Finset.sum_eq_zero
-  intro i _
-  exact palindromeLevelCost_gt _ _ (by omega) _ _
-
-end Thorp
-
-end ThorpNine.HighTail
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.HighTail.ColoredEmbedding
+
+namespace OAI
+
+namespace ThorpNine.HighTail
+
+namespace Thorp
+
+noncomputable def cutoffMass (H : ℕ) : (d : ℕ) → {ι : Type*} → [Fintype ι] →
+    (ι ↪ Card d) → BenesCoins d → ℕ
+  | 0, _, _, _, _ => 0
+  | d+1, ι, _, e, ω =>
+      if d+1 ≤ H then crowdedMass (Fintype.card ι) else
+      let σ := benesStepEquiv d ω
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x σ.2.1
+      let hc := PairRouting.colors_compatible x σ.2.1
+      cutoffMass H d (PairRouting.childEmbedding x c hc false) σ.1.1 +
+        cutoffMass H d (PairRouting.childEmbedding x c hc true) σ.1.2
+
+lemma palindromeLowCost_eq (H d : ℕ) (h : d ≤ H) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) :
+    palindromeLowCost H d e ω = palindromeCost d e ω := by
+  classical
+  induction d generalizing ι with
+  | zero => rfl
+  | succ d ih =>
+    rw [palindromeLowCost,palindromeCost,ite_eq_left h]
+    congr 2 <;> exact ih (by omega) _ _
+
+lemma palindromeLowCost_le_cutoff (H d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) :
+    palindromeLowCost H d e ω ≤ H*cutoffMass H d e ω := by
+  classical
+  induction d generalizing ι with
+  | zero => simp [palindromeLowCost,cutoffMass]
+  | succ d ih =>
+    by_cases h : d+1 ≤ H
+    · rw [palindromeLowCost_eq H (d+1) h,cutoffMass,ite_eq_left h]
+      exact (palindromeCost_le_crowded (d+1) e ω).trans (Nat.mul_le_mul_right _ h)
+    · rw [palindromeLowCost,cutoffMass,ite_eq_right h,ite_eq_right h,zero_add,Nat.mul_add]
+      exact Nat.add_le_add (ih _ _) (ih _ _)
+
+
+noncomputable def centralBlock (r H : ℕ) (b : Card r) : Finset (Card (H+r)) :=
+  Finset.univ.image (fun z : Card H => (cardSplit r H).symm (b,z))
+
+lemma mem_centralBlock (r H : ℕ) (b : Card r) (x : Card (H+r)) :
+    x ∈ centralBlock r H b ↔ (cardSplit r H x).1 = b := by
+  classical
+  constructor
+  · intro h
+    obtain ⟨z,_,rfl⟩ := Finset.mem_image.mp h
+    simp only [Equiv.apply_symm_apply]
+  · intro h
+    apply Finset.mem_image.mpr
+    refine ⟨(cardSplit r H x).2,Finset.mem_univ _,?_⟩
+    rw [←h,Prod.mk.eta,Equiv.symm_apply_apply]
+
+lemma centralBlock_card (r H : ℕ) (b : Card r) : (centralBlock r H b).card = 2^H := by
+  classical
+  rw [centralBlock,Finset.card_image_of_injective]
+  · exact card_positions H
+  · intro x y h
+    exact congrArg Prod.snd ((cardSplit r H).symm.injective h)
+
+lemma centralBlock_disjoint (r H : ℕ) :
+    Pairwise (fun b b' => Disjoint (centralBlock r H b) (centralBlock r H b')) := by
+  classical
+  intro b b' h
+  apply Finset.disjoint_left.mpr
+  intro x hx hy
+  exact h (((mem_centralBlock r H b x).mp hx).symm.trans
+    ((mem_centralBlock r H b' x).mp hy))
+
+lemma centralBlock_zero (H : ℕ) (b : Card 0) : centralBlock 0 H b = Finset.univ := by
+  classical
+  ext x
+  simp only [mem_centralBlock,Finset.mem_univ,iff_true]
+  exact Subsingleton.elim _ _
+
+lemma centralBlock_step (r H : ℕ) (b : Bool) (c : Card r) :
+    centralBlock (r+1) H (Fin.cons b c) = (centralBlock r H c).image (Fin.cons b) := by
+  classical
+  simp only [centralBlock,Finset.image_image,cardSplit,Equiv.coe_fn_symm_mk,
+    Fin.cons_zero,Fin.tail_cons,Function.comp_def]
+
+lemma block_inter_step (r H : ℕ) (b : Bool) (c : Card r) (A : Finset (Card (H+(r+1)))) :
+    centralBlock (r+1) H (Fin.cons b c) ∩ A =
+      (centralBlock r H c ∩ childMarks A b).image (Fin.cons b) := by
+  classical
+  rw [centralBlock_step]
+  ext x
+  constructor
+  · intro hx
+    obtain ⟨hxB,hxA⟩ := Finset.mem_inter.mp hx
+    obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hxB
+    exact Finset.mem_image.mpr ⟨y,Finset.mem_inter.mpr ⟨hy,(mem_childMarks A b y).mpr hxA⟩,rfl⟩
+  · intro hx
+    obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
+    exact Finset.mem_inter.mpr ⟨Finset.mem_image.mpr ⟨y,(Finset.mem_inter.mp hy).1,rfl⟩,
+      (mem_childMarks A b y).mp (Finset.mem_inter.mp hy).2⟩
+
+lemma sparseMass_central_zero (H : ℕ) (A : Finset (Card H)) :
+    sparseMass (centralBlock 0 H) A = crowdedMass A.card := by
+  classical
+  simp only [sparseMass,centralBlock_zero,Finset.univ_inter,crowdedMass,
+    Finset.sum_const,Finset.card_univ,card_positions,pow_zero,one_nsmul]
+
+lemma sparseMass_central_step (r H : ℕ) (A : Finset (Card (H+(r+1)))) :
+    sparseMass (centralBlock (r+1) H) A =
+      sparseMass (centralBlock r H) (childMarks A false) +
+        sparseMass (centralBlock r H) (childMarks A true) := by
+  classical
+  unfold sparseMass
+  rw [←Equiv.sum_comp (headTailEquiv r).symm,Fintype.sum_prod_type]
+  have ht (b : Bool) (c : Card r) : (headTailEquiv r).symm (b,c) = Fin.cons b c := rfl
+  simp only [Fintype.sum_bool,ht]
+  have hc (b : Bool) (c : Card r) :
+      (centralBlock (r+1) H (Fin.cons b c) ∩ A).card =
+      (centralBlock r H c ∩ childMarks A b).card := by
+    rw [block_inter_step]
+    apply Finset.card_image_of_injective
+    intro x y h
+    simpa only [Fin.tail_cons] using congrArg Fin.tail h
+  simp only [hc]
+  exact add_comm _ _
+
+
+lemma inverseButterfly_coordinates_step (d : ℕ)
+    (c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (ξ : Card d → Bool)
+    (x : Card (d+1)) :
+    headTailEquiv d ((butterflyPerm (d+1)
+      (decodeButterfly (d+1) ((coinStepEquiv d).symm (c,ξ)))).symm x) =
+      (Bool.xor (x 0) (ξ (Fin.tail x)),
+        (butterflyPerm d (decodeButterfly d
+          (if Bool.xor (x 0) (ξ (Fin.tail x)) then c.2 else c.1))).symm (Fin.tail x)) := by
+  rw [butterflyPerm_step]
+  rfl
+
+lemma inverseButterflyMarks_child (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (d+1))
+    (c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (ξ : Card d → Bool) (b : Bool) :
+    childMarks (inverseButterflyMarks (d+1) (Finset.univ.image e)
+      ((coinStepEquiv d).symm (c,ξ))) b =
+      inverseButterflyMarks d
+        (Finset.univ.image (PairRouting.childEmbedding (e.trans (headTailEquiv d).toEmbedding)
+          (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) ξ)
+            (PairRouting.colors_compatible _ _) b)) (if b then c.2 else c.1) := by
+  classical
+  let x := e.trans (headTailEquiv d).toEmbedding
+  let col := PairRouting.colors x ξ
+  let child := PairRouting.childEmbedding x col (PairRouting.colors_compatible x ξ) b
+  have hp (i : ι) : headTailEquiv d ((butterflyPerm (d+1)
+      (decodeButterfly (d+1) ((coinStepEquiv d).symm (c,ξ)))).symm (e i)) =
+      (col i,(butterflyPerm d (decodeButterfly d (if col i then c.2 else c.1))).symm (x i).2) :=
+    inverseButterfly_coordinates_step d c ξ (e i)
+  ext y
+  simp only [mem_childMarks,inverseButterflyMarks,Finset.image_image,
+    Finset.mem_image,Finset.mem_univ,true_and,Function.comp_def]
+  constructor
+  · rintro ⟨i,hi⟩
+    have he := congrArg (headTailEquiv d) hi
+    rw [hp i] at he
+    change (col i,_) = (b,y) at he
+    have hb := congrArg Prod.fst he
+    have hy := congrArg Prod.snd he
+    dsimp only at hb hy
+    refine ⟨⟨i,hb⟩,?_⟩
+    change (butterflyPerm d (decodeButterfly d (if b then c.2 else c.1))).symm (x i).2 = y
+    simpa only [hb] using hy
+  · rintro ⟨i,hi⟩
+    refine ⟨i.val,?_⟩
+    apply (headTailEquiv d).injective
+    rw [hp]
+    change (col i.val,_) = (b,y)
+    have hb : col i.val = b := i.property
+    rw [hb]
+    exact Prod.ext rfl hi
+
+lemma inverseButterflyMarks_card (d : ℕ) (A : Finset (Card d)) (ω : SwitchIndex d → Bool) :
+    (inverseButterflyMarks d A ω).card = A.card := by
+  classical
+  exact Finset.card_image_of_injective _ (butterflyPerm d (decodeButterfly d ω)).symm.injective
+
+lemma benes_input_coordinates (d : ℕ) (ω : BenesCoins (d+1)) :
+    ω.2 = (coinStepEquiv d).symm
+      (((benesStepEquiv d ω).1.1.2,(benesStepEquiv d ω).1.2.2),(benesStepEquiv d ω).2.1) := by
+  change ω.2 = (coinStepEquiv d).symm (coinStepEquiv d ω.2)
+  exact ((coinStepEquiv d).symm_apply_apply ω.2).symm
+
+lemma cutoffMass_central (r H : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (H+r)) (ω : BenesCoins (H+r)) :
+    cutoffMass H (H+r) e ω =
+      sparseMass (centralBlock r H) (inverseButterflyMarks (H+r) (Finset.univ.image e) ω.2) := by
+  classical
+  induction r generalizing ι with
+  | zero =>
+    rw [sparseMass_central_zero,inverseButterflyMarks_card,
+      Finset.card_image_of_injective _ e.injective,Finset.card_univ]
+    cases H with
+    | zero =>
+      have hcard : Fintype.card ι ≤ 1 := by
+        have hh := Fintype.card_le_of_injective e e.injective
+        simpa only [Nat.add_zero,card_positions,pow_zero] using hh
+      simp [cutoffMass,crowdedMass,show ¬2≤Fintype.card ι by omega]
+    | succ H => simp only [cutoffMass,le_refl,ite_true]
+  | succ r ih =>
+    simp only [Nat.add_succ] at *
+    rw [cutoffMass,ite_eq_right (by omega : ¬H+r+1≤H)]
+    let σ := benesStepEquiv (H+r) ω
+    let x := e.trans (headTailEquiv (H+r)).toEmbedding
+    let col := PairRouting.colors x σ.2.1
+    let hc := PairRouting.colors_compatible x σ.2.1
+    change cutoffMass H (H+r) (PairRouting.childEmbedding x col hc false) σ.1.1 +
+      cutoffMass H (H+r) (PairRouting.childEmbedding x col hc true) σ.1.2 = _
+    rw [ih,ih,sparseMass_central_step,benes_input_coordinates (H+r) ω]
+    rw [inverseButterflyMarks_child,inverseButterflyMarks_child]
+    rfl
+
+
+theorem palindrome_low_mgf (r H : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (H+r)) (q : ℝ) (hq : 1 ≤ q)
+    (hr : (2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*q^H) ≤ 1/2) :
+    finiteMean (fun ω : BenesCoins (H+r) => q^(palindromeLowCost H (H+r) e ω)) ≤
+      Real.exp ((2:ℝ)^r * (2 * ((2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*q^H))^2)) := by
+  classical
+  have hec : (Finset.univ.image e).card = Fintype.card ι := by
+    rw [Finset.card_image_of_injective _ e.injective,Finset.card_univ]
+  have hm := inverseButterfly_sparse_mgf (H+r) (centralBlock r H)
+    (centralBlock_disjoint r H) (Finset.univ.image e) (q^H)
+    (pow_nonneg (by linarith) _) (fun b => by
+      simpa only [centralBlock_card,hec,Nat.cast_pow,Nat.cast_ofNat] using hr)
+  calc
+    _ ≤ finiteMean (fun ω : BenesCoins (H+r) => (q^H)^(cutoffMass H (H+r) e ω)) := by
+      apply finiteMean_mono
+      intro ω
+      rw [←pow_mul]
+      exact pow_le_pow_right₀ hq (palindromeLowCost_le_cutoff H (H+r) e ω)
+    _ = finiteMean (fun ω : SwitchIndex (H+r) → Bool =>
+        (q^H)^(sparseMass (centralBlock r H)
+          (inverseButterflyMarks (H+r) (Finset.univ.image e) ω))) := by
+      simp_rw [cutoffMass_central]
+      simp only [finiteMean_prod,finiteMean_const]
+    _ ≤ _ := by
+      simpa only [centralBlock_card,hec,Nat.cast_pow,Nat.cast_ofNat,Finset.sum_const,
+        Finset.card_univ,card_positions,nsmul_eq_mul] using hm
+
+variable {α B : Type*} {r s : ℕ}
+
+def parallelFreshPre (pre : α ≃ B × Card (s+r))
+    (lo : B → Card r × SwitchIndex s → Bool) : α ≃ (B × Card s) × Card r :=
+  ((pre.trans (Equiv.prodCongr (Equiv.refl B) (cardSplit r s))).trans
+    (Equiv.prodCongrRight (fun b =>
+      (parallelPerm (lo b)).trans (Equiv.prodComm _ _)))).trans
+    (Equiv.prodAssoc _ _ _).symm
+
+def parallelFreshPost (post : B × Card (s+r) ≃ α) : (B × Card s) × Card r ≃ α :=
+  (((Equiv.prodAssoc _ _ _).trans (Equiv.prodCongr (Equiv.refl B)
+    (Equiv.prodComm _ _))).trans (Equiv.prodCongr (Equiv.refl B) (cardSplit r s).symm)).trans post
+
+def parallelAssembled (lo : B → Card r × SwitchIndex s → Bool)
+    (hi : (B × Card s) × SwitchIndex r → Bool) : B × SwitchIndex (s+r) → Bool :=
+  fun i => assembleBits r s (lo i.1) (fun j => hi ((i.1,j.1),j.2)) i.2
+
+lemma parallel_exposed_perm (pre : α ≃ B × Card (s+r)) (post : B × Card (s+r) ≃ α)
+    (lo : B → Card r × SwitchIndex s → Bool)
+    (hi : (B × Card s) × SwitchIndex r → Bool) :
+    relabeledPerm pre post (parallelAssembled lo hi) =
+      relabeledPerm (parallelFreshPre pre lo) (parallelFreshPost post) hi := by
+  ext x
+  have h := butterfly_split r s (lo (pre x).1)
+    (fun j => hi (((pre x).1,j.1),j.2)) (pre x).2
+  have hh := congrArg (fun y => post ((pre x).1,(cardSplit r s).symm y)) h
+  simpa only [relabeledPerm,parallelAssembled,parallelPerm,parallelFreshPre,parallelFreshPost,
+    Equiv.trans_apply,Equiv.prodCongr_apply,Equiv.prodCongrRight,Equiv.coe_fn_mk,
+    Equiv.refl_apply,Equiv.prodComm_apply,Prod.swap,Equiv.prodAssoc_apply,
+    Equiv.prodAssoc_symm_apply,Prod.map,Equiv.symm_apply_apply] using hh
+
+theorem parallel_partially_exposed_log_mgf [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B]
+    (pre : α ≃ B × Card (s+r)) (post : B × Card (s+r) ≃ α)
+    (lo : B → Card r × SwitchIndex s → Bool) (S : Finset α)
+    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    Real.log (finiteMean (fun hi : (B × Card s) × SwitchIndex r → Bool =>
+      q^(Fintype.card (RoutingNetwork.SelectedCycle
+        (relabeledPerm pre post (parallelAssembled lo hi)) S)))) ≤
+      (S.card:ℝ)/J*Real.log q + 2*q*S.card*Real.sqrt J/Real.sqrt ((2:ℝ)^r) := by
+  simp_rw [parallel_exposed_perm]
+  exact parallel_cycle_log_mgf (parallelFreshPre pre lo) (parallelFreshPost post) S J hJ q hq
+
+end Thorp
+namespace Thorp
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+lemma orbitSet_inverse (p : Equiv.Perm α) (x : α) : orbitSet p⁻¹ x = orbitSet p x := by
+  ext y
+  simp only [mem_orbitSet,Equiv.Perm.sameCycle_inv]
+
+lemma selectedCycle_inverse (p : Equiv.Perm α) (S : Finset α) :
+    Fintype.card (RoutingNetwork.SelectedCycle p⁻¹ S) =
+      Fintype.card (RoutingNetwork.SelectedCycle p S) := by
+  apply Fintype.card_congr
+  refine { toFun := fun C => ⟨C.val,?_,C.property.2⟩
+           invFun := fun C => ⟨C.val,?_,C.property.2⟩
+           left_inv := fun _ => rfl
+           right_inv := fun _ => rfl }
+  · obtain ⟨x,hx⟩ := C.property.1
+    exact ⟨x,hx.trans (orbitSet_inverse p x)⟩
+  · obtain ⟨x,hx⟩ := C.property.1
+    exact ⟨x,hx.trans (orbitSet_inverse p x).symm⟩
+
+lemma orbitSet_conjugate (p g : Equiv.Perm α) (x : α) :
+    orbitSet (g*p*g⁻¹) (g x) = (orbitSet p x).image g := by
+  ext y
+  simp only [mem_orbitSet,Finset.mem_image]
+  rw [Equiv.Perm.sameCycle_conj]
+  change p.SameCycle (g.symm (g x)) (g.symm y) ↔ ∃ z, p.SameCycle x z ∧ g z = y
+  rw [Equiv.symm_apply_apply]
+  constructor
+  · intro h
+    exact ⟨g.symm y,h,g.apply_symm_apply y⟩
+  · rintro ⟨z,hz,rfl⟩
+    simpa only [Equiv.symm_apply_apply] using hz
+
+noncomputable def conjugateCycle (p g : Equiv.Perm α) (S : Finset α)
+    (C : RoutingNetwork.SelectedCycle p S) :
+    RoutingNetwork.SelectedCycle (g*p*g⁻¹) (S.image g) := by
+  refine ⟨C.val.image g,?_,Finset.image_subset_image C.property.2⟩
+  obtain ⟨x,hx⟩ := C.property.1
+  exact ⟨g x,by rw [hx,orbitSet_conjugate]⟩
+
+lemma conjugateCycle_injective (p g : Equiv.Perm α) (S : Finset α) :
+    Function.Injective (conjugateCycle p g S) := by
+  intro C D he
+  apply Subtype.ext
+  exact (Finset.image_injective g.injective) (congrArg Subtype.val he)
+
+lemma selectedCycle_conjugate (p g : Equiv.Perm α) (S : Finset α) :
+    Fintype.card (RoutingNetwork.SelectedCycle (g*p*g⁻¹) (S.image g)) =
+      Fintype.card (RoutingNetwork.SelectedCycle p S) := by
+  apply Nat.le_antisymm
+  · have h := Fintype.card_le_of_injective _
+      (conjugateCycle_injective (g*p*g⁻¹) g⁻¹ (S.image g))
+    have hp : g⁻¹*(g*p*g⁻¹)*g = p := by group
+    rw [inv_inv,hp] at h
+    have hs : (S.image g).image g.symm = S := by
+      ext x
+      simp only [Finset.mem_image]
+      constructor
+      · rintro ⟨y,⟨z,hz,rfl⟩,he⟩
+        have hz' : z = x := by simpa only [Equiv.symm_apply_apply] using he
+        exact hz' ▸ hz
+      · intro hx
+        exact ⟨g x,⟨x,hx,rfl⟩,g.symm_apply_apply x⟩
+    change _ ≤ Fintype.card (RoutingNetwork.SelectedCycle p ((S.image g).image g.symm)) at h
+    rw [hs] at h
+    exact h
+  · exact Fintype.card_le_of_injective _ (conjugateCycle_injective p g S)
+
+lemma alternatingCycles_le_projection {ι : Type*} [Fintype ι]
+    (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    PairRouting.alternatingCycles e p ≤
+      Fintype.card (RoutingNetwork.SelectedCycle (PairRouting.alternatingProjection p)
+        (boolFiber (PairRouting.labelSet e) false)) := by
+  have h := CycleColoring.card_alternating_cycles_le (PairRouting.alternatingProjection p)
+    (boolFiber (PairRouting.labelSet e) false) (boolFiber (PairRouting.labelSet e) true)
+  simpa only [PairRouting.alternatingCycles,childJoin_boolFiber] using h
+
+lemma projectionCycles_forward (X₀ X₁ Y₀ Y₁ : Equiv.Perm α) (S : Finset α) :
+    Fintype.card (RoutingNetwork.SelectedCycle
+      ((Y₀*X₀⁻¹)⁻¹*(Y₁*X₁⁻¹)) S) =
+    Fintype.card (RoutingNetwork.SelectedCycle
+      (X₀⁻¹*X₁*Y₁⁻¹*Y₀) (S.image X₀.symm)) := by
+  let p := (Y₀*X₀⁻¹)⁻¹*(Y₁*X₁⁻¹)
+  calc
+    _ = Fintype.card (RoutingNetwork.SelectedCycle (X₀⁻¹*p*X₀)
+        (S.image X₀.symm)) := by
+      change Fintype.card (RoutingNetwork.SelectedCycle p S) = _
+      have h := (selectedCycle_conjugate p X₀⁻¹ S).symm
+      rw [inv_inv] at h
+      exact h
+    _ = Fintype.card (RoutingNetwork.SelectedCycle (X₀⁻¹*p*X₀)⁻¹
+        (S.image X₀.symm)) := (selectedCycle_inverse _ _).symm
+    _ = _ := by
+      have he : (X₀⁻¹*p*X₀)⁻¹ = X₀⁻¹*X₁*Y₁⁻¹*Y₀ := by dsimp only [p]; group
+      rw [he]
+
+end Thorp
+namespace Thorp
+
+noncomputable def cycleCoefficient (r J : ℕ) (q : ℝ) : ℝ :=
+  Real.log q / J + 2*q*Real.sqrt J/Real.sqrt ((2:ℝ)^r)
+
+lemma cycleCoefficient_nonneg (r J : ℕ) (q : ℝ) (hq : 1 ≤ q) :
+    0 ≤ cycleCoefficient r J q := by
+  unfold cycleCoefficient
+  exact add_nonneg (div_nonneg (Real.log_nonneg hq) (Nat.cast_nonneg _))
+    (div_nonneg (mul_nonneg (by linarith) (Real.sqrt_nonneg _)) (Real.sqrt_nonneg _))
+
+lemma projectionSide_card_le {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι]
+    (e : ι ↪ Bool × α) : (boolFiber (PairRouting.labelSet e) false).card ≤ Fintype.card ι := by
+  have h := card_childJoin (boolFiber (PairRouting.labelSet e) false)
+    (boolFiber (PairRouting.labelSet e) true)
+  rw [childJoin_boolFiber,PairRouting.card_labelSet] at h
+  omega
+
+theorem node_partial_mgf (r s : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Bool × Card (s+r)) (X₀ X₁ Y₁ : Equiv.Perm (Card (s+r)))
+    (lo : Card r × SwitchIndex s → Bool) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(PairRouting.alternatingCycles e (fun b => if b then Y₁*X₁⁻¹ else
+        butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi))*X₀⁻¹))) ≤
+      Real.exp (Fintype.card ι * cycleCoefficient r J q) := by
+  classical
+  let S := boolFiber (PairRouting.labelSet e) false
+  let S' := S.image X₀.symm
+  let post := X₀⁻¹*X₁*Y₁⁻¹
+  let count := fun hi : Card s × SwitchIndex r → Bool =>
+    Fintype.card (RoutingNetwork.SelectedCycle
+      ((Equiv.refl _).trans (butterflyPerm (s+r)
+        (decodeButterfly (s+r) (assembleBits r s lo hi))) |>.trans post) S')
+  have hpoint (hi : Card s × SwitchIndex r → Bool) :
+      PairRouting.alternatingCycles e (fun b => if b then Y₁*X₁⁻¹ else
+        butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi))*X₀⁻¹) ≤ count hi := by
+    refine (alternatingCycles_le_projection e _).trans_eq ?_
+    simp only [PairRouting.alternatingProjection,Bool.false_eq_true,↓reduceIte]
+    change Fintype.card (RoutingNetwork.SelectedCycle
+      ((butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi))*X₀⁻¹)⁻¹ * (Y₁*X₁⁻¹)) S) = count hi
+    exact projectionCycles_forward X₀ X₁ _ Y₁ S
+  have hm := partially_exposed_cycle_log_mgf (Equiv.refl (Card (s+r))) post lo S' J hJ q hq
+  have hp : 0 < finiteMean (fun hi : Card s × SwitchIndex r → Bool => q^(count hi)) :=
+    finiteMean_pos (fun _ => pow_pos (by linarith) _)
+  have hb : finiteMean (fun hi : Card s × SwitchIndex r → Bool => q^(count hi)) ≤
+      Real.exp (S'.card * cycleCoefficient r J q) := by
+    rw [←Real.exp_log hp]
+    apply Real.exp_le_exp.mpr
+    calc
+      _ ≤ _ := hm
+      _ = _ := by unfold cycleCoefficient; ring
+  have hcard : S'.card ≤ Fintype.card ι := by
+    rw [show S'.card = S.card from Finset.card_image_of_injective _ X₀.symm.injective]
+    exact projectionSide_card_le e
+  calc
+    _ ≤ finiteMean (fun hi : Card s × SwitchIndex r → Bool => q^(count hi)) :=
+      finiteMean_mono (fun hi => pow_le_pow_right₀ hq (hpoint hi))
+    _ ≤ Real.exp (S'.card * cycleCoefficient r J q) := hb
+    _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right
+      (by exact_mod_cast hcard) (cycleCoefficient_nonneg r J q hq))
+
+
+def highStepEquiv (r s : ℕ) : (Card s × SwitchIndex (r+1) → Bool) ≃
+    (((Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool)) ×
+      (Card s × Card r → Bool)) where
+  toFun hi := ((fun j => hi (j.1,Sum.inr (false,j.2)),
+    fun j => hi (j.1,Sum.inr (true,j.2))),fun j => hi (j.1,Sum.inl j.2))
+  invFun c := fun j => match j.2 with
+    | Sum.inl y => c.2 (j.1,y)
+    | Sum.inr (b,i) => if b then c.1.2 (j.1,i) else c.1.1 (j.1,i)
+  left_inv hi := by
+    funext j
+    rcases j with ⟨z,y|⟨b,i⟩⟩
+    · rfl
+    · cases b <;> rfl
+  right_inv c := by
+    rcases c with ⟨⟨h₀,h₁⟩,η⟩
+    rfl
+
+lemma assembleBits_step (r s : ℕ) (lo : Card (r+1) × SwitchIndex s → Bool)
+    (c : ((Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool)) ×
+      (Card s × Card r → Bool)) :
+    assembleBits (r+1) s lo ((highStepEquiv r s).symm c) =
+      (coinStepEquiv (s+r)).symm
+        ((assembleBits r s (fun j => lo (Fin.cons false j.1,j.2)) c.1.1,
+          assembleBits r s (fun j => lo (Fin.cons true j.1,j.2)) c.1.2),
+        fun y => c.2 ((cardSplit r s y).2,(cardSplit r s y).1)) := by
+  funext i
+  rcases i with y | ⟨b,i⟩
+  · rfl
+  · cases b <;> rfl
+
+noncomputable def palindromeLevelCost (t : ℕ) : (d : ℕ) → {ι : Type*} → [Fintype ι] →
+    (ι ↪ Card d) → BenesCoins d → ℕ
+  | 0, _, _, _, _ => 0
+  | d+1, _, _, e, ω =>
+      let σ := benesStepEquiv d ω
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x σ.2.1
+      let hc := PairRouting.colors_compatible x σ.2.1
+      if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
+          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) else
+        palindromeLevelCost t d (PairRouting.childEmbedding x c hc false) σ.1.1 +
+          palindromeLevelCost t d (PairRouting.childEmbedding x c hc true) σ.1.2
+
+lemma palindromeLevelCost_step (t d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (d+1)) (X Y : (SwitchIndex d → Bool) × (SwitchIndex d → Bool))
+    (ξ η : Card d → Bool) :
+    palindromeLevelCost t (d+1) e
+      ((coinStepEquiv d).symm (Y,η),(coinStepEquiv d).symm (X,ξ)) =
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x ξ
+      let hc := PairRouting.colors_compatible x ξ
+      if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x ξ)
+        (fun b => palindromePerm d (if b then (Y.2,X.2) else (Y.1,X.1))) else
+        palindromeLevelCost t d (PairRouting.childEmbedding x c hc false) (Y.1,X.1) +
+        palindromeLevelCost t d (PairRouting.childEmbedding x c hc true) (Y.2,X.2) := by
+  rw [palindromeLevelCost]
+  rfl
+
+lemma palindromeLevelCost_gt (t d : ℕ) (h : d < t) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) : palindromeLevelCost t d e ω = 0 := by
+  classical
+  induction d generalizing ι with
+  | zero => rfl
+  | succ d ih =>
+    rw [palindromeLevelCost,ite_eq_right (by omega : t ≠ d+1),ih (by omega),ih (by omega),Nat.zero_add]
+
+
+theorem palindrome_level_partial_mgf (r s u : ℕ) (hu : u < r)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+r)) (X : SwitchIndex (s+r) → Bool)
+    (lo : Card r × SwitchIndex s → Bool) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(palindromeLevelCost (s+u+1) (s+r) e (assembleBits r s lo hi,X))) ≤
+      Real.exp (Fintype.card ι * cycleCoefficient u J q) := by
+  classical
+  induction r generalizing u ι with
+  | zero => omega
+  | succ r ih =>
+    simp only [Nat.add_succ] at *
+    let χ := coinStepEquiv (s+r) X
+    let x := e.trans (headTailEquiv (s+r)).toEmbedding
+    let c := PairRouting.colors x χ.2
+    let hc := PairRouting.colors_compatible x χ.2
+    let e₀ := PairRouting.childEmbedding x c hc false
+    let e₁ := PairRouting.childEmbedding x c hc true
+    let e' := PairRouting.switchedEmbedding x χ.2
+    let lo₀ : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons false j.1,j.2)
+    let lo₁ : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons true j.1,j.2)
+    let Y₀ := fun hi : Card s × SwitchIndex r → Bool => assembleBits r s lo₀ hi
+    let Y₁ := fun hi : Card s × SwitchIndex r → Bool => assembleBits r s lo₁ hi
+    let N := fun hi : (Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool) =>
+      PairRouting.alternatingCycles e' (fun b => if b then palindromePerm (s+r) (Y₁ hi.2,χ.1.2)
+        else palindromePerm (s+r) (Y₀ hi.1,χ.1.1))
+    let L₀ := fun hi : Card s × SwitchIndex r → Bool =>
+      palindromeLevelCost (s+u+1) (s+r) e₀ (Y₀ hi,χ.1.1)
+    let L₁ := fun hi : Card s × SwitchIndex r → Bool =>
+      palindromeLevelCost (s+u+1) (s+r) e₁ (Y₁ hi,χ.1.2)
+    have hs (hi : (Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool))
+        (η : Card s × Card r → Bool) :
+        palindromeLevelCost (s+u+1) (s+r+1) e
+          (assembleBits (r+1) s lo ((highStepEquiv r s).symm (hi,η)),X) =
+          if u = r then N hi else L₀ hi.1 + L₁ hi.2 := by
+      rw [assembleBits_step]
+      conv_lhs => rw [←(coinStepEquiv (s+r)).symm_apply_apply X]
+      rw [palindromeLevelCost_step]
+      simp only [Nat.add_right_cancel_iff,Nat.add_left_cancel_iff]
+      dsimp only [χ,x,c,hc,e₀,e₁,e',lo₀,lo₁,Y₀,Y₁,N,L₀,L₁]
+      congr 2
+      funext b
+      cases b <;> rfl
+    rw [finiteMean_equiv (highStepEquiv r s),finiteMean_prod]
+    simp_rw [hs,finiteMean_const]
+    by_cases hur : u = r
+    · subst u
+      simp only [ite_true]
+      rw [finiteMean_prod,finiteMean_comm]
+      calc
+        _ ≤ finiteMean (fun _hi : Card s × SwitchIndex r → Bool =>
+          Real.exp (Fintype.card ι * cycleCoefficient r J q)) := by
+          apply finiteMean_mono
+          intro hi
+          have hn := node_partial_mgf r s e'
+            (butterflyPerm (s+r) (decodeButterfly (s+r) χ.1.1))
+            (butterflyPerm (s+r) (decodeButterfly (s+r) χ.1.2))
+            (butterflyPerm (s+r) (decodeButterfly (s+r) (Y₁ hi))) lo₀ J hJ q hq
+          exact hn
+        _ = _ := finiteMean_const _
+    · simp only [ite_eq_right hur,pow_add]
+      rw [finiteMean_prod_mul (fun hi => q ^ L₀ hi) (fun hi => q ^ L₁ hi)]
+      have hu' : u < r := by omega
+      have h₀ := ih u hu' e₀ χ.1.1 lo₀
+      have h₁ := ih u hu' e₁ χ.1.2 lo₁
+      have hp := mul_le_mul h₀ h₁ (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
+        (Real.exp_nonneg _)
+      refine hp.trans_eq ?_
+      rw [←Real.exp_add]
+      congr 1
+      have hcount := PairRouting.color_card_add c
+      have hcount' : (Fintype.card {i : ι // c i = false} : ℝ) +
+          Fintype.card {i : ι // c i = true} = Fintype.card ι := by exact_mod_cast hcount
+      change (Fintype.card {i : ι // c i = false} : ℝ) * cycleCoefficient u J q +
+          Fintype.card {i : ι // c i = true} * cycleCoefficient u J q = _
+      rw [←add_mul,hcount']
+
+
+def switchHeight : (d : ℕ) → SwitchIndex d → ℕ
+  | 0, i => nomatch i
+  | d+1, Sum.inl _ => d+1
+  | d+1, Sum.inr (_,i) => switchHeight d i
+
+lemma switchHeight_le (d : ℕ) (i : SwitchIndex d) : switchHeight d i ≤ d := by
+  induction d with
+  | zero => exact Empty.elim i
+  | succ d ih =>
+    rcases i with y | ⟨b,i⟩
+    · exact le_refl _
+    · exact (ih i).trans (Nat.le_succ d)
+
+def AgreeThrough (d s : ℕ) (Y Y' : SwitchIndex d → Bool) : Prop :=
+  ∀ i, switchHeight d i ≤ s → Y i = Y' i
+
+def LayerAdapted (d s : ℕ) {β : Type*} (f : (SwitchIndex d → Bool) → β) : Prop :=
+  ∀ Y Y', AgreeThrough d s Y Y' → f Y = f Y'
+
+lemma agreeThrough_mono {d s t : ℕ} (h : s ≤ t) {Y Y' : SwitchIndex d → Bool}
+    (hh : AgreeThrough d t Y Y') : AgreeThrough d s Y Y' :=
+  fun i hi => hh i (hi.trans h)
+
+lemma layerAdapted_mono {d s t : ℕ} {β : Type*} {f : (SwitchIndex d → Bool) → β}
+    (hf : LayerAdapted d s f) (h : s ≤ t) : LayerAdapted d t f :=
+  fun Y Y' hh => hf Y Y' (agreeThrough_mono h hh)
+
+lemma layerAdapted_comp {d s : ℕ} {α β : Type*} {f : (SwitchIndex d → Bool) → α}
+    (hf : LayerAdapted d s f) (g : α → β) : LayerAdapted d s (g ∘ f) :=
+  fun Y Y' h => congrArg g (hf Y Y' h)
+
+lemma layerAdapted_sum {d s : ℕ} {ι : Type*} (A : Finset ι)
+    (f : ι → (SwitchIndex d → Bool) → ℕ)
+    (hf : ∀ i ∈ A, LayerAdapted d s (f i)) :
+    LayerAdapted d s (fun Y => ∑ i ∈ A, f i Y) := by
+  intro Y Y' h
+  exact Finset.sum_congr rfl (fun i hi => hf i hi Y Y' h)
+
+lemma assembleBits_agree (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
+    (hi hi' : Card s × SwitchIndex r → Bool) :
+    AgreeThrough (s+r) s (assembleBits r s lo hi) (assembleBits r s lo hi') := by
+  induction r with
+  | zero => intro i _; rfl
+  | succ r ih =>
+    intro i h
+    rcases i with y | ⟨b,i⟩
+    · have : s+r+1 ≤ s := h
+      omega
+    · exact ih (fun j => lo (Fin.cons b j.1,j.2))
+        (fun j => hi (j.1,Sum.inr (b,j.2))) (fun j => hi' (j.1,Sum.inr (b,j.2))) i h
+
+lemma palindromeLevelCost_adapted (t d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) :
+    LayerAdapted d (t-1) (fun Y => palindromeLevelCost t d e (Y,X)) := by
+  classical
+  induction d generalizing ι with
+  | zero => intro Y Y' _; rfl
+  | succ d ih =>
+    intro Y Y' h
+    change palindromeLevelCost t (d+1) e (Y,X) = palindromeLevelCost t (d+1) e (Y',X)
+    conv_lhs => rw [←(coinStepEquiv d).symm_apply_apply Y,←(coinStepEquiv d).symm_apply_apply X]
+    conv_rhs => rw [←(coinStepEquiv d).symm_apply_apply Y',←(coinStepEquiv d).symm_apply_apply X]
+    rw [palindromeLevelCost_step,palindromeLevelCost_step]
+    by_cases ht : t = d+1
+    · subst t
+      rw [ite_eq_left rfl,ite_eq_left rfl]
+      have he₀ : (coinStepEquiv d Y).1.1 = (coinStepEquiv d Y').1.1 := by
+        funext i
+        exact h (Sum.inr (false,i)) (by simpa only [switchHeight,Nat.add_sub_cancel] using switchHeight_le d i)
+      have he₁ : (coinStepEquiv d Y).1.2 = (coinStepEquiv d Y').1.2 := by
+        funext i
+        exact h (Sum.inr (true,i)) (by simpa only [switchHeight,Nat.add_sub_cancel] using switchHeight_le d i)
+      rw [he₀,he₁]
+    · rw [ite_eq_right ht,ite_eq_right ht]
+      congr 1
+      · apply ih
+        intro i hi
+        exact h (Sum.inr (false,i)) hi
+      · apply ih
+        intro i hi
+        exact h (Sum.inr (true,i)) hi
+
+lemma adapted_mul_mean_le (r s : ℕ) (F G : (SwitchIndex (s+r) → Bool) → ℝ)
+    (hF : ∀ Y, 0 ≤ F Y) (hA : LayerAdapted (s+r) s F) (B : ℝ)
+    (hG : ∀ lo : Card r × SwitchIndex s → Bool,
+      finiteMean (fun hi : Card s × SwitchIndex r → Bool => G (assembleBits r s lo hi)) ≤ B) :
+    finiteMean (fun Y => F Y*G Y) ≤ finiteMean F * B := by
+  rw [finiteMean_split r s,finiteMean_split r s F,←finiteMean_mul_const]
+  apply finiteMean_mono
+  intro lo
+  let H : Card s × SwitchIndex r → Bool := fun _ => false
+  have hh (hi : Card s × SwitchIndex r → Bool) :
+      F (assembleBits r s lo hi) = F (assembleBits r s lo H) :=
+    hA _ _ (assembleBits_agree r s lo hi H)
+  simp_rw [hh,finiteMean_const]
+  rw [show (fun hi => F (assembleBits r s lo H) * G (assembleBits r s lo hi)) =
+      (fun hi => G (assembleBits r s lo hi) * F (assembleBits r s lo H)) by
+        funext hi; exact mul_comm _ _,finiteMean_mul_const]
+  simpa only [mul_comm] using mul_le_mul_of_nonneg_left (hG lo) (hF _)
+
+
+theorem cycleCoefficient_decay (l r : ℕ) (hr : (l:ℝ)/2 ≤ r)
+    (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ (l:ℝ)/250) :
+    cycleCoefficient r ⌈Real.exp ((l:ℝ)/32)⌉₊ q ≤
+      68 * Real.exp (-(l:ℝ)/64) := by
+  let E := Real.exp ((l:ℝ)/32)
+  let J := ⌈E⌉₊
+  have hl : (0:ℝ) ≤ l := Nat.cast_nonneg _
+  have hE : 1 ≤ E := Real.one_le_exp_iff.mpr (by positivity)
+  have hEpos : 0 < E := Real.exp_pos _
+  have hEJ : E ≤ (J:ℝ) := Nat.le_ceil E
+  have hJ : (1:ℝ) ≤ J := hE.trans hEJ
+  have hJpos : (0:ℝ) < J := lt_of_lt_of_le (by norm_num) hJ
+  have hJE : (J:ℝ) ≤ 2*E := by
+    have h := Nat.ceil_lt_add_one (le_of_lt hEpos)
+    change (J:ℝ) < E+1 at h
+    linarith
+  have hsqJ : Real.sqrt J ≤ (J:ℝ) := by
+    apply Real.sqrt_le_iff.mpr
+    constructor
+    · positivity
+    · nlinarith
+  have hqp : 0 < q := lt_of_lt_of_le (by norm_num) hq
+  have hqE : q ≤ Real.exp ((l:ℝ)/250) := by
+    rw [←Real.exp_log hqp]
+    exact Real.exp_le_exp.mpr hlog
+  have hln : (1/2:ℝ) ≤ Real.log 2 := by
+    have hh := Real.one_sub_inv_le_log_of_pos (by norm_num : (0:ℝ)<2)
+    norm_num at hh ⊢
+    exact hh
+  have hden : Real.exp ((l:ℝ)/8) ≤ Real.sqrt ((2:ℝ)^r) := by
+    have he : (2:ℝ)^r = Real.exp ((r:ℝ)*Real.log 2) := by
+      rw [Real.exp_nat_mul,Real.exp_log (by norm_num)]
+    rw [he,Real.sqrt_eq_rpow,Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp]
+    apply Real.exp_le_exp.mpr
+    have hrn : (0:ℝ) ≤ r := Nat.cast_nonneg _
+    nlinarith
+  have hlog' : Real.log q ≤ (l:ℝ) := by linarith
+  have h₁ : Real.log q / (J:ℝ) ≤ 64*Real.exp (-(l:ℝ)/64) := by
+    calc
+      _ ≤ (l:ℝ)/E := div_le_div₀ (by positivity) hlog' hEpos hEJ
+      _ ≤ (64*Real.exp ((l:ℝ)/64))/E := by
+        apply div_le_div_of_nonneg_right _ (le_of_lt hEpos)
+        have hh := Real.add_one_le_exp ((l:ℝ)/64)
+        linarith
+      _ = _ := by
+        dsimp [E]
+        rw [mul_div_assoc,←Real.exp_sub]
+        congr 2
+        ring
+  have h₂ : 2*q*Real.sqrt J / Real.sqrt ((2:ℝ)^r) ≤
+      4*Real.exp (-(l:ℝ)/64) := by
+    calc
+      _ ≤ (4*Real.exp ((l:ℝ)/250)*E)/Real.exp ((l:ℝ)/8) := by
+        apply div_le_div₀ (by positivity) _ (Real.exp_pos _) hden
+        have hs := hsqJ.trans hJE
+        have hm := mul_le_mul hqE hs (Real.sqrt_nonneg _) (Real.exp_nonneg _)
+        nlinarith
+      _ = 4*Real.exp ((l:ℝ)/250+(l:ℝ)/32-(l:ℝ)/8) := by
+        dsimp [E]
+        rw [mul_assoc,←Real.exp_add,mul_div_assoc,←Real.exp_sub]
+      _ ≤ _ := by
+        apply mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr _) (by norm_num)
+        linarith
+  change Real.log q / (J:ℝ) + 2*q*Real.sqrt J / Real.sqrt ((2:ℝ)^r) ≤ _
+  linarith
+
+lemma cycleCutoff_pos (l : ℕ) : 0 < ⌈Real.exp ((l:ℝ)/32)⌉₊ := by
+  have h := Nat.le_ceil (Real.exp ((l:ℝ)/32))
+  have he := Real.exp_pos ((l:ℝ)/32)
+  have : (0:ℝ) < ⌈Real.exp ((l:ℝ)/32)⌉₊ := lt_of_lt_of_le he h
+  exact_mod_cast this
+
+
+lemma pow_sum_le_mean_powers {ι : Type*} [Fintype ι] [Nonempty ι]
+    (z : ι → ℕ) (q : ℝ) (hq : 0 < q) :
+    q^(∑ i, z i) ≤ finiteMean (fun i => q^(Fintype.card ι*z i)) := by
+  have hn : (Fintype.card ι:ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
+  have hm : finiteMean (fun i => (Fintype.card ι*z i:ℕ)*Real.log q) =
+      (∑ i, z i:ℕ)*Real.log q := by
+    simp only [finiteMean,Nat.cast_mul,Nat.cast_sum,←Finset.mul_sum,←Finset.sum_mul]
+    field_simp
+  have he := exp_finiteMean_le (fun i => (Fintype.card ι*z i:ℕ)*Real.log q)
+  rw [hm,Real.exp_nat_mul,Real.exp_log hq] at he
+  simpa only [Real.exp_nat_mul,Real.exp_log hq] using he
+
+noncomputable def palindromeSlabCost (l d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) : ℕ :=
+  ∑ i : Fin l, palindromeLevelCost (l+i.1+1) d e ω
+
+lemma palindromeSlabCost_adapted (l d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) :
+    LayerAdapted d (2*l) (fun Y => palindromeSlabCost l d e (Y,X)) := by
+  apply layerAdapted_sum
+  intro i _
+  exact layerAdapted_mono (palindromeLevelCost_adapted (l+i.1+1) d e X) (by omega)
+
+theorem palindrome_slab_partial_mgf (r s l : ℕ) (hl : 0 < l) (hs : 2*s ≤ l)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+r)) (X : SwitchIndex (s+r) → Bool)
+    (lo : Card r × SwitchIndex s → Bool) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(palindromeSlabCost l (s+r) e (assembleBits r s lo hi,X))) ≤
+      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
+  classical
+  let : NeZero l := ⟨ne_of_gt hl⟩
+  have hqp : 0 < q := lt_of_lt_of_le (by norm_num) hq
+  calc
+    _ ≤ finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      finiteMean (fun i : Fin l => q^(l*palindromeLevelCost (l+i.1+1) (s+r) e
+        (assembleBits r s lo hi,X)))) := by
+      apply finiteMean_mono
+      intro hi
+      simpa only [palindromeSlabCost,Fintype.card_fin] using
+        pow_sum_le_mean_powers (fun i : Fin l => palindromeLevelCost (l+i.1+1) (s+r) e
+          (assembleBits r s lo hi,X)) q hqp
+    _ = finiteMean (fun i : Fin l => finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(l*palindromeLevelCost (l+i.1+1) (s+r) e (assembleBits r s lo hi,X)))) := finiteMean_comm _
+    _ ≤ finiteMean (fun _i : Fin l =>
+      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64)))) := by
+      apply finiteMean_mono
+      intro i
+      by_cases ht : l+i.1+1 ≤ s+r
+      · have hs' : s ≤ l+i.1 := by omega
+        have he : l+i.1+1 = s+(l+i.1-s)+1 := by omega
+        have hu : l+i.1-s < r := by omega
+        have hm := palindrome_level_partial_mgf r s (l+i.1-s) hu e X lo
+          ⌈Real.exp ((l:ℝ)/32)⌉₊ (cycleCutoff_pos l) (q^l) (one_le_pow₀ hq)
+        have hc := cycleCoefficient_decay l (l+i.1-s) (by
+          rw [Nat.cast_sub hs',Nat.cast_add]
+          have hsr : (2:ℝ)*s ≤ l := by exact_mod_cast hs
+          have hi : (0:ℝ) ≤ i.1 := Nat.cast_nonneg _
+          linarith) (q^l) (one_le_pow₀ hq) (by
+            rw [Real.log_pow]
+            nlinarith [mul_le_mul_of_nonneg_left hlog (Nat.cast_nonneg l)])
+        simp_rw [he,pow_mul]
+        exact hm.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left hc (Nat.cast_nonneg _)))
+      · have hz := palindromeLevelCost_gt (l+i.1+1) (s+r) (by omega) e
+        simp only [hz,Nat.mul_zero,pow_zero,finiteMean_const]
+        exact Real.one_le_exp_iff.mpr (by positivity)
+    _ = _ := finiteMean_const _
+
+
+lemma palindromeSlabCost_gt (l d : ℕ) (h : d ≤ l) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) : palindromeSlabCost l d e ω = 0 := by
+  apply Finset.sum_eq_zero
+  intro i _
+  exact palindromeLevelCost_gt _ _ (by omega) _ _
+
+end Thorp
+
+end ThorpNine.HighTail
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/HighTail/ExposedCoins.lean b/lean/OAI/Probability/ThorpRouting/HighTail/ExposedCoins.lean
index 8e1fcc04e14215b6da6b1c655793738b466ea0e2..339440b233bf5a40bd8792c75064a6e53498baf0 100644
--- a/lean/OAI/Probability/ThorpRouting/HighTail/ExposedCoins.lean
+++ b/lean/OAI/Probability/ThorpRouting/HighTail/ExposedCoins.lean
@@ -1,860 +1,860 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.HighTail.Coins
-
-namespace OAI
-
-namespace ThorpNine.HighTail
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-
-lemma card_routeDomain (d : ℕ) (x y : Card d) : (routeDomain d x y).card = d := by
-  classical
-  induction d with
-  | zero => rfl
-  | succ d ih =>
-    change (insert (Sum.inl (Fin.tail y))
-      ((routeDomain d (Fin.tail x) (Fin.tail y)).image
-        (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) :
-      Finset (Sum (Card d) (Bool × SwitchIndex d))).card = d + 1
-    rw [Finset.card_insert_of_notMem (by simp)]
-    rw [Finset.card_image_of_injective _ (by intro i j h; cases h; rfl), ih]
-
-lemma xor_eq_iff (a b c : Bool) : Bool.xor a b = c ↔ b = Bool.xor a c := by
-  cases a <;> cases b <;> cases c <;> decide
-
-lemma butterfly_endpoint_iff (d : ℕ) (ω : SwitchIndex d → Bool) (x y : Card d) :
-    butterflyPerm d (decodeButterfly d ω) x = y ↔
-      ∀ i ∈ routeDomain d x y, ω i = routeValue d x y i := by
-  classical
-  induction d with
-  | zero =>
-    have he : butterflyPerm 0 (decodeButterfly 0 ω) x = y := Subsingleton.elim _ _
-    simp [he, routeDomain]
-  | succ d ih =>
-    let v := butterflyPerm d (decodeButterfly d (fun i => ω (Sum.inr (x 0,i)))) (Fin.tail x)
-    change Fin.cons (Bool.xor (x 0) (ω (Sum.inl v))) v = y ↔ _
-    have hv : v = Fin.tail y ↔
-        ∀ i ∈ routeDomain d (Fin.tail x) (Fin.tail y),
-          ω (Sum.inr (x 0,i)) = routeValue d (Fin.tail x) (Fin.tail y) i :=
-      ih _ _ _
-    constructor
-    · intro h i hi
-      have ht : v = Fin.tail y := by simpa using congrArg Fin.tail h
-      have hh : ω (Sum.inl (Fin.tail y)) = Bool.xor (x 0) (y 0) := by
-        have he := congrFun h 0
-        simp only [Fin.cons_zero, ht] at he
-        exact (xor_eq_iff _ _ _).mp he
-      rcases Finset.mem_insert.mp hi with htop | hchild
-      · subst i
-        exact hh
-      · obtain ⟨j, hj, rfl⟩ := Finset.mem_image.mp hchild
-        exact hv.mp ht j hj
-    · intro h
-      have hh := h (Sum.inl (Fin.tail y)) (Finset.mem_insert_self _ _)
-      have ht : v = Fin.tail y := hv.mpr (fun i hi =>
-        h (Sum.inr (x 0,i)) (Finset.mem_insert_of_mem (Finset.mem_image_of_mem _ hi)))
-      have hh' : Bool.xor (x 0) (ω (Sum.inl (Fin.tail y))) = y 0 :=
-        (xor_eq_iff _ _ _).mpr hh
-      rw [ht, hh', Fin.cons_self_tail]
-
-lemma finiteMean_congr {Ω : Type*} [Fintype Ω] {f g : Ω → ℝ}
-    (h : ∀ ω, f ω = g ω) : finiteMean f = finiteMean g := by
-  unfold finiteMean
-  congr 1
-  exact Finset.sum_congr rfl (fun ω _ => h ω)
-
-lemma mean_coinCylinder {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (s : Finset ι) (v : ι → Bool) :
-    finiteMean (fun ω : ι → Bool => if ∀ i ∈ s, ω i = v i then (1 : ℝ) else 0) =
-      1 / (2 : ℝ) ^ s.card := by
-  induction s using Finset.induction_on with
-  | empty => simp [finiteMean]
-  | @insert i s hi ih =>
-    let f := fun ω : ι → Bool => if ∀ j ∈ s, ω j = v j then (1 : ℝ) else 0
-    have hf : ∀ ω, f (QueryTree.flipCoin i ω) = f ω := by
-      intro ω
-      have he : (∀ j ∈ s, QueryTree.flipCoin i ω j = v j) ↔ ∀ j ∈ s, ω j = v j := by
-        apply forall₂_congr
-        intro j hj
-        rw [QueryTree.flipCoin_apply_ne i j (fun h => hi (h ▸ hj))]
-      dsimp [f]
-      simp only [he]
-    have hm := QueryTree.mean_indicator_bit f i (v i) hf
-    have he : (fun ω : ι → Bool => if ∀ j ∈ insert i s, ω j = v j then (1 : ℝ) else 0) =
-        (fun ω => if ω i = v i then f ω else 0) := by
-      funext ω
-      by_cases h : ω i = v i <;> simp [f, h]
-    rw [he, hm]
-    change finiteMean (fun ω : ι → Bool => if ∀ j ∈ s, ω j = v j then (1 : ℝ) else 0) / 2 = _
-    rw [ih, Finset.card_insert_of_notMem hi, pow_succ]
-    ring
-
-def exposedCoins {ι : Type*} [DecidableEq ι] (R : Finset ι) (η ω : ι → Bool) : ι → Bool :=
-  fun i => if i ∈ R then η i else ω i
-
-lemma mean_coinCylinder_exposed {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (s R : Finset ι) (v η : ι → Bool) :
-    finiteMean (fun ω : ι → Bool =>
-      if ∀ i ∈ s, exposedCoins R η ω i = v i then (1 : ℝ) else 0) =
-    if (∀ i ∈ s ∩ R, η i = v i) then 1 / (2 : ℝ) ^ (s \ R).card else 0 := by
-  have he (ω : ι → Bool) : (∀ i ∈ s, exposedCoins R η ω i = v i) ↔
-      (∀ i ∈ s ∩ R, η i = v i) ∧ (∀ i ∈ s \ R, ω i = v i) := by
-    constructor
-    · intro h
-      constructor
-      · intro i hi
-        simpa [exposedCoins, (Finset.mem_inter.mp hi).2] using h i (Finset.mem_inter.mp hi).1
-      · intro i hi
-        simpa [exposedCoins, (Finset.mem_sdiff.mp hi).2] using h i (Finset.mem_sdiff.mp hi).1
-    · rintro ⟨ho, hf⟩ i hi
-      by_cases hR : i ∈ R
-      · simpa [exposedCoins, hR] using ho i (Finset.mem_inter.mpr ⟨hi,hR⟩)
-      · simpa [exposedCoins, hR] using hf i (Finset.mem_sdiff.mpr ⟨hi,hR⟩)
-  have hm : finiteMean (fun ω : ι → Bool =>
-      if ∀ i ∈ s, exposedCoins R η ω i = v i then (1 : ℝ) else 0) =
-      finiteMean (fun ω : ι → Bool =>
-        if (∀ i ∈ s ∩ R, η i = v i) ∧ (∀ i ∈ s \ R, ω i = v i) then (1 : ℝ) else 0) :=
-    finiteMean_congr (fun ω => by simp only [he ω])
-  rw [hm]
-  by_cases h : ∀ i ∈ s ∩ R, η i = v i
-  · have hp : (∀ i ∈ s ∩ R, η i = v i) ↔ True := iff_true_intro h
-    simp only [hp, true_and, ↓reduceIte]
-    exact mean_coinCylinder (s \ R) v
-  · have hp : (∀ i ∈ s ∩ R, η i = v i) ↔ False := iff_false_intro h
-    simp only [hp, false_and, ↓reduceIte]
-    simp [finiteMean]
-
-lemma butterfly_closure_probability (d : ℕ) (x y : Card d)
-    (R : Finset (SwitchIndex d)) (η : SwitchIndex d → Bool) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      if butterflyPerm d (decodeButterfly d (exposedCoins R η ω)) x = y then (1 : ℝ) else 0) =
-      if (∀ i ∈ routeDomain d x y ∩ R, η i = routeValue d x y i) then
-        (2 : ℝ) ^ (routeDomain d x y ∩ R).card / (2 : ℝ) ^ d else 0 := by
-  have he : finiteMean (fun ω : SwitchIndex d → Bool =>
-      if butterflyPerm d (decodeButterfly d (exposedCoins R η ω)) x = y then (1 : ℝ) else 0) =
-      finiteMean (fun ω : SwitchIndex d → Bool =>
-        if ∀ i ∈ routeDomain d x y, exposedCoins R η ω i = routeValue d x y i then (1 : ℝ) else 0) :=
-    finiteMean_congr (fun ω => by simp only [butterfly_endpoint_iff])
-  rw [he, mean_coinCylinder_exposed]
-  congr 1
-  have hc : (routeDomain d x y \ R).card + (routeDomain d x y ∩ R).card = d := by
-    rw [Finset.card_sdiff_add_card_inter, card_routeDomain]
-  have hp : (2 : ℝ) ^ d = (2 : ℝ) ^ (routeDomain d x y \ R).card *
-      (2 : ℝ) ^ (routeDomain d x y ∩ R).card := by
-    rw [← pow_add]
-    exact congrArg (fun n => (2 : ℝ) ^ n) hc.symm
-  rw [hp]
-  have hn : (2 : ℝ) ^ (routeDomain d x y ∩ R).card ≠ 0 := pow_ne_zero _ (by norm_num)
-  field_simp
-
-lemma butterfly_single_card_uniform (d : ℕ) (x y : Card d) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      if butterflyPerm d (decodeButterfly d ω) x = y then (1 : ℝ) else 0) = 1 / (2 : ℝ) ^ d := by
-  have he : finiteMean (fun ω : SwitchIndex d → Bool =>
-      if butterflyPerm d (decodeButterfly d ω) x = y then (1 : ℝ) else 0) =
-      finiteMean (fun ω : SwitchIndex d → Bool =>
-        if ∀ i ∈ routeDomain d x y, ω i = routeValue d x y i then (1 : ℝ) else 0) :=
-    finiteMean_congr (fun ω => by simp only [butterfly_endpoint_iff])
-  rw [he, mean_coinCylinder, card_routeDomain]
-
-def switchUsers : (d : ℕ) → Butterfly d → SwitchIndex d → Card d × Card d
-  | 0, _, i => nomatch i
-  | d + 1, b, Sum.inl y =>
-      (Fin.cons false ((butterflyPerm d (b.2 false)).symm y),
-       Fin.cons true ((butterflyPerm d (b.2 true)).symm y))
-  | d + 1, b, Sum.inr (ε,i) =>
-      (Fin.cons ε (switchUsers d (b.2 ε) i).1,
-       Fin.cons ε (switchUsers d (b.2 ε) i).2)
-
-lemma switchUsers_ne (d : ℕ) (b : Butterfly d) (i : SwitchIndex d) :
-    (switchUsers d b i).1 ≠ (switchUsers d b i).2 := by
-  induction d with
-  | zero => exact Empty.elim i
-  | succ d ih =>
-    cases i with
-    | inl y =>
-      intro h
-      have := congrFun h 0
-      simp only [switchUsers, Fin.cons_zero] at this
-      exact Bool.false_ne_true this
-    | inr z =>
-      intro h
-      exact ih (b.2 z.1) z.2 (by simpa only [switchUsers, Fin.tail_cons] using congrArg Fin.tail h)
-
-lemma cons_eq_iff {d : ℕ} (ε : Bool) (a : Card d) (x : Card (d + 1)) :
-    Fin.cons ε a = x ↔ ε = x 0 ∧ a = Fin.tail x := by
-  constructor
-  · intro h
-    exact ⟨by simpa only [Fin.cons_zero] using congrFun h 0,
-      by simpa only [Fin.tail_cons] using congrArg Fin.tail h⟩
-  · rintro ⟨rfl,rfl⟩
-    exact Fin.cons_self_tail x
-
-lemma mem_routeDomain_iff_user (d : ℕ) (b : Butterfly d) (x : Card d) (i : SwitchIndex d) :
-    i ∈ routeDomain d x (butterflyPerm d b x) ↔
-      (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x := by
-  classical
-  induction d with
-  | zero => exact Empty.elim i
-  | succ d ih =>
-    cases i with
-    | inl y =>
-      change Sum.inl y ∈ insert (Sum.inl (Fin.tail (butterflyPerm (d+1) b x)))
-        ((routeDomain d (Fin.tail x) (Fin.tail (butterflyPerm (d+1) b x))).image
-          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) ↔ _
-      simp only [Finset.mem_insert, Sum.inl.injEq, Finset.mem_image,
-        Sum.inr_ne_inl, and_false, exists_false, or_false]
-      change y = butterflyPerm d (b.2 (x 0)) (Fin.tail x) ↔ _
-      simp only [switchUsers, cons_eq_iff]
-      cases x 0 <;> simp [Equiv.symm_apply_eq]
-    | inr z =>
-      obtain ⟨ε,i⟩ := z
-      change Sum.inr (ε,i) ∈ insert (Sum.inl (Fin.tail (butterflyPerm (d+1) b x)))
-        ((routeDomain d (Fin.tail x) (Fin.tail (butterflyPerm (d+1) b x))).image
-          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) ↔ _
-      simp only [Finset.mem_insert, Sum.inr_ne_inl, false_or, Finset.mem_image,
-        Sum.inr.injEq, Prod.mk.injEq]
-      change (∃ j, j ∈ routeDomain d (Fin.tail x) (butterflyPerm d (b.2 (x 0)) (Fin.tail x)) ∧
-        x 0 = ε ∧ j = i) ↔ _
-      simp only [switchUsers, cons_eq_iff]
-      constructor
-      · rintro ⟨j,hj,hε,rfl⟩
-        subst ε
-        rcases (ih (b.2 (x 0)) (Fin.tail x) j).mp hj with h | h
-        · exact Or.inl ⟨rfl,h⟩
-        · exact Or.inr ⟨rfl,h⟩
-      · rintro (⟨hε,h⟩ | ⟨hε,h⟩)
-        · subst ε
-          exact ⟨i,(ih (b.2 (x 0)) (Fin.tail x) i).mpr (Or.inl h),rfl,rfl⟩
-        · subst ε
-          exact ⟨i,(ih (b.2 (x 0)) (Fin.tail x) i).mpr (Or.inr h),rfl,rfl⟩
-
-lemma card_switches_using (d : ℕ) (b : Butterfly d) (x : Card d) :
-    (Finset.univ.filter (fun i : SwitchIndex d =>
-      (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x)).card = d := by
-  classical
-  have he : Finset.univ.filter (fun i : SwitchIndex d =>
-      (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x) =
-      routeDomain d x (butterflyPerm d b x) := by
-    ext i
-    simp only [Finset.mem_filter, Finset.mem_univ, true_and, mem_routeDomain_iff_user]
-  rw [he,card_routeDomain]
-
-lemma switch_incidence_sum (d : ℕ) (b : Butterfly d) (w : Card d → ℝ) :
-    ∑ i : SwitchIndex d, (w (switchUsers d b i).1 + w (switchUsers d b i).2) =
-      (d : ℝ) * ∑ x : Card d, w x := by
-  classical
-  calc
-    _ = ∑ i : SwitchIndex d, ∑ x : Card d,
-        if (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x then w x else 0 := by
-      apply Finset.sum_congr rfl
-      intro i _
-      rw [Finset.sum_ite]
-      simp only [Finset.sum_const_zero, add_zero]
-      have he : Finset.univ.filter (fun x : Card d =>
-          (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x) =
-          {(switchUsers d b i).1, (switchUsers d b i).2} := by
-        ext x; simp [eq_comm]
-      rw [he, Finset.sum_pair (switchUsers_ne d b i)]
-    _ = ∑ x : Card d, ∑ i : SwitchIndex d,
-        if (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x then w x else 0 :=
-      Finset.sum_comm
-    _ = ∑ x : Card d, (d : ℝ) * w x := by
-      apply Finset.sum_congr rfl
-      intro x _
-      rw [Finset.sum_ite]
-      simp only [Finset.sum_const_zero, add_zero, Finset.sum_const, nsmul_eq_mul,
-        card_switches_using]
-    _ = _ := by rw [Finset.mul_sum]
-
-lemma weighted_switch_min_bound (d : ℕ) (b : Butterfly d) (w : Card d → ℝ) :
-    ∑ i : SwitchIndex d, min (w (switchUsers d b i).1) (w (switchUsers d b i).2) ≤
-      (d : ℝ) / 2 * ∑ x : Card d, w x := by
-  have h : 2 * (∑ i : SwitchIndex d,
-      min (w (switchUsers d b i).1) (w (switchUsers d b i).2)) ≤
-      ∑ i : SwitchIndex d, (w (switchUsers d b i).1 + w (switchUsers d b i).2) := by
-    rw [Finset.mul_sum]
-    apply Finset.sum_le_sum
-    intro i _
-    linarith [min_le_left (w (switchUsers d b i).1) (w (switchUsers d b i).2),
-      min_le_right (w (switchUsers d b i).1) (w (switchUsers d b i).2)]
-  rw [switch_incidence_sum] at h
-  linarith
-
-lemma mem_childMarks {d : ℕ} (s : Finset (Card (d + 1))) (ε : Bool) (x : Card d) :
-    x ∈ childMarks s ε ↔ Fin.cons ε x ∈ s := by
-  classical
-  constructor
-  · intro h
-    obtain ⟨y,hy,hxy⟩ := Finset.mem_image.mp h
-    obtain ⟨hys,hyε⟩ := Finset.mem_filter.mp hy
-    rw [← hxy, ← hyε, Fin.cons_self_tail]
-    exact hys
-  · intro h
-    exact Finset.mem_image.mpr ⟨Fin.cons ε x,
-      Finset.mem_filter.mpr ⟨h,by simp⟩,by simp⟩
-
-lemma mem_topMeetings {d : ℕ} (b : Butterfly (d + 1))
-    (s : Finset (Card (d + 1))) (y : Card d) :
-    y ∈ topMeetings b s ↔
-      (switchUsers (d+1) b (Sum.inl y)).1 ∈ s ∧
-      (switchUsers (d+1) b (Sum.inl y)).2 ∈ s := by
-  classical
-  have he (ε : Bool) : y ∈ (childMarks s ε).image (butterflyPerm d (b.2 ε)) ↔
-      (butterflyPerm d (b.2 ε)).symm y ∈ childMarks s ε := by
-    constructor
-    · intro h
-      obtain ⟨x,hx,rfl⟩ := Finset.mem_image.mp h
-      simpa only [Equiv.symm_apply_apply] using hx
-    · intro h
-      exact Finset.mem_image.mpr ⟨_,h,Equiv.apply_symm_apply _ _⟩
-  simp only [topMeetings, Finset.mem_inter, he, mem_childMarks, switchUsers]
-
-lemma encounters_eq_switch_sum (d : ℕ) (b : Butterfly d) (s : Finset (Card d)) :
-    encounters d b s = ∑ i : SwitchIndex d,
-      if (switchUsers d b i).1 ∈ s ∧ (switchUsers d b i).2 ∈ s then 1 else 0 := by
-  classical
-  induction d with
-  | zero =>
-    simp only [encounters]
-    symm
-    apply Finset.sum_eq_zero
-    intro i _
-    exact Empty.elim i
-  | succ d ih =>
-    change encounters d (b.2 false) (childMarks s false) +
-      encounters d (b.2 true) (childMarks s true) + (topMeetings b s).card =
-        ∑ i : Sum (Card d) (Bool × SwitchIndex d), _
-    rw [Fintype.sum_sum_type, Fintype.sum_prod_type]
-    simp only [Fintype.sum_bool]
-    rw [ih,ih]
-    have ht : (∑ y : Card d,
-        if (switchUsers (d+1) b (Sum.inl y)).1 ∈ s ∧
-          (switchUsers (d+1) b (Sum.inl y)).2 ∈ s then 1 else 0) =
-        (topMeetings b s).card := by
-      simp only [← mem_topMeetings]
-      simp
-    rw [ht]
-    simp only [mem_childMarks, switchUsers]
-    ac_rfl
-
-lemma marked_switch_entropy_bound (d : ℕ) (b : Butterfly d) (s : Finset (Card d)) :
-    (∑ i : SwitchIndex d,
-      if (switchUsers d b i).1 ∈ s ∧ (switchUsers d b i).2 ∈ s then (1:ℝ) else 0) ≤
-      (s.card : ℝ) * Real.log s.card / (2 * Real.log 2) := by
-  classical
-  have he : (encounters d b s : ℝ) =
-      ∑ i : SwitchIndex d,
-        if (switchUsers d b i).1 ∈ s ∧ (switchUsers d b i).2 ∈ s then (1:ℝ) else 0 := by
-    rw [encounters_eq_switch_sum, Nat.cast_sum]
-    apply Finset.sum_congr rfl
-    intro i _
-    split_ifs <;> simp
-  rw [← he]
-  exact butterfly_encounters d b s
-
-namespace QueryTree
-variable {ι : Type*} [DecidableEq ι]
-
-def memo : QueryTree ι → Finset ι → (ι → Bool) → QueryTree ι
-  | reject, _, _ => reject
-  | accept, _, _ => accept
-  | branch i zero one, R, η =>
-      if i ∈ R then (if η i then memo one R η else memo zero R η) else
-      branch i (memo zero (insert i R) (Function.update η i false))
-        (memo one (insert i R) (Function.update η i true))
-  | force i b next, R, η =>
-      if i ∈ R then (if η i = b then memo next R η else reject) else
-      force i b (memo next (insert i R) (Function.update η i b))
-
-lemma memo_queries_disjoint (T : QueryTree ι) (R : Finset ι) (η : ι → Bool) :
-    Disjoint (memo T R η).queries R := by
-  induction T generalizing R η with
-  | reject => simp [memo,queries]
-  | accept => simp [memo,queries]
-  | branch i zero one ih₀ ih₁ =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte]
-      cases η i <;> simp only [Bool.false_eq_true,↓reduceIte]
-      · exact ih₀ R η
-      · exact ih₁ R η
-    · simp only [memo,hi,↓reduceIte,queries,Finset.disjoint_insert_left,Finset.disjoint_union_left]
-      exact ⟨by simp, (ih₀ (insert i R) _).mono_right (Finset.subset_insert _ _),
-        (ih₁ (insert i R) _).mono_right (Finset.subset_insert _ _)⟩
-  | force i b next ih =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte]
-      by_cases hb : η i = b
-      · simpa only [hb,↓reduceIte] using ih R η
-      · simp [hb,queries]
-    · simp only [memo,hi,↓reduceIte,queries,Finset.disjoint_insert_left]
-      exact ⟨by simp,(ih (insert i R) _).mono_right (Finset.subset_insert _ _)⟩
-
-lemma memo_fresh (T : QueryTree ι) (R : Finset ι) (η : ι → Bool) :
-    (memo T R η).Fresh := by
-  induction T generalizing R η with
-  | reject => trivial
-  | accept => trivial
-  | branch i zero one ih₀ ih₁ =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte]
-      cases η i <;> simp only [Bool.false_eq_true,↓reduceIte]
-      · exact ih₀ R η
-      · exact ih₁ R η
-    · simp only [memo,hi,↓reduceIte,Fresh]
-      exact ⟨fun hm => Finset.disjoint_left.mp (memo_queries_disjoint zero (insert i R) _) hm
-          (Finset.mem_insert_self _ _),
-        fun hm => Finset.disjoint_left.mp (memo_queries_disjoint one (insert i R) _) hm
-          (Finset.mem_insert_self _ _), ih₀ _ _,ih₁ _ _⟩
-  | force i b next ih =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte]
-      by_cases hb : η i = b
-      · simpa only [hb,↓reduceIte] using ih R η
-      · simp [hb,Fresh]
-    · simp only [memo,hi,↓reduceIte,Fresh]
-      exact ⟨fun hm => Finset.disjoint_left.mp (memo_queries_disjoint next (insert i R) _) hm
-        (Finset.mem_insert_self _ _),ih _ _⟩
-
-lemma update_agrees {R : Finset ι} {η ω : ι → Bool}
-    (h : ∀ i ∈ R, η i = ω i) (i : ι) :
-    ∀ j ∈ insert i R, Function.update η i (ω i) j = ω j := by
-  intro j hj
-  rcases Finset.mem_insert.mp hj with rfl | hj
-  · simp
-  · by_cases hji : j = i
-    · subst j; simp
-    · rw [Function.update_of_ne hji]
-      exact h j hj
-
-lemma memo_succeeds (T : QueryTree ι) (R : Finset ι) (η ω : ι → Bool)
-    (h : ∀ i ∈ R, η i = ω i) : (memo T R η).succeeds ω = T.succeeds ω := by
-  induction T generalizing R η with
-  | reject => rfl
-  | accept => rfl
-  | branch i zero one ih₀ ih₁ =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte,succeeds,h i hi]
-      cases ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
-      · exact ih₀ R η h
-      · exact ih₁ R η h
-    · simp only [memo,hi,↓reduceIte,succeeds]
-      cases hb : ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
-      · exact ih₀ _ _ (by simpa only [hb] using update_agrees h i)
-      · exact ih₁ _ _ (by simpa only [hb] using update_agrees h i)
-  | force i b next ih =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte,h i hi]
-      by_cases hb : ω i = b
-      · simp only [hb,↓reduceIte,succeeds]
-        exact ih R η h
-      · simp [hb,succeeds]
-    · simp only [memo,hi,↓reduceIte,succeeds]
-      by_cases hb : ω i = b
-      · simp only [hb,↓reduceIte]
-        exact ih _ _ (by simpa only [hb] using update_agrees h i)
-      · simp [hb]
-
-def freshCharge : QueryTree ι → Finset ι → (ι → Bool) → ℕ
-  | reject, _, _ => 0
-  | accept, _, _ => 0
-  | branch i zero one, R, ω => if ω i then freshCharge one (insert i R) ω else freshCharge zero (insert i R) ω
-  | force i b next, R, ω => if ω i = b then
-      freshCharge next (insert i R) ω + (if i ∈ R then 0 else 1) else 0
-
-lemma memo_charge (T : QueryTree ι) (R : Finset ι) (η ω : ι → Bool)
-    (h : ∀ i ∈ R, η i = ω i) : (memo T R η).charge ω = freshCharge T R ω := by
-  induction T generalizing R η with
-  | reject => rfl
-  | accept => rfl
-  | branch i zero one ih₀ ih₁ =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte,freshCharge,h i hi,Finset.insert_eq_of_mem hi]
-      cases ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
-      · exact ih₀ R η h
-      · exact ih₁ R η h
-    · simp only [memo,hi,↓reduceIte,charge,freshCharge]
-      cases hb : ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
-      · exact ih₀ _ _ (by simpa only [hb] using update_agrees h i)
-      · exact ih₁ _ _ (by simpa only [hb] using update_agrees h i)
-  | force i b next ih =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte,h i hi,freshCharge,Finset.insert_eq_of_mem hi,Nat.add_zero]
-      by_cases hb : ω i = b
-      · simp only [hb,↓reduceIte]
-        exact ih R η h
-      · simp [hb,charge]
-    · simp only [memo,hi,↓reduceIte,charge,freshCharge]
-      by_cases hb : ω i = b
-      · simp only [hb,↓reduceIte]
-        rw [ih _ _ (by simpa only [hb] using update_agrees h i)]
-      · simp [hb]
-
-lemma mean_freshCharge_le_one [Fintype ι] (T : QueryTree ι) :
-    finiteMean (fun ω : ι → Bool =>
-      if T.succeeds ω then (2:ℝ) ^ T.freshCharge ∅ ω else 0) ≤ 1 := by
-  let η : ι → Bool := fun _ => false
-  have hm := mean_weighted_le_one (memo T ∅ η) (memo_fresh T ∅ η)
-  have he : finiteMean (memo T ∅ η).weighted =
-      finiteMean (fun ω : ι → Bool =>
-        if T.succeeds ω then (2:ℝ) ^ T.freshCharge ∅ ω else 0) := by
-    apply finiteMean_congr
-    intro ω
-    rw [weighted_eq, memo_succeeds _ _ _ _ (by simp), memo_charge _ _ _ _ (by simp)]
-  rw [he] at hm
-  exact hm
-
-
-def followPath : (d : ℕ) → (SwitchIndex d → ι) → Card d →
-    (Card d → QueryTree ι) → QueryTree ι
-  | 0, _, x, k => k x
-  | d + 1, e, x, k => followPath d (fun i => e (Sum.inr (x 0,i))) (Fin.tail x)
-      (fun y => branch (e (Sum.inl y))
-        (k (Fin.cons (Bool.xor (x 0) false) y))
-        (k (Fin.cons (Bool.xor (x 0) true) y)))
-
-omit [DecidableEq ι] in
-lemma followPath_succeeds (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
-    (k : Card d → QueryTree ι) (ω : ι → Bool) :
-    (followPath d e x k).succeeds ω =
-      (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).succeeds ω := by
-  induction d with
-  | zero => rfl
-  | succ d ih =>
-    rw [followPath, ih]
-    let y := butterflyPerm d
-      (decodeButterfly d (fun i => ω (e (Sum.inr (x 0,i))))) (Fin.tail x)
-    change (if ω (e (Sum.inl y)) then
-      (k (Fin.cons (Bool.xor (x 0) true) y)).succeeds ω else
-      (k (Fin.cons (Bool.xor (x 0) false) y)).succeeds ω) =
-      (k (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).succeeds ω
-    cases ω (e (Sum.inl y)) <;> rfl
-
-lemma followPath_charge (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
-    (k : Card d → QueryTree ι) (R : Finset ι) (ω : ι → Bool) :
-    freshCharge (followPath d e x k) R ω =
-      freshCharge (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x))
-        (R ∪ (routeDomain d x (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).image e) ω := by
-  classical
-  induction d generalizing R with
-  | zero => simp only [followPath, routeDomain, Finset.image_empty, Finset.union_empty]; rfl
-  | succ d ih =>
-    rw [followPath, ih]
-    let y := butterflyPerm d
-      (decodeButterfly d (fun i => ω (e (Sum.inr (x 0,i))))) (Fin.tail x)
-    let S := (routeDomain d (Fin.tail x) y).image (fun i => e (Sum.inr (x 0,i)))
-    change (if ω (e (Sum.inl y)) then
-      freshCharge (k (Fin.cons (Bool.xor (x 0) true) y)) (insert (e (Sum.inl y)) (R ∪ S)) ω else
-      freshCharge (k (Fin.cons (Bool.xor (x 0) false) y)) (insert (e (Sum.inl y)) (R ∪ S)) ω) =
-      freshCharge (k (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y))
-        (R ∪ (routeDomain (d+1) x
-          (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).image e) ω
-    have hs : (routeDomain (d+1) x
-        (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).image e =
-        insert (e (Sum.inl y)) S := by
-      let e' : Sum (Card d) (Bool × SwitchIndex d) → ι := e
-      change (insert (Sum.inl y)
-        ((routeDomain d (Fin.tail x) y).image
-          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) :
-          Finset (Sum (Card d) (Bool × SwitchIndex d))).image e' = _
-      rw [Finset.image_insert, Finset.image_image]
-      rfl
-    rw [hs,Finset.union_insert]
-    cases ω (e (Sum.inl y)) <;> rfl
-
-def forceList : List ι → (ι → Bool) → QueryTree ι → QueryTree ι
-  | [], _, k => k
-  | i :: L, v, k => force i (v i) (forceList L v k)
-
-omit [DecidableEq ι] in
-lemma forceList_succeeds (L : List ι) (v : ι → Bool) (k : QueryTree ι) (ω : ι → Bool) :
-    (forceList L v k).succeeds ω =
-      if ∀ i ∈ L, ω i = v i then k.succeeds ω else false := by
-  induction L with
-  | nil => simp [forceList]
-  | cons i L ih =>
-    simp only [forceList,succeeds,ih,List.mem_cons,forall_eq_or_imp]
-    by_cases h : ω i = v i <;> simp [h]
-
-lemma forceList_charge (L : List ι) (v : ι → Bool) (k : QueryTree ι)
-    (R : Finset ι) (ω : ι → Bool) (h : ∀ i ∈ L, ω i = v i) :
-    freshCharge (forceList L v k) R ω =
-      freshCharge k (R ∪ L.toFinset) ω + (L.toFinset \ R).card := by
-  induction L generalizing R with
-  | nil => simp [forceList]
-  | cons i L ih =>
-    rw [forceList,freshCharge,ite_eq_left (h i List.mem_cons_self),
-      ih (insert i R) (fun j hj => h j (List.mem_cons_of_mem _ hj))]
-    simp only [List.toFinset_cons]
-    have hs : insert i R ∪ L.toFinset = R ∪ insert i L.toFinset := by ext j; simp
-    rw [hs]
-    by_cases hi : i ∈ R
-    · rw [ite_eq_left hi]
-      have he : L.toFinset \ insert i R = insert i L.toFinset \ R := by
-        ext j
-        by_cases hji : j = i
-        · subst j; simp [hi]
-        · simp [hji]
-      rw [he,Nat.add_zero]
-    · rw [ite_eq_right hi]
-      have he : insert i L.toFinset \ R = insert i (L.toFinset \ insert i R) := by
-        ext j
-        by_cases hji : j = i
-        · subst j; simp [hi]
-        · simp [hji]
-      rw [he, Finset.card_insert_of_notMem (by simp)]
-      omega
-
-
-def staleCharge : QueryTree ι → Finset ι → (ι → Bool) → ℕ
-  | reject, _, _ => 0
-  | accept, _, _ => 0
-  | branch i zero one, R, ω => if ω i then staleCharge one (insert i R) ω
-      else staleCharge zero (insert i R) ω
-  | force i b next, R, ω => if ω i = b then
-      staleCharge next (insert i R) ω + (if i ∈ R then 1 else 0) else 0
-
-lemma fresh_add_stale (T : QueryTree ι) (R : Finset ι) (ω : ι → Bool) :
-    T.freshCharge R ω + T.staleCharge R ω = T.charge ω := by
-  induction T generalizing R with
-  | reject => rfl
-  | accept => rfl
-  | branch i zero one ih₀ ih₁ =>
-    cases hb : ω i <;> simp [freshCharge,staleCharge,charge,hb,ih₀,ih₁]
-  | force i b next ih =>
-    by_cases hb : ω i = b
-    · simp only [freshCharge,staleCharge,charge,hb,↓reduceIte]
-      have he := ih (insert i R)
-      by_cases hi : i ∈ R <;> simp only [hi,↓reduceIte] <;> omega
-    · simp [freshCharge,staleCharge,charge,hb]
-
-omit [DecidableEq ι] in
-lemma followPath_raw_charge (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
-    (k : Card d → QueryTree ι) (ω : ι → Bool) :
-    (followPath d e x k).charge ω =
-      (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).charge ω := by
-  induction d with
-  | zero => rfl
-  | succ d ih =>
-    rw [followPath, ih]
-    let y := butterflyPerm d
-      (decodeButterfly d (fun i => ω (e (Sum.inr (x 0,i))))) (Fin.tail x)
-    change (if ω (e (Sum.inl y)) then
-      (k (Fin.cons (Bool.xor (x 0) true) y)).charge ω else
-      (k (Fin.cons (Bool.xor (x 0) false) y)).charge ω) =
-      (k (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).charge ω
-    cases ω (e (Sum.inl y)) <;> rfl
-
-omit [DecidableEq ι] in
-lemma forceList_raw_charge (L : List ι) (v : ι → Bool) (k : QueryTree ι) (ω : ι → Bool)
-    (h : ∀ i ∈ L, ω i = v i) :
-    (forceList L v k).charge ω = k.charge ω + L.length := by
-  induction L with
-  | nil => simp [forceList]
-  | cons i L ih =>
-    rw [forceList,charge,ite_eq_left (h i List.mem_cons_self),
-      ih (fun j hj => h j (List.mem_cons_of_mem _ hj))]
-    simp only [List.length_cons,Nat.add_assoc]
-
-noncomputable def closePath (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d)) :
-    QueryTree (SwitchIndex d) :=
-  forceList (routeDomain d x y).toList (routeValue d x y) k
-
-lemma closePath_succeeds (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
-    (ω : SwitchIndex d → Bool) :
-    (closePath d x y k).succeeds ω =
-      if butterflyPerm d (decodeButterfly d ω) x = y then k.succeeds ω else false := by
-  unfold closePath
-  rw [forceList_succeeds]
-  simp only [Finset.mem_toList, ← butterfly_endpoint_iff]
-
-lemma closePath_raw_charge (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
-    (ω : SwitchIndex d → Bool) (h : butterflyPerm d (decodeButterfly d ω) x = y) :
-    (closePath d x y k).charge ω = k.charge ω + d := by
-  unfold closePath
-  rw [forceList_raw_charge _ _ _ _ (by simpa only [Finset.mem_toList] using
-    (butterfly_endpoint_iff d ω x y).mp h), Finset.length_toList, card_routeDomain]
-
-lemma closePath_fresh_charge (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
-    (R : Finset (SwitchIndex d)) (ω : SwitchIndex d → Bool)
-    (h : butterflyPerm d (decodeButterfly d ω) x = y) :
-    (closePath d x y k).freshCharge R ω =
-      k.freshCharge (R ∪ routeDomain d x y) ω + (routeDomain d x y \ R).card := by
-  unfold closePath
-  rw [forceList_charge _ _ _ _ _ (by simpa only [Finset.mem_toList] using
-    (butterfly_endpoint_iff d ω x y).mp h), Finset.toList_toFinset]
-
-lemma closePath_stale_charge (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
-    (R : Finset (SwitchIndex d)) (ω : SwitchIndex d → Bool)
-    (h : butterflyPerm d (decodeButterfly d ω) x = y) :
-    (closePath d x y k).staleCharge R ω =
-      k.staleCharge (R ∪ routeDomain d x y) ω + (routeDomain d x y ∩ R).card := by
-  have h₀ := fresh_add_stale (closePath d x y k) R ω
-  have h₁ := fresh_add_stale k (R ∪ routeDomain d x y) ω
-  rw [closePath_raw_charge _ _ _ _ _ h,closePath_fresh_charge _ _ _ _ _ _ h] at h₀
-  have h₂ : (routeDomain d x y \ R).card + (routeDomain d x y ∩ R).card = d := by
-    rw [Finset.card_sdiff_add_card_inter,card_routeDomain]
-  omega
-
-noncomputable def seekCycle (d : ℕ) (S : Finset (Card d)) (a : Card d) :
-    ℕ → Card d → Finset (Card d) → (Finset (Card d) → QueryTree (SwitchIndex d)) →
-      QueryTree (SwitchIndex d)
-  | 0, x, V, k => if x ∈ S ∧ x ∉ V then closePath d x a (k (insert x V)) else reject
-  | r + 1, x, V, k => if x ∈ S ∧ x ∉ V then followPath d id x
-      (fun y => seekCycle d S a r y (insert x V) k) else reject
-
-lemma seekCycle_raw_charge (d : ℕ) (S : Finset (Card d)) (a : Card d) (r : ℕ)
-    (x : Card d) (V : Finset (Card d)) (k : Finset (Card d) → QueryTree (SwitchIndex d))
-    (ω : SwitchIndex d → Bool) (K : ℕ)
-    (hK : ∀ V, (k V).succeeds ω = true → (k V).charge ω = K)
-    (hs : (seekCycle d S a r x V k).succeeds ω = true) :
-    (seekCycle d S a r x V k).charge ω = K + d := by
-  induction r generalizing x V with
-  | zero =>
-    by_cases hv : x ∈ S ∧ x ∉ V
-    · rw [seekCycle,ite_eq_left hv,closePath_succeeds] at hs
-      split_ifs at hs with he
-      · rw [seekCycle,ite_eq_left hv,closePath_raw_charge _ _ _ _ _ he,hK _ hs]
-    · simp [seekCycle,hv,succeeds] at hs
-  | succ r ih =>
-    by_cases hv : x ∈ S ∧ x ∉ V
-    · rw [seekCycle,ite_eq_left hv,followPath_succeeds] at hs
-      rw [seekCycle,ite_eq_left hv,followPath_raw_charge]
-      exact ih _ _ hs
-    · simp [seekCycle,hv,succeeds] at hs
-
-noncomputable def seekCycles (d : ℕ) (S : Finset (Card d)) :
-    List (Card d × ℕ) → Finset (Card d) → QueryTree (SwitchIndex d)
-  | [], _ => accept
-  | (a,r) :: L, V => seekCycle d S a r a V (fun V => seekCycles d S L V)
-
-lemma seekCycles_raw_charge (d : ℕ) (S : Finset (Card d)) (L : List (Card d × ℕ))
-    (V : Finset (Card d)) (ω : SwitchIndex d → Bool)
-    (hs : (seekCycles d S L V).succeeds ω = true) :
-    (seekCycles d S L V).charge ω = L.length * d := by
-  induction L generalizing V with
-  | nil => simp [seekCycles,charge]
-  | cons ar L ih =>
-    obtain ⟨a,r⟩ := ar
-    rw [seekCycles]
-    rw [seekCycle_raw_charge d S a r a V (fun V => seekCycles d S L V) ω (L.length*d)
-      (fun V hs => ih V hs) hs]
-    simp [Nat.add_mul]
-
-lemma cycles_weighted_success (d : ℕ) (S : Finset (Card d)) (L : List (Card d × ℕ)) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      if (seekCycles d S L ∅).succeeds ω then
-        ((2:ℝ)^d)^L.length / (2:ℝ)^(seekCycles d S L ∅).staleCharge ∅ ω else 0) ≤ 1 := by
-  let T := seekCycles d S L ∅
-  have hh := mean_freshCharge_le_one T
-  have he : finiteMean (fun ω : SwitchIndex d → Bool =>
-      if T.succeeds ω then (2:ℝ)^T.freshCharge ∅ ω else 0) =
-      finiteMean (fun ω : SwitchIndex d → Bool =>
-        if T.succeeds ω then ((2:ℝ)^d)^L.length / (2:ℝ)^T.staleCharge ∅ ω else 0) := by
-    apply finiteMean_congr
-    intro ω
-    cases hs : T.succeeds ω
-    · simp
-    · simp only [↓reduceIte]
-      have hc := fresh_add_stale T ∅ ω
-      rw [seekCycles_raw_charge d S L ∅ ω hs] at hc
-      have hp : ((2:ℝ)^d)^L.length = (2:ℝ)^T.freshCharge ∅ ω * (2:ℝ)^T.staleCharge ∅ ω := by
-        rw [←pow_add,hc,←pow_mul,Nat.mul_comm]
-      rw [hp,mul_div_cancel_right₀ _ (pow_ne_zero _ (by norm_num))]
-  rw [← he]
-  exact hh
-
-end QueryTree
-
-lemma finiteMean_add {Ω : Type*} [Fintype Ω] (f g : Ω → ℝ) :
-    finiteMean (fun ω => f ω + g ω) = finiteMean f + finiteMean g := by
-  simp only [finiteMean,Finset.sum_add_distrib,add_div]
-
-lemma finiteMean_mul_const {Ω : Type*} [Fintype Ω] (f : Ω → ℝ) (c : ℝ) :
-    finiteMean (fun ω => f ω * c) = finiteMean f * c := by
-  unfold finiteMean
-  rw [← Finset.sum_mul]
-  ring
-
-lemma finiteMean_sum {Ω α : Type*} [Fintype Ω] (s : Finset α) (f : α → Ω → ℝ) :
-    finiteMean (fun ω => ∑ i ∈ s, f i ω) = ∑ i ∈ s, finiteMean (f i) := by
-  simp only [finiteMean,Finset.sum_div]
-  exact Finset.sum_comm
-
-lemma finiteMean_mono {Ω : Type*} [Fintype Ω] {f g : Ω → ℝ}
-    (h : ∀ ω, f ω ≤ g ω) : finiteMean f ≤ finiteMean g := by
-  exact div_le_div_of_nonneg_right (Finset.sum_le_sum (fun ω _ => h ω)) (Nat.cast_nonneg _)
-
-lemma finiteMean_equiv {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
-    (e : Ω ≃ Ω') (f : Ω → ℝ) :
-    finiteMean f = finiteMean (fun ω => f (e.symm ω)) := by
-  unfold finiteMean
-  rw [Fintype.card_congr e]
-  congr 1
-  exact Fintype.sum_equiv e _ _ (fun ω => by simp)
-
-lemma finiteMean_prod_left {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] [Nonempty Ω']
-    (f : Ω → ℝ) : finiteMean (fun ω : Ω × Ω' => f ω.1) = finiteMean f := by
-  simp only [finiteMean,Fintype.sum_prod_type,Finset.sum_const,nsmul_eq_mul,
-    ← Finset.mul_sum,Fintype.card_prod,Nat.cast_mul,Finset.card_univ]
-  have hn : (Fintype.card Ω' : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
-  field_simp
-
-lemma finiteMean_pi_eval {C : Type*} [Fintype C] [DecidableEq C]
-    {A : C → Type*} [∀ c, Fintype (A c)] [∀ c, Nonempty (A c)]
-    (c : C) (g : A c → ℝ) :
-    finiteMean (fun z : ∀ c, A c => g (z c)) = finiteMean g := by
-  rw [finiteMean_equiv (Equiv.piSplitAt c A)]
-  have he : (fun z : A c × ((j : {j // j ≠ c}) → A j) =>
-      g ((Equiv.piSplitAt c A).symm z c)) = (fun z => g z.1) := by
-    funext z
-    simp [Equiv.piSplitAt]
-  rw [he,finiteMean_prod_left]
-
-lemma finiteMean_eq_indicator {Ω : Type*} [Fintype Ω] [DecidableEq Ω] (a : Ω) :
-    finiteMean (fun ω : Ω => if ω = a then (1:ℝ) else 0) = 1 / Fintype.card Ω := by
-  simp [finiteMean]
-
-lemma exp_finiteMean_le {Ω : Type*} [Fintype Ω] [Nonempty Ω] (f : Ω → ℝ) :
-    Real.exp (finiteMean f) ≤ finiteMean (fun ω => Real.exp (f ω)) := by
-  have hn : (Fintype.card Ω : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
-  have hw : ∑ ω : Ω, (1 / (Fintype.card Ω : ℝ)) = 1 := by simp [hn]
-  have hj := convexOn_exp.map_sum_le (t := Finset.univ)
-    (w := fun _ : Ω => 1 / (Fintype.card Ω : ℝ)) (p := f)
-    (fun _ _ => by positivity) hw (fun _ _ => Set.mem_univ _)
-  simp only [smul_eq_mul, ← Finset.mul_sum] at hj
-  simpa only [finiteMean, div_eq_mul_inv, one_div, mul_comm, mul_one, one_mul] using hj
-
-variable {d A : ℕ}
-
-def cycleFiber (o : Card d → Option (Fin A)) (i : Fin A) : Finset (Card d) := by
-  classical
-  exact Finset.univ.filter (fun x => o x = some i)
-
-@[simp] lemma mem_cycleFiber (o : Card d → Option (Fin A)) (i : Fin A) (x : Card d) :
-    x ∈ cycleFiber o i ↔ o x = some i := by
-  classical
-  simp [cycleFiber]
-
-abbrev Terminal (o : Card d → Option (Fin A)) := ∀ i : Fin A, {x : Card d // o x = some i}
-
-noncomputable def slotWeight (o : Card d → Option (Fin A)) (i : Fin A) : ℝ :=
-  1 / (cycleFiber o i).card
-
-noncomputable def pathWeight (o : Card d → Option (Fin A)) (x : Card d) : ℝ :=
-  match o x with
-  | none => 0
-  | some i => slotWeight o i
-
-noncomputable def withinWeight (o : Card d → Option (Fin A)) (u v : Card d) : ℝ :=
-  match o u, o v with
-  | some i, some j => if i = j then slotWeight o i else 0
-  | _, _ => 0
-
-end Thorp
-
-end ThorpNine.HighTail
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.HighTail.Coins
+
+namespace OAI
+
+namespace ThorpNine.HighTail
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+
+lemma card_routeDomain (d : ℕ) (x y : Card d) : (routeDomain d x y).card = d := by
+  classical
+  induction d with
+  | zero => rfl
+  | succ d ih =>
+    change (insert (Sum.inl (Fin.tail y))
+      ((routeDomain d (Fin.tail x) (Fin.tail y)).image
+        (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) :
+      Finset (Sum (Card d) (Bool × SwitchIndex d))).card = d + 1
+    rw [Finset.card_insert_of_notMem (by simp)]
+    rw [Finset.card_image_of_injective _ (by intro i j h; cases h; rfl), ih]
+
+lemma xor_eq_iff (a b c : Bool) : Bool.xor a b = c ↔ b = Bool.xor a c := by
+  cases a <;> cases b <;> cases c <;> decide
+
+lemma butterfly_endpoint_iff (d : ℕ) (ω : SwitchIndex d → Bool) (x y : Card d) :
+    butterflyPerm d (decodeButterfly d ω) x = y ↔
+      ∀ i ∈ routeDomain d x y, ω i = routeValue d x y i := by
+  classical
+  induction d with
+  | zero =>
+    have he : butterflyPerm 0 (decodeButterfly 0 ω) x = y := Subsingleton.elim _ _
+    simp [he, routeDomain]
+  | succ d ih =>
+    let v := butterflyPerm d (decodeButterfly d (fun i => ω (Sum.inr (x 0,i)))) (Fin.tail x)
+    change Fin.cons (Bool.xor (x 0) (ω (Sum.inl v))) v = y ↔ _
+    have hv : v = Fin.tail y ↔
+        ∀ i ∈ routeDomain d (Fin.tail x) (Fin.tail y),
+          ω (Sum.inr (x 0,i)) = routeValue d (Fin.tail x) (Fin.tail y) i :=
+      ih _ _ _
+    constructor
+    · intro h i hi
+      have ht : v = Fin.tail y := by simpa using congrArg Fin.tail h
+      have hh : ω (Sum.inl (Fin.tail y)) = Bool.xor (x 0) (y 0) := by
+        have he := congrFun h 0
+        simp only [Fin.cons_zero, ht] at he
+        exact (xor_eq_iff _ _ _).mp he
+      rcases Finset.mem_insert.mp hi with htop | hchild
+      · subst i
+        exact hh
+      · obtain ⟨j, hj, rfl⟩ := Finset.mem_image.mp hchild
+        exact hv.mp ht j hj
+    · intro h
+      have hh := h (Sum.inl (Fin.tail y)) (Finset.mem_insert_self _ _)
+      have ht : v = Fin.tail y := hv.mpr (fun i hi =>
+        h (Sum.inr (x 0,i)) (Finset.mem_insert_of_mem (Finset.mem_image_of_mem _ hi)))
+      have hh' : Bool.xor (x 0) (ω (Sum.inl (Fin.tail y))) = y 0 :=
+        (xor_eq_iff _ _ _).mpr hh
+      rw [ht, hh', Fin.cons_self_tail]
+
+lemma finiteMean_congr {Ω : Type*} [Fintype Ω] {f g : Ω → ℝ}
+    (h : ∀ ω, f ω = g ω) : finiteMean f = finiteMean g := by
+  unfold finiteMean
+  congr 1
+  exact Finset.sum_congr rfl (fun ω _ => h ω)
+
+lemma mean_coinCylinder {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (s : Finset ι) (v : ι → Bool) :
+    finiteMean (fun ω : ι → Bool => if ∀ i ∈ s, ω i = v i then (1 : ℝ) else 0) =
+      1 / (2 : ℝ) ^ s.card := by
+  induction s using Finset.induction_on with
+  | empty => simp [finiteMean]
+  | @insert i s hi ih =>
+    let f := fun ω : ι → Bool => if ∀ j ∈ s, ω j = v j then (1 : ℝ) else 0
+    have hf : ∀ ω, f (QueryTree.flipCoin i ω) = f ω := by
+      intro ω
+      have he : (∀ j ∈ s, QueryTree.flipCoin i ω j = v j) ↔ ∀ j ∈ s, ω j = v j := by
+        apply forall₂_congr
+        intro j hj
+        rw [QueryTree.flipCoin_apply_ne i j (fun h => hi (h ▸ hj))]
+      dsimp [f]
+      simp only [he]
+    have hm := QueryTree.mean_indicator_bit f i (v i) hf
+    have he : (fun ω : ι → Bool => if ∀ j ∈ insert i s, ω j = v j then (1 : ℝ) else 0) =
+        (fun ω => if ω i = v i then f ω else 0) := by
+      funext ω
+      by_cases h : ω i = v i <;> simp [f, h]
+    rw [he, hm]
+    change finiteMean (fun ω : ι → Bool => if ∀ j ∈ s, ω j = v j then (1 : ℝ) else 0) / 2 = _
+    rw [ih, Finset.card_insert_of_notMem hi, pow_succ]
+    ring
+
+def exposedCoins {ι : Type*} [DecidableEq ι] (R : Finset ι) (η ω : ι → Bool) : ι → Bool :=
+  fun i => if i ∈ R then η i else ω i
+
+lemma mean_coinCylinder_exposed {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (s R : Finset ι) (v η : ι → Bool) :
+    finiteMean (fun ω : ι → Bool =>
+      if ∀ i ∈ s, exposedCoins R η ω i = v i then (1 : ℝ) else 0) =
+    if (∀ i ∈ s ∩ R, η i = v i) then 1 / (2 : ℝ) ^ (s \ R).card else 0 := by
+  have he (ω : ι → Bool) : (∀ i ∈ s, exposedCoins R η ω i = v i) ↔
+      (∀ i ∈ s ∩ R, η i = v i) ∧ (∀ i ∈ s \ R, ω i = v i) := by
+    constructor
+    · intro h
+      constructor
+      · intro i hi
+        simpa [exposedCoins, (Finset.mem_inter.mp hi).2] using h i (Finset.mem_inter.mp hi).1
+      · intro i hi
+        simpa [exposedCoins, (Finset.mem_sdiff.mp hi).2] using h i (Finset.mem_sdiff.mp hi).1
+    · rintro ⟨ho, hf⟩ i hi
+      by_cases hR : i ∈ R
+      · simpa [exposedCoins, hR] using ho i (Finset.mem_inter.mpr ⟨hi,hR⟩)
+      · simpa [exposedCoins, hR] using hf i (Finset.mem_sdiff.mpr ⟨hi,hR⟩)
+  have hm : finiteMean (fun ω : ι → Bool =>
+      if ∀ i ∈ s, exposedCoins R η ω i = v i then (1 : ℝ) else 0) =
+      finiteMean (fun ω : ι → Bool =>
+        if (∀ i ∈ s ∩ R, η i = v i) ∧ (∀ i ∈ s \ R, ω i = v i) then (1 : ℝ) else 0) :=
+    finiteMean_congr (fun ω => by simp only [he ω])
+  rw [hm]
+  by_cases h : ∀ i ∈ s ∩ R, η i = v i
+  · have hp : (∀ i ∈ s ∩ R, η i = v i) ↔ True := iff_true_intro h
+    simp only [hp, true_and, ↓reduceIte]
+    exact mean_coinCylinder (s \ R) v
+  · have hp : (∀ i ∈ s ∩ R, η i = v i) ↔ False := iff_false_intro h
+    simp only [hp, false_and, ↓reduceIte]
+    simp [finiteMean]
+
+lemma butterfly_closure_probability (d : ℕ) (x y : Card d)
+    (R : Finset (SwitchIndex d)) (η : SwitchIndex d → Bool) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      if butterflyPerm d (decodeButterfly d (exposedCoins R η ω)) x = y then (1 : ℝ) else 0) =
+      if (∀ i ∈ routeDomain d x y ∩ R, η i = routeValue d x y i) then
+        (2 : ℝ) ^ (routeDomain d x y ∩ R).card / (2 : ℝ) ^ d else 0 := by
+  have he : finiteMean (fun ω : SwitchIndex d → Bool =>
+      if butterflyPerm d (decodeButterfly d (exposedCoins R η ω)) x = y then (1 : ℝ) else 0) =
+      finiteMean (fun ω : SwitchIndex d → Bool =>
+        if ∀ i ∈ routeDomain d x y, exposedCoins R η ω i = routeValue d x y i then (1 : ℝ) else 0) :=
+    finiteMean_congr (fun ω => by simp only [butterfly_endpoint_iff])
+  rw [he, mean_coinCylinder_exposed]
+  congr 1
+  have hc : (routeDomain d x y \ R).card + (routeDomain d x y ∩ R).card = d := by
+    rw [Finset.card_sdiff_add_card_inter, card_routeDomain]
+  have hp : (2 : ℝ) ^ d = (2 : ℝ) ^ (routeDomain d x y \ R).card *
+      (2 : ℝ) ^ (routeDomain d x y ∩ R).card := by
+    rw [← pow_add]
+    exact congrArg (fun n => (2 : ℝ) ^ n) hc.symm
+  rw [hp]
+  have hn : (2 : ℝ) ^ (routeDomain d x y ∩ R).card ≠ 0 := pow_ne_zero _ (by norm_num)
+  field_simp
+
+lemma butterfly_single_card_uniform (d : ℕ) (x y : Card d) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      if butterflyPerm d (decodeButterfly d ω) x = y then (1 : ℝ) else 0) = 1 / (2 : ℝ) ^ d := by
+  have he : finiteMean (fun ω : SwitchIndex d → Bool =>
+      if butterflyPerm d (decodeButterfly d ω) x = y then (1 : ℝ) else 0) =
+      finiteMean (fun ω : SwitchIndex d → Bool =>
+        if ∀ i ∈ routeDomain d x y, ω i = routeValue d x y i then (1 : ℝ) else 0) :=
+    finiteMean_congr (fun ω => by simp only [butterfly_endpoint_iff])
+  rw [he, mean_coinCylinder, card_routeDomain]
+
+def switchUsers : (d : ℕ) → Butterfly d → SwitchIndex d → Card d × Card d
+  | 0, _, i => nomatch i
+  | d + 1, b, Sum.inl y =>
+      (Fin.cons false ((butterflyPerm d (b.2 false)).symm y),
+       Fin.cons true ((butterflyPerm d (b.2 true)).symm y))
+  | d + 1, b, Sum.inr (ε,i) =>
+      (Fin.cons ε (switchUsers d (b.2 ε) i).1,
+       Fin.cons ε (switchUsers d (b.2 ε) i).2)
+
+lemma switchUsers_ne (d : ℕ) (b : Butterfly d) (i : SwitchIndex d) :
+    (switchUsers d b i).1 ≠ (switchUsers d b i).2 := by
+  induction d with
+  | zero => exact Empty.elim i
+  | succ d ih =>
+    cases i with
+    | inl y =>
+      intro h
+      have := congrFun h 0
+      simp only [switchUsers, Fin.cons_zero] at this
+      exact Bool.false_ne_true this
+    | inr z =>
+      intro h
+      exact ih (b.2 z.1) z.2 (by simpa only [switchUsers, Fin.tail_cons] using congrArg Fin.tail h)
+
+lemma cons_eq_iff {d : ℕ} (ε : Bool) (a : Card d) (x : Card (d + 1)) :
+    Fin.cons ε a = x ↔ ε = x 0 ∧ a = Fin.tail x := by
+  constructor
+  · intro h
+    exact ⟨by simpa only [Fin.cons_zero] using congrFun h 0,
+      by simpa only [Fin.tail_cons] using congrArg Fin.tail h⟩
+  · rintro ⟨rfl,rfl⟩
+    exact Fin.cons_self_tail x
+
+lemma mem_routeDomain_iff_user (d : ℕ) (b : Butterfly d) (x : Card d) (i : SwitchIndex d) :
+    i ∈ routeDomain d x (butterflyPerm d b x) ↔
+      (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x := by
+  classical
+  induction d with
+  | zero => exact Empty.elim i
+  | succ d ih =>
+    cases i with
+    | inl y =>
+      change Sum.inl y ∈ insert (Sum.inl (Fin.tail (butterflyPerm (d+1) b x)))
+        ((routeDomain d (Fin.tail x) (Fin.tail (butterflyPerm (d+1) b x))).image
+          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) ↔ _
+      simp only [Finset.mem_insert, Sum.inl.injEq, Finset.mem_image,
+        Sum.inr_ne_inl, and_false, exists_false, or_false]
+      change y = butterflyPerm d (b.2 (x 0)) (Fin.tail x) ↔ _
+      simp only [switchUsers, cons_eq_iff]
+      cases x 0 <;> simp [Equiv.symm_apply_eq]
+    | inr z =>
+      obtain ⟨ε,i⟩ := z
+      change Sum.inr (ε,i) ∈ insert (Sum.inl (Fin.tail (butterflyPerm (d+1) b x)))
+        ((routeDomain d (Fin.tail x) (Fin.tail (butterflyPerm (d+1) b x))).image
+          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) ↔ _
+      simp only [Finset.mem_insert, Sum.inr_ne_inl, false_or, Finset.mem_image,
+        Sum.inr.injEq, Prod.mk.injEq]
+      change (∃ j, j ∈ routeDomain d (Fin.tail x) (butterflyPerm d (b.2 (x 0)) (Fin.tail x)) ∧
+        x 0 = ε ∧ j = i) ↔ _
+      simp only [switchUsers, cons_eq_iff]
+      constructor
+      · rintro ⟨j,hj,hε,rfl⟩
+        subst ε
+        rcases (ih (b.2 (x 0)) (Fin.tail x) j).mp hj with h | h
+        · exact Or.inl ⟨rfl,h⟩
+        · exact Or.inr ⟨rfl,h⟩
+      · rintro (⟨hε,h⟩ | ⟨hε,h⟩)
+        · subst ε
+          exact ⟨i,(ih (b.2 (x 0)) (Fin.tail x) i).mpr (Or.inl h),rfl,rfl⟩
+        · subst ε
+          exact ⟨i,(ih (b.2 (x 0)) (Fin.tail x) i).mpr (Or.inr h),rfl,rfl⟩
+
+lemma card_switches_using (d : ℕ) (b : Butterfly d) (x : Card d) :
+    (Finset.univ.filter (fun i : SwitchIndex d =>
+      (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x)).card = d := by
+  classical
+  have he : Finset.univ.filter (fun i : SwitchIndex d =>
+      (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x) =
+      routeDomain d x (butterflyPerm d b x) := by
+    ext i
+    simp only [Finset.mem_filter, Finset.mem_univ, true_and, mem_routeDomain_iff_user]
+  rw [he,card_routeDomain]
+
+lemma switch_incidence_sum (d : ℕ) (b : Butterfly d) (w : Card d → ℝ) :
+    ∑ i : SwitchIndex d, (w (switchUsers d b i).1 + w (switchUsers d b i).2) =
+      (d : ℝ) * ∑ x : Card d, w x := by
+  classical
+  calc
+    _ = ∑ i : SwitchIndex d, ∑ x : Card d,
+        if (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x then w x else 0 := by
+      apply Finset.sum_congr rfl
+      intro i _
+      rw [Finset.sum_ite]
+      simp only [Finset.sum_const_zero, add_zero]
+      have he : Finset.univ.filter (fun x : Card d =>
+          (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x) =
+          {(switchUsers d b i).1, (switchUsers d b i).2} := by
+        ext x; simp [eq_comm]
+      rw [he, Finset.sum_pair (switchUsers_ne d b i)]
+    _ = ∑ x : Card d, ∑ i : SwitchIndex d,
+        if (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x then w x else 0 :=
+      Finset.sum_comm
+    _ = ∑ x : Card d, (d : ℝ) * w x := by
+      apply Finset.sum_congr rfl
+      intro x _
+      rw [Finset.sum_ite]
+      simp only [Finset.sum_const_zero, add_zero, Finset.sum_const, nsmul_eq_mul,
+        card_switches_using]
+    _ = _ := by rw [Finset.mul_sum]
+
+lemma weighted_switch_min_bound (d : ℕ) (b : Butterfly d) (w : Card d → ℝ) :
+    ∑ i : SwitchIndex d, min (w (switchUsers d b i).1) (w (switchUsers d b i).2) ≤
+      (d : ℝ) / 2 * ∑ x : Card d, w x := by
+  have h : 2 * (∑ i : SwitchIndex d,
+      min (w (switchUsers d b i).1) (w (switchUsers d b i).2)) ≤
+      ∑ i : SwitchIndex d, (w (switchUsers d b i).1 + w (switchUsers d b i).2) := by
+    rw [Finset.mul_sum]
+    apply Finset.sum_le_sum
+    intro i _
+    linarith [min_le_left (w (switchUsers d b i).1) (w (switchUsers d b i).2),
+      min_le_right (w (switchUsers d b i).1) (w (switchUsers d b i).2)]
+  rw [switch_incidence_sum] at h
+  linarith
+
+lemma mem_childMarks {d : ℕ} (s : Finset (Card (d + 1))) (ε : Bool) (x : Card d) :
+    x ∈ childMarks s ε ↔ Fin.cons ε x ∈ s := by
+  classical
+  constructor
+  · intro h
+    obtain ⟨y,hy,hxy⟩ := Finset.mem_image.mp h
+    obtain ⟨hys,hyε⟩ := Finset.mem_filter.mp hy
+    rw [← hxy, ← hyε, Fin.cons_self_tail]
+    exact hys
+  · intro h
+    exact Finset.mem_image.mpr ⟨Fin.cons ε x,
+      Finset.mem_filter.mpr ⟨h,by simp⟩,by simp⟩
+
+lemma mem_topMeetings {d : ℕ} (b : Butterfly (d + 1))
+    (s : Finset (Card (d + 1))) (y : Card d) :
+    y ∈ topMeetings b s ↔
+      (switchUsers (d+1) b (Sum.inl y)).1 ∈ s ∧
+      (switchUsers (d+1) b (Sum.inl y)).2 ∈ s := by
+  classical
+  have he (ε : Bool) : y ∈ (childMarks s ε).image (butterflyPerm d (b.2 ε)) ↔
+      (butterflyPerm d (b.2 ε)).symm y ∈ childMarks s ε := by
+    constructor
+    · intro h
+      obtain ⟨x,hx,rfl⟩ := Finset.mem_image.mp h
+      simpa only [Equiv.symm_apply_apply] using hx
+    · intro h
+      exact Finset.mem_image.mpr ⟨_,h,Equiv.apply_symm_apply _ _⟩
+  simp only [topMeetings, Finset.mem_inter, he, mem_childMarks, switchUsers]
+
+lemma encounters_eq_switch_sum (d : ℕ) (b : Butterfly d) (s : Finset (Card d)) :
+    encounters d b s = ∑ i : SwitchIndex d,
+      if (switchUsers d b i).1 ∈ s ∧ (switchUsers d b i).2 ∈ s then 1 else 0 := by
+  classical
+  induction d with
+  | zero =>
+    simp only [encounters]
+    symm
+    apply Finset.sum_eq_zero
+    intro i _
+    exact Empty.elim i
+  | succ d ih =>
+    change encounters d (b.2 false) (childMarks s false) +
+      encounters d (b.2 true) (childMarks s true) + (topMeetings b s).card =
+        ∑ i : Sum (Card d) (Bool × SwitchIndex d), _
+    rw [Fintype.sum_sum_type, Fintype.sum_prod_type]
+    simp only [Fintype.sum_bool]
+    rw [ih,ih]
+    have ht : (∑ y : Card d,
+        if (switchUsers (d+1) b (Sum.inl y)).1 ∈ s ∧
+          (switchUsers (d+1) b (Sum.inl y)).2 ∈ s then 1 else 0) =
+        (topMeetings b s).card := by
+      simp only [← mem_topMeetings]
+      simp
+    rw [ht]
+    simp only [mem_childMarks, switchUsers]
+    ac_rfl
+
+lemma marked_switch_entropy_bound (d : ℕ) (b : Butterfly d) (s : Finset (Card d)) :
+    (∑ i : SwitchIndex d,
+      if (switchUsers d b i).1 ∈ s ∧ (switchUsers d b i).2 ∈ s then (1:ℝ) else 0) ≤
+      (s.card : ℝ) * Real.log s.card / (2 * Real.log 2) := by
+  classical
+  have he : (encounters d b s : ℝ) =
+      ∑ i : SwitchIndex d,
+        if (switchUsers d b i).1 ∈ s ∧ (switchUsers d b i).2 ∈ s then (1:ℝ) else 0 := by
+    rw [encounters_eq_switch_sum, Nat.cast_sum]
+    apply Finset.sum_congr rfl
+    intro i _
+    split_ifs <;> simp
+  rw [← he]
+  exact butterfly_encounters d b s
+
+namespace QueryTree
+variable {ι : Type*} [DecidableEq ι]
+
+def memo : QueryTree ι → Finset ι → (ι → Bool) → QueryTree ι
+  | reject, _, _ => reject
+  | accept, _, _ => accept
+  | branch i zero one, R, η =>
+      if i ∈ R then (if η i then memo one R η else memo zero R η) else
+      branch i (memo zero (insert i R) (Function.update η i false))
+        (memo one (insert i R) (Function.update η i true))
+  | force i b next, R, η =>
+      if i ∈ R then (if η i = b then memo next R η else reject) else
+      force i b (memo next (insert i R) (Function.update η i b))
+
+lemma memo_queries_disjoint (T : QueryTree ι) (R : Finset ι) (η : ι → Bool) :
+    Disjoint (memo T R η).queries R := by
+  induction T generalizing R η with
+  | reject => simp [memo,queries]
+  | accept => simp [memo,queries]
+  | branch i zero one ih₀ ih₁ =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte]
+      cases η i <;> simp only [Bool.false_eq_true,↓reduceIte]
+      · exact ih₀ R η
+      · exact ih₁ R η
+    · simp only [memo,hi,↓reduceIte,queries,Finset.disjoint_insert_left,Finset.disjoint_union_left]
+      exact ⟨by simp, (ih₀ (insert i R) _).mono_right (Finset.subset_insert _ _),
+        (ih₁ (insert i R) _).mono_right (Finset.subset_insert _ _)⟩
+  | force i b next ih =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte]
+      by_cases hb : η i = b
+      · simpa only [hb,↓reduceIte] using ih R η
+      · simp [hb,queries]
+    · simp only [memo,hi,↓reduceIte,queries,Finset.disjoint_insert_left]
+      exact ⟨by simp,(ih (insert i R) _).mono_right (Finset.subset_insert _ _)⟩
+
+lemma memo_fresh (T : QueryTree ι) (R : Finset ι) (η : ι → Bool) :
+    (memo T R η).Fresh := by
+  induction T generalizing R η with
+  | reject => trivial
+  | accept => trivial
+  | branch i zero one ih₀ ih₁ =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte]
+      cases η i <;> simp only [Bool.false_eq_true,↓reduceIte]
+      · exact ih₀ R η
+      · exact ih₁ R η
+    · simp only [memo,hi,↓reduceIte,Fresh]
+      exact ⟨fun hm => Finset.disjoint_left.mp (memo_queries_disjoint zero (insert i R) _) hm
+          (Finset.mem_insert_self _ _),
+        fun hm => Finset.disjoint_left.mp (memo_queries_disjoint one (insert i R) _) hm
+          (Finset.mem_insert_self _ _), ih₀ _ _,ih₁ _ _⟩
+  | force i b next ih =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte]
+      by_cases hb : η i = b
+      · simpa only [hb,↓reduceIte] using ih R η
+      · simp [hb,Fresh]
+    · simp only [memo,hi,↓reduceIte,Fresh]
+      exact ⟨fun hm => Finset.disjoint_left.mp (memo_queries_disjoint next (insert i R) _) hm
+        (Finset.mem_insert_self _ _),ih _ _⟩
+
+lemma update_agrees {R : Finset ι} {η ω : ι → Bool}
+    (h : ∀ i ∈ R, η i = ω i) (i : ι) :
+    ∀ j ∈ insert i R, Function.update η i (ω i) j = ω j := by
+  intro j hj
+  rcases Finset.mem_insert.mp hj with rfl | hj
+  · simp
+  · by_cases hji : j = i
+    · subst j; simp
+    · rw [Function.update_of_ne hji]
+      exact h j hj
+
+lemma memo_succeeds (T : QueryTree ι) (R : Finset ι) (η ω : ι → Bool)
+    (h : ∀ i ∈ R, η i = ω i) : (memo T R η).succeeds ω = T.succeeds ω := by
+  induction T generalizing R η with
+  | reject => rfl
+  | accept => rfl
+  | branch i zero one ih₀ ih₁ =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte,succeeds,h i hi]
+      cases ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
+      · exact ih₀ R η h
+      · exact ih₁ R η h
+    · simp only [memo,hi,↓reduceIte,succeeds]
+      cases hb : ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
+      · exact ih₀ _ _ (by simpa only [hb] using update_agrees h i)
+      · exact ih₁ _ _ (by simpa only [hb] using update_agrees h i)
+  | force i b next ih =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte,h i hi]
+      by_cases hb : ω i = b
+      · simp only [hb,↓reduceIte,succeeds]
+        exact ih R η h
+      · simp [hb,succeeds]
+    · simp only [memo,hi,↓reduceIte,succeeds]
+      by_cases hb : ω i = b
+      · simp only [hb,↓reduceIte]
+        exact ih _ _ (by simpa only [hb] using update_agrees h i)
+      · simp [hb]
+
+def freshCharge : QueryTree ι → Finset ι → (ι → Bool) → ℕ
+  | reject, _, _ => 0
+  | accept, _, _ => 0
+  | branch i zero one, R, ω => if ω i then freshCharge one (insert i R) ω else freshCharge zero (insert i R) ω
+  | force i b next, R, ω => if ω i = b then
+      freshCharge next (insert i R) ω + (if i ∈ R then 0 else 1) else 0
+
+lemma memo_charge (T : QueryTree ι) (R : Finset ι) (η ω : ι → Bool)
+    (h : ∀ i ∈ R, η i = ω i) : (memo T R η).charge ω = freshCharge T R ω := by
+  induction T generalizing R η with
+  | reject => rfl
+  | accept => rfl
+  | branch i zero one ih₀ ih₁ =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte,freshCharge,h i hi,Finset.insert_eq_of_mem hi]
+      cases ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
+      · exact ih₀ R η h
+      · exact ih₁ R η h
+    · simp only [memo,hi,↓reduceIte,charge,freshCharge]
+      cases hb : ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
+      · exact ih₀ _ _ (by simpa only [hb] using update_agrees h i)
+      · exact ih₁ _ _ (by simpa only [hb] using update_agrees h i)
+  | force i b next ih =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte,h i hi,freshCharge,Finset.insert_eq_of_mem hi,Nat.add_zero]
+      by_cases hb : ω i = b
+      · simp only [hb,↓reduceIte]
+        exact ih R η h
+      · simp [hb,charge]
+    · simp only [memo,hi,↓reduceIte,charge,freshCharge]
+      by_cases hb : ω i = b
+      · simp only [hb,↓reduceIte]
+        rw [ih _ _ (by simpa only [hb] using update_agrees h i)]
+      · simp [hb]
+
+lemma mean_freshCharge_le_one [Fintype ι] (T : QueryTree ι) :
+    finiteMean (fun ω : ι → Bool =>
+      if T.succeeds ω then (2:ℝ) ^ T.freshCharge ∅ ω else 0) ≤ 1 := by
+  let η : ι → Bool := fun _ => false
+  have hm := mean_weighted_le_one (memo T ∅ η) (memo_fresh T ∅ η)
+  have he : finiteMean (memo T ∅ η).weighted =
+      finiteMean (fun ω : ι → Bool =>
+        if T.succeeds ω then (2:ℝ) ^ T.freshCharge ∅ ω else 0) := by
+    apply finiteMean_congr
+    intro ω
+    rw [weighted_eq, memo_succeeds _ _ _ _ (by simp), memo_charge _ _ _ _ (by simp)]
+  rw [he] at hm
+  exact hm
+
+
+def followPath : (d : ℕ) → (SwitchIndex d → ι) → Card d →
+    (Card d → QueryTree ι) → QueryTree ι
+  | 0, _, x, k => k x
+  | d + 1, e, x, k => followPath d (fun i => e (Sum.inr (x 0,i))) (Fin.tail x)
+      (fun y => branch (e (Sum.inl y))
+        (k (Fin.cons (Bool.xor (x 0) false) y))
+        (k (Fin.cons (Bool.xor (x 0) true) y)))
+
+omit [DecidableEq ι] in
+lemma followPath_succeeds (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
+    (k : Card d → QueryTree ι) (ω : ι → Bool) :
+    (followPath d e x k).succeeds ω =
+      (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).succeeds ω := by
+  induction d with
+  | zero => rfl
+  | succ d ih =>
+    rw [followPath, ih]
+    let y := butterflyPerm d
+      (decodeButterfly d (fun i => ω (e (Sum.inr (x 0,i))))) (Fin.tail x)
+    change (if ω (e (Sum.inl y)) then
+      (k (Fin.cons (Bool.xor (x 0) true) y)).succeeds ω else
+      (k (Fin.cons (Bool.xor (x 0) false) y)).succeeds ω) =
+      (k (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).succeeds ω
+    cases ω (e (Sum.inl y)) <;> rfl
+
+lemma followPath_charge (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
+    (k : Card d → QueryTree ι) (R : Finset ι) (ω : ι → Bool) :
+    freshCharge (followPath d e x k) R ω =
+      freshCharge (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x))
+        (R ∪ (routeDomain d x (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).image e) ω := by
+  classical
+  induction d generalizing R with
+  | zero => simp only [followPath, routeDomain, Finset.image_empty, Finset.union_empty]; rfl
+  | succ d ih =>
+    rw [followPath, ih]
+    let y := butterflyPerm d
+      (decodeButterfly d (fun i => ω (e (Sum.inr (x 0,i))))) (Fin.tail x)
+    let S := (routeDomain d (Fin.tail x) y).image (fun i => e (Sum.inr (x 0,i)))
+    change (if ω (e (Sum.inl y)) then
+      freshCharge (k (Fin.cons (Bool.xor (x 0) true) y)) (insert (e (Sum.inl y)) (R ∪ S)) ω else
+      freshCharge (k (Fin.cons (Bool.xor (x 0) false) y)) (insert (e (Sum.inl y)) (R ∪ S)) ω) =
+      freshCharge (k (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y))
+        (R ∪ (routeDomain (d+1) x
+          (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).image e) ω
+    have hs : (routeDomain (d+1) x
+        (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).image e =
+        insert (e (Sum.inl y)) S := by
+      let e' : Sum (Card d) (Bool × SwitchIndex d) → ι := e
+      change (insert (Sum.inl y)
+        ((routeDomain d (Fin.tail x) y).image
+          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) :
+          Finset (Sum (Card d) (Bool × SwitchIndex d))).image e' = _
+      rw [Finset.image_insert, Finset.image_image]
+      rfl
+    rw [hs,Finset.union_insert]
+    cases ω (e (Sum.inl y)) <;> rfl
+
+def forceList : List ι → (ι → Bool) → QueryTree ι → QueryTree ι
+  | [], _, k => k
+  | i :: L, v, k => force i (v i) (forceList L v k)
+
+omit [DecidableEq ι] in
+lemma forceList_succeeds (L : List ι) (v : ι → Bool) (k : QueryTree ι) (ω : ι → Bool) :
+    (forceList L v k).succeeds ω =
+      if ∀ i ∈ L, ω i = v i then k.succeeds ω else false := by
+  induction L with
+  | nil => simp [forceList]
+  | cons i L ih =>
+    simp only [forceList,succeeds,ih,List.mem_cons,forall_eq_or_imp]
+    by_cases h : ω i = v i <;> simp [h]
+
+lemma forceList_charge (L : List ι) (v : ι → Bool) (k : QueryTree ι)
+    (R : Finset ι) (ω : ι → Bool) (h : ∀ i ∈ L, ω i = v i) :
+    freshCharge (forceList L v k) R ω =
+      freshCharge k (R ∪ L.toFinset) ω + (L.toFinset \ R).card := by
+  induction L generalizing R with
+  | nil => simp [forceList]
+  | cons i L ih =>
+    rw [forceList,freshCharge,ite_eq_left (h i List.mem_cons_self),
+      ih (insert i R) (fun j hj => h j (List.mem_cons_of_mem _ hj))]
+    simp only [List.toFinset_cons]
+    have hs : insert i R ∪ L.toFinset = R ∪ insert i L.toFinset := by ext j; simp
+    rw [hs]
+    by_cases hi : i ∈ R
+    · rw [ite_eq_left hi]
+      have he : L.toFinset \ insert i R = insert i L.toFinset \ R := by
+        ext j
+        by_cases hji : j = i
+        · subst j; simp [hi]
+        · simp [hji]
+      rw [he,Nat.add_zero]
+    · rw [ite_eq_right hi]
+      have he : insert i L.toFinset \ R = insert i (L.toFinset \ insert i R) := by
+        ext j
+        by_cases hji : j = i
+        · subst j; simp [hi]
+        · simp [hji]
+      rw [he, Finset.card_insert_of_notMem (by simp)]
+      omega
+
+
+def staleCharge : QueryTree ι → Finset ι → (ι → Bool) → ℕ
+  | reject, _, _ => 0
+  | accept, _, _ => 0
+  | branch i zero one, R, ω => if ω i then staleCharge one (insert i R) ω
+      else staleCharge zero (insert i R) ω
+  | force i b next, R, ω => if ω i = b then
+      staleCharge next (insert i R) ω + (if i ∈ R then 1 else 0) else 0
+
+lemma fresh_add_stale (T : QueryTree ι) (R : Finset ι) (ω : ι → Bool) :
+    T.freshCharge R ω + T.staleCharge R ω = T.charge ω := by
+  induction T generalizing R with
+  | reject => rfl
+  | accept => rfl
+  | branch i zero one ih₀ ih₁ =>
+    cases hb : ω i <;> simp [freshCharge,staleCharge,charge,hb,ih₀,ih₁]
+  | force i b next ih =>
+    by_cases hb : ω i = b
+    · simp only [freshCharge,staleCharge,charge,hb,↓reduceIte]
+      have he := ih (insert i R)
+      by_cases hi : i ∈ R <;> simp only [hi,↓reduceIte] <;> omega
+    · simp [freshCharge,staleCharge,charge,hb]
+
+omit [DecidableEq ι] in
+lemma followPath_raw_charge (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
+    (k : Card d → QueryTree ι) (ω : ι → Bool) :
+    (followPath d e x k).charge ω =
+      (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).charge ω := by
+  induction d with
+  | zero => rfl
+  | succ d ih =>
+    rw [followPath, ih]
+    let y := butterflyPerm d
+      (decodeButterfly d (fun i => ω (e (Sum.inr (x 0,i))))) (Fin.tail x)
+    change (if ω (e (Sum.inl y)) then
+      (k (Fin.cons (Bool.xor (x 0) true) y)).charge ω else
+      (k (Fin.cons (Bool.xor (x 0) false) y)).charge ω) =
+      (k (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).charge ω
+    cases ω (e (Sum.inl y)) <;> rfl
+
+omit [DecidableEq ι] in
+lemma forceList_raw_charge (L : List ι) (v : ι → Bool) (k : QueryTree ι) (ω : ι → Bool)
+    (h : ∀ i ∈ L, ω i = v i) :
+    (forceList L v k).charge ω = k.charge ω + L.length := by
+  induction L with
+  | nil => simp [forceList]
+  | cons i L ih =>
+    rw [forceList,charge,ite_eq_left (h i List.mem_cons_self),
+      ih (fun j hj => h j (List.mem_cons_of_mem _ hj))]
+    simp only [List.length_cons,Nat.add_assoc]
+
+noncomputable def closePath (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d)) :
+    QueryTree (SwitchIndex d) :=
+  forceList (routeDomain d x y).toList (routeValue d x y) k
+
+lemma closePath_succeeds (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
+    (ω : SwitchIndex d → Bool) :
+    (closePath d x y k).succeeds ω =
+      if butterflyPerm d (decodeButterfly d ω) x = y then k.succeeds ω else false := by
+  unfold closePath
+  rw [forceList_succeeds]
+  simp only [Finset.mem_toList, ← butterfly_endpoint_iff]
+
+lemma closePath_raw_charge (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
+    (ω : SwitchIndex d → Bool) (h : butterflyPerm d (decodeButterfly d ω) x = y) :
+    (closePath d x y k).charge ω = k.charge ω + d := by
+  unfold closePath
+  rw [forceList_raw_charge _ _ _ _ (by simpa only [Finset.mem_toList] using
+    (butterfly_endpoint_iff d ω x y).mp h), Finset.length_toList, card_routeDomain]
+
+lemma closePath_fresh_charge (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
+    (R : Finset (SwitchIndex d)) (ω : SwitchIndex d → Bool)
+    (h : butterflyPerm d (decodeButterfly d ω) x = y) :
+    (closePath d x y k).freshCharge R ω =
+      k.freshCharge (R ∪ routeDomain d x y) ω + (routeDomain d x y \ R).card := by
+  unfold closePath
+  rw [forceList_charge _ _ _ _ _ (by simpa only [Finset.mem_toList] using
+    (butterfly_endpoint_iff d ω x y).mp h), Finset.toList_toFinset]
+
+lemma closePath_stale_charge (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
+    (R : Finset (SwitchIndex d)) (ω : SwitchIndex d → Bool)
+    (h : butterflyPerm d (decodeButterfly d ω) x = y) :
+    (closePath d x y k).staleCharge R ω =
+      k.staleCharge (R ∪ routeDomain d x y) ω + (routeDomain d x y ∩ R).card := by
+  have h₀ := fresh_add_stale (closePath d x y k) R ω
+  have h₁ := fresh_add_stale k (R ∪ routeDomain d x y) ω
+  rw [closePath_raw_charge _ _ _ _ _ h,closePath_fresh_charge _ _ _ _ _ _ h] at h₀
+  have h₂ : (routeDomain d x y \ R).card + (routeDomain d x y ∩ R).card = d := by
+    rw [Finset.card_sdiff_add_card_inter,card_routeDomain]
+  omega
+
+noncomputable def seekCycle (d : ℕ) (S : Finset (Card d)) (a : Card d) :
+    ℕ → Card d → Finset (Card d) → (Finset (Card d) → QueryTree (SwitchIndex d)) →
+      QueryTree (SwitchIndex d)
+  | 0, x, V, k => if x ∈ S ∧ x ∉ V then closePath d x a (k (insert x V)) else reject
+  | r + 1, x, V, k => if x ∈ S ∧ x ∉ V then followPath d id x
+      (fun y => seekCycle d S a r y (insert x V) k) else reject
+
+lemma seekCycle_raw_charge (d : ℕ) (S : Finset (Card d)) (a : Card d) (r : ℕ)
+    (x : Card d) (V : Finset (Card d)) (k : Finset (Card d) → QueryTree (SwitchIndex d))
+    (ω : SwitchIndex d → Bool) (K : ℕ)
+    (hK : ∀ V, (k V).succeeds ω = true → (k V).charge ω = K)
+    (hs : (seekCycle d S a r x V k).succeeds ω = true) :
+    (seekCycle d S a r x V k).charge ω = K + d := by
+  induction r generalizing x V with
+  | zero =>
+    by_cases hv : x ∈ S ∧ x ∉ V
+    · rw [seekCycle,ite_eq_left hv,closePath_succeeds] at hs
+      split_ifs at hs with he
+      · rw [seekCycle,ite_eq_left hv,closePath_raw_charge _ _ _ _ _ he,hK _ hs]
+    · simp [seekCycle,hv,succeeds] at hs
+  | succ r ih =>
+    by_cases hv : x ∈ S ∧ x ∉ V
+    · rw [seekCycle,ite_eq_left hv,followPath_succeeds] at hs
+      rw [seekCycle,ite_eq_left hv,followPath_raw_charge]
+      exact ih _ _ hs
+    · simp [seekCycle,hv,succeeds] at hs
+
+noncomputable def seekCycles (d : ℕ) (S : Finset (Card d)) :
+    List (Card d × ℕ) → Finset (Card d) → QueryTree (SwitchIndex d)
+  | [], _ => accept
+  | (a,r) :: L, V => seekCycle d S a r a V (fun V => seekCycles d S L V)
+
+lemma seekCycles_raw_charge (d : ℕ) (S : Finset (Card d)) (L : List (Card d × ℕ))
+    (V : Finset (Card d)) (ω : SwitchIndex d → Bool)
+    (hs : (seekCycles d S L V).succeeds ω = true) :
+    (seekCycles d S L V).charge ω = L.length * d := by
+  induction L generalizing V with
+  | nil => simp [seekCycles,charge]
+  | cons ar L ih =>
+    obtain ⟨a,r⟩ := ar
+    rw [seekCycles]
+    rw [seekCycle_raw_charge d S a r a V (fun V => seekCycles d S L V) ω (L.length*d)
+      (fun V hs => ih V hs) hs]
+    simp [Nat.add_mul]
+
+lemma cycles_weighted_success (d : ℕ) (S : Finset (Card d)) (L : List (Card d × ℕ)) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      if (seekCycles d S L ∅).succeeds ω then
+        ((2:ℝ)^d)^L.length / (2:ℝ)^(seekCycles d S L ∅).staleCharge ∅ ω else 0) ≤ 1 := by
+  let T := seekCycles d S L ∅
+  have hh := mean_freshCharge_le_one T
+  have he : finiteMean (fun ω : SwitchIndex d → Bool =>
+      if T.succeeds ω then (2:ℝ)^T.freshCharge ∅ ω else 0) =
+      finiteMean (fun ω : SwitchIndex d → Bool =>
+        if T.succeeds ω then ((2:ℝ)^d)^L.length / (2:ℝ)^T.staleCharge ∅ ω else 0) := by
+    apply finiteMean_congr
+    intro ω
+    cases hs : T.succeeds ω
+    · simp
+    · simp only [↓reduceIte]
+      have hc := fresh_add_stale T ∅ ω
+      rw [seekCycles_raw_charge d S L ∅ ω hs] at hc
+      have hp : ((2:ℝ)^d)^L.length = (2:ℝ)^T.freshCharge ∅ ω * (2:ℝ)^T.staleCharge ∅ ω := by
+        rw [←pow_add,hc,←pow_mul,Nat.mul_comm]
+      rw [hp,mul_div_cancel_right₀ _ (pow_ne_zero _ (by norm_num))]
+  rw [← he]
+  exact hh
+
+end QueryTree
+
+lemma finiteMean_add {Ω : Type*} [Fintype Ω] (f g : Ω → ℝ) :
+    finiteMean (fun ω => f ω + g ω) = finiteMean f + finiteMean g := by
+  simp only [finiteMean,Finset.sum_add_distrib,add_div]
+
+lemma finiteMean_mul_const {Ω : Type*} [Fintype Ω] (f : Ω → ℝ) (c : ℝ) :
+    finiteMean (fun ω => f ω * c) = finiteMean f * c := by
+  unfold finiteMean
+  rw [← Finset.sum_mul]
+  ring
+
+lemma finiteMean_sum {Ω α : Type*} [Fintype Ω] (s : Finset α) (f : α → Ω → ℝ) :
+    finiteMean (fun ω => ∑ i ∈ s, f i ω) = ∑ i ∈ s, finiteMean (f i) := by
+  simp only [finiteMean,Finset.sum_div]
+  exact Finset.sum_comm
+
+lemma finiteMean_mono {Ω : Type*} [Fintype Ω] {f g : Ω → ℝ}
+    (h : ∀ ω, f ω ≤ g ω) : finiteMean f ≤ finiteMean g := by
+  exact div_le_div_of_nonneg_right (Finset.sum_le_sum (fun ω _ => h ω)) (Nat.cast_nonneg _)
+
+lemma finiteMean_equiv {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
+    (e : Ω ≃ Ω') (f : Ω → ℝ) :
+    finiteMean f = finiteMean (fun ω => f (e.symm ω)) := by
+  unfold finiteMean
+  rw [Fintype.card_congr e]
+  congr 1
+  exact Fintype.sum_equiv e _ _ (fun ω => by simp)
+
+lemma finiteMean_prod_left {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] [Nonempty Ω']
+    (f : Ω → ℝ) : finiteMean (fun ω : Ω × Ω' => f ω.1) = finiteMean f := by
+  simp only [finiteMean,Fintype.sum_prod_type,Finset.sum_const,nsmul_eq_mul,
+    ← Finset.mul_sum,Fintype.card_prod,Nat.cast_mul,Finset.card_univ]
+  have hn : (Fintype.card Ω' : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
+  field_simp
+
+lemma finiteMean_pi_eval {C : Type*} [Fintype C] [DecidableEq C]
+    {A : C → Type*} [∀ c, Fintype (A c)] [∀ c, Nonempty (A c)]
+    (c : C) (g : A c → ℝ) :
+    finiteMean (fun z : ∀ c, A c => g (z c)) = finiteMean g := by
+  rw [finiteMean_equiv (Equiv.piSplitAt c A)]
+  have he : (fun z : A c × ((j : {j // j ≠ c}) → A j) =>
+      g ((Equiv.piSplitAt c A).symm z c)) = (fun z => g z.1) := by
+    funext z
+    simp [Equiv.piSplitAt]
+  rw [he,finiteMean_prod_left]
+
+lemma finiteMean_eq_indicator {Ω : Type*} [Fintype Ω] [DecidableEq Ω] (a : Ω) :
+    finiteMean (fun ω : Ω => if ω = a then (1:ℝ) else 0) = 1 / Fintype.card Ω := by
+  simp [finiteMean]
+
+lemma exp_finiteMean_le {Ω : Type*} [Fintype Ω] [Nonempty Ω] (f : Ω → ℝ) :
+    Real.exp (finiteMean f) ≤ finiteMean (fun ω => Real.exp (f ω)) := by
+  have hn : (Fintype.card Ω : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
+  have hw : ∑ ω : Ω, (1 / (Fintype.card Ω : ℝ)) = 1 := by simp [hn]
+  have hj := convexOn_exp.map_sum_le (t := Finset.univ)
+    (w := fun _ : Ω => 1 / (Fintype.card Ω : ℝ)) (p := f)
+    (fun _ _ => by positivity) hw (fun _ _ => Set.mem_univ _)
+  simp only [smul_eq_mul, ← Finset.mul_sum] at hj
+  simpa only [finiteMean, div_eq_mul_inv, one_div, mul_comm, mul_one, one_mul] using hj
+
+variable {d A : ℕ}
+
+def cycleFiber (o : Card d → Option (Fin A)) (i : Fin A) : Finset (Card d) := by
+  classical
+  exact Finset.univ.filter (fun x => o x = some i)
+
+@[simp] lemma mem_cycleFiber (o : Card d → Option (Fin A)) (i : Fin A) (x : Card d) :
+    x ∈ cycleFiber o i ↔ o x = some i := by
+  classical
+  simp [cycleFiber]
+
+abbrev Terminal (o : Card d → Option (Fin A)) := ∀ i : Fin A, {x : Card d // o x = some i}
+
+noncomputable def slotWeight (o : Card d → Option (Fin A)) (i : Fin A) : ℝ :=
+  1 / (cycleFiber o i).card
+
+noncomputable def pathWeight (o : Card d → Option (Fin A)) (x : Card d) : ℝ :=
+  match o x with
+  | none => 0
+  | some i => slotWeight o i
+
+noncomputable def withinWeight (o : Card d → Option (Fin A)) (u v : Card d) : ℝ :=
+  match o u, o v with
+  | some i, some j => if i = j then slotWeight o i else 0
+  | _, _ => 0
+
+end Thorp
+
+end ThorpNine.HighTail
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/HighTail/Main.lean b/lean/OAI/Probability/ThorpRouting/HighTail/Main.lean
index bce9727fd188c083d50d9e23c61a17b23587c422..a29ed4798fef0dc6fb6bdf0d73ea617ae0320d04 100644
--- a/lean/OAI/Probability/ThorpRouting/HighTail/Main.lean
+++ b/lean/OAI/Probability/ThorpRouting/HighTail/Main.lean
@@ -1,927 +1,927 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.HighTail.CutoffMass
-
-namespace OAI
-
-namespace ThorpNine.HighTail
-
-namespace Thorp
-
-lemma palindrome_slab_adapted_mul_le (d s l : ℕ) (hsd : s ≤ d) (hl : 0 < l)
-    (hs : 2*s ≤ l) {ι : Type*} [Fintype ι] (e : ι ↪ Card d)
-    (X : SwitchIndex d → Bool) (F : (SwitchIndex d → Bool) → ℝ)
-    (hF : ∀ Y, 0 ≤ F Y) (hA : LayerAdapted d s F) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun Y => F Y*q^(palindromeSlabCost l d e (Y,X))) ≤
-      finiteMean F * Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
-  obtain ⟨r,rfl⟩ := Nat.exists_eq_add_of_le hsd
-  exact adapted_mul_mean_le r s F _ hF hA _
-    (fun lo => palindrome_slab_partial_mgf r s l hl hs e X lo q hq hlog)
-
-def familyEnd (L n : ℕ) : ℕ := if n = 0 then 0 else 2*(L*4^(n-1))
-
-lemma familyEnd_two (L n : ℕ) : 2*familyEnd L n ≤ L*4^n := by
-  cases n with
-  | zero => simp [familyEnd]
-  | succ n => simp only [familyEnd,Nat.succ_ne_zero,ite_false,Nat.add_one_sub_one,pow_succ]; nlinarith
-
-lemma familyEnd_ge (L n j : ℕ) (hj : j < n) : 2*(L*4^j) ≤ familyEnd L n := by
-  cases n with
-  | zero => omega
-  | succ n =>
-    simp only [familyEnd,Nat.succ_ne_zero,ite_false,Nat.add_one_sub_one]
-    apply Nat.mul_le_mul_left
-    exact Nat.mul_le_mul_left L (Nat.pow_le_pow_right (by decide) (by omega : j ≤ n))
-
-noncomputable def palindromeFamilyCost (L n d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) : ℕ :=
-  ∑ j ∈ Finset.range n, palindromeSlabCost (L*4^j) d e ω
-
-lemma palindromeFamilyCost_adapted (L n d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) :
-    LayerAdapted d (familyEnd L n) (fun Y => palindromeFamilyCost L n d e (Y,X)) := by
-  apply layerAdapted_sum
-  intro j hj
-  exact layerAdapted_mono (palindromeSlabCost_adapted (L*4^j) d e X)
-    (familyEnd_ge L n j (Finset.mem_range.mp hj))
-
-theorem palindrome_family_mgf (L n d : ℕ) (hL : 0 < L) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) ≤
-      Real.exp (Fintype.card ι * (68*∑ j ∈ Finset.range n,
-        Real.exp (-((L*4^j:ℕ):ℝ)/64))) := by
-  induction n with
-  | zero => simp only [palindromeFamilyCost,Finset.range_zero,Finset.sum_empty,pow_zero,
-      finiteMean_const,mul_zero,Real.exp_zero,le_refl]
-  | succ n ih =>
-    let l := L*4^n
-    have hl : 0 < l := Nat.mul_pos hL (pow_pos (by decide) _)
-    have hs : 2*familyEnd L n ≤ l := familyEnd_two L n
-    have hc : ∀ Y, palindromeFamilyCost L (n+1) d e (Y,X) =
-        palindromeFamilyCost L n d e (Y,X)+palindromeSlabCost l d e (Y,X) := by
-      intro Y
-      exact Finset.sum_range_succ _ _
-    have hm : finiteMean (fun Y => q^(palindromeFamilyCost L (n+1) d e (Y,X))) ≤
-        finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) *
-          Real.exp (Fintype.card ι*(68*Real.exp (-(l:ℝ)/64))) := by
-      by_cases hld : l ≤ d
-      · simp_rw [hc,pow_add]
-        exact palindrome_slab_adapted_mul_le d (familyEnd L n) l (by omega) hl hs e X _
-          (fun _ => pow_nonneg (by linarith) _)
-          (layerAdapted_comp (palindromeFamilyCost_adapted L n d e X) (fun c => q^c))
-          q hq hlog
-      · simp_rw [hc,palindromeSlabCost_gt l d (by omega) e,Nat.add_zero]
-        exact le_mul_of_one_le_right
-          (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
-          (Real.one_le_exp_iff.mpr (by positivity))
-    refine (hm.trans (mul_le_mul_of_nonneg_right ih (Real.exp_nonneg _))).trans_eq ?_
-    rw [←Real.exp_add,Finset.sum_range_succ]
-    congr 1
-    dsimp [l]
-    ring
-
-lemma pow_four_ge (j : ℕ) : j+1 ≤ 4^j := by
-  induction j with
-  | zero => norm_num
-  | succ j ih => rw [pow_succ]; omega
-
-lemma sum_family_decay (L n : ℕ) (hL : 0 < L) :
-    (∑ j ∈ Finset.range n, Real.exp (-((L*4^j:ℕ):ℝ)/64)) ≤
-      Real.exp (-(L:ℝ)/64)/(1-Real.exp (-(1:ℝ)/64)) := by
-  let r := Real.exp (-(1:ℝ)/64)
-  have hr : 0 < r := Real.exp_pos _
-  have hr1 : r < 1 := Real.exp_lt_one_iff.mpr (by norm_num)
-  have hg : (∑ j ∈ Finset.range n, r^j) ≤ 1/(1-r) := by
-    apply (le_div_iff₀ (by linarith : (0:ℝ)<1-r)).mpr
-    have hh := geom_sum_mul r n
-    nlinarith [pow_nonneg (le_of_lt hr) n]
-  calc
-    _ ≤ ∑ j ∈ Finset.range n, Real.exp (-(L:ℝ)/64)*r^j := by
-      apply Finset.sum_le_sum
-      intro j _
-      have hj := pow_four_ge j
-      have hlj : L+j ≤ L*4^j := by nlinarith
-      have hlj' : (L:ℝ)+j ≤ (L*4^j:ℕ) := by exact_mod_cast hlj
-      rw [←Real.exp_nat_mul,←Real.exp_add]
-      exact Real.exp_le_exp.mpr (by linarith)
-    _ = Real.exp (-(L:ℝ)/64)*(∑ j ∈ Finset.range n, r^j) := by rw [Finset.mul_sum]
-    _ ≤ Real.exp (-(L:ℝ)/64)*(1/(1-r)) :=
-      mul_le_mul_of_nonneg_left hg (Real.exp_nonneg _)
-    _ = _ := by ring
-
-
-lemma palindromeLevelCost_step_add (t d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (d+1)) (ω : BenesCoins (d+1)) :
-    palindromeLevelCost t (d+1) e ω =
-      let σ := benesStepEquiv d ω
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x σ.2.1
-      let hc := PairRouting.colors_compatible x σ.2.1
-      (if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
-          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) else 0) +
-        palindromeLevelCost t d (PairRouting.childEmbedding x c hc false) σ.1.1 +
-        palindromeLevelCost t d (PairRouting.childEmbedding x c hc true) σ.1.2 := by
-  classical
-  rw [palindromeLevelCost]
-  dsimp only
-  split_ifs with h
-  · simp only [palindromeLevelCost_gt t d (by omega),Nat.add_zero]
-  · simp only [Nat.zero_add]
-
-lemma palindromeLowCost_sum (H d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) :
-    palindromeLowCost H d e ω = ∑ i ∈ Finset.range H, palindromeLevelCost (i+1) d e ω := by
-  classical
-  induction d generalizing ι with
-  | zero => simp only [palindromeLowCost,palindromeLevelCost,Finset.sum_const_zero]
-  | succ d ih =>
-    rw [palindromeLowCost]
-    simp only [palindromeLevelCost_step_add,Finset.sum_add_distrib]
-    rw [←ih,←ih]
-    congr 2
-    simp only [Nat.add_right_cancel_iff,Finset.sum_ite_eq',Finset.mem_range,Nat.succ_le_iff]
-
-lemma palindromeLowCost_slab (L d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) :
-    palindromeLowCost L d e ω + palindromeSlabCost L d e ω = palindromeLowCost (2*L) d e ω := by
-  rw [palindromeLowCost_sum,palindromeLowCost_sum,show 2*L=L+L by omega,Finset.sum_range_add]
-  congr 1
-  exact Fin.sum_univ_eq_sum_range (fun i => palindromeLevelCost (L+i+1) d e ω) L
-
-lemma palindromeCost_family (L n d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) :
-    palindromeLowCost L d e ω + palindromeFamilyCost L n d e ω +
-      palindromeFamilyCost (2*L) n d e ω = palindromeLowCost (L*4^n) d e ω := by
-  induction n with
-  | zero => simp only [palindromeFamilyCost,Finset.range_zero,Finset.sum_empty,Nat.add_zero,
-      pow_zero,mul_one]
-  | succ n ih =>
-    have he : (2*L)*4^n = 2*(L*4^n) := by ring
-    have he' : L*4^(n+1) = 2*(2*(L*4^n)) := by rw [pow_succ]; ring
-    simp only [palindromeFamilyCost,Finset.sum_range_succ] at ih ⊢
-    rw [he,he',←palindromeLowCost_slab,←palindromeLowCost_slab,←ih]
-    omega
-
-
-lemma finiteMean_mul_sq_le {Ω : Type*} [Fintype Ω] (f g : Ω → ℝ) :
-    (finiteMean (fun ω => f ω*g ω))^2 ≤ finiteMean (fun ω => (f ω)^2)*finiteMean (fun ω => (g ω)^2) := by
-  have h := Finset.sum_mul_sq_le_sq_mul_sq Finset.univ f g
-  simp only [finiteMean,div_pow,div_mul_div_comm,←pow_two]
-  exact div_le_div_of_nonneg_right h (sq_nonneg _)
-
-lemma finiteMean_mul_exp_le {Ω : Type*} [Fintype Ω] (f g : Ω → ℝ) (A B : ℝ)
-    (hf : finiteMean (fun ω => (f ω)^2) ≤ Real.exp A)
-    (hg : finiteMean (fun ω => (g ω)^2) ≤ Real.exp B) :
-    finiteMean (fun ω => f ω*g ω) ≤ Real.exp ((A+B)/2) := by
-  have hc := finiteMean_mul_sq_le f g
-  have hn : 0 ≤ finiteMean (fun ω => (g ω)^2) := finiteMean_nonneg (fun _ => sq_nonneg _)
-  have hu := mul_le_mul hf hg hn (Real.exp_nonneg _)
-  have he : Real.exp A*Real.exp B = (Real.exp ((A+B)/2))^2 := by
-    rw [←Real.exp_add,←Real.exp_nat_mul]
-    congr 1
-    push_cast
-    ring
-  rw [he] at hu
-  have hh := hc.trans hu
-  have hep := Real.exp_pos ((A+B)/2)
-  nlinarith
-
-
-noncomputable def heightDecay (H : ℕ) : ℝ :=
-  68*Real.exp (-(H:ℝ)/64)/(1-Real.exp (-(1:ℝ)/64))
-
-lemma heightDecay_nonneg (H : ℕ) : 0 ≤ heightDecay H := by
-  have h : Real.exp (-(1:ℝ)/64) < 1 := Real.exp_lt_one_iff.mpr (by norm_num)
-  apply div_nonneg (by positivity) (by linarith)
-
-lemma heightDecay_antitone : Antitone heightDecay := by
-  intro L H h
-  have hden : 0 ≤ 1-Real.exp (-(1:ℝ)/64) := by
-    have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
-    linarith
-  apply div_le_div_of_nonneg_right _ hden
-  apply mul_le_mul_of_nonneg_left _ (by norm_num)
-  apply Real.exp_le_exp.mpr
-  have h' : (L:ℝ) ≤ H := by exact_mod_cast h
-  linarith
-
-lemma palindrome_family_mgf_decay (L n d : ℕ) (hL : 0 < L) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) ≤
-      Real.exp (Fintype.card ι*heightDecay L) := by
-  apply (palindrome_family_mgf L n d hL e X q hq hlog).trans
-  apply Real.exp_le_exp.mpr
-  apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
-  have h := mul_le_mul_of_nonneg_left (sum_family_decay L n hL) (by norm_num : (0:ℝ)≤68)
-  simpa only [heightDecay,mul_div_assoc] using h
-
-theorem palindrome_high_mgf (H n d : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/500) :
-    finiteMean (fun Y => q^(palindromeFamilyCost H n d e (Y,X)+
-      palindromeFamilyCost (2*H) n d e (Y,X))) ≤ Real.exp (Fintype.card ι*heightDecay H) := by
-  have hq2 := one_le_pow₀ (n:=2) hq
-  have hlog2 : Real.log (q^2) ≤ 1/250 := by rw [Real.log_pow]; norm_num; linarith
-  have h₀ := palindrome_family_mgf_decay H n d hH e X (q^2) hq2 hlog2
-  have h₁ := palindrome_family_mgf_decay (2*H) n d (by omega) e X (q^2) hq2 hlog2
-  have h₁' := h₁.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left
-    (heightDecay_antitone (show H ≤ 2*H by omega)) (Nat.cast_nonneg _)))
-  have hh := finiteMean_mul_exp_le (fun Y => q^(palindromeFamilyCost H n d e (Y,X)))
-    (fun Y => q^(palindromeFamilyCost (2*H) n d e (Y,X)))
-    (Fintype.card ι*heightDecay H) (Fintype.card ι*heightDecay H)
-    (by simpa only [pow_right_comm] using h₀) (by simpa only [pow_right_comm] using h₁')
-  simpa only [←pow_add,add_self_div_two] using hh
-
-lemma palindrome_high_full_mgf (H n d : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/500) :
-    finiteMean (fun ω : BenesCoins d => q^(palindromeFamilyCost H n d e ω+
-      palindromeFamilyCost (2*H) n d e ω)) ≤ Real.exp (Fintype.card ι*heightDecay H) := by
-  rw [finiteMean_prod,finiteMean_comm]
-  calc
-    _ ≤ finiteMean (fun _X : SwitchIndex d → Bool => Real.exp (Fintype.card ι*heightDecay H)) := by
-      apply finiteMean_mono
-      intro X
-      exact palindrome_high_mgf H n d hH e X q hq hlog
-    _ = _ := finiteMean_const _
-
-theorem palindrome_cost_mgf_cutoff (r H : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (H+r)) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/1000)
-    (hr : (2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*(q^2)^H) ≤ 1/2) :
-    finiteMean (fun ω : BenesCoins (H+r) => q^(palindromeCost (H+r) e ω)) ≤
-      Real.exp (((2:ℝ)^r*(2*((2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*(q^2)^H))^2)+
-        Fintype.card ι*heightDecay H)/2) := by
-  let d := H+r
-  have hd : d ≤ H*4^d := by
-    have hh := pow_four_ge d
-    nlinarith
-  have hcost (ω : BenesCoins d) : palindromeCost d e ω = palindromeLowCost H d e ω+
-      (palindromeFamilyCost H d d e ω+palindromeFamilyCost (2*H) d d e ω) := by
-    have hh := palindromeCost_family H d d e ω
-    rw [palindromeLowCost_eq _ _ hd] at hh
-    simpa only [Nat.add_assoc] using hh.symm
-  have hq2 := one_le_pow₀ (n:=2) hq
-  have hlog2 : Real.log (q^2) ≤ 1/500 := by rw [Real.log_pow]; norm_num; linarith
-  have hlow := palindrome_low_mgf r H e (q^2) hq2 hr
-  have hhigh := palindrome_high_full_mgf H d d hH e (q^2) hq2 hlog2
-  have hh := finiteMean_mul_exp_le (fun ω : BenesCoins d => q^(palindromeLowCost H d e ω))
-    (fun ω : BenesCoins d => q^(palindromeFamilyCost H d d e ω+palindromeFamilyCost (2*H) d d e ω))
-    _ _ (by simpa only [pow_right_comm] using hlow) (by simpa only [pow_right_comm] using hhigh)
-  change finiteMean (fun ω : BenesCoins d => q^(palindromeCost d e ω)) ≤ _
-  simpa only [hcost,pow_add,pow_right_comm] using hh
-
-
-lemma cutoffMass_le (H d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (ω : BenesCoins d) :
-    cutoffMass H d e ω ≤ Fintype.card ι := by
-  classical
-  induction d generalizing ι with
-  | zero => exact Nat.zero_le _
-  | succ d ih =>
-    rw [cutoffMass]
-    split_ifs
-    · exact crowdedMass_le _
-    · exact (Nat.add_le_add (ih _ _) (ih _ _)).trans_eq (PairRouting.color_card_add _)
-
-lemma palindromeLowCost_le (H d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) : palindromeLowCost H d e ω ≤ H*Fintype.card ι :=
-  (palindromeLowCost_le_cutoff H d e ω).trans (Nat.mul_le_mul_left H (cutoffMass_le H d e ω))
-
-theorem palindrome_cost_mgf_coarse (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/1000) :
-    finiteMean (fun ω : BenesCoins d => q^(palindromeCost d e ω)) ≤
-      Real.exp (Fintype.card ι*(1+heightDecay 1)) := by
-  have hd : d ≤ 1*4^d := by have hh := pow_four_ge d; omega
-  have hcost (ω : BenesCoins d) : palindromeCost d e ω = palindromeLowCost 1 d e ω+
-      (palindromeFamilyCost 1 d d e ω+palindromeFamilyCost 2 d d e ω) := by
-    have hh := palindromeCost_family 1 d d e ω
-    rw [palindromeLowCost_eq _ _ hd] at hh
-    simpa only [Nat.add_assoc] using hh.symm
-  have hq2 := one_le_pow₀ (n:=2) hq
-  have hlog2 : Real.log (q^2) ≤ 1/500 := by rw [Real.log_pow]; norm_num; linarith
-  have hlow : finiteMean (fun ω : BenesCoins d => (q^2)^(palindromeLowCost 1 d e ω)) ≤
-      Real.exp (Fintype.card ι) := by
-    calc
-      _ ≤ finiteMean (fun _ω : BenesCoins d => (q^2)^(Fintype.card ι)) := by
-        apply finiteMean_mono
-        intro ω
-        apply pow_le_pow_right₀ hq2
-        simpa only [one_mul] using palindromeLowCost_le 1 d e ω
-      _ = (q^2)^(Fintype.card ι) := finiteMean_const _
-      _ = Real.exp (Fintype.card ι * Real.log (q^2)) := by
-        rw [Real.exp_nat_mul,Real.exp_log (by positivity : 0<q^2)]
-      _ ≤ _ := Real.exp_le_exp.mpr (by
-        have hh := mul_le_mul_of_nonneg_left hlog2 (Nat.cast_nonneg (Fintype.card ι) : (0:ℝ)≤_)
-        nlinarith [Nat.cast_nonneg (α:=ℝ) (Fintype.card ι)])
-  have hhigh := palindrome_high_full_mgf 1 d d (by decide) e (q^2) hq2 hlog2
-  have hh := finiteMean_mul_exp_le (fun ω : BenesCoins d => q^(palindromeLowCost 1 d e ω))
-    (fun ω : BenesCoins d => q^(palindromeFamilyCost 1 d d e ω+palindromeFamilyCost 2 d d e ω))
-    (Fintype.card ι) (Fintype.card ι*heightDecay 1)
-    (by simpa only [pow_right_comm] using hlow) (by simpa only [pow_right_comm] using hhigh)
-  simp only [←pow_add,←hcost] at hh
-  apply hh.trans (Real.exp_le_exp.mpr _)
-  have hk : (0:ℝ) ≤ Fintype.card ι := Nat.cast_nonneg _
-  have hD := heightDecay_nonneg 1
-  nlinarith
-
-noncomputable def momentBase : ℝ := (2:ℝ)^(1/1000:ℝ)
-
-lemma momentBase_one_le : 1 ≤ momentBase := by
-  exact Real.one_le_rpow (by norm_num) (by norm_num)
-
-lemma momentBase_log : Real.log momentBase ≤ 1/1000 := by
-  rw [momentBase,Real.log_rpow (by norm_num)]
-  have hh := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
-  norm_num at hh ⊢
-  linarith
-
-lemma momentBase_pow (c : ℕ) : momentBase^c = (2:ℝ)^((c:ℝ)*(1/1000:ℝ)) := by
-  rw [mul_comm,Real.rpow_mul_natCast (by norm_num)]
-  rfl
-
-lemma palindromeRowMoment_le_cost (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) :
-    palindromeRowMoment d e (1/1000) ≤
-      finiteMean (fun ω : BenesCoins d => momentBase^(palindromeCost d e ω)) := by
-  have hn : (0:ℝ) < ((2^d:ℕ):ℝ)^(Fintype.card ι) := by positivity
-  have hn0 : (0:ℝ) ≤ ((2^d).descFactorial (Fintype.card ι):ℝ) /
-      ((2^d:ℕ):ℝ)^(Fintype.card ι) := by positivity
-  have hn1 : ((2^d).descFactorial (Fintype.card ι):ℝ) /
-      ((2^d:ℕ):ℝ)^(Fintype.card ι) ≤ 1 := by
-    apply (div_le_one hn).mpr
-    exact_mod_cast Nat.descFactorial_le_pow (2^d) (Fintype.card ι)
-  have hpow := Real.rpow_le_one hn0 hn1 (by norm_num : (0:ℝ)≤1/1000)
-  have hm := palindrome_row_density_moment d e (1/1000) (by norm_num)
-  simp_rw [←momentBase_pow] at hm
-  exact hm.trans (mul_le_of_le_one_left (finiteMean_nonneg (fun _ =>
-    pow_nonneg (le_trans (by norm_num) momentBase_one_le) _)) hpow)
-
-theorem palindrome_row_moment_coarse (d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) :
-    palindromeRowMoment d e (1/1000) ≤ Real.exp (Fintype.card ι*(1+heightDecay 1)) :=
-  (palindromeRowMoment_le_cost d e).trans
-    (palindrome_cost_mgf_coarse d e momentBase momentBase_one_le momentBase_log)
-
-
-noncomputable def densityCutoff (p : ℝ) : ℕ := ⌊-Real.log p/(4*Real.log 2)⌋₊
-
-lemma log_small_density {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
-    Real.log p ≤ -8*Real.log 2 := by
-  have he : Real.log (1/256:ℝ) = -8*Real.log 2 := by
-    rw [show (1/256:ℝ) = 1/2^8 by norm_num,
-      Real.log_div (by norm_num) (by positivity),Real.log_one,Real.log_pow]
-    ring
-  exact (Real.log_le_log hp hs).trans_eq he
-
-lemma densityCutoff_bounds {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
-    0 < densityCutoff p ∧
-    (densityCutoff p:ℝ)*(4*Real.log 2) ≤ -Real.log p ∧
-    -Real.log p < ((densityCutoff p:ℝ)+1)*(4*Real.log 2) := by
-  have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
-  have hd : 0 < 4*Real.log 2 := by positivity
-  have hh := log_small_density hp hs
-  have hx : 1 ≤ -Real.log p/(4*Real.log 2) := by
-    apply (le_div_iff₀ hd).mpr
-    nlinarith
-  refine ⟨Nat.floor_pos.mpr hx,?_,?_⟩
-  · exact (le_div_iff₀ hd).mp (Nat.floor_le (by linarith : 0 ≤ -Real.log p/(4*Real.log 2)))
-  · exact (div_lt_iff₀ hd).mp (Nat.lt_floor_add_one (-Real.log p/(4*Real.log 2)))
-
-lemma densityCutoff_decay {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
-    Real.exp (-(densityCutoff p:ℝ)/64) ≤ Real.exp (1/64) * p^(1/512:ℝ) := by
-  have hh := (densityCutoff_bounds hp hs).2.2
-  have hb : Real.log 2 ≤ 2 := by
-    have := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
-    linarith
-  have hH : 0 ≤ (densityCutoff p:ℝ)+1 := by positivity
-  have hu := mul_le_mul_of_nonneg_left hb hH
-  rw [Real.rpow_def_of_pos hp,←Real.exp_add]
-  apply Real.exp_le_exp.mpr
-  nlinarith
-
-lemma densityCutoff_power {p q : ℝ} (hp : 0 < p) (hs : p ≤ 1/256)
-    (hq : 0 < q) (hlog : Real.log q ≤ Real.log 2/4) (n : ℕ) (hn : n ≤ 4)
-    (hq1 : 1 ≤ q) :
-    p*(2:ℝ)^(densityCutoff p)*q^(n*densityCutoff p) ≤ p^(1/2:ℝ) := by
-  have hcut := (densityCutoff_bounds hp hs).2.1
-  have hH : 0 ≤ (densityCutoff p:ℝ) := Nat.cast_nonneg _
-  have hn' : (n:ℝ) ≤ 4 := by exact_mod_cast hn
-  have hln := Real.log_nonneg hq1
-  have hl := mul_le_mul_of_nonneg_right hn' hln
-  have hnl : (n:ℝ)*Real.log q ≤ Real.log 2 := by linarith
-  have hm := mul_le_mul_of_nonneg_left hnl hH
-  have heq : p*(2:ℝ)^(densityCutoff p)*q^(n*densityCutoff p) =
-      Real.exp (Real.log p+(densityCutoff p:ℝ)*Real.log 2+
-        (n*densityCutoff p:ℕ)*Real.log q) := by
-    rw [Real.exp_add,Real.exp_add,Real.exp_nat_mul,Real.exp_nat_mul,
-      Real.exp_log hp,Real.exp_log hq,Real.exp_log (by norm_num : (0:ℝ)<2)]
-  rw [heq,Real.rpow_def_of_pos hp]
-  apply Real.exp_le_exp.mpr
-  push_cast
-  nlinarith
-
-lemma densityCutoff_sparse {p q : ℝ} (hp : 0 < p) (hs : p ≤ 1/256)
-    (hq : 0 < q) (hlog : Real.log q ≤ Real.log 2/4) (hq1 : 1 ≤ q) :
-    (2:ℝ)^(densityCutoff p)*(p*(q^2)^(densityCutoff p)) ≤ 1/2 := by
-  have hh := densityCutoff_power hp hs hq hlog 2 (by decide) hq1
-  rw [←pow_mul] at ⊢
-  have hle : p^(1/2:ℝ) ≤ 1/2 := by
-    have hlogp := log_small_density hp hs
-    have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
-    rw [Real.rpow_def_of_pos hp]
-    calc
-      _ ≤ Real.exp (-Real.log 2) := by apply Real.exp_le_exp.mpr; nlinarith
-      _ = 1/2 := by rw [Real.exp_neg,Real.exp_log (by norm_num : (0:ℝ)<2)]; norm_num
-  nlinarith [hh.trans hle]
-
-
-lemma momentBase_log_exact : Real.log momentBase = Real.log 2/1000 := by
-  rw [momentBase,Real.log_rpow (by norm_num)]
-  ring
-
-lemma densityCutoff_le_height (d k : ℕ) (hk : 0 < k) :
-    densityCutoff ((k:ℝ)/(2:ℝ)^d) ≤ d := by
-  have hk' : (1:ℝ) ≤ k := by exact_mod_cast hk
-  have hp : 0 < (k:ℝ)/(2:ℝ)^d := by positivity
-  have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
-  have hlogk := Real.log_nonneg hk'
-  by_cases hx : 0 ≤ -Real.log ((k:ℝ)/(2:ℝ)^d)/(4*Real.log 2)
-  · have hh := (le_div_iff₀ (by positivity : 0 < 4*Real.log 2)).mp (Nat.floor_le hx)
-    have hd : (0:ℝ) ≤ d := Nat.cast_nonneg _
-    have hH : (0:ℝ) ≤ densityCutoff ((k:ℝ)/(2:ℝ)^d) := Nat.cast_nonneg _
-    change (densityCutoff ((k:ℝ)/(2:ℝ)^d):ℝ)*(4*Real.log 2) ≤ -Real.log ((k:ℝ)/(2:ℝ)^d) at hh
-    rw [Real.log_div (by positivity) (by positivity),Real.log_pow] at hh
-    have hle : (densityCutoff ((k:ℝ)/(2:ℝ)^d):ℝ) ≤ d := by nlinarith
-    exact_mod_cast hle
-  · have hz := Nat.floor_of_nonpos (le_of_not_ge hx)
-    simp only [densityCutoff,hz,Nat.zero_le]
-
-noncomputable def densityTailConstant : ℝ :=
-  68*Real.exp (1/64)/(1-Real.exp (-(1:ℝ)/64))
-
-lemma densityTailConstant_nonneg : 0 ≤ densityTailConstant := by
-  have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
-  apply div_nonneg (by positivity) (by linarith)
-
-lemma heightDecay_cutoff {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
-    heightDecay (densityCutoff p) ≤ densityTailConstant*p^(1/512:ℝ) := by
-  have hden : 0 ≤ 1-Real.exp (-(1:ℝ)/64) := by
-    have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
-    linarith
-  have hh := div_le_div_of_nonneg_right
-    (mul_le_mul_of_nonneg_left (densityCutoff_decay hp hs) (by norm_num : (0:ℝ)≤68)) hden
-  simpa only [heightDecay,densityTailConstant,mul_assoc,div_mul_eq_mul_div] using hh
-
-lemma sparse_low_algebra (r H k : ℕ) (q : ℝ) :
-    (2:ℝ)^r*(2*((2:ℝ)^H*(((k:ℝ)/(2:ℝ)^(H+r))*(q^2)^H))^2) =
-    2*k*(((k:ℝ)/(2:ℝ)^(H+r))*(2:ℝ)^H*q^(4*H)) := by
-  rw [pow_add,←pow_mul,show 4*H = (2*H)*2 by omega,pow_mul]
-  have hH : (2:ℝ)^H ≠ 0 := by positivity
-  have hr : (2:ℝ)^r ≠ 0 := by positivity
-  field_simp
-  ring
-
-lemma palindrome_row_moment_small (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι)
-    (hs : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1/256) :
-    palindromeRowMoment d e (1/1000) ≤
-      Real.exp (Fintype.card ι*(1+densityTailConstant)*
-        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ)) := by
-  generalize hpdef : ((Fintype.card ι:ℝ)/(2:ℝ)^d) = p at hs ⊢
-  have hp : 0 < p := by rw [←hpdef]; positivity
-  have hle := densityCutoff_le_height d (Fintype.card ι) hk
-  rw [hpdef] at hle
-  obtain ⟨r,hr⟩ := Nat.exists_eq_add_of_le hle
-  subst d
-  have hH := (densityCutoff_bounds hp hs).1
-  have hq := momentBase_one_le
-  have hqp : 0 < momentBase := by linarith
-  have hlog : Real.log momentBase ≤ Real.log 2/4 := by
-    rw [momentBase_log_exact]
-    have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
-    linarith
-  have hsp := densityCutoff_sparse hp hs hqp hlog hq
-  rw [←hpdef] at hsp
-  have hh := palindrome_cost_mgf_cutoff r (densityCutoff p) hH e momentBase hq momentBase_log
-    (by simpa only [hpdef] using hsp)
-  apply (palindromeRowMoment_le_cost _ e).trans (hh.trans (Real.exp_le_exp.mpr _))
-  rw [sparse_low_algebra,hpdef]
-  have hlow := densityCutoff_power hp hs hqp hlog 4 (by decide) hq
-  have hp1 : p ≤ 1 := by linarith
-  have he := Real.rpow_le_rpow_of_exponent_ge hp hp1 (by norm_num : (1:ℝ)/512 ≤ 1/2)
-  have hlow' := mul_le_mul_of_nonneg_left (hlow.trans he)
-    (by positivity : (0:ℝ)≤2*Fintype.card ι)
-  have hhigh := mul_le_mul_of_nonneg_left (heightDecay_cutoff hp hs)
-    (Nat.cast_nonneg (Fintype.card ι): (0:ℝ)≤_)
-  have hnon : 0 ≤ Fintype.card ι*densityTailConstant*p^(1/512:ℝ) :=
-    mul_nonneg (mul_nonneg (Nat.cast_nonneg _) densityTailConstant_nonneg)
-      (Real.rpow_nonneg (le_of_lt hp) _)
-  nlinarith
-
-noncomputable def densityConstant : ℝ :=
-  256*(1+heightDecay 1)+1+densityTailConstant
-
-lemma densityConstant_pos : 0 < densityConstant := by
-  dsimp [densityConstant]
-  have hh := heightDecay_nonneg 1
-  have ht := densityTailConstant_nonneg
-  linarith
-
-theorem palindrome_row_moment_bound (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι) :
-    palindromeRowMoment d e (1/1000) ≤ Real.exp (densityConstant*Fintype.card ι*
-        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ)) := by
-  have hp : 0 < (Fintype.card ι:ℝ)/(2:ℝ)^d := by positivity
-  have hp1 : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1 := by
-    apply (div_le_one (by positivity)).mpr
-    have h := Fintype.card_le_of_injective e e.injective
-    rw [card_positions] at h
-    exact_mod_cast h
-  have hk0 : (0:ℝ) ≤ Fintype.card ι := Nat.cast_nonneg _
-  have hpow := Real.rpow_nonneg (le_of_lt hp) (1/512:ℝ)
-  by_cases hs : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1/256
-  · apply (palindrome_row_moment_small d e hk hs).trans (Real.exp_le_exp.mpr _)
-    have hD : 1+densityTailConstant ≤ densityConstant := by
-      dsimp [densityConstant]
-      have := heightDecay_nonneg 1
-      linarith
-    have hh := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hD hk0) hpow
-    nlinarith
-  · have hlow : (1:ℝ)/256 ≤ ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ) := by
-      have he := Real.rpow_le_rpow_of_exponent_ge hp hp1 (by norm_num : (1:ℝ)/512 ≤ 1)
-      rw [Real.rpow_one] at he
-      exact (le_of_not_ge hs).trans he
-    have hD : 256*(1+heightDecay 1) ≤ densityConstant := by
-      dsimp [densityConstant]
-      have := densityTailConstant_nonneg
-      linarith
-    apply (palindrome_row_moment_coarse d e).trans (Real.exp_le_exp.mpr _)
-    have hD0 : 0 ≤ 1+heightDecay 1 := by have := heightDecay_nonneg 1; linarith
-    have ha := mul_le_mul_of_nonneg_left hlow (mul_nonneg (by norm_num : (0:ℝ)≤256) hD0)
-    have hb := mul_le_mul_of_nonneg_right hD hpow
-    have hc := mul_le_mul_of_nonneg_right (ha.trans hb) hk0
-    nlinarith
-
-lemma palindromeRowMoment_nonneg (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (a : ℝ) : 0 ≤ palindromeRowMoment d e a := by
-  apply finiteMean_nonneg
-  intro f
-  exact Real.rpow_nonneg (mul_nonneg (Nat.cast_nonneg _)
-    (PairRouting.tupleProbability_nonneg _ _ _)) _
-
-theorem palindrome_log_row_moment (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι) :
-    Real.log (palindromeRowMoment d e (1/1000)) ≤ densityConstant*Fintype.card ι*
-        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ) := by
-  by_cases hz : palindromeRowMoment d e (1/1000) = 0
-  · rw [hz,Real.log_zero]
-    exact mul_nonneg (mul_nonneg (le_of_lt densityConstant_pos) (Nat.cast_nonneg _))
-      (Real.rpow_nonneg (by positivity) _)
-  · exact (Real.log_le_iff_le_exp (lt_of_le_of_ne (palindromeRowMoment_nonneg _ _ _) (Ne.symm hz))).mpr
-      (palindrome_row_moment_bound d e hk)
-
-end Thorp
-
-
-namespace Thorp.HighHeight
-open scoped BigOperators Classical
-
-lemma family_weighted (L n d s : ℕ) (hL : 0<L) (hs : 2*s≤L)
-    {ι : Type*} [Fintype ι] (x : ι ↪ Card d) (X : SwitchIndex d → Bool)
-    (F : (SwitchIndex d → Bool) → ℝ) (hF : ∀ Y, 0≤F Y) (hA : LayerAdapted d s F)
-    (q : ℝ) (hq : 1≤q) (hlog : Real.log q≤1/250) :
-    finiteMean (fun Y => F Y*q^(palindromeFamilyCost L n d x (Y,X))) ≤
-      finiteMean F*Real.exp (Fintype.card ι*(68*∑ j ∈ Finset.range n,
-        Real.exp (-((L*4^j:ℕ):ℝ)/64))) := by
-  induction n with
-  | zero =>
-    simp only [palindromeFamilyCost,Finset.range_zero,Finset.sum_empty,pow_zero,
-      mul_one,mul_zero,Real.exp_zero]
-    exact le_rfl
-  | succ n ih =>
-    let l := L*4^n
-    have hl : 0<l := Nat.mul_pos hL (pow_pos (by decide) _)
-    have hLl : L≤l := by dsimp [l]; exact Nat.le_mul_of_pos_right L (by positivity)
-    have hs' : 2*max s (familyEnd L n)≤l := by
-      rcases le_total s (familyEnd L n) with h | h
-      · simpa only [max_eq_right h] using familyEnd_two L n
-      · simpa only [max_eq_left h] using hs.trans hLl
-    have hc (Y) : palindromeFamilyCost L (n+1) d x (Y,X)=
-        palindromeFamilyCost L n d x (Y,X)+palindromeSlabCost l d x (Y,X) :=
-      Finset.sum_range_succ _ _
-    have hwt (Y) : 0≤F Y*q^(palindromeFamilyCost L n d x (Y,X)) :=
-      mul_nonneg (hF Y) (pow_nonneg (by linarith) _)
-    have had : LayerAdapted d (max s (familyEnd L n))
-        (fun Y => F Y*q^(palindromeFamilyCost L n d x (Y,X))) := by
-      intro Y Y' hh
-      dsimp only
-      rw [hA Y Y' (agreeThrough_mono (le_max_left _ _) hh)]
-      congr 2
-      exact palindromeFamilyCost_adapted L n d x X Y Y' (agreeThrough_mono (le_max_right _ _) hh)
-    have hm : finiteMean (fun Y => F Y*q^(palindromeFamilyCost L (n+1) d x (Y,X))) ≤
-        finiteMean (fun Y => F Y*q^(palindromeFamilyCost L n d x (Y,X)))*
-          Real.exp (Fintype.card ι*(68*Real.exp (-(l:ℝ)/64))) := by
-      by_cases hld : l≤d
-      · simp_rw [hc,pow_add,←mul_assoc]
-        simpa only [mul_assoc] using palindrome_slab_adapted_mul_le d (max s (familyEnd L n)) l (by omega)
-          hl hs' x X _ hwt had q hq hlog
-      · simp_rw [hc,palindromeSlabCost_gt l d (by omega) x,Nat.add_zero]
-        exact le_mul_of_one_le_right (finiteMean_nonneg hwt)
-          (Real.one_le_exp_iff.mpr (by positivity))
-    refine (hm.trans (mul_le_mul_of_nonneg_right ih (Real.exp_nonneg _))).trans_eq ?_
-    rw [mul_assoc,←Real.exp_add,Finset.sum_range_succ]
-    congr 2
-    dsimp [l]
-    ring
-
-lemma low_switch_height (r s : ℕ) (j : Card r × SwitchIndex s) :
-    switchHeight (s+r) ((splitSwitch r s).symm (Sum.inl j)) ≤ s := by
-  induction r with
-  | zero => exact switchHeight_le s j.2
-  | succ r ih =>
-    exact ih (Fin.tail j.1,j.2)
-
-def lowCoins (r s : ℕ) (Y : SwitchIndex (s+r) → Bool) :
-    Card r × SwitchIndex s → Bool :=
-  fun j => Y ((splitSwitch r s).symm (Sum.inl j))
-
-lemma lowCoins_adapted (r s : ℕ) : LayerAdapted (s+r) s (lowCoins r s) := by
-  intro Y Y' hh
-  funext j
-  exact hh _ (low_switch_height r s j)
-
-lemma lowCoins_assemble (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
-    (hi : Card s × SwitchIndex r → Bool) : lowCoins r s (assembleBits r s lo hi)=lo := by
-  funext j
-  simp only [lowCoins,assembleBits_eq,Equiv.apply_symm_apply,Sum.elim_inl]
-
-noncomputable def lowEvent (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
-    (Y : SwitchIndex (s+r) → Bool) : ℝ := if lowCoins r s Y=lo then 1 else 0
-
-lemma lowEvent_nonneg (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
-    (Y : SwitchIndex (s+r) → Bool) : 0≤lowEvent r s lo Y := by
-  unfold lowEvent; split_ifs <;> norm_num
-
-lemma lowEvent_adapted (r s : ℕ) (lo : Card r × SwitchIndex s → Bool) :
-    LayerAdapted (s+r) s (lowEvent r s lo) :=
-  layerAdapted_comp (lowCoins_adapted r s) (fun v => if v=lo then (1:ℝ) else 0)
-
-lemma lowEvent_mean_mul (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
-    (G : (SwitchIndex (s+r) → Bool) → ℝ) :
-    finiteMean (fun Y => lowEvent r s lo Y*G Y)=
-      finiteMean (fun hi : Card s × SwitchIndex r → Bool => G (assembleBits r s lo hi))/
-        (Fintype.card (Card r × SwitchIndex s → Bool):ℝ) := by
-  rw [finiteMean_split]
-  simp only [lowEvent,lowCoins_assemble,ite_mul,one_mul,zero_mul,finiteMean]
-  simp_rw [Finset.sum_ite_irrel,Finset.sum_const_zero,ite_div,zero_div]
-  simp
-
-lemma lowEvent_mean (r s : ℕ) (lo : Card r × SwitchIndex s → Bool) :
-    finiteMean (lowEvent r s lo)=1/(Fintype.card (Card r × SwitchIndex s → Bool):ℝ) := by
-  simpa only [mul_one,finiteMean_const] using lowEvent_mean_mul r s lo (fun _ => 1)
-
-lemma family_partial (L n r s : ℕ) (hL : 0<L) (hs : 2*s≤L)
-    {ι : Type*} [Fintype ι] (x : ι ↪ Card (s+r)) (X : SwitchIndex (s+r) → Bool)
-    (lo : Card r × SwitchIndex s → Bool) (q : ℝ) (hq : 1≤q) (hlog : Real.log q≤1/250) :
-    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(palindromeFamilyCost L n (s+r) x (assembleBits r s lo hi,X))) ≤
-        Real.exp (Fintype.card ι*heightDecay L) := by
-  have hh:=family_weighted L n (s+r) s hL hs x X (lowEvent r s lo)
-    (lowEvent_nonneg r s lo) (lowEvent_adapted r s lo) q hq hlog
-  rw [lowEvent_mean_mul,lowEvent_mean,one_div,mul_comm] at hh
-  have hp : (0:ℝ)<Fintype.card (Card r × SwitchIndex s → Bool) := by
-    exact_mod_cast Fintype.card_pos
-  have hh': _≤Real.exp (Fintype.card ι*(68*∑ j ∈ Finset.range n,
-      Real.exp (-((L*4^j:ℕ):ℝ)/64))) := (div_le_div_iff_of_pos_right hp).mp hh
-  apply hh'.trans (Real.exp_le_exp.mpr _)
-  apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
-  have he:=mul_le_mul_of_nonneg_left (sum_family_decay L n hL) (by norm_num : (0:ℝ)≤68)
-  simpa only [heightDecay,mul_div_assoc] using he
-
-
-lemma highCost_family (J d : ℕ) (hJ : 2≤J) {ι : Type*} [Fintype ι]
-    (x : ι ↪ Card d) (ω : BenesCoins d) :
-    highCost J d x ω = palindromeFamilyCost (J-1) d d x ω+
-      palindromeFamilyCost (2*(J-1)) d d x ω := by
-  have hd : d≤(J-1)*4^d := by
-    have hh:=pow_four_ge d
-    have hp : 1≤J-1 := by omega
-    nlinarith
-  have hh:=palindromeCost_family (J-1) d d x ω
-  rw [palindromeLowCost_eq _ _ hd] at hh
-  unfold highCost
-  omega
-
-lemma highCost_levels (J d : ℕ) (hJ : 1≤J) {ι : Type*} [Fintype ι]
-    (x : ι ↪ Card d) (ω : BenesCoins d) :
-    highCost J d x ω = ∑ i ∈ Finset.range (d-(J-1)), palindromeLevelCost (J+i) d x ω := by
-  by_cases hd : J-1≤d
-  · have he : d=(J-1)+(d-(J-1)) := by omega
-    have hh := palindromeLowCost_sum d d x ω
-    rw [palindromeLowCost_eq _ _ le_rfl] at hh
-    rw [show Finset.range d=Finset.range ((J-1)+(d-(J-1))) by rw [←he],
-      Finset.sum_range_add] at hh
-    rw [←palindromeLowCost_sum] at hh
-    have he' (i : ℕ) : J-1+i+1=J+i := by omega
-    simp_rw [he'] at hh
-    unfold highCost
-    omega
-  · rw [highCost,palindromeLowCost_eq _ _ (by omega),Nat.sub_self,
-      Nat.sub_eq_zero_of_le (by omega : d≤J-1),Finset.range_zero,Finset.sum_empty]
-
-lemma high_partial (r s J : ℕ) (hJ : 2≤J) (hs : 2*s<J)
-    {ι : Type*} [Fintype ι] (x : ι ↪ Card (s+r)) (X : SwitchIndex (s+r) → Bool)
-    (lo : Card r × SwitchIndex s → Bool) (q : ℝ) (hq : 1≤q) (hlog : Real.log q≤1/500) :
-    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(highCost J (s+r) x (assembleBits r s lo hi,X))) ≤
-        Real.exp (Fintype.card ι*heightDecay (J-1)) := by
-  have hq2:=one_le_pow₀ (n:=2) hq
-  have hlog2 : Real.log (q^2)≤1/250 := by rw [Real.log_pow]; norm_num; linarith
-  have h₀:=family_partial (J-1) (s+r) r s (by omega) (by omega) x X lo (q^2) hq2 hlog2
-  have h₁:=family_partial (2*(J-1)) (s+r) r s (by omega) (by omega) x X lo (q^2) hq2 hlog2
-  have h₁':=h₁.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left
-    (heightDecay_antitone (show J-1≤2*(J-1) by omega)) (Nat.cast_nonneg _)))
-  have hh:=finiteMean_mul_exp_le
-    (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(palindromeFamilyCost (J-1) (s+r) (s+r) x (assembleBits r s lo hi,X)))
-    (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(palindromeFamilyCost (2*(J-1)) (s+r) (s+r) x (assembleBits r s lo hi,X)))
-    (Fintype.card ι*heightDecay (J-1)) (Fintype.card ι*heightDecay (J-1))
-    (by simpa only [pow_right_comm] using h₀) (by simpa only [pow_right_comm] using h₁')
-  simpa only [highCost_family J _ hJ,←pow_add,add_self_div_two] using hh
-
-lemma high_unconditional (d J : ℕ) (hJ : 2≤J) {ι : Type*} [Fintype ι]
-    (x : ι ↪ Card d) (q : ℝ) (hq : 1≤q) (hlog : Real.log q≤1/500) :
-    finiteMean (fun ω : BenesCoins d => q^(highCost J d x ω)) ≤
-      Real.exp (Fintype.card ι*heightDecay (J-1)) := by
-  simpa only [highCost_family J _ hJ] using
-    palindrome_high_full_mgf (J-1) d d (by omega) x q hq hlog
-
-
-lemma cost_small (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι)
-    (hs : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1/256) :
-    finiteMean (fun ω : BenesCoins d => momentBase^(palindromeCost d e ω)) ≤
-      Real.exp (Fintype.card ι*(1+densityTailConstant)*
-        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ)) := by
-  generalize hpdef : ((Fintype.card ι:ℝ)/(2:ℝ)^d) = p at hs ⊢
-  have hp : 0 < p := by rw [←hpdef]; positivity
-  have hle := densityCutoff_le_height d (Fintype.card ι) hk
-  rw [hpdef] at hle
-  obtain ⟨r,hr⟩ := Nat.exists_eq_add_of_le hle
-  subst d
-  have hH := (densityCutoff_bounds hp hs).1
-  have hq := momentBase_one_le
-  have hqp : 0 < momentBase := by linarith
-  have hlog : Real.log momentBase ≤ Real.log 2/4 := by
-    rw [momentBase_log_exact]
-    have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
-    linarith
-  have hsp := densityCutoff_sparse hp hs hqp hlog hq
-  rw [←hpdef] at hsp
-  have hh := palindrome_cost_mgf_cutoff r (densityCutoff p) hH e momentBase hq momentBase_log
-    (by simpa only [hpdef] using hsp)
-  apply hh.trans (Real.exp_le_exp.mpr _)
-  rw [sparse_low_algebra,hpdef]
-  have hlow := densityCutoff_power hp hs hqp hlog 4 (by decide) hq
-  have hp1 : p ≤ 1 := by linarith
-  have he := Real.rpow_le_rpow_of_exponent_ge hp hp1 (by norm_num : (1:ℝ)/512 ≤ 1/2)
-  have hlow' := mul_le_mul_of_nonneg_left (hlow.trans he)
-    (by positivity : (0:ℝ)≤2*Fintype.card ι)
-  have hhigh := mul_le_mul_of_nonneg_left (heightDecay_cutoff hp hs)
-    (Nat.cast_nonneg (Fintype.card ι): (0:ℝ)≤_)
-  have hnon : 0 ≤ Fintype.card ι*densityTailConstant*p^(1/512:ℝ) :=
-    mul_nonneg (mul_nonneg (Nat.cast_nonneg _) densityTailConstant_nonneg)
-      (Real.rpow_nonneg (le_of_lt hp) _)
-  nlinarith
-
-theorem cost_bound_pos (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι) :
-    finiteMean (fun ω : BenesCoins d => momentBase^(palindromeCost d e ω)) ≤ Real.exp (densityConstant*Fintype.card ι*
-        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ)) := by
-  have hp : 0 < (Fintype.card ι:ℝ)/(2:ℝ)^d := by positivity
-  have hp1 : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1 := by
-    apply (div_le_one (by positivity)).mpr
-    have h := Fintype.card_le_of_injective e e.injective
-    rw [card_positions] at h
-    exact_mod_cast h
-  have hk0 : (0:ℝ) ≤ Fintype.card ι := Nat.cast_nonneg _
-  have hpow := Real.rpow_nonneg (le_of_lt hp) (1/512:ℝ)
-  by_cases hs : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1/256
-  · apply (cost_small d e hk hs).trans (Real.exp_le_exp.mpr _)
-    have hD : 1+densityTailConstant ≤ densityConstant := by
-      dsimp [densityConstant]
-      have := heightDecay_nonneg 1
-      linarith
-    have hh := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hD hk0) hpow
-    nlinarith
-  · have hlow : (1:ℝ)/256 ≤ ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ) := by
-      have he := Real.rpow_le_rpow_of_exponent_ge hp hp1 (by norm_num : (1:ℝ)/512 ≤ 1)
-      rw [Real.rpow_one] at he
-      exact (le_of_not_ge hs).trans he
-    have hD : 256*(1+heightDecay 1) ≤ densityConstant := by
-      dsimp [densityConstant]
-      have := densityTailConstant_nonneg
-      linarith
-    apply (palindrome_cost_mgf_coarse d e momentBase momentBase_one_le momentBase_log).trans (Real.exp_le_exp.mpr _)
-    have hD0 : 0 ≤ 1+heightDecay 1 := by have := heightDecay_nonneg 1; linarith
-    have ha := mul_le_mul_of_nonneg_left hlow (mul_nonneg (by norm_num : (0:ℝ)≤256) hD0)
-    have hb := mul_le_mul_of_nonneg_right hD hpow
-    have hc := mul_le_mul_of_nonneg_right (ha.trans hb) hk0
-    nlinarith
-
-lemma cost_bound (d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) :
-    finiteMean (fun ω : BenesCoins d => momentBase^(palindromeCost d e ω)) ≤
-      Real.exp (densityConstant*Fintype.card ι*((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ)) := by
-  by_cases hk : 0<Fintype.card ι
-  · exact cost_bound_pos d e hk
-  · have hz : Fintype.card ι=0 := by omega
-    simpa only [hz,Nat.cast_zero,mul_zero,zero_mul,Real.exp_zero] using
-      palindrome_cost_mgf_coarse d e momentBase momentBase_one_le momentBase_log
-
-
-noncomputable def tailConstant : ℝ :=
-  68*Real.exp (1/64)/(1-Real.exp (-1/64))
-
-lemma tailConstant_pos : 0<tailConstant := by
-  have hh:=Real.exp_lt_one_iff.mpr (by norm_num : (-1:ℝ)/64<0)
-  exact div_pos (by positivity) (by linarith)
-
-lemma decay_bound (J : ℕ) (hJ : 1≤J) :
-    heightDecay (J-1) ≤ tailConstant*(2:ℝ)^(-(1/1000:ℝ)*J) := by
-  have hd : 0<1-Real.exp (-1/64:ℝ) := by
-    have h:=Real.exp_lt_one_iff.mpr (by norm_num : (-1:ℝ)/64<0)
-    linarith
-  have hlog : Real.log 2≤1 := by
-    have hh:=Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
-    norm_num at hh ⊢
-    exact hh
-  rw [heightDecay,tailConstant,div_mul_eq_mul_div,mul_assoc,
-    Real.rpow_def_of_pos (by norm_num : (0:ℝ)<2)]
-  apply div_le_div_of_nonneg_right _ hd.le
-  apply mul_le_mul_of_nonneg_left _ (by norm_num : (0:ℝ)≤68)
-  rw [←Real.exp_add]
-  apply Real.exp_le_exp.mpr
-  rw [Nat.cast_sub hJ,Nat.cast_one]
-  nlinarith [Nat.cast_nonneg (α:=ℝ) J,
-    mul_le_mul_of_nonneg_right hlog (Nat.cast_nonneg J)]
-
-noncomputable def constant : ℝ := (tailConstant+densityConstant)/Real.log 2
-
-lemma constant_pos : 0<constant := by
-  exact div_pos (add_pos tailConstant_pos densityConstant_pos) (Real.log_pos (by norm_num))
-
-lemma log_mgf_le {Ω : Type*} [Fintype Ω] [Nonempty Ω] (f : Ω → ℕ) (A : ℝ)
-    (h : finiteMean (fun ω => momentBase^(f ω))≤Real.exp A) :
-    Real.log (finiteMean (fun ω => momentBase^(f ω)))≤A := by
-  apply (Real.log_le_iff_le_exp (finiteMean_pos (fun ω => pow_pos
-    (lt_of_lt_of_le zero_lt_one momentBase_one_le) _))).mpr h
-
-lemma high_log {Ω : Type*} [Fintype Ω] [Nonempty Ω] (f : Ω → ℕ) (k J : ℕ)
-    (hJ : 1≤J) (h : finiteMean (fun ω => momentBase^(f ω))≤Real.exp (k*heightDecay (J-1))) :
-    Real.log (finiteMean (fun ω => momentBase^(f ω)))/Real.log 2 ≤
-      constant*k*(2:ℝ)^(-(1/1000:ℝ)*J) := by
-  have hl:=log_mgf_le f _ h
-  have hp : 0<Real.log 2 := Real.log_pos (by norm_num)
-  apply (div_le_iff₀ hp).mpr
-  have hh:=mul_le_mul_of_nonneg_left (decay_bound J hJ) (Nat.cast_nonneg k : (0:ℝ)≤_)
-  have hinc : (k:ℝ)*tailConstant*(2:ℝ)^(-(1/1000:ℝ)*J) ≤
-      (tailConstant+densityConstant)*k*(2:ℝ)^(-(1/1000:ℝ)*J) := by
-    have hnon : 0≤densityConstant*k*(2:ℝ)^(-(1/1000:ℝ)*J) := by
-      exact mul_nonneg (mul_nonneg densityConstant_pos.le (Nat.cast_nonneg _)) (Real.rpow_nonneg (by norm_num) _)
-    nlinarith
-  have hfinal:=hl.trans (hh.trans (by simpa only [mul_assoc] using hinc))
-  refine hfinal.trans_eq ?_
-  unfold constant
-  field_simp
-
-lemma sparse_log (d k : ℕ) (x : Fin k ↪ Card d) :
-    Real.log (finiteMean (fun ω : BenesCoins d => momentBase^(palindromeCost d x ω)))/Real.log 2 ≤
-      constant*k*((k:ℝ)/(2:ℝ)^d)^(1/512:ℝ) := by
-  have h:=log_mgf_le (fun ω : BenesCoins d => palindromeCost d x ω) _ (cost_bound d x)
-  simp only [Fintype.card_fin] at h
-  have hp : 0<Real.log 2 := Real.log_pos (by norm_num)
-  apply (div_le_iff₀ hp).mpr
-  have hn : 0≤tailConstant*k*((k:ℝ)/(2:ℝ)^d)^(1/512:ℝ) := by
-    exact mul_nonneg (mul_nonneg tailConstant_pos.le (Nat.cast_nonneg _)) (Real.rpow_nonneg (by positivity) _)
-  unfold constant
-  field_simp
-  nlinarith
-
-theorem high_tail : MainStatement := by
-  refine ⟨1/1000,1/512,constant,by norm_num,by norm_num,constant_pos,2,by decide,?_,?_,?_⟩
-  · intro d k x J hJ
-    apply high_log _ k J (by omega)
-    simpa only [Fintype.card_fin] using high_unconditional d J hJ x momentBase momentBase_one_le
-      (by have hh:=momentBase_log; linarith)
-  · intro d k x
-    exact sparse_log d k x
-  · intro r s k x J hJ hs X lo
-    apply high_log _ k J (by omega)
-    simpa only [Fintype.card_fin] using high_partial r s J hJ hs x X lo momentBase momentBase_one_le
-      (by have hh:=momentBase_log; linarith)
-
-end Thorp.HighHeight
-
-end ThorpNine.HighTail
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.HighTail.CutoffMass
+
+namespace OAI
+
+namespace ThorpNine.HighTail
+
+namespace Thorp
+
+lemma palindrome_slab_adapted_mul_le (d s l : ℕ) (hsd : s ≤ d) (hl : 0 < l)
+    (hs : 2*s ≤ l) {ι : Type*} [Fintype ι] (e : ι ↪ Card d)
+    (X : SwitchIndex d → Bool) (F : (SwitchIndex d → Bool) → ℝ)
+    (hF : ∀ Y, 0 ≤ F Y) (hA : LayerAdapted d s F) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun Y => F Y*q^(palindromeSlabCost l d e (Y,X))) ≤
+      finiteMean F * Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
+  obtain ⟨r,rfl⟩ := Nat.exists_eq_add_of_le hsd
+  exact adapted_mul_mean_le r s F _ hF hA _
+    (fun lo => palindrome_slab_partial_mgf r s l hl hs e X lo q hq hlog)
+
+def familyEnd (L n : ℕ) : ℕ := if n = 0 then 0 else 2*(L*4^(n-1))
+
+lemma familyEnd_two (L n : ℕ) : 2*familyEnd L n ≤ L*4^n := by
+  cases n with
+  | zero => simp [familyEnd]
+  | succ n => simp only [familyEnd,Nat.succ_ne_zero,ite_false,Nat.add_one_sub_one,pow_succ]; nlinarith
+
+lemma familyEnd_ge (L n j : ℕ) (hj : j < n) : 2*(L*4^j) ≤ familyEnd L n := by
+  cases n with
+  | zero => omega
+  | succ n =>
+    simp only [familyEnd,Nat.succ_ne_zero,ite_false,Nat.add_one_sub_one]
+    apply Nat.mul_le_mul_left
+    exact Nat.mul_le_mul_left L (Nat.pow_le_pow_right (by decide) (by omega : j ≤ n))
+
+noncomputable def palindromeFamilyCost (L n d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) : ℕ :=
+  ∑ j ∈ Finset.range n, palindromeSlabCost (L*4^j) d e ω
+
+lemma palindromeFamilyCost_adapted (L n d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) :
+    LayerAdapted d (familyEnd L n) (fun Y => palindromeFamilyCost L n d e (Y,X)) := by
+  apply layerAdapted_sum
+  intro j hj
+  exact layerAdapted_mono (palindromeSlabCost_adapted (L*4^j) d e X)
+    (familyEnd_ge L n j (Finset.mem_range.mp hj))
+
+theorem palindrome_family_mgf (L n d : ℕ) (hL : 0 < L) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) ≤
+      Real.exp (Fintype.card ι * (68*∑ j ∈ Finset.range n,
+        Real.exp (-((L*4^j:ℕ):ℝ)/64))) := by
+  induction n with
+  | zero => simp only [palindromeFamilyCost,Finset.range_zero,Finset.sum_empty,pow_zero,
+      finiteMean_const,mul_zero,Real.exp_zero,le_refl]
+  | succ n ih =>
+    let l := L*4^n
+    have hl : 0 < l := Nat.mul_pos hL (pow_pos (by decide) _)
+    have hs : 2*familyEnd L n ≤ l := familyEnd_two L n
+    have hc : ∀ Y, palindromeFamilyCost L (n+1) d e (Y,X) =
+        palindromeFamilyCost L n d e (Y,X)+palindromeSlabCost l d e (Y,X) := by
+      intro Y
+      exact Finset.sum_range_succ _ _
+    have hm : finiteMean (fun Y => q^(palindromeFamilyCost L (n+1) d e (Y,X))) ≤
+        finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) *
+          Real.exp (Fintype.card ι*(68*Real.exp (-(l:ℝ)/64))) := by
+      by_cases hld : l ≤ d
+      · simp_rw [hc,pow_add]
+        exact palindrome_slab_adapted_mul_le d (familyEnd L n) l (by omega) hl hs e X _
+          (fun _ => pow_nonneg (by linarith) _)
+          (layerAdapted_comp (palindromeFamilyCost_adapted L n d e X) (fun c => q^c))
+          q hq hlog
+      · simp_rw [hc,palindromeSlabCost_gt l d (by omega) e,Nat.add_zero]
+        exact le_mul_of_one_le_right
+          (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
+          (Real.one_le_exp_iff.mpr (by positivity))
+    refine (hm.trans (mul_le_mul_of_nonneg_right ih (Real.exp_nonneg _))).trans_eq ?_
+    rw [←Real.exp_add,Finset.sum_range_succ]
+    congr 1
+    dsimp [l]
+    ring
+
+lemma pow_four_ge (j : ℕ) : j+1 ≤ 4^j := by
+  induction j with
+  | zero => norm_num
+  | succ j ih => rw [pow_succ]; omega
+
+lemma sum_family_decay (L n : ℕ) (hL : 0 < L) :
+    (∑ j ∈ Finset.range n, Real.exp (-((L*4^j:ℕ):ℝ)/64)) ≤
+      Real.exp (-(L:ℝ)/64)/(1-Real.exp (-(1:ℝ)/64)) := by
+  let r := Real.exp (-(1:ℝ)/64)
+  have hr : 0 < r := Real.exp_pos _
+  have hr1 : r < 1 := Real.exp_lt_one_iff.mpr (by norm_num)
+  have hg : (∑ j ∈ Finset.range n, r^j) ≤ 1/(1-r) := by
+    apply (le_div_iff₀ (by linarith : (0:ℝ)<1-r)).mpr
+    have hh := geom_sum_mul r n
+    nlinarith [pow_nonneg (le_of_lt hr) n]
+  calc
+    _ ≤ ∑ j ∈ Finset.range n, Real.exp (-(L:ℝ)/64)*r^j := by
+      apply Finset.sum_le_sum
+      intro j _
+      have hj := pow_four_ge j
+      have hlj : L+j ≤ L*4^j := by nlinarith
+      have hlj' : (L:ℝ)+j ≤ (L*4^j:ℕ) := by exact_mod_cast hlj
+      rw [←Real.exp_nat_mul,←Real.exp_add]
+      exact Real.exp_le_exp.mpr (by linarith)
+    _ = Real.exp (-(L:ℝ)/64)*(∑ j ∈ Finset.range n, r^j) := by rw [Finset.mul_sum]
+    _ ≤ Real.exp (-(L:ℝ)/64)*(1/(1-r)) :=
+      mul_le_mul_of_nonneg_left hg (Real.exp_nonneg _)
+    _ = _ := by ring
+
+
+lemma palindromeLevelCost_step_add (t d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (d+1)) (ω : BenesCoins (d+1)) :
+    palindromeLevelCost t (d+1) e ω =
+      let σ := benesStepEquiv d ω
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x σ.2.1
+      let hc := PairRouting.colors_compatible x σ.2.1
+      (if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
+          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) else 0) +
+        palindromeLevelCost t d (PairRouting.childEmbedding x c hc false) σ.1.1 +
+        palindromeLevelCost t d (PairRouting.childEmbedding x c hc true) σ.1.2 := by
+  classical
+  rw [palindromeLevelCost]
+  dsimp only
+  split_ifs with h
+  · simp only [palindromeLevelCost_gt t d (by omega),Nat.add_zero]
+  · simp only [Nat.zero_add]
+
+lemma palindromeLowCost_sum (H d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) :
+    palindromeLowCost H d e ω = ∑ i ∈ Finset.range H, palindromeLevelCost (i+1) d e ω := by
+  classical
+  induction d generalizing ι with
+  | zero => simp only [palindromeLowCost,palindromeLevelCost,Finset.sum_const_zero]
+  | succ d ih =>
+    rw [palindromeLowCost]
+    simp only [palindromeLevelCost_step_add,Finset.sum_add_distrib]
+    rw [←ih,←ih]
+    congr 2
+    simp only [Nat.add_right_cancel_iff,Finset.sum_ite_eq',Finset.mem_range,Nat.succ_le_iff]
+
+lemma palindromeLowCost_slab (L d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) :
+    palindromeLowCost L d e ω + palindromeSlabCost L d e ω = palindromeLowCost (2*L) d e ω := by
+  rw [palindromeLowCost_sum,palindromeLowCost_sum,show 2*L=L+L by omega,Finset.sum_range_add]
+  congr 1
+  exact Fin.sum_univ_eq_sum_range (fun i => palindromeLevelCost (L+i+1) d e ω) L
+
+lemma palindromeCost_family (L n d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) :
+    palindromeLowCost L d e ω + palindromeFamilyCost L n d e ω +
+      palindromeFamilyCost (2*L) n d e ω = palindromeLowCost (L*4^n) d e ω := by
+  induction n with
+  | zero => simp only [palindromeFamilyCost,Finset.range_zero,Finset.sum_empty,Nat.add_zero,
+      pow_zero,mul_one]
+  | succ n ih =>
+    have he : (2*L)*4^n = 2*(L*4^n) := by ring
+    have he' : L*4^(n+1) = 2*(2*(L*4^n)) := by rw [pow_succ]; ring
+    simp only [palindromeFamilyCost,Finset.sum_range_succ] at ih ⊢
+    rw [he,he',←palindromeLowCost_slab,←palindromeLowCost_slab,←ih]
+    omega
+
+
+lemma finiteMean_mul_sq_le {Ω : Type*} [Fintype Ω] (f g : Ω → ℝ) :
+    (finiteMean (fun ω => f ω*g ω))^2 ≤ finiteMean (fun ω => (f ω)^2)*finiteMean (fun ω => (g ω)^2) := by
+  have h := Finset.sum_mul_sq_le_sq_mul_sq Finset.univ f g
+  simp only [finiteMean,div_pow,div_mul_div_comm,←pow_two]
+  exact div_le_div_of_nonneg_right h (sq_nonneg _)
+
+lemma finiteMean_mul_exp_le {Ω : Type*} [Fintype Ω] (f g : Ω → ℝ) (A B : ℝ)
+    (hf : finiteMean (fun ω => (f ω)^2) ≤ Real.exp A)
+    (hg : finiteMean (fun ω => (g ω)^2) ≤ Real.exp B) :
+    finiteMean (fun ω => f ω*g ω) ≤ Real.exp ((A+B)/2) := by
+  have hc := finiteMean_mul_sq_le f g
+  have hn : 0 ≤ finiteMean (fun ω => (g ω)^2) := finiteMean_nonneg (fun _ => sq_nonneg _)
+  have hu := mul_le_mul hf hg hn (Real.exp_nonneg _)
+  have he : Real.exp A*Real.exp B = (Real.exp ((A+B)/2))^2 := by
+    rw [←Real.exp_add,←Real.exp_nat_mul]
+    congr 1
+    push_cast
+    ring
+  rw [he] at hu
+  have hh := hc.trans hu
+  have hep := Real.exp_pos ((A+B)/2)
+  nlinarith
+
+
+noncomputable def heightDecay (H : ℕ) : ℝ :=
+  68*Real.exp (-(H:ℝ)/64)/(1-Real.exp (-(1:ℝ)/64))
+
+lemma heightDecay_nonneg (H : ℕ) : 0 ≤ heightDecay H := by
+  have h : Real.exp (-(1:ℝ)/64) < 1 := Real.exp_lt_one_iff.mpr (by norm_num)
+  apply div_nonneg (by positivity) (by linarith)
+
+lemma heightDecay_antitone : Antitone heightDecay := by
+  intro L H h
+  have hden : 0 ≤ 1-Real.exp (-(1:ℝ)/64) := by
+    have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
+    linarith
+  apply div_le_div_of_nonneg_right _ hden
+  apply mul_le_mul_of_nonneg_left _ (by norm_num)
+  apply Real.exp_le_exp.mpr
+  have h' : (L:ℝ) ≤ H := by exact_mod_cast h
+  linarith
+
+lemma palindrome_family_mgf_decay (L n d : ℕ) (hL : 0 < L) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) ≤
+      Real.exp (Fintype.card ι*heightDecay L) := by
+  apply (palindrome_family_mgf L n d hL e X q hq hlog).trans
+  apply Real.exp_le_exp.mpr
+  apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
+  have h := mul_le_mul_of_nonneg_left (sum_family_decay L n hL) (by norm_num : (0:ℝ)≤68)
+  simpa only [heightDecay,mul_div_assoc] using h
+
+theorem palindrome_high_mgf (H n d : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/500) :
+    finiteMean (fun Y => q^(palindromeFamilyCost H n d e (Y,X)+
+      palindromeFamilyCost (2*H) n d e (Y,X))) ≤ Real.exp (Fintype.card ι*heightDecay H) := by
+  have hq2 := one_le_pow₀ (n:=2) hq
+  have hlog2 : Real.log (q^2) ≤ 1/250 := by rw [Real.log_pow]; norm_num; linarith
+  have h₀ := palindrome_family_mgf_decay H n d hH e X (q^2) hq2 hlog2
+  have h₁ := palindrome_family_mgf_decay (2*H) n d (by omega) e X (q^2) hq2 hlog2
+  have h₁' := h₁.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left
+    (heightDecay_antitone (show H ≤ 2*H by omega)) (Nat.cast_nonneg _)))
+  have hh := finiteMean_mul_exp_le (fun Y => q^(palindromeFamilyCost H n d e (Y,X)))
+    (fun Y => q^(palindromeFamilyCost (2*H) n d e (Y,X)))
+    (Fintype.card ι*heightDecay H) (Fintype.card ι*heightDecay H)
+    (by simpa only [pow_right_comm] using h₀) (by simpa only [pow_right_comm] using h₁')
+  simpa only [←pow_add,add_self_div_two] using hh
+
+lemma palindrome_high_full_mgf (H n d : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/500) :
+    finiteMean (fun ω : BenesCoins d => q^(palindromeFamilyCost H n d e ω+
+      palindromeFamilyCost (2*H) n d e ω)) ≤ Real.exp (Fintype.card ι*heightDecay H) := by
+  rw [finiteMean_prod,finiteMean_comm]
+  calc
+    _ ≤ finiteMean (fun _X : SwitchIndex d → Bool => Real.exp (Fintype.card ι*heightDecay H)) := by
+      apply finiteMean_mono
+      intro X
+      exact palindrome_high_mgf H n d hH e X q hq hlog
+    _ = _ := finiteMean_const _
+
+theorem palindrome_cost_mgf_cutoff (r H : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (H+r)) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/1000)
+    (hr : (2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*(q^2)^H) ≤ 1/2) :
+    finiteMean (fun ω : BenesCoins (H+r) => q^(palindromeCost (H+r) e ω)) ≤
+      Real.exp (((2:ℝ)^r*(2*((2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*(q^2)^H))^2)+
+        Fintype.card ι*heightDecay H)/2) := by
+  let d := H+r
+  have hd : d ≤ H*4^d := by
+    have hh := pow_four_ge d
+    nlinarith
+  have hcost (ω : BenesCoins d) : palindromeCost d e ω = palindromeLowCost H d e ω+
+      (palindromeFamilyCost H d d e ω+palindromeFamilyCost (2*H) d d e ω) := by
+    have hh := palindromeCost_family H d d e ω
+    rw [palindromeLowCost_eq _ _ hd] at hh
+    simpa only [Nat.add_assoc] using hh.symm
+  have hq2 := one_le_pow₀ (n:=2) hq
+  have hlog2 : Real.log (q^2) ≤ 1/500 := by rw [Real.log_pow]; norm_num; linarith
+  have hlow := palindrome_low_mgf r H e (q^2) hq2 hr
+  have hhigh := palindrome_high_full_mgf H d d hH e (q^2) hq2 hlog2
+  have hh := finiteMean_mul_exp_le (fun ω : BenesCoins d => q^(palindromeLowCost H d e ω))
+    (fun ω : BenesCoins d => q^(palindromeFamilyCost H d d e ω+palindromeFamilyCost (2*H) d d e ω))
+    _ _ (by simpa only [pow_right_comm] using hlow) (by simpa only [pow_right_comm] using hhigh)
+  change finiteMean (fun ω : BenesCoins d => q^(palindromeCost d e ω)) ≤ _
+  simpa only [hcost,pow_add,pow_right_comm] using hh
+
+
+lemma cutoffMass_le (H d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (ω : BenesCoins d) :
+    cutoffMass H d e ω ≤ Fintype.card ι := by
+  classical
+  induction d generalizing ι with
+  | zero => exact Nat.zero_le _
+  | succ d ih =>
+    rw [cutoffMass]
+    split_ifs
+    · exact crowdedMass_le _
+    · exact (Nat.add_le_add (ih _ _) (ih _ _)).trans_eq (PairRouting.color_card_add _)
+
+lemma palindromeLowCost_le (H d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) : palindromeLowCost H d e ω ≤ H*Fintype.card ι :=
+  (palindromeLowCost_le_cutoff H d e ω).trans (Nat.mul_le_mul_left H (cutoffMass_le H d e ω))
+
+theorem palindrome_cost_mgf_coarse (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/1000) :
+    finiteMean (fun ω : BenesCoins d => q^(palindromeCost d e ω)) ≤
+      Real.exp (Fintype.card ι*(1+heightDecay 1)) := by
+  have hd : d ≤ 1*4^d := by have hh := pow_four_ge d; omega
+  have hcost (ω : BenesCoins d) : palindromeCost d e ω = palindromeLowCost 1 d e ω+
+      (palindromeFamilyCost 1 d d e ω+palindromeFamilyCost 2 d d e ω) := by
+    have hh := palindromeCost_family 1 d d e ω
+    rw [palindromeLowCost_eq _ _ hd] at hh
+    simpa only [Nat.add_assoc] using hh.symm
+  have hq2 := one_le_pow₀ (n:=2) hq
+  have hlog2 : Real.log (q^2) ≤ 1/500 := by rw [Real.log_pow]; norm_num; linarith
+  have hlow : finiteMean (fun ω : BenesCoins d => (q^2)^(palindromeLowCost 1 d e ω)) ≤
+      Real.exp (Fintype.card ι) := by
+    calc
+      _ ≤ finiteMean (fun _ω : BenesCoins d => (q^2)^(Fintype.card ι)) := by
+        apply finiteMean_mono
+        intro ω
+        apply pow_le_pow_right₀ hq2
+        simpa only [one_mul] using palindromeLowCost_le 1 d e ω
+      _ = (q^2)^(Fintype.card ι) := finiteMean_const _
+      _ = Real.exp (Fintype.card ι * Real.log (q^2)) := by
+        rw [Real.exp_nat_mul,Real.exp_log (by positivity : 0<q^2)]
+      _ ≤ _ := Real.exp_le_exp.mpr (by
+        have hh := mul_le_mul_of_nonneg_left hlog2 (Nat.cast_nonneg (Fintype.card ι) : (0:ℝ)≤_)
+        nlinarith [Nat.cast_nonneg (α:=ℝ) (Fintype.card ι)])
+  have hhigh := palindrome_high_full_mgf 1 d d (by decide) e (q^2) hq2 hlog2
+  have hh := finiteMean_mul_exp_le (fun ω : BenesCoins d => q^(palindromeLowCost 1 d e ω))
+    (fun ω : BenesCoins d => q^(palindromeFamilyCost 1 d d e ω+palindromeFamilyCost 2 d d e ω))
+    (Fintype.card ι) (Fintype.card ι*heightDecay 1)
+    (by simpa only [pow_right_comm] using hlow) (by simpa only [pow_right_comm] using hhigh)
+  simp only [←pow_add,←hcost] at hh
+  apply hh.trans (Real.exp_le_exp.mpr _)
+  have hk : (0:ℝ) ≤ Fintype.card ι := Nat.cast_nonneg _
+  have hD := heightDecay_nonneg 1
+  nlinarith
+
+noncomputable def momentBase : ℝ := (2:ℝ)^(1/1000:ℝ)
+
+lemma momentBase_one_le : 1 ≤ momentBase := by
+  exact Real.one_le_rpow (by norm_num) (by norm_num)
+
+lemma momentBase_log : Real.log momentBase ≤ 1/1000 := by
+  rw [momentBase,Real.log_rpow (by norm_num)]
+  have hh := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
+  norm_num at hh ⊢
+  linarith
+
+lemma momentBase_pow (c : ℕ) : momentBase^c = (2:ℝ)^((c:ℝ)*(1/1000:ℝ)) := by
+  rw [mul_comm,Real.rpow_mul_natCast (by norm_num)]
+  rfl
+
+lemma palindromeRowMoment_le_cost (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) :
+    palindromeRowMoment d e (1/1000) ≤
+      finiteMean (fun ω : BenesCoins d => momentBase^(palindromeCost d e ω)) := by
+  have hn : (0:ℝ) < ((2^d:ℕ):ℝ)^(Fintype.card ι) := by positivity
+  have hn0 : (0:ℝ) ≤ ((2^d).descFactorial (Fintype.card ι):ℝ) /
+      ((2^d:ℕ):ℝ)^(Fintype.card ι) := by positivity
+  have hn1 : ((2^d).descFactorial (Fintype.card ι):ℝ) /
+      ((2^d:ℕ):ℝ)^(Fintype.card ι) ≤ 1 := by
+    apply (div_le_one hn).mpr
+    exact_mod_cast Nat.descFactorial_le_pow (2^d) (Fintype.card ι)
+  have hpow := Real.rpow_le_one hn0 hn1 (by norm_num : (0:ℝ)≤1/1000)
+  have hm := palindrome_row_density_moment d e (1/1000) (by norm_num)
+  simp_rw [←momentBase_pow] at hm
+  exact hm.trans (mul_le_of_le_one_left (finiteMean_nonneg (fun _ =>
+    pow_nonneg (le_trans (by norm_num) momentBase_one_le) _)) hpow)
+
+theorem palindrome_row_moment_coarse (d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) :
+    palindromeRowMoment d e (1/1000) ≤ Real.exp (Fintype.card ι*(1+heightDecay 1)) :=
+  (palindromeRowMoment_le_cost d e).trans
+    (palindrome_cost_mgf_coarse d e momentBase momentBase_one_le momentBase_log)
+
+
+noncomputable def densityCutoff (p : ℝ) : ℕ := ⌊-Real.log p/(4*Real.log 2)⌋₊
+
+lemma log_small_density {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
+    Real.log p ≤ -8*Real.log 2 := by
+  have he : Real.log (1/256:ℝ) = -8*Real.log 2 := by
+    rw [show (1/256:ℝ) = 1/2^8 by norm_num,
+      Real.log_div (by norm_num) (by positivity),Real.log_one,Real.log_pow]
+    ring
+  exact (Real.log_le_log hp hs).trans_eq he
+
+lemma densityCutoff_bounds {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
+    0 < densityCutoff p ∧
+    (densityCutoff p:ℝ)*(4*Real.log 2) ≤ -Real.log p ∧
+    -Real.log p < ((densityCutoff p:ℝ)+1)*(4*Real.log 2) := by
+  have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
+  have hd : 0 < 4*Real.log 2 := by positivity
+  have hh := log_small_density hp hs
+  have hx : 1 ≤ -Real.log p/(4*Real.log 2) := by
+    apply (le_div_iff₀ hd).mpr
+    nlinarith
+  refine ⟨Nat.floor_pos.mpr hx,?_,?_⟩
+  · exact (le_div_iff₀ hd).mp (Nat.floor_le (by linarith : 0 ≤ -Real.log p/(4*Real.log 2)))
+  · exact (div_lt_iff₀ hd).mp (Nat.lt_floor_add_one (-Real.log p/(4*Real.log 2)))
+
+lemma densityCutoff_decay {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
+    Real.exp (-(densityCutoff p:ℝ)/64) ≤ Real.exp (1/64) * p^(1/512:ℝ) := by
+  have hh := (densityCutoff_bounds hp hs).2.2
+  have hb : Real.log 2 ≤ 2 := by
+    have := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
+    linarith
+  have hH : 0 ≤ (densityCutoff p:ℝ)+1 := by positivity
+  have hu := mul_le_mul_of_nonneg_left hb hH
+  rw [Real.rpow_def_of_pos hp,←Real.exp_add]
+  apply Real.exp_le_exp.mpr
+  nlinarith
+
+lemma densityCutoff_power {p q : ℝ} (hp : 0 < p) (hs : p ≤ 1/256)
+    (hq : 0 < q) (hlog : Real.log q ≤ Real.log 2/4) (n : ℕ) (hn : n ≤ 4)
+    (hq1 : 1 ≤ q) :
+    p*(2:ℝ)^(densityCutoff p)*q^(n*densityCutoff p) ≤ p^(1/2:ℝ) := by
+  have hcut := (densityCutoff_bounds hp hs).2.1
+  have hH : 0 ≤ (densityCutoff p:ℝ) := Nat.cast_nonneg _
+  have hn' : (n:ℝ) ≤ 4 := by exact_mod_cast hn
+  have hln := Real.log_nonneg hq1
+  have hl := mul_le_mul_of_nonneg_right hn' hln
+  have hnl : (n:ℝ)*Real.log q ≤ Real.log 2 := by linarith
+  have hm := mul_le_mul_of_nonneg_left hnl hH
+  have heq : p*(2:ℝ)^(densityCutoff p)*q^(n*densityCutoff p) =
+      Real.exp (Real.log p+(densityCutoff p:ℝ)*Real.log 2+
+        (n*densityCutoff p:ℕ)*Real.log q) := by
+    rw [Real.exp_add,Real.exp_add,Real.exp_nat_mul,Real.exp_nat_mul,
+      Real.exp_log hp,Real.exp_log hq,Real.exp_log (by norm_num : (0:ℝ)<2)]
+  rw [heq,Real.rpow_def_of_pos hp]
+  apply Real.exp_le_exp.mpr
+  push_cast
+  nlinarith
+
+lemma densityCutoff_sparse {p q : ℝ} (hp : 0 < p) (hs : p ≤ 1/256)
+    (hq : 0 < q) (hlog : Real.log q ≤ Real.log 2/4) (hq1 : 1 ≤ q) :
+    (2:ℝ)^(densityCutoff p)*(p*(q^2)^(densityCutoff p)) ≤ 1/2 := by
+  have hh := densityCutoff_power hp hs hq hlog 2 (by decide) hq1
+  rw [←pow_mul] at ⊢
+  have hle : p^(1/2:ℝ) ≤ 1/2 := by
+    have hlogp := log_small_density hp hs
+    have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
+    rw [Real.rpow_def_of_pos hp]
+    calc
+      _ ≤ Real.exp (-Real.log 2) := by apply Real.exp_le_exp.mpr; nlinarith
+      _ = 1/2 := by rw [Real.exp_neg,Real.exp_log (by norm_num : (0:ℝ)<2)]; norm_num
+  nlinarith [hh.trans hle]
+
+
+lemma momentBase_log_exact : Real.log momentBase = Real.log 2/1000 := by
+  rw [momentBase,Real.log_rpow (by norm_num)]
+  ring
+
+lemma densityCutoff_le_height (d k : ℕ) (hk : 0 < k) :
+    densityCutoff ((k:ℝ)/(2:ℝ)^d) ≤ d := by
+  have hk' : (1:ℝ) ≤ k := by exact_mod_cast hk
+  have hp : 0 < (k:ℝ)/(2:ℝ)^d := by positivity
+  have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
+  have hlogk := Real.log_nonneg hk'
+  by_cases hx : 0 ≤ -Real.log ((k:ℝ)/(2:ℝ)^d)/(4*Real.log 2)
+  · have hh := (le_div_iff₀ (by positivity : 0 < 4*Real.log 2)).mp (Nat.floor_le hx)
+    have hd : (0:ℝ) ≤ d := Nat.cast_nonneg _
+    have hH : (0:ℝ) ≤ densityCutoff ((k:ℝ)/(2:ℝ)^d) := Nat.cast_nonneg _
+    change (densityCutoff ((k:ℝ)/(2:ℝ)^d):ℝ)*(4*Real.log 2) ≤ -Real.log ((k:ℝ)/(2:ℝ)^d) at hh
+    rw [Real.log_div (by positivity) (by positivity),Real.log_pow] at hh
+    have hle : (densityCutoff ((k:ℝ)/(2:ℝ)^d):ℝ) ≤ d := by nlinarith
+    exact_mod_cast hle
+  · have hz := Nat.floor_of_nonpos (le_of_not_ge hx)
+    simp only [densityCutoff,hz,Nat.zero_le]
+
+noncomputable def densityTailConstant : ℝ :=
+  68*Real.exp (1/64)/(1-Real.exp (-(1:ℝ)/64))
+
+lemma densityTailConstant_nonneg : 0 ≤ densityTailConstant := by
+  have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
+  apply div_nonneg (by positivity) (by linarith)
+
+lemma heightDecay_cutoff {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
+    heightDecay (densityCutoff p) ≤ densityTailConstant*p^(1/512:ℝ) := by
+  have hden : 0 ≤ 1-Real.exp (-(1:ℝ)/64) := by
+    have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
+    linarith
+  have hh := div_le_div_of_nonneg_right
+    (mul_le_mul_of_nonneg_left (densityCutoff_decay hp hs) (by norm_num : (0:ℝ)≤68)) hden
+  simpa only [heightDecay,densityTailConstant,mul_assoc,div_mul_eq_mul_div] using hh
+
+lemma sparse_low_algebra (r H k : ℕ) (q : ℝ) :
+    (2:ℝ)^r*(2*((2:ℝ)^H*(((k:ℝ)/(2:ℝ)^(H+r))*(q^2)^H))^2) =
+    2*k*(((k:ℝ)/(2:ℝ)^(H+r))*(2:ℝ)^H*q^(4*H)) := by
+  rw [pow_add,←pow_mul,show 4*H = (2*H)*2 by omega,pow_mul]
+  have hH : (2:ℝ)^H ≠ 0 := by positivity
+  have hr : (2:ℝ)^r ≠ 0 := by positivity
+  field_simp
+  ring
+
+lemma palindrome_row_moment_small (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι)
+    (hs : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1/256) :
+    palindromeRowMoment d e (1/1000) ≤
+      Real.exp (Fintype.card ι*(1+densityTailConstant)*
+        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ)) := by
+  generalize hpdef : ((Fintype.card ι:ℝ)/(2:ℝ)^d) = p at hs ⊢
+  have hp : 0 < p := by rw [←hpdef]; positivity
+  have hle := densityCutoff_le_height d (Fintype.card ι) hk
+  rw [hpdef] at hle
+  obtain ⟨r,hr⟩ := Nat.exists_eq_add_of_le hle
+  subst d
+  have hH := (densityCutoff_bounds hp hs).1
+  have hq := momentBase_one_le
+  have hqp : 0 < momentBase := by linarith
+  have hlog : Real.log momentBase ≤ Real.log 2/4 := by
+    rw [momentBase_log_exact]
+    have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
+    linarith
+  have hsp := densityCutoff_sparse hp hs hqp hlog hq
+  rw [←hpdef] at hsp
+  have hh := palindrome_cost_mgf_cutoff r (densityCutoff p) hH e momentBase hq momentBase_log
+    (by simpa only [hpdef] using hsp)
+  apply (palindromeRowMoment_le_cost _ e).trans (hh.trans (Real.exp_le_exp.mpr _))
+  rw [sparse_low_algebra,hpdef]
+  have hlow := densityCutoff_power hp hs hqp hlog 4 (by decide) hq
+  have hp1 : p ≤ 1 := by linarith
+  have he := Real.rpow_le_rpow_of_exponent_ge hp hp1 (by norm_num : (1:ℝ)/512 ≤ 1/2)
+  have hlow' := mul_le_mul_of_nonneg_left (hlow.trans he)
+    (by positivity : (0:ℝ)≤2*Fintype.card ι)
+  have hhigh := mul_le_mul_of_nonneg_left (heightDecay_cutoff hp hs)
+    (Nat.cast_nonneg (Fintype.card ι): (0:ℝ)≤_)
+  have hnon : 0 ≤ Fintype.card ι*densityTailConstant*p^(1/512:ℝ) :=
+    mul_nonneg (mul_nonneg (Nat.cast_nonneg _) densityTailConstant_nonneg)
+      (Real.rpow_nonneg (le_of_lt hp) _)
+  nlinarith
+
+noncomputable def densityConstant : ℝ :=
+  256*(1+heightDecay 1)+1+densityTailConstant
+
+lemma densityConstant_pos : 0 < densityConstant := by
+  dsimp [densityConstant]
+  have hh := heightDecay_nonneg 1
+  have ht := densityTailConstant_nonneg
+  linarith
+
+theorem palindrome_row_moment_bound (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι) :
+    palindromeRowMoment d e (1/1000) ≤ Real.exp (densityConstant*Fintype.card ι*
+        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ)) := by
+  have hp : 0 < (Fintype.card ι:ℝ)/(2:ℝ)^d := by positivity
+  have hp1 : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1 := by
+    apply (div_le_one (by positivity)).mpr
+    have h := Fintype.card_le_of_injective e e.injective
+    rw [card_positions] at h
+    exact_mod_cast h
+  have hk0 : (0:ℝ) ≤ Fintype.card ι := Nat.cast_nonneg _
+  have hpow := Real.rpow_nonneg (le_of_lt hp) (1/512:ℝ)
+  by_cases hs : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1/256
+  · apply (palindrome_row_moment_small d e hk hs).trans (Real.exp_le_exp.mpr _)
+    have hD : 1+densityTailConstant ≤ densityConstant := by
+      dsimp [densityConstant]
+      have := heightDecay_nonneg 1
+      linarith
+    have hh := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hD hk0) hpow
+    nlinarith
+  · have hlow : (1:ℝ)/256 ≤ ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ) := by
+      have he := Real.rpow_le_rpow_of_exponent_ge hp hp1 (by norm_num : (1:ℝ)/512 ≤ 1)
+      rw [Real.rpow_one] at he
+      exact (le_of_not_ge hs).trans he
+    have hD : 256*(1+heightDecay 1) ≤ densityConstant := by
+      dsimp [densityConstant]
+      have := densityTailConstant_nonneg
+      linarith
+    apply (palindrome_row_moment_coarse d e).trans (Real.exp_le_exp.mpr _)
+    have hD0 : 0 ≤ 1+heightDecay 1 := by have := heightDecay_nonneg 1; linarith
+    have ha := mul_le_mul_of_nonneg_left hlow (mul_nonneg (by norm_num : (0:ℝ)≤256) hD0)
+    have hb := mul_le_mul_of_nonneg_right hD hpow
+    have hc := mul_le_mul_of_nonneg_right (ha.trans hb) hk0
+    nlinarith
+
+lemma palindromeRowMoment_nonneg (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (a : ℝ) : 0 ≤ palindromeRowMoment d e a := by
+  apply finiteMean_nonneg
+  intro f
+  exact Real.rpow_nonneg (mul_nonneg (Nat.cast_nonneg _)
+    (PairRouting.tupleProbability_nonneg _ _ _)) _
+
+theorem palindrome_log_row_moment (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι) :
+    Real.log (palindromeRowMoment d e (1/1000)) ≤ densityConstant*Fintype.card ι*
+        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ) := by
+  by_cases hz : palindromeRowMoment d e (1/1000) = 0
+  · rw [hz,Real.log_zero]
+    exact mul_nonneg (mul_nonneg (le_of_lt densityConstant_pos) (Nat.cast_nonneg _))
+      (Real.rpow_nonneg (by positivity) _)
+  · exact (Real.log_le_iff_le_exp (lt_of_le_of_ne (palindromeRowMoment_nonneg _ _ _) (Ne.symm hz))).mpr
+      (palindrome_row_moment_bound d e hk)
+
+end Thorp
+
+
+namespace Thorp.HighHeight
+open scoped BigOperators Classical
+
+lemma family_weighted (L n d s : ℕ) (hL : 0<L) (hs : 2*s≤L)
+    {ι : Type*} [Fintype ι] (x : ι ↪ Card d) (X : SwitchIndex d → Bool)
+    (F : (SwitchIndex d → Bool) → ℝ) (hF : ∀ Y, 0≤F Y) (hA : LayerAdapted d s F)
+    (q : ℝ) (hq : 1≤q) (hlog : Real.log q≤1/250) :
+    finiteMean (fun Y => F Y*q^(palindromeFamilyCost L n d x (Y,X))) ≤
+      finiteMean F*Real.exp (Fintype.card ι*(68*∑ j ∈ Finset.range n,
+        Real.exp (-((L*4^j:ℕ):ℝ)/64))) := by
+  induction n with
+  | zero =>
+    simp only [palindromeFamilyCost,Finset.range_zero,Finset.sum_empty,pow_zero,
+      mul_one,mul_zero,Real.exp_zero]
+    exact le_rfl
+  | succ n ih =>
+    let l := L*4^n
+    have hl : 0<l := Nat.mul_pos hL (pow_pos (by decide) _)
+    have hLl : L≤l := by dsimp [l]; exact Nat.le_mul_of_pos_right L (by positivity)
+    have hs' : 2*max s (familyEnd L n)≤l := by
+      rcases le_total s (familyEnd L n) with h | h
+      · simpa only [max_eq_right h] using familyEnd_two L n
+      · simpa only [max_eq_left h] using hs.trans hLl
+    have hc (Y) : palindromeFamilyCost L (n+1) d x (Y,X)=
+        palindromeFamilyCost L n d x (Y,X)+palindromeSlabCost l d x (Y,X) :=
+      Finset.sum_range_succ _ _
+    have hwt (Y) : 0≤F Y*q^(palindromeFamilyCost L n d x (Y,X)) :=
+      mul_nonneg (hF Y) (pow_nonneg (by linarith) _)
+    have had : LayerAdapted d (max s (familyEnd L n))
+        (fun Y => F Y*q^(palindromeFamilyCost L n d x (Y,X))) := by
+      intro Y Y' hh
+      dsimp only
+      rw [hA Y Y' (agreeThrough_mono (le_max_left _ _) hh)]
+      congr 2
+      exact palindromeFamilyCost_adapted L n d x X Y Y' (agreeThrough_mono (le_max_right _ _) hh)
+    have hm : finiteMean (fun Y => F Y*q^(palindromeFamilyCost L (n+1) d x (Y,X))) ≤
+        finiteMean (fun Y => F Y*q^(palindromeFamilyCost L n d x (Y,X)))*
+          Real.exp (Fintype.card ι*(68*Real.exp (-(l:ℝ)/64))) := by
+      by_cases hld : l≤d
+      · simp_rw [hc,pow_add,←mul_assoc]
+        simpa only [mul_assoc] using palindrome_slab_adapted_mul_le d (max s (familyEnd L n)) l (by omega)
+          hl hs' x X _ hwt had q hq hlog
+      · simp_rw [hc,palindromeSlabCost_gt l d (by omega) x,Nat.add_zero]
+        exact le_mul_of_one_le_right (finiteMean_nonneg hwt)
+          (Real.one_le_exp_iff.mpr (by positivity))
+    refine (hm.trans (mul_le_mul_of_nonneg_right ih (Real.exp_nonneg _))).trans_eq ?_
+    rw [mul_assoc,←Real.exp_add,Finset.sum_range_succ]
+    congr 2
+    dsimp [l]
+    ring
+
+lemma low_switch_height (r s : ℕ) (j : Card r × SwitchIndex s) :
+    switchHeight (s+r) ((splitSwitch r s).symm (Sum.inl j)) ≤ s := by
+  induction r with
+  | zero => exact switchHeight_le s j.2
+  | succ r ih =>
+    exact ih (Fin.tail j.1,j.2)
+
+def lowCoins (r s : ℕ) (Y : SwitchIndex (s+r) → Bool) :
+    Card r × SwitchIndex s → Bool :=
+  fun j => Y ((splitSwitch r s).symm (Sum.inl j))
+
+lemma lowCoins_adapted (r s : ℕ) : LayerAdapted (s+r) s (lowCoins r s) := by
+  intro Y Y' hh
+  funext j
+  exact hh _ (low_switch_height r s j)
+
+lemma lowCoins_assemble (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
+    (hi : Card s × SwitchIndex r → Bool) : lowCoins r s (assembleBits r s lo hi)=lo := by
+  funext j
+  simp only [lowCoins,assembleBits_eq,Equiv.apply_symm_apply,Sum.elim_inl]
+
+noncomputable def lowEvent (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
+    (Y : SwitchIndex (s+r) → Bool) : ℝ := if lowCoins r s Y=lo then 1 else 0
+
+lemma lowEvent_nonneg (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
+    (Y : SwitchIndex (s+r) → Bool) : 0≤lowEvent r s lo Y := by
+  unfold lowEvent; split_ifs <;> norm_num
+
+lemma lowEvent_adapted (r s : ℕ) (lo : Card r × SwitchIndex s → Bool) :
+    LayerAdapted (s+r) s (lowEvent r s lo) :=
+  layerAdapted_comp (lowCoins_adapted r s) (fun v => if v=lo then (1:ℝ) else 0)
+
+lemma lowEvent_mean_mul (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
+    (G : (SwitchIndex (s+r) → Bool) → ℝ) :
+    finiteMean (fun Y => lowEvent r s lo Y*G Y)=
+      finiteMean (fun hi : Card s × SwitchIndex r → Bool => G (assembleBits r s lo hi))/
+        (Fintype.card (Card r × SwitchIndex s → Bool):ℝ) := by
+  rw [finiteMean_split]
+  simp only [lowEvent,lowCoins_assemble,ite_mul,one_mul,zero_mul,finiteMean]
+  simp_rw [Finset.sum_ite_irrel,Finset.sum_const_zero,ite_div,zero_div]
+  simp
+
+lemma lowEvent_mean (r s : ℕ) (lo : Card r × SwitchIndex s → Bool) :
+    finiteMean (lowEvent r s lo)=1/(Fintype.card (Card r × SwitchIndex s → Bool):ℝ) := by
+  simpa only [mul_one,finiteMean_const] using lowEvent_mean_mul r s lo (fun _ => 1)
+
+lemma family_partial (L n r s : ℕ) (hL : 0<L) (hs : 2*s≤L)
+    {ι : Type*} [Fintype ι] (x : ι ↪ Card (s+r)) (X : SwitchIndex (s+r) → Bool)
+    (lo : Card r × SwitchIndex s → Bool) (q : ℝ) (hq : 1≤q) (hlog : Real.log q≤1/250) :
+    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(palindromeFamilyCost L n (s+r) x (assembleBits r s lo hi,X))) ≤
+        Real.exp (Fintype.card ι*heightDecay L) := by
+  have hh:=family_weighted L n (s+r) s hL hs x X (lowEvent r s lo)
+    (lowEvent_nonneg r s lo) (lowEvent_adapted r s lo) q hq hlog
+  rw [lowEvent_mean_mul,lowEvent_mean,one_div,mul_comm] at hh
+  have hp : (0:ℝ)<Fintype.card (Card r × SwitchIndex s → Bool) := by
+    exact_mod_cast Fintype.card_pos
+  have hh': _≤Real.exp (Fintype.card ι*(68*∑ j ∈ Finset.range n,
+      Real.exp (-((L*4^j:ℕ):ℝ)/64))) := (div_le_div_iff_of_pos_right hp).mp hh
+  apply hh'.trans (Real.exp_le_exp.mpr _)
+  apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
+  have he:=mul_le_mul_of_nonneg_left (sum_family_decay L n hL) (by norm_num : (0:ℝ)≤68)
+  simpa only [heightDecay,mul_div_assoc] using he
+
+
+lemma highCost_family (J d : ℕ) (hJ : 2≤J) {ι : Type*} [Fintype ι]
+    (x : ι ↪ Card d) (ω : BenesCoins d) :
+    highCost J d x ω = palindromeFamilyCost (J-1) d d x ω+
+      palindromeFamilyCost (2*(J-1)) d d x ω := by
+  have hd : d≤(J-1)*4^d := by
+    have hh:=pow_four_ge d
+    have hp : 1≤J-1 := by omega
+    nlinarith
+  have hh:=palindromeCost_family (J-1) d d x ω
+  rw [palindromeLowCost_eq _ _ hd] at hh
+  unfold highCost
+  omega
+
+lemma highCost_levels (J d : ℕ) (hJ : 1≤J) {ι : Type*} [Fintype ι]
+    (x : ι ↪ Card d) (ω : BenesCoins d) :
+    highCost J d x ω = ∑ i ∈ Finset.range (d-(J-1)), palindromeLevelCost (J+i) d x ω := by
+  by_cases hd : J-1≤d
+  · have he : d=(J-1)+(d-(J-1)) := by omega
+    have hh := palindromeLowCost_sum d d x ω
+    rw [palindromeLowCost_eq _ _ le_rfl] at hh
+    rw [show Finset.range d=Finset.range ((J-1)+(d-(J-1))) by rw [←he],
+      Finset.sum_range_add] at hh
+    rw [←palindromeLowCost_sum] at hh
+    have he' (i : ℕ) : J-1+i+1=J+i := by omega
+    simp_rw [he'] at hh
+    unfold highCost
+    omega
+  · rw [highCost,palindromeLowCost_eq _ _ (by omega),Nat.sub_self,
+      Nat.sub_eq_zero_of_le (by omega : d≤J-1),Finset.range_zero,Finset.sum_empty]
+
+lemma high_partial (r s J : ℕ) (hJ : 2≤J) (hs : 2*s<J)
+    {ι : Type*} [Fintype ι] (x : ι ↪ Card (s+r)) (X : SwitchIndex (s+r) → Bool)
+    (lo : Card r × SwitchIndex s → Bool) (q : ℝ) (hq : 1≤q) (hlog : Real.log q≤1/500) :
+    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(highCost J (s+r) x (assembleBits r s lo hi,X))) ≤
+        Real.exp (Fintype.card ι*heightDecay (J-1)) := by
+  have hq2:=one_le_pow₀ (n:=2) hq
+  have hlog2 : Real.log (q^2)≤1/250 := by rw [Real.log_pow]; norm_num; linarith
+  have h₀:=family_partial (J-1) (s+r) r s (by omega) (by omega) x X lo (q^2) hq2 hlog2
+  have h₁:=family_partial (2*(J-1)) (s+r) r s (by omega) (by omega) x X lo (q^2) hq2 hlog2
+  have h₁':=h₁.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left
+    (heightDecay_antitone (show J-1≤2*(J-1) by omega)) (Nat.cast_nonneg _)))
+  have hh:=finiteMean_mul_exp_le
+    (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(palindromeFamilyCost (J-1) (s+r) (s+r) x (assembleBits r s lo hi,X)))
+    (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(palindromeFamilyCost (2*(J-1)) (s+r) (s+r) x (assembleBits r s lo hi,X)))
+    (Fintype.card ι*heightDecay (J-1)) (Fintype.card ι*heightDecay (J-1))
+    (by simpa only [pow_right_comm] using h₀) (by simpa only [pow_right_comm] using h₁')
+  simpa only [highCost_family J _ hJ,←pow_add,add_self_div_two] using hh
+
+lemma high_unconditional (d J : ℕ) (hJ : 2≤J) {ι : Type*} [Fintype ι]
+    (x : ι ↪ Card d) (q : ℝ) (hq : 1≤q) (hlog : Real.log q≤1/500) :
+    finiteMean (fun ω : BenesCoins d => q^(highCost J d x ω)) ≤
+      Real.exp (Fintype.card ι*heightDecay (J-1)) := by
+  simpa only [highCost_family J _ hJ] using
+    palindrome_high_full_mgf (J-1) d d (by omega) x q hq hlog
+
+
+lemma cost_small (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι)
+    (hs : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1/256) :
+    finiteMean (fun ω : BenesCoins d => momentBase^(palindromeCost d e ω)) ≤
+      Real.exp (Fintype.card ι*(1+densityTailConstant)*
+        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ)) := by
+  generalize hpdef : ((Fintype.card ι:ℝ)/(2:ℝ)^d) = p at hs ⊢
+  have hp : 0 < p := by rw [←hpdef]; positivity
+  have hle := densityCutoff_le_height d (Fintype.card ι) hk
+  rw [hpdef] at hle
+  obtain ⟨r,hr⟩ := Nat.exists_eq_add_of_le hle
+  subst d
+  have hH := (densityCutoff_bounds hp hs).1
+  have hq := momentBase_one_le
+  have hqp : 0 < momentBase := by linarith
+  have hlog : Real.log momentBase ≤ Real.log 2/4 := by
+    rw [momentBase_log_exact]
+    have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
+    linarith
+  have hsp := densityCutoff_sparse hp hs hqp hlog hq
+  rw [←hpdef] at hsp
+  have hh := palindrome_cost_mgf_cutoff r (densityCutoff p) hH e momentBase hq momentBase_log
+    (by simpa only [hpdef] using hsp)
+  apply hh.trans (Real.exp_le_exp.mpr _)
+  rw [sparse_low_algebra,hpdef]
+  have hlow := densityCutoff_power hp hs hqp hlog 4 (by decide) hq
+  have hp1 : p ≤ 1 := by linarith
+  have he := Real.rpow_le_rpow_of_exponent_ge hp hp1 (by norm_num : (1:ℝ)/512 ≤ 1/2)
+  have hlow' := mul_le_mul_of_nonneg_left (hlow.trans he)
+    (by positivity : (0:ℝ)≤2*Fintype.card ι)
+  have hhigh := mul_le_mul_of_nonneg_left (heightDecay_cutoff hp hs)
+    (Nat.cast_nonneg (Fintype.card ι): (0:ℝ)≤_)
+  have hnon : 0 ≤ Fintype.card ι*densityTailConstant*p^(1/512:ℝ) :=
+    mul_nonneg (mul_nonneg (Nat.cast_nonneg _) densityTailConstant_nonneg)
+      (Real.rpow_nonneg (le_of_lt hp) _)
+  nlinarith
+
+theorem cost_bound_pos (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι) :
+    finiteMean (fun ω : BenesCoins d => momentBase^(palindromeCost d e ω)) ≤ Real.exp (densityConstant*Fintype.card ι*
+        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ)) := by
+  have hp : 0 < (Fintype.card ι:ℝ)/(2:ℝ)^d := by positivity
+  have hp1 : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1 := by
+    apply (div_le_one (by positivity)).mpr
+    have h := Fintype.card_le_of_injective e e.injective
+    rw [card_positions] at h
+    exact_mod_cast h
+  have hk0 : (0:ℝ) ≤ Fintype.card ι := Nat.cast_nonneg _
+  have hpow := Real.rpow_nonneg (le_of_lt hp) (1/512:ℝ)
+  by_cases hs : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1/256
+  · apply (cost_small d e hk hs).trans (Real.exp_le_exp.mpr _)
+    have hD : 1+densityTailConstant ≤ densityConstant := by
+      dsimp [densityConstant]
+      have := heightDecay_nonneg 1
+      linarith
+    have hh := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hD hk0) hpow
+    nlinarith
+  · have hlow : (1:ℝ)/256 ≤ ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ) := by
+      have he := Real.rpow_le_rpow_of_exponent_ge hp hp1 (by norm_num : (1:ℝ)/512 ≤ 1)
+      rw [Real.rpow_one] at he
+      exact (le_of_not_ge hs).trans he
+    have hD : 256*(1+heightDecay 1) ≤ densityConstant := by
+      dsimp [densityConstant]
+      have := densityTailConstant_nonneg
+      linarith
+    apply (palindrome_cost_mgf_coarse d e momentBase momentBase_one_le momentBase_log).trans (Real.exp_le_exp.mpr _)
+    have hD0 : 0 ≤ 1+heightDecay 1 := by have := heightDecay_nonneg 1; linarith
+    have ha := mul_le_mul_of_nonneg_left hlow (mul_nonneg (by norm_num : (0:ℝ)≤256) hD0)
+    have hb := mul_le_mul_of_nonneg_right hD hpow
+    have hc := mul_le_mul_of_nonneg_right (ha.trans hb) hk0
+    nlinarith
+
+lemma cost_bound (d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) :
+    finiteMean (fun ω : BenesCoins d => momentBase^(palindromeCost d e ω)) ≤
+      Real.exp (densityConstant*Fintype.card ι*((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ)) := by
+  by_cases hk : 0<Fintype.card ι
+  · exact cost_bound_pos d e hk
+  · have hz : Fintype.card ι=0 := by omega
+    simpa only [hz,Nat.cast_zero,mul_zero,zero_mul,Real.exp_zero] using
+      palindrome_cost_mgf_coarse d e momentBase momentBase_one_le momentBase_log
+
+
+noncomputable def tailConstant : ℝ :=
+  68*Real.exp (1/64)/(1-Real.exp (-1/64))
+
+lemma tailConstant_pos : 0<tailConstant := by
+  have hh:=Real.exp_lt_one_iff.mpr (by norm_num : (-1:ℝ)/64<0)
+  exact div_pos (by positivity) (by linarith)
+
+lemma decay_bound (J : ℕ) (hJ : 1≤J) :
+    heightDecay (J-1) ≤ tailConstant*(2:ℝ)^(-(1/1000:ℝ)*J) := by
+  have hd : 0<1-Real.exp (-1/64:ℝ) := by
+    have h:=Real.exp_lt_one_iff.mpr (by norm_num : (-1:ℝ)/64<0)
+    linarith
+  have hlog : Real.log 2≤1 := by
+    have hh:=Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
+    norm_num at hh ⊢
+    exact hh
+  rw [heightDecay,tailConstant,div_mul_eq_mul_div,mul_assoc,
+    Real.rpow_def_of_pos (by norm_num : (0:ℝ)<2)]
+  apply div_le_div_of_nonneg_right _ hd.le
+  apply mul_le_mul_of_nonneg_left _ (by norm_num : (0:ℝ)≤68)
+  rw [←Real.exp_add]
+  apply Real.exp_le_exp.mpr
+  rw [Nat.cast_sub hJ,Nat.cast_one]
+  nlinarith [Nat.cast_nonneg (α:=ℝ) J,
+    mul_le_mul_of_nonneg_right hlog (Nat.cast_nonneg J)]
+
+noncomputable def constant : ℝ := (tailConstant+densityConstant)/Real.log 2
+
+lemma constant_pos : 0<constant := by
+  exact div_pos (add_pos tailConstant_pos densityConstant_pos) (Real.log_pos (by norm_num))
+
+lemma log_mgf_le {Ω : Type*} [Fintype Ω] [Nonempty Ω] (f : Ω → ℕ) (A : ℝ)
+    (h : finiteMean (fun ω => momentBase^(f ω))≤Real.exp A) :
+    Real.log (finiteMean (fun ω => momentBase^(f ω)))≤A := by
+  apply (Real.log_le_iff_le_exp (finiteMean_pos (fun ω => pow_pos
+    (lt_of_lt_of_le zero_lt_one momentBase_one_le) _))).mpr h
+
+lemma high_log {Ω : Type*} [Fintype Ω] [Nonempty Ω] (f : Ω → ℕ) (k J : ℕ)
+    (hJ : 1≤J) (h : finiteMean (fun ω => momentBase^(f ω))≤Real.exp (k*heightDecay (J-1))) :
+    Real.log (finiteMean (fun ω => momentBase^(f ω)))/Real.log 2 ≤
+      constant*k*(2:ℝ)^(-(1/1000:ℝ)*J) := by
+  have hl:=log_mgf_le f _ h
+  have hp : 0<Real.log 2 := Real.log_pos (by norm_num)
+  apply (div_le_iff₀ hp).mpr
+  have hh:=mul_le_mul_of_nonneg_left (decay_bound J hJ) (Nat.cast_nonneg k : (0:ℝ)≤_)
+  have hinc : (k:ℝ)*tailConstant*(2:ℝ)^(-(1/1000:ℝ)*J) ≤
+      (tailConstant+densityConstant)*k*(2:ℝ)^(-(1/1000:ℝ)*J) := by
+    have hnon : 0≤densityConstant*k*(2:ℝ)^(-(1/1000:ℝ)*J) := by
+      exact mul_nonneg (mul_nonneg densityConstant_pos.le (Nat.cast_nonneg _)) (Real.rpow_nonneg (by norm_num) _)
+    nlinarith
+  have hfinal:=hl.trans (hh.trans (by simpa only [mul_assoc] using hinc))
+  refine hfinal.trans_eq ?_
+  unfold constant
+  field_simp
+
+lemma sparse_log (d k : ℕ) (x : Fin k ↪ Card d) :
+    Real.log (finiteMean (fun ω : BenesCoins d => momentBase^(palindromeCost d x ω)))/Real.log 2 ≤
+      constant*k*((k:ℝ)/(2:ℝ)^d)^(1/512:ℝ) := by
+  have h:=log_mgf_le (fun ω : BenesCoins d => palindromeCost d x ω) _ (cost_bound d x)
+  simp only [Fintype.card_fin] at h
+  have hp : 0<Real.log 2 := Real.log_pos (by norm_num)
+  apply (div_le_iff₀ hp).mpr
+  have hn : 0≤tailConstant*k*((k:ℝ)/(2:ℝ)^d)^(1/512:ℝ) := by
+    exact mul_nonneg (mul_nonneg tailConstant_pos.le (Nat.cast_nonneg _)) (Real.rpow_nonneg (by positivity) _)
+  unfold constant
+  field_simp
+  nlinarith
+
+theorem high_tail : MainStatement := by
+  refine ⟨1/1000,1/512,constant,by norm_num,by norm_num,constant_pos,2,by decide,?_,?_,?_⟩
+  · intro d k x J hJ
+    apply high_log _ k J (by omega)
+    simpa only [Fintype.card_fin] using high_unconditional d J hJ x momentBase momentBase_one_le
+      (by have hh:=momentBase_log; linarith)
+  · intro d k x
+    exact sparse_log d k x
+  · intro r s k x J hJ hs X lo
+    apply high_log _ k J (by omega)
+    simpa only [Fintype.card_fin] using high_partial r s J hJ hs x X lo momentBase momentBase_one_le
+      (by have hh:=momentBase_log; linarith)
+
+end Thorp.HighHeight
+
+end ThorpNine.HighTail
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/HighTail/Model.lean b/lean/OAI/Probability/ThorpRouting/HighTail/Model.lean
index 7e275655ad3a0965e9b5c0f12416579515c297ed..1aefa483712b30459b9c99ad4c1b3c47147204dd 100644
--- a/lean/OAI/Probability/ThorpRouting/HighTail/Model.lean
+++ b/lean/OAI/Probability/ThorpRouting/HighTail/Model.lean
@@ -1,298 +1,298 @@
-import Mathlib
-
-namespace OAI
-
-namespace ThorpNine.HighTail
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-
-abbrev Card (d : ℕ) := Fin d → Bool
-
-def switchFun {d : ℕ} (ξ : Card d → Bool) (x : Card (d + 1)) : Card (d + 1) :=
-  Fin.cons (Bool.xor (x 0) (ξ (Fin.tail x))) (Fin.tail x)
-
-theorem switchFun_involutive {d : ℕ} (ξ : Card d → Bool) :
-    Function.Involutive (switchFun ξ) := by
-  intro x
-  funext i
-  refine Fin.cases ?_ (fun j => ?_) i
-  · simp only [switchFun, Fin.cons_zero, Fin.tail_cons]
-    cases x 0 <;> cases ξ (Fin.tail x) <;> rfl
-  · simp [switchFun, Fin.tail]
-
-def pairSwitch {d : ℕ} (ξ : Card d → Bool) : Equiv.Perm (Card (d + 1)) :=
-  { toFun := switchFun ξ
-    invFun := switchFun ξ
-    left_inv := switchFun_involutive ξ
-    right_inv := switchFun_involutive ξ }
-
-def Butterfly : ℕ → Type
-  | 0 => Unit
-  | d + 1 => (Card d → Bool) × (Bool → Butterfly d)
-
-def childLift {d : ℕ} (p : Bool → Equiv.Perm (Card d)) : Equiv.Perm (Card (d + 1)) where
-  toFun x := Fin.cons (x 0) (p (x 0) (Fin.tail x))
-  invFun x := Fin.cons (x 0) ((p (x 0)).symm (Fin.tail x))
-  left_inv x := by simp only [Fin.cons_zero, Fin.tail_cons, Equiv.symm_apply_apply, Fin.cons_self_tail]
-  right_inv x := by simp only [Fin.cons_zero, Fin.tail_cons, Equiv.apply_symm_apply, Fin.cons_self_tail]
-
-def butterflyPerm : (d : ℕ) → Butterfly d → Equiv.Perm (Card d)
-  | 0, _ => 1
-  | d + 1, b => pairSwitch b.1 * childLift (fun ε => butterflyPerm d (b.2 ε))
-
-noncomputable def finiteMean {Ω : Type*} [Fintype Ω] (f : Ω → ℝ) : ℝ :=
-  (∑ ω, f ω) / Fintype.card Ω
-
-def SwitchIndex : ℕ → Type
-  | 0 => Empty
-  | d + 1 => Sum (Card d) (Bool × SwitchIndex d)
-
-instance (d : ℕ) : Fintype (SwitchIndex d) := by
-  induction d with
-  | zero => exact inferInstanceAs (Fintype Empty)
-  | succ d ih => exact inferInstanceAs (Fintype (Sum (Card d) (Bool × SwitchIndex d)))
-
-instance (d : ℕ) : DecidableEq (SwitchIndex d) := by
-  induction d with
-  | zero => exact inferInstanceAs (DecidableEq Empty)
-  | succ d ih => exact inferInstanceAs (DecidableEq (Sum (Card d) (Bool × SwitchIndex d)))
-
-def decodeButterfly : (d : ℕ) → (SwitchIndex d → Bool) → Butterfly d
-  | 0, _ => ()
-  | d + 1, ω => (fun y => ω (Sum.inl y), fun ε =>
-      decodeButterfly d (fun i => ω (Sum.inr (ε,i))))
-
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-noncomputable def orbitSet (p : Equiv.Perm α) (x : α) : Finset α :=
-  Finset.univ.filter (p.SameCycle x)
-
-end Thorp
-
-namespace Thorp.RoutingNetwork
-open scoped BigOperators
-open Filter
-variable {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι] [DecidableEq ι]
-variable {A : ℕ}
-
-abbrev SelectedCycle (p : Equiv.Perm (α)) (S : Finset (α)) :=
-  {C : Finset (α) // (∃ x, C = orbitSet p x) ∧ C ⊆ S}
-
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) :
-    Fintype (SelectedCycle p S) := Fintype.ofFinite _
-
-end Thorp.RoutingNetwork
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-
-def cardSplit : (r s : ℕ) → Card (s+r) ≃ Card r × Card s
-  | 0, s =>
-    { toFun := fun x => ((fun i => Fin.elim0 i), x)
-      invFun := fun x => x.2
-      left_inv := fun _ => rfl
-      right_inv := fun x => by
-        apply Prod.ext
-        · exact Subsingleton.elim _ _
-        · rfl }
-  | r+1, s =>
-    { toFun := fun x => (Fin.cons (x 0) ((cardSplit r s (Fin.tail x)).1),
-        (cardSplit r s (Fin.tail x)).2)
-      invFun := fun x => Fin.cons (x.1 0) ((cardSplit r s).symm (Fin.tail x.1,x.2))
-      left_inv := by
-        intro x
-        simp only [Fin.cons_zero, Fin.tail_cons, Prod.mk.eta, Equiv.symm_apply_apply, Fin.cons_self_tail]
-      right_inv := by
-        intro x
-        simp only [Fin.cons_zero, Fin.tail_cons, Equiv.apply_symm_apply, Fin.cons_self_tail] }
-
-def assembleBits : (r s : ℕ) → (Card r × SwitchIndex s → Bool) →
-    (Card s × SwitchIndex r → Bool) → SwitchIndex (s+r) → Bool
-  | 0, _, lo, _, i => lo ((fun j => Fin.elim0 j), i)
-  | r+1, s, _, hi, Sum.inl y => hi ((cardSplit r s y).2,Sum.inl (cardSplit r s y).1)
-  | r+1, s, lo, hi, Sum.inr (b,i) => assembleBits r s
-      (fun j => lo (Fin.cons b j.1,j.2)) (fun j => hi (j.1,Sum.inr (b,j.2))) i
-
-
-section
-variable {β : Type*} [Fintype β] [DecidableEq β]
-
-def pairLayer (ξ : β → Bool) : Equiv.Perm (Bool × β) where
-  toFun x := (Bool.xor x.1 (ξ x.2),x.2)
-  invFun x := (Bool.xor x.1 (ξ x.2),x.2)
-  left_inv x := by
-    rcases x with ⟨b,x⟩
-    change (Bool.xor (Bool.xor b (ξ x)) (ξ x),x) = (b,x)
-    cases b <;> cases ξ x <;> rfl
-  right_inv x := by
-    rcases x with ⟨b,x⟩
-    change (Bool.xor (Bool.xor b (ξ x)) (ξ x),x) = (b,x)
-    cases b <;> cases ξ x <;> rfl
-
-end
-
-def coinStepEquiv (d : ℕ) : (SwitchIndex (d+1) → Bool) ≃
-    ((SwitchIndex d → Bool) × (SwitchIndex d → Bool)) × (Card d → Bool) where
-  toFun ω := ((fun i => ω (Sum.inr (false,i)),fun i => ω (Sum.inr (true,i))),
-    fun x => ω (Sum.inl x))
-  invFun ω := Sum.elim ω.2 (fun bi => if bi.1 then ω.1.2 bi.2 else ω.1.1 bi.2)
-  left_inv ω := by
-    funext i
-    cases i with
-    | inl x => rfl
-    | inr bi => cases bi with | mk b i => cases b <;> rfl
-  right_inv ω := by
-    rcases ω with ⟨⟨ω₀,ω₁⟩,ξ⟩
-    rfl
-
-def headTailEquiv (d : ℕ) : Card (d+1) ≃ Bool × Card d :=
-  (Fin.consEquiv (fun _ : Fin (d+1) => Bool)).symm
-
-end Thorp
-
-namespace Thorp.CycleColoring
-open scoped BigOperators
-open Filter
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-def alternatingPerm (p : Equiv.Perm α) : Equiv.Perm (Bool × α) where
-  toFun x := if x.1 then (false,p x.2) else (true,x.2)
-  invFun x := if x.1 then (false,x.2) else (true,p.symm x.2)
-  left_inv x := by rcases x with ⟨b,x⟩; cases b <;> simp
-  right_inv x := by rcases x with ⟨b,x⟩; cases b <;> simp
-
-end Thorp.CycleColoring
-
-namespace Thorp.PairRouting
-open scoped BigOperators
-open Filter
-variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
-
-def required (x : ι → Bool × α) (c : ι → Bool) (i : ι) : Bool := Bool.xor (x i).1 (c i)
-
-def Compatible (x : ι → Bool × α) (c : ι → Bool) : Prop :=
-  ∀ i j, (x i).2 = (x j).2 → required x c i = required x c j
-
-def colors (x : ι → Bool × α) (ξ : α → Bool) (i : ι) : Bool :=
-  Bool.xor (x i).1 (ξ (x i).2)
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma required_colors (x : ι → Bool × α) (ξ : α → Bool) (i : ι) :
-    required x (colors x ξ) i = ξ (x i).2 := by
-  simp only [required,colors]
-  cases (x i).1 <;> cases ξ (x i).2 <;> rfl
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma colors_compatible (x : ι → Bool × α) (ξ : α → Bool) : Compatible x (colors x ξ) := by
-  intro i j he
-  simp only [required_colors,he]
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma Compatible.tail_injective {x : ι ↪ Bool × α} {c : ι → Bool} (hc : Compatible x c) (b : Bool) :
-    Function.Injective (fun i : {i : ι // c i = b} => (x i.val).2) := by
-  intro i j he
-  apply Subtype.ext
-  apply x.injective
-  apply Prod.ext
-  · have hh := hc i j he
-    simp only [required,i.property,j.property] at hh
-    cases h₀ : (x i.val).1 <;> cases h₁ : (x j.val).1 <;> cases b <;> simp_all
-  · exact he
-
-def childEmbedding (x : ι ↪ Bool × α) (c : ι → Bool) (hc : Compatible x c) (b : Bool) :
-    {i : ι // c i = b} ↪ α := ⟨fun i => (x i.val).2,hc.tail_injective b⟩
-
-noncomputable def labelSet (e : ι ↪ Bool × α) : Finset (Bool × α) := Finset.univ.image e
-
-def alternatingProjection (p : Bool → Equiv.Perm α) : Equiv.Perm α := (p false).symm * p true
-
-noncomputable def alternatingCycles (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) : ℕ :=
-  Fintype.card (RoutingNetwork.SelectedCycle
-    (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e))
-
-def switchedEmbedding (e : ι ↪ Bool × α) (ξ : α → Bool) : ι ↪ Bool × α :=
-  e.trans (pairLayer ξ).toEmbedding
-
-end Thorp.PairRouting
-
-namespace Thorp
-
-section
-open scoped BigOperators
-open Filter
-
-abbrev BenesCoins (d : ℕ) := (SwitchIndex d → Bool) × (SwitchIndex d → Bool)
-
-def palindromePerm (d : ℕ) (ω : BenesCoins d) : Equiv.Perm (Card d) :=
-  butterflyPerm d (decodeButterfly d ω.1) * (butterflyPerm d (decodeButterfly d ω.2)).symm
-
-def sandwichShuffle {A B C D E F : Type*} :
-    ((A × B) × C) × ((D × E) × F) ≃ ((A × D) × (B × E)) × (F × C) where
-  toFun x := (((x.1.1.1,x.2.1.1),(x.1.1.2,x.2.1.2)),(x.2.2,x.1.2))
-  invFun x := (((x.1.1.1,x.1.2.1),x.2.2),((x.1.1.2,x.1.2.2),x.2.1))
-  left_inv _ := rfl
-  right_inv _ := rfl
-
-def benesStepEquiv (d : ℕ) : BenesCoins (d+1) ≃
-    (BenesCoins d × BenesCoins d) × ((Card d → Bool) × (Card d → Bool)) :=
-  (Equiv.prodCongr (coinStepEquiv d) (coinStepEquiv d)).trans sandwichShuffle
-
-noncomputable def palindromeCost : (d : ℕ) → {ι : Type*} → [Fintype ι] →
-    (ι ↪ Card d) → BenesCoins d → ℕ
-  | 0, _, _, _, _ => 0
-  | d+1, _, _, e, ω =>
-      let σ := benesStepEquiv d ω
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x σ.2.1
-      let hc := PairRouting.colors_compatible x σ.2.1
-      PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
-          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) +
-        palindromeCost d (PairRouting.childEmbedding x c hc false) σ.1.1 +
-        palindromeCost d (PairRouting.childEmbedding x c hc true) σ.1.2
-
-end
-
-noncomputable def palindromeLowCost (H : ℕ) : (d : ℕ) → {ι : Type*} → [Fintype ι] →
-    (ι ↪ Card d) → BenesCoins d → ℕ
-  | 0, _, _, _, _ => 0
-  | d+1, _, _, e, ω =>
-      let σ := benesStepEquiv d ω
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x σ.2.1
-      let hc := PairRouting.colors_compatible x σ.2.1
-      (if d+1 ≤ H then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
-          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) else 0) +
-        palindromeLowCost H d (PairRouting.childEmbedding x c hc false) σ.1.1 +
-        palindromeLowCost H d (PairRouting.childEmbedding x c hc true) σ.1.2
-
-end Thorp
-
-namespace Thorp.HighHeight
-open scoped BigOperators Classical
-
-noncomputable def highCost (J d : ℕ) {ι : Type*} [Fintype ι]
-    (x : ι ↪ Card d) (ω : BenesCoins d) : ℕ :=
-  palindromeCost d x ω-palindromeLowCost (J-1) d x ω
-
-def MainStatement : Prop :=
-  ∃ b δ C : ℝ, 0<b ∧ 0<δ ∧ 0<C ∧ ∃ J₀ : ℕ, 1≤J₀ ∧
-    (∀ d k : ℕ, ∀ x : Fin k ↪ Card d, ∀ J : ℕ, J₀≤J →
-      Real.log (finiteMean (fun ω : BenesCoins d => ((2:ℝ)^b)^(highCost J d x ω)))/Real.log 2 ≤
-        C*k*(2:ℝ)^(-b*J)) ∧
-    (∀ d k : ℕ, ∀ x : Fin k ↪ Card d,
-      Real.log (finiteMean (fun ω : BenesCoins d => ((2:ℝ)^b)^(palindromeCost d x ω)))/Real.log 2 ≤
-        C*k*((k:ℝ)/(2:ℝ)^d)^δ) ∧
-    (∀ r s k : ℕ, ∀ x : Fin k ↪ Card (s+r), ∀ J : ℕ, J₀≤J → 2*s<J →
-      ∀ (X : SwitchIndex (s+r) → Bool) (lo : Card r × SwitchIndex s → Bool),
-      Real.log (finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-        ((2:ℝ)^b)^(highCost J (s+r) x (assembleBits r s lo hi,X))))/Real.log 2 ≤
-          C*k*(2:ℝ)^(-b*J))
-
-end Thorp.HighHeight
-
-end ThorpNine.HighTail
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+namespace ThorpNine.HighTail
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+
+abbrev Card (d : ℕ) := Fin d → Bool
+
+def switchFun {d : ℕ} (ξ : Card d → Bool) (x : Card (d + 1)) : Card (d + 1) :=
+  Fin.cons (Bool.xor (x 0) (ξ (Fin.tail x))) (Fin.tail x)
+
+theorem switchFun_involutive {d : ℕ} (ξ : Card d → Bool) :
+    Function.Involutive (switchFun ξ) := by
+  intro x
+  funext i
+  refine Fin.cases ?_ (fun j => ?_) i
+  · simp only [switchFun, Fin.cons_zero, Fin.tail_cons]
+    cases x 0 <;> cases ξ (Fin.tail x) <;> rfl
+  · simp [switchFun, Fin.tail]
+
+def pairSwitch {d : ℕ} (ξ : Card d → Bool) : Equiv.Perm (Card (d + 1)) :=
+  { toFun := switchFun ξ
+    invFun := switchFun ξ
+    left_inv := switchFun_involutive ξ
+    right_inv := switchFun_involutive ξ }
+
+def Butterfly : ℕ → Type
+  | 0 => Unit
+  | d + 1 => (Card d → Bool) × (Bool → Butterfly d)
+
+def childLift {d : ℕ} (p : Bool → Equiv.Perm (Card d)) : Equiv.Perm (Card (d + 1)) where
+  toFun x := Fin.cons (x 0) (p (x 0) (Fin.tail x))
+  invFun x := Fin.cons (x 0) ((p (x 0)).symm (Fin.tail x))
+  left_inv x := by simp only [Fin.cons_zero, Fin.tail_cons, Equiv.symm_apply_apply, Fin.cons_self_tail]
+  right_inv x := by simp only [Fin.cons_zero, Fin.tail_cons, Equiv.apply_symm_apply, Fin.cons_self_tail]
+
+def butterflyPerm : (d : ℕ) → Butterfly d → Equiv.Perm (Card d)
+  | 0, _ => 1
+  | d + 1, b => pairSwitch b.1 * childLift (fun ε => butterflyPerm d (b.2 ε))
+
+noncomputable def finiteMean {Ω : Type*} [Fintype Ω] (f : Ω → ℝ) : ℝ :=
+  (∑ ω, f ω) / Fintype.card Ω
+
+def SwitchIndex : ℕ → Type
+  | 0 => Empty
+  | d + 1 => Sum (Card d) (Bool × SwitchIndex d)
+
+instance (d : ℕ) : Fintype (SwitchIndex d) := by
+  induction d with
+  | zero => exact inferInstanceAs (Fintype Empty)
+  | succ d ih => exact inferInstanceAs (Fintype (Sum (Card d) (Bool × SwitchIndex d)))
+
+instance (d : ℕ) : DecidableEq (SwitchIndex d) := by
+  induction d with
+  | zero => exact inferInstanceAs (DecidableEq Empty)
+  | succ d ih => exact inferInstanceAs (DecidableEq (Sum (Card d) (Bool × SwitchIndex d)))
+
+def decodeButterfly : (d : ℕ) → (SwitchIndex d → Bool) → Butterfly d
+  | 0, _ => ()
+  | d + 1, ω => (fun y => ω (Sum.inl y), fun ε =>
+      decodeButterfly d (fun i => ω (Sum.inr (ε,i))))
+
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+noncomputable def orbitSet (p : Equiv.Perm α) (x : α) : Finset α :=
+  Finset.univ.filter (p.SameCycle x)
+
+end Thorp
+
+namespace Thorp.RoutingNetwork
+open scoped BigOperators
+open Filter
+variable {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι] [DecidableEq ι]
+variable {A : ℕ}
+
+abbrev SelectedCycle (p : Equiv.Perm (α)) (S : Finset (α)) :=
+  {C : Finset (α) // (∃ x, C = orbitSet p x) ∧ C ⊆ S}
+
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) :
+    Fintype (SelectedCycle p S) := Fintype.ofFinite _
+
+end Thorp.RoutingNetwork
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+
+def cardSplit : (r s : ℕ) → Card (s+r) ≃ Card r × Card s
+  | 0, s =>
+    { toFun := fun x => ((fun i => Fin.elim0 i), x)
+      invFun := fun x => x.2
+      left_inv := fun _ => rfl
+      right_inv := fun x => by
+        apply Prod.ext
+        · exact Subsingleton.elim _ _
+        · rfl }
+  | r+1, s =>
+    { toFun := fun x => (Fin.cons (x 0) ((cardSplit r s (Fin.tail x)).1),
+        (cardSplit r s (Fin.tail x)).2)
+      invFun := fun x => Fin.cons (x.1 0) ((cardSplit r s).symm (Fin.tail x.1,x.2))
+      left_inv := by
+        intro x
+        simp only [Fin.cons_zero, Fin.tail_cons, Prod.mk.eta, Equiv.symm_apply_apply, Fin.cons_self_tail]
+      right_inv := by
+        intro x
+        simp only [Fin.cons_zero, Fin.tail_cons, Equiv.apply_symm_apply, Fin.cons_self_tail] }
+
+def assembleBits : (r s : ℕ) → (Card r × SwitchIndex s → Bool) →
+    (Card s × SwitchIndex r → Bool) → SwitchIndex (s+r) → Bool
+  | 0, _, lo, _, i => lo ((fun j => Fin.elim0 j), i)
+  | r+1, s, _, hi, Sum.inl y => hi ((cardSplit r s y).2,Sum.inl (cardSplit r s y).1)
+  | r+1, s, lo, hi, Sum.inr (b,i) => assembleBits r s
+      (fun j => lo (Fin.cons b j.1,j.2)) (fun j => hi (j.1,Sum.inr (b,j.2))) i
+
+
+section
+variable {β : Type*} [Fintype β] [DecidableEq β]
+
+def pairLayer (ξ : β → Bool) : Equiv.Perm (Bool × β) where
+  toFun x := (Bool.xor x.1 (ξ x.2),x.2)
+  invFun x := (Bool.xor x.1 (ξ x.2),x.2)
+  left_inv x := by
+    rcases x with ⟨b,x⟩
+    change (Bool.xor (Bool.xor b (ξ x)) (ξ x),x) = (b,x)
+    cases b <;> cases ξ x <;> rfl
+  right_inv x := by
+    rcases x with ⟨b,x⟩
+    change (Bool.xor (Bool.xor b (ξ x)) (ξ x),x) = (b,x)
+    cases b <;> cases ξ x <;> rfl
+
+end
+
+def coinStepEquiv (d : ℕ) : (SwitchIndex (d+1) → Bool) ≃
+    ((SwitchIndex d → Bool) × (SwitchIndex d → Bool)) × (Card d → Bool) where
+  toFun ω := ((fun i => ω (Sum.inr (false,i)),fun i => ω (Sum.inr (true,i))),
+    fun x => ω (Sum.inl x))
+  invFun ω := Sum.elim ω.2 (fun bi => if bi.1 then ω.1.2 bi.2 else ω.1.1 bi.2)
+  left_inv ω := by
+    funext i
+    cases i with
+    | inl x => rfl
+    | inr bi => cases bi with | mk b i => cases b <;> rfl
+  right_inv ω := by
+    rcases ω with ⟨⟨ω₀,ω₁⟩,ξ⟩
+    rfl
+
+def headTailEquiv (d : ℕ) : Card (d+1) ≃ Bool × Card d :=
+  (Fin.consEquiv (fun _ : Fin (d+1) => Bool)).symm
+
+end Thorp
+
+namespace Thorp.CycleColoring
+open scoped BigOperators
+open Filter
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+def alternatingPerm (p : Equiv.Perm α) : Equiv.Perm (Bool × α) where
+  toFun x := if x.1 then (false,p x.2) else (true,x.2)
+  invFun x := if x.1 then (false,x.2) else (true,p.symm x.2)
+  left_inv x := by rcases x with ⟨b,x⟩; cases b <;> simp
+  right_inv x := by rcases x with ⟨b,x⟩; cases b <;> simp
+
+end Thorp.CycleColoring
+
+namespace Thorp.PairRouting
+open scoped BigOperators
+open Filter
+variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
+
+def required (x : ι → Bool × α) (c : ι → Bool) (i : ι) : Bool := Bool.xor (x i).1 (c i)
+
+def Compatible (x : ι → Bool × α) (c : ι → Bool) : Prop :=
+  ∀ i j, (x i).2 = (x j).2 → required x c i = required x c j
+
+def colors (x : ι → Bool × α) (ξ : α → Bool) (i : ι) : Bool :=
+  Bool.xor (x i).1 (ξ (x i).2)
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma required_colors (x : ι → Bool × α) (ξ : α → Bool) (i : ι) :
+    required x (colors x ξ) i = ξ (x i).2 := by
+  simp only [required,colors]
+  cases (x i).1 <;> cases ξ (x i).2 <;> rfl
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma colors_compatible (x : ι → Bool × α) (ξ : α → Bool) : Compatible x (colors x ξ) := by
+  intro i j he
+  simp only [required_colors,he]
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma Compatible.tail_injective {x : ι ↪ Bool × α} {c : ι → Bool} (hc : Compatible x c) (b : Bool) :
+    Function.Injective (fun i : {i : ι // c i = b} => (x i.val).2) := by
+  intro i j he
+  apply Subtype.ext
+  apply x.injective
+  apply Prod.ext
+  · have hh := hc i j he
+    simp only [required,i.property,j.property] at hh
+    cases h₀ : (x i.val).1 <;> cases h₁ : (x j.val).1 <;> cases b <;> simp_all
+  · exact he
+
+def childEmbedding (x : ι ↪ Bool × α) (c : ι → Bool) (hc : Compatible x c) (b : Bool) :
+    {i : ι // c i = b} ↪ α := ⟨fun i => (x i.val).2,hc.tail_injective b⟩
+
+noncomputable def labelSet (e : ι ↪ Bool × α) : Finset (Bool × α) := Finset.univ.image e
+
+def alternatingProjection (p : Bool → Equiv.Perm α) : Equiv.Perm α := (p false).symm * p true
+
+noncomputable def alternatingCycles (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) : ℕ :=
+  Fintype.card (RoutingNetwork.SelectedCycle
+    (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e))
+
+def switchedEmbedding (e : ι ↪ Bool × α) (ξ : α → Bool) : ι ↪ Bool × α :=
+  e.trans (pairLayer ξ).toEmbedding
+
+end Thorp.PairRouting
+
+namespace Thorp
+
+section
+open scoped BigOperators
+open Filter
+
+abbrev BenesCoins (d : ℕ) := (SwitchIndex d → Bool) × (SwitchIndex d → Bool)
+
+def palindromePerm (d : ℕ) (ω : BenesCoins d) : Equiv.Perm (Card d) :=
+  butterflyPerm d (decodeButterfly d ω.1) * (butterflyPerm d (decodeButterfly d ω.2)).symm
+
+def sandwichShuffle {A B C D E F : Type*} :
+    ((A × B) × C) × ((D × E) × F) ≃ ((A × D) × (B × E)) × (F × C) where
+  toFun x := (((x.1.1.1,x.2.1.1),(x.1.1.2,x.2.1.2)),(x.2.2,x.1.2))
+  invFun x := (((x.1.1.1,x.1.2.1),x.2.2),((x.1.1.2,x.1.2.2),x.2.1))
+  left_inv _ := rfl
+  right_inv _ := rfl
+
+def benesStepEquiv (d : ℕ) : BenesCoins (d+1) ≃
+    (BenesCoins d × BenesCoins d) × ((Card d → Bool) × (Card d → Bool)) :=
+  (Equiv.prodCongr (coinStepEquiv d) (coinStepEquiv d)).trans sandwichShuffle
+
+noncomputable def palindromeCost : (d : ℕ) → {ι : Type*} → [Fintype ι] →
+    (ι ↪ Card d) → BenesCoins d → ℕ
+  | 0, _, _, _, _ => 0
+  | d+1, _, _, e, ω =>
+      let σ := benesStepEquiv d ω
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x σ.2.1
+      let hc := PairRouting.colors_compatible x σ.2.1
+      PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
+          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) +
+        palindromeCost d (PairRouting.childEmbedding x c hc false) σ.1.1 +
+        palindromeCost d (PairRouting.childEmbedding x c hc true) σ.1.2
+
+end
+
+noncomputable def palindromeLowCost (H : ℕ) : (d : ℕ) → {ι : Type*} → [Fintype ι] →
+    (ι ↪ Card d) → BenesCoins d → ℕ
+  | 0, _, _, _, _ => 0
+  | d+1, _, _, e, ω =>
+      let σ := benesStepEquiv d ω
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x σ.2.1
+      let hc := PairRouting.colors_compatible x σ.2.1
+      (if d+1 ≤ H then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
+          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) else 0) +
+        palindromeLowCost H d (PairRouting.childEmbedding x c hc false) σ.1.1 +
+        palindromeLowCost H d (PairRouting.childEmbedding x c hc true) σ.1.2
+
+end Thorp
+
+namespace Thorp.HighHeight
+open scoped BigOperators Classical
+
+noncomputable def highCost (J d : ℕ) {ι : Type*} [Fintype ι]
+    (x : ι ↪ Card d) (ω : BenesCoins d) : ℕ :=
+  palindromeCost d x ω-palindromeLowCost (J-1) d x ω
+
+def MainStatement : Prop :=
+  ∃ b δ C : ℝ, 0<b ∧ 0<δ ∧ 0<C ∧ ∃ J₀ : ℕ, 1≤J₀ ∧
+    (∀ d k : ℕ, ∀ x : Fin k ↪ Card d, ∀ J : ℕ, J₀≤J →
+      Real.log (finiteMean (fun ω : BenesCoins d => ((2:ℝ)^b)^(highCost J d x ω)))/Real.log 2 ≤
+        C*k*(2:ℝ)^(-b*J)) ∧
+    (∀ d k : ℕ, ∀ x : Fin k ↪ Card d,
+      Real.log (finiteMean (fun ω : BenesCoins d => ((2:ℝ)^b)^(palindromeCost d x ω)))/Real.log 2 ≤
+        C*k*((k:ℝ)/(2:ℝ)^d)^δ) ∧
+    (∀ r s k : ℕ, ∀ x : Fin k ↪ Card (s+r), ∀ J : ℕ, J₀≤J → 2*s<J →
+      ∀ (X : SwitchIndex (s+r) → Bool) (lo : Card r × SwitchIndex s → Bool),
+      Real.log (finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+        ((2:ℝ)^b)^(highCost J (s+r) x (assembleBits r s lo hi,X))))/Real.log 2 ≤
+          C*k*(2:ℝ)^(-b*J))
+
+end Thorp.HighHeight
+
+end ThorpNine.HighTail
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/HighTail/ParallelFollow.lean b/lean/OAI/Probability/ThorpRouting/HighTail/ParallelFollow.lean
index 978862fb02d85789e651e412399b357207788179..7ce00a822a82fc1472658f110633541e77528646 100644
--- a/lean/OAI/Probability/ThorpRouting/HighTail/ParallelFollow.lean
+++ b/lean/OAI/Probability/ThorpRouting/HighTail/ParallelFollow.lean
@@ -1,827 +1,827 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.HighTail.PriorPaths
-
-namespace OAI
-
-namespace ThorpNine.HighTail
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-section ParallelButterfly
-variable {α B : Type*} [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] {d : ℕ}
-omit [DecidableEq B] in
-lemma parallel_entropy (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool) (S : Finset α) :
-    (∑ i : B × SwitchIndex d,
-      if (parallelUsers pre ω i).1 ∈ S ∧ (parallelUsers pre ω i).2 ∈ S then (1:ℝ) else 0) ≤
-      (S.card : ℝ) * Real.log S.card / (2 * Real.log 2) := by
-  classical
-  rw [Fintype.sum_prod_type]
-  calc
-    _ ≤ ∑ b : B, (blockMarks pre S b).card * Real.log (blockMarks pre S b).card / (2*Real.log 2) := by
-      apply Finset.sum_le_sum
-      intro b _
-      simpa only [←mem_blockMarks,parallelUsers] using
-        marked_switch_entropy_bound d (decodeButterfly d (fun i => ω (b,i))) (blockMarks pre S b)
-    _ ≤ ∑ b : B, (blockMarks pre S b).card * Real.log S.card / (2*Real.log 2) := by
-      apply Finset.sum_le_sum
-      intro b _
-      by_cases hz : (blockMarks pre S b).card = 0
-      · simp [hz]
-      · apply div_le_div_of_nonneg_right _ (by positivity)
-        apply mul_le_mul_of_nonneg_left _ (by positivity)
-        exact Real.log_le_log (by exact_mod_cast Nat.pos_of_ne_zero hz)
-          (by exact_mod_cast card_blockMarks_le pre S b)
-    _ = _ := by rw [←Finset.sum_div,←Finset.sum_mul,←Nat.cast_sum,sum_card_blockMarks]
-
-noncomputable def parallelFollow (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x : α) (k : α → QueryTree (B × SwitchIndex d)) : QueryTree (B × SwitchIndex d) :=
-  QueryTree.followPath d (fun i => ((pre x).1,i)) (pre x).2 (fun y => k (post ((pre x).1,y)))
-
-omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
-lemma parallelFollow_succeeds (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
-    (parallelFollow pre post x k).succeeds ω = (k (relabeledPerm pre post ω x)).succeeds ω := by
-  exact QueryTree.followPath_succeeds _ _ _ _ _
-
-omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
-lemma parallelFollow_raw_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
-    (parallelFollow pre post x k).charge ω = (k (relabeledPerm pre post ω x)).charge ω := by
-  exact QueryTree.followPath_raw_charge _ _ _ _ _
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma parallelFollow_stale_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
-    (ω : B × SwitchIndex d → Bool) :
-    (parallelFollow pre post x k).staleCharge R ω =
-      (k (relabeledPerm pre post ω x)).staleCharge (R ∪ parallelPath pre ω x) ω := by
-  exact QueryTree.followPath_stale_charge _ _ _ _ _ _
-
-noncomputable def closingDomain (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) : Finset (B × SwitchIndex d) :=
-  (routeDomain d (pre x).2 (post.symm y).2).image (fun i => ((pre x).1,i))
-
-def closingValue (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (i : B × SwitchIndex d) : Bool := routeValue d (pre x).2 (post.symm y).2 i.2
-
-omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
-lemma relabeled_endpoint_pair (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (ω : B × SwitchIndex d → Bool) (x y : α) :
-    relabeledPerm pre post ω x = y ↔
-      (pre x).1 = (post.symm y).1 ∧
-      butterflyPerm d (decodeButterfly d (fun i => ω ((pre x).1,i))) (pre x).2 = (post.symm y).2 := by
-  change post ((pre x).1,_) = y ↔ _
-  rw [←post.eq_symm_apply,Prod.mk.injEq]
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma relabeled_endpoint_iff (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (ω : B × SwitchIndex d → Bool) (x y : α) :
-    relabeledPerm pre post ω x = y ↔
-      (pre x).1 = (post.symm y).1 ∧
-        ∀ i ∈ closingDomain pre post x y, ω i = closingValue pre post x y i := by
-  classical
-  rw [relabeled_endpoint_pair,butterfly_endpoint_iff]
-  simp only [closingDomain,Finset.forall_mem_image,closingValue]
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma closingDomain_eq (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (ω : B × SwitchIndex d → Bool) (x y : α) (h : relabeledPerm pre post ω x = y) :
-    closingDomain pre post x y = parallelPath pre ω x := by
-  rw [closingDomain,parallelPath,(relabeled_endpoint_pair _ _ _ _ _).mp h |>.2]
-
-noncomputable def parallelClose (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (k : QueryTree (B × SwitchIndex d)) : QueryTree (B × SwitchIndex d) :=
-  if (pre x).1 = (post.symm y).1 then
-    QueryTree.forceList (closingDomain pre post x y).toList (closingValue pre post x y) k
-  else QueryTree.reject
-
-omit [Fintype α] in
-lemma parallelClose_succeeds (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (k : QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
-    (parallelClose pre post x y k).succeeds ω =
-      if relabeledPerm pre post ω x = y then k.succeeds ω else false := by
-  classical
-  simp only [relabeled_endpoint_iff,parallelClose]
-  by_cases hb : (pre x).1 = (post.symm y).1
-  · rw [ite_eq_left hb,QueryTree.forceList_succeeds]
-    simp only [Finset.mem_toList,hb,true_and]
-  · simp [hb,QueryTree.succeeds]
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma parallelClose_raw_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (k : QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool)
-    (h : relabeledPerm pre post ω x = y) :
-    (parallelClose pre post x y k).charge ω = k.charge ω + d := by
-  have hh := (relabeled_endpoint_iff _ _ _ _ _).mp h
-  rw [parallelClose,ite_eq_left hh.1,QueryTree.forceList_raw_charge _ _ _ _
-    (by simpa only [Finset.mem_toList] using hh.2),Finset.length_toList,
-    closingDomain_eq _ _ _ _ _ h,card_parallelPath]
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma parallelClose_fresh_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (k : QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
-    (ω : B × SwitchIndex d → Bool) (h : relabeledPerm pre post ω x = y) :
-    (parallelClose pre post x y k).freshCharge R ω =
-      k.freshCharge (R ∪ parallelPath pre ω x) ω + (parallelPath pre ω x \ R).card := by
-  have hh := (relabeled_endpoint_iff _ _ _ _ _).mp h
-  rw [parallelClose,ite_eq_left hh.1,QueryTree.forceList_charge _ _ _ _ _
-    (by simpa only [Finset.mem_toList] using hh.2),Finset.toList_toFinset,
-    closingDomain_eq _ _ _ _ _ h]
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma parallelClose_stale_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (k : QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
-    (ω : B × SwitchIndex d → Bool) (h : relabeledPerm pre post ω x = y) :
-    (parallelClose pre post x y k).staleCharge R ω =
-      k.staleCharge (R ∪ parallelPath pre ω x) ω + (parallelPath pre ω x ∩ R).card := by
-  have h₀ := QueryTree.fresh_add_stale (parallelClose pre post x y k) R ω
-  have h₁ := QueryTree.fresh_add_stale k (R ∪ parallelPath pre ω x) ω
-  rw [parallelClose_raw_charge _ _ _ _ _ _ h,parallelClose_fresh_charge _ _ _ _ _ _ _ h] at h₀
-  have h₂ : (parallelPath pre ω x \ R).card + (parallelPath pre ω x ∩ R).card = d := by
-    rw [Finset.card_sdiff_add_card_inter,card_parallelPath]
-  omega
-
-noncomputable def parallelNetwork (pre : α ≃ B × Card d) (post : B × Card d ≃ α) :
-    RoutingNetwork α (B × SwitchIndex d) d where
-  perm := relabeledPerm pre post
-  path := parallelPath pre
-  users := parallelUsers pre
-  users_ne := parallelUsers_ne pre
-  mem_path := mem_parallelPath pre
-  card_path := card_parallelPath pre
-  entropy := parallel_entropy pre
-  follow := parallelFollow pre post
-  close := parallelClose pre post
-  follow_succeeds := parallelFollow_succeeds pre post
-  follow_raw_charge := parallelFollow_raw_charge pre post
-  follow_stale_charge := parallelFollow_stale_charge pre post
-  close_succeeds := parallelClose_succeeds pre post
-  close_raw_charge := parallelClose_raw_charge pre post
-  close_stale_charge := parallelClose_stale_charge pre post
-
-lemma parallel_cycle_log_mgf (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (S : Finset α) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    Real.log (finiteMean (fun ω : B × SwitchIndex d → Bool =>
-      q^(Fintype.card (RoutingNetwork.SelectedCycle (relabeledPerm pre post ω) S)))) ≤
-      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
-  exact (parallelNetwork pre post).butterfly_cycle_log_mgf S J hJ q hq
-
-end ParallelButterfly
-
-lemma butterfly_split (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
-    (hi : Card s × SwitchIndex r → Bool) (x : Card (s+r)) :
-    cardSplit r s (butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi)) x) =
-      let y := cardSplit r s x
-      let z := butterflyPerm s (decodeButterfly s (fun i => lo (y.1,i))) y.2
-      (butterflyPerm r (decodeButterfly r (fun i => hi (z,i))) y.1,z) := by
-  induction r with
-  | zero =>
-    apply Prod.ext
-    · funext i
-      exact Fin.elim0 i
-    · rfl
-  | succ r ih =>
-    let lo' : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons (x 0) j.1,j.2)
-    let hi' : Card s × SwitchIndex r → Bool := fun j => hi (j.1,Sum.inr (x 0,j.2))
-    have hh := ih lo' hi' (Fin.tail x)
-    let u := butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo' hi')) (Fin.tail x)
-    let y := cardSplit r s (Fin.tail x)
-    let z := butterflyPerm s (decodeButterfly s (fun i => lo' (y.1,i))) y.2
-    let w := butterflyPerm r (decodeButterfly r (fun i => hi' (z,i))) y.1
-    change cardSplit r s u = (w,z) at hh
-    change ((Fin.cons (Bool.xor (x 0) (hi ((cardSplit r s u).2,Sum.inl (cardSplit r s u).1)))
-        ((cardSplit r s u).1) : Card (r+1)), (cardSplit r s u).2) =
-      ((Fin.cons (Bool.xor (x 0) (hi (z,Sum.inl w))) w : Card (r+1)),z)
-    rw [hh]
-
-def splitSwitch : (r s : ℕ) → SwitchIndex (s+r) ≃
-    (Card r × SwitchIndex s) ⊕ (Card s × SwitchIndex r)
-  | 0, _ =>
-    { toFun := fun i => Sum.inl ((fun j => Fin.elim0 j),i)
-      invFun := fun i => match i with
-        | Sum.inl (_,j) => j
-        | Sum.inr (_,j) => nomatch j
-      left_inv := fun _ => rfl
-      right_inv := by
-        intro i
-        rcases i with ⟨x,j⟩ | ⟨x,j⟩
-        · exact congrArg Sum.inl (Prod.ext (Subsingleton.elim _ _) rfl)
-        · exact Empty.elim j }
-  | r+1, s =>
-    { toFun := fun i => match i with
-        | Sum.inl y => Sum.inr ((cardSplit r s y).2,Sum.inl (cardSplit r s y).1)
-        | Sum.inr (b,j) => match splitSwitch r s j with
-          | Sum.inl (p,j) => Sum.inl (Fin.cons b p,j)
-          | Sum.inr (q,j) => Sum.inr (q,Sum.inr (b,j))
-      invFun := fun i => match i with
-        | Sum.inl (p,j) => Sum.inr (p 0,(splitSwitch r s).symm (Sum.inl (Fin.tail p,j)))
-        | Sum.inr (q,Sum.inl p) => Sum.inl ((cardSplit r s).symm (p,q))
-        | Sum.inr (q,Sum.inr (b,j)) => Sum.inr (b,(splitSwitch r s).symm (Sum.inr (q,j)))
-      left_inv := by
-        intro i
-        rcases i with y | ⟨b,j⟩
-        · dsimp only
-          rw [Prod.mk.eta, Equiv.symm_apply_apply]
-        · dsimp only
-          cases hz : splitSwitch r s j with
-          | inl z =>
-            rcases z with ⟨p,k⟩
-            simp only [Fin.cons_zero, Fin.tail_cons]
-            rw [←hz,Equiv.symm_apply_apply]
-          | inr z =>
-            rcases z with ⟨q,k⟩
-            dsimp only
-            rw [←hz,Equiv.symm_apply_apply]
-      right_inv := by
-        intro i
-        rcases i with ⟨p,j⟩ | ⟨q,j⟩
-        · simp only [Equiv.apply_symm_apply, Fin.cons_self_tail]
-        · rcases j with p | ⟨b,j⟩ <;> simp only [Equiv.apply_symm_apply] }
-
-lemma assembleBits_eq (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
-    (hi : Card s × SwitchIndex r → Bool) (i : SwitchIndex (s+r)) :
-    assembleBits r s lo hi i = Sum.elim lo hi (splitSwitch r s i) := by
-  induction r with
-  | zero => rfl
-  | succ r ih =>
-    rcases i with y | ⟨b,j⟩
-    · rfl
-    · simp only [assembleBits,splitSwitch,Equiv.coe_fn_mk]
-      rw [ih]
-      cases splitSwitch r s j <;> rfl
-
-def splitCoinEquiv (r s : ℕ) :
-    ((Card r × SwitchIndex s → Bool) × (Card s × SwitchIndex r → Bool)) ≃
-      (SwitchIndex (s+r) → Bool) :=
-  (Equiv.sumArrowEquivProdArrow _ _ Bool).symm.trans
-    (Equiv.arrowCongr (splitSwitch r s).symm (Equiv.refl Bool))
-
-lemma splitCoinEquiv_apply (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
-    (hi : Card s × SwitchIndex r → Bool) :
-    splitCoinEquiv r s (lo,hi) = assembleBits r s lo hi := by
-  funext i
-  exact (assembleBits_eq r s lo hi i).symm
-
-lemma finiteMean_prod {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] (f : Ω × Ω' → ℝ) :
-    finiteMean f = finiteMean (fun x => finiteMean (fun y => f (x,y))) := by
-  simp only [finiteMean, Fintype.sum_prod_type, Fintype.card_prod, Nat.cast_mul,
-    ←Finset.sum_div, div_div, mul_comm]
-
-lemma finiteMean_split (r s : ℕ) (f : (SwitchIndex (s+r) → Bool) → ℝ) :
-    finiteMean f = finiteMean (fun lo : Card r × SwitchIndex s → Bool =>
-      finiteMean (fun hi : Card s × SwitchIndex r → Bool => f (assembleBits r s lo hi))) := by
-  rw [finiteMean_equiv (splitCoinEquiv r s).symm]
-  simp only [Equiv.symm_symm]
-  rw [finiteMean_prod]
-  apply finiteMean_congr
-  intro lo
-  apply finiteMean_congr
-  intro hi
-  rw [splitCoinEquiv_apply]
-
-section PartiallyExposed
-variable {α : Type*} {r s : ℕ}
-
-def freshPre (pre : α ≃ Card (s+r)) (lo : Card r × SwitchIndex s → Bool) :
-    α ≃ Card s × Card r :=
-  ((pre.trans (cardSplit r s)).trans (parallelPerm lo)).trans (Equiv.prodComm _ _)
-
-def freshPost (post : Card (s+r) ≃ α) : Card s × Card r ≃ α :=
-  ((Equiv.prodComm _ _).trans (cardSplit r s).symm).trans post
-
-lemma exposed_perm_eq (pre : α ≃ Card (s+r)) (post : Card (s+r) ≃ α)
-    (lo : Card r × SwitchIndex s → Bool) (hi : Card s × SwitchIndex r → Bool) :
-    (pre.trans (butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi)))).trans post =
-      relabeledPerm (freshPre pre lo) (freshPost post) hi := by
-  ext x
-  simpa only [relabeledPerm, freshPre, freshPost, parallelPerm, Equiv.trans_apply,
-    Equiv.coe_fn_mk, Equiv.prodComm_apply, Prod.swap, Equiv.symm_apply_apply] using
-    congrArg (fun y => post ((cardSplit r s).symm y)) (butterfly_split r s lo hi (pre x))
-
-lemma partially_exposed_cycle_log_mgf [Fintype α] [DecidableEq α]
-    (pre : α ≃ Card (s+r)) (post : Card (s+r) ≃ α)
-    (lo : Card r × SwitchIndex s → Bool) (S : Finset α)
-    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    Real.log (finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(Fintype.card (RoutingNetwork.SelectedCycle
-        ((pre.trans (butterflyPerm (s+r)
-          (decodeButterfly (s+r) (assembleBits r s lo hi)))).trans post) S)))) ≤
-      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^r) := by
-  simp only [exposed_perm_eq]
-  exact parallel_cycle_log_mgf (freshPre pre lo) (freshPost post) S J hJ q hq
-
-end PartiallyExposed
-
-lemma inclusionMoment_eq {α Ω : Type*} [DecidableEq α] [Fintype Ω]
-    (F : Ω → Finset α) (S : Finset α) :
-    inclusionMoment F S = finiteMean (fun ω => if S ⊆ F ω then (1:ℝ) else 0) := by
-  unfold inclusionMoment
-  apply finiteMean_congr
-  intro ω
-  split_ifs <;> rfl
-
-lemma finiteMean_bool (f : Bool → ℝ) : finiteMean f = (f false+f true)/2 := by
-  simp only [finiteMean, Fintype.sum_bool, Fintype.card_bool, Nat.cast_ofNat]
-  ring
-
-lemma finiteMean_const {Ω : Type*} [Fintype Ω] [Nonempty Ω] (c : ℝ) :
-    finiteMean (fun _ : Ω => c) = c := by
-  simp only [finiteMean,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
-  exact mul_div_cancel_left₀ c (Nat.cast_ne_zero.mpr Fintype.card_ne_zero)
-
-lemma finiteMean_pi_prod {ι : Type*} [Fintype ι] [DecidableEq ι] {Ω : ι → Type*}
-    [∀ i, Fintype (Ω i)] (f : (i : ι) → Ω i → ℝ) :
-    finiteMean (fun ω : (i : ι) → Ω i => ∏ i, f i (ω i)) = ∏ i, finiteMean (f i) := by
-  simp only [finiteMean,←Fintype.prod_sum,Fintype.card_pi,Nat.cast_prod,Finset.prod_div_distrib]
-
-section ParallelFairLayer
-variable {β : Type*} [Fintype β] [DecidableEq β]
-
-noncomputable def pairFiber (S : Finset (Bool × β)) (b : β) : Finset Bool :=
-  Finset.univ.filter (fun ε => (ε,b) ∈ S)
-
-lemma prod_pairFiber (S : Finset (Bool × β)) (f : Bool × β → ℝ) :
-    (∏ x ∈ S, f x) = ∏ b : β, ∏ ε ∈ pairFiber S b, f (ε,b) := by
-  classical
-  simp only [pairFiber,Finset.prod_filter]
-  rw [Finset.prod_comm,←Fintype.prod_prod_type (fun x : Bool × β => if x ∈ S then f x else 1)]
-  rw [←Finset.prod_filter]
-  congr 1
-  ext x
-  simp
-
-lemma fiber_average_bound (S : Finset Bool) (p : Bool → ℝ) (hp : ∀ b, 0 ≤ p b) :
-    finiteMean (fun c : Bool => ∏ ε ∈ S, p (Bool.xor ε c)) ≤
-      ∏ ε ∈ S, (p ε + p (!ε))/2 := by
-  rw [finiteMean_bool]
-  have hxor : ∀ ε : Bool, Bool.xor ε true = Equiv.swap false true ε := by decide
-  simp only [Bool.xor_false,hxor]
-  calc
-    _ ≤ ∏ ε ∈ S, (p ε+p (Equiv.swap false true ε))/2 :=
-      swap_average_product S p hp false true
-    _ = _ := Finset.prod_congr rfl (fun ε _ => by cases ε <;> rfl)
-
-lemma CylinderBound.fair_parallel {m : Finset (Bool × β) → ℝ} {p : Bool × β → ℝ}
-    (h : CylinderBound m p) (hp : ∀ x, 0 ≤ p x) :
-    CylinderBound (fun S => finiteMean (fun ξ : β → Bool => m (S.image (pairLayer ξ))))
-      (fun x => (p x+p (!x.1,x.2))/2) := by
-  intro S
-  calc
-    _ ≤ finiteMean (fun ξ : β → Bool => ∏ x ∈ S, p (pairLayer ξ x)) := by
-      apply finiteMean_mono
-      intro ξ
-      simpa only [Finset.prod_image (pairLayer ξ).injective.injOn] using h (S.image (pairLayer ξ))
-    _ = ∏ b : β, finiteMean (fun c : Bool => ∏ ε ∈ pairFiber S b, p (Bool.xor ε c,b)) := by
-      calc
-        _ = finiteMean (fun ξ : β → Bool => ∏ b : β,
-            ∏ ε ∈ pairFiber S b, p (Bool.xor ε (ξ b),b)) :=
-          finiteMean_congr (fun ξ => prod_pairFiber S (fun x => p (pairLayer ξ x)))
-        _ = _ := finiteMean_pi_prod (fun b c => ∏ ε ∈ pairFiber S b, p (Bool.xor ε c,b))
-    _ ≤ ∏ b : β, ∏ ε ∈ pairFiber S b, (p (ε,b)+p (!ε,b))/2 := by
-      apply Finset.prod_le_prod₀
-      · intro b _
-        exact div_nonneg (Finset.sum_nonneg (fun c _ =>
-          Finset.prod_nonneg (fun ε _ => hp (Bool.xor ε c,b)))) (Nat.cast_nonneg _)
-      · intro b _
-        exact fiber_average_bound (pairFiber S b) (fun ε => p (ε,b)) (fun ε => hp (ε,b))
-    _ = _ := (prod_pairFiber S (fun x => (p x+p (!x.1,x.2))/2)).symm
-
-omit [Fintype β] in
-lemma subset_pairLayer_image_iff (S A : Finset (Bool × β)) (ξ : β → Bool) :
-    S ⊆ A.image (pairLayer ξ) ↔ S.image (pairLayer ξ) ⊆ A := by
-  have hi (x : Bool × β) : pairLayer ξ (pairLayer ξ x) = x := (pairLayer ξ).left_inv x
-  constructor
-  · intro h
-    have h' := Finset.image_subset_image h (f := pairLayer ξ)
-    simpa only [Finset.image_image,Function.comp_def,hi,Finset.image_id'] using h'
-  · intro h
-    have h' := Finset.image_subset_image h (f := pairLayer ξ)
-    simpa only [Finset.image_image,Function.comp_def,hi,Finset.image_id'] using h'
-
-noncomputable def fairSetLayer {Ω : Type*} (F : Ω → Finset (Bool × β)) :
-    Ω × (β → Bool) → Finset (Bool × β) := fun ω => (F ω.1).image (pairLayer ω.2)
-
-lemma finiteMean_comm {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] (f : Ω → Ω' → ℝ) :
-    finiteMean (fun x => finiteMean (f x)) = finiteMean (fun y => finiteMean (fun x => f x y)) := by
-  simp only [finiteMean,←Finset.sum_div,div_div]
-  rw [Finset.sum_comm,mul_comm]
-
-lemma inclusionMoment_fairSetLayer {Ω : Type*} [Fintype Ω]
-    (F : Ω → Finset (Bool × β)) (S : Finset (Bool × β)) :
-    inclusionMoment (fairSetLayer F) S =
-      finiteMean (fun ξ : β → Bool => inclusionMoment F (S.image (pairLayer ξ))) := by
-  classical
-  simp only [inclusionMoment_eq]
-  have he : (fun ω : Ω × (β → Bool) => if S ⊆ fairSetLayer F ω then (1:ℝ) else 0) =
-      (fun ω : Ω × (β → Bool) => if S.image (pairLayer ω.2) ⊆ F ω.1 then (1:ℝ) else 0) := by
-    funext ω
-    simp only [fairSetLayer,subset_pairLayer_image_iff]
-  apply (congrArg finiteMean he).trans
-  exact (finiteMean_prod (fun ω : Ω × (β → Bool) =>
-    if S.image (pairLayer ω.2) ⊆ F ω.1 then (1:ℝ) else 0)).trans
-      (finiteMean_comm (fun ω ξ => if S.image (pairLayer ξ) ⊆ F ω then (1:ℝ) else 0))
-
-lemma inclusionMoment_fairSetLayer_singleton {Ω : Type*} [Fintype Ω]
-    (F : Ω → Finset (Bool × β)) (x : Bool × β) :
-    inclusionMoment (fairSetLayer F) {x} =
-      (inclusionMoment F {x}+inclusionMoment F {(!x.1,x.2)})/2 := by
-  rw [inclusionMoment_fairSetLayer]
-  simp only [Finset.image_singleton,pairLayer,Equiv.coe_fn_mk]
-  rw [finiteMean_pi_eval (A := fun _ : β => Bool) x.2 (fun c : Bool => inclusionMoment F {(Bool.xor x.1 c,x.2)})]
-  rw [finiteMean_bool]
-  rcases x with ⟨b,x⟩
-  cases b <;> simp only [Bool.xor_false,Bool.false_xor,Bool.true_xor, Bool.not_false,Bool.not_true]
-
-lemma cylinder_fairSetLayer {Ω : Type*} [Fintype Ω]
-    (F : Ω → Finset (Bool × β))
-    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
-    CylinderBound (inclusionMoment (fairSetLayer F))
-      (fun x => inclusionMoment (fairSetLayer F) {x}) := by
-  have hh := h.fair_parallel (fun x => inclusionMoment_nonneg F {x})
-  intro S
-  rw [inclusionMoment_fairSetLayer]
-  simp only [inclusionMoment_fairSetLayer_singleton]
-  exact hh S
-
-end ParallelFairLayer
-
-section RelabelMarks
-variable {α β Ω Ω' : Type*} [DecidableEq α] [DecidableEq β] [Fintype Ω] [Fintype Ω']
-
-omit [Fintype Ω] [Fintype Ω'] in
-lemma subset_image_equiv_iff (e : α ≃ β) (A : Finset α) (S : Finset β) :
-    S ⊆ A.image e ↔ S.image e.symm ⊆ A := by
-  constructor
-  · intro h x hx
-    obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
-    obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp (h hy)
-    simpa only [←hxy,Equiv.symm_apply_apply] using hx
-  · intro h y hy
-    exact Finset.mem_image.mpr ⟨e.symm y,h (Finset.mem_image_of_mem _ hy),e.apply_symm_apply y⟩
-
-omit [Fintype Ω'] in
-lemma inclusionMoment_image_equiv (e : α ≃ β) (F : Ω → Finset α) (S : Finset β) :
-    inclusionMoment (fun ω => (F ω).image e) S = inclusionMoment F (S.image e.symm) := by
-  classical
-  simp only [inclusionMoment_eq]
-  apply finiteMean_congr
-  intro ω
-  simp only [subset_image_equiv_iff]
-
-omit [Fintype Ω'] in
-lemma cylinder_image_equiv (e : α ≃ β) (F : Ω → Finset α)
-    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
-    CylinderBound (inclusionMoment (fun ω => (F ω).image e))
-      (fun x => inclusionMoment (fun ω => (F ω).image e) {x}) := by
-  intro S
-  simp only [inclusionMoment_image_equiv,Finset.image_singleton]
-  have hh := h (S.image e.symm)
-  simpa only [Finset.prod_image e.symm.injective.injOn] using hh
-
-omit [DecidableEq β] in
-lemma inclusionMoment_comp_equiv (e : Ω ≃ Ω') (F : Ω' → Finset α) (S : Finset α) :
-    inclusionMoment (fun ω => F (e ω)) S = inclusionMoment F S := by
-  classical
-  simp only [inclusionMoment_eq]
-  simpa only [Equiv.apply_symm_apply] using
-    finiteMean_equiv e (fun ω => if S ⊆ F (e ω) then (1 : ℝ) else 0)
-
-omit [DecidableEq β] in
-lemma cylinder_comp_equiv (e : Ω ≃ Ω') (F : Ω' → Finset α)
-    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
-    CylinderBound (inclusionMoment (fun ω => F (e ω)))
-      (fun x => inclusionMoment (fun ω => F (e ω)) {x}) := by
-  intro S
-  simpa only [inclusionMoment_comp_equiv] using h S
-end RelabelMarks
-
-lemma finiteMean_prod_mul {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
-    (f : Ω → ℝ) (g : Ω' → ℝ) :
-    finiteMean (fun ω : Ω × Ω' => f ω.1*g ω.2) = finiteMean f * finiteMean g := by
-  simp only [finiteMean,Fintype.sum_prod_type,Fintype.card_prod,Nat.cast_mul,
-    ←Finset.mul_sum,←Finset.sum_mul]
-  ring
-
-section JoinChildren
-variable {β : Type*} [DecidableEq β]
-
-noncomputable def childJoin (A B : Finset β) : Finset (Bool × β) :=
-  A.image (fun x => (false,x)) ∪ B.image (fun x => (true,x))
-
-lemma mem_childJoin (A B : Finset β) (b : Bool) (x : β) :
-    (b,x) ∈ childJoin A B ↔ (if b then x ∈ B else x ∈ A) := by
-  cases b <;> simp [childJoin]
-
-noncomputable def boolFiber (S : Finset (Bool × β)) (b : Bool) : Finset β :=
-  (S.filter (fun x => x.1 = b)).image Prod.snd
-
-lemma mem_boolFiber (S : Finset (Bool × β)) (b : Bool) (x : β) :
-    x ∈ boolFiber S b ↔ (b,x) ∈ S := by
-  constructor
-  · intro h
-    obtain ⟨⟨c,y⟩,hy,hxy⟩ := Finset.mem_image.mp h
-    obtain ⟨hy,hc⟩ := Finset.mem_filter.mp hy
-    change c = b at hc
-    change y = x at hxy
-    simpa only [hc,hxy] using hy
-  · intro h
-    exact Finset.mem_image.mpr ⟨(b,x),Finset.mem_filter.mpr ⟨h,rfl⟩,rfl⟩
-
-lemma childJoin_boolFiber (S : Finset (Bool × β)) :
-    childJoin (boolFiber S false) (boolFiber S true) = S := by
-  ext ⟨b,x⟩
-  rw [mem_childJoin]
-  cases b <;> simp only [Bool.false_eq_true,↓reduceIte,mem_boolFiber]
-
-lemma subset_childJoin_iff (S : Finset (Bool × β)) (A B : Finset β) :
-    S ⊆ childJoin A B ↔ boolFiber S false ⊆ A ∧ boolFiber S true ⊆ B := by
-  constructor
-  · intro h
-    constructor
-    · intro x hx
-      exact (mem_childJoin A B false x).mp (h ((mem_boolFiber S false x).mp hx))
-    · intro x hx
-      exact (mem_childJoin A B true x).mp (h ((mem_boolFiber S true x).mp hx))
-  · rintro ⟨h₀,h₁⟩ ⟨b,x⟩ hx
-    rw [mem_childJoin]
-    cases b
-    · exact h₀ ((mem_boolFiber S false x).mpr hx)
-    · exact h₁ ((mem_boolFiber S true x).mpr hx)
-
-lemma prod_childJoin (A B : Finset β) (f : Bool × β → ℝ) :
-    (∏ x ∈ childJoin A B, f x) = (∏ x ∈ A, f (false,x)) * ∏ x ∈ B, f (true,x) := by
-  classical
-  have hd : Disjoint (A.image (fun x => (false,x))) (B.image (fun x => (true,x))) := by
-    apply Finset.disjoint_left.mpr
-    rintro x hx hy
-    obtain ⟨a,ha,rfl⟩ := Finset.mem_image.mp hx
-    obtain ⟨b,hb,hba⟩ := Finset.mem_image.mp hy
-    exact Bool.false_ne_true (congrArg Prod.fst hba).symm
-  rw [childJoin,Finset.prod_union hd,
-    Finset.prod_image (fun _ _ _ _ h => congrArg Prod.snd h),
-    Finset.prod_image (fun _ _ _ _ h => congrArg Prod.snd h)]
-
-lemma prod_boolFibers (S : Finset (Bool × β)) (f : Bool × β → ℝ) :
-    (∏ x ∈ S, f x) = (∏ x ∈ boolFiber S false, f (false,x)) *
-      ∏ x ∈ boolFiber S true, f (true,x) := by
-  rw [←prod_childJoin,childJoin_boolFiber]
-
-lemma inclusionMoment_childJoin {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
-    (F : Ω → Finset β) (G : Ω' → Finset β) (S : Finset (Bool × β)) :
-    inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) S =
-      inclusionMoment F (boolFiber S false) * inclusionMoment G (boolFiber S true) := by
-  classical
-  simp only [inclusionMoment_eq]
-  rw [←finiteMean_prod_mul]
-  apply finiteMean_congr
-  intro ω
-  simp only [subset_childJoin_iff]
-  split_ifs <;> simp_all
-
-lemma cylinder_childJoin {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] [Nonempty Ω] [Nonempty Ω']
-    (F : Ω → Finset β) (G : Ω' → Finset β)
-    (hF : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x}))
-    (hG : CylinderBound (inclusionMoment G) (fun x => inclusionMoment G {x})) :
-    CylinderBound (inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)))
-      (fun x => inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {x}) := by
-  intro S
-  rw [inclusionMoment_childJoin,prod_boolFibers]
-  have hf (b : Bool) (x : β) : boolFiber {(b,x)} b = {x} := by
-    ext y
-    simp only [mem_boolFiber,Finset.mem_singleton,Prod.mk.injEq,true_and]
-  have hn (b : Bool) (x : β) : boolFiber {(!b,x)} b = ∅ := by
-    ext y
-    cases b <;> simp [mem_boolFiber]
-  have h₀ (x : β) : inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {(false,x)} =
-      inclusionMoment F {x} * inclusionMoment G ∅ := by
-    rw [inclusionMoment_childJoin,hf]
-    have hh := hn true x
-    exact congrArg (fun T => inclusionMoment F {x} * inclusionMoment G T) hh
-  have h₁ (x : β) : inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {(true,x)} =
-      inclusionMoment F ∅ * inclusionMoment G {x} := by
-    rw [inclusionMoment_childJoin,hf]
-    have hh := hn false x
-    exact congrArg (fun T => inclusionMoment F T * inclusionMoment G {x}) hh
-  have emF : inclusionMoment F ∅ = 1 := by simp [inclusionMoment,finiteMean_const]
-  have emG : inclusionMoment G ∅ = 1 := by simp [inclusionMoment,finiteMean_const]
-  simp only [h₀,h₁,emF,emG,mul_one,one_mul]
-  exact mul_le_mul (hF _) (hG _) (inclusionMoment_nonneg _ _)
-    (Finset.prod_nonneg (fun x _ => inclusionMoment_nonneg F {x}))
-end JoinChildren
-
-lemma childMarks_join (d : ℕ) (A : Finset (Card (d+1))) :
-    A = (childMarks A false).image (Fin.cons false) ∪
-      (childMarks A true).image (Fin.cons true) := by
-  classical
-  ext x
-  constructor
-  · intro hx
-    cases hb : x 0
-    · exact Finset.mem_union_left _ (Finset.mem_image.mpr
-        ⟨Fin.tail x,(mem_childMarks _ _ _).mpr (by simpa only [←hb,Fin.cons_self_tail] using hx),
-          by simp only [←hb,Fin.cons_self_tail]⟩)
-    · exact Finset.mem_union_right _ (Finset.mem_image.mpr
-        ⟨Fin.tail x,(mem_childMarks _ _ _).mpr (by simpa only [←hb,Fin.cons_self_tail] using hx),
-          by simp only [←hb,Fin.cons_self_tail]⟩)
-  · intro hx
-    rcases Finset.mem_union.mp hx with hx | hx
-    · obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
-      exact (mem_childMarks _ _ _).mp hy
-    · obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
-      exact (mem_childMarks _ _ _).mp hy
-
-noncomputable def butterflyMarks (d : ℕ) (A : Finset (Card d))
-    (ω : SwitchIndex d → Bool) : Finset (Card d) :=
-  A.image (butterflyPerm d (decodeButterfly d ω))
-
-lemma butterflyMarks_succ (d : ℕ) (A : Finset (Card (d+1))) (ω : SwitchIndex (d+1) → Bool) :
-    butterflyMarks (d+1) A ω =
-      (fairSetLayer (fun p : (SwitchIndex d → Bool) × (SwitchIndex d → Bool) =>
-        childJoin (butterflyMarks d (childMarks A false) p.1)
-          (butterflyMarks d (childMarks A true) p.2)) (coinStepEquiv d ω)).image
-            (headTailEquiv d).symm := by
-  classical
-  unfold butterflyMarks fairSetLayer childJoin
-  rw [Finset.image_union,Finset.image_union]
-  simp only [Finset.image_image]
-  conv_lhs => rw [childMarks_join d A]
-  rw [Finset.image_union]
-  simp only [Finset.image_image]
-  apply congrArg₂ (· ∪ ·)
-  · apply Finset.image_congr
-    intro x _
-    simp only [Function.comp_def,butterflyPerm,decodeButterfly,Equiv.Perm.coe_mul,
-      childLift,pairSwitch,switchFun,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
-      coinStepEquiv,headTailEquiv,Equiv.symm_symm,pairLayer]
-    rfl
-  · apply Finset.image_congr
-    intro x _
-    simp only [Function.comp_def,butterflyPerm,decodeButterfly,Equiv.Perm.coe_mul,
-      childLift,pairSwitch,switchFun,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
-      coinStepEquiv,headTailEquiv,Equiv.symm_symm,pairLayer]
-    rfl
-
-lemma butterflyMarks_cylinder (d : ℕ) (A : Finset (Card d)) :
-    CylinderBound (inclusionMoment (butterflyMarks d A))
-      (fun x => inclusionMoment (butterflyMarks d A) {x}) := by
-  induction d with
-  | zero =>
-    have he : butterflyMarks 0 A = fun _ : SwitchIndex 0 → Bool => A := by
-      funext ω
-      simp [butterflyMarks,butterflyPerm]
-    rw [he]
-    exact cylinder_const A
-  | succ d ih =>
-    have h := cylinder_childJoin (butterflyMarks d (childMarks A false))
-      (butterflyMarks d (childMarks A true)) (ih _) (ih _)
-    have hh := cylinder_image_equiv (headTailEquiv d).symm _ (cylinder_fairSetLayer _ h)
-    have hs := cylinder_comp_equiv (coinStepEquiv d) _ hh
-    simpa only [←butterflyMarks_succ] using hs
-
-lemma indicator_mem_image_perm {α : Type*} [DecidableEq α]
-    (A : Finset α) (p : Equiv.Perm α) (y : α) :
-    (if y ∈ A.image p then (1 : ℝ) else 0) = ∑ x ∈ A, if p x = y then 1 else 0 := by
-  classical
-  by_cases h : y ∈ A.image p
-  · obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp h
-    rw [ite_eq_left (Finset.mem_image.mpr ⟨x,hx,hxy⟩)]
-    symm
-    calc
-      _ = (if p x = y then (1:ℝ) else 0) := Finset.sum_eq_single_of_mem x hx (by
-        intro z _ hzx
-        exact ite_eq_right (fun hz => hzx (p.injective (hz.trans hxy.symm))))
-      _ = 1 := ite_eq_left hxy
-  · rw [ite_eq_right h]
-    symm
-    apply Finset.sum_eq_zero
-    intro x hx
-    exact ite_eq_right (fun hxy => h (Finset.mem_image.mpr ⟨x,hx,hxy⟩))
-
-lemma butterflyMarks_singleton (d : ℕ) (A : Finset (Card d)) (y : Card d) :
-    inclusionMoment (butterflyMarks d A) {y} = (A.card : ℝ)/(2:ℝ)^d := by
-  classical
-  unfold inclusionMoment
-  simp only [Finset.singleton_subset_iff,butterflyMarks,indicator_mem_image_perm]
-  rw [finiteMean_sum]
-  simp only [butterfly_single_card_uniform,Finset.sum_const,nsmul_eq_mul]
-  ring
-
-lemma butterflyMarks_inclusion (d : ℕ) (A S : Finset (Card d)) :
-    inclusionMoment (butterflyMarks d A) S ≤ ((A.card : ℝ)/(2:ℝ)^d)^S.card := by
-  have h := butterflyMarks_cylinder d A S
-  simpa only [butterflyMarks_singleton,Finset.prod_const] using h
-
-namespace CycleColoring
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-abbrev Coloring (p : Equiv.Perm α) (S : Finset α) :=
-  {f : α → Bool // (∀ x, x ∉ S → f x = false) ∧
-    ∀ x ∈ S, p x ∈ S → f (p x) = f x}
-
-noncomputable instance (p : Equiv.Perm α) (S : Finset α) : Fintype (Coloring p S) :=
-  Fintype.ofFinite _
-
-noncomputable def boundary (p : Equiv.Perm α) (S : Finset α) : Finset α :=
-  S.filter (fun x => p.symm x ∉ S)
-
-noncomputable def root {p : Equiv.Perm α} {S : Finset α}
-    (C : RoutingNetwork.SelectedCycle p S) : α := C.nonempty.choose
-
-lemma root_mem {p : Equiv.Perm α} {S : Finset α} (C : RoutingNetwork.SelectedCycle p S) :
-    root C ∈ C.val := C.nonempty.choose_spec
-
-abbrev Seed (p : Equiv.Perm α) (S : Finset α) :=
-  {x // x ∈ boundary p S} ⊕ RoutingNetwork.SelectedCycle p S
-
-noncomputable def record {p : Equiv.Perm α} {S : Finset α}
-    (f : Coloring p S) : Seed p S → Bool :=
-  Sum.elim (fun x => f.val x.val) (fun C => f.val (root C))
-
-lemma orbit_subset_of_predecessor_closed (p : Equiv.Perm α) (D : Finset α)
-    (hD : ∀ x ∈ D, p.symm x ∈ D) {x : α} (hx : x ∈ D) : orbitSet p x ⊆ D := by
-  intro y hy
-  have hh := ((mem_orbitSet p x y).mp hy).inv
-  obtain ⟨n,hn⟩ := hh.exists_nat_pow_eq
-  have hp (k : ℕ) : (p⁻¹ ^ k) x ∈ D := by
-    induction k with
-    | zero => simpa using hx
-    | succ k ih => simpa only [pow_succ',Equiv.Perm.mul_apply,Equiv.Perm.coe_inv] using hD _ ih
-  exact hn ▸ hp n
-
-lemma record_injective (p : Equiv.Perm α) (S : Finset α) :
-    Function.Injective (record (p := p) (S := S)) := by
-  classical
-  intro f g hfg
-  apply Subtype.ext
-  funext x
-  by_contra hne
-  have hx : x ∈ S := by
-    by_contra h
-    exact hne ((f.property.1 x h).trans (g.property.1 x h).symm)
-  let D := S.filter (fun y => f.val y ≠ g.val y)
-  have hxD : x ∈ D := Finset.mem_filter.mpr ⟨hx,hne⟩
-  have hprev (y : α) (hy : y ∈ D) : p.symm y ∈ D := by
-    obtain ⟨hyS,hyne⟩ := Finset.mem_filter.mp hy
-    have hpS : p.symm y ∈ S := by
-      by_contra hn
-      have hseed : y ∈ boundary p S := Finset.mem_filter.mpr ⟨hyS,hn⟩
-      exact hyne (congrFun hfg (Sum.inl ⟨y,hseed⟩))
-    refine Finset.mem_filter.mpr ⟨hpS,?_⟩
-    have hf := f.property.2 (p.symm y) hpS (by simpa only [Equiv.apply_symm_apply] using hyS)
-    have hg := g.property.2 (p.symm y) hpS (by simpa only [Equiv.apply_symm_apply] using hyS)
-    simp only [Equiv.apply_symm_apply] at hf hg
-    intro he
-    exact hyne (hf.trans (he.trans hg.symm))
-  have horb := orbit_subset_of_predecessor_closed p D hprev hxD
-  let C : RoutingNetwork.SelectedCycle p S :=
-    ⟨orbitSet p x,⟨x,rfl⟩,fun y hy => (Finset.mem_filter.mp (horb hy)).1⟩
-  have hrD : root C ∈ D := horb (root_mem C)
-  have hrne := (Finset.mem_filter.mp hrD).2
-  exact hrne (congrFun hfg (Sum.inr C))
-
-lemma card_coloring_le (p : Equiv.Perm α) (S : Finset α) :
-    Fintype.card (Coloring p S) ≤
-      2 ^ ((boundary p S).card + Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
-  have h := Fintype.card_le_of_injective (record (p := p) (S := S)) (record_injective p S)
-  simpa only [Fintype.card_fun,Fintype.card_sum,Fintype.card_coe,Fintype.card_bool] using h
-
-omit [Fintype α] in
-lemma boundary_card (p : Equiv.Perm α) (S : Finset α) :
-    (boundary p S).card = S.card - (S.filter (fun x => p.symm x ∈ S)).card := by
-  classical
-  have h := Finset.card_filter_add_card_filter_not (s := S) (fun x => p.symm x ∈ S)
-  unfold boundary
-  omega
-
-lemma card_coloring_le_edges (p : Equiv.Perm α) (S : Finset α) :
-    Fintype.card (Coloring p S) ≤
-      2 ^ (S.card - (S.filter (fun x => p.symm x ∈ S)).card +
-        Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
-  simpa only [boundary_card] using card_coloring_le p S
-
-
-abbrev OppositeColoring (p : Equiv.Perm α) (S : Finset α) :=
-  {f : α → Bool // (∀ x, x ∉ S → f x = false) ∧
-    ∀ x ∈ S, p x ∈ S → f (p x) = !(f x)}
-
-noncomputable instance (p : Equiv.Perm α) (S : Finset α) : Fintype (OppositeColoring p S) :=
-  Fintype.ofFinite _
-
-noncomputable def bipartitionFlip (p : Equiv.Perm α) (S : Finset α)
-    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x))
-    (f : OppositeColoring p S) : Coloring p S := by
-  classical
-  refine ⟨(fun x => if x ∈ S then Bool.xor (f.val x) (χ x) else false),?_,?_⟩
-  · intro x hx
-    exact ite_eq_right hx
-  · intro x hx hpx
-    simp only [ite_eq_left hx,ite_eq_left hpx,f.property.2 x hx hpx,hχ x]
-    cases f.val x <;> cases χ x <;> rfl
-
-end CycleColoring
-end Thorp
-
-end ThorpNine.HighTail
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.HighTail.PriorPaths
+
+namespace OAI
+
+namespace ThorpNine.HighTail
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+section ParallelButterfly
+variable {α B : Type*} [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] {d : ℕ}
+omit [DecidableEq B] in
+lemma parallel_entropy (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool) (S : Finset α) :
+    (∑ i : B × SwitchIndex d,
+      if (parallelUsers pre ω i).1 ∈ S ∧ (parallelUsers pre ω i).2 ∈ S then (1:ℝ) else 0) ≤
+      (S.card : ℝ) * Real.log S.card / (2 * Real.log 2) := by
+  classical
+  rw [Fintype.sum_prod_type]
+  calc
+    _ ≤ ∑ b : B, (blockMarks pre S b).card * Real.log (blockMarks pre S b).card / (2*Real.log 2) := by
+      apply Finset.sum_le_sum
+      intro b _
+      simpa only [←mem_blockMarks,parallelUsers] using
+        marked_switch_entropy_bound d (decodeButterfly d (fun i => ω (b,i))) (blockMarks pre S b)
+    _ ≤ ∑ b : B, (blockMarks pre S b).card * Real.log S.card / (2*Real.log 2) := by
+      apply Finset.sum_le_sum
+      intro b _
+      by_cases hz : (blockMarks pre S b).card = 0
+      · simp [hz]
+      · apply div_le_div_of_nonneg_right _ (by positivity)
+        apply mul_le_mul_of_nonneg_left _ (by positivity)
+        exact Real.log_le_log (by exact_mod_cast Nat.pos_of_ne_zero hz)
+          (by exact_mod_cast card_blockMarks_le pre S b)
+    _ = _ := by rw [←Finset.sum_div,←Finset.sum_mul,←Nat.cast_sum,sum_card_blockMarks]
+
+noncomputable def parallelFollow (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x : α) (k : α → QueryTree (B × SwitchIndex d)) : QueryTree (B × SwitchIndex d) :=
+  QueryTree.followPath d (fun i => ((pre x).1,i)) (pre x).2 (fun y => k (post ((pre x).1,y)))
+
+omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
+lemma parallelFollow_succeeds (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
+    (parallelFollow pre post x k).succeeds ω = (k (relabeledPerm pre post ω x)).succeeds ω := by
+  exact QueryTree.followPath_succeeds _ _ _ _ _
+
+omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
+lemma parallelFollow_raw_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
+    (parallelFollow pre post x k).charge ω = (k (relabeledPerm pre post ω x)).charge ω := by
+  exact QueryTree.followPath_raw_charge _ _ _ _ _
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma parallelFollow_stale_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
+    (ω : B × SwitchIndex d → Bool) :
+    (parallelFollow pre post x k).staleCharge R ω =
+      (k (relabeledPerm pre post ω x)).staleCharge (R ∪ parallelPath pre ω x) ω := by
+  exact QueryTree.followPath_stale_charge _ _ _ _ _ _
+
+noncomputable def closingDomain (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) : Finset (B × SwitchIndex d) :=
+  (routeDomain d (pre x).2 (post.symm y).2).image (fun i => ((pre x).1,i))
+
+def closingValue (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (i : B × SwitchIndex d) : Bool := routeValue d (pre x).2 (post.symm y).2 i.2
+
+omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
+lemma relabeled_endpoint_pair (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (ω : B × SwitchIndex d → Bool) (x y : α) :
+    relabeledPerm pre post ω x = y ↔
+      (pre x).1 = (post.symm y).1 ∧
+      butterflyPerm d (decodeButterfly d (fun i => ω ((pre x).1,i))) (pre x).2 = (post.symm y).2 := by
+  change post ((pre x).1,_) = y ↔ _
+  rw [←post.eq_symm_apply,Prod.mk.injEq]
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma relabeled_endpoint_iff (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (ω : B × SwitchIndex d → Bool) (x y : α) :
+    relabeledPerm pre post ω x = y ↔
+      (pre x).1 = (post.symm y).1 ∧
+        ∀ i ∈ closingDomain pre post x y, ω i = closingValue pre post x y i := by
+  classical
+  rw [relabeled_endpoint_pair,butterfly_endpoint_iff]
+  simp only [closingDomain,Finset.forall_mem_image,closingValue]
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma closingDomain_eq (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (ω : B × SwitchIndex d → Bool) (x y : α) (h : relabeledPerm pre post ω x = y) :
+    closingDomain pre post x y = parallelPath pre ω x := by
+  rw [closingDomain,parallelPath,(relabeled_endpoint_pair _ _ _ _ _).mp h |>.2]
+
+noncomputable def parallelClose (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (k : QueryTree (B × SwitchIndex d)) : QueryTree (B × SwitchIndex d) :=
+  if (pre x).1 = (post.symm y).1 then
+    QueryTree.forceList (closingDomain pre post x y).toList (closingValue pre post x y) k
+  else QueryTree.reject
+
+omit [Fintype α] in
+lemma parallelClose_succeeds (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (k : QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
+    (parallelClose pre post x y k).succeeds ω =
+      if relabeledPerm pre post ω x = y then k.succeeds ω else false := by
+  classical
+  simp only [relabeled_endpoint_iff,parallelClose]
+  by_cases hb : (pre x).1 = (post.symm y).1
+  · rw [ite_eq_left hb,QueryTree.forceList_succeeds]
+    simp only [Finset.mem_toList,hb,true_and]
+  · simp [hb,QueryTree.succeeds]
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma parallelClose_raw_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (k : QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool)
+    (h : relabeledPerm pre post ω x = y) :
+    (parallelClose pre post x y k).charge ω = k.charge ω + d := by
+  have hh := (relabeled_endpoint_iff _ _ _ _ _).mp h
+  rw [parallelClose,ite_eq_left hh.1,QueryTree.forceList_raw_charge _ _ _ _
+    (by simpa only [Finset.mem_toList] using hh.2),Finset.length_toList,
+    closingDomain_eq _ _ _ _ _ h,card_parallelPath]
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma parallelClose_fresh_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (k : QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
+    (ω : B × SwitchIndex d → Bool) (h : relabeledPerm pre post ω x = y) :
+    (parallelClose pre post x y k).freshCharge R ω =
+      k.freshCharge (R ∪ parallelPath pre ω x) ω + (parallelPath pre ω x \ R).card := by
+  have hh := (relabeled_endpoint_iff _ _ _ _ _).mp h
+  rw [parallelClose,ite_eq_left hh.1,QueryTree.forceList_charge _ _ _ _ _
+    (by simpa only [Finset.mem_toList] using hh.2),Finset.toList_toFinset,
+    closingDomain_eq _ _ _ _ _ h]
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma parallelClose_stale_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (k : QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
+    (ω : B × SwitchIndex d → Bool) (h : relabeledPerm pre post ω x = y) :
+    (parallelClose pre post x y k).staleCharge R ω =
+      k.staleCharge (R ∪ parallelPath pre ω x) ω + (parallelPath pre ω x ∩ R).card := by
+  have h₀ := QueryTree.fresh_add_stale (parallelClose pre post x y k) R ω
+  have h₁ := QueryTree.fresh_add_stale k (R ∪ parallelPath pre ω x) ω
+  rw [parallelClose_raw_charge _ _ _ _ _ _ h,parallelClose_fresh_charge _ _ _ _ _ _ _ h] at h₀
+  have h₂ : (parallelPath pre ω x \ R).card + (parallelPath pre ω x ∩ R).card = d := by
+    rw [Finset.card_sdiff_add_card_inter,card_parallelPath]
+  omega
+
+noncomputable def parallelNetwork (pre : α ≃ B × Card d) (post : B × Card d ≃ α) :
+    RoutingNetwork α (B × SwitchIndex d) d where
+  perm := relabeledPerm pre post
+  path := parallelPath pre
+  users := parallelUsers pre
+  users_ne := parallelUsers_ne pre
+  mem_path := mem_parallelPath pre
+  card_path := card_parallelPath pre
+  entropy := parallel_entropy pre
+  follow := parallelFollow pre post
+  close := parallelClose pre post
+  follow_succeeds := parallelFollow_succeeds pre post
+  follow_raw_charge := parallelFollow_raw_charge pre post
+  follow_stale_charge := parallelFollow_stale_charge pre post
+  close_succeeds := parallelClose_succeeds pre post
+  close_raw_charge := parallelClose_raw_charge pre post
+  close_stale_charge := parallelClose_stale_charge pre post
+
+lemma parallel_cycle_log_mgf (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (S : Finset α) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    Real.log (finiteMean (fun ω : B × SwitchIndex d → Bool =>
+      q^(Fintype.card (RoutingNetwork.SelectedCycle (relabeledPerm pre post ω) S)))) ≤
+      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
+  exact (parallelNetwork pre post).butterfly_cycle_log_mgf S J hJ q hq
+
+end ParallelButterfly
+
+lemma butterfly_split (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
+    (hi : Card s × SwitchIndex r → Bool) (x : Card (s+r)) :
+    cardSplit r s (butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi)) x) =
+      let y := cardSplit r s x
+      let z := butterflyPerm s (decodeButterfly s (fun i => lo (y.1,i))) y.2
+      (butterflyPerm r (decodeButterfly r (fun i => hi (z,i))) y.1,z) := by
+  induction r with
+  | zero =>
+    apply Prod.ext
+    · funext i
+      exact Fin.elim0 i
+    · rfl
+  | succ r ih =>
+    let lo' : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons (x 0) j.1,j.2)
+    let hi' : Card s × SwitchIndex r → Bool := fun j => hi (j.1,Sum.inr (x 0,j.2))
+    have hh := ih lo' hi' (Fin.tail x)
+    let u := butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo' hi')) (Fin.tail x)
+    let y := cardSplit r s (Fin.tail x)
+    let z := butterflyPerm s (decodeButterfly s (fun i => lo' (y.1,i))) y.2
+    let w := butterflyPerm r (decodeButterfly r (fun i => hi' (z,i))) y.1
+    change cardSplit r s u = (w,z) at hh
+    change ((Fin.cons (Bool.xor (x 0) (hi ((cardSplit r s u).2,Sum.inl (cardSplit r s u).1)))
+        ((cardSplit r s u).1) : Card (r+1)), (cardSplit r s u).2) =
+      ((Fin.cons (Bool.xor (x 0) (hi (z,Sum.inl w))) w : Card (r+1)),z)
+    rw [hh]
+
+def splitSwitch : (r s : ℕ) → SwitchIndex (s+r) ≃
+    (Card r × SwitchIndex s) ⊕ (Card s × SwitchIndex r)
+  | 0, _ =>
+    { toFun := fun i => Sum.inl ((fun j => Fin.elim0 j),i)
+      invFun := fun i => match i with
+        | Sum.inl (_,j) => j
+        | Sum.inr (_,j) => nomatch j
+      left_inv := fun _ => rfl
+      right_inv := by
+        intro i
+        rcases i with ⟨x,j⟩ | ⟨x,j⟩
+        · exact congrArg Sum.inl (Prod.ext (Subsingleton.elim _ _) rfl)
+        · exact Empty.elim j }
+  | r+1, s =>
+    { toFun := fun i => match i with
+        | Sum.inl y => Sum.inr ((cardSplit r s y).2,Sum.inl (cardSplit r s y).1)
+        | Sum.inr (b,j) => match splitSwitch r s j with
+          | Sum.inl (p,j) => Sum.inl (Fin.cons b p,j)
+          | Sum.inr (q,j) => Sum.inr (q,Sum.inr (b,j))
+      invFun := fun i => match i with
+        | Sum.inl (p,j) => Sum.inr (p 0,(splitSwitch r s).symm (Sum.inl (Fin.tail p,j)))
+        | Sum.inr (q,Sum.inl p) => Sum.inl ((cardSplit r s).symm (p,q))
+        | Sum.inr (q,Sum.inr (b,j)) => Sum.inr (b,(splitSwitch r s).symm (Sum.inr (q,j)))
+      left_inv := by
+        intro i
+        rcases i with y | ⟨b,j⟩
+        · dsimp only
+          rw [Prod.mk.eta, Equiv.symm_apply_apply]
+        · dsimp only
+          cases hz : splitSwitch r s j with
+          | inl z =>
+            rcases z with ⟨p,k⟩
+            simp only [Fin.cons_zero, Fin.tail_cons]
+            rw [←hz,Equiv.symm_apply_apply]
+          | inr z =>
+            rcases z with ⟨q,k⟩
+            dsimp only
+            rw [←hz,Equiv.symm_apply_apply]
+      right_inv := by
+        intro i
+        rcases i with ⟨p,j⟩ | ⟨q,j⟩
+        · simp only [Equiv.apply_symm_apply, Fin.cons_self_tail]
+        · rcases j with p | ⟨b,j⟩ <;> simp only [Equiv.apply_symm_apply] }
+
+lemma assembleBits_eq (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
+    (hi : Card s × SwitchIndex r → Bool) (i : SwitchIndex (s+r)) :
+    assembleBits r s lo hi i = Sum.elim lo hi (splitSwitch r s i) := by
+  induction r with
+  | zero => rfl
+  | succ r ih =>
+    rcases i with y | ⟨b,j⟩
+    · rfl
+    · simp only [assembleBits,splitSwitch,Equiv.coe_fn_mk]
+      rw [ih]
+      cases splitSwitch r s j <;> rfl
+
+def splitCoinEquiv (r s : ℕ) :
+    ((Card r × SwitchIndex s → Bool) × (Card s × SwitchIndex r → Bool)) ≃
+      (SwitchIndex (s+r) → Bool) :=
+  (Equiv.sumArrowEquivProdArrow _ _ Bool).symm.trans
+    (Equiv.arrowCongr (splitSwitch r s).symm (Equiv.refl Bool))
+
+lemma splitCoinEquiv_apply (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
+    (hi : Card s × SwitchIndex r → Bool) :
+    splitCoinEquiv r s (lo,hi) = assembleBits r s lo hi := by
+  funext i
+  exact (assembleBits_eq r s lo hi i).symm
+
+lemma finiteMean_prod {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] (f : Ω × Ω' → ℝ) :
+    finiteMean f = finiteMean (fun x => finiteMean (fun y => f (x,y))) := by
+  simp only [finiteMean, Fintype.sum_prod_type, Fintype.card_prod, Nat.cast_mul,
+    ←Finset.sum_div, div_div, mul_comm]
+
+lemma finiteMean_split (r s : ℕ) (f : (SwitchIndex (s+r) → Bool) → ℝ) :
+    finiteMean f = finiteMean (fun lo : Card r × SwitchIndex s → Bool =>
+      finiteMean (fun hi : Card s × SwitchIndex r → Bool => f (assembleBits r s lo hi))) := by
+  rw [finiteMean_equiv (splitCoinEquiv r s).symm]
+  simp only [Equiv.symm_symm]
+  rw [finiteMean_prod]
+  apply finiteMean_congr
+  intro lo
+  apply finiteMean_congr
+  intro hi
+  rw [splitCoinEquiv_apply]
+
+section PartiallyExposed
+variable {α : Type*} {r s : ℕ}
+
+def freshPre (pre : α ≃ Card (s+r)) (lo : Card r × SwitchIndex s → Bool) :
+    α ≃ Card s × Card r :=
+  ((pre.trans (cardSplit r s)).trans (parallelPerm lo)).trans (Equiv.prodComm _ _)
+
+def freshPost (post : Card (s+r) ≃ α) : Card s × Card r ≃ α :=
+  ((Equiv.prodComm _ _).trans (cardSplit r s).symm).trans post
+
+lemma exposed_perm_eq (pre : α ≃ Card (s+r)) (post : Card (s+r) ≃ α)
+    (lo : Card r × SwitchIndex s → Bool) (hi : Card s × SwitchIndex r → Bool) :
+    (pre.trans (butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi)))).trans post =
+      relabeledPerm (freshPre pre lo) (freshPost post) hi := by
+  ext x
+  simpa only [relabeledPerm, freshPre, freshPost, parallelPerm, Equiv.trans_apply,
+    Equiv.coe_fn_mk, Equiv.prodComm_apply, Prod.swap, Equiv.symm_apply_apply] using
+    congrArg (fun y => post ((cardSplit r s).symm y)) (butterfly_split r s lo hi (pre x))
+
+lemma partially_exposed_cycle_log_mgf [Fintype α] [DecidableEq α]
+    (pre : α ≃ Card (s+r)) (post : Card (s+r) ≃ α)
+    (lo : Card r × SwitchIndex s → Bool) (S : Finset α)
+    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    Real.log (finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(Fintype.card (RoutingNetwork.SelectedCycle
+        ((pre.trans (butterflyPerm (s+r)
+          (decodeButterfly (s+r) (assembleBits r s lo hi)))).trans post) S)))) ≤
+      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^r) := by
+  simp only [exposed_perm_eq]
+  exact parallel_cycle_log_mgf (freshPre pre lo) (freshPost post) S J hJ q hq
+
+end PartiallyExposed
+
+lemma inclusionMoment_eq {α Ω : Type*} [DecidableEq α] [Fintype Ω]
+    (F : Ω → Finset α) (S : Finset α) :
+    inclusionMoment F S = finiteMean (fun ω => if S ⊆ F ω then (1:ℝ) else 0) := by
+  unfold inclusionMoment
+  apply finiteMean_congr
+  intro ω
+  split_ifs <;> rfl
+
+lemma finiteMean_bool (f : Bool → ℝ) : finiteMean f = (f false+f true)/2 := by
+  simp only [finiteMean, Fintype.sum_bool, Fintype.card_bool, Nat.cast_ofNat]
+  ring
+
+lemma finiteMean_const {Ω : Type*} [Fintype Ω] [Nonempty Ω] (c : ℝ) :
+    finiteMean (fun _ : Ω => c) = c := by
+  simp only [finiteMean,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
+  exact mul_div_cancel_left₀ c (Nat.cast_ne_zero.mpr Fintype.card_ne_zero)
+
+lemma finiteMean_pi_prod {ι : Type*} [Fintype ι] [DecidableEq ι] {Ω : ι → Type*}
+    [∀ i, Fintype (Ω i)] (f : (i : ι) → Ω i → ℝ) :
+    finiteMean (fun ω : (i : ι) → Ω i => ∏ i, f i (ω i)) = ∏ i, finiteMean (f i) := by
+  simp only [finiteMean,←Fintype.prod_sum,Fintype.card_pi,Nat.cast_prod,Finset.prod_div_distrib]
+
+section ParallelFairLayer
+variable {β : Type*} [Fintype β] [DecidableEq β]
+
+noncomputable def pairFiber (S : Finset (Bool × β)) (b : β) : Finset Bool :=
+  Finset.univ.filter (fun ε => (ε,b) ∈ S)
+
+lemma prod_pairFiber (S : Finset (Bool × β)) (f : Bool × β → ℝ) :
+    (∏ x ∈ S, f x) = ∏ b : β, ∏ ε ∈ pairFiber S b, f (ε,b) := by
+  classical
+  simp only [pairFiber,Finset.prod_filter]
+  rw [Finset.prod_comm,←Fintype.prod_prod_type (fun x : Bool × β => if x ∈ S then f x else 1)]
+  rw [←Finset.prod_filter]
+  congr 1
+  ext x
+  simp
+
+lemma fiber_average_bound (S : Finset Bool) (p : Bool → ℝ) (hp : ∀ b, 0 ≤ p b) :
+    finiteMean (fun c : Bool => ∏ ε ∈ S, p (Bool.xor ε c)) ≤
+      ∏ ε ∈ S, (p ε + p (!ε))/2 := by
+  rw [finiteMean_bool]
+  have hxor : ∀ ε : Bool, Bool.xor ε true = Equiv.swap false true ε := by decide
+  simp only [Bool.xor_false,hxor]
+  calc
+    _ ≤ ∏ ε ∈ S, (p ε+p (Equiv.swap false true ε))/2 :=
+      swap_average_product S p hp false true
+    _ = _ := Finset.prod_congr rfl (fun ε _ => by cases ε <;> rfl)
+
+lemma CylinderBound.fair_parallel {m : Finset (Bool × β) → ℝ} {p : Bool × β → ℝ}
+    (h : CylinderBound m p) (hp : ∀ x, 0 ≤ p x) :
+    CylinderBound (fun S => finiteMean (fun ξ : β → Bool => m (S.image (pairLayer ξ))))
+      (fun x => (p x+p (!x.1,x.2))/2) := by
+  intro S
+  calc
+    _ ≤ finiteMean (fun ξ : β → Bool => ∏ x ∈ S, p (pairLayer ξ x)) := by
+      apply finiteMean_mono
+      intro ξ
+      simpa only [Finset.prod_image (pairLayer ξ).injective.injOn] using h (S.image (pairLayer ξ))
+    _ = ∏ b : β, finiteMean (fun c : Bool => ∏ ε ∈ pairFiber S b, p (Bool.xor ε c,b)) := by
+      calc
+        _ = finiteMean (fun ξ : β → Bool => ∏ b : β,
+            ∏ ε ∈ pairFiber S b, p (Bool.xor ε (ξ b),b)) :=
+          finiteMean_congr (fun ξ => prod_pairFiber S (fun x => p (pairLayer ξ x)))
+        _ = _ := finiteMean_pi_prod (fun b c => ∏ ε ∈ pairFiber S b, p (Bool.xor ε c,b))
+    _ ≤ ∏ b : β, ∏ ε ∈ pairFiber S b, (p (ε,b)+p (!ε,b))/2 := by
+      apply Finset.prod_le_prod₀
+      · intro b _
+        exact div_nonneg (Finset.sum_nonneg (fun c _ =>
+          Finset.prod_nonneg (fun ε _ => hp (Bool.xor ε c,b)))) (Nat.cast_nonneg _)
+      · intro b _
+        exact fiber_average_bound (pairFiber S b) (fun ε => p (ε,b)) (fun ε => hp (ε,b))
+    _ = _ := (prod_pairFiber S (fun x => (p x+p (!x.1,x.2))/2)).symm
+
+omit [Fintype β] in
+lemma subset_pairLayer_image_iff (S A : Finset (Bool × β)) (ξ : β → Bool) :
+    S ⊆ A.image (pairLayer ξ) ↔ S.image (pairLayer ξ) ⊆ A := by
+  have hi (x : Bool × β) : pairLayer ξ (pairLayer ξ x) = x := (pairLayer ξ).left_inv x
+  constructor
+  · intro h
+    have h' := Finset.image_subset_image h (f := pairLayer ξ)
+    simpa only [Finset.image_image,Function.comp_def,hi,Finset.image_id'] using h'
+  · intro h
+    have h' := Finset.image_subset_image h (f := pairLayer ξ)
+    simpa only [Finset.image_image,Function.comp_def,hi,Finset.image_id'] using h'
+
+noncomputable def fairSetLayer {Ω : Type*} (F : Ω → Finset (Bool × β)) :
+    Ω × (β → Bool) → Finset (Bool × β) := fun ω => (F ω.1).image (pairLayer ω.2)
+
+lemma finiteMean_comm {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] (f : Ω → Ω' → ℝ) :
+    finiteMean (fun x => finiteMean (f x)) = finiteMean (fun y => finiteMean (fun x => f x y)) := by
+  simp only [finiteMean,←Finset.sum_div,div_div]
+  rw [Finset.sum_comm,mul_comm]
+
+lemma inclusionMoment_fairSetLayer {Ω : Type*} [Fintype Ω]
+    (F : Ω → Finset (Bool × β)) (S : Finset (Bool × β)) :
+    inclusionMoment (fairSetLayer F) S =
+      finiteMean (fun ξ : β → Bool => inclusionMoment F (S.image (pairLayer ξ))) := by
+  classical
+  simp only [inclusionMoment_eq]
+  have he : (fun ω : Ω × (β → Bool) => if S ⊆ fairSetLayer F ω then (1:ℝ) else 0) =
+      (fun ω : Ω × (β → Bool) => if S.image (pairLayer ω.2) ⊆ F ω.1 then (1:ℝ) else 0) := by
+    funext ω
+    simp only [fairSetLayer,subset_pairLayer_image_iff]
+  apply (congrArg finiteMean he).trans
+  exact (finiteMean_prod (fun ω : Ω × (β → Bool) =>
+    if S.image (pairLayer ω.2) ⊆ F ω.1 then (1:ℝ) else 0)).trans
+      (finiteMean_comm (fun ω ξ => if S.image (pairLayer ξ) ⊆ F ω then (1:ℝ) else 0))
+
+lemma inclusionMoment_fairSetLayer_singleton {Ω : Type*} [Fintype Ω]
+    (F : Ω → Finset (Bool × β)) (x : Bool × β) :
+    inclusionMoment (fairSetLayer F) {x} =
+      (inclusionMoment F {x}+inclusionMoment F {(!x.1,x.2)})/2 := by
+  rw [inclusionMoment_fairSetLayer]
+  simp only [Finset.image_singleton,pairLayer,Equiv.coe_fn_mk]
+  rw [finiteMean_pi_eval (A := fun _ : β => Bool) x.2 (fun c : Bool => inclusionMoment F {(Bool.xor x.1 c,x.2)})]
+  rw [finiteMean_bool]
+  rcases x with ⟨b,x⟩
+  cases b <;> simp only [Bool.xor_false,Bool.false_xor,Bool.true_xor, Bool.not_false,Bool.not_true]
+
+lemma cylinder_fairSetLayer {Ω : Type*} [Fintype Ω]
+    (F : Ω → Finset (Bool × β))
+    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
+    CylinderBound (inclusionMoment (fairSetLayer F))
+      (fun x => inclusionMoment (fairSetLayer F) {x}) := by
+  have hh := h.fair_parallel (fun x => inclusionMoment_nonneg F {x})
+  intro S
+  rw [inclusionMoment_fairSetLayer]
+  simp only [inclusionMoment_fairSetLayer_singleton]
+  exact hh S
+
+end ParallelFairLayer
+
+section RelabelMarks
+variable {α β Ω Ω' : Type*} [DecidableEq α] [DecidableEq β] [Fintype Ω] [Fintype Ω']
+
+omit [Fintype Ω] [Fintype Ω'] in
+lemma subset_image_equiv_iff (e : α ≃ β) (A : Finset α) (S : Finset β) :
+    S ⊆ A.image e ↔ S.image e.symm ⊆ A := by
+  constructor
+  · intro h x hx
+    obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
+    obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp (h hy)
+    simpa only [←hxy,Equiv.symm_apply_apply] using hx
+  · intro h y hy
+    exact Finset.mem_image.mpr ⟨e.symm y,h (Finset.mem_image_of_mem _ hy),e.apply_symm_apply y⟩
+
+omit [Fintype Ω'] in
+lemma inclusionMoment_image_equiv (e : α ≃ β) (F : Ω → Finset α) (S : Finset β) :
+    inclusionMoment (fun ω => (F ω).image e) S = inclusionMoment F (S.image e.symm) := by
+  classical
+  simp only [inclusionMoment_eq]
+  apply finiteMean_congr
+  intro ω
+  simp only [subset_image_equiv_iff]
+
+omit [Fintype Ω'] in
+lemma cylinder_image_equiv (e : α ≃ β) (F : Ω → Finset α)
+    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
+    CylinderBound (inclusionMoment (fun ω => (F ω).image e))
+      (fun x => inclusionMoment (fun ω => (F ω).image e) {x}) := by
+  intro S
+  simp only [inclusionMoment_image_equiv,Finset.image_singleton]
+  have hh := h (S.image e.symm)
+  simpa only [Finset.prod_image e.symm.injective.injOn] using hh
+
+omit [DecidableEq β] in
+lemma inclusionMoment_comp_equiv (e : Ω ≃ Ω') (F : Ω' → Finset α) (S : Finset α) :
+    inclusionMoment (fun ω => F (e ω)) S = inclusionMoment F S := by
+  classical
+  simp only [inclusionMoment_eq]
+  simpa only [Equiv.apply_symm_apply] using
+    finiteMean_equiv e (fun ω => if S ⊆ F (e ω) then (1 : ℝ) else 0)
+
+omit [DecidableEq β] in
+lemma cylinder_comp_equiv (e : Ω ≃ Ω') (F : Ω' → Finset α)
+    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
+    CylinderBound (inclusionMoment (fun ω => F (e ω)))
+      (fun x => inclusionMoment (fun ω => F (e ω)) {x}) := by
+  intro S
+  simpa only [inclusionMoment_comp_equiv] using h S
+end RelabelMarks
+
+lemma finiteMean_prod_mul {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
+    (f : Ω → ℝ) (g : Ω' → ℝ) :
+    finiteMean (fun ω : Ω × Ω' => f ω.1*g ω.2) = finiteMean f * finiteMean g := by
+  simp only [finiteMean,Fintype.sum_prod_type,Fintype.card_prod,Nat.cast_mul,
+    ←Finset.mul_sum,←Finset.sum_mul]
+  ring
+
+section JoinChildren
+variable {β : Type*} [DecidableEq β]
+
+noncomputable def childJoin (A B : Finset β) : Finset (Bool × β) :=
+  A.image (fun x => (false,x)) ∪ B.image (fun x => (true,x))
+
+lemma mem_childJoin (A B : Finset β) (b : Bool) (x : β) :
+    (b,x) ∈ childJoin A B ↔ (if b then x ∈ B else x ∈ A) := by
+  cases b <;> simp [childJoin]
+
+noncomputable def boolFiber (S : Finset (Bool × β)) (b : Bool) : Finset β :=
+  (S.filter (fun x => x.1 = b)).image Prod.snd
+
+lemma mem_boolFiber (S : Finset (Bool × β)) (b : Bool) (x : β) :
+    x ∈ boolFiber S b ↔ (b,x) ∈ S := by
+  constructor
+  · intro h
+    obtain ⟨⟨c,y⟩,hy,hxy⟩ := Finset.mem_image.mp h
+    obtain ⟨hy,hc⟩ := Finset.mem_filter.mp hy
+    change c = b at hc
+    change y = x at hxy
+    simpa only [hc,hxy] using hy
+  · intro h
+    exact Finset.mem_image.mpr ⟨(b,x),Finset.mem_filter.mpr ⟨h,rfl⟩,rfl⟩
+
+lemma childJoin_boolFiber (S : Finset (Bool × β)) :
+    childJoin (boolFiber S false) (boolFiber S true) = S := by
+  ext ⟨b,x⟩
+  rw [mem_childJoin]
+  cases b <;> simp only [Bool.false_eq_true,↓reduceIte,mem_boolFiber]
+
+lemma subset_childJoin_iff (S : Finset (Bool × β)) (A B : Finset β) :
+    S ⊆ childJoin A B ↔ boolFiber S false ⊆ A ∧ boolFiber S true ⊆ B := by
+  constructor
+  · intro h
+    constructor
+    · intro x hx
+      exact (mem_childJoin A B false x).mp (h ((mem_boolFiber S false x).mp hx))
+    · intro x hx
+      exact (mem_childJoin A B true x).mp (h ((mem_boolFiber S true x).mp hx))
+  · rintro ⟨h₀,h₁⟩ ⟨b,x⟩ hx
+    rw [mem_childJoin]
+    cases b
+    · exact h₀ ((mem_boolFiber S false x).mpr hx)
+    · exact h₁ ((mem_boolFiber S true x).mpr hx)
+
+lemma prod_childJoin (A B : Finset β) (f : Bool × β → ℝ) :
+    (∏ x ∈ childJoin A B, f x) = (∏ x ∈ A, f (false,x)) * ∏ x ∈ B, f (true,x) := by
+  classical
+  have hd : Disjoint (A.image (fun x => (false,x))) (B.image (fun x => (true,x))) := by
+    apply Finset.disjoint_left.mpr
+    rintro x hx hy
+    obtain ⟨a,ha,rfl⟩ := Finset.mem_image.mp hx
+    obtain ⟨b,hb,hba⟩ := Finset.mem_image.mp hy
+    exact Bool.false_ne_true (congrArg Prod.fst hba).symm
+  rw [childJoin,Finset.prod_union hd,
+    Finset.prod_image (fun _ _ _ _ h => congrArg Prod.snd h),
+    Finset.prod_image (fun _ _ _ _ h => congrArg Prod.snd h)]
+
+lemma prod_boolFibers (S : Finset (Bool × β)) (f : Bool × β → ℝ) :
+    (∏ x ∈ S, f x) = (∏ x ∈ boolFiber S false, f (false,x)) *
+      ∏ x ∈ boolFiber S true, f (true,x) := by
+  rw [←prod_childJoin,childJoin_boolFiber]
+
+lemma inclusionMoment_childJoin {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
+    (F : Ω → Finset β) (G : Ω' → Finset β) (S : Finset (Bool × β)) :
+    inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) S =
+      inclusionMoment F (boolFiber S false) * inclusionMoment G (boolFiber S true) := by
+  classical
+  simp only [inclusionMoment_eq]
+  rw [←finiteMean_prod_mul]
+  apply finiteMean_congr
+  intro ω
+  simp only [subset_childJoin_iff]
+  split_ifs <;> simp_all
+
+lemma cylinder_childJoin {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] [Nonempty Ω] [Nonempty Ω']
+    (F : Ω → Finset β) (G : Ω' → Finset β)
+    (hF : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x}))
+    (hG : CylinderBound (inclusionMoment G) (fun x => inclusionMoment G {x})) :
+    CylinderBound (inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)))
+      (fun x => inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {x}) := by
+  intro S
+  rw [inclusionMoment_childJoin,prod_boolFibers]
+  have hf (b : Bool) (x : β) : boolFiber {(b,x)} b = {x} := by
+    ext y
+    simp only [mem_boolFiber,Finset.mem_singleton,Prod.mk.injEq,true_and]
+  have hn (b : Bool) (x : β) : boolFiber {(!b,x)} b = ∅ := by
+    ext y
+    cases b <;> simp [mem_boolFiber]
+  have h₀ (x : β) : inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {(false,x)} =
+      inclusionMoment F {x} * inclusionMoment G ∅ := by
+    rw [inclusionMoment_childJoin,hf]
+    have hh := hn true x
+    exact congrArg (fun T => inclusionMoment F {x} * inclusionMoment G T) hh
+  have h₁ (x : β) : inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {(true,x)} =
+      inclusionMoment F ∅ * inclusionMoment G {x} := by
+    rw [inclusionMoment_childJoin,hf]
+    have hh := hn false x
+    exact congrArg (fun T => inclusionMoment F T * inclusionMoment G {x}) hh
+  have emF : inclusionMoment F ∅ = 1 := by simp [inclusionMoment,finiteMean_const]
+  have emG : inclusionMoment G ∅ = 1 := by simp [inclusionMoment,finiteMean_const]
+  simp only [h₀,h₁,emF,emG,mul_one,one_mul]
+  exact mul_le_mul (hF _) (hG _) (inclusionMoment_nonneg _ _)
+    (Finset.prod_nonneg (fun x _ => inclusionMoment_nonneg F {x}))
+end JoinChildren
+
+lemma childMarks_join (d : ℕ) (A : Finset (Card (d+1))) :
+    A = (childMarks A false).image (Fin.cons false) ∪
+      (childMarks A true).image (Fin.cons true) := by
+  classical
+  ext x
+  constructor
+  · intro hx
+    cases hb : x 0
+    · exact Finset.mem_union_left _ (Finset.mem_image.mpr
+        ⟨Fin.tail x,(mem_childMarks _ _ _).mpr (by simpa only [←hb,Fin.cons_self_tail] using hx),
+          by simp only [←hb,Fin.cons_self_tail]⟩)
+    · exact Finset.mem_union_right _ (Finset.mem_image.mpr
+        ⟨Fin.tail x,(mem_childMarks _ _ _).mpr (by simpa only [←hb,Fin.cons_self_tail] using hx),
+          by simp only [←hb,Fin.cons_self_tail]⟩)
+  · intro hx
+    rcases Finset.mem_union.mp hx with hx | hx
+    · obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
+      exact (mem_childMarks _ _ _).mp hy
+    · obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
+      exact (mem_childMarks _ _ _).mp hy
+
+noncomputable def butterflyMarks (d : ℕ) (A : Finset (Card d))
+    (ω : SwitchIndex d → Bool) : Finset (Card d) :=
+  A.image (butterflyPerm d (decodeButterfly d ω))
+
+lemma butterflyMarks_succ (d : ℕ) (A : Finset (Card (d+1))) (ω : SwitchIndex (d+1) → Bool) :
+    butterflyMarks (d+1) A ω =
+      (fairSetLayer (fun p : (SwitchIndex d → Bool) × (SwitchIndex d → Bool) =>
+        childJoin (butterflyMarks d (childMarks A false) p.1)
+          (butterflyMarks d (childMarks A true) p.2)) (coinStepEquiv d ω)).image
+            (headTailEquiv d).symm := by
+  classical
+  unfold butterflyMarks fairSetLayer childJoin
+  rw [Finset.image_union,Finset.image_union]
+  simp only [Finset.image_image]
+  conv_lhs => rw [childMarks_join d A]
+  rw [Finset.image_union]
+  simp only [Finset.image_image]
+  apply congrArg₂ (· ∪ ·)
+  · apply Finset.image_congr
+    intro x _
+    simp only [Function.comp_def,butterflyPerm,decodeButterfly,Equiv.Perm.coe_mul,
+      childLift,pairSwitch,switchFun,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
+      coinStepEquiv,headTailEquiv,Equiv.symm_symm,pairLayer]
+    rfl
+  · apply Finset.image_congr
+    intro x _
+    simp only [Function.comp_def,butterflyPerm,decodeButterfly,Equiv.Perm.coe_mul,
+      childLift,pairSwitch,switchFun,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
+      coinStepEquiv,headTailEquiv,Equiv.symm_symm,pairLayer]
+    rfl
+
+lemma butterflyMarks_cylinder (d : ℕ) (A : Finset (Card d)) :
+    CylinderBound (inclusionMoment (butterflyMarks d A))
+      (fun x => inclusionMoment (butterflyMarks d A) {x}) := by
+  induction d with
+  | zero =>
+    have he : butterflyMarks 0 A = fun _ : SwitchIndex 0 → Bool => A := by
+      funext ω
+      simp [butterflyMarks,butterflyPerm]
+    rw [he]
+    exact cylinder_const A
+  | succ d ih =>
+    have h := cylinder_childJoin (butterflyMarks d (childMarks A false))
+      (butterflyMarks d (childMarks A true)) (ih _) (ih _)
+    have hh := cylinder_image_equiv (headTailEquiv d).symm _ (cylinder_fairSetLayer _ h)
+    have hs := cylinder_comp_equiv (coinStepEquiv d) _ hh
+    simpa only [←butterflyMarks_succ] using hs
+
+lemma indicator_mem_image_perm {α : Type*} [DecidableEq α]
+    (A : Finset α) (p : Equiv.Perm α) (y : α) :
+    (if y ∈ A.image p then (1 : ℝ) else 0) = ∑ x ∈ A, if p x = y then 1 else 0 := by
+  classical
+  by_cases h : y ∈ A.image p
+  · obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp h
+    rw [ite_eq_left (Finset.mem_image.mpr ⟨x,hx,hxy⟩)]
+    symm
+    calc
+      _ = (if p x = y then (1:ℝ) else 0) := Finset.sum_eq_single_of_mem x hx (by
+        intro z _ hzx
+        exact ite_eq_right (fun hz => hzx (p.injective (hz.trans hxy.symm))))
+      _ = 1 := ite_eq_left hxy
+  · rw [ite_eq_right h]
+    symm
+    apply Finset.sum_eq_zero
+    intro x hx
+    exact ite_eq_right (fun hxy => h (Finset.mem_image.mpr ⟨x,hx,hxy⟩))
+
+lemma butterflyMarks_singleton (d : ℕ) (A : Finset (Card d)) (y : Card d) :
+    inclusionMoment (butterflyMarks d A) {y} = (A.card : ℝ)/(2:ℝ)^d := by
+  classical
+  unfold inclusionMoment
+  simp only [Finset.singleton_subset_iff,butterflyMarks,indicator_mem_image_perm]
+  rw [finiteMean_sum]
+  simp only [butterfly_single_card_uniform,Finset.sum_const,nsmul_eq_mul]
+  ring
+
+lemma butterflyMarks_inclusion (d : ℕ) (A S : Finset (Card d)) :
+    inclusionMoment (butterflyMarks d A) S ≤ ((A.card : ℝ)/(2:ℝ)^d)^S.card := by
+  have h := butterflyMarks_cylinder d A S
+  simpa only [butterflyMarks_singleton,Finset.prod_const] using h
+
+namespace CycleColoring
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+abbrev Coloring (p : Equiv.Perm α) (S : Finset α) :=
+  {f : α → Bool // (∀ x, x ∉ S → f x = false) ∧
+    ∀ x ∈ S, p x ∈ S → f (p x) = f x}
+
+noncomputable instance (p : Equiv.Perm α) (S : Finset α) : Fintype (Coloring p S) :=
+  Fintype.ofFinite _
+
+noncomputable def boundary (p : Equiv.Perm α) (S : Finset α) : Finset α :=
+  S.filter (fun x => p.symm x ∉ S)
+
+noncomputable def root {p : Equiv.Perm α} {S : Finset α}
+    (C : RoutingNetwork.SelectedCycle p S) : α := C.nonempty.choose
+
+lemma root_mem {p : Equiv.Perm α} {S : Finset α} (C : RoutingNetwork.SelectedCycle p S) :
+    root C ∈ C.val := C.nonempty.choose_spec
+
+abbrev Seed (p : Equiv.Perm α) (S : Finset α) :=
+  {x // x ∈ boundary p S} ⊕ RoutingNetwork.SelectedCycle p S
+
+noncomputable def record {p : Equiv.Perm α} {S : Finset α}
+    (f : Coloring p S) : Seed p S → Bool :=
+  Sum.elim (fun x => f.val x.val) (fun C => f.val (root C))
+
+lemma orbit_subset_of_predecessor_closed (p : Equiv.Perm α) (D : Finset α)
+    (hD : ∀ x ∈ D, p.symm x ∈ D) {x : α} (hx : x ∈ D) : orbitSet p x ⊆ D := by
+  intro y hy
+  have hh := ((mem_orbitSet p x y).mp hy).inv
+  obtain ⟨n,hn⟩ := hh.exists_nat_pow_eq
+  have hp (k : ℕ) : (p⁻¹ ^ k) x ∈ D := by
+    induction k with
+    | zero => simpa using hx
+    | succ k ih => simpa only [pow_succ',Equiv.Perm.mul_apply,Equiv.Perm.coe_inv] using hD _ ih
+  exact hn ▸ hp n
+
+lemma record_injective (p : Equiv.Perm α) (S : Finset α) :
+    Function.Injective (record (p := p) (S := S)) := by
+  classical
+  intro f g hfg
+  apply Subtype.ext
+  funext x
+  by_contra hne
+  have hx : x ∈ S := by
+    by_contra h
+    exact hne ((f.property.1 x h).trans (g.property.1 x h).symm)
+  let D := S.filter (fun y => f.val y ≠ g.val y)
+  have hxD : x ∈ D := Finset.mem_filter.mpr ⟨hx,hne⟩
+  have hprev (y : α) (hy : y ∈ D) : p.symm y ∈ D := by
+    obtain ⟨hyS,hyne⟩ := Finset.mem_filter.mp hy
+    have hpS : p.symm y ∈ S := by
+      by_contra hn
+      have hseed : y ∈ boundary p S := Finset.mem_filter.mpr ⟨hyS,hn⟩
+      exact hyne (congrFun hfg (Sum.inl ⟨y,hseed⟩))
+    refine Finset.mem_filter.mpr ⟨hpS,?_⟩
+    have hf := f.property.2 (p.symm y) hpS (by simpa only [Equiv.apply_symm_apply] using hyS)
+    have hg := g.property.2 (p.symm y) hpS (by simpa only [Equiv.apply_symm_apply] using hyS)
+    simp only [Equiv.apply_symm_apply] at hf hg
+    intro he
+    exact hyne (hf.trans (he.trans hg.symm))
+  have horb := orbit_subset_of_predecessor_closed p D hprev hxD
+  let C : RoutingNetwork.SelectedCycle p S :=
+    ⟨orbitSet p x,⟨x,rfl⟩,fun y hy => (Finset.mem_filter.mp (horb hy)).1⟩
+  have hrD : root C ∈ D := horb (root_mem C)
+  have hrne := (Finset.mem_filter.mp hrD).2
+  exact hrne (congrFun hfg (Sum.inr C))
+
+lemma card_coloring_le (p : Equiv.Perm α) (S : Finset α) :
+    Fintype.card (Coloring p S) ≤
+      2 ^ ((boundary p S).card + Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
+  have h := Fintype.card_le_of_injective (record (p := p) (S := S)) (record_injective p S)
+  simpa only [Fintype.card_fun,Fintype.card_sum,Fintype.card_coe,Fintype.card_bool] using h
+
+omit [Fintype α] in
+lemma boundary_card (p : Equiv.Perm α) (S : Finset α) :
+    (boundary p S).card = S.card - (S.filter (fun x => p.symm x ∈ S)).card := by
+  classical
+  have h := Finset.card_filter_add_card_filter_not (s := S) (fun x => p.symm x ∈ S)
+  unfold boundary
+  omega
+
+lemma card_coloring_le_edges (p : Equiv.Perm α) (S : Finset α) :
+    Fintype.card (Coloring p S) ≤
+      2 ^ (S.card - (S.filter (fun x => p.symm x ∈ S)).card +
+        Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
+  simpa only [boundary_card] using card_coloring_le p S
+
+
+abbrev OppositeColoring (p : Equiv.Perm α) (S : Finset α) :=
+  {f : α → Bool // (∀ x, x ∉ S → f x = false) ∧
+    ∀ x ∈ S, p x ∈ S → f (p x) = !(f x)}
+
+noncomputable instance (p : Equiv.Perm α) (S : Finset α) : Fintype (OppositeColoring p S) :=
+  Fintype.ofFinite _
+
+noncomputable def bipartitionFlip (p : Equiv.Perm α) (S : Finset α)
+    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x))
+    (f : OppositeColoring p S) : Coloring p S := by
+  classical
+  refine ⟨(fun x => if x ∈ S then Bool.xor (f.val x) (χ x) else false),?_,?_⟩
+  · intro x hx
+    exact ite_eq_right hx
+  · intro x hx hpx
+    simp only [ite_eq_left hx,ite_eq_left hpx,f.property.2 x hx hpx,hχ x]
+    cases f.val x <;> cases χ x <;> rfl
+
+end CycleColoring
+end Thorp
+
+end ThorpNine.HighTail
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/HighTail/PriorPaths.lean b/lean/OAI/Probability/ThorpRouting/HighTail/PriorPaths.lean
index 98e5d6503b2717dc348fda467aee6295b6a9bf19..fb78ac4a63d7894f5416aa39ffbf4060b8d1dca8 100644
--- a/lean/OAI/Probability/ThorpRouting/HighTail/PriorPaths.lean
+++ b/lean/OAI/Probability/ThorpRouting/HighTail/PriorPaths.lean
@@ -1,876 +1,876 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.HighTail.AssignmentSlots
-
-namespace OAI
-
-namespace ThorpNine.HighTail
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-namespace RoutingNetwork
-variable {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι] [DecidableEq ι]
-  {d : ℕ} (R : RoutingNetwork α ι d)
-lemma mem_exposedPaths (b : ι → Bool) (V : Finset (α)) (s : ι) :
-    s ∈ exposedPaths R b V ↔ (R.users b s).1 ∈ V ∨ (R.users b s).2 ∈ V := by
-  classical
-  simp only [exposedPaths,Finset.mem_biUnion,R.mem_path]
-  constructor
-  · rintro ⟨x,hx,h | h⟩
-    · exact Or.inl (h ▸ hx)
-    · exact Or.inr (h ▸ hx)
-  · rintro (h | h)
-    · exact ⟨_,h,Or.inl rfl⟩
-    · exact ⟨_,h,Or.inr rfl⟩
-
-lemma terminalCharge_sum (b : ι → Bool) (z : α) (V : Finset (α))
-    (hz : z ∉ V) :
-    (terminalCharge R b z V : ℝ) = ∑ s : ι,
-      ((if (R.users b s).1 = z ∧ (R.users b s).2 ∈ V then (1:ℝ) else 0) +
-      (if (R.users b s).2 = z ∧ (R.users b s).1 ∈ V then (1:ℝ) else 0)) := by
-  classical
-  have he : R.path b z ∩ exposedPaths R b V =
-      Finset.univ.filter (fun s =>
-        ((R.users b s).1 = z ∧ (R.users b s).2 ∈ V) ∨
-        ((R.users b s).2 = z ∧ (R.users b s).1 ∈ V)) := by
-    ext s
-    simp only [Finset.mem_inter,R.mem_path,mem_exposedPaths,
-      Finset.mem_filter,Finset.mem_univ,true_and]
-    by_cases h₁ : (R.users b s).1 = z <;>
-      by_cases h₂ : (R.users b s).2 = z <;> simp [h₁,h₂,hz]
-  rw [terminalCharge,he]
-  simp only [← Finset.sum_boole]
-  apply Finset.sum_congr rfl
-  intro s _
-  by_cases h₁ : (R.users b s).1 = z <;>
-    by_cases h₂ : (R.users b s).2 = z <;> simp [h₁,h₂,hz]
-
-noncomputable def priorPaths {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) : Finset (α) :=
-  Finset.univ.filter (fun x => ∃ i : Fin A, o x = some i ∧ i.val < t)
-
-omit [DecidableEq α] in
-@[simp] lemma mem_priorPaths {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) (x : α) :
-    x ∈ priorPaths o t ↔ ∃ i : Fin A, o x = some i ∧ i.val < t := by
-  classical
-  simp [priorPaths]
-
-lemma terminalCharge_prior_sum {A : ℕ} (o : α → Option (Fin A))
-    (b : ι → Bool) (z : Terminal o) (i : Fin A) :
-    (terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) =
-      ∑ s : ι,
-        ((if (z i).val = (R.users b s).1 ∧ (R.users b s).2 ∈ priorPaths o (i.val+1)
-          then (1:ℝ) else 0) +
-        (if (z i).val = (R.users b s).2 ∧ (R.users b s).1 ∈ priorPaths o (i.val+1)
-          then (1:ℝ) else 0)) := by
-  classical
-  rw [terminalCharge_sum _ _ _ _ (Finset.notMem_erase _ _)]
-  apply Finset.sum_congr rfl
-  intro s _
-  have hn := R.users_ne b s
-  simp only [Finset.mem_erase]
-  by_cases h₁ : (z i).val = (R.users b s).1 <;>
-    by_cases h₂ : (z i).val = (R.users b s).2 <;> simp_all [eq_comm]
-
-lemma rotationMeeting_eq_slot_sum {A : ℕ} (o : α → Option (Fin A))
-    (z : Terminal o) (u v : α) :
-    rotationMeeting o z u v = ∑ i : Fin A,
-      ((if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) +
-      (if (z i).val = v ∧ u ∈ priorPaths o (i.val+1) then (1:ℝ) else 0)) := by
-  classical
-  have only_owner (u : α) (a : Fin A) (hou : o u = some a)
-      (i : Fin A) (hia : i ≠ a) : (z i).val ≠ u := by
-    intro h
-    have hi := (z i).property
-    rw [h,hou] at hi
-    exact hia (Option.some.inj hi).symm
-  have sum_one (u v : α) (a : Fin A) (hou : o u = some a) :
-      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) =
-      if (z a).val = u ∧ v ∈ priorPaths o (a.val+1) then 1 else 0 := by
-    apply Finset.sum_eq_single a
-    · intro i _ hia
-      simp [only_owner u a hou i hia]
-    · simp
-  have sum_none (u v : α) (hou : o u = none) :
-      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) = 0 := by
-    apply Finset.sum_eq_zero
-    intro i _
-    have hn : (z i).val ≠ u := by
-      intro h
-      have hi := (z i).property
-      rw [h,hou] at hi
-      contradiction
-    simp [hn]
-  rw [Finset.sum_add_distrib]
-  cases hou : o u with
-  | none =>
-    rw [sum_none u v hou]
-    simp [rotationMeeting,hou,mem_priorPaths]
-  | some a =>
-    rw [sum_one u v a hou]
-    cases hov : o v with
-    | none =>
-      rw [sum_none v u hov]
-      simp [rotationMeeting,hou,hov,mem_priorPaths]
-    | some b =>
-      rw [sum_one v u b hov]
-      simp only [rotationMeeting,hou,hov,mem_priorPaths,Option.some.injEq]
-      rcases lt_trichotomy a b with hab | rfl | hba
-      · have hn : ¬ b.val < a.val+1 := by omega
-        have hy : a.val < b.val+1 := by omega
-        simp [hab,hn,hy]
-      · simp
-      · have hn : ¬ a.val < b.val+1 := by omega
-        have hy : b.val < a.val+1 := by omega
-        simp [hba,not_lt_of_ge (le_of_lt hba),hn,hy]
-
-lemma rotationCharge_eq_terminal_sum {A : ℕ} (o : α → Option (Fin A))
-    (b : ι → Bool) (z : Terminal o) :
-    rotationCharge R o b z = ∑ i : Fin A,
-      (terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) := by
-  simp only [rotationCharge,rotationMeeting_eq_slot_sum,terminalCharge_prior_sum]
-  exact Finset.sum_comm
-
-omit [DecidableEq α] in
-@[simp] lemma priorPaths_zero {A : ℕ} (o : α → Option (Fin A)) : priorPaths o 0 = ∅ := by
-  classical
-  ext x
-  simp
-
-lemma priorPaths_succ {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) (ht : t < A) :
-    priorPaths o (t+1) = priorPaths o t ∪ cycleFiber o ⟨t,ht⟩ := by
-  classical
-  ext x
-  simp only [mem_priorPaths,Finset.mem_union,mem_cycleFiber]
-  constructor
-  · rintro ⟨i,hi,hit⟩
-    by_cases h : i.val < t
-    · exact Or.inl ⟨i,hi,h⟩
-    · have he : i = ⟨t,ht⟩ := Fin.ext (by change i.val = t; omega)
-      exact Or.inr (he ▸ hi)
-  · rintro (⟨i,hi,hit⟩ | hi)
-    · exact ⟨i,hi,by omega⟩
-    · exact ⟨⟨t,ht⟩,hi,by change t < t+1; omega⟩
-
-omit [DecidableEq α] in
-lemma priorPaths_disjoint {A : ℕ} (o : α → Option (Fin A)) (i : Fin A) :
-    Disjoint (priorPaths o i.val) (cycleFiber o i) := by
-  classical
-  apply Finset.disjoint_left.mpr
-  intro x hx hx'
-  obtain ⟨j,hj,hji⟩ := (mem_priorPaths _ _ _).mp hx
-  have he : j = i := Option.some.inj (hj.symm.trans ((mem_cycleFiber _ _ _).mp hx'))
-  subst j
-  exact (lt_irrefl _ hji)
-
-namespace QueryTree
-open Thorp.QueryTree
-
-lemma seekCycles_owned {A : ℕ} (S : Finset (α)) (r : Fin A → ℕ)
-    (o : α → Option (Fin A)) (z : Terminal o) (ω : ι → Bool)
-    (hS : ∀ i, cycleFiber o i ⊆ S)
-    (hO : ∀ i, orbitSet (R.perm ω) (z i).val = cycleFiber o i)
-    (hC : ∀ i, (cycleFiber o i).card = r i+1) :
-    let p := R.perm ω
-    let L := List.ofFn (fun i : Fin A => (p (z i).val,r i))
-    (seekCycles R S L ∅).succeeds ω = true ∧
-      ((seekCycles R S L ∅).staleCharge ∅ ω : ℝ) = rotationCharge R o (ω) z := by
-  classical
-  let b := ω
-  let p := R.perm b
-  let f (i : Fin A) := (p (z i).val,r i)
-  let c (i : Fin A) : ℝ := terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val)
-  have hc (i : Fin A) : Function.minimalPeriod p (z i).val - 1 = r i := by
-    have he := orbitSet_card p (z i).val
-    rw [hO,hC] at he
-    omega
-  have hs (t : ℕ) (ht : t ≤ A) :
-      (seekCycles R S ((List.ofFn f).drop t) (priorPaths o t)).succeeds ω = true ∧
-        ((seekCycles R S ((List.ofFn f).drop t) (priorPaths o t)).staleCharge (exposedPaths R b (priorPaths o t)) ω : ℝ) =
-          ((List.ofFn c).drop t).sum := by
-    induction ht using Nat.decreasingInduction with
-    | self => simp only [List.length_ofFn,le_refl,List.drop_eq_nil_of_le,List.sum_nil,seekCycles,Thorp.QueryTree.succeeds,staleCharge,Nat.cast_zero,and_self]
-    | of_succ t ht ih =>
-      have htf : t < (List.ofFn f).length := by simpa using ht
-      have htc : t < (List.ofFn c).length := by simpa using ht
-      rw [List.drop_eq_getElem_cons htf,List.drop_eq_getElem_cons htc]
-      simp only [List.getElem_ofFn,List.sum_cons,seekCycles]
-      have h := seekCycle_orbit R S (priorPaths o t) (z ⟨t,ht⟩).val
-        (fun V => seekCycles R S ((List.ofFn f).drop (t+1)) V) ω
-        (by rw [hO]; exact hS _) (by rw [hO]; exact priorPaths_disjoint o ⟨t,ht⟩)
-      dsimp only at h
-      rw [hc,hO,←priorPaths_succ o t ht] at h
-      dsimp only [f]
-      constructor
-      · exact h.1.trans ih.1
-      · rw [h.2,Nat.cast_add,ih.2]
-        exact add_comm _ _
-  have h := hs 0 (Nat.zero_le _)
-  simp only [List.drop_zero,priorPaths_zero,exposedPaths_empty,List.sum_ofFn] at h
-  dsimp only
-  exact ⟨h.1,h.2.trans (rotationCharge_eq_terminal_sum R o b z).symm⟩
-
-end QueryTree
-section Collections
-variable {A : ℕ}
-
-def ValidCycleSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ)
-    (o : α → Option (Fin A)) : Prop :=
-  (∀ i, (cycleFiber o i).card = r i + 1) ∧
-  (∀ i, cycleFiber o i ⊆ S) ∧
-  (∀ i x, o x = some i → orbitSet p x = cycleFiber o i)
-
-abbrev CycleSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
-  {o : α → Option (Fin A) // ValidCycleSlots p S r o}
-
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
-    Fintype (CycleSlots p S r) := Fintype.ofFinite _
-
-lemma CycleSlots.terminal_nonempty {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (c : CycleSlots p S r) (i : Fin A) : Nonempty {x : α // c.val x = some i} := by
-  have hn : (cycleFiber c.val i).Nonempty := Finset.card_pos.mp (by rw [c.property.1]; omega)
-  obtain ⟨x,hx⟩ := hn
-  exact ⟨x,(mem_cycleFiber _ _ _).mp hx⟩
-
-omit [DecidableEq α] in
-lemma owner_ext {o o' : α → Option (Fin A)}
-    (h : ∀ i, cycleFiber o i = cycleFiber o' i) : o = o' := by
-  funext x
-  have he (i : Fin A) : o x = some i ↔ o' x = some i := by
-    simpa only [mem_cycleFiber] using (Finset.ext_iff.mp (h i) x)
-  cases ho : o x with
-  | some i => exact ((he i).mp ho).symm
-  | none =>
-    cases ho' : o' x with
-    | none => rfl
-    | some i => have hn := (he i).mpr ho'; rw [ho] at hn; contradiction
-
-abbrev RootedSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
-  Σ c : CycleSlots p S r, Terminal c.val
-
-noncomputable def rootedStart {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (c : RootedSlots p S r) : Fin A → S := fun i =>
-  ⟨p (c.2 i).val,c.1.property.2.1 i (by
-    rw [←c.1.property.2.2 i (c.2 i).val (c.2 i).property]
-    exact (mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.apply_right (Equiv.Perm.SameCycle.refl p _)))⟩
-
-lemma rootedStart_injective {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ} :
-    Function.Injective (rootedStart (p:=p) (S:=S) (r:=r)) := by
-  rintro ⟨c,z⟩ ⟨c',z'⟩ h
-  have hz (i : Fin A) : (z i).val = (z' i).val :=
-    p.injective (congrArg Subtype.val (congrFun h i))
-  have hc : c = c' := Subtype.ext (owner_ext (fun i => by
-    rw [←c.property.2.2 i (z i).val (z i).property,
-      ←c'.property.2.2 i (z' i).val (z' i).property,hz]))
-  subst c'
-  have hzz : z = z' := funext (fun i => Subtype.ext (hz i))
-  subst z'
-  rfl
-
-lemma rooted_weight_sum_le (S : Finset (α)) (r : Fin A → ℕ) (ω : ι → Bool) :
-    (∑ c : RootedSlots (R.perm ω) S r,
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2)) ≤
-    ∑ a : Fin A → S,
-      if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-        ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-      else 0 := by
-  classical
-  let p := R.perm ω
-  let g (a : Fin A → S) : ℝ :=
-    if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-    else 0
-  have he (c : RootedSlots p S r) :
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2) =
-      g (rootedStart c) := by
-    have hs := QueryTree.seekCycles_owned R S r c.1.val c.2 ω c.1.property.2.1
-      (fun i => c.1.property.2.2 i (c.2 i).val (c.2 i).property) c.1.property.1
-    dsimp only at hs
-    unfold g
-    rw [ite_eq_left (show (QueryTree.seekCycles R S (List.ofFn (fun i => ((rootedStart c i).val,r i))) ∅).succeeds ω = true from hs.1)]
-    rw [← hs.2,neg_mul,Real.exp_neg,mul_comm (Real.log 2),Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
-    rfl
-  simp_rw [he]
-  exact sum_comp_injective_le rootedStart rootedStart_injective g (fun a => by dsimp [g]; split_ifs <;> positivity)
-
-lemma collection_rotation_lower (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r)
-    (ω : ι → Bool) (c : CycleSlots (R.perm ω) S r) :
-    collectionWeight d r ≤ ∑ z : Terminal c.val,
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z) := by
-  classical
-  let (i : Fin A) : Nonempty {x : α // c.val x = some i} := c.terminal_nonempty i
-  have hm : Monotone (fun i => (cycleFiber c.val i).card) := by
-    intro i j hij
-    dsimp only
-    rw [c.property.1,c.property.1]
-    exact Nat.add_le_add_right (hr hij) 1
-  have hb := rotation_exp_bound R hm (ω)
-  have hcard : (Fintype.card (Terminal c.val) : ℝ) = ∏ i : Fin A, (r i+1 : ℝ) := by
-    rw [Fintype.card_pi,Nat.cast_prod]
-    apply Finset.prod_congr rfl
-    intro i _
-    rw [card_cycleFiber,c.property.1,Nat.cast_add,Nat.cast_one]
-  have hp : (0:ℝ) < Fintype.card (Terminal c.val) := Nat.cast_pos.mpr Fintype.card_pos
-  have hl : rotationLower d r ≤ finiteMean (fun z : Terminal c.val => Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z)) := by
-    simpa only [rotationLower,c.property.1,Nat.cast_add,Nat.cast_one] using hb
-  have hh := (le_div_iff₀ hp).mp hl
-  rw [hcard] at hh
-  unfold collectionWeight
-  rw [← Finset.mul_sum]
-  calc
-    _ = ((2:ℝ)^d)^A * (rotationLower d r * ∏ i : Fin A, (r i+1 : ℝ)) := by ring
-    _ ≤ _ := mul_le_mul_of_nonneg_left hh (by positivity)
-
-lemma cycleSlots_weighted_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : ι → Bool =>
-      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) *
-      collectionWeight d r ≤ (S.card : ℝ)^A := by
-  classical
-  let g (a : Fin A → S) (ω : ι → Bool) : ℝ :=
-    if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-    else 0
-  have hω (ω : ι → Bool) :
-      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ) * collectionWeight d r ≤
-        ∑ a : Fin A → S, g a ω := by
-    calc
-      _ = ∑ _c : CycleSlots (R.perm ω) S r, collectionWeight d r := by simp
-      _ ≤ ∑ c : CycleSlots (R.perm ω) S r,
-          ∑ z : Terminal c.val, ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z) :=
-        Finset.sum_le_sum (fun c _ => collection_rotation_lower R S r hr ω c)
-      _ = ∑ c : RootedSlots (R.perm ω) S r,
-          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2) :=
-        (Fintype.sum_sigma (fun c : RootedSlots (R.perm ω) S r =>
-          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2))).symm
-      _ ≤ _ := rooted_weight_sum_le R S r ω
-  have hb := finiteMean_mono hω
-  rw [finiteMean_mul_const,finiteMean_sum] at hb
-  calc
-    _ ≤ ∑ a : Fin A → S, finiteMean (g a) := hb
-    _ ≤ ∑ _a : Fin A → S, (1:ℝ) := by
-      apply Finset.sum_le_sum
-      intro a _
-      have hh := QueryTree.cycles_weighted_success R S (List.ofFn (fun i : Fin A => ((a i).val,r i)))
-      simpa only [List.length_ofFn] using hh
-    _ = _ := by simp
-
-lemma cycleSlots_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : ι → Bool =>
-      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
-        (S.card : ℝ)^A / collectionWeight d r := by
-  exact (le_div_iff₀ (collectionWeight_pos d r)).mpr (cycleSlots_weighted_moment R S r hr)
-
-lemma butterfly_cycleSlots_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : ι → Bool =>
-      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
-        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
-  have h := cycleSlots_moment R S r hr
-  rw [collectionWeight_eq_sqrt,div_mul_eq_div_mul_one_div] at h
-  rw [div_pow]
-  simpa only [one_div,Finset.prod_inv_distrib] using h
-
-end Collections
-
-section CycleCounting
-variable {A : ℕ}
-
-lemma SelectedCycle.nonempty {p : Equiv.Perm (α)} {S : Finset (α)}
-    (C : SelectedCycle p S) : C.val.Nonempty := by
-  obtain ⟨x,hx⟩ := C.property.1
-  rw [hx]
-  exact orbitSet_nonempty p x
-
-lemma SelectedCycle.orbit_eq {p : Equiv.Perm (α)} {S : Finset (α)}
-    (C : SelectedCycle p S) {x : α} (hx : x ∈ C.val) : orbitSet p x = C.val := by
-  obtain ⟨y,hy⟩ := C.property.1
-  rw [hy] at hx ⊢
-  exact (orbitSet_eq p ((mem_orbitSet _ _ _).mp hx)).symm
-
-lemma SelectedCycle.eq_of_mem {p : Equiv.Perm (α)} {S : Finset (α)}
-    (C D : SelectedCycle p S) {x : α} (hC : x ∈ C.val) (hD : x ∈ D.val) : C = D := by
-  apply Subtype.ext
-  exact (C.orbit_eq hC).symm.trans (D.orbit_eq hD)
-
-abbrev SlotAssignment (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
-  {f : Fin A → SelectedCycle p S // Function.Injective f ∧ ∀ i, (f i).val.card = r i+1}
-
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
-    Fintype (SlotAssignment p S r) := Fintype.ofFinite _
-
-noncomputable def slotsToAssignment {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (c : CycleSlots p S r) : SlotAssignment p S r := by
-  classical
-  let f (i : Fin A) : SelectedCycle p S := ⟨cycleFiber c.val i,by
-    obtain ⟨x,hx⟩ := c.terminal_nonempty i
-    exact ⟨⟨x,(c.property.2.2 i x hx).symm⟩,c.property.2.1 i⟩⟩
-  refine ⟨f,?_,fun i => c.property.1 i⟩
-  intro i j hij
-  have he : cycleFiber c.val i = cycleFiber c.val j := congrArg Subtype.val hij
-  obtain ⟨x,hx⟩ := c.terminal_nonempty i
-  have hxi : x ∈ cycleFiber c.val i := (mem_cycleFiber _ _ _).mpr hx
-  rw [he] at hxi
-  exact Option.some.inj (hx.symm.trans ((mem_cycleFiber _ _ _).mp hxi))
-
-noncomputable def assignmentOwner {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) (x : α) : Option (Fin A) := by
-  classical
-  exact if h : ∃ i, x ∈ (f.val i).val then some (Classical.choose h) else none
-
-lemma assignmentOwner_eq_some {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) (x : α) (i : Fin A) :
-    assignmentOwner f x = some i ↔ x ∈ (f.val i).val := by
-  classical
-  unfold assignmentOwner
-  split_ifs with h
-  · constructor
-    · intro hi
-      have hx := Classical.choose_spec h
-      rw [Option.some.inj hi] at hx
-      exact hx
-    · intro hx
-      have he := (f.val (Classical.choose h)).eq_of_mem (f.val i) (Classical.choose_spec h) hx
-      exact congrArg some (f.property.1 he)
-  · constructor
-    · intro hi; contradiction
-    · intro hx; exact (h ⟨i,hx⟩).elim
-
-lemma assignmentOwner_fiber {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) (i : Fin A) : cycleFiber (assignmentOwner f) i = (f.val i).val := by
-  ext x
-  simp only [mem_cycleFiber,assignmentOwner_eq_some]
-
-noncomputable def assignmentToSlots {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) : CycleSlots p S r :=
-  ⟨assignmentOwner f,by
-    refine ⟨?_,?_,?_⟩
-    · intro i
-      rw [assignmentOwner_fiber]
-      exact f.property.2 i
-    · intro i
-      rw [assignmentOwner_fiber]
-      exact (f.val i).property.2
-    · intro i x hx
-      rw [assignmentOwner_fiber]
-      exact (f.val i).orbit_eq ((assignmentOwner_eq_some f x i).mp hx)⟩
-
-noncomputable def cycleSlotsEquivAssignments (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
-    CycleSlots p S r ≃ SlotAssignment p S r where
-  toFun := slotsToAssignment
-  invFun := assignmentToSlots
-  left_inv c := by
-    apply Subtype.ext
-    apply owner_ext
-    intro i
-    rw [assignmentToSlots,assignmentOwner_fiber]
-    rfl
-  right_inv f := by
-    apply Subtype.ext
-    funext i
-    apply Subtype.ext
-    exact assignmentOwner_fiber f i
-
-end CycleCounting
-
-section GroupedCounting
-variable {A : ℕ}
-
-abbrev LengthClass (r : Fin A → ℕ) := {j : ℕ // j ∈ Finset.univ.image (fun i => r i + 1)}
-
-def slotClass (r : Fin A → ℕ) (i : Fin A) : LengthClass r :=
-  ⟨r i+1, Finset.mem_image.mpr ⟨i, Finset.mem_univ _, rfl⟩⟩
-
-abbrev SlotsOfLength (r : Fin A → ℕ) (j : LengthClass r) := {i : Fin A // slotClass r i = j}
-abbrev CyclesOfLength (p : Equiv.Perm (α)) (S : Finset (α)) (j : ℕ) :=
-  {C : SelectedCycle p S // C.val.card = j}
-
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (j : ℕ) :
-    Fintype (CyclesOfLength p S j) := Fintype.ofFinite _
-
-abbrev GroupedAssignments (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
-  ∀ j : LengthClass r, SlotsOfLength r j ↪ CyclesOfLength p S j.val
-
-noncomputable def assignmentToGroups {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) : GroupedAssignments p S r := fun _j =>
-  { toFun := fun i => ⟨f.val i.val, (f.property.2 i.val).trans (congrArg Subtype.val i.property)⟩
-    inj' := fun _i _k h => Subtype.ext (f.property.1 (congrArg Subtype.val h)) }
-
-noncomputable def groupsEmbedding {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (F : GroupedAssignments p S r) : Fin A ↪ SelectedCycle p S :=
-  ((Equiv.sigmaFiberEquiv (slotClass r)).symm.toEmbedding.trans
-    (Function.Embedding.sigmaMap (Function.Embedding.refl _) F)).trans
-    ((Function.Embedding.sigmaMap (Function.Embedding.subtype _) (fun _ => Function.Embedding.refl _)).trans
-      (Equiv.sigmaFiberEquiv (fun C : SelectedCycle p S => C.val.card)).toEmbedding)
-
-noncomputable def groupsToAssignment {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (F : GroupedAssignments p S r) : SlotAssignment p S r :=
-  ⟨groupsEmbedding F, (groupsEmbedding F).injective,
-    fun i => (F (slotClass r i) ⟨i,rfl⟩).property⟩
-
-noncomputable def assignmentEquivGroups (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
-    SlotAssignment p S r ≃ GroupedAssignments p S r where
-  toFun := assignmentToGroups
-  invFun := groupsToAssignment
-  left_inv f := by
-    apply Subtype.ext
-    rfl
-  right_inv F := by
-    funext j
-    apply Function.Embedding.ext
-    intro i
-    rcases i with ⟨i,hi⟩
-    subst j
-    rfl
-
-lemma card_cycleSlots_factorial (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
-    Fintype.card (CycleSlots p S r) =
-      ∏ j : LengthClass r, (Fintype.card (CyclesOfLength p S j.val)).descFactorial
-        (Fintype.card (SlotsOfLength r j)) := by
-  classical
-  rw [Fintype.card_congr (cycleSlotsEquivAssignments p S r),
-    Fintype.card_congr (assignmentEquivGroups p S r), Fintype.card_pi]
-  simp only [Fintype.card_embedding_eq]
-
-lemma butterfly_factorial_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : ι → Bool =>
-       (∏ j : LengthClass r,
-         (Fintype.card (CyclesOfLength (R.perm ω) S j.val)).descFactorial
-           (Fintype.card (SlotsOfLength r j)) : ℕ)) ≤
-      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-        ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
-  simpa only [←card_cycleSlots_factorial] using butterfly_cycleSlots_moment R S r hr
-
-end GroupedCounting
-
-section AllLengths
-variable {A : ℕ}
-
-noncomputable def assignmentRelabel (p : Equiv.Perm (α)) (S : Finset (α))
-    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
-    SlotAssignment p S r ≃ SlotAssignment p S (r ∘ e) where
-  toFun f := ⟨f.val ∘ e, f.property.1.comp e.injective, fun i => f.property.2 (e i)⟩
-  invFun f := ⟨f.val ∘ e.symm, f.property.1.comp e.symm.injective, fun i => by
-    simpa only [Function.comp_apply,Equiv.apply_symm_apply] using f.property.2 (e.symm i)⟩
-  left_inv f := by apply Subtype.ext; funext i; simp
-  right_inv f := by apply Subtype.ext; funext i; simp
-
-lemma card_cycleSlots_relabel (p : Equiv.Perm (α)) (S : Finset (α))
-    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
-    Fintype.card (CycleSlots p S r) = Fintype.card (CycleSlots p S (r ∘ e)) := by
-  exact Fintype.card_congr ((cycleSlotsEquivAssignments p S r).trans
-    ((assignmentRelabel p S r e).trans (cycleSlotsEquivAssignments p S (r ∘ e)).symm))
-
-lemma butterfly_cycleSlots_moment_all (S : Finset (α)) (r : Fin A → ℕ) :
-    finiteMean (fun ω : ι → Bool =>
-      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
-        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
-  have h := butterfly_cycleSlots_moment R S (r ∘ Tuple.sort r) (Tuple.monotone_sort r)
-  simp_rw [←card_cycleSlots_relabel] at h
-  have hp := Equiv.prod_comp (Tuple.sort r) (fun i : Fin A => (1 / Real.sqrt (r i+1) : ℝ))
-  simpa only [Function.comp_apply,hp] using h
-
-abbrev ShortCycle (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :=
-  {C : SelectedCycle p S // C.val.card ≤ J}
-
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    Fintype (ShortCycle p S J) := Fintype.ofFinite _
-
-noncomputable def shortCycleLength {p : Equiv.Perm (α)} {S : Finset (α)} {J : ℕ}
-    (C : ShortCycle p S J) : Fin J :=
-  ⟨C.val.val.card-1, by have := C.val.nonempty.card_pos; have := C.property; omega⟩
-
-lemma shortCycleLength_add_one {p : Equiv.Perm (α)} {S : Finset (α)} {J : ℕ}
-    (C : ShortCycle p S J) : (shortCycleLength C : ℕ)+1 = C.val.val.card := by
-  have := C.val.nonempty.card_pos
-  simp only [shortCycleLength]
-  omega
-
-noncomputable def shortEmbeddingEquiv (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    (Fin A ↪ ShortCycle p S J) ≃
-      Σ r : Fin A → Fin J, SlotAssignment p S (fun i => (r i : ℕ)) where
-  toFun f := ⟨fun i => shortCycleLength (f i),
-    ⟨fun i => (f i).val, fun i j h => f.injective (Subtype.ext h),
-      fun i => (shortCycleLength_add_one (f i)).symm⟩⟩
-  invFun c :=
-    { toFun := fun i => ⟨c.2.val i, by rw [c.2.property.2]; exact (c.1 i).isLt⟩
-      inj' := fun _i _j h => c.2.property.1 (congrArg Subtype.val h) }
-  left_inv f := by
-    apply Function.Embedding.ext
-    intro i
-    rfl
-  right_inv c := by
-    rcases c with ⟨r,f⟩
-    have he : (fun i => shortCycleLength (⟨f.val i, by rw [f.property.2]; exact (r i).isLt⟩ : ShortCycle p S J)) = r := by
-      funext i
-      apply Fin.ext
-      simp only [shortCycleLength,f.property.2,Nat.add_sub_cancel]
-    apply Sigma.ext he
-    apply (Subtype.heq_iff_coe_eq (by
-      intro x
-      change (Function.Injective x ∧ ∀ i, (x i).val.card = (f.val i).val.card - 1 + 1) ↔ _
-      simp only [f.property.2,Nat.add_sub_cancel])).mpr
-    rfl
-
-lemma shortCycle_factorial_eq (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    (Fintype.card (ShortCycle p S J)).descFactorial A =
-      ∑ r : Fin A → Fin J, Fintype.card (CycleSlots p S (fun i => (r i : ℕ))) := by
-  classical
-  have hc := Fintype.card_embedding_eq (α := Fin A) (β := ShortCycle p S J)
-  simp only [Fintype.card_fin] at hc
-  rw [←hc, Fintype.card_congr (shortEmbeddingEquiv p S J),Fintype.card_sigma]
-  apply Finset.sum_congr rfl
-  intro r _
-  exact (Fintype.card_congr (cycleSlotsEquivAssignments p S (fun i => (r i : ℕ)))).symm
-
-end AllLengths
-
-section CycleMGF
-
-lemma butterfly_shortCycle_moment (S : Finset (α)) (J A : ℕ) :
-    finiteMean (fun ω : ι → Bool =>
-      ((Fintype.card (ShortCycle (R.perm ω) S J)).descFactorial A : ℝ)) ≤
-      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))^A := by
-  classical
-  simp_rw [shortCycle_factorial_eq,Nat.cast_sum,finiteMean_sum]
-  calc
-    _ ≤ ∑ r : Fin A → Fin J,
-        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-          ∏ i : Fin A, (1 / Real.sqrt ((r i : ℕ)+1)) := by
-      apply Finset.sum_le_sum
-      intro r _
-      exact butterfly_cycleSlots_moment_all R S (fun i => (r i : ℕ))
-    _ = _ := by
-      rw [←Finset.mul_sum,mul_pow]
-      congr 1
-      rw [←Fintype.prod_sum (fun (_ : Fin A) (j : Fin J) => (1 / Real.sqrt ((j:ℕ)+1) : ℝ))]
-      simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fin]
-
-noncomputable def cycleMass (p : Equiv.Perm (α)) (S : Finset (α)) : ℕ :=
-  ∑ C : SelectedCycle p S, C.val.card
-
-lemma cycleMass_le (p : Equiv.Perm (α)) (S : Finset (α)) : cycleMass p S ≤ S.card := by
-  classical
-  let f : (Σ C : SelectedCycle p S, {x : α // x ∈ C.val}) → S :=
-    fun c => ⟨c.2.val,c.1.property.2 c.2.property⟩
-  have hi : Function.Injective f := by
-    intro c e h
-    have hx : c.2.val = e.2.val := congrArg Subtype.val h
-    have hC : c.1 = e.1 := c.1.eq_of_mem e.1 c.2.property (hx ▸ e.2.property)
-    rcases c with ⟨c,x⟩
-    rcases e with ⟨e,y⟩
-    dsimp only at hC
-    subst e
-    have he : x=y := Subtype.ext hx
-    cases he
-    rfl
-  have hh := Fintype.card_le_of_injective f hi
-  simpa only [Fintype.card_sigma,Fintype.card_coe,cycleMass] using hh
-
-lemma shortCycle_card_le (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    Fintype.card (ShortCycle p S J) ≤ S.card := by
-  classical
-  calc
-    _ ≤ Fintype.card (SelectedCycle p S) := Fintype.card_subtype_le _
-    _ = ∑ _ : SelectedCycle p S, 1 := by simp
-    _ ≤ cycleMass p S := Finset.sum_le_sum (fun C _ => C.nonempty.card_pos)
-    _ ≤ _ := cycleMass_le p S
-
-lemma butterfly_shortCycle_mgf (S : Finset (α)) (J : ℕ) (q : ℝ) (hq : 1 ≤ q) :
-    finiteMean (fun ω : ι → Bool =>
-      q^(Fintype.card (ShortCycle (R.perm ω) S J))) ≤
-      Real.exp ((q-1)*((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))) := by
-  apply factorial_mgf _ S.card _ q (fun ω => shortCycle_card_le _ _ _)
-  · positivity
-  · exact hq
-  · exact butterfly_shortCycle_moment R S J
-
-end CycleMGF
-
-
-abbrev LongCycle (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :=
-  {C : SelectedCycle p S // ¬ C.val.card ≤ J}
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    Fintype (LongCycle p S J) := Fintype.ofFinite _
-
-lemma selectedCycle_card_split (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    Fintype.card (SelectedCycle p S) =
-      Fintype.card (ShortCycle p S J) + Fintype.card (LongCycle p S J) := by
-  classical
-  rw [←Fintype.card_congr (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J)),Fintype.card_sum]
-
-lemma longCycle_card_bound (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    J * Fintype.card (LongCycle p S J) ≤ S.card := by
-  classical
-  have hs := Equiv.sum_comp (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J))
-    (fun C : SelectedCycle p S => C.val.card)
-  rw [Fintype.sum_sum_type] at hs
-  have h0 : J * Fintype.card (LongCycle p S J) ≤
-      ∑ C : LongCycle p S J, C.val.val.card := by
-    calc
-      _ = ∑ _C : LongCycle p S J, J := by simp [mul_comm]
-      _ ≤ _ := Finset.sum_le_sum (fun C _ => Nat.le_of_lt (Nat.lt_of_not_ge C.property))
-  have hh := cycleMass_le p S
-  unfold cycleMass at hh
-  rw [←hs] at hh
-  exact h0.trans ((Nat.le_add_left _ _).trans hh)
-
-lemma longCycle_pow_bound (p : Equiv.Perm (α)) (S : Finset (α))
-    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    q^(Fintype.card (LongCycle p S J)) ≤ Real.exp ((S.card : ℝ)/J * Real.log q) := by
-  have hq0 : 0 < q := lt_of_lt_of_le zero_lt_one hq
-  have hc : (Fintype.card (LongCycle p S J) : ℝ) ≤ (S.card : ℝ)/J := by
-    apply (le_div_iff₀ (by exact_mod_cast hJ)).mpr
-    have hh := longCycle_card_bound p S J
-    exact_mod_cast (by simpa only [mul_comm] using hh)
-  calc
-    _ = Real.exp ((Fintype.card (LongCycle p S J) : ℝ)*Real.log q) := by
-      rw [Real.exp_nat_mul,Real.exp_log hq0]
-    _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right hc (Real.log_nonneg hq))
-
-lemma butterfly_cycle_log_mgf (S : Finset (α)) (J : ℕ) (hJ : 0 < J)
-    (q : ℝ) (hq : 1 ≤ q) :
-    Real.log (finiteMean (fun ω : ι → Bool =>
-      q^(Fintype.card (SelectedCycle (R.perm ω) S)))) ≤
-      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
-  let lam : ℝ := (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1)
-  let c : ℝ := (S.card : ℝ)/J * Real.log q
-  have hnum : (q-1)*lam ≤ 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
-    have hsum := sum_inv_sqrt J
-    have hp : 0 ≤ (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) := by positivity
-    have h0 : 0 ≤ 2*Real.sqrt J := by positivity
-    have hh := mul_le_mul_of_nonneg_left hsum hp
-    have hq1 : 0 ≤ q-1 := sub_nonneg.mpr hq
-    calc
-      _ ≤ (q-1)*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
-        mul_le_mul_of_nonneg_left hh hq1
-      _ ≤ q*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
-        mul_le_mul_of_nonneg_right (by linarith) (mul_nonneg hp h0)
-      _ = _ := by ring
-  have hm : finiteMean (fun ω : ι → Bool =>
-      q^(Fintype.card (SelectedCycle (R.perm ω) S))) ≤
-      Real.exp (c+(q-1)*lam) := by
-    calc
-      _ ≤ finiteMean (fun ω : ι → Bool =>
-          q^(Fintype.card (ShortCycle (R.perm ω) S J))*Real.exp c) := by
-        apply finiteMean_mono
-        intro ω
-        rw [selectedCycle_card_split _ _ J,pow_add]
-        exact mul_le_mul_of_nonneg_left (longCycle_pow_bound _ S J hJ q hq) (by positivity)
-      _ = finiteMean (fun ω : ι → Bool =>
-          q^(Fintype.card (ShortCycle (R.perm ω) S J))) * Real.exp c :=
-        finiteMean_mul_const _ _
-      _ ≤ Real.exp ((q-1)*lam) * Real.exp c :=
-        mul_le_mul_of_nonneg_right (butterfly_shortCycle_mgf R S J q hq) (Real.exp_nonneg _)
-      _ = _ := by rw [←Real.exp_add,add_comm]
-  have hpos : 0 < finiteMean (fun ω : ι → Bool =>
-      q^(Fintype.card (SelectedCycle (R.perm ω) S))) := by
-    apply finiteMean_pos
-    intro ω
-    positivity
-  have hlog := Real.log_le_log hpos hm
-  rw [Real.log_exp] at hlog
-  dsimp [c] at hlog
-  linarith
-
-
-end RoutingNetwork
-
-variable {α B : Type*} [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] {d : ℕ}
-
-def parallelPerm (ω : B × SwitchIndex d → Bool) : Equiv.Perm (B × Card d) where
-  toFun x := (x.1,butterflyPerm d (decodeButterfly d (fun i => ω (x.1,i))) x.2)
-  invFun y := (y.1,(butterflyPerm d (decodeButterfly d (fun i => ω (y.1,i)))).symm y.2)
-  left_inv x := by simp
-  right_inv y := by simp
-
-def relabeledPerm (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (ω : B × SwitchIndex d → Bool) : Equiv.Perm α :=
-  (pre.trans (parallelPerm ω)).trans post
-
-noncomputable def parallelPath (pre : α ≃ B × Card d)
-    (ω : B × SwitchIndex d → Bool) (x : α) : Finset (B × SwitchIndex d) :=
-  (routeDomain d (pre x).2
-    (butterflyPerm d (decodeButterfly d (fun i => ω ((pre x).1,i))) (pre x).2)).image
-      (fun i => ((pre x).1,i))
-
-def parallelUsers (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
-    (i : B × SwitchIndex d) : α × α :=
-  let u := switchUsers d (decodeButterfly d (fun j => ω (i.1,j))) i.2
-  (pre.symm (i.1,u.1),pre.symm (i.1,u.2))
-
-omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
-lemma parallelUsers_ne (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
-    (i : B × SwitchIndex d) : (parallelUsers pre ω i).1 ≠ (parallelUsers pre ω i).2 := by
-  intro h
-  exact switchUsers_ne d _ _ (congrArg Prod.snd (pre.symm.injective h))
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma mem_parallelPath (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
-    (x : α) (i : B × SwitchIndex d) :
-    i ∈ parallelPath pre ω x ↔ (parallelUsers pre ω i).1 = x ∨ (parallelUsers pre ω i).2 = x := by
-  classical
-  obtain ⟨b,j⟩ := i
-  simp only [parallelPath,Finset.mem_image,Prod.mk.injEq]
-  constructor
-  · rintro ⟨k,hk,hb,rfl⟩
-    subst b
-    rcases (mem_routeDomain_iff_user d _ _ _).mp hk with h | h
-    · left
-      apply pre.injective
-      change pre (pre.symm ((pre x).1,_)) = pre x
-      rw [Equiv.apply_symm_apply]
-      exact Prod.ext rfl h
-    · right
-      apply pre.injective
-      change pre (pre.symm ((pre x).1,_)) = pre x
-      rw [Equiv.apply_symm_apply]
-      exact Prod.ext rfl h
-  · intro h
-    have hh : (b,(switchUsers d (decodeButterfly d (fun j => ω (b,j))) j).1) = pre x ∨
-        (b,(switchUsers d (decodeButterfly d (fun j => ω (b,j))) j).2) = pre x := by
-      rcases h with h | h
-      · exact Or.inl (by simpa only [parallelUsers,Equiv.apply_symm_apply] using congrArg pre h)
-      · exact Or.inr (by simpa only [parallelUsers,Equiv.apply_symm_apply] using congrArg pre h)
-    have hb : b = (pre x).1 := hh.elim (congrArg Prod.fst) (congrArg Prod.fst)
-    subst b
-    refine ⟨j,?_,rfl,rfl⟩
-    apply (mem_routeDomain_iff_user d _ _ _).mpr
-    exact hh.elim (fun h => Or.inl (congrArg Prod.snd h)) (fun h => Or.inr (congrArg Prod.snd h))
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma card_parallelPath (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool) (x : α) :
-    (parallelPath pre ω x).card = d := by
-  classical
-  rw [parallelPath,Finset.card_image_of_injective _ (fun _ _ h => Prod.mk.inj h |>.2),card_routeDomain]
-
-noncomputable def blockMarks (pre : α ≃ B × Card d) (S : Finset α) (b : B) : Finset (Card d) :=
-  Finset.univ.filter (fun x => pre.symm (b,x) ∈ S)
-
-omit [Fintype α] [Fintype B] [DecidableEq B] in
-@[simp] lemma mem_blockMarks (pre : α ≃ B × Card d) (S : Finset α) (b : B) (x : Card d) :
-    x ∈ blockMarks pre S b ↔ pre.symm (b,x) ∈ S := by
-  classical
-  simp [blockMarks]
-
-omit [DecidableEq B] in
-lemma sum_card_blockMarks (pre : α ≃ B × Card d) (S : Finset α) :
-    ∑ b : B, (blockMarks pre S b).card = S.card := by
-  classical
-  calc
-    _ = ∑ b : B, ∑ x : Card d, if pre.symm (b,x) ∈ S then 1 else 0 := by
-      simp only [blockMarks,Finset.card_filter]
-    _ = ∑ u : B × Card d, if pre.symm u ∈ S then 1 else 0 := (Fintype.sum_prod_type (fun u : B × Card d => if pre.symm u ∈ S then (1:ℕ) else 0)).symm
-    _ = ∑ x : α, if x ∈ S then 1 else 0 := Equiv.sum_comp pre.symm (fun x : α => if x ∈ S then (1:ℕ) else 0)
-    _ = _ := by simp
-
-omit [DecidableEq B] in
-lemma card_blockMarks_le (pre : α ≃ B × Card d) (S : Finset α) (b : B) :
-    (blockMarks pre S b).card ≤ S.card := by
-  rw [←sum_card_blockMarks pre S]
-  exact Finset.single_le_sum (f := fun b : B => (blockMarks pre S b).card) (fun _ _ => Nat.zero_le _) (Finset.mem_univ b)
-
-end Thorp
-
-end ThorpNine.HighTail
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.HighTail.AssignmentSlots
+
+namespace OAI
+
+namespace ThorpNine.HighTail
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+namespace RoutingNetwork
+variable {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι] [DecidableEq ι]
+  {d : ℕ} (R : RoutingNetwork α ι d)
+lemma mem_exposedPaths (b : ι → Bool) (V : Finset (α)) (s : ι) :
+    s ∈ exposedPaths R b V ↔ (R.users b s).1 ∈ V ∨ (R.users b s).2 ∈ V := by
+  classical
+  simp only [exposedPaths,Finset.mem_biUnion,R.mem_path]
+  constructor
+  · rintro ⟨x,hx,h | h⟩
+    · exact Or.inl (h ▸ hx)
+    · exact Or.inr (h ▸ hx)
+  · rintro (h | h)
+    · exact ⟨_,h,Or.inl rfl⟩
+    · exact ⟨_,h,Or.inr rfl⟩
+
+lemma terminalCharge_sum (b : ι → Bool) (z : α) (V : Finset (α))
+    (hz : z ∉ V) :
+    (terminalCharge R b z V : ℝ) = ∑ s : ι,
+      ((if (R.users b s).1 = z ∧ (R.users b s).2 ∈ V then (1:ℝ) else 0) +
+      (if (R.users b s).2 = z ∧ (R.users b s).1 ∈ V then (1:ℝ) else 0)) := by
+  classical
+  have he : R.path b z ∩ exposedPaths R b V =
+      Finset.univ.filter (fun s =>
+        ((R.users b s).1 = z ∧ (R.users b s).2 ∈ V) ∨
+        ((R.users b s).2 = z ∧ (R.users b s).1 ∈ V)) := by
+    ext s
+    simp only [Finset.mem_inter,R.mem_path,mem_exposedPaths,
+      Finset.mem_filter,Finset.mem_univ,true_and]
+    by_cases h₁ : (R.users b s).1 = z <;>
+      by_cases h₂ : (R.users b s).2 = z <;> simp [h₁,h₂,hz]
+  rw [terminalCharge,he]
+  simp only [← Finset.sum_boole]
+  apply Finset.sum_congr rfl
+  intro s _
+  by_cases h₁ : (R.users b s).1 = z <;>
+    by_cases h₂ : (R.users b s).2 = z <;> simp [h₁,h₂,hz]
+
+noncomputable def priorPaths {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) : Finset (α) :=
+  Finset.univ.filter (fun x => ∃ i : Fin A, o x = some i ∧ i.val < t)
+
+omit [DecidableEq α] in
+@[simp] lemma mem_priorPaths {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) (x : α) :
+    x ∈ priorPaths o t ↔ ∃ i : Fin A, o x = some i ∧ i.val < t := by
+  classical
+  simp [priorPaths]
+
+lemma terminalCharge_prior_sum {A : ℕ} (o : α → Option (Fin A))
+    (b : ι → Bool) (z : Terminal o) (i : Fin A) :
+    (terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) =
+      ∑ s : ι,
+        ((if (z i).val = (R.users b s).1 ∧ (R.users b s).2 ∈ priorPaths o (i.val+1)
+          then (1:ℝ) else 0) +
+        (if (z i).val = (R.users b s).2 ∧ (R.users b s).1 ∈ priorPaths o (i.val+1)
+          then (1:ℝ) else 0)) := by
+  classical
+  rw [terminalCharge_sum _ _ _ _ (Finset.notMem_erase _ _)]
+  apply Finset.sum_congr rfl
+  intro s _
+  have hn := R.users_ne b s
+  simp only [Finset.mem_erase]
+  by_cases h₁ : (z i).val = (R.users b s).1 <;>
+    by_cases h₂ : (z i).val = (R.users b s).2 <;> simp_all [eq_comm]
+
+lemma rotationMeeting_eq_slot_sum {A : ℕ} (o : α → Option (Fin A))
+    (z : Terminal o) (u v : α) :
+    rotationMeeting o z u v = ∑ i : Fin A,
+      ((if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) +
+      (if (z i).val = v ∧ u ∈ priorPaths o (i.val+1) then (1:ℝ) else 0)) := by
+  classical
+  have only_owner (u : α) (a : Fin A) (hou : o u = some a)
+      (i : Fin A) (hia : i ≠ a) : (z i).val ≠ u := by
+    intro h
+    have hi := (z i).property
+    rw [h,hou] at hi
+    exact hia (Option.some.inj hi).symm
+  have sum_one (u v : α) (a : Fin A) (hou : o u = some a) :
+      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) =
+      if (z a).val = u ∧ v ∈ priorPaths o (a.val+1) then 1 else 0 := by
+    apply Finset.sum_eq_single a
+    · intro i _ hia
+      simp [only_owner u a hou i hia]
+    · simp
+  have sum_none (u v : α) (hou : o u = none) :
+      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) = 0 := by
+    apply Finset.sum_eq_zero
+    intro i _
+    have hn : (z i).val ≠ u := by
+      intro h
+      have hi := (z i).property
+      rw [h,hou] at hi
+      contradiction
+    simp [hn]
+  rw [Finset.sum_add_distrib]
+  cases hou : o u with
+  | none =>
+    rw [sum_none u v hou]
+    simp [rotationMeeting,hou,mem_priorPaths]
+  | some a =>
+    rw [sum_one u v a hou]
+    cases hov : o v with
+    | none =>
+      rw [sum_none v u hov]
+      simp [rotationMeeting,hou,hov,mem_priorPaths]
+    | some b =>
+      rw [sum_one v u b hov]
+      simp only [rotationMeeting,hou,hov,mem_priorPaths,Option.some.injEq]
+      rcases lt_trichotomy a b with hab | rfl | hba
+      · have hn : ¬ b.val < a.val+1 := by omega
+        have hy : a.val < b.val+1 := by omega
+        simp [hab,hn,hy]
+      · simp
+      · have hn : ¬ a.val < b.val+1 := by omega
+        have hy : b.val < a.val+1 := by omega
+        simp [hba,not_lt_of_ge (le_of_lt hba),hn,hy]
+
+lemma rotationCharge_eq_terminal_sum {A : ℕ} (o : α → Option (Fin A))
+    (b : ι → Bool) (z : Terminal o) :
+    rotationCharge R o b z = ∑ i : Fin A,
+      (terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) := by
+  simp only [rotationCharge,rotationMeeting_eq_slot_sum,terminalCharge_prior_sum]
+  exact Finset.sum_comm
+
+omit [DecidableEq α] in
+@[simp] lemma priorPaths_zero {A : ℕ} (o : α → Option (Fin A)) : priorPaths o 0 = ∅ := by
+  classical
+  ext x
+  simp
+
+lemma priorPaths_succ {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) (ht : t < A) :
+    priorPaths o (t+1) = priorPaths o t ∪ cycleFiber o ⟨t,ht⟩ := by
+  classical
+  ext x
+  simp only [mem_priorPaths,Finset.mem_union,mem_cycleFiber]
+  constructor
+  · rintro ⟨i,hi,hit⟩
+    by_cases h : i.val < t
+    · exact Or.inl ⟨i,hi,h⟩
+    · have he : i = ⟨t,ht⟩ := Fin.ext (by change i.val = t; omega)
+      exact Or.inr (he ▸ hi)
+  · rintro (⟨i,hi,hit⟩ | hi)
+    · exact ⟨i,hi,by omega⟩
+    · exact ⟨⟨t,ht⟩,hi,by change t < t+1; omega⟩
+
+omit [DecidableEq α] in
+lemma priorPaths_disjoint {A : ℕ} (o : α → Option (Fin A)) (i : Fin A) :
+    Disjoint (priorPaths o i.val) (cycleFiber o i) := by
+  classical
+  apply Finset.disjoint_left.mpr
+  intro x hx hx'
+  obtain ⟨j,hj,hji⟩ := (mem_priorPaths _ _ _).mp hx
+  have he : j = i := Option.some.inj (hj.symm.trans ((mem_cycleFiber _ _ _).mp hx'))
+  subst j
+  exact (lt_irrefl _ hji)
+
+namespace QueryTree
+open Thorp.QueryTree
+
+lemma seekCycles_owned {A : ℕ} (S : Finset (α)) (r : Fin A → ℕ)
+    (o : α → Option (Fin A)) (z : Terminal o) (ω : ι → Bool)
+    (hS : ∀ i, cycleFiber o i ⊆ S)
+    (hO : ∀ i, orbitSet (R.perm ω) (z i).val = cycleFiber o i)
+    (hC : ∀ i, (cycleFiber o i).card = r i+1) :
+    let p := R.perm ω
+    let L := List.ofFn (fun i : Fin A => (p (z i).val,r i))
+    (seekCycles R S L ∅).succeeds ω = true ∧
+      ((seekCycles R S L ∅).staleCharge ∅ ω : ℝ) = rotationCharge R o (ω) z := by
+  classical
+  let b := ω
+  let p := R.perm b
+  let f (i : Fin A) := (p (z i).val,r i)
+  let c (i : Fin A) : ℝ := terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val)
+  have hc (i : Fin A) : Function.minimalPeriod p (z i).val - 1 = r i := by
+    have he := orbitSet_card p (z i).val
+    rw [hO,hC] at he
+    omega
+  have hs (t : ℕ) (ht : t ≤ A) :
+      (seekCycles R S ((List.ofFn f).drop t) (priorPaths o t)).succeeds ω = true ∧
+        ((seekCycles R S ((List.ofFn f).drop t) (priorPaths o t)).staleCharge (exposedPaths R b (priorPaths o t)) ω : ℝ) =
+          ((List.ofFn c).drop t).sum := by
+    induction ht using Nat.decreasingInduction with
+    | self => simp only [List.length_ofFn,le_refl,List.drop_eq_nil_of_le,List.sum_nil,seekCycles,Thorp.QueryTree.succeeds,staleCharge,Nat.cast_zero,and_self]
+    | of_succ t ht ih =>
+      have htf : t < (List.ofFn f).length := by simpa using ht
+      have htc : t < (List.ofFn c).length := by simpa using ht
+      rw [List.drop_eq_getElem_cons htf,List.drop_eq_getElem_cons htc]
+      simp only [List.getElem_ofFn,List.sum_cons,seekCycles]
+      have h := seekCycle_orbit R S (priorPaths o t) (z ⟨t,ht⟩).val
+        (fun V => seekCycles R S ((List.ofFn f).drop (t+1)) V) ω
+        (by rw [hO]; exact hS _) (by rw [hO]; exact priorPaths_disjoint o ⟨t,ht⟩)
+      dsimp only at h
+      rw [hc,hO,←priorPaths_succ o t ht] at h
+      dsimp only [f]
+      constructor
+      · exact h.1.trans ih.1
+      · rw [h.2,Nat.cast_add,ih.2]
+        exact add_comm _ _
+  have h := hs 0 (Nat.zero_le _)
+  simp only [List.drop_zero,priorPaths_zero,exposedPaths_empty,List.sum_ofFn] at h
+  dsimp only
+  exact ⟨h.1,h.2.trans (rotationCharge_eq_terminal_sum R o b z).symm⟩
+
+end QueryTree
+section Collections
+variable {A : ℕ}
+
+def ValidCycleSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ)
+    (o : α → Option (Fin A)) : Prop :=
+  (∀ i, (cycleFiber o i).card = r i + 1) ∧
+  (∀ i, cycleFiber o i ⊆ S) ∧
+  (∀ i x, o x = some i → orbitSet p x = cycleFiber o i)
+
+abbrev CycleSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
+  {o : α → Option (Fin A) // ValidCycleSlots p S r o}
+
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
+    Fintype (CycleSlots p S r) := Fintype.ofFinite _
+
+lemma CycleSlots.terminal_nonempty {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (c : CycleSlots p S r) (i : Fin A) : Nonempty {x : α // c.val x = some i} := by
+  have hn : (cycleFiber c.val i).Nonempty := Finset.card_pos.mp (by rw [c.property.1]; omega)
+  obtain ⟨x,hx⟩ := hn
+  exact ⟨x,(mem_cycleFiber _ _ _).mp hx⟩
+
+omit [DecidableEq α] in
+lemma owner_ext {o o' : α → Option (Fin A)}
+    (h : ∀ i, cycleFiber o i = cycleFiber o' i) : o = o' := by
+  funext x
+  have he (i : Fin A) : o x = some i ↔ o' x = some i := by
+    simpa only [mem_cycleFiber] using (Finset.ext_iff.mp (h i) x)
+  cases ho : o x with
+  | some i => exact ((he i).mp ho).symm
+  | none =>
+    cases ho' : o' x with
+    | none => rfl
+    | some i => have hn := (he i).mpr ho'; rw [ho] at hn; contradiction
+
+abbrev RootedSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
+  Σ c : CycleSlots p S r, Terminal c.val
+
+noncomputable def rootedStart {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (c : RootedSlots p S r) : Fin A → S := fun i =>
+  ⟨p (c.2 i).val,c.1.property.2.1 i (by
+    rw [←c.1.property.2.2 i (c.2 i).val (c.2 i).property]
+    exact (mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.apply_right (Equiv.Perm.SameCycle.refl p _)))⟩
+
+lemma rootedStart_injective {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ} :
+    Function.Injective (rootedStart (p:=p) (S:=S) (r:=r)) := by
+  rintro ⟨c,z⟩ ⟨c',z'⟩ h
+  have hz (i : Fin A) : (z i).val = (z' i).val :=
+    p.injective (congrArg Subtype.val (congrFun h i))
+  have hc : c = c' := Subtype.ext (owner_ext (fun i => by
+    rw [←c.property.2.2 i (z i).val (z i).property,
+      ←c'.property.2.2 i (z' i).val (z' i).property,hz]))
+  subst c'
+  have hzz : z = z' := funext (fun i => Subtype.ext (hz i))
+  subst z'
+  rfl
+
+lemma rooted_weight_sum_le (S : Finset (α)) (r : Fin A → ℕ) (ω : ι → Bool) :
+    (∑ c : RootedSlots (R.perm ω) S r,
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2)) ≤
+    ∑ a : Fin A → S,
+      if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+        ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+      else 0 := by
+  classical
+  let p := R.perm ω
+  let g (a : Fin A → S) : ℝ :=
+    if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+    else 0
+  have he (c : RootedSlots p S r) :
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2) =
+      g (rootedStart c) := by
+    have hs := QueryTree.seekCycles_owned R S r c.1.val c.2 ω c.1.property.2.1
+      (fun i => c.1.property.2.2 i (c.2 i).val (c.2 i).property) c.1.property.1
+    dsimp only at hs
+    unfold g
+    rw [ite_eq_left (show (QueryTree.seekCycles R S (List.ofFn (fun i => ((rootedStart c i).val,r i))) ∅).succeeds ω = true from hs.1)]
+    rw [← hs.2,neg_mul,Real.exp_neg,mul_comm (Real.log 2),Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
+    rfl
+  simp_rw [he]
+  exact sum_comp_injective_le rootedStart rootedStart_injective g (fun a => by dsimp [g]; split_ifs <;> positivity)
+
+lemma collection_rotation_lower (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r)
+    (ω : ι → Bool) (c : CycleSlots (R.perm ω) S r) :
+    collectionWeight d r ≤ ∑ z : Terminal c.val,
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z) := by
+  classical
+  let (i : Fin A) : Nonempty {x : α // c.val x = some i} := c.terminal_nonempty i
+  have hm : Monotone (fun i => (cycleFiber c.val i).card) := by
+    intro i j hij
+    dsimp only
+    rw [c.property.1,c.property.1]
+    exact Nat.add_le_add_right (hr hij) 1
+  have hb := rotation_exp_bound R hm (ω)
+  have hcard : (Fintype.card (Terminal c.val) : ℝ) = ∏ i : Fin A, (r i+1 : ℝ) := by
+    rw [Fintype.card_pi,Nat.cast_prod]
+    apply Finset.prod_congr rfl
+    intro i _
+    rw [card_cycleFiber,c.property.1,Nat.cast_add,Nat.cast_one]
+  have hp : (0:ℝ) < Fintype.card (Terminal c.val) := Nat.cast_pos.mpr Fintype.card_pos
+  have hl : rotationLower d r ≤ finiteMean (fun z : Terminal c.val => Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z)) := by
+    simpa only [rotationLower,c.property.1,Nat.cast_add,Nat.cast_one] using hb
+  have hh := (le_div_iff₀ hp).mp hl
+  rw [hcard] at hh
+  unfold collectionWeight
+  rw [← Finset.mul_sum]
+  calc
+    _ = ((2:ℝ)^d)^A * (rotationLower d r * ∏ i : Fin A, (r i+1 : ℝ)) := by ring
+    _ ≤ _ := mul_le_mul_of_nonneg_left hh (by positivity)
+
+lemma cycleSlots_weighted_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : ι → Bool =>
+      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) *
+      collectionWeight d r ≤ (S.card : ℝ)^A := by
+  classical
+  let g (a : Fin A → S) (ω : ι → Bool) : ℝ :=
+    if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+    else 0
+  have hω (ω : ι → Bool) :
+      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ) * collectionWeight d r ≤
+        ∑ a : Fin A → S, g a ω := by
+    calc
+      _ = ∑ _c : CycleSlots (R.perm ω) S r, collectionWeight d r := by simp
+      _ ≤ ∑ c : CycleSlots (R.perm ω) S r,
+          ∑ z : Terminal c.val, ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z) :=
+        Finset.sum_le_sum (fun c _ => collection_rotation_lower R S r hr ω c)
+      _ = ∑ c : RootedSlots (R.perm ω) S r,
+          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2) :=
+        (Fintype.sum_sigma (fun c : RootedSlots (R.perm ω) S r =>
+          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2))).symm
+      _ ≤ _ := rooted_weight_sum_le R S r ω
+  have hb := finiteMean_mono hω
+  rw [finiteMean_mul_const,finiteMean_sum] at hb
+  calc
+    _ ≤ ∑ a : Fin A → S, finiteMean (g a) := hb
+    _ ≤ ∑ _a : Fin A → S, (1:ℝ) := by
+      apply Finset.sum_le_sum
+      intro a _
+      have hh := QueryTree.cycles_weighted_success R S (List.ofFn (fun i : Fin A => ((a i).val,r i)))
+      simpa only [List.length_ofFn] using hh
+    _ = _ := by simp
+
+lemma cycleSlots_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : ι → Bool =>
+      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
+        (S.card : ℝ)^A / collectionWeight d r := by
+  exact (le_div_iff₀ (collectionWeight_pos d r)).mpr (cycleSlots_weighted_moment R S r hr)
+
+lemma butterfly_cycleSlots_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : ι → Bool =>
+      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
+        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
+  have h := cycleSlots_moment R S r hr
+  rw [collectionWeight_eq_sqrt,div_mul_eq_div_mul_one_div] at h
+  rw [div_pow]
+  simpa only [one_div,Finset.prod_inv_distrib] using h
+
+end Collections
+
+section CycleCounting
+variable {A : ℕ}
+
+lemma SelectedCycle.nonempty {p : Equiv.Perm (α)} {S : Finset (α)}
+    (C : SelectedCycle p S) : C.val.Nonempty := by
+  obtain ⟨x,hx⟩ := C.property.1
+  rw [hx]
+  exact orbitSet_nonempty p x
+
+lemma SelectedCycle.orbit_eq {p : Equiv.Perm (α)} {S : Finset (α)}
+    (C : SelectedCycle p S) {x : α} (hx : x ∈ C.val) : orbitSet p x = C.val := by
+  obtain ⟨y,hy⟩ := C.property.1
+  rw [hy] at hx ⊢
+  exact (orbitSet_eq p ((mem_orbitSet _ _ _).mp hx)).symm
+
+lemma SelectedCycle.eq_of_mem {p : Equiv.Perm (α)} {S : Finset (α)}
+    (C D : SelectedCycle p S) {x : α} (hC : x ∈ C.val) (hD : x ∈ D.val) : C = D := by
+  apply Subtype.ext
+  exact (C.orbit_eq hC).symm.trans (D.orbit_eq hD)
+
+abbrev SlotAssignment (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
+  {f : Fin A → SelectedCycle p S // Function.Injective f ∧ ∀ i, (f i).val.card = r i+1}
+
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
+    Fintype (SlotAssignment p S r) := Fintype.ofFinite _
+
+noncomputable def slotsToAssignment {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (c : CycleSlots p S r) : SlotAssignment p S r := by
+  classical
+  let f (i : Fin A) : SelectedCycle p S := ⟨cycleFiber c.val i,by
+    obtain ⟨x,hx⟩ := c.terminal_nonempty i
+    exact ⟨⟨x,(c.property.2.2 i x hx).symm⟩,c.property.2.1 i⟩⟩
+  refine ⟨f,?_,fun i => c.property.1 i⟩
+  intro i j hij
+  have he : cycleFiber c.val i = cycleFiber c.val j := congrArg Subtype.val hij
+  obtain ⟨x,hx⟩ := c.terminal_nonempty i
+  have hxi : x ∈ cycleFiber c.val i := (mem_cycleFiber _ _ _).mpr hx
+  rw [he] at hxi
+  exact Option.some.inj (hx.symm.trans ((mem_cycleFiber _ _ _).mp hxi))
+
+noncomputable def assignmentOwner {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) (x : α) : Option (Fin A) := by
+  classical
+  exact if h : ∃ i, x ∈ (f.val i).val then some (Classical.choose h) else none
+
+lemma assignmentOwner_eq_some {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) (x : α) (i : Fin A) :
+    assignmentOwner f x = some i ↔ x ∈ (f.val i).val := by
+  classical
+  unfold assignmentOwner
+  split_ifs with h
+  · constructor
+    · intro hi
+      have hx := Classical.choose_spec h
+      rw [Option.some.inj hi] at hx
+      exact hx
+    · intro hx
+      have he := (f.val (Classical.choose h)).eq_of_mem (f.val i) (Classical.choose_spec h) hx
+      exact congrArg some (f.property.1 he)
+  · constructor
+    · intro hi; contradiction
+    · intro hx; exact (h ⟨i,hx⟩).elim
+
+lemma assignmentOwner_fiber {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) (i : Fin A) : cycleFiber (assignmentOwner f) i = (f.val i).val := by
+  ext x
+  simp only [mem_cycleFiber,assignmentOwner_eq_some]
+
+noncomputable def assignmentToSlots {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) : CycleSlots p S r :=
+  ⟨assignmentOwner f,by
+    refine ⟨?_,?_,?_⟩
+    · intro i
+      rw [assignmentOwner_fiber]
+      exact f.property.2 i
+    · intro i
+      rw [assignmentOwner_fiber]
+      exact (f.val i).property.2
+    · intro i x hx
+      rw [assignmentOwner_fiber]
+      exact (f.val i).orbit_eq ((assignmentOwner_eq_some f x i).mp hx)⟩
+
+noncomputable def cycleSlotsEquivAssignments (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
+    CycleSlots p S r ≃ SlotAssignment p S r where
+  toFun := slotsToAssignment
+  invFun := assignmentToSlots
+  left_inv c := by
+    apply Subtype.ext
+    apply owner_ext
+    intro i
+    rw [assignmentToSlots,assignmentOwner_fiber]
+    rfl
+  right_inv f := by
+    apply Subtype.ext
+    funext i
+    apply Subtype.ext
+    exact assignmentOwner_fiber f i
+
+end CycleCounting
+
+section GroupedCounting
+variable {A : ℕ}
+
+abbrev LengthClass (r : Fin A → ℕ) := {j : ℕ // j ∈ Finset.univ.image (fun i => r i + 1)}
+
+def slotClass (r : Fin A → ℕ) (i : Fin A) : LengthClass r :=
+  ⟨r i+1, Finset.mem_image.mpr ⟨i, Finset.mem_univ _, rfl⟩⟩
+
+abbrev SlotsOfLength (r : Fin A → ℕ) (j : LengthClass r) := {i : Fin A // slotClass r i = j}
+abbrev CyclesOfLength (p : Equiv.Perm (α)) (S : Finset (α)) (j : ℕ) :=
+  {C : SelectedCycle p S // C.val.card = j}
+
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (j : ℕ) :
+    Fintype (CyclesOfLength p S j) := Fintype.ofFinite _
+
+abbrev GroupedAssignments (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
+  ∀ j : LengthClass r, SlotsOfLength r j ↪ CyclesOfLength p S j.val
+
+noncomputable def assignmentToGroups {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) : GroupedAssignments p S r := fun _j =>
+  { toFun := fun i => ⟨f.val i.val, (f.property.2 i.val).trans (congrArg Subtype.val i.property)⟩
+    inj' := fun _i _k h => Subtype.ext (f.property.1 (congrArg Subtype.val h)) }
+
+noncomputable def groupsEmbedding {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (F : GroupedAssignments p S r) : Fin A ↪ SelectedCycle p S :=
+  ((Equiv.sigmaFiberEquiv (slotClass r)).symm.toEmbedding.trans
+    (Function.Embedding.sigmaMap (Function.Embedding.refl _) F)).trans
+    ((Function.Embedding.sigmaMap (Function.Embedding.subtype _) (fun _ => Function.Embedding.refl _)).trans
+      (Equiv.sigmaFiberEquiv (fun C : SelectedCycle p S => C.val.card)).toEmbedding)
+
+noncomputable def groupsToAssignment {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (F : GroupedAssignments p S r) : SlotAssignment p S r :=
+  ⟨groupsEmbedding F, (groupsEmbedding F).injective,
+    fun i => (F (slotClass r i) ⟨i,rfl⟩).property⟩
+
+noncomputable def assignmentEquivGroups (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
+    SlotAssignment p S r ≃ GroupedAssignments p S r where
+  toFun := assignmentToGroups
+  invFun := groupsToAssignment
+  left_inv f := by
+    apply Subtype.ext
+    rfl
+  right_inv F := by
+    funext j
+    apply Function.Embedding.ext
+    intro i
+    rcases i with ⟨i,hi⟩
+    subst j
+    rfl
+
+lemma card_cycleSlots_factorial (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
+    Fintype.card (CycleSlots p S r) =
+      ∏ j : LengthClass r, (Fintype.card (CyclesOfLength p S j.val)).descFactorial
+        (Fintype.card (SlotsOfLength r j)) := by
+  classical
+  rw [Fintype.card_congr (cycleSlotsEquivAssignments p S r),
+    Fintype.card_congr (assignmentEquivGroups p S r), Fintype.card_pi]
+  simp only [Fintype.card_embedding_eq]
+
+lemma butterfly_factorial_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : ι → Bool =>
+       (∏ j : LengthClass r,
+         (Fintype.card (CyclesOfLength (R.perm ω) S j.val)).descFactorial
+           (Fintype.card (SlotsOfLength r j)) : ℕ)) ≤
+      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+        ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
+  simpa only [←card_cycleSlots_factorial] using butterfly_cycleSlots_moment R S r hr
+
+end GroupedCounting
+
+section AllLengths
+variable {A : ℕ}
+
+noncomputable def assignmentRelabel (p : Equiv.Perm (α)) (S : Finset (α))
+    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
+    SlotAssignment p S r ≃ SlotAssignment p S (r ∘ e) where
+  toFun f := ⟨f.val ∘ e, f.property.1.comp e.injective, fun i => f.property.2 (e i)⟩
+  invFun f := ⟨f.val ∘ e.symm, f.property.1.comp e.symm.injective, fun i => by
+    simpa only [Function.comp_apply,Equiv.apply_symm_apply] using f.property.2 (e.symm i)⟩
+  left_inv f := by apply Subtype.ext; funext i; simp
+  right_inv f := by apply Subtype.ext; funext i; simp
+
+lemma card_cycleSlots_relabel (p : Equiv.Perm (α)) (S : Finset (α))
+    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
+    Fintype.card (CycleSlots p S r) = Fintype.card (CycleSlots p S (r ∘ e)) := by
+  exact Fintype.card_congr ((cycleSlotsEquivAssignments p S r).trans
+    ((assignmentRelabel p S r e).trans (cycleSlotsEquivAssignments p S (r ∘ e)).symm))
+
+lemma butterfly_cycleSlots_moment_all (S : Finset (α)) (r : Fin A → ℕ) :
+    finiteMean (fun ω : ι → Bool =>
+      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
+        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
+  have h := butterfly_cycleSlots_moment R S (r ∘ Tuple.sort r) (Tuple.monotone_sort r)
+  simp_rw [←card_cycleSlots_relabel] at h
+  have hp := Equiv.prod_comp (Tuple.sort r) (fun i : Fin A => (1 / Real.sqrt (r i+1) : ℝ))
+  simpa only [Function.comp_apply,hp] using h
+
+abbrev ShortCycle (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :=
+  {C : SelectedCycle p S // C.val.card ≤ J}
+
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    Fintype (ShortCycle p S J) := Fintype.ofFinite _
+
+noncomputable def shortCycleLength {p : Equiv.Perm (α)} {S : Finset (α)} {J : ℕ}
+    (C : ShortCycle p S J) : Fin J :=
+  ⟨C.val.val.card-1, by have := C.val.nonempty.card_pos; have := C.property; omega⟩
+
+lemma shortCycleLength_add_one {p : Equiv.Perm (α)} {S : Finset (α)} {J : ℕ}
+    (C : ShortCycle p S J) : (shortCycleLength C : ℕ)+1 = C.val.val.card := by
+  have := C.val.nonempty.card_pos
+  simp only [shortCycleLength]
+  omega
+
+noncomputable def shortEmbeddingEquiv (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    (Fin A ↪ ShortCycle p S J) ≃
+      Σ r : Fin A → Fin J, SlotAssignment p S (fun i => (r i : ℕ)) where
+  toFun f := ⟨fun i => shortCycleLength (f i),
+    ⟨fun i => (f i).val, fun i j h => f.injective (Subtype.ext h),
+      fun i => (shortCycleLength_add_one (f i)).symm⟩⟩
+  invFun c :=
+    { toFun := fun i => ⟨c.2.val i, by rw [c.2.property.2]; exact (c.1 i).isLt⟩
+      inj' := fun _i _j h => c.2.property.1 (congrArg Subtype.val h) }
+  left_inv f := by
+    apply Function.Embedding.ext
+    intro i
+    rfl
+  right_inv c := by
+    rcases c with ⟨r,f⟩
+    have he : (fun i => shortCycleLength (⟨f.val i, by rw [f.property.2]; exact (r i).isLt⟩ : ShortCycle p S J)) = r := by
+      funext i
+      apply Fin.ext
+      simp only [shortCycleLength,f.property.2,Nat.add_sub_cancel]
+    apply Sigma.ext he
+    apply (Subtype.heq_iff_coe_eq (by
+      intro x
+      change (Function.Injective x ∧ ∀ i, (x i).val.card = (f.val i).val.card - 1 + 1) ↔ _
+      simp only [f.property.2,Nat.add_sub_cancel])).mpr
+    rfl
+
+lemma shortCycle_factorial_eq (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    (Fintype.card (ShortCycle p S J)).descFactorial A =
+      ∑ r : Fin A → Fin J, Fintype.card (CycleSlots p S (fun i => (r i : ℕ))) := by
+  classical
+  have hc := Fintype.card_embedding_eq (α := Fin A) (β := ShortCycle p S J)
+  simp only [Fintype.card_fin] at hc
+  rw [←hc, Fintype.card_congr (shortEmbeddingEquiv p S J),Fintype.card_sigma]
+  apply Finset.sum_congr rfl
+  intro r _
+  exact (Fintype.card_congr (cycleSlotsEquivAssignments p S (fun i => (r i : ℕ)))).symm
+
+end AllLengths
+
+section CycleMGF
+
+lemma butterfly_shortCycle_moment (S : Finset (α)) (J A : ℕ) :
+    finiteMean (fun ω : ι → Bool =>
+      ((Fintype.card (ShortCycle (R.perm ω) S J)).descFactorial A : ℝ)) ≤
+      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))^A := by
+  classical
+  simp_rw [shortCycle_factorial_eq,Nat.cast_sum,finiteMean_sum]
+  calc
+    _ ≤ ∑ r : Fin A → Fin J,
+        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+          ∏ i : Fin A, (1 / Real.sqrt ((r i : ℕ)+1)) := by
+      apply Finset.sum_le_sum
+      intro r _
+      exact butterfly_cycleSlots_moment_all R S (fun i => (r i : ℕ))
+    _ = _ := by
+      rw [←Finset.mul_sum,mul_pow]
+      congr 1
+      rw [←Fintype.prod_sum (fun (_ : Fin A) (j : Fin J) => (1 / Real.sqrt ((j:ℕ)+1) : ℝ))]
+      simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fin]
+
+noncomputable def cycleMass (p : Equiv.Perm (α)) (S : Finset (α)) : ℕ :=
+  ∑ C : SelectedCycle p S, C.val.card
+
+lemma cycleMass_le (p : Equiv.Perm (α)) (S : Finset (α)) : cycleMass p S ≤ S.card := by
+  classical
+  let f : (Σ C : SelectedCycle p S, {x : α // x ∈ C.val}) → S :=
+    fun c => ⟨c.2.val,c.1.property.2 c.2.property⟩
+  have hi : Function.Injective f := by
+    intro c e h
+    have hx : c.2.val = e.2.val := congrArg Subtype.val h
+    have hC : c.1 = e.1 := c.1.eq_of_mem e.1 c.2.property (hx ▸ e.2.property)
+    rcases c with ⟨c,x⟩
+    rcases e with ⟨e,y⟩
+    dsimp only at hC
+    subst e
+    have he : x=y := Subtype.ext hx
+    cases he
+    rfl
+  have hh := Fintype.card_le_of_injective f hi
+  simpa only [Fintype.card_sigma,Fintype.card_coe,cycleMass] using hh
+
+lemma shortCycle_card_le (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    Fintype.card (ShortCycle p S J) ≤ S.card := by
+  classical
+  calc
+    _ ≤ Fintype.card (SelectedCycle p S) := Fintype.card_subtype_le _
+    _ = ∑ _ : SelectedCycle p S, 1 := by simp
+    _ ≤ cycleMass p S := Finset.sum_le_sum (fun C _ => C.nonempty.card_pos)
+    _ ≤ _ := cycleMass_le p S
+
+lemma butterfly_shortCycle_mgf (S : Finset (α)) (J : ℕ) (q : ℝ) (hq : 1 ≤ q) :
+    finiteMean (fun ω : ι → Bool =>
+      q^(Fintype.card (ShortCycle (R.perm ω) S J))) ≤
+      Real.exp ((q-1)*((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))) := by
+  apply factorial_mgf _ S.card _ q (fun ω => shortCycle_card_le _ _ _)
+  · positivity
+  · exact hq
+  · exact butterfly_shortCycle_moment R S J
+
+end CycleMGF
+
+
+abbrev LongCycle (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :=
+  {C : SelectedCycle p S // ¬ C.val.card ≤ J}
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    Fintype (LongCycle p S J) := Fintype.ofFinite _
+
+lemma selectedCycle_card_split (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    Fintype.card (SelectedCycle p S) =
+      Fintype.card (ShortCycle p S J) + Fintype.card (LongCycle p S J) := by
+  classical
+  rw [←Fintype.card_congr (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J)),Fintype.card_sum]
+
+lemma longCycle_card_bound (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    J * Fintype.card (LongCycle p S J) ≤ S.card := by
+  classical
+  have hs := Equiv.sum_comp (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J))
+    (fun C : SelectedCycle p S => C.val.card)
+  rw [Fintype.sum_sum_type] at hs
+  have h0 : J * Fintype.card (LongCycle p S J) ≤
+      ∑ C : LongCycle p S J, C.val.val.card := by
+    calc
+      _ = ∑ _C : LongCycle p S J, J := by simp [mul_comm]
+      _ ≤ _ := Finset.sum_le_sum (fun C _ => Nat.le_of_lt (Nat.lt_of_not_ge C.property))
+  have hh := cycleMass_le p S
+  unfold cycleMass at hh
+  rw [←hs] at hh
+  exact h0.trans ((Nat.le_add_left _ _).trans hh)
+
+lemma longCycle_pow_bound (p : Equiv.Perm (α)) (S : Finset (α))
+    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    q^(Fintype.card (LongCycle p S J)) ≤ Real.exp ((S.card : ℝ)/J * Real.log q) := by
+  have hq0 : 0 < q := lt_of_lt_of_le zero_lt_one hq
+  have hc : (Fintype.card (LongCycle p S J) : ℝ) ≤ (S.card : ℝ)/J := by
+    apply (le_div_iff₀ (by exact_mod_cast hJ)).mpr
+    have hh := longCycle_card_bound p S J
+    exact_mod_cast (by simpa only [mul_comm] using hh)
+  calc
+    _ = Real.exp ((Fintype.card (LongCycle p S J) : ℝ)*Real.log q) := by
+      rw [Real.exp_nat_mul,Real.exp_log hq0]
+    _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right hc (Real.log_nonneg hq))
+
+lemma butterfly_cycle_log_mgf (S : Finset (α)) (J : ℕ) (hJ : 0 < J)
+    (q : ℝ) (hq : 1 ≤ q) :
+    Real.log (finiteMean (fun ω : ι → Bool =>
+      q^(Fintype.card (SelectedCycle (R.perm ω) S)))) ≤
+      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
+  let lam : ℝ := (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1)
+  let c : ℝ := (S.card : ℝ)/J * Real.log q
+  have hnum : (q-1)*lam ≤ 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
+    have hsum := sum_inv_sqrt J
+    have hp : 0 ≤ (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) := by positivity
+    have h0 : 0 ≤ 2*Real.sqrt J := by positivity
+    have hh := mul_le_mul_of_nonneg_left hsum hp
+    have hq1 : 0 ≤ q-1 := sub_nonneg.mpr hq
+    calc
+      _ ≤ (q-1)*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
+        mul_le_mul_of_nonneg_left hh hq1
+      _ ≤ q*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
+        mul_le_mul_of_nonneg_right (by linarith) (mul_nonneg hp h0)
+      _ = _ := by ring
+  have hm : finiteMean (fun ω : ι → Bool =>
+      q^(Fintype.card (SelectedCycle (R.perm ω) S))) ≤
+      Real.exp (c+(q-1)*lam) := by
+    calc
+      _ ≤ finiteMean (fun ω : ι → Bool =>
+          q^(Fintype.card (ShortCycle (R.perm ω) S J))*Real.exp c) := by
+        apply finiteMean_mono
+        intro ω
+        rw [selectedCycle_card_split _ _ J,pow_add]
+        exact mul_le_mul_of_nonneg_left (longCycle_pow_bound _ S J hJ q hq) (by positivity)
+      _ = finiteMean (fun ω : ι → Bool =>
+          q^(Fintype.card (ShortCycle (R.perm ω) S J))) * Real.exp c :=
+        finiteMean_mul_const _ _
+      _ ≤ Real.exp ((q-1)*lam) * Real.exp c :=
+        mul_le_mul_of_nonneg_right (butterfly_shortCycle_mgf R S J q hq) (Real.exp_nonneg _)
+      _ = _ := by rw [←Real.exp_add,add_comm]
+  have hpos : 0 < finiteMean (fun ω : ι → Bool =>
+      q^(Fintype.card (SelectedCycle (R.perm ω) S))) := by
+    apply finiteMean_pos
+    intro ω
+    positivity
+  have hlog := Real.log_le_log hpos hm
+  rw [Real.log_exp] at hlog
+  dsimp [c] at hlog
+  linarith
+
+
+end RoutingNetwork
+
+variable {α B : Type*} [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] {d : ℕ}
+
+def parallelPerm (ω : B × SwitchIndex d → Bool) : Equiv.Perm (B × Card d) where
+  toFun x := (x.1,butterflyPerm d (decodeButterfly d (fun i => ω (x.1,i))) x.2)
+  invFun y := (y.1,(butterflyPerm d (decodeButterfly d (fun i => ω (y.1,i)))).symm y.2)
+  left_inv x := by simp
+  right_inv y := by simp
+
+def relabeledPerm (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (ω : B × SwitchIndex d → Bool) : Equiv.Perm α :=
+  (pre.trans (parallelPerm ω)).trans post
+
+noncomputable def parallelPath (pre : α ≃ B × Card d)
+    (ω : B × SwitchIndex d → Bool) (x : α) : Finset (B × SwitchIndex d) :=
+  (routeDomain d (pre x).2
+    (butterflyPerm d (decodeButterfly d (fun i => ω ((pre x).1,i))) (pre x).2)).image
+      (fun i => ((pre x).1,i))
+
+def parallelUsers (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
+    (i : B × SwitchIndex d) : α × α :=
+  let u := switchUsers d (decodeButterfly d (fun j => ω (i.1,j))) i.2
+  (pre.symm (i.1,u.1),pre.symm (i.1,u.2))
+
+omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
+lemma parallelUsers_ne (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
+    (i : B × SwitchIndex d) : (parallelUsers pre ω i).1 ≠ (parallelUsers pre ω i).2 := by
+  intro h
+  exact switchUsers_ne d _ _ (congrArg Prod.snd (pre.symm.injective h))
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma mem_parallelPath (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
+    (x : α) (i : B × SwitchIndex d) :
+    i ∈ parallelPath pre ω x ↔ (parallelUsers pre ω i).1 = x ∨ (parallelUsers pre ω i).2 = x := by
+  classical
+  obtain ⟨b,j⟩ := i
+  simp only [parallelPath,Finset.mem_image,Prod.mk.injEq]
+  constructor
+  · rintro ⟨k,hk,hb,rfl⟩
+    subst b
+    rcases (mem_routeDomain_iff_user d _ _ _).mp hk with h | h
+    · left
+      apply pre.injective
+      change pre (pre.symm ((pre x).1,_)) = pre x
+      rw [Equiv.apply_symm_apply]
+      exact Prod.ext rfl h
+    · right
+      apply pre.injective
+      change pre (pre.symm ((pre x).1,_)) = pre x
+      rw [Equiv.apply_symm_apply]
+      exact Prod.ext rfl h
+  · intro h
+    have hh : (b,(switchUsers d (decodeButterfly d (fun j => ω (b,j))) j).1) = pre x ∨
+        (b,(switchUsers d (decodeButterfly d (fun j => ω (b,j))) j).2) = pre x := by
+      rcases h with h | h
+      · exact Or.inl (by simpa only [parallelUsers,Equiv.apply_symm_apply] using congrArg pre h)
+      · exact Or.inr (by simpa only [parallelUsers,Equiv.apply_symm_apply] using congrArg pre h)
+    have hb : b = (pre x).1 := hh.elim (congrArg Prod.fst) (congrArg Prod.fst)
+    subst b
+    refine ⟨j,?_,rfl,rfl⟩
+    apply (mem_routeDomain_iff_user d _ _ _).mpr
+    exact hh.elim (fun h => Or.inl (congrArg Prod.snd h)) (fun h => Or.inr (congrArg Prod.snd h))
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma card_parallelPath (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool) (x : α) :
+    (parallelPath pre ω x).card = d := by
+  classical
+  rw [parallelPath,Finset.card_image_of_injective _ (fun _ _ h => Prod.mk.inj h |>.2),card_routeDomain]
+
+noncomputable def blockMarks (pre : α ≃ B × Card d) (S : Finset α) (b : B) : Finset (Card d) :=
+  Finset.univ.filter (fun x => pre.symm (b,x) ∈ S)
+
+omit [Fintype α] [Fintype B] [DecidableEq B] in
+@[simp] lemma mem_blockMarks (pre : α ≃ B × Card d) (S : Finset α) (b : B) (x : Card d) :
+    x ∈ blockMarks pre S b ↔ pre.symm (b,x) ∈ S := by
+  classical
+  simp [blockMarks]
+
+omit [DecidableEq B] in
+lemma sum_card_blockMarks (pre : α ≃ B × Card d) (S : Finset α) :
+    ∑ b : B, (blockMarks pre S b).card = S.card := by
+  classical
+  calc
+    _ = ∑ b : B, ∑ x : Card d, if pre.symm (b,x) ∈ S then 1 else 0 := by
+      simp only [blockMarks,Finset.card_filter]
+    _ = ∑ u : B × Card d, if pre.symm u ∈ S then 1 else 0 := (Fintype.sum_prod_type (fun u : B × Card d => if pre.symm u ∈ S then (1:ℕ) else 0)).symm
+    _ = ∑ x : α, if x ∈ S then 1 else 0 := Equiv.sum_comp pre.symm (fun x : α => if x ∈ S then (1:ℕ) else 0)
+    _ = _ := by simp
+
+omit [DecidableEq B] in
+lemma card_blockMarks_le (pre : α ≃ B × Card d) (S : Finset α) (b : B) :
+    (blockMarks pre S b).card ≤ S.card := by
+  rw [←sum_card_blockMarks pre S]
+  exact Finset.single_le_sum (f := fun b : B => (blockMarks pre S b).card) (fun _ _ => Nat.zero_le _) (Finset.mem_univ b)
+
+end Thorp
+
+end ThorpNine.HighTail
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/HighTail/ProjectCycle.lean b/lean/OAI/Probability/ThorpRouting/HighTail/ProjectCycle.lean
index a7d80640eafe98f06a1be038592f005c8aea4f32..9aa268aa0b1ddebcaf5d2d0356abee6db1b63379 100644
--- a/lean/OAI/Probability/ThorpRouting/HighTail/ProjectCycle.lean
+++ b/lean/OAI/Probability/ThorpRouting/HighTail/ProjectCycle.lean
@@ -1,811 +1,811 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.HighTail.ParallelFollow
-
-namespace OAI
-
-namespace ThorpNine.HighTail
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-namespace CycleColoring
-variable {α : Type*} [Fintype α] [DecidableEq α]
-omit [Fintype α] in
-lemma bipartitionFlip_injective (p : Equiv.Perm α) (S : Finset α)
-    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x)) :
-    Function.Injective (bipartitionFlip p S χ hχ) := by
-  classical
-  intro f g he
-  apply Subtype.ext
-  funext x
-  by_cases hx : x ∈ S
-  · have hh := congrFun (congrArg Subtype.val he) x
-    change (if x ∈ S then Bool.xor (f.val x) (χ x) else false) =
-      (if x ∈ S then Bool.xor (g.val x) (χ x) else false) at hh
-    simp only [ite_eq_left hx] at hh
-    cases hfx : f.val x <;> cases hgx : g.val x <;> cases hcx : χ x <;> simp_all
-  · exact (f.property.1 x hx).trans (g.property.1 x hx).symm
-
-lemma card_opposite_le (p : Equiv.Perm α) (S : Finset α)
-    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x)) :
-    Fintype.card (OppositeColoring p S) ≤
-      2 ^ ((boundary p S).card + Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
-  exact (Fintype.card_le_of_injective _ (bipartitionFlip_injective p S χ hχ)).trans
-    (card_coloring_le p S)
-
-omit [Fintype α] [DecidableEq α] in
-lemma alternatingPerm_flip (p : Equiv.Perm α) (x : Bool × α) :
-    (alternatingPerm p x).1 = !x.1 := by
-  rcases x with ⟨b,x⟩
-  cases b <;> rfl
-
-omit [Fintype α] [DecidableEq α] in
-lemma alternating_step_sameCycle (p : Equiv.Perm α) (x : Bool × α) :
-    p.SameCycle x.2 (alternatingPerm p x).2 := by
-  rcases x with ⟨b,x⟩
-  cases b
-  · exact Equiv.Perm.SameCycle.refl p x
-  · exact (Equiv.Perm.SameCycle.refl p x).apply_right
-
-omit [DecidableEq α] in
-lemma alternating_sameCycle_snd (p : Equiv.Perm α) {x y : Bool × α}
-    (h : (alternatingPerm p).SameCycle x y) : p.SameCycle x.2 y.2 := by
-  obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
-  have hp (k : ℕ) : p.SameCycle x.2 (((alternatingPerm p)^k) x).2 := by
-    induction k with
-    | zero => exact Equiv.Perm.SameCycle.refl p x.2
-    | succ k ih =>
-      rw [pow_succ',Equiv.Perm.mul_apply]
-      exact ih.trans (alternating_step_sameCycle p _)
-  exact hn ▸ hp n
-
-omit [DecidableEq α] in
-lemma alternating_zero_sameCycle (p : Equiv.Perm α) {x y : α}
-    (h : p.SameCycle x y) : (alternatingPerm p).SameCycle (false,x) (false,y) := by
-  obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
-  have hp (k : ℕ) : (alternatingPerm p).SameCycle (false,x) (false,(p^k) x) := by
-    induction k with
-    | zero => exact Equiv.Perm.SameCycle.refl _ _
-    | succ k ih =>
-      have hs := ih.apply_right.apply_right
-      simpa only [pow_succ',Equiv.Perm.mul_apply,alternatingPerm,Equiv.coe_fn_mk,
-        Bool.false_eq_true,↓reduceIte] using hs
-  simpa only [hn] using hp n
-
-omit [Fintype α] [DecidableEq α] in
-lemma alternating_sameCycle_zero (p : Equiv.Perm α) (x : Bool × α) :
-    (alternatingPerm p).SameCycle x (false,x.2) := by
-  rcases x with ⟨b,x⟩
-  cases b
-  · exact Equiv.Perm.SameCycle.refl _ _
-  · exact ((Equiv.Perm.SameCycle.refl (alternatingPerm p) (false,x)).apply_right).symm
-
-omit [DecidableEq α] in
-lemma alternating_sameCycle_iff (p : Equiv.Perm α) (x y : Bool × α) :
-    (alternatingPerm p).SameCycle x y ↔ p.SameCycle x.2 y.2 := by
-  constructor
-  · exact alternating_sameCycle_snd p
-  · intro h
-    exact (alternating_sameCycle_zero p x).trans
-      ((alternating_zero_sameCycle p h).trans (alternating_sameCycle_zero p y).symm)
-
-lemma orbit_alternating (p : Equiv.Perm α) (x : Bool × α) :
-    orbitSet (alternatingPerm p) x = childJoin (orbitSet p x.2) (orbitSet p x.2) := by
-  ext ⟨b,y⟩
-  simp only [mem_orbitSet,alternating_sameCycle_iff,mem_childJoin,ite_self]
-
-omit [Fintype α] in
-lemma boolFiber_childJoin (A B : Finset α) (b : Bool) :
-    boolFiber (childJoin A B) b = if b then B else A := by
-  ext x
-  cases b <;> simp only [mem_boolFiber,mem_childJoin,Bool.false_eq_true,↓reduceIte]
-
-noncomputable def projectCycle (p : Equiv.Perm α) (A B : Finset α)
-    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) :
-    RoutingNetwork.SelectedCycle p A := by
-  refine ⟨boolFiber C.val false,?_,?_⟩
-  · obtain ⟨x,hx⟩ := C.property.1
-    refine ⟨x.2,?_⟩
-    rw [hx,orbit_alternating,boolFiber_childJoin]
-    rfl
-  · intro x hx
-    exact (mem_childJoin A B false x).mp (C.property.2 ((mem_boolFiber _ _ _).mp hx))
-
-lemma selected_alternating_eq (p : Equiv.Perm α) (A B : Finset α)
-    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) :
-    C.val = childJoin (projectCycle p A B C).val (projectCycle p A B C).val := by
-  change C.val = childJoin (boolFiber C.val false) (boolFiber C.val false)
-  obtain ⟨x,hx⟩ := C.property.1
-  rw [hx,orbit_alternating,boolFiber_childJoin]
-  rfl
-
-lemma projectCycle_injective (p : Equiv.Perm α) (A B : Finset α) :
-    Function.Injective (projectCycle p A B) := by
-  intro C D he
-  apply Subtype.ext
-  rw [selected_alternating_eq p A B C,selected_alternating_eq p A B D,he]
-
-lemma card_alternating_cycles_le (p : Equiv.Perm α) (A B : Finset α) :
-    Fintype.card (RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) ≤
-      Fintype.card (RoutingNetwork.SelectedCycle p A) :=
-  Fintype.card_le_of_injective _ (projectCycle_injective p A B)
-
-end CycleColoring
-
-section PairAssignmentCount
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-omit [Fintype α] in
-lemma mem_image_perm_iff (p : Equiv.Perm α) (A : Finset α) (y : α) :
-    y ∈ A.image p ↔ p.symm y ∈ A := by
-  constructor
-  · intro h
-    obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp h
-    simpa only [←hxy,Equiv.symm_apply_apply] using hx
-  · intro h
-    exact Finset.mem_image.mpr ⟨p.symm y,h,p.apply_symm_apply y⟩
-
-omit [Fintype α] in
-lemma card_childJoin (A B : Finset α) : (childJoin A B).card = A.card+B.card := by
-  have hd : Disjoint (A.image (fun x => (false,x))) (B.image (fun x => (true,x))) := by
-    apply Finset.disjoint_left.mpr
-    rintro x hx hy
-    obtain ⟨a,ha,rfl⟩ := Finset.mem_image.mp hx
-    obtain ⟨b,hb,hba⟩ := Finset.mem_image.mp hy
-    exact Bool.false_ne_true (congrArg Prod.fst hba).symm
-  have hi (b : Bool) : Function.Injective (fun x : α => (b,x)) :=
-    fun _ _ h => congrArg Prod.snd h
-  rw [childJoin,Finset.card_union_of_disjoint hd,
-    Finset.card_image_of_injective A (hi false),Finset.card_image_of_injective B (hi true)]
-
-omit [Fintype α] in
-lemma alternating_internal (p : Equiv.Perm α) (A B : Finset α) :
-    ((childJoin A B).filter (fun x => (CycleColoring.alternatingPerm p).symm x ∈ childJoin A B)) =
-      childJoin (A ∩ B.image p) (B ∩ A) := by
-  ext ⟨b,x⟩
-  rw [Finset.mem_filter]
-  cases b <;> simp only [mem_childJoin,CycleColoring.alternatingPerm,
-    Equiv.symm_mk,Equiv.coe_fn_mk,Bool.false_eq_true,↓reduceIte,Finset.mem_inter,mem_image_perm_iff]
-
-omit [Fintype α] in
-lemma alternating_boundary_card (p : Equiv.Perm α) (A B : Finset α) :
-    (CycleColoring.boundary (CycleColoring.alternatingPerm p) (childJoin A B)).card =
-      (A ∪ B).card + (A ∪ B.image p).card - (A.card+B.card) := by
-  rw [CycleColoring.boundary_card,alternating_internal,card_childJoin,card_childJoin]
-  have h₀ := Finset.card_union_add_card_inter A B
-  have h₁ := Finset.card_union_add_card_inter A (B.image p)
-  rw [Finset.card_image_of_injective _ p.injective] at h₁
-  have h₂ := Finset.inter_comm B A
-  rw [h₂]
-  omega
-
-lemma compatible_assignment_bound (p : Equiv.Perm α) (A B : Finset α) :
-    Fintype.card (CycleColoring.OppositeColoring (CycleColoring.alternatingPerm p) (childJoin A B)) ≤
-      2 ^ ((A ∪ B).card + (A ∪ B.image p).card - (A.card+B.card) +
-        Fintype.card (RoutingNetwork.SelectedCycle p A)) := by
-  have h := CycleColoring.card_opposite_le (CycleColoring.alternatingPerm p) (childJoin A B)
-    Prod.fst (CycleColoring.alternatingPerm_flip p)
-  rw [alternating_boundary_card] at h
-  exact h.trans (Nat.pow_le_pow_right (by norm_num)
-    (Nat.add_le_add_left (CycleColoring.card_alternating_cycles_le p A B) _))
-
-end PairAssignmentCount
-
-lemma finiteMean_nonneg {Ω : Type*} [Fintype Ω] {f : Ω → ℝ}
-    (h : ∀ ω, 0 ≤ f ω) : 0 ≤ finiteMean f := by
-  exact div_nonneg (Finset.sum_nonneg (fun ω _ => h ω)) (Nat.cast_nonneg _)
-
-lemma boolFiber_headTail (d : ℕ) (S : Finset (Card (d+1))) (b : Bool) :
-    boolFiber (S.image (headTailEquiv d)) b = childMarks S b := by
-  classical
-  ext x
-  rw [mem_boolFiber,mem_childMarks]
-  constructor
-  · intro h
-    obtain ⟨y,hy,he⟩ := Finset.mem_image.mp h
-    have hh := congrArg (headTailEquiv d).symm he
-    rw [Equiv.symm_apply_apply] at hh
-    change y = Fin.cons b x at hh
-    exact hh ▸ hy
-  · intro h
-    exact Finset.mem_image.mpr ⟨Fin.cons b x,h,by rfl⟩
-
-lemma prod_childMarks (d : ℕ) (S : Finset (Card (d+1))) (f : Card (d+1) → ℝ) :
-    (∏ x ∈ S, f x) =
-      (∏ x ∈ childMarks S false, f (Fin.cons false x)) *
-      ∏ x ∈ childMarks S true, f (Fin.cons true x) := by
-  classical
-  have h := prod_boolFibers (S.image (headTailEquiv d)) (fun x => f ((headTailEquiv d).symm x))
-  rw [Finset.prod_image (headTailEquiv d).injective.injOn,
-    boolFiber_headTail,boolFiber_headTail] at h
-  simp only [Equiv.symm_apply_apply] at h
-  exact h
-
-noncomputable def headAverage {d : ℕ} (p : Card (d+1) → ℝ) (y : Card d) : ℝ :=
-  (p (Fin.cons false y)+p (Fin.cons true y))/2
-
-lemma finiteMean_headAverage (d : ℕ) (p : Card (d+1) → ℝ) :
-    finiteMean (headAverage p) = finiteMean p := by
-  rw [finiteMean_equiv (headTailEquiv d),finiteMean_prod,finiteMean_comm]
-  apply finiteMean_congr
-  intro y
-  rw [finiteMean_bool]
-  rfl
-
-lemma pairSwitch_product_bound (d : ℕ) (S : Finset (Card (d+1)))
-    (p : Card (d+1) → ℝ) (hp : ∀ x, 0 ≤ p x) :
-    finiteMean (fun ξ : Card d → Bool => ∏ x ∈ S, p (pairSwitch ξ x)) ≤
-      ∏ x ∈ S, headAverage p (Fin.tail x) := by
-  classical
-  have h : CylinderBound (fun T => ∏ x ∈ T, p ((headTailEquiv d).symm x))
-      (fun x => p ((headTailEquiv d).symm x)) := fun _ => le_rfl
-  have hh := h.fair_parallel (fun x => hp _) (S.image (headTailEquiv d))
-  simp only [Finset.prod_image (pairLayer _).injective.injOn,
-    Finset.prod_image (headTailEquiv d).injective.injOn] at hh
-  convert hh using 1
-  · apply finiteMean_congr
-    intro ξ
-    apply Finset.prod_congr rfl
-    intro x _
-    rfl
-  · apply Finset.prod_congr rfl
-    intro x _
-    change (p (Fin.cons false (Fin.tail x))+p (Fin.cons true (Fin.tail x)))/2 =
-      (p (Fin.cons (x 0) (Fin.tail x))+p (Fin.cons (!(x 0)) (Fin.tail x)))/2
-    cases x 0
-    · rfl
-    · simp only [Bool.not_true]
-      ring
-
-lemma butterflyPerm_step (d : ℕ)
-    (c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (ξ : Card d → Bool) :
-    butterflyPerm (d+1) (decodeButterfly (d+1) ((coinStepEquiv d).symm (c,ξ))) =
-    pairSwitch ξ * childLift (fun b => butterflyPerm d (decodeButterfly d (if b then c.2 else c.1))) := by
-  change pairSwitch (fun y => (coinStepEquiv d).symm (c,ξ) (Sum.inl y)) *
-    childLift (fun b => butterflyPerm d (decodeButterfly d
-      (fun i => (coinStepEquiv d).symm (c,ξ) (Sum.inr (b,i))))) = _
-  apply congrArg₂ (· * ·)
-  · rfl
-  · apply congrArg childLift
-    funext b
-    apply congrArg (fun ω => butterflyPerm d (decodeButterfly d ω))
-    funext i
-    cases b <;> rfl
-
-lemma butterfly_product_bound (d : ℕ) (S : Finset (Card d))
-    (p : Card d → ℝ) (hp : ∀ x, 0 ≤ p x) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      ∏ x ∈ S, p (butterflyPerm d (decodeButterfly d ω) x)) ≤ (finiteMean p)^S.card := by
-  classical
-  induction d with
-  | zero =>
-    have he : p = fun _ => p (fun i => Fin.elim0 i) := by
-      funext x
-      exact congrArg p (Subsingleton.elim _ _)
-    rw [he]
-    simp only [Finset.prod_const,finiteMean_const,le_refl]
-  | succ d ih =>
-    let g : Card d → ℝ := headAverage p
-    have hg : ∀ x, 0 ≤ g x := fun x => div_nonneg (add_nonneg (hp _) (hp _)) (by norm_num)
-    rw [finiteMean_equiv (coinStepEquiv d),finiteMean_prod]
-    calc
-      _ ≤ finiteMean (fun c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool) =>
-        (∏ x ∈ childMarks S false, g (butterflyPerm d (decodeButterfly d c.1) x)) *
-        ∏ x ∈ childMarks S true, g (butterflyPerm d (decodeButterfly d c.2) x)) := by
-        apply finiteMean_mono
-        intro c
-        let f : Equiv.Perm (Card (d+1)) := childLift (fun b =>
-          butterflyPerm d (decodeButterfly d (if b then c.2 else c.1)))
-        have hh := pairSwitch_product_bound d (S.image f) p hp
-        simp only [Finset.prod_image f.injective.injOn] at hh
-        convert hh using 1
-        · apply finiteMean_congr
-          intro ξ
-          apply Finset.prod_congr rfl
-          intro x _
-          rw [butterflyPerm_step]
-          rfl
-        · rw [prod_childMarks]
-          simp only [f,childLift,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
-            Bool.false_eq_true,↓reduceIte]
-          rfl
-      _ = (finiteMean (fun c : SwitchIndex d → Bool =>
-        ∏ x ∈ childMarks S false, g (butterflyPerm d (decodeButterfly d c) x))) *
-        finiteMean (fun c : SwitchIndex d → Bool =>
-        ∏ x ∈ childMarks S true, g (butterflyPerm d (decodeButterfly d c) x)) :=
-          finiteMean_prod_mul
-            (fun c : SwitchIndex d → Bool => ∏ x ∈ childMarks S false,
-              g (butterflyPerm d (decodeButterfly d c) x))
-            (fun c : SwitchIndex d → Bool => ∏ x ∈ childMarks S true,
-              g (butterflyPerm d (decodeButterfly d c) x))
-      _ ≤ (finiteMean g)^(childMarks S false).card * (finiteMean g)^(childMarks S true).card :=
-        mul_le_mul (ih (childMarks S false) g hg) (ih (childMarks S true) g hg)
-          (finiteMean_nonneg fun c => Finset.prod_nonneg (fun x _ => hg _))
-          (pow_nonneg (finiteMean_nonneg hg) _)
-      _ = (finiteMean p)^S.card := by
-        rw [←pow_add,card_childMarks_add,finiteMean_headAverage]
-
-noncomputable def inverseButterflyMarks (d : ℕ) (A : Finset (Card d))
-    (ω : SwitchIndex d → Bool) : Finset (Card d) :=
-  A.image (butterflyPerm d (decodeButterfly d ω)).symm
-
-lemma inverseButterflyMarks_inclusion (d : ℕ) (A S : Finset (Card d)) :
-    inclusionMoment (inverseButterflyMarks d A) S ≤
-      ((A.card : ℝ)/(2:ℝ)^d)^S.card := by
-  classical
-  let p : Card d → ℝ := fun x => if x ∈ A then 1 else 0
-  have hm : finiteMean p = (A.card : ℝ)/(2:ℝ)^d := by
-    simp only [finiteMean,p,card_positions,Nat.cast_pow,Nat.cast_ofNat]
-    congr 1
-    simp
-  have he : inclusionMoment (inverseButterflyMarks d A) S =
-      finiteMean (fun ω : SwitchIndex d → Bool =>
-        ∏ x ∈ S, p (butterflyPerm d (decodeButterfly d ω) x)) := by
-    rw [inclusionMoment_eq]
-    apply finiteMean_congr
-    intro ω
-    simp only [inverseButterflyMarks,subset_image_equiv_iff,Equiv.symm_symm,indicator_subset_prod,
-      Finset.prod_image (butterflyPerm d (decodeButterfly d ω)).injective.injOn,p]
-  rw [he]
-  simpa only [hm] using butterfly_product_bound d S p (fun x => by dsimp [p]; split_ifs <;> norm_num)
-
-namespace PairRouting
-
-section
-variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
-
-noncomputable def occupied (x : ι → Bool × α) : Finset α :=
-  Finset.univ.image (fun i => (x i).2)
-
-omit [Fintype α] in
-lemma mem_occupied (x : ι → Bool × α) (y : α) : y ∈ occupied x ↔ ∃ i, (x i).2 = y := by
-  simp [occupied]
-
-noncomputable def chosenCoin (x : ι → Bool × α) (c : ι → Bool) (y : α) : Bool := by
-  classical
-  exact if h : ∃ i, (x i).2 = y then required x c (Classical.choose h) else false
-
-omit [Fintype α] in
-lemma chosenCoin_apply (x : ι → Bool × α) (c : ι → Bool) (hc : Compatible x c) (i : ι) :
-    chosenCoin x c (x i).2 = required x c i := by
-  classical
-  simp only [chosenCoin,dite_eq_left (show ∃ j, (x j).2 = (x i).2 from ⟨i,rfl⟩)]
-  exact hc _ i (Classical.choose_spec (show ∃ j, (x j).2 = (x i).2 from ⟨i,rfl⟩))
-
-omit [Fintype α] in
-lemma colors_eq_iff (x : ι → Bool × α) (c : ι → Bool) (hc : Compatible x c) (ξ : α → Bool) :
-    colors x ξ = c ↔ ∀ y ∈ occupied x, ξ y = chosenCoin x c y := by
-  constructor
-  · intro he y hy
-    obtain ⟨i,rfl⟩ := (mem_occupied x y).mp hy
-    rw [chosenCoin_apply x c hc i]
-    rw [←he,required_colors]
-  · intro he
-    funext i
-    apply (xor_eq_iff _ _ _).mpr
-    exact (he _ ((mem_occupied x _).mpr ⟨i,rfl⟩)).trans (chosenCoin_apply x c hc i)
-
-noncomputable instance (x : ι → Bool × α) (c : ι → Bool) : Decidable (Compatible x c) :=
-  Classical.propDecidable _
-
-lemma colors_probability (x : ι → Bool × α) (c : ι → Bool) :
-    finiteMean (fun ξ : α → Bool => if colors x ξ = c then (1 : ℝ) else 0) =
-      if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0 := by
-  classical
-  by_cases hc : Compatible x c
-  · rw [ite_eq_left hc]
-    have he := fun ξ => colors_eq_iff x c hc ξ
-    simp only [he]
-    exact mean_coinCylinder (occupied x) (chosenCoin x c)
-  · rw [ite_eq_right hc]
-    have he : ∀ ξ, colors x ξ ≠ c := fun ξ hh => hc (hh ▸ colors_compatible x ξ)
-    simp only [he,ite_false,finiteMean_const]
-
-
-def lift (p : Bool → Equiv.Perm α) : Equiv.Perm (Bool × α) where
-  toFun x := (x.1,p x.1 x.2)
-  invFun x := (x.1,(p x.1).symm x.2)
-  left_inv x := by simp
-  right_inv x := by simp
-
-def sandwich (ξ η : α → Bool) (p : Bool → Equiv.Perm α) : Equiv.Perm (Bool × α) :=
-  pairLayer η * lift p * pairLayer ξ
-
-def ChildMatches (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) : Prop :=
-  ∀ i, p (c i) (x i).2 = (y i).2
-
-noncomputable instance (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
-    Decidable (ChildMatches p x y c) := Classical.propDecidable _
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma sandwich_matches_iff (ξ η : α → Bool) (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) :
-    (∀ i, sandwich ξ η p (x i) = y i) ↔
-      colors x ξ = colors y η ∧ ChildMatches p x y (colors x ξ) := by
-  have he (i : ι) : sandwich ξ η p (x i) = y i ↔
-      colors x ξ i = colors y η i ∧ p (colors x ξ i) (x i).2 = (y i).2 := by
-    change pairLayer η (lift p (pairLayer ξ (x i))) = y i ↔ _
-    rw [←Equiv.eq_symm_apply]
-    change (colors x ξ i,p (colors x ξ i) (x i).2) = (colors y η i,(y i).2) ↔ _
-    exact ⟨Prod.mk.inj,fun h => Prod.ext h.1 h.2⟩
-  simp only [he,forall_and,←funext_iff,ChildMatches]
-
-lemma sandwich_probability [DecidableEq ι] (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) :
-    finiteMean (fun coins : (α → Bool) × (α → Bool) =>
-      if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then (1:ℝ) else 0) =
-    ∑ c : ι → Bool,
-      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
-      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) *
-      (if ChildMatches p x y c then 1 else 0) := by
-  classical
-  have he (coins : (α → Bool) × (α → Bool)) :
-      (if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then (1:ℝ) else 0) =
-      ∑ c : ι → Bool, (if colors x coins.1 = c then (1:ℝ) else 0) *
-        (if colors y coins.2 = c then 1 else 0) * (if ChildMatches p x y c then 1 else 0) := by
-    rw [Finset.sum_eq_single (colors x coins.1)]
-    · by_cases hc : colors x coins.1 = colors y coins.2
-      · simp only [sandwich_matches_iff,hc,true_and,↓reduceIte,one_mul]
-      · simp only [sandwich_matches_iff,hc,false_and,↓reduceIte,one_mul,Ne.symm hc,zero_mul]
-    · intro c _ hc
-      simp only [Ne.symm hc,ite_false,zero_mul]
-    · simp
-  simp_rw [he]
-  rw [finiteMean_sum]
-  apply Finset.sum_congr rfl
-  intro c _
-  rw [finiteMean_mul_const]
-  rw [finiteMean_prod_mul
-    (fun ξ : α → Bool => if colors x ξ = c then (1:ℝ) else 0)
-    (fun η : α → Bool => if colors y η = c then (1:ℝ) else 0),
-    colors_probability,colors_probability]
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma childMatches_split (p₀ p₁ : Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
-    ChildMatches (fun b => if b then p₁ else p₀) x y c ↔
-      (∀ i : {i // c i = false}, p₀ (x i.val).2 = (y i.val).2) ∧
-      (∀ i : {i // c i = true}, p₁ (x i.val).2 = (y i.val).2) := by
-  constructor
-  · intro h
-    constructor
-    · intro i
-      simpa only [i.property,Bool.false_eq_true,↓reduceIte] using h i.val
-    · intro i
-      simpa only [i.property,↓reduceIte] using h i.val
-  · rintro ⟨h₀,h₁⟩ i
-    cases hi : c i
-    · simpa only [hi,Bool.false_eq_true,↓reduceIte] using h₀ ⟨i,hi⟩
-    · simpa only [hi,↓reduceIte] using h₁ ⟨i,hi⟩
-
-noncomputable def tupleProbability {Ω β : Type*} [Fintype Ω] [DecidableEq β]
-    (P : Ω → Equiv.Perm β) (x y : ι → β) : ℝ := by
-  classical
-  exact finiteMean (fun ω => if ∀ i, P ω (x i) = y i then 1 else 0)
-
-omit [Fintype α] in
-lemma childMatches_mean {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
-    (P : Ω → Equiv.Perm α) (Q : Ω' → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
-    finiteMean (fun ω : Ω × Ω' =>
-      if ChildMatches (fun b => if b then Q ω.2 else P ω.1) x y c then (1:ℝ) else 0) =
-      tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
-        (fun i : {i // c i = false} => (y i.val).2) *
-      tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
-        (fun i : {i // c i = true} => (y i.val).2) := by
-  classical
-  let f : Ω → ℝ := fun ω => if ∀ i : {i // c i = false}, P ω (x i.val).2 = (y i.val).2 then 1 else 0
-  let g : Ω' → ℝ := fun ω => if ∀ i : {i // c i = true}, Q ω (x i.val).2 = (y i.val).2 then 1 else 0
-  have he (ω : Ω × Ω') :
-      (if ChildMatches (fun b => if b then Q ω.2 else P ω.1) x y c then (1:ℝ) else 0) = f ω.1*g ω.2 := by
-    simp only [childMatches_split,f,g]
-    split_ifs <;> simp_all
-  simp_rw [he]
-  exact finiteMean_prod_mul f g
-
-lemma sandwich_mean [DecidableEq ι] {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
-    (P : Ω → Equiv.Perm α) (Q : Ω' → Equiv.Perm α) (x y : ι → Bool × α) :
-    finiteMean (fun ω : Ω × Ω' => finiteMean (fun coins : (α → Bool) × (α → Bool) =>
-      if ∀ i, sandwich coins.1 coins.2 (fun b => if b then Q ω.2 else P ω.1) (x i) = y i
-      then (1:ℝ) else 0)) =
-    ∑ c : ι → Bool,
-      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
-      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) *
-      (tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
-        (fun i : {i // c i = false} => (y i.val).2) *
-      tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
-        (fun i : {i // c i = true} => (y i.val).2)) := by
-  classical
-  simp_rw [sandwich_probability]
-  rw [finiteMean_sum]
-  apply Finset.sum_congr rfl
-  intro c _
-  simp only [mul_comm ((if Compatible x c then _ else _) * _),finiteMean_mul_const,
-    childMatches_mean]
-
-
-abbrev BothAssignments (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :=
-  {c : ι → Bool // Compatible e c ∧ Compatible (fun i => lift p (e i)) c}
-
-noncomputable instance (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    Fintype (BothAssignments e p) := Fintype.ofFinite _
-
-noncomputable def assignmentColoring (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α)
-    (c : BothAssignments e p) : CycleColoring.OppositeColoring
-      (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e) := by
-  classical
-  refine ⟨Function.extend e c.val (fun _ => false),?_,?_⟩
-  · intro z hz
-    apply Function.extend_apply'
-    rintro ⟨i,rfl⟩
-    exact hz (Finset.mem_image.mpr ⟨i,Finset.mem_univ _,rfl⟩)
-  · intro z hz hnext
-    obtain ⟨i,_,hi⟩ := Finset.mem_image.mp hz
-    obtain ⟨j,_,hj⟩ := Finset.mem_image.mp hnext
-    have hfi := e.injective.extend_apply c.val (fun _ => false) i
-    have hfj := e.injective.extend_apply c.val (fun _ => false) j
-    rw [hi] at hfi
-    rw [hj] at hfj
-    rw [hfi,hfj]
-    rcases z with ⟨b,a⟩
-    cases b
-    · have hj' : e j = (true,a) := hj
-      have ht : (e i).2 = (e j).2 := by rw [hi,hj']
-      have hh := c.property.1 i j ht
-      simp only [required,hi,hj'] at hh
-      revert hh
-      generalize c.val i = bi
-      generalize c.val j = bj
-      cases bi <;> cases bj <;> decide
-    · have hj' : e j = (false,alternatingProjection p a) := hj
-      have ht : (lift p (e i)).2 = (lift p (e j)).2 := by
-        simp only [hi,hj',lift,Equiv.coe_fn_mk,alternatingProjection,Equiv.Perm.coe_mul,
-          Function.comp_apply,Equiv.apply_symm_apply]
-      have hh := c.property.2 i j ht
-      simp only [required,hi,hj',lift,Equiv.coe_fn_mk] at hh
-      revert hh
-      generalize c.val i = bi
-      generalize c.val j = bj
-      cases bi <;> cases bj <;> decide
-
-omit [Fintype α] in
-lemma assignmentColoring_injective (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    Function.Injective (assignmentColoring e p) := by
-  intro c c' he
-  apply Subtype.ext
-  funext i
-  have hh := congrFun (congrArg Subtype.val he) (e i)
-  change Function.extend e c.val (fun _ => false) (e i) =
-    Function.extend e c'.val (fun _ => false) (e i) at hh
-  simpa only [e.injective.extend_apply] using hh
-
-lemma bothAssignments_card (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    Fintype.card (BothAssignments e p) ≤
-      2 ^ ((boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true).card +
-        (boolFiber (labelSet e) false ∪ (boolFiber (labelSet e) true).image
-          (alternatingProjection p)).card -
-        ((boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card) +
-        Fintype.card (RoutingNetwork.SelectedCycle (alternatingProjection p) (boolFiber (labelSet e) false))) := by
-  have h := compatible_assignment_bound (alternatingProjection p)
-    (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
-  rw [childJoin_boolFiber] at h
-  exact (Fintype.card_le_of_injective _ (assignmentColoring_injective e p)).trans h
-
-omit [Fintype α] in
-lemma occupied_labelSet (e : ι ↪ Bool × α) : occupied e = (labelSet e).image Prod.snd := by
-  classical
-  simp only [occupied,labelSet,Finset.image_image,Function.comp_def]
-
-omit [Fintype ι] [Fintype α] in
-lemma image_snd_childJoin (A B : Finset α) : (childJoin A B).image Prod.snd = A ∪ B := by
-  classical
-  simp only [childJoin,Finset.image_union,Finset.image_image,Function.comp_def,Finset.image_id']
-
-omit [Fintype ι] [Fintype α] in
-lemma image_lift_childJoin (p : Bool → Equiv.Perm α) (A B : Finset α) :
-    (childJoin A B).image (fun x => (lift p x).2) = A.image (p false) ∪ B.image (p true) := by
-  classical
-  simp only [childJoin,Finset.image_union,Finset.image_image,Function.comp_def,lift,Equiv.coe_fn_mk]
-
-omit [Fintype α] in
-lemma occupied_lift_card (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    (occupied (fun i => lift p (e i))).card =
-      (boolFiber (labelSet e) false ∪ (boolFiber (labelSet e) true).image (alternatingProjection p)).card := by
-  classical
-  have he : occupied (fun i => lift p (e i)) = (labelSet e).image (fun x => (lift p x).2) := by
-    simp only [occupied,labelSet,Finset.image_image,Function.comp_def]
-  rw [he,←childJoin_boolFiber (labelSet e),image_lift_childJoin]
-  simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
-  let A := boolFiber (labelSet e) false
-  let B := boolFiber (labelSet e) true
-  change (A.image (p false) ∪ B.image (p true)).card = (A ∪ B.image (alternatingProjection p)).card
-  have hh : A.image (p false) ∪ B.image (p true) =
-      (A ∪ B.image (alternatingProjection p)).image (p false) := by
-    simp only [Finset.image_union,Finset.image_image,Function.comp_def,
-      alternatingProjection,Equiv.Perm.coe_mul,Equiv.apply_symm_apply]
-  rw [hh,Finset.card_image_of_injective _ (p false).injective]
-
-lemma bothAssignments_occupied_bound (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    Fintype.card (BothAssignments e p) ≤
-      2 ^ ((occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι +
-        Fintype.card (RoutingNetwork.SelectedCycle (alternatingProjection p) (boolFiber (labelSet e) false))) := by
-  have he : (occupied e).card =
-      (boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true).card := by
-    rw [occupied_labelSet,←childJoin_boolFiber (labelSet e),image_snd_childJoin]
-    simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
-  have hc : (boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card = Fintype.card ι := by
-    rw [←card_childJoin,childJoin_boolFiber,labelSet,Finset.card_image_of_injective _ e.injective,
-      Finset.card_univ]
-  simpa only [he,occupied_lift_card,←hc] using bothAssignments_card e p
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma compatible_pairLayer_iff (x : ι → Bool × α) (c : ι → Bool) (ξ : α → Bool) :
-    Compatible (fun i => pairLayer ξ (x i)) c ↔ Compatible x c := by
-  have he (i j : ι) (ht : (x i).2 = (x j).2) :
-      required (fun i => pairLayer ξ (x i)) c i = required (fun i => pairLayer ξ (x i)) c j ↔
-      required x c i = required x c j := by
-    simp only [required,pairLayer,Equiv.coe_fn_mk,ht]
-    cases (x i).1 <;> cases (x j).1 <;> cases c i <;> cases c j <;> cases ξ (x j).2 <;> decide
-  constructor
-  · intro h i j ht
-    exact (he i j ht).mp (h i j ht)
-  · intro h i j ht
-    exact (he i j ht).mpr (h i j ht)
-
-
-omit [Fintype α] in
-lemma occupied_eq_fibers (e : ι ↪ Bool × α) :
-    occupied e = boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true := by
-  rw [occupied_labelSet,←childJoin_boolFiber (labelSet e),image_snd_childJoin]
-  simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
-
-omit [Fintype α] in
-lemma card_labelSet (e : ι ↪ Bool × α) : (labelSet e).card = Fintype.card ι := by
-  classical
-  exact (Finset.card_image_of_injective _ e.injective).trans (Finset.card_univ)
-
-omit [Fintype α] in
-lemma occupied_card_double (e : ι ↪ Bool × α) : Fintype.card ι ≤ 2*(occupied e).card := by
-  have h₀ : (boolFiber (labelSet e) false).card ≤ (occupied e).card := by
-    rw [occupied_eq_fibers]
-    exact Finset.card_le_card Finset.subset_union_left
-  have h₁ : (boolFiber (labelSet e) true).card ≤ (occupied e).card := by
-    rw [occupied_eq_fibers]
-    exact Finset.card_le_card Finset.subset_union_right
-  have he : (boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card = Fintype.card ι := by
-    rw [←card_childJoin,childJoin_boolFiber,card_labelSet]
-  omega
-
-lemma bothAssignments_exact_bound (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    Fintype.card (BothAssignments e p) ≤
-      2 ^ ((occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι +
-        alternatingCycles e p) := by
-  have h := CycleColoring.card_opposite_le
-    (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e)
-    Prod.fst (CycleColoring.alternatingPerm_flip (alternatingProjection p))
-  have hb : (CycleColoring.boundary (CycleColoring.alternatingPerm (alternatingProjection p))
-      (labelSet e)).card =
-      (occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι := by
-    have hh := alternating_boundary_card (alternatingProjection p)
-      (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
-    rw [childJoin_boolFiber] at hh
-    rw [occupied_eq_fibers,occupied_lift_card]
-    have he := card_childJoin (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
-    rw [childJoin_boolFiber,card_labelSet] at he
-    simpa only [he] using hh
-  rw [hb] at h
-  exact (Fintype.card_le_of_injective _ (assignmentColoring_injective e p)).trans h
-
-abbrev Assignments (x y : ι → Bool × α) :=
-  {c : ι → Bool // Compatible x c ∧ Compatible y c}
-
-noncomputable instance (x y : ι → Bool × α) : Fintype (Assignments x y) := Fintype.ofFinite _
-
-def routedEmbedding (e : ι ↪ Bool × α) (ξ η : α → Bool) (p : Bool → Equiv.Perm α) :
-    ι ↪ Bool × α := e.trans (sandwich ξ η p).toEmbedding
-
-omit [Fintype α] in
-lemma occupied_pairLayer (x : ι → Bool × α) (ξ : α → Bool) :
-    occupied (fun i => pairLayer ξ (x i)) = occupied x := rfl
-
-noncomputable def assignmentsEquiv (e : ι ↪ Bool × α) (ξ η : α → Bool)
-    (p : Bool → Equiv.Perm α) :
-    Assignments e (routedEmbedding e ξ η p) ≃ BothAssignments (switchedEmbedding e ξ) p := by
-  apply Equiv.subtypeEquivRight
-  intro c
-  change Compatible e c ∧ Compatible (fun i => pairLayer η (lift p (pairLayer ξ (e i)))) c ↔ _
-  rw [compatible_pairLayer_iff]
-  exact and_congr (compatible_pairLayer_iff e c ξ).symm Iff.rfl
-
-lemma assignments_actual_bound (e : ι ↪ Bool × α) (ξ η : α → Bool) (p : Bool → Equiv.Perm α) :
-    Fintype.card (Assignments e (routedEmbedding e ξ η p)) ≤
-      2 ^ ((occupied e).card + (occupied (routedEmbedding e ξ η p)).card - Fintype.card ι +
-        alternatingCycles (switchedEmbedding e ξ) p) := by
-  rw [Fintype.card_congr (assignmentsEquiv e ξ η p)]
-  exact bothAssignments_exact_bound (switchedEmbedding e ξ) p
-
-omit [Fintype α] in
-lemma assignments_actual_exponent_nonneg (e : ι ↪ Bool × α) (ξ η : α → Bool)
-    (p : Bool → Equiv.Perm α) :
-    Fintype.card ι ≤ (occupied e).card + (occupied (routedEmbedding e ξ η p)).card := by
-  have h₀ := occupied_card_double e
-  have h₁ := occupied_card_double (routedEmbedding e ξ η p)
-  omega
-
-end
-variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α] [DecidableEq ι]
-
-noncomputable def compatibleSet (x y : ι → Bool × α) : Finset (ι → Bool) := by
-  classical
-  exact Finset.univ.filter (fun c => Compatible x c ∧ Compatible y c)
-
-omit [Fintype α] [DecidableEq α] in
-lemma mem_compatibleSet (x y : ι → Bool × α) (c : ι → Bool) :
-    c ∈ compatibleSet x y ↔ Compatible x c ∧ Compatible y c := by
-  classical
-  simp only [compatibleSet,Finset.mem_filter,Finset.mem_univ,true_and]
-
-omit [Fintype α] [DecidableEq α] in
-lemma card_compatibleSet (x y : ι → Bool × α) :
-    (compatibleSet x y).card = Fintype.card (Assignments x y) := by
-  classical
-  exact (Fintype.card_subtype _).symm
-
-noncomputable def outerWeight (x y : ι → Bool × α) : ℝ :=
-  1/(2:ℝ)^((occupied x).card+(occupied y).card)
-
-omit [Fintype α] [DecidableEq ι] in
-lemma outerWeight_pos (x y : ι → Bool × α) : 0 < outerWeight x y := by
-  unfold outerWeight
-  positivity
-
-omit [Fintype α] in
-lemma sum_colors_factor (x y : ι → Bool × α) (f : (ι → Bool) → ℝ) :
-    (∑ c : ι → Bool,
-      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
-      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) * f c) =
-      outerWeight x y * ∑ c ∈ compatibleSet x y, f c := by
-  classical
-  rw [compatibleSet,Finset.sum_filter,Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro c _
-  by_cases hx : Compatible x c <;> by_cases hy : Compatible y c
-  · simp only [hx,hy,true_and,↓reduceIte,outerWeight,pow_add,one_div_mul_one_div]
-  · simp only [hx,hy,and_false,↓reduceIte,mul_zero,zero_mul]
-  · simp only [hx,hy,false_and,↓reduceIte,mul_zero,zero_mul]
-  · simp only [hx,hy,false_and,↓reduceIte,mul_zero,zero_mul]
-
-lemma sandwich_weighted_probability (p : Bool → Equiv.Perm α) (x y : ι → Bool × α)
-    (W : (ι → Bool) → ℝ) :
-    finiteMean (fun coins : (α → Bool) × (α → Bool) =>
-      if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then W (colors x coins.1) else 0) =
-    outerWeight x y * ∑ c ∈ compatibleSet x y, if ChildMatches p x y c then W c else 0 := by
-  classical
-  have he (coins : (α → Bool) × (α → Bool)) :
-      (if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then W (colors x coins.1) else 0) =
-      ∑ c : ι → Bool, (if colors x coins.1 = c then (1:ℝ) else 0) *
-        (if colors y coins.2 = c then 1 else 0) * (if ChildMatches p x y c then W c else 0) := by
-    rw [Finset.sum_eq_single (colors x coins.1)]
-    · by_cases hc : colors x coins.1 = colors y coins.2
-      · simp only [sandwich_matches_iff,hc,true_and,↓reduceIte,one_mul]
-      · simp only [sandwich_matches_iff,hc,false_and,↓reduceIte,one_mul,Ne.symm hc,zero_mul]
-    · intro c _ hc
-      simp only [Ne.symm hc,ite_false,zero_mul]
-    · simp
-  simp_rw [he]
-  rw [finiteMean_sum,←sum_colors_factor]
-  apply Finset.sum_congr rfl
-  intro c _
-  rw [finiteMean_mul_const,finiteMean_prod_mul
-    (fun ξ : α → Bool => if colors x ξ = c then (1:ℝ) else 0)
-    (fun η : α → Bool => if colors y η = c then (1:ℝ) else 0),
-    colors_probability,colors_probability]
-
-omit [Fintype α] [DecidableEq ι] in
-lemma tupleProbability_nonneg {Ω β : Type*} [Fintype Ω] [DecidableEq β]
-    (P : Ω → Equiv.Perm β) (x y : ι → β) : 0 ≤ tupleProbability P x y := by
-  classical
-  exact finiteMean_nonneg (fun ω => by split_ifs <;> norm_num)
-
-end PairRouting
-end Thorp
-
-end ThorpNine.HighTail
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.HighTail.ParallelFollow
+
+namespace OAI
+
+namespace ThorpNine.HighTail
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+namespace CycleColoring
+variable {α : Type*} [Fintype α] [DecidableEq α]
+omit [Fintype α] in
+lemma bipartitionFlip_injective (p : Equiv.Perm α) (S : Finset α)
+    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x)) :
+    Function.Injective (bipartitionFlip p S χ hχ) := by
+  classical
+  intro f g he
+  apply Subtype.ext
+  funext x
+  by_cases hx : x ∈ S
+  · have hh := congrFun (congrArg Subtype.val he) x
+    change (if x ∈ S then Bool.xor (f.val x) (χ x) else false) =
+      (if x ∈ S then Bool.xor (g.val x) (χ x) else false) at hh
+    simp only [ite_eq_left hx] at hh
+    cases hfx : f.val x <;> cases hgx : g.val x <;> cases hcx : χ x <;> simp_all
+  · exact (f.property.1 x hx).trans (g.property.1 x hx).symm
+
+lemma card_opposite_le (p : Equiv.Perm α) (S : Finset α)
+    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x)) :
+    Fintype.card (OppositeColoring p S) ≤
+      2 ^ ((boundary p S).card + Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
+  exact (Fintype.card_le_of_injective _ (bipartitionFlip_injective p S χ hχ)).trans
+    (card_coloring_le p S)
+
+omit [Fintype α] [DecidableEq α] in
+lemma alternatingPerm_flip (p : Equiv.Perm α) (x : Bool × α) :
+    (alternatingPerm p x).1 = !x.1 := by
+  rcases x with ⟨b,x⟩
+  cases b <;> rfl
+
+omit [Fintype α] [DecidableEq α] in
+lemma alternating_step_sameCycle (p : Equiv.Perm α) (x : Bool × α) :
+    p.SameCycle x.2 (alternatingPerm p x).2 := by
+  rcases x with ⟨b,x⟩
+  cases b
+  · exact Equiv.Perm.SameCycle.refl p x
+  · exact (Equiv.Perm.SameCycle.refl p x).apply_right
+
+omit [DecidableEq α] in
+lemma alternating_sameCycle_snd (p : Equiv.Perm α) {x y : Bool × α}
+    (h : (alternatingPerm p).SameCycle x y) : p.SameCycle x.2 y.2 := by
+  obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
+  have hp (k : ℕ) : p.SameCycle x.2 (((alternatingPerm p)^k) x).2 := by
+    induction k with
+    | zero => exact Equiv.Perm.SameCycle.refl p x.2
+    | succ k ih =>
+      rw [pow_succ',Equiv.Perm.mul_apply]
+      exact ih.trans (alternating_step_sameCycle p _)
+  exact hn ▸ hp n
+
+omit [DecidableEq α] in
+lemma alternating_zero_sameCycle (p : Equiv.Perm α) {x y : α}
+    (h : p.SameCycle x y) : (alternatingPerm p).SameCycle (false,x) (false,y) := by
+  obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
+  have hp (k : ℕ) : (alternatingPerm p).SameCycle (false,x) (false,(p^k) x) := by
+    induction k with
+    | zero => exact Equiv.Perm.SameCycle.refl _ _
+    | succ k ih =>
+      have hs := ih.apply_right.apply_right
+      simpa only [pow_succ',Equiv.Perm.mul_apply,alternatingPerm,Equiv.coe_fn_mk,
+        Bool.false_eq_true,↓reduceIte] using hs
+  simpa only [hn] using hp n
+
+omit [Fintype α] [DecidableEq α] in
+lemma alternating_sameCycle_zero (p : Equiv.Perm α) (x : Bool × α) :
+    (alternatingPerm p).SameCycle x (false,x.2) := by
+  rcases x with ⟨b,x⟩
+  cases b
+  · exact Equiv.Perm.SameCycle.refl _ _
+  · exact ((Equiv.Perm.SameCycle.refl (alternatingPerm p) (false,x)).apply_right).symm
+
+omit [DecidableEq α] in
+lemma alternating_sameCycle_iff (p : Equiv.Perm α) (x y : Bool × α) :
+    (alternatingPerm p).SameCycle x y ↔ p.SameCycle x.2 y.2 := by
+  constructor
+  · exact alternating_sameCycle_snd p
+  · intro h
+    exact (alternating_sameCycle_zero p x).trans
+      ((alternating_zero_sameCycle p h).trans (alternating_sameCycle_zero p y).symm)
+
+lemma orbit_alternating (p : Equiv.Perm α) (x : Bool × α) :
+    orbitSet (alternatingPerm p) x = childJoin (orbitSet p x.2) (orbitSet p x.2) := by
+  ext ⟨b,y⟩
+  simp only [mem_orbitSet,alternating_sameCycle_iff,mem_childJoin,ite_self]
+
+omit [Fintype α] in
+lemma boolFiber_childJoin (A B : Finset α) (b : Bool) :
+    boolFiber (childJoin A B) b = if b then B else A := by
+  ext x
+  cases b <;> simp only [mem_boolFiber,mem_childJoin,Bool.false_eq_true,↓reduceIte]
+
+noncomputable def projectCycle (p : Equiv.Perm α) (A B : Finset α)
+    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) :
+    RoutingNetwork.SelectedCycle p A := by
+  refine ⟨boolFiber C.val false,?_,?_⟩
+  · obtain ⟨x,hx⟩ := C.property.1
+    refine ⟨x.2,?_⟩
+    rw [hx,orbit_alternating,boolFiber_childJoin]
+    rfl
+  · intro x hx
+    exact (mem_childJoin A B false x).mp (C.property.2 ((mem_boolFiber _ _ _).mp hx))
+
+lemma selected_alternating_eq (p : Equiv.Perm α) (A B : Finset α)
+    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) :
+    C.val = childJoin (projectCycle p A B C).val (projectCycle p A B C).val := by
+  change C.val = childJoin (boolFiber C.val false) (boolFiber C.val false)
+  obtain ⟨x,hx⟩ := C.property.1
+  rw [hx,orbit_alternating,boolFiber_childJoin]
+  rfl
+
+lemma projectCycle_injective (p : Equiv.Perm α) (A B : Finset α) :
+    Function.Injective (projectCycle p A B) := by
+  intro C D he
+  apply Subtype.ext
+  rw [selected_alternating_eq p A B C,selected_alternating_eq p A B D,he]
+
+lemma card_alternating_cycles_le (p : Equiv.Perm α) (A B : Finset α) :
+    Fintype.card (RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) ≤
+      Fintype.card (RoutingNetwork.SelectedCycle p A) :=
+  Fintype.card_le_of_injective _ (projectCycle_injective p A B)
+
+end CycleColoring
+
+section PairAssignmentCount
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+omit [Fintype α] in
+lemma mem_image_perm_iff (p : Equiv.Perm α) (A : Finset α) (y : α) :
+    y ∈ A.image p ↔ p.symm y ∈ A := by
+  constructor
+  · intro h
+    obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp h
+    simpa only [←hxy,Equiv.symm_apply_apply] using hx
+  · intro h
+    exact Finset.mem_image.mpr ⟨p.symm y,h,p.apply_symm_apply y⟩
+
+omit [Fintype α] in
+lemma card_childJoin (A B : Finset α) : (childJoin A B).card = A.card+B.card := by
+  have hd : Disjoint (A.image (fun x => (false,x))) (B.image (fun x => (true,x))) := by
+    apply Finset.disjoint_left.mpr
+    rintro x hx hy
+    obtain ⟨a,ha,rfl⟩ := Finset.mem_image.mp hx
+    obtain ⟨b,hb,hba⟩ := Finset.mem_image.mp hy
+    exact Bool.false_ne_true (congrArg Prod.fst hba).symm
+  have hi (b : Bool) : Function.Injective (fun x : α => (b,x)) :=
+    fun _ _ h => congrArg Prod.snd h
+  rw [childJoin,Finset.card_union_of_disjoint hd,
+    Finset.card_image_of_injective A (hi false),Finset.card_image_of_injective B (hi true)]
+
+omit [Fintype α] in
+lemma alternating_internal (p : Equiv.Perm α) (A B : Finset α) :
+    ((childJoin A B).filter (fun x => (CycleColoring.alternatingPerm p).symm x ∈ childJoin A B)) =
+      childJoin (A ∩ B.image p) (B ∩ A) := by
+  ext ⟨b,x⟩
+  rw [Finset.mem_filter]
+  cases b <;> simp only [mem_childJoin,CycleColoring.alternatingPerm,
+    Equiv.symm_mk,Equiv.coe_fn_mk,Bool.false_eq_true,↓reduceIte,Finset.mem_inter,mem_image_perm_iff]
+
+omit [Fintype α] in
+lemma alternating_boundary_card (p : Equiv.Perm α) (A B : Finset α) :
+    (CycleColoring.boundary (CycleColoring.alternatingPerm p) (childJoin A B)).card =
+      (A ∪ B).card + (A ∪ B.image p).card - (A.card+B.card) := by
+  rw [CycleColoring.boundary_card,alternating_internal,card_childJoin,card_childJoin]
+  have h₀ := Finset.card_union_add_card_inter A B
+  have h₁ := Finset.card_union_add_card_inter A (B.image p)
+  rw [Finset.card_image_of_injective _ p.injective] at h₁
+  have h₂ := Finset.inter_comm B A
+  rw [h₂]
+  omega
+
+lemma compatible_assignment_bound (p : Equiv.Perm α) (A B : Finset α) :
+    Fintype.card (CycleColoring.OppositeColoring (CycleColoring.alternatingPerm p) (childJoin A B)) ≤
+      2 ^ ((A ∪ B).card + (A ∪ B.image p).card - (A.card+B.card) +
+        Fintype.card (RoutingNetwork.SelectedCycle p A)) := by
+  have h := CycleColoring.card_opposite_le (CycleColoring.alternatingPerm p) (childJoin A B)
+    Prod.fst (CycleColoring.alternatingPerm_flip p)
+  rw [alternating_boundary_card] at h
+  exact h.trans (Nat.pow_le_pow_right (by norm_num)
+    (Nat.add_le_add_left (CycleColoring.card_alternating_cycles_le p A B) _))
+
+end PairAssignmentCount
+
+lemma finiteMean_nonneg {Ω : Type*} [Fintype Ω] {f : Ω → ℝ}
+    (h : ∀ ω, 0 ≤ f ω) : 0 ≤ finiteMean f := by
+  exact div_nonneg (Finset.sum_nonneg (fun ω _ => h ω)) (Nat.cast_nonneg _)
+
+lemma boolFiber_headTail (d : ℕ) (S : Finset (Card (d+1))) (b : Bool) :
+    boolFiber (S.image (headTailEquiv d)) b = childMarks S b := by
+  classical
+  ext x
+  rw [mem_boolFiber,mem_childMarks]
+  constructor
+  · intro h
+    obtain ⟨y,hy,he⟩ := Finset.mem_image.mp h
+    have hh := congrArg (headTailEquiv d).symm he
+    rw [Equiv.symm_apply_apply] at hh
+    change y = Fin.cons b x at hh
+    exact hh ▸ hy
+  · intro h
+    exact Finset.mem_image.mpr ⟨Fin.cons b x,h,by rfl⟩
+
+lemma prod_childMarks (d : ℕ) (S : Finset (Card (d+1))) (f : Card (d+1) → ℝ) :
+    (∏ x ∈ S, f x) =
+      (∏ x ∈ childMarks S false, f (Fin.cons false x)) *
+      ∏ x ∈ childMarks S true, f (Fin.cons true x) := by
+  classical
+  have h := prod_boolFibers (S.image (headTailEquiv d)) (fun x => f ((headTailEquiv d).symm x))
+  rw [Finset.prod_image (headTailEquiv d).injective.injOn,
+    boolFiber_headTail,boolFiber_headTail] at h
+  simp only [Equiv.symm_apply_apply] at h
+  exact h
+
+noncomputable def headAverage {d : ℕ} (p : Card (d+1) → ℝ) (y : Card d) : ℝ :=
+  (p (Fin.cons false y)+p (Fin.cons true y))/2
+
+lemma finiteMean_headAverage (d : ℕ) (p : Card (d+1) → ℝ) :
+    finiteMean (headAverage p) = finiteMean p := by
+  rw [finiteMean_equiv (headTailEquiv d),finiteMean_prod,finiteMean_comm]
+  apply finiteMean_congr
+  intro y
+  rw [finiteMean_bool]
+  rfl
+
+lemma pairSwitch_product_bound (d : ℕ) (S : Finset (Card (d+1)))
+    (p : Card (d+1) → ℝ) (hp : ∀ x, 0 ≤ p x) :
+    finiteMean (fun ξ : Card d → Bool => ∏ x ∈ S, p (pairSwitch ξ x)) ≤
+      ∏ x ∈ S, headAverage p (Fin.tail x) := by
+  classical
+  have h : CylinderBound (fun T => ∏ x ∈ T, p ((headTailEquiv d).symm x))
+      (fun x => p ((headTailEquiv d).symm x)) := fun _ => le_rfl
+  have hh := h.fair_parallel (fun x => hp _) (S.image (headTailEquiv d))
+  simp only [Finset.prod_image (pairLayer _).injective.injOn,
+    Finset.prod_image (headTailEquiv d).injective.injOn] at hh
+  convert hh using 1
+  · apply finiteMean_congr
+    intro ξ
+    apply Finset.prod_congr rfl
+    intro x _
+    rfl
+  · apply Finset.prod_congr rfl
+    intro x _
+    change (p (Fin.cons false (Fin.tail x))+p (Fin.cons true (Fin.tail x)))/2 =
+      (p (Fin.cons (x 0) (Fin.tail x))+p (Fin.cons (!(x 0)) (Fin.tail x)))/2
+    cases x 0
+    · rfl
+    · simp only [Bool.not_true]
+      ring
+
+lemma butterflyPerm_step (d : ℕ)
+    (c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (ξ : Card d → Bool) :
+    butterflyPerm (d+1) (decodeButterfly (d+1) ((coinStepEquiv d).symm (c,ξ))) =
+    pairSwitch ξ * childLift (fun b => butterflyPerm d (decodeButterfly d (if b then c.2 else c.1))) := by
+  change pairSwitch (fun y => (coinStepEquiv d).symm (c,ξ) (Sum.inl y)) *
+    childLift (fun b => butterflyPerm d (decodeButterfly d
+      (fun i => (coinStepEquiv d).symm (c,ξ) (Sum.inr (b,i))))) = _
+  apply congrArg₂ (· * ·)
+  · rfl
+  · apply congrArg childLift
+    funext b
+    apply congrArg (fun ω => butterflyPerm d (decodeButterfly d ω))
+    funext i
+    cases b <;> rfl
+
+lemma butterfly_product_bound (d : ℕ) (S : Finset (Card d))
+    (p : Card d → ℝ) (hp : ∀ x, 0 ≤ p x) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      ∏ x ∈ S, p (butterflyPerm d (decodeButterfly d ω) x)) ≤ (finiteMean p)^S.card := by
+  classical
+  induction d with
+  | zero =>
+    have he : p = fun _ => p (fun i => Fin.elim0 i) := by
+      funext x
+      exact congrArg p (Subsingleton.elim _ _)
+    rw [he]
+    simp only [Finset.prod_const,finiteMean_const,le_refl]
+  | succ d ih =>
+    let g : Card d → ℝ := headAverage p
+    have hg : ∀ x, 0 ≤ g x := fun x => div_nonneg (add_nonneg (hp _) (hp _)) (by norm_num)
+    rw [finiteMean_equiv (coinStepEquiv d),finiteMean_prod]
+    calc
+      _ ≤ finiteMean (fun c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool) =>
+        (∏ x ∈ childMarks S false, g (butterflyPerm d (decodeButterfly d c.1) x)) *
+        ∏ x ∈ childMarks S true, g (butterflyPerm d (decodeButterfly d c.2) x)) := by
+        apply finiteMean_mono
+        intro c
+        let f : Equiv.Perm (Card (d+1)) := childLift (fun b =>
+          butterflyPerm d (decodeButterfly d (if b then c.2 else c.1)))
+        have hh := pairSwitch_product_bound d (S.image f) p hp
+        simp only [Finset.prod_image f.injective.injOn] at hh
+        convert hh using 1
+        · apply finiteMean_congr
+          intro ξ
+          apply Finset.prod_congr rfl
+          intro x _
+          rw [butterflyPerm_step]
+          rfl
+        · rw [prod_childMarks]
+          simp only [f,childLift,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
+            Bool.false_eq_true,↓reduceIte]
+          rfl
+      _ = (finiteMean (fun c : SwitchIndex d → Bool =>
+        ∏ x ∈ childMarks S false, g (butterflyPerm d (decodeButterfly d c) x))) *
+        finiteMean (fun c : SwitchIndex d → Bool =>
+        ∏ x ∈ childMarks S true, g (butterflyPerm d (decodeButterfly d c) x)) :=
+          finiteMean_prod_mul
+            (fun c : SwitchIndex d → Bool => ∏ x ∈ childMarks S false,
+              g (butterflyPerm d (decodeButterfly d c) x))
+            (fun c : SwitchIndex d → Bool => ∏ x ∈ childMarks S true,
+              g (butterflyPerm d (decodeButterfly d c) x))
+      _ ≤ (finiteMean g)^(childMarks S false).card * (finiteMean g)^(childMarks S true).card :=
+        mul_le_mul (ih (childMarks S false) g hg) (ih (childMarks S true) g hg)
+          (finiteMean_nonneg fun c => Finset.prod_nonneg (fun x _ => hg _))
+          (pow_nonneg (finiteMean_nonneg hg) _)
+      _ = (finiteMean p)^S.card := by
+        rw [←pow_add,card_childMarks_add,finiteMean_headAverage]
+
+noncomputable def inverseButterflyMarks (d : ℕ) (A : Finset (Card d))
+    (ω : SwitchIndex d → Bool) : Finset (Card d) :=
+  A.image (butterflyPerm d (decodeButterfly d ω)).symm
+
+lemma inverseButterflyMarks_inclusion (d : ℕ) (A S : Finset (Card d)) :
+    inclusionMoment (inverseButterflyMarks d A) S ≤
+      ((A.card : ℝ)/(2:ℝ)^d)^S.card := by
+  classical
+  let p : Card d → ℝ := fun x => if x ∈ A then 1 else 0
+  have hm : finiteMean p = (A.card : ℝ)/(2:ℝ)^d := by
+    simp only [finiteMean,p,card_positions,Nat.cast_pow,Nat.cast_ofNat]
+    congr 1
+    simp
+  have he : inclusionMoment (inverseButterflyMarks d A) S =
+      finiteMean (fun ω : SwitchIndex d → Bool =>
+        ∏ x ∈ S, p (butterflyPerm d (decodeButterfly d ω) x)) := by
+    rw [inclusionMoment_eq]
+    apply finiteMean_congr
+    intro ω
+    simp only [inverseButterflyMarks,subset_image_equiv_iff,Equiv.symm_symm,indicator_subset_prod,
+      Finset.prod_image (butterflyPerm d (decodeButterfly d ω)).injective.injOn,p]
+  rw [he]
+  simpa only [hm] using butterfly_product_bound d S p (fun x => by dsimp [p]; split_ifs <;> norm_num)
+
+namespace PairRouting
+
+section
+variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
+
+noncomputable def occupied (x : ι → Bool × α) : Finset α :=
+  Finset.univ.image (fun i => (x i).2)
+
+omit [Fintype α] in
+lemma mem_occupied (x : ι → Bool × α) (y : α) : y ∈ occupied x ↔ ∃ i, (x i).2 = y := by
+  simp [occupied]
+
+noncomputable def chosenCoin (x : ι → Bool × α) (c : ι → Bool) (y : α) : Bool := by
+  classical
+  exact if h : ∃ i, (x i).2 = y then required x c (Classical.choose h) else false
+
+omit [Fintype α] in
+lemma chosenCoin_apply (x : ι → Bool × α) (c : ι → Bool) (hc : Compatible x c) (i : ι) :
+    chosenCoin x c (x i).2 = required x c i := by
+  classical
+  simp only [chosenCoin,dite_eq_left (show ∃ j, (x j).2 = (x i).2 from ⟨i,rfl⟩)]
+  exact hc _ i (Classical.choose_spec (show ∃ j, (x j).2 = (x i).2 from ⟨i,rfl⟩))
+
+omit [Fintype α] in
+lemma colors_eq_iff (x : ι → Bool × α) (c : ι → Bool) (hc : Compatible x c) (ξ : α → Bool) :
+    colors x ξ = c ↔ ∀ y ∈ occupied x, ξ y = chosenCoin x c y := by
+  constructor
+  · intro he y hy
+    obtain ⟨i,rfl⟩ := (mem_occupied x y).mp hy
+    rw [chosenCoin_apply x c hc i]
+    rw [←he,required_colors]
+  · intro he
+    funext i
+    apply (xor_eq_iff _ _ _).mpr
+    exact (he _ ((mem_occupied x _).mpr ⟨i,rfl⟩)).trans (chosenCoin_apply x c hc i)
+
+noncomputable instance (x : ι → Bool × α) (c : ι → Bool) : Decidable (Compatible x c) :=
+  Classical.propDecidable _
+
+lemma colors_probability (x : ι → Bool × α) (c : ι → Bool) :
+    finiteMean (fun ξ : α → Bool => if colors x ξ = c then (1 : ℝ) else 0) =
+      if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0 := by
+  classical
+  by_cases hc : Compatible x c
+  · rw [ite_eq_left hc]
+    have he := fun ξ => colors_eq_iff x c hc ξ
+    simp only [he]
+    exact mean_coinCylinder (occupied x) (chosenCoin x c)
+  · rw [ite_eq_right hc]
+    have he : ∀ ξ, colors x ξ ≠ c := fun ξ hh => hc (hh ▸ colors_compatible x ξ)
+    simp only [he,ite_false,finiteMean_const]
+
+
+def lift (p : Bool → Equiv.Perm α) : Equiv.Perm (Bool × α) where
+  toFun x := (x.1,p x.1 x.2)
+  invFun x := (x.1,(p x.1).symm x.2)
+  left_inv x := by simp
+  right_inv x := by simp
+
+def sandwich (ξ η : α → Bool) (p : Bool → Equiv.Perm α) : Equiv.Perm (Bool × α) :=
+  pairLayer η * lift p * pairLayer ξ
+
+def ChildMatches (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) : Prop :=
+  ∀ i, p (c i) (x i).2 = (y i).2
+
+noncomputable instance (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
+    Decidable (ChildMatches p x y c) := Classical.propDecidable _
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma sandwich_matches_iff (ξ η : α → Bool) (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) :
+    (∀ i, sandwich ξ η p (x i) = y i) ↔
+      colors x ξ = colors y η ∧ ChildMatches p x y (colors x ξ) := by
+  have he (i : ι) : sandwich ξ η p (x i) = y i ↔
+      colors x ξ i = colors y η i ∧ p (colors x ξ i) (x i).2 = (y i).2 := by
+    change pairLayer η (lift p (pairLayer ξ (x i))) = y i ↔ _
+    rw [←Equiv.eq_symm_apply]
+    change (colors x ξ i,p (colors x ξ i) (x i).2) = (colors y η i,(y i).2) ↔ _
+    exact ⟨Prod.mk.inj,fun h => Prod.ext h.1 h.2⟩
+  simp only [he,forall_and,←funext_iff,ChildMatches]
+
+lemma sandwich_probability [DecidableEq ι] (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) :
+    finiteMean (fun coins : (α → Bool) × (α → Bool) =>
+      if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then (1:ℝ) else 0) =
+    ∑ c : ι → Bool,
+      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
+      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) *
+      (if ChildMatches p x y c then 1 else 0) := by
+  classical
+  have he (coins : (α → Bool) × (α → Bool)) :
+      (if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then (1:ℝ) else 0) =
+      ∑ c : ι → Bool, (if colors x coins.1 = c then (1:ℝ) else 0) *
+        (if colors y coins.2 = c then 1 else 0) * (if ChildMatches p x y c then 1 else 0) := by
+    rw [Finset.sum_eq_single (colors x coins.1)]
+    · by_cases hc : colors x coins.1 = colors y coins.2
+      · simp only [sandwich_matches_iff,hc,true_and,↓reduceIte,one_mul]
+      · simp only [sandwich_matches_iff,hc,false_and,↓reduceIte,one_mul,Ne.symm hc,zero_mul]
+    · intro c _ hc
+      simp only [Ne.symm hc,ite_false,zero_mul]
+    · simp
+  simp_rw [he]
+  rw [finiteMean_sum]
+  apply Finset.sum_congr rfl
+  intro c _
+  rw [finiteMean_mul_const]
+  rw [finiteMean_prod_mul
+    (fun ξ : α → Bool => if colors x ξ = c then (1:ℝ) else 0)
+    (fun η : α → Bool => if colors y η = c then (1:ℝ) else 0),
+    colors_probability,colors_probability]
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma childMatches_split (p₀ p₁ : Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
+    ChildMatches (fun b => if b then p₁ else p₀) x y c ↔
+      (∀ i : {i // c i = false}, p₀ (x i.val).2 = (y i.val).2) ∧
+      (∀ i : {i // c i = true}, p₁ (x i.val).2 = (y i.val).2) := by
+  constructor
+  · intro h
+    constructor
+    · intro i
+      simpa only [i.property,Bool.false_eq_true,↓reduceIte] using h i.val
+    · intro i
+      simpa only [i.property,↓reduceIte] using h i.val
+  · rintro ⟨h₀,h₁⟩ i
+    cases hi : c i
+    · simpa only [hi,Bool.false_eq_true,↓reduceIte] using h₀ ⟨i,hi⟩
+    · simpa only [hi,↓reduceIte] using h₁ ⟨i,hi⟩
+
+noncomputable def tupleProbability {Ω β : Type*} [Fintype Ω] [DecidableEq β]
+    (P : Ω → Equiv.Perm β) (x y : ι → β) : ℝ := by
+  classical
+  exact finiteMean (fun ω => if ∀ i, P ω (x i) = y i then 1 else 0)
+
+omit [Fintype α] in
+lemma childMatches_mean {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
+    (P : Ω → Equiv.Perm α) (Q : Ω' → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
+    finiteMean (fun ω : Ω × Ω' =>
+      if ChildMatches (fun b => if b then Q ω.2 else P ω.1) x y c then (1:ℝ) else 0) =
+      tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
+        (fun i : {i // c i = false} => (y i.val).2) *
+      tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
+        (fun i : {i // c i = true} => (y i.val).2) := by
+  classical
+  let f : Ω → ℝ := fun ω => if ∀ i : {i // c i = false}, P ω (x i.val).2 = (y i.val).2 then 1 else 0
+  let g : Ω' → ℝ := fun ω => if ∀ i : {i // c i = true}, Q ω (x i.val).2 = (y i.val).2 then 1 else 0
+  have he (ω : Ω × Ω') :
+      (if ChildMatches (fun b => if b then Q ω.2 else P ω.1) x y c then (1:ℝ) else 0) = f ω.1*g ω.2 := by
+    simp only [childMatches_split,f,g]
+    split_ifs <;> simp_all
+  simp_rw [he]
+  exact finiteMean_prod_mul f g
+
+lemma sandwich_mean [DecidableEq ι] {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
+    (P : Ω → Equiv.Perm α) (Q : Ω' → Equiv.Perm α) (x y : ι → Bool × α) :
+    finiteMean (fun ω : Ω × Ω' => finiteMean (fun coins : (α → Bool) × (α → Bool) =>
+      if ∀ i, sandwich coins.1 coins.2 (fun b => if b then Q ω.2 else P ω.1) (x i) = y i
+      then (1:ℝ) else 0)) =
+    ∑ c : ι → Bool,
+      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
+      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) *
+      (tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
+        (fun i : {i // c i = false} => (y i.val).2) *
+      tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
+        (fun i : {i // c i = true} => (y i.val).2)) := by
+  classical
+  simp_rw [sandwich_probability]
+  rw [finiteMean_sum]
+  apply Finset.sum_congr rfl
+  intro c _
+  simp only [mul_comm ((if Compatible x c then _ else _) * _),finiteMean_mul_const,
+    childMatches_mean]
+
+
+abbrev BothAssignments (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :=
+  {c : ι → Bool // Compatible e c ∧ Compatible (fun i => lift p (e i)) c}
+
+noncomputable instance (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    Fintype (BothAssignments e p) := Fintype.ofFinite _
+
+noncomputable def assignmentColoring (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α)
+    (c : BothAssignments e p) : CycleColoring.OppositeColoring
+      (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e) := by
+  classical
+  refine ⟨Function.extend e c.val (fun _ => false),?_,?_⟩
+  · intro z hz
+    apply Function.extend_apply'
+    rintro ⟨i,rfl⟩
+    exact hz (Finset.mem_image.mpr ⟨i,Finset.mem_univ _,rfl⟩)
+  · intro z hz hnext
+    obtain ⟨i,_,hi⟩ := Finset.mem_image.mp hz
+    obtain ⟨j,_,hj⟩ := Finset.mem_image.mp hnext
+    have hfi := e.injective.extend_apply c.val (fun _ => false) i
+    have hfj := e.injective.extend_apply c.val (fun _ => false) j
+    rw [hi] at hfi
+    rw [hj] at hfj
+    rw [hfi,hfj]
+    rcases z with ⟨b,a⟩
+    cases b
+    · have hj' : e j = (true,a) := hj
+      have ht : (e i).2 = (e j).2 := by rw [hi,hj']
+      have hh := c.property.1 i j ht
+      simp only [required,hi,hj'] at hh
+      revert hh
+      generalize c.val i = bi
+      generalize c.val j = bj
+      cases bi <;> cases bj <;> decide
+    · have hj' : e j = (false,alternatingProjection p a) := hj
+      have ht : (lift p (e i)).2 = (lift p (e j)).2 := by
+        simp only [hi,hj',lift,Equiv.coe_fn_mk,alternatingProjection,Equiv.Perm.coe_mul,
+          Function.comp_apply,Equiv.apply_symm_apply]
+      have hh := c.property.2 i j ht
+      simp only [required,hi,hj',lift,Equiv.coe_fn_mk] at hh
+      revert hh
+      generalize c.val i = bi
+      generalize c.val j = bj
+      cases bi <;> cases bj <;> decide
+
+omit [Fintype α] in
+lemma assignmentColoring_injective (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    Function.Injective (assignmentColoring e p) := by
+  intro c c' he
+  apply Subtype.ext
+  funext i
+  have hh := congrFun (congrArg Subtype.val he) (e i)
+  change Function.extend e c.val (fun _ => false) (e i) =
+    Function.extend e c'.val (fun _ => false) (e i) at hh
+  simpa only [e.injective.extend_apply] using hh
+
+lemma bothAssignments_card (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    Fintype.card (BothAssignments e p) ≤
+      2 ^ ((boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true).card +
+        (boolFiber (labelSet e) false ∪ (boolFiber (labelSet e) true).image
+          (alternatingProjection p)).card -
+        ((boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card) +
+        Fintype.card (RoutingNetwork.SelectedCycle (alternatingProjection p) (boolFiber (labelSet e) false))) := by
+  have h := compatible_assignment_bound (alternatingProjection p)
+    (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
+  rw [childJoin_boolFiber] at h
+  exact (Fintype.card_le_of_injective _ (assignmentColoring_injective e p)).trans h
+
+omit [Fintype α] in
+lemma occupied_labelSet (e : ι ↪ Bool × α) : occupied e = (labelSet e).image Prod.snd := by
+  classical
+  simp only [occupied,labelSet,Finset.image_image,Function.comp_def]
+
+omit [Fintype ι] [Fintype α] in
+lemma image_snd_childJoin (A B : Finset α) : (childJoin A B).image Prod.snd = A ∪ B := by
+  classical
+  simp only [childJoin,Finset.image_union,Finset.image_image,Function.comp_def,Finset.image_id']
+
+omit [Fintype ι] [Fintype α] in
+lemma image_lift_childJoin (p : Bool → Equiv.Perm α) (A B : Finset α) :
+    (childJoin A B).image (fun x => (lift p x).2) = A.image (p false) ∪ B.image (p true) := by
+  classical
+  simp only [childJoin,Finset.image_union,Finset.image_image,Function.comp_def,lift,Equiv.coe_fn_mk]
+
+omit [Fintype α] in
+lemma occupied_lift_card (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    (occupied (fun i => lift p (e i))).card =
+      (boolFiber (labelSet e) false ∪ (boolFiber (labelSet e) true).image (alternatingProjection p)).card := by
+  classical
+  have he : occupied (fun i => lift p (e i)) = (labelSet e).image (fun x => (lift p x).2) := by
+    simp only [occupied,labelSet,Finset.image_image,Function.comp_def]
+  rw [he,←childJoin_boolFiber (labelSet e),image_lift_childJoin]
+  simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
+  let A := boolFiber (labelSet e) false
+  let B := boolFiber (labelSet e) true
+  change (A.image (p false) ∪ B.image (p true)).card = (A ∪ B.image (alternatingProjection p)).card
+  have hh : A.image (p false) ∪ B.image (p true) =
+      (A ∪ B.image (alternatingProjection p)).image (p false) := by
+    simp only [Finset.image_union,Finset.image_image,Function.comp_def,
+      alternatingProjection,Equiv.Perm.coe_mul,Equiv.apply_symm_apply]
+  rw [hh,Finset.card_image_of_injective _ (p false).injective]
+
+lemma bothAssignments_occupied_bound (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    Fintype.card (BothAssignments e p) ≤
+      2 ^ ((occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι +
+        Fintype.card (RoutingNetwork.SelectedCycle (alternatingProjection p) (boolFiber (labelSet e) false))) := by
+  have he : (occupied e).card =
+      (boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true).card := by
+    rw [occupied_labelSet,←childJoin_boolFiber (labelSet e),image_snd_childJoin]
+    simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
+  have hc : (boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card = Fintype.card ι := by
+    rw [←card_childJoin,childJoin_boolFiber,labelSet,Finset.card_image_of_injective _ e.injective,
+      Finset.card_univ]
+  simpa only [he,occupied_lift_card,←hc] using bothAssignments_card e p
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma compatible_pairLayer_iff (x : ι → Bool × α) (c : ι → Bool) (ξ : α → Bool) :
+    Compatible (fun i => pairLayer ξ (x i)) c ↔ Compatible x c := by
+  have he (i j : ι) (ht : (x i).2 = (x j).2) :
+      required (fun i => pairLayer ξ (x i)) c i = required (fun i => pairLayer ξ (x i)) c j ↔
+      required x c i = required x c j := by
+    simp only [required,pairLayer,Equiv.coe_fn_mk,ht]
+    cases (x i).1 <;> cases (x j).1 <;> cases c i <;> cases c j <;> cases ξ (x j).2 <;> decide
+  constructor
+  · intro h i j ht
+    exact (he i j ht).mp (h i j ht)
+  · intro h i j ht
+    exact (he i j ht).mpr (h i j ht)
+
+
+omit [Fintype α] in
+lemma occupied_eq_fibers (e : ι ↪ Bool × α) :
+    occupied e = boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true := by
+  rw [occupied_labelSet,←childJoin_boolFiber (labelSet e),image_snd_childJoin]
+  simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
+
+omit [Fintype α] in
+lemma card_labelSet (e : ι ↪ Bool × α) : (labelSet e).card = Fintype.card ι := by
+  classical
+  exact (Finset.card_image_of_injective _ e.injective).trans (Finset.card_univ)
+
+omit [Fintype α] in
+lemma occupied_card_double (e : ι ↪ Bool × α) : Fintype.card ι ≤ 2*(occupied e).card := by
+  have h₀ : (boolFiber (labelSet e) false).card ≤ (occupied e).card := by
+    rw [occupied_eq_fibers]
+    exact Finset.card_le_card Finset.subset_union_left
+  have h₁ : (boolFiber (labelSet e) true).card ≤ (occupied e).card := by
+    rw [occupied_eq_fibers]
+    exact Finset.card_le_card Finset.subset_union_right
+  have he : (boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card = Fintype.card ι := by
+    rw [←card_childJoin,childJoin_boolFiber,card_labelSet]
+  omega
+
+lemma bothAssignments_exact_bound (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    Fintype.card (BothAssignments e p) ≤
+      2 ^ ((occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι +
+        alternatingCycles e p) := by
+  have h := CycleColoring.card_opposite_le
+    (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e)
+    Prod.fst (CycleColoring.alternatingPerm_flip (alternatingProjection p))
+  have hb : (CycleColoring.boundary (CycleColoring.alternatingPerm (alternatingProjection p))
+      (labelSet e)).card =
+      (occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι := by
+    have hh := alternating_boundary_card (alternatingProjection p)
+      (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
+    rw [childJoin_boolFiber] at hh
+    rw [occupied_eq_fibers,occupied_lift_card]
+    have he := card_childJoin (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
+    rw [childJoin_boolFiber,card_labelSet] at he
+    simpa only [he] using hh
+  rw [hb] at h
+  exact (Fintype.card_le_of_injective _ (assignmentColoring_injective e p)).trans h
+
+abbrev Assignments (x y : ι → Bool × α) :=
+  {c : ι → Bool // Compatible x c ∧ Compatible y c}
+
+noncomputable instance (x y : ι → Bool × α) : Fintype (Assignments x y) := Fintype.ofFinite _
+
+def routedEmbedding (e : ι ↪ Bool × α) (ξ η : α → Bool) (p : Bool → Equiv.Perm α) :
+    ι ↪ Bool × α := e.trans (sandwich ξ η p).toEmbedding
+
+omit [Fintype α] in
+lemma occupied_pairLayer (x : ι → Bool × α) (ξ : α → Bool) :
+    occupied (fun i => pairLayer ξ (x i)) = occupied x := rfl
+
+noncomputable def assignmentsEquiv (e : ι ↪ Bool × α) (ξ η : α → Bool)
+    (p : Bool → Equiv.Perm α) :
+    Assignments e (routedEmbedding e ξ η p) ≃ BothAssignments (switchedEmbedding e ξ) p := by
+  apply Equiv.subtypeEquivRight
+  intro c
+  change Compatible e c ∧ Compatible (fun i => pairLayer η (lift p (pairLayer ξ (e i)))) c ↔ _
+  rw [compatible_pairLayer_iff]
+  exact and_congr (compatible_pairLayer_iff e c ξ).symm Iff.rfl
+
+lemma assignments_actual_bound (e : ι ↪ Bool × α) (ξ η : α → Bool) (p : Bool → Equiv.Perm α) :
+    Fintype.card (Assignments e (routedEmbedding e ξ η p)) ≤
+      2 ^ ((occupied e).card + (occupied (routedEmbedding e ξ η p)).card - Fintype.card ι +
+        alternatingCycles (switchedEmbedding e ξ) p) := by
+  rw [Fintype.card_congr (assignmentsEquiv e ξ η p)]
+  exact bothAssignments_exact_bound (switchedEmbedding e ξ) p
+
+omit [Fintype α] in
+lemma assignments_actual_exponent_nonneg (e : ι ↪ Bool × α) (ξ η : α → Bool)
+    (p : Bool → Equiv.Perm α) :
+    Fintype.card ι ≤ (occupied e).card + (occupied (routedEmbedding e ξ η p)).card := by
+  have h₀ := occupied_card_double e
+  have h₁ := occupied_card_double (routedEmbedding e ξ η p)
+  omega
+
+end
+variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α] [DecidableEq ι]
+
+noncomputable def compatibleSet (x y : ι → Bool × α) : Finset (ι → Bool) := by
+  classical
+  exact Finset.univ.filter (fun c => Compatible x c ∧ Compatible y c)
+
+omit [Fintype α] [DecidableEq α] in
+lemma mem_compatibleSet (x y : ι → Bool × α) (c : ι → Bool) :
+    c ∈ compatibleSet x y ↔ Compatible x c ∧ Compatible y c := by
+  classical
+  simp only [compatibleSet,Finset.mem_filter,Finset.mem_univ,true_and]
+
+omit [Fintype α] [DecidableEq α] in
+lemma card_compatibleSet (x y : ι → Bool × α) :
+    (compatibleSet x y).card = Fintype.card (Assignments x y) := by
+  classical
+  exact (Fintype.card_subtype _).symm
+
+noncomputable def outerWeight (x y : ι → Bool × α) : ℝ :=
+  1/(2:ℝ)^((occupied x).card+(occupied y).card)
+
+omit [Fintype α] [DecidableEq ι] in
+lemma outerWeight_pos (x y : ι → Bool × α) : 0 < outerWeight x y := by
+  unfold outerWeight
+  positivity
+
+omit [Fintype α] in
+lemma sum_colors_factor (x y : ι → Bool × α) (f : (ι → Bool) → ℝ) :
+    (∑ c : ι → Bool,
+      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
+      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) * f c) =
+      outerWeight x y * ∑ c ∈ compatibleSet x y, f c := by
+  classical
+  rw [compatibleSet,Finset.sum_filter,Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro c _
+  by_cases hx : Compatible x c <;> by_cases hy : Compatible y c
+  · simp only [hx,hy,true_and,↓reduceIte,outerWeight,pow_add,one_div_mul_one_div]
+  · simp only [hx,hy,and_false,↓reduceIte,mul_zero,zero_mul]
+  · simp only [hx,hy,false_and,↓reduceIte,mul_zero,zero_mul]
+  · simp only [hx,hy,false_and,↓reduceIte,mul_zero,zero_mul]
+
+lemma sandwich_weighted_probability (p : Bool → Equiv.Perm α) (x y : ι → Bool × α)
+    (W : (ι → Bool) → ℝ) :
+    finiteMean (fun coins : (α → Bool) × (α → Bool) =>
+      if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then W (colors x coins.1) else 0) =
+    outerWeight x y * ∑ c ∈ compatibleSet x y, if ChildMatches p x y c then W c else 0 := by
+  classical
+  have he (coins : (α → Bool) × (α → Bool)) :
+      (if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then W (colors x coins.1) else 0) =
+      ∑ c : ι → Bool, (if colors x coins.1 = c then (1:ℝ) else 0) *
+        (if colors y coins.2 = c then 1 else 0) * (if ChildMatches p x y c then W c else 0) := by
+    rw [Finset.sum_eq_single (colors x coins.1)]
+    · by_cases hc : colors x coins.1 = colors y coins.2
+      · simp only [sandwich_matches_iff,hc,true_and,↓reduceIte,one_mul]
+      · simp only [sandwich_matches_iff,hc,false_and,↓reduceIte,one_mul,Ne.symm hc,zero_mul]
+    · intro c _ hc
+      simp only [Ne.symm hc,ite_false,zero_mul]
+    · simp
+  simp_rw [he]
+  rw [finiteMean_sum,←sum_colors_factor]
+  apply Finset.sum_congr rfl
+  intro c _
+  rw [finiteMean_mul_const,finiteMean_prod_mul
+    (fun ξ : α → Bool => if colors x ξ = c then (1:ℝ) else 0)
+    (fun η : α → Bool => if colors y η = c then (1:ℝ) else 0),
+    colors_probability,colors_probability]
+
+omit [Fintype α] [DecidableEq ι] in
+lemma tupleProbability_nonneg {Ω β : Type*} [Fintype Ω] [DecidableEq β]
+    (P : Ω → Equiv.Perm β) (x y : ι → β) : 0 ≤ tupleProbability P x y := by
+  classical
+  exact finiteMean_nonneg (fun ω => by split_ifs <;> norm_num)
+
+end PairRouting
+end Thorp
+
+end ThorpNine.HighTail
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/HighTail/RotationMeeting.lean b/lean/OAI/Probability/ThorpRouting/HighTail/RotationMeeting.lean
index 9872557b8cbec7ce3299c57ff88835aa68fe5873..3a64b6d886ce3749dc195ddb8a9d5e67243d0aec 100644
--- a/lean/OAI/Probability/ThorpRouting/HighTail/RotationMeeting.lean
+++ b/lean/OAI/Probability/ThorpRouting/HighTail/RotationMeeting.lean
@@ -1,878 +1,878 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.HighTail.ExposedCoins
-
-namespace OAI
-
-namespace ThorpNine.HighTail
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-section Rotation
-variable {d A : ℕ}
-
-noncomputable def rotationMeeting (o : Card d → Option (Fin A)) (z : Terminal o)
-    (u v : Card d) : ℝ :=
-  match o u, o v with
-  | some i, some j => if i < j then (if (z j).val = v then 1 else 0)
-      else if j < i then (if (z i).val = u then 1 else 0)
-      else (if (z i).val = u then 1 else 0) + (if (z i).val = v then 1 else 0)
-  | _, _ => 0
-
-lemma card_cycleFiber (o : Card d → Option (Fin A)) (i : Fin A) :
-    Fintype.card {x : Card d // o x = some i} = (cycleFiber o i).card := by
-  classical
-  simp [cycleFiber,Fintype.card_subtype]
-
-lemma terminal_mean {o : Card d → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
-    (i : Fin A) (x : Card d) (hx : o x = some i) :
-    finiteMean (fun z : Terminal o => if (z i).val = x then (1:ℝ) else 0) = slotWeight o i := by
-  classical
-  have hev := finiteMean_pi_eval (A := fun i : Fin A => {x : Card d // o x = some i})
-    i (fun z => if z.val = x then (1:ℝ) else 0)
-  rw [hev]
-  have he : (fun z : {x : Card d // o x = some i} => if z.val = x then (1:ℝ) else 0) =
-      (fun z => if z = ⟨x,hx⟩ then 1 else 0) := by
-    funext z
-    simp only [Subtype.ext_iff]
-  rw [he,finiteMean_eq_indicator,card_cycleFiber,slotWeight]
-
-lemma slotWeight_antitone (o : Card d → Option (Fin A))
-    (hn : ∀ i, (cycleFiber o i).Nonempty)
-    (hm : Monotone (fun i => (cycleFiber o i).card)) : Antitone (slotWeight o) := by
-  intro i j hij
-  unfold slotWeight
-  apply one_div_le_one_div_of_le
-  · exact_mod_cast Finset.card_pos.mpr (hn i)
-  · exact_mod_cast hm hij
-
-lemma rotationMeeting_mean {o : Card d → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
-    (hm : Monotone (fun i => (cycleFiber o i).card)) (u v : Card d) :
-    finiteMean (fun z : Terminal o => rotationMeeting o z u v) =
-      min (pathWeight o u) (pathWeight o v) + withinWeight o u v := by
-  classical
-  have hn : ∀ i, (cycleFiber o i).Nonempty := by
-    intro i
-    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : Card d // o x = some i})
-    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
-  have hw := slotWeight_antitone o hn hm
-  cases hu : o u with
-  | none =>
-    cases hv : o v with
-    | none => simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean]
-    | some j =>
-      have hj : 0 ≤ slotWeight o j := by unfold slotWeight; positivity
-      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_left hj]
-  | some i =>
-    cases hv : o v with
-    | none =>
-      have hi : 0 ≤ slotWeight o i := by unfold slotWeight; positivity
-      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_right hi]
-    | some j =>
-      simp only [rotationMeeting,pathWeight,withinWeight,hu,hv]
-      rcases lt_trichotomy i j with hij | rfl | hji
-      · simp only [hij,↓reduceIte,ne_of_lt hij]
-        rw [terminal_mean j v hv,min_eq_right (hw (le_of_lt hij)),add_zero]
-      · simp only [lt_self_iff_false,↓reduceIte,min_self]
-        rw [finiteMean_add,terminal_mean i u hu,terminal_mean i v hv]
-      · simp only [not_lt_of_ge (le_of_lt hji),hji,↓reduceIte,ne_of_gt hji]
-        rw [terminal_mean i u hu,min_eq_left (hw (le_of_lt hji)),add_zero]
-
-lemma pathWeight_sum (o : Card d → Option (Fin A)) (hn : ∀ i, (cycleFiber o i).Nonempty) :
-    ∑ x : Card d, pathWeight o x = (A : ℝ) := by
-  classical
-  have he (x : Card d) : pathWeight o x =
-      ∑ i : Fin A, if o x = some i then slotWeight o i else 0 := by
-    cases hx : o x with
-    | none => simp [pathWeight,hx]
-    | some j => simp [pathWeight,hx]
-  simp only [he]
-  rw [Finset.sum_comm]
-  have hf (i : Fin A) : (∑ x : Card d, if o x = some i then slotWeight o i else 0) = 1 := by
-    rw [← Finset.sum_filter]
-    change (∑ _x ∈ cycleFiber o i, slotWeight o i) = 1
-    simp only [Finset.sum_const,nsmul_eq_mul,slotWeight]
-    exact mul_one_div_cancel (by exact_mod_cast (Finset.card_pos.mpr (hn i)).ne')
-  simp only [hf,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,mul_one]
-
-lemma withinWeight_sum_bound (o : Card d → Option (Fin A))
-    (hn : ∀ i, (cycleFiber o i).Nonempty) (b : Butterfly d) :
-    (∑ s : SwitchIndex d, withinWeight o (switchUsers d b s).1 (switchUsers d b s).2) ≤
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
-  classical
-  have he (u v : Card d) : withinWeight o u v =
-      ∑ i : Fin A, (if u ∈ cycleFiber o i ∧ v ∈ cycleFiber o i then (1:ℝ) else 0) * slotWeight o i := by
-    simp only [mem_cycleFiber]
-    cases hu : o u <;> cases hv : o v <;> simp [withinWeight,hu,hv,ite_and,eq_comm]
-  simp only [he]
-  rw [Finset.sum_comm]
-  apply Finset.sum_le_sum
-  intro i _
-  rw [← Finset.sum_mul]
-  have hi := marked_switch_entropy_bound d b (cycleFiber o i)
-  have hpos : (0:ℝ) < (cycleFiber o i).card := by exact_mod_cast Finset.card_pos.mpr (hn i)
-  have hmul := mul_le_mul_of_nonneg_right hi (show 0 ≤ slotWeight o i by unfold slotWeight; positivity)
-  calc
-    _ ≤ ((cycleFiber o i).card : ℝ) * Real.log (cycleFiber o i).card /
-        (2 * Real.log 2) * slotWeight o i := hmul
-    _ = _ := by unfold slotWeight; field_simp
-
-noncomputable def rotationCharge (o : Card d → Option (Fin A)) (b : Butterfly d)
-    (z : Terminal o) : ℝ :=
-  ∑ s : SwitchIndex d, rotationMeeting o z (switchUsers d b s).1 (switchUsers d b s).2
-
-lemma rotationCharge_mean_bound {o : Card d → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
-    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : Butterfly d) :
-    finiteMean (rotationCharge o b) ≤ (A:ℝ) * d / 2 +
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
-  classical
-  have hn : ∀ i, (cycleFiber o i).Nonempty := by
-    intro i
-    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : Card d // o x = some i})
-    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
-  unfold rotationCharge
-  rw [finiteMean_sum]
-  simp only [rotationMeeting_mean hm,Finset.sum_add_distrib]
-  have h₁ := weighted_switch_min_bound d b (pathWeight o)
-  rw [pathWeight_sum o hn] at h₁
-  have h₂ := withinWeight_sum_bound o hn b
-  linarith
-
-lemma rotation_exp_bound {o : Card d → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
-    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : Butterfly d) :
-    Real.exp (-Real.log 2 * ((A:ℝ)*d/2 +
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2))) ≤
-      finiteMean (fun z : Terminal o => Real.exp (-Real.log 2 * rotationCharge o b z)) := by
-  have hb := rotationCharge_mean_bound hm b
-  have hlog : 0 ≤ Real.log 2 := Real.log_nonneg (by norm_num)
-  have hm' : -Real.log 2 * ((A:ℝ)*d/2 +
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2)) ≤
-      finiteMean (fun z : Terminal o => -Real.log 2 * rotationCharge o b z) := by
-    have he := finiteMean_mul_const (rotationCharge o b) (-Real.log 2)
-    simp only [mul_comm] at he
-    rw [he]
-    exact mul_le_mul_of_nonpos_left hb (neg_nonpos.mpr hlog)
-  exact (Real.exp_le_exp.mpr hm').trans (exp_finiteMean_le _)
-
-end Rotation
-
-section OrbitCore
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-@[simp] lemma mem_orbitSet (p : Equiv.Perm α) (x y : α) :
-    y ∈ orbitSet p x ↔ p.SameCycle x y := by
-  classical
-  simp [orbitSet]
-
-lemma orbitSet_nonempty (p : Equiv.Perm α) (x : α) : (orbitSet p x).Nonempty :=
-  ⟨x,(mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.refl _ _)⟩
-
-lemma orbitSet_eq (p : Equiv.Perm α) {x y : α} (h : p.SameCycle x y) :
-    orbitSet p x = orbitSet p y := by
-  ext z
-  simp only [mem_orbitSet]
-  exact ⟨fun hz => h.symm.trans hz,fun hz => h.trans hz⟩
-
-@[simp] lemma orbitSet_apply (p : Equiv.Perm α) (x : α) : orbitSet p (p x) = orbitSet p x := by
-  ext y
-  simp only [mem_orbitSet,Equiv.Perm.sameCycle_apply_left]
-
-lemma orbitSet_eq_or_disjoint (p : Equiv.Perm α) (x y : α) :
-    orbitSet p x = orbitSet p y ∨ Disjoint (orbitSet p x) (orbitSet p y) := by
-  classical
-  by_cases h : p.SameCycle x y
-  · exact Or.inl (orbitSet_eq p h)
-  · right
-    apply Finset.disjoint_left.mpr
-    intro z hz hz'
-    exact h (((mem_orbitSet _ _ _).mp hz).trans ((mem_orbitSet _ _ _).mp hz').symm)
-
-omit [DecidableEq α] in
-lemma perm_minimalPeriod_pos (p : Equiv.Perm α) (x : α) :
-    0 < Function.minimalPeriod p x :=
-  Function.minimalPeriod_pos_of_mem_periodicPts (p.injective.mem_periodicPts x)
-
-lemma orbitSet_card (p : Equiv.Perm α) (x : α) :
-    (orbitSet p x).card = Function.minimalPeriod p x := by
-  classical
-  let f : Fin (Function.minimalPeriod p x) → {y // p.SameCycle x y} := fun n =>
-    ⟨p^[n.val] x,by rw [Equiv.Perm.iterate_eq_pow]; exact ⟨(n.val:ℤ),by simp⟩⟩
-  have hf : Function.Bijective f := by
-    constructor
-    · intro i j hij
-      apply Fin.ext
-      exact Function.iterate_injOn_Iio_minimalPeriod i.isLt j.isLt (congrArg Subtype.val hij)
-    · intro y
-      obtain ⟨n,hn⟩ := y.property.exists_nat_pow_eq
-      refine ⟨⟨n % Function.minimalPeriod p x, Nat.mod_lt _ (perm_minimalPeriod_pos p x)⟩,?_⟩
-      apply Subtype.ext
-      change p^[n % Function.minimalPeriod p x] x = y.val
-      rw [Function.iterate_mod_minimalPeriod_eq,Equiv.Perm.iterate_eq_pow,hn]
-  have hc := Fintype.card_congr (Equiv.ofBijective f hf)
-  simpa [orbitSet,Fintype.card_subtype] using hc.symm
-
-def walkSet (p : Equiv.Perm α) (r : ℕ) (x : α) : Finset α :=
-  (Finset.range r).image (fun n => p^[n] x)
-
-omit [Fintype α] in
-@[simp] lemma mem_walkSet (p : Equiv.Perm α) (r : ℕ) (x y : α) :
-    y ∈ walkSet p r x ↔ ∃ n < r, p^[n] x = y := by
-  simp [walkSet]
-
-omit [Fintype α] in
-@[simp] lemma walkSet_zero (p : Equiv.Perm α) (x : α) : walkSet p 0 x = ∅ := by
-  simp [walkSet]
-
-omit [Fintype α] in
-lemma walkSet_succ_last (p : Equiv.Perm α) (r : ℕ) (x : α) :
-    walkSet p (r+1) x = insert (p^[r] x) (walkSet p r x) := by
-  simp [walkSet,Finset.range_add_one]
-
-omit [Fintype α] in
-lemma walkSet_succ_first (p : Equiv.Perm α) (r : ℕ) (x : α) :
-    walkSet p (r+1) x = insert x (walkSet p r (p x)) := by
-  ext y
-  simp only [mem_walkSet,Finset.mem_insert]
-  constructor
-  · rintro ⟨n,hn,hny⟩
-    cases n with
-    | zero => exact Or.inl (by simpa using hny.symm)
-    | succ n => exact Or.inr ⟨n,by omega,by simpa only [Function.iterate_succ_apply] using hny⟩
-  · rintro (rfl | ⟨n,hn,hny⟩)
-    · exact ⟨0,by omega,rfl⟩
-    · exact ⟨n+1,by omega,by simpa only [Function.iterate_succ_apply] using hny⟩
-
-omit [Fintype α] in
-lemma not_mem_short_walk (p : Equiv.Perm α) (x : α) {r : ℕ}
-    (hr : r < Function.minimalPeriod p x) : x ∉ walkSet p r (p x) := by
-  rintro hx
-  obtain ⟨n,hn,hnx⟩ := (mem_walkSet _ _ _ _).mp hx
-  have hperiod : Function.IsPeriodicPt p (n+1) x := by
-    simpa only [Function.IsPeriodicPt,Function.IsFixedPt,Function.iterate_succ_apply] using hnx
-  have hp := hperiod.minimalPeriod_le (by omega : 0 < n+1)
-  omega
-
-lemma walkSet_period (p : Equiv.Perm α) (x : α) :
-    walkSet p (Function.minimalPeriod p x) x = orbitSet p x := by
-  ext y
-  simp only [mem_walkSet,mem_orbitSet]
-  constructor
-  · rintro ⟨n,_,rfl⟩
-    rw [Equiv.Perm.iterate_eq_pow]
-    exact ⟨(n:ℤ),by simp⟩
-  · intro h
-    obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
-    refine ⟨n % Function.minimalPeriod p x,Nat.mod_lt _ (perm_minimalPeriod_pos p x),?_⟩
-    rw [Function.iterate_mod_minimalPeriod_eq,Equiv.Perm.iterate_eq_pow,hn]
-
-omit [Fintype α] in
-lemma walkSet_card (p : Equiv.Perm α) (x : α) {r : ℕ}
-    (hr : r ≤ Function.minimalPeriod p x) : (walkSet p r x).card = r := by
-  rw [walkSet,Finset.card_image_of_injOn]
-  · exact Finset.card_range _
-  · intro i hi j hj hij
-    exact Function.iterate_injOn_Iio_minimalPeriod
-      (lt_of_lt_of_le (Finset.mem_range.mp hi) hr)
-      (lt_of_lt_of_le (Finset.mem_range.mp hj) hr) hij
-
-lemma walkSet_subset_orbit (p : Equiv.Perm α) (r : ℕ) (x : α) :
-    walkSet p r x ⊆ orbitSet p x := by
-  intro y hy
-  obtain ⟨n,_,rfl⟩ := (mem_walkSet _ _ _ _).mp hy
-  rw [mem_orbitSet,Equiv.Perm.iterate_eq_pow]
-  exact ⟨(n:ℤ),by simp⟩
-
-def cycleTrace (p : Equiv.Perm α) (S : Finset α) (a : α) :
-    ℕ → α → Finset α → Option (Finset α × α)
-  | 0, x, V => if x ∈ S ∧ x ∉ V ∧ p x = a then some (insert x V,x) else none
-  | r+1, x, V => if x ∈ S ∧ x ∉ V then cycleTrace p S a r (p x) (insert x V) else none
-
-lemma cycleTrace_walk (p : Equiv.Perm α) (S : Finset α) (r : ℕ) (x : α) (V : Finset α)
-    (hr : r < Function.minimalPeriod p x) (hS : walkSet p (r+1) x ⊆ S)
-    (hV : Disjoint V (walkSet p (r+1) x)) :
-    cycleTrace p S (p^[r+1] x) r x V = some (V ∪ walkSet p (r+1) x,p^[r] x) := by
-  induction r generalizing x V with
-  | zero =>
-    have hxS : x ∈ S := hS (by simp [walkSet_succ_first])
-    have hxV : x ∉ V := fun h => Finset.disjoint_left.mp hV h (by simp [walkSet_succ_first])
-    simp [cycleTrace,hxS,hxV,walkSet_succ_first]
-  | succ r ih =>
-    have hxS : x ∈ S := hS (by rw [walkSet_succ_first]; exact Finset.mem_insert_self _ _)
-    have hxV : x ∉ V := fun h => Finset.disjoint_left.mp hV h
-      (by rw [walkSet_succ_first]; exact Finset.mem_insert_self _ _)
-    have hr' : r < Function.minimalPeriod p (p x) := by
-      rw [Function.minimalPeriod_apply (p.injective.mem_periodicPts x)]
-      omega
-    have hS' : walkSet p (r+1) (p x) ⊆ S := by
-      intro y hy
-      apply hS
-      rw [walkSet_succ_first]
-      exact Finset.mem_insert_of_mem hy
-    have hV' : Disjoint (insert x V) (walkSet p (r+1) (p x)) := by
-      apply Finset.disjoint_left.mpr
-      intro y hy hy'
-      rcases Finset.mem_insert.mp hy with hy | hy
-      · subst y
-        exact not_mem_short_walk p x hr hy'
-      · apply Finset.disjoint_left.mp hV hy
-        rw [walkSet_succ_first]
-        exact Finset.mem_insert_of_mem hy'
-    rw [cycleTrace,ite_eq_left ⟨hxS,hxV⟩,Function.iterate_succ_apply p (r+1) x,
-      ih _ _ hr' hS' hV',walkSet_succ_first p (r+1) x,Function.iterate_succ_apply]
-    simp only [Finset.insert_union,Finset.union_insert]
-
-lemma cycleTrace_orbit (p : Equiv.Perm α) (S V : Finset α) (z : α)
-    (hS : orbitSet p z ⊆ S) (hV : Disjoint V (orbitSet p z)) :
-    cycleTrace p S (p z) (Function.minimalPeriod p z - 1) (p z) V =
-      some (V ∪ orbitSet p z,z) := by
-  have hpos := perm_minimalPeriod_pos p z
-  have he : Function.minimalPeriod p (p z) = Function.minimalPeriod p z :=
-    Function.minimalPeriod_apply (p.injective.mem_periodicPts z)
-  have hp : Function.minimalPeriod p z - 1 + 1 = Function.minimalPeriod p z := by omega
-  have hw : walkSet p (Function.minimalPeriod p z - 1 + 1) (p z) = orbitSet p z := by
-    rw [hp,←he,walkSet_period,orbitSet_apply]
-  have hr : Function.minimalPeriod p z - 1 < Function.minimalPeriod p (p z) := by rw [he]; omega
-  have h := cycleTrace_walk p S (Function.minimalPeriod p z-1) (p z) V hr
-    (by simpa only [hw] using hS) (by simpa only [hw] using hV)
-  have ht : p^[Function.minimalPeriod p z-1] (p z) = z := by
-    rw [← Function.iterate_succ_apply,show (Function.minimalPeriod p z-1).succ =
-      Function.minimalPeriod p z by omega,Function.iterate_minimalPeriod]
-  have ha : p^[Function.minimalPeriod p z-1+1] (p z) = p z := by
-    rw [hp,←he,Function.iterate_minimalPeriod]
-  simpa only [ha,ht,hw] using h
-
-end OrbitCore
-
-noncomputable def exposedPaths (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) :
-    Finset (SwitchIndex d) :=
-  V.biUnion (fun x => routeDomain d x (butterflyPerm d b x))
-
-@[simp] lemma exposedPaths_empty (d : ℕ) (b : Butterfly d) : exposedPaths d b ∅ = ∅ := by
-  classical
-  simp [exposedPaths]
-
-lemma exposedPaths_insert (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) (x : Card d) :
-    exposedPaths d b (insert x V) = exposedPaths d b V ∪ routeDomain d x (butterflyPerm d b x) := by
-  classical
-  simp [exposedPaths,Finset.union_comm]
-
-noncomputable def terminalCharge (d : ℕ) (b : Butterfly d) (z : Card d) (V : Finset (Card d)) : ℕ :=
-  (routeDomain d z (butterflyPerm d b z) ∩ exposedPaths d b V).card
-
-namespace QueryTree
-
-lemma followPath_stale_charge {ι : Type*} [DecidableEq ι]
-    (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
-    (k : Card d → QueryTree ι) (R : Finset ι) (ω : ι → Bool) :
-    staleCharge (followPath d e x k) R ω =
-      staleCharge (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x))
-        (R ∪ (routeDomain d x (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).image e) ω := by
-  have h₀ := fresh_add_stale (followPath d e x k) R ω
-  have h₁ := fresh_add_stale (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x))
-    (R ∪ (routeDomain d x (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).image e) ω
-  rw [followPath_charge,followPath_raw_charge] at h₀
-  omega
-
-lemma seekCycle_semantics (d : ℕ) (S : Finset (Card d)) (a : Card d) (r : ℕ)
-    (x : Card d) (V W : Finset (Card d)) (z : Card d)
-    (k : Finset (Card d) → QueryTree (SwitchIndex d)) (ω : SwitchIndex d → Bool)
-    (h : cycleTrace (butterflyPerm d (decodeButterfly d ω)) S a r x V = some (W,z)) :
-    (seekCycle d S a r x V k).succeeds ω = (k W).succeeds ω ∧
-      (seekCycle d S a r x V k).staleCharge (exposedPaths d (decodeButterfly d ω) V) ω =
-        (k W).staleCharge (exposedPaths d (decodeButterfly d ω) W) ω +
-          terminalCharge d (decodeButterfly d ω) z (W.erase z) := by
-  classical
-  induction r generalizing x V with
-  | zero =>
-    simp only [cycleTrace] at h
-    split_ifs at h with hv
-    obtain ⟨hxS,hxV,hxa⟩ := hv
-    obtain ⟨rfl,rfl⟩ := Prod.mk.inj (Option.some.inj h)
-    simp only [seekCycle,ite_eq_left (show x ∈ S ∧ x ∉ V from ⟨hxS,hxV⟩),closePath_succeeds,ite_eq_left hxa]
-    constructor
-    · trivial
-    · rw [closePath_stale_charge _ _ _ _ _ _ hxa,exposedPaths_insert,
-        Finset.erase_insert hxV]
-      simp only [terminalCharge,hxa]
-  | succ r ih =>
-    simp only [cycleTrace] at h
-    split_ifs at h with hv
-    obtain ⟨hxS,hxV⟩ := hv
-    rw [seekCycle,ite_eq_left ⟨hxS,hxV⟩,followPath_succeeds,followPath_stale_charge]
-    simp only [Function.comp_id,Finset.image_id]
-    rw [← exposedPaths_insert]
-    exact ih _ _ h
-
-lemma seekCycle_orbit (d : ℕ) (S V : Finset (Card d)) (z : Card d)
-    (k : Finset (Card d) → QueryTree (SwitchIndex d)) (ω : SwitchIndex d → Bool)
-    (hS : orbitSet (butterflyPerm d (decodeButterfly d ω)) z ⊆ S)
-    (hV : Disjoint V (orbitSet (butterflyPerm d (decodeButterfly d ω)) z)) :
-    let p := butterflyPerm d (decodeButterfly d ω)
-    let W := V ∪ orbitSet p z
-    (seekCycle d S (p z) (Function.minimalPeriod p z-1) (p z) V k).succeeds ω = (k W).succeeds ω ∧
-      (seekCycle d S (p z) (Function.minimalPeriod p z-1) (p z) V k).staleCharge
-          (exposedPaths d (decodeButterfly d ω) V) ω =
-        (k W).staleCharge (exposedPaths d (decodeButterfly d ω) W) ω +
-          terminalCharge d (decodeButterfly d ω) z (W.erase z) := by
-  exact seekCycle_semantics _ _ _ _ _ _ _ _ _ _ (cycleTrace_orbit _ _ _ _ hS hV)
-
-end QueryTree
-
-lemma mem_exposedPaths (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) (s : SwitchIndex d) :
-    s ∈ exposedPaths d b V ↔ (switchUsers d b s).1 ∈ V ∨ (switchUsers d b s).2 ∈ V := by
-  classical
-  simp only [exposedPaths,Finset.mem_biUnion,mem_routeDomain_iff_user]
-  constructor
-  · rintro ⟨x,hx,h | h⟩
-    · exact Or.inl (h ▸ hx)
-    · exact Or.inr (h ▸ hx)
-  · rintro (h | h)
-    · exact ⟨_,h,Or.inl rfl⟩
-    · exact ⟨_,h,Or.inr rfl⟩
-
-lemma terminalCharge_sum (d : ℕ) (b : Butterfly d) (z : Card d) (V : Finset (Card d))
-    (hz : z ∉ V) :
-    (terminalCharge d b z V : ℝ) = ∑ s : SwitchIndex d,
-      ((if (switchUsers d b s).1 = z ∧ (switchUsers d b s).2 ∈ V then (1:ℝ) else 0) +
-      (if (switchUsers d b s).2 = z ∧ (switchUsers d b s).1 ∈ V then (1:ℝ) else 0)) := by
-  classical
-  have he : routeDomain d z (butterflyPerm d b z) ∩ exposedPaths d b V =
-      Finset.univ.filter (fun s =>
-        ((switchUsers d b s).1 = z ∧ (switchUsers d b s).2 ∈ V) ∨
-        ((switchUsers d b s).2 = z ∧ (switchUsers d b s).1 ∈ V)) := by
-    ext s
-    simp only [Finset.mem_inter,mem_routeDomain_iff_user,mem_exposedPaths,
-      Finset.mem_filter,Finset.mem_univ,true_and]
-    by_cases h₁ : (switchUsers d b s).1 = z <;>
-      by_cases h₂ : (switchUsers d b s).2 = z <;> simp [h₁,h₂,hz]
-  rw [terminalCharge,he]
-  simp only [← Finset.sum_boole]
-  apply Finset.sum_congr rfl
-  intro s _
-  by_cases h₁ : (switchUsers d b s).1 = z <;>
-    by_cases h₂ : (switchUsers d b s).2 = z <;> simp [h₁,h₂,hz]
-
-noncomputable def priorPaths {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) : Finset (Card d) :=
-  Finset.univ.filter (fun x => ∃ i : Fin A, o x = some i ∧ i.val < t)
-
-@[simp] lemma mem_priorPaths {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) (x : Card d) :
-    x ∈ priorPaths o t ↔ ∃ i : Fin A, o x = some i ∧ i.val < t := by
-  classical
-  simp [priorPaths]
-
-lemma terminalCharge_prior_sum {d A : ℕ} (o : Card d → Option (Fin A))
-    (b : Butterfly d) (z : Terminal o) (i : Fin A) :
-    (terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) =
-      ∑ s : SwitchIndex d,
-        ((if (z i).val = (switchUsers d b s).1 ∧ (switchUsers d b s).2 ∈ priorPaths o (i.val+1)
-          then (1:ℝ) else 0) +
-        (if (z i).val = (switchUsers d b s).2 ∧ (switchUsers d b s).1 ∈ priorPaths o (i.val+1)
-          then (1:ℝ) else 0)) := by
-  classical
-  rw [terminalCharge_sum _ _ _ _ (Finset.notMem_erase _ _)]
-  apply Finset.sum_congr rfl
-  intro s _
-  have hn := switchUsers_ne d b s
-  simp only [Finset.mem_erase]
-  by_cases h₁ : (z i).val = (switchUsers d b s).1 <;>
-    by_cases h₂ : (z i).val = (switchUsers d b s).2 <;> simp_all [eq_comm]
-
-lemma rotationMeeting_eq_slot_sum {d A : ℕ} (o : Card d → Option (Fin A))
-    (z : Terminal o) (u v : Card d) :
-    rotationMeeting o z u v = ∑ i : Fin A,
-      ((if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) +
-      (if (z i).val = v ∧ u ∈ priorPaths o (i.val+1) then (1:ℝ) else 0)) := by
-  classical
-  have only_owner (u : Card d) (a : Fin A) (hou : o u = some a)
-      (i : Fin A) (hia : i ≠ a) : (z i).val ≠ u := by
-    intro h
-    have hi := (z i).property
-    rw [h,hou] at hi
-    exact hia (Option.some.inj hi).symm
-  have sum_one (u v : Card d) (a : Fin A) (hou : o u = some a) :
-      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) =
-      if (z a).val = u ∧ v ∈ priorPaths o (a.val+1) then 1 else 0 := by
-    apply Finset.sum_eq_single a
-    · intro i _ hia
-      simp [only_owner u a hou i hia]
-    · simp
-  have sum_none (u v : Card d) (hou : o u = none) :
-      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) = 0 := by
-    apply Finset.sum_eq_zero
-    intro i _
-    have hn : (z i).val ≠ u := by
-      intro h
-      have hi := (z i).property
-      rw [h,hou] at hi
-      contradiction
-    simp [hn]
-  rw [Finset.sum_add_distrib]
-  cases hou : o u with
-  | none =>
-    rw [sum_none u v hou]
-    simp [rotationMeeting,hou,mem_priorPaths]
-  | some a =>
-    rw [sum_one u v a hou]
-    cases hov : o v with
-    | none =>
-      rw [sum_none v u hov]
-      simp [rotationMeeting,hou,hov,mem_priorPaths]
-    | some b =>
-      rw [sum_one v u b hov]
-      simp only [rotationMeeting,hou,hov,mem_priorPaths,Option.some.injEq]
-      rcases lt_trichotomy a b with hab | rfl | hba
-      · have hn : ¬ b.val < a.val+1 := by omega
-        have hy : a.val < b.val+1 := by omega
-        simp [hab,hn,hy]
-      · simp
-      · have hn : ¬ a.val < b.val+1 := by omega
-        have hy : b.val < a.val+1 := by omega
-        simp [hba,not_lt_of_ge (le_of_lt hba),hn,hy]
-
-lemma rotationCharge_eq_terminal_sum {d A : ℕ} (o : Card d → Option (Fin A))
-    (b : Butterfly d) (z : Terminal o) :
-    rotationCharge o b z = ∑ i : Fin A,
-      (terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) := by
-  simp only [rotationCharge,rotationMeeting_eq_slot_sum,terminalCharge_prior_sum]
-  exact Finset.sum_comm
-
-@[simp] lemma priorPaths_zero {d A : ℕ} (o : Card d → Option (Fin A)) : priorPaths o 0 = ∅ := by
-  classical
-  ext x
-  simp
-
-lemma priorPaths_succ {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) (ht : t < A) :
-    priorPaths o (t+1) = priorPaths o t ∪ cycleFiber o ⟨t,ht⟩ := by
-  classical
-  ext x
-  simp only [mem_priorPaths,Finset.mem_union,mem_cycleFiber]
-  constructor
-  · rintro ⟨i,hi,hit⟩
-    by_cases h : i.val < t
-    · exact Or.inl ⟨i,hi,h⟩
-    · have he : i = ⟨t,ht⟩ := Fin.ext (by change i.val = t; omega)
-      exact Or.inr (he ▸ hi)
-  · rintro (⟨i,hi,hit⟩ | hi)
-    · exact ⟨i,hi,by omega⟩
-    · exact ⟨⟨t,ht⟩,hi,by change t < t+1; omega⟩
-
-lemma priorPaths_disjoint {d A : ℕ} (o : Card d → Option (Fin A)) (i : Fin A) :
-    Disjoint (priorPaths o i.val) (cycleFiber o i) := by
-  classical
-  apply Finset.disjoint_left.mpr
-  intro x hx hx'
-  obtain ⟨j,hj,hji⟩ := (mem_priorPaths _ _ _).mp hx
-  have he : j = i := Option.some.inj (hj.symm.trans ((mem_cycleFiber _ _ _).mp hx'))
-  subst j
-  exact (lt_irrefl _ hji)
-
-namespace QueryTree
-
-lemma seekCycles_owned {d A : ℕ} (S : Finset (Card d)) (r : Fin A → ℕ)
-    (o : Card d → Option (Fin A)) (z : Terminal o) (ω : SwitchIndex d → Bool)
-    (hS : ∀ i, cycleFiber o i ⊆ S)
-    (hO : ∀ i, orbitSet (butterflyPerm d (decodeButterfly d ω)) (z i).val = cycleFiber o i)
-    (hC : ∀ i, (cycleFiber o i).card = r i+1) :
-    let p := butterflyPerm d (decodeButterfly d ω)
-    let L := List.ofFn (fun i : Fin A => (p (z i).val,r i))
-    (seekCycles d S L ∅).succeeds ω = true ∧
-      ((seekCycles d S L ∅).staleCharge ∅ ω : ℝ) = rotationCharge o (decodeButterfly d ω) z := by
-  classical
-  let b := decodeButterfly d ω
-  let p := butterflyPerm d b
-  let f (i : Fin A) := (p (z i).val,r i)
-  let c (i : Fin A) : ℝ := terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val)
-  have hc (i : Fin A) : Function.minimalPeriod p (z i).val - 1 = r i := by
-    have he := orbitSet_card p (z i).val
-    rw [hO,hC] at he
-    omega
-  have hs (t : ℕ) (ht : t ≤ A) :
-      (seekCycles d S ((List.ofFn f).drop t) (priorPaths o t)).succeeds ω = true ∧
-        ((seekCycles d S ((List.ofFn f).drop t) (priorPaths o t)).staleCharge (exposedPaths d b (priorPaths o t)) ω : ℝ) =
-          ((List.ofFn c).drop t).sum := by
-    induction ht using Nat.decreasingInduction with
-    | self => simp only [List.length_ofFn,le_refl,List.drop_eq_nil_of_le,List.sum_nil,seekCycles,succeeds,staleCharge,Nat.cast_zero,and_self]
-    | of_succ t ht ih =>
-      have htf : t < (List.ofFn f).length := by simpa using ht
-      have htc : t < (List.ofFn c).length := by simpa using ht
-      rw [List.drop_eq_getElem_cons htf,List.drop_eq_getElem_cons htc]
-      simp only [List.getElem_ofFn,List.sum_cons,seekCycles]
-      have h := seekCycle_orbit d S (priorPaths o t) (z ⟨t,ht⟩).val
-        (fun V => seekCycles d S ((List.ofFn f).drop (t+1)) V) ω
-        (by rw [hO]; exact hS _) (by rw [hO]; exact priorPaths_disjoint o ⟨t,ht⟩)
-      dsimp only at h
-      rw [hc,hO,←priorPaths_succ o t ht] at h
-      dsimp only [f]
-      constructor
-      · exact h.1.trans ih.1
-      · rw [h.2,Nat.cast_add,ih.2]
-        exact add_comm _ _
-  have h := hs 0 (Nat.zero_le _)
-  simp only [List.drop_zero,priorPaths_zero,exposedPaths_empty,List.sum_ofFn] at h
-  dsimp only
-  exact ⟨h.1,h.2.trans (rotationCharge_eq_terminal_sum o b z).symm⟩
-
-end QueryTree
-
-section Collections
-variable {d A : ℕ}
-
-def ValidCycleSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ)
-    (o : Card d → Option (Fin A)) : Prop :=
-  (∀ i, (cycleFiber o i).card = r i + 1) ∧
-  (∀ i, cycleFiber o i ⊆ S) ∧
-  (∀ i x, o x = some i → orbitSet p x = cycleFiber o i)
-
-abbrev CycleSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
-  {o : Card d → Option (Fin A) // ValidCycleSlots p S r o}
-
-noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
-    Fintype (CycleSlots p S r) := Fintype.ofFinite _
-
-lemma CycleSlots.terminal_nonempty {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (c : CycleSlots p S r) (i : Fin A) : Nonempty {x : Card d // c.val x = some i} := by
-  have hn : (cycleFiber c.val i).Nonempty := Finset.card_pos.mp (by rw [c.property.1]; omega)
-  obtain ⟨x,hx⟩ := hn
-  exact ⟨x,(mem_cycleFiber _ _ _).mp hx⟩
-
-lemma owner_ext {o o' : Card d → Option (Fin A)}
-    (h : ∀ i, cycleFiber o i = cycleFiber o' i) : o = o' := by
-  funext x
-  have he (i : Fin A) : o x = some i ↔ o' x = some i := by
-    simpa only [mem_cycleFiber] using (Finset.ext_iff.mp (h i) x)
-  cases ho : o x with
-  | some i => exact ((he i).mp ho).symm
-  | none =>
-    cases ho' : o' x with
-    | none => rfl
-    | some i => have hn := (he i).mpr ho'; rw [ho] at hn; contradiction
-
-abbrev RootedSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
-  Σ c : CycleSlots p S r, Terminal c.val
-
-noncomputable def rootedStart {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (c : RootedSlots p S r) : Fin A → S := fun i =>
-  ⟨p (c.2 i).val,c.1.property.2.1 i (by
-    rw [←c.1.property.2.2 i (c.2 i).val (c.2 i).property]
-    exact (mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.apply_right (Equiv.Perm.SameCycle.refl p _)))⟩
-
-lemma rootedStart_injective {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ} :
-    Function.Injective (rootedStart (p:=p) (S:=S) (r:=r)) := by
-  rintro ⟨c,z⟩ ⟨c',z'⟩ h
-  have hz (i : Fin A) : (z i).val = (z' i).val :=
-    p.injective (congrArg Subtype.val (congrFun h i))
-  have hc : c = c' := Subtype.ext (owner_ext (fun i => by
-    rw [←c.property.2.2 i (z i).val (z i).property,
-      ←c'.property.2.2 i (z' i).val (z' i).property,hz]))
-  subst c'
-  have hzz : z = z' := funext (fun i => Subtype.ext (hz i))
-  subst z'
-  rfl
-
-lemma sum_comp_injective_le {α β : Type*} [Fintype α] [Fintype β]
-    (e : α → β) (he : Function.Injective e) (f : β → ℝ) (hf : ∀ b, 0 ≤ f b) :
-    ∑ a, f (e a) ≤ ∑ b, f b := by
-  classical
-  calc
-    _ = ∑ b ∈ Finset.univ.image e, f b := (Finset.sum_image (fun _ _ _ _ h => he h)).symm
-    _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _) (fun b _ _ => hf b)
-
-lemma rooted_weight_sum_le (S : Finset (Card d)) (r : Fin A → ℕ) (ω : SwitchIndex d → Bool) :
-    (∑ c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r,
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2)) ≤
-    ∑ a : Fin A → S,
-      if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-        ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-      else 0 := by
-  classical
-  let p := butterflyPerm d (decodeButterfly d ω)
-  let g (a : Fin A → S) : ℝ :=
-    if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-    else 0
-  have he (c : RootedSlots p S r) :
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2) =
-      g (rootedStart c) := by
-    have hs := QueryTree.seekCycles_owned S r c.1.val c.2 ω c.1.property.2.1
-      (fun i => c.1.property.2.2 i (c.2 i).val (c.2 i).property) c.1.property.1
-    dsimp only at hs
-    unfold g
-    rw [ite_eq_left (show (QueryTree.seekCycles d S (List.ofFn (fun i => ((rootedStart c i).val,r i))) ∅).succeeds ω = true from hs.1)]
-    rw [← hs.2,neg_mul,Real.exp_neg,mul_comm (Real.log 2),Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
-    rfl
-  simp_rw [he]
-  exact sum_comp_injective_le rootedStart rootedStart_injective g (fun a => by dsimp [g]; split_ifs <;> positivity)
-
-noncomputable def rotationLower (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : ℝ :=
-  Real.exp (-Real.log 2 * ((A:ℝ)*d/2 + ∑ i : Fin A, Real.log (r i+1) / (2 * Real.log 2)))
-
-noncomputable def collectionWeight (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : ℝ :=
-  ((2:ℝ)^d)^A * (∏ i : Fin A, (r i+1 : ℝ)) * rotationLower d r
-
-lemma collectionWeight_pos (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : 0 < collectionWeight d r := by
-  unfold collectionWeight rotationLower
-  positivity
-
-lemma collection_rotation_lower (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r)
-    (ω : SwitchIndex d → Bool) (c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) :
-    collectionWeight d r ≤ ∑ z : Terminal c.val,
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z) := by
-  classical
-  let (i : Fin A) : Nonempty {x : Card d // c.val x = some i} := c.terminal_nonempty i
-  have hm : Monotone (fun i => (cycleFiber c.val i).card) := by
-    intro i j hij
-    dsimp only
-    rw [c.property.1,c.property.1]
-    exact Nat.add_le_add_right (hr hij) 1
-  have hb := rotation_exp_bound hm (decodeButterfly d ω)
-  have hcard : (Fintype.card (Terminal c.val) : ℝ) = ∏ i : Fin A, (r i+1 : ℝ) := by
-    rw [Fintype.card_pi,Nat.cast_prod]
-    apply Finset.prod_congr rfl
-    intro i _
-    rw [card_cycleFiber,c.property.1,Nat.cast_add,Nat.cast_one]
-  have hp : (0:ℝ) < Fintype.card (Terminal c.val) := Nat.cast_pos.mpr Fintype.card_pos
-  have hl : rotationLower d r ≤ finiteMean (fun z : Terminal c.val => Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z)) := by
-    simpa only [rotationLower,c.property.1,Nat.cast_add,Nat.cast_one] using hb
-  have hh := (le_div_iff₀ hp).mp hl
-  rw [hcard] at hh
-  unfold collectionWeight
-  rw [← Finset.mul_sum]
-  calc
-    _ = ((2:ℝ)^d)^A * (rotationLower d r * ∏ i : Fin A, (r i+1 : ℝ)) := by ring
-    _ ≤ _ := mul_le_mul_of_nonneg_left hh (by positivity)
-
-lemma cycleSlots_weighted_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) *
-      collectionWeight d r ≤ (S.card : ℝ)^A := by
-  classical
-  let g (a : Fin A → S) (ω : SwitchIndex d → Bool) : ℝ :=
-    if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-    else 0
-  have hω (ω : SwitchIndex d → Bool) :
-      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ) * collectionWeight d r ≤
-        ∑ a : Fin A → S, g a ω := by
-    calc
-      _ = ∑ _c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r, collectionWeight d r := by simp
-      _ ≤ ∑ c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r,
-          ∑ z : Terminal c.val, ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z) :=
-        Finset.sum_le_sum (fun c _ => collection_rotation_lower S r hr ω c)
-      _ = ∑ c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r,
-          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2) :=
-        (Fintype.sum_sigma (fun c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r =>
-          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2))).symm
-      _ ≤ _ := rooted_weight_sum_le S r ω
-  have hb := finiteMean_mono hω
-  rw [finiteMean_mul_const,finiteMean_sum] at hb
-  calc
-    _ ≤ ∑ a : Fin A → S, finiteMean (g a) := hb
-    _ ≤ ∑ _a : Fin A → S, (1:ℝ) := by
-      apply Finset.sum_le_sum
-      intro a _
-      have hh := QueryTree.cycles_weighted_success d S (List.ofFn (fun i : Fin A => ((a i).val,r i)))
-      simpa only [List.length_ofFn] using hh
-    _ = _ := by simp
-
-lemma cycleSlots_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) ≤
-        (S.card : ℝ)^A / collectionWeight d r := by
-  exact (le_div_iff₀ (collectionWeight_pos d r)).mpr (cycleSlots_weighted_moment S r hr)
-
-lemma collectionWeight_eq_sqrt (d : ℕ) {A : ℕ} (r : Fin A → ℕ) :
-    collectionWeight d r = (Real.sqrt ((2:ℝ)^d))^A * ∏ i : Fin A, Real.sqrt (r i+1) := by
-  have hn : ((2:ℝ)^d)^A = Real.exp ((A:ℝ)*d*Real.log 2) := by
-    rw [mul_assoc,Real.exp_nat_mul,Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
-  have hp : (∏ i : Fin A, (r i+1 : ℝ)) = Real.exp (∑ i : Fin A, Real.log (r i+1)) := by
-    rw [Real.exp_sum]
-    apply Finset.prod_congr rfl
-    intro i _
-    exact (Real.exp_log (by positivity)).symm
-  have hs : Real.sqrt ((2:ℝ)^d) = Real.exp ((d:ℝ)*Real.log 2/2) := by
-    rw [Real.sqrt_eq_rpow,Real.rpow_def_of_pos (by positivity),Real.log_pow]
-    congr 1
-    ring
-  have hj (i : Fin A) : Real.sqrt (r i+1) = Real.exp (Real.log (r i+1)/2) := by
-    rw [Real.sqrt_eq_rpow,Real.rpow_def_of_pos (by positivity)]
-    congr 1
-    ring
-  simp_rw [collectionWeight,hn,hp,rotationLower,hs,hj]
-  rw [←Real.exp_add,←Real.exp_add,←Real.exp_nat_mul,←Real.exp_sum,←Real.exp_add]
-  congr 1
-  rw [←Finset.sum_div,←Finset.sum_div]
-  have hl : Real.log 2 ≠ 0 := ne_of_gt (Real.log_pos (by norm_num))
-  field_simp
-  ring
-
-lemma butterfly_cycleSlots_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) ≤
-        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
-  have h := cycleSlots_moment S r hr
-  rw [collectionWeight_eq_sqrt,div_mul_eq_div_mul_one_div] at h
-  rw [div_pow]
-  simpa only [one_div,Finset.prod_inv_distrib] using h
-
-end Collections
-
-variable {d A : ℕ}
-
-abbrev SelectedCycle (p : Equiv.Perm (Card d)) (S : Finset (Card d)) :=
-  {C : Finset (Card d) // (∃ x, C = orbitSet p x) ∧ C ⊆ S}
-
-noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) :
-    Fintype (SelectedCycle p S) := Fintype.ofFinite _
-
-lemma SelectedCycle.nonempty {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
-    (C : SelectedCycle p S) : C.val.Nonempty := by
-  obtain ⟨x,hx⟩ := C.property.1
-  rw [hx]
-  exact orbitSet_nonempty p x
-
-lemma SelectedCycle.orbit_eq {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
-    (C : SelectedCycle p S) {x : Card d} (hx : x ∈ C.val) : orbitSet p x = C.val := by
-  obtain ⟨y,hy⟩ := C.property.1
-  rw [hy] at hx ⊢
-  exact (orbitSet_eq p ((mem_orbitSet _ _ _).mp hx)).symm
-
-lemma SelectedCycle.eq_of_mem {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
-    (C D : SelectedCycle p S) {x : Card d} (hC : x ∈ C.val) (hD : x ∈ D.val) : C = D := by
-  apply Subtype.ext
-  exact (C.orbit_eq hC).symm.trans (D.orbit_eq hD)
-
-abbrev SlotAssignment (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
-  {f : Fin A → SelectedCycle p S // Function.Injective f ∧ ∀ i, (f i).val.card = r i+1}
-
-noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
-    Fintype (SlotAssignment p S r) := Fintype.ofFinite _
-
-noncomputable def slotsToAssignment {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (c : CycleSlots p S r) : SlotAssignment p S r := by
-  classical
-  let f (i : Fin A) : SelectedCycle p S := ⟨cycleFiber c.val i,by
-    obtain ⟨x,hx⟩ := c.terminal_nonempty i
-    exact ⟨⟨x,(c.property.2.2 i x hx).symm⟩,c.property.2.1 i⟩⟩
-  refine ⟨f,?_,fun i => c.property.1 i⟩
-  intro i j hij
-  have he : cycleFiber c.val i = cycleFiber c.val j := congrArg Subtype.val hij
-  obtain ⟨x,hx⟩ := c.terminal_nonempty i
-  have hxi : x ∈ cycleFiber c.val i := (mem_cycleFiber _ _ _).mpr hx
-  rw [he] at hxi
-  exact Option.some.inj (hx.symm.trans ((mem_cycleFiber _ _ _).mp hxi))
-
-noncomputable def assignmentOwner {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) (x : Card d) : Option (Fin A) := by
-  classical
-  exact if h : ∃ i, x ∈ (f.val i).val then some (Classical.choose h) else none
-
-end Thorp
-
-end ThorpNine.HighTail
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.HighTail.ExposedCoins
+
+namespace OAI
+
+namespace ThorpNine.HighTail
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+section Rotation
+variable {d A : ℕ}
+
+noncomputable def rotationMeeting (o : Card d → Option (Fin A)) (z : Terminal o)
+    (u v : Card d) : ℝ :=
+  match o u, o v with
+  | some i, some j => if i < j then (if (z j).val = v then 1 else 0)
+      else if j < i then (if (z i).val = u then 1 else 0)
+      else (if (z i).val = u then 1 else 0) + (if (z i).val = v then 1 else 0)
+  | _, _ => 0
+
+lemma card_cycleFiber (o : Card d → Option (Fin A)) (i : Fin A) :
+    Fintype.card {x : Card d // o x = some i} = (cycleFiber o i).card := by
+  classical
+  simp [cycleFiber,Fintype.card_subtype]
+
+lemma terminal_mean {o : Card d → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
+    (i : Fin A) (x : Card d) (hx : o x = some i) :
+    finiteMean (fun z : Terminal o => if (z i).val = x then (1:ℝ) else 0) = slotWeight o i := by
+  classical
+  have hev := finiteMean_pi_eval (A := fun i : Fin A => {x : Card d // o x = some i})
+    i (fun z => if z.val = x then (1:ℝ) else 0)
+  rw [hev]
+  have he : (fun z : {x : Card d // o x = some i} => if z.val = x then (1:ℝ) else 0) =
+      (fun z => if z = ⟨x,hx⟩ then 1 else 0) := by
+    funext z
+    simp only [Subtype.ext_iff]
+  rw [he,finiteMean_eq_indicator,card_cycleFiber,slotWeight]
+
+lemma slotWeight_antitone (o : Card d → Option (Fin A))
+    (hn : ∀ i, (cycleFiber o i).Nonempty)
+    (hm : Monotone (fun i => (cycleFiber o i).card)) : Antitone (slotWeight o) := by
+  intro i j hij
+  unfold slotWeight
+  apply one_div_le_one_div_of_le
+  · exact_mod_cast Finset.card_pos.mpr (hn i)
+  · exact_mod_cast hm hij
+
+lemma rotationMeeting_mean {o : Card d → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
+    (hm : Monotone (fun i => (cycleFiber o i).card)) (u v : Card d) :
+    finiteMean (fun z : Terminal o => rotationMeeting o z u v) =
+      min (pathWeight o u) (pathWeight o v) + withinWeight o u v := by
+  classical
+  have hn : ∀ i, (cycleFiber o i).Nonempty := by
+    intro i
+    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : Card d // o x = some i})
+    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
+  have hw := slotWeight_antitone o hn hm
+  cases hu : o u with
+  | none =>
+    cases hv : o v with
+    | none => simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean]
+    | some j =>
+      have hj : 0 ≤ slotWeight o j := by unfold slotWeight; positivity
+      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_left hj]
+  | some i =>
+    cases hv : o v with
+    | none =>
+      have hi : 0 ≤ slotWeight o i := by unfold slotWeight; positivity
+      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_right hi]
+    | some j =>
+      simp only [rotationMeeting,pathWeight,withinWeight,hu,hv]
+      rcases lt_trichotomy i j with hij | rfl | hji
+      · simp only [hij,↓reduceIte,ne_of_lt hij]
+        rw [terminal_mean j v hv,min_eq_right (hw (le_of_lt hij)),add_zero]
+      · simp only [lt_self_iff_false,↓reduceIte,min_self]
+        rw [finiteMean_add,terminal_mean i u hu,terminal_mean i v hv]
+      · simp only [not_lt_of_ge (le_of_lt hji),hji,↓reduceIte,ne_of_gt hji]
+        rw [terminal_mean i u hu,min_eq_left (hw (le_of_lt hji)),add_zero]
+
+lemma pathWeight_sum (o : Card d → Option (Fin A)) (hn : ∀ i, (cycleFiber o i).Nonempty) :
+    ∑ x : Card d, pathWeight o x = (A : ℝ) := by
+  classical
+  have he (x : Card d) : pathWeight o x =
+      ∑ i : Fin A, if o x = some i then slotWeight o i else 0 := by
+    cases hx : o x with
+    | none => simp [pathWeight,hx]
+    | some j => simp [pathWeight,hx]
+  simp only [he]
+  rw [Finset.sum_comm]
+  have hf (i : Fin A) : (∑ x : Card d, if o x = some i then slotWeight o i else 0) = 1 := by
+    rw [← Finset.sum_filter]
+    change (∑ _x ∈ cycleFiber o i, slotWeight o i) = 1
+    simp only [Finset.sum_const,nsmul_eq_mul,slotWeight]
+    exact mul_one_div_cancel (by exact_mod_cast (Finset.card_pos.mpr (hn i)).ne')
+  simp only [hf,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,mul_one]
+
+lemma withinWeight_sum_bound (o : Card d → Option (Fin A))
+    (hn : ∀ i, (cycleFiber o i).Nonempty) (b : Butterfly d) :
+    (∑ s : SwitchIndex d, withinWeight o (switchUsers d b s).1 (switchUsers d b s).2) ≤
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
+  classical
+  have he (u v : Card d) : withinWeight o u v =
+      ∑ i : Fin A, (if u ∈ cycleFiber o i ∧ v ∈ cycleFiber o i then (1:ℝ) else 0) * slotWeight o i := by
+    simp only [mem_cycleFiber]
+    cases hu : o u <;> cases hv : o v <;> simp [withinWeight,hu,hv,ite_and,eq_comm]
+  simp only [he]
+  rw [Finset.sum_comm]
+  apply Finset.sum_le_sum
+  intro i _
+  rw [← Finset.sum_mul]
+  have hi := marked_switch_entropy_bound d b (cycleFiber o i)
+  have hpos : (0:ℝ) < (cycleFiber o i).card := by exact_mod_cast Finset.card_pos.mpr (hn i)
+  have hmul := mul_le_mul_of_nonneg_right hi (show 0 ≤ slotWeight o i by unfold slotWeight; positivity)
+  calc
+    _ ≤ ((cycleFiber o i).card : ℝ) * Real.log (cycleFiber o i).card /
+        (2 * Real.log 2) * slotWeight o i := hmul
+    _ = _ := by unfold slotWeight; field_simp
+
+noncomputable def rotationCharge (o : Card d → Option (Fin A)) (b : Butterfly d)
+    (z : Terminal o) : ℝ :=
+  ∑ s : SwitchIndex d, rotationMeeting o z (switchUsers d b s).1 (switchUsers d b s).2
+
+lemma rotationCharge_mean_bound {o : Card d → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
+    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : Butterfly d) :
+    finiteMean (rotationCharge o b) ≤ (A:ℝ) * d / 2 +
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
+  classical
+  have hn : ∀ i, (cycleFiber o i).Nonempty := by
+    intro i
+    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : Card d // o x = some i})
+    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
+  unfold rotationCharge
+  rw [finiteMean_sum]
+  simp only [rotationMeeting_mean hm,Finset.sum_add_distrib]
+  have h₁ := weighted_switch_min_bound d b (pathWeight o)
+  rw [pathWeight_sum o hn] at h₁
+  have h₂ := withinWeight_sum_bound o hn b
+  linarith
+
+lemma rotation_exp_bound {o : Card d → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
+    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : Butterfly d) :
+    Real.exp (-Real.log 2 * ((A:ℝ)*d/2 +
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2))) ≤
+      finiteMean (fun z : Terminal o => Real.exp (-Real.log 2 * rotationCharge o b z)) := by
+  have hb := rotationCharge_mean_bound hm b
+  have hlog : 0 ≤ Real.log 2 := Real.log_nonneg (by norm_num)
+  have hm' : -Real.log 2 * ((A:ℝ)*d/2 +
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2)) ≤
+      finiteMean (fun z : Terminal o => -Real.log 2 * rotationCharge o b z) := by
+    have he := finiteMean_mul_const (rotationCharge o b) (-Real.log 2)
+    simp only [mul_comm] at he
+    rw [he]
+    exact mul_le_mul_of_nonpos_left hb (neg_nonpos.mpr hlog)
+  exact (Real.exp_le_exp.mpr hm').trans (exp_finiteMean_le _)
+
+end Rotation
+
+section OrbitCore
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+@[simp] lemma mem_orbitSet (p : Equiv.Perm α) (x y : α) :
+    y ∈ orbitSet p x ↔ p.SameCycle x y := by
+  classical
+  simp [orbitSet]
+
+lemma orbitSet_nonempty (p : Equiv.Perm α) (x : α) : (orbitSet p x).Nonempty :=
+  ⟨x,(mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.refl _ _)⟩
+
+lemma orbitSet_eq (p : Equiv.Perm α) {x y : α} (h : p.SameCycle x y) :
+    orbitSet p x = orbitSet p y := by
+  ext z
+  simp only [mem_orbitSet]
+  exact ⟨fun hz => h.symm.trans hz,fun hz => h.trans hz⟩
+
+@[simp] lemma orbitSet_apply (p : Equiv.Perm α) (x : α) : orbitSet p (p x) = orbitSet p x := by
+  ext y
+  simp only [mem_orbitSet,Equiv.Perm.sameCycle_apply_left]
+
+lemma orbitSet_eq_or_disjoint (p : Equiv.Perm α) (x y : α) :
+    orbitSet p x = orbitSet p y ∨ Disjoint (orbitSet p x) (orbitSet p y) := by
+  classical
+  by_cases h : p.SameCycle x y
+  · exact Or.inl (orbitSet_eq p h)
+  · right
+    apply Finset.disjoint_left.mpr
+    intro z hz hz'
+    exact h (((mem_orbitSet _ _ _).mp hz).trans ((mem_orbitSet _ _ _).mp hz').symm)
+
+omit [DecidableEq α] in
+lemma perm_minimalPeriod_pos (p : Equiv.Perm α) (x : α) :
+    0 < Function.minimalPeriod p x :=
+  Function.minimalPeriod_pos_of_mem_periodicPts (p.injective.mem_periodicPts x)
+
+lemma orbitSet_card (p : Equiv.Perm α) (x : α) :
+    (orbitSet p x).card = Function.minimalPeriod p x := by
+  classical
+  let f : Fin (Function.minimalPeriod p x) → {y // p.SameCycle x y} := fun n =>
+    ⟨p^[n.val] x,by rw [Equiv.Perm.iterate_eq_pow]; exact ⟨(n.val:ℤ),by simp⟩⟩
+  have hf : Function.Bijective f := by
+    constructor
+    · intro i j hij
+      apply Fin.ext
+      exact Function.iterate_injOn_Iio_minimalPeriod i.isLt j.isLt (congrArg Subtype.val hij)
+    · intro y
+      obtain ⟨n,hn⟩ := y.property.exists_nat_pow_eq
+      refine ⟨⟨n % Function.minimalPeriod p x, Nat.mod_lt _ (perm_minimalPeriod_pos p x)⟩,?_⟩
+      apply Subtype.ext
+      change p^[n % Function.minimalPeriod p x] x = y.val
+      rw [Function.iterate_mod_minimalPeriod_eq,Equiv.Perm.iterate_eq_pow,hn]
+  have hc := Fintype.card_congr (Equiv.ofBijective f hf)
+  simpa [orbitSet,Fintype.card_subtype] using hc.symm
+
+def walkSet (p : Equiv.Perm α) (r : ℕ) (x : α) : Finset α :=
+  (Finset.range r).image (fun n => p^[n] x)
+
+omit [Fintype α] in
+@[simp] lemma mem_walkSet (p : Equiv.Perm α) (r : ℕ) (x y : α) :
+    y ∈ walkSet p r x ↔ ∃ n < r, p^[n] x = y := by
+  simp [walkSet]
+
+omit [Fintype α] in
+@[simp] lemma walkSet_zero (p : Equiv.Perm α) (x : α) : walkSet p 0 x = ∅ := by
+  simp [walkSet]
+
+omit [Fintype α] in
+lemma walkSet_succ_last (p : Equiv.Perm α) (r : ℕ) (x : α) :
+    walkSet p (r+1) x = insert (p^[r] x) (walkSet p r x) := by
+  simp [walkSet,Finset.range_add_one]
+
+omit [Fintype α] in
+lemma walkSet_succ_first (p : Equiv.Perm α) (r : ℕ) (x : α) :
+    walkSet p (r+1) x = insert x (walkSet p r (p x)) := by
+  ext y
+  simp only [mem_walkSet,Finset.mem_insert]
+  constructor
+  · rintro ⟨n,hn,hny⟩
+    cases n with
+    | zero => exact Or.inl (by simpa using hny.symm)
+    | succ n => exact Or.inr ⟨n,by omega,by simpa only [Function.iterate_succ_apply] using hny⟩
+  · rintro (rfl | ⟨n,hn,hny⟩)
+    · exact ⟨0,by omega,rfl⟩
+    · exact ⟨n+1,by omega,by simpa only [Function.iterate_succ_apply] using hny⟩
+
+omit [Fintype α] in
+lemma not_mem_short_walk (p : Equiv.Perm α) (x : α) {r : ℕ}
+    (hr : r < Function.minimalPeriod p x) : x ∉ walkSet p r (p x) := by
+  rintro hx
+  obtain ⟨n,hn,hnx⟩ := (mem_walkSet _ _ _ _).mp hx
+  have hperiod : Function.IsPeriodicPt p (n+1) x := by
+    simpa only [Function.IsPeriodicPt,Function.IsFixedPt,Function.iterate_succ_apply] using hnx
+  have hp := hperiod.minimalPeriod_le (by omega : 0 < n+1)
+  omega
+
+lemma walkSet_period (p : Equiv.Perm α) (x : α) :
+    walkSet p (Function.minimalPeriod p x) x = orbitSet p x := by
+  ext y
+  simp only [mem_walkSet,mem_orbitSet]
+  constructor
+  · rintro ⟨n,_,rfl⟩
+    rw [Equiv.Perm.iterate_eq_pow]
+    exact ⟨(n:ℤ),by simp⟩
+  · intro h
+    obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
+    refine ⟨n % Function.minimalPeriod p x,Nat.mod_lt _ (perm_minimalPeriod_pos p x),?_⟩
+    rw [Function.iterate_mod_minimalPeriod_eq,Equiv.Perm.iterate_eq_pow,hn]
+
+omit [Fintype α] in
+lemma walkSet_card (p : Equiv.Perm α) (x : α) {r : ℕ}
+    (hr : r ≤ Function.minimalPeriod p x) : (walkSet p r x).card = r := by
+  rw [walkSet,Finset.card_image_of_injOn]
+  · exact Finset.card_range _
+  · intro i hi j hj hij
+    exact Function.iterate_injOn_Iio_minimalPeriod
+      (lt_of_lt_of_le (Finset.mem_range.mp hi) hr)
+      (lt_of_lt_of_le (Finset.mem_range.mp hj) hr) hij
+
+lemma walkSet_subset_orbit (p : Equiv.Perm α) (r : ℕ) (x : α) :
+    walkSet p r x ⊆ orbitSet p x := by
+  intro y hy
+  obtain ⟨n,_,rfl⟩ := (mem_walkSet _ _ _ _).mp hy
+  rw [mem_orbitSet,Equiv.Perm.iterate_eq_pow]
+  exact ⟨(n:ℤ),by simp⟩
+
+def cycleTrace (p : Equiv.Perm α) (S : Finset α) (a : α) :
+    ℕ → α → Finset α → Option (Finset α × α)
+  | 0, x, V => if x ∈ S ∧ x ∉ V ∧ p x = a then some (insert x V,x) else none
+  | r+1, x, V => if x ∈ S ∧ x ∉ V then cycleTrace p S a r (p x) (insert x V) else none
+
+lemma cycleTrace_walk (p : Equiv.Perm α) (S : Finset α) (r : ℕ) (x : α) (V : Finset α)
+    (hr : r < Function.minimalPeriod p x) (hS : walkSet p (r+1) x ⊆ S)
+    (hV : Disjoint V (walkSet p (r+1) x)) :
+    cycleTrace p S (p^[r+1] x) r x V = some (V ∪ walkSet p (r+1) x,p^[r] x) := by
+  induction r generalizing x V with
+  | zero =>
+    have hxS : x ∈ S := hS (by simp [walkSet_succ_first])
+    have hxV : x ∉ V := fun h => Finset.disjoint_left.mp hV h (by simp [walkSet_succ_first])
+    simp [cycleTrace,hxS,hxV,walkSet_succ_first]
+  | succ r ih =>
+    have hxS : x ∈ S := hS (by rw [walkSet_succ_first]; exact Finset.mem_insert_self _ _)
+    have hxV : x ∉ V := fun h => Finset.disjoint_left.mp hV h
+      (by rw [walkSet_succ_first]; exact Finset.mem_insert_self _ _)
+    have hr' : r < Function.minimalPeriod p (p x) := by
+      rw [Function.minimalPeriod_apply (p.injective.mem_periodicPts x)]
+      omega
+    have hS' : walkSet p (r+1) (p x) ⊆ S := by
+      intro y hy
+      apply hS
+      rw [walkSet_succ_first]
+      exact Finset.mem_insert_of_mem hy
+    have hV' : Disjoint (insert x V) (walkSet p (r+1) (p x)) := by
+      apply Finset.disjoint_left.mpr
+      intro y hy hy'
+      rcases Finset.mem_insert.mp hy with hy | hy
+      · subst y
+        exact not_mem_short_walk p x hr hy'
+      · apply Finset.disjoint_left.mp hV hy
+        rw [walkSet_succ_first]
+        exact Finset.mem_insert_of_mem hy'
+    rw [cycleTrace,ite_eq_left ⟨hxS,hxV⟩,Function.iterate_succ_apply p (r+1) x,
+      ih _ _ hr' hS' hV',walkSet_succ_first p (r+1) x,Function.iterate_succ_apply]
+    simp only [Finset.insert_union,Finset.union_insert]
+
+lemma cycleTrace_orbit (p : Equiv.Perm α) (S V : Finset α) (z : α)
+    (hS : orbitSet p z ⊆ S) (hV : Disjoint V (orbitSet p z)) :
+    cycleTrace p S (p z) (Function.minimalPeriod p z - 1) (p z) V =
+      some (V ∪ orbitSet p z,z) := by
+  have hpos := perm_minimalPeriod_pos p z
+  have he : Function.minimalPeriod p (p z) = Function.minimalPeriod p z :=
+    Function.minimalPeriod_apply (p.injective.mem_periodicPts z)
+  have hp : Function.minimalPeriod p z - 1 + 1 = Function.minimalPeriod p z := by omega
+  have hw : walkSet p (Function.minimalPeriod p z - 1 + 1) (p z) = orbitSet p z := by
+    rw [hp,←he,walkSet_period,orbitSet_apply]
+  have hr : Function.minimalPeriod p z - 1 < Function.minimalPeriod p (p z) := by rw [he]; omega
+  have h := cycleTrace_walk p S (Function.minimalPeriod p z-1) (p z) V hr
+    (by simpa only [hw] using hS) (by simpa only [hw] using hV)
+  have ht : p^[Function.minimalPeriod p z-1] (p z) = z := by
+    rw [← Function.iterate_succ_apply,show (Function.minimalPeriod p z-1).succ =
+      Function.minimalPeriod p z by omega,Function.iterate_minimalPeriod]
+  have ha : p^[Function.minimalPeriod p z-1+1] (p z) = p z := by
+    rw [hp,←he,Function.iterate_minimalPeriod]
+  simpa only [ha,ht,hw] using h
+
+end OrbitCore
+
+noncomputable def exposedPaths (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) :
+    Finset (SwitchIndex d) :=
+  V.biUnion (fun x => routeDomain d x (butterflyPerm d b x))
+
+@[simp] lemma exposedPaths_empty (d : ℕ) (b : Butterfly d) : exposedPaths d b ∅ = ∅ := by
+  classical
+  simp [exposedPaths]
+
+lemma exposedPaths_insert (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) (x : Card d) :
+    exposedPaths d b (insert x V) = exposedPaths d b V ∪ routeDomain d x (butterflyPerm d b x) := by
+  classical
+  simp [exposedPaths,Finset.union_comm]
+
+noncomputable def terminalCharge (d : ℕ) (b : Butterfly d) (z : Card d) (V : Finset (Card d)) : ℕ :=
+  (routeDomain d z (butterflyPerm d b z) ∩ exposedPaths d b V).card
+
+namespace QueryTree
+
+lemma followPath_stale_charge {ι : Type*} [DecidableEq ι]
+    (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
+    (k : Card d → QueryTree ι) (R : Finset ι) (ω : ι → Bool) :
+    staleCharge (followPath d e x k) R ω =
+      staleCharge (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x))
+        (R ∪ (routeDomain d x (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).image e) ω := by
+  have h₀ := fresh_add_stale (followPath d e x k) R ω
+  have h₁ := fresh_add_stale (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x))
+    (R ∪ (routeDomain d x (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).image e) ω
+  rw [followPath_charge,followPath_raw_charge] at h₀
+  omega
+
+lemma seekCycle_semantics (d : ℕ) (S : Finset (Card d)) (a : Card d) (r : ℕ)
+    (x : Card d) (V W : Finset (Card d)) (z : Card d)
+    (k : Finset (Card d) → QueryTree (SwitchIndex d)) (ω : SwitchIndex d → Bool)
+    (h : cycleTrace (butterflyPerm d (decodeButterfly d ω)) S a r x V = some (W,z)) :
+    (seekCycle d S a r x V k).succeeds ω = (k W).succeeds ω ∧
+      (seekCycle d S a r x V k).staleCharge (exposedPaths d (decodeButterfly d ω) V) ω =
+        (k W).staleCharge (exposedPaths d (decodeButterfly d ω) W) ω +
+          terminalCharge d (decodeButterfly d ω) z (W.erase z) := by
+  classical
+  induction r generalizing x V with
+  | zero =>
+    simp only [cycleTrace] at h
+    split_ifs at h with hv
+    obtain ⟨hxS,hxV,hxa⟩ := hv
+    obtain ⟨rfl,rfl⟩ := Prod.mk.inj (Option.some.inj h)
+    simp only [seekCycle,ite_eq_left (show x ∈ S ∧ x ∉ V from ⟨hxS,hxV⟩),closePath_succeeds,ite_eq_left hxa]
+    constructor
+    · trivial
+    · rw [closePath_stale_charge _ _ _ _ _ _ hxa,exposedPaths_insert,
+        Finset.erase_insert hxV]
+      simp only [terminalCharge,hxa]
+  | succ r ih =>
+    simp only [cycleTrace] at h
+    split_ifs at h with hv
+    obtain ⟨hxS,hxV⟩ := hv
+    rw [seekCycle,ite_eq_left ⟨hxS,hxV⟩,followPath_succeeds,followPath_stale_charge]
+    simp only [Function.comp_id,Finset.image_id]
+    rw [← exposedPaths_insert]
+    exact ih _ _ h
+
+lemma seekCycle_orbit (d : ℕ) (S V : Finset (Card d)) (z : Card d)
+    (k : Finset (Card d) → QueryTree (SwitchIndex d)) (ω : SwitchIndex d → Bool)
+    (hS : orbitSet (butterflyPerm d (decodeButterfly d ω)) z ⊆ S)
+    (hV : Disjoint V (orbitSet (butterflyPerm d (decodeButterfly d ω)) z)) :
+    let p := butterflyPerm d (decodeButterfly d ω)
+    let W := V ∪ orbitSet p z
+    (seekCycle d S (p z) (Function.minimalPeriod p z-1) (p z) V k).succeeds ω = (k W).succeeds ω ∧
+      (seekCycle d S (p z) (Function.minimalPeriod p z-1) (p z) V k).staleCharge
+          (exposedPaths d (decodeButterfly d ω) V) ω =
+        (k W).staleCharge (exposedPaths d (decodeButterfly d ω) W) ω +
+          terminalCharge d (decodeButterfly d ω) z (W.erase z) := by
+  exact seekCycle_semantics _ _ _ _ _ _ _ _ _ _ (cycleTrace_orbit _ _ _ _ hS hV)
+
+end QueryTree
+
+lemma mem_exposedPaths (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) (s : SwitchIndex d) :
+    s ∈ exposedPaths d b V ↔ (switchUsers d b s).1 ∈ V ∨ (switchUsers d b s).2 ∈ V := by
+  classical
+  simp only [exposedPaths,Finset.mem_biUnion,mem_routeDomain_iff_user]
+  constructor
+  · rintro ⟨x,hx,h | h⟩
+    · exact Or.inl (h ▸ hx)
+    · exact Or.inr (h ▸ hx)
+  · rintro (h | h)
+    · exact ⟨_,h,Or.inl rfl⟩
+    · exact ⟨_,h,Or.inr rfl⟩
+
+lemma terminalCharge_sum (d : ℕ) (b : Butterfly d) (z : Card d) (V : Finset (Card d))
+    (hz : z ∉ V) :
+    (terminalCharge d b z V : ℝ) = ∑ s : SwitchIndex d,
+      ((if (switchUsers d b s).1 = z ∧ (switchUsers d b s).2 ∈ V then (1:ℝ) else 0) +
+      (if (switchUsers d b s).2 = z ∧ (switchUsers d b s).1 ∈ V then (1:ℝ) else 0)) := by
+  classical
+  have he : routeDomain d z (butterflyPerm d b z) ∩ exposedPaths d b V =
+      Finset.univ.filter (fun s =>
+        ((switchUsers d b s).1 = z ∧ (switchUsers d b s).2 ∈ V) ∨
+        ((switchUsers d b s).2 = z ∧ (switchUsers d b s).1 ∈ V)) := by
+    ext s
+    simp only [Finset.mem_inter,mem_routeDomain_iff_user,mem_exposedPaths,
+      Finset.mem_filter,Finset.mem_univ,true_and]
+    by_cases h₁ : (switchUsers d b s).1 = z <;>
+      by_cases h₂ : (switchUsers d b s).2 = z <;> simp [h₁,h₂,hz]
+  rw [terminalCharge,he]
+  simp only [← Finset.sum_boole]
+  apply Finset.sum_congr rfl
+  intro s _
+  by_cases h₁ : (switchUsers d b s).1 = z <;>
+    by_cases h₂ : (switchUsers d b s).2 = z <;> simp [h₁,h₂,hz]
+
+noncomputable def priorPaths {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) : Finset (Card d) :=
+  Finset.univ.filter (fun x => ∃ i : Fin A, o x = some i ∧ i.val < t)
+
+@[simp] lemma mem_priorPaths {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) (x : Card d) :
+    x ∈ priorPaths o t ↔ ∃ i : Fin A, o x = some i ∧ i.val < t := by
+  classical
+  simp [priorPaths]
+
+lemma terminalCharge_prior_sum {d A : ℕ} (o : Card d → Option (Fin A))
+    (b : Butterfly d) (z : Terminal o) (i : Fin A) :
+    (terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) =
+      ∑ s : SwitchIndex d,
+        ((if (z i).val = (switchUsers d b s).1 ∧ (switchUsers d b s).2 ∈ priorPaths o (i.val+1)
+          then (1:ℝ) else 0) +
+        (if (z i).val = (switchUsers d b s).2 ∧ (switchUsers d b s).1 ∈ priorPaths o (i.val+1)
+          then (1:ℝ) else 0)) := by
+  classical
+  rw [terminalCharge_sum _ _ _ _ (Finset.notMem_erase _ _)]
+  apply Finset.sum_congr rfl
+  intro s _
+  have hn := switchUsers_ne d b s
+  simp only [Finset.mem_erase]
+  by_cases h₁ : (z i).val = (switchUsers d b s).1 <;>
+    by_cases h₂ : (z i).val = (switchUsers d b s).2 <;> simp_all [eq_comm]
+
+lemma rotationMeeting_eq_slot_sum {d A : ℕ} (o : Card d → Option (Fin A))
+    (z : Terminal o) (u v : Card d) :
+    rotationMeeting o z u v = ∑ i : Fin A,
+      ((if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) +
+      (if (z i).val = v ∧ u ∈ priorPaths o (i.val+1) then (1:ℝ) else 0)) := by
+  classical
+  have only_owner (u : Card d) (a : Fin A) (hou : o u = some a)
+      (i : Fin A) (hia : i ≠ a) : (z i).val ≠ u := by
+    intro h
+    have hi := (z i).property
+    rw [h,hou] at hi
+    exact hia (Option.some.inj hi).symm
+  have sum_one (u v : Card d) (a : Fin A) (hou : o u = some a) :
+      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) =
+      if (z a).val = u ∧ v ∈ priorPaths o (a.val+1) then 1 else 0 := by
+    apply Finset.sum_eq_single a
+    · intro i _ hia
+      simp [only_owner u a hou i hia]
+    · simp
+  have sum_none (u v : Card d) (hou : o u = none) :
+      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) = 0 := by
+    apply Finset.sum_eq_zero
+    intro i _
+    have hn : (z i).val ≠ u := by
+      intro h
+      have hi := (z i).property
+      rw [h,hou] at hi
+      contradiction
+    simp [hn]
+  rw [Finset.sum_add_distrib]
+  cases hou : o u with
+  | none =>
+    rw [sum_none u v hou]
+    simp [rotationMeeting,hou,mem_priorPaths]
+  | some a =>
+    rw [sum_one u v a hou]
+    cases hov : o v with
+    | none =>
+      rw [sum_none v u hov]
+      simp [rotationMeeting,hou,hov,mem_priorPaths]
+    | some b =>
+      rw [sum_one v u b hov]
+      simp only [rotationMeeting,hou,hov,mem_priorPaths,Option.some.injEq]
+      rcases lt_trichotomy a b with hab | rfl | hba
+      · have hn : ¬ b.val < a.val+1 := by omega
+        have hy : a.val < b.val+1 := by omega
+        simp [hab,hn,hy]
+      · simp
+      · have hn : ¬ a.val < b.val+1 := by omega
+        have hy : b.val < a.val+1 := by omega
+        simp [hba,not_lt_of_ge (le_of_lt hba),hn,hy]
+
+lemma rotationCharge_eq_terminal_sum {d A : ℕ} (o : Card d → Option (Fin A))
+    (b : Butterfly d) (z : Terminal o) :
+    rotationCharge o b z = ∑ i : Fin A,
+      (terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) := by
+  simp only [rotationCharge,rotationMeeting_eq_slot_sum,terminalCharge_prior_sum]
+  exact Finset.sum_comm
+
+@[simp] lemma priorPaths_zero {d A : ℕ} (o : Card d → Option (Fin A)) : priorPaths o 0 = ∅ := by
+  classical
+  ext x
+  simp
+
+lemma priorPaths_succ {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) (ht : t < A) :
+    priorPaths o (t+1) = priorPaths o t ∪ cycleFiber o ⟨t,ht⟩ := by
+  classical
+  ext x
+  simp only [mem_priorPaths,Finset.mem_union,mem_cycleFiber]
+  constructor
+  · rintro ⟨i,hi,hit⟩
+    by_cases h : i.val < t
+    · exact Or.inl ⟨i,hi,h⟩
+    · have he : i = ⟨t,ht⟩ := Fin.ext (by change i.val = t; omega)
+      exact Or.inr (he ▸ hi)
+  · rintro (⟨i,hi,hit⟩ | hi)
+    · exact ⟨i,hi,by omega⟩
+    · exact ⟨⟨t,ht⟩,hi,by change t < t+1; omega⟩
+
+lemma priorPaths_disjoint {d A : ℕ} (o : Card d → Option (Fin A)) (i : Fin A) :
+    Disjoint (priorPaths o i.val) (cycleFiber o i) := by
+  classical
+  apply Finset.disjoint_left.mpr
+  intro x hx hx'
+  obtain ⟨j,hj,hji⟩ := (mem_priorPaths _ _ _).mp hx
+  have he : j = i := Option.some.inj (hj.symm.trans ((mem_cycleFiber _ _ _).mp hx'))
+  subst j
+  exact (lt_irrefl _ hji)
+
+namespace QueryTree
+
+lemma seekCycles_owned {d A : ℕ} (S : Finset (Card d)) (r : Fin A → ℕ)
+    (o : Card d → Option (Fin A)) (z : Terminal o) (ω : SwitchIndex d → Bool)
+    (hS : ∀ i, cycleFiber o i ⊆ S)
+    (hO : ∀ i, orbitSet (butterflyPerm d (decodeButterfly d ω)) (z i).val = cycleFiber o i)
+    (hC : ∀ i, (cycleFiber o i).card = r i+1) :
+    let p := butterflyPerm d (decodeButterfly d ω)
+    let L := List.ofFn (fun i : Fin A => (p (z i).val,r i))
+    (seekCycles d S L ∅).succeeds ω = true ∧
+      ((seekCycles d S L ∅).staleCharge ∅ ω : ℝ) = rotationCharge o (decodeButterfly d ω) z := by
+  classical
+  let b := decodeButterfly d ω
+  let p := butterflyPerm d b
+  let f (i : Fin A) := (p (z i).val,r i)
+  let c (i : Fin A) : ℝ := terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val)
+  have hc (i : Fin A) : Function.minimalPeriod p (z i).val - 1 = r i := by
+    have he := orbitSet_card p (z i).val
+    rw [hO,hC] at he
+    omega
+  have hs (t : ℕ) (ht : t ≤ A) :
+      (seekCycles d S ((List.ofFn f).drop t) (priorPaths o t)).succeeds ω = true ∧
+        ((seekCycles d S ((List.ofFn f).drop t) (priorPaths o t)).staleCharge (exposedPaths d b (priorPaths o t)) ω : ℝ) =
+          ((List.ofFn c).drop t).sum := by
+    induction ht using Nat.decreasingInduction with
+    | self => simp only [List.length_ofFn,le_refl,List.drop_eq_nil_of_le,List.sum_nil,seekCycles,succeeds,staleCharge,Nat.cast_zero,and_self]
+    | of_succ t ht ih =>
+      have htf : t < (List.ofFn f).length := by simpa using ht
+      have htc : t < (List.ofFn c).length := by simpa using ht
+      rw [List.drop_eq_getElem_cons htf,List.drop_eq_getElem_cons htc]
+      simp only [List.getElem_ofFn,List.sum_cons,seekCycles]
+      have h := seekCycle_orbit d S (priorPaths o t) (z ⟨t,ht⟩).val
+        (fun V => seekCycles d S ((List.ofFn f).drop (t+1)) V) ω
+        (by rw [hO]; exact hS _) (by rw [hO]; exact priorPaths_disjoint o ⟨t,ht⟩)
+      dsimp only at h
+      rw [hc,hO,←priorPaths_succ o t ht] at h
+      dsimp only [f]
+      constructor
+      · exact h.1.trans ih.1
+      · rw [h.2,Nat.cast_add,ih.2]
+        exact add_comm _ _
+  have h := hs 0 (Nat.zero_le _)
+  simp only [List.drop_zero,priorPaths_zero,exposedPaths_empty,List.sum_ofFn] at h
+  dsimp only
+  exact ⟨h.1,h.2.trans (rotationCharge_eq_terminal_sum o b z).symm⟩
+
+end QueryTree
+
+section Collections
+variable {d A : ℕ}
+
+def ValidCycleSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ)
+    (o : Card d → Option (Fin A)) : Prop :=
+  (∀ i, (cycleFiber o i).card = r i + 1) ∧
+  (∀ i, cycleFiber o i ⊆ S) ∧
+  (∀ i x, o x = some i → orbitSet p x = cycleFiber o i)
+
+abbrev CycleSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
+  {o : Card d → Option (Fin A) // ValidCycleSlots p S r o}
+
+noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
+    Fintype (CycleSlots p S r) := Fintype.ofFinite _
+
+lemma CycleSlots.terminal_nonempty {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (c : CycleSlots p S r) (i : Fin A) : Nonempty {x : Card d // c.val x = some i} := by
+  have hn : (cycleFiber c.val i).Nonempty := Finset.card_pos.mp (by rw [c.property.1]; omega)
+  obtain ⟨x,hx⟩ := hn
+  exact ⟨x,(mem_cycleFiber _ _ _).mp hx⟩
+
+lemma owner_ext {o o' : Card d → Option (Fin A)}
+    (h : ∀ i, cycleFiber o i = cycleFiber o' i) : o = o' := by
+  funext x
+  have he (i : Fin A) : o x = some i ↔ o' x = some i := by
+    simpa only [mem_cycleFiber] using (Finset.ext_iff.mp (h i) x)
+  cases ho : o x with
+  | some i => exact ((he i).mp ho).symm
+  | none =>
+    cases ho' : o' x with
+    | none => rfl
+    | some i => have hn := (he i).mpr ho'; rw [ho] at hn; contradiction
+
+abbrev RootedSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
+  Σ c : CycleSlots p S r, Terminal c.val
+
+noncomputable def rootedStart {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (c : RootedSlots p S r) : Fin A → S := fun i =>
+  ⟨p (c.2 i).val,c.1.property.2.1 i (by
+    rw [←c.1.property.2.2 i (c.2 i).val (c.2 i).property]
+    exact (mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.apply_right (Equiv.Perm.SameCycle.refl p _)))⟩
+
+lemma rootedStart_injective {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ} :
+    Function.Injective (rootedStart (p:=p) (S:=S) (r:=r)) := by
+  rintro ⟨c,z⟩ ⟨c',z'⟩ h
+  have hz (i : Fin A) : (z i).val = (z' i).val :=
+    p.injective (congrArg Subtype.val (congrFun h i))
+  have hc : c = c' := Subtype.ext (owner_ext (fun i => by
+    rw [←c.property.2.2 i (z i).val (z i).property,
+      ←c'.property.2.2 i (z' i).val (z' i).property,hz]))
+  subst c'
+  have hzz : z = z' := funext (fun i => Subtype.ext (hz i))
+  subst z'
+  rfl
+
+lemma sum_comp_injective_le {α β : Type*} [Fintype α] [Fintype β]
+    (e : α → β) (he : Function.Injective e) (f : β → ℝ) (hf : ∀ b, 0 ≤ f b) :
+    ∑ a, f (e a) ≤ ∑ b, f b := by
+  classical
+  calc
+    _ = ∑ b ∈ Finset.univ.image e, f b := (Finset.sum_image (fun _ _ _ _ h => he h)).symm
+    _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _) (fun b _ _ => hf b)
+
+lemma rooted_weight_sum_le (S : Finset (Card d)) (r : Fin A → ℕ) (ω : SwitchIndex d → Bool) :
+    (∑ c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r,
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2)) ≤
+    ∑ a : Fin A → S,
+      if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+        ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+      else 0 := by
+  classical
+  let p := butterflyPerm d (decodeButterfly d ω)
+  let g (a : Fin A → S) : ℝ :=
+    if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+    else 0
+  have he (c : RootedSlots p S r) :
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2) =
+      g (rootedStart c) := by
+    have hs := QueryTree.seekCycles_owned S r c.1.val c.2 ω c.1.property.2.1
+      (fun i => c.1.property.2.2 i (c.2 i).val (c.2 i).property) c.1.property.1
+    dsimp only at hs
+    unfold g
+    rw [ite_eq_left (show (QueryTree.seekCycles d S (List.ofFn (fun i => ((rootedStart c i).val,r i))) ∅).succeeds ω = true from hs.1)]
+    rw [← hs.2,neg_mul,Real.exp_neg,mul_comm (Real.log 2),Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
+    rfl
+  simp_rw [he]
+  exact sum_comp_injective_le rootedStart rootedStart_injective g (fun a => by dsimp [g]; split_ifs <;> positivity)
+
+noncomputable def rotationLower (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : ℝ :=
+  Real.exp (-Real.log 2 * ((A:ℝ)*d/2 + ∑ i : Fin A, Real.log (r i+1) / (2 * Real.log 2)))
+
+noncomputable def collectionWeight (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : ℝ :=
+  ((2:ℝ)^d)^A * (∏ i : Fin A, (r i+1 : ℝ)) * rotationLower d r
+
+lemma collectionWeight_pos (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : 0 < collectionWeight d r := by
+  unfold collectionWeight rotationLower
+  positivity
+
+lemma collection_rotation_lower (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r)
+    (ω : SwitchIndex d → Bool) (c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) :
+    collectionWeight d r ≤ ∑ z : Terminal c.val,
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z) := by
+  classical
+  let (i : Fin A) : Nonempty {x : Card d // c.val x = some i} := c.terminal_nonempty i
+  have hm : Monotone (fun i => (cycleFiber c.val i).card) := by
+    intro i j hij
+    dsimp only
+    rw [c.property.1,c.property.1]
+    exact Nat.add_le_add_right (hr hij) 1
+  have hb := rotation_exp_bound hm (decodeButterfly d ω)
+  have hcard : (Fintype.card (Terminal c.val) : ℝ) = ∏ i : Fin A, (r i+1 : ℝ) := by
+    rw [Fintype.card_pi,Nat.cast_prod]
+    apply Finset.prod_congr rfl
+    intro i _
+    rw [card_cycleFiber,c.property.1,Nat.cast_add,Nat.cast_one]
+  have hp : (0:ℝ) < Fintype.card (Terminal c.val) := Nat.cast_pos.mpr Fintype.card_pos
+  have hl : rotationLower d r ≤ finiteMean (fun z : Terminal c.val => Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z)) := by
+    simpa only [rotationLower,c.property.1,Nat.cast_add,Nat.cast_one] using hb
+  have hh := (le_div_iff₀ hp).mp hl
+  rw [hcard] at hh
+  unfold collectionWeight
+  rw [← Finset.mul_sum]
+  calc
+    _ = ((2:ℝ)^d)^A * (rotationLower d r * ∏ i : Fin A, (r i+1 : ℝ)) := by ring
+    _ ≤ _ := mul_le_mul_of_nonneg_left hh (by positivity)
+
+lemma cycleSlots_weighted_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) *
+      collectionWeight d r ≤ (S.card : ℝ)^A := by
+  classical
+  let g (a : Fin A → S) (ω : SwitchIndex d → Bool) : ℝ :=
+    if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+    else 0
+  have hω (ω : SwitchIndex d → Bool) :
+      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ) * collectionWeight d r ≤
+        ∑ a : Fin A → S, g a ω := by
+    calc
+      _ = ∑ _c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r, collectionWeight d r := by simp
+      _ ≤ ∑ c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r,
+          ∑ z : Terminal c.val, ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z) :=
+        Finset.sum_le_sum (fun c _ => collection_rotation_lower S r hr ω c)
+      _ = ∑ c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r,
+          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2) :=
+        (Fintype.sum_sigma (fun c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r =>
+          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2))).symm
+      _ ≤ _ := rooted_weight_sum_le S r ω
+  have hb := finiteMean_mono hω
+  rw [finiteMean_mul_const,finiteMean_sum] at hb
+  calc
+    _ ≤ ∑ a : Fin A → S, finiteMean (g a) := hb
+    _ ≤ ∑ _a : Fin A → S, (1:ℝ) := by
+      apply Finset.sum_le_sum
+      intro a _
+      have hh := QueryTree.cycles_weighted_success d S (List.ofFn (fun i : Fin A => ((a i).val,r i)))
+      simpa only [List.length_ofFn] using hh
+    _ = _ := by simp
+
+lemma cycleSlots_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) ≤
+        (S.card : ℝ)^A / collectionWeight d r := by
+  exact (le_div_iff₀ (collectionWeight_pos d r)).mpr (cycleSlots_weighted_moment S r hr)
+
+lemma collectionWeight_eq_sqrt (d : ℕ) {A : ℕ} (r : Fin A → ℕ) :
+    collectionWeight d r = (Real.sqrt ((2:ℝ)^d))^A * ∏ i : Fin A, Real.sqrt (r i+1) := by
+  have hn : ((2:ℝ)^d)^A = Real.exp ((A:ℝ)*d*Real.log 2) := by
+    rw [mul_assoc,Real.exp_nat_mul,Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
+  have hp : (∏ i : Fin A, (r i+1 : ℝ)) = Real.exp (∑ i : Fin A, Real.log (r i+1)) := by
+    rw [Real.exp_sum]
+    apply Finset.prod_congr rfl
+    intro i _
+    exact (Real.exp_log (by positivity)).symm
+  have hs : Real.sqrt ((2:ℝ)^d) = Real.exp ((d:ℝ)*Real.log 2/2) := by
+    rw [Real.sqrt_eq_rpow,Real.rpow_def_of_pos (by positivity),Real.log_pow]
+    congr 1
+    ring
+  have hj (i : Fin A) : Real.sqrt (r i+1) = Real.exp (Real.log (r i+1)/2) := by
+    rw [Real.sqrt_eq_rpow,Real.rpow_def_of_pos (by positivity)]
+    congr 1
+    ring
+  simp_rw [collectionWeight,hn,hp,rotationLower,hs,hj]
+  rw [←Real.exp_add,←Real.exp_add,←Real.exp_nat_mul,←Real.exp_sum,←Real.exp_add]
+  congr 1
+  rw [←Finset.sum_div,←Finset.sum_div]
+  have hl : Real.log 2 ≠ 0 := ne_of_gt (Real.log_pos (by norm_num))
+  field_simp
+  ring
+
+lemma butterfly_cycleSlots_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) ≤
+        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
+  have h := cycleSlots_moment S r hr
+  rw [collectionWeight_eq_sqrt,div_mul_eq_div_mul_one_div] at h
+  rw [div_pow]
+  simpa only [one_div,Finset.prod_inv_distrib] using h
+
+end Collections
+
+variable {d A : ℕ}
+
+abbrev SelectedCycle (p : Equiv.Perm (Card d)) (S : Finset (Card d)) :=
+  {C : Finset (Card d) // (∃ x, C = orbitSet p x) ∧ C ⊆ S}
+
+noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) :
+    Fintype (SelectedCycle p S) := Fintype.ofFinite _
+
+lemma SelectedCycle.nonempty {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
+    (C : SelectedCycle p S) : C.val.Nonempty := by
+  obtain ⟨x,hx⟩ := C.property.1
+  rw [hx]
+  exact orbitSet_nonempty p x
+
+lemma SelectedCycle.orbit_eq {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
+    (C : SelectedCycle p S) {x : Card d} (hx : x ∈ C.val) : orbitSet p x = C.val := by
+  obtain ⟨y,hy⟩ := C.property.1
+  rw [hy] at hx ⊢
+  exact (orbitSet_eq p ((mem_orbitSet _ _ _).mp hx)).symm
+
+lemma SelectedCycle.eq_of_mem {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
+    (C D : SelectedCycle p S) {x : Card d} (hC : x ∈ C.val) (hD : x ∈ D.val) : C = D := by
+  apply Subtype.ext
+  exact (C.orbit_eq hC).symm.trans (D.orbit_eq hD)
+
+abbrev SlotAssignment (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
+  {f : Fin A → SelectedCycle p S // Function.Injective f ∧ ∀ i, (f i).val.card = r i+1}
+
+noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
+    Fintype (SlotAssignment p S r) := Fintype.ofFinite _
+
+noncomputable def slotsToAssignment {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (c : CycleSlots p S r) : SlotAssignment p S r := by
+  classical
+  let f (i : Fin A) : SelectedCycle p S := ⟨cycleFiber c.val i,by
+    obtain ⟨x,hx⟩ := c.terminal_nonempty i
+    exact ⟨⟨x,(c.property.2.2 i x hx).symm⟩,c.property.2.1 i⟩⟩
+  refine ⟨f,?_,fun i => c.property.1 i⟩
+  intro i j hij
+  have he : cycleFiber c.val i = cycleFiber c.val j := congrArg Subtype.val hij
+  obtain ⟨x,hx⟩ := c.terminal_nonempty i
+  have hxi : x ∈ cycleFiber c.val i := (mem_cycleFiber _ _ _).mpr hx
+  rw [he] at hxi
+  exact Option.some.inj (hx.symm.trans ((mem_cycleFiber _ _ _).mp hxi))
+
+noncomputable def assignmentOwner {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) (x : Card d) : Option (Fin A) := by
+  classical
+  exact if h : ∃ i, x ∈ (f.val i).val then some (Classical.choose h) else none
+
+end Thorp
+
+end ThorpNine.HighTail
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Main.lean b/lean/OAI/Probability/ThorpRouting/Main.lean
index 706a757ddee8c42360e0209ba1bd268a707340b3..94d3657952538ab7f7f080b0d86d7bd7623f3205 100644
--- a/lean/OAI/Probability/ThorpRouting/Main.lean
+++ b/lean/OAI/Probability/ThorpRouting/Main.lean
@@ -1,33 +1,33 @@
-import OAI.Probability.ThorpRouting.Adaptive.Main
-import OAI.Probability.ThorpRouting.Casimir.Main
-import OAI.Probability.ThorpRouting.Contact.Main
-import OAI.Probability.ThorpRouting.Dense.Main
-import OAI.Probability.ThorpRouting.Harmonic.Main
-import OAI.Probability.ThorpRouting.HighTail.Main
-import OAI.Probability.ThorpRouting.Smoothing.Main
-import OAI.Probability.ThorpRouting.SparseSaving.Main
-import OAI.Probability.ThorpRouting.Tail.Main
-
-namespace OAI
-
-theorem ThorpNine.main :
-    ThorpNine.Adaptive.Thorp.AdaptiveBounds.MainStatement ∧
-    ThorpNine.Casimir.Thorp.CasimirBounds.MainStatement ∧
-    ThorpNine.Dense.Thorp.DenseTruncation.MainStatement ∧
-    ThorpNine.Contact.Thorp.SparseContact.MainStatement ∧
-    ThorpNine.Harmonic.Thorp.HarmonicBounds.MainStatement ∧
-    ThorpNine.Smoothing.Thorp.StrongSmoothing.MainStatement ∧
-    ThorpNine.Tail.Thorp.StrongTail.MainStatement ∧
-    ThorpNine.HighTail.Thorp.HighHeight.MainStatement ∧
-    ThorpNine.SparseSaving.Thorp.SparseRegime.MainStatement := by
-  exact ⟨ThorpNine.Adaptive.Thorp.AdaptiveBounds.adaptive_main,
-    ThorpNine.Casimir.Thorp.CasimirBounds.casimir_main,
-    ThorpNine.Dense.Thorp.DenseTruncation.dense_truncation,
-    ThorpNine.Contact.Thorp.SparseContact.uniform_sparse_contact,
-    ThorpNine.Harmonic.Thorp.HarmonicBounds.harmonic_cycle_main,
-    ThorpNine.Smoothing.Thorp.StrongSmoothing.strong_smoothing,
-    ThorpNine.Tail.Thorp.StrongTail.strong_tail,
-    ThorpNine.HighTail.Thorp.HighHeight.high_tail,
-    ThorpNine.SparseSaving.Thorp.SparseRegime.exponential_sparse_saving⟩
-
-end OAI
+import OAI.Probability.ThorpRouting.Adaptive.Main
+import OAI.Probability.ThorpRouting.Casimir.Main
+import OAI.Probability.ThorpRouting.Contact.Main
+import OAI.Probability.ThorpRouting.Dense.Main
+import OAI.Probability.ThorpRouting.Harmonic.Main
+import OAI.Probability.ThorpRouting.HighTail.Main
+import OAI.Probability.ThorpRouting.Smoothing.Main
+import OAI.Probability.ThorpRouting.SparseSaving.Main
+import OAI.Probability.ThorpRouting.Tail.Main
+
+namespace OAI
+
+theorem ThorpNine.main :
+    ThorpNine.Adaptive.Thorp.AdaptiveBounds.MainStatement ∧
+    ThorpNine.Casimir.Thorp.CasimirBounds.MainStatement ∧
+    ThorpNine.Dense.Thorp.DenseTruncation.MainStatement ∧
+    ThorpNine.Contact.Thorp.SparseContact.MainStatement ∧
+    ThorpNine.Harmonic.Thorp.HarmonicBounds.MainStatement ∧
+    ThorpNine.Smoothing.Thorp.StrongSmoothing.MainStatement ∧
+    ThorpNine.Tail.Thorp.StrongTail.MainStatement ∧
+    ThorpNine.HighTail.Thorp.HighHeight.MainStatement ∧
+    ThorpNine.SparseSaving.Thorp.SparseRegime.MainStatement := by
+  exact ⟨ThorpNine.Adaptive.Thorp.AdaptiveBounds.adaptive_main,
+    ThorpNine.Casimir.Thorp.CasimirBounds.casimir_main,
+    ThorpNine.Dense.Thorp.DenseTruncation.dense_truncation,
+    ThorpNine.Contact.Thorp.SparseContact.uniform_sparse_contact,
+    ThorpNine.Harmonic.Thorp.HarmonicBounds.harmonic_cycle_main,
+    ThorpNine.Smoothing.Thorp.StrongSmoothing.strong_smoothing,
+    ThorpNine.Tail.Thorp.StrongTail.strong_tail,
+    ThorpNine.HighTail.Thorp.HighHeight.high_tail,
+    ThorpNine.SparseSaving.Thorp.SparseRegime.exponential_sparse_saving⟩
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/RegularTrace.lean b/lean/OAI/Probability/ThorpRouting/RegularTrace.lean
index 20b7287cab08ac366e8da5a8cc94cd4e352b4811..eb45fab55711a5ecd8f849e15ee190937dbeedfb 100644
--- a/lean/OAI/Probability/ThorpRouting/RegularTrace.lean
+++ b/lean/OAI/Probability/ThorpRouting/RegularTrace.lean
@@ -1,93 +1,93 @@
-import OAI.Probability.ThorpRouting.TraceBound
-
-namespace OAI
-
-namespace ThorpNine.Harmonic
-
-namespace Thorp.LowPlanes
-open scoped BigOperators Classical
-open Casimir Specht UnitaryFinite
-
-attribute [local instance] hsNorm hsInner hsFinite
-
-lemma sharp_physical_fourier (d t : ℕ) :
-    (2 * distance d t)^2 ≤ ∑ μ : Shapes (2^d),
-      (Module.finrank ℂ (hilbertSpace μ.1) : ℝ)^2 *
-        ‖sampleOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) (run d t) -
-          uniformOperator (V := hilbertSpace μ.1) (cardRepresentation d μ)‖^2 := by
-  let V : Shapes (2^d) → Type := fun μ => hilbertSpace μ.1
-  let : ∀ μ, NormedAddCommGroup (V μ) := fun _ => inferInstance
-  let : ∀ μ, InnerProductSpace ℂ (V μ) := fun _ => inferInstance
-  let : ∀ μ, FiniteDimensional ℂ (V μ) := fun _ => inferInstance
-  let p := fun g : Equiv.Perm (Card d) => law d t g - 1/(Nat.factorial (2^d) : ℝ)
-  have hb := FiniteFourier.l1_fourier_bound V (cardRepresentation d)
-    (cardRepresentation_unitary d) (cardRepresentation_distinct d) (card_conjugacy_shapes d) p
-  have he (μ : Shapes (2^d)) : weightedOperator (V := V μ) (cardRepresentation d μ) p =
-      sampleOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) (run d t) -
-        uniformOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) := by
-    rw [sampleOperator_weighted, sampleLaw_run]
-    dsimp only [p]
-    rw [weighted_sub]
-    congr 1
-    unfold uniformOperator
-    rw [card_permutations]
-    simp only [one_div]
-  simp_rw [he] at hb
-  convert hb using 1
-  change (2 * ((1/2 : ℝ) * ∑ g, |p g|))^2 = (∑ g, |p g|)^2
-  ring
-
-lemma forward_bound (d v : ℕ) (hv : 0 < v) :
-    (2 * distance d (v*d))^2 ≤ tailBound d v := by
-  apply (sharp_physical_fourier d (v*d)).trans
-  apply Finset.sum_le_sum
-  intro μ _
-  by_cases hz : μ.1.card - μ.1.rowLen 0 = 0
-  · simp [hz, card_fourier_full_row d (v*d) μ hz]
-  · simp only [hz, ite_false]
-    have hu : uniformOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) = 0 :=
-      nontrivial_uniform_zero d μ.1 (cardLabels d μ) (by omega)
-    rw [hu, sub_zero]
-    exact mul_le_mul_of_nonneg_left
-      (physical_power_bound d v hv (cardRepresentation d μ) (cardRepresentation_unitary d μ))
-      (sq_nonneg _)
-
-lemma distance_of_tail_bound (d v : ℕ) (hv : 0 < v)
-    (ht : tailBound d v ≤ ((2 : ℝ)^d)^(-10 : ℤ)) :
-    distance d (v*d) ≤ (1/2 : ℝ) * ((2 : ℝ)^d)^(-5 : ℤ) := by
-  have hb := (forward_bound d v hv).trans ht
-  have hn : 0 < (2 : ℝ)^d := by positivity
-  have he : (((2 : ℝ)^d)^(-5 : ℤ))^2 = ((2 : ℝ)^d)^(-10 : ℤ) := by
-    rw [pow_two, ←zpow_add₀ hn.ne']
-    norm_num
-  have hp : 0 < ((2 : ℝ)^d)^(-5 : ℤ) := zpow_pos hn _
-  rw [←he] at hb
-  have hh : 2 * distance d (v*d) ≤ ((2 : ℝ)^d)^(-5 : ℤ) := by
-    nlinarith [distance_nonneg d (v*d)]
-  linarith
-
-theorem regular_trace_smoothing : MainStatement := by
-  obtain ⟨u, hu, htail⟩ := uniform_tail_bound
-  refine ⟨2 * (u : ℝ), ?_, ?_⟩
-  · have h : (1 : ℝ) ≤ u := by exact_mod_cast hu
-    linarith
-  · intro d hd
-    refine ⟨?_, ?_⟩
-    · exact (regularTrace_le d u (by omega)).trans
-        (add_le_add le_rfl (htail d hd u le_rfl))
-    · intro v hv σ
-      have hvR : (u : ℝ) ≤ v := by linarith
-      have hvZ : (u : ℤ) ≤ v := by exact_mod_cast hvR
-      have hv0 : 0 ≤ v := (Int.natCast_nonneg u).trans hvZ
-      have hvn : u ≤ v.toNat := by
-        rw [←Int.toNat_of_nonneg hv0] at hvZ
-        exact_mod_cast hvZ
-      change fromDistance d (v.toNat*d) σ ≤ _
-      rw [fromDistance_eq]
-      exact distance_of_tail_bound d v.toNat (by omega) (htail d hd v.toNat hvn)
-
-end Thorp.LowPlanes
-
-end ThorpNine.Harmonic
-
-end OAI
+import OAI.Probability.ThorpRouting.TraceBound
+
+namespace OAI
+
+namespace ThorpNine.Harmonic
+
+namespace Thorp.LowPlanes
+open scoped BigOperators Classical
+open Casimir Specht UnitaryFinite
+
+attribute [local instance] hsNorm hsInner hsFinite
+
+lemma sharp_physical_fourier (d t : ℕ) :
+    (2 * distance d t)^2 ≤ ∑ μ : Shapes (2^d),
+      (Module.finrank ℂ (hilbertSpace μ.1) : ℝ)^2 *
+        ‖sampleOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) (run d t) -
+          uniformOperator (V := hilbertSpace μ.1) (cardRepresentation d μ)‖^2 := by
+  let V : Shapes (2^d) → Type := fun μ => hilbertSpace μ.1
+  let : ∀ μ, NormedAddCommGroup (V μ) := fun _ => inferInstance
+  let : ∀ μ, InnerProductSpace ℂ (V μ) := fun _ => inferInstance
+  let : ∀ μ, FiniteDimensional ℂ (V μ) := fun _ => inferInstance
+  let p := fun g : Equiv.Perm (Card d) => law d t g - 1/(Nat.factorial (2^d) : ℝ)
+  have hb := FiniteFourier.l1_fourier_bound V (cardRepresentation d)
+    (cardRepresentation_unitary d) (cardRepresentation_distinct d) (card_conjugacy_shapes d) p
+  have he (μ : Shapes (2^d)) : weightedOperator (V := V μ) (cardRepresentation d μ) p =
+      sampleOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) (run d t) -
+        uniformOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) := by
+    rw [sampleOperator_weighted, sampleLaw_run]
+    dsimp only [p]
+    rw [weighted_sub]
+    congr 1
+    unfold uniformOperator
+    rw [card_permutations]
+    simp only [one_div]
+  simp_rw [he] at hb
+  convert hb using 1
+  change (2 * ((1/2 : ℝ) * ∑ g, |p g|))^2 = (∑ g, |p g|)^2
+  ring
+
+lemma forward_bound (d v : ℕ) (hv : 0 < v) :
+    (2 * distance d (v*d))^2 ≤ tailBound d v := by
+  apply (sharp_physical_fourier d (v*d)).trans
+  apply Finset.sum_le_sum
+  intro μ _
+  by_cases hz : μ.1.card - μ.1.rowLen 0 = 0
+  · simp [hz, card_fourier_full_row d (v*d) μ hz]
+  · simp only [hz, ite_false]
+    have hu : uniformOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) = 0 :=
+      nontrivial_uniform_zero d μ.1 (cardLabels d μ) (by omega)
+    rw [hu, sub_zero]
+    exact mul_le_mul_of_nonneg_left
+      (physical_power_bound d v hv (cardRepresentation d μ) (cardRepresentation_unitary d μ))
+      (sq_nonneg _)
+
+lemma distance_of_tail_bound (d v : ℕ) (hv : 0 < v)
+    (ht : tailBound d v ≤ ((2 : ℝ)^d)^(-10 : ℤ)) :
+    distance d (v*d) ≤ (1/2 : ℝ) * ((2 : ℝ)^d)^(-5 : ℤ) := by
+  have hb := (forward_bound d v hv).trans ht
+  have hn : 0 < (2 : ℝ)^d := by positivity
+  have he : (((2 : ℝ)^d)^(-5 : ℤ))^2 = ((2 : ℝ)^d)^(-10 : ℤ) := by
+    rw [pow_two, ←zpow_add₀ hn.ne']
+    norm_num
+  have hp : 0 < ((2 : ℝ)^d)^(-5 : ℤ) := zpow_pos hn _
+  rw [←he] at hb
+  have hh : 2 * distance d (v*d) ≤ ((2 : ℝ)^d)^(-5 : ℤ) := by
+    nlinarith [distance_nonneg d (v*d)]
+  linarith
+
+theorem regular_trace_smoothing : MainStatement := by
+  obtain ⟨u, hu, htail⟩ := uniform_tail_bound
+  refine ⟨2 * (u : ℝ), ?_, ?_⟩
+  · have h : (1 : ℝ) ≤ u := by exact_mod_cast hu
+    linarith
+  · intro d hd
+    refine ⟨?_, ?_⟩
+    · exact (regularTrace_le d u (by omega)).trans
+        (add_le_add le_rfl (htail d hd u le_rfl))
+    · intro v hv σ
+      have hvR : (u : ℝ) ≤ v := by linarith
+      have hvZ : (u : ℤ) ≤ v := by exact_mod_cast hvR
+      have hv0 : 0 ≤ v := (Int.natCast_nonneg u).trans hvZ
+      have hvn : u ≤ v.toNat := by
+        rw [←Int.toNat_of_nonneg hv0] at hvZ
+        exact_mod_cast hvZ
+      change fromDistance d (v.toNat*d) σ ≤ _
+      rw [fromDistance_eq]
+      exact distance_of_tail_bound d v.toNat (by omega) (htail d hd v.toNat hvn)
+
+end Thorp.LowPlanes
+
+end ThorpNine.Harmonic
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Smoothing/AlternatingPerm.lean b/lean/OAI/Probability/ThorpRouting/Smoothing/AlternatingPerm.lean
index fffe85ba9a21a6ab29659941ed33b0b29bd2cab3..ef568ce2835c2778ace6279dd90027c5f57f7f1b 100644
--- a/lean/OAI/Probability/ThorpRouting/Smoothing/AlternatingPerm.lean
+++ b/lean/OAI/Probability/ThorpRouting/Smoothing/AlternatingPerm.lean
@@ -1,895 +1,895 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Smoothing.ParallelFollow
-
-namespace OAI
-
-namespace ThorpNine.Smoothing
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-namespace CycleColoring
-variable {α : Type*} [Fintype α] [DecidableEq α]
-omit [Fintype α] in
-lemma bipartitionFlip_injective (p : Equiv.Perm α) (S : Finset α)
-    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x)) :
-    Function.Injective (bipartitionFlip p S χ hχ) := by
-  classical
-  intro f g he
-  apply Subtype.ext
-  funext x
-  by_cases hx : x ∈ S
-  · have hh := congrFun (congrArg Subtype.val he) x
-    change (if x ∈ S then Bool.xor (f.val x) (χ x) else false) =
-      (if x ∈ S then Bool.xor (g.val x) (χ x) else false) at hh
-    simp only [ite_eq_left hx] at hh
-    cases hfx : f.val x <;> cases hgx : g.val x <;> cases hcx : χ x <;> simp_all
-  · exact (f.property.1 x hx).trans (g.property.1 x hx).symm
-
-lemma card_opposite_le (p : Equiv.Perm α) (S : Finset α)
-    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x)) :
-    Fintype.card (OppositeColoring p S) ≤
-      2 ^ ((boundary p S).card + Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
-  exact (Fintype.card_le_of_injective _ (bipartitionFlip_injective p S χ hχ)).trans
-    (card_coloring_le p S)
-
-def alternatingPerm (p : Equiv.Perm α) : Equiv.Perm (Bool × α) where
-  toFun x := if x.1 then (false,p x.2) else (true,x.2)
-  invFun x := if x.1 then (false,x.2) else (true,p.symm x.2)
-  left_inv x := by rcases x with ⟨b,x⟩; cases b <;> simp
-  right_inv x := by rcases x with ⟨b,x⟩; cases b <;> simp
-
-omit [Fintype α] [DecidableEq α] in
-lemma alternatingPerm_flip (p : Equiv.Perm α) (x : Bool × α) :
-    (alternatingPerm p x).1 = !x.1 := by
-  rcases x with ⟨b,x⟩
-  cases b <;> rfl
-
-omit [Fintype α] [DecidableEq α] in
-lemma alternating_step_sameCycle (p : Equiv.Perm α) (x : Bool × α) :
-    p.SameCycle x.2 (alternatingPerm p x).2 := by
-  rcases x with ⟨b,x⟩
-  cases b
-  · exact Equiv.Perm.SameCycle.refl p x
-  · exact (Equiv.Perm.SameCycle.refl p x).apply_right
-
-omit [DecidableEq α] in
-lemma alternating_sameCycle_snd (p : Equiv.Perm α) {x y : Bool × α}
-    (h : (alternatingPerm p).SameCycle x y) : p.SameCycle x.2 y.2 := by
-  obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
-  have hp (k : ℕ) : p.SameCycle x.2 (((alternatingPerm p)^k) x).2 := by
-    induction k with
-    | zero => exact Equiv.Perm.SameCycle.refl p x.2
-    | succ k ih =>
-      rw [pow_succ',Equiv.Perm.mul_apply]
-      exact ih.trans (alternating_step_sameCycle p _)
-  exact hn ▸ hp n
-
-omit [DecidableEq α] in
-lemma alternating_zero_sameCycle (p : Equiv.Perm α) {x y : α}
-    (h : p.SameCycle x y) : (alternatingPerm p).SameCycle (false,x) (false,y) := by
-  obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
-  have hp (k : ℕ) : (alternatingPerm p).SameCycle (false,x) (false,(p^k) x) := by
-    induction k with
-    | zero => exact Equiv.Perm.SameCycle.refl _ _
-    | succ k ih =>
-      have hs := ih.apply_right.apply_right
-      simpa only [pow_succ',Equiv.Perm.mul_apply,alternatingPerm,Equiv.coe_fn_mk,
-        Bool.false_eq_true,↓reduceIte] using hs
-  simpa only [hn] using hp n
-
-omit [Fintype α] [DecidableEq α] in
-lemma alternating_sameCycle_zero (p : Equiv.Perm α) (x : Bool × α) :
-    (alternatingPerm p).SameCycle x (false,x.2) := by
-  rcases x with ⟨b,x⟩
-  cases b
-  · exact Equiv.Perm.SameCycle.refl _ _
-  · exact ((Equiv.Perm.SameCycle.refl (alternatingPerm p) (false,x)).apply_right).symm
-
-omit [DecidableEq α] in
-lemma alternating_sameCycle_iff (p : Equiv.Perm α) (x y : Bool × α) :
-    (alternatingPerm p).SameCycle x y ↔ p.SameCycle x.2 y.2 := by
-  constructor
-  · exact alternating_sameCycle_snd p
-  · intro h
-    exact (alternating_sameCycle_zero p x).trans
-      ((alternating_zero_sameCycle p h).trans (alternating_sameCycle_zero p y).symm)
-
-lemma orbit_alternating (p : Equiv.Perm α) (x : Bool × α) :
-    orbitSet (alternatingPerm p) x = childJoin (orbitSet p x.2) (orbitSet p x.2) := by
-  ext ⟨b,y⟩
-  simp only [mem_orbitSet,alternating_sameCycle_iff,mem_childJoin,ite_self]
-
-omit [Fintype α] in
-lemma boolFiber_childJoin (A B : Finset α) (b : Bool) :
-    boolFiber (childJoin A B) b = if b then B else A := by
-  ext x
-  cases b <;> simp only [mem_boolFiber,mem_childJoin,Bool.false_eq_true,↓reduceIte]
-
-noncomputable def projectCycle (p : Equiv.Perm α) (A B : Finset α)
-    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) :
-    RoutingNetwork.SelectedCycle p A := by
-  refine ⟨boolFiber C.val false,?_,?_⟩
-  · obtain ⟨x,hx⟩ := C.property.1
-    refine ⟨x.2,?_⟩
-    rw [hx,orbit_alternating,boolFiber_childJoin]
-    rfl
-  · intro x hx
-    exact (mem_childJoin A B false x).mp (C.property.2 ((mem_boolFiber _ _ _).mp hx))
-
-lemma selected_alternating_eq (p : Equiv.Perm α) (A B : Finset α)
-    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) :
-    C.val = childJoin (projectCycle p A B C).val (projectCycle p A B C).val := by
-  change C.val = childJoin (boolFiber C.val false) (boolFiber C.val false)
-  obtain ⟨x,hx⟩ := C.property.1
-  rw [hx,orbit_alternating,boolFiber_childJoin]
-  rfl
-
-lemma projectCycle_injective (p : Equiv.Perm α) (A B : Finset α) :
-    Function.Injective (projectCycle p A B) := by
-  intro C D he
-  apply Subtype.ext
-  rw [selected_alternating_eq p A B C,selected_alternating_eq p A B D,he]
-
-lemma card_alternating_cycles_le (p : Equiv.Perm α) (A B : Finset α) :
-    Fintype.card (RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) ≤
-      Fintype.card (RoutingNetwork.SelectedCycle p A) :=
-  Fintype.card_le_of_injective _ (projectCycle_injective p A B)
-
-end CycleColoring
-
-section PairAssignmentCount
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-omit [Fintype α] in
-lemma mem_image_perm_iff (p : Equiv.Perm α) (A : Finset α) (y : α) :
-    y ∈ A.image p ↔ p.symm y ∈ A := by
-  constructor
-  · intro h
-    obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp h
-    simpa only [←hxy,Equiv.symm_apply_apply] using hx
-  · intro h
-    exact Finset.mem_image.mpr ⟨p.symm y,h,p.apply_symm_apply y⟩
-
-omit [Fintype α] in
-lemma card_childJoin (A B : Finset α) : (childJoin A B).card = A.card+B.card := by
-  have hd : Disjoint (A.image (fun x => (false,x))) (B.image (fun x => (true,x))) := by
-    apply Finset.disjoint_left.mpr
-    rintro x hx hy
-    obtain ⟨a,ha,rfl⟩ := Finset.mem_image.mp hx
-    obtain ⟨b,hb,hba⟩ := Finset.mem_image.mp hy
-    exact Bool.false_ne_true (congrArg Prod.fst hba).symm
-  have hi (b : Bool) : Function.Injective (fun x : α => (b,x)) :=
-    fun _ _ h => congrArg Prod.snd h
-  rw [childJoin,Finset.card_union_of_disjoint hd,
-    Finset.card_image_of_injective A (hi false),Finset.card_image_of_injective B (hi true)]
-
-omit [Fintype α] in
-lemma alternating_internal (p : Equiv.Perm α) (A B : Finset α) :
-    ((childJoin A B).filter (fun x => (CycleColoring.alternatingPerm p).symm x ∈ childJoin A B)) =
-      childJoin (A ∩ B.image p) (B ∩ A) := by
-  ext ⟨b,x⟩
-  rw [Finset.mem_filter]
-  cases b <;> simp only [mem_childJoin,CycleColoring.alternatingPerm,
-    Equiv.symm_mk,Equiv.coe_fn_mk,Bool.false_eq_true,↓reduceIte,Finset.mem_inter,mem_image_perm_iff]
-
-omit [Fintype α] in
-lemma alternating_boundary_card (p : Equiv.Perm α) (A B : Finset α) :
-    (CycleColoring.boundary (CycleColoring.alternatingPerm p) (childJoin A B)).card =
-      (A ∪ B).card + (A ∪ B.image p).card - (A.card+B.card) := by
-  rw [CycleColoring.boundary_card,alternating_internal,card_childJoin,card_childJoin]
-  have h₀ := Finset.card_union_add_card_inter A B
-  have h₁ := Finset.card_union_add_card_inter A (B.image p)
-  rw [Finset.card_image_of_injective _ p.injective] at h₁
-  have h₂ := Finset.inter_comm B A
-  rw [h₂]
-  omega
-
-lemma compatible_assignment_bound (p : Equiv.Perm α) (A B : Finset α) :
-    Fintype.card (CycleColoring.OppositeColoring (CycleColoring.alternatingPerm p) (childJoin A B)) ≤
-      2 ^ ((A ∪ B).card + (A ∪ B.image p).card - (A.card+B.card) +
-        Fintype.card (RoutingNetwork.SelectedCycle p A)) := by
-  have h := CycleColoring.card_opposite_le (CycleColoring.alternatingPerm p) (childJoin A B)
-    Prod.fst (CycleColoring.alternatingPerm_flip p)
-  rw [alternating_boundary_card] at h
-  exact h.trans (Nat.pow_le_pow_right (by norm_num)
-    (Nat.add_le_add_left (CycleColoring.card_alternating_cycles_le p A B) _))
-
-end PairAssignmentCount
-
-lemma finiteMean_nonneg {Ω : Type*} [Fintype Ω] {f : Ω → ℝ}
-    (h : ∀ ω, 0 ≤ f ω) : 0 ≤ finiteMean f := by
-  exact div_nonneg (Finset.sum_nonneg (fun ω _ => h ω)) (Nat.cast_nonneg _)
-
-lemma boolFiber_headTail (d : ℕ) (S : Finset (Card (d+1))) (b : Bool) :
-    boolFiber (S.image (headTailEquiv d)) b = childMarks S b := by
-  classical
-  ext x
-  rw [mem_boolFiber,mem_childMarks]
-  constructor
-  · intro h
-    obtain ⟨y,hy,he⟩ := Finset.mem_image.mp h
-    have hh := congrArg (headTailEquiv d).symm he
-    rw [Equiv.symm_apply_apply] at hh
-    change y = Fin.cons b x at hh
-    exact hh ▸ hy
-  · intro h
-    exact Finset.mem_image.mpr ⟨Fin.cons b x,h,by rfl⟩
-
-lemma prod_childMarks (d : ℕ) (S : Finset (Card (d+1))) (f : Card (d+1) → ℝ) :
-    (∏ x ∈ S, f x) =
-      (∏ x ∈ childMarks S false, f (Fin.cons false x)) *
-      ∏ x ∈ childMarks S true, f (Fin.cons true x) := by
-  classical
-  have h := prod_boolFibers (S.image (headTailEquiv d)) (fun x => f ((headTailEquiv d).symm x))
-  rw [Finset.prod_image (headTailEquiv d).injective.injOn,
-    boolFiber_headTail,boolFiber_headTail] at h
-  simp only [Equiv.symm_apply_apply] at h
-  exact h
-
-noncomputable def headAverage {d : ℕ} (p : Card (d+1) → ℝ) (y : Card d) : ℝ :=
-  (p (Fin.cons false y)+p (Fin.cons true y))/2
-
-lemma finiteMean_headAverage (d : ℕ) (p : Card (d+1) → ℝ) :
-    finiteMean (headAverage p) = finiteMean p := by
-  rw [finiteMean_equiv (headTailEquiv d),finiteMean_prod,finiteMean_comm]
-  apply finiteMean_congr
-  intro y
-  rw [finiteMean_bool]
-  rfl
-
-lemma pairSwitch_product_bound (d : ℕ) (S : Finset (Card (d+1)))
-    (p : Card (d+1) → ℝ) (hp : ∀ x, 0 ≤ p x) :
-    finiteMean (fun ξ : Card d → Bool => ∏ x ∈ S, p (pairSwitch ξ x)) ≤
-      ∏ x ∈ S, headAverage p (Fin.tail x) := by
-  classical
-  have h : CylinderBound (fun T => ∏ x ∈ T, p ((headTailEquiv d).symm x))
-      (fun x => p ((headTailEquiv d).symm x)) := fun _ => le_rfl
-  have hh := h.fair_parallel (fun x => hp _) (S.image (headTailEquiv d))
-  simp only [Finset.prod_image (pairLayer _).injective.injOn,
-    Finset.prod_image (headTailEquiv d).injective.injOn] at hh
-  convert hh using 1
-  · apply finiteMean_congr
-    intro ξ
-    apply Finset.prod_congr rfl
-    intro x _
-    rfl
-  · apply Finset.prod_congr rfl
-    intro x _
-    change (p (Fin.cons false (Fin.tail x))+p (Fin.cons true (Fin.tail x)))/2 =
-      (p (Fin.cons (x 0) (Fin.tail x))+p (Fin.cons (!(x 0)) (Fin.tail x)))/2
-    cases x 0
-    · rfl
-    · simp only [Bool.not_true]
-      ring
-
-lemma butterflyPerm_step (d : ℕ)
-    (c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (ξ : Card d → Bool) :
-    butterflyPerm (d+1) (decodeButterfly (d+1) ((coinStepEquiv d).symm (c,ξ))) =
-    pairSwitch ξ * childLift (fun b => butterflyPerm d (decodeButterfly d (if b then c.2 else c.1))) := by
-  change pairSwitch (fun y => (coinStepEquiv d).symm (c,ξ) (Sum.inl y)) *
-    childLift (fun b => butterflyPerm d (decodeButterfly d
-      (fun i => (coinStepEquiv d).symm (c,ξ) (Sum.inr (b,i))))) = _
-  apply congrArg₂ (· * ·)
-  · rfl
-  · apply congrArg childLift
-    funext b
-    apply congrArg (fun ω => butterflyPerm d (decodeButterfly d ω))
-    funext i
-    cases b <;> rfl
-
-lemma butterfly_product_bound (d : ℕ) (S : Finset (Card d))
-    (p : Card d → ℝ) (hp : ∀ x, 0 ≤ p x) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      ∏ x ∈ S, p (butterflyPerm d (decodeButterfly d ω) x)) ≤ (finiteMean p)^S.card := by
-  classical
-  induction d with
-  | zero =>
-    have he : p = fun _ => p (fun i => Fin.elim0 i) := by
-      funext x
-      exact congrArg p (Subsingleton.elim _ _)
-    rw [he]
-    simp only [Finset.prod_const,finiteMean_const,le_refl]
-  | succ d ih =>
-    let g : Card d → ℝ := headAverage p
-    have hg : ∀ x, 0 ≤ g x := fun x => div_nonneg (add_nonneg (hp _) (hp _)) (by norm_num)
-    rw [finiteMean_equiv (coinStepEquiv d),finiteMean_prod]
-    calc
-      _ ≤ finiteMean (fun c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool) =>
-        (∏ x ∈ childMarks S false, g (butterflyPerm d (decodeButterfly d c.1) x)) *
-        ∏ x ∈ childMarks S true, g (butterflyPerm d (decodeButterfly d c.2) x)) := by
-        apply finiteMean_mono
-        intro c
-        let f : Equiv.Perm (Card (d+1)) := childLift (fun b =>
-          butterflyPerm d (decodeButterfly d (if b then c.2 else c.1)))
-        have hh := pairSwitch_product_bound d (S.image f) p hp
-        simp only [Finset.prod_image f.injective.injOn] at hh
-        convert hh using 1
-        · apply finiteMean_congr
-          intro ξ
-          apply Finset.prod_congr rfl
-          intro x _
-          rw [butterflyPerm_step]
-          rfl
-        · rw [prod_childMarks]
-          simp only [f,childLift,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
-            Bool.false_eq_true,↓reduceIte]
-          rfl
-      _ = (finiteMean (fun c : SwitchIndex d → Bool =>
-        ∏ x ∈ childMarks S false, g (butterflyPerm d (decodeButterfly d c) x))) *
-        finiteMean (fun c : SwitchIndex d → Bool =>
-        ∏ x ∈ childMarks S true, g (butterflyPerm d (decodeButterfly d c) x)) :=
-          finiteMean_prod_mul
-            (fun c : SwitchIndex d → Bool => ∏ x ∈ childMarks S false,
-              g (butterflyPerm d (decodeButterfly d c) x))
-            (fun c : SwitchIndex d → Bool => ∏ x ∈ childMarks S true,
-              g (butterflyPerm d (decodeButterfly d c) x))
-      _ ≤ (finiteMean g)^(childMarks S false).card * (finiteMean g)^(childMarks S true).card :=
-        mul_le_mul (ih (childMarks S false) g hg) (ih (childMarks S true) g hg)
-          (finiteMean_nonneg fun c => Finset.prod_nonneg (fun x _ => hg _))
-          (pow_nonneg (finiteMean_nonneg hg) _)
-      _ = (finiteMean p)^S.card := by
-        rw [←pow_add,card_childMarks_add,finiteMean_headAverage]
-
-noncomputable def inverseButterflyMarks (d : ℕ) (A : Finset (Card d))
-    (ω : SwitchIndex d → Bool) : Finset (Card d) :=
-  A.image (butterflyPerm d (decodeButterfly d ω)).symm
-
-lemma inverseButterflyMarks_inclusion (d : ℕ) (A S : Finset (Card d)) :
-    inclusionMoment (inverseButterflyMarks d A) S ≤
-      ((A.card : ℝ)/(2:ℝ)^d)^S.card := by
-  classical
-  let p : Card d → ℝ := fun x => if x ∈ A then 1 else 0
-  have hm : finiteMean p = (A.card : ℝ)/(2:ℝ)^d := by
-    simp only [finiteMean,p,card_positions,Nat.cast_pow,Nat.cast_ofNat]
-    congr 1
-    simp
-  have he : inclusionMoment (inverseButterflyMarks d A) S =
-      finiteMean (fun ω : SwitchIndex d → Bool =>
-        ∏ x ∈ S, p (butterflyPerm d (decodeButterfly d ω) x)) := by
-    rw [inclusionMoment_eq]
-    apply finiteMean_congr
-    intro ω
-    simp only [inverseButterflyMarks,subset_image_equiv_iff,Equiv.symm_symm,indicator_subset_prod,
-      Finset.prod_image (butterflyPerm d (decodeButterfly d ω)).injective.injOn,p]
-  rw [he]
-  simpa only [hm] using butterfly_product_bound d S p (fun x => by dsimp [p]; split_ifs <;> norm_num)
-
-namespace PairRouting
-
-section
-variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
-
-noncomputable def occupied (x : ι → Bool × α) : Finset α :=
-  Finset.univ.image (fun i => (x i).2)
-
-omit [Fintype α] in
-lemma mem_occupied (x : ι → Bool × α) (y : α) : y ∈ occupied x ↔ ∃ i, (x i).2 = y := by
-  simp [occupied]
-
-def required (x : ι → Bool × α) (c : ι → Bool) (i : ι) : Bool := Bool.xor (x i).1 (c i)
-
-def Compatible (x : ι → Bool × α) (c : ι → Bool) : Prop :=
-  ∀ i j, (x i).2 = (x j).2 → required x c i = required x c j
-
-def colors (x : ι → Bool × α) (ξ : α → Bool) (i : ι) : Bool :=
-  Bool.xor (x i).1 (ξ (x i).2)
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma required_colors (x : ι → Bool × α) (ξ : α → Bool) (i : ι) :
-    required x (colors x ξ) i = ξ (x i).2 := by
-  simp only [required,colors]
-  cases (x i).1 <;> cases ξ (x i).2 <;> rfl
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma colors_compatible (x : ι → Bool × α) (ξ : α → Bool) : Compatible x (colors x ξ) := by
-  intro i j he
-  simp only [required_colors,he]
-
-noncomputable def chosenCoin (x : ι → Bool × α) (c : ι → Bool) (y : α) : Bool := by
-  classical
-  exact if h : ∃ i, (x i).2 = y then required x c (Classical.choose h) else false
-
-omit [Fintype α] in
-lemma chosenCoin_apply (x : ι → Bool × α) (c : ι → Bool) (hc : Compatible x c) (i : ι) :
-    chosenCoin x c (x i).2 = required x c i := by
-  classical
-  simp only [chosenCoin,dite_eq_left (show ∃ j, (x j).2 = (x i).2 from ⟨i,rfl⟩)]
-  exact hc _ i (Classical.choose_spec (show ∃ j, (x j).2 = (x i).2 from ⟨i,rfl⟩))
-
-omit [Fintype α] in
-lemma colors_eq_iff (x : ι → Bool × α) (c : ι → Bool) (hc : Compatible x c) (ξ : α → Bool) :
-    colors x ξ = c ↔ ∀ y ∈ occupied x, ξ y = chosenCoin x c y := by
-  constructor
-  · intro he y hy
-    obtain ⟨i,rfl⟩ := (mem_occupied x y).mp hy
-    rw [chosenCoin_apply x c hc i]
-    rw [←he,required_colors]
-  · intro he
-    funext i
-    apply (xor_eq_iff _ _ _).mpr
-    exact (he _ ((mem_occupied x _).mpr ⟨i,rfl⟩)).trans (chosenCoin_apply x c hc i)
-
-noncomputable instance (x : ι → Bool × α) (c : ι → Bool) : Decidable (Compatible x c) :=
-  Classical.propDecidable _
-
-lemma colors_probability (x : ι → Bool × α) (c : ι → Bool) :
-    finiteMean (fun ξ : α → Bool => if colors x ξ = c then (1 : ℝ) else 0) =
-      if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0 := by
-  classical
-  by_cases hc : Compatible x c
-  · rw [ite_eq_left hc]
-    have he := fun ξ => colors_eq_iff x c hc ξ
-    simp only [he]
-    exact mean_coinCylinder (occupied x) (chosenCoin x c)
-  · rw [ite_eq_right hc]
-    have he : ∀ ξ, colors x ξ ≠ c := fun ξ hh => hc (hh ▸ colors_compatible x ξ)
-    simp only [he,ite_false,finiteMean_const]
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma Compatible.tail_injective {x : ι ↪ Bool × α} {c : ι → Bool} (hc : Compatible x c) (b : Bool) :
-    Function.Injective (fun i : {i : ι // c i = b} => (x i.val).2) := by
-  intro i j he
-  apply Subtype.ext
-  apply x.injective
-  apply Prod.ext
-  · have hh := hc i j he
-    simp only [required,i.property,j.property] at hh
-    cases h₀ : (x i.val).1 <;> cases h₁ : (x j.val).1 <;> cases b <;> simp_all
-  · exact he
-
-def childEmbedding (x : ι ↪ Bool × α) (c : ι → Bool) (hc : Compatible x c) (b : Bool) :
-    {i : ι // c i = b} ↪ α := ⟨fun i => (x i.val).2,hc.tail_injective b⟩
-
-
-def lift (p : Bool → Equiv.Perm α) : Equiv.Perm (Bool × α) where
-  toFun x := (x.1,p x.1 x.2)
-  invFun x := (x.1,(p x.1).symm x.2)
-  left_inv x := by simp
-  right_inv x := by simp
-
-def sandwich (ξ η : α → Bool) (p : Bool → Equiv.Perm α) : Equiv.Perm (Bool × α) :=
-  pairLayer η * lift p * pairLayer ξ
-
-def ChildMatches (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) : Prop :=
-  ∀ i, p (c i) (x i).2 = (y i).2
-
-noncomputable instance (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
-    Decidable (ChildMatches p x y c) := Classical.propDecidable _
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma sandwich_matches_iff (ξ η : α → Bool) (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) :
-    (∀ i, sandwich ξ η p (x i) = y i) ↔
-      colors x ξ = colors y η ∧ ChildMatches p x y (colors x ξ) := by
-  have he (i : ι) : sandwich ξ η p (x i) = y i ↔
-      colors x ξ i = colors y η i ∧ p (colors x ξ i) (x i).2 = (y i).2 := by
-    change pairLayer η (lift p (pairLayer ξ (x i))) = y i ↔ _
-    rw [←Equiv.eq_symm_apply]
-    change (colors x ξ i,p (colors x ξ i) (x i).2) = (colors y η i,(y i).2) ↔ _
-    exact ⟨Prod.mk.inj,fun h => Prod.ext h.1 h.2⟩
-  simp only [he,forall_and,←funext_iff,ChildMatches]
-
-lemma sandwich_probability [DecidableEq ι] (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) :
-    finiteMean (fun coins : (α → Bool) × (α → Bool) =>
-      if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then (1:ℝ) else 0) =
-    ∑ c : ι → Bool,
-      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
-      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) *
-      (if ChildMatches p x y c then 1 else 0) := by
-  classical
-  have he (coins : (α → Bool) × (α → Bool)) :
-      (if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then (1:ℝ) else 0) =
-      ∑ c : ι → Bool, (if colors x coins.1 = c then (1:ℝ) else 0) *
-        (if colors y coins.2 = c then 1 else 0) * (if ChildMatches p x y c then 1 else 0) := by
-    rw [Finset.sum_eq_single (colors x coins.1)]
-    · by_cases hc : colors x coins.1 = colors y coins.2
-      · simp only [sandwich_matches_iff,hc,true_and,↓reduceIte,one_mul]
-      · simp only [sandwich_matches_iff,hc,false_and,↓reduceIte,one_mul,Ne.symm hc,zero_mul]
-    · intro c _ hc
-      simp only [Ne.symm hc,ite_false,zero_mul]
-    · simp
-  simp_rw [he]
-  rw [finiteMean_sum]
-  apply Finset.sum_congr rfl
-  intro c _
-  rw [finiteMean_mul_const]
-  rw [finiteMean_prod_mul
-    (fun ξ : α → Bool => if colors x ξ = c then (1:ℝ) else 0)
-    (fun η : α → Bool => if colors y η = c then (1:ℝ) else 0),
-    colors_probability,colors_probability]
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma childMatches_split (p₀ p₁ : Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
-    ChildMatches (fun b => if b then p₁ else p₀) x y c ↔
-      (∀ i : {i // c i = false}, p₀ (x i.val).2 = (y i.val).2) ∧
-      (∀ i : {i // c i = true}, p₁ (x i.val).2 = (y i.val).2) := by
-  constructor
-  · intro h
-    constructor
-    · intro i
-      simpa only [i.property,Bool.false_eq_true,↓reduceIte] using h i.val
-    · intro i
-      simpa only [i.property,↓reduceIte] using h i.val
-  · rintro ⟨h₀,h₁⟩ i
-    cases hi : c i
-    · simpa only [hi,Bool.false_eq_true,↓reduceIte] using h₀ ⟨i,hi⟩
-    · simpa only [hi,↓reduceIte] using h₁ ⟨i,hi⟩
-
-omit [Fintype α] in
-lemma childMatches_mean {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
-    (P : Ω → Equiv.Perm α) (Q : Ω' → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
-    finiteMean (fun ω : Ω × Ω' =>
-      if ChildMatches (fun b => if b then Q ω.2 else P ω.1) x y c then (1:ℝ) else 0) =
-      tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
-        (fun i : {i // c i = false} => (y i.val).2) *
-      tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
-        (fun i : {i // c i = true} => (y i.val).2) := by
-  classical
-  let f : Ω → ℝ := fun ω => if ∀ i : {i // c i = false}, P ω (x i.val).2 = (y i.val).2 then 1 else 0
-  let g : Ω' → ℝ := fun ω => if ∀ i : {i // c i = true}, Q ω (x i.val).2 = (y i.val).2 then 1 else 0
-  have he (ω : Ω × Ω') :
-      (if ChildMatches (fun b => if b then Q ω.2 else P ω.1) x y c then (1:ℝ) else 0) = f ω.1*g ω.2 := by
-    simp only [childMatches_split,f,g]
-    split_ifs <;> simp_all
-  simp_rw [he]
-  exact finiteMean_prod_mul f g
-
-lemma sandwich_mean [DecidableEq ι] {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
-    (P : Ω → Equiv.Perm α) (Q : Ω' → Equiv.Perm α) (x y : ι → Bool × α) :
-    finiteMean (fun ω : Ω × Ω' => finiteMean (fun coins : (α → Bool) × (α → Bool) =>
-      if ∀ i, sandwich coins.1 coins.2 (fun b => if b then Q ω.2 else P ω.1) (x i) = y i
-      then (1:ℝ) else 0)) =
-    ∑ c : ι → Bool,
-      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
-      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) *
-      (tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
-        (fun i : {i // c i = false} => (y i.val).2) *
-      tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
-        (fun i : {i // c i = true} => (y i.val).2)) := by
-  classical
-  simp_rw [sandwich_probability]
-  rw [finiteMean_sum]
-  apply Finset.sum_congr rfl
-  intro c _
-  simp only [mul_comm ((if Compatible x c then _ else _) * _),finiteMean_mul_const,
-    childMatches_mean]
-
-
-abbrev BothAssignments (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :=
-  {c : ι → Bool // Compatible e c ∧ Compatible (fun i => lift p (e i)) c}
-
-noncomputable instance (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    Fintype (BothAssignments e p) := Fintype.ofFinite _
-
-noncomputable def labelSet (e : ι ↪ Bool × α) : Finset (Bool × α) := Finset.univ.image e
-
-def alternatingProjection (p : Bool → Equiv.Perm α) : Equiv.Perm α := (p false).symm * p true
-
-noncomputable def assignmentColoring (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α)
-    (c : BothAssignments e p) : CycleColoring.OppositeColoring
-      (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e) := by
-  classical
-  refine ⟨Function.extend e c.val (fun _ => false),?_,?_⟩
-  · intro z hz
-    apply Function.extend_apply'
-    rintro ⟨i,rfl⟩
-    exact hz (Finset.mem_image.mpr ⟨i,Finset.mem_univ _,rfl⟩)
-  · intro z hz hnext
-    obtain ⟨i,_,hi⟩ := Finset.mem_image.mp hz
-    obtain ⟨j,_,hj⟩ := Finset.mem_image.mp hnext
-    have hfi := e.injective.extend_apply c.val (fun _ => false) i
-    have hfj := e.injective.extend_apply c.val (fun _ => false) j
-    rw [hi] at hfi
-    rw [hj] at hfj
-    rw [hfi,hfj]
-    rcases z with ⟨b,a⟩
-    cases b
-    · have hj' : e j = (true,a) := hj
-      have ht : (e i).2 = (e j).2 := by rw [hi,hj']
-      have hh := c.property.1 i j ht
-      simp only [required,hi,hj'] at hh
-      revert hh
-      generalize c.val i = bi
-      generalize c.val j = bj
-      cases bi <;> cases bj <;> decide
-    · have hj' : e j = (false,alternatingProjection p a) := hj
-      have ht : (lift p (e i)).2 = (lift p (e j)).2 := by
-        simp only [hi,hj',lift,Equiv.coe_fn_mk,alternatingProjection,Equiv.Perm.coe_mul,
-          Function.comp_apply,Equiv.apply_symm_apply]
-      have hh := c.property.2 i j ht
-      simp only [required,hi,hj',lift,Equiv.coe_fn_mk] at hh
-      revert hh
-      generalize c.val i = bi
-      generalize c.val j = bj
-      cases bi <;> cases bj <;> decide
-
-omit [Fintype α] in
-lemma assignmentColoring_injective (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    Function.Injective (assignmentColoring e p) := by
-  intro c c' he
-  apply Subtype.ext
-  funext i
-  have hh := congrFun (congrArg Subtype.val he) (e i)
-  change Function.extend e c.val (fun _ => false) (e i) =
-    Function.extend e c'.val (fun _ => false) (e i) at hh
-  simpa only [e.injective.extend_apply] using hh
-
-lemma bothAssignments_card (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    Fintype.card (BothAssignments e p) ≤
-      2 ^ ((boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true).card +
-        (boolFiber (labelSet e) false ∪ (boolFiber (labelSet e) true).image
-          (alternatingProjection p)).card -
-        ((boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card) +
-        Fintype.card (RoutingNetwork.SelectedCycle (alternatingProjection p) (boolFiber (labelSet e) false))) := by
-  have h := compatible_assignment_bound (alternatingProjection p)
-    (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
-  rw [childJoin_boolFiber] at h
-  exact (Fintype.card_le_of_injective _ (assignmentColoring_injective e p)).trans h
-
-omit [Fintype α] in
-lemma occupied_labelSet (e : ι ↪ Bool × α) : occupied e = (labelSet e).image Prod.snd := by
-  classical
-  simp only [occupied,labelSet,Finset.image_image,Function.comp_def]
-
-omit [Fintype ι] [Fintype α] in
-lemma image_snd_childJoin (A B : Finset α) : (childJoin A B).image Prod.snd = A ∪ B := by
-  classical
-  simp only [childJoin,Finset.image_union,Finset.image_image,Function.comp_def,Finset.image_id']
-
-omit [Fintype ι] [Fintype α] in
-lemma image_lift_childJoin (p : Bool → Equiv.Perm α) (A B : Finset α) :
-    (childJoin A B).image (fun x => (lift p x).2) = A.image (p false) ∪ B.image (p true) := by
-  classical
-  simp only [childJoin,Finset.image_union,Finset.image_image,Function.comp_def,lift,Equiv.coe_fn_mk]
-
-omit [Fintype α] in
-lemma occupied_lift_card (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    (occupied (fun i => lift p (e i))).card =
-      (boolFiber (labelSet e) false ∪ (boolFiber (labelSet e) true).image (alternatingProjection p)).card := by
-  classical
-  have he : occupied (fun i => lift p (e i)) = (labelSet e).image (fun x => (lift p x).2) := by
-    simp only [occupied,labelSet,Finset.image_image,Function.comp_def]
-  rw [he,←childJoin_boolFiber (labelSet e),image_lift_childJoin]
-  simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
-  let A := boolFiber (labelSet e) false
-  let B := boolFiber (labelSet e) true
-  change (A.image (p false) ∪ B.image (p true)).card = (A ∪ B.image (alternatingProjection p)).card
-  have hh : A.image (p false) ∪ B.image (p true) =
-      (A ∪ B.image (alternatingProjection p)).image (p false) := by
-    simp only [Finset.image_union,Finset.image_image,Function.comp_def,
-      alternatingProjection,Equiv.Perm.coe_mul,Equiv.apply_symm_apply]
-  rw [hh,Finset.card_image_of_injective _ (p false).injective]
-
-lemma bothAssignments_occupied_bound (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    Fintype.card (BothAssignments e p) ≤
-      2 ^ ((occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι +
-        Fintype.card (RoutingNetwork.SelectedCycle (alternatingProjection p) (boolFiber (labelSet e) false))) := by
-  have he : (occupied e).card =
-      (boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true).card := by
-    rw [occupied_labelSet,←childJoin_boolFiber (labelSet e),image_snd_childJoin]
-    simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
-  have hc : (boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card = Fintype.card ι := by
-    rw [←card_childJoin,childJoin_boolFiber,labelSet,Finset.card_image_of_injective _ e.injective,
-      Finset.card_univ]
-  simpa only [he,occupied_lift_card,←hc] using bothAssignments_card e p
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma compatible_pairLayer_iff (x : ι → Bool × α) (c : ι → Bool) (ξ : α → Bool) :
-    Compatible (fun i => pairLayer ξ (x i)) c ↔ Compatible x c := by
-  have he (i j : ι) (ht : (x i).2 = (x j).2) :
-      required (fun i => pairLayer ξ (x i)) c i = required (fun i => pairLayer ξ (x i)) c j ↔
-      required x c i = required x c j := by
-    simp only [required,pairLayer,Equiv.coe_fn_mk,ht]
-    cases (x i).1 <;> cases (x j).1 <;> cases c i <;> cases c j <;> cases ξ (x j).2 <;> decide
-  constructor
-  · intro h i j ht
-    exact (he i j ht).mp (h i j ht)
-  · intro h i j ht
-    exact (he i j ht).mpr (h i j ht)
-
-
-noncomputable def alternatingCycles (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) : ℕ :=
-  Fintype.card (RoutingNetwork.SelectedCycle
-    (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e))
-
-omit [Fintype α] in
-lemma occupied_eq_fibers (e : ι ↪ Bool × α) :
-    occupied e = boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true := by
-  rw [occupied_labelSet,←childJoin_boolFiber (labelSet e),image_snd_childJoin]
-  simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
-
-omit [Fintype α] in
-lemma card_labelSet (e : ι ↪ Bool × α) : (labelSet e).card = Fintype.card ι := by
-  classical
-  exact (Finset.card_image_of_injective _ e.injective).trans (Finset.card_univ)
-
-omit [Fintype α] in
-lemma occupied_card_double (e : ι ↪ Bool × α) : Fintype.card ι ≤ 2*(occupied e).card := by
-  have h₀ : (boolFiber (labelSet e) false).card ≤ (occupied e).card := by
-    rw [occupied_eq_fibers]
-    exact Finset.card_le_card Finset.subset_union_left
-  have h₁ : (boolFiber (labelSet e) true).card ≤ (occupied e).card := by
-    rw [occupied_eq_fibers]
-    exact Finset.card_le_card Finset.subset_union_right
-  have he : (boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card = Fintype.card ι := by
-    rw [←card_childJoin,childJoin_boolFiber,card_labelSet]
-  omega
-
-lemma bothAssignments_exact_bound (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    Fintype.card (BothAssignments e p) ≤
-      2 ^ ((occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι +
-        alternatingCycles e p) := by
-  have h := CycleColoring.card_opposite_le
-    (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e)
-    Prod.fst (CycleColoring.alternatingPerm_flip (alternatingProjection p))
-  have hb : (CycleColoring.boundary (CycleColoring.alternatingPerm (alternatingProjection p))
-      (labelSet e)).card =
-      (occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι := by
-    have hh := alternating_boundary_card (alternatingProjection p)
-      (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
-    rw [childJoin_boolFiber] at hh
-    rw [occupied_eq_fibers,occupied_lift_card]
-    have he := card_childJoin (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
-    rw [childJoin_boolFiber,card_labelSet] at he
-    simpa only [he] using hh
-  rw [hb] at h
-  exact (Fintype.card_le_of_injective _ (assignmentColoring_injective e p)).trans h
-
-abbrev Assignments (x y : ι → Bool × α) :=
-  {c : ι → Bool // Compatible x c ∧ Compatible y c}
-
-noncomputable instance (x y : ι → Bool × α) : Fintype (Assignments x y) := Fintype.ofFinite _
-
-def switchedEmbedding (e : ι ↪ Bool × α) (ξ : α → Bool) : ι ↪ Bool × α :=
-  e.trans (pairLayer ξ).toEmbedding
-
-def routedEmbedding (e : ι ↪ Bool × α) (ξ η : α → Bool) (p : Bool → Equiv.Perm α) :
-    ι ↪ Bool × α := e.trans (sandwich ξ η p).toEmbedding
-
-omit [Fintype α] in
-lemma occupied_pairLayer (x : ι → Bool × α) (ξ : α → Bool) :
-    occupied (fun i => pairLayer ξ (x i)) = occupied x := rfl
-
-noncomputable def assignmentsEquiv (e : ι ↪ Bool × α) (ξ η : α → Bool)
-    (p : Bool → Equiv.Perm α) :
-    Assignments e (routedEmbedding e ξ η p) ≃ BothAssignments (switchedEmbedding e ξ) p := by
-  apply Equiv.subtypeEquivRight
-  intro c
-  change Compatible e c ∧ Compatible (fun i => pairLayer η (lift p (pairLayer ξ (e i)))) c ↔ _
-  rw [compatible_pairLayer_iff]
-  exact and_congr (compatible_pairLayer_iff e c ξ).symm Iff.rfl
-
-lemma assignments_actual_bound (e : ι ↪ Bool × α) (ξ η : α → Bool) (p : Bool → Equiv.Perm α) :
-    Fintype.card (Assignments e (routedEmbedding e ξ η p)) ≤
-      2 ^ ((occupied e).card + (occupied (routedEmbedding e ξ η p)).card - Fintype.card ι +
-        alternatingCycles (switchedEmbedding e ξ) p) := by
-  rw [Fintype.card_congr (assignmentsEquiv e ξ η p)]
-  exact bothAssignments_exact_bound (switchedEmbedding e ξ) p
-
-omit [Fintype α] in
-lemma assignments_actual_exponent_nonneg (e : ι ↪ Bool × α) (ξ η : α → Bool)
-    (p : Bool → Equiv.Perm α) :
-    Fintype.card ι ≤ (occupied e).card + (occupied (routedEmbedding e ξ η p)).card := by
-  have h₀ := occupied_card_double e
-  have h₁ := occupied_card_double (routedEmbedding e ξ η p)
-  omega
-
-end
-variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α] [DecidableEq ι]
-
-noncomputable def compatibleSet (x y : ι → Bool × α) : Finset (ι → Bool) := by
-  classical
-  exact Finset.univ.filter (fun c => Compatible x c ∧ Compatible y c)
-
-omit [Fintype α] [DecidableEq α] in
-lemma mem_compatibleSet (x y : ι → Bool × α) (c : ι → Bool) :
-    c ∈ compatibleSet x y ↔ Compatible x c ∧ Compatible y c := by
-  classical
-  simp only [compatibleSet,Finset.mem_filter,Finset.mem_univ,true_and]
-
-omit [Fintype α] [DecidableEq α] in
-lemma card_compatibleSet (x y : ι → Bool × α) :
-    (compatibleSet x y).card = Fintype.card (Assignments x y) := by
-  classical
-  exact (Fintype.card_subtype _).symm
-
-noncomputable def outerWeight (x y : ι → Bool × α) : ℝ :=
-  1/(2:ℝ)^((occupied x).card+(occupied y).card)
-
-omit [Fintype α] [DecidableEq ι] in
-lemma outerWeight_pos (x y : ι → Bool × α) : 0 < outerWeight x y := by
-  unfold outerWeight
-  positivity
-
-omit [Fintype α] in
-lemma sum_colors_factor (x y : ι → Bool × α) (f : (ι → Bool) → ℝ) :
-    (∑ c : ι → Bool,
-      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
-      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) * f c) =
-      outerWeight x y * ∑ c ∈ compatibleSet x y, f c := by
-  classical
-  rw [compatibleSet,Finset.sum_filter,Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro c _
-  by_cases hx : Compatible x c <;> by_cases hy : Compatible y c
-  · simp only [hx,hy,true_and,↓reduceIte,outerWeight,pow_add,one_div_mul_one_div]
-  · simp only [hx,hy,and_false,↓reduceIte,mul_zero,zero_mul]
-  · simp only [hx,hy,false_and,↓reduceIte,mul_zero,zero_mul]
-  · simp only [hx,hy,false_and,↓reduceIte,mul_zero,zero_mul]
-
-lemma sandwich_weighted_probability (p : Bool → Equiv.Perm α) (x y : ι → Bool × α)
-    (W : (ι → Bool) → ℝ) :
-    finiteMean (fun coins : (α → Bool) × (α → Bool) =>
-      if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then W (colors x coins.1) else 0) =
-    outerWeight x y * ∑ c ∈ compatibleSet x y, if ChildMatches p x y c then W c else 0 := by
-  classical
-  have he (coins : (α → Bool) × (α → Bool)) :
-      (if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then W (colors x coins.1) else 0) =
-      ∑ c : ι → Bool, (if colors x coins.1 = c then (1:ℝ) else 0) *
-        (if colors y coins.2 = c then 1 else 0) * (if ChildMatches p x y c then W c else 0) := by
-    rw [Finset.sum_eq_single (colors x coins.1)]
-    · by_cases hc : colors x coins.1 = colors y coins.2
-      · simp only [sandwich_matches_iff,hc,true_and,↓reduceIte,one_mul]
-      · simp only [sandwich_matches_iff,hc,false_and,↓reduceIte,one_mul,Ne.symm hc,zero_mul]
-    · intro c _ hc
-      simp only [Ne.symm hc,ite_false,zero_mul]
-    · simp
-  simp_rw [he]
-  rw [finiteMean_sum,←sum_colors_factor]
-  apply Finset.sum_congr rfl
-  intro c _
-  rw [finiteMean_mul_const,finiteMean_prod_mul
-    (fun ξ : α → Bool => if colors x ξ = c then (1:ℝ) else 0)
-    (fun η : α → Bool => if colors y η = c then (1:ℝ) else 0),
-    colors_probability,colors_probability]
-
-omit [Fintype α] [DecidableEq ι] in
-lemma tupleProbability_nonneg {Ω β : Type*} [Fintype Ω] [DecidableEq β]
-    (P : Ω → Equiv.Perm β) (x y : ι → β) : 0 ≤ tupleProbability P x y := by
-  classical
-  exact finiteMean_nonneg (fun ω => by split_ifs <;> norm_num)
-
-lemma sandwich_power_bound {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
-    (P : Ω → Equiv.Perm α) (Q : Ω' → Equiv.Perm α) (x y : ι → Bool × α)
-    (a : ℝ) (ha : 0 ≤ a) :
-    (finiteMean (fun ω : Ω × Ω' => finiteMean (fun coins : (α → Bool) × (α → Bool) =>
-      if ∀ i, sandwich coins.1 coins.2 (fun b => if b then Q ω.2 else P ω.1) (x i) = y i
-      then (1:ℝ) else 0)))^(1+a) ≤
-    (outerWeight x y)^(1+a) * ((compatibleSet x y).card:ℝ)^a *
-      ∑ c ∈ compatibleSet x y,
-      (tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
-        (fun i : {i // c i = false} => (y i.val).2))^(1+a) *
-      (tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
-        (fun i : {i // c i = true} => (y i.val).2))^(1+a) := by
-  classical
-  rw [sandwich_mean,sum_colors_factor]
-  let f (c : ι → Bool) : ℝ :=
-      tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
-        (fun i : {i // c i = false} => (y i.val).2) *
-      tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
-        (fun i : {i // c i = true} => (y i.val).2)
-  have hf (c : ι → Bool) : 0 ≤ f c := mul_nonneg (tupleProbability_nonneg _ _ _)
-    (tupleProbability_nonneg _ _ _)
-  rw [Real.mul_rpow (le_of_lt (outerWeight_pos _ _)) (Finset.sum_nonneg (fun c _ => hf c))]
-  have hh := Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg (compatibleSet x y)
-    (show 1 ≤ 1+a by linarith) (fun c _ => hf c)
-  have he : 1+a-1=a := by ring
-  rw [he] at hh
-  calc
-    _ ≤ (outerWeight x y)^(1+a) *
-        (((compatibleSet x y).card:ℝ)^a * ∑ c ∈ compatibleSet x y, (f c)^(1+a)) :=
-      mul_le_mul_of_nonneg_left hh (Real.rpow_nonneg (le_of_lt (outerWeight_pos _ _)) _)
-    _ = _ := by
-      rw [←mul_assoc]
-      congr 1
-      apply Finset.sum_congr rfl
-      intro c _
-      exact Real.mul_rpow (tupleProbability_nonneg _ _ _) (tupleProbability_nonneg _ _ _)
-
-end PairRouting
-
-end Thorp
-
-end ThorpNine.Smoothing
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Smoothing.ParallelFollow
+
+namespace OAI
+
+namespace ThorpNine.Smoothing
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+namespace CycleColoring
+variable {α : Type*} [Fintype α] [DecidableEq α]
+omit [Fintype α] in
+lemma bipartitionFlip_injective (p : Equiv.Perm α) (S : Finset α)
+    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x)) :
+    Function.Injective (bipartitionFlip p S χ hχ) := by
+  classical
+  intro f g he
+  apply Subtype.ext
+  funext x
+  by_cases hx : x ∈ S
+  · have hh := congrFun (congrArg Subtype.val he) x
+    change (if x ∈ S then Bool.xor (f.val x) (χ x) else false) =
+      (if x ∈ S then Bool.xor (g.val x) (χ x) else false) at hh
+    simp only [ite_eq_left hx] at hh
+    cases hfx : f.val x <;> cases hgx : g.val x <;> cases hcx : χ x <;> simp_all
+  · exact (f.property.1 x hx).trans (g.property.1 x hx).symm
+
+lemma card_opposite_le (p : Equiv.Perm α) (S : Finset α)
+    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x)) :
+    Fintype.card (OppositeColoring p S) ≤
+      2 ^ ((boundary p S).card + Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
+  exact (Fintype.card_le_of_injective _ (bipartitionFlip_injective p S χ hχ)).trans
+    (card_coloring_le p S)
+
+def alternatingPerm (p : Equiv.Perm α) : Equiv.Perm (Bool × α) where
+  toFun x := if x.1 then (false,p x.2) else (true,x.2)
+  invFun x := if x.1 then (false,x.2) else (true,p.symm x.2)
+  left_inv x := by rcases x with ⟨b,x⟩; cases b <;> simp
+  right_inv x := by rcases x with ⟨b,x⟩; cases b <;> simp
+
+omit [Fintype α] [DecidableEq α] in
+lemma alternatingPerm_flip (p : Equiv.Perm α) (x : Bool × α) :
+    (alternatingPerm p x).1 = !x.1 := by
+  rcases x with ⟨b,x⟩
+  cases b <;> rfl
+
+omit [Fintype α] [DecidableEq α] in
+lemma alternating_step_sameCycle (p : Equiv.Perm α) (x : Bool × α) :
+    p.SameCycle x.2 (alternatingPerm p x).2 := by
+  rcases x with ⟨b,x⟩
+  cases b
+  · exact Equiv.Perm.SameCycle.refl p x
+  · exact (Equiv.Perm.SameCycle.refl p x).apply_right
+
+omit [DecidableEq α] in
+lemma alternating_sameCycle_snd (p : Equiv.Perm α) {x y : Bool × α}
+    (h : (alternatingPerm p).SameCycle x y) : p.SameCycle x.2 y.2 := by
+  obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
+  have hp (k : ℕ) : p.SameCycle x.2 (((alternatingPerm p)^k) x).2 := by
+    induction k with
+    | zero => exact Equiv.Perm.SameCycle.refl p x.2
+    | succ k ih =>
+      rw [pow_succ',Equiv.Perm.mul_apply]
+      exact ih.trans (alternating_step_sameCycle p _)
+  exact hn ▸ hp n
+
+omit [DecidableEq α] in
+lemma alternating_zero_sameCycle (p : Equiv.Perm α) {x y : α}
+    (h : p.SameCycle x y) : (alternatingPerm p).SameCycle (false,x) (false,y) := by
+  obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
+  have hp (k : ℕ) : (alternatingPerm p).SameCycle (false,x) (false,(p^k) x) := by
+    induction k with
+    | zero => exact Equiv.Perm.SameCycle.refl _ _
+    | succ k ih =>
+      have hs := ih.apply_right.apply_right
+      simpa only [pow_succ',Equiv.Perm.mul_apply,alternatingPerm,Equiv.coe_fn_mk,
+        Bool.false_eq_true,↓reduceIte] using hs
+  simpa only [hn] using hp n
+
+omit [Fintype α] [DecidableEq α] in
+lemma alternating_sameCycle_zero (p : Equiv.Perm α) (x : Bool × α) :
+    (alternatingPerm p).SameCycle x (false,x.2) := by
+  rcases x with ⟨b,x⟩
+  cases b
+  · exact Equiv.Perm.SameCycle.refl _ _
+  · exact ((Equiv.Perm.SameCycle.refl (alternatingPerm p) (false,x)).apply_right).symm
+
+omit [DecidableEq α] in
+lemma alternating_sameCycle_iff (p : Equiv.Perm α) (x y : Bool × α) :
+    (alternatingPerm p).SameCycle x y ↔ p.SameCycle x.2 y.2 := by
+  constructor
+  · exact alternating_sameCycle_snd p
+  · intro h
+    exact (alternating_sameCycle_zero p x).trans
+      ((alternating_zero_sameCycle p h).trans (alternating_sameCycle_zero p y).symm)
+
+lemma orbit_alternating (p : Equiv.Perm α) (x : Bool × α) :
+    orbitSet (alternatingPerm p) x = childJoin (orbitSet p x.2) (orbitSet p x.2) := by
+  ext ⟨b,y⟩
+  simp only [mem_orbitSet,alternating_sameCycle_iff,mem_childJoin,ite_self]
+
+omit [Fintype α] in
+lemma boolFiber_childJoin (A B : Finset α) (b : Bool) :
+    boolFiber (childJoin A B) b = if b then B else A := by
+  ext x
+  cases b <;> simp only [mem_boolFiber,mem_childJoin,Bool.false_eq_true,↓reduceIte]
+
+noncomputable def projectCycle (p : Equiv.Perm α) (A B : Finset α)
+    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) :
+    RoutingNetwork.SelectedCycle p A := by
+  refine ⟨boolFiber C.val false,?_,?_⟩
+  · obtain ⟨x,hx⟩ := C.property.1
+    refine ⟨x.2,?_⟩
+    rw [hx,orbit_alternating,boolFiber_childJoin]
+    rfl
+  · intro x hx
+    exact (mem_childJoin A B false x).mp (C.property.2 ((mem_boolFiber _ _ _).mp hx))
+
+lemma selected_alternating_eq (p : Equiv.Perm α) (A B : Finset α)
+    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) :
+    C.val = childJoin (projectCycle p A B C).val (projectCycle p A B C).val := by
+  change C.val = childJoin (boolFiber C.val false) (boolFiber C.val false)
+  obtain ⟨x,hx⟩ := C.property.1
+  rw [hx,orbit_alternating,boolFiber_childJoin]
+  rfl
+
+lemma projectCycle_injective (p : Equiv.Perm α) (A B : Finset α) :
+    Function.Injective (projectCycle p A B) := by
+  intro C D he
+  apply Subtype.ext
+  rw [selected_alternating_eq p A B C,selected_alternating_eq p A B D,he]
+
+lemma card_alternating_cycles_le (p : Equiv.Perm α) (A B : Finset α) :
+    Fintype.card (RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) ≤
+      Fintype.card (RoutingNetwork.SelectedCycle p A) :=
+  Fintype.card_le_of_injective _ (projectCycle_injective p A B)
+
+end CycleColoring
+
+section PairAssignmentCount
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+omit [Fintype α] in
+lemma mem_image_perm_iff (p : Equiv.Perm α) (A : Finset α) (y : α) :
+    y ∈ A.image p ↔ p.symm y ∈ A := by
+  constructor
+  · intro h
+    obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp h
+    simpa only [←hxy,Equiv.symm_apply_apply] using hx
+  · intro h
+    exact Finset.mem_image.mpr ⟨p.symm y,h,p.apply_symm_apply y⟩
+
+omit [Fintype α] in
+lemma card_childJoin (A B : Finset α) : (childJoin A B).card = A.card+B.card := by
+  have hd : Disjoint (A.image (fun x => (false,x))) (B.image (fun x => (true,x))) := by
+    apply Finset.disjoint_left.mpr
+    rintro x hx hy
+    obtain ⟨a,ha,rfl⟩ := Finset.mem_image.mp hx
+    obtain ⟨b,hb,hba⟩ := Finset.mem_image.mp hy
+    exact Bool.false_ne_true (congrArg Prod.fst hba).symm
+  have hi (b : Bool) : Function.Injective (fun x : α => (b,x)) :=
+    fun _ _ h => congrArg Prod.snd h
+  rw [childJoin,Finset.card_union_of_disjoint hd,
+    Finset.card_image_of_injective A (hi false),Finset.card_image_of_injective B (hi true)]
+
+omit [Fintype α] in
+lemma alternating_internal (p : Equiv.Perm α) (A B : Finset α) :
+    ((childJoin A B).filter (fun x => (CycleColoring.alternatingPerm p).symm x ∈ childJoin A B)) =
+      childJoin (A ∩ B.image p) (B ∩ A) := by
+  ext ⟨b,x⟩
+  rw [Finset.mem_filter]
+  cases b <;> simp only [mem_childJoin,CycleColoring.alternatingPerm,
+    Equiv.symm_mk,Equiv.coe_fn_mk,Bool.false_eq_true,↓reduceIte,Finset.mem_inter,mem_image_perm_iff]
+
+omit [Fintype α] in
+lemma alternating_boundary_card (p : Equiv.Perm α) (A B : Finset α) :
+    (CycleColoring.boundary (CycleColoring.alternatingPerm p) (childJoin A B)).card =
+      (A ∪ B).card + (A ∪ B.image p).card - (A.card+B.card) := by
+  rw [CycleColoring.boundary_card,alternating_internal,card_childJoin,card_childJoin]
+  have h₀ := Finset.card_union_add_card_inter A B
+  have h₁ := Finset.card_union_add_card_inter A (B.image p)
+  rw [Finset.card_image_of_injective _ p.injective] at h₁
+  have h₂ := Finset.inter_comm B A
+  rw [h₂]
+  omega
+
+lemma compatible_assignment_bound (p : Equiv.Perm α) (A B : Finset α) :
+    Fintype.card (CycleColoring.OppositeColoring (CycleColoring.alternatingPerm p) (childJoin A B)) ≤
+      2 ^ ((A ∪ B).card + (A ∪ B.image p).card - (A.card+B.card) +
+        Fintype.card (RoutingNetwork.SelectedCycle p A)) := by
+  have h := CycleColoring.card_opposite_le (CycleColoring.alternatingPerm p) (childJoin A B)
+    Prod.fst (CycleColoring.alternatingPerm_flip p)
+  rw [alternating_boundary_card] at h
+  exact h.trans (Nat.pow_le_pow_right (by norm_num)
+    (Nat.add_le_add_left (CycleColoring.card_alternating_cycles_le p A B) _))
+
+end PairAssignmentCount
+
+lemma finiteMean_nonneg {Ω : Type*} [Fintype Ω] {f : Ω → ℝ}
+    (h : ∀ ω, 0 ≤ f ω) : 0 ≤ finiteMean f := by
+  exact div_nonneg (Finset.sum_nonneg (fun ω _ => h ω)) (Nat.cast_nonneg _)
+
+lemma boolFiber_headTail (d : ℕ) (S : Finset (Card (d+1))) (b : Bool) :
+    boolFiber (S.image (headTailEquiv d)) b = childMarks S b := by
+  classical
+  ext x
+  rw [mem_boolFiber,mem_childMarks]
+  constructor
+  · intro h
+    obtain ⟨y,hy,he⟩ := Finset.mem_image.mp h
+    have hh := congrArg (headTailEquiv d).symm he
+    rw [Equiv.symm_apply_apply] at hh
+    change y = Fin.cons b x at hh
+    exact hh ▸ hy
+  · intro h
+    exact Finset.mem_image.mpr ⟨Fin.cons b x,h,by rfl⟩
+
+lemma prod_childMarks (d : ℕ) (S : Finset (Card (d+1))) (f : Card (d+1) → ℝ) :
+    (∏ x ∈ S, f x) =
+      (∏ x ∈ childMarks S false, f (Fin.cons false x)) *
+      ∏ x ∈ childMarks S true, f (Fin.cons true x) := by
+  classical
+  have h := prod_boolFibers (S.image (headTailEquiv d)) (fun x => f ((headTailEquiv d).symm x))
+  rw [Finset.prod_image (headTailEquiv d).injective.injOn,
+    boolFiber_headTail,boolFiber_headTail] at h
+  simp only [Equiv.symm_apply_apply] at h
+  exact h
+
+noncomputable def headAverage {d : ℕ} (p : Card (d+1) → ℝ) (y : Card d) : ℝ :=
+  (p (Fin.cons false y)+p (Fin.cons true y))/2
+
+lemma finiteMean_headAverage (d : ℕ) (p : Card (d+1) → ℝ) :
+    finiteMean (headAverage p) = finiteMean p := by
+  rw [finiteMean_equiv (headTailEquiv d),finiteMean_prod,finiteMean_comm]
+  apply finiteMean_congr
+  intro y
+  rw [finiteMean_bool]
+  rfl
+
+lemma pairSwitch_product_bound (d : ℕ) (S : Finset (Card (d+1)))
+    (p : Card (d+1) → ℝ) (hp : ∀ x, 0 ≤ p x) :
+    finiteMean (fun ξ : Card d → Bool => ∏ x ∈ S, p (pairSwitch ξ x)) ≤
+      ∏ x ∈ S, headAverage p (Fin.tail x) := by
+  classical
+  have h : CylinderBound (fun T => ∏ x ∈ T, p ((headTailEquiv d).symm x))
+      (fun x => p ((headTailEquiv d).symm x)) := fun _ => le_rfl
+  have hh := h.fair_parallel (fun x => hp _) (S.image (headTailEquiv d))
+  simp only [Finset.prod_image (pairLayer _).injective.injOn,
+    Finset.prod_image (headTailEquiv d).injective.injOn] at hh
+  convert hh using 1
+  · apply finiteMean_congr
+    intro ξ
+    apply Finset.prod_congr rfl
+    intro x _
+    rfl
+  · apply Finset.prod_congr rfl
+    intro x _
+    change (p (Fin.cons false (Fin.tail x))+p (Fin.cons true (Fin.tail x)))/2 =
+      (p (Fin.cons (x 0) (Fin.tail x))+p (Fin.cons (!(x 0)) (Fin.tail x)))/2
+    cases x 0
+    · rfl
+    · simp only [Bool.not_true]
+      ring
+
+lemma butterflyPerm_step (d : ℕ)
+    (c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (ξ : Card d → Bool) :
+    butterflyPerm (d+1) (decodeButterfly (d+1) ((coinStepEquiv d).symm (c,ξ))) =
+    pairSwitch ξ * childLift (fun b => butterflyPerm d (decodeButterfly d (if b then c.2 else c.1))) := by
+  change pairSwitch (fun y => (coinStepEquiv d).symm (c,ξ) (Sum.inl y)) *
+    childLift (fun b => butterflyPerm d (decodeButterfly d
+      (fun i => (coinStepEquiv d).symm (c,ξ) (Sum.inr (b,i))))) = _
+  apply congrArg₂ (· * ·)
+  · rfl
+  · apply congrArg childLift
+    funext b
+    apply congrArg (fun ω => butterflyPerm d (decodeButterfly d ω))
+    funext i
+    cases b <;> rfl
+
+lemma butterfly_product_bound (d : ℕ) (S : Finset (Card d))
+    (p : Card d → ℝ) (hp : ∀ x, 0 ≤ p x) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      ∏ x ∈ S, p (butterflyPerm d (decodeButterfly d ω) x)) ≤ (finiteMean p)^S.card := by
+  classical
+  induction d with
+  | zero =>
+    have he : p = fun _ => p (fun i => Fin.elim0 i) := by
+      funext x
+      exact congrArg p (Subsingleton.elim _ _)
+    rw [he]
+    simp only [Finset.prod_const,finiteMean_const,le_refl]
+  | succ d ih =>
+    let g : Card d → ℝ := headAverage p
+    have hg : ∀ x, 0 ≤ g x := fun x => div_nonneg (add_nonneg (hp _) (hp _)) (by norm_num)
+    rw [finiteMean_equiv (coinStepEquiv d),finiteMean_prod]
+    calc
+      _ ≤ finiteMean (fun c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool) =>
+        (∏ x ∈ childMarks S false, g (butterflyPerm d (decodeButterfly d c.1) x)) *
+        ∏ x ∈ childMarks S true, g (butterflyPerm d (decodeButterfly d c.2) x)) := by
+        apply finiteMean_mono
+        intro c
+        let f : Equiv.Perm (Card (d+1)) := childLift (fun b =>
+          butterflyPerm d (decodeButterfly d (if b then c.2 else c.1)))
+        have hh := pairSwitch_product_bound d (S.image f) p hp
+        simp only [Finset.prod_image f.injective.injOn] at hh
+        convert hh using 1
+        · apply finiteMean_congr
+          intro ξ
+          apply Finset.prod_congr rfl
+          intro x _
+          rw [butterflyPerm_step]
+          rfl
+        · rw [prod_childMarks]
+          simp only [f,childLift,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
+            Bool.false_eq_true,↓reduceIte]
+          rfl
+      _ = (finiteMean (fun c : SwitchIndex d → Bool =>
+        ∏ x ∈ childMarks S false, g (butterflyPerm d (decodeButterfly d c) x))) *
+        finiteMean (fun c : SwitchIndex d → Bool =>
+        ∏ x ∈ childMarks S true, g (butterflyPerm d (decodeButterfly d c) x)) :=
+          finiteMean_prod_mul
+            (fun c : SwitchIndex d → Bool => ∏ x ∈ childMarks S false,
+              g (butterflyPerm d (decodeButterfly d c) x))
+            (fun c : SwitchIndex d → Bool => ∏ x ∈ childMarks S true,
+              g (butterflyPerm d (decodeButterfly d c) x))
+      _ ≤ (finiteMean g)^(childMarks S false).card * (finiteMean g)^(childMarks S true).card :=
+        mul_le_mul (ih (childMarks S false) g hg) (ih (childMarks S true) g hg)
+          (finiteMean_nonneg fun c => Finset.prod_nonneg (fun x _ => hg _))
+          (pow_nonneg (finiteMean_nonneg hg) _)
+      _ = (finiteMean p)^S.card := by
+        rw [←pow_add,card_childMarks_add,finiteMean_headAverage]
+
+noncomputable def inverseButterflyMarks (d : ℕ) (A : Finset (Card d))
+    (ω : SwitchIndex d → Bool) : Finset (Card d) :=
+  A.image (butterflyPerm d (decodeButterfly d ω)).symm
+
+lemma inverseButterflyMarks_inclusion (d : ℕ) (A S : Finset (Card d)) :
+    inclusionMoment (inverseButterflyMarks d A) S ≤
+      ((A.card : ℝ)/(2:ℝ)^d)^S.card := by
+  classical
+  let p : Card d → ℝ := fun x => if x ∈ A then 1 else 0
+  have hm : finiteMean p = (A.card : ℝ)/(2:ℝ)^d := by
+    simp only [finiteMean,p,card_positions,Nat.cast_pow,Nat.cast_ofNat]
+    congr 1
+    simp
+  have he : inclusionMoment (inverseButterflyMarks d A) S =
+      finiteMean (fun ω : SwitchIndex d → Bool =>
+        ∏ x ∈ S, p (butterflyPerm d (decodeButterfly d ω) x)) := by
+    rw [inclusionMoment_eq]
+    apply finiteMean_congr
+    intro ω
+    simp only [inverseButterflyMarks,subset_image_equiv_iff,Equiv.symm_symm,indicator_subset_prod,
+      Finset.prod_image (butterflyPerm d (decodeButterfly d ω)).injective.injOn,p]
+  rw [he]
+  simpa only [hm] using butterfly_product_bound d S p (fun x => by dsimp [p]; split_ifs <;> norm_num)
+
+namespace PairRouting
+
+section
+variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
+
+noncomputable def occupied (x : ι → Bool × α) : Finset α :=
+  Finset.univ.image (fun i => (x i).2)
+
+omit [Fintype α] in
+lemma mem_occupied (x : ι → Bool × α) (y : α) : y ∈ occupied x ↔ ∃ i, (x i).2 = y := by
+  simp [occupied]
+
+def required (x : ι → Bool × α) (c : ι → Bool) (i : ι) : Bool := Bool.xor (x i).1 (c i)
+
+def Compatible (x : ι → Bool × α) (c : ι → Bool) : Prop :=
+  ∀ i j, (x i).2 = (x j).2 → required x c i = required x c j
+
+def colors (x : ι → Bool × α) (ξ : α → Bool) (i : ι) : Bool :=
+  Bool.xor (x i).1 (ξ (x i).2)
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma required_colors (x : ι → Bool × α) (ξ : α → Bool) (i : ι) :
+    required x (colors x ξ) i = ξ (x i).2 := by
+  simp only [required,colors]
+  cases (x i).1 <;> cases ξ (x i).2 <;> rfl
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma colors_compatible (x : ι → Bool × α) (ξ : α → Bool) : Compatible x (colors x ξ) := by
+  intro i j he
+  simp only [required_colors,he]
+
+noncomputable def chosenCoin (x : ι → Bool × α) (c : ι → Bool) (y : α) : Bool := by
+  classical
+  exact if h : ∃ i, (x i).2 = y then required x c (Classical.choose h) else false
+
+omit [Fintype α] in
+lemma chosenCoin_apply (x : ι → Bool × α) (c : ι → Bool) (hc : Compatible x c) (i : ι) :
+    chosenCoin x c (x i).2 = required x c i := by
+  classical
+  simp only [chosenCoin,dite_eq_left (show ∃ j, (x j).2 = (x i).2 from ⟨i,rfl⟩)]
+  exact hc _ i (Classical.choose_spec (show ∃ j, (x j).2 = (x i).2 from ⟨i,rfl⟩))
+
+omit [Fintype α] in
+lemma colors_eq_iff (x : ι → Bool × α) (c : ι → Bool) (hc : Compatible x c) (ξ : α → Bool) :
+    colors x ξ = c ↔ ∀ y ∈ occupied x, ξ y = chosenCoin x c y := by
+  constructor
+  · intro he y hy
+    obtain ⟨i,rfl⟩ := (mem_occupied x y).mp hy
+    rw [chosenCoin_apply x c hc i]
+    rw [←he,required_colors]
+  · intro he
+    funext i
+    apply (xor_eq_iff _ _ _).mpr
+    exact (he _ ((mem_occupied x _).mpr ⟨i,rfl⟩)).trans (chosenCoin_apply x c hc i)
+
+noncomputable instance (x : ι → Bool × α) (c : ι → Bool) : Decidable (Compatible x c) :=
+  Classical.propDecidable _
+
+lemma colors_probability (x : ι → Bool × α) (c : ι → Bool) :
+    finiteMean (fun ξ : α → Bool => if colors x ξ = c then (1 : ℝ) else 0) =
+      if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0 := by
+  classical
+  by_cases hc : Compatible x c
+  · rw [ite_eq_left hc]
+    have he := fun ξ => colors_eq_iff x c hc ξ
+    simp only [he]
+    exact mean_coinCylinder (occupied x) (chosenCoin x c)
+  · rw [ite_eq_right hc]
+    have he : ∀ ξ, colors x ξ ≠ c := fun ξ hh => hc (hh ▸ colors_compatible x ξ)
+    simp only [he,ite_false,finiteMean_const]
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma Compatible.tail_injective {x : ι ↪ Bool × α} {c : ι → Bool} (hc : Compatible x c) (b : Bool) :
+    Function.Injective (fun i : {i : ι // c i = b} => (x i.val).2) := by
+  intro i j he
+  apply Subtype.ext
+  apply x.injective
+  apply Prod.ext
+  · have hh := hc i j he
+    simp only [required,i.property,j.property] at hh
+    cases h₀ : (x i.val).1 <;> cases h₁ : (x j.val).1 <;> cases b <;> simp_all
+  · exact he
+
+def childEmbedding (x : ι ↪ Bool × α) (c : ι → Bool) (hc : Compatible x c) (b : Bool) :
+    {i : ι // c i = b} ↪ α := ⟨fun i => (x i.val).2,hc.tail_injective b⟩
+
+
+def lift (p : Bool → Equiv.Perm α) : Equiv.Perm (Bool × α) where
+  toFun x := (x.1,p x.1 x.2)
+  invFun x := (x.1,(p x.1).symm x.2)
+  left_inv x := by simp
+  right_inv x := by simp
+
+def sandwich (ξ η : α → Bool) (p : Bool → Equiv.Perm α) : Equiv.Perm (Bool × α) :=
+  pairLayer η * lift p * pairLayer ξ
+
+def ChildMatches (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) : Prop :=
+  ∀ i, p (c i) (x i).2 = (y i).2
+
+noncomputable instance (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
+    Decidable (ChildMatches p x y c) := Classical.propDecidable _
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma sandwich_matches_iff (ξ η : α → Bool) (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) :
+    (∀ i, sandwich ξ η p (x i) = y i) ↔
+      colors x ξ = colors y η ∧ ChildMatches p x y (colors x ξ) := by
+  have he (i : ι) : sandwich ξ η p (x i) = y i ↔
+      colors x ξ i = colors y η i ∧ p (colors x ξ i) (x i).2 = (y i).2 := by
+    change pairLayer η (lift p (pairLayer ξ (x i))) = y i ↔ _
+    rw [←Equiv.eq_symm_apply]
+    change (colors x ξ i,p (colors x ξ i) (x i).2) = (colors y η i,(y i).2) ↔ _
+    exact ⟨Prod.mk.inj,fun h => Prod.ext h.1 h.2⟩
+  simp only [he,forall_and,←funext_iff,ChildMatches]
+
+lemma sandwich_probability [DecidableEq ι] (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) :
+    finiteMean (fun coins : (α → Bool) × (α → Bool) =>
+      if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then (1:ℝ) else 0) =
+    ∑ c : ι → Bool,
+      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
+      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) *
+      (if ChildMatches p x y c then 1 else 0) := by
+  classical
+  have he (coins : (α → Bool) × (α → Bool)) :
+      (if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then (1:ℝ) else 0) =
+      ∑ c : ι → Bool, (if colors x coins.1 = c then (1:ℝ) else 0) *
+        (if colors y coins.2 = c then 1 else 0) * (if ChildMatches p x y c then 1 else 0) := by
+    rw [Finset.sum_eq_single (colors x coins.1)]
+    · by_cases hc : colors x coins.1 = colors y coins.2
+      · simp only [sandwich_matches_iff,hc,true_and,↓reduceIte,one_mul]
+      · simp only [sandwich_matches_iff,hc,false_and,↓reduceIte,one_mul,Ne.symm hc,zero_mul]
+    · intro c _ hc
+      simp only [Ne.symm hc,ite_false,zero_mul]
+    · simp
+  simp_rw [he]
+  rw [finiteMean_sum]
+  apply Finset.sum_congr rfl
+  intro c _
+  rw [finiteMean_mul_const]
+  rw [finiteMean_prod_mul
+    (fun ξ : α → Bool => if colors x ξ = c then (1:ℝ) else 0)
+    (fun η : α → Bool => if colors y η = c then (1:ℝ) else 0),
+    colors_probability,colors_probability]
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma childMatches_split (p₀ p₁ : Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
+    ChildMatches (fun b => if b then p₁ else p₀) x y c ↔
+      (∀ i : {i // c i = false}, p₀ (x i.val).2 = (y i.val).2) ∧
+      (∀ i : {i // c i = true}, p₁ (x i.val).2 = (y i.val).2) := by
+  constructor
+  · intro h
+    constructor
+    · intro i
+      simpa only [i.property,Bool.false_eq_true,↓reduceIte] using h i.val
+    · intro i
+      simpa only [i.property,↓reduceIte] using h i.val
+  · rintro ⟨h₀,h₁⟩ i
+    cases hi : c i
+    · simpa only [hi,Bool.false_eq_true,↓reduceIte] using h₀ ⟨i,hi⟩
+    · simpa only [hi,↓reduceIte] using h₁ ⟨i,hi⟩
+
+omit [Fintype α] in
+lemma childMatches_mean {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
+    (P : Ω → Equiv.Perm α) (Q : Ω' → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
+    finiteMean (fun ω : Ω × Ω' =>
+      if ChildMatches (fun b => if b then Q ω.2 else P ω.1) x y c then (1:ℝ) else 0) =
+      tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
+        (fun i : {i // c i = false} => (y i.val).2) *
+      tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
+        (fun i : {i // c i = true} => (y i.val).2) := by
+  classical
+  let f : Ω → ℝ := fun ω => if ∀ i : {i // c i = false}, P ω (x i.val).2 = (y i.val).2 then 1 else 0
+  let g : Ω' → ℝ := fun ω => if ∀ i : {i // c i = true}, Q ω (x i.val).2 = (y i.val).2 then 1 else 0
+  have he (ω : Ω × Ω') :
+      (if ChildMatches (fun b => if b then Q ω.2 else P ω.1) x y c then (1:ℝ) else 0) = f ω.1*g ω.2 := by
+    simp only [childMatches_split,f,g]
+    split_ifs <;> simp_all
+  simp_rw [he]
+  exact finiteMean_prod_mul f g
+
+lemma sandwich_mean [DecidableEq ι] {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
+    (P : Ω → Equiv.Perm α) (Q : Ω' → Equiv.Perm α) (x y : ι → Bool × α) :
+    finiteMean (fun ω : Ω × Ω' => finiteMean (fun coins : (α → Bool) × (α → Bool) =>
+      if ∀ i, sandwich coins.1 coins.2 (fun b => if b then Q ω.2 else P ω.1) (x i) = y i
+      then (1:ℝ) else 0)) =
+    ∑ c : ι → Bool,
+      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
+      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) *
+      (tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
+        (fun i : {i // c i = false} => (y i.val).2) *
+      tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
+        (fun i : {i // c i = true} => (y i.val).2)) := by
+  classical
+  simp_rw [sandwich_probability]
+  rw [finiteMean_sum]
+  apply Finset.sum_congr rfl
+  intro c _
+  simp only [mul_comm ((if Compatible x c then _ else _) * _),finiteMean_mul_const,
+    childMatches_mean]
+
+
+abbrev BothAssignments (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :=
+  {c : ι → Bool // Compatible e c ∧ Compatible (fun i => lift p (e i)) c}
+
+noncomputable instance (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    Fintype (BothAssignments e p) := Fintype.ofFinite _
+
+noncomputable def labelSet (e : ι ↪ Bool × α) : Finset (Bool × α) := Finset.univ.image e
+
+def alternatingProjection (p : Bool → Equiv.Perm α) : Equiv.Perm α := (p false).symm * p true
+
+noncomputable def assignmentColoring (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α)
+    (c : BothAssignments e p) : CycleColoring.OppositeColoring
+      (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e) := by
+  classical
+  refine ⟨Function.extend e c.val (fun _ => false),?_,?_⟩
+  · intro z hz
+    apply Function.extend_apply'
+    rintro ⟨i,rfl⟩
+    exact hz (Finset.mem_image.mpr ⟨i,Finset.mem_univ _,rfl⟩)
+  · intro z hz hnext
+    obtain ⟨i,_,hi⟩ := Finset.mem_image.mp hz
+    obtain ⟨j,_,hj⟩ := Finset.mem_image.mp hnext
+    have hfi := e.injective.extend_apply c.val (fun _ => false) i
+    have hfj := e.injective.extend_apply c.val (fun _ => false) j
+    rw [hi] at hfi
+    rw [hj] at hfj
+    rw [hfi,hfj]
+    rcases z with ⟨b,a⟩
+    cases b
+    · have hj' : e j = (true,a) := hj
+      have ht : (e i).2 = (e j).2 := by rw [hi,hj']
+      have hh := c.property.1 i j ht
+      simp only [required,hi,hj'] at hh
+      revert hh
+      generalize c.val i = bi
+      generalize c.val j = bj
+      cases bi <;> cases bj <;> decide
+    · have hj' : e j = (false,alternatingProjection p a) := hj
+      have ht : (lift p (e i)).2 = (lift p (e j)).2 := by
+        simp only [hi,hj',lift,Equiv.coe_fn_mk,alternatingProjection,Equiv.Perm.coe_mul,
+          Function.comp_apply,Equiv.apply_symm_apply]
+      have hh := c.property.2 i j ht
+      simp only [required,hi,hj',lift,Equiv.coe_fn_mk] at hh
+      revert hh
+      generalize c.val i = bi
+      generalize c.val j = bj
+      cases bi <;> cases bj <;> decide
+
+omit [Fintype α] in
+lemma assignmentColoring_injective (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    Function.Injective (assignmentColoring e p) := by
+  intro c c' he
+  apply Subtype.ext
+  funext i
+  have hh := congrFun (congrArg Subtype.val he) (e i)
+  change Function.extend e c.val (fun _ => false) (e i) =
+    Function.extend e c'.val (fun _ => false) (e i) at hh
+  simpa only [e.injective.extend_apply] using hh
+
+lemma bothAssignments_card (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    Fintype.card (BothAssignments e p) ≤
+      2 ^ ((boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true).card +
+        (boolFiber (labelSet e) false ∪ (boolFiber (labelSet e) true).image
+          (alternatingProjection p)).card -
+        ((boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card) +
+        Fintype.card (RoutingNetwork.SelectedCycle (alternatingProjection p) (boolFiber (labelSet e) false))) := by
+  have h := compatible_assignment_bound (alternatingProjection p)
+    (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
+  rw [childJoin_boolFiber] at h
+  exact (Fintype.card_le_of_injective _ (assignmentColoring_injective e p)).trans h
+
+omit [Fintype α] in
+lemma occupied_labelSet (e : ι ↪ Bool × α) : occupied e = (labelSet e).image Prod.snd := by
+  classical
+  simp only [occupied,labelSet,Finset.image_image,Function.comp_def]
+
+omit [Fintype ι] [Fintype α] in
+lemma image_snd_childJoin (A B : Finset α) : (childJoin A B).image Prod.snd = A ∪ B := by
+  classical
+  simp only [childJoin,Finset.image_union,Finset.image_image,Function.comp_def,Finset.image_id']
+
+omit [Fintype ι] [Fintype α] in
+lemma image_lift_childJoin (p : Bool → Equiv.Perm α) (A B : Finset α) :
+    (childJoin A B).image (fun x => (lift p x).2) = A.image (p false) ∪ B.image (p true) := by
+  classical
+  simp only [childJoin,Finset.image_union,Finset.image_image,Function.comp_def,lift,Equiv.coe_fn_mk]
+
+omit [Fintype α] in
+lemma occupied_lift_card (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    (occupied (fun i => lift p (e i))).card =
+      (boolFiber (labelSet e) false ∪ (boolFiber (labelSet e) true).image (alternatingProjection p)).card := by
+  classical
+  have he : occupied (fun i => lift p (e i)) = (labelSet e).image (fun x => (lift p x).2) := by
+    simp only [occupied,labelSet,Finset.image_image,Function.comp_def]
+  rw [he,←childJoin_boolFiber (labelSet e),image_lift_childJoin]
+  simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
+  let A := boolFiber (labelSet e) false
+  let B := boolFiber (labelSet e) true
+  change (A.image (p false) ∪ B.image (p true)).card = (A ∪ B.image (alternatingProjection p)).card
+  have hh : A.image (p false) ∪ B.image (p true) =
+      (A ∪ B.image (alternatingProjection p)).image (p false) := by
+    simp only [Finset.image_union,Finset.image_image,Function.comp_def,
+      alternatingProjection,Equiv.Perm.coe_mul,Equiv.apply_symm_apply]
+  rw [hh,Finset.card_image_of_injective _ (p false).injective]
+
+lemma bothAssignments_occupied_bound (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    Fintype.card (BothAssignments e p) ≤
+      2 ^ ((occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι +
+        Fintype.card (RoutingNetwork.SelectedCycle (alternatingProjection p) (boolFiber (labelSet e) false))) := by
+  have he : (occupied e).card =
+      (boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true).card := by
+    rw [occupied_labelSet,←childJoin_boolFiber (labelSet e),image_snd_childJoin]
+    simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
+  have hc : (boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card = Fintype.card ι := by
+    rw [←card_childJoin,childJoin_boolFiber,labelSet,Finset.card_image_of_injective _ e.injective,
+      Finset.card_univ]
+  simpa only [he,occupied_lift_card,←hc] using bothAssignments_card e p
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma compatible_pairLayer_iff (x : ι → Bool × α) (c : ι → Bool) (ξ : α → Bool) :
+    Compatible (fun i => pairLayer ξ (x i)) c ↔ Compatible x c := by
+  have he (i j : ι) (ht : (x i).2 = (x j).2) :
+      required (fun i => pairLayer ξ (x i)) c i = required (fun i => pairLayer ξ (x i)) c j ↔
+      required x c i = required x c j := by
+    simp only [required,pairLayer,Equiv.coe_fn_mk,ht]
+    cases (x i).1 <;> cases (x j).1 <;> cases c i <;> cases c j <;> cases ξ (x j).2 <;> decide
+  constructor
+  · intro h i j ht
+    exact (he i j ht).mp (h i j ht)
+  · intro h i j ht
+    exact (he i j ht).mpr (h i j ht)
+
+
+noncomputable def alternatingCycles (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) : ℕ :=
+  Fintype.card (RoutingNetwork.SelectedCycle
+    (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e))
+
+omit [Fintype α] in
+lemma occupied_eq_fibers (e : ι ↪ Bool × α) :
+    occupied e = boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true := by
+  rw [occupied_labelSet,←childJoin_boolFiber (labelSet e),image_snd_childJoin]
+  simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
+
+omit [Fintype α] in
+lemma card_labelSet (e : ι ↪ Bool × α) : (labelSet e).card = Fintype.card ι := by
+  classical
+  exact (Finset.card_image_of_injective _ e.injective).trans (Finset.card_univ)
+
+omit [Fintype α] in
+lemma occupied_card_double (e : ι ↪ Bool × α) : Fintype.card ι ≤ 2*(occupied e).card := by
+  have h₀ : (boolFiber (labelSet e) false).card ≤ (occupied e).card := by
+    rw [occupied_eq_fibers]
+    exact Finset.card_le_card Finset.subset_union_left
+  have h₁ : (boolFiber (labelSet e) true).card ≤ (occupied e).card := by
+    rw [occupied_eq_fibers]
+    exact Finset.card_le_card Finset.subset_union_right
+  have he : (boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card = Fintype.card ι := by
+    rw [←card_childJoin,childJoin_boolFiber,card_labelSet]
+  omega
+
+lemma bothAssignments_exact_bound (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    Fintype.card (BothAssignments e p) ≤
+      2 ^ ((occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι +
+        alternatingCycles e p) := by
+  have h := CycleColoring.card_opposite_le
+    (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e)
+    Prod.fst (CycleColoring.alternatingPerm_flip (alternatingProjection p))
+  have hb : (CycleColoring.boundary (CycleColoring.alternatingPerm (alternatingProjection p))
+      (labelSet e)).card =
+      (occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι := by
+    have hh := alternating_boundary_card (alternatingProjection p)
+      (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
+    rw [childJoin_boolFiber] at hh
+    rw [occupied_eq_fibers,occupied_lift_card]
+    have he := card_childJoin (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
+    rw [childJoin_boolFiber,card_labelSet] at he
+    simpa only [he] using hh
+  rw [hb] at h
+  exact (Fintype.card_le_of_injective _ (assignmentColoring_injective e p)).trans h
+
+abbrev Assignments (x y : ι → Bool × α) :=
+  {c : ι → Bool // Compatible x c ∧ Compatible y c}
+
+noncomputable instance (x y : ι → Bool × α) : Fintype (Assignments x y) := Fintype.ofFinite _
+
+def switchedEmbedding (e : ι ↪ Bool × α) (ξ : α → Bool) : ι ↪ Bool × α :=
+  e.trans (pairLayer ξ).toEmbedding
+
+def routedEmbedding (e : ι ↪ Bool × α) (ξ η : α → Bool) (p : Bool → Equiv.Perm α) :
+    ι ↪ Bool × α := e.trans (sandwich ξ η p).toEmbedding
+
+omit [Fintype α] in
+lemma occupied_pairLayer (x : ι → Bool × α) (ξ : α → Bool) :
+    occupied (fun i => pairLayer ξ (x i)) = occupied x := rfl
+
+noncomputable def assignmentsEquiv (e : ι ↪ Bool × α) (ξ η : α → Bool)
+    (p : Bool → Equiv.Perm α) :
+    Assignments e (routedEmbedding e ξ η p) ≃ BothAssignments (switchedEmbedding e ξ) p := by
+  apply Equiv.subtypeEquivRight
+  intro c
+  change Compatible e c ∧ Compatible (fun i => pairLayer η (lift p (pairLayer ξ (e i)))) c ↔ _
+  rw [compatible_pairLayer_iff]
+  exact and_congr (compatible_pairLayer_iff e c ξ).symm Iff.rfl
+
+lemma assignments_actual_bound (e : ι ↪ Bool × α) (ξ η : α → Bool) (p : Bool → Equiv.Perm α) :
+    Fintype.card (Assignments e (routedEmbedding e ξ η p)) ≤
+      2 ^ ((occupied e).card + (occupied (routedEmbedding e ξ η p)).card - Fintype.card ι +
+        alternatingCycles (switchedEmbedding e ξ) p) := by
+  rw [Fintype.card_congr (assignmentsEquiv e ξ η p)]
+  exact bothAssignments_exact_bound (switchedEmbedding e ξ) p
+
+omit [Fintype α] in
+lemma assignments_actual_exponent_nonneg (e : ι ↪ Bool × α) (ξ η : α → Bool)
+    (p : Bool → Equiv.Perm α) :
+    Fintype.card ι ≤ (occupied e).card + (occupied (routedEmbedding e ξ η p)).card := by
+  have h₀ := occupied_card_double e
+  have h₁ := occupied_card_double (routedEmbedding e ξ η p)
+  omega
+
+end
+variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α] [DecidableEq ι]
+
+noncomputable def compatibleSet (x y : ι → Bool × α) : Finset (ι → Bool) := by
+  classical
+  exact Finset.univ.filter (fun c => Compatible x c ∧ Compatible y c)
+
+omit [Fintype α] [DecidableEq α] in
+lemma mem_compatibleSet (x y : ι → Bool × α) (c : ι → Bool) :
+    c ∈ compatibleSet x y ↔ Compatible x c ∧ Compatible y c := by
+  classical
+  simp only [compatibleSet,Finset.mem_filter,Finset.mem_univ,true_and]
+
+omit [Fintype α] [DecidableEq α] in
+lemma card_compatibleSet (x y : ι → Bool × α) :
+    (compatibleSet x y).card = Fintype.card (Assignments x y) := by
+  classical
+  exact (Fintype.card_subtype _).symm
+
+noncomputable def outerWeight (x y : ι → Bool × α) : ℝ :=
+  1/(2:ℝ)^((occupied x).card+(occupied y).card)
+
+omit [Fintype α] [DecidableEq ι] in
+lemma outerWeight_pos (x y : ι → Bool × α) : 0 < outerWeight x y := by
+  unfold outerWeight
+  positivity
+
+omit [Fintype α] in
+lemma sum_colors_factor (x y : ι → Bool × α) (f : (ι → Bool) → ℝ) :
+    (∑ c : ι → Bool,
+      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
+      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) * f c) =
+      outerWeight x y * ∑ c ∈ compatibleSet x y, f c := by
+  classical
+  rw [compatibleSet,Finset.sum_filter,Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro c _
+  by_cases hx : Compatible x c <;> by_cases hy : Compatible y c
+  · simp only [hx,hy,true_and,↓reduceIte,outerWeight,pow_add,one_div_mul_one_div]
+  · simp only [hx,hy,and_false,↓reduceIte,mul_zero,zero_mul]
+  · simp only [hx,hy,false_and,↓reduceIte,mul_zero,zero_mul]
+  · simp only [hx,hy,false_and,↓reduceIte,mul_zero,zero_mul]
+
+lemma sandwich_weighted_probability (p : Bool → Equiv.Perm α) (x y : ι → Bool × α)
+    (W : (ι → Bool) → ℝ) :
+    finiteMean (fun coins : (α → Bool) × (α → Bool) =>
+      if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then W (colors x coins.1) else 0) =
+    outerWeight x y * ∑ c ∈ compatibleSet x y, if ChildMatches p x y c then W c else 0 := by
+  classical
+  have he (coins : (α → Bool) × (α → Bool)) :
+      (if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then W (colors x coins.1) else 0) =
+      ∑ c : ι → Bool, (if colors x coins.1 = c then (1:ℝ) else 0) *
+        (if colors y coins.2 = c then 1 else 0) * (if ChildMatches p x y c then W c else 0) := by
+    rw [Finset.sum_eq_single (colors x coins.1)]
+    · by_cases hc : colors x coins.1 = colors y coins.2
+      · simp only [sandwich_matches_iff,hc,true_and,↓reduceIte,one_mul]
+      · simp only [sandwich_matches_iff,hc,false_and,↓reduceIte,one_mul,Ne.symm hc,zero_mul]
+    · intro c _ hc
+      simp only [Ne.symm hc,ite_false,zero_mul]
+    · simp
+  simp_rw [he]
+  rw [finiteMean_sum,←sum_colors_factor]
+  apply Finset.sum_congr rfl
+  intro c _
+  rw [finiteMean_mul_const,finiteMean_prod_mul
+    (fun ξ : α → Bool => if colors x ξ = c then (1:ℝ) else 0)
+    (fun η : α → Bool => if colors y η = c then (1:ℝ) else 0),
+    colors_probability,colors_probability]
+
+omit [Fintype α] [DecidableEq ι] in
+lemma tupleProbability_nonneg {Ω β : Type*} [Fintype Ω] [DecidableEq β]
+    (P : Ω → Equiv.Perm β) (x y : ι → β) : 0 ≤ tupleProbability P x y := by
+  classical
+  exact finiteMean_nonneg (fun ω => by split_ifs <;> norm_num)
+
+lemma sandwich_power_bound {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
+    (P : Ω → Equiv.Perm α) (Q : Ω' → Equiv.Perm α) (x y : ι → Bool × α)
+    (a : ℝ) (ha : 0 ≤ a) :
+    (finiteMean (fun ω : Ω × Ω' => finiteMean (fun coins : (α → Bool) × (α → Bool) =>
+      if ∀ i, sandwich coins.1 coins.2 (fun b => if b then Q ω.2 else P ω.1) (x i) = y i
+      then (1:ℝ) else 0)))^(1+a) ≤
+    (outerWeight x y)^(1+a) * ((compatibleSet x y).card:ℝ)^a *
+      ∑ c ∈ compatibleSet x y,
+      (tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
+        (fun i : {i // c i = false} => (y i.val).2))^(1+a) *
+      (tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
+        (fun i : {i // c i = true} => (y i.val).2))^(1+a) := by
+  classical
+  rw [sandwich_mean,sum_colors_factor]
+  let f (c : ι → Bool) : ℝ :=
+      tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
+        (fun i : {i // c i = false} => (y i.val).2) *
+      tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
+        (fun i : {i // c i = true} => (y i.val).2)
+  have hf (c : ι → Bool) : 0 ≤ f c := mul_nonneg (tupleProbability_nonneg _ _ _)
+    (tupleProbability_nonneg _ _ _)
+  rw [Real.mul_rpow (le_of_lt (outerWeight_pos _ _)) (Finset.sum_nonneg (fun c _ => hf c))]
+  have hh := Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg (compatibleSet x y)
+    (show 1 ≤ 1+a by linarith) (fun c _ => hf c)
+  have he : 1+a-1=a := by ring
+  rw [he] at hh
+  calc
+    _ ≤ (outerWeight x y)^(1+a) *
+        (((compatibleSet x y).card:ℝ)^a * ∑ c ∈ compatibleSet x y, (f c)^(1+a)) :=
+      mul_le_mul_of_nonneg_left hh (Real.rpow_nonneg (le_of_lt (outerWeight_pos _ _)) _)
+    _ = _ := by
+      rw [←mul_assoc]
+      congr 1
+      apply Finset.sum_congr rfl
+      intro c _
+      exact Real.mul_rpow (tupleProbability_nonneg _ _ _) (tupleProbability_nonneg _ _ _)
+
+end PairRouting
+
+end Thorp
+
+end ThorpNine.Smoothing
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Smoothing/AssignmentSlots.lean b/lean/OAI/Probability/ThorpRouting/Smoothing/AssignmentSlots.lean
index da2eee18e266ec26aa55bd43acc53e2b4b01ed1a..4801d88454d98fb554db331deb0d5f64d033bdab 100644
--- a/lean/OAI/Probability/ThorpRouting/Smoothing/AssignmentSlots.lean
+++ b/lean/OAI/Probability/ThorpRouting/Smoothing/AssignmentSlots.lean
@@ -1,816 +1,816 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Smoothing.RotationCharge
-
-namespace OAI
-
-namespace ThorpNine.Smoothing
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-section CycleCounting
-variable {d A : ℕ}
-
-lemma assignmentOwner_fiber {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) (i : Fin A) : cycleFiber (assignmentOwner f) i = (f.val i).val := by
-  ext x
-  simp only [mem_cycleFiber,assignmentOwner_eq_some]
-
-noncomputable def assignmentToSlots {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) : CycleSlots p S r :=
-  ⟨assignmentOwner f,by
-    refine ⟨?_,?_,?_⟩
-    · intro i
-      rw [assignmentOwner_fiber]
-      exact f.property.2 i
-    · intro i
-      rw [assignmentOwner_fiber]
-      exact (f.val i).property.2
-    · intro i x hx
-      rw [assignmentOwner_fiber]
-      exact (f.val i).orbit_eq ((assignmentOwner_eq_some f x i).mp hx)⟩
-
-noncomputable def cycleSlotsEquivAssignments (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
-    CycleSlots p S r ≃ SlotAssignment p S r where
-  toFun := slotsToAssignment
-  invFun := assignmentToSlots
-  left_inv c := by
-    apply Subtype.ext
-    apply owner_ext
-    intro i
-    rw [assignmentToSlots,assignmentOwner_fiber]
-    rfl
-  right_inv f := by
-    apply Subtype.ext
-    funext i
-    apply Subtype.ext
-    exact assignmentOwner_fiber f i
-
-end CycleCounting
-
-section GroupedCounting
-variable {d A : ℕ}
-
-abbrev LengthClass (r : Fin A → ℕ) := {j : ℕ // j ∈ Finset.univ.image (fun i => r i + 1)}
-
-def slotClass (r : Fin A → ℕ) (i : Fin A) : LengthClass r :=
-  ⟨r i+1, Finset.mem_image.mpr ⟨i, Finset.mem_univ _, rfl⟩⟩
-
-abbrev SlotsOfLength (r : Fin A → ℕ) (j : LengthClass r) := {i : Fin A // slotClass r i = j}
-abbrev CyclesOfLength (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (j : ℕ) :=
-  {C : SelectedCycle p S // C.val.card = j}
-
-noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (j : ℕ) :
-    Fintype (CyclesOfLength p S j) := Fintype.ofFinite _
-
-abbrev GroupedAssignments (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
-  ∀ j : LengthClass r, SlotsOfLength r j ↪ CyclesOfLength p S j.val
-
-noncomputable def assignmentToGroups {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) : GroupedAssignments p S r := fun _j =>
-  { toFun := fun i => ⟨f.val i.val, (f.property.2 i.val).trans (congrArg Subtype.val i.property)⟩
-    inj' := fun _i _k h => Subtype.ext (f.property.1 (congrArg Subtype.val h)) }
-
-noncomputable def groupsEmbedding {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (F : GroupedAssignments p S r) : Fin A ↪ SelectedCycle p S :=
-  ((Equiv.sigmaFiberEquiv (slotClass r)).symm.toEmbedding.trans
-    (Function.Embedding.sigmaMap (Function.Embedding.refl _) F)).trans
-    ((Function.Embedding.sigmaMap (Function.Embedding.subtype _) (fun _ => Function.Embedding.refl _)).trans
-      (Equiv.sigmaFiberEquiv (fun C : SelectedCycle p S => C.val.card)).toEmbedding)
-
-noncomputable def groupsToAssignment {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (F : GroupedAssignments p S r) : SlotAssignment p S r :=
-  ⟨groupsEmbedding F, (groupsEmbedding F).injective,
-    fun i => (F (slotClass r i) ⟨i,rfl⟩).property⟩
-
-noncomputable def assignmentEquivGroups (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
-    SlotAssignment p S r ≃ GroupedAssignments p S r where
-  toFun := assignmentToGroups
-  invFun := groupsToAssignment
-  left_inv f := by
-    apply Subtype.ext
-    rfl
-  right_inv F := by
-    funext j
-    apply Function.Embedding.ext
-    intro i
-    rcases i with ⟨i,hi⟩
-    subst j
-    rfl
-
-lemma card_cycleSlots_factorial (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
-    Fintype.card (CycleSlots p S r) =
-      ∏ j : LengthClass r, (Fintype.card (CyclesOfLength p S j.val)).descFactorial
-        (Fintype.card (SlotsOfLength r j)) := by
-  classical
-  rw [Fintype.card_congr (cycleSlotsEquivAssignments p S r),
-    Fintype.card_congr (assignmentEquivGroups p S r), Fintype.card_pi]
-  simp only [Fintype.card_embedding_eq]
-
-lemma butterfly_factorial_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-       (∏ j : LengthClass r,
-         (Fintype.card (CyclesOfLength (butterflyPerm d (decodeButterfly d ω)) S j.val)).descFactorial
-           (Fintype.card (SlotsOfLength r j)) : ℕ)) ≤
-      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-        ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
-  simpa only [←card_cycleSlots_factorial] using butterfly_cycleSlots_moment S r hr
-
-end GroupedCounting
-
-section AllLengths
-variable {d A : ℕ}
-
-noncomputable def assignmentRelabel (p : Equiv.Perm (Card d)) (S : Finset (Card d))
-    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
-    SlotAssignment p S r ≃ SlotAssignment p S (r ∘ e) where
-  toFun f := ⟨f.val ∘ e, f.property.1.comp e.injective, fun i => f.property.2 (e i)⟩
-  invFun f := ⟨f.val ∘ e.symm, f.property.1.comp e.symm.injective, fun i => by
-    simpa only [Function.comp_apply,Equiv.apply_symm_apply] using f.property.2 (e.symm i)⟩
-  left_inv f := by apply Subtype.ext; funext i; simp
-  right_inv f := by apply Subtype.ext; funext i; simp
-
-lemma card_cycleSlots_relabel (p : Equiv.Perm (Card d)) (S : Finset (Card d))
-    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
-    Fintype.card (CycleSlots p S r) = Fintype.card (CycleSlots p S (r ∘ e)) := by
-  exact Fintype.card_congr ((cycleSlotsEquivAssignments p S r).trans
-    ((assignmentRelabel p S r e).trans (cycleSlotsEquivAssignments p S (r ∘ e)).symm))
-
-lemma butterfly_cycleSlots_moment_all (S : Finset (Card d)) (r : Fin A → ℕ) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) ≤
-        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
-  have h := butterfly_cycleSlots_moment S (r ∘ Tuple.sort r) (Tuple.monotone_sort r)
-  simp_rw [←card_cycleSlots_relabel] at h
-  have hp := Equiv.prod_comp (Tuple.sort r) (fun i : Fin A => (1 / Real.sqrt (r i+1) : ℝ))
-  simpa only [Function.comp_apply,hp] using h
-
-abbrev ShortCycle (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :=
-  {C : SelectedCycle p S // C.val.card ≤ J}
-
-noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
-    Fintype (ShortCycle p S J) := Fintype.ofFinite _
-
-noncomputable def shortCycleLength {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {J : ℕ}
-    (C : ShortCycle p S J) : Fin J :=
-  ⟨C.val.val.card-1, by have := C.val.nonempty.card_pos; have := C.property; omega⟩
-
-lemma shortCycleLength_add_one {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {J : ℕ}
-    (C : ShortCycle p S J) : (shortCycleLength C : ℕ)+1 = C.val.val.card := by
-  have := C.val.nonempty.card_pos
-  simp only [shortCycleLength]
-  omega
-
-noncomputable def shortEmbeddingEquiv (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
-    (Fin A ↪ ShortCycle p S J) ≃
-      Σ r : Fin A → Fin J, SlotAssignment p S (fun i => (r i : ℕ)) where
-  toFun f := ⟨fun i => shortCycleLength (f i),
-    ⟨fun i => (f i).val, fun i j h => f.injective (Subtype.ext h),
-      fun i => (shortCycleLength_add_one (f i)).symm⟩⟩
-  invFun c :=
-    { toFun := fun i => ⟨c.2.val i, by rw [c.2.property.2]; exact (c.1 i).isLt⟩
-      inj' := fun _i _j h => c.2.property.1 (congrArg Subtype.val h) }
-  left_inv f := by
-    apply Function.Embedding.ext
-    intro i
-    rfl
-  right_inv c := by
-    rcases c with ⟨r,f⟩
-    have he : (fun i => shortCycleLength (⟨f.val i, by rw [f.property.2]; exact (r i).isLt⟩ : ShortCycle p S J)) = r := by
-      funext i
-      apply Fin.ext
-      simp only [shortCycleLength,f.property.2,Nat.add_sub_cancel]
-    apply Sigma.ext he
-    apply (Subtype.heq_iff_coe_eq (by
-      intro x
-      change (Function.Injective x ∧ ∀ i, (x i).val.card = (f.val i).val.card - 1 + 1) ↔ _
-      simp only [f.property.2,Nat.add_sub_cancel])).mpr
-    rfl
-
-lemma shortCycle_factorial_eq (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
-    (Fintype.card (ShortCycle p S J)).descFactorial A =
-      ∑ r : Fin A → Fin J, Fintype.card (CycleSlots p S (fun i => (r i : ℕ))) := by
-  classical
-  have hc := Fintype.card_embedding_eq (α := Fin A) (β := ShortCycle p S J)
-  simp only [Fintype.card_fin] at hc
-  rw [←hc, Fintype.card_congr (shortEmbeddingEquiv p S J),Fintype.card_sigma]
-  apply Finset.sum_congr rfl
-  intro r _
-  exact (Fintype.card_congr (cycleSlotsEquivAssignments p S (fun i => (r i : ℕ)))).symm
-
-end AllLengths
-
-lemma factorial_mgf {Ω : Type*} [Fintype Ω] (F : Ω → ℕ) (k : ℕ) (Λ q : ℝ)
-    (hF : ∀ ω, F ω ≤ k) (hΛ : 0 ≤ Λ) (hq : 1 ≤ q)
-    (hm : ∀ a, finiteMean (fun ω => ((F ω).descFactorial a : ℝ)) ≤ Λ^a) :
-    finiteMean (fun ω => q^(F ω)) ≤ Real.exp ((q-1)*Λ) := by
-  have hc (n a : ℕ) : (n.choose a : ℝ) = (n.descFactorial a : ℝ) / (a.factorial : ℝ) := by
-    rw [Nat.descFactorial_eq_factorial_mul_choose,Nat.cast_mul]
-    exact (mul_div_cancel_left₀ _ (by exact_mod_cast Nat.factorial_ne_zero a)).symm
-  have hex (ω : Ω) : q^(F ω) = ∑ a ∈ Finset.range (k+1),
-      ((F ω).descFactorial a : ℝ) * ((q-1)^a / (a.factorial : ℝ)) := by
-    calc
-      _ = ((q-1)+1)^(F ω) := by congr 1; ring
-      _ = ∑ a ∈ Finset.range (F ω+1), (q-1)^a * (F ω).choose a := by
-        rw [add_pow]
-        simp only [one_pow,mul_one]
-      _ = ∑ a ∈ Finset.range (k+1), (q-1)^a * (F ω).choose a := by
-        apply Finset.sum_subset (Finset.range_mono (Nat.add_le_add_right (hF ω) _))
-        intro a _ ha
-        have hn : F ω < a := by have := (Finset.mem_range.not.mp ha); omega
-        rw [Nat.choose_eq_zero_of_lt hn,Nat.cast_zero,mul_zero]
-      _ = _ := by
-        apply Finset.sum_congr rfl
-        intro a _
-        rw [hc]
-        ring
-  simp_rw [hex,finiteMean_sum,finiteMean_mul_const]
-  calc
-    _ ≤ ∑ a ∈ Finset.range (k+1), Λ^a * ((q-1)^a / (a.factorial : ℝ)) := by
-      apply Finset.sum_le_sum
-      intro a _
-      exact mul_le_mul_of_nonneg_right (hm a) (by positivity)
-    _ = ∑ a ∈ Finset.range (k+1), ((q-1)*Λ)^a / (a.factorial : ℝ) := by
-      apply Finset.sum_congr rfl
-      intro a _
-      rw [mul_pow]
-      ring
-    _ ≤ _ := Real.sum_le_exp_of_nonneg (mul_nonneg (sub_nonneg.mpr hq) hΛ) _
-
-lemma inv_sqrt_step (j : ℕ) :
-    1 / Real.sqrt (j+1) ≤ 2*(Real.sqrt (j+1)-Real.sqrt j) := by
-  have hp : 0 < Real.sqrt (j+1) := Real.sqrt_pos.mpr (by positivity)
-  apply (div_le_iff₀ hp).mpr
-  have h1 := Real.sq_sqrt (show (0:ℝ) ≤ j+1 by positivity)
-  have h0 := Real.sq_sqrt (show (0:ℝ) ≤ j by positivity)
-  nlinarith [sq_nonneg (Real.sqrt (j+1)-Real.sqrt j)]
-
-lemma sum_inv_sqrt (J : ℕ) :
-    (∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1)) ≤ 2*Real.sqrt J := by
-  have h (n : ℕ) : (∑ j ∈ Finset.range n, 1 / Real.sqrt (j+1)) ≤ 2*Real.sqrt n := by
-    induction n with
-    | zero => simp
-    | succ n ih =>
-      rw [Finset.sum_range_succ]
-      have hs := inv_sqrt_step n
-      push_cast
-      linarith
-  rw [Fin.sum_univ_eq_sum_range (fun j : ℕ => (1 / Real.sqrt (j+1) : ℝ)) J]
-  exact h J
-section CycleMGF
-variable {d : ℕ}
-
-lemma butterfly_shortCycle_moment (S : Finset (Card d)) (J A : ℕ) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      ((Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J)).descFactorial A : ℝ)) ≤
-      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))^A := by
-  classical
-  simp_rw [shortCycle_factorial_eq,Nat.cast_sum,finiteMean_sum]
-  calc
-    _ ≤ ∑ r : Fin A → Fin J,
-        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-          ∏ i : Fin A, (1 / Real.sqrt ((r i : ℕ)+1)) := by
-      apply Finset.sum_le_sum
-      intro r _
-      exact butterfly_cycleSlots_moment_all S (fun i => (r i : ℕ))
-    _ = _ := by
-      rw [←Finset.mul_sum,mul_pow]
-      congr 1
-      rw [←Fintype.prod_sum (fun (_ : Fin A) (j : Fin J) => (1 / Real.sqrt ((j:ℕ)+1) : ℝ))]
-      simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fin]
-
-noncomputable def cycleMass (p : Equiv.Perm (Card d)) (S : Finset (Card d)) : ℕ :=
-  ∑ C : SelectedCycle p S, C.val.card
-
-lemma cycleMass_le (p : Equiv.Perm (Card d)) (S : Finset (Card d)) : cycleMass p S ≤ S.card := by
-  classical
-  let f : (Σ C : SelectedCycle p S, {x : Card d // x ∈ C.val}) → S :=
-    fun c => ⟨c.2.val,c.1.property.2 c.2.property⟩
-  have hi : Function.Injective f := by
-    intro c e h
-    have hx : c.2.val = e.2.val := congrArg Subtype.val h
-    have hC : c.1 = e.1 := c.1.eq_of_mem e.1 c.2.property (hx ▸ e.2.property)
-    rcases c with ⟨c,x⟩
-    rcases e with ⟨e,y⟩
-    dsimp only at hC
-    subst e
-    have he : x=y := Subtype.ext hx
-    cases he
-    rfl
-  have hh := Fintype.card_le_of_injective f hi
-  simpa only [Fintype.card_sigma,Fintype.card_coe,cycleMass] using hh
-
-lemma shortCycle_card_le (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
-    Fintype.card (ShortCycle p S J) ≤ S.card := by
-  classical
-  calc
-    _ ≤ Fintype.card (SelectedCycle p S) := Fintype.card_subtype_le _
-    _ = ∑ _ : SelectedCycle p S, 1 := by simp
-    _ ≤ cycleMass p S := Finset.sum_le_sum (fun C _ => C.nonempty.card_pos)
-    _ ≤ _ := cycleMass_le p S
-
-lemma butterfly_shortCycle_mgf (S : Finset (Card d)) (J : ℕ) (q : ℝ) (hq : 1 ≤ q) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      q^(Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J))) ≤
-      Real.exp ((q-1)*((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))) := by
-  apply factorial_mgf _ S.card _ q (fun ω => shortCycle_card_le _ _ _)
-  · positivity
-  · exact hq
-  · exact butterfly_shortCycle_moment S J
-
-end CycleMGF
-
-section FullCycleMGF
-variable {d : ℕ}
-
-abbrev LongCycle (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :=
-  {C : SelectedCycle p S // ¬ C.val.card ≤ J}
-noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
-    Fintype (LongCycle p S J) := Fintype.ofFinite _
-
-lemma selectedCycle_card_split (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
-    Fintype.card (SelectedCycle p S) =
-      Fintype.card (ShortCycle p S J) + Fintype.card (LongCycle p S J) := by
-  classical
-  rw [←Fintype.card_congr (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J)),Fintype.card_sum]
-
-lemma longCycle_card_bound (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
-    J * Fintype.card (LongCycle p S J) ≤ S.card := by
-  classical
-  have hs := Equiv.sum_comp (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J))
-    (fun C : SelectedCycle p S => C.val.card)
-  rw [Fintype.sum_sum_type] at hs
-  have h0 : J * Fintype.card (LongCycle p S J) ≤
-      ∑ C : LongCycle p S J, C.val.val.card := by
-    calc
-      _ = ∑ _C : LongCycle p S J, J := by simp [mul_comm]
-      _ ≤ _ := Finset.sum_le_sum (fun C _ => Nat.le_of_lt (Nat.lt_of_not_ge C.property))
-  have hh := cycleMass_le p S
-  unfold cycleMass at hh
-  rw [←hs] at hh
-  exact h0.trans ((Nat.le_add_left _ _).trans hh)
-
-lemma finiteMean_pos {Ω : Type*} [Fintype Ω] [Nonempty Ω] {f : Ω → ℝ}
-    (h : ∀ ω, 0 < f ω) : 0 < finiteMean f := by
-  apply div_pos
-  · exact Finset.sum_pos (fun ω _ => h ω) Finset.univ_nonempty
-  · exact Nat.cast_pos.mpr Fintype.card_pos
-
-lemma longCycle_pow_bound (p : Equiv.Perm (Card d)) (S : Finset (Card d))
-    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    q^(Fintype.card (LongCycle p S J)) ≤ Real.exp ((S.card : ℝ)/J * Real.log q) := by
-  have hq0 : 0 < q := lt_of_lt_of_le zero_lt_one hq
-  have hc : (Fintype.card (LongCycle p S J) : ℝ) ≤ (S.card : ℝ)/J := by
-    apply (le_div_iff₀ (by exact_mod_cast hJ)).mpr
-    have hh := longCycle_card_bound p S J
-    exact_mod_cast (by simpa only [mul_comm] using hh)
-  calc
-    _ = Real.exp ((Fintype.card (LongCycle p S J) : ℝ)*Real.log q) := by
-      rw [Real.exp_nat_mul,Real.exp_log hq0]
-    _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right hc (Real.log_nonneg hq))
-
-lemma butterfly_cycle_log_mgf (S : Finset (Card d)) (J : ℕ) (hJ : 0 < J)
-    (q : ℝ) (hq : 1 ≤ q) :
-    Real.log (finiteMean (fun ω : SwitchIndex d → Bool =>
-      q^(Fintype.card (SelectedCycle (butterflyPerm d (decodeButterfly d ω)) S)))) ≤
-      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
-  let lam : ℝ := (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1)
-  let c : ℝ := (S.card : ℝ)/J * Real.log q
-  have hnum : (q-1)*lam ≤ 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
-    have hsum := sum_inv_sqrt J
-    have hp : 0 ≤ (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) := by positivity
-    have h0 : 0 ≤ 2*Real.sqrt J := by positivity
-    have hh := mul_le_mul_of_nonneg_left hsum hp
-    have hq1 : 0 ≤ q-1 := sub_nonneg.mpr hq
-    calc
-      _ ≤ (q-1)*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
-        mul_le_mul_of_nonneg_left hh hq1
-      _ ≤ q*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
-        mul_le_mul_of_nonneg_right (by linarith) (mul_nonneg hp h0)
-      _ = _ := by ring
-  have hm : finiteMean (fun ω : SwitchIndex d → Bool =>
-      q^(Fintype.card (SelectedCycle (butterflyPerm d (decodeButterfly d ω)) S))) ≤
-      Real.exp (c+(q-1)*lam) := by
-    calc
-      _ ≤ finiteMean (fun ω : SwitchIndex d → Bool =>
-          q^(Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J))*Real.exp c) := by
-        apply finiteMean_mono
-        intro ω
-        rw [selectedCycle_card_split _ _ J,pow_add]
-        exact mul_le_mul_of_nonneg_left (longCycle_pow_bound _ S J hJ q hq) (by positivity)
-      _ = finiteMean (fun ω : SwitchIndex d → Bool =>
-          q^(Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J))) * Real.exp c :=
-        finiteMean_mul_const _ _
-      _ ≤ Real.exp ((q-1)*lam) * Real.exp c :=
-        mul_le_mul_of_nonneg_right (butterfly_shortCycle_mgf S J q hq) (Real.exp_nonneg _)
-      _ = _ := by rw [←Real.exp_add,add_comm]
-  have hpos : 0 < finiteMean (fun ω : SwitchIndex d → Bool =>
-      q^(Fintype.card (SelectedCycle (butterflyPerm d (decodeButterfly d ω)) S))) := by
-    apply finiteMean_pos
-    intro ω
-    positivity
-  have hlog := Real.log_le_log hpos hm
-  rw [Real.log_exp] at hlog
-  dsimp [c] at hlog
-  linarith
-
-end FullCycleMGF
-
-structure RoutingNetwork (α ι : Type*) [Fintype α] [DecidableEq α]
-    [Fintype ι] [DecidableEq ι] (d : ℕ) where
-  perm : (ι → Bool) → Equiv.Perm α
-  path : (ι → Bool) → α → Finset ι
-  users : (ι → Bool) → ι → α × α
-  users_ne : ∀ ω i, (users ω i).1 ≠ (users ω i).2
-  mem_path : ∀ ω x i, i ∈ path ω x ↔ (users ω i).1 = x ∨ (users ω i).2 = x
-  card_path : ∀ ω x, (path ω x).card = d
-  entropy : ∀ ω (S : Finset α),
-    (∑ i : ι, if (users ω i).1 ∈ S ∧ (users ω i).2 ∈ S then (1:ℝ) else 0) ≤
-      (S.card : ℝ) * Real.log S.card / (2 * Real.log 2)
-  follow : α → (α → QueryTree ι) → QueryTree ι
-  close : α → α → QueryTree ι → QueryTree ι
-  follow_succeeds : ∀ x k ω, (follow x k).succeeds ω = (k (perm ω x)).succeeds ω
-  follow_raw_charge : ∀ x k ω, (follow x k).charge ω = (k (perm ω x)).charge ω
-  follow_stale_charge : ∀ x k R ω,
-    (follow x k).staleCharge R ω = (k (perm ω x)).staleCharge (R ∪ path ω x) ω
-  close_succeeds : ∀ x y k ω,
-    (close x y k).succeeds ω = if perm ω x = y then k.succeeds ω else false
-  close_raw_charge : ∀ x y k ω, perm ω x = y → (close x y k).charge ω = k.charge ω + d
-  close_stale_charge : ∀ x y k R ω, perm ω x = y →
-    (close x y k).staleCharge R ω = k.staleCharge (R ∪ path ω x) ω + (path ω x ∩ R).card
-
-namespace RoutingNetwork
-variable {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι] [DecidableEq ι]
-  {d : ℕ} (R : RoutingNetwork α ι d)
-
-lemma card_switches_using (ω : ι → Bool) (x : α) :
-    (Finset.univ.filter (fun i : ι =>
-      (R.users ω i).1 = x ∨ (R.users ω i).2 = x)).card = d := by
-  classical
-  have he : Finset.univ.filter (fun i : ι =>
-      (R.users ω i).1 = x ∨ (R.users ω i).2 = x) = R.path ω x := by
-    ext i
-    simp only [Finset.mem_filter,Finset.mem_univ,true_and,R.mem_path]
-  rw [he,R.card_path]
-lemma switch_incidence_sum (b : ι → Bool) (w : α → ℝ) :
-    ∑ i : ι, (w (R.users b i).1 + w (R.users b i).2) =
-      (d : ℝ) * ∑ x : α, w x := by
-  classical
-  calc
-    _ = ∑ i : ι, ∑ x : α,
-        if (R.users b i).1 = x ∨ (R.users b i).2 = x then w x else 0 := by
-      apply Finset.sum_congr rfl
-      intro i _
-      rw [Finset.sum_ite]
-      simp only [Finset.sum_const_zero, add_zero]
-      have he : Finset.univ.filter (fun x : α =>
-          (R.users b i).1 = x ∨ (R.users b i).2 = x) =
-          {(R.users b i).1, (R.users b i).2} := by
-        ext x; simp [eq_comm]
-      rw [he, Finset.sum_pair (R.users_ne b i)]
-    _ = ∑ x : α, ∑ i : ι,
-        if (R.users b i).1 = x ∨ (R.users b i).2 = x then w x else 0 :=
-      Finset.sum_comm
-    _ = ∑ x : α, (d : ℝ) * w x := by
-      apply Finset.sum_congr rfl
-      intro x _
-      rw [Finset.sum_ite]
-      simp only [Finset.sum_const_zero, add_zero, Finset.sum_const, nsmul_eq_mul,
-        card_switches_using]
-    _ = _ := by rw [Finset.mul_sum]
-
-lemma weighted_switch_min_bound (b : ι → Bool) (w : α → ℝ) :
-    ∑ i : ι, min (w (R.users b i).1) (w (R.users b i).2) ≤
-      (d : ℝ) / 2 * ∑ x : α, w x := by
-  have h : 2 * (∑ i : ι,
-      min (w (R.users b i).1) (w (R.users b i).2)) ≤
-      ∑ i : ι, (w (R.users b i).1 + w (R.users b i).2) := by
-    rw [Finset.mul_sum]
-    apply Finset.sum_le_sum
-    intro i _
-    linarith [min_le_left (w (R.users b i).1) (w (R.users b i).2),
-      min_le_right (w (R.users b i).1) (w (R.users b i).2)]
-  rw [switch_incidence_sum] at h
-  linarith
-section Rotation
-variable {A : ℕ}
-
-def cycleFiber (o : α → Option (Fin A)) (i : Fin A) : Finset (α) := by
-  classical
-  exact Finset.univ.filter (fun x => o x = some i)
-
-omit [DecidableEq α] in
-@[simp] lemma mem_cycleFiber (o : α → Option (Fin A)) (i : Fin A) (x : α) :
-    x ∈ cycleFiber o i ↔ o x = some i := by
-  classical
-  simp [cycleFiber]
-
-abbrev Terminal (o : α → Option (Fin A)) := ∀ i : Fin A, {x : α // o x = some i}
-
-noncomputable def slotWeight (o : α → Option (Fin A)) (i : Fin A) : ℝ :=
-  1 / (cycleFiber o i).card
-
-noncomputable def pathWeight (o : α → Option (Fin A)) (x : α) : ℝ :=
-  match o x with
-  | none => 0
-  | some i => slotWeight o i
-
-noncomputable def withinWeight (o : α → Option (Fin A)) (u v : α) : ℝ :=
-  match o u, o v with
-  | some i, some j => if i = j then slotWeight o i else 0
-  | _, _ => 0
-
-noncomputable def rotationMeeting (o : α → Option (Fin A)) (z : Terminal o)
-    (u v : α) : ℝ :=
-  match o u, o v with
-  | some i, some j => if i < j then (if (z j).val = v then 1 else 0)
-      else if j < i then (if (z i).val = u then 1 else 0)
-      else (if (z i).val = u then 1 else 0) + (if (z i).val = v then 1 else 0)
-  | _, _ => 0
-
-omit [DecidableEq α] in
-lemma card_cycleFiber (o : α → Option (Fin A)) (i : Fin A) :
-    Fintype.card {x : α // o x = some i} = (cycleFiber o i).card := by
-  classical
-  simp [cycleFiber,Fintype.card_subtype]
-
-lemma terminal_mean {o : α → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
-    (i : Fin A) (x : α) (hx : o x = some i) :
-    finiteMean (fun z : Terminal o => if (z i).val = x then (1:ℝ) else 0) = slotWeight o i := by
-  classical
-  have hev := finiteMean_pi_eval (A := fun i : Fin A => {x : α // o x = some i})
-    i (fun z => if z.val = x then (1:ℝ) else 0)
-  rw [hev]
-  have he : (fun z : {x : α // o x = some i} => if z.val = x then (1:ℝ) else 0) =
-      (fun z => if z = ⟨x,hx⟩ then 1 else 0) := by
-    funext z
-    simp only [Subtype.ext_iff]
-  rw [he,finiteMean_eq_indicator,card_cycleFiber,slotWeight]
-
-omit [DecidableEq α] in
-lemma slotWeight_antitone (o : α → Option (Fin A))
-    (hn : ∀ i, (cycleFiber o i).Nonempty)
-    (hm : Monotone (fun i => (cycleFiber o i).card)) : Antitone (slotWeight o) := by
-  intro i j hij
-  unfold slotWeight
-  apply one_div_le_one_div_of_le
-  · exact_mod_cast Finset.card_pos.mpr (hn i)
-  · exact_mod_cast hm hij
-
-lemma rotationMeeting_mean {o : α → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
-    (hm : Monotone (fun i => (cycleFiber o i).card)) (u v : α) :
-    finiteMean (fun z : Terminal o => rotationMeeting o z u v) =
-      min (pathWeight o u) (pathWeight o v) + withinWeight o u v := by
-  classical
-  have hn : ∀ i, (cycleFiber o i).Nonempty := by
-    intro i
-    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : α // o x = some i})
-    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
-  have hw := slotWeight_antitone o hn hm
-  cases hu : o u with
-  | none =>
-    cases hv : o v with
-    | none => simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean]
-    | some j =>
-      have hj : 0 ≤ slotWeight o j := by unfold slotWeight; positivity
-      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_left hj]
-  | some i =>
-    cases hv : o v with
-    | none =>
-      have hi : 0 ≤ slotWeight o i := by unfold slotWeight; positivity
-      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_right hi]
-    | some j =>
-      simp only [rotationMeeting,pathWeight,withinWeight,hu,hv]
-      rcases lt_trichotomy i j with hij | rfl | hji
-      · simp only [hij,↓reduceIte,ne_of_lt hij]
-        rw [terminal_mean j v hv,min_eq_right (hw (le_of_lt hij)),add_zero]
-      · simp only [lt_self_iff_false,↓reduceIte,min_self]
-        rw [finiteMean_add,terminal_mean i u hu,terminal_mean i v hv]
-      · simp only [not_lt_of_ge (le_of_lt hji),hji,↓reduceIte,ne_of_gt hji]
-        rw [terminal_mean i u hu,min_eq_left (hw (le_of_lt hji)),add_zero]
-
-omit [DecidableEq α] in
-lemma pathWeight_sum (o : α → Option (Fin A)) (hn : ∀ i, (cycleFiber o i).Nonempty) :
-    ∑ x : α, pathWeight o x = (A : ℝ) := by
-  classical
-  have he (x : α) : pathWeight o x =
-      ∑ i : Fin A, if o x = some i then slotWeight o i else 0 := by
-    cases hx : o x with
-    | none => simp [pathWeight,hx]
-    | some j => simp [pathWeight,hx]
-  simp only [he]
-  rw [Finset.sum_comm]
-  have hf (i : Fin A) : (∑ x : α, if o x = some i then slotWeight o i else 0) = 1 := by
-    rw [← Finset.sum_filter]
-    change (∑ _x ∈ cycleFiber o i, slotWeight o i) = 1
-    simp only [Finset.sum_const,nsmul_eq_mul,slotWeight]
-    exact mul_one_div_cancel (by exact_mod_cast (Finset.card_pos.mpr (hn i)).ne')
-  simp only [hf,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,mul_one]
-
-lemma withinWeight_sum_bound (o : α → Option (Fin A))
-    (hn : ∀ i, (cycleFiber o i).Nonempty) (b : ι → Bool) :
-    (∑ s : ι, withinWeight o (R.users b s).1 (R.users b s).2) ≤
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
-  classical
-  have he (u v : α) : withinWeight o u v =
-      ∑ i : Fin A, (if u ∈ cycleFiber o i ∧ v ∈ cycleFiber o i then (1:ℝ) else 0) * slotWeight o i := by
-    simp only [mem_cycleFiber]
-    cases hu : o u <;> cases hv : o v <;> simp [withinWeight,hu,hv,ite_and,eq_comm]
-  simp only [he]
-  rw [Finset.sum_comm]
-  apply Finset.sum_le_sum
-  intro i _
-  rw [← Finset.sum_mul]
-  have hi := R.entropy b (cycleFiber o i)
-  have hpos : (0:ℝ) < (cycleFiber o i).card := by exact_mod_cast Finset.card_pos.mpr (hn i)
-  have hmul := mul_le_mul_of_nonneg_right hi (show 0 ≤ slotWeight o i by unfold slotWeight; positivity)
-  calc
-    _ ≤ ((cycleFiber o i).card : ℝ) * Real.log (cycleFiber o i).card /
-        (2 * Real.log 2) * slotWeight o i := hmul
-    _ = _ := by unfold slotWeight; field_simp
-
-noncomputable def rotationCharge (o : α → Option (Fin A)) (b : ι → Bool)
-    (z : Terminal o) : ℝ :=
-  ∑ s : ι, rotationMeeting o z (R.users b s).1 (R.users b s).2
-
-lemma rotationCharge_mean_bound {o : α → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
-    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : ι → Bool) :
-    finiteMean (rotationCharge R o b) ≤ (A:ℝ) * d / 2 +
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
-  classical
-  have hn : ∀ i, (cycleFiber o i).Nonempty := by
-    intro i
-    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : α // o x = some i})
-    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
-  unfold rotationCharge
-  rw [finiteMean_sum]
-  simp only [rotationMeeting_mean hm,Finset.sum_add_distrib]
-  have h₁ := weighted_switch_min_bound R b (pathWeight o)
-  rw [pathWeight_sum o hn] at h₁
-  have h₂ := withinWeight_sum_bound R o hn b
-  linarith
-
-lemma rotation_exp_bound {o : α → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
-    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : ι → Bool) :
-    Real.exp (-Real.log 2 * ((A:ℝ)*d/2 +
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2))) ≤
-      finiteMean (fun z : Terminal o => Real.exp (-Real.log 2 * rotationCharge R o b z)) := by
-  have hb := rotationCharge_mean_bound R hm b
-  have hlog : 0 ≤ Real.log 2 := Real.log_nonneg (by norm_num)
-  have hm' : -Real.log 2 * ((A:ℝ)*d/2 +
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2)) ≤
-      finiteMean (fun z : Terminal o => -Real.log 2 * rotationCharge R o b z) := by
-    have he := finiteMean_mul_const (rotationCharge R o b) (-Real.log 2)
-    simp only [mul_comm] at he
-    rw [he]
-    exact mul_le_mul_of_nonpos_left hb (neg_nonpos.mpr hlog)
-  exact (Real.exp_le_exp.mpr hm').trans (exp_finiteMean_le _)
-
-end Rotation
-
-namespace QueryTree
-open Thorp.QueryTree
-
-noncomputable def seekCycle (R : RoutingNetwork α ι d) (S : Finset (α)) (a : α) :
-    ℕ → α → Finset (α) → (Finset (α) → QueryTree (ι)) →
-      QueryTree (ι)
-  | 0, x, V, k => if x ∈ S ∧ x ∉ V then R.close x a (k (insert x V)) else reject
-  | r + 1, x, V, k => if x ∈ S ∧ x ∉ V then R.follow x
-      (fun y => seekCycle R S a r y (insert x V) k) else reject
-
-lemma seekCycle_raw_charge (S : Finset (α)) (a : α) (r : ℕ)
-    (x : α) (V : Finset (α)) (k : Finset (α) → QueryTree (ι))
-    (ω : ι → Bool) (K : ℕ)
-    (hK : ∀ V, (k V).succeeds ω = true → (k V).charge ω = K)
-    (hs : (seekCycle R S a r x V k).succeeds ω = true) :
-    (seekCycle R S a r x V k).charge ω = K + d := by
-  induction r generalizing x V with
-  | zero =>
-    by_cases hv : x ∈ S ∧ x ∉ V
-    · rw [seekCycle,ite_eq_left hv,R.close_succeeds] at hs
-      split_ifs at hs with he
-      · rw [seekCycle,ite_eq_left hv,R.close_raw_charge _ _ _ _ he,hK _ hs]
-    · simp [seekCycle,hv,Thorp.QueryTree.succeeds] at hs
-  | succ r ih =>
-    by_cases hv : x ∈ S ∧ x ∉ V
-    · rw [seekCycle,ite_eq_left hv,R.follow_succeeds] at hs
-      rw [seekCycle,ite_eq_left hv,R.follow_raw_charge]
-      exact ih _ _ hs
-    · simp [seekCycle,hv,Thorp.QueryTree.succeeds] at hs
-
-noncomputable def seekCycles (R : RoutingNetwork α ι d) (S : Finset (α)) :
-    List (α × ℕ) → Finset (α) → QueryTree (ι)
-  | [], _ => accept
-  | (a,r) :: L, V => seekCycle R S a r a V (fun V => seekCycles R S L V)
-
-lemma seekCycles_raw_charge (S : Finset (α)) (L : List (α × ℕ))
-    (V : Finset (α)) (ω : ι → Bool)
-    (hs : (seekCycles R S L V).succeeds ω = true) :
-    (seekCycles R S L V).charge ω = L.length * d := by
-  induction L generalizing V with
-  | nil => simp [seekCycles,charge]
-  | cons ar L ih =>
-    obtain ⟨a,r⟩ := ar
-    rw [seekCycles]
-    rw [seekCycle_raw_charge R S a r a V (fun V => seekCycles R S L V) ω (L.length*d)
-      (fun V hs => ih V hs) hs]
-    simp [Nat.add_mul]
-
-lemma cycles_weighted_success (S : Finset (α)) (L : List (α × ℕ)) :
-    finiteMean (fun ω : ι → Bool =>
-      if (seekCycles R S L ∅).succeeds ω then
-        ((2:ℝ)^d)^L.length / (2:ℝ)^(seekCycles R S L ∅).staleCharge ∅ ω else 0) ≤ 1 := by
-  let T := seekCycles R S L ∅
-  have hh := mean_freshCharge_le_one T
-  have he : finiteMean (fun ω : ι → Bool =>
-      if T.succeeds ω then (2:ℝ)^T.freshCharge ∅ ω else 0) =
-      finiteMean (fun ω : ι → Bool =>
-        if T.succeeds ω then ((2:ℝ)^d)^L.length / (2:ℝ)^T.staleCharge ∅ ω else 0) := by
-    apply finiteMean_congr
-    intro ω
-    cases hs : T.succeeds ω
-    · simp
-    · simp only [↓reduceIte]
-      have hc := fresh_add_stale T ∅ ω
-      rw [seekCycles_raw_charge R S L ∅ ω hs] at hc
-      have hp : ((2:ℝ)^d)^L.length = (2:ℝ)^T.freshCharge ∅ ω * (2:ℝ)^T.staleCharge ∅ ω := by
-        rw [←pow_add,hc,←pow_mul,Nat.mul_comm]
-      rw [hp,mul_div_cancel_right₀ _ (pow_ne_zero _ (by norm_num))]
-  rw [← he]
-  exact hh
-
-end QueryTree
-noncomputable def exposedPaths (b : ι → Bool) (V : Finset (α)) :
-    Finset (ι) :=
-  V.biUnion (fun x => R.path b x)
-
-@[simp] lemma exposedPaths_empty (b : ι → Bool) : exposedPaths R b ∅ = ∅ := by
-  classical
-  simp [exposedPaths]
-
-lemma exposedPaths_insert (b : ι → Bool) (V : Finset (α)) (x : α) :
-    exposedPaths R b (insert x V) = exposedPaths R b V ∪ R.path b x := by
-  classical
-  simp [exposedPaths,Finset.union_comm]
-
-noncomputable def terminalCharge (b : ι → Bool) (z : α) (V : Finset (α)) : ℕ :=
-  (R.path b z ∩ exposedPaths R b V).card
-
-namespace QueryTree
-open Thorp.QueryTree
-
-lemma seekCycle_semantics (S : Finset (α)) (a : α) (r : ℕ)
-    (x : α) (V W : Finset (α)) (z : α)
-    (k : Finset (α) → QueryTree (ι)) (ω : ι → Bool)
-    (h : cycleTrace (R.perm ω) S a r x V = some (W,z)) :
-    (seekCycle R S a r x V k).succeeds ω = (k W).succeeds ω ∧
-      (seekCycle R S a r x V k).staleCharge (exposedPaths R (ω) V) ω =
-        (k W).staleCharge (exposedPaths R (ω) W) ω +
-          terminalCharge R (ω) z (W.erase z) := by
-  classical
-  induction r generalizing x V with
-  | zero =>
-    simp only [cycleTrace] at h
-    split_ifs at h with hv
-    obtain ⟨hxS,hxV,hxa⟩ := hv
-    obtain ⟨rfl,rfl⟩ := Prod.mk.inj (Option.some.inj h)
-    simp only [seekCycle,ite_eq_left (show x ∈ S ∧ x ∉ V from ⟨hxS,hxV⟩),R.close_succeeds,ite_eq_left hxa]
-    constructor
-    · trivial
-    · rw [R.close_stale_charge _ _ _ _ _ hxa,exposedPaths_insert,
-        Finset.erase_insert hxV]
-      simp only [terminalCharge]
-  | succ r ih =>
-    simp only [cycleTrace] at h
-    split_ifs at h with hv
-    obtain ⟨hxS,hxV⟩ := hv
-    rw [seekCycle,ite_eq_left ⟨hxS,hxV⟩,R.follow_succeeds,R.follow_stale_charge]
-    rw [← exposedPaths_insert]
-    exact ih _ _ h
-
-lemma seekCycle_orbit (S V : Finset (α)) (z : α)
-    (k : Finset (α) → QueryTree (ι)) (ω : ι → Bool)
-    (hS : orbitSet (R.perm ω) z ⊆ S)
-    (hV : Disjoint V (orbitSet (R.perm ω) z)) :
-    let p := R.perm ω
-    let W := V ∪ orbitSet p z
-    (seekCycle R S (p z) (Function.minimalPeriod p z-1) (p z) V k).succeeds ω = (k W).succeeds ω ∧
-      (seekCycle R S (p z) (Function.minimalPeriod p z-1) (p z) V k).staleCharge
-          (exposedPaths R (ω) V) ω =
-        (k W).staleCharge (exposedPaths R (ω) W) ω +
-          terminalCharge R (ω) z (W.erase z) := by
-  exact seekCycle_semantics _ _ _ _ _ _ _ _ _ _ (cycleTrace_orbit _ _ _ _ hS hV)
-
-end QueryTree
-
-end RoutingNetwork
-end Thorp
-
-end ThorpNine.Smoothing
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Smoothing.RotationCharge
+
+namespace OAI
+
+namespace ThorpNine.Smoothing
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+section CycleCounting
+variable {d A : ℕ}
+
+lemma assignmentOwner_fiber {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) (i : Fin A) : cycleFiber (assignmentOwner f) i = (f.val i).val := by
+  ext x
+  simp only [mem_cycleFiber,assignmentOwner_eq_some]
+
+noncomputable def assignmentToSlots {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) : CycleSlots p S r :=
+  ⟨assignmentOwner f,by
+    refine ⟨?_,?_,?_⟩
+    · intro i
+      rw [assignmentOwner_fiber]
+      exact f.property.2 i
+    · intro i
+      rw [assignmentOwner_fiber]
+      exact (f.val i).property.2
+    · intro i x hx
+      rw [assignmentOwner_fiber]
+      exact (f.val i).orbit_eq ((assignmentOwner_eq_some f x i).mp hx)⟩
+
+noncomputable def cycleSlotsEquivAssignments (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
+    CycleSlots p S r ≃ SlotAssignment p S r where
+  toFun := slotsToAssignment
+  invFun := assignmentToSlots
+  left_inv c := by
+    apply Subtype.ext
+    apply owner_ext
+    intro i
+    rw [assignmentToSlots,assignmentOwner_fiber]
+    rfl
+  right_inv f := by
+    apply Subtype.ext
+    funext i
+    apply Subtype.ext
+    exact assignmentOwner_fiber f i
+
+end CycleCounting
+
+section GroupedCounting
+variable {d A : ℕ}
+
+abbrev LengthClass (r : Fin A → ℕ) := {j : ℕ // j ∈ Finset.univ.image (fun i => r i + 1)}
+
+def slotClass (r : Fin A → ℕ) (i : Fin A) : LengthClass r :=
+  ⟨r i+1, Finset.mem_image.mpr ⟨i, Finset.mem_univ _, rfl⟩⟩
+
+abbrev SlotsOfLength (r : Fin A → ℕ) (j : LengthClass r) := {i : Fin A // slotClass r i = j}
+abbrev CyclesOfLength (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (j : ℕ) :=
+  {C : SelectedCycle p S // C.val.card = j}
+
+noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (j : ℕ) :
+    Fintype (CyclesOfLength p S j) := Fintype.ofFinite _
+
+abbrev GroupedAssignments (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
+  ∀ j : LengthClass r, SlotsOfLength r j ↪ CyclesOfLength p S j.val
+
+noncomputable def assignmentToGroups {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) : GroupedAssignments p S r := fun _j =>
+  { toFun := fun i => ⟨f.val i.val, (f.property.2 i.val).trans (congrArg Subtype.val i.property)⟩
+    inj' := fun _i _k h => Subtype.ext (f.property.1 (congrArg Subtype.val h)) }
+
+noncomputable def groupsEmbedding {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (F : GroupedAssignments p S r) : Fin A ↪ SelectedCycle p S :=
+  ((Equiv.sigmaFiberEquiv (slotClass r)).symm.toEmbedding.trans
+    (Function.Embedding.sigmaMap (Function.Embedding.refl _) F)).trans
+    ((Function.Embedding.sigmaMap (Function.Embedding.subtype _) (fun _ => Function.Embedding.refl _)).trans
+      (Equiv.sigmaFiberEquiv (fun C : SelectedCycle p S => C.val.card)).toEmbedding)
+
+noncomputable def groupsToAssignment {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (F : GroupedAssignments p S r) : SlotAssignment p S r :=
+  ⟨groupsEmbedding F, (groupsEmbedding F).injective,
+    fun i => (F (slotClass r i) ⟨i,rfl⟩).property⟩
+
+noncomputable def assignmentEquivGroups (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
+    SlotAssignment p S r ≃ GroupedAssignments p S r where
+  toFun := assignmentToGroups
+  invFun := groupsToAssignment
+  left_inv f := by
+    apply Subtype.ext
+    rfl
+  right_inv F := by
+    funext j
+    apply Function.Embedding.ext
+    intro i
+    rcases i with ⟨i,hi⟩
+    subst j
+    rfl
+
+lemma card_cycleSlots_factorial (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
+    Fintype.card (CycleSlots p S r) =
+      ∏ j : LengthClass r, (Fintype.card (CyclesOfLength p S j.val)).descFactorial
+        (Fintype.card (SlotsOfLength r j)) := by
+  classical
+  rw [Fintype.card_congr (cycleSlotsEquivAssignments p S r),
+    Fintype.card_congr (assignmentEquivGroups p S r), Fintype.card_pi]
+  simp only [Fintype.card_embedding_eq]
+
+lemma butterfly_factorial_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+       (∏ j : LengthClass r,
+         (Fintype.card (CyclesOfLength (butterflyPerm d (decodeButterfly d ω)) S j.val)).descFactorial
+           (Fintype.card (SlotsOfLength r j)) : ℕ)) ≤
+      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+        ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
+  simpa only [←card_cycleSlots_factorial] using butterfly_cycleSlots_moment S r hr
+
+end GroupedCounting
+
+section AllLengths
+variable {d A : ℕ}
+
+noncomputable def assignmentRelabel (p : Equiv.Perm (Card d)) (S : Finset (Card d))
+    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
+    SlotAssignment p S r ≃ SlotAssignment p S (r ∘ e) where
+  toFun f := ⟨f.val ∘ e, f.property.1.comp e.injective, fun i => f.property.2 (e i)⟩
+  invFun f := ⟨f.val ∘ e.symm, f.property.1.comp e.symm.injective, fun i => by
+    simpa only [Function.comp_apply,Equiv.apply_symm_apply] using f.property.2 (e.symm i)⟩
+  left_inv f := by apply Subtype.ext; funext i; simp
+  right_inv f := by apply Subtype.ext; funext i; simp
+
+lemma card_cycleSlots_relabel (p : Equiv.Perm (Card d)) (S : Finset (Card d))
+    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
+    Fintype.card (CycleSlots p S r) = Fintype.card (CycleSlots p S (r ∘ e)) := by
+  exact Fintype.card_congr ((cycleSlotsEquivAssignments p S r).trans
+    ((assignmentRelabel p S r e).trans (cycleSlotsEquivAssignments p S (r ∘ e)).symm))
+
+lemma butterfly_cycleSlots_moment_all (S : Finset (Card d)) (r : Fin A → ℕ) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) ≤
+        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
+  have h := butterfly_cycleSlots_moment S (r ∘ Tuple.sort r) (Tuple.monotone_sort r)
+  simp_rw [←card_cycleSlots_relabel] at h
+  have hp := Equiv.prod_comp (Tuple.sort r) (fun i : Fin A => (1 / Real.sqrt (r i+1) : ℝ))
+  simpa only [Function.comp_apply,hp] using h
+
+abbrev ShortCycle (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :=
+  {C : SelectedCycle p S // C.val.card ≤ J}
+
+noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
+    Fintype (ShortCycle p S J) := Fintype.ofFinite _
+
+noncomputable def shortCycleLength {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {J : ℕ}
+    (C : ShortCycle p S J) : Fin J :=
+  ⟨C.val.val.card-1, by have := C.val.nonempty.card_pos; have := C.property; omega⟩
+
+lemma shortCycleLength_add_one {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {J : ℕ}
+    (C : ShortCycle p S J) : (shortCycleLength C : ℕ)+1 = C.val.val.card := by
+  have := C.val.nonempty.card_pos
+  simp only [shortCycleLength]
+  omega
+
+noncomputable def shortEmbeddingEquiv (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
+    (Fin A ↪ ShortCycle p S J) ≃
+      Σ r : Fin A → Fin J, SlotAssignment p S (fun i => (r i : ℕ)) where
+  toFun f := ⟨fun i => shortCycleLength (f i),
+    ⟨fun i => (f i).val, fun i j h => f.injective (Subtype.ext h),
+      fun i => (shortCycleLength_add_one (f i)).symm⟩⟩
+  invFun c :=
+    { toFun := fun i => ⟨c.2.val i, by rw [c.2.property.2]; exact (c.1 i).isLt⟩
+      inj' := fun _i _j h => c.2.property.1 (congrArg Subtype.val h) }
+  left_inv f := by
+    apply Function.Embedding.ext
+    intro i
+    rfl
+  right_inv c := by
+    rcases c with ⟨r,f⟩
+    have he : (fun i => shortCycleLength (⟨f.val i, by rw [f.property.2]; exact (r i).isLt⟩ : ShortCycle p S J)) = r := by
+      funext i
+      apply Fin.ext
+      simp only [shortCycleLength,f.property.2,Nat.add_sub_cancel]
+    apply Sigma.ext he
+    apply (Subtype.heq_iff_coe_eq (by
+      intro x
+      change (Function.Injective x ∧ ∀ i, (x i).val.card = (f.val i).val.card - 1 + 1) ↔ _
+      simp only [f.property.2,Nat.add_sub_cancel])).mpr
+    rfl
+
+lemma shortCycle_factorial_eq (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
+    (Fintype.card (ShortCycle p S J)).descFactorial A =
+      ∑ r : Fin A → Fin J, Fintype.card (CycleSlots p S (fun i => (r i : ℕ))) := by
+  classical
+  have hc := Fintype.card_embedding_eq (α := Fin A) (β := ShortCycle p S J)
+  simp only [Fintype.card_fin] at hc
+  rw [←hc, Fintype.card_congr (shortEmbeddingEquiv p S J),Fintype.card_sigma]
+  apply Finset.sum_congr rfl
+  intro r _
+  exact (Fintype.card_congr (cycleSlotsEquivAssignments p S (fun i => (r i : ℕ)))).symm
+
+end AllLengths
+
+lemma factorial_mgf {Ω : Type*} [Fintype Ω] (F : Ω → ℕ) (k : ℕ) (Λ q : ℝ)
+    (hF : ∀ ω, F ω ≤ k) (hΛ : 0 ≤ Λ) (hq : 1 ≤ q)
+    (hm : ∀ a, finiteMean (fun ω => ((F ω).descFactorial a : ℝ)) ≤ Λ^a) :
+    finiteMean (fun ω => q^(F ω)) ≤ Real.exp ((q-1)*Λ) := by
+  have hc (n a : ℕ) : (n.choose a : ℝ) = (n.descFactorial a : ℝ) / (a.factorial : ℝ) := by
+    rw [Nat.descFactorial_eq_factorial_mul_choose,Nat.cast_mul]
+    exact (mul_div_cancel_left₀ _ (by exact_mod_cast Nat.factorial_ne_zero a)).symm
+  have hex (ω : Ω) : q^(F ω) = ∑ a ∈ Finset.range (k+1),
+      ((F ω).descFactorial a : ℝ) * ((q-1)^a / (a.factorial : ℝ)) := by
+    calc
+      _ = ((q-1)+1)^(F ω) := by congr 1; ring
+      _ = ∑ a ∈ Finset.range (F ω+1), (q-1)^a * (F ω).choose a := by
+        rw [add_pow]
+        simp only [one_pow,mul_one]
+      _ = ∑ a ∈ Finset.range (k+1), (q-1)^a * (F ω).choose a := by
+        apply Finset.sum_subset (Finset.range_mono (Nat.add_le_add_right (hF ω) _))
+        intro a _ ha
+        have hn : F ω < a := by have := (Finset.mem_range.not.mp ha); omega
+        rw [Nat.choose_eq_zero_of_lt hn,Nat.cast_zero,mul_zero]
+      _ = _ := by
+        apply Finset.sum_congr rfl
+        intro a _
+        rw [hc]
+        ring
+  simp_rw [hex,finiteMean_sum,finiteMean_mul_const]
+  calc
+    _ ≤ ∑ a ∈ Finset.range (k+1), Λ^a * ((q-1)^a / (a.factorial : ℝ)) := by
+      apply Finset.sum_le_sum
+      intro a _
+      exact mul_le_mul_of_nonneg_right (hm a) (by positivity)
+    _ = ∑ a ∈ Finset.range (k+1), ((q-1)*Λ)^a / (a.factorial : ℝ) := by
+      apply Finset.sum_congr rfl
+      intro a _
+      rw [mul_pow]
+      ring
+    _ ≤ _ := Real.sum_le_exp_of_nonneg (mul_nonneg (sub_nonneg.mpr hq) hΛ) _
+
+lemma inv_sqrt_step (j : ℕ) :
+    1 / Real.sqrt (j+1) ≤ 2*(Real.sqrt (j+1)-Real.sqrt j) := by
+  have hp : 0 < Real.sqrt (j+1) := Real.sqrt_pos.mpr (by positivity)
+  apply (div_le_iff₀ hp).mpr
+  have h1 := Real.sq_sqrt (show (0:ℝ) ≤ j+1 by positivity)
+  have h0 := Real.sq_sqrt (show (0:ℝ) ≤ j by positivity)
+  nlinarith [sq_nonneg (Real.sqrt (j+1)-Real.sqrt j)]
+
+lemma sum_inv_sqrt (J : ℕ) :
+    (∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1)) ≤ 2*Real.sqrt J := by
+  have h (n : ℕ) : (∑ j ∈ Finset.range n, 1 / Real.sqrt (j+1)) ≤ 2*Real.sqrt n := by
+    induction n with
+    | zero => simp
+    | succ n ih =>
+      rw [Finset.sum_range_succ]
+      have hs := inv_sqrt_step n
+      push_cast
+      linarith
+  rw [Fin.sum_univ_eq_sum_range (fun j : ℕ => (1 / Real.sqrt (j+1) : ℝ)) J]
+  exact h J
+section CycleMGF
+variable {d : ℕ}
+
+lemma butterfly_shortCycle_moment (S : Finset (Card d)) (J A : ℕ) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      ((Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J)).descFactorial A : ℝ)) ≤
+      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))^A := by
+  classical
+  simp_rw [shortCycle_factorial_eq,Nat.cast_sum,finiteMean_sum]
+  calc
+    _ ≤ ∑ r : Fin A → Fin J,
+        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+          ∏ i : Fin A, (1 / Real.sqrt ((r i : ℕ)+1)) := by
+      apply Finset.sum_le_sum
+      intro r _
+      exact butterfly_cycleSlots_moment_all S (fun i => (r i : ℕ))
+    _ = _ := by
+      rw [←Finset.mul_sum,mul_pow]
+      congr 1
+      rw [←Fintype.prod_sum (fun (_ : Fin A) (j : Fin J) => (1 / Real.sqrt ((j:ℕ)+1) : ℝ))]
+      simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fin]
+
+noncomputable def cycleMass (p : Equiv.Perm (Card d)) (S : Finset (Card d)) : ℕ :=
+  ∑ C : SelectedCycle p S, C.val.card
+
+lemma cycleMass_le (p : Equiv.Perm (Card d)) (S : Finset (Card d)) : cycleMass p S ≤ S.card := by
+  classical
+  let f : (Σ C : SelectedCycle p S, {x : Card d // x ∈ C.val}) → S :=
+    fun c => ⟨c.2.val,c.1.property.2 c.2.property⟩
+  have hi : Function.Injective f := by
+    intro c e h
+    have hx : c.2.val = e.2.val := congrArg Subtype.val h
+    have hC : c.1 = e.1 := c.1.eq_of_mem e.1 c.2.property (hx ▸ e.2.property)
+    rcases c with ⟨c,x⟩
+    rcases e with ⟨e,y⟩
+    dsimp only at hC
+    subst e
+    have he : x=y := Subtype.ext hx
+    cases he
+    rfl
+  have hh := Fintype.card_le_of_injective f hi
+  simpa only [Fintype.card_sigma,Fintype.card_coe,cycleMass] using hh
+
+lemma shortCycle_card_le (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
+    Fintype.card (ShortCycle p S J) ≤ S.card := by
+  classical
+  calc
+    _ ≤ Fintype.card (SelectedCycle p S) := Fintype.card_subtype_le _
+    _ = ∑ _ : SelectedCycle p S, 1 := by simp
+    _ ≤ cycleMass p S := Finset.sum_le_sum (fun C _ => C.nonempty.card_pos)
+    _ ≤ _ := cycleMass_le p S
+
+lemma butterfly_shortCycle_mgf (S : Finset (Card d)) (J : ℕ) (q : ℝ) (hq : 1 ≤ q) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      q^(Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J))) ≤
+      Real.exp ((q-1)*((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))) := by
+  apply factorial_mgf _ S.card _ q (fun ω => shortCycle_card_le _ _ _)
+  · positivity
+  · exact hq
+  · exact butterfly_shortCycle_moment S J
+
+end CycleMGF
+
+section FullCycleMGF
+variable {d : ℕ}
+
+abbrev LongCycle (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :=
+  {C : SelectedCycle p S // ¬ C.val.card ≤ J}
+noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
+    Fintype (LongCycle p S J) := Fintype.ofFinite _
+
+lemma selectedCycle_card_split (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
+    Fintype.card (SelectedCycle p S) =
+      Fintype.card (ShortCycle p S J) + Fintype.card (LongCycle p S J) := by
+  classical
+  rw [←Fintype.card_congr (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J)),Fintype.card_sum]
+
+lemma longCycle_card_bound (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
+    J * Fintype.card (LongCycle p S J) ≤ S.card := by
+  classical
+  have hs := Equiv.sum_comp (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J))
+    (fun C : SelectedCycle p S => C.val.card)
+  rw [Fintype.sum_sum_type] at hs
+  have h0 : J * Fintype.card (LongCycle p S J) ≤
+      ∑ C : LongCycle p S J, C.val.val.card := by
+    calc
+      _ = ∑ _C : LongCycle p S J, J := by simp [mul_comm]
+      _ ≤ _ := Finset.sum_le_sum (fun C _ => Nat.le_of_lt (Nat.lt_of_not_ge C.property))
+  have hh := cycleMass_le p S
+  unfold cycleMass at hh
+  rw [←hs] at hh
+  exact h0.trans ((Nat.le_add_left _ _).trans hh)
+
+lemma finiteMean_pos {Ω : Type*} [Fintype Ω] [Nonempty Ω] {f : Ω → ℝ}
+    (h : ∀ ω, 0 < f ω) : 0 < finiteMean f := by
+  apply div_pos
+  · exact Finset.sum_pos (fun ω _ => h ω) Finset.univ_nonempty
+  · exact Nat.cast_pos.mpr Fintype.card_pos
+
+lemma longCycle_pow_bound (p : Equiv.Perm (Card d)) (S : Finset (Card d))
+    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    q^(Fintype.card (LongCycle p S J)) ≤ Real.exp ((S.card : ℝ)/J * Real.log q) := by
+  have hq0 : 0 < q := lt_of_lt_of_le zero_lt_one hq
+  have hc : (Fintype.card (LongCycle p S J) : ℝ) ≤ (S.card : ℝ)/J := by
+    apply (le_div_iff₀ (by exact_mod_cast hJ)).mpr
+    have hh := longCycle_card_bound p S J
+    exact_mod_cast (by simpa only [mul_comm] using hh)
+  calc
+    _ = Real.exp ((Fintype.card (LongCycle p S J) : ℝ)*Real.log q) := by
+      rw [Real.exp_nat_mul,Real.exp_log hq0]
+    _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right hc (Real.log_nonneg hq))
+
+lemma butterfly_cycle_log_mgf (S : Finset (Card d)) (J : ℕ) (hJ : 0 < J)
+    (q : ℝ) (hq : 1 ≤ q) :
+    Real.log (finiteMean (fun ω : SwitchIndex d → Bool =>
+      q^(Fintype.card (SelectedCycle (butterflyPerm d (decodeButterfly d ω)) S)))) ≤
+      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
+  let lam : ℝ := (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1)
+  let c : ℝ := (S.card : ℝ)/J * Real.log q
+  have hnum : (q-1)*lam ≤ 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
+    have hsum := sum_inv_sqrt J
+    have hp : 0 ≤ (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) := by positivity
+    have h0 : 0 ≤ 2*Real.sqrt J := by positivity
+    have hh := mul_le_mul_of_nonneg_left hsum hp
+    have hq1 : 0 ≤ q-1 := sub_nonneg.mpr hq
+    calc
+      _ ≤ (q-1)*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
+        mul_le_mul_of_nonneg_left hh hq1
+      _ ≤ q*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
+        mul_le_mul_of_nonneg_right (by linarith) (mul_nonneg hp h0)
+      _ = _ := by ring
+  have hm : finiteMean (fun ω : SwitchIndex d → Bool =>
+      q^(Fintype.card (SelectedCycle (butterflyPerm d (decodeButterfly d ω)) S))) ≤
+      Real.exp (c+(q-1)*lam) := by
+    calc
+      _ ≤ finiteMean (fun ω : SwitchIndex d → Bool =>
+          q^(Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J))*Real.exp c) := by
+        apply finiteMean_mono
+        intro ω
+        rw [selectedCycle_card_split _ _ J,pow_add]
+        exact mul_le_mul_of_nonneg_left (longCycle_pow_bound _ S J hJ q hq) (by positivity)
+      _ = finiteMean (fun ω : SwitchIndex d → Bool =>
+          q^(Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J))) * Real.exp c :=
+        finiteMean_mul_const _ _
+      _ ≤ Real.exp ((q-1)*lam) * Real.exp c :=
+        mul_le_mul_of_nonneg_right (butterfly_shortCycle_mgf S J q hq) (Real.exp_nonneg _)
+      _ = _ := by rw [←Real.exp_add,add_comm]
+  have hpos : 0 < finiteMean (fun ω : SwitchIndex d → Bool =>
+      q^(Fintype.card (SelectedCycle (butterflyPerm d (decodeButterfly d ω)) S))) := by
+    apply finiteMean_pos
+    intro ω
+    positivity
+  have hlog := Real.log_le_log hpos hm
+  rw [Real.log_exp] at hlog
+  dsimp [c] at hlog
+  linarith
+
+end FullCycleMGF
+
+structure RoutingNetwork (α ι : Type*) [Fintype α] [DecidableEq α]
+    [Fintype ι] [DecidableEq ι] (d : ℕ) where
+  perm : (ι → Bool) → Equiv.Perm α
+  path : (ι → Bool) → α → Finset ι
+  users : (ι → Bool) → ι → α × α
+  users_ne : ∀ ω i, (users ω i).1 ≠ (users ω i).2
+  mem_path : ∀ ω x i, i ∈ path ω x ↔ (users ω i).1 = x ∨ (users ω i).2 = x
+  card_path : ∀ ω x, (path ω x).card = d
+  entropy : ∀ ω (S : Finset α),
+    (∑ i : ι, if (users ω i).1 ∈ S ∧ (users ω i).2 ∈ S then (1:ℝ) else 0) ≤
+      (S.card : ℝ) * Real.log S.card / (2 * Real.log 2)
+  follow : α → (α → QueryTree ι) → QueryTree ι
+  close : α → α → QueryTree ι → QueryTree ι
+  follow_succeeds : ∀ x k ω, (follow x k).succeeds ω = (k (perm ω x)).succeeds ω
+  follow_raw_charge : ∀ x k ω, (follow x k).charge ω = (k (perm ω x)).charge ω
+  follow_stale_charge : ∀ x k R ω,
+    (follow x k).staleCharge R ω = (k (perm ω x)).staleCharge (R ∪ path ω x) ω
+  close_succeeds : ∀ x y k ω,
+    (close x y k).succeeds ω = if perm ω x = y then k.succeeds ω else false
+  close_raw_charge : ∀ x y k ω, perm ω x = y → (close x y k).charge ω = k.charge ω + d
+  close_stale_charge : ∀ x y k R ω, perm ω x = y →
+    (close x y k).staleCharge R ω = k.staleCharge (R ∪ path ω x) ω + (path ω x ∩ R).card
+
+namespace RoutingNetwork
+variable {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι] [DecidableEq ι]
+  {d : ℕ} (R : RoutingNetwork α ι d)
+
+lemma card_switches_using (ω : ι → Bool) (x : α) :
+    (Finset.univ.filter (fun i : ι =>
+      (R.users ω i).1 = x ∨ (R.users ω i).2 = x)).card = d := by
+  classical
+  have he : Finset.univ.filter (fun i : ι =>
+      (R.users ω i).1 = x ∨ (R.users ω i).2 = x) = R.path ω x := by
+    ext i
+    simp only [Finset.mem_filter,Finset.mem_univ,true_and,R.mem_path]
+  rw [he,R.card_path]
+lemma switch_incidence_sum (b : ι → Bool) (w : α → ℝ) :
+    ∑ i : ι, (w (R.users b i).1 + w (R.users b i).2) =
+      (d : ℝ) * ∑ x : α, w x := by
+  classical
+  calc
+    _ = ∑ i : ι, ∑ x : α,
+        if (R.users b i).1 = x ∨ (R.users b i).2 = x then w x else 0 := by
+      apply Finset.sum_congr rfl
+      intro i _
+      rw [Finset.sum_ite]
+      simp only [Finset.sum_const_zero, add_zero]
+      have he : Finset.univ.filter (fun x : α =>
+          (R.users b i).1 = x ∨ (R.users b i).2 = x) =
+          {(R.users b i).1, (R.users b i).2} := by
+        ext x; simp [eq_comm]
+      rw [he, Finset.sum_pair (R.users_ne b i)]
+    _ = ∑ x : α, ∑ i : ι,
+        if (R.users b i).1 = x ∨ (R.users b i).2 = x then w x else 0 :=
+      Finset.sum_comm
+    _ = ∑ x : α, (d : ℝ) * w x := by
+      apply Finset.sum_congr rfl
+      intro x _
+      rw [Finset.sum_ite]
+      simp only [Finset.sum_const_zero, add_zero, Finset.sum_const, nsmul_eq_mul,
+        card_switches_using]
+    _ = _ := by rw [Finset.mul_sum]
+
+lemma weighted_switch_min_bound (b : ι → Bool) (w : α → ℝ) :
+    ∑ i : ι, min (w (R.users b i).1) (w (R.users b i).2) ≤
+      (d : ℝ) / 2 * ∑ x : α, w x := by
+  have h : 2 * (∑ i : ι,
+      min (w (R.users b i).1) (w (R.users b i).2)) ≤
+      ∑ i : ι, (w (R.users b i).1 + w (R.users b i).2) := by
+    rw [Finset.mul_sum]
+    apply Finset.sum_le_sum
+    intro i _
+    linarith [min_le_left (w (R.users b i).1) (w (R.users b i).2),
+      min_le_right (w (R.users b i).1) (w (R.users b i).2)]
+  rw [switch_incidence_sum] at h
+  linarith
+section Rotation
+variable {A : ℕ}
+
+def cycleFiber (o : α → Option (Fin A)) (i : Fin A) : Finset (α) := by
+  classical
+  exact Finset.univ.filter (fun x => o x = some i)
+
+omit [DecidableEq α] in
+@[simp] lemma mem_cycleFiber (o : α → Option (Fin A)) (i : Fin A) (x : α) :
+    x ∈ cycleFiber o i ↔ o x = some i := by
+  classical
+  simp [cycleFiber]
+
+abbrev Terminal (o : α → Option (Fin A)) := ∀ i : Fin A, {x : α // o x = some i}
+
+noncomputable def slotWeight (o : α → Option (Fin A)) (i : Fin A) : ℝ :=
+  1 / (cycleFiber o i).card
+
+noncomputable def pathWeight (o : α → Option (Fin A)) (x : α) : ℝ :=
+  match o x with
+  | none => 0
+  | some i => slotWeight o i
+
+noncomputable def withinWeight (o : α → Option (Fin A)) (u v : α) : ℝ :=
+  match o u, o v with
+  | some i, some j => if i = j then slotWeight o i else 0
+  | _, _ => 0
+
+noncomputable def rotationMeeting (o : α → Option (Fin A)) (z : Terminal o)
+    (u v : α) : ℝ :=
+  match o u, o v with
+  | some i, some j => if i < j then (if (z j).val = v then 1 else 0)
+      else if j < i then (if (z i).val = u then 1 else 0)
+      else (if (z i).val = u then 1 else 0) + (if (z i).val = v then 1 else 0)
+  | _, _ => 0
+
+omit [DecidableEq α] in
+lemma card_cycleFiber (o : α → Option (Fin A)) (i : Fin A) :
+    Fintype.card {x : α // o x = some i} = (cycleFiber o i).card := by
+  classical
+  simp [cycleFiber,Fintype.card_subtype]
+
+lemma terminal_mean {o : α → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
+    (i : Fin A) (x : α) (hx : o x = some i) :
+    finiteMean (fun z : Terminal o => if (z i).val = x then (1:ℝ) else 0) = slotWeight o i := by
+  classical
+  have hev := finiteMean_pi_eval (A := fun i : Fin A => {x : α // o x = some i})
+    i (fun z => if z.val = x then (1:ℝ) else 0)
+  rw [hev]
+  have he : (fun z : {x : α // o x = some i} => if z.val = x then (1:ℝ) else 0) =
+      (fun z => if z = ⟨x,hx⟩ then 1 else 0) := by
+    funext z
+    simp only [Subtype.ext_iff]
+  rw [he,finiteMean_eq_indicator,card_cycleFiber,slotWeight]
+
+omit [DecidableEq α] in
+lemma slotWeight_antitone (o : α → Option (Fin A))
+    (hn : ∀ i, (cycleFiber o i).Nonempty)
+    (hm : Monotone (fun i => (cycleFiber o i).card)) : Antitone (slotWeight o) := by
+  intro i j hij
+  unfold slotWeight
+  apply one_div_le_one_div_of_le
+  · exact_mod_cast Finset.card_pos.mpr (hn i)
+  · exact_mod_cast hm hij
+
+lemma rotationMeeting_mean {o : α → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
+    (hm : Monotone (fun i => (cycleFiber o i).card)) (u v : α) :
+    finiteMean (fun z : Terminal o => rotationMeeting o z u v) =
+      min (pathWeight o u) (pathWeight o v) + withinWeight o u v := by
+  classical
+  have hn : ∀ i, (cycleFiber o i).Nonempty := by
+    intro i
+    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : α // o x = some i})
+    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
+  have hw := slotWeight_antitone o hn hm
+  cases hu : o u with
+  | none =>
+    cases hv : o v with
+    | none => simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean]
+    | some j =>
+      have hj : 0 ≤ slotWeight o j := by unfold slotWeight; positivity
+      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_left hj]
+  | some i =>
+    cases hv : o v with
+    | none =>
+      have hi : 0 ≤ slotWeight o i := by unfold slotWeight; positivity
+      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_right hi]
+    | some j =>
+      simp only [rotationMeeting,pathWeight,withinWeight,hu,hv]
+      rcases lt_trichotomy i j with hij | rfl | hji
+      · simp only [hij,↓reduceIte,ne_of_lt hij]
+        rw [terminal_mean j v hv,min_eq_right (hw (le_of_lt hij)),add_zero]
+      · simp only [lt_self_iff_false,↓reduceIte,min_self]
+        rw [finiteMean_add,terminal_mean i u hu,terminal_mean i v hv]
+      · simp only [not_lt_of_ge (le_of_lt hji),hji,↓reduceIte,ne_of_gt hji]
+        rw [terminal_mean i u hu,min_eq_left (hw (le_of_lt hji)),add_zero]
+
+omit [DecidableEq α] in
+lemma pathWeight_sum (o : α → Option (Fin A)) (hn : ∀ i, (cycleFiber o i).Nonempty) :
+    ∑ x : α, pathWeight o x = (A : ℝ) := by
+  classical
+  have he (x : α) : pathWeight o x =
+      ∑ i : Fin A, if o x = some i then slotWeight o i else 0 := by
+    cases hx : o x with
+    | none => simp [pathWeight,hx]
+    | some j => simp [pathWeight,hx]
+  simp only [he]
+  rw [Finset.sum_comm]
+  have hf (i : Fin A) : (∑ x : α, if o x = some i then slotWeight o i else 0) = 1 := by
+    rw [← Finset.sum_filter]
+    change (∑ _x ∈ cycleFiber o i, slotWeight o i) = 1
+    simp only [Finset.sum_const,nsmul_eq_mul,slotWeight]
+    exact mul_one_div_cancel (by exact_mod_cast (Finset.card_pos.mpr (hn i)).ne')
+  simp only [hf,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,mul_one]
+
+lemma withinWeight_sum_bound (o : α → Option (Fin A))
+    (hn : ∀ i, (cycleFiber o i).Nonempty) (b : ι → Bool) :
+    (∑ s : ι, withinWeight o (R.users b s).1 (R.users b s).2) ≤
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
+  classical
+  have he (u v : α) : withinWeight o u v =
+      ∑ i : Fin A, (if u ∈ cycleFiber o i ∧ v ∈ cycleFiber o i then (1:ℝ) else 0) * slotWeight o i := by
+    simp only [mem_cycleFiber]
+    cases hu : o u <;> cases hv : o v <;> simp [withinWeight,hu,hv,ite_and,eq_comm]
+  simp only [he]
+  rw [Finset.sum_comm]
+  apply Finset.sum_le_sum
+  intro i _
+  rw [← Finset.sum_mul]
+  have hi := R.entropy b (cycleFiber o i)
+  have hpos : (0:ℝ) < (cycleFiber o i).card := by exact_mod_cast Finset.card_pos.mpr (hn i)
+  have hmul := mul_le_mul_of_nonneg_right hi (show 0 ≤ slotWeight o i by unfold slotWeight; positivity)
+  calc
+    _ ≤ ((cycleFiber o i).card : ℝ) * Real.log (cycleFiber o i).card /
+        (2 * Real.log 2) * slotWeight o i := hmul
+    _ = _ := by unfold slotWeight; field_simp
+
+noncomputable def rotationCharge (o : α → Option (Fin A)) (b : ι → Bool)
+    (z : Terminal o) : ℝ :=
+  ∑ s : ι, rotationMeeting o z (R.users b s).1 (R.users b s).2
+
+lemma rotationCharge_mean_bound {o : α → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
+    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : ι → Bool) :
+    finiteMean (rotationCharge R o b) ≤ (A:ℝ) * d / 2 +
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
+  classical
+  have hn : ∀ i, (cycleFiber o i).Nonempty := by
+    intro i
+    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : α // o x = some i})
+    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
+  unfold rotationCharge
+  rw [finiteMean_sum]
+  simp only [rotationMeeting_mean hm,Finset.sum_add_distrib]
+  have h₁ := weighted_switch_min_bound R b (pathWeight o)
+  rw [pathWeight_sum o hn] at h₁
+  have h₂ := withinWeight_sum_bound R o hn b
+  linarith
+
+lemma rotation_exp_bound {o : α → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
+    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : ι → Bool) :
+    Real.exp (-Real.log 2 * ((A:ℝ)*d/2 +
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2))) ≤
+      finiteMean (fun z : Terminal o => Real.exp (-Real.log 2 * rotationCharge R o b z)) := by
+  have hb := rotationCharge_mean_bound R hm b
+  have hlog : 0 ≤ Real.log 2 := Real.log_nonneg (by norm_num)
+  have hm' : -Real.log 2 * ((A:ℝ)*d/2 +
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2)) ≤
+      finiteMean (fun z : Terminal o => -Real.log 2 * rotationCharge R o b z) := by
+    have he := finiteMean_mul_const (rotationCharge R o b) (-Real.log 2)
+    simp only [mul_comm] at he
+    rw [he]
+    exact mul_le_mul_of_nonpos_left hb (neg_nonpos.mpr hlog)
+  exact (Real.exp_le_exp.mpr hm').trans (exp_finiteMean_le _)
+
+end Rotation
+
+namespace QueryTree
+open Thorp.QueryTree
+
+noncomputable def seekCycle (R : RoutingNetwork α ι d) (S : Finset (α)) (a : α) :
+    ℕ → α → Finset (α) → (Finset (α) → QueryTree (ι)) →
+      QueryTree (ι)
+  | 0, x, V, k => if x ∈ S ∧ x ∉ V then R.close x a (k (insert x V)) else reject
+  | r + 1, x, V, k => if x ∈ S ∧ x ∉ V then R.follow x
+      (fun y => seekCycle R S a r y (insert x V) k) else reject
+
+lemma seekCycle_raw_charge (S : Finset (α)) (a : α) (r : ℕ)
+    (x : α) (V : Finset (α)) (k : Finset (α) → QueryTree (ι))
+    (ω : ι → Bool) (K : ℕ)
+    (hK : ∀ V, (k V).succeeds ω = true → (k V).charge ω = K)
+    (hs : (seekCycle R S a r x V k).succeeds ω = true) :
+    (seekCycle R S a r x V k).charge ω = K + d := by
+  induction r generalizing x V with
+  | zero =>
+    by_cases hv : x ∈ S ∧ x ∉ V
+    · rw [seekCycle,ite_eq_left hv,R.close_succeeds] at hs
+      split_ifs at hs with he
+      · rw [seekCycle,ite_eq_left hv,R.close_raw_charge _ _ _ _ he,hK _ hs]
+    · simp [seekCycle,hv,Thorp.QueryTree.succeeds] at hs
+  | succ r ih =>
+    by_cases hv : x ∈ S ∧ x ∉ V
+    · rw [seekCycle,ite_eq_left hv,R.follow_succeeds] at hs
+      rw [seekCycle,ite_eq_left hv,R.follow_raw_charge]
+      exact ih _ _ hs
+    · simp [seekCycle,hv,Thorp.QueryTree.succeeds] at hs
+
+noncomputable def seekCycles (R : RoutingNetwork α ι d) (S : Finset (α)) :
+    List (α × ℕ) → Finset (α) → QueryTree (ι)
+  | [], _ => accept
+  | (a,r) :: L, V => seekCycle R S a r a V (fun V => seekCycles R S L V)
+
+lemma seekCycles_raw_charge (S : Finset (α)) (L : List (α × ℕ))
+    (V : Finset (α)) (ω : ι → Bool)
+    (hs : (seekCycles R S L V).succeeds ω = true) :
+    (seekCycles R S L V).charge ω = L.length * d := by
+  induction L generalizing V with
+  | nil => simp [seekCycles,charge]
+  | cons ar L ih =>
+    obtain ⟨a,r⟩ := ar
+    rw [seekCycles]
+    rw [seekCycle_raw_charge R S a r a V (fun V => seekCycles R S L V) ω (L.length*d)
+      (fun V hs => ih V hs) hs]
+    simp [Nat.add_mul]
+
+lemma cycles_weighted_success (S : Finset (α)) (L : List (α × ℕ)) :
+    finiteMean (fun ω : ι → Bool =>
+      if (seekCycles R S L ∅).succeeds ω then
+        ((2:ℝ)^d)^L.length / (2:ℝ)^(seekCycles R S L ∅).staleCharge ∅ ω else 0) ≤ 1 := by
+  let T := seekCycles R S L ∅
+  have hh := mean_freshCharge_le_one T
+  have he : finiteMean (fun ω : ι → Bool =>
+      if T.succeeds ω then (2:ℝ)^T.freshCharge ∅ ω else 0) =
+      finiteMean (fun ω : ι → Bool =>
+        if T.succeeds ω then ((2:ℝ)^d)^L.length / (2:ℝ)^T.staleCharge ∅ ω else 0) := by
+    apply finiteMean_congr
+    intro ω
+    cases hs : T.succeeds ω
+    · simp
+    · simp only [↓reduceIte]
+      have hc := fresh_add_stale T ∅ ω
+      rw [seekCycles_raw_charge R S L ∅ ω hs] at hc
+      have hp : ((2:ℝ)^d)^L.length = (2:ℝ)^T.freshCharge ∅ ω * (2:ℝ)^T.staleCharge ∅ ω := by
+        rw [←pow_add,hc,←pow_mul,Nat.mul_comm]
+      rw [hp,mul_div_cancel_right₀ _ (pow_ne_zero _ (by norm_num))]
+  rw [← he]
+  exact hh
+
+end QueryTree
+noncomputable def exposedPaths (b : ι → Bool) (V : Finset (α)) :
+    Finset (ι) :=
+  V.biUnion (fun x => R.path b x)
+
+@[simp] lemma exposedPaths_empty (b : ι → Bool) : exposedPaths R b ∅ = ∅ := by
+  classical
+  simp [exposedPaths]
+
+lemma exposedPaths_insert (b : ι → Bool) (V : Finset (α)) (x : α) :
+    exposedPaths R b (insert x V) = exposedPaths R b V ∪ R.path b x := by
+  classical
+  simp [exposedPaths,Finset.union_comm]
+
+noncomputable def terminalCharge (b : ι → Bool) (z : α) (V : Finset (α)) : ℕ :=
+  (R.path b z ∩ exposedPaths R b V).card
+
+namespace QueryTree
+open Thorp.QueryTree
+
+lemma seekCycle_semantics (S : Finset (α)) (a : α) (r : ℕ)
+    (x : α) (V W : Finset (α)) (z : α)
+    (k : Finset (α) → QueryTree (ι)) (ω : ι → Bool)
+    (h : cycleTrace (R.perm ω) S a r x V = some (W,z)) :
+    (seekCycle R S a r x V k).succeeds ω = (k W).succeeds ω ∧
+      (seekCycle R S a r x V k).staleCharge (exposedPaths R (ω) V) ω =
+        (k W).staleCharge (exposedPaths R (ω) W) ω +
+          terminalCharge R (ω) z (W.erase z) := by
+  classical
+  induction r generalizing x V with
+  | zero =>
+    simp only [cycleTrace] at h
+    split_ifs at h with hv
+    obtain ⟨hxS,hxV,hxa⟩ := hv
+    obtain ⟨rfl,rfl⟩ := Prod.mk.inj (Option.some.inj h)
+    simp only [seekCycle,ite_eq_left (show x ∈ S ∧ x ∉ V from ⟨hxS,hxV⟩),R.close_succeeds,ite_eq_left hxa]
+    constructor
+    · trivial
+    · rw [R.close_stale_charge _ _ _ _ _ hxa,exposedPaths_insert,
+        Finset.erase_insert hxV]
+      simp only [terminalCharge]
+  | succ r ih =>
+    simp only [cycleTrace] at h
+    split_ifs at h with hv
+    obtain ⟨hxS,hxV⟩ := hv
+    rw [seekCycle,ite_eq_left ⟨hxS,hxV⟩,R.follow_succeeds,R.follow_stale_charge]
+    rw [← exposedPaths_insert]
+    exact ih _ _ h
+
+lemma seekCycle_orbit (S V : Finset (α)) (z : α)
+    (k : Finset (α) → QueryTree (ι)) (ω : ι → Bool)
+    (hS : orbitSet (R.perm ω) z ⊆ S)
+    (hV : Disjoint V (orbitSet (R.perm ω) z)) :
+    let p := R.perm ω
+    let W := V ∪ orbitSet p z
+    (seekCycle R S (p z) (Function.minimalPeriod p z-1) (p z) V k).succeeds ω = (k W).succeeds ω ∧
+      (seekCycle R S (p z) (Function.minimalPeriod p z-1) (p z) V k).staleCharge
+          (exposedPaths R (ω) V) ω =
+        (k W).staleCharge (exposedPaths R (ω) W) ω +
+          terminalCharge R (ω) z (W.erase z) := by
+  exact seekCycle_semantics _ _ _ _ _ _ _ _ _ _ (cycleTrace_orbit _ _ _ _ hS hV)
+
+end QueryTree
+
+end RoutingNetwork
+end Thorp
+
+end ThorpNine.Smoothing
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Smoothing/BenesCoins.lean b/lean/OAI/Probability/ThorpRouting/Smoothing/BenesCoins.lean
index e9c32deb1fdd76f667bac296719bfa20b3be4a7a..27dcf9de9975f75d0833c6491dadb82d223336bd 100644
--- a/lean/OAI/Probability/ThorpRouting/Smoothing/BenesCoins.lean
+++ b/lean/OAI/Probability/ThorpRouting/Smoothing/BenesCoins.lean
@@ -1,882 +1,882 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Smoothing.AlternatingPerm
-
-namespace OAI
-
-namespace ThorpNine.Smoothing
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-
-abbrev BenesCoins (d : ℕ) := (SwitchIndex d → Bool) × (SwitchIndex d → Bool)
-
-def palindromePerm (d : ℕ) (ω : BenesCoins d) : Equiv.Perm (Card d) :=
-  butterflyPerm d (decodeButterfly d ω.1) * (butterflyPerm d (decodeButterfly d ω.2)).symm
-
-def sandwichShuffle {A B C D E F : Type*} :
-    ((A × B) × C) × ((D × E) × F) ≃ ((A × D) × (B × E)) × (F × C) where
-  toFun x := (((x.1.1.1,x.2.1.1),(x.1.1.2,x.2.1.2)),(x.2.2,x.1.2))
-  invFun x := (((x.1.1.1,x.1.2.1),x.2.2),((x.1.1.2,x.1.2.2),x.2.1))
-  left_inv _ := rfl
-  right_inv _ := rfl
-
-def benesStepEquiv (d : ℕ) : BenesCoins (d+1) ≃
-    (BenesCoins d × BenesCoins d) × ((Card d → Bool) × (Card d → Bool)) :=
-  (Equiv.prodCongr (coinStepEquiv d) (coinStepEquiv d)).trans sandwichShuffle
-
-lemma childLift_mul (d : ℕ) (p q : Bool → Equiv.Perm (Card d)) :
-    childLift p * childLift q = childLift (fun b => p b * q b) := by
-  ext x i
-  simp only [Equiv.Perm.mul_apply,childLift,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons]
-
-lemma childLift_inv (d : ℕ) (p : Bool → Equiv.Perm (Card d)) :
-    (childLift p)⁻¹ = childLift (fun b => (p b)⁻¹) := rfl
-
-lemma pairSwitch_inv (d : ℕ) (ξ : Card d → Bool) : (pairSwitch ξ)⁻¹ = pairSwitch ξ := rfl
-
-lemma palindromePerm_step (d : ℕ) (c : BenesCoins d × BenesCoins d)
-    (coins : (Card d → Bool) × (Card d → Bool)) :
-    palindromePerm (d+1) ((benesStepEquiv d).symm (c,coins)) =
-      pairSwitch coins.2 * childLift (fun b => palindromePerm d (if b then c.2 else c.1)) *
-        pairSwitch coins.1 := by
-  change (butterflyPerm (d+1) (decodeButterfly (d+1)
-      ((coinStepEquiv d).symm ((c.1.1,c.2.1),coins.2)))) *
-    (butterflyPerm (d+1) (decodeButterfly (d+1)
-      ((coinStepEquiv d).symm ((c.1.2,c.2.2),coins.1))))⁻¹ = _
-  rw [butterflyPerm_step,butterflyPerm_step,mul_inv_rev,childLift_inv,pairSwitch_inv]
-  rw [mul_assoc (pairSwitch coins.2),←mul_assoc (childLift _),childLift_mul,←mul_assoc]
-  congr 2
-  apply congrArg childLift
-  funext b
-  cases b <;> rfl
-
-lemma headTail_switch (d : ℕ) (ξ : Card d → Bool) (x : Card (d+1)) :
-    headTailEquiv d (pairSwitch ξ x) = pairLayer ξ (headTailEquiv d x) := rfl
-
-lemma headTail_child (d : ℕ) (p : Bool → Equiv.Perm (Card d)) (x : Card (d+1)) :
-    headTailEquiv d (childLift p x) = PairRouting.lift p (headTailEquiv d x) := rfl
-
-lemma palindromePerm_coordinates (d : ℕ) (c : BenesCoins d × BenesCoins d)
-    (coins : (Card d → Bool) × (Card d → Bool)) (x : Card (d+1)) :
-    headTailEquiv d (palindromePerm (d+1) ((benesStepEquiv d).symm (c,coins)) x) =
-      PairRouting.sandwich coins.1 coins.2 (fun b => palindromePerm d (if b then c.2 else c.1))
-        (headTailEquiv d x) := by
-  rw [palindromePerm_step]
-  simp only [Equiv.Perm.mul_apply,headTail_switch,headTail_child]
-  rfl
-
-noncomputable def palindromeCost : (d : ℕ) → {ι : Type*} → [Fintype ι] →
-    (ι ↪ Card d) → BenesCoins d → ℕ
-  | 0, _, _, _, _ => 0
-  | d+1, _, _, e, ω =>
-      let σ := benesStepEquiv d ω
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x σ.2.1
-      let hc := PairRouting.colors_compatible x σ.2.1
-      PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
-          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) +
-        palindromeCost d (PairRouting.childEmbedding x c hc false) σ.1.1 +
-        palindromeCost d (PairRouting.childEmbedding x c hc true) σ.1.2
-
-namespace CycleColoring
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-lemma alternating_cycle_two (p : Equiv.Perm α) (S : Finset (Bool × α))
-    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) S) : 2 ≤ C.val.card := by
-  obtain ⟨x,hx⟩ := C.nonempty
-  have hpx : alternatingPerm p x ∈ C.val := by
-    rw [←C.orbit_eq hx,mem_orbitSet]
-    exact (Equiv.Perm.SameCycle.refl _ _).apply_right
-  have hne : x ≠ alternatingPerm p x := by
-    intro he
-    have hh := congrArg Prod.fst he
-    rw [alternatingPerm_flip] at hh
-    cases hb : x.1 <;> simp only [hb, Bool.not_false, Bool.not_true] at hh <;> cases hh
-  have hs : {x,alternatingPerm p x} ⊆ C.val := by
-    intro y hy
-    simp only [Finset.mem_insert,Finset.mem_singleton] at hy
-    rcases hy with rfl | rfl <;> assumption
-  have hh := Finset.card_le_card hs
-  simpa only [Finset.card_pair hne] using hh
-
-lemma alternating_cycle_count_double (p : Equiv.Perm α) (S : Finset (Bool × α)) :
-    2 * Fintype.card (RoutingNetwork.SelectedCycle (alternatingPerm p) S) ≤ S.card := by
-  classical
-  calc
-    _ = ∑ _C : RoutingNetwork.SelectedCycle (alternatingPerm p) S, 2 := by simp [Nat.mul_comm]
-    _ ≤ RoutingNetwork.cycleMass (alternatingPerm p) S :=
-      Finset.sum_le_sum (fun C _ => alternating_cycle_two p S C)
-    _ ≤ _ := RoutingNetwork.cycleMass_le _ _
-
-lemma alternating_cycle_count_zero (p : Equiv.Perm α) (S : Finset (Bool × α))
-    (hS : S.card ≤ 1) : Fintype.card (RoutingNetwork.SelectedCycle (alternatingPerm p) S) = 0 := by
-  have h := alternating_cycle_count_double p S
-  omega
-
-end CycleColoring
-
-namespace PairRouting
-
-section
-variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
-
-lemma alternatingCycles_double (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    2 * alternatingCycles e p ≤ Fintype.card ι := by
-  have hh := CycleColoring.alternating_cycle_count_double (alternatingProjection p) (labelSet e)
-  simpa only [alternatingCycles, card_labelSet] using hh
-
-lemma alternatingCycles_zero (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α)
-    (hι : Fintype.card ι ≤ 1) : alternatingCycles e p = 0 := by
-  have hh := alternatingCycles_double e p
-  omega
-
-end
-variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α] [DecidableEq ι]
-
-omit [Fintype α] [DecidableEq ι] in
-lemma colors_chosenCoin (x : ι → Bool × α) (c : ι → Bool) (hc : Compatible x c) :
-    colors x (chosenCoin x c) = c := by
-  funext i
-  simp only [colors,chosenCoin_apply x c hc i,required]
-  cases (x i).1 <;> cases c i <;> rfl
-
-noncomputable def coloredEmbedding (x : ι ↪ Bool × α) (c : ι → Bool) (_hc : Compatible x c) :
-    ι ↪ Bool × α := switchedEmbedding x (chosenCoin x c)
-
-omit [Fintype α] [DecidableEq ι] in
-lemma coloredEmbedding_apply (x : ι ↪ Bool × α) (c : ι → Bool) (hc : Compatible x c) (i : ι) :
-    coloredEmbedding x c hc i = (c i,(x i).2) := by
-  apply Prod.ext
-  · exact congrFun (colors_chosenCoin x c hc) i
-  · rfl
-
-omit [Fintype α] [DecidableEq ι] in
-lemma switchedEmbedding_eq_colored (x : ι ↪ Bool × α) (c : ι → Bool) (hc : Compatible x c)
-    (ξ : α → Bool) (he : colors x ξ = c) :
-    switchedEmbedding x ξ = coloredEmbedding x c hc := by
-  ext i : 1
-  rw [coloredEmbedding_apply]
-  apply Prod.ext
-  · exact congrFun he i
-  · rfl
-
-lemma assignments_match_bound (x y : ι ↪ Bool × α) (c : ι → Bool)
-    (hc : c ∈ compatibleSet x y) (p : Bool → Equiv.Perm α) (hm : ChildMatches p x y c) :
-    (compatibleSet x y).card ≤
-      2 ^ ((occupied x).card + (occupied y).card - Fintype.card ι +
-        alternatingCycles (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p) := by
-  obtain ⟨hcx,hcy⟩ := (mem_compatibleSet x y c).mp hc
-  let ξ := chosenCoin x c
-  let η := chosenCoin y c
-  have he : routedEmbedding x ξ η p = y := by
-    ext i : 1
-    have h := (sandwich_matches_iff ξ η p x y).mpr
-      ⟨(colors_chosenCoin x c hcx).trans (colors_chosenCoin y c hcy).symm,by
-        simpa only [ξ,colors_chosenCoin x c hcx] using hm⟩
-    exact h i
-  have hh := assignments_actual_bound x ξ η p
-  rw [he] at hh
-  rw [card_compatibleSet]
-  exact hh
-
-omit [Fintype α] [DecidableEq ι] in
-lemma occupied_sum_ge (x y : ι ↪ Bool × α) :
-    Fintype.card ι ≤ (occupied x).card + (occupied y).card := by
-  have h₀ := occupied_card_double x
-  have h₁ := occupied_card_double y
-  omega
-
-lemma assignment_weight_bound (x y : ι ↪ Bool × α) (c : ι → Bool)
-    (hc : c ∈ compatibleSet x y) (p : Bool → Equiv.Perm α) (hm : ChildMatches p x y c) :
-    ((compatibleSet x y).card:ℝ) * outerWeight x y ≤
-      (2:ℝ)^((alternatingCycles
-        (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p : ℝ) - Fintype.card ι) := by
-  let z := alternatingCycles (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p
-  let b := (occupied x).card+(occupied y).card
-  let h := Fintype.card ι
-  have hb : h ≤ b := occupied_sum_ge x y
-  have hn : ((compatibleSet x y).card:ℝ) ≤ (2:ℝ)^(b-h+z) := by
-    exact_mod_cast assignments_match_bound x y c hc p hm
-  change ((compatibleSet x y).card:ℝ) * (1/(2:ℝ)^b) ≤ (2:ℝ)^((z:ℝ)-h)
-  rw [mul_one_div]
-  calc
-    _ ≤ (2:ℝ)^(b-h+z)/(2:ℝ)^b := div_le_div_of_nonneg_right hn (by positivity)
-    _ = _ := by
-      rw [←Real.rpow_natCast,←Real.rpow_natCast,←Real.rpow_sub (by norm_num : (0:ℝ)<2)]
-      congr 1
-      rw [Nat.cast_add,Nat.cast_sub hb]
-      ring
-
-lemma assignment_factor_bound (x y : ι ↪ Bool × α) (c : ι → Bool)
-    (hc : c ∈ compatibleSet x y) (p : Bool → Equiv.Perm α) (hm : ChildMatches p x y c)
-    (a : ℝ) (ha : 0 ≤ a) :
-    (outerWeight x y)^a * ((compatibleSet x y).card:ℝ)^a ≤
-      (2:ℝ)^(((alternatingCycles
-        (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p : ℝ) - Fintype.card ι)*a) := by
-  have h := Real.rpow_le_rpow (mul_nonneg (Nat.cast_nonneg _) (le_of_lt (outerWeight_pos x y)))
-    (assignment_weight_bound x y c hc p hm) ha
-  rw [Real.mul_rpow (Nat.cast_nonneg _) (le_of_lt (outerWeight_pos x y)),
-    ←Real.rpow_mul (by norm_num : (0:ℝ)≤2)] at h
-  simpa only [mul_comm] using h
-
-
-lemma sandwich_weighted_probability_subtype (p : Bool → Equiv.Perm α) (x y : ι → Bool × α)
-    (W : (ι → Bool) → ℝ) :
-    finiteMean (fun coins : (α → Bool) × (α → Bool) =>
-      if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then W (colors x coins.1) else 0) =
-    outerWeight x y * ∑ c : {c // c ∈ compatibleSet x y},
-      if ChildMatches p x y c.val then W c.val else 0 := by
-  rw [sandwich_weighted_probability]
-  congr 1
-  rw [Finset.sum_subtype (compatibleSet x y) (fun _ => Iff.rfl)]
-
-omit [Fintype α] [DecidableEq α] [DecidableEq ι] in
-lemma color_card_add (c : ι → Bool) :
-    Fintype.card {i // c i = false} + Fintype.card {i // c i = true} = Fintype.card ι := by
-  have he : {i // c i = true} ≃ {i // ¬c i = false} := Equiv.subtypeEquivRight fun i => by
-    cases c i <;> simp
-  rw [Fintype.card_congr he]
-  rw [Fintype.card_subtype_compl]
-  exact Nat.add_sub_of_le (Fintype.card_subtype_le _)
-
-end PairRouting
-
-noncomputable def assignmentCost (d : ℕ) {ι : Type*} [Fintype ι]
-    (x : ι ↪ Bool × Card d) (c : ι → Bool) (hc : PairRouting.Compatible x c)
-    (ω : BenesCoins d × BenesCoins d) : ℕ :=
-  PairRouting.alternatingCycles (PairRouting.coloredEmbedding x c hc)
-      (fun b => palindromePerm d (if b then ω.2 else ω.1)) +
-    palindromeCost d (PairRouting.childEmbedding x c hc false) ω.1 +
-    palindromeCost d (PairRouting.childEmbedding x c hc true) ω.2
-
-lemma palindromeCost_eq (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (d+1)) (σ : BenesCoins (d+1)) :
-    palindromeCost (d+1) e σ =
-      assignmentCost d (e.trans (headTailEquiv d).toEmbedding)
-        (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) (benesStepEquiv d σ).2.1)
-        (PairRouting.colors_compatible _ _) (benesStepEquiv d σ).1 := by
-  unfold palindromeCost assignmentCost
-  dsimp only
-  rw [PairRouting.switchedEmbedding_eq_colored _ _ (PairRouting.colors_compatible _ _) _ rfl]
-
-lemma palindromeCost_step (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (d+1)) (ω : BenesCoins d × BenesCoins d)
-    (coins : (Card d → Bool) × (Card d → Bool)) :
-    palindromeCost (d+1) e ((benesStepEquiv d).symm (ω,coins)) =
-      assignmentCost d (e.trans (headTailEquiv d).toEmbedding)
-        (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) coins.1)
-        (PairRouting.colors_compatible _ _) ω := by
-  rw [palindromeCost_eq]
-  simp only [Equiv.apply_symm_apply]
-
-noncomputable def routingMoment (d : ℕ) {ι : Type*} [Fintype ι]
-    (e f : ι ↪ Card d) (a : ℝ) : ℝ := by
-  classical
-  exact finiteMean (fun ω : BenesCoins d =>
-    if ∀ i, palindromePerm d ω (e i) = f i then (2:ℝ)^((palindromeCost d e ω:ℝ)*a) else 0)
-
-lemma routingMoment_nonneg (d : ℕ) {ι : Type*} [Fintype ι]
-    (e f : ι ↪ Card d) (a : ℝ) : 0 ≤ routingMoment d e f a := by
-  classical
-  exact finiteMean_nonneg (fun _ => by split_ifs <;> positivity)
-
-lemma palindrome_matches_step (d : ℕ) {ι : Type*}
-    (e f : ι ↪ Card (d+1)) (ω : BenesCoins d × BenesCoins d)
-    (coins : (Card d → Bool) × (Card d → Bool)) :
-    (∀ i, palindromePerm (d+1) ((benesStepEquiv d).symm (ω,coins)) (e i) = f i) ↔
-    (∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => palindromePerm d (if b then ω.2 else ω.1))
-      ((e.trans (headTailEquiv d).toEmbedding) i) = ((f.trans (headTailEquiv d).toEmbedding) i)) := by
-  apply forall_congr'
-  intro i
-  rw [←(headTailEquiv d).injective.eq_iff,palindromePerm_coordinates]
-  rfl
-
-lemma routingMoment_step (d : ℕ) {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (e f : ι ↪ Card (d+1)) (a : ℝ) :
-    let x := e.trans (headTailEquiv d).toEmbedding
-    let y := f.trans (headTailEquiv d).toEmbedding
-    routingMoment (d+1) e f a = PairRouting.outerWeight x y *
-      ∑ c : {c // c ∈ PairRouting.compatibleSet x y},
-        finiteMean (fun ω : BenesCoins d × BenesCoins d =>
-          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c.val
-          then (2:ℝ)^((assignmentCost d x c.val
-            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0) := by
-  classical
-  intro x y
-  rw [routingMoment,finiteMean_equiv (benesStepEquiv d),finiteMean_prod]
-  have he (ω : BenesCoins d × BenesCoins d) :
-      finiteMean (fun coins : (Card d → Bool) × (Card d → Bool) =>
-        if ∀ i, palindromePerm (d+1) ((benesStepEquiv d).symm (ω,coins)) (e i) = f i
-        then (2:ℝ)^((palindromeCost (d+1) e ((benesStepEquiv d).symm (ω,coins)):ℝ)*a) else 0) =
-      PairRouting.outerWeight x y *
-        ∑ c : {c // c ∈ PairRouting.compatibleSet x y},
-          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c.val
-          then (2:ℝ)^((assignmentCost d x c.val
-            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0 := by
-    let W : (ι → Bool) → ℝ := fun c =>
-      if hc : PairRouting.Compatible x c then (2:ℝ)^((assignmentCost d x c hc ω:ℝ)*a) else 0
-    calc
-      _ = finiteMean (fun coins : (Card d → Bool) × (Card d → Bool) =>
-        if ∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => palindromePerm d (if b then ω.2 else ω.1))
-          (x i) = y i then W (PairRouting.colors x coins.1) else 0) := by
-        apply finiteMean_congr
-        intro coins
-        simp only [palindrome_matches_step,palindromeCost_step]
-        simp only [W,PairRouting.colors_compatible,↓reduceDIte]
-        rfl
-      _ = _ := by
-        rw [PairRouting.sandwich_weighted_probability_subtype]
-        congr 1
-        apply Finset.sum_congr rfl
-        intro c _
-        simp only [W,((PairRouting.mem_compatibleSet x y c.val).mp c.property).1,↓reduceDIte]
-  simp_rw [he]
-  simp_rw [mul_comm (PairRouting.outerWeight x y)]
-  rw [finiteMean_mul_const,finiteMean_sum,mul_comm]
-
-lemma child_moment_product (d : ℕ) {ι : Type*} [Fintype ι]
-    (x y : ι ↪ Bool × Card d) (c : ι → Bool)
-    (hcx : PairRouting.Compatible x c) (hcy : PairRouting.Compatible y c) (a : ℝ) :
-    routingMoment d (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a *
-      routingMoment d (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a =
-    finiteMean (fun ω : BenesCoins d × BenesCoins d =>
-      if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
-      then (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
-        palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0) := by
-  classical
-  unfold routingMoment
-  rw [←finiteMean_prod_mul]
-  apply finiteMean_congr
-  intro ω
-  let M₀ := ∀ i : {i // c i = false}, palindromePerm d ω.1 (x i.val).2 = (y i.val).2
-  let M₁ := ∀ i : {i // c i = true}, palindromePerm d ω.2 (x i.val).2 = (y i.val).2
-  have hp : (fun b : Bool => palindromePerm d (if b then ω.2 else ω.1)) =
-      (fun b => if b then palindromePerm d ω.2 else palindromePerm d ω.1) := by
-    funext b
-    cases b <;> rfl
-  have hs : PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c ↔
-      M₀ ∧ M₁ := by rw [hp]; exact PairRouting.childMatches_split _ _ _ _ _
-  change (if M₀ then _ else 0) * (if M₁ then _ else 0) = _
-  by_cases h₀ : M₀ <;> by_cases h₁ : M₁
-  · simp only [hs,h₀,h₁,true_and,↓reduceIte]
-    rw [←Real.rpow_add (by norm_num : (0:ℝ)<2)]
-    congr 1
-    ring
-  · simp only [hs,h₀,h₁,and_false,↓reduceIte,mul_zero]
-  · simp only [hs,h₀,h₁,false_and,↓reduceIte,zero_mul]
-  · simp only [hs,h₀,h₁,false_and,↓reduceIte,mul_zero]
-
-lemma palindrome_probability_step (d : ℕ) {ι : Type*} [Fintype ι]
-    (e f : ι ↪ Card (d+1)) :
-    PairRouting.tupleProbability (palindromePerm (d+1)) e f =
-    finiteMean (fun ω : BenesCoins d × BenesCoins d =>
-      finiteMean (fun coins : (Card d → Bool) × (Card d → Bool) =>
-        if ∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => if b then palindromePerm d ω.2 else palindromePerm d ω.1)
-          ((e.trans (headTailEquiv d).toEmbedding) i) = ((f.trans (headTailEquiv d).toEmbedding) i)
-        then (1:ℝ) else 0)) := by
-  classical
-  rw [PairRouting.tupleProbability,finiteMean_equiv (benesStepEquiv d),finiteMean_prod]
-  have hp (b : Bool) (p q : BenesCoins d) :
-      palindromePerm d (if b then p else q) = (if b then palindromePerm d p else palindromePerm d q) := by
-    cases b <;> rfl
-  simp only [palindrome_matches_step,hp]
-
-lemma child_power_product_bound (d : ℕ) {ι : Type*} [Fintype ι]
-    (x y : ι ↪ Bool × Card d) (c : ι → Bool)
-    (hcx : PairRouting.Compatible x c) (hcy : PairRouting.Compatible y c) (a : ℝ)
-    (hfalse : (PairRouting.tupleProbability (palindromePerm d)
-        (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false))^(1+a) ≤
-      (2:ℝ)^(-(d:ℝ)*Fintype.card {i // c i = false}*a) *
-        routingMoment d (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a)
-    (htrue : (PairRouting.tupleProbability (palindromePerm d)
-        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true))^(1+a) ≤
-      (2:ℝ)^(-(d:ℝ)*Fintype.card {i // c i = true}*a) *
-        routingMoment d (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a) :
-    (PairRouting.tupleProbability (palindromePerm d)
-      (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false))^(1+a) *
-      (PairRouting.tupleProbability (palindromePerm d)
-        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true))^(1+a) ≤
-    (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) *
-      finiteMean (fun ω : BenesCoins d × BenesCoins d =>
-        if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
-        then (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
-          palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0) := by
-  have hh := mul_le_mul hfalse htrue
-    (Real.rpow_nonneg (PairRouting.tupleProbability_nonneg _ _ _) _)
-    (mul_nonneg (by positivity) (routingMoment_nonneg _ _ _ _))
-  rw [mul_mul_mul_comm,child_moment_product,←Real.rpow_add (by norm_num : (0:ℝ)<2)] at hh
-  have he : -(d:ℝ)*Fintype.card {i // c i = false}*a + -(d:ℝ)*Fintype.card {i // c i = true}*a =
-      -(d:ℝ)*Fintype.card ι*a := by
-    rw [←add_mul,←mul_add,←Nat.cast_add,PairRouting.color_card_add]
-  rwa [he] at hh
-
-lemma assignment_moment_bound (d : ℕ) {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (x y : ι ↪ Bool × Card d) (c : ι → Bool) (hc : c ∈ PairRouting.compatibleSet x y)
-    (a : ℝ) (ha : 0 ≤ a) :
-    let hcx := ((PairRouting.mem_compatibleSet x y c).mp hc).1
-    PairRouting.outerWeight x y ^ a * ((PairRouting.compatibleSet x y).card:ℝ)^a *
-      ((2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) *
-        finiteMean (fun ω : BenesCoins d × BenesCoins d =>
-          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
-          then (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
-            palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0)) ≤
-    (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) *
-      finiteMean (fun ω : BenesCoins d × BenesCoins d =>
-        if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
-        then (2:ℝ)^((assignmentCost d x c hcx ω:ℝ)*a) else 0) := by
-  classical
-  intro hcx
-  have hscale (q : ℝ) (F : BenesCoins d × BenesCoins d → ℝ) :
-      q * finiteMean F = finiteMean (fun ω => q * F ω) := by
-    simp_rw [mul_comm q,finiteMean_mul_const]
-  rw [←mul_assoc,hscale,hscale]
-  apply finiteMean_mono
-  intro ω
-  split_ifs with hm
-  · have hh := PairRouting.assignment_factor_bound x y c hc
-      (fun b => palindromePerm d (if b then ω.2 else ω.1)) hm a ha
-    calc
-      _ ≤ ((2:ℝ)^(((PairRouting.alternatingCycles (PairRouting.coloredEmbedding x c hcx)
-          (fun b => palindromePerm d (if b then ω.2 else ω.1)):ℝ)-Fintype.card ι)*a) *
-          (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a)) *
-          (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
-            palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) :=
-        mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hh (by positivity)) (by positivity)
-      _ = _ := by
-        simp only [←Real.rpow_add (by norm_num : (0:ℝ)<2),assignmentCost,Nat.cast_add]
-        congr 1
-        ring
-  · simp only [mul_zero]
-    exact le_rfl
-
-theorem palindrome_pointwise_moment (d : ℕ) {ι : Type*} [Fintype ι]
-    (e f : ι ↪ Card d) (a : ℝ) (ha : 0 ≤ a) :
-    (PairRouting.tupleProbability (palindromePerm d) e f)^(1+a) ≤
-      (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) * routingMoment d e f a := by
-  classical
-  induction d generalizing ι with
-  | zero =>
-      have he : ∀ (ω : BenesCoins 0) (i : ι), palindromePerm 0 ω (e i) = f i :=
-        fun _ _ => Subsingleton.elim _ _
-      simp [PairRouting.tupleProbability,routingMoment,he,palindromeCost,finiteMean_const]
-  | succ d ih =>
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let y := f.trans (headTailEquiv d).toEmbedding
-      let W := PairRouting.outerWeight x y
-      let C := PairRouting.compatibleSet x y
-      let p : (ι → Bool) → ℝ := fun c =>
-        (PairRouting.tupleProbability (palindromePerm d)
-          (fun i : {i // c i = false} => (x i.val).2)
-          (fun i : {i // c i = false} => (y i.val).2))^(1+a) *
-        (PairRouting.tupleProbability (palindromePerm d)
-          (fun i : {i // c i = true} => (x i.val).2)
-          (fun i : {i // c i = true} => (y i.val).2))^(1+a)
-      let m : {c // c ∈ C} → ℝ := fun c =>
-        finiteMean (fun ω : BenesCoins d × BenesCoins d =>
-          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c.val
-          then (2:ℝ)^((assignmentCost d x c.val
-            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0)
-      have hp (c : {c // c ∈ C}) : W^a * (C.card:ℝ)^a * p c.val ≤
-          (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * m c := by
-        obtain ⟨hcx,hcy⟩ := (PairRouting.mem_compatibleSet x y c.val).mp c.property
-        have h₀ := ih (PairRouting.childEmbedding x c.val hcx false)
-          (PairRouting.childEmbedding y c.val hcy false)
-        have h₁ := ih (PairRouting.childEmbedding x c.val hcx true)
-          (PairRouting.childEmbedding y c.val hcy true)
-        have hh := child_power_product_bound d x y c.val hcx hcy a h₀ h₁
-        have hh' := mul_le_mul_of_nonneg_left hh
-          (mul_nonneg (Real.rpow_nonneg (le_of_lt (PairRouting.outerWeight_pos x y)) a)
-            (Real.rpow_nonneg (Nat.cast_nonneg C.card) a))
-        exact hh'.trans (assignment_moment_bound d x y c.val c.property a ha)
-      have hnode := PairRouting.sandwich_power_bound (palindromePerm d) (palindromePerm d) x y a ha
-      rw [←palindrome_probability_step d e f] at hnode
-      change (PairRouting.tupleProbability (palindromePerm (d+1)) e f)^(1+a) ≤
-        W^(1+a) * (C.card:ℝ)^a * ∑ c ∈ C, p c at hnode
-      have hsum : ∑ c ∈ C, p c = ∑ c : {c // c ∈ C}, p c.val := by
-        rw [Finset.sum_subtype C (fun _ => Iff.rfl)]
-      have hW : 0 < W := PairRouting.outerWeight_pos x y
-      calc
-        _ ≤ W^(1+a) * (C.card:ℝ)^a * ∑ c ∈ C, p c := hnode
-        _ = W * ∑ c : {c // c ∈ C}, W^a * (C.card:ℝ)^a * p c.val := by
-          rw [Real.rpow_add hW,Real.rpow_one,hsum,←Finset.mul_sum]
-          ring
-        _ ≤ W * ∑ c : {c // c ∈ C}, (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * m c :=
-          mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun c _ => hp c)) (le_of_lt hW)
-        _ = _ := by
-          rw [←Finset.mul_sum,routingMoment_step]
-          simp only [Nat.cast_add,Nat.cast_one]
-          change W * ((2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * ∑ c, m c) =
-            (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * (W * ∑ c, m c)
-          ring
-
-lemma routingMoment_sum (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (a : ℝ) :
-    (∑ f : ι ↪ Card d, routingMoment d e f a) =
-      finiteMean (fun ω : BenesCoins d => (2:ℝ)^((palindromeCost d e ω:ℝ)*a)) := by
-  classical
-  unfold routingMoment
-  rw [←finiteMean_sum]
-  apply finiteMean_congr
-  intro ω
-  let f₀ : ι ↪ Card d := e.trans (palindromePerm d ω).toEmbedding
-  rw [Finset.sum_eq_single f₀]
-  · have hm : ∀ i, palindromePerm d ω (e i) = f₀ i := fun _ => rfl
-    simp only [hm,implies_true,↓reduceIte]
-  · intro f _ hf
-    have hm : ¬ ∀ i, palindromePerm d ω (e i) = f i := by
-      intro hh
-      apply hf
-      ext i : 1
-      exact (hh i).symm
-    exact ite_eq_right hm
-  · simp
-
-lemma finite_row_power {X : Type*} [Fintype X] [Nonempty X]
-    (p : X → ℝ) (hp : ∀ x, 0 ≤ p x) (a : ℝ) :
-    finiteMean (fun x => ((Fintype.card X:ℝ)*p x)^(1+a)) =
-      (Fintype.card X:ℝ)^a * ∑ x, (p x)^(1+a) := by
-  have hn : (0:ℝ) < Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
-  simp only [finiteMean,Real.mul_rpow (le_of_lt hn) (hp _),←Finset.mul_sum]
-  rw [Real.rpow_add hn,Real.rpow_one]
-  field_simp
-
-noncomputable def palindromeRowMoment (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (a : ℝ) : ℝ :=
-  finiteMean (fun f : ι ↪ Card d =>
-    ((Fintype.card (ι ↪ Card d):ℝ)*PairRouting.tupleProbability (palindromePerm d) e f)^(1+a))
-
-lemma density_normalization (d h n : ℕ) (a : ℝ) :
-    (n:ℝ)^a * (2:ℝ)^(-(d:ℝ)*h*a) = ((n:ℝ)/((2:ℝ)^d)^h)^a := by
-  rw [Real.div_rpow (Nat.cast_nonneg _) (by positivity),←pow_mul,
-    ←Real.rpow_natCast,←Real.rpow_mul (by norm_num : (0:ℝ) ≤ 2),Nat.cast_mul]
-  rw [show -(d:ℝ)*h*a = -((d:ℝ)*h*a) by ring,
-    Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2),div_eq_mul_inv]
-
-theorem palindrome_row_density_moment (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (a : ℝ) (ha : 0 ≤ a) :
-    palindromeRowMoment d e a ≤
-      (((2^d).descFactorial (Fintype.card ι):ℝ)/((2^d:ℕ):ℝ)^(Fintype.card ι))^a *
-        finiteMean (fun ω : BenesCoins d => (2:ℝ)^((palindromeCost d e ω:ℝ)*a)) := by
-  classical
-  have : Nonempty (ι ↪ Card d) := ⟨e⟩
-  rw [palindromeRowMoment,finite_row_power _ (fun f => PairRouting.tupleProbability_nonneg _ _ _)]
-  calc
-    _ ≤ (Fintype.card (ι ↪ Card d):ℝ)^a *
-      ∑ f : ι ↪ Card d, (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) * routingMoment d e f a :=
-      mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun f _ => palindrome_pointwise_moment d e f a ha))
-        (Real.rpow_nonneg (Nat.cast_nonneg _) _)
-    _ = _ := by
-      rw [←Finset.mul_sum,routingMoment_sum,←mul_assoc,density_normalization,
-        Fintype.card_embedding_eq,card_positions]
-      norm_cast
-
-end Thorp
-namespace Thorp
-
-lemma subset_biUnion_indicator {α β : Type*} [DecidableEq α] [Fintype β]
-    (A : Finset α) (T : β → Finset α) :
-    (if Finset.univ.biUnion T ⊆ A then (1:ℝ) else 0) =
-      ∏ b, if T b ⊆ A then (1:ℝ) else 0 := by
-  classical
-  by_cases h : Finset.univ.biUnion T ⊆ A
-  · rw [ite_eq_left h]
-    symm
-    apply Finset.prod_eq_one
-    intro b _
-    rw [ite_eq_left]
-    exact (Finset.subset_biUnion_of_mem T (Finset.mem_univ b)).trans h
-  · rw [ite_eq_right h]
-    obtain ⟨x,hx,hxA⟩ := Finset.not_subset.mp h
-    obtain ⟨b,_,hxb⟩ := Finset.mem_biUnion.mp hx
-    symm
-    apply Finset.prod_eq_zero (Finset.mem_univ b)
-    rw [ite_eq_right]
-    exact fun hb => hxA (hb hxb)
-
-lemma cylinder_block_polynomial {α β Ω : Type*} [DecidableEq α] [Fintype β]
-    [Fintype Ω] (τ : β → Type*) [∀ b, Fintype (τ b)]
-    (F : Ω → Finset α) (p : α → ℝ)
-    (hF : CylinderBound (inclusionMoment F) p)
-    (T : ∀ b, τ b → Finset α) (c : ∀ b, τ b → ℝ)
-    (hc : ∀ b s, 0 ≤ c b s)
-    (hd : ∀ b b', b ≠ b' → ∀ s s', Disjoint (T b s) (T b' s')) :
-    finiteMean (fun ω => ∏ b, ∑ s, c b s * if T b s ⊆ F ω then 1 else 0) ≤
-      ∏ b, ∑ s, c b s * ∏ x ∈ T b s, p x := by
-  classical
-  simp_rw [Fintype.prod_sum]
-  rw [finiteMean_sum]
-  apply Finset.sum_le_sum
-  intro s _
-  let U := Finset.univ.biUnion (fun b => T b (s b))
-  have he : (fun ω => ∏ b, c b (s b) * if T b (s b) ⊆ F ω then 1 else 0) =
-      fun ω => (if U ⊆ F ω then 1 else 0) * ∏ b, c b (s b) := by
-    funext ω
-    rw [Finset.prod_mul_distrib,←subset_biUnion_indicator]
-    exact mul_comm _ _
-  rw [he,finiteMean_mul_const]
-  have hdis : ((Finset.univ : Finset β) : Set β).PairwiseDisjoint (fun b => T b (s b)) := by
-    intro b _ b' _ hbb
-    exact hd b b' hbb (s b) (s b')
-  have hmean : finiteMean (fun ω => if U ⊆ F ω then 1 else 0) = inclusionMoment F U := by
-    unfold inclusionMoment
-    apply finiteMean_congr
-    intro ω
-    by_cases hh : U ⊆ F ω <;> simp [hh]
-  calc
-    _ ≤ (∏ x ∈ U, p x) * ∏ b, c b (s b) :=
-      mul_le_mul_of_nonneg_right (hmean.le.trans (hF U)) (Finset.prod_nonneg (fun b _ => hc b (s b)))
-    _ = _ := by
-      dsimp only [U]
-      rw [Finset.prod_biUnion hdis,←Finset.prod_mul_distrib]
-      apply Finset.prod_congr rfl
-      intro b _
-      exact mul_comm _ _
-
-
-noncomputable def sparseCoefficient {α : Type*} (w : ℝ) (T : Finset α) : ℝ :=
-  if T.card = 1 then 0 else w^T.card
-
-lemma sparseCoefficient_nonneg {α : Type*} (w : ℝ) (hw : 0 ≤ w) (T : Finset α) :
-    0 ≤ sparseCoefficient w T := by
-  unfold sparseCoefficient
-  split_ifs <;> positivity
-
-lemma sparse_block_pointwise {α : Type*} [DecidableEq α] (A D : Finset α)
-    (w : ℝ) (hw : 0 ≤ w) :
-    w^(if 2 ≤ (D ∩ A).card then (D ∩ A).card else 0) ≤
-      ∑ T : ↑D.powerset, sparseCoefficient w T.val * if T.val ⊆ A then 1 else 0 := by
-  classical
-  let V := D ∩ A
-  let T : ↑D.powerset := if 2 ≤ V.card then
-    ⟨V,Finset.mem_powerset.mpr Finset.inter_subset_left⟩ else
-    ⟨∅,Finset.mem_powerset.mpr (Finset.empty_subset D)⟩
-  calc
-    _ = sparseCoefficient w T.val * (if T.val ⊆ A then 1 else 0) := by
-      by_cases h : 2 ≤ V.card
-      · have hn : V.card ≠ 1 := by omega
-        simp [T,ite_eq_left h,sparseCoefficient,hn,show V ⊆ A from Finset.inter_subset_right,V]
-      · simp [T,ite_eq_right h,sparseCoefficient,V] at *
-    _ ≤ _ := Finset.single_le_sum (f := fun S : ↑D.powerset =>
-      sparseCoefficient w S.val * (if S.val ⊆ A then 1 else 0))
-      (fun S _ => mul_nonneg (sparseCoefficient_nonneg w hw S.val)
-        (by split_ifs <;> norm_num)) (Finset.mem_univ T)
-
-lemma geometric_sum_half (n : ℕ) {x : ℝ} (hx : 0 ≤ x) (hhalf : x ≤ 1/2) :
-    ∑ i ∈ Finset.range n, x^i ≤ 2 := by
-  induction n with
-  | zero => simp
-  | succ n ih =>
-    rw [Finset.sum_range_succ',pow_zero]
-    simp only [pow_succ',←Finset.mul_sum]
-    nlinarith
-
-lemma sparse_binomial_bound (R : ℕ) (z : ℝ) (hz : 0 ≤ z) (hr : (R:ℝ)*z ≤ 1/2) :
-    ∑ i ∈ Finset.range (R+1), (R.choose i:ℝ) * (if i = 1 then 0 else z^i) ≤
-      1 + 2*((R:ℝ)*z)^2 := by
-  classical
-  let x := (R:ℝ)*z
-  have hx : 0 ≤ x := mul_nonneg (Nat.cast_nonneg _) hz
-  have hbound (i : ℕ) : (R.choose i:ℝ)*(if i=1 then 0 else z^i) ≤
-      if i=1 then 0 else x^i := by
-    by_cases h : i=1
-    · simp [h]
-    · simp only [ite_eq_right h]
-      dsimp [x]
-      rw [mul_pow]
-      apply mul_le_mul_of_nonneg_right _ (pow_nonneg hz i)
-      exact_mod_cast Nat.choose_le_pow R i
-  calc
-    _ ≤ ∑ i ∈ Finset.range (R+1), if i=1 then 0 else x^i :=
-      Finset.sum_le_sum (fun i _ => hbound i)
-    _ ≤ 1+2*x^2 := by
-      rcases R with _ | R
-      · simp [x]
-      rcases R with _ | R
-      · norm_num [Finset.sum_range_succ]
-        positivity
-      have he : ∑ i ∈ Finset.range (R+2+1), (if i=1 then 0 else x^i) =
-          1 + x^2*∑ i ∈ Finset.range (R+1), x^i := by
-        rw [show R+2+1=2+(R+1) by omega,Finset.sum_range_add]
-        have hn (i : ℕ) : 2+i ≠ 1 := by omega
-        simp only [hn,ite_false,pow_add,←Finset.mul_sum]
-        norm_num [Finset.sum_range_succ]
-      rw [he]
-      have hg := geometric_sum_half (R+1) hx hr
-      nlinarith [sq_nonneg x]
-
-lemma sparse_block_product_bound {α : Type*} [DecidableEq α]
-    (D : Finset α) (p w : ℝ) (hp : 0 ≤ p) (hw : 0 ≤ w)
-    (hr : (D.card:ℝ)*(p*w) ≤ 1/2) :
-    ∑ T : ↑D.powerset, sparseCoefficient w T.val * p^T.val.card ≤
-      Real.exp (2*((D.card:ℝ)*(p*w))^2) := by
-  classical
-  have he : ∑ T : ↑D.powerset, sparseCoefficient w T.val * p^T.val.card =
-      ∑ i ∈ Finset.range (D.card+1), (D.card.choose i:ℝ)*(if i=1 then 0 else (p*w)^i) := by
-    rw [Finset.sum_coe_sort D.powerset (fun T => sparseCoefficient w T*p^T.card)]
-    have hh : (fun T : Finset α => sparseCoefficient w T*p^T.card) =
-        fun T => if T.card=1 then 0 else (p*w)^T.card := by
-      funext T
-      unfold sparseCoefficient
-      split_ifs <;> simp [mul_pow,mul_comm]
-    change (∑ T ∈ D.powerset, (fun T : Finset α => sparseCoefficient w T*p^T.card) T) = _
-    rw [hh,Finset.sum_powerset_apply_card (fun i => if i=1 then (0:ℝ) else (p*w)^i)]
-    simp only [nsmul_eq_mul]
-  rw [he]
-  exact (sparse_binomial_bound D.card (p*w) (mul_nonneg hp hw) hr).trans
-    (by simpa only [add_comm] using Real.add_one_le_exp (2*((D.card:ℝ)*(p*w))^2))
-
-
-noncomputable def sparseMass {α β : Type*} [DecidableEq α] [Fintype β]
-    (D : β → Finset α) (A : Finset α) : ℕ :=
-  ∑ b, if 2 ≤ (D b ∩ A).card then (D b ∩ A).card else 0
-
-theorem sparse_block_mgf {α β Ω : Type*} [DecidableEq α] [Fintype β] [Fintype Ω]
-    (D : β → Finset α) (hd : Pairwise (fun b b' => Disjoint (D b) (D b')))
-    (F : Ω → Finset α) (p w : ℝ) (hp : 0 ≤ p) (hw : 0 ≤ w)
-    (hF : ∀ T, inclusionMoment F T ≤ p^T.card)
-    (hr : ∀ b, ((D b).card:ℝ)*(p*w) ≤ 1/2) :
-    finiteMean (fun ω => w^(sparseMass D (F ω))) ≤
-      Real.exp (∑ b, 2*(((D b).card:ℝ)*(p*w))^2) := by
-  classical
-  have hc : CylinderBound (inclusionMoment F) (fun _ => p) := by
-    intro T
-    simpa only [Finset.prod_const] using hF T
-  have hb := cylinder_block_polynomial (fun b => ↑(D b).powerset) F (fun _ => p) hc
-    (fun _ S => S.val) (fun _ S => sparseCoefficient w S.val)
-    (fun _ S => sparseCoefficient_nonneg w hw S.val)
-    (fun b b' h S S' => (hd h).mono
-      (Finset.mem_powerset.mp S.property) (Finset.mem_powerset.mp S'.property))
-  calc
-    _ ≤ finiteMean (fun ω => ∏ b, ∑ S : ↑(D b).powerset,
-        sparseCoefficient w S.val * if S.val ⊆ F ω then 1 else 0) := by
-      apply finiteMean_mono
-      intro ω
-      rw [sparseMass,←Finset.prod_pow_eq_pow_sum]
-      apply Finset.prod_le_prod₀ (fun _ _ => pow_nonneg hw _)
-      intro b _
-      exact sparse_block_pointwise (F ω) (D b) w hw
-    _ ≤ ∏ b, ∑ S : ↑(D b).powerset, sparseCoefficient w S.val * p^S.val.card := by
-      simpa only [Finset.prod_const] using hb
-    _ ≤ _ := by
-      rw [Real.exp_sum]
-      apply Finset.prod_le_prod₀
-      · intro b _
-        exact Finset.sum_nonneg (fun S _ => mul_nonneg
-          (sparseCoefficient_nonneg w hw S.val) (pow_nonneg hp _))
-      · intro b _
-        exact sparse_block_product_bound (D b) p w hp hw (hr b)
-
-theorem inverseButterfly_sparse_mgf (d : ℕ) {β : Type*} [Fintype β]
-    (D : β → Finset (Card d)) (hd : Pairwise (fun b b' => Disjoint (D b) (D b')))
-    (A : Finset (Card d)) (w : ℝ) (hw : 0 ≤ w)
-    (hr : ∀ b, ((D b).card:ℝ)*((A.card/(2:ℝ)^d)*w) ≤ 1/2) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      w^(sparseMass D (inverseButterflyMarks d A ω))) ≤
-      Real.exp (∑ b, 2*(((D b).card:ℝ)*((A.card/(2:ℝ)^d)*w))^2) := by
-  apply sparse_block_mgf D hd (inverseButterflyMarks d A) (A.card/(2:ℝ)^d) w
-    (by positivity) hw
-  · exact inverseButterflyMarks_inclusion d A
-  · exact hr
-
-
-def crowdedMass (h : ℕ) : ℕ := if 2 ≤ h then h else 0
-
-lemma crowdedMass_le (h : ℕ) : crowdedMass h ≤ h := by
-  unfold crowdedMass
-  split_ifs <;> omega
-
-lemma crowdedMass_add (u v : ℕ) : crowdedMass u + crowdedMass v ≤ crowdedMass (u+v) := by
-  unfold crowdedMass
-  split_ifs <;> omega
-
-lemma alternatingCycles_le_crowded {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
-    (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    PairRouting.alternatingCycles e p ≤ crowdedMass (Fintype.card ι) := by
-  unfold crowdedMass
-  split_ifs with h
-  · have hh := PairRouting.alternatingCycles_double e p
-    omega
-  · rw [PairRouting.alternatingCycles_zero e p (by omega)]
-
-lemma palindromeCost_le_crowded (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) :
-    palindromeCost d e ω ≤ d*crowdedMass (Fintype.card ι) := by
-  classical
-  induction d generalizing ι with
-  | zero => simp [palindromeCost]
-  | succ d ih =>
-    let σ := benesStepEquiv d ω
-    let x := e.trans (headTailEquiv d).toEmbedding
-    let c := PairRouting.colors x σ.2.1
-    let hc := PairRouting.colors_compatible x σ.2.1
-    have h₀ := ih (PairRouting.childEmbedding x c hc false) σ.1.1
-    have h₁ := ih (PairRouting.childEmbedding x c hc true) σ.1.2
-    have hz := alternatingCycles_le_crowded (PairRouting.switchedEmbedding x σ.2.1)
-      (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1))
-    have hs := crowdedMass_add (Fintype.card {i // c i = false})
-      (Fintype.card {i // c i = true})
-    rw [PairRouting.color_card_add] at hs
-    change PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
-        (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) +
-        palindromeCost d (PairRouting.childEmbedding x c hc false) σ.1.1 +
-        palindromeCost d (PairRouting.childEmbedding x c hc true) σ.1.2 ≤ _
-    nlinarith
-
-noncomputable def palindromeLowCost (H : ℕ) : (d : ℕ) → {ι : Type*} → [Fintype ι] →
-    (ι ↪ Card d) → BenesCoins d → ℕ
-  | 0, _, _, _, _ => 0
-  | d+1, _, _, e, ω =>
-      let σ := benesStepEquiv d ω
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x σ.2.1
-      let hc := PairRouting.colors_compatible x σ.2.1
-      (if d+1 ≤ H then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
-          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) else 0) +
-        palindromeLowCost H d (PairRouting.childEmbedding x c hc false) σ.1.1 +
-        palindromeLowCost H d (PairRouting.childEmbedding x c hc true) σ.1.2
-
-noncomputable def cutoffMass (H : ℕ) : (d : ℕ) → {ι : Type*} → [Fintype ι] →
-    (ι ↪ Card d) → BenesCoins d → ℕ
-  | 0, _, _, _, _ => 0
-  | d+1, ι, _, e, ω =>
-      if d+1 ≤ H then crowdedMass (Fintype.card ι) else
-      let σ := benesStepEquiv d ω
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x σ.2.1
-      let hc := PairRouting.colors_compatible x σ.2.1
-      cutoffMass H d (PairRouting.childEmbedding x c hc false) σ.1.1 +
-        cutoffMass H d (PairRouting.childEmbedding x c hc true) σ.1.2
-
-lemma palindromeLowCost_eq (H d : ℕ) (h : d ≤ H) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) :
-    palindromeLowCost H d e ω = palindromeCost d e ω := by
-  classical
-  induction d generalizing ι with
-  | zero => rfl
-  | succ d ih =>
-    rw [palindromeLowCost,palindromeCost,ite_eq_left h]
-    congr 2 <;> exact ih (by omega) _ _
-
-lemma palindromeLowCost_le_cutoff (H d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) :
-    palindromeLowCost H d e ω ≤ H*cutoffMass H d e ω := by
-  classical
-  induction d generalizing ι with
-  | zero => simp [palindromeLowCost,cutoffMass]
-  | succ d ih =>
-    by_cases h : d+1 ≤ H
-    · rw [palindromeLowCost_eq H (d+1) h,cutoffMass,ite_eq_left h]
-      exact (palindromeCost_le_crowded (d+1) e ω).trans (Nat.mul_le_mul_right _ h)
-    · rw [palindromeLowCost,cutoffMass,ite_eq_right h,ite_eq_right h,zero_add,Nat.mul_add]
-      exact Nat.add_le_add (ih _ _) (ih _ _)
-
-
-noncomputable def centralBlock (r H : ℕ) (b : Card r) : Finset (Card (H+r)) :=
-  Finset.univ.image (fun z : Card H => (cardSplit r H).symm (b,z))
-
-end Thorp
-
-end ThorpNine.Smoothing
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Smoothing.AlternatingPerm
+
+namespace OAI
+
+namespace ThorpNine.Smoothing
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+
+abbrev BenesCoins (d : ℕ) := (SwitchIndex d → Bool) × (SwitchIndex d → Bool)
+
+def palindromePerm (d : ℕ) (ω : BenesCoins d) : Equiv.Perm (Card d) :=
+  butterflyPerm d (decodeButterfly d ω.1) * (butterflyPerm d (decodeButterfly d ω.2)).symm
+
+def sandwichShuffle {A B C D E F : Type*} :
+    ((A × B) × C) × ((D × E) × F) ≃ ((A × D) × (B × E)) × (F × C) where
+  toFun x := (((x.1.1.1,x.2.1.1),(x.1.1.2,x.2.1.2)),(x.2.2,x.1.2))
+  invFun x := (((x.1.1.1,x.1.2.1),x.2.2),((x.1.1.2,x.1.2.2),x.2.1))
+  left_inv _ := rfl
+  right_inv _ := rfl
+
+def benesStepEquiv (d : ℕ) : BenesCoins (d+1) ≃
+    (BenesCoins d × BenesCoins d) × ((Card d → Bool) × (Card d → Bool)) :=
+  (Equiv.prodCongr (coinStepEquiv d) (coinStepEquiv d)).trans sandwichShuffle
+
+lemma childLift_mul (d : ℕ) (p q : Bool → Equiv.Perm (Card d)) :
+    childLift p * childLift q = childLift (fun b => p b * q b) := by
+  ext x i
+  simp only [Equiv.Perm.mul_apply,childLift,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons]
+
+lemma childLift_inv (d : ℕ) (p : Bool → Equiv.Perm (Card d)) :
+    (childLift p)⁻¹ = childLift (fun b => (p b)⁻¹) := rfl
+
+lemma pairSwitch_inv (d : ℕ) (ξ : Card d → Bool) : (pairSwitch ξ)⁻¹ = pairSwitch ξ := rfl
+
+lemma palindromePerm_step (d : ℕ) (c : BenesCoins d × BenesCoins d)
+    (coins : (Card d → Bool) × (Card d → Bool)) :
+    palindromePerm (d+1) ((benesStepEquiv d).symm (c,coins)) =
+      pairSwitch coins.2 * childLift (fun b => palindromePerm d (if b then c.2 else c.1)) *
+        pairSwitch coins.1 := by
+  change (butterflyPerm (d+1) (decodeButterfly (d+1)
+      ((coinStepEquiv d).symm ((c.1.1,c.2.1),coins.2)))) *
+    (butterflyPerm (d+1) (decodeButterfly (d+1)
+      ((coinStepEquiv d).symm ((c.1.2,c.2.2),coins.1))))⁻¹ = _
+  rw [butterflyPerm_step,butterflyPerm_step,mul_inv_rev,childLift_inv,pairSwitch_inv]
+  rw [mul_assoc (pairSwitch coins.2),←mul_assoc (childLift _),childLift_mul,←mul_assoc]
+  congr 2
+  apply congrArg childLift
+  funext b
+  cases b <;> rfl
+
+lemma headTail_switch (d : ℕ) (ξ : Card d → Bool) (x : Card (d+1)) :
+    headTailEquiv d (pairSwitch ξ x) = pairLayer ξ (headTailEquiv d x) := rfl
+
+lemma headTail_child (d : ℕ) (p : Bool → Equiv.Perm (Card d)) (x : Card (d+1)) :
+    headTailEquiv d (childLift p x) = PairRouting.lift p (headTailEquiv d x) := rfl
+
+lemma palindromePerm_coordinates (d : ℕ) (c : BenesCoins d × BenesCoins d)
+    (coins : (Card d → Bool) × (Card d → Bool)) (x : Card (d+1)) :
+    headTailEquiv d (palindromePerm (d+1) ((benesStepEquiv d).symm (c,coins)) x) =
+      PairRouting.sandwich coins.1 coins.2 (fun b => palindromePerm d (if b then c.2 else c.1))
+        (headTailEquiv d x) := by
+  rw [palindromePerm_step]
+  simp only [Equiv.Perm.mul_apply,headTail_switch,headTail_child]
+  rfl
+
+noncomputable def palindromeCost : (d : ℕ) → {ι : Type*} → [Fintype ι] →
+    (ι ↪ Card d) → BenesCoins d → ℕ
+  | 0, _, _, _, _ => 0
+  | d+1, _, _, e, ω =>
+      let σ := benesStepEquiv d ω
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x σ.2.1
+      let hc := PairRouting.colors_compatible x σ.2.1
+      PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
+          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) +
+        palindromeCost d (PairRouting.childEmbedding x c hc false) σ.1.1 +
+        palindromeCost d (PairRouting.childEmbedding x c hc true) σ.1.2
+
+namespace CycleColoring
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+lemma alternating_cycle_two (p : Equiv.Perm α) (S : Finset (Bool × α))
+    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) S) : 2 ≤ C.val.card := by
+  obtain ⟨x,hx⟩ := C.nonempty
+  have hpx : alternatingPerm p x ∈ C.val := by
+    rw [←C.orbit_eq hx,mem_orbitSet]
+    exact (Equiv.Perm.SameCycle.refl _ _).apply_right
+  have hne : x ≠ alternatingPerm p x := by
+    intro he
+    have hh := congrArg Prod.fst he
+    rw [alternatingPerm_flip] at hh
+    cases hb : x.1 <;> simp only [hb, Bool.not_false, Bool.not_true] at hh <;> cases hh
+  have hs : {x,alternatingPerm p x} ⊆ C.val := by
+    intro y hy
+    simp only [Finset.mem_insert,Finset.mem_singleton] at hy
+    rcases hy with rfl | rfl <;> assumption
+  have hh := Finset.card_le_card hs
+  simpa only [Finset.card_pair hne] using hh
+
+lemma alternating_cycle_count_double (p : Equiv.Perm α) (S : Finset (Bool × α)) :
+    2 * Fintype.card (RoutingNetwork.SelectedCycle (alternatingPerm p) S) ≤ S.card := by
+  classical
+  calc
+    _ = ∑ _C : RoutingNetwork.SelectedCycle (alternatingPerm p) S, 2 := by simp [Nat.mul_comm]
+    _ ≤ RoutingNetwork.cycleMass (alternatingPerm p) S :=
+      Finset.sum_le_sum (fun C _ => alternating_cycle_two p S C)
+    _ ≤ _ := RoutingNetwork.cycleMass_le _ _
+
+lemma alternating_cycle_count_zero (p : Equiv.Perm α) (S : Finset (Bool × α))
+    (hS : S.card ≤ 1) : Fintype.card (RoutingNetwork.SelectedCycle (alternatingPerm p) S) = 0 := by
+  have h := alternating_cycle_count_double p S
+  omega
+
+end CycleColoring
+
+namespace PairRouting
+
+section
+variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
+
+lemma alternatingCycles_double (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    2 * alternatingCycles e p ≤ Fintype.card ι := by
+  have hh := CycleColoring.alternating_cycle_count_double (alternatingProjection p) (labelSet e)
+  simpa only [alternatingCycles, card_labelSet] using hh
+
+lemma alternatingCycles_zero (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α)
+    (hι : Fintype.card ι ≤ 1) : alternatingCycles e p = 0 := by
+  have hh := alternatingCycles_double e p
+  omega
+
+end
+variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α] [DecidableEq ι]
+
+omit [Fintype α] [DecidableEq ι] in
+lemma colors_chosenCoin (x : ι → Bool × α) (c : ι → Bool) (hc : Compatible x c) :
+    colors x (chosenCoin x c) = c := by
+  funext i
+  simp only [colors,chosenCoin_apply x c hc i,required]
+  cases (x i).1 <;> cases c i <;> rfl
+
+noncomputable def coloredEmbedding (x : ι ↪ Bool × α) (c : ι → Bool) (_hc : Compatible x c) :
+    ι ↪ Bool × α := switchedEmbedding x (chosenCoin x c)
+
+omit [Fintype α] [DecidableEq ι] in
+lemma coloredEmbedding_apply (x : ι ↪ Bool × α) (c : ι → Bool) (hc : Compatible x c) (i : ι) :
+    coloredEmbedding x c hc i = (c i,(x i).2) := by
+  apply Prod.ext
+  · exact congrFun (colors_chosenCoin x c hc) i
+  · rfl
+
+omit [Fintype α] [DecidableEq ι] in
+lemma switchedEmbedding_eq_colored (x : ι ↪ Bool × α) (c : ι → Bool) (hc : Compatible x c)
+    (ξ : α → Bool) (he : colors x ξ = c) :
+    switchedEmbedding x ξ = coloredEmbedding x c hc := by
+  ext i : 1
+  rw [coloredEmbedding_apply]
+  apply Prod.ext
+  · exact congrFun he i
+  · rfl
+
+lemma assignments_match_bound (x y : ι ↪ Bool × α) (c : ι → Bool)
+    (hc : c ∈ compatibleSet x y) (p : Bool → Equiv.Perm α) (hm : ChildMatches p x y c) :
+    (compatibleSet x y).card ≤
+      2 ^ ((occupied x).card + (occupied y).card - Fintype.card ι +
+        alternatingCycles (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p) := by
+  obtain ⟨hcx,hcy⟩ := (mem_compatibleSet x y c).mp hc
+  let ξ := chosenCoin x c
+  let η := chosenCoin y c
+  have he : routedEmbedding x ξ η p = y := by
+    ext i : 1
+    have h := (sandwich_matches_iff ξ η p x y).mpr
+      ⟨(colors_chosenCoin x c hcx).trans (colors_chosenCoin y c hcy).symm,by
+        simpa only [ξ,colors_chosenCoin x c hcx] using hm⟩
+    exact h i
+  have hh := assignments_actual_bound x ξ η p
+  rw [he] at hh
+  rw [card_compatibleSet]
+  exact hh
+
+omit [Fintype α] [DecidableEq ι] in
+lemma occupied_sum_ge (x y : ι ↪ Bool × α) :
+    Fintype.card ι ≤ (occupied x).card + (occupied y).card := by
+  have h₀ := occupied_card_double x
+  have h₁ := occupied_card_double y
+  omega
+
+lemma assignment_weight_bound (x y : ι ↪ Bool × α) (c : ι → Bool)
+    (hc : c ∈ compatibleSet x y) (p : Bool → Equiv.Perm α) (hm : ChildMatches p x y c) :
+    ((compatibleSet x y).card:ℝ) * outerWeight x y ≤
+      (2:ℝ)^((alternatingCycles
+        (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p : ℝ) - Fintype.card ι) := by
+  let z := alternatingCycles (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p
+  let b := (occupied x).card+(occupied y).card
+  let h := Fintype.card ι
+  have hb : h ≤ b := occupied_sum_ge x y
+  have hn : ((compatibleSet x y).card:ℝ) ≤ (2:ℝ)^(b-h+z) := by
+    exact_mod_cast assignments_match_bound x y c hc p hm
+  change ((compatibleSet x y).card:ℝ) * (1/(2:ℝ)^b) ≤ (2:ℝ)^((z:ℝ)-h)
+  rw [mul_one_div]
+  calc
+    _ ≤ (2:ℝ)^(b-h+z)/(2:ℝ)^b := div_le_div_of_nonneg_right hn (by positivity)
+    _ = _ := by
+      rw [←Real.rpow_natCast,←Real.rpow_natCast,←Real.rpow_sub (by norm_num : (0:ℝ)<2)]
+      congr 1
+      rw [Nat.cast_add,Nat.cast_sub hb]
+      ring
+
+lemma assignment_factor_bound (x y : ι ↪ Bool × α) (c : ι → Bool)
+    (hc : c ∈ compatibleSet x y) (p : Bool → Equiv.Perm α) (hm : ChildMatches p x y c)
+    (a : ℝ) (ha : 0 ≤ a) :
+    (outerWeight x y)^a * ((compatibleSet x y).card:ℝ)^a ≤
+      (2:ℝ)^(((alternatingCycles
+        (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p : ℝ) - Fintype.card ι)*a) := by
+  have h := Real.rpow_le_rpow (mul_nonneg (Nat.cast_nonneg _) (le_of_lt (outerWeight_pos x y)))
+    (assignment_weight_bound x y c hc p hm) ha
+  rw [Real.mul_rpow (Nat.cast_nonneg _) (le_of_lt (outerWeight_pos x y)),
+    ←Real.rpow_mul (by norm_num : (0:ℝ)≤2)] at h
+  simpa only [mul_comm] using h
+
+
+lemma sandwich_weighted_probability_subtype (p : Bool → Equiv.Perm α) (x y : ι → Bool × α)
+    (W : (ι → Bool) → ℝ) :
+    finiteMean (fun coins : (α → Bool) × (α → Bool) =>
+      if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then W (colors x coins.1) else 0) =
+    outerWeight x y * ∑ c : {c // c ∈ compatibleSet x y},
+      if ChildMatches p x y c.val then W c.val else 0 := by
+  rw [sandwich_weighted_probability]
+  congr 1
+  rw [Finset.sum_subtype (compatibleSet x y) (fun _ => Iff.rfl)]
+
+omit [Fintype α] [DecidableEq α] [DecidableEq ι] in
+lemma color_card_add (c : ι → Bool) :
+    Fintype.card {i // c i = false} + Fintype.card {i // c i = true} = Fintype.card ι := by
+  have he : {i // c i = true} ≃ {i // ¬c i = false} := Equiv.subtypeEquivRight fun i => by
+    cases c i <;> simp
+  rw [Fintype.card_congr he]
+  rw [Fintype.card_subtype_compl]
+  exact Nat.add_sub_of_le (Fintype.card_subtype_le _)
+
+end PairRouting
+
+noncomputable def assignmentCost (d : ℕ) {ι : Type*} [Fintype ι]
+    (x : ι ↪ Bool × Card d) (c : ι → Bool) (hc : PairRouting.Compatible x c)
+    (ω : BenesCoins d × BenesCoins d) : ℕ :=
+  PairRouting.alternatingCycles (PairRouting.coloredEmbedding x c hc)
+      (fun b => palindromePerm d (if b then ω.2 else ω.1)) +
+    palindromeCost d (PairRouting.childEmbedding x c hc false) ω.1 +
+    palindromeCost d (PairRouting.childEmbedding x c hc true) ω.2
+
+lemma palindromeCost_eq (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (d+1)) (σ : BenesCoins (d+1)) :
+    palindromeCost (d+1) e σ =
+      assignmentCost d (e.trans (headTailEquiv d).toEmbedding)
+        (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) (benesStepEquiv d σ).2.1)
+        (PairRouting.colors_compatible _ _) (benesStepEquiv d σ).1 := by
+  unfold palindromeCost assignmentCost
+  dsimp only
+  rw [PairRouting.switchedEmbedding_eq_colored _ _ (PairRouting.colors_compatible _ _) _ rfl]
+
+lemma palindromeCost_step (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (d+1)) (ω : BenesCoins d × BenesCoins d)
+    (coins : (Card d → Bool) × (Card d → Bool)) :
+    palindromeCost (d+1) e ((benesStepEquiv d).symm (ω,coins)) =
+      assignmentCost d (e.trans (headTailEquiv d).toEmbedding)
+        (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) coins.1)
+        (PairRouting.colors_compatible _ _) ω := by
+  rw [palindromeCost_eq]
+  simp only [Equiv.apply_symm_apply]
+
+noncomputable def routingMoment (d : ℕ) {ι : Type*} [Fintype ι]
+    (e f : ι ↪ Card d) (a : ℝ) : ℝ := by
+  classical
+  exact finiteMean (fun ω : BenesCoins d =>
+    if ∀ i, palindromePerm d ω (e i) = f i then (2:ℝ)^((palindromeCost d e ω:ℝ)*a) else 0)
+
+lemma routingMoment_nonneg (d : ℕ) {ι : Type*} [Fintype ι]
+    (e f : ι ↪ Card d) (a : ℝ) : 0 ≤ routingMoment d e f a := by
+  classical
+  exact finiteMean_nonneg (fun _ => by split_ifs <;> positivity)
+
+lemma palindrome_matches_step (d : ℕ) {ι : Type*}
+    (e f : ι ↪ Card (d+1)) (ω : BenesCoins d × BenesCoins d)
+    (coins : (Card d → Bool) × (Card d → Bool)) :
+    (∀ i, palindromePerm (d+1) ((benesStepEquiv d).symm (ω,coins)) (e i) = f i) ↔
+    (∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => palindromePerm d (if b then ω.2 else ω.1))
+      ((e.trans (headTailEquiv d).toEmbedding) i) = ((f.trans (headTailEquiv d).toEmbedding) i)) := by
+  apply forall_congr'
+  intro i
+  rw [←(headTailEquiv d).injective.eq_iff,palindromePerm_coordinates]
+  rfl
+
+lemma routingMoment_step (d : ℕ) {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (e f : ι ↪ Card (d+1)) (a : ℝ) :
+    let x := e.trans (headTailEquiv d).toEmbedding
+    let y := f.trans (headTailEquiv d).toEmbedding
+    routingMoment (d+1) e f a = PairRouting.outerWeight x y *
+      ∑ c : {c // c ∈ PairRouting.compatibleSet x y},
+        finiteMean (fun ω : BenesCoins d × BenesCoins d =>
+          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c.val
+          then (2:ℝ)^((assignmentCost d x c.val
+            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0) := by
+  classical
+  intro x y
+  rw [routingMoment,finiteMean_equiv (benesStepEquiv d),finiteMean_prod]
+  have he (ω : BenesCoins d × BenesCoins d) :
+      finiteMean (fun coins : (Card d → Bool) × (Card d → Bool) =>
+        if ∀ i, palindromePerm (d+1) ((benesStepEquiv d).symm (ω,coins)) (e i) = f i
+        then (2:ℝ)^((palindromeCost (d+1) e ((benesStepEquiv d).symm (ω,coins)):ℝ)*a) else 0) =
+      PairRouting.outerWeight x y *
+        ∑ c : {c // c ∈ PairRouting.compatibleSet x y},
+          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c.val
+          then (2:ℝ)^((assignmentCost d x c.val
+            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0 := by
+    let W : (ι → Bool) → ℝ := fun c =>
+      if hc : PairRouting.Compatible x c then (2:ℝ)^((assignmentCost d x c hc ω:ℝ)*a) else 0
+    calc
+      _ = finiteMean (fun coins : (Card d → Bool) × (Card d → Bool) =>
+        if ∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => palindromePerm d (if b then ω.2 else ω.1))
+          (x i) = y i then W (PairRouting.colors x coins.1) else 0) := by
+        apply finiteMean_congr
+        intro coins
+        simp only [palindrome_matches_step,palindromeCost_step]
+        simp only [W,PairRouting.colors_compatible,↓reduceDIte]
+        rfl
+      _ = _ := by
+        rw [PairRouting.sandwich_weighted_probability_subtype]
+        congr 1
+        apply Finset.sum_congr rfl
+        intro c _
+        simp only [W,((PairRouting.mem_compatibleSet x y c.val).mp c.property).1,↓reduceDIte]
+  simp_rw [he]
+  simp_rw [mul_comm (PairRouting.outerWeight x y)]
+  rw [finiteMean_mul_const,finiteMean_sum,mul_comm]
+
+lemma child_moment_product (d : ℕ) {ι : Type*} [Fintype ι]
+    (x y : ι ↪ Bool × Card d) (c : ι → Bool)
+    (hcx : PairRouting.Compatible x c) (hcy : PairRouting.Compatible y c) (a : ℝ) :
+    routingMoment d (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a *
+      routingMoment d (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a =
+    finiteMean (fun ω : BenesCoins d × BenesCoins d =>
+      if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
+      then (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
+        palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0) := by
+  classical
+  unfold routingMoment
+  rw [←finiteMean_prod_mul]
+  apply finiteMean_congr
+  intro ω
+  let M₀ := ∀ i : {i // c i = false}, palindromePerm d ω.1 (x i.val).2 = (y i.val).2
+  let M₁ := ∀ i : {i // c i = true}, palindromePerm d ω.2 (x i.val).2 = (y i.val).2
+  have hp : (fun b : Bool => palindromePerm d (if b then ω.2 else ω.1)) =
+      (fun b => if b then palindromePerm d ω.2 else palindromePerm d ω.1) := by
+    funext b
+    cases b <;> rfl
+  have hs : PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c ↔
+      M₀ ∧ M₁ := by rw [hp]; exact PairRouting.childMatches_split _ _ _ _ _
+  change (if M₀ then _ else 0) * (if M₁ then _ else 0) = _
+  by_cases h₀ : M₀ <;> by_cases h₁ : M₁
+  · simp only [hs,h₀,h₁,true_and,↓reduceIte]
+    rw [←Real.rpow_add (by norm_num : (0:ℝ)<2)]
+    congr 1
+    ring
+  · simp only [hs,h₀,h₁,and_false,↓reduceIte,mul_zero]
+  · simp only [hs,h₀,h₁,false_and,↓reduceIte,zero_mul]
+  · simp only [hs,h₀,h₁,false_and,↓reduceIte,mul_zero]
+
+lemma palindrome_probability_step (d : ℕ) {ι : Type*} [Fintype ι]
+    (e f : ι ↪ Card (d+1)) :
+    PairRouting.tupleProbability (palindromePerm (d+1)) e f =
+    finiteMean (fun ω : BenesCoins d × BenesCoins d =>
+      finiteMean (fun coins : (Card d → Bool) × (Card d → Bool) =>
+        if ∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => if b then palindromePerm d ω.2 else palindromePerm d ω.1)
+          ((e.trans (headTailEquiv d).toEmbedding) i) = ((f.trans (headTailEquiv d).toEmbedding) i)
+        then (1:ℝ) else 0)) := by
+  classical
+  rw [PairRouting.tupleProbability,finiteMean_equiv (benesStepEquiv d),finiteMean_prod]
+  have hp (b : Bool) (p q : BenesCoins d) :
+      palindromePerm d (if b then p else q) = (if b then palindromePerm d p else palindromePerm d q) := by
+    cases b <;> rfl
+  simp only [palindrome_matches_step,hp]
+
+lemma child_power_product_bound (d : ℕ) {ι : Type*} [Fintype ι]
+    (x y : ι ↪ Bool × Card d) (c : ι → Bool)
+    (hcx : PairRouting.Compatible x c) (hcy : PairRouting.Compatible y c) (a : ℝ)
+    (hfalse : (PairRouting.tupleProbability (palindromePerm d)
+        (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false))^(1+a) ≤
+      (2:ℝ)^(-(d:ℝ)*Fintype.card {i // c i = false}*a) *
+        routingMoment d (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a)
+    (htrue : (PairRouting.tupleProbability (palindromePerm d)
+        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true))^(1+a) ≤
+      (2:ℝ)^(-(d:ℝ)*Fintype.card {i // c i = true}*a) *
+        routingMoment d (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a) :
+    (PairRouting.tupleProbability (palindromePerm d)
+      (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false))^(1+a) *
+      (PairRouting.tupleProbability (palindromePerm d)
+        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true))^(1+a) ≤
+    (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) *
+      finiteMean (fun ω : BenesCoins d × BenesCoins d =>
+        if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
+        then (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
+          palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0) := by
+  have hh := mul_le_mul hfalse htrue
+    (Real.rpow_nonneg (PairRouting.tupleProbability_nonneg _ _ _) _)
+    (mul_nonneg (by positivity) (routingMoment_nonneg _ _ _ _))
+  rw [mul_mul_mul_comm,child_moment_product,←Real.rpow_add (by norm_num : (0:ℝ)<2)] at hh
+  have he : -(d:ℝ)*Fintype.card {i // c i = false}*a + -(d:ℝ)*Fintype.card {i // c i = true}*a =
+      -(d:ℝ)*Fintype.card ι*a := by
+    rw [←add_mul,←mul_add,←Nat.cast_add,PairRouting.color_card_add]
+  rwa [he] at hh
+
+lemma assignment_moment_bound (d : ℕ) {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (x y : ι ↪ Bool × Card d) (c : ι → Bool) (hc : c ∈ PairRouting.compatibleSet x y)
+    (a : ℝ) (ha : 0 ≤ a) :
+    let hcx := ((PairRouting.mem_compatibleSet x y c).mp hc).1
+    PairRouting.outerWeight x y ^ a * ((PairRouting.compatibleSet x y).card:ℝ)^a *
+      ((2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) *
+        finiteMean (fun ω : BenesCoins d × BenesCoins d =>
+          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
+          then (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
+            palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0)) ≤
+    (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) *
+      finiteMean (fun ω : BenesCoins d × BenesCoins d =>
+        if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
+        then (2:ℝ)^((assignmentCost d x c hcx ω:ℝ)*a) else 0) := by
+  classical
+  intro hcx
+  have hscale (q : ℝ) (F : BenesCoins d × BenesCoins d → ℝ) :
+      q * finiteMean F = finiteMean (fun ω => q * F ω) := by
+    simp_rw [mul_comm q,finiteMean_mul_const]
+  rw [←mul_assoc,hscale,hscale]
+  apply finiteMean_mono
+  intro ω
+  split_ifs with hm
+  · have hh := PairRouting.assignment_factor_bound x y c hc
+      (fun b => palindromePerm d (if b then ω.2 else ω.1)) hm a ha
+    calc
+      _ ≤ ((2:ℝ)^(((PairRouting.alternatingCycles (PairRouting.coloredEmbedding x c hcx)
+          (fun b => palindromePerm d (if b then ω.2 else ω.1)):ℝ)-Fintype.card ι)*a) *
+          (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a)) *
+          (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
+            palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) :=
+        mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hh (by positivity)) (by positivity)
+      _ = _ := by
+        simp only [←Real.rpow_add (by norm_num : (0:ℝ)<2),assignmentCost,Nat.cast_add]
+        congr 1
+        ring
+  · simp only [mul_zero]
+    exact le_rfl
+
+theorem palindrome_pointwise_moment (d : ℕ) {ι : Type*} [Fintype ι]
+    (e f : ι ↪ Card d) (a : ℝ) (ha : 0 ≤ a) :
+    (PairRouting.tupleProbability (palindromePerm d) e f)^(1+a) ≤
+      (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) * routingMoment d e f a := by
+  classical
+  induction d generalizing ι with
+  | zero =>
+      have he : ∀ (ω : BenesCoins 0) (i : ι), palindromePerm 0 ω (e i) = f i :=
+        fun _ _ => Subsingleton.elim _ _
+      simp [PairRouting.tupleProbability,routingMoment,he,palindromeCost,finiteMean_const]
+  | succ d ih =>
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let y := f.trans (headTailEquiv d).toEmbedding
+      let W := PairRouting.outerWeight x y
+      let C := PairRouting.compatibleSet x y
+      let p : (ι → Bool) → ℝ := fun c =>
+        (PairRouting.tupleProbability (palindromePerm d)
+          (fun i : {i // c i = false} => (x i.val).2)
+          (fun i : {i // c i = false} => (y i.val).2))^(1+a) *
+        (PairRouting.tupleProbability (palindromePerm d)
+          (fun i : {i // c i = true} => (x i.val).2)
+          (fun i : {i // c i = true} => (y i.val).2))^(1+a)
+      let m : {c // c ∈ C} → ℝ := fun c =>
+        finiteMean (fun ω : BenesCoins d × BenesCoins d =>
+          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c.val
+          then (2:ℝ)^((assignmentCost d x c.val
+            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0)
+      have hp (c : {c // c ∈ C}) : W^a * (C.card:ℝ)^a * p c.val ≤
+          (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * m c := by
+        obtain ⟨hcx,hcy⟩ := (PairRouting.mem_compatibleSet x y c.val).mp c.property
+        have h₀ := ih (PairRouting.childEmbedding x c.val hcx false)
+          (PairRouting.childEmbedding y c.val hcy false)
+        have h₁ := ih (PairRouting.childEmbedding x c.val hcx true)
+          (PairRouting.childEmbedding y c.val hcy true)
+        have hh := child_power_product_bound d x y c.val hcx hcy a h₀ h₁
+        have hh' := mul_le_mul_of_nonneg_left hh
+          (mul_nonneg (Real.rpow_nonneg (le_of_lt (PairRouting.outerWeight_pos x y)) a)
+            (Real.rpow_nonneg (Nat.cast_nonneg C.card) a))
+        exact hh'.trans (assignment_moment_bound d x y c.val c.property a ha)
+      have hnode := PairRouting.sandwich_power_bound (palindromePerm d) (palindromePerm d) x y a ha
+      rw [←palindrome_probability_step d e f] at hnode
+      change (PairRouting.tupleProbability (palindromePerm (d+1)) e f)^(1+a) ≤
+        W^(1+a) * (C.card:ℝ)^a * ∑ c ∈ C, p c at hnode
+      have hsum : ∑ c ∈ C, p c = ∑ c : {c // c ∈ C}, p c.val := by
+        rw [Finset.sum_subtype C (fun _ => Iff.rfl)]
+      have hW : 0 < W := PairRouting.outerWeight_pos x y
+      calc
+        _ ≤ W^(1+a) * (C.card:ℝ)^a * ∑ c ∈ C, p c := hnode
+        _ = W * ∑ c : {c // c ∈ C}, W^a * (C.card:ℝ)^a * p c.val := by
+          rw [Real.rpow_add hW,Real.rpow_one,hsum,←Finset.mul_sum]
+          ring
+        _ ≤ W * ∑ c : {c // c ∈ C}, (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * m c :=
+          mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun c _ => hp c)) (le_of_lt hW)
+        _ = _ := by
+          rw [←Finset.mul_sum,routingMoment_step]
+          simp only [Nat.cast_add,Nat.cast_one]
+          change W * ((2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * ∑ c, m c) =
+            (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * (W * ∑ c, m c)
+          ring
+
+lemma routingMoment_sum (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (a : ℝ) :
+    (∑ f : ι ↪ Card d, routingMoment d e f a) =
+      finiteMean (fun ω : BenesCoins d => (2:ℝ)^((palindromeCost d e ω:ℝ)*a)) := by
+  classical
+  unfold routingMoment
+  rw [←finiteMean_sum]
+  apply finiteMean_congr
+  intro ω
+  let f₀ : ι ↪ Card d := e.trans (palindromePerm d ω).toEmbedding
+  rw [Finset.sum_eq_single f₀]
+  · have hm : ∀ i, palindromePerm d ω (e i) = f₀ i := fun _ => rfl
+    simp only [hm,implies_true,↓reduceIte]
+  · intro f _ hf
+    have hm : ¬ ∀ i, palindromePerm d ω (e i) = f i := by
+      intro hh
+      apply hf
+      ext i : 1
+      exact (hh i).symm
+    exact ite_eq_right hm
+  · simp
+
+lemma finite_row_power {X : Type*} [Fintype X] [Nonempty X]
+    (p : X → ℝ) (hp : ∀ x, 0 ≤ p x) (a : ℝ) :
+    finiteMean (fun x => ((Fintype.card X:ℝ)*p x)^(1+a)) =
+      (Fintype.card X:ℝ)^a * ∑ x, (p x)^(1+a) := by
+  have hn : (0:ℝ) < Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
+  simp only [finiteMean,Real.mul_rpow (le_of_lt hn) (hp _),←Finset.mul_sum]
+  rw [Real.rpow_add hn,Real.rpow_one]
+  field_simp
+
+noncomputable def palindromeRowMoment (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (a : ℝ) : ℝ :=
+  finiteMean (fun f : ι ↪ Card d =>
+    ((Fintype.card (ι ↪ Card d):ℝ)*PairRouting.tupleProbability (palindromePerm d) e f)^(1+a))
+
+lemma density_normalization (d h n : ℕ) (a : ℝ) :
+    (n:ℝ)^a * (2:ℝ)^(-(d:ℝ)*h*a) = ((n:ℝ)/((2:ℝ)^d)^h)^a := by
+  rw [Real.div_rpow (Nat.cast_nonneg _) (by positivity),←pow_mul,
+    ←Real.rpow_natCast,←Real.rpow_mul (by norm_num : (0:ℝ) ≤ 2),Nat.cast_mul]
+  rw [show -(d:ℝ)*h*a = -((d:ℝ)*h*a) by ring,
+    Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2),div_eq_mul_inv]
+
+theorem palindrome_row_density_moment (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (a : ℝ) (ha : 0 ≤ a) :
+    palindromeRowMoment d e a ≤
+      (((2^d).descFactorial (Fintype.card ι):ℝ)/((2^d:ℕ):ℝ)^(Fintype.card ι))^a *
+        finiteMean (fun ω : BenesCoins d => (2:ℝ)^((palindromeCost d e ω:ℝ)*a)) := by
+  classical
+  have : Nonempty (ι ↪ Card d) := ⟨e⟩
+  rw [palindromeRowMoment,finite_row_power _ (fun f => PairRouting.tupleProbability_nonneg _ _ _)]
+  calc
+    _ ≤ (Fintype.card (ι ↪ Card d):ℝ)^a *
+      ∑ f : ι ↪ Card d, (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) * routingMoment d e f a :=
+      mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun f _ => palindrome_pointwise_moment d e f a ha))
+        (Real.rpow_nonneg (Nat.cast_nonneg _) _)
+    _ = _ := by
+      rw [←Finset.mul_sum,routingMoment_sum,←mul_assoc,density_normalization,
+        Fintype.card_embedding_eq,card_positions]
+      norm_cast
+
+end Thorp
+namespace Thorp
+
+lemma subset_biUnion_indicator {α β : Type*} [DecidableEq α] [Fintype β]
+    (A : Finset α) (T : β → Finset α) :
+    (if Finset.univ.biUnion T ⊆ A then (1:ℝ) else 0) =
+      ∏ b, if T b ⊆ A then (1:ℝ) else 0 := by
+  classical
+  by_cases h : Finset.univ.biUnion T ⊆ A
+  · rw [ite_eq_left h]
+    symm
+    apply Finset.prod_eq_one
+    intro b _
+    rw [ite_eq_left]
+    exact (Finset.subset_biUnion_of_mem T (Finset.mem_univ b)).trans h
+  · rw [ite_eq_right h]
+    obtain ⟨x,hx,hxA⟩ := Finset.not_subset.mp h
+    obtain ⟨b,_,hxb⟩ := Finset.mem_biUnion.mp hx
+    symm
+    apply Finset.prod_eq_zero (Finset.mem_univ b)
+    rw [ite_eq_right]
+    exact fun hb => hxA (hb hxb)
+
+lemma cylinder_block_polynomial {α β Ω : Type*} [DecidableEq α] [Fintype β]
+    [Fintype Ω] (τ : β → Type*) [∀ b, Fintype (τ b)]
+    (F : Ω → Finset α) (p : α → ℝ)
+    (hF : CylinderBound (inclusionMoment F) p)
+    (T : ∀ b, τ b → Finset α) (c : ∀ b, τ b → ℝ)
+    (hc : ∀ b s, 0 ≤ c b s)
+    (hd : ∀ b b', b ≠ b' → ∀ s s', Disjoint (T b s) (T b' s')) :
+    finiteMean (fun ω => ∏ b, ∑ s, c b s * if T b s ⊆ F ω then 1 else 0) ≤
+      ∏ b, ∑ s, c b s * ∏ x ∈ T b s, p x := by
+  classical
+  simp_rw [Fintype.prod_sum]
+  rw [finiteMean_sum]
+  apply Finset.sum_le_sum
+  intro s _
+  let U := Finset.univ.biUnion (fun b => T b (s b))
+  have he : (fun ω => ∏ b, c b (s b) * if T b (s b) ⊆ F ω then 1 else 0) =
+      fun ω => (if U ⊆ F ω then 1 else 0) * ∏ b, c b (s b) := by
+    funext ω
+    rw [Finset.prod_mul_distrib,←subset_biUnion_indicator]
+    exact mul_comm _ _
+  rw [he,finiteMean_mul_const]
+  have hdis : ((Finset.univ : Finset β) : Set β).PairwiseDisjoint (fun b => T b (s b)) := by
+    intro b _ b' _ hbb
+    exact hd b b' hbb (s b) (s b')
+  have hmean : finiteMean (fun ω => if U ⊆ F ω then 1 else 0) = inclusionMoment F U := by
+    unfold inclusionMoment
+    apply finiteMean_congr
+    intro ω
+    by_cases hh : U ⊆ F ω <;> simp [hh]
+  calc
+    _ ≤ (∏ x ∈ U, p x) * ∏ b, c b (s b) :=
+      mul_le_mul_of_nonneg_right (hmean.le.trans (hF U)) (Finset.prod_nonneg (fun b _ => hc b (s b)))
+    _ = _ := by
+      dsimp only [U]
+      rw [Finset.prod_biUnion hdis,←Finset.prod_mul_distrib]
+      apply Finset.prod_congr rfl
+      intro b _
+      exact mul_comm _ _
+
+
+noncomputable def sparseCoefficient {α : Type*} (w : ℝ) (T : Finset α) : ℝ :=
+  if T.card = 1 then 0 else w^T.card
+
+lemma sparseCoefficient_nonneg {α : Type*} (w : ℝ) (hw : 0 ≤ w) (T : Finset α) :
+    0 ≤ sparseCoefficient w T := by
+  unfold sparseCoefficient
+  split_ifs <;> positivity
+
+lemma sparse_block_pointwise {α : Type*} [DecidableEq α] (A D : Finset α)
+    (w : ℝ) (hw : 0 ≤ w) :
+    w^(if 2 ≤ (D ∩ A).card then (D ∩ A).card else 0) ≤
+      ∑ T : ↑D.powerset, sparseCoefficient w T.val * if T.val ⊆ A then 1 else 0 := by
+  classical
+  let V := D ∩ A
+  let T : ↑D.powerset := if 2 ≤ V.card then
+    ⟨V,Finset.mem_powerset.mpr Finset.inter_subset_left⟩ else
+    ⟨∅,Finset.mem_powerset.mpr (Finset.empty_subset D)⟩
+  calc
+    _ = sparseCoefficient w T.val * (if T.val ⊆ A then 1 else 0) := by
+      by_cases h : 2 ≤ V.card
+      · have hn : V.card ≠ 1 := by omega
+        simp [T,ite_eq_left h,sparseCoefficient,hn,show V ⊆ A from Finset.inter_subset_right,V]
+      · simp [T,ite_eq_right h,sparseCoefficient,V] at *
+    _ ≤ _ := Finset.single_le_sum (f := fun S : ↑D.powerset =>
+      sparseCoefficient w S.val * (if S.val ⊆ A then 1 else 0))
+      (fun S _ => mul_nonneg (sparseCoefficient_nonneg w hw S.val)
+        (by split_ifs <;> norm_num)) (Finset.mem_univ T)
+
+lemma geometric_sum_half (n : ℕ) {x : ℝ} (hx : 0 ≤ x) (hhalf : x ≤ 1/2) :
+    ∑ i ∈ Finset.range n, x^i ≤ 2 := by
+  induction n with
+  | zero => simp
+  | succ n ih =>
+    rw [Finset.sum_range_succ',pow_zero]
+    simp only [pow_succ',←Finset.mul_sum]
+    nlinarith
+
+lemma sparse_binomial_bound (R : ℕ) (z : ℝ) (hz : 0 ≤ z) (hr : (R:ℝ)*z ≤ 1/2) :
+    ∑ i ∈ Finset.range (R+1), (R.choose i:ℝ) * (if i = 1 then 0 else z^i) ≤
+      1 + 2*((R:ℝ)*z)^2 := by
+  classical
+  let x := (R:ℝ)*z
+  have hx : 0 ≤ x := mul_nonneg (Nat.cast_nonneg _) hz
+  have hbound (i : ℕ) : (R.choose i:ℝ)*(if i=1 then 0 else z^i) ≤
+      if i=1 then 0 else x^i := by
+    by_cases h : i=1
+    · simp [h]
+    · simp only [ite_eq_right h]
+      dsimp [x]
+      rw [mul_pow]
+      apply mul_le_mul_of_nonneg_right _ (pow_nonneg hz i)
+      exact_mod_cast Nat.choose_le_pow R i
+  calc
+    _ ≤ ∑ i ∈ Finset.range (R+1), if i=1 then 0 else x^i :=
+      Finset.sum_le_sum (fun i _ => hbound i)
+    _ ≤ 1+2*x^2 := by
+      rcases R with _ | R
+      · simp [x]
+      rcases R with _ | R
+      · norm_num [Finset.sum_range_succ]
+        positivity
+      have he : ∑ i ∈ Finset.range (R+2+1), (if i=1 then 0 else x^i) =
+          1 + x^2*∑ i ∈ Finset.range (R+1), x^i := by
+        rw [show R+2+1=2+(R+1) by omega,Finset.sum_range_add]
+        have hn (i : ℕ) : 2+i ≠ 1 := by omega
+        simp only [hn,ite_false,pow_add,←Finset.mul_sum]
+        norm_num [Finset.sum_range_succ]
+      rw [he]
+      have hg := geometric_sum_half (R+1) hx hr
+      nlinarith [sq_nonneg x]
+
+lemma sparse_block_product_bound {α : Type*} [DecidableEq α]
+    (D : Finset α) (p w : ℝ) (hp : 0 ≤ p) (hw : 0 ≤ w)
+    (hr : (D.card:ℝ)*(p*w) ≤ 1/2) :
+    ∑ T : ↑D.powerset, sparseCoefficient w T.val * p^T.val.card ≤
+      Real.exp (2*((D.card:ℝ)*(p*w))^2) := by
+  classical
+  have he : ∑ T : ↑D.powerset, sparseCoefficient w T.val * p^T.val.card =
+      ∑ i ∈ Finset.range (D.card+1), (D.card.choose i:ℝ)*(if i=1 then 0 else (p*w)^i) := by
+    rw [Finset.sum_coe_sort D.powerset (fun T => sparseCoefficient w T*p^T.card)]
+    have hh : (fun T : Finset α => sparseCoefficient w T*p^T.card) =
+        fun T => if T.card=1 then 0 else (p*w)^T.card := by
+      funext T
+      unfold sparseCoefficient
+      split_ifs <;> simp [mul_pow,mul_comm]
+    change (∑ T ∈ D.powerset, (fun T : Finset α => sparseCoefficient w T*p^T.card) T) = _
+    rw [hh,Finset.sum_powerset_apply_card (fun i => if i=1 then (0:ℝ) else (p*w)^i)]
+    simp only [nsmul_eq_mul]
+  rw [he]
+  exact (sparse_binomial_bound D.card (p*w) (mul_nonneg hp hw) hr).trans
+    (by simpa only [add_comm] using Real.add_one_le_exp (2*((D.card:ℝ)*(p*w))^2))
+
+
+noncomputable def sparseMass {α β : Type*} [DecidableEq α] [Fintype β]
+    (D : β → Finset α) (A : Finset α) : ℕ :=
+  ∑ b, if 2 ≤ (D b ∩ A).card then (D b ∩ A).card else 0
+
+theorem sparse_block_mgf {α β Ω : Type*} [DecidableEq α] [Fintype β] [Fintype Ω]
+    (D : β → Finset α) (hd : Pairwise (fun b b' => Disjoint (D b) (D b')))
+    (F : Ω → Finset α) (p w : ℝ) (hp : 0 ≤ p) (hw : 0 ≤ w)
+    (hF : ∀ T, inclusionMoment F T ≤ p^T.card)
+    (hr : ∀ b, ((D b).card:ℝ)*(p*w) ≤ 1/2) :
+    finiteMean (fun ω => w^(sparseMass D (F ω))) ≤
+      Real.exp (∑ b, 2*(((D b).card:ℝ)*(p*w))^2) := by
+  classical
+  have hc : CylinderBound (inclusionMoment F) (fun _ => p) := by
+    intro T
+    simpa only [Finset.prod_const] using hF T
+  have hb := cylinder_block_polynomial (fun b => ↑(D b).powerset) F (fun _ => p) hc
+    (fun _ S => S.val) (fun _ S => sparseCoefficient w S.val)
+    (fun _ S => sparseCoefficient_nonneg w hw S.val)
+    (fun b b' h S S' => (hd h).mono
+      (Finset.mem_powerset.mp S.property) (Finset.mem_powerset.mp S'.property))
+  calc
+    _ ≤ finiteMean (fun ω => ∏ b, ∑ S : ↑(D b).powerset,
+        sparseCoefficient w S.val * if S.val ⊆ F ω then 1 else 0) := by
+      apply finiteMean_mono
+      intro ω
+      rw [sparseMass,←Finset.prod_pow_eq_pow_sum]
+      apply Finset.prod_le_prod₀ (fun _ _ => pow_nonneg hw _)
+      intro b _
+      exact sparse_block_pointwise (F ω) (D b) w hw
+    _ ≤ ∏ b, ∑ S : ↑(D b).powerset, sparseCoefficient w S.val * p^S.val.card := by
+      simpa only [Finset.prod_const] using hb
+    _ ≤ _ := by
+      rw [Real.exp_sum]
+      apply Finset.prod_le_prod₀
+      · intro b _
+        exact Finset.sum_nonneg (fun S _ => mul_nonneg
+          (sparseCoefficient_nonneg w hw S.val) (pow_nonneg hp _))
+      · intro b _
+        exact sparse_block_product_bound (D b) p w hp hw (hr b)
+
+theorem inverseButterfly_sparse_mgf (d : ℕ) {β : Type*} [Fintype β]
+    (D : β → Finset (Card d)) (hd : Pairwise (fun b b' => Disjoint (D b) (D b')))
+    (A : Finset (Card d)) (w : ℝ) (hw : 0 ≤ w)
+    (hr : ∀ b, ((D b).card:ℝ)*((A.card/(2:ℝ)^d)*w) ≤ 1/2) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      w^(sparseMass D (inverseButterflyMarks d A ω))) ≤
+      Real.exp (∑ b, 2*(((D b).card:ℝ)*((A.card/(2:ℝ)^d)*w))^2) := by
+  apply sparse_block_mgf D hd (inverseButterflyMarks d A) (A.card/(2:ℝ)^d) w
+    (by positivity) hw
+  · exact inverseButterflyMarks_inclusion d A
+  · exact hr
+
+
+def crowdedMass (h : ℕ) : ℕ := if 2 ≤ h then h else 0
+
+lemma crowdedMass_le (h : ℕ) : crowdedMass h ≤ h := by
+  unfold crowdedMass
+  split_ifs <;> omega
+
+lemma crowdedMass_add (u v : ℕ) : crowdedMass u + crowdedMass v ≤ crowdedMass (u+v) := by
+  unfold crowdedMass
+  split_ifs <;> omega
+
+lemma alternatingCycles_le_crowded {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
+    (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    PairRouting.alternatingCycles e p ≤ crowdedMass (Fintype.card ι) := by
+  unfold crowdedMass
+  split_ifs with h
+  · have hh := PairRouting.alternatingCycles_double e p
+    omega
+  · rw [PairRouting.alternatingCycles_zero e p (by omega)]
+
+lemma palindromeCost_le_crowded (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) :
+    palindromeCost d e ω ≤ d*crowdedMass (Fintype.card ι) := by
+  classical
+  induction d generalizing ι with
+  | zero => simp [palindromeCost]
+  | succ d ih =>
+    let σ := benesStepEquiv d ω
+    let x := e.trans (headTailEquiv d).toEmbedding
+    let c := PairRouting.colors x σ.2.1
+    let hc := PairRouting.colors_compatible x σ.2.1
+    have h₀ := ih (PairRouting.childEmbedding x c hc false) σ.1.1
+    have h₁ := ih (PairRouting.childEmbedding x c hc true) σ.1.2
+    have hz := alternatingCycles_le_crowded (PairRouting.switchedEmbedding x σ.2.1)
+      (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1))
+    have hs := crowdedMass_add (Fintype.card {i // c i = false})
+      (Fintype.card {i // c i = true})
+    rw [PairRouting.color_card_add] at hs
+    change PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
+        (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) +
+        palindromeCost d (PairRouting.childEmbedding x c hc false) σ.1.1 +
+        palindromeCost d (PairRouting.childEmbedding x c hc true) σ.1.2 ≤ _
+    nlinarith
+
+noncomputable def palindromeLowCost (H : ℕ) : (d : ℕ) → {ι : Type*} → [Fintype ι] →
+    (ι ↪ Card d) → BenesCoins d → ℕ
+  | 0, _, _, _, _ => 0
+  | d+1, _, _, e, ω =>
+      let σ := benesStepEquiv d ω
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x σ.2.1
+      let hc := PairRouting.colors_compatible x σ.2.1
+      (if d+1 ≤ H then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
+          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) else 0) +
+        palindromeLowCost H d (PairRouting.childEmbedding x c hc false) σ.1.1 +
+        palindromeLowCost H d (PairRouting.childEmbedding x c hc true) σ.1.2
+
+noncomputable def cutoffMass (H : ℕ) : (d : ℕ) → {ι : Type*} → [Fintype ι] →
+    (ι ↪ Card d) → BenesCoins d → ℕ
+  | 0, _, _, _, _ => 0
+  | d+1, ι, _, e, ω =>
+      if d+1 ≤ H then crowdedMass (Fintype.card ι) else
+      let σ := benesStepEquiv d ω
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x σ.2.1
+      let hc := PairRouting.colors_compatible x σ.2.1
+      cutoffMass H d (PairRouting.childEmbedding x c hc false) σ.1.1 +
+        cutoffMass H d (PairRouting.childEmbedding x c hc true) σ.1.2
+
+lemma palindromeLowCost_eq (H d : ℕ) (h : d ≤ H) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) :
+    palindromeLowCost H d e ω = palindromeCost d e ω := by
+  classical
+  induction d generalizing ι with
+  | zero => rfl
+  | succ d ih =>
+    rw [palindromeLowCost,palindromeCost,ite_eq_left h]
+    congr 2 <;> exact ih (by omega) _ _
+
+lemma palindromeLowCost_le_cutoff (H d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) :
+    palindromeLowCost H d e ω ≤ H*cutoffMass H d e ω := by
+  classical
+  induction d generalizing ι with
+  | zero => simp [palindromeLowCost,cutoffMass]
+  | succ d ih =>
+    by_cases h : d+1 ≤ H
+    · rw [palindromeLowCost_eq H (d+1) h,cutoffMass,ite_eq_left h]
+      exact (palindromeCost_le_crowded (d+1) e ω).trans (Nat.mul_le_mul_right _ h)
+    · rw [palindromeLowCost,cutoffMass,ite_eq_right h,ite_eq_right h,zero_add,Nat.mul_add]
+      exact Nat.add_le_add (ih _ _) (ih _ _)
+
+
+noncomputable def centralBlock (r H : ℕ) (b : Card r) : Finset (Card (H+r)) :=
+  Finset.univ.image (fun z : Card H => (cardSplit r H).symm (b,z))
+
+end Thorp
+
+end ThorpNine.Smoothing
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Smoothing/Coins.lean b/lean/OAI/Probability/ThorpRouting/Smoothing/Coins.lean
index b6d239e6e5e5e68f2c275333babc1a1f148f300c..64c1936bbcbc7cfa2f85039e5c1018a41fb5c731 100644
--- a/lean/OAI/Probability/ThorpRouting/Smoothing/Coins.lean
+++ b/lean/OAI/Probability/ThorpRouting/Smoothing/Coins.lean
@@ -1,776 +1,776 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Smoothing.Model
-
-namespace OAI
-
-namespace ThorpNine.Smoothing
-
-namespace Thorp
-
-open scoped BigOperators
-open Filter
-
-def Coins : ℕ → Type
-  | 0 => Unit
-  | d + 1 => Card d → Bool
-
-instance (d : ℕ) : Fintype (Coins d) := by
-  cases d <;> dsimp [Coins] <;> infer_instance
-
-instance (d : ℕ) : Nonempty (Coins d) := by
-  cases d <;> dsimp [Coins] <;> infer_instance
-
-def rotate (d : ℕ) : Equiv.Perm (Card d) where
-  toFun x i := x (finRotate d i)
-  invFun x i := x ((finRotate d).symm i)
-  left_inv x := by funext i; simp only [Equiv.apply_symm_apply]
-  right_inv x := by funext i; simp only [Equiv.symm_apply_apply]
-
-def step : (d : ℕ) → Coins d → Equiv.Perm (Card d)
-  | 0, _ => 1
-  | d + 1, ξ => rotate (d + 1) * pairSwitch ξ
-
-def run (d : ℕ) : (t : ℕ) → (Fin t → Coins d) → Equiv.Perm (Card d)
-  | 0, _ => 1
-  | t + 1, ω => step d (ω (Fin.last t)) * run d t (fun i => ω i.castSucc)
-
-noncomputable def law (d t : ℕ) (g : Equiv.Perm (Card d)) : ℝ :=
-  (Fintype.card {ω : Fin t → Coins d // run d t ω = g} : ℝ) /
-    (Fintype.card (Fin t → Coins d) : ℝ)
-
-noncomputable def distance (d t : ℕ) : ℝ :=
-  (1 / 2 : ℝ) * ∑ g : Equiv.Perm (Card d),
-    |law d t g - 1 / (Nat.factorial (2 ^ d) : ℝ)|
-
-noncomputable def mixingTime (d : ℕ) : ℕ :=
-  sInf {t : ℕ | distance d t ≤ (1 / 4 : ℝ)}
-
-noncomputable def supportTimeBound (d : ℕ) : ℤ :=
-  ⌈(2 / (2 : ℝ) ^ d) *
-    (Real.log ((3 / 4 : ℝ) * (Nat.factorial (2 ^ d) : ℝ)) / Real.log 2)⌉
-
-def OptimalOrder : Prop :=
-  ∃ C : ℕ, 0 < C ∧
-    Tendsto (fun d : ℕ => distance d (C * d)) atTop (nhds 0) ∧
-    (∀ d t : ℕ, 1 - (2 : ℝ) ^ ((t : ℝ) * (2 : ℝ) ^ d / 2) /
-      (Nat.factorial (2 ^ d) : ℝ) ≤ distance d t) ∧
-    (∀ᶠ d : ℕ in atTop,
-      supportTimeBound d ≤ (mixingTime d : ℤ) ∧ mixingTime d ≤ C * d) ∧
-    (∃ K : ℝ, ∀ᶠ d : ℕ in atTop, 2 * (d : ℝ) - K ≤ (mixingTime d : ℝ)) ∧
-    Asymptotics.IsTheta atTop (fun d : ℕ => (mixingTime d : ℝ)) (fun d : ℕ => (d : ℝ))
-
-noncomputable def finiteLaw {α β : Type*} [Fintype α] (f : α → β) (b : β) : ℝ := by
-  classical
-  exact (Fintype.card {a : α // f a = b} : ℝ) / (Fintype.card α : ℝ)
-
-lemma sum_card_fibers {α β : Type*} [Fintype α] [Fintype β] [DecidableEq β] (f : α → β) :
-    ∑ b, Fintype.card {a : α // f a = b} = Fintype.card α := by
-  classical
-  rw [← Fintype.card_sigma]
-  exact Fintype.card_congr (Equiv.sigmaFiberEquiv f)
-
-lemma finiteLaw_nonneg {α β : Type*} [Fintype α] (f : α → β) (b : β) :
-    0 ≤ finiteLaw f b := by
-  exact div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _)
-
-lemma finiteLaw_sum {α β : Type*} [Fintype α] [Nonempty α] [Fintype β] (f : α → β) :
-    ∑ b, finiteLaw f b = 1 := by
-  classical
-  simp only [finiteLaw, ← Finset.sum_div]
-  have hc : (∑ b, (Fintype.card {a : α // f a = b} : ℝ)) = (Fintype.card α : ℝ) := by
-    exact_mod_cast sum_card_fibers f
-  rw [hc]
-  exact div_self (Nat.cast_ne_zero.mpr Fintype.card_ne_zero)
-
-lemma finiteLaw_eq_zero_of_not_mem {α β : Type*} [Fintype α] [DecidableEq β]
-    (f : α → β) {b : β} (hb : b ∉ Finset.univ.image f) : finiteLaw f b = 0 := by
-  classical
-  have : IsEmpty {a : α // f a = b} := ⟨fun a => hb (Finset.mem_image.mpr
-    ⟨a.val, Finset.mem_univ _, a.property⟩)⟩
-  simp [finiteLaw]
-
-lemma event_le_totalVariation {β : Type*} [Fintype β] [DecidableEq β]
-    (p q : β → ℝ) (hpq : ∑ b, (p b - q b) = 0) (s : Finset β) :
-    (∑ b ∈ s, p b) - (∑ b ∈ s, q b) ≤ (1 / 2 : ℝ) * ∑ b, |p b - q b| := by
-  have h₁ : (∑ b ∈ s, (p b - q b)) ≤ ∑ b ∈ s, |p b - q b| :=
-    Finset.sum_le_sum (fun b _ => le_abs_self (p b - q b))
-  have h₂ : -(∑ b ∈ sᶜ, (p b - q b)) ≤ ∑ b ∈ sᶜ, |p b - q b| := by
-    rw [← Finset.sum_neg_distrib]
-    exact Finset.sum_le_sum (fun b _ => neg_le_abs (p b - q b))
-  have hz := Finset.sum_add_sum_compl s (fun b => p b - q b)
-  have ha := Finset.sum_add_sum_compl s (fun b => |p b - q b|)
-  rw [hpq] at hz
-  rw [← Finset.sum_sub_distrib]
-  linarith
-
-lemma finiteLaw_support_bound {α β : Type*} [Fintype α] [Nonempty α]
-    [Fintype β] [Nonempty β] (f : α → β) :
-    1 - (Fintype.card α : ℝ) / (Fintype.card β : ℝ) ≤
-      (1 / 2 : ℝ) * ∑ b, |finiteLaw f b - 1 / (Fintype.card β : ℝ)| := by
-  classical
-  let s := Finset.univ.image f
-  have hs : ∑ b ∈ s, finiteLaw f b = 1 := by
-    have h := Finset.sum_subset (Finset.subset_univ s)
-      (fun b _ hb => finiteLaw_eq_zero_of_not_mem f hb)
-    simpa only [finiteLaw_sum] using h
-  have hβ : (Fintype.card β : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
-  have hq : (∑ _b : β, 1 / (Fintype.card β : ℝ)) = 1 := by simp [hβ]
-  have hd : (∑ b, (finiteLaw f b - 1 / (Fintype.card β : ℝ))) = 0 := by
-    rw [Finset.sum_sub_distrib, finiteLaw_sum, hq, sub_self]
-  have he := event_le_totalVariation (finiteLaw f) (fun _ => 1 / (Fintype.card β : ℝ)) hd s
-  rw [hs] at he
-  have hsq : (∑ _b ∈ s, 1 / (Fintype.card β : ℝ)) = (s.card : ℝ) / Fintype.card β := by
-    simp [div_eq_mul_inv]
-  rw [hsq] at he
-  have hcard : (s.card : ℝ) ≤ Fintype.card α := by
-    exact_mod_cast (Finset.card_image_le (s := (Finset.univ : Finset α)) (f := f))
-  have hdiv := div_le_div_of_nonneg_right hcard (Nat.cast_nonneg (Fintype.card β) : (0 : ℝ) ≤ _)
-  linarith
-
-lemma card_positions (d : ℕ) : Fintype.card (Card d) = 2 ^ d := by
-  simp [Card]
-
-lemma card_permutations (d : ℕ) : Fintype.card (Equiv.Perm (Card d)) = Nat.factorial (2 ^ d) := by
-  rw [Fintype.card_perm, card_positions]
-
-lemma card_coins_succ (d : ℕ) : Fintype.card (Coins (d + 1)) = 2 ^ (2 ^ d) := by
-  simp [Coins]
-
-lemma card_tapes_succ (d t : ℕ) : Fintype.card (Fin t → Coins (d + 1)) = 2 ^ (t * 2 ^ d) := by
-  rw [Fintype.card_fun, card_coins_succ, Fintype.card_fin, ← pow_mul, Nat.mul_comm]
-
-lemma law_eq_finiteLaw (d t : ℕ) (g : Equiv.Perm (Card d)) :
-    law d t g = finiteLaw (run d t) g := by
-  unfold law finiteLaw
-  apply congrArg (fun n : ℕ => (n : ℝ) / (Fintype.card (Fin t → Coins d) : ℝ))
-  exact congrArg (@Fintype.card _) (Subsingleton.elim _ _)
-
-lemma law_nonneg (d t : ℕ) (g : Equiv.Perm (Card d)) : 0 ≤ law d t g := by
-  rw [law_eq_finiteLaw]
-  exact finiteLaw_nonneg (run d t) g
-
-lemma law_sum (d t : ℕ) : ∑ g, law d t g = 1 := by
-  simp only [law_eq_finiteLaw]
-  exact finiteLaw_sum (run d t)
-
-lemma physical_support_bound_succ (d t : ℕ) :
-    1 - (2 : ℝ) ^ (t * 2 ^ d) / (Nat.factorial (2 ^ (d + 1)) : ℝ) ≤ distance (d + 1) t := by
-  have : Nonempty (Equiv.Perm (Card (d + 1))) := ⟨1⟩
-  have h := finiteLaw_support_bound (run (d + 1) t)
-  rw [card_tapes_succ, card_permutations] at h
-  simpa only [Nat.cast_pow, Nat.cast_ofNat, distance, law_eq_finiteLaw] using h
-
-lemma step_succ_formula (d : ℕ) (ξ : Coins (d + 1)) (x : Card (d + 1)) :
-    step (d + 1) ξ x = Fin.snoc (Fin.tail x) (Bool.xor (x 0) (ξ (Fin.tail x))) := by
-  change (fun i => switchFun ξ x (finRotate (d + 1) i)) = _
-  rw [Fin.snoc_eq_cons_rotate]
-  rfl
-
-lemma distance_nonneg (d t : ℕ) : 0 ≤ distance d t := by
-  unfold distance
-  exact mul_nonneg (by norm_num) (Finset.sum_nonneg (fun _ _ => abs_nonneg _))
-
-lemma physical_support_bound (d t : ℕ) :
-    1 - (2 : ℝ) ^ ((t : ℝ) * (2 : ℝ) ^ d / 2) /
-      (Nat.factorial (2 ^ d) : ℝ) ≤ distance d t := by
-  cases d with
-  | zero =>
-    have h : 1 ≤ (2 : ℝ) ^ ((t : ℝ) / 2) := Real.one_le_rpow (by norm_num) (by positivity)
-    simpa only [pow_zero, Nat.factorial_one, Nat.cast_one, div_one, mul_one] using
-      le_trans (sub_nonpos.mpr h) (distance_nonneg 0 t)
-  | succ d =>
-    have he : (t : ℝ) * (2 : ℝ) ^ (d + 1) / 2 = (t * 2 ^ d : ℕ) := by
-      push_cast
-      rw [pow_succ]
-      ring
-    rw [he, Real.rpow_natCast]
-    exact physical_support_bound_succ d t
-
-lemma supportTimeBound_le_of_distance_le (d t : ℕ) (ht : distance d t ≤ (1 / 4 : ℝ)) :
-    supportTimeBound d ≤ (t : ℤ) := by
-  have hF : (0 : ℝ) < Nat.factorial (2 ^ d) := by positivity
-  have hN : (0 : ℝ) < (2 : ℝ) ^ d := by positivity
-  have hL : (0 : ℝ) < Real.log 2 := Real.log_pos (by norm_num)
-  have hb := physical_support_bound d t
-  have hp : (3 / 4 : ℝ) ≤
-      (2 : ℝ) ^ ((t : ℝ) * (2 : ℝ) ^ d / 2) / (Nat.factorial (2 ^ d) : ℝ) := by
-    linarith
-  have hr := (le_div_iff₀ hF).mp hp
-  have hlog := Real.log_le_log (show (0 : ℝ) < (3 / 4 : ℝ) * Nat.factorial (2 ^ d) by positivity) hr
-  rw [Real.log_rpow (by norm_num : (0 : ℝ) < 2)] at hlog
-  unfold supportTimeBound
-  apply Int.ceil_le.mpr
-  simp only [Int.cast_natCast]
-  calc
-    2 / (2 : ℝ) ^ d * (Real.log ((3 / 4 : ℝ) * Nat.factorial (2 ^ d)) / Real.log 2) =
-        Real.log ((3 / 4 : ℝ) * Nat.factorial (2 ^ d)) / ((2 : ℝ) ^ d / 2 * Real.log 2) := by
-      field_simp
-    _ ≤ (t : ℝ) := (div_le_iff₀ (mul_pos (div_pos hN (by norm_num)) hL)).mpr (by
-      nlinarith [hlog])
-
-lemma supportTimeBound_le_mixingTime {d : ℕ}
-    (h : ({t : ℕ | distance d t ≤ (1 / 4 : ℝ)} : Set ℕ).Nonempty) :
-    supportTimeBound d ≤ (mixingTime d : ℤ) := by
-  exact supportTimeBound_le_of_distance_le d (mixingTime d) (Nat.sInf_mem h)
-
-lemma log_factorial_lower {n : ℕ} (hn : 1 ≤ n) :
-    (n : ℝ) * Real.log n - n ≤ Real.log (Nat.factorial n) := by
-  have h := Stirling.le_log_factorial_stirling (show n ≠ 0 by omega)
-  have hln : 0 ≤ Real.log (n : ℝ) := Real.log_nonneg (by exact_mod_cast hn)
-  have hlp : 0 ≤ Real.log (2 * Real.pi) := Real.log_nonneg (by
-    have := Real.pi_gt_three
-    linarith)
-  linarith
-
-lemma log_three_quarters_lower : -(1 : ℝ) ≤ Real.log (3 / 4 : ℝ) := by
-  have h := Real.log_le_sub_one_of_pos (show (0 : ℝ) < 4 / 3 by norm_num)
-  have he : Real.log (4 / 3 : ℝ) = - Real.log (3 / 4 : ℝ) := by
-    rw [← Real.log_inv]
-    congr 1
-    norm_num
-  rw [he] at h
-  linarith
-
-lemma linear_le_supportTimeBound (d : ℕ) :
-    2 * (d : ℝ) - 4 / Real.log 2 ≤ (supportTimeBound d : ℝ) := by
-  have hN : 1 ≤ (2 : ℝ) ^ d := one_le_pow₀ (by norm_num)
-  have hNp : 0 < (2 : ℝ) ^ d := by positivity
-  have hL : 0 < Real.log 2 := Real.log_pos (by norm_num)
-  have hf := log_factorial_lower (show 1 ≤ 2 ^ d by exact one_le_pow₀ (by norm_num))
-  have hlog : (2 : ℝ) ^ d * ((d : ℝ) * Real.log 2) - (2 : ℝ) ^ d - 1 ≤
-      Real.log ((3 / 4 : ℝ) * (Nat.factorial (2 ^ d) : ℝ)) := by
-    rw [Real.log_mul (by norm_num) (by positivity)]
-    push_cast at hf
-    rw [Real.log_pow] at hf
-    linarith [log_three_quarters_lower]
-  apply le_trans _ (Int.le_ceil _)
-  have hmul : (2 * (d : ℝ) - 4 / Real.log 2) *
-      ((2 : ℝ) ^ d / 2 * Real.log 2) ≤
-      Real.log ((3 / 4 : ℝ) * (Nat.factorial (2 ^ d) : ℝ)) := by
-    have hcancel : (2 * (d : ℝ) - 4 / Real.log 2) *
-        ((2 : ℝ) ^ d / 2 * Real.log 2) =
-        (2 : ℝ) ^ d * ((d : ℝ) * Real.log 2) - 2 * (2 : ℝ) ^ d := by
-      field_simp
-      ring
-    rw [hcancel]
-    linarith
-  convert (le_div_iff₀ (mul_pos (div_pos hNp (by norm_num)) hL)).mpr hmul using 1
-  field_simp
-
-lemma linear_le_time_of_distance_le (d t : ℕ)
-    (ht : distance d t ≤ (1 / 4 : ℝ)) :
-    2 * (d : ℝ) - 4 / Real.log 2 ≤ (t : ℝ) := by
-  have h := supportTimeBound_le_of_distance_le d t ht
-  have hr : (supportTimeBound d : ℝ) ≤ (t : ℝ) := by exact_mod_cast h
-  exact (linear_le_supportTimeBound d).trans hr
-
-lemma entropy_chord (p : ℝ) (hp : 0 ≤ p) (hq : p ≤ 1 / 2) :
-    2 * p * Real.log 2 ≤ Real.binEntropy p := by
-  have hc := Real.strictConcave_binEntropy.concaveOn.2
-    (show (0 : ℝ) ∈ Set.Icc (0 : ℝ) 1 by norm_num)
-    (show (2 : ℝ)⁻¹ ∈ Set.Icc (0 : ℝ) 1 by norm_num)
-    (show 0 ≤ 1 - 2 * p by linarith)
-    (show 0 ≤ 2 * p by positivity)
-    (show (1 - 2 * p) + 2 * p = 1 by ring)
-  simpa only [Real.binEntropy_zero, Real.binEntropy_two_inv,
-    smul_eq_mul, mul_zero, zero_add, show 2 * p * (2 : ℝ)⁻¹ = p by ring] using hc
-
-lemma split_entropy_lower (a b : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) :
-    2 * min a b * Real.log 2 ≤
-      (a + b) * Real.log (a + b) - a * Real.log a - b * Real.log b := by
-  wlog hab : a ≤ b generalizing a b
-  · have hba := this b a hb ha (le_of_not_ge hab)
-    simpa only [min_comm, add_comm b a] using hba.trans_eq (show (b + a) * Real.log (b + a) - b * Real.log b - a * Real.log a =
-      (a + b) * Real.log (a + b) - a * Real.log a - b * Real.log b by rw [add_comm b a]; ring)
-  rw [min_eq_left hab]
-  by_cases hz : a = 0
-  · subst a
-    simp
-  have hap : 0 < a := lt_of_le_of_ne ha (Ne.symm hz)
-  have hbp : 0 < b := hap.trans_le hab
-  have hs : 0 < a + b := add_pos hap hbp
-  have hp : 0 ≤ a / (a + b) := div_nonneg ha hs.le
-  have hq : a / (a + b) ≤ 1 / 2 := (div_le_iff₀ hs).mpr (by linarith)
-  have hc := mul_le_mul_of_nonneg_left (entropy_chord (a / (a + b)) hp hq) hs.le
-  have he : (a + b) * Real.binEntropy (a / (a + b)) =
-      (a + b) * Real.log (a + b) - a * Real.log a - b * Real.log b := by
-    unfold Real.binEntropy
-    rw [show 1 - a / (a + b) = b / (a + b) by field_simp; ring]
-    simp only [Real.log_inv]
-    rw [Real.log_div (ne_of_gt hap) (ne_of_gt hs),
-      Real.log_div (ne_of_gt hbp) (ne_of_gt hs)]
-    field_simp
-    ring
-  rw [he] at hc
-  calc
-    2 * a * Real.log 2 = (a + b) * (2 * (a / (a + b)) * Real.log 2) := by
-      field_simp
-    _ ≤ _ := hc
-
-noncomputable def childMarks {d : ℕ} (s : Finset (Card (d + 1))) (ε : Bool) :
-    Finset (Card d) := (s.filter (fun x => x 0 = ε)).image Fin.tail
-
-lemma card_childMarks {d : ℕ} (s : Finset (Card (d + 1))) (ε : Bool) :
-    (childMarks s ε).card = (s.filter (fun x => x 0 = ε)).card := by
-  classical
-  unfold childMarks
-  apply Finset.card_image_iff.mpr
-  intro x hx y hy hxy
-  have hx0 := (Finset.mem_filter.mp hx).2
-  have hy0 := (Finset.mem_filter.mp hy).2
-  rw [← Fin.cons_self_tail x, ← Fin.cons_self_tail y, hx0, hy0, hxy]
-
-lemma card_childMarks_add {d : ℕ} (s : Finset (Card (d + 1))) :
-    (childMarks s false).card + (childMarks s true).card = s.card := by
-  classical
-  rw [card_childMarks, card_childMarks]
-  simpa only [Bool.not_eq_false] using
-    (Finset.card_filter_add_card_filter_not (s := s) (fun x => x 0 = false))
-
-noncomputable def topMeetings {d : ℕ} (b : Butterfly (d + 1))
-    (s : Finset (Card (d + 1))) : Finset (Card d) :=
-  ((childMarks s false).image (butterflyPerm d (b.2 false))) ∩
-    ((childMarks s true).image (butterflyPerm d (b.2 true)))
-
-lemma card_topMeetings_le {d : ℕ} (b : Butterfly (d + 1)) (s : Finset (Card (d + 1))) :
-    (topMeetings b s).card ≤ min (childMarks s false).card (childMarks s true).card := by
-  classical
-  apply le_min
-  · exact (Finset.card_le_card Finset.inter_subset_left).trans_eq
-      (Finset.card_image_of_injective _ (butterflyPerm d (b.2 false)).injective)
-  · exact (Finset.card_le_card Finset.inter_subset_right).trans_eq
-      (Finset.card_image_of_injective _ (butterflyPerm d (b.2 true)).injective)
-
-noncomputable def encounters : (d : ℕ) → Butterfly d → Finset (Card d) → ℕ
-  | 0, _, _ => 0
-  | d + 1, b, s => encounters d (b.2 false) (childMarks s false) +
-      encounters d (b.2 true) (childMarks s true) + (topMeetings b s).card
-
-lemma butterfly_encounters (d : ℕ) (b : Butterfly d) (s : Finset (Card d)) :
-    (encounters d b s : ℝ) ≤ (s.card : ℝ) * Real.log s.card / (2 * Real.log 2) := by
-  induction d with
-  | zero =>
-    simp only [encounters, Nat.cast_zero]
-    have hs : s.card ≤ 1 := by
-      simpa [Card] using Finset.card_le_univ s
-    interval_cases s.card <;> simp
-  | succ d ih =>
-    have h₀ := ih (b.2 false) (childMarks s false)
-    have h₁ := ih (b.2 true) (childMarks s true)
-    have ht : ((topMeetings b s).card : ℝ) ≤
-        min ((childMarks s false).card : ℝ) ((childMarks s true).card : ℝ) := by
-      exact_mod_cast card_topMeetings_le b s
-    have he := split_entropy_lower ((childMarks s false).card : ℝ)
-      ((childMarks s true).card : ℝ) (by positivity) (by positivity)
-    have hs : ((childMarks s false).card : ℝ) + (childMarks s true).card = s.card := by
-      exact_mod_cast card_childMarks_add s
-    rw [hs] at he
-    have hL : 0 < 2 * Real.log 2 := by positivity
-    rw [le_div_iff₀ hL] at h₀ h₁ ⊢
-    simp only [encounters, Nat.cast_add]
-    nlinarith [mul_le_mul_of_nonneg_right ht hL.le]
-
-lemma prod_split_at {α : Type*} [DecidableEq α] (s : Finset α) (f : α → ℝ) (a : α) :
-    ∏ i ∈ s, f i = (if a ∈ s then f a else 1) * ∏ i ∈ s.erase a, f i := by
-  by_cases ha : a ∈ s
-  · rw [ite_eq_left ha]
-    exact (Finset.mul_prod_erase s f ha).symm
-  · simp [ha]
-
-lemma prod_split_two {α : Type*} [DecidableEq α] (s : Finset α) (f : α → ℝ)
-    (a b : α) (hab : a ≠ b) :
-    ∏ i ∈ s, f i = (if a ∈ s then f a else 1) * (if b ∈ s then f b else 1) *
-      ∏ i ∈ (s.erase a).erase b, f i := by
-  calc
-    ∏ i ∈ s, f i = (if a ∈ s then f a else 1) * ∏ i ∈ s.erase a, f i :=
-      prod_split_at s f a
-    _ = _ := by
-      rw [prod_split_at (s.erase a) f b]
-      have hm : (b ∈ s.erase a) ↔ b ∈ s :=
-        ⟨fun h => (Finset.mem_erase.mp h).2, fun h => Finset.mem_erase.mpr ⟨Ne.symm hab, h⟩⟩
-      simp only [hm, mul_assoc]
-
-lemma swap_average_product {α : Type*} [DecidableEq α] (s : Finset α) (p : α → ℝ)
-    (hp : ∀ i, 0 ≤ p i) (a b : α) :
-    ((∏ i ∈ s, p i) + ∏ i ∈ s, p (Equiv.swap a b i)) / 2 ≤
-      ∏ i ∈ s, (p i + p (Equiv.swap a b i)) / 2 := by
-  by_cases hab : a = b
-  · subst b
-    simp only [Equiv.swap_self, Equiv.refl_apply]
-    have h : (fun i => (p i + p i) / 2) = p := by funext i; ring
-    rw [h]
-    linarith
-  let t := (s.erase a).erase b
-  have hfix : ∀ i ∈ t, Equiv.swap a b i = i := by
-    intro i hi
-    exact Equiv.swap_apply_of_ne_of_ne (Finset.mem_erase.mp (Finset.mem_erase.mp hi).2).1
-      (Finset.mem_erase.mp hi).1
-  have hprod : (∏ i ∈ t, p (Equiv.swap a b i)) = ∏ i ∈ t, p i :=
-    Finset.prod_congr rfl (fun i hi => by rw [hfix i hi])
-  have havg : (∏ i ∈ t, (p i + p (Equiv.swap a b i)) / 2) = ∏ i ∈ t, p i :=
-    Finset.prod_congr rfl (fun i hi => by rw [hfix i hi]; ring)
-  rw [prod_split_two s p a b hab,
-    prod_split_two s (fun i => p (Equiv.swap a b i)) a b hab,
-    prod_split_two s (fun i => (p i + p (Equiv.swap a b i)) / 2) a b hab]
-  change ((if a ∈ s then p a else 1) * (if b ∈ s then p b else 1) * (∏ i ∈ t, p i) +
-    (if a ∈ s then p (Equiv.swap a b a) else 1) *
-      (if b ∈ s then p (Equiv.swap a b b) else 1) * (∏ i ∈ t, p (Equiv.swap a b i))) / 2 ≤ _
-  rw [hprod]
-  change _ ≤ (if a ∈ s then (p a + p (Equiv.swap a b a)) / 2 else 1) *
-    (if b ∈ s then (p b + p (Equiv.swap a b b)) / 2 else 1) *
-      (∏ i ∈ t, (p i + p (Equiv.swap a b i)) / 2)
-  rw [havg, Equiv.swap_apply_left, Equiv.swap_apply_right]
-  have ht : 0 ≤ ∏ i ∈ t, p i := Finset.prod_nonneg (fun i _ => hp i)
-  by_cases ha : a ∈ s <;> by_cases hb : b ∈ s <;> simp only [ha, hb, ↓reduceIte]
-  · have hsq := mul_nonneg (sq_nonneg (p a - p b)) ht
-    nlinarith
-  all_goals ring_nf; rfl
-
-def CylinderBound {α : Type*} (m : Finset α → ℝ) (p : α → ℝ) : Prop :=
-  ∀ s, m s ≤ ∏ i ∈ s, p i
-
-lemma CylinderBound.fair_swap {α : Type*} [DecidableEq α]
-    {m : Finset α → ℝ} {p : α → ℝ} (h : CylinderBound m p)
-    (hp : ∀ i, 0 ≤ p i) (a b : α) :
-    CylinderBound (fun s => (m s + m (s.image (Equiv.swap a b))) / 2)
-      (fun i => (p i + p (Equiv.swap a b i)) / 2) := by
-  intro s
-  have h₀ := h s
-  have h₁ := h (s.image (Equiv.swap a b))
-  rw [Finset.prod_image (Equiv.swap a b).injective.injOn] at h₁
-  exact (show (m s + m (s.image (Equiv.swap a b))) / 2 ≤
-      ((∏ i ∈ s, p i) + ∏ i ∈ s, p (Equiv.swap a b i)) / 2 by linarith).trans
-    (swap_average_product s p hp a b)
-
-noncomputable def inclusionMoment {α Ω : Type*} [Fintype Ω]
-    (F : Ω → Finset α) (s : Finset α) : ℝ := by
-  classical
-  exact finiteMean (fun ω => if s ⊆ F ω then 1 else 0)
-
-lemma inclusionMoment_nonneg {α Ω : Type*} [Fintype Ω]
-    (F : Ω → Finset α) (s : Finset α) : 0 ≤ inclusionMoment F s := by
-  classical
-  unfold inclusionMoment finiteMean
-  positivity
-
-lemma indicator_subset_prod {α : Type*} [DecidableEq α] (A s : Finset α) :
-    (if s ⊆ A then (1 : ℝ) else 0) = ∏ i ∈ s, (if i ∈ A then 1 else 0) := by
-  by_cases h : s ⊆ A
-  · rw [ite_eq_left h]
-    exact (Finset.prod_eq_one (fun i hi => ite_eq_left (h hi))).symm
-  · rw [ite_eq_right h]
-    obtain ⟨i, hi, hiA⟩ := Finset.not_subset.mp h
-    exact (Finset.prod_eq_zero hi (ite_eq_right hiA)).symm
-
-lemma inclusionMoment_const {α Ω : Type*} [DecidableEq α] [Fintype Ω] [Nonempty Ω]
-    (A s : Finset α) : inclusionMoment (fun _ : Ω => A) s =
-      if s ⊆ A then 1 else 0 := by
-  classical
-  have hn : (Fintype.card Ω : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
-  by_cases h : s ⊆ A <;> simp [inclusionMoment, finiteMean, h, hn]
-
-lemma cylinder_const {α Ω : Type*} [Fintype Ω] [Nonempty Ω] (A : Finset α) :
-    CylinderBound (inclusionMoment (fun _ : Ω => A))
-      (fun i => inclusionMoment (fun _ : Ω => A) {i}) := by
-  classical
-  intro s
-  simp only [inclusionMoment_const, Finset.singleton_subset_iff]
-  exact (indicator_subset_prod A s).le
-
-noncomputable def fairSetSwap {α Ω : Type*} [DecidableEq α]
-    (F : Ω → Finset α) (a b : α) : Ω × Bool → Finset α :=
-  fun ω => if ω.2 then (F ω.1).image (Equiv.swap a b) else F ω.1
-
-lemma subset_swap_image_iff {α : Type*} [DecidableEq α]
-    (A s : Finset α) (a b : α) :
-    s ⊆ A.image (Equiv.swap a b) ↔ s.image (Equiv.swap a b) ⊆ A := by
-  constructor
-  · intro h
-    have h' := Finset.image_subset_image h (f := Equiv.swap a b)
-    simpa only [Finset.image_image, Function.comp_def, Equiv.swap_apply_self, Finset.image_id'] using h'
-  · intro h
-    have h' := Finset.image_subset_image h (f := Equiv.swap a b)
-    simpa only [Finset.image_image, Function.comp_def, Equiv.swap_apply_self, Finset.image_id'] using h'
-
-lemma inclusionMoment_fairSetSwap {α Ω : Type*} [DecidableEq α] [Fintype Ω]
-    (F : Ω → Finset α) (a b : α) (s : Finset α) :
-    inclusionMoment (fairSetSwap F a b) s =
-      (inclusionMoment F s + inclusionMoment F (s.image (Equiv.swap a b))) / 2 := by
-  classical
-  simp only [inclusionMoment, finiteMean, Fintype.sum_prod_type, Fintype.sum_bool,
-    fairSetSwap, Bool.false_eq_true, ↓reduceIte, Fintype.card_prod, Fintype.card_bool,
-    Nat.cast_mul, Nat.cast_ofNat, subset_swap_image_iff, Finset.sum_add_distrib]
-  rw [div_mul_eq_div_div, add_div]
-  ring_nf
-
-lemma cylinder_fairSetSwap {α Ω : Type*} [DecidableEq α] [Fintype Ω]
-    (F : Ω → Finset α) (a b : α)
-    (h : CylinderBound (inclusionMoment F) (fun i => inclusionMoment F {i})) :
-    CylinderBound (inclusionMoment (fairSetSwap F a b))
-      (fun i => inclusionMoment (fairSetSwap F a b) {i}) := by
-  classical
-  have h' := h.fair_swap (fun i => inclusionMoment_nonneg F {i}) a b
-  intro s
-  simpa only [inclusionMoment_fairSetSwap, Finset.image_singleton] using h' s
-
-def SwitchSamples : ℕ → Type
-  | 0 => Unit
-  | n + 1 => SwitchSamples n × Bool
-
-instance (n : ℕ) : Fintype (SwitchSamples n) := by
-  induction n with
-  | zero => exact inferInstanceAs (Fintype Unit)
-  | succ n ih => exact inferInstanceAs (Fintype (SwitchSamples n × Bool))
-
-instance (n : ℕ) : Nonempty (SwitchSamples n) := by
-  induction n with
-  | zero => exact inferInstanceAs (Nonempty Unit)
-  | succ n ih => exact inferInstanceAs (Nonempty (SwitchSamples n × Bool))
-
-noncomputable def switchedSet {α : Type*} [DecidableEq α] (A : Finset α) :
-    (L : List (α × α)) → SwitchSamples L.length → Finset α
-  | [], _ => A
-  | (a,b) :: L, ω => fairSetSwap (switchedSet A L) a b ω
-
-lemma switchedSet_cylinder {α : Type*} [DecidableEq α] (A : Finset α)
-    (L : List (α × α)) :
-    CylinderBound (inclusionMoment (switchedSet A L))
-      (fun i => inclusionMoment (switchedSet A L) {i}) := by
-  induction L with
-  | nil => exact cylinder_const A
-  | cons e L ih => exact cylinder_fairSetSwap (switchedSet A L) e.1 e.2 ih
-
-inductive QueryTree (ι : Type*)
-  | reject : QueryTree ι
-  | accept : QueryTree ι
-  | branch (i : ι) (zero one : QueryTree ι) : QueryTree ι
-  | force (i : ι) (b : Bool) (next : QueryTree ι) : QueryTree ι
-
-namespace QueryTree
-
-variable {ι : Type*} [DecidableEq ι]
-
-def queries : QueryTree ι → Finset ι
-  | reject => ∅
-  | accept => ∅
-  | branch i zero one => insert i (queries zero ∪ queries one)
-  | force i _ next => insert i (queries next)
-
-def Fresh : QueryTree ι → Prop
-  | reject => True
-  | accept => True
-  | branch i zero one => i ∉ queries zero ∧ i ∉ queries one ∧ Fresh zero ∧ Fresh one
-  | force i _ next => i ∉ queries next ∧ Fresh next
-
-def weighted : QueryTree ι → (ι → Bool) → ℝ
-  | reject, _ => 0
-  | accept, _ => 1
-  | branch i zero one, ω => if ω i then weighted one ω else weighted zero ω
-  | force i b next, ω => if ω i = b then 2 * weighted next ω else 0
-
-omit [DecidableEq ι] in
-lemma weighted_nonneg (T : QueryTree ι) (ω : ι → Bool) : 0 ≤ T.weighted ω := by
-  induction T with
-  | reject => exact le_rfl
-  | accept => exact zero_le_one
-  | branch i zero one ih₀ ih₁ => simpa only [weighted] using ite_nonneg ih₁ ih₀
-  | force i b next ih => simpa only [weighted] using ite_nonneg (mul_nonneg (by norm_num) ih) le_rfl
-
-lemma weighted_congr (T : QueryTree ι) {ω η : ι → Bool}
-    (h : ∀ i ∈ T.queries, ω i = η i) : T.weighted ω = T.weighted η := by
-  induction T with
-  | reject => rfl
-  | accept => rfl
-  | branch i zero one ih₀ ih₁ =>
-    have hi := h i (Finset.mem_insert_self _ _)
-    simp only [weighted, hi]
-    rw [ih₀ (fun j hj => h j (Finset.mem_insert_of_mem (Finset.mem_union_left _ hj))),
-      ih₁ (fun j hj => h j (Finset.mem_insert_of_mem (Finset.mem_union_right _ hj)))]
-  | force i b next ih =>
-    have hi := h i (Finset.mem_insert_self _ _)
-    simp only [weighted, hi]
-    rw [ih (fun j hj => h j (Finset.mem_insert_of_mem hj))]
-
-def flipCoin (i : ι) : Equiv.Perm (ι → Bool) :=
-  (show Function.Involutive (fun ω : ι → Bool => Function.update ω i (!(ω i))) from by
-    intro ω
-    funext j
-    by_cases h : j = i
-    · subst j; simp
-    · simp [Function.update_of_ne h]).toPerm
-
-lemma flipCoin_apply_eq (i : ι) (ω : ι → Bool) : flipCoin i ω i = !(ω i) := by
-  change Function.update ω i (!(ω i)) i = _
-  exact Function.update_self _ _ _
-
-lemma flipCoin_apply_ne (i j : ι) (h : j ≠ i) (ω : ι → Bool) :
-    flipCoin i ω j = ω j := by
-  change Function.update ω i (!(ω i)) j = _
-  exact Function.update_of_ne h _ _
-
-lemma weighted_flip_of_not_mem (T : QueryTree ι) (i : ι) (hi : i ∉ T.queries)
-    (ω : ι → Bool) : T.weighted (flipCoin i ω) = T.weighted ω := by
-  apply weighted_congr
-  intro j hj
-  exact flipCoin_apply_ne i j (fun h => hi (h ▸ hj)) ω
-
-lemma mean_indicator_bit [Fintype ι] (f : (ι → Bool) → ℝ) (i : ι) (b : Bool)
-    (hf : ∀ ω, f (flipCoin i ω) = f ω) :
-    finiteMean (fun ω => if ω i = b then f ω else 0) = finiteMean f / 2 := by
-  let g := fun ω : ι → Bool => if ω i = b then f ω else 0
-  have hp (ω : ι → Bool) : g ω + g (flipCoin i ω) = f ω := by
-    dsimp [g]
-    rw [flipCoin_apply_eq, hf]
-    cases hω : ω i <;> cases b <;> simp
-  have hs := Equiv.sum_comp (flipCoin i) g
-  have hadd : (∑ ω, g ω) + (∑ ω, g (flipCoin i ω)) = ∑ ω, f ω := by
-    rw [← Finset.sum_add_distrib]
-    exact Finset.sum_congr rfl (fun ω _ => hp ω)
-  unfold finiteMean
-  change (∑ ω, g ω) / _ = _
-  rw [hs] at hadd
-  have hh : (∑ ω, g ω) = (∑ ω, f ω) / 2 := by linarith
-  rw [hh]
-  ring
-
-lemma mean_weighted_le_one [Fintype ι] (T : QueryTree ι) (hT : T.Fresh) :
-    finiteMean T.weighted ≤ 1 := by
-  induction T with
-  | reject => simp [weighted, finiteMean]
-  | accept => simp [weighted, finiteMean]
-  | branch i zero one ih₀ ih₁ =>
-    obtain ⟨hz, ho, hfz, hfo⟩ := hT
-    have h₀ := mean_indicator_bit zero.weighted i false (zero.weighted_flip_of_not_mem i hz)
-    have h₁ := mean_indicator_bit one.weighted i true (one.weighted_flip_of_not_mem i ho)
-    have he : finiteMean (branch i zero one).weighted =
-        finiteMean (fun ω => if ω i = false then zero.weighted ω else 0) +
-        finiteMean (fun ω => if ω i = true then one.weighted ω else 0) := by
-      unfold finiteMean
-      rw [← add_div, ← Finset.sum_add_distrib]
-      congr 1
-      apply Finset.sum_congr rfl
-      intro ω _
-      cases hω : ω i <;> simp [weighted, hω]
-    rw [he, h₀, h₁]
-    linarith [ih₀ hfz, ih₁ hfo]
-  | force i b next ih =>
-    obtain ⟨hi, hn⟩ := hT
-    have hm := mean_indicator_bit next.weighted i b (next.weighted_flip_of_not_mem i hi)
-    have he : finiteMean (force i b next).weighted =
-        2 * finiteMean (fun ω => if ω i = b then next.weighted ω else 0) := by
-      unfold finiteMean
-      rw [← mul_div_assoc, Finset.mul_sum]
-      congr 1
-      apply Finset.sum_congr rfl
-      intro ω _
-      by_cases h : ω i = b <;> simp [weighted, h]
-    rw [he, hm]
-    linarith [ih hn]
-
-def succeeds : QueryTree ι → (ι → Bool) → Bool
-  | reject, _ => false
-  | accept, _ => true
-  | branch i zero one, ω => if ω i then succeeds one ω else succeeds zero ω
-  | force i b next, ω => if ω i = b then succeeds next ω else false
-
-def charge : QueryTree ι → (ι → Bool) → ℕ
-  | reject, _ => 0
-  | accept, _ => 0
-  | branch i zero one, ω => if ω i then charge one ω else charge zero ω
-  | force i b next, ω => if ω i = b then charge next ω + 1 else 0
-
-omit [DecidableEq ι] in
-lemma weighted_eq (T : QueryTree ι) (ω : ι → Bool) :
-    T.weighted ω = if T.succeeds ω then (2 : ℝ) ^ T.charge ω else 0 := by
-  induction T with
-  | reject => rfl
-  | accept => norm_num [weighted, succeeds, charge]
-  | branch i zero one ih₀ ih₁ =>
-    cases h : ω i <;> simp [weighted, succeeds, charge, h, ih₀, ih₁]
-  | force i b next ih =>
-    by_cases h : ω i = b
-    · simp only [weighted, succeeds, charge, h, ↓reduceIte, ih]
-      cases hs : next.succeeds ω <;> simp [pow_succ, mul_comm]
-    · simp [weighted, succeeds, h]
-
-lemma success_probability_le [Fintype ι] (T : QueryTree ι) (hT : T.Fresh)
-    (K : ℕ) (hK : ∀ ω, T.succeeds ω = true → K ≤ T.charge ω) :
-    finiteMean (fun ω => if T.succeeds ω then (1 : ℝ) else 0) ≤ 1 / (2 : ℝ) ^ K := by
-  have hw (ω : ι → Bool) :
-      (2 : ℝ) ^ K * (if T.succeeds ω then (1 : ℝ) else 0) ≤ T.weighted ω := by
-    rw [weighted_eq]
-    cases hs : T.succeeds ω
-    · simp
-    · simpa using pow_le_pow_right₀ (show (1 : ℝ) ≤ 2 by norm_num) (hK ω hs)
-  have hn : 0 ≤ (Fintype.card (ι → Bool) : ℝ) := Nat.cast_nonneg _
-  have hm := div_le_div_of_nonneg_right
-    (Finset.sum_le_sum (fun ω (_ : ω ∈ Finset.univ) => hw ω)) hn
-  rw [← Finset.mul_sum, mul_div_assoc] at hm
-  change (2 : ℝ) ^ K * finiteMean (fun ω => if T.succeeds ω then 1 else 0) ≤
-    finiteMean T.weighted at hm
-  exact (le_div_iff₀ (pow_pos (by norm_num) K)).mpr (by
-    rw [mul_comm]
-    exact hm.trans (mean_weighted_le_one T hT))
-
-end QueryTree
-
-def encodeButterfly : (d : ℕ) → Butterfly d → SwitchIndex d → Bool
-  | 0, _, i => nomatch i
-  | _d + 1, b, Sum.inl y => b.1 y
-  | d + 1, b, Sum.inr (ε,i) => encodeButterfly d (b.2 ε) i
-
-lemma decode_encodeButterfly (d : ℕ) (b : Butterfly d) :
-    decodeButterfly d (encodeButterfly d b) = b := by
-  induction d with
-  | zero => exact Unit.ext _ _
-  | succ d ih =>
-    apply Prod.ext
-    · rfl
-    · funext ε
-      exact ih (b.2 ε)
-
-lemma encode_decodeButterfly (d : ℕ) (ω : SwitchIndex d → Bool) :
-    encodeButterfly d (decodeButterfly d ω) = ω := by
-  induction d with
-  | zero => funext i; exact Empty.elim i
-  | succ d ih =>
-    funext i
-    cases i with
-    | inl y => rfl
-    | inr z => exact congrFun (ih (fun i => ω (Sum.inr (z.1,i)))) z.2
-
-def butterflyCoinEquiv (d : ℕ) : (SwitchIndex d → Bool) ≃ Butterfly d where
-  toFun := decodeButterfly d
-  invFun := encodeButterfly d
-  left_inv := encode_decodeButterfly d
-  right_inv := decode_encodeButterfly d
-
-noncomputable def routeDomain : (d : ℕ) → Card d → Card d → Finset (SwitchIndex d)
-  | 0, _, _ => ∅
-  | d + 1, x, y => by
-      classical
-      exact insert (Sum.inl (Fin.tail y))
-        ((routeDomain d (Fin.tail x) (Fin.tail y)).image (fun i => Sum.inr (x 0,i)))
-
-def routeValue : (d : ℕ) → Card d → Card d → SwitchIndex d → Bool
-  | 0, _, _, i => nomatch i
-  | _d + 1, x, y, Sum.inl _ => Bool.xor (x 0) (y 0)
-  | d + 1, x, y, Sum.inr (_,i) => routeValue d (Fin.tail x) (Fin.tail y) i
-
-lemma card_routeDomain (d : ℕ) (x y : Card d) : (routeDomain d x y).card = d := by
-  classical
-  induction d with
-  | zero => rfl
-  | succ d ih =>
-    change (insert (Sum.inl (Fin.tail y))
-      ((routeDomain d (Fin.tail x) (Fin.tail y)).image
-        (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) :
-      Finset (Sum (Card d) (Bool × SwitchIndex d))).card = d + 1
-    rw [Finset.card_insert_of_notMem (by simp)]
-    rw [Finset.card_image_of_injective _ (by intro i j h; cases h; rfl), ih]
-
-end Thorp
-
-end ThorpNine.Smoothing
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Smoothing.Model
+
+namespace OAI
+
+namespace ThorpNine.Smoothing
+
+namespace Thorp
+
+open scoped BigOperators
+open Filter
+
+def Coins : ℕ → Type
+  | 0 => Unit
+  | d + 1 => Card d → Bool
+
+instance (d : ℕ) : Fintype (Coins d) := by
+  cases d <;> dsimp [Coins] <;> infer_instance
+
+instance (d : ℕ) : Nonempty (Coins d) := by
+  cases d <;> dsimp [Coins] <;> infer_instance
+
+def rotate (d : ℕ) : Equiv.Perm (Card d) where
+  toFun x i := x (finRotate d i)
+  invFun x i := x ((finRotate d).symm i)
+  left_inv x := by funext i; simp only [Equiv.apply_symm_apply]
+  right_inv x := by funext i; simp only [Equiv.symm_apply_apply]
+
+def step : (d : ℕ) → Coins d → Equiv.Perm (Card d)
+  | 0, _ => 1
+  | d + 1, ξ => rotate (d + 1) * pairSwitch ξ
+
+def run (d : ℕ) : (t : ℕ) → (Fin t → Coins d) → Equiv.Perm (Card d)
+  | 0, _ => 1
+  | t + 1, ω => step d (ω (Fin.last t)) * run d t (fun i => ω i.castSucc)
+
+noncomputable def law (d t : ℕ) (g : Equiv.Perm (Card d)) : ℝ :=
+  (Fintype.card {ω : Fin t → Coins d // run d t ω = g} : ℝ) /
+    (Fintype.card (Fin t → Coins d) : ℝ)
+
+noncomputable def distance (d t : ℕ) : ℝ :=
+  (1 / 2 : ℝ) * ∑ g : Equiv.Perm (Card d),
+    |law d t g - 1 / (Nat.factorial (2 ^ d) : ℝ)|
+
+noncomputable def mixingTime (d : ℕ) : ℕ :=
+  sInf {t : ℕ | distance d t ≤ (1 / 4 : ℝ)}
+
+noncomputable def supportTimeBound (d : ℕ) : ℤ :=
+  ⌈(2 / (2 : ℝ) ^ d) *
+    (Real.log ((3 / 4 : ℝ) * (Nat.factorial (2 ^ d) : ℝ)) / Real.log 2)⌉
+
+def OptimalOrder : Prop :=
+  ∃ C : ℕ, 0 < C ∧
+    Tendsto (fun d : ℕ => distance d (C * d)) atTop (nhds 0) ∧
+    (∀ d t : ℕ, 1 - (2 : ℝ) ^ ((t : ℝ) * (2 : ℝ) ^ d / 2) /
+      (Nat.factorial (2 ^ d) : ℝ) ≤ distance d t) ∧
+    (∀ᶠ d : ℕ in atTop,
+      supportTimeBound d ≤ (mixingTime d : ℤ) ∧ mixingTime d ≤ C * d) ∧
+    (∃ K : ℝ, ∀ᶠ d : ℕ in atTop, 2 * (d : ℝ) - K ≤ (mixingTime d : ℝ)) ∧
+    Asymptotics.IsTheta atTop (fun d : ℕ => (mixingTime d : ℝ)) (fun d : ℕ => (d : ℝ))
+
+noncomputable def finiteLaw {α β : Type*} [Fintype α] (f : α → β) (b : β) : ℝ := by
+  classical
+  exact (Fintype.card {a : α // f a = b} : ℝ) / (Fintype.card α : ℝ)
+
+lemma sum_card_fibers {α β : Type*} [Fintype α] [Fintype β] [DecidableEq β] (f : α → β) :
+    ∑ b, Fintype.card {a : α // f a = b} = Fintype.card α := by
+  classical
+  rw [← Fintype.card_sigma]
+  exact Fintype.card_congr (Equiv.sigmaFiberEquiv f)
+
+lemma finiteLaw_nonneg {α β : Type*} [Fintype α] (f : α → β) (b : β) :
+    0 ≤ finiteLaw f b := by
+  exact div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _)
+
+lemma finiteLaw_sum {α β : Type*} [Fintype α] [Nonempty α] [Fintype β] (f : α → β) :
+    ∑ b, finiteLaw f b = 1 := by
+  classical
+  simp only [finiteLaw, ← Finset.sum_div]
+  have hc : (∑ b, (Fintype.card {a : α // f a = b} : ℝ)) = (Fintype.card α : ℝ) := by
+    exact_mod_cast sum_card_fibers f
+  rw [hc]
+  exact div_self (Nat.cast_ne_zero.mpr Fintype.card_ne_zero)
+
+lemma finiteLaw_eq_zero_of_not_mem {α β : Type*} [Fintype α] [DecidableEq β]
+    (f : α → β) {b : β} (hb : b ∉ Finset.univ.image f) : finiteLaw f b = 0 := by
+  classical
+  have : IsEmpty {a : α // f a = b} := ⟨fun a => hb (Finset.mem_image.mpr
+    ⟨a.val, Finset.mem_univ _, a.property⟩)⟩
+  simp [finiteLaw]
+
+lemma event_le_totalVariation {β : Type*} [Fintype β] [DecidableEq β]
+    (p q : β → ℝ) (hpq : ∑ b, (p b - q b) = 0) (s : Finset β) :
+    (∑ b ∈ s, p b) - (∑ b ∈ s, q b) ≤ (1 / 2 : ℝ) * ∑ b, |p b - q b| := by
+  have h₁ : (∑ b ∈ s, (p b - q b)) ≤ ∑ b ∈ s, |p b - q b| :=
+    Finset.sum_le_sum (fun b _ => le_abs_self (p b - q b))
+  have h₂ : -(∑ b ∈ sᶜ, (p b - q b)) ≤ ∑ b ∈ sᶜ, |p b - q b| := by
+    rw [← Finset.sum_neg_distrib]
+    exact Finset.sum_le_sum (fun b _ => neg_le_abs (p b - q b))
+  have hz := Finset.sum_add_sum_compl s (fun b => p b - q b)
+  have ha := Finset.sum_add_sum_compl s (fun b => |p b - q b|)
+  rw [hpq] at hz
+  rw [← Finset.sum_sub_distrib]
+  linarith
+
+lemma finiteLaw_support_bound {α β : Type*} [Fintype α] [Nonempty α]
+    [Fintype β] [Nonempty β] (f : α → β) :
+    1 - (Fintype.card α : ℝ) / (Fintype.card β : ℝ) ≤
+      (1 / 2 : ℝ) * ∑ b, |finiteLaw f b - 1 / (Fintype.card β : ℝ)| := by
+  classical
+  let s := Finset.univ.image f
+  have hs : ∑ b ∈ s, finiteLaw f b = 1 := by
+    have h := Finset.sum_subset (Finset.subset_univ s)
+      (fun b _ hb => finiteLaw_eq_zero_of_not_mem f hb)
+    simpa only [finiteLaw_sum] using h
+  have hβ : (Fintype.card β : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
+  have hq : (∑ _b : β, 1 / (Fintype.card β : ℝ)) = 1 := by simp [hβ]
+  have hd : (∑ b, (finiteLaw f b - 1 / (Fintype.card β : ℝ))) = 0 := by
+    rw [Finset.sum_sub_distrib, finiteLaw_sum, hq, sub_self]
+  have he := event_le_totalVariation (finiteLaw f) (fun _ => 1 / (Fintype.card β : ℝ)) hd s
+  rw [hs] at he
+  have hsq : (∑ _b ∈ s, 1 / (Fintype.card β : ℝ)) = (s.card : ℝ) / Fintype.card β := by
+    simp [div_eq_mul_inv]
+  rw [hsq] at he
+  have hcard : (s.card : ℝ) ≤ Fintype.card α := by
+    exact_mod_cast (Finset.card_image_le (s := (Finset.univ : Finset α)) (f := f))
+  have hdiv := div_le_div_of_nonneg_right hcard (Nat.cast_nonneg (Fintype.card β) : (0 : ℝ) ≤ _)
+  linarith
+
+lemma card_positions (d : ℕ) : Fintype.card (Card d) = 2 ^ d := by
+  simp [Card]
+
+lemma card_permutations (d : ℕ) : Fintype.card (Equiv.Perm (Card d)) = Nat.factorial (2 ^ d) := by
+  rw [Fintype.card_perm, card_positions]
+
+lemma card_coins_succ (d : ℕ) : Fintype.card (Coins (d + 1)) = 2 ^ (2 ^ d) := by
+  simp [Coins]
+
+lemma card_tapes_succ (d t : ℕ) : Fintype.card (Fin t → Coins (d + 1)) = 2 ^ (t * 2 ^ d) := by
+  rw [Fintype.card_fun, card_coins_succ, Fintype.card_fin, ← pow_mul, Nat.mul_comm]
+
+lemma law_eq_finiteLaw (d t : ℕ) (g : Equiv.Perm (Card d)) :
+    law d t g = finiteLaw (run d t) g := by
+  unfold law finiteLaw
+  apply congrArg (fun n : ℕ => (n : ℝ) / (Fintype.card (Fin t → Coins d) : ℝ))
+  exact congrArg (@Fintype.card _) (Subsingleton.elim _ _)
+
+lemma law_nonneg (d t : ℕ) (g : Equiv.Perm (Card d)) : 0 ≤ law d t g := by
+  rw [law_eq_finiteLaw]
+  exact finiteLaw_nonneg (run d t) g
+
+lemma law_sum (d t : ℕ) : ∑ g, law d t g = 1 := by
+  simp only [law_eq_finiteLaw]
+  exact finiteLaw_sum (run d t)
+
+lemma physical_support_bound_succ (d t : ℕ) :
+    1 - (2 : ℝ) ^ (t * 2 ^ d) / (Nat.factorial (2 ^ (d + 1)) : ℝ) ≤ distance (d + 1) t := by
+  have : Nonempty (Equiv.Perm (Card (d + 1))) := ⟨1⟩
+  have h := finiteLaw_support_bound (run (d + 1) t)
+  rw [card_tapes_succ, card_permutations] at h
+  simpa only [Nat.cast_pow, Nat.cast_ofNat, distance, law_eq_finiteLaw] using h
+
+lemma step_succ_formula (d : ℕ) (ξ : Coins (d + 1)) (x : Card (d + 1)) :
+    step (d + 1) ξ x = Fin.snoc (Fin.tail x) (Bool.xor (x 0) (ξ (Fin.tail x))) := by
+  change (fun i => switchFun ξ x (finRotate (d + 1) i)) = _
+  rw [Fin.snoc_eq_cons_rotate]
+  rfl
+
+lemma distance_nonneg (d t : ℕ) : 0 ≤ distance d t := by
+  unfold distance
+  exact mul_nonneg (by norm_num) (Finset.sum_nonneg (fun _ _ => abs_nonneg _))
+
+lemma physical_support_bound (d t : ℕ) :
+    1 - (2 : ℝ) ^ ((t : ℝ) * (2 : ℝ) ^ d / 2) /
+      (Nat.factorial (2 ^ d) : ℝ) ≤ distance d t := by
+  cases d with
+  | zero =>
+    have h : 1 ≤ (2 : ℝ) ^ ((t : ℝ) / 2) := Real.one_le_rpow (by norm_num) (by positivity)
+    simpa only [pow_zero, Nat.factorial_one, Nat.cast_one, div_one, mul_one] using
+      le_trans (sub_nonpos.mpr h) (distance_nonneg 0 t)
+  | succ d =>
+    have he : (t : ℝ) * (2 : ℝ) ^ (d + 1) / 2 = (t * 2 ^ d : ℕ) := by
+      push_cast
+      rw [pow_succ]
+      ring
+    rw [he, Real.rpow_natCast]
+    exact physical_support_bound_succ d t
+
+lemma supportTimeBound_le_of_distance_le (d t : ℕ) (ht : distance d t ≤ (1 / 4 : ℝ)) :
+    supportTimeBound d ≤ (t : ℤ) := by
+  have hF : (0 : ℝ) < Nat.factorial (2 ^ d) := by positivity
+  have hN : (0 : ℝ) < (2 : ℝ) ^ d := by positivity
+  have hL : (0 : ℝ) < Real.log 2 := Real.log_pos (by norm_num)
+  have hb := physical_support_bound d t
+  have hp : (3 / 4 : ℝ) ≤
+      (2 : ℝ) ^ ((t : ℝ) * (2 : ℝ) ^ d / 2) / (Nat.factorial (2 ^ d) : ℝ) := by
+    linarith
+  have hr := (le_div_iff₀ hF).mp hp
+  have hlog := Real.log_le_log (show (0 : ℝ) < (3 / 4 : ℝ) * Nat.factorial (2 ^ d) by positivity) hr
+  rw [Real.log_rpow (by norm_num : (0 : ℝ) < 2)] at hlog
+  unfold supportTimeBound
+  apply Int.ceil_le.mpr
+  simp only [Int.cast_natCast]
+  calc
+    2 / (2 : ℝ) ^ d * (Real.log ((3 / 4 : ℝ) * Nat.factorial (2 ^ d)) / Real.log 2) =
+        Real.log ((3 / 4 : ℝ) * Nat.factorial (2 ^ d)) / ((2 : ℝ) ^ d / 2 * Real.log 2) := by
+      field_simp
+    _ ≤ (t : ℝ) := (div_le_iff₀ (mul_pos (div_pos hN (by norm_num)) hL)).mpr (by
+      nlinarith [hlog])
+
+lemma supportTimeBound_le_mixingTime {d : ℕ}
+    (h : ({t : ℕ | distance d t ≤ (1 / 4 : ℝ)} : Set ℕ).Nonempty) :
+    supportTimeBound d ≤ (mixingTime d : ℤ) := by
+  exact supportTimeBound_le_of_distance_le d (mixingTime d) (Nat.sInf_mem h)
+
+lemma log_factorial_lower {n : ℕ} (hn : 1 ≤ n) :
+    (n : ℝ) * Real.log n - n ≤ Real.log (Nat.factorial n) := by
+  have h := Stirling.le_log_factorial_stirling (show n ≠ 0 by omega)
+  have hln : 0 ≤ Real.log (n : ℝ) := Real.log_nonneg (by exact_mod_cast hn)
+  have hlp : 0 ≤ Real.log (2 * Real.pi) := Real.log_nonneg (by
+    have := Real.pi_gt_three
+    linarith)
+  linarith
+
+lemma log_three_quarters_lower : -(1 : ℝ) ≤ Real.log (3 / 4 : ℝ) := by
+  have h := Real.log_le_sub_one_of_pos (show (0 : ℝ) < 4 / 3 by norm_num)
+  have he : Real.log (4 / 3 : ℝ) = - Real.log (3 / 4 : ℝ) := by
+    rw [← Real.log_inv]
+    congr 1
+    norm_num
+  rw [he] at h
+  linarith
+
+lemma linear_le_supportTimeBound (d : ℕ) :
+    2 * (d : ℝ) - 4 / Real.log 2 ≤ (supportTimeBound d : ℝ) := by
+  have hN : 1 ≤ (2 : ℝ) ^ d := one_le_pow₀ (by norm_num)
+  have hNp : 0 < (2 : ℝ) ^ d := by positivity
+  have hL : 0 < Real.log 2 := Real.log_pos (by norm_num)
+  have hf := log_factorial_lower (show 1 ≤ 2 ^ d by exact one_le_pow₀ (by norm_num))
+  have hlog : (2 : ℝ) ^ d * ((d : ℝ) * Real.log 2) - (2 : ℝ) ^ d - 1 ≤
+      Real.log ((3 / 4 : ℝ) * (Nat.factorial (2 ^ d) : ℝ)) := by
+    rw [Real.log_mul (by norm_num) (by positivity)]
+    push_cast at hf
+    rw [Real.log_pow] at hf
+    linarith [log_three_quarters_lower]
+  apply le_trans _ (Int.le_ceil _)
+  have hmul : (2 * (d : ℝ) - 4 / Real.log 2) *
+      ((2 : ℝ) ^ d / 2 * Real.log 2) ≤
+      Real.log ((3 / 4 : ℝ) * (Nat.factorial (2 ^ d) : ℝ)) := by
+    have hcancel : (2 * (d : ℝ) - 4 / Real.log 2) *
+        ((2 : ℝ) ^ d / 2 * Real.log 2) =
+        (2 : ℝ) ^ d * ((d : ℝ) * Real.log 2) - 2 * (2 : ℝ) ^ d := by
+      field_simp
+      ring
+    rw [hcancel]
+    linarith
+  convert (le_div_iff₀ (mul_pos (div_pos hNp (by norm_num)) hL)).mpr hmul using 1
+  field_simp
+
+lemma linear_le_time_of_distance_le (d t : ℕ)
+    (ht : distance d t ≤ (1 / 4 : ℝ)) :
+    2 * (d : ℝ) - 4 / Real.log 2 ≤ (t : ℝ) := by
+  have h := supportTimeBound_le_of_distance_le d t ht
+  have hr : (supportTimeBound d : ℝ) ≤ (t : ℝ) := by exact_mod_cast h
+  exact (linear_le_supportTimeBound d).trans hr
+
+lemma entropy_chord (p : ℝ) (hp : 0 ≤ p) (hq : p ≤ 1 / 2) :
+    2 * p * Real.log 2 ≤ Real.binEntropy p := by
+  have hc := Real.strictConcave_binEntropy.concaveOn.2
+    (show (0 : ℝ) ∈ Set.Icc (0 : ℝ) 1 by norm_num)
+    (show (2 : ℝ)⁻¹ ∈ Set.Icc (0 : ℝ) 1 by norm_num)
+    (show 0 ≤ 1 - 2 * p by linarith)
+    (show 0 ≤ 2 * p by positivity)
+    (show (1 - 2 * p) + 2 * p = 1 by ring)
+  simpa only [Real.binEntropy_zero, Real.binEntropy_two_inv,
+    smul_eq_mul, mul_zero, zero_add, show 2 * p * (2 : ℝ)⁻¹ = p by ring] using hc
+
+lemma split_entropy_lower (a b : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) :
+    2 * min a b * Real.log 2 ≤
+      (a + b) * Real.log (a + b) - a * Real.log a - b * Real.log b := by
+  wlog hab : a ≤ b generalizing a b
+  · have hba := this b a hb ha (le_of_not_ge hab)
+    simpa only [min_comm, add_comm b a] using hba.trans_eq (show (b + a) * Real.log (b + a) - b * Real.log b - a * Real.log a =
+      (a + b) * Real.log (a + b) - a * Real.log a - b * Real.log b by rw [add_comm b a]; ring)
+  rw [min_eq_left hab]
+  by_cases hz : a = 0
+  · subst a
+    simp
+  have hap : 0 < a := lt_of_le_of_ne ha (Ne.symm hz)
+  have hbp : 0 < b := hap.trans_le hab
+  have hs : 0 < a + b := add_pos hap hbp
+  have hp : 0 ≤ a / (a + b) := div_nonneg ha hs.le
+  have hq : a / (a + b) ≤ 1 / 2 := (div_le_iff₀ hs).mpr (by linarith)
+  have hc := mul_le_mul_of_nonneg_left (entropy_chord (a / (a + b)) hp hq) hs.le
+  have he : (a + b) * Real.binEntropy (a / (a + b)) =
+      (a + b) * Real.log (a + b) - a * Real.log a - b * Real.log b := by
+    unfold Real.binEntropy
+    rw [show 1 - a / (a + b) = b / (a + b) by field_simp; ring]
+    simp only [Real.log_inv]
+    rw [Real.log_div (ne_of_gt hap) (ne_of_gt hs),
+      Real.log_div (ne_of_gt hbp) (ne_of_gt hs)]
+    field_simp
+    ring
+  rw [he] at hc
+  calc
+    2 * a * Real.log 2 = (a + b) * (2 * (a / (a + b)) * Real.log 2) := by
+      field_simp
+    _ ≤ _ := hc
+
+noncomputable def childMarks {d : ℕ} (s : Finset (Card (d + 1))) (ε : Bool) :
+    Finset (Card d) := (s.filter (fun x => x 0 = ε)).image Fin.tail
+
+lemma card_childMarks {d : ℕ} (s : Finset (Card (d + 1))) (ε : Bool) :
+    (childMarks s ε).card = (s.filter (fun x => x 0 = ε)).card := by
+  classical
+  unfold childMarks
+  apply Finset.card_image_iff.mpr
+  intro x hx y hy hxy
+  have hx0 := (Finset.mem_filter.mp hx).2
+  have hy0 := (Finset.mem_filter.mp hy).2
+  rw [← Fin.cons_self_tail x, ← Fin.cons_self_tail y, hx0, hy0, hxy]
+
+lemma card_childMarks_add {d : ℕ} (s : Finset (Card (d + 1))) :
+    (childMarks s false).card + (childMarks s true).card = s.card := by
+  classical
+  rw [card_childMarks, card_childMarks]
+  simpa only [Bool.not_eq_false] using
+    (Finset.card_filter_add_card_filter_not (s := s) (fun x => x 0 = false))
+
+noncomputable def topMeetings {d : ℕ} (b : Butterfly (d + 1))
+    (s : Finset (Card (d + 1))) : Finset (Card d) :=
+  ((childMarks s false).image (butterflyPerm d (b.2 false))) ∩
+    ((childMarks s true).image (butterflyPerm d (b.2 true)))
+
+lemma card_topMeetings_le {d : ℕ} (b : Butterfly (d + 1)) (s : Finset (Card (d + 1))) :
+    (topMeetings b s).card ≤ min (childMarks s false).card (childMarks s true).card := by
+  classical
+  apply le_min
+  · exact (Finset.card_le_card Finset.inter_subset_left).trans_eq
+      (Finset.card_image_of_injective _ (butterflyPerm d (b.2 false)).injective)
+  · exact (Finset.card_le_card Finset.inter_subset_right).trans_eq
+      (Finset.card_image_of_injective _ (butterflyPerm d (b.2 true)).injective)
+
+noncomputable def encounters : (d : ℕ) → Butterfly d → Finset (Card d) → ℕ
+  | 0, _, _ => 0
+  | d + 1, b, s => encounters d (b.2 false) (childMarks s false) +
+      encounters d (b.2 true) (childMarks s true) + (topMeetings b s).card
+
+lemma butterfly_encounters (d : ℕ) (b : Butterfly d) (s : Finset (Card d)) :
+    (encounters d b s : ℝ) ≤ (s.card : ℝ) * Real.log s.card / (2 * Real.log 2) := by
+  induction d with
+  | zero =>
+    simp only [encounters, Nat.cast_zero]
+    have hs : s.card ≤ 1 := by
+      simpa [Card] using Finset.card_le_univ s
+    interval_cases s.card <;> simp
+  | succ d ih =>
+    have h₀ := ih (b.2 false) (childMarks s false)
+    have h₁ := ih (b.2 true) (childMarks s true)
+    have ht : ((topMeetings b s).card : ℝ) ≤
+        min ((childMarks s false).card : ℝ) ((childMarks s true).card : ℝ) := by
+      exact_mod_cast card_topMeetings_le b s
+    have he := split_entropy_lower ((childMarks s false).card : ℝ)
+      ((childMarks s true).card : ℝ) (by positivity) (by positivity)
+    have hs : ((childMarks s false).card : ℝ) + (childMarks s true).card = s.card := by
+      exact_mod_cast card_childMarks_add s
+    rw [hs] at he
+    have hL : 0 < 2 * Real.log 2 := by positivity
+    rw [le_div_iff₀ hL] at h₀ h₁ ⊢
+    simp only [encounters, Nat.cast_add]
+    nlinarith [mul_le_mul_of_nonneg_right ht hL.le]
+
+lemma prod_split_at {α : Type*} [DecidableEq α] (s : Finset α) (f : α → ℝ) (a : α) :
+    ∏ i ∈ s, f i = (if a ∈ s then f a else 1) * ∏ i ∈ s.erase a, f i := by
+  by_cases ha : a ∈ s
+  · rw [ite_eq_left ha]
+    exact (Finset.mul_prod_erase s f ha).symm
+  · simp [ha]
+
+lemma prod_split_two {α : Type*} [DecidableEq α] (s : Finset α) (f : α → ℝ)
+    (a b : α) (hab : a ≠ b) :
+    ∏ i ∈ s, f i = (if a ∈ s then f a else 1) * (if b ∈ s then f b else 1) *
+      ∏ i ∈ (s.erase a).erase b, f i := by
+  calc
+    ∏ i ∈ s, f i = (if a ∈ s then f a else 1) * ∏ i ∈ s.erase a, f i :=
+      prod_split_at s f a
+    _ = _ := by
+      rw [prod_split_at (s.erase a) f b]
+      have hm : (b ∈ s.erase a) ↔ b ∈ s :=
+        ⟨fun h => (Finset.mem_erase.mp h).2, fun h => Finset.mem_erase.mpr ⟨Ne.symm hab, h⟩⟩
+      simp only [hm, mul_assoc]
+
+lemma swap_average_product {α : Type*} [DecidableEq α] (s : Finset α) (p : α → ℝ)
+    (hp : ∀ i, 0 ≤ p i) (a b : α) :
+    ((∏ i ∈ s, p i) + ∏ i ∈ s, p (Equiv.swap a b i)) / 2 ≤
+      ∏ i ∈ s, (p i + p (Equiv.swap a b i)) / 2 := by
+  by_cases hab : a = b
+  · subst b
+    simp only [Equiv.swap_self, Equiv.refl_apply]
+    have h : (fun i => (p i + p i) / 2) = p := by funext i; ring
+    rw [h]
+    linarith
+  let t := (s.erase a).erase b
+  have hfix : ∀ i ∈ t, Equiv.swap a b i = i := by
+    intro i hi
+    exact Equiv.swap_apply_of_ne_of_ne (Finset.mem_erase.mp (Finset.mem_erase.mp hi).2).1
+      (Finset.mem_erase.mp hi).1
+  have hprod : (∏ i ∈ t, p (Equiv.swap a b i)) = ∏ i ∈ t, p i :=
+    Finset.prod_congr rfl (fun i hi => by rw [hfix i hi])
+  have havg : (∏ i ∈ t, (p i + p (Equiv.swap a b i)) / 2) = ∏ i ∈ t, p i :=
+    Finset.prod_congr rfl (fun i hi => by rw [hfix i hi]; ring)
+  rw [prod_split_two s p a b hab,
+    prod_split_two s (fun i => p (Equiv.swap a b i)) a b hab,
+    prod_split_two s (fun i => (p i + p (Equiv.swap a b i)) / 2) a b hab]
+  change ((if a ∈ s then p a else 1) * (if b ∈ s then p b else 1) * (∏ i ∈ t, p i) +
+    (if a ∈ s then p (Equiv.swap a b a) else 1) *
+      (if b ∈ s then p (Equiv.swap a b b) else 1) * (∏ i ∈ t, p (Equiv.swap a b i))) / 2 ≤ _
+  rw [hprod]
+  change _ ≤ (if a ∈ s then (p a + p (Equiv.swap a b a)) / 2 else 1) *
+    (if b ∈ s then (p b + p (Equiv.swap a b b)) / 2 else 1) *
+      (∏ i ∈ t, (p i + p (Equiv.swap a b i)) / 2)
+  rw [havg, Equiv.swap_apply_left, Equiv.swap_apply_right]
+  have ht : 0 ≤ ∏ i ∈ t, p i := Finset.prod_nonneg (fun i _ => hp i)
+  by_cases ha : a ∈ s <;> by_cases hb : b ∈ s <;> simp only [ha, hb, ↓reduceIte]
+  · have hsq := mul_nonneg (sq_nonneg (p a - p b)) ht
+    nlinarith
+  all_goals ring_nf; rfl
+
+def CylinderBound {α : Type*} (m : Finset α → ℝ) (p : α → ℝ) : Prop :=
+  ∀ s, m s ≤ ∏ i ∈ s, p i
+
+lemma CylinderBound.fair_swap {α : Type*} [DecidableEq α]
+    {m : Finset α → ℝ} {p : α → ℝ} (h : CylinderBound m p)
+    (hp : ∀ i, 0 ≤ p i) (a b : α) :
+    CylinderBound (fun s => (m s + m (s.image (Equiv.swap a b))) / 2)
+      (fun i => (p i + p (Equiv.swap a b i)) / 2) := by
+  intro s
+  have h₀ := h s
+  have h₁ := h (s.image (Equiv.swap a b))
+  rw [Finset.prod_image (Equiv.swap a b).injective.injOn] at h₁
+  exact (show (m s + m (s.image (Equiv.swap a b))) / 2 ≤
+      ((∏ i ∈ s, p i) + ∏ i ∈ s, p (Equiv.swap a b i)) / 2 by linarith).trans
+    (swap_average_product s p hp a b)
+
+noncomputable def inclusionMoment {α Ω : Type*} [Fintype Ω]
+    (F : Ω → Finset α) (s : Finset α) : ℝ := by
+  classical
+  exact finiteMean (fun ω => if s ⊆ F ω then 1 else 0)
+
+lemma inclusionMoment_nonneg {α Ω : Type*} [Fintype Ω]
+    (F : Ω → Finset α) (s : Finset α) : 0 ≤ inclusionMoment F s := by
+  classical
+  unfold inclusionMoment finiteMean
+  positivity
+
+lemma indicator_subset_prod {α : Type*} [DecidableEq α] (A s : Finset α) :
+    (if s ⊆ A then (1 : ℝ) else 0) = ∏ i ∈ s, (if i ∈ A then 1 else 0) := by
+  by_cases h : s ⊆ A
+  · rw [ite_eq_left h]
+    exact (Finset.prod_eq_one (fun i hi => ite_eq_left (h hi))).symm
+  · rw [ite_eq_right h]
+    obtain ⟨i, hi, hiA⟩ := Finset.not_subset.mp h
+    exact (Finset.prod_eq_zero hi (ite_eq_right hiA)).symm
+
+lemma inclusionMoment_const {α Ω : Type*} [DecidableEq α] [Fintype Ω] [Nonempty Ω]
+    (A s : Finset α) : inclusionMoment (fun _ : Ω => A) s =
+      if s ⊆ A then 1 else 0 := by
+  classical
+  have hn : (Fintype.card Ω : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
+  by_cases h : s ⊆ A <;> simp [inclusionMoment, finiteMean, h, hn]
+
+lemma cylinder_const {α Ω : Type*} [Fintype Ω] [Nonempty Ω] (A : Finset α) :
+    CylinderBound (inclusionMoment (fun _ : Ω => A))
+      (fun i => inclusionMoment (fun _ : Ω => A) {i}) := by
+  classical
+  intro s
+  simp only [inclusionMoment_const, Finset.singleton_subset_iff]
+  exact (indicator_subset_prod A s).le
+
+noncomputable def fairSetSwap {α Ω : Type*} [DecidableEq α]
+    (F : Ω → Finset α) (a b : α) : Ω × Bool → Finset α :=
+  fun ω => if ω.2 then (F ω.1).image (Equiv.swap a b) else F ω.1
+
+lemma subset_swap_image_iff {α : Type*} [DecidableEq α]
+    (A s : Finset α) (a b : α) :
+    s ⊆ A.image (Equiv.swap a b) ↔ s.image (Equiv.swap a b) ⊆ A := by
+  constructor
+  · intro h
+    have h' := Finset.image_subset_image h (f := Equiv.swap a b)
+    simpa only [Finset.image_image, Function.comp_def, Equiv.swap_apply_self, Finset.image_id'] using h'
+  · intro h
+    have h' := Finset.image_subset_image h (f := Equiv.swap a b)
+    simpa only [Finset.image_image, Function.comp_def, Equiv.swap_apply_self, Finset.image_id'] using h'
+
+lemma inclusionMoment_fairSetSwap {α Ω : Type*} [DecidableEq α] [Fintype Ω]
+    (F : Ω → Finset α) (a b : α) (s : Finset α) :
+    inclusionMoment (fairSetSwap F a b) s =
+      (inclusionMoment F s + inclusionMoment F (s.image (Equiv.swap a b))) / 2 := by
+  classical
+  simp only [inclusionMoment, finiteMean, Fintype.sum_prod_type, Fintype.sum_bool,
+    fairSetSwap, Bool.false_eq_true, ↓reduceIte, Fintype.card_prod, Fintype.card_bool,
+    Nat.cast_mul, Nat.cast_ofNat, subset_swap_image_iff, Finset.sum_add_distrib]
+  rw [div_mul_eq_div_div, add_div]
+  ring_nf
+
+lemma cylinder_fairSetSwap {α Ω : Type*} [DecidableEq α] [Fintype Ω]
+    (F : Ω → Finset α) (a b : α)
+    (h : CylinderBound (inclusionMoment F) (fun i => inclusionMoment F {i})) :
+    CylinderBound (inclusionMoment (fairSetSwap F a b))
+      (fun i => inclusionMoment (fairSetSwap F a b) {i}) := by
+  classical
+  have h' := h.fair_swap (fun i => inclusionMoment_nonneg F {i}) a b
+  intro s
+  simpa only [inclusionMoment_fairSetSwap, Finset.image_singleton] using h' s
+
+def SwitchSamples : ℕ → Type
+  | 0 => Unit
+  | n + 1 => SwitchSamples n × Bool
+
+instance (n : ℕ) : Fintype (SwitchSamples n) := by
+  induction n with
+  | zero => exact inferInstanceAs (Fintype Unit)
+  | succ n ih => exact inferInstanceAs (Fintype (SwitchSamples n × Bool))
+
+instance (n : ℕ) : Nonempty (SwitchSamples n) := by
+  induction n with
+  | zero => exact inferInstanceAs (Nonempty Unit)
+  | succ n ih => exact inferInstanceAs (Nonempty (SwitchSamples n × Bool))
+
+noncomputable def switchedSet {α : Type*} [DecidableEq α] (A : Finset α) :
+    (L : List (α × α)) → SwitchSamples L.length → Finset α
+  | [], _ => A
+  | (a,b) :: L, ω => fairSetSwap (switchedSet A L) a b ω
+
+lemma switchedSet_cylinder {α : Type*} [DecidableEq α] (A : Finset α)
+    (L : List (α × α)) :
+    CylinderBound (inclusionMoment (switchedSet A L))
+      (fun i => inclusionMoment (switchedSet A L) {i}) := by
+  induction L with
+  | nil => exact cylinder_const A
+  | cons e L ih => exact cylinder_fairSetSwap (switchedSet A L) e.1 e.2 ih
+
+inductive QueryTree (ι : Type*)
+  | reject : QueryTree ι
+  | accept : QueryTree ι
+  | branch (i : ι) (zero one : QueryTree ι) : QueryTree ι
+  | force (i : ι) (b : Bool) (next : QueryTree ι) : QueryTree ι
+
+namespace QueryTree
+
+variable {ι : Type*} [DecidableEq ι]
+
+def queries : QueryTree ι → Finset ι
+  | reject => ∅
+  | accept => ∅
+  | branch i zero one => insert i (queries zero ∪ queries one)
+  | force i _ next => insert i (queries next)
+
+def Fresh : QueryTree ι → Prop
+  | reject => True
+  | accept => True
+  | branch i zero one => i ∉ queries zero ∧ i ∉ queries one ∧ Fresh zero ∧ Fresh one
+  | force i _ next => i ∉ queries next ∧ Fresh next
+
+def weighted : QueryTree ι → (ι → Bool) → ℝ
+  | reject, _ => 0
+  | accept, _ => 1
+  | branch i zero one, ω => if ω i then weighted one ω else weighted zero ω
+  | force i b next, ω => if ω i = b then 2 * weighted next ω else 0
+
+omit [DecidableEq ι] in
+lemma weighted_nonneg (T : QueryTree ι) (ω : ι → Bool) : 0 ≤ T.weighted ω := by
+  induction T with
+  | reject => exact le_rfl
+  | accept => exact zero_le_one
+  | branch i zero one ih₀ ih₁ => simpa only [weighted] using ite_nonneg ih₁ ih₀
+  | force i b next ih => simpa only [weighted] using ite_nonneg (mul_nonneg (by norm_num) ih) le_rfl
+
+lemma weighted_congr (T : QueryTree ι) {ω η : ι → Bool}
+    (h : ∀ i ∈ T.queries, ω i = η i) : T.weighted ω = T.weighted η := by
+  induction T with
+  | reject => rfl
+  | accept => rfl
+  | branch i zero one ih₀ ih₁ =>
+    have hi := h i (Finset.mem_insert_self _ _)
+    simp only [weighted, hi]
+    rw [ih₀ (fun j hj => h j (Finset.mem_insert_of_mem (Finset.mem_union_left _ hj))),
+      ih₁ (fun j hj => h j (Finset.mem_insert_of_mem (Finset.mem_union_right _ hj)))]
+  | force i b next ih =>
+    have hi := h i (Finset.mem_insert_self _ _)
+    simp only [weighted, hi]
+    rw [ih (fun j hj => h j (Finset.mem_insert_of_mem hj))]
+
+def flipCoin (i : ι) : Equiv.Perm (ι → Bool) :=
+  (show Function.Involutive (fun ω : ι → Bool => Function.update ω i (!(ω i))) from by
+    intro ω
+    funext j
+    by_cases h : j = i
+    · subst j; simp
+    · simp [Function.update_of_ne h]).toPerm
+
+lemma flipCoin_apply_eq (i : ι) (ω : ι → Bool) : flipCoin i ω i = !(ω i) := by
+  change Function.update ω i (!(ω i)) i = _
+  exact Function.update_self _ _ _
+
+lemma flipCoin_apply_ne (i j : ι) (h : j ≠ i) (ω : ι → Bool) :
+    flipCoin i ω j = ω j := by
+  change Function.update ω i (!(ω i)) j = _
+  exact Function.update_of_ne h _ _
+
+lemma weighted_flip_of_not_mem (T : QueryTree ι) (i : ι) (hi : i ∉ T.queries)
+    (ω : ι → Bool) : T.weighted (flipCoin i ω) = T.weighted ω := by
+  apply weighted_congr
+  intro j hj
+  exact flipCoin_apply_ne i j (fun h => hi (h ▸ hj)) ω
+
+lemma mean_indicator_bit [Fintype ι] (f : (ι → Bool) → ℝ) (i : ι) (b : Bool)
+    (hf : ∀ ω, f (flipCoin i ω) = f ω) :
+    finiteMean (fun ω => if ω i = b then f ω else 0) = finiteMean f / 2 := by
+  let g := fun ω : ι → Bool => if ω i = b then f ω else 0
+  have hp (ω : ι → Bool) : g ω + g (flipCoin i ω) = f ω := by
+    dsimp [g]
+    rw [flipCoin_apply_eq, hf]
+    cases hω : ω i <;> cases b <;> simp
+  have hs := Equiv.sum_comp (flipCoin i) g
+  have hadd : (∑ ω, g ω) + (∑ ω, g (flipCoin i ω)) = ∑ ω, f ω := by
+    rw [← Finset.sum_add_distrib]
+    exact Finset.sum_congr rfl (fun ω _ => hp ω)
+  unfold finiteMean
+  change (∑ ω, g ω) / _ = _
+  rw [hs] at hadd
+  have hh : (∑ ω, g ω) = (∑ ω, f ω) / 2 := by linarith
+  rw [hh]
+  ring
+
+lemma mean_weighted_le_one [Fintype ι] (T : QueryTree ι) (hT : T.Fresh) :
+    finiteMean T.weighted ≤ 1 := by
+  induction T with
+  | reject => simp [weighted, finiteMean]
+  | accept => simp [weighted, finiteMean]
+  | branch i zero one ih₀ ih₁ =>
+    obtain ⟨hz, ho, hfz, hfo⟩ := hT
+    have h₀ := mean_indicator_bit zero.weighted i false (zero.weighted_flip_of_not_mem i hz)
+    have h₁ := mean_indicator_bit one.weighted i true (one.weighted_flip_of_not_mem i ho)
+    have he : finiteMean (branch i zero one).weighted =
+        finiteMean (fun ω => if ω i = false then zero.weighted ω else 0) +
+        finiteMean (fun ω => if ω i = true then one.weighted ω else 0) := by
+      unfold finiteMean
+      rw [← add_div, ← Finset.sum_add_distrib]
+      congr 1
+      apply Finset.sum_congr rfl
+      intro ω _
+      cases hω : ω i <;> simp [weighted, hω]
+    rw [he, h₀, h₁]
+    linarith [ih₀ hfz, ih₁ hfo]
+  | force i b next ih =>
+    obtain ⟨hi, hn⟩ := hT
+    have hm := mean_indicator_bit next.weighted i b (next.weighted_flip_of_not_mem i hi)
+    have he : finiteMean (force i b next).weighted =
+        2 * finiteMean (fun ω => if ω i = b then next.weighted ω else 0) := by
+      unfold finiteMean
+      rw [← mul_div_assoc, Finset.mul_sum]
+      congr 1
+      apply Finset.sum_congr rfl
+      intro ω _
+      by_cases h : ω i = b <;> simp [weighted, h]
+    rw [he, hm]
+    linarith [ih hn]
+
+def succeeds : QueryTree ι → (ι → Bool) → Bool
+  | reject, _ => false
+  | accept, _ => true
+  | branch i zero one, ω => if ω i then succeeds one ω else succeeds zero ω
+  | force i b next, ω => if ω i = b then succeeds next ω else false
+
+def charge : QueryTree ι → (ι → Bool) → ℕ
+  | reject, _ => 0
+  | accept, _ => 0
+  | branch i zero one, ω => if ω i then charge one ω else charge zero ω
+  | force i b next, ω => if ω i = b then charge next ω + 1 else 0
+
+omit [DecidableEq ι] in
+lemma weighted_eq (T : QueryTree ι) (ω : ι → Bool) :
+    T.weighted ω = if T.succeeds ω then (2 : ℝ) ^ T.charge ω else 0 := by
+  induction T with
+  | reject => rfl
+  | accept => norm_num [weighted, succeeds, charge]
+  | branch i zero one ih₀ ih₁ =>
+    cases h : ω i <;> simp [weighted, succeeds, charge, h, ih₀, ih₁]
+  | force i b next ih =>
+    by_cases h : ω i = b
+    · simp only [weighted, succeeds, charge, h, ↓reduceIte, ih]
+      cases hs : next.succeeds ω <;> simp [pow_succ, mul_comm]
+    · simp [weighted, succeeds, h]
+
+lemma success_probability_le [Fintype ι] (T : QueryTree ι) (hT : T.Fresh)
+    (K : ℕ) (hK : ∀ ω, T.succeeds ω = true → K ≤ T.charge ω) :
+    finiteMean (fun ω => if T.succeeds ω then (1 : ℝ) else 0) ≤ 1 / (2 : ℝ) ^ K := by
+  have hw (ω : ι → Bool) :
+      (2 : ℝ) ^ K * (if T.succeeds ω then (1 : ℝ) else 0) ≤ T.weighted ω := by
+    rw [weighted_eq]
+    cases hs : T.succeeds ω
+    · simp
+    · simpa using pow_le_pow_right₀ (show (1 : ℝ) ≤ 2 by norm_num) (hK ω hs)
+  have hn : 0 ≤ (Fintype.card (ι → Bool) : ℝ) := Nat.cast_nonneg _
+  have hm := div_le_div_of_nonneg_right
+    (Finset.sum_le_sum (fun ω (_ : ω ∈ Finset.univ) => hw ω)) hn
+  rw [← Finset.mul_sum, mul_div_assoc] at hm
+  change (2 : ℝ) ^ K * finiteMean (fun ω => if T.succeeds ω then 1 else 0) ≤
+    finiteMean T.weighted at hm
+  exact (le_div_iff₀ (pow_pos (by norm_num) K)).mpr (by
+    rw [mul_comm]
+    exact hm.trans (mean_weighted_le_one T hT))
+
+end QueryTree
+
+def encodeButterfly : (d : ℕ) → Butterfly d → SwitchIndex d → Bool
+  | 0, _, i => nomatch i
+  | _d + 1, b, Sum.inl y => b.1 y
+  | d + 1, b, Sum.inr (ε,i) => encodeButterfly d (b.2 ε) i
+
+lemma decode_encodeButterfly (d : ℕ) (b : Butterfly d) :
+    decodeButterfly d (encodeButterfly d b) = b := by
+  induction d with
+  | zero => exact Unit.ext _ _
+  | succ d ih =>
+    apply Prod.ext
+    · rfl
+    · funext ε
+      exact ih (b.2 ε)
+
+lemma encode_decodeButterfly (d : ℕ) (ω : SwitchIndex d → Bool) :
+    encodeButterfly d (decodeButterfly d ω) = ω := by
+  induction d with
+  | zero => funext i; exact Empty.elim i
+  | succ d ih =>
+    funext i
+    cases i with
+    | inl y => rfl
+    | inr z => exact congrFun (ih (fun i => ω (Sum.inr (z.1,i)))) z.2
+
+def butterflyCoinEquiv (d : ℕ) : (SwitchIndex d → Bool) ≃ Butterfly d where
+  toFun := decodeButterfly d
+  invFun := encodeButterfly d
+  left_inv := encode_decodeButterfly d
+  right_inv := decode_encodeButterfly d
+
+noncomputable def routeDomain : (d : ℕ) → Card d → Card d → Finset (SwitchIndex d)
+  | 0, _, _ => ∅
+  | d + 1, x, y => by
+      classical
+      exact insert (Sum.inl (Fin.tail y))
+        ((routeDomain d (Fin.tail x) (Fin.tail y)).image (fun i => Sum.inr (x 0,i)))
+
+def routeValue : (d : ℕ) → Card d → Card d → SwitchIndex d → Bool
+  | 0, _, _, i => nomatch i
+  | _d + 1, x, y, Sum.inl _ => Bool.xor (x 0) (y 0)
+  | d + 1, x, y, Sum.inr (_,i) => routeValue d (Fin.tail x) (Fin.tail y) i
+
+lemma card_routeDomain (d : ℕ) (x y : Card d) : (routeDomain d x y).card = d := by
+  classical
+  induction d with
+  | zero => rfl
+  | succ d ih =>
+    change (insert (Sum.inl (Fin.tail y))
+      ((routeDomain d (Fin.tail x) (Fin.tail y)).image
+        (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) :
+      Finset (Sum (Card d) (Bool × SwitchIndex d))).card = d + 1
+    rw [Finset.card_insert_of_notMem (by simp)]
+    rw [Finset.card_image_of_injective _ (by intro i j h; cases h; rfl), ih]
+
+end Thorp
+
+end ThorpNine.Smoothing
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Smoothing/ExposedCoins.lean b/lean/OAI/Probability/ThorpRouting/Smoothing/ExposedCoins.lean
index 18af04399089ab4737630dd6cfd9095b66700f0f..3571af9b45fee74cc18105a5f7ece56649a8c70a 100644
--- a/lean/OAI/Probability/ThorpRouting/Smoothing/ExposedCoins.lean
+++ b/lean/OAI/Probability/ThorpRouting/Smoothing/ExposedCoins.lean
@@ -1,856 +1,856 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Smoothing.Coins
-
-namespace OAI
-
-namespace ThorpNine.Smoothing
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-
-lemma xor_eq_iff (a b c : Bool) : Bool.xor a b = c ↔ b = Bool.xor a c := by
-  cases a <;> cases b <;> cases c <;> decide
-
-lemma butterfly_endpoint_iff (d : ℕ) (ω : SwitchIndex d → Bool) (x y : Card d) :
-    butterflyPerm d (decodeButterfly d ω) x = y ↔
-      ∀ i ∈ routeDomain d x y, ω i = routeValue d x y i := by
-  classical
-  induction d with
-  | zero =>
-    have he : butterflyPerm 0 (decodeButterfly 0 ω) x = y := Subsingleton.elim _ _
-    simp [he, routeDomain]
-  | succ d ih =>
-    let v := butterflyPerm d (decodeButterfly d (fun i => ω (Sum.inr (x 0,i)))) (Fin.tail x)
-    change Fin.cons (Bool.xor (x 0) (ω (Sum.inl v))) v = y ↔ _
-    have hv : v = Fin.tail y ↔
-        ∀ i ∈ routeDomain d (Fin.tail x) (Fin.tail y),
-          ω (Sum.inr (x 0,i)) = routeValue d (Fin.tail x) (Fin.tail y) i :=
-      ih _ _ _
-    constructor
-    · intro h i hi
-      have ht : v = Fin.tail y := by simpa using congrArg Fin.tail h
-      have hh : ω (Sum.inl (Fin.tail y)) = Bool.xor (x 0) (y 0) := by
-        have he := congrFun h 0
-        simp only [Fin.cons_zero, ht] at he
-        exact (xor_eq_iff _ _ _).mp he
-      rcases Finset.mem_insert.mp hi with htop | hchild
-      · subst i
-        exact hh
-      · obtain ⟨j, hj, rfl⟩ := Finset.mem_image.mp hchild
-        exact hv.mp ht j hj
-    · intro h
-      have hh := h (Sum.inl (Fin.tail y)) (Finset.mem_insert_self _ _)
-      have ht : v = Fin.tail y := hv.mpr (fun i hi =>
-        h (Sum.inr (x 0,i)) (Finset.mem_insert_of_mem (Finset.mem_image_of_mem _ hi)))
-      have hh' : Bool.xor (x 0) (ω (Sum.inl (Fin.tail y))) = y 0 :=
-        (xor_eq_iff _ _ _).mpr hh
-      rw [ht, hh', Fin.cons_self_tail]
-
-lemma finiteMean_congr {Ω : Type*} [Fintype Ω] {f g : Ω → ℝ}
-    (h : ∀ ω, f ω = g ω) : finiteMean f = finiteMean g := by
-  unfold finiteMean
-  congr 1
-  exact Finset.sum_congr rfl (fun ω _ => h ω)
-
-lemma mean_coinCylinder {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (s : Finset ι) (v : ι → Bool) :
-    finiteMean (fun ω : ι → Bool => if ∀ i ∈ s, ω i = v i then (1 : ℝ) else 0) =
-      1 / (2 : ℝ) ^ s.card := by
-  induction s using Finset.induction_on with
-  | empty => simp [finiteMean]
-  | @insert i s hi ih =>
-    let f := fun ω : ι → Bool => if ∀ j ∈ s, ω j = v j then (1 : ℝ) else 0
-    have hf : ∀ ω, f (QueryTree.flipCoin i ω) = f ω := by
-      intro ω
-      have he : (∀ j ∈ s, QueryTree.flipCoin i ω j = v j) ↔ ∀ j ∈ s, ω j = v j := by
-        apply forall₂_congr
-        intro j hj
-        rw [QueryTree.flipCoin_apply_ne i j (fun h => hi (h ▸ hj))]
-      dsimp [f]
-      simp only [he]
-    have hm := QueryTree.mean_indicator_bit f i (v i) hf
-    have he : (fun ω : ι → Bool => if ∀ j ∈ insert i s, ω j = v j then (1 : ℝ) else 0) =
-        (fun ω => if ω i = v i then f ω else 0) := by
-      funext ω
-      by_cases h : ω i = v i <;> simp [f, h]
-    rw [he, hm]
-    change finiteMean (fun ω : ι → Bool => if ∀ j ∈ s, ω j = v j then (1 : ℝ) else 0) / 2 = _
-    rw [ih, Finset.card_insert_of_notMem hi, pow_succ]
-    ring
-
-def exposedCoins {ι : Type*} [DecidableEq ι] (R : Finset ι) (η ω : ι → Bool) : ι → Bool :=
-  fun i => if i ∈ R then η i else ω i
-
-lemma mean_coinCylinder_exposed {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (s R : Finset ι) (v η : ι → Bool) :
-    finiteMean (fun ω : ι → Bool =>
-      if ∀ i ∈ s, exposedCoins R η ω i = v i then (1 : ℝ) else 0) =
-    if (∀ i ∈ s ∩ R, η i = v i) then 1 / (2 : ℝ) ^ (s \ R).card else 0 := by
-  have he (ω : ι → Bool) : (∀ i ∈ s, exposedCoins R η ω i = v i) ↔
-      (∀ i ∈ s ∩ R, η i = v i) ∧ (∀ i ∈ s \ R, ω i = v i) := by
-    constructor
-    · intro h
-      constructor
-      · intro i hi
-        simpa [exposedCoins, (Finset.mem_inter.mp hi).2] using h i (Finset.mem_inter.mp hi).1
-      · intro i hi
-        simpa [exposedCoins, (Finset.mem_sdiff.mp hi).2] using h i (Finset.mem_sdiff.mp hi).1
-    · rintro ⟨ho, hf⟩ i hi
-      by_cases hR : i ∈ R
-      · simpa [exposedCoins, hR] using ho i (Finset.mem_inter.mpr ⟨hi,hR⟩)
-      · simpa [exposedCoins, hR] using hf i (Finset.mem_sdiff.mpr ⟨hi,hR⟩)
-  have hm : finiteMean (fun ω : ι → Bool =>
-      if ∀ i ∈ s, exposedCoins R η ω i = v i then (1 : ℝ) else 0) =
-      finiteMean (fun ω : ι → Bool =>
-        if (∀ i ∈ s ∩ R, η i = v i) ∧ (∀ i ∈ s \ R, ω i = v i) then (1 : ℝ) else 0) :=
-    finiteMean_congr (fun ω => by simp only [he ω])
-  rw [hm]
-  by_cases h : ∀ i ∈ s ∩ R, η i = v i
-  · have hp : (∀ i ∈ s ∩ R, η i = v i) ↔ True := iff_true_intro h
-    simp only [hp, true_and, ↓reduceIte]
-    exact mean_coinCylinder (s \ R) v
-  · have hp : (∀ i ∈ s ∩ R, η i = v i) ↔ False := iff_false_intro h
-    simp only [hp, false_and, ↓reduceIte]
-    simp [finiteMean]
-
-lemma butterfly_closure_probability (d : ℕ) (x y : Card d)
-    (R : Finset (SwitchIndex d)) (η : SwitchIndex d → Bool) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      if butterflyPerm d (decodeButterfly d (exposedCoins R η ω)) x = y then (1 : ℝ) else 0) =
-      if (∀ i ∈ routeDomain d x y ∩ R, η i = routeValue d x y i) then
-        (2 : ℝ) ^ (routeDomain d x y ∩ R).card / (2 : ℝ) ^ d else 0 := by
-  have he : finiteMean (fun ω : SwitchIndex d → Bool =>
-      if butterflyPerm d (decodeButterfly d (exposedCoins R η ω)) x = y then (1 : ℝ) else 0) =
-      finiteMean (fun ω : SwitchIndex d → Bool =>
-        if ∀ i ∈ routeDomain d x y, exposedCoins R η ω i = routeValue d x y i then (1 : ℝ) else 0) :=
-    finiteMean_congr (fun ω => by simp only [butterfly_endpoint_iff])
-  rw [he, mean_coinCylinder_exposed]
-  congr 1
-  have hc : (routeDomain d x y \ R).card + (routeDomain d x y ∩ R).card = d := by
-    rw [Finset.card_sdiff_add_card_inter, card_routeDomain]
-  have hp : (2 : ℝ) ^ d = (2 : ℝ) ^ (routeDomain d x y \ R).card *
-      (2 : ℝ) ^ (routeDomain d x y ∩ R).card := by
-    rw [← pow_add]
-    exact congrArg (fun n => (2 : ℝ) ^ n) hc.symm
-  rw [hp]
-  have hn : (2 : ℝ) ^ (routeDomain d x y ∩ R).card ≠ 0 := pow_ne_zero _ (by norm_num)
-  field_simp
-
-lemma butterfly_single_card_uniform (d : ℕ) (x y : Card d) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      if butterflyPerm d (decodeButterfly d ω) x = y then (1 : ℝ) else 0) = 1 / (2 : ℝ) ^ d := by
-  have he : finiteMean (fun ω : SwitchIndex d → Bool =>
-      if butterflyPerm d (decodeButterfly d ω) x = y then (1 : ℝ) else 0) =
-      finiteMean (fun ω : SwitchIndex d → Bool =>
-        if ∀ i ∈ routeDomain d x y, ω i = routeValue d x y i then (1 : ℝ) else 0) :=
-    finiteMean_congr (fun ω => by simp only [butterfly_endpoint_iff])
-  rw [he, mean_coinCylinder, card_routeDomain]
-
-def switchUsers : (d : ℕ) → Butterfly d → SwitchIndex d → Card d × Card d
-  | 0, _, i => nomatch i
-  | d + 1, b, Sum.inl y =>
-      (Fin.cons false ((butterflyPerm d (b.2 false)).symm y),
-       Fin.cons true ((butterflyPerm d (b.2 true)).symm y))
-  | d + 1, b, Sum.inr (ε,i) =>
-      (Fin.cons ε (switchUsers d (b.2 ε) i).1,
-       Fin.cons ε (switchUsers d (b.2 ε) i).2)
-
-lemma switchUsers_ne (d : ℕ) (b : Butterfly d) (i : SwitchIndex d) :
-    (switchUsers d b i).1 ≠ (switchUsers d b i).2 := by
-  induction d with
-  | zero => exact Empty.elim i
-  | succ d ih =>
-    cases i with
-    | inl y =>
-      intro h
-      have := congrFun h 0
-      simp only [switchUsers, Fin.cons_zero] at this
-      exact Bool.false_ne_true this
-    | inr z =>
-      intro h
-      exact ih (b.2 z.1) z.2 (by simpa only [switchUsers, Fin.tail_cons] using congrArg Fin.tail h)
-
-lemma cons_eq_iff {d : ℕ} (ε : Bool) (a : Card d) (x : Card (d + 1)) :
-    Fin.cons ε a = x ↔ ε = x 0 ∧ a = Fin.tail x := by
-  constructor
-  · intro h
-    exact ⟨by simpa only [Fin.cons_zero] using congrFun h 0,
-      by simpa only [Fin.tail_cons] using congrArg Fin.tail h⟩
-  · rintro ⟨rfl,rfl⟩
-    exact Fin.cons_self_tail x
-
-lemma mem_routeDomain_iff_user (d : ℕ) (b : Butterfly d) (x : Card d) (i : SwitchIndex d) :
-    i ∈ routeDomain d x (butterflyPerm d b x) ↔
-      (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x := by
-  classical
-  induction d with
-  | zero => exact Empty.elim i
-  | succ d ih =>
-    cases i with
-    | inl y =>
-      change Sum.inl y ∈ insert (Sum.inl (Fin.tail (butterflyPerm (d+1) b x)))
-        ((routeDomain d (Fin.tail x) (Fin.tail (butterflyPerm (d+1) b x))).image
-          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) ↔ _
-      simp only [Finset.mem_insert, Sum.inl.injEq, Finset.mem_image,
-        Sum.inr_ne_inl, and_false, exists_false, or_false]
-      change y = butterflyPerm d (b.2 (x 0)) (Fin.tail x) ↔ _
-      simp only [switchUsers, cons_eq_iff]
-      cases x 0 <;> simp [Equiv.symm_apply_eq]
-    | inr z =>
-      obtain ⟨ε,i⟩ := z
-      change Sum.inr (ε,i) ∈ insert (Sum.inl (Fin.tail (butterflyPerm (d+1) b x)))
-        ((routeDomain d (Fin.tail x) (Fin.tail (butterflyPerm (d+1) b x))).image
-          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) ↔ _
-      simp only [Finset.mem_insert, Sum.inr_ne_inl, false_or, Finset.mem_image,
-        Sum.inr.injEq, Prod.mk.injEq]
-      change (∃ j, j ∈ routeDomain d (Fin.tail x) (butterflyPerm d (b.2 (x 0)) (Fin.tail x)) ∧
-        x 0 = ε ∧ j = i) ↔ _
-      simp only [switchUsers, cons_eq_iff]
-      constructor
-      · rintro ⟨j,hj,hε,rfl⟩
-        subst ε
-        rcases (ih (b.2 (x 0)) (Fin.tail x) j).mp hj with h | h
-        · exact Or.inl ⟨rfl,h⟩
-        · exact Or.inr ⟨rfl,h⟩
-      · rintro (⟨hε,h⟩ | ⟨hε,h⟩)
-        · subst ε
-          exact ⟨i,(ih (b.2 (x 0)) (Fin.tail x) i).mpr (Or.inl h),rfl,rfl⟩
-        · subst ε
-          exact ⟨i,(ih (b.2 (x 0)) (Fin.tail x) i).mpr (Or.inr h),rfl,rfl⟩
-
-lemma card_switches_using (d : ℕ) (b : Butterfly d) (x : Card d) :
-    (Finset.univ.filter (fun i : SwitchIndex d =>
-      (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x)).card = d := by
-  classical
-  have he : Finset.univ.filter (fun i : SwitchIndex d =>
-      (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x) =
-      routeDomain d x (butterflyPerm d b x) := by
-    ext i
-    simp only [Finset.mem_filter, Finset.mem_univ, true_and, mem_routeDomain_iff_user]
-  rw [he,card_routeDomain]
-
-lemma switch_incidence_sum (d : ℕ) (b : Butterfly d) (w : Card d → ℝ) :
-    ∑ i : SwitchIndex d, (w (switchUsers d b i).1 + w (switchUsers d b i).2) =
-      (d : ℝ) * ∑ x : Card d, w x := by
-  classical
-  calc
-    _ = ∑ i : SwitchIndex d, ∑ x : Card d,
-        if (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x then w x else 0 := by
-      apply Finset.sum_congr rfl
-      intro i _
-      rw [Finset.sum_ite]
-      simp only [Finset.sum_const_zero, add_zero]
-      have he : Finset.univ.filter (fun x : Card d =>
-          (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x) =
-          {(switchUsers d b i).1, (switchUsers d b i).2} := by
-        ext x; simp [eq_comm]
-      rw [he, Finset.sum_pair (switchUsers_ne d b i)]
-    _ = ∑ x : Card d, ∑ i : SwitchIndex d,
-        if (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x then w x else 0 :=
-      Finset.sum_comm
-    _ = ∑ x : Card d, (d : ℝ) * w x := by
-      apply Finset.sum_congr rfl
-      intro x _
-      rw [Finset.sum_ite]
-      simp only [Finset.sum_const_zero, add_zero, Finset.sum_const, nsmul_eq_mul,
-        card_switches_using]
-    _ = _ := by rw [Finset.mul_sum]
-
-lemma weighted_switch_min_bound (d : ℕ) (b : Butterfly d) (w : Card d → ℝ) :
-    ∑ i : SwitchIndex d, min (w (switchUsers d b i).1) (w (switchUsers d b i).2) ≤
-      (d : ℝ) / 2 * ∑ x : Card d, w x := by
-  have h : 2 * (∑ i : SwitchIndex d,
-      min (w (switchUsers d b i).1) (w (switchUsers d b i).2)) ≤
-      ∑ i : SwitchIndex d, (w (switchUsers d b i).1 + w (switchUsers d b i).2) := by
-    rw [Finset.mul_sum]
-    apply Finset.sum_le_sum
-    intro i _
-    linarith [min_le_left (w (switchUsers d b i).1) (w (switchUsers d b i).2),
-      min_le_right (w (switchUsers d b i).1) (w (switchUsers d b i).2)]
-  rw [switch_incidence_sum] at h
-  linarith
-
-lemma mem_childMarks {d : ℕ} (s : Finset (Card (d + 1))) (ε : Bool) (x : Card d) :
-    x ∈ childMarks s ε ↔ Fin.cons ε x ∈ s := by
-  classical
-  constructor
-  · intro h
-    obtain ⟨y,hy,hxy⟩ := Finset.mem_image.mp h
-    obtain ⟨hys,hyε⟩ := Finset.mem_filter.mp hy
-    rw [← hxy, ← hyε, Fin.cons_self_tail]
-    exact hys
-  · intro h
-    exact Finset.mem_image.mpr ⟨Fin.cons ε x,
-      Finset.mem_filter.mpr ⟨h,by simp⟩,by simp⟩
-
-lemma mem_topMeetings {d : ℕ} (b : Butterfly (d + 1))
-    (s : Finset (Card (d + 1))) (y : Card d) :
-    y ∈ topMeetings b s ↔
-      (switchUsers (d+1) b (Sum.inl y)).1 ∈ s ∧
-      (switchUsers (d+1) b (Sum.inl y)).2 ∈ s := by
-  classical
-  have he (ε : Bool) : y ∈ (childMarks s ε).image (butterflyPerm d (b.2 ε)) ↔
-      (butterflyPerm d (b.2 ε)).symm y ∈ childMarks s ε := by
-    constructor
-    · intro h
-      obtain ⟨x,hx,rfl⟩ := Finset.mem_image.mp h
-      simpa only [Equiv.symm_apply_apply] using hx
-    · intro h
-      exact Finset.mem_image.mpr ⟨_,h,Equiv.apply_symm_apply _ _⟩
-  simp only [topMeetings, Finset.mem_inter, he, mem_childMarks, switchUsers]
-
-lemma encounters_eq_switch_sum (d : ℕ) (b : Butterfly d) (s : Finset (Card d)) :
-    encounters d b s = ∑ i : SwitchIndex d,
-      if (switchUsers d b i).1 ∈ s ∧ (switchUsers d b i).2 ∈ s then 1 else 0 := by
-  classical
-  induction d with
-  | zero =>
-    simp only [encounters]
-    symm
-    apply Finset.sum_eq_zero
-    intro i _
-    exact Empty.elim i
-  | succ d ih =>
-    change encounters d (b.2 false) (childMarks s false) +
-      encounters d (b.2 true) (childMarks s true) + (topMeetings b s).card =
-        ∑ i : Sum (Card d) (Bool × SwitchIndex d), _
-    rw [Fintype.sum_sum_type, Fintype.sum_prod_type]
-    simp only [Fintype.sum_bool]
-    rw [ih,ih]
-    have ht : (∑ y : Card d,
-        if (switchUsers (d+1) b (Sum.inl y)).1 ∈ s ∧
-          (switchUsers (d+1) b (Sum.inl y)).2 ∈ s then 1 else 0) =
-        (topMeetings b s).card := by
-      simp only [← mem_topMeetings]
-      simp
-    rw [ht]
-    simp only [mem_childMarks, switchUsers]
-    ac_rfl
-
-lemma marked_switch_entropy_bound (d : ℕ) (b : Butterfly d) (s : Finset (Card d)) :
-    (∑ i : SwitchIndex d,
-      if (switchUsers d b i).1 ∈ s ∧ (switchUsers d b i).2 ∈ s then (1:ℝ) else 0) ≤
-      (s.card : ℝ) * Real.log s.card / (2 * Real.log 2) := by
-  classical
-  have he : (encounters d b s : ℝ) =
-      ∑ i : SwitchIndex d,
-        if (switchUsers d b i).1 ∈ s ∧ (switchUsers d b i).2 ∈ s then (1:ℝ) else 0 := by
-    rw [encounters_eq_switch_sum, Nat.cast_sum]
-    apply Finset.sum_congr rfl
-    intro i _
-    split_ifs <;> simp
-  rw [← he]
-  exact butterfly_encounters d b s
-
-namespace QueryTree
-variable {ι : Type*} [DecidableEq ι]
-
-def memo : QueryTree ι → Finset ι → (ι → Bool) → QueryTree ι
-  | reject, _, _ => reject
-  | accept, _, _ => accept
-  | branch i zero one, R, η =>
-      if i ∈ R then (if η i then memo one R η else memo zero R η) else
-      branch i (memo zero (insert i R) (Function.update η i false))
-        (memo one (insert i R) (Function.update η i true))
-  | force i b next, R, η =>
-      if i ∈ R then (if η i = b then memo next R η else reject) else
-      force i b (memo next (insert i R) (Function.update η i b))
-
-lemma memo_queries_disjoint (T : QueryTree ι) (R : Finset ι) (η : ι → Bool) :
-    Disjoint (memo T R η).queries R := by
-  induction T generalizing R η with
-  | reject => simp [memo,queries]
-  | accept => simp [memo,queries]
-  | branch i zero one ih₀ ih₁ =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte]
-      cases η i <;> simp only [Bool.false_eq_true,↓reduceIte]
-      · exact ih₀ R η
-      · exact ih₁ R η
-    · simp only [memo,hi,↓reduceIte,queries,Finset.disjoint_insert_left,Finset.disjoint_union_left]
-      exact ⟨by simp, (ih₀ (insert i R) _).mono_right (Finset.subset_insert _ _),
-        (ih₁ (insert i R) _).mono_right (Finset.subset_insert _ _)⟩
-  | force i b next ih =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte]
-      by_cases hb : η i = b
-      · simpa only [hb,↓reduceIte] using ih R η
-      · simp [hb,queries]
-    · simp only [memo,hi,↓reduceIte,queries,Finset.disjoint_insert_left]
-      exact ⟨by simp,(ih (insert i R) _).mono_right (Finset.subset_insert _ _)⟩
-
-lemma memo_fresh (T : QueryTree ι) (R : Finset ι) (η : ι → Bool) :
-    (memo T R η).Fresh := by
-  induction T generalizing R η with
-  | reject => trivial
-  | accept => trivial
-  | branch i zero one ih₀ ih₁ =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte]
-      cases η i <;> simp only [Bool.false_eq_true,↓reduceIte]
-      · exact ih₀ R η
-      · exact ih₁ R η
-    · simp only [memo,hi,↓reduceIte,Fresh]
-      exact ⟨fun hm => Finset.disjoint_left.mp (memo_queries_disjoint zero (insert i R) _) hm
-          (Finset.mem_insert_self _ _),
-        fun hm => Finset.disjoint_left.mp (memo_queries_disjoint one (insert i R) _) hm
-          (Finset.mem_insert_self _ _), ih₀ _ _,ih₁ _ _⟩
-  | force i b next ih =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte]
-      by_cases hb : η i = b
-      · simpa only [hb,↓reduceIte] using ih R η
-      · simp [hb,Fresh]
-    · simp only [memo,hi,↓reduceIte,Fresh]
-      exact ⟨fun hm => Finset.disjoint_left.mp (memo_queries_disjoint next (insert i R) _) hm
-        (Finset.mem_insert_self _ _),ih _ _⟩
-
-lemma update_agrees {R : Finset ι} {η ω : ι → Bool}
-    (h : ∀ i ∈ R, η i = ω i) (i : ι) :
-    ∀ j ∈ insert i R, Function.update η i (ω i) j = ω j := by
-  intro j hj
-  rcases Finset.mem_insert.mp hj with rfl | hj
-  · simp
-  · by_cases hji : j = i
-    · subst j; simp
-    · rw [Function.update_of_ne hji]
-      exact h j hj
-
-lemma memo_succeeds (T : QueryTree ι) (R : Finset ι) (η ω : ι → Bool)
-    (h : ∀ i ∈ R, η i = ω i) : (memo T R η).succeeds ω = T.succeeds ω := by
-  induction T generalizing R η with
-  | reject => rfl
-  | accept => rfl
-  | branch i zero one ih₀ ih₁ =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte,succeeds,h i hi]
-      cases ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
-      · exact ih₀ R η h
-      · exact ih₁ R η h
-    · simp only [memo,hi,↓reduceIte,succeeds]
-      cases hb : ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
-      · exact ih₀ _ _ (by simpa only [hb] using update_agrees h i)
-      · exact ih₁ _ _ (by simpa only [hb] using update_agrees h i)
-  | force i b next ih =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte,h i hi]
-      by_cases hb : ω i = b
-      · simp only [hb,↓reduceIte,succeeds]
-        exact ih R η h
-      · simp [hb,succeeds]
-    · simp only [memo,hi,↓reduceIte,succeeds]
-      by_cases hb : ω i = b
-      · simp only [hb,↓reduceIte]
-        exact ih _ _ (by simpa only [hb] using update_agrees h i)
-      · simp [hb]
-
-def freshCharge : QueryTree ι → Finset ι → (ι → Bool) → ℕ
-  | reject, _, _ => 0
-  | accept, _, _ => 0
-  | branch i zero one, R, ω => if ω i then freshCharge one (insert i R) ω else freshCharge zero (insert i R) ω
-  | force i b next, R, ω => if ω i = b then
-      freshCharge next (insert i R) ω + (if i ∈ R then 0 else 1) else 0
-
-lemma memo_charge (T : QueryTree ι) (R : Finset ι) (η ω : ι → Bool)
-    (h : ∀ i ∈ R, η i = ω i) : (memo T R η).charge ω = freshCharge T R ω := by
-  induction T generalizing R η with
-  | reject => rfl
-  | accept => rfl
-  | branch i zero one ih₀ ih₁ =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte,freshCharge,h i hi,Finset.insert_eq_of_mem hi]
-      cases ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
-      · exact ih₀ R η h
-      · exact ih₁ R η h
-    · simp only [memo,hi,↓reduceIte,charge,freshCharge]
-      cases hb : ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
-      · exact ih₀ _ _ (by simpa only [hb] using update_agrees h i)
-      · exact ih₁ _ _ (by simpa only [hb] using update_agrees h i)
-  | force i b next ih =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte,h i hi,freshCharge,Finset.insert_eq_of_mem hi,Nat.add_zero]
-      by_cases hb : ω i = b
-      · simp only [hb,↓reduceIte]
-        exact ih R η h
-      · simp [hb,charge]
-    · simp only [memo,hi,↓reduceIte,charge,freshCharge]
-      by_cases hb : ω i = b
-      · simp only [hb,↓reduceIte]
-        rw [ih _ _ (by simpa only [hb] using update_agrees h i)]
-      · simp [hb]
-
-lemma mean_freshCharge_le_one [Fintype ι] (T : QueryTree ι) :
-    finiteMean (fun ω : ι → Bool =>
-      if T.succeeds ω then (2:ℝ) ^ T.freshCharge ∅ ω else 0) ≤ 1 := by
-  let η : ι → Bool := fun _ => false
-  have hm := mean_weighted_le_one (memo T ∅ η) (memo_fresh T ∅ η)
-  have he : finiteMean (memo T ∅ η).weighted =
-      finiteMean (fun ω : ι → Bool =>
-        if T.succeeds ω then (2:ℝ) ^ T.freshCharge ∅ ω else 0) := by
-    apply finiteMean_congr
-    intro ω
-    rw [weighted_eq, memo_succeeds _ _ _ _ (by simp), memo_charge _ _ _ _ (by simp)]
-  rw [he] at hm
-  exact hm
-
-
-def followPath : (d : ℕ) → (SwitchIndex d → ι) → Card d →
-    (Card d → QueryTree ι) → QueryTree ι
-  | 0, _, x, k => k x
-  | d + 1, e, x, k => followPath d (fun i => e (Sum.inr (x 0,i))) (Fin.tail x)
-      (fun y => branch (e (Sum.inl y))
-        (k (Fin.cons (Bool.xor (x 0) false) y))
-        (k (Fin.cons (Bool.xor (x 0) true) y)))
-
-omit [DecidableEq ι] in
-lemma followPath_succeeds (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
-    (k : Card d → QueryTree ι) (ω : ι → Bool) :
-    (followPath d e x k).succeeds ω =
-      (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).succeeds ω := by
-  induction d with
-  | zero => rfl
-  | succ d ih =>
-    rw [followPath, ih]
-    let y := butterflyPerm d
-      (decodeButterfly d (fun i => ω (e (Sum.inr (x 0,i))))) (Fin.tail x)
-    change (if ω (e (Sum.inl y)) then
-      (k (Fin.cons (Bool.xor (x 0) true) y)).succeeds ω else
-      (k (Fin.cons (Bool.xor (x 0) false) y)).succeeds ω) =
-      (k (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).succeeds ω
-    cases ω (e (Sum.inl y)) <;> rfl
-
-lemma followPath_charge (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
-    (k : Card d → QueryTree ι) (R : Finset ι) (ω : ι → Bool) :
-    freshCharge (followPath d e x k) R ω =
-      freshCharge (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x))
-        (R ∪ (routeDomain d x (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).image e) ω := by
-  classical
-  induction d generalizing R with
-  | zero => simp only [followPath, routeDomain, Finset.image_empty, Finset.union_empty]; rfl
-  | succ d ih =>
-    rw [followPath, ih]
-    let y := butterflyPerm d
-      (decodeButterfly d (fun i => ω (e (Sum.inr (x 0,i))))) (Fin.tail x)
-    let S := (routeDomain d (Fin.tail x) y).image (fun i => e (Sum.inr (x 0,i)))
-    change (if ω (e (Sum.inl y)) then
-      freshCharge (k (Fin.cons (Bool.xor (x 0) true) y)) (insert (e (Sum.inl y)) (R ∪ S)) ω else
-      freshCharge (k (Fin.cons (Bool.xor (x 0) false) y)) (insert (e (Sum.inl y)) (R ∪ S)) ω) =
-      freshCharge (k (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y))
-        (R ∪ (routeDomain (d+1) x
-          (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).image e) ω
-    have hs : (routeDomain (d+1) x
-        (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).image e =
-        insert (e (Sum.inl y)) S := by
-      let e' : Sum (Card d) (Bool × SwitchIndex d) → ι := e
-      change (insert (Sum.inl y)
-        ((routeDomain d (Fin.tail x) y).image
-          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) :
-          Finset (Sum (Card d) (Bool × SwitchIndex d))).image e' = _
-      rw [Finset.image_insert, Finset.image_image]
-      rfl
-    rw [hs,Finset.union_insert]
-    cases ω (e (Sum.inl y)) <;> rfl
-
-def forceList : List ι → (ι → Bool) → QueryTree ι → QueryTree ι
-  | [], _, k => k
-  | i :: L, v, k => force i (v i) (forceList L v k)
-
-omit [DecidableEq ι] in
-lemma forceList_succeeds (L : List ι) (v : ι → Bool) (k : QueryTree ι) (ω : ι → Bool) :
-    (forceList L v k).succeeds ω =
-      if ∀ i ∈ L, ω i = v i then k.succeeds ω else false := by
-  induction L with
-  | nil => simp [forceList]
-  | cons i L ih =>
-    simp only [forceList,succeeds,ih,List.mem_cons,forall_eq_or_imp]
-    by_cases h : ω i = v i <;> simp [h]
-
-lemma forceList_charge (L : List ι) (v : ι → Bool) (k : QueryTree ι)
-    (R : Finset ι) (ω : ι → Bool) (h : ∀ i ∈ L, ω i = v i) :
-    freshCharge (forceList L v k) R ω =
-      freshCharge k (R ∪ L.toFinset) ω + (L.toFinset \ R).card := by
-  induction L generalizing R with
-  | nil => simp [forceList]
-  | cons i L ih =>
-    rw [forceList,freshCharge,ite_eq_left (h i List.mem_cons_self),
-      ih (insert i R) (fun j hj => h j (List.mem_cons_of_mem _ hj))]
-    simp only [List.toFinset_cons]
-    have hs : insert i R ∪ L.toFinset = R ∪ insert i L.toFinset := by ext j; simp
-    rw [hs]
-    by_cases hi : i ∈ R
-    · rw [ite_eq_left hi]
-      have he : L.toFinset \ insert i R = insert i L.toFinset \ R := by
-        ext j
-        by_cases hji : j = i
-        · subst j; simp [hi]
-        · simp [hji]
-      rw [he,Nat.add_zero]
-    · rw [ite_eq_right hi]
-      have he : insert i L.toFinset \ R = insert i (L.toFinset \ insert i R) := by
-        ext j
-        by_cases hji : j = i
-        · subst j; simp [hi]
-        · simp [hji]
-      rw [he, Finset.card_insert_of_notMem (by simp)]
-      omega
-
-
-def staleCharge : QueryTree ι → Finset ι → (ι → Bool) → ℕ
-  | reject, _, _ => 0
-  | accept, _, _ => 0
-  | branch i zero one, R, ω => if ω i then staleCharge one (insert i R) ω
-      else staleCharge zero (insert i R) ω
-  | force i b next, R, ω => if ω i = b then
-      staleCharge next (insert i R) ω + (if i ∈ R then 1 else 0) else 0
-
-lemma fresh_add_stale (T : QueryTree ι) (R : Finset ι) (ω : ι → Bool) :
-    T.freshCharge R ω + T.staleCharge R ω = T.charge ω := by
-  induction T generalizing R with
-  | reject => rfl
-  | accept => rfl
-  | branch i zero one ih₀ ih₁ =>
-    cases hb : ω i <;> simp [freshCharge,staleCharge,charge,hb,ih₀,ih₁]
-  | force i b next ih =>
-    by_cases hb : ω i = b
-    · simp only [freshCharge,staleCharge,charge,hb,↓reduceIte]
-      have he := ih (insert i R)
-      by_cases hi : i ∈ R <;> simp only [hi,↓reduceIte] <;> omega
-    · simp [freshCharge,staleCharge,charge,hb]
-
-omit [DecidableEq ι] in
-lemma followPath_raw_charge (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
-    (k : Card d → QueryTree ι) (ω : ι → Bool) :
-    (followPath d e x k).charge ω =
-      (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).charge ω := by
-  induction d with
-  | zero => rfl
-  | succ d ih =>
-    rw [followPath, ih]
-    let y := butterflyPerm d
-      (decodeButterfly d (fun i => ω (e (Sum.inr (x 0,i))))) (Fin.tail x)
-    change (if ω (e (Sum.inl y)) then
-      (k (Fin.cons (Bool.xor (x 0) true) y)).charge ω else
-      (k (Fin.cons (Bool.xor (x 0) false) y)).charge ω) =
-      (k (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).charge ω
-    cases ω (e (Sum.inl y)) <;> rfl
-
-omit [DecidableEq ι] in
-lemma forceList_raw_charge (L : List ι) (v : ι → Bool) (k : QueryTree ι) (ω : ι → Bool)
-    (h : ∀ i ∈ L, ω i = v i) :
-    (forceList L v k).charge ω = k.charge ω + L.length := by
-  induction L with
-  | nil => simp [forceList]
-  | cons i L ih =>
-    rw [forceList,charge,ite_eq_left (h i List.mem_cons_self),
-      ih (fun j hj => h j (List.mem_cons_of_mem _ hj))]
-    simp only [List.length_cons,Nat.add_assoc]
-
-noncomputable def closePath (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d)) :
-    QueryTree (SwitchIndex d) :=
-  forceList (routeDomain d x y).toList (routeValue d x y) k
-
-lemma closePath_succeeds (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
-    (ω : SwitchIndex d → Bool) :
-    (closePath d x y k).succeeds ω =
-      if butterflyPerm d (decodeButterfly d ω) x = y then k.succeeds ω else false := by
-  unfold closePath
-  rw [forceList_succeeds]
-  simp only [Finset.mem_toList, ← butterfly_endpoint_iff]
-
-lemma closePath_raw_charge (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
-    (ω : SwitchIndex d → Bool) (h : butterflyPerm d (decodeButterfly d ω) x = y) :
-    (closePath d x y k).charge ω = k.charge ω + d := by
-  unfold closePath
-  rw [forceList_raw_charge _ _ _ _ (by simpa only [Finset.mem_toList] using
-    (butterfly_endpoint_iff d ω x y).mp h), Finset.length_toList, card_routeDomain]
-
-lemma closePath_fresh_charge (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
-    (R : Finset (SwitchIndex d)) (ω : SwitchIndex d → Bool)
-    (h : butterflyPerm d (decodeButterfly d ω) x = y) :
-    (closePath d x y k).freshCharge R ω =
-      k.freshCharge (R ∪ routeDomain d x y) ω + (routeDomain d x y \ R).card := by
-  unfold closePath
-  rw [forceList_charge _ _ _ _ _ (by simpa only [Finset.mem_toList] using
-    (butterfly_endpoint_iff d ω x y).mp h), Finset.toList_toFinset]
-
-lemma closePath_stale_charge (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
-    (R : Finset (SwitchIndex d)) (ω : SwitchIndex d → Bool)
-    (h : butterflyPerm d (decodeButterfly d ω) x = y) :
-    (closePath d x y k).staleCharge R ω =
-      k.staleCharge (R ∪ routeDomain d x y) ω + (routeDomain d x y ∩ R).card := by
-  have h₀ := fresh_add_stale (closePath d x y k) R ω
-  have h₁ := fresh_add_stale k (R ∪ routeDomain d x y) ω
-  rw [closePath_raw_charge _ _ _ _ _ h,closePath_fresh_charge _ _ _ _ _ _ h] at h₀
-  have h₂ : (routeDomain d x y \ R).card + (routeDomain d x y ∩ R).card = d := by
-    rw [Finset.card_sdiff_add_card_inter,card_routeDomain]
-  omega
-
-noncomputable def seekCycle (d : ℕ) (S : Finset (Card d)) (a : Card d) :
-    ℕ → Card d → Finset (Card d) → (Finset (Card d) → QueryTree (SwitchIndex d)) →
-      QueryTree (SwitchIndex d)
-  | 0, x, V, k => if x ∈ S ∧ x ∉ V then closePath d x a (k (insert x V)) else reject
-  | r + 1, x, V, k => if x ∈ S ∧ x ∉ V then followPath d id x
-      (fun y => seekCycle d S a r y (insert x V) k) else reject
-
-lemma seekCycle_raw_charge (d : ℕ) (S : Finset (Card d)) (a : Card d) (r : ℕ)
-    (x : Card d) (V : Finset (Card d)) (k : Finset (Card d) → QueryTree (SwitchIndex d))
-    (ω : SwitchIndex d → Bool) (K : ℕ)
-    (hK : ∀ V, (k V).succeeds ω = true → (k V).charge ω = K)
-    (hs : (seekCycle d S a r x V k).succeeds ω = true) :
-    (seekCycle d S a r x V k).charge ω = K + d := by
-  induction r generalizing x V with
-  | zero =>
-    by_cases hv : x ∈ S ∧ x ∉ V
-    · rw [seekCycle,ite_eq_left hv,closePath_succeeds] at hs
-      split_ifs at hs with he
-      · rw [seekCycle,ite_eq_left hv,closePath_raw_charge _ _ _ _ _ he,hK _ hs]
-    · simp [seekCycle,hv,succeeds] at hs
-  | succ r ih =>
-    by_cases hv : x ∈ S ∧ x ∉ V
-    · rw [seekCycle,ite_eq_left hv,followPath_succeeds] at hs
-      rw [seekCycle,ite_eq_left hv,followPath_raw_charge]
-      exact ih _ _ hs
-    · simp [seekCycle,hv,succeeds] at hs
-
-noncomputable def seekCycles (d : ℕ) (S : Finset (Card d)) :
-    List (Card d × ℕ) → Finset (Card d) → QueryTree (SwitchIndex d)
-  | [], _ => accept
-  | (a,r) :: L, V => seekCycle d S a r a V (fun V => seekCycles d S L V)
-
-lemma seekCycles_raw_charge (d : ℕ) (S : Finset (Card d)) (L : List (Card d × ℕ))
-    (V : Finset (Card d)) (ω : SwitchIndex d → Bool)
-    (hs : (seekCycles d S L V).succeeds ω = true) :
-    (seekCycles d S L V).charge ω = L.length * d := by
-  induction L generalizing V with
-  | nil => simp [seekCycles,charge]
-  | cons ar L ih =>
-    obtain ⟨a,r⟩ := ar
-    rw [seekCycles]
-    rw [seekCycle_raw_charge d S a r a V (fun V => seekCycles d S L V) ω (L.length*d)
-      (fun V hs => ih V hs) hs]
-    simp [Nat.add_mul]
-
-lemma cycles_weighted_success (d : ℕ) (S : Finset (Card d)) (L : List (Card d × ℕ)) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      if (seekCycles d S L ∅).succeeds ω then
-        ((2:ℝ)^d)^L.length / (2:ℝ)^(seekCycles d S L ∅).staleCharge ∅ ω else 0) ≤ 1 := by
-  let T := seekCycles d S L ∅
-  have hh := mean_freshCharge_le_one T
-  have he : finiteMean (fun ω : SwitchIndex d → Bool =>
-      if T.succeeds ω then (2:ℝ)^T.freshCharge ∅ ω else 0) =
-      finiteMean (fun ω : SwitchIndex d → Bool =>
-        if T.succeeds ω then ((2:ℝ)^d)^L.length / (2:ℝ)^T.staleCharge ∅ ω else 0) := by
-    apply finiteMean_congr
-    intro ω
-    cases hs : T.succeeds ω
-    · simp
-    · simp only [↓reduceIte]
-      have hc := fresh_add_stale T ∅ ω
-      rw [seekCycles_raw_charge d S L ∅ ω hs] at hc
-      have hp : ((2:ℝ)^d)^L.length = (2:ℝ)^T.freshCharge ∅ ω * (2:ℝ)^T.staleCharge ∅ ω := by
-        rw [←pow_add,hc,←pow_mul,Nat.mul_comm]
-      rw [hp,mul_div_cancel_right₀ _ (pow_ne_zero _ (by norm_num))]
-  rw [← he]
-  exact hh
-
-end QueryTree
-
-lemma finiteMean_add {Ω : Type*} [Fintype Ω] (f g : Ω → ℝ) :
-    finiteMean (fun ω => f ω + g ω) = finiteMean f + finiteMean g := by
-  simp only [finiteMean,Finset.sum_add_distrib,add_div]
-
-lemma finiteMean_mul_const {Ω : Type*} [Fintype Ω] (f : Ω → ℝ) (c : ℝ) :
-    finiteMean (fun ω => f ω * c) = finiteMean f * c := by
-  unfold finiteMean
-  rw [← Finset.sum_mul]
-  ring
-
-lemma finiteMean_sum {Ω α : Type*} [Fintype Ω] (s : Finset α) (f : α → Ω → ℝ) :
-    finiteMean (fun ω => ∑ i ∈ s, f i ω) = ∑ i ∈ s, finiteMean (f i) := by
-  simp only [finiteMean,Finset.sum_div]
-  exact Finset.sum_comm
-
-lemma finiteMean_mono {Ω : Type*} [Fintype Ω] {f g : Ω → ℝ}
-    (h : ∀ ω, f ω ≤ g ω) : finiteMean f ≤ finiteMean g := by
-  exact div_le_div_of_nonneg_right (Finset.sum_le_sum (fun ω _ => h ω)) (Nat.cast_nonneg _)
-
-lemma finiteMean_equiv {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
-    (e : Ω ≃ Ω') (f : Ω → ℝ) :
-    finiteMean f = finiteMean (fun ω => f (e.symm ω)) := by
-  unfold finiteMean
-  rw [Fintype.card_congr e]
-  congr 1
-  exact Fintype.sum_equiv e _ _ (fun ω => by simp)
-
-lemma finiteMean_prod_left {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] [Nonempty Ω']
-    (f : Ω → ℝ) : finiteMean (fun ω : Ω × Ω' => f ω.1) = finiteMean f := by
-  simp only [finiteMean,Fintype.sum_prod_type,Finset.sum_const,nsmul_eq_mul,
-    ← Finset.mul_sum,Fintype.card_prod,Nat.cast_mul,Finset.card_univ]
-  have hn : (Fintype.card Ω' : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
-  field_simp
-
-lemma finiteMean_pi_eval {C : Type*} [Fintype C] [DecidableEq C]
-    {A : C → Type*} [∀ c, Fintype (A c)] [∀ c, Nonempty (A c)]
-    (c : C) (g : A c → ℝ) :
-    finiteMean (fun z : ∀ c, A c => g (z c)) = finiteMean g := by
-  rw [finiteMean_equiv (Equiv.piSplitAt c A)]
-  have he : (fun z : A c × ((j : {j // j ≠ c}) → A j) =>
-      g ((Equiv.piSplitAt c A).symm z c)) = (fun z => g z.1) := by
-    funext z
-    simp [Equiv.piSplitAt]
-  rw [he,finiteMean_prod_left]
-
-lemma finiteMean_eq_indicator {Ω : Type*} [Fintype Ω] [DecidableEq Ω] (a : Ω) :
-    finiteMean (fun ω : Ω => if ω = a then (1:ℝ) else 0) = 1 / Fintype.card Ω := by
-  simp [finiteMean]
-
-lemma exp_finiteMean_le {Ω : Type*} [Fintype Ω] [Nonempty Ω] (f : Ω → ℝ) :
-    Real.exp (finiteMean f) ≤ finiteMean (fun ω => Real.exp (f ω)) := by
-  have hn : (Fintype.card Ω : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
-  have hw : ∑ ω : Ω, (1 / (Fintype.card Ω : ℝ)) = 1 := by simp [hn]
-  have hj := convexOn_exp.map_sum_le (t := Finset.univ)
-    (w := fun _ : Ω => 1 / (Fintype.card Ω : ℝ)) (p := f)
-    (fun _ _ => by positivity) hw (fun _ _ => Set.mem_univ _)
-  simp only [smul_eq_mul, ← Finset.mul_sum] at hj
-  simpa only [finiteMean, div_eq_mul_inv, one_div, mul_comm, mul_one, one_mul] using hj
-
-variable {d A : ℕ}
-
-def cycleFiber (o : Card d → Option (Fin A)) (i : Fin A) : Finset (Card d) := by
-  classical
-  exact Finset.univ.filter (fun x => o x = some i)
-
-@[simp] lemma mem_cycleFiber (o : Card d → Option (Fin A)) (i : Fin A) (x : Card d) :
-    x ∈ cycleFiber o i ↔ o x = some i := by
-  classical
-  simp [cycleFiber]
-
-abbrev Terminal (o : Card d → Option (Fin A)) := ∀ i : Fin A, {x : Card d // o x = some i}
-
-noncomputable def slotWeight (o : Card d → Option (Fin A)) (i : Fin A) : ℝ :=
-  1 / (cycleFiber o i).card
-
-noncomputable def pathWeight (o : Card d → Option (Fin A)) (x : Card d) : ℝ :=
-  match o x with
-  | none => 0
-  | some i => slotWeight o i
-
-noncomputable def withinWeight (o : Card d → Option (Fin A)) (u v : Card d) : ℝ :=
-  match o u, o v with
-  | some i, some j => if i = j then slotWeight o i else 0
-  | _, _ => 0
-
-noncomputable def rotationMeeting (o : Card d → Option (Fin A)) (z : Terminal o)
-    (u v : Card d) : ℝ :=
-  match o u, o v with
-  | some i, some j => if i < j then (if (z j).val = v then 1 else 0)
-      else if j < i then (if (z i).val = u then 1 else 0)
-      else (if (z i).val = u then 1 else 0) + (if (z i).val = v then 1 else 0)
-  | _, _ => 0
-
-end Thorp
-
-end ThorpNine.Smoothing
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Smoothing.Coins
+
+namespace OAI
+
+namespace ThorpNine.Smoothing
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+
+lemma xor_eq_iff (a b c : Bool) : Bool.xor a b = c ↔ b = Bool.xor a c := by
+  cases a <;> cases b <;> cases c <;> decide
+
+lemma butterfly_endpoint_iff (d : ℕ) (ω : SwitchIndex d → Bool) (x y : Card d) :
+    butterflyPerm d (decodeButterfly d ω) x = y ↔
+      ∀ i ∈ routeDomain d x y, ω i = routeValue d x y i := by
+  classical
+  induction d with
+  | zero =>
+    have he : butterflyPerm 0 (decodeButterfly 0 ω) x = y := Subsingleton.elim _ _
+    simp [he, routeDomain]
+  | succ d ih =>
+    let v := butterflyPerm d (decodeButterfly d (fun i => ω (Sum.inr (x 0,i)))) (Fin.tail x)
+    change Fin.cons (Bool.xor (x 0) (ω (Sum.inl v))) v = y ↔ _
+    have hv : v = Fin.tail y ↔
+        ∀ i ∈ routeDomain d (Fin.tail x) (Fin.tail y),
+          ω (Sum.inr (x 0,i)) = routeValue d (Fin.tail x) (Fin.tail y) i :=
+      ih _ _ _
+    constructor
+    · intro h i hi
+      have ht : v = Fin.tail y := by simpa using congrArg Fin.tail h
+      have hh : ω (Sum.inl (Fin.tail y)) = Bool.xor (x 0) (y 0) := by
+        have he := congrFun h 0
+        simp only [Fin.cons_zero, ht] at he
+        exact (xor_eq_iff _ _ _).mp he
+      rcases Finset.mem_insert.mp hi with htop | hchild
+      · subst i
+        exact hh
+      · obtain ⟨j, hj, rfl⟩ := Finset.mem_image.mp hchild
+        exact hv.mp ht j hj
+    · intro h
+      have hh := h (Sum.inl (Fin.tail y)) (Finset.mem_insert_self _ _)
+      have ht : v = Fin.tail y := hv.mpr (fun i hi =>
+        h (Sum.inr (x 0,i)) (Finset.mem_insert_of_mem (Finset.mem_image_of_mem _ hi)))
+      have hh' : Bool.xor (x 0) (ω (Sum.inl (Fin.tail y))) = y 0 :=
+        (xor_eq_iff _ _ _).mpr hh
+      rw [ht, hh', Fin.cons_self_tail]
+
+lemma finiteMean_congr {Ω : Type*} [Fintype Ω] {f g : Ω → ℝ}
+    (h : ∀ ω, f ω = g ω) : finiteMean f = finiteMean g := by
+  unfold finiteMean
+  congr 1
+  exact Finset.sum_congr rfl (fun ω _ => h ω)
+
+lemma mean_coinCylinder {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (s : Finset ι) (v : ι → Bool) :
+    finiteMean (fun ω : ι → Bool => if ∀ i ∈ s, ω i = v i then (1 : ℝ) else 0) =
+      1 / (2 : ℝ) ^ s.card := by
+  induction s using Finset.induction_on with
+  | empty => simp [finiteMean]
+  | @insert i s hi ih =>
+    let f := fun ω : ι → Bool => if ∀ j ∈ s, ω j = v j then (1 : ℝ) else 0
+    have hf : ∀ ω, f (QueryTree.flipCoin i ω) = f ω := by
+      intro ω
+      have he : (∀ j ∈ s, QueryTree.flipCoin i ω j = v j) ↔ ∀ j ∈ s, ω j = v j := by
+        apply forall₂_congr
+        intro j hj
+        rw [QueryTree.flipCoin_apply_ne i j (fun h => hi (h ▸ hj))]
+      dsimp [f]
+      simp only [he]
+    have hm := QueryTree.mean_indicator_bit f i (v i) hf
+    have he : (fun ω : ι → Bool => if ∀ j ∈ insert i s, ω j = v j then (1 : ℝ) else 0) =
+        (fun ω => if ω i = v i then f ω else 0) := by
+      funext ω
+      by_cases h : ω i = v i <;> simp [f, h]
+    rw [he, hm]
+    change finiteMean (fun ω : ι → Bool => if ∀ j ∈ s, ω j = v j then (1 : ℝ) else 0) / 2 = _
+    rw [ih, Finset.card_insert_of_notMem hi, pow_succ]
+    ring
+
+def exposedCoins {ι : Type*} [DecidableEq ι] (R : Finset ι) (η ω : ι → Bool) : ι → Bool :=
+  fun i => if i ∈ R then η i else ω i
+
+lemma mean_coinCylinder_exposed {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (s R : Finset ι) (v η : ι → Bool) :
+    finiteMean (fun ω : ι → Bool =>
+      if ∀ i ∈ s, exposedCoins R η ω i = v i then (1 : ℝ) else 0) =
+    if (∀ i ∈ s ∩ R, η i = v i) then 1 / (2 : ℝ) ^ (s \ R).card else 0 := by
+  have he (ω : ι → Bool) : (∀ i ∈ s, exposedCoins R η ω i = v i) ↔
+      (∀ i ∈ s ∩ R, η i = v i) ∧ (∀ i ∈ s \ R, ω i = v i) := by
+    constructor
+    · intro h
+      constructor
+      · intro i hi
+        simpa [exposedCoins, (Finset.mem_inter.mp hi).2] using h i (Finset.mem_inter.mp hi).1
+      · intro i hi
+        simpa [exposedCoins, (Finset.mem_sdiff.mp hi).2] using h i (Finset.mem_sdiff.mp hi).1
+    · rintro ⟨ho, hf⟩ i hi
+      by_cases hR : i ∈ R
+      · simpa [exposedCoins, hR] using ho i (Finset.mem_inter.mpr ⟨hi,hR⟩)
+      · simpa [exposedCoins, hR] using hf i (Finset.mem_sdiff.mpr ⟨hi,hR⟩)
+  have hm : finiteMean (fun ω : ι → Bool =>
+      if ∀ i ∈ s, exposedCoins R η ω i = v i then (1 : ℝ) else 0) =
+      finiteMean (fun ω : ι → Bool =>
+        if (∀ i ∈ s ∩ R, η i = v i) ∧ (∀ i ∈ s \ R, ω i = v i) then (1 : ℝ) else 0) :=
+    finiteMean_congr (fun ω => by simp only [he ω])
+  rw [hm]
+  by_cases h : ∀ i ∈ s ∩ R, η i = v i
+  · have hp : (∀ i ∈ s ∩ R, η i = v i) ↔ True := iff_true_intro h
+    simp only [hp, true_and, ↓reduceIte]
+    exact mean_coinCylinder (s \ R) v
+  · have hp : (∀ i ∈ s ∩ R, η i = v i) ↔ False := iff_false_intro h
+    simp only [hp, false_and, ↓reduceIte]
+    simp [finiteMean]
+
+lemma butterfly_closure_probability (d : ℕ) (x y : Card d)
+    (R : Finset (SwitchIndex d)) (η : SwitchIndex d → Bool) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      if butterflyPerm d (decodeButterfly d (exposedCoins R η ω)) x = y then (1 : ℝ) else 0) =
+      if (∀ i ∈ routeDomain d x y ∩ R, η i = routeValue d x y i) then
+        (2 : ℝ) ^ (routeDomain d x y ∩ R).card / (2 : ℝ) ^ d else 0 := by
+  have he : finiteMean (fun ω : SwitchIndex d → Bool =>
+      if butterflyPerm d (decodeButterfly d (exposedCoins R η ω)) x = y then (1 : ℝ) else 0) =
+      finiteMean (fun ω : SwitchIndex d → Bool =>
+        if ∀ i ∈ routeDomain d x y, exposedCoins R η ω i = routeValue d x y i then (1 : ℝ) else 0) :=
+    finiteMean_congr (fun ω => by simp only [butterfly_endpoint_iff])
+  rw [he, mean_coinCylinder_exposed]
+  congr 1
+  have hc : (routeDomain d x y \ R).card + (routeDomain d x y ∩ R).card = d := by
+    rw [Finset.card_sdiff_add_card_inter, card_routeDomain]
+  have hp : (2 : ℝ) ^ d = (2 : ℝ) ^ (routeDomain d x y \ R).card *
+      (2 : ℝ) ^ (routeDomain d x y ∩ R).card := by
+    rw [← pow_add]
+    exact congrArg (fun n => (2 : ℝ) ^ n) hc.symm
+  rw [hp]
+  have hn : (2 : ℝ) ^ (routeDomain d x y ∩ R).card ≠ 0 := pow_ne_zero _ (by norm_num)
+  field_simp
+
+lemma butterfly_single_card_uniform (d : ℕ) (x y : Card d) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      if butterflyPerm d (decodeButterfly d ω) x = y then (1 : ℝ) else 0) = 1 / (2 : ℝ) ^ d := by
+  have he : finiteMean (fun ω : SwitchIndex d → Bool =>
+      if butterflyPerm d (decodeButterfly d ω) x = y then (1 : ℝ) else 0) =
+      finiteMean (fun ω : SwitchIndex d → Bool =>
+        if ∀ i ∈ routeDomain d x y, ω i = routeValue d x y i then (1 : ℝ) else 0) :=
+    finiteMean_congr (fun ω => by simp only [butterfly_endpoint_iff])
+  rw [he, mean_coinCylinder, card_routeDomain]
+
+def switchUsers : (d : ℕ) → Butterfly d → SwitchIndex d → Card d × Card d
+  | 0, _, i => nomatch i
+  | d + 1, b, Sum.inl y =>
+      (Fin.cons false ((butterflyPerm d (b.2 false)).symm y),
+       Fin.cons true ((butterflyPerm d (b.2 true)).symm y))
+  | d + 1, b, Sum.inr (ε,i) =>
+      (Fin.cons ε (switchUsers d (b.2 ε) i).1,
+       Fin.cons ε (switchUsers d (b.2 ε) i).2)
+
+lemma switchUsers_ne (d : ℕ) (b : Butterfly d) (i : SwitchIndex d) :
+    (switchUsers d b i).1 ≠ (switchUsers d b i).2 := by
+  induction d with
+  | zero => exact Empty.elim i
+  | succ d ih =>
+    cases i with
+    | inl y =>
+      intro h
+      have := congrFun h 0
+      simp only [switchUsers, Fin.cons_zero] at this
+      exact Bool.false_ne_true this
+    | inr z =>
+      intro h
+      exact ih (b.2 z.1) z.2 (by simpa only [switchUsers, Fin.tail_cons] using congrArg Fin.tail h)
+
+lemma cons_eq_iff {d : ℕ} (ε : Bool) (a : Card d) (x : Card (d + 1)) :
+    Fin.cons ε a = x ↔ ε = x 0 ∧ a = Fin.tail x := by
+  constructor
+  · intro h
+    exact ⟨by simpa only [Fin.cons_zero] using congrFun h 0,
+      by simpa only [Fin.tail_cons] using congrArg Fin.tail h⟩
+  · rintro ⟨rfl,rfl⟩
+    exact Fin.cons_self_tail x
+
+lemma mem_routeDomain_iff_user (d : ℕ) (b : Butterfly d) (x : Card d) (i : SwitchIndex d) :
+    i ∈ routeDomain d x (butterflyPerm d b x) ↔
+      (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x := by
+  classical
+  induction d with
+  | zero => exact Empty.elim i
+  | succ d ih =>
+    cases i with
+    | inl y =>
+      change Sum.inl y ∈ insert (Sum.inl (Fin.tail (butterflyPerm (d+1) b x)))
+        ((routeDomain d (Fin.tail x) (Fin.tail (butterflyPerm (d+1) b x))).image
+          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) ↔ _
+      simp only [Finset.mem_insert, Sum.inl.injEq, Finset.mem_image,
+        Sum.inr_ne_inl, and_false, exists_false, or_false]
+      change y = butterflyPerm d (b.2 (x 0)) (Fin.tail x) ↔ _
+      simp only [switchUsers, cons_eq_iff]
+      cases x 0 <;> simp [Equiv.symm_apply_eq]
+    | inr z =>
+      obtain ⟨ε,i⟩ := z
+      change Sum.inr (ε,i) ∈ insert (Sum.inl (Fin.tail (butterflyPerm (d+1) b x)))
+        ((routeDomain d (Fin.tail x) (Fin.tail (butterflyPerm (d+1) b x))).image
+          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) ↔ _
+      simp only [Finset.mem_insert, Sum.inr_ne_inl, false_or, Finset.mem_image,
+        Sum.inr.injEq, Prod.mk.injEq]
+      change (∃ j, j ∈ routeDomain d (Fin.tail x) (butterflyPerm d (b.2 (x 0)) (Fin.tail x)) ∧
+        x 0 = ε ∧ j = i) ↔ _
+      simp only [switchUsers, cons_eq_iff]
+      constructor
+      · rintro ⟨j,hj,hε,rfl⟩
+        subst ε
+        rcases (ih (b.2 (x 0)) (Fin.tail x) j).mp hj with h | h
+        · exact Or.inl ⟨rfl,h⟩
+        · exact Or.inr ⟨rfl,h⟩
+      · rintro (⟨hε,h⟩ | ⟨hε,h⟩)
+        · subst ε
+          exact ⟨i,(ih (b.2 (x 0)) (Fin.tail x) i).mpr (Or.inl h),rfl,rfl⟩
+        · subst ε
+          exact ⟨i,(ih (b.2 (x 0)) (Fin.tail x) i).mpr (Or.inr h),rfl,rfl⟩
+
+lemma card_switches_using (d : ℕ) (b : Butterfly d) (x : Card d) :
+    (Finset.univ.filter (fun i : SwitchIndex d =>
+      (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x)).card = d := by
+  classical
+  have he : Finset.univ.filter (fun i : SwitchIndex d =>
+      (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x) =
+      routeDomain d x (butterflyPerm d b x) := by
+    ext i
+    simp only [Finset.mem_filter, Finset.mem_univ, true_and, mem_routeDomain_iff_user]
+  rw [he,card_routeDomain]
+
+lemma switch_incidence_sum (d : ℕ) (b : Butterfly d) (w : Card d → ℝ) :
+    ∑ i : SwitchIndex d, (w (switchUsers d b i).1 + w (switchUsers d b i).2) =
+      (d : ℝ) * ∑ x : Card d, w x := by
+  classical
+  calc
+    _ = ∑ i : SwitchIndex d, ∑ x : Card d,
+        if (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x then w x else 0 := by
+      apply Finset.sum_congr rfl
+      intro i _
+      rw [Finset.sum_ite]
+      simp only [Finset.sum_const_zero, add_zero]
+      have he : Finset.univ.filter (fun x : Card d =>
+          (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x) =
+          {(switchUsers d b i).1, (switchUsers d b i).2} := by
+        ext x; simp [eq_comm]
+      rw [he, Finset.sum_pair (switchUsers_ne d b i)]
+    _ = ∑ x : Card d, ∑ i : SwitchIndex d,
+        if (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x then w x else 0 :=
+      Finset.sum_comm
+    _ = ∑ x : Card d, (d : ℝ) * w x := by
+      apply Finset.sum_congr rfl
+      intro x _
+      rw [Finset.sum_ite]
+      simp only [Finset.sum_const_zero, add_zero, Finset.sum_const, nsmul_eq_mul,
+        card_switches_using]
+    _ = _ := by rw [Finset.mul_sum]
+
+lemma weighted_switch_min_bound (d : ℕ) (b : Butterfly d) (w : Card d → ℝ) :
+    ∑ i : SwitchIndex d, min (w (switchUsers d b i).1) (w (switchUsers d b i).2) ≤
+      (d : ℝ) / 2 * ∑ x : Card d, w x := by
+  have h : 2 * (∑ i : SwitchIndex d,
+      min (w (switchUsers d b i).1) (w (switchUsers d b i).2)) ≤
+      ∑ i : SwitchIndex d, (w (switchUsers d b i).1 + w (switchUsers d b i).2) := by
+    rw [Finset.mul_sum]
+    apply Finset.sum_le_sum
+    intro i _
+    linarith [min_le_left (w (switchUsers d b i).1) (w (switchUsers d b i).2),
+      min_le_right (w (switchUsers d b i).1) (w (switchUsers d b i).2)]
+  rw [switch_incidence_sum] at h
+  linarith
+
+lemma mem_childMarks {d : ℕ} (s : Finset (Card (d + 1))) (ε : Bool) (x : Card d) :
+    x ∈ childMarks s ε ↔ Fin.cons ε x ∈ s := by
+  classical
+  constructor
+  · intro h
+    obtain ⟨y,hy,hxy⟩ := Finset.mem_image.mp h
+    obtain ⟨hys,hyε⟩ := Finset.mem_filter.mp hy
+    rw [← hxy, ← hyε, Fin.cons_self_tail]
+    exact hys
+  · intro h
+    exact Finset.mem_image.mpr ⟨Fin.cons ε x,
+      Finset.mem_filter.mpr ⟨h,by simp⟩,by simp⟩
+
+lemma mem_topMeetings {d : ℕ} (b : Butterfly (d + 1))
+    (s : Finset (Card (d + 1))) (y : Card d) :
+    y ∈ topMeetings b s ↔
+      (switchUsers (d+1) b (Sum.inl y)).1 ∈ s ∧
+      (switchUsers (d+1) b (Sum.inl y)).2 ∈ s := by
+  classical
+  have he (ε : Bool) : y ∈ (childMarks s ε).image (butterflyPerm d (b.2 ε)) ↔
+      (butterflyPerm d (b.2 ε)).symm y ∈ childMarks s ε := by
+    constructor
+    · intro h
+      obtain ⟨x,hx,rfl⟩ := Finset.mem_image.mp h
+      simpa only [Equiv.symm_apply_apply] using hx
+    · intro h
+      exact Finset.mem_image.mpr ⟨_,h,Equiv.apply_symm_apply _ _⟩
+  simp only [topMeetings, Finset.mem_inter, he, mem_childMarks, switchUsers]
+
+lemma encounters_eq_switch_sum (d : ℕ) (b : Butterfly d) (s : Finset (Card d)) :
+    encounters d b s = ∑ i : SwitchIndex d,
+      if (switchUsers d b i).1 ∈ s ∧ (switchUsers d b i).2 ∈ s then 1 else 0 := by
+  classical
+  induction d with
+  | zero =>
+    simp only [encounters]
+    symm
+    apply Finset.sum_eq_zero
+    intro i _
+    exact Empty.elim i
+  | succ d ih =>
+    change encounters d (b.2 false) (childMarks s false) +
+      encounters d (b.2 true) (childMarks s true) + (topMeetings b s).card =
+        ∑ i : Sum (Card d) (Bool × SwitchIndex d), _
+    rw [Fintype.sum_sum_type, Fintype.sum_prod_type]
+    simp only [Fintype.sum_bool]
+    rw [ih,ih]
+    have ht : (∑ y : Card d,
+        if (switchUsers (d+1) b (Sum.inl y)).1 ∈ s ∧
+          (switchUsers (d+1) b (Sum.inl y)).2 ∈ s then 1 else 0) =
+        (topMeetings b s).card := by
+      simp only [← mem_topMeetings]
+      simp
+    rw [ht]
+    simp only [mem_childMarks, switchUsers]
+    ac_rfl
+
+lemma marked_switch_entropy_bound (d : ℕ) (b : Butterfly d) (s : Finset (Card d)) :
+    (∑ i : SwitchIndex d,
+      if (switchUsers d b i).1 ∈ s ∧ (switchUsers d b i).2 ∈ s then (1:ℝ) else 0) ≤
+      (s.card : ℝ) * Real.log s.card / (2 * Real.log 2) := by
+  classical
+  have he : (encounters d b s : ℝ) =
+      ∑ i : SwitchIndex d,
+        if (switchUsers d b i).1 ∈ s ∧ (switchUsers d b i).2 ∈ s then (1:ℝ) else 0 := by
+    rw [encounters_eq_switch_sum, Nat.cast_sum]
+    apply Finset.sum_congr rfl
+    intro i _
+    split_ifs <;> simp
+  rw [← he]
+  exact butterfly_encounters d b s
+
+namespace QueryTree
+variable {ι : Type*} [DecidableEq ι]
+
+def memo : QueryTree ι → Finset ι → (ι → Bool) → QueryTree ι
+  | reject, _, _ => reject
+  | accept, _, _ => accept
+  | branch i zero one, R, η =>
+      if i ∈ R then (if η i then memo one R η else memo zero R η) else
+      branch i (memo zero (insert i R) (Function.update η i false))
+        (memo one (insert i R) (Function.update η i true))
+  | force i b next, R, η =>
+      if i ∈ R then (if η i = b then memo next R η else reject) else
+      force i b (memo next (insert i R) (Function.update η i b))
+
+lemma memo_queries_disjoint (T : QueryTree ι) (R : Finset ι) (η : ι → Bool) :
+    Disjoint (memo T R η).queries R := by
+  induction T generalizing R η with
+  | reject => simp [memo,queries]
+  | accept => simp [memo,queries]
+  | branch i zero one ih₀ ih₁ =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte]
+      cases η i <;> simp only [Bool.false_eq_true,↓reduceIte]
+      · exact ih₀ R η
+      · exact ih₁ R η
+    · simp only [memo,hi,↓reduceIte,queries,Finset.disjoint_insert_left,Finset.disjoint_union_left]
+      exact ⟨by simp, (ih₀ (insert i R) _).mono_right (Finset.subset_insert _ _),
+        (ih₁ (insert i R) _).mono_right (Finset.subset_insert _ _)⟩
+  | force i b next ih =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte]
+      by_cases hb : η i = b
+      · simpa only [hb,↓reduceIte] using ih R η
+      · simp [hb,queries]
+    · simp only [memo,hi,↓reduceIte,queries,Finset.disjoint_insert_left]
+      exact ⟨by simp,(ih (insert i R) _).mono_right (Finset.subset_insert _ _)⟩
+
+lemma memo_fresh (T : QueryTree ι) (R : Finset ι) (η : ι → Bool) :
+    (memo T R η).Fresh := by
+  induction T generalizing R η with
+  | reject => trivial
+  | accept => trivial
+  | branch i zero one ih₀ ih₁ =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte]
+      cases η i <;> simp only [Bool.false_eq_true,↓reduceIte]
+      · exact ih₀ R η
+      · exact ih₁ R η
+    · simp only [memo,hi,↓reduceIte,Fresh]
+      exact ⟨fun hm => Finset.disjoint_left.mp (memo_queries_disjoint zero (insert i R) _) hm
+          (Finset.mem_insert_self _ _),
+        fun hm => Finset.disjoint_left.mp (memo_queries_disjoint one (insert i R) _) hm
+          (Finset.mem_insert_self _ _), ih₀ _ _,ih₁ _ _⟩
+  | force i b next ih =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte]
+      by_cases hb : η i = b
+      · simpa only [hb,↓reduceIte] using ih R η
+      · simp [hb,Fresh]
+    · simp only [memo,hi,↓reduceIte,Fresh]
+      exact ⟨fun hm => Finset.disjoint_left.mp (memo_queries_disjoint next (insert i R) _) hm
+        (Finset.mem_insert_self _ _),ih _ _⟩
+
+lemma update_agrees {R : Finset ι} {η ω : ι → Bool}
+    (h : ∀ i ∈ R, η i = ω i) (i : ι) :
+    ∀ j ∈ insert i R, Function.update η i (ω i) j = ω j := by
+  intro j hj
+  rcases Finset.mem_insert.mp hj with rfl | hj
+  · simp
+  · by_cases hji : j = i
+    · subst j; simp
+    · rw [Function.update_of_ne hji]
+      exact h j hj
+
+lemma memo_succeeds (T : QueryTree ι) (R : Finset ι) (η ω : ι → Bool)
+    (h : ∀ i ∈ R, η i = ω i) : (memo T R η).succeeds ω = T.succeeds ω := by
+  induction T generalizing R η with
+  | reject => rfl
+  | accept => rfl
+  | branch i zero one ih₀ ih₁ =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte,succeeds,h i hi]
+      cases ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
+      · exact ih₀ R η h
+      · exact ih₁ R η h
+    · simp only [memo,hi,↓reduceIte,succeeds]
+      cases hb : ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
+      · exact ih₀ _ _ (by simpa only [hb] using update_agrees h i)
+      · exact ih₁ _ _ (by simpa only [hb] using update_agrees h i)
+  | force i b next ih =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte,h i hi]
+      by_cases hb : ω i = b
+      · simp only [hb,↓reduceIte,succeeds]
+        exact ih R η h
+      · simp [hb,succeeds]
+    · simp only [memo,hi,↓reduceIte,succeeds]
+      by_cases hb : ω i = b
+      · simp only [hb,↓reduceIte]
+        exact ih _ _ (by simpa only [hb] using update_agrees h i)
+      · simp [hb]
+
+def freshCharge : QueryTree ι → Finset ι → (ι → Bool) → ℕ
+  | reject, _, _ => 0
+  | accept, _, _ => 0
+  | branch i zero one, R, ω => if ω i then freshCharge one (insert i R) ω else freshCharge zero (insert i R) ω
+  | force i b next, R, ω => if ω i = b then
+      freshCharge next (insert i R) ω + (if i ∈ R then 0 else 1) else 0
+
+lemma memo_charge (T : QueryTree ι) (R : Finset ι) (η ω : ι → Bool)
+    (h : ∀ i ∈ R, η i = ω i) : (memo T R η).charge ω = freshCharge T R ω := by
+  induction T generalizing R η with
+  | reject => rfl
+  | accept => rfl
+  | branch i zero one ih₀ ih₁ =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte,freshCharge,h i hi,Finset.insert_eq_of_mem hi]
+      cases ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
+      · exact ih₀ R η h
+      · exact ih₁ R η h
+    · simp only [memo,hi,↓reduceIte,charge,freshCharge]
+      cases hb : ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
+      · exact ih₀ _ _ (by simpa only [hb] using update_agrees h i)
+      · exact ih₁ _ _ (by simpa only [hb] using update_agrees h i)
+  | force i b next ih =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte,h i hi,freshCharge,Finset.insert_eq_of_mem hi,Nat.add_zero]
+      by_cases hb : ω i = b
+      · simp only [hb,↓reduceIte]
+        exact ih R η h
+      · simp [hb,charge]
+    · simp only [memo,hi,↓reduceIte,charge,freshCharge]
+      by_cases hb : ω i = b
+      · simp only [hb,↓reduceIte]
+        rw [ih _ _ (by simpa only [hb] using update_agrees h i)]
+      · simp [hb]
+
+lemma mean_freshCharge_le_one [Fintype ι] (T : QueryTree ι) :
+    finiteMean (fun ω : ι → Bool =>
+      if T.succeeds ω then (2:ℝ) ^ T.freshCharge ∅ ω else 0) ≤ 1 := by
+  let η : ι → Bool := fun _ => false
+  have hm := mean_weighted_le_one (memo T ∅ η) (memo_fresh T ∅ η)
+  have he : finiteMean (memo T ∅ η).weighted =
+      finiteMean (fun ω : ι → Bool =>
+        if T.succeeds ω then (2:ℝ) ^ T.freshCharge ∅ ω else 0) := by
+    apply finiteMean_congr
+    intro ω
+    rw [weighted_eq, memo_succeeds _ _ _ _ (by simp), memo_charge _ _ _ _ (by simp)]
+  rw [he] at hm
+  exact hm
+
+
+def followPath : (d : ℕ) → (SwitchIndex d → ι) → Card d →
+    (Card d → QueryTree ι) → QueryTree ι
+  | 0, _, x, k => k x
+  | d + 1, e, x, k => followPath d (fun i => e (Sum.inr (x 0,i))) (Fin.tail x)
+      (fun y => branch (e (Sum.inl y))
+        (k (Fin.cons (Bool.xor (x 0) false) y))
+        (k (Fin.cons (Bool.xor (x 0) true) y)))
+
+omit [DecidableEq ι] in
+lemma followPath_succeeds (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
+    (k : Card d → QueryTree ι) (ω : ι → Bool) :
+    (followPath d e x k).succeeds ω =
+      (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).succeeds ω := by
+  induction d with
+  | zero => rfl
+  | succ d ih =>
+    rw [followPath, ih]
+    let y := butterflyPerm d
+      (decodeButterfly d (fun i => ω (e (Sum.inr (x 0,i))))) (Fin.tail x)
+    change (if ω (e (Sum.inl y)) then
+      (k (Fin.cons (Bool.xor (x 0) true) y)).succeeds ω else
+      (k (Fin.cons (Bool.xor (x 0) false) y)).succeeds ω) =
+      (k (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).succeeds ω
+    cases ω (e (Sum.inl y)) <;> rfl
+
+lemma followPath_charge (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
+    (k : Card d → QueryTree ι) (R : Finset ι) (ω : ι → Bool) :
+    freshCharge (followPath d e x k) R ω =
+      freshCharge (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x))
+        (R ∪ (routeDomain d x (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).image e) ω := by
+  classical
+  induction d generalizing R with
+  | zero => simp only [followPath, routeDomain, Finset.image_empty, Finset.union_empty]; rfl
+  | succ d ih =>
+    rw [followPath, ih]
+    let y := butterflyPerm d
+      (decodeButterfly d (fun i => ω (e (Sum.inr (x 0,i))))) (Fin.tail x)
+    let S := (routeDomain d (Fin.tail x) y).image (fun i => e (Sum.inr (x 0,i)))
+    change (if ω (e (Sum.inl y)) then
+      freshCharge (k (Fin.cons (Bool.xor (x 0) true) y)) (insert (e (Sum.inl y)) (R ∪ S)) ω else
+      freshCharge (k (Fin.cons (Bool.xor (x 0) false) y)) (insert (e (Sum.inl y)) (R ∪ S)) ω) =
+      freshCharge (k (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y))
+        (R ∪ (routeDomain (d+1) x
+          (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).image e) ω
+    have hs : (routeDomain (d+1) x
+        (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).image e =
+        insert (e (Sum.inl y)) S := by
+      let e' : Sum (Card d) (Bool × SwitchIndex d) → ι := e
+      change (insert (Sum.inl y)
+        ((routeDomain d (Fin.tail x) y).image
+          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) :
+          Finset (Sum (Card d) (Bool × SwitchIndex d))).image e' = _
+      rw [Finset.image_insert, Finset.image_image]
+      rfl
+    rw [hs,Finset.union_insert]
+    cases ω (e (Sum.inl y)) <;> rfl
+
+def forceList : List ι → (ι → Bool) → QueryTree ι → QueryTree ι
+  | [], _, k => k
+  | i :: L, v, k => force i (v i) (forceList L v k)
+
+omit [DecidableEq ι] in
+lemma forceList_succeeds (L : List ι) (v : ι → Bool) (k : QueryTree ι) (ω : ι → Bool) :
+    (forceList L v k).succeeds ω =
+      if ∀ i ∈ L, ω i = v i then k.succeeds ω else false := by
+  induction L with
+  | nil => simp [forceList]
+  | cons i L ih =>
+    simp only [forceList,succeeds,ih,List.mem_cons,forall_eq_or_imp]
+    by_cases h : ω i = v i <;> simp [h]
+
+lemma forceList_charge (L : List ι) (v : ι → Bool) (k : QueryTree ι)
+    (R : Finset ι) (ω : ι → Bool) (h : ∀ i ∈ L, ω i = v i) :
+    freshCharge (forceList L v k) R ω =
+      freshCharge k (R ∪ L.toFinset) ω + (L.toFinset \ R).card := by
+  induction L generalizing R with
+  | nil => simp [forceList]
+  | cons i L ih =>
+    rw [forceList,freshCharge,ite_eq_left (h i List.mem_cons_self),
+      ih (insert i R) (fun j hj => h j (List.mem_cons_of_mem _ hj))]
+    simp only [List.toFinset_cons]
+    have hs : insert i R ∪ L.toFinset = R ∪ insert i L.toFinset := by ext j; simp
+    rw [hs]
+    by_cases hi : i ∈ R
+    · rw [ite_eq_left hi]
+      have he : L.toFinset \ insert i R = insert i L.toFinset \ R := by
+        ext j
+        by_cases hji : j = i
+        · subst j; simp [hi]
+        · simp [hji]
+      rw [he,Nat.add_zero]
+    · rw [ite_eq_right hi]
+      have he : insert i L.toFinset \ R = insert i (L.toFinset \ insert i R) := by
+        ext j
+        by_cases hji : j = i
+        · subst j; simp [hi]
+        · simp [hji]
+      rw [he, Finset.card_insert_of_notMem (by simp)]
+      omega
+
+
+def staleCharge : QueryTree ι → Finset ι → (ι → Bool) → ℕ
+  | reject, _, _ => 0
+  | accept, _, _ => 0
+  | branch i zero one, R, ω => if ω i then staleCharge one (insert i R) ω
+      else staleCharge zero (insert i R) ω
+  | force i b next, R, ω => if ω i = b then
+      staleCharge next (insert i R) ω + (if i ∈ R then 1 else 0) else 0
+
+lemma fresh_add_stale (T : QueryTree ι) (R : Finset ι) (ω : ι → Bool) :
+    T.freshCharge R ω + T.staleCharge R ω = T.charge ω := by
+  induction T generalizing R with
+  | reject => rfl
+  | accept => rfl
+  | branch i zero one ih₀ ih₁ =>
+    cases hb : ω i <;> simp [freshCharge,staleCharge,charge,hb,ih₀,ih₁]
+  | force i b next ih =>
+    by_cases hb : ω i = b
+    · simp only [freshCharge,staleCharge,charge,hb,↓reduceIte]
+      have he := ih (insert i R)
+      by_cases hi : i ∈ R <;> simp only [hi,↓reduceIte] <;> omega
+    · simp [freshCharge,staleCharge,charge,hb]
+
+omit [DecidableEq ι] in
+lemma followPath_raw_charge (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
+    (k : Card d → QueryTree ι) (ω : ι → Bool) :
+    (followPath d e x k).charge ω =
+      (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).charge ω := by
+  induction d with
+  | zero => rfl
+  | succ d ih =>
+    rw [followPath, ih]
+    let y := butterflyPerm d
+      (decodeButterfly d (fun i => ω (e (Sum.inr (x 0,i))))) (Fin.tail x)
+    change (if ω (e (Sum.inl y)) then
+      (k (Fin.cons (Bool.xor (x 0) true) y)).charge ω else
+      (k (Fin.cons (Bool.xor (x 0) false) y)).charge ω) =
+      (k (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).charge ω
+    cases ω (e (Sum.inl y)) <;> rfl
+
+omit [DecidableEq ι] in
+lemma forceList_raw_charge (L : List ι) (v : ι → Bool) (k : QueryTree ι) (ω : ι → Bool)
+    (h : ∀ i ∈ L, ω i = v i) :
+    (forceList L v k).charge ω = k.charge ω + L.length := by
+  induction L with
+  | nil => simp [forceList]
+  | cons i L ih =>
+    rw [forceList,charge,ite_eq_left (h i List.mem_cons_self),
+      ih (fun j hj => h j (List.mem_cons_of_mem _ hj))]
+    simp only [List.length_cons,Nat.add_assoc]
+
+noncomputable def closePath (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d)) :
+    QueryTree (SwitchIndex d) :=
+  forceList (routeDomain d x y).toList (routeValue d x y) k
+
+lemma closePath_succeeds (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
+    (ω : SwitchIndex d → Bool) :
+    (closePath d x y k).succeeds ω =
+      if butterflyPerm d (decodeButterfly d ω) x = y then k.succeeds ω else false := by
+  unfold closePath
+  rw [forceList_succeeds]
+  simp only [Finset.mem_toList, ← butterfly_endpoint_iff]
+
+lemma closePath_raw_charge (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
+    (ω : SwitchIndex d → Bool) (h : butterflyPerm d (decodeButterfly d ω) x = y) :
+    (closePath d x y k).charge ω = k.charge ω + d := by
+  unfold closePath
+  rw [forceList_raw_charge _ _ _ _ (by simpa only [Finset.mem_toList] using
+    (butterfly_endpoint_iff d ω x y).mp h), Finset.length_toList, card_routeDomain]
+
+lemma closePath_fresh_charge (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
+    (R : Finset (SwitchIndex d)) (ω : SwitchIndex d → Bool)
+    (h : butterflyPerm d (decodeButterfly d ω) x = y) :
+    (closePath d x y k).freshCharge R ω =
+      k.freshCharge (R ∪ routeDomain d x y) ω + (routeDomain d x y \ R).card := by
+  unfold closePath
+  rw [forceList_charge _ _ _ _ _ (by simpa only [Finset.mem_toList] using
+    (butterfly_endpoint_iff d ω x y).mp h), Finset.toList_toFinset]
+
+lemma closePath_stale_charge (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
+    (R : Finset (SwitchIndex d)) (ω : SwitchIndex d → Bool)
+    (h : butterflyPerm d (decodeButterfly d ω) x = y) :
+    (closePath d x y k).staleCharge R ω =
+      k.staleCharge (R ∪ routeDomain d x y) ω + (routeDomain d x y ∩ R).card := by
+  have h₀ := fresh_add_stale (closePath d x y k) R ω
+  have h₁ := fresh_add_stale k (R ∪ routeDomain d x y) ω
+  rw [closePath_raw_charge _ _ _ _ _ h,closePath_fresh_charge _ _ _ _ _ _ h] at h₀
+  have h₂ : (routeDomain d x y \ R).card + (routeDomain d x y ∩ R).card = d := by
+    rw [Finset.card_sdiff_add_card_inter,card_routeDomain]
+  omega
+
+noncomputable def seekCycle (d : ℕ) (S : Finset (Card d)) (a : Card d) :
+    ℕ → Card d → Finset (Card d) → (Finset (Card d) → QueryTree (SwitchIndex d)) →
+      QueryTree (SwitchIndex d)
+  | 0, x, V, k => if x ∈ S ∧ x ∉ V then closePath d x a (k (insert x V)) else reject
+  | r + 1, x, V, k => if x ∈ S ∧ x ∉ V then followPath d id x
+      (fun y => seekCycle d S a r y (insert x V) k) else reject
+
+lemma seekCycle_raw_charge (d : ℕ) (S : Finset (Card d)) (a : Card d) (r : ℕ)
+    (x : Card d) (V : Finset (Card d)) (k : Finset (Card d) → QueryTree (SwitchIndex d))
+    (ω : SwitchIndex d → Bool) (K : ℕ)
+    (hK : ∀ V, (k V).succeeds ω = true → (k V).charge ω = K)
+    (hs : (seekCycle d S a r x V k).succeeds ω = true) :
+    (seekCycle d S a r x V k).charge ω = K + d := by
+  induction r generalizing x V with
+  | zero =>
+    by_cases hv : x ∈ S ∧ x ∉ V
+    · rw [seekCycle,ite_eq_left hv,closePath_succeeds] at hs
+      split_ifs at hs with he
+      · rw [seekCycle,ite_eq_left hv,closePath_raw_charge _ _ _ _ _ he,hK _ hs]
+    · simp [seekCycle,hv,succeeds] at hs
+  | succ r ih =>
+    by_cases hv : x ∈ S ∧ x ∉ V
+    · rw [seekCycle,ite_eq_left hv,followPath_succeeds] at hs
+      rw [seekCycle,ite_eq_left hv,followPath_raw_charge]
+      exact ih _ _ hs
+    · simp [seekCycle,hv,succeeds] at hs
+
+noncomputable def seekCycles (d : ℕ) (S : Finset (Card d)) :
+    List (Card d × ℕ) → Finset (Card d) → QueryTree (SwitchIndex d)
+  | [], _ => accept
+  | (a,r) :: L, V => seekCycle d S a r a V (fun V => seekCycles d S L V)
+
+lemma seekCycles_raw_charge (d : ℕ) (S : Finset (Card d)) (L : List (Card d × ℕ))
+    (V : Finset (Card d)) (ω : SwitchIndex d → Bool)
+    (hs : (seekCycles d S L V).succeeds ω = true) :
+    (seekCycles d S L V).charge ω = L.length * d := by
+  induction L generalizing V with
+  | nil => simp [seekCycles,charge]
+  | cons ar L ih =>
+    obtain ⟨a,r⟩ := ar
+    rw [seekCycles]
+    rw [seekCycle_raw_charge d S a r a V (fun V => seekCycles d S L V) ω (L.length*d)
+      (fun V hs => ih V hs) hs]
+    simp [Nat.add_mul]
+
+lemma cycles_weighted_success (d : ℕ) (S : Finset (Card d)) (L : List (Card d × ℕ)) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      if (seekCycles d S L ∅).succeeds ω then
+        ((2:ℝ)^d)^L.length / (2:ℝ)^(seekCycles d S L ∅).staleCharge ∅ ω else 0) ≤ 1 := by
+  let T := seekCycles d S L ∅
+  have hh := mean_freshCharge_le_one T
+  have he : finiteMean (fun ω : SwitchIndex d → Bool =>
+      if T.succeeds ω then (2:ℝ)^T.freshCharge ∅ ω else 0) =
+      finiteMean (fun ω : SwitchIndex d → Bool =>
+        if T.succeeds ω then ((2:ℝ)^d)^L.length / (2:ℝ)^T.staleCharge ∅ ω else 0) := by
+    apply finiteMean_congr
+    intro ω
+    cases hs : T.succeeds ω
+    · simp
+    · simp only [↓reduceIte]
+      have hc := fresh_add_stale T ∅ ω
+      rw [seekCycles_raw_charge d S L ∅ ω hs] at hc
+      have hp : ((2:ℝ)^d)^L.length = (2:ℝ)^T.freshCharge ∅ ω * (2:ℝ)^T.staleCharge ∅ ω := by
+        rw [←pow_add,hc,←pow_mul,Nat.mul_comm]
+      rw [hp,mul_div_cancel_right₀ _ (pow_ne_zero _ (by norm_num))]
+  rw [← he]
+  exact hh
+
+end QueryTree
+
+lemma finiteMean_add {Ω : Type*} [Fintype Ω] (f g : Ω → ℝ) :
+    finiteMean (fun ω => f ω + g ω) = finiteMean f + finiteMean g := by
+  simp only [finiteMean,Finset.sum_add_distrib,add_div]
+
+lemma finiteMean_mul_const {Ω : Type*} [Fintype Ω] (f : Ω → ℝ) (c : ℝ) :
+    finiteMean (fun ω => f ω * c) = finiteMean f * c := by
+  unfold finiteMean
+  rw [← Finset.sum_mul]
+  ring
+
+lemma finiteMean_sum {Ω α : Type*} [Fintype Ω] (s : Finset α) (f : α → Ω → ℝ) :
+    finiteMean (fun ω => ∑ i ∈ s, f i ω) = ∑ i ∈ s, finiteMean (f i) := by
+  simp only [finiteMean,Finset.sum_div]
+  exact Finset.sum_comm
+
+lemma finiteMean_mono {Ω : Type*} [Fintype Ω] {f g : Ω → ℝ}
+    (h : ∀ ω, f ω ≤ g ω) : finiteMean f ≤ finiteMean g := by
+  exact div_le_div_of_nonneg_right (Finset.sum_le_sum (fun ω _ => h ω)) (Nat.cast_nonneg _)
+
+lemma finiteMean_equiv {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
+    (e : Ω ≃ Ω') (f : Ω → ℝ) :
+    finiteMean f = finiteMean (fun ω => f (e.symm ω)) := by
+  unfold finiteMean
+  rw [Fintype.card_congr e]
+  congr 1
+  exact Fintype.sum_equiv e _ _ (fun ω => by simp)
+
+lemma finiteMean_prod_left {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] [Nonempty Ω']
+    (f : Ω → ℝ) : finiteMean (fun ω : Ω × Ω' => f ω.1) = finiteMean f := by
+  simp only [finiteMean,Fintype.sum_prod_type,Finset.sum_const,nsmul_eq_mul,
+    ← Finset.mul_sum,Fintype.card_prod,Nat.cast_mul,Finset.card_univ]
+  have hn : (Fintype.card Ω' : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
+  field_simp
+
+lemma finiteMean_pi_eval {C : Type*} [Fintype C] [DecidableEq C]
+    {A : C → Type*} [∀ c, Fintype (A c)] [∀ c, Nonempty (A c)]
+    (c : C) (g : A c → ℝ) :
+    finiteMean (fun z : ∀ c, A c => g (z c)) = finiteMean g := by
+  rw [finiteMean_equiv (Equiv.piSplitAt c A)]
+  have he : (fun z : A c × ((j : {j // j ≠ c}) → A j) =>
+      g ((Equiv.piSplitAt c A).symm z c)) = (fun z => g z.1) := by
+    funext z
+    simp [Equiv.piSplitAt]
+  rw [he,finiteMean_prod_left]
+
+lemma finiteMean_eq_indicator {Ω : Type*} [Fintype Ω] [DecidableEq Ω] (a : Ω) :
+    finiteMean (fun ω : Ω => if ω = a then (1:ℝ) else 0) = 1 / Fintype.card Ω := by
+  simp [finiteMean]
+
+lemma exp_finiteMean_le {Ω : Type*} [Fintype Ω] [Nonempty Ω] (f : Ω → ℝ) :
+    Real.exp (finiteMean f) ≤ finiteMean (fun ω => Real.exp (f ω)) := by
+  have hn : (Fintype.card Ω : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
+  have hw : ∑ ω : Ω, (1 / (Fintype.card Ω : ℝ)) = 1 := by simp [hn]
+  have hj := convexOn_exp.map_sum_le (t := Finset.univ)
+    (w := fun _ : Ω => 1 / (Fintype.card Ω : ℝ)) (p := f)
+    (fun _ _ => by positivity) hw (fun _ _ => Set.mem_univ _)
+  simp only [smul_eq_mul, ← Finset.mul_sum] at hj
+  simpa only [finiteMean, div_eq_mul_inv, one_div, mul_comm, mul_one, one_mul] using hj
+
+variable {d A : ℕ}
+
+def cycleFiber (o : Card d → Option (Fin A)) (i : Fin A) : Finset (Card d) := by
+  classical
+  exact Finset.univ.filter (fun x => o x = some i)
+
+@[simp] lemma mem_cycleFiber (o : Card d → Option (Fin A)) (i : Fin A) (x : Card d) :
+    x ∈ cycleFiber o i ↔ o x = some i := by
+  classical
+  simp [cycleFiber]
+
+abbrev Terminal (o : Card d → Option (Fin A)) := ∀ i : Fin A, {x : Card d // o x = some i}
+
+noncomputable def slotWeight (o : Card d → Option (Fin A)) (i : Fin A) : ℝ :=
+  1 / (cycleFiber o i).card
+
+noncomputable def pathWeight (o : Card d → Option (Fin A)) (x : Card d) : ℝ :=
+  match o x with
+  | none => 0
+  | some i => slotWeight o i
+
+noncomputable def withinWeight (o : Card d → Option (Fin A)) (u v : Card d) : ℝ :=
+  match o u, o v with
+  | some i, some j => if i = j then slotWeight o i else 0
+  | _, _ => 0
+
+noncomputable def rotationMeeting (o : Card d → Option (Fin A)) (z : Terminal o)
+    (u v : Card d) : ℝ :=
+  match o u, o v with
+  | some i, some j => if i < j then (if (z j).val = v then 1 else 0)
+      else if j < i then (if (z i).val = u then 1 else 0)
+      else (if (z i).val = u then 1 else 0) + (if (z i).val = v then 1 else 0)
+  | _, _ => 0
+
+end Thorp
+
+end ThorpNine.Smoothing
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Smoothing/Main.lean b/lean/OAI/Probability/ThorpRouting/Smoothing/Main.lean
index ff1722b085337de6043209b6bf987960b3ffb836..514ebb302310d161394410426ce554bf2f2b1b17 100644
--- a/lean/OAI/Probability/ThorpRouting/Smoothing/Main.lean
+++ b/lean/OAI/Probability/ThorpRouting/Smoothing/Main.lean
@@ -1,1064 +1,1064 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Smoothing.ParallelFreshPre
-
-namespace OAI
-
-namespace ThorpNine.Smoothing
-
-namespace Thorp
-
-theorem palindrome_family_mgf (L n d : ℕ) (hL : 0 < L) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) ≤
-      Real.exp (Fintype.card ι * (68*∑ j ∈ Finset.range n,
-        Real.exp (-((L*4^j:ℕ):ℝ)/64))) := by
-  induction n with
-  | zero => simp only [palindromeFamilyCost,Finset.range_zero,Finset.sum_empty,pow_zero,
-      finiteMean_const,mul_zero,Real.exp_zero,le_refl]
-  | succ n ih =>
-    let l := L*4^n
-    have hl : 0 < l := Nat.mul_pos hL (pow_pos (by decide) _)
-    have hs : 2*familyEnd L n ≤ l := familyEnd_two L n
-    have hc : ∀ Y, palindromeFamilyCost L (n+1) d e (Y,X) =
-        palindromeFamilyCost L n d e (Y,X)+palindromeSlabCost l d e (Y,X) := by
-      intro Y
-      exact Finset.sum_range_succ _ _
-    have hm : finiteMean (fun Y => q^(palindromeFamilyCost L (n+1) d e (Y,X))) ≤
-        finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) *
-          Real.exp (Fintype.card ι*(68*Real.exp (-(l:ℝ)/64))) := by
-      by_cases hld : l ≤ d
-      · simp_rw [hc,pow_add]
-        exact palindrome_slab_adapted_mul_le d (familyEnd L n) l (by omega) hl hs e X _
-          (fun _ => pow_nonneg (by linarith) _)
-          (layerAdapted_comp (palindromeFamilyCost_adapted L n d e X) (fun c => q^c))
-          q hq hlog
-      · simp_rw [hc,palindromeSlabCost_gt l d (by omega) e,Nat.add_zero]
-        exact le_mul_of_one_le_right
-          (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
-          (Real.one_le_exp_iff.mpr (by positivity))
-    refine (hm.trans (mul_le_mul_of_nonneg_right ih (Real.exp_nonneg _))).trans_eq ?_
-    rw [←Real.exp_add,Finset.sum_range_succ]
-    congr 1
-    dsimp [l]
-    ring
-
-lemma pow_four_ge (j : ℕ) : j+1 ≤ 4^j := by
-  induction j with
-  | zero => norm_num
-  | succ j ih => rw [pow_succ]; omega
-
-lemma sum_family_decay (L n : ℕ) (hL : 0 < L) :
-    (∑ j ∈ Finset.range n, Real.exp (-((L*4^j:ℕ):ℝ)/64)) ≤
-      Real.exp (-(L:ℝ)/64)/(1-Real.exp (-(1:ℝ)/64)) := by
-  let r := Real.exp (-(1:ℝ)/64)
-  have hr : 0 < r := Real.exp_pos _
-  have hr1 : r < 1 := Real.exp_lt_one_iff.mpr (by norm_num)
-  have hg : (∑ j ∈ Finset.range n, r^j) ≤ 1/(1-r) := by
-    apply (le_div_iff₀ (by linarith : (0:ℝ)<1-r)).mpr
-    have hh := geom_sum_mul r n
-    nlinarith [pow_nonneg (le_of_lt hr) n]
-  calc
-    _ ≤ ∑ j ∈ Finset.range n, Real.exp (-(L:ℝ)/64)*r^j := by
-      apply Finset.sum_le_sum
-      intro j _
-      have hj := pow_four_ge j
-      have hlj : L+j ≤ L*4^j := by nlinarith
-      have hlj' : (L:ℝ)+j ≤ (L*4^j:ℕ) := by exact_mod_cast hlj
-      rw [←Real.exp_nat_mul,←Real.exp_add]
-      exact Real.exp_le_exp.mpr (by linarith)
-    _ = Real.exp (-(L:ℝ)/64)*(∑ j ∈ Finset.range n, r^j) := by rw [Finset.mul_sum]
-    _ ≤ Real.exp (-(L:ℝ)/64)*(1/(1-r)) :=
-      mul_le_mul_of_nonneg_left hg (Real.exp_nonneg _)
-    _ = _ := by ring
-
-
-lemma palindromeLevelCost_step_add (t d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (d+1)) (ω : BenesCoins (d+1)) :
-    palindromeLevelCost t (d+1) e ω =
-      let σ := benesStepEquiv d ω
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x σ.2.1
-      let hc := PairRouting.colors_compatible x σ.2.1
-      (if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
-          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) else 0) +
-        palindromeLevelCost t d (PairRouting.childEmbedding x c hc false) σ.1.1 +
-        palindromeLevelCost t d (PairRouting.childEmbedding x c hc true) σ.1.2 := by
-  classical
-  rw [palindromeLevelCost]
-  dsimp only
-  split_ifs with h
-  · simp only [palindromeLevelCost_gt t d (by omega),Nat.add_zero]
-  · simp only [Nat.zero_add]
-
-lemma palindromeLowCost_sum (H d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) :
-    palindromeLowCost H d e ω = ∑ i ∈ Finset.range H, palindromeLevelCost (i+1) d e ω := by
-  classical
-  induction d generalizing ι with
-  | zero => simp only [palindromeLowCost,palindromeLevelCost,Finset.sum_const_zero]
-  | succ d ih =>
-    rw [palindromeLowCost]
-    simp only [palindromeLevelCost_step_add,Finset.sum_add_distrib]
-    rw [←ih,←ih]
-    congr 2
-    simp only [Nat.add_right_cancel_iff,Finset.sum_ite_eq',Finset.mem_range,Nat.succ_le_iff]
-
-lemma palindromeLowCost_slab (L d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) :
-    palindromeLowCost L d e ω + palindromeSlabCost L d e ω = palindromeLowCost (2*L) d e ω := by
-  rw [palindromeLowCost_sum,palindromeLowCost_sum,show 2*L=L+L by omega,Finset.sum_range_add]
-  congr 1
-  exact Fin.sum_univ_eq_sum_range (fun i => palindromeLevelCost (L+i+1) d e ω) L
-
-lemma palindromeCost_family (L n d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) :
-    palindromeLowCost L d e ω + palindromeFamilyCost L n d e ω +
-      palindromeFamilyCost (2*L) n d e ω = palindromeLowCost (L*4^n) d e ω := by
-  induction n with
-  | zero => simp only [palindromeFamilyCost,Finset.range_zero,Finset.sum_empty,Nat.add_zero,
-      pow_zero,mul_one]
-  | succ n ih =>
-    have he : (2*L)*4^n = 2*(L*4^n) := by ring
-    have he' : L*4^(n+1) = 2*(2*(L*4^n)) := by rw [pow_succ]; ring
-    simp only [palindromeFamilyCost,Finset.sum_range_succ] at ih ⊢
-    rw [he,he',←palindromeLowCost_slab,←palindromeLowCost_slab,←ih]
-    omega
-
-
-lemma finiteMean_mul_sq_le {Ω : Type*} [Fintype Ω] (f g : Ω → ℝ) :
-    (finiteMean (fun ω => f ω*g ω))^2 ≤ finiteMean (fun ω => (f ω)^2)*finiteMean (fun ω => (g ω)^2) := by
-  have h := Finset.sum_mul_sq_le_sq_mul_sq Finset.univ f g
-  simp only [finiteMean,div_pow,div_mul_div_comm,←pow_two]
-  exact div_le_div_of_nonneg_right h (sq_nonneg _)
-
-lemma finiteMean_mul_exp_le {Ω : Type*} [Fintype Ω] (f g : Ω → ℝ) (A B : ℝ)
-    (hf : finiteMean (fun ω => (f ω)^2) ≤ Real.exp A)
-    (hg : finiteMean (fun ω => (g ω)^2) ≤ Real.exp B) :
-    finiteMean (fun ω => f ω*g ω) ≤ Real.exp ((A+B)/2) := by
-  have hc := finiteMean_mul_sq_le f g
-  have hn : 0 ≤ finiteMean (fun ω => (g ω)^2) := finiteMean_nonneg (fun _ => sq_nonneg _)
-  have hu := mul_le_mul hf hg hn (Real.exp_nonneg _)
-  have he : Real.exp A*Real.exp B = (Real.exp ((A+B)/2))^2 := by
-    rw [←Real.exp_add,←Real.exp_nat_mul]
-    congr 1
-    push_cast
-    ring
-  rw [he] at hu
-  have hh := hc.trans hu
-  have hep := Real.exp_pos ((A+B)/2)
-  nlinarith
-
-
-noncomputable def heightDecay (H : ℕ) : ℝ :=
-  68*Real.exp (-(H:ℝ)/64)/(1-Real.exp (-(1:ℝ)/64))
-
-lemma heightDecay_nonneg (H : ℕ) : 0 ≤ heightDecay H := by
-  have h : Real.exp (-(1:ℝ)/64) < 1 := Real.exp_lt_one_iff.mpr (by norm_num)
-  apply div_nonneg (by positivity) (by linarith)
-
-lemma heightDecay_antitone : Antitone heightDecay := by
-  intro L H h
-  have hden : 0 ≤ 1-Real.exp (-(1:ℝ)/64) := by
-    have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
-    linarith
-  apply div_le_div_of_nonneg_right _ hden
-  apply mul_le_mul_of_nonneg_left _ (by norm_num)
-  apply Real.exp_le_exp.mpr
-  have h' : (L:ℝ) ≤ H := by exact_mod_cast h
-  linarith
-
-lemma palindrome_family_mgf_decay (L n d : ℕ) (hL : 0 < L) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) ≤
-      Real.exp (Fintype.card ι*heightDecay L) := by
-  apply (palindrome_family_mgf L n d hL e X q hq hlog).trans
-  apply Real.exp_le_exp.mpr
-  apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
-  have h := mul_le_mul_of_nonneg_left (sum_family_decay L n hL) (by norm_num : (0:ℝ)≤68)
-  simpa only [heightDecay,mul_div_assoc] using h
-
-theorem palindrome_high_mgf (H n d : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/500) :
-    finiteMean (fun Y => q^(palindromeFamilyCost H n d e (Y,X)+
-      palindromeFamilyCost (2*H) n d e (Y,X))) ≤ Real.exp (Fintype.card ι*heightDecay H) := by
-  have hq2 := one_le_pow₀ (n:=2) hq
-  have hlog2 : Real.log (q^2) ≤ 1/250 := by rw [Real.log_pow]; norm_num; linarith
-  have h₀ := palindrome_family_mgf_decay H n d hH e X (q^2) hq2 hlog2
-  have h₁ := palindrome_family_mgf_decay (2*H) n d (by omega) e X (q^2) hq2 hlog2
-  have h₁' := h₁.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left
-    (heightDecay_antitone (show H ≤ 2*H by omega)) (Nat.cast_nonneg _)))
-  have hh := finiteMean_mul_exp_le (fun Y => q^(palindromeFamilyCost H n d e (Y,X)))
-    (fun Y => q^(palindromeFamilyCost (2*H) n d e (Y,X)))
-    (Fintype.card ι*heightDecay H) (Fintype.card ι*heightDecay H)
-    (by simpa only [pow_right_comm] using h₀) (by simpa only [pow_right_comm] using h₁')
-  simpa only [←pow_add,add_self_div_two] using hh
-
-lemma palindrome_high_full_mgf (H n d : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/500) :
-    finiteMean (fun ω : BenesCoins d => q^(palindromeFamilyCost H n d e ω+
-      palindromeFamilyCost (2*H) n d e ω)) ≤ Real.exp (Fintype.card ι*heightDecay H) := by
-  rw [finiteMean_prod,finiteMean_comm]
-  calc
-    _ ≤ finiteMean (fun _X : SwitchIndex d → Bool => Real.exp (Fintype.card ι*heightDecay H)) := by
-      apply finiteMean_mono
-      intro X
-      exact palindrome_high_mgf H n d hH e X q hq hlog
-    _ = _ := finiteMean_const _
-
-theorem palindrome_cost_mgf_cutoff (r H : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (H+r)) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/1000)
-    (hr : (2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*(q^2)^H) ≤ 1/2) :
-    finiteMean (fun ω : BenesCoins (H+r) => q^(palindromeCost (H+r) e ω)) ≤
-      Real.exp (((2:ℝ)^r*(2*((2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*(q^2)^H))^2)+
-        Fintype.card ι*heightDecay H)/2) := by
-  let d := H+r
-  have hd : d ≤ H*4^d := by
-    have hh := pow_four_ge d
-    nlinarith
-  have hcost (ω : BenesCoins d) : palindromeCost d e ω = palindromeLowCost H d e ω+
-      (palindromeFamilyCost H d d e ω+palindromeFamilyCost (2*H) d d e ω) := by
-    have hh := palindromeCost_family H d d e ω
-    rw [palindromeLowCost_eq _ _ hd] at hh
-    simpa only [Nat.add_assoc] using hh.symm
-  have hq2 := one_le_pow₀ (n:=2) hq
-  have hlog2 : Real.log (q^2) ≤ 1/500 := by rw [Real.log_pow]; norm_num; linarith
-  have hlow := palindrome_low_mgf r H e (q^2) hq2 hr
-  have hhigh := palindrome_high_full_mgf H d d hH e (q^2) hq2 hlog2
-  have hh := finiteMean_mul_exp_le (fun ω : BenesCoins d => q^(palindromeLowCost H d e ω))
-    (fun ω : BenesCoins d => q^(palindromeFamilyCost H d d e ω+palindromeFamilyCost (2*H) d d e ω))
-    _ _ (by simpa only [pow_right_comm] using hlow) (by simpa only [pow_right_comm] using hhigh)
-  change finiteMean (fun ω : BenesCoins d => q^(palindromeCost d e ω)) ≤ _
-  simpa only [hcost,pow_add,pow_right_comm] using hh
-
-
-lemma cutoffMass_le (H d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (ω : BenesCoins d) :
-    cutoffMass H d e ω ≤ Fintype.card ι := by
-  classical
-  induction d generalizing ι with
-  | zero => exact Nat.zero_le _
-  | succ d ih =>
-    rw [cutoffMass]
-    split_ifs
-    · exact crowdedMass_le _
-    · exact (Nat.add_le_add (ih _ _) (ih _ _)).trans_eq (PairRouting.color_card_add _)
-
-lemma palindromeLowCost_le (H d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) : palindromeLowCost H d e ω ≤ H*Fintype.card ι :=
-  (palindromeLowCost_le_cutoff H d e ω).trans (Nat.mul_le_mul_left H (cutoffMass_le H d e ω))
-
-theorem palindrome_cost_mgf_coarse (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/1000) :
-    finiteMean (fun ω : BenesCoins d => q^(palindromeCost d e ω)) ≤
-      Real.exp (Fintype.card ι*(1+heightDecay 1)) := by
-  have hd : d ≤ 1*4^d := by have hh := pow_four_ge d; omega
-  have hcost (ω : BenesCoins d) : palindromeCost d e ω = palindromeLowCost 1 d e ω+
-      (palindromeFamilyCost 1 d d e ω+palindromeFamilyCost 2 d d e ω) := by
-    have hh := palindromeCost_family 1 d d e ω
-    rw [palindromeLowCost_eq _ _ hd] at hh
-    simpa only [Nat.add_assoc] using hh.symm
-  have hq2 := one_le_pow₀ (n:=2) hq
-  have hlog2 : Real.log (q^2) ≤ 1/500 := by rw [Real.log_pow]; norm_num; linarith
-  have hlow : finiteMean (fun ω : BenesCoins d => (q^2)^(palindromeLowCost 1 d e ω)) ≤
-      Real.exp (Fintype.card ι) := by
-    calc
-      _ ≤ finiteMean (fun _ω : BenesCoins d => (q^2)^(Fintype.card ι)) := by
-        apply finiteMean_mono
-        intro ω
-        apply pow_le_pow_right₀ hq2
-        simpa only [one_mul] using palindromeLowCost_le 1 d e ω
-      _ = (q^2)^(Fintype.card ι) := finiteMean_const _
-      _ = Real.exp (Fintype.card ι * Real.log (q^2)) := by
-        rw [Real.exp_nat_mul,Real.exp_log (by positivity : 0<q^2)]
-      _ ≤ _ := Real.exp_le_exp.mpr (by
-        have hh := mul_le_mul_of_nonneg_left hlog2 (Nat.cast_nonneg (Fintype.card ι) : (0:ℝ)≤_)
-        nlinarith [Nat.cast_nonneg (α:=ℝ) (Fintype.card ι)])
-  have hhigh := palindrome_high_full_mgf 1 d d (by decide) e (q^2) hq2 hlog2
-  have hh := finiteMean_mul_exp_le (fun ω : BenesCoins d => q^(palindromeLowCost 1 d e ω))
-    (fun ω : BenesCoins d => q^(palindromeFamilyCost 1 d d e ω+palindromeFamilyCost 2 d d e ω))
-    (Fintype.card ι) (Fintype.card ι*heightDecay 1)
-    (by simpa only [pow_right_comm] using hlow) (by simpa only [pow_right_comm] using hhigh)
-  simp only [←pow_add,←hcost] at hh
-  apply hh.trans (Real.exp_le_exp.mpr _)
-  have hk : (0:ℝ) ≤ Fintype.card ι := Nat.cast_nonneg _
-  have hD := heightDecay_nonneg 1
-  nlinarith
-
-noncomputable def momentBase : ℝ := (2:ℝ)^(1/1000:ℝ)
-
-lemma momentBase_one_le : 1 ≤ momentBase := by
-  exact Real.one_le_rpow (by norm_num) (by norm_num)
-
-lemma momentBase_log : Real.log momentBase ≤ 1/1000 := by
-  rw [momentBase,Real.log_rpow (by norm_num)]
-  have hh := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
-  norm_num at hh ⊢
-  linarith
-
-lemma momentBase_pow (c : ℕ) : momentBase^c = (2:ℝ)^((c:ℝ)*(1/1000:ℝ)) := by
-  rw [mul_comm,Real.rpow_mul_natCast (by norm_num)]
-  rfl
-
-lemma palindromeRowMoment_le_cost (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) :
-    palindromeRowMoment d e (1/1000) ≤
-      finiteMean (fun ω : BenesCoins d => momentBase^(palindromeCost d e ω)) := by
-  have hn : (0:ℝ) < ((2^d:ℕ):ℝ)^(Fintype.card ι) := by positivity
-  have hn0 : (0:ℝ) ≤ ((2^d).descFactorial (Fintype.card ι):ℝ) /
-      ((2^d:ℕ):ℝ)^(Fintype.card ι) := by positivity
-  have hn1 : ((2^d).descFactorial (Fintype.card ι):ℝ) /
-      ((2^d:ℕ):ℝ)^(Fintype.card ι) ≤ 1 := by
-    apply (div_le_one hn).mpr
-    exact_mod_cast Nat.descFactorial_le_pow (2^d) (Fintype.card ι)
-  have hpow := Real.rpow_le_one hn0 hn1 (by norm_num : (0:ℝ)≤1/1000)
-  have hm := palindrome_row_density_moment d e (1/1000) (by norm_num)
-  simp_rw [←momentBase_pow] at hm
-  exact hm.trans (mul_le_of_le_one_left (finiteMean_nonneg (fun _ =>
-    pow_nonneg (le_trans (by norm_num) momentBase_one_le) _)) hpow)
-
-theorem palindrome_row_moment_coarse (d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) :
-    palindromeRowMoment d e (1/1000) ≤ Real.exp (Fintype.card ι*(1+heightDecay 1)) :=
-  (palindromeRowMoment_le_cost d e).trans
-    (palindrome_cost_mgf_coarse d e momentBase momentBase_one_le momentBase_log)
-
-
-noncomputable def densityCutoff (p : ℝ) : ℕ := ⌊-Real.log p/(4*Real.log 2)⌋₊
-
-lemma log_small_density {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
-    Real.log p ≤ -8*Real.log 2 := by
-  have he : Real.log (1/256:ℝ) = -8*Real.log 2 := by
-    rw [show (1/256:ℝ) = 1/2^8 by norm_num,
-      Real.log_div (by norm_num) (by positivity),Real.log_one,Real.log_pow]
-    ring
-  exact (Real.log_le_log hp hs).trans_eq he
-
-lemma densityCutoff_bounds {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
-    0 < densityCutoff p ∧
-    (densityCutoff p:ℝ)*(4*Real.log 2) ≤ -Real.log p ∧
-    -Real.log p < ((densityCutoff p:ℝ)+1)*(4*Real.log 2) := by
-  have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
-  have hd : 0 < 4*Real.log 2 := by positivity
-  have hh := log_small_density hp hs
-  have hx : 1 ≤ -Real.log p/(4*Real.log 2) := by
-    apply (le_div_iff₀ hd).mpr
-    nlinarith
-  refine ⟨Nat.floor_pos.mpr hx,?_,?_⟩
-  · exact (le_div_iff₀ hd).mp (Nat.floor_le (by linarith : 0 ≤ -Real.log p/(4*Real.log 2)))
-  · exact (div_lt_iff₀ hd).mp (Nat.lt_floor_add_one (-Real.log p/(4*Real.log 2)))
-
-lemma densityCutoff_decay {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
-    Real.exp (-(densityCutoff p:ℝ)/64) ≤ Real.exp (1/64) * p^(1/512:ℝ) := by
-  have hh := (densityCutoff_bounds hp hs).2.2
-  have hb : Real.log 2 ≤ 2 := by
-    have := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
-    linarith
-  have hH : 0 ≤ (densityCutoff p:ℝ)+1 := by positivity
-  have hu := mul_le_mul_of_nonneg_left hb hH
-  rw [Real.rpow_def_of_pos hp,←Real.exp_add]
-  apply Real.exp_le_exp.mpr
-  nlinarith
-
-lemma densityCutoff_power {p q : ℝ} (hp : 0 < p) (hs : p ≤ 1/256)
-    (hq : 0 < q) (hlog : Real.log q ≤ Real.log 2/4) (n : ℕ) (hn : n ≤ 4)
-    (hq1 : 1 ≤ q) :
-    p*(2:ℝ)^(densityCutoff p)*q^(n*densityCutoff p) ≤ p^(1/2:ℝ) := by
-  have hcut := (densityCutoff_bounds hp hs).2.1
-  have hH : 0 ≤ (densityCutoff p:ℝ) := Nat.cast_nonneg _
-  have hn' : (n:ℝ) ≤ 4 := by exact_mod_cast hn
-  have hln := Real.log_nonneg hq1
-  have hl := mul_le_mul_of_nonneg_right hn' hln
-  have hnl : (n:ℝ)*Real.log q ≤ Real.log 2 := by linarith
-  have hm := mul_le_mul_of_nonneg_left hnl hH
-  have heq : p*(2:ℝ)^(densityCutoff p)*q^(n*densityCutoff p) =
-      Real.exp (Real.log p+(densityCutoff p:ℝ)*Real.log 2+
-        (n*densityCutoff p:ℕ)*Real.log q) := by
-    rw [Real.exp_add,Real.exp_add,Real.exp_nat_mul,Real.exp_nat_mul,
-      Real.exp_log hp,Real.exp_log hq,Real.exp_log (by norm_num : (0:ℝ)<2)]
-  rw [heq,Real.rpow_def_of_pos hp]
-  apply Real.exp_le_exp.mpr
-  push_cast
-  nlinarith
-
-lemma densityCutoff_sparse {p q : ℝ} (hp : 0 < p) (hs : p ≤ 1/256)
-    (hq : 0 < q) (hlog : Real.log q ≤ Real.log 2/4) (hq1 : 1 ≤ q) :
-    (2:ℝ)^(densityCutoff p)*(p*(q^2)^(densityCutoff p)) ≤ 1/2 := by
-  have hh := densityCutoff_power hp hs hq hlog 2 (by decide) hq1
-  rw [←pow_mul] at ⊢
-  have hle : p^(1/2:ℝ) ≤ 1/2 := by
-    have hlogp := log_small_density hp hs
-    have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
-    rw [Real.rpow_def_of_pos hp]
-    calc
-      _ ≤ Real.exp (-Real.log 2) := by apply Real.exp_le_exp.mpr; nlinarith
-      _ = 1/2 := by rw [Real.exp_neg,Real.exp_log (by norm_num : (0:ℝ)<2)]; norm_num
-  nlinarith [hh.trans hle]
-
-
-lemma momentBase_log_exact : Real.log momentBase = Real.log 2/1000 := by
-  rw [momentBase,Real.log_rpow (by norm_num)]
-  ring
-
-lemma densityCutoff_le_height (d k : ℕ) (hk : 0 < k) :
-    densityCutoff ((k:ℝ)/(2:ℝ)^d) ≤ d := by
-  have hk' : (1:ℝ) ≤ k := by exact_mod_cast hk
-  have hp : 0 < (k:ℝ)/(2:ℝ)^d := by positivity
-  have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
-  have hlogk := Real.log_nonneg hk'
-  by_cases hx : 0 ≤ -Real.log ((k:ℝ)/(2:ℝ)^d)/(4*Real.log 2)
-  · have hh := (le_div_iff₀ (by positivity : 0 < 4*Real.log 2)).mp (Nat.floor_le hx)
-    have hd : (0:ℝ) ≤ d := Nat.cast_nonneg _
-    have hH : (0:ℝ) ≤ densityCutoff ((k:ℝ)/(2:ℝ)^d) := Nat.cast_nonneg _
-    change (densityCutoff ((k:ℝ)/(2:ℝ)^d):ℝ)*(4*Real.log 2) ≤ -Real.log ((k:ℝ)/(2:ℝ)^d) at hh
-    rw [Real.log_div (by positivity) (by positivity),Real.log_pow] at hh
-    have hle : (densityCutoff ((k:ℝ)/(2:ℝ)^d):ℝ) ≤ d := by nlinarith
-    exact_mod_cast hle
-  · have hz := Nat.floor_of_nonpos (le_of_not_ge hx)
-    simp only [densityCutoff,hz,Nat.zero_le]
-
-noncomputable def densityTailConstant : ℝ :=
-  68*Real.exp (1/64)/(1-Real.exp (-(1:ℝ)/64))
-
-lemma densityTailConstant_nonneg : 0 ≤ densityTailConstant := by
-  have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
-  apply div_nonneg (by positivity) (by linarith)
-
-lemma heightDecay_cutoff {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
-    heightDecay (densityCutoff p) ≤ densityTailConstant*p^(1/512:ℝ) := by
-  have hden : 0 ≤ 1-Real.exp (-(1:ℝ)/64) := by
-    have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
-    linarith
-  have hh := div_le_div_of_nonneg_right
-    (mul_le_mul_of_nonneg_left (densityCutoff_decay hp hs) (by norm_num : (0:ℝ)≤68)) hden
-  simpa only [heightDecay,densityTailConstant,mul_assoc,div_mul_eq_mul_div] using hh
-
-lemma sparse_low_algebra (r H k : ℕ) (q : ℝ) :
-    (2:ℝ)^r*(2*((2:ℝ)^H*(((k:ℝ)/(2:ℝ)^(H+r))*(q^2)^H))^2) =
-    2*k*(((k:ℝ)/(2:ℝ)^(H+r))*(2:ℝ)^H*q^(4*H)) := by
-  rw [pow_add,←pow_mul,show 4*H = (2*H)*2 by omega,pow_mul]
-  have hH : (2:ℝ)^H ≠ 0 := by positivity
-  have hr : (2:ℝ)^r ≠ 0 := by positivity
-  field_simp
-  ring
-
-lemma palindrome_row_moment_small (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι)
-    (hs : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1/256) :
-    palindromeRowMoment d e (1/1000) ≤
-      Real.exp (Fintype.card ι*(1+densityTailConstant)*
-        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ)) := by
-  generalize hpdef : ((Fintype.card ι:ℝ)/(2:ℝ)^d) = p at hs ⊢
-  have hp : 0 < p := by rw [←hpdef]; positivity
-  have hle := densityCutoff_le_height d (Fintype.card ι) hk
-  rw [hpdef] at hle
-  obtain ⟨r,hr⟩ := Nat.exists_eq_add_of_le hle
-  subst d
-  have hH := (densityCutoff_bounds hp hs).1
-  have hq := momentBase_one_le
-  have hqp : 0 < momentBase := by linarith
-  have hlog : Real.log momentBase ≤ Real.log 2/4 := by
-    rw [momentBase_log_exact]
-    have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
-    linarith
-  have hsp := densityCutoff_sparse hp hs hqp hlog hq
-  rw [←hpdef] at hsp
-  have hh := palindrome_cost_mgf_cutoff r (densityCutoff p) hH e momentBase hq momentBase_log
-    (by simpa only [hpdef] using hsp)
-  apply (palindromeRowMoment_le_cost _ e).trans (hh.trans (Real.exp_le_exp.mpr _))
-  rw [sparse_low_algebra,hpdef]
-  have hlow := densityCutoff_power hp hs hqp hlog 4 (by decide) hq
-  have hp1 : p ≤ 1 := by linarith
-  have he := Real.rpow_le_rpow_of_exponent_ge hp hp1 (by norm_num : (1:ℝ)/512 ≤ 1/2)
-  have hlow' := mul_le_mul_of_nonneg_left (hlow.trans he)
-    (by positivity : (0:ℝ)≤2*Fintype.card ι)
-  have hhigh := mul_le_mul_of_nonneg_left (heightDecay_cutoff hp hs)
-    (Nat.cast_nonneg (Fintype.card ι): (0:ℝ)≤_)
-  have hnon : 0 ≤ Fintype.card ι*densityTailConstant*p^(1/512:ℝ) :=
-    mul_nonneg (mul_nonneg (Nat.cast_nonneg _) densityTailConstant_nonneg)
-      (Real.rpow_nonneg (le_of_lt hp) _)
-  nlinarith
-
-noncomputable def densityConstant : ℝ :=
-  256*(1+heightDecay 1)+1+densityTailConstant
-
-lemma densityConstant_pos : 0 < densityConstant := by
-  dsimp [densityConstant]
-  have hh := heightDecay_nonneg 1
-  have ht := densityTailConstant_nonneg
-  linarith
-
-theorem palindrome_row_moment_bound (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι) :
-    palindromeRowMoment d e (1/1000) ≤ Real.exp (densityConstant*Fintype.card ι*
-        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ)) := by
-  have hp : 0 < (Fintype.card ι:ℝ)/(2:ℝ)^d := by positivity
-  have hp1 : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1 := by
-    apply (div_le_one (by positivity)).mpr
-    have h := Fintype.card_le_of_injective e e.injective
-    rw [card_positions] at h
-    exact_mod_cast h
-  have hk0 : (0:ℝ) ≤ Fintype.card ι := Nat.cast_nonneg _
-  have hpow := Real.rpow_nonneg (le_of_lt hp) (1/512:ℝ)
-  by_cases hs : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1/256
-  · apply (palindrome_row_moment_small d e hk hs).trans (Real.exp_le_exp.mpr _)
-    have hD : 1+densityTailConstant ≤ densityConstant := by
-      dsimp [densityConstant]
-      have := heightDecay_nonneg 1
-      linarith
-    have hh := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hD hk0) hpow
-    nlinarith
-  · have hlow : (1:ℝ)/256 ≤ ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ) := by
-      have he := Real.rpow_le_rpow_of_exponent_ge hp hp1 (by norm_num : (1:ℝ)/512 ≤ 1)
-      rw [Real.rpow_one] at he
-      exact (le_of_not_ge hs).trans he
-    have hD : 256*(1+heightDecay 1) ≤ densityConstant := by
-      dsimp [densityConstant]
-      have := densityTailConstant_nonneg
-      linarith
-    apply (palindrome_row_moment_coarse d e).trans (Real.exp_le_exp.mpr _)
-    have hD0 : 0 ≤ 1+heightDecay 1 := by have := heightDecay_nonneg 1; linarith
-    have ha := mul_le_mul_of_nonneg_left hlow (mul_nonneg (by norm_num : (0:ℝ)≤256) hD0)
-    have hb := mul_le_mul_of_nonneg_right hD hpow
-    have hc := mul_le_mul_of_nonneg_right (ha.trans hb) hk0
-    nlinarith
-
-lemma palindromeRowMoment_nonneg (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (a : ℝ) : 0 ≤ palindromeRowMoment d e a := by
-  apply finiteMean_nonneg
-  intro f
-  exact Real.rpow_nonneg (mul_nonneg (Nat.cast_nonneg _)
-    (PairRouting.tupleProbability_nonneg _ _ _)) _
-
-theorem palindrome_log_row_moment (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι) :
-    Real.log (palindromeRowMoment d e (1/1000)) ≤ densityConstant*Fintype.card ι*
-        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ) := by
-  by_cases hz : palindromeRowMoment d e (1/1000) = 0
-  · rw [hz,Real.log_zero]
-    exact mul_nonneg (mul_nonneg (le_of_lt densityConstant_pos) (Nat.cast_nonneg _))
-      (Real.rpow_nonneg (by positivity) _)
-  · exact (Real.log_le_iff_le_exp (lt_of_le_of_ne (palindromeRowMoment_nonneg _ _ _) (Ne.symm hz))).mpr
-      (palindrome_row_moment_bound d e hk)
-
-end Thorp
-
-
-namespace Thorp.StrongSmoothing
-open scoped BigOperators
-
-variable {X : Type*} [Fintype X] [Nonempty X]
-
-omit [Nonempty X] in
-lemma mean_nonneg (f : X → ℝ) (hf : ∀ x, 0 ≤ f x) : 0 ≤ finiteMean f :=
-  finiteMean_nonneg hf
-
-lemma mean_rpow_concave (f : X → ℝ) (hf : ∀ x, 0 ≤ f x)
-    {a : ℝ} (ha : 0 ≤ a) (ha1 : a ≤ 1) :
-    finiteMean (fun x => (f x)^a) ≤ (finiteMean f)^a := by
-  have hN : (0:ℝ) < Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
-  have hh := (Real.concaveOn_rpow ha ha1).le_map_sum
-    (t := Finset.univ) (w := fun _ : X => 1/(Fintype.card X:ℝ))
-    (p := f) (fun _ _ => by positivity)
-    (by simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]; field_simp)
-    (fun x _ => hf x)
-  simpa only [smul_eq_mul,one_div,←Finset.mul_sum,mul_comm _ (∑ _ : X,_),
-    ←div_eq_mul_inv,finiteMean] using hh
-
-noncomputable def act (K : Matrix X X ℝ) (f : X → ℝ) (y : X) : ℝ := ∑ x, K x y*f x
-
-omit [Nonempty X] in
-lemma act_nonneg (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
-    (f : X → ℝ) (hf : ∀ x, 0 ≤ f x) (y : X) : 0 ≤ act K f y :=
-  Finset.sum_nonneg (fun x _ => mul_nonneg (hK x y) (hf x))
-
-omit [Nonempty X] in
-lemma act_rpow_le (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
-    (hcol : ∀ y, ∑ x, K x y = 1) (f : X → ℝ) (hf : ∀ x, 0 ≤ f x)
-    {b : ℝ} (hb : 1 ≤ b) (y : X) :
-    (act K f y)^b ≤ act K (fun x => (f x)^b) y := by
-  simpa only [smul_eq_mul,act] using
-    (convexOn_rpow hb).map_sum_le (t := Finset.univ) (w := fun x => K x y)
-      (p := f) (fun x _ => hK x y) (hcol y) (fun x _ => hf x)
-
-lemma act_one_to_p (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
-    {p A : ℝ} (hp : 1 ≤ p)
-    (hm : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) ≤ Real.exp A)
-    (f : X → ℝ) (hf : ∀ x, 0 ≤ f x) :
-    finiteMean (fun y => (act K f y)^p) ≤ Real.exp A*(finiteMean f)^p := by
-  classical
-  let s : ℝ := ∑ x, f x
-  have hs : 0 ≤ s := Finset.sum_nonneg (fun x _ => hf x)
-  have hN : (0:ℝ) < Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
-  by_cases hz : s=0
-  · have hfz : ∀ x, f x=0 := by
-      intro x
-      exact (Finset.sum_eq_zero_iff_of_nonneg (fun x _ => hf x)).mp hz x (Finset.mem_univ _)
-    simp only [act,hfz,mul_zero,Finset.sum_const_zero,finiteMean,zero_div,
-      Real.zero_rpow (show p ≠ 0 by linarith),mul_zero,le_refl]
-  · have hsp : 0 < s := lt_of_le_of_ne hs (Ne.symm hz)
-    let w : X → ℝ := fun x => f x/s
-    have hw : ∀ x, 0 ≤ w x := fun x => div_nonneg (hf x) hs
-    have hws : ∑ x, w x = 1 := by simp only [w,←Finset.sum_div]; exact div_self hz
-    have hact (y : X) : act K f y = finiteMean f * ∑ x, w x*((Fintype.card X:ℝ)*K x y) := by
-      simp only [act,w,finiteMean,Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro x _
-      have hz' : (∑ x, f x) ≠ 0 := hz
-      dsimp [s]
-      field_simp [hz',ne_of_gt hN]
-    have hj (y : X) : (act K f y)^p ≤ (finiteMean f)^p*
-        ∑ x, w x*((Fintype.card X:ℝ)*K x y)^p := by
-      rw [hact,Real.mul_rpow (mean_nonneg f hf)
-        (Finset.sum_nonneg (fun x _ => mul_nonneg (hw x) (mul_nonneg hN.le (hK x y))))]
-      apply mul_le_mul_of_nonneg_left _ (Real.rpow_nonneg (mean_nonneg f hf) _)
-      exact (convexOn_rpow hp).map_sum_le
-        (fun x _ => hw x) hws (fun x _ => mul_nonneg hN.le (hK x y))
-    have hh := finiteMean_mono hj
-    have he : finiteMean (fun y => (finiteMean f)^p*
-        ∑ x, w x*((Fintype.card X:ℝ)*K x y)^p) =
-        (finiteMean f)^p * ∑ x, w x*finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) := by
-      simp_rw [mul_comm ((finiteMean f)^p), finiteMean_mul_const,
-        finiteMean_sum, mul_comm (w _), finiteMean_mul_const]
-    rw [he] at hh
-    apply hh.trans
-    have hsum : (∑ x, w x*finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p)) ≤ Real.exp A := by
-      calc
-        _ ≤ ∑ x, w x*Real.exp A := Finset.sum_le_sum
-          (fun x _ => mul_le_mul_of_nonneg_left (hm x) (hw x))
-        _ = Real.exp A := by rw [←Finset.sum_mul,hws,one_mul]
-    simpa only [mul_comm (Real.exp A)] using
-      mul_le_mul_of_nonneg_left hsum (Real.rpow_nonneg (mean_nonneg f hf) _)
-
-lemma act_b_to_pb (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
-    (hcol : ∀ y, ∑ x, K x y = 1) {p A b : ℝ} (hp : 1 ≤ p) (hb : 1 ≤ b)
-    (hm : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) ≤ Real.exp A)
-    (f : X → ℝ) (hf : ∀ x, 0 ≤ f x) :
-    finiteMean (fun y => (act K f y)^(b*p)) ≤
-      Real.exp A*(finiteMean (fun x => (f x)^b))^p := by
-  calc
-    _ ≤ finiteMean (fun y => (act K (fun x => (f x)^b) y)^p) := by
-      apply finiteMean_mono
-      intro y
-      rw [Real.rpow_mul (act_nonneg K hK f hf y)]
-      exact Real.rpow_le_rpow (Real.rpow_nonneg (act_nonneg K hK f hf y) _) (act_rpow_le K hK hcol f hf hb y) (by linarith)
-    _ ≤ _ := act_one_to_p K hK hp hm _ (fun x => Real.rpow_nonneg (hf x) _)
-
-variable [DecidableEq X]
-
-omit [Nonempty X] in
-lemma kernel_pow_nonneg (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y) (r : ℕ) :
-    ∀ x y, 0 ≤ (K^r) x y := by
-  classical
-  induction r with
-  | zero => intro x y; simp only [pow_zero,Matrix.one_apply]; split_ifs <;> norm_num
-  | succ r ih =>
-    intro x y
-    rw [pow_succ,Matrix.mul_apply]
-    exact Finset.sum_nonneg (fun z _ => mul_nonneg (ih x z) (hK z y))
-
-omit [Nonempty X] in
-lemma kernel_pow_row (K : Matrix X X ℝ) (hrow : ∀ x, ∑ y, K x y = 1) (r : ℕ) :
-    ∀ x, ∑ y, (K^r) x y = 1 := by
-  classical
-  induction r with
-  | zero => intro x; simp [Matrix.one_apply]
-  | succ r ih =>
-    intro x
-    simp only [pow_succ,Matrix.mul_apply]
-    rw [Finset.sum_comm]
-    simp only [←Finset.mul_sum,hrow,mul_one,ih]
-
-noncomputable def rowDensity (K : Matrix X X ℝ) (r : ℕ) (x y : X) : ℝ :=
-  (Fintype.card X:ℝ)*(K^r) x y
-
-omit [Nonempty X] in
-lemma rowDensity_nonneg (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y) (r : ℕ) (x y : X) :
-    0 ≤ rowDensity K r x y := mul_nonneg (Nat.cast_nonneg _) (kernel_pow_nonneg K hK r x y)
-
-omit [Nonempty X] in
-lemma rowDensity_succ (K : Matrix X X ℝ) (r : ℕ) (x y : X) :
-    rowDensity K (r+1) x y = act K (rowDensity K r x) y := by
-  simp only [rowDensity,pow_succ,Matrix.mul_apply,act,Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro z _
-  ring
-
-lemma rowDensity_mean (K : Matrix X X ℝ) (hrow : ∀ x, ∑ y, K x y = 1) (r : ℕ) (x : X) :
-    finiteMean (rowDensity K r x) = 1 := by
-  have hN : (Fintype.card X:ℝ) ≠ 0 := ne_of_gt (Nat.cast_pos.mpr Fintype.card_pos)
-  simp only [finiteMean,rowDensity,←Finset.mul_sum,kernel_pow_row K hrow,mul_one,div_self hN]
-
-lemma rowDensity_moment_iter (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
-    (hrow : ∀ x, ∑ y, K x y = 1) (hcol : ∀ y, ∑ x, K x y = 1)
-    {p A : ℝ} (hp : 1 ≤ p) (hA : 0 ≤ A)
-    (hm : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) ≤ Real.exp A)
-    (r : ℕ) (x : X) :
-    finiteMean (fun y => (rowDensity K r x y)^(p^r)) ≤ Real.exp (A*r*p^r) := by
-  induction r with
-  | zero => simp [rowDensity_mean K hrow]
-  | succ r ih =>
-    have hpr : 1 ≤ p^r := one_le_pow₀ hp
-    have hps : 1 ≤ p^(r+1) := one_le_pow₀ hp
-    simp_rw [rowDensity_succ]
-    calc
-      _ ≤ Real.exp A*(finiteMean (fun y => (rowDensity K r x y)^(p^r)))^p := by
-        rw [pow_succ]
-        exact act_b_to_pb K hK hcol hp hpr hm _ (rowDensity_nonneg K hK r x)
-      _ ≤ Real.exp A*(Real.exp (A*r*p^r))^p := by
-        apply mul_le_mul_of_nonneg_left _ (Real.exp_pos _).le
-        exact Real.rpow_le_rpow (mean_nonneg _ (fun y => Real.rpow_nonneg
-          (rowDensity_nonneg K hK r x y) _)) ih (le_trans zero_le_one hp)
-      _ = Real.exp (A+A*r*p^(r+1)) := by rw [←Real.exp_mul,←Real.exp_add,pow_succ]; congr 1; ring
-      _ ≤ _ := by
-        apply Real.exp_le_exp.mpr
-        have hh := mul_le_mul_of_nonneg_left hps hA
-        push_cast
-        nlinarith
-
-lemma rowDensity_second_moment (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
-    (hrow : ∀ x, ∑ y, K x y = 1) (hcol : ∀ y, ∑ x, K x y = 1)
-    {p A : ℝ} (hp : 1 ≤ p) (hA : 0 ≤ A)
-    (hm : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) ≤ Real.exp A)
-    (r : ℕ) (hr : 2 ≤ p^r) (x : X) :
-    finiteMean (fun y => (rowDensity K r x y)^2) ≤ Real.exp (2*A*r) := by
-  have hpos : 0 < p^r := by linarith
-  have hj := mean_rpow_concave (fun y => (rowDensity K r x y)^(p^r))
-    (fun y => Real.rpow_nonneg (rowDensity_nonneg K hK r x y) _)
-    (a := 2/(p^r)) (by positivity) ((div_le_one hpos).mpr hr)
-  have he (y : X) : ((rowDensity K r x y)^(p^r))^(2/(p^r)) = (rowDensity K r x y)^2 := by
-    rw [←Real.rpow_mul (rowDensity_nonneg K hK r x y)]
-    rw [mul_div_cancel₀ _ (ne_of_gt hpos),Real.rpow_two]
-  simp_rw [he] at hj
-  apply hj.trans
-  calc
-    _ ≤ (Real.exp (A*r*p^r))^(2/(p^r)) :=
-      Real.rpow_le_rpow (mean_nonneg _ (fun y => Real.rpow_nonneg
-        (rowDensity_nonneg K hK r x y) _)) (rowDensity_moment_iter K hK hrow hcol hp hA hm r x)
-        (by positivity)
-    _ = _ := by rw [←Real.exp_mul]; congr 1; field_simp [ne_of_gt hpos]
-
-lemma symmetric_kernel_smoothing (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
-    (hrow : ∀ x, ∑ y, K x y = 1) (hcol : ∀ y, ∑ x, K x y = 1)
-    (hsym : K.IsSymm) {p A : ℝ} (hp : 1 ≤ p) (hA : 0 ≤ A)
-    (hm : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) ≤ Real.exp A)
-    (r : ℕ) (hr : 2 ≤ p^r) (x y : X) :
-    (K^(2*r)) x y ≤ Real.exp (2*A*r)/(Fintype.card X:ℝ) := by
-  have hN : (0:ℝ) < Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
-  have hs : ∀ z, (K^r) z y = (K^r) y z := fun z => (hsym.pow r).apply y z
-  have he : (Fintype.card X:ℝ)*(K^(2*r)) x y =
-      finiteMean (fun z => rowDensity K r x z*rowDensity K r y z) := by
-    simp only [finiteMean,rowDensity,two_mul,pow_add,Matrix.mul_apply,hs,Finset.mul_sum]
-    rw [Finset.sum_div]
-    apply Finset.sum_congr rfl
-    intro z _
-    field_simp
-  have hh : finiteMean (fun z => rowDensity K r x z*rowDensity K r y z) ≤
-      Real.exp (2*A*r) := by
-    calc
-      _ ≤ finiteMean (fun z => ((rowDensity K r x z)^2+(rowDensity K r y z)^2)/2) := by
-        apply finiteMean_mono
-        intro z
-        nlinarith [sq_nonneg (rowDensity K r x z-rowDensity K r y z)]
-      _ = (finiteMean (fun z => (rowDensity K r x z)^2)+
-          finiteMean (fun z => (rowDensity K r y z)^2))/2 := by
-        simp only [div_eq_mul_inv,finiteMean_mul_const,finiteMean_add]
-      _ ≤ _ := by
-        have hx := rowDensity_second_moment K hK hrow hcol hp hA hm r hr x
-        have hy := rowDensity_second_moment K hK hrow hcol hp hA hm r hr y
-        linarith
-  apply (le_div_iff₀ hN).mpr
-  simpa only [←he,mul_comm (Fintype.card X:ℝ)] using hh
-
-lemma density_exponent_numeric : 2 ≤ (1+(1/1000:ℝ))^(1000:ℕ) := by
-  calc
-    (2:ℝ) = 1+(1000:ℝ)*(1/1000) := by norm_num
-    _ ≤ _ := by simpa only [Nat.cast_ofNat] using
-      (one_add_mul_le_pow (a := (1/1000:ℝ)) (by norm_num) 1000)
-
-end Thorp.StrongSmoothing
-
-namespace Thorp.StrongSmoothing.PermLaw
-open scoped BigOperators
-variable {X Ω Ω' : Type*} [Fintype X] [DecidableEq X] [Fintype Ω] [Fintype Ω']
-
-noncomputable def kernel (P : Ω → Equiv.Perm X) : Matrix X X ℝ :=
-  fun x y => finiteMean (fun ω => if P ω x = y then (1:ℝ) else 0)
-
-omit [Fintype X] in
-lemma nonneg (P : Ω → Equiv.Perm X) (x y : X) : 0 ≤ kernel P x y := by
-  exact finiteMean_nonneg (fun ω => by split_ifs <;> norm_num)
-
-lemma row [Nonempty Ω] (P : Ω → Equiv.Perm X) (x : X) : ∑ y, kernel P x y = 1 := by
-  simp only [kernel,←finiteMean_sum]
-  simp only [Finset.sum_ite_eq,Finset.mem_univ,↓reduceIte,finiteMean_const]
-
-omit [Fintype X] in
-lemma inverse (P : Ω → Equiv.Perm X) : kernel (fun ω => (P ω)⁻¹) = (kernel P).transpose := by
-  ext x y
-  apply finiteMean_congr
-  intro ω
-  have he : (P ω)⁻¹ x=y ↔ P ω y=x := by
-    change (P ω).symm x=y ↔ P ω y=x
-    exact (Equiv.symm_apply_eq (P ω)).trans eq_comm
-  simp only [he]
-
-lemma col [Nonempty Ω] (P : Ω → Equiv.Perm X) (y : X) : ∑ x, kernel P x y = 1 := by
-  have hh := row (fun ω => (P ω)⁻¹) y
-  rw [inverse] at hh
-  exact hh
-
-lemma mul (P : Ω → Equiv.Perm X) (Q : Ω' → Equiv.Perm X) :
-    kernel (fun ω : Ω × Ω' => P ω.1*Q ω.2) = kernel Q*kernel P := by
-  ext x y
-  simp only [Matrix.mul_apply,kernel]
-  have he (ω : Ω × Ω') :
-      (if (P ω.1*Q ω.2) x=y then (1:ℝ) else 0) =
-        ∑ z : X, (if P ω.1 z=y then (1:ℝ) else 0)*(if Q ω.2 x=z then (1:ℝ) else 0) := by
-    simp only [mul_ite,mul_one,mul_zero,Finset.sum_ite_eq,Finset.mem_univ,↓reduceIte]
-    rfl
-  change finiteMean (fun ω : Ω × Ω' => if (P ω.1*Q ω.2) x=y then (1:ℝ) else 0) = _
-  simp_rw [he]
-  rw [finiteMean_sum]
-  apply Finset.sum_congr rfl
-  intro z _
-  exact (finiteMean_prod_mul (fun ω : Ω => if P ω z=y then (1:ℝ) else 0)
-    (fun ω : Ω' => if Q ω x=z then (1:ℝ) else 0)).trans (mul_comm _ _)
-
-omit [Fintype X] in
-lemma conjugate (P : Ω → Equiv.Perm X) (e : Equiv.Perm X) (x y : X) :
-    kernel (fun ω => e*P ω*e⁻¹) x y = kernel P (e.symm x) (e.symm y) := by
-  apply finiteMean_congr
-  intro ω
-  have he : (e*P ω*e⁻¹) x=y ↔ P ω (e.symm x)=e.symm y := by
-    change e (P ω (e.symm x))=y ↔ _
-    exact (Equiv.eq_symm_apply e).symm
-  simp only [he]
-
-end Thorp.StrongSmoothing.PermLaw
-
-namespace Thorp.StrongSmoothing
-open scoped BigOperators
-
-section MatrixHelpers
-variable {X : Type*} [Fintype X] [DecidableEq X]
-
-omit [DecidableEq X] in
-lemma mul_nonneg_entries (A B : Matrix X X ℝ) (hA : ∀ x y, 0 ≤ A x y)
-    (hB : ∀ x y, 0 ≤ B x y) (x y : X) : 0 ≤ (A*B) x y :=
-  Finset.sum_nonneg (fun z _ => mul_nonneg (hA x z) (hB z y))
-
-omit [DecidableEq X] in
-lemma mul_row (A B : Matrix X X ℝ) (hA : ∀ x, ∑ y, A x y = 1)
-    (hB : ∀ x, ∑ y, B x y = 1) (x : X) : ∑ y, (A*B) x y = 1 := by
-  simp only [Matrix.mul_apply]
-  rw [Finset.sum_comm]
-  simp only [←Finset.mul_sum,hB,mul_one,hA]
-
-omit [DecidableEq X] in
-lemma mul_col (A B : Matrix X X ℝ) (hA : ∀ y, ∑ x, A x y = 1)
-    (hB : ∀ y, ∑ x, B x y = 1) (y : X) : ∑ x, (A*B) x y = 1 := by
-  simp only [Matrix.mul_apply]
-  rw [Finset.sum_comm]
-  simp only [←Finset.sum_mul,hA,one_mul,hB]
-
-omit [DecidableEq X] in
-lemma bound_mul_left (A B : Matrix X X ℝ) (hA : ∀ x y, 0 ≤ A x y)
-    (hrow : ∀ x, ∑ y, A x y = 1) {D : ℝ} (hB : ∀ x y, B x y ≤ D) :
-    ∀ x y, (A*B) x y ≤ D := by
-  intro x y
-  calc
-    _ ≤ ∑ z, A x z*D := Finset.sum_le_sum (fun z _ => mul_le_mul_of_nonneg_left (hB z y) (hA x z))
-    _ = D := by rw [←Finset.sum_mul,hrow,one_mul]
-
-omit [DecidableEq X] in
-lemma bound_mul_right (A B : Matrix X X ℝ) (hB : ∀ x y, 0 ≤ B x y)
-    (hcol : ∀ y, ∑ x, B x y = 1) {D : ℝ} (hA : ∀ x y, A x y ≤ D) :
-    ∀ x y, (A*B) x y ≤ D := by
-  intro x y
-  calc
-    _ ≤ ∑ z, D*B z y := Finset.sum_le_sum (fun z _ => mul_le_mul_of_nonneg_right (hA x z) (hB z y))
-    _ = D := by rw [←Finset.mul_sum,hcol,mul_one]
-
-lemma sandwich_power (A B : Matrix X X ℝ) (r : ℕ) : A*(B*A)^r*B = (A*B)^(r+1) := by
-  induction r with
-  | zero => simp
-  | succ r ih =>
-    calc
-      _ = (A*(B*A)^r*B)*(A*B) := by rw [pow_succ]; noncomm_ring
-      _ = (A*B)^(r+1)*(A*B) := by rw [ih]
-      _ = _ := (pow_succ (A*B) (r+1)).symm
-
-end MatrixHelpers
-
-section TupleAction
-variable {ι β Ω : Type*} [Fintype ι] [Fintype β] [DecidableEq β] [Fintype Ω]
-
-def tuplePerm (P : Equiv.Perm β) : Equiv.Perm (ι ↪ β) where
-  toFun x := x.trans P.toEmbedding
-  invFun x := x.trans P.symm.toEmbedding
-  left_inv x := by ext i; simp
-  right_inv x := by ext i; simp
-
-omit [Fintype ι] [Fintype β] [DecidableEq β] in
-@[simp] lemma tuplePerm_apply (P : Equiv.Perm β) (x : ι ↪ β) (i : ι) :
-    tuplePerm P x i = P (x i) := rfl
-
-omit [Fintype ι] [Fintype β] [DecidableEq β] in
-lemma tuplePerm_mul (P Q : Equiv.Perm β) :
-    tuplePerm (ι := ι) (P*Q) = tuplePerm P*tuplePerm Q := by
-  ext x i
-  rfl
-
-omit [Fintype ι] [Fintype β] [DecidableEq β] in
-lemma tuplePerm_inverse (P : Equiv.Perm β) :
-    tuplePerm (ι := ι) P⁻¹ = (tuplePerm P)⁻¹ := by
-  ext x i
-  rfl
-
-omit [Fintype β] in
-lemma tupleProbability_kernel (P : Ω → Equiv.Perm β) (x y : ι ↪ β) :
-    PairRouting.tupleProbability P x y = PermLaw.kernel (fun ω => tuplePerm (P ω)) x y := by
-  classical
-  apply finiteMean_congr
-  intro ω
-  have he : (∀ i, P ω (x i) = y i) ↔ tuplePerm (P ω) x = y := by
-    constructor
-    · intro h; ext i; exact h i
-    · intro h i; exact congrArg (fun z : ι ↪ β => z i) h
-  simp only [he]
-end TupleAction
-
-lemma sweepKernel_law (d l : ℕ) (order : Equiv.Perm (Fin d)) :
-    sweepKernel d l order = PermLaw.kernel (fun ω => tuplePerm (ι := Fin l) (sweep d order ω)) := by
-  ext x y
-  exact tupleProbability_kernel _ x y
-
-lemma reflectedKernel_conjugate (d l : ℕ) (order : Equiv.Perm (Fin d)) (x y : Tuples d l) :
-    reflectedKernel d l order false x y =
-      PairRouting.tupleProbability (palindromePerm d)
-        ((tuplePerm (coordinateRelabel d order)).symm x)
-        ((tuplePerm (coordinateRelabel d order)).symm y) := by
-  classical
-  let e : Equiv.Perm (Tuples d l) := tuplePerm (coordinateRelabel d order)
-  let P := fun ω => tuplePerm (ι := Fin l) (butterflyPerm d (decodeButterfly d ω))
-  have hA : sweepKernel d l order = PermLaw.kernel (fun ω => e*P ω*e⁻¹) := by
-    rw [sweepKernel_law]
-    simp only [sweep,tuplePerm_mul,tuplePerm_inverse]
-    rfl
-  rw [reflectedKernel,ite_eq_right Bool.false_ne_true,hA]
-  rw [←PermLaw.inverse,←PermLaw.mul]
-  have hprod : (fun ω : BenesCoins d => (e*P ω.1*e⁻¹)*((e*P ω.2*e⁻¹)⁻¹)) =
-      fun ω : BenesCoins d => e*tuplePerm (ι := Fin l) (palindromePerm d ω)*e⁻¹ := by
-    funext ω
-    simp only [mul_inv_rev,inv_inv,palindromePerm,tuplePerm_mul]
-    change (e*P ω.1*e⁻¹)*(e*(P ω.2)⁻¹*e⁻¹) = e*(P ω.1*(P ω.2)⁻¹)*e⁻¹
-    group
-  rw [hprod,PermLaw.conjugate,tupleProbability_kernel]
-
-
-lemma sweepKernel_nonneg (d l : ℕ) (order : Equiv.Perm (Fin d)) (x y : Tuples d l) :
-    0 ≤ sweepKernel d l order x y := PairRouting.tupleProbability_nonneg _ _ _
-
-lemma sweepKernel_row (d l : ℕ) (order : Equiv.Perm (Fin d)) (x : Tuples d l) :
-    ∑ y, sweepKernel d l order x y = 1 := by
-  rw [sweepKernel_law]
-  exact PermLaw.row _ _
-
-lemma sweepKernel_col (d l : ℕ) (order : Equiv.Perm (Fin d)) (y : Tuples d l) :
-    ∑ x, sweepKernel d l order x y = 1 := by
-  rw [sweepKernel_law]
-  exact PermLaw.col _ _
-
-lemma reflected_nonneg (d l : ℕ) (order : Equiv.Perm (Fin d)) (x y : Tuples d l) :
-    0 ≤ reflectedKernel d l order false x y := by
-  exact mul_nonneg_entries _ _ (fun x y => sweepKernel_nonneg d l order y x)
-    (sweepKernel_nonneg d l order) x y
-
-lemma reflected_row (d l : ℕ) (order : Equiv.Perm (Fin d)) (x : Tuples d l) :
-    ∑ y, reflectedKernel d l order false x y = 1 :=
-  mul_row _ _ (sweepKernel_col d l order) (sweepKernel_row d l order) x
-
-lemma reflected_col (d l : ℕ) (order : Equiv.Perm (Fin d)) (y : Tuples d l) :
-    ∑ x, reflectedKernel d l order false x y = 1 :=
-  mul_col _ _ (sweepKernel_row d l order) (sweepKernel_col d l order) y
-
-lemma reflected_symmetric (d l : ℕ) (order : Equiv.Perm (Fin d)) :
-    (reflectedKernel d l order false).IsSymm := Matrix.isSymm_transpose_mul_self _
-
-lemma reflected_moment (d l : ℕ) (order : Equiv.Perm (Fin d)) (x : Tuples d l) :
-    finiteMean (fun y : Tuples d l =>
-      ((Fintype.card (Tuples d l):ℝ)*reflectedKernel d l order false x y)^(1+(1/1000:ℝ))) ≤
-      Real.exp (l*(1+heightDecay 1)) := by
-  classical
-  let e : Equiv.Perm (Tuples d l) := tuplePerm (coordinateRelabel d order)
-  simp_rw [reflectedKernel_conjugate]
-  have he := finiteMean_equiv e (fun y : Tuples d l =>
-    ((Fintype.card (Tuples d l):ℝ)*PairRouting.tupleProbability (palindromePerm d) (e.symm x) y)^
-      (1+(1/1000:ℝ)))
-  change finiteMean (fun y =>
-    ((Fintype.card (Tuples d l):ℝ)*PairRouting.tupleProbability (palindromePerm d) (e.symm x) (e.symm y))^
-      (1+(1/1000:ℝ))) ≤ _
-  rw [←he]
-  simpa only [palindromeRowMoment,Fintype.card_fin] using palindrome_row_moment_coarse d (e.symm x)
-
-lemma tuples_card (d l : ℕ) : Fintype.card (Tuples d l) = (2^d).descFactorial l := by
-  simp only [Tuples,Fintype.card_embedding_eq,Fintype.card_fin,card_positions]
-
-lemma canonical_pointwise (d l : ℕ) (order : Equiv.Perm (Fin d)) (x y : Tuples d l) :
-    ((reflectedKernel d l order false)^2000) x y ≤
-      Real.exp ((2000*(1+heightDecay 1))*l)/((2^d).descFactorial l:ℝ) := by
-  let : Nonempty (Tuples d l) := ⟨x⟩
-  have hh := symmetric_kernel_smoothing (reflectedKernel d l order false)
-    (reflected_nonneg d l order) (reflected_row d l order) (reflected_col d l order)
-    (reflected_symmetric d l order) (p := 1+(1/1000:ℝ)) (A := l*(1+heightDecay 1))
-    (by norm_num) (mul_nonneg (Nat.cast_nonneg _) (by linarith [heightDecay_nonneg 1]))
-    (reflected_moment d l order) 1000 density_exponent_numeric x y
-  have he : 2*((l:ℝ)*(1+heightDecay 1))*(1000:ℝ) = (2000*(1+heightDecay 1))*l := by ring
-  norm_num only [show 2*1000=2000 by norm_num,Nat.cast_ofNat] at hh
-  simpa only [he,tuples_card] using hh
-
-lemma both_pointwise (d l : ℕ) (order : Equiv.Perm (Fin d)) (orientation : Bool)
-    (x y : Tuples d l) :
-    ((reflectedKernel d l order orientation)^2001) x y ≤
-      Real.exp ((2000*(1+heightDecay 1))*l)/((2^d).descFactorial l:ℝ) := by
-  cases orientation
-  · change ((reflectedKernel d l order false)^(2000+1)) x y ≤ _
-    rw [pow_succ]
-    exact bound_mul_right _ _ (reflected_nonneg d l order) (reflected_col d l order)
-      (canonical_pointwise d l order) x y
-  · let A := sweepKernel d l order
-    change ((A*A.transpose)^(2000+1)) x y ≤ _
-    rw [←sandwich_power]
-    apply bound_mul_right _ _ (fun x y => sweepKernel_nonneg d l order y x)
-      (sweepKernel_row d l order)
-    exact bound_mul_left _ _ (sweepKernel_nonneg d l order) (sweepKernel_row d l order)
-      (canonical_pointwise d l order)
-
-lemma regular_trace_bound (d : ℕ) (order : Equiv.Perm (Fin d)) :
-    regularAbsoluteEvenTrace d order 2001 ≤ Real.exp ((2000*(1+heightDecay 1))*(2:ℝ)^d) := by
-  have hn : 0 < Fintype.card (Tuples d (2^d)) := by
-    rw [tuples_card,Nat.descFactorial_self]
-    exact Nat.factorial_pos _
-  have hn' : (Fintype.card (Tuples d (2^d)):ℝ) ≠ 0 := Nat.cast_ne_zero.mpr (ne_of_gt hn)
-  calc
-    _ ≤ ∑ x : Tuples d (2^d), Real.exp ((2000*(1+heightDecay 1))*(2^d:ℕ)) /
-        ((2^d).descFactorial (2^d):ℝ) := by
-      apply Finset.sum_le_sum
-      intro x _
-      exact both_pointwise d (2^d) order false x x
-    _ = _ := by
-      simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,←tuples_card]
-      push_cast
-      field_simp
-
-theorem strong_smoothing : MainStatement := by
-  refine ⟨2001,by norm_num,2000*(1+heightDecay 1),?_,regular_trace_bound⟩
-  intro d l _ order orientation x y
-  exact both_pointwise d l order orientation x y
-
-end Thorp.StrongSmoothing
-
-end ThorpNine.Smoothing
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Smoothing.ParallelFreshPre
+
+namespace OAI
+
+namespace ThorpNine.Smoothing
+
+namespace Thorp
+
+theorem palindrome_family_mgf (L n d : ℕ) (hL : 0 < L) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) ≤
+      Real.exp (Fintype.card ι * (68*∑ j ∈ Finset.range n,
+        Real.exp (-((L*4^j:ℕ):ℝ)/64))) := by
+  induction n with
+  | zero => simp only [palindromeFamilyCost,Finset.range_zero,Finset.sum_empty,pow_zero,
+      finiteMean_const,mul_zero,Real.exp_zero,le_refl]
+  | succ n ih =>
+    let l := L*4^n
+    have hl : 0 < l := Nat.mul_pos hL (pow_pos (by decide) _)
+    have hs : 2*familyEnd L n ≤ l := familyEnd_two L n
+    have hc : ∀ Y, palindromeFamilyCost L (n+1) d e (Y,X) =
+        palindromeFamilyCost L n d e (Y,X)+palindromeSlabCost l d e (Y,X) := by
+      intro Y
+      exact Finset.sum_range_succ _ _
+    have hm : finiteMean (fun Y => q^(palindromeFamilyCost L (n+1) d e (Y,X))) ≤
+        finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) *
+          Real.exp (Fintype.card ι*(68*Real.exp (-(l:ℝ)/64))) := by
+      by_cases hld : l ≤ d
+      · simp_rw [hc,pow_add]
+        exact palindrome_slab_adapted_mul_le d (familyEnd L n) l (by omega) hl hs e X _
+          (fun _ => pow_nonneg (by linarith) _)
+          (layerAdapted_comp (palindromeFamilyCost_adapted L n d e X) (fun c => q^c))
+          q hq hlog
+      · simp_rw [hc,palindromeSlabCost_gt l d (by omega) e,Nat.add_zero]
+        exact le_mul_of_one_le_right
+          (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
+          (Real.one_le_exp_iff.mpr (by positivity))
+    refine (hm.trans (mul_le_mul_of_nonneg_right ih (Real.exp_nonneg _))).trans_eq ?_
+    rw [←Real.exp_add,Finset.sum_range_succ]
+    congr 1
+    dsimp [l]
+    ring
+
+lemma pow_four_ge (j : ℕ) : j+1 ≤ 4^j := by
+  induction j with
+  | zero => norm_num
+  | succ j ih => rw [pow_succ]; omega
+
+lemma sum_family_decay (L n : ℕ) (hL : 0 < L) :
+    (∑ j ∈ Finset.range n, Real.exp (-((L*4^j:ℕ):ℝ)/64)) ≤
+      Real.exp (-(L:ℝ)/64)/(1-Real.exp (-(1:ℝ)/64)) := by
+  let r := Real.exp (-(1:ℝ)/64)
+  have hr : 0 < r := Real.exp_pos _
+  have hr1 : r < 1 := Real.exp_lt_one_iff.mpr (by norm_num)
+  have hg : (∑ j ∈ Finset.range n, r^j) ≤ 1/(1-r) := by
+    apply (le_div_iff₀ (by linarith : (0:ℝ)<1-r)).mpr
+    have hh := geom_sum_mul r n
+    nlinarith [pow_nonneg (le_of_lt hr) n]
+  calc
+    _ ≤ ∑ j ∈ Finset.range n, Real.exp (-(L:ℝ)/64)*r^j := by
+      apply Finset.sum_le_sum
+      intro j _
+      have hj := pow_four_ge j
+      have hlj : L+j ≤ L*4^j := by nlinarith
+      have hlj' : (L:ℝ)+j ≤ (L*4^j:ℕ) := by exact_mod_cast hlj
+      rw [←Real.exp_nat_mul,←Real.exp_add]
+      exact Real.exp_le_exp.mpr (by linarith)
+    _ = Real.exp (-(L:ℝ)/64)*(∑ j ∈ Finset.range n, r^j) := by rw [Finset.mul_sum]
+    _ ≤ Real.exp (-(L:ℝ)/64)*(1/(1-r)) :=
+      mul_le_mul_of_nonneg_left hg (Real.exp_nonneg _)
+    _ = _ := by ring
+
+
+lemma palindromeLevelCost_step_add (t d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (d+1)) (ω : BenesCoins (d+1)) :
+    palindromeLevelCost t (d+1) e ω =
+      let σ := benesStepEquiv d ω
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x σ.2.1
+      let hc := PairRouting.colors_compatible x σ.2.1
+      (if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
+          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) else 0) +
+        palindromeLevelCost t d (PairRouting.childEmbedding x c hc false) σ.1.1 +
+        palindromeLevelCost t d (PairRouting.childEmbedding x c hc true) σ.1.2 := by
+  classical
+  rw [palindromeLevelCost]
+  dsimp only
+  split_ifs with h
+  · simp only [palindromeLevelCost_gt t d (by omega),Nat.add_zero]
+  · simp only [Nat.zero_add]
+
+lemma palindromeLowCost_sum (H d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) :
+    palindromeLowCost H d e ω = ∑ i ∈ Finset.range H, palindromeLevelCost (i+1) d e ω := by
+  classical
+  induction d generalizing ι with
+  | zero => simp only [palindromeLowCost,palindromeLevelCost,Finset.sum_const_zero]
+  | succ d ih =>
+    rw [palindromeLowCost]
+    simp only [palindromeLevelCost_step_add,Finset.sum_add_distrib]
+    rw [←ih,←ih]
+    congr 2
+    simp only [Nat.add_right_cancel_iff,Finset.sum_ite_eq',Finset.mem_range,Nat.succ_le_iff]
+
+lemma palindromeLowCost_slab (L d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) :
+    palindromeLowCost L d e ω + palindromeSlabCost L d e ω = palindromeLowCost (2*L) d e ω := by
+  rw [palindromeLowCost_sum,palindromeLowCost_sum,show 2*L=L+L by omega,Finset.sum_range_add]
+  congr 1
+  exact Fin.sum_univ_eq_sum_range (fun i => palindromeLevelCost (L+i+1) d e ω) L
+
+lemma palindromeCost_family (L n d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) :
+    palindromeLowCost L d e ω + palindromeFamilyCost L n d e ω +
+      palindromeFamilyCost (2*L) n d e ω = palindromeLowCost (L*4^n) d e ω := by
+  induction n with
+  | zero => simp only [palindromeFamilyCost,Finset.range_zero,Finset.sum_empty,Nat.add_zero,
+      pow_zero,mul_one]
+  | succ n ih =>
+    have he : (2*L)*4^n = 2*(L*4^n) := by ring
+    have he' : L*4^(n+1) = 2*(2*(L*4^n)) := by rw [pow_succ]; ring
+    simp only [palindromeFamilyCost,Finset.sum_range_succ] at ih ⊢
+    rw [he,he',←palindromeLowCost_slab,←palindromeLowCost_slab,←ih]
+    omega
+
+
+lemma finiteMean_mul_sq_le {Ω : Type*} [Fintype Ω] (f g : Ω → ℝ) :
+    (finiteMean (fun ω => f ω*g ω))^2 ≤ finiteMean (fun ω => (f ω)^2)*finiteMean (fun ω => (g ω)^2) := by
+  have h := Finset.sum_mul_sq_le_sq_mul_sq Finset.univ f g
+  simp only [finiteMean,div_pow,div_mul_div_comm,←pow_two]
+  exact div_le_div_of_nonneg_right h (sq_nonneg _)
+
+lemma finiteMean_mul_exp_le {Ω : Type*} [Fintype Ω] (f g : Ω → ℝ) (A B : ℝ)
+    (hf : finiteMean (fun ω => (f ω)^2) ≤ Real.exp A)
+    (hg : finiteMean (fun ω => (g ω)^2) ≤ Real.exp B) :
+    finiteMean (fun ω => f ω*g ω) ≤ Real.exp ((A+B)/2) := by
+  have hc := finiteMean_mul_sq_le f g
+  have hn : 0 ≤ finiteMean (fun ω => (g ω)^2) := finiteMean_nonneg (fun _ => sq_nonneg _)
+  have hu := mul_le_mul hf hg hn (Real.exp_nonneg _)
+  have he : Real.exp A*Real.exp B = (Real.exp ((A+B)/2))^2 := by
+    rw [←Real.exp_add,←Real.exp_nat_mul]
+    congr 1
+    push_cast
+    ring
+  rw [he] at hu
+  have hh := hc.trans hu
+  have hep := Real.exp_pos ((A+B)/2)
+  nlinarith
+
+
+noncomputable def heightDecay (H : ℕ) : ℝ :=
+  68*Real.exp (-(H:ℝ)/64)/(1-Real.exp (-(1:ℝ)/64))
+
+lemma heightDecay_nonneg (H : ℕ) : 0 ≤ heightDecay H := by
+  have h : Real.exp (-(1:ℝ)/64) < 1 := Real.exp_lt_one_iff.mpr (by norm_num)
+  apply div_nonneg (by positivity) (by linarith)
+
+lemma heightDecay_antitone : Antitone heightDecay := by
+  intro L H h
+  have hden : 0 ≤ 1-Real.exp (-(1:ℝ)/64) := by
+    have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
+    linarith
+  apply div_le_div_of_nonneg_right _ hden
+  apply mul_le_mul_of_nonneg_left _ (by norm_num)
+  apply Real.exp_le_exp.mpr
+  have h' : (L:ℝ) ≤ H := by exact_mod_cast h
+  linarith
+
+lemma palindrome_family_mgf_decay (L n d : ℕ) (hL : 0 < L) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) ≤
+      Real.exp (Fintype.card ι*heightDecay L) := by
+  apply (palindrome_family_mgf L n d hL e X q hq hlog).trans
+  apply Real.exp_le_exp.mpr
+  apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
+  have h := mul_le_mul_of_nonneg_left (sum_family_decay L n hL) (by norm_num : (0:ℝ)≤68)
+  simpa only [heightDecay,mul_div_assoc] using h
+
+theorem palindrome_high_mgf (H n d : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/500) :
+    finiteMean (fun Y => q^(palindromeFamilyCost H n d e (Y,X)+
+      palindromeFamilyCost (2*H) n d e (Y,X))) ≤ Real.exp (Fintype.card ι*heightDecay H) := by
+  have hq2 := one_le_pow₀ (n:=2) hq
+  have hlog2 : Real.log (q^2) ≤ 1/250 := by rw [Real.log_pow]; norm_num; linarith
+  have h₀ := palindrome_family_mgf_decay H n d hH e X (q^2) hq2 hlog2
+  have h₁ := palindrome_family_mgf_decay (2*H) n d (by omega) e X (q^2) hq2 hlog2
+  have h₁' := h₁.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left
+    (heightDecay_antitone (show H ≤ 2*H by omega)) (Nat.cast_nonneg _)))
+  have hh := finiteMean_mul_exp_le (fun Y => q^(palindromeFamilyCost H n d e (Y,X)))
+    (fun Y => q^(palindromeFamilyCost (2*H) n d e (Y,X)))
+    (Fintype.card ι*heightDecay H) (Fintype.card ι*heightDecay H)
+    (by simpa only [pow_right_comm] using h₀) (by simpa only [pow_right_comm] using h₁')
+  simpa only [←pow_add,add_self_div_two] using hh
+
+lemma palindrome_high_full_mgf (H n d : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/500) :
+    finiteMean (fun ω : BenesCoins d => q^(palindromeFamilyCost H n d e ω+
+      palindromeFamilyCost (2*H) n d e ω)) ≤ Real.exp (Fintype.card ι*heightDecay H) := by
+  rw [finiteMean_prod,finiteMean_comm]
+  calc
+    _ ≤ finiteMean (fun _X : SwitchIndex d → Bool => Real.exp (Fintype.card ι*heightDecay H)) := by
+      apply finiteMean_mono
+      intro X
+      exact palindrome_high_mgf H n d hH e X q hq hlog
+    _ = _ := finiteMean_const _
+
+theorem palindrome_cost_mgf_cutoff (r H : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (H+r)) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/1000)
+    (hr : (2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*(q^2)^H) ≤ 1/2) :
+    finiteMean (fun ω : BenesCoins (H+r) => q^(palindromeCost (H+r) e ω)) ≤
+      Real.exp (((2:ℝ)^r*(2*((2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*(q^2)^H))^2)+
+        Fintype.card ι*heightDecay H)/2) := by
+  let d := H+r
+  have hd : d ≤ H*4^d := by
+    have hh := pow_four_ge d
+    nlinarith
+  have hcost (ω : BenesCoins d) : palindromeCost d e ω = palindromeLowCost H d e ω+
+      (palindromeFamilyCost H d d e ω+palindromeFamilyCost (2*H) d d e ω) := by
+    have hh := palindromeCost_family H d d e ω
+    rw [palindromeLowCost_eq _ _ hd] at hh
+    simpa only [Nat.add_assoc] using hh.symm
+  have hq2 := one_le_pow₀ (n:=2) hq
+  have hlog2 : Real.log (q^2) ≤ 1/500 := by rw [Real.log_pow]; norm_num; linarith
+  have hlow := palindrome_low_mgf r H e (q^2) hq2 hr
+  have hhigh := palindrome_high_full_mgf H d d hH e (q^2) hq2 hlog2
+  have hh := finiteMean_mul_exp_le (fun ω : BenesCoins d => q^(palindromeLowCost H d e ω))
+    (fun ω : BenesCoins d => q^(palindromeFamilyCost H d d e ω+palindromeFamilyCost (2*H) d d e ω))
+    _ _ (by simpa only [pow_right_comm] using hlow) (by simpa only [pow_right_comm] using hhigh)
+  change finiteMean (fun ω : BenesCoins d => q^(palindromeCost d e ω)) ≤ _
+  simpa only [hcost,pow_add,pow_right_comm] using hh
+
+
+lemma cutoffMass_le (H d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (ω : BenesCoins d) :
+    cutoffMass H d e ω ≤ Fintype.card ι := by
+  classical
+  induction d generalizing ι with
+  | zero => exact Nat.zero_le _
+  | succ d ih =>
+    rw [cutoffMass]
+    split_ifs
+    · exact crowdedMass_le _
+    · exact (Nat.add_le_add (ih _ _) (ih _ _)).trans_eq (PairRouting.color_card_add _)
+
+lemma palindromeLowCost_le (H d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) : palindromeLowCost H d e ω ≤ H*Fintype.card ι :=
+  (palindromeLowCost_le_cutoff H d e ω).trans (Nat.mul_le_mul_left H (cutoffMass_le H d e ω))
+
+theorem palindrome_cost_mgf_coarse (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/1000) :
+    finiteMean (fun ω : BenesCoins d => q^(palindromeCost d e ω)) ≤
+      Real.exp (Fintype.card ι*(1+heightDecay 1)) := by
+  have hd : d ≤ 1*4^d := by have hh := pow_four_ge d; omega
+  have hcost (ω : BenesCoins d) : palindromeCost d e ω = palindromeLowCost 1 d e ω+
+      (palindromeFamilyCost 1 d d e ω+palindromeFamilyCost 2 d d e ω) := by
+    have hh := palindromeCost_family 1 d d e ω
+    rw [palindromeLowCost_eq _ _ hd] at hh
+    simpa only [Nat.add_assoc] using hh.symm
+  have hq2 := one_le_pow₀ (n:=2) hq
+  have hlog2 : Real.log (q^2) ≤ 1/500 := by rw [Real.log_pow]; norm_num; linarith
+  have hlow : finiteMean (fun ω : BenesCoins d => (q^2)^(palindromeLowCost 1 d e ω)) ≤
+      Real.exp (Fintype.card ι) := by
+    calc
+      _ ≤ finiteMean (fun _ω : BenesCoins d => (q^2)^(Fintype.card ι)) := by
+        apply finiteMean_mono
+        intro ω
+        apply pow_le_pow_right₀ hq2
+        simpa only [one_mul] using palindromeLowCost_le 1 d e ω
+      _ = (q^2)^(Fintype.card ι) := finiteMean_const _
+      _ = Real.exp (Fintype.card ι * Real.log (q^2)) := by
+        rw [Real.exp_nat_mul,Real.exp_log (by positivity : 0<q^2)]
+      _ ≤ _ := Real.exp_le_exp.mpr (by
+        have hh := mul_le_mul_of_nonneg_left hlog2 (Nat.cast_nonneg (Fintype.card ι) : (0:ℝ)≤_)
+        nlinarith [Nat.cast_nonneg (α:=ℝ) (Fintype.card ι)])
+  have hhigh := palindrome_high_full_mgf 1 d d (by decide) e (q^2) hq2 hlog2
+  have hh := finiteMean_mul_exp_le (fun ω : BenesCoins d => q^(palindromeLowCost 1 d e ω))
+    (fun ω : BenesCoins d => q^(palindromeFamilyCost 1 d d e ω+palindromeFamilyCost 2 d d e ω))
+    (Fintype.card ι) (Fintype.card ι*heightDecay 1)
+    (by simpa only [pow_right_comm] using hlow) (by simpa only [pow_right_comm] using hhigh)
+  simp only [←pow_add,←hcost] at hh
+  apply hh.trans (Real.exp_le_exp.mpr _)
+  have hk : (0:ℝ) ≤ Fintype.card ι := Nat.cast_nonneg _
+  have hD := heightDecay_nonneg 1
+  nlinarith
+
+noncomputable def momentBase : ℝ := (2:ℝ)^(1/1000:ℝ)
+
+lemma momentBase_one_le : 1 ≤ momentBase := by
+  exact Real.one_le_rpow (by norm_num) (by norm_num)
+
+lemma momentBase_log : Real.log momentBase ≤ 1/1000 := by
+  rw [momentBase,Real.log_rpow (by norm_num)]
+  have hh := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
+  norm_num at hh ⊢
+  linarith
+
+lemma momentBase_pow (c : ℕ) : momentBase^c = (2:ℝ)^((c:ℝ)*(1/1000:ℝ)) := by
+  rw [mul_comm,Real.rpow_mul_natCast (by norm_num)]
+  rfl
+
+lemma palindromeRowMoment_le_cost (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) :
+    palindromeRowMoment d e (1/1000) ≤
+      finiteMean (fun ω : BenesCoins d => momentBase^(palindromeCost d e ω)) := by
+  have hn : (0:ℝ) < ((2^d:ℕ):ℝ)^(Fintype.card ι) := by positivity
+  have hn0 : (0:ℝ) ≤ ((2^d).descFactorial (Fintype.card ι):ℝ) /
+      ((2^d:ℕ):ℝ)^(Fintype.card ι) := by positivity
+  have hn1 : ((2^d).descFactorial (Fintype.card ι):ℝ) /
+      ((2^d:ℕ):ℝ)^(Fintype.card ι) ≤ 1 := by
+    apply (div_le_one hn).mpr
+    exact_mod_cast Nat.descFactorial_le_pow (2^d) (Fintype.card ι)
+  have hpow := Real.rpow_le_one hn0 hn1 (by norm_num : (0:ℝ)≤1/1000)
+  have hm := palindrome_row_density_moment d e (1/1000) (by norm_num)
+  simp_rw [←momentBase_pow] at hm
+  exact hm.trans (mul_le_of_le_one_left (finiteMean_nonneg (fun _ =>
+    pow_nonneg (le_trans (by norm_num) momentBase_one_le) _)) hpow)
+
+theorem palindrome_row_moment_coarse (d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) :
+    palindromeRowMoment d e (1/1000) ≤ Real.exp (Fintype.card ι*(1+heightDecay 1)) :=
+  (palindromeRowMoment_le_cost d e).trans
+    (palindrome_cost_mgf_coarse d e momentBase momentBase_one_le momentBase_log)
+
+
+noncomputable def densityCutoff (p : ℝ) : ℕ := ⌊-Real.log p/(4*Real.log 2)⌋₊
+
+lemma log_small_density {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
+    Real.log p ≤ -8*Real.log 2 := by
+  have he : Real.log (1/256:ℝ) = -8*Real.log 2 := by
+    rw [show (1/256:ℝ) = 1/2^8 by norm_num,
+      Real.log_div (by norm_num) (by positivity),Real.log_one,Real.log_pow]
+    ring
+  exact (Real.log_le_log hp hs).trans_eq he
+
+lemma densityCutoff_bounds {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
+    0 < densityCutoff p ∧
+    (densityCutoff p:ℝ)*(4*Real.log 2) ≤ -Real.log p ∧
+    -Real.log p < ((densityCutoff p:ℝ)+1)*(4*Real.log 2) := by
+  have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
+  have hd : 0 < 4*Real.log 2 := by positivity
+  have hh := log_small_density hp hs
+  have hx : 1 ≤ -Real.log p/(4*Real.log 2) := by
+    apply (le_div_iff₀ hd).mpr
+    nlinarith
+  refine ⟨Nat.floor_pos.mpr hx,?_,?_⟩
+  · exact (le_div_iff₀ hd).mp (Nat.floor_le (by linarith : 0 ≤ -Real.log p/(4*Real.log 2)))
+  · exact (div_lt_iff₀ hd).mp (Nat.lt_floor_add_one (-Real.log p/(4*Real.log 2)))
+
+lemma densityCutoff_decay {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
+    Real.exp (-(densityCutoff p:ℝ)/64) ≤ Real.exp (1/64) * p^(1/512:ℝ) := by
+  have hh := (densityCutoff_bounds hp hs).2.2
+  have hb : Real.log 2 ≤ 2 := by
+    have := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
+    linarith
+  have hH : 0 ≤ (densityCutoff p:ℝ)+1 := by positivity
+  have hu := mul_le_mul_of_nonneg_left hb hH
+  rw [Real.rpow_def_of_pos hp,←Real.exp_add]
+  apply Real.exp_le_exp.mpr
+  nlinarith
+
+lemma densityCutoff_power {p q : ℝ} (hp : 0 < p) (hs : p ≤ 1/256)
+    (hq : 0 < q) (hlog : Real.log q ≤ Real.log 2/4) (n : ℕ) (hn : n ≤ 4)
+    (hq1 : 1 ≤ q) :
+    p*(2:ℝ)^(densityCutoff p)*q^(n*densityCutoff p) ≤ p^(1/2:ℝ) := by
+  have hcut := (densityCutoff_bounds hp hs).2.1
+  have hH : 0 ≤ (densityCutoff p:ℝ) := Nat.cast_nonneg _
+  have hn' : (n:ℝ) ≤ 4 := by exact_mod_cast hn
+  have hln := Real.log_nonneg hq1
+  have hl := mul_le_mul_of_nonneg_right hn' hln
+  have hnl : (n:ℝ)*Real.log q ≤ Real.log 2 := by linarith
+  have hm := mul_le_mul_of_nonneg_left hnl hH
+  have heq : p*(2:ℝ)^(densityCutoff p)*q^(n*densityCutoff p) =
+      Real.exp (Real.log p+(densityCutoff p:ℝ)*Real.log 2+
+        (n*densityCutoff p:ℕ)*Real.log q) := by
+    rw [Real.exp_add,Real.exp_add,Real.exp_nat_mul,Real.exp_nat_mul,
+      Real.exp_log hp,Real.exp_log hq,Real.exp_log (by norm_num : (0:ℝ)<2)]
+  rw [heq,Real.rpow_def_of_pos hp]
+  apply Real.exp_le_exp.mpr
+  push_cast
+  nlinarith
+
+lemma densityCutoff_sparse {p q : ℝ} (hp : 0 < p) (hs : p ≤ 1/256)
+    (hq : 0 < q) (hlog : Real.log q ≤ Real.log 2/4) (hq1 : 1 ≤ q) :
+    (2:ℝ)^(densityCutoff p)*(p*(q^2)^(densityCutoff p)) ≤ 1/2 := by
+  have hh := densityCutoff_power hp hs hq hlog 2 (by decide) hq1
+  rw [←pow_mul] at ⊢
+  have hle : p^(1/2:ℝ) ≤ 1/2 := by
+    have hlogp := log_small_density hp hs
+    have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
+    rw [Real.rpow_def_of_pos hp]
+    calc
+      _ ≤ Real.exp (-Real.log 2) := by apply Real.exp_le_exp.mpr; nlinarith
+      _ = 1/2 := by rw [Real.exp_neg,Real.exp_log (by norm_num : (0:ℝ)<2)]; norm_num
+  nlinarith [hh.trans hle]
+
+
+lemma momentBase_log_exact : Real.log momentBase = Real.log 2/1000 := by
+  rw [momentBase,Real.log_rpow (by norm_num)]
+  ring
+
+lemma densityCutoff_le_height (d k : ℕ) (hk : 0 < k) :
+    densityCutoff ((k:ℝ)/(2:ℝ)^d) ≤ d := by
+  have hk' : (1:ℝ) ≤ k := by exact_mod_cast hk
+  have hp : 0 < (k:ℝ)/(2:ℝ)^d := by positivity
+  have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
+  have hlogk := Real.log_nonneg hk'
+  by_cases hx : 0 ≤ -Real.log ((k:ℝ)/(2:ℝ)^d)/(4*Real.log 2)
+  · have hh := (le_div_iff₀ (by positivity : 0 < 4*Real.log 2)).mp (Nat.floor_le hx)
+    have hd : (0:ℝ) ≤ d := Nat.cast_nonneg _
+    have hH : (0:ℝ) ≤ densityCutoff ((k:ℝ)/(2:ℝ)^d) := Nat.cast_nonneg _
+    change (densityCutoff ((k:ℝ)/(2:ℝ)^d):ℝ)*(4*Real.log 2) ≤ -Real.log ((k:ℝ)/(2:ℝ)^d) at hh
+    rw [Real.log_div (by positivity) (by positivity),Real.log_pow] at hh
+    have hle : (densityCutoff ((k:ℝ)/(2:ℝ)^d):ℝ) ≤ d := by nlinarith
+    exact_mod_cast hle
+  · have hz := Nat.floor_of_nonpos (le_of_not_ge hx)
+    simp only [densityCutoff,hz,Nat.zero_le]
+
+noncomputable def densityTailConstant : ℝ :=
+  68*Real.exp (1/64)/(1-Real.exp (-(1:ℝ)/64))
+
+lemma densityTailConstant_nonneg : 0 ≤ densityTailConstant := by
+  have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
+  apply div_nonneg (by positivity) (by linarith)
+
+lemma heightDecay_cutoff {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
+    heightDecay (densityCutoff p) ≤ densityTailConstant*p^(1/512:ℝ) := by
+  have hden : 0 ≤ 1-Real.exp (-(1:ℝ)/64) := by
+    have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
+    linarith
+  have hh := div_le_div_of_nonneg_right
+    (mul_le_mul_of_nonneg_left (densityCutoff_decay hp hs) (by norm_num : (0:ℝ)≤68)) hden
+  simpa only [heightDecay,densityTailConstant,mul_assoc,div_mul_eq_mul_div] using hh
+
+lemma sparse_low_algebra (r H k : ℕ) (q : ℝ) :
+    (2:ℝ)^r*(2*((2:ℝ)^H*(((k:ℝ)/(2:ℝ)^(H+r))*(q^2)^H))^2) =
+    2*k*(((k:ℝ)/(2:ℝ)^(H+r))*(2:ℝ)^H*q^(4*H)) := by
+  rw [pow_add,←pow_mul,show 4*H = (2*H)*2 by omega,pow_mul]
+  have hH : (2:ℝ)^H ≠ 0 := by positivity
+  have hr : (2:ℝ)^r ≠ 0 := by positivity
+  field_simp
+  ring
+
+lemma palindrome_row_moment_small (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι)
+    (hs : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1/256) :
+    palindromeRowMoment d e (1/1000) ≤
+      Real.exp (Fintype.card ι*(1+densityTailConstant)*
+        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ)) := by
+  generalize hpdef : ((Fintype.card ι:ℝ)/(2:ℝ)^d) = p at hs ⊢
+  have hp : 0 < p := by rw [←hpdef]; positivity
+  have hle := densityCutoff_le_height d (Fintype.card ι) hk
+  rw [hpdef] at hle
+  obtain ⟨r,hr⟩ := Nat.exists_eq_add_of_le hle
+  subst d
+  have hH := (densityCutoff_bounds hp hs).1
+  have hq := momentBase_one_le
+  have hqp : 0 < momentBase := by linarith
+  have hlog : Real.log momentBase ≤ Real.log 2/4 := by
+    rw [momentBase_log_exact]
+    have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
+    linarith
+  have hsp := densityCutoff_sparse hp hs hqp hlog hq
+  rw [←hpdef] at hsp
+  have hh := palindrome_cost_mgf_cutoff r (densityCutoff p) hH e momentBase hq momentBase_log
+    (by simpa only [hpdef] using hsp)
+  apply (palindromeRowMoment_le_cost _ e).trans (hh.trans (Real.exp_le_exp.mpr _))
+  rw [sparse_low_algebra,hpdef]
+  have hlow := densityCutoff_power hp hs hqp hlog 4 (by decide) hq
+  have hp1 : p ≤ 1 := by linarith
+  have he := Real.rpow_le_rpow_of_exponent_ge hp hp1 (by norm_num : (1:ℝ)/512 ≤ 1/2)
+  have hlow' := mul_le_mul_of_nonneg_left (hlow.trans he)
+    (by positivity : (0:ℝ)≤2*Fintype.card ι)
+  have hhigh := mul_le_mul_of_nonneg_left (heightDecay_cutoff hp hs)
+    (Nat.cast_nonneg (Fintype.card ι): (0:ℝ)≤_)
+  have hnon : 0 ≤ Fintype.card ι*densityTailConstant*p^(1/512:ℝ) :=
+    mul_nonneg (mul_nonneg (Nat.cast_nonneg _) densityTailConstant_nonneg)
+      (Real.rpow_nonneg (le_of_lt hp) _)
+  nlinarith
+
+noncomputable def densityConstant : ℝ :=
+  256*(1+heightDecay 1)+1+densityTailConstant
+
+lemma densityConstant_pos : 0 < densityConstant := by
+  dsimp [densityConstant]
+  have hh := heightDecay_nonneg 1
+  have ht := densityTailConstant_nonneg
+  linarith
+
+theorem palindrome_row_moment_bound (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι) :
+    palindromeRowMoment d e (1/1000) ≤ Real.exp (densityConstant*Fintype.card ι*
+        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ)) := by
+  have hp : 0 < (Fintype.card ι:ℝ)/(2:ℝ)^d := by positivity
+  have hp1 : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1 := by
+    apply (div_le_one (by positivity)).mpr
+    have h := Fintype.card_le_of_injective e e.injective
+    rw [card_positions] at h
+    exact_mod_cast h
+  have hk0 : (0:ℝ) ≤ Fintype.card ι := Nat.cast_nonneg _
+  have hpow := Real.rpow_nonneg (le_of_lt hp) (1/512:ℝ)
+  by_cases hs : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1/256
+  · apply (palindrome_row_moment_small d e hk hs).trans (Real.exp_le_exp.mpr _)
+    have hD : 1+densityTailConstant ≤ densityConstant := by
+      dsimp [densityConstant]
+      have := heightDecay_nonneg 1
+      linarith
+    have hh := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hD hk0) hpow
+    nlinarith
+  · have hlow : (1:ℝ)/256 ≤ ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ) := by
+      have he := Real.rpow_le_rpow_of_exponent_ge hp hp1 (by norm_num : (1:ℝ)/512 ≤ 1)
+      rw [Real.rpow_one] at he
+      exact (le_of_not_ge hs).trans he
+    have hD : 256*(1+heightDecay 1) ≤ densityConstant := by
+      dsimp [densityConstant]
+      have := densityTailConstant_nonneg
+      linarith
+    apply (palindrome_row_moment_coarse d e).trans (Real.exp_le_exp.mpr _)
+    have hD0 : 0 ≤ 1+heightDecay 1 := by have := heightDecay_nonneg 1; linarith
+    have ha := mul_le_mul_of_nonneg_left hlow (mul_nonneg (by norm_num : (0:ℝ)≤256) hD0)
+    have hb := mul_le_mul_of_nonneg_right hD hpow
+    have hc := mul_le_mul_of_nonneg_right (ha.trans hb) hk0
+    nlinarith
+
+lemma palindromeRowMoment_nonneg (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (a : ℝ) : 0 ≤ palindromeRowMoment d e a := by
+  apply finiteMean_nonneg
+  intro f
+  exact Real.rpow_nonneg (mul_nonneg (Nat.cast_nonneg _)
+    (PairRouting.tupleProbability_nonneg _ _ _)) _
+
+theorem palindrome_log_row_moment (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι) :
+    Real.log (palindromeRowMoment d e (1/1000)) ≤ densityConstant*Fintype.card ι*
+        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ) := by
+  by_cases hz : palindromeRowMoment d e (1/1000) = 0
+  · rw [hz,Real.log_zero]
+    exact mul_nonneg (mul_nonneg (le_of_lt densityConstant_pos) (Nat.cast_nonneg _))
+      (Real.rpow_nonneg (by positivity) _)
+  · exact (Real.log_le_iff_le_exp (lt_of_le_of_ne (palindromeRowMoment_nonneg _ _ _) (Ne.symm hz))).mpr
+      (palindrome_row_moment_bound d e hk)
+
+end Thorp
+
+
+namespace Thorp.StrongSmoothing
+open scoped BigOperators
+
+variable {X : Type*} [Fintype X] [Nonempty X]
+
+omit [Nonempty X] in
+lemma mean_nonneg (f : X → ℝ) (hf : ∀ x, 0 ≤ f x) : 0 ≤ finiteMean f :=
+  finiteMean_nonneg hf
+
+lemma mean_rpow_concave (f : X → ℝ) (hf : ∀ x, 0 ≤ f x)
+    {a : ℝ} (ha : 0 ≤ a) (ha1 : a ≤ 1) :
+    finiteMean (fun x => (f x)^a) ≤ (finiteMean f)^a := by
+  have hN : (0:ℝ) < Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
+  have hh := (Real.concaveOn_rpow ha ha1).le_map_sum
+    (t := Finset.univ) (w := fun _ : X => 1/(Fintype.card X:ℝ))
+    (p := f) (fun _ _ => by positivity)
+    (by simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]; field_simp)
+    (fun x _ => hf x)
+  simpa only [smul_eq_mul,one_div,←Finset.mul_sum,mul_comm _ (∑ _ : X,_),
+    ←div_eq_mul_inv,finiteMean] using hh
+
+noncomputable def act (K : Matrix X X ℝ) (f : X → ℝ) (y : X) : ℝ := ∑ x, K x y*f x
+
+omit [Nonempty X] in
+lemma act_nonneg (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
+    (f : X → ℝ) (hf : ∀ x, 0 ≤ f x) (y : X) : 0 ≤ act K f y :=
+  Finset.sum_nonneg (fun x _ => mul_nonneg (hK x y) (hf x))
+
+omit [Nonempty X] in
+lemma act_rpow_le (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
+    (hcol : ∀ y, ∑ x, K x y = 1) (f : X → ℝ) (hf : ∀ x, 0 ≤ f x)
+    {b : ℝ} (hb : 1 ≤ b) (y : X) :
+    (act K f y)^b ≤ act K (fun x => (f x)^b) y := by
+  simpa only [smul_eq_mul,act] using
+    (convexOn_rpow hb).map_sum_le (t := Finset.univ) (w := fun x => K x y)
+      (p := f) (fun x _ => hK x y) (hcol y) (fun x _ => hf x)
+
+lemma act_one_to_p (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
+    {p A : ℝ} (hp : 1 ≤ p)
+    (hm : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) ≤ Real.exp A)
+    (f : X → ℝ) (hf : ∀ x, 0 ≤ f x) :
+    finiteMean (fun y => (act K f y)^p) ≤ Real.exp A*(finiteMean f)^p := by
+  classical
+  let s : ℝ := ∑ x, f x
+  have hs : 0 ≤ s := Finset.sum_nonneg (fun x _ => hf x)
+  have hN : (0:ℝ) < Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
+  by_cases hz : s=0
+  · have hfz : ∀ x, f x=0 := by
+      intro x
+      exact (Finset.sum_eq_zero_iff_of_nonneg (fun x _ => hf x)).mp hz x (Finset.mem_univ _)
+    simp only [act,hfz,mul_zero,Finset.sum_const_zero,finiteMean,zero_div,
+      Real.zero_rpow (show p ≠ 0 by linarith),mul_zero,le_refl]
+  · have hsp : 0 < s := lt_of_le_of_ne hs (Ne.symm hz)
+    let w : X → ℝ := fun x => f x/s
+    have hw : ∀ x, 0 ≤ w x := fun x => div_nonneg (hf x) hs
+    have hws : ∑ x, w x = 1 := by simp only [w,←Finset.sum_div]; exact div_self hz
+    have hact (y : X) : act K f y = finiteMean f * ∑ x, w x*((Fintype.card X:ℝ)*K x y) := by
+      simp only [act,w,finiteMean,Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro x _
+      have hz' : (∑ x, f x) ≠ 0 := hz
+      dsimp [s]
+      field_simp [hz',ne_of_gt hN]
+    have hj (y : X) : (act K f y)^p ≤ (finiteMean f)^p*
+        ∑ x, w x*((Fintype.card X:ℝ)*K x y)^p := by
+      rw [hact,Real.mul_rpow (mean_nonneg f hf)
+        (Finset.sum_nonneg (fun x _ => mul_nonneg (hw x) (mul_nonneg hN.le (hK x y))))]
+      apply mul_le_mul_of_nonneg_left _ (Real.rpow_nonneg (mean_nonneg f hf) _)
+      exact (convexOn_rpow hp).map_sum_le
+        (fun x _ => hw x) hws (fun x _ => mul_nonneg hN.le (hK x y))
+    have hh := finiteMean_mono hj
+    have he : finiteMean (fun y => (finiteMean f)^p*
+        ∑ x, w x*((Fintype.card X:ℝ)*K x y)^p) =
+        (finiteMean f)^p * ∑ x, w x*finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) := by
+      simp_rw [mul_comm ((finiteMean f)^p), finiteMean_mul_const,
+        finiteMean_sum, mul_comm (w _), finiteMean_mul_const]
+    rw [he] at hh
+    apply hh.trans
+    have hsum : (∑ x, w x*finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p)) ≤ Real.exp A := by
+      calc
+        _ ≤ ∑ x, w x*Real.exp A := Finset.sum_le_sum
+          (fun x _ => mul_le_mul_of_nonneg_left (hm x) (hw x))
+        _ = Real.exp A := by rw [←Finset.sum_mul,hws,one_mul]
+    simpa only [mul_comm (Real.exp A)] using
+      mul_le_mul_of_nonneg_left hsum (Real.rpow_nonneg (mean_nonneg f hf) _)
+
+lemma act_b_to_pb (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
+    (hcol : ∀ y, ∑ x, K x y = 1) {p A b : ℝ} (hp : 1 ≤ p) (hb : 1 ≤ b)
+    (hm : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) ≤ Real.exp A)
+    (f : X → ℝ) (hf : ∀ x, 0 ≤ f x) :
+    finiteMean (fun y => (act K f y)^(b*p)) ≤
+      Real.exp A*(finiteMean (fun x => (f x)^b))^p := by
+  calc
+    _ ≤ finiteMean (fun y => (act K (fun x => (f x)^b) y)^p) := by
+      apply finiteMean_mono
+      intro y
+      rw [Real.rpow_mul (act_nonneg K hK f hf y)]
+      exact Real.rpow_le_rpow (Real.rpow_nonneg (act_nonneg K hK f hf y) _) (act_rpow_le K hK hcol f hf hb y) (by linarith)
+    _ ≤ _ := act_one_to_p K hK hp hm _ (fun x => Real.rpow_nonneg (hf x) _)
+
+variable [DecidableEq X]
+
+omit [Nonempty X] in
+lemma kernel_pow_nonneg (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y) (r : ℕ) :
+    ∀ x y, 0 ≤ (K^r) x y := by
+  classical
+  induction r with
+  | zero => intro x y; simp only [pow_zero,Matrix.one_apply]; split_ifs <;> norm_num
+  | succ r ih =>
+    intro x y
+    rw [pow_succ,Matrix.mul_apply]
+    exact Finset.sum_nonneg (fun z _ => mul_nonneg (ih x z) (hK z y))
+
+omit [Nonempty X] in
+lemma kernel_pow_row (K : Matrix X X ℝ) (hrow : ∀ x, ∑ y, K x y = 1) (r : ℕ) :
+    ∀ x, ∑ y, (K^r) x y = 1 := by
+  classical
+  induction r with
+  | zero => intro x; simp [Matrix.one_apply]
+  | succ r ih =>
+    intro x
+    simp only [pow_succ,Matrix.mul_apply]
+    rw [Finset.sum_comm]
+    simp only [←Finset.mul_sum,hrow,mul_one,ih]
+
+noncomputable def rowDensity (K : Matrix X X ℝ) (r : ℕ) (x y : X) : ℝ :=
+  (Fintype.card X:ℝ)*(K^r) x y
+
+omit [Nonempty X] in
+lemma rowDensity_nonneg (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y) (r : ℕ) (x y : X) :
+    0 ≤ rowDensity K r x y := mul_nonneg (Nat.cast_nonneg _) (kernel_pow_nonneg K hK r x y)
+
+omit [Nonempty X] in
+lemma rowDensity_succ (K : Matrix X X ℝ) (r : ℕ) (x y : X) :
+    rowDensity K (r+1) x y = act K (rowDensity K r x) y := by
+  simp only [rowDensity,pow_succ,Matrix.mul_apply,act,Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro z _
+  ring
+
+lemma rowDensity_mean (K : Matrix X X ℝ) (hrow : ∀ x, ∑ y, K x y = 1) (r : ℕ) (x : X) :
+    finiteMean (rowDensity K r x) = 1 := by
+  have hN : (Fintype.card X:ℝ) ≠ 0 := ne_of_gt (Nat.cast_pos.mpr Fintype.card_pos)
+  simp only [finiteMean,rowDensity,←Finset.mul_sum,kernel_pow_row K hrow,mul_one,div_self hN]
+
+lemma rowDensity_moment_iter (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
+    (hrow : ∀ x, ∑ y, K x y = 1) (hcol : ∀ y, ∑ x, K x y = 1)
+    {p A : ℝ} (hp : 1 ≤ p) (hA : 0 ≤ A)
+    (hm : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) ≤ Real.exp A)
+    (r : ℕ) (x : X) :
+    finiteMean (fun y => (rowDensity K r x y)^(p^r)) ≤ Real.exp (A*r*p^r) := by
+  induction r with
+  | zero => simp [rowDensity_mean K hrow]
+  | succ r ih =>
+    have hpr : 1 ≤ p^r := one_le_pow₀ hp
+    have hps : 1 ≤ p^(r+1) := one_le_pow₀ hp
+    simp_rw [rowDensity_succ]
+    calc
+      _ ≤ Real.exp A*(finiteMean (fun y => (rowDensity K r x y)^(p^r)))^p := by
+        rw [pow_succ]
+        exact act_b_to_pb K hK hcol hp hpr hm _ (rowDensity_nonneg K hK r x)
+      _ ≤ Real.exp A*(Real.exp (A*r*p^r))^p := by
+        apply mul_le_mul_of_nonneg_left _ (Real.exp_pos _).le
+        exact Real.rpow_le_rpow (mean_nonneg _ (fun y => Real.rpow_nonneg
+          (rowDensity_nonneg K hK r x y) _)) ih (le_trans zero_le_one hp)
+      _ = Real.exp (A+A*r*p^(r+1)) := by rw [←Real.exp_mul,←Real.exp_add,pow_succ]; congr 1; ring
+      _ ≤ _ := by
+        apply Real.exp_le_exp.mpr
+        have hh := mul_le_mul_of_nonneg_left hps hA
+        push_cast
+        nlinarith
+
+lemma rowDensity_second_moment (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
+    (hrow : ∀ x, ∑ y, K x y = 1) (hcol : ∀ y, ∑ x, K x y = 1)
+    {p A : ℝ} (hp : 1 ≤ p) (hA : 0 ≤ A)
+    (hm : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) ≤ Real.exp A)
+    (r : ℕ) (hr : 2 ≤ p^r) (x : X) :
+    finiteMean (fun y => (rowDensity K r x y)^2) ≤ Real.exp (2*A*r) := by
+  have hpos : 0 < p^r := by linarith
+  have hj := mean_rpow_concave (fun y => (rowDensity K r x y)^(p^r))
+    (fun y => Real.rpow_nonneg (rowDensity_nonneg K hK r x y) _)
+    (a := 2/(p^r)) (by positivity) ((div_le_one hpos).mpr hr)
+  have he (y : X) : ((rowDensity K r x y)^(p^r))^(2/(p^r)) = (rowDensity K r x y)^2 := by
+    rw [←Real.rpow_mul (rowDensity_nonneg K hK r x y)]
+    rw [mul_div_cancel₀ _ (ne_of_gt hpos),Real.rpow_two]
+  simp_rw [he] at hj
+  apply hj.trans
+  calc
+    _ ≤ (Real.exp (A*r*p^r))^(2/(p^r)) :=
+      Real.rpow_le_rpow (mean_nonneg _ (fun y => Real.rpow_nonneg
+        (rowDensity_nonneg K hK r x y) _)) (rowDensity_moment_iter K hK hrow hcol hp hA hm r x)
+        (by positivity)
+    _ = _ := by rw [←Real.exp_mul]; congr 1; field_simp [ne_of_gt hpos]
+
+lemma symmetric_kernel_smoothing (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
+    (hrow : ∀ x, ∑ y, K x y = 1) (hcol : ∀ y, ∑ x, K x y = 1)
+    (hsym : K.IsSymm) {p A : ℝ} (hp : 1 ≤ p) (hA : 0 ≤ A)
+    (hm : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) ≤ Real.exp A)
+    (r : ℕ) (hr : 2 ≤ p^r) (x y : X) :
+    (K^(2*r)) x y ≤ Real.exp (2*A*r)/(Fintype.card X:ℝ) := by
+  have hN : (0:ℝ) < Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
+  have hs : ∀ z, (K^r) z y = (K^r) y z := fun z => (hsym.pow r).apply y z
+  have he : (Fintype.card X:ℝ)*(K^(2*r)) x y =
+      finiteMean (fun z => rowDensity K r x z*rowDensity K r y z) := by
+    simp only [finiteMean,rowDensity,two_mul,pow_add,Matrix.mul_apply,hs,Finset.mul_sum]
+    rw [Finset.sum_div]
+    apply Finset.sum_congr rfl
+    intro z _
+    field_simp
+  have hh : finiteMean (fun z => rowDensity K r x z*rowDensity K r y z) ≤
+      Real.exp (2*A*r) := by
+    calc
+      _ ≤ finiteMean (fun z => ((rowDensity K r x z)^2+(rowDensity K r y z)^2)/2) := by
+        apply finiteMean_mono
+        intro z
+        nlinarith [sq_nonneg (rowDensity K r x z-rowDensity K r y z)]
+      _ = (finiteMean (fun z => (rowDensity K r x z)^2)+
+          finiteMean (fun z => (rowDensity K r y z)^2))/2 := by
+        simp only [div_eq_mul_inv,finiteMean_mul_const,finiteMean_add]
+      _ ≤ _ := by
+        have hx := rowDensity_second_moment K hK hrow hcol hp hA hm r hr x
+        have hy := rowDensity_second_moment K hK hrow hcol hp hA hm r hr y
+        linarith
+  apply (le_div_iff₀ hN).mpr
+  simpa only [←he,mul_comm (Fintype.card X:ℝ)] using hh
+
+lemma density_exponent_numeric : 2 ≤ (1+(1/1000:ℝ))^(1000:ℕ) := by
+  calc
+    (2:ℝ) = 1+(1000:ℝ)*(1/1000) := by norm_num
+    _ ≤ _ := by simpa only [Nat.cast_ofNat] using
+      (one_add_mul_le_pow (a := (1/1000:ℝ)) (by norm_num) 1000)
+
+end Thorp.StrongSmoothing
+
+namespace Thorp.StrongSmoothing.PermLaw
+open scoped BigOperators
+variable {X Ω Ω' : Type*} [Fintype X] [DecidableEq X] [Fintype Ω] [Fintype Ω']
+
+noncomputable def kernel (P : Ω → Equiv.Perm X) : Matrix X X ℝ :=
+  fun x y => finiteMean (fun ω => if P ω x = y then (1:ℝ) else 0)
+
+omit [Fintype X] in
+lemma nonneg (P : Ω → Equiv.Perm X) (x y : X) : 0 ≤ kernel P x y := by
+  exact finiteMean_nonneg (fun ω => by split_ifs <;> norm_num)
+
+lemma row [Nonempty Ω] (P : Ω → Equiv.Perm X) (x : X) : ∑ y, kernel P x y = 1 := by
+  simp only [kernel,←finiteMean_sum]
+  simp only [Finset.sum_ite_eq,Finset.mem_univ,↓reduceIte,finiteMean_const]
+
+omit [Fintype X] in
+lemma inverse (P : Ω → Equiv.Perm X) : kernel (fun ω => (P ω)⁻¹) = (kernel P).transpose := by
+  ext x y
+  apply finiteMean_congr
+  intro ω
+  have he : (P ω)⁻¹ x=y ↔ P ω y=x := by
+    change (P ω).symm x=y ↔ P ω y=x
+    exact (Equiv.symm_apply_eq (P ω)).trans eq_comm
+  simp only [he]
+
+lemma col [Nonempty Ω] (P : Ω → Equiv.Perm X) (y : X) : ∑ x, kernel P x y = 1 := by
+  have hh := row (fun ω => (P ω)⁻¹) y
+  rw [inverse] at hh
+  exact hh
+
+lemma mul (P : Ω → Equiv.Perm X) (Q : Ω' → Equiv.Perm X) :
+    kernel (fun ω : Ω × Ω' => P ω.1*Q ω.2) = kernel Q*kernel P := by
+  ext x y
+  simp only [Matrix.mul_apply,kernel]
+  have he (ω : Ω × Ω') :
+      (if (P ω.1*Q ω.2) x=y then (1:ℝ) else 0) =
+        ∑ z : X, (if P ω.1 z=y then (1:ℝ) else 0)*(if Q ω.2 x=z then (1:ℝ) else 0) := by
+    simp only [mul_ite,mul_one,mul_zero,Finset.sum_ite_eq,Finset.mem_univ,↓reduceIte]
+    rfl
+  change finiteMean (fun ω : Ω × Ω' => if (P ω.1*Q ω.2) x=y then (1:ℝ) else 0) = _
+  simp_rw [he]
+  rw [finiteMean_sum]
+  apply Finset.sum_congr rfl
+  intro z _
+  exact (finiteMean_prod_mul (fun ω : Ω => if P ω z=y then (1:ℝ) else 0)
+    (fun ω : Ω' => if Q ω x=z then (1:ℝ) else 0)).trans (mul_comm _ _)
+
+omit [Fintype X] in
+lemma conjugate (P : Ω → Equiv.Perm X) (e : Equiv.Perm X) (x y : X) :
+    kernel (fun ω => e*P ω*e⁻¹) x y = kernel P (e.symm x) (e.symm y) := by
+  apply finiteMean_congr
+  intro ω
+  have he : (e*P ω*e⁻¹) x=y ↔ P ω (e.symm x)=e.symm y := by
+    change e (P ω (e.symm x))=y ↔ _
+    exact (Equiv.eq_symm_apply e).symm
+  simp only [he]
+
+end Thorp.StrongSmoothing.PermLaw
+
+namespace Thorp.StrongSmoothing
+open scoped BigOperators
+
+section MatrixHelpers
+variable {X : Type*} [Fintype X] [DecidableEq X]
+
+omit [DecidableEq X] in
+lemma mul_nonneg_entries (A B : Matrix X X ℝ) (hA : ∀ x y, 0 ≤ A x y)
+    (hB : ∀ x y, 0 ≤ B x y) (x y : X) : 0 ≤ (A*B) x y :=
+  Finset.sum_nonneg (fun z _ => mul_nonneg (hA x z) (hB z y))
+
+omit [DecidableEq X] in
+lemma mul_row (A B : Matrix X X ℝ) (hA : ∀ x, ∑ y, A x y = 1)
+    (hB : ∀ x, ∑ y, B x y = 1) (x : X) : ∑ y, (A*B) x y = 1 := by
+  simp only [Matrix.mul_apply]
+  rw [Finset.sum_comm]
+  simp only [←Finset.mul_sum,hB,mul_one,hA]
+
+omit [DecidableEq X] in
+lemma mul_col (A B : Matrix X X ℝ) (hA : ∀ y, ∑ x, A x y = 1)
+    (hB : ∀ y, ∑ x, B x y = 1) (y : X) : ∑ x, (A*B) x y = 1 := by
+  simp only [Matrix.mul_apply]
+  rw [Finset.sum_comm]
+  simp only [←Finset.sum_mul,hA,one_mul,hB]
+
+omit [DecidableEq X] in
+lemma bound_mul_left (A B : Matrix X X ℝ) (hA : ∀ x y, 0 ≤ A x y)
+    (hrow : ∀ x, ∑ y, A x y = 1) {D : ℝ} (hB : ∀ x y, B x y ≤ D) :
+    ∀ x y, (A*B) x y ≤ D := by
+  intro x y
+  calc
+    _ ≤ ∑ z, A x z*D := Finset.sum_le_sum (fun z _ => mul_le_mul_of_nonneg_left (hB z y) (hA x z))
+    _ = D := by rw [←Finset.sum_mul,hrow,one_mul]
+
+omit [DecidableEq X] in
+lemma bound_mul_right (A B : Matrix X X ℝ) (hB : ∀ x y, 0 ≤ B x y)
+    (hcol : ∀ y, ∑ x, B x y = 1) {D : ℝ} (hA : ∀ x y, A x y ≤ D) :
+    ∀ x y, (A*B) x y ≤ D := by
+  intro x y
+  calc
+    _ ≤ ∑ z, D*B z y := Finset.sum_le_sum (fun z _ => mul_le_mul_of_nonneg_right (hA x z) (hB z y))
+    _ = D := by rw [←Finset.mul_sum,hcol,mul_one]
+
+lemma sandwich_power (A B : Matrix X X ℝ) (r : ℕ) : A*(B*A)^r*B = (A*B)^(r+1) := by
+  induction r with
+  | zero => simp
+  | succ r ih =>
+    calc
+      _ = (A*(B*A)^r*B)*(A*B) := by rw [pow_succ]; noncomm_ring
+      _ = (A*B)^(r+1)*(A*B) := by rw [ih]
+      _ = _ := (pow_succ (A*B) (r+1)).symm
+
+end MatrixHelpers
+
+section TupleAction
+variable {ι β Ω : Type*} [Fintype ι] [Fintype β] [DecidableEq β] [Fintype Ω]
+
+def tuplePerm (P : Equiv.Perm β) : Equiv.Perm (ι ↪ β) where
+  toFun x := x.trans P.toEmbedding
+  invFun x := x.trans P.symm.toEmbedding
+  left_inv x := by ext i; simp
+  right_inv x := by ext i; simp
+
+omit [Fintype ι] [Fintype β] [DecidableEq β] in
+@[simp] lemma tuplePerm_apply (P : Equiv.Perm β) (x : ι ↪ β) (i : ι) :
+    tuplePerm P x i = P (x i) := rfl
+
+omit [Fintype ι] [Fintype β] [DecidableEq β] in
+lemma tuplePerm_mul (P Q : Equiv.Perm β) :
+    tuplePerm (ι := ι) (P*Q) = tuplePerm P*tuplePerm Q := by
+  ext x i
+  rfl
+
+omit [Fintype ι] [Fintype β] [DecidableEq β] in
+lemma tuplePerm_inverse (P : Equiv.Perm β) :
+    tuplePerm (ι := ι) P⁻¹ = (tuplePerm P)⁻¹ := by
+  ext x i
+  rfl
+
+omit [Fintype β] in
+lemma tupleProbability_kernel (P : Ω → Equiv.Perm β) (x y : ι ↪ β) :
+    PairRouting.tupleProbability P x y = PermLaw.kernel (fun ω => tuplePerm (P ω)) x y := by
+  classical
+  apply finiteMean_congr
+  intro ω
+  have he : (∀ i, P ω (x i) = y i) ↔ tuplePerm (P ω) x = y := by
+    constructor
+    · intro h; ext i; exact h i
+    · intro h i; exact congrArg (fun z : ι ↪ β => z i) h
+  simp only [he]
+end TupleAction
+
+lemma sweepKernel_law (d l : ℕ) (order : Equiv.Perm (Fin d)) :
+    sweepKernel d l order = PermLaw.kernel (fun ω => tuplePerm (ι := Fin l) (sweep d order ω)) := by
+  ext x y
+  exact tupleProbability_kernel _ x y
+
+lemma reflectedKernel_conjugate (d l : ℕ) (order : Equiv.Perm (Fin d)) (x y : Tuples d l) :
+    reflectedKernel d l order false x y =
+      PairRouting.tupleProbability (palindromePerm d)
+        ((tuplePerm (coordinateRelabel d order)).symm x)
+        ((tuplePerm (coordinateRelabel d order)).symm y) := by
+  classical
+  let e : Equiv.Perm (Tuples d l) := tuplePerm (coordinateRelabel d order)
+  let P := fun ω => tuplePerm (ι := Fin l) (butterflyPerm d (decodeButterfly d ω))
+  have hA : sweepKernel d l order = PermLaw.kernel (fun ω => e*P ω*e⁻¹) := by
+    rw [sweepKernel_law]
+    simp only [sweep,tuplePerm_mul,tuplePerm_inverse]
+    rfl
+  rw [reflectedKernel,ite_eq_right Bool.false_ne_true,hA]
+  rw [←PermLaw.inverse,←PermLaw.mul]
+  have hprod : (fun ω : BenesCoins d => (e*P ω.1*e⁻¹)*((e*P ω.2*e⁻¹)⁻¹)) =
+      fun ω : BenesCoins d => e*tuplePerm (ι := Fin l) (palindromePerm d ω)*e⁻¹ := by
+    funext ω
+    simp only [mul_inv_rev,inv_inv,palindromePerm,tuplePerm_mul]
+    change (e*P ω.1*e⁻¹)*(e*(P ω.2)⁻¹*e⁻¹) = e*(P ω.1*(P ω.2)⁻¹)*e⁻¹
+    group
+  rw [hprod,PermLaw.conjugate,tupleProbability_kernel]
+
+
+lemma sweepKernel_nonneg (d l : ℕ) (order : Equiv.Perm (Fin d)) (x y : Tuples d l) :
+    0 ≤ sweepKernel d l order x y := PairRouting.tupleProbability_nonneg _ _ _
+
+lemma sweepKernel_row (d l : ℕ) (order : Equiv.Perm (Fin d)) (x : Tuples d l) :
+    ∑ y, sweepKernel d l order x y = 1 := by
+  rw [sweepKernel_law]
+  exact PermLaw.row _ _
+
+lemma sweepKernel_col (d l : ℕ) (order : Equiv.Perm (Fin d)) (y : Tuples d l) :
+    ∑ x, sweepKernel d l order x y = 1 := by
+  rw [sweepKernel_law]
+  exact PermLaw.col _ _
+
+lemma reflected_nonneg (d l : ℕ) (order : Equiv.Perm (Fin d)) (x y : Tuples d l) :
+    0 ≤ reflectedKernel d l order false x y := by
+  exact mul_nonneg_entries _ _ (fun x y => sweepKernel_nonneg d l order y x)
+    (sweepKernel_nonneg d l order) x y
+
+lemma reflected_row (d l : ℕ) (order : Equiv.Perm (Fin d)) (x : Tuples d l) :
+    ∑ y, reflectedKernel d l order false x y = 1 :=
+  mul_row _ _ (sweepKernel_col d l order) (sweepKernel_row d l order) x
+
+lemma reflected_col (d l : ℕ) (order : Equiv.Perm (Fin d)) (y : Tuples d l) :
+    ∑ x, reflectedKernel d l order false x y = 1 :=
+  mul_col _ _ (sweepKernel_row d l order) (sweepKernel_col d l order) y
+
+lemma reflected_symmetric (d l : ℕ) (order : Equiv.Perm (Fin d)) :
+    (reflectedKernel d l order false).IsSymm := Matrix.isSymm_transpose_mul_self _
+
+lemma reflected_moment (d l : ℕ) (order : Equiv.Perm (Fin d)) (x : Tuples d l) :
+    finiteMean (fun y : Tuples d l =>
+      ((Fintype.card (Tuples d l):ℝ)*reflectedKernel d l order false x y)^(1+(1/1000:ℝ))) ≤
+      Real.exp (l*(1+heightDecay 1)) := by
+  classical
+  let e : Equiv.Perm (Tuples d l) := tuplePerm (coordinateRelabel d order)
+  simp_rw [reflectedKernel_conjugate]
+  have he := finiteMean_equiv e (fun y : Tuples d l =>
+    ((Fintype.card (Tuples d l):ℝ)*PairRouting.tupleProbability (palindromePerm d) (e.symm x) y)^
+      (1+(1/1000:ℝ)))
+  change finiteMean (fun y =>
+    ((Fintype.card (Tuples d l):ℝ)*PairRouting.tupleProbability (palindromePerm d) (e.symm x) (e.symm y))^
+      (1+(1/1000:ℝ))) ≤ _
+  rw [←he]
+  simpa only [palindromeRowMoment,Fintype.card_fin] using palindrome_row_moment_coarse d (e.symm x)
+
+lemma tuples_card (d l : ℕ) : Fintype.card (Tuples d l) = (2^d).descFactorial l := by
+  simp only [Tuples,Fintype.card_embedding_eq,Fintype.card_fin,card_positions]
+
+lemma canonical_pointwise (d l : ℕ) (order : Equiv.Perm (Fin d)) (x y : Tuples d l) :
+    ((reflectedKernel d l order false)^2000) x y ≤
+      Real.exp ((2000*(1+heightDecay 1))*l)/((2^d).descFactorial l:ℝ) := by
+  let : Nonempty (Tuples d l) := ⟨x⟩
+  have hh := symmetric_kernel_smoothing (reflectedKernel d l order false)
+    (reflected_nonneg d l order) (reflected_row d l order) (reflected_col d l order)
+    (reflected_symmetric d l order) (p := 1+(1/1000:ℝ)) (A := l*(1+heightDecay 1))
+    (by norm_num) (mul_nonneg (Nat.cast_nonneg _) (by linarith [heightDecay_nonneg 1]))
+    (reflected_moment d l order) 1000 density_exponent_numeric x y
+  have he : 2*((l:ℝ)*(1+heightDecay 1))*(1000:ℝ) = (2000*(1+heightDecay 1))*l := by ring
+  norm_num only [show 2*1000=2000 by norm_num,Nat.cast_ofNat] at hh
+  simpa only [he,tuples_card] using hh
+
+lemma both_pointwise (d l : ℕ) (order : Equiv.Perm (Fin d)) (orientation : Bool)
+    (x y : Tuples d l) :
+    ((reflectedKernel d l order orientation)^2001) x y ≤
+      Real.exp ((2000*(1+heightDecay 1))*l)/((2^d).descFactorial l:ℝ) := by
+  cases orientation
+  · change ((reflectedKernel d l order false)^(2000+1)) x y ≤ _
+    rw [pow_succ]
+    exact bound_mul_right _ _ (reflected_nonneg d l order) (reflected_col d l order)
+      (canonical_pointwise d l order) x y
+  · let A := sweepKernel d l order
+    change ((A*A.transpose)^(2000+1)) x y ≤ _
+    rw [←sandwich_power]
+    apply bound_mul_right _ _ (fun x y => sweepKernel_nonneg d l order y x)
+      (sweepKernel_row d l order)
+    exact bound_mul_left _ _ (sweepKernel_nonneg d l order) (sweepKernel_row d l order)
+      (canonical_pointwise d l order)
+
+lemma regular_trace_bound (d : ℕ) (order : Equiv.Perm (Fin d)) :
+    regularAbsoluteEvenTrace d order 2001 ≤ Real.exp ((2000*(1+heightDecay 1))*(2:ℝ)^d) := by
+  have hn : 0 < Fintype.card (Tuples d (2^d)) := by
+    rw [tuples_card,Nat.descFactorial_self]
+    exact Nat.factorial_pos _
+  have hn' : (Fintype.card (Tuples d (2^d)):ℝ) ≠ 0 := Nat.cast_ne_zero.mpr (ne_of_gt hn)
+  calc
+    _ ≤ ∑ x : Tuples d (2^d), Real.exp ((2000*(1+heightDecay 1))*(2^d:ℕ)) /
+        ((2^d).descFactorial (2^d):ℝ) := by
+      apply Finset.sum_le_sum
+      intro x _
+      exact both_pointwise d (2^d) order false x x
+    _ = _ := by
+      simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,←tuples_card]
+      push_cast
+      field_simp
+
+theorem strong_smoothing : MainStatement := by
+  refine ⟨2001,by norm_num,2000*(1+heightDecay 1),?_,regular_trace_bound⟩
+  intro d l _ order orientation x y
+  exact both_pointwise d l order orientation x y
+
+end Thorp.StrongSmoothing
+
+end ThorpNine.Smoothing
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Smoothing/Model.lean b/lean/OAI/Probability/ThorpRouting/Smoothing/Model.lean
index 40f2a245f6a803005ace3fbdc2472bdef8eead74..032fe596f5154b195c8e1f4b870644ef32c8f717 100644
--- a/lean/OAI/Probability/ThorpRouting/Smoothing/Model.lean
+++ b/lean/OAI/Probability/ThorpRouting/Smoothing/Model.lean
@@ -1,121 +1,121 @@
-import Mathlib
-
-namespace OAI
-
-namespace ThorpNine.Smoothing
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-
-abbrev Card (d : ℕ) := Fin d → Bool
-
-def switchFun {d : ℕ} (ξ : Card d → Bool) (x : Card (d + 1)) : Card (d + 1) :=
-  Fin.cons (Bool.xor (x 0) (ξ (Fin.tail x))) (Fin.tail x)
-
-theorem switchFun_involutive {d : ℕ} (ξ : Card d → Bool) :
-    Function.Involutive (switchFun ξ) := by
-  intro x
-  funext i
-  refine Fin.cases ?_ (fun j => ?_) i
-  · simp only [switchFun, Fin.cons_zero, Fin.tail_cons]
-    cases x 0 <;> cases ξ (Fin.tail x) <;> rfl
-  · simp [switchFun, Fin.tail]
-
-def pairSwitch {d : ℕ} (ξ : Card d → Bool) : Equiv.Perm (Card (d + 1)) :=
-  { toFun := switchFun ξ
-    invFun := switchFun ξ
-    left_inv := switchFun_involutive ξ
-    right_inv := switchFun_involutive ξ }
-
-def Butterfly : ℕ → Type
-  | 0 => Unit
-  | d + 1 => (Card d → Bool) × (Bool → Butterfly d)
-
-def childLift {d : ℕ} (p : Bool → Equiv.Perm (Card d)) : Equiv.Perm (Card (d + 1)) where
-  toFun x := Fin.cons (x 0) (p (x 0) (Fin.tail x))
-  invFun x := Fin.cons (x 0) ((p (x 0)).symm (Fin.tail x))
-  left_inv x := by simp only [Fin.cons_zero, Fin.tail_cons, Equiv.symm_apply_apply, Fin.cons_self_tail]
-  right_inv x := by simp only [Fin.cons_zero, Fin.tail_cons, Equiv.apply_symm_apply, Fin.cons_self_tail]
-
-def butterflyPerm : (d : ℕ) → Butterfly d → Equiv.Perm (Card d)
-  | 0, _ => 1
-  | d + 1, b => pairSwitch b.1 * childLift (fun ε => butterflyPerm d (b.2 ε))
-
-noncomputable def finiteMean {Ω : Type*} [Fintype Ω] (f : Ω → ℝ) : ℝ :=
-  (∑ ω, f ω) / Fintype.card Ω
-
-def SwitchIndex : ℕ → Type
-  | 0 => Empty
-  | d + 1 => Sum (Card d) (Bool × SwitchIndex d)
-
-instance (d : ℕ) : Fintype (SwitchIndex d) := by
-  induction d with
-  | zero => exact inferInstanceAs (Fintype Empty)
-  | succ d ih => exact inferInstanceAs (Fintype (Sum (Card d) (Bool × SwitchIndex d)))
-
-instance (d : ℕ) : DecidableEq (SwitchIndex d) := by
-  induction d with
-  | zero => exact inferInstanceAs (DecidableEq Empty)
-  | succ d ih => exact inferInstanceAs (DecidableEq (Sum (Card d) (Bool × SwitchIndex d)))
-
-def decodeButterfly : (d : ℕ) → (SwitchIndex d → Bool) → Butterfly d
-  | 0, _ => ()
-  | d + 1, ω => (fun y => ω (Sum.inl y), fun ε =>
-      decodeButterfly d (fun i => ω (Sum.inr (ε,i))))
-
-end Thorp
-
-namespace Thorp.PairRouting
-open scoped BigOperators
-open Filter
-variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
-
-noncomputable def tupleProbability {Ω β : Type*} [Fintype Ω] [DecidableEq β]
-    (P : Ω → Equiv.Perm β) (x y : ι → β) : ℝ := by
-  classical
-  exact finiteMean (fun ω => if ∀ i, P ω (x i) = y i then 1 else 0)
-
-end Thorp.PairRouting
-
-namespace Thorp.StrongSmoothing
-
-def coordinateRelabel (d : ℕ) (order : Equiv.Perm (Fin d)) : Equiv.Perm (Card d) where
-  toFun x := fun i => x (order.symm i)
-  invFun x := fun i => x (order i)
-  left_inv x := by funext i; simp
-  right_inv x := by funext i; simp
-
-def sweep (d : ℕ) (order : Equiv.Perm (Fin d)) (ω : SwitchIndex d → Bool) :
-    Equiv.Perm (Card d) :=
-  coordinateRelabel d order * butterflyPerm d (decodeButterfly d ω) *
-    (coordinateRelabel d order)⁻¹
-
-abbrev Tuples (d l : ℕ) := Fin l ↪ Card d
-
-noncomputable def sweepKernel (d l : ℕ) (order : Equiv.Perm (Fin d)) :
-    Matrix (Tuples d l) (Tuples d l) ℝ :=
-  fun x y => PairRouting.tupleProbability (sweep d order) x y
-
-noncomputable def reflectedKernel (d l : ℕ) (order : Equiv.Perm (Fin d)) (orientation : Bool) :
-    Matrix (Tuples d l) (Tuples d l) ℝ :=
-  let A := sweepKernel d l order
-  if orientation then A * A.transpose else A.transpose * A
-
-noncomputable def regularAbsoluteEvenTrace (d : ℕ) (order : Equiv.Perm (Fin d)) (u : ℕ) : ℝ :=
-  Matrix.trace ((reflectedKernel d (2^d) order false)^u)
-
-def MainStatement : Prop :=
-  ∃ u : ℕ, 1 ≤ u ∧ ∃ C : ℝ,
-    (∀ d l : ℕ, l ≤ 2^d → ∀ order : Equiv.Perm (Fin d), ∀ orientation : Bool,
-      ∀ x y : Tuples d l,
-        ((reflectedKernel d l order orientation)^u) x y ≤
-          Real.exp (C*l)/((2^d).descFactorial l:ℝ)) ∧
-    (∀ d : ℕ, ∀ order : Equiv.Perm (Fin d),
-      regularAbsoluteEvenTrace d order u ≤ Real.exp (C*(2:ℝ)^d))
-
-end Thorp.StrongSmoothing
-
-end ThorpNine.Smoothing
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+namespace ThorpNine.Smoothing
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+
+abbrev Card (d : ℕ) := Fin d → Bool
+
+def switchFun {d : ℕ} (ξ : Card d → Bool) (x : Card (d + 1)) : Card (d + 1) :=
+  Fin.cons (Bool.xor (x 0) (ξ (Fin.tail x))) (Fin.tail x)
+
+theorem switchFun_involutive {d : ℕ} (ξ : Card d → Bool) :
+    Function.Involutive (switchFun ξ) := by
+  intro x
+  funext i
+  refine Fin.cases ?_ (fun j => ?_) i
+  · simp only [switchFun, Fin.cons_zero, Fin.tail_cons]
+    cases x 0 <;> cases ξ (Fin.tail x) <;> rfl
+  · simp [switchFun, Fin.tail]
+
+def pairSwitch {d : ℕ} (ξ : Card d → Bool) : Equiv.Perm (Card (d + 1)) :=
+  { toFun := switchFun ξ
+    invFun := switchFun ξ
+    left_inv := switchFun_involutive ξ
+    right_inv := switchFun_involutive ξ }
+
+def Butterfly : ℕ → Type
+  | 0 => Unit
+  | d + 1 => (Card d → Bool) × (Bool → Butterfly d)
+
+def childLift {d : ℕ} (p : Bool → Equiv.Perm (Card d)) : Equiv.Perm (Card (d + 1)) where
+  toFun x := Fin.cons (x 0) (p (x 0) (Fin.tail x))
+  invFun x := Fin.cons (x 0) ((p (x 0)).symm (Fin.tail x))
+  left_inv x := by simp only [Fin.cons_zero, Fin.tail_cons, Equiv.symm_apply_apply, Fin.cons_self_tail]
+  right_inv x := by simp only [Fin.cons_zero, Fin.tail_cons, Equiv.apply_symm_apply, Fin.cons_self_tail]
+
+def butterflyPerm : (d : ℕ) → Butterfly d → Equiv.Perm (Card d)
+  | 0, _ => 1
+  | d + 1, b => pairSwitch b.1 * childLift (fun ε => butterflyPerm d (b.2 ε))
+
+noncomputable def finiteMean {Ω : Type*} [Fintype Ω] (f : Ω → ℝ) : ℝ :=
+  (∑ ω, f ω) / Fintype.card Ω
+
+def SwitchIndex : ℕ → Type
+  | 0 => Empty
+  | d + 1 => Sum (Card d) (Bool × SwitchIndex d)
+
+instance (d : ℕ) : Fintype (SwitchIndex d) := by
+  induction d with
+  | zero => exact inferInstanceAs (Fintype Empty)
+  | succ d ih => exact inferInstanceAs (Fintype (Sum (Card d) (Bool × SwitchIndex d)))
+
+instance (d : ℕ) : DecidableEq (SwitchIndex d) := by
+  induction d with
+  | zero => exact inferInstanceAs (DecidableEq Empty)
+  | succ d ih => exact inferInstanceAs (DecidableEq (Sum (Card d) (Bool × SwitchIndex d)))
+
+def decodeButterfly : (d : ℕ) → (SwitchIndex d → Bool) → Butterfly d
+  | 0, _ => ()
+  | d + 1, ω => (fun y => ω (Sum.inl y), fun ε =>
+      decodeButterfly d (fun i => ω (Sum.inr (ε,i))))
+
+end Thorp
+
+namespace Thorp.PairRouting
+open scoped BigOperators
+open Filter
+variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
+
+noncomputable def tupleProbability {Ω β : Type*} [Fintype Ω] [DecidableEq β]
+    (P : Ω → Equiv.Perm β) (x y : ι → β) : ℝ := by
+  classical
+  exact finiteMean (fun ω => if ∀ i, P ω (x i) = y i then 1 else 0)
+
+end Thorp.PairRouting
+
+namespace Thorp.StrongSmoothing
+
+def coordinateRelabel (d : ℕ) (order : Equiv.Perm (Fin d)) : Equiv.Perm (Card d) where
+  toFun x := fun i => x (order.symm i)
+  invFun x := fun i => x (order i)
+  left_inv x := by funext i; simp
+  right_inv x := by funext i; simp
+
+def sweep (d : ℕ) (order : Equiv.Perm (Fin d)) (ω : SwitchIndex d → Bool) :
+    Equiv.Perm (Card d) :=
+  coordinateRelabel d order * butterflyPerm d (decodeButterfly d ω) *
+    (coordinateRelabel d order)⁻¹
+
+abbrev Tuples (d l : ℕ) := Fin l ↪ Card d
+
+noncomputable def sweepKernel (d l : ℕ) (order : Equiv.Perm (Fin d)) :
+    Matrix (Tuples d l) (Tuples d l) ℝ :=
+  fun x y => PairRouting.tupleProbability (sweep d order) x y
+
+noncomputable def reflectedKernel (d l : ℕ) (order : Equiv.Perm (Fin d)) (orientation : Bool) :
+    Matrix (Tuples d l) (Tuples d l) ℝ :=
+  let A := sweepKernel d l order
+  if orientation then A * A.transpose else A.transpose * A
+
+noncomputable def regularAbsoluteEvenTrace (d : ℕ) (order : Equiv.Perm (Fin d)) (u : ℕ) : ℝ :=
+  Matrix.trace ((reflectedKernel d (2^d) order false)^u)
+
+def MainStatement : Prop :=
+  ∃ u : ℕ, 1 ≤ u ∧ ∃ C : ℝ,
+    (∀ d l : ℕ, l ≤ 2^d → ∀ order : Equiv.Perm (Fin d), ∀ orientation : Bool,
+      ∀ x y : Tuples d l,
+        ((reflectedKernel d l order orientation)^u) x y ≤
+          Real.exp (C*l)/((2^d).descFactorial l:ℝ)) ∧
+    (∀ d : ℕ, ∀ order : Equiv.Perm (Fin d),
+      regularAbsoluteEvenTrace d order u ≤ Real.exp (C*(2:ℝ)^d))
+
+end Thorp.StrongSmoothing
+
+end ThorpNine.Smoothing
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Smoothing/ParallelFollow.lean b/lean/OAI/Probability/ThorpRouting/Smoothing/ParallelFollow.lean
index 38b17d7bb9de543ae51d3f54d785c6201b3fa64f..59cca4fb8b3461a70a07b0247cefffadc9755de1 100644
--- a/lean/OAI/Probability/ThorpRouting/Smoothing/ParallelFollow.lean
+++ b/lean/OAI/Probability/ThorpRouting/Smoothing/ParallelFollow.lean
@@ -1,883 +1,883 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Smoothing.PriorPaths
-
-namespace OAI
-
-namespace ThorpNine.Smoothing
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-section ParallelButterfly
-variable {α B : Type*} [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] {d : ℕ}
-omit [DecidableEq B] in
-lemma parallel_entropy (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool) (S : Finset α) :
-    (∑ i : B × SwitchIndex d,
-      if (parallelUsers pre ω i).1 ∈ S ∧ (parallelUsers pre ω i).2 ∈ S then (1:ℝ) else 0) ≤
-      (S.card : ℝ) * Real.log S.card / (2 * Real.log 2) := by
-  classical
-  rw [Fintype.sum_prod_type]
-  calc
-    _ ≤ ∑ b : B, (blockMarks pre S b).card * Real.log (blockMarks pre S b).card / (2*Real.log 2) := by
-      apply Finset.sum_le_sum
-      intro b _
-      simpa only [←mem_blockMarks,parallelUsers] using
-        marked_switch_entropy_bound d (decodeButterfly d (fun i => ω (b,i))) (blockMarks pre S b)
-    _ ≤ ∑ b : B, (blockMarks pre S b).card * Real.log S.card / (2*Real.log 2) := by
-      apply Finset.sum_le_sum
-      intro b _
-      by_cases hz : (blockMarks pre S b).card = 0
-      · simp [hz]
-      · apply div_le_div_of_nonneg_right _ (by positivity)
-        apply mul_le_mul_of_nonneg_left _ (by positivity)
-        exact Real.log_le_log (by exact_mod_cast Nat.pos_of_ne_zero hz)
-          (by exact_mod_cast card_blockMarks_le pre S b)
-    _ = _ := by rw [←Finset.sum_div,←Finset.sum_mul,←Nat.cast_sum,sum_card_blockMarks]
-
-noncomputable def parallelFollow (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x : α) (k : α → QueryTree (B × SwitchIndex d)) : QueryTree (B × SwitchIndex d) :=
-  QueryTree.followPath d (fun i => ((pre x).1,i)) (pre x).2 (fun y => k (post ((pre x).1,y)))
-
-omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
-lemma parallelFollow_succeeds (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
-    (parallelFollow pre post x k).succeeds ω = (k (relabeledPerm pre post ω x)).succeeds ω := by
-  exact QueryTree.followPath_succeeds _ _ _ _ _
-
-omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
-lemma parallelFollow_raw_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
-    (parallelFollow pre post x k).charge ω = (k (relabeledPerm pre post ω x)).charge ω := by
-  exact QueryTree.followPath_raw_charge _ _ _ _ _
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma parallelFollow_stale_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
-    (ω : B × SwitchIndex d → Bool) :
-    (parallelFollow pre post x k).staleCharge R ω =
-      (k (relabeledPerm pre post ω x)).staleCharge (R ∪ parallelPath pre ω x) ω := by
-  exact QueryTree.followPath_stale_charge _ _ _ _ _ _
-
-noncomputable def closingDomain (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) : Finset (B × SwitchIndex d) :=
-  (routeDomain d (pre x).2 (post.symm y).2).image (fun i => ((pre x).1,i))
-
-def closingValue (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (i : B × SwitchIndex d) : Bool := routeValue d (pre x).2 (post.symm y).2 i.2
-
-omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
-lemma relabeled_endpoint_pair (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (ω : B × SwitchIndex d → Bool) (x y : α) :
-    relabeledPerm pre post ω x = y ↔
-      (pre x).1 = (post.symm y).1 ∧
-      butterflyPerm d (decodeButterfly d (fun i => ω ((pre x).1,i))) (pre x).2 = (post.symm y).2 := by
-  change post ((pre x).1,_) = y ↔ _
-  rw [←post.eq_symm_apply,Prod.mk.injEq]
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma relabeled_endpoint_iff (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (ω : B × SwitchIndex d → Bool) (x y : α) :
-    relabeledPerm pre post ω x = y ↔
-      (pre x).1 = (post.symm y).1 ∧
-        ∀ i ∈ closingDomain pre post x y, ω i = closingValue pre post x y i := by
-  classical
-  rw [relabeled_endpoint_pair,butterfly_endpoint_iff]
-  simp only [closingDomain,Finset.forall_mem_image,closingValue]
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma closingDomain_eq (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (ω : B × SwitchIndex d → Bool) (x y : α) (h : relabeledPerm pre post ω x = y) :
-    closingDomain pre post x y = parallelPath pre ω x := by
-  rw [closingDomain,parallelPath,(relabeled_endpoint_pair _ _ _ _ _).mp h |>.2]
-
-noncomputable def parallelClose (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (k : QueryTree (B × SwitchIndex d)) : QueryTree (B × SwitchIndex d) :=
-  if (pre x).1 = (post.symm y).1 then
-    QueryTree.forceList (closingDomain pre post x y).toList (closingValue pre post x y) k
-  else QueryTree.reject
-
-omit [Fintype α] in
-lemma parallelClose_succeeds (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (k : QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
-    (parallelClose pre post x y k).succeeds ω =
-      if relabeledPerm pre post ω x = y then k.succeeds ω else false := by
-  classical
-  simp only [relabeled_endpoint_iff,parallelClose]
-  by_cases hb : (pre x).1 = (post.symm y).1
-  · rw [ite_eq_left hb,QueryTree.forceList_succeeds]
-    simp only [Finset.mem_toList,hb,true_and]
-  · simp [hb,QueryTree.succeeds]
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma parallelClose_raw_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (k : QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool)
-    (h : relabeledPerm pre post ω x = y) :
-    (parallelClose pre post x y k).charge ω = k.charge ω + d := by
-  have hh := (relabeled_endpoint_iff _ _ _ _ _).mp h
-  rw [parallelClose,ite_eq_left hh.1,QueryTree.forceList_raw_charge _ _ _ _
-    (by simpa only [Finset.mem_toList] using hh.2),Finset.length_toList,
-    closingDomain_eq _ _ _ _ _ h,card_parallelPath]
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma parallelClose_fresh_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (k : QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
-    (ω : B × SwitchIndex d → Bool) (h : relabeledPerm pre post ω x = y) :
-    (parallelClose pre post x y k).freshCharge R ω =
-      k.freshCharge (R ∪ parallelPath pre ω x) ω + (parallelPath pre ω x \ R).card := by
-  have hh := (relabeled_endpoint_iff _ _ _ _ _).mp h
-  rw [parallelClose,ite_eq_left hh.1,QueryTree.forceList_charge _ _ _ _ _
-    (by simpa only [Finset.mem_toList] using hh.2),Finset.toList_toFinset,
-    closingDomain_eq _ _ _ _ _ h]
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma parallelClose_stale_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (k : QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
-    (ω : B × SwitchIndex d → Bool) (h : relabeledPerm pre post ω x = y) :
-    (parallelClose pre post x y k).staleCharge R ω =
-      k.staleCharge (R ∪ parallelPath pre ω x) ω + (parallelPath pre ω x ∩ R).card := by
-  have h₀ := QueryTree.fresh_add_stale (parallelClose pre post x y k) R ω
-  have h₁ := QueryTree.fresh_add_stale k (R ∪ parallelPath pre ω x) ω
-  rw [parallelClose_raw_charge _ _ _ _ _ _ h,parallelClose_fresh_charge _ _ _ _ _ _ _ h] at h₀
-  have h₂ : (parallelPath pre ω x \ R).card + (parallelPath pre ω x ∩ R).card = d := by
-    rw [Finset.card_sdiff_add_card_inter,card_parallelPath]
-  omega
-
-noncomputable def parallelNetwork (pre : α ≃ B × Card d) (post : B × Card d ≃ α) :
-    RoutingNetwork α (B × SwitchIndex d) d where
-  perm := relabeledPerm pre post
-  path := parallelPath pre
-  users := parallelUsers pre
-  users_ne := parallelUsers_ne pre
-  mem_path := mem_parallelPath pre
-  card_path := card_parallelPath pre
-  entropy := parallel_entropy pre
-  follow := parallelFollow pre post
-  close := parallelClose pre post
-  follow_succeeds := parallelFollow_succeeds pre post
-  follow_raw_charge := parallelFollow_raw_charge pre post
-  follow_stale_charge := parallelFollow_stale_charge pre post
-  close_succeeds := parallelClose_succeeds pre post
-  close_raw_charge := parallelClose_raw_charge pre post
-  close_stale_charge := parallelClose_stale_charge pre post
-
-lemma parallel_cycle_log_mgf (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (S : Finset α) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    Real.log (finiteMean (fun ω : B × SwitchIndex d → Bool =>
-      q^(Fintype.card (RoutingNetwork.SelectedCycle (relabeledPerm pre post ω) S)))) ≤
-      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
-  exact (parallelNetwork pre post).butterfly_cycle_log_mgf S J hJ q hq
-
-end ParallelButterfly
-
-def cardSplit : (r s : ℕ) → Card (s+r) ≃ Card r × Card s
-  | 0, s =>
-    { toFun := fun x => ((fun i => Fin.elim0 i), x)
-      invFun := fun x => x.2
-      left_inv := fun _ => rfl
-      right_inv := fun x => by
-        apply Prod.ext
-        · exact Subsingleton.elim _ _
-        · rfl }
-  | r+1, s =>
-    { toFun := fun x => (Fin.cons (x 0) ((cardSplit r s (Fin.tail x)).1),
-        (cardSplit r s (Fin.tail x)).2)
-      invFun := fun x => Fin.cons (x.1 0) ((cardSplit r s).symm (Fin.tail x.1,x.2))
-      left_inv := by
-        intro x
-        simp only [Fin.cons_zero, Fin.tail_cons, Prod.mk.eta, Equiv.symm_apply_apply, Fin.cons_self_tail]
-      right_inv := by
-        intro x
-        simp only [Fin.cons_zero, Fin.tail_cons, Equiv.apply_symm_apply, Fin.cons_self_tail] }
-
-def assembleBits : (r s : ℕ) → (Card r × SwitchIndex s → Bool) →
-    (Card s × SwitchIndex r → Bool) → SwitchIndex (s+r) → Bool
-  | 0, _, lo, _, i => lo ((fun j => Fin.elim0 j), i)
-  | r+1, s, _, hi, Sum.inl y => hi ((cardSplit r s y).2,Sum.inl (cardSplit r s y).1)
-  | r+1, s, lo, hi, Sum.inr (b,i) => assembleBits r s
-      (fun j => lo (Fin.cons b j.1,j.2)) (fun j => hi (j.1,Sum.inr (b,j.2))) i
-
-lemma butterfly_split (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
-    (hi : Card s × SwitchIndex r → Bool) (x : Card (s+r)) :
-    cardSplit r s (butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi)) x) =
-      let y := cardSplit r s x
-      let z := butterflyPerm s (decodeButterfly s (fun i => lo (y.1,i))) y.2
-      (butterflyPerm r (decodeButterfly r (fun i => hi (z,i))) y.1,z) := by
-  induction r with
-  | zero =>
-    apply Prod.ext
-    · funext i
-      exact Fin.elim0 i
-    · rfl
-  | succ r ih =>
-    let lo' : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons (x 0) j.1,j.2)
-    let hi' : Card s × SwitchIndex r → Bool := fun j => hi (j.1,Sum.inr (x 0,j.2))
-    have hh := ih lo' hi' (Fin.tail x)
-    let u := butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo' hi')) (Fin.tail x)
-    let y := cardSplit r s (Fin.tail x)
-    let z := butterflyPerm s (decodeButterfly s (fun i => lo' (y.1,i))) y.2
-    let w := butterflyPerm r (decodeButterfly r (fun i => hi' (z,i))) y.1
-    change cardSplit r s u = (w,z) at hh
-    change ((Fin.cons (Bool.xor (x 0) (hi ((cardSplit r s u).2,Sum.inl (cardSplit r s u).1)))
-        ((cardSplit r s u).1) : Card (r+1)), (cardSplit r s u).2) =
-      ((Fin.cons (Bool.xor (x 0) (hi (z,Sum.inl w))) w : Card (r+1)),z)
-    rw [hh]
-
-def splitSwitch : (r s : ℕ) → SwitchIndex (s+r) ≃
-    (Card r × SwitchIndex s) ⊕ (Card s × SwitchIndex r)
-  | 0, _ =>
-    { toFun := fun i => Sum.inl ((fun j => Fin.elim0 j),i)
-      invFun := fun i => match i with
-        | Sum.inl (_,j) => j
-        | Sum.inr (_,j) => nomatch j
-      left_inv := fun _ => rfl
-      right_inv := by
-        intro i
-        rcases i with ⟨x,j⟩ | ⟨x,j⟩
-        · exact congrArg Sum.inl (Prod.ext (Subsingleton.elim _ _) rfl)
-        · exact Empty.elim j }
-  | r+1, s =>
-    { toFun := fun i => match i with
-        | Sum.inl y => Sum.inr ((cardSplit r s y).2,Sum.inl (cardSplit r s y).1)
-        | Sum.inr (b,j) => match splitSwitch r s j with
-          | Sum.inl (p,j) => Sum.inl (Fin.cons b p,j)
-          | Sum.inr (q,j) => Sum.inr (q,Sum.inr (b,j))
-      invFun := fun i => match i with
-        | Sum.inl (p,j) => Sum.inr (p 0,(splitSwitch r s).symm (Sum.inl (Fin.tail p,j)))
-        | Sum.inr (q,Sum.inl p) => Sum.inl ((cardSplit r s).symm (p,q))
-        | Sum.inr (q,Sum.inr (b,j)) => Sum.inr (b,(splitSwitch r s).symm (Sum.inr (q,j)))
-      left_inv := by
-        intro i
-        rcases i with y | ⟨b,j⟩
-        · dsimp only
-          rw [Prod.mk.eta, Equiv.symm_apply_apply]
-        · dsimp only
-          cases hz : splitSwitch r s j with
-          | inl z =>
-            rcases z with ⟨p,k⟩
-            simp only [Fin.cons_zero, Fin.tail_cons]
-            rw [←hz,Equiv.symm_apply_apply]
-          | inr z =>
-            rcases z with ⟨q,k⟩
-            dsimp only
-            rw [←hz,Equiv.symm_apply_apply]
-      right_inv := by
-        intro i
-        rcases i with ⟨p,j⟩ | ⟨q,j⟩
-        · simp only [Equiv.apply_symm_apply, Fin.cons_self_tail]
-        · rcases j with p | ⟨b,j⟩ <;> simp only [Equiv.apply_symm_apply] }
-
-lemma assembleBits_eq (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
-    (hi : Card s × SwitchIndex r → Bool) (i : SwitchIndex (s+r)) :
-    assembleBits r s lo hi i = Sum.elim lo hi (splitSwitch r s i) := by
-  induction r with
-  | zero => rfl
-  | succ r ih =>
-    rcases i with y | ⟨b,j⟩
-    · rfl
-    · simp only [assembleBits,splitSwitch,Equiv.coe_fn_mk]
-      rw [ih]
-      cases splitSwitch r s j <;> rfl
-
-def splitCoinEquiv (r s : ℕ) :
-    ((Card r × SwitchIndex s → Bool) × (Card s × SwitchIndex r → Bool)) ≃
-      (SwitchIndex (s+r) → Bool) :=
-  (Equiv.sumArrowEquivProdArrow _ _ Bool).symm.trans
-    (Equiv.arrowCongr (splitSwitch r s).symm (Equiv.refl Bool))
-
-lemma splitCoinEquiv_apply (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
-    (hi : Card s × SwitchIndex r → Bool) :
-    splitCoinEquiv r s (lo,hi) = assembleBits r s lo hi := by
-  funext i
-  exact (assembleBits_eq r s lo hi i).symm
-
-lemma finiteMean_prod {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] (f : Ω × Ω' → ℝ) :
-    finiteMean f = finiteMean (fun x => finiteMean (fun y => f (x,y))) := by
-  simp only [finiteMean, Fintype.sum_prod_type, Fintype.card_prod, Nat.cast_mul,
-    ←Finset.sum_div, div_div, mul_comm]
-
-lemma finiteMean_split (r s : ℕ) (f : (SwitchIndex (s+r) → Bool) → ℝ) :
-    finiteMean f = finiteMean (fun lo : Card r × SwitchIndex s → Bool =>
-      finiteMean (fun hi : Card s × SwitchIndex r → Bool => f (assembleBits r s lo hi))) := by
-  rw [finiteMean_equiv (splitCoinEquiv r s).symm]
-  simp only [Equiv.symm_symm]
-  rw [finiteMean_prod]
-  apply finiteMean_congr
-  intro lo
-  apply finiteMean_congr
-  intro hi
-  rw [splitCoinEquiv_apply]
-
-section PartiallyExposed
-variable {α : Type*} {r s : ℕ}
-
-def freshPre (pre : α ≃ Card (s+r)) (lo : Card r × SwitchIndex s → Bool) :
-    α ≃ Card s × Card r :=
-  ((pre.trans (cardSplit r s)).trans (parallelPerm lo)).trans (Equiv.prodComm _ _)
-
-def freshPost (post : Card (s+r) ≃ α) : Card s × Card r ≃ α :=
-  ((Equiv.prodComm _ _).trans (cardSplit r s).symm).trans post
-
-lemma exposed_perm_eq (pre : α ≃ Card (s+r)) (post : Card (s+r) ≃ α)
-    (lo : Card r × SwitchIndex s → Bool) (hi : Card s × SwitchIndex r → Bool) :
-    (pre.trans (butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi)))).trans post =
-      relabeledPerm (freshPre pre lo) (freshPost post) hi := by
-  ext x
-  simpa only [relabeledPerm, freshPre, freshPost, parallelPerm, Equiv.trans_apply,
-    Equiv.coe_fn_mk, Equiv.prodComm_apply, Prod.swap, Equiv.symm_apply_apply] using
-    congrArg (fun y => post ((cardSplit r s).symm y)) (butterfly_split r s lo hi (pre x))
-
-lemma partially_exposed_cycle_log_mgf [Fintype α] [DecidableEq α]
-    (pre : α ≃ Card (s+r)) (post : Card (s+r) ≃ α)
-    (lo : Card r × SwitchIndex s → Bool) (S : Finset α)
-    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    Real.log (finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(Fintype.card (RoutingNetwork.SelectedCycle
-        ((pre.trans (butterflyPerm (s+r)
-          (decodeButterfly (s+r) (assembleBits r s lo hi)))).trans post) S)))) ≤
-      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^r) := by
-  simp only [exposed_perm_eq]
-  exact parallel_cycle_log_mgf (freshPre pre lo) (freshPost post) S J hJ q hq
-
-end PartiallyExposed
-
-lemma inclusionMoment_eq {α Ω : Type*} [DecidableEq α] [Fintype Ω]
-    (F : Ω → Finset α) (S : Finset α) :
-    inclusionMoment F S = finiteMean (fun ω => if S ⊆ F ω then (1:ℝ) else 0) := by
-  unfold inclusionMoment
-  apply finiteMean_congr
-  intro ω
-  split_ifs <;> rfl
-
-lemma finiteMean_bool (f : Bool → ℝ) : finiteMean f = (f false+f true)/2 := by
-  simp only [finiteMean, Fintype.sum_bool, Fintype.card_bool, Nat.cast_ofNat]
-  ring
-
-lemma finiteMean_const {Ω : Type*} [Fintype Ω] [Nonempty Ω] (c : ℝ) :
-    finiteMean (fun _ : Ω => c) = c := by
-  simp only [finiteMean,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
-  exact mul_div_cancel_left₀ c (Nat.cast_ne_zero.mpr Fintype.card_ne_zero)
-
-lemma finiteMean_pi_prod {ι : Type*} [Fintype ι] [DecidableEq ι] {Ω : ι → Type*}
-    [∀ i, Fintype (Ω i)] (f : (i : ι) → Ω i → ℝ) :
-    finiteMean (fun ω : (i : ι) → Ω i => ∏ i, f i (ω i)) = ∏ i, finiteMean (f i) := by
-  simp only [finiteMean,←Fintype.prod_sum,Fintype.card_pi,Nat.cast_prod,Finset.prod_div_distrib]
-
-section ParallelFairLayer
-variable {β : Type*} [Fintype β] [DecidableEq β]
-
-def pairLayer (ξ : β → Bool) : Equiv.Perm (Bool × β) where
-  toFun x := (Bool.xor x.1 (ξ x.2),x.2)
-  invFun x := (Bool.xor x.1 (ξ x.2),x.2)
-  left_inv x := by
-    rcases x with ⟨b,x⟩
-    change (Bool.xor (Bool.xor b (ξ x)) (ξ x),x) = (b,x)
-    cases b <;> cases ξ x <;> rfl
-  right_inv x := by
-    rcases x with ⟨b,x⟩
-    change (Bool.xor (Bool.xor b (ξ x)) (ξ x),x) = (b,x)
-    cases b <;> cases ξ x <;> rfl
-
-noncomputable def pairFiber (S : Finset (Bool × β)) (b : β) : Finset Bool :=
-  Finset.univ.filter (fun ε => (ε,b) ∈ S)
-
-lemma prod_pairFiber (S : Finset (Bool × β)) (f : Bool × β → ℝ) :
-    (∏ x ∈ S, f x) = ∏ b : β, ∏ ε ∈ pairFiber S b, f (ε,b) := by
-  classical
-  simp only [pairFiber,Finset.prod_filter]
-  rw [Finset.prod_comm,←Fintype.prod_prod_type (fun x : Bool × β => if x ∈ S then f x else 1)]
-  rw [←Finset.prod_filter]
-  congr 1
-  ext x
-  simp
-
-lemma fiber_average_bound (S : Finset Bool) (p : Bool → ℝ) (hp : ∀ b, 0 ≤ p b) :
-    finiteMean (fun c : Bool => ∏ ε ∈ S, p (Bool.xor ε c)) ≤
-      ∏ ε ∈ S, (p ε + p (!ε))/2 := by
-  rw [finiteMean_bool]
-  have hxor : ∀ ε : Bool, Bool.xor ε true = Equiv.swap false true ε := by decide
-  simp only [Bool.xor_false,hxor]
-  calc
-    _ ≤ ∏ ε ∈ S, (p ε+p (Equiv.swap false true ε))/2 :=
-      swap_average_product S p hp false true
-    _ = _ := Finset.prod_congr rfl (fun ε _ => by cases ε <;> rfl)
-
-lemma CylinderBound.fair_parallel {m : Finset (Bool × β) → ℝ} {p : Bool × β → ℝ}
-    (h : CylinderBound m p) (hp : ∀ x, 0 ≤ p x) :
-    CylinderBound (fun S => finiteMean (fun ξ : β → Bool => m (S.image (pairLayer ξ))))
-      (fun x => (p x+p (!x.1,x.2))/2) := by
-  intro S
-  calc
-    _ ≤ finiteMean (fun ξ : β → Bool => ∏ x ∈ S, p (pairLayer ξ x)) := by
-      apply finiteMean_mono
-      intro ξ
-      simpa only [Finset.prod_image (pairLayer ξ).injective.injOn] using h (S.image (pairLayer ξ))
-    _ = ∏ b : β, finiteMean (fun c : Bool => ∏ ε ∈ pairFiber S b, p (Bool.xor ε c,b)) := by
-      calc
-        _ = finiteMean (fun ξ : β → Bool => ∏ b : β,
-            ∏ ε ∈ pairFiber S b, p (Bool.xor ε (ξ b),b)) :=
-          finiteMean_congr (fun ξ => prod_pairFiber S (fun x => p (pairLayer ξ x)))
-        _ = _ := finiteMean_pi_prod (fun b c => ∏ ε ∈ pairFiber S b, p (Bool.xor ε c,b))
-    _ ≤ ∏ b : β, ∏ ε ∈ pairFiber S b, (p (ε,b)+p (!ε,b))/2 := by
-      apply Finset.prod_le_prod₀
-      · intro b _
-        exact div_nonneg (Finset.sum_nonneg (fun c _ =>
-          Finset.prod_nonneg (fun ε _ => hp (Bool.xor ε c,b)))) (Nat.cast_nonneg _)
-      · intro b _
-        exact fiber_average_bound (pairFiber S b) (fun ε => p (ε,b)) (fun ε => hp (ε,b))
-    _ = _ := (prod_pairFiber S (fun x => (p x+p (!x.1,x.2))/2)).symm
-
-omit [Fintype β] in
-lemma subset_pairLayer_image_iff (S A : Finset (Bool × β)) (ξ : β → Bool) :
-    S ⊆ A.image (pairLayer ξ) ↔ S.image (pairLayer ξ) ⊆ A := by
-  have hi (x : Bool × β) : pairLayer ξ (pairLayer ξ x) = x := (pairLayer ξ).left_inv x
-  constructor
-  · intro h
-    have h' := Finset.image_subset_image h (f := pairLayer ξ)
-    simpa only [Finset.image_image,Function.comp_def,hi,Finset.image_id'] using h'
-  · intro h
-    have h' := Finset.image_subset_image h (f := pairLayer ξ)
-    simpa only [Finset.image_image,Function.comp_def,hi,Finset.image_id'] using h'
-
-noncomputable def fairSetLayer {Ω : Type*} (F : Ω → Finset (Bool × β)) :
-    Ω × (β → Bool) → Finset (Bool × β) := fun ω => (F ω.1).image (pairLayer ω.2)
-
-lemma finiteMean_comm {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] (f : Ω → Ω' → ℝ) :
-    finiteMean (fun x => finiteMean (f x)) = finiteMean (fun y => finiteMean (fun x => f x y)) := by
-  simp only [finiteMean,←Finset.sum_div,div_div]
-  rw [Finset.sum_comm,mul_comm]
-
-lemma inclusionMoment_fairSetLayer {Ω : Type*} [Fintype Ω]
-    (F : Ω → Finset (Bool × β)) (S : Finset (Bool × β)) :
-    inclusionMoment (fairSetLayer F) S =
-      finiteMean (fun ξ : β → Bool => inclusionMoment F (S.image (pairLayer ξ))) := by
-  classical
-  simp only [inclusionMoment_eq]
-  have he : (fun ω : Ω × (β → Bool) => if S ⊆ fairSetLayer F ω then (1:ℝ) else 0) =
-      (fun ω : Ω × (β → Bool) => if S.image (pairLayer ω.2) ⊆ F ω.1 then (1:ℝ) else 0) := by
-    funext ω
-    simp only [fairSetLayer,subset_pairLayer_image_iff]
-  apply (congrArg finiteMean he).trans
-  exact (finiteMean_prod (fun ω : Ω × (β → Bool) =>
-    if S.image (pairLayer ω.2) ⊆ F ω.1 then (1:ℝ) else 0)).trans
-      (finiteMean_comm (fun ω ξ => if S.image (pairLayer ξ) ⊆ F ω then (1:ℝ) else 0))
-
-lemma inclusionMoment_fairSetLayer_singleton {Ω : Type*} [Fintype Ω]
-    (F : Ω → Finset (Bool × β)) (x : Bool × β) :
-    inclusionMoment (fairSetLayer F) {x} =
-      (inclusionMoment F {x}+inclusionMoment F {(!x.1,x.2)})/2 := by
-  rw [inclusionMoment_fairSetLayer]
-  simp only [Finset.image_singleton,pairLayer,Equiv.coe_fn_mk]
-  rw [finiteMean_pi_eval (A := fun _ : β => Bool) x.2 (fun c : Bool => inclusionMoment F {(Bool.xor x.1 c,x.2)})]
-  rw [finiteMean_bool]
-  rcases x with ⟨b,x⟩
-  cases b <;> simp only [Bool.xor_false,Bool.false_xor,Bool.true_xor, Bool.not_false,Bool.not_true]
-
-lemma cylinder_fairSetLayer {Ω : Type*} [Fintype Ω]
-    (F : Ω → Finset (Bool × β))
-    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
-    CylinderBound (inclusionMoment (fairSetLayer F))
-      (fun x => inclusionMoment (fairSetLayer F) {x}) := by
-  have hh := h.fair_parallel (fun x => inclusionMoment_nonneg F {x})
-  intro S
-  rw [inclusionMoment_fairSetLayer]
-  simp only [inclusionMoment_fairSetLayer_singleton]
-  exact hh S
-
-end ParallelFairLayer
-
-section RelabelMarks
-variable {α β Ω Ω' : Type*} [DecidableEq α] [DecidableEq β] [Fintype Ω] [Fintype Ω']
-
-omit [Fintype Ω] [Fintype Ω'] in
-lemma subset_image_equiv_iff (e : α ≃ β) (A : Finset α) (S : Finset β) :
-    S ⊆ A.image e ↔ S.image e.symm ⊆ A := by
-  constructor
-  · intro h x hx
-    obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
-    obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp (h hy)
-    simpa only [←hxy,Equiv.symm_apply_apply] using hx
-  · intro h y hy
-    exact Finset.mem_image.mpr ⟨e.symm y,h (Finset.mem_image_of_mem _ hy),e.apply_symm_apply y⟩
-
-omit [Fintype Ω'] in
-lemma inclusionMoment_image_equiv (e : α ≃ β) (F : Ω → Finset α) (S : Finset β) :
-    inclusionMoment (fun ω => (F ω).image e) S = inclusionMoment F (S.image e.symm) := by
-  classical
-  simp only [inclusionMoment_eq]
-  apply finiteMean_congr
-  intro ω
-  simp only [subset_image_equiv_iff]
-
-omit [Fintype Ω'] in
-lemma cylinder_image_equiv (e : α ≃ β) (F : Ω → Finset α)
-    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
-    CylinderBound (inclusionMoment (fun ω => (F ω).image e))
-      (fun x => inclusionMoment (fun ω => (F ω).image e) {x}) := by
-  intro S
-  simp only [inclusionMoment_image_equiv,Finset.image_singleton]
-  have hh := h (S.image e.symm)
-  simpa only [Finset.prod_image e.symm.injective.injOn] using hh
-
-omit [DecidableEq β] in
-lemma inclusionMoment_comp_equiv (e : Ω ≃ Ω') (F : Ω' → Finset α) (S : Finset α) :
-    inclusionMoment (fun ω => F (e ω)) S = inclusionMoment F S := by
-  classical
-  simp only [inclusionMoment_eq]
-  simpa only [Equiv.apply_symm_apply] using
-    finiteMean_equiv e (fun ω => if S ⊆ F (e ω) then (1 : ℝ) else 0)
-
-omit [DecidableEq β] in
-lemma cylinder_comp_equiv (e : Ω ≃ Ω') (F : Ω' → Finset α)
-    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
-    CylinderBound (inclusionMoment (fun ω => F (e ω)))
-      (fun x => inclusionMoment (fun ω => F (e ω)) {x}) := by
-  intro S
-  simpa only [inclusionMoment_comp_equiv] using h S
-end RelabelMarks
-
-lemma finiteMean_prod_mul {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
-    (f : Ω → ℝ) (g : Ω' → ℝ) :
-    finiteMean (fun ω : Ω × Ω' => f ω.1*g ω.2) = finiteMean f * finiteMean g := by
-  simp only [finiteMean,Fintype.sum_prod_type,Fintype.card_prod,Nat.cast_mul,
-    ←Finset.mul_sum,←Finset.sum_mul]
-  ring
-
-section JoinChildren
-variable {β : Type*} [DecidableEq β]
-
-noncomputable def childJoin (A B : Finset β) : Finset (Bool × β) :=
-  A.image (fun x => (false,x)) ∪ B.image (fun x => (true,x))
-
-lemma mem_childJoin (A B : Finset β) (b : Bool) (x : β) :
-    (b,x) ∈ childJoin A B ↔ (if b then x ∈ B else x ∈ A) := by
-  cases b <;> simp [childJoin]
-
-noncomputable def boolFiber (S : Finset (Bool × β)) (b : Bool) : Finset β :=
-  (S.filter (fun x => x.1 = b)).image Prod.snd
-
-lemma mem_boolFiber (S : Finset (Bool × β)) (b : Bool) (x : β) :
-    x ∈ boolFiber S b ↔ (b,x) ∈ S := by
-  constructor
-  · intro h
-    obtain ⟨⟨c,y⟩,hy,hxy⟩ := Finset.mem_image.mp h
-    obtain ⟨hy,hc⟩ := Finset.mem_filter.mp hy
-    change c = b at hc
-    change y = x at hxy
-    simpa only [hc,hxy] using hy
-  · intro h
-    exact Finset.mem_image.mpr ⟨(b,x),Finset.mem_filter.mpr ⟨h,rfl⟩,rfl⟩
-
-lemma childJoin_boolFiber (S : Finset (Bool × β)) :
-    childJoin (boolFiber S false) (boolFiber S true) = S := by
-  ext ⟨b,x⟩
-  rw [mem_childJoin]
-  cases b <;> simp only [Bool.false_eq_true,↓reduceIte,mem_boolFiber]
-
-lemma subset_childJoin_iff (S : Finset (Bool × β)) (A B : Finset β) :
-    S ⊆ childJoin A B ↔ boolFiber S false ⊆ A ∧ boolFiber S true ⊆ B := by
-  constructor
-  · intro h
-    constructor
-    · intro x hx
-      exact (mem_childJoin A B false x).mp (h ((mem_boolFiber S false x).mp hx))
-    · intro x hx
-      exact (mem_childJoin A B true x).mp (h ((mem_boolFiber S true x).mp hx))
-  · rintro ⟨h₀,h₁⟩ ⟨b,x⟩ hx
-    rw [mem_childJoin]
-    cases b
-    · exact h₀ ((mem_boolFiber S false x).mpr hx)
-    · exact h₁ ((mem_boolFiber S true x).mpr hx)
-
-lemma prod_childJoin (A B : Finset β) (f : Bool × β → ℝ) :
-    (∏ x ∈ childJoin A B, f x) = (∏ x ∈ A, f (false,x)) * ∏ x ∈ B, f (true,x) := by
-  classical
-  have hd : Disjoint (A.image (fun x => (false,x))) (B.image (fun x => (true,x))) := by
-    apply Finset.disjoint_left.mpr
-    rintro x hx hy
-    obtain ⟨a,ha,rfl⟩ := Finset.mem_image.mp hx
-    obtain ⟨b,hb,hba⟩ := Finset.mem_image.mp hy
-    exact Bool.false_ne_true (congrArg Prod.fst hba).symm
-  rw [childJoin,Finset.prod_union hd,
-    Finset.prod_image (fun _ _ _ _ h => congrArg Prod.snd h),
-    Finset.prod_image (fun _ _ _ _ h => congrArg Prod.snd h)]
-
-lemma prod_boolFibers (S : Finset (Bool × β)) (f : Bool × β → ℝ) :
-    (∏ x ∈ S, f x) = (∏ x ∈ boolFiber S false, f (false,x)) *
-      ∏ x ∈ boolFiber S true, f (true,x) := by
-  rw [←prod_childJoin,childJoin_boolFiber]
-
-lemma inclusionMoment_childJoin {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
-    (F : Ω → Finset β) (G : Ω' → Finset β) (S : Finset (Bool × β)) :
-    inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) S =
-      inclusionMoment F (boolFiber S false) * inclusionMoment G (boolFiber S true) := by
-  classical
-  simp only [inclusionMoment_eq]
-  rw [←finiteMean_prod_mul]
-  apply finiteMean_congr
-  intro ω
-  simp only [subset_childJoin_iff]
-  split_ifs <;> simp_all
-
-lemma cylinder_childJoin {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] [Nonempty Ω] [Nonempty Ω']
-    (F : Ω → Finset β) (G : Ω' → Finset β)
-    (hF : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x}))
-    (hG : CylinderBound (inclusionMoment G) (fun x => inclusionMoment G {x})) :
-    CylinderBound (inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)))
-      (fun x => inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {x}) := by
-  intro S
-  rw [inclusionMoment_childJoin,prod_boolFibers]
-  have hf (b : Bool) (x : β) : boolFiber {(b,x)} b = {x} := by
-    ext y
-    simp only [mem_boolFiber,Finset.mem_singleton,Prod.mk.injEq,true_and]
-  have hn (b : Bool) (x : β) : boolFiber {(!b,x)} b = ∅ := by
-    ext y
-    cases b <;> simp [mem_boolFiber]
-  have h₀ (x : β) : inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {(false,x)} =
-      inclusionMoment F {x} * inclusionMoment G ∅ := by
-    rw [inclusionMoment_childJoin,hf]
-    have hh := hn true x
-    exact congrArg (fun T => inclusionMoment F {x} * inclusionMoment G T) hh
-  have h₁ (x : β) : inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {(true,x)} =
-      inclusionMoment F ∅ * inclusionMoment G {x} := by
-    rw [inclusionMoment_childJoin,hf]
-    have hh := hn false x
-    exact congrArg (fun T => inclusionMoment F T * inclusionMoment G {x}) hh
-  have emF : inclusionMoment F ∅ = 1 := by simp [inclusionMoment,finiteMean_const]
-  have emG : inclusionMoment G ∅ = 1 := by simp [inclusionMoment,finiteMean_const]
-  simp only [h₀,h₁,emF,emG,mul_one,one_mul]
-  exact mul_le_mul (hF _) (hG _) (inclusionMoment_nonneg _ _)
-    (Finset.prod_nonneg (fun x _ => inclusionMoment_nonneg F {x}))
-end JoinChildren
-
-def coinStepEquiv (d : ℕ) : (SwitchIndex (d+1) → Bool) ≃
-    ((SwitchIndex d → Bool) × (SwitchIndex d → Bool)) × (Card d → Bool) where
-  toFun ω := ((fun i => ω (Sum.inr (false,i)),fun i => ω (Sum.inr (true,i))),
-    fun x => ω (Sum.inl x))
-  invFun ω := Sum.elim ω.2 (fun bi => if bi.1 then ω.1.2 bi.2 else ω.1.1 bi.2)
-  left_inv ω := by
-    funext i
-    cases i with
-    | inl x => rfl
-    | inr bi => cases bi with | mk b i => cases b <;> rfl
-  right_inv ω := by
-    rcases ω with ⟨⟨ω₀,ω₁⟩,ξ⟩
-    rfl
-
-def headTailEquiv (d : ℕ) : Card (d+1) ≃ Bool × Card d :=
-  (Fin.consEquiv (fun _ : Fin (d+1) => Bool)).symm
-
-lemma childMarks_join (d : ℕ) (A : Finset (Card (d+1))) :
-    A = (childMarks A false).image (Fin.cons false) ∪
-      (childMarks A true).image (Fin.cons true) := by
-  classical
-  ext x
-  constructor
-  · intro hx
-    cases hb : x 0
-    · exact Finset.mem_union_left _ (Finset.mem_image.mpr
-        ⟨Fin.tail x,(mem_childMarks _ _ _).mpr (by simpa only [←hb,Fin.cons_self_tail] using hx),
-          by simp only [←hb,Fin.cons_self_tail]⟩)
-    · exact Finset.mem_union_right _ (Finset.mem_image.mpr
-        ⟨Fin.tail x,(mem_childMarks _ _ _).mpr (by simpa only [←hb,Fin.cons_self_tail] using hx),
-          by simp only [←hb,Fin.cons_self_tail]⟩)
-  · intro hx
-    rcases Finset.mem_union.mp hx with hx | hx
-    · obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
-      exact (mem_childMarks _ _ _).mp hy
-    · obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
-      exact (mem_childMarks _ _ _).mp hy
-
-noncomputable def butterflyMarks (d : ℕ) (A : Finset (Card d))
-    (ω : SwitchIndex d → Bool) : Finset (Card d) :=
-  A.image (butterflyPerm d (decodeButterfly d ω))
-
-lemma butterflyMarks_succ (d : ℕ) (A : Finset (Card (d+1))) (ω : SwitchIndex (d+1) → Bool) :
-    butterflyMarks (d+1) A ω =
-      (fairSetLayer (fun p : (SwitchIndex d → Bool) × (SwitchIndex d → Bool) =>
-        childJoin (butterflyMarks d (childMarks A false) p.1)
-          (butterflyMarks d (childMarks A true) p.2)) (coinStepEquiv d ω)).image
-            (headTailEquiv d).symm := by
-  classical
-  unfold butterflyMarks fairSetLayer childJoin
-  rw [Finset.image_union,Finset.image_union]
-  simp only [Finset.image_image]
-  conv_lhs => rw [childMarks_join d A]
-  rw [Finset.image_union]
-  simp only [Finset.image_image]
-  apply congrArg₂ (· ∪ ·)
-  · apply Finset.image_congr
-    intro x _
-    simp only [Function.comp_def,butterflyPerm,decodeButterfly,Equiv.Perm.coe_mul,
-      childLift,pairSwitch,switchFun,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
-      coinStepEquiv,headTailEquiv,Equiv.symm_symm,pairLayer]
-    rfl
-  · apply Finset.image_congr
-    intro x _
-    simp only [Function.comp_def,butterflyPerm,decodeButterfly,Equiv.Perm.coe_mul,
-      childLift,pairSwitch,switchFun,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
-      coinStepEquiv,headTailEquiv,Equiv.symm_symm,pairLayer]
-    rfl
-
-lemma butterflyMarks_cylinder (d : ℕ) (A : Finset (Card d)) :
-    CylinderBound (inclusionMoment (butterflyMarks d A))
-      (fun x => inclusionMoment (butterflyMarks d A) {x}) := by
-  induction d with
-  | zero =>
-    have he : butterflyMarks 0 A = fun _ : SwitchIndex 0 → Bool => A := by
-      funext ω
-      simp [butterflyMarks,butterflyPerm]
-    rw [he]
-    exact cylinder_const A
-  | succ d ih =>
-    have h := cylinder_childJoin (butterflyMarks d (childMarks A false))
-      (butterflyMarks d (childMarks A true)) (ih _) (ih _)
-    have hh := cylinder_image_equiv (headTailEquiv d).symm _ (cylinder_fairSetLayer _ h)
-    have hs := cylinder_comp_equiv (coinStepEquiv d) _ hh
-    simpa only [←butterflyMarks_succ] using hs
-
-lemma indicator_mem_image_perm {α : Type*} [DecidableEq α]
-    (A : Finset α) (p : Equiv.Perm α) (y : α) :
-    (if y ∈ A.image p then (1 : ℝ) else 0) = ∑ x ∈ A, if p x = y then 1 else 0 := by
-  classical
-  by_cases h : y ∈ A.image p
-  · obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp h
-    rw [ite_eq_left (Finset.mem_image.mpr ⟨x,hx,hxy⟩)]
-    symm
-    calc
-      _ = (if p x = y then (1:ℝ) else 0) := Finset.sum_eq_single_of_mem x hx (by
-        intro z _ hzx
-        exact ite_eq_right (fun hz => hzx (p.injective (hz.trans hxy.symm))))
-      _ = 1 := ite_eq_left hxy
-  · rw [ite_eq_right h]
-    symm
-    apply Finset.sum_eq_zero
-    intro x hx
-    exact ite_eq_right (fun hxy => h (Finset.mem_image.mpr ⟨x,hx,hxy⟩))
-
-lemma butterflyMarks_singleton (d : ℕ) (A : Finset (Card d)) (y : Card d) :
-    inclusionMoment (butterflyMarks d A) {y} = (A.card : ℝ)/(2:ℝ)^d := by
-  classical
-  unfold inclusionMoment
-  simp only [Finset.singleton_subset_iff,butterflyMarks,indicator_mem_image_perm]
-  rw [finiteMean_sum]
-  simp only [butterfly_single_card_uniform,Finset.sum_const,nsmul_eq_mul]
-  ring
-
-lemma butterflyMarks_inclusion (d : ℕ) (A S : Finset (Card d)) :
-    inclusionMoment (butterflyMarks d A) S ≤ ((A.card : ℝ)/(2:ℝ)^d)^S.card := by
-  have h := butterflyMarks_cylinder d A S
-  simpa only [butterflyMarks_singleton,Finset.prod_const] using h
-
-namespace CycleColoring
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-abbrev Coloring (p : Equiv.Perm α) (S : Finset α) :=
-  {f : α → Bool // (∀ x, x ∉ S → f x = false) ∧
-    ∀ x ∈ S, p x ∈ S → f (p x) = f x}
-
-noncomputable instance (p : Equiv.Perm α) (S : Finset α) : Fintype (Coloring p S) :=
-  Fintype.ofFinite _
-
-noncomputable def boundary (p : Equiv.Perm α) (S : Finset α) : Finset α :=
-  S.filter (fun x => p.symm x ∉ S)
-
-noncomputable def root {p : Equiv.Perm α} {S : Finset α}
-    (C : RoutingNetwork.SelectedCycle p S) : α := C.nonempty.choose
-
-lemma root_mem {p : Equiv.Perm α} {S : Finset α} (C : RoutingNetwork.SelectedCycle p S) :
-    root C ∈ C.val := C.nonempty.choose_spec
-
-abbrev Seed (p : Equiv.Perm α) (S : Finset α) :=
-  {x // x ∈ boundary p S} ⊕ RoutingNetwork.SelectedCycle p S
-
-noncomputable def record {p : Equiv.Perm α} {S : Finset α}
-    (f : Coloring p S) : Seed p S → Bool :=
-  Sum.elim (fun x => f.val x.val) (fun C => f.val (root C))
-
-lemma orbit_subset_of_predecessor_closed (p : Equiv.Perm α) (D : Finset α)
-    (hD : ∀ x ∈ D, p.symm x ∈ D) {x : α} (hx : x ∈ D) : orbitSet p x ⊆ D := by
-  intro y hy
-  have hh := ((mem_orbitSet p x y).mp hy).inv
-  obtain ⟨n,hn⟩ := hh.exists_nat_pow_eq
-  have hp (k : ℕ) : (p⁻¹ ^ k) x ∈ D := by
-    induction k with
-    | zero => simpa using hx
-    | succ k ih => simpa only [pow_succ',Equiv.Perm.mul_apply,Equiv.Perm.coe_inv] using hD _ ih
-  exact hn ▸ hp n
-
-lemma record_injective (p : Equiv.Perm α) (S : Finset α) :
-    Function.Injective (record (p := p) (S := S)) := by
-  classical
-  intro f g hfg
-  apply Subtype.ext
-  funext x
-  by_contra hne
-  have hx : x ∈ S := by
-    by_contra h
-    exact hne ((f.property.1 x h).trans (g.property.1 x h).symm)
-  let D := S.filter (fun y => f.val y ≠ g.val y)
-  have hxD : x ∈ D := Finset.mem_filter.mpr ⟨hx,hne⟩
-  have hprev (y : α) (hy : y ∈ D) : p.symm y ∈ D := by
-    obtain ⟨hyS,hyne⟩ := Finset.mem_filter.mp hy
-    have hpS : p.symm y ∈ S := by
-      by_contra hn
-      have hseed : y ∈ boundary p S := Finset.mem_filter.mpr ⟨hyS,hn⟩
-      exact hyne (congrFun hfg (Sum.inl ⟨y,hseed⟩))
-    refine Finset.mem_filter.mpr ⟨hpS,?_⟩
-    have hf := f.property.2 (p.symm y) hpS (by simpa only [Equiv.apply_symm_apply] using hyS)
-    have hg := g.property.2 (p.symm y) hpS (by simpa only [Equiv.apply_symm_apply] using hyS)
-    simp only [Equiv.apply_symm_apply] at hf hg
-    intro he
-    exact hyne (hf.trans (he.trans hg.symm))
-  have horb := orbit_subset_of_predecessor_closed p D hprev hxD
-  let C : RoutingNetwork.SelectedCycle p S :=
-    ⟨orbitSet p x,⟨x,rfl⟩,fun y hy => (Finset.mem_filter.mp (horb hy)).1⟩
-  have hrD : root C ∈ D := horb (root_mem C)
-  have hrne := (Finset.mem_filter.mp hrD).2
-  exact hrne (congrFun hfg (Sum.inr C))
-
-lemma card_coloring_le (p : Equiv.Perm α) (S : Finset α) :
-    Fintype.card (Coloring p S) ≤
-      2 ^ ((boundary p S).card + Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
-  have h := Fintype.card_le_of_injective (record (p := p) (S := S)) (record_injective p S)
-  simpa only [Fintype.card_fun,Fintype.card_sum,Fintype.card_coe,Fintype.card_bool] using h
-
-omit [Fintype α] in
-lemma boundary_card (p : Equiv.Perm α) (S : Finset α) :
-    (boundary p S).card = S.card - (S.filter (fun x => p.symm x ∈ S)).card := by
-  classical
-  have h := Finset.card_filter_add_card_filter_not (s := S) (fun x => p.symm x ∈ S)
-  unfold boundary
-  omega
-
-lemma card_coloring_le_edges (p : Equiv.Perm α) (S : Finset α) :
-    Fintype.card (Coloring p S) ≤
-      2 ^ (S.card - (S.filter (fun x => p.symm x ∈ S)).card +
-        Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
-  simpa only [boundary_card] using card_coloring_le p S
-
-
-abbrev OppositeColoring (p : Equiv.Perm α) (S : Finset α) :=
-  {f : α → Bool // (∀ x, x ∉ S → f x = false) ∧
-    ∀ x ∈ S, p x ∈ S → f (p x) = !(f x)}
-
-noncomputable instance (p : Equiv.Perm α) (S : Finset α) : Fintype (OppositeColoring p S) :=
-  Fintype.ofFinite _
-
-noncomputable def bipartitionFlip (p : Equiv.Perm α) (S : Finset α)
-    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x))
-    (f : OppositeColoring p S) : Coloring p S := by
-  classical
-  refine ⟨(fun x => if x ∈ S then Bool.xor (f.val x) (χ x) else false),?_,?_⟩
-  · intro x hx
-    exact ite_eq_right hx
-  · intro x hx hpx
-    simp only [ite_eq_left hx,ite_eq_left hpx,f.property.2 x hx hpx,hχ x]
-    cases f.val x <;> cases χ x <;> rfl
-
-end CycleColoring
-end Thorp
-
-end ThorpNine.Smoothing
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Smoothing.PriorPaths
+
+namespace OAI
+
+namespace ThorpNine.Smoothing
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+section ParallelButterfly
+variable {α B : Type*} [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] {d : ℕ}
+omit [DecidableEq B] in
+lemma parallel_entropy (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool) (S : Finset α) :
+    (∑ i : B × SwitchIndex d,
+      if (parallelUsers pre ω i).1 ∈ S ∧ (parallelUsers pre ω i).2 ∈ S then (1:ℝ) else 0) ≤
+      (S.card : ℝ) * Real.log S.card / (2 * Real.log 2) := by
+  classical
+  rw [Fintype.sum_prod_type]
+  calc
+    _ ≤ ∑ b : B, (blockMarks pre S b).card * Real.log (blockMarks pre S b).card / (2*Real.log 2) := by
+      apply Finset.sum_le_sum
+      intro b _
+      simpa only [←mem_blockMarks,parallelUsers] using
+        marked_switch_entropy_bound d (decodeButterfly d (fun i => ω (b,i))) (blockMarks pre S b)
+    _ ≤ ∑ b : B, (blockMarks pre S b).card * Real.log S.card / (2*Real.log 2) := by
+      apply Finset.sum_le_sum
+      intro b _
+      by_cases hz : (blockMarks pre S b).card = 0
+      · simp [hz]
+      · apply div_le_div_of_nonneg_right _ (by positivity)
+        apply mul_le_mul_of_nonneg_left _ (by positivity)
+        exact Real.log_le_log (by exact_mod_cast Nat.pos_of_ne_zero hz)
+          (by exact_mod_cast card_blockMarks_le pre S b)
+    _ = _ := by rw [←Finset.sum_div,←Finset.sum_mul,←Nat.cast_sum,sum_card_blockMarks]
+
+noncomputable def parallelFollow (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x : α) (k : α → QueryTree (B × SwitchIndex d)) : QueryTree (B × SwitchIndex d) :=
+  QueryTree.followPath d (fun i => ((pre x).1,i)) (pre x).2 (fun y => k (post ((pre x).1,y)))
+
+omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
+lemma parallelFollow_succeeds (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
+    (parallelFollow pre post x k).succeeds ω = (k (relabeledPerm pre post ω x)).succeeds ω := by
+  exact QueryTree.followPath_succeeds _ _ _ _ _
+
+omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
+lemma parallelFollow_raw_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
+    (parallelFollow pre post x k).charge ω = (k (relabeledPerm pre post ω x)).charge ω := by
+  exact QueryTree.followPath_raw_charge _ _ _ _ _
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma parallelFollow_stale_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
+    (ω : B × SwitchIndex d → Bool) :
+    (parallelFollow pre post x k).staleCharge R ω =
+      (k (relabeledPerm pre post ω x)).staleCharge (R ∪ parallelPath pre ω x) ω := by
+  exact QueryTree.followPath_stale_charge _ _ _ _ _ _
+
+noncomputable def closingDomain (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) : Finset (B × SwitchIndex d) :=
+  (routeDomain d (pre x).2 (post.symm y).2).image (fun i => ((pre x).1,i))
+
+def closingValue (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (i : B × SwitchIndex d) : Bool := routeValue d (pre x).2 (post.symm y).2 i.2
+
+omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
+lemma relabeled_endpoint_pair (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (ω : B × SwitchIndex d → Bool) (x y : α) :
+    relabeledPerm pre post ω x = y ↔
+      (pre x).1 = (post.symm y).1 ∧
+      butterflyPerm d (decodeButterfly d (fun i => ω ((pre x).1,i))) (pre x).2 = (post.symm y).2 := by
+  change post ((pre x).1,_) = y ↔ _
+  rw [←post.eq_symm_apply,Prod.mk.injEq]
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma relabeled_endpoint_iff (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (ω : B × SwitchIndex d → Bool) (x y : α) :
+    relabeledPerm pre post ω x = y ↔
+      (pre x).1 = (post.symm y).1 ∧
+        ∀ i ∈ closingDomain pre post x y, ω i = closingValue pre post x y i := by
+  classical
+  rw [relabeled_endpoint_pair,butterfly_endpoint_iff]
+  simp only [closingDomain,Finset.forall_mem_image,closingValue]
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma closingDomain_eq (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (ω : B × SwitchIndex d → Bool) (x y : α) (h : relabeledPerm pre post ω x = y) :
+    closingDomain pre post x y = parallelPath pre ω x := by
+  rw [closingDomain,parallelPath,(relabeled_endpoint_pair _ _ _ _ _).mp h |>.2]
+
+noncomputable def parallelClose (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (k : QueryTree (B × SwitchIndex d)) : QueryTree (B × SwitchIndex d) :=
+  if (pre x).1 = (post.symm y).1 then
+    QueryTree.forceList (closingDomain pre post x y).toList (closingValue pre post x y) k
+  else QueryTree.reject
+
+omit [Fintype α] in
+lemma parallelClose_succeeds (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (k : QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
+    (parallelClose pre post x y k).succeeds ω =
+      if relabeledPerm pre post ω x = y then k.succeeds ω else false := by
+  classical
+  simp only [relabeled_endpoint_iff,parallelClose]
+  by_cases hb : (pre x).1 = (post.symm y).1
+  · rw [ite_eq_left hb,QueryTree.forceList_succeeds]
+    simp only [Finset.mem_toList,hb,true_and]
+  · simp [hb,QueryTree.succeeds]
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma parallelClose_raw_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (k : QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool)
+    (h : relabeledPerm pre post ω x = y) :
+    (parallelClose pre post x y k).charge ω = k.charge ω + d := by
+  have hh := (relabeled_endpoint_iff _ _ _ _ _).mp h
+  rw [parallelClose,ite_eq_left hh.1,QueryTree.forceList_raw_charge _ _ _ _
+    (by simpa only [Finset.mem_toList] using hh.2),Finset.length_toList,
+    closingDomain_eq _ _ _ _ _ h,card_parallelPath]
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma parallelClose_fresh_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (k : QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
+    (ω : B × SwitchIndex d → Bool) (h : relabeledPerm pre post ω x = y) :
+    (parallelClose pre post x y k).freshCharge R ω =
+      k.freshCharge (R ∪ parallelPath pre ω x) ω + (parallelPath pre ω x \ R).card := by
+  have hh := (relabeled_endpoint_iff _ _ _ _ _).mp h
+  rw [parallelClose,ite_eq_left hh.1,QueryTree.forceList_charge _ _ _ _ _
+    (by simpa only [Finset.mem_toList] using hh.2),Finset.toList_toFinset,
+    closingDomain_eq _ _ _ _ _ h]
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma parallelClose_stale_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (k : QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
+    (ω : B × SwitchIndex d → Bool) (h : relabeledPerm pre post ω x = y) :
+    (parallelClose pre post x y k).staleCharge R ω =
+      k.staleCharge (R ∪ parallelPath pre ω x) ω + (parallelPath pre ω x ∩ R).card := by
+  have h₀ := QueryTree.fresh_add_stale (parallelClose pre post x y k) R ω
+  have h₁ := QueryTree.fresh_add_stale k (R ∪ parallelPath pre ω x) ω
+  rw [parallelClose_raw_charge _ _ _ _ _ _ h,parallelClose_fresh_charge _ _ _ _ _ _ _ h] at h₀
+  have h₂ : (parallelPath pre ω x \ R).card + (parallelPath pre ω x ∩ R).card = d := by
+    rw [Finset.card_sdiff_add_card_inter,card_parallelPath]
+  omega
+
+noncomputable def parallelNetwork (pre : α ≃ B × Card d) (post : B × Card d ≃ α) :
+    RoutingNetwork α (B × SwitchIndex d) d where
+  perm := relabeledPerm pre post
+  path := parallelPath pre
+  users := parallelUsers pre
+  users_ne := parallelUsers_ne pre
+  mem_path := mem_parallelPath pre
+  card_path := card_parallelPath pre
+  entropy := parallel_entropy pre
+  follow := parallelFollow pre post
+  close := parallelClose pre post
+  follow_succeeds := parallelFollow_succeeds pre post
+  follow_raw_charge := parallelFollow_raw_charge pre post
+  follow_stale_charge := parallelFollow_stale_charge pre post
+  close_succeeds := parallelClose_succeeds pre post
+  close_raw_charge := parallelClose_raw_charge pre post
+  close_stale_charge := parallelClose_stale_charge pre post
+
+lemma parallel_cycle_log_mgf (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (S : Finset α) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    Real.log (finiteMean (fun ω : B × SwitchIndex d → Bool =>
+      q^(Fintype.card (RoutingNetwork.SelectedCycle (relabeledPerm pre post ω) S)))) ≤
+      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
+  exact (parallelNetwork pre post).butterfly_cycle_log_mgf S J hJ q hq
+
+end ParallelButterfly
+
+def cardSplit : (r s : ℕ) → Card (s+r) ≃ Card r × Card s
+  | 0, s =>
+    { toFun := fun x => ((fun i => Fin.elim0 i), x)
+      invFun := fun x => x.2
+      left_inv := fun _ => rfl
+      right_inv := fun x => by
+        apply Prod.ext
+        · exact Subsingleton.elim _ _
+        · rfl }
+  | r+1, s =>
+    { toFun := fun x => (Fin.cons (x 0) ((cardSplit r s (Fin.tail x)).1),
+        (cardSplit r s (Fin.tail x)).2)
+      invFun := fun x => Fin.cons (x.1 0) ((cardSplit r s).symm (Fin.tail x.1,x.2))
+      left_inv := by
+        intro x
+        simp only [Fin.cons_zero, Fin.tail_cons, Prod.mk.eta, Equiv.symm_apply_apply, Fin.cons_self_tail]
+      right_inv := by
+        intro x
+        simp only [Fin.cons_zero, Fin.tail_cons, Equiv.apply_symm_apply, Fin.cons_self_tail] }
+
+def assembleBits : (r s : ℕ) → (Card r × SwitchIndex s → Bool) →
+    (Card s × SwitchIndex r → Bool) → SwitchIndex (s+r) → Bool
+  | 0, _, lo, _, i => lo ((fun j => Fin.elim0 j), i)
+  | r+1, s, _, hi, Sum.inl y => hi ((cardSplit r s y).2,Sum.inl (cardSplit r s y).1)
+  | r+1, s, lo, hi, Sum.inr (b,i) => assembleBits r s
+      (fun j => lo (Fin.cons b j.1,j.2)) (fun j => hi (j.1,Sum.inr (b,j.2))) i
+
+lemma butterfly_split (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
+    (hi : Card s × SwitchIndex r → Bool) (x : Card (s+r)) :
+    cardSplit r s (butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi)) x) =
+      let y := cardSplit r s x
+      let z := butterflyPerm s (decodeButterfly s (fun i => lo (y.1,i))) y.2
+      (butterflyPerm r (decodeButterfly r (fun i => hi (z,i))) y.1,z) := by
+  induction r with
+  | zero =>
+    apply Prod.ext
+    · funext i
+      exact Fin.elim0 i
+    · rfl
+  | succ r ih =>
+    let lo' : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons (x 0) j.1,j.2)
+    let hi' : Card s × SwitchIndex r → Bool := fun j => hi (j.1,Sum.inr (x 0,j.2))
+    have hh := ih lo' hi' (Fin.tail x)
+    let u := butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo' hi')) (Fin.tail x)
+    let y := cardSplit r s (Fin.tail x)
+    let z := butterflyPerm s (decodeButterfly s (fun i => lo' (y.1,i))) y.2
+    let w := butterflyPerm r (decodeButterfly r (fun i => hi' (z,i))) y.1
+    change cardSplit r s u = (w,z) at hh
+    change ((Fin.cons (Bool.xor (x 0) (hi ((cardSplit r s u).2,Sum.inl (cardSplit r s u).1)))
+        ((cardSplit r s u).1) : Card (r+1)), (cardSplit r s u).2) =
+      ((Fin.cons (Bool.xor (x 0) (hi (z,Sum.inl w))) w : Card (r+1)),z)
+    rw [hh]
+
+def splitSwitch : (r s : ℕ) → SwitchIndex (s+r) ≃
+    (Card r × SwitchIndex s) ⊕ (Card s × SwitchIndex r)
+  | 0, _ =>
+    { toFun := fun i => Sum.inl ((fun j => Fin.elim0 j),i)
+      invFun := fun i => match i with
+        | Sum.inl (_,j) => j
+        | Sum.inr (_,j) => nomatch j
+      left_inv := fun _ => rfl
+      right_inv := by
+        intro i
+        rcases i with ⟨x,j⟩ | ⟨x,j⟩
+        · exact congrArg Sum.inl (Prod.ext (Subsingleton.elim _ _) rfl)
+        · exact Empty.elim j }
+  | r+1, s =>
+    { toFun := fun i => match i with
+        | Sum.inl y => Sum.inr ((cardSplit r s y).2,Sum.inl (cardSplit r s y).1)
+        | Sum.inr (b,j) => match splitSwitch r s j with
+          | Sum.inl (p,j) => Sum.inl (Fin.cons b p,j)
+          | Sum.inr (q,j) => Sum.inr (q,Sum.inr (b,j))
+      invFun := fun i => match i with
+        | Sum.inl (p,j) => Sum.inr (p 0,(splitSwitch r s).symm (Sum.inl (Fin.tail p,j)))
+        | Sum.inr (q,Sum.inl p) => Sum.inl ((cardSplit r s).symm (p,q))
+        | Sum.inr (q,Sum.inr (b,j)) => Sum.inr (b,(splitSwitch r s).symm (Sum.inr (q,j)))
+      left_inv := by
+        intro i
+        rcases i with y | ⟨b,j⟩
+        · dsimp only
+          rw [Prod.mk.eta, Equiv.symm_apply_apply]
+        · dsimp only
+          cases hz : splitSwitch r s j with
+          | inl z =>
+            rcases z with ⟨p,k⟩
+            simp only [Fin.cons_zero, Fin.tail_cons]
+            rw [←hz,Equiv.symm_apply_apply]
+          | inr z =>
+            rcases z with ⟨q,k⟩
+            dsimp only
+            rw [←hz,Equiv.symm_apply_apply]
+      right_inv := by
+        intro i
+        rcases i with ⟨p,j⟩ | ⟨q,j⟩
+        · simp only [Equiv.apply_symm_apply, Fin.cons_self_tail]
+        · rcases j with p | ⟨b,j⟩ <;> simp only [Equiv.apply_symm_apply] }
+
+lemma assembleBits_eq (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
+    (hi : Card s × SwitchIndex r → Bool) (i : SwitchIndex (s+r)) :
+    assembleBits r s lo hi i = Sum.elim lo hi (splitSwitch r s i) := by
+  induction r with
+  | zero => rfl
+  | succ r ih =>
+    rcases i with y | ⟨b,j⟩
+    · rfl
+    · simp only [assembleBits,splitSwitch,Equiv.coe_fn_mk]
+      rw [ih]
+      cases splitSwitch r s j <;> rfl
+
+def splitCoinEquiv (r s : ℕ) :
+    ((Card r × SwitchIndex s → Bool) × (Card s × SwitchIndex r → Bool)) ≃
+      (SwitchIndex (s+r) → Bool) :=
+  (Equiv.sumArrowEquivProdArrow _ _ Bool).symm.trans
+    (Equiv.arrowCongr (splitSwitch r s).symm (Equiv.refl Bool))
+
+lemma splitCoinEquiv_apply (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
+    (hi : Card s × SwitchIndex r → Bool) :
+    splitCoinEquiv r s (lo,hi) = assembleBits r s lo hi := by
+  funext i
+  exact (assembleBits_eq r s lo hi i).symm
+
+lemma finiteMean_prod {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] (f : Ω × Ω' → ℝ) :
+    finiteMean f = finiteMean (fun x => finiteMean (fun y => f (x,y))) := by
+  simp only [finiteMean, Fintype.sum_prod_type, Fintype.card_prod, Nat.cast_mul,
+    ←Finset.sum_div, div_div, mul_comm]
+
+lemma finiteMean_split (r s : ℕ) (f : (SwitchIndex (s+r) → Bool) → ℝ) :
+    finiteMean f = finiteMean (fun lo : Card r × SwitchIndex s → Bool =>
+      finiteMean (fun hi : Card s × SwitchIndex r → Bool => f (assembleBits r s lo hi))) := by
+  rw [finiteMean_equiv (splitCoinEquiv r s).symm]
+  simp only [Equiv.symm_symm]
+  rw [finiteMean_prod]
+  apply finiteMean_congr
+  intro lo
+  apply finiteMean_congr
+  intro hi
+  rw [splitCoinEquiv_apply]
+
+section PartiallyExposed
+variable {α : Type*} {r s : ℕ}
+
+def freshPre (pre : α ≃ Card (s+r)) (lo : Card r × SwitchIndex s → Bool) :
+    α ≃ Card s × Card r :=
+  ((pre.trans (cardSplit r s)).trans (parallelPerm lo)).trans (Equiv.prodComm _ _)
+
+def freshPost (post : Card (s+r) ≃ α) : Card s × Card r ≃ α :=
+  ((Equiv.prodComm _ _).trans (cardSplit r s).symm).trans post
+
+lemma exposed_perm_eq (pre : α ≃ Card (s+r)) (post : Card (s+r) ≃ α)
+    (lo : Card r × SwitchIndex s → Bool) (hi : Card s × SwitchIndex r → Bool) :
+    (pre.trans (butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi)))).trans post =
+      relabeledPerm (freshPre pre lo) (freshPost post) hi := by
+  ext x
+  simpa only [relabeledPerm, freshPre, freshPost, parallelPerm, Equiv.trans_apply,
+    Equiv.coe_fn_mk, Equiv.prodComm_apply, Prod.swap, Equiv.symm_apply_apply] using
+    congrArg (fun y => post ((cardSplit r s).symm y)) (butterfly_split r s lo hi (pre x))
+
+lemma partially_exposed_cycle_log_mgf [Fintype α] [DecidableEq α]
+    (pre : α ≃ Card (s+r)) (post : Card (s+r) ≃ α)
+    (lo : Card r × SwitchIndex s → Bool) (S : Finset α)
+    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    Real.log (finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(Fintype.card (RoutingNetwork.SelectedCycle
+        ((pre.trans (butterflyPerm (s+r)
+          (decodeButterfly (s+r) (assembleBits r s lo hi)))).trans post) S)))) ≤
+      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^r) := by
+  simp only [exposed_perm_eq]
+  exact parallel_cycle_log_mgf (freshPre pre lo) (freshPost post) S J hJ q hq
+
+end PartiallyExposed
+
+lemma inclusionMoment_eq {α Ω : Type*} [DecidableEq α] [Fintype Ω]
+    (F : Ω → Finset α) (S : Finset α) :
+    inclusionMoment F S = finiteMean (fun ω => if S ⊆ F ω then (1:ℝ) else 0) := by
+  unfold inclusionMoment
+  apply finiteMean_congr
+  intro ω
+  split_ifs <;> rfl
+
+lemma finiteMean_bool (f : Bool → ℝ) : finiteMean f = (f false+f true)/2 := by
+  simp only [finiteMean, Fintype.sum_bool, Fintype.card_bool, Nat.cast_ofNat]
+  ring
+
+lemma finiteMean_const {Ω : Type*} [Fintype Ω] [Nonempty Ω] (c : ℝ) :
+    finiteMean (fun _ : Ω => c) = c := by
+  simp only [finiteMean,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
+  exact mul_div_cancel_left₀ c (Nat.cast_ne_zero.mpr Fintype.card_ne_zero)
+
+lemma finiteMean_pi_prod {ι : Type*} [Fintype ι] [DecidableEq ι] {Ω : ι → Type*}
+    [∀ i, Fintype (Ω i)] (f : (i : ι) → Ω i → ℝ) :
+    finiteMean (fun ω : (i : ι) → Ω i => ∏ i, f i (ω i)) = ∏ i, finiteMean (f i) := by
+  simp only [finiteMean,←Fintype.prod_sum,Fintype.card_pi,Nat.cast_prod,Finset.prod_div_distrib]
+
+section ParallelFairLayer
+variable {β : Type*} [Fintype β] [DecidableEq β]
+
+def pairLayer (ξ : β → Bool) : Equiv.Perm (Bool × β) where
+  toFun x := (Bool.xor x.1 (ξ x.2),x.2)
+  invFun x := (Bool.xor x.1 (ξ x.2),x.2)
+  left_inv x := by
+    rcases x with ⟨b,x⟩
+    change (Bool.xor (Bool.xor b (ξ x)) (ξ x),x) = (b,x)
+    cases b <;> cases ξ x <;> rfl
+  right_inv x := by
+    rcases x with ⟨b,x⟩
+    change (Bool.xor (Bool.xor b (ξ x)) (ξ x),x) = (b,x)
+    cases b <;> cases ξ x <;> rfl
+
+noncomputable def pairFiber (S : Finset (Bool × β)) (b : β) : Finset Bool :=
+  Finset.univ.filter (fun ε => (ε,b) ∈ S)
+
+lemma prod_pairFiber (S : Finset (Bool × β)) (f : Bool × β → ℝ) :
+    (∏ x ∈ S, f x) = ∏ b : β, ∏ ε ∈ pairFiber S b, f (ε,b) := by
+  classical
+  simp only [pairFiber,Finset.prod_filter]
+  rw [Finset.prod_comm,←Fintype.prod_prod_type (fun x : Bool × β => if x ∈ S then f x else 1)]
+  rw [←Finset.prod_filter]
+  congr 1
+  ext x
+  simp
+
+lemma fiber_average_bound (S : Finset Bool) (p : Bool → ℝ) (hp : ∀ b, 0 ≤ p b) :
+    finiteMean (fun c : Bool => ∏ ε ∈ S, p (Bool.xor ε c)) ≤
+      ∏ ε ∈ S, (p ε + p (!ε))/2 := by
+  rw [finiteMean_bool]
+  have hxor : ∀ ε : Bool, Bool.xor ε true = Equiv.swap false true ε := by decide
+  simp only [Bool.xor_false,hxor]
+  calc
+    _ ≤ ∏ ε ∈ S, (p ε+p (Equiv.swap false true ε))/2 :=
+      swap_average_product S p hp false true
+    _ = _ := Finset.prod_congr rfl (fun ε _ => by cases ε <;> rfl)
+
+lemma CylinderBound.fair_parallel {m : Finset (Bool × β) → ℝ} {p : Bool × β → ℝ}
+    (h : CylinderBound m p) (hp : ∀ x, 0 ≤ p x) :
+    CylinderBound (fun S => finiteMean (fun ξ : β → Bool => m (S.image (pairLayer ξ))))
+      (fun x => (p x+p (!x.1,x.2))/2) := by
+  intro S
+  calc
+    _ ≤ finiteMean (fun ξ : β → Bool => ∏ x ∈ S, p (pairLayer ξ x)) := by
+      apply finiteMean_mono
+      intro ξ
+      simpa only [Finset.prod_image (pairLayer ξ).injective.injOn] using h (S.image (pairLayer ξ))
+    _ = ∏ b : β, finiteMean (fun c : Bool => ∏ ε ∈ pairFiber S b, p (Bool.xor ε c,b)) := by
+      calc
+        _ = finiteMean (fun ξ : β → Bool => ∏ b : β,
+            ∏ ε ∈ pairFiber S b, p (Bool.xor ε (ξ b),b)) :=
+          finiteMean_congr (fun ξ => prod_pairFiber S (fun x => p (pairLayer ξ x)))
+        _ = _ := finiteMean_pi_prod (fun b c => ∏ ε ∈ pairFiber S b, p (Bool.xor ε c,b))
+    _ ≤ ∏ b : β, ∏ ε ∈ pairFiber S b, (p (ε,b)+p (!ε,b))/2 := by
+      apply Finset.prod_le_prod₀
+      · intro b _
+        exact div_nonneg (Finset.sum_nonneg (fun c _ =>
+          Finset.prod_nonneg (fun ε _ => hp (Bool.xor ε c,b)))) (Nat.cast_nonneg _)
+      · intro b _
+        exact fiber_average_bound (pairFiber S b) (fun ε => p (ε,b)) (fun ε => hp (ε,b))
+    _ = _ := (prod_pairFiber S (fun x => (p x+p (!x.1,x.2))/2)).symm
+
+omit [Fintype β] in
+lemma subset_pairLayer_image_iff (S A : Finset (Bool × β)) (ξ : β → Bool) :
+    S ⊆ A.image (pairLayer ξ) ↔ S.image (pairLayer ξ) ⊆ A := by
+  have hi (x : Bool × β) : pairLayer ξ (pairLayer ξ x) = x := (pairLayer ξ).left_inv x
+  constructor
+  · intro h
+    have h' := Finset.image_subset_image h (f := pairLayer ξ)
+    simpa only [Finset.image_image,Function.comp_def,hi,Finset.image_id'] using h'
+  · intro h
+    have h' := Finset.image_subset_image h (f := pairLayer ξ)
+    simpa only [Finset.image_image,Function.comp_def,hi,Finset.image_id'] using h'
+
+noncomputable def fairSetLayer {Ω : Type*} (F : Ω → Finset (Bool × β)) :
+    Ω × (β → Bool) → Finset (Bool × β) := fun ω => (F ω.1).image (pairLayer ω.2)
+
+lemma finiteMean_comm {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] (f : Ω → Ω' → ℝ) :
+    finiteMean (fun x => finiteMean (f x)) = finiteMean (fun y => finiteMean (fun x => f x y)) := by
+  simp only [finiteMean,←Finset.sum_div,div_div]
+  rw [Finset.sum_comm,mul_comm]
+
+lemma inclusionMoment_fairSetLayer {Ω : Type*} [Fintype Ω]
+    (F : Ω → Finset (Bool × β)) (S : Finset (Bool × β)) :
+    inclusionMoment (fairSetLayer F) S =
+      finiteMean (fun ξ : β → Bool => inclusionMoment F (S.image (pairLayer ξ))) := by
+  classical
+  simp only [inclusionMoment_eq]
+  have he : (fun ω : Ω × (β → Bool) => if S ⊆ fairSetLayer F ω then (1:ℝ) else 0) =
+      (fun ω : Ω × (β → Bool) => if S.image (pairLayer ω.2) ⊆ F ω.1 then (1:ℝ) else 0) := by
+    funext ω
+    simp only [fairSetLayer,subset_pairLayer_image_iff]
+  apply (congrArg finiteMean he).trans
+  exact (finiteMean_prod (fun ω : Ω × (β → Bool) =>
+    if S.image (pairLayer ω.2) ⊆ F ω.1 then (1:ℝ) else 0)).trans
+      (finiteMean_comm (fun ω ξ => if S.image (pairLayer ξ) ⊆ F ω then (1:ℝ) else 0))
+
+lemma inclusionMoment_fairSetLayer_singleton {Ω : Type*} [Fintype Ω]
+    (F : Ω → Finset (Bool × β)) (x : Bool × β) :
+    inclusionMoment (fairSetLayer F) {x} =
+      (inclusionMoment F {x}+inclusionMoment F {(!x.1,x.2)})/2 := by
+  rw [inclusionMoment_fairSetLayer]
+  simp only [Finset.image_singleton,pairLayer,Equiv.coe_fn_mk]
+  rw [finiteMean_pi_eval (A := fun _ : β => Bool) x.2 (fun c : Bool => inclusionMoment F {(Bool.xor x.1 c,x.2)})]
+  rw [finiteMean_bool]
+  rcases x with ⟨b,x⟩
+  cases b <;> simp only [Bool.xor_false,Bool.false_xor,Bool.true_xor, Bool.not_false,Bool.not_true]
+
+lemma cylinder_fairSetLayer {Ω : Type*} [Fintype Ω]
+    (F : Ω → Finset (Bool × β))
+    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
+    CylinderBound (inclusionMoment (fairSetLayer F))
+      (fun x => inclusionMoment (fairSetLayer F) {x}) := by
+  have hh := h.fair_parallel (fun x => inclusionMoment_nonneg F {x})
+  intro S
+  rw [inclusionMoment_fairSetLayer]
+  simp only [inclusionMoment_fairSetLayer_singleton]
+  exact hh S
+
+end ParallelFairLayer
+
+section RelabelMarks
+variable {α β Ω Ω' : Type*} [DecidableEq α] [DecidableEq β] [Fintype Ω] [Fintype Ω']
+
+omit [Fintype Ω] [Fintype Ω'] in
+lemma subset_image_equiv_iff (e : α ≃ β) (A : Finset α) (S : Finset β) :
+    S ⊆ A.image e ↔ S.image e.symm ⊆ A := by
+  constructor
+  · intro h x hx
+    obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
+    obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp (h hy)
+    simpa only [←hxy,Equiv.symm_apply_apply] using hx
+  · intro h y hy
+    exact Finset.mem_image.mpr ⟨e.symm y,h (Finset.mem_image_of_mem _ hy),e.apply_symm_apply y⟩
+
+omit [Fintype Ω'] in
+lemma inclusionMoment_image_equiv (e : α ≃ β) (F : Ω → Finset α) (S : Finset β) :
+    inclusionMoment (fun ω => (F ω).image e) S = inclusionMoment F (S.image e.symm) := by
+  classical
+  simp only [inclusionMoment_eq]
+  apply finiteMean_congr
+  intro ω
+  simp only [subset_image_equiv_iff]
+
+omit [Fintype Ω'] in
+lemma cylinder_image_equiv (e : α ≃ β) (F : Ω → Finset α)
+    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
+    CylinderBound (inclusionMoment (fun ω => (F ω).image e))
+      (fun x => inclusionMoment (fun ω => (F ω).image e) {x}) := by
+  intro S
+  simp only [inclusionMoment_image_equiv,Finset.image_singleton]
+  have hh := h (S.image e.symm)
+  simpa only [Finset.prod_image e.symm.injective.injOn] using hh
+
+omit [DecidableEq β] in
+lemma inclusionMoment_comp_equiv (e : Ω ≃ Ω') (F : Ω' → Finset α) (S : Finset α) :
+    inclusionMoment (fun ω => F (e ω)) S = inclusionMoment F S := by
+  classical
+  simp only [inclusionMoment_eq]
+  simpa only [Equiv.apply_symm_apply] using
+    finiteMean_equiv e (fun ω => if S ⊆ F (e ω) then (1 : ℝ) else 0)
+
+omit [DecidableEq β] in
+lemma cylinder_comp_equiv (e : Ω ≃ Ω') (F : Ω' → Finset α)
+    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
+    CylinderBound (inclusionMoment (fun ω => F (e ω)))
+      (fun x => inclusionMoment (fun ω => F (e ω)) {x}) := by
+  intro S
+  simpa only [inclusionMoment_comp_equiv] using h S
+end RelabelMarks
+
+lemma finiteMean_prod_mul {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
+    (f : Ω → ℝ) (g : Ω' → ℝ) :
+    finiteMean (fun ω : Ω × Ω' => f ω.1*g ω.2) = finiteMean f * finiteMean g := by
+  simp only [finiteMean,Fintype.sum_prod_type,Fintype.card_prod,Nat.cast_mul,
+    ←Finset.mul_sum,←Finset.sum_mul]
+  ring
+
+section JoinChildren
+variable {β : Type*} [DecidableEq β]
+
+noncomputable def childJoin (A B : Finset β) : Finset (Bool × β) :=
+  A.image (fun x => (false,x)) ∪ B.image (fun x => (true,x))
+
+lemma mem_childJoin (A B : Finset β) (b : Bool) (x : β) :
+    (b,x) ∈ childJoin A B ↔ (if b then x ∈ B else x ∈ A) := by
+  cases b <;> simp [childJoin]
+
+noncomputable def boolFiber (S : Finset (Bool × β)) (b : Bool) : Finset β :=
+  (S.filter (fun x => x.1 = b)).image Prod.snd
+
+lemma mem_boolFiber (S : Finset (Bool × β)) (b : Bool) (x : β) :
+    x ∈ boolFiber S b ↔ (b,x) ∈ S := by
+  constructor
+  · intro h
+    obtain ⟨⟨c,y⟩,hy,hxy⟩ := Finset.mem_image.mp h
+    obtain ⟨hy,hc⟩ := Finset.mem_filter.mp hy
+    change c = b at hc
+    change y = x at hxy
+    simpa only [hc,hxy] using hy
+  · intro h
+    exact Finset.mem_image.mpr ⟨(b,x),Finset.mem_filter.mpr ⟨h,rfl⟩,rfl⟩
+
+lemma childJoin_boolFiber (S : Finset (Bool × β)) :
+    childJoin (boolFiber S false) (boolFiber S true) = S := by
+  ext ⟨b,x⟩
+  rw [mem_childJoin]
+  cases b <;> simp only [Bool.false_eq_true,↓reduceIte,mem_boolFiber]
+
+lemma subset_childJoin_iff (S : Finset (Bool × β)) (A B : Finset β) :
+    S ⊆ childJoin A B ↔ boolFiber S false ⊆ A ∧ boolFiber S true ⊆ B := by
+  constructor
+  · intro h
+    constructor
+    · intro x hx
+      exact (mem_childJoin A B false x).mp (h ((mem_boolFiber S false x).mp hx))
+    · intro x hx
+      exact (mem_childJoin A B true x).mp (h ((mem_boolFiber S true x).mp hx))
+  · rintro ⟨h₀,h₁⟩ ⟨b,x⟩ hx
+    rw [mem_childJoin]
+    cases b
+    · exact h₀ ((mem_boolFiber S false x).mpr hx)
+    · exact h₁ ((mem_boolFiber S true x).mpr hx)
+
+lemma prod_childJoin (A B : Finset β) (f : Bool × β → ℝ) :
+    (∏ x ∈ childJoin A B, f x) = (∏ x ∈ A, f (false,x)) * ∏ x ∈ B, f (true,x) := by
+  classical
+  have hd : Disjoint (A.image (fun x => (false,x))) (B.image (fun x => (true,x))) := by
+    apply Finset.disjoint_left.mpr
+    rintro x hx hy
+    obtain ⟨a,ha,rfl⟩ := Finset.mem_image.mp hx
+    obtain ⟨b,hb,hba⟩ := Finset.mem_image.mp hy
+    exact Bool.false_ne_true (congrArg Prod.fst hba).symm
+  rw [childJoin,Finset.prod_union hd,
+    Finset.prod_image (fun _ _ _ _ h => congrArg Prod.snd h),
+    Finset.prod_image (fun _ _ _ _ h => congrArg Prod.snd h)]
+
+lemma prod_boolFibers (S : Finset (Bool × β)) (f : Bool × β → ℝ) :
+    (∏ x ∈ S, f x) = (∏ x ∈ boolFiber S false, f (false,x)) *
+      ∏ x ∈ boolFiber S true, f (true,x) := by
+  rw [←prod_childJoin,childJoin_boolFiber]
+
+lemma inclusionMoment_childJoin {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
+    (F : Ω → Finset β) (G : Ω' → Finset β) (S : Finset (Bool × β)) :
+    inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) S =
+      inclusionMoment F (boolFiber S false) * inclusionMoment G (boolFiber S true) := by
+  classical
+  simp only [inclusionMoment_eq]
+  rw [←finiteMean_prod_mul]
+  apply finiteMean_congr
+  intro ω
+  simp only [subset_childJoin_iff]
+  split_ifs <;> simp_all
+
+lemma cylinder_childJoin {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] [Nonempty Ω] [Nonempty Ω']
+    (F : Ω → Finset β) (G : Ω' → Finset β)
+    (hF : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x}))
+    (hG : CylinderBound (inclusionMoment G) (fun x => inclusionMoment G {x})) :
+    CylinderBound (inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)))
+      (fun x => inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {x}) := by
+  intro S
+  rw [inclusionMoment_childJoin,prod_boolFibers]
+  have hf (b : Bool) (x : β) : boolFiber {(b,x)} b = {x} := by
+    ext y
+    simp only [mem_boolFiber,Finset.mem_singleton,Prod.mk.injEq,true_and]
+  have hn (b : Bool) (x : β) : boolFiber {(!b,x)} b = ∅ := by
+    ext y
+    cases b <;> simp [mem_boolFiber]
+  have h₀ (x : β) : inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {(false,x)} =
+      inclusionMoment F {x} * inclusionMoment G ∅ := by
+    rw [inclusionMoment_childJoin,hf]
+    have hh := hn true x
+    exact congrArg (fun T => inclusionMoment F {x} * inclusionMoment G T) hh
+  have h₁ (x : β) : inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {(true,x)} =
+      inclusionMoment F ∅ * inclusionMoment G {x} := by
+    rw [inclusionMoment_childJoin,hf]
+    have hh := hn false x
+    exact congrArg (fun T => inclusionMoment F T * inclusionMoment G {x}) hh
+  have emF : inclusionMoment F ∅ = 1 := by simp [inclusionMoment,finiteMean_const]
+  have emG : inclusionMoment G ∅ = 1 := by simp [inclusionMoment,finiteMean_const]
+  simp only [h₀,h₁,emF,emG,mul_one,one_mul]
+  exact mul_le_mul (hF _) (hG _) (inclusionMoment_nonneg _ _)
+    (Finset.prod_nonneg (fun x _ => inclusionMoment_nonneg F {x}))
+end JoinChildren
+
+def coinStepEquiv (d : ℕ) : (SwitchIndex (d+1) → Bool) ≃
+    ((SwitchIndex d → Bool) × (SwitchIndex d → Bool)) × (Card d → Bool) where
+  toFun ω := ((fun i => ω (Sum.inr (false,i)),fun i => ω (Sum.inr (true,i))),
+    fun x => ω (Sum.inl x))
+  invFun ω := Sum.elim ω.2 (fun bi => if bi.1 then ω.1.2 bi.2 else ω.1.1 bi.2)
+  left_inv ω := by
+    funext i
+    cases i with
+    | inl x => rfl
+    | inr bi => cases bi with | mk b i => cases b <;> rfl
+  right_inv ω := by
+    rcases ω with ⟨⟨ω₀,ω₁⟩,ξ⟩
+    rfl
+
+def headTailEquiv (d : ℕ) : Card (d+1) ≃ Bool × Card d :=
+  (Fin.consEquiv (fun _ : Fin (d+1) => Bool)).symm
+
+lemma childMarks_join (d : ℕ) (A : Finset (Card (d+1))) :
+    A = (childMarks A false).image (Fin.cons false) ∪
+      (childMarks A true).image (Fin.cons true) := by
+  classical
+  ext x
+  constructor
+  · intro hx
+    cases hb : x 0
+    · exact Finset.mem_union_left _ (Finset.mem_image.mpr
+        ⟨Fin.tail x,(mem_childMarks _ _ _).mpr (by simpa only [←hb,Fin.cons_self_tail] using hx),
+          by simp only [←hb,Fin.cons_self_tail]⟩)
+    · exact Finset.mem_union_right _ (Finset.mem_image.mpr
+        ⟨Fin.tail x,(mem_childMarks _ _ _).mpr (by simpa only [←hb,Fin.cons_self_tail] using hx),
+          by simp only [←hb,Fin.cons_self_tail]⟩)
+  · intro hx
+    rcases Finset.mem_union.mp hx with hx | hx
+    · obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
+      exact (mem_childMarks _ _ _).mp hy
+    · obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
+      exact (mem_childMarks _ _ _).mp hy
+
+noncomputable def butterflyMarks (d : ℕ) (A : Finset (Card d))
+    (ω : SwitchIndex d → Bool) : Finset (Card d) :=
+  A.image (butterflyPerm d (decodeButterfly d ω))
+
+lemma butterflyMarks_succ (d : ℕ) (A : Finset (Card (d+1))) (ω : SwitchIndex (d+1) → Bool) :
+    butterflyMarks (d+1) A ω =
+      (fairSetLayer (fun p : (SwitchIndex d → Bool) × (SwitchIndex d → Bool) =>
+        childJoin (butterflyMarks d (childMarks A false) p.1)
+          (butterflyMarks d (childMarks A true) p.2)) (coinStepEquiv d ω)).image
+            (headTailEquiv d).symm := by
+  classical
+  unfold butterflyMarks fairSetLayer childJoin
+  rw [Finset.image_union,Finset.image_union]
+  simp only [Finset.image_image]
+  conv_lhs => rw [childMarks_join d A]
+  rw [Finset.image_union]
+  simp only [Finset.image_image]
+  apply congrArg₂ (· ∪ ·)
+  · apply Finset.image_congr
+    intro x _
+    simp only [Function.comp_def,butterflyPerm,decodeButterfly,Equiv.Perm.coe_mul,
+      childLift,pairSwitch,switchFun,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
+      coinStepEquiv,headTailEquiv,Equiv.symm_symm,pairLayer]
+    rfl
+  · apply Finset.image_congr
+    intro x _
+    simp only [Function.comp_def,butterflyPerm,decodeButterfly,Equiv.Perm.coe_mul,
+      childLift,pairSwitch,switchFun,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
+      coinStepEquiv,headTailEquiv,Equiv.symm_symm,pairLayer]
+    rfl
+
+lemma butterflyMarks_cylinder (d : ℕ) (A : Finset (Card d)) :
+    CylinderBound (inclusionMoment (butterflyMarks d A))
+      (fun x => inclusionMoment (butterflyMarks d A) {x}) := by
+  induction d with
+  | zero =>
+    have he : butterflyMarks 0 A = fun _ : SwitchIndex 0 → Bool => A := by
+      funext ω
+      simp [butterflyMarks,butterflyPerm]
+    rw [he]
+    exact cylinder_const A
+  | succ d ih =>
+    have h := cylinder_childJoin (butterflyMarks d (childMarks A false))
+      (butterflyMarks d (childMarks A true)) (ih _) (ih _)
+    have hh := cylinder_image_equiv (headTailEquiv d).symm _ (cylinder_fairSetLayer _ h)
+    have hs := cylinder_comp_equiv (coinStepEquiv d) _ hh
+    simpa only [←butterflyMarks_succ] using hs
+
+lemma indicator_mem_image_perm {α : Type*} [DecidableEq α]
+    (A : Finset α) (p : Equiv.Perm α) (y : α) :
+    (if y ∈ A.image p then (1 : ℝ) else 0) = ∑ x ∈ A, if p x = y then 1 else 0 := by
+  classical
+  by_cases h : y ∈ A.image p
+  · obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp h
+    rw [ite_eq_left (Finset.mem_image.mpr ⟨x,hx,hxy⟩)]
+    symm
+    calc
+      _ = (if p x = y then (1:ℝ) else 0) := Finset.sum_eq_single_of_mem x hx (by
+        intro z _ hzx
+        exact ite_eq_right (fun hz => hzx (p.injective (hz.trans hxy.symm))))
+      _ = 1 := ite_eq_left hxy
+  · rw [ite_eq_right h]
+    symm
+    apply Finset.sum_eq_zero
+    intro x hx
+    exact ite_eq_right (fun hxy => h (Finset.mem_image.mpr ⟨x,hx,hxy⟩))
+
+lemma butterflyMarks_singleton (d : ℕ) (A : Finset (Card d)) (y : Card d) :
+    inclusionMoment (butterflyMarks d A) {y} = (A.card : ℝ)/(2:ℝ)^d := by
+  classical
+  unfold inclusionMoment
+  simp only [Finset.singleton_subset_iff,butterflyMarks,indicator_mem_image_perm]
+  rw [finiteMean_sum]
+  simp only [butterfly_single_card_uniform,Finset.sum_const,nsmul_eq_mul]
+  ring
+
+lemma butterflyMarks_inclusion (d : ℕ) (A S : Finset (Card d)) :
+    inclusionMoment (butterflyMarks d A) S ≤ ((A.card : ℝ)/(2:ℝ)^d)^S.card := by
+  have h := butterflyMarks_cylinder d A S
+  simpa only [butterflyMarks_singleton,Finset.prod_const] using h
+
+namespace CycleColoring
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+abbrev Coloring (p : Equiv.Perm α) (S : Finset α) :=
+  {f : α → Bool // (∀ x, x ∉ S → f x = false) ∧
+    ∀ x ∈ S, p x ∈ S → f (p x) = f x}
+
+noncomputable instance (p : Equiv.Perm α) (S : Finset α) : Fintype (Coloring p S) :=
+  Fintype.ofFinite _
+
+noncomputable def boundary (p : Equiv.Perm α) (S : Finset α) : Finset α :=
+  S.filter (fun x => p.symm x ∉ S)
+
+noncomputable def root {p : Equiv.Perm α} {S : Finset α}
+    (C : RoutingNetwork.SelectedCycle p S) : α := C.nonempty.choose
+
+lemma root_mem {p : Equiv.Perm α} {S : Finset α} (C : RoutingNetwork.SelectedCycle p S) :
+    root C ∈ C.val := C.nonempty.choose_spec
+
+abbrev Seed (p : Equiv.Perm α) (S : Finset α) :=
+  {x // x ∈ boundary p S} ⊕ RoutingNetwork.SelectedCycle p S
+
+noncomputable def record {p : Equiv.Perm α} {S : Finset α}
+    (f : Coloring p S) : Seed p S → Bool :=
+  Sum.elim (fun x => f.val x.val) (fun C => f.val (root C))
+
+lemma orbit_subset_of_predecessor_closed (p : Equiv.Perm α) (D : Finset α)
+    (hD : ∀ x ∈ D, p.symm x ∈ D) {x : α} (hx : x ∈ D) : orbitSet p x ⊆ D := by
+  intro y hy
+  have hh := ((mem_orbitSet p x y).mp hy).inv
+  obtain ⟨n,hn⟩ := hh.exists_nat_pow_eq
+  have hp (k : ℕ) : (p⁻¹ ^ k) x ∈ D := by
+    induction k with
+    | zero => simpa using hx
+    | succ k ih => simpa only [pow_succ',Equiv.Perm.mul_apply,Equiv.Perm.coe_inv] using hD _ ih
+  exact hn ▸ hp n
+
+lemma record_injective (p : Equiv.Perm α) (S : Finset α) :
+    Function.Injective (record (p := p) (S := S)) := by
+  classical
+  intro f g hfg
+  apply Subtype.ext
+  funext x
+  by_contra hne
+  have hx : x ∈ S := by
+    by_contra h
+    exact hne ((f.property.1 x h).trans (g.property.1 x h).symm)
+  let D := S.filter (fun y => f.val y ≠ g.val y)
+  have hxD : x ∈ D := Finset.mem_filter.mpr ⟨hx,hne⟩
+  have hprev (y : α) (hy : y ∈ D) : p.symm y ∈ D := by
+    obtain ⟨hyS,hyne⟩ := Finset.mem_filter.mp hy
+    have hpS : p.symm y ∈ S := by
+      by_contra hn
+      have hseed : y ∈ boundary p S := Finset.mem_filter.mpr ⟨hyS,hn⟩
+      exact hyne (congrFun hfg (Sum.inl ⟨y,hseed⟩))
+    refine Finset.mem_filter.mpr ⟨hpS,?_⟩
+    have hf := f.property.2 (p.symm y) hpS (by simpa only [Equiv.apply_symm_apply] using hyS)
+    have hg := g.property.2 (p.symm y) hpS (by simpa only [Equiv.apply_symm_apply] using hyS)
+    simp only [Equiv.apply_symm_apply] at hf hg
+    intro he
+    exact hyne (hf.trans (he.trans hg.symm))
+  have horb := orbit_subset_of_predecessor_closed p D hprev hxD
+  let C : RoutingNetwork.SelectedCycle p S :=
+    ⟨orbitSet p x,⟨x,rfl⟩,fun y hy => (Finset.mem_filter.mp (horb hy)).1⟩
+  have hrD : root C ∈ D := horb (root_mem C)
+  have hrne := (Finset.mem_filter.mp hrD).2
+  exact hrne (congrFun hfg (Sum.inr C))
+
+lemma card_coloring_le (p : Equiv.Perm α) (S : Finset α) :
+    Fintype.card (Coloring p S) ≤
+      2 ^ ((boundary p S).card + Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
+  have h := Fintype.card_le_of_injective (record (p := p) (S := S)) (record_injective p S)
+  simpa only [Fintype.card_fun,Fintype.card_sum,Fintype.card_coe,Fintype.card_bool] using h
+
+omit [Fintype α] in
+lemma boundary_card (p : Equiv.Perm α) (S : Finset α) :
+    (boundary p S).card = S.card - (S.filter (fun x => p.symm x ∈ S)).card := by
+  classical
+  have h := Finset.card_filter_add_card_filter_not (s := S) (fun x => p.symm x ∈ S)
+  unfold boundary
+  omega
+
+lemma card_coloring_le_edges (p : Equiv.Perm α) (S : Finset α) :
+    Fintype.card (Coloring p S) ≤
+      2 ^ (S.card - (S.filter (fun x => p.symm x ∈ S)).card +
+        Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
+  simpa only [boundary_card] using card_coloring_le p S
+
+
+abbrev OppositeColoring (p : Equiv.Perm α) (S : Finset α) :=
+  {f : α → Bool // (∀ x, x ∉ S → f x = false) ∧
+    ∀ x ∈ S, p x ∈ S → f (p x) = !(f x)}
+
+noncomputable instance (p : Equiv.Perm α) (S : Finset α) : Fintype (OppositeColoring p S) :=
+  Fintype.ofFinite _
+
+noncomputable def bipartitionFlip (p : Equiv.Perm α) (S : Finset α)
+    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x))
+    (f : OppositeColoring p S) : Coloring p S := by
+  classical
+  refine ⟨(fun x => if x ∈ S then Bool.xor (f.val x) (χ x) else false),?_,?_⟩
+  · intro x hx
+    exact ite_eq_right hx
+  · intro x hx hpx
+    simp only [ite_eq_left hx,ite_eq_left hpx,f.property.2 x hx hpx,hχ x]
+    cases f.val x <;> cases χ x <;> rfl
+
+end CycleColoring
+end Thorp
+
+end ThorpNine.Smoothing
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Smoothing/ParallelFreshPre.lean b/lean/OAI/Probability/ThorpRouting/Smoothing/ParallelFreshPre.lean
index f9c2cda61f1f0d5b8936e4d210297d7b731b4f7b..0e87038e88285a03b7e6437d850492c71b359bfc 100644
--- a/lean/OAI/Probability/ThorpRouting/Smoothing/ParallelFreshPre.lean
+++ b/lean/OAI/Probability/ThorpRouting/Smoothing/ParallelFreshPre.lean
@@ -1,848 +1,848 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Smoothing.BenesCoins
-
-namespace OAI
-
-namespace ThorpNine.Smoothing
-
-namespace Thorp
-
-lemma mem_centralBlock (r H : ℕ) (b : Card r) (x : Card (H+r)) :
-    x ∈ centralBlock r H b ↔ (cardSplit r H x).1 = b := by
-  classical
-  constructor
-  · intro h
-    obtain ⟨z,_,rfl⟩ := Finset.mem_image.mp h
-    simp only [Equiv.apply_symm_apply]
-  · intro h
-    apply Finset.mem_image.mpr
-    refine ⟨(cardSplit r H x).2,Finset.mem_univ _,?_⟩
-    rw [←h,Prod.mk.eta,Equiv.symm_apply_apply]
-
-lemma centralBlock_card (r H : ℕ) (b : Card r) : (centralBlock r H b).card = 2^H := by
-  classical
-  rw [centralBlock,Finset.card_image_of_injective]
-  · exact card_positions H
-  · intro x y h
-    exact congrArg Prod.snd ((cardSplit r H).symm.injective h)
-
-lemma centralBlock_disjoint (r H : ℕ) :
-    Pairwise (fun b b' => Disjoint (centralBlock r H b) (centralBlock r H b')) := by
-  classical
-  intro b b' h
-  apply Finset.disjoint_left.mpr
-  intro x hx hy
-  exact h (((mem_centralBlock r H b x).mp hx).symm.trans
-    ((mem_centralBlock r H b' x).mp hy))
-
-lemma centralBlock_zero (H : ℕ) (b : Card 0) : centralBlock 0 H b = Finset.univ := by
-  classical
-  ext x
-  simp only [mem_centralBlock,Finset.mem_univ,iff_true]
-  exact Subsingleton.elim _ _
-
-lemma centralBlock_step (r H : ℕ) (b : Bool) (c : Card r) :
-    centralBlock (r+1) H (Fin.cons b c) = (centralBlock r H c).image (Fin.cons b) := by
-  classical
-  simp only [centralBlock,Finset.image_image,cardSplit,Equiv.coe_fn_symm_mk,
-    Fin.cons_zero,Fin.tail_cons,Function.comp_def]
-
-lemma block_inter_step (r H : ℕ) (b : Bool) (c : Card r) (A : Finset (Card (H+(r+1)))) :
-    centralBlock (r+1) H (Fin.cons b c) ∩ A =
-      (centralBlock r H c ∩ childMarks A b).image (Fin.cons b) := by
-  classical
-  rw [centralBlock_step]
-  ext x
-  constructor
-  · intro hx
-    obtain ⟨hxB,hxA⟩ := Finset.mem_inter.mp hx
-    obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hxB
-    exact Finset.mem_image.mpr ⟨y,Finset.mem_inter.mpr ⟨hy,(mem_childMarks A b y).mpr hxA⟩,rfl⟩
-  · intro hx
-    obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
-    exact Finset.mem_inter.mpr ⟨Finset.mem_image.mpr ⟨y,(Finset.mem_inter.mp hy).1,rfl⟩,
-      (mem_childMarks A b y).mp (Finset.mem_inter.mp hy).2⟩
-
-lemma sparseMass_central_zero (H : ℕ) (A : Finset (Card H)) :
-    sparseMass (centralBlock 0 H) A = crowdedMass A.card := by
-  classical
-  simp only [sparseMass,centralBlock_zero,Finset.univ_inter,crowdedMass,
-    Finset.sum_const,Finset.card_univ,card_positions,pow_zero,one_nsmul]
-
-lemma sparseMass_central_step (r H : ℕ) (A : Finset (Card (H+(r+1)))) :
-    sparseMass (centralBlock (r+1) H) A =
-      sparseMass (centralBlock r H) (childMarks A false) +
-        sparseMass (centralBlock r H) (childMarks A true) := by
-  classical
-  unfold sparseMass
-  rw [←Equiv.sum_comp (headTailEquiv r).symm,Fintype.sum_prod_type]
-  have ht (b : Bool) (c : Card r) : (headTailEquiv r).symm (b,c) = Fin.cons b c := rfl
-  simp only [Fintype.sum_bool,ht]
-  have hc (b : Bool) (c : Card r) :
-      (centralBlock (r+1) H (Fin.cons b c) ∩ A).card =
-      (centralBlock r H c ∩ childMarks A b).card := by
-    rw [block_inter_step]
-    apply Finset.card_image_of_injective
-    intro x y h
-    simpa only [Fin.tail_cons] using congrArg Fin.tail h
-  simp only [hc]
-  exact add_comm _ _
-
-
-lemma inverseButterfly_coordinates_step (d : ℕ)
-    (c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (ξ : Card d → Bool)
-    (x : Card (d+1)) :
-    headTailEquiv d ((butterflyPerm (d+1)
-      (decodeButterfly (d+1) ((coinStepEquiv d).symm (c,ξ)))).symm x) =
-      (Bool.xor (x 0) (ξ (Fin.tail x)),
-        (butterflyPerm d (decodeButterfly d
-          (if Bool.xor (x 0) (ξ (Fin.tail x)) then c.2 else c.1))).symm (Fin.tail x)) := by
-  rw [butterflyPerm_step]
-  rfl
-
-lemma inverseButterflyMarks_child (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (d+1))
-    (c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (ξ : Card d → Bool) (b : Bool) :
-    childMarks (inverseButterflyMarks (d+1) (Finset.univ.image e)
-      ((coinStepEquiv d).symm (c,ξ))) b =
-      inverseButterflyMarks d
-        (Finset.univ.image (PairRouting.childEmbedding (e.trans (headTailEquiv d).toEmbedding)
-          (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) ξ)
-            (PairRouting.colors_compatible _ _) b)) (if b then c.2 else c.1) := by
-  classical
-  let x := e.trans (headTailEquiv d).toEmbedding
-  let col := PairRouting.colors x ξ
-  let child := PairRouting.childEmbedding x col (PairRouting.colors_compatible x ξ) b
-  have hp (i : ι) : headTailEquiv d ((butterflyPerm (d+1)
-      (decodeButterfly (d+1) ((coinStepEquiv d).symm (c,ξ)))).symm (e i)) =
-      (col i,(butterflyPerm d (decodeButterfly d (if col i then c.2 else c.1))).symm (x i).2) :=
-    inverseButterfly_coordinates_step d c ξ (e i)
-  ext y
-  simp only [mem_childMarks,inverseButterflyMarks,Finset.image_image,
-    Finset.mem_image,Finset.mem_univ,true_and,Function.comp_def]
-  constructor
-  · rintro ⟨i,hi⟩
-    have he := congrArg (headTailEquiv d) hi
-    rw [hp i] at he
-    change (col i,_) = (b,y) at he
-    have hb := congrArg Prod.fst he
-    have hy := congrArg Prod.snd he
-    dsimp only at hb hy
-    refine ⟨⟨i,hb⟩,?_⟩
-    change (butterflyPerm d (decodeButterfly d (if b then c.2 else c.1))).symm (x i).2 = y
-    simpa only [hb] using hy
-  · rintro ⟨i,hi⟩
-    refine ⟨i.val,?_⟩
-    apply (headTailEquiv d).injective
-    rw [hp]
-    change (col i.val,_) = (b,y)
-    have hb : col i.val = b := i.property
-    rw [hb]
-    exact Prod.ext rfl hi
-
-lemma inverseButterflyMarks_card (d : ℕ) (A : Finset (Card d)) (ω : SwitchIndex d → Bool) :
-    (inverseButterflyMarks d A ω).card = A.card := by
-  classical
-  exact Finset.card_image_of_injective _ (butterflyPerm d (decodeButterfly d ω)).symm.injective
-
-lemma benes_input_coordinates (d : ℕ) (ω : BenesCoins (d+1)) :
-    ω.2 = (coinStepEquiv d).symm
-      (((benesStepEquiv d ω).1.1.2,(benesStepEquiv d ω).1.2.2),(benesStepEquiv d ω).2.1) := by
-  change ω.2 = (coinStepEquiv d).symm (coinStepEquiv d ω.2)
-  exact ((coinStepEquiv d).symm_apply_apply ω.2).symm
-
-lemma cutoffMass_central (r H : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (H+r)) (ω : BenesCoins (H+r)) :
-    cutoffMass H (H+r) e ω =
-      sparseMass (centralBlock r H) (inverseButterflyMarks (H+r) (Finset.univ.image e) ω.2) := by
-  classical
-  induction r generalizing ι with
-  | zero =>
-    rw [sparseMass_central_zero,inverseButterflyMarks_card,
-      Finset.card_image_of_injective _ e.injective,Finset.card_univ]
-    cases H with
-    | zero =>
-      have hcard : Fintype.card ι ≤ 1 := by
-        have hh := Fintype.card_le_of_injective e e.injective
-        simpa only [Nat.add_zero,card_positions,pow_zero] using hh
-      simp [cutoffMass,crowdedMass,show ¬2≤Fintype.card ι by omega]
-    | succ H => simp only [cutoffMass,le_refl,ite_true]
-  | succ r ih =>
-    simp only [Nat.add_succ] at *
-    rw [cutoffMass,ite_eq_right (by omega : ¬H+r+1≤H)]
-    let σ := benesStepEquiv (H+r) ω
-    let x := e.trans (headTailEquiv (H+r)).toEmbedding
-    let col := PairRouting.colors x σ.2.1
-    let hc := PairRouting.colors_compatible x σ.2.1
-    change cutoffMass H (H+r) (PairRouting.childEmbedding x col hc false) σ.1.1 +
-      cutoffMass H (H+r) (PairRouting.childEmbedding x col hc true) σ.1.2 = _
-    rw [ih,ih,sparseMass_central_step,benes_input_coordinates (H+r) ω]
-    rw [inverseButterflyMarks_child,inverseButterflyMarks_child]
-    rfl
-
-
-theorem palindrome_low_mgf (r H : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (H+r)) (q : ℝ) (hq : 1 ≤ q)
-    (hr : (2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*q^H) ≤ 1/2) :
-    finiteMean (fun ω : BenesCoins (H+r) => q^(palindromeLowCost H (H+r) e ω)) ≤
-      Real.exp ((2:ℝ)^r * (2 * ((2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*q^H))^2)) := by
-  classical
-  have hec : (Finset.univ.image e).card = Fintype.card ι := by
-    rw [Finset.card_image_of_injective _ e.injective,Finset.card_univ]
-  have hm := inverseButterfly_sparse_mgf (H+r) (centralBlock r H)
-    (centralBlock_disjoint r H) (Finset.univ.image e) (q^H)
-    (pow_nonneg (by linarith) _) (fun b => by
-      simpa only [centralBlock_card,hec,Nat.cast_pow,Nat.cast_ofNat] using hr)
-  calc
-    _ ≤ finiteMean (fun ω : BenesCoins (H+r) => (q^H)^(cutoffMass H (H+r) e ω)) := by
-      apply finiteMean_mono
-      intro ω
-      rw [←pow_mul]
-      exact pow_le_pow_right₀ hq (palindromeLowCost_le_cutoff H (H+r) e ω)
-    _ = finiteMean (fun ω : SwitchIndex (H+r) → Bool =>
-        (q^H)^(sparseMass (centralBlock r H)
-          (inverseButterflyMarks (H+r) (Finset.univ.image e) ω))) := by
-      simp_rw [cutoffMass_central]
-      simp only [finiteMean_prod,finiteMean_const]
-    _ ≤ _ := by
-      simpa only [centralBlock_card,hec,Nat.cast_pow,Nat.cast_ofNat,Finset.sum_const,
-        Finset.card_univ,card_positions,nsmul_eq_mul] using hm
-
-variable {α B : Type*} {r s : ℕ}
-
-def parallelFreshPre (pre : α ≃ B × Card (s+r))
-    (lo : B → Card r × SwitchIndex s → Bool) : α ≃ (B × Card s) × Card r :=
-  ((pre.trans (Equiv.prodCongr (Equiv.refl B) (cardSplit r s))).trans
-    (Equiv.prodCongrRight (fun b =>
-      (parallelPerm (lo b)).trans (Equiv.prodComm _ _)))).trans
-    (Equiv.prodAssoc _ _ _).symm
-
-def parallelFreshPost (post : B × Card (s+r) ≃ α) : (B × Card s) × Card r ≃ α :=
-  (((Equiv.prodAssoc _ _ _).trans (Equiv.prodCongr (Equiv.refl B)
-    (Equiv.prodComm _ _))).trans (Equiv.prodCongr (Equiv.refl B) (cardSplit r s).symm)).trans post
-
-def parallelAssembled (lo : B → Card r × SwitchIndex s → Bool)
-    (hi : (B × Card s) × SwitchIndex r → Bool) : B × SwitchIndex (s+r) → Bool :=
-  fun i => assembleBits r s (lo i.1) (fun j => hi ((i.1,j.1),j.2)) i.2
-
-lemma parallel_exposed_perm (pre : α ≃ B × Card (s+r)) (post : B × Card (s+r) ≃ α)
-    (lo : B → Card r × SwitchIndex s → Bool)
-    (hi : (B × Card s) × SwitchIndex r → Bool) :
-    relabeledPerm pre post (parallelAssembled lo hi) =
-      relabeledPerm (parallelFreshPre pre lo) (parallelFreshPost post) hi := by
-  ext x
-  have h := butterfly_split r s (lo (pre x).1)
-    (fun j => hi (((pre x).1,j.1),j.2)) (pre x).2
-  have hh := congrArg (fun y => post ((pre x).1,(cardSplit r s).symm y)) h
-  simpa only [relabeledPerm,parallelAssembled,parallelPerm,parallelFreshPre,parallelFreshPost,
-    Equiv.trans_apply,Equiv.prodCongr_apply,Equiv.prodCongrRight,Equiv.coe_fn_mk,
-    Equiv.refl_apply,Equiv.prodComm_apply,Prod.swap,Equiv.prodAssoc_apply,
-    Equiv.prodAssoc_symm_apply,Prod.map,Equiv.symm_apply_apply] using hh
-
-theorem parallel_partially_exposed_log_mgf [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B]
-    (pre : α ≃ B × Card (s+r)) (post : B × Card (s+r) ≃ α)
-    (lo : B → Card r × SwitchIndex s → Bool) (S : Finset α)
-    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    Real.log (finiteMean (fun hi : (B × Card s) × SwitchIndex r → Bool =>
-      q^(Fintype.card (RoutingNetwork.SelectedCycle
-        (relabeledPerm pre post (parallelAssembled lo hi)) S)))) ≤
-      (S.card:ℝ)/J*Real.log q + 2*q*S.card*Real.sqrt J/Real.sqrt ((2:ℝ)^r) := by
-  simp_rw [parallel_exposed_perm]
-  exact parallel_cycle_log_mgf (parallelFreshPre pre lo) (parallelFreshPost post) S J hJ q hq
-
-end Thorp
-namespace Thorp
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-lemma orbitSet_inverse (p : Equiv.Perm α) (x : α) : orbitSet p⁻¹ x = orbitSet p x := by
-  ext y
-  simp only [mem_orbitSet,Equiv.Perm.sameCycle_inv]
-
-lemma selectedCycle_inverse (p : Equiv.Perm α) (S : Finset α) :
-    Fintype.card (RoutingNetwork.SelectedCycle p⁻¹ S) =
-      Fintype.card (RoutingNetwork.SelectedCycle p S) := by
-  apply Fintype.card_congr
-  refine { toFun := fun C => ⟨C.val,?_,C.property.2⟩
-           invFun := fun C => ⟨C.val,?_,C.property.2⟩
-           left_inv := fun _ => rfl
-           right_inv := fun _ => rfl }
-  · obtain ⟨x,hx⟩ := C.property.1
-    exact ⟨x,hx.trans (orbitSet_inverse p x)⟩
-  · obtain ⟨x,hx⟩ := C.property.1
-    exact ⟨x,hx.trans (orbitSet_inverse p x).symm⟩
-
-lemma orbitSet_conjugate (p g : Equiv.Perm α) (x : α) :
-    orbitSet (g*p*g⁻¹) (g x) = (orbitSet p x).image g := by
-  ext y
-  simp only [mem_orbitSet,Finset.mem_image]
-  rw [Equiv.Perm.sameCycle_conj]
-  change p.SameCycle (g.symm (g x)) (g.symm y) ↔ ∃ z, p.SameCycle x z ∧ g z = y
-  rw [Equiv.symm_apply_apply]
-  constructor
-  · intro h
-    exact ⟨g.symm y,h,g.apply_symm_apply y⟩
-  · rintro ⟨z,hz,rfl⟩
-    simpa only [Equiv.symm_apply_apply] using hz
-
-noncomputable def conjugateCycle (p g : Equiv.Perm α) (S : Finset α)
-    (C : RoutingNetwork.SelectedCycle p S) :
-    RoutingNetwork.SelectedCycle (g*p*g⁻¹) (S.image g) := by
-  refine ⟨C.val.image g,?_,Finset.image_subset_image C.property.2⟩
-  obtain ⟨x,hx⟩ := C.property.1
-  exact ⟨g x,by rw [hx,orbitSet_conjugate]⟩
-
-lemma conjugateCycle_injective (p g : Equiv.Perm α) (S : Finset α) :
-    Function.Injective (conjugateCycle p g S) := by
-  intro C D he
-  apply Subtype.ext
-  exact (Finset.image_injective g.injective) (congrArg Subtype.val he)
-
-lemma selectedCycle_conjugate (p g : Equiv.Perm α) (S : Finset α) :
-    Fintype.card (RoutingNetwork.SelectedCycle (g*p*g⁻¹) (S.image g)) =
-      Fintype.card (RoutingNetwork.SelectedCycle p S) := by
-  apply Nat.le_antisymm
-  · have h := Fintype.card_le_of_injective _
-      (conjugateCycle_injective (g*p*g⁻¹) g⁻¹ (S.image g))
-    have hp : g⁻¹*(g*p*g⁻¹)*g = p := by group
-    rw [inv_inv,hp] at h
-    have hs : (S.image g).image g.symm = S := by
-      ext x
-      simp only [Finset.mem_image]
-      constructor
-      · rintro ⟨y,⟨z,hz,rfl⟩,he⟩
-        have hz' : z = x := by simpa only [Equiv.symm_apply_apply] using he
-        exact hz' ▸ hz
-      · intro hx
-        exact ⟨g x,⟨x,hx,rfl⟩,g.symm_apply_apply x⟩
-    change _ ≤ Fintype.card (RoutingNetwork.SelectedCycle p ((S.image g).image g.symm)) at h
-    rw [hs] at h
-    exact h
-  · exact Fintype.card_le_of_injective _ (conjugateCycle_injective p g S)
-
-lemma alternatingCycles_le_projection {ι : Type*} [Fintype ι]
-    (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    PairRouting.alternatingCycles e p ≤
-      Fintype.card (RoutingNetwork.SelectedCycle (PairRouting.alternatingProjection p)
-        (boolFiber (PairRouting.labelSet e) false)) := by
-  have h := CycleColoring.card_alternating_cycles_le (PairRouting.alternatingProjection p)
-    (boolFiber (PairRouting.labelSet e) false) (boolFiber (PairRouting.labelSet e) true)
-  simpa only [PairRouting.alternatingCycles,childJoin_boolFiber] using h
-
-lemma projectionCycles_forward (X₀ X₁ Y₀ Y₁ : Equiv.Perm α) (S : Finset α) :
-    Fintype.card (RoutingNetwork.SelectedCycle
-      ((Y₀*X₀⁻¹)⁻¹*(Y₁*X₁⁻¹)) S) =
-    Fintype.card (RoutingNetwork.SelectedCycle
-      (X₀⁻¹*X₁*Y₁⁻¹*Y₀) (S.image X₀.symm)) := by
-  let p := (Y₀*X₀⁻¹)⁻¹*(Y₁*X₁⁻¹)
-  calc
-    _ = Fintype.card (RoutingNetwork.SelectedCycle (X₀⁻¹*p*X₀)
-        (S.image X₀.symm)) := by
-      change Fintype.card (RoutingNetwork.SelectedCycle p S) = _
-      have h := (selectedCycle_conjugate p X₀⁻¹ S).symm
-      rw [inv_inv] at h
-      exact h
-    _ = Fintype.card (RoutingNetwork.SelectedCycle (X₀⁻¹*p*X₀)⁻¹
-        (S.image X₀.symm)) := (selectedCycle_inverse _ _).symm
-    _ = _ := by
-      have he : (X₀⁻¹*p*X₀)⁻¹ = X₀⁻¹*X₁*Y₁⁻¹*Y₀ := by dsimp only [p]; group
-      rw [he]
-
-end Thorp
-namespace Thorp
-
-noncomputable def cycleCoefficient (r J : ℕ) (q : ℝ) : ℝ :=
-  Real.log q / J + 2*q*Real.sqrt J/Real.sqrt ((2:ℝ)^r)
-
-lemma cycleCoefficient_nonneg (r J : ℕ) (q : ℝ) (hq : 1 ≤ q) :
-    0 ≤ cycleCoefficient r J q := by
-  unfold cycleCoefficient
-  exact add_nonneg (div_nonneg (Real.log_nonneg hq) (Nat.cast_nonneg _))
-    (div_nonneg (mul_nonneg (by linarith) (Real.sqrt_nonneg _)) (Real.sqrt_nonneg _))
-
-lemma projectionSide_card_le {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι]
-    (e : ι ↪ Bool × α) : (boolFiber (PairRouting.labelSet e) false).card ≤ Fintype.card ι := by
-  have h := card_childJoin (boolFiber (PairRouting.labelSet e) false)
-    (boolFiber (PairRouting.labelSet e) true)
-  rw [childJoin_boolFiber,PairRouting.card_labelSet] at h
-  omega
-
-theorem node_partial_mgf (r s : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Bool × Card (s+r)) (X₀ X₁ Y₁ : Equiv.Perm (Card (s+r)))
-    (lo : Card r × SwitchIndex s → Bool) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(PairRouting.alternatingCycles e (fun b => if b then Y₁*X₁⁻¹ else
-        butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi))*X₀⁻¹))) ≤
-      Real.exp (Fintype.card ι * cycleCoefficient r J q) := by
-  classical
-  let S := boolFiber (PairRouting.labelSet e) false
-  let S' := S.image X₀.symm
-  let post := X₀⁻¹*X₁*Y₁⁻¹
-  let count := fun hi : Card s × SwitchIndex r → Bool =>
-    Fintype.card (RoutingNetwork.SelectedCycle
-      ((Equiv.refl _).trans (butterflyPerm (s+r)
-        (decodeButterfly (s+r) (assembleBits r s lo hi))) |>.trans post) S')
-  have hpoint (hi : Card s × SwitchIndex r → Bool) :
-      PairRouting.alternatingCycles e (fun b => if b then Y₁*X₁⁻¹ else
-        butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi))*X₀⁻¹) ≤ count hi := by
-    refine (alternatingCycles_le_projection e _).trans_eq ?_
-    simp only [PairRouting.alternatingProjection,Bool.false_eq_true,↓reduceIte]
-    change Fintype.card (RoutingNetwork.SelectedCycle
-      ((butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi))*X₀⁻¹)⁻¹ * (Y₁*X₁⁻¹)) S) = count hi
-    exact projectionCycles_forward X₀ X₁ _ Y₁ S
-  have hm := partially_exposed_cycle_log_mgf (Equiv.refl (Card (s+r))) post lo S' J hJ q hq
-  have hp : 0 < finiteMean (fun hi : Card s × SwitchIndex r → Bool => q^(count hi)) :=
-    finiteMean_pos (fun _ => pow_pos (by linarith) _)
-  have hb : finiteMean (fun hi : Card s × SwitchIndex r → Bool => q^(count hi)) ≤
-      Real.exp (S'.card * cycleCoefficient r J q) := by
-    rw [←Real.exp_log hp]
-    apply Real.exp_le_exp.mpr
-    calc
-      _ ≤ _ := hm
-      _ = _ := by unfold cycleCoefficient; ring
-  have hcard : S'.card ≤ Fintype.card ι := by
-    rw [show S'.card = S.card from Finset.card_image_of_injective _ X₀.symm.injective]
-    exact projectionSide_card_le e
-  calc
-    _ ≤ finiteMean (fun hi : Card s × SwitchIndex r → Bool => q^(count hi)) :=
-      finiteMean_mono (fun hi => pow_le_pow_right₀ hq (hpoint hi))
-    _ ≤ Real.exp (S'.card * cycleCoefficient r J q) := hb
-    _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right
-      (by exact_mod_cast hcard) (cycleCoefficient_nonneg r J q hq))
-
-
-def highStepEquiv (r s : ℕ) : (Card s × SwitchIndex (r+1) → Bool) ≃
-    (((Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool)) ×
-      (Card s × Card r → Bool)) where
-  toFun hi := ((fun j => hi (j.1,Sum.inr (false,j.2)),
-    fun j => hi (j.1,Sum.inr (true,j.2))),fun j => hi (j.1,Sum.inl j.2))
-  invFun c := fun j => match j.2 with
-    | Sum.inl y => c.2 (j.1,y)
-    | Sum.inr (b,i) => if b then c.1.2 (j.1,i) else c.1.1 (j.1,i)
-  left_inv hi := by
-    funext j
-    rcases j with ⟨z,y|⟨b,i⟩⟩
-    · rfl
-    · cases b <;> rfl
-  right_inv c := by
-    rcases c with ⟨⟨h₀,h₁⟩,η⟩
-    rfl
-
-lemma assembleBits_step (r s : ℕ) (lo : Card (r+1) × SwitchIndex s → Bool)
-    (c : ((Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool)) ×
-      (Card s × Card r → Bool)) :
-    assembleBits (r+1) s lo ((highStepEquiv r s).symm c) =
-      (coinStepEquiv (s+r)).symm
-        ((assembleBits r s (fun j => lo (Fin.cons false j.1,j.2)) c.1.1,
-          assembleBits r s (fun j => lo (Fin.cons true j.1,j.2)) c.1.2),
-        fun y => c.2 ((cardSplit r s y).2,(cardSplit r s y).1)) := by
-  funext i
-  rcases i with y | ⟨b,i⟩
-  · rfl
-  · cases b <;> rfl
-
-noncomputable def palindromeLevelCost (t : ℕ) : (d : ℕ) → {ι : Type*} → [Fintype ι] →
-    (ι ↪ Card d) → BenesCoins d → ℕ
-  | 0, _, _, _, _ => 0
-  | d+1, _, _, e, ω =>
-      let σ := benesStepEquiv d ω
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x σ.2.1
-      let hc := PairRouting.colors_compatible x σ.2.1
-      if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
-          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) else
-        palindromeLevelCost t d (PairRouting.childEmbedding x c hc false) σ.1.1 +
-          palindromeLevelCost t d (PairRouting.childEmbedding x c hc true) σ.1.2
-
-lemma palindromeLevelCost_step (t d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (d+1)) (X Y : (SwitchIndex d → Bool) × (SwitchIndex d → Bool))
-    (ξ η : Card d → Bool) :
-    palindromeLevelCost t (d+1) e
-      ((coinStepEquiv d).symm (Y,η),(coinStepEquiv d).symm (X,ξ)) =
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x ξ
-      let hc := PairRouting.colors_compatible x ξ
-      if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x ξ)
-        (fun b => palindromePerm d (if b then (Y.2,X.2) else (Y.1,X.1))) else
-        palindromeLevelCost t d (PairRouting.childEmbedding x c hc false) (Y.1,X.1) +
-        palindromeLevelCost t d (PairRouting.childEmbedding x c hc true) (Y.2,X.2) := by
-  rw [palindromeLevelCost]
-  rfl
-
-lemma palindromeLevelCost_gt (t d : ℕ) (h : d < t) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) : palindromeLevelCost t d e ω = 0 := by
-  classical
-  induction d generalizing ι with
-  | zero => rfl
-  | succ d ih =>
-    rw [palindromeLevelCost,ite_eq_right (by omega : t ≠ d+1),ih (by omega),ih (by omega),Nat.zero_add]
-
-
-theorem palindrome_level_partial_mgf (r s u : ℕ) (hu : u < r)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+r)) (X : SwitchIndex (s+r) → Bool)
-    (lo : Card r × SwitchIndex s → Bool) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(palindromeLevelCost (s+u+1) (s+r) e (assembleBits r s lo hi,X))) ≤
-      Real.exp (Fintype.card ι * cycleCoefficient u J q) := by
-  classical
-  induction r generalizing u ι with
-  | zero => omega
-  | succ r ih =>
-    simp only [Nat.add_succ] at *
-    let χ := coinStepEquiv (s+r) X
-    let x := e.trans (headTailEquiv (s+r)).toEmbedding
-    let c := PairRouting.colors x χ.2
-    let hc := PairRouting.colors_compatible x χ.2
-    let e₀ := PairRouting.childEmbedding x c hc false
-    let e₁ := PairRouting.childEmbedding x c hc true
-    let e' := PairRouting.switchedEmbedding x χ.2
-    let lo₀ : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons false j.1,j.2)
-    let lo₁ : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons true j.1,j.2)
-    let Y₀ := fun hi : Card s × SwitchIndex r → Bool => assembleBits r s lo₀ hi
-    let Y₁ := fun hi : Card s × SwitchIndex r → Bool => assembleBits r s lo₁ hi
-    let N := fun hi : (Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool) =>
-      PairRouting.alternatingCycles e' (fun b => if b then palindromePerm (s+r) (Y₁ hi.2,χ.1.2)
-        else palindromePerm (s+r) (Y₀ hi.1,χ.1.1))
-    let L₀ := fun hi : Card s × SwitchIndex r → Bool =>
-      palindromeLevelCost (s+u+1) (s+r) e₀ (Y₀ hi,χ.1.1)
-    let L₁ := fun hi : Card s × SwitchIndex r → Bool =>
-      palindromeLevelCost (s+u+1) (s+r) e₁ (Y₁ hi,χ.1.2)
-    have hs (hi : (Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool))
-        (η : Card s × Card r → Bool) :
-        palindromeLevelCost (s+u+1) (s+r+1) e
-          (assembleBits (r+1) s lo ((highStepEquiv r s).symm (hi,η)),X) =
-          if u = r then N hi else L₀ hi.1 + L₁ hi.2 := by
-      rw [assembleBits_step]
-      conv_lhs => rw [←(coinStepEquiv (s+r)).symm_apply_apply X]
-      rw [palindromeLevelCost_step]
-      simp only [Nat.add_right_cancel_iff,Nat.add_left_cancel_iff]
-      dsimp only [χ,x,c,hc,e₀,e₁,e',lo₀,lo₁,Y₀,Y₁,N,L₀,L₁]
-      congr 2
-      funext b
-      cases b <;> rfl
-    rw [finiteMean_equiv (highStepEquiv r s),finiteMean_prod]
-    simp_rw [hs,finiteMean_const]
-    by_cases hur : u = r
-    · subst u
-      simp only [ite_true]
-      rw [finiteMean_prod,finiteMean_comm]
-      calc
-        _ ≤ finiteMean (fun _hi : Card s × SwitchIndex r → Bool =>
-          Real.exp (Fintype.card ι * cycleCoefficient r J q)) := by
-          apply finiteMean_mono
-          intro hi
-          have hn := node_partial_mgf r s e'
-            (butterflyPerm (s+r) (decodeButterfly (s+r) χ.1.1))
-            (butterflyPerm (s+r) (decodeButterfly (s+r) χ.1.2))
-            (butterflyPerm (s+r) (decodeButterfly (s+r) (Y₁ hi))) lo₀ J hJ q hq
-          exact hn
-        _ = _ := finiteMean_const _
-    · simp only [ite_eq_right hur,pow_add]
-      rw [finiteMean_prod_mul (fun hi => q ^ L₀ hi) (fun hi => q ^ L₁ hi)]
-      have hu' : u < r := by omega
-      have h₀ := ih u hu' e₀ χ.1.1 lo₀
-      have h₁ := ih u hu' e₁ χ.1.2 lo₁
-      have hp := mul_le_mul h₀ h₁ (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
-        (Real.exp_nonneg _)
-      refine hp.trans_eq ?_
-      rw [←Real.exp_add]
-      congr 1
-      have hcount := PairRouting.color_card_add c
-      have hcount' : (Fintype.card {i : ι // c i = false} : ℝ) +
-          Fintype.card {i : ι // c i = true} = Fintype.card ι := by exact_mod_cast hcount
-      change (Fintype.card {i : ι // c i = false} : ℝ) * cycleCoefficient u J q +
-          Fintype.card {i : ι // c i = true} * cycleCoefficient u J q = _
-      rw [←add_mul,hcount']
-
-
-def switchHeight : (d : ℕ) → SwitchIndex d → ℕ
-  | 0, i => nomatch i
-  | d+1, Sum.inl _ => d+1
-  | d+1, Sum.inr (_,i) => switchHeight d i
-
-lemma switchHeight_le (d : ℕ) (i : SwitchIndex d) : switchHeight d i ≤ d := by
-  induction d with
-  | zero => exact Empty.elim i
-  | succ d ih =>
-    rcases i with y | ⟨b,i⟩
-    · exact le_refl _
-    · exact (ih i).trans (Nat.le_succ d)
-
-def AgreeThrough (d s : ℕ) (Y Y' : SwitchIndex d → Bool) : Prop :=
-  ∀ i, switchHeight d i ≤ s → Y i = Y' i
-
-def LayerAdapted (d s : ℕ) {β : Type*} (f : (SwitchIndex d → Bool) → β) : Prop :=
-  ∀ Y Y', AgreeThrough d s Y Y' → f Y = f Y'
-
-lemma agreeThrough_mono {d s t : ℕ} (h : s ≤ t) {Y Y' : SwitchIndex d → Bool}
-    (hh : AgreeThrough d t Y Y') : AgreeThrough d s Y Y' :=
-  fun i hi => hh i (hi.trans h)
-
-lemma layerAdapted_mono {d s t : ℕ} {β : Type*} {f : (SwitchIndex d → Bool) → β}
-    (hf : LayerAdapted d s f) (h : s ≤ t) : LayerAdapted d t f :=
-  fun Y Y' hh => hf Y Y' (agreeThrough_mono h hh)
-
-lemma layerAdapted_comp {d s : ℕ} {α β : Type*} {f : (SwitchIndex d → Bool) → α}
-    (hf : LayerAdapted d s f) (g : α → β) : LayerAdapted d s (g ∘ f) :=
-  fun Y Y' h => congrArg g (hf Y Y' h)
-
-lemma layerAdapted_sum {d s : ℕ} {ι : Type*} (A : Finset ι)
-    (f : ι → (SwitchIndex d → Bool) → ℕ)
-    (hf : ∀ i ∈ A, LayerAdapted d s (f i)) :
-    LayerAdapted d s (fun Y => ∑ i ∈ A, f i Y) := by
-  intro Y Y' h
-  exact Finset.sum_congr rfl (fun i hi => hf i hi Y Y' h)
-
-lemma assembleBits_agree (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
-    (hi hi' : Card s × SwitchIndex r → Bool) :
-    AgreeThrough (s+r) s (assembleBits r s lo hi) (assembleBits r s lo hi') := by
-  induction r with
-  | zero => intro i _; rfl
-  | succ r ih =>
-    intro i h
-    rcases i with y | ⟨b,i⟩
-    · have : s+r+1 ≤ s := h
-      omega
-    · exact ih (fun j => lo (Fin.cons b j.1,j.2))
-        (fun j => hi (j.1,Sum.inr (b,j.2))) (fun j => hi' (j.1,Sum.inr (b,j.2))) i h
-
-lemma palindromeLevelCost_adapted (t d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) :
-    LayerAdapted d (t-1) (fun Y => palindromeLevelCost t d e (Y,X)) := by
-  classical
-  induction d generalizing ι with
-  | zero => intro Y Y' _; rfl
-  | succ d ih =>
-    intro Y Y' h
-    change palindromeLevelCost t (d+1) e (Y,X) = palindromeLevelCost t (d+1) e (Y',X)
-    conv_lhs => rw [←(coinStepEquiv d).symm_apply_apply Y,←(coinStepEquiv d).symm_apply_apply X]
-    conv_rhs => rw [←(coinStepEquiv d).symm_apply_apply Y',←(coinStepEquiv d).symm_apply_apply X]
-    rw [palindromeLevelCost_step,palindromeLevelCost_step]
-    by_cases ht : t = d+1
-    · subst t
-      rw [ite_eq_left rfl,ite_eq_left rfl]
-      have he₀ : (coinStepEquiv d Y).1.1 = (coinStepEquiv d Y').1.1 := by
-        funext i
-        exact h (Sum.inr (false,i)) (by simpa only [switchHeight,Nat.add_sub_cancel] using switchHeight_le d i)
-      have he₁ : (coinStepEquiv d Y).1.2 = (coinStepEquiv d Y').1.2 := by
-        funext i
-        exact h (Sum.inr (true,i)) (by simpa only [switchHeight,Nat.add_sub_cancel] using switchHeight_le d i)
-      rw [he₀,he₁]
-    · rw [ite_eq_right ht,ite_eq_right ht]
-      congr 1
-      · apply ih
-        intro i hi
-        exact h (Sum.inr (false,i)) hi
-      · apply ih
-        intro i hi
-        exact h (Sum.inr (true,i)) hi
-
-lemma adapted_mul_mean_le (r s : ℕ) (F G : (SwitchIndex (s+r) → Bool) → ℝ)
-    (hF : ∀ Y, 0 ≤ F Y) (hA : LayerAdapted (s+r) s F) (B : ℝ)
-    (hG : ∀ lo : Card r × SwitchIndex s → Bool,
-      finiteMean (fun hi : Card s × SwitchIndex r → Bool => G (assembleBits r s lo hi)) ≤ B) :
-    finiteMean (fun Y => F Y*G Y) ≤ finiteMean F * B := by
-  rw [finiteMean_split r s,finiteMean_split r s F,←finiteMean_mul_const]
-  apply finiteMean_mono
-  intro lo
-  let H : Card s × SwitchIndex r → Bool := fun _ => false
-  have hh (hi : Card s × SwitchIndex r → Bool) :
-      F (assembleBits r s lo hi) = F (assembleBits r s lo H) :=
-    hA _ _ (assembleBits_agree r s lo hi H)
-  simp_rw [hh,finiteMean_const]
-  rw [show (fun hi => F (assembleBits r s lo H) * G (assembleBits r s lo hi)) =
-      (fun hi => G (assembleBits r s lo hi) * F (assembleBits r s lo H)) by
-        funext hi; exact mul_comm _ _,finiteMean_mul_const]
-  simpa only [mul_comm] using mul_le_mul_of_nonneg_left (hG lo) (hF _)
-
-
-theorem cycleCoefficient_decay (l r : ℕ) (hr : (l:ℝ)/2 ≤ r)
-    (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ (l:ℝ)/250) :
-    cycleCoefficient r ⌈Real.exp ((l:ℝ)/32)⌉₊ q ≤
-      68 * Real.exp (-(l:ℝ)/64) := by
-  let E := Real.exp ((l:ℝ)/32)
-  let J := ⌈E⌉₊
-  have hl : (0:ℝ) ≤ l := Nat.cast_nonneg _
-  have hE : 1 ≤ E := Real.one_le_exp_iff.mpr (by positivity)
-  have hEpos : 0 < E := Real.exp_pos _
-  have hEJ : E ≤ (J:ℝ) := Nat.le_ceil E
-  have hJ : (1:ℝ) ≤ J := hE.trans hEJ
-  have hJpos : (0:ℝ) < J := lt_of_lt_of_le (by norm_num) hJ
-  have hJE : (J:ℝ) ≤ 2*E := by
-    have h := Nat.ceil_lt_add_one (le_of_lt hEpos)
-    change (J:ℝ) < E+1 at h
-    linarith
-  have hsqJ : Real.sqrt J ≤ (J:ℝ) := by
-    apply Real.sqrt_le_iff.mpr
-    constructor
-    · positivity
-    · nlinarith
-  have hqp : 0 < q := lt_of_lt_of_le (by norm_num) hq
-  have hqE : q ≤ Real.exp ((l:ℝ)/250) := by
-    rw [←Real.exp_log hqp]
-    exact Real.exp_le_exp.mpr hlog
-  have hln : (1/2:ℝ) ≤ Real.log 2 := by
-    have hh := Real.one_sub_inv_le_log_of_pos (by norm_num : (0:ℝ)<2)
-    norm_num at hh ⊢
-    exact hh
-  have hden : Real.exp ((l:ℝ)/8) ≤ Real.sqrt ((2:ℝ)^r) := by
-    have he : (2:ℝ)^r = Real.exp ((r:ℝ)*Real.log 2) := by
-      rw [Real.exp_nat_mul,Real.exp_log (by norm_num)]
-    rw [he,Real.sqrt_eq_rpow,Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp]
-    apply Real.exp_le_exp.mpr
-    have hrn : (0:ℝ) ≤ r := Nat.cast_nonneg _
-    nlinarith
-  have hlog' : Real.log q ≤ (l:ℝ) := by linarith
-  have h₁ : Real.log q / (J:ℝ) ≤ 64*Real.exp (-(l:ℝ)/64) := by
-    calc
-      _ ≤ (l:ℝ)/E := div_le_div₀ (by positivity) hlog' hEpos hEJ
-      _ ≤ (64*Real.exp ((l:ℝ)/64))/E := by
-        apply div_le_div_of_nonneg_right _ (le_of_lt hEpos)
-        have hh := Real.add_one_le_exp ((l:ℝ)/64)
-        linarith
-      _ = _ := by
-        dsimp [E]
-        rw [mul_div_assoc,←Real.exp_sub]
-        congr 2
-        ring
-  have h₂ : 2*q*Real.sqrt J / Real.sqrt ((2:ℝ)^r) ≤
-      4*Real.exp (-(l:ℝ)/64) := by
-    calc
-      _ ≤ (4*Real.exp ((l:ℝ)/250)*E)/Real.exp ((l:ℝ)/8) := by
-        apply div_le_div₀ (by positivity) _ (Real.exp_pos _) hden
-        have hs := hsqJ.trans hJE
-        have hm := mul_le_mul hqE hs (Real.sqrt_nonneg _) (Real.exp_nonneg _)
-        nlinarith
-      _ = 4*Real.exp ((l:ℝ)/250+(l:ℝ)/32-(l:ℝ)/8) := by
-        dsimp [E]
-        rw [mul_assoc,←Real.exp_add,mul_div_assoc,←Real.exp_sub]
-      _ ≤ _ := by
-        apply mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr _) (by norm_num)
-        linarith
-  change Real.log q / (J:ℝ) + 2*q*Real.sqrt J / Real.sqrt ((2:ℝ)^r) ≤ _
-  linarith
-
-lemma cycleCutoff_pos (l : ℕ) : 0 < ⌈Real.exp ((l:ℝ)/32)⌉₊ := by
-  have h := Nat.le_ceil (Real.exp ((l:ℝ)/32))
-  have he := Real.exp_pos ((l:ℝ)/32)
-  have : (0:ℝ) < ⌈Real.exp ((l:ℝ)/32)⌉₊ := lt_of_lt_of_le he h
-  exact_mod_cast this
-
-
-lemma pow_sum_le_mean_powers {ι : Type*} [Fintype ι] [Nonempty ι]
-    (z : ι → ℕ) (q : ℝ) (hq : 0 < q) :
-    q^(∑ i, z i) ≤ finiteMean (fun i => q^(Fintype.card ι*z i)) := by
-  have hn : (Fintype.card ι:ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
-  have hm : finiteMean (fun i => (Fintype.card ι*z i:ℕ)*Real.log q) =
-      (∑ i, z i:ℕ)*Real.log q := by
-    simp only [finiteMean,Nat.cast_mul,Nat.cast_sum,←Finset.mul_sum,←Finset.sum_mul]
-    field_simp
-  have he := exp_finiteMean_le (fun i => (Fintype.card ι*z i:ℕ)*Real.log q)
-  rw [hm,Real.exp_nat_mul,Real.exp_log hq] at he
-  simpa only [Real.exp_nat_mul,Real.exp_log hq] using he
-
-noncomputable def palindromeSlabCost (l d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) : ℕ :=
-  ∑ i : Fin l, palindromeLevelCost (l+i.1+1) d e ω
-
-lemma palindromeSlabCost_adapted (l d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) :
-    LayerAdapted d (2*l) (fun Y => palindromeSlabCost l d e (Y,X)) := by
-  apply layerAdapted_sum
-  intro i _
-  exact layerAdapted_mono (palindromeLevelCost_adapted (l+i.1+1) d e X) (by omega)
-
-theorem palindrome_slab_partial_mgf (r s l : ℕ) (hl : 0 < l) (hs : 2*s ≤ l)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+r)) (X : SwitchIndex (s+r) → Bool)
-    (lo : Card r × SwitchIndex s → Bool) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(palindromeSlabCost l (s+r) e (assembleBits r s lo hi,X))) ≤
-      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
-  classical
-  let : NeZero l := ⟨ne_of_gt hl⟩
-  have hqp : 0 < q := lt_of_lt_of_le (by norm_num) hq
-  calc
-    _ ≤ finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      finiteMean (fun i : Fin l => q^(l*palindromeLevelCost (l+i.1+1) (s+r) e
-        (assembleBits r s lo hi,X)))) := by
-      apply finiteMean_mono
-      intro hi
-      simpa only [palindromeSlabCost,Fintype.card_fin] using
-        pow_sum_le_mean_powers (fun i : Fin l => palindromeLevelCost (l+i.1+1) (s+r) e
-          (assembleBits r s lo hi,X)) q hqp
-    _ = finiteMean (fun i : Fin l => finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(l*palindromeLevelCost (l+i.1+1) (s+r) e (assembleBits r s lo hi,X)))) := finiteMean_comm _
-    _ ≤ finiteMean (fun _i : Fin l =>
-      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64)))) := by
-      apply finiteMean_mono
-      intro i
-      by_cases ht : l+i.1+1 ≤ s+r
-      · have hs' : s ≤ l+i.1 := by omega
-        have he : l+i.1+1 = s+(l+i.1-s)+1 := by omega
-        have hu : l+i.1-s < r := by omega
-        have hm := palindrome_level_partial_mgf r s (l+i.1-s) hu e X lo
-          ⌈Real.exp ((l:ℝ)/32)⌉₊ (cycleCutoff_pos l) (q^l) (one_le_pow₀ hq)
-        have hc := cycleCoefficient_decay l (l+i.1-s) (by
-          rw [Nat.cast_sub hs',Nat.cast_add]
-          have hsr : (2:ℝ)*s ≤ l := by exact_mod_cast hs
-          have hi : (0:ℝ) ≤ i.1 := Nat.cast_nonneg _
-          linarith) (q^l) (one_le_pow₀ hq) (by
-            rw [Real.log_pow]
-            nlinarith [mul_le_mul_of_nonneg_left hlog (Nat.cast_nonneg l)])
-        simp_rw [he,pow_mul]
-        exact hm.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left hc (Nat.cast_nonneg _)))
-      · have hz := palindromeLevelCost_gt (l+i.1+1) (s+r) (by omega) e
-        simp only [hz,Nat.mul_zero,pow_zero,finiteMean_const]
-        exact Real.one_le_exp_iff.mpr (by positivity)
-    _ = _ := finiteMean_const _
-
-
-lemma palindromeSlabCost_gt (l d : ℕ) (h : d ≤ l) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) : palindromeSlabCost l d e ω = 0 := by
-  apply Finset.sum_eq_zero
-  intro i _
-  exact palindromeLevelCost_gt _ _ (by omega) _ _
-
-lemma palindrome_slab_adapted_mul_le (d s l : ℕ) (hsd : s ≤ d) (hl : 0 < l)
-    (hs : 2*s ≤ l) {ι : Type*} [Fintype ι] (e : ι ↪ Card d)
-    (X : SwitchIndex d → Bool) (F : (SwitchIndex d → Bool) → ℝ)
-    (hF : ∀ Y, 0 ≤ F Y) (hA : LayerAdapted d s F) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun Y => F Y*q^(palindromeSlabCost l d e (Y,X))) ≤
-      finiteMean F * Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
-  obtain ⟨r,rfl⟩ := Nat.exists_eq_add_of_le hsd
-  exact adapted_mul_mean_le r s F _ hF hA _
-    (fun lo => palindrome_slab_partial_mgf r s l hl hs e X lo q hq hlog)
-
-def familyEnd (L n : ℕ) : ℕ := if n = 0 then 0 else 2*(L*4^(n-1))
-
-lemma familyEnd_two (L n : ℕ) : 2*familyEnd L n ≤ L*4^n := by
-  cases n with
-  | zero => simp [familyEnd]
-  | succ n => simp only [familyEnd,Nat.succ_ne_zero,ite_false,Nat.add_one_sub_one,pow_succ]; nlinarith
-
-lemma familyEnd_ge (L n j : ℕ) (hj : j < n) : 2*(L*4^j) ≤ familyEnd L n := by
-  cases n with
-  | zero => omega
-  | succ n =>
-    simp only [familyEnd,Nat.succ_ne_zero,ite_false,Nat.add_one_sub_one]
-    apply Nat.mul_le_mul_left
-    exact Nat.mul_le_mul_left L (Nat.pow_le_pow_right (by decide) (by omega : j ≤ n))
-
-noncomputable def palindromeFamilyCost (L n d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) : ℕ :=
-  ∑ j ∈ Finset.range n, palindromeSlabCost (L*4^j) d e ω
-
-lemma palindromeFamilyCost_adapted (L n d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) :
-    LayerAdapted d (familyEnd L n) (fun Y => palindromeFamilyCost L n d e (Y,X)) := by
-  apply layerAdapted_sum
-  intro j hj
-  exact layerAdapted_mono (palindromeSlabCost_adapted (L*4^j) d e X)
-    (familyEnd_ge L n j (Finset.mem_range.mp hj))
-
-end Thorp
-
-end ThorpNine.Smoothing
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Smoothing.BenesCoins
+
+namespace OAI
+
+namespace ThorpNine.Smoothing
+
+namespace Thorp
+
+lemma mem_centralBlock (r H : ℕ) (b : Card r) (x : Card (H+r)) :
+    x ∈ centralBlock r H b ↔ (cardSplit r H x).1 = b := by
+  classical
+  constructor
+  · intro h
+    obtain ⟨z,_,rfl⟩ := Finset.mem_image.mp h
+    simp only [Equiv.apply_symm_apply]
+  · intro h
+    apply Finset.mem_image.mpr
+    refine ⟨(cardSplit r H x).2,Finset.mem_univ _,?_⟩
+    rw [←h,Prod.mk.eta,Equiv.symm_apply_apply]
+
+lemma centralBlock_card (r H : ℕ) (b : Card r) : (centralBlock r H b).card = 2^H := by
+  classical
+  rw [centralBlock,Finset.card_image_of_injective]
+  · exact card_positions H
+  · intro x y h
+    exact congrArg Prod.snd ((cardSplit r H).symm.injective h)
+
+lemma centralBlock_disjoint (r H : ℕ) :
+    Pairwise (fun b b' => Disjoint (centralBlock r H b) (centralBlock r H b')) := by
+  classical
+  intro b b' h
+  apply Finset.disjoint_left.mpr
+  intro x hx hy
+  exact h (((mem_centralBlock r H b x).mp hx).symm.trans
+    ((mem_centralBlock r H b' x).mp hy))
+
+lemma centralBlock_zero (H : ℕ) (b : Card 0) : centralBlock 0 H b = Finset.univ := by
+  classical
+  ext x
+  simp only [mem_centralBlock,Finset.mem_univ,iff_true]
+  exact Subsingleton.elim _ _
+
+lemma centralBlock_step (r H : ℕ) (b : Bool) (c : Card r) :
+    centralBlock (r+1) H (Fin.cons b c) = (centralBlock r H c).image (Fin.cons b) := by
+  classical
+  simp only [centralBlock,Finset.image_image,cardSplit,Equiv.coe_fn_symm_mk,
+    Fin.cons_zero,Fin.tail_cons,Function.comp_def]
+
+lemma block_inter_step (r H : ℕ) (b : Bool) (c : Card r) (A : Finset (Card (H+(r+1)))) :
+    centralBlock (r+1) H (Fin.cons b c) ∩ A =
+      (centralBlock r H c ∩ childMarks A b).image (Fin.cons b) := by
+  classical
+  rw [centralBlock_step]
+  ext x
+  constructor
+  · intro hx
+    obtain ⟨hxB,hxA⟩ := Finset.mem_inter.mp hx
+    obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hxB
+    exact Finset.mem_image.mpr ⟨y,Finset.mem_inter.mpr ⟨hy,(mem_childMarks A b y).mpr hxA⟩,rfl⟩
+  · intro hx
+    obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
+    exact Finset.mem_inter.mpr ⟨Finset.mem_image.mpr ⟨y,(Finset.mem_inter.mp hy).1,rfl⟩,
+      (mem_childMarks A b y).mp (Finset.mem_inter.mp hy).2⟩
+
+lemma sparseMass_central_zero (H : ℕ) (A : Finset (Card H)) :
+    sparseMass (centralBlock 0 H) A = crowdedMass A.card := by
+  classical
+  simp only [sparseMass,centralBlock_zero,Finset.univ_inter,crowdedMass,
+    Finset.sum_const,Finset.card_univ,card_positions,pow_zero,one_nsmul]
+
+lemma sparseMass_central_step (r H : ℕ) (A : Finset (Card (H+(r+1)))) :
+    sparseMass (centralBlock (r+1) H) A =
+      sparseMass (centralBlock r H) (childMarks A false) +
+        sparseMass (centralBlock r H) (childMarks A true) := by
+  classical
+  unfold sparseMass
+  rw [←Equiv.sum_comp (headTailEquiv r).symm,Fintype.sum_prod_type]
+  have ht (b : Bool) (c : Card r) : (headTailEquiv r).symm (b,c) = Fin.cons b c := rfl
+  simp only [Fintype.sum_bool,ht]
+  have hc (b : Bool) (c : Card r) :
+      (centralBlock (r+1) H (Fin.cons b c) ∩ A).card =
+      (centralBlock r H c ∩ childMarks A b).card := by
+    rw [block_inter_step]
+    apply Finset.card_image_of_injective
+    intro x y h
+    simpa only [Fin.tail_cons] using congrArg Fin.tail h
+  simp only [hc]
+  exact add_comm _ _
+
+
+lemma inverseButterfly_coordinates_step (d : ℕ)
+    (c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (ξ : Card d → Bool)
+    (x : Card (d+1)) :
+    headTailEquiv d ((butterflyPerm (d+1)
+      (decodeButterfly (d+1) ((coinStepEquiv d).symm (c,ξ)))).symm x) =
+      (Bool.xor (x 0) (ξ (Fin.tail x)),
+        (butterflyPerm d (decodeButterfly d
+          (if Bool.xor (x 0) (ξ (Fin.tail x)) then c.2 else c.1))).symm (Fin.tail x)) := by
+  rw [butterflyPerm_step]
+  rfl
+
+lemma inverseButterflyMarks_child (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (d+1))
+    (c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (ξ : Card d → Bool) (b : Bool) :
+    childMarks (inverseButterflyMarks (d+1) (Finset.univ.image e)
+      ((coinStepEquiv d).symm (c,ξ))) b =
+      inverseButterflyMarks d
+        (Finset.univ.image (PairRouting.childEmbedding (e.trans (headTailEquiv d).toEmbedding)
+          (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) ξ)
+            (PairRouting.colors_compatible _ _) b)) (if b then c.2 else c.1) := by
+  classical
+  let x := e.trans (headTailEquiv d).toEmbedding
+  let col := PairRouting.colors x ξ
+  let child := PairRouting.childEmbedding x col (PairRouting.colors_compatible x ξ) b
+  have hp (i : ι) : headTailEquiv d ((butterflyPerm (d+1)
+      (decodeButterfly (d+1) ((coinStepEquiv d).symm (c,ξ)))).symm (e i)) =
+      (col i,(butterflyPerm d (decodeButterfly d (if col i then c.2 else c.1))).symm (x i).2) :=
+    inverseButterfly_coordinates_step d c ξ (e i)
+  ext y
+  simp only [mem_childMarks,inverseButterflyMarks,Finset.image_image,
+    Finset.mem_image,Finset.mem_univ,true_and,Function.comp_def]
+  constructor
+  · rintro ⟨i,hi⟩
+    have he := congrArg (headTailEquiv d) hi
+    rw [hp i] at he
+    change (col i,_) = (b,y) at he
+    have hb := congrArg Prod.fst he
+    have hy := congrArg Prod.snd he
+    dsimp only at hb hy
+    refine ⟨⟨i,hb⟩,?_⟩
+    change (butterflyPerm d (decodeButterfly d (if b then c.2 else c.1))).symm (x i).2 = y
+    simpa only [hb] using hy
+  · rintro ⟨i,hi⟩
+    refine ⟨i.val,?_⟩
+    apply (headTailEquiv d).injective
+    rw [hp]
+    change (col i.val,_) = (b,y)
+    have hb : col i.val = b := i.property
+    rw [hb]
+    exact Prod.ext rfl hi
+
+lemma inverseButterflyMarks_card (d : ℕ) (A : Finset (Card d)) (ω : SwitchIndex d → Bool) :
+    (inverseButterflyMarks d A ω).card = A.card := by
+  classical
+  exact Finset.card_image_of_injective _ (butterflyPerm d (decodeButterfly d ω)).symm.injective
+
+lemma benes_input_coordinates (d : ℕ) (ω : BenesCoins (d+1)) :
+    ω.2 = (coinStepEquiv d).symm
+      (((benesStepEquiv d ω).1.1.2,(benesStepEquiv d ω).1.2.2),(benesStepEquiv d ω).2.1) := by
+  change ω.2 = (coinStepEquiv d).symm (coinStepEquiv d ω.2)
+  exact ((coinStepEquiv d).symm_apply_apply ω.2).symm
+
+lemma cutoffMass_central (r H : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (H+r)) (ω : BenesCoins (H+r)) :
+    cutoffMass H (H+r) e ω =
+      sparseMass (centralBlock r H) (inverseButterflyMarks (H+r) (Finset.univ.image e) ω.2) := by
+  classical
+  induction r generalizing ι with
+  | zero =>
+    rw [sparseMass_central_zero,inverseButterflyMarks_card,
+      Finset.card_image_of_injective _ e.injective,Finset.card_univ]
+    cases H with
+    | zero =>
+      have hcard : Fintype.card ι ≤ 1 := by
+        have hh := Fintype.card_le_of_injective e e.injective
+        simpa only [Nat.add_zero,card_positions,pow_zero] using hh
+      simp [cutoffMass,crowdedMass,show ¬2≤Fintype.card ι by omega]
+    | succ H => simp only [cutoffMass,le_refl,ite_true]
+  | succ r ih =>
+    simp only [Nat.add_succ] at *
+    rw [cutoffMass,ite_eq_right (by omega : ¬H+r+1≤H)]
+    let σ := benesStepEquiv (H+r) ω
+    let x := e.trans (headTailEquiv (H+r)).toEmbedding
+    let col := PairRouting.colors x σ.2.1
+    let hc := PairRouting.colors_compatible x σ.2.1
+    change cutoffMass H (H+r) (PairRouting.childEmbedding x col hc false) σ.1.1 +
+      cutoffMass H (H+r) (PairRouting.childEmbedding x col hc true) σ.1.2 = _
+    rw [ih,ih,sparseMass_central_step,benes_input_coordinates (H+r) ω]
+    rw [inverseButterflyMarks_child,inverseButterflyMarks_child]
+    rfl
+
+
+theorem palindrome_low_mgf (r H : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (H+r)) (q : ℝ) (hq : 1 ≤ q)
+    (hr : (2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*q^H) ≤ 1/2) :
+    finiteMean (fun ω : BenesCoins (H+r) => q^(palindromeLowCost H (H+r) e ω)) ≤
+      Real.exp ((2:ℝ)^r * (2 * ((2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*q^H))^2)) := by
+  classical
+  have hec : (Finset.univ.image e).card = Fintype.card ι := by
+    rw [Finset.card_image_of_injective _ e.injective,Finset.card_univ]
+  have hm := inverseButterfly_sparse_mgf (H+r) (centralBlock r H)
+    (centralBlock_disjoint r H) (Finset.univ.image e) (q^H)
+    (pow_nonneg (by linarith) _) (fun b => by
+      simpa only [centralBlock_card,hec,Nat.cast_pow,Nat.cast_ofNat] using hr)
+  calc
+    _ ≤ finiteMean (fun ω : BenesCoins (H+r) => (q^H)^(cutoffMass H (H+r) e ω)) := by
+      apply finiteMean_mono
+      intro ω
+      rw [←pow_mul]
+      exact pow_le_pow_right₀ hq (palindromeLowCost_le_cutoff H (H+r) e ω)
+    _ = finiteMean (fun ω : SwitchIndex (H+r) → Bool =>
+        (q^H)^(sparseMass (centralBlock r H)
+          (inverseButterflyMarks (H+r) (Finset.univ.image e) ω))) := by
+      simp_rw [cutoffMass_central]
+      simp only [finiteMean_prod,finiteMean_const]
+    _ ≤ _ := by
+      simpa only [centralBlock_card,hec,Nat.cast_pow,Nat.cast_ofNat,Finset.sum_const,
+        Finset.card_univ,card_positions,nsmul_eq_mul] using hm
+
+variable {α B : Type*} {r s : ℕ}
+
+def parallelFreshPre (pre : α ≃ B × Card (s+r))
+    (lo : B → Card r × SwitchIndex s → Bool) : α ≃ (B × Card s) × Card r :=
+  ((pre.trans (Equiv.prodCongr (Equiv.refl B) (cardSplit r s))).trans
+    (Equiv.prodCongrRight (fun b =>
+      (parallelPerm (lo b)).trans (Equiv.prodComm _ _)))).trans
+    (Equiv.prodAssoc _ _ _).symm
+
+def parallelFreshPost (post : B × Card (s+r) ≃ α) : (B × Card s) × Card r ≃ α :=
+  (((Equiv.prodAssoc _ _ _).trans (Equiv.prodCongr (Equiv.refl B)
+    (Equiv.prodComm _ _))).trans (Equiv.prodCongr (Equiv.refl B) (cardSplit r s).symm)).trans post
+
+def parallelAssembled (lo : B → Card r × SwitchIndex s → Bool)
+    (hi : (B × Card s) × SwitchIndex r → Bool) : B × SwitchIndex (s+r) → Bool :=
+  fun i => assembleBits r s (lo i.1) (fun j => hi ((i.1,j.1),j.2)) i.2
+
+lemma parallel_exposed_perm (pre : α ≃ B × Card (s+r)) (post : B × Card (s+r) ≃ α)
+    (lo : B → Card r × SwitchIndex s → Bool)
+    (hi : (B × Card s) × SwitchIndex r → Bool) :
+    relabeledPerm pre post (parallelAssembled lo hi) =
+      relabeledPerm (parallelFreshPre pre lo) (parallelFreshPost post) hi := by
+  ext x
+  have h := butterfly_split r s (lo (pre x).1)
+    (fun j => hi (((pre x).1,j.1),j.2)) (pre x).2
+  have hh := congrArg (fun y => post ((pre x).1,(cardSplit r s).symm y)) h
+  simpa only [relabeledPerm,parallelAssembled,parallelPerm,parallelFreshPre,parallelFreshPost,
+    Equiv.trans_apply,Equiv.prodCongr_apply,Equiv.prodCongrRight,Equiv.coe_fn_mk,
+    Equiv.refl_apply,Equiv.prodComm_apply,Prod.swap,Equiv.prodAssoc_apply,
+    Equiv.prodAssoc_symm_apply,Prod.map,Equiv.symm_apply_apply] using hh
+
+theorem parallel_partially_exposed_log_mgf [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B]
+    (pre : α ≃ B × Card (s+r)) (post : B × Card (s+r) ≃ α)
+    (lo : B → Card r × SwitchIndex s → Bool) (S : Finset α)
+    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    Real.log (finiteMean (fun hi : (B × Card s) × SwitchIndex r → Bool =>
+      q^(Fintype.card (RoutingNetwork.SelectedCycle
+        (relabeledPerm pre post (parallelAssembled lo hi)) S)))) ≤
+      (S.card:ℝ)/J*Real.log q + 2*q*S.card*Real.sqrt J/Real.sqrt ((2:ℝ)^r) := by
+  simp_rw [parallel_exposed_perm]
+  exact parallel_cycle_log_mgf (parallelFreshPre pre lo) (parallelFreshPost post) S J hJ q hq
+
+end Thorp
+namespace Thorp
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+lemma orbitSet_inverse (p : Equiv.Perm α) (x : α) : orbitSet p⁻¹ x = orbitSet p x := by
+  ext y
+  simp only [mem_orbitSet,Equiv.Perm.sameCycle_inv]
+
+lemma selectedCycle_inverse (p : Equiv.Perm α) (S : Finset α) :
+    Fintype.card (RoutingNetwork.SelectedCycle p⁻¹ S) =
+      Fintype.card (RoutingNetwork.SelectedCycle p S) := by
+  apply Fintype.card_congr
+  refine { toFun := fun C => ⟨C.val,?_,C.property.2⟩
+           invFun := fun C => ⟨C.val,?_,C.property.2⟩
+           left_inv := fun _ => rfl
+           right_inv := fun _ => rfl }
+  · obtain ⟨x,hx⟩ := C.property.1
+    exact ⟨x,hx.trans (orbitSet_inverse p x)⟩
+  · obtain ⟨x,hx⟩ := C.property.1
+    exact ⟨x,hx.trans (orbitSet_inverse p x).symm⟩
+
+lemma orbitSet_conjugate (p g : Equiv.Perm α) (x : α) :
+    orbitSet (g*p*g⁻¹) (g x) = (orbitSet p x).image g := by
+  ext y
+  simp only [mem_orbitSet,Finset.mem_image]
+  rw [Equiv.Perm.sameCycle_conj]
+  change p.SameCycle (g.symm (g x)) (g.symm y) ↔ ∃ z, p.SameCycle x z ∧ g z = y
+  rw [Equiv.symm_apply_apply]
+  constructor
+  · intro h
+    exact ⟨g.symm y,h,g.apply_symm_apply y⟩
+  · rintro ⟨z,hz,rfl⟩
+    simpa only [Equiv.symm_apply_apply] using hz
+
+noncomputable def conjugateCycle (p g : Equiv.Perm α) (S : Finset α)
+    (C : RoutingNetwork.SelectedCycle p S) :
+    RoutingNetwork.SelectedCycle (g*p*g⁻¹) (S.image g) := by
+  refine ⟨C.val.image g,?_,Finset.image_subset_image C.property.2⟩
+  obtain ⟨x,hx⟩ := C.property.1
+  exact ⟨g x,by rw [hx,orbitSet_conjugate]⟩
+
+lemma conjugateCycle_injective (p g : Equiv.Perm α) (S : Finset α) :
+    Function.Injective (conjugateCycle p g S) := by
+  intro C D he
+  apply Subtype.ext
+  exact (Finset.image_injective g.injective) (congrArg Subtype.val he)
+
+lemma selectedCycle_conjugate (p g : Equiv.Perm α) (S : Finset α) :
+    Fintype.card (RoutingNetwork.SelectedCycle (g*p*g⁻¹) (S.image g)) =
+      Fintype.card (RoutingNetwork.SelectedCycle p S) := by
+  apply Nat.le_antisymm
+  · have h := Fintype.card_le_of_injective _
+      (conjugateCycle_injective (g*p*g⁻¹) g⁻¹ (S.image g))
+    have hp : g⁻¹*(g*p*g⁻¹)*g = p := by group
+    rw [inv_inv,hp] at h
+    have hs : (S.image g).image g.symm = S := by
+      ext x
+      simp only [Finset.mem_image]
+      constructor
+      · rintro ⟨y,⟨z,hz,rfl⟩,he⟩
+        have hz' : z = x := by simpa only [Equiv.symm_apply_apply] using he
+        exact hz' ▸ hz
+      · intro hx
+        exact ⟨g x,⟨x,hx,rfl⟩,g.symm_apply_apply x⟩
+    change _ ≤ Fintype.card (RoutingNetwork.SelectedCycle p ((S.image g).image g.symm)) at h
+    rw [hs] at h
+    exact h
+  · exact Fintype.card_le_of_injective _ (conjugateCycle_injective p g S)
+
+lemma alternatingCycles_le_projection {ι : Type*} [Fintype ι]
+    (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    PairRouting.alternatingCycles e p ≤
+      Fintype.card (RoutingNetwork.SelectedCycle (PairRouting.alternatingProjection p)
+        (boolFiber (PairRouting.labelSet e) false)) := by
+  have h := CycleColoring.card_alternating_cycles_le (PairRouting.alternatingProjection p)
+    (boolFiber (PairRouting.labelSet e) false) (boolFiber (PairRouting.labelSet e) true)
+  simpa only [PairRouting.alternatingCycles,childJoin_boolFiber] using h
+
+lemma projectionCycles_forward (X₀ X₁ Y₀ Y₁ : Equiv.Perm α) (S : Finset α) :
+    Fintype.card (RoutingNetwork.SelectedCycle
+      ((Y₀*X₀⁻¹)⁻¹*(Y₁*X₁⁻¹)) S) =
+    Fintype.card (RoutingNetwork.SelectedCycle
+      (X₀⁻¹*X₁*Y₁⁻¹*Y₀) (S.image X₀.symm)) := by
+  let p := (Y₀*X₀⁻¹)⁻¹*(Y₁*X₁⁻¹)
+  calc
+    _ = Fintype.card (RoutingNetwork.SelectedCycle (X₀⁻¹*p*X₀)
+        (S.image X₀.symm)) := by
+      change Fintype.card (RoutingNetwork.SelectedCycle p S) = _
+      have h := (selectedCycle_conjugate p X₀⁻¹ S).symm
+      rw [inv_inv] at h
+      exact h
+    _ = Fintype.card (RoutingNetwork.SelectedCycle (X₀⁻¹*p*X₀)⁻¹
+        (S.image X₀.symm)) := (selectedCycle_inverse _ _).symm
+    _ = _ := by
+      have he : (X₀⁻¹*p*X₀)⁻¹ = X₀⁻¹*X₁*Y₁⁻¹*Y₀ := by dsimp only [p]; group
+      rw [he]
+
+end Thorp
+namespace Thorp
+
+noncomputable def cycleCoefficient (r J : ℕ) (q : ℝ) : ℝ :=
+  Real.log q / J + 2*q*Real.sqrt J/Real.sqrt ((2:ℝ)^r)
+
+lemma cycleCoefficient_nonneg (r J : ℕ) (q : ℝ) (hq : 1 ≤ q) :
+    0 ≤ cycleCoefficient r J q := by
+  unfold cycleCoefficient
+  exact add_nonneg (div_nonneg (Real.log_nonneg hq) (Nat.cast_nonneg _))
+    (div_nonneg (mul_nonneg (by linarith) (Real.sqrt_nonneg _)) (Real.sqrt_nonneg _))
+
+lemma projectionSide_card_le {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι]
+    (e : ι ↪ Bool × α) : (boolFiber (PairRouting.labelSet e) false).card ≤ Fintype.card ι := by
+  have h := card_childJoin (boolFiber (PairRouting.labelSet e) false)
+    (boolFiber (PairRouting.labelSet e) true)
+  rw [childJoin_boolFiber,PairRouting.card_labelSet] at h
+  omega
+
+theorem node_partial_mgf (r s : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Bool × Card (s+r)) (X₀ X₁ Y₁ : Equiv.Perm (Card (s+r)))
+    (lo : Card r × SwitchIndex s → Bool) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(PairRouting.alternatingCycles e (fun b => if b then Y₁*X₁⁻¹ else
+        butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi))*X₀⁻¹))) ≤
+      Real.exp (Fintype.card ι * cycleCoefficient r J q) := by
+  classical
+  let S := boolFiber (PairRouting.labelSet e) false
+  let S' := S.image X₀.symm
+  let post := X₀⁻¹*X₁*Y₁⁻¹
+  let count := fun hi : Card s × SwitchIndex r → Bool =>
+    Fintype.card (RoutingNetwork.SelectedCycle
+      ((Equiv.refl _).trans (butterflyPerm (s+r)
+        (decodeButterfly (s+r) (assembleBits r s lo hi))) |>.trans post) S')
+  have hpoint (hi : Card s × SwitchIndex r → Bool) :
+      PairRouting.alternatingCycles e (fun b => if b then Y₁*X₁⁻¹ else
+        butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi))*X₀⁻¹) ≤ count hi := by
+    refine (alternatingCycles_le_projection e _).trans_eq ?_
+    simp only [PairRouting.alternatingProjection,Bool.false_eq_true,↓reduceIte]
+    change Fintype.card (RoutingNetwork.SelectedCycle
+      ((butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi))*X₀⁻¹)⁻¹ * (Y₁*X₁⁻¹)) S) = count hi
+    exact projectionCycles_forward X₀ X₁ _ Y₁ S
+  have hm := partially_exposed_cycle_log_mgf (Equiv.refl (Card (s+r))) post lo S' J hJ q hq
+  have hp : 0 < finiteMean (fun hi : Card s × SwitchIndex r → Bool => q^(count hi)) :=
+    finiteMean_pos (fun _ => pow_pos (by linarith) _)
+  have hb : finiteMean (fun hi : Card s × SwitchIndex r → Bool => q^(count hi)) ≤
+      Real.exp (S'.card * cycleCoefficient r J q) := by
+    rw [←Real.exp_log hp]
+    apply Real.exp_le_exp.mpr
+    calc
+      _ ≤ _ := hm
+      _ = _ := by unfold cycleCoefficient; ring
+  have hcard : S'.card ≤ Fintype.card ι := by
+    rw [show S'.card = S.card from Finset.card_image_of_injective _ X₀.symm.injective]
+    exact projectionSide_card_le e
+  calc
+    _ ≤ finiteMean (fun hi : Card s × SwitchIndex r → Bool => q^(count hi)) :=
+      finiteMean_mono (fun hi => pow_le_pow_right₀ hq (hpoint hi))
+    _ ≤ Real.exp (S'.card * cycleCoefficient r J q) := hb
+    _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right
+      (by exact_mod_cast hcard) (cycleCoefficient_nonneg r J q hq))
+
+
+def highStepEquiv (r s : ℕ) : (Card s × SwitchIndex (r+1) → Bool) ≃
+    (((Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool)) ×
+      (Card s × Card r → Bool)) where
+  toFun hi := ((fun j => hi (j.1,Sum.inr (false,j.2)),
+    fun j => hi (j.1,Sum.inr (true,j.2))),fun j => hi (j.1,Sum.inl j.2))
+  invFun c := fun j => match j.2 with
+    | Sum.inl y => c.2 (j.1,y)
+    | Sum.inr (b,i) => if b then c.1.2 (j.1,i) else c.1.1 (j.1,i)
+  left_inv hi := by
+    funext j
+    rcases j with ⟨z,y|⟨b,i⟩⟩
+    · rfl
+    · cases b <;> rfl
+  right_inv c := by
+    rcases c with ⟨⟨h₀,h₁⟩,η⟩
+    rfl
+
+lemma assembleBits_step (r s : ℕ) (lo : Card (r+1) × SwitchIndex s → Bool)
+    (c : ((Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool)) ×
+      (Card s × Card r → Bool)) :
+    assembleBits (r+1) s lo ((highStepEquiv r s).symm c) =
+      (coinStepEquiv (s+r)).symm
+        ((assembleBits r s (fun j => lo (Fin.cons false j.1,j.2)) c.1.1,
+          assembleBits r s (fun j => lo (Fin.cons true j.1,j.2)) c.1.2),
+        fun y => c.2 ((cardSplit r s y).2,(cardSplit r s y).1)) := by
+  funext i
+  rcases i with y | ⟨b,i⟩
+  · rfl
+  · cases b <;> rfl
+
+noncomputable def palindromeLevelCost (t : ℕ) : (d : ℕ) → {ι : Type*} → [Fintype ι] →
+    (ι ↪ Card d) → BenesCoins d → ℕ
+  | 0, _, _, _, _ => 0
+  | d+1, _, _, e, ω =>
+      let σ := benesStepEquiv d ω
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x σ.2.1
+      let hc := PairRouting.colors_compatible x σ.2.1
+      if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
+          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) else
+        palindromeLevelCost t d (PairRouting.childEmbedding x c hc false) σ.1.1 +
+          palindromeLevelCost t d (PairRouting.childEmbedding x c hc true) σ.1.2
+
+lemma palindromeLevelCost_step (t d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (d+1)) (X Y : (SwitchIndex d → Bool) × (SwitchIndex d → Bool))
+    (ξ η : Card d → Bool) :
+    palindromeLevelCost t (d+1) e
+      ((coinStepEquiv d).symm (Y,η),(coinStepEquiv d).symm (X,ξ)) =
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x ξ
+      let hc := PairRouting.colors_compatible x ξ
+      if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x ξ)
+        (fun b => palindromePerm d (if b then (Y.2,X.2) else (Y.1,X.1))) else
+        palindromeLevelCost t d (PairRouting.childEmbedding x c hc false) (Y.1,X.1) +
+        palindromeLevelCost t d (PairRouting.childEmbedding x c hc true) (Y.2,X.2) := by
+  rw [palindromeLevelCost]
+  rfl
+
+lemma palindromeLevelCost_gt (t d : ℕ) (h : d < t) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) : palindromeLevelCost t d e ω = 0 := by
+  classical
+  induction d generalizing ι with
+  | zero => rfl
+  | succ d ih =>
+    rw [palindromeLevelCost,ite_eq_right (by omega : t ≠ d+1),ih (by omega),ih (by omega),Nat.zero_add]
+
+
+theorem palindrome_level_partial_mgf (r s u : ℕ) (hu : u < r)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+r)) (X : SwitchIndex (s+r) → Bool)
+    (lo : Card r × SwitchIndex s → Bool) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(palindromeLevelCost (s+u+1) (s+r) e (assembleBits r s lo hi,X))) ≤
+      Real.exp (Fintype.card ι * cycleCoefficient u J q) := by
+  classical
+  induction r generalizing u ι with
+  | zero => omega
+  | succ r ih =>
+    simp only [Nat.add_succ] at *
+    let χ := coinStepEquiv (s+r) X
+    let x := e.trans (headTailEquiv (s+r)).toEmbedding
+    let c := PairRouting.colors x χ.2
+    let hc := PairRouting.colors_compatible x χ.2
+    let e₀ := PairRouting.childEmbedding x c hc false
+    let e₁ := PairRouting.childEmbedding x c hc true
+    let e' := PairRouting.switchedEmbedding x χ.2
+    let lo₀ : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons false j.1,j.2)
+    let lo₁ : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons true j.1,j.2)
+    let Y₀ := fun hi : Card s × SwitchIndex r → Bool => assembleBits r s lo₀ hi
+    let Y₁ := fun hi : Card s × SwitchIndex r → Bool => assembleBits r s lo₁ hi
+    let N := fun hi : (Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool) =>
+      PairRouting.alternatingCycles e' (fun b => if b then palindromePerm (s+r) (Y₁ hi.2,χ.1.2)
+        else palindromePerm (s+r) (Y₀ hi.1,χ.1.1))
+    let L₀ := fun hi : Card s × SwitchIndex r → Bool =>
+      palindromeLevelCost (s+u+1) (s+r) e₀ (Y₀ hi,χ.1.1)
+    let L₁ := fun hi : Card s × SwitchIndex r → Bool =>
+      palindromeLevelCost (s+u+1) (s+r) e₁ (Y₁ hi,χ.1.2)
+    have hs (hi : (Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool))
+        (η : Card s × Card r → Bool) :
+        palindromeLevelCost (s+u+1) (s+r+1) e
+          (assembleBits (r+1) s lo ((highStepEquiv r s).symm (hi,η)),X) =
+          if u = r then N hi else L₀ hi.1 + L₁ hi.2 := by
+      rw [assembleBits_step]
+      conv_lhs => rw [←(coinStepEquiv (s+r)).symm_apply_apply X]
+      rw [palindromeLevelCost_step]
+      simp only [Nat.add_right_cancel_iff,Nat.add_left_cancel_iff]
+      dsimp only [χ,x,c,hc,e₀,e₁,e',lo₀,lo₁,Y₀,Y₁,N,L₀,L₁]
+      congr 2
+      funext b
+      cases b <;> rfl
+    rw [finiteMean_equiv (highStepEquiv r s),finiteMean_prod]
+    simp_rw [hs,finiteMean_const]
+    by_cases hur : u = r
+    · subst u
+      simp only [ite_true]
+      rw [finiteMean_prod,finiteMean_comm]
+      calc
+        _ ≤ finiteMean (fun _hi : Card s × SwitchIndex r → Bool =>
+          Real.exp (Fintype.card ι * cycleCoefficient r J q)) := by
+          apply finiteMean_mono
+          intro hi
+          have hn := node_partial_mgf r s e'
+            (butterflyPerm (s+r) (decodeButterfly (s+r) χ.1.1))
+            (butterflyPerm (s+r) (decodeButterfly (s+r) χ.1.2))
+            (butterflyPerm (s+r) (decodeButterfly (s+r) (Y₁ hi))) lo₀ J hJ q hq
+          exact hn
+        _ = _ := finiteMean_const _
+    · simp only [ite_eq_right hur,pow_add]
+      rw [finiteMean_prod_mul (fun hi => q ^ L₀ hi) (fun hi => q ^ L₁ hi)]
+      have hu' : u < r := by omega
+      have h₀ := ih u hu' e₀ χ.1.1 lo₀
+      have h₁ := ih u hu' e₁ χ.1.2 lo₁
+      have hp := mul_le_mul h₀ h₁ (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
+        (Real.exp_nonneg _)
+      refine hp.trans_eq ?_
+      rw [←Real.exp_add]
+      congr 1
+      have hcount := PairRouting.color_card_add c
+      have hcount' : (Fintype.card {i : ι // c i = false} : ℝ) +
+          Fintype.card {i : ι // c i = true} = Fintype.card ι := by exact_mod_cast hcount
+      change (Fintype.card {i : ι // c i = false} : ℝ) * cycleCoefficient u J q +
+          Fintype.card {i : ι // c i = true} * cycleCoefficient u J q = _
+      rw [←add_mul,hcount']
+
+
+def switchHeight : (d : ℕ) → SwitchIndex d → ℕ
+  | 0, i => nomatch i
+  | d+1, Sum.inl _ => d+1
+  | d+1, Sum.inr (_,i) => switchHeight d i
+
+lemma switchHeight_le (d : ℕ) (i : SwitchIndex d) : switchHeight d i ≤ d := by
+  induction d with
+  | zero => exact Empty.elim i
+  | succ d ih =>
+    rcases i with y | ⟨b,i⟩
+    · exact le_refl _
+    · exact (ih i).trans (Nat.le_succ d)
+
+def AgreeThrough (d s : ℕ) (Y Y' : SwitchIndex d → Bool) : Prop :=
+  ∀ i, switchHeight d i ≤ s → Y i = Y' i
+
+def LayerAdapted (d s : ℕ) {β : Type*} (f : (SwitchIndex d → Bool) → β) : Prop :=
+  ∀ Y Y', AgreeThrough d s Y Y' → f Y = f Y'
+
+lemma agreeThrough_mono {d s t : ℕ} (h : s ≤ t) {Y Y' : SwitchIndex d → Bool}
+    (hh : AgreeThrough d t Y Y') : AgreeThrough d s Y Y' :=
+  fun i hi => hh i (hi.trans h)
+
+lemma layerAdapted_mono {d s t : ℕ} {β : Type*} {f : (SwitchIndex d → Bool) → β}
+    (hf : LayerAdapted d s f) (h : s ≤ t) : LayerAdapted d t f :=
+  fun Y Y' hh => hf Y Y' (agreeThrough_mono h hh)
+
+lemma layerAdapted_comp {d s : ℕ} {α β : Type*} {f : (SwitchIndex d → Bool) → α}
+    (hf : LayerAdapted d s f) (g : α → β) : LayerAdapted d s (g ∘ f) :=
+  fun Y Y' h => congrArg g (hf Y Y' h)
+
+lemma layerAdapted_sum {d s : ℕ} {ι : Type*} (A : Finset ι)
+    (f : ι → (SwitchIndex d → Bool) → ℕ)
+    (hf : ∀ i ∈ A, LayerAdapted d s (f i)) :
+    LayerAdapted d s (fun Y => ∑ i ∈ A, f i Y) := by
+  intro Y Y' h
+  exact Finset.sum_congr rfl (fun i hi => hf i hi Y Y' h)
+
+lemma assembleBits_agree (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
+    (hi hi' : Card s × SwitchIndex r → Bool) :
+    AgreeThrough (s+r) s (assembleBits r s lo hi) (assembleBits r s lo hi') := by
+  induction r with
+  | zero => intro i _; rfl
+  | succ r ih =>
+    intro i h
+    rcases i with y | ⟨b,i⟩
+    · have : s+r+1 ≤ s := h
+      omega
+    · exact ih (fun j => lo (Fin.cons b j.1,j.2))
+        (fun j => hi (j.1,Sum.inr (b,j.2))) (fun j => hi' (j.1,Sum.inr (b,j.2))) i h
+
+lemma palindromeLevelCost_adapted (t d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) :
+    LayerAdapted d (t-1) (fun Y => palindromeLevelCost t d e (Y,X)) := by
+  classical
+  induction d generalizing ι with
+  | zero => intro Y Y' _; rfl
+  | succ d ih =>
+    intro Y Y' h
+    change palindromeLevelCost t (d+1) e (Y,X) = palindromeLevelCost t (d+1) e (Y',X)
+    conv_lhs => rw [←(coinStepEquiv d).symm_apply_apply Y,←(coinStepEquiv d).symm_apply_apply X]
+    conv_rhs => rw [←(coinStepEquiv d).symm_apply_apply Y',←(coinStepEquiv d).symm_apply_apply X]
+    rw [palindromeLevelCost_step,palindromeLevelCost_step]
+    by_cases ht : t = d+1
+    · subst t
+      rw [ite_eq_left rfl,ite_eq_left rfl]
+      have he₀ : (coinStepEquiv d Y).1.1 = (coinStepEquiv d Y').1.1 := by
+        funext i
+        exact h (Sum.inr (false,i)) (by simpa only [switchHeight,Nat.add_sub_cancel] using switchHeight_le d i)
+      have he₁ : (coinStepEquiv d Y).1.2 = (coinStepEquiv d Y').1.2 := by
+        funext i
+        exact h (Sum.inr (true,i)) (by simpa only [switchHeight,Nat.add_sub_cancel] using switchHeight_le d i)
+      rw [he₀,he₁]
+    · rw [ite_eq_right ht,ite_eq_right ht]
+      congr 1
+      · apply ih
+        intro i hi
+        exact h (Sum.inr (false,i)) hi
+      · apply ih
+        intro i hi
+        exact h (Sum.inr (true,i)) hi
+
+lemma adapted_mul_mean_le (r s : ℕ) (F G : (SwitchIndex (s+r) → Bool) → ℝ)
+    (hF : ∀ Y, 0 ≤ F Y) (hA : LayerAdapted (s+r) s F) (B : ℝ)
+    (hG : ∀ lo : Card r × SwitchIndex s → Bool,
+      finiteMean (fun hi : Card s × SwitchIndex r → Bool => G (assembleBits r s lo hi)) ≤ B) :
+    finiteMean (fun Y => F Y*G Y) ≤ finiteMean F * B := by
+  rw [finiteMean_split r s,finiteMean_split r s F,←finiteMean_mul_const]
+  apply finiteMean_mono
+  intro lo
+  let H : Card s × SwitchIndex r → Bool := fun _ => false
+  have hh (hi : Card s × SwitchIndex r → Bool) :
+      F (assembleBits r s lo hi) = F (assembleBits r s lo H) :=
+    hA _ _ (assembleBits_agree r s lo hi H)
+  simp_rw [hh,finiteMean_const]
+  rw [show (fun hi => F (assembleBits r s lo H) * G (assembleBits r s lo hi)) =
+      (fun hi => G (assembleBits r s lo hi) * F (assembleBits r s lo H)) by
+        funext hi; exact mul_comm _ _,finiteMean_mul_const]
+  simpa only [mul_comm] using mul_le_mul_of_nonneg_left (hG lo) (hF _)
+
+
+theorem cycleCoefficient_decay (l r : ℕ) (hr : (l:ℝ)/2 ≤ r)
+    (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ (l:ℝ)/250) :
+    cycleCoefficient r ⌈Real.exp ((l:ℝ)/32)⌉₊ q ≤
+      68 * Real.exp (-(l:ℝ)/64) := by
+  let E := Real.exp ((l:ℝ)/32)
+  let J := ⌈E⌉₊
+  have hl : (0:ℝ) ≤ l := Nat.cast_nonneg _
+  have hE : 1 ≤ E := Real.one_le_exp_iff.mpr (by positivity)
+  have hEpos : 0 < E := Real.exp_pos _
+  have hEJ : E ≤ (J:ℝ) := Nat.le_ceil E
+  have hJ : (1:ℝ) ≤ J := hE.trans hEJ
+  have hJpos : (0:ℝ) < J := lt_of_lt_of_le (by norm_num) hJ
+  have hJE : (J:ℝ) ≤ 2*E := by
+    have h := Nat.ceil_lt_add_one (le_of_lt hEpos)
+    change (J:ℝ) < E+1 at h
+    linarith
+  have hsqJ : Real.sqrt J ≤ (J:ℝ) := by
+    apply Real.sqrt_le_iff.mpr
+    constructor
+    · positivity
+    · nlinarith
+  have hqp : 0 < q := lt_of_lt_of_le (by norm_num) hq
+  have hqE : q ≤ Real.exp ((l:ℝ)/250) := by
+    rw [←Real.exp_log hqp]
+    exact Real.exp_le_exp.mpr hlog
+  have hln : (1/2:ℝ) ≤ Real.log 2 := by
+    have hh := Real.one_sub_inv_le_log_of_pos (by norm_num : (0:ℝ)<2)
+    norm_num at hh ⊢
+    exact hh
+  have hden : Real.exp ((l:ℝ)/8) ≤ Real.sqrt ((2:ℝ)^r) := by
+    have he : (2:ℝ)^r = Real.exp ((r:ℝ)*Real.log 2) := by
+      rw [Real.exp_nat_mul,Real.exp_log (by norm_num)]
+    rw [he,Real.sqrt_eq_rpow,Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp]
+    apply Real.exp_le_exp.mpr
+    have hrn : (0:ℝ) ≤ r := Nat.cast_nonneg _
+    nlinarith
+  have hlog' : Real.log q ≤ (l:ℝ) := by linarith
+  have h₁ : Real.log q / (J:ℝ) ≤ 64*Real.exp (-(l:ℝ)/64) := by
+    calc
+      _ ≤ (l:ℝ)/E := div_le_div₀ (by positivity) hlog' hEpos hEJ
+      _ ≤ (64*Real.exp ((l:ℝ)/64))/E := by
+        apply div_le_div_of_nonneg_right _ (le_of_lt hEpos)
+        have hh := Real.add_one_le_exp ((l:ℝ)/64)
+        linarith
+      _ = _ := by
+        dsimp [E]
+        rw [mul_div_assoc,←Real.exp_sub]
+        congr 2
+        ring
+  have h₂ : 2*q*Real.sqrt J / Real.sqrt ((2:ℝ)^r) ≤
+      4*Real.exp (-(l:ℝ)/64) := by
+    calc
+      _ ≤ (4*Real.exp ((l:ℝ)/250)*E)/Real.exp ((l:ℝ)/8) := by
+        apply div_le_div₀ (by positivity) _ (Real.exp_pos _) hden
+        have hs := hsqJ.trans hJE
+        have hm := mul_le_mul hqE hs (Real.sqrt_nonneg _) (Real.exp_nonneg _)
+        nlinarith
+      _ = 4*Real.exp ((l:ℝ)/250+(l:ℝ)/32-(l:ℝ)/8) := by
+        dsimp [E]
+        rw [mul_assoc,←Real.exp_add,mul_div_assoc,←Real.exp_sub]
+      _ ≤ _ := by
+        apply mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr _) (by norm_num)
+        linarith
+  change Real.log q / (J:ℝ) + 2*q*Real.sqrt J / Real.sqrt ((2:ℝ)^r) ≤ _
+  linarith
+
+lemma cycleCutoff_pos (l : ℕ) : 0 < ⌈Real.exp ((l:ℝ)/32)⌉₊ := by
+  have h := Nat.le_ceil (Real.exp ((l:ℝ)/32))
+  have he := Real.exp_pos ((l:ℝ)/32)
+  have : (0:ℝ) < ⌈Real.exp ((l:ℝ)/32)⌉₊ := lt_of_lt_of_le he h
+  exact_mod_cast this
+
+
+lemma pow_sum_le_mean_powers {ι : Type*} [Fintype ι] [Nonempty ι]
+    (z : ι → ℕ) (q : ℝ) (hq : 0 < q) :
+    q^(∑ i, z i) ≤ finiteMean (fun i => q^(Fintype.card ι*z i)) := by
+  have hn : (Fintype.card ι:ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
+  have hm : finiteMean (fun i => (Fintype.card ι*z i:ℕ)*Real.log q) =
+      (∑ i, z i:ℕ)*Real.log q := by
+    simp only [finiteMean,Nat.cast_mul,Nat.cast_sum,←Finset.mul_sum,←Finset.sum_mul]
+    field_simp
+  have he := exp_finiteMean_le (fun i => (Fintype.card ι*z i:ℕ)*Real.log q)
+  rw [hm,Real.exp_nat_mul,Real.exp_log hq] at he
+  simpa only [Real.exp_nat_mul,Real.exp_log hq] using he
+
+noncomputable def palindromeSlabCost (l d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) : ℕ :=
+  ∑ i : Fin l, palindromeLevelCost (l+i.1+1) d e ω
+
+lemma palindromeSlabCost_adapted (l d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) :
+    LayerAdapted d (2*l) (fun Y => palindromeSlabCost l d e (Y,X)) := by
+  apply layerAdapted_sum
+  intro i _
+  exact layerAdapted_mono (palindromeLevelCost_adapted (l+i.1+1) d e X) (by omega)
+
+theorem palindrome_slab_partial_mgf (r s l : ℕ) (hl : 0 < l) (hs : 2*s ≤ l)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+r)) (X : SwitchIndex (s+r) → Bool)
+    (lo : Card r × SwitchIndex s → Bool) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(palindromeSlabCost l (s+r) e (assembleBits r s lo hi,X))) ≤
+      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
+  classical
+  let : NeZero l := ⟨ne_of_gt hl⟩
+  have hqp : 0 < q := lt_of_lt_of_le (by norm_num) hq
+  calc
+    _ ≤ finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      finiteMean (fun i : Fin l => q^(l*palindromeLevelCost (l+i.1+1) (s+r) e
+        (assembleBits r s lo hi,X)))) := by
+      apply finiteMean_mono
+      intro hi
+      simpa only [palindromeSlabCost,Fintype.card_fin] using
+        pow_sum_le_mean_powers (fun i : Fin l => palindromeLevelCost (l+i.1+1) (s+r) e
+          (assembleBits r s lo hi,X)) q hqp
+    _ = finiteMean (fun i : Fin l => finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(l*palindromeLevelCost (l+i.1+1) (s+r) e (assembleBits r s lo hi,X)))) := finiteMean_comm _
+    _ ≤ finiteMean (fun _i : Fin l =>
+      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64)))) := by
+      apply finiteMean_mono
+      intro i
+      by_cases ht : l+i.1+1 ≤ s+r
+      · have hs' : s ≤ l+i.1 := by omega
+        have he : l+i.1+1 = s+(l+i.1-s)+1 := by omega
+        have hu : l+i.1-s < r := by omega
+        have hm := palindrome_level_partial_mgf r s (l+i.1-s) hu e X lo
+          ⌈Real.exp ((l:ℝ)/32)⌉₊ (cycleCutoff_pos l) (q^l) (one_le_pow₀ hq)
+        have hc := cycleCoefficient_decay l (l+i.1-s) (by
+          rw [Nat.cast_sub hs',Nat.cast_add]
+          have hsr : (2:ℝ)*s ≤ l := by exact_mod_cast hs
+          have hi : (0:ℝ) ≤ i.1 := Nat.cast_nonneg _
+          linarith) (q^l) (one_le_pow₀ hq) (by
+            rw [Real.log_pow]
+            nlinarith [mul_le_mul_of_nonneg_left hlog (Nat.cast_nonneg l)])
+        simp_rw [he,pow_mul]
+        exact hm.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left hc (Nat.cast_nonneg _)))
+      · have hz := palindromeLevelCost_gt (l+i.1+1) (s+r) (by omega) e
+        simp only [hz,Nat.mul_zero,pow_zero,finiteMean_const]
+        exact Real.one_le_exp_iff.mpr (by positivity)
+    _ = _ := finiteMean_const _
+
+
+lemma palindromeSlabCost_gt (l d : ℕ) (h : d ≤ l) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) : palindromeSlabCost l d e ω = 0 := by
+  apply Finset.sum_eq_zero
+  intro i _
+  exact palindromeLevelCost_gt _ _ (by omega) _ _
+
+lemma palindrome_slab_adapted_mul_le (d s l : ℕ) (hsd : s ≤ d) (hl : 0 < l)
+    (hs : 2*s ≤ l) {ι : Type*} [Fintype ι] (e : ι ↪ Card d)
+    (X : SwitchIndex d → Bool) (F : (SwitchIndex d → Bool) → ℝ)
+    (hF : ∀ Y, 0 ≤ F Y) (hA : LayerAdapted d s F) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun Y => F Y*q^(palindromeSlabCost l d e (Y,X))) ≤
+      finiteMean F * Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
+  obtain ⟨r,rfl⟩ := Nat.exists_eq_add_of_le hsd
+  exact adapted_mul_mean_le r s F _ hF hA _
+    (fun lo => palindrome_slab_partial_mgf r s l hl hs e X lo q hq hlog)
+
+def familyEnd (L n : ℕ) : ℕ := if n = 0 then 0 else 2*(L*4^(n-1))
+
+lemma familyEnd_two (L n : ℕ) : 2*familyEnd L n ≤ L*4^n := by
+  cases n with
+  | zero => simp [familyEnd]
+  | succ n => simp only [familyEnd,Nat.succ_ne_zero,ite_false,Nat.add_one_sub_one,pow_succ]; nlinarith
+
+lemma familyEnd_ge (L n j : ℕ) (hj : j < n) : 2*(L*4^j) ≤ familyEnd L n := by
+  cases n with
+  | zero => omega
+  | succ n =>
+    simp only [familyEnd,Nat.succ_ne_zero,ite_false,Nat.add_one_sub_one]
+    apply Nat.mul_le_mul_left
+    exact Nat.mul_le_mul_left L (Nat.pow_le_pow_right (by decide) (by omega : j ≤ n))
+
+noncomputable def palindromeFamilyCost (L n d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) : ℕ :=
+  ∑ j ∈ Finset.range n, palindromeSlabCost (L*4^j) d e ω
+
+lemma palindromeFamilyCost_adapted (L n d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) :
+    LayerAdapted d (familyEnd L n) (fun Y => palindromeFamilyCost L n d e (Y,X)) := by
+  apply layerAdapted_sum
+  intro j hj
+  exact layerAdapted_mono (palindromeSlabCost_adapted (L*4^j) d e X)
+    (familyEnd_ge L n j (Finset.mem_range.mp hj))
+
+end Thorp
+
+end ThorpNine.Smoothing
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Smoothing/PriorPaths.lean b/lean/OAI/Probability/ThorpRouting/Smoothing/PriorPaths.lean
index 1dcb9a7d8fe48c51b79e98360af426bcb5e14afe..dbbfaf88e7e2f24fa0c2152c16c97979ee0dc14f 100644
--- a/lean/OAI/Probability/ThorpRouting/Smoothing/PriorPaths.lean
+++ b/lean/OAI/Probability/ThorpRouting/Smoothing/PriorPaths.lean
@@ -1,882 +1,882 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Smoothing.AssignmentSlots
-
-namespace OAI
-
-namespace ThorpNine.Smoothing
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-namespace RoutingNetwork
-variable {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι] [DecidableEq ι]
-  {d : ℕ} (R : RoutingNetwork α ι d)
-lemma mem_exposedPaths (b : ι → Bool) (V : Finset (α)) (s : ι) :
-    s ∈ exposedPaths R b V ↔ (R.users b s).1 ∈ V ∨ (R.users b s).2 ∈ V := by
-  classical
-  simp only [exposedPaths,Finset.mem_biUnion,R.mem_path]
-  constructor
-  · rintro ⟨x,hx,h | h⟩
-    · exact Or.inl (h ▸ hx)
-    · exact Or.inr (h ▸ hx)
-  · rintro (h | h)
-    · exact ⟨_,h,Or.inl rfl⟩
-    · exact ⟨_,h,Or.inr rfl⟩
-
-lemma terminalCharge_sum (b : ι → Bool) (z : α) (V : Finset (α))
-    (hz : z ∉ V) :
-    (terminalCharge R b z V : ℝ) = ∑ s : ι,
-      ((if (R.users b s).1 = z ∧ (R.users b s).2 ∈ V then (1:ℝ) else 0) +
-      (if (R.users b s).2 = z ∧ (R.users b s).1 ∈ V then (1:ℝ) else 0)) := by
-  classical
-  have he : R.path b z ∩ exposedPaths R b V =
-      Finset.univ.filter (fun s =>
-        ((R.users b s).1 = z ∧ (R.users b s).2 ∈ V) ∨
-        ((R.users b s).2 = z ∧ (R.users b s).1 ∈ V)) := by
-    ext s
-    simp only [Finset.mem_inter,R.mem_path,mem_exposedPaths,
-      Finset.mem_filter,Finset.mem_univ,true_and]
-    by_cases h₁ : (R.users b s).1 = z <;>
-      by_cases h₂ : (R.users b s).2 = z <;> simp [h₁,h₂,hz]
-  rw [terminalCharge,he]
-  simp only [← Finset.sum_boole]
-  apply Finset.sum_congr rfl
-  intro s _
-  by_cases h₁ : (R.users b s).1 = z <;>
-    by_cases h₂ : (R.users b s).2 = z <;> simp [h₁,h₂,hz]
-
-noncomputable def priorPaths {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) : Finset (α) :=
-  Finset.univ.filter (fun x => ∃ i : Fin A, o x = some i ∧ i.val < t)
-
-omit [DecidableEq α] in
-@[simp] lemma mem_priorPaths {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) (x : α) :
-    x ∈ priorPaths o t ↔ ∃ i : Fin A, o x = some i ∧ i.val < t := by
-  classical
-  simp [priorPaths]
-
-lemma terminalCharge_prior_sum {A : ℕ} (o : α → Option (Fin A))
-    (b : ι → Bool) (z : Terminal o) (i : Fin A) :
-    (terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) =
-      ∑ s : ι,
-        ((if (z i).val = (R.users b s).1 ∧ (R.users b s).2 ∈ priorPaths o (i.val+1)
-          then (1:ℝ) else 0) +
-        (if (z i).val = (R.users b s).2 ∧ (R.users b s).1 ∈ priorPaths o (i.val+1)
-          then (1:ℝ) else 0)) := by
-  classical
-  rw [terminalCharge_sum _ _ _ _ (Finset.notMem_erase _ _)]
-  apply Finset.sum_congr rfl
-  intro s _
-  have hn := R.users_ne b s
-  simp only [Finset.mem_erase]
-  by_cases h₁ : (z i).val = (R.users b s).1 <;>
-    by_cases h₂ : (z i).val = (R.users b s).2 <;> simp_all [eq_comm]
-
-lemma rotationMeeting_eq_slot_sum {A : ℕ} (o : α → Option (Fin A))
-    (z : Terminal o) (u v : α) :
-    rotationMeeting o z u v = ∑ i : Fin A,
-      ((if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) +
-      (if (z i).val = v ∧ u ∈ priorPaths o (i.val+1) then (1:ℝ) else 0)) := by
-  classical
-  have only_owner (u : α) (a : Fin A) (hou : o u = some a)
-      (i : Fin A) (hia : i ≠ a) : (z i).val ≠ u := by
-    intro h
-    have hi := (z i).property
-    rw [h,hou] at hi
-    exact hia (Option.some.inj hi).symm
-  have sum_one (u v : α) (a : Fin A) (hou : o u = some a) :
-      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) =
-      if (z a).val = u ∧ v ∈ priorPaths o (a.val+1) then 1 else 0 := by
-    apply Finset.sum_eq_single a
-    · intro i _ hia
-      simp [only_owner u a hou i hia]
-    · simp
-  have sum_none (u v : α) (hou : o u = none) :
-      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) = 0 := by
-    apply Finset.sum_eq_zero
-    intro i _
-    have hn : (z i).val ≠ u := by
-      intro h
-      have hi := (z i).property
-      rw [h,hou] at hi
-      contradiction
-    simp [hn]
-  rw [Finset.sum_add_distrib]
-  cases hou : o u with
-  | none =>
-    rw [sum_none u v hou]
-    simp [rotationMeeting,hou,mem_priorPaths]
-  | some a =>
-    rw [sum_one u v a hou]
-    cases hov : o v with
-    | none =>
-      rw [sum_none v u hov]
-      simp [rotationMeeting,hou,hov,mem_priorPaths]
-    | some b =>
-      rw [sum_one v u b hov]
-      simp only [rotationMeeting,hou,hov,mem_priorPaths,Option.some.injEq]
-      rcases lt_trichotomy a b with hab | rfl | hba
-      · have hn : ¬ b.val < a.val+1 := by omega
-        have hy : a.val < b.val+1 := by omega
-        simp [hab,hn,hy]
-      · simp
-      · have hn : ¬ a.val < b.val+1 := by omega
-        have hy : b.val < a.val+1 := by omega
-        simp [hba,not_lt_of_ge (le_of_lt hba),hn,hy]
-
-lemma rotationCharge_eq_terminal_sum {A : ℕ} (o : α → Option (Fin A))
-    (b : ι → Bool) (z : Terminal o) :
-    rotationCharge R o b z = ∑ i : Fin A,
-      (terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) := by
-  simp only [rotationCharge,rotationMeeting_eq_slot_sum,terminalCharge_prior_sum]
-  exact Finset.sum_comm
-
-omit [DecidableEq α] in
-@[simp] lemma priorPaths_zero {A : ℕ} (o : α → Option (Fin A)) : priorPaths o 0 = ∅ := by
-  classical
-  ext x
-  simp
-
-lemma priorPaths_succ {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) (ht : t < A) :
-    priorPaths o (t+1) = priorPaths o t ∪ cycleFiber o ⟨t,ht⟩ := by
-  classical
-  ext x
-  simp only [mem_priorPaths,Finset.mem_union,mem_cycleFiber]
-  constructor
-  · rintro ⟨i,hi,hit⟩
-    by_cases h : i.val < t
-    · exact Or.inl ⟨i,hi,h⟩
-    · have he : i = ⟨t,ht⟩ := Fin.ext (by change i.val = t; omega)
-      exact Or.inr (he ▸ hi)
-  · rintro (⟨i,hi,hit⟩ | hi)
-    · exact ⟨i,hi,by omega⟩
-    · exact ⟨⟨t,ht⟩,hi,by change t < t+1; omega⟩
-
-omit [DecidableEq α] in
-lemma priorPaths_disjoint {A : ℕ} (o : α → Option (Fin A)) (i : Fin A) :
-    Disjoint (priorPaths o i.val) (cycleFiber o i) := by
-  classical
-  apply Finset.disjoint_left.mpr
-  intro x hx hx'
-  obtain ⟨j,hj,hji⟩ := (mem_priorPaths _ _ _).mp hx
-  have he : j = i := Option.some.inj (hj.symm.trans ((mem_cycleFiber _ _ _).mp hx'))
-  subst j
-  exact (lt_irrefl _ hji)
-
-namespace QueryTree
-open Thorp.QueryTree
-
-lemma seekCycles_owned {A : ℕ} (S : Finset (α)) (r : Fin A → ℕ)
-    (o : α → Option (Fin A)) (z : Terminal o) (ω : ι → Bool)
-    (hS : ∀ i, cycleFiber o i ⊆ S)
-    (hO : ∀ i, orbitSet (R.perm ω) (z i).val = cycleFiber o i)
-    (hC : ∀ i, (cycleFiber o i).card = r i+1) :
-    let p := R.perm ω
-    let L := List.ofFn (fun i : Fin A => (p (z i).val,r i))
-    (seekCycles R S L ∅).succeeds ω = true ∧
-      ((seekCycles R S L ∅).staleCharge ∅ ω : ℝ) = rotationCharge R o (ω) z := by
-  classical
-  let b := ω
-  let p := R.perm b
-  let f (i : Fin A) := (p (z i).val,r i)
-  let c (i : Fin A) : ℝ := terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val)
-  have hc (i : Fin A) : Function.minimalPeriod p (z i).val - 1 = r i := by
-    have he := orbitSet_card p (z i).val
-    rw [hO,hC] at he
-    omega
-  have hs (t : ℕ) (ht : t ≤ A) :
-      (seekCycles R S ((List.ofFn f).drop t) (priorPaths o t)).succeeds ω = true ∧
-        ((seekCycles R S ((List.ofFn f).drop t) (priorPaths o t)).staleCharge (exposedPaths R b (priorPaths o t)) ω : ℝ) =
-          ((List.ofFn c).drop t).sum := by
-    induction ht using Nat.decreasingInduction with
-    | self => simp only [List.length_ofFn,le_refl,List.drop_eq_nil_of_le,List.sum_nil,seekCycles,Thorp.QueryTree.succeeds,staleCharge,Nat.cast_zero,and_self]
-    | of_succ t ht ih =>
-      have htf : t < (List.ofFn f).length := by simpa using ht
-      have htc : t < (List.ofFn c).length := by simpa using ht
-      rw [List.drop_eq_getElem_cons htf,List.drop_eq_getElem_cons htc]
-      simp only [List.getElem_ofFn,List.sum_cons,seekCycles]
-      have h := seekCycle_orbit R S (priorPaths o t) (z ⟨t,ht⟩).val
-        (fun V => seekCycles R S ((List.ofFn f).drop (t+1)) V) ω
-        (by rw [hO]; exact hS _) (by rw [hO]; exact priorPaths_disjoint o ⟨t,ht⟩)
-      dsimp only at h
-      rw [hc,hO,←priorPaths_succ o t ht] at h
-      dsimp only [f]
-      constructor
-      · exact h.1.trans ih.1
-      · rw [h.2,Nat.cast_add,ih.2]
-        exact add_comm _ _
-  have h := hs 0 (Nat.zero_le _)
-  simp only [List.drop_zero,priorPaths_zero,exposedPaths_empty,List.sum_ofFn] at h
-  dsimp only
-  exact ⟨h.1,h.2.trans (rotationCharge_eq_terminal_sum R o b z).symm⟩
-
-end QueryTree
-section Collections
-variable {A : ℕ}
-
-def ValidCycleSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ)
-    (o : α → Option (Fin A)) : Prop :=
-  (∀ i, (cycleFiber o i).card = r i + 1) ∧
-  (∀ i, cycleFiber o i ⊆ S) ∧
-  (∀ i x, o x = some i → orbitSet p x = cycleFiber o i)
-
-abbrev CycleSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
-  {o : α → Option (Fin A) // ValidCycleSlots p S r o}
-
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
-    Fintype (CycleSlots p S r) := Fintype.ofFinite _
-
-lemma CycleSlots.terminal_nonempty {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (c : CycleSlots p S r) (i : Fin A) : Nonempty {x : α // c.val x = some i} := by
-  have hn : (cycleFiber c.val i).Nonempty := Finset.card_pos.mp (by rw [c.property.1]; omega)
-  obtain ⟨x,hx⟩ := hn
-  exact ⟨x,(mem_cycleFiber _ _ _).mp hx⟩
-
-omit [DecidableEq α] in
-lemma owner_ext {o o' : α → Option (Fin A)}
-    (h : ∀ i, cycleFiber o i = cycleFiber o' i) : o = o' := by
-  funext x
-  have he (i : Fin A) : o x = some i ↔ o' x = some i := by
-    simpa only [mem_cycleFiber] using (Finset.ext_iff.mp (h i) x)
-  cases ho : o x with
-  | some i => exact ((he i).mp ho).symm
-  | none =>
-    cases ho' : o' x with
-    | none => rfl
-    | some i => have hn := (he i).mpr ho'; rw [ho] at hn; contradiction
-
-abbrev RootedSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
-  Σ c : CycleSlots p S r, Terminal c.val
-
-noncomputable def rootedStart {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (c : RootedSlots p S r) : Fin A → S := fun i =>
-  ⟨p (c.2 i).val,c.1.property.2.1 i (by
-    rw [←c.1.property.2.2 i (c.2 i).val (c.2 i).property]
-    exact (mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.apply_right (Equiv.Perm.SameCycle.refl p _)))⟩
-
-lemma rootedStart_injective {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ} :
-    Function.Injective (rootedStart (p:=p) (S:=S) (r:=r)) := by
-  rintro ⟨c,z⟩ ⟨c',z'⟩ h
-  have hz (i : Fin A) : (z i).val = (z' i).val :=
-    p.injective (congrArg Subtype.val (congrFun h i))
-  have hc : c = c' := Subtype.ext (owner_ext (fun i => by
-    rw [←c.property.2.2 i (z i).val (z i).property,
-      ←c'.property.2.2 i (z' i).val (z' i).property,hz]))
-  subst c'
-  have hzz : z = z' := funext (fun i => Subtype.ext (hz i))
-  subst z'
-  rfl
-
-lemma rooted_weight_sum_le (S : Finset (α)) (r : Fin A → ℕ) (ω : ι → Bool) :
-    (∑ c : RootedSlots (R.perm ω) S r,
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2)) ≤
-    ∑ a : Fin A → S,
-      if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-        ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-      else 0 := by
-  classical
-  let p := R.perm ω
-  let g (a : Fin A → S) : ℝ :=
-    if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-    else 0
-  have he (c : RootedSlots p S r) :
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2) =
-      g (rootedStart c) := by
-    have hs := QueryTree.seekCycles_owned R S r c.1.val c.2 ω c.1.property.2.1
-      (fun i => c.1.property.2.2 i (c.2 i).val (c.2 i).property) c.1.property.1
-    dsimp only at hs
-    unfold g
-    rw [ite_eq_left (show (QueryTree.seekCycles R S (List.ofFn (fun i => ((rootedStart c i).val,r i))) ∅).succeeds ω = true from hs.1)]
-    rw [← hs.2,neg_mul,Real.exp_neg,mul_comm (Real.log 2),Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
-    rfl
-  simp_rw [he]
-  exact sum_comp_injective_le rootedStart rootedStart_injective g (fun a => by dsimp [g]; split_ifs <;> positivity)
-
-lemma collection_rotation_lower (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r)
-    (ω : ι → Bool) (c : CycleSlots (R.perm ω) S r) :
-    collectionWeight d r ≤ ∑ z : Terminal c.val,
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z) := by
-  classical
-  let (i : Fin A) : Nonempty {x : α // c.val x = some i} := c.terminal_nonempty i
-  have hm : Monotone (fun i => (cycleFiber c.val i).card) := by
-    intro i j hij
-    dsimp only
-    rw [c.property.1,c.property.1]
-    exact Nat.add_le_add_right (hr hij) 1
-  have hb := rotation_exp_bound R hm (ω)
-  have hcard : (Fintype.card (Terminal c.val) : ℝ) = ∏ i : Fin A, (r i+1 : ℝ) := by
-    rw [Fintype.card_pi,Nat.cast_prod]
-    apply Finset.prod_congr rfl
-    intro i _
-    rw [card_cycleFiber,c.property.1,Nat.cast_add,Nat.cast_one]
-  have hp : (0:ℝ) < Fintype.card (Terminal c.val) := Nat.cast_pos.mpr Fintype.card_pos
-  have hl : rotationLower d r ≤ finiteMean (fun z : Terminal c.val => Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z)) := by
-    simpa only [rotationLower,c.property.1,Nat.cast_add,Nat.cast_one] using hb
-  have hh := (le_div_iff₀ hp).mp hl
-  rw [hcard] at hh
-  unfold collectionWeight
-  rw [← Finset.mul_sum]
-  calc
-    _ = ((2:ℝ)^d)^A * (rotationLower d r * ∏ i : Fin A, (r i+1 : ℝ)) := by ring
-    _ ≤ _ := mul_le_mul_of_nonneg_left hh (by positivity)
-
-lemma cycleSlots_weighted_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : ι → Bool =>
-      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) *
-      collectionWeight d r ≤ (S.card : ℝ)^A := by
-  classical
-  let g (a : Fin A → S) (ω : ι → Bool) : ℝ :=
-    if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-    else 0
-  have hω (ω : ι → Bool) :
-      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ) * collectionWeight d r ≤
-        ∑ a : Fin A → S, g a ω := by
-    calc
-      _ = ∑ _c : CycleSlots (R.perm ω) S r, collectionWeight d r := by simp
-      _ ≤ ∑ c : CycleSlots (R.perm ω) S r,
-          ∑ z : Terminal c.val, ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z) :=
-        Finset.sum_le_sum (fun c _ => collection_rotation_lower R S r hr ω c)
-      _ = ∑ c : RootedSlots (R.perm ω) S r,
-          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2) :=
-        (Fintype.sum_sigma (fun c : RootedSlots (R.perm ω) S r =>
-          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2))).symm
-      _ ≤ _ := rooted_weight_sum_le R S r ω
-  have hb := finiteMean_mono hω
-  rw [finiteMean_mul_const,finiteMean_sum] at hb
-  calc
-    _ ≤ ∑ a : Fin A → S, finiteMean (g a) := hb
-    _ ≤ ∑ _a : Fin A → S, (1:ℝ) := by
-      apply Finset.sum_le_sum
-      intro a _
-      have hh := QueryTree.cycles_weighted_success R S (List.ofFn (fun i : Fin A => ((a i).val,r i)))
-      simpa only [List.length_ofFn] using hh
-    _ = _ := by simp
-
-lemma cycleSlots_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : ι → Bool =>
-      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
-        (S.card : ℝ)^A / collectionWeight d r := by
-  exact (le_div_iff₀ (collectionWeight_pos d r)).mpr (cycleSlots_weighted_moment R S r hr)
-
-lemma butterfly_cycleSlots_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : ι → Bool =>
-      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
-        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
-  have h := cycleSlots_moment R S r hr
-  rw [collectionWeight_eq_sqrt,div_mul_eq_div_mul_one_div] at h
-  rw [div_pow]
-  simpa only [one_div,Finset.prod_inv_distrib] using h
-
-end Collections
-
-section CycleCounting
-variable {A : ℕ}
-
-abbrev SelectedCycle (p : Equiv.Perm (α)) (S : Finset (α)) :=
-  {C : Finset (α) // (∃ x, C = orbitSet p x) ∧ C ⊆ S}
-
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) :
-    Fintype (SelectedCycle p S) := Fintype.ofFinite _
-
-lemma SelectedCycle.nonempty {p : Equiv.Perm (α)} {S : Finset (α)}
-    (C : SelectedCycle p S) : C.val.Nonempty := by
-  obtain ⟨x,hx⟩ := C.property.1
-  rw [hx]
-  exact orbitSet_nonempty p x
-
-lemma SelectedCycle.orbit_eq {p : Equiv.Perm (α)} {S : Finset (α)}
-    (C : SelectedCycle p S) {x : α} (hx : x ∈ C.val) : orbitSet p x = C.val := by
-  obtain ⟨y,hy⟩ := C.property.1
-  rw [hy] at hx ⊢
-  exact (orbitSet_eq p ((mem_orbitSet _ _ _).mp hx)).symm
-
-lemma SelectedCycle.eq_of_mem {p : Equiv.Perm (α)} {S : Finset (α)}
-    (C D : SelectedCycle p S) {x : α} (hC : x ∈ C.val) (hD : x ∈ D.val) : C = D := by
-  apply Subtype.ext
-  exact (C.orbit_eq hC).symm.trans (D.orbit_eq hD)
-
-abbrev SlotAssignment (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
-  {f : Fin A → SelectedCycle p S // Function.Injective f ∧ ∀ i, (f i).val.card = r i+1}
-
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
-    Fintype (SlotAssignment p S r) := Fintype.ofFinite _
-
-noncomputable def slotsToAssignment {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (c : CycleSlots p S r) : SlotAssignment p S r := by
-  classical
-  let f (i : Fin A) : SelectedCycle p S := ⟨cycleFiber c.val i,by
-    obtain ⟨x,hx⟩ := c.terminal_nonempty i
-    exact ⟨⟨x,(c.property.2.2 i x hx).symm⟩,c.property.2.1 i⟩⟩
-  refine ⟨f,?_,fun i => c.property.1 i⟩
-  intro i j hij
-  have he : cycleFiber c.val i = cycleFiber c.val j := congrArg Subtype.val hij
-  obtain ⟨x,hx⟩ := c.terminal_nonempty i
-  have hxi : x ∈ cycleFiber c.val i := (mem_cycleFiber _ _ _).mpr hx
-  rw [he] at hxi
-  exact Option.some.inj (hx.symm.trans ((mem_cycleFiber _ _ _).mp hxi))
-
-noncomputable def assignmentOwner {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) (x : α) : Option (Fin A) := by
-  classical
-  exact if h : ∃ i, x ∈ (f.val i).val then some (Classical.choose h) else none
-
-lemma assignmentOwner_eq_some {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) (x : α) (i : Fin A) :
-    assignmentOwner f x = some i ↔ x ∈ (f.val i).val := by
-  classical
-  unfold assignmentOwner
-  split_ifs with h
-  · constructor
-    · intro hi
-      have hx := Classical.choose_spec h
-      rw [Option.some.inj hi] at hx
-      exact hx
-    · intro hx
-      have he := (f.val (Classical.choose h)).eq_of_mem (f.val i) (Classical.choose_spec h) hx
-      exact congrArg some (f.property.1 he)
-  · constructor
-    · intro hi; contradiction
-    · intro hx; exact (h ⟨i,hx⟩).elim
-
-lemma assignmentOwner_fiber {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) (i : Fin A) : cycleFiber (assignmentOwner f) i = (f.val i).val := by
-  ext x
-  simp only [mem_cycleFiber,assignmentOwner_eq_some]
-
-noncomputable def assignmentToSlots {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) : CycleSlots p S r :=
-  ⟨assignmentOwner f,by
-    refine ⟨?_,?_,?_⟩
-    · intro i
-      rw [assignmentOwner_fiber]
-      exact f.property.2 i
-    · intro i
-      rw [assignmentOwner_fiber]
-      exact (f.val i).property.2
-    · intro i x hx
-      rw [assignmentOwner_fiber]
-      exact (f.val i).orbit_eq ((assignmentOwner_eq_some f x i).mp hx)⟩
-
-noncomputable def cycleSlotsEquivAssignments (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
-    CycleSlots p S r ≃ SlotAssignment p S r where
-  toFun := slotsToAssignment
-  invFun := assignmentToSlots
-  left_inv c := by
-    apply Subtype.ext
-    apply owner_ext
-    intro i
-    rw [assignmentToSlots,assignmentOwner_fiber]
-    rfl
-  right_inv f := by
-    apply Subtype.ext
-    funext i
-    apply Subtype.ext
-    exact assignmentOwner_fiber f i
-
-end CycleCounting
-
-section GroupedCounting
-variable {A : ℕ}
-
-abbrev LengthClass (r : Fin A → ℕ) := {j : ℕ // j ∈ Finset.univ.image (fun i => r i + 1)}
-
-def slotClass (r : Fin A → ℕ) (i : Fin A) : LengthClass r :=
-  ⟨r i+1, Finset.mem_image.mpr ⟨i, Finset.mem_univ _, rfl⟩⟩
-
-abbrev SlotsOfLength (r : Fin A → ℕ) (j : LengthClass r) := {i : Fin A // slotClass r i = j}
-abbrev CyclesOfLength (p : Equiv.Perm (α)) (S : Finset (α)) (j : ℕ) :=
-  {C : SelectedCycle p S // C.val.card = j}
-
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (j : ℕ) :
-    Fintype (CyclesOfLength p S j) := Fintype.ofFinite _
-
-abbrev GroupedAssignments (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
-  ∀ j : LengthClass r, SlotsOfLength r j ↪ CyclesOfLength p S j.val
-
-noncomputable def assignmentToGroups {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) : GroupedAssignments p S r := fun _j =>
-  { toFun := fun i => ⟨f.val i.val, (f.property.2 i.val).trans (congrArg Subtype.val i.property)⟩
-    inj' := fun _i _k h => Subtype.ext (f.property.1 (congrArg Subtype.val h)) }
-
-noncomputable def groupsEmbedding {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (F : GroupedAssignments p S r) : Fin A ↪ SelectedCycle p S :=
-  ((Equiv.sigmaFiberEquiv (slotClass r)).symm.toEmbedding.trans
-    (Function.Embedding.sigmaMap (Function.Embedding.refl _) F)).trans
-    ((Function.Embedding.sigmaMap (Function.Embedding.subtype _) (fun _ => Function.Embedding.refl _)).trans
-      (Equiv.sigmaFiberEquiv (fun C : SelectedCycle p S => C.val.card)).toEmbedding)
-
-noncomputable def groupsToAssignment {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (F : GroupedAssignments p S r) : SlotAssignment p S r :=
-  ⟨groupsEmbedding F, (groupsEmbedding F).injective,
-    fun i => (F (slotClass r i) ⟨i,rfl⟩).property⟩
-
-noncomputable def assignmentEquivGroups (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
-    SlotAssignment p S r ≃ GroupedAssignments p S r where
-  toFun := assignmentToGroups
-  invFun := groupsToAssignment
-  left_inv f := by
-    apply Subtype.ext
-    rfl
-  right_inv F := by
-    funext j
-    apply Function.Embedding.ext
-    intro i
-    rcases i with ⟨i,hi⟩
-    subst j
-    rfl
-
-lemma card_cycleSlots_factorial (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
-    Fintype.card (CycleSlots p S r) =
-      ∏ j : LengthClass r, (Fintype.card (CyclesOfLength p S j.val)).descFactorial
-        (Fintype.card (SlotsOfLength r j)) := by
-  classical
-  rw [Fintype.card_congr (cycleSlotsEquivAssignments p S r),
-    Fintype.card_congr (assignmentEquivGroups p S r), Fintype.card_pi]
-  simp only [Fintype.card_embedding_eq]
-
-lemma butterfly_factorial_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : ι → Bool =>
-       (∏ j : LengthClass r,
-         (Fintype.card (CyclesOfLength (R.perm ω) S j.val)).descFactorial
-           (Fintype.card (SlotsOfLength r j)) : ℕ)) ≤
-      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-        ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
-  simpa only [←card_cycleSlots_factorial] using butterfly_cycleSlots_moment R S r hr
-
-end GroupedCounting
-
-section AllLengths
-variable {A : ℕ}
-
-noncomputable def assignmentRelabel (p : Equiv.Perm (α)) (S : Finset (α))
-    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
-    SlotAssignment p S r ≃ SlotAssignment p S (r ∘ e) where
-  toFun f := ⟨f.val ∘ e, f.property.1.comp e.injective, fun i => f.property.2 (e i)⟩
-  invFun f := ⟨f.val ∘ e.symm, f.property.1.comp e.symm.injective, fun i => by
-    simpa only [Function.comp_apply,Equiv.apply_symm_apply] using f.property.2 (e.symm i)⟩
-  left_inv f := by apply Subtype.ext; funext i; simp
-  right_inv f := by apply Subtype.ext; funext i; simp
-
-lemma card_cycleSlots_relabel (p : Equiv.Perm (α)) (S : Finset (α))
-    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
-    Fintype.card (CycleSlots p S r) = Fintype.card (CycleSlots p S (r ∘ e)) := by
-  exact Fintype.card_congr ((cycleSlotsEquivAssignments p S r).trans
-    ((assignmentRelabel p S r e).trans (cycleSlotsEquivAssignments p S (r ∘ e)).symm))
-
-lemma butterfly_cycleSlots_moment_all (S : Finset (α)) (r : Fin A → ℕ) :
-    finiteMean (fun ω : ι → Bool =>
-      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
-        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
-  have h := butterfly_cycleSlots_moment R S (r ∘ Tuple.sort r) (Tuple.monotone_sort r)
-  simp_rw [←card_cycleSlots_relabel] at h
-  have hp := Equiv.prod_comp (Tuple.sort r) (fun i : Fin A => (1 / Real.sqrt (r i+1) : ℝ))
-  simpa only [Function.comp_apply,hp] using h
-
-abbrev ShortCycle (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :=
-  {C : SelectedCycle p S // C.val.card ≤ J}
-
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    Fintype (ShortCycle p S J) := Fintype.ofFinite _
-
-noncomputable def shortCycleLength {p : Equiv.Perm (α)} {S : Finset (α)} {J : ℕ}
-    (C : ShortCycle p S J) : Fin J :=
-  ⟨C.val.val.card-1, by have := C.val.nonempty.card_pos; have := C.property; omega⟩
-
-lemma shortCycleLength_add_one {p : Equiv.Perm (α)} {S : Finset (α)} {J : ℕ}
-    (C : ShortCycle p S J) : (shortCycleLength C : ℕ)+1 = C.val.val.card := by
-  have := C.val.nonempty.card_pos
-  simp only [shortCycleLength]
-  omega
-
-noncomputable def shortEmbeddingEquiv (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    (Fin A ↪ ShortCycle p S J) ≃
-      Σ r : Fin A → Fin J, SlotAssignment p S (fun i => (r i : ℕ)) where
-  toFun f := ⟨fun i => shortCycleLength (f i),
-    ⟨fun i => (f i).val, fun i j h => f.injective (Subtype.ext h),
-      fun i => (shortCycleLength_add_one (f i)).symm⟩⟩
-  invFun c :=
-    { toFun := fun i => ⟨c.2.val i, by rw [c.2.property.2]; exact (c.1 i).isLt⟩
-      inj' := fun _i _j h => c.2.property.1 (congrArg Subtype.val h) }
-  left_inv f := by
-    apply Function.Embedding.ext
-    intro i
-    rfl
-  right_inv c := by
-    rcases c with ⟨r,f⟩
-    have he : (fun i => shortCycleLength (⟨f.val i, by rw [f.property.2]; exact (r i).isLt⟩ : ShortCycle p S J)) = r := by
-      funext i
-      apply Fin.ext
-      simp only [shortCycleLength,f.property.2,Nat.add_sub_cancel]
-    apply Sigma.ext he
-    apply (Subtype.heq_iff_coe_eq (by
-      intro x
-      change (Function.Injective x ∧ ∀ i, (x i).val.card = (f.val i).val.card - 1 + 1) ↔ _
-      simp only [f.property.2,Nat.add_sub_cancel])).mpr
-    rfl
-
-lemma shortCycle_factorial_eq (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    (Fintype.card (ShortCycle p S J)).descFactorial A =
-      ∑ r : Fin A → Fin J, Fintype.card (CycleSlots p S (fun i => (r i : ℕ))) := by
-  classical
-  have hc := Fintype.card_embedding_eq (α := Fin A) (β := ShortCycle p S J)
-  simp only [Fintype.card_fin] at hc
-  rw [←hc, Fintype.card_congr (shortEmbeddingEquiv p S J),Fintype.card_sigma]
-  apply Finset.sum_congr rfl
-  intro r _
-  exact (Fintype.card_congr (cycleSlotsEquivAssignments p S (fun i => (r i : ℕ)))).symm
-
-end AllLengths
-
-section CycleMGF
-
-lemma butterfly_shortCycle_moment (S : Finset (α)) (J A : ℕ) :
-    finiteMean (fun ω : ι → Bool =>
-      ((Fintype.card (ShortCycle (R.perm ω) S J)).descFactorial A : ℝ)) ≤
-      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))^A := by
-  classical
-  simp_rw [shortCycle_factorial_eq,Nat.cast_sum,finiteMean_sum]
-  calc
-    _ ≤ ∑ r : Fin A → Fin J,
-        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-          ∏ i : Fin A, (1 / Real.sqrt ((r i : ℕ)+1)) := by
-      apply Finset.sum_le_sum
-      intro r _
-      exact butterfly_cycleSlots_moment_all R S (fun i => (r i : ℕ))
-    _ = _ := by
-      rw [←Finset.mul_sum,mul_pow]
-      congr 1
-      rw [←Fintype.prod_sum (fun (_ : Fin A) (j : Fin J) => (1 / Real.sqrt ((j:ℕ)+1) : ℝ))]
-      simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fin]
-
-noncomputable def cycleMass (p : Equiv.Perm (α)) (S : Finset (α)) : ℕ :=
-  ∑ C : SelectedCycle p S, C.val.card
-
-lemma cycleMass_le (p : Equiv.Perm (α)) (S : Finset (α)) : cycleMass p S ≤ S.card := by
-  classical
-  let f : (Σ C : SelectedCycle p S, {x : α // x ∈ C.val}) → S :=
-    fun c => ⟨c.2.val,c.1.property.2 c.2.property⟩
-  have hi : Function.Injective f := by
-    intro c e h
-    have hx : c.2.val = e.2.val := congrArg Subtype.val h
-    have hC : c.1 = e.1 := c.1.eq_of_mem e.1 c.2.property (hx ▸ e.2.property)
-    rcases c with ⟨c,x⟩
-    rcases e with ⟨e,y⟩
-    dsimp only at hC
-    subst e
-    have he : x=y := Subtype.ext hx
-    cases he
-    rfl
-  have hh := Fintype.card_le_of_injective f hi
-  simpa only [Fintype.card_sigma,Fintype.card_coe,cycleMass] using hh
-
-lemma shortCycle_card_le (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    Fintype.card (ShortCycle p S J) ≤ S.card := by
-  classical
-  calc
-    _ ≤ Fintype.card (SelectedCycle p S) := Fintype.card_subtype_le _
-    _ = ∑ _ : SelectedCycle p S, 1 := by simp
-    _ ≤ cycleMass p S := Finset.sum_le_sum (fun C _ => C.nonempty.card_pos)
-    _ ≤ _ := cycleMass_le p S
-
-lemma butterfly_shortCycle_mgf (S : Finset (α)) (J : ℕ) (q : ℝ) (hq : 1 ≤ q) :
-    finiteMean (fun ω : ι → Bool =>
-      q^(Fintype.card (ShortCycle (R.perm ω) S J))) ≤
-      Real.exp ((q-1)*((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))) := by
-  apply factorial_mgf _ S.card _ q (fun ω => shortCycle_card_le _ _ _)
-  · positivity
-  · exact hq
-  · exact butterfly_shortCycle_moment R S J
-
-end CycleMGF
-
-
-abbrev LongCycle (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :=
-  {C : SelectedCycle p S // ¬ C.val.card ≤ J}
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    Fintype (LongCycle p S J) := Fintype.ofFinite _
-
-lemma selectedCycle_card_split (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    Fintype.card (SelectedCycle p S) =
-      Fintype.card (ShortCycle p S J) + Fintype.card (LongCycle p S J) := by
-  classical
-  rw [←Fintype.card_congr (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J)),Fintype.card_sum]
-
-lemma longCycle_card_bound (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    J * Fintype.card (LongCycle p S J) ≤ S.card := by
-  classical
-  have hs := Equiv.sum_comp (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J))
-    (fun C : SelectedCycle p S => C.val.card)
-  rw [Fintype.sum_sum_type] at hs
-  have h0 : J * Fintype.card (LongCycle p S J) ≤
-      ∑ C : LongCycle p S J, C.val.val.card := by
-    calc
-      _ = ∑ _C : LongCycle p S J, J := by simp [mul_comm]
-      _ ≤ _ := Finset.sum_le_sum (fun C _ => Nat.le_of_lt (Nat.lt_of_not_ge C.property))
-  have hh := cycleMass_le p S
-  unfold cycleMass at hh
-  rw [←hs] at hh
-  exact h0.trans ((Nat.le_add_left _ _).trans hh)
-
-lemma longCycle_pow_bound (p : Equiv.Perm (α)) (S : Finset (α))
-    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    q^(Fintype.card (LongCycle p S J)) ≤ Real.exp ((S.card : ℝ)/J * Real.log q) := by
-  have hq0 : 0 < q := lt_of_lt_of_le zero_lt_one hq
-  have hc : (Fintype.card (LongCycle p S J) : ℝ) ≤ (S.card : ℝ)/J := by
-    apply (le_div_iff₀ (by exact_mod_cast hJ)).mpr
-    have hh := longCycle_card_bound p S J
-    exact_mod_cast (by simpa only [mul_comm] using hh)
-  calc
-    _ = Real.exp ((Fintype.card (LongCycle p S J) : ℝ)*Real.log q) := by
-      rw [Real.exp_nat_mul,Real.exp_log hq0]
-    _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right hc (Real.log_nonneg hq))
-
-lemma butterfly_cycle_log_mgf (S : Finset (α)) (J : ℕ) (hJ : 0 < J)
-    (q : ℝ) (hq : 1 ≤ q) :
-    Real.log (finiteMean (fun ω : ι → Bool =>
-      q^(Fintype.card (SelectedCycle (R.perm ω) S)))) ≤
-      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
-  let lam : ℝ := (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1)
-  let c : ℝ := (S.card : ℝ)/J * Real.log q
-  have hnum : (q-1)*lam ≤ 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
-    have hsum := sum_inv_sqrt J
-    have hp : 0 ≤ (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) := by positivity
-    have h0 : 0 ≤ 2*Real.sqrt J := by positivity
-    have hh := mul_le_mul_of_nonneg_left hsum hp
-    have hq1 : 0 ≤ q-1 := sub_nonneg.mpr hq
-    calc
-      _ ≤ (q-1)*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
-        mul_le_mul_of_nonneg_left hh hq1
-      _ ≤ q*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
-        mul_le_mul_of_nonneg_right (by linarith) (mul_nonneg hp h0)
-      _ = _ := by ring
-  have hm : finiteMean (fun ω : ι → Bool =>
-      q^(Fintype.card (SelectedCycle (R.perm ω) S))) ≤
-      Real.exp (c+(q-1)*lam) := by
-    calc
-      _ ≤ finiteMean (fun ω : ι → Bool =>
-          q^(Fintype.card (ShortCycle (R.perm ω) S J))*Real.exp c) := by
-        apply finiteMean_mono
-        intro ω
-        rw [selectedCycle_card_split _ _ J,pow_add]
-        exact mul_le_mul_of_nonneg_left (longCycle_pow_bound _ S J hJ q hq) (by positivity)
-      _ = finiteMean (fun ω : ι → Bool =>
-          q^(Fintype.card (ShortCycle (R.perm ω) S J))) * Real.exp c :=
-        finiteMean_mul_const _ _
-      _ ≤ Real.exp ((q-1)*lam) * Real.exp c :=
-        mul_le_mul_of_nonneg_right (butterfly_shortCycle_mgf R S J q hq) (Real.exp_nonneg _)
-      _ = _ := by rw [←Real.exp_add,add_comm]
-  have hpos : 0 < finiteMean (fun ω : ι → Bool =>
-      q^(Fintype.card (SelectedCycle (R.perm ω) S))) := by
-    apply finiteMean_pos
-    intro ω
-    positivity
-  have hlog := Real.log_le_log hpos hm
-  rw [Real.log_exp] at hlog
-  dsimp [c] at hlog
-  linarith
-
-
-end RoutingNetwork
-
-variable {α B : Type*} [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] {d : ℕ}
-
-def parallelPerm (ω : B × SwitchIndex d → Bool) : Equiv.Perm (B × Card d) where
-  toFun x := (x.1,butterflyPerm d (decodeButterfly d (fun i => ω (x.1,i))) x.2)
-  invFun y := (y.1,(butterflyPerm d (decodeButterfly d (fun i => ω (y.1,i)))).symm y.2)
-  left_inv x := by simp
-  right_inv y := by simp
-
-def relabeledPerm (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (ω : B × SwitchIndex d → Bool) : Equiv.Perm α :=
-  (pre.trans (parallelPerm ω)).trans post
-
-noncomputable def parallelPath (pre : α ≃ B × Card d)
-    (ω : B × SwitchIndex d → Bool) (x : α) : Finset (B × SwitchIndex d) :=
-  (routeDomain d (pre x).2
-    (butterflyPerm d (decodeButterfly d (fun i => ω ((pre x).1,i))) (pre x).2)).image
-      (fun i => ((pre x).1,i))
-
-def parallelUsers (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
-    (i : B × SwitchIndex d) : α × α :=
-  let u := switchUsers d (decodeButterfly d (fun j => ω (i.1,j))) i.2
-  (pre.symm (i.1,u.1),pre.symm (i.1,u.2))
-
-omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
-lemma parallelUsers_ne (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
-    (i : B × SwitchIndex d) : (parallelUsers pre ω i).1 ≠ (parallelUsers pre ω i).2 := by
-  intro h
-  exact switchUsers_ne d _ _ (congrArg Prod.snd (pre.symm.injective h))
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma mem_parallelPath (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
-    (x : α) (i : B × SwitchIndex d) :
-    i ∈ parallelPath pre ω x ↔ (parallelUsers pre ω i).1 = x ∨ (parallelUsers pre ω i).2 = x := by
-  classical
-  obtain ⟨b,j⟩ := i
-  simp only [parallelPath,Finset.mem_image,Prod.mk.injEq]
-  constructor
-  · rintro ⟨k,hk,hb,rfl⟩
-    subst b
-    rcases (mem_routeDomain_iff_user d _ _ _).mp hk with h | h
-    · left
-      apply pre.injective
-      change pre (pre.symm ((pre x).1,_)) = pre x
-      rw [Equiv.apply_symm_apply]
-      exact Prod.ext rfl h
-    · right
-      apply pre.injective
-      change pre (pre.symm ((pre x).1,_)) = pre x
-      rw [Equiv.apply_symm_apply]
-      exact Prod.ext rfl h
-  · intro h
-    have hh : (b,(switchUsers d (decodeButterfly d (fun j => ω (b,j))) j).1) = pre x ∨
-        (b,(switchUsers d (decodeButterfly d (fun j => ω (b,j))) j).2) = pre x := by
-      rcases h with h | h
-      · exact Or.inl (by simpa only [parallelUsers,Equiv.apply_symm_apply] using congrArg pre h)
-      · exact Or.inr (by simpa only [parallelUsers,Equiv.apply_symm_apply] using congrArg pre h)
-    have hb : b = (pre x).1 := hh.elim (congrArg Prod.fst) (congrArg Prod.fst)
-    subst b
-    refine ⟨j,?_,rfl,rfl⟩
-    apply (mem_routeDomain_iff_user d _ _ _).mpr
-    exact hh.elim (fun h => Or.inl (congrArg Prod.snd h)) (fun h => Or.inr (congrArg Prod.snd h))
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma card_parallelPath (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool) (x : α) :
-    (parallelPath pre ω x).card = d := by
-  classical
-  rw [parallelPath,Finset.card_image_of_injective _ (fun _ _ h => Prod.mk.inj h |>.2),card_routeDomain]
-
-noncomputable def blockMarks (pre : α ≃ B × Card d) (S : Finset α) (b : B) : Finset (Card d) :=
-  Finset.univ.filter (fun x => pre.symm (b,x) ∈ S)
-
-omit [Fintype α] [Fintype B] [DecidableEq B] in
-@[simp] lemma mem_blockMarks (pre : α ≃ B × Card d) (S : Finset α) (b : B) (x : Card d) :
-    x ∈ blockMarks pre S b ↔ pre.symm (b,x) ∈ S := by
-  classical
-  simp [blockMarks]
-
-omit [DecidableEq B] in
-lemma sum_card_blockMarks (pre : α ≃ B × Card d) (S : Finset α) :
-    ∑ b : B, (blockMarks pre S b).card = S.card := by
-  classical
-  calc
-    _ = ∑ b : B, ∑ x : Card d, if pre.symm (b,x) ∈ S then 1 else 0 := by
-      simp only [blockMarks,Finset.card_filter]
-    _ = ∑ u : B × Card d, if pre.symm u ∈ S then 1 else 0 := (Fintype.sum_prod_type (fun u : B × Card d => if pre.symm u ∈ S then (1:ℕ) else 0)).symm
-    _ = ∑ x : α, if x ∈ S then 1 else 0 := Equiv.sum_comp pre.symm (fun x : α => if x ∈ S then (1:ℕ) else 0)
-    _ = _ := by simp
-
-omit [DecidableEq B] in
-lemma card_blockMarks_le (pre : α ≃ B × Card d) (S : Finset α) (b : B) :
-    (blockMarks pre S b).card ≤ S.card := by
-  rw [←sum_card_blockMarks pre S]
-  exact Finset.single_le_sum (f := fun b : B => (blockMarks pre S b).card) (fun _ _ => Nat.zero_le _) (Finset.mem_univ b)
-
-end Thorp
-
-end ThorpNine.Smoothing
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Smoothing.AssignmentSlots
+
+namespace OAI
+
+namespace ThorpNine.Smoothing
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+namespace RoutingNetwork
+variable {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι] [DecidableEq ι]
+  {d : ℕ} (R : RoutingNetwork α ι d)
+lemma mem_exposedPaths (b : ι → Bool) (V : Finset (α)) (s : ι) :
+    s ∈ exposedPaths R b V ↔ (R.users b s).1 ∈ V ∨ (R.users b s).2 ∈ V := by
+  classical
+  simp only [exposedPaths,Finset.mem_biUnion,R.mem_path]
+  constructor
+  · rintro ⟨x,hx,h | h⟩
+    · exact Or.inl (h ▸ hx)
+    · exact Or.inr (h ▸ hx)
+  · rintro (h | h)
+    · exact ⟨_,h,Or.inl rfl⟩
+    · exact ⟨_,h,Or.inr rfl⟩
+
+lemma terminalCharge_sum (b : ι → Bool) (z : α) (V : Finset (α))
+    (hz : z ∉ V) :
+    (terminalCharge R b z V : ℝ) = ∑ s : ι,
+      ((if (R.users b s).1 = z ∧ (R.users b s).2 ∈ V then (1:ℝ) else 0) +
+      (if (R.users b s).2 = z ∧ (R.users b s).1 ∈ V then (1:ℝ) else 0)) := by
+  classical
+  have he : R.path b z ∩ exposedPaths R b V =
+      Finset.univ.filter (fun s =>
+        ((R.users b s).1 = z ∧ (R.users b s).2 ∈ V) ∨
+        ((R.users b s).2 = z ∧ (R.users b s).1 ∈ V)) := by
+    ext s
+    simp only [Finset.mem_inter,R.mem_path,mem_exposedPaths,
+      Finset.mem_filter,Finset.mem_univ,true_and]
+    by_cases h₁ : (R.users b s).1 = z <;>
+      by_cases h₂ : (R.users b s).2 = z <;> simp [h₁,h₂,hz]
+  rw [terminalCharge,he]
+  simp only [← Finset.sum_boole]
+  apply Finset.sum_congr rfl
+  intro s _
+  by_cases h₁ : (R.users b s).1 = z <;>
+    by_cases h₂ : (R.users b s).2 = z <;> simp [h₁,h₂,hz]
+
+noncomputable def priorPaths {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) : Finset (α) :=
+  Finset.univ.filter (fun x => ∃ i : Fin A, o x = some i ∧ i.val < t)
+
+omit [DecidableEq α] in
+@[simp] lemma mem_priorPaths {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) (x : α) :
+    x ∈ priorPaths o t ↔ ∃ i : Fin A, o x = some i ∧ i.val < t := by
+  classical
+  simp [priorPaths]
+
+lemma terminalCharge_prior_sum {A : ℕ} (o : α → Option (Fin A))
+    (b : ι → Bool) (z : Terminal o) (i : Fin A) :
+    (terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) =
+      ∑ s : ι,
+        ((if (z i).val = (R.users b s).1 ∧ (R.users b s).2 ∈ priorPaths o (i.val+1)
+          then (1:ℝ) else 0) +
+        (if (z i).val = (R.users b s).2 ∧ (R.users b s).1 ∈ priorPaths o (i.val+1)
+          then (1:ℝ) else 0)) := by
+  classical
+  rw [terminalCharge_sum _ _ _ _ (Finset.notMem_erase _ _)]
+  apply Finset.sum_congr rfl
+  intro s _
+  have hn := R.users_ne b s
+  simp only [Finset.mem_erase]
+  by_cases h₁ : (z i).val = (R.users b s).1 <;>
+    by_cases h₂ : (z i).val = (R.users b s).2 <;> simp_all [eq_comm]
+
+lemma rotationMeeting_eq_slot_sum {A : ℕ} (o : α → Option (Fin A))
+    (z : Terminal o) (u v : α) :
+    rotationMeeting o z u v = ∑ i : Fin A,
+      ((if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) +
+      (if (z i).val = v ∧ u ∈ priorPaths o (i.val+1) then (1:ℝ) else 0)) := by
+  classical
+  have only_owner (u : α) (a : Fin A) (hou : o u = some a)
+      (i : Fin A) (hia : i ≠ a) : (z i).val ≠ u := by
+    intro h
+    have hi := (z i).property
+    rw [h,hou] at hi
+    exact hia (Option.some.inj hi).symm
+  have sum_one (u v : α) (a : Fin A) (hou : o u = some a) :
+      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) =
+      if (z a).val = u ∧ v ∈ priorPaths o (a.val+1) then 1 else 0 := by
+    apply Finset.sum_eq_single a
+    · intro i _ hia
+      simp [only_owner u a hou i hia]
+    · simp
+  have sum_none (u v : α) (hou : o u = none) :
+      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) = 0 := by
+    apply Finset.sum_eq_zero
+    intro i _
+    have hn : (z i).val ≠ u := by
+      intro h
+      have hi := (z i).property
+      rw [h,hou] at hi
+      contradiction
+    simp [hn]
+  rw [Finset.sum_add_distrib]
+  cases hou : o u with
+  | none =>
+    rw [sum_none u v hou]
+    simp [rotationMeeting,hou,mem_priorPaths]
+  | some a =>
+    rw [sum_one u v a hou]
+    cases hov : o v with
+    | none =>
+      rw [sum_none v u hov]
+      simp [rotationMeeting,hou,hov,mem_priorPaths]
+    | some b =>
+      rw [sum_one v u b hov]
+      simp only [rotationMeeting,hou,hov,mem_priorPaths,Option.some.injEq]
+      rcases lt_trichotomy a b with hab | rfl | hba
+      · have hn : ¬ b.val < a.val+1 := by omega
+        have hy : a.val < b.val+1 := by omega
+        simp [hab,hn,hy]
+      · simp
+      · have hn : ¬ a.val < b.val+1 := by omega
+        have hy : b.val < a.val+1 := by omega
+        simp [hba,not_lt_of_ge (le_of_lt hba),hn,hy]
+
+lemma rotationCharge_eq_terminal_sum {A : ℕ} (o : α → Option (Fin A))
+    (b : ι → Bool) (z : Terminal o) :
+    rotationCharge R o b z = ∑ i : Fin A,
+      (terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) := by
+  simp only [rotationCharge,rotationMeeting_eq_slot_sum,terminalCharge_prior_sum]
+  exact Finset.sum_comm
+
+omit [DecidableEq α] in
+@[simp] lemma priorPaths_zero {A : ℕ} (o : α → Option (Fin A)) : priorPaths o 0 = ∅ := by
+  classical
+  ext x
+  simp
+
+lemma priorPaths_succ {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) (ht : t < A) :
+    priorPaths o (t+1) = priorPaths o t ∪ cycleFiber o ⟨t,ht⟩ := by
+  classical
+  ext x
+  simp only [mem_priorPaths,Finset.mem_union,mem_cycleFiber]
+  constructor
+  · rintro ⟨i,hi,hit⟩
+    by_cases h : i.val < t
+    · exact Or.inl ⟨i,hi,h⟩
+    · have he : i = ⟨t,ht⟩ := Fin.ext (by change i.val = t; omega)
+      exact Or.inr (he ▸ hi)
+  · rintro (⟨i,hi,hit⟩ | hi)
+    · exact ⟨i,hi,by omega⟩
+    · exact ⟨⟨t,ht⟩,hi,by change t < t+1; omega⟩
+
+omit [DecidableEq α] in
+lemma priorPaths_disjoint {A : ℕ} (o : α → Option (Fin A)) (i : Fin A) :
+    Disjoint (priorPaths o i.val) (cycleFiber o i) := by
+  classical
+  apply Finset.disjoint_left.mpr
+  intro x hx hx'
+  obtain ⟨j,hj,hji⟩ := (mem_priorPaths _ _ _).mp hx
+  have he : j = i := Option.some.inj (hj.symm.trans ((mem_cycleFiber _ _ _).mp hx'))
+  subst j
+  exact (lt_irrefl _ hji)
+
+namespace QueryTree
+open Thorp.QueryTree
+
+lemma seekCycles_owned {A : ℕ} (S : Finset (α)) (r : Fin A → ℕ)
+    (o : α → Option (Fin A)) (z : Terminal o) (ω : ι → Bool)
+    (hS : ∀ i, cycleFiber o i ⊆ S)
+    (hO : ∀ i, orbitSet (R.perm ω) (z i).val = cycleFiber o i)
+    (hC : ∀ i, (cycleFiber o i).card = r i+1) :
+    let p := R.perm ω
+    let L := List.ofFn (fun i : Fin A => (p (z i).val,r i))
+    (seekCycles R S L ∅).succeeds ω = true ∧
+      ((seekCycles R S L ∅).staleCharge ∅ ω : ℝ) = rotationCharge R o (ω) z := by
+  classical
+  let b := ω
+  let p := R.perm b
+  let f (i : Fin A) := (p (z i).val,r i)
+  let c (i : Fin A) : ℝ := terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val)
+  have hc (i : Fin A) : Function.minimalPeriod p (z i).val - 1 = r i := by
+    have he := orbitSet_card p (z i).val
+    rw [hO,hC] at he
+    omega
+  have hs (t : ℕ) (ht : t ≤ A) :
+      (seekCycles R S ((List.ofFn f).drop t) (priorPaths o t)).succeeds ω = true ∧
+        ((seekCycles R S ((List.ofFn f).drop t) (priorPaths o t)).staleCharge (exposedPaths R b (priorPaths o t)) ω : ℝ) =
+          ((List.ofFn c).drop t).sum := by
+    induction ht using Nat.decreasingInduction with
+    | self => simp only [List.length_ofFn,le_refl,List.drop_eq_nil_of_le,List.sum_nil,seekCycles,Thorp.QueryTree.succeeds,staleCharge,Nat.cast_zero,and_self]
+    | of_succ t ht ih =>
+      have htf : t < (List.ofFn f).length := by simpa using ht
+      have htc : t < (List.ofFn c).length := by simpa using ht
+      rw [List.drop_eq_getElem_cons htf,List.drop_eq_getElem_cons htc]
+      simp only [List.getElem_ofFn,List.sum_cons,seekCycles]
+      have h := seekCycle_orbit R S (priorPaths o t) (z ⟨t,ht⟩).val
+        (fun V => seekCycles R S ((List.ofFn f).drop (t+1)) V) ω
+        (by rw [hO]; exact hS _) (by rw [hO]; exact priorPaths_disjoint o ⟨t,ht⟩)
+      dsimp only at h
+      rw [hc,hO,←priorPaths_succ o t ht] at h
+      dsimp only [f]
+      constructor
+      · exact h.1.trans ih.1
+      · rw [h.2,Nat.cast_add,ih.2]
+        exact add_comm _ _
+  have h := hs 0 (Nat.zero_le _)
+  simp only [List.drop_zero,priorPaths_zero,exposedPaths_empty,List.sum_ofFn] at h
+  dsimp only
+  exact ⟨h.1,h.2.trans (rotationCharge_eq_terminal_sum R o b z).symm⟩
+
+end QueryTree
+section Collections
+variable {A : ℕ}
+
+def ValidCycleSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ)
+    (o : α → Option (Fin A)) : Prop :=
+  (∀ i, (cycleFiber o i).card = r i + 1) ∧
+  (∀ i, cycleFiber o i ⊆ S) ∧
+  (∀ i x, o x = some i → orbitSet p x = cycleFiber o i)
+
+abbrev CycleSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
+  {o : α → Option (Fin A) // ValidCycleSlots p S r o}
+
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
+    Fintype (CycleSlots p S r) := Fintype.ofFinite _
+
+lemma CycleSlots.terminal_nonempty {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (c : CycleSlots p S r) (i : Fin A) : Nonempty {x : α // c.val x = some i} := by
+  have hn : (cycleFiber c.val i).Nonempty := Finset.card_pos.mp (by rw [c.property.1]; omega)
+  obtain ⟨x,hx⟩ := hn
+  exact ⟨x,(mem_cycleFiber _ _ _).mp hx⟩
+
+omit [DecidableEq α] in
+lemma owner_ext {o o' : α → Option (Fin A)}
+    (h : ∀ i, cycleFiber o i = cycleFiber o' i) : o = o' := by
+  funext x
+  have he (i : Fin A) : o x = some i ↔ o' x = some i := by
+    simpa only [mem_cycleFiber] using (Finset.ext_iff.mp (h i) x)
+  cases ho : o x with
+  | some i => exact ((he i).mp ho).symm
+  | none =>
+    cases ho' : o' x with
+    | none => rfl
+    | some i => have hn := (he i).mpr ho'; rw [ho] at hn; contradiction
+
+abbrev RootedSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
+  Σ c : CycleSlots p S r, Terminal c.val
+
+noncomputable def rootedStart {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (c : RootedSlots p S r) : Fin A → S := fun i =>
+  ⟨p (c.2 i).val,c.1.property.2.1 i (by
+    rw [←c.1.property.2.2 i (c.2 i).val (c.2 i).property]
+    exact (mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.apply_right (Equiv.Perm.SameCycle.refl p _)))⟩
+
+lemma rootedStart_injective {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ} :
+    Function.Injective (rootedStart (p:=p) (S:=S) (r:=r)) := by
+  rintro ⟨c,z⟩ ⟨c',z'⟩ h
+  have hz (i : Fin A) : (z i).val = (z' i).val :=
+    p.injective (congrArg Subtype.val (congrFun h i))
+  have hc : c = c' := Subtype.ext (owner_ext (fun i => by
+    rw [←c.property.2.2 i (z i).val (z i).property,
+      ←c'.property.2.2 i (z' i).val (z' i).property,hz]))
+  subst c'
+  have hzz : z = z' := funext (fun i => Subtype.ext (hz i))
+  subst z'
+  rfl
+
+lemma rooted_weight_sum_le (S : Finset (α)) (r : Fin A → ℕ) (ω : ι → Bool) :
+    (∑ c : RootedSlots (R.perm ω) S r,
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2)) ≤
+    ∑ a : Fin A → S,
+      if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+        ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+      else 0 := by
+  classical
+  let p := R.perm ω
+  let g (a : Fin A → S) : ℝ :=
+    if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+    else 0
+  have he (c : RootedSlots p S r) :
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2) =
+      g (rootedStart c) := by
+    have hs := QueryTree.seekCycles_owned R S r c.1.val c.2 ω c.1.property.2.1
+      (fun i => c.1.property.2.2 i (c.2 i).val (c.2 i).property) c.1.property.1
+    dsimp only at hs
+    unfold g
+    rw [ite_eq_left (show (QueryTree.seekCycles R S (List.ofFn (fun i => ((rootedStart c i).val,r i))) ∅).succeeds ω = true from hs.1)]
+    rw [← hs.2,neg_mul,Real.exp_neg,mul_comm (Real.log 2),Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
+    rfl
+  simp_rw [he]
+  exact sum_comp_injective_le rootedStart rootedStart_injective g (fun a => by dsimp [g]; split_ifs <;> positivity)
+
+lemma collection_rotation_lower (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r)
+    (ω : ι → Bool) (c : CycleSlots (R.perm ω) S r) :
+    collectionWeight d r ≤ ∑ z : Terminal c.val,
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z) := by
+  classical
+  let (i : Fin A) : Nonempty {x : α // c.val x = some i} := c.terminal_nonempty i
+  have hm : Monotone (fun i => (cycleFiber c.val i).card) := by
+    intro i j hij
+    dsimp only
+    rw [c.property.1,c.property.1]
+    exact Nat.add_le_add_right (hr hij) 1
+  have hb := rotation_exp_bound R hm (ω)
+  have hcard : (Fintype.card (Terminal c.val) : ℝ) = ∏ i : Fin A, (r i+1 : ℝ) := by
+    rw [Fintype.card_pi,Nat.cast_prod]
+    apply Finset.prod_congr rfl
+    intro i _
+    rw [card_cycleFiber,c.property.1,Nat.cast_add,Nat.cast_one]
+  have hp : (0:ℝ) < Fintype.card (Terminal c.val) := Nat.cast_pos.mpr Fintype.card_pos
+  have hl : rotationLower d r ≤ finiteMean (fun z : Terminal c.val => Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z)) := by
+    simpa only [rotationLower,c.property.1,Nat.cast_add,Nat.cast_one] using hb
+  have hh := (le_div_iff₀ hp).mp hl
+  rw [hcard] at hh
+  unfold collectionWeight
+  rw [← Finset.mul_sum]
+  calc
+    _ = ((2:ℝ)^d)^A * (rotationLower d r * ∏ i : Fin A, (r i+1 : ℝ)) := by ring
+    _ ≤ _ := mul_le_mul_of_nonneg_left hh (by positivity)
+
+lemma cycleSlots_weighted_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : ι → Bool =>
+      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) *
+      collectionWeight d r ≤ (S.card : ℝ)^A := by
+  classical
+  let g (a : Fin A → S) (ω : ι → Bool) : ℝ :=
+    if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+    else 0
+  have hω (ω : ι → Bool) :
+      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ) * collectionWeight d r ≤
+        ∑ a : Fin A → S, g a ω := by
+    calc
+      _ = ∑ _c : CycleSlots (R.perm ω) S r, collectionWeight d r := by simp
+      _ ≤ ∑ c : CycleSlots (R.perm ω) S r,
+          ∑ z : Terminal c.val, ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z) :=
+        Finset.sum_le_sum (fun c _ => collection_rotation_lower R S r hr ω c)
+      _ = ∑ c : RootedSlots (R.perm ω) S r,
+          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2) :=
+        (Fintype.sum_sigma (fun c : RootedSlots (R.perm ω) S r =>
+          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2))).symm
+      _ ≤ _ := rooted_weight_sum_le R S r ω
+  have hb := finiteMean_mono hω
+  rw [finiteMean_mul_const,finiteMean_sum] at hb
+  calc
+    _ ≤ ∑ a : Fin A → S, finiteMean (g a) := hb
+    _ ≤ ∑ _a : Fin A → S, (1:ℝ) := by
+      apply Finset.sum_le_sum
+      intro a _
+      have hh := QueryTree.cycles_weighted_success R S (List.ofFn (fun i : Fin A => ((a i).val,r i)))
+      simpa only [List.length_ofFn] using hh
+    _ = _ := by simp
+
+lemma cycleSlots_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : ι → Bool =>
+      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
+        (S.card : ℝ)^A / collectionWeight d r := by
+  exact (le_div_iff₀ (collectionWeight_pos d r)).mpr (cycleSlots_weighted_moment R S r hr)
+
+lemma butterfly_cycleSlots_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : ι → Bool =>
+      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
+        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
+  have h := cycleSlots_moment R S r hr
+  rw [collectionWeight_eq_sqrt,div_mul_eq_div_mul_one_div] at h
+  rw [div_pow]
+  simpa only [one_div,Finset.prod_inv_distrib] using h
+
+end Collections
+
+section CycleCounting
+variable {A : ℕ}
+
+abbrev SelectedCycle (p : Equiv.Perm (α)) (S : Finset (α)) :=
+  {C : Finset (α) // (∃ x, C = orbitSet p x) ∧ C ⊆ S}
+
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) :
+    Fintype (SelectedCycle p S) := Fintype.ofFinite _
+
+lemma SelectedCycle.nonempty {p : Equiv.Perm (α)} {S : Finset (α)}
+    (C : SelectedCycle p S) : C.val.Nonempty := by
+  obtain ⟨x,hx⟩ := C.property.1
+  rw [hx]
+  exact orbitSet_nonempty p x
+
+lemma SelectedCycle.orbit_eq {p : Equiv.Perm (α)} {S : Finset (α)}
+    (C : SelectedCycle p S) {x : α} (hx : x ∈ C.val) : orbitSet p x = C.val := by
+  obtain ⟨y,hy⟩ := C.property.1
+  rw [hy] at hx ⊢
+  exact (orbitSet_eq p ((mem_orbitSet _ _ _).mp hx)).symm
+
+lemma SelectedCycle.eq_of_mem {p : Equiv.Perm (α)} {S : Finset (α)}
+    (C D : SelectedCycle p S) {x : α} (hC : x ∈ C.val) (hD : x ∈ D.val) : C = D := by
+  apply Subtype.ext
+  exact (C.orbit_eq hC).symm.trans (D.orbit_eq hD)
+
+abbrev SlotAssignment (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
+  {f : Fin A → SelectedCycle p S // Function.Injective f ∧ ∀ i, (f i).val.card = r i+1}
+
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
+    Fintype (SlotAssignment p S r) := Fintype.ofFinite _
+
+noncomputable def slotsToAssignment {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (c : CycleSlots p S r) : SlotAssignment p S r := by
+  classical
+  let f (i : Fin A) : SelectedCycle p S := ⟨cycleFiber c.val i,by
+    obtain ⟨x,hx⟩ := c.terminal_nonempty i
+    exact ⟨⟨x,(c.property.2.2 i x hx).symm⟩,c.property.2.1 i⟩⟩
+  refine ⟨f,?_,fun i => c.property.1 i⟩
+  intro i j hij
+  have he : cycleFiber c.val i = cycleFiber c.val j := congrArg Subtype.val hij
+  obtain ⟨x,hx⟩ := c.terminal_nonempty i
+  have hxi : x ∈ cycleFiber c.val i := (mem_cycleFiber _ _ _).mpr hx
+  rw [he] at hxi
+  exact Option.some.inj (hx.symm.trans ((mem_cycleFiber _ _ _).mp hxi))
+
+noncomputable def assignmentOwner {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) (x : α) : Option (Fin A) := by
+  classical
+  exact if h : ∃ i, x ∈ (f.val i).val then some (Classical.choose h) else none
+
+lemma assignmentOwner_eq_some {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) (x : α) (i : Fin A) :
+    assignmentOwner f x = some i ↔ x ∈ (f.val i).val := by
+  classical
+  unfold assignmentOwner
+  split_ifs with h
+  · constructor
+    · intro hi
+      have hx := Classical.choose_spec h
+      rw [Option.some.inj hi] at hx
+      exact hx
+    · intro hx
+      have he := (f.val (Classical.choose h)).eq_of_mem (f.val i) (Classical.choose_spec h) hx
+      exact congrArg some (f.property.1 he)
+  · constructor
+    · intro hi; contradiction
+    · intro hx; exact (h ⟨i,hx⟩).elim
+
+lemma assignmentOwner_fiber {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) (i : Fin A) : cycleFiber (assignmentOwner f) i = (f.val i).val := by
+  ext x
+  simp only [mem_cycleFiber,assignmentOwner_eq_some]
+
+noncomputable def assignmentToSlots {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) : CycleSlots p S r :=
+  ⟨assignmentOwner f,by
+    refine ⟨?_,?_,?_⟩
+    · intro i
+      rw [assignmentOwner_fiber]
+      exact f.property.2 i
+    · intro i
+      rw [assignmentOwner_fiber]
+      exact (f.val i).property.2
+    · intro i x hx
+      rw [assignmentOwner_fiber]
+      exact (f.val i).orbit_eq ((assignmentOwner_eq_some f x i).mp hx)⟩
+
+noncomputable def cycleSlotsEquivAssignments (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
+    CycleSlots p S r ≃ SlotAssignment p S r where
+  toFun := slotsToAssignment
+  invFun := assignmentToSlots
+  left_inv c := by
+    apply Subtype.ext
+    apply owner_ext
+    intro i
+    rw [assignmentToSlots,assignmentOwner_fiber]
+    rfl
+  right_inv f := by
+    apply Subtype.ext
+    funext i
+    apply Subtype.ext
+    exact assignmentOwner_fiber f i
+
+end CycleCounting
+
+section GroupedCounting
+variable {A : ℕ}
+
+abbrev LengthClass (r : Fin A → ℕ) := {j : ℕ // j ∈ Finset.univ.image (fun i => r i + 1)}
+
+def slotClass (r : Fin A → ℕ) (i : Fin A) : LengthClass r :=
+  ⟨r i+1, Finset.mem_image.mpr ⟨i, Finset.mem_univ _, rfl⟩⟩
+
+abbrev SlotsOfLength (r : Fin A → ℕ) (j : LengthClass r) := {i : Fin A // slotClass r i = j}
+abbrev CyclesOfLength (p : Equiv.Perm (α)) (S : Finset (α)) (j : ℕ) :=
+  {C : SelectedCycle p S // C.val.card = j}
+
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (j : ℕ) :
+    Fintype (CyclesOfLength p S j) := Fintype.ofFinite _
+
+abbrev GroupedAssignments (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
+  ∀ j : LengthClass r, SlotsOfLength r j ↪ CyclesOfLength p S j.val
+
+noncomputable def assignmentToGroups {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) : GroupedAssignments p S r := fun _j =>
+  { toFun := fun i => ⟨f.val i.val, (f.property.2 i.val).trans (congrArg Subtype.val i.property)⟩
+    inj' := fun _i _k h => Subtype.ext (f.property.1 (congrArg Subtype.val h)) }
+
+noncomputable def groupsEmbedding {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (F : GroupedAssignments p S r) : Fin A ↪ SelectedCycle p S :=
+  ((Equiv.sigmaFiberEquiv (slotClass r)).symm.toEmbedding.trans
+    (Function.Embedding.sigmaMap (Function.Embedding.refl _) F)).trans
+    ((Function.Embedding.sigmaMap (Function.Embedding.subtype _) (fun _ => Function.Embedding.refl _)).trans
+      (Equiv.sigmaFiberEquiv (fun C : SelectedCycle p S => C.val.card)).toEmbedding)
+
+noncomputable def groupsToAssignment {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (F : GroupedAssignments p S r) : SlotAssignment p S r :=
+  ⟨groupsEmbedding F, (groupsEmbedding F).injective,
+    fun i => (F (slotClass r i) ⟨i,rfl⟩).property⟩
+
+noncomputable def assignmentEquivGroups (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
+    SlotAssignment p S r ≃ GroupedAssignments p S r where
+  toFun := assignmentToGroups
+  invFun := groupsToAssignment
+  left_inv f := by
+    apply Subtype.ext
+    rfl
+  right_inv F := by
+    funext j
+    apply Function.Embedding.ext
+    intro i
+    rcases i with ⟨i,hi⟩
+    subst j
+    rfl
+
+lemma card_cycleSlots_factorial (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
+    Fintype.card (CycleSlots p S r) =
+      ∏ j : LengthClass r, (Fintype.card (CyclesOfLength p S j.val)).descFactorial
+        (Fintype.card (SlotsOfLength r j)) := by
+  classical
+  rw [Fintype.card_congr (cycleSlotsEquivAssignments p S r),
+    Fintype.card_congr (assignmentEquivGroups p S r), Fintype.card_pi]
+  simp only [Fintype.card_embedding_eq]
+
+lemma butterfly_factorial_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : ι → Bool =>
+       (∏ j : LengthClass r,
+         (Fintype.card (CyclesOfLength (R.perm ω) S j.val)).descFactorial
+           (Fintype.card (SlotsOfLength r j)) : ℕ)) ≤
+      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+        ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
+  simpa only [←card_cycleSlots_factorial] using butterfly_cycleSlots_moment R S r hr
+
+end GroupedCounting
+
+section AllLengths
+variable {A : ℕ}
+
+noncomputable def assignmentRelabel (p : Equiv.Perm (α)) (S : Finset (α))
+    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
+    SlotAssignment p S r ≃ SlotAssignment p S (r ∘ e) where
+  toFun f := ⟨f.val ∘ e, f.property.1.comp e.injective, fun i => f.property.2 (e i)⟩
+  invFun f := ⟨f.val ∘ e.symm, f.property.1.comp e.symm.injective, fun i => by
+    simpa only [Function.comp_apply,Equiv.apply_symm_apply] using f.property.2 (e.symm i)⟩
+  left_inv f := by apply Subtype.ext; funext i; simp
+  right_inv f := by apply Subtype.ext; funext i; simp
+
+lemma card_cycleSlots_relabel (p : Equiv.Perm (α)) (S : Finset (α))
+    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
+    Fintype.card (CycleSlots p S r) = Fintype.card (CycleSlots p S (r ∘ e)) := by
+  exact Fintype.card_congr ((cycleSlotsEquivAssignments p S r).trans
+    ((assignmentRelabel p S r e).trans (cycleSlotsEquivAssignments p S (r ∘ e)).symm))
+
+lemma butterfly_cycleSlots_moment_all (S : Finset (α)) (r : Fin A → ℕ) :
+    finiteMean (fun ω : ι → Bool =>
+      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
+        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
+  have h := butterfly_cycleSlots_moment R S (r ∘ Tuple.sort r) (Tuple.monotone_sort r)
+  simp_rw [←card_cycleSlots_relabel] at h
+  have hp := Equiv.prod_comp (Tuple.sort r) (fun i : Fin A => (1 / Real.sqrt (r i+1) : ℝ))
+  simpa only [Function.comp_apply,hp] using h
+
+abbrev ShortCycle (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :=
+  {C : SelectedCycle p S // C.val.card ≤ J}
+
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    Fintype (ShortCycle p S J) := Fintype.ofFinite _
+
+noncomputable def shortCycleLength {p : Equiv.Perm (α)} {S : Finset (α)} {J : ℕ}
+    (C : ShortCycle p S J) : Fin J :=
+  ⟨C.val.val.card-1, by have := C.val.nonempty.card_pos; have := C.property; omega⟩
+
+lemma shortCycleLength_add_one {p : Equiv.Perm (α)} {S : Finset (α)} {J : ℕ}
+    (C : ShortCycle p S J) : (shortCycleLength C : ℕ)+1 = C.val.val.card := by
+  have := C.val.nonempty.card_pos
+  simp only [shortCycleLength]
+  omega
+
+noncomputable def shortEmbeddingEquiv (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    (Fin A ↪ ShortCycle p S J) ≃
+      Σ r : Fin A → Fin J, SlotAssignment p S (fun i => (r i : ℕ)) where
+  toFun f := ⟨fun i => shortCycleLength (f i),
+    ⟨fun i => (f i).val, fun i j h => f.injective (Subtype.ext h),
+      fun i => (shortCycleLength_add_one (f i)).symm⟩⟩
+  invFun c :=
+    { toFun := fun i => ⟨c.2.val i, by rw [c.2.property.2]; exact (c.1 i).isLt⟩
+      inj' := fun _i _j h => c.2.property.1 (congrArg Subtype.val h) }
+  left_inv f := by
+    apply Function.Embedding.ext
+    intro i
+    rfl
+  right_inv c := by
+    rcases c with ⟨r,f⟩
+    have he : (fun i => shortCycleLength (⟨f.val i, by rw [f.property.2]; exact (r i).isLt⟩ : ShortCycle p S J)) = r := by
+      funext i
+      apply Fin.ext
+      simp only [shortCycleLength,f.property.2,Nat.add_sub_cancel]
+    apply Sigma.ext he
+    apply (Subtype.heq_iff_coe_eq (by
+      intro x
+      change (Function.Injective x ∧ ∀ i, (x i).val.card = (f.val i).val.card - 1 + 1) ↔ _
+      simp only [f.property.2,Nat.add_sub_cancel])).mpr
+    rfl
+
+lemma shortCycle_factorial_eq (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    (Fintype.card (ShortCycle p S J)).descFactorial A =
+      ∑ r : Fin A → Fin J, Fintype.card (CycleSlots p S (fun i => (r i : ℕ))) := by
+  classical
+  have hc := Fintype.card_embedding_eq (α := Fin A) (β := ShortCycle p S J)
+  simp only [Fintype.card_fin] at hc
+  rw [←hc, Fintype.card_congr (shortEmbeddingEquiv p S J),Fintype.card_sigma]
+  apply Finset.sum_congr rfl
+  intro r _
+  exact (Fintype.card_congr (cycleSlotsEquivAssignments p S (fun i => (r i : ℕ)))).symm
+
+end AllLengths
+
+section CycleMGF
+
+lemma butterfly_shortCycle_moment (S : Finset (α)) (J A : ℕ) :
+    finiteMean (fun ω : ι → Bool =>
+      ((Fintype.card (ShortCycle (R.perm ω) S J)).descFactorial A : ℝ)) ≤
+      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))^A := by
+  classical
+  simp_rw [shortCycle_factorial_eq,Nat.cast_sum,finiteMean_sum]
+  calc
+    _ ≤ ∑ r : Fin A → Fin J,
+        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+          ∏ i : Fin A, (1 / Real.sqrt ((r i : ℕ)+1)) := by
+      apply Finset.sum_le_sum
+      intro r _
+      exact butterfly_cycleSlots_moment_all R S (fun i => (r i : ℕ))
+    _ = _ := by
+      rw [←Finset.mul_sum,mul_pow]
+      congr 1
+      rw [←Fintype.prod_sum (fun (_ : Fin A) (j : Fin J) => (1 / Real.sqrt ((j:ℕ)+1) : ℝ))]
+      simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fin]
+
+noncomputable def cycleMass (p : Equiv.Perm (α)) (S : Finset (α)) : ℕ :=
+  ∑ C : SelectedCycle p S, C.val.card
+
+lemma cycleMass_le (p : Equiv.Perm (α)) (S : Finset (α)) : cycleMass p S ≤ S.card := by
+  classical
+  let f : (Σ C : SelectedCycle p S, {x : α // x ∈ C.val}) → S :=
+    fun c => ⟨c.2.val,c.1.property.2 c.2.property⟩
+  have hi : Function.Injective f := by
+    intro c e h
+    have hx : c.2.val = e.2.val := congrArg Subtype.val h
+    have hC : c.1 = e.1 := c.1.eq_of_mem e.1 c.2.property (hx ▸ e.2.property)
+    rcases c with ⟨c,x⟩
+    rcases e with ⟨e,y⟩
+    dsimp only at hC
+    subst e
+    have he : x=y := Subtype.ext hx
+    cases he
+    rfl
+  have hh := Fintype.card_le_of_injective f hi
+  simpa only [Fintype.card_sigma,Fintype.card_coe,cycleMass] using hh
+
+lemma shortCycle_card_le (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    Fintype.card (ShortCycle p S J) ≤ S.card := by
+  classical
+  calc
+    _ ≤ Fintype.card (SelectedCycle p S) := Fintype.card_subtype_le _
+    _ = ∑ _ : SelectedCycle p S, 1 := by simp
+    _ ≤ cycleMass p S := Finset.sum_le_sum (fun C _ => C.nonempty.card_pos)
+    _ ≤ _ := cycleMass_le p S
+
+lemma butterfly_shortCycle_mgf (S : Finset (α)) (J : ℕ) (q : ℝ) (hq : 1 ≤ q) :
+    finiteMean (fun ω : ι → Bool =>
+      q^(Fintype.card (ShortCycle (R.perm ω) S J))) ≤
+      Real.exp ((q-1)*((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))) := by
+  apply factorial_mgf _ S.card _ q (fun ω => shortCycle_card_le _ _ _)
+  · positivity
+  · exact hq
+  · exact butterfly_shortCycle_moment R S J
+
+end CycleMGF
+
+
+abbrev LongCycle (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :=
+  {C : SelectedCycle p S // ¬ C.val.card ≤ J}
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    Fintype (LongCycle p S J) := Fintype.ofFinite _
+
+lemma selectedCycle_card_split (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    Fintype.card (SelectedCycle p S) =
+      Fintype.card (ShortCycle p S J) + Fintype.card (LongCycle p S J) := by
+  classical
+  rw [←Fintype.card_congr (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J)),Fintype.card_sum]
+
+lemma longCycle_card_bound (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    J * Fintype.card (LongCycle p S J) ≤ S.card := by
+  classical
+  have hs := Equiv.sum_comp (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J))
+    (fun C : SelectedCycle p S => C.val.card)
+  rw [Fintype.sum_sum_type] at hs
+  have h0 : J * Fintype.card (LongCycle p S J) ≤
+      ∑ C : LongCycle p S J, C.val.val.card := by
+    calc
+      _ = ∑ _C : LongCycle p S J, J := by simp [mul_comm]
+      _ ≤ _ := Finset.sum_le_sum (fun C _ => Nat.le_of_lt (Nat.lt_of_not_ge C.property))
+  have hh := cycleMass_le p S
+  unfold cycleMass at hh
+  rw [←hs] at hh
+  exact h0.trans ((Nat.le_add_left _ _).trans hh)
+
+lemma longCycle_pow_bound (p : Equiv.Perm (α)) (S : Finset (α))
+    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    q^(Fintype.card (LongCycle p S J)) ≤ Real.exp ((S.card : ℝ)/J * Real.log q) := by
+  have hq0 : 0 < q := lt_of_lt_of_le zero_lt_one hq
+  have hc : (Fintype.card (LongCycle p S J) : ℝ) ≤ (S.card : ℝ)/J := by
+    apply (le_div_iff₀ (by exact_mod_cast hJ)).mpr
+    have hh := longCycle_card_bound p S J
+    exact_mod_cast (by simpa only [mul_comm] using hh)
+  calc
+    _ = Real.exp ((Fintype.card (LongCycle p S J) : ℝ)*Real.log q) := by
+      rw [Real.exp_nat_mul,Real.exp_log hq0]
+    _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right hc (Real.log_nonneg hq))
+
+lemma butterfly_cycle_log_mgf (S : Finset (α)) (J : ℕ) (hJ : 0 < J)
+    (q : ℝ) (hq : 1 ≤ q) :
+    Real.log (finiteMean (fun ω : ι → Bool =>
+      q^(Fintype.card (SelectedCycle (R.perm ω) S)))) ≤
+      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
+  let lam : ℝ := (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1)
+  let c : ℝ := (S.card : ℝ)/J * Real.log q
+  have hnum : (q-1)*lam ≤ 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
+    have hsum := sum_inv_sqrt J
+    have hp : 0 ≤ (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) := by positivity
+    have h0 : 0 ≤ 2*Real.sqrt J := by positivity
+    have hh := mul_le_mul_of_nonneg_left hsum hp
+    have hq1 : 0 ≤ q-1 := sub_nonneg.mpr hq
+    calc
+      _ ≤ (q-1)*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
+        mul_le_mul_of_nonneg_left hh hq1
+      _ ≤ q*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
+        mul_le_mul_of_nonneg_right (by linarith) (mul_nonneg hp h0)
+      _ = _ := by ring
+  have hm : finiteMean (fun ω : ι → Bool =>
+      q^(Fintype.card (SelectedCycle (R.perm ω) S))) ≤
+      Real.exp (c+(q-1)*lam) := by
+    calc
+      _ ≤ finiteMean (fun ω : ι → Bool =>
+          q^(Fintype.card (ShortCycle (R.perm ω) S J))*Real.exp c) := by
+        apply finiteMean_mono
+        intro ω
+        rw [selectedCycle_card_split _ _ J,pow_add]
+        exact mul_le_mul_of_nonneg_left (longCycle_pow_bound _ S J hJ q hq) (by positivity)
+      _ = finiteMean (fun ω : ι → Bool =>
+          q^(Fintype.card (ShortCycle (R.perm ω) S J))) * Real.exp c :=
+        finiteMean_mul_const _ _
+      _ ≤ Real.exp ((q-1)*lam) * Real.exp c :=
+        mul_le_mul_of_nonneg_right (butterfly_shortCycle_mgf R S J q hq) (Real.exp_nonneg _)
+      _ = _ := by rw [←Real.exp_add,add_comm]
+  have hpos : 0 < finiteMean (fun ω : ι → Bool =>
+      q^(Fintype.card (SelectedCycle (R.perm ω) S))) := by
+    apply finiteMean_pos
+    intro ω
+    positivity
+  have hlog := Real.log_le_log hpos hm
+  rw [Real.log_exp] at hlog
+  dsimp [c] at hlog
+  linarith
+
+
+end RoutingNetwork
+
+variable {α B : Type*} [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] {d : ℕ}
+
+def parallelPerm (ω : B × SwitchIndex d → Bool) : Equiv.Perm (B × Card d) where
+  toFun x := (x.1,butterflyPerm d (decodeButterfly d (fun i => ω (x.1,i))) x.2)
+  invFun y := (y.1,(butterflyPerm d (decodeButterfly d (fun i => ω (y.1,i)))).symm y.2)
+  left_inv x := by simp
+  right_inv y := by simp
+
+def relabeledPerm (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (ω : B × SwitchIndex d → Bool) : Equiv.Perm α :=
+  (pre.trans (parallelPerm ω)).trans post
+
+noncomputable def parallelPath (pre : α ≃ B × Card d)
+    (ω : B × SwitchIndex d → Bool) (x : α) : Finset (B × SwitchIndex d) :=
+  (routeDomain d (pre x).2
+    (butterflyPerm d (decodeButterfly d (fun i => ω ((pre x).1,i))) (pre x).2)).image
+      (fun i => ((pre x).1,i))
+
+def parallelUsers (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
+    (i : B × SwitchIndex d) : α × α :=
+  let u := switchUsers d (decodeButterfly d (fun j => ω (i.1,j))) i.2
+  (pre.symm (i.1,u.1),pre.symm (i.1,u.2))
+
+omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
+lemma parallelUsers_ne (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
+    (i : B × SwitchIndex d) : (parallelUsers pre ω i).1 ≠ (parallelUsers pre ω i).2 := by
+  intro h
+  exact switchUsers_ne d _ _ (congrArg Prod.snd (pre.symm.injective h))
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma mem_parallelPath (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
+    (x : α) (i : B × SwitchIndex d) :
+    i ∈ parallelPath pre ω x ↔ (parallelUsers pre ω i).1 = x ∨ (parallelUsers pre ω i).2 = x := by
+  classical
+  obtain ⟨b,j⟩ := i
+  simp only [parallelPath,Finset.mem_image,Prod.mk.injEq]
+  constructor
+  · rintro ⟨k,hk,hb,rfl⟩
+    subst b
+    rcases (mem_routeDomain_iff_user d _ _ _).mp hk with h | h
+    · left
+      apply pre.injective
+      change pre (pre.symm ((pre x).1,_)) = pre x
+      rw [Equiv.apply_symm_apply]
+      exact Prod.ext rfl h
+    · right
+      apply pre.injective
+      change pre (pre.symm ((pre x).1,_)) = pre x
+      rw [Equiv.apply_symm_apply]
+      exact Prod.ext rfl h
+  · intro h
+    have hh : (b,(switchUsers d (decodeButterfly d (fun j => ω (b,j))) j).1) = pre x ∨
+        (b,(switchUsers d (decodeButterfly d (fun j => ω (b,j))) j).2) = pre x := by
+      rcases h with h | h
+      · exact Or.inl (by simpa only [parallelUsers,Equiv.apply_symm_apply] using congrArg pre h)
+      · exact Or.inr (by simpa only [parallelUsers,Equiv.apply_symm_apply] using congrArg pre h)
+    have hb : b = (pre x).1 := hh.elim (congrArg Prod.fst) (congrArg Prod.fst)
+    subst b
+    refine ⟨j,?_,rfl,rfl⟩
+    apply (mem_routeDomain_iff_user d _ _ _).mpr
+    exact hh.elim (fun h => Or.inl (congrArg Prod.snd h)) (fun h => Or.inr (congrArg Prod.snd h))
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma card_parallelPath (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool) (x : α) :
+    (parallelPath pre ω x).card = d := by
+  classical
+  rw [parallelPath,Finset.card_image_of_injective _ (fun _ _ h => Prod.mk.inj h |>.2),card_routeDomain]
+
+noncomputable def blockMarks (pre : α ≃ B × Card d) (S : Finset α) (b : B) : Finset (Card d) :=
+  Finset.univ.filter (fun x => pre.symm (b,x) ∈ S)
+
+omit [Fintype α] [Fintype B] [DecidableEq B] in
+@[simp] lemma mem_blockMarks (pre : α ≃ B × Card d) (S : Finset α) (b : B) (x : Card d) :
+    x ∈ blockMarks pre S b ↔ pre.symm (b,x) ∈ S := by
+  classical
+  simp [blockMarks]
+
+omit [DecidableEq B] in
+lemma sum_card_blockMarks (pre : α ≃ B × Card d) (S : Finset α) :
+    ∑ b : B, (blockMarks pre S b).card = S.card := by
+  classical
+  calc
+    _ = ∑ b : B, ∑ x : Card d, if pre.symm (b,x) ∈ S then 1 else 0 := by
+      simp only [blockMarks,Finset.card_filter]
+    _ = ∑ u : B × Card d, if pre.symm u ∈ S then 1 else 0 := (Fintype.sum_prod_type (fun u : B × Card d => if pre.symm u ∈ S then (1:ℕ) else 0)).symm
+    _ = ∑ x : α, if x ∈ S then 1 else 0 := Equiv.sum_comp pre.symm (fun x : α => if x ∈ S then (1:ℕ) else 0)
+    _ = _ := by simp
+
+omit [DecidableEq B] in
+lemma card_blockMarks_le (pre : α ≃ B × Card d) (S : Finset α) (b : B) :
+    (blockMarks pre S b).card ≤ S.card := by
+  rw [←sum_card_blockMarks pre S]
+  exact Finset.single_le_sum (f := fun b : B => (blockMarks pre S b).card) (fun _ _ => Nat.zero_le _) (Finset.mem_univ b)
+
+end Thorp
+
+end ThorpNine.Smoothing
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Smoothing/RotationCharge.lean b/lean/OAI/Probability/ThorpRouting/Smoothing/RotationCharge.lean
index 96377350588dfe520c71106be9424dc66fc356c6..bb1bc4d9028fa5952e6650d9883235b490d4ad0f 100644
--- a/lean/OAI/Probability/ThorpRouting/Smoothing/RotationCharge.lean
+++ b/lean/OAI/Probability/ThorpRouting/Smoothing/RotationCharge.lean
@@ -1,891 +1,891 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Smoothing.ExposedCoins
-
-namespace OAI
-
-namespace ThorpNine.Smoothing
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-section Rotation
-variable {d A : ℕ}
-
-lemma card_cycleFiber (o : Card d → Option (Fin A)) (i : Fin A) :
-    Fintype.card {x : Card d // o x = some i} = (cycleFiber o i).card := by
-  classical
-  simp [cycleFiber,Fintype.card_subtype]
-
-lemma terminal_mean {o : Card d → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
-    (i : Fin A) (x : Card d) (hx : o x = some i) :
-    finiteMean (fun z : Terminal o => if (z i).val = x then (1:ℝ) else 0) = slotWeight o i := by
-  classical
-  have hev := finiteMean_pi_eval (A := fun i : Fin A => {x : Card d // o x = some i})
-    i (fun z => if z.val = x then (1:ℝ) else 0)
-  rw [hev]
-  have he : (fun z : {x : Card d // o x = some i} => if z.val = x then (1:ℝ) else 0) =
-      (fun z => if z = ⟨x,hx⟩ then 1 else 0) := by
-    funext z
-    simp only [Subtype.ext_iff]
-  rw [he,finiteMean_eq_indicator,card_cycleFiber,slotWeight]
-
-lemma slotWeight_antitone (o : Card d → Option (Fin A))
-    (hn : ∀ i, (cycleFiber o i).Nonempty)
-    (hm : Monotone (fun i => (cycleFiber o i).card)) : Antitone (slotWeight o) := by
-  intro i j hij
-  unfold slotWeight
-  apply one_div_le_one_div_of_le
-  · exact_mod_cast Finset.card_pos.mpr (hn i)
-  · exact_mod_cast hm hij
-
-lemma rotationMeeting_mean {o : Card d → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
-    (hm : Monotone (fun i => (cycleFiber o i).card)) (u v : Card d) :
-    finiteMean (fun z : Terminal o => rotationMeeting o z u v) =
-      min (pathWeight o u) (pathWeight o v) + withinWeight o u v := by
-  classical
-  have hn : ∀ i, (cycleFiber o i).Nonempty := by
-    intro i
-    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : Card d // o x = some i})
-    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
-  have hw := slotWeight_antitone o hn hm
-  cases hu : o u with
-  | none =>
-    cases hv : o v with
-    | none => simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean]
-    | some j =>
-      have hj : 0 ≤ slotWeight o j := by unfold slotWeight; positivity
-      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_left hj]
-  | some i =>
-    cases hv : o v with
-    | none =>
-      have hi : 0 ≤ slotWeight o i := by unfold slotWeight; positivity
-      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_right hi]
-    | some j =>
-      simp only [rotationMeeting,pathWeight,withinWeight,hu,hv]
-      rcases lt_trichotomy i j with hij | rfl | hji
-      · simp only [hij,↓reduceIte,ne_of_lt hij]
-        rw [terminal_mean j v hv,min_eq_right (hw (le_of_lt hij)),add_zero]
-      · simp only [lt_self_iff_false,↓reduceIte,min_self]
-        rw [finiteMean_add,terminal_mean i u hu,terminal_mean i v hv]
-      · simp only [not_lt_of_ge (le_of_lt hji),hji,↓reduceIte,ne_of_gt hji]
-        rw [terminal_mean i u hu,min_eq_left (hw (le_of_lt hji)),add_zero]
-
-lemma pathWeight_sum (o : Card d → Option (Fin A)) (hn : ∀ i, (cycleFiber o i).Nonempty) :
-    ∑ x : Card d, pathWeight o x = (A : ℝ) := by
-  classical
-  have he (x : Card d) : pathWeight o x =
-      ∑ i : Fin A, if o x = some i then slotWeight o i else 0 := by
-    cases hx : o x with
-    | none => simp [pathWeight,hx]
-    | some j => simp [pathWeight,hx]
-  simp only [he]
-  rw [Finset.sum_comm]
-  have hf (i : Fin A) : (∑ x : Card d, if o x = some i then slotWeight o i else 0) = 1 := by
-    rw [← Finset.sum_filter]
-    change (∑ _x ∈ cycleFiber o i, slotWeight o i) = 1
-    simp only [Finset.sum_const,nsmul_eq_mul,slotWeight]
-    exact mul_one_div_cancel (by exact_mod_cast (Finset.card_pos.mpr (hn i)).ne')
-  simp only [hf,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,mul_one]
-
-lemma withinWeight_sum_bound (o : Card d → Option (Fin A))
-    (hn : ∀ i, (cycleFiber o i).Nonempty) (b : Butterfly d) :
-    (∑ s : SwitchIndex d, withinWeight o (switchUsers d b s).1 (switchUsers d b s).2) ≤
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
-  classical
-  have he (u v : Card d) : withinWeight o u v =
-      ∑ i : Fin A, (if u ∈ cycleFiber o i ∧ v ∈ cycleFiber o i then (1:ℝ) else 0) * slotWeight o i := by
-    simp only [mem_cycleFiber]
-    cases hu : o u <;> cases hv : o v <;> simp [withinWeight,hu,hv,ite_and,eq_comm]
-  simp only [he]
-  rw [Finset.sum_comm]
-  apply Finset.sum_le_sum
-  intro i _
-  rw [← Finset.sum_mul]
-  have hi := marked_switch_entropy_bound d b (cycleFiber o i)
-  have hpos : (0:ℝ) < (cycleFiber o i).card := by exact_mod_cast Finset.card_pos.mpr (hn i)
-  have hmul := mul_le_mul_of_nonneg_right hi (show 0 ≤ slotWeight o i by unfold slotWeight; positivity)
-  calc
-    _ ≤ ((cycleFiber o i).card : ℝ) * Real.log (cycleFiber o i).card /
-        (2 * Real.log 2) * slotWeight o i := hmul
-    _ = _ := by unfold slotWeight; field_simp
-
-noncomputable def rotationCharge (o : Card d → Option (Fin A)) (b : Butterfly d)
-    (z : Terminal o) : ℝ :=
-  ∑ s : SwitchIndex d, rotationMeeting o z (switchUsers d b s).1 (switchUsers d b s).2
-
-lemma rotationCharge_mean_bound {o : Card d → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
-    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : Butterfly d) :
-    finiteMean (rotationCharge o b) ≤ (A:ℝ) * d / 2 +
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
-  classical
-  have hn : ∀ i, (cycleFiber o i).Nonempty := by
-    intro i
-    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : Card d // o x = some i})
-    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
-  unfold rotationCharge
-  rw [finiteMean_sum]
-  simp only [rotationMeeting_mean hm,Finset.sum_add_distrib]
-  have h₁ := weighted_switch_min_bound d b (pathWeight o)
-  rw [pathWeight_sum o hn] at h₁
-  have h₂ := withinWeight_sum_bound o hn b
-  linarith
-
-lemma rotation_exp_bound {o : Card d → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
-    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : Butterfly d) :
-    Real.exp (-Real.log 2 * ((A:ℝ)*d/2 +
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2))) ≤
-      finiteMean (fun z : Terminal o => Real.exp (-Real.log 2 * rotationCharge o b z)) := by
-  have hb := rotationCharge_mean_bound hm b
-  have hlog : 0 ≤ Real.log 2 := Real.log_nonneg (by norm_num)
-  have hm' : -Real.log 2 * ((A:ℝ)*d/2 +
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2)) ≤
-      finiteMean (fun z : Terminal o => -Real.log 2 * rotationCharge o b z) := by
-    have he := finiteMean_mul_const (rotationCharge o b) (-Real.log 2)
-    simp only [mul_comm] at he
-    rw [he]
-    exact mul_le_mul_of_nonpos_left hb (neg_nonpos.mpr hlog)
-  exact (Real.exp_le_exp.mpr hm').trans (exp_finiteMean_le _)
-
-end Rotation
-
-section OrbitCore
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-noncomputable def orbitSet (p : Equiv.Perm α) (x : α) : Finset α :=
-  Finset.univ.filter (p.SameCycle x)
-
-@[simp] lemma mem_orbitSet (p : Equiv.Perm α) (x y : α) :
-    y ∈ orbitSet p x ↔ p.SameCycle x y := by
-  classical
-  simp [orbitSet]
-
-lemma orbitSet_nonempty (p : Equiv.Perm α) (x : α) : (orbitSet p x).Nonempty :=
-  ⟨x,(mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.refl _ _)⟩
-
-lemma orbitSet_eq (p : Equiv.Perm α) {x y : α} (h : p.SameCycle x y) :
-    orbitSet p x = orbitSet p y := by
-  ext z
-  simp only [mem_orbitSet]
-  exact ⟨fun hz => h.symm.trans hz,fun hz => h.trans hz⟩
-
-@[simp] lemma orbitSet_apply (p : Equiv.Perm α) (x : α) : orbitSet p (p x) = orbitSet p x := by
-  ext y
-  simp only [mem_orbitSet,Equiv.Perm.sameCycle_apply_left]
-
-lemma orbitSet_eq_or_disjoint (p : Equiv.Perm α) (x y : α) :
-    orbitSet p x = orbitSet p y ∨ Disjoint (orbitSet p x) (orbitSet p y) := by
-  classical
-  by_cases h : p.SameCycle x y
-  · exact Or.inl (orbitSet_eq p h)
-  · right
-    apply Finset.disjoint_left.mpr
-    intro z hz hz'
-    exact h (((mem_orbitSet _ _ _).mp hz).trans ((mem_orbitSet _ _ _).mp hz').symm)
-
-omit [DecidableEq α] in
-lemma perm_minimalPeriod_pos (p : Equiv.Perm α) (x : α) :
-    0 < Function.minimalPeriod p x :=
-  Function.minimalPeriod_pos_of_mem_periodicPts (p.injective.mem_periodicPts x)
-
-lemma orbitSet_card (p : Equiv.Perm α) (x : α) :
-    (orbitSet p x).card = Function.minimalPeriod p x := by
-  classical
-  let f : Fin (Function.minimalPeriod p x) → {y // p.SameCycle x y} := fun n =>
-    ⟨p^[n.val] x,by rw [Equiv.Perm.iterate_eq_pow]; exact ⟨(n.val:ℤ),by simp⟩⟩
-  have hf : Function.Bijective f := by
-    constructor
-    · intro i j hij
-      apply Fin.ext
-      exact Function.iterate_injOn_Iio_minimalPeriod i.isLt j.isLt (congrArg Subtype.val hij)
-    · intro y
-      obtain ⟨n,hn⟩ := y.property.exists_nat_pow_eq
-      refine ⟨⟨n % Function.minimalPeriod p x, Nat.mod_lt _ (perm_minimalPeriod_pos p x)⟩,?_⟩
-      apply Subtype.ext
-      change p^[n % Function.minimalPeriod p x] x = y.val
-      rw [Function.iterate_mod_minimalPeriod_eq,Equiv.Perm.iterate_eq_pow,hn]
-  have hc := Fintype.card_congr (Equiv.ofBijective f hf)
-  simpa [orbitSet,Fintype.card_subtype] using hc.symm
-
-def walkSet (p : Equiv.Perm α) (r : ℕ) (x : α) : Finset α :=
-  (Finset.range r).image (fun n => p^[n] x)
-
-omit [Fintype α] in
-@[simp] lemma mem_walkSet (p : Equiv.Perm α) (r : ℕ) (x y : α) :
-    y ∈ walkSet p r x ↔ ∃ n < r, p^[n] x = y := by
-  simp [walkSet]
-
-omit [Fintype α] in
-@[simp] lemma walkSet_zero (p : Equiv.Perm α) (x : α) : walkSet p 0 x = ∅ := by
-  simp [walkSet]
-
-omit [Fintype α] in
-lemma walkSet_succ_last (p : Equiv.Perm α) (r : ℕ) (x : α) :
-    walkSet p (r+1) x = insert (p^[r] x) (walkSet p r x) := by
-  simp [walkSet,Finset.range_add_one]
-
-omit [Fintype α] in
-lemma walkSet_succ_first (p : Equiv.Perm α) (r : ℕ) (x : α) :
-    walkSet p (r+1) x = insert x (walkSet p r (p x)) := by
-  ext y
-  simp only [mem_walkSet,Finset.mem_insert]
-  constructor
-  · rintro ⟨n,hn,hny⟩
-    cases n with
-    | zero => exact Or.inl (by simpa using hny.symm)
-    | succ n => exact Or.inr ⟨n,by omega,by simpa only [Function.iterate_succ_apply] using hny⟩
-  · rintro (rfl | ⟨n,hn,hny⟩)
-    · exact ⟨0,by omega,rfl⟩
-    · exact ⟨n+1,by omega,by simpa only [Function.iterate_succ_apply] using hny⟩
-
-omit [Fintype α] in
-lemma not_mem_short_walk (p : Equiv.Perm α) (x : α) {r : ℕ}
-    (hr : r < Function.minimalPeriod p x) : x ∉ walkSet p r (p x) := by
-  rintro hx
-  obtain ⟨n,hn,hnx⟩ := (mem_walkSet _ _ _ _).mp hx
-  have hperiod : Function.IsPeriodicPt p (n+1) x := by
-    simpa only [Function.IsPeriodicPt,Function.IsFixedPt,Function.iterate_succ_apply] using hnx
-  have hp := hperiod.minimalPeriod_le (by omega : 0 < n+1)
-  omega
-
-lemma walkSet_period (p : Equiv.Perm α) (x : α) :
-    walkSet p (Function.minimalPeriod p x) x = orbitSet p x := by
-  ext y
-  simp only [mem_walkSet,mem_orbitSet]
-  constructor
-  · rintro ⟨n,_,rfl⟩
-    rw [Equiv.Perm.iterate_eq_pow]
-    exact ⟨(n:ℤ),by simp⟩
-  · intro h
-    obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
-    refine ⟨n % Function.minimalPeriod p x,Nat.mod_lt _ (perm_minimalPeriod_pos p x),?_⟩
-    rw [Function.iterate_mod_minimalPeriod_eq,Equiv.Perm.iterate_eq_pow,hn]
-
-omit [Fintype α] in
-lemma walkSet_card (p : Equiv.Perm α) (x : α) {r : ℕ}
-    (hr : r ≤ Function.minimalPeriod p x) : (walkSet p r x).card = r := by
-  rw [walkSet,Finset.card_image_of_injOn]
-  · exact Finset.card_range _
-  · intro i hi j hj hij
-    exact Function.iterate_injOn_Iio_minimalPeriod
-      (lt_of_lt_of_le (Finset.mem_range.mp hi) hr)
-      (lt_of_lt_of_le (Finset.mem_range.mp hj) hr) hij
-
-lemma walkSet_subset_orbit (p : Equiv.Perm α) (r : ℕ) (x : α) :
-    walkSet p r x ⊆ orbitSet p x := by
-  intro y hy
-  obtain ⟨n,_,rfl⟩ := (mem_walkSet _ _ _ _).mp hy
-  rw [mem_orbitSet,Equiv.Perm.iterate_eq_pow]
-  exact ⟨(n:ℤ),by simp⟩
-
-def cycleTrace (p : Equiv.Perm α) (S : Finset α) (a : α) :
-    ℕ → α → Finset α → Option (Finset α × α)
-  | 0, x, V => if x ∈ S ∧ x ∉ V ∧ p x = a then some (insert x V,x) else none
-  | r+1, x, V => if x ∈ S ∧ x ∉ V then cycleTrace p S a r (p x) (insert x V) else none
-
-lemma cycleTrace_walk (p : Equiv.Perm α) (S : Finset α) (r : ℕ) (x : α) (V : Finset α)
-    (hr : r < Function.minimalPeriod p x) (hS : walkSet p (r+1) x ⊆ S)
-    (hV : Disjoint V (walkSet p (r+1) x)) :
-    cycleTrace p S (p^[r+1] x) r x V = some (V ∪ walkSet p (r+1) x,p^[r] x) := by
-  induction r generalizing x V with
-  | zero =>
-    have hxS : x ∈ S := hS (by simp [walkSet_succ_first])
-    have hxV : x ∉ V := fun h => Finset.disjoint_left.mp hV h (by simp [walkSet_succ_first])
-    simp [cycleTrace,hxS,hxV,walkSet_succ_first]
-  | succ r ih =>
-    have hxS : x ∈ S := hS (by rw [walkSet_succ_first]; exact Finset.mem_insert_self _ _)
-    have hxV : x ∉ V := fun h => Finset.disjoint_left.mp hV h
-      (by rw [walkSet_succ_first]; exact Finset.mem_insert_self _ _)
-    have hr' : r < Function.minimalPeriod p (p x) := by
-      rw [Function.minimalPeriod_apply (p.injective.mem_periodicPts x)]
-      omega
-    have hS' : walkSet p (r+1) (p x) ⊆ S := by
-      intro y hy
-      apply hS
-      rw [walkSet_succ_first]
-      exact Finset.mem_insert_of_mem hy
-    have hV' : Disjoint (insert x V) (walkSet p (r+1) (p x)) := by
-      apply Finset.disjoint_left.mpr
-      intro y hy hy'
-      rcases Finset.mem_insert.mp hy with hy | hy
-      · subst y
-        exact not_mem_short_walk p x hr hy'
-      · apply Finset.disjoint_left.mp hV hy
-        rw [walkSet_succ_first]
-        exact Finset.mem_insert_of_mem hy'
-    rw [cycleTrace,ite_eq_left ⟨hxS,hxV⟩,Function.iterate_succ_apply p (r+1) x,
-      ih _ _ hr' hS' hV',walkSet_succ_first p (r+1) x,Function.iterate_succ_apply]
-    simp only [Finset.insert_union,Finset.union_insert]
-
-lemma cycleTrace_orbit (p : Equiv.Perm α) (S V : Finset α) (z : α)
-    (hS : orbitSet p z ⊆ S) (hV : Disjoint V (orbitSet p z)) :
-    cycleTrace p S (p z) (Function.minimalPeriod p z - 1) (p z) V =
-      some (V ∪ orbitSet p z,z) := by
-  have hpos := perm_minimalPeriod_pos p z
-  have he : Function.minimalPeriod p (p z) = Function.minimalPeriod p z :=
-    Function.minimalPeriod_apply (p.injective.mem_periodicPts z)
-  have hp : Function.minimalPeriod p z - 1 + 1 = Function.minimalPeriod p z := by omega
-  have hw : walkSet p (Function.minimalPeriod p z - 1 + 1) (p z) = orbitSet p z := by
-    rw [hp,←he,walkSet_period,orbitSet_apply]
-  have hr : Function.minimalPeriod p z - 1 < Function.minimalPeriod p (p z) := by rw [he]; omega
-  have h := cycleTrace_walk p S (Function.minimalPeriod p z-1) (p z) V hr
-    (by simpa only [hw] using hS) (by simpa only [hw] using hV)
-  have ht : p^[Function.minimalPeriod p z-1] (p z) = z := by
-    rw [← Function.iterate_succ_apply,show (Function.minimalPeriod p z-1).succ =
-      Function.minimalPeriod p z by omega,Function.iterate_minimalPeriod]
-  have ha : p^[Function.minimalPeriod p z-1+1] (p z) = p z := by
-    rw [hp,←he,Function.iterate_minimalPeriod]
-  simpa only [ha,ht,hw] using h
-
-end OrbitCore
-
-noncomputable def exposedPaths (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) :
-    Finset (SwitchIndex d) :=
-  V.biUnion (fun x => routeDomain d x (butterflyPerm d b x))
-
-@[simp] lemma exposedPaths_empty (d : ℕ) (b : Butterfly d) : exposedPaths d b ∅ = ∅ := by
-  classical
-  simp [exposedPaths]
-
-lemma exposedPaths_insert (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) (x : Card d) :
-    exposedPaths d b (insert x V) = exposedPaths d b V ∪ routeDomain d x (butterflyPerm d b x) := by
-  classical
-  simp [exposedPaths,Finset.union_comm]
-
-noncomputable def terminalCharge (d : ℕ) (b : Butterfly d) (z : Card d) (V : Finset (Card d)) : ℕ :=
-  (routeDomain d z (butterflyPerm d b z) ∩ exposedPaths d b V).card
-
-namespace QueryTree
-
-lemma followPath_stale_charge {ι : Type*} [DecidableEq ι]
-    (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
-    (k : Card d → QueryTree ι) (R : Finset ι) (ω : ι → Bool) :
-    staleCharge (followPath d e x k) R ω =
-      staleCharge (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x))
-        (R ∪ (routeDomain d x (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).image e) ω := by
-  have h₀ := fresh_add_stale (followPath d e x k) R ω
-  have h₁ := fresh_add_stale (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x))
-    (R ∪ (routeDomain d x (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).image e) ω
-  rw [followPath_charge,followPath_raw_charge] at h₀
-  omega
-
-lemma seekCycle_semantics (d : ℕ) (S : Finset (Card d)) (a : Card d) (r : ℕ)
-    (x : Card d) (V W : Finset (Card d)) (z : Card d)
-    (k : Finset (Card d) → QueryTree (SwitchIndex d)) (ω : SwitchIndex d → Bool)
-    (h : cycleTrace (butterflyPerm d (decodeButterfly d ω)) S a r x V = some (W,z)) :
-    (seekCycle d S a r x V k).succeeds ω = (k W).succeeds ω ∧
-      (seekCycle d S a r x V k).staleCharge (exposedPaths d (decodeButterfly d ω) V) ω =
-        (k W).staleCharge (exposedPaths d (decodeButterfly d ω) W) ω +
-          terminalCharge d (decodeButterfly d ω) z (W.erase z) := by
-  classical
-  induction r generalizing x V with
-  | zero =>
-    simp only [cycleTrace] at h
-    split_ifs at h with hv
-    obtain ⟨hxS,hxV,hxa⟩ := hv
-    obtain ⟨rfl,rfl⟩ := Prod.mk.inj (Option.some.inj h)
-    simp only [seekCycle,ite_eq_left (show x ∈ S ∧ x ∉ V from ⟨hxS,hxV⟩),closePath_succeeds,ite_eq_left hxa]
-    constructor
-    · trivial
-    · rw [closePath_stale_charge _ _ _ _ _ _ hxa,exposedPaths_insert,
-        Finset.erase_insert hxV]
-      simp only [terminalCharge,hxa]
-  | succ r ih =>
-    simp only [cycleTrace] at h
-    split_ifs at h with hv
-    obtain ⟨hxS,hxV⟩ := hv
-    rw [seekCycle,ite_eq_left ⟨hxS,hxV⟩,followPath_succeeds,followPath_stale_charge]
-    simp only [Function.comp_id,Finset.image_id]
-    rw [← exposedPaths_insert]
-    exact ih _ _ h
-
-lemma seekCycle_orbit (d : ℕ) (S V : Finset (Card d)) (z : Card d)
-    (k : Finset (Card d) → QueryTree (SwitchIndex d)) (ω : SwitchIndex d → Bool)
-    (hS : orbitSet (butterflyPerm d (decodeButterfly d ω)) z ⊆ S)
-    (hV : Disjoint V (orbitSet (butterflyPerm d (decodeButterfly d ω)) z)) :
-    let p := butterflyPerm d (decodeButterfly d ω)
-    let W := V ∪ orbitSet p z
-    (seekCycle d S (p z) (Function.minimalPeriod p z-1) (p z) V k).succeeds ω = (k W).succeeds ω ∧
-      (seekCycle d S (p z) (Function.minimalPeriod p z-1) (p z) V k).staleCharge
-          (exposedPaths d (decodeButterfly d ω) V) ω =
-        (k W).staleCharge (exposedPaths d (decodeButterfly d ω) W) ω +
-          terminalCharge d (decodeButterfly d ω) z (W.erase z) := by
-  exact seekCycle_semantics _ _ _ _ _ _ _ _ _ _ (cycleTrace_orbit _ _ _ _ hS hV)
-
-end QueryTree
-
-lemma mem_exposedPaths (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) (s : SwitchIndex d) :
-    s ∈ exposedPaths d b V ↔ (switchUsers d b s).1 ∈ V ∨ (switchUsers d b s).2 ∈ V := by
-  classical
-  simp only [exposedPaths,Finset.mem_biUnion,mem_routeDomain_iff_user]
-  constructor
-  · rintro ⟨x,hx,h | h⟩
-    · exact Or.inl (h ▸ hx)
-    · exact Or.inr (h ▸ hx)
-  · rintro (h | h)
-    · exact ⟨_,h,Or.inl rfl⟩
-    · exact ⟨_,h,Or.inr rfl⟩
-
-lemma terminalCharge_sum (d : ℕ) (b : Butterfly d) (z : Card d) (V : Finset (Card d))
-    (hz : z ∉ V) :
-    (terminalCharge d b z V : ℝ) = ∑ s : SwitchIndex d,
-      ((if (switchUsers d b s).1 = z ∧ (switchUsers d b s).2 ∈ V then (1:ℝ) else 0) +
-      (if (switchUsers d b s).2 = z ∧ (switchUsers d b s).1 ∈ V then (1:ℝ) else 0)) := by
-  classical
-  have he : routeDomain d z (butterflyPerm d b z) ∩ exposedPaths d b V =
-      Finset.univ.filter (fun s =>
-        ((switchUsers d b s).1 = z ∧ (switchUsers d b s).2 ∈ V) ∨
-        ((switchUsers d b s).2 = z ∧ (switchUsers d b s).1 ∈ V)) := by
-    ext s
-    simp only [Finset.mem_inter,mem_routeDomain_iff_user,mem_exposedPaths,
-      Finset.mem_filter,Finset.mem_univ,true_and]
-    by_cases h₁ : (switchUsers d b s).1 = z <;>
-      by_cases h₂ : (switchUsers d b s).2 = z <;> simp [h₁,h₂,hz]
-  rw [terminalCharge,he]
-  simp only [← Finset.sum_boole]
-  apply Finset.sum_congr rfl
-  intro s _
-  by_cases h₁ : (switchUsers d b s).1 = z <;>
-    by_cases h₂ : (switchUsers d b s).2 = z <;> simp [h₁,h₂,hz]
-
-noncomputable def priorPaths {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) : Finset (Card d) :=
-  Finset.univ.filter (fun x => ∃ i : Fin A, o x = some i ∧ i.val < t)
-
-@[simp] lemma mem_priorPaths {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) (x : Card d) :
-    x ∈ priorPaths o t ↔ ∃ i : Fin A, o x = some i ∧ i.val < t := by
-  classical
-  simp [priorPaths]
-
-lemma terminalCharge_prior_sum {d A : ℕ} (o : Card d → Option (Fin A))
-    (b : Butterfly d) (z : Terminal o) (i : Fin A) :
-    (terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) =
-      ∑ s : SwitchIndex d,
-        ((if (z i).val = (switchUsers d b s).1 ∧ (switchUsers d b s).2 ∈ priorPaths o (i.val+1)
-          then (1:ℝ) else 0) +
-        (if (z i).val = (switchUsers d b s).2 ∧ (switchUsers d b s).1 ∈ priorPaths o (i.val+1)
-          then (1:ℝ) else 0)) := by
-  classical
-  rw [terminalCharge_sum _ _ _ _ (Finset.notMem_erase _ _)]
-  apply Finset.sum_congr rfl
-  intro s _
-  have hn := switchUsers_ne d b s
-  simp only [Finset.mem_erase]
-  by_cases h₁ : (z i).val = (switchUsers d b s).1 <;>
-    by_cases h₂ : (z i).val = (switchUsers d b s).2 <;> simp_all [eq_comm]
-
-lemma rotationMeeting_eq_slot_sum {d A : ℕ} (o : Card d → Option (Fin A))
-    (z : Terminal o) (u v : Card d) :
-    rotationMeeting o z u v = ∑ i : Fin A,
-      ((if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) +
-      (if (z i).val = v ∧ u ∈ priorPaths o (i.val+1) then (1:ℝ) else 0)) := by
-  classical
-  have only_owner (u : Card d) (a : Fin A) (hou : o u = some a)
-      (i : Fin A) (hia : i ≠ a) : (z i).val ≠ u := by
-    intro h
-    have hi := (z i).property
-    rw [h,hou] at hi
-    exact hia (Option.some.inj hi).symm
-  have sum_one (u v : Card d) (a : Fin A) (hou : o u = some a) :
-      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) =
-      if (z a).val = u ∧ v ∈ priorPaths o (a.val+1) then 1 else 0 := by
-    apply Finset.sum_eq_single a
-    · intro i _ hia
-      simp [only_owner u a hou i hia]
-    · simp
-  have sum_none (u v : Card d) (hou : o u = none) :
-      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) = 0 := by
-    apply Finset.sum_eq_zero
-    intro i _
-    have hn : (z i).val ≠ u := by
-      intro h
-      have hi := (z i).property
-      rw [h,hou] at hi
-      contradiction
-    simp [hn]
-  rw [Finset.sum_add_distrib]
-  cases hou : o u with
-  | none =>
-    rw [sum_none u v hou]
-    simp [rotationMeeting,hou,mem_priorPaths]
-  | some a =>
-    rw [sum_one u v a hou]
-    cases hov : o v with
-    | none =>
-      rw [sum_none v u hov]
-      simp [rotationMeeting,hou,hov,mem_priorPaths]
-    | some b =>
-      rw [sum_one v u b hov]
-      simp only [rotationMeeting,hou,hov,mem_priorPaths,Option.some.injEq]
-      rcases lt_trichotomy a b with hab | rfl | hba
-      · have hn : ¬ b.val < a.val+1 := by omega
-        have hy : a.val < b.val+1 := by omega
-        simp [hab,hn,hy]
-      · simp
-      · have hn : ¬ a.val < b.val+1 := by omega
-        have hy : b.val < a.val+1 := by omega
-        simp [hba,not_lt_of_ge (le_of_lt hba),hn,hy]
-
-lemma rotationCharge_eq_terminal_sum {d A : ℕ} (o : Card d → Option (Fin A))
-    (b : Butterfly d) (z : Terminal o) :
-    rotationCharge o b z = ∑ i : Fin A,
-      (terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) := by
-  simp only [rotationCharge,rotationMeeting_eq_slot_sum,terminalCharge_prior_sum]
-  exact Finset.sum_comm
-
-@[simp] lemma priorPaths_zero {d A : ℕ} (o : Card d → Option (Fin A)) : priorPaths o 0 = ∅ := by
-  classical
-  ext x
-  simp
-
-lemma priorPaths_succ {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) (ht : t < A) :
-    priorPaths o (t+1) = priorPaths o t ∪ cycleFiber o ⟨t,ht⟩ := by
-  classical
-  ext x
-  simp only [mem_priorPaths,Finset.mem_union,mem_cycleFiber]
-  constructor
-  · rintro ⟨i,hi,hit⟩
-    by_cases h : i.val < t
-    · exact Or.inl ⟨i,hi,h⟩
-    · have he : i = ⟨t,ht⟩ := Fin.ext (by change i.val = t; omega)
-      exact Or.inr (he ▸ hi)
-  · rintro (⟨i,hi,hit⟩ | hi)
-    · exact ⟨i,hi,by omega⟩
-    · exact ⟨⟨t,ht⟩,hi,by change t < t+1; omega⟩
-
-lemma priorPaths_disjoint {d A : ℕ} (o : Card d → Option (Fin A)) (i : Fin A) :
-    Disjoint (priorPaths o i.val) (cycleFiber o i) := by
-  classical
-  apply Finset.disjoint_left.mpr
-  intro x hx hx'
-  obtain ⟨j,hj,hji⟩ := (mem_priorPaths _ _ _).mp hx
-  have he : j = i := Option.some.inj (hj.symm.trans ((mem_cycleFiber _ _ _).mp hx'))
-  subst j
-  exact (lt_irrefl _ hji)
-
-namespace QueryTree
-
-lemma seekCycles_owned {d A : ℕ} (S : Finset (Card d)) (r : Fin A → ℕ)
-    (o : Card d → Option (Fin A)) (z : Terminal o) (ω : SwitchIndex d → Bool)
-    (hS : ∀ i, cycleFiber o i ⊆ S)
-    (hO : ∀ i, orbitSet (butterflyPerm d (decodeButterfly d ω)) (z i).val = cycleFiber o i)
-    (hC : ∀ i, (cycleFiber o i).card = r i+1) :
-    let p := butterflyPerm d (decodeButterfly d ω)
-    let L := List.ofFn (fun i : Fin A => (p (z i).val,r i))
-    (seekCycles d S L ∅).succeeds ω = true ∧
-      ((seekCycles d S L ∅).staleCharge ∅ ω : ℝ) = rotationCharge o (decodeButterfly d ω) z := by
-  classical
-  let b := decodeButterfly d ω
-  let p := butterflyPerm d b
-  let f (i : Fin A) := (p (z i).val,r i)
-  let c (i : Fin A) : ℝ := terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val)
-  have hc (i : Fin A) : Function.minimalPeriod p (z i).val - 1 = r i := by
-    have he := orbitSet_card p (z i).val
-    rw [hO,hC] at he
-    omega
-  have hs (t : ℕ) (ht : t ≤ A) :
-      (seekCycles d S ((List.ofFn f).drop t) (priorPaths o t)).succeeds ω = true ∧
-        ((seekCycles d S ((List.ofFn f).drop t) (priorPaths o t)).staleCharge (exposedPaths d b (priorPaths o t)) ω : ℝ) =
-          ((List.ofFn c).drop t).sum := by
-    induction ht using Nat.decreasingInduction with
-    | self => simp only [List.length_ofFn,le_refl,List.drop_eq_nil_of_le,List.sum_nil,seekCycles,succeeds,staleCharge,Nat.cast_zero,and_self]
-    | of_succ t ht ih =>
-      have htf : t < (List.ofFn f).length := by simpa using ht
-      have htc : t < (List.ofFn c).length := by simpa using ht
-      rw [List.drop_eq_getElem_cons htf,List.drop_eq_getElem_cons htc]
-      simp only [List.getElem_ofFn,List.sum_cons,seekCycles]
-      have h := seekCycle_orbit d S (priorPaths o t) (z ⟨t,ht⟩).val
-        (fun V => seekCycles d S ((List.ofFn f).drop (t+1)) V) ω
-        (by rw [hO]; exact hS _) (by rw [hO]; exact priorPaths_disjoint o ⟨t,ht⟩)
-      dsimp only at h
-      rw [hc,hO,←priorPaths_succ o t ht] at h
-      dsimp only [f]
-      constructor
-      · exact h.1.trans ih.1
-      · rw [h.2,Nat.cast_add,ih.2]
-        exact add_comm _ _
-  have h := hs 0 (Nat.zero_le _)
-  simp only [List.drop_zero,priorPaths_zero,exposedPaths_empty,List.sum_ofFn] at h
-  dsimp only
-  exact ⟨h.1,h.2.trans (rotationCharge_eq_terminal_sum o b z).symm⟩
-
-end QueryTree
-
-section Collections
-variable {d A : ℕ}
-
-def ValidCycleSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ)
-    (o : Card d → Option (Fin A)) : Prop :=
-  (∀ i, (cycleFiber o i).card = r i + 1) ∧
-  (∀ i, cycleFiber o i ⊆ S) ∧
-  (∀ i x, o x = some i → orbitSet p x = cycleFiber o i)
-
-abbrev CycleSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
-  {o : Card d → Option (Fin A) // ValidCycleSlots p S r o}
-
-noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
-    Fintype (CycleSlots p S r) := Fintype.ofFinite _
-
-lemma CycleSlots.terminal_nonempty {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (c : CycleSlots p S r) (i : Fin A) : Nonempty {x : Card d // c.val x = some i} := by
-  have hn : (cycleFiber c.val i).Nonempty := Finset.card_pos.mp (by rw [c.property.1]; omega)
-  obtain ⟨x,hx⟩ := hn
-  exact ⟨x,(mem_cycleFiber _ _ _).mp hx⟩
-
-lemma owner_ext {o o' : Card d → Option (Fin A)}
-    (h : ∀ i, cycleFiber o i = cycleFiber o' i) : o = o' := by
-  funext x
-  have he (i : Fin A) : o x = some i ↔ o' x = some i := by
-    simpa only [mem_cycleFiber] using (Finset.ext_iff.mp (h i) x)
-  cases ho : o x with
-  | some i => exact ((he i).mp ho).symm
-  | none =>
-    cases ho' : o' x with
-    | none => rfl
-    | some i => have hn := (he i).mpr ho'; rw [ho] at hn; contradiction
-
-abbrev RootedSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
-  Σ c : CycleSlots p S r, Terminal c.val
-
-noncomputable def rootedStart {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (c : RootedSlots p S r) : Fin A → S := fun i =>
-  ⟨p (c.2 i).val,c.1.property.2.1 i (by
-    rw [←c.1.property.2.2 i (c.2 i).val (c.2 i).property]
-    exact (mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.apply_right (Equiv.Perm.SameCycle.refl p _)))⟩
-
-lemma rootedStart_injective {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ} :
-    Function.Injective (rootedStart (p:=p) (S:=S) (r:=r)) := by
-  rintro ⟨c,z⟩ ⟨c',z'⟩ h
-  have hz (i : Fin A) : (z i).val = (z' i).val :=
-    p.injective (congrArg Subtype.val (congrFun h i))
-  have hc : c = c' := Subtype.ext (owner_ext (fun i => by
-    rw [←c.property.2.2 i (z i).val (z i).property,
-      ←c'.property.2.2 i (z' i).val (z' i).property,hz]))
-  subst c'
-  have hzz : z = z' := funext (fun i => Subtype.ext (hz i))
-  subst z'
-  rfl
-
-lemma sum_comp_injective_le {α β : Type*} [Fintype α] [Fintype β]
-    (e : α → β) (he : Function.Injective e) (f : β → ℝ) (hf : ∀ b, 0 ≤ f b) :
-    ∑ a, f (e a) ≤ ∑ b, f b := by
-  classical
-  calc
-    _ = ∑ b ∈ Finset.univ.image e, f b := (Finset.sum_image (fun _ _ _ _ h => he h)).symm
-    _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _) (fun b _ _ => hf b)
-
-lemma rooted_weight_sum_le (S : Finset (Card d)) (r : Fin A → ℕ) (ω : SwitchIndex d → Bool) :
-    (∑ c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r,
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2)) ≤
-    ∑ a : Fin A → S,
-      if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-        ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-      else 0 := by
-  classical
-  let p := butterflyPerm d (decodeButterfly d ω)
-  let g (a : Fin A → S) : ℝ :=
-    if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-    else 0
-  have he (c : RootedSlots p S r) :
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2) =
-      g (rootedStart c) := by
-    have hs := QueryTree.seekCycles_owned S r c.1.val c.2 ω c.1.property.2.1
-      (fun i => c.1.property.2.2 i (c.2 i).val (c.2 i).property) c.1.property.1
-    dsimp only at hs
-    unfold g
-    rw [ite_eq_left (show (QueryTree.seekCycles d S (List.ofFn (fun i => ((rootedStart c i).val,r i))) ∅).succeeds ω = true from hs.1)]
-    rw [← hs.2,neg_mul,Real.exp_neg,mul_comm (Real.log 2),Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
-    rfl
-  simp_rw [he]
-  exact sum_comp_injective_le rootedStart rootedStart_injective g (fun a => by dsimp [g]; split_ifs <;> positivity)
-
-noncomputable def rotationLower (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : ℝ :=
-  Real.exp (-Real.log 2 * ((A:ℝ)*d/2 + ∑ i : Fin A, Real.log (r i+1) / (2 * Real.log 2)))
-
-noncomputable def collectionWeight (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : ℝ :=
-  ((2:ℝ)^d)^A * (∏ i : Fin A, (r i+1 : ℝ)) * rotationLower d r
-
-lemma collectionWeight_pos (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : 0 < collectionWeight d r := by
-  unfold collectionWeight rotationLower
-  positivity
-
-lemma collection_rotation_lower (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r)
-    (ω : SwitchIndex d → Bool) (c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) :
-    collectionWeight d r ≤ ∑ z : Terminal c.val,
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z) := by
-  classical
-  let (i : Fin A) : Nonempty {x : Card d // c.val x = some i} := c.terminal_nonempty i
-  have hm : Monotone (fun i => (cycleFiber c.val i).card) := by
-    intro i j hij
-    dsimp only
-    rw [c.property.1,c.property.1]
-    exact Nat.add_le_add_right (hr hij) 1
-  have hb := rotation_exp_bound hm (decodeButterfly d ω)
-  have hcard : (Fintype.card (Terminal c.val) : ℝ) = ∏ i : Fin A, (r i+1 : ℝ) := by
-    rw [Fintype.card_pi,Nat.cast_prod]
-    apply Finset.prod_congr rfl
-    intro i _
-    rw [card_cycleFiber,c.property.1,Nat.cast_add,Nat.cast_one]
-  have hp : (0:ℝ) < Fintype.card (Terminal c.val) := Nat.cast_pos.mpr Fintype.card_pos
-  have hl : rotationLower d r ≤ finiteMean (fun z : Terminal c.val => Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z)) := by
-    simpa only [rotationLower,c.property.1,Nat.cast_add,Nat.cast_one] using hb
-  have hh := (le_div_iff₀ hp).mp hl
-  rw [hcard] at hh
-  unfold collectionWeight
-  rw [← Finset.mul_sum]
-  calc
-    _ = ((2:ℝ)^d)^A * (rotationLower d r * ∏ i : Fin A, (r i+1 : ℝ)) := by ring
-    _ ≤ _ := mul_le_mul_of_nonneg_left hh (by positivity)
-
-lemma cycleSlots_weighted_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) *
-      collectionWeight d r ≤ (S.card : ℝ)^A := by
-  classical
-  let g (a : Fin A → S) (ω : SwitchIndex d → Bool) : ℝ :=
-    if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-    else 0
-  have hω (ω : SwitchIndex d → Bool) :
-      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ) * collectionWeight d r ≤
-        ∑ a : Fin A → S, g a ω := by
-    calc
-      _ = ∑ _c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r, collectionWeight d r := by simp
-      _ ≤ ∑ c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r,
-          ∑ z : Terminal c.val, ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z) :=
-        Finset.sum_le_sum (fun c _ => collection_rotation_lower S r hr ω c)
-      _ = ∑ c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r,
-          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2) :=
-        (Fintype.sum_sigma (fun c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r =>
-          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2))).symm
-      _ ≤ _ := rooted_weight_sum_le S r ω
-  have hb := finiteMean_mono hω
-  rw [finiteMean_mul_const,finiteMean_sum] at hb
-  calc
-    _ ≤ ∑ a : Fin A → S, finiteMean (g a) := hb
-    _ ≤ ∑ _a : Fin A → S, (1:ℝ) := by
-      apply Finset.sum_le_sum
-      intro a _
-      have hh := QueryTree.cycles_weighted_success d S (List.ofFn (fun i : Fin A => ((a i).val,r i)))
-      simpa only [List.length_ofFn] using hh
-    _ = _ := by simp
-
-lemma cycleSlots_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) ≤
-        (S.card : ℝ)^A / collectionWeight d r := by
-  exact (le_div_iff₀ (collectionWeight_pos d r)).mpr (cycleSlots_weighted_moment S r hr)
-
-lemma collectionWeight_eq_sqrt (d : ℕ) {A : ℕ} (r : Fin A → ℕ) :
-    collectionWeight d r = (Real.sqrt ((2:ℝ)^d))^A * ∏ i : Fin A, Real.sqrt (r i+1) := by
-  have hn : ((2:ℝ)^d)^A = Real.exp ((A:ℝ)*d*Real.log 2) := by
-    rw [mul_assoc,Real.exp_nat_mul,Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
-  have hp : (∏ i : Fin A, (r i+1 : ℝ)) = Real.exp (∑ i : Fin A, Real.log (r i+1)) := by
-    rw [Real.exp_sum]
-    apply Finset.prod_congr rfl
-    intro i _
-    exact (Real.exp_log (by positivity)).symm
-  have hs : Real.sqrt ((2:ℝ)^d) = Real.exp ((d:ℝ)*Real.log 2/2) := by
-    rw [Real.sqrt_eq_rpow,Real.rpow_def_of_pos (by positivity),Real.log_pow]
-    congr 1
-    ring
-  have hj (i : Fin A) : Real.sqrt (r i+1) = Real.exp (Real.log (r i+1)/2) := by
-    rw [Real.sqrt_eq_rpow,Real.rpow_def_of_pos (by positivity)]
-    congr 1
-    ring
-  simp_rw [collectionWeight,hn,hp,rotationLower,hs,hj]
-  rw [←Real.exp_add,←Real.exp_add,←Real.exp_nat_mul,←Real.exp_sum,←Real.exp_add]
-  congr 1
-  rw [←Finset.sum_div,←Finset.sum_div]
-  have hl : Real.log 2 ≠ 0 := ne_of_gt (Real.log_pos (by norm_num))
-  field_simp
-  ring
-
-lemma butterfly_cycleSlots_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) ≤
-        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
-  have h := cycleSlots_moment S r hr
-  rw [collectionWeight_eq_sqrt,div_mul_eq_div_mul_one_div] at h
-  rw [div_pow]
-  simpa only [one_div,Finset.prod_inv_distrib] using h
-
-end Collections
-
-variable {d A : ℕ}
-
-abbrev SelectedCycle (p : Equiv.Perm (Card d)) (S : Finset (Card d)) :=
-  {C : Finset (Card d) // (∃ x, C = orbitSet p x) ∧ C ⊆ S}
-
-noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) :
-    Fintype (SelectedCycle p S) := Fintype.ofFinite _
-
-lemma SelectedCycle.nonempty {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
-    (C : SelectedCycle p S) : C.val.Nonempty := by
-  obtain ⟨x,hx⟩ := C.property.1
-  rw [hx]
-  exact orbitSet_nonempty p x
-
-lemma SelectedCycle.orbit_eq {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
-    (C : SelectedCycle p S) {x : Card d} (hx : x ∈ C.val) : orbitSet p x = C.val := by
-  obtain ⟨y,hy⟩ := C.property.1
-  rw [hy] at hx ⊢
-  exact (orbitSet_eq p ((mem_orbitSet _ _ _).mp hx)).symm
-
-lemma SelectedCycle.eq_of_mem {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
-    (C D : SelectedCycle p S) {x : Card d} (hC : x ∈ C.val) (hD : x ∈ D.val) : C = D := by
-  apply Subtype.ext
-  exact (C.orbit_eq hC).symm.trans (D.orbit_eq hD)
-
-abbrev SlotAssignment (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
-  {f : Fin A → SelectedCycle p S // Function.Injective f ∧ ∀ i, (f i).val.card = r i+1}
-
-noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
-    Fintype (SlotAssignment p S r) := Fintype.ofFinite _
-
-noncomputable def slotsToAssignment {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (c : CycleSlots p S r) : SlotAssignment p S r := by
-  classical
-  let f (i : Fin A) : SelectedCycle p S := ⟨cycleFiber c.val i,by
-    obtain ⟨x,hx⟩ := c.terminal_nonempty i
-    exact ⟨⟨x,(c.property.2.2 i x hx).symm⟩,c.property.2.1 i⟩⟩
-  refine ⟨f,?_,fun i => c.property.1 i⟩
-  intro i j hij
-  have he : cycleFiber c.val i = cycleFiber c.val j := congrArg Subtype.val hij
-  obtain ⟨x,hx⟩ := c.terminal_nonempty i
-  have hxi : x ∈ cycleFiber c.val i := (mem_cycleFiber _ _ _).mpr hx
-  rw [he] at hxi
-  exact Option.some.inj (hx.symm.trans ((mem_cycleFiber _ _ _).mp hxi))
-
-noncomputable def assignmentOwner {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) (x : Card d) : Option (Fin A) := by
-  classical
-  exact if h : ∃ i, x ∈ (f.val i).val then some (Classical.choose h) else none
-
-lemma assignmentOwner_eq_some {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) (x : Card d) (i : Fin A) :
-    assignmentOwner f x = some i ↔ x ∈ (f.val i).val := by
-  classical
-  unfold assignmentOwner
-  split_ifs with h
-  · constructor
-    · intro hi
-      have hx := Classical.choose_spec h
-      rw [Option.some.inj hi] at hx
-      exact hx
-    · intro hx
-      have he := (f.val (Classical.choose h)).eq_of_mem (f.val i) (Classical.choose_spec h) hx
-      exact congrArg some (f.property.1 he)
-  · constructor
-    · intro hi; contradiction
-    · intro hx; exact (h ⟨i,hx⟩).elim
-
-end Thorp
-
-end ThorpNine.Smoothing
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Smoothing.ExposedCoins
+
+namespace OAI
+
+namespace ThorpNine.Smoothing
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+section Rotation
+variable {d A : ℕ}
+
+lemma card_cycleFiber (o : Card d → Option (Fin A)) (i : Fin A) :
+    Fintype.card {x : Card d // o x = some i} = (cycleFiber o i).card := by
+  classical
+  simp [cycleFiber,Fintype.card_subtype]
+
+lemma terminal_mean {o : Card d → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
+    (i : Fin A) (x : Card d) (hx : o x = some i) :
+    finiteMean (fun z : Terminal o => if (z i).val = x then (1:ℝ) else 0) = slotWeight o i := by
+  classical
+  have hev := finiteMean_pi_eval (A := fun i : Fin A => {x : Card d // o x = some i})
+    i (fun z => if z.val = x then (1:ℝ) else 0)
+  rw [hev]
+  have he : (fun z : {x : Card d // o x = some i} => if z.val = x then (1:ℝ) else 0) =
+      (fun z => if z = ⟨x,hx⟩ then 1 else 0) := by
+    funext z
+    simp only [Subtype.ext_iff]
+  rw [he,finiteMean_eq_indicator,card_cycleFiber,slotWeight]
+
+lemma slotWeight_antitone (o : Card d → Option (Fin A))
+    (hn : ∀ i, (cycleFiber o i).Nonempty)
+    (hm : Monotone (fun i => (cycleFiber o i).card)) : Antitone (slotWeight o) := by
+  intro i j hij
+  unfold slotWeight
+  apply one_div_le_one_div_of_le
+  · exact_mod_cast Finset.card_pos.mpr (hn i)
+  · exact_mod_cast hm hij
+
+lemma rotationMeeting_mean {o : Card d → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
+    (hm : Monotone (fun i => (cycleFiber o i).card)) (u v : Card d) :
+    finiteMean (fun z : Terminal o => rotationMeeting o z u v) =
+      min (pathWeight o u) (pathWeight o v) + withinWeight o u v := by
+  classical
+  have hn : ∀ i, (cycleFiber o i).Nonempty := by
+    intro i
+    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : Card d // o x = some i})
+    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
+  have hw := slotWeight_antitone o hn hm
+  cases hu : o u with
+  | none =>
+    cases hv : o v with
+    | none => simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean]
+    | some j =>
+      have hj : 0 ≤ slotWeight o j := by unfold slotWeight; positivity
+      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_left hj]
+  | some i =>
+    cases hv : o v with
+    | none =>
+      have hi : 0 ≤ slotWeight o i := by unfold slotWeight; positivity
+      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_right hi]
+    | some j =>
+      simp only [rotationMeeting,pathWeight,withinWeight,hu,hv]
+      rcases lt_trichotomy i j with hij | rfl | hji
+      · simp only [hij,↓reduceIte,ne_of_lt hij]
+        rw [terminal_mean j v hv,min_eq_right (hw (le_of_lt hij)),add_zero]
+      · simp only [lt_self_iff_false,↓reduceIte,min_self]
+        rw [finiteMean_add,terminal_mean i u hu,terminal_mean i v hv]
+      · simp only [not_lt_of_ge (le_of_lt hji),hji,↓reduceIte,ne_of_gt hji]
+        rw [terminal_mean i u hu,min_eq_left (hw (le_of_lt hji)),add_zero]
+
+lemma pathWeight_sum (o : Card d → Option (Fin A)) (hn : ∀ i, (cycleFiber o i).Nonempty) :
+    ∑ x : Card d, pathWeight o x = (A : ℝ) := by
+  classical
+  have he (x : Card d) : pathWeight o x =
+      ∑ i : Fin A, if o x = some i then slotWeight o i else 0 := by
+    cases hx : o x with
+    | none => simp [pathWeight,hx]
+    | some j => simp [pathWeight,hx]
+  simp only [he]
+  rw [Finset.sum_comm]
+  have hf (i : Fin A) : (∑ x : Card d, if o x = some i then slotWeight o i else 0) = 1 := by
+    rw [← Finset.sum_filter]
+    change (∑ _x ∈ cycleFiber o i, slotWeight o i) = 1
+    simp only [Finset.sum_const,nsmul_eq_mul,slotWeight]
+    exact mul_one_div_cancel (by exact_mod_cast (Finset.card_pos.mpr (hn i)).ne')
+  simp only [hf,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,mul_one]
+
+lemma withinWeight_sum_bound (o : Card d → Option (Fin A))
+    (hn : ∀ i, (cycleFiber o i).Nonempty) (b : Butterfly d) :
+    (∑ s : SwitchIndex d, withinWeight o (switchUsers d b s).1 (switchUsers d b s).2) ≤
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
+  classical
+  have he (u v : Card d) : withinWeight o u v =
+      ∑ i : Fin A, (if u ∈ cycleFiber o i ∧ v ∈ cycleFiber o i then (1:ℝ) else 0) * slotWeight o i := by
+    simp only [mem_cycleFiber]
+    cases hu : o u <;> cases hv : o v <;> simp [withinWeight,hu,hv,ite_and,eq_comm]
+  simp only [he]
+  rw [Finset.sum_comm]
+  apply Finset.sum_le_sum
+  intro i _
+  rw [← Finset.sum_mul]
+  have hi := marked_switch_entropy_bound d b (cycleFiber o i)
+  have hpos : (0:ℝ) < (cycleFiber o i).card := by exact_mod_cast Finset.card_pos.mpr (hn i)
+  have hmul := mul_le_mul_of_nonneg_right hi (show 0 ≤ slotWeight o i by unfold slotWeight; positivity)
+  calc
+    _ ≤ ((cycleFiber o i).card : ℝ) * Real.log (cycleFiber o i).card /
+        (2 * Real.log 2) * slotWeight o i := hmul
+    _ = _ := by unfold slotWeight; field_simp
+
+noncomputable def rotationCharge (o : Card d → Option (Fin A)) (b : Butterfly d)
+    (z : Terminal o) : ℝ :=
+  ∑ s : SwitchIndex d, rotationMeeting o z (switchUsers d b s).1 (switchUsers d b s).2
+
+lemma rotationCharge_mean_bound {o : Card d → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
+    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : Butterfly d) :
+    finiteMean (rotationCharge o b) ≤ (A:ℝ) * d / 2 +
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
+  classical
+  have hn : ∀ i, (cycleFiber o i).Nonempty := by
+    intro i
+    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : Card d // o x = some i})
+    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
+  unfold rotationCharge
+  rw [finiteMean_sum]
+  simp only [rotationMeeting_mean hm,Finset.sum_add_distrib]
+  have h₁ := weighted_switch_min_bound d b (pathWeight o)
+  rw [pathWeight_sum o hn] at h₁
+  have h₂ := withinWeight_sum_bound o hn b
+  linarith
+
+lemma rotation_exp_bound {o : Card d → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
+    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : Butterfly d) :
+    Real.exp (-Real.log 2 * ((A:ℝ)*d/2 +
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2))) ≤
+      finiteMean (fun z : Terminal o => Real.exp (-Real.log 2 * rotationCharge o b z)) := by
+  have hb := rotationCharge_mean_bound hm b
+  have hlog : 0 ≤ Real.log 2 := Real.log_nonneg (by norm_num)
+  have hm' : -Real.log 2 * ((A:ℝ)*d/2 +
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2)) ≤
+      finiteMean (fun z : Terminal o => -Real.log 2 * rotationCharge o b z) := by
+    have he := finiteMean_mul_const (rotationCharge o b) (-Real.log 2)
+    simp only [mul_comm] at he
+    rw [he]
+    exact mul_le_mul_of_nonpos_left hb (neg_nonpos.mpr hlog)
+  exact (Real.exp_le_exp.mpr hm').trans (exp_finiteMean_le _)
+
+end Rotation
+
+section OrbitCore
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+noncomputable def orbitSet (p : Equiv.Perm α) (x : α) : Finset α :=
+  Finset.univ.filter (p.SameCycle x)
+
+@[simp] lemma mem_orbitSet (p : Equiv.Perm α) (x y : α) :
+    y ∈ orbitSet p x ↔ p.SameCycle x y := by
+  classical
+  simp [orbitSet]
+
+lemma orbitSet_nonempty (p : Equiv.Perm α) (x : α) : (orbitSet p x).Nonempty :=
+  ⟨x,(mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.refl _ _)⟩
+
+lemma orbitSet_eq (p : Equiv.Perm α) {x y : α} (h : p.SameCycle x y) :
+    orbitSet p x = orbitSet p y := by
+  ext z
+  simp only [mem_orbitSet]
+  exact ⟨fun hz => h.symm.trans hz,fun hz => h.trans hz⟩
+
+@[simp] lemma orbitSet_apply (p : Equiv.Perm α) (x : α) : orbitSet p (p x) = orbitSet p x := by
+  ext y
+  simp only [mem_orbitSet,Equiv.Perm.sameCycle_apply_left]
+
+lemma orbitSet_eq_or_disjoint (p : Equiv.Perm α) (x y : α) :
+    orbitSet p x = orbitSet p y ∨ Disjoint (orbitSet p x) (orbitSet p y) := by
+  classical
+  by_cases h : p.SameCycle x y
+  · exact Or.inl (orbitSet_eq p h)
+  · right
+    apply Finset.disjoint_left.mpr
+    intro z hz hz'
+    exact h (((mem_orbitSet _ _ _).mp hz).trans ((mem_orbitSet _ _ _).mp hz').symm)
+
+omit [DecidableEq α] in
+lemma perm_minimalPeriod_pos (p : Equiv.Perm α) (x : α) :
+    0 < Function.minimalPeriod p x :=
+  Function.minimalPeriod_pos_of_mem_periodicPts (p.injective.mem_periodicPts x)
+
+lemma orbitSet_card (p : Equiv.Perm α) (x : α) :
+    (orbitSet p x).card = Function.minimalPeriod p x := by
+  classical
+  let f : Fin (Function.minimalPeriod p x) → {y // p.SameCycle x y} := fun n =>
+    ⟨p^[n.val] x,by rw [Equiv.Perm.iterate_eq_pow]; exact ⟨(n.val:ℤ),by simp⟩⟩
+  have hf : Function.Bijective f := by
+    constructor
+    · intro i j hij
+      apply Fin.ext
+      exact Function.iterate_injOn_Iio_minimalPeriod i.isLt j.isLt (congrArg Subtype.val hij)
+    · intro y
+      obtain ⟨n,hn⟩ := y.property.exists_nat_pow_eq
+      refine ⟨⟨n % Function.minimalPeriod p x, Nat.mod_lt _ (perm_minimalPeriod_pos p x)⟩,?_⟩
+      apply Subtype.ext
+      change p^[n % Function.minimalPeriod p x] x = y.val
+      rw [Function.iterate_mod_minimalPeriod_eq,Equiv.Perm.iterate_eq_pow,hn]
+  have hc := Fintype.card_congr (Equiv.ofBijective f hf)
+  simpa [orbitSet,Fintype.card_subtype] using hc.symm
+
+def walkSet (p : Equiv.Perm α) (r : ℕ) (x : α) : Finset α :=
+  (Finset.range r).image (fun n => p^[n] x)
+
+omit [Fintype α] in
+@[simp] lemma mem_walkSet (p : Equiv.Perm α) (r : ℕ) (x y : α) :
+    y ∈ walkSet p r x ↔ ∃ n < r, p^[n] x = y := by
+  simp [walkSet]
+
+omit [Fintype α] in
+@[simp] lemma walkSet_zero (p : Equiv.Perm α) (x : α) : walkSet p 0 x = ∅ := by
+  simp [walkSet]
+
+omit [Fintype α] in
+lemma walkSet_succ_last (p : Equiv.Perm α) (r : ℕ) (x : α) :
+    walkSet p (r+1) x = insert (p^[r] x) (walkSet p r x) := by
+  simp [walkSet,Finset.range_add_one]
+
+omit [Fintype α] in
+lemma walkSet_succ_first (p : Equiv.Perm α) (r : ℕ) (x : α) :
+    walkSet p (r+1) x = insert x (walkSet p r (p x)) := by
+  ext y
+  simp only [mem_walkSet,Finset.mem_insert]
+  constructor
+  · rintro ⟨n,hn,hny⟩
+    cases n with
+    | zero => exact Or.inl (by simpa using hny.symm)
+    | succ n => exact Or.inr ⟨n,by omega,by simpa only [Function.iterate_succ_apply] using hny⟩
+  · rintro (rfl | ⟨n,hn,hny⟩)
+    · exact ⟨0,by omega,rfl⟩
+    · exact ⟨n+1,by omega,by simpa only [Function.iterate_succ_apply] using hny⟩
+
+omit [Fintype α] in
+lemma not_mem_short_walk (p : Equiv.Perm α) (x : α) {r : ℕ}
+    (hr : r < Function.minimalPeriod p x) : x ∉ walkSet p r (p x) := by
+  rintro hx
+  obtain ⟨n,hn,hnx⟩ := (mem_walkSet _ _ _ _).mp hx
+  have hperiod : Function.IsPeriodicPt p (n+1) x := by
+    simpa only [Function.IsPeriodicPt,Function.IsFixedPt,Function.iterate_succ_apply] using hnx
+  have hp := hperiod.minimalPeriod_le (by omega : 0 < n+1)
+  omega
+
+lemma walkSet_period (p : Equiv.Perm α) (x : α) :
+    walkSet p (Function.minimalPeriod p x) x = orbitSet p x := by
+  ext y
+  simp only [mem_walkSet,mem_orbitSet]
+  constructor
+  · rintro ⟨n,_,rfl⟩
+    rw [Equiv.Perm.iterate_eq_pow]
+    exact ⟨(n:ℤ),by simp⟩
+  · intro h
+    obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
+    refine ⟨n % Function.minimalPeriod p x,Nat.mod_lt _ (perm_minimalPeriod_pos p x),?_⟩
+    rw [Function.iterate_mod_minimalPeriod_eq,Equiv.Perm.iterate_eq_pow,hn]
+
+omit [Fintype α] in
+lemma walkSet_card (p : Equiv.Perm α) (x : α) {r : ℕ}
+    (hr : r ≤ Function.minimalPeriod p x) : (walkSet p r x).card = r := by
+  rw [walkSet,Finset.card_image_of_injOn]
+  · exact Finset.card_range _
+  · intro i hi j hj hij
+    exact Function.iterate_injOn_Iio_minimalPeriod
+      (lt_of_lt_of_le (Finset.mem_range.mp hi) hr)
+      (lt_of_lt_of_le (Finset.mem_range.mp hj) hr) hij
+
+lemma walkSet_subset_orbit (p : Equiv.Perm α) (r : ℕ) (x : α) :
+    walkSet p r x ⊆ orbitSet p x := by
+  intro y hy
+  obtain ⟨n,_,rfl⟩ := (mem_walkSet _ _ _ _).mp hy
+  rw [mem_orbitSet,Equiv.Perm.iterate_eq_pow]
+  exact ⟨(n:ℤ),by simp⟩
+
+def cycleTrace (p : Equiv.Perm α) (S : Finset α) (a : α) :
+    ℕ → α → Finset α → Option (Finset α × α)
+  | 0, x, V => if x ∈ S ∧ x ∉ V ∧ p x = a then some (insert x V,x) else none
+  | r+1, x, V => if x ∈ S ∧ x ∉ V then cycleTrace p S a r (p x) (insert x V) else none
+
+lemma cycleTrace_walk (p : Equiv.Perm α) (S : Finset α) (r : ℕ) (x : α) (V : Finset α)
+    (hr : r < Function.minimalPeriod p x) (hS : walkSet p (r+1) x ⊆ S)
+    (hV : Disjoint V (walkSet p (r+1) x)) :
+    cycleTrace p S (p^[r+1] x) r x V = some (V ∪ walkSet p (r+1) x,p^[r] x) := by
+  induction r generalizing x V with
+  | zero =>
+    have hxS : x ∈ S := hS (by simp [walkSet_succ_first])
+    have hxV : x ∉ V := fun h => Finset.disjoint_left.mp hV h (by simp [walkSet_succ_first])
+    simp [cycleTrace,hxS,hxV,walkSet_succ_first]
+  | succ r ih =>
+    have hxS : x ∈ S := hS (by rw [walkSet_succ_first]; exact Finset.mem_insert_self _ _)
+    have hxV : x ∉ V := fun h => Finset.disjoint_left.mp hV h
+      (by rw [walkSet_succ_first]; exact Finset.mem_insert_self _ _)
+    have hr' : r < Function.minimalPeriod p (p x) := by
+      rw [Function.minimalPeriod_apply (p.injective.mem_periodicPts x)]
+      omega
+    have hS' : walkSet p (r+1) (p x) ⊆ S := by
+      intro y hy
+      apply hS
+      rw [walkSet_succ_first]
+      exact Finset.mem_insert_of_mem hy
+    have hV' : Disjoint (insert x V) (walkSet p (r+1) (p x)) := by
+      apply Finset.disjoint_left.mpr
+      intro y hy hy'
+      rcases Finset.mem_insert.mp hy with hy | hy
+      · subst y
+        exact not_mem_short_walk p x hr hy'
+      · apply Finset.disjoint_left.mp hV hy
+        rw [walkSet_succ_first]
+        exact Finset.mem_insert_of_mem hy'
+    rw [cycleTrace,ite_eq_left ⟨hxS,hxV⟩,Function.iterate_succ_apply p (r+1) x,
+      ih _ _ hr' hS' hV',walkSet_succ_first p (r+1) x,Function.iterate_succ_apply]
+    simp only [Finset.insert_union,Finset.union_insert]
+
+lemma cycleTrace_orbit (p : Equiv.Perm α) (S V : Finset α) (z : α)
+    (hS : orbitSet p z ⊆ S) (hV : Disjoint V (orbitSet p z)) :
+    cycleTrace p S (p z) (Function.minimalPeriod p z - 1) (p z) V =
+      some (V ∪ orbitSet p z,z) := by
+  have hpos := perm_minimalPeriod_pos p z
+  have he : Function.minimalPeriod p (p z) = Function.minimalPeriod p z :=
+    Function.minimalPeriod_apply (p.injective.mem_periodicPts z)
+  have hp : Function.minimalPeriod p z - 1 + 1 = Function.minimalPeriod p z := by omega
+  have hw : walkSet p (Function.minimalPeriod p z - 1 + 1) (p z) = orbitSet p z := by
+    rw [hp,←he,walkSet_period,orbitSet_apply]
+  have hr : Function.minimalPeriod p z - 1 < Function.minimalPeriod p (p z) := by rw [he]; omega
+  have h := cycleTrace_walk p S (Function.minimalPeriod p z-1) (p z) V hr
+    (by simpa only [hw] using hS) (by simpa only [hw] using hV)
+  have ht : p^[Function.minimalPeriod p z-1] (p z) = z := by
+    rw [← Function.iterate_succ_apply,show (Function.minimalPeriod p z-1).succ =
+      Function.minimalPeriod p z by omega,Function.iterate_minimalPeriod]
+  have ha : p^[Function.minimalPeriod p z-1+1] (p z) = p z := by
+    rw [hp,←he,Function.iterate_minimalPeriod]
+  simpa only [ha,ht,hw] using h
+
+end OrbitCore
+
+noncomputable def exposedPaths (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) :
+    Finset (SwitchIndex d) :=
+  V.biUnion (fun x => routeDomain d x (butterflyPerm d b x))
+
+@[simp] lemma exposedPaths_empty (d : ℕ) (b : Butterfly d) : exposedPaths d b ∅ = ∅ := by
+  classical
+  simp [exposedPaths]
+
+lemma exposedPaths_insert (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) (x : Card d) :
+    exposedPaths d b (insert x V) = exposedPaths d b V ∪ routeDomain d x (butterflyPerm d b x) := by
+  classical
+  simp [exposedPaths,Finset.union_comm]
+
+noncomputable def terminalCharge (d : ℕ) (b : Butterfly d) (z : Card d) (V : Finset (Card d)) : ℕ :=
+  (routeDomain d z (butterflyPerm d b z) ∩ exposedPaths d b V).card
+
+namespace QueryTree
+
+lemma followPath_stale_charge {ι : Type*} [DecidableEq ι]
+    (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
+    (k : Card d → QueryTree ι) (R : Finset ι) (ω : ι → Bool) :
+    staleCharge (followPath d e x k) R ω =
+      staleCharge (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x))
+        (R ∪ (routeDomain d x (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).image e) ω := by
+  have h₀ := fresh_add_stale (followPath d e x k) R ω
+  have h₁ := fresh_add_stale (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x))
+    (R ∪ (routeDomain d x (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).image e) ω
+  rw [followPath_charge,followPath_raw_charge] at h₀
+  omega
+
+lemma seekCycle_semantics (d : ℕ) (S : Finset (Card d)) (a : Card d) (r : ℕ)
+    (x : Card d) (V W : Finset (Card d)) (z : Card d)
+    (k : Finset (Card d) → QueryTree (SwitchIndex d)) (ω : SwitchIndex d → Bool)
+    (h : cycleTrace (butterflyPerm d (decodeButterfly d ω)) S a r x V = some (W,z)) :
+    (seekCycle d S a r x V k).succeeds ω = (k W).succeeds ω ∧
+      (seekCycle d S a r x V k).staleCharge (exposedPaths d (decodeButterfly d ω) V) ω =
+        (k W).staleCharge (exposedPaths d (decodeButterfly d ω) W) ω +
+          terminalCharge d (decodeButterfly d ω) z (W.erase z) := by
+  classical
+  induction r generalizing x V with
+  | zero =>
+    simp only [cycleTrace] at h
+    split_ifs at h with hv
+    obtain ⟨hxS,hxV,hxa⟩ := hv
+    obtain ⟨rfl,rfl⟩ := Prod.mk.inj (Option.some.inj h)
+    simp only [seekCycle,ite_eq_left (show x ∈ S ∧ x ∉ V from ⟨hxS,hxV⟩),closePath_succeeds,ite_eq_left hxa]
+    constructor
+    · trivial
+    · rw [closePath_stale_charge _ _ _ _ _ _ hxa,exposedPaths_insert,
+        Finset.erase_insert hxV]
+      simp only [terminalCharge,hxa]
+  | succ r ih =>
+    simp only [cycleTrace] at h
+    split_ifs at h with hv
+    obtain ⟨hxS,hxV⟩ := hv
+    rw [seekCycle,ite_eq_left ⟨hxS,hxV⟩,followPath_succeeds,followPath_stale_charge]
+    simp only [Function.comp_id,Finset.image_id]
+    rw [← exposedPaths_insert]
+    exact ih _ _ h
+
+lemma seekCycle_orbit (d : ℕ) (S V : Finset (Card d)) (z : Card d)
+    (k : Finset (Card d) → QueryTree (SwitchIndex d)) (ω : SwitchIndex d → Bool)
+    (hS : orbitSet (butterflyPerm d (decodeButterfly d ω)) z ⊆ S)
+    (hV : Disjoint V (orbitSet (butterflyPerm d (decodeButterfly d ω)) z)) :
+    let p := butterflyPerm d (decodeButterfly d ω)
+    let W := V ∪ orbitSet p z
+    (seekCycle d S (p z) (Function.minimalPeriod p z-1) (p z) V k).succeeds ω = (k W).succeeds ω ∧
+      (seekCycle d S (p z) (Function.minimalPeriod p z-1) (p z) V k).staleCharge
+          (exposedPaths d (decodeButterfly d ω) V) ω =
+        (k W).staleCharge (exposedPaths d (decodeButterfly d ω) W) ω +
+          terminalCharge d (decodeButterfly d ω) z (W.erase z) := by
+  exact seekCycle_semantics _ _ _ _ _ _ _ _ _ _ (cycleTrace_orbit _ _ _ _ hS hV)
+
+end QueryTree
+
+lemma mem_exposedPaths (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) (s : SwitchIndex d) :
+    s ∈ exposedPaths d b V ↔ (switchUsers d b s).1 ∈ V ∨ (switchUsers d b s).2 ∈ V := by
+  classical
+  simp only [exposedPaths,Finset.mem_biUnion,mem_routeDomain_iff_user]
+  constructor
+  · rintro ⟨x,hx,h | h⟩
+    · exact Or.inl (h ▸ hx)
+    · exact Or.inr (h ▸ hx)
+  · rintro (h | h)
+    · exact ⟨_,h,Or.inl rfl⟩
+    · exact ⟨_,h,Or.inr rfl⟩
+
+lemma terminalCharge_sum (d : ℕ) (b : Butterfly d) (z : Card d) (V : Finset (Card d))
+    (hz : z ∉ V) :
+    (terminalCharge d b z V : ℝ) = ∑ s : SwitchIndex d,
+      ((if (switchUsers d b s).1 = z ∧ (switchUsers d b s).2 ∈ V then (1:ℝ) else 0) +
+      (if (switchUsers d b s).2 = z ∧ (switchUsers d b s).1 ∈ V then (1:ℝ) else 0)) := by
+  classical
+  have he : routeDomain d z (butterflyPerm d b z) ∩ exposedPaths d b V =
+      Finset.univ.filter (fun s =>
+        ((switchUsers d b s).1 = z ∧ (switchUsers d b s).2 ∈ V) ∨
+        ((switchUsers d b s).2 = z ∧ (switchUsers d b s).1 ∈ V)) := by
+    ext s
+    simp only [Finset.mem_inter,mem_routeDomain_iff_user,mem_exposedPaths,
+      Finset.mem_filter,Finset.mem_univ,true_and]
+    by_cases h₁ : (switchUsers d b s).1 = z <;>
+      by_cases h₂ : (switchUsers d b s).2 = z <;> simp [h₁,h₂,hz]
+  rw [terminalCharge,he]
+  simp only [← Finset.sum_boole]
+  apply Finset.sum_congr rfl
+  intro s _
+  by_cases h₁ : (switchUsers d b s).1 = z <;>
+    by_cases h₂ : (switchUsers d b s).2 = z <;> simp [h₁,h₂,hz]
+
+noncomputable def priorPaths {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) : Finset (Card d) :=
+  Finset.univ.filter (fun x => ∃ i : Fin A, o x = some i ∧ i.val < t)
+
+@[simp] lemma mem_priorPaths {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) (x : Card d) :
+    x ∈ priorPaths o t ↔ ∃ i : Fin A, o x = some i ∧ i.val < t := by
+  classical
+  simp [priorPaths]
+
+lemma terminalCharge_prior_sum {d A : ℕ} (o : Card d → Option (Fin A))
+    (b : Butterfly d) (z : Terminal o) (i : Fin A) :
+    (terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) =
+      ∑ s : SwitchIndex d,
+        ((if (z i).val = (switchUsers d b s).1 ∧ (switchUsers d b s).2 ∈ priorPaths o (i.val+1)
+          then (1:ℝ) else 0) +
+        (if (z i).val = (switchUsers d b s).2 ∧ (switchUsers d b s).1 ∈ priorPaths o (i.val+1)
+          then (1:ℝ) else 0)) := by
+  classical
+  rw [terminalCharge_sum _ _ _ _ (Finset.notMem_erase _ _)]
+  apply Finset.sum_congr rfl
+  intro s _
+  have hn := switchUsers_ne d b s
+  simp only [Finset.mem_erase]
+  by_cases h₁ : (z i).val = (switchUsers d b s).1 <;>
+    by_cases h₂ : (z i).val = (switchUsers d b s).2 <;> simp_all [eq_comm]
+
+lemma rotationMeeting_eq_slot_sum {d A : ℕ} (o : Card d → Option (Fin A))
+    (z : Terminal o) (u v : Card d) :
+    rotationMeeting o z u v = ∑ i : Fin A,
+      ((if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) +
+      (if (z i).val = v ∧ u ∈ priorPaths o (i.val+1) then (1:ℝ) else 0)) := by
+  classical
+  have only_owner (u : Card d) (a : Fin A) (hou : o u = some a)
+      (i : Fin A) (hia : i ≠ a) : (z i).val ≠ u := by
+    intro h
+    have hi := (z i).property
+    rw [h,hou] at hi
+    exact hia (Option.some.inj hi).symm
+  have sum_one (u v : Card d) (a : Fin A) (hou : o u = some a) :
+      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) =
+      if (z a).val = u ∧ v ∈ priorPaths o (a.val+1) then 1 else 0 := by
+    apply Finset.sum_eq_single a
+    · intro i _ hia
+      simp [only_owner u a hou i hia]
+    · simp
+  have sum_none (u v : Card d) (hou : o u = none) :
+      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) = 0 := by
+    apply Finset.sum_eq_zero
+    intro i _
+    have hn : (z i).val ≠ u := by
+      intro h
+      have hi := (z i).property
+      rw [h,hou] at hi
+      contradiction
+    simp [hn]
+  rw [Finset.sum_add_distrib]
+  cases hou : o u with
+  | none =>
+    rw [sum_none u v hou]
+    simp [rotationMeeting,hou,mem_priorPaths]
+  | some a =>
+    rw [sum_one u v a hou]
+    cases hov : o v with
+    | none =>
+      rw [sum_none v u hov]
+      simp [rotationMeeting,hou,hov,mem_priorPaths]
+    | some b =>
+      rw [sum_one v u b hov]
+      simp only [rotationMeeting,hou,hov,mem_priorPaths,Option.some.injEq]
+      rcases lt_trichotomy a b with hab | rfl | hba
+      · have hn : ¬ b.val < a.val+1 := by omega
+        have hy : a.val < b.val+1 := by omega
+        simp [hab,hn,hy]
+      · simp
+      · have hn : ¬ a.val < b.val+1 := by omega
+        have hy : b.val < a.val+1 := by omega
+        simp [hba,not_lt_of_ge (le_of_lt hba),hn,hy]
+
+lemma rotationCharge_eq_terminal_sum {d A : ℕ} (o : Card d → Option (Fin A))
+    (b : Butterfly d) (z : Terminal o) :
+    rotationCharge o b z = ∑ i : Fin A,
+      (terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) := by
+  simp only [rotationCharge,rotationMeeting_eq_slot_sum,terminalCharge_prior_sum]
+  exact Finset.sum_comm
+
+@[simp] lemma priorPaths_zero {d A : ℕ} (o : Card d → Option (Fin A)) : priorPaths o 0 = ∅ := by
+  classical
+  ext x
+  simp
+
+lemma priorPaths_succ {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) (ht : t < A) :
+    priorPaths o (t+1) = priorPaths o t ∪ cycleFiber o ⟨t,ht⟩ := by
+  classical
+  ext x
+  simp only [mem_priorPaths,Finset.mem_union,mem_cycleFiber]
+  constructor
+  · rintro ⟨i,hi,hit⟩
+    by_cases h : i.val < t
+    · exact Or.inl ⟨i,hi,h⟩
+    · have he : i = ⟨t,ht⟩ := Fin.ext (by change i.val = t; omega)
+      exact Or.inr (he ▸ hi)
+  · rintro (⟨i,hi,hit⟩ | hi)
+    · exact ⟨i,hi,by omega⟩
+    · exact ⟨⟨t,ht⟩,hi,by change t < t+1; omega⟩
+
+lemma priorPaths_disjoint {d A : ℕ} (o : Card d → Option (Fin A)) (i : Fin A) :
+    Disjoint (priorPaths o i.val) (cycleFiber o i) := by
+  classical
+  apply Finset.disjoint_left.mpr
+  intro x hx hx'
+  obtain ⟨j,hj,hji⟩ := (mem_priorPaths _ _ _).mp hx
+  have he : j = i := Option.some.inj (hj.symm.trans ((mem_cycleFiber _ _ _).mp hx'))
+  subst j
+  exact (lt_irrefl _ hji)
+
+namespace QueryTree
+
+lemma seekCycles_owned {d A : ℕ} (S : Finset (Card d)) (r : Fin A → ℕ)
+    (o : Card d → Option (Fin A)) (z : Terminal o) (ω : SwitchIndex d → Bool)
+    (hS : ∀ i, cycleFiber o i ⊆ S)
+    (hO : ∀ i, orbitSet (butterflyPerm d (decodeButterfly d ω)) (z i).val = cycleFiber o i)
+    (hC : ∀ i, (cycleFiber o i).card = r i+1) :
+    let p := butterflyPerm d (decodeButterfly d ω)
+    let L := List.ofFn (fun i : Fin A => (p (z i).val,r i))
+    (seekCycles d S L ∅).succeeds ω = true ∧
+      ((seekCycles d S L ∅).staleCharge ∅ ω : ℝ) = rotationCharge o (decodeButterfly d ω) z := by
+  classical
+  let b := decodeButterfly d ω
+  let p := butterflyPerm d b
+  let f (i : Fin A) := (p (z i).val,r i)
+  let c (i : Fin A) : ℝ := terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val)
+  have hc (i : Fin A) : Function.minimalPeriod p (z i).val - 1 = r i := by
+    have he := orbitSet_card p (z i).val
+    rw [hO,hC] at he
+    omega
+  have hs (t : ℕ) (ht : t ≤ A) :
+      (seekCycles d S ((List.ofFn f).drop t) (priorPaths o t)).succeeds ω = true ∧
+        ((seekCycles d S ((List.ofFn f).drop t) (priorPaths o t)).staleCharge (exposedPaths d b (priorPaths o t)) ω : ℝ) =
+          ((List.ofFn c).drop t).sum := by
+    induction ht using Nat.decreasingInduction with
+    | self => simp only [List.length_ofFn,le_refl,List.drop_eq_nil_of_le,List.sum_nil,seekCycles,succeeds,staleCharge,Nat.cast_zero,and_self]
+    | of_succ t ht ih =>
+      have htf : t < (List.ofFn f).length := by simpa using ht
+      have htc : t < (List.ofFn c).length := by simpa using ht
+      rw [List.drop_eq_getElem_cons htf,List.drop_eq_getElem_cons htc]
+      simp only [List.getElem_ofFn,List.sum_cons,seekCycles]
+      have h := seekCycle_orbit d S (priorPaths o t) (z ⟨t,ht⟩).val
+        (fun V => seekCycles d S ((List.ofFn f).drop (t+1)) V) ω
+        (by rw [hO]; exact hS _) (by rw [hO]; exact priorPaths_disjoint o ⟨t,ht⟩)
+      dsimp only at h
+      rw [hc,hO,←priorPaths_succ o t ht] at h
+      dsimp only [f]
+      constructor
+      · exact h.1.trans ih.1
+      · rw [h.2,Nat.cast_add,ih.2]
+        exact add_comm _ _
+  have h := hs 0 (Nat.zero_le _)
+  simp only [List.drop_zero,priorPaths_zero,exposedPaths_empty,List.sum_ofFn] at h
+  dsimp only
+  exact ⟨h.1,h.2.trans (rotationCharge_eq_terminal_sum o b z).symm⟩
+
+end QueryTree
+
+section Collections
+variable {d A : ℕ}
+
+def ValidCycleSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ)
+    (o : Card d → Option (Fin A)) : Prop :=
+  (∀ i, (cycleFiber o i).card = r i + 1) ∧
+  (∀ i, cycleFiber o i ⊆ S) ∧
+  (∀ i x, o x = some i → orbitSet p x = cycleFiber o i)
+
+abbrev CycleSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
+  {o : Card d → Option (Fin A) // ValidCycleSlots p S r o}
+
+noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
+    Fintype (CycleSlots p S r) := Fintype.ofFinite _
+
+lemma CycleSlots.terminal_nonempty {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (c : CycleSlots p S r) (i : Fin A) : Nonempty {x : Card d // c.val x = some i} := by
+  have hn : (cycleFiber c.val i).Nonempty := Finset.card_pos.mp (by rw [c.property.1]; omega)
+  obtain ⟨x,hx⟩ := hn
+  exact ⟨x,(mem_cycleFiber _ _ _).mp hx⟩
+
+lemma owner_ext {o o' : Card d → Option (Fin A)}
+    (h : ∀ i, cycleFiber o i = cycleFiber o' i) : o = o' := by
+  funext x
+  have he (i : Fin A) : o x = some i ↔ o' x = some i := by
+    simpa only [mem_cycleFiber] using (Finset.ext_iff.mp (h i) x)
+  cases ho : o x with
+  | some i => exact ((he i).mp ho).symm
+  | none =>
+    cases ho' : o' x with
+    | none => rfl
+    | some i => have hn := (he i).mpr ho'; rw [ho] at hn; contradiction
+
+abbrev RootedSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
+  Σ c : CycleSlots p S r, Terminal c.val
+
+noncomputable def rootedStart {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (c : RootedSlots p S r) : Fin A → S := fun i =>
+  ⟨p (c.2 i).val,c.1.property.2.1 i (by
+    rw [←c.1.property.2.2 i (c.2 i).val (c.2 i).property]
+    exact (mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.apply_right (Equiv.Perm.SameCycle.refl p _)))⟩
+
+lemma rootedStart_injective {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ} :
+    Function.Injective (rootedStart (p:=p) (S:=S) (r:=r)) := by
+  rintro ⟨c,z⟩ ⟨c',z'⟩ h
+  have hz (i : Fin A) : (z i).val = (z' i).val :=
+    p.injective (congrArg Subtype.val (congrFun h i))
+  have hc : c = c' := Subtype.ext (owner_ext (fun i => by
+    rw [←c.property.2.2 i (z i).val (z i).property,
+      ←c'.property.2.2 i (z' i).val (z' i).property,hz]))
+  subst c'
+  have hzz : z = z' := funext (fun i => Subtype.ext (hz i))
+  subst z'
+  rfl
+
+lemma sum_comp_injective_le {α β : Type*} [Fintype α] [Fintype β]
+    (e : α → β) (he : Function.Injective e) (f : β → ℝ) (hf : ∀ b, 0 ≤ f b) :
+    ∑ a, f (e a) ≤ ∑ b, f b := by
+  classical
+  calc
+    _ = ∑ b ∈ Finset.univ.image e, f b := (Finset.sum_image (fun _ _ _ _ h => he h)).symm
+    _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _) (fun b _ _ => hf b)
+
+lemma rooted_weight_sum_le (S : Finset (Card d)) (r : Fin A → ℕ) (ω : SwitchIndex d → Bool) :
+    (∑ c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r,
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2)) ≤
+    ∑ a : Fin A → S,
+      if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+        ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+      else 0 := by
+  classical
+  let p := butterflyPerm d (decodeButterfly d ω)
+  let g (a : Fin A → S) : ℝ :=
+    if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+    else 0
+  have he (c : RootedSlots p S r) :
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2) =
+      g (rootedStart c) := by
+    have hs := QueryTree.seekCycles_owned S r c.1.val c.2 ω c.1.property.2.1
+      (fun i => c.1.property.2.2 i (c.2 i).val (c.2 i).property) c.1.property.1
+    dsimp only at hs
+    unfold g
+    rw [ite_eq_left (show (QueryTree.seekCycles d S (List.ofFn (fun i => ((rootedStart c i).val,r i))) ∅).succeeds ω = true from hs.1)]
+    rw [← hs.2,neg_mul,Real.exp_neg,mul_comm (Real.log 2),Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
+    rfl
+  simp_rw [he]
+  exact sum_comp_injective_le rootedStart rootedStart_injective g (fun a => by dsimp [g]; split_ifs <;> positivity)
+
+noncomputable def rotationLower (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : ℝ :=
+  Real.exp (-Real.log 2 * ((A:ℝ)*d/2 + ∑ i : Fin A, Real.log (r i+1) / (2 * Real.log 2)))
+
+noncomputable def collectionWeight (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : ℝ :=
+  ((2:ℝ)^d)^A * (∏ i : Fin A, (r i+1 : ℝ)) * rotationLower d r
+
+lemma collectionWeight_pos (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : 0 < collectionWeight d r := by
+  unfold collectionWeight rotationLower
+  positivity
+
+lemma collection_rotation_lower (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r)
+    (ω : SwitchIndex d → Bool) (c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) :
+    collectionWeight d r ≤ ∑ z : Terminal c.val,
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z) := by
+  classical
+  let (i : Fin A) : Nonempty {x : Card d // c.val x = some i} := c.terminal_nonempty i
+  have hm : Monotone (fun i => (cycleFiber c.val i).card) := by
+    intro i j hij
+    dsimp only
+    rw [c.property.1,c.property.1]
+    exact Nat.add_le_add_right (hr hij) 1
+  have hb := rotation_exp_bound hm (decodeButterfly d ω)
+  have hcard : (Fintype.card (Terminal c.val) : ℝ) = ∏ i : Fin A, (r i+1 : ℝ) := by
+    rw [Fintype.card_pi,Nat.cast_prod]
+    apply Finset.prod_congr rfl
+    intro i _
+    rw [card_cycleFiber,c.property.1,Nat.cast_add,Nat.cast_one]
+  have hp : (0:ℝ) < Fintype.card (Terminal c.val) := Nat.cast_pos.mpr Fintype.card_pos
+  have hl : rotationLower d r ≤ finiteMean (fun z : Terminal c.val => Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z)) := by
+    simpa only [rotationLower,c.property.1,Nat.cast_add,Nat.cast_one] using hb
+  have hh := (le_div_iff₀ hp).mp hl
+  rw [hcard] at hh
+  unfold collectionWeight
+  rw [← Finset.mul_sum]
+  calc
+    _ = ((2:ℝ)^d)^A * (rotationLower d r * ∏ i : Fin A, (r i+1 : ℝ)) := by ring
+    _ ≤ _ := mul_le_mul_of_nonneg_left hh (by positivity)
+
+lemma cycleSlots_weighted_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) *
+      collectionWeight d r ≤ (S.card : ℝ)^A := by
+  classical
+  let g (a : Fin A → S) (ω : SwitchIndex d → Bool) : ℝ :=
+    if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+    else 0
+  have hω (ω : SwitchIndex d → Bool) :
+      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ) * collectionWeight d r ≤
+        ∑ a : Fin A → S, g a ω := by
+    calc
+      _ = ∑ _c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r, collectionWeight d r := by simp
+      _ ≤ ∑ c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r,
+          ∑ z : Terminal c.val, ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z) :=
+        Finset.sum_le_sum (fun c _ => collection_rotation_lower S r hr ω c)
+      _ = ∑ c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r,
+          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2) :=
+        (Fintype.sum_sigma (fun c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r =>
+          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2))).symm
+      _ ≤ _ := rooted_weight_sum_le S r ω
+  have hb := finiteMean_mono hω
+  rw [finiteMean_mul_const,finiteMean_sum] at hb
+  calc
+    _ ≤ ∑ a : Fin A → S, finiteMean (g a) := hb
+    _ ≤ ∑ _a : Fin A → S, (1:ℝ) := by
+      apply Finset.sum_le_sum
+      intro a _
+      have hh := QueryTree.cycles_weighted_success d S (List.ofFn (fun i : Fin A => ((a i).val,r i)))
+      simpa only [List.length_ofFn] using hh
+    _ = _ := by simp
+
+lemma cycleSlots_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) ≤
+        (S.card : ℝ)^A / collectionWeight d r := by
+  exact (le_div_iff₀ (collectionWeight_pos d r)).mpr (cycleSlots_weighted_moment S r hr)
+
+lemma collectionWeight_eq_sqrt (d : ℕ) {A : ℕ} (r : Fin A → ℕ) :
+    collectionWeight d r = (Real.sqrt ((2:ℝ)^d))^A * ∏ i : Fin A, Real.sqrt (r i+1) := by
+  have hn : ((2:ℝ)^d)^A = Real.exp ((A:ℝ)*d*Real.log 2) := by
+    rw [mul_assoc,Real.exp_nat_mul,Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
+  have hp : (∏ i : Fin A, (r i+1 : ℝ)) = Real.exp (∑ i : Fin A, Real.log (r i+1)) := by
+    rw [Real.exp_sum]
+    apply Finset.prod_congr rfl
+    intro i _
+    exact (Real.exp_log (by positivity)).symm
+  have hs : Real.sqrt ((2:ℝ)^d) = Real.exp ((d:ℝ)*Real.log 2/2) := by
+    rw [Real.sqrt_eq_rpow,Real.rpow_def_of_pos (by positivity),Real.log_pow]
+    congr 1
+    ring
+  have hj (i : Fin A) : Real.sqrt (r i+1) = Real.exp (Real.log (r i+1)/2) := by
+    rw [Real.sqrt_eq_rpow,Real.rpow_def_of_pos (by positivity)]
+    congr 1
+    ring
+  simp_rw [collectionWeight,hn,hp,rotationLower,hs,hj]
+  rw [←Real.exp_add,←Real.exp_add,←Real.exp_nat_mul,←Real.exp_sum,←Real.exp_add]
+  congr 1
+  rw [←Finset.sum_div,←Finset.sum_div]
+  have hl : Real.log 2 ≠ 0 := ne_of_gt (Real.log_pos (by norm_num))
+  field_simp
+  ring
+
+lemma butterfly_cycleSlots_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) ≤
+        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
+  have h := cycleSlots_moment S r hr
+  rw [collectionWeight_eq_sqrt,div_mul_eq_div_mul_one_div] at h
+  rw [div_pow]
+  simpa only [one_div,Finset.prod_inv_distrib] using h
+
+end Collections
+
+variable {d A : ℕ}
+
+abbrev SelectedCycle (p : Equiv.Perm (Card d)) (S : Finset (Card d)) :=
+  {C : Finset (Card d) // (∃ x, C = orbitSet p x) ∧ C ⊆ S}
+
+noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) :
+    Fintype (SelectedCycle p S) := Fintype.ofFinite _
+
+lemma SelectedCycle.nonempty {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
+    (C : SelectedCycle p S) : C.val.Nonempty := by
+  obtain ⟨x,hx⟩ := C.property.1
+  rw [hx]
+  exact orbitSet_nonempty p x
+
+lemma SelectedCycle.orbit_eq {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
+    (C : SelectedCycle p S) {x : Card d} (hx : x ∈ C.val) : orbitSet p x = C.val := by
+  obtain ⟨y,hy⟩ := C.property.1
+  rw [hy] at hx ⊢
+  exact (orbitSet_eq p ((mem_orbitSet _ _ _).mp hx)).symm
+
+lemma SelectedCycle.eq_of_mem {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
+    (C D : SelectedCycle p S) {x : Card d} (hC : x ∈ C.val) (hD : x ∈ D.val) : C = D := by
+  apply Subtype.ext
+  exact (C.orbit_eq hC).symm.trans (D.orbit_eq hD)
+
+abbrev SlotAssignment (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
+  {f : Fin A → SelectedCycle p S // Function.Injective f ∧ ∀ i, (f i).val.card = r i+1}
+
+noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
+    Fintype (SlotAssignment p S r) := Fintype.ofFinite _
+
+noncomputable def slotsToAssignment {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (c : CycleSlots p S r) : SlotAssignment p S r := by
+  classical
+  let f (i : Fin A) : SelectedCycle p S := ⟨cycleFiber c.val i,by
+    obtain ⟨x,hx⟩ := c.terminal_nonempty i
+    exact ⟨⟨x,(c.property.2.2 i x hx).symm⟩,c.property.2.1 i⟩⟩
+  refine ⟨f,?_,fun i => c.property.1 i⟩
+  intro i j hij
+  have he : cycleFiber c.val i = cycleFiber c.val j := congrArg Subtype.val hij
+  obtain ⟨x,hx⟩ := c.terminal_nonempty i
+  have hxi : x ∈ cycleFiber c.val i := (mem_cycleFiber _ _ _).mpr hx
+  rw [he] at hxi
+  exact Option.some.inj (hx.symm.trans ((mem_cycleFiber _ _ _).mp hxi))
+
+noncomputable def assignmentOwner {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) (x : Card d) : Option (Fin A) := by
+  classical
+  exact if h : ∃ i, x ∈ (f.val i).val then some (Classical.choose h) else none
+
+lemma assignmentOwner_eq_some {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) (x : Card d) (i : Fin A) :
+    assignmentOwner f x = some i ↔ x ∈ (f.val i).val := by
+  classical
+  unfold assignmentOwner
+  split_ifs with h
+  · constructor
+    · intro hi
+      have hx := Classical.choose_spec h
+      rw [Option.some.inj hi] at hx
+      exact hx
+    · intro hx
+      have he := (f.val (Classical.choose h)).eq_of_mem (f.val i) (Classical.choose_spec h) hx
+      exact congrArg some (f.property.1 he)
+  · constructor
+    · intro hi; contradiction
+    · intro hx; exact (h ⟨i,hx⟩).elim
+
+end Thorp
+
+end ThorpNine.Smoothing
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/SparseSaving/AlternatingPerm.lean b/lean/OAI/Probability/ThorpRouting/SparseSaving/AlternatingPerm.lean
index 9bffc02be908ba4454443eeae83b19ff3c551ca4..fab51c46b39f32f5be343e86e4203a016cf59d06 100644
--- a/lean/OAI/Probability/ThorpRouting/SparseSaving/AlternatingPerm.lean
+++ b/lean/OAI/Probability/ThorpRouting/SparseSaving/AlternatingPerm.lean
@@ -1,900 +1,900 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.SparseSaving.ParallelFollow
-
-namespace OAI
-
-namespace ThorpNine.SparseSaving
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-namespace CycleColoring
-variable {α : Type*} [Fintype α] [DecidableEq α]
-omit [Fintype α] in
-lemma bipartitionFlip_injective (p : Equiv.Perm α) (S : Finset α)
-    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x)) :
-    Function.Injective (bipartitionFlip p S χ hχ) := by
-  classical
-  intro f g he
-  apply Subtype.ext
-  funext x
-  by_cases hx : x ∈ S
-  · have hh := congrFun (congrArg Subtype.val he) x
-    change (if x ∈ S then Bool.xor (f.val x) (χ x) else false) =
-      (if x ∈ S then Bool.xor (g.val x) (χ x) else false) at hh
-    simp only [ite_eq_left hx] at hh
-    cases hfx : f.val x <;> cases hgx : g.val x <;> cases hcx : χ x <;> simp_all
-  · exact (f.property.1 x hx).trans (g.property.1 x hx).symm
-
-lemma card_opposite_le (p : Equiv.Perm α) (S : Finset α)
-    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x)) :
-    Fintype.card (OppositeColoring p S) ≤
-      2 ^ ((boundary p S).card + Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
-  exact (Fintype.card_le_of_injective _ (bipartitionFlip_injective p S χ hχ)).trans
-    (card_coloring_le p S)
-
-def alternatingPerm (p : Equiv.Perm α) : Equiv.Perm (Bool × α) where
-  toFun x := if x.1 then (false,p x.2) else (true,x.2)
-  invFun x := if x.1 then (false,x.2) else (true,p.symm x.2)
-  left_inv x := by rcases x with ⟨b,x⟩; cases b <;> simp
-  right_inv x := by rcases x with ⟨b,x⟩; cases b <;> simp
-
-omit [Fintype α] [DecidableEq α] in
-lemma alternatingPerm_flip (p : Equiv.Perm α) (x : Bool × α) :
-    (alternatingPerm p x).1 = !x.1 := by
-  rcases x with ⟨b,x⟩
-  cases b <;> rfl
-
-omit [Fintype α] [DecidableEq α] in
-lemma alternating_step_sameCycle (p : Equiv.Perm α) (x : Bool × α) :
-    p.SameCycle x.2 (alternatingPerm p x).2 := by
-  rcases x with ⟨b,x⟩
-  cases b
-  · exact Equiv.Perm.SameCycle.refl p x
-  · exact (Equiv.Perm.SameCycle.refl p x).apply_right
-
-omit [DecidableEq α] in
-lemma alternating_sameCycle_snd (p : Equiv.Perm α) {x y : Bool × α}
-    (h : (alternatingPerm p).SameCycle x y) : p.SameCycle x.2 y.2 := by
-  obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
-  have hp (k : ℕ) : p.SameCycle x.2 (((alternatingPerm p)^k) x).2 := by
-    induction k with
-    | zero => exact Equiv.Perm.SameCycle.refl p x.2
-    | succ k ih =>
-      rw [pow_succ',Equiv.Perm.mul_apply]
-      exact ih.trans (alternating_step_sameCycle p _)
-  exact hn ▸ hp n
-
-omit [DecidableEq α] in
-lemma alternating_zero_sameCycle (p : Equiv.Perm α) {x y : α}
-    (h : p.SameCycle x y) : (alternatingPerm p).SameCycle (false,x) (false,y) := by
-  obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
-  have hp (k : ℕ) : (alternatingPerm p).SameCycle (false,x) (false,(p^k) x) := by
-    induction k with
-    | zero => exact Equiv.Perm.SameCycle.refl _ _
-    | succ k ih =>
-      have hs := ih.apply_right.apply_right
-      simpa only [pow_succ',Equiv.Perm.mul_apply,alternatingPerm,Equiv.coe_fn_mk,
-        Bool.false_eq_true,↓reduceIte] using hs
-  simpa only [hn] using hp n
-
-omit [Fintype α] [DecidableEq α] in
-lemma alternating_sameCycle_zero (p : Equiv.Perm α) (x : Bool × α) :
-    (alternatingPerm p).SameCycle x (false,x.2) := by
-  rcases x with ⟨b,x⟩
-  cases b
-  · exact Equiv.Perm.SameCycle.refl _ _
-  · exact ((Equiv.Perm.SameCycle.refl (alternatingPerm p) (false,x)).apply_right).symm
-
-omit [DecidableEq α] in
-lemma alternating_sameCycle_iff (p : Equiv.Perm α) (x y : Bool × α) :
-    (alternatingPerm p).SameCycle x y ↔ p.SameCycle x.2 y.2 := by
-  constructor
-  · exact alternating_sameCycle_snd p
-  · intro h
-    exact (alternating_sameCycle_zero p x).trans
-      ((alternating_zero_sameCycle p h).trans (alternating_sameCycle_zero p y).symm)
-
-lemma orbit_alternating (p : Equiv.Perm α) (x : Bool × α) :
-    orbitSet (alternatingPerm p) x = childJoin (orbitSet p x.2) (orbitSet p x.2) := by
-  ext ⟨b,y⟩
-  simp only [mem_orbitSet,alternating_sameCycle_iff,mem_childJoin,ite_self]
-
-omit [Fintype α] in
-lemma boolFiber_childJoin (A B : Finset α) (b : Bool) :
-    boolFiber (childJoin A B) b = if b then B else A := by
-  ext x
-  cases b <;> simp only [mem_boolFiber,mem_childJoin,Bool.false_eq_true,↓reduceIte]
-
-noncomputable def projectCycle (p : Equiv.Perm α) (A B : Finset α)
-    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) :
-    RoutingNetwork.SelectedCycle p A := by
-  refine ⟨boolFiber C.val false,?_,?_⟩
-  · obtain ⟨x,hx⟩ := C.property.1
-    refine ⟨x.2,?_⟩
-    rw [hx,orbit_alternating,boolFiber_childJoin]
-    rfl
-  · intro x hx
-    exact (mem_childJoin A B false x).mp (C.property.2 ((mem_boolFiber _ _ _).mp hx))
-
-lemma selected_alternating_eq (p : Equiv.Perm α) (A B : Finset α)
-    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) :
-    C.val = childJoin (projectCycle p A B C).val (projectCycle p A B C).val := by
-  change C.val = childJoin (boolFiber C.val false) (boolFiber C.val false)
-  obtain ⟨x,hx⟩ := C.property.1
-  rw [hx,orbit_alternating,boolFiber_childJoin]
-  rfl
-
-lemma projectCycle_injective (p : Equiv.Perm α) (A B : Finset α) :
-    Function.Injective (projectCycle p A B) := by
-  intro C D he
-  apply Subtype.ext
-  rw [selected_alternating_eq p A B C,selected_alternating_eq p A B D,he]
-
-lemma card_alternating_cycles_le (p : Equiv.Perm α) (A B : Finset α) :
-    Fintype.card (RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) ≤
-      Fintype.card (RoutingNetwork.SelectedCycle p A) :=
-  Fintype.card_le_of_injective _ (projectCycle_injective p A B)
-
-end CycleColoring
-
-section PairAssignmentCount
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-omit [Fintype α] in
-lemma mem_image_perm_iff (p : Equiv.Perm α) (A : Finset α) (y : α) :
-    y ∈ A.image p ↔ p.symm y ∈ A := by
-  constructor
-  · intro h
-    obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp h
-    simpa only [←hxy,Equiv.symm_apply_apply] using hx
-  · intro h
-    exact Finset.mem_image.mpr ⟨p.symm y,h,p.apply_symm_apply y⟩
-
-omit [Fintype α] in
-lemma card_childJoin (A B : Finset α) : (childJoin A B).card = A.card+B.card := by
-  have hd : Disjoint (A.image (fun x => (false,x))) (B.image (fun x => (true,x))) := by
-    apply Finset.disjoint_left.mpr
-    rintro x hx hy
-    obtain ⟨a,ha,rfl⟩ := Finset.mem_image.mp hx
-    obtain ⟨b,hb,hba⟩ := Finset.mem_image.mp hy
-    exact Bool.false_ne_true (congrArg Prod.fst hba).symm
-  have hi (b : Bool) : Function.Injective (fun x : α => (b,x)) :=
-    fun _ _ h => congrArg Prod.snd h
-  rw [childJoin,Finset.card_union_of_disjoint hd,
-    Finset.card_image_of_injective A (hi false),Finset.card_image_of_injective B (hi true)]
-
-omit [Fintype α] in
-lemma alternating_internal (p : Equiv.Perm α) (A B : Finset α) :
-    ((childJoin A B).filter (fun x => (CycleColoring.alternatingPerm p).symm x ∈ childJoin A B)) =
-      childJoin (A ∩ B.image p) (B ∩ A) := by
-  ext ⟨b,x⟩
-  rw [Finset.mem_filter]
-  cases b <;> simp only [mem_childJoin,CycleColoring.alternatingPerm,
-    Equiv.symm_mk,Equiv.coe_fn_mk,Bool.false_eq_true,↓reduceIte,Finset.mem_inter,mem_image_perm_iff]
-
-omit [Fintype α] in
-lemma alternating_boundary_card (p : Equiv.Perm α) (A B : Finset α) :
-    (CycleColoring.boundary (CycleColoring.alternatingPerm p) (childJoin A B)).card =
-      (A ∪ B).card + (A ∪ B.image p).card - (A.card+B.card) := by
-  rw [CycleColoring.boundary_card,alternating_internal,card_childJoin,card_childJoin]
-  have h₀ := Finset.card_union_add_card_inter A B
-  have h₁ := Finset.card_union_add_card_inter A (B.image p)
-  rw [Finset.card_image_of_injective _ p.injective] at h₁
-  have h₂ := Finset.inter_comm B A
-  rw [h₂]
-  omega
-
-lemma compatible_assignment_bound (p : Equiv.Perm α) (A B : Finset α) :
-    Fintype.card (CycleColoring.OppositeColoring (CycleColoring.alternatingPerm p) (childJoin A B)) ≤
-      2 ^ ((A ∪ B).card + (A ∪ B.image p).card - (A.card+B.card) +
-        Fintype.card (RoutingNetwork.SelectedCycle p A)) := by
-  have h := CycleColoring.card_opposite_le (CycleColoring.alternatingPerm p) (childJoin A B)
-    Prod.fst (CycleColoring.alternatingPerm_flip p)
-  rw [alternating_boundary_card] at h
-  exact h.trans (Nat.pow_le_pow_right (by norm_num)
-    (Nat.add_le_add_left (CycleColoring.card_alternating_cycles_le p A B) _))
-
-end PairAssignmentCount
-
-lemma finiteMean_nonneg {Ω : Type*} [Fintype Ω] {f : Ω → ℝ}
-    (h : ∀ ω, 0 ≤ f ω) : 0 ≤ finiteMean f := by
-  exact div_nonneg (Finset.sum_nonneg (fun ω _ => h ω)) (Nat.cast_nonneg _)
-
-lemma boolFiber_headTail (d : ℕ) (S : Finset (Card (d+1))) (b : Bool) :
-    boolFiber (S.image (headTailEquiv d)) b = childMarks S b := by
-  classical
-  ext x
-  rw [mem_boolFiber,mem_childMarks]
-  constructor
-  · intro h
-    obtain ⟨y,hy,he⟩ := Finset.mem_image.mp h
-    have hh := congrArg (headTailEquiv d).symm he
-    rw [Equiv.symm_apply_apply] at hh
-    change y = Fin.cons b x at hh
-    exact hh ▸ hy
-  · intro h
-    exact Finset.mem_image.mpr ⟨Fin.cons b x,h,by rfl⟩
-
-lemma prod_childMarks (d : ℕ) (S : Finset (Card (d+1))) (f : Card (d+1) → ℝ) :
-    (∏ x ∈ S, f x) =
-      (∏ x ∈ childMarks S false, f (Fin.cons false x)) *
-      ∏ x ∈ childMarks S true, f (Fin.cons true x) := by
-  classical
-  have h := prod_boolFibers (S.image (headTailEquiv d)) (fun x => f ((headTailEquiv d).symm x))
-  rw [Finset.prod_image (headTailEquiv d).injective.injOn,
-    boolFiber_headTail,boolFiber_headTail] at h
-  simp only [Equiv.symm_apply_apply] at h
-  exact h
-
-noncomputable def headAverage {d : ℕ} (p : Card (d+1) → ℝ) (y : Card d) : ℝ :=
-  (p (Fin.cons false y)+p (Fin.cons true y))/2
-
-lemma finiteMean_headAverage (d : ℕ) (p : Card (d+1) → ℝ) :
-    finiteMean (headAverage p) = finiteMean p := by
-  rw [finiteMean_equiv (headTailEquiv d),finiteMean_prod,finiteMean_comm]
-  apply finiteMean_congr
-  intro y
-  rw [finiteMean_bool]
-  rfl
-
-lemma pairSwitch_product_bound (d : ℕ) (S : Finset (Card (d+1)))
-    (p : Card (d+1) → ℝ) (hp : ∀ x, 0 ≤ p x) :
-    finiteMean (fun ξ : Card d → Bool => ∏ x ∈ S, p (pairSwitch ξ x)) ≤
-      ∏ x ∈ S, headAverage p (Fin.tail x) := by
-  classical
-  have h : CylinderBound (fun T => ∏ x ∈ T, p ((headTailEquiv d).symm x))
-      (fun x => p ((headTailEquiv d).symm x)) := fun _ => le_rfl
-  have hh := h.fair_parallel (fun x => hp _) (S.image (headTailEquiv d))
-  simp only [Finset.prod_image (pairLayer _).injective.injOn,
-    Finset.prod_image (headTailEquiv d).injective.injOn] at hh
-  convert hh using 1
-  · apply finiteMean_congr
-    intro ξ
-    apply Finset.prod_congr rfl
-    intro x _
-    rfl
-  · apply Finset.prod_congr rfl
-    intro x _
-    change (p (Fin.cons false (Fin.tail x))+p (Fin.cons true (Fin.tail x)))/2 =
-      (p (Fin.cons (x 0) (Fin.tail x))+p (Fin.cons (!(x 0)) (Fin.tail x)))/2
-    cases x 0
-    · rfl
-    · simp only [Bool.not_true]
-      ring
-
-lemma butterflyPerm_step (d : ℕ)
-    (c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (ξ : Card d → Bool) :
-    butterflyPerm (d+1) (decodeButterfly (d+1) ((coinStepEquiv d).symm (c,ξ))) =
-    pairSwitch ξ * childLift (fun b => butterflyPerm d (decodeButterfly d (if b then c.2 else c.1))) := by
-  change pairSwitch (fun y => (coinStepEquiv d).symm (c,ξ) (Sum.inl y)) *
-    childLift (fun b => butterflyPerm d (decodeButterfly d
-      (fun i => (coinStepEquiv d).symm (c,ξ) (Sum.inr (b,i))))) = _
-  apply congrArg₂ (· * ·)
-  · rfl
-  · apply congrArg childLift
-    funext b
-    apply congrArg (fun ω => butterflyPerm d (decodeButterfly d ω))
-    funext i
-    cases b <;> rfl
-
-lemma butterfly_product_bound (d : ℕ) (S : Finset (Card d))
-    (p : Card d → ℝ) (hp : ∀ x, 0 ≤ p x) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      ∏ x ∈ S, p (butterflyPerm d (decodeButterfly d ω) x)) ≤ (finiteMean p)^S.card := by
-  classical
-  induction d with
-  | zero =>
-    have he : p = fun _ => p (fun i => Fin.elim0 i) := by
-      funext x
-      exact congrArg p (Subsingleton.elim _ _)
-    rw [he]
-    simp only [Finset.prod_const,finiteMean_const,le_refl]
-  | succ d ih =>
-    let g : Card d → ℝ := headAverage p
-    have hg : ∀ x, 0 ≤ g x := fun x => div_nonneg (add_nonneg (hp _) (hp _)) (by norm_num)
-    rw [finiteMean_equiv (coinStepEquiv d),finiteMean_prod]
-    calc
-      _ ≤ finiteMean (fun c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool) =>
-        (∏ x ∈ childMarks S false, g (butterflyPerm d (decodeButterfly d c.1) x)) *
-        ∏ x ∈ childMarks S true, g (butterflyPerm d (decodeButterfly d c.2) x)) := by
-        apply finiteMean_mono
-        intro c
-        let f : Equiv.Perm (Card (d+1)) := childLift (fun b =>
-          butterflyPerm d (decodeButterfly d (if b then c.2 else c.1)))
-        have hh := pairSwitch_product_bound d (S.image f) p hp
-        simp only [Finset.prod_image f.injective.injOn] at hh
-        convert hh using 1
-        · apply finiteMean_congr
-          intro ξ
-          apply Finset.prod_congr rfl
-          intro x _
-          rw [butterflyPerm_step]
-          rfl
-        · rw [prod_childMarks]
-          simp only [f,childLift,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
-            Bool.false_eq_true,↓reduceIte]
-          rfl
-      _ = (finiteMean (fun c : SwitchIndex d → Bool =>
-        ∏ x ∈ childMarks S false, g (butterflyPerm d (decodeButterfly d c) x))) *
-        finiteMean (fun c : SwitchIndex d → Bool =>
-        ∏ x ∈ childMarks S true, g (butterflyPerm d (decodeButterfly d c) x)) :=
-          finiteMean_prod_mul
-            (fun c : SwitchIndex d → Bool => ∏ x ∈ childMarks S false,
-              g (butterflyPerm d (decodeButterfly d c) x))
-            (fun c : SwitchIndex d → Bool => ∏ x ∈ childMarks S true,
-              g (butterflyPerm d (decodeButterfly d c) x))
-      _ ≤ (finiteMean g)^(childMarks S false).card * (finiteMean g)^(childMarks S true).card :=
-        mul_le_mul (ih (childMarks S false) g hg) (ih (childMarks S true) g hg)
-          (finiteMean_nonneg fun c => Finset.prod_nonneg (fun x _ => hg _))
-          (pow_nonneg (finiteMean_nonneg hg) _)
-      _ = (finiteMean p)^S.card := by
-        rw [←pow_add,card_childMarks_add,finiteMean_headAverage]
-
-noncomputable def inverseButterflyMarks (d : ℕ) (A : Finset (Card d))
-    (ω : SwitchIndex d → Bool) : Finset (Card d) :=
-  A.image (butterflyPerm d (decodeButterfly d ω)).symm
-
-lemma inverseButterflyMarks_inclusion (d : ℕ) (A S : Finset (Card d)) :
-    inclusionMoment (inverseButterflyMarks d A) S ≤
-      ((A.card : ℝ)/(2:ℝ)^d)^S.card := by
-  classical
-  let p : Card d → ℝ := fun x => if x ∈ A then 1 else 0
-  have hm : finiteMean p = (A.card : ℝ)/(2:ℝ)^d := by
-    simp only [finiteMean,p,card_positions,Nat.cast_pow,Nat.cast_ofNat]
-    congr 1
-    simp
-  have he : inclusionMoment (inverseButterflyMarks d A) S =
-      finiteMean (fun ω : SwitchIndex d → Bool =>
-        ∏ x ∈ S, p (butterflyPerm d (decodeButterfly d ω) x)) := by
-    rw [inclusionMoment_eq]
-    apply finiteMean_congr
-    intro ω
-    simp only [inverseButterflyMarks,subset_image_equiv_iff,Equiv.symm_symm,indicator_subset_prod,
-      Finset.prod_image (butterflyPerm d (decodeButterfly d ω)).injective.injOn,p]
-  rw [he]
-  simpa only [hm] using butterfly_product_bound d S p (fun x => by dsimp [p]; split_ifs <;> norm_num)
-
-namespace PairRouting
-
-section
-variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
-
-noncomputable def occupied (x : ι → Bool × α) : Finset α :=
-  Finset.univ.image (fun i => (x i).2)
-
-omit [Fintype α] in
-lemma mem_occupied (x : ι → Bool × α) (y : α) : y ∈ occupied x ↔ ∃ i, (x i).2 = y := by
-  simp [occupied]
-
-def required (x : ι → Bool × α) (c : ι → Bool) (i : ι) : Bool := Bool.xor (x i).1 (c i)
-
-def Compatible (x : ι → Bool × α) (c : ι → Bool) : Prop :=
-  ∀ i j, (x i).2 = (x j).2 → required x c i = required x c j
-
-def colors (x : ι → Bool × α) (ξ : α → Bool) (i : ι) : Bool :=
-  Bool.xor (x i).1 (ξ (x i).2)
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma required_colors (x : ι → Bool × α) (ξ : α → Bool) (i : ι) :
-    required x (colors x ξ) i = ξ (x i).2 := by
-  simp only [required,colors]
-  cases (x i).1 <;> cases ξ (x i).2 <;> rfl
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma colors_compatible (x : ι → Bool × α) (ξ : α → Bool) : Compatible x (colors x ξ) := by
-  intro i j he
-  simp only [required_colors,he]
-
-noncomputable def chosenCoin (x : ι → Bool × α) (c : ι → Bool) (y : α) : Bool := by
-  classical
-  exact if h : ∃ i, (x i).2 = y then required x c (Classical.choose h) else false
-
-omit [Fintype α] in
-lemma chosenCoin_apply (x : ι → Bool × α) (c : ι → Bool) (hc : Compatible x c) (i : ι) :
-    chosenCoin x c (x i).2 = required x c i := by
-  classical
-  simp only [chosenCoin,dite_eq_left (show ∃ j, (x j).2 = (x i).2 from ⟨i,rfl⟩)]
-  exact hc _ i (Classical.choose_spec (show ∃ j, (x j).2 = (x i).2 from ⟨i,rfl⟩))
-
-omit [Fintype α] in
-lemma colors_eq_iff (x : ι → Bool × α) (c : ι → Bool) (hc : Compatible x c) (ξ : α → Bool) :
-    colors x ξ = c ↔ ∀ y ∈ occupied x, ξ y = chosenCoin x c y := by
-  constructor
-  · intro he y hy
-    obtain ⟨i,rfl⟩ := (mem_occupied x y).mp hy
-    rw [chosenCoin_apply x c hc i]
-    rw [←he,required_colors]
-  · intro he
-    funext i
-    apply (xor_eq_iff _ _ _).mpr
-    exact (he _ ((mem_occupied x _).mpr ⟨i,rfl⟩)).trans (chosenCoin_apply x c hc i)
-
-noncomputable instance (x : ι → Bool × α) (c : ι → Bool) : Decidable (Compatible x c) :=
-  Classical.propDecidable _
-
-lemma colors_probability (x : ι → Bool × α) (c : ι → Bool) :
-    finiteMean (fun ξ : α → Bool => if colors x ξ = c then (1 : ℝ) else 0) =
-      if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0 := by
-  classical
-  by_cases hc : Compatible x c
-  · rw [ite_eq_left hc]
-    have he := fun ξ => colors_eq_iff x c hc ξ
-    simp only [he]
-    exact mean_coinCylinder (occupied x) (chosenCoin x c)
-  · rw [ite_eq_right hc]
-    have he : ∀ ξ, colors x ξ ≠ c := fun ξ hh => hc (hh ▸ colors_compatible x ξ)
-    simp only [he,ite_false,finiteMean_const]
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma Compatible.tail_injective {x : ι ↪ Bool × α} {c : ι → Bool} (hc : Compatible x c) (b : Bool) :
-    Function.Injective (fun i : {i : ι // c i = b} => (x i.val).2) := by
-  intro i j he
-  apply Subtype.ext
-  apply x.injective
-  apply Prod.ext
-  · have hh := hc i j he
-    simp only [required,i.property,j.property] at hh
-    cases h₀ : (x i.val).1 <;> cases h₁ : (x j.val).1 <;> cases b <;> simp_all
-  · exact he
-
-def childEmbedding (x : ι ↪ Bool × α) (c : ι → Bool) (hc : Compatible x c) (b : Bool) :
-    {i : ι // c i = b} ↪ α := ⟨fun i => (x i.val).2,hc.tail_injective b⟩
-
-
-def lift (p : Bool → Equiv.Perm α) : Equiv.Perm (Bool × α) where
-  toFun x := (x.1,p x.1 x.2)
-  invFun x := (x.1,(p x.1).symm x.2)
-  left_inv x := by simp
-  right_inv x := by simp
-
-def sandwich (ξ η : α → Bool) (p : Bool → Equiv.Perm α) : Equiv.Perm (Bool × α) :=
-  pairLayer η * lift p * pairLayer ξ
-
-def ChildMatches (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) : Prop :=
-  ∀ i, p (c i) (x i).2 = (y i).2
-
-noncomputable instance (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
-    Decidable (ChildMatches p x y c) := Classical.propDecidable _
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma sandwich_matches_iff (ξ η : α → Bool) (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) :
-    (∀ i, sandwich ξ η p (x i) = y i) ↔
-      colors x ξ = colors y η ∧ ChildMatches p x y (colors x ξ) := by
-  have he (i : ι) : sandwich ξ η p (x i) = y i ↔
-      colors x ξ i = colors y η i ∧ p (colors x ξ i) (x i).2 = (y i).2 := by
-    change pairLayer η (lift p (pairLayer ξ (x i))) = y i ↔ _
-    rw [←Equiv.eq_symm_apply]
-    change (colors x ξ i,p (colors x ξ i) (x i).2) = (colors y η i,(y i).2) ↔ _
-    exact ⟨Prod.mk.inj,fun h => Prod.ext h.1 h.2⟩
-  simp only [he,forall_and,←funext_iff,ChildMatches]
-
-lemma sandwich_probability [DecidableEq ι] (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) :
-    finiteMean (fun coins : (α → Bool) × (α → Bool) =>
-      if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then (1:ℝ) else 0) =
-    ∑ c : ι → Bool,
-      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
-      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) *
-      (if ChildMatches p x y c then 1 else 0) := by
-  classical
-  have he (coins : (α → Bool) × (α → Bool)) :
-      (if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then (1:ℝ) else 0) =
-      ∑ c : ι → Bool, (if colors x coins.1 = c then (1:ℝ) else 0) *
-        (if colors y coins.2 = c then 1 else 0) * (if ChildMatches p x y c then 1 else 0) := by
-    rw [Finset.sum_eq_single (colors x coins.1)]
-    · by_cases hc : colors x coins.1 = colors y coins.2
-      · simp only [sandwich_matches_iff,hc,true_and,↓reduceIte,one_mul]
-      · simp only [sandwich_matches_iff,hc,false_and,↓reduceIte,one_mul,Ne.symm hc,zero_mul]
-    · intro c _ hc
-      simp only [Ne.symm hc,ite_false,zero_mul]
-    · simp
-  simp_rw [he]
-  rw [finiteMean_sum]
-  apply Finset.sum_congr rfl
-  intro c _
-  rw [finiteMean_mul_const]
-  rw [finiteMean_prod_mul
-    (fun ξ : α → Bool => if colors x ξ = c then (1:ℝ) else 0)
-    (fun η : α → Bool => if colors y η = c then (1:ℝ) else 0),
-    colors_probability,colors_probability]
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma childMatches_split (p₀ p₁ : Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
-    ChildMatches (fun b => if b then p₁ else p₀) x y c ↔
-      (∀ i : {i // c i = false}, p₀ (x i.val).2 = (y i.val).2) ∧
-      (∀ i : {i // c i = true}, p₁ (x i.val).2 = (y i.val).2) := by
-  constructor
-  · intro h
-    constructor
-    · intro i
-      simpa only [i.property,Bool.false_eq_true,↓reduceIte] using h i.val
-    · intro i
-      simpa only [i.property,↓reduceIte] using h i.val
-  · rintro ⟨h₀,h₁⟩ i
-    cases hi : c i
-    · simpa only [hi,Bool.false_eq_true,↓reduceIte] using h₀ ⟨i,hi⟩
-    · simpa only [hi,↓reduceIte] using h₁ ⟨i,hi⟩
-
-noncomputable def tupleProbability {Ω β : Type*} [Fintype Ω] [DecidableEq β]
-    (P : Ω → Equiv.Perm β) (x y : ι → β) : ℝ := by
-  classical
-  exact finiteMean (fun ω => if ∀ i, P ω (x i) = y i then 1 else 0)
-
-omit [Fintype α] in
-lemma childMatches_mean {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
-    (P : Ω → Equiv.Perm α) (Q : Ω' → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
-    finiteMean (fun ω : Ω × Ω' =>
-      if ChildMatches (fun b => if b then Q ω.2 else P ω.1) x y c then (1:ℝ) else 0) =
-      tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
-        (fun i : {i // c i = false} => (y i.val).2) *
-      tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
-        (fun i : {i // c i = true} => (y i.val).2) := by
-  classical
-  let f : Ω → ℝ := fun ω => if ∀ i : {i // c i = false}, P ω (x i.val).2 = (y i.val).2 then 1 else 0
-  let g : Ω' → ℝ := fun ω => if ∀ i : {i // c i = true}, Q ω (x i.val).2 = (y i.val).2 then 1 else 0
-  have he (ω : Ω × Ω') :
-      (if ChildMatches (fun b => if b then Q ω.2 else P ω.1) x y c then (1:ℝ) else 0) = f ω.1*g ω.2 := by
-    simp only [childMatches_split,f,g]
-    split_ifs <;> simp_all
-  simp_rw [he]
-  exact finiteMean_prod_mul f g
-
-lemma sandwich_mean [DecidableEq ι] {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
-    (P : Ω → Equiv.Perm α) (Q : Ω' → Equiv.Perm α) (x y : ι → Bool × α) :
-    finiteMean (fun ω : Ω × Ω' => finiteMean (fun coins : (α → Bool) × (α → Bool) =>
-      if ∀ i, sandwich coins.1 coins.2 (fun b => if b then Q ω.2 else P ω.1) (x i) = y i
-      then (1:ℝ) else 0)) =
-    ∑ c : ι → Bool,
-      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
-      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) *
-      (tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
-        (fun i : {i // c i = false} => (y i.val).2) *
-      tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
-        (fun i : {i // c i = true} => (y i.val).2)) := by
-  classical
-  simp_rw [sandwich_probability]
-  rw [finiteMean_sum]
-  apply Finset.sum_congr rfl
-  intro c _
-  simp only [mul_comm ((if Compatible x c then _ else _) * _),finiteMean_mul_const,
-    childMatches_mean]
-
-
-abbrev BothAssignments (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :=
-  {c : ι → Bool // Compatible e c ∧ Compatible (fun i => lift p (e i)) c}
-
-noncomputable instance (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    Fintype (BothAssignments e p) := Fintype.ofFinite _
-
-noncomputable def labelSet (e : ι ↪ Bool × α) : Finset (Bool × α) := Finset.univ.image e
-
-def alternatingProjection (p : Bool → Equiv.Perm α) : Equiv.Perm α := (p false).symm * p true
-
-noncomputable def assignmentColoring (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α)
-    (c : BothAssignments e p) : CycleColoring.OppositeColoring
-      (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e) := by
-  classical
-  refine ⟨Function.extend e c.val (fun _ => false),?_,?_⟩
-  · intro z hz
-    apply Function.extend_apply'
-    rintro ⟨i,rfl⟩
-    exact hz (Finset.mem_image.mpr ⟨i,Finset.mem_univ _,rfl⟩)
-  · intro z hz hnext
-    obtain ⟨i,_,hi⟩ := Finset.mem_image.mp hz
-    obtain ⟨j,_,hj⟩ := Finset.mem_image.mp hnext
-    have hfi := e.injective.extend_apply c.val (fun _ => false) i
-    have hfj := e.injective.extend_apply c.val (fun _ => false) j
-    rw [hi] at hfi
-    rw [hj] at hfj
-    rw [hfi,hfj]
-    rcases z with ⟨b,a⟩
-    cases b
-    · have hj' : e j = (true,a) := hj
-      have ht : (e i).2 = (e j).2 := by rw [hi,hj']
-      have hh := c.property.1 i j ht
-      simp only [required,hi,hj'] at hh
-      revert hh
-      generalize c.val i = bi
-      generalize c.val j = bj
-      cases bi <;> cases bj <;> decide
-    · have hj' : e j = (false,alternatingProjection p a) := hj
-      have ht : (lift p (e i)).2 = (lift p (e j)).2 := by
-        simp only [hi,hj',lift,Equiv.coe_fn_mk,alternatingProjection,Equiv.Perm.coe_mul,
-          Function.comp_apply,Equiv.apply_symm_apply]
-      have hh := c.property.2 i j ht
-      simp only [required,hi,hj',lift,Equiv.coe_fn_mk] at hh
-      revert hh
-      generalize c.val i = bi
-      generalize c.val j = bj
-      cases bi <;> cases bj <;> decide
-
-omit [Fintype α] in
-lemma assignmentColoring_injective (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    Function.Injective (assignmentColoring e p) := by
-  intro c c' he
-  apply Subtype.ext
-  funext i
-  have hh := congrFun (congrArg Subtype.val he) (e i)
-  change Function.extend e c.val (fun _ => false) (e i) =
-    Function.extend e c'.val (fun _ => false) (e i) at hh
-  simpa only [e.injective.extend_apply] using hh
-
-lemma bothAssignments_card (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    Fintype.card (BothAssignments e p) ≤
-      2 ^ ((boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true).card +
-        (boolFiber (labelSet e) false ∪ (boolFiber (labelSet e) true).image
-          (alternatingProjection p)).card -
-        ((boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card) +
-        Fintype.card (RoutingNetwork.SelectedCycle (alternatingProjection p) (boolFiber (labelSet e) false))) := by
-  have h := compatible_assignment_bound (alternatingProjection p)
-    (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
-  rw [childJoin_boolFiber] at h
-  exact (Fintype.card_le_of_injective _ (assignmentColoring_injective e p)).trans h
-
-omit [Fintype α] in
-lemma occupied_labelSet (e : ι ↪ Bool × α) : occupied e = (labelSet e).image Prod.snd := by
-  classical
-  simp only [occupied,labelSet,Finset.image_image,Function.comp_def]
-
-omit [Fintype ι] [Fintype α] in
-lemma image_snd_childJoin (A B : Finset α) : (childJoin A B).image Prod.snd = A ∪ B := by
-  classical
-  simp only [childJoin,Finset.image_union,Finset.image_image,Function.comp_def,Finset.image_id']
-
-omit [Fintype ι] [Fintype α] in
-lemma image_lift_childJoin (p : Bool → Equiv.Perm α) (A B : Finset α) :
-    (childJoin A B).image (fun x => (lift p x).2) = A.image (p false) ∪ B.image (p true) := by
-  classical
-  simp only [childJoin,Finset.image_union,Finset.image_image,Function.comp_def,lift,Equiv.coe_fn_mk]
-
-omit [Fintype α] in
-lemma occupied_lift_card (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    (occupied (fun i => lift p (e i))).card =
-      (boolFiber (labelSet e) false ∪ (boolFiber (labelSet e) true).image (alternatingProjection p)).card := by
-  classical
-  have he : occupied (fun i => lift p (e i)) = (labelSet e).image (fun x => (lift p x).2) := by
-    simp only [occupied,labelSet,Finset.image_image,Function.comp_def]
-  rw [he,←childJoin_boolFiber (labelSet e),image_lift_childJoin]
-  simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
-  let A := boolFiber (labelSet e) false
-  let B := boolFiber (labelSet e) true
-  change (A.image (p false) ∪ B.image (p true)).card = (A ∪ B.image (alternatingProjection p)).card
-  have hh : A.image (p false) ∪ B.image (p true) =
-      (A ∪ B.image (alternatingProjection p)).image (p false) := by
-    simp only [Finset.image_union,Finset.image_image,Function.comp_def,
-      alternatingProjection,Equiv.Perm.coe_mul,Equiv.apply_symm_apply]
-  rw [hh,Finset.card_image_of_injective _ (p false).injective]
-
-lemma bothAssignments_occupied_bound (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    Fintype.card (BothAssignments e p) ≤
-      2 ^ ((occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι +
-        Fintype.card (RoutingNetwork.SelectedCycle (alternatingProjection p) (boolFiber (labelSet e) false))) := by
-  have he : (occupied e).card =
-      (boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true).card := by
-    rw [occupied_labelSet,←childJoin_boolFiber (labelSet e),image_snd_childJoin]
-    simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
-  have hc : (boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card = Fintype.card ι := by
-    rw [←card_childJoin,childJoin_boolFiber,labelSet,Finset.card_image_of_injective _ e.injective,
-      Finset.card_univ]
-  simpa only [he,occupied_lift_card,←hc] using bothAssignments_card e p
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma compatible_pairLayer_iff (x : ι → Bool × α) (c : ι → Bool) (ξ : α → Bool) :
-    Compatible (fun i => pairLayer ξ (x i)) c ↔ Compatible x c := by
-  have he (i j : ι) (ht : (x i).2 = (x j).2) :
-      required (fun i => pairLayer ξ (x i)) c i = required (fun i => pairLayer ξ (x i)) c j ↔
-      required x c i = required x c j := by
-    simp only [required,pairLayer,Equiv.coe_fn_mk,ht]
-    cases (x i).1 <;> cases (x j).1 <;> cases c i <;> cases c j <;> cases ξ (x j).2 <;> decide
-  constructor
-  · intro h i j ht
-    exact (he i j ht).mp (h i j ht)
-  · intro h i j ht
-    exact (he i j ht).mpr (h i j ht)
-
-
-noncomputable def alternatingCycles (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) : ℕ :=
-  Fintype.card (RoutingNetwork.SelectedCycle
-    (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e))
-
-omit [Fintype α] in
-lemma occupied_eq_fibers (e : ι ↪ Bool × α) :
-    occupied e = boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true := by
-  rw [occupied_labelSet,←childJoin_boolFiber (labelSet e),image_snd_childJoin]
-  simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
-
-omit [Fintype α] in
-lemma card_labelSet (e : ι ↪ Bool × α) : (labelSet e).card = Fintype.card ι := by
-  classical
-  exact (Finset.card_image_of_injective _ e.injective).trans (Finset.card_univ)
-
-omit [Fintype α] in
-lemma occupied_card_double (e : ι ↪ Bool × α) : Fintype.card ι ≤ 2*(occupied e).card := by
-  have h₀ : (boolFiber (labelSet e) false).card ≤ (occupied e).card := by
-    rw [occupied_eq_fibers]
-    exact Finset.card_le_card Finset.subset_union_left
-  have h₁ : (boolFiber (labelSet e) true).card ≤ (occupied e).card := by
-    rw [occupied_eq_fibers]
-    exact Finset.card_le_card Finset.subset_union_right
-  have he : (boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card = Fintype.card ι := by
-    rw [←card_childJoin,childJoin_boolFiber,card_labelSet]
-  omega
-
-lemma bothAssignments_exact_bound (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    Fintype.card (BothAssignments e p) ≤
-      2 ^ ((occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι +
-        alternatingCycles e p) := by
-  have h := CycleColoring.card_opposite_le
-    (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e)
-    Prod.fst (CycleColoring.alternatingPerm_flip (alternatingProjection p))
-  have hb : (CycleColoring.boundary (CycleColoring.alternatingPerm (alternatingProjection p))
-      (labelSet e)).card =
-      (occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι := by
-    have hh := alternating_boundary_card (alternatingProjection p)
-      (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
-    rw [childJoin_boolFiber] at hh
-    rw [occupied_eq_fibers,occupied_lift_card]
-    have he := card_childJoin (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
-    rw [childJoin_boolFiber,card_labelSet] at he
-    simpa only [he] using hh
-  rw [hb] at h
-  exact (Fintype.card_le_of_injective _ (assignmentColoring_injective e p)).trans h
-
-abbrev Assignments (x y : ι → Bool × α) :=
-  {c : ι → Bool // Compatible x c ∧ Compatible y c}
-
-noncomputable instance (x y : ι → Bool × α) : Fintype (Assignments x y) := Fintype.ofFinite _
-
-def switchedEmbedding (e : ι ↪ Bool × α) (ξ : α → Bool) : ι ↪ Bool × α :=
-  e.trans (pairLayer ξ).toEmbedding
-
-def routedEmbedding (e : ι ↪ Bool × α) (ξ η : α → Bool) (p : Bool → Equiv.Perm α) :
-    ι ↪ Bool × α := e.trans (sandwich ξ η p).toEmbedding
-
-omit [Fintype α] in
-lemma occupied_pairLayer (x : ι → Bool × α) (ξ : α → Bool) :
-    occupied (fun i => pairLayer ξ (x i)) = occupied x := rfl
-
-noncomputable def assignmentsEquiv (e : ι ↪ Bool × α) (ξ η : α → Bool)
-    (p : Bool → Equiv.Perm α) :
-    Assignments e (routedEmbedding e ξ η p) ≃ BothAssignments (switchedEmbedding e ξ) p := by
-  apply Equiv.subtypeEquivRight
-  intro c
-  change Compatible e c ∧ Compatible (fun i => pairLayer η (lift p (pairLayer ξ (e i)))) c ↔ _
-  rw [compatible_pairLayer_iff]
-  exact and_congr (compatible_pairLayer_iff e c ξ).symm Iff.rfl
-
-lemma assignments_actual_bound (e : ι ↪ Bool × α) (ξ η : α → Bool) (p : Bool → Equiv.Perm α) :
-    Fintype.card (Assignments e (routedEmbedding e ξ η p)) ≤
-      2 ^ ((occupied e).card + (occupied (routedEmbedding e ξ η p)).card - Fintype.card ι +
-        alternatingCycles (switchedEmbedding e ξ) p) := by
-  rw [Fintype.card_congr (assignmentsEquiv e ξ η p)]
-  exact bothAssignments_exact_bound (switchedEmbedding e ξ) p
-
-omit [Fintype α] in
-lemma assignments_actual_exponent_nonneg (e : ι ↪ Bool × α) (ξ η : α → Bool)
-    (p : Bool → Equiv.Perm α) :
-    Fintype.card ι ≤ (occupied e).card + (occupied (routedEmbedding e ξ η p)).card := by
-  have h₀ := occupied_card_double e
-  have h₁ := occupied_card_double (routedEmbedding e ξ η p)
-  omega
-
-end
-variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α] [DecidableEq ι]
-
-noncomputable def compatibleSet (x y : ι → Bool × α) : Finset (ι → Bool) := by
-  classical
-  exact Finset.univ.filter (fun c => Compatible x c ∧ Compatible y c)
-
-omit [Fintype α] [DecidableEq α] in
-lemma mem_compatibleSet (x y : ι → Bool × α) (c : ι → Bool) :
-    c ∈ compatibleSet x y ↔ Compatible x c ∧ Compatible y c := by
-  classical
-  simp only [compatibleSet,Finset.mem_filter,Finset.mem_univ,true_and]
-
-omit [Fintype α] [DecidableEq α] in
-lemma card_compatibleSet (x y : ι → Bool × α) :
-    (compatibleSet x y).card = Fintype.card (Assignments x y) := by
-  classical
-  exact (Fintype.card_subtype _).symm
-
-noncomputable def outerWeight (x y : ι → Bool × α) : ℝ :=
-  1/(2:ℝ)^((occupied x).card+(occupied y).card)
-
-omit [Fintype α] [DecidableEq ι] in
-lemma outerWeight_pos (x y : ι → Bool × α) : 0 < outerWeight x y := by
-  unfold outerWeight
-  positivity
-
-omit [Fintype α] in
-lemma sum_colors_factor (x y : ι → Bool × α) (f : (ι → Bool) → ℝ) :
-    (∑ c : ι → Bool,
-      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
-      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) * f c) =
-      outerWeight x y * ∑ c ∈ compatibleSet x y, f c := by
-  classical
-  rw [compatibleSet,Finset.sum_filter,Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro c _
-  by_cases hx : Compatible x c <;> by_cases hy : Compatible y c
-  · simp only [hx,hy,true_and,↓reduceIte,outerWeight,pow_add,one_div_mul_one_div]
-  · simp only [hx,hy,and_false,↓reduceIte,mul_zero,zero_mul]
-  · simp only [hx,hy,false_and,↓reduceIte,mul_zero,zero_mul]
-  · simp only [hx,hy,false_and,↓reduceIte,mul_zero,zero_mul]
-
-lemma sandwich_weighted_probability (p : Bool → Equiv.Perm α) (x y : ι → Bool × α)
-    (W : (ι → Bool) → ℝ) :
-    finiteMean (fun coins : (α → Bool) × (α → Bool) =>
-      if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then W (colors x coins.1) else 0) =
-    outerWeight x y * ∑ c ∈ compatibleSet x y, if ChildMatches p x y c then W c else 0 := by
-  classical
-  have he (coins : (α → Bool) × (α → Bool)) :
-      (if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then W (colors x coins.1) else 0) =
-      ∑ c : ι → Bool, (if colors x coins.1 = c then (1:ℝ) else 0) *
-        (if colors y coins.2 = c then 1 else 0) * (if ChildMatches p x y c then W c else 0) := by
-    rw [Finset.sum_eq_single (colors x coins.1)]
-    · by_cases hc : colors x coins.1 = colors y coins.2
-      · simp only [sandwich_matches_iff,hc,true_and,↓reduceIte,one_mul]
-      · simp only [sandwich_matches_iff,hc,false_and,↓reduceIte,one_mul,Ne.symm hc,zero_mul]
-    · intro c _ hc
-      simp only [Ne.symm hc,ite_false,zero_mul]
-    · simp
-  simp_rw [he]
-  rw [finiteMean_sum,←sum_colors_factor]
-  apply Finset.sum_congr rfl
-  intro c _
-  rw [finiteMean_mul_const,finiteMean_prod_mul
-    (fun ξ : α → Bool => if colors x ξ = c then (1:ℝ) else 0)
-    (fun η : α → Bool => if colors y η = c then (1:ℝ) else 0),
-    colors_probability,colors_probability]
-
-omit [Fintype α] [DecidableEq ι] in
-lemma tupleProbability_nonneg {Ω β : Type*} [Fintype Ω] [DecidableEq β]
-    (P : Ω → Equiv.Perm β) (x y : ι → β) : 0 ≤ tupleProbability P x y := by
-  classical
-  exact finiteMean_nonneg (fun ω => by split_ifs <;> norm_num)
-
-lemma sandwich_power_bound {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
-    (P : Ω → Equiv.Perm α) (Q : Ω' → Equiv.Perm α) (x y : ι → Bool × α)
-    (a : ℝ) (ha : 0 ≤ a) :
-    (finiteMean (fun ω : Ω × Ω' => finiteMean (fun coins : (α → Bool) × (α → Bool) =>
-      if ∀ i, sandwich coins.1 coins.2 (fun b => if b then Q ω.2 else P ω.1) (x i) = y i
-      then (1:ℝ) else 0)))^(1+a) ≤
-    (outerWeight x y)^(1+a) * ((compatibleSet x y).card:ℝ)^a *
-      ∑ c ∈ compatibleSet x y,
-      (tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
-        (fun i : {i // c i = false} => (y i.val).2))^(1+a) *
-      (tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
-        (fun i : {i // c i = true} => (y i.val).2))^(1+a) := by
-  classical
-  rw [sandwich_mean,sum_colors_factor]
-  let f (c : ι → Bool) : ℝ :=
-      tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
-        (fun i : {i // c i = false} => (y i.val).2) *
-      tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
-        (fun i : {i // c i = true} => (y i.val).2)
-  have hf (c : ι → Bool) : 0 ≤ f c := mul_nonneg (tupleProbability_nonneg _ _ _)
-    (tupleProbability_nonneg _ _ _)
-  rw [Real.mul_rpow (le_of_lt (outerWeight_pos _ _)) (Finset.sum_nonneg (fun c _ => hf c))]
-  have hh := Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg (compatibleSet x y)
-    (show 1 ≤ 1+a by linarith) (fun c _ => hf c)
-  have he : 1+a-1=a := by ring
-  rw [he] at hh
-  calc
-    _ ≤ (outerWeight x y)^(1+a) *
-        (((compatibleSet x y).card:ℝ)^a * ∑ c ∈ compatibleSet x y, (f c)^(1+a)) :=
-      mul_le_mul_of_nonneg_left hh (Real.rpow_nonneg (le_of_lt (outerWeight_pos _ _)) _)
-    _ = _ := by
-      rw [←mul_assoc]
-      congr 1
-      apply Finset.sum_congr rfl
-      intro c _
-      exact Real.mul_rpow (tupleProbability_nonneg _ _ _) (tupleProbability_nonneg _ _ _)
-
-end PairRouting
-
-end Thorp
-
-end ThorpNine.SparseSaving
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.SparseSaving.ParallelFollow
+
+namespace OAI
+
+namespace ThorpNine.SparseSaving
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+namespace CycleColoring
+variable {α : Type*} [Fintype α] [DecidableEq α]
+omit [Fintype α] in
+lemma bipartitionFlip_injective (p : Equiv.Perm α) (S : Finset α)
+    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x)) :
+    Function.Injective (bipartitionFlip p S χ hχ) := by
+  classical
+  intro f g he
+  apply Subtype.ext
+  funext x
+  by_cases hx : x ∈ S
+  · have hh := congrFun (congrArg Subtype.val he) x
+    change (if x ∈ S then Bool.xor (f.val x) (χ x) else false) =
+      (if x ∈ S then Bool.xor (g.val x) (χ x) else false) at hh
+    simp only [ite_eq_left hx] at hh
+    cases hfx : f.val x <;> cases hgx : g.val x <;> cases hcx : χ x <;> simp_all
+  · exact (f.property.1 x hx).trans (g.property.1 x hx).symm
+
+lemma card_opposite_le (p : Equiv.Perm α) (S : Finset α)
+    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x)) :
+    Fintype.card (OppositeColoring p S) ≤
+      2 ^ ((boundary p S).card + Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
+  exact (Fintype.card_le_of_injective _ (bipartitionFlip_injective p S χ hχ)).trans
+    (card_coloring_le p S)
+
+def alternatingPerm (p : Equiv.Perm α) : Equiv.Perm (Bool × α) where
+  toFun x := if x.1 then (false,p x.2) else (true,x.2)
+  invFun x := if x.1 then (false,x.2) else (true,p.symm x.2)
+  left_inv x := by rcases x with ⟨b,x⟩; cases b <;> simp
+  right_inv x := by rcases x with ⟨b,x⟩; cases b <;> simp
+
+omit [Fintype α] [DecidableEq α] in
+lemma alternatingPerm_flip (p : Equiv.Perm α) (x : Bool × α) :
+    (alternatingPerm p x).1 = !x.1 := by
+  rcases x with ⟨b,x⟩
+  cases b <;> rfl
+
+omit [Fintype α] [DecidableEq α] in
+lemma alternating_step_sameCycle (p : Equiv.Perm α) (x : Bool × α) :
+    p.SameCycle x.2 (alternatingPerm p x).2 := by
+  rcases x with ⟨b,x⟩
+  cases b
+  · exact Equiv.Perm.SameCycle.refl p x
+  · exact (Equiv.Perm.SameCycle.refl p x).apply_right
+
+omit [DecidableEq α] in
+lemma alternating_sameCycle_snd (p : Equiv.Perm α) {x y : Bool × α}
+    (h : (alternatingPerm p).SameCycle x y) : p.SameCycle x.2 y.2 := by
+  obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
+  have hp (k : ℕ) : p.SameCycle x.2 (((alternatingPerm p)^k) x).2 := by
+    induction k with
+    | zero => exact Equiv.Perm.SameCycle.refl p x.2
+    | succ k ih =>
+      rw [pow_succ',Equiv.Perm.mul_apply]
+      exact ih.trans (alternating_step_sameCycle p _)
+  exact hn ▸ hp n
+
+omit [DecidableEq α] in
+lemma alternating_zero_sameCycle (p : Equiv.Perm α) {x y : α}
+    (h : p.SameCycle x y) : (alternatingPerm p).SameCycle (false,x) (false,y) := by
+  obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
+  have hp (k : ℕ) : (alternatingPerm p).SameCycle (false,x) (false,(p^k) x) := by
+    induction k with
+    | zero => exact Equiv.Perm.SameCycle.refl _ _
+    | succ k ih =>
+      have hs := ih.apply_right.apply_right
+      simpa only [pow_succ',Equiv.Perm.mul_apply,alternatingPerm,Equiv.coe_fn_mk,
+        Bool.false_eq_true,↓reduceIte] using hs
+  simpa only [hn] using hp n
+
+omit [Fintype α] [DecidableEq α] in
+lemma alternating_sameCycle_zero (p : Equiv.Perm α) (x : Bool × α) :
+    (alternatingPerm p).SameCycle x (false,x.2) := by
+  rcases x with ⟨b,x⟩
+  cases b
+  · exact Equiv.Perm.SameCycle.refl _ _
+  · exact ((Equiv.Perm.SameCycle.refl (alternatingPerm p) (false,x)).apply_right).symm
+
+omit [DecidableEq α] in
+lemma alternating_sameCycle_iff (p : Equiv.Perm α) (x y : Bool × α) :
+    (alternatingPerm p).SameCycle x y ↔ p.SameCycle x.2 y.2 := by
+  constructor
+  · exact alternating_sameCycle_snd p
+  · intro h
+    exact (alternating_sameCycle_zero p x).trans
+      ((alternating_zero_sameCycle p h).trans (alternating_sameCycle_zero p y).symm)
+
+lemma orbit_alternating (p : Equiv.Perm α) (x : Bool × α) :
+    orbitSet (alternatingPerm p) x = childJoin (orbitSet p x.2) (orbitSet p x.2) := by
+  ext ⟨b,y⟩
+  simp only [mem_orbitSet,alternating_sameCycle_iff,mem_childJoin,ite_self]
+
+omit [Fintype α] in
+lemma boolFiber_childJoin (A B : Finset α) (b : Bool) :
+    boolFiber (childJoin A B) b = if b then B else A := by
+  ext x
+  cases b <;> simp only [mem_boolFiber,mem_childJoin,Bool.false_eq_true,↓reduceIte]
+
+noncomputable def projectCycle (p : Equiv.Perm α) (A B : Finset α)
+    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) :
+    RoutingNetwork.SelectedCycle p A := by
+  refine ⟨boolFiber C.val false,?_,?_⟩
+  · obtain ⟨x,hx⟩ := C.property.1
+    refine ⟨x.2,?_⟩
+    rw [hx,orbit_alternating,boolFiber_childJoin]
+    rfl
+  · intro x hx
+    exact (mem_childJoin A B false x).mp (C.property.2 ((mem_boolFiber _ _ _).mp hx))
+
+lemma selected_alternating_eq (p : Equiv.Perm α) (A B : Finset α)
+    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) :
+    C.val = childJoin (projectCycle p A B C).val (projectCycle p A B C).val := by
+  change C.val = childJoin (boolFiber C.val false) (boolFiber C.val false)
+  obtain ⟨x,hx⟩ := C.property.1
+  rw [hx,orbit_alternating,boolFiber_childJoin]
+  rfl
+
+lemma projectCycle_injective (p : Equiv.Perm α) (A B : Finset α) :
+    Function.Injective (projectCycle p A B) := by
+  intro C D he
+  apply Subtype.ext
+  rw [selected_alternating_eq p A B C,selected_alternating_eq p A B D,he]
+
+lemma card_alternating_cycles_le (p : Equiv.Perm α) (A B : Finset α) :
+    Fintype.card (RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) ≤
+      Fintype.card (RoutingNetwork.SelectedCycle p A) :=
+  Fintype.card_le_of_injective _ (projectCycle_injective p A B)
+
+end CycleColoring
+
+section PairAssignmentCount
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+omit [Fintype α] in
+lemma mem_image_perm_iff (p : Equiv.Perm α) (A : Finset α) (y : α) :
+    y ∈ A.image p ↔ p.symm y ∈ A := by
+  constructor
+  · intro h
+    obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp h
+    simpa only [←hxy,Equiv.symm_apply_apply] using hx
+  · intro h
+    exact Finset.mem_image.mpr ⟨p.symm y,h,p.apply_symm_apply y⟩
+
+omit [Fintype α] in
+lemma card_childJoin (A B : Finset α) : (childJoin A B).card = A.card+B.card := by
+  have hd : Disjoint (A.image (fun x => (false,x))) (B.image (fun x => (true,x))) := by
+    apply Finset.disjoint_left.mpr
+    rintro x hx hy
+    obtain ⟨a,ha,rfl⟩ := Finset.mem_image.mp hx
+    obtain ⟨b,hb,hba⟩ := Finset.mem_image.mp hy
+    exact Bool.false_ne_true (congrArg Prod.fst hba).symm
+  have hi (b : Bool) : Function.Injective (fun x : α => (b,x)) :=
+    fun _ _ h => congrArg Prod.snd h
+  rw [childJoin,Finset.card_union_of_disjoint hd,
+    Finset.card_image_of_injective A (hi false),Finset.card_image_of_injective B (hi true)]
+
+omit [Fintype α] in
+lemma alternating_internal (p : Equiv.Perm α) (A B : Finset α) :
+    ((childJoin A B).filter (fun x => (CycleColoring.alternatingPerm p).symm x ∈ childJoin A B)) =
+      childJoin (A ∩ B.image p) (B ∩ A) := by
+  ext ⟨b,x⟩
+  rw [Finset.mem_filter]
+  cases b <;> simp only [mem_childJoin,CycleColoring.alternatingPerm,
+    Equiv.symm_mk,Equiv.coe_fn_mk,Bool.false_eq_true,↓reduceIte,Finset.mem_inter,mem_image_perm_iff]
+
+omit [Fintype α] in
+lemma alternating_boundary_card (p : Equiv.Perm α) (A B : Finset α) :
+    (CycleColoring.boundary (CycleColoring.alternatingPerm p) (childJoin A B)).card =
+      (A ∪ B).card + (A ∪ B.image p).card - (A.card+B.card) := by
+  rw [CycleColoring.boundary_card,alternating_internal,card_childJoin,card_childJoin]
+  have h₀ := Finset.card_union_add_card_inter A B
+  have h₁ := Finset.card_union_add_card_inter A (B.image p)
+  rw [Finset.card_image_of_injective _ p.injective] at h₁
+  have h₂ := Finset.inter_comm B A
+  rw [h₂]
+  omega
+
+lemma compatible_assignment_bound (p : Equiv.Perm α) (A B : Finset α) :
+    Fintype.card (CycleColoring.OppositeColoring (CycleColoring.alternatingPerm p) (childJoin A B)) ≤
+      2 ^ ((A ∪ B).card + (A ∪ B.image p).card - (A.card+B.card) +
+        Fintype.card (RoutingNetwork.SelectedCycle p A)) := by
+  have h := CycleColoring.card_opposite_le (CycleColoring.alternatingPerm p) (childJoin A B)
+    Prod.fst (CycleColoring.alternatingPerm_flip p)
+  rw [alternating_boundary_card] at h
+  exact h.trans (Nat.pow_le_pow_right (by norm_num)
+    (Nat.add_le_add_left (CycleColoring.card_alternating_cycles_le p A B) _))
+
+end PairAssignmentCount
+
+lemma finiteMean_nonneg {Ω : Type*} [Fintype Ω] {f : Ω → ℝ}
+    (h : ∀ ω, 0 ≤ f ω) : 0 ≤ finiteMean f := by
+  exact div_nonneg (Finset.sum_nonneg (fun ω _ => h ω)) (Nat.cast_nonneg _)
+
+lemma boolFiber_headTail (d : ℕ) (S : Finset (Card (d+1))) (b : Bool) :
+    boolFiber (S.image (headTailEquiv d)) b = childMarks S b := by
+  classical
+  ext x
+  rw [mem_boolFiber,mem_childMarks]
+  constructor
+  · intro h
+    obtain ⟨y,hy,he⟩ := Finset.mem_image.mp h
+    have hh := congrArg (headTailEquiv d).symm he
+    rw [Equiv.symm_apply_apply] at hh
+    change y = Fin.cons b x at hh
+    exact hh ▸ hy
+  · intro h
+    exact Finset.mem_image.mpr ⟨Fin.cons b x,h,by rfl⟩
+
+lemma prod_childMarks (d : ℕ) (S : Finset (Card (d+1))) (f : Card (d+1) → ℝ) :
+    (∏ x ∈ S, f x) =
+      (∏ x ∈ childMarks S false, f (Fin.cons false x)) *
+      ∏ x ∈ childMarks S true, f (Fin.cons true x) := by
+  classical
+  have h := prod_boolFibers (S.image (headTailEquiv d)) (fun x => f ((headTailEquiv d).symm x))
+  rw [Finset.prod_image (headTailEquiv d).injective.injOn,
+    boolFiber_headTail,boolFiber_headTail] at h
+  simp only [Equiv.symm_apply_apply] at h
+  exact h
+
+noncomputable def headAverage {d : ℕ} (p : Card (d+1) → ℝ) (y : Card d) : ℝ :=
+  (p (Fin.cons false y)+p (Fin.cons true y))/2
+
+lemma finiteMean_headAverage (d : ℕ) (p : Card (d+1) → ℝ) :
+    finiteMean (headAverage p) = finiteMean p := by
+  rw [finiteMean_equiv (headTailEquiv d),finiteMean_prod,finiteMean_comm]
+  apply finiteMean_congr
+  intro y
+  rw [finiteMean_bool]
+  rfl
+
+lemma pairSwitch_product_bound (d : ℕ) (S : Finset (Card (d+1)))
+    (p : Card (d+1) → ℝ) (hp : ∀ x, 0 ≤ p x) :
+    finiteMean (fun ξ : Card d → Bool => ∏ x ∈ S, p (pairSwitch ξ x)) ≤
+      ∏ x ∈ S, headAverage p (Fin.tail x) := by
+  classical
+  have h : CylinderBound (fun T => ∏ x ∈ T, p ((headTailEquiv d).symm x))
+      (fun x => p ((headTailEquiv d).symm x)) := fun _ => le_rfl
+  have hh := h.fair_parallel (fun x => hp _) (S.image (headTailEquiv d))
+  simp only [Finset.prod_image (pairLayer _).injective.injOn,
+    Finset.prod_image (headTailEquiv d).injective.injOn] at hh
+  convert hh using 1
+  · apply finiteMean_congr
+    intro ξ
+    apply Finset.prod_congr rfl
+    intro x _
+    rfl
+  · apply Finset.prod_congr rfl
+    intro x _
+    change (p (Fin.cons false (Fin.tail x))+p (Fin.cons true (Fin.tail x)))/2 =
+      (p (Fin.cons (x 0) (Fin.tail x))+p (Fin.cons (!(x 0)) (Fin.tail x)))/2
+    cases x 0
+    · rfl
+    · simp only [Bool.not_true]
+      ring
+
+lemma butterflyPerm_step (d : ℕ)
+    (c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (ξ : Card d → Bool) :
+    butterflyPerm (d+1) (decodeButterfly (d+1) ((coinStepEquiv d).symm (c,ξ))) =
+    pairSwitch ξ * childLift (fun b => butterflyPerm d (decodeButterfly d (if b then c.2 else c.1))) := by
+  change pairSwitch (fun y => (coinStepEquiv d).symm (c,ξ) (Sum.inl y)) *
+    childLift (fun b => butterflyPerm d (decodeButterfly d
+      (fun i => (coinStepEquiv d).symm (c,ξ) (Sum.inr (b,i))))) = _
+  apply congrArg₂ (· * ·)
+  · rfl
+  · apply congrArg childLift
+    funext b
+    apply congrArg (fun ω => butterflyPerm d (decodeButterfly d ω))
+    funext i
+    cases b <;> rfl
+
+lemma butterfly_product_bound (d : ℕ) (S : Finset (Card d))
+    (p : Card d → ℝ) (hp : ∀ x, 0 ≤ p x) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      ∏ x ∈ S, p (butterflyPerm d (decodeButterfly d ω) x)) ≤ (finiteMean p)^S.card := by
+  classical
+  induction d with
+  | zero =>
+    have he : p = fun _ => p (fun i => Fin.elim0 i) := by
+      funext x
+      exact congrArg p (Subsingleton.elim _ _)
+    rw [he]
+    simp only [Finset.prod_const,finiteMean_const,le_refl]
+  | succ d ih =>
+    let g : Card d → ℝ := headAverage p
+    have hg : ∀ x, 0 ≤ g x := fun x => div_nonneg (add_nonneg (hp _) (hp _)) (by norm_num)
+    rw [finiteMean_equiv (coinStepEquiv d),finiteMean_prod]
+    calc
+      _ ≤ finiteMean (fun c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool) =>
+        (∏ x ∈ childMarks S false, g (butterflyPerm d (decodeButterfly d c.1) x)) *
+        ∏ x ∈ childMarks S true, g (butterflyPerm d (decodeButterfly d c.2) x)) := by
+        apply finiteMean_mono
+        intro c
+        let f : Equiv.Perm (Card (d+1)) := childLift (fun b =>
+          butterflyPerm d (decodeButterfly d (if b then c.2 else c.1)))
+        have hh := pairSwitch_product_bound d (S.image f) p hp
+        simp only [Finset.prod_image f.injective.injOn] at hh
+        convert hh using 1
+        · apply finiteMean_congr
+          intro ξ
+          apply Finset.prod_congr rfl
+          intro x _
+          rw [butterflyPerm_step]
+          rfl
+        · rw [prod_childMarks]
+          simp only [f,childLift,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
+            Bool.false_eq_true,↓reduceIte]
+          rfl
+      _ = (finiteMean (fun c : SwitchIndex d → Bool =>
+        ∏ x ∈ childMarks S false, g (butterflyPerm d (decodeButterfly d c) x))) *
+        finiteMean (fun c : SwitchIndex d → Bool =>
+        ∏ x ∈ childMarks S true, g (butterflyPerm d (decodeButterfly d c) x)) :=
+          finiteMean_prod_mul
+            (fun c : SwitchIndex d → Bool => ∏ x ∈ childMarks S false,
+              g (butterflyPerm d (decodeButterfly d c) x))
+            (fun c : SwitchIndex d → Bool => ∏ x ∈ childMarks S true,
+              g (butterflyPerm d (decodeButterfly d c) x))
+      _ ≤ (finiteMean g)^(childMarks S false).card * (finiteMean g)^(childMarks S true).card :=
+        mul_le_mul (ih (childMarks S false) g hg) (ih (childMarks S true) g hg)
+          (finiteMean_nonneg fun c => Finset.prod_nonneg (fun x _ => hg _))
+          (pow_nonneg (finiteMean_nonneg hg) _)
+      _ = (finiteMean p)^S.card := by
+        rw [←pow_add,card_childMarks_add,finiteMean_headAverage]
+
+noncomputable def inverseButterflyMarks (d : ℕ) (A : Finset (Card d))
+    (ω : SwitchIndex d → Bool) : Finset (Card d) :=
+  A.image (butterflyPerm d (decodeButterfly d ω)).symm
+
+lemma inverseButterflyMarks_inclusion (d : ℕ) (A S : Finset (Card d)) :
+    inclusionMoment (inverseButterflyMarks d A) S ≤
+      ((A.card : ℝ)/(2:ℝ)^d)^S.card := by
+  classical
+  let p : Card d → ℝ := fun x => if x ∈ A then 1 else 0
+  have hm : finiteMean p = (A.card : ℝ)/(2:ℝ)^d := by
+    simp only [finiteMean,p,card_positions,Nat.cast_pow,Nat.cast_ofNat]
+    congr 1
+    simp
+  have he : inclusionMoment (inverseButterflyMarks d A) S =
+      finiteMean (fun ω : SwitchIndex d → Bool =>
+        ∏ x ∈ S, p (butterflyPerm d (decodeButterfly d ω) x)) := by
+    rw [inclusionMoment_eq]
+    apply finiteMean_congr
+    intro ω
+    simp only [inverseButterflyMarks,subset_image_equiv_iff,Equiv.symm_symm,indicator_subset_prod,
+      Finset.prod_image (butterflyPerm d (decodeButterfly d ω)).injective.injOn,p]
+  rw [he]
+  simpa only [hm] using butterfly_product_bound d S p (fun x => by dsimp [p]; split_ifs <;> norm_num)
+
+namespace PairRouting
+
+section
+variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
+
+noncomputable def occupied (x : ι → Bool × α) : Finset α :=
+  Finset.univ.image (fun i => (x i).2)
+
+omit [Fintype α] in
+lemma mem_occupied (x : ι → Bool × α) (y : α) : y ∈ occupied x ↔ ∃ i, (x i).2 = y := by
+  simp [occupied]
+
+def required (x : ι → Bool × α) (c : ι → Bool) (i : ι) : Bool := Bool.xor (x i).1 (c i)
+
+def Compatible (x : ι → Bool × α) (c : ι → Bool) : Prop :=
+  ∀ i j, (x i).2 = (x j).2 → required x c i = required x c j
+
+def colors (x : ι → Bool × α) (ξ : α → Bool) (i : ι) : Bool :=
+  Bool.xor (x i).1 (ξ (x i).2)
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma required_colors (x : ι → Bool × α) (ξ : α → Bool) (i : ι) :
+    required x (colors x ξ) i = ξ (x i).2 := by
+  simp only [required,colors]
+  cases (x i).1 <;> cases ξ (x i).2 <;> rfl
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma colors_compatible (x : ι → Bool × α) (ξ : α → Bool) : Compatible x (colors x ξ) := by
+  intro i j he
+  simp only [required_colors,he]
+
+noncomputable def chosenCoin (x : ι → Bool × α) (c : ι → Bool) (y : α) : Bool := by
+  classical
+  exact if h : ∃ i, (x i).2 = y then required x c (Classical.choose h) else false
+
+omit [Fintype α] in
+lemma chosenCoin_apply (x : ι → Bool × α) (c : ι → Bool) (hc : Compatible x c) (i : ι) :
+    chosenCoin x c (x i).2 = required x c i := by
+  classical
+  simp only [chosenCoin,dite_eq_left (show ∃ j, (x j).2 = (x i).2 from ⟨i,rfl⟩)]
+  exact hc _ i (Classical.choose_spec (show ∃ j, (x j).2 = (x i).2 from ⟨i,rfl⟩))
+
+omit [Fintype α] in
+lemma colors_eq_iff (x : ι → Bool × α) (c : ι → Bool) (hc : Compatible x c) (ξ : α → Bool) :
+    colors x ξ = c ↔ ∀ y ∈ occupied x, ξ y = chosenCoin x c y := by
+  constructor
+  · intro he y hy
+    obtain ⟨i,rfl⟩ := (mem_occupied x y).mp hy
+    rw [chosenCoin_apply x c hc i]
+    rw [←he,required_colors]
+  · intro he
+    funext i
+    apply (xor_eq_iff _ _ _).mpr
+    exact (he _ ((mem_occupied x _).mpr ⟨i,rfl⟩)).trans (chosenCoin_apply x c hc i)
+
+noncomputable instance (x : ι → Bool × α) (c : ι → Bool) : Decidable (Compatible x c) :=
+  Classical.propDecidable _
+
+lemma colors_probability (x : ι → Bool × α) (c : ι → Bool) :
+    finiteMean (fun ξ : α → Bool => if colors x ξ = c then (1 : ℝ) else 0) =
+      if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0 := by
+  classical
+  by_cases hc : Compatible x c
+  · rw [ite_eq_left hc]
+    have he := fun ξ => colors_eq_iff x c hc ξ
+    simp only [he]
+    exact mean_coinCylinder (occupied x) (chosenCoin x c)
+  · rw [ite_eq_right hc]
+    have he : ∀ ξ, colors x ξ ≠ c := fun ξ hh => hc (hh ▸ colors_compatible x ξ)
+    simp only [he,ite_false,finiteMean_const]
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma Compatible.tail_injective {x : ι ↪ Bool × α} {c : ι → Bool} (hc : Compatible x c) (b : Bool) :
+    Function.Injective (fun i : {i : ι // c i = b} => (x i.val).2) := by
+  intro i j he
+  apply Subtype.ext
+  apply x.injective
+  apply Prod.ext
+  · have hh := hc i j he
+    simp only [required,i.property,j.property] at hh
+    cases h₀ : (x i.val).1 <;> cases h₁ : (x j.val).1 <;> cases b <;> simp_all
+  · exact he
+
+def childEmbedding (x : ι ↪ Bool × α) (c : ι → Bool) (hc : Compatible x c) (b : Bool) :
+    {i : ι // c i = b} ↪ α := ⟨fun i => (x i.val).2,hc.tail_injective b⟩
+
+
+def lift (p : Bool → Equiv.Perm α) : Equiv.Perm (Bool × α) where
+  toFun x := (x.1,p x.1 x.2)
+  invFun x := (x.1,(p x.1).symm x.2)
+  left_inv x := by simp
+  right_inv x := by simp
+
+def sandwich (ξ η : α → Bool) (p : Bool → Equiv.Perm α) : Equiv.Perm (Bool × α) :=
+  pairLayer η * lift p * pairLayer ξ
+
+def ChildMatches (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) : Prop :=
+  ∀ i, p (c i) (x i).2 = (y i).2
+
+noncomputable instance (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
+    Decidable (ChildMatches p x y c) := Classical.propDecidable _
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma sandwich_matches_iff (ξ η : α → Bool) (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) :
+    (∀ i, sandwich ξ η p (x i) = y i) ↔
+      colors x ξ = colors y η ∧ ChildMatches p x y (colors x ξ) := by
+  have he (i : ι) : sandwich ξ η p (x i) = y i ↔
+      colors x ξ i = colors y η i ∧ p (colors x ξ i) (x i).2 = (y i).2 := by
+    change pairLayer η (lift p (pairLayer ξ (x i))) = y i ↔ _
+    rw [←Equiv.eq_symm_apply]
+    change (colors x ξ i,p (colors x ξ i) (x i).2) = (colors y η i,(y i).2) ↔ _
+    exact ⟨Prod.mk.inj,fun h => Prod.ext h.1 h.2⟩
+  simp only [he,forall_and,←funext_iff,ChildMatches]
+
+lemma sandwich_probability [DecidableEq ι] (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) :
+    finiteMean (fun coins : (α → Bool) × (α → Bool) =>
+      if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then (1:ℝ) else 0) =
+    ∑ c : ι → Bool,
+      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
+      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) *
+      (if ChildMatches p x y c then 1 else 0) := by
+  classical
+  have he (coins : (α → Bool) × (α → Bool)) :
+      (if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then (1:ℝ) else 0) =
+      ∑ c : ι → Bool, (if colors x coins.1 = c then (1:ℝ) else 0) *
+        (if colors y coins.2 = c then 1 else 0) * (if ChildMatches p x y c then 1 else 0) := by
+    rw [Finset.sum_eq_single (colors x coins.1)]
+    · by_cases hc : colors x coins.1 = colors y coins.2
+      · simp only [sandwich_matches_iff,hc,true_and,↓reduceIte,one_mul]
+      · simp only [sandwich_matches_iff,hc,false_and,↓reduceIte,one_mul,Ne.symm hc,zero_mul]
+    · intro c _ hc
+      simp only [Ne.symm hc,ite_false,zero_mul]
+    · simp
+  simp_rw [he]
+  rw [finiteMean_sum]
+  apply Finset.sum_congr rfl
+  intro c _
+  rw [finiteMean_mul_const]
+  rw [finiteMean_prod_mul
+    (fun ξ : α → Bool => if colors x ξ = c then (1:ℝ) else 0)
+    (fun η : α → Bool => if colors y η = c then (1:ℝ) else 0),
+    colors_probability,colors_probability]
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma childMatches_split (p₀ p₁ : Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
+    ChildMatches (fun b => if b then p₁ else p₀) x y c ↔
+      (∀ i : {i // c i = false}, p₀ (x i.val).2 = (y i.val).2) ∧
+      (∀ i : {i // c i = true}, p₁ (x i.val).2 = (y i.val).2) := by
+  constructor
+  · intro h
+    constructor
+    · intro i
+      simpa only [i.property,Bool.false_eq_true,↓reduceIte] using h i.val
+    · intro i
+      simpa only [i.property,↓reduceIte] using h i.val
+  · rintro ⟨h₀,h₁⟩ i
+    cases hi : c i
+    · simpa only [hi,Bool.false_eq_true,↓reduceIte] using h₀ ⟨i,hi⟩
+    · simpa only [hi,↓reduceIte] using h₁ ⟨i,hi⟩
+
+noncomputable def tupleProbability {Ω β : Type*} [Fintype Ω] [DecidableEq β]
+    (P : Ω → Equiv.Perm β) (x y : ι → β) : ℝ := by
+  classical
+  exact finiteMean (fun ω => if ∀ i, P ω (x i) = y i then 1 else 0)
+
+omit [Fintype α] in
+lemma childMatches_mean {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
+    (P : Ω → Equiv.Perm α) (Q : Ω' → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
+    finiteMean (fun ω : Ω × Ω' =>
+      if ChildMatches (fun b => if b then Q ω.2 else P ω.1) x y c then (1:ℝ) else 0) =
+      tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
+        (fun i : {i // c i = false} => (y i.val).2) *
+      tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
+        (fun i : {i // c i = true} => (y i.val).2) := by
+  classical
+  let f : Ω → ℝ := fun ω => if ∀ i : {i // c i = false}, P ω (x i.val).2 = (y i.val).2 then 1 else 0
+  let g : Ω' → ℝ := fun ω => if ∀ i : {i // c i = true}, Q ω (x i.val).2 = (y i.val).2 then 1 else 0
+  have he (ω : Ω × Ω') :
+      (if ChildMatches (fun b => if b then Q ω.2 else P ω.1) x y c then (1:ℝ) else 0) = f ω.1*g ω.2 := by
+    simp only [childMatches_split,f,g]
+    split_ifs <;> simp_all
+  simp_rw [he]
+  exact finiteMean_prod_mul f g
+
+lemma sandwich_mean [DecidableEq ι] {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
+    (P : Ω → Equiv.Perm α) (Q : Ω' → Equiv.Perm α) (x y : ι → Bool × α) :
+    finiteMean (fun ω : Ω × Ω' => finiteMean (fun coins : (α → Bool) × (α → Bool) =>
+      if ∀ i, sandwich coins.1 coins.2 (fun b => if b then Q ω.2 else P ω.1) (x i) = y i
+      then (1:ℝ) else 0)) =
+    ∑ c : ι → Bool,
+      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
+      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) *
+      (tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
+        (fun i : {i // c i = false} => (y i.val).2) *
+      tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
+        (fun i : {i // c i = true} => (y i.val).2)) := by
+  classical
+  simp_rw [sandwich_probability]
+  rw [finiteMean_sum]
+  apply Finset.sum_congr rfl
+  intro c _
+  simp only [mul_comm ((if Compatible x c then _ else _) * _),finiteMean_mul_const,
+    childMatches_mean]
+
+
+abbrev BothAssignments (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :=
+  {c : ι → Bool // Compatible e c ∧ Compatible (fun i => lift p (e i)) c}
+
+noncomputable instance (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    Fintype (BothAssignments e p) := Fintype.ofFinite _
+
+noncomputable def labelSet (e : ι ↪ Bool × α) : Finset (Bool × α) := Finset.univ.image e
+
+def alternatingProjection (p : Bool → Equiv.Perm α) : Equiv.Perm α := (p false).symm * p true
+
+noncomputable def assignmentColoring (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α)
+    (c : BothAssignments e p) : CycleColoring.OppositeColoring
+      (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e) := by
+  classical
+  refine ⟨Function.extend e c.val (fun _ => false),?_,?_⟩
+  · intro z hz
+    apply Function.extend_apply'
+    rintro ⟨i,rfl⟩
+    exact hz (Finset.mem_image.mpr ⟨i,Finset.mem_univ _,rfl⟩)
+  · intro z hz hnext
+    obtain ⟨i,_,hi⟩ := Finset.mem_image.mp hz
+    obtain ⟨j,_,hj⟩ := Finset.mem_image.mp hnext
+    have hfi := e.injective.extend_apply c.val (fun _ => false) i
+    have hfj := e.injective.extend_apply c.val (fun _ => false) j
+    rw [hi] at hfi
+    rw [hj] at hfj
+    rw [hfi,hfj]
+    rcases z with ⟨b,a⟩
+    cases b
+    · have hj' : e j = (true,a) := hj
+      have ht : (e i).2 = (e j).2 := by rw [hi,hj']
+      have hh := c.property.1 i j ht
+      simp only [required,hi,hj'] at hh
+      revert hh
+      generalize c.val i = bi
+      generalize c.val j = bj
+      cases bi <;> cases bj <;> decide
+    · have hj' : e j = (false,alternatingProjection p a) := hj
+      have ht : (lift p (e i)).2 = (lift p (e j)).2 := by
+        simp only [hi,hj',lift,Equiv.coe_fn_mk,alternatingProjection,Equiv.Perm.coe_mul,
+          Function.comp_apply,Equiv.apply_symm_apply]
+      have hh := c.property.2 i j ht
+      simp only [required,hi,hj',lift,Equiv.coe_fn_mk] at hh
+      revert hh
+      generalize c.val i = bi
+      generalize c.val j = bj
+      cases bi <;> cases bj <;> decide
+
+omit [Fintype α] in
+lemma assignmentColoring_injective (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    Function.Injective (assignmentColoring e p) := by
+  intro c c' he
+  apply Subtype.ext
+  funext i
+  have hh := congrFun (congrArg Subtype.val he) (e i)
+  change Function.extend e c.val (fun _ => false) (e i) =
+    Function.extend e c'.val (fun _ => false) (e i) at hh
+  simpa only [e.injective.extend_apply] using hh
+
+lemma bothAssignments_card (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    Fintype.card (BothAssignments e p) ≤
+      2 ^ ((boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true).card +
+        (boolFiber (labelSet e) false ∪ (boolFiber (labelSet e) true).image
+          (alternatingProjection p)).card -
+        ((boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card) +
+        Fintype.card (RoutingNetwork.SelectedCycle (alternatingProjection p) (boolFiber (labelSet e) false))) := by
+  have h := compatible_assignment_bound (alternatingProjection p)
+    (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
+  rw [childJoin_boolFiber] at h
+  exact (Fintype.card_le_of_injective _ (assignmentColoring_injective e p)).trans h
+
+omit [Fintype α] in
+lemma occupied_labelSet (e : ι ↪ Bool × α) : occupied e = (labelSet e).image Prod.snd := by
+  classical
+  simp only [occupied,labelSet,Finset.image_image,Function.comp_def]
+
+omit [Fintype ι] [Fintype α] in
+lemma image_snd_childJoin (A B : Finset α) : (childJoin A B).image Prod.snd = A ∪ B := by
+  classical
+  simp only [childJoin,Finset.image_union,Finset.image_image,Function.comp_def,Finset.image_id']
+
+omit [Fintype ι] [Fintype α] in
+lemma image_lift_childJoin (p : Bool → Equiv.Perm α) (A B : Finset α) :
+    (childJoin A B).image (fun x => (lift p x).2) = A.image (p false) ∪ B.image (p true) := by
+  classical
+  simp only [childJoin,Finset.image_union,Finset.image_image,Function.comp_def,lift,Equiv.coe_fn_mk]
+
+omit [Fintype α] in
+lemma occupied_lift_card (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    (occupied (fun i => lift p (e i))).card =
+      (boolFiber (labelSet e) false ∪ (boolFiber (labelSet e) true).image (alternatingProjection p)).card := by
+  classical
+  have he : occupied (fun i => lift p (e i)) = (labelSet e).image (fun x => (lift p x).2) := by
+    simp only [occupied,labelSet,Finset.image_image,Function.comp_def]
+  rw [he,←childJoin_boolFiber (labelSet e),image_lift_childJoin]
+  simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
+  let A := boolFiber (labelSet e) false
+  let B := boolFiber (labelSet e) true
+  change (A.image (p false) ∪ B.image (p true)).card = (A ∪ B.image (alternatingProjection p)).card
+  have hh : A.image (p false) ∪ B.image (p true) =
+      (A ∪ B.image (alternatingProjection p)).image (p false) := by
+    simp only [Finset.image_union,Finset.image_image,Function.comp_def,
+      alternatingProjection,Equiv.Perm.coe_mul,Equiv.apply_symm_apply]
+  rw [hh,Finset.card_image_of_injective _ (p false).injective]
+
+lemma bothAssignments_occupied_bound (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    Fintype.card (BothAssignments e p) ≤
+      2 ^ ((occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι +
+        Fintype.card (RoutingNetwork.SelectedCycle (alternatingProjection p) (boolFiber (labelSet e) false))) := by
+  have he : (occupied e).card =
+      (boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true).card := by
+    rw [occupied_labelSet,←childJoin_boolFiber (labelSet e),image_snd_childJoin]
+    simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
+  have hc : (boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card = Fintype.card ι := by
+    rw [←card_childJoin,childJoin_boolFiber,labelSet,Finset.card_image_of_injective _ e.injective,
+      Finset.card_univ]
+  simpa only [he,occupied_lift_card,←hc] using bothAssignments_card e p
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma compatible_pairLayer_iff (x : ι → Bool × α) (c : ι → Bool) (ξ : α → Bool) :
+    Compatible (fun i => pairLayer ξ (x i)) c ↔ Compatible x c := by
+  have he (i j : ι) (ht : (x i).2 = (x j).2) :
+      required (fun i => pairLayer ξ (x i)) c i = required (fun i => pairLayer ξ (x i)) c j ↔
+      required x c i = required x c j := by
+    simp only [required,pairLayer,Equiv.coe_fn_mk,ht]
+    cases (x i).1 <;> cases (x j).1 <;> cases c i <;> cases c j <;> cases ξ (x j).2 <;> decide
+  constructor
+  · intro h i j ht
+    exact (he i j ht).mp (h i j ht)
+  · intro h i j ht
+    exact (he i j ht).mpr (h i j ht)
+
+
+noncomputable def alternatingCycles (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) : ℕ :=
+  Fintype.card (RoutingNetwork.SelectedCycle
+    (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e))
+
+omit [Fintype α] in
+lemma occupied_eq_fibers (e : ι ↪ Bool × α) :
+    occupied e = boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true := by
+  rw [occupied_labelSet,←childJoin_boolFiber (labelSet e),image_snd_childJoin]
+  simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
+
+omit [Fintype α] in
+lemma card_labelSet (e : ι ↪ Bool × α) : (labelSet e).card = Fintype.card ι := by
+  classical
+  exact (Finset.card_image_of_injective _ e.injective).trans (Finset.card_univ)
+
+omit [Fintype α] in
+lemma occupied_card_double (e : ι ↪ Bool × α) : Fintype.card ι ≤ 2*(occupied e).card := by
+  have h₀ : (boolFiber (labelSet e) false).card ≤ (occupied e).card := by
+    rw [occupied_eq_fibers]
+    exact Finset.card_le_card Finset.subset_union_left
+  have h₁ : (boolFiber (labelSet e) true).card ≤ (occupied e).card := by
+    rw [occupied_eq_fibers]
+    exact Finset.card_le_card Finset.subset_union_right
+  have he : (boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card = Fintype.card ι := by
+    rw [←card_childJoin,childJoin_boolFiber,card_labelSet]
+  omega
+
+lemma bothAssignments_exact_bound (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    Fintype.card (BothAssignments e p) ≤
+      2 ^ ((occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι +
+        alternatingCycles e p) := by
+  have h := CycleColoring.card_opposite_le
+    (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e)
+    Prod.fst (CycleColoring.alternatingPerm_flip (alternatingProjection p))
+  have hb : (CycleColoring.boundary (CycleColoring.alternatingPerm (alternatingProjection p))
+      (labelSet e)).card =
+      (occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι := by
+    have hh := alternating_boundary_card (alternatingProjection p)
+      (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
+    rw [childJoin_boolFiber] at hh
+    rw [occupied_eq_fibers,occupied_lift_card]
+    have he := card_childJoin (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
+    rw [childJoin_boolFiber,card_labelSet] at he
+    simpa only [he] using hh
+  rw [hb] at h
+  exact (Fintype.card_le_of_injective _ (assignmentColoring_injective e p)).trans h
+
+abbrev Assignments (x y : ι → Bool × α) :=
+  {c : ι → Bool // Compatible x c ∧ Compatible y c}
+
+noncomputable instance (x y : ι → Bool × α) : Fintype (Assignments x y) := Fintype.ofFinite _
+
+def switchedEmbedding (e : ι ↪ Bool × α) (ξ : α → Bool) : ι ↪ Bool × α :=
+  e.trans (pairLayer ξ).toEmbedding
+
+def routedEmbedding (e : ι ↪ Bool × α) (ξ η : α → Bool) (p : Bool → Equiv.Perm α) :
+    ι ↪ Bool × α := e.trans (sandwich ξ η p).toEmbedding
+
+omit [Fintype α] in
+lemma occupied_pairLayer (x : ι → Bool × α) (ξ : α → Bool) :
+    occupied (fun i => pairLayer ξ (x i)) = occupied x := rfl
+
+noncomputable def assignmentsEquiv (e : ι ↪ Bool × α) (ξ η : α → Bool)
+    (p : Bool → Equiv.Perm α) :
+    Assignments e (routedEmbedding e ξ η p) ≃ BothAssignments (switchedEmbedding e ξ) p := by
+  apply Equiv.subtypeEquivRight
+  intro c
+  change Compatible e c ∧ Compatible (fun i => pairLayer η (lift p (pairLayer ξ (e i)))) c ↔ _
+  rw [compatible_pairLayer_iff]
+  exact and_congr (compatible_pairLayer_iff e c ξ).symm Iff.rfl
+
+lemma assignments_actual_bound (e : ι ↪ Bool × α) (ξ η : α → Bool) (p : Bool → Equiv.Perm α) :
+    Fintype.card (Assignments e (routedEmbedding e ξ η p)) ≤
+      2 ^ ((occupied e).card + (occupied (routedEmbedding e ξ η p)).card - Fintype.card ι +
+        alternatingCycles (switchedEmbedding e ξ) p) := by
+  rw [Fintype.card_congr (assignmentsEquiv e ξ η p)]
+  exact bothAssignments_exact_bound (switchedEmbedding e ξ) p
+
+omit [Fintype α] in
+lemma assignments_actual_exponent_nonneg (e : ι ↪ Bool × α) (ξ η : α → Bool)
+    (p : Bool → Equiv.Perm α) :
+    Fintype.card ι ≤ (occupied e).card + (occupied (routedEmbedding e ξ η p)).card := by
+  have h₀ := occupied_card_double e
+  have h₁ := occupied_card_double (routedEmbedding e ξ η p)
+  omega
+
+end
+variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α] [DecidableEq ι]
+
+noncomputable def compatibleSet (x y : ι → Bool × α) : Finset (ι → Bool) := by
+  classical
+  exact Finset.univ.filter (fun c => Compatible x c ∧ Compatible y c)
+
+omit [Fintype α] [DecidableEq α] in
+lemma mem_compatibleSet (x y : ι → Bool × α) (c : ι → Bool) :
+    c ∈ compatibleSet x y ↔ Compatible x c ∧ Compatible y c := by
+  classical
+  simp only [compatibleSet,Finset.mem_filter,Finset.mem_univ,true_and]
+
+omit [Fintype α] [DecidableEq α] in
+lemma card_compatibleSet (x y : ι → Bool × α) :
+    (compatibleSet x y).card = Fintype.card (Assignments x y) := by
+  classical
+  exact (Fintype.card_subtype _).symm
+
+noncomputable def outerWeight (x y : ι → Bool × α) : ℝ :=
+  1/(2:ℝ)^((occupied x).card+(occupied y).card)
+
+omit [Fintype α] [DecidableEq ι] in
+lemma outerWeight_pos (x y : ι → Bool × α) : 0 < outerWeight x y := by
+  unfold outerWeight
+  positivity
+
+omit [Fintype α] in
+lemma sum_colors_factor (x y : ι → Bool × α) (f : (ι → Bool) → ℝ) :
+    (∑ c : ι → Bool,
+      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
+      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) * f c) =
+      outerWeight x y * ∑ c ∈ compatibleSet x y, f c := by
+  classical
+  rw [compatibleSet,Finset.sum_filter,Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro c _
+  by_cases hx : Compatible x c <;> by_cases hy : Compatible y c
+  · simp only [hx,hy,true_and,↓reduceIte,outerWeight,pow_add,one_div_mul_one_div]
+  · simp only [hx,hy,and_false,↓reduceIte,mul_zero,zero_mul]
+  · simp only [hx,hy,false_and,↓reduceIte,mul_zero,zero_mul]
+  · simp only [hx,hy,false_and,↓reduceIte,mul_zero,zero_mul]
+
+lemma sandwich_weighted_probability (p : Bool → Equiv.Perm α) (x y : ι → Bool × α)
+    (W : (ι → Bool) → ℝ) :
+    finiteMean (fun coins : (α → Bool) × (α → Bool) =>
+      if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then W (colors x coins.1) else 0) =
+    outerWeight x y * ∑ c ∈ compatibleSet x y, if ChildMatches p x y c then W c else 0 := by
+  classical
+  have he (coins : (α → Bool) × (α → Bool)) :
+      (if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then W (colors x coins.1) else 0) =
+      ∑ c : ι → Bool, (if colors x coins.1 = c then (1:ℝ) else 0) *
+        (if colors y coins.2 = c then 1 else 0) * (if ChildMatches p x y c then W c else 0) := by
+    rw [Finset.sum_eq_single (colors x coins.1)]
+    · by_cases hc : colors x coins.1 = colors y coins.2
+      · simp only [sandwich_matches_iff,hc,true_and,↓reduceIte,one_mul]
+      · simp only [sandwich_matches_iff,hc,false_and,↓reduceIte,one_mul,Ne.symm hc,zero_mul]
+    · intro c _ hc
+      simp only [Ne.symm hc,ite_false,zero_mul]
+    · simp
+  simp_rw [he]
+  rw [finiteMean_sum,←sum_colors_factor]
+  apply Finset.sum_congr rfl
+  intro c _
+  rw [finiteMean_mul_const,finiteMean_prod_mul
+    (fun ξ : α → Bool => if colors x ξ = c then (1:ℝ) else 0)
+    (fun η : α → Bool => if colors y η = c then (1:ℝ) else 0),
+    colors_probability,colors_probability]
+
+omit [Fintype α] [DecidableEq ι] in
+lemma tupleProbability_nonneg {Ω β : Type*} [Fintype Ω] [DecidableEq β]
+    (P : Ω → Equiv.Perm β) (x y : ι → β) : 0 ≤ tupleProbability P x y := by
+  classical
+  exact finiteMean_nonneg (fun ω => by split_ifs <;> norm_num)
+
+lemma sandwich_power_bound {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
+    (P : Ω → Equiv.Perm α) (Q : Ω' → Equiv.Perm α) (x y : ι → Bool × α)
+    (a : ℝ) (ha : 0 ≤ a) :
+    (finiteMean (fun ω : Ω × Ω' => finiteMean (fun coins : (α → Bool) × (α → Bool) =>
+      if ∀ i, sandwich coins.1 coins.2 (fun b => if b then Q ω.2 else P ω.1) (x i) = y i
+      then (1:ℝ) else 0)))^(1+a) ≤
+    (outerWeight x y)^(1+a) * ((compatibleSet x y).card:ℝ)^a *
+      ∑ c ∈ compatibleSet x y,
+      (tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
+        (fun i : {i // c i = false} => (y i.val).2))^(1+a) *
+      (tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
+        (fun i : {i // c i = true} => (y i.val).2))^(1+a) := by
+  classical
+  rw [sandwich_mean,sum_colors_factor]
+  let f (c : ι → Bool) : ℝ :=
+      tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
+        (fun i : {i // c i = false} => (y i.val).2) *
+      tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
+        (fun i : {i // c i = true} => (y i.val).2)
+  have hf (c : ι → Bool) : 0 ≤ f c := mul_nonneg (tupleProbability_nonneg _ _ _)
+    (tupleProbability_nonneg _ _ _)
+  rw [Real.mul_rpow (le_of_lt (outerWeight_pos _ _)) (Finset.sum_nonneg (fun c _ => hf c))]
+  have hh := Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg (compatibleSet x y)
+    (show 1 ≤ 1+a by linarith) (fun c _ => hf c)
+  have he : 1+a-1=a := by ring
+  rw [he] at hh
+  calc
+    _ ≤ (outerWeight x y)^(1+a) *
+        (((compatibleSet x y).card:ℝ)^a * ∑ c ∈ compatibleSet x y, (f c)^(1+a)) :=
+      mul_le_mul_of_nonneg_left hh (Real.rpow_nonneg (le_of_lt (outerWeight_pos _ _)) _)
+    _ = _ := by
+      rw [←mul_assoc]
+      congr 1
+      apply Finset.sum_congr rfl
+      intro c _
+      exact Real.mul_rpow (tupleProbability_nonneg _ _ _) (tupleProbability_nonneg _ _ _)
+
+end PairRouting
+
+end Thorp
+
+end ThorpNine.SparseSaving
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/SparseSaving/AssignmentSlots.lean b/lean/OAI/Probability/ThorpRouting/SparseSaving/AssignmentSlots.lean
index f3fb711c3367c6963ceb86eda0b9d7a61bd07a3e..91001448c15e8603ebe8997c68ff98ee8e6e0e7e 100644
--- a/lean/OAI/Probability/ThorpRouting/SparseSaving/AssignmentSlots.lean
+++ b/lean/OAI/Probability/ThorpRouting/SparseSaving/AssignmentSlots.lean
@@ -1,834 +1,834 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.SparseSaving.RotationMeeting
-
-namespace OAI
-
-namespace ThorpNine.SparseSaving
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-section CycleCounting
-variable {d A : ℕ}
-
-lemma assignmentOwner_eq_some {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) (x : Card d) (i : Fin A) :
-    assignmentOwner f x = some i ↔ x ∈ (f.val i).val := by
-  classical
-  unfold assignmentOwner
-  split_ifs with h
-  · constructor
-    · intro hi
-      have hx := Classical.choose_spec h
-      rw [Option.some.inj hi] at hx
-      exact hx
-    · intro hx
-      have he := (f.val (Classical.choose h)).eq_of_mem (f.val i) (Classical.choose_spec h) hx
-      exact congrArg some (f.property.1 he)
-  · constructor
-    · intro hi; contradiction
-    · intro hx; exact (h ⟨i,hx⟩).elim
-
-lemma assignmentOwner_fiber {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) (i : Fin A) : cycleFiber (assignmentOwner f) i = (f.val i).val := by
-  ext x
-  simp only [mem_cycleFiber,assignmentOwner_eq_some]
-
-noncomputable def assignmentToSlots {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) : CycleSlots p S r :=
-  ⟨assignmentOwner f,by
-    refine ⟨?_,?_,?_⟩
-    · intro i
-      rw [assignmentOwner_fiber]
-      exact f.property.2 i
-    · intro i
-      rw [assignmentOwner_fiber]
-      exact (f.val i).property.2
-    · intro i x hx
-      rw [assignmentOwner_fiber]
-      exact (f.val i).orbit_eq ((assignmentOwner_eq_some f x i).mp hx)⟩
-
-noncomputable def cycleSlotsEquivAssignments (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
-    CycleSlots p S r ≃ SlotAssignment p S r where
-  toFun := slotsToAssignment
-  invFun := assignmentToSlots
-  left_inv c := by
-    apply Subtype.ext
-    apply owner_ext
-    intro i
-    rw [assignmentToSlots,assignmentOwner_fiber]
-    rfl
-  right_inv f := by
-    apply Subtype.ext
-    funext i
-    apply Subtype.ext
-    exact assignmentOwner_fiber f i
-
-end CycleCounting
-
-section GroupedCounting
-variable {d A : ℕ}
-
-abbrev LengthClass (r : Fin A → ℕ) := {j : ℕ // j ∈ Finset.univ.image (fun i => r i + 1)}
-
-def slotClass (r : Fin A → ℕ) (i : Fin A) : LengthClass r :=
-  ⟨r i+1, Finset.mem_image.mpr ⟨i, Finset.mem_univ _, rfl⟩⟩
-
-abbrev SlotsOfLength (r : Fin A → ℕ) (j : LengthClass r) := {i : Fin A // slotClass r i = j}
-abbrev CyclesOfLength (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (j : ℕ) :=
-  {C : SelectedCycle p S // C.val.card = j}
-
-noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (j : ℕ) :
-    Fintype (CyclesOfLength p S j) := Fintype.ofFinite _
-
-abbrev GroupedAssignments (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
-  ∀ j : LengthClass r, SlotsOfLength r j ↪ CyclesOfLength p S j.val
-
-noncomputable def assignmentToGroups {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) : GroupedAssignments p S r := fun _j =>
-  { toFun := fun i => ⟨f.val i.val, (f.property.2 i.val).trans (congrArg Subtype.val i.property)⟩
-    inj' := fun _i _k h => Subtype.ext (f.property.1 (congrArg Subtype.val h)) }
-
-noncomputable def groupsEmbedding {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (F : GroupedAssignments p S r) : Fin A ↪ SelectedCycle p S :=
-  ((Equiv.sigmaFiberEquiv (slotClass r)).symm.toEmbedding.trans
-    (Function.Embedding.sigmaMap (Function.Embedding.refl _) F)).trans
-    ((Function.Embedding.sigmaMap (Function.Embedding.subtype _) (fun _ => Function.Embedding.refl _)).trans
-      (Equiv.sigmaFiberEquiv (fun C : SelectedCycle p S => C.val.card)).toEmbedding)
-
-noncomputable def groupsToAssignment {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (F : GroupedAssignments p S r) : SlotAssignment p S r :=
-  ⟨groupsEmbedding F, (groupsEmbedding F).injective,
-    fun i => (F (slotClass r i) ⟨i,rfl⟩).property⟩
-
-noncomputable def assignmentEquivGroups (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
-    SlotAssignment p S r ≃ GroupedAssignments p S r where
-  toFun := assignmentToGroups
-  invFun := groupsToAssignment
-  left_inv f := by
-    apply Subtype.ext
-    rfl
-  right_inv F := by
-    funext j
-    apply Function.Embedding.ext
-    intro i
-    rcases i with ⟨i,hi⟩
-    subst j
-    rfl
-
-lemma card_cycleSlots_factorial (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
-    Fintype.card (CycleSlots p S r) =
-      ∏ j : LengthClass r, (Fintype.card (CyclesOfLength p S j.val)).descFactorial
-        (Fintype.card (SlotsOfLength r j)) := by
-  classical
-  rw [Fintype.card_congr (cycleSlotsEquivAssignments p S r),
-    Fintype.card_congr (assignmentEquivGroups p S r), Fintype.card_pi]
-  simp only [Fintype.card_embedding_eq]
-
-lemma butterfly_factorial_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-       (∏ j : LengthClass r,
-         (Fintype.card (CyclesOfLength (butterflyPerm d (decodeButterfly d ω)) S j.val)).descFactorial
-           (Fintype.card (SlotsOfLength r j)) : ℕ)) ≤
-      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-        ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
-  simpa only [←card_cycleSlots_factorial] using butterfly_cycleSlots_moment S r hr
-
-end GroupedCounting
-
-section AllLengths
-variable {d A : ℕ}
-
-noncomputable def assignmentRelabel (p : Equiv.Perm (Card d)) (S : Finset (Card d))
-    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
-    SlotAssignment p S r ≃ SlotAssignment p S (r ∘ e) where
-  toFun f := ⟨f.val ∘ e, f.property.1.comp e.injective, fun i => f.property.2 (e i)⟩
-  invFun f := ⟨f.val ∘ e.symm, f.property.1.comp e.symm.injective, fun i => by
-    simpa only [Function.comp_apply,Equiv.apply_symm_apply] using f.property.2 (e.symm i)⟩
-  left_inv f := by apply Subtype.ext; funext i; simp
-  right_inv f := by apply Subtype.ext; funext i; simp
-
-lemma card_cycleSlots_relabel (p : Equiv.Perm (Card d)) (S : Finset (Card d))
-    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
-    Fintype.card (CycleSlots p S r) = Fintype.card (CycleSlots p S (r ∘ e)) := by
-  exact Fintype.card_congr ((cycleSlotsEquivAssignments p S r).trans
-    ((assignmentRelabel p S r e).trans (cycleSlotsEquivAssignments p S (r ∘ e)).symm))
-
-lemma butterfly_cycleSlots_moment_all (S : Finset (Card d)) (r : Fin A → ℕ) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) ≤
-        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
-  have h := butterfly_cycleSlots_moment S (r ∘ Tuple.sort r) (Tuple.monotone_sort r)
-  simp_rw [←card_cycleSlots_relabel] at h
-  have hp := Equiv.prod_comp (Tuple.sort r) (fun i : Fin A => (1 / Real.sqrt (r i+1) : ℝ))
-  simpa only [Function.comp_apply,hp] using h
-
-abbrev ShortCycle (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :=
-  {C : SelectedCycle p S // C.val.card ≤ J}
-
-noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
-    Fintype (ShortCycle p S J) := Fintype.ofFinite _
-
-noncomputable def shortCycleLength {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {J : ℕ}
-    (C : ShortCycle p S J) : Fin J :=
-  ⟨C.val.val.card-1, by have := C.val.nonempty.card_pos; have := C.property; omega⟩
-
-lemma shortCycleLength_add_one {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {J : ℕ}
-    (C : ShortCycle p S J) : (shortCycleLength C : ℕ)+1 = C.val.val.card := by
-  have := C.val.nonempty.card_pos
-  simp only [shortCycleLength]
-  omega
-
-noncomputable def shortEmbeddingEquiv (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
-    (Fin A ↪ ShortCycle p S J) ≃
-      Σ r : Fin A → Fin J, SlotAssignment p S (fun i => (r i : ℕ)) where
-  toFun f := ⟨fun i => shortCycleLength (f i),
-    ⟨fun i => (f i).val, fun i j h => f.injective (Subtype.ext h),
-      fun i => (shortCycleLength_add_one (f i)).symm⟩⟩
-  invFun c :=
-    { toFun := fun i => ⟨c.2.val i, by rw [c.2.property.2]; exact (c.1 i).isLt⟩
-      inj' := fun _i _j h => c.2.property.1 (congrArg Subtype.val h) }
-  left_inv f := by
-    apply Function.Embedding.ext
-    intro i
-    rfl
-  right_inv c := by
-    rcases c with ⟨r,f⟩
-    have he : (fun i => shortCycleLength (⟨f.val i, by rw [f.property.2]; exact (r i).isLt⟩ : ShortCycle p S J)) = r := by
-      funext i
-      apply Fin.ext
-      simp only [shortCycleLength,f.property.2,Nat.add_sub_cancel]
-    apply Sigma.ext he
-    apply (Subtype.heq_iff_coe_eq (by
-      intro x
-      change (Function.Injective x ∧ ∀ i, (x i).val.card = (f.val i).val.card - 1 + 1) ↔ _
-      simp only [f.property.2,Nat.add_sub_cancel])).mpr
-    rfl
-
-lemma shortCycle_factorial_eq (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
-    (Fintype.card (ShortCycle p S J)).descFactorial A =
-      ∑ r : Fin A → Fin J, Fintype.card (CycleSlots p S (fun i => (r i : ℕ))) := by
-  classical
-  have hc := Fintype.card_embedding_eq (α := Fin A) (β := ShortCycle p S J)
-  simp only [Fintype.card_fin] at hc
-  rw [←hc, Fintype.card_congr (shortEmbeddingEquiv p S J),Fintype.card_sigma]
-  apply Finset.sum_congr rfl
-  intro r _
-  exact (Fintype.card_congr (cycleSlotsEquivAssignments p S (fun i => (r i : ℕ)))).symm
-
-end AllLengths
-
-lemma factorial_mgf {Ω : Type*} [Fintype Ω] (F : Ω → ℕ) (k : ℕ) (Λ q : ℝ)
-    (hF : ∀ ω, F ω ≤ k) (hΛ : 0 ≤ Λ) (hq : 1 ≤ q)
-    (hm : ∀ a, finiteMean (fun ω => ((F ω).descFactorial a : ℝ)) ≤ Λ^a) :
-    finiteMean (fun ω => q^(F ω)) ≤ Real.exp ((q-1)*Λ) := by
-  have hc (n a : ℕ) : (n.choose a : ℝ) = (n.descFactorial a : ℝ) / (a.factorial : ℝ) := by
-    rw [Nat.descFactorial_eq_factorial_mul_choose,Nat.cast_mul]
-    exact (mul_div_cancel_left₀ _ (by exact_mod_cast Nat.factorial_ne_zero a)).symm
-  have hex (ω : Ω) : q^(F ω) = ∑ a ∈ Finset.range (k+1),
-      ((F ω).descFactorial a : ℝ) * ((q-1)^a / (a.factorial : ℝ)) := by
-    calc
-      _ = ((q-1)+1)^(F ω) := by congr 1; ring
-      _ = ∑ a ∈ Finset.range (F ω+1), (q-1)^a * (F ω).choose a := by
-        rw [add_pow]
-        simp only [one_pow,mul_one]
-      _ = ∑ a ∈ Finset.range (k+1), (q-1)^a * (F ω).choose a := by
-        apply Finset.sum_subset (Finset.range_mono (Nat.add_le_add_right (hF ω) _))
-        intro a _ ha
-        have hn : F ω < a := by have := (Finset.mem_range.not.mp ha); omega
-        rw [Nat.choose_eq_zero_of_lt hn,Nat.cast_zero,mul_zero]
-      _ = _ := by
-        apply Finset.sum_congr rfl
-        intro a _
-        rw [hc]
-        ring
-  simp_rw [hex,finiteMean_sum,finiteMean_mul_const]
-  calc
-    _ ≤ ∑ a ∈ Finset.range (k+1), Λ^a * ((q-1)^a / (a.factorial : ℝ)) := by
-      apply Finset.sum_le_sum
-      intro a _
-      exact mul_le_mul_of_nonneg_right (hm a) (by positivity)
-    _ = ∑ a ∈ Finset.range (k+1), ((q-1)*Λ)^a / (a.factorial : ℝ) := by
-      apply Finset.sum_congr rfl
-      intro a _
-      rw [mul_pow]
-      ring
-    _ ≤ _ := Real.sum_le_exp_of_nonneg (mul_nonneg (sub_nonneg.mpr hq) hΛ) _
-
-lemma inv_sqrt_step (j : ℕ) :
-    1 / Real.sqrt (j+1) ≤ 2*(Real.sqrt (j+1)-Real.sqrt j) := by
-  have hp : 0 < Real.sqrt (j+1) := Real.sqrt_pos.mpr (by positivity)
-  apply (div_le_iff₀ hp).mpr
-  have h1 := Real.sq_sqrt (show (0:ℝ) ≤ j+1 by positivity)
-  have h0 := Real.sq_sqrt (show (0:ℝ) ≤ j by positivity)
-  nlinarith [sq_nonneg (Real.sqrt (j+1)-Real.sqrt j)]
-
-lemma sum_inv_sqrt (J : ℕ) :
-    (∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1)) ≤ 2*Real.sqrt J := by
-  have h (n : ℕ) : (∑ j ∈ Finset.range n, 1 / Real.sqrt (j+1)) ≤ 2*Real.sqrt n := by
-    induction n with
-    | zero => simp
-    | succ n ih =>
-      rw [Finset.sum_range_succ]
-      have hs := inv_sqrt_step n
-      push_cast
-      linarith
-  rw [Fin.sum_univ_eq_sum_range (fun j : ℕ => (1 / Real.sqrt (j+1) : ℝ)) J]
-  exact h J
-section CycleMGF
-variable {d : ℕ}
-
-lemma butterfly_shortCycle_moment (S : Finset (Card d)) (J A : ℕ) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      ((Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J)).descFactorial A : ℝ)) ≤
-      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))^A := by
-  classical
-  simp_rw [shortCycle_factorial_eq,Nat.cast_sum,finiteMean_sum]
-  calc
-    _ ≤ ∑ r : Fin A → Fin J,
-        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-          ∏ i : Fin A, (1 / Real.sqrt ((r i : ℕ)+1)) := by
-      apply Finset.sum_le_sum
-      intro r _
-      exact butterfly_cycleSlots_moment_all S (fun i => (r i : ℕ))
-    _ = _ := by
-      rw [←Finset.mul_sum,mul_pow]
-      congr 1
-      rw [←Fintype.prod_sum (fun (_ : Fin A) (j : Fin J) => (1 / Real.sqrt ((j:ℕ)+1) : ℝ))]
-      simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fin]
-
-noncomputable def cycleMass (p : Equiv.Perm (Card d)) (S : Finset (Card d)) : ℕ :=
-  ∑ C : SelectedCycle p S, C.val.card
-
-lemma cycleMass_le (p : Equiv.Perm (Card d)) (S : Finset (Card d)) : cycleMass p S ≤ S.card := by
-  classical
-  let f : (Σ C : SelectedCycle p S, {x : Card d // x ∈ C.val}) → S :=
-    fun c => ⟨c.2.val,c.1.property.2 c.2.property⟩
-  have hi : Function.Injective f := by
-    intro c e h
-    have hx : c.2.val = e.2.val := congrArg Subtype.val h
-    have hC : c.1 = e.1 := c.1.eq_of_mem e.1 c.2.property (hx ▸ e.2.property)
-    rcases c with ⟨c,x⟩
-    rcases e with ⟨e,y⟩
-    dsimp only at hC
-    subst e
-    have he : x=y := Subtype.ext hx
-    cases he
-    rfl
-  have hh := Fintype.card_le_of_injective f hi
-  simpa only [Fintype.card_sigma,Fintype.card_coe,cycleMass] using hh
-
-lemma shortCycle_card_le (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
-    Fintype.card (ShortCycle p S J) ≤ S.card := by
-  classical
-  calc
-    _ ≤ Fintype.card (SelectedCycle p S) := Fintype.card_subtype_le _
-    _ = ∑ _ : SelectedCycle p S, 1 := by simp
-    _ ≤ cycleMass p S := Finset.sum_le_sum (fun C _ => C.nonempty.card_pos)
-    _ ≤ _ := cycleMass_le p S
-
-lemma butterfly_shortCycle_mgf (S : Finset (Card d)) (J : ℕ) (q : ℝ) (hq : 1 ≤ q) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      q^(Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J))) ≤
-      Real.exp ((q-1)*((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))) := by
-  apply factorial_mgf _ S.card _ q (fun ω => shortCycle_card_le _ _ _)
-  · positivity
-  · exact hq
-  · exact butterfly_shortCycle_moment S J
-
-end CycleMGF
-
-section FullCycleMGF
-variable {d : ℕ}
-
-abbrev LongCycle (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :=
-  {C : SelectedCycle p S // ¬ C.val.card ≤ J}
-noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
-    Fintype (LongCycle p S J) := Fintype.ofFinite _
-
-lemma selectedCycle_card_split (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
-    Fintype.card (SelectedCycle p S) =
-      Fintype.card (ShortCycle p S J) + Fintype.card (LongCycle p S J) := by
-  classical
-  rw [←Fintype.card_congr (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J)),Fintype.card_sum]
-
-lemma longCycle_card_bound (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
-    J * Fintype.card (LongCycle p S J) ≤ S.card := by
-  classical
-  have hs := Equiv.sum_comp (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J))
-    (fun C : SelectedCycle p S => C.val.card)
-  rw [Fintype.sum_sum_type] at hs
-  have h0 : J * Fintype.card (LongCycle p S J) ≤
-      ∑ C : LongCycle p S J, C.val.val.card := by
-    calc
-      _ = ∑ _C : LongCycle p S J, J := by simp [mul_comm]
-      _ ≤ _ := Finset.sum_le_sum (fun C _ => Nat.le_of_lt (Nat.lt_of_not_ge C.property))
-  have hh := cycleMass_le p S
-  unfold cycleMass at hh
-  rw [←hs] at hh
-  exact h0.trans ((Nat.le_add_left _ _).trans hh)
-
-lemma finiteMean_pos {Ω : Type*} [Fintype Ω] [Nonempty Ω] {f : Ω → ℝ}
-    (h : ∀ ω, 0 < f ω) : 0 < finiteMean f := by
-  apply div_pos
-  · exact Finset.sum_pos (fun ω _ => h ω) Finset.univ_nonempty
-  · exact Nat.cast_pos.mpr Fintype.card_pos
-
-lemma longCycle_pow_bound (p : Equiv.Perm (Card d)) (S : Finset (Card d))
-    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    q^(Fintype.card (LongCycle p S J)) ≤ Real.exp ((S.card : ℝ)/J * Real.log q) := by
-  have hq0 : 0 < q := lt_of_lt_of_le zero_lt_one hq
-  have hc : (Fintype.card (LongCycle p S J) : ℝ) ≤ (S.card : ℝ)/J := by
-    apply (le_div_iff₀ (by exact_mod_cast hJ)).mpr
-    have hh := longCycle_card_bound p S J
-    exact_mod_cast (by simpa only [mul_comm] using hh)
-  calc
-    _ = Real.exp ((Fintype.card (LongCycle p S J) : ℝ)*Real.log q) := by
-      rw [Real.exp_nat_mul,Real.exp_log hq0]
-    _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right hc (Real.log_nonneg hq))
-
-lemma butterfly_cycle_log_mgf (S : Finset (Card d)) (J : ℕ) (hJ : 0 < J)
-    (q : ℝ) (hq : 1 ≤ q) :
-    Real.log (finiteMean (fun ω : SwitchIndex d → Bool =>
-      q^(Fintype.card (SelectedCycle (butterflyPerm d (decodeButterfly d ω)) S)))) ≤
-      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
-  let lam : ℝ := (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1)
-  let c : ℝ := (S.card : ℝ)/J * Real.log q
-  have hnum : (q-1)*lam ≤ 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
-    have hsum := sum_inv_sqrt J
-    have hp : 0 ≤ (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) := by positivity
-    have h0 : 0 ≤ 2*Real.sqrt J := by positivity
-    have hh := mul_le_mul_of_nonneg_left hsum hp
-    have hq1 : 0 ≤ q-1 := sub_nonneg.mpr hq
-    calc
-      _ ≤ (q-1)*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
-        mul_le_mul_of_nonneg_left hh hq1
-      _ ≤ q*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
-        mul_le_mul_of_nonneg_right (by linarith) (mul_nonneg hp h0)
-      _ = _ := by ring
-  have hm : finiteMean (fun ω : SwitchIndex d → Bool =>
-      q^(Fintype.card (SelectedCycle (butterflyPerm d (decodeButterfly d ω)) S))) ≤
-      Real.exp (c+(q-1)*lam) := by
-    calc
-      _ ≤ finiteMean (fun ω : SwitchIndex d → Bool =>
-          q^(Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J))*Real.exp c) := by
-        apply finiteMean_mono
-        intro ω
-        rw [selectedCycle_card_split _ _ J,pow_add]
-        exact mul_le_mul_of_nonneg_left (longCycle_pow_bound _ S J hJ q hq) (by positivity)
-      _ = finiteMean (fun ω : SwitchIndex d → Bool =>
-          q^(Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J))) * Real.exp c :=
-        finiteMean_mul_const _ _
-      _ ≤ Real.exp ((q-1)*lam) * Real.exp c :=
-        mul_le_mul_of_nonneg_right (butterfly_shortCycle_mgf S J q hq) (Real.exp_nonneg _)
-      _ = _ := by rw [←Real.exp_add,add_comm]
-  have hpos : 0 < finiteMean (fun ω : SwitchIndex d → Bool =>
-      q^(Fintype.card (SelectedCycle (butterflyPerm d (decodeButterfly d ω)) S))) := by
-    apply finiteMean_pos
-    intro ω
-    positivity
-  have hlog := Real.log_le_log hpos hm
-  rw [Real.log_exp] at hlog
-  dsimp [c] at hlog
-  linarith
-
-end FullCycleMGF
-
-structure RoutingNetwork (α ι : Type*) [Fintype α] [DecidableEq α]
-    [Fintype ι] [DecidableEq ι] (d : ℕ) where
-  perm : (ι → Bool) → Equiv.Perm α
-  path : (ι → Bool) → α → Finset ι
-  users : (ι → Bool) → ι → α × α
-  users_ne : ∀ ω i, (users ω i).1 ≠ (users ω i).2
-  mem_path : ∀ ω x i, i ∈ path ω x ↔ (users ω i).1 = x ∨ (users ω i).2 = x
-  card_path : ∀ ω x, (path ω x).card = d
-  entropy : ∀ ω (S : Finset α),
-    (∑ i : ι, if (users ω i).1 ∈ S ∧ (users ω i).2 ∈ S then (1:ℝ) else 0) ≤
-      (S.card : ℝ) * Real.log S.card / (2 * Real.log 2)
-  follow : α → (α → QueryTree ι) → QueryTree ι
-  close : α → α → QueryTree ι → QueryTree ι
-  follow_succeeds : ∀ x k ω, (follow x k).succeeds ω = (k (perm ω x)).succeeds ω
-  follow_raw_charge : ∀ x k ω, (follow x k).charge ω = (k (perm ω x)).charge ω
-  follow_stale_charge : ∀ x k R ω,
-    (follow x k).staleCharge R ω = (k (perm ω x)).staleCharge (R ∪ path ω x) ω
-  close_succeeds : ∀ x y k ω,
-    (close x y k).succeeds ω = if perm ω x = y then k.succeeds ω else false
-  close_raw_charge : ∀ x y k ω, perm ω x = y → (close x y k).charge ω = k.charge ω + d
-  close_stale_charge : ∀ x y k R ω, perm ω x = y →
-    (close x y k).staleCharge R ω = k.staleCharge (R ∪ path ω x) ω + (path ω x ∩ R).card
-
-namespace RoutingNetwork
-variable {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι] [DecidableEq ι]
-  {d : ℕ} (R : RoutingNetwork α ι d)
-
-lemma card_switches_using (ω : ι → Bool) (x : α) :
-    (Finset.univ.filter (fun i : ι =>
-      (R.users ω i).1 = x ∨ (R.users ω i).2 = x)).card = d := by
-  classical
-  have he : Finset.univ.filter (fun i : ι =>
-      (R.users ω i).1 = x ∨ (R.users ω i).2 = x) = R.path ω x := by
-    ext i
-    simp only [Finset.mem_filter,Finset.mem_univ,true_and,R.mem_path]
-  rw [he,R.card_path]
-lemma switch_incidence_sum (b : ι → Bool) (w : α → ℝ) :
-    ∑ i : ι, (w (R.users b i).1 + w (R.users b i).2) =
-      (d : ℝ) * ∑ x : α, w x := by
-  classical
-  calc
-    _ = ∑ i : ι, ∑ x : α,
-        if (R.users b i).1 = x ∨ (R.users b i).2 = x then w x else 0 := by
-      apply Finset.sum_congr rfl
-      intro i _
-      rw [Finset.sum_ite]
-      simp only [Finset.sum_const_zero, add_zero]
-      have he : Finset.univ.filter (fun x : α =>
-          (R.users b i).1 = x ∨ (R.users b i).2 = x) =
-          {(R.users b i).1, (R.users b i).2} := by
-        ext x; simp [eq_comm]
-      rw [he, Finset.sum_pair (R.users_ne b i)]
-    _ = ∑ x : α, ∑ i : ι,
-        if (R.users b i).1 = x ∨ (R.users b i).2 = x then w x else 0 :=
-      Finset.sum_comm
-    _ = ∑ x : α, (d : ℝ) * w x := by
-      apply Finset.sum_congr rfl
-      intro x _
-      rw [Finset.sum_ite]
-      simp only [Finset.sum_const_zero, add_zero, Finset.sum_const, nsmul_eq_mul,
-        card_switches_using]
-    _ = _ := by rw [Finset.mul_sum]
-
-lemma weighted_switch_min_bound (b : ι → Bool) (w : α → ℝ) :
-    ∑ i : ι, min (w (R.users b i).1) (w (R.users b i).2) ≤
-      (d : ℝ) / 2 * ∑ x : α, w x := by
-  have h : 2 * (∑ i : ι,
-      min (w (R.users b i).1) (w (R.users b i).2)) ≤
-      ∑ i : ι, (w (R.users b i).1 + w (R.users b i).2) := by
-    rw [Finset.mul_sum]
-    apply Finset.sum_le_sum
-    intro i _
-    linarith [min_le_left (w (R.users b i).1) (w (R.users b i).2),
-      min_le_right (w (R.users b i).1) (w (R.users b i).2)]
-  rw [switch_incidence_sum] at h
-  linarith
-section Rotation
-variable {A : ℕ}
-
-def cycleFiber (o : α → Option (Fin A)) (i : Fin A) : Finset (α) := by
-  classical
-  exact Finset.univ.filter (fun x => o x = some i)
-
-omit [DecidableEq α] in
-@[simp] lemma mem_cycleFiber (o : α → Option (Fin A)) (i : Fin A) (x : α) :
-    x ∈ cycleFiber o i ↔ o x = some i := by
-  classical
-  simp [cycleFiber]
-
-abbrev Terminal (o : α → Option (Fin A)) := ∀ i : Fin A, {x : α // o x = some i}
-
-noncomputable def slotWeight (o : α → Option (Fin A)) (i : Fin A) : ℝ :=
-  1 / (cycleFiber o i).card
-
-noncomputable def pathWeight (o : α → Option (Fin A)) (x : α) : ℝ :=
-  match o x with
-  | none => 0
-  | some i => slotWeight o i
-
-noncomputable def withinWeight (o : α → Option (Fin A)) (u v : α) : ℝ :=
-  match o u, o v with
-  | some i, some j => if i = j then slotWeight o i else 0
-  | _, _ => 0
-
-noncomputable def rotationMeeting (o : α → Option (Fin A)) (z : Terminal o)
-    (u v : α) : ℝ :=
-  match o u, o v with
-  | some i, some j => if i < j then (if (z j).val = v then 1 else 0)
-      else if j < i then (if (z i).val = u then 1 else 0)
-      else (if (z i).val = u then 1 else 0) + (if (z i).val = v then 1 else 0)
-  | _, _ => 0
-
-omit [DecidableEq α] in
-lemma card_cycleFiber (o : α → Option (Fin A)) (i : Fin A) :
-    Fintype.card {x : α // o x = some i} = (cycleFiber o i).card := by
-  classical
-  simp [cycleFiber,Fintype.card_subtype]
-
-lemma terminal_mean {o : α → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
-    (i : Fin A) (x : α) (hx : o x = some i) :
-    finiteMean (fun z : Terminal o => if (z i).val = x then (1:ℝ) else 0) = slotWeight o i := by
-  classical
-  have hev := finiteMean_pi_eval (A := fun i : Fin A => {x : α // o x = some i})
-    i (fun z => if z.val = x then (1:ℝ) else 0)
-  rw [hev]
-  have he : (fun z : {x : α // o x = some i} => if z.val = x then (1:ℝ) else 0) =
-      (fun z => if z = ⟨x,hx⟩ then 1 else 0) := by
-    funext z
-    simp only [Subtype.ext_iff]
-  rw [he,finiteMean_eq_indicator,card_cycleFiber,slotWeight]
-
-omit [DecidableEq α] in
-lemma slotWeight_antitone (o : α → Option (Fin A))
-    (hn : ∀ i, (cycleFiber o i).Nonempty)
-    (hm : Monotone (fun i => (cycleFiber o i).card)) : Antitone (slotWeight o) := by
-  intro i j hij
-  unfold slotWeight
-  apply one_div_le_one_div_of_le
-  · exact_mod_cast Finset.card_pos.mpr (hn i)
-  · exact_mod_cast hm hij
-
-lemma rotationMeeting_mean {o : α → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
-    (hm : Monotone (fun i => (cycleFiber o i).card)) (u v : α) :
-    finiteMean (fun z : Terminal o => rotationMeeting o z u v) =
-      min (pathWeight o u) (pathWeight o v) + withinWeight o u v := by
-  classical
-  have hn : ∀ i, (cycleFiber o i).Nonempty := by
-    intro i
-    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : α // o x = some i})
-    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
-  have hw := slotWeight_antitone o hn hm
-  cases hu : o u with
-  | none =>
-    cases hv : o v with
-    | none => simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean]
-    | some j =>
-      have hj : 0 ≤ slotWeight o j := by unfold slotWeight; positivity
-      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_left hj]
-  | some i =>
-    cases hv : o v with
-    | none =>
-      have hi : 0 ≤ slotWeight o i := by unfold slotWeight; positivity
-      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_right hi]
-    | some j =>
-      simp only [rotationMeeting,pathWeight,withinWeight,hu,hv]
-      rcases lt_trichotomy i j with hij | rfl | hji
-      · simp only [hij,↓reduceIte,ne_of_lt hij]
-        rw [terminal_mean j v hv,min_eq_right (hw (le_of_lt hij)),add_zero]
-      · simp only [lt_self_iff_false,↓reduceIte,min_self]
-        rw [finiteMean_add,terminal_mean i u hu,terminal_mean i v hv]
-      · simp only [not_lt_of_ge (le_of_lt hji),hji,↓reduceIte,ne_of_gt hji]
-        rw [terminal_mean i u hu,min_eq_left (hw (le_of_lt hji)),add_zero]
-
-omit [DecidableEq α] in
-lemma pathWeight_sum (o : α → Option (Fin A)) (hn : ∀ i, (cycleFiber o i).Nonempty) :
-    ∑ x : α, pathWeight o x = (A : ℝ) := by
-  classical
-  have he (x : α) : pathWeight o x =
-      ∑ i : Fin A, if o x = some i then slotWeight o i else 0 := by
-    cases hx : o x with
-    | none => simp [pathWeight,hx]
-    | some j => simp [pathWeight,hx]
-  simp only [he]
-  rw [Finset.sum_comm]
-  have hf (i : Fin A) : (∑ x : α, if o x = some i then slotWeight o i else 0) = 1 := by
-    rw [← Finset.sum_filter]
-    change (∑ _x ∈ cycleFiber o i, slotWeight o i) = 1
-    simp only [Finset.sum_const,nsmul_eq_mul,slotWeight]
-    exact mul_one_div_cancel (by exact_mod_cast (Finset.card_pos.mpr (hn i)).ne')
-  simp only [hf,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,mul_one]
-
-lemma withinWeight_sum_bound (o : α → Option (Fin A))
-    (hn : ∀ i, (cycleFiber o i).Nonempty) (b : ι → Bool) :
-    (∑ s : ι, withinWeight o (R.users b s).1 (R.users b s).2) ≤
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
-  classical
-  have he (u v : α) : withinWeight o u v =
-      ∑ i : Fin A, (if u ∈ cycleFiber o i ∧ v ∈ cycleFiber o i then (1:ℝ) else 0) * slotWeight o i := by
-    simp only [mem_cycleFiber]
-    cases hu : o u <;> cases hv : o v <;> simp [withinWeight,hu,hv,ite_and,eq_comm]
-  simp only [he]
-  rw [Finset.sum_comm]
-  apply Finset.sum_le_sum
-  intro i _
-  rw [← Finset.sum_mul]
-  have hi := R.entropy b (cycleFiber o i)
-  have hpos : (0:ℝ) < (cycleFiber o i).card := by exact_mod_cast Finset.card_pos.mpr (hn i)
-  have hmul := mul_le_mul_of_nonneg_right hi (show 0 ≤ slotWeight o i by unfold slotWeight; positivity)
-  calc
-    _ ≤ ((cycleFiber o i).card : ℝ) * Real.log (cycleFiber o i).card /
-        (2 * Real.log 2) * slotWeight o i := hmul
-    _ = _ := by unfold slotWeight; field_simp
-
-noncomputable def rotationCharge (o : α → Option (Fin A)) (b : ι → Bool)
-    (z : Terminal o) : ℝ :=
-  ∑ s : ι, rotationMeeting o z (R.users b s).1 (R.users b s).2
-
-lemma rotationCharge_mean_bound {o : α → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
-    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : ι → Bool) :
-    finiteMean (rotationCharge R o b) ≤ (A:ℝ) * d / 2 +
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
-  classical
-  have hn : ∀ i, (cycleFiber o i).Nonempty := by
-    intro i
-    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : α // o x = some i})
-    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
-  unfold rotationCharge
-  rw [finiteMean_sum]
-  simp only [rotationMeeting_mean hm,Finset.sum_add_distrib]
-  have h₁ := weighted_switch_min_bound R b (pathWeight o)
-  rw [pathWeight_sum o hn] at h₁
-  have h₂ := withinWeight_sum_bound R o hn b
-  linarith
-
-lemma rotation_exp_bound {o : α → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
-    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : ι → Bool) :
-    Real.exp (-Real.log 2 * ((A:ℝ)*d/2 +
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2))) ≤
-      finiteMean (fun z : Terminal o => Real.exp (-Real.log 2 * rotationCharge R o b z)) := by
-  have hb := rotationCharge_mean_bound R hm b
-  have hlog : 0 ≤ Real.log 2 := Real.log_nonneg (by norm_num)
-  have hm' : -Real.log 2 * ((A:ℝ)*d/2 +
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2)) ≤
-      finiteMean (fun z : Terminal o => -Real.log 2 * rotationCharge R o b z) := by
-    have he := finiteMean_mul_const (rotationCharge R o b) (-Real.log 2)
-    simp only [mul_comm] at he
-    rw [he]
-    exact mul_le_mul_of_nonpos_left hb (neg_nonpos.mpr hlog)
-  exact (Real.exp_le_exp.mpr hm').trans (exp_finiteMean_le _)
-
-end Rotation
-
-namespace QueryTree
-open Thorp.QueryTree
-
-noncomputable def seekCycle (R : RoutingNetwork α ι d) (S : Finset (α)) (a : α) :
-    ℕ → α → Finset (α) → (Finset (α) → QueryTree (ι)) →
-      QueryTree (ι)
-  | 0, x, V, k => if x ∈ S ∧ x ∉ V then R.close x a (k (insert x V)) else reject
-  | r + 1, x, V, k => if x ∈ S ∧ x ∉ V then R.follow x
-      (fun y => seekCycle R S a r y (insert x V) k) else reject
-
-lemma seekCycle_raw_charge (S : Finset (α)) (a : α) (r : ℕ)
-    (x : α) (V : Finset (α)) (k : Finset (α) → QueryTree (ι))
-    (ω : ι → Bool) (K : ℕ)
-    (hK : ∀ V, (k V).succeeds ω = true → (k V).charge ω = K)
-    (hs : (seekCycle R S a r x V k).succeeds ω = true) :
-    (seekCycle R S a r x V k).charge ω = K + d := by
-  induction r generalizing x V with
-  | zero =>
-    by_cases hv : x ∈ S ∧ x ∉ V
-    · rw [seekCycle,ite_eq_left hv,R.close_succeeds] at hs
-      split_ifs at hs with he
-      · rw [seekCycle,ite_eq_left hv,R.close_raw_charge _ _ _ _ he,hK _ hs]
-    · simp [seekCycle,hv,Thorp.QueryTree.succeeds] at hs
-  | succ r ih =>
-    by_cases hv : x ∈ S ∧ x ∉ V
-    · rw [seekCycle,ite_eq_left hv,R.follow_succeeds] at hs
-      rw [seekCycle,ite_eq_left hv,R.follow_raw_charge]
-      exact ih _ _ hs
-    · simp [seekCycle,hv,Thorp.QueryTree.succeeds] at hs
-
-noncomputable def seekCycles (R : RoutingNetwork α ι d) (S : Finset (α)) :
-    List (α × ℕ) → Finset (α) → QueryTree (ι)
-  | [], _ => accept
-  | (a,r) :: L, V => seekCycle R S a r a V (fun V => seekCycles R S L V)
-
-lemma seekCycles_raw_charge (S : Finset (α)) (L : List (α × ℕ))
-    (V : Finset (α)) (ω : ι → Bool)
-    (hs : (seekCycles R S L V).succeeds ω = true) :
-    (seekCycles R S L V).charge ω = L.length * d := by
-  induction L generalizing V with
-  | nil => simp [seekCycles,charge]
-  | cons ar L ih =>
-    obtain ⟨a,r⟩ := ar
-    rw [seekCycles]
-    rw [seekCycle_raw_charge R S a r a V (fun V => seekCycles R S L V) ω (L.length*d)
-      (fun V hs => ih V hs) hs]
-    simp [Nat.add_mul]
-
-lemma cycles_weighted_success (S : Finset (α)) (L : List (α × ℕ)) :
-    finiteMean (fun ω : ι → Bool =>
-      if (seekCycles R S L ∅).succeeds ω then
-        ((2:ℝ)^d)^L.length / (2:ℝ)^(seekCycles R S L ∅).staleCharge ∅ ω else 0) ≤ 1 := by
-  let T := seekCycles R S L ∅
-  have hh := mean_freshCharge_le_one T
-  have he : finiteMean (fun ω : ι → Bool =>
-      if T.succeeds ω then (2:ℝ)^T.freshCharge ∅ ω else 0) =
-      finiteMean (fun ω : ι → Bool =>
-        if T.succeeds ω then ((2:ℝ)^d)^L.length / (2:ℝ)^T.staleCharge ∅ ω else 0) := by
-    apply finiteMean_congr
-    intro ω
-    cases hs : T.succeeds ω
-    · simp
-    · simp only [↓reduceIte]
-      have hc := fresh_add_stale T ∅ ω
-      rw [seekCycles_raw_charge R S L ∅ ω hs] at hc
-      have hp : ((2:ℝ)^d)^L.length = (2:ℝ)^T.freshCharge ∅ ω * (2:ℝ)^T.staleCharge ∅ ω := by
-        rw [←pow_add,hc,←pow_mul,Nat.mul_comm]
-      rw [hp,mul_div_cancel_right₀ _ (pow_ne_zero _ (by norm_num))]
-  rw [← he]
-  exact hh
-
-end QueryTree
-noncomputable def exposedPaths (b : ι → Bool) (V : Finset (α)) :
-    Finset (ι) :=
-  V.biUnion (fun x => R.path b x)
-
-@[simp] lemma exposedPaths_empty (b : ι → Bool) : exposedPaths R b ∅ = ∅ := by
-  classical
-  simp [exposedPaths]
-
-lemma exposedPaths_insert (b : ι → Bool) (V : Finset (α)) (x : α) :
-    exposedPaths R b (insert x V) = exposedPaths R b V ∪ R.path b x := by
-  classical
-  simp [exposedPaths,Finset.union_comm]
-
-noncomputable def terminalCharge (b : ι → Bool) (z : α) (V : Finset (α)) : ℕ :=
-  (R.path b z ∩ exposedPaths R b V).card
-
-namespace QueryTree
-open Thorp.QueryTree
-
-lemma seekCycle_semantics (S : Finset (α)) (a : α) (r : ℕ)
-    (x : α) (V W : Finset (α)) (z : α)
-    (k : Finset (α) → QueryTree (ι)) (ω : ι → Bool)
-    (h : cycleTrace (R.perm ω) S a r x V = some (W,z)) :
-    (seekCycle R S a r x V k).succeeds ω = (k W).succeeds ω ∧
-      (seekCycle R S a r x V k).staleCharge (exposedPaths R (ω) V) ω =
-        (k W).staleCharge (exposedPaths R (ω) W) ω +
-          terminalCharge R (ω) z (W.erase z) := by
-  classical
-  induction r generalizing x V with
-  | zero =>
-    simp only [cycleTrace] at h
-    split_ifs at h with hv
-    obtain ⟨hxS,hxV,hxa⟩ := hv
-    obtain ⟨rfl,rfl⟩ := Prod.mk.inj (Option.some.inj h)
-    simp only [seekCycle,ite_eq_left (show x ∈ S ∧ x ∉ V from ⟨hxS,hxV⟩),R.close_succeeds,ite_eq_left hxa]
-    constructor
-    · trivial
-    · rw [R.close_stale_charge _ _ _ _ _ hxa,exposedPaths_insert,
-        Finset.erase_insert hxV]
-      simp only [terminalCharge]
-  | succ r ih =>
-    simp only [cycleTrace] at h
-    split_ifs at h with hv
-    obtain ⟨hxS,hxV⟩ := hv
-    rw [seekCycle,ite_eq_left ⟨hxS,hxV⟩,R.follow_succeeds,R.follow_stale_charge]
-    rw [← exposedPaths_insert]
-    exact ih _ _ h
-
-lemma seekCycle_orbit (S V : Finset (α)) (z : α)
-    (k : Finset (α) → QueryTree (ι)) (ω : ι → Bool)
-    (hS : orbitSet (R.perm ω) z ⊆ S)
-    (hV : Disjoint V (orbitSet (R.perm ω) z)) :
-    let p := R.perm ω
-    let W := V ∪ orbitSet p z
-    (seekCycle R S (p z) (Function.minimalPeriod p z-1) (p z) V k).succeeds ω = (k W).succeeds ω ∧
-      (seekCycle R S (p z) (Function.minimalPeriod p z-1) (p z) V k).staleCharge
-          (exposedPaths R (ω) V) ω =
-        (k W).staleCharge (exposedPaths R (ω) W) ω +
-          terminalCharge R (ω) z (W.erase z) := by
-  exact seekCycle_semantics _ _ _ _ _ _ _ _ _ _ (cycleTrace_orbit _ _ _ _ hS hV)
-
-end QueryTree
-
-end RoutingNetwork
-end Thorp
-
-end ThorpNine.SparseSaving
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.SparseSaving.RotationMeeting
+
+namespace OAI
+
+namespace ThorpNine.SparseSaving
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+section CycleCounting
+variable {d A : ℕ}
+
+lemma assignmentOwner_eq_some {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) (x : Card d) (i : Fin A) :
+    assignmentOwner f x = some i ↔ x ∈ (f.val i).val := by
+  classical
+  unfold assignmentOwner
+  split_ifs with h
+  · constructor
+    · intro hi
+      have hx := Classical.choose_spec h
+      rw [Option.some.inj hi] at hx
+      exact hx
+    · intro hx
+      have he := (f.val (Classical.choose h)).eq_of_mem (f.val i) (Classical.choose_spec h) hx
+      exact congrArg some (f.property.1 he)
+  · constructor
+    · intro hi; contradiction
+    · intro hx; exact (h ⟨i,hx⟩).elim
+
+lemma assignmentOwner_fiber {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) (i : Fin A) : cycleFiber (assignmentOwner f) i = (f.val i).val := by
+  ext x
+  simp only [mem_cycleFiber,assignmentOwner_eq_some]
+
+noncomputable def assignmentToSlots {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) : CycleSlots p S r :=
+  ⟨assignmentOwner f,by
+    refine ⟨?_,?_,?_⟩
+    · intro i
+      rw [assignmentOwner_fiber]
+      exact f.property.2 i
+    · intro i
+      rw [assignmentOwner_fiber]
+      exact (f.val i).property.2
+    · intro i x hx
+      rw [assignmentOwner_fiber]
+      exact (f.val i).orbit_eq ((assignmentOwner_eq_some f x i).mp hx)⟩
+
+noncomputable def cycleSlotsEquivAssignments (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
+    CycleSlots p S r ≃ SlotAssignment p S r where
+  toFun := slotsToAssignment
+  invFun := assignmentToSlots
+  left_inv c := by
+    apply Subtype.ext
+    apply owner_ext
+    intro i
+    rw [assignmentToSlots,assignmentOwner_fiber]
+    rfl
+  right_inv f := by
+    apply Subtype.ext
+    funext i
+    apply Subtype.ext
+    exact assignmentOwner_fiber f i
+
+end CycleCounting
+
+section GroupedCounting
+variable {d A : ℕ}
+
+abbrev LengthClass (r : Fin A → ℕ) := {j : ℕ // j ∈ Finset.univ.image (fun i => r i + 1)}
+
+def slotClass (r : Fin A → ℕ) (i : Fin A) : LengthClass r :=
+  ⟨r i+1, Finset.mem_image.mpr ⟨i, Finset.mem_univ _, rfl⟩⟩
+
+abbrev SlotsOfLength (r : Fin A → ℕ) (j : LengthClass r) := {i : Fin A // slotClass r i = j}
+abbrev CyclesOfLength (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (j : ℕ) :=
+  {C : SelectedCycle p S // C.val.card = j}
+
+noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (j : ℕ) :
+    Fintype (CyclesOfLength p S j) := Fintype.ofFinite _
+
+abbrev GroupedAssignments (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
+  ∀ j : LengthClass r, SlotsOfLength r j ↪ CyclesOfLength p S j.val
+
+noncomputable def assignmentToGroups {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) : GroupedAssignments p S r := fun _j =>
+  { toFun := fun i => ⟨f.val i.val, (f.property.2 i.val).trans (congrArg Subtype.val i.property)⟩
+    inj' := fun _i _k h => Subtype.ext (f.property.1 (congrArg Subtype.val h)) }
+
+noncomputable def groupsEmbedding {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (F : GroupedAssignments p S r) : Fin A ↪ SelectedCycle p S :=
+  ((Equiv.sigmaFiberEquiv (slotClass r)).symm.toEmbedding.trans
+    (Function.Embedding.sigmaMap (Function.Embedding.refl _) F)).trans
+    ((Function.Embedding.sigmaMap (Function.Embedding.subtype _) (fun _ => Function.Embedding.refl _)).trans
+      (Equiv.sigmaFiberEquiv (fun C : SelectedCycle p S => C.val.card)).toEmbedding)
+
+noncomputable def groupsToAssignment {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (F : GroupedAssignments p S r) : SlotAssignment p S r :=
+  ⟨groupsEmbedding F, (groupsEmbedding F).injective,
+    fun i => (F (slotClass r i) ⟨i,rfl⟩).property⟩
+
+noncomputable def assignmentEquivGroups (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
+    SlotAssignment p S r ≃ GroupedAssignments p S r where
+  toFun := assignmentToGroups
+  invFun := groupsToAssignment
+  left_inv f := by
+    apply Subtype.ext
+    rfl
+  right_inv F := by
+    funext j
+    apply Function.Embedding.ext
+    intro i
+    rcases i with ⟨i,hi⟩
+    subst j
+    rfl
+
+lemma card_cycleSlots_factorial (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
+    Fintype.card (CycleSlots p S r) =
+      ∏ j : LengthClass r, (Fintype.card (CyclesOfLength p S j.val)).descFactorial
+        (Fintype.card (SlotsOfLength r j)) := by
+  classical
+  rw [Fintype.card_congr (cycleSlotsEquivAssignments p S r),
+    Fintype.card_congr (assignmentEquivGroups p S r), Fintype.card_pi]
+  simp only [Fintype.card_embedding_eq]
+
+lemma butterfly_factorial_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+       (∏ j : LengthClass r,
+         (Fintype.card (CyclesOfLength (butterflyPerm d (decodeButterfly d ω)) S j.val)).descFactorial
+           (Fintype.card (SlotsOfLength r j)) : ℕ)) ≤
+      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+        ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
+  simpa only [←card_cycleSlots_factorial] using butterfly_cycleSlots_moment S r hr
+
+end GroupedCounting
+
+section AllLengths
+variable {d A : ℕ}
+
+noncomputable def assignmentRelabel (p : Equiv.Perm (Card d)) (S : Finset (Card d))
+    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
+    SlotAssignment p S r ≃ SlotAssignment p S (r ∘ e) where
+  toFun f := ⟨f.val ∘ e, f.property.1.comp e.injective, fun i => f.property.2 (e i)⟩
+  invFun f := ⟨f.val ∘ e.symm, f.property.1.comp e.symm.injective, fun i => by
+    simpa only [Function.comp_apply,Equiv.apply_symm_apply] using f.property.2 (e.symm i)⟩
+  left_inv f := by apply Subtype.ext; funext i; simp
+  right_inv f := by apply Subtype.ext; funext i; simp
+
+lemma card_cycleSlots_relabel (p : Equiv.Perm (Card d)) (S : Finset (Card d))
+    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
+    Fintype.card (CycleSlots p S r) = Fintype.card (CycleSlots p S (r ∘ e)) := by
+  exact Fintype.card_congr ((cycleSlotsEquivAssignments p S r).trans
+    ((assignmentRelabel p S r e).trans (cycleSlotsEquivAssignments p S (r ∘ e)).symm))
+
+lemma butterfly_cycleSlots_moment_all (S : Finset (Card d)) (r : Fin A → ℕ) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) ≤
+        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
+  have h := butterfly_cycleSlots_moment S (r ∘ Tuple.sort r) (Tuple.monotone_sort r)
+  simp_rw [←card_cycleSlots_relabel] at h
+  have hp := Equiv.prod_comp (Tuple.sort r) (fun i : Fin A => (1 / Real.sqrt (r i+1) : ℝ))
+  simpa only [Function.comp_apply,hp] using h
+
+abbrev ShortCycle (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :=
+  {C : SelectedCycle p S // C.val.card ≤ J}
+
+noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
+    Fintype (ShortCycle p S J) := Fintype.ofFinite _
+
+noncomputable def shortCycleLength {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {J : ℕ}
+    (C : ShortCycle p S J) : Fin J :=
+  ⟨C.val.val.card-1, by have := C.val.nonempty.card_pos; have := C.property; omega⟩
+
+lemma shortCycleLength_add_one {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {J : ℕ}
+    (C : ShortCycle p S J) : (shortCycleLength C : ℕ)+1 = C.val.val.card := by
+  have := C.val.nonempty.card_pos
+  simp only [shortCycleLength]
+  omega
+
+noncomputable def shortEmbeddingEquiv (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
+    (Fin A ↪ ShortCycle p S J) ≃
+      Σ r : Fin A → Fin J, SlotAssignment p S (fun i => (r i : ℕ)) where
+  toFun f := ⟨fun i => shortCycleLength (f i),
+    ⟨fun i => (f i).val, fun i j h => f.injective (Subtype.ext h),
+      fun i => (shortCycleLength_add_one (f i)).symm⟩⟩
+  invFun c :=
+    { toFun := fun i => ⟨c.2.val i, by rw [c.2.property.2]; exact (c.1 i).isLt⟩
+      inj' := fun _i _j h => c.2.property.1 (congrArg Subtype.val h) }
+  left_inv f := by
+    apply Function.Embedding.ext
+    intro i
+    rfl
+  right_inv c := by
+    rcases c with ⟨r,f⟩
+    have he : (fun i => shortCycleLength (⟨f.val i, by rw [f.property.2]; exact (r i).isLt⟩ : ShortCycle p S J)) = r := by
+      funext i
+      apply Fin.ext
+      simp only [shortCycleLength,f.property.2,Nat.add_sub_cancel]
+    apply Sigma.ext he
+    apply (Subtype.heq_iff_coe_eq (by
+      intro x
+      change (Function.Injective x ∧ ∀ i, (x i).val.card = (f.val i).val.card - 1 + 1) ↔ _
+      simp only [f.property.2,Nat.add_sub_cancel])).mpr
+    rfl
+
+lemma shortCycle_factorial_eq (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
+    (Fintype.card (ShortCycle p S J)).descFactorial A =
+      ∑ r : Fin A → Fin J, Fintype.card (CycleSlots p S (fun i => (r i : ℕ))) := by
+  classical
+  have hc := Fintype.card_embedding_eq (α := Fin A) (β := ShortCycle p S J)
+  simp only [Fintype.card_fin] at hc
+  rw [←hc, Fintype.card_congr (shortEmbeddingEquiv p S J),Fintype.card_sigma]
+  apply Finset.sum_congr rfl
+  intro r _
+  exact (Fintype.card_congr (cycleSlotsEquivAssignments p S (fun i => (r i : ℕ)))).symm
+
+end AllLengths
+
+lemma factorial_mgf {Ω : Type*} [Fintype Ω] (F : Ω → ℕ) (k : ℕ) (Λ q : ℝ)
+    (hF : ∀ ω, F ω ≤ k) (hΛ : 0 ≤ Λ) (hq : 1 ≤ q)
+    (hm : ∀ a, finiteMean (fun ω => ((F ω).descFactorial a : ℝ)) ≤ Λ^a) :
+    finiteMean (fun ω => q^(F ω)) ≤ Real.exp ((q-1)*Λ) := by
+  have hc (n a : ℕ) : (n.choose a : ℝ) = (n.descFactorial a : ℝ) / (a.factorial : ℝ) := by
+    rw [Nat.descFactorial_eq_factorial_mul_choose,Nat.cast_mul]
+    exact (mul_div_cancel_left₀ _ (by exact_mod_cast Nat.factorial_ne_zero a)).symm
+  have hex (ω : Ω) : q^(F ω) = ∑ a ∈ Finset.range (k+1),
+      ((F ω).descFactorial a : ℝ) * ((q-1)^a / (a.factorial : ℝ)) := by
+    calc
+      _ = ((q-1)+1)^(F ω) := by congr 1; ring
+      _ = ∑ a ∈ Finset.range (F ω+1), (q-1)^a * (F ω).choose a := by
+        rw [add_pow]
+        simp only [one_pow,mul_one]
+      _ = ∑ a ∈ Finset.range (k+1), (q-1)^a * (F ω).choose a := by
+        apply Finset.sum_subset (Finset.range_mono (Nat.add_le_add_right (hF ω) _))
+        intro a _ ha
+        have hn : F ω < a := by have := (Finset.mem_range.not.mp ha); omega
+        rw [Nat.choose_eq_zero_of_lt hn,Nat.cast_zero,mul_zero]
+      _ = _ := by
+        apply Finset.sum_congr rfl
+        intro a _
+        rw [hc]
+        ring
+  simp_rw [hex,finiteMean_sum,finiteMean_mul_const]
+  calc
+    _ ≤ ∑ a ∈ Finset.range (k+1), Λ^a * ((q-1)^a / (a.factorial : ℝ)) := by
+      apply Finset.sum_le_sum
+      intro a _
+      exact mul_le_mul_of_nonneg_right (hm a) (by positivity)
+    _ = ∑ a ∈ Finset.range (k+1), ((q-1)*Λ)^a / (a.factorial : ℝ) := by
+      apply Finset.sum_congr rfl
+      intro a _
+      rw [mul_pow]
+      ring
+    _ ≤ _ := Real.sum_le_exp_of_nonneg (mul_nonneg (sub_nonneg.mpr hq) hΛ) _
+
+lemma inv_sqrt_step (j : ℕ) :
+    1 / Real.sqrt (j+1) ≤ 2*(Real.sqrt (j+1)-Real.sqrt j) := by
+  have hp : 0 < Real.sqrt (j+1) := Real.sqrt_pos.mpr (by positivity)
+  apply (div_le_iff₀ hp).mpr
+  have h1 := Real.sq_sqrt (show (0:ℝ) ≤ j+1 by positivity)
+  have h0 := Real.sq_sqrt (show (0:ℝ) ≤ j by positivity)
+  nlinarith [sq_nonneg (Real.sqrt (j+1)-Real.sqrt j)]
+
+lemma sum_inv_sqrt (J : ℕ) :
+    (∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1)) ≤ 2*Real.sqrt J := by
+  have h (n : ℕ) : (∑ j ∈ Finset.range n, 1 / Real.sqrt (j+1)) ≤ 2*Real.sqrt n := by
+    induction n with
+    | zero => simp
+    | succ n ih =>
+      rw [Finset.sum_range_succ]
+      have hs := inv_sqrt_step n
+      push_cast
+      linarith
+  rw [Fin.sum_univ_eq_sum_range (fun j : ℕ => (1 / Real.sqrt (j+1) : ℝ)) J]
+  exact h J
+section CycleMGF
+variable {d : ℕ}
+
+lemma butterfly_shortCycle_moment (S : Finset (Card d)) (J A : ℕ) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      ((Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J)).descFactorial A : ℝ)) ≤
+      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))^A := by
+  classical
+  simp_rw [shortCycle_factorial_eq,Nat.cast_sum,finiteMean_sum]
+  calc
+    _ ≤ ∑ r : Fin A → Fin J,
+        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+          ∏ i : Fin A, (1 / Real.sqrt ((r i : ℕ)+1)) := by
+      apply Finset.sum_le_sum
+      intro r _
+      exact butterfly_cycleSlots_moment_all S (fun i => (r i : ℕ))
+    _ = _ := by
+      rw [←Finset.mul_sum,mul_pow]
+      congr 1
+      rw [←Fintype.prod_sum (fun (_ : Fin A) (j : Fin J) => (1 / Real.sqrt ((j:ℕ)+1) : ℝ))]
+      simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fin]
+
+noncomputable def cycleMass (p : Equiv.Perm (Card d)) (S : Finset (Card d)) : ℕ :=
+  ∑ C : SelectedCycle p S, C.val.card
+
+lemma cycleMass_le (p : Equiv.Perm (Card d)) (S : Finset (Card d)) : cycleMass p S ≤ S.card := by
+  classical
+  let f : (Σ C : SelectedCycle p S, {x : Card d // x ∈ C.val}) → S :=
+    fun c => ⟨c.2.val,c.1.property.2 c.2.property⟩
+  have hi : Function.Injective f := by
+    intro c e h
+    have hx : c.2.val = e.2.val := congrArg Subtype.val h
+    have hC : c.1 = e.1 := c.1.eq_of_mem e.1 c.2.property (hx ▸ e.2.property)
+    rcases c with ⟨c,x⟩
+    rcases e with ⟨e,y⟩
+    dsimp only at hC
+    subst e
+    have he : x=y := Subtype.ext hx
+    cases he
+    rfl
+  have hh := Fintype.card_le_of_injective f hi
+  simpa only [Fintype.card_sigma,Fintype.card_coe,cycleMass] using hh
+
+lemma shortCycle_card_le (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
+    Fintype.card (ShortCycle p S J) ≤ S.card := by
+  classical
+  calc
+    _ ≤ Fintype.card (SelectedCycle p S) := Fintype.card_subtype_le _
+    _ = ∑ _ : SelectedCycle p S, 1 := by simp
+    _ ≤ cycleMass p S := Finset.sum_le_sum (fun C _ => C.nonempty.card_pos)
+    _ ≤ _ := cycleMass_le p S
+
+lemma butterfly_shortCycle_mgf (S : Finset (Card d)) (J : ℕ) (q : ℝ) (hq : 1 ≤ q) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      q^(Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J))) ≤
+      Real.exp ((q-1)*((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))) := by
+  apply factorial_mgf _ S.card _ q (fun ω => shortCycle_card_le _ _ _)
+  · positivity
+  · exact hq
+  · exact butterfly_shortCycle_moment S J
+
+end CycleMGF
+
+section FullCycleMGF
+variable {d : ℕ}
+
+abbrev LongCycle (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :=
+  {C : SelectedCycle p S // ¬ C.val.card ≤ J}
+noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
+    Fintype (LongCycle p S J) := Fintype.ofFinite _
+
+lemma selectedCycle_card_split (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
+    Fintype.card (SelectedCycle p S) =
+      Fintype.card (ShortCycle p S J) + Fintype.card (LongCycle p S J) := by
+  classical
+  rw [←Fintype.card_congr (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J)),Fintype.card_sum]
+
+lemma longCycle_card_bound (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
+    J * Fintype.card (LongCycle p S J) ≤ S.card := by
+  classical
+  have hs := Equiv.sum_comp (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J))
+    (fun C : SelectedCycle p S => C.val.card)
+  rw [Fintype.sum_sum_type] at hs
+  have h0 : J * Fintype.card (LongCycle p S J) ≤
+      ∑ C : LongCycle p S J, C.val.val.card := by
+    calc
+      _ = ∑ _C : LongCycle p S J, J := by simp [mul_comm]
+      _ ≤ _ := Finset.sum_le_sum (fun C _ => Nat.le_of_lt (Nat.lt_of_not_ge C.property))
+  have hh := cycleMass_le p S
+  unfold cycleMass at hh
+  rw [←hs] at hh
+  exact h0.trans ((Nat.le_add_left _ _).trans hh)
+
+lemma finiteMean_pos {Ω : Type*} [Fintype Ω] [Nonempty Ω] {f : Ω → ℝ}
+    (h : ∀ ω, 0 < f ω) : 0 < finiteMean f := by
+  apply div_pos
+  · exact Finset.sum_pos (fun ω _ => h ω) Finset.univ_nonempty
+  · exact Nat.cast_pos.mpr Fintype.card_pos
+
+lemma longCycle_pow_bound (p : Equiv.Perm (Card d)) (S : Finset (Card d))
+    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    q^(Fintype.card (LongCycle p S J)) ≤ Real.exp ((S.card : ℝ)/J * Real.log q) := by
+  have hq0 : 0 < q := lt_of_lt_of_le zero_lt_one hq
+  have hc : (Fintype.card (LongCycle p S J) : ℝ) ≤ (S.card : ℝ)/J := by
+    apply (le_div_iff₀ (by exact_mod_cast hJ)).mpr
+    have hh := longCycle_card_bound p S J
+    exact_mod_cast (by simpa only [mul_comm] using hh)
+  calc
+    _ = Real.exp ((Fintype.card (LongCycle p S J) : ℝ)*Real.log q) := by
+      rw [Real.exp_nat_mul,Real.exp_log hq0]
+    _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right hc (Real.log_nonneg hq))
+
+lemma butterfly_cycle_log_mgf (S : Finset (Card d)) (J : ℕ) (hJ : 0 < J)
+    (q : ℝ) (hq : 1 ≤ q) :
+    Real.log (finiteMean (fun ω : SwitchIndex d → Bool =>
+      q^(Fintype.card (SelectedCycle (butterflyPerm d (decodeButterfly d ω)) S)))) ≤
+      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
+  let lam : ℝ := (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1)
+  let c : ℝ := (S.card : ℝ)/J * Real.log q
+  have hnum : (q-1)*lam ≤ 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
+    have hsum := sum_inv_sqrt J
+    have hp : 0 ≤ (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) := by positivity
+    have h0 : 0 ≤ 2*Real.sqrt J := by positivity
+    have hh := mul_le_mul_of_nonneg_left hsum hp
+    have hq1 : 0 ≤ q-1 := sub_nonneg.mpr hq
+    calc
+      _ ≤ (q-1)*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
+        mul_le_mul_of_nonneg_left hh hq1
+      _ ≤ q*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
+        mul_le_mul_of_nonneg_right (by linarith) (mul_nonneg hp h0)
+      _ = _ := by ring
+  have hm : finiteMean (fun ω : SwitchIndex d → Bool =>
+      q^(Fintype.card (SelectedCycle (butterflyPerm d (decodeButterfly d ω)) S))) ≤
+      Real.exp (c+(q-1)*lam) := by
+    calc
+      _ ≤ finiteMean (fun ω : SwitchIndex d → Bool =>
+          q^(Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J))*Real.exp c) := by
+        apply finiteMean_mono
+        intro ω
+        rw [selectedCycle_card_split _ _ J,pow_add]
+        exact mul_le_mul_of_nonneg_left (longCycle_pow_bound _ S J hJ q hq) (by positivity)
+      _ = finiteMean (fun ω : SwitchIndex d → Bool =>
+          q^(Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J))) * Real.exp c :=
+        finiteMean_mul_const _ _
+      _ ≤ Real.exp ((q-1)*lam) * Real.exp c :=
+        mul_le_mul_of_nonneg_right (butterfly_shortCycle_mgf S J q hq) (Real.exp_nonneg _)
+      _ = _ := by rw [←Real.exp_add,add_comm]
+  have hpos : 0 < finiteMean (fun ω : SwitchIndex d → Bool =>
+      q^(Fintype.card (SelectedCycle (butterflyPerm d (decodeButterfly d ω)) S))) := by
+    apply finiteMean_pos
+    intro ω
+    positivity
+  have hlog := Real.log_le_log hpos hm
+  rw [Real.log_exp] at hlog
+  dsimp [c] at hlog
+  linarith
+
+end FullCycleMGF
+
+structure RoutingNetwork (α ι : Type*) [Fintype α] [DecidableEq α]
+    [Fintype ι] [DecidableEq ι] (d : ℕ) where
+  perm : (ι → Bool) → Equiv.Perm α
+  path : (ι → Bool) → α → Finset ι
+  users : (ι → Bool) → ι → α × α
+  users_ne : ∀ ω i, (users ω i).1 ≠ (users ω i).2
+  mem_path : ∀ ω x i, i ∈ path ω x ↔ (users ω i).1 = x ∨ (users ω i).2 = x
+  card_path : ∀ ω x, (path ω x).card = d
+  entropy : ∀ ω (S : Finset α),
+    (∑ i : ι, if (users ω i).1 ∈ S ∧ (users ω i).2 ∈ S then (1:ℝ) else 0) ≤
+      (S.card : ℝ) * Real.log S.card / (2 * Real.log 2)
+  follow : α → (α → QueryTree ι) → QueryTree ι
+  close : α → α → QueryTree ι → QueryTree ι
+  follow_succeeds : ∀ x k ω, (follow x k).succeeds ω = (k (perm ω x)).succeeds ω
+  follow_raw_charge : ∀ x k ω, (follow x k).charge ω = (k (perm ω x)).charge ω
+  follow_stale_charge : ∀ x k R ω,
+    (follow x k).staleCharge R ω = (k (perm ω x)).staleCharge (R ∪ path ω x) ω
+  close_succeeds : ∀ x y k ω,
+    (close x y k).succeeds ω = if perm ω x = y then k.succeeds ω else false
+  close_raw_charge : ∀ x y k ω, perm ω x = y → (close x y k).charge ω = k.charge ω + d
+  close_stale_charge : ∀ x y k R ω, perm ω x = y →
+    (close x y k).staleCharge R ω = k.staleCharge (R ∪ path ω x) ω + (path ω x ∩ R).card
+
+namespace RoutingNetwork
+variable {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι] [DecidableEq ι]
+  {d : ℕ} (R : RoutingNetwork α ι d)
+
+lemma card_switches_using (ω : ι → Bool) (x : α) :
+    (Finset.univ.filter (fun i : ι =>
+      (R.users ω i).1 = x ∨ (R.users ω i).2 = x)).card = d := by
+  classical
+  have he : Finset.univ.filter (fun i : ι =>
+      (R.users ω i).1 = x ∨ (R.users ω i).2 = x) = R.path ω x := by
+    ext i
+    simp only [Finset.mem_filter,Finset.mem_univ,true_and,R.mem_path]
+  rw [he,R.card_path]
+lemma switch_incidence_sum (b : ι → Bool) (w : α → ℝ) :
+    ∑ i : ι, (w (R.users b i).1 + w (R.users b i).2) =
+      (d : ℝ) * ∑ x : α, w x := by
+  classical
+  calc
+    _ = ∑ i : ι, ∑ x : α,
+        if (R.users b i).1 = x ∨ (R.users b i).2 = x then w x else 0 := by
+      apply Finset.sum_congr rfl
+      intro i _
+      rw [Finset.sum_ite]
+      simp only [Finset.sum_const_zero, add_zero]
+      have he : Finset.univ.filter (fun x : α =>
+          (R.users b i).1 = x ∨ (R.users b i).2 = x) =
+          {(R.users b i).1, (R.users b i).2} := by
+        ext x; simp [eq_comm]
+      rw [he, Finset.sum_pair (R.users_ne b i)]
+    _ = ∑ x : α, ∑ i : ι,
+        if (R.users b i).1 = x ∨ (R.users b i).2 = x then w x else 0 :=
+      Finset.sum_comm
+    _ = ∑ x : α, (d : ℝ) * w x := by
+      apply Finset.sum_congr rfl
+      intro x _
+      rw [Finset.sum_ite]
+      simp only [Finset.sum_const_zero, add_zero, Finset.sum_const, nsmul_eq_mul,
+        card_switches_using]
+    _ = _ := by rw [Finset.mul_sum]
+
+lemma weighted_switch_min_bound (b : ι → Bool) (w : α → ℝ) :
+    ∑ i : ι, min (w (R.users b i).1) (w (R.users b i).2) ≤
+      (d : ℝ) / 2 * ∑ x : α, w x := by
+  have h : 2 * (∑ i : ι,
+      min (w (R.users b i).1) (w (R.users b i).2)) ≤
+      ∑ i : ι, (w (R.users b i).1 + w (R.users b i).2) := by
+    rw [Finset.mul_sum]
+    apply Finset.sum_le_sum
+    intro i _
+    linarith [min_le_left (w (R.users b i).1) (w (R.users b i).2),
+      min_le_right (w (R.users b i).1) (w (R.users b i).2)]
+  rw [switch_incidence_sum] at h
+  linarith
+section Rotation
+variable {A : ℕ}
+
+def cycleFiber (o : α → Option (Fin A)) (i : Fin A) : Finset (α) := by
+  classical
+  exact Finset.univ.filter (fun x => o x = some i)
+
+omit [DecidableEq α] in
+@[simp] lemma mem_cycleFiber (o : α → Option (Fin A)) (i : Fin A) (x : α) :
+    x ∈ cycleFiber o i ↔ o x = some i := by
+  classical
+  simp [cycleFiber]
+
+abbrev Terminal (o : α → Option (Fin A)) := ∀ i : Fin A, {x : α // o x = some i}
+
+noncomputable def slotWeight (o : α → Option (Fin A)) (i : Fin A) : ℝ :=
+  1 / (cycleFiber o i).card
+
+noncomputable def pathWeight (o : α → Option (Fin A)) (x : α) : ℝ :=
+  match o x with
+  | none => 0
+  | some i => slotWeight o i
+
+noncomputable def withinWeight (o : α → Option (Fin A)) (u v : α) : ℝ :=
+  match o u, o v with
+  | some i, some j => if i = j then slotWeight o i else 0
+  | _, _ => 0
+
+noncomputable def rotationMeeting (o : α → Option (Fin A)) (z : Terminal o)
+    (u v : α) : ℝ :=
+  match o u, o v with
+  | some i, some j => if i < j then (if (z j).val = v then 1 else 0)
+      else if j < i then (if (z i).val = u then 1 else 0)
+      else (if (z i).val = u then 1 else 0) + (if (z i).val = v then 1 else 0)
+  | _, _ => 0
+
+omit [DecidableEq α] in
+lemma card_cycleFiber (o : α → Option (Fin A)) (i : Fin A) :
+    Fintype.card {x : α // o x = some i} = (cycleFiber o i).card := by
+  classical
+  simp [cycleFiber,Fintype.card_subtype]
+
+lemma terminal_mean {o : α → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
+    (i : Fin A) (x : α) (hx : o x = some i) :
+    finiteMean (fun z : Terminal o => if (z i).val = x then (1:ℝ) else 0) = slotWeight o i := by
+  classical
+  have hev := finiteMean_pi_eval (A := fun i : Fin A => {x : α // o x = some i})
+    i (fun z => if z.val = x then (1:ℝ) else 0)
+  rw [hev]
+  have he : (fun z : {x : α // o x = some i} => if z.val = x then (1:ℝ) else 0) =
+      (fun z => if z = ⟨x,hx⟩ then 1 else 0) := by
+    funext z
+    simp only [Subtype.ext_iff]
+  rw [he,finiteMean_eq_indicator,card_cycleFiber,slotWeight]
+
+omit [DecidableEq α] in
+lemma slotWeight_antitone (o : α → Option (Fin A))
+    (hn : ∀ i, (cycleFiber o i).Nonempty)
+    (hm : Monotone (fun i => (cycleFiber o i).card)) : Antitone (slotWeight o) := by
+  intro i j hij
+  unfold slotWeight
+  apply one_div_le_one_div_of_le
+  · exact_mod_cast Finset.card_pos.mpr (hn i)
+  · exact_mod_cast hm hij
+
+lemma rotationMeeting_mean {o : α → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
+    (hm : Monotone (fun i => (cycleFiber o i).card)) (u v : α) :
+    finiteMean (fun z : Terminal o => rotationMeeting o z u v) =
+      min (pathWeight o u) (pathWeight o v) + withinWeight o u v := by
+  classical
+  have hn : ∀ i, (cycleFiber o i).Nonempty := by
+    intro i
+    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : α // o x = some i})
+    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
+  have hw := slotWeight_antitone o hn hm
+  cases hu : o u with
+  | none =>
+    cases hv : o v with
+    | none => simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean]
+    | some j =>
+      have hj : 0 ≤ slotWeight o j := by unfold slotWeight; positivity
+      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_left hj]
+  | some i =>
+    cases hv : o v with
+    | none =>
+      have hi : 0 ≤ slotWeight o i := by unfold slotWeight; positivity
+      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_right hi]
+    | some j =>
+      simp only [rotationMeeting,pathWeight,withinWeight,hu,hv]
+      rcases lt_trichotomy i j with hij | rfl | hji
+      · simp only [hij,↓reduceIte,ne_of_lt hij]
+        rw [terminal_mean j v hv,min_eq_right (hw (le_of_lt hij)),add_zero]
+      · simp only [lt_self_iff_false,↓reduceIte,min_self]
+        rw [finiteMean_add,terminal_mean i u hu,terminal_mean i v hv]
+      · simp only [not_lt_of_ge (le_of_lt hji),hji,↓reduceIte,ne_of_gt hji]
+        rw [terminal_mean i u hu,min_eq_left (hw (le_of_lt hji)),add_zero]
+
+omit [DecidableEq α] in
+lemma pathWeight_sum (o : α → Option (Fin A)) (hn : ∀ i, (cycleFiber o i).Nonempty) :
+    ∑ x : α, pathWeight o x = (A : ℝ) := by
+  classical
+  have he (x : α) : pathWeight o x =
+      ∑ i : Fin A, if o x = some i then slotWeight o i else 0 := by
+    cases hx : o x with
+    | none => simp [pathWeight,hx]
+    | some j => simp [pathWeight,hx]
+  simp only [he]
+  rw [Finset.sum_comm]
+  have hf (i : Fin A) : (∑ x : α, if o x = some i then slotWeight o i else 0) = 1 := by
+    rw [← Finset.sum_filter]
+    change (∑ _x ∈ cycleFiber o i, slotWeight o i) = 1
+    simp only [Finset.sum_const,nsmul_eq_mul,slotWeight]
+    exact mul_one_div_cancel (by exact_mod_cast (Finset.card_pos.mpr (hn i)).ne')
+  simp only [hf,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,mul_one]
+
+lemma withinWeight_sum_bound (o : α → Option (Fin A))
+    (hn : ∀ i, (cycleFiber o i).Nonempty) (b : ι → Bool) :
+    (∑ s : ι, withinWeight o (R.users b s).1 (R.users b s).2) ≤
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
+  classical
+  have he (u v : α) : withinWeight o u v =
+      ∑ i : Fin A, (if u ∈ cycleFiber o i ∧ v ∈ cycleFiber o i then (1:ℝ) else 0) * slotWeight o i := by
+    simp only [mem_cycleFiber]
+    cases hu : o u <;> cases hv : o v <;> simp [withinWeight,hu,hv,ite_and,eq_comm]
+  simp only [he]
+  rw [Finset.sum_comm]
+  apply Finset.sum_le_sum
+  intro i _
+  rw [← Finset.sum_mul]
+  have hi := R.entropy b (cycleFiber o i)
+  have hpos : (0:ℝ) < (cycleFiber o i).card := by exact_mod_cast Finset.card_pos.mpr (hn i)
+  have hmul := mul_le_mul_of_nonneg_right hi (show 0 ≤ slotWeight o i by unfold slotWeight; positivity)
+  calc
+    _ ≤ ((cycleFiber o i).card : ℝ) * Real.log (cycleFiber o i).card /
+        (2 * Real.log 2) * slotWeight o i := hmul
+    _ = _ := by unfold slotWeight; field_simp
+
+noncomputable def rotationCharge (o : α → Option (Fin A)) (b : ι → Bool)
+    (z : Terminal o) : ℝ :=
+  ∑ s : ι, rotationMeeting o z (R.users b s).1 (R.users b s).2
+
+lemma rotationCharge_mean_bound {o : α → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
+    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : ι → Bool) :
+    finiteMean (rotationCharge R o b) ≤ (A:ℝ) * d / 2 +
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
+  classical
+  have hn : ∀ i, (cycleFiber o i).Nonempty := by
+    intro i
+    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : α // o x = some i})
+    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
+  unfold rotationCharge
+  rw [finiteMean_sum]
+  simp only [rotationMeeting_mean hm,Finset.sum_add_distrib]
+  have h₁ := weighted_switch_min_bound R b (pathWeight o)
+  rw [pathWeight_sum o hn] at h₁
+  have h₂ := withinWeight_sum_bound R o hn b
+  linarith
+
+lemma rotation_exp_bound {o : α → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
+    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : ι → Bool) :
+    Real.exp (-Real.log 2 * ((A:ℝ)*d/2 +
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2))) ≤
+      finiteMean (fun z : Terminal o => Real.exp (-Real.log 2 * rotationCharge R o b z)) := by
+  have hb := rotationCharge_mean_bound R hm b
+  have hlog : 0 ≤ Real.log 2 := Real.log_nonneg (by norm_num)
+  have hm' : -Real.log 2 * ((A:ℝ)*d/2 +
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2)) ≤
+      finiteMean (fun z : Terminal o => -Real.log 2 * rotationCharge R o b z) := by
+    have he := finiteMean_mul_const (rotationCharge R o b) (-Real.log 2)
+    simp only [mul_comm] at he
+    rw [he]
+    exact mul_le_mul_of_nonpos_left hb (neg_nonpos.mpr hlog)
+  exact (Real.exp_le_exp.mpr hm').trans (exp_finiteMean_le _)
+
+end Rotation
+
+namespace QueryTree
+open Thorp.QueryTree
+
+noncomputable def seekCycle (R : RoutingNetwork α ι d) (S : Finset (α)) (a : α) :
+    ℕ → α → Finset (α) → (Finset (α) → QueryTree (ι)) →
+      QueryTree (ι)
+  | 0, x, V, k => if x ∈ S ∧ x ∉ V then R.close x a (k (insert x V)) else reject
+  | r + 1, x, V, k => if x ∈ S ∧ x ∉ V then R.follow x
+      (fun y => seekCycle R S a r y (insert x V) k) else reject
+
+lemma seekCycle_raw_charge (S : Finset (α)) (a : α) (r : ℕ)
+    (x : α) (V : Finset (α)) (k : Finset (α) → QueryTree (ι))
+    (ω : ι → Bool) (K : ℕ)
+    (hK : ∀ V, (k V).succeeds ω = true → (k V).charge ω = K)
+    (hs : (seekCycle R S a r x V k).succeeds ω = true) :
+    (seekCycle R S a r x V k).charge ω = K + d := by
+  induction r generalizing x V with
+  | zero =>
+    by_cases hv : x ∈ S ∧ x ∉ V
+    · rw [seekCycle,ite_eq_left hv,R.close_succeeds] at hs
+      split_ifs at hs with he
+      · rw [seekCycle,ite_eq_left hv,R.close_raw_charge _ _ _ _ he,hK _ hs]
+    · simp [seekCycle,hv,Thorp.QueryTree.succeeds] at hs
+  | succ r ih =>
+    by_cases hv : x ∈ S ∧ x ∉ V
+    · rw [seekCycle,ite_eq_left hv,R.follow_succeeds] at hs
+      rw [seekCycle,ite_eq_left hv,R.follow_raw_charge]
+      exact ih _ _ hs
+    · simp [seekCycle,hv,Thorp.QueryTree.succeeds] at hs
+
+noncomputable def seekCycles (R : RoutingNetwork α ι d) (S : Finset (α)) :
+    List (α × ℕ) → Finset (α) → QueryTree (ι)
+  | [], _ => accept
+  | (a,r) :: L, V => seekCycle R S a r a V (fun V => seekCycles R S L V)
+
+lemma seekCycles_raw_charge (S : Finset (α)) (L : List (α × ℕ))
+    (V : Finset (α)) (ω : ι → Bool)
+    (hs : (seekCycles R S L V).succeeds ω = true) :
+    (seekCycles R S L V).charge ω = L.length * d := by
+  induction L generalizing V with
+  | nil => simp [seekCycles,charge]
+  | cons ar L ih =>
+    obtain ⟨a,r⟩ := ar
+    rw [seekCycles]
+    rw [seekCycle_raw_charge R S a r a V (fun V => seekCycles R S L V) ω (L.length*d)
+      (fun V hs => ih V hs) hs]
+    simp [Nat.add_mul]
+
+lemma cycles_weighted_success (S : Finset (α)) (L : List (α × ℕ)) :
+    finiteMean (fun ω : ι → Bool =>
+      if (seekCycles R S L ∅).succeeds ω then
+        ((2:ℝ)^d)^L.length / (2:ℝ)^(seekCycles R S L ∅).staleCharge ∅ ω else 0) ≤ 1 := by
+  let T := seekCycles R S L ∅
+  have hh := mean_freshCharge_le_one T
+  have he : finiteMean (fun ω : ι → Bool =>
+      if T.succeeds ω then (2:ℝ)^T.freshCharge ∅ ω else 0) =
+      finiteMean (fun ω : ι → Bool =>
+        if T.succeeds ω then ((2:ℝ)^d)^L.length / (2:ℝ)^T.staleCharge ∅ ω else 0) := by
+    apply finiteMean_congr
+    intro ω
+    cases hs : T.succeeds ω
+    · simp
+    · simp only [↓reduceIte]
+      have hc := fresh_add_stale T ∅ ω
+      rw [seekCycles_raw_charge R S L ∅ ω hs] at hc
+      have hp : ((2:ℝ)^d)^L.length = (2:ℝ)^T.freshCharge ∅ ω * (2:ℝ)^T.staleCharge ∅ ω := by
+        rw [←pow_add,hc,←pow_mul,Nat.mul_comm]
+      rw [hp,mul_div_cancel_right₀ _ (pow_ne_zero _ (by norm_num))]
+  rw [← he]
+  exact hh
+
+end QueryTree
+noncomputable def exposedPaths (b : ι → Bool) (V : Finset (α)) :
+    Finset (ι) :=
+  V.biUnion (fun x => R.path b x)
+
+@[simp] lemma exposedPaths_empty (b : ι → Bool) : exposedPaths R b ∅ = ∅ := by
+  classical
+  simp [exposedPaths]
+
+lemma exposedPaths_insert (b : ι → Bool) (V : Finset (α)) (x : α) :
+    exposedPaths R b (insert x V) = exposedPaths R b V ∪ R.path b x := by
+  classical
+  simp [exposedPaths,Finset.union_comm]
+
+noncomputable def terminalCharge (b : ι → Bool) (z : α) (V : Finset (α)) : ℕ :=
+  (R.path b z ∩ exposedPaths R b V).card
+
+namespace QueryTree
+open Thorp.QueryTree
+
+lemma seekCycle_semantics (S : Finset (α)) (a : α) (r : ℕ)
+    (x : α) (V W : Finset (α)) (z : α)
+    (k : Finset (α) → QueryTree (ι)) (ω : ι → Bool)
+    (h : cycleTrace (R.perm ω) S a r x V = some (W,z)) :
+    (seekCycle R S a r x V k).succeeds ω = (k W).succeeds ω ∧
+      (seekCycle R S a r x V k).staleCharge (exposedPaths R (ω) V) ω =
+        (k W).staleCharge (exposedPaths R (ω) W) ω +
+          terminalCharge R (ω) z (W.erase z) := by
+  classical
+  induction r generalizing x V with
+  | zero =>
+    simp only [cycleTrace] at h
+    split_ifs at h with hv
+    obtain ⟨hxS,hxV,hxa⟩ := hv
+    obtain ⟨rfl,rfl⟩ := Prod.mk.inj (Option.some.inj h)
+    simp only [seekCycle,ite_eq_left (show x ∈ S ∧ x ∉ V from ⟨hxS,hxV⟩),R.close_succeeds,ite_eq_left hxa]
+    constructor
+    · trivial
+    · rw [R.close_stale_charge _ _ _ _ _ hxa,exposedPaths_insert,
+        Finset.erase_insert hxV]
+      simp only [terminalCharge]
+  | succ r ih =>
+    simp only [cycleTrace] at h
+    split_ifs at h with hv
+    obtain ⟨hxS,hxV⟩ := hv
+    rw [seekCycle,ite_eq_left ⟨hxS,hxV⟩,R.follow_succeeds,R.follow_stale_charge]
+    rw [← exposedPaths_insert]
+    exact ih _ _ h
+
+lemma seekCycle_orbit (S V : Finset (α)) (z : α)
+    (k : Finset (α) → QueryTree (ι)) (ω : ι → Bool)
+    (hS : orbitSet (R.perm ω) z ⊆ S)
+    (hV : Disjoint V (orbitSet (R.perm ω) z)) :
+    let p := R.perm ω
+    let W := V ∪ orbitSet p z
+    (seekCycle R S (p z) (Function.minimalPeriod p z-1) (p z) V k).succeeds ω = (k W).succeeds ω ∧
+      (seekCycle R S (p z) (Function.minimalPeriod p z-1) (p z) V k).staleCharge
+          (exposedPaths R (ω) V) ω =
+        (k W).staleCharge (exposedPaths R (ω) W) ω +
+          terminalCharge R (ω) z (W.erase z) := by
+  exact seekCycle_semantics _ _ _ _ _ _ _ _ _ _ (cycleTrace_orbit _ _ _ _ hS hV)
+
+end QueryTree
+
+end RoutingNetwork
+end Thorp
+
+end ThorpNine.SparseSaving
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/SparseSaving/BenesCoins.lean b/lean/OAI/Probability/ThorpRouting/SparseSaving/BenesCoins.lean
index 8464dd8548ca2fed1284e2b1e402a55590491714..6f2905ba4afa5afd25bd3c063ccb6a1b6b4fa5a3 100644
--- a/lean/OAI/Probability/ThorpRouting/SparseSaving/BenesCoins.lean
+++ b/lean/OAI/Probability/ThorpRouting/SparseSaving/BenesCoins.lean
@@ -1,882 +1,882 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.SparseSaving.AlternatingPerm
-
-namespace OAI
-
-namespace ThorpNine.SparseSaving
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-
-abbrev BenesCoins (d : ℕ) := (SwitchIndex d → Bool) × (SwitchIndex d → Bool)
-
-def palindromePerm (d : ℕ) (ω : BenesCoins d) : Equiv.Perm (Card d) :=
-  butterflyPerm d (decodeButterfly d ω.1) * (butterflyPerm d (decodeButterfly d ω.2)).symm
-
-def sandwichShuffle {A B C D E F : Type*} :
-    ((A × B) × C) × ((D × E) × F) ≃ ((A × D) × (B × E)) × (F × C) where
-  toFun x := (((x.1.1.1,x.2.1.1),(x.1.1.2,x.2.1.2)),(x.2.2,x.1.2))
-  invFun x := (((x.1.1.1,x.1.2.1),x.2.2),((x.1.1.2,x.1.2.2),x.2.1))
-  left_inv _ := rfl
-  right_inv _ := rfl
-
-def benesStepEquiv (d : ℕ) : BenesCoins (d+1) ≃
-    (BenesCoins d × BenesCoins d) × ((Card d → Bool) × (Card d → Bool)) :=
-  (Equiv.prodCongr (coinStepEquiv d) (coinStepEquiv d)).trans sandwichShuffle
-
-lemma childLift_mul (d : ℕ) (p q : Bool → Equiv.Perm (Card d)) :
-    childLift p * childLift q = childLift (fun b => p b * q b) := by
-  ext x i
-  simp only [Equiv.Perm.mul_apply,childLift,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons]
-
-lemma childLift_inv (d : ℕ) (p : Bool → Equiv.Perm (Card d)) :
-    (childLift p)⁻¹ = childLift (fun b => (p b)⁻¹) := rfl
-
-lemma pairSwitch_inv (d : ℕ) (ξ : Card d → Bool) : (pairSwitch ξ)⁻¹ = pairSwitch ξ := rfl
-
-lemma palindromePerm_step (d : ℕ) (c : BenesCoins d × BenesCoins d)
-    (coins : (Card d → Bool) × (Card d → Bool)) :
-    palindromePerm (d+1) ((benesStepEquiv d).symm (c,coins)) =
-      pairSwitch coins.2 * childLift (fun b => palindromePerm d (if b then c.2 else c.1)) *
-        pairSwitch coins.1 := by
-  change (butterflyPerm (d+1) (decodeButterfly (d+1)
-      ((coinStepEquiv d).symm ((c.1.1,c.2.1),coins.2)))) *
-    (butterflyPerm (d+1) (decodeButterfly (d+1)
-      ((coinStepEquiv d).symm ((c.1.2,c.2.2),coins.1))))⁻¹ = _
-  rw [butterflyPerm_step,butterflyPerm_step,mul_inv_rev,childLift_inv,pairSwitch_inv]
-  rw [mul_assoc (pairSwitch coins.2),←mul_assoc (childLift _),childLift_mul,←mul_assoc]
-  congr 2
-  apply congrArg childLift
-  funext b
-  cases b <;> rfl
-
-lemma headTail_switch (d : ℕ) (ξ : Card d → Bool) (x : Card (d+1)) :
-    headTailEquiv d (pairSwitch ξ x) = pairLayer ξ (headTailEquiv d x) := rfl
-
-lemma headTail_child (d : ℕ) (p : Bool → Equiv.Perm (Card d)) (x : Card (d+1)) :
-    headTailEquiv d (childLift p x) = PairRouting.lift p (headTailEquiv d x) := rfl
-
-lemma palindromePerm_coordinates (d : ℕ) (c : BenesCoins d × BenesCoins d)
-    (coins : (Card d → Bool) × (Card d → Bool)) (x : Card (d+1)) :
-    headTailEquiv d (palindromePerm (d+1) ((benesStepEquiv d).symm (c,coins)) x) =
-      PairRouting.sandwich coins.1 coins.2 (fun b => palindromePerm d (if b then c.2 else c.1))
-        (headTailEquiv d x) := by
-  rw [palindromePerm_step]
-  simp only [Equiv.Perm.mul_apply,headTail_switch,headTail_child]
-  rfl
-
-noncomputable def palindromeCost : (d : ℕ) → {ι : Type*} → [Fintype ι] →
-    (ι ↪ Card d) → BenesCoins d → ℕ
-  | 0, _, _, _, _ => 0
-  | d+1, _, _, e, ω =>
-      let σ := benesStepEquiv d ω
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x σ.2.1
-      let hc := PairRouting.colors_compatible x σ.2.1
-      PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
-          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) +
-        palindromeCost d (PairRouting.childEmbedding x c hc false) σ.1.1 +
-        palindromeCost d (PairRouting.childEmbedding x c hc true) σ.1.2
-
-namespace CycleColoring
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-lemma alternating_cycle_two (p : Equiv.Perm α) (S : Finset (Bool × α))
-    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) S) : 2 ≤ C.val.card := by
-  obtain ⟨x,hx⟩ := C.nonempty
-  have hpx : alternatingPerm p x ∈ C.val := by
-    rw [←C.orbit_eq hx,mem_orbitSet]
-    exact (Equiv.Perm.SameCycle.refl _ _).apply_right
-  have hne : x ≠ alternatingPerm p x := by
-    intro he
-    have hh := congrArg Prod.fst he
-    rw [alternatingPerm_flip] at hh
-    cases hb : x.1 <;> simp only [hb, Bool.not_false, Bool.not_true] at hh <;> cases hh
-  have hs : {x,alternatingPerm p x} ⊆ C.val := by
-    intro y hy
-    simp only [Finset.mem_insert,Finset.mem_singleton] at hy
-    rcases hy with rfl | rfl <;> assumption
-  have hh := Finset.card_le_card hs
-  simpa only [Finset.card_pair hne] using hh
-
-lemma alternating_cycle_count_double (p : Equiv.Perm α) (S : Finset (Bool × α)) :
-    2 * Fintype.card (RoutingNetwork.SelectedCycle (alternatingPerm p) S) ≤ S.card := by
-  classical
-  calc
-    _ = ∑ _C : RoutingNetwork.SelectedCycle (alternatingPerm p) S, 2 := by simp [Nat.mul_comm]
-    _ ≤ RoutingNetwork.cycleMass (alternatingPerm p) S :=
-      Finset.sum_le_sum (fun C _ => alternating_cycle_two p S C)
-    _ ≤ _ := RoutingNetwork.cycleMass_le _ _
-
-lemma alternating_cycle_count_zero (p : Equiv.Perm α) (S : Finset (Bool × α))
-    (hS : S.card ≤ 1) : Fintype.card (RoutingNetwork.SelectedCycle (alternatingPerm p) S) = 0 := by
-  have h := alternating_cycle_count_double p S
-  omega
-
-end CycleColoring
-
-namespace PairRouting
-
-section
-variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
-
-lemma alternatingCycles_double (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    2 * alternatingCycles e p ≤ Fintype.card ι := by
-  have hh := CycleColoring.alternating_cycle_count_double (alternatingProjection p) (labelSet e)
-  simpa only [alternatingCycles, card_labelSet] using hh
-
-lemma alternatingCycles_zero (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α)
-    (hι : Fintype.card ι ≤ 1) : alternatingCycles e p = 0 := by
-  have hh := alternatingCycles_double e p
-  omega
-
-end
-variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α] [DecidableEq ι]
-
-omit [Fintype α] [DecidableEq ι] in
-lemma colors_chosenCoin (x : ι → Bool × α) (c : ι → Bool) (hc : Compatible x c) :
-    colors x (chosenCoin x c) = c := by
-  funext i
-  simp only [colors,chosenCoin_apply x c hc i,required]
-  cases (x i).1 <;> cases c i <;> rfl
-
-noncomputable def coloredEmbedding (x : ι ↪ Bool × α) (c : ι → Bool) (_hc : Compatible x c) :
-    ι ↪ Bool × α := switchedEmbedding x (chosenCoin x c)
-
-omit [Fintype α] [DecidableEq ι] in
-lemma coloredEmbedding_apply (x : ι ↪ Bool × α) (c : ι → Bool) (hc : Compatible x c) (i : ι) :
-    coloredEmbedding x c hc i = (c i,(x i).2) := by
-  apply Prod.ext
-  · exact congrFun (colors_chosenCoin x c hc) i
-  · rfl
-
-omit [Fintype α] [DecidableEq ι] in
-lemma switchedEmbedding_eq_colored (x : ι ↪ Bool × α) (c : ι → Bool) (hc : Compatible x c)
-    (ξ : α → Bool) (he : colors x ξ = c) :
-    switchedEmbedding x ξ = coloredEmbedding x c hc := by
-  ext i : 1
-  rw [coloredEmbedding_apply]
-  apply Prod.ext
-  · exact congrFun he i
-  · rfl
-
-lemma assignments_match_bound (x y : ι ↪ Bool × α) (c : ι → Bool)
-    (hc : c ∈ compatibleSet x y) (p : Bool → Equiv.Perm α) (hm : ChildMatches p x y c) :
-    (compatibleSet x y).card ≤
-      2 ^ ((occupied x).card + (occupied y).card - Fintype.card ι +
-        alternatingCycles (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p) := by
-  obtain ⟨hcx,hcy⟩ := (mem_compatibleSet x y c).mp hc
-  let ξ := chosenCoin x c
-  let η := chosenCoin y c
-  have he : routedEmbedding x ξ η p = y := by
-    ext i : 1
-    have h := (sandwich_matches_iff ξ η p x y).mpr
-      ⟨(colors_chosenCoin x c hcx).trans (colors_chosenCoin y c hcy).symm,by
-        simpa only [ξ,colors_chosenCoin x c hcx] using hm⟩
-    exact h i
-  have hh := assignments_actual_bound x ξ η p
-  rw [he] at hh
-  rw [card_compatibleSet]
-  exact hh
-
-omit [Fintype α] [DecidableEq ι] in
-lemma occupied_sum_ge (x y : ι ↪ Bool × α) :
-    Fintype.card ι ≤ (occupied x).card + (occupied y).card := by
-  have h₀ := occupied_card_double x
-  have h₁ := occupied_card_double y
-  omega
-
-lemma assignment_weight_bound (x y : ι ↪ Bool × α) (c : ι → Bool)
-    (hc : c ∈ compatibleSet x y) (p : Bool → Equiv.Perm α) (hm : ChildMatches p x y c) :
-    ((compatibleSet x y).card:ℝ) * outerWeight x y ≤
-      (2:ℝ)^((alternatingCycles
-        (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p : ℝ) - Fintype.card ι) := by
-  let z := alternatingCycles (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p
-  let b := (occupied x).card+(occupied y).card
-  let h := Fintype.card ι
-  have hb : h ≤ b := occupied_sum_ge x y
-  have hn : ((compatibleSet x y).card:ℝ) ≤ (2:ℝ)^(b-h+z) := by
-    exact_mod_cast assignments_match_bound x y c hc p hm
-  change ((compatibleSet x y).card:ℝ) * (1/(2:ℝ)^b) ≤ (2:ℝ)^((z:ℝ)-h)
-  rw [mul_one_div]
-  calc
-    _ ≤ (2:ℝ)^(b-h+z)/(2:ℝ)^b := div_le_div_of_nonneg_right hn (by positivity)
-    _ = _ := by
-      rw [←Real.rpow_natCast,←Real.rpow_natCast,←Real.rpow_sub (by norm_num : (0:ℝ)<2)]
-      congr 1
-      rw [Nat.cast_add,Nat.cast_sub hb]
-      ring
-
-lemma assignment_factor_bound (x y : ι ↪ Bool × α) (c : ι → Bool)
-    (hc : c ∈ compatibleSet x y) (p : Bool → Equiv.Perm α) (hm : ChildMatches p x y c)
-    (a : ℝ) (ha : 0 ≤ a) :
-    (outerWeight x y)^a * ((compatibleSet x y).card:ℝ)^a ≤
-      (2:ℝ)^(((alternatingCycles
-        (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p : ℝ) - Fintype.card ι)*a) := by
-  have h := Real.rpow_le_rpow (mul_nonneg (Nat.cast_nonneg _) (le_of_lt (outerWeight_pos x y)))
-    (assignment_weight_bound x y c hc p hm) ha
-  rw [Real.mul_rpow (Nat.cast_nonneg _) (le_of_lt (outerWeight_pos x y)),
-    ←Real.rpow_mul (by norm_num : (0:ℝ)≤2)] at h
-  simpa only [mul_comm] using h
-
-
-lemma sandwich_weighted_probability_subtype (p : Bool → Equiv.Perm α) (x y : ι → Bool × α)
-    (W : (ι → Bool) → ℝ) :
-    finiteMean (fun coins : (α → Bool) × (α → Bool) =>
-      if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then W (colors x coins.1) else 0) =
-    outerWeight x y * ∑ c : {c // c ∈ compatibleSet x y},
-      if ChildMatches p x y c.val then W c.val else 0 := by
-  rw [sandwich_weighted_probability]
-  congr 1
-  rw [Finset.sum_subtype (compatibleSet x y) (fun _ => Iff.rfl)]
-
-omit [Fintype α] [DecidableEq α] [DecidableEq ι] in
-lemma color_card_add (c : ι → Bool) :
-    Fintype.card {i // c i = false} + Fintype.card {i // c i = true} = Fintype.card ι := by
-  have he : {i // c i = true} ≃ {i // ¬c i = false} := Equiv.subtypeEquivRight fun i => by
-    cases c i <;> simp
-  rw [Fintype.card_congr he]
-  rw [Fintype.card_subtype_compl]
-  exact Nat.add_sub_of_le (Fintype.card_subtype_le _)
-
-end PairRouting
-
-noncomputable def assignmentCost (d : ℕ) {ι : Type*} [Fintype ι]
-    (x : ι ↪ Bool × Card d) (c : ι → Bool) (hc : PairRouting.Compatible x c)
-    (ω : BenesCoins d × BenesCoins d) : ℕ :=
-  PairRouting.alternatingCycles (PairRouting.coloredEmbedding x c hc)
-      (fun b => palindromePerm d (if b then ω.2 else ω.1)) +
-    palindromeCost d (PairRouting.childEmbedding x c hc false) ω.1 +
-    palindromeCost d (PairRouting.childEmbedding x c hc true) ω.2
-
-lemma palindromeCost_eq (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (d+1)) (σ : BenesCoins (d+1)) :
-    palindromeCost (d+1) e σ =
-      assignmentCost d (e.trans (headTailEquiv d).toEmbedding)
-        (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) (benesStepEquiv d σ).2.1)
-        (PairRouting.colors_compatible _ _) (benesStepEquiv d σ).1 := by
-  unfold palindromeCost assignmentCost
-  dsimp only
-  rw [PairRouting.switchedEmbedding_eq_colored _ _ (PairRouting.colors_compatible _ _) _ rfl]
-
-lemma palindromeCost_step (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (d+1)) (ω : BenesCoins d × BenesCoins d)
-    (coins : (Card d → Bool) × (Card d → Bool)) :
-    palindromeCost (d+1) e ((benesStepEquiv d).symm (ω,coins)) =
-      assignmentCost d (e.trans (headTailEquiv d).toEmbedding)
-        (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) coins.1)
-        (PairRouting.colors_compatible _ _) ω := by
-  rw [palindromeCost_eq]
-  simp only [Equiv.apply_symm_apply]
-
-noncomputable def routingMoment (d : ℕ) {ι : Type*} [Fintype ι]
-    (e f : ι ↪ Card d) (a : ℝ) : ℝ := by
-  classical
-  exact finiteMean (fun ω : BenesCoins d =>
-    if ∀ i, palindromePerm d ω (e i) = f i then (2:ℝ)^((palindromeCost d e ω:ℝ)*a) else 0)
-
-lemma routingMoment_nonneg (d : ℕ) {ι : Type*} [Fintype ι]
-    (e f : ι ↪ Card d) (a : ℝ) : 0 ≤ routingMoment d e f a := by
-  classical
-  exact finiteMean_nonneg (fun _ => by split_ifs <;> positivity)
-
-lemma palindrome_matches_step (d : ℕ) {ι : Type*}
-    (e f : ι ↪ Card (d+1)) (ω : BenesCoins d × BenesCoins d)
-    (coins : (Card d → Bool) × (Card d → Bool)) :
-    (∀ i, palindromePerm (d+1) ((benesStepEquiv d).symm (ω,coins)) (e i) = f i) ↔
-    (∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => palindromePerm d (if b then ω.2 else ω.1))
-      ((e.trans (headTailEquiv d).toEmbedding) i) = ((f.trans (headTailEquiv d).toEmbedding) i)) := by
-  apply forall_congr'
-  intro i
-  rw [←(headTailEquiv d).injective.eq_iff,palindromePerm_coordinates]
-  rfl
-
-lemma routingMoment_step (d : ℕ) {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (e f : ι ↪ Card (d+1)) (a : ℝ) :
-    let x := e.trans (headTailEquiv d).toEmbedding
-    let y := f.trans (headTailEquiv d).toEmbedding
-    routingMoment (d+1) e f a = PairRouting.outerWeight x y *
-      ∑ c : {c // c ∈ PairRouting.compatibleSet x y},
-        finiteMean (fun ω : BenesCoins d × BenesCoins d =>
-          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c.val
-          then (2:ℝ)^((assignmentCost d x c.val
-            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0) := by
-  classical
-  intro x y
-  rw [routingMoment,finiteMean_equiv (benesStepEquiv d),finiteMean_prod]
-  have he (ω : BenesCoins d × BenesCoins d) :
-      finiteMean (fun coins : (Card d → Bool) × (Card d → Bool) =>
-        if ∀ i, palindromePerm (d+1) ((benesStepEquiv d).symm (ω,coins)) (e i) = f i
-        then (2:ℝ)^((palindromeCost (d+1) e ((benesStepEquiv d).symm (ω,coins)):ℝ)*a) else 0) =
-      PairRouting.outerWeight x y *
-        ∑ c : {c // c ∈ PairRouting.compatibleSet x y},
-          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c.val
-          then (2:ℝ)^((assignmentCost d x c.val
-            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0 := by
-    let W : (ι → Bool) → ℝ := fun c =>
-      if hc : PairRouting.Compatible x c then (2:ℝ)^((assignmentCost d x c hc ω:ℝ)*a) else 0
-    calc
-      _ = finiteMean (fun coins : (Card d → Bool) × (Card d → Bool) =>
-        if ∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => palindromePerm d (if b then ω.2 else ω.1))
-          (x i) = y i then W (PairRouting.colors x coins.1) else 0) := by
-        apply finiteMean_congr
-        intro coins
-        simp only [palindrome_matches_step,palindromeCost_step]
-        simp only [W,PairRouting.colors_compatible,↓reduceDIte]
-        rfl
-      _ = _ := by
-        rw [PairRouting.sandwich_weighted_probability_subtype]
-        congr 1
-        apply Finset.sum_congr rfl
-        intro c _
-        simp only [W,((PairRouting.mem_compatibleSet x y c.val).mp c.property).1,↓reduceDIte]
-  simp_rw [he]
-  simp_rw [mul_comm (PairRouting.outerWeight x y)]
-  rw [finiteMean_mul_const,finiteMean_sum,mul_comm]
-
-lemma child_moment_product (d : ℕ) {ι : Type*} [Fintype ι]
-    (x y : ι ↪ Bool × Card d) (c : ι → Bool)
-    (hcx : PairRouting.Compatible x c) (hcy : PairRouting.Compatible y c) (a : ℝ) :
-    routingMoment d (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a *
-      routingMoment d (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a =
-    finiteMean (fun ω : BenesCoins d × BenesCoins d =>
-      if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
-      then (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
-        palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0) := by
-  classical
-  unfold routingMoment
-  rw [←finiteMean_prod_mul]
-  apply finiteMean_congr
-  intro ω
-  let M₀ := ∀ i : {i // c i = false}, palindromePerm d ω.1 (x i.val).2 = (y i.val).2
-  let M₁ := ∀ i : {i // c i = true}, palindromePerm d ω.2 (x i.val).2 = (y i.val).2
-  have hp : (fun b : Bool => palindromePerm d (if b then ω.2 else ω.1)) =
-      (fun b => if b then palindromePerm d ω.2 else palindromePerm d ω.1) := by
-    funext b
-    cases b <;> rfl
-  have hs : PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c ↔
-      M₀ ∧ M₁ := by rw [hp]; exact PairRouting.childMatches_split _ _ _ _ _
-  change (if M₀ then _ else 0) * (if M₁ then _ else 0) = _
-  by_cases h₀ : M₀ <;> by_cases h₁ : M₁
-  · simp only [hs,h₀,h₁,true_and,↓reduceIte]
-    rw [←Real.rpow_add (by norm_num : (0:ℝ)<2)]
-    congr 1
-    ring
-  · simp only [hs,h₀,h₁,and_false,↓reduceIte,mul_zero]
-  · simp only [hs,h₀,h₁,false_and,↓reduceIte,zero_mul]
-  · simp only [hs,h₀,h₁,false_and,↓reduceIte,mul_zero]
-
-lemma palindrome_probability_step (d : ℕ) {ι : Type*} [Fintype ι]
-    (e f : ι ↪ Card (d+1)) :
-    PairRouting.tupleProbability (palindromePerm (d+1)) e f =
-    finiteMean (fun ω : BenesCoins d × BenesCoins d =>
-      finiteMean (fun coins : (Card d → Bool) × (Card d → Bool) =>
-        if ∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => if b then palindromePerm d ω.2 else palindromePerm d ω.1)
-          ((e.trans (headTailEquiv d).toEmbedding) i) = ((f.trans (headTailEquiv d).toEmbedding) i)
-        then (1:ℝ) else 0)) := by
-  classical
-  rw [PairRouting.tupleProbability,finiteMean_equiv (benesStepEquiv d),finiteMean_prod]
-  have hp (b : Bool) (p q : BenesCoins d) :
-      palindromePerm d (if b then p else q) = (if b then palindromePerm d p else palindromePerm d q) := by
-    cases b <;> rfl
-  simp only [palindrome_matches_step,hp]
-
-lemma child_power_product_bound (d : ℕ) {ι : Type*} [Fintype ι]
-    (x y : ι ↪ Bool × Card d) (c : ι → Bool)
-    (hcx : PairRouting.Compatible x c) (hcy : PairRouting.Compatible y c) (a : ℝ)
-    (hfalse : (PairRouting.tupleProbability (palindromePerm d)
-        (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false))^(1+a) ≤
-      (2:ℝ)^(-(d:ℝ)*Fintype.card {i // c i = false}*a) *
-        routingMoment d (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a)
-    (htrue : (PairRouting.tupleProbability (palindromePerm d)
-        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true))^(1+a) ≤
-      (2:ℝ)^(-(d:ℝ)*Fintype.card {i // c i = true}*a) *
-        routingMoment d (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a) :
-    (PairRouting.tupleProbability (palindromePerm d)
-      (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false))^(1+a) *
-      (PairRouting.tupleProbability (palindromePerm d)
-        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true))^(1+a) ≤
-    (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) *
-      finiteMean (fun ω : BenesCoins d × BenesCoins d =>
-        if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
-        then (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
-          palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0) := by
-  have hh := mul_le_mul hfalse htrue
-    (Real.rpow_nonneg (PairRouting.tupleProbability_nonneg _ _ _) _)
-    (mul_nonneg (by positivity) (routingMoment_nonneg _ _ _ _))
-  rw [mul_mul_mul_comm,child_moment_product,←Real.rpow_add (by norm_num : (0:ℝ)<2)] at hh
-  have he : -(d:ℝ)*Fintype.card {i // c i = false}*a + -(d:ℝ)*Fintype.card {i // c i = true}*a =
-      -(d:ℝ)*Fintype.card ι*a := by
-    rw [←add_mul,←mul_add,←Nat.cast_add,PairRouting.color_card_add]
-  rwa [he] at hh
-
-lemma assignment_moment_bound (d : ℕ) {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (x y : ι ↪ Bool × Card d) (c : ι → Bool) (hc : c ∈ PairRouting.compatibleSet x y)
-    (a : ℝ) (ha : 0 ≤ a) :
-    let hcx := ((PairRouting.mem_compatibleSet x y c).mp hc).1
-    PairRouting.outerWeight x y ^ a * ((PairRouting.compatibleSet x y).card:ℝ)^a *
-      ((2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) *
-        finiteMean (fun ω : BenesCoins d × BenesCoins d =>
-          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
-          then (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
-            palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0)) ≤
-    (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) *
-      finiteMean (fun ω : BenesCoins d × BenesCoins d =>
-        if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
-        then (2:ℝ)^((assignmentCost d x c hcx ω:ℝ)*a) else 0) := by
-  classical
-  intro hcx
-  have hscale (q : ℝ) (F : BenesCoins d × BenesCoins d → ℝ) :
-      q * finiteMean F = finiteMean (fun ω => q * F ω) := by
-    simp_rw [mul_comm q,finiteMean_mul_const]
-  rw [←mul_assoc,hscale,hscale]
-  apply finiteMean_mono
-  intro ω
-  split_ifs with hm
-  · have hh := PairRouting.assignment_factor_bound x y c hc
-      (fun b => palindromePerm d (if b then ω.2 else ω.1)) hm a ha
-    calc
-      _ ≤ ((2:ℝ)^(((PairRouting.alternatingCycles (PairRouting.coloredEmbedding x c hcx)
-          (fun b => palindromePerm d (if b then ω.2 else ω.1)):ℝ)-Fintype.card ι)*a) *
-          (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a)) *
-          (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
-            palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) :=
-        mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hh (by positivity)) (by positivity)
-      _ = _ := by
-        simp only [←Real.rpow_add (by norm_num : (0:ℝ)<2),assignmentCost,Nat.cast_add]
-        congr 1
-        ring
-  · simp only [mul_zero]
-    exact le_rfl
-
-theorem palindrome_pointwise_moment (d : ℕ) {ι : Type*} [Fintype ι]
-    (e f : ι ↪ Card d) (a : ℝ) (ha : 0 ≤ a) :
-    (PairRouting.tupleProbability (palindromePerm d) e f)^(1+a) ≤
-      (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) * routingMoment d e f a := by
-  classical
-  induction d generalizing ι with
-  | zero =>
-      have he : ∀ (ω : BenesCoins 0) (i : ι), palindromePerm 0 ω (e i) = f i :=
-        fun _ _ => Subsingleton.elim _ _
-      simp [PairRouting.tupleProbability,routingMoment,he,palindromeCost,finiteMean_const]
-  | succ d ih =>
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let y := f.trans (headTailEquiv d).toEmbedding
-      let W := PairRouting.outerWeight x y
-      let C := PairRouting.compatibleSet x y
-      let p : (ι → Bool) → ℝ := fun c =>
-        (PairRouting.tupleProbability (palindromePerm d)
-          (fun i : {i // c i = false} => (x i.val).2)
-          (fun i : {i // c i = false} => (y i.val).2))^(1+a) *
-        (PairRouting.tupleProbability (palindromePerm d)
-          (fun i : {i // c i = true} => (x i.val).2)
-          (fun i : {i // c i = true} => (y i.val).2))^(1+a)
-      let m : {c // c ∈ C} → ℝ := fun c =>
-        finiteMean (fun ω : BenesCoins d × BenesCoins d =>
-          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c.val
-          then (2:ℝ)^((assignmentCost d x c.val
-            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0)
-      have hp (c : {c // c ∈ C}) : W^a * (C.card:ℝ)^a * p c.val ≤
-          (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * m c := by
-        obtain ⟨hcx,hcy⟩ := (PairRouting.mem_compatibleSet x y c.val).mp c.property
-        have h₀ := ih (PairRouting.childEmbedding x c.val hcx false)
-          (PairRouting.childEmbedding y c.val hcy false)
-        have h₁ := ih (PairRouting.childEmbedding x c.val hcx true)
-          (PairRouting.childEmbedding y c.val hcy true)
-        have hh := child_power_product_bound d x y c.val hcx hcy a h₀ h₁
-        have hh' := mul_le_mul_of_nonneg_left hh
-          (mul_nonneg (Real.rpow_nonneg (le_of_lt (PairRouting.outerWeight_pos x y)) a)
-            (Real.rpow_nonneg (Nat.cast_nonneg C.card) a))
-        exact hh'.trans (assignment_moment_bound d x y c.val c.property a ha)
-      have hnode := PairRouting.sandwich_power_bound (palindromePerm d) (palindromePerm d) x y a ha
-      rw [←palindrome_probability_step d e f] at hnode
-      change (PairRouting.tupleProbability (palindromePerm (d+1)) e f)^(1+a) ≤
-        W^(1+a) * (C.card:ℝ)^a * ∑ c ∈ C, p c at hnode
-      have hsum : ∑ c ∈ C, p c = ∑ c : {c // c ∈ C}, p c.val := by
-        rw [Finset.sum_subtype C (fun _ => Iff.rfl)]
-      have hW : 0 < W := PairRouting.outerWeight_pos x y
-      calc
-        _ ≤ W^(1+a) * (C.card:ℝ)^a * ∑ c ∈ C, p c := hnode
-        _ = W * ∑ c : {c // c ∈ C}, W^a * (C.card:ℝ)^a * p c.val := by
-          rw [Real.rpow_add hW,Real.rpow_one,hsum,←Finset.mul_sum]
-          ring
-        _ ≤ W * ∑ c : {c // c ∈ C}, (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * m c :=
-          mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun c _ => hp c)) (le_of_lt hW)
-        _ = _ := by
-          rw [←Finset.mul_sum,routingMoment_step]
-          simp only [Nat.cast_add,Nat.cast_one]
-          change W * ((2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * ∑ c, m c) =
-            (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * (W * ∑ c, m c)
-          ring
-
-lemma routingMoment_sum (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (a : ℝ) :
-    (∑ f : ι ↪ Card d, routingMoment d e f a) =
-      finiteMean (fun ω : BenesCoins d => (2:ℝ)^((palindromeCost d e ω:ℝ)*a)) := by
-  classical
-  unfold routingMoment
-  rw [←finiteMean_sum]
-  apply finiteMean_congr
-  intro ω
-  let f₀ : ι ↪ Card d := e.trans (palindromePerm d ω).toEmbedding
-  rw [Finset.sum_eq_single f₀]
-  · have hm : ∀ i, palindromePerm d ω (e i) = f₀ i := fun _ => rfl
-    simp only [hm,implies_true,↓reduceIte]
-  · intro f _ hf
-    have hm : ¬ ∀ i, palindromePerm d ω (e i) = f i := by
-      intro hh
-      apply hf
-      ext i : 1
-      exact (hh i).symm
-    exact ite_eq_right hm
-  · simp
-
-lemma finite_row_power {X : Type*} [Fintype X] [Nonempty X]
-    (p : X → ℝ) (hp : ∀ x, 0 ≤ p x) (a : ℝ) :
-    finiteMean (fun x => ((Fintype.card X:ℝ)*p x)^(1+a)) =
-      (Fintype.card X:ℝ)^a * ∑ x, (p x)^(1+a) := by
-  have hn : (0:ℝ) < Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
-  simp only [finiteMean,Real.mul_rpow (le_of_lt hn) (hp _),←Finset.mul_sum]
-  rw [Real.rpow_add hn,Real.rpow_one]
-  field_simp
-
-noncomputable def palindromeRowMoment (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (a : ℝ) : ℝ :=
-  finiteMean (fun f : ι ↪ Card d =>
-    ((Fintype.card (ι ↪ Card d):ℝ)*PairRouting.tupleProbability (palindromePerm d) e f)^(1+a))
-
-lemma density_normalization (d h n : ℕ) (a : ℝ) :
-    (n:ℝ)^a * (2:ℝ)^(-(d:ℝ)*h*a) = ((n:ℝ)/((2:ℝ)^d)^h)^a := by
-  rw [Real.div_rpow (Nat.cast_nonneg _) (by positivity),←pow_mul,
-    ←Real.rpow_natCast,←Real.rpow_mul (by norm_num : (0:ℝ) ≤ 2),Nat.cast_mul]
-  rw [show -(d:ℝ)*h*a = -((d:ℝ)*h*a) by ring,
-    Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2),div_eq_mul_inv]
-
-theorem palindrome_row_density_moment (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (a : ℝ) (ha : 0 ≤ a) :
-    palindromeRowMoment d e a ≤
-      (((2^d).descFactorial (Fintype.card ι):ℝ)/((2^d:ℕ):ℝ)^(Fintype.card ι))^a *
-        finiteMean (fun ω : BenesCoins d => (2:ℝ)^((palindromeCost d e ω:ℝ)*a)) := by
-  classical
-  have : Nonempty (ι ↪ Card d) := ⟨e⟩
-  rw [palindromeRowMoment,finite_row_power _ (fun f => PairRouting.tupleProbability_nonneg _ _ _)]
-  calc
-    _ ≤ (Fintype.card (ι ↪ Card d):ℝ)^a *
-      ∑ f : ι ↪ Card d, (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) * routingMoment d e f a :=
-      mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun f _ => palindrome_pointwise_moment d e f a ha))
-        (Real.rpow_nonneg (Nat.cast_nonneg _) _)
-    _ = _ := by
-      rw [←Finset.mul_sum,routingMoment_sum,←mul_assoc,density_normalization,
-        Fintype.card_embedding_eq,card_positions]
-      norm_cast
-
-end Thorp
-namespace Thorp
-
-lemma subset_biUnion_indicator {α β : Type*} [DecidableEq α] [Fintype β]
-    (A : Finset α) (T : β → Finset α) :
-    (if Finset.univ.biUnion T ⊆ A then (1:ℝ) else 0) =
-      ∏ b, if T b ⊆ A then (1:ℝ) else 0 := by
-  classical
-  by_cases h : Finset.univ.biUnion T ⊆ A
-  · rw [ite_eq_left h]
-    symm
-    apply Finset.prod_eq_one
-    intro b _
-    rw [ite_eq_left]
-    exact (Finset.subset_biUnion_of_mem T (Finset.mem_univ b)).trans h
-  · rw [ite_eq_right h]
-    obtain ⟨x,hx,hxA⟩ := Finset.not_subset.mp h
-    obtain ⟨b,_,hxb⟩ := Finset.mem_biUnion.mp hx
-    symm
-    apply Finset.prod_eq_zero (Finset.mem_univ b)
-    rw [ite_eq_right]
-    exact fun hb => hxA (hb hxb)
-
-lemma cylinder_block_polynomial {α β Ω : Type*} [DecidableEq α] [Fintype β]
-    [Fintype Ω] (τ : β → Type*) [∀ b, Fintype (τ b)]
-    (F : Ω → Finset α) (p : α → ℝ)
-    (hF : CylinderBound (inclusionMoment F) p)
-    (T : ∀ b, τ b → Finset α) (c : ∀ b, τ b → ℝ)
-    (hc : ∀ b s, 0 ≤ c b s)
-    (hd : ∀ b b', b ≠ b' → ∀ s s', Disjoint (T b s) (T b' s')) :
-    finiteMean (fun ω => ∏ b, ∑ s, c b s * if T b s ⊆ F ω then 1 else 0) ≤
-      ∏ b, ∑ s, c b s * ∏ x ∈ T b s, p x := by
-  classical
-  simp_rw [Fintype.prod_sum]
-  rw [finiteMean_sum]
-  apply Finset.sum_le_sum
-  intro s _
-  let U := Finset.univ.biUnion (fun b => T b (s b))
-  have he : (fun ω => ∏ b, c b (s b) * if T b (s b) ⊆ F ω then 1 else 0) =
-      fun ω => (if U ⊆ F ω then 1 else 0) * ∏ b, c b (s b) := by
-    funext ω
-    rw [Finset.prod_mul_distrib,←subset_biUnion_indicator]
-    exact mul_comm _ _
-  rw [he,finiteMean_mul_const]
-  have hdis : ((Finset.univ : Finset β) : Set β).PairwiseDisjoint (fun b => T b (s b)) := by
-    intro b _ b' _ hbb
-    exact hd b b' hbb (s b) (s b')
-  have hmean : finiteMean (fun ω => if U ⊆ F ω then 1 else 0) = inclusionMoment F U := by
-    unfold inclusionMoment
-    apply finiteMean_congr
-    intro ω
-    by_cases hh : U ⊆ F ω <;> simp [hh]
-  calc
-    _ ≤ (∏ x ∈ U, p x) * ∏ b, c b (s b) :=
-      mul_le_mul_of_nonneg_right (hmean.le.trans (hF U)) (Finset.prod_nonneg (fun b _ => hc b (s b)))
-    _ = _ := by
-      dsimp only [U]
-      rw [Finset.prod_biUnion hdis,←Finset.prod_mul_distrib]
-      apply Finset.prod_congr rfl
-      intro b _
-      exact mul_comm _ _
-
-
-noncomputable def sparseCoefficient {α : Type*} (w : ℝ) (T : Finset α) : ℝ :=
-  if T.card = 1 then 0 else w^T.card
-
-lemma sparseCoefficient_nonneg {α : Type*} (w : ℝ) (hw : 0 ≤ w) (T : Finset α) :
-    0 ≤ sparseCoefficient w T := by
-  unfold sparseCoefficient
-  split_ifs <;> positivity
-
-lemma sparse_block_pointwise {α : Type*} [DecidableEq α] (A D : Finset α)
-    (w : ℝ) (hw : 0 ≤ w) :
-    w^(if 2 ≤ (D ∩ A).card then (D ∩ A).card else 0) ≤
-      ∑ T : ↑D.powerset, sparseCoefficient w T.val * if T.val ⊆ A then 1 else 0 := by
-  classical
-  let V := D ∩ A
-  let T : ↑D.powerset := if 2 ≤ V.card then
-    ⟨V,Finset.mem_powerset.mpr Finset.inter_subset_left⟩ else
-    ⟨∅,Finset.mem_powerset.mpr (Finset.empty_subset D)⟩
-  calc
-    _ = sparseCoefficient w T.val * (if T.val ⊆ A then 1 else 0) := by
-      by_cases h : 2 ≤ V.card
-      · have hn : V.card ≠ 1 := by omega
-        simp [T,ite_eq_left h,sparseCoefficient,hn,show V ⊆ A from Finset.inter_subset_right,V]
-      · simp [T,ite_eq_right h,sparseCoefficient,V] at *
-    _ ≤ _ := Finset.single_le_sum (f := fun S : ↑D.powerset =>
-      sparseCoefficient w S.val * (if S.val ⊆ A then 1 else 0))
-      (fun S _ => mul_nonneg (sparseCoefficient_nonneg w hw S.val)
-        (by split_ifs <;> norm_num)) (Finset.mem_univ T)
-
-lemma geometric_sum_half (n : ℕ) {x : ℝ} (hx : 0 ≤ x) (hhalf : x ≤ 1/2) :
-    ∑ i ∈ Finset.range n, x^i ≤ 2 := by
-  induction n with
-  | zero => simp
-  | succ n ih =>
-    rw [Finset.sum_range_succ',pow_zero]
-    simp only [pow_succ',←Finset.mul_sum]
-    nlinarith
-
-lemma sparse_binomial_bound (R : ℕ) (z : ℝ) (hz : 0 ≤ z) (hr : (R:ℝ)*z ≤ 1/2) :
-    ∑ i ∈ Finset.range (R+1), (R.choose i:ℝ) * (if i = 1 then 0 else z^i) ≤
-      1 + 2*((R:ℝ)*z)^2 := by
-  classical
-  let x := (R:ℝ)*z
-  have hx : 0 ≤ x := mul_nonneg (Nat.cast_nonneg _) hz
-  have hbound (i : ℕ) : (R.choose i:ℝ)*(if i=1 then 0 else z^i) ≤
-      if i=1 then 0 else x^i := by
-    by_cases h : i=1
-    · simp [h]
-    · simp only [ite_eq_right h]
-      dsimp [x]
-      rw [mul_pow]
-      apply mul_le_mul_of_nonneg_right _ (pow_nonneg hz i)
-      exact_mod_cast Nat.choose_le_pow R i
-  calc
-    _ ≤ ∑ i ∈ Finset.range (R+1), if i=1 then 0 else x^i :=
-      Finset.sum_le_sum (fun i _ => hbound i)
-    _ ≤ 1+2*x^2 := by
-      rcases R with _ | R
-      · simp [x]
-      rcases R with _ | R
-      · norm_num [Finset.sum_range_succ]
-        positivity
-      have he : ∑ i ∈ Finset.range (R+2+1), (if i=1 then 0 else x^i) =
-          1 + x^2*∑ i ∈ Finset.range (R+1), x^i := by
-        rw [show R+2+1=2+(R+1) by omega,Finset.sum_range_add]
-        have hn (i : ℕ) : 2+i ≠ 1 := by omega
-        simp only [hn,ite_false,pow_add,←Finset.mul_sum]
-        norm_num [Finset.sum_range_succ]
-      rw [he]
-      have hg := geometric_sum_half (R+1) hx hr
-      nlinarith [sq_nonneg x]
-
-lemma sparse_block_product_bound {α : Type*} [DecidableEq α]
-    (D : Finset α) (p w : ℝ) (hp : 0 ≤ p) (hw : 0 ≤ w)
-    (hr : (D.card:ℝ)*(p*w) ≤ 1/2) :
-    ∑ T : ↑D.powerset, sparseCoefficient w T.val * p^T.val.card ≤
-      Real.exp (2*((D.card:ℝ)*(p*w))^2) := by
-  classical
-  have he : ∑ T : ↑D.powerset, sparseCoefficient w T.val * p^T.val.card =
-      ∑ i ∈ Finset.range (D.card+1), (D.card.choose i:ℝ)*(if i=1 then 0 else (p*w)^i) := by
-    rw [Finset.sum_coe_sort D.powerset (fun T => sparseCoefficient w T*p^T.card)]
-    have hh : (fun T : Finset α => sparseCoefficient w T*p^T.card) =
-        fun T => if T.card=1 then 0 else (p*w)^T.card := by
-      funext T
-      unfold sparseCoefficient
-      split_ifs <;> simp [mul_pow,mul_comm]
-    change (∑ T ∈ D.powerset, (fun T : Finset α => sparseCoefficient w T*p^T.card) T) = _
-    rw [hh,Finset.sum_powerset_apply_card (fun i => if i=1 then (0:ℝ) else (p*w)^i)]
-    simp only [nsmul_eq_mul]
-  rw [he]
-  exact (sparse_binomial_bound D.card (p*w) (mul_nonneg hp hw) hr).trans
-    (by simpa only [add_comm] using Real.add_one_le_exp (2*((D.card:ℝ)*(p*w))^2))
-
-
-noncomputable def sparseMass {α β : Type*} [DecidableEq α] [Fintype β]
-    (D : β → Finset α) (A : Finset α) : ℕ :=
-  ∑ b, if 2 ≤ (D b ∩ A).card then (D b ∩ A).card else 0
-
-theorem sparse_block_mgf {α β Ω : Type*} [DecidableEq α] [Fintype β] [Fintype Ω]
-    (D : β → Finset α) (hd : Pairwise (fun b b' => Disjoint (D b) (D b')))
-    (F : Ω → Finset α) (p w : ℝ) (hp : 0 ≤ p) (hw : 0 ≤ w)
-    (hF : ∀ T, inclusionMoment F T ≤ p^T.card)
-    (hr : ∀ b, ((D b).card:ℝ)*(p*w) ≤ 1/2) :
-    finiteMean (fun ω => w^(sparseMass D (F ω))) ≤
-      Real.exp (∑ b, 2*(((D b).card:ℝ)*(p*w))^2) := by
-  classical
-  have hc : CylinderBound (inclusionMoment F) (fun _ => p) := by
-    intro T
-    simpa only [Finset.prod_const] using hF T
-  have hb := cylinder_block_polynomial (fun b => ↑(D b).powerset) F (fun _ => p) hc
-    (fun _ S => S.val) (fun _ S => sparseCoefficient w S.val)
-    (fun _ S => sparseCoefficient_nonneg w hw S.val)
-    (fun b b' h S S' => (hd h).mono
-      (Finset.mem_powerset.mp S.property) (Finset.mem_powerset.mp S'.property))
-  calc
-    _ ≤ finiteMean (fun ω => ∏ b, ∑ S : ↑(D b).powerset,
-        sparseCoefficient w S.val * if S.val ⊆ F ω then 1 else 0) := by
-      apply finiteMean_mono
-      intro ω
-      rw [sparseMass,←Finset.prod_pow_eq_pow_sum]
-      apply Finset.prod_le_prod₀ (fun _ _ => pow_nonneg hw _)
-      intro b _
-      exact sparse_block_pointwise (F ω) (D b) w hw
-    _ ≤ ∏ b, ∑ S : ↑(D b).powerset, sparseCoefficient w S.val * p^S.val.card := by
-      simpa only [Finset.prod_const] using hb
-    _ ≤ _ := by
-      rw [Real.exp_sum]
-      apply Finset.prod_le_prod₀
-      · intro b _
-        exact Finset.sum_nonneg (fun S _ => mul_nonneg
-          (sparseCoefficient_nonneg w hw S.val) (pow_nonneg hp _))
-      · intro b _
-        exact sparse_block_product_bound (D b) p w hp hw (hr b)
-
-theorem inverseButterfly_sparse_mgf (d : ℕ) {β : Type*} [Fintype β]
-    (D : β → Finset (Card d)) (hd : Pairwise (fun b b' => Disjoint (D b) (D b')))
-    (A : Finset (Card d)) (w : ℝ) (hw : 0 ≤ w)
-    (hr : ∀ b, ((D b).card:ℝ)*((A.card/(2:ℝ)^d)*w) ≤ 1/2) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      w^(sparseMass D (inverseButterflyMarks d A ω))) ≤
-      Real.exp (∑ b, 2*(((D b).card:ℝ)*((A.card/(2:ℝ)^d)*w))^2) := by
-  apply sparse_block_mgf D hd (inverseButterflyMarks d A) (A.card/(2:ℝ)^d) w
-    (by positivity) hw
-  · exact inverseButterflyMarks_inclusion d A
-  · exact hr
-
-
-def crowdedMass (h : ℕ) : ℕ := if 2 ≤ h then h else 0
-
-lemma crowdedMass_le (h : ℕ) : crowdedMass h ≤ h := by
-  unfold crowdedMass
-  split_ifs <;> omega
-
-lemma crowdedMass_add (u v : ℕ) : crowdedMass u + crowdedMass v ≤ crowdedMass (u+v) := by
-  unfold crowdedMass
-  split_ifs <;> omega
-
-lemma alternatingCycles_le_crowded {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
-    (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    PairRouting.alternatingCycles e p ≤ crowdedMass (Fintype.card ι) := by
-  unfold crowdedMass
-  split_ifs with h
-  · have hh := PairRouting.alternatingCycles_double e p
-    omega
-  · rw [PairRouting.alternatingCycles_zero e p (by omega)]
-
-lemma palindromeCost_le_crowded (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) :
-    palindromeCost d e ω ≤ d*crowdedMass (Fintype.card ι) := by
-  classical
-  induction d generalizing ι with
-  | zero => simp [palindromeCost]
-  | succ d ih =>
-    let σ := benesStepEquiv d ω
-    let x := e.trans (headTailEquiv d).toEmbedding
-    let c := PairRouting.colors x σ.2.1
-    let hc := PairRouting.colors_compatible x σ.2.1
-    have h₀ := ih (PairRouting.childEmbedding x c hc false) σ.1.1
-    have h₁ := ih (PairRouting.childEmbedding x c hc true) σ.1.2
-    have hz := alternatingCycles_le_crowded (PairRouting.switchedEmbedding x σ.2.1)
-      (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1))
-    have hs := crowdedMass_add (Fintype.card {i // c i = false})
-      (Fintype.card {i // c i = true})
-    rw [PairRouting.color_card_add] at hs
-    change PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
-        (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) +
-        palindromeCost d (PairRouting.childEmbedding x c hc false) σ.1.1 +
-        palindromeCost d (PairRouting.childEmbedding x c hc true) σ.1.2 ≤ _
-    nlinarith
-
-noncomputable def palindromeLowCost (H : ℕ) : (d : ℕ) → {ι : Type*} → [Fintype ι] →
-    (ι ↪ Card d) → BenesCoins d → ℕ
-  | 0, _, _, _, _ => 0
-  | d+1, _, _, e, ω =>
-      let σ := benesStepEquiv d ω
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x σ.2.1
-      let hc := PairRouting.colors_compatible x σ.2.1
-      (if d+1 ≤ H then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
-          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) else 0) +
-        palindromeLowCost H d (PairRouting.childEmbedding x c hc false) σ.1.1 +
-        palindromeLowCost H d (PairRouting.childEmbedding x c hc true) σ.1.2
-
-noncomputable def cutoffMass (H : ℕ) : (d : ℕ) → {ι : Type*} → [Fintype ι] →
-    (ι ↪ Card d) → BenesCoins d → ℕ
-  | 0, _, _, _, _ => 0
-  | d+1, ι, _, e, ω =>
-      if d+1 ≤ H then crowdedMass (Fintype.card ι) else
-      let σ := benesStepEquiv d ω
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x σ.2.1
-      let hc := PairRouting.colors_compatible x σ.2.1
-      cutoffMass H d (PairRouting.childEmbedding x c hc false) σ.1.1 +
-        cutoffMass H d (PairRouting.childEmbedding x c hc true) σ.1.2
-
-lemma palindromeLowCost_eq (H d : ℕ) (h : d ≤ H) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) :
-    palindromeLowCost H d e ω = palindromeCost d e ω := by
-  classical
-  induction d generalizing ι with
-  | zero => rfl
-  | succ d ih =>
-    rw [palindromeLowCost,palindromeCost,ite_eq_left h]
-    congr 2 <;> exact ih (by omega) _ _
-
-lemma palindromeLowCost_le_cutoff (H d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) :
-    palindromeLowCost H d e ω ≤ H*cutoffMass H d e ω := by
-  classical
-  induction d generalizing ι with
-  | zero => simp [palindromeLowCost,cutoffMass]
-  | succ d ih =>
-    by_cases h : d+1 ≤ H
-    · rw [palindromeLowCost_eq H (d+1) h,cutoffMass,ite_eq_left h]
-      exact (palindromeCost_le_crowded (d+1) e ω).trans (Nat.mul_le_mul_right _ h)
-    · rw [palindromeLowCost,cutoffMass,ite_eq_right h,ite_eq_right h,zero_add,Nat.mul_add]
-      exact Nat.add_le_add (ih _ _) (ih _ _)
-
-
-noncomputable def centralBlock (r H : ℕ) (b : Card r) : Finset (Card (H+r)) :=
-  Finset.univ.image (fun z : Card H => (cardSplit r H).symm (b,z))
-
-end Thorp
-
-end ThorpNine.SparseSaving
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.SparseSaving.AlternatingPerm
+
+namespace OAI
+
+namespace ThorpNine.SparseSaving
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+
+abbrev BenesCoins (d : ℕ) := (SwitchIndex d → Bool) × (SwitchIndex d → Bool)
+
+def palindromePerm (d : ℕ) (ω : BenesCoins d) : Equiv.Perm (Card d) :=
+  butterflyPerm d (decodeButterfly d ω.1) * (butterflyPerm d (decodeButterfly d ω.2)).symm
+
+def sandwichShuffle {A B C D E F : Type*} :
+    ((A × B) × C) × ((D × E) × F) ≃ ((A × D) × (B × E)) × (F × C) where
+  toFun x := (((x.1.1.1,x.2.1.1),(x.1.1.2,x.2.1.2)),(x.2.2,x.1.2))
+  invFun x := (((x.1.1.1,x.1.2.1),x.2.2),((x.1.1.2,x.1.2.2),x.2.1))
+  left_inv _ := rfl
+  right_inv _ := rfl
+
+def benesStepEquiv (d : ℕ) : BenesCoins (d+1) ≃
+    (BenesCoins d × BenesCoins d) × ((Card d → Bool) × (Card d → Bool)) :=
+  (Equiv.prodCongr (coinStepEquiv d) (coinStepEquiv d)).trans sandwichShuffle
+
+lemma childLift_mul (d : ℕ) (p q : Bool → Equiv.Perm (Card d)) :
+    childLift p * childLift q = childLift (fun b => p b * q b) := by
+  ext x i
+  simp only [Equiv.Perm.mul_apply,childLift,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons]
+
+lemma childLift_inv (d : ℕ) (p : Bool → Equiv.Perm (Card d)) :
+    (childLift p)⁻¹ = childLift (fun b => (p b)⁻¹) := rfl
+
+lemma pairSwitch_inv (d : ℕ) (ξ : Card d → Bool) : (pairSwitch ξ)⁻¹ = pairSwitch ξ := rfl
+
+lemma palindromePerm_step (d : ℕ) (c : BenesCoins d × BenesCoins d)
+    (coins : (Card d → Bool) × (Card d → Bool)) :
+    palindromePerm (d+1) ((benesStepEquiv d).symm (c,coins)) =
+      pairSwitch coins.2 * childLift (fun b => palindromePerm d (if b then c.2 else c.1)) *
+        pairSwitch coins.1 := by
+  change (butterflyPerm (d+1) (decodeButterfly (d+1)
+      ((coinStepEquiv d).symm ((c.1.1,c.2.1),coins.2)))) *
+    (butterflyPerm (d+1) (decodeButterfly (d+1)
+      ((coinStepEquiv d).symm ((c.1.2,c.2.2),coins.1))))⁻¹ = _
+  rw [butterflyPerm_step,butterflyPerm_step,mul_inv_rev,childLift_inv,pairSwitch_inv]
+  rw [mul_assoc (pairSwitch coins.2),←mul_assoc (childLift _),childLift_mul,←mul_assoc]
+  congr 2
+  apply congrArg childLift
+  funext b
+  cases b <;> rfl
+
+lemma headTail_switch (d : ℕ) (ξ : Card d → Bool) (x : Card (d+1)) :
+    headTailEquiv d (pairSwitch ξ x) = pairLayer ξ (headTailEquiv d x) := rfl
+
+lemma headTail_child (d : ℕ) (p : Bool → Equiv.Perm (Card d)) (x : Card (d+1)) :
+    headTailEquiv d (childLift p x) = PairRouting.lift p (headTailEquiv d x) := rfl
+
+lemma palindromePerm_coordinates (d : ℕ) (c : BenesCoins d × BenesCoins d)
+    (coins : (Card d → Bool) × (Card d → Bool)) (x : Card (d+1)) :
+    headTailEquiv d (palindromePerm (d+1) ((benesStepEquiv d).symm (c,coins)) x) =
+      PairRouting.sandwich coins.1 coins.2 (fun b => palindromePerm d (if b then c.2 else c.1))
+        (headTailEquiv d x) := by
+  rw [palindromePerm_step]
+  simp only [Equiv.Perm.mul_apply,headTail_switch,headTail_child]
+  rfl
+
+noncomputable def palindromeCost : (d : ℕ) → {ι : Type*} → [Fintype ι] →
+    (ι ↪ Card d) → BenesCoins d → ℕ
+  | 0, _, _, _, _ => 0
+  | d+1, _, _, e, ω =>
+      let σ := benesStepEquiv d ω
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x σ.2.1
+      let hc := PairRouting.colors_compatible x σ.2.1
+      PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
+          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) +
+        palindromeCost d (PairRouting.childEmbedding x c hc false) σ.1.1 +
+        palindromeCost d (PairRouting.childEmbedding x c hc true) σ.1.2
+
+namespace CycleColoring
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+lemma alternating_cycle_two (p : Equiv.Perm α) (S : Finset (Bool × α))
+    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) S) : 2 ≤ C.val.card := by
+  obtain ⟨x,hx⟩ := C.nonempty
+  have hpx : alternatingPerm p x ∈ C.val := by
+    rw [←C.orbit_eq hx,mem_orbitSet]
+    exact (Equiv.Perm.SameCycle.refl _ _).apply_right
+  have hne : x ≠ alternatingPerm p x := by
+    intro he
+    have hh := congrArg Prod.fst he
+    rw [alternatingPerm_flip] at hh
+    cases hb : x.1 <;> simp only [hb, Bool.not_false, Bool.not_true] at hh <;> cases hh
+  have hs : {x,alternatingPerm p x} ⊆ C.val := by
+    intro y hy
+    simp only [Finset.mem_insert,Finset.mem_singleton] at hy
+    rcases hy with rfl | rfl <;> assumption
+  have hh := Finset.card_le_card hs
+  simpa only [Finset.card_pair hne] using hh
+
+lemma alternating_cycle_count_double (p : Equiv.Perm α) (S : Finset (Bool × α)) :
+    2 * Fintype.card (RoutingNetwork.SelectedCycle (alternatingPerm p) S) ≤ S.card := by
+  classical
+  calc
+    _ = ∑ _C : RoutingNetwork.SelectedCycle (alternatingPerm p) S, 2 := by simp [Nat.mul_comm]
+    _ ≤ RoutingNetwork.cycleMass (alternatingPerm p) S :=
+      Finset.sum_le_sum (fun C _ => alternating_cycle_two p S C)
+    _ ≤ _ := RoutingNetwork.cycleMass_le _ _
+
+lemma alternating_cycle_count_zero (p : Equiv.Perm α) (S : Finset (Bool × α))
+    (hS : S.card ≤ 1) : Fintype.card (RoutingNetwork.SelectedCycle (alternatingPerm p) S) = 0 := by
+  have h := alternating_cycle_count_double p S
+  omega
+
+end CycleColoring
+
+namespace PairRouting
+
+section
+variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
+
+lemma alternatingCycles_double (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    2 * alternatingCycles e p ≤ Fintype.card ι := by
+  have hh := CycleColoring.alternating_cycle_count_double (alternatingProjection p) (labelSet e)
+  simpa only [alternatingCycles, card_labelSet] using hh
+
+lemma alternatingCycles_zero (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α)
+    (hι : Fintype.card ι ≤ 1) : alternatingCycles e p = 0 := by
+  have hh := alternatingCycles_double e p
+  omega
+
+end
+variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α] [DecidableEq ι]
+
+omit [Fintype α] [DecidableEq ι] in
+lemma colors_chosenCoin (x : ι → Bool × α) (c : ι → Bool) (hc : Compatible x c) :
+    colors x (chosenCoin x c) = c := by
+  funext i
+  simp only [colors,chosenCoin_apply x c hc i,required]
+  cases (x i).1 <;> cases c i <;> rfl
+
+noncomputable def coloredEmbedding (x : ι ↪ Bool × α) (c : ι → Bool) (_hc : Compatible x c) :
+    ι ↪ Bool × α := switchedEmbedding x (chosenCoin x c)
+
+omit [Fintype α] [DecidableEq ι] in
+lemma coloredEmbedding_apply (x : ι ↪ Bool × α) (c : ι → Bool) (hc : Compatible x c) (i : ι) :
+    coloredEmbedding x c hc i = (c i,(x i).2) := by
+  apply Prod.ext
+  · exact congrFun (colors_chosenCoin x c hc) i
+  · rfl
+
+omit [Fintype α] [DecidableEq ι] in
+lemma switchedEmbedding_eq_colored (x : ι ↪ Bool × α) (c : ι → Bool) (hc : Compatible x c)
+    (ξ : α → Bool) (he : colors x ξ = c) :
+    switchedEmbedding x ξ = coloredEmbedding x c hc := by
+  ext i : 1
+  rw [coloredEmbedding_apply]
+  apply Prod.ext
+  · exact congrFun he i
+  · rfl
+
+lemma assignments_match_bound (x y : ι ↪ Bool × α) (c : ι → Bool)
+    (hc : c ∈ compatibleSet x y) (p : Bool → Equiv.Perm α) (hm : ChildMatches p x y c) :
+    (compatibleSet x y).card ≤
+      2 ^ ((occupied x).card + (occupied y).card - Fintype.card ι +
+        alternatingCycles (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p) := by
+  obtain ⟨hcx,hcy⟩ := (mem_compatibleSet x y c).mp hc
+  let ξ := chosenCoin x c
+  let η := chosenCoin y c
+  have he : routedEmbedding x ξ η p = y := by
+    ext i : 1
+    have h := (sandwich_matches_iff ξ η p x y).mpr
+      ⟨(colors_chosenCoin x c hcx).trans (colors_chosenCoin y c hcy).symm,by
+        simpa only [ξ,colors_chosenCoin x c hcx] using hm⟩
+    exact h i
+  have hh := assignments_actual_bound x ξ η p
+  rw [he] at hh
+  rw [card_compatibleSet]
+  exact hh
+
+omit [Fintype α] [DecidableEq ι] in
+lemma occupied_sum_ge (x y : ι ↪ Bool × α) :
+    Fintype.card ι ≤ (occupied x).card + (occupied y).card := by
+  have h₀ := occupied_card_double x
+  have h₁ := occupied_card_double y
+  omega
+
+lemma assignment_weight_bound (x y : ι ↪ Bool × α) (c : ι → Bool)
+    (hc : c ∈ compatibleSet x y) (p : Bool → Equiv.Perm α) (hm : ChildMatches p x y c) :
+    ((compatibleSet x y).card:ℝ) * outerWeight x y ≤
+      (2:ℝ)^((alternatingCycles
+        (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p : ℝ) - Fintype.card ι) := by
+  let z := alternatingCycles (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p
+  let b := (occupied x).card+(occupied y).card
+  let h := Fintype.card ι
+  have hb : h ≤ b := occupied_sum_ge x y
+  have hn : ((compatibleSet x y).card:ℝ) ≤ (2:ℝ)^(b-h+z) := by
+    exact_mod_cast assignments_match_bound x y c hc p hm
+  change ((compatibleSet x y).card:ℝ) * (1/(2:ℝ)^b) ≤ (2:ℝ)^((z:ℝ)-h)
+  rw [mul_one_div]
+  calc
+    _ ≤ (2:ℝ)^(b-h+z)/(2:ℝ)^b := div_le_div_of_nonneg_right hn (by positivity)
+    _ = _ := by
+      rw [←Real.rpow_natCast,←Real.rpow_natCast,←Real.rpow_sub (by norm_num : (0:ℝ)<2)]
+      congr 1
+      rw [Nat.cast_add,Nat.cast_sub hb]
+      ring
+
+lemma assignment_factor_bound (x y : ι ↪ Bool × α) (c : ι → Bool)
+    (hc : c ∈ compatibleSet x y) (p : Bool → Equiv.Perm α) (hm : ChildMatches p x y c)
+    (a : ℝ) (ha : 0 ≤ a) :
+    (outerWeight x y)^a * ((compatibleSet x y).card:ℝ)^a ≤
+      (2:ℝ)^(((alternatingCycles
+        (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p : ℝ) - Fintype.card ι)*a) := by
+  have h := Real.rpow_le_rpow (mul_nonneg (Nat.cast_nonneg _) (le_of_lt (outerWeight_pos x y)))
+    (assignment_weight_bound x y c hc p hm) ha
+  rw [Real.mul_rpow (Nat.cast_nonneg _) (le_of_lt (outerWeight_pos x y)),
+    ←Real.rpow_mul (by norm_num : (0:ℝ)≤2)] at h
+  simpa only [mul_comm] using h
+
+
+lemma sandwich_weighted_probability_subtype (p : Bool → Equiv.Perm α) (x y : ι → Bool × α)
+    (W : (ι → Bool) → ℝ) :
+    finiteMean (fun coins : (α → Bool) × (α → Bool) =>
+      if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then W (colors x coins.1) else 0) =
+    outerWeight x y * ∑ c : {c // c ∈ compatibleSet x y},
+      if ChildMatches p x y c.val then W c.val else 0 := by
+  rw [sandwich_weighted_probability]
+  congr 1
+  rw [Finset.sum_subtype (compatibleSet x y) (fun _ => Iff.rfl)]
+
+omit [Fintype α] [DecidableEq α] [DecidableEq ι] in
+lemma color_card_add (c : ι → Bool) :
+    Fintype.card {i // c i = false} + Fintype.card {i // c i = true} = Fintype.card ι := by
+  have he : {i // c i = true} ≃ {i // ¬c i = false} := Equiv.subtypeEquivRight fun i => by
+    cases c i <;> simp
+  rw [Fintype.card_congr he]
+  rw [Fintype.card_subtype_compl]
+  exact Nat.add_sub_of_le (Fintype.card_subtype_le _)
+
+end PairRouting
+
+noncomputable def assignmentCost (d : ℕ) {ι : Type*} [Fintype ι]
+    (x : ι ↪ Bool × Card d) (c : ι → Bool) (hc : PairRouting.Compatible x c)
+    (ω : BenesCoins d × BenesCoins d) : ℕ :=
+  PairRouting.alternatingCycles (PairRouting.coloredEmbedding x c hc)
+      (fun b => palindromePerm d (if b then ω.2 else ω.1)) +
+    palindromeCost d (PairRouting.childEmbedding x c hc false) ω.1 +
+    palindromeCost d (PairRouting.childEmbedding x c hc true) ω.2
+
+lemma palindromeCost_eq (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (d+1)) (σ : BenesCoins (d+1)) :
+    palindromeCost (d+1) e σ =
+      assignmentCost d (e.trans (headTailEquiv d).toEmbedding)
+        (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) (benesStepEquiv d σ).2.1)
+        (PairRouting.colors_compatible _ _) (benesStepEquiv d σ).1 := by
+  unfold palindromeCost assignmentCost
+  dsimp only
+  rw [PairRouting.switchedEmbedding_eq_colored _ _ (PairRouting.colors_compatible _ _) _ rfl]
+
+lemma palindromeCost_step (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (d+1)) (ω : BenesCoins d × BenesCoins d)
+    (coins : (Card d → Bool) × (Card d → Bool)) :
+    palindromeCost (d+1) e ((benesStepEquiv d).symm (ω,coins)) =
+      assignmentCost d (e.trans (headTailEquiv d).toEmbedding)
+        (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) coins.1)
+        (PairRouting.colors_compatible _ _) ω := by
+  rw [palindromeCost_eq]
+  simp only [Equiv.apply_symm_apply]
+
+noncomputable def routingMoment (d : ℕ) {ι : Type*} [Fintype ι]
+    (e f : ι ↪ Card d) (a : ℝ) : ℝ := by
+  classical
+  exact finiteMean (fun ω : BenesCoins d =>
+    if ∀ i, palindromePerm d ω (e i) = f i then (2:ℝ)^((palindromeCost d e ω:ℝ)*a) else 0)
+
+lemma routingMoment_nonneg (d : ℕ) {ι : Type*} [Fintype ι]
+    (e f : ι ↪ Card d) (a : ℝ) : 0 ≤ routingMoment d e f a := by
+  classical
+  exact finiteMean_nonneg (fun _ => by split_ifs <;> positivity)
+
+lemma palindrome_matches_step (d : ℕ) {ι : Type*}
+    (e f : ι ↪ Card (d+1)) (ω : BenesCoins d × BenesCoins d)
+    (coins : (Card d → Bool) × (Card d → Bool)) :
+    (∀ i, palindromePerm (d+1) ((benesStepEquiv d).symm (ω,coins)) (e i) = f i) ↔
+    (∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => palindromePerm d (if b then ω.2 else ω.1))
+      ((e.trans (headTailEquiv d).toEmbedding) i) = ((f.trans (headTailEquiv d).toEmbedding) i)) := by
+  apply forall_congr'
+  intro i
+  rw [←(headTailEquiv d).injective.eq_iff,palindromePerm_coordinates]
+  rfl
+
+lemma routingMoment_step (d : ℕ) {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (e f : ι ↪ Card (d+1)) (a : ℝ) :
+    let x := e.trans (headTailEquiv d).toEmbedding
+    let y := f.trans (headTailEquiv d).toEmbedding
+    routingMoment (d+1) e f a = PairRouting.outerWeight x y *
+      ∑ c : {c // c ∈ PairRouting.compatibleSet x y},
+        finiteMean (fun ω : BenesCoins d × BenesCoins d =>
+          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c.val
+          then (2:ℝ)^((assignmentCost d x c.val
+            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0) := by
+  classical
+  intro x y
+  rw [routingMoment,finiteMean_equiv (benesStepEquiv d),finiteMean_prod]
+  have he (ω : BenesCoins d × BenesCoins d) :
+      finiteMean (fun coins : (Card d → Bool) × (Card d → Bool) =>
+        if ∀ i, palindromePerm (d+1) ((benesStepEquiv d).symm (ω,coins)) (e i) = f i
+        then (2:ℝ)^((palindromeCost (d+1) e ((benesStepEquiv d).symm (ω,coins)):ℝ)*a) else 0) =
+      PairRouting.outerWeight x y *
+        ∑ c : {c // c ∈ PairRouting.compatibleSet x y},
+          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c.val
+          then (2:ℝ)^((assignmentCost d x c.val
+            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0 := by
+    let W : (ι → Bool) → ℝ := fun c =>
+      if hc : PairRouting.Compatible x c then (2:ℝ)^((assignmentCost d x c hc ω:ℝ)*a) else 0
+    calc
+      _ = finiteMean (fun coins : (Card d → Bool) × (Card d → Bool) =>
+        if ∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => palindromePerm d (if b then ω.2 else ω.1))
+          (x i) = y i then W (PairRouting.colors x coins.1) else 0) := by
+        apply finiteMean_congr
+        intro coins
+        simp only [palindrome_matches_step,palindromeCost_step]
+        simp only [W,PairRouting.colors_compatible,↓reduceDIte]
+        rfl
+      _ = _ := by
+        rw [PairRouting.sandwich_weighted_probability_subtype]
+        congr 1
+        apply Finset.sum_congr rfl
+        intro c _
+        simp only [W,((PairRouting.mem_compatibleSet x y c.val).mp c.property).1,↓reduceDIte]
+  simp_rw [he]
+  simp_rw [mul_comm (PairRouting.outerWeight x y)]
+  rw [finiteMean_mul_const,finiteMean_sum,mul_comm]
+
+lemma child_moment_product (d : ℕ) {ι : Type*} [Fintype ι]
+    (x y : ι ↪ Bool × Card d) (c : ι → Bool)
+    (hcx : PairRouting.Compatible x c) (hcy : PairRouting.Compatible y c) (a : ℝ) :
+    routingMoment d (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a *
+      routingMoment d (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a =
+    finiteMean (fun ω : BenesCoins d × BenesCoins d =>
+      if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
+      then (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
+        palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0) := by
+  classical
+  unfold routingMoment
+  rw [←finiteMean_prod_mul]
+  apply finiteMean_congr
+  intro ω
+  let M₀ := ∀ i : {i // c i = false}, palindromePerm d ω.1 (x i.val).2 = (y i.val).2
+  let M₁ := ∀ i : {i // c i = true}, palindromePerm d ω.2 (x i.val).2 = (y i.val).2
+  have hp : (fun b : Bool => palindromePerm d (if b then ω.2 else ω.1)) =
+      (fun b => if b then palindromePerm d ω.2 else palindromePerm d ω.1) := by
+    funext b
+    cases b <;> rfl
+  have hs : PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c ↔
+      M₀ ∧ M₁ := by rw [hp]; exact PairRouting.childMatches_split _ _ _ _ _
+  change (if M₀ then _ else 0) * (if M₁ then _ else 0) = _
+  by_cases h₀ : M₀ <;> by_cases h₁ : M₁
+  · simp only [hs,h₀,h₁,true_and,↓reduceIte]
+    rw [←Real.rpow_add (by norm_num : (0:ℝ)<2)]
+    congr 1
+    ring
+  · simp only [hs,h₀,h₁,and_false,↓reduceIte,mul_zero]
+  · simp only [hs,h₀,h₁,false_and,↓reduceIte,zero_mul]
+  · simp only [hs,h₀,h₁,false_and,↓reduceIte,mul_zero]
+
+lemma palindrome_probability_step (d : ℕ) {ι : Type*} [Fintype ι]
+    (e f : ι ↪ Card (d+1)) :
+    PairRouting.tupleProbability (palindromePerm (d+1)) e f =
+    finiteMean (fun ω : BenesCoins d × BenesCoins d =>
+      finiteMean (fun coins : (Card d → Bool) × (Card d → Bool) =>
+        if ∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => if b then palindromePerm d ω.2 else palindromePerm d ω.1)
+          ((e.trans (headTailEquiv d).toEmbedding) i) = ((f.trans (headTailEquiv d).toEmbedding) i)
+        then (1:ℝ) else 0)) := by
+  classical
+  rw [PairRouting.tupleProbability,finiteMean_equiv (benesStepEquiv d),finiteMean_prod]
+  have hp (b : Bool) (p q : BenesCoins d) :
+      palindromePerm d (if b then p else q) = (if b then palindromePerm d p else palindromePerm d q) := by
+    cases b <;> rfl
+  simp only [palindrome_matches_step,hp]
+
+lemma child_power_product_bound (d : ℕ) {ι : Type*} [Fintype ι]
+    (x y : ι ↪ Bool × Card d) (c : ι → Bool)
+    (hcx : PairRouting.Compatible x c) (hcy : PairRouting.Compatible y c) (a : ℝ)
+    (hfalse : (PairRouting.tupleProbability (palindromePerm d)
+        (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false))^(1+a) ≤
+      (2:ℝ)^(-(d:ℝ)*Fintype.card {i // c i = false}*a) *
+        routingMoment d (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a)
+    (htrue : (PairRouting.tupleProbability (palindromePerm d)
+        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true))^(1+a) ≤
+      (2:ℝ)^(-(d:ℝ)*Fintype.card {i // c i = true}*a) *
+        routingMoment d (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a) :
+    (PairRouting.tupleProbability (palindromePerm d)
+      (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false))^(1+a) *
+      (PairRouting.tupleProbability (palindromePerm d)
+        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true))^(1+a) ≤
+    (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) *
+      finiteMean (fun ω : BenesCoins d × BenesCoins d =>
+        if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
+        then (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
+          palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0) := by
+  have hh := mul_le_mul hfalse htrue
+    (Real.rpow_nonneg (PairRouting.tupleProbability_nonneg _ _ _) _)
+    (mul_nonneg (by positivity) (routingMoment_nonneg _ _ _ _))
+  rw [mul_mul_mul_comm,child_moment_product,←Real.rpow_add (by norm_num : (0:ℝ)<2)] at hh
+  have he : -(d:ℝ)*Fintype.card {i // c i = false}*a + -(d:ℝ)*Fintype.card {i // c i = true}*a =
+      -(d:ℝ)*Fintype.card ι*a := by
+    rw [←add_mul,←mul_add,←Nat.cast_add,PairRouting.color_card_add]
+  rwa [he] at hh
+
+lemma assignment_moment_bound (d : ℕ) {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (x y : ι ↪ Bool × Card d) (c : ι → Bool) (hc : c ∈ PairRouting.compatibleSet x y)
+    (a : ℝ) (ha : 0 ≤ a) :
+    let hcx := ((PairRouting.mem_compatibleSet x y c).mp hc).1
+    PairRouting.outerWeight x y ^ a * ((PairRouting.compatibleSet x y).card:ℝ)^a *
+      ((2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) *
+        finiteMean (fun ω : BenesCoins d × BenesCoins d =>
+          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
+          then (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
+            palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0)) ≤
+    (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) *
+      finiteMean (fun ω : BenesCoins d × BenesCoins d =>
+        if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
+        then (2:ℝ)^((assignmentCost d x c hcx ω:ℝ)*a) else 0) := by
+  classical
+  intro hcx
+  have hscale (q : ℝ) (F : BenesCoins d × BenesCoins d → ℝ) :
+      q * finiteMean F = finiteMean (fun ω => q * F ω) := by
+    simp_rw [mul_comm q,finiteMean_mul_const]
+  rw [←mul_assoc,hscale,hscale]
+  apply finiteMean_mono
+  intro ω
+  split_ifs with hm
+  · have hh := PairRouting.assignment_factor_bound x y c hc
+      (fun b => palindromePerm d (if b then ω.2 else ω.1)) hm a ha
+    calc
+      _ ≤ ((2:ℝ)^(((PairRouting.alternatingCycles (PairRouting.coloredEmbedding x c hcx)
+          (fun b => palindromePerm d (if b then ω.2 else ω.1)):ℝ)-Fintype.card ι)*a) *
+          (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a)) *
+          (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
+            palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) :=
+        mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hh (by positivity)) (by positivity)
+      _ = _ := by
+        simp only [←Real.rpow_add (by norm_num : (0:ℝ)<2),assignmentCost,Nat.cast_add]
+        congr 1
+        ring
+  · simp only [mul_zero]
+    exact le_rfl
+
+theorem palindrome_pointwise_moment (d : ℕ) {ι : Type*} [Fintype ι]
+    (e f : ι ↪ Card d) (a : ℝ) (ha : 0 ≤ a) :
+    (PairRouting.tupleProbability (palindromePerm d) e f)^(1+a) ≤
+      (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) * routingMoment d e f a := by
+  classical
+  induction d generalizing ι with
+  | zero =>
+      have he : ∀ (ω : BenesCoins 0) (i : ι), palindromePerm 0 ω (e i) = f i :=
+        fun _ _ => Subsingleton.elim _ _
+      simp [PairRouting.tupleProbability,routingMoment,he,palindromeCost,finiteMean_const]
+  | succ d ih =>
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let y := f.trans (headTailEquiv d).toEmbedding
+      let W := PairRouting.outerWeight x y
+      let C := PairRouting.compatibleSet x y
+      let p : (ι → Bool) → ℝ := fun c =>
+        (PairRouting.tupleProbability (palindromePerm d)
+          (fun i : {i // c i = false} => (x i.val).2)
+          (fun i : {i // c i = false} => (y i.val).2))^(1+a) *
+        (PairRouting.tupleProbability (palindromePerm d)
+          (fun i : {i // c i = true} => (x i.val).2)
+          (fun i : {i // c i = true} => (y i.val).2))^(1+a)
+      let m : {c // c ∈ C} → ℝ := fun c =>
+        finiteMean (fun ω : BenesCoins d × BenesCoins d =>
+          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c.val
+          then (2:ℝ)^((assignmentCost d x c.val
+            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0)
+      have hp (c : {c // c ∈ C}) : W^a * (C.card:ℝ)^a * p c.val ≤
+          (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * m c := by
+        obtain ⟨hcx,hcy⟩ := (PairRouting.mem_compatibleSet x y c.val).mp c.property
+        have h₀ := ih (PairRouting.childEmbedding x c.val hcx false)
+          (PairRouting.childEmbedding y c.val hcy false)
+        have h₁ := ih (PairRouting.childEmbedding x c.val hcx true)
+          (PairRouting.childEmbedding y c.val hcy true)
+        have hh := child_power_product_bound d x y c.val hcx hcy a h₀ h₁
+        have hh' := mul_le_mul_of_nonneg_left hh
+          (mul_nonneg (Real.rpow_nonneg (le_of_lt (PairRouting.outerWeight_pos x y)) a)
+            (Real.rpow_nonneg (Nat.cast_nonneg C.card) a))
+        exact hh'.trans (assignment_moment_bound d x y c.val c.property a ha)
+      have hnode := PairRouting.sandwich_power_bound (palindromePerm d) (palindromePerm d) x y a ha
+      rw [←palindrome_probability_step d e f] at hnode
+      change (PairRouting.tupleProbability (palindromePerm (d+1)) e f)^(1+a) ≤
+        W^(1+a) * (C.card:ℝ)^a * ∑ c ∈ C, p c at hnode
+      have hsum : ∑ c ∈ C, p c = ∑ c : {c // c ∈ C}, p c.val := by
+        rw [Finset.sum_subtype C (fun _ => Iff.rfl)]
+      have hW : 0 < W := PairRouting.outerWeight_pos x y
+      calc
+        _ ≤ W^(1+a) * (C.card:ℝ)^a * ∑ c ∈ C, p c := hnode
+        _ = W * ∑ c : {c // c ∈ C}, W^a * (C.card:ℝ)^a * p c.val := by
+          rw [Real.rpow_add hW,Real.rpow_one,hsum,←Finset.mul_sum]
+          ring
+        _ ≤ W * ∑ c : {c // c ∈ C}, (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * m c :=
+          mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun c _ => hp c)) (le_of_lt hW)
+        _ = _ := by
+          rw [←Finset.mul_sum,routingMoment_step]
+          simp only [Nat.cast_add,Nat.cast_one]
+          change W * ((2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * ∑ c, m c) =
+            (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * (W * ∑ c, m c)
+          ring
+
+lemma routingMoment_sum (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (a : ℝ) :
+    (∑ f : ι ↪ Card d, routingMoment d e f a) =
+      finiteMean (fun ω : BenesCoins d => (2:ℝ)^((palindromeCost d e ω:ℝ)*a)) := by
+  classical
+  unfold routingMoment
+  rw [←finiteMean_sum]
+  apply finiteMean_congr
+  intro ω
+  let f₀ : ι ↪ Card d := e.trans (palindromePerm d ω).toEmbedding
+  rw [Finset.sum_eq_single f₀]
+  · have hm : ∀ i, palindromePerm d ω (e i) = f₀ i := fun _ => rfl
+    simp only [hm,implies_true,↓reduceIte]
+  · intro f _ hf
+    have hm : ¬ ∀ i, palindromePerm d ω (e i) = f i := by
+      intro hh
+      apply hf
+      ext i : 1
+      exact (hh i).symm
+    exact ite_eq_right hm
+  · simp
+
+lemma finite_row_power {X : Type*} [Fintype X] [Nonempty X]
+    (p : X → ℝ) (hp : ∀ x, 0 ≤ p x) (a : ℝ) :
+    finiteMean (fun x => ((Fintype.card X:ℝ)*p x)^(1+a)) =
+      (Fintype.card X:ℝ)^a * ∑ x, (p x)^(1+a) := by
+  have hn : (0:ℝ) < Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
+  simp only [finiteMean,Real.mul_rpow (le_of_lt hn) (hp _),←Finset.mul_sum]
+  rw [Real.rpow_add hn,Real.rpow_one]
+  field_simp
+
+noncomputable def palindromeRowMoment (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (a : ℝ) : ℝ :=
+  finiteMean (fun f : ι ↪ Card d =>
+    ((Fintype.card (ι ↪ Card d):ℝ)*PairRouting.tupleProbability (palindromePerm d) e f)^(1+a))
+
+lemma density_normalization (d h n : ℕ) (a : ℝ) :
+    (n:ℝ)^a * (2:ℝ)^(-(d:ℝ)*h*a) = ((n:ℝ)/((2:ℝ)^d)^h)^a := by
+  rw [Real.div_rpow (Nat.cast_nonneg _) (by positivity),←pow_mul,
+    ←Real.rpow_natCast,←Real.rpow_mul (by norm_num : (0:ℝ) ≤ 2),Nat.cast_mul]
+  rw [show -(d:ℝ)*h*a = -((d:ℝ)*h*a) by ring,
+    Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2),div_eq_mul_inv]
+
+theorem palindrome_row_density_moment (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (a : ℝ) (ha : 0 ≤ a) :
+    palindromeRowMoment d e a ≤
+      (((2^d).descFactorial (Fintype.card ι):ℝ)/((2^d:ℕ):ℝ)^(Fintype.card ι))^a *
+        finiteMean (fun ω : BenesCoins d => (2:ℝ)^((palindromeCost d e ω:ℝ)*a)) := by
+  classical
+  have : Nonempty (ι ↪ Card d) := ⟨e⟩
+  rw [palindromeRowMoment,finite_row_power _ (fun f => PairRouting.tupleProbability_nonneg _ _ _)]
+  calc
+    _ ≤ (Fintype.card (ι ↪ Card d):ℝ)^a *
+      ∑ f : ι ↪ Card d, (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) * routingMoment d e f a :=
+      mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun f _ => palindrome_pointwise_moment d e f a ha))
+        (Real.rpow_nonneg (Nat.cast_nonneg _) _)
+    _ = _ := by
+      rw [←Finset.mul_sum,routingMoment_sum,←mul_assoc,density_normalization,
+        Fintype.card_embedding_eq,card_positions]
+      norm_cast
+
+end Thorp
+namespace Thorp
+
+lemma subset_biUnion_indicator {α β : Type*} [DecidableEq α] [Fintype β]
+    (A : Finset α) (T : β → Finset α) :
+    (if Finset.univ.biUnion T ⊆ A then (1:ℝ) else 0) =
+      ∏ b, if T b ⊆ A then (1:ℝ) else 0 := by
+  classical
+  by_cases h : Finset.univ.biUnion T ⊆ A
+  · rw [ite_eq_left h]
+    symm
+    apply Finset.prod_eq_one
+    intro b _
+    rw [ite_eq_left]
+    exact (Finset.subset_biUnion_of_mem T (Finset.mem_univ b)).trans h
+  · rw [ite_eq_right h]
+    obtain ⟨x,hx,hxA⟩ := Finset.not_subset.mp h
+    obtain ⟨b,_,hxb⟩ := Finset.mem_biUnion.mp hx
+    symm
+    apply Finset.prod_eq_zero (Finset.mem_univ b)
+    rw [ite_eq_right]
+    exact fun hb => hxA (hb hxb)
+
+lemma cylinder_block_polynomial {α β Ω : Type*} [DecidableEq α] [Fintype β]
+    [Fintype Ω] (τ : β → Type*) [∀ b, Fintype (τ b)]
+    (F : Ω → Finset α) (p : α → ℝ)
+    (hF : CylinderBound (inclusionMoment F) p)
+    (T : ∀ b, τ b → Finset α) (c : ∀ b, τ b → ℝ)
+    (hc : ∀ b s, 0 ≤ c b s)
+    (hd : ∀ b b', b ≠ b' → ∀ s s', Disjoint (T b s) (T b' s')) :
+    finiteMean (fun ω => ∏ b, ∑ s, c b s * if T b s ⊆ F ω then 1 else 0) ≤
+      ∏ b, ∑ s, c b s * ∏ x ∈ T b s, p x := by
+  classical
+  simp_rw [Fintype.prod_sum]
+  rw [finiteMean_sum]
+  apply Finset.sum_le_sum
+  intro s _
+  let U := Finset.univ.biUnion (fun b => T b (s b))
+  have he : (fun ω => ∏ b, c b (s b) * if T b (s b) ⊆ F ω then 1 else 0) =
+      fun ω => (if U ⊆ F ω then 1 else 0) * ∏ b, c b (s b) := by
+    funext ω
+    rw [Finset.prod_mul_distrib,←subset_biUnion_indicator]
+    exact mul_comm _ _
+  rw [he,finiteMean_mul_const]
+  have hdis : ((Finset.univ : Finset β) : Set β).PairwiseDisjoint (fun b => T b (s b)) := by
+    intro b _ b' _ hbb
+    exact hd b b' hbb (s b) (s b')
+  have hmean : finiteMean (fun ω => if U ⊆ F ω then 1 else 0) = inclusionMoment F U := by
+    unfold inclusionMoment
+    apply finiteMean_congr
+    intro ω
+    by_cases hh : U ⊆ F ω <;> simp [hh]
+  calc
+    _ ≤ (∏ x ∈ U, p x) * ∏ b, c b (s b) :=
+      mul_le_mul_of_nonneg_right (hmean.le.trans (hF U)) (Finset.prod_nonneg (fun b _ => hc b (s b)))
+    _ = _ := by
+      dsimp only [U]
+      rw [Finset.prod_biUnion hdis,←Finset.prod_mul_distrib]
+      apply Finset.prod_congr rfl
+      intro b _
+      exact mul_comm _ _
+
+
+noncomputable def sparseCoefficient {α : Type*} (w : ℝ) (T : Finset α) : ℝ :=
+  if T.card = 1 then 0 else w^T.card
+
+lemma sparseCoefficient_nonneg {α : Type*} (w : ℝ) (hw : 0 ≤ w) (T : Finset α) :
+    0 ≤ sparseCoefficient w T := by
+  unfold sparseCoefficient
+  split_ifs <;> positivity
+
+lemma sparse_block_pointwise {α : Type*} [DecidableEq α] (A D : Finset α)
+    (w : ℝ) (hw : 0 ≤ w) :
+    w^(if 2 ≤ (D ∩ A).card then (D ∩ A).card else 0) ≤
+      ∑ T : ↑D.powerset, sparseCoefficient w T.val * if T.val ⊆ A then 1 else 0 := by
+  classical
+  let V := D ∩ A
+  let T : ↑D.powerset := if 2 ≤ V.card then
+    ⟨V,Finset.mem_powerset.mpr Finset.inter_subset_left⟩ else
+    ⟨∅,Finset.mem_powerset.mpr (Finset.empty_subset D)⟩
+  calc
+    _ = sparseCoefficient w T.val * (if T.val ⊆ A then 1 else 0) := by
+      by_cases h : 2 ≤ V.card
+      · have hn : V.card ≠ 1 := by omega
+        simp [T,ite_eq_left h,sparseCoefficient,hn,show V ⊆ A from Finset.inter_subset_right,V]
+      · simp [T,ite_eq_right h,sparseCoefficient,V] at *
+    _ ≤ _ := Finset.single_le_sum (f := fun S : ↑D.powerset =>
+      sparseCoefficient w S.val * (if S.val ⊆ A then 1 else 0))
+      (fun S _ => mul_nonneg (sparseCoefficient_nonneg w hw S.val)
+        (by split_ifs <;> norm_num)) (Finset.mem_univ T)
+
+lemma geometric_sum_half (n : ℕ) {x : ℝ} (hx : 0 ≤ x) (hhalf : x ≤ 1/2) :
+    ∑ i ∈ Finset.range n, x^i ≤ 2 := by
+  induction n with
+  | zero => simp
+  | succ n ih =>
+    rw [Finset.sum_range_succ',pow_zero]
+    simp only [pow_succ',←Finset.mul_sum]
+    nlinarith
+
+lemma sparse_binomial_bound (R : ℕ) (z : ℝ) (hz : 0 ≤ z) (hr : (R:ℝ)*z ≤ 1/2) :
+    ∑ i ∈ Finset.range (R+1), (R.choose i:ℝ) * (if i = 1 then 0 else z^i) ≤
+      1 + 2*((R:ℝ)*z)^2 := by
+  classical
+  let x := (R:ℝ)*z
+  have hx : 0 ≤ x := mul_nonneg (Nat.cast_nonneg _) hz
+  have hbound (i : ℕ) : (R.choose i:ℝ)*(if i=1 then 0 else z^i) ≤
+      if i=1 then 0 else x^i := by
+    by_cases h : i=1
+    · simp [h]
+    · simp only [ite_eq_right h]
+      dsimp [x]
+      rw [mul_pow]
+      apply mul_le_mul_of_nonneg_right _ (pow_nonneg hz i)
+      exact_mod_cast Nat.choose_le_pow R i
+  calc
+    _ ≤ ∑ i ∈ Finset.range (R+1), if i=1 then 0 else x^i :=
+      Finset.sum_le_sum (fun i _ => hbound i)
+    _ ≤ 1+2*x^2 := by
+      rcases R with _ | R
+      · simp [x]
+      rcases R with _ | R
+      · norm_num [Finset.sum_range_succ]
+        positivity
+      have he : ∑ i ∈ Finset.range (R+2+1), (if i=1 then 0 else x^i) =
+          1 + x^2*∑ i ∈ Finset.range (R+1), x^i := by
+        rw [show R+2+1=2+(R+1) by omega,Finset.sum_range_add]
+        have hn (i : ℕ) : 2+i ≠ 1 := by omega
+        simp only [hn,ite_false,pow_add,←Finset.mul_sum]
+        norm_num [Finset.sum_range_succ]
+      rw [he]
+      have hg := geometric_sum_half (R+1) hx hr
+      nlinarith [sq_nonneg x]
+
+lemma sparse_block_product_bound {α : Type*} [DecidableEq α]
+    (D : Finset α) (p w : ℝ) (hp : 0 ≤ p) (hw : 0 ≤ w)
+    (hr : (D.card:ℝ)*(p*w) ≤ 1/2) :
+    ∑ T : ↑D.powerset, sparseCoefficient w T.val * p^T.val.card ≤
+      Real.exp (2*((D.card:ℝ)*(p*w))^2) := by
+  classical
+  have he : ∑ T : ↑D.powerset, sparseCoefficient w T.val * p^T.val.card =
+      ∑ i ∈ Finset.range (D.card+1), (D.card.choose i:ℝ)*(if i=1 then 0 else (p*w)^i) := by
+    rw [Finset.sum_coe_sort D.powerset (fun T => sparseCoefficient w T*p^T.card)]
+    have hh : (fun T : Finset α => sparseCoefficient w T*p^T.card) =
+        fun T => if T.card=1 then 0 else (p*w)^T.card := by
+      funext T
+      unfold sparseCoefficient
+      split_ifs <;> simp [mul_pow,mul_comm]
+    change (∑ T ∈ D.powerset, (fun T : Finset α => sparseCoefficient w T*p^T.card) T) = _
+    rw [hh,Finset.sum_powerset_apply_card (fun i => if i=1 then (0:ℝ) else (p*w)^i)]
+    simp only [nsmul_eq_mul]
+  rw [he]
+  exact (sparse_binomial_bound D.card (p*w) (mul_nonneg hp hw) hr).trans
+    (by simpa only [add_comm] using Real.add_one_le_exp (2*((D.card:ℝ)*(p*w))^2))
+
+
+noncomputable def sparseMass {α β : Type*} [DecidableEq α] [Fintype β]
+    (D : β → Finset α) (A : Finset α) : ℕ :=
+  ∑ b, if 2 ≤ (D b ∩ A).card then (D b ∩ A).card else 0
+
+theorem sparse_block_mgf {α β Ω : Type*} [DecidableEq α] [Fintype β] [Fintype Ω]
+    (D : β → Finset α) (hd : Pairwise (fun b b' => Disjoint (D b) (D b')))
+    (F : Ω → Finset α) (p w : ℝ) (hp : 0 ≤ p) (hw : 0 ≤ w)
+    (hF : ∀ T, inclusionMoment F T ≤ p^T.card)
+    (hr : ∀ b, ((D b).card:ℝ)*(p*w) ≤ 1/2) :
+    finiteMean (fun ω => w^(sparseMass D (F ω))) ≤
+      Real.exp (∑ b, 2*(((D b).card:ℝ)*(p*w))^2) := by
+  classical
+  have hc : CylinderBound (inclusionMoment F) (fun _ => p) := by
+    intro T
+    simpa only [Finset.prod_const] using hF T
+  have hb := cylinder_block_polynomial (fun b => ↑(D b).powerset) F (fun _ => p) hc
+    (fun _ S => S.val) (fun _ S => sparseCoefficient w S.val)
+    (fun _ S => sparseCoefficient_nonneg w hw S.val)
+    (fun b b' h S S' => (hd h).mono
+      (Finset.mem_powerset.mp S.property) (Finset.mem_powerset.mp S'.property))
+  calc
+    _ ≤ finiteMean (fun ω => ∏ b, ∑ S : ↑(D b).powerset,
+        sparseCoefficient w S.val * if S.val ⊆ F ω then 1 else 0) := by
+      apply finiteMean_mono
+      intro ω
+      rw [sparseMass,←Finset.prod_pow_eq_pow_sum]
+      apply Finset.prod_le_prod₀ (fun _ _ => pow_nonneg hw _)
+      intro b _
+      exact sparse_block_pointwise (F ω) (D b) w hw
+    _ ≤ ∏ b, ∑ S : ↑(D b).powerset, sparseCoefficient w S.val * p^S.val.card := by
+      simpa only [Finset.prod_const] using hb
+    _ ≤ _ := by
+      rw [Real.exp_sum]
+      apply Finset.prod_le_prod₀
+      · intro b _
+        exact Finset.sum_nonneg (fun S _ => mul_nonneg
+          (sparseCoefficient_nonneg w hw S.val) (pow_nonneg hp _))
+      · intro b _
+        exact sparse_block_product_bound (D b) p w hp hw (hr b)
+
+theorem inverseButterfly_sparse_mgf (d : ℕ) {β : Type*} [Fintype β]
+    (D : β → Finset (Card d)) (hd : Pairwise (fun b b' => Disjoint (D b) (D b')))
+    (A : Finset (Card d)) (w : ℝ) (hw : 0 ≤ w)
+    (hr : ∀ b, ((D b).card:ℝ)*((A.card/(2:ℝ)^d)*w) ≤ 1/2) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      w^(sparseMass D (inverseButterflyMarks d A ω))) ≤
+      Real.exp (∑ b, 2*(((D b).card:ℝ)*((A.card/(2:ℝ)^d)*w))^2) := by
+  apply sparse_block_mgf D hd (inverseButterflyMarks d A) (A.card/(2:ℝ)^d) w
+    (by positivity) hw
+  · exact inverseButterflyMarks_inclusion d A
+  · exact hr
+
+
+def crowdedMass (h : ℕ) : ℕ := if 2 ≤ h then h else 0
+
+lemma crowdedMass_le (h : ℕ) : crowdedMass h ≤ h := by
+  unfold crowdedMass
+  split_ifs <;> omega
+
+lemma crowdedMass_add (u v : ℕ) : crowdedMass u + crowdedMass v ≤ crowdedMass (u+v) := by
+  unfold crowdedMass
+  split_ifs <;> omega
+
+lemma alternatingCycles_le_crowded {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
+    (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    PairRouting.alternatingCycles e p ≤ crowdedMass (Fintype.card ι) := by
+  unfold crowdedMass
+  split_ifs with h
+  · have hh := PairRouting.alternatingCycles_double e p
+    omega
+  · rw [PairRouting.alternatingCycles_zero e p (by omega)]
+
+lemma palindromeCost_le_crowded (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) :
+    palindromeCost d e ω ≤ d*crowdedMass (Fintype.card ι) := by
+  classical
+  induction d generalizing ι with
+  | zero => simp [palindromeCost]
+  | succ d ih =>
+    let σ := benesStepEquiv d ω
+    let x := e.trans (headTailEquiv d).toEmbedding
+    let c := PairRouting.colors x σ.2.1
+    let hc := PairRouting.colors_compatible x σ.2.1
+    have h₀ := ih (PairRouting.childEmbedding x c hc false) σ.1.1
+    have h₁ := ih (PairRouting.childEmbedding x c hc true) σ.1.2
+    have hz := alternatingCycles_le_crowded (PairRouting.switchedEmbedding x σ.2.1)
+      (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1))
+    have hs := crowdedMass_add (Fintype.card {i // c i = false})
+      (Fintype.card {i // c i = true})
+    rw [PairRouting.color_card_add] at hs
+    change PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
+        (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) +
+        palindromeCost d (PairRouting.childEmbedding x c hc false) σ.1.1 +
+        palindromeCost d (PairRouting.childEmbedding x c hc true) σ.1.2 ≤ _
+    nlinarith
+
+noncomputable def palindromeLowCost (H : ℕ) : (d : ℕ) → {ι : Type*} → [Fintype ι] →
+    (ι ↪ Card d) → BenesCoins d → ℕ
+  | 0, _, _, _, _ => 0
+  | d+1, _, _, e, ω =>
+      let σ := benesStepEquiv d ω
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x σ.2.1
+      let hc := PairRouting.colors_compatible x σ.2.1
+      (if d+1 ≤ H then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
+          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) else 0) +
+        palindromeLowCost H d (PairRouting.childEmbedding x c hc false) σ.1.1 +
+        palindromeLowCost H d (PairRouting.childEmbedding x c hc true) σ.1.2
+
+noncomputable def cutoffMass (H : ℕ) : (d : ℕ) → {ι : Type*} → [Fintype ι] →
+    (ι ↪ Card d) → BenesCoins d → ℕ
+  | 0, _, _, _, _ => 0
+  | d+1, ι, _, e, ω =>
+      if d+1 ≤ H then crowdedMass (Fintype.card ι) else
+      let σ := benesStepEquiv d ω
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x σ.2.1
+      let hc := PairRouting.colors_compatible x σ.2.1
+      cutoffMass H d (PairRouting.childEmbedding x c hc false) σ.1.1 +
+        cutoffMass H d (PairRouting.childEmbedding x c hc true) σ.1.2
+
+lemma palindromeLowCost_eq (H d : ℕ) (h : d ≤ H) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) :
+    palindromeLowCost H d e ω = palindromeCost d e ω := by
+  classical
+  induction d generalizing ι with
+  | zero => rfl
+  | succ d ih =>
+    rw [palindromeLowCost,palindromeCost,ite_eq_left h]
+    congr 2 <;> exact ih (by omega) _ _
+
+lemma palindromeLowCost_le_cutoff (H d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) :
+    palindromeLowCost H d e ω ≤ H*cutoffMass H d e ω := by
+  classical
+  induction d generalizing ι with
+  | zero => simp [palindromeLowCost,cutoffMass]
+  | succ d ih =>
+    by_cases h : d+1 ≤ H
+    · rw [palindromeLowCost_eq H (d+1) h,cutoffMass,ite_eq_left h]
+      exact (palindromeCost_le_crowded (d+1) e ω).trans (Nat.mul_le_mul_right _ h)
+    · rw [palindromeLowCost,cutoffMass,ite_eq_right h,ite_eq_right h,zero_add,Nat.mul_add]
+      exact Nat.add_le_add (ih _ _) (ih _ _)
+
+
+noncomputable def centralBlock (r H : ℕ) (b : Card r) : Finset (Card (H+r)) :=
+  Finset.univ.image (fun z : Card H => (cardSplit r H).symm (b,z))
+
+end Thorp
+
+end ThorpNine.SparseSaving
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/SparseSaving/BlockProduct.lean b/lean/OAI/Probability/ThorpRouting/SparseSaving/BlockProduct.lean
index ec4091a6cc0fcddbcff9d4084e12056a4b18e56e..b8ad19552c2a1b8c7d04686ddbdc265042d3e7f8 100644
--- a/lean/OAI/Probability/ThorpRouting/SparseSaving/BlockProduct.lean
+++ b/lean/OAI/Probability/ThorpRouting/SparseSaving/BlockProduct.lean
@@ -1,894 +1,894 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.SparseSaving.Representative
-
-namespace OAI
-
-namespace ThorpNine.SparseSaving
-
-namespace Thorp.CommutingBlocks
-open scoped BigOperators Classical
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
-variable {ι : Type*} [Fintype ι]
-variable [DecidableEq ι]
-variable (P : ι → V →L[ℂ] V) (hcomm : ∀ i j, Commute (P i) (P j))
-include hcomm
-omit [CompleteSpace V] [Fintype ι] in
-lemma sector_sum (s : Finset ι) : ∑ E∈s.powerset, sector P hcomm s E=1 := by
-  induction s using Finset.induction_on with
-  | empty =>
-    simp only [Finset.powerset_empty,Finset.sum_singleton]
-    exact Finset.noncommProd_empty _ _
-  | @insert a s ha ih =>
-    rw [Finset.sum_powerset_insert ha]
-    have h₁ (E : Finset ι) (hE : E∈s.powerset) :
-        sector P hcomm (insert a s) E=P a*sector P hcomm s E := by
-      rw [sector_insert P hcomm s E a ha]
-      congr 1
-      exact ite_eq_right (fun h => ha ((Finset.mem_powerset.mp hE) h))
-    have h₂ (E : Finset ι) (_hE : E∈s.powerset) :
-        sector P hcomm (insert a s) (insert a E)=(1-P a)*sector P hcomm s E := by
-      rw [sector_insert P hcomm s (insert a E) a ha]
-      simp only [choice,Finset.mem_insert_self,↓reduceIte]
-      congr 1
-      apply sector_congr
-      intro i hi
-      simp [ne_of_mem_of_not_mem hi ha]
-    rw [Finset.sum_congr rfl h₁,Finset.sum_congr rfl h₂,
-      ←Finset.mul_sum,←Finset.mul_sum,ih]
-    simp
-
-omit [Fintype ι] in
-lemma sector_absorb_factor (s E : Finset ι) (i : ι) (hi : i∈s)
-    (hP : ∀ i, IsStarProjection (P i)) :
-    choice P E i*sector P hcomm s E=sector P hcomm s E := by
-  have hc : IsIdempotentElem (choice P E i) := by
-    unfold choice
-    split_ifs
-    · exact (hP i).one_sub.isIdempotentElem
-    · exact (hP i).isIdempotentElem
-  have hsplit := Finset.mul_noncommProd_erase s hi (choice P E)
-    (fun i _ j _ _ => choice_commute P hcomm E E i j)
-  change choice P E i*sector P hcomm s E=sector P hcomm s E
-  change _=sector P hcomm s E at hsplit
-  nth_rw 1 [←hsplit]
-  rw [←mul_assoc,hc.eq,hsplit]
-
-omit [Fintype ι] in
-lemma sector_orthogonal (s E F : Finset ι) (hE : E⊆s) (hF : F⊆s) (hEF : E≠F)
-    (hP : ∀ i, IsStarProjection (P i)) :
-    sector P hcomm s E*sector P hcomm s F=0 := by
-  have hdiff : ∃ i∈s, ¬(i∈E ↔ i∈F) := by
-    by_contra h
-    push Not at h
-    exact hEF (Finset.ext fun i => by
-      by_cases hi : i∈s
-      · exact h i hi
-      · exact iff_of_false (fun h => hi (hE h)) (fun h => hi (hF h)))
-  obtain ⟨i,hi,hdiff⟩ := hdiff
-  have hz : choice P E i*choice P F i=0 := by
-    by_cases he : i∈E <;> by_cases hf : i∈F
-    · exact (hdiff (iff_of_true he hf)).elim
-    · simpa only [choice,ite_eq_left he,ite_eq_right hf] using (hP i).one_sub_mul_self
-    · simpa only [choice,ite_eq_right he,ite_eq_left hf] using (hP i).mul_one_sub_self
-    · exact (hdiff (iff_of_false he hf)).elim
-  have hEi := sector_absorb_factor P hcomm s E i hi hP
-  have hFi := sector_absorb_factor P hcomm s F i hi hP
-  calc
-    _ = sector P hcomm s E*(choice P E i*choice P F i)*sector P hcomm s F := by
-      simp only [mul_assoc]
-      rw [hFi]
-      rw [←mul_assoc,(sector_factor_commute P hcomm s E E i).eq.symm,hEi]
-    _ = 0 := by rw [hz,mul_zero,zero_mul]
-
-noncomputable def blockProduct (A : ι → V →L[ℂ] V)
-    (hA : ∀ i j, Commute (A i) (A j)) (s : Finset ι) : V →L[ℂ] V :=
-  s.noncommProd A (fun i _ j _ _ => hA i j)
-
-omit hcomm [Fintype ι] in
-lemma norm_noncommProd_le (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
-    (s : Finset ι) : ‖blockProduct A hA s‖≤∏ i∈s,‖A i‖ := by
-  induction s using Finset.induction_on with
-  | empty => exact ContinuousLinearMap.norm_id_le
-  | @insert a s ha ih =>
-    have he : blockProduct A hA (insert a s)=A a*blockProduct A hA s :=
-      Finset.noncommProd_insert_of_notMem _ _ _ _ ha
-    rw [he,Finset.prod_insert ha]
-    exact (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_left ih (norm_nonneg _))
-
-noncomputable def blocks (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j)) : V →L[ℂ] V :=
-  blockProduct A hA Finset.univ
-
-omit hcomm in
-lemma blocks_norm_le_one (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
-    (hn : ∀ i,‖A i‖≤1) : ‖blocks A hA‖≤1 :=
-  (norm_noncommProd_le A hA Finset.univ).trans (Finset.prod_le_one₀ (fun _ _ => norm_nonneg _) (fun i _ => hn i))
-
-omit [CompleteSpace V] in
-lemma blocks_sector_commute (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
-    (hAP : ∀ i j, Commute (A i) (P j)) (E : Finset ι) :
-    Commute (blocks A hA) (sector P hcomm Finset.univ E) := by
-  unfold blocks blockProduct
-  apply (Finset.noncommProd_commute _ _ (fun i _ j _ _ => hA i j) _ _).symm
-  intro i _
-  unfold sector
-  apply (Finset.noncommProd_commute _ _ (fun i _ j _ _ => choice_commute P hcomm E E i j) _ _).symm
-  intro j _
-  unfold choice
-  split_ifs
-  · exact (Commute.one_right _).sub_right (hAP i j)
-  · exact hAP i j
-
-lemma blocks_sector_norm (hP : ∀ i, IsStarProjection (P i))
-    (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
-    (hAP : ∀ i j, Commute (A i) (P j)) (hn : ∀ i,‖A i‖≤1)
-    (θ : ℝ) (_hθ : 0≤θ) (hgap : ∀ i,‖A i*(1-P i)‖≤θ) (E : Finset ι) :
-    ‖blocks A hA*sector P hcomm Finset.univ E‖≤θ^E.card := by
-  have hAC (i j : ι) : Commute (A i) (choice P E j) := by
-    unfold choice
-    split_ifs
-    · exact (Commute.one_right _).sub_right (hAP i j)
-    · exact hAP i j
-  have hprod := Finset.noncommProd_mul_distrib (s := Finset.univ) A (choice P E)
-    (fun i _ j _ _ => hA i j) (fun i _ j _ _ => choice_commute P hcomm E E i j)
-    (fun i _ j _ _ => (hAC j i).symm)
-  change _=blocks A hA*sector P hcomm Finset.univ E at hprod
-  rw [←hprod]
-  have hc (i j : ι) : Commute ((A*choice P E) i) ((A*choice P E) j) :=
-    ((hA i j).mul_right (hAC i j)).mul_left ((hAC j i).symm.mul_right (choice_commute P hcomm E E i j))
-  apply (norm_noncommProd_le _ hc Finset.univ).trans
-  calc
-    _ ≤ ∏ i : ι, if i∈E then θ else 1 := by
-      apply Finset.prod_le_prod₀ (fun i _ => norm_nonneg _)
-      intro i _
-      dsimp only [Pi.mul_apply]
-      unfold choice
-      split_ifs with hi
-      · exact hgap i
-      · exact (norm_mul_le _ _).trans (by nlinarith [hn i,(hP i).norm_le (P i),norm_nonneg (A i),norm_nonneg (P i)])
-    _ = _ := by rw [Fintype.prod_ite_mem,Finset.prod_const]
-
-noncomputable def coarse (E : Finset ι) : V →L[ℂ] V :=
-  Eᶜ.noncommProd P (fun i _ j _ _ => hcomm i j)
-
-lemma coarse_projection (hP : ∀ i, IsStarProjection (P i)) (E : Finset ι) :
-    IsStarProjection (coarse P hcomm E) := by
-  have he : coarse P hcomm E=sector P hcomm Eᶜ ∅ := by
-    apply Finset.noncommProd_congr rfl
-    intro i _
-    simp [choice]
-  rw [he]
-  exact sector_projection P hcomm hP _ _
-
-lemma sector_coarse (hP : ∀ i, IsStarProjection (P i)) (E : Finset ι) :
-    sector P hcomm Finset.univ E*coarse P hcomm E=sector P hcomm Finset.univ E := by
-  have hpi (i : ι) (hi : i∈Eᶜ) : sector P hcomm Finset.univ E*P i=sector P hcomm Finset.univ E := by
-    have h := sector_absorb_factor P hcomm Finset.univ E i (Finset.mem_univ i) hP
-    rw [(sector_factor_commute P hcomm Finset.univ E E i).eq] at h
-    simpa only [choice,ite_eq_right (Finset.mem_compl.mp hi)] using h
-  have hp (s : Finset ι) (hs : ∀ i∈s,
-      sector P hcomm Finset.univ E*P i=sector P hcomm Finset.univ E) :
-      sector P hcomm Finset.univ E*blockProduct P hcomm s=sector P hcomm Finset.univ E := by
-    induction s using Finset.induction_on with
-    | empty => exact mul_one _
-    | @insert a s ha ih =>
-      have he : blockProduct P hcomm (insert a s)=P a*blockProduct P hcomm s :=
-        Finset.noncommProd_insert_of_notMem _ _ _ _ ha
-      rw [he,←mul_assoc,hs a (Finset.mem_insert_self _ _)]
-      exact ih (fun i hi => hs i (Finset.mem_insert_of_mem hi))
-  exact hp Eᶜ hpi
-
-theorem blocks_exception_bound (hP : ∀ i, IsStarProjection (P i))
-    (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
-    (hAP : ∀ i j, Commute (A i) (P j)) (hn : ∀ i,‖A i‖≤1)
-    (θ : ℝ) (hθ : 0≤θ) (hθ1 : θ≤1) (hgap : ∀ i,‖A i*(1-P i)‖≤θ)
-    (B : V →L[ℂ] V) (hB : ‖B‖≤1) (n : ℕ) :
-    ‖B.adjoint*blocks A hA*B‖≤θ^n+
-      ∑ E : Finset ι, if E.card<n then ‖B.adjoint*coarse P hcomm E*B‖ else 0 := by
-  let T : Finset ι → V →L[ℂ] V := sector P hcomm Finset.univ
-  have hsum : ∑ E,T E=1 := by
-    simpa only [Finset.powerset_univ] using sector_sum P hcomm Finset.univ
-  have hbound := resolution_exception_bound T (coarse P hcomm)
-    (fun E => sector_projection P hcomm hP _ E)
-    (fun E F hEF => sector_orthogonal P hcomm _ E F (Finset.subset_univ _) (Finset.subset_univ _) hEF hP)
-    hsum (coarse_projection P hcomm hP) (sector_coarse P hcomm hP)
-    (blocks A hA) B (blocks_sector_commute P hcomm A hA hAP)
-    (blocks_norm_le_one A hA hn) hB (Finset.univ.filter fun E : Finset ι => E.card<n)
-    (θ^n) (pow_nonneg hθ _) (by
-      intro E hE
-      have hEn : n≤E.card := by simpa using hE
-      exact (blocks_sector_norm P hcomm hP A hA hAP hn θ hθ hgap E).trans
-        (pow_le_pow_of_le_one hθ hθ1 hEn))
-  simpa only [Finset.sum_filter] using hbound
-
-
-end Thorp.CommutingBlocks
-
-namespace Thorp.UnitaryFinite
-open scoped BigOperators Classical
-
-variable {G : Type*} [Group G] [Fintype G]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-variable {Ω Ξ : Type*} [Fintype Ω] [Fintype Ξ]
-
-lemma sampleOperator_weighted (ρ : Representation ℂ G V) (P : Ω → G) :
-    sampleOperator ρ P=weightedOperator ρ (sampleLaw P) := by
-  unfold sampleOperator weightedOperator sampleLaw finiteMean
-  simp only [Complex.ofReal_sum,div_eq_mul_inv,Finset.sum_mul,Finset.sum_smul]
-  rw [Finset.sum_comm]
-  simp only [Finset.smul_sum]
-  apply Finset.sum_congr rfl
-  intro ω _
-  calc
-    _ = ∑ g, if P ω=g then (Fintype.card Ω:ℂ)⁻¹ • continuousRepresentation ρ g else 0 := by
-      symm
-      simp only [Finset.sum_ite_eq,Finset.mem_univ,↓reduceIte]
-    _ = _ := by
-      apply Finset.sum_congr rfl
-      intro g _
-      split_ifs <;> simp_all
-
-lemma sampleOperator_norm_le [Nonempty Ω] (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (P : Ω → G) :
-    ‖sampleOperator ρ P‖≤1 := by
-  rw [sampleOperator_weighted]
-  exact (weightedOperator_norm_le ρ hρ _ (sampleLaw_nonneg _)).trans_eq (sampleLaw_sum P)
-
-omit [Fintype G] in
-lemma sampleOperator_equiv (ρ : Representation ℂ G V) (P : Ω → G) (e : Ξ ≃ Ω) :
-    sampleOperator ρ (P ∘ e)=sampleOperator ρ P := by
-  simp only [sampleOperator,Fintype.card_congr e,Function.comp_apply]
-  rw [Equiv.sum_comp e (fun ω => continuousRepresentation ρ (P ω))]
-
-omit [Fintype G] in
-lemma sampleOperator_prod (ρ : Representation ℂ G V) (P : Ω → G) (Q : Ξ → G) :
-    sampleOperator ρ (fun ω : Ω × Ξ => P ω.1*Q ω.2)=sampleOperator ρ P*sampleOperator ρ Q := by
-  simp only [sampleOperator,Fintype.card_prod,Nat.cast_mul,mul_inv_rev,map_mul,Fintype.sum_prod_type,
-    Finset.sum_mul,Finset.mul_sum,smul_mul_smul,Finset.smul_sum]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro ω _
-  apply Finset.sum_congr rfl
-  intro ξ _
-  rw [mul_comm]
-
-omit [Fintype G] in
-lemma sampleOperator_inv (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (P : Ω → G) :
-    sampleOperator ρ (fun ω => (P ω)⁻¹)=(sampleOperator ρ P).adjoint := by
-  simp only [sampleOperator,map_smulₛₗ,map_sum,map_inv₀,map_natCast,
-    continuousRepresentation_adjoint ρ hρ]
-
-lemma sampleOperator_palindrome (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V)
-    (hρ : IsUnitary ρ) :
-    sampleOperator ρ (palindromePerm d)=
-      sampleOperator ρ (fun ω => butterflyPerm d (decodeButterfly d ω))*
-      (sampleOperator ρ (fun ω => butterflyPerm d (decodeButterfly d ω))).adjoint := by
-  rw [←sampleOperator_inv ρ hρ,←sampleOperator_prod]
-  rfl
-
-omit [Fintype G] in
-lemma sampleOperator_one [Nonempty Ω] (ρ : Representation ℂ G V) :
-    sampleOperator ρ (fun _ : Ω => 1)=1 := by
-  simp only [sampleOperator,map_one,Finset.sum_const,Finset.card_univ,←Nat.cast_smul_eq_nsmul ℂ,
-    smul_smul]
-  rw [inv_mul_cancel₀ (by exact_mod_cast Fintype.card_ne_zero),one_smul]
-
-lemma sampleOperator_uniform (ρ : Representation ℂ G V) : sampleOperator ρ (id : G → G)=uniformOperator ρ := by
-  simp only [sampleOperator,uniformOperator,weightedOperator,Complex.ofReal_inv,Complex.ofReal_natCast,
-    Finset.smul_sum,id_eq]
-
-lemma sampleOperator_uniform_mul [Nonempty Ω] (ρ : Representation ℂ G V) (P : Ω → G) :
-    sampleOperator ρ (fun ω : Ω × G => P ω.1*ω.2⁻¹)=uniformOperator ρ := by
-  rw [sampleOperator_prod]
-  have he : sampleOperator ρ (fun g : G => g⁻¹)=uniformOperator ρ := by
-    rw [←sampleOperator_uniform]
-    exact sampleOperator_equiv ρ id (Equiv.inv G)
-  rw [he,sampleOperator_weighted,weighted_uniform_left,sampleLaw_sum,one_smul]
-
-end Thorp.UnitaryFinite
-
-namespace Thorp.FiniteAverage
-open scoped BigOperators Classical
-variable {I Ω A : Type*} [Fintype I] [DecidableEq I] [Fintype Ω]
-    [Ring A] [Algebra ℂ A]
-
-noncomputable def orderedProduct : MultilinearMap ℂ (fun _ : I => A) A :=
-  MultilinearMap.domDomCongr (Fintype.equivFin I).symm
-    (MultilinearMap.mkPiAlgebraFin ℂ (Fintype.card I) A)
-
-lemma orderedProduct_eq (f : I → A) (hc : Pairwise fun i j => Commute (f i) (f j)) :
-    orderedProduct f=Finset.univ.noncommProd f (fun _ _ _ _ hij => hc hij) := by
-  unfold orderedProduct
-  rw [MultilinearMap.domDomCongr_apply,MultilinearMap.mkPiAlgebraFin_apply]
-  let l := List.ofFn (Fintype.equivFin I).symm
-  have hl : l.Nodup := List.nodup_ofFn.mpr (Fintype.equivFin I).symm.injective
-  have hu : l.toFinset=Finset.univ := by
-    ext i
-    simp only [List.mem_toFinset,List.mem_ofFn,Finset.mem_univ,iff_true,l]
-    exact (Fintype.equivFin I).symm.surjective i
-  have he := Finset.noncommProd_toFinset l f (by intro i hi j hj hij; exact hc hij) hl
-  rw [hu] at he
-  rw [he]
-  congr 1
-  exact List.map_ofFn.symm
-
-lemma multilinear_average (F : MultilinearMap ℂ (fun _ : I => A) A) (x : I → Ω → A) :
-    F (fun i => (Fintype.card Ω:ℂ)⁻¹ • ∑ ω,x i ω)=
-      (Fintype.card (I → Ω):ℂ)⁻¹ • ∑ ω : I → Ω,F (fun i => x i (ω i)) := by
-  rw [F.map_smul_univ,F.map_sum]
-  simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fun,Nat.cast_pow,inv_pow]
-
-lemma ordered_average (x : I → Ω → A)
-    (hc : ∀ ω, Pairwise fun i j => Commute (x i (ω i)) (x j (ω j)))
-    (havg : Pairwise fun i j => Commute ((Fintype.card Ω:ℂ)⁻¹ • ∑ ω,x i ω)
-      ((Fintype.card Ω:ℂ)⁻¹ • ∑ ω,x j ω)) :
-    (Finset.univ.noncommProd (fun i => (Fintype.card Ω:ℂ)⁻¹ • ∑ ω,x i ω)
-      (fun _ _ _ _ hij => havg hij))=
-    (Fintype.card (I → Ω):ℂ)⁻¹ • ∑ ω : I → Ω,
-      Finset.univ.noncommProd (fun i => x i (ω i)) (fun _ _ _ _ hij => hc ω hij) := by
-  rw [←orderedProduct_eq _ havg,multilinear_average]
-  simp only [orderedProduct_eq _ (hc _)]
-
-end Thorp.FiniteAverage
-namespace Thorp.FiniteCharacters
-open Module
-open scoped BigOperators Classical
-
-variable {G ι : Type*} [Group G] [Fintype G] [Fintype ι]
-variable (V : ι → Type*) [∀ i, AddCommGroup (V i)] [∀ i, Module ℂ (V i)]
-  [∀ i, FiniteDimensional ℂ (V i)]
-variable (ρ : ∀ i, Representation ℂ G (V i)) [∀ i, Representation.IsIrreducible (ρ i)]
-variable (hneq : ∀ i j, i≠j → ¬Nonempty (Representation.Equiv (ρ i) (ρ j)))
-  (hcard : Nat.card (ConjClasses G)≤Fintype.card ι)
-include hneq hcard
-
-lemma character_expansion (f : ConjClasses G → ℂ) :
-    f=∑ i, characterDual (ρ i) f • characterClass (ρ i) := by
-  let : Fintype (ConjClasses G) := Fintype.ofFinite _
-  have horth (i j : ι) : characterDual (ρ i) (characterClass (ρ j))=if i=j then 1 else 0 := by
-    rw [characterDual_characterClass]
-    by_cases h : i=j
-    · subst j; rw [ite_eq_left ⟨Representation.Equiv.refl _⟩,ite_eq_left rfl]
-    · rw [ite_eq_right h,ite_eq_right (hneq i j h)]
-  have hLI : LinearIndependent ℂ (fun i => characterClass (ρ i)) := by
-    apply LinearIndependent.of_pairwise_dual_eq_zero_one _ (fun i => characterDual (ρ i))
-    · intro i j hij; exact (horth i j).trans (ite_eq_right hij)
-    · intro i; simpa using horth i i
-  have heq : Fintype.card ι=Module.finrank ℂ (ConjClasses G → ℂ) := by
-    apply Nat.le_antisymm hLI.fintype_card_le_finrank
-    simpa only [Module.finrank_pi,Module.finrank_self,Finset.sum_const,Finset.card_univ,
-      smul_eq_mul,mul_one,Nat.card_eq_fintype_card] using hcard
-  let b := Basis.mk hLI (by rw [hLI.span_eq_top_of_card_eq_finrank' heq])
-  have hb (i : ι) : b i=characterClass (ρ i) := Basis.mk_apply _ _ _
-  have hd (i : ι) : b.coord i=characterDual (ρ i) := by
-    apply b.ext
-    intro j
-    rw [hb,horth]
-    simp only [←hb,Basis.coord_apply,Basis.repr_self,Finsupp.single_apply]
-    split_ifs <;> simp_all
-  have h := b.sum_repr f
-  simpa only [hb,show ∀ i,b.repr f i=characterDual (ρ i) f from fun i => congrArg (fun l => l f) (hd i)] using h.symm
-
-theorem sum_dimension_sq : ∑ i, (Module.finrank ℂ (V i))^2=Fintype.card G := by
-  let f : ConjClasses G → ℂ := fun c => if c=ConjClasses.mk (1:G) then 1 else 0
-  have hf (g : G) : f (ConjClasses.mk g)=if g=1 then 1 else 0 := by
-    simp only [f,ConjClasses.mk_eq_mk_iff_isConj,isConj_one_left]
-  have hd (i : ι) : characterDual (ρ i) f=(Fintype.card G:ℂ)⁻¹*(Module.finrank ℂ (V i):ℂ) := by
-    change (Nat.card G:ℂ)⁻¹ * ∑ g, f (ConjClasses.mk g)*(ρ i).character g⁻¹=_
-    simp only [hf,ite_mul,one_mul,zero_mul,Finset.sum_ite_eq',Finset.mem_univ,↓reduceIte,inv_one,
-      Representation.char_one,Nat.card_eq_fintype_card]
-  have he := congrFun (character_expansion V ρ hneq hcard f) (ConjClasses.mk (1:G))
-  rw [hf,ite_eq_left rfl] at he
-  simp only [Finset.sum_apply,Pi.smul_apply,smul_eq_mul,hd,characterClass_mk,Representation.char_one] at he
-  have hN : (Fintype.card G:ℂ)≠0 := by exact_mod_cast Fintype.card_ne_zero
-  have hs : (∑ i,(Module.finrank ℂ (V i):ℂ)^2)=(Fintype.card G:ℂ) := by
-    calc
-      _ = (Fintype.card G:ℂ) * ∑ i,(Fintype.card G:ℂ)⁻¹ *
-          (Module.finrank ℂ (V i):ℂ)*(Module.finrank ℂ (V i):ℂ) := by
-        rw [Finset.mul_sum]
-        apply Finset.sum_congr rfl
-        intro i _
-        field_simp
-      _ = _ := by rw [←he,mul_one]
-  exact_mod_cast hs
-
-end Thorp.FiniteCharacters
-
-namespace Thorp.UnitaryFinite
-open scoped BigOperators ComplexConjugate Classical
-
-variable {G : Type*} [Group G] [Fintype G]
-variable {V W : Type*} [NormedAddCommGroup V] [NormedAddCommGroup W]
-  [InnerProductSpace ℂ V] [InnerProductSpace ℂ W]
-  [FiniteDimensional ℂ V] [FiniteDimensional ℂ W]
-
-omit [FiniteDimensional ℂ V] [FiniteDimensional ℂ W] in
-lemma averageMap_zero (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
-    [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ]
-    (hne : ¬Nonempty (Representation.Equiv ρ σ)) (A : V →ₗ[ℂ] W) :
-    averageMap ρ σ A=0 := by
-  rcases Representation.IsIrreducible.bijective_or_eq_zero (averageMap ρ σ A) with h|h
-  · exact (hne ⟨Representation.IntertwiningMap.ofBijective _ h⟩).elim
-  · exact h
-
-omit [FiniteDimensional ℂ V] [FiniteDimensional ℂ W] in
-lemma coefficient_cross_orthogonality (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
-    [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ]
-    (hρ : IsUnitary ρ) (hne : ¬Nonempty (Representation.Equiv ρ σ))
-    (u z : W) (v w : V) :
-    ∑ g, inner ℂ u (σ g z)*inner ℂ (ρ g w) v=0 := by
-  have he := averageMap_zero ρ σ hne (InnerProductSpace.rankOne ℂ z w).toLinearMap
-  have hv := congrArg (fun T : Representation.IntertwiningMap ρ σ => inner ℂ u (T v)) he
-  change inner ℂ u ((∑ g, (σ g).comp ((InnerProductSpace.rankOne ℂ z w).toLinearMap.comp (ρ g⁻¹))) v)=_ at hv
-  simpa only [LinearMap.sum_apply,LinearMap.comp_apply,ContinuousLinearMap.coe_coe,
-    InnerProductSpace.rankOne_apply,map_smul,unitary_inner_inv ρ hρ,inner_sum,
-    inner_smul_right,Representation.IntertwiningMap.coe_zero,Pi.zero_apply,inner_zero_right,mul_comm] using hv
-
-end Thorp.UnitaryFinite
-namespace Thorp.UnitaryFinite
-open scoped BigOperators ComplexConjugate Classical
-
-variable {G : Type*} [Group G] [Fintype G]
-
-def regularAction : G →* Equiv.Perm G where
-  toFun := Equiv.mulLeft
-  map_one' := by ext; simp
-  map_mul' _ _ := by ext; simp
-
-omit [Fintype G] in
-lemma regular_transitive (x : G) : ∃ g,regularAction g (1:G)=x := ⟨x,mul_one x⟩
-
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-omit [Fintype G] [FiniteDimensional ℂ V] in
-lemma regular_fixed (ρ : Representation ℂ G V) (w : V) : IsFixed regularAction (1:G) ρ w := by
-  intro g hg
-  have he : g=1 := by simpa only [regularAction,MonoidHom.coe_mk,OneHom.coe_mk,Equiv.coe_mulLeft,mul_one] using hg
-  subst g
-  simp
-
-noncomputable def regularCoefficient (ρ : Representation ℂ G V) (w v : V) : EuclideanSpace ℂ G :=
-  normalizedCoefficient regularAction 1 regular_transitive ρ w v
-
-omit [FiniteDimensional ℂ V] in
-lemma regularCoefficient_apply (ρ : Representation ℂ G V) (w v : V) (g : G) :
-    regularCoefficient ρ w v g = (Real.sqrt ((Module.finrank ℂ V:ℝ)/(Fintype.card G:ℝ)):ℂ)*inner ℂ (ρ g w) v := by
-  unfold regularCoefficient normalizedCoefficient
-  simp only [PiLp.smul_apply,smul_eq_mul]
-  congr 1
-  have he := coefficient_at regularAction 1 regular_transitive ρ w (regular_fixed ρ w) v g
-  simpa only [regularAction,MonoidHom.coe_mk,OneHom.coe_mk,Equiv.coe_mulLeft,mul_one] using he
-
-lemma regularCoefficient_inner (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
-    (hρ : IsUnitary ρ) (w z v t : V) :
-    inner ℂ (regularCoefficient ρ w v) (regularCoefficient ρ z t)=inner ℂ z w*inner ℂ v t :=
-  normalizedCoefficient_inner regularAction 1 regular_transitive ρ hρ w z
-    (regular_fixed ρ w) (regular_fixed ρ z) v t
-
-variable {W : Type*} [NormedAddCommGroup W] [InnerProductSpace ℂ W] [FiniteDimensional ℂ W]
-
-omit [FiniteDimensional ℂ V] [FiniteDimensional ℂ W] in
-lemma regularCoefficient_cross (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
-    [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ]
-    (hσ : IsUnitary σ) (hne : ¬Nonempty (Representation.Equiv σ ρ))
-    (w v : V) (z t : W) : inner ℂ (regularCoefficient ρ w v) (regularCoefficient σ z t)=0 := by
-  rw [PiLp.inner_apply]
-  simp only [RCLike.inner_apply,regularCoefficient_apply,map_mul,Complex.conj_ofReal,
-    inner_conj_symm]
-  have he := coefficient_cross_orthogonality σ ρ hσ hne v w t z
-  calc
-    _ = ((Real.sqrt ((Module.finrank ℂ V:ℝ)/(Fintype.card G:ℝ)):ℂ)*
-        (Real.sqrt ((Module.finrank ℂ W:ℝ)/(Fintype.card G:ℝ)):ℂ)) *
-        ∑ g,inner ℂ v (ρ g w)*inner ℂ (σ g z) t := by
-      rw [Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro g _
-      ring
-    _ = 0 := by rw [he,mul_zero]
-
-end Thorp.UnitaryFinite
-
-namespace Thorp.FiniteFourier
-open scoped BigOperators ComplexConjugate Classical
-open UnitaryFinite
-variable {G ι : Type*} [Group G] [Fintype G] [Fintype ι]
-variable (V : ι → Type*) [∀ i, NormedAddCommGroup (V i)] [∀ i, InnerProductSpace ℂ (V i)]
-  [∀ i, FiniteDimensional ℂ (V i)]
-variable (ρ : ∀ i, Representation ℂ G (V i)) [∀ i, Representation.IsIrreducible (ρ i)]
-variable (hρ : ∀ i, IsUnitary (ρ i))
-variable (hneq : ∀ i j, i≠j → ¬Nonempty (Representation.Equiv (ρ i) (ρ j)))
-  (hcard : Nat.card (ConjClasses G)≤Fintype.card ι)
-
-
-abbrev CoefficientIndex := Σ i,Fin (Module.finrank ℂ (V i)) × Fin (Module.finrank ℂ (V i))
-
-noncomputable def coefficientFamily (a : CoefficientIndex V) : EuclideanSpace ℂ G :=
-  regularCoefficient (ρ a.1) (stdOrthonormalBasis ℂ (V a.1) a.2.1) (stdOrthonormalBasis ℂ (V a.1) a.2.2)
-
-omit [Fintype ι] in
-include hρ hneq in
-lemma coefficientFamily_orthonormal : Orthonormal ℂ (coefficientFamily V ρ) := by
-  rw [orthonormal_iff_ite]
-  rintro ⟨i,a,b⟩ ⟨j,c,d⟩
-  dsimp only [coefficientFamily]
-  by_cases hij : i=j
-  · subst j
-    rw [regularCoefficient_inner _ (hρ i)]
-    simp only [(stdOrthonormalBasis ℂ (V i)).inner_eq_ite]
-    by_cases hac : a=c <;> by_cases hbd : b=d <;> simp [hac,hbd,eq_comm]
-  · rw [regularCoefficient_cross _ _ (hρ j) (hneq j i (Ne.symm hij))]
-    rw [ite_eq_right (by intro h; exact hij (congrArg Sigma.fst h))]
-
-include ρ hneq hcard in
-lemma coefficientIndex_card : Fintype.card (CoefficientIndex V)=Fintype.card G := by
-  simp only [CoefficientIndex,Fintype.card_sigma,Fintype.card_prod,Fintype.card_fin,←pow_two]
-  exact FiniteCharacters.sum_dimension_sq V ρ hneq hcard
-
-noncomputable def coefficientBasis : OrthonormalBasis (CoefficientIndex V) ℂ (EuclideanSpace ℂ G) :=
-  OrthonormalBasis.mk (coefficientFamily_orthonormal V ρ hρ hneq)
-    (by rw [(coefficientFamily_orthonormal V ρ hρ hneq).linearIndependent.span_eq_top_of_card_eq_finrank'
-          (by rw [coefficientIndex_card V ρ hneq hcard,finrank_euclideanSpace])])
-
-lemma coefficientBasis_apply (a : CoefficientIndex V) :
-    coefficientBasis V ρ hρ hneq hcard a=coefficientFamily V ρ a := congrFun (OrthonormalBasis.coe_mk _ _) a
-
-include hρ hneq hcard in
-
-theorem parseval (f : EuclideanSpace ℂ G) :
-    ∑ a : CoefficientIndex V, ‖inner ℂ (coefficientFamily V ρ a) f‖^2=‖f‖^2 := by
-  simpa only [coefficientBasis_apply] using
-    (coefficientBasis V ρ hρ hneq hcard).sum_sq_norm_inner_right f
-
-
-noncomputable def realVector (p : G → ℝ) : EuclideanSpace ℂ G := WithLp.toLp 2 (fun g => (p g:ℂ))
-
-omit [Fintype ι] [∀ i, Representation.IsIrreducible (ρ i)] in
-lemma coefficient_realVector (p : G → ℝ) (a : CoefficientIndex V) :
-    inner ℂ (coefficientFamily V ρ a) (realVector p)=
-      (Real.sqrt ((Module.finrank ℂ (V a.1):ℝ)/(Fintype.card G:ℝ)):ℂ)*
-      inner ℂ (stdOrthonormalBasis ℂ (V a.1) a.2.2)
-        (weightedOperator (ρ a.1) p (stdOrthonormalBasis ℂ (V a.1) a.2.1)) := by
-  simp only [PiLp.inner_apply,coefficientFamily,regularCoefficient_apply,realVector,
-    RCLike.inner_apply,map_mul,Complex.conj_ofReal,inner_conj_symm,
-    weightedOperator,sum_apply,smul_apply,
-    continuousRepresentation_apply,inner_sum,inner_smul_right,Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro g _
-  ring
-
-omit [Fintype ι] [∀ i, Representation.IsIrreducible (ρ i)] in
-lemma coefficient_realVector_sq (p : G → ℝ) (a : CoefficientIndex V) :
-    ‖inner ℂ (coefficientFamily V ρ a) (realVector p)‖^2=
-      ((Module.finrank ℂ (V a.1):ℝ)/(Fintype.card G:ℝ))*
-      ‖inner ℂ (stdOrthonormalBasis ℂ (V a.1) a.2.2)
-        (weightedOperator (ρ a.1) p (stdOrthonormalBasis ℂ (V a.1) a.2.1))‖^2 := by
-  rw [coefficient_realVector,norm_mul,Complex.norm_real,Real.norm_eq_abs,
-    abs_of_nonneg (Real.sqrt_nonneg _),mul_pow,Real.sq_sqrt (div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _))]
-
-include hρ hneq hcard in
-lemma parseval_operator (p : G → ℝ) :
-    (Fintype.card G:ℝ)*‖realVector p‖^2=
-      ∑ i,(Module.finrank ℂ (V i):ℝ)*∑ a : Fin (Module.finrank ℂ (V i)),
-        ‖weightedOperator (ρ i) p (stdOrthonormalBasis ℂ (V i) a)‖^2 := by
-  rw [←parseval V ρ hρ hneq hcard (realVector p)]
-  simp only [Fintype.sum_sigma,Fintype.sum_prod_type,coefficient_realVector_sq]
-  have hN : (Fintype.card G:ℝ)≠0 := by exact_mod_cast Fintype.card_ne_zero
-  rw [Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro i _
-  have he (a : Fin (Module.finrank ℂ (V i))) :
-      (∑ b : Fin (Module.finrank ℂ (V i)),
-        (Module.finrank ℂ (V i):ℝ)/(Fintype.card G:ℝ)*
-          ‖inner ℂ (stdOrthonormalBasis ℂ (V i) b)
-            (weightedOperator (ρ i) p (stdOrthonormalBasis ℂ (V i) a))‖^2)=
-        (Module.finrank ℂ (V i):ℝ)/(Fintype.card G:ℝ)*
-          ‖weightedOperator (ρ i) p (stdOrthonormalBasis ℂ (V i) a)‖^2 := by
-    rw [←Finset.mul_sum,(stdOrthonormalBasis ℂ (V i)).sum_sq_norm_inner_right]
-  simp_rw [he]
-  rw [←Finset.mul_sum]
-  field_simp
-
-include hρ hneq hcard in
-lemma realVector_bound (p : G → ℝ) :
-    (Fintype.card G:ℝ)*‖realVector p‖^2≤
-      ∑ i,(Module.finrank ℂ (V i):ℝ)^2*‖weightedOperator (ρ i) p‖^2 := by
-  rw [parseval_operator V ρ hρ hneq hcard]
-  apply Finset.sum_le_sum
-  intro i _
-  calc
-    _ ≤ (Module.finrank ℂ (V i):ℝ)*∑ _a : Fin (Module.finrank ℂ (V i)),‖weightedOperator (ρ i) p‖^2 := by
-      apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
-      apply Finset.sum_le_sum
-      intro a _
-      apply pow_le_pow_left₀ (norm_nonneg _)
-      simpa only [(stdOrthonormalBasis ℂ (V i)).norm_eq_one,mul_one] using
-        (weightedOperator (ρ i) p).le_opNorm (stdOrthonormalBasis ℂ (V i) a)
-    _ = _ := by simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]; ring
-
-include hρ hneq hcard in
-
-theorem l1_fourier_bound (p : G → ℝ) :
-    (∑ g,|p g|)^2≤∑ i,(Module.finrank ℂ (V i):ℝ)^2*‖weightedOperator (ρ i) p‖^2 := by
-  apply le_trans _ (realVector_bound V ρ hρ hneq hcard p)
-  have he : ‖realVector p‖^2=∑ g,|p g|^2 := by
-    rw [EuclideanSpace.norm_sq_eq]
-    simp only [realVector,PiLp.toLp_apply,Complex.norm_real,Real.norm_eq_abs]
-  rw [he]
-  simpa only [Finset.card_univ] using sq_sum_le_card_mul_sum_sq (s := Finset.univ) (f := fun g => |p g|)
-
-end Thorp.FiniteFourier
-
-namespace Thorp.ParallelBlocks
-open scoped BigOperators Classical
-variable {B X : Type*}
-
-def lift : (B → Equiv.Perm X) →* Equiv.Perm (B × X) where
-  toFun p :=
-    { toFun := fun x => (x.1,p x.1 x.2)
-      invFun := fun x => (x.1,(p x.1).symm x.2)
-      left_inv := fun x => by simp
-      right_inv := fun x => by simp }
-  map_one' := rfl
-  map_mul' _ _ := rfl
-
-@[simp] lemma lift_apply (p : B → Equiv.Perm X) (x : B × X) : lift p x=(x.1,p x.1 x.2) := rfl
-
-noncomputable def injection (b : B) : Equiv.Perm X →* Equiv.Perm (B × X) :=
-  lift.comp
-    { toFun := fun p c => if c=b then p else 1
-      map_one' := by funext c; simp
-      map_mul' := by intro p q; funext c; by_cases h : c=b <;> simp [h] }
-
-@[simp] lemma injection_apply (b : B) (p : Equiv.Perm X) (x : B × X) :
-    injection b p x=(x.1,if x.1=b then p x.2 else x.2) := by
-  simp only [injection,MonoidHom.comp_apply,lift_apply]
-  by_cases h : x.1=b <;> simp [h]
-
-lemma injection_commute {b c : B} (h : b≠c) (p q : Equiv.Perm X) :
-    Commute (injection b p) (injection c q) := by
-  apply Equiv.ext
-  intro x
-  simp only [Equiv.Perm.mul_apply,injection_apply]
-  by_cases hb : x.1=b <;> by_cases hc : x.1=c <;> simp_all
-
-variable [Fintype B]
-
-lemma lift_product (p : B → Equiv.Perm X) :
-    lift p=Finset.univ.noncommProd (fun b => injection b (p b))
-      (fun _ _ _ _ h => injection_commute h _ _) := by
-  have he (s : Finset B) (x : B × X) :
-      s.noncommProd (fun b => injection b (p b))
-        (fun _ _ _ _ h => injection_commute h _ _) x=
-          (x.1,if x.1∈s then p x.1 x.2 else x.2) := by
-    induction s using Finset.induction_on with
-    | empty =>
-      have hz : Finset.noncommProd (∅ : Finset B) (fun b => injection b (p b))
-          (fun _ _ _ _ h => injection_commute h _ _)=1 := Finset.noncommProd_empty _ _
-      rw [hz]; simp
-    | @insert a s ha ih =>
-      have hz : (insert a s).noncommProd (fun b => injection b (p b))
-          (fun _ _ _ _ h => injection_commute h _ _)=injection a (p a)*
-          s.noncommProd (fun b => injection b (p b))
-            (fun _ _ _ _ h => injection_commute h _ _) :=
-        Finset.noncommProd_insert_of_notMem _ _ _ _ ha
-      rw [hz,Equiv.Perm.mul_apply,ih,injection_apply]
-      by_cases h : x.1=a
-      · subst a; simp [ha]
-      · simp [h]
-  apply Equiv.ext
-  intro x
-  rw [he]
-  simp
-
-end Thorp.ParallelBlocks
-namespace Thorp.UnitaryFinite
-open scoped BigOperators Classical
-
-variable {G H : Type*} [Group G] [Group H]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-variable {Ω Ξ : Type*} [Fintype Ω] [Fintype Ξ]
-
-lemma continuousRepresentation_comp (ρ : Representation ℂ G V) (φ : H →* G) (h : H) :
-    continuousRepresentation (ρ.comp φ) h=continuousRepresentation ρ (φ h) := rfl
-
-lemma sampleOperator_comp (ρ : Representation ℂ G V) (φ : H →* G) (P : Ω → H) :
-    sampleOperator (ρ.comp φ) P=sampleOperator ρ (φ ∘ P) := rfl
-
-omit [FiniteDimensional ℂ V] in
-lemma unitary_comp (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (φ : H →* G) :
-    IsUnitary (ρ.comp φ) := fun h v w => hρ (φ h) v w
-
-lemma sampleOperator_commute (ρ : Representation ℂ G V) (P : Ω → G) (Q : Ξ → G)
-    (h : ∀ ω ξ, Commute (P ω) (Q ξ)) :
-    Commute (sampleOperator ρ P) (sampleOperator ρ Q) := by
-  unfold sampleOperator
-  apply Commute.smul_left
-  apply Commute.smul_right
-  apply Commute.sum_left
-  intro ω _
-  apply Commute.sum_right
-  intro ξ _
-  exact (h ω ξ).map (continuousRepresentation ρ)
-
-lemma sampleOperator_uniform_absorb [Fintype G] [Nonempty Ω]
-    (ρ : Representation ℂ G V) (P : Ω → G) :
-    sampleOperator ρ P*uniformOperator ρ=uniformOperator ρ := by
-  rw [sampleOperator_weighted,weighted_uniform_left,sampleLaw_sum,one_smul]
-
-end Thorp.UnitaryFinite
-namespace Thorp.ParallelBlocks
-open scoped BigOperators Classical
-open UnitaryFinite
-variable {B X Ω : Type*} [Fintype B] [DecidableEq B] [Fintype Ω]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-omit [Fintype B] [DecidableEq B] in
-lemma local_average_commute (ρ : Representation ℂ (Equiv.Perm (B × X)) V)
-    (P : B → Ω → Equiv.Perm X) (b c : B) :
-    Commute (sampleOperator (ρ.comp (injection b)) (P b))
-      (sampleOperator (ρ.comp (injection c)) (P c)) := by
-  by_cases h : b=c
-  · subst c; exact Commute.refl _
-  · rw [sampleOperator_comp,sampleOperator_comp]
-    exact sampleOperator_commute ρ _ _ (fun _ _ => injection_commute h _ _)
-
-lemma sampleOperator_parallel (ρ : Representation ℂ (Equiv.Perm (B × X)) V)
-    (P : B → Ω → Equiv.Perm X) :
-    sampleOperator ρ (fun ω : B → Ω => lift (fun b => P b (ω b)))=
-      CommutingBlocks.blocks (fun b => sampleOperator (ρ.comp (injection b)) (P b))
-        (local_average_commute ρ P) := by
-  let A (b : B) (w : Ω) := continuousRepresentation ρ (injection b (P b w))
-  have hc : ∀ ω : B → Ω, Pairwise fun b c => Commute (A b (ω b)) (A c (ω c)) := by
-    intro ω b c h
-    exact (injection_commute h _ _).map (continuousRepresentation ρ)
-  have havg : Pairwise fun b c => Commute ((Fintype.card Ω:ℂ)⁻¹ • ∑ w,A b w)
-      ((Fintype.card Ω:ℂ)⁻¹ • ∑ w,A c w) := by
-    intro b c _
-    exact local_average_commute ρ P b c
-  have he (ω : B → Ω) : continuousRepresentation ρ (lift (fun b => P b (ω b)))=
-      Finset.univ.noncommProd (fun b => A b (ω b)) (fun _ _ _ _ h => hc ω h) := by
-    rw [lift_product]
-    exact Finset.map_noncommProd _ _ _ (continuousRepresentation ρ)
-  unfold sampleOperator
-  simp_rw [he]
-  exact (FiniteAverage.ordered_average A hc havg).symm
-
-end Thorp.ParallelBlocks
-namespace Thorp
-open scoped BigOperators Classical
-
-noncomputable def splitPerm (r L : ℕ) : Equiv.Perm (Card (L+r)) ≃* Equiv.Perm (Card r × Card L) :=
-  (cardSplit r L).permCongrHom
-
-noncomputable def outerPerm (r L : ℕ) (hi : Card L × SwitchIndex r → Bool) :
-    Equiv.Perm (Card r × Card L) :=
-  (Equiv.prodComm (Card L) (Card r)).permCongrHom
-    (ParallelBlocks.lift (fun x => butterflyPerm r (decodeButterfly r (fun i => hi (x,i)))))
-
-lemma outerPerm_apply (r L : ℕ) (hi : Card L × SwitchIndex r → Bool) (x : Card r × Card L) :
-    outerPerm r L hi x=(butterflyPerm r (decodeButterfly r (fun i => hi (x.2,i))) x.1,x.2) := rfl
-
-lemma splitPerm_butterfly (r L : ℕ) (lo : Card r × SwitchIndex L → Bool)
-    (hi : Card L × SwitchIndex r → Bool) :
-    splitPerm r L (butterflyPerm (L+r) (decodeButterfly (L+r) (assembleBits r L lo hi)))=
-      outerPerm r L hi*ParallelBlocks.lift
-        (fun b => butterflyPerm L (decodeButterfly L (fun i => lo (b,i)))) := by
-  apply Equiv.ext
-  intro x
-  change cardSplit r L (butterflyPerm (L+r) (decodeButterfly (L+r) (assembleBits r L lo hi))
-    ((cardSplit r L).symm x))=_
-  rw [butterfly_split,Equiv.apply_symm_apply]
-  rfl
-
-lemma boundaryTree_node_raw (L r : ℕ) (U : Card (r+1) → Equiv.Perm (Card L))
-    (X : SwitchIndex (L+r+1) → Bool) :
-    boundaryTree L (r+1) U X=InputTree.node (fun y => X (Sum.inl y))
-      (fun b => boundaryTree L r (fun u => U (Fin.cons b u)) (fun i => X (Sum.inr (b,i)))) := by
-  conv_lhs => unfold boundaryTree
-  congr 1
-  funext b
-  cases b <;> simp only [Bool.false_eq_true,ite_false,ite_true] <;> rfl
-
-lemma boundaryOutput_node_raw (L r : ℕ) (Y : SwitchIndex (L+r+1) → Bool) (b : Card (r+1)) :
-    boundaryOutput L (r+1) Y b=
-      boundaryOutput L r (fun i => Y (Sum.inr (b 0,i))) (Fin.tail b) := by
-  conv_lhs => unfold boundaryOutput
-  cases b 0 <;> simp only [Bool.false_eq_true,ite_false,ite_true] <;> rfl
-
-lemma boundaryTree_split (L r : ℕ) (U : Card r → Equiv.Perm (Card L))
-    (lo : Card r × SwitchIndex L → Bool) (hi : Card L × SwitchIndex r → Bool) (x : Card (L+r)) :
-    cardSplit r L ((boundaryTree L r U (assembleBits r L lo hi)).perm x)=
-      let y := cardSplit r L x
-      let z := U y.1 y.2
-      (butterflyPerm r (decodeButterfly r (fun i => hi (z,i))) y.1,z) := by
-  induction r with
-  | zero =>
-    apply Prod.ext
-    · exact Subsingleton.elim _ _
-    · rfl
-  | succ r ih =>
-    let lo' : Card r × SwitchIndex L → Bool := fun j => lo (Fin.cons (x 0) j.1,j.2)
-    let hi' : Card L × SwitchIndex r → Bool := fun j => hi (j.1,Sum.inr (x 0,j.2))
-    let U' : Card r → Equiv.Perm (Card L) := fun b => U (Fin.cons (x 0) b)
-    have hh := ih U' lo' hi' (Fin.tail x)
-    let u := (boundaryTree L r U' (assembleBits r L lo' hi')).perm (Fin.tail x)
-    let y := cardSplit r L (Fin.tail x)
-    let z := U' y.1 y.2
-    let w := butterflyPerm r (decodeButterfly r (fun i => hi' (z,i))) y.1
-    change cardSplit r L u=(w,z) at hh
-    rw [boundaryTree_node_raw]
-    change ((Fin.cons (Bool.xor (x 0) (hi ((cardSplit r L u).2,Sum.inl (cardSplit r L u).1)))
-      ((cardSplit r L u).1) : Card (r+1)),(cardSplit r L u).2)=
-        ((Fin.cons (Bool.xor (x 0) (hi (z,Sum.inl w))) w : Card (r+1)),z)
-    rw [hh]
-
-lemma splitPerm_boundaryTree (L r : ℕ) (U : Card r → Equiv.Perm (Card L))
-    (lo : Card r × SwitchIndex L → Bool) (hi : Card L × SwitchIndex r → Bool) :
-    splitPerm r L (boundaryTree L r U (assembleBits r L lo hi)).perm=
-      outerPerm r L hi*ParallelBlocks.lift U := by
-  apply Equiv.ext
-  intro x
-  change cardSplit r L ((boundaryTree L r U (assembleBits r L lo hi)).perm
-    ((cardSplit r L).symm x))=_
-  rw [boundaryTree_split,Equiv.apply_symm_apply]
-  rfl
-
-lemma boundaryOutput_split (L r : ℕ) (lo : Card r × SwitchIndex L → Bool)
-    (hi : Card L × SwitchIndex r → Bool) (b : Card r) :
-    boundaryOutput L r (assembleBits r L lo hi) b=
-      butterflyPerm L (decodeButterfly L (fun i => lo (b,i))) := by
-  induction r with
-  | zero =>
-    rw [Subsingleton.elim b (fun j => Fin.elim0 j)]
-    rfl
-  | succ r ih =>
-    rw [boundaryOutput_node_raw]
-    change boundaryOutput L r (assembleBits r L (fun j => lo (Fin.cons (b 0) j.1,j.2))
-      (fun j => hi (j.1,Sum.inr (b 0,j.2)))) (Fin.tail b)=_
-    rw [ih]
-    simp only [Fin.cons_self_tail]
-
-
-abbrev OuterCoins (r L : ℕ) := Card L × SwitchIndex r → Bool
-abbrev LocalCoins (L : ℕ) := SwitchIndex L → Bool
-
-def parallelSplitCoinEquiv (r L : ℕ) :
-    ((Card r → LocalCoins L) × OuterCoins r L) ≃ LocalCoins (L+r) :=
-  (Equiv.prodCongr (Equiv.curry (Card r) (SwitchIndex L) Bool).symm (Equiv.refl _)).trans
-    (splitCoinEquiv r L)
-
-lemma parallelSplitCoinEquiv_apply (r L : ℕ) (c : Card r → LocalCoins L) (h : OuterCoins r L) :
-    parallelSplitCoinEquiv r L (c,h)=assembleBits r L (fun x => c x.1 x.2) h := by
-  exact splitCoinEquiv_apply _ _ _ _
-
-def pairAssembleEquiv (B C H : Type*) :
-    ((H × (B → C × C)) × H) ≃ (((B → C) × H) × ((B → C) × H)) where
-  toFun w := (((fun b => (w.1.2 b).1),w.1.1),((fun b => (w.1.2 b).2),w.2))
-  invFun w := ((w.1.2,fun b => (w.1.1 b,w.2.1 b)),w.2.2)
-  left_inv _w := rfl
-  right_inv _w := rfl
-
-def palindromeStageEquiv (r L : ℕ) :
-    ((OuterCoins r L × (Card r → BenesCoins L)) × OuterCoins r L) ≃ BenesCoins (L+r) :=
-  (pairAssembleEquiv _ _ _).trans
-    (Equiv.prodCongr (parallelSplitCoinEquiv r L) (parallelSplitCoinEquiv r L))
-
-lemma palindromeStageEquiv_apply (r L : ℕ)
-    (w : (OuterCoins r L × (Card r → BenesCoins L)) × OuterCoins r L) :
-    palindromeStageEquiv r L w=
-      (assembleBits r L (fun x => (w.1.2 x.1).1 x.2) w.1.1,
-       assembleBits r L (fun x => (w.1.2 x.1).2 x.2) w.2) := by
-  simp only [palindromeStageEquiv,Equiv.trans_apply,Equiv.prodCongr_apply,pairAssembleEquiv,
-    Equiv.coe_fn_mk,Prod.map_apply,parallelSplitCoinEquiv_apply]
-
-def modifiedAssembleEquiv (B C H U : Type*) :
-    ((H × (B → U × (C × C))) × H) ≃ (B → U) × (((B → C) × H) × ((B → C) × H)) where
-  toFun w := ((fun b => (w.1.2 b).1),
-    (((fun b => (w.1.2 b).2.1),w.1.1),((fun b => (w.1.2 b).2.2),w.2)))
-  invFun w := ((w.2.1.2,fun b => (w.1 b,(w.2.1.1 b,w.2.2.1 b))),w.2.2.2)
-  left_inv _w := rfl
-  right_inv _w := rfl
-
-end Thorp
-
-end ThorpNine.SparseSaving
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.SparseSaving.Representative
+
+namespace OAI
+
+namespace ThorpNine.SparseSaving
+
+namespace Thorp.CommutingBlocks
+open scoped BigOperators Classical
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
+variable {ι : Type*} [Fintype ι]
+variable [DecidableEq ι]
+variable (P : ι → V →L[ℂ] V) (hcomm : ∀ i j, Commute (P i) (P j))
+include hcomm
+omit [CompleteSpace V] [Fintype ι] in
+lemma sector_sum (s : Finset ι) : ∑ E∈s.powerset, sector P hcomm s E=1 := by
+  induction s using Finset.induction_on with
+  | empty =>
+    simp only [Finset.powerset_empty,Finset.sum_singleton]
+    exact Finset.noncommProd_empty _ _
+  | @insert a s ha ih =>
+    rw [Finset.sum_powerset_insert ha]
+    have h₁ (E : Finset ι) (hE : E∈s.powerset) :
+        sector P hcomm (insert a s) E=P a*sector P hcomm s E := by
+      rw [sector_insert P hcomm s E a ha]
+      congr 1
+      exact ite_eq_right (fun h => ha ((Finset.mem_powerset.mp hE) h))
+    have h₂ (E : Finset ι) (_hE : E∈s.powerset) :
+        sector P hcomm (insert a s) (insert a E)=(1-P a)*sector P hcomm s E := by
+      rw [sector_insert P hcomm s (insert a E) a ha]
+      simp only [choice,Finset.mem_insert_self,↓reduceIte]
+      congr 1
+      apply sector_congr
+      intro i hi
+      simp [ne_of_mem_of_not_mem hi ha]
+    rw [Finset.sum_congr rfl h₁,Finset.sum_congr rfl h₂,
+      ←Finset.mul_sum,←Finset.mul_sum,ih]
+    simp
+
+omit [Fintype ι] in
+lemma sector_absorb_factor (s E : Finset ι) (i : ι) (hi : i∈s)
+    (hP : ∀ i, IsStarProjection (P i)) :
+    choice P E i*sector P hcomm s E=sector P hcomm s E := by
+  have hc : IsIdempotentElem (choice P E i) := by
+    unfold choice
+    split_ifs
+    · exact (hP i).one_sub.isIdempotentElem
+    · exact (hP i).isIdempotentElem
+  have hsplit := Finset.mul_noncommProd_erase s hi (choice P E)
+    (fun i _ j _ _ => choice_commute P hcomm E E i j)
+  change choice P E i*sector P hcomm s E=sector P hcomm s E
+  change _=sector P hcomm s E at hsplit
+  nth_rw 1 [←hsplit]
+  rw [←mul_assoc,hc.eq,hsplit]
+
+omit [Fintype ι] in
+lemma sector_orthogonal (s E F : Finset ι) (hE : E⊆s) (hF : F⊆s) (hEF : E≠F)
+    (hP : ∀ i, IsStarProjection (P i)) :
+    sector P hcomm s E*sector P hcomm s F=0 := by
+  have hdiff : ∃ i∈s, ¬(i∈E ↔ i∈F) := by
+    by_contra h
+    push Not at h
+    exact hEF (Finset.ext fun i => by
+      by_cases hi : i∈s
+      · exact h i hi
+      · exact iff_of_false (fun h => hi (hE h)) (fun h => hi (hF h)))
+  obtain ⟨i,hi,hdiff⟩ := hdiff
+  have hz : choice P E i*choice P F i=0 := by
+    by_cases he : i∈E <;> by_cases hf : i∈F
+    · exact (hdiff (iff_of_true he hf)).elim
+    · simpa only [choice,ite_eq_left he,ite_eq_right hf] using (hP i).one_sub_mul_self
+    · simpa only [choice,ite_eq_right he,ite_eq_left hf] using (hP i).mul_one_sub_self
+    · exact (hdiff (iff_of_false he hf)).elim
+  have hEi := sector_absorb_factor P hcomm s E i hi hP
+  have hFi := sector_absorb_factor P hcomm s F i hi hP
+  calc
+    _ = sector P hcomm s E*(choice P E i*choice P F i)*sector P hcomm s F := by
+      simp only [mul_assoc]
+      rw [hFi]
+      rw [←mul_assoc,(sector_factor_commute P hcomm s E E i).eq.symm,hEi]
+    _ = 0 := by rw [hz,mul_zero,zero_mul]
+
+noncomputable def blockProduct (A : ι → V →L[ℂ] V)
+    (hA : ∀ i j, Commute (A i) (A j)) (s : Finset ι) : V →L[ℂ] V :=
+  s.noncommProd A (fun i _ j _ _ => hA i j)
+
+omit hcomm [Fintype ι] in
+lemma norm_noncommProd_le (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
+    (s : Finset ι) : ‖blockProduct A hA s‖≤∏ i∈s,‖A i‖ := by
+  induction s using Finset.induction_on with
+  | empty => exact ContinuousLinearMap.norm_id_le
+  | @insert a s ha ih =>
+    have he : blockProduct A hA (insert a s)=A a*blockProduct A hA s :=
+      Finset.noncommProd_insert_of_notMem _ _ _ _ ha
+    rw [he,Finset.prod_insert ha]
+    exact (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_left ih (norm_nonneg _))
+
+noncomputable def blocks (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j)) : V →L[ℂ] V :=
+  blockProduct A hA Finset.univ
+
+omit hcomm in
+lemma blocks_norm_le_one (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
+    (hn : ∀ i,‖A i‖≤1) : ‖blocks A hA‖≤1 :=
+  (norm_noncommProd_le A hA Finset.univ).trans (Finset.prod_le_one₀ (fun _ _ => norm_nonneg _) (fun i _ => hn i))
+
+omit [CompleteSpace V] in
+lemma blocks_sector_commute (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
+    (hAP : ∀ i j, Commute (A i) (P j)) (E : Finset ι) :
+    Commute (blocks A hA) (sector P hcomm Finset.univ E) := by
+  unfold blocks blockProduct
+  apply (Finset.noncommProd_commute _ _ (fun i _ j _ _ => hA i j) _ _).symm
+  intro i _
+  unfold sector
+  apply (Finset.noncommProd_commute _ _ (fun i _ j _ _ => choice_commute P hcomm E E i j) _ _).symm
+  intro j _
+  unfold choice
+  split_ifs
+  · exact (Commute.one_right _).sub_right (hAP i j)
+  · exact hAP i j
+
+lemma blocks_sector_norm (hP : ∀ i, IsStarProjection (P i))
+    (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
+    (hAP : ∀ i j, Commute (A i) (P j)) (hn : ∀ i,‖A i‖≤1)
+    (θ : ℝ) (_hθ : 0≤θ) (hgap : ∀ i,‖A i*(1-P i)‖≤θ) (E : Finset ι) :
+    ‖blocks A hA*sector P hcomm Finset.univ E‖≤θ^E.card := by
+  have hAC (i j : ι) : Commute (A i) (choice P E j) := by
+    unfold choice
+    split_ifs
+    · exact (Commute.one_right _).sub_right (hAP i j)
+    · exact hAP i j
+  have hprod := Finset.noncommProd_mul_distrib (s := Finset.univ) A (choice P E)
+    (fun i _ j _ _ => hA i j) (fun i _ j _ _ => choice_commute P hcomm E E i j)
+    (fun i _ j _ _ => (hAC j i).symm)
+  change _=blocks A hA*sector P hcomm Finset.univ E at hprod
+  rw [←hprod]
+  have hc (i j : ι) : Commute ((A*choice P E) i) ((A*choice P E) j) :=
+    ((hA i j).mul_right (hAC i j)).mul_left ((hAC j i).symm.mul_right (choice_commute P hcomm E E i j))
+  apply (norm_noncommProd_le _ hc Finset.univ).trans
+  calc
+    _ ≤ ∏ i : ι, if i∈E then θ else 1 := by
+      apply Finset.prod_le_prod₀ (fun i _ => norm_nonneg _)
+      intro i _
+      dsimp only [Pi.mul_apply]
+      unfold choice
+      split_ifs with hi
+      · exact hgap i
+      · exact (norm_mul_le _ _).trans (by nlinarith [hn i,(hP i).norm_le (P i),norm_nonneg (A i),norm_nonneg (P i)])
+    _ = _ := by rw [Fintype.prod_ite_mem,Finset.prod_const]
+
+noncomputable def coarse (E : Finset ι) : V →L[ℂ] V :=
+  Eᶜ.noncommProd P (fun i _ j _ _ => hcomm i j)
+
+lemma coarse_projection (hP : ∀ i, IsStarProjection (P i)) (E : Finset ι) :
+    IsStarProjection (coarse P hcomm E) := by
+  have he : coarse P hcomm E=sector P hcomm Eᶜ ∅ := by
+    apply Finset.noncommProd_congr rfl
+    intro i _
+    simp [choice]
+  rw [he]
+  exact sector_projection P hcomm hP _ _
+
+lemma sector_coarse (hP : ∀ i, IsStarProjection (P i)) (E : Finset ι) :
+    sector P hcomm Finset.univ E*coarse P hcomm E=sector P hcomm Finset.univ E := by
+  have hpi (i : ι) (hi : i∈Eᶜ) : sector P hcomm Finset.univ E*P i=sector P hcomm Finset.univ E := by
+    have h := sector_absorb_factor P hcomm Finset.univ E i (Finset.mem_univ i) hP
+    rw [(sector_factor_commute P hcomm Finset.univ E E i).eq] at h
+    simpa only [choice,ite_eq_right (Finset.mem_compl.mp hi)] using h
+  have hp (s : Finset ι) (hs : ∀ i∈s,
+      sector P hcomm Finset.univ E*P i=sector P hcomm Finset.univ E) :
+      sector P hcomm Finset.univ E*blockProduct P hcomm s=sector P hcomm Finset.univ E := by
+    induction s using Finset.induction_on with
+    | empty => exact mul_one _
+    | @insert a s ha ih =>
+      have he : blockProduct P hcomm (insert a s)=P a*blockProduct P hcomm s :=
+        Finset.noncommProd_insert_of_notMem _ _ _ _ ha
+      rw [he,←mul_assoc,hs a (Finset.mem_insert_self _ _)]
+      exact ih (fun i hi => hs i (Finset.mem_insert_of_mem hi))
+  exact hp Eᶜ hpi
+
+theorem blocks_exception_bound (hP : ∀ i, IsStarProjection (P i))
+    (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
+    (hAP : ∀ i j, Commute (A i) (P j)) (hn : ∀ i,‖A i‖≤1)
+    (θ : ℝ) (hθ : 0≤θ) (hθ1 : θ≤1) (hgap : ∀ i,‖A i*(1-P i)‖≤θ)
+    (B : V →L[ℂ] V) (hB : ‖B‖≤1) (n : ℕ) :
+    ‖B.adjoint*blocks A hA*B‖≤θ^n+
+      ∑ E : Finset ι, if E.card<n then ‖B.adjoint*coarse P hcomm E*B‖ else 0 := by
+  let T : Finset ι → V →L[ℂ] V := sector P hcomm Finset.univ
+  have hsum : ∑ E,T E=1 := by
+    simpa only [Finset.powerset_univ] using sector_sum P hcomm Finset.univ
+  have hbound := resolution_exception_bound T (coarse P hcomm)
+    (fun E => sector_projection P hcomm hP _ E)
+    (fun E F hEF => sector_orthogonal P hcomm _ E F (Finset.subset_univ _) (Finset.subset_univ _) hEF hP)
+    hsum (coarse_projection P hcomm hP) (sector_coarse P hcomm hP)
+    (blocks A hA) B (blocks_sector_commute P hcomm A hA hAP)
+    (blocks_norm_le_one A hA hn) hB (Finset.univ.filter fun E : Finset ι => E.card<n)
+    (θ^n) (pow_nonneg hθ _) (by
+      intro E hE
+      have hEn : n≤E.card := by simpa using hE
+      exact (blocks_sector_norm P hcomm hP A hA hAP hn θ hθ hgap E).trans
+        (pow_le_pow_of_le_one hθ hθ1 hEn))
+  simpa only [Finset.sum_filter] using hbound
+
+
+end Thorp.CommutingBlocks
+
+namespace Thorp.UnitaryFinite
+open scoped BigOperators Classical
+
+variable {G : Type*} [Group G] [Fintype G]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+variable {Ω Ξ : Type*} [Fintype Ω] [Fintype Ξ]
+
+lemma sampleOperator_weighted (ρ : Representation ℂ G V) (P : Ω → G) :
+    sampleOperator ρ P=weightedOperator ρ (sampleLaw P) := by
+  unfold sampleOperator weightedOperator sampleLaw finiteMean
+  simp only [Complex.ofReal_sum,div_eq_mul_inv,Finset.sum_mul,Finset.sum_smul]
+  rw [Finset.sum_comm]
+  simp only [Finset.smul_sum]
+  apply Finset.sum_congr rfl
+  intro ω _
+  calc
+    _ = ∑ g, if P ω=g then (Fintype.card Ω:ℂ)⁻¹ • continuousRepresentation ρ g else 0 := by
+      symm
+      simp only [Finset.sum_ite_eq,Finset.mem_univ,↓reduceIte]
+    _ = _ := by
+      apply Finset.sum_congr rfl
+      intro g _
+      split_ifs <;> simp_all
+
+lemma sampleOperator_norm_le [Nonempty Ω] (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (P : Ω → G) :
+    ‖sampleOperator ρ P‖≤1 := by
+  rw [sampleOperator_weighted]
+  exact (weightedOperator_norm_le ρ hρ _ (sampleLaw_nonneg _)).trans_eq (sampleLaw_sum P)
+
+omit [Fintype G] in
+lemma sampleOperator_equiv (ρ : Representation ℂ G V) (P : Ω → G) (e : Ξ ≃ Ω) :
+    sampleOperator ρ (P ∘ e)=sampleOperator ρ P := by
+  simp only [sampleOperator,Fintype.card_congr e,Function.comp_apply]
+  rw [Equiv.sum_comp e (fun ω => continuousRepresentation ρ (P ω))]
+
+omit [Fintype G] in
+lemma sampleOperator_prod (ρ : Representation ℂ G V) (P : Ω → G) (Q : Ξ → G) :
+    sampleOperator ρ (fun ω : Ω × Ξ => P ω.1*Q ω.2)=sampleOperator ρ P*sampleOperator ρ Q := by
+  simp only [sampleOperator,Fintype.card_prod,Nat.cast_mul,mul_inv_rev,map_mul,Fintype.sum_prod_type,
+    Finset.sum_mul,Finset.mul_sum,smul_mul_smul,Finset.smul_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro ω _
+  apply Finset.sum_congr rfl
+  intro ξ _
+  rw [mul_comm]
+
+omit [Fintype G] in
+lemma sampleOperator_inv (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (P : Ω → G) :
+    sampleOperator ρ (fun ω => (P ω)⁻¹)=(sampleOperator ρ P).adjoint := by
+  simp only [sampleOperator,map_smulₛₗ,map_sum,map_inv₀,map_natCast,
+    continuousRepresentation_adjoint ρ hρ]
+
+lemma sampleOperator_palindrome (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V)
+    (hρ : IsUnitary ρ) :
+    sampleOperator ρ (palindromePerm d)=
+      sampleOperator ρ (fun ω => butterflyPerm d (decodeButterfly d ω))*
+      (sampleOperator ρ (fun ω => butterflyPerm d (decodeButterfly d ω))).adjoint := by
+  rw [←sampleOperator_inv ρ hρ,←sampleOperator_prod]
+  rfl
+
+omit [Fintype G] in
+lemma sampleOperator_one [Nonempty Ω] (ρ : Representation ℂ G V) :
+    sampleOperator ρ (fun _ : Ω => 1)=1 := by
+  simp only [sampleOperator,map_one,Finset.sum_const,Finset.card_univ,←Nat.cast_smul_eq_nsmul ℂ,
+    smul_smul]
+  rw [inv_mul_cancel₀ (by exact_mod_cast Fintype.card_ne_zero),one_smul]
+
+lemma sampleOperator_uniform (ρ : Representation ℂ G V) : sampleOperator ρ (id : G → G)=uniformOperator ρ := by
+  simp only [sampleOperator,uniformOperator,weightedOperator,Complex.ofReal_inv,Complex.ofReal_natCast,
+    Finset.smul_sum,id_eq]
+
+lemma sampleOperator_uniform_mul [Nonempty Ω] (ρ : Representation ℂ G V) (P : Ω → G) :
+    sampleOperator ρ (fun ω : Ω × G => P ω.1*ω.2⁻¹)=uniformOperator ρ := by
+  rw [sampleOperator_prod]
+  have he : sampleOperator ρ (fun g : G => g⁻¹)=uniformOperator ρ := by
+    rw [←sampleOperator_uniform]
+    exact sampleOperator_equiv ρ id (Equiv.inv G)
+  rw [he,sampleOperator_weighted,weighted_uniform_left,sampleLaw_sum,one_smul]
+
+end Thorp.UnitaryFinite
+
+namespace Thorp.FiniteAverage
+open scoped BigOperators Classical
+variable {I Ω A : Type*} [Fintype I] [DecidableEq I] [Fintype Ω]
+    [Ring A] [Algebra ℂ A]
+
+noncomputable def orderedProduct : MultilinearMap ℂ (fun _ : I => A) A :=
+  MultilinearMap.domDomCongr (Fintype.equivFin I).symm
+    (MultilinearMap.mkPiAlgebraFin ℂ (Fintype.card I) A)
+
+lemma orderedProduct_eq (f : I → A) (hc : Pairwise fun i j => Commute (f i) (f j)) :
+    orderedProduct f=Finset.univ.noncommProd f (fun _ _ _ _ hij => hc hij) := by
+  unfold orderedProduct
+  rw [MultilinearMap.domDomCongr_apply,MultilinearMap.mkPiAlgebraFin_apply]
+  let l := List.ofFn (Fintype.equivFin I).symm
+  have hl : l.Nodup := List.nodup_ofFn.mpr (Fintype.equivFin I).symm.injective
+  have hu : l.toFinset=Finset.univ := by
+    ext i
+    simp only [List.mem_toFinset,List.mem_ofFn,Finset.mem_univ,iff_true,l]
+    exact (Fintype.equivFin I).symm.surjective i
+  have he := Finset.noncommProd_toFinset l f (by intro i hi j hj hij; exact hc hij) hl
+  rw [hu] at he
+  rw [he]
+  congr 1
+  exact List.map_ofFn.symm
+
+lemma multilinear_average (F : MultilinearMap ℂ (fun _ : I => A) A) (x : I → Ω → A) :
+    F (fun i => (Fintype.card Ω:ℂ)⁻¹ • ∑ ω,x i ω)=
+      (Fintype.card (I → Ω):ℂ)⁻¹ • ∑ ω : I → Ω,F (fun i => x i (ω i)) := by
+  rw [F.map_smul_univ,F.map_sum]
+  simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fun,Nat.cast_pow,inv_pow]
+
+lemma ordered_average (x : I → Ω → A)
+    (hc : ∀ ω, Pairwise fun i j => Commute (x i (ω i)) (x j (ω j)))
+    (havg : Pairwise fun i j => Commute ((Fintype.card Ω:ℂ)⁻¹ • ∑ ω,x i ω)
+      ((Fintype.card Ω:ℂ)⁻¹ • ∑ ω,x j ω)) :
+    (Finset.univ.noncommProd (fun i => (Fintype.card Ω:ℂ)⁻¹ • ∑ ω,x i ω)
+      (fun _ _ _ _ hij => havg hij))=
+    (Fintype.card (I → Ω):ℂ)⁻¹ • ∑ ω : I → Ω,
+      Finset.univ.noncommProd (fun i => x i (ω i)) (fun _ _ _ _ hij => hc ω hij) := by
+  rw [←orderedProduct_eq _ havg,multilinear_average]
+  simp only [orderedProduct_eq _ (hc _)]
+
+end Thorp.FiniteAverage
+namespace Thorp.FiniteCharacters
+open Module
+open scoped BigOperators Classical
+
+variable {G ι : Type*} [Group G] [Fintype G] [Fintype ι]
+variable (V : ι → Type*) [∀ i, AddCommGroup (V i)] [∀ i, Module ℂ (V i)]
+  [∀ i, FiniteDimensional ℂ (V i)]
+variable (ρ : ∀ i, Representation ℂ G (V i)) [∀ i, Representation.IsIrreducible (ρ i)]
+variable (hneq : ∀ i j, i≠j → ¬Nonempty (Representation.Equiv (ρ i) (ρ j)))
+  (hcard : Nat.card (ConjClasses G)≤Fintype.card ι)
+include hneq hcard
+
+lemma character_expansion (f : ConjClasses G → ℂ) :
+    f=∑ i, characterDual (ρ i) f • characterClass (ρ i) := by
+  let : Fintype (ConjClasses G) := Fintype.ofFinite _
+  have horth (i j : ι) : characterDual (ρ i) (characterClass (ρ j))=if i=j then 1 else 0 := by
+    rw [characterDual_characterClass]
+    by_cases h : i=j
+    · subst j; rw [ite_eq_left ⟨Representation.Equiv.refl _⟩,ite_eq_left rfl]
+    · rw [ite_eq_right h,ite_eq_right (hneq i j h)]
+  have hLI : LinearIndependent ℂ (fun i => characterClass (ρ i)) := by
+    apply LinearIndependent.of_pairwise_dual_eq_zero_one _ (fun i => characterDual (ρ i))
+    · intro i j hij; exact (horth i j).trans (ite_eq_right hij)
+    · intro i; simpa using horth i i
+  have heq : Fintype.card ι=Module.finrank ℂ (ConjClasses G → ℂ) := by
+    apply Nat.le_antisymm hLI.fintype_card_le_finrank
+    simpa only [Module.finrank_pi,Module.finrank_self,Finset.sum_const,Finset.card_univ,
+      smul_eq_mul,mul_one,Nat.card_eq_fintype_card] using hcard
+  let b := Basis.mk hLI (by rw [hLI.span_eq_top_of_card_eq_finrank' heq])
+  have hb (i : ι) : b i=characterClass (ρ i) := Basis.mk_apply _ _ _
+  have hd (i : ι) : b.coord i=characterDual (ρ i) := by
+    apply b.ext
+    intro j
+    rw [hb,horth]
+    simp only [←hb,Basis.coord_apply,Basis.repr_self,Finsupp.single_apply]
+    split_ifs <;> simp_all
+  have h := b.sum_repr f
+  simpa only [hb,show ∀ i,b.repr f i=characterDual (ρ i) f from fun i => congrArg (fun l => l f) (hd i)] using h.symm
+
+theorem sum_dimension_sq : ∑ i, (Module.finrank ℂ (V i))^2=Fintype.card G := by
+  let f : ConjClasses G → ℂ := fun c => if c=ConjClasses.mk (1:G) then 1 else 0
+  have hf (g : G) : f (ConjClasses.mk g)=if g=1 then 1 else 0 := by
+    simp only [f,ConjClasses.mk_eq_mk_iff_isConj,isConj_one_left]
+  have hd (i : ι) : characterDual (ρ i) f=(Fintype.card G:ℂ)⁻¹*(Module.finrank ℂ (V i):ℂ) := by
+    change (Nat.card G:ℂ)⁻¹ * ∑ g, f (ConjClasses.mk g)*(ρ i).character g⁻¹=_
+    simp only [hf,ite_mul,one_mul,zero_mul,Finset.sum_ite_eq',Finset.mem_univ,↓reduceIte,inv_one,
+      Representation.char_one,Nat.card_eq_fintype_card]
+  have he := congrFun (character_expansion V ρ hneq hcard f) (ConjClasses.mk (1:G))
+  rw [hf,ite_eq_left rfl] at he
+  simp only [Finset.sum_apply,Pi.smul_apply,smul_eq_mul,hd,characterClass_mk,Representation.char_one] at he
+  have hN : (Fintype.card G:ℂ)≠0 := by exact_mod_cast Fintype.card_ne_zero
+  have hs : (∑ i,(Module.finrank ℂ (V i):ℂ)^2)=(Fintype.card G:ℂ) := by
+    calc
+      _ = (Fintype.card G:ℂ) * ∑ i,(Fintype.card G:ℂ)⁻¹ *
+          (Module.finrank ℂ (V i):ℂ)*(Module.finrank ℂ (V i):ℂ) := by
+        rw [Finset.mul_sum]
+        apply Finset.sum_congr rfl
+        intro i _
+        field_simp
+      _ = _ := by rw [←he,mul_one]
+  exact_mod_cast hs
+
+end Thorp.FiniteCharacters
+
+namespace Thorp.UnitaryFinite
+open scoped BigOperators ComplexConjugate Classical
+
+variable {G : Type*} [Group G] [Fintype G]
+variable {V W : Type*} [NormedAddCommGroup V] [NormedAddCommGroup W]
+  [InnerProductSpace ℂ V] [InnerProductSpace ℂ W]
+  [FiniteDimensional ℂ V] [FiniteDimensional ℂ W]
+
+omit [FiniteDimensional ℂ V] [FiniteDimensional ℂ W] in
+lemma averageMap_zero (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
+    [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ]
+    (hne : ¬Nonempty (Representation.Equiv ρ σ)) (A : V →ₗ[ℂ] W) :
+    averageMap ρ σ A=0 := by
+  rcases Representation.IsIrreducible.bijective_or_eq_zero (averageMap ρ σ A) with h|h
+  · exact (hne ⟨Representation.IntertwiningMap.ofBijective _ h⟩).elim
+  · exact h
+
+omit [FiniteDimensional ℂ V] [FiniteDimensional ℂ W] in
+lemma coefficient_cross_orthogonality (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
+    [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ]
+    (hρ : IsUnitary ρ) (hne : ¬Nonempty (Representation.Equiv ρ σ))
+    (u z : W) (v w : V) :
+    ∑ g, inner ℂ u (σ g z)*inner ℂ (ρ g w) v=0 := by
+  have he := averageMap_zero ρ σ hne (InnerProductSpace.rankOne ℂ z w).toLinearMap
+  have hv := congrArg (fun T : Representation.IntertwiningMap ρ σ => inner ℂ u (T v)) he
+  change inner ℂ u ((∑ g, (σ g).comp ((InnerProductSpace.rankOne ℂ z w).toLinearMap.comp (ρ g⁻¹))) v)=_ at hv
+  simpa only [LinearMap.sum_apply,LinearMap.comp_apply,ContinuousLinearMap.coe_coe,
+    InnerProductSpace.rankOne_apply,map_smul,unitary_inner_inv ρ hρ,inner_sum,
+    inner_smul_right,Representation.IntertwiningMap.coe_zero,Pi.zero_apply,inner_zero_right,mul_comm] using hv
+
+end Thorp.UnitaryFinite
+namespace Thorp.UnitaryFinite
+open scoped BigOperators ComplexConjugate Classical
+
+variable {G : Type*} [Group G] [Fintype G]
+
+def regularAction : G →* Equiv.Perm G where
+  toFun := Equiv.mulLeft
+  map_one' := by ext; simp
+  map_mul' _ _ := by ext; simp
+
+omit [Fintype G] in
+lemma regular_transitive (x : G) : ∃ g,regularAction g (1:G)=x := ⟨x,mul_one x⟩
+
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+omit [Fintype G] [FiniteDimensional ℂ V] in
+lemma regular_fixed (ρ : Representation ℂ G V) (w : V) : IsFixed regularAction (1:G) ρ w := by
+  intro g hg
+  have he : g=1 := by simpa only [regularAction,MonoidHom.coe_mk,OneHom.coe_mk,Equiv.coe_mulLeft,mul_one] using hg
+  subst g
+  simp
+
+noncomputable def regularCoefficient (ρ : Representation ℂ G V) (w v : V) : EuclideanSpace ℂ G :=
+  normalizedCoefficient regularAction 1 regular_transitive ρ w v
+
+omit [FiniteDimensional ℂ V] in
+lemma regularCoefficient_apply (ρ : Representation ℂ G V) (w v : V) (g : G) :
+    regularCoefficient ρ w v g = (Real.sqrt ((Module.finrank ℂ V:ℝ)/(Fintype.card G:ℝ)):ℂ)*inner ℂ (ρ g w) v := by
+  unfold regularCoefficient normalizedCoefficient
+  simp only [PiLp.smul_apply,smul_eq_mul]
+  congr 1
+  have he := coefficient_at regularAction 1 regular_transitive ρ w (regular_fixed ρ w) v g
+  simpa only [regularAction,MonoidHom.coe_mk,OneHom.coe_mk,Equiv.coe_mulLeft,mul_one] using he
+
+lemma regularCoefficient_inner (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
+    (hρ : IsUnitary ρ) (w z v t : V) :
+    inner ℂ (regularCoefficient ρ w v) (regularCoefficient ρ z t)=inner ℂ z w*inner ℂ v t :=
+  normalizedCoefficient_inner regularAction 1 regular_transitive ρ hρ w z
+    (regular_fixed ρ w) (regular_fixed ρ z) v t
+
+variable {W : Type*} [NormedAddCommGroup W] [InnerProductSpace ℂ W] [FiniteDimensional ℂ W]
+
+omit [FiniteDimensional ℂ V] [FiniteDimensional ℂ W] in
+lemma regularCoefficient_cross (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
+    [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ]
+    (hσ : IsUnitary σ) (hne : ¬Nonempty (Representation.Equiv σ ρ))
+    (w v : V) (z t : W) : inner ℂ (regularCoefficient ρ w v) (regularCoefficient σ z t)=0 := by
+  rw [PiLp.inner_apply]
+  simp only [RCLike.inner_apply,regularCoefficient_apply,map_mul,Complex.conj_ofReal,
+    inner_conj_symm]
+  have he := coefficient_cross_orthogonality σ ρ hσ hne v w t z
+  calc
+    _ = ((Real.sqrt ((Module.finrank ℂ V:ℝ)/(Fintype.card G:ℝ)):ℂ)*
+        (Real.sqrt ((Module.finrank ℂ W:ℝ)/(Fintype.card G:ℝ)):ℂ)) *
+        ∑ g,inner ℂ v (ρ g w)*inner ℂ (σ g z) t := by
+      rw [Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro g _
+      ring
+    _ = 0 := by rw [he,mul_zero]
+
+end Thorp.UnitaryFinite
+
+namespace Thorp.FiniteFourier
+open scoped BigOperators ComplexConjugate Classical
+open UnitaryFinite
+variable {G ι : Type*} [Group G] [Fintype G] [Fintype ι]
+variable (V : ι → Type*) [∀ i, NormedAddCommGroup (V i)] [∀ i, InnerProductSpace ℂ (V i)]
+  [∀ i, FiniteDimensional ℂ (V i)]
+variable (ρ : ∀ i, Representation ℂ G (V i)) [∀ i, Representation.IsIrreducible (ρ i)]
+variable (hρ : ∀ i, IsUnitary (ρ i))
+variable (hneq : ∀ i j, i≠j → ¬Nonempty (Representation.Equiv (ρ i) (ρ j)))
+  (hcard : Nat.card (ConjClasses G)≤Fintype.card ι)
+
+
+abbrev CoefficientIndex := Σ i,Fin (Module.finrank ℂ (V i)) × Fin (Module.finrank ℂ (V i))
+
+noncomputable def coefficientFamily (a : CoefficientIndex V) : EuclideanSpace ℂ G :=
+  regularCoefficient (ρ a.1) (stdOrthonormalBasis ℂ (V a.1) a.2.1) (stdOrthonormalBasis ℂ (V a.1) a.2.2)
+
+omit [Fintype ι] in
+include hρ hneq in
+lemma coefficientFamily_orthonormal : Orthonormal ℂ (coefficientFamily V ρ) := by
+  rw [orthonormal_iff_ite]
+  rintro ⟨i,a,b⟩ ⟨j,c,d⟩
+  dsimp only [coefficientFamily]
+  by_cases hij : i=j
+  · subst j
+    rw [regularCoefficient_inner _ (hρ i)]
+    simp only [(stdOrthonormalBasis ℂ (V i)).inner_eq_ite]
+    by_cases hac : a=c <;> by_cases hbd : b=d <;> simp [hac,hbd,eq_comm]
+  · rw [regularCoefficient_cross _ _ (hρ j) (hneq j i (Ne.symm hij))]
+    rw [ite_eq_right (by intro h; exact hij (congrArg Sigma.fst h))]
+
+include ρ hneq hcard in
+lemma coefficientIndex_card : Fintype.card (CoefficientIndex V)=Fintype.card G := by
+  simp only [CoefficientIndex,Fintype.card_sigma,Fintype.card_prod,Fintype.card_fin,←pow_two]
+  exact FiniteCharacters.sum_dimension_sq V ρ hneq hcard
+
+noncomputable def coefficientBasis : OrthonormalBasis (CoefficientIndex V) ℂ (EuclideanSpace ℂ G) :=
+  OrthonormalBasis.mk (coefficientFamily_orthonormal V ρ hρ hneq)
+    (by rw [(coefficientFamily_orthonormal V ρ hρ hneq).linearIndependent.span_eq_top_of_card_eq_finrank'
+          (by rw [coefficientIndex_card V ρ hneq hcard,finrank_euclideanSpace])])
+
+lemma coefficientBasis_apply (a : CoefficientIndex V) :
+    coefficientBasis V ρ hρ hneq hcard a=coefficientFamily V ρ a := congrFun (OrthonormalBasis.coe_mk _ _) a
+
+include hρ hneq hcard in
+
+theorem parseval (f : EuclideanSpace ℂ G) :
+    ∑ a : CoefficientIndex V, ‖inner ℂ (coefficientFamily V ρ a) f‖^2=‖f‖^2 := by
+  simpa only [coefficientBasis_apply] using
+    (coefficientBasis V ρ hρ hneq hcard).sum_sq_norm_inner_right f
+
+
+noncomputable def realVector (p : G → ℝ) : EuclideanSpace ℂ G := WithLp.toLp 2 (fun g => (p g:ℂ))
+
+omit [Fintype ι] [∀ i, Representation.IsIrreducible (ρ i)] in
+lemma coefficient_realVector (p : G → ℝ) (a : CoefficientIndex V) :
+    inner ℂ (coefficientFamily V ρ a) (realVector p)=
+      (Real.sqrt ((Module.finrank ℂ (V a.1):ℝ)/(Fintype.card G:ℝ)):ℂ)*
+      inner ℂ (stdOrthonormalBasis ℂ (V a.1) a.2.2)
+        (weightedOperator (ρ a.1) p (stdOrthonormalBasis ℂ (V a.1) a.2.1)) := by
+  simp only [PiLp.inner_apply,coefficientFamily,regularCoefficient_apply,realVector,
+    RCLike.inner_apply,map_mul,Complex.conj_ofReal,inner_conj_symm,
+    weightedOperator,sum_apply,smul_apply,
+    continuousRepresentation_apply,inner_sum,inner_smul_right,Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro g _
+  ring
+
+omit [Fintype ι] [∀ i, Representation.IsIrreducible (ρ i)] in
+lemma coefficient_realVector_sq (p : G → ℝ) (a : CoefficientIndex V) :
+    ‖inner ℂ (coefficientFamily V ρ a) (realVector p)‖^2=
+      ((Module.finrank ℂ (V a.1):ℝ)/(Fintype.card G:ℝ))*
+      ‖inner ℂ (stdOrthonormalBasis ℂ (V a.1) a.2.2)
+        (weightedOperator (ρ a.1) p (stdOrthonormalBasis ℂ (V a.1) a.2.1))‖^2 := by
+  rw [coefficient_realVector,norm_mul,Complex.norm_real,Real.norm_eq_abs,
+    abs_of_nonneg (Real.sqrt_nonneg _),mul_pow,Real.sq_sqrt (div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _))]
+
+include hρ hneq hcard in
+lemma parseval_operator (p : G → ℝ) :
+    (Fintype.card G:ℝ)*‖realVector p‖^2=
+      ∑ i,(Module.finrank ℂ (V i):ℝ)*∑ a : Fin (Module.finrank ℂ (V i)),
+        ‖weightedOperator (ρ i) p (stdOrthonormalBasis ℂ (V i) a)‖^2 := by
+  rw [←parseval V ρ hρ hneq hcard (realVector p)]
+  simp only [Fintype.sum_sigma,Fintype.sum_prod_type,coefficient_realVector_sq]
+  have hN : (Fintype.card G:ℝ)≠0 := by exact_mod_cast Fintype.card_ne_zero
+  rw [Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro i _
+  have he (a : Fin (Module.finrank ℂ (V i))) :
+      (∑ b : Fin (Module.finrank ℂ (V i)),
+        (Module.finrank ℂ (V i):ℝ)/(Fintype.card G:ℝ)*
+          ‖inner ℂ (stdOrthonormalBasis ℂ (V i) b)
+            (weightedOperator (ρ i) p (stdOrthonormalBasis ℂ (V i) a))‖^2)=
+        (Module.finrank ℂ (V i):ℝ)/(Fintype.card G:ℝ)*
+          ‖weightedOperator (ρ i) p (stdOrthonormalBasis ℂ (V i) a)‖^2 := by
+    rw [←Finset.mul_sum,(stdOrthonormalBasis ℂ (V i)).sum_sq_norm_inner_right]
+  simp_rw [he]
+  rw [←Finset.mul_sum]
+  field_simp
+
+include hρ hneq hcard in
+lemma realVector_bound (p : G → ℝ) :
+    (Fintype.card G:ℝ)*‖realVector p‖^2≤
+      ∑ i,(Module.finrank ℂ (V i):ℝ)^2*‖weightedOperator (ρ i) p‖^2 := by
+  rw [parseval_operator V ρ hρ hneq hcard]
+  apply Finset.sum_le_sum
+  intro i _
+  calc
+    _ ≤ (Module.finrank ℂ (V i):ℝ)*∑ _a : Fin (Module.finrank ℂ (V i)),‖weightedOperator (ρ i) p‖^2 := by
+      apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
+      apply Finset.sum_le_sum
+      intro a _
+      apply pow_le_pow_left₀ (norm_nonneg _)
+      simpa only [(stdOrthonormalBasis ℂ (V i)).norm_eq_one,mul_one] using
+        (weightedOperator (ρ i) p).le_opNorm (stdOrthonormalBasis ℂ (V i) a)
+    _ = _ := by simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]; ring
+
+include hρ hneq hcard in
+
+theorem l1_fourier_bound (p : G → ℝ) :
+    (∑ g,|p g|)^2≤∑ i,(Module.finrank ℂ (V i):ℝ)^2*‖weightedOperator (ρ i) p‖^2 := by
+  apply le_trans _ (realVector_bound V ρ hρ hneq hcard p)
+  have he : ‖realVector p‖^2=∑ g,|p g|^2 := by
+    rw [EuclideanSpace.norm_sq_eq]
+    simp only [realVector,PiLp.toLp_apply,Complex.norm_real,Real.norm_eq_abs]
+  rw [he]
+  simpa only [Finset.card_univ] using sq_sum_le_card_mul_sum_sq (s := Finset.univ) (f := fun g => |p g|)
+
+end Thorp.FiniteFourier
+
+namespace Thorp.ParallelBlocks
+open scoped BigOperators Classical
+variable {B X : Type*}
+
+def lift : (B → Equiv.Perm X) →* Equiv.Perm (B × X) where
+  toFun p :=
+    { toFun := fun x => (x.1,p x.1 x.2)
+      invFun := fun x => (x.1,(p x.1).symm x.2)
+      left_inv := fun x => by simp
+      right_inv := fun x => by simp }
+  map_one' := rfl
+  map_mul' _ _ := rfl
+
+@[simp] lemma lift_apply (p : B → Equiv.Perm X) (x : B × X) : lift p x=(x.1,p x.1 x.2) := rfl
+
+noncomputable def injection (b : B) : Equiv.Perm X →* Equiv.Perm (B × X) :=
+  lift.comp
+    { toFun := fun p c => if c=b then p else 1
+      map_one' := by funext c; simp
+      map_mul' := by intro p q; funext c; by_cases h : c=b <;> simp [h] }
+
+@[simp] lemma injection_apply (b : B) (p : Equiv.Perm X) (x : B × X) :
+    injection b p x=(x.1,if x.1=b then p x.2 else x.2) := by
+  simp only [injection,MonoidHom.comp_apply,lift_apply]
+  by_cases h : x.1=b <;> simp [h]
+
+lemma injection_commute {b c : B} (h : b≠c) (p q : Equiv.Perm X) :
+    Commute (injection b p) (injection c q) := by
+  apply Equiv.ext
+  intro x
+  simp only [Equiv.Perm.mul_apply,injection_apply]
+  by_cases hb : x.1=b <;> by_cases hc : x.1=c <;> simp_all
+
+variable [Fintype B]
+
+lemma lift_product (p : B → Equiv.Perm X) :
+    lift p=Finset.univ.noncommProd (fun b => injection b (p b))
+      (fun _ _ _ _ h => injection_commute h _ _) := by
+  have he (s : Finset B) (x : B × X) :
+      s.noncommProd (fun b => injection b (p b))
+        (fun _ _ _ _ h => injection_commute h _ _) x=
+          (x.1,if x.1∈s then p x.1 x.2 else x.2) := by
+    induction s using Finset.induction_on with
+    | empty =>
+      have hz : Finset.noncommProd (∅ : Finset B) (fun b => injection b (p b))
+          (fun _ _ _ _ h => injection_commute h _ _)=1 := Finset.noncommProd_empty _ _
+      rw [hz]; simp
+    | @insert a s ha ih =>
+      have hz : (insert a s).noncommProd (fun b => injection b (p b))
+          (fun _ _ _ _ h => injection_commute h _ _)=injection a (p a)*
+          s.noncommProd (fun b => injection b (p b))
+            (fun _ _ _ _ h => injection_commute h _ _) :=
+        Finset.noncommProd_insert_of_notMem _ _ _ _ ha
+      rw [hz,Equiv.Perm.mul_apply,ih,injection_apply]
+      by_cases h : x.1=a
+      · subst a; simp [ha]
+      · simp [h]
+  apply Equiv.ext
+  intro x
+  rw [he]
+  simp
+
+end Thorp.ParallelBlocks
+namespace Thorp.UnitaryFinite
+open scoped BigOperators Classical
+
+variable {G H : Type*} [Group G] [Group H]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+variable {Ω Ξ : Type*} [Fintype Ω] [Fintype Ξ]
+
+lemma continuousRepresentation_comp (ρ : Representation ℂ G V) (φ : H →* G) (h : H) :
+    continuousRepresentation (ρ.comp φ) h=continuousRepresentation ρ (φ h) := rfl
+
+lemma sampleOperator_comp (ρ : Representation ℂ G V) (φ : H →* G) (P : Ω → H) :
+    sampleOperator (ρ.comp φ) P=sampleOperator ρ (φ ∘ P) := rfl
+
+omit [FiniteDimensional ℂ V] in
+lemma unitary_comp (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (φ : H →* G) :
+    IsUnitary (ρ.comp φ) := fun h v w => hρ (φ h) v w
+
+lemma sampleOperator_commute (ρ : Representation ℂ G V) (P : Ω → G) (Q : Ξ → G)
+    (h : ∀ ω ξ, Commute (P ω) (Q ξ)) :
+    Commute (sampleOperator ρ P) (sampleOperator ρ Q) := by
+  unfold sampleOperator
+  apply Commute.smul_left
+  apply Commute.smul_right
+  apply Commute.sum_left
+  intro ω _
+  apply Commute.sum_right
+  intro ξ _
+  exact (h ω ξ).map (continuousRepresentation ρ)
+
+lemma sampleOperator_uniform_absorb [Fintype G] [Nonempty Ω]
+    (ρ : Representation ℂ G V) (P : Ω → G) :
+    sampleOperator ρ P*uniformOperator ρ=uniformOperator ρ := by
+  rw [sampleOperator_weighted,weighted_uniform_left,sampleLaw_sum,one_smul]
+
+end Thorp.UnitaryFinite
+namespace Thorp.ParallelBlocks
+open scoped BigOperators Classical
+open UnitaryFinite
+variable {B X Ω : Type*} [Fintype B] [DecidableEq B] [Fintype Ω]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+omit [Fintype B] [DecidableEq B] in
+lemma local_average_commute (ρ : Representation ℂ (Equiv.Perm (B × X)) V)
+    (P : B → Ω → Equiv.Perm X) (b c : B) :
+    Commute (sampleOperator (ρ.comp (injection b)) (P b))
+      (sampleOperator (ρ.comp (injection c)) (P c)) := by
+  by_cases h : b=c
+  · subst c; exact Commute.refl _
+  · rw [sampleOperator_comp,sampleOperator_comp]
+    exact sampleOperator_commute ρ _ _ (fun _ _ => injection_commute h _ _)
+
+lemma sampleOperator_parallel (ρ : Representation ℂ (Equiv.Perm (B × X)) V)
+    (P : B → Ω → Equiv.Perm X) :
+    sampleOperator ρ (fun ω : B → Ω => lift (fun b => P b (ω b)))=
+      CommutingBlocks.blocks (fun b => sampleOperator (ρ.comp (injection b)) (P b))
+        (local_average_commute ρ P) := by
+  let A (b : B) (w : Ω) := continuousRepresentation ρ (injection b (P b w))
+  have hc : ∀ ω : B → Ω, Pairwise fun b c => Commute (A b (ω b)) (A c (ω c)) := by
+    intro ω b c h
+    exact (injection_commute h _ _).map (continuousRepresentation ρ)
+  have havg : Pairwise fun b c => Commute ((Fintype.card Ω:ℂ)⁻¹ • ∑ w,A b w)
+      ((Fintype.card Ω:ℂ)⁻¹ • ∑ w,A c w) := by
+    intro b c _
+    exact local_average_commute ρ P b c
+  have he (ω : B → Ω) : continuousRepresentation ρ (lift (fun b => P b (ω b)))=
+      Finset.univ.noncommProd (fun b => A b (ω b)) (fun _ _ _ _ h => hc ω h) := by
+    rw [lift_product]
+    exact Finset.map_noncommProd _ _ _ (continuousRepresentation ρ)
+  unfold sampleOperator
+  simp_rw [he]
+  exact (FiniteAverage.ordered_average A hc havg).symm
+
+end Thorp.ParallelBlocks
+namespace Thorp
+open scoped BigOperators Classical
+
+noncomputable def splitPerm (r L : ℕ) : Equiv.Perm (Card (L+r)) ≃* Equiv.Perm (Card r × Card L) :=
+  (cardSplit r L).permCongrHom
+
+noncomputable def outerPerm (r L : ℕ) (hi : Card L × SwitchIndex r → Bool) :
+    Equiv.Perm (Card r × Card L) :=
+  (Equiv.prodComm (Card L) (Card r)).permCongrHom
+    (ParallelBlocks.lift (fun x => butterflyPerm r (decodeButterfly r (fun i => hi (x,i)))))
+
+lemma outerPerm_apply (r L : ℕ) (hi : Card L × SwitchIndex r → Bool) (x : Card r × Card L) :
+    outerPerm r L hi x=(butterflyPerm r (decodeButterfly r (fun i => hi (x.2,i))) x.1,x.2) := rfl
+
+lemma splitPerm_butterfly (r L : ℕ) (lo : Card r × SwitchIndex L → Bool)
+    (hi : Card L × SwitchIndex r → Bool) :
+    splitPerm r L (butterflyPerm (L+r) (decodeButterfly (L+r) (assembleBits r L lo hi)))=
+      outerPerm r L hi*ParallelBlocks.lift
+        (fun b => butterflyPerm L (decodeButterfly L (fun i => lo (b,i)))) := by
+  apply Equiv.ext
+  intro x
+  change cardSplit r L (butterflyPerm (L+r) (decodeButterfly (L+r) (assembleBits r L lo hi))
+    ((cardSplit r L).symm x))=_
+  rw [butterfly_split,Equiv.apply_symm_apply]
+  rfl
+
+lemma boundaryTree_node_raw (L r : ℕ) (U : Card (r+1) → Equiv.Perm (Card L))
+    (X : SwitchIndex (L+r+1) → Bool) :
+    boundaryTree L (r+1) U X=InputTree.node (fun y => X (Sum.inl y))
+      (fun b => boundaryTree L r (fun u => U (Fin.cons b u)) (fun i => X (Sum.inr (b,i)))) := by
+  conv_lhs => unfold boundaryTree
+  congr 1
+  funext b
+  cases b <;> simp only [Bool.false_eq_true,ite_false,ite_true] <;> rfl
+
+lemma boundaryOutput_node_raw (L r : ℕ) (Y : SwitchIndex (L+r+1) → Bool) (b : Card (r+1)) :
+    boundaryOutput L (r+1) Y b=
+      boundaryOutput L r (fun i => Y (Sum.inr (b 0,i))) (Fin.tail b) := by
+  conv_lhs => unfold boundaryOutput
+  cases b 0 <;> simp only [Bool.false_eq_true,ite_false,ite_true] <;> rfl
+
+lemma boundaryTree_split (L r : ℕ) (U : Card r → Equiv.Perm (Card L))
+    (lo : Card r × SwitchIndex L → Bool) (hi : Card L × SwitchIndex r → Bool) (x : Card (L+r)) :
+    cardSplit r L ((boundaryTree L r U (assembleBits r L lo hi)).perm x)=
+      let y := cardSplit r L x
+      let z := U y.1 y.2
+      (butterflyPerm r (decodeButterfly r (fun i => hi (z,i))) y.1,z) := by
+  induction r with
+  | zero =>
+    apply Prod.ext
+    · exact Subsingleton.elim _ _
+    · rfl
+  | succ r ih =>
+    let lo' : Card r × SwitchIndex L → Bool := fun j => lo (Fin.cons (x 0) j.1,j.2)
+    let hi' : Card L × SwitchIndex r → Bool := fun j => hi (j.1,Sum.inr (x 0,j.2))
+    let U' : Card r → Equiv.Perm (Card L) := fun b => U (Fin.cons (x 0) b)
+    have hh := ih U' lo' hi' (Fin.tail x)
+    let u := (boundaryTree L r U' (assembleBits r L lo' hi')).perm (Fin.tail x)
+    let y := cardSplit r L (Fin.tail x)
+    let z := U' y.1 y.2
+    let w := butterflyPerm r (decodeButterfly r (fun i => hi' (z,i))) y.1
+    change cardSplit r L u=(w,z) at hh
+    rw [boundaryTree_node_raw]
+    change ((Fin.cons (Bool.xor (x 0) (hi ((cardSplit r L u).2,Sum.inl (cardSplit r L u).1)))
+      ((cardSplit r L u).1) : Card (r+1)),(cardSplit r L u).2)=
+        ((Fin.cons (Bool.xor (x 0) (hi (z,Sum.inl w))) w : Card (r+1)),z)
+    rw [hh]
+
+lemma splitPerm_boundaryTree (L r : ℕ) (U : Card r → Equiv.Perm (Card L))
+    (lo : Card r × SwitchIndex L → Bool) (hi : Card L × SwitchIndex r → Bool) :
+    splitPerm r L (boundaryTree L r U (assembleBits r L lo hi)).perm=
+      outerPerm r L hi*ParallelBlocks.lift U := by
+  apply Equiv.ext
+  intro x
+  change cardSplit r L ((boundaryTree L r U (assembleBits r L lo hi)).perm
+    ((cardSplit r L).symm x))=_
+  rw [boundaryTree_split,Equiv.apply_symm_apply]
+  rfl
+
+lemma boundaryOutput_split (L r : ℕ) (lo : Card r × SwitchIndex L → Bool)
+    (hi : Card L × SwitchIndex r → Bool) (b : Card r) :
+    boundaryOutput L r (assembleBits r L lo hi) b=
+      butterflyPerm L (decodeButterfly L (fun i => lo (b,i))) := by
+  induction r with
+  | zero =>
+    rw [Subsingleton.elim b (fun j => Fin.elim0 j)]
+    rfl
+  | succ r ih =>
+    rw [boundaryOutput_node_raw]
+    change boundaryOutput L r (assembleBits r L (fun j => lo (Fin.cons (b 0) j.1,j.2))
+      (fun j => hi (j.1,Sum.inr (b 0,j.2)))) (Fin.tail b)=_
+    rw [ih]
+    simp only [Fin.cons_self_tail]
+
+
+abbrev OuterCoins (r L : ℕ) := Card L × SwitchIndex r → Bool
+abbrev LocalCoins (L : ℕ) := SwitchIndex L → Bool
+
+def parallelSplitCoinEquiv (r L : ℕ) :
+    ((Card r → LocalCoins L) × OuterCoins r L) ≃ LocalCoins (L+r) :=
+  (Equiv.prodCongr (Equiv.curry (Card r) (SwitchIndex L) Bool).symm (Equiv.refl _)).trans
+    (splitCoinEquiv r L)
+
+lemma parallelSplitCoinEquiv_apply (r L : ℕ) (c : Card r → LocalCoins L) (h : OuterCoins r L) :
+    parallelSplitCoinEquiv r L (c,h)=assembleBits r L (fun x => c x.1 x.2) h := by
+  exact splitCoinEquiv_apply _ _ _ _
+
+def pairAssembleEquiv (B C H : Type*) :
+    ((H × (B → C × C)) × H) ≃ (((B → C) × H) × ((B → C) × H)) where
+  toFun w := (((fun b => (w.1.2 b).1),w.1.1),((fun b => (w.1.2 b).2),w.2))
+  invFun w := ((w.1.2,fun b => (w.1.1 b,w.2.1 b)),w.2.2)
+  left_inv _w := rfl
+  right_inv _w := rfl
+
+def palindromeStageEquiv (r L : ℕ) :
+    ((OuterCoins r L × (Card r → BenesCoins L)) × OuterCoins r L) ≃ BenesCoins (L+r) :=
+  (pairAssembleEquiv _ _ _).trans
+    (Equiv.prodCongr (parallelSplitCoinEquiv r L) (parallelSplitCoinEquiv r L))
+
+lemma palindromeStageEquiv_apply (r L : ℕ)
+    (w : (OuterCoins r L × (Card r → BenesCoins L)) × OuterCoins r L) :
+    palindromeStageEquiv r L w=
+      (assembleBits r L (fun x => (w.1.2 x.1).1 x.2) w.1.1,
+       assembleBits r L (fun x => (w.1.2 x.1).2 x.2) w.2) := by
+  simp only [palindromeStageEquiv,Equiv.trans_apply,Equiv.prodCongr_apply,pairAssembleEquiv,
+    Equiv.coe_fn_mk,Prod.map_apply,parallelSplitCoinEquiv_apply]
+
+def modifiedAssembleEquiv (B C H U : Type*) :
+    ((H × (B → U × (C × C))) × H) ≃ (B → U) × (((B → C) × H) × ((B → C) × H)) where
+  toFun w := ((fun b => (w.1.2 b).1),
+    (((fun b => (w.1.2 b).2.1),w.1.1),((fun b => (w.1.2 b).2.2),w.2)))
+  invFun w := ((w.2.1.2,fun b => (w.1 b,(w.2.1.1 b,w.2.2.1 b))),w.2.2.2)
+  left_inv _w := rfl
+  right_inv _w := rfl
+
+end Thorp
+
+end ThorpNine.SparseSaving
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/SparseSaving/Coins.lean b/lean/OAI/Probability/ThorpRouting/SparseSaving/Coins.lean
index cbf202901ae6254a459188acb806a1e557372a2d..34618abd78f0363c3f030d1b4e2d037c108fb29c 100644
--- a/lean/OAI/Probability/ThorpRouting/SparseSaving/Coins.lean
+++ b/lean/OAI/Probability/ThorpRouting/SparseSaving/Coins.lean
@@ -1,767 +1,767 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.SparseSaving.Model
-
-namespace OAI
-
-namespace ThorpNine.SparseSaving
-
-namespace Thorp
-
-open scoped BigOperators
-open Filter
-
-def Coins : ℕ → Type
-  | 0 => Unit
-  | d + 1 => Card d → Bool
-
-instance (d : ℕ) : Fintype (Coins d) := by
-  cases d <;> dsimp [Coins] <;> infer_instance
-
-instance (d : ℕ) : Nonempty (Coins d) := by
-  cases d <;> dsimp [Coins] <;> infer_instance
-
-def rotate (d : ℕ) : Equiv.Perm (Card d) where
-  toFun x i := x (finRotate d i)
-  invFun x i := x ((finRotate d).symm i)
-  left_inv x := by funext i; simp only [Equiv.apply_symm_apply]
-  right_inv x := by funext i; simp only [Equiv.symm_apply_apply]
-
-def step : (d : ℕ) → Coins d → Equiv.Perm (Card d)
-  | 0, _ => 1
-  | d + 1, ξ => rotate (d + 1) * pairSwitch ξ
-
-def run (d : ℕ) : (t : ℕ) → (Fin t → Coins d) → Equiv.Perm (Card d)
-  | 0, _ => 1
-  | t + 1, ω => step d (ω (Fin.last t)) * run d t (fun i => ω i.castSucc)
-
-noncomputable def law (d t : ℕ) (g : Equiv.Perm (Card d)) : ℝ :=
-  (Fintype.card {ω : Fin t → Coins d // run d t ω = g} : ℝ) /
-    (Fintype.card (Fin t → Coins d) : ℝ)
-
-noncomputable def distance (d t : ℕ) : ℝ :=
-  (1 / 2 : ℝ) * ∑ g : Equiv.Perm (Card d),
-    |law d t g - 1 / (Nat.factorial (2 ^ d) : ℝ)|
-
-noncomputable def mixingTime (d : ℕ) : ℕ :=
-  sInf {t : ℕ | distance d t ≤ (1 / 4 : ℝ)}
-
-noncomputable def supportTimeBound (d : ℕ) : ℤ :=
-  ⌈(2 / (2 : ℝ) ^ d) *
-    (Real.log ((3 / 4 : ℝ) * (Nat.factorial (2 ^ d) : ℝ)) / Real.log 2)⌉
-
-def OptimalOrder : Prop :=
-  ∃ C : ℕ, 0 < C ∧
-    Tendsto (fun d : ℕ => distance d (C * d)) atTop (nhds 0) ∧
-    (∀ d t : ℕ, 1 - (2 : ℝ) ^ ((t : ℝ) * (2 : ℝ) ^ d / 2) /
-      (Nat.factorial (2 ^ d) : ℝ) ≤ distance d t) ∧
-    (∀ᶠ d : ℕ in atTop,
-      supportTimeBound d ≤ (mixingTime d : ℤ) ∧ mixingTime d ≤ C * d) ∧
-    (∃ K : ℝ, ∀ᶠ d : ℕ in atTop, 2 * (d : ℝ) - K ≤ (mixingTime d : ℝ)) ∧
-    Asymptotics.IsTheta atTop (fun d : ℕ => (mixingTime d : ℝ)) (fun d : ℕ => (d : ℝ))
-
-noncomputable def finiteLaw {α β : Type*} [Fintype α] (f : α → β) (b : β) : ℝ := by
-  classical
-  exact (Fintype.card {a : α // f a = b} : ℝ) / (Fintype.card α : ℝ)
-
-lemma sum_card_fibers {α β : Type*} [Fintype α] [Fintype β] [DecidableEq β] (f : α → β) :
-    ∑ b, Fintype.card {a : α // f a = b} = Fintype.card α := by
-  classical
-  rw [← Fintype.card_sigma]
-  exact Fintype.card_congr (Equiv.sigmaFiberEquiv f)
-
-lemma finiteLaw_nonneg {α β : Type*} [Fintype α] (f : α → β) (b : β) :
-    0 ≤ finiteLaw f b := by
-  exact div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _)
-
-lemma finiteLaw_sum {α β : Type*} [Fintype α] [Nonempty α] [Fintype β] (f : α → β) :
-    ∑ b, finiteLaw f b = 1 := by
-  classical
-  simp only [finiteLaw, ← Finset.sum_div]
-  have hc : (∑ b, (Fintype.card {a : α // f a = b} : ℝ)) = (Fintype.card α : ℝ) := by
-    exact_mod_cast sum_card_fibers f
-  rw [hc]
-  exact div_self (Nat.cast_ne_zero.mpr Fintype.card_ne_zero)
-
-lemma finiteLaw_eq_zero_of_not_mem {α β : Type*} [Fintype α] [DecidableEq β]
-    (f : α → β) {b : β} (hb : b ∉ Finset.univ.image f) : finiteLaw f b = 0 := by
-  classical
-  have : IsEmpty {a : α // f a = b} := ⟨fun a => hb (Finset.mem_image.mpr
-    ⟨a.val, Finset.mem_univ _, a.property⟩)⟩
-  simp [finiteLaw]
-
-lemma event_le_totalVariation {β : Type*} [Fintype β] [DecidableEq β]
-    (p q : β → ℝ) (hpq : ∑ b, (p b - q b) = 0) (s : Finset β) :
-    (∑ b ∈ s, p b) - (∑ b ∈ s, q b) ≤ (1 / 2 : ℝ) * ∑ b, |p b - q b| := by
-  have h₁ : (∑ b ∈ s, (p b - q b)) ≤ ∑ b ∈ s, |p b - q b| :=
-    Finset.sum_le_sum (fun b _ => le_abs_self (p b - q b))
-  have h₂ : -(∑ b ∈ sᶜ, (p b - q b)) ≤ ∑ b ∈ sᶜ, |p b - q b| := by
-    rw [← Finset.sum_neg_distrib]
-    exact Finset.sum_le_sum (fun b _ => neg_le_abs (p b - q b))
-  have hz := Finset.sum_add_sum_compl s (fun b => p b - q b)
-  have ha := Finset.sum_add_sum_compl s (fun b => |p b - q b|)
-  rw [hpq] at hz
-  rw [← Finset.sum_sub_distrib]
-  linarith
-
-lemma finiteLaw_support_bound {α β : Type*} [Fintype α] [Nonempty α]
-    [Fintype β] [Nonempty β] (f : α → β) :
-    1 - (Fintype.card α : ℝ) / (Fintype.card β : ℝ) ≤
-      (1 / 2 : ℝ) * ∑ b, |finiteLaw f b - 1 / (Fintype.card β : ℝ)| := by
-  classical
-  let s := Finset.univ.image f
-  have hs : ∑ b ∈ s, finiteLaw f b = 1 := by
-    have h := Finset.sum_subset (Finset.subset_univ s)
-      (fun b _ hb => finiteLaw_eq_zero_of_not_mem f hb)
-    simpa only [finiteLaw_sum] using h
-  have hβ : (Fintype.card β : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
-  have hq : (∑ _b : β, 1 / (Fintype.card β : ℝ)) = 1 := by simp [hβ]
-  have hd : (∑ b, (finiteLaw f b - 1 / (Fintype.card β : ℝ))) = 0 := by
-    rw [Finset.sum_sub_distrib, finiteLaw_sum, hq, sub_self]
-  have he := event_le_totalVariation (finiteLaw f) (fun _ => 1 / (Fintype.card β : ℝ)) hd s
-  rw [hs] at he
-  have hsq : (∑ _b ∈ s, 1 / (Fintype.card β : ℝ)) = (s.card : ℝ) / Fintype.card β := by
-    simp [div_eq_mul_inv]
-  rw [hsq] at he
-  have hcard : (s.card : ℝ) ≤ Fintype.card α := by
-    exact_mod_cast (Finset.card_image_le (s := (Finset.univ : Finset α)) (f := f))
-  have hdiv := div_le_div_of_nonneg_right hcard (Nat.cast_nonneg (Fintype.card β) : (0 : ℝ) ≤ _)
-  linarith
-
-lemma card_positions (d : ℕ) : Fintype.card (Card d) = 2 ^ d := by
-  simp [Card]
-
-lemma card_permutations (d : ℕ) : Fintype.card (Equiv.Perm (Card d)) = Nat.factorial (2 ^ d) := by
-  rw [Fintype.card_perm, card_positions]
-
-lemma card_coins_succ (d : ℕ) : Fintype.card (Coins (d + 1)) = 2 ^ (2 ^ d) := by
-  simp [Coins]
-
-lemma card_tapes_succ (d t : ℕ) : Fintype.card (Fin t → Coins (d + 1)) = 2 ^ (t * 2 ^ d) := by
-  rw [Fintype.card_fun, card_coins_succ, Fintype.card_fin, ← pow_mul, Nat.mul_comm]
-
-lemma law_eq_finiteLaw (d t : ℕ) (g : Equiv.Perm (Card d)) :
-    law d t g = finiteLaw (run d t) g := by
-  unfold law finiteLaw
-  apply congrArg (fun n : ℕ => (n : ℝ) / (Fintype.card (Fin t → Coins d) : ℝ))
-  exact congrArg (@Fintype.card _) (Subsingleton.elim _ _)
-
-lemma law_nonneg (d t : ℕ) (g : Equiv.Perm (Card d)) : 0 ≤ law d t g := by
-  rw [law_eq_finiteLaw]
-  exact finiteLaw_nonneg (run d t) g
-
-lemma law_sum (d t : ℕ) : ∑ g, law d t g = 1 := by
-  simp only [law_eq_finiteLaw]
-  exact finiteLaw_sum (run d t)
-
-lemma physical_support_bound_succ (d t : ℕ) :
-    1 - (2 : ℝ) ^ (t * 2 ^ d) / (Nat.factorial (2 ^ (d + 1)) : ℝ) ≤ distance (d + 1) t := by
-  have : Nonempty (Equiv.Perm (Card (d + 1))) := ⟨1⟩
-  have h := finiteLaw_support_bound (run (d + 1) t)
-  rw [card_tapes_succ, card_permutations] at h
-  simpa only [Nat.cast_pow, Nat.cast_ofNat, distance, law_eq_finiteLaw] using h
-
-lemma step_succ_formula (d : ℕ) (ξ : Coins (d + 1)) (x : Card (d + 1)) :
-    step (d + 1) ξ x = Fin.snoc (Fin.tail x) (Bool.xor (x 0) (ξ (Fin.tail x))) := by
-  change (fun i => switchFun ξ x (finRotate (d + 1) i)) = _
-  rw [Fin.snoc_eq_cons_rotate]
-  rfl
-
-lemma distance_nonneg (d t : ℕ) : 0 ≤ distance d t := by
-  unfold distance
-  exact mul_nonneg (by norm_num) (Finset.sum_nonneg (fun _ _ => abs_nonneg _))
-
-lemma physical_support_bound (d t : ℕ) :
-    1 - (2 : ℝ) ^ ((t : ℝ) * (2 : ℝ) ^ d / 2) /
-      (Nat.factorial (2 ^ d) : ℝ) ≤ distance d t := by
-  cases d with
-  | zero =>
-    have h : 1 ≤ (2 : ℝ) ^ ((t : ℝ) / 2) := Real.one_le_rpow (by norm_num) (by positivity)
-    simpa only [pow_zero, Nat.factorial_one, Nat.cast_one, div_one, mul_one] using
-      le_trans (sub_nonpos.mpr h) (distance_nonneg 0 t)
-  | succ d =>
-    have he : (t : ℝ) * (2 : ℝ) ^ (d + 1) / 2 = (t * 2 ^ d : ℕ) := by
-      push_cast
-      rw [pow_succ]
-      ring
-    rw [he, Real.rpow_natCast]
-    exact physical_support_bound_succ d t
-
-lemma supportTimeBound_le_of_distance_le (d t : ℕ) (ht : distance d t ≤ (1 / 4 : ℝ)) :
-    supportTimeBound d ≤ (t : ℤ) := by
-  have hF : (0 : ℝ) < Nat.factorial (2 ^ d) := by positivity
-  have hN : (0 : ℝ) < (2 : ℝ) ^ d := by positivity
-  have hL : (0 : ℝ) < Real.log 2 := Real.log_pos (by norm_num)
-  have hb := physical_support_bound d t
-  have hp : (3 / 4 : ℝ) ≤
-      (2 : ℝ) ^ ((t : ℝ) * (2 : ℝ) ^ d / 2) / (Nat.factorial (2 ^ d) : ℝ) := by
-    linarith
-  have hr := (le_div_iff₀ hF).mp hp
-  have hlog := Real.log_le_log (show (0 : ℝ) < (3 / 4 : ℝ) * Nat.factorial (2 ^ d) by positivity) hr
-  rw [Real.log_rpow (by norm_num : (0 : ℝ) < 2)] at hlog
-  unfold supportTimeBound
-  apply Int.ceil_le.mpr
-  simp only [Int.cast_natCast]
-  calc
-    2 / (2 : ℝ) ^ d * (Real.log ((3 / 4 : ℝ) * Nat.factorial (2 ^ d)) / Real.log 2) =
-        Real.log ((3 / 4 : ℝ) * Nat.factorial (2 ^ d)) / ((2 : ℝ) ^ d / 2 * Real.log 2) := by
-      field_simp
-    _ ≤ (t : ℝ) := (div_le_iff₀ (mul_pos (div_pos hN (by norm_num)) hL)).mpr (by
-      nlinarith [hlog])
-
-lemma supportTimeBound_le_mixingTime {d : ℕ}
-    (h : ({t : ℕ | distance d t ≤ (1 / 4 : ℝ)} : Set ℕ).Nonempty) :
-    supportTimeBound d ≤ (mixingTime d : ℤ) := by
-  exact supportTimeBound_le_of_distance_le d (mixingTime d) (Nat.sInf_mem h)
-
-lemma log_factorial_lower {n : ℕ} (hn : 1 ≤ n) :
-    (n : ℝ) * Real.log n - n ≤ Real.log (Nat.factorial n) := by
-  have h := Stirling.le_log_factorial_stirling (show n ≠ 0 by omega)
-  have hln : 0 ≤ Real.log (n : ℝ) := Real.log_nonneg (by exact_mod_cast hn)
-  have hlp : 0 ≤ Real.log (2 * Real.pi) := Real.log_nonneg (by
-    have := Real.pi_gt_three
-    linarith)
-  linarith
-
-lemma log_three_quarters_lower : -(1 : ℝ) ≤ Real.log (3 / 4 : ℝ) := by
-  have h := Real.log_le_sub_one_of_pos (show (0 : ℝ) < 4 / 3 by norm_num)
-  have he : Real.log (4 / 3 : ℝ) = - Real.log (3 / 4 : ℝ) := by
-    rw [← Real.log_inv]
-    congr 1
-    norm_num
-  rw [he] at h
-  linarith
-
-lemma linear_le_supportTimeBound (d : ℕ) :
-    2 * (d : ℝ) - 4 / Real.log 2 ≤ (supportTimeBound d : ℝ) := by
-  have hN : 1 ≤ (2 : ℝ) ^ d := one_le_pow₀ (by norm_num)
-  have hNp : 0 < (2 : ℝ) ^ d := by positivity
-  have hL : 0 < Real.log 2 := Real.log_pos (by norm_num)
-  have hf := log_factorial_lower (show 1 ≤ 2 ^ d by exact one_le_pow₀ (by norm_num))
-  have hlog : (2 : ℝ) ^ d * ((d : ℝ) * Real.log 2) - (2 : ℝ) ^ d - 1 ≤
-      Real.log ((3 / 4 : ℝ) * (Nat.factorial (2 ^ d) : ℝ)) := by
-    rw [Real.log_mul (by norm_num) (by positivity)]
-    push_cast at hf
-    rw [Real.log_pow] at hf
-    linarith [log_three_quarters_lower]
-  apply le_trans _ (Int.le_ceil _)
-  have hmul : (2 * (d : ℝ) - 4 / Real.log 2) *
-      ((2 : ℝ) ^ d / 2 * Real.log 2) ≤
-      Real.log ((3 / 4 : ℝ) * (Nat.factorial (2 ^ d) : ℝ)) := by
-    have hcancel : (2 * (d : ℝ) - 4 / Real.log 2) *
-        ((2 : ℝ) ^ d / 2 * Real.log 2) =
-        (2 : ℝ) ^ d * ((d : ℝ) * Real.log 2) - 2 * (2 : ℝ) ^ d := by
-      field_simp
-      ring
-    rw [hcancel]
-    linarith
-  convert (le_div_iff₀ (mul_pos (div_pos hNp (by norm_num)) hL)).mpr hmul using 1
-  field_simp
-
-lemma linear_le_time_of_distance_le (d t : ℕ)
-    (ht : distance d t ≤ (1 / 4 : ℝ)) :
-    2 * (d : ℝ) - 4 / Real.log 2 ≤ (t : ℝ) := by
-  have h := supportTimeBound_le_of_distance_le d t ht
-  have hr : (supportTimeBound d : ℝ) ≤ (t : ℝ) := by exact_mod_cast h
-  exact (linear_le_supportTimeBound d).trans hr
-
-lemma entropy_chord (p : ℝ) (hp : 0 ≤ p) (hq : p ≤ 1 / 2) :
-    2 * p * Real.log 2 ≤ Real.binEntropy p := by
-  have hc := Real.strictConcave_binEntropy.concaveOn.2
-    (show (0 : ℝ) ∈ Set.Icc (0 : ℝ) 1 by norm_num)
-    (show (2 : ℝ)⁻¹ ∈ Set.Icc (0 : ℝ) 1 by norm_num)
-    (show 0 ≤ 1 - 2 * p by linarith)
-    (show 0 ≤ 2 * p by positivity)
-    (show (1 - 2 * p) + 2 * p = 1 by ring)
-  simpa only [Real.binEntropy_zero, Real.binEntropy_two_inv,
-    smul_eq_mul, mul_zero, zero_add, show 2 * p * (2 : ℝ)⁻¹ = p by ring] using hc
-
-lemma split_entropy_lower (a b : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) :
-    2 * min a b * Real.log 2 ≤
-      (a + b) * Real.log (a + b) - a * Real.log a - b * Real.log b := by
-  wlog hab : a ≤ b generalizing a b
-  · have hba := this b a hb ha (le_of_not_ge hab)
-    simpa only [min_comm, add_comm b a] using hba.trans_eq (show (b + a) * Real.log (b + a) - b * Real.log b - a * Real.log a =
-      (a + b) * Real.log (a + b) - a * Real.log a - b * Real.log b by rw [add_comm b a]; ring)
-  rw [min_eq_left hab]
-  by_cases hz : a = 0
-  · subst a
-    simp
-  have hap : 0 < a := lt_of_le_of_ne ha (Ne.symm hz)
-  have hbp : 0 < b := hap.trans_le hab
-  have hs : 0 < a + b := add_pos hap hbp
-  have hp : 0 ≤ a / (a + b) := div_nonneg ha hs.le
-  have hq : a / (a + b) ≤ 1 / 2 := (div_le_iff₀ hs).mpr (by linarith)
-  have hc := mul_le_mul_of_nonneg_left (entropy_chord (a / (a + b)) hp hq) hs.le
-  have he : (a + b) * Real.binEntropy (a / (a + b)) =
-      (a + b) * Real.log (a + b) - a * Real.log a - b * Real.log b := by
-    unfold Real.binEntropy
-    rw [show 1 - a / (a + b) = b / (a + b) by field_simp; ring]
-    simp only [Real.log_inv]
-    rw [Real.log_div (ne_of_gt hap) (ne_of_gt hs),
-      Real.log_div (ne_of_gt hbp) (ne_of_gt hs)]
-    field_simp
-    ring
-  rw [he] at hc
-  calc
-    2 * a * Real.log 2 = (a + b) * (2 * (a / (a + b)) * Real.log 2) := by
-      field_simp
-    _ ≤ _ := hc
-
-noncomputable def childMarks {d : ℕ} (s : Finset (Card (d + 1))) (ε : Bool) :
-    Finset (Card d) := (s.filter (fun x => x 0 = ε)).image Fin.tail
-
-lemma card_childMarks {d : ℕ} (s : Finset (Card (d + 1))) (ε : Bool) :
-    (childMarks s ε).card = (s.filter (fun x => x 0 = ε)).card := by
-  classical
-  unfold childMarks
-  apply Finset.card_image_iff.mpr
-  intro x hx y hy hxy
-  have hx0 := (Finset.mem_filter.mp hx).2
-  have hy0 := (Finset.mem_filter.mp hy).2
-  rw [← Fin.cons_self_tail x, ← Fin.cons_self_tail y, hx0, hy0, hxy]
-
-lemma card_childMarks_add {d : ℕ} (s : Finset (Card (d + 1))) :
-    (childMarks s false).card + (childMarks s true).card = s.card := by
-  classical
-  rw [card_childMarks, card_childMarks]
-  simpa only [Bool.not_eq_false] using
-    (Finset.card_filter_add_card_filter_not (s := s) (fun x => x 0 = false))
-
-noncomputable def topMeetings {d : ℕ} (b : Butterfly (d + 1))
-    (s : Finset (Card (d + 1))) : Finset (Card d) :=
-  ((childMarks s false).image (butterflyPerm d (b.2 false))) ∩
-    ((childMarks s true).image (butterflyPerm d (b.2 true)))
-
-lemma card_topMeetings_le {d : ℕ} (b : Butterfly (d + 1)) (s : Finset (Card (d + 1))) :
-    (topMeetings b s).card ≤ min (childMarks s false).card (childMarks s true).card := by
-  classical
-  apply le_min
-  · exact (Finset.card_le_card Finset.inter_subset_left).trans_eq
-      (Finset.card_image_of_injective _ (butterflyPerm d (b.2 false)).injective)
-  · exact (Finset.card_le_card Finset.inter_subset_right).trans_eq
-      (Finset.card_image_of_injective _ (butterflyPerm d (b.2 true)).injective)
-
-noncomputable def encounters : (d : ℕ) → Butterfly d → Finset (Card d) → ℕ
-  | 0, _, _ => 0
-  | d + 1, b, s => encounters d (b.2 false) (childMarks s false) +
-      encounters d (b.2 true) (childMarks s true) + (topMeetings b s).card
-
-lemma butterfly_encounters (d : ℕ) (b : Butterfly d) (s : Finset (Card d)) :
-    (encounters d b s : ℝ) ≤ (s.card : ℝ) * Real.log s.card / (2 * Real.log 2) := by
-  induction d with
-  | zero =>
-    simp only [encounters, Nat.cast_zero]
-    have hs : s.card ≤ 1 := by
-      simpa [Card] using Finset.card_le_univ s
-    interval_cases s.card <;> simp
-  | succ d ih =>
-    have h₀ := ih (b.2 false) (childMarks s false)
-    have h₁ := ih (b.2 true) (childMarks s true)
-    have ht : ((topMeetings b s).card : ℝ) ≤
-        min ((childMarks s false).card : ℝ) ((childMarks s true).card : ℝ) := by
-      exact_mod_cast card_topMeetings_le b s
-    have he := split_entropy_lower ((childMarks s false).card : ℝ)
-      ((childMarks s true).card : ℝ) (by positivity) (by positivity)
-    have hs : ((childMarks s false).card : ℝ) + (childMarks s true).card = s.card := by
-      exact_mod_cast card_childMarks_add s
-    rw [hs] at he
-    have hL : 0 < 2 * Real.log 2 := by positivity
-    rw [le_div_iff₀ hL] at h₀ h₁ ⊢
-    simp only [encounters, Nat.cast_add]
-    nlinarith [mul_le_mul_of_nonneg_right ht hL.le]
-
-lemma prod_split_at {α : Type*} [DecidableEq α] (s : Finset α) (f : α → ℝ) (a : α) :
-    ∏ i ∈ s, f i = (if a ∈ s then f a else 1) * ∏ i ∈ s.erase a, f i := by
-  by_cases ha : a ∈ s
-  · rw [ite_eq_left ha]
-    exact (Finset.mul_prod_erase s f ha).symm
-  · simp [ha]
-
-lemma prod_split_two {α : Type*} [DecidableEq α] (s : Finset α) (f : α → ℝ)
-    (a b : α) (hab : a ≠ b) :
-    ∏ i ∈ s, f i = (if a ∈ s then f a else 1) * (if b ∈ s then f b else 1) *
-      ∏ i ∈ (s.erase a).erase b, f i := by
-  calc
-    ∏ i ∈ s, f i = (if a ∈ s then f a else 1) * ∏ i ∈ s.erase a, f i :=
-      prod_split_at s f a
-    _ = _ := by
-      rw [prod_split_at (s.erase a) f b]
-      have hm : (b ∈ s.erase a) ↔ b ∈ s :=
-        ⟨fun h => (Finset.mem_erase.mp h).2, fun h => Finset.mem_erase.mpr ⟨Ne.symm hab, h⟩⟩
-      simp only [hm, mul_assoc]
-
-lemma swap_average_product {α : Type*} [DecidableEq α] (s : Finset α) (p : α → ℝ)
-    (hp : ∀ i, 0 ≤ p i) (a b : α) :
-    ((∏ i ∈ s, p i) + ∏ i ∈ s, p (Equiv.swap a b i)) / 2 ≤
-      ∏ i ∈ s, (p i + p (Equiv.swap a b i)) / 2 := by
-  by_cases hab : a = b
-  · subst b
-    simp only [Equiv.swap_self, Equiv.refl_apply]
-    have h : (fun i => (p i + p i) / 2) = p := by funext i; ring
-    rw [h]
-    linarith
-  let t := (s.erase a).erase b
-  have hfix : ∀ i ∈ t, Equiv.swap a b i = i := by
-    intro i hi
-    exact Equiv.swap_apply_of_ne_of_ne (Finset.mem_erase.mp (Finset.mem_erase.mp hi).2).1
-      (Finset.mem_erase.mp hi).1
-  have hprod : (∏ i ∈ t, p (Equiv.swap a b i)) = ∏ i ∈ t, p i :=
-    Finset.prod_congr rfl (fun i hi => by rw [hfix i hi])
-  have havg : (∏ i ∈ t, (p i + p (Equiv.swap a b i)) / 2) = ∏ i ∈ t, p i :=
-    Finset.prod_congr rfl (fun i hi => by rw [hfix i hi]; ring)
-  rw [prod_split_two s p a b hab,
-    prod_split_two s (fun i => p (Equiv.swap a b i)) a b hab,
-    prod_split_two s (fun i => (p i + p (Equiv.swap a b i)) / 2) a b hab]
-  change ((if a ∈ s then p a else 1) * (if b ∈ s then p b else 1) * (∏ i ∈ t, p i) +
-    (if a ∈ s then p (Equiv.swap a b a) else 1) *
-      (if b ∈ s then p (Equiv.swap a b b) else 1) * (∏ i ∈ t, p (Equiv.swap a b i))) / 2 ≤ _
-  rw [hprod]
-  change _ ≤ (if a ∈ s then (p a + p (Equiv.swap a b a)) / 2 else 1) *
-    (if b ∈ s then (p b + p (Equiv.swap a b b)) / 2 else 1) *
-      (∏ i ∈ t, (p i + p (Equiv.swap a b i)) / 2)
-  rw [havg, Equiv.swap_apply_left, Equiv.swap_apply_right]
-  have ht : 0 ≤ ∏ i ∈ t, p i := Finset.prod_nonneg (fun i _ => hp i)
-  by_cases ha : a ∈ s <;> by_cases hb : b ∈ s <;> simp only [ha, hb, ↓reduceIte]
-  · have hsq := mul_nonneg (sq_nonneg (p a - p b)) ht
-    nlinarith
-  all_goals ring_nf; rfl
-
-def CylinderBound {α : Type*} (m : Finset α → ℝ) (p : α → ℝ) : Prop :=
-  ∀ s, m s ≤ ∏ i ∈ s, p i
-
-lemma CylinderBound.fair_swap {α : Type*} [DecidableEq α]
-    {m : Finset α → ℝ} {p : α → ℝ} (h : CylinderBound m p)
-    (hp : ∀ i, 0 ≤ p i) (a b : α) :
-    CylinderBound (fun s => (m s + m (s.image (Equiv.swap a b))) / 2)
-      (fun i => (p i + p (Equiv.swap a b i)) / 2) := by
-  intro s
-  have h₀ := h s
-  have h₁ := h (s.image (Equiv.swap a b))
-  rw [Finset.prod_image (Equiv.swap a b).injective.injOn] at h₁
-  exact (show (m s + m (s.image (Equiv.swap a b))) / 2 ≤
-      ((∏ i ∈ s, p i) + ∏ i ∈ s, p (Equiv.swap a b i)) / 2 by linarith).trans
-    (swap_average_product s p hp a b)
-
-noncomputable def finiteMean {Ω : Type*} [Fintype Ω] (f : Ω → ℝ) : ℝ :=
-  (∑ ω, f ω) / Fintype.card Ω
-
-noncomputable def inclusionMoment {α Ω : Type*} [Fintype Ω]
-    (F : Ω → Finset α) (s : Finset α) : ℝ := by
-  classical
-  exact finiteMean (fun ω => if s ⊆ F ω then 1 else 0)
-
-lemma inclusionMoment_nonneg {α Ω : Type*} [Fintype Ω]
-    (F : Ω → Finset α) (s : Finset α) : 0 ≤ inclusionMoment F s := by
-  classical
-  unfold inclusionMoment finiteMean
-  positivity
-
-lemma indicator_subset_prod {α : Type*} [DecidableEq α] (A s : Finset α) :
-    (if s ⊆ A then (1 : ℝ) else 0) = ∏ i ∈ s, (if i ∈ A then 1 else 0) := by
-  by_cases h : s ⊆ A
-  · rw [ite_eq_left h]
-    exact (Finset.prod_eq_one (fun i hi => ite_eq_left (h hi))).symm
-  · rw [ite_eq_right h]
-    obtain ⟨i, hi, hiA⟩ := Finset.not_subset.mp h
-    exact (Finset.prod_eq_zero hi (ite_eq_right hiA)).symm
-
-lemma inclusionMoment_const {α Ω : Type*} [DecidableEq α] [Fintype Ω] [Nonempty Ω]
-    (A s : Finset α) : inclusionMoment (fun _ : Ω => A) s =
-      if s ⊆ A then 1 else 0 := by
-  classical
-  have hn : (Fintype.card Ω : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
-  by_cases h : s ⊆ A <;> simp [inclusionMoment, finiteMean, h, hn]
-
-lemma cylinder_const {α Ω : Type*} [Fintype Ω] [Nonempty Ω] (A : Finset α) :
-    CylinderBound (inclusionMoment (fun _ : Ω => A))
-      (fun i => inclusionMoment (fun _ : Ω => A) {i}) := by
-  classical
-  intro s
-  simp only [inclusionMoment_const, Finset.singleton_subset_iff]
-  exact (indicator_subset_prod A s).le
-
-noncomputable def fairSetSwap {α Ω : Type*} [DecidableEq α]
-    (F : Ω → Finset α) (a b : α) : Ω × Bool → Finset α :=
-  fun ω => if ω.2 then (F ω.1).image (Equiv.swap a b) else F ω.1
-
-lemma subset_swap_image_iff {α : Type*} [DecidableEq α]
-    (A s : Finset α) (a b : α) :
-    s ⊆ A.image (Equiv.swap a b) ↔ s.image (Equiv.swap a b) ⊆ A := by
-  constructor
-  · intro h
-    have h' := Finset.image_subset_image h (f := Equiv.swap a b)
-    simpa only [Finset.image_image, Function.comp_def, Equiv.swap_apply_self, Finset.image_id'] using h'
-  · intro h
-    have h' := Finset.image_subset_image h (f := Equiv.swap a b)
-    simpa only [Finset.image_image, Function.comp_def, Equiv.swap_apply_self, Finset.image_id'] using h'
-
-lemma inclusionMoment_fairSetSwap {α Ω : Type*} [DecidableEq α] [Fintype Ω]
-    (F : Ω → Finset α) (a b : α) (s : Finset α) :
-    inclusionMoment (fairSetSwap F a b) s =
-      (inclusionMoment F s + inclusionMoment F (s.image (Equiv.swap a b))) / 2 := by
-  classical
-  simp only [inclusionMoment, finiteMean, Fintype.sum_prod_type, Fintype.sum_bool,
-    fairSetSwap, Bool.false_eq_true, ↓reduceIte, Fintype.card_prod, Fintype.card_bool,
-    Nat.cast_mul, Nat.cast_ofNat, subset_swap_image_iff, Finset.sum_add_distrib]
-  rw [div_mul_eq_div_div, add_div]
-  ring_nf
-
-lemma cylinder_fairSetSwap {α Ω : Type*} [DecidableEq α] [Fintype Ω]
-    (F : Ω → Finset α) (a b : α)
-    (h : CylinderBound (inclusionMoment F) (fun i => inclusionMoment F {i})) :
-    CylinderBound (inclusionMoment (fairSetSwap F a b))
-      (fun i => inclusionMoment (fairSetSwap F a b) {i}) := by
-  classical
-  have h' := h.fair_swap (fun i => inclusionMoment_nonneg F {i}) a b
-  intro s
-  simpa only [inclusionMoment_fairSetSwap, Finset.image_singleton] using h' s
-
-def SwitchSamples : ℕ → Type
-  | 0 => Unit
-  | n + 1 => SwitchSamples n × Bool
-
-instance (n : ℕ) : Fintype (SwitchSamples n) := by
-  induction n with
-  | zero => exact inferInstanceAs (Fintype Unit)
-  | succ n ih => exact inferInstanceAs (Fintype (SwitchSamples n × Bool))
-
-instance (n : ℕ) : Nonempty (SwitchSamples n) := by
-  induction n with
-  | zero => exact inferInstanceAs (Nonempty Unit)
-  | succ n ih => exact inferInstanceAs (Nonempty (SwitchSamples n × Bool))
-
-noncomputable def switchedSet {α : Type*} [DecidableEq α] (A : Finset α) :
-    (L : List (α × α)) → SwitchSamples L.length → Finset α
-  | [], _ => A
-  | (a,b) :: L, ω => fairSetSwap (switchedSet A L) a b ω
-
-lemma switchedSet_cylinder {α : Type*} [DecidableEq α] (A : Finset α)
-    (L : List (α × α)) :
-    CylinderBound (inclusionMoment (switchedSet A L))
-      (fun i => inclusionMoment (switchedSet A L) {i}) := by
-  induction L with
-  | nil => exact cylinder_const A
-  | cons e L ih => exact cylinder_fairSetSwap (switchedSet A L) e.1 e.2 ih
-
-inductive QueryTree (ι : Type*)
-  | reject : QueryTree ι
-  | accept : QueryTree ι
-  | branch (i : ι) (zero one : QueryTree ι) : QueryTree ι
-  | force (i : ι) (b : Bool) (next : QueryTree ι) : QueryTree ι
-
-namespace QueryTree
-
-variable {ι : Type*} [DecidableEq ι]
-
-def queries : QueryTree ι → Finset ι
-  | reject => ∅
-  | accept => ∅
-  | branch i zero one => insert i (queries zero ∪ queries one)
-  | force i _ next => insert i (queries next)
-
-def Fresh : QueryTree ι → Prop
-  | reject => True
-  | accept => True
-  | branch i zero one => i ∉ queries zero ∧ i ∉ queries one ∧ Fresh zero ∧ Fresh one
-  | force i _ next => i ∉ queries next ∧ Fresh next
-
-def weighted : QueryTree ι → (ι → Bool) → ℝ
-  | reject, _ => 0
-  | accept, _ => 1
-  | branch i zero one, ω => if ω i then weighted one ω else weighted zero ω
-  | force i b next, ω => if ω i = b then 2 * weighted next ω else 0
-
-omit [DecidableEq ι] in
-lemma weighted_nonneg (T : QueryTree ι) (ω : ι → Bool) : 0 ≤ T.weighted ω := by
-  induction T with
-  | reject => exact le_rfl
-  | accept => exact zero_le_one
-  | branch i zero one ih₀ ih₁ => simpa only [weighted] using ite_nonneg ih₁ ih₀
-  | force i b next ih => simpa only [weighted] using ite_nonneg (mul_nonneg (by norm_num) ih) le_rfl
-
-lemma weighted_congr (T : QueryTree ι) {ω η : ι → Bool}
-    (h : ∀ i ∈ T.queries, ω i = η i) : T.weighted ω = T.weighted η := by
-  induction T with
-  | reject => rfl
-  | accept => rfl
-  | branch i zero one ih₀ ih₁ =>
-    have hi := h i (Finset.mem_insert_self _ _)
-    simp only [weighted, hi]
-    rw [ih₀ (fun j hj => h j (Finset.mem_insert_of_mem (Finset.mem_union_left _ hj))),
-      ih₁ (fun j hj => h j (Finset.mem_insert_of_mem (Finset.mem_union_right _ hj)))]
-  | force i b next ih =>
-    have hi := h i (Finset.mem_insert_self _ _)
-    simp only [weighted, hi]
-    rw [ih (fun j hj => h j (Finset.mem_insert_of_mem hj))]
-
-def flipCoin (i : ι) : Equiv.Perm (ι → Bool) :=
-  (show Function.Involutive (fun ω : ι → Bool => Function.update ω i (!(ω i))) from by
-    intro ω
-    funext j
-    by_cases h : j = i
-    · subst j; simp
-    · simp [Function.update_of_ne h]).toPerm
-
-lemma flipCoin_apply_eq (i : ι) (ω : ι → Bool) : flipCoin i ω i = !(ω i) := by
-  change Function.update ω i (!(ω i)) i = _
-  exact Function.update_self _ _ _
-
-lemma flipCoin_apply_ne (i j : ι) (h : j ≠ i) (ω : ι → Bool) :
-    flipCoin i ω j = ω j := by
-  change Function.update ω i (!(ω i)) j = _
-  exact Function.update_of_ne h _ _
-
-lemma weighted_flip_of_not_mem (T : QueryTree ι) (i : ι) (hi : i ∉ T.queries)
-    (ω : ι → Bool) : T.weighted (flipCoin i ω) = T.weighted ω := by
-  apply weighted_congr
-  intro j hj
-  exact flipCoin_apply_ne i j (fun h => hi (h ▸ hj)) ω
-
-lemma mean_indicator_bit [Fintype ι] (f : (ι → Bool) → ℝ) (i : ι) (b : Bool)
-    (hf : ∀ ω, f (flipCoin i ω) = f ω) :
-    finiteMean (fun ω => if ω i = b then f ω else 0) = finiteMean f / 2 := by
-  let g := fun ω : ι → Bool => if ω i = b then f ω else 0
-  have hp (ω : ι → Bool) : g ω + g (flipCoin i ω) = f ω := by
-    dsimp [g]
-    rw [flipCoin_apply_eq, hf]
-    cases hω : ω i <;> cases b <;> simp
-  have hs := Equiv.sum_comp (flipCoin i) g
-  have hadd : (∑ ω, g ω) + (∑ ω, g (flipCoin i ω)) = ∑ ω, f ω := by
-    rw [← Finset.sum_add_distrib]
-    exact Finset.sum_congr rfl (fun ω _ => hp ω)
-  unfold finiteMean
-  change (∑ ω, g ω) / _ = _
-  rw [hs] at hadd
-  have hh : (∑ ω, g ω) = (∑ ω, f ω) / 2 := by linarith
-  rw [hh]
-  ring
-
-lemma mean_weighted_le_one [Fintype ι] (T : QueryTree ι) (hT : T.Fresh) :
-    finiteMean T.weighted ≤ 1 := by
-  induction T with
-  | reject => simp [weighted, finiteMean]
-  | accept => simp [weighted, finiteMean]
-  | branch i zero one ih₀ ih₁ =>
-    obtain ⟨hz, ho, hfz, hfo⟩ := hT
-    have h₀ := mean_indicator_bit zero.weighted i false (zero.weighted_flip_of_not_mem i hz)
-    have h₁ := mean_indicator_bit one.weighted i true (one.weighted_flip_of_not_mem i ho)
-    have he : finiteMean (branch i zero one).weighted =
-        finiteMean (fun ω => if ω i = false then zero.weighted ω else 0) +
-        finiteMean (fun ω => if ω i = true then one.weighted ω else 0) := by
-      unfold finiteMean
-      rw [← add_div, ← Finset.sum_add_distrib]
-      congr 1
-      apply Finset.sum_congr rfl
-      intro ω _
-      cases hω : ω i <;> simp [weighted, hω]
-    rw [he, h₀, h₁]
-    linarith [ih₀ hfz, ih₁ hfo]
-  | force i b next ih =>
-    obtain ⟨hi, hn⟩ := hT
-    have hm := mean_indicator_bit next.weighted i b (next.weighted_flip_of_not_mem i hi)
-    have he : finiteMean (force i b next).weighted =
-        2 * finiteMean (fun ω => if ω i = b then next.weighted ω else 0) := by
-      unfold finiteMean
-      rw [← mul_div_assoc, Finset.mul_sum]
-      congr 1
-      apply Finset.sum_congr rfl
-      intro ω _
-      by_cases h : ω i = b <;> simp [weighted, h]
-    rw [he, hm]
-    linarith [ih hn]
-
-def succeeds : QueryTree ι → (ι → Bool) → Bool
-  | reject, _ => false
-  | accept, _ => true
-  | branch i zero one, ω => if ω i then succeeds one ω else succeeds zero ω
-  | force i b next, ω => if ω i = b then succeeds next ω else false
-
-def charge : QueryTree ι → (ι → Bool) → ℕ
-  | reject, _ => 0
-  | accept, _ => 0
-  | branch i zero one, ω => if ω i then charge one ω else charge zero ω
-  | force i b next, ω => if ω i = b then charge next ω + 1 else 0
-
-omit [DecidableEq ι] in
-lemma weighted_eq (T : QueryTree ι) (ω : ι → Bool) :
-    T.weighted ω = if T.succeeds ω then (2 : ℝ) ^ T.charge ω else 0 := by
-  induction T with
-  | reject => rfl
-  | accept => norm_num [weighted, succeeds, charge]
-  | branch i zero one ih₀ ih₁ =>
-    cases h : ω i <;> simp [weighted, succeeds, charge, h, ih₀, ih₁]
-  | force i b next ih =>
-    by_cases h : ω i = b
-    · simp only [weighted, succeeds, charge, h, ↓reduceIte, ih]
-      cases hs : next.succeeds ω <;> simp [pow_succ, mul_comm]
-    · simp [weighted, succeeds, h]
-
-lemma success_probability_le [Fintype ι] (T : QueryTree ι) (hT : T.Fresh)
-    (K : ℕ) (hK : ∀ ω, T.succeeds ω = true → K ≤ T.charge ω) :
-    finiteMean (fun ω => if T.succeeds ω then (1 : ℝ) else 0) ≤ 1 / (2 : ℝ) ^ K := by
-  have hw (ω : ι → Bool) :
-      (2 : ℝ) ^ K * (if T.succeeds ω then (1 : ℝ) else 0) ≤ T.weighted ω := by
-    rw [weighted_eq]
-    cases hs : T.succeeds ω
-    · simp
-    · simpa using pow_le_pow_right₀ (show (1 : ℝ) ≤ 2 by norm_num) (hK ω hs)
-  have hn : 0 ≤ (Fintype.card (ι → Bool) : ℝ) := Nat.cast_nonneg _
-  have hm := div_le_div_of_nonneg_right
-    (Finset.sum_le_sum (fun ω (_ : ω ∈ Finset.univ) => hw ω)) hn
-  rw [← Finset.mul_sum, mul_div_assoc] at hm
-  change (2 : ℝ) ^ K * finiteMean (fun ω => if T.succeeds ω then 1 else 0) ≤
-    finiteMean T.weighted at hm
-  exact (le_div_iff₀ (pow_pos (by norm_num) K)).mpr (by
-    rw [mul_comm]
-    exact hm.trans (mean_weighted_le_one T hT))
-
-end QueryTree
-
-def encodeButterfly : (d : ℕ) → Butterfly d → SwitchIndex d → Bool
-  | 0, _, i => nomatch i
-  | _d + 1, b, Sum.inl y => b.1 y
-  | d + 1, b, Sum.inr (ε,i) => encodeButterfly d (b.2 ε) i
-
-lemma decode_encodeButterfly (d : ℕ) (b : Butterfly d) :
-    decodeButterfly d (encodeButterfly d b) = b := by
-  induction d with
-  | zero => exact Unit.ext _ _
-  | succ d ih =>
-    apply Prod.ext
-    · rfl
-    · funext ε
-      exact ih (b.2 ε)
-
-lemma encode_decodeButterfly (d : ℕ) (ω : SwitchIndex d → Bool) :
-    encodeButterfly d (decodeButterfly d ω) = ω := by
-  induction d with
-  | zero => funext i; exact Empty.elim i
-  | succ d ih =>
-    funext i
-    cases i with
-    | inl y => rfl
-    | inr z => exact congrFun (ih (fun i => ω (Sum.inr (z.1,i)))) z.2
-
-def butterflyCoinEquiv (d : ℕ) : (SwitchIndex d → Bool) ≃ Butterfly d where
-  toFun := decodeButterfly d
-  invFun := encodeButterfly d
-  left_inv := encode_decodeButterfly d
-  right_inv := decode_encodeButterfly d
-
-noncomputable def routeDomain : (d : ℕ) → Card d → Card d → Finset (SwitchIndex d)
-  | 0, _, _ => ∅
-  | d + 1, x, y => by
-      classical
-      exact insert (Sum.inl (Fin.tail y))
-        ((routeDomain d (Fin.tail x) (Fin.tail y)).image (fun i => Sum.inr (x 0,i)))
-
-def routeValue : (d : ℕ) → Card d → Card d → SwitchIndex d → Bool
-  | 0, _, _, i => nomatch i
-  | _d + 1, x, y, Sum.inl _ => Bool.xor (x 0) (y 0)
-  | d + 1, x, y, Sum.inr (_,i) => routeValue d (Fin.tail x) (Fin.tail y) i
-
-end Thorp
-
-end ThorpNine.SparseSaving
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.SparseSaving.Model
+
+namespace OAI
+
+namespace ThorpNine.SparseSaving
+
+namespace Thorp
+
+open scoped BigOperators
+open Filter
+
+def Coins : ℕ → Type
+  | 0 => Unit
+  | d + 1 => Card d → Bool
+
+instance (d : ℕ) : Fintype (Coins d) := by
+  cases d <;> dsimp [Coins] <;> infer_instance
+
+instance (d : ℕ) : Nonempty (Coins d) := by
+  cases d <;> dsimp [Coins] <;> infer_instance
+
+def rotate (d : ℕ) : Equiv.Perm (Card d) where
+  toFun x i := x (finRotate d i)
+  invFun x i := x ((finRotate d).symm i)
+  left_inv x := by funext i; simp only [Equiv.apply_symm_apply]
+  right_inv x := by funext i; simp only [Equiv.symm_apply_apply]
+
+def step : (d : ℕ) → Coins d → Equiv.Perm (Card d)
+  | 0, _ => 1
+  | d + 1, ξ => rotate (d + 1) * pairSwitch ξ
+
+def run (d : ℕ) : (t : ℕ) → (Fin t → Coins d) → Equiv.Perm (Card d)
+  | 0, _ => 1
+  | t + 1, ω => step d (ω (Fin.last t)) * run d t (fun i => ω i.castSucc)
+
+noncomputable def law (d t : ℕ) (g : Equiv.Perm (Card d)) : ℝ :=
+  (Fintype.card {ω : Fin t → Coins d // run d t ω = g} : ℝ) /
+    (Fintype.card (Fin t → Coins d) : ℝ)
+
+noncomputable def distance (d t : ℕ) : ℝ :=
+  (1 / 2 : ℝ) * ∑ g : Equiv.Perm (Card d),
+    |law d t g - 1 / (Nat.factorial (2 ^ d) : ℝ)|
+
+noncomputable def mixingTime (d : ℕ) : ℕ :=
+  sInf {t : ℕ | distance d t ≤ (1 / 4 : ℝ)}
+
+noncomputable def supportTimeBound (d : ℕ) : ℤ :=
+  ⌈(2 / (2 : ℝ) ^ d) *
+    (Real.log ((3 / 4 : ℝ) * (Nat.factorial (2 ^ d) : ℝ)) / Real.log 2)⌉
+
+def OptimalOrder : Prop :=
+  ∃ C : ℕ, 0 < C ∧
+    Tendsto (fun d : ℕ => distance d (C * d)) atTop (nhds 0) ∧
+    (∀ d t : ℕ, 1 - (2 : ℝ) ^ ((t : ℝ) * (2 : ℝ) ^ d / 2) /
+      (Nat.factorial (2 ^ d) : ℝ) ≤ distance d t) ∧
+    (∀ᶠ d : ℕ in atTop,
+      supportTimeBound d ≤ (mixingTime d : ℤ) ∧ mixingTime d ≤ C * d) ∧
+    (∃ K : ℝ, ∀ᶠ d : ℕ in atTop, 2 * (d : ℝ) - K ≤ (mixingTime d : ℝ)) ∧
+    Asymptotics.IsTheta atTop (fun d : ℕ => (mixingTime d : ℝ)) (fun d : ℕ => (d : ℝ))
+
+noncomputable def finiteLaw {α β : Type*} [Fintype α] (f : α → β) (b : β) : ℝ := by
+  classical
+  exact (Fintype.card {a : α // f a = b} : ℝ) / (Fintype.card α : ℝ)
+
+lemma sum_card_fibers {α β : Type*} [Fintype α] [Fintype β] [DecidableEq β] (f : α → β) :
+    ∑ b, Fintype.card {a : α // f a = b} = Fintype.card α := by
+  classical
+  rw [← Fintype.card_sigma]
+  exact Fintype.card_congr (Equiv.sigmaFiberEquiv f)
+
+lemma finiteLaw_nonneg {α β : Type*} [Fintype α] (f : α → β) (b : β) :
+    0 ≤ finiteLaw f b := by
+  exact div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _)
+
+lemma finiteLaw_sum {α β : Type*} [Fintype α] [Nonempty α] [Fintype β] (f : α → β) :
+    ∑ b, finiteLaw f b = 1 := by
+  classical
+  simp only [finiteLaw, ← Finset.sum_div]
+  have hc : (∑ b, (Fintype.card {a : α // f a = b} : ℝ)) = (Fintype.card α : ℝ) := by
+    exact_mod_cast sum_card_fibers f
+  rw [hc]
+  exact div_self (Nat.cast_ne_zero.mpr Fintype.card_ne_zero)
+
+lemma finiteLaw_eq_zero_of_not_mem {α β : Type*} [Fintype α] [DecidableEq β]
+    (f : α → β) {b : β} (hb : b ∉ Finset.univ.image f) : finiteLaw f b = 0 := by
+  classical
+  have : IsEmpty {a : α // f a = b} := ⟨fun a => hb (Finset.mem_image.mpr
+    ⟨a.val, Finset.mem_univ _, a.property⟩)⟩
+  simp [finiteLaw]
+
+lemma event_le_totalVariation {β : Type*} [Fintype β] [DecidableEq β]
+    (p q : β → ℝ) (hpq : ∑ b, (p b - q b) = 0) (s : Finset β) :
+    (∑ b ∈ s, p b) - (∑ b ∈ s, q b) ≤ (1 / 2 : ℝ) * ∑ b, |p b - q b| := by
+  have h₁ : (∑ b ∈ s, (p b - q b)) ≤ ∑ b ∈ s, |p b - q b| :=
+    Finset.sum_le_sum (fun b _ => le_abs_self (p b - q b))
+  have h₂ : -(∑ b ∈ sᶜ, (p b - q b)) ≤ ∑ b ∈ sᶜ, |p b - q b| := by
+    rw [← Finset.sum_neg_distrib]
+    exact Finset.sum_le_sum (fun b _ => neg_le_abs (p b - q b))
+  have hz := Finset.sum_add_sum_compl s (fun b => p b - q b)
+  have ha := Finset.sum_add_sum_compl s (fun b => |p b - q b|)
+  rw [hpq] at hz
+  rw [← Finset.sum_sub_distrib]
+  linarith
+
+lemma finiteLaw_support_bound {α β : Type*} [Fintype α] [Nonempty α]
+    [Fintype β] [Nonempty β] (f : α → β) :
+    1 - (Fintype.card α : ℝ) / (Fintype.card β : ℝ) ≤
+      (1 / 2 : ℝ) * ∑ b, |finiteLaw f b - 1 / (Fintype.card β : ℝ)| := by
+  classical
+  let s := Finset.univ.image f
+  have hs : ∑ b ∈ s, finiteLaw f b = 1 := by
+    have h := Finset.sum_subset (Finset.subset_univ s)
+      (fun b _ hb => finiteLaw_eq_zero_of_not_mem f hb)
+    simpa only [finiteLaw_sum] using h
+  have hβ : (Fintype.card β : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
+  have hq : (∑ _b : β, 1 / (Fintype.card β : ℝ)) = 1 := by simp [hβ]
+  have hd : (∑ b, (finiteLaw f b - 1 / (Fintype.card β : ℝ))) = 0 := by
+    rw [Finset.sum_sub_distrib, finiteLaw_sum, hq, sub_self]
+  have he := event_le_totalVariation (finiteLaw f) (fun _ => 1 / (Fintype.card β : ℝ)) hd s
+  rw [hs] at he
+  have hsq : (∑ _b ∈ s, 1 / (Fintype.card β : ℝ)) = (s.card : ℝ) / Fintype.card β := by
+    simp [div_eq_mul_inv]
+  rw [hsq] at he
+  have hcard : (s.card : ℝ) ≤ Fintype.card α := by
+    exact_mod_cast (Finset.card_image_le (s := (Finset.univ : Finset α)) (f := f))
+  have hdiv := div_le_div_of_nonneg_right hcard (Nat.cast_nonneg (Fintype.card β) : (0 : ℝ) ≤ _)
+  linarith
+
+lemma card_positions (d : ℕ) : Fintype.card (Card d) = 2 ^ d := by
+  simp [Card]
+
+lemma card_permutations (d : ℕ) : Fintype.card (Equiv.Perm (Card d)) = Nat.factorial (2 ^ d) := by
+  rw [Fintype.card_perm, card_positions]
+
+lemma card_coins_succ (d : ℕ) : Fintype.card (Coins (d + 1)) = 2 ^ (2 ^ d) := by
+  simp [Coins]
+
+lemma card_tapes_succ (d t : ℕ) : Fintype.card (Fin t → Coins (d + 1)) = 2 ^ (t * 2 ^ d) := by
+  rw [Fintype.card_fun, card_coins_succ, Fintype.card_fin, ← pow_mul, Nat.mul_comm]
+
+lemma law_eq_finiteLaw (d t : ℕ) (g : Equiv.Perm (Card d)) :
+    law d t g = finiteLaw (run d t) g := by
+  unfold law finiteLaw
+  apply congrArg (fun n : ℕ => (n : ℝ) / (Fintype.card (Fin t → Coins d) : ℝ))
+  exact congrArg (@Fintype.card _) (Subsingleton.elim _ _)
+
+lemma law_nonneg (d t : ℕ) (g : Equiv.Perm (Card d)) : 0 ≤ law d t g := by
+  rw [law_eq_finiteLaw]
+  exact finiteLaw_nonneg (run d t) g
+
+lemma law_sum (d t : ℕ) : ∑ g, law d t g = 1 := by
+  simp only [law_eq_finiteLaw]
+  exact finiteLaw_sum (run d t)
+
+lemma physical_support_bound_succ (d t : ℕ) :
+    1 - (2 : ℝ) ^ (t * 2 ^ d) / (Nat.factorial (2 ^ (d + 1)) : ℝ) ≤ distance (d + 1) t := by
+  have : Nonempty (Equiv.Perm (Card (d + 1))) := ⟨1⟩
+  have h := finiteLaw_support_bound (run (d + 1) t)
+  rw [card_tapes_succ, card_permutations] at h
+  simpa only [Nat.cast_pow, Nat.cast_ofNat, distance, law_eq_finiteLaw] using h
+
+lemma step_succ_formula (d : ℕ) (ξ : Coins (d + 1)) (x : Card (d + 1)) :
+    step (d + 1) ξ x = Fin.snoc (Fin.tail x) (Bool.xor (x 0) (ξ (Fin.tail x))) := by
+  change (fun i => switchFun ξ x (finRotate (d + 1) i)) = _
+  rw [Fin.snoc_eq_cons_rotate]
+  rfl
+
+lemma distance_nonneg (d t : ℕ) : 0 ≤ distance d t := by
+  unfold distance
+  exact mul_nonneg (by norm_num) (Finset.sum_nonneg (fun _ _ => abs_nonneg _))
+
+lemma physical_support_bound (d t : ℕ) :
+    1 - (2 : ℝ) ^ ((t : ℝ) * (2 : ℝ) ^ d / 2) /
+      (Nat.factorial (2 ^ d) : ℝ) ≤ distance d t := by
+  cases d with
+  | zero =>
+    have h : 1 ≤ (2 : ℝ) ^ ((t : ℝ) / 2) := Real.one_le_rpow (by norm_num) (by positivity)
+    simpa only [pow_zero, Nat.factorial_one, Nat.cast_one, div_one, mul_one] using
+      le_trans (sub_nonpos.mpr h) (distance_nonneg 0 t)
+  | succ d =>
+    have he : (t : ℝ) * (2 : ℝ) ^ (d + 1) / 2 = (t * 2 ^ d : ℕ) := by
+      push_cast
+      rw [pow_succ]
+      ring
+    rw [he, Real.rpow_natCast]
+    exact physical_support_bound_succ d t
+
+lemma supportTimeBound_le_of_distance_le (d t : ℕ) (ht : distance d t ≤ (1 / 4 : ℝ)) :
+    supportTimeBound d ≤ (t : ℤ) := by
+  have hF : (0 : ℝ) < Nat.factorial (2 ^ d) := by positivity
+  have hN : (0 : ℝ) < (2 : ℝ) ^ d := by positivity
+  have hL : (0 : ℝ) < Real.log 2 := Real.log_pos (by norm_num)
+  have hb := physical_support_bound d t
+  have hp : (3 / 4 : ℝ) ≤
+      (2 : ℝ) ^ ((t : ℝ) * (2 : ℝ) ^ d / 2) / (Nat.factorial (2 ^ d) : ℝ) := by
+    linarith
+  have hr := (le_div_iff₀ hF).mp hp
+  have hlog := Real.log_le_log (show (0 : ℝ) < (3 / 4 : ℝ) * Nat.factorial (2 ^ d) by positivity) hr
+  rw [Real.log_rpow (by norm_num : (0 : ℝ) < 2)] at hlog
+  unfold supportTimeBound
+  apply Int.ceil_le.mpr
+  simp only [Int.cast_natCast]
+  calc
+    2 / (2 : ℝ) ^ d * (Real.log ((3 / 4 : ℝ) * Nat.factorial (2 ^ d)) / Real.log 2) =
+        Real.log ((3 / 4 : ℝ) * Nat.factorial (2 ^ d)) / ((2 : ℝ) ^ d / 2 * Real.log 2) := by
+      field_simp
+    _ ≤ (t : ℝ) := (div_le_iff₀ (mul_pos (div_pos hN (by norm_num)) hL)).mpr (by
+      nlinarith [hlog])
+
+lemma supportTimeBound_le_mixingTime {d : ℕ}
+    (h : ({t : ℕ | distance d t ≤ (1 / 4 : ℝ)} : Set ℕ).Nonempty) :
+    supportTimeBound d ≤ (mixingTime d : ℤ) := by
+  exact supportTimeBound_le_of_distance_le d (mixingTime d) (Nat.sInf_mem h)
+
+lemma log_factorial_lower {n : ℕ} (hn : 1 ≤ n) :
+    (n : ℝ) * Real.log n - n ≤ Real.log (Nat.factorial n) := by
+  have h := Stirling.le_log_factorial_stirling (show n ≠ 0 by omega)
+  have hln : 0 ≤ Real.log (n : ℝ) := Real.log_nonneg (by exact_mod_cast hn)
+  have hlp : 0 ≤ Real.log (2 * Real.pi) := Real.log_nonneg (by
+    have := Real.pi_gt_three
+    linarith)
+  linarith
+
+lemma log_three_quarters_lower : -(1 : ℝ) ≤ Real.log (3 / 4 : ℝ) := by
+  have h := Real.log_le_sub_one_of_pos (show (0 : ℝ) < 4 / 3 by norm_num)
+  have he : Real.log (4 / 3 : ℝ) = - Real.log (3 / 4 : ℝ) := by
+    rw [← Real.log_inv]
+    congr 1
+    norm_num
+  rw [he] at h
+  linarith
+
+lemma linear_le_supportTimeBound (d : ℕ) :
+    2 * (d : ℝ) - 4 / Real.log 2 ≤ (supportTimeBound d : ℝ) := by
+  have hN : 1 ≤ (2 : ℝ) ^ d := one_le_pow₀ (by norm_num)
+  have hNp : 0 < (2 : ℝ) ^ d := by positivity
+  have hL : 0 < Real.log 2 := Real.log_pos (by norm_num)
+  have hf := log_factorial_lower (show 1 ≤ 2 ^ d by exact one_le_pow₀ (by norm_num))
+  have hlog : (2 : ℝ) ^ d * ((d : ℝ) * Real.log 2) - (2 : ℝ) ^ d - 1 ≤
+      Real.log ((3 / 4 : ℝ) * (Nat.factorial (2 ^ d) : ℝ)) := by
+    rw [Real.log_mul (by norm_num) (by positivity)]
+    push_cast at hf
+    rw [Real.log_pow] at hf
+    linarith [log_three_quarters_lower]
+  apply le_trans _ (Int.le_ceil _)
+  have hmul : (2 * (d : ℝ) - 4 / Real.log 2) *
+      ((2 : ℝ) ^ d / 2 * Real.log 2) ≤
+      Real.log ((3 / 4 : ℝ) * (Nat.factorial (2 ^ d) : ℝ)) := by
+    have hcancel : (2 * (d : ℝ) - 4 / Real.log 2) *
+        ((2 : ℝ) ^ d / 2 * Real.log 2) =
+        (2 : ℝ) ^ d * ((d : ℝ) * Real.log 2) - 2 * (2 : ℝ) ^ d := by
+      field_simp
+      ring
+    rw [hcancel]
+    linarith
+  convert (le_div_iff₀ (mul_pos (div_pos hNp (by norm_num)) hL)).mpr hmul using 1
+  field_simp
+
+lemma linear_le_time_of_distance_le (d t : ℕ)
+    (ht : distance d t ≤ (1 / 4 : ℝ)) :
+    2 * (d : ℝ) - 4 / Real.log 2 ≤ (t : ℝ) := by
+  have h := supportTimeBound_le_of_distance_le d t ht
+  have hr : (supportTimeBound d : ℝ) ≤ (t : ℝ) := by exact_mod_cast h
+  exact (linear_le_supportTimeBound d).trans hr
+
+lemma entropy_chord (p : ℝ) (hp : 0 ≤ p) (hq : p ≤ 1 / 2) :
+    2 * p * Real.log 2 ≤ Real.binEntropy p := by
+  have hc := Real.strictConcave_binEntropy.concaveOn.2
+    (show (0 : ℝ) ∈ Set.Icc (0 : ℝ) 1 by norm_num)
+    (show (2 : ℝ)⁻¹ ∈ Set.Icc (0 : ℝ) 1 by norm_num)
+    (show 0 ≤ 1 - 2 * p by linarith)
+    (show 0 ≤ 2 * p by positivity)
+    (show (1 - 2 * p) + 2 * p = 1 by ring)
+  simpa only [Real.binEntropy_zero, Real.binEntropy_two_inv,
+    smul_eq_mul, mul_zero, zero_add, show 2 * p * (2 : ℝ)⁻¹ = p by ring] using hc
+
+lemma split_entropy_lower (a b : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) :
+    2 * min a b * Real.log 2 ≤
+      (a + b) * Real.log (a + b) - a * Real.log a - b * Real.log b := by
+  wlog hab : a ≤ b generalizing a b
+  · have hba := this b a hb ha (le_of_not_ge hab)
+    simpa only [min_comm, add_comm b a] using hba.trans_eq (show (b + a) * Real.log (b + a) - b * Real.log b - a * Real.log a =
+      (a + b) * Real.log (a + b) - a * Real.log a - b * Real.log b by rw [add_comm b a]; ring)
+  rw [min_eq_left hab]
+  by_cases hz : a = 0
+  · subst a
+    simp
+  have hap : 0 < a := lt_of_le_of_ne ha (Ne.symm hz)
+  have hbp : 0 < b := hap.trans_le hab
+  have hs : 0 < a + b := add_pos hap hbp
+  have hp : 0 ≤ a / (a + b) := div_nonneg ha hs.le
+  have hq : a / (a + b) ≤ 1 / 2 := (div_le_iff₀ hs).mpr (by linarith)
+  have hc := mul_le_mul_of_nonneg_left (entropy_chord (a / (a + b)) hp hq) hs.le
+  have he : (a + b) * Real.binEntropy (a / (a + b)) =
+      (a + b) * Real.log (a + b) - a * Real.log a - b * Real.log b := by
+    unfold Real.binEntropy
+    rw [show 1 - a / (a + b) = b / (a + b) by field_simp; ring]
+    simp only [Real.log_inv]
+    rw [Real.log_div (ne_of_gt hap) (ne_of_gt hs),
+      Real.log_div (ne_of_gt hbp) (ne_of_gt hs)]
+    field_simp
+    ring
+  rw [he] at hc
+  calc
+    2 * a * Real.log 2 = (a + b) * (2 * (a / (a + b)) * Real.log 2) := by
+      field_simp
+    _ ≤ _ := hc
+
+noncomputable def childMarks {d : ℕ} (s : Finset (Card (d + 1))) (ε : Bool) :
+    Finset (Card d) := (s.filter (fun x => x 0 = ε)).image Fin.tail
+
+lemma card_childMarks {d : ℕ} (s : Finset (Card (d + 1))) (ε : Bool) :
+    (childMarks s ε).card = (s.filter (fun x => x 0 = ε)).card := by
+  classical
+  unfold childMarks
+  apply Finset.card_image_iff.mpr
+  intro x hx y hy hxy
+  have hx0 := (Finset.mem_filter.mp hx).2
+  have hy0 := (Finset.mem_filter.mp hy).2
+  rw [← Fin.cons_self_tail x, ← Fin.cons_self_tail y, hx0, hy0, hxy]
+
+lemma card_childMarks_add {d : ℕ} (s : Finset (Card (d + 1))) :
+    (childMarks s false).card + (childMarks s true).card = s.card := by
+  classical
+  rw [card_childMarks, card_childMarks]
+  simpa only [Bool.not_eq_false] using
+    (Finset.card_filter_add_card_filter_not (s := s) (fun x => x 0 = false))
+
+noncomputable def topMeetings {d : ℕ} (b : Butterfly (d + 1))
+    (s : Finset (Card (d + 1))) : Finset (Card d) :=
+  ((childMarks s false).image (butterflyPerm d (b.2 false))) ∩
+    ((childMarks s true).image (butterflyPerm d (b.2 true)))
+
+lemma card_topMeetings_le {d : ℕ} (b : Butterfly (d + 1)) (s : Finset (Card (d + 1))) :
+    (topMeetings b s).card ≤ min (childMarks s false).card (childMarks s true).card := by
+  classical
+  apply le_min
+  · exact (Finset.card_le_card Finset.inter_subset_left).trans_eq
+      (Finset.card_image_of_injective _ (butterflyPerm d (b.2 false)).injective)
+  · exact (Finset.card_le_card Finset.inter_subset_right).trans_eq
+      (Finset.card_image_of_injective _ (butterflyPerm d (b.2 true)).injective)
+
+noncomputable def encounters : (d : ℕ) → Butterfly d → Finset (Card d) → ℕ
+  | 0, _, _ => 0
+  | d + 1, b, s => encounters d (b.2 false) (childMarks s false) +
+      encounters d (b.2 true) (childMarks s true) + (topMeetings b s).card
+
+lemma butterfly_encounters (d : ℕ) (b : Butterfly d) (s : Finset (Card d)) :
+    (encounters d b s : ℝ) ≤ (s.card : ℝ) * Real.log s.card / (2 * Real.log 2) := by
+  induction d with
+  | zero =>
+    simp only [encounters, Nat.cast_zero]
+    have hs : s.card ≤ 1 := by
+      simpa [Card] using Finset.card_le_univ s
+    interval_cases s.card <;> simp
+  | succ d ih =>
+    have h₀ := ih (b.2 false) (childMarks s false)
+    have h₁ := ih (b.2 true) (childMarks s true)
+    have ht : ((topMeetings b s).card : ℝ) ≤
+        min ((childMarks s false).card : ℝ) ((childMarks s true).card : ℝ) := by
+      exact_mod_cast card_topMeetings_le b s
+    have he := split_entropy_lower ((childMarks s false).card : ℝ)
+      ((childMarks s true).card : ℝ) (by positivity) (by positivity)
+    have hs : ((childMarks s false).card : ℝ) + (childMarks s true).card = s.card := by
+      exact_mod_cast card_childMarks_add s
+    rw [hs] at he
+    have hL : 0 < 2 * Real.log 2 := by positivity
+    rw [le_div_iff₀ hL] at h₀ h₁ ⊢
+    simp only [encounters, Nat.cast_add]
+    nlinarith [mul_le_mul_of_nonneg_right ht hL.le]
+
+lemma prod_split_at {α : Type*} [DecidableEq α] (s : Finset α) (f : α → ℝ) (a : α) :
+    ∏ i ∈ s, f i = (if a ∈ s then f a else 1) * ∏ i ∈ s.erase a, f i := by
+  by_cases ha : a ∈ s
+  · rw [ite_eq_left ha]
+    exact (Finset.mul_prod_erase s f ha).symm
+  · simp [ha]
+
+lemma prod_split_two {α : Type*} [DecidableEq α] (s : Finset α) (f : α → ℝ)
+    (a b : α) (hab : a ≠ b) :
+    ∏ i ∈ s, f i = (if a ∈ s then f a else 1) * (if b ∈ s then f b else 1) *
+      ∏ i ∈ (s.erase a).erase b, f i := by
+  calc
+    ∏ i ∈ s, f i = (if a ∈ s then f a else 1) * ∏ i ∈ s.erase a, f i :=
+      prod_split_at s f a
+    _ = _ := by
+      rw [prod_split_at (s.erase a) f b]
+      have hm : (b ∈ s.erase a) ↔ b ∈ s :=
+        ⟨fun h => (Finset.mem_erase.mp h).2, fun h => Finset.mem_erase.mpr ⟨Ne.symm hab, h⟩⟩
+      simp only [hm, mul_assoc]
+
+lemma swap_average_product {α : Type*} [DecidableEq α] (s : Finset α) (p : α → ℝ)
+    (hp : ∀ i, 0 ≤ p i) (a b : α) :
+    ((∏ i ∈ s, p i) + ∏ i ∈ s, p (Equiv.swap a b i)) / 2 ≤
+      ∏ i ∈ s, (p i + p (Equiv.swap a b i)) / 2 := by
+  by_cases hab : a = b
+  · subst b
+    simp only [Equiv.swap_self, Equiv.refl_apply]
+    have h : (fun i => (p i + p i) / 2) = p := by funext i; ring
+    rw [h]
+    linarith
+  let t := (s.erase a).erase b
+  have hfix : ∀ i ∈ t, Equiv.swap a b i = i := by
+    intro i hi
+    exact Equiv.swap_apply_of_ne_of_ne (Finset.mem_erase.mp (Finset.mem_erase.mp hi).2).1
+      (Finset.mem_erase.mp hi).1
+  have hprod : (∏ i ∈ t, p (Equiv.swap a b i)) = ∏ i ∈ t, p i :=
+    Finset.prod_congr rfl (fun i hi => by rw [hfix i hi])
+  have havg : (∏ i ∈ t, (p i + p (Equiv.swap a b i)) / 2) = ∏ i ∈ t, p i :=
+    Finset.prod_congr rfl (fun i hi => by rw [hfix i hi]; ring)
+  rw [prod_split_two s p a b hab,
+    prod_split_two s (fun i => p (Equiv.swap a b i)) a b hab,
+    prod_split_two s (fun i => (p i + p (Equiv.swap a b i)) / 2) a b hab]
+  change ((if a ∈ s then p a else 1) * (if b ∈ s then p b else 1) * (∏ i ∈ t, p i) +
+    (if a ∈ s then p (Equiv.swap a b a) else 1) *
+      (if b ∈ s then p (Equiv.swap a b b) else 1) * (∏ i ∈ t, p (Equiv.swap a b i))) / 2 ≤ _
+  rw [hprod]
+  change _ ≤ (if a ∈ s then (p a + p (Equiv.swap a b a)) / 2 else 1) *
+    (if b ∈ s then (p b + p (Equiv.swap a b b)) / 2 else 1) *
+      (∏ i ∈ t, (p i + p (Equiv.swap a b i)) / 2)
+  rw [havg, Equiv.swap_apply_left, Equiv.swap_apply_right]
+  have ht : 0 ≤ ∏ i ∈ t, p i := Finset.prod_nonneg (fun i _ => hp i)
+  by_cases ha : a ∈ s <;> by_cases hb : b ∈ s <;> simp only [ha, hb, ↓reduceIte]
+  · have hsq := mul_nonneg (sq_nonneg (p a - p b)) ht
+    nlinarith
+  all_goals ring_nf; rfl
+
+def CylinderBound {α : Type*} (m : Finset α → ℝ) (p : α → ℝ) : Prop :=
+  ∀ s, m s ≤ ∏ i ∈ s, p i
+
+lemma CylinderBound.fair_swap {α : Type*} [DecidableEq α]
+    {m : Finset α → ℝ} {p : α → ℝ} (h : CylinderBound m p)
+    (hp : ∀ i, 0 ≤ p i) (a b : α) :
+    CylinderBound (fun s => (m s + m (s.image (Equiv.swap a b))) / 2)
+      (fun i => (p i + p (Equiv.swap a b i)) / 2) := by
+  intro s
+  have h₀ := h s
+  have h₁ := h (s.image (Equiv.swap a b))
+  rw [Finset.prod_image (Equiv.swap a b).injective.injOn] at h₁
+  exact (show (m s + m (s.image (Equiv.swap a b))) / 2 ≤
+      ((∏ i ∈ s, p i) + ∏ i ∈ s, p (Equiv.swap a b i)) / 2 by linarith).trans
+    (swap_average_product s p hp a b)
+
+noncomputable def finiteMean {Ω : Type*} [Fintype Ω] (f : Ω → ℝ) : ℝ :=
+  (∑ ω, f ω) / Fintype.card Ω
+
+noncomputable def inclusionMoment {α Ω : Type*} [Fintype Ω]
+    (F : Ω → Finset α) (s : Finset α) : ℝ := by
+  classical
+  exact finiteMean (fun ω => if s ⊆ F ω then 1 else 0)
+
+lemma inclusionMoment_nonneg {α Ω : Type*} [Fintype Ω]
+    (F : Ω → Finset α) (s : Finset α) : 0 ≤ inclusionMoment F s := by
+  classical
+  unfold inclusionMoment finiteMean
+  positivity
+
+lemma indicator_subset_prod {α : Type*} [DecidableEq α] (A s : Finset α) :
+    (if s ⊆ A then (1 : ℝ) else 0) = ∏ i ∈ s, (if i ∈ A then 1 else 0) := by
+  by_cases h : s ⊆ A
+  · rw [ite_eq_left h]
+    exact (Finset.prod_eq_one (fun i hi => ite_eq_left (h hi))).symm
+  · rw [ite_eq_right h]
+    obtain ⟨i, hi, hiA⟩ := Finset.not_subset.mp h
+    exact (Finset.prod_eq_zero hi (ite_eq_right hiA)).symm
+
+lemma inclusionMoment_const {α Ω : Type*} [DecidableEq α] [Fintype Ω] [Nonempty Ω]
+    (A s : Finset α) : inclusionMoment (fun _ : Ω => A) s =
+      if s ⊆ A then 1 else 0 := by
+  classical
+  have hn : (Fintype.card Ω : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
+  by_cases h : s ⊆ A <;> simp [inclusionMoment, finiteMean, h, hn]
+
+lemma cylinder_const {α Ω : Type*} [Fintype Ω] [Nonempty Ω] (A : Finset α) :
+    CylinderBound (inclusionMoment (fun _ : Ω => A))
+      (fun i => inclusionMoment (fun _ : Ω => A) {i}) := by
+  classical
+  intro s
+  simp only [inclusionMoment_const, Finset.singleton_subset_iff]
+  exact (indicator_subset_prod A s).le
+
+noncomputable def fairSetSwap {α Ω : Type*} [DecidableEq α]
+    (F : Ω → Finset α) (a b : α) : Ω × Bool → Finset α :=
+  fun ω => if ω.2 then (F ω.1).image (Equiv.swap a b) else F ω.1
+
+lemma subset_swap_image_iff {α : Type*} [DecidableEq α]
+    (A s : Finset α) (a b : α) :
+    s ⊆ A.image (Equiv.swap a b) ↔ s.image (Equiv.swap a b) ⊆ A := by
+  constructor
+  · intro h
+    have h' := Finset.image_subset_image h (f := Equiv.swap a b)
+    simpa only [Finset.image_image, Function.comp_def, Equiv.swap_apply_self, Finset.image_id'] using h'
+  · intro h
+    have h' := Finset.image_subset_image h (f := Equiv.swap a b)
+    simpa only [Finset.image_image, Function.comp_def, Equiv.swap_apply_self, Finset.image_id'] using h'
+
+lemma inclusionMoment_fairSetSwap {α Ω : Type*} [DecidableEq α] [Fintype Ω]
+    (F : Ω → Finset α) (a b : α) (s : Finset α) :
+    inclusionMoment (fairSetSwap F a b) s =
+      (inclusionMoment F s + inclusionMoment F (s.image (Equiv.swap a b))) / 2 := by
+  classical
+  simp only [inclusionMoment, finiteMean, Fintype.sum_prod_type, Fintype.sum_bool,
+    fairSetSwap, Bool.false_eq_true, ↓reduceIte, Fintype.card_prod, Fintype.card_bool,
+    Nat.cast_mul, Nat.cast_ofNat, subset_swap_image_iff, Finset.sum_add_distrib]
+  rw [div_mul_eq_div_div, add_div]
+  ring_nf
+
+lemma cylinder_fairSetSwap {α Ω : Type*} [DecidableEq α] [Fintype Ω]
+    (F : Ω → Finset α) (a b : α)
+    (h : CylinderBound (inclusionMoment F) (fun i => inclusionMoment F {i})) :
+    CylinderBound (inclusionMoment (fairSetSwap F a b))
+      (fun i => inclusionMoment (fairSetSwap F a b) {i}) := by
+  classical
+  have h' := h.fair_swap (fun i => inclusionMoment_nonneg F {i}) a b
+  intro s
+  simpa only [inclusionMoment_fairSetSwap, Finset.image_singleton] using h' s
+
+def SwitchSamples : ℕ → Type
+  | 0 => Unit
+  | n + 1 => SwitchSamples n × Bool
+
+instance (n : ℕ) : Fintype (SwitchSamples n) := by
+  induction n with
+  | zero => exact inferInstanceAs (Fintype Unit)
+  | succ n ih => exact inferInstanceAs (Fintype (SwitchSamples n × Bool))
+
+instance (n : ℕ) : Nonempty (SwitchSamples n) := by
+  induction n with
+  | zero => exact inferInstanceAs (Nonempty Unit)
+  | succ n ih => exact inferInstanceAs (Nonempty (SwitchSamples n × Bool))
+
+noncomputable def switchedSet {α : Type*} [DecidableEq α] (A : Finset α) :
+    (L : List (α × α)) → SwitchSamples L.length → Finset α
+  | [], _ => A
+  | (a,b) :: L, ω => fairSetSwap (switchedSet A L) a b ω
+
+lemma switchedSet_cylinder {α : Type*} [DecidableEq α] (A : Finset α)
+    (L : List (α × α)) :
+    CylinderBound (inclusionMoment (switchedSet A L))
+      (fun i => inclusionMoment (switchedSet A L) {i}) := by
+  induction L with
+  | nil => exact cylinder_const A
+  | cons e L ih => exact cylinder_fairSetSwap (switchedSet A L) e.1 e.2 ih
+
+inductive QueryTree (ι : Type*)
+  | reject : QueryTree ι
+  | accept : QueryTree ι
+  | branch (i : ι) (zero one : QueryTree ι) : QueryTree ι
+  | force (i : ι) (b : Bool) (next : QueryTree ι) : QueryTree ι
+
+namespace QueryTree
+
+variable {ι : Type*} [DecidableEq ι]
+
+def queries : QueryTree ι → Finset ι
+  | reject => ∅
+  | accept => ∅
+  | branch i zero one => insert i (queries zero ∪ queries one)
+  | force i _ next => insert i (queries next)
+
+def Fresh : QueryTree ι → Prop
+  | reject => True
+  | accept => True
+  | branch i zero one => i ∉ queries zero ∧ i ∉ queries one ∧ Fresh zero ∧ Fresh one
+  | force i _ next => i ∉ queries next ∧ Fresh next
+
+def weighted : QueryTree ι → (ι → Bool) → ℝ
+  | reject, _ => 0
+  | accept, _ => 1
+  | branch i zero one, ω => if ω i then weighted one ω else weighted zero ω
+  | force i b next, ω => if ω i = b then 2 * weighted next ω else 0
+
+omit [DecidableEq ι] in
+lemma weighted_nonneg (T : QueryTree ι) (ω : ι → Bool) : 0 ≤ T.weighted ω := by
+  induction T with
+  | reject => exact le_rfl
+  | accept => exact zero_le_one
+  | branch i zero one ih₀ ih₁ => simpa only [weighted] using ite_nonneg ih₁ ih₀
+  | force i b next ih => simpa only [weighted] using ite_nonneg (mul_nonneg (by norm_num) ih) le_rfl
+
+lemma weighted_congr (T : QueryTree ι) {ω η : ι → Bool}
+    (h : ∀ i ∈ T.queries, ω i = η i) : T.weighted ω = T.weighted η := by
+  induction T with
+  | reject => rfl
+  | accept => rfl
+  | branch i zero one ih₀ ih₁ =>
+    have hi := h i (Finset.mem_insert_self _ _)
+    simp only [weighted, hi]
+    rw [ih₀ (fun j hj => h j (Finset.mem_insert_of_mem (Finset.mem_union_left _ hj))),
+      ih₁ (fun j hj => h j (Finset.mem_insert_of_mem (Finset.mem_union_right _ hj)))]
+  | force i b next ih =>
+    have hi := h i (Finset.mem_insert_self _ _)
+    simp only [weighted, hi]
+    rw [ih (fun j hj => h j (Finset.mem_insert_of_mem hj))]
+
+def flipCoin (i : ι) : Equiv.Perm (ι → Bool) :=
+  (show Function.Involutive (fun ω : ι → Bool => Function.update ω i (!(ω i))) from by
+    intro ω
+    funext j
+    by_cases h : j = i
+    · subst j; simp
+    · simp [Function.update_of_ne h]).toPerm
+
+lemma flipCoin_apply_eq (i : ι) (ω : ι → Bool) : flipCoin i ω i = !(ω i) := by
+  change Function.update ω i (!(ω i)) i = _
+  exact Function.update_self _ _ _
+
+lemma flipCoin_apply_ne (i j : ι) (h : j ≠ i) (ω : ι → Bool) :
+    flipCoin i ω j = ω j := by
+  change Function.update ω i (!(ω i)) j = _
+  exact Function.update_of_ne h _ _
+
+lemma weighted_flip_of_not_mem (T : QueryTree ι) (i : ι) (hi : i ∉ T.queries)
+    (ω : ι → Bool) : T.weighted (flipCoin i ω) = T.weighted ω := by
+  apply weighted_congr
+  intro j hj
+  exact flipCoin_apply_ne i j (fun h => hi (h ▸ hj)) ω
+
+lemma mean_indicator_bit [Fintype ι] (f : (ι → Bool) → ℝ) (i : ι) (b : Bool)
+    (hf : ∀ ω, f (flipCoin i ω) = f ω) :
+    finiteMean (fun ω => if ω i = b then f ω else 0) = finiteMean f / 2 := by
+  let g := fun ω : ι → Bool => if ω i = b then f ω else 0
+  have hp (ω : ι → Bool) : g ω + g (flipCoin i ω) = f ω := by
+    dsimp [g]
+    rw [flipCoin_apply_eq, hf]
+    cases hω : ω i <;> cases b <;> simp
+  have hs := Equiv.sum_comp (flipCoin i) g
+  have hadd : (∑ ω, g ω) + (∑ ω, g (flipCoin i ω)) = ∑ ω, f ω := by
+    rw [← Finset.sum_add_distrib]
+    exact Finset.sum_congr rfl (fun ω _ => hp ω)
+  unfold finiteMean
+  change (∑ ω, g ω) / _ = _
+  rw [hs] at hadd
+  have hh : (∑ ω, g ω) = (∑ ω, f ω) / 2 := by linarith
+  rw [hh]
+  ring
+
+lemma mean_weighted_le_one [Fintype ι] (T : QueryTree ι) (hT : T.Fresh) :
+    finiteMean T.weighted ≤ 1 := by
+  induction T with
+  | reject => simp [weighted, finiteMean]
+  | accept => simp [weighted, finiteMean]
+  | branch i zero one ih₀ ih₁ =>
+    obtain ⟨hz, ho, hfz, hfo⟩ := hT
+    have h₀ := mean_indicator_bit zero.weighted i false (zero.weighted_flip_of_not_mem i hz)
+    have h₁ := mean_indicator_bit one.weighted i true (one.weighted_flip_of_not_mem i ho)
+    have he : finiteMean (branch i zero one).weighted =
+        finiteMean (fun ω => if ω i = false then zero.weighted ω else 0) +
+        finiteMean (fun ω => if ω i = true then one.weighted ω else 0) := by
+      unfold finiteMean
+      rw [← add_div, ← Finset.sum_add_distrib]
+      congr 1
+      apply Finset.sum_congr rfl
+      intro ω _
+      cases hω : ω i <;> simp [weighted, hω]
+    rw [he, h₀, h₁]
+    linarith [ih₀ hfz, ih₁ hfo]
+  | force i b next ih =>
+    obtain ⟨hi, hn⟩ := hT
+    have hm := mean_indicator_bit next.weighted i b (next.weighted_flip_of_not_mem i hi)
+    have he : finiteMean (force i b next).weighted =
+        2 * finiteMean (fun ω => if ω i = b then next.weighted ω else 0) := by
+      unfold finiteMean
+      rw [← mul_div_assoc, Finset.mul_sum]
+      congr 1
+      apply Finset.sum_congr rfl
+      intro ω _
+      by_cases h : ω i = b <;> simp [weighted, h]
+    rw [he, hm]
+    linarith [ih hn]
+
+def succeeds : QueryTree ι → (ι → Bool) → Bool
+  | reject, _ => false
+  | accept, _ => true
+  | branch i zero one, ω => if ω i then succeeds one ω else succeeds zero ω
+  | force i b next, ω => if ω i = b then succeeds next ω else false
+
+def charge : QueryTree ι → (ι → Bool) → ℕ
+  | reject, _ => 0
+  | accept, _ => 0
+  | branch i zero one, ω => if ω i then charge one ω else charge zero ω
+  | force i b next, ω => if ω i = b then charge next ω + 1 else 0
+
+omit [DecidableEq ι] in
+lemma weighted_eq (T : QueryTree ι) (ω : ι → Bool) :
+    T.weighted ω = if T.succeeds ω then (2 : ℝ) ^ T.charge ω else 0 := by
+  induction T with
+  | reject => rfl
+  | accept => norm_num [weighted, succeeds, charge]
+  | branch i zero one ih₀ ih₁ =>
+    cases h : ω i <;> simp [weighted, succeeds, charge, h, ih₀, ih₁]
+  | force i b next ih =>
+    by_cases h : ω i = b
+    · simp only [weighted, succeeds, charge, h, ↓reduceIte, ih]
+      cases hs : next.succeeds ω <;> simp [pow_succ, mul_comm]
+    · simp [weighted, succeeds, h]
+
+lemma success_probability_le [Fintype ι] (T : QueryTree ι) (hT : T.Fresh)
+    (K : ℕ) (hK : ∀ ω, T.succeeds ω = true → K ≤ T.charge ω) :
+    finiteMean (fun ω => if T.succeeds ω then (1 : ℝ) else 0) ≤ 1 / (2 : ℝ) ^ K := by
+  have hw (ω : ι → Bool) :
+      (2 : ℝ) ^ K * (if T.succeeds ω then (1 : ℝ) else 0) ≤ T.weighted ω := by
+    rw [weighted_eq]
+    cases hs : T.succeeds ω
+    · simp
+    · simpa using pow_le_pow_right₀ (show (1 : ℝ) ≤ 2 by norm_num) (hK ω hs)
+  have hn : 0 ≤ (Fintype.card (ι → Bool) : ℝ) := Nat.cast_nonneg _
+  have hm := div_le_div_of_nonneg_right
+    (Finset.sum_le_sum (fun ω (_ : ω ∈ Finset.univ) => hw ω)) hn
+  rw [← Finset.mul_sum, mul_div_assoc] at hm
+  change (2 : ℝ) ^ K * finiteMean (fun ω => if T.succeeds ω then 1 else 0) ≤
+    finiteMean T.weighted at hm
+  exact (le_div_iff₀ (pow_pos (by norm_num) K)).mpr (by
+    rw [mul_comm]
+    exact hm.trans (mean_weighted_le_one T hT))
+
+end QueryTree
+
+def encodeButterfly : (d : ℕ) → Butterfly d → SwitchIndex d → Bool
+  | 0, _, i => nomatch i
+  | _d + 1, b, Sum.inl y => b.1 y
+  | d + 1, b, Sum.inr (ε,i) => encodeButterfly d (b.2 ε) i
+
+lemma decode_encodeButterfly (d : ℕ) (b : Butterfly d) :
+    decodeButterfly d (encodeButterfly d b) = b := by
+  induction d with
+  | zero => exact Unit.ext _ _
+  | succ d ih =>
+    apply Prod.ext
+    · rfl
+    · funext ε
+      exact ih (b.2 ε)
+
+lemma encode_decodeButterfly (d : ℕ) (ω : SwitchIndex d → Bool) :
+    encodeButterfly d (decodeButterfly d ω) = ω := by
+  induction d with
+  | zero => funext i; exact Empty.elim i
+  | succ d ih =>
+    funext i
+    cases i with
+    | inl y => rfl
+    | inr z => exact congrFun (ih (fun i => ω (Sum.inr (z.1,i)))) z.2
+
+def butterflyCoinEquiv (d : ℕ) : (SwitchIndex d → Bool) ≃ Butterfly d where
+  toFun := decodeButterfly d
+  invFun := encodeButterfly d
+  left_inv := encode_decodeButterfly d
+  right_inv := decode_encodeButterfly d
+
+noncomputable def routeDomain : (d : ℕ) → Card d → Card d → Finset (SwitchIndex d)
+  | 0, _, _ => ∅
+  | d + 1, x, y => by
+      classical
+      exact insert (Sum.inl (Fin.tail y))
+        ((routeDomain d (Fin.tail x) (Fin.tail y)).image (fun i => Sum.inr (x 0,i)))
+
+def routeValue : (d : ℕ) → Card d → Card d → SwitchIndex d → Bool
+  | 0, _, _, i => nomatch i
+  | _d + 1, x, y, Sum.inl _ => Bool.xor (x 0) (y 0)
+  | d + 1, x, y, Sum.inr (_,i) => routeValue d (Fin.tail x) (Fin.tail y) i
+
+end Thorp
+
+end ThorpNine.SparseSaving
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/SparseSaving/ExposedCoins.lean b/lean/OAI/Probability/ThorpRouting/SparseSaving/ExposedCoins.lean
index 563496f360f623dd12c81ead97e3e7101a11cd91..b55fe6ae9178a0a86c35a7746ae866215c2ae66c 100644
--- a/lean/OAI/Probability/ThorpRouting/SparseSaving/ExposedCoins.lean
+++ b/lean/OAI/Probability/ThorpRouting/SparseSaving/ExposedCoins.lean
@@ -1,860 +1,860 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.SparseSaving.Coins
-
-namespace OAI
-
-namespace ThorpNine.SparseSaving
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-
-lemma card_routeDomain (d : ℕ) (x y : Card d) : (routeDomain d x y).card = d := by
-  classical
-  induction d with
-  | zero => rfl
-  | succ d ih =>
-    change (insert (Sum.inl (Fin.tail y))
-      ((routeDomain d (Fin.tail x) (Fin.tail y)).image
-        (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) :
-      Finset (Sum (Card d) (Bool × SwitchIndex d))).card = d + 1
-    rw [Finset.card_insert_of_notMem (by simp)]
-    rw [Finset.card_image_of_injective _ (by intro i j h; cases h; rfl), ih]
-
-lemma xor_eq_iff (a b c : Bool) : Bool.xor a b = c ↔ b = Bool.xor a c := by
-  cases a <;> cases b <;> cases c <;> decide
-
-lemma butterfly_endpoint_iff (d : ℕ) (ω : SwitchIndex d → Bool) (x y : Card d) :
-    butterflyPerm d (decodeButterfly d ω) x = y ↔
-      ∀ i ∈ routeDomain d x y, ω i = routeValue d x y i := by
-  classical
-  induction d with
-  | zero =>
-    have he : butterflyPerm 0 (decodeButterfly 0 ω) x = y := Subsingleton.elim _ _
-    simp [he, routeDomain]
-  | succ d ih =>
-    let v := butterflyPerm d (decodeButterfly d (fun i => ω (Sum.inr (x 0,i)))) (Fin.tail x)
-    change Fin.cons (Bool.xor (x 0) (ω (Sum.inl v))) v = y ↔ _
-    have hv : v = Fin.tail y ↔
-        ∀ i ∈ routeDomain d (Fin.tail x) (Fin.tail y),
-          ω (Sum.inr (x 0,i)) = routeValue d (Fin.tail x) (Fin.tail y) i :=
-      ih _ _ _
-    constructor
-    · intro h i hi
-      have ht : v = Fin.tail y := by simpa using congrArg Fin.tail h
-      have hh : ω (Sum.inl (Fin.tail y)) = Bool.xor (x 0) (y 0) := by
-        have he := congrFun h 0
-        simp only [Fin.cons_zero, ht] at he
-        exact (xor_eq_iff _ _ _).mp he
-      rcases Finset.mem_insert.mp hi with htop | hchild
-      · subst i
-        exact hh
-      · obtain ⟨j, hj, rfl⟩ := Finset.mem_image.mp hchild
-        exact hv.mp ht j hj
-    · intro h
-      have hh := h (Sum.inl (Fin.tail y)) (Finset.mem_insert_self _ _)
-      have ht : v = Fin.tail y := hv.mpr (fun i hi =>
-        h (Sum.inr (x 0,i)) (Finset.mem_insert_of_mem (Finset.mem_image_of_mem _ hi)))
-      have hh' : Bool.xor (x 0) (ω (Sum.inl (Fin.tail y))) = y 0 :=
-        (xor_eq_iff _ _ _).mpr hh
-      rw [ht, hh', Fin.cons_self_tail]
-
-lemma finiteMean_congr {Ω : Type*} [Fintype Ω] {f g : Ω → ℝ}
-    (h : ∀ ω, f ω = g ω) : finiteMean f = finiteMean g := by
-  unfold finiteMean
-  congr 1
-  exact Finset.sum_congr rfl (fun ω _ => h ω)
-
-lemma mean_coinCylinder {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (s : Finset ι) (v : ι → Bool) :
-    finiteMean (fun ω : ι → Bool => if ∀ i ∈ s, ω i = v i then (1 : ℝ) else 0) =
-      1 / (2 : ℝ) ^ s.card := by
-  induction s using Finset.induction_on with
-  | empty => simp [finiteMean]
-  | @insert i s hi ih =>
-    let f := fun ω : ι → Bool => if ∀ j ∈ s, ω j = v j then (1 : ℝ) else 0
-    have hf : ∀ ω, f (QueryTree.flipCoin i ω) = f ω := by
-      intro ω
-      have he : (∀ j ∈ s, QueryTree.flipCoin i ω j = v j) ↔ ∀ j ∈ s, ω j = v j := by
-        apply forall₂_congr
-        intro j hj
-        rw [QueryTree.flipCoin_apply_ne i j (fun h => hi (h ▸ hj))]
-      dsimp [f]
-      simp only [he]
-    have hm := QueryTree.mean_indicator_bit f i (v i) hf
-    have he : (fun ω : ι → Bool => if ∀ j ∈ insert i s, ω j = v j then (1 : ℝ) else 0) =
-        (fun ω => if ω i = v i then f ω else 0) := by
-      funext ω
-      by_cases h : ω i = v i <;> simp [f, h]
-    rw [he, hm]
-    change finiteMean (fun ω : ι → Bool => if ∀ j ∈ s, ω j = v j then (1 : ℝ) else 0) / 2 = _
-    rw [ih, Finset.card_insert_of_notMem hi, pow_succ]
-    ring
-
-def exposedCoins {ι : Type*} [DecidableEq ι] (R : Finset ι) (η ω : ι → Bool) : ι → Bool :=
-  fun i => if i ∈ R then η i else ω i
-
-lemma mean_coinCylinder_exposed {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (s R : Finset ι) (v η : ι → Bool) :
-    finiteMean (fun ω : ι → Bool =>
-      if ∀ i ∈ s, exposedCoins R η ω i = v i then (1 : ℝ) else 0) =
-    if (∀ i ∈ s ∩ R, η i = v i) then 1 / (2 : ℝ) ^ (s \ R).card else 0 := by
-  have he (ω : ι → Bool) : (∀ i ∈ s, exposedCoins R η ω i = v i) ↔
-      (∀ i ∈ s ∩ R, η i = v i) ∧ (∀ i ∈ s \ R, ω i = v i) := by
-    constructor
-    · intro h
-      constructor
-      · intro i hi
-        simpa [exposedCoins, (Finset.mem_inter.mp hi).2] using h i (Finset.mem_inter.mp hi).1
-      · intro i hi
-        simpa [exposedCoins, (Finset.mem_sdiff.mp hi).2] using h i (Finset.mem_sdiff.mp hi).1
-    · rintro ⟨ho, hf⟩ i hi
-      by_cases hR : i ∈ R
-      · simpa [exposedCoins, hR] using ho i (Finset.mem_inter.mpr ⟨hi,hR⟩)
-      · simpa [exposedCoins, hR] using hf i (Finset.mem_sdiff.mpr ⟨hi,hR⟩)
-  have hm : finiteMean (fun ω : ι → Bool =>
-      if ∀ i ∈ s, exposedCoins R η ω i = v i then (1 : ℝ) else 0) =
-      finiteMean (fun ω : ι → Bool =>
-        if (∀ i ∈ s ∩ R, η i = v i) ∧ (∀ i ∈ s \ R, ω i = v i) then (1 : ℝ) else 0) :=
-    finiteMean_congr (fun ω => by simp only [he ω])
-  rw [hm]
-  by_cases h : ∀ i ∈ s ∩ R, η i = v i
-  · have hp : (∀ i ∈ s ∩ R, η i = v i) ↔ True := iff_true_intro h
-    simp only [hp, true_and, ↓reduceIte]
-    exact mean_coinCylinder (s \ R) v
-  · have hp : (∀ i ∈ s ∩ R, η i = v i) ↔ False := iff_false_intro h
-    simp only [hp, false_and, ↓reduceIte]
-    simp [finiteMean]
-
-lemma butterfly_closure_probability (d : ℕ) (x y : Card d)
-    (R : Finset (SwitchIndex d)) (η : SwitchIndex d → Bool) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      if butterflyPerm d (decodeButterfly d (exposedCoins R η ω)) x = y then (1 : ℝ) else 0) =
-      if (∀ i ∈ routeDomain d x y ∩ R, η i = routeValue d x y i) then
-        (2 : ℝ) ^ (routeDomain d x y ∩ R).card / (2 : ℝ) ^ d else 0 := by
-  have he : finiteMean (fun ω : SwitchIndex d → Bool =>
-      if butterflyPerm d (decodeButterfly d (exposedCoins R η ω)) x = y then (1 : ℝ) else 0) =
-      finiteMean (fun ω : SwitchIndex d → Bool =>
-        if ∀ i ∈ routeDomain d x y, exposedCoins R η ω i = routeValue d x y i then (1 : ℝ) else 0) :=
-    finiteMean_congr (fun ω => by simp only [butterfly_endpoint_iff])
-  rw [he, mean_coinCylinder_exposed]
-  congr 1
-  have hc : (routeDomain d x y \ R).card + (routeDomain d x y ∩ R).card = d := by
-    rw [Finset.card_sdiff_add_card_inter, card_routeDomain]
-  have hp : (2 : ℝ) ^ d = (2 : ℝ) ^ (routeDomain d x y \ R).card *
-      (2 : ℝ) ^ (routeDomain d x y ∩ R).card := by
-    rw [← pow_add]
-    exact congrArg (fun n => (2 : ℝ) ^ n) hc.symm
-  rw [hp]
-  have hn : (2 : ℝ) ^ (routeDomain d x y ∩ R).card ≠ 0 := pow_ne_zero _ (by norm_num)
-  field_simp
-
-lemma butterfly_single_card_uniform (d : ℕ) (x y : Card d) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      if butterflyPerm d (decodeButterfly d ω) x = y then (1 : ℝ) else 0) = 1 / (2 : ℝ) ^ d := by
-  have he : finiteMean (fun ω : SwitchIndex d → Bool =>
-      if butterflyPerm d (decodeButterfly d ω) x = y then (1 : ℝ) else 0) =
-      finiteMean (fun ω : SwitchIndex d → Bool =>
-        if ∀ i ∈ routeDomain d x y, ω i = routeValue d x y i then (1 : ℝ) else 0) :=
-    finiteMean_congr (fun ω => by simp only [butterfly_endpoint_iff])
-  rw [he, mean_coinCylinder, card_routeDomain]
-
-def switchUsers : (d : ℕ) → Butterfly d → SwitchIndex d → Card d × Card d
-  | 0, _, i => nomatch i
-  | d + 1, b, Sum.inl y =>
-      (Fin.cons false ((butterflyPerm d (b.2 false)).symm y),
-       Fin.cons true ((butterflyPerm d (b.2 true)).symm y))
-  | d + 1, b, Sum.inr (ε,i) =>
-      (Fin.cons ε (switchUsers d (b.2 ε) i).1,
-       Fin.cons ε (switchUsers d (b.2 ε) i).2)
-
-lemma switchUsers_ne (d : ℕ) (b : Butterfly d) (i : SwitchIndex d) :
-    (switchUsers d b i).1 ≠ (switchUsers d b i).2 := by
-  induction d with
-  | zero => exact Empty.elim i
-  | succ d ih =>
-    cases i with
-    | inl y =>
-      intro h
-      have := congrFun h 0
-      simp only [switchUsers, Fin.cons_zero] at this
-      exact Bool.false_ne_true this
-    | inr z =>
-      intro h
-      exact ih (b.2 z.1) z.2 (by simpa only [switchUsers, Fin.tail_cons] using congrArg Fin.tail h)
-
-lemma cons_eq_iff {d : ℕ} (ε : Bool) (a : Card d) (x : Card (d + 1)) :
-    Fin.cons ε a = x ↔ ε = x 0 ∧ a = Fin.tail x := by
-  constructor
-  · intro h
-    exact ⟨by simpa only [Fin.cons_zero] using congrFun h 0,
-      by simpa only [Fin.tail_cons] using congrArg Fin.tail h⟩
-  · rintro ⟨rfl,rfl⟩
-    exact Fin.cons_self_tail x
-
-lemma mem_routeDomain_iff_user (d : ℕ) (b : Butterfly d) (x : Card d) (i : SwitchIndex d) :
-    i ∈ routeDomain d x (butterflyPerm d b x) ↔
-      (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x := by
-  classical
-  induction d with
-  | zero => exact Empty.elim i
-  | succ d ih =>
-    cases i with
-    | inl y =>
-      change Sum.inl y ∈ insert (Sum.inl (Fin.tail (butterflyPerm (d+1) b x)))
-        ((routeDomain d (Fin.tail x) (Fin.tail (butterflyPerm (d+1) b x))).image
-          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) ↔ _
-      simp only [Finset.mem_insert, Sum.inl.injEq, Finset.mem_image,
-        Sum.inr_ne_inl, and_false, exists_false, or_false]
-      change y = butterflyPerm d (b.2 (x 0)) (Fin.tail x) ↔ _
-      simp only [switchUsers, cons_eq_iff]
-      cases x 0 <;> simp [Equiv.symm_apply_eq]
-    | inr z =>
-      obtain ⟨ε,i⟩ := z
-      change Sum.inr (ε,i) ∈ insert (Sum.inl (Fin.tail (butterflyPerm (d+1) b x)))
-        ((routeDomain d (Fin.tail x) (Fin.tail (butterflyPerm (d+1) b x))).image
-          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) ↔ _
-      simp only [Finset.mem_insert, Sum.inr_ne_inl, false_or, Finset.mem_image,
-        Sum.inr.injEq, Prod.mk.injEq]
-      change (∃ j, j ∈ routeDomain d (Fin.tail x) (butterflyPerm d (b.2 (x 0)) (Fin.tail x)) ∧
-        x 0 = ε ∧ j = i) ↔ _
-      simp only [switchUsers, cons_eq_iff]
-      constructor
-      · rintro ⟨j,hj,hε,rfl⟩
-        subst ε
-        rcases (ih (b.2 (x 0)) (Fin.tail x) j).mp hj with h | h
-        · exact Or.inl ⟨rfl,h⟩
-        · exact Or.inr ⟨rfl,h⟩
-      · rintro (⟨hε,h⟩ | ⟨hε,h⟩)
-        · subst ε
-          exact ⟨i,(ih (b.2 (x 0)) (Fin.tail x) i).mpr (Or.inl h),rfl,rfl⟩
-        · subst ε
-          exact ⟨i,(ih (b.2 (x 0)) (Fin.tail x) i).mpr (Or.inr h),rfl,rfl⟩
-
-lemma card_switches_using (d : ℕ) (b : Butterfly d) (x : Card d) :
-    (Finset.univ.filter (fun i : SwitchIndex d =>
-      (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x)).card = d := by
-  classical
-  have he : Finset.univ.filter (fun i : SwitchIndex d =>
-      (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x) =
-      routeDomain d x (butterflyPerm d b x) := by
-    ext i
-    simp only [Finset.mem_filter, Finset.mem_univ, true_and, mem_routeDomain_iff_user]
-  rw [he,card_routeDomain]
-
-lemma switch_incidence_sum (d : ℕ) (b : Butterfly d) (w : Card d → ℝ) :
-    ∑ i : SwitchIndex d, (w (switchUsers d b i).1 + w (switchUsers d b i).2) =
-      (d : ℝ) * ∑ x : Card d, w x := by
-  classical
-  calc
-    _ = ∑ i : SwitchIndex d, ∑ x : Card d,
-        if (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x then w x else 0 := by
-      apply Finset.sum_congr rfl
-      intro i _
-      rw [Finset.sum_ite]
-      simp only [Finset.sum_const_zero, add_zero]
-      have he : Finset.univ.filter (fun x : Card d =>
-          (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x) =
-          {(switchUsers d b i).1, (switchUsers d b i).2} := by
-        ext x; simp [eq_comm]
-      rw [he, Finset.sum_pair (switchUsers_ne d b i)]
-    _ = ∑ x : Card d, ∑ i : SwitchIndex d,
-        if (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x then w x else 0 :=
-      Finset.sum_comm
-    _ = ∑ x : Card d, (d : ℝ) * w x := by
-      apply Finset.sum_congr rfl
-      intro x _
-      rw [Finset.sum_ite]
-      simp only [Finset.sum_const_zero, add_zero, Finset.sum_const, nsmul_eq_mul,
-        card_switches_using]
-    _ = _ := by rw [Finset.mul_sum]
-
-lemma weighted_switch_min_bound (d : ℕ) (b : Butterfly d) (w : Card d → ℝ) :
-    ∑ i : SwitchIndex d, min (w (switchUsers d b i).1) (w (switchUsers d b i).2) ≤
-      (d : ℝ) / 2 * ∑ x : Card d, w x := by
-  have h : 2 * (∑ i : SwitchIndex d,
-      min (w (switchUsers d b i).1) (w (switchUsers d b i).2)) ≤
-      ∑ i : SwitchIndex d, (w (switchUsers d b i).1 + w (switchUsers d b i).2) := by
-    rw [Finset.mul_sum]
-    apply Finset.sum_le_sum
-    intro i _
-    linarith [min_le_left (w (switchUsers d b i).1) (w (switchUsers d b i).2),
-      min_le_right (w (switchUsers d b i).1) (w (switchUsers d b i).2)]
-  rw [switch_incidence_sum] at h
-  linarith
-
-lemma mem_childMarks {d : ℕ} (s : Finset (Card (d + 1))) (ε : Bool) (x : Card d) :
-    x ∈ childMarks s ε ↔ Fin.cons ε x ∈ s := by
-  classical
-  constructor
-  · intro h
-    obtain ⟨y,hy,hxy⟩ := Finset.mem_image.mp h
-    obtain ⟨hys,hyε⟩ := Finset.mem_filter.mp hy
-    rw [← hxy, ← hyε, Fin.cons_self_tail]
-    exact hys
-  · intro h
-    exact Finset.mem_image.mpr ⟨Fin.cons ε x,
-      Finset.mem_filter.mpr ⟨h,by simp⟩,by simp⟩
-
-lemma mem_topMeetings {d : ℕ} (b : Butterfly (d + 1))
-    (s : Finset (Card (d + 1))) (y : Card d) :
-    y ∈ topMeetings b s ↔
-      (switchUsers (d+1) b (Sum.inl y)).1 ∈ s ∧
-      (switchUsers (d+1) b (Sum.inl y)).2 ∈ s := by
-  classical
-  have he (ε : Bool) : y ∈ (childMarks s ε).image (butterflyPerm d (b.2 ε)) ↔
-      (butterflyPerm d (b.2 ε)).symm y ∈ childMarks s ε := by
-    constructor
-    · intro h
-      obtain ⟨x,hx,rfl⟩ := Finset.mem_image.mp h
-      simpa only [Equiv.symm_apply_apply] using hx
-    · intro h
-      exact Finset.mem_image.mpr ⟨_,h,Equiv.apply_symm_apply _ _⟩
-  simp only [topMeetings, Finset.mem_inter, he, mem_childMarks, switchUsers]
-
-lemma encounters_eq_switch_sum (d : ℕ) (b : Butterfly d) (s : Finset (Card d)) :
-    encounters d b s = ∑ i : SwitchIndex d,
-      if (switchUsers d b i).1 ∈ s ∧ (switchUsers d b i).2 ∈ s then 1 else 0 := by
-  classical
-  induction d with
-  | zero =>
-    simp only [encounters]
-    symm
-    apply Finset.sum_eq_zero
-    intro i _
-    exact Empty.elim i
-  | succ d ih =>
-    change encounters d (b.2 false) (childMarks s false) +
-      encounters d (b.2 true) (childMarks s true) + (topMeetings b s).card =
-        ∑ i : Sum (Card d) (Bool × SwitchIndex d), _
-    rw [Fintype.sum_sum_type, Fintype.sum_prod_type]
-    simp only [Fintype.sum_bool]
-    rw [ih,ih]
-    have ht : (∑ y : Card d,
-        if (switchUsers (d+1) b (Sum.inl y)).1 ∈ s ∧
-          (switchUsers (d+1) b (Sum.inl y)).2 ∈ s then 1 else 0) =
-        (topMeetings b s).card := by
-      simp only [← mem_topMeetings]
-      simp
-    rw [ht]
-    simp only [mem_childMarks, switchUsers]
-    ac_rfl
-
-lemma marked_switch_entropy_bound (d : ℕ) (b : Butterfly d) (s : Finset (Card d)) :
-    (∑ i : SwitchIndex d,
-      if (switchUsers d b i).1 ∈ s ∧ (switchUsers d b i).2 ∈ s then (1:ℝ) else 0) ≤
-      (s.card : ℝ) * Real.log s.card / (2 * Real.log 2) := by
-  classical
-  have he : (encounters d b s : ℝ) =
-      ∑ i : SwitchIndex d,
-        if (switchUsers d b i).1 ∈ s ∧ (switchUsers d b i).2 ∈ s then (1:ℝ) else 0 := by
-    rw [encounters_eq_switch_sum, Nat.cast_sum]
-    apply Finset.sum_congr rfl
-    intro i _
-    split_ifs <;> simp
-  rw [← he]
-  exact butterfly_encounters d b s
-
-namespace QueryTree
-variable {ι : Type*} [DecidableEq ι]
-
-def memo : QueryTree ι → Finset ι → (ι → Bool) → QueryTree ι
-  | reject, _, _ => reject
-  | accept, _, _ => accept
-  | branch i zero one, R, η =>
-      if i ∈ R then (if η i then memo one R η else memo zero R η) else
-      branch i (memo zero (insert i R) (Function.update η i false))
-        (memo one (insert i R) (Function.update η i true))
-  | force i b next, R, η =>
-      if i ∈ R then (if η i = b then memo next R η else reject) else
-      force i b (memo next (insert i R) (Function.update η i b))
-
-lemma memo_queries_disjoint (T : QueryTree ι) (R : Finset ι) (η : ι → Bool) :
-    Disjoint (memo T R η).queries R := by
-  induction T generalizing R η with
-  | reject => simp [memo,queries]
-  | accept => simp [memo,queries]
-  | branch i zero one ih₀ ih₁ =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte]
-      cases η i <;> simp only [Bool.false_eq_true,↓reduceIte]
-      · exact ih₀ R η
-      · exact ih₁ R η
-    · simp only [memo,hi,↓reduceIte,queries,Finset.disjoint_insert_left,Finset.disjoint_union_left]
-      exact ⟨by simp, (ih₀ (insert i R) _).mono_right (Finset.subset_insert _ _),
-        (ih₁ (insert i R) _).mono_right (Finset.subset_insert _ _)⟩
-  | force i b next ih =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte]
-      by_cases hb : η i = b
-      · simpa only [hb,↓reduceIte] using ih R η
-      · simp [hb,queries]
-    · simp only [memo,hi,↓reduceIte,queries,Finset.disjoint_insert_left]
-      exact ⟨by simp,(ih (insert i R) _).mono_right (Finset.subset_insert _ _)⟩
-
-lemma memo_fresh (T : QueryTree ι) (R : Finset ι) (η : ι → Bool) :
-    (memo T R η).Fresh := by
-  induction T generalizing R η with
-  | reject => trivial
-  | accept => trivial
-  | branch i zero one ih₀ ih₁ =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte]
-      cases η i <;> simp only [Bool.false_eq_true,↓reduceIte]
-      · exact ih₀ R η
-      · exact ih₁ R η
-    · simp only [memo,hi,↓reduceIte,Fresh]
-      exact ⟨fun hm => Finset.disjoint_left.mp (memo_queries_disjoint zero (insert i R) _) hm
-          (Finset.mem_insert_self _ _),
-        fun hm => Finset.disjoint_left.mp (memo_queries_disjoint one (insert i R) _) hm
-          (Finset.mem_insert_self _ _), ih₀ _ _,ih₁ _ _⟩
-  | force i b next ih =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte]
-      by_cases hb : η i = b
-      · simpa only [hb,↓reduceIte] using ih R η
-      · simp [hb,Fresh]
-    · simp only [memo,hi,↓reduceIte,Fresh]
-      exact ⟨fun hm => Finset.disjoint_left.mp (memo_queries_disjoint next (insert i R) _) hm
-        (Finset.mem_insert_self _ _),ih _ _⟩
-
-lemma update_agrees {R : Finset ι} {η ω : ι → Bool}
-    (h : ∀ i ∈ R, η i = ω i) (i : ι) :
-    ∀ j ∈ insert i R, Function.update η i (ω i) j = ω j := by
-  intro j hj
-  rcases Finset.mem_insert.mp hj with rfl | hj
-  · simp
-  · by_cases hji : j = i
-    · subst j; simp
-    · rw [Function.update_of_ne hji]
-      exact h j hj
-
-lemma memo_succeeds (T : QueryTree ι) (R : Finset ι) (η ω : ι → Bool)
-    (h : ∀ i ∈ R, η i = ω i) : (memo T R η).succeeds ω = T.succeeds ω := by
-  induction T generalizing R η with
-  | reject => rfl
-  | accept => rfl
-  | branch i zero one ih₀ ih₁ =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte,succeeds,h i hi]
-      cases ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
-      · exact ih₀ R η h
-      · exact ih₁ R η h
-    · simp only [memo,hi,↓reduceIte,succeeds]
-      cases hb : ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
-      · exact ih₀ _ _ (by simpa only [hb] using update_agrees h i)
-      · exact ih₁ _ _ (by simpa only [hb] using update_agrees h i)
-  | force i b next ih =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte,h i hi]
-      by_cases hb : ω i = b
-      · simp only [hb,↓reduceIte,succeeds]
-        exact ih R η h
-      · simp [hb,succeeds]
-    · simp only [memo,hi,↓reduceIte,succeeds]
-      by_cases hb : ω i = b
-      · simp only [hb,↓reduceIte]
-        exact ih _ _ (by simpa only [hb] using update_agrees h i)
-      · simp [hb]
-
-def freshCharge : QueryTree ι → Finset ι → (ι → Bool) → ℕ
-  | reject, _, _ => 0
-  | accept, _, _ => 0
-  | branch i zero one, R, ω => if ω i then freshCharge one (insert i R) ω else freshCharge zero (insert i R) ω
-  | force i b next, R, ω => if ω i = b then
-      freshCharge next (insert i R) ω + (if i ∈ R then 0 else 1) else 0
-
-lemma memo_charge (T : QueryTree ι) (R : Finset ι) (η ω : ι → Bool)
-    (h : ∀ i ∈ R, η i = ω i) : (memo T R η).charge ω = freshCharge T R ω := by
-  induction T generalizing R η with
-  | reject => rfl
-  | accept => rfl
-  | branch i zero one ih₀ ih₁ =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte,freshCharge,h i hi,Finset.insert_eq_of_mem hi]
-      cases ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
-      · exact ih₀ R η h
-      · exact ih₁ R η h
-    · simp only [memo,hi,↓reduceIte,charge,freshCharge]
-      cases hb : ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
-      · exact ih₀ _ _ (by simpa only [hb] using update_agrees h i)
-      · exact ih₁ _ _ (by simpa only [hb] using update_agrees h i)
-  | force i b next ih =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte,h i hi,freshCharge,Finset.insert_eq_of_mem hi,Nat.add_zero]
-      by_cases hb : ω i = b
-      · simp only [hb,↓reduceIte]
-        exact ih R η h
-      · simp [hb,charge]
-    · simp only [memo,hi,↓reduceIte,charge,freshCharge]
-      by_cases hb : ω i = b
-      · simp only [hb,↓reduceIte]
-        rw [ih _ _ (by simpa only [hb] using update_agrees h i)]
-      · simp [hb]
-
-lemma mean_freshCharge_le_one [Fintype ι] (T : QueryTree ι) :
-    finiteMean (fun ω : ι → Bool =>
-      if T.succeeds ω then (2:ℝ) ^ T.freshCharge ∅ ω else 0) ≤ 1 := by
-  let η : ι → Bool := fun _ => false
-  have hm := mean_weighted_le_one (memo T ∅ η) (memo_fresh T ∅ η)
-  have he : finiteMean (memo T ∅ η).weighted =
-      finiteMean (fun ω : ι → Bool =>
-        if T.succeeds ω then (2:ℝ) ^ T.freshCharge ∅ ω else 0) := by
-    apply finiteMean_congr
-    intro ω
-    rw [weighted_eq, memo_succeeds _ _ _ _ (by simp), memo_charge _ _ _ _ (by simp)]
-  rw [he] at hm
-  exact hm
-
-
-def followPath : (d : ℕ) → (SwitchIndex d → ι) → Card d →
-    (Card d → QueryTree ι) → QueryTree ι
-  | 0, _, x, k => k x
-  | d + 1, e, x, k => followPath d (fun i => e (Sum.inr (x 0,i))) (Fin.tail x)
-      (fun y => branch (e (Sum.inl y))
-        (k (Fin.cons (Bool.xor (x 0) false) y))
-        (k (Fin.cons (Bool.xor (x 0) true) y)))
-
-omit [DecidableEq ι] in
-lemma followPath_succeeds (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
-    (k : Card d → QueryTree ι) (ω : ι → Bool) :
-    (followPath d e x k).succeeds ω =
-      (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).succeeds ω := by
-  induction d with
-  | zero => rfl
-  | succ d ih =>
-    rw [followPath, ih]
-    let y := butterflyPerm d
-      (decodeButterfly d (fun i => ω (e (Sum.inr (x 0,i))))) (Fin.tail x)
-    change (if ω (e (Sum.inl y)) then
-      (k (Fin.cons (Bool.xor (x 0) true) y)).succeeds ω else
-      (k (Fin.cons (Bool.xor (x 0) false) y)).succeeds ω) =
-      (k (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).succeeds ω
-    cases ω (e (Sum.inl y)) <;> rfl
-
-lemma followPath_charge (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
-    (k : Card d → QueryTree ι) (R : Finset ι) (ω : ι → Bool) :
-    freshCharge (followPath d e x k) R ω =
-      freshCharge (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x))
-        (R ∪ (routeDomain d x (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).image e) ω := by
-  classical
-  induction d generalizing R with
-  | zero => simp only [followPath, routeDomain, Finset.image_empty, Finset.union_empty]; rfl
-  | succ d ih =>
-    rw [followPath, ih]
-    let y := butterflyPerm d
-      (decodeButterfly d (fun i => ω (e (Sum.inr (x 0,i))))) (Fin.tail x)
-    let S := (routeDomain d (Fin.tail x) y).image (fun i => e (Sum.inr (x 0,i)))
-    change (if ω (e (Sum.inl y)) then
-      freshCharge (k (Fin.cons (Bool.xor (x 0) true) y)) (insert (e (Sum.inl y)) (R ∪ S)) ω else
-      freshCharge (k (Fin.cons (Bool.xor (x 0) false) y)) (insert (e (Sum.inl y)) (R ∪ S)) ω) =
-      freshCharge (k (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y))
-        (R ∪ (routeDomain (d+1) x
-          (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).image e) ω
-    have hs : (routeDomain (d+1) x
-        (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).image e =
-        insert (e (Sum.inl y)) S := by
-      let e' : Sum (Card d) (Bool × SwitchIndex d) → ι := e
-      change (insert (Sum.inl y)
-        ((routeDomain d (Fin.tail x) y).image
-          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) :
-          Finset (Sum (Card d) (Bool × SwitchIndex d))).image e' = _
-      rw [Finset.image_insert, Finset.image_image]
-      rfl
-    rw [hs,Finset.union_insert]
-    cases ω (e (Sum.inl y)) <;> rfl
-
-def forceList : List ι → (ι → Bool) → QueryTree ι → QueryTree ι
-  | [], _, k => k
-  | i :: L, v, k => force i (v i) (forceList L v k)
-
-omit [DecidableEq ι] in
-lemma forceList_succeeds (L : List ι) (v : ι → Bool) (k : QueryTree ι) (ω : ι → Bool) :
-    (forceList L v k).succeeds ω =
-      if ∀ i ∈ L, ω i = v i then k.succeeds ω else false := by
-  induction L with
-  | nil => simp [forceList]
-  | cons i L ih =>
-    simp only [forceList,succeeds,ih,List.mem_cons,forall_eq_or_imp]
-    by_cases h : ω i = v i <;> simp [h]
-
-lemma forceList_charge (L : List ι) (v : ι → Bool) (k : QueryTree ι)
-    (R : Finset ι) (ω : ι → Bool) (h : ∀ i ∈ L, ω i = v i) :
-    freshCharge (forceList L v k) R ω =
-      freshCharge k (R ∪ L.toFinset) ω + (L.toFinset \ R).card := by
-  induction L generalizing R with
-  | nil => simp [forceList]
-  | cons i L ih =>
-    rw [forceList,freshCharge,ite_eq_left (h i List.mem_cons_self),
-      ih (insert i R) (fun j hj => h j (List.mem_cons_of_mem _ hj))]
-    simp only [List.toFinset_cons]
-    have hs : insert i R ∪ L.toFinset = R ∪ insert i L.toFinset := by ext j; simp
-    rw [hs]
-    by_cases hi : i ∈ R
-    · rw [ite_eq_left hi]
-      have he : L.toFinset \ insert i R = insert i L.toFinset \ R := by
-        ext j
-        by_cases hji : j = i
-        · subst j; simp [hi]
-        · simp [hji]
-      rw [he,Nat.add_zero]
-    · rw [ite_eq_right hi]
-      have he : insert i L.toFinset \ R = insert i (L.toFinset \ insert i R) := by
-        ext j
-        by_cases hji : j = i
-        · subst j; simp [hi]
-        · simp [hji]
-      rw [he, Finset.card_insert_of_notMem (by simp)]
-      omega
-
-
-def staleCharge : QueryTree ι → Finset ι → (ι → Bool) → ℕ
-  | reject, _, _ => 0
-  | accept, _, _ => 0
-  | branch i zero one, R, ω => if ω i then staleCharge one (insert i R) ω
-      else staleCharge zero (insert i R) ω
-  | force i b next, R, ω => if ω i = b then
-      staleCharge next (insert i R) ω + (if i ∈ R then 1 else 0) else 0
-
-lemma fresh_add_stale (T : QueryTree ι) (R : Finset ι) (ω : ι → Bool) :
-    T.freshCharge R ω + T.staleCharge R ω = T.charge ω := by
-  induction T generalizing R with
-  | reject => rfl
-  | accept => rfl
-  | branch i zero one ih₀ ih₁ =>
-    cases hb : ω i <;> simp [freshCharge,staleCharge,charge,hb,ih₀,ih₁]
-  | force i b next ih =>
-    by_cases hb : ω i = b
-    · simp only [freshCharge,staleCharge,charge,hb,↓reduceIte]
-      have he := ih (insert i R)
-      by_cases hi : i ∈ R <;> simp only [hi,↓reduceIte] <;> omega
-    · simp [freshCharge,staleCharge,charge,hb]
-
-omit [DecidableEq ι] in
-lemma followPath_raw_charge (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
-    (k : Card d → QueryTree ι) (ω : ι → Bool) :
-    (followPath d e x k).charge ω =
-      (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).charge ω := by
-  induction d with
-  | zero => rfl
-  | succ d ih =>
-    rw [followPath, ih]
-    let y := butterflyPerm d
-      (decodeButterfly d (fun i => ω (e (Sum.inr (x 0,i))))) (Fin.tail x)
-    change (if ω (e (Sum.inl y)) then
-      (k (Fin.cons (Bool.xor (x 0) true) y)).charge ω else
-      (k (Fin.cons (Bool.xor (x 0) false) y)).charge ω) =
-      (k (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).charge ω
-    cases ω (e (Sum.inl y)) <;> rfl
-
-omit [DecidableEq ι] in
-lemma forceList_raw_charge (L : List ι) (v : ι → Bool) (k : QueryTree ι) (ω : ι → Bool)
-    (h : ∀ i ∈ L, ω i = v i) :
-    (forceList L v k).charge ω = k.charge ω + L.length := by
-  induction L with
-  | nil => simp [forceList]
-  | cons i L ih =>
-    rw [forceList,charge,ite_eq_left (h i List.mem_cons_self),
-      ih (fun j hj => h j (List.mem_cons_of_mem _ hj))]
-    simp only [List.length_cons,Nat.add_assoc]
-
-noncomputable def closePath (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d)) :
-    QueryTree (SwitchIndex d) :=
-  forceList (routeDomain d x y).toList (routeValue d x y) k
-
-lemma closePath_succeeds (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
-    (ω : SwitchIndex d → Bool) :
-    (closePath d x y k).succeeds ω =
-      if butterflyPerm d (decodeButterfly d ω) x = y then k.succeeds ω else false := by
-  unfold closePath
-  rw [forceList_succeeds]
-  simp only [Finset.mem_toList, ← butterfly_endpoint_iff]
-
-lemma closePath_raw_charge (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
-    (ω : SwitchIndex d → Bool) (h : butterflyPerm d (decodeButterfly d ω) x = y) :
-    (closePath d x y k).charge ω = k.charge ω + d := by
-  unfold closePath
-  rw [forceList_raw_charge _ _ _ _ (by simpa only [Finset.mem_toList] using
-    (butterfly_endpoint_iff d ω x y).mp h), Finset.length_toList, card_routeDomain]
-
-lemma closePath_fresh_charge (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
-    (R : Finset (SwitchIndex d)) (ω : SwitchIndex d → Bool)
-    (h : butterflyPerm d (decodeButterfly d ω) x = y) :
-    (closePath d x y k).freshCharge R ω =
-      k.freshCharge (R ∪ routeDomain d x y) ω + (routeDomain d x y \ R).card := by
-  unfold closePath
-  rw [forceList_charge _ _ _ _ _ (by simpa only [Finset.mem_toList] using
-    (butterfly_endpoint_iff d ω x y).mp h), Finset.toList_toFinset]
-
-lemma closePath_stale_charge (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
-    (R : Finset (SwitchIndex d)) (ω : SwitchIndex d → Bool)
-    (h : butterflyPerm d (decodeButterfly d ω) x = y) :
-    (closePath d x y k).staleCharge R ω =
-      k.staleCharge (R ∪ routeDomain d x y) ω + (routeDomain d x y ∩ R).card := by
-  have h₀ := fresh_add_stale (closePath d x y k) R ω
-  have h₁ := fresh_add_stale k (R ∪ routeDomain d x y) ω
-  rw [closePath_raw_charge _ _ _ _ _ h,closePath_fresh_charge _ _ _ _ _ _ h] at h₀
-  have h₂ : (routeDomain d x y \ R).card + (routeDomain d x y ∩ R).card = d := by
-    rw [Finset.card_sdiff_add_card_inter,card_routeDomain]
-  omega
-
-noncomputable def seekCycle (d : ℕ) (S : Finset (Card d)) (a : Card d) :
-    ℕ → Card d → Finset (Card d) → (Finset (Card d) → QueryTree (SwitchIndex d)) →
-      QueryTree (SwitchIndex d)
-  | 0, x, V, k => if x ∈ S ∧ x ∉ V then closePath d x a (k (insert x V)) else reject
-  | r + 1, x, V, k => if x ∈ S ∧ x ∉ V then followPath d id x
-      (fun y => seekCycle d S a r y (insert x V) k) else reject
-
-lemma seekCycle_raw_charge (d : ℕ) (S : Finset (Card d)) (a : Card d) (r : ℕ)
-    (x : Card d) (V : Finset (Card d)) (k : Finset (Card d) → QueryTree (SwitchIndex d))
-    (ω : SwitchIndex d → Bool) (K : ℕ)
-    (hK : ∀ V, (k V).succeeds ω = true → (k V).charge ω = K)
-    (hs : (seekCycle d S a r x V k).succeeds ω = true) :
-    (seekCycle d S a r x V k).charge ω = K + d := by
-  induction r generalizing x V with
-  | zero =>
-    by_cases hv : x ∈ S ∧ x ∉ V
-    · rw [seekCycle,ite_eq_left hv,closePath_succeeds] at hs
-      split_ifs at hs with he
-      · rw [seekCycle,ite_eq_left hv,closePath_raw_charge _ _ _ _ _ he,hK _ hs]
-    · simp [seekCycle,hv,succeeds] at hs
-  | succ r ih =>
-    by_cases hv : x ∈ S ∧ x ∉ V
-    · rw [seekCycle,ite_eq_left hv,followPath_succeeds] at hs
-      rw [seekCycle,ite_eq_left hv,followPath_raw_charge]
-      exact ih _ _ hs
-    · simp [seekCycle,hv,succeeds] at hs
-
-noncomputable def seekCycles (d : ℕ) (S : Finset (Card d)) :
-    List (Card d × ℕ) → Finset (Card d) → QueryTree (SwitchIndex d)
-  | [], _ => accept
-  | (a,r) :: L, V => seekCycle d S a r a V (fun V => seekCycles d S L V)
-
-lemma seekCycles_raw_charge (d : ℕ) (S : Finset (Card d)) (L : List (Card d × ℕ))
-    (V : Finset (Card d)) (ω : SwitchIndex d → Bool)
-    (hs : (seekCycles d S L V).succeeds ω = true) :
-    (seekCycles d S L V).charge ω = L.length * d := by
-  induction L generalizing V with
-  | nil => simp [seekCycles,charge]
-  | cons ar L ih =>
-    obtain ⟨a,r⟩ := ar
-    rw [seekCycles]
-    rw [seekCycle_raw_charge d S a r a V (fun V => seekCycles d S L V) ω (L.length*d)
-      (fun V hs => ih V hs) hs]
-    simp [Nat.add_mul]
-
-lemma cycles_weighted_success (d : ℕ) (S : Finset (Card d)) (L : List (Card d × ℕ)) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      if (seekCycles d S L ∅).succeeds ω then
-        ((2:ℝ)^d)^L.length / (2:ℝ)^(seekCycles d S L ∅).staleCharge ∅ ω else 0) ≤ 1 := by
-  let T := seekCycles d S L ∅
-  have hh := mean_freshCharge_le_one T
-  have he : finiteMean (fun ω : SwitchIndex d → Bool =>
-      if T.succeeds ω then (2:ℝ)^T.freshCharge ∅ ω else 0) =
-      finiteMean (fun ω : SwitchIndex d → Bool =>
-        if T.succeeds ω then ((2:ℝ)^d)^L.length / (2:ℝ)^T.staleCharge ∅ ω else 0) := by
-    apply finiteMean_congr
-    intro ω
-    cases hs : T.succeeds ω
-    · simp
-    · simp only [↓reduceIte]
-      have hc := fresh_add_stale T ∅ ω
-      rw [seekCycles_raw_charge d S L ∅ ω hs] at hc
-      have hp : ((2:ℝ)^d)^L.length = (2:ℝ)^T.freshCharge ∅ ω * (2:ℝ)^T.staleCharge ∅ ω := by
-        rw [←pow_add,hc,←pow_mul,Nat.mul_comm]
-      rw [hp,mul_div_cancel_right₀ _ (pow_ne_zero _ (by norm_num))]
-  rw [← he]
-  exact hh
-
-end QueryTree
-
-lemma finiteMean_add {Ω : Type*} [Fintype Ω] (f g : Ω → ℝ) :
-    finiteMean (fun ω => f ω + g ω) = finiteMean f + finiteMean g := by
-  simp only [finiteMean,Finset.sum_add_distrib,add_div]
-
-lemma finiteMean_mul_const {Ω : Type*} [Fintype Ω] (f : Ω → ℝ) (c : ℝ) :
-    finiteMean (fun ω => f ω * c) = finiteMean f * c := by
-  unfold finiteMean
-  rw [← Finset.sum_mul]
-  ring
-
-lemma finiteMean_sum {Ω α : Type*} [Fintype Ω] (s : Finset α) (f : α → Ω → ℝ) :
-    finiteMean (fun ω => ∑ i ∈ s, f i ω) = ∑ i ∈ s, finiteMean (f i) := by
-  simp only [finiteMean,Finset.sum_div]
-  exact Finset.sum_comm
-
-lemma finiteMean_mono {Ω : Type*} [Fintype Ω] {f g : Ω → ℝ}
-    (h : ∀ ω, f ω ≤ g ω) : finiteMean f ≤ finiteMean g := by
-  exact div_le_div_of_nonneg_right (Finset.sum_le_sum (fun ω _ => h ω)) (Nat.cast_nonneg _)
-
-lemma finiteMean_equiv {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
-    (e : Ω ≃ Ω') (f : Ω → ℝ) :
-    finiteMean f = finiteMean (fun ω => f (e.symm ω)) := by
-  unfold finiteMean
-  rw [Fintype.card_congr e]
-  congr 1
-  exact Fintype.sum_equiv e _ _ (fun ω => by simp)
-
-lemma finiteMean_prod_left {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] [Nonempty Ω']
-    (f : Ω → ℝ) : finiteMean (fun ω : Ω × Ω' => f ω.1) = finiteMean f := by
-  simp only [finiteMean,Fintype.sum_prod_type,Finset.sum_const,nsmul_eq_mul,
-    ← Finset.mul_sum,Fintype.card_prod,Nat.cast_mul,Finset.card_univ]
-  have hn : (Fintype.card Ω' : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
-  field_simp
-
-lemma finiteMean_pi_eval {C : Type*} [Fintype C] [DecidableEq C]
-    {A : C → Type*} [∀ c, Fintype (A c)] [∀ c, Nonempty (A c)]
-    (c : C) (g : A c → ℝ) :
-    finiteMean (fun z : ∀ c, A c => g (z c)) = finiteMean g := by
-  rw [finiteMean_equiv (Equiv.piSplitAt c A)]
-  have he : (fun z : A c × ((j : {j // j ≠ c}) → A j) =>
-      g ((Equiv.piSplitAt c A).symm z c)) = (fun z => g z.1) := by
-    funext z
-    simp [Equiv.piSplitAt]
-  rw [he,finiteMean_prod_left]
-
-lemma finiteMean_eq_indicator {Ω : Type*} [Fintype Ω] [DecidableEq Ω] (a : Ω) :
-    finiteMean (fun ω : Ω => if ω = a then (1:ℝ) else 0) = 1 / Fintype.card Ω := by
-  simp [finiteMean]
-
-lemma exp_finiteMean_le {Ω : Type*} [Fintype Ω] [Nonempty Ω] (f : Ω → ℝ) :
-    Real.exp (finiteMean f) ≤ finiteMean (fun ω => Real.exp (f ω)) := by
-  have hn : (Fintype.card Ω : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
-  have hw : ∑ ω : Ω, (1 / (Fintype.card Ω : ℝ)) = 1 := by simp [hn]
-  have hj := convexOn_exp.map_sum_le (t := Finset.univ)
-    (w := fun _ : Ω => 1 / (Fintype.card Ω : ℝ)) (p := f)
-    (fun _ _ => by positivity) hw (fun _ _ => Set.mem_univ _)
-  simp only [smul_eq_mul, ← Finset.mul_sum] at hj
-  simpa only [finiteMean, div_eq_mul_inv, one_div, mul_comm, mul_one, one_mul] using hj
-
-variable {d A : ℕ}
-
-def cycleFiber (o : Card d → Option (Fin A)) (i : Fin A) : Finset (Card d) := by
-  classical
-  exact Finset.univ.filter (fun x => o x = some i)
-
-@[simp] lemma mem_cycleFiber (o : Card d → Option (Fin A)) (i : Fin A) (x : Card d) :
-    x ∈ cycleFiber o i ↔ o x = some i := by
-  classical
-  simp [cycleFiber]
-
-abbrev Terminal (o : Card d → Option (Fin A)) := ∀ i : Fin A, {x : Card d // o x = some i}
-
-noncomputable def slotWeight (o : Card d → Option (Fin A)) (i : Fin A) : ℝ :=
-  1 / (cycleFiber o i).card
-
-noncomputable def pathWeight (o : Card d → Option (Fin A)) (x : Card d) : ℝ :=
-  match o x with
-  | none => 0
-  | some i => slotWeight o i
-
-noncomputable def withinWeight (o : Card d → Option (Fin A)) (u v : Card d) : ℝ :=
-  match o u, o v with
-  | some i, some j => if i = j then slotWeight o i else 0
-  | _, _ => 0
-
-end Thorp
-
-end ThorpNine.SparseSaving
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.SparseSaving.Coins
+
+namespace OAI
+
+namespace ThorpNine.SparseSaving
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+
+lemma card_routeDomain (d : ℕ) (x y : Card d) : (routeDomain d x y).card = d := by
+  classical
+  induction d with
+  | zero => rfl
+  | succ d ih =>
+    change (insert (Sum.inl (Fin.tail y))
+      ((routeDomain d (Fin.tail x) (Fin.tail y)).image
+        (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) :
+      Finset (Sum (Card d) (Bool × SwitchIndex d))).card = d + 1
+    rw [Finset.card_insert_of_notMem (by simp)]
+    rw [Finset.card_image_of_injective _ (by intro i j h; cases h; rfl), ih]
+
+lemma xor_eq_iff (a b c : Bool) : Bool.xor a b = c ↔ b = Bool.xor a c := by
+  cases a <;> cases b <;> cases c <;> decide
+
+lemma butterfly_endpoint_iff (d : ℕ) (ω : SwitchIndex d → Bool) (x y : Card d) :
+    butterflyPerm d (decodeButterfly d ω) x = y ↔
+      ∀ i ∈ routeDomain d x y, ω i = routeValue d x y i := by
+  classical
+  induction d with
+  | zero =>
+    have he : butterflyPerm 0 (decodeButterfly 0 ω) x = y := Subsingleton.elim _ _
+    simp [he, routeDomain]
+  | succ d ih =>
+    let v := butterflyPerm d (decodeButterfly d (fun i => ω (Sum.inr (x 0,i)))) (Fin.tail x)
+    change Fin.cons (Bool.xor (x 0) (ω (Sum.inl v))) v = y ↔ _
+    have hv : v = Fin.tail y ↔
+        ∀ i ∈ routeDomain d (Fin.tail x) (Fin.tail y),
+          ω (Sum.inr (x 0,i)) = routeValue d (Fin.tail x) (Fin.tail y) i :=
+      ih _ _ _
+    constructor
+    · intro h i hi
+      have ht : v = Fin.tail y := by simpa using congrArg Fin.tail h
+      have hh : ω (Sum.inl (Fin.tail y)) = Bool.xor (x 0) (y 0) := by
+        have he := congrFun h 0
+        simp only [Fin.cons_zero, ht] at he
+        exact (xor_eq_iff _ _ _).mp he
+      rcases Finset.mem_insert.mp hi with htop | hchild
+      · subst i
+        exact hh
+      · obtain ⟨j, hj, rfl⟩ := Finset.mem_image.mp hchild
+        exact hv.mp ht j hj
+    · intro h
+      have hh := h (Sum.inl (Fin.tail y)) (Finset.mem_insert_self _ _)
+      have ht : v = Fin.tail y := hv.mpr (fun i hi =>
+        h (Sum.inr (x 0,i)) (Finset.mem_insert_of_mem (Finset.mem_image_of_mem _ hi)))
+      have hh' : Bool.xor (x 0) (ω (Sum.inl (Fin.tail y))) = y 0 :=
+        (xor_eq_iff _ _ _).mpr hh
+      rw [ht, hh', Fin.cons_self_tail]
+
+lemma finiteMean_congr {Ω : Type*} [Fintype Ω] {f g : Ω → ℝ}
+    (h : ∀ ω, f ω = g ω) : finiteMean f = finiteMean g := by
+  unfold finiteMean
+  congr 1
+  exact Finset.sum_congr rfl (fun ω _ => h ω)
+
+lemma mean_coinCylinder {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (s : Finset ι) (v : ι → Bool) :
+    finiteMean (fun ω : ι → Bool => if ∀ i ∈ s, ω i = v i then (1 : ℝ) else 0) =
+      1 / (2 : ℝ) ^ s.card := by
+  induction s using Finset.induction_on with
+  | empty => simp [finiteMean]
+  | @insert i s hi ih =>
+    let f := fun ω : ι → Bool => if ∀ j ∈ s, ω j = v j then (1 : ℝ) else 0
+    have hf : ∀ ω, f (QueryTree.flipCoin i ω) = f ω := by
+      intro ω
+      have he : (∀ j ∈ s, QueryTree.flipCoin i ω j = v j) ↔ ∀ j ∈ s, ω j = v j := by
+        apply forall₂_congr
+        intro j hj
+        rw [QueryTree.flipCoin_apply_ne i j (fun h => hi (h ▸ hj))]
+      dsimp [f]
+      simp only [he]
+    have hm := QueryTree.mean_indicator_bit f i (v i) hf
+    have he : (fun ω : ι → Bool => if ∀ j ∈ insert i s, ω j = v j then (1 : ℝ) else 0) =
+        (fun ω => if ω i = v i then f ω else 0) := by
+      funext ω
+      by_cases h : ω i = v i <;> simp [f, h]
+    rw [he, hm]
+    change finiteMean (fun ω : ι → Bool => if ∀ j ∈ s, ω j = v j then (1 : ℝ) else 0) / 2 = _
+    rw [ih, Finset.card_insert_of_notMem hi, pow_succ]
+    ring
+
+def exposedCoins {ι : Type*} [DecidableEq ι] (R : Finset ι) (η ω : ι → Bool) : ι → Bool :=
+  fun i => if i ∈ R then η i else ω i
+
+lemma mean_coinCylinder_exposed {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (s R : Finset ι) (v η : ι → Bool) :
+    finiteMean (fun ω : ι → Bool =>
+      if ∀ i ∈ s, exposedCoins R η ω i = v i then (1 : ℝ) else 0) =
+    if (∀ i ∈ s ∩ R, η i = v i) then 1 / (2 : ℝ) ^ (s \ R).card else 0 := by
+  have he (ω : ι → Bool) : (∀ i ∈ s, exposedCoins R η ω i = v i) ↔
+      (∀ i ∈ s ∩ R, η i = v i) ∧ (∀ i ∈ s \ R, ω i = v i) := by
+    constructor
+    · intro h
+      constructor
+      · intro i hi
+        simpa [exposedCoins, (Finset.mem_inter.mp hi).2] using h i (Finset.mem_inter.mp hi).1
+      · intro i hi
+        simpa [exposedCoins, (Finset.mem_sdiff.mp hi).2] using h i (Finset.mem_sdiff.mp hi).1
+    · rintro ⟨ho, hf⟩ i hi
+      by_cases hR : i ∈ R
+      · simpa [exposedCoins, hR] using ho i (Finset.mem_inter.mpr ⟨hi,hR⟩)
+      · simpa [exposedCoins, hR] using hf i (Finset.mem_sdiff.mpr ⟨hi,hR⟩)
+  have hm : finiteMean (fun ω : ι → Bool =>
+      if ∀ i ∈ s, exposedCoins R η ω i = v i then (1 : ℝ) else 0) =
+      finiteMean (fun ω : ι → Bool =>
+        if (∀ i ∈ s ∩ R, η i = v i) ∧ (∀ i ∈ s \ R, ω i = v i) then (1 : ℝ) else 0) :=
+    finiteMean_congr (fun ω => by simp only [he ω])
+  rw [hm]
+  by_cases h : ∀ i ∈ s ∩ R, η i = v i
+  · have hp : (∀ i ∈ s ∩ R, η i = v i) ↔ True := iff_true_intro h
+    simp only [hp, true_and, ↓reduceIte]
+    exact mean_coinCylinder (s \ R) v
+  · have hp : (∀ i ∈ s ∩ R, η i = v i) ↔ False := iff_false_intro h
+    simp only [hp, false_and, ↓reduceIte]
+    simp [finiteMean]
+
+lemma butterfly_closure_probability (d : ℕ) (x y : Card d)
+    (R : Finset (SwitchIndex d)) (η : SwitchIndex d → Bool) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      if butterflyPerm d (decodeButterfly d (exposedCoins R η ω)) x = y then (1 : ℝ) else 0) =
+      if (∀ i ∈ routeDomain d x y ∩ R, η i = routeValue d x y i) then
+        (2 : ℝ) ^ (routeDomain d x y ∩ R).card / (2 : ℝ) ^ d else 0 := by
+  have he : finiteMean (fun ω : SwitchIndex d → Bool =>
+      if butterflyPerm d (decodeButterfly d (exposedCoins R η ω)) x = y then (1 : ℝ) else 0) =
+      finiteMean (fun ω : SwitchIndex d → Bool =>
+        if ∀ i ∈ routeDomain d x y, exposedCoins R η ω i = routeValue d x y i then (1 : ℝ) else 0) :=
+    finiteMean_congr (fun ω => by simp only [butterfly_endpoint_iff])
+  rw [he, mean_coinCylinder_exposed]
+  congr 1
+  have hc : (routeDomain d x y \ R).card + (routeDomain d x y ∩ R).card = d := by
+    rw [Finset.card_sdiff_add_card_inter, card_routeDomain]
+  have hp : (2 : ℝ) ^ d = (2 : ℝ) ^ (routeDomain d x y \ R).card *
+      (2 : ℝ) ^ (routeDomain d x y ∩ R).card := by
+    rw [← pow_add]
+    exact congrArg (fun n => (2 : ℝ) ^ n) hc.symm
+  rw [hp]
+  have hn : (2 : ℝ) ^ (routeDomain d x y ∩ R).card ≠ 0 := pow_ne_zero _ (by norm_num)
+  field_simp
+
+lemma butterfly_single_card_uniform (d : ℕ) (x y : Card d) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      if butterflyPerm d (decodeButterfly d ω) x = y then (1 : ℝ) else 0) = 1 / (2 : ℝ) ^ d := by
+  have he : finiteMean (fun ω : SwitchIndex d → Bool =>
+      if butterflyPerm d (decodeButterfly d ω) x = y then (1 : ℝ) else 0) =
+      finiteMean (fun ω : SwitchIndex d → Bool =>
+        if ∀ i ∈ routeDomain d x y, ω i = routeValue d x y i then (1 : ℝ) else 0) :=
+    finiteMean_congr (fun ω => by simp only [butterfly_endpoint_iff])
+  rw [he, mean_coinCylinder, card_routeDomain]
+
+def switchUsers : (d : ℕ) → Butterfly d → SwitchIndex d → Card d × Card d
+  | 0, _, i => nomatch i
+  | d + 1, b, Sum.inl y =>
+      (Fin.cons false ((butterflyPerm d (b.2 false)).symm y),
+       Fin.cons true ((butterflyPerm d (b.2 true)).symm y))
+  | d + 1, b, Sum.inr (ε,i) =>
+      (Fin.cons ε (switchUsers d (b.2 ε) i).1,
+       Fin.cons ε (switchUsers d (b.2 ε) i).2)
+
+lemma switchUsers_ne (d : ℕ) (b : Butterfly d) (i : SwitchIndex d) :
+    (switchUsers d b i).1 ≠ (switchUsers d b i).2 := by
+  induction d with
+  | zero => exact Empty.elim i
+  | succ d ih =>
+    cases i with
+    | inl y =>
+      intro h
+      have := congrFun h 0
+      simp only [switchUsers, Fin.cons_zero] at this
+      exact Bool.false_ne_true this
+    | inr z =>
+      intro h
+      exact ih (b.2 z.1) z.2 (by simpa only [switchUsers, Fin.tail_cons] using congrArg Fin.tail h)
+
+lemma cons_eq_iff {d : ℕ} (ε : Bool) (a : Card d) (x : Card (d + 1)) :
+    Fin.cons ε a = x ↔ ε = x 0 ∧ a = Fin.tail x := by
+  constructor
+  · intro h
+    exact ⟨by simpa only [Fin.cons_zero] using congrFun h 0,
+      by simpa only [Fin.tail_cons] using congrArg Fin.tail h⟩
+  · rintro ⟨rfl,rfl⟩
+    exact Fin.cons_self_tail x
+
+lemma mem_routeDomain_iff_user (d : ℕ) (b : Butterfly d) (x : Card d) (i : SwitchIndex d) :
+    i ∈ routeDomain d x (butterflyPerm d b x) ↔
+      (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x := by
+  classical
+  induction d with
+  | zero => exact Empty.elim i
+  | succ d ih =>
+    cases i with
+    | inl y =>
+      change Sum.inl y ∈ insert (Sum.inl (Fin.tail (butterflyPerm (d+1) b x)))
+        ((routeDomain d (Fin.tail x) (Fin.tail (butterflyPerm (d+1) b x))).image
+          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) ↔ _
+      simp only [Finset.mem_insert, Sum.inl.injEq, Finset.mem_image,
+        Sum.inr_ne_inl, and_false, exists_false, or_false]
+      change y = butterflyPerm d (b.2 (x 0)) (Fin.tail x) ↔ _
+      simp only [switchUsers, cons_eq_iff]
+      cases x 0 <;> simp [Equiv.symm_apply_eq]
+    | inr z =>
+      obtain ⟨ε,i⟩ := z
+      change Sum.inr (ε,i) ∈ insert (Sum.inl (Fin.tail (butterflyPerm (d+1) b x)))
+        ((routeDomain d (Fin.tail x) (Fin.tail (butterflyPerm (d+1) b x))).image
+          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) ↔ _
+      simp only [Finset.mem_insert, Sum.inr_ne_inl, false_or, Finset.mem_image,
+        Sum.inr.injEq, Prod.mk.injEq]
+      change (∃ j, j ∈ routeDomain d (Fin.tail x) (butterflyPerm d (b.2 (x 0)) (Fin.tail x)) ∧
+        x 0 = ε ∧ j = i) ↔ _
+      simp only [switchUsers, cons_eq_iff]
+      constructor
+      · rintro ⟨j,hj,hε,rfl⟩
+        subst ε
+        rcases (ih (b.2 (x 0)) (Fin.tail x) j).mp hj with h | h
+        · exact Or.inl ⟨rfl,h⟩
+        · exact Or.inr ⟨rfl,h⟩
+      · rintro (⟨hε,h⟩ | ⟨hε,h⟩)
+        · subst ε
+          exact ⟨i,(ih (b.2 (x 0)) (Fin.tail x) i).mpr (Or.inl h),rfl,rfl⟩
+        · subst ε
+          exact ⟨i,(ih (b.2 (x 0)) (Fin.tail x) i).mpr (Or.inr h),rfl,rfl⟩
+
+lemma card_switches_using (d : ℕ) (b : Butterfly d) (x : Card d) :
+    (Finset.univ.filter (fun i : SwitchIndex d =>
+      (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x)).card = d := by
+  classical
+  have he : Finset.univ.filter (fun i : SwitchIndex d =>
+      (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x) =
+      routeDomain d x (butterflyPerm d b x) := by
+    ext i
+    simp only [Finset.mem_filter, Finset.mem_univ, true_and, mem_routeDomain_iff_user]
+  rw [he,card_routeDomain]
+
+lemma switch_incidence_sum (d : ℕ) (b : Butterfly d) (w : Card d → ℝ) :
+    ∑ i : SwitchIndex d, (w (switchUsers d b i).1 + w (switchUsers d b i).2) =
+      (d : ℝ) * ∑ x : Card d, w x := by
+  classical
+  calc
+    _ = ∑ i : SwitchIndex d, ∑ x : Card d,
+        if (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x then w x else 0 := by
+      apply Finset.sum_congr rfl
+      intro i _
+      rw [Finset.sum_ite]
+      simp only [Finset.sum_const_zero, add_zero]
+      have he : Finset.univ.filter (fun x : Card d =>
+          (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x) =
+          {(switchUsers d b i).1, (switchUsers d b i).2} := by
+        ext x; simp [eq_comm]
+      rw [he, Finset.sum_pair (switchUsers_ne d b i)]
+    _ = ∑ x : Card d, ∑ i : SwitchIndex d,
+        if (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x then w x else 0 :=
+      Finset.sum_comm
+    _ = ∑ x : Card d, (d : ℝ) * w x := by
+      apply Finset.sum_congr rfl
+      intro x _
+      rw [Finset.sum_ite]
+      simp only [Finset.sum_const_zero, add_zero, Finset.sum_const, nsmul_eq_mul,
+        card_switches_using]
+    _ = _ := by rw [Finset.mul_sum]
+
+lemma weighted_switch_min_bound (d : ℕ) (b : Butterfly d) (w : Card d → ℝ) :
+    ∑ i : SwitchIndex d, min (w (switchUsers d b i).1) (w (switchUsers d b i).2) ≤
+      (d : ℝ) / 2 * ∑ x : Card d, w x := by
+  have h : 2 * (∑ i : SwitchIndex d,
+      min (w (switchUsers d b i).1) (w (switchUsers d b i).2)) ≤
+      ∑ i : SwitchIndex d, (w (switchUsers d b i).1 + w (switchUsers d b i).2) := by
+    rw [Finset.mul_sum]
+    apply Finset.sum_le_sum
+    intro i _
+    linarith [min_le_left (w (switchUsers d b i).1) (w (switchUsers d b i).2),
+      min_le_right (w (switchUsers d b i).1) (w (switchUsers d b i).2)]
+  rw [switch_incidence_sum] at h
+  linarith
+
+lemma mem_childMarks {d : ℕ} (s : Finset (Card (d + 1))) (ε : Bool) (x : Card d) :
+    x ∈ childMarks s ε ↔ Fin.cons ε x ∈ s := by
+  classical
+  constructor
+  · intro h
+    obtain ⟨y,hy,hxy⟩ := Finset.mem_image.mp h
+    obtain ⟨hys,hyε⟩ := Finset.mem_filter.mp hy
+    rw [← hxy, ← hyε, Fin.cons_self_tail]
+    exact hys
+  · intro h
+    exact Finset.mem_image.mpr ⟨Fin.cons ε x,
+      Finset.mem_filter.mpr ⟨h,by simp⟩,by simp⟩
+
+lemma mem_topMeetings {d : ℕ} (b : Butterfly (d + 1))
+    (s : Finset (Card (d + 1))) (y : Card d) :
+    y ∈ topMeetings b s ↔
+      (switchUsers (d+1) b (Sum.inl y)).1 ∈ s ∧
+      (switchUsers (d+1) b (Sum.inl y)).2 ∈ s := by
+  classical
+  have he (ε : Bool) : y ∈ (childMarks s ε).image (butterflyPerm d (b.2 ε)) ↔
+      (butterflyPerm d (b.2 ε)).symm y ∈ childMarks s ε := by
+    constructor
+    · intro h
+      obtain ⟨x,hx,rfl⟩ := Finset.mem_image.mp h
+      simpa only [Equiv.symm_apply_apply] using hx
+    · intro h
+      exact Finset.mem_image.mpr ⟨_,h,Equiv.apply_symm_apply _ _⟩
+  simp only [topMeetings, Finset.mem_inter, he, mem_childMarks, switchUsers]
+
+lemma encounters_eq_switch_sum (d : ℕ) (b : Butterfly d) (s : Finset (Card d)) :
+    encounters d b s = ∑ i : SwitchIndex d,
+      if (switchUsers d b i).1 ∈ s ∧ (switchUsers d b i).2 ∈ s then 1 else 0 := by
+  classical
+  induction d with
+  | zero =>
+    simp only [encounters]
+    symm
+    apply Finset.sum_eq_zero
+    intro i _
+    exact Empty.elim i
+  | succ d ih =>
+    change encounters d (b.2 false) (childMarks s false) +
+      encounters d (b.2 true) (childMarks s true) + (topMeetings b s).card =
+        ∑ i : Sum (Card d) (Bool × SwitchIndex d), _
+    rw [Fintype.sum_sum_type, Fintype.sum_prod_type]
+    simp only [Fintype.sum_bool]
+    rw [ih,ih]
+    have ht : (∑ y : Card d,
+        if (switchUsers (d+1) b (Sum.inl y)).1 ∈ s ∧
+          (switchUsers (d+1) b (Sum.inl y)).2 ∈ s then 1 else 0) =
+        (topMeetings b s).card := by
+      simp only [← mem_topMeetings]
+      simp
+    rw [ht]
+    simp only [mem_childMarks, switchUsers]
+    ac_rfl
+
+lemma marked_switch_entropy_bound (d : ℕ) (b : Butterfly d) (s : Finset (Card d)) :
+    (∑ i : SwitchIndex d,
+      if (switchUsers d b i).1 ∈ s ∧ (switchUsers d b i).2 ∈ s then (1:ℝ) else 0) ≤
+      (s.card : ℝ) * Real.log s.card / (2 * Real.log 2) := by
+  classical
+  have he : (encounters d b s : ℝ) =
+      ∑ i : SwitchIndex d,
+        if (switchUsers d b i).1 ∈ s ∧ (switchUsers d b i).2 ∈ s then (1:ℝ) else 0 := by
+    rw [encounters_eq_switch_sum, Nat.cast_sum]
+    apply Finset.sum_congr rfl
+    intro i _
+    split_ifs <;> simp
+  rw [← he]
+  exact butterfly_encounters d b s
+
+namespace QueryTree
+variable {ι : Type*} [DecidableEq ι]
+
+def memo : QueryTree ι → Finset ι → (ι → Bool) → QueryTree ι
+  | reject, _, _ => reject
+  | accept, _, _ => accept
+  | branch i zero one, R, η =>
+      if i ∈ R then (if η i then memo one R η else memo zero R η) else
+      branch i (memo zero (insert i R) (Function.update η i false))
+        (memo one (insert i R) (Function.update η i true))
+  | force i b next, R, η =>
+      if i ∈ R then (if η i = b then memo next R η else reject) else
+      force i b (memo next (insert i R) (Function.update η i b))
+
+lemma memo_queries_disjoint (T : QueryTree ι) (R : Finset ι) (η : ι → Bool) :
+    Disjoint (memo T R η).queries R := by
+  induction T generalizing R η with
+  | reject => simp [memo,queries]
+  | accept => simp [memo,queries]
+  | branch i zero one ih₀ ih₁ =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte]
+      cases η i <;> simp only [Bool.false_eq_true,↓reduceIte]
+      · exact ih₀ R η
+      · exact ih₁ R η
+    · simp only [memo,hi,↓reduceIte,queries,Finset.disjoint_insert_left,Finset.disjoint_union_left]
+      exact ⟨by simp, (ih₀ (insert i R) _).mono_right (Finset.subset_insert _ _),
+        (ih₁ (insert i R) _).mono_right (Finset.subset_insert _ _)⟩
+  | force i b next ih =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte]
+      by_cases hb : η i = b
+      · simpa only [hb,↓reduceIte] using ih R η
+      · simp [hb,queries]
+    · simp only [memo,hi,↓reduceIte,queries,Finset.disjoint_insert_left]
+      exact ⟨by simp,(ih (insert i R) _).mono_right (Finset.subset_insert _ _)⟩
+
+lemma memo_fresh (T : QueryTree ι) (R : Finset ι) (η : ι → Bool) :
+    (memo T R η).Fresh := by
+  induction T generalizing R η with
+  | reject => trivial
+  | accept => trivial
+  | branch i zero one ih₀ ih₁ =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte]
+      cases η i <;> simp only [Bool.false_eq_true,↓reduceIte]
+      · exact ih₀ R η
+      · exact ih₁ R η
+    · simp only [memo,hi,↓reduceIte,Fresh]
+      exact ⟨fun hm => Finset.disjoint_left.mp (memo_queries_disjoint zero (insert i R) _) hm
+          (Finset.mem_insert_self _ _),
+        fun hm => Finset.disjoint_left.mp (memo_queries_disjoint one (insert i R) _) hm
+          (Finset.mem_insert_self _ _), ih₀ _ _,ih₁ _ _⟩
+  | force i b next ih =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte]
+      by_cases hb : η i = b
+      · simpa only [hb,↓reduceIte] using ih R η
+      · simp [hb,Fresh]
+    · simp only [memo,hi,↓reduceIte,Fresh]
+      exact ⟨fun hm => Finset.disjoint_left.mp (memo_queries_disjoint next (insert i R) _) hm
+        (Finset.mem_insert_self _ _),ih _ _⟩
+
+lemma update_agrees {R : Finset ι} {η ω : ι → Bool}
+    (h : ∀ i ∈ R, η i = ω i) (i : ι) :
+    ∀ j ∈ insert i R, Function.update η i (ω i) j = ω j := by
+  intro j hj
+  rcases Finset.mem_insert.mp hj with rfl | hj
+  · simp
+  · by_cases hji : j = i
+    · subst j; simp
+    · rw [Function.update_of_ne hji]
+      exact h j hj
+
+lemma memo_succeeds (T : QueryTree ι) (R : Finset ι) (η ω : ι → Bool)
+    (h : ∀ i ∈ R, η i = ω i) : (memo T R η).succeeds ω = T.succeeds ω := by
+  induction T generalizing R η with
+  | reject => rfl
+  | accept => rfl
+  | branch i zero one ih₀ ih₁ =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte,succeeds,h i hi]
+      cases ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
+      · exact ih₀ R η h
+      · exact ih₁ R η h
+    · simp only [memo,hi,↓reduceIte,succeeds]
+      cases hb : ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
+      · exact ih₀ _ _ (by simpa only [hb] using update_agrees h i)
+      · exact ih₁ _ _ (by simpa only [hb] using update_agrees h i)
+  | force i b next ih =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte,h i hi]
+      by_cases hb : ω i = b
+      · simp only [hb,↓reduceIte,succeeds]
+        exact ih R η h
+      · simp [hb,succeeds]
+    · simp only [memo,hi,↓reduceIte,succeeds]
+      by_cases hb : ω i = b
+      · simp only [hb,↓reduceIte]
+        exact ih _ _ (by simpa only [hb] using update_agrees h i)
+      · simp [hb]
+
+def freshCharge : QueryTree ι → Finset ι → (ι → Bool) → ℕ
+  | reject, _, _ => 0
+  | accept, _, _ => 0
+  | branch i zero one, R, ω => if ω i then freshCharge one (insert i R) ω else freshCharge zero (insert i R) ω
+  | force i b next, R, ω => if ω i = b then
+      freshCharge next (insert i R) ω + (if i ∈ R then 0 else 1) else 0
+
+lemma memo_charge (T : QueryTree ι) (R : Finset ι) (η ω : ι → Bool)
+    (h : ∀ i ∈ R, η i = ω i) : (memo T R η).charge ω = freshCharge T R ω := by
+  induction T generalizing R η with
+  | reject => rfl
+  | accept => rfl
+  | branch i zero one ih₀ ih₁ =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte,freshCharge,h i hi,Finset.insert_eq_of_mem hi]
+      cases ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
+      · exact ih₀ R η h
+      · exact ih₁ R η h
+    · simp only [memo,hi,↓reduceIte,charge,freshCharge]
+      cases hb : ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
+      · exact ih₀ _ _ (by simpa only [hb] using update_agrees h i)
+      · exact ih₁ _ _ (by simpa only [hb] using update_agrees h i)
+  | force i b next ih =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte,h i hi,freshCharge,Finset.insert_eq_of_mem hi,Nat.add_zero]
+      by_cases hb : ω i = b
+      · simp only [hb,↓reduceIte]
+        exact ih R η h
+      · simp [hb,charge]
+    · simp only [memo,hi,↓reduceIte,charge,freshCharge]
+      by_cases hb : ω i = b
+      · simp only [hb,↓reduceIte]
+        rw [ih _ _ (by simpa only [hb] using update_agrees h i)]
+      · simp [hb]
+
+lemma mean_freshCharge_le_one [Fintype ι] (T : QueryTree ι) :
+    finiteMean (fun ω : ι → Bool =>
+      if T.succeeds ω then (2:ℝ) ^ T.freshCharge ∅ ω else 0) ≤ 1 := by
+  let η : ι → Bool := fun _ => false
+  have hm := mean_weighted_le_one (memo T ∅ η) (memo_fresh T ∅ η)
+  have he : finiteMean (memo T ∅ η).weighted =
+      finiteMean (fun ω : ι → Bool =>
+        if T.succeeds ω then (2:ℝ) ^ T.freshCharge ∅ ω else 0) := by
+    apply finiteMean_congr
+    intro ω
+    rw [weighted_eq, memo_succeeds _ _ _ _ (by simp), memo_charge _ _ _ _ (by simp)]
+  rw [he] at hm
+  exact hm
+
+
+def followPath : (d : ℕ) → (SwitchIndex d → ι) → Card d →
+    (Card d → QueryTree ι) → QueryTree ι
+  | 0, _, x, k => k x
+  | d + 1, e, x, k => followPath d (fun i => e (Sum.inr (x 0,i))) (Fin.tail x)
+      (fun y => branch (e (Sum.inl y))
+        (k (Fin.cons (Bool.xor (x 0) false) y))
+        (k (Fin.cons (Bool.xor (x 0) true) y)))
+
+omit [DecidableEq ι] in
+lemma followPath_succeeds (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
+    (k : Card d → QueryTree ι) (ω : ι → Bool) :
+    (followPath d e x k).succeeds ω =
+      (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).succeeds ω := by
+  induction d with
+  | zero => rfl
+  | succ d ih =>
+    rw [followPath, ih]
+    let y := butterflyPerm d
+      (decodeButterfly d (fun i => ω (e (Sum.inr (x 0,i))))) (Fin.tail x)
+    change (if ω (e (Sum.inl y)) then
+      (k (Fin.cons (Bool.xor (x 0) true) y)).succeeds ω else
+      (k (Fin.cons (Bool.xor (x 0) false) y)).succeeds ω) =
+      (k (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).succeeds ω
+    cases ω (e (Sum.inl y)) <;> rfl
+
+lemma followPath_charge (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
+    (k : Card d → QueryTree ι) (R : Finset ι) (ω : ι → Bool) :
+    freshCharge (followPath d e x k) R ω =
+      freshCharge (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x))
+        (R ∪ (routeDomain d x (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).image e) ω := by
+  classical
+  induction d generalizing R with
+  | zero => simp only [followPath, routeDomain, Finset.image_empty, Finset.union_empty]; rfl
+  | succ d ih =>
+    rw [followPath, ih]
+    let y := butterflyPerm d
+      (decodeButterfly d (fun i => ω (e (Sum.inr (x 0,i))))) (Fin.tail x)
+    let S := (routeDomain d (Fin.tail x) y).image (fun i => e (Sum.inr (x 0,i)))
+    change (if ω (e (Sum.inl y)) then
+      freshCharge (k (Fin.cons (Bool.xor (x 0) true) y)) (insert (e (Sum.inl y)) (R ∪ S)) ω else
+      freshCharge (k (Fin.cons (Bool.xor (x 0) false) y)) (insert (e (Sum.inl y)) (R ∪ S)) ω) =
+      freshCharge (k (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y))
+        (R ∪ (routeDomain (d+1) x
+          (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).image e) ω
+    have hs : (routeDomain (d+1) x
+        (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).image e =
+        insert (e (Sum.inl y)) S := by
+      let e' : Sum (Card d) (Bool × SwitchIndex d) → ι := e
+      change (insert (Sum.inl y)
+        ((routeDomain d (Fin.tail x) y).image
+          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) :
+          Finset (Sum (Card d) (Bool × SwitchIndex d))).image e' = _
+      rw [Finset.image_insert, Finset.image_image]
+      rfl
+    rw [hs,Finset.union_insert]
+    cases ω (e (Sum.inl y)) <;> rfl
+
+def forceList : List ι → (ι → Bool) → QueryTree ι → QueryTree ι
+  | [], _, k => k
+  | i :: L, v, k => force i (v i) (forceList L v k)
+
+omit [DecidableEq ι] in
+lemma forceList_succeeds (L : List ι) (v : ι → Bool) (k : QueryTree ι) (ω : ι → Bool) :
+    (forceList L v k).succeeds ω =
+      if ∀ i ∈ L, ω i = v i then k.succeeds ω else false := by
+  induction L with
+  | nil => simp [forceList]
+  | cons i L ih =>
+    simp only [forceList,succeeds,ih,List.mem_cons,forall_eq_or_imp]
+    by_cases h : ω i = v i <;> simp [h]
+
+lemma forceList_charge (L : List ι) (v : ι → Bool) (k : QueryTree ι)
+    (R : Finset ι) (ω : ι → Bool) (h : ∀ i ∈ L, ω i = v i) :
+    freshCharge (forceList L v k) R ω =
+      freshCharge k (R ∪ L.toFinset) ω + (L.toFinset \ R).card := by
+  induction L generalizing R with
+  | nil => simp [forceList]
+  | cons i L ih =>
+    rw [forceList,freshCharge,ite_eq_left (h i List.mem_cons_self),
+      ih (insert i R) (fun j hj => h j (List.mem_cons_of_mem _ hj))]
+    simp only [List.toFinset_cons]
+    have hs : insert i R ∪ L.toFinset = R ∪ insert i L.toFinset := by ext j; simp
+    rw [hs]
+    by_cases hi : i ∈ R
+    · rw [ite_eq_left hi]
+      have he : L.toFinset \ insert i R = insert i L.toFinset \ R := by
+        ext j
+        by_cases hji : j = i
+        · subst j; simp [hi]
+        · simp [hji]
+      rw [he,Nat.add_zero]
+    · rw [ite_eq_right hi]
+      have he : insert i L.toFinset \ R = insert i (L.toFinset \ insert i R) := by
+        ext j
+        by_cases hji : j = i
+        · subst j; simp [hi]
+        · simp [hji]
+      rw [he, Finset.card_insert_of_notMem (by simp)]
+      omega
+
+
+def staleCharge : QueryTree ι → Finset ι → (ι → Bool) → ℕ
+  | reject, _, _ => 0
+  | accept, _, _ => 0
+  | branch i zero one, R, ω => if ω i then staleCharge one (insert i R) ω
+      else staleCharge zero (insert i R) ω
+  | force i b next, R, ω => if ω i = b then
+      staleCharge next (insert i R) ω + (if i ∈ R then 1 else 0) else 0
+
+lemma fresh_add_stale (T : QueryTree ι) (R : Finset ι) (ω : ι → Bool) :
+    T.freshCharge R ω + T.staleCharge R ω = T.charge ω := by
+  induction T generalizing R with
+  | reject => rfl
+  | accept => rfl
+  | branch i zero one ih₀ ih₁ =>
+    cases hb : ω i <;> simp [freshCharge,staleCharge,charge,hb,ih₀,ih₁]
+  | force i b next ih =>
+    by_cases hb : ω i = b
+    · simp only [freshCharge,staleCharge,charge,hb,↓reduceIte]
+      have he := ih (insert i R)
+      by_cases hi : i ∈ R <;> simp only [hi,↓reduceIte] <;> omega
+    · simp [freshCharge,staleCharge,charge,hb]
+
+omit [DecidableEq ι] in
+lemma followPath_raw_charge (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
+    (k : Card d → QueryTree ι) (ω : ι → Bool) :
+    (followPath d e x k).charge ω =
+      (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).charge ω := by
+  induction d with
+  | zero => rfl
+  | succ d ih =>
+    rw [followPath, ih]
+    let y := butterflyPerm d
+      (decodeButterfly d (fun i => ω (e (Sum.inr (x 0,i))))) (Fin.tail x)
+    change (if ω (e (Sum.inl y)) then
+      (k (Fin.cons (Bool.xor (x 0) true) y)).charge ω else
+      (k (Fin.cons (Bool.xor (x 0) false) y)).charge ω) =
+      (k (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).charge ω
+    cases ω (e (Sum.inl y)) <;> rfl
+
+omit [DecidableEq ι] in
+lemma forceList_raw_charge (L : List ι) (v : ι → Bool) (k : QueryTree ι) (ω : ι → Bool)
+    (h : ∀ i ∈ L, ω i = v i) :
+    (forceList L v k).charge ω = k.charge ω + L.length := by
+  induction L with
+  | nil => simp [forceList]
+  | cons i L ih =>
+    rw [forceList,charge,ite_eq_left (h i List.mem_cons_self),
+      ih (fun j hj => h j (List.mem_cons_of_mem _ hj))]
+    simp only [List.length_cons,Nat.add_assoc]
+
+noncomputable def closePath (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d)) :
+    QueryTree (SwitchIndex d) :=
+  forceList (routeDomain d x y).toList (routeValue d x y) k
+
+lemma closePath_succeeds (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
+    (ω : SwitchIndex d → Bool) :
+    (closePath d x y k).succeeds ω =
+      if butterflyPerm d (decodeButterfly d ω) x = y then k.succeeds ω else false := by
+  unfold closePath
+  rw [forceList_succeeds]
+  simp only [Finset.mem_toList, ← butterfly_endpoint_iff]
+
+lemma closePath_raw_charge (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
+    (ω : SwitchIndex d → Bool) (h : butterflyPerm d (decodeButterfly d ω) x = y) :
+    (closePath d x y k).charge ω = k.charge ω + d := by
+  unfold closePath
+  rw [forceList_raw_charge _ _ _ _ (by simpa only [Finset.mem_toList] using
+    (butterfly_endpoint_iff d ω x y).mp h), Finset.length_toList, card_routeDomain]
+
+lemma closePath_fresh_charge (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
+    (R : Finset (SwitchIndex d)) (ω : SwitchIndex d → Bool)
+    (h : butterflyPerm d (decodeButterfly d ω) x = y) :
+    (closePath d x y k).freshCharge R ω =
+      k.freshCharge (R ∪ routeDomain d x y) ω + (routeDomain d x y \ R).card := by
+  unfold closePath
+  rw [forceList_charge _ _ _ _ _ (by simpa only [Finset.mem_toList] using
+    (butterfly_endpoint_iff d ω x y).mp h), Finset.toList_toFinset]
+
+lemma closePath_stale_charge (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
+    (R : Finset (SwitchIndex d)) (ω : SwitchIndex d → Bool)
+    (h : butterflyPerm d (decodeButterfly d ω) x = y) :
+    (closePath d x y k).staleCharge R ω =
+      k.staleCharge (R ∪ routeDomain d x y) ω + (routeDomain d x y ∩ R).card := by
+  have h₀ := fresh_add_stale (closePath d x y k) R ω
+  have h₁ := fresh_add_stale k (R ∪ routeDomain d x y) ω
+  rw [closePath_raw_charge _ _ _ _ _ h,closePath_fresh_charge _ _ _ _ _ _ h] at h₀
+  have h₂ : (routeDomain d x y \ R).card + (routeDomain d x y ∩ R).card = d := by
+    rw [Finset.card_sdiff_add_card_inter,card_routeDomain]
+  omega
+
+noncomputable def seekCycle (d : ℕ) (S : Finset (Card d)) (a : Card d) :
+    ℕ → Card d → Finset (Card d) → (Finset (Card d) → QueryTree (SwitchIndex d)) →
+      QueryTree (SwitchIndex d)
+  | 0, x, V, k => if x ∈ S ∧ x ∉ V then closePath d x a (k (insert x V)) else reject
+  | r + 1, x, V, k => if x ∈ S ∧ x ∉ V then followPath d id x
+      (fun y => seekCycle d S a r y (insert x V) k) else reject
+
+lemma seekCycle_raw_charge (d : ℕ) (S : Finset (Card d)) (a : Card d) (r : ℕ)
+    (x : Card d) (V : Finset (Card d)) (k : Finset (Card d) → QueryTree (SwitchIndex d))
+    (ω : SwitchIndex d → Bool) (K : ℕ)
+    (hK : ∀ V, (k V).succeeds ω = true → (k V).charge ω = K)
+    (hs : (seekCycle d S a r x V k).succeeds ω = true) :
+    (seekCycle d S a r x V k).charge ω = K + d := by
+  induction r generalizing x V with
+  | zero =>
+    by_cases hv : x ∈ S ∧ x ∉ V
+    · rw [seekCycle,ite_eq_left hv,closePath_succeeds] at hs
+      split_ifs at hs with he
+      · rw [seekCycle,ite_eq_left hv,closePath_raw_charge _ _ _ _ _ he,hK _ hs]
+    · simp [seekCycle,hv,succeeds] at hs
+  | succ r ih =>
+    by_cases hv : x ∈ S ∧ x ∉ V
+    · rw [seekCycle,ite_eq_left hv,followPath_succeeds] at hs
+      rw [seekCycle,ite_eq_left hv,followPath_raw_charge]
+      exact ih _ _ hs
+    · simp [seekCycle,hv,succeeds] at hs
+
+noncomputable def seekCycles (d : ℕ) (S : Finset (Card d)) :
+    List (Card d × ℕ) → Finset (Card d) → QueryTree (SwitchIndex d)
+  | [], _ => accept
+  | (a,r) :: L, V => seekCycle d S a r a V (fun V => seekCycles d S L V)
+
+lemma seekCycles_raw_charge (d : ℕ) (S : Finset (Card d)) (L : List (Card d × ℕ))
+    (V : Finset (Card d)) (ω : SwitchIndex d → Bool)
+    (hs : (seekCycles d S L V).succeeds ω = true) :
+    (seekCycles d S L V).charge ω = L.length * d := by
+  induction L generalizing V with
+  | nil => simp [seekCycles,charge]
+  | cons ar L ih =>
+    obtain ⟨a,r⟩ := ar
+    rw [seekCycles]
+    rw [seekCycle_raw_charge d S a r a V (fun V => seekCycles d S L V) ω (L.length*d)
+      (fun V hs => ih V hs) hs]
+    simp [Nat.add_mul]
+
+lemma cycles_weighted_success (d : ℕ) (S : Finset (Card d)) (L : List (Card d × ℕ)) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      if (seekCycles d S L ∅).succeeds ω then
+        ((2:ℝ)^d)^L.length / (2:ℝ)^(seekCycles d S L ∅).staleCharge ∅ ω else 0) ≤ 1 := by
+  let T := seekCycles d S L ∅
+  have hh := mean_freshCharge_le_one T
+  have he : finiteMean (fun ω : SwitchIndex d → Bool =>
+      if T.succeeds ω then (2:ℝ)^T.freshCharge ∅ ω else 0) =
+      finiteMean (fun ω : SwitchIndex d → Bool =>
+        if T.succeeds ω then ((2:ℝ)^d)^L.length / (2:ℝ)^T.staleCharge ∅ ω else 0) := by
+    apply finiteMean_congr
+    intro ω
+    cases hs : T.succeeds ω
+    · simp
+    · simp only [↓reduceIte]
+      have hc := fresh_add_stale T ∅ ω
+      rw [seekCycles_raw_charge d S L ∅ ω hs] at hc
+      have hp : ((2:ℝ)^d)^L.length = (2:ℝ)^T.freshCharge ∅ ω * (2:ℝ)^T.staleCharge ∅ ω := by
+        rw [←pow_add,hc,←pow_mul,Nat.mul_comm]
+      rw [hp,mul_div_cancel_right₀ _ (pow_ne_zero _ (by norm_num))]
+  rw [← he]
+  exact hh
+
+end QueryTree
+
+lemma finiteMean_add {Ω : Type*} [Fintype Ω] (f g : Ω → ℝ) :
+    finiteMean (fun ω => f ω + g ω) = finiteMean f + finiteMean g := by
+  simp only [finiteMean,Finset.sum_add_distrib,add_div]
+
+lemma finiteMean_mul_const {Ω : Type*} [Fintype Ω] (f : Ω → ℝ) (c : ℝ) :
+    finiteMean (fun ω => f ω * c) = finiteMean f * c := by
+  unfold finiteMean
+  rw [← Finset.sum_mul]
+  ring
+
+lemma finiteMean_sum {Ω α : Type*} [Fintype Ω] (s : Finset α) (f : α → Ω → ℝ) :
+    finiteMean (fun ω => ∑ i ∈ s, f i ω) = ∑ i ∈ s, finiteMean (f i) := by
+  simp only [finiteMean,Finset.sum_div]
+  exact Finset.sum_comm
+
+lemma finiteMean_mono {Ω : Type*} [Fintype Ω] {f g : Ω → ℝ}
+    (h : ∀ ω, f ω ≤ g ω) : finiteMean f ≤ finiteMean g := by
+  exact div_le_div_of_nonneg_right (Finset.sum_le_sum (fun ω _ => h ω)) (Nat.cast_nonneg _)
+
+lemma finiteMean_equiv {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
+    (e : Ω ≃ Ω') (f : Ω → ℝ) :
+    finiteMean f = finiteMean (fun ω => f (e.symm ω)) := by
+  unfold finiteMean
+  rw [Fintype.card_congr e]
+  congr 1
+  exact Fintype.sum_equiv e _ _ (fun ω => by simp)
+
+lemma finiteMean_prod_left {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] [Nonempty Ω']
+    (f : Ω → ℝ) : finiteMean (fun ω : Ω × Ω' => f ω.1) = finiteMean f := by
+  simp only [finiteMean,Fintype.sum_prod_type,Finset.sum_const,nsmul_eq_mul,
+    ← Finset.mul_sum,Fintype.card_prod,Nat.cast_mul,Finset.card_univ]
+  have hn : (Fintype.card Ω' : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
+  field_simp
+
+lemma finiteMean_pi_eval {C : Type*} [Fintype C] [DecidableEq C]
+    {A : C → Type*} [∀ c, Fintype (A c)] [∀ c, Nonempty (A c)]
+    (c : C) (g : A c → ℝ) :
+    finiteMean (fun z : ∀ c, A c => g (z c)) = finiteMean g := by
+  rw [finiteMean_equiv (Equiv.piSplitAt c A)]
+  have he : (fun z : A c × ((j : {j // j ≠ c}) → A j) =>
+      g ((Equiv.piSplitAt c A).symm z c)) = (fun z => g z.1) := by
+    funext z
+    simp [Equiv.piSplitAt]
+  rw [he,finiteMean_prod_left]
+
+lemma finiteMean_eq_indicator {Ω : Type*} [Fintype Ω] [DecidableEq Ω] (a : Ω) :
+    finiteMean (fun ω : Ω => if ω = a then (1:ℝ) else 0) = 1 / Fintype.card Ω := by
+  simp [finiteMean]
+
+lemma exp_finiteMean_le {Ω : Type*} [Fintype Ω] [Nonempty Ω] (f : Ω → ℝ) :
+    Real.exp (finiteMean f) ≤ finiteMean (fun ω => Real.exp (f ω)) := by
+  have hn : (Fintype.card Ω : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
+  have hw : ∑ ω : Ω, (1 / (Fintype.card Ω : ℝ)) = 1 := by simp [hn]
+  have hj := convexOn_exp.map_sum_le (t := Finset.univ)
+    (w := fun _ : Ω => 1 / (Fintype.card Ω : ℝ)) (p := f)
+    (fun _ _ => by positivity) hw (fun _ _ => Set.mem_univ _)
+  simp only [smul_eq_mul, ← Finset.mul_sum] at hj
+  simpa only [finiteMean, div_eq_mul_inv, one_div, mul_comm, mul_one, one_mul] using hj
+
+variable {d A : ℕ}
+
+def cycleFiber (o : Card d → Option (Fin A)) (i : Fin A) : Finset (Card d) := by
+  classical
+  exact Finset.univ.filter (fun x => o x = some i)
+
+@[simp] lemma mem_cycleFiber (o : Card d → Option (Fin A)) (i : Fin A) (x : Card d) :
+    x ∈ cycleFiber o i ↔ o x = some i := by
+  classical
+  simp [cycleFiber]
+
+abbrev Terminal (o : Card d → Option (Fin A)) := ∀ i : Fin A, {x : Card d // o x = some i}
+
+noncomputable def slotWeight (o : Card d → Option (Fin A)) (i : Fin A) : ℝ :=
+  1 / (cycleFiber o i).card
+
+noncomputable def pathWeight (o : Card d → Option (Fin A)) (x : Card d) : ℝ :=
+  match o x with
+  | none => 0
+  | some i => slotWeight o i
+
+noncomputable def withinWeight (o : Card d → Option (Fin A)) (u v : Card d) : ℝ :=
+  match o u, o v with
+  | some i, some j => if i = j then slotWeight o i else 0
+  | _, _ => 0
+
+end Thorp
+
+end ThorpNine.SparseSaving
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/SparseSaving/HeightDecay.lean b/lean/OAI/Probability/ThorpRouting/SparseSaving/HeightDecay.lean
index 3d9b5207f0f5b3e0face7aa37dce5d354016901d..bc9b1acc4a3eb464b61a34e69c85214b83a57b80 100644
--- a/lean/OAI/Probability/ThorpRouting/SparseSaving/HeightDecay.lean
+++ b/lean/OAI/Probability/ThorpRouting/SparseSaving/HeightDecay.lean
@@ -1,863 +1,863 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.SparseSaving.ParallelFreshPre
-
-namespace OAI
-
-namespace ThorpNine.SparseSaving
-
-namespace Thorp
-
-theorem palindrome_family_mgf (L n d : ℕ) (hL : 0 < L) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) ≤
-      Real.exp (Fintype.card ι * (68*∑ j ∈ Finset.range n,
-        Real.exp (-((L*4^j:ℕ):ℝ)/64))) := by
-  induction n with
-  | zero => simp only [palindromeFamilyCost,Finset.range_zero,Finset.sum_empty,pow_zero,
-      finiteMean_const,mul_zero,Real.exp_zero,le_refl]
-  | succ n ih =>
-    let l := L*4^n
-    have hl : 0 < l := Nat.mul_pos hL (pow_pos (by decide) _)
-    have hs : 2*familyEnd L n ≤ l := familyEnd_two L n
-    have hc : ∀ Y, palindromeFamilyCost L (n+1) d e (Y,X) =
-        palindromeFamilyCost L n d e (Y,X)+palindromeSlabCost l d e (Y,X) := by
-      intro Y
-      exact Finset.sum_range_succ _ _
-    have hm : finiteMean (fun Y => q^(palindromeFamilyCost L (n+1) d e (Y,X))) ≤
-        finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) *
-          Real.exp (Fintype.card ι*(68*Real.exp (-(l:ℝ)/64))) := by
-      by_cases hld : l ≤ d
-      · simp_rw [hc,pow_add]
-        exact palindrome_slab_adapted_mul_le d (familyEnd L n) l (by omega) hl hs e X _
-          (fun _ => pow_nonneg (by linarith) _)
-          (layerAdapted_comp (palindromeFamilyCost_adapted L n d e X) (fun c => q^c))
-          q hq hlog
-      · simp_rw [hc,palindromeSlabCost_gt l d (by omega) e,Nat.add_zero]
-        exact le_mul_of_one_le_right
-          (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
-          (Real.one_le_exp_iff.mpr (by positivity))
-    refine (hm.trans (mul_le_mul_of_nonneg_right ih (Real.exp_nonneg _))).trans_eq ?_
-    rw [←Real.exp_add,Finset.sum_range_succ]
-    congr 1
-    dsimp [l]
-    ring
-
-lemma pow_four_ge (j : ℕ) : j+1 ≤ 4^j := by
-  induction j with
-  | zero => norm_num
-  | succ j ih => rw [pow_succ]; omega
-
-lemma sum_family_decay (L n : ℕ) (hL : 0 < L) :
-    (∑ j ∈ Finset.range n, Real.exp (-((L*4^j:ℕ):ℝ)/64)) ≤
-      Real.exp (-(L:ℝ)/64)/(1-Real.exp (-(1:ℝ)/64)) := by
-  let r := Real.exp (-(1:ℝ)/64)
-  have hr : 0 < r := Real.exp_pos _
-  have hr1 : r < 1 := Real.exp_lt_one_iff.mpr (by norm_num)
-  have hg : (∑ j ∈ Finset.range n, r^j) ≤ 1/(1-r) := by
-    apply (le_div_iff₀ (by linarith : (0:ℝ)<1-r)).mpr
-    have hh := geom_sum_mul r n
-    nlinarith [pow_nonneg (le_of_lt hr) n]
-  calc
-    _ ≤ ∑ j ∈ Finset.range n, Real.exp (-(L:ℝ)/64)*r^j := by
-      apply Finset.sum_le_sum
-      intro j _
-      have hj := pow_four_ge j
-      have hlj : L+j ≤ L*4^j := by nlinarith
-      have hlj' : (L:ℝ)+j ≤ (L*4^j:ℕ) := by exact_mod_cast hlj
-      rw [←Real.exp_nat_mul,←Real.exp_add]
-      exact Real.exp_le_exp.mpr (by linarith)
-    _ = Real.exp (-(L:ℝ)/64)*(∑ j ∈ Finset.range n, r^j) := by rw [Finset.mul_sum]
-    _ ≤ Real.exp (-(L:ℝ)/64)*(1/(1-r)) :=
-      mul_le_mul_of_nonneg_left hg (Real.exp_nonneg _)
-    _ = _ := by ring
-
-
-lemma palindromeLevelCost_step_add (t d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (d+1)) (ω : BenesCoins (d+1)) :
-    palindromeLevelCost t (d+1) e ω =
-      let σ := benesStepEquiv d ω
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x σ.2.1
-      let hc := PairRouting.colors_compatible x σ.2.1
-      (if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
-          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) else 0) +
-        palindromeLevelCost t d (PairRouting.childEmbedding x c hc false) σ.1.1 +
-        palindromeLevelCost t d (PairRouting.childEmbedding x c hc true) σ.1.2 := by
-  classical
-  rw [palindromeLevelCost]
-  dsimp only
-  split_ifs with h
-  · simp only [palindromeLevelCost_gt t d (by omega),Nat.add_zero]
-  · simp only [Nat.zero_add]
-
-lemma palindromeLowCost_sum (H d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) :
-    palindromeLowCost H d e ω = ∑ i ∈ Finset.range H, palindromeLevelCost (i+1) d e ω := by
-  classical
-  induction d generalizing ι with
-  | zero => simp only [palindromeLowCost,palindromeLevelCost,Finset.sum_const_zero]
-  | succ d ih =>
-    rw [palindromeLowCost]
-    simp only [palindromeLevelCost_step_add,Finset.sum_add_distrib]
-    rw [←ih,←ih]
-    congr 2
-    simp only [Nat.add_right_cancel_iff,Finset.sum_ite_eq',Finset.mem_range,Nat.succ_le_iff]
-
-lemma palindromeLowCost_slab (L d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) :
-    palindromeLowCost L d e ω + palindromeSlabCost L d e ω = palindromeLowCost (2*L) d e ω := by
-  rw [palindromeLowCost_sum,palindromeLowCost_sum,show 2*L=L+L by omega,Finset.sum_range_add]
-  congr 1
-  exact Fin.sum_univ_eq_sum_range (fun i => palindromeLevelCost (L+i+1) d e ω) L
-
-lemma palindromeCost_family (L n d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) :
-    palindromeLowCost L d e ω + palindromeFamilyCost L n d e ω +
-      palindromeFamilyCost (2*L) n d e ω = palindromeLowCost (L*4^n) d e ω := by
-  induction n with
-  | zero => simp only [palindromeFamilyCost,Finset.range_zero,Finset.sum_empty,Nat.add_zero,
-      pow_zero,mul_one]
-  | succ n ih =>
-    have he : (2*L)*4^n = 2*(L*4^n) := by ring
-    have he' : L*4^(n+1) = 2*(2*(L*4^n)) := by rw [pow_succ]; ring
-    simp only [palindromeFamilyCost,Finset.sum_range_succ] at ih ⊢
-    rw [he,he',←palindromeLowCost_slab,←palindromeLowCost_slab,←ih]
-    omega
-
-
-lemma finiteMean_mul_sq_le {Ω : Type*} [Fintype Ω] (f g : Ω → ℝ) :
-    (finiteMean (fun ω => f ω*g ω))^2 ≤ finiteMean (fun ω => (f ω)^2)*finiteMean (fun ω => (g ω)^2) := by
-  have h := Finset.sum_mul_sq_le_sq_mul_sq Finset.univ f g
-  simp only [finiteMean,div_pow,div_mul_div_comm,←pow_two]
-  exact div_le_div_of_nonneg_right h (sq_nonneg _)
-
-lemma finiteMean_mul_exp_le {Ω : Type*} [Fintype Ω] (f g : Ω → ℝ) (A B : ℝ)
-    (hf : finiteMean (fun ω => (f ω)^2) ≤ Real.exp A)
-    (hg : finiteMean (fun ω => (g ω)^2) ≤ Real.exp B) :
-    finiteMean (fun ω => f ω*g ω) ≤ Real.exp ((A+B)/2) := by
-  have hc := finiteMean_mul_sq_le f g
-  have hn : 0 ≤ finiteMean (fun ω => (g ω)^2) := finiteMean_nonneg (fun _ => sq_nonneg _)
-  have hu := mul_le_mul hf hg hn (Real.exp_nonneg _)
-  have he : Real.exp A*Real.exp B = (Real.exp ((A+B)/2))^2 := by
-    rw [←Real.exp_add,←Real.exp_nat_mul]
-    congr 1
-    push_cast
-    ring
-  rw [he] at hu
-  have hh := hc.trans hu
-  have hep := Real.exp_pos ((A+B)/2)
-  nlinarith
-
-
-noncomputable def heightDecay (H : ℕ) : ℝ :=
-  68*Real.exp (-(H:ℝ)/64)/(1-Real.exp (-(1:ℝ)/64))
-
-lemma heightDecay_nonneg (H : ℕ) : 0 ≤ heightDecay H := by
-  have h : Real.exp (-(1:ℝ)/64) < 1 := Real.exp_lt_one_iff.mpr (by norm_num)
-  apply div_nonneg (by positivity) (by linarith)
-
-lemma heightDecay_antitone : Antitone heightDecay := by
-  intro L H h
-  have hden : 0 ≤ 1-Real.exp (-(1:ℝ)/64) := by
-    have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
-    linarith
-  apply div_le_div_of_nonneg_right _ hden
-  apply mul_le_mul_of_nonneg_left _ (by norm_num)
-  apply Real.exp_le_exp.mpr
-  have h' : (L:ℝ) ≤ H := by exact_mod_cast h
-  linarith
-
-lemma palindrome_family_mgf_decay (L n d : ℕ) (hL : 0 < L) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) ≤
-      Real.exp (Fintype.card ι*heightDecay L) := by
-  apply (palindrome_family_mgf L n d hL e X q hq hlog).trans
-  apply Real.exp_le_exp.mpr
-  apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
-  have h := mul_le_mul_of_nonneg_left (sum_family_decay L n hL) (by norm_num : (0:ℝ)≤68)
-  simpa only [heightDecay,mul_div_assoc] using h
-
-theorem palindrome_high_mgf (H n d : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/500) :
-    finiteMean (fun Y => q^(palindromeFamilyCost H n d e (Y,X)+
-      palindromeFamilyCost (2*H) n d e (Y,X))) ≤ Real.exp (Fintype.card ι*heightDecay H) := by
-  have hq2 := one_le_pow₀ (n:=2) hq
-  have hlog2 : Real.log (q^2) ≤ 1/250 := by rw [Real.log_pow]; norm_num; linarith
-  have h₀ := palindrome_family_mgf_decay H n d hH e X (q^2) hq2 hlog2
-  have h₁ := palindrome_family_mgf_decay (2*H) n d (by omega) e X (q^2) hq2 hlog2
-  have h₁' := h₁.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left
-    (heightDecay_antitone (show H ≤ 2*H by omega)) (Nat.cast_nonneg _)))
-  have hh := finiteMean_mul_exp_le (fun Y => q^(palindromeFamilyCost H n d e (Y,X)))
-    (fun Y => q^(palindromeFamilyCost (2*H) n d e (Y,X)))
-    (Fintype.card ι*heightDecay H) (Fintype.card ι*heightDecay H)
-    (by simpa only [pow_right_comm] using h₀) (by simpa only [pow_right_comm] using h₁')
-  simpa only [←pow_add,add_self_div_two] using hh
-
-lemma palindrome_high_full_mgf (H n d : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/500) :
-    finiteMean (fun ω : BenesCoins d => q^(palindromeFamilyCost H n d e ω+
-      palindromeFamilyCost (2*H) n d e ω)) ≤ Real.exp (Fintype.card ι*heightDecay H) := by
-  rw [finiteMean_prod,finiteMean_comm]
-  calc
-    _ ≤ finiteMean (fun _X : SwitchIndex d → Bool => Real.exp (Fintype.card ι*heightDecay H)) := by
-      apply finiteMean_mono
-      intro X
-      exact palindrome_high_mgf H n d hH e X q hq hlog
-    _ = _ := finiteMean_const _
-
-theorem palindrome_cost_mgf_cutoff (r H : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (H+r)) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/1000)
-    (hr : (2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*(q^2)^H) ≤ 1/2) :
-    finiteMean (fun ω : BenesCoins (H+r) => q^(palindromeCost (H+r) e ω)) ≤
-      Real.exp (((2:ℝ)^r*(2*((2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*(q^2)^H))^2)+
-        Fintype.card ι*heightDecay H)/2) := by
-  let d := H+r
-  have hd : d ≤ H*4^d := by
-    have hh := pow_four_ge d
-    nlinarith
-  have hcost (ω : BenesCoins d) : palindromeCost d e ω = palindromeLowCost H d e ω+
-      (palindromeFamilyCost H d d e ω+palindromeFamilyCost (2*H) d d e ω) := by
-    have hh := palindromeCost_family H d d e ω
-    rw [palindromeLowCost_eq _ _ hd] at hh
-    simpa only [Nat.add_assoc] using hh.symm
-  have hq2 := one_le_pow₀ (n:=2) hq
-  have hlog2 : Real.log (q^2) ≤ 1/500 := by rw [Real.log_pow]; norm_num; linarith
-  have hlow := palindrome_low_mgf r H e (q^2) hq2 hr
-  have hhigh := palindrome_high_full_mgf H d d hH e (q^2) hq2 hlog2
-  have hh := finiteMean_mul_exp_le (fun ω : BenesCoins d => q^(palindromeLowCost H d e ω))
-    (fun ω : BenesCoins d => q^(palindromeFamilyCost H d d e ω+palindromeFamilyCost (2*H) d d e ω))
-    _ _ (by simpa only [pow_right_comm] using hlow) (by simpa only [pow_right_comm] using hhigh)
-  change finiteMean (fun ω : BenesCoins d => q^(palindromeCost d e ω)) ≤ _
-  simpa only [hcost,pow_add,pow_right_comm] using hh
-
-
-lemma cutoffMass_le (H d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (ω : BenesCoins d) :
-    cutoffMass H d e ω ≤ Fintype.card ι := by
-  classical
-  induction d generalizing ι with
-  | zero => exact Nat.zero_le _
-  | succ d ih =>
-    rw [cutoffMass]
-    split_ifs
-    · exact crowdedMass_le _
-    · exact (Nat.add_le_add (ih _ _) (ih _ _)).trans_eq (PairRouting.color_card_add _)
-
-lemma palindromeLowCost_le (H d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) : palindromeLowCost H d e ω ≤ H*Fintype.card ι :=
-  (palindromeLowCost_le_cutoff H d e ω).trans (Nat.mul_le_mul_left H (cutoffMass_le H d e ω))
-
-theorem palindrome_cost_mgf_coarse (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/1000) :
-    finiteMean (fun ω : BenesCoins d => q^(palindromeCost d e ω)) ≤
-      Real.exp (Fintype.card ι*(1+heightDecay 1)) := by
-  have hd : d ≤ 1*4^d := by have hh := pow_four_ge d; omega
-  have hcost (ω : BenesCoins d) : palindromeCost d e ω = palindromeLowCost 1 d e ω+
-      (palindromeFamilyCost 1 d d e ω+palindromeFamilyCost 2 d d e ω) := by
-    have hh := palindromeCost_family 1 d d e ω
-    rw [palindromeLowCost_eq _ _ hd] at hh
-    simpa only [Nat.add_assoc] using hh.symm
-  have hq2 := one_le_pow₀ (n:=2) hq
-  have hlog2 : Real.log (q^2) ≤ 1/500 := by rw [Real.log_pow]; norm_num; linarith
-  have hlow : finiteMean (fun ω : BenesCoins d => (q^2)^(palindromeLowCost 1 d e ω)) ≤
-      Real.exp (Fintype.card ι) := by
-    calc
-      _ ≤ finiteMean (fun _ω : BenesCoins d => (q^2)^(Fintype.card ι)) := by
-        apply finiteMean_mono
-        intro ω
-        apply pow_le_pow_right₀ hq2
-        simpa only [one_mul] using palindromeLowCost_le 1 d e ω
-      _ = (q^2)^(Fintype.card ι) := finiteMean_const _
-      _ = Real.exp (Fintype.card ι * Real.log (q^2)) := by
-        rw [Real.exp_nat_mul,Real.exp_log (by positivity : 0<q^2)]
-      _ ≤ _ := Real.exp_le_exp.mpr (by
-        have hh := mul_le_mul_of_nonneg_left hlog2 (Nat.cast_nonneg (Fintype.card ι) : (0:ℝ)≤_)
-        nlinarith [Nat.cast_nonneg (α:=ℝ) (Fintype.card ι)])
-  have hhigh := palindrome_high_full_mgf 1 d d (by decide) e (q^2) hq2 hlog2
-  have hh := finiteMean_mul_exp_le (fun ω : BenesCoins d => q^(palindromeLowCost 1 d e ω))
-    (fun ω : BenesCoins d => q^(palindromeFamilyCost 1 d d e ω+palindromeFamilyCost 2 d d e ω))
-    (Fintype.card ι) (Fintype.card ι*heightDecay 1)
-    (by simpa only [pow_right_comm] using hlow) (by simpa only [pow_right_comm] using hhigh)
-  simp only [←pow_add,←hcost] at hh
-  apply hh.trans (Real.exp_le_exp.mpr _)
-  have hk : (0:ℝ) ≤ Fintype.card ι := Nat.cast_nonneg _
-  have hD := heightDecay_nonneg 1
-  nlinarith
-
-noncomputable def momentBase : ℝ := (2:ℝ)^(1/1000:ℝ)
-
-lemma momentBase_one_le : 1 ≤ momentBase := by
-  exact Real.one_le_rpow (by norm_num) (by norm_num)
-
-lemma momentBase_log : Real.log momentBase ≤ 1/1000 := by
-  rw [momentBase,Real.log_rpow (by norm_num)]
-  have hh := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
-  norm_num at hh ⊢
-  linarith
-
-lemma momentBase_pow (c : ℕ) : momentBase^c = (2:ℝ)^((c:ℝ)*(1/1000:ℝ)) := by
-  rw [mul_comm,Real.rpow_mul_natCast (by norm_num)]
-  rfl
-
-lemma palindromeRowMoment_le_cost (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) :
-    palindromeRowMoment d e (1/1000) ≤
-      finiteMean (fun ω : BenesCoins d => momentBase^(palindromeCost d e ω)) := by
-  have hn : (0:ℝ) < ((2^d:ℕ):ℝ)^(Fintype.card ι) := by positivity
-  have hn0 : (0:ℝ) ≤ ((2^d).descFactorial (Fintype.card ι):ℝ) /
-      ((2^d:ℕ):ℝ)^(Fintype.card ι) := by positivity
-  have hn1 : ((2^d).descFactorial (Fintype.card ι):ℝ) /
-      ((2^d:ℕ):ℝ)^(Fintype.card ι) ≤ 1 := by
-    apply (div_le_one hn).mpr
-    exact_mod_cast Nat.descFactorial_le_pow (2^d) (Fintype.card ι)
-  have hpow := Real.rpow_le_one hn0 hn1 (by norm_num : (0:ℝ)≤1/1000)
-  have hm := palindrome_row_density_moment d e (1/1000) (by norm_num)
-  simp_rw [←momentBase_pow] at hm
-  exact hm.trans (mul_le_of_le_one_left (finiteMean_nonneg (fun _ =>
-    pow_nonneg (le_trans (by norm_num) momentBase_one_le) _)) hpow)
-
-theorem palindrome_row_moment_coarse (d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) :
-    palindromeRowMoment d e (1/1000) ≤ Real.exp (Fintype.card ι*(1+heightDecay 1)) :=
-  (palindromeRowMoment_le_cost d e).trans
-    (palindrome_cost_mgf_coarse d e momentBase momentBase_one_le momentBase_log)
-
-
-noncomputable def densityCutoff (p : ℝ) : ℕ := ⌊-Real.log p/(4*Real.log 2)⌋₊
-
-lemma log_small_density {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
-    Real.log p ≤ -8*Real.log 2 := by
-  have he : Real.log (1/256:ℝ) = -8*Real.log 2 := by
-    rw [show (1/256:ℝ) = 1/2^8 by norm_num,
-      Real.log_div (by norm_num) (by positivity),Real.log_one,Real.log_pow]
-    ring
-  exact (Real.log_le_log hp hs).trans_eq he
-
-lemma densityCutoff_bounds {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
-    0 < densityCutoff p ∧
-    (densityCutoff p:ℝ)*(4*Real.log 2) ≤ -Real.log p ∧
-    -Real.log p < ((densityCutoff p:ℝ)+1)*(4*Real.log 2) := by
-  have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
-  have hd : 0 < 4*Real.log 2 := by positivity
-  have hh := log_small_density hp hs
-  have hx : 1 ≤ -Real.log p/(4*Real.log 2) := by
-    apply (le_div_iff₀ hd).mpr
-    nlinarith
-  refine ⟨Nat.floor_pos.mpr hx,?_,?_⟩
-  · exact (le_div_iff₀ hd).mp (Nat.floor_le (by linarith : 0 ≤ -Real.log p/(4*Real.log 2)))
-  · exact (div_lt_iff₀ hd).mp (Nat.lt_floor_add_one (-Real.log p/(4*Real.log 2)))
-
-lemma densityCutoff_decay {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
-    Real.exp (-(densityCutoff p:ℝ)/64) ≤ Real.exp (1/64) * p^(1/512:ℝ) := by
-  have hh := (densityCutoff_bounds hp hs).2.2
-  have hb : Real.log 2 ≤ 2 := by
-    have := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
-    linarith
-  have hH : 0 ≤ (densityCutoff p:ℝ)+1 := by positivity
-  have hu := mul_le_mul_of_nonneg_left hb hH
-  rw [Real.rpow_def_of_pos hp,←Real.exp_add]
-  apply Real.exp_le_exp.mpr
-  nlinarith
-
-lemma densityCutoff_power {p q : ℝ} (hp : 0 < p) (hs : p ≤ 1/256)
-    (hq : 0 < q) (hlog : Real.log q ≤ Real.log 2/4) (n : ℕ) (hn : n ≤ 4)
-    (hq1 : 1 ≤ q) :
-    p*(2:ℝ)^(densityCutoff p)*q^(n*densityCutoff p) ≤ p^(1/2:ℝ) := by
-  have hcut := (densityCutoff_bounds hp hs).2.1
-  have hH : 0 ≤ (densityCutoff p:ℝ) := Nat.cast_nonneg _
-  have hn' : (n:ℝ) ≤ 4 := by exact_mod_cast hn
-  have hln := Real.log_nonneg hq1
-  have hl := mul_le_mul_of_nonneg_right hn' hln
-  have hnl : (n:ℝ)*Real.log q ≤ Real.log 2 := by linarith
-  have hm := mul_le_mul_of_nonneg_left hnl hH
-  have heq : p*(2:ℝ)^(densityCutoff p)*q^(n*densityCutoff p) =
-      Real.exp (Real.log p+(densityCutoff p:ℝ)*Real.log 2+
-        (n*densityCutoff p:ℕ)*Real.log q) := by
-    rw [Real.exp_add,Real.exp_add,Real.exp_nat_mul,Real.exp_nat_mul,
-      Real.exp_log hp,Real.exp_log hq,Real.exp_log (by norm_num : (0:ℝ)<2)]
-  rw [heq,Real.rpow_def_of_pos hp]
-  apply Real.exp_le_exp.mpr
-  push_cast
-  nlinarith
-
-lemma densityCutoff_sparse {p q : ℝ} (hp : 0 < p) (hs : p ≤ 1/256)
-    (hq : 0 < q) (hlog : Real.log q ≤ Real.log 2/4) (hq1 : 1 ≤ q) :
-    (2:ℝ)^(densityCutoff p)*(p*(q^2)^(densityCutoff p)) ≤ 1/2 := by
-  have hh := densityCutoff_power hp hs hq hlog 2 (by decide) hq1
-  rw [←pow_mul] at ⊢
-  have hle : p^(1/2:ℝ) ≤ 1/2 := by
-    have hlogp := log_small_density hp hs
-    have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
-    rw [Real.rpow_def_of_pos hp]
-    calc
-      _ ≤ Real.exp (-Real.log 2) := by apply Real.exp_le_exp.mpr; nlinarith
-      _ = 1/2 := by rw [Real.exp_neg,Real.exp_log (by norm_num : (0:ℝ)<2)]; norm_num
-  nlinarith [hh.trans hle]
-
-
-lemma momentBase_log_exact : Real.log momentBase = Real.log 2/1000 := by
-  rw [momentBase,Real.log_rpow (by norm_num)]
-  ring
-
-lemma densityCutoff_le_height (d k : ℕ) (hk : 0 < k) :
-    densityCutoff ((k:ℝ)/(2:ℝ)^d) ≤ d := by
-  have hk' : (1:ℝ) ≤ k := by exact_mod_cast hk
-  have hp : 0 < (k:ℝ)/(2:ℝ)^d := by positivity
-  have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
-  have hlogk := Real.log_nonneg hk'
-  by_cases hx : 0 ≤ -Real.log ((k:ℝ)/(2:ℝ)^d)/(4*Real.log 2)
-  · have hh := (le_div_iff₀ (by positivity : 0 < 4*Real.log 2)).mp (Nat.floor_le hx)
-    have hd : (0:ℝ) ≤ d := Nat.cast_nonneg _
-    have hH : (0:ℝ) ≤ densityCutoff ((k:ℝ)/(2:ℝ)^d) := Nat.cast_nonneg _
-    change (densityCutoff ((k:ℝ)/(2:ℝ)^d):ℝ)*(4*Real.log 2) ≤ -Real.log ((k:ℝ)/(2:ℝ)^d) at hh
-    rw [Real.log_div (by positivity) (by positivity),Real.log_pow] at hh
-    have hle : (densityCutoff ((k:ℝ)/(2:ℝ)^d):ℝ) ≤ d := by nlinarith
-    exact_mod_cast hle
-  · have hz := Nat.floor_of_nonpos (le_of_not_ge hx)
-    simp only [densityCutoff,hz,Nat.zero_le]
-
-noncomputable def densityTailConstant : ℝ :=
-  68*Real.exp (1/64)/(1-Real.exp (-(1:ℝ)/64))
-
-lemma densityTailConstant_nonneg : 0 ≤ densityTailConstant := by
-  have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
-  apply div_nonneg (by positivity) (by linarith)
-
-lemma heightDecay_cutoff {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
-    heightDecay (densityCutoff p) ≤ densityTailConstant*p^(1/512:ℝ) := by
-  have hden : 0 ≤ 1-Real.exp (-(1:ℝ)/64) := by
-    have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
-    linarith
-  have hh := div_le_div_of_nonneg_right
-    (mul_le_mul_of_nonneg_left (densityCutoff_decay hp hs) (by norm_num : (0:ℝ)≤68)) hden
-  simpa only [heightDecay,densityTailConstant,mul_assoc,div_mul_eq_mul_div] using hh
-
-lemma sparse_low_algebra (r H k : ℕ) (q : ℝ) :
-    (2:ℝ)^r*(2*((2:ℝ)^H*(((k:ℝ)/(2:ℝ)^(H+r))*(q^2)^H))^2) =
-    2*k*(((k:ℝ)/(2:ℝ)^(H+r))*(2:ℝ)^H*q^(4*H)) := by
-  rw [pow_add,←pow_mul,show 4*H = (2*H)*2 by omega,pow_mul]
-  have hH : (2:ℝ)^H ≠ 0 := by positivity
-  have hr : (2:ℝ)^r ≠ 0 := by positivity
-  field_simp
-  ring
-
-lemma palindrome_row_moment_small (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι)
-    (hs : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1/256) :
-    palindromeRowMoment d e (1/1000) ≤
-      Real.exp (Fintype.card ι*(1+densityTailConstant)*
-        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ)) := by
-  generalize hpdef : ((Fintype.card ι:ℝ)/(2:ℝ)^d) = p at hs ⊢
-  have hp : 0 < p := by rw [←hpdef]; positivity
-  have hle := densityCutoff_le_height d (Fintype.card ι) hk
-  rw [hpdef] at hle
-  obtain ⟨r,hr⟩ := Nat.exists_eq_add_of_le hle
-  subst d
-  have hH := (densityCutoff_bounds hp hs).1
-  have hq := momentBase_one_le
-  have hqp : 0 < momentBase := by linarith
-  have hlog : Real.log momentBase ≤ Real.log 2/4 := by
-    rw [momentBase_log_exact]
-    have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
-    linarith
-  have hsp := densityCutoff_sparse hp hs hqp hlog hq
-  rw [←hpdef] at hsp
-  have hh := palindrome_cost_mgf_cutoff r (densityCutoff p) hH e momentBase hq momentBase_log
-    (by simpa only [hpdef] using hsp)
-  apply (palindromeRowMoment_le_cost _ e).trans (hh.trans (Real.exp_le_exp.mpr _))
-  rw [sparse_low_algebra,hpdef]
-  have hlow := densityCutoff_power hp hs hqp hlog 4 (by decide) hq
-  have hp1 : p ≤ 1 := by linarith
-  have he := Real.rpow_le_rpow_of_exponent_ge hp hp1 (by norm_num : (1:ℝ)/512 ≤ 1/2)
-  have hlow' := mul_le_mul_of_nonneg_left (hlow.trans he)
-    (by positivity : (0:ℝ)≤2*Fintype.card ι)
-  have hhigh := mul_le_mul_of_nonneg_left (heightDecay_cutoff hp hs)
-    (Nat.cast_nonneg (Fintype.card ι): (0:ℝ)≤_)
-  have hnon : 0 ≤ Fintype.card ι*densityTailConstant*p^(1/512:ℝ) :=
-    mul_nonneg (mul_nonneg (Nat.cast_nonneg _) densityTailConstant_nonneg)
-      (Real.rpow_nonneg (le_of_lt hp) _)
-  nlinarith
-
-noncomputable def densityConstant : ℝ :=
-  256*(1+heightDecay 1)+1+densityTailConstant
-
-lemma densityConstant_pos : 0 < densityConstant := by
-  dsimp [densityConstant]
-  have hh := heightDecay_nonneg 1
-  have ht := densityTailConstant_nonneg
-  linarith
-
-theorem palindrome_row_moment_bound (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι) :
-    palindromeRowMoment d e (1/1000) ≤ Real.exp (densityConstant*Fintype.card ι*
-        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ)) := by
-  have hp : 0 < (Fintype.card ι:ℝ)/(2:ℝ)^d := by positivity
-  have hp1 : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1 := by
-    apply (div_le_one (by positivity)).mpr
-    have h := Fintype.card_le_of_injective e e.injective
-    rw [card_positions] at h
-    exact_mod_cast h
-  have hk0 : (0:ℝ) ≤ Fintype.card ι := Nat.cast_nonneg _
-  have hpow := Real.rpow_nonneg (le_of_lt hp) (1/512:ℝ)
-  by_cases hs : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1/256
-  · apply (palindrome_row_moment_small d e hk hs).trans (Real.exp_le_exp.mpr _)
-    have hD : 1+densityTailConstant ≤ densityConstant := by
-      dsimp [densityConstant]
-      have := heightDecay_nonneg 1
-      linarith
-    have hh := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hD hk0) hpow
-    nlinarith
-  · have hlow : (1:ℝ)/256 ≤ ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ) := by
-      have he := Real.rpow_le_rpow_of_exponent_ge hp hp1 (by norm_num : (1:ℝ)/512 ≤ 1)
-      rw [Real.rpow_one] at he
-      exact (le_of_not_ge hs).trans he
-    have hD : 256*(1+heightDecay 1) ≤ densityConstant := by
-      dsimp [densityConstant]
-      have := densityTailConstant_nonneg
-      linarith
-    apply (palindrome_row_moment_coarse d e).trans (Real.exp_le_exp.mpr _)
-    have hD0 : 0 ≤ 1+heightDecay 1 := by have := heightDecay_nonneg 1; linarith
-    have ha := mul_le_mul_of_nonneg_left hlow (mul_nonneg (by norm_num : (0:ℝ)≤256) hD0)
-    have hb := mul_le_mul_of_nonneg_right hD hpow
-    have hc := mul_le_mul_of_nonneg_right (ha.trans hb) hk0
-    nlinarith
-
-lemma palindromeRowMoment_nonneg (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (a : ℝ) : 0 ≤ palindromeRowMoment d e a := by
-  apply finiteMean_nonneg
-  intro f
-  exact Real.rpow_nonneg (mul_nonneg (Nat.cast_nonneg _)
-    (PairRouting.tupleProbability_nonneg _ _ _)) _
-
-theorem palindrome_log_row_moment (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι) :
-    Real.log (palindromeRowMoment d e (1/1000)) ≤ densityConstant*Fintype.card ι*
-        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ) := by
-  by_cases hz : palindromeRowMoment d e (1/1000) = 0
-  · rw [hz,Real.log_zero]
-    exact mul_nonneg (mul_nonneg (le_of_lt densityConstant_pos) (Nat.cast_nonneg _))
-      (Real.rpow_nonneg (by positivity) _)
-  · exact (Real.log_le_iff_le_exp (lt_of_le_of_ne (palindromeRowMoment_nonneg _ _ _) (Ne.symm hz))).mpr
-      (palindrome_row_moment_bound d e hk)
-
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-lemma orbitSet_eq_of_agree (p q : Equiv.Perm α) (x : α)
-    (h : ∀ z ∈ orbitSet p x, p z = q z) : orbitSet p x = orbitSet q x := by
-  have hp (n : ℕ) : (p^n) x ∈ orbitSet p x := by
-    apply (mem_orbitSet _ _ _).mpr
-    exact Equiv.Perm.sameCycle_pow_right.mpr (Equiv.Perm.SameCycle.refl _ _)
-  have he (n : ℕ) : (p^n) x = (q^n) x := by
-    induction n with
-    | zero => rfl
-    | succ n ih =>
-      rw [pow_succ',pow_succ',Equiv.Perm.mul_apply,Equiv.Perm.mul_apply,←ih]
-      exact h _ (hp n)
-  ext y
-  simp only [mem_orbitSet]
-  constructor
-  · intro hy
-    obtain ⟨n,hn⟩ := hy.exists_nat_pow_eq
-    rw [←hn,he]
-    exact Equiv.Perm.sameCycle_pow_right.mpr (Equiv.Perm.SameCycle.refl _ _)
-  · intro hy
-    obtain ⟨n,hn⟩ := hy.exists_nat_pow_eq
-    rw [←hn,←he]
-    exact Equiv.Perm.sameCycle_pow_right.mpr (Equiv.Perm.SameCycle.refl _ _)
-
-noncomputable def survivingCycle (p q : Equiv.Perm α) (S B : Finset α)
-    (h : ∀ x ∉ B, p x = q x) (C : RoutingNetwork.SelectedCycle p S)
-    (hC : ∀ x ∈ C.val, x ∉ B) : RoutingNetwork.SelectedCycle q S := by
-  refine ⟨C.val,?_,C.property.2⟩
-  obtain ⟨x,hx⟩ := C.property.1
-  refine ⟨x,hx.trans (orbitSet_eq_of_agree p q x ?_)⟩
-  intro z hz
-  exact h z (hC z (hx ▸ hz))
-
-noncomputable def perturbCycleMap (p q : Equiv.Perm α) (S B : Finset α)
-    (h : ∀ x ∉ B, p x = q x) (C : RoutingNetwork.SelectedCycle p S) :
-    RoutingNetwork.SelectedCycle q S ⊕ {x // x ∈ B} := by
-  classical
-  by_cases hb : ∃ x ∈ C.val, x ∈ B
-  · exact Sum.inr ⟨hb.choose,hb.choose_spec.2⟩
-  · exact Sum.inl (survivingCycle p q S B h C (by simpa only [not_exists,not_and] using hb))
-
-lemma perturbCycleMap_injective (p q : Equiv.Perm α) (S B : Finset α)
-    (h : ∀ x ∉ B, p x = q x) : Function.Injective (perturbCycleMap p q S B h) := by
-  classical
-  intro C D he
-  by_cases hC : ∃ x ∈ C.val, x ∈ B <;>
-    by_cases hD : ∃ x ∈ D.val, x ∈ B
-  · simp only [perturbCycleMap,dite_eq_left hC,dite_eq_left hD,Sum.inr.injEq] at he
-    have hx : hC.choose = hD.choose := congrArg Subtype.val he
-    exact RoutingNetwork.SelectedCycle.eq_of_mem C D hC.choose_spec.1
-      (hx.symm ▸ hD.choose_spec.1)
-  · simp only [perturbCycleMap,dite_eq_left hC,dite_eq_right hD] at he
-    cases he
-  · simp only [perturbCycleMap,dite_eq_right hC,dite_eq_left hD] at he
-    cases he
-  · simp only [perturbCycleMap,dite_eq_right hC,dite_eq_right hD,Sum.inl.injEq] at he
-    have hval : C.val = D.val := congrArg (fun Z : RoutingNetwork.SelectedCycle q S => Z.val) he
-    exact Subtype.ext hval
-
-theorem selectedCycle_perturb_le (p q : Equiv.Perm α) (S B : Finset α)
-    (h : ∀ x ∉ B, p x = q x) :
-    Fintype.card (RoutingNetwork.SelectedCycle p S) ≤
-      Fintype.card (RoutingNetwork.SelectedCycle q S)+B.card := by
-  have hh := Fintype.card_le_of_injective _ (perturbCycleMap_injective p q S B h)
-  simpa only [Fintype.card_sum,Fintype.card_coe] using hh
-
-
-noncomputable def permDistance (p q : Equiv.Perm α) : ℕ :=
-  (Finset.univ.filter (fun x => p x ≠ q x)).card
-
-lemma permDistance_sum (p q : Equiv.Perm α) :
-    permDistance p q = ∑ x : α, if p x = q x then 0 else 1 := by
-  classical
-  simp [permDistance,Finset.card_filter]
-
-lemma permDistance_refl (p : Equiv.Perm α) : permDistance p p = 0 := by
-  simp [permDistance]
-
-lemma permDistance_le (p q : Equiv.Perm α) : permDistance p q ≤ Fintype.card α :=
-  (Finset.card_filter_le _ _).trans_eq (Finset.card_univ)
-
-lemma permDistance_symm (p q : Equiv.Perm α) : permDistance p q = permDistance q p := by
-  simp only [permDistance,ne_comm]
-
-lemma permDistance_triangle (p q r : Equiv.Perm α) :
-    permDistance p r ≤ permDistance p q+permDistance q r := by
-  classical
-  simp only [permDistance_sum,←Finset.sum_add_distrib]
-  apply Finset.sum_le_sum
-  intro x _
-  by_cases hpq : p x = q x <;> by_cases hqr : q x = r x <;>
-    by_cases hpr : p x = r x <;> simp only [hpq,hqr,hpr,↓reduceIte] <;>
-    omega
-
-lemma permDistance_mul_left (p q g : Equiv.Perm α) :
-    permDistance (g*p) (g*q) = permDistance p q := by
-  classical
-  unfold permDistance
-  congr 1
-  ext x
-  simp only [Finset.mem_filter,Finset.mem_univ,true_and]
-  exact not_congr g.injective.eq_iff
-
-lemma permDistance_mul_right (p q g : Equiv.Perm α) :
-    permDistance (p*g) (q*g) = permDistance p q := by
-  classical
-  simp only [permDistance_sum,Equiv.Perm.mul_apply]
-  exact Fintype.sum_equiv g _ _ (fun _ => rfl)
-
-lemma permDistance_inv (p q : Equiv.Perm α) : permDistance p⁻¹ q⁻¹ = permDistance p q := by
-  have hh := permDistance_mul_left p⁻¹ q⁻¹ q
-  have hhh := permDistance_mul_right (q*p⁻¹) 1 p
-  simp only [mul_inv_cancel,one_mul,inv_mul_cancel_right] at hh hhh
-  rw [←hhh] at hh
-  rw [←hh,permDistance_symm]
-
-lemma permDistance_mul_le (p q g h : Equiv.Perm α) :
-    permDistance (p*g) (q*h) ≤ permDistance p q+permDistance g h := by
-  have hh := permDistance_triangle (p*g) (q*g) (q*h)
-  simpa only [permDistance_mul_left,permDistance_mul_right] using hh
-
-lemma selectedCycle_distance_le (p q : Equiv.Perm α) (S : Finset α) :
-    Fintype.card (RoutingNetwork.SelectedCycle p S) ≤
-      Fintype.card (RoutingNetwork.SelectedCycle q S)+permDistance p q := by
-  apply selectedCycle_perturb_le
-  intro x hx
-  simpa only [Finset.mem_filter,Finset.mem_univ,true_and,not_not] using hx
-
-lemma permDistance_alternating (p q : Equiv.Perm α) :
-    permDistance (CycleColoring.alternatingPerm p) (CycleColoring.alternatingPerm q) = permDistance p q := by
-  classical
-  simp only [permDistance_sum,Fintype.sum_prod_type,Fintype.sum_bool]
-  simp [CycleColoring.alternatingPerm]
-
-lemma alternatingCycles_distance_le {ι : Type*} [Fintype ι]
-    (e : ι ↪ Bool × α) (p q : Bool → Equiv.Perm α) :
-    PairRouting.alternatingCycles e p ≤ PairRouting.alternatingCycles e q+
-      (permDistance (p false) (q false)+permDistance (p true) (q true)) := by
-  have hh := selectedCycle_distance_le
-    (CycleColoring.alternatingPerm (PairRouting.alternatingProjection p))
-    (CycleColoring.alternatingPerm (PairRouting.alternatingProjection q)) (PairRouting.labelSet e)
-  rw [permDistance_alternating] at hh
-  apply hh.trans
-  apply Nat.add_le_add_left
-  dsimp only [PairRouting.alternatingProjection]
-  exact (permDistance_mul_le _ _ _ _).trans_eq (by
-    change permDistance (p false)⁻¹ (q false)⁻¹ + _ = _
-    rw [permDistance_inv])
-
-end Thorp
-namespace Thorp
-
-def centralBlockPerm (L : ℕ) : (r : ℕ) → (Card r → Equiv.Perm (Card L)) →
-    Equiv.Perm (Card (L+r))
-  | 0, C => C (fun j => Fin.elim0 j)
-  | r+1, C => childLift (fun b => centralBlockPerm L r (fun u => C (Fin.cons b u)))
-
-def centralPalindromePerm (L r : ℕ) (C : Card r → Equiv.Perm (Card L))
-    (ω : BenesCoins (L+r)) : Equiv.Perm (Card (L+r)) :=
-  butterflyPerm (L+r) (decodeButterfly (L+r) ω.1) * centralBlockPerm L r C *
-    (butterflyPerm (L+r) (decodeButterfly (L+r) ω.2))⁻¹
-
-lemma centralPalindromePerm_step (L r : ℕ) (C : Card (r+1) → Equiv.Perm (Card L))
-    (c : BenesCoins (L+r) × BenesCoins (L+r))
-    (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
-    centralPalindromePerm L (r+1) C ((benesStepEquiv (L+r)).symm (c,coins)) =
-      pairSwitch coins.2 * childLift (fun b => centralPalindromePerm L r
-        (fun u => C (Fin.cons b u)) (if b then c.2 else c.1)) * pairSwitch coins.1 := by
-  change ((butterflyPerm (L+r+1) (decodeButterfly (L+r+1)
-      ((coinStepEquiv (L+r)).symm ((c.1.1,c.2.1),coins.2))))) *
-    childLift (fun b => centralBlockPerm L r (fun u => C (Fin.cons b u))) *
-    (butterflyPerm (L+r+1) (decodeButterfly (L+r+1)
-      ((coinStepEquiv (L+r)).symm ((c.1.2,c.2.2),coins.1))))⁻¹ = _
-  rw [butterflyPerm_step,butterflyPerm_step,mul_inv_rev,childLift_inv,pairSwitch_inv]
-  simp only [mul_assoc]
-  rw [←mul_assoc (childLift _),childLift_mul,←mul_assoc (childLift _),childLift_mul]
-  congr 2
-  apply congrArg childLift
-  funext b
-  cases b <;> rfl
-
-noncomputable def centralLevelCost (t L : ℕ) : (r : ℕ) →
-    (Card r → Equiv.Perm (Card L)) → {ι : Type*} → [Fintype ι] →
-    (ι ↪ Card (L+r)) → BenesCoins (L+r) → ℕ
-  | 0, _, _, _, _, _ => 0
-  | r+1, C, _, _, e, ω =>
-    let σ := benesStepEquiv (L+r) ω
-    let x := e.trans (headTailEquiv (L+r)).toEmbedding
-    let c := PairRouting.colors x σ.2.1
-    let hc := PairRouting.colors_compatible x σ.2.1
-    if t = L+r+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
-      (fun b => centralPalindromePerm L r (fun u => C (Fin.cons b u)) (if b then σ.1.2 else σ.1.1))
-    else centralLevelCost t L r (fun u => C (Fin.cons false u))
-      (PairRouting.childEmbedding x c hc false) σ.1.1 +
-      centralLevelCost t L r (fun u => C (Fin.cons true u))
-      (PairRouting.childEmbedding x c hc true) σ.1.2
-
-noncomputable def centralSlabCost (l L r : ℕ) (C : Card r → Equiv.Perm (Card L))
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (ω : BenesCoins (L+r)) : ℕ :=
-  ∑ i : Fin l, centralLevelCost (l+i.val+1) L r C e ω
-
-lemma permDistance_childLift (d : ℕ) (p q : Bool → Equiv.Perm (Card d)) :
-    permDistance (childLift p) (childLift q) =
-      permDistance (p false) (q false)+permDistance (p true) (q true) := by
-  classical
-  rw [permDistance_sum]
-  have hh : (∑ x : Card (d+1), if childLift p x = childLift q x then 0 else 1) =
-      ∑ x : Bool × Card d, if p x.1 x.2 = q x.1 x.2 then 0 else 1 := by
-    apply Fintype.sum_equiv (headTailEquiv d)
-    intro x
-    have hx : (headTailEquiv d) x = (x 0,Fin.tail x) := rfl
-    rw [hx]
-    congr 1
-    apply propext
-    constructor
-    · intro h
-      have ht := congrArg Fin.tail h
-      simpa only [childLift,Equiv.coe_fn_mk,Fin.tail_cons] using ht
-    · intro h
-      dsimp only [childLift,Equiv.coe_fn_mk]
-      exact congrArg (fun z : Card d => (Fin.cons (x 0) z : Card (d+1))) h
-  rw [hh]
-  simp only [Fintype.sum_prod_type,Fintype.sum_bool,permDistance_sum]
-  omega
-
-noncomputable def centralDistance (L r : ℕ) (C D : Card r → Equiv.Perm (Card L)) : ℕ :=
-  ∑ u : Card r, permDistance (C u) (D u)
-
-lemma centralDistance_step (L r : ℕ) (C D : Card (r+1) → Equiv.Perm (Card L)) :
-    centralDistance L (r+1) C D =
-      centralDistance L r (fun u => C (Fin.cons false u)) (fun u => D (Fin.cons false u))+
-      centralDistance L r (fun u => C (Fin.cons true u)) (fun u => D (Fin.cons true u)) := by
-  classical
-  unfold centralDistance
-  rw [Fintype.sum_equiv (headTailEquiv r) _
-    (fun x : Bool × Card r => permDistance (C (Fin.cons x.1 x.2)) (D (Fin.cons x.1 x.2)))
-      (fun x => by
-        change permDistance (C x) (D x) =
-          permDistance (C (Fin.cons (x 0) (Fin.tail x))) (D (Fin.cons (x 0) (Fin.tail x)))
-        rw [Fin.cons_self_tail])]
-  simp only [Fintype.sum_prod_type,Fintype.sum_bool]
-  omega
-
-lemma permDistance_centralBlock (L r : ℕ) (C D : Card r → Equiv.Perm (Card L)) :
-    permDistance (centralBlockPerm L r C) (centralBlockPerm L r D) = centralDistance L r C D := by
-  induction r with
-  | zero =>
-    simp only [centralBlockPerm,centralDistance,Fintype.sum_unique]
-    congr 1
-  | succ r ih =>
-    rw [centralBlockPerm,centralBlockPerm,permDistance_childLift,centralDistance_step,ih,ih]
-
-lemma permDistance_centralPalindrome (L r : ℕ) (C D : Card r → Equiv.Perm (Card L))
-    (ω : BenesCoins (L+r)) :
-    permDistance (centralPalindromePerm L r C ω) (centralPalindromePerm L r D ω) =
-      centralDistance L r C D := by
-  rw [centralPalindromePerm,centralPalindromePerm,permDistance_mul_right,
-    permDistance_mul_left,permDistance_centralBlock]
-
-lemma centralLevelCost_step (t L r : ℕ) (C : Card (r+1) → Equiv.Perm (Card L))
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r+1))
-    (ω : BenesCoins (L+r) × BenesCoins (L+r))
-    (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
-    centralLevelCost t L (r+1) C e ((benesStepEquiv (L+r)).symm (ω,coins)) =
-      let x := e.trans (headTailEquiv (L+r)).toEmbedding
-      let c := PairRouting.colors x coins.1
-      let hc := PairRouting.colors_compatible x coins.1
-      if t = L+r+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x coins.1)
-        (fun b => centralPalindromePerm L r (fun u => C (Fin.cons b u)) (if b then ω.2 else ω.1))
-      else centralLevelCost t L r (fun u => C (Fin.cons false u))
-        (PairRouting.childEmbedding x c hc false) ω.1 +
-        centralLevelCost t L r (fun u => C (Fin.cons true u))
-        (PairRouting.childEmbedding x c hc true) ω.2 := by
-  rw [centralLevelCost]
-  rfl
-
-lemma centralLevelCost_perturb (t L r : ℕ) (C D : Card r → Equiv.Perm (Card L))
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (ω : BenesCoins (L+r)) :
-    centralLevelCost t L r C e ω ≤ centralLevelCost t L r D e ω+centralDistance L r C D := by
-  induction r generalizing ι with
-  | zero => simp only [centralLevelCost,Nat.zero_add,Nat.zero_le]
-  | succ r ih =>
-    rw [←(benesStepEquiv (L+r)).symm_apply_apply ω,centralLevelCost_step,centralLevelCost_step]
-    dsimp only
-    split_ifs with ht
-    · have hh := alternatingCycles_distance_le
-        (PairRouting.switchedEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
-          (benesStepEquiv (L+r) ω).2.1)
-        (fun b => centralPalindromePerm L r (fun u => C (Fin.cons b u))
-          (if b then (benesStepEquiv (L+r) ω).1.2 else (benesStepEquiv (L+r) ω).1.1))
-        (fun b => centralPalindromePerm L r (fun u => D (Fin.cons b u))
-          (if b then (benesStepEquiv (L+r) ω).1.2 else (benesStepEquiv (L+r) ω).1.1))
-      simpa only [Bool.false_eq_true,↓reduceIte,permDistance_centralPalindrome,←centralDistance_step] using hh
-    · rw [centralDistance_step]
-      have h₀ := ih (fun u => C (Fin.cons false u)) (fun u => D (Fin.cons false u))
-        (PairRouting.childEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
-          (PairRouting.colors (e.trans (headTailEquiv (L+r)).toEmbedding) (benesStepEquiv (L+r) ω).2.1)
-          (PairRouting.colors_compatible _ _) false)
-        (benesStepEquiv (L+r) ω).1.1
-      have h₁ := ih (fun u => C (Fin.cons true u)) (fun u => D (Fin.cons true u))
-        (PairRouting.childEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
-          (PairRouting.colors (e.trans (headTailEquiv (L+r)).toEmbedding) (benesStepEquiv (L+r) ω).2.1)
-          (PairRouting.colors_compatible _ _) true)
-        (benesStepEquiv (L+r) ω).1.2
-      omega
-
-lemma centralSlabCost_perturb (l L r : ℕ) (C D : Card r → Equiv.Perm (Card L))
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (ω : BenesCoins (L+r)) :
-    centralSlabCost l L r C e ω ≤ centralSlabCost l L r D e ω+l*centralDistance L r C D := by
-  have hh := Finset.sum_le_sum (fun (i : Fin l) (_ : i ∈ Finset.univ) =>
-    centralLevelCost_perturb (l+i.val+1) L r C D e ω)
-  simpa only [Finset.sum_add_distrib,Finset.sum_const,Finset.card_univ,Fintype.card_fin,
-    nsmul_eq_mul,Nat.cast_id,centralSlabCost] using hh
-
-end Thorp
-
-end ThorpNine.SparseSaving
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.SparseSaving.ParallelFreshPre
+
+namespace OAI
+
+namespace ThorpNine.SparseSaving
+
+namespace Thorp
+
+theorem palindrome_family_mgf (L n d : ℕ) (hL : 0 < L) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) ≤
+      Real.exp (Fintype.card ι * (68*∑ j ∈ Finset.range n,
+        Real.exp (-((L*4^j:ℕ):ℝ)/64))) := by
+  induction n with
+  | zero => simp only [palindromeFamilyCost,Finset.range_zero,Finset.sum_empty,pow_zero,
+      finiteMean_const,mul_zero,Real.exp_zero,le_refl]
+  | succ n ih =>
+    let l := L*4^n
+    have hl : 0 < l := Nat.mul_pos hL (pow_pos (by decide) _)
+    have hs : 2*familyEnd L n ≤ l := familyEnd_two L n
+    have hc : ∀ Y, palindromeFamilyCost L (n+1) d e (Y,X) =
+        palindromeFamilyCost L n d e (Y,X)+palindromeSlabCost l d e (Y,X) := by
+      intro Y
+      exact Finset.sum_range_succ _ _
+    have hm : finiteMean (fun Y => q^(palindromeFamilyCost L (n+1) d e (Y,X))) ≤
+        finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) *
+          Real.exp (Fintype.card ι*(68*Real.exp (-(l:ℝ)/64))) := by
+      by_cases hld : l ≤ d
+      · simp_rw [hc,pow_add]
+        exact palindrome_slab_adapted_mul_le d (familyEnd L n) l (by omega) hl hs e X _
+          (fun _ => pow_nonneg (by linarith) _)
+          (layerAdapted_comp (palindromeFamilyCost_adapted L n d e X) (fun c => q^c))
+          q hq hlog
+      · simp_rw [hc,palindromeSlabCost_gt l d (by omega) e,Nat.add_zero]
+        exact le_mul_of_one_le_right
+          (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
+          (Real.one_le_exp_iff.mpr (by positivity))
+    refine (hm.trans (mul_le_mul_of_nonneg_right ih (Real.exp_nonneg _))).trans_eq ?_
+    rw [←Real.exp_add,Finset.sum_range_succ]
+    congr 1
+    dsimp [l]
+    ring
+
+lemma pow_four_ge (j : ℕ) : j+1 ≤ 4^j := by
+  induction j with
+  | zero => norm_num
+  | succ j ih => rw [pow_succ]; omega
+
+lemma sum_family_decay (L n : ℕ) (hL : 0 < L) :
+    (∑ j ∈ Finset.range n, Real.exp (-((L*4^j:ℕ):ℝ)/64)) ≤
+      Real.exp (-(L:ℝ)/64)/(1-Real.exp (-(1:ℝ)/64)) := by
+  let r := Real.exp (-(1:ℝ)/64)
+  have hr : 0 < r := Real.exp_pos _
+  have hr1 : r < 1 := Real.exp_lt_one_iff.mpr (by norm_num)
+  have hg : (∑ j ∈ Finset.range n, r^j) ≤ 1/(1-r) := by
+    apply (le_div_iff₀ (by linarith : (0:ℝ)<1-r)).mpr
+    have hh := geom_sum_mul r n
+    nlinarith [pow_nonneg (le_of_lt hr) n]
+  calc
+    _ ≤ ∑ j ∈ Finset.range n, Real.exp (-(L:ℝ)/64)*r^j := by
+      apply Finset.sum_le_sum
+      intro j _
+      have hj := pow_four_ge j
+      have hlj : L+j ≤ L*4^j := by nlinarith
+      have hlj' : (L:ℝ)+j ≤ (L*4^j:ℕ) := by exact_mod_cast hlj
+      rw [←Real.exp_nat_mul,←Real.exp_add]
+      exact Real.exp_le_exp.mpr (by linarith)
+    _ = Real.exp (-(L:ℝ)/64)*(∑ j ∈ Finset.range n, r^j) := by rw [Finset.mul_sum]
+    _ ≤ Real.exp (-(L:ℝ)/64)*(1/(1-r)) :=
+      mul_le_mul_of_nonneg_left hg (Real.exp_nonneg _)
+    _ = _ := by ring
+
+
+lemma palindromeLevelCost_step_add (t d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (d+1)) (ω : BenesCoins (d+1)) :
+    palindromeLevelCost t (d+1) e ω =
+      let σ := benesStepEquiv d ω
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x σ.2.1
+      let hc := PairRouting.colors_compatible x σ.2.1
+      (if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
+          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) else 0) +
+        palindromeLevelCost t d (PairRouting.childEmbedding x c hc false) σ.1.1 +
+        palindromeLevelCost t d (PairRouting.childEmbedding x c hc true) σ.1.2 := by
+  classical
+  rw [palindromeLevelCost]
+  dsimp only
+  split_ifs with h
+  · simp only [palindromeLevelCost_gt t d (by omega),Nat.add_zero]
+  · simp only [Nat.zero_add]
+
+lemma palindromeLowCost_sum (H d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) :
+    palindromeLowCost H d e ω = ∑ i ∈ Finset.range H, palindromeLevelCost (i+1) d e ω := by
+  classical
+  induction d generalizing ι with
+  | zero => simp only [palindromeLowCost,palindromeLevelCost,Finset.sum_const_zero]
+  | succ d ih =>
+    rw [palindromeLowCost]
+    simp only [palindromeLevelCost_step_add,Finset.sum_add_distrib]
+    rw [←ih,←ih]
+    congr 2
+    simp only [Nat.add_right_cancel_iff,Finset.sum_ite_eq',Finset.mem_range,Nat.succ_le_iff]
+
+lemma palindromeLowCost_slab (L d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) :
+    palindromeLowCost L d e ω + palindromeSlabCost L d e ω = palindromeLowCost (2*L) d e ω := by
+  rw [palindromeLowCost_sum,palindromeLowCost_sum,show 2*L=L+L by omega,Finset.sum_range_add]
+  congr 1
+  exact Fin.sum_univ_eq_sum_range (fun i => palindromeLevelCost (L+i+1) d e ω) L
+
+lemma palindromeCost_family (L n d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) :
+    palindromeLowCost L d e ω + palindromeFamilyCost L n d e ω +
+      palindromeFamilyCost (2*L) n d e ω = palindromeLowCost (L*4^n) d e ω := by
+  induction n with
+  | zero => simp only [palindromeFamilyCost,Finset.range_zero,Finset.sum_empty,Nat.add_zero,
+      pow_zero,mul_one]
+  | succ n ih =>
+    have he : (2*L)*4^n = 2*(L*4^n) := by ring
+    have he' : L*4^(n+1) = 2*(2*(L*4^n)) := by rw [pow_succ]; ring
+    simp only [palindromeFamilyCost,Finset.sum_range_succ] at ih ⊢
+    rw [he,he',←palindromeLowCost_slab,←palindromeLowCost_slab,←ih]
+    omega
+
+
+lemma finiteMean_mul_sq_le {Ω : Type*} [Fintype Ω] (f g : Ω → ℝ) :
+    (finiteMean (fun ω => f ω*g ω))^2 ≤ finiteMean (fun ω => (f ω)^2)*finiteMean (fun ω => (g ω)^2) := by
+  have h := Finset.sum_mul_sq_le_sq_mul_sq Finset.univ f g
+  simp only [finiteMean,div_pow,div_mul_div_comm,←pow_two]
+  exact div_le_div_of_nonneg_right h (sq_nonneg _)
+
+lemma finiteMean_mul_exp_le {Ω : Type*} [Fintype Ω] (f g : Ω → ℝ) (A B : ℝ)
+    (hf : finiteMean (fun ω => (f ω)^2) ≤ Real.exp A)
+    (hg : finiteMean (fun ω => (g ω)^2) ≤ Real.exp B) :
+    finiteMean (fun ω => f ω*g ω) ≤ Real.exp ((A+B)/2) := by
+  have hc := finiteMean_mul_sq_le f g
+  have hn : 0 ≤ finiteMean (fun ω => (g ω)^2) := finiteMean_nonneg (fun _ => sq_nonneg _)
+  have hu := mul_le_mul hf hg hn (Real.exp_nonneg _)
+  have he : Real.exp A*Real.exp B = (Real.exp ((A+B)/2))^2 := by
+    rw [←Real.exp_add,←Real.exp_nat_mul]
+    congr 1
+    push_cast
+    ring
+  rw [he] at hu
+  have hh := hc.trans hu
+  have hep := Real.exp_pos ((A+B)/2)
+  nlinarith
+
+
+noncomputable def heightDecay (H : ℕ) : ℝ :=
+  68*Real.exp (-(H:ℝ)/64)/(1-Real.exp (-(1:ℝ)/64))
+
+lemma heightDecay_nonneg (H : ℕ) : 0 ≤ heightDecay H := by
+  have h : Real.exp (-(1:ℝ)/64) < 1 := Real.exp_lt_one_iff.mpr (by norm_num)
+  apply div_nonneg (by positivity) (by linarith)
+
+lemma heightDecay_antitone : Antitone heightDecay := by
+  intro L H h
+  have hden : 0 ≤ 1-Real.exp (-(1:ℝ)/64) := by
+    have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
+    linarith
+  apply div_le_div_of_nonneg_right _ hden
+  apply mul_le_mul_of_nonneg_left _ (by norm_num)
+  apply Real.exp_le_exp.mpr
+  have h' : (L:ℝ) ≤ H := by exact_mod_cast h
+  linarith
+
+lemma palindrome_family_mgf_decay (L n d : ℕ) (hL : 0 < L) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) ≤
+      Real.exp (Fintype.card ι*heightDecay L) := by
+  apply (palindrome_family_mgf L n d hL e X q hq hlog).trans
+  apply Real.exp_le_exp.mpr
+  apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
+  have h := mul_le_mul_of_nonneg_left (sum_family_decay L n hL) (by norm_num : (0:ℝ)≤68)
+  simpa only [heightDecay,mul_div_assoc] using h
+
+theorem palindrome_high_mgf (H n d : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/500) :
+    finiteMean (fun Y => q^(palindromeFamilyCost H n d e (Y,X)+
+      palindromeFamilyCost (2*H) n d e (Y,X))) ≤ Real.exp (Fintype.card ι*heightDecay H) := by
+  have hq2 := one_le_pow₀ (n:=2) hq
+  have hlog2 : Real.log (q^2) ≤ 1/250 := by rw [Real.log_pow]; norm_num; linarith
+  have h₀ := palindrome_family_mgf_decay H n d hH e X (q^2) hq2 hlog2
+  have h₁ := palindrome_family_mgf_decay (2*H) n d (by omega) e X (q^2) hq2 hlog2
+  have h₁' := h₁.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left
+    (heightDecay_antitone (show H ≤ 2*H by omega)) (Nat.cast_nonneg _)))
+  have hh := finiteMean_mul_exp_le (fun Y => q^(palindromeFamilyCost H n d e (Y,X)))
+    (fun Y => q^(palindromeFamilyCost (2*H) n d e (Y,X)))
+    (Fintype.card ι*heightDecay H) (Fintype.card ι*heightDecay H)
+    (by simpa only [pow_right_comm] using h₀) (by simpa only [pow_right_comm] using h₁')
+  simpa only [←pow_add,add_self_div_two] using hh
+
+lemma palindrome_high_full_mgf (H n d : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/500) :
+    finiteMean (fun ω : BenesCoins d => q^(palindromeFamilyCost H n d e ω+
+      palindromeFamilyCost (2*H) n d e ω)) ≤ Real.exp (Fintype.card ι*heightDecay H) := by
+  rw [finiteMean_prod,finiteMean_comm]
+  calc
+    _ ≤ finiteMean (fun _X : SwitchIndex d → Bool => Real.exp (Fintype.card ι*heightDecay H)) := by
+      apply finiteMean_mono
+      intro X
+      exact palindrome_high_mgf H n d hH e X q hq hlog
+    _ = _ := finiteMean_const _
+
+theorem palindrome_cost_mgf_cutoff (r H : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (H+r)) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/1000)
+    (hr : (2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*(q^2)^H) ≤ 1/2) :
+    finiteMean (fun ω : BenesCoins (H+r) => q^(palindromeCost (H+r) e ω)) ≤
+      Real.exp (((2:ℝ)^r*(2*((2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*(q^2)^H))^2)+
+        Fintype.card ι*heightDecay H)/2) := by
+  let d := H+r
+  have hd : d ≤ H*4^d := by
+    have hh := pow_four_ge d
+    nlinarith
+  have hcost (ω : BenesCoins d) : palindromeCost d e ω = palindromeLowCost H d e ω+
+      (palindromeFamilyCost H d d e ω+palindromeFamilyCost (2*H) d d e ω) := by
+    have hh := palindromeCost_family H d d e ω
+    rw [palindromeLowCost_eq _ _ hd] at hh
+    simpa only [Nat.add_assoc] using hh.symm
+  have hq2 := one_le_pow₀ (n:=2) hq
+  have hlog2 : Real.log (q^2) ≤ 1/500 := by rw [Real.log_pow]; norm_num; linarith
+  have hlow := palindrome_low_mgf r H e (q^2) hq2 hr
+  have hhigh := palindrome_high_full_mgf H d d hH e (q^2) hq2 hlog2
+  have hh := finiteMean_mul_exp_le (fun ω : BenesCoins d => q^(palindromeLowCost H d e ω))
+    (fun ω : BenesCoins d => q^(palindromeFamilyCost H d d e ω+palindromeFamilyCost (2*H) d d e ω))
+    _ _ (by simpa only [pow_right_comm] using hlow) (by simpa only [pow_right_comm] using hhigh)
+  change finiteMean (fun ω : BenesCoins d => q^(palindromeCost d e ω)) ≤ _
+  simpa only [hcost,pow_add,pow_right_comm] using hh
+
+
+lemma cutoffMass_le (H d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (ω : BenesCoins d) :
+    cutoffMass H d e ω ≤ Fintype.card ι := by
+  classical
+  induction d generalizing ι with
+  | zero => exact Nat.zero_le _
+  | succ d ih =>
+    rw [cutoffMass]
+    split_ifs
+    · exact crowdedMass_le _
+    · exact (Nat.add_le_add (ih _ _) (ih _ _)).trans_eq (PairRouting.color_card_add _)
+
+lemma palindromeLowCost_le (H d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) : palindromeLowCost H d e ω ≤ H*Fintype.card ι :=
+  (palindromeLowCost_le_cutoff H d e ω).trans (Nat.mul_le_mul_left H (cutoffMass_le H d e ω))
+
+theorem palindrome_cost_mgf_coarse (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/1000) :
+    finiteMean (fun ω : BenesCoins d => q^(palindromeCost d e ω)) ≤
+      Real.exp (Fintype.card ι*(1+heightDecay 1)) := by
+  have hd : d ≤ 1*4^d := by have hh := pow_four_ge d; omega
+  have hcost (ω : BenesCoins d) : palindromeCost d e ω = palindromeLowCost 1 d e ω+
+      (palindromeFamilyCost 1 d d e ω+palindromeFamilyCost 2 d d e ω) := by
+    have hh := palindromeCost_family 1 d d e ω
+    rw [palindromeLowCost_eq _ _ hd] at hh
+    simpa only [Nat.add_assoc] using hh.symm
+  have hq2 := one_le_pow₀ (n:=2) hq
+  have hlog2 : Real.log (q^2) ≤ 1/500 := by rw [Real.log_pow]; norm_num; linarith
+  have hlow : finiteMean (fun ω : BenesCoins d => (q^2)^(palindromeLowCost 1 d e ω)) ≤
+      Real.exp (Fintype.card ι) := by
+    calc
+      _ ≤ finiteMean (fun _ω : BenesCoins d => (q^2)^(Fintype.card ι)) := by
+        apply finiteMean_mono
+        intro ω
+        apply pow_le_pow_right₀ hq2
+        simpa only [one_mul] using palindromeLowCost_le 1 d e ω
+      _ = (q^2)^(Fintype.card ι) := finiteMean_const _
+      _ = Real.exp (Fintype.card ι * Real.log (q^2)) := by
+        rw [Real.exp_nat_mul,Real.exp_log (by positivity : 0<q^2)]
+      _ ≤ _ := Real.exp_le_exp.mpr (by
+        have hh := mul_le_mul_of_nonneg_left hlog2 (Nat.cast_nonneg (Fintype.card ι) : (0:ℝ)≤_)
+        nlinarith [Nat.cast_nonneg (α:=ℝ) (Fintype.card ι)])
+  have hhigh := palindrome_high_full_mgf 1 d d (by decide) e (q^2) hq2 hlog2
+  have hh := finiteMean_mul_exp_le (fun ω : BenesCoins d => q^(palindromeLowCost 1 d e ω))
+    (fun ω : BenesCoins d => q^(palindromeFamilyCost 1 d d e ω+palindromeFamilyCost 2 d d e ω))
+    (Fintype.card ι) (Fintype.card ι*heightDecay 1)
+    (by simpa only [pow_right_comm] using hlow) (by simpa only [pow_right_comm] using hhigh)
+  simp only [←pow_add,←hcost] at hh
+  apply hh.trans (Real.exp_le_exp.mpr _)
+  have hk : (0:ℝ) ≤ Fintype.card ι := Nat.cast_nonneg _
+  have hD := heightDecay_nonneg 1
+  nlinarith
+
+noncomputable def momentBase : ℝ := (2:ℝ)^(1/1000:ℝ)
+
+lemma momentBase_one_le : 1 ≤ momentBase := by
+  exact Real.one_le_rpow (by norm_num) (by norm_num)
+
+lemma momentBase_log : Real.log momentBase ≤ 1/1000 := by
+  rw [momentBase,Real.log_rpow (by norm_num)]
+  have hh := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
+  norm_num at hh ⊢
+  linarith
+
+lemma momentBase_pow (c : ℕ) : momentBase^c = (2:ℝ)^((c:ℝ)*(1/1000:ℝ)) := by
+  rw [mul_comm,Real.rpow_mul_natCast (by norm_num)]
+  rfl
+
+lemma palindromeRowMoment_le_cost (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) :
+    palindromeRowMoment d e (1/1000) ≤
+      finiteMean (fun ω : BenesCoins d => momentBase^(palindromeCost d e ω)) := by
+  have hn : (0:ℝ) < ((2^d:ℕ):ℝ)^(Fintype.card ι) := by positivity
+  have hn0 : (0:ℝ) ≤ ((2^d).descFactorial (Fintype.card ι):ℝ) /
+      ((2^d:ℕ):ℝ)^(Fintype.card ι) := by positivity
+  have hn1 : ((2^d).descFactorial (Fintype.card ι):ℝ) /
+      ((2^d:ℕ):ℝ)^(Fintype.card ι) ≤ 1 := by
+    apply (div_le_one hn).mpr
+    exact_mod_cast Nat.descFactorial_le_pow (2^d) (Fintype.card ι)
+  have hpow := Real.rpow_le_one hn0 hn1 (by norm_num : (0:ℝ)≤1/1000)
+  have hm := palindrome_row_density_moment d e (1/1000) (by norm_num)
+  simp_rw [←momentBase_pow] at hm
+  exact hm.trans (mul_le_of_le_one_left (finiteMean_nonneg (fun _ =>
+    pow_nonneg (le_trans (by norm_num) momentBase_one_le) _)) hpow)
+
+theorem palindrome_row_moment_coarse (d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) :
+    palindromeRowMoment d e (1/1000) ≤ Real.exp (Fintype.card ι*(1+heightDecay 1)) :=
+  (palindromeRowMoment_le_cost d e).trans
+    (palindrome_cost_mgf_coarse d e momentBase momentBase_one_le momentBase_log)
+
+
+noncomputable def densityCutoff (p : ℝ) : ℕ := ⌊-Real.log p/(4*Real.log 2)⌋₊
+
+lemma log_small_density {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
+    Real.log p ≤ -8*Real.log 2 := by
+  have he : Real.log (1/256:ℝ) = -8*Real.log 2 := by
+    rw [show (1/256:ℝ) = 1/2^8 by norm_num,
+      Real.log_div (by norm_num) (by positivity),Real.log_one,Real.log_pow]
+    ring
+  exact (Real.log_le_log hp hs).trans_eq he
+
+lemma densityCutoff_bounds {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
+    0 < densityCutoff p ∧
+    (densityCutoff p:ℝ)*(4*Real.log 2) ≤ -Real.log p ∧
+    -Real.log p < ((densityCutoff p:ℝ)+1)*(4*Real.log 2) := by
+  have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
+  have hd : 0 < 4*Real.log 2 := by positivity
+  have hh := log_small_density hp hs
+  have hx : 1 ≤ -Real.log p/(4*Real.log 2) := by
+    apply (le_div_iff₀ hd).mpr
+    nlinarith
+  refine ⟨Nat.floor_pos.mpr hx,?_,?_⟩
+  · exact (le_div_iff₀ hd).mp (Nat.floor_le (by linarith : 0 ≤ -Real.log p/(4*Real.log 2)))
+  · exact (div_lt_iff₀ hd).mp (Nat.lt_floor_add_one (-Real.log p/(4*Real.log 2)))
+
+lemma densityCutoff_decay {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
+    Real.exp (-(densityCutoff p:ℝ)/64) ≤ Real.exp (1/64) * p^(1/512:ℝ) := by
+  have hh := (densityCutoff_bounds hp hs).2.2
+  have hb : Real.log 2 ≤ 2 := by
+    have := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
+    linarith
+  have hH : 0 ≤ (densityCutoff p:ℝ)+1 := by positivity
+  have hu := mul_le_mul_of_nonneg_left hb hH
+  rw [Real.rpow_def_of_pos hp,←Real.exp_add]
+  apply Real.exp_le_exp.mpr
+  nlinarith
+
+lemma densityCutoff_power {p q : ℝ} (hp : 0 < p) (hs : p ≤ 1/256)
+    (hq : 0 < q) (hlog : Real.log q ≤ Real.log 2/4) (n : ℕ) (hn : n ≤ 4)
+    (hq1 : 1 ≤ q) :
+    p*(2:ℝ)^(densityCutoff p)*q^(n*densityCutoff p) ≤ p^(1/2:ℝ) := by
+  have hcut := (densityCutoff_bounds hp hs).2.1
+  have hH : 0 ≤ (densityCutoff p:ℝ) := Nat.cast_nonneg _
+  have hn' : (n:ℝ) ≤ 4 := by exact_mod_cast hn
+  have hln := Real.log_nonneg hq1
+  have hl := mul_le_mul_of_nonneg_right hn' hln
+  have hnl : (n:ℝ)*Real.log q ≤ Real.log 2 := by linarith
+  have hm := mul_le_mul_of_nonneg_left hnl hH
+  have heq : p*(2:ℝ)^(densityCutoff p)*q^(n*densityCutoff p) =
+      Real.exp (Real.log p+(densityCutoff p:ℝ)*Real.log 2+
+        (n*densityCutoff p:ℕ)*Real.log q) := by
+    rw [Real.exp_add,Real.exp_add,Real.exp_nat_mul,Real.exp_nat_mul,
+      Real.exp_log hp,Real.exp_log hq,Real.exp_log (by norm_num : (0:ℝ)<2)]
+  rw [heq,Real.rpow_def_of_pos hp]
+  apply Real.exp_le_exp.mpr
+  push_cast
+  nlinarith
+
+lemma densityCutoff_sparse {p q : ℝ} (hp : 0 < p) (hs : p ≤ 1/256)
+    (hq : 0 < q) (hlog : Real.log q ≤ Real.log 2/4) (hq1 : 1 ≤ q) :
+    (2:ℝ)^(densityCutoff p)*(p*(q^2)^(densityCutoff p)) ≤ 1/2 := by
+  have hh := densityCutoff_power hp hs hq hlog 2 (by decide) hq1
+  rw [←pow_mul] at ⊢
+  have hle : p^(1/2:ℝ) ≤ 1/2 := by
+    have hlogp := log_small_density hp hs
+    have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
+    rw [Real.rpow_def_of_pos hp]
+    calc
+      _ ≤ Real.exp (-Real.log 2) := by apply Real.exp_le_exp.mpr; nlinarith
+      _ = 1/2 := by rw [Real.exp_neg,Real.exp_log (by norm_num : (0:ℝ)<2)]; norm_num
+  nlinarith [hh.trans hle]
+
+
+lemma momentBase_log_exact : Real.log momentBase = Real.log 2/1000 := by
+  rw [momentBase,Real.log_rpow (by norm_num)]
+  ring
+
+lemma densityCutoff_le_height (d k : ℕ) (hk : 0 < k) :
+    densityCutoff ((k:ℝ)/(2:ℝ)^d) ≤ d := by
+  have hk' : (1:ℝ) ≤ k := by exact_mod_cast hk
+  have hp : 0 < (k:ℝ)/(2:ℝ)^d := by positivity
+  have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
+  have hlogk := Real.log_nonneg hk'
+  by_cases hx : 0 ≤ -Real.log ((k:ℝ)/(2:ℝ)^d)/(4*Real.log 2)
+  · have hh := (le_div_iff₀ (by positivity : 0 < 4*Real.log 2)).mp (Nat.floor_le hx)
+    have hd : (0:ℝ) ≤ d := Nat.cast_nonneg _
+    have hH : (0:ℝ) ≤ densityCutoff ((k:ℝ)/(2:ℝ)^d) := Nat.cast_nonneg _
+    change (densityCutoff ((k:ℝ)/(2:ℝ)^d):ℝ)*(4*Real.log 2) ≤ -Real.log ((k:ℝ)/(2:ℝ)^d) at hh
+    rw [Real.log_div (by positivity) (by positivity),Real.log_pow] at hh
+    have hle : (densityCutoff ((k:ℝ)/(2:ℝ)^d):ℝ) ≤ d := by nlinarith
+    exact_mod_cast hle
+  · have hz := Nat.floor_of_nonpos (le_of_not_ge hx)
+    simp only [densityCutoff,hz,Nat.zero_le]
+
+noncomputable def densityTailConstant : ℝ :=
+  68*Real.exp (1/64)/(1-Real.exp (-(1:ℝ)/64))
+
+lemma densityTailConstant_nonneg : 0 ≤ densityTailConstant := by
+  have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
+  apply div_nonneg (by positivity) (by linarith)
+
+lemma heightDecay_cutoff {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
+    heightDecay (densityCutoff p) ≤ densityTailConstant*p^(1/512:ℝ) := by
+  have hden : 0 ≤ 1-Real.exp (-(1:ℝ)/64) := by
+    have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
+    linarith
+  have hh := div_le_div_of_nonneg_right
+    (mul_le_mul_of_nonneg_left (densityCutoff_decay hp hs) (by norm_num : (0:ℝ)≤68)) hden
+  simpa only [heightDecay,densityTailConstant,mul_assoc,div_mul_eq_mul_div] using hh
+
+lemma sparse_low_algebra (r H k : ℕ) (q : ℝ) :
+    (2:ℝ)^r*(2*((2:ℝ)^H*(((k:ℝ)/(2:ℝ)^(H+r))*(q^2)^H))^2) =
+    2*k*(((k:ℝ)/(2:ℝ)^(H+r))*(2:ℝ)^H*q^(4*H)) := by
+  rw [pow_add,←pow_mul,show 4*H = (2*H)*2 by omega,pow_mul]
+  have hH : (2:ℝ)^H ≠ 0 := by positivity
+  have hr : (2:ℝ)^r ≠ 0 := by positivity
+  field_simp
+  ring
+
+lemma palindrome_row_moment_small (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι)
+    (hs : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1/256) :
+    palindromeRowMoment d e (1/1000) ≤
+      Real.exp (Fintype.card ι*(1+densityTailConstant)*
+        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ)) := by
+  generalize hpdef : ((Fintype.card ι:ℝ)/(2:ℝ)^d) = p at hs ⊢
+  have hp : 0 < p := by rw [←hpdef]; positivity
+  have hle := densityCutoff_le_height d (Fintype.card ι) hk
+  rw [hpdef] at hle
+  obtain ⟨r,hr⟩ := Nat.exists_eq_add_of_le hle
+  subst d
+  have hH := (densityCutoff_bounds hp hs).1
+  have hq := momentBase_one_le
+  have hqp : 0 < momentBase := by linarith
+  have hlog : Real.log momentBase ≤ Real.log 2/4 := by
+    rw [momentBase_log_exact]
+    have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
+    linarith
+  have hsp := densityCutoff_sparse hp hs hqp hlog hq
+  rw [←hpdef] at hsp
+  have hh := palindrome_cost_mgf_cutoff r (densityCutoff p) hH e momentBase hq momentBase_log
+    (by simpa only [hpdef] using hsp)
+  apply (palindromeRowMoment_le_cost _ e).trans (hh.trans (Real.exp_le_exp.mpr _))
+  rw [sparse_low_algebra,hpdef]
+  have hlow := densityCutoff_power hp hs hqp hlog 4 (by decide) hq
+  have hp1 : p ≤ 1 := by linarith
+  have he := Real.rpow_le_rpow_of_exponent_ge hp hp1 (by norm_num : (1:ℝ)/512 ≤ 1/2)
+  have hlow' := mul_le_mul_of_nonneg_left (hlow.trans he)
+    (by positivity : (0:ℝ)≤2*Fintype.card ι)
+  have hhigh := mul_le_mul_of_nonneg_left (heightDecay_cutoff hp hs)
+    (Nat.cast_nonneg (Fintype.card ι): (0:ℝ)≤_)
+  have hnon : 0 ≤ Fintype.card ι*densityTailConstant*p^(1/512:ℝ) :=
+    mul_nonneg (mul_nonneg (Nat.cast_nonneg _) densityTailConstant_nonneg)
+      (Real.rpow_nonneg (le_of_lt hp) _)
+  nlinarith
+
+noncomputable def densityConstant : ℝ :=
+  256*(1+heightDecay 1)+1+densityTailConstant
+
+lemma densityConstant_pos : 0 < densityConstant := by
+  dsimp [densityConstant]
+  have hh := heightDecay_nonneg 1
+  have ht := densityTailConstant_nonneg
+  linarith
+
+theorem palindrome_row_moment_bound (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι) :
+    palindromeRowMoment d e (1/1000) ≤ Real.exp (densityConstant*Fintype.card ι*
+        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ)) := by
+  have hp : 0 < (Fintype.card ι:ℝ)/(2:ℝ)^d := by positivity
+  have hp1 : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1 := by
+    apply (div_le_one (by positivity)).mpr
+    have h := Fintype.card_le_of_injective e e.injective
+    rw [card_positions] at h
+    exact_mod_cast h
+  have hk0 : (0:ℝ) ≤ Fintype.card ι := Nat.cast_nonneg _
+  have hpow := Real.rpow_nonneg (le_of_lt hp) (1/512:ℝ)
+  by_cases hs : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1/256
+  · apply (palindrome_row_moment_small d e hk hs).trans (Real.exp_le_exp.mpr _)
+    have hD : 1+densityTailConstant ≤ densityConstant := by
+      dsimp [densityConstant]
+      have := heightDecay_nonneg 1
+      linarith
+    have hh := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hD hk0) hpow
+    nlinarith
+  · have hlow : (1:ℝ)/256 ≤ ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ) := by
+      have he := Real.rpow_le_rpow_of_exponent_ge hp hp1 (by norm_num : (1:ℝ)/512 ≤ 1)
+      rw [Real.rpow_one] at he
+      exact (le_of_not_ge hs).trans he
+    have hD : 256*(1+heightDecay 1) ≤ densityConstant := by
+      dsimp [densityConstant]
+      have := densityTailConstant_nonneg
+      linarith
+    apply (palindrome_row_moment_coarse d e).trans (Real.exp_le_exp.mpr _)
+    have hD0 : 0 ≤ 1+heightDecay 1 := by have := heightDecay_nonneg 1; linarith
+    have ha := mul_le_mul_of_nonneg_left hlow (mul_nonneg (by norm_num : (0:ℝ)≤256) hD0)
+    have hb := mul_le_mul_of_nonneg_right hD hpow
+    have hc := mul_le_mul_of_nonneg_right (ha.trans hb) hk0
+    nlinarith
+
+lemma palindromeRowMoment_nonneg (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (a : ℝ) : 0 ≤ palindromeRowMoment d e a := by
+  apply finiteMean_nonneg
+  intro f
+  exact Real.rpow_nonneg (mul_nonneg (Nat.cast_nonneg _)
+    (PairRouting.tupleProbability_nonneg _ _ _)) _
+
+theorem palindrome_log_row_moment (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι) :
+    Real.log (palindromeRowMoment d e (1/1000)) ≤ densityConstant*Fintype.card ι*
+        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ) := by
+  by_cases hz : palindromeRowMoment d e (1/1000) = 0
+  · rw [hz,Real.log_zero]
+    exact mul_nonneg (mul_nonneg (le_of_lt densityConstant_pos) (Nat.cast_nonneg _))
+      (Real.rpow_nonneg (by positivity) _)
+  · exact (Real.log_le_iff_le_exp (lt_of_le_of_ne (palindromeRowMoment_nonneg _ _ _) (Ne.symm hz))).mpr
+      (palindrome_row_moment_bound d e hk)
+
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+lemma orbitSet_eq_of_agree (p q : Equiv.Perm α) (x : α)
+    (h : ∀ z ∈ orbitSet p x, p z = q z) : orbitSet p x = orbitSet q x := by
+  have hp (n : ℕ) : (p^n) x ∈ orbitSet p x := by
+    apply (mem_orbitSet _ _ _).mpr
+    exact Equiv.Perm.sameCycle_pow_right.mpr (Equiv.Perm.SameCycle.refl _ _)
+  have he (n : ℕ) : (p^n) x = (q^n) x := by
+    induction n with
+    | zero => rfl
+    | succ n ih =>
+      rw [pow_succ',pow_succ',Equiv.Perm.mul_apply,Equiv.Perm.mul_apply,←ih]
+      exact h _ (hp n)
+  ext y
+  simp only [mem_orbitSet]
+  constructor
+  · intro hy
+    obtain ⟨n,hn⟩ := hy.exists_nat_pow_eq
+    rw [←hn,he]
+    exact Equiv.Perm.sameCycle_pow_right.mpr (Equiv.Perm.SameCycle.refl _ _)
+  · intro hy
+    obtain ⟨n,hn⟩ := hy.exists_nat_pow_eq
+    rw [←hn,←he]
+    exact Equiv.Perm.sameCycle_pow_right.mpr (Equiv.Perm.SameCycle.refl _ _)
+
+noncomputable def survivingCycle (p q : Equiv.Perm α) (S B : Finset α)
+    (h : ∀ x ∉ B, p x = q x) (C : RoutingNetwork.SelectedCycle p S)
+    (hC : ∀ x ∈ C.val, x ∉ B) : RoutingNetwork.SelectedCycle q S := by
+  refine ⟨C.val,?_,C.property.2⟩
+  obtain ⟨x,hx⟩ := C.property.1
+  refine ⟨x,hx.trans (orbitSet_eq_of_agree p q x ?_)⟩
+  intro z hz
+  exact h z (hC z (hx ▸ hz))
+
+noncomputable def perturbCycleMap (p q : Equiv.Perm α) (S B : Finset α)
+    (h : ∀ x ∉ B, p x = q x) (C : RoutingNetwork.SelectedCycle p S) :
+    RoutingNetwork.SelectedCycle q S ⊕ {x // x ∈ B} := by
+  classical
+  by_cases hb : ∃ x ∈ C.val, x ∈ B
+  · exact Sum.inr ⟨hb.choose,hb.choose_spec.2⟩
+  · exact Sum.inl (survivingCycle p q S B h C (by simpa only [not_exists,not_and] using hb))
+
+lemma perturbCycleMap_injective (p q : Equiv.Perm α) (S B : Finset α)
+    (h : ∀ x ∉ B, p x = q x) : Function.Injective (perturbCycleMap p q S B h) := by
+  classical
+  intro C D he
+  by_cases hC : ∃ x ∈ C.val, x ∈ B <;>
+    by_cases hD : ∃ x ∈ D.val, x ∈ B
+  · simp only [perturbCycleMap,dite_eq_left hC,dite_eq_left hD,Sum.inr.injEq] at he
+    have hx : hC.choose = hD.choose := congrArg Subtype.val he
+    exact RoutingNetwork.SelectedCycle.eq_of_mem C D hC.choose_spec.1
+      (hx.symm ▸ hD.choose_spec.1)
+  · simp only [perturbCycleMap,dite_eq_left hC,dite_eq_right hD] at he
+    cases he
+  · simp only [perturbCycleMap,dite_eq_right hC,dite_eq_left hD] at he
+    cases he
+  · simp only [perturbCycleMap,dite_eq_right hC,dite_eq_right hD,Sum.inl.injEq] at he
+    have hval : C.val = D.val := congrArg (fun Z : RoutingNetwork.SelectedCycle q S => Z.val) he
+    exact Subtype.ext hval
+
+theorem selectedCycle_perturb_le (p q : Equiv.Perm α) (S B : Finset α)
+    (h : ∀ x ∉ B, p x = q x) :
+    Fintype.card (RoutingNetwork.SelectedCycle p S) ≤
+      Fintype.card (RoutingNetwork.SelectedCycle q S)+B.card := by
+  have hh := Fintype.card_le_of_injective _ (perturbCycleMap_injective p q S B h)
+  simpa only [Fintype.card_sum,Fintype.card_coe] using hh
+
+
+noncomputable def permDistance (p q : Equiv.Perm α) : ℕ :=
+  (Finset.univ.filter (fun x => p x ≠ q x)).card
+
+lemma permDistance_sum (p q : Equiv.Perm α) :
+    permDistance p q = ∑ x : α, if p x = q x then 0 else 1 := by
+  classical
+  simp [permDistance,Finset.card_filter]
+
+lemma permDistance_refl (p : Equiv.Perm α) : permDistance p p = 0 := by
+  simp [permDistance]
+
+lemma permDistance_le (p q : Equiv.Perm α) : permDistance p q ≤ Fintype.card α :=
+  (Finset.card_filter_le _ _).trans_eq (Finset.card_univ)
+
+lemma permDistance_symm (p q : Equiv.Perm α) : permDistance p q = permDistance q p := by
+  simp only [permDistance,ne_comm]
+
+lemma permDistance_triangle (p q r : Equiv.Perm α) :
+    permDistance p r ≤ permDistance p q+permDistance q r := by
+  classical
+  simp only [permDistance_sum,←Finset.sum_add_distrib]
+  apply Finset.sum_le_sum
+  intro x _
+  by_cases hpq : p x = q x <;> by_cases hqr : q x = r x <;>
+    by_cases hpr : p x = r x <;> simp only [hpq,hqr,hpr,↓reduceIte] <;>
+    omega
+
+lemma permDistance_mul_left (p q g : Equiv.Perm α) :
+    permDistance (g*p) (g*q) = permDistance p q := by
+  classical
+  unfold permDistance
+  congr 1
+  ext x
+  simp only [Finset.mem_filter,Finset.mem_univ,true_and]
+  exact not_congr g.injective.eq_iff
+
+lemma permDistance_mul_right (p q g : Equiv.Perm α) :
+    permDistance (p*g) (q*g) = permDistance p q := by
+  classical
+  simp only [permDistance_sum,Equiv.Perm.mul_apply]
+  exact Fintype.sum_equiv g _ _ (fun _ => rfl)
+
+lemma permDistance_inv (p q : Equiv.Perm α) : permDistance p⁻¹ q⁻¹ = permDistance p q := by
+  have hh := permDistance_mul_left p⁻¹ q⁻¹ q
+  have hhh := permDistance_mul_right (q*p⁻¹) 1 p
+  simp only [mul_inv_cancel,one_mul,inv_mul_cancel_right] at hh hhh
+  rw [←hhh] at hh
+  rw [←hh,permDistance_symm]
+
+lemma permDistance_mul_le (p q g h : Equiv.Perm α) :
+    permDistance (p*g) (q*h) ≤ permDistance p q+permDistance g h := by
+  have hh := permDistance_triangle (p*g) (q*g) (q*h)
+  simpa only [permDistance_mul_left,permDistance_mul_right] using hh
+
+lemma selectedCycle_distance_le (p q : Equiv.Perm α) (S : Finset α) :
+    Fintype.card (RoutingNetwork.SelectedCycle p S) ≤
+      Fintype.card (RoutingNetwork.SelectedCycle q S)+permDistance p q := by
+  apply selectedCycle_perturb_le
+  intro x hx
+  simpa only [Finset.mem_filter,Finset.mem_univ,true_and,not_not] using hx
+
+lemma permDistance_alternating (p q : Equiv.Perm α) :
+    permDistance (CycleColoring.alternatingPerm p) (CycleColoring.alternatingPerm q) = permDistance p q := by
+  classical
+  simp only [permDistance_sum,Fintype.sum_prod_type,Fintype.sum_bool]
+  simp [CycleColoring.alternatingPerm]
+
+lemma alternatingCycles_distance_le {ι : Type*} [Fintype ι]
+    (e : ι ↪ Bool × α) (p q : Bool → Equiv.Perm α) :
+    PairRouting.alternatingCycles e p ≤ PairRouting.alternatingCycles e q+
+      (permDistance (p false) (q false)+permDistance (p true) (q true)) := by
+  have hh := selectedCycle_distance_le
+    (CycleColoring.alternatingPerm (PairRouting.alternatingProjection p))
+    (CycleColoring.alternatingPerm (PairRouting.alternatingProjection q)) (PairRouting.labelSet e)
+  rw [permDistance_alternating] at hh
+  apply hh.trans
+  apply Nat.add_le_add_left
+  dsimp only [PairRouting.alternatingProjection]
+  exact (permDistance_mul_le _ _ _ _).trans_eq (by
+    change permDistance (p false)⁻¹ (q false)⁻¹ + _ = _
+    rw [permDistance_inv])
+
+end Thorp
+namespace Thorp
+
+def centralBlockPerm (L : ℕ) : (r : ℕ) → (Card r → Equiv.Perm (Card L)) →
+    Equiv.Perm (Card (L+r))
+  | 0, C => C (fun j => Fin.elim0 j)
+  | r+1, C => childLift (fun b => centralBlockPerm L r (fun u => C (Fin.cons b u)))
+
+def centralPalindromePerm (L r : ℕ) (C : Card r → Equiv.Perm (Card L))
+    (ω : BenesCoins (L+r)) : Equiv.Perm (Card (L+r)) :=
+  butterflyPerm (L+r) (decodeButterfly (L+r) ω.1) * centralBlockPerm L r C *
+    (butterflyPerm (L+r) (decodeButterfly (L+r) ω.2))⁻¹
+
+lemma centralPalindromePerm_step (L r : ℕ) (C : Card (r+1) → Equiv.Perm (Card L))
+    (c : BenesCoins (L+r) × BenesCoins (L+r))
+    (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
+    centralPalindromePerm L (r+1) C ((benesStepEquiv (L+r)).symm (c,coins)) =
+      pairSwitch coins.2 * childLift (fun b => centralPalindromePerm L r
+        (fun u => C (Fin.cons b u)) (if b then c.2 else c.1)) * pairSwitch coins.1 := by
+  change ((butterflyPerm (L+r+1) (decodeButterfly (L+r+1)
+      ((coinStepEquiv (L+r)).symm ((c.1.1,c.2.1),coins.2))))) *
+    childLift (fun b => centralBlockPerm L r (fun u => C (Fin.cons b u))) *
+    (butterflyPerm (L+r+1) (decodeButterfly (L+r+1)
+      ((coinStepEquiv (L+r)).symm ((c.1.2,c.2.2),coins.1))))⁻¹ = _
+  rw [butterflyPerm_step,butterflyPerm_step,mul_inv_rev,childLift_inv,pairSwitch_inv]
+  simp only [mul_assoc]
+  rw [←mul_assoc (childLift _),childLift_mul,←mul_assoc (childLift _),childLift_mul]
+  congr 2
+  apply congrArg childLift
+  funext b
+  cases b <;> rfl
+
+noncomputable def centralLevelCost (t L : ℕ) : (r : ℕ) →
+    (Card r → Equiv.Perm (Card L)) → {ι : Type*} → [Fintype ι] →
+    (ι ↪ Card (L+r)) → BenesCoins (L+r) → ℕ
+  | 0, _, _, _, _, _ => 0
+  | r+1, C, _, _, e, ω =>
+    let σ := benesStepEquiv (L+r) ω
+    let x := e.trans (headTailEquiv (L+r)).toEmbedding
+    let c := PairRouting.colors x σ.2.1
+    let hc := PairRouting.colors_compatible x σ.2.1
+    if t = L+r+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
+      (fun b => centralPalindromePerm L r (fun u => C (Fin.cons b u)) (if b then σ.1.2 else σ.1.1))
+    else centralLevelCost t L r (fun u => C (Fin.cons false u))
+      (PairRouting.childEmbedding x c hc false) σ.1.1 +
+      centralLevelCost t L r (fun u => C (Fin.cons true u))
+      (PairRouting.childEmbedding x c hc true) σ.1.2
+
+noncomputable def centralSlabCost (l L r : ℕ) (C : Card r → Equiv.Perm (Card L))
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (ω : BenesCoins (L+r)) : ℕ :=
+  ∑ i : Fin l, centralLevelCost (l+i.val+1) L r C e ω
+
+lemma permDistance_childLift (d : ℕ) (p q : Bool → Equiv.Perm (Card d)) :
+    permDistance (childLift p) (childLift q) =
+      permDistance (p false) (q false)+permDistance (p true) (q true) := by
+  classical
+  rw [permDistance_sum]
+  have hh : (∑ x : Card (d+1), if childLift p x = childLift q x then 0 else 1) =
+      ∑ x : Bool × Card d, if p x.1 x.2 = q x.1 x.2 then 0 else 1 := by
+    apply Fintype.sum_equiv (headTailEquiv d)
+    intro x
+    have hx : (headTailEquiv d) x = (x 0,Fin.tail x) := rfl
+    rw [hx]
+    congr 1
+    apply propext
+    constructor
+    · intro h
+      have ht := congrArg Fin.tail h
+      simpa only [childLift,Equiv.coe_fn_mk,Fin.tail_cons] using ht
+    · intro h
+      dsimp only [childLift,Equiv.coe_fn_mk]
+      exact congrArg (fun z : Card d => (Fin.cons (x 0) z : Card (d+1))) h
+  rw [hh]
+  simp only [Fintype.sum_prod_type,Fintype.sum_bool,permDistance_sum]
+  omega
+
+noncomputable def centralDistance (L r : ℕ) (C D : Card r → Equiv.Perm (Card L)) : ℕ :=
+  ∑ u : Card r, permDistance (C u) (D u)
+
+lemma centralDistance_step (L r : ℕ) (C D : Card (r+1) → Equiv.Perm (Card L)) :
+    centralDistance L (r+1) C D =
+      centralDistance L r (fun u => C (Fin.cons false u)) (fun u => D (Fin.cons false u))+
+      centralDistance L r (fun u => C (Fin.cons true u)) (fun u => D (Fin.cons true u)) := by
+  classical
+  unfold centralDistance
+  rw [Fintype.sum_equiv (headTailEquiv r) _
+    (fun x : Bool × Card r => permDistance (C (Fin.cons x.1 x.2)) (D (Fin.cons x.1 x.2)))
+      (fun x => by
+        change permDistance (C x) (D x) =
+          permDistance (C (Fin.cons (x 0) (Fin.tail x))) (D (Fin.cons (x 0) (Fin.tail x)))
+        rw [Fin.cons_self_tail])]
+  simp only [Fintype.sum_prod_type,Fintype.sum_bool]
+  omega
+
+lemma permDistance_centralBlock (L r : ℕ) (C D : Card r → Equiv.Perm (Card L)) :
+    permDistance (centralBlockPerm L r C) (centralBlockPerm L r D) = centralDistance L r C D := by
+  induction r with
+  | zero =>
+    simp only [centralBlockPerm,centralDistance,Fintype.sum_unique]
+    congr 1
+  | succ r ih =>
+    rw [centralBlockPerm,centralBlockPerm,permDistance_childLift,centralDistance_step,ih,ih]
+
+lemma permDistance_centralPalindrome (L r : ℕ) (C D : Card r → Equiv.Perm (Card L))
+    (ω : BenesCoins (L+r)) :
+    permDistance (centralPalindromePerm L r C ω) (centralPalindromePerm L r D ω) =
+      centralDistance L r C D := by
+  rw [centralPalindromePerm,centralPalindromePerm,permDistance_mul_right,
+    permDistance_mul_left,permDistance_centralBlock]
+
+lemma centralLevelCost_step (t L r : ℕ) (C : Card (r+1) → Equiv.Perm (Card L))
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r+1))
+    (ω : BenesCoins (L+r) × BenesCoins (L+r))
+    (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
+    centralLevelCost t L (r+1) C e ((benesStepEquiv (L+r)).symm (ω,coins)) =
+      let x := e.trans (headTailEquiv (L+r)).toEmbedding
+      let c := PairRouting.colors x coins.1
+      let hc := PairRouting.colors_compatible x coins.1
+      if t = L+r+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x coins.1)
+        (fun b => centralPalindromePerm L r (fun u => C (Fin.cons b u)) (if b then ω.2 else ω.1))
+      else centralLevelCost t L r (fun u => C (Fin.cons false u))
+        (PairRouting.childEmbedding x c hc false) ω.1 +
+        centralLevelCost t L r (fun u => C (Fin.cons true u))
+        (PairRouting.childEmbedding x c hc true) ω.2 := by
+  rw [centralLevelCost]
+  rfl
+
+lemma centralLevelCost_perturb (t L r : ℕ) (C D : Card r → Equiv.Perm (Card L))
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (ω : BenesCoins (L+r)) :
+    centralLevelCost t L r C e ω ≤ centralLevelCost t L r D e ω+centralDistance L r C D := by
+  induction r generalizing ι with
+  | zero => simp only [centralLevelCost,Nat.zero_add,Nat.zero_le]
+  | succ r ih =>
+    rw [←(benesStepEquiv (L+r)).symm_apply_apply ω,centralLevelCost_step,centralLevelCost_step]
+    dsimp only
+    split_ifs with ht
+    · have hh := alternatingCycles_distance_le
+        (PairRouting.switchedEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
+          (benesStepEquiv (L+r) ω).2.1)
+        (fun b => centralPalindromePerm L r (fun u => C (Fin.cons b u))
+          (if b then (benesStepEquiv (L+r) ω).1.2 else (benesStepEquiv (L+r) ω).1.1))
+        (fun b => centralPalindromePerm L r (fun u => D (Fin.cons b u))
+          (if b then (benesStepEquiv (L+r) ω).1.2 else (benesStepEquiv (L+r) ω).1.1))
+      simpa only [Bool.false_eq_true,↓reduceIte,permDistance_centralPalindrome,←centralDistance_step] using hh
+    · rw [centralDistance_step]
+      have h₀ := ih (fun u => C (Fin.cons false u)) (fun u => D (Fin.cons false u))
+        (PairRouting.childEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
+          (PairRouting.colors (e.trans (headTailEquiv (L+r)).toEmbedding) (benesStepEquiv (L+r) ω).2.1)
+          (PairRouting.colors_compatible _ _) false)
+        (benesStepEquiv (L+r) ω).1.1
+      have h₁ := ih (fun u => C (Fin.cons true u)) (fun u => D (Fin.cons true u))
+        (PairRouting.childEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
+          (PairRouting.colors (e.trans (headTailEquiv (L+r)).toEmbedding) (benesStepEquiv (L+r) ω).2.1)
+          (PairRouting.colors_compatible _ _) true)
+        (benesStepEquiv (L+r) ω).1.2
+      omega
+
+lemma centralSlabCost_perturb (l L r : ℕ) (C D : Card r → Equiv.Perm (Card L))
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (ω : BenesCoins (L+r)) :
+    centralSlabCost l L r C e ω ≤ centralSlabCost l L r D e ω+l*centralDistance L r C D := by
+  have hh := Finset.sum_le_sum (fun (i : Fin l) (_ : i ∈ Finset.univ) =>
+    centralLevelCost_perturb (l+i.val+1) L r C D e ω)
+  simpa only [Finset.sum_add_distrib,Finset.sum_const,Finset.card_univ,Fintype.card_fin,
+    nsmul_eq_mul,Nat.cast_id,centralSlabCost] using hh
+
+end Thorp
+
+end ThorpNine.SparseSaving
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/SparseSaving/InputTree.lean b/lean/OAI/Probability/ThorpRouting/SparseSaving/InputTree.lean
index f79677262b89ef0bf36dd7bf709f12c352edd8db..2f9db3c5f9b45ac69c5921b5c99972392a96075e 100644
--- a/lean/OAI/Probability/ThorpRouting/SparseSaving/InputTree.lean
+++ b/lean/OAI/Probability/ThorpRouting/SparseSaving/InputTree.lean
@@ -1,884 +1,884 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.SparseSaving.HeightDecay
-
-namespace OAI
-
-namespace ThorpNine.SparseSaving
-
-namespace Thorp
-
-lemma centralDistance_exceptional (L r : ℕ) (C D : Card r → Equiv.Perm (Card L))
-    (E : Finset (Card r)) (h : ∀ u ∉ E, C u = D u) :
-    centralDistance L r C D ≤ E.card*2^L := by
-  classical
-  calc
-    _ ≤ ∑ u : Card r, if u ∈ E then 2^L else 0 := by
-      apply Finset.sum_le_sum
-      intro u _
-      split_ifs with hu
-      · simpa only [card_positions] using permDistance_le (C u) (D u)
-      · rw [h u hu,permDistance_refl]
-    _ = _ := by simp
-
-
-inductive InputTree : ℕ → Type
-  | tip {d : ℕ} (p : Equiv.Perm (Card d)) : InputTree d
-  | node {d : ℕ} (ξ : Card d → Bool) (T : Bool → InputTree d) : InputTree (d+1)
-
-def InputTree.perm : {d : ℕ} → InputTree d → Equiv.Perm (Card d)
-  | _, .tip p => p
-  | _, .node ξ T => pairSwitch ξ * childLift (fun b => (T b).perm)
-
-noncomputable def InputTree.levelCost (t : ℕ) : {d : ℕ} → InputTree d →
-    {ι : Type*} → [Fintype ι] → (ι ↪ Card d) → (SwitchIndex d → Bool) → ℕ
-  | _, .tip _, _, _, _, _ => 0
-  | d+1, .node ξ T, _, _, e, Y =>
-    let x := e.trans (headTailEquiv d).toEmbedding
-    let c := PairRouting.colors x ξ
-    let hc := PairRouting.colors_compatible x ξ
-    let σ := coinStepEquiv d Y
-    if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x ξ)
-      (fun b => if b then butterflyPerm d (decodeButterfly d σ.1.2) * (T true).perm⁻¹
-        else butterflyPerm d (decodeButterfly d σ.1.1) * (T false).perm⁻¹)
-    else (T false).levelCost t (PairRouting.childEmbedding x c hc false) σ.1.1 +
-      (T true).levelCost t (PairRouting.childEmbedding x c hc true) σ.1.2
-
-lemma InputTree.levelCost_node (t d : ℕ) (ξ : Card d → Bool) (T : Bool → InputTree d)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (d+1))
-    (Y : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (η : Card d → Bool) :
-    (InputTree.node ξ T).levelCost t e ((coinStepEquiv d).symm (Y,η)) =
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x ξ
-      let hc := PairRouting.colors_compatible x ξ
-      if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x ξ)
-        (fun b => if b then butterflyPerm d (decodeButterfly d Y.2) * (T true).perm⁻¹
-          else butterflyPerm d (decodeButterfly d Y.1) * (T false).perm⁻¹)
-      else (T false).levelCost t (PairRouting.childEmbedding x c hc false) Y.1 +
-        (T true).levelCost t (PairRouting.childEmbedding x c hc true) Y.2 := by
-  rw [InputTree.levelCost]
-  rfl
-
-lemma InputTree.levelCost_gt (t d : ℕ) (T : InputTree d) (ht : d < t)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) :
-    T.levelCost t e Y = 0 := by
-  induction T generalizing ι with
-  | tip p => rfl
-  | @node d ξ T ih =>
-    rw [InputTree.levelCost,ite_eq_right (by omega : t ≠ d+1),ih false (by omega),ih true (by omega)]
-
-lemma InputTree.levelCost_adapted (t d : ℕ) (T : InputTree d)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) :
-    LayerAdapted d (t-1) (fun Y => T.levelCost t e Y) := by
-  induction T generalizing ι with
-  | tip p => intro Y Y' _; rfl
-  | @node d ξ T ih =>
-    intro Y Y' h
-    dsimp only
-    conv_lhs => rw [←(coinStepEquiv d).symm_apply_apply Y]
-    conv_rhs => rw [←(coinStepEquiv d).symm_apply_apply Y']
-    rw [InputTree.levelCost_node t d ξ T e ((coinStepEquiv d Y).1) ((coinStepEquiv d Y).2),
-      InputTree.levelCost_node t d ξ T e ((coinStepEquiv d Y').1) ((coinStepEquiv d Y').2)]
-    by_cases ht : t = d+1
-    · subst t
-      rw [ite_eq_left rfl,ite_eq_left rfl]
-      have he₀ : (coinStepEquiv d Y).1.1 = (coinStepEquiv d Y').1.1 := by
-        funext i
-        exact h (Sum.inr (false,i)) (by simpa only [switchHeight,Nat.add_sub_cancel] using switchHeight_le d i)
-      have he₁ : (coinStepEquiv d Y).1.2 = (coinStepEquiv d Y').1.2 := by
-        funext i
-        exact h (Sum.inr (true,i)) (by simpa only [switchHeight,Nat.add_sub_cancel] using switchHeight_le d i)
-      rw [he₀,he₁]
-    · rw [ite_eq_right ht,ite_eq_right ht]
-      congr 1
-      · apply ih false
-        intro i hi
-        exact h (Sum.inr (false,i)) hi
-      · apply ih true
-        intro i hi
-        exact h (Sum.inr (true,i)) hi
-
-theorem InputTree.level_partial_mgf (r s u : ℕ) (hu : u < r)
-    (T : InputTree (s+r)) {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+r))
-    (lo : Card r × SwitchIndex s → Bool) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(T.levelCost (s+u+1) e (assembleBits r s lo hi))) ≤
-      Real.exp (Fintype.card ι * cycleCoefficient u J q) := by
-  classical
-  induction r generalizing u ι with
-  | zero => omega
-  | succ r ih =>
-    simp only [Nat.add_succ] at *
-    cases T with
-    | tip p =>
-      simp only [InputTree.levelCost,pow_zero,finiteMean_const]
-      exact Real.one_le_exp (mul_nonneg (Nat.cast_nonneg _) (cycleCoefficient_nonneg u J q hq))
-    | node ξ T =>
-      let x := e.trans (headTailEquiv (s+r)).toEmbedding
-      let c := PairRouting.colors x ξ
-      let hc := PairRouting.colors_compatible x ξ
-      let e₀ := PairRouting.childEmbedding x c hc false
-      let e₁ := PairRouting.childEmbedding x c hc true
-      let e' := PairRouting.switchedEmbedding x ξ
-      let lo₀ : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons false j.1,j.2)
-      let lo₁ : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons true j.1,j.2)
-      let Y₀ := fun hi : Card s × SwitchIndex r → Bool => assembleBits r s lo₀ hi
-      let Y₁ := fun hi : Card s × SwitchIndex r → Bool => assembleBits r s lo₁ hi
-      let N := fun hi : (Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool) =>
-        PairRouting.alternatingCycles e' (fun b =>
-          if b then butterflyPerm (s+r) (decodeButterfly (s+r) (Y₁ hi.2)) * (T true).perm⁻¹
-          else butterflyPerm (s+r) (decodeButterfly (s+r) (Y₀ hi.1)) * (T false).perm⁻¹)
-      let L₀ := fun hi : Card s × SwitchIndex r → Bool =>
-        (T false).levelCost (s+u+1) e₀ (Y₀ hi)
-      let L₁ := fun hi : Card s × SwitchIndex r → Bool =>
-        (T true).levelCost (s+u+1) e₁ (Y₁ hi)
-      have hs (hi : (Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool))
-          (η : Card s × Card r → Bool) :
-          (InputTree.node ξ T).levelCost (s+u+1) e
-            (assembleBits (r+1) s lo ((highStepEquiv r s).symm (hi,η))) =
-            if u = r then N hi else L₀ hi.1 + L₁ hi.2 := by
-        rw [assembleBits_step]
-        rw [InputTree.levelCost_node (s+u+1) (s+r) ξ T e (Y₀ hi.1,Y₁ hi.2) (fun y => η (((cardSplit r s) y).2,((cardSplit r s) y).1))]
-        simp only [Nat.add_right_cancel_iff,Nat.add_left_cancel_iff]
-        rfl
-      rw [finiteMean_equiv (highStepEquiv r s),finiteMean_prod]
-      simp_rw [hs,finiteMean_const]
-      by_cases hur : u = r
-      · subst u
-        simp only [ite_true]
-        rw [finiteMean_prod,finiteMean_comm]
-        calc
-          _ ≤ finiteMean (fun _hi : Card s × SwitchIndex r → Bool =>
-            Real.exp (Fintype.card ι * cycleCoefficient r J q)) := by
-            apply finiteMean_mono
-            intro hi
-            have hn := node_partial_mgf r s e' (T false).perm (T true).perm
-              (butterflyPerm (s+r) (decodeButterfly (s+r) (Y₁ hi))) lo₀ J hJ q hq
-            exact hn
-          _ = _ := finiteMean_const _
-      · simp only [ite_eq_right hur,pow_add]
-        rw [finiteMean_prod_mul (fun hi => q ^ L₀ hi) (fun hi => q ^ L₁ hi)]
-        have hu' : u < r := by omega
-        have h₀ := ih u hu' (T false) e₀ lo₀
-        have h₁ := ih u hu' (T true) e₁ lo₁
-        have hp := mul_le_mul h₀ h₁ (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
-          (Real.exp_nonneg _)
-        refine hp.trans_eq ?_
-        rw [←Real.exp_add]
-        congr 1
-        have hcount := PairRouting.color_card_add c
-        have hcount' : (Fintype.card {i : ι // c i = false} : ℝ) +
-            Fintype.card {i : ι // c i = true} = Fintype.card ι := by exact_mod_cast hcount
-        change (Fintype.card {i : ι // c i = false} : ℝ) * cycleCoefficient u J q +
-            Fintype.card {i : ι // c i = true} * cycleCoefficient u J q = _
-        rw [←add_mul,hcount']
-
-
-noncomputable def InputTree.slabCost (l d : ℕ) (T : InputTree d) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) : ℕ :=
-  ∑ i : Fin l, T.levelCost (l+i.1+1) e Y
-
-lemma InputTree.slabCost_adapted (l d : ℕ) (T : InputTree d) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) :
-    LayerAdapted d (2*l) (fun Y => InputTree.slabCost l d T e Y) := by
-  apply layerAdapted_sum
-  intro i _
-  exact layerAdapted_mono (InputTree.levelCost_adapted (l+i.1+1) d T e) (by omega)
-
-theorem InputTree.slab_partial_mgf (r s l : ℕ) (hl : 0 < l) (hs : 2*s ≤ l) (T : InputTree (s+r))
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+r))
-    (lo : Card r × SwitchIndex s → Bool) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(InputTree.slabCost l (s+r) T e (assembleBits r s lo hi))) ≤
-      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
-  classical
-  let : NeZero l := ⟨ne_of_gt hl⟩
-  have hqp : 0 < q := lt_of_lt_of_le (by norm_num) hq
-  calc
-    _ ≤ finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      finiteMean (fun i : Fin l => q^(l*T.levelCost (l+i.1+1) e
-        (assembleBits r s lo hi)))) := by
-      apply finiteMean_mono
-      intro hi
-      simpa only [InputTree.slabCost,Fintype.card_fin] using
-        pow_sum_le_mean_powers (fun i : Fin l => T.levelCost (l+i.1+1) e
-          (assembleBits r s lo hi)) q hqp
-    _ = finiteMean (fun i : Fin l => finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(l*T.levelCost (l+i.1+1) e (assembleBits r s lo hi)))) := finiteMean_comm _
-    _ ≤ finiteMean (fun _i : Fin l =>
-      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64)))) := by
-      apply finiteMean_mono
-      intro i
-      by_cases ht : l+i.1+1 ≤ s+r
-      · have hs' : s ≤ l+i.1 := by omega
-        have he : l+i.1+1 = s+(l+i.1-s)+1 := by omega
-        have hu : l+i.1-s < r := by omega
-        have hm := InputTree.level_partial_mgf r s (l+i.1-s) hu T e lo
-          ⌈Real.exp ((l:ℝ)/32)⌉₊ (cycleCutoff_pos l) (q^l) (one_le_pow₀ hq)
-        have hc := cycleCoefficient_decay l (l+i.1-s) (by
-          rw [Nat.cast_sub hs',Nat.cast_add]
-          have hsr : (2:ℝ)*s ≤ l := by exact_mod_cast hs
-          have hi : (0:ℝ) ≤ i.1 := Nat.cast_nonneg _
-          linarith) (q^l) (one_le_pow₀ hq) (by
-            rw [Real.log_pow]
-            nlinarith [mul_le_mul_of_nonneg_left hlog (Nat.cast_nonneg l)])
-        simp_rw [he,pow_mul]
-        exact hm.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left hc (Nat.cast_nonneg _)))
-      · have hz := InputTree.levelCost_gt (l+i.1+1) (s+r) T (by omega) e
-        simp only [hz,Nat.mul_zero,pow_zero,finiteMean_const]
-        exact Real.one_le_exp_iff.mpr (by positivity)
-    _ = _ := finiteMean_const _
-
-
-def boundaryTree (L : ℕ) : (r : ℕ) → (Card r → Equiv.Perm (Card L)) →
-    (SwitchIndex (L+r) → Bool) → InputTree (L+r)
-  | 0, U, _ => .tip (U (fun j => Fin.elim0 j))
-  | r+1, U, X => .node (coinStepEquiv (L+r) X).2 (fun b =>
-      boundaryTree L r (fun u => U (Fin.cons b u))
-        (if b then (coinStepEquiv (L+r) X).1.2 else (coinStepEquiv (L+r) X).1.1))
-
-def boundaryOutput (L : ℕ) : (r : ℕ) → (SwitchIndex (L+r) → Bool) →
-    Card r → Equiv.Perm (Card L)
-  | 0, Y, _ => butterflyPerm L (decodeButterfly L Y)
-  | r+1, Y, u => boundaryOutput L r
-      (if u 0 then (coinStepEquiv (L+r) Y).1.2 else (coinStepEquiv (L+r) Y).1.1) (Fin.tail u)
-
-lemma boundaryOutput_adapted (L r : ℕ) :
-    LayerAdapted (L+r) L (boundaryOutput L r) := by
-  induction r with
-  | zero =>
-    intro Y Y' h
-    have he : Y = Y' := funext (fun i => h i (by simpa only [Nat.add_zero] using switchHeight_le L i))
-    rw [he]
-  | succ r ih =>
-    intro Y Y' h
-    funext u
-    dsimp only [boundaryOutput]
-    cases hb : u 0 <;> simp only [Bool.false_eq_true,↓reduceIte]
-    · exact congrFun (ih _ _ (fun i hi => h (Sum.inr (false,i)) hi)) (Fin.tail u)
-    · exact congrFun (ih _ _ (fun i hi => h (Sum.inr (true,i)) hi)) (Fin.tail u)
-
-noncomputable def modifiedBoundary (L r : ℕ) (E : Finset (Card r))
-    (U : Card r → Equiv.Perm (Card L)) (Y : SwitchIndex (L+r) → Bool) :
-    Card r → Equiv.Perm (Card L) := fun u => if u ∈ E then boundaryOutput L r Y u else U u
-
-lemma modifiedBoundary_adapted (L r : ℕ) (E : Finset (Card r))
-    (U : Card r → Equiv.Perm (Card L)) :
-    LayerAdapted (L+r) L (modifiedBoundary L r E U) := by
-  intro Y Y' h
-  unfold modifiedBoundary
-  rw [boundaryOutput_adapted L r Y Y' h]
-
-lemma permDistance_boundaryTree (L r : ℕ) (U V : Card r → Equiv.Perm (Card L))
-    (X : SwitchIndex (L+r) → Bool) :
-    permDistance (boundaryTree L r U X).perm (boundaryTree L r V X).perm =
-      centralDistance L r U V := by
-  induction r with
-  | zero =>
-    simp only [boundaryTree,InputTree.perm,centralDistance,Fintype.sum_unique]
-    congr 1
-  | succ r ih =>
-    simp only [boundaryTree,InputTree.perm,permDistance_mul_left]
-    rw [permDistance_childLift,centralDistance_step,ih,ih]
-
-lemma boundaryTree_levelCost_perturb (t L r : ℕ) (U V : Card r → Equiv.Perm (Card L))
-    (X : SwitchIndex (L+r) → Bool) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (L+r)) (Y : SwitchIndex (L+r) → Bool) :
-    (boundaryTree L r U X).levelCost t e Y ≤
-      (boundaryTree L r V X).levelCost t e Y + centralDistance L r U V := by
-  induction r generalizing ι with
-  | zero => simp only [boundaryTree,InputTree.levelCost,Nat.zero_add,Nat.zero_le]
-  | succ r ih =>
-    rw [boundaryTree,boundaryTree,InputTree.levelCost,InputTree.levelCost]
-    by_cases ht : t = L+r+1
-    · simp only [ite_eq_left ht]
-      have hh := alternatingCycles_distance_le
-        (PairRouting.switchedEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
-          (coinStepEquiv (L+r) X).2)
-        (fun b => if b then butterflyPerm (L+r) (decodeButterfly (L+r) (coinStepEquiv (L+r) Y).1.2) *
-            (boundaryTree L r (fun u => U (Fin.cons true u)) (coinStepEquiv (L+r) X).1.2).perm⁻¹
-          else butterflyPerm (L+r) (decodeButterfly (L+r) (coinStepEquiv (L+r) Y).1.1) *
-            (boundaryTree L r (fun u => U (Fin.cons false u)) (coinStepEquiv (L+r) X).1.1).perm⁻¹)
-        (fun b => if b then butterflyPerm (L+r) (decodeButterfly (L+r) (coinStepEquiv (L+r) Y).1.2) *
-            (boundaryTree L r (fun u => V (Fin.cons true u)) (coinStepEquiv (L+r) X).1.2).perm⁻¹
-          else butterflyPerm (L+r) (decodeButterfly (L+r) (coinStepEquiv (L+r) Y).1.1) *
-            (boundaryTree L r (fun u => V (Fin.cons false u)) (coinStepEquiv (L+r) X).1.1).perm⁻¹)
-      simpa only [Bool.false_eq_true,↓reduceIte,permDistance_mul_left,permDistance_inv,
-        permDistance_boundaryTree,←centralDistance_step] using hh
-    · simp only [ite_eq_right ht,Bool.false_eq_true,↓reduceIte]
-      rw [centralDistance_step]
-      have h₀ := ih (fun u => U (Fin.cons false u)) (fun u => V (Fin.cons false u))
-        (coinStepEquiv (L+r) X).1.1
-        (PairRouting.childEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
-          (PairRouting.colors (e.trans (headTailEquiv (L+r)).toEmbedding) (coinStepEquiv (L+r) X).2)
-          (PairRouting.colors_compatible _ _) false) (coinStepEquiv (L+r) Y).1.1
-      have h₁ := ih (fun u => U (Fin.cons true u)) (fun u => V (Fin.cons true u))
-        (coinStepEquiv (L+r) X).1.2
-        (PairRouting.childEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
-          (PairRouting.colors (e.trans (headTailEquiv (L+r)).toEmbedding) (coinStepEquiv (L+r) X).2)
-          (PairRouting.colors_compatible _ _) true) (coinStepEquiv (L+r) Y).1.2
-      omega
-
-lemma boundaryTree_slabCost_perturb (l L r : ℕ) (U V : Card r → Equiv.Perm (Card L))
-    (X : SwitchIndex (L+r) → Bool) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (L+r)) (Y : SwitchIndex (L+r) → Bool) :
-    InputTree.slabCost l (L+r) (boundaryTree L r U X) e Y ≤
-      InputTree.slabCost l (L+r) (boundaryTree L r V X) e Y + l*centralDistance L r U V := by
-  have hh := Finset.sum_le_sum (fun (i : Fin l) (_ : i ∈ Finset.univ) =>
-    boundaryTree_levelCost_perturb (l+i.val+1) L r U V X e Y)
-  simpa only [Finset.sum_add_distrib,Finset.sum_const,Finset.card_univ,Fintype.card_fin,
-    nsmul_eq_mul,Nat.cast_id,InputTree.slabCost] using hh
-
-lemma modifiedBoundary_distance_le (L r : ℕ) (E : Finset (Card r))
-    (U : Card r → Equiv.Perm (Card L)) (Y : SwitchIndex (L+r) → Bool) :
-    centralDistance L r (modifiedBoundary L r E U Y) U ≤ E.card*2^L := by
-  apply centralDistance_exceptional
-  intro u hu
-  exact ite_eq_right hu
-
-
-lemma InputTree.slabCost_gt (l d : ℕ) (T : InputTree d) (hd : d ≤ l)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) :
-    InputTree.slabCost l d T e Y = 0 := by
-  apply Finset.sum_eq_zero
-  intro i _
-  exact InputTree.levelCost_gt _ _ T (by omega) _ _
-
-lemma InputTree.random_slab_adapted_mul_le (d s l : ℕ) (hsd : s ≤ d) (hl : 0 < l)
-    (hs : 2*s ≤ l) (T : (SwitchIndex d → Bool) → InputTree d)
-    (hT : LayerAdapted d s T) {ι : Type*} [Fintype ι] (e : ι ↪ Card d)
-    (F : (SwitchIndex d → Bool) → ℝ) (hF : ∀ Y, 0 ≤ F Y) (hA : LayerAdapted d s F)
-    (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun Y => F Y*q^(InputTree.slabCost l d (T Y) e Y)) ≤
-      finiteMean F * Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
-  obtain ⟨r,rfl⟩ := Nat.exists_eq_add_of_le hsd
-  apply adapted_mul_mean_le r s F _ hF hA
-  intro lo
-  let hi₀ : Card s × SwitchIndex r → Bool := fun _ => false
-  have he (hi : Card s × SwitchIndex r → Bool) :
-      T (assembleBits r s lo hi) = T (assembleBits r s lo hi₀) :=
-    hT _ _ (assembleBits_agree r s lo hi hi₀)
-  simp_rw [he]
-  exact InputTree.slab_partial_mgf r s l hl hs _ e lo q hq hlog
-
-lemma InputTree.slab_mgf (l d : ℕ) (hl : 0 < l) (T : InputTree d)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun Y => q^(InputTree.slabCost l d T e Y)) ≤
-      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
-  have hh := InputTree.random_slab_adapted_mul_le d 0 l (Nat.zero_le _) hl (Nat.zero_le _)
-    (fun _ => T) (fun _ _ _ => rfl) e (fun _ => 1) (fun _ => by norm_num)
-    (fun _ _ _ => rfl) q hq hlog
-  simpa only [one_mul,finiteMean_const] using hh
-
-noncomputable def InputTree.familyCost (L n d : ℕ)
-    (T : (SwitchIndex d → Bool) → InputTree d) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) : ℕ :=
-  ∑ j ∈ Finset.range n, InputTree.slabCost (L*4^j) d (T Y) e Y
-
-lemma InputTree.random_slabCost_adapted (l d s : ℕ) (T : (SwitchIndex d → Bool) → InputTree d)
-    (hT : LayerAdapted d s T) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) :
-    LayerAdapted d (max s (2*l)) (fun Y => InputTree.slabCost l d (T Y) e Y) := by
-  intro Y Y' h
-  dsimp only
-  rw [hT _ _ (agreeThrough_mono (le_max_left _ _) h)]
-  exact InputTree.slabCost_adapted l d (T Y') e Y Y'
-    (agreeThrough_mono (le_max_right _ _) h)
-
-lemma InputTree.familyCost_adapted (L n d s : ℕ) (T : (SwitchIndex d → Bool) → InputTree d)
-    (hT : LayerAdapted d s T) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) :
-    LayerAdapted d (max s (familyEnd L n)) (fun Y => InputTree.familyCost L n d T e Y) := by
-  apply layerAdapted_sum
-  intro j hj
-  exact layerAdapted_mono (InputTree.random_slabCost_adapted (L*4^j) d s T hT e)
-    (max_le_max_left s (familyEnd_ge L n j (Finset.mem_range.mp hj)))
-
-theorem InputTree.family_mgf_first (L n d s : ℕ) (hL : 0 < L) (hs : s ≤ L)
-    (T : (SwitchIndex d → Bool) → InputTree d) (hT : LayerAdapted d s T)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) (p : ℝ) (hp : 0 ≤ p)
-    (hfirst : finiteMean (fun Y => q^(InputTree.slabCost L d (T Y) e Y)) ≤
-      Real.exp (p+Fintype.card ι*(68*Real.exp (-(L:ℝ)/64)))) :
-    finiteMean (fun Y => q^(InputTree.familyCost L n d T e Y)) ≤
-      Real.exp (p+Fintype.card ι*(68*∑ j ∈ Finset.range n,
-        Real.exp (-((L*4^j:ℕ):ℝ)/64))) := by
-  induction n with
-  | zero =>
-    simp only [InputTree.familyCost,Finset.range_zero,Finset.sum_empty,pow_zero,
-      finiteMean_const,mul_zero,add_zero]
-    exact Real.one_le_exp hp
-  | succ n ih =>
-    by_cases hn : n = 0
-    · subst n
-      simpa only [InputTree.familyCost,Nat.zero_add,Finset.sum_range_one,pow_zero,mul_one] using hfirst
-    · let l := L*4^n
-      have hl : 0 < l := Nat.mul_pos hL (pow_pos (by decide) _)
-      have hsl : 2*familyEnd L n ≤ l := familyEnd_two L n
-      have hs' : s ≤ familyEnd L n := by
-        have hh := familyEnd_ge L n 0 (Nat.pos_of_ne_zero hn)
-        simp only [pow_zero,mul_one] at hh
-        omega
-      have hc : ∀ Y, InputTree.familyCost L (n+1) d T e Y =
-          InputTree.familyCost L n d T e Y+InputTree.slabCost l d (T Y) e Y := by
-        intro Y
-        exact Finset.sum_range_succ _ _
-      have hm : finiteMean (fun Y => q^(InputTree.familyCost L (n+1) d T e Y)) ≤
-          finiteMean (fun Y => q^(InputTree.familyCost L n d T e Y)) *
-            Real.exp (Fintype.card ι*(68*Real.exp (-(l:ℝ)/64))) := by
-        by_cases hld : l ≤ d
-        · simp_rw [hc,pow_add]
-          apply InputTree.random_slab_adapted_mul_le d (familyEnd L n) l (by omega) hl hsl T
-            (layerAdapted_mono hT hs') e _ (fun _ => pow_nonneg (by linarith) _)
-          · apply layerAdapted_comp
-            exact layerAdapted_mono (InputTree.familyCost_adapted L n d s T hT e) (max_le hs' le_rfl)
-          · exact hq
-          · exact hlog
-        · simp_rw [hc,InputTree.slabCost_gt l d _ (by omega),Nat.add_zero]
-          exact le_mul_of_one_le_right
-            (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
-            (Real.one_le_exp_iff.mpr (by positivity))
-      refine (hm.trans (mul_le_mul_of_nonneg_right ih (Real.exp_nonneg _))).trans_eq ?_
-      rw [←Real.exp_add,Finset.sum_range_succ]
-      congr 1
-      dsimp only [l]
-      ring
-
-lemma InputTree.family_mgf_first_decay (L n d s : ℕ) (hL : 0 < L) (hs : s ≤ L)
-    (T : (SwitchIndex d → Bool) → InputTree d) (hT : LayerAdapted d s T)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) (p : ℝ) (hp : 0 ≤ p)
-    (hfirst : finiteMean (fun Y => q^(InputTree.slabCost L d (T Y) e Y)) ≤
-      Real.exp (p+Fintype.card ι*(68*Real.exp (-(L:ℝ)/64)))) :
-    finiteMean (fun Y => q^(InputTree.familyCost L n d T e Y)) ≤
-      Real.exp (p+Fintype.card ι*heightDecay L) := by
-  apply (InputTree.family_mgf_first L n d s hL hs T hT e q hq hlog p hp hfirst).trans
-  apply Real.exp_le_exp.mpr
-  apply add_le_add_right
-  apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
-  have hh := mul_le_mul_of_nonneg_left (sum_family_decay L n hL) (by norm_num : (0:ℝ)≤68)
-  simpa only [heightDecay,mul_div_assoc] using hh
-
-
-noncomputable def modifiedInputTree (L r : ℕ) (E : Finset (Card r))
-    (U : Card r → Equiv.Perm (Card L)) (X Y : SwitchIndex (L+r) → Bool) : InputTree (L+r) :=
-  boundaryTree L r (modifiedBoundary L r E U Y) X
-
-lemma modifiedInputTree_adapted (L r : ℕ) (E : Finset (Card r))
-    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool) :
-    LayerAdapted (L+r) L (modifiedInputTree L r E U X) := by
-  intro Y Y' h
-  unfold modifiedInputTree
-  rw [modifiedBoundary_adapted L r E U Y Y' h]
-
-lemma modified_slab_mgf (l L r : ℕ) (hl : 0 < l) (E : Finset (Card r))
-    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun Y => q^(InputTree.slabCost l (L+r) (modifiedInputTree L r E U X Y) e Y)) ≤
-      Real.exp (((l*E.card*2^L:ℕ):ℝ)*Real.log q+
-        Fintype.card ι*(68*Real.exp (-(l:ℝ)/64))) := by
-  have hqp : 0 < q := by linarith
-  have hpoint (Y : SwitchIndex (L+r) → Bool) :
-      InputTree.slabCost l (L+r) (modifiedInputTree L r E U X Y) e Y ≤
-        InputTree.slabCost l (L+r) (boundaryTree L r U X) e Y+l*E.card*2^L := by
-    apply (boundaryTree_slabCost_perturb l L r (modifiedBoundary L r E U Y) U X e Y).trans
-    apply Nat.add_le_add_left
-    simpa only [Nat.mul_assoc] using Nat.mul_le_mul_left l (modifiedBoundary_distance_le L r E U Y)
-  calc
-    _ ≤ finiteMean (fun Y => q^(InputTree.slabCost l (L+r) (boundaryTree L r U X) e Y+l*E.card*2^L)) :=
-      finiteMean_mono (fun Y => pow_le_pow_right₀ hq (hpoint Y))
-    _ = finiteMean (fun Y => q^(InputTree.slabCost l (L+r) (boundaryTree L r U X) e Y)) *
-        q^(l*E.card*2^L) := by simp_rw [pow_add]; exact finiteMean_mul_const _ _
-    _ ≤ Real.exp (Fintype.card ι*(68*Real.exp (-(l:ℝ)/64))) * q^(l*E.card*2^L) :=
-      mul_le_mul_of_nonneg_right (InputTree.slab_mgf l (L+r) hl _ e q hq hlog) (pow_nonneg (by linarith) _)
-    _ = _ := by
-      have he : Real.exp (((l*E.card*2^L:ℕ):ℝ)*Real.log q) = q^(l*E.card*2^L) := by
-        rw [Real.exp_nat_mul,Real.exp_log hqp]
-      rw [Real.exp_add,he,mul_comm]
-
-lemma modified_family_mgf (l n L r : ℕ) (hl : 0 < l) (hLl : L ≤ l) (E : Finset (Card r))
-    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun Y => q^(InputTree.familyCost l n (L+r) (modifiedInputTree L r E U X) e Y)) ≤
-      Real.exp (((l*E.card*2^L:ℕ):ℝ)*Real.log q+Fintype.card ι*heightDecay l) := by
-  exact InputTree.family_mgf_first_decay l n (L+r) L hl hLl _
-    (modifiedInputTree_adapted L r E U X) e q hq hlog _
-    (mul_nonneg (Nat.cast_nonneg _) (Real.log_nonneg hq))
-    (modified_slab_mgf l L r hl E U X e q hq hlog)
-
-theorem modified_high_mgf (L n r : ℕ) (hL : 0 < L) (E : Finset (Card r))
-    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/500) :
-    finiteMean (fun Y => q^(InputTree.familyCost L n (L+r) (modifiedInputTree L r E U X) e Y+
-      InputTree.familyCost (2*L) n (L+r) (modifiedInputTree L r E U X) e Y)) ≤
-      Real.exp (Fintype.card ι*heightDecay L +
-        3*((L*E.card*2^L:ℕ):ℝ)*Real.log q) := by
-  have hq2 := one_le_pow₀ (n:=2) hq
-  have hlog2 : Real.log (q^2) ≤ 1/250 := by rw [Real.log_pow]; norm_num; linarith
-  have h₀ := modified_family_mgf L n L r hL le_rfl E U X e (q^2) hq2 hlog2
-  have h₁ := modified_family_mgf (2*L) n L r (by omega) (by omega) E U X e (q^2) hq2 hlog2
-  have h₁' : finiteMean (fun Y => (q^2)^(InputTree.familyCost (2*L) n (L+r)
-      (modifiedInputTree L r E U X) e Y)) ≤
-      Real.exp ((((2*L)*E.card*2^L:ℕ):ℝ)*Real.log (q^2)+Fintype.card ι*heightDecay L) := by
-    apply h₁.trans
-    apply Real.exp_le_exp.mpr
-    exact add_le_add_right (mul_le_mul_of_nonneg_left
-      (heightDecay_antitone (show L ≤ 2*L by omega)) (Nat.cast_nonneg _)) _
-  have hh := finiteMean_mul_exp_le
-    (fun Y => q^(InputTree.familyCost L n (L+r) (modifiedInputTree L r E U X) e Y))
-    (fun Y => q^(InputTree.familyCost (2*L) n (L+r) (modifiedInputTree L r E U X) e Y))
-    _ _ (by simpa only [pow_right_comm] using h₀) (by simpa only [pow_right_comm] using h₁')
-  simp only [←pow_add] at hh
-  refine hh.trans_eq ?_
-  congr 1
-  simp only [Real.log_pow,Nat.cast_mul,Nat.cast_ofNat]
-  ring
-
-noncomputable def InputTree.lowCost (H d : ℕ) (T : InputTree d)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) : ℕ :=
-  ∑ i ∈ Finset.range H, T.levelCost (i+1) e Y
-
-lemma InputTree.lowCost_slab (L d : ℕ) (T : InputTree d)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) :
-    InputTree.lowCost L d T e Y+InputTree.slabCost L d T e Y = InputTree.lowCost (2*L) d T e Y := by
-  rw [InputTree.lowCost,InputTree.lowCost,show 2*L=L+L by omega,Finset.sum_range_add]
-  congr 1
-  exact Fin.sum_univ_eq_sum_range (fun i => T.levelCost (L+i+1) e Y) L
-
-lemma InputTree.cost_family (L n d : ℕ) (T : (SwitchIndex d → Bool) → InputTree d)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) :
-    InputTree.lowCost L d (T Y) e Y + InputTree.familyCost L n d T e Y +
-      InputTree.familyCost (2*L) n d T e Y = InputTree.lowCost (L*4^n) d (T Y) e Y := by
-  induction n with
-  | zero => simp only [InputTree.familyCost,Finset.range_zero,Finset.sum_empty,Nat.add_zero,
-      pow_zero,mul_one]
-  | succ n ih =>
-    have he : (2*L)*4^n = 2*(L*4^n) := by ring
-    have he' : L*4^(n+1) = 2*(2*(L*4^n)) := by rw [pow_succ]; ring
-    simp only [InputTree.familyCost,Finset.sum_range_succ] at ih ⊢
-    rw [he,he',←InputTree.lowCost_slab,←InputTree.lowCost_slab,←ih]
-    omega
-
-lemma InputTree.lowCost_ge (H d : ℕ) (hd : d ≤ H) (T : InputTree d)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) :
-    InputTree.lowCost H d T e Y = InputTree.lowCost d d T e Y := by
-  symm
-  apply Finset.sum_subset (Finset.range_mono hd)
-  intro i _ hi
-  have hi' : d ≤ i := by simpa only [Finset.mem_range,not_lt] using hi
-  exact InputTree.levelCost_gt (i+1) d T (by omega) e Y
-
-lemma boundaryTree_levelCost_low (t L r : ℕ) (ht : t ≤ L)
-    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (Y : SwitchIndex (L+r) → Bool) :
-    (boundaryTree L r U X).levelCost t e Y = 0 := by
-  induction r generalizing ι with
-  | zero => rfl
-  | succ r ih =>
-    rw [boundaryTree,InputTree.levelCost,ite_eq_right (by omega : t ≠ L+r+1),ih,ih]
-
-lemma boundaryTree_lowCost_zero (L r : ℕ) (U : Card r → Equiv.Perm (Card L))
-    (X : SwitchIndex (L+r) → Bool) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (L+r)) (Y : SwitchIndex (L+r) → Bool) :
-    InputTree.lowCost L (L+r) (boundaryTree L r U X) e Y = 0 := by
-  apply Finset.sum_eq_zero
-  intro i hi
-  exact boundaryTree_levelCost_low (i+1) L r (by simpa only [Finset.mem_range,Nat.lt_iff_add_one_le] using hi) U X e Y
-
-theorem modified_cost_mgf (L r : ℕ) (hL : 0 < L) (E : Finset (Card r))
-    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/500) :
-    finiteMean (fun Y => q^(InputTree.lowCost (L+r) (L+r) (modifiedInputTree L r E U X Y) e Y)) ≤
-      Real.exp (Fintype.card ι*heightDecay L +
-        3*((L*E.card*2^L:ℕ):ℝ)*Real.log q) := by
-  have hbound : L+r ≤ L*4^(L+r) := by
-    have h₀ := pow_four_ge (L+r)
-    have h₁ := Nat.mul_le_mul_right (4^(L+r)) hL
-    nlinarith
-  have he (Y : SwitchIndex (L+r) → Bool) :=
-    InputTree.cost_family L (L+r) (L+r) (modifiedInputTree L r E U X) e Y
-  have hz (Y : SwitchIndex (L+r) → Bool) :
-      InputTree.lowCost L (L+r) (modifiedInputTree L r E U X Y) e Y = 0 :=
-    boundaryTree_lowCost_zero L r _ X e Y
-  simp only [hz,Nat.zero_add,InputTree.lowCost_ge _ _ hbound] at he
-  simp_rw [←he]
-  exact modified_high_mgf L (L+r) r hL E U X e q hq hlog
-
-
-noncomputable def exceptionChild {r : ℕ} (E : Finset (Card (r+1))) (b : Bool) : Finset (Card r) :=
-  Finset.univ.filter (fun u => Fin.cons b u ∈ E)
-
-lemma mem_exceptionChild {r : ℕ} (E : Finset (Card (r+1))) (b : Bool) (u : Card r) :
-    u ∈ exceptionChild E b ↔ Fin.cons b u ∈ E := by
-  classical
-  simp only [exceptionChild,Finset.mem_filter,Finset.mem_univ,true_and]
-
-lemma exceptionChild_card_add {r : ℕ} (E : Finset (Card (r+1))) :
-    (exceptionChild E false).card+(exceptionChild E true).card = E.card := by
-  classical
-  have he : (∑ u : Card (r+1), if u ∈ E then 1 else 0) = E.card := by simp
-  rw [Fintype.sum_equiv (headTailEquiv r) _
-    (fun x : Bool × Card r => if Fin.cons x.1 x.2 ∈ E then 1 else 0)
-    (fun x => by change (if x ∈ E then 1 else 0) = (if Fin.cons (x 0) (Fin.tail x) ∈ E then 1 else 0); rw [Fin.cons_self_tail])] at he
-  simp only [Fintype.sum_prod_type,Fintype.sum_bool] at he
-  simpa only [exceptionChild,Finset.card_filter,add_comm] using he
-
-def seedSplitEquiv (r : ℕ) (A : Type*) : (Card (r+1) → A) ≃ (Card r → A) × (Card r → A) where
-  toFun U := (fun u => U (Fin.cons false u),fun u => U (Fin.cons true u))
-  invFun U := fun u => if u 0 then U.2 (Fin.tail u) else U.1 (Fin.tail u)
-  left_inv U := by
-    funext u
-    dsimp only
-    split_ifs with h
-    · rw [←h,Fin.cons_self_tail]
-    · have h' : u 0 = false := by cases hh : u 0 <;> simp_all
-      rw [←h',Fin.cons_self_tail]
-  right_inv U := by simp only [Fin.cons_zero,Fin.tail_cons,Bool.false_eq_true,↓reduceIte]
-
-def seedSandwichShuffle {A B C D E : Type*} :
-    (A × B) × ((C × D) × E) ≃ ((A × C) × (B × D)) × E where
-  toFun x := (((x.1.1,x.2.1.1),(x.1.2,x.2.1.2)),x.2.2)
-  invFun x := ((x.1.1.1,x.1.2.1),((x.1.1.2,x.1.2.2),x.2))
-  left_inv _ := rfl
-  right_inv _ := rfl
-
-abbrev ModifiedCoins (L r : ℕ) := (Card r → Equiv.Perm (Card L)) × BenesCoins (L+r)
-
-def modifiedStepEquiv (L r : ℕ) : ModifiedCoins L (r+1) ≃
-    (ModifiedCoins L r × ModifiedCoins L r) × ((Card (L+r) → Bool) × (Card (L+r) → Bool)) :=
-  (Equiv.prodCongr (seedSplitEquiv r (Equiv.Perm (Card L))) (benesStepEquiv (L+r))).trans
-    seedSandwichShuffle
-
-noncomputable def modifiedPerm (L r : ℕ) (E : Finset (Card r)) (ω : ModifiedCoins L r) :
-    Equiv.Perm (Card (L+r)) :=
-  butterflyPerm (L+r) (decodeButterfly (L+r) ω.2.1) *
-    (modifiedInputTree L r E ω.1 ω.2.2 ω.2.1).perm⁻¹
-
-noncomputable def modifiedCost (L r : ℕ) (E : Finset (Card r))
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (ω : ModifiedCoins L r) : ℕ :=
-  InputTree.lowCost (L+r) (L+r) (modifiedInputTree L r E ω.1 ω.2.2 ω.2.1) e ω.2.1
-
-lemma modifiedBoundary_cons (L r : ℕ) (E : Finset (Card (r+1)))
-    (U : Card (r+1) → Equiv.Perm (Card L)) (Y : SwitchIndex (L+r+1) → Bool) (b : Bool) :
-    (fun u => modifiedBoundary L (r+1) E U Y (Fin.cons b u)) =
-    modifiedBoundary L r (exceptionChild E b) (fun u => U (Fin.cons b u))
-      (if b then (coinStepEquiv (L+r) Y).1.2 else (coinStepEquiv (L+r) Y).1.1) := by
-  funext u
-  simp only [modifiedBoundary,mem_exceptionChild,boundaryOutput,Fin.cons_zero,Fin.tail_cons]
-
-lemma modifiedInputTree_node (L r : ℕ) (E : Finset (Card (r+1)))
-    (ω : ModifiedCoins L r × ModifiedCoins L r) (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
-    let z := (modifiedStepEquiv L r).symm (ω,coins)
-    modifiedInputTree L (r+1) E z.1 z.2.2 z.2.1 =
-      InputTree.node coins.1 (fun b =>
-        let w := if b then ω.2 else ω.1
-        modifiedInputTree L r (exceptionChild E b) w.1 w.2.2 w.2.1) := by
-  dsimp only
-  unfold modifiedInputTree
-  rw [boundaryTree]
-  change InputTree.node coins.1 (fun b => boundaryTree L r _ _) = _
-  congr 1
-  funext b
-  rw [modifiedBoundary_cons]
-  cases b <;> rfl
-
-lemma modifiedPerm_step (L r : ℕ) (E : Finset (Card (r+1)))
-    (ω : ModifiedCoins L r × ModifiedCoins L r) (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
-    modifiedPerm L (r+1) E ((modifiedStepEquiv L r).symm (ω,coins)) =
-      pairSwitch coins.2 * childLift (fun b => modifiedPerm L r (exceptionChild E b)
-        (if b then ω.2 else ω.1)) * pairSwitch coins.1 := by
-  rw [modifiedPerm,modifiedInputTree_node,InputTree.perm,mul_inv_rev,pairSwitch_inv,childLift_inv]
-  change (butterflyPerm (L+r+1) (decodeButterfly (L+r+1)
-    ((coinStepEquiv (L+r)).symm ((ω.1.2.1,ω.2.2.1),coins.2)))) * _ = _
-  rw [butterflyPerm_step,mul_assoc,←mul_assoc (childLift _),childLift_mul,←mul_assoc]
-  congr 2
-  apply congrArg childLift
-  funext b
-  cases b <;> rfl
-
-lemma modifiedPerm_coordinates (L r : ℕ) (E : Finset (Card (r+1)))
-    (ω : ModifiedCoins L r × ModifiedCoins L r) (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool))
-    (x : Card (L+r+1)) :
-    headTailEquiv (L+r) (modifiedPerm L (r+1) E ((modifiedStepEquiv L r).symm (ω,coins)) x) =
-      PairRouting.sandwich coins.1 coins.2 (fun b => modifiedPerm L r (exceptionChild E b)
-        (if b then ω.2 else ω.1)) (headTailEquiv (L+r) x) := by
-  rw [modifiedPerm_step]
-  simp only [Equiv.Perm.mul_apply,headTail_switch,headTail_child]
-  rfl
-
-lemma InputTree.lowCost_node (d : ℕ) (ξ : Card d → Bool) (T : Bool → InputTree d)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (d+1)) (Y : (SwitchIndex d → Bool) × (SwitchIndex d → Bool))
-    (η : Card d → Bool) :
-    InputTree.lowCost (d+1) (d+1) (.node ξ T) e ((coinStepEquiv d).symm (Y,η)) =
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x ξ
-      let hc := PairRouting.colors_compatible x ξ
-      PairRouting.alternatingCycles (PairRouting.switchedEmbedding x ξ)
-        (fun b => if b then butterflyPerm d (decodeButterfly d Y.2) * (T true).perm⁻¹
-          else butterflyPerm d (decodeButterfly d Y.1) * (T false).perm⁻¹) +
-        InputTree.lowCost d d (T false) (PairRouting.childEmbedding x c hc false) Y.1 +
-        InputTree.lowCost d d (T true) (PairRouting.childEmbedding x c hc true) Y.2 := by
-  dsimp only
-  rw [InputTree.lowCost,Finset.sum_range_succ,InputTree.levelCost_node,ite_eq_left rfl]
-  have he : (∑ i ∈ Finset.range d,
-      (.node ξ T : InputTree (d+1)).levelCost (i+1) e ((coinStepEquiv d).symm (Y,η))) =
-      InputTree.lowCost d d (T false)
-        (PairRouting.childEmbedding (e.trans (headTailEquiv d).toEmbedding)
-          (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) ξ) (PairRouting.colors_compatible _ _) false) Y.1 +
-      InputTree.lowCost d d (T true)
-        (PairRouting.childEmbedding (e.trans (headTailEquiv d).toEmbedding)
-          (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) ξ) (PairRouting.colors_compatible _ _) true) Y.2 := by
-    rw [InputTree.lowCost,InputTree.lowCost,←Finset.sum_add_distrib]
-    apply Finset.sum_congr rfl
-    intro i hi
-    rw [InputTree.levelCost_node,ite_eq_right (by simp only [Finset.mem_range] at hi; omega : i+1 ≠ d+1)]
-  rw [he]
-  omega
-
-noncomputable def modifiedAssignmentCost (L r : ℕ) (E : Finset (Card (r+1)))
-    {ι : Type*} [Fintype ι] (x : ι ↪ Bool × Card (L+r)) (c : ι → Bool)
-    (hc : PairRouting.Compatible x c) (ω : ModifiedCoins L r × ModifiedCoins L r) : ℕ :=
-  PairRouting.alternatingCycles (PairRouting.coloredEmbedding x c hc)
-    (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) +
-  modifiedCost L r (exceptionChild E false) (PairRouting.childEmbedding x c hc false) ω.1 +
-  modifiedCost L r (exceptionChild E true) (PairRouting.childEmbedding x c hc true) ω.2
-
-lemma modifiedCost_step (L r : ℕ) (E : Finset (Card (r+1)))
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r+1))
-    (ω : ModifiedCoins L r × ModifiedCoins L r) (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
-    modifiedCost L (r+1) E e ((modifiedStepEquiv L r).symm (ω,coins)) =
-      modifiedAssignmentCost L r E (e.trans (headTailEquiv (L+r)).toEmbedding)
-        (PairRouting.colors (e.trans (headTailEquiv (L+r)).toEmbedding) coins.1)
-        (PairRouting.colors_compatible _ _) ω := by
-  rw [modifiedCost,modifiedInputTree_node]
-  change InputTree.lowCost (L+r+1) (L+r+1) _ e
-    ((coinStepEquiv (L+r)).symm ((ω.1.2.1,ω.2.2.1),coins.2)) = _
-  rw [InputTree.lowCost_node]
-  unfold modifiedAssignmentCost
-  dsimp only
-  simp only [Bool.false_eq_true,↓reduceIte]
-  rw [PairRouting.switchedEmbedding_eq_colored _ _ (PairRouting.colors_compatible _ _) _ rfl]
-  congr 2
-
-
-namespace PairRouting
-variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α] [DecidableEq ι]
-
-omit [Fintype α] [DecidableEq ι] in
-lemma childEmbedding_surjective (x : ι ↪ Bool × α) (hx : Function.Surjective x)
-    (c : ι → Bool) (hc : Compatible x c) (b : Bool) :
-    Function.Surjective (childEmbedding x c hc b) := by
-  intro y
-  have hs : Function.Surjective (coloredEmbedding x c hc) :=
-    (pairLayer (chosenCoin x c)).surjective.comp hx
-  obtain ⟨i,hi⟩ := hs (b,y)
-  rw [coloredEmbedding_apply] at hi
-  exact ⟨⟨i,congrArg Prod.fst hi⟩,congrArg Prod.snd hi⟩
-
-omit [DecidableEq ι] in
-lemma uniformPerm_tuple_upper (e f : ι ↪ α) (he : Function.Surjective e) (p : Equiv.Perm α) :
-    finiteMean (fun u : Equiv.Perm α => if ∀ i, (p*u⁻¹) (e i) = f i then (1:ℝ) else 0) ≤
-      1/(Fintype.card (Equiv.Perm α):ℝ) := by
-  classical
-  let M : Equiv.Perm α → Prop := fun u => ∀ i, (p*u⁻¹) (e i) = f i
-  have huniq (u v : Equiv.Perm α) (hu : M u) (hv : M v) : u = v := by
-    have hh : p*u⁻¹ = p*v⁻¹ := by
-      ext x
-      obtain ⟨i,rfl⟩ := he x
-      exact (hu i).trans (hv i).symm
-    exact inv_injective (mul_left_cancel hh)
-  unfold finiteMean
-  apply div_le_div_of_nonneg_right _ (Nat.cast_nonneg _)
-  change (∑ u, if M u then (1:ℝ) else 0) ≤ 1
-  by_cases h : ∃ u, M u
-  · obtain ⟨u,hu⟩ := h
-    rw [Finset.sum_eq_single u]
-    · simp only [hu,↓reduceIte,le_refl]
-    · intro v _ hv
-      exact ite_eq_right (fun hh => hv (huniq v u hh hu))
-    · simp
-  · push Not at h
-    simp only [h,ite_false,Finset.sum_const_zero,zero_le_one]
-
-end PairRouting
-
-noncomputable def centralFactor (L r : ℕ) (E : Finset (Card r)) (a : ℝ) : ℝ :=
-  (Nat.factorial (2^L):ℝ)^(-((2:ℝ)^r-E.card)*a)
-
-lemma centralFactor_nonneg (L r : ℕ) (E : Finset (Card r)) (a : ℝ) :
-    0 ≤ centralFactor L r E a := Real.rpow_nonneg (Nat.cast_nonneg _) _
-
-lemma centralFactor_step (L r : ℕ) (E : Finset (Card (r+1))) (a : ℝ) :
-    centralFactor L (r+1) E a =
-      centralFactor L r (exceptionChild E false) a * centralFactor L r (exceptionChild E true) a := by
-  unfold centralFactor
-  rw [←Real.rpow_add (Nat.cast_pos.mpr (Nat.factorial_pos _))]
-  congr 1
-  have he : ((exceptionChild E false).card:ℝ)+(exceptionChild E true).card = E.card := by
-    exact_mod_cast exceptionChild_card_add E
-  rw [pow_succ,←he]
-  ring
-
-lemma modifiedPerm_zero (L : ℕ) (E : Finset (Card 0)) (ω : ModifiedCoins L 0) :
-    modifiedPerm L 0 E ω =
-      if (fun j => Fin.elim0 j : Card 0) ∈ E then 1
-      else butterflyPerm L (decodeButterfly L ω.2.1)*(ω.1 (fun j => Fin.elim0 j))⁻¹ := by
-  unfold modifiedPerm modifiedInputTree boundaryTree InputTree.perm modifiedBoundary boundaryOutput
-  split_ifs
-  · exact mul_inv_cancel _
-  · rfl
-
-lemma modifiedCost_zero (L : ℕ) (E : Finset (Card 0)) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card L) (ω : ModifiedCoins L 0) : modifiedCost L 0 E e ω = 0 := by
-  simp only [modifiedCost,modifiedInputTree,boundaryTree,InputTree.lowCost,
-    InputTree.levelCost,Finset.sum_const_zero]
-
-lemma centralFactor_zero (L : ℕ) (E : Finset (Card 0)) (a : ℝ) :
-    centralFactor L 0 E a =
-      if (fun j => Fin.elim0 j : Card 0) ∈ E then 1 else (Nat.factorial (2^L):ℝ)^(-a) := by
-  classical
-  let z : Card 0 := fun j => Fin.elim0 j
-  have he : (E.card:ℝ) = if z ∈ E then 1 else 0 := by
-    by_cases hz : z ∈ E
-    · have he : E = {z} := by
-        ext u
-        rw [Subsingleton.elim u z]
-        simp only [hz,Finset.mem_singleton]
-      rw [he,ite_eq_left (by simpa only [he] using hz),Finset.card_singleton,Nat.cast_one]
-    · have he : E = ∅ := Finset.eq_empty_iff_forall_notMem.mpr (fun u hu => hz (Subsingleton.elim u z ▸ hu))
-      rw [he,Finset.card_empty,Nat.cast_zero,ite_eq_right (by simp)]
-  unfold centralFactor
-  rw [pow_zero,he]
-  split_ifs <;> simp
-
-lemma modified_probability_zero_le (L : ℕ) (E : Finset (Card 0))
-    {ι : Type*} [Fintype ι] (e f : ι ↪ Card L) (he : Function.Surjective e) :
-    PairRouting.tupleProbability (modifiedPerm L 0 E) e f ≤
-      if (fun j => Fin.elim0 j : Card 0) ∈ E then 1 else 1/(Nat.factorial (2^L):ℝ) := by
-  classical
-  rw [PairRouting.tupleProbability,finiteMean_prod,finiteMean_comm]
-  let B : ℝ := if (fun j => Fin.elim0 j : Card 0) ∈ E then 1 else 1/(Nat.factorial (2^L):ℝ)
-  change _ ≤ B
-  calc
-    _ ≤ finiteMean (fun _ω : BenesCoins L => B) := by
-      apply finiteMean_mono
-      intro ω
-      simp only [modifiedPerm_zero]
-      by_cases h : (fun j => Fin.elim0 j : Card 0) ∈ E
-      · simp only [h,↓reduceIte,Equiv.Perm.one_apply,finiteMean_const,B]
-        split_ifs <;> norm_num
-      · simp only [h,↓reduceIte,B]
-        rw [finiteMean_pi_eval (A := fun _ : Card 0 => Equiv.Perm (Card L)) (fun j => Fin.elim0 j) (fun u : Equiv.Perm (Card L) =>
-          if ∀ i, (butterflyPerm L (decodeButterfly L ω.1)*u⁻¹) (e i) = f i then (1:ℝ) else 0)]
-        convert PairRouting.uniformPerm_tuple_upper e f he (butterflyPerm L (decodeButterfly L ω.1)) using 1
-        rw [Fintype.card_perm,card_positions]
-    _ = _ := finiteMean_const B
-
-end Thorp
-
-end ThorpNine.SparseSaving
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.SparseSaving.HeightDecay
+
+namespace OAI
+
+namespace ThorpNine.SparseSaving
+
+namespace Thorp
+
+lemma centralDistance_exceptional (L r : ℕ) (C D : Card r → Equiv.Perm (Card L))
+    (E : Finset (Card r)) (h : ∀ u ∉ E, C u = D u) :
+    centralDistance L r C D ≤ E.card*2^L := by
+  classical
+  calc
+    _ ≤ ∑ u : Card r, if u ∈ E then 2^L else 0 := by
+      apply Finset.sum_le_sum
+      intro u _
+      split_ifs with hu
+      · simpa only [card_positions] using permDistance_le (C u) (D u)
+      · rw [h u hu,permDistance_refl]
+    _ = _ := by simp
+
+
+inductive InputTree : ℕ → Type
+  | tip {d : ℕ} (p : Equiv.Perm (Card d)) : InputTree d
+  | node {d : ℕ} (ξ : Card d → Bool) (T : Bool → InputTree d) : InputTree (d+1)
+
+def InputTree.perm : {d : ℕ} → InputTree d → Equiv.Perm (Card d)
+  | _, .tip p => p
+  | _, .node ξ T => pairSwitch ξ * childLift (fun b => (T b).perm)
+
+noncomputable def InputTree.levelCost (t : ℕ) : {d : ℕ} → InputTree d →
+    {ι : Type*} → [Fintype ι] → (ι ↪ Card d) → (SwitchIndex d → Bool) → ℕ
+  | _, .tip _, _, _, _, _ => 0
+  | d+1, .node ξ T, _, _, e, Y =>
+    let x := e.trans (headTailEquiv d).toEmbedding
+    let c := PairRouting.colors x ξ
+    let hc := PairRouting.colors_compatible x ξ
+    let σ := coinStepEquiv d Y
+    if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x ξ)
+      (fun b => if b then butterflyPerm d (decodeButterfly d σ.1.2) * (T true).perm⁻¹
+        else butterflyPerm d (decodeButterfly d σ.1.1) * (T false).perm⁻¹)
+    else (T false).levelCost t (PairRouting.childEmbedding x c hc false) σ.1.1 +
+      (T true).levelCost t (PairRouting.childEmbedding x c hc true) σ.1.2
+
+lemma InputTree.levelCost_node (t d : ℕ) (ξ : Card d → Bool) (T : Bool → InputTree d)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (d+1))
+    (Y : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (η : Card d → Bool) :
+    (InputTree.node ξ T).levelCost t e ((coinStepEquiv d).symm (Y,η)) =
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x ξ
+      let hc := PairRouting.colors_compatible x ξ
+      if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x ξ)
+        (fun b => if b then butterflyPerm d (decodeButterfly d Y.2) * (T true).perm⁻¹
+          else butterflyPerm d (decodeButterfly d Y.1) * (T false).perm⁻¹)
+      else (T false).levelCost t (PairRouting.childEmbedding x c hc false) Y.1 +
+        (T true).levelCost t (PairRouting.childEmbedding x c hc true) Y.2 := by
+  rw [InputTree.levelCost]
+  rfl
+
+lemma InputTree.levelCost_gt (t d : ℕ) (T : InputTree d) (ht : d < t)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) :
+    T.levelCost t e Y = 0 := by
+  induction T generalizing ι with
+  | tip p => rfl
+  | @node d ξ T ih =>
+    rw [InputTree.levelCost,ite_eq_right (by omega : t ≠ d+1),ih false (by omega),ih true (by omega)]
+
+lemma InputTree.levelCost_adapted (t d : ℕ) (T : InputTree d)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) :
+    LayerAdapted d (t-1) (fun Y => T.levelCost t e Y) := by
+  induction T generalizing ι with
+  | tip p => intro Y Y' _; rfl
+  | @node d ξ T ih =>
+    intro Y Y' h
+    dsimp only
+    conv_lhs => rw [←(coinStepEquiv d).symm_apply_apply Y]
+    conv_rhs => rw [←(coinStepEquiv d).symm_apply_apply Y']
+    rw [InputTree.levelCost_node t d ξ T e ((coinStepEquiv d Y).1) ((coinStepEquiv d Y).2),
+      InputTree.levelCost_node t d ξ T e ((coinStepEquiv d Y').1) ((coinStepEquiv d Y').2)]
+    by_cases ht : t = d+1
+    · subst t
+      rw [ite_eq_left rfl,ite_eq_left rfl]
+      have he₀ : (coinStepEquiv d Y).1.1 = (coinStepEquiv d Y').1.1 := by
+        funext i
+        exact h (Sum.inr (false,i)) (by simpa only [switchHeight,Nat.add_sub_cancel] using switchHeight_le d i)
+      have he₁ : (coinStepEquiv d Y).1.2 = (coinStepEquiv d Y').1.2 := by
+        funext i
+        exact h (Sum.inr (true,i)) (by simpa only [switchHeight,Nat.add_sub_cancel] using switchHeight_le d i)
+      rw [he₀,he₁]
+    · rw [ite_eq_right ht,ite_eq_right ht]
+      congr 1
+      · apply ih false
+        intro i hi
+        exact h (Sum.inr (false,i)) hi
+      · apply ih true
+        intro i hi
+        exact h (Sum.inr (true,i)) hi
+
+theorem InputTree.level_partial_mgf (r s u : ℕ) (hu : u < r)
+    (T : InputTree (s+r)) {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+r))
+    (lo : Card r × SwitchIndex s → Bool) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(T.levelCost (s+u+1) e (assembleBits r s lo hi))) ≤
+      Real.exp (Fintype.card ι * cycleCoefficient u J q) := by
+  classical
+  induction r generalizing u ι with
+  | zero => omega
+  | succ r ih =>
+    simp only [Nat.add_succ] at *
+    cases T with
+    | tip p =>
+      simp only [InputTree.levelCost,pow_zero,finiteMean_const]
+      exact Real.one_le_exp (mul_nonneg (Nat.cast_nonneg _) (cycleCoefficient_nonneg u J q hq))
+    | node ξ T =>
+      let x := e.trans (headTailEquiv (s+r)).toEmbedding
+      let c := PairRouting.colors x ξ
+      let hc := PairRouting.colors_compatible x ξ
+      let e₀ := PairRouting.childEmbedding x c hc false
+      let e₁ := PairRouting.childEmbedding x c hc true
+      let e' := PairRouting.switchedEmbedding x ξ
+      let lo₀ : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons false j.1,j.2)
+      let lo₁ : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons true j.1,j.2)
+      let Y₀ := fun hi : Card s × SwitchIndex r → Bool => assembleBits r s lo₀ hi
+      let Y₁ := fun hi : Card s × SwitchIndex r → Bool => assembleBits r s lo₁ hi
+      let N := fun hi : (Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool) =>
+        PairRouting.alternatingCycles e' (fun b =>
+          if b then butterflyPerm (s+r) (decodeButterfly (s+r) (Y₁ hi.2)) * (T true).perm⁻¹
+          else butterflyPerm (s+r) (decodeButterfly (s+r) (Y₀ hi.1)) * (T false).perm⁻¹)
+      let L₀ := fun hi : Card s × SwitchIndex r → Bool =>
+        (T false).levelCost (s+u+1) e₀ (Y₀ hi)
+      let L₁ := fun hi : Card s × SwitchIndex r → Bool =>
+        (T true).levelCost (s+u+1) e₁ (Y₁ hi)
+      have hs (hi : (Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool))
+          (η : Card s × Card r → Bool) :
+          (InputTree.node ξ T).levelCost (s+u+1) e
+            (assembleBits (r+1) s lo ((highStepEquiv r s).symm (hi,η))) =
+            if u = r then N hi else L₀ hi.1 + L₁ hi.2 := by
+        rw [assembleBits_step]
+        rw [InputTree.levelCost_node (s+u+1) (s+r) ξ T e (Y₀ hi.1,Y₁ hi.2) (fun y => η (((cardSplit r s) y).2,((cardSplit r s) y).1))]
+        simp only [Nat.add_right_cancel_iff,Nat.add_left_cancel_iff]
+        rfl
+      rw [finiteMean_equiv (highStepEquiv r s),finiteMean_prod]
+      simp_rw [hs,finiteMean_const]
+      by_cases hur : u = r
+      · subst u
+        simp only [ite_true]
+        rw [finiteMean_prod,finiteMean_comm]
+        calc
+          _ ≤ finiteMean (fun _hi : Card s × SwitchIndex r → Bool =>
+            Real.exp (Fintype.card ι * cycleCoefficient r J q)) := by
+            apply finiteMean_mono
+            intro hi
+            have hn := node_partial_mgf r s e' (T false).perm (T true).perm
+              (butterflyPerm (s+r) (decodeButterfly (s+r) (Y₁ hi))) lo₀ J hJ q hq
+            exact hn
+          _ = _ := finiteMean_const _
+      · simp only [ite_eq_right hur,pow_add]
+        rw [finiteMean_prod_mul (fun hi => q ^ L₀ hi) (fun hi => q ^ L₁ hi)]
+        have hu' : u < r := by omega
+        have h₀ := ih u hu' (T false) e₀ lo₀
+        have h₁ := ih u hu' (T true) e₁ lo₁
+        have hp := mul_le_mul h₀ h₁ (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
+          (Real.exp_nonneg _)
+        refine hp.trans_eq ?_
+        rw [←Real.exp_add]
+        congr 1
+        have hcount := PairRouting.color_card_add c
+        have hcount' : (Fintype.card {i : ι // c i = false} : ℝ) +
+            Fintype.card {i : ι // c i = true} = Fintype.card ι := by exact_mod_cast hcount
+        change (Fintype.card {i : ι // c i = false} : ℝ) * cycleCoefficient u J q +
+            Fintype.card {i : ι // c i = true} * cycleCoefficient u J q = _
+        rw [←add_mul,hcount']
+
+
+noncomputable def InputTree.slabCost (l d : ℕ) (T : InputTree d) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) : ℕ :=
+  ∑ i : Fin l, T.levelCost (l+i.1+1) e Y
+
+lemma InputTree.slabCost_adapted (l d : ℕ) (T : InputTree d) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) :
+    LayerAdapted d (2*l) (fun Y => InputTree.slabCost l d T e Y) := by
+  apply layerAdapted_sum
+  intro i _
+  exact layerAdapted_mono (InputTree.levelCost_adapted (l+i.1+1) d T e) (by omega)
+
+theorem InputTree.slab_partial_mgf (r s l : ℕ) (hl : 0 < l) (hs : 2*s ≤ l) (T : InputTree (s+r))
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+r))
+    (lo : Card r × SwitchIndex s → Bool) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(InputTree.slabCost l (s+r) T e (assembleBits r s lo hi))) ≤
+      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
+  classical
+  let : NeZero l := ⟨ne_of_gt hl⟩
+  have hqp : 0 < q := lt_of_lt_of_le (by norm_num) hq
+  calc
+    _ ≤ finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      finiteMean (fun i : Fin l => q^(l*T.levelCost (l+i.1+1) e
+        (assembleBits r s lo hi)))) := by
+      apply finiteMean_mono
+      intro hi
+      simpa only [InputTree.slabCost,Fintype.card_fin] using
+        pow_sum_le_mean_powers (fun i : Fin l => T.levelCost (l+i.1+1) e
+          (assembleBits r s lo hi)) q hqp
+    _ = finiteMean (fun i : Fin l => finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(l*T.levelCost (l+i.1+1) e (assembleBits r s lo hi)))) := finiteMean_comm _
+    _ ≤ finiteMean (fun _i : Fin l =>
+      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64)))) := by
+      apply finiteMean_mono
+      intro i
+      by_cases ht : l+i.1+1 ≤ s+r
+      · have hs' : s ≤ l+i.1 := by omega
+        have he : l+i.1+1 = s+(l+i.1-s)+1 := by omega
+        have hu : l+i.1-s < r := by omega
+        have hm := InputTree.level_partial_mgf r s (l+i.1-s) hu T e lo
+          ⌈Real.exp ((l:ℝ)/32)⌉₊ (cycleCutoff_pos l) (q^l) (one_le_pow₀ hq)
+        have hc := cycleCoefficient_decay l (l+i.1-s) (by
+          rw [Nat.cast_sub hs',Nat.cast_add]
+          have hsr : (2:ℝ)*s ≤ l := by exact_mod_cast hs
+          have hi : (0:ℝ) ≤ i.1 := Nat.cast_nonneg _
+          linarith) (q^l) (one_le_pow₀ hq) (by
+            rw [Real.log_pow]
+            nlinarith [mul_le_mul_of_nonneg_left hlog (Nat.cast_nonneg l)])
+        simp_rw [he,pow_mul]
+        exact hm.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left hc (Nat.cast_nonneg _)))
+      · have hz := InputTree.levelCost_gt (l+i.1+1) (s+r) T (by omega) e
+        simp only [hz,Nat.mul_zero,pow_zero,finiteMean_const]
+        exact Real.one_le_exp_iff.mpr (by positivity)
+    _ = _ := finiteMean_const _
+
+
+def boundaryTree (L : ℕ) : (r : ℕ) → (Card r → Equiv.Perm (Card L)) →
+    (SwitchIndex (L+r) → Bool) → InputTree (L+r)
+  | 0, U, _ => .tip (U (fun j => Fin.elim0 j))
+  | r+1, U, X => .node (coinStepEquiv (L+r) X).2 (fun b =>
+      boundaryTree L r (fun u => U (Fin.cons b u))
+        (if b then (coinStepEquiv (L+r) X).1.2 else (coinStepEquiv (L+r) X).1.1))
+
+def boundaryOutput (L : ℕ) : (r : ℕ) → (SwitchIndex (L+r) → Bool) →
+    Card r → Equiv.Perm (Card L)
+  | 0, Y, _ => butterflyPerm L (decodeButterfly L Y)
+  | r+1, Y, u => boundaryOutput L r
+      (if u 0 then (coinStepEquiv (L+r) Y).1.2 else (coinStepEquiv (L+r) Y).1.1) (Fin.tail u)
+
+lemma boundaryOutput_adapted (L r : ℕ) :
+    LayerAdapted (L+r) L (boundaryOutput L r) := by
+  induction r with
+  | zero =>
+    intro Y Y' h
+    have he : Y = Y' := funext (fun i => h i (by simpa only [Nat.add_zero] using switchHeight_le L i))
+    rw [he]
+  | succ r ih =>
+    intro Y Y' h
+    funext u
+    dsimp only [boundaryOutput]
+    cases hb : u 0 <;> simp only [Bool.false_eq_true,↓reduceIte]
+    · exact congrFun (ih _ _ (fun i hi => h (Sum.inr (false,i)) hi)) (Fin.tail u)
+    · exact congrFun (ih _ _ (fun i hi => h (Sum.inr (true,i)) hi)) (Fin.tail u)
+
+noncomputable def modifiedBoundary (L r : ℕ) (E : Finset (Card r))
+    (U : Card r → Equiv.Perm (Card L)) (Y : SwitchIndex (L+r) → Bool) :
+    Card r → Equiv.Perm (Card L) := fun u => if u ∈ E then boundaryOutput L r Y u else U u
+
+lemma modifiedBoundary_adapted (L r : ℕ) (E : Finset (Card r))
+    (U : Card r → Equiv.Perm (Card L)) :
+    LayerAdapted (L+r) L (modifiedBoundary L r E U) := by
+  intro Y Y' h
+  unfold modifiedBoundary
+  rw [boundaryOutput_adapted L r Y Y' h]
+
+lemma permDistance_boundaryTree (L r : ℕ) (U V : Card r → Equiv.Perm (Card L))
+    (X : SwitchIndex (L+r) → Bool) :
+    permDistance (boundaryTree L r U X).perm (boundaryTree L r V X).perm =
+      centralDistance L r U V := by
+  induction r with
+  | zero =>
+    simp only [boundaryTree,InputTree.perm,centralDistance,Fintype.sum_unique]
+    congr 1
+  | succ r ih =>
+    simp only [boundaryTree,InputTree.perm,permDistance_mul_left]
+    rw [permDistance_childLift,centralDistance_step,ih,ih]
+
+lemma boundaryTree_levelCost_perturb (t L r : ℕ) (U V : Card r → Equiv.Perm (Card L))
+    (X : SwitchIndex (L+r) → Bool) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (L+r)) (Y : SwitchIndex (L+r) → Bool) :
+    (boundaryTree L r U X).levelCost t e Y ≤
+      (boundaryTree L r V X).levelCost t e Y + centralDistance L r U V := by
+  induction r generalizing ι with
+  | zero => simp only [boundaryTree,InputTree.levelCost,Nat.zero_add,Nat.zero_le]
+  | succ r ih =>
+    rw [boundaryTree,boundaryTree,InputTree.levelCost,InputTree.levelCost]
+    by_cases ht : t = L+r+1
+    · simp only [ite_eq_left ht]
+      have hh := alternatingCycles_distance_le
+        (PairRouting.switchedEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
+          (coinStepEquiv (L+r) X).2)
+        (fun b => if b then butterflyPerm (L+r) (decodeButterfly (L+r) (coinStepEquiv (L+r) Y).1.2) *
+            (boundaryTree L r (fun u => U (Fin.cons true u)) (coinStepEquiv (L+r) X).1.2).perm⁻¹
+          else butterflyPerm (L+r) (decodeButterfly (L+r) (coinStepEquiv (L+r) Y).1.1) *
+            (boundaryTree L r (fun u => U (Fin.cons false u)) (coinStepEquiv (L+r) X).1.1).perm⁻¹)
+        (fun b => if b then butterflyPerm (L+r) (decodeButterfly (L+r) (coinStepEquiv (L+r) Y).1.2) *
+            (boundaryTree L r (fun u => V (Fin.cons true u)) (coinStepEquiv (L+r) X).1.2).perm⁻¹
+          else butterflyPerm (L+r) (decodeButterfly (L+r) (coinStepEquiv (L+r) Y).1.1) *
+            (boundaryTree L r (fun u => V (Fin.cons false u)) (coinStepEquiv (L+r) X).1.1).perm⁻¹)
+      simpa only [Bool.false_eq_true,↓reduceIte,permDistance_mul_left,permDistance_inv,
+        permDistance_boundaryTree,←centralDistance_step] using hh
+    · simp only [ite_eq_right ht,Bool.false_eq_true,↓reduceIte]
+      rw [centralDistance_step]
+      have h₀ := ih (fun u => U (Fin.cons false u)) (fun u => V (Fin.cons false u))
+        (coinStepEquiv (L+r) X).1.1
+        (PairRouting.childEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
+          (PairRouting.colors (e.trans (headTailEquiv (L+r)).toEmbedding) (coinStepEquiv (L+r) X).2)
+          (PairRouting.colors_compatible _ _) false) (coinStepEquiv (L+r) Y).1.1
+      have h₁ := ih (fun u => U (Fin.cons true u)) (fun u => V (Fin.cons true u))
+        (coinStepEquiv (L+r) X).1.2
+        (PairRouting.childEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
+          (PairRouting.colors (e.trans (headTailEquiv (L+r)).toEmbedding) (coinStepEquiv (L+r) X).2)
+          (PairRouting.colors_compatible _ _) true) (coinStepEquiv (L+r) Y).1.2
+      omega
+
+lemma boundaryTree_slabCost_perturb (l L r : ℕ) (U V : Card r → Equiv.Perm (Card L))
+    (X : SwitchIndex (L+r) → Bool) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (L+r)) (Y : SwitchIndex (L+r) → Bool) :
+    InputTree.slabCost l (L+r) (boundaryTree L r U X) e Y ≤
+      InputTree.slabCost l (L+r) (boundaryTree L r V X) e Y + l*centralDistance L r U V := by
+  have hh := Finset.sum_le_sum (fun (i : Fin l) (_ : i ∈ Finset.univ) =>
+    boundaryTree_levelCost_perturb (l+i.val+1) L r U V X e Y)
+  simpa only [Finset.sum_add_distrib,Finset.sum_const,Finset.card_univ,Fintype.card_fin,
+    nsmul_eq_mul,Nat.cast_id,InputTree.slabCost] using hh
+
+lemma modifiedBoundary_distance_le (L r : ℕ) (E : Finset (Card r))
+    (U : Card r → Equiv.Perm (Card L)) (Y : SwitchIndex (L+r) → Bool) :
+    centralDistance L r (modifiedBoundary L r E U Y) U ≤ E.card*2^L := by
+  apply centralDistance_exceptional
+  intro u hu
+  exact ite_eq_right hu
+
+
+lemma InputTree.slabCost_gt (l d : ℕ) (T : InputTree d) (hd : d ≤ l)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) :
+    InputTree.slabCost l d T e Y = 0 := by
+  apply Finset.sum_eq_zero
+  intro i _
+  exact InputTree.levelCost_gt _ _ T (by omega) _ _
+
+lemma InputTree.random_slab_adapted_mul_le (d s l : ℕ) (hsd : s ≤ d) (hl : 0 < l)
+    (hs : 2*s ≤ l) (T : (SwitchIndex d → Bool) → InputTree d)
+    (hT : LayerAdapted d s T) {ι : Type*} [Fintype ι] (e : ι ↪ Card d)
+    (F : (SwitchIndex d → Bool) → ℝ) (hF : ∀ Y, 0 ≤ F Y) (hA : LayerAdapted d s F)
+    (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun Y => F Y*q^(InputTree.slabCost l d (T Y) e Y)) ≤
+      finiteMean F * Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
+  obtain ⟨r,rfl⟩ := Nat.exists_eq_add_of_le hsd
+  apply adapted_mul_mean_le r s F _ hF hA
+  intro lo
+  let hi₀ : Card s × SwitchIndex r → Bool := fun _ => false
+  have he (hi : Card s × SwitchIndex r → Bool) :
+      T (assembleBits r s lo hi) = T (assembleBits r s lo hi₀) :=
+    hT _ _ (assembleBits_agree r s lo hi hi₀)
+  simp_rw [he]
+  exact InputTree.slab_partial_mgf r s l hl hs _ e lo q hq hlog
+
+lemma InputTree.slab_mgf (l d : ℕ) (hl : 0 < l) (T : InputTree d)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun Y => q^(InputTree.slabCost l d T e Y)) ≤
+      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
+  have hh := InputTree.random_slab_adapted_mul_le d 0 l (Nat.zero_le _) hl (Nat.zero_le _)
+    (fun _ => T) (fun _ _ _ => rfl) e (fun _ => 1) (fun _ => by norm_num)
+    (fun _ _ _ => rfl) q hq hlog
+  simpa only [one_mul,finiteMean_const] using hh
+
+noncomputable def InputTree.familyCost (L n d : ℕ)
+    (T : (SwitchIndex d → Bool) → InputTree d) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) : ℕ :=
+  ∑ j ∈ Finset.range n, InputTree.slabCost (L*4^j) d (T Y) e Y
+
+lemma InputTree.random_slabCost_adapted (l d s : ℕ) (T : (SwitchIndex d → Bool) → InputTree d)
+    (hT : LayerAdapted d s T) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) :
+    LayerAdapted d (max s (2*l)) (fun Y => InputTree.slabCost l d (T Y) e Y) := by
+  intro Y Y' h
+  dsimp only
+  rw [hT _ _ (agreeThrough_mono (le_max_left _ _) h)]
+  exact InputTree.slabCost_adapted l d (T Y') e Y Y'
+    (agreeThrough_mono (le_max_right _ _) h)
+
+lemma InputTree.familyCost_adapted (L n d s : ℕ) (T : (SwitchIndex d → Bool) → InputTree d)
+    (hT : LayerAdapted d s T) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) :
+    LayerAdapted d (max s (familyEnd L n)) (fun Y => InputTree.familyCost L n d T e Y) := by
+  apply layerAdapted_sum
+  intro j hj
+  exact layerAdapted_mono (InputTree.random_slabCost_adapted (L*4^j) d s T hT e)
+    (max_le_max_left s (familyEnd_ge L n j (Finset.mem_range.mp hj)))
+
+theorem InputTree.family_mgf_first (L n d s : ℕ) (hL : 0 < L) (hs : s ≤ L)
+    (T : (SwitchIndex d → Bool) → InputTree d) (hT : LayerAdapted d s T)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) (p : ℝ) (hp : 0 ≤ p)
+    (hfirst : finiteMean (fun Y => q^(InputTree.slabCost L d (T Y) e Y)) ≤
+      Real.exp (p+Fintype.card ι*(68*Real.exp (-(L:ℝ)/64)))) :
+    finiteMean (fun Y => q^(InputTree.familyCost L n d T e Y)) ≤
+      Real.exp (p+Fintype.card ι*(68*∑ j ∈ Finset.range n,
+        Real.exp (-((L*4^j:ℕ):ℝ)/64))) := by
+  induction n with
+  | zero =>
+    simp only [InputTree.familyCost,Finset.range_zero,Finset.sum_empty,pow_zero,
+      finiteMean_const,mul_zero,add_zero]
+    exact Real.one_le_exp hp
+  | succ n ih =>
+    by_cases hn : n = 0
+    · subst n
+      simpa only [InputTree.familyCost,Nat.zero_add,Finset.sum_range_one,pow_zero,mul_one] using hfirst
+    · let l := L*4^n
+      have hl : 0 < l := Nat.mul_pos hL (pow_pos (by decide) _)
+      have hsl : 2*familyEnd L n ≤ l := familyEnd_two L n
+      have hs' : s ≤ familyEnd L n := by
+        have hh := familyEnd_ge L n 0 (Nat.pos_of_ne_zero hn)
+        simp only [pow_zero,mul_one] at hh
+        omega
+      have hc : ∀ Y, InputTree.familyCost L (n+1) d T e Y =
+          InputTree.familyCost L n d T e Y+InputTree.slabCost l d (T Y) e Y := by
+        intro Y
+        exact Finset.sum_range_succ _ _
+      have hm : finiteMean (fun Y => q^(InputTree.familyCost L (n+1) d T e Y)) ≤
+          finiteMean (fun Y => q^(InputTree.familyCost L n d T e Y)) *
+            Real.exp (Fintype.card ι*(68*Real.exp (-(l:ℝ)/64))) := by
+        by_cases hld : l ≤ d
+        · simp_rw [hc,pow_add]
+          apply InputTree.random_slab_adapted_mul_le d (familyEnd L n) l (by omega) hl hsl T
+            (layerAdapted_mono hT hs') e _ (fun _ => pow_nonneg (by linarith) _)
+          · apply layerAdapted_comp
+            exact layerAdapted_mono (InputTree.familyCost_adapted L n d s T hT e) (max_le hs' le_rfl)
+          · exact hq
+          · exact hlog
+        · simp_rw [hc,InputTree.slabCost_gt l d _ (by omega),Nat.add_zero]
+          exact le_mul_of_one_le_right
+            (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
+            (Real.one_le_exp_iff.mpr (by positivity))
+      refine (hm.trans (mul_le_mul_of_nonneg_right ih (Real.exp_nonneg _))).trans_eq ?_
+      rw [←Real.exp_add,Finset.sum_range_succ]
+      congr 1
+      dsimp only [l]
+      ring
+
+lemma InputTree.family_mgf_first_decay (L n d s : ℕ) (hL : 0 < L) (hs : s ≤ L)
+    (T : (SwitchIndex d → Bool) → InputTree d) (hT : LayerAdapted d s T)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) (p : ℝ) (hp : 0 ≤ p)
+    (hfirst : finiteMean (fun Y => q^(InputTree.slabCost L d (T Y) e Y)) ≤
+      Real.exp (p+Fintype.card ι*(68*Real.exp (-(L:ℝ)/64)))) :
+    finiteMean (fun Y => q^(InputTree.familyCost L n d T e Y)) ≤
+      Real.exp (p+Fintype.card ι*heightDecay L) := by
+  apply (InputTree.family_mgf_first L n d s hL hs T hT e q hq hlog p hp hfirst).trans
+  apply Real.exp_le_exp.mpr
+  apply add_le_add_right
+  apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
+  have hh := mul_le_mul_of_nonneg_left (sum_family_decay L n hL) (by norm_num : (0:ℝ)≤68)
+  simpa only [heightDecay,mul_div_assoc] using hh
+
+
+noncomputable def modifiedInputTree (L r : ℕ) (E : Finset (Card r))
+    (U : Card r → Equiv.Perm (Card L)) (X Y : SwitchIndex (L+r) → Bool) : InputTree (L+r) :=
+  boundaryTree L r (modifiedBoundary L r E U Y) X
+
+lemma modifiedInputTree_adapted (L r : ℕ) (E : Finset (Card r))
+    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool) :
+    LayerAdapted (L+r) L (modifiedInputTree L r E U X) := by
+  intro Y Y' h
+  unfold modifiedInputTree
+  rw [modifiedBoundary_adapted L r E U Y Y' h]
+
+lemma modified_slab_mgf (l L r : ℕ) (hl : 0 < l) (E : Finset (Card r))
+    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun Y => q^(InputTree.slabCost l (L+r) (modifiedInputTree L r E U X Y) e Y)) ≤
+      Real.exp (((l*E.card*2^L:ℕ):ℝ)*Real.log q+
+        Fintype.card ι*(68*Real.exp (-(l:ℝ)/64))) := by
+  have hqp : 0 < q := by linarith
+  have hpoint (Y : SwitchIndex (L+r) → Bool) :
+      InputTree.slabCost l (L+r) (modifiedInputTree L r E U X Y) e Y ≤
+        InputTree.slabCost l (L+r) (boundaryTree L r U X) e Y+l*E.card*2^L := by
+    apply (boundaryTree_slabCost_perturb l L r (modifiedBoundary L r E U Y) U X e Y).trans
+    apply Nat.add_le_add_left
+    simpa only [Nat.mul_assoc] using Nat.mul_le_mul_left l (modifiedBoundary_distance_le L r E U Y)
+  calc
+    _ ≤ finiteMean (fun Y => q^(InputTree.slabCost l (L+r) (boundaryTree L r U X) e Y+l*E.card*2^L)) :=
+      finiteMean_mono (fun Y => pow_le_pow_right₀ hq (hpoint Y))
+    _ = finiteMean (fun Y => q^(InputTree.slabCost l (L+r) (boundaryTree L r U X) e Y)) *
+        q^(l*E.card*2^L) := by simp_rw [pow_add]; exact finiteMean_mul_const _ _
+    _ ≤ Real.exp (Fintype.card ι*(68*Real.exp (-(l:ℝ)/64))) * q^(l*E.card*2^L) :=
+      mul_le_mul_of_nonneg_right (InputTree.slab_mgf l (L+r) hl _ e q hq hlog) (pow_nonneg (by linarith) _)
+    _ = _ := by
+      have he : Real.exp (((l*E.card*2^L:ℕ):ℝ)*Real.log q) = q^(l*E.card*2^L) := by
+        rw [Real.exp_nat_mul,Real.exp_log hqp]
+      rw [Real.exp_add,he,mul_comm]
+
+lemma modified_family_mgf (l n L r : ℕ) (hl : 0 < l) (hLl : L ≤ l) (E : Finset (Card r))
+    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun Y => q^(InputTree.familyCost l n (L+r) (modifiedInputTree L r E U X) e Y)) ≤
+      Real.exp (((l*E.card*2^L:ℕ):ℝ)*Real.log q+Fintype.card ι*heightDecay l) := by
+  exact InputTree.family_mgf_first_decay l n (L+r) L hl hLl _
+    (modifiedInputTree_adapted L r E U X) e q hq hlog _
+    (mul_nonneg (Nat.cast_nonneg _) (Real.log_nonneg hq))
+    (modified_slab_mgf l L r hl E U X e q hq hlog)
+
+theorem modified_high_mgf (L n r : ℕ) (hL : 0 < L) (E : Finset (Card r))
+    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/500) :
+    finiteMean (fun Y => q^(InputTree.familyCost L n (L+r) (modifiedInputTree L r E U X) e Y+
+      InputTree.familyCost (2*L) n (L+r) (modifiedInputTree L r E U X) e Y)) ≤
+      Real.exp (Fintype.card ι*heightDecay L +
+        3*((L*E.card*2^L:ℕ):ℝ)*Real.log q) := by
+  have hq2 := one_le_pow₀ (n:=2) hq
+  have hlog2 : Real.log (q^2) ≤ 1/250 := by rw [Real.log_pow]; norm_num; linarith
+  have h₀ := modified_family_mgf L n L r hL le_rfl E U X e (q^2) hq2 hlog2
+  have h₁ := modified_family_mgf (2*L) n L r (by omega) (by omega) E U X e (q^2) hq2 hlog2
+  have h₁' : finiteMean (fun Y => (q^2)^(InputTree.familyCost (2*L) n (L+r)
+      (modifiedInputTree L r E U X) e Y)) ≤
+      Real.exp ((((2*L)*E.card*2^L:ℕ):ℝ)*Real.log (q^2)+Fintype.card ι*heightDecay L) := by
+    apply h₁.trans
+    apply Real.exp_le_exp.mpr
+    exact add_le_add_right (mul_le_mul_of_nonneg_left
+      (heightDecay_antitone (show L ≤ 2*L by omega)) (Nat.cast_nonneg _)) _
+  have hh := finiteMean_mul_exp_le
+    (fun Y => q^(InputTree.familyCost L n (L+r) (modifiedInputTree L r E U X) e Y))
+    (fun Y => q^(InputTree.familyCost (2*L) n (L+r) (modifiedInputTree L r E U X) e Y))
+    _ _ (by simpa only [pow_right_comm] using h₀) (by simpa only [pow_right_comm] using h₁')
+  simp only [←pow_add] at hh
+  refine hh.trans_eq ?_
+  congr 1
+  simp only [Real.log_pow,Nat.cast_mul,Nat.cast_ofNat]
+  ring
+
+noncomputable def InputTree.lowCost (H d : ℕ) (T : InputTree d)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) : ℕ :=
+  ∑ i ∈ Finset.range H, T.levelCost (i+1) e Y
+
+lemma InputTree.lowCost_slab (L d : ℕ) (T : InputTree d)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) :
+    InputTree.lowCost L d T e Y+InputTree.slabCost L d T e Y = InputTree.lowCost (2*L) d T e Y := by
+  rw [InputTree.lowCost,InputTree.lowCost,show 2*L=L+L by omega,Finset.sum_range_add]
+  congr 1
+  exact Fin.sum_univ_eq_sum_range (fun i => T.levelCost (L+i+1) e Y) L
+
+lemma InputTree.cost_family (L n d : ℕ) (T : (SwitchIndex d → Bool) → InputTree d)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) :
+    InputTree.lowCost L d (T Y) e Y + InputTree.familyCost L n d T e Y +
+      InputTree.familyCost (2*L) n d T e Y = InputTree.lowCost (L*4^n) d (T Y) e Y := by
+  induction n with
+  | zero => simp only [InputTree.familyCost,Finset.range_zero,Finset.sum_empty,Nat.add_zero,
+      pow_zero,mul_one]
+  | succ n ih =>
+    have he : (2*L)*4^n = 2*(L*4^n) := by ring
+    have he' : L*4^(n+1) = 2*(2*(L*4^n)) := by rw [pow_succ]; ring
+    simp only [InputTree.familyCost,Finset.sum_range_succ] at ih ⊢
+    rw [he,he',←InputTree.lowCost_slab,←InputTree.lowCost_slab,←ih]
+    omega
+
+lemma InputTree.lowCost_ge (H d : ℕ) (hd : d ≤ H) (T : InputTree d)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) :
+    InputTree.lowCost H d T e Y = InputTree.lowCost d d T e Y := by
+  symm
+  apply Finset.sum_subset (Finset.range_mono hd)
+  intro i _ hi
+  have hi' : d ≤ i := by simpa only [Finset.mem_range,not_lt] using hi
+  exact InputTree.levelCost_gt (i+1) d T (by omega) e Y
+
+lemma boundaryTree_levelCost_low (t L r : ℕ) (ht : t ≤ L)
+    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (Y : SwitchIndex (L+r) → Bool) :
+    (boundaryTree L r U X).levelCost t e Y = 0 := by
+  induction r generalizing ι with
+  | zero => rfl
+  | succ r ih =>
+    rw [boundaryTree,InputTree.levelCost,ite_eq_right (by omega : t ≠ L+r+1),ih,ih]
+
+lemma boundaryTree_lowCost_zero (L r : ℕ) (U : Card r → Equiv.Perm (Card L))
+    (X : SwitchIndex (L+r) → Bool) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (L+r)) (Y : SwitchIndex (L+r) → Bool) :
+    InputTree.lowCost L (L+r) (boundaryTree L r U X) e Y = 0 := by
+  apply Finset.sum_eq_zero
+  intro i hi
+  exact boundaryTree_levelCost_low (i+1) L r (by simpa only [Finset.mem_range,Nat.lt_iff_add_one_le] using hi) U X e Y
+
+theorem modified_cost_mgf (L r : ℕ) (hL : 0 < L) (E : Finset (Card r))
+    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/500) :
+    finiteMean (fun Y => q^(InputTree.lowCost (L+r) (L+r) (modifiedInputTree L r E U X Y) e Y)) ≤
+      Real.exp (Fintype.card ι*heightDecay L +
+        3*((L*E.card*2^L:ℕ):ℝ)*Real.log q) := by
+  have hbound : L+r ≤ L*4^(L+r) := by
+    have h₀ := pow_four_ge (L+r)
+    have h₁ := Nat.mul_le_mul_right (4^(L+r)) hL
+    nlinarith
+  have he (Y : SwitchIndex (L+r) → Bool) :=
+    InputTree.cost_family L (L+r) (L+r) (modifiedInputTree L r E U X) e Y
+  have hz (Y : SwitchIndex (L+r) → Bool) :
+      InputTree.lowCost L (L+r) (modifiedInputTree L r E U X Y) e Y = 0 :=
+    boundaryTree_lowCost_zero L r _ X e Y
+  simp only [hz,Nat.zero_add,InputTree.lowCost_ge _ _ hbound] at he
+  simp_rw [←he]
+  exact modified_high_mgf L (L+r) r hL E U X e q hq hlog
+
+
+noncomputable def exceptionChild {r : ℕ} (E : Finset (Card (r+1))) (b : Bool) : Finset (Card r) :=
+  Finset.univ.filter (fun u => Fin.cons b u ∈ E)
+
+lemma mem_exceptionChild {r : ℕ} (E : Finset (Card (r+1))) (b : Bool) (u : Card r) :
+    u ∈ exceptionChild E b ↔ Fin.cons b u ∈ E := by
+  classical
+  simp only [exceptionChild,Finset.mem_filter,Finset.mem_univ,true_and]
+
+lemma exceptionChild_card_add {r : ℕ} (E : Finset (Card (r+1))) :
+    (exceptionChild E false).card+(exceptionChild E true).card = E.card := by
+  classical
+  have he : (∑ u : Card (r+1), if u ∈ E then 1 else 0) = E.card := by simp
+  rw [Fintype.sum_equiv (headTailEquiv r) _
+    (fun x : Bool × Card r => if Fin.cons x.1 x.2 ∈ E then 1 else 0)
+    (fun x => by change (if x ∈ E then 1 else 0) = (if Fin.cons (x 0) (Fin.tail x) ∈ E then 1 else 0); rw [Fin.cons_self_tail])] at he
+  simp only [Fintype.sum_prod_type,Fintype.sum_bool] at he
+  simpa only [exceptionChild,Finset.card_filter,add_comm] using he
+
+def seedSplitEquiv (r : ℕ) (A : Type*) : (Card (r+1) → A) ≃ (Card r → A) × (Card r → A) where
+  toFun U := (fun u => U (Fin.cons false u),fun u => U (Fin.cons true u))
+  invFun U := fun u => if u 0 then U.2 (Fin.tail u) else U.1 (Fin.tail u)
+  left_inv U := by
+    funext u
+    dsimp only
+    split_ifs with h
+    · rw [←h,Fin.cons_self_tail]
+    · have h' : u 0 = false := by cases hh : u 0 <;> simp_all
+      rw [←h',Fin.cons_self_tail]
+  right_inv U := by simp only [Fin.cons_zero,Fin.tail_cons,Bool.false_eq_true,↓reduceIte]
+
+def seedSandwichShuffle {A B C D E : Type*} :
+    (A × B) × ((C × D) × E) ≃ ((A × C) × (B × D)) × E where
+  toFun x := (((x.1.1,x.2.1.1),(x.1.2,x.2.1.2)),x.2.2)
+  invFun x := ((x.1.1.1,x.1.2.1),((x.1.1.2,x.1.2.2),x.2))
+  left_inv _ := rfl
+  right_inv _ := rfl
+
+abbrev ModifiedCoins (L r : ℕ) := (Card r → Equiv.Perm (Card L)) × BenesCoins (L+r)
+
+def modifiedStepEquiv (L r : ℕ) : ModifiedCoins L (r+1) ≃
+    (ModifiedCoins L r × ModifiedCoins L r) × ((Card (L+r) → Bool) × (Card (L+r) → Bool)) :=
+  (Equiv.prodCongr (seedSplitEquiv r (Equiv.Perm (Card L))) (benesStepEquiv (L+r))).trans
+    seedSandwichShuffle
+
+noncomputable def modifiedPerm (L r : ℕ) (E : Finset (Card r)) (ω : ModifiedCoins L r) :
+    Equiv.Perm (Card (L+r)) :=
+  butterflyPerm (L+r) (decodeButterfly (L+r) ω.2.1) *
+    (modifiedInputTree L r E ω.1 ω.2.2 ω.2.1).perm⁻¹
+
+noncomputable def modifiedCost (L r : ℕ) (E : Finset (Card r))
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (ω : ModifiedCoins L r) : ℕ :=
+  InputTree.lowCost (L+r) (L+r) (modifiedInputTree L r E ω.1 ω.2.2 ω.2.1) e ω.2.1
+
+lemma modifiedBoundary_cons (L r : ℕ) (E : Finset (Card (r+1)))
+    (U : Card (r+1) → Equiv.Perm (Card L)) (Y : SwitchIndex (L+r+1) → Bool) (b : Bool) :
+    (fun u => modifiedBoundary L (r+1) E U Y (Fin.cons b u)) =
+    modifiedBoundary L r (exceptionChild E b) (fun u => U (Fin.cons b u))
+      (if b then (coinStepEquiv (L+r) Y).1.2 else (coinStepEquiv (L+r) Y).1.1) := by
+  funext u
+  simp only [modifiedBoundary,mem_exceptionChild,boundaryOutput,Fin.cons_zero,Fin.tail_cons]
+
+lemma modifiedInputTree_node (L r : ℕ) (E : Finset (Card (r+1)))
+    (ω : ModifiedCoins L r × ModifiedCoins L r) (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
+    let z := (modifiedStepEquiv L r).symm (ω,coins)
+    modifiedInputTree L (r+1) E z.1 z.2.2 z.2.1 =
+      InputTree.node coins.1 (fun b =>
+        let w := if b then ω.2 else ω.1
+        modifiedInputTree L r (exceptionChild E b) w.1 w.2.2 w.2.1) := by
+  dsimp only
+  unfold modifiedInputTree
+  rw [boundaryTree]
+  change InputTree.node coins.1 (fun b => boundaryTree L r _ _) = _
+  congr 1
+  funext b
+  rw [modifiedBoundary_cons]
+  cases b <;> rfl
+
+lemma modifiedPerm_step (L r : ℕ) (E : Finset (Card (r+1)))
+    (ω : ModifiedCoins L r × ModifiedCoins L r) (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
+    modifiedPerm L (r+1) E ((modifiedStepEquiv L r).symm (ω,coins)) =
+      pairSwitch coins.2 * childLift (fun b => modifiedPerm L r (exceptionChild E b)
+        (if b then ω.2 else ω.1)) * pairSwitch coins.1 := by
+  rw [modifiedPerm,modifiedInputTree_node,InputTree.perm,mul_inv_rev,pairSwitch_inv,childLift_inv]
+  change (butterflyPerm (L+r+1) (decodeButterfly (L+r+1)
+    ((coinStepEquiv (L+r)).symm ((ω.1.2.1,ω.2.2.1),coins.2)))) * _ = _
+  rw [butterflyPerm_step,mul_assoc,←mul_assoc (childLift _),childLift_mul,←mul_assoc]
+  congr 2
+  apply congrArg childLift
+  funext b
+  cases b <;> rfl
+
+lemma modifiedPerm_coordinates (L r : ℕ) (E : Finset (Card (r+1)))
+    (ω : ModifiedCoins L r × ModifiedCoins L r) (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool))
+    (x : Card (L+r+1)) :
+    headTailEquiv (L+r) (modifiedPerm L (r+1) E ((modifiedStepEquiv L r).symm (ω,coins)) x) =
+      PairRouting.sandwich coins.1 coins.2 (fun b => modifiedPerm L r (exceptionChild E b)
+        (if b then ω.2 else ω.1)) (headTailEquiv (L+r) x) := by
+  rw [modifiedPerm_step]
+  simp only [Equiv.Perm.mul_apply,headTail_switch,headTail_child]
+  rfl
+
+lemma InputTree.lowCost_node (d : ℕ) (ξ : Card d → Bool) (T : Bool → InputTree d)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (d+1)) (Y : (SwitchIndex d → Bool) × (SwitchIndex d → Bool))
+    (η : Card d → Bool) :
+    InputTree.lowCost (d+1) (d+1) (.node ξ T) e ((coinStepEquiv d).symm (Y,η)) =
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x ξ
+      let hc := PairRouting.colors_compatible x ξ
+      PairRouting.alternatingCycles (PairRouting.switchedEmbedding x ξ)
+        (fun b => if b then butterflyPerm d (decodeButterfly d Y.2) * (T true).perm⁻¹
+          else butterflyPerm d (decodeButterfly d Y.1) * (T false).perm⁻¹) +
+        InputTree.lowCost d d (T false) (PairRouting.childEmbedding x c hc false) Y.1 +
+        InputTree.lowCost d d (T true) (PairRouting.childEmbedding x c hc true) Y.2 := by
+  dsimp only
+  rw [InputTree.lowCost,Finset.sum_range_succ,InputTree.levelCost_node,ite_eq_left rfl]
+  have he : (∑ i ∈ Finset.range d,
+      (.node ξ T : InputTree (d+1)).levelCost (i+1) e ((coinStepEquiv d).symm (Y,η))) =
+      InputTree.lowCost d d (T false)
+        (PairRouting.childEmbedding (e.trans (headTailEquiv d).toEmbedding)
+          (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) ξ) (PairRouting.colors_compatible _ _) false) Y.1 +
+      InputTree.lowCost d d (T true)
+        (PairRouting.childEmbedding (e.trans (headTailEquiv d).toEmbedding)
+          (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) ξ) (PairRouting.colors_compatible _ _) true) Y.2 := by
+    rw [InputTree.lowCost,InputTree.lowCost,←Finset.sum_add_distrib]
+    apply Finset.sum_congr rfl
+    intro i hi
+    rw [InputTree.levelCost_node,ite_eq_right (by simp only [Finset.mem_range] at hi; omega : i+1 ≠ d+1)]
+  rw [he]
+  omega
+
+noncomputable def modifiedAssignmentCost (L r : ℕ) (E : Finset (Card (r+1)))
+    {ι : Type*} [Fintype ι] (x : ι ↪ Bool × Card (L+r)) (c : ι → Bool)
+    (hc : PairRouting.Compatible x c) (ω : ModifiedCoins L r × ModifiedCoins L r) : ℕ :=
+  PairRouting.alternatingCycles (PairRouting.coloredEmbedding x c hc)
+    (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) +
+  modifiedCost L r (exceptionChild E false) (PairRouting.childEmbedding x c hc false) ω.1 +
+  modifiedCost L r (exceptionChild E true) (PairRouting.childEmbedding x c hc true) ω.2
+
+lemma modifiedCost_step (L r : ℕ) (E : Finset (Card (r+1)))
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r+1))
+    (ω : ModifiedCoins L r × ModifiedCoins L r) (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
+    modifiedCost L (r+1) E e ((modifiedStepEquiv L r).symm (ω,coins)) =
+      modifiedAssignmentCost L r E (e.trans (headTailEquiv (L+r)).toEmbedding)
+        (PairRouting.colors (e.trans (headTailEquiv (L+r)).toEmbedding) coins.1)
+        (PairRouting.colors_compatible _ _) ω := by
+  rw [modifiedCost,modifiedInputTree_node]
+  change InputTree.lowCost (L+r+1) (L+r+1) _ e
+    ((coinStepEquiv (L+r)).symm ((ω.1.2.1,ω.2.2.1),coins.2)) = _
+  rw [InputTree.lowCost_node]
+  unfold modifiedAssignmentCost
+  dsimp only
+  simp only [Bool.false_eq_true,↓reduceIte]
+  rw [PairRouting.switchedEmbedding_eq_colored _ _ (PairRouting.colors_compatible _ _) _ rfl]
+  congr 2
+
+
+namespace PairRouting
+variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α] [DecidableEq ι]
+
+omit [Fintype α] [DecidableEq ι] in
+lemma childEmbedding_surjective (x : ι ↪ Bool × α) (hx : Function.Surjective x)
+    (c : ι → Bool) (hc : Compatible x c) (b : Bool) :
+    Function.Surjective (childEmbedding x c hc b) := by
+  intro y
+  have hs : Function.Surjective (coloredEmbedding x c hc) :=
+    (pairLayer (chosenCoin x c)).surjective.comp hx
+  obtain ⟨i,hi⟩ := hs (b,y)
+  rw [coloredEmbedding_apply] at hi
+  exact ⟨⟨i,congrArg Prod.fst hi⟩,congrArg Prod.snd hi⟩
+
+omit [DecidableEq ι] in
+lemma uniformPerm_tuple_upper (e f : ι ↪ α) (he : Function.Surjective e) (p : Equiv.Perm α) :
+    finiteMean (fun u : Equiv.Perm α => if ∀ i, (p*u⁻¹) (e i) = f i then (1:ℝ) else 0) ≤
+      1/(Fintype.card (Equiv.Perm α):ℝ) := by
+  classical
+  let M : Equiv.Perm α → Prop := fun u => ∀ i, (p*u⁻¹) (e i) = f i
+  have huniq (u v : Equiv.Perm α) (hu : M u) (hv : M v) : u = v := by
+    have hh : p*u⁻¹ = p*v⁻¹ := by
+      ext x
+      obtain ⟨i,rfl⟩ := he x
+      exact (hu i).trans (hv i).symm
+    exact inv_injective (mul_left_cancel hh)
+  unfold finiteMean
+  apply div_le_div_of_nonneg_right _ (Nat.cast_nonneg _)
+  change (∑ u, if M u then (1:ℝ) else 0) ≤ 1
+  by_cases h : ∃ u, M u
+  · obtain ⟨u,hu⟩ := h
+    rw [Finset.sum_eq_single u]
+    · simp only [hu,↓reduceIte,le_refl]
+    · intro v _ hv
+      exact ite_eq_right (fun hh => hv (huniq v u hh hu))
+    · simp
+  · push Not at h
+    simp only [h,ite_false,Finset.sum_const_zero,zero_le_one]
+
+end PairRouting
+
+noncomputable def centralFactor (L r : ℕ) (E : Finset (Card r)) (a : ℝ) : ℝ :=
+  (Nat.factorial (2^L):ℝ)^(-((2:ℝ)^r-E.card)*a)
+
+lemma centralFactor_nonneg (L r : ℕ) (E : Finset (Card r)) (a : ℝ) :
+    0 ≤ centralFactor L r E a := Real.rpow_nonneg (Nat.cast_nonneg _) _
+
+lemma centralFactor_step (L r : ℕ) (E : Finset (Card (r+1))) (a : ℝ) :
+    centralFactor L (r+1) E a =
+      centralFactor L r (exceptionChild E false) a * centralFactor L r (exceptionChild E true) a := by
+  unfold centralFactor
+  rw [←Real.rpow_add (Nat.cast_pos.mpr (Nat.factorial_pos _))]
+  congr 1
+  have he : ((exceptionChild E false).card:ℝ)+(exceptionChild E true).card = E.card := by
+    exact_mod_cast exceptionChild_card_add E
+  rw [pow_succ,←he]
+  ring
+
+lemma modifiedPerm_zero (L : ℕ) (E : Finset (Card 0)) (ω : ModifiedCoins L 0) :
+    modifiedPerm L 0 E ω =
+      if (fun j => Fin.elim0 j : Card 0) ∈ E then 1
+      else butterflyPerm L (decodeButterfly L ω.2.1)*(ω.1 (fun j => Fin.elim0 j))⁻¹ := by
+  unfold modifiedPerm modifiedInputTree boundaryTree InputTree.perm modifiedBoundary boundaryOutput
+  split_ifs
+  · exact mul_inv_cancel _
+  · rfl
+
+lemma modifiedCost_zero (L : ℕ) (E : Finset (Card 0)) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card L) (ω : ModifiedCoins L 0) : modifiedCost L 0 E e ω = 0 := by
+  simp only [modifiedCost,modifiedInputTree,boundaryTree,InputTree.lowCost,
+    InputTree.levelCost,Finset.sum_const_zero]
+
+lemma centralFactor_zero (L : ℕ) (E : Finset (Card 0)) (a : ℝ) :
+    centralFactor L 0 E a =
+      if (fun j => Fin.elim0 j : Card 0) ∈ E then 1 else (Nat.factorial (2^L):ℝ)^(-a) := by
+  classical
+  let z : Card 0 := fun j => Fin.elim0 j
+  have he : (E.card:ℝ) = if z ∈ E then 1 else 0 := by
+    by_cases hz : z ∈ E
+    · have he : E = {z} := by
+        ext u
+        rw [Subsingleton.elim u z]
+        simp only [hz,Finset.mem_singleton]
+      rw [he,ite_eq_left (by simpa only [he] using hz),Finset.card_singleton,Nat.cast_one]
+    · have he : E = ∅ := Finset.eq_empty_iff_forall_notMem.mpr (fun u hu => hz (Subsingleton.elim u z ▸ hu))
+      rw [he,Finset.card_empty,Nat.cast_zero,ite_eq_right (by simp)]
+  unfold centralFactor
+  rw [pow_zero,he]
+  split_ifs <;> simp
+
+lemma modified_probability_zero_le (L : ℕ) (E : Finset (Card 0))
+    {ι : Type*} [Fintype ι] (e f : ι ↪ Card L) (he : Function.Surjective e) :
+    PairRouting.tupleProbability (modifiedPerm L 0 E) e f ≤
+      if (fun j => Fin.elim0 j : Card 0) ∈ E then 1 else 1/(Nat.factorial (2^L):ℝ) := by
+  classical
+  rw [PairRouting.tupleProbability,finiteMean_prod,finiteMean_comm]
+  let B : ℝ := if (fun j => Fin.elim0 j : Card 0) ∈ E then 1 else 1/(Nat.factorial (2^L):ℝ)
+  change _ ≤ B
+  calc
+    _ ≤ finiteMean (fun _ω : BenesCoins L => B) := by
+      apply finiteMean_mono
+      intro ω
+      simp only [modifiedPerm_zero]
+      by_cases h : (fun j => Fin.elim0 j : Card 0) ∈ E
+      · simp only [h,↓reduceIte,Equiv.Perm.one_apply,finiteMean_const,B]
+        split_ifs <;> norm_num
+      · simp only [h,↓reduceIte,B]
+        rw [finiteMean_pi_eval (A := fun _ : Card 0 => Equiv.Perm (Card L)) (fun j => Fin.elim0 j) (fun u : Equiv.Perm (Card L) =>
+          if ∀ i, (butterflyPerm L (decodeButterfly L ω.1)*u⁻¹) (e i) = f i then (1:ℝ) else 0)]
+        convert PairRouting.uniformPerm_tuple_upper e f he (butterflyPerm L (decodeButterfly L ω.1)) using 1
+        rw [Fintype.card_perm,card_positions]
+    _ = _ := finiteMean_const B
+
+end Thorp
+
+end ThorpNine.SparseSaving
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/SparseSaving/Main.lean b/lean/OAI/Probability/ThorpRouting/SparseSaving/Main.lean
index c8e869a704176f71d6ecb53e9bb8a18fe4bb79f8..06a1e04c79ab3277b505143b8d6d5af348584d2d 100644
--- a/lean/OAI/Probability/ThorpRouting/SparseSaving/Main.lean
+++ b/lean/OAI/Probability/ThorpRouting/SparseSaving/Main.lean
@@ -1,1333 +1,1333 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.SparseSaving.ModifiedStageEquiv
-
-namespace OAI
-
-namespace ThorpNine.SparseSaving
-
-namespace Thorp.SparseContact
-open scoped BigOperators Classical
-
-noncomputable def typedMarks {N : Type*} [Fintype N] (d : ℕ)
-    (A : N → Finset (Card d)) (ω : N → (SwitchIndex d → Bool)) : Finset (N × Card d) :=
-  typedSet (fun t => butterflyMarks d (A t) (ω t))
-
-lemma typedMarks_card {N : Type*} [Fintype N] (d : ℕ)
-    (A : N → Finset (Card d)) (ω : N → (SwitchIndex d → Bool)) :
-    (typedMarks d A ω).card = ∑ t, (A t).card := by
-  classical
-  rw [typedMarks, typedSet_card]
-  apply Finset.sum_congr rfl
-  intro t _
-  exact Finset.card_image_of_injective _ (butterflyPerm d (decodeButterfly d (ω t))).injective
-
-lemma typedMarks_inclusion {N : Type*} [Fintype N] (d : ℕ)
-    (A : N → Finset (Card d)) (D : Finset (N × Card d)) :
-    inclusionMoment (typedMarks d A) D ≤
-      ∏ v ∈ D, ((A v.1).card:ℝ)/(2:ℝ)^d := by
-  classical
-  have he : inclusionMoment (typedMarks d A) D =
-      ∏ t, inclusionMoment (butterflyMarks d (A t)) (fiber D t) := by
-    rw [inclusionMoment_eq]
-    have hh (ω : N → (SwitchIndex d → Bool)) :
-        (if D ⊆ typedMarks d A ω then (1:ℝ) else 0) =
-        ∏ t, if fiber D t ⊆ butterflyMarks d (A t) (ω t) then (1:ℝ) else 0 := by
-      rw [Fintype.prod_boole]
-      simp only [typedMarks, subset_typedSet]
-    rw [finiteMean_congr hh]
-    simp only [inclusionMoment_eq]
-    exact finiteMean_pi_prod (ι := N) (Ω := fun _ => SwitchIndex d → Bool)
-      (fun t (ω : SwitchIndex d → Bool) =>
-        if fiber D t ⊆ butterflyMarks d (A t) ω then (1:ℝ) else 0)
-  have hprod : (∏ v ∈ D, ((A v.1).card:ℝ)/(2:ℝ)^d) =
-      ∏ t, ∏ x ∈ fiber D t, ((A t).card:ℝ)/(2:ℝ)^d := by
-    conv_lhs => rw [← typedSet_fiber D]
-    exact typedSet_prod _ _
-  rw [he, hprod]
-  apply Finset.prod_le_prod₀
-  · intro t _; exact inclusionMoment_nonneg _ _
-  · intro t _
-    simpa only [Finset.prod_const] using butterflyMarks_inclusion d (A t) (fiber D t)
-
-noncomputable def returnRoute {N : Type*} (d : ℕ) (b : N → Butterfly d)
-    (v : N × Card d) : Finset (SwitchIndex d) :=
-  routeDomain d ((butterflyPerm d (b v.1)).symm v.2) v.2
-
-noncomputable def returnGraph {N : Type*} (d : ℕ) (b : N → Butterfly d) :
-    SimpleGraph (N × Card d) where
-  Adj v w := v ≠ w ∧ ¬Disjoint (returnRoute d b v) (returnRoute d b w)
-  symm := ⟨fun _ _ h => ⟨h.1.symm, fun hh => h.2 hh.symm⟩⟩
-  loopless := ⟨fun _ h => h.1 rfl⟩
-
-lemma returnRoute_card {N : Type*} (d : ℕ) (b : N → Butterfly d) (v : N × Card d) :
-    (returnRoute d b v).card = d := card_routeDomain _ _ _
-
-lemma mem_returnRoute {N : Type*} (d : ℕ) (b : N → Butterfly d)
-    (t : N) (y : Card d) (i : SwitchIndex d) :
-    i ∈ returnRoute d b (t,y) ↔
-      y = butterflyPerm d (b t) (switchUsers d (b t) i).1 ∨
-      y = butterflyPerm d (b t) (switchUsers d (b t) i).2 := by
-  have h := mem_routeDomain_iff_user d (b t) ((butterflyPerm d (b t)).symm y) i
-  simpa only [Equiv.apply_symm_apply, returnRoute, Equiv.eq_symm_apply, eq_comm] using h
-
-lemma returnRoute_incidence {N : Type*} [Fintype N] (d : ℕ) (b : N → Butterfly d)
-    (p : N → ℝ) (i : SwitchIndex d) :
-    (∑ v : N × Card d, if i ∈ returnRoute d b v then p v.1 else 0) =
-      2 * ∑ t, p t := by
-  classical
-  rw [Fintype.sum_prod_type, Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro t _
-  have hne : butterflyPerm d (b t) (switchUsers d (b t) i).1 ≠
-      butterflyPerm d (b t) (switchUsers d (b t) i).2 :=
-    (butterflyPerm d (b t)).injective.ne (switchUsers_ne d (b t) i)
-  simp only [mem_returnRoute]
-  have he (y : Card d) :
-      (if y = butterflyPerm d (b t) (switchUsers d (b t) i).1 ∨
-        y = butterflyPerm d (b t) (switchUsers d (b t) i).2 then p t else 0) =
-      (if y = butterflyPerm d (b t) (switchUsers d (b t) i).1 then p t else 0) +
-      (if y = butterflyPerm d (b t) (switchUsers d (b t) i).2 then p t else 0) := by
-    split_ifs <;> simp_all
-  simp only [he, Finset.sum_add_distrib, Finset.sum_ite_eq', Finset.mem_univ, ite_true]
-  ring
-
-lemma returnGraph_degree {N : Type*} [Fintype N] (d : ℕ) (b : N → Butterfly d)
-    (p : N → ℝ) (hp : ∀ t, 0 ≤ p t) (v : N × Card d) :
-    (∑ w, if (returnGraph d b).Adj v w then p w.1 else 0) ≤
-      2*d*∑ t, p t := by
-  classical
-  calc
-    _ ≤ ∑ w, ∑ i ∈ returnRoute d b v, if i ∈ returnRoute d b w then p w.1 else 0 := by
-      apply Finset.sum_le_sum
-      intro w _
-      split_ifs with h
-      · obtain ⟨i, hi, hw⟩ := Finset.not_disjoint_iff.mp h.2
-        have hh := Finset.single_le_sum
-          (f := fun i => if i ∈ returnRoute d b w then p w.1 else 0)
-          (fun i _ => by split_ifs; exact hp _; exact le_rfl) hi
-        simpa only [ite_eq_left hw] using hh
-      · exact Finset.sum_nonneg (fun i _ => by split_ifs; exact hp _; exact le_rfl)
-    _ = ∑ i ∈ returnRoute d b v, ∑ w, if i ∈ returnRoute d b w then p w.1 else 0 :=
-      Finset.sum_comm
-    _ = _ := by simp only [returnRoute_incidence, Finset.sum_const, returnRoute_card,
-        nsmul_eq_mul]; ring
-
-lemma typed_weight_sum {N : Type*} [Fintype N] (d : ℕ)
-    (A : N → Finset (Card d)) :
-    (∑ v : N × Card d, ((A v.1).card:ℝ)/(2:ℝ)^d) = ∑ t, ((A t).card:ℝ) := by
-  rw [Fintype.sum_prod_type]
-  apply Finset.sum_congr rfl
-  intro t _
-  simp only [Finset.sum_const, Finset.card_univ, card_positions, Nat.cast_pow,
-    Nat.cast_ofNat, nsmul_eq_mul]
-  field_simp
-
-lemma mixed_return_noIsolated {N : Type*} [Fintype N] (d : ℕ)
-    (A : N → Finset (Card d)) (b : N → Butterfly d) (k : ℕ)
-    (hk : ∑ t, (A t).card = k)
-    (hδ : 0 < 2*(d:ℝ)*k/(2:ℝ)^d) (hδ1 : 2*(d:ℝ)*k/(2:ℝ)^d ≤ 1) :
-    finiteMean (fun ω : N → (SwitchIndex d → Bool) =>
-      if NoIsolated (returnGraph d b) (typedMarks d A ω) then (1:ℝ) else 0) ≤
-      (2*(d:ℝ)*k/(2:ℝ)^d)^((k:ℝ)/2) * Real.exp (Real.exp 1*k) := by
-  have hs : (∑ t, ((A t).card:ℝ)) = k := by exact_mod_cast hk
-  have hm := noIsolated_probability_le (returnGraph d b) (typedMarks d A) k
-    (fun ω => (typedMarks_card d A ω).trans hk)
-    (fun v => ((A v.1).card:ℝ)/(2:ℝ)^d) (fun v => by positivity)
-    (typedMarks_inclusion d A) _ hδ hδ1
-    (fun v => (returnGraph_degree d b (fun t => ((A t).card:ℝ)/(2:ℝ)^d)
-      (fun t => by positivity) v).trans_eq (by rw [← Finset.sum_div, hs]; ring))
-  simpa only [typed_weight_sum, hs] using hm
-
-
-lemma mean_fresh_cylinder {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (s : Finset ι) (v : ι → Bool) (f : (ι → Bool) → ℝ)
-    (hf : ∀ i ∈ s, ∀ ω, f (QueryTree.flipCoin i ω) = f ω) :
-    finiteMean (fun ω => if ∀ i ∈ s, ω i = v i then f ω else 0) =
-      finiteMean f / (2:ℝ)^s.card := by
-  induction s using Finset.induction_on with
-  | empty => simp [finiteMean]
-  | @insert i s hi ih =>
-    let g := fun ω => if ∀ j ∈ s, ω j = v j then f ω else 0
-    have hg : ∀ ω, g (QueryTree.flipCoin i ω) = g ω := by
-      intro ω
-      have he : (∀ j ∈ s, QueryTree.flipCoin i ω j = v j) ↔ ∀ j ∈ s, ω j = v j := by
-        apply forall₂_congr
-        intro j hj
-        rw [QueryTree.flipCoin_apply_ne i j (fun h => hi (h ▸ hj))]
-      simp only [g, he, hf i (Finset.mem_insert_self _ _) ω]
-    have he (ω : ι → Bool) :
-        (if ∀ j ∈ insert i s, ω j = v j then f ω else 0) =
-        (if ω i = v i then g ω else 0) := by
-      by_cases h : ω i = v i <;> simp [h,g]
-    rw [finiteMean_congr he, QueryTree.mean_indicator_bit g i (v i) hg]
-    have hs := ih (fun j hj => hf j (Finset.mem_insert_of_mem hj))
-    change finiteMean (fun ω => if ∀ j ∈ s, ω j = v j then f ω else 0) / 2 = _
-    rw [hs, Finset.card_insert_of_notMem hi, pow_succ]
-    ring
-
-noncomputable def endpointProb {L : Type*} (d : ℕ) (A : Finset L)
-    (x y : L → Card d) : ℝ := by
-  classical
-  exact finiteMean (fun ω : SwitchIndex d → Bool =>
-    if ∀ i ∈ A, butterflyPerm d (decodeButterfly d ω) (x i) = y i then (1:ℝ) else 0)
-
-lemma endpointProb_nonneg {L : Type*} (d : ℕ) (A : Finset L)
-    (x y : L → Card d) : 0 ≤ endpointProb d A x y := by
-  classical
-  exact finiteMean_nonneg (fun _ => by positivity)
-
-lemma endpointProb_insert_isolated {L : Type*} (d : ℕ) (A : Finset L)
-    (x y : L → Card d) (i : L)
-    (hiso : ∀ j ∈ A, Disjoint (routeDomain d (x i) (y i)) (routeDomain d (x j) (y j))) :
-    endpointProb d (insert i A) x y = endpointProb d A x y / (2:ℝ)^d := by
-  classical
-  let f := fun ω : SwitchIndex d → Bool =>
-    if ∀ j ∈ A, butterflyPerm d (decodeButterfly d ω) (x j) = y j then (1:ℝ) else 0
-  have hf : ∀ t ∈ routeDomain d (x i) (y i), ∀ ω, f (QueryTree.flipCoin t ω) = f ω := by
-    intro t ht ω
-    have he : (∀ j ∈ A, butterflyPerm d (decodeButterfly d (QueryTree.flipCoin t ω)) (x j) = y j) ↔
-        ∀ j ∈ A, butterflyPerm d (decodeButterfly d ω) (x j) = y j := by
-      apply forall₂_congr
-      intro j hj
-      rw [butterfly_endpoint_iff, butterfly_endpoint_iff]
-      apply forall₂_congr
-      intro s hs
-      rw [QueryTree.flipCoin_apply_ne t s (fun h =>
-        Finset.disjoint_left.mp (hiso j hj) ht (h ▸ hs))]
-    simp only [f, he]
-  have he (ω : SwitchIndex d → Bool) :
-      (if ∀ j ∈ insert i A, butterflyPerm d (decodeButterfly d ω) (x j) = y j then (1:ℝ) else 0) =
-      (if ∀ t ∈ routeDomain d (x i) (y i), ω t = routeValue d (x i) (y i) t then f ω else 0) := by
-    by_cases h : butterflyPerm d (decodeButterfly d ω) (x i) = y i
-    · have hh := (butterfly_endpoint_iff d ω (x i) (y i)).mp h
-      rw [ite_eq_left hh]
-      simp only [Finset.forall_mem_insert, h, true_and, f]
-    · have hh := mt (butterfly_endpoint_iff d ω (x i) (y i)).mpr h
-      simp only [Finset.forall_mem_insert, h, false_and, hh, ite_false]
-  change finiteMean _ = _
-  rw [finiteMean_congr he, mean_fresh_cylinder _ _ f hf, card_routeDomain]
-  rfl
-
-noncomputable def partialKernel {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x y : L → Card d) : ℝ :=
-  ((2:ℝ)^d)^A.card / ((2:ℝ)^d)^(Fintype.card L) * endpointProb d A x y
-
-lemma partialKernel_insert_isolated {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x y : L → Card d) (i : L) (hi : i ∉ A)
-    (hiso : ∀ j ∈ A, Disjoint (routeDomain d (x i) (y i)) (routeDomain d (x j) (y j))) :
-    partialKernel d (insert i A) x y = partialKernel d A x y := by
-  classical
-  rw [partialKernel, endpointProb_insert_isolated d A x y i hiso,
-    Finset.card_insert_of_notMem hi, pow_succ, partialKernel]
-  field_simp
-
-def ContactEvent {L : Type*} (d : ℕ) (x y : L → Card d) : Prop :=
-  ∀ i, ∃ j, i ≠ j ∧ ¬Disjoint (routeDomain d (x i) (y i)) (routeDomain d (x j) (y j))
-
-lemma alternating_cube_cancel {L : Type*} [Fintype L] (f : Finset L → ℝ)
-    (i : L) (hf : ∀ A, i ∉ A → f (insert i A) = f A) :
-    (∑ A : Finset L, (-1:ℝ)^A.card * f A) = 0 := by
-  classical
-  have hi : i ∈ (Finset.univ : Finset L) := Finset.mem_univ _
-  rw [← Finset.powerset_univ, ← Finset.insert_erase hi, Finset.sum_powerset_insert]
-  · rw [← Finset.sum_add_distrib]
-    apply Finset.sum_eq_zero
-    intro A hA
-    have hiA : i ∉ A := fun hh => Finset.notMem_erase i Finset.univ
-      ((Finset.mem_powerset.mp hA) hh)
-    rw [hf A hiA, Finset.card_insert_of_notMem hiA, pow_succ]
-    ring
-  · exact Finset.notMem_erase _ _
-
-lemma alternating_kernel_cancel {L : Type*} [Fintype L] (d : ℕ)
-    (x y : L → Card d) (h : ¬ContactEvent d x y) :
-    (∑ A : Finset L, (-1:ℝ)^A.card * partialKernel d A x y) = 0 := by
-  classical
-  unfold ContactEvent at h
-  push Not at h
-  obtain ⟨i, hi⟩ := h
-  apply alternating_cube_cancel _ i
-  intro A hiA
-  apply partialKernel_insert_isolated d A x y i hiA
-  intro j hj
-  exact hi j (fun hij => hiA (hij ▸ hj))
-
-
-noncomputable def networkOf {L : Type*} (A : Finset L) (i : L) : Option L := by
-  classical
-  exact if i ∈ A then none else some i
-
-noncomputable def mixedTuple {L : Type*} (d : ℕ) (A : Finset L)
-    (ω : Option L → (SwitchIndex d → Bool)) (x : L → Card d) : L → Card d :=
-  fun i => butterflyPerm d (decodeButterfly d (ω (networkOf A i))) (x i)
-
-noncomputable def mixedReturn {L : Type*} (d : ℕ) (A : Finset L)
-    (ω : Option L → (SwitchIndex d → Bool)) (y : L → Card d) : L → Card d :=
-  fun i => (butterflyPerm d (decodeButterfly d (ω (networkOf A i)))).symm (y i)
-
-noncomputable def labelsOf {L : Type*} (A : Finset L) : Option L → Finset L := by
-  classical
-  exact fun t => match t with
-    | none => A
-    | some i => if i ∈ A then ∅ else {i}
-
-lemma mem_labelsOf {L : Type*} (A : Finset L) (i : L) (t : Option L) :
-    i ∈ labelsOf A t ↔ networkOf A i = t := by
-  classical
-  cases t with
-  | none => simp [labelsOf, networkOf]
-  | some j => by_cases hj : j ∈ A <;> by_cases hi : i ∈ A <;> simp_all [labelsOf, networkOf] <;> intro h <;> subst j <;> contradiction
-
-lemma mixedTuple_eq_iff {L : Type*} (d : ℕ) (A : Finset L)
-    (ω : Option L → (SwitchIndex d → Bool)) (x y : L → Card d) :
-    mixedTuple d A ω x = y ↔ ∀ t, ∀ i ∈ labelsOf A t,
-      butterflyPerm d (decodeButterfly d (ω t)) (x i) = y i := by
-  constructor
-  · intro h t i hi
-    have he := congrFun h i
-    simpa only [mixedTuple, mem_labelsOf A i t |>.mp hi] using he
-  · intro h
-    funext i
-    exact h (networkOf A i) i (mem_labelsOf A i _ |>.mpr rfl)
-
-lemma mixedReturn_eq_iff {L : Type*} (d : ℕ) (A : Finset L)
-    (ω : Option L → (SwitchIndex d → Bool)) (y z : L → Card d) :
-    mixedReturn d A ω y = z ↔ mixedTuple d A ω z = y := by
-  constructor
-  · intro h
-    funext i
-    exact (Equiv.symm_apply_eq _).mp (congrFun h i) |>.symm
-  · intro h
-    funext i
-    exact (Equiv.symm_apply_eq _).mpr (congrFun h i).symm
-
-lemma endpointProb_singleton {L : Type*} (d : ℕ) (i : L) (x y : L → Card d) :
-    endpointProb d {i} x y = 1/(2:ℝ)^d := by
-  classical
-  simpa [endpointProb] using butterfly_single_card_uniform d (x i) (y i)
-
-lemma mixedTuple_mass {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x y : L → Card d) :
-    finiteMean (fun ω : Option L → (SwitchIndex d → Bool) =>
-      if mixedTuple d A ω x = y then (1:ℝ) else 0) = partialKernel d A x y := by
-  classical
-  have he (ω : Option L → (SwitchIndex d → Bool)) :
-      (if mixedTuple d A ω x = y then (1:ℝ) else 0) =
-      ∏ t, if ∀ i ∈ labelsOf A t, butterflyPerm d (decodeButterfly d (ω t)) (x i) = y i
-        then (1:ℝ) else 0 := by
-    simp only [Fintype.prod_boole, mixedTuple_eq_iff]
-  rw [finiteMean_congr he]
-  rw [finiteMean_pi_prod (fun t (w : SwitchIndex d → Bool) =>
-    if ∀ i ∈ labelsOf A t, butterflyPerm d (decodeButterfly d w) (x i) = y i then (1:ℝ) else 0)]
-  have hfac (t : Option L) : (finiteMean (fun w : SwitchIndex d → Bool =>
-      if ∀ i ∈ labelsOf A t, butterflyPerm d (decodeButterfly d w) (x i) = y i then (1:ℝ) else 0)) =
-      endpointProb d (labelsOf A t) x y := by
-        unfold endpointProb
-        apply finiteMean_congr
-        intro w
-        split_ifs <;> rfl
-  simp only [hfac]
-  rw [Fintype.prod_option]
-  have hi (i : L) : endpointProb d (labelsOf A (some i)) x y =
-      if i ∈ A then 1 else 1/(2:ℝ)^d := by
-    by_cases h : i ∈ A
-    · simp [labelsOf, h, endpointProb, finiteMean_const]
-    · simp only [labelsOf, ite_eq_right h, endpointProb_singleton]
-  rw [show labelsOf A none = A from rfl]
-  simp only [hi]
-  have hp : (∏ i : L, if i ∈ A then (1:ℝ) else 1/(2:ℝ)^d) =
-      (1/(2:ℝ)^d)^((Fintype.card L) - A.card) := by
-    rw [Finset.prod_ite]
-    simp only [Finset.prod_const_one, one_mul]
-    simp only [Finset.filter_not, Finset.filter_mem_eq_inter, Finset.univ_inter,
-      Finset.prod_const, Finset.card_sdiff, Finset.inter_univ, Finset.card_univ]
-  rw [hp, partialKernel, one_div, inv_pow]
-  have hc : A.card ≤ Fintype.card L := Finset.card_le_univ _
-  have hn : (((2:ℝ)^d)^A.card) * (((2:ℝ)^d)^(Fintype.card L - A.card)) =
-      ((2:ℝ)^d)^(Fintype.card L) := by rw [← pow_add, Nat.add_sub_of_le hc]
-  have hnz : ((2:ℝ)^d)^(Fintype.card L) ≠ 0 := by positivity
-  have hmz : ((2:ℝ)^d)^(Fintype.card L - A.card) ≠ 0 := by positivity
-  field_simp
-  rw [← hn]
-  ring
-
-lemma mixedReturn_mass {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (y z : L → Card d) :
-    finiteMean (fun ω : Option L → (SwitchIndex d → Bool) =>
-      if mixedReturn d A ω y = z then (1:ℝ) else 0) = partialKernel d A z y := by
-  simpa only [mixedReturn_eq_iff] using mixedTuple_mass d A z y
-
-noncomputable def mixedInitial {L : Type*} (d : ℕ) (A : Finset L)
-    (x : L → Card d) (t : Option L) : Finset (Card d) := by
-  classical
-  exact (labelsOf A t).image x
-
-lemma typedMarks_mixedInitial {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x : L → Card d) (ω : Option L → (SwitchIndex d → Bool)) :
-    typedMarks d (mixedInitial d A x) ω = Finset.univ.image
-      (fun i => (networkOf A i, mixedTuple d A ω x i)) := by
-  classical
-  ext v
-  simp only [typedMarks, mem_typedSet, butterflyMarks, mixedInitial,
-    Finset.mem_image, Finset.mem_univ, true_and]
-  constructor
-  · rintro ⟨z, ⟨i,hi,rfl⟩, hv⟩
-    have ht := mem_labelsOf A i v.1 |>.mp hi
-    exact ⟨i, Prod.ext ht (by simpa only [mixedTuple, ht] using hv)⟩
-  · rintro ⟨i, hi⟩
-    have ht := congrArg Prod.fst hi
-    have hv := congrArg Prod.snd hi
-    change networkOf A i = v.1 at ht
-    change mixedTuple d A ω x i = v.2 at hv
-    exact ⟨x i, ⟨i, mem_labelsOf A i v.1 |>.mpr ht, rfl⟩,
-      by simpa only [mixedTuple, ht] using hv⟩
-
-lemma typedTuple_injective {L : Type*} (d : ℕ) (A : Finset L)
-    (x : L → Card d) (hx : Function.Injective x)
-    (ω : Option L → (SwitchIndex d → Bool)) :
-    Function.Injective (fun i => (networkOf A i, mixedTuple d A ω x i)) := by
-  intro i j h
-  have ht := congrArg Prod.fst h
-  have hv := congrArg Prod.snd h
-  change networkOf A i = networkOf A j at ht
-  dsimp only [mixedTuple] at hv
-  rw [ht] at hv
-  exact hx ((butterflyPerm d (decodeButterfly d (ω (networkOf A j)))).injective hv)
-
-lemma mixedInitial_card {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x : L → Card d) (hx : Function.Injective x) :
-    (∑ t, (mixedInitial d A x t).card) = Fintype.card L := by
-  classical
-  let ω : Option L → (SwitchIndex d → Bool) := fun _ _ => false
-  rw [← typedMarks_card d (mixedInitial d A x) ω, typedMarks_mixedInitial,
-    Finset.card_image_of_injective _ (typedTuple_injective d A x hx ω), Finset.card_univ]
-
-lemma contactEvent_mixedReturn_iff {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x : L → Card d) (hx : Function.Injective x)
-    (ω b : Option L → (SwitchIndex d → Bool)) :
-    ContactEvent d (mixedReturn d A b (mixedTuple d A ω x)) (mixedTuple d A ω x) ↔
-      NoIsolated (returnGraph d (fun t => decodeButterfly d (b t)))
-        (typedMarks d (mixedInitial d A x) ω) := by
-  classical
-  rw [typedMarks_mixedInitial]
-  let f := fun i => (networkOf A i, mixedTuple d A ω x i)
-  have hf : Function.Injective f := typedTuple_injective d A x hx ω
-  constructor
-  · intro h v hv
-    obtain ⟨i,_,rfl⟩ := Finset.mem_image.mp hv
-    obtain ⟨j,hij,hc⟩ := h i
-    exact ⟨f j, Finset.mem_image.mpr ⟨j, Finset.mem_univ _,rfl⟩,
-      hf.ne hij, hc⟩
-  · intro h i
-    obtain ⟨v,hv,hij,hc⟩ := h (f i) (Finset.mem_image.mpr ⟨i,Finset.mem_univ _,rfl⟩)
-    obtain ⟨j,_,rfl⟩ := Finset.mem_image.mp hv
-    exact ⟨j, fun he => hij (congrArg f he), hc⟩
-
-lemma probability_mixed_contact {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x : L → Card d) (hx : Function.Injective x)
-    (b : Option L → (SwitchIndex d → Bool))
-    (hδ : 0 < 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)
-    (hδ1 : 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d ≤ 1) :
-    finiteMean (fun ω : Option L → (SwitchIndex d → Bool) =>
-      if ContactEvent d (mixedReturn d A b (mixedTuple d A ω x)) (mixedTuple d A ω x)
-        then (1:ℝ) else 0) ≤
-      (2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
-        Real.exp (Real.exp 1*(Fintype.card L)) := by
-  classical
-  simp only [contactEvent_mixedReturn_iff d A x hx]
-  convert mixed_return_noIsolated d (mixedInitial d A x) (fun t => decodeButterfly d (b t))
-      (Fintype.card L) (mixedInitial_card d A x hx) hδ hδ1 using 1
-  congr 1
-  exact Subsingleton.elim _ _
-
-
-lemma finiteMean_pushforward {Ω X : Type*} [Fintype Ω] [Fintype X]
-    (g : Ω → X) (f : X → ℝ) :
-    finiteMean (fun ω => f (g ω)) =
-      ∑ y, finiteMean (fun ω => if g ω = y then (1:ℝ) else 0) * f y := by
-  classical
-  simp_rw [← finiteMean_mul_const, ← finiteMean_sum]
-  apply finiteMean_congr
-  intro ω
-  simp
-
-lemma mixedTuple_expectation {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x : L → Card d) (f : (L → Card d) → ℝ) :
-    finiteMean (fun ω : Option L → (SwitchIndex d → Bool) => f (mixedTuple d A ω x)) =
-      ∑ y, partialKernel d A x y * f y := by
-  classical
-  rw [finiteMean_pushforward (fun ω : Option L → (SwitchIndex d → Bool) => mixedTuple d A ω x) f]
-  apply Finset.sum_congr rfl
-  intro y _
-  congr 1
-  convert mixedTuple_mass d A x y using 1
-  congr 1
-  funext ω
-  split_ifs <;> rfl
-
-lemma mixedReturn_expectation {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (y : L → Card d) (f : (L → Card d) → ℝ) :
-    finiteMean (fun ω : Option L → (SwitchIndex d → Bool) => f (mixedReturn d A ω y)) =
-      ∑ z, partialKernel d A z y * f z := by
-  classical
-  rw [finiteMean_pushforward (fun ω : Option L → (SwitchIndex d → Bool) => mixedReturn d A ω y) f]
-  apply Finset.sum_congr rfl
-  intro z _
-  congr 1
-  convert mixedReturn_mass d A y z using 1
-  congr 1
-  funext ω
-  split_ifs <;> rfl
-
-lemma partialKernel_nonneg {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x y : L → Card d) : 0 ≤ partialKernel d A x y :=
-  mul_nonneg (by positivity) (endpointProb_nonneg d A x y)
-
-noncomputable def retainedKernel {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x : L ↪ Card d) (y : L → Card d) : ℝ := by
-  classical
-  exact if ContactEvent d x y then partialKernel d A x y else 0
-
-lemma retainedKernel_nonneg {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x : L ↪ Card d) (y : L → Card d) : 0 ≤ retainedKernel d A x y := by
-  classical
-  unfold retainedKernel
-  split_ifs
-  · exact partialKernel_nonneg d A x y
-  · exact le_rfl
-
-lemma retainedKernel_le {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x : L ↪ Card d) (y : L → Card d) : retainedKernel d A x y ≤ partialKernel d A x y := by
-  classical
-  unfold retainedKernel
-  split_ifs
-  · exact le_rfl
-  · exact partialKernel_nonneg d A x y
-
-lemma retained_square_row {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x : L ↪ Card d)
-    (hδ : 0 < 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)
-    (hδ1 : 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d ≤ 1) :
-    (∑ x' : L ↪ Card d, ∑ y : L → Card d, retainedKernel d A x y * retainedKernel d A x' y) ≤
-      (2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
-        Real.exp (Real.exp 1*(Fintype.card L)) := by
-  classical
-  let f := fun y z : L → Card d => if ContactEvent d z y then (1:ℝ) else 0
-  have hsub (y : L → Card d) : (∑ x' : L ↪ Card d, retainedKernel d A x' y) ≤
-      ∑ z : L → Card d, partialKernel d A z y * f y z := by
-    have he (x' : L ↪ Card d) : retainedKernel d A x' y = partialKernel d A x' y * f y x' := by
-      unfold retainedKernel f
-      split_ifs <;> ring
-    simp_rw [he]
-    rw [← Finset.sum_image (f := fun z => partialKernel d A z y * f y z)
-      (Function.Embedding.coe_injective.injOn)]
-    apply Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _)
-    intro z _ _
-    exact mul_nonneg (partialKernel_nonneg d A z y) (by dsimp [f]; positivity)
-  calc
-    _ = ∑ y, retainedKernel d A x y * ∑ x' : L ↪ Card d, retainedKernel d A x' y := by
-      rw [Finset.sum_comm]
-      simp only [Finset.mul_sum]
-    _ ≤ ∑ y, partialKernel d A x y * ∑ z : L → Card d, partialKernel d A z y * f y z := by
-      apply Finset.sum_le_sum
-      intro y _
-      exact mul_le_mul (retainedKernel_le d A x y) (hsub y)
-        (Finset.sum_nonneg (fun x' _ => retainedKernel_nonneg d A x' y)) (partialKernel_nonneg d A x y)
-    _ = finiteMean (fun ω : Option L → (SwitchIndex d → Bool) =>
-        finiteMean (fun b : Option L → (SwitchIndex d → Bool) =>
-          f (mixedTuple d A ω x) (mixedReturn d A b (mixedTuple d A ω x)))) := by
-      rw [mixedTuple_expectation d A x (fun y =>
-        finiteMean (fun b : Option L → (SwitchIndex d → Bool) => f y (mixedReturn d A b y)))]
-      simp only [mixedReturn_expectation]
-    _ = finiteMean (fun b : Option L → (SwitchIndex d → Bool) =>
-        finiteMean (fun ω : Option L → (SwitchIndex d → Bool) =>
-          f (mixedTuple d A ω x) (mixedReturn d A b (mixedTuple d A ω x)))) := finiteMean_comm _
-    _ ≤ _ := by
-      apply (finiteMean_mono (fun b => probability_mixed_contact d A x x.injective b hδ hδ1)).trans_eq
-      exact finiteMean_const _
-
-noncomputable def rectOperator {X Y : Type*} [Fintype X] [Fintype Y]
-    (P : X → Y → ℝ) : EuclideanSpace ℂ Y →L[ℂ] EuclideanSpace ℂ X :=
-  LinearMap.toContinuousLinearMap
-    { toFun := fun v => WithLp.toLp 2 (fun x => ∑ y, (P x y : ℂ)*v y)
-      map_add' := by
-        intro v w
-        ext x
-        simp only [PiLp.add_apply, mul_add, Finset.sum_add_distrib]
-      map_smul' := by
-        intro c v
-        ext x
-        simp only [PiLp.smul_apply, smul_eq_mul, RingHom.id_apply]
-        change (∑ y, (P x y : ℂ)*(c*v y)) = c*∑ y, (P x y : ℂ)*v y
-        rw [Finset.mul_sum]
-        apply Finset.sum_congr rfl
-        intro y _
-        ring }
-
-@[simp] lemma rectOperator_apply {X Y : Type*} [Fintype X] [Fintype Y]
-    (P : X → Y → ℝ) (v : EuclideanSpace ℂ Y) (x : X) :
-    rectOperator P v x = ∑ y, (P x y:ℂ)*v y := rfl
-
-lemma rectOperator_adjoint {X Y : Type*} [Fintype X] [Fintype Y]
-    (P : X → Y → ℝ) : ContinuousLinearMap.adjoint (rectOperator P) =
-      rectOperator (fun y x => P x y) := by
-  apply Eq.symm
-  rw [ContinuousLinearMap.eq_adjoint_iff]
-  intro v w
-  simp only [PiLp.inner_apply, RCLike.inner_apply, rectOperator_apply,
-    map_sum, map_mul, Complex.conj_ofReal, Finset.sum_mul, Finset.mul_sum]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro x _
-  apply Finset.sum_congr rfl
-  intro y _
-  ring
-
-lemma rectOperator_comp_transpose {X Y : Type*} [Fintype X] [Fintype Y]
-    (P : X → Y → ℝ) : (rectOperator P).comp (rectOperator (fun y x => P x y)) =
-      rectOperator (fun x x' => ∑ y, P x y * P x' y) := by
-  ext v x
-  simp only [ContinuousLinearMap.comp_apply, rectOperator_apply, Finset.mul_sum,
-    Complex.ofReal_sum, Complex.ofReal_mul, Finset.sum_mul]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro x' _
-  apply Finset.sum_congr rfl
-  intro y _
-  ring
-
-lemma rectOperator_norm_sq_le {X Y : Type*} [Fintype X] [Fintype Y]
-    (P : X → Y → ℝ) (hP : ∀ x y, 0 ≤ P x y) (B : ℝ) (hB : 0 ≤ B)
-    (hrow : ∀ x, (∑ x', ∑ y, P x y * P x' y) ≤ B) :
-    ‖rectOperator P‖^2 ≤ B := by
-  let S := fun x x' => ∑ y, P x y * P x' y
-  have hs : ∀ x x', S x x' = S x' x := by
-    intro x x'
-    unfold S
-    apply Finset.sum_congr rfl
-    intro y _
-    ring
-  have hbound : ‖rectOperator S‖ ≤ B := by
-    apply ContinuousLinearMap.opNorm_le_bound _ hB
-    intro v
-    exact DensityTransfer.schur_bound S
-      (fun x x' => Finset.sum_nonneg (fun y _ => mul_nonneg (hP x y) (hP x' y))) B hB
-      hrow (fun x' => by
-        calc (∑ x, S x x') = ∑ x, S x' x := Finset.sum_congr rfl (fun x _ => hs x x')
-             _ ≤ B := hrow x') v
-  have hn := ContinuousLinearMap.norm_adjoint_comp_self
-    (ContinuousLinearMap.adjoint (rectOperator P))
-  rw [ContinuousLinearMap.adjoint_adjoint, (ContinuousLinearMap.adjoint.norm_map _),
-    rectOperator_adjoint, rectOperator_comp_transpose] at hn
-  rw [pow_two, ← hn]
-  exact hbound
-
-lemma retained_norm_sq {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (hδ : 0 < 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)
-    (hδ1 : 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d ≤ 1) :
-    ‖rectOperator (retainedKernel d A)‖^2 ≤
-      (2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
-        Real.exp (Real.exp 1*(Fintype.card L)) :=
-  rectOperator_norm_sq_le _ (retainedKernel_nonneg d A) _ (by positivity)
-    (fun x => retained_square_row d A x hδ hδ1)
-
-
-noncomputable def pointStabilizer {α : Type*} (A : Finset α) : Subgroup (Equiv.Perm α) where
-  carrier := {p | ∀ x ∈ A, p x = x}
-  one_mem' := by intro x _; rfl
-  mul_mem' := by intro p q hp hq x hx; change p (q x) = x; rw [hq x hx, hp x hx]
-  inv_mem' := by intro p hp x hx; exact (Equiv.Perm.inv_eq_iff_eq).mpr (hp x hx).symm
-
-lemma swap_mem_pointStabilizer {α : Type*} [DecidableEq α] (A : Finset α)
-    (x y : α) (hx : x ∉ A) (hy : y ∉ A) : Equiv.swap x y ∈ pointStabilizer A := by
-  intro z hz
-  exact Equiv.swap_apply_of_ne_of_ne (fun h => hx (h ▸ hz)) (fun h => hy (h ▸ hz))
-
-lemma point_symmetrizer_zero (μ : YoungDiagram) (A : Finset (Specht.Cell μ))
-    (hA : A.card < μ.card - μ.rowLen 0) (v : Specht.Tabloid μ → ℂ) (hv : v ∈ Specht.space μ) :
-    Specht.symmetrizer (pointStabilizer A) (Specht.tabloidRep μ) v = 0 := by
-  classical
-  induction hv using Submodule.span_induction with
-  | mem v hv =>
-    obtain ⟨p,rfl⟩ := hv
-    let e : {x : Specht.Cell μ // p x ∉ A} ≃ {x : Specht.Cell μ // x ∉ A} :=
-      { toFun := fun x => ⟨p x.1,x.2⟩
-        invFun := fun x => ⟨p.symm x.1, by simpa only [p.apply_symm_apply] using x.2⟩
-        left_inv := fun x => by apply Subtype.ext; exact p.symm_apply_apply _
-        right_inv := fun x => by apply Subtype.ext; exact p.apply_symm_apply _ }
-    have hc : μ.rowLen 0 < Fintype.card {x : Specht.Cell μ // p x ∉ A} := by
-      rw [Fintype.card_congr e, Fintype.card_subtype_compl, Specht.card_cell, Fintype.card_coe]
-      have hr := Specht.rowLen_zero_le_card μ
-      omega
-    obtain ⟨x,y,hne,hcol⟩ := Fintype.exists_ne_map_eq_of_card_lt
-      (fun x : {x : Specht.Cell μ // p x ∉ A} => Specht.colIndex x.1)
-      (by simpa only [Fintype.card_fin] using hc)
-    have hxy : x.1 ≠ y.1 := fun h => hne (Subtype.ext h)
-    let h : pointStabilizer A := ⟨Equiv.swap (p x.1) (p y.1),
-      swap_mem_pointStabilizer A _ _ x.2 y.2⟩
-    apply Specht.symmetrizer_eq_zero_of_neg _ _ h
-    change Specht.tabloidRep μ (Equiv.swap (p x.1) (p y.1))
-      (Specht.tabloidRep μ p (Specht.polytabloid μ)) = -Specht.tabloidRep μ p (Specht.polytabloid μ)
-    rw [←Module.End.mul_apply, ←map_mul, ←Equiv.mul_swap_eq_swap_mul]
-    rw [map_mul, Module.End.mul_apply,
-      Specht.swap_polytabloid x.1 y.1 hxy (congrArg Fin.val hcol), map_neg]
-  | zero => exact map_zero _
-  | add v w hv hw ihv ihw => simp only [map_add,ihv,ihw,add_zero]
-  | smul a v hv ih => simp only [map_smul,ih,smul_zero]
-
-lemma point_fixed_zero (μ : YoungDiagram) (A : Finset (Specht.Cell μ))
-    (hA : A.card < μ.card - μ.rowLen 0) (v : Specht.Tabloid μ → ℂ) (hv : v ∈ Specht.space μ)
-    (hfix : ∀ p, p ∈ pointStabilizer A → Specht.tabloidRep μ p v = v) : v = 0 := by
-  classical
-  have hz := point_symmetrizer_zero μ A hA v hv
-  have he : Specht.symmetrizer (pointStabilizer A) (Specht.tabloidRep μ) v =
-      (Fintype.card (pointStabilizer A) : ℂ) • v := by
-    simp only [Specht.symmetrizer, LinearMap.sum_apply, hfix _ (Subtype.prop _),
-      Finset.sum_const, Finset.card_univ, nsmul_eq_mul, Nat.cast_smul_eq_nsmul]
-  rw [he] at hz
-  exact (smul_eq_zero.mp hz).resolve_left (by exact_mod_cast Fintype.card_ne_zero)
-
-lemma point_unitary_fixed_zero (μ : YoungDiagram) (A : Finset (Specht.Cell μ))
-    (hA : A.card < μ.card - μ.rowLen 0) (v : Specht.hilbertSpace μ)
-    (hfix : ∀ p, p ∈ pointStabilizer A → Specht.unitaryRepresentation μ p v = v) : v = 0 := by
-  let w := (Specht.spaceHilbertEquiv μ).symm v
-  have hw : ∀ p, p ∈ pointStabilizer A → Specht.tabloidRep μ p w.val = w.val := by
-    intro p hp
-    have he := congrArg ((Specht.spaceHilbertEquiv μ).symm) (hfix p hp)
-    change Specht.representation μ p w = w at he
-    exact congrArg Subtype.val he
-  have hz := point_fixed_zero μ A hA w.val w.prop hw
-  have hw0 : w = 0 := Subtype.ext hz
-  have he := congrArg (Specht.spaceHilbertEquiv μ) hw0
-  simpa only [w, LinearEquiv.apply_symm_apply, map_zero] using he
-
-lemma point_relabelled_fixed_zero {α : Type*} [Fintype α] (μ : YoungDiagram)
-    (e : α ≃ Specht.Cell μ) (A : Finset α) (hA : A.card < μ.card - μ.rowLen 0)
-    (v : Specht.hilbertSpace μ)
-    (hfix : ∀ p, p ∈ pointStabilizer A → Specht.relabelledUnitary μ e p v = v) : v = 0 := by
-  classical
-  apply point_unitary_fixed_zero μ (A.image e) (by simpa only [Finset.card_image_of_injective _ e.injective] using hA) v
-  intro p hp
-  have hf : e.symm.permCongr p ∈ pointStabilizer A := by
-    intro x hx
-    have hh := hp (e x) (Finset.mem_image.mpr ⟨x,hx,rfl⟩)
-    simp only [Equiv.permCongr_apply, Equiv.symm_symm, hh, e.symm_apply_apply]
-  have hh := hfix _ hf
-  change Specht.unitaryRepresentation μ (e.permCongr (e.symm.permCongr p)) v = v at hh
-  have he : e.permCongr (e.symm.permCongr p) = p := by
-    apply Equiv.ext
-    intro x
-    simp only [Equiv.permCongr_apply, Equiv.symm_symm, e.apply_symm_apply]
-  simpa only [he] using hh
-
-
-section Orbit
-variable {G X V : Type*} [Group G] [Fintype G] [Fintype X]
-    [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-    (a : G →* Equiv.Perm X) (x₀ : X) (ha : ∀ x, ∃ g, a g x₀ = x)
-    (ρ : Representation ℂ G V) (w : V) (hw : UnitaryFinite.IsFixed a x₀ ρ w)
-
-noncomputable def orbitVector (x : X) : V := ρ (UnitaryFinite.representative a x₀ ha x) w
-
-include hw in
-omit [Fintype G] [Fintype X] [FiniteDimensional ℂ V] in
-lemma orbitVector_at (g : G) : orbitVector a x₀ ha ρ w (a g x₀) = ρ g w := by
-  apply ext_inner_right ℂ
-  intro v
-  exact UnitaryFinite.coefficient_at a x₀ ha ρ w hw v g
-
-include hw in
-omit [Fintype G] [Fintype X] [FiniteDimensional ℂ V] in
-lemma orbitVector_action (g : G) (x : X) :
-    orbitVector a x₀ ha ρ w (a g x) = ρ g (orbitVector a x₀ ha ρ w x) := by
-  obtain ⟨h,rfl⟩ := ha x
-  rw [←Equiv.Perm.mul_apply, ←map_mul, orbitVector_at a x₀ ha ρ w hw,
-    orbitVector_at a x₀ ha ρ w hw, map_mul, Module.End.mul_apply]
-end Orbit
-
-def Harmonic {L α : Type*} [Fintype L] [Fintype α] (f : EuclideanSpace ℂ (L ↪ α)) : Prop :=
-  ∀ (A : Finset L), A ≠ Finset.univ → ∀ z : L → α,
-    (∑ y : L ↪ α, if ∀ i ∈ A, y i = z i then f y else 0) = 0
-
-lemma normalizedCoefficient_harmonic {L α : Type*} [Fintype L] [Fintype α]
-    (μ : YoungDiagram) (e : α ≃ Specht.Cell μ)
-    (hk : Fintype.card L = μ.card - μ.rowLen 0) (x₀ : L ↪ α)
-    (w : Specht.hilbertSpace μ)
-    (hw : UnitaryFinite.IsFixed UnitaryFinite.tupleAction x₀ (Specht.relabelledUnitary μ e) w)
-    (v : Specht.hilbertSpace μ) :
-    Harmonic (UnitaryFinite.normalizedCoefficient UnitaryFinite.tupleAction x₀
-      (UnitaryFinite.tupleAction_transitive x₀) (Specht.relabelledUnitary μ e) w v) := by
-  classical
-  intro A hA z
-  let ρ := Specht.relabelledUnitary μ e
-  let F := orbitVector UnitaryFinite.tupleAction x₀ (UnitaryFinite.tupleAction_transitive x₀) ρ w
-  let S : Specht.hilbertSpace μ := ∑ y : L ↪ α, if ∀ i ∈ A, y i = z i then F y else 0
-  have hcard : (A.image z).card < μ.card - μ.rowLen 0 := by
-    apply (Finset.card_image_le).trans_lt
-    rw [← hk]
-    exact Finset.card_lt_card (Finset.ssubset_univ_iff.mpr hA)
-  have hz : S = 0 := by
-    apply point_relabelled_fixed_zero μ e (A.image z) hcard
-    intro p hp
-    change ρ p S = S
-    simp only [S, map_sum, apply_ite, map_zero]
-    apply Fintype.sum_equiv (UnitaryFinite.tupleAction p)
-    intro y
-    have he : (∀ i ∈ A, UnitaryFinite.tupleAction p y i = z i) ↔ ∀ i ∈ A, y i = z i := by
-      apply forall₂_congr
-      intro i hi
-      have hpi : p (z i) = z i := hp _ (Finset.mem_image.mpr ⟨i, hi,rfl⟩)
-      change p (y i) = z i ↔ y i = z i
-      exact (show (p (y i) = z i) ↔ p (y i) = p (z i) by rw [hpi]).trans p.injective.eq_iff
-    simp only [he]
-    split_ifs
-    · exact (orbitVector_action UnitaryFinite.tupleAction x₀
-        (UnitaryFinite.tupleAction_transitive x₀) ρ w hw p y).symm
-    · rfl
-  let c : ℂ := Real.sqrt ((Module.finrank ℂ (Specht.hilbertSpace μ):ℝ)/(Fintype.card (L ↪ α):ℝ))
-  calc
-    _ = c * inner ℂ S v := by
-      dsimp only [S]
-      rw [sum_inner (𝕜 := ℂ) Finset.univ _ v, Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro y _
-      split_ifs <;> simp_all only [UnitaryFinite.normalizedCoefficient, PiLp.smul_apply,
-        smul_eq_mul, UnitaryFinite.coefficient, F, orbitVector, inner_zero_left (𝕜 := ℂ), mul_zero, c, ρ]
-    _ = 0 := by rw [hz, inner_zero_left (𝕜 := ℂ) v, mul_zero]
-
-end Thorp.SparseContact
-
-namespace Thorp.SparseContact
-
-section
-open scoped BigOperators Classical
-
-lemma sum_injections {L α M : Type*} [Fintype L] [Fintype α] [AddCommMonoid M]
-    (F : (L → α) → M) (hF : ∀ y, ¬Function.Injective y → F y = 0) :
-    (∑ y : L → α, F y) = ∑ y : L ↪ α, F y := by
-  classical
-  rw [← Finset.sum_image Function.Embedding.coe_injective.injOn]
-  apply Eq.symm
-  apply Finset.sum_subset (Finset.subset_univ _)
-  intro y _ hy
-  apply hF y
-  intro h
-  exact hy (Finset.mem_image.mpr ⟨⟨y,h⟩, Finset.mem_univ _, rfl⟩)
-
-noncomputable def zeroExtend {L α : Type*} [Fintype L] [Fintype α]
-    (f : EuclideanSpace ℂ (L ↪ α)) : EuclideanSpace ℂ (L → α) := by
-  classical
-  exact WithLp.toLp 2 (fun y => if h : Function.Injective y then f ⟨y,h⟩ else 0)
-
-@[simp] lemma zeroExtend_inj {L α : Type*} [Fintype L] [Fintype α]
-    (f : EuclideanSpace ℂ (L ↪ α)) (y : L ↪ α) : zeroExtend f y = f y := by
-  classical
-  simp only [zeroExtend, WithLp.ofLp_toLp, dite_eq_left y.injective]
-  congr 1
-
-lemma zeroExtend_noninj {L α : Type*} [Fintype L] [Fintype α]
-    (f : EuclideanSpace ℂ (L ↪ α)) (y : L → α) (hy : ¬Function.Injective y) : zeroExtend f y = 0 := by
-  classical
-  simp only [zeroExtend, WithLp.ofLp_toLp, dite_eq_right hy]
-
-lemma sum_zeroExtend_mul {L α : Type*} [Fintype L] [Fintype α]
-    (f : EuclideanSpace ℂ (L ↪ α)) (g : (L → α) → ℂ) :
-    (∑ y : L → α, g y * zeroExtend f y) = ∑ y : L ↪ α, g y * f y := by
-  rw [sum_injections _ (fun y hy => by rw [zeroExtend_noninj f y hy, mul_zero])]
-  simp only [zeroExtend_inj]
-
-lemma norm_zeroExtend {L α : Type*} [Fintype L] [Fintype α]
-    (f : EuclideanSpace ℂ (L ↪ α)) : ‖zeroExtend f‖ = ‖f‖ := by
-  have hs : ‖zeroExtend f‖^2 = ‖f‖^2 := by
-    rw [EuclideanSpace.norm_sq_eq, EuclideanSpace.norm_sq_eq,
-      sum_injections _ (fun y hy => by rw [zeroExtend_noninj f y hy]; simp)]
-    simp only [zeroExtend_inj]
-  nlinarith [norm_nonneg (zeroExtend f), norm_nonneg f]
-
-lemma partial_proper_zero {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (hA : A ≠ Finset.univ) (f : EuclideanSpace ℂ (L ↪ Card d)) (hf : Harmonic f) :
-    rectOperator (fun (x : L ↪ Card d) (y : L → Card d) => partialKernel d A x y) (zeroExtend f) = 0 := by
-  classical
-  ext x
-  rw [rectOperator_apply, sum_zeroExtend_mul]
-  change (∑ y : L ↪ Card d, (partialKernel d A x y : ℂ) * f y) = 0
-  simp only [partialKernel, endpointProb, finiteMean, Complex.ofReal_mul, Complex.ofReal_div,
-    Complex.ofReal_sum, Complex.ofReal_natCast]
-  simp_rw [div_eq_mul_inv, mul_assoc, ← Finset.mul_sum]
-  rw [mul_eq_zero]
-  right
-  simp only [Finset.sum_mul]
-  rw [Finset.sum_comm]
-  apply mul_eq_zero_of_right
-  apply Finset.sum_eq_zero
-  intro ω _
-  calc
-    _ = ∑ y : L ↪ Card d, (↑(Fintype.card (SwitchIndex d → Bool)) : ℂ)⁻¹ *
-        (if ∀ i ∈ A, y i = butterflyPerm d (decodeButterfly d ω) (x i) then f y else 0) :=
-      by
-        apply Finset.sum_congr rfl
-        intro y _
-        have hh : (∀ i ∈ A, butterflyPerm d (decodeButterfly d ω) (x i) = y i) ↔
-            (∀ i ∈ A, y i = butterflyPerm d (decodeButterfly d ω) (x i)) :=
-          forall₂_congr fun _ _ => eq_comm
-        by_cases h : ∀ i ∈ A, butterflyPerm d (decodeButterfly d ω) (x i) = y i
-        · simp only [ite_eq_left h, ite_eq_left (hh.mp h), Complex.ofReal_one, one_mul]
-        · simp only [ite_eq_right h, ite_eq_right (mt hh.mpr h), Complex.ofReal_zero, zero_mul, mul_zero]
-    _ = 0 := by
-      rw [← Finset.mul_sum]
-      apply mul_eq_zero_of_right
-      convert hf A hA (fun i => butterflyPerm d (decodeButterfly d ω) (x i)) using 1
-      apply Finset.sum_congr rfl
-      intro y _
-      split_ifs <;> rfl
-
-lemma partial_full_action {L : Type*} [Fintype L] (d : ℕ)
-    (f : EuclideanSpace ℂ (L ↪ Card d)) :
-    rectOperator (fun (x : L ↪ Card d) (y : L → Card d) => partialKernel d Finset.univ x y) (zeroExtend f) =
-      DensityTransfer.applyKernel (UnitaryFinite.actionKernel UnitaryFinite.tupleAction
-        (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))) f := by
-  classical
-  ext x
-  rw [rectOperator_apply, sum_zeroExtend_mul]
-  apply Finset.sum_congr rfl
-  intro y _
-  congr 2
-  simp only [partialKernel, Finset.card_univ, div_self (by positivity : ((2:ℝ)^d)^(Fintype.card L) ≠ 0),
-    one_mul, endpointProb, UnitaryFinite.actionKernel]
-  apply finiteMean_congr
-  intro ω
-  congr 1
-  simp only [Finset.mem_univ, forall_const, Function.Embedding.ext_iff]
-  rfl
-
-lemma alternating_retained_eq {L : Type*} [Fintype L] (d : ℕ)
-    (x : L ↪ Card d) (y : L → Card d) :
-    (∑ A : Finset L, (-1:ℝ)^A.card * retainedKernel d A x y) =
-      ∑ A : Finset L, (-1:ℝ)^A.card * partialKernel d A x y := by
-  classical
-  by_cases h : ContactEvent d x y
-  · simp only [retainedKernel, ite_eq_left h]
-  · simp only [retainedKernel, ite_eq_right h, mul_zero, Finset.sum_const_zero]
-    exact (alternating_kernel_cancel d x y h).symm
-
-lemma alternating_harmonic_action {L : Type*} [Fintype L] (d : ℕ)
-    (f : EuclideanSpace ℂ (L ↪ Card d)) (hf : Harmonic f) :
-    (∑ A : Finset L, ((-1:ℂ)^A.card) • rectOperator (retainedKernel d A) (zeroExtend f)) =
-      ((-1:ℂ)^(Fintype.card L)) • DensityTransfer.applyKernel
-        (UnitaryFinite.actionKernel UnitaryFinite.tupleAction
-          (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))) f := by
-  classical
-  calc
-    _ = ∑ A : Finset L, ((-1:ℂ)^A.card) •
-        rectOperator (fun (x : L ↪ Card d) y => partialKernel d A x y) (zeroExtend f) := by
-      ext x
-      simp only [WithLp.ofLp_sum, Finset.sum_apply, PiLp.smul_apply, rectOperator_apply, smul_eq_mul, Finset.mul_sum]
-      rw [Finset.sum_comm, Finset.sum_comm (s := (Finset.univ : Finset (Finset L)))]
-      apply Finset.sum_congr rfl
-      intro y _
-      simp_rw [← mul_assoc, ← Finset.sum_mul]
-      congr 1
-      have he := congrArg (fun t : ℝ => (t:ℂ)) (alternating_retained_eq d x y)
-      simpa only [Complex.ofReal_sum, Complex.ofReal_mul, Complex.ofReal_pow, Complex.ofReal_neg,
-        Complex.ofReal_one] using he
-    _ = ((-1:ℂ)^(Fintype.card L)) •
-        rectOperator (fun (x : L ↪ Card d) y => partialKernel d Finset.univ x y) (zeroExtend f) := by
-      rw [Finset.sum_eq_single (Finset.univ : Finset L)]
-      · simp only [Finset.card_univ]
-      · intro A _ hA
-        rw [partial_proper_zero d A hA f hf, smul_zero]
-      · simp
-    _ = _ := by rw [partial_full_action]
-
-lemma harmonic_action_bound {L : Type*} [Fintype L] (d : ℕ)
-    (f : EuclideanSpace ℂ (L ↪ Card d)) (hf : Harmonic f)
-    (hδ : 0 < 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)
-    (hδ1 : 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d ≤ 1) :
-    ‖DensityTransfer.applyKernel (UnitaryFinite.actionKernel UnitaryFinite.tupleAction
-      (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))) f‖ ≤
-      (2:ℝ)^(Fintype.card L) *
-        Real.sqrt ((2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
-          Real.exp (Real.exp 1*(Fintype.card L))) * ‖f‖ := by
-  classical
-  let B := (2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
-          Real.exp (Real.exp 1*(Fintype.card L))
-  have hB : 0 ≤ B := by dsimp [B]; positivity
-  have hnorm (A : Finset L) : ‖rectOperator (retainedKernel d A)‖ ≤ Real.sqrt B := by
-    have hh := retained_norm_sq d A hδ hδ1
-    have hs := Real.sq_sqrt hB
-    have hn := Real.sqrt_nonneg B
-    nlinarith
-  calc
-    _ = ‖((-1:ℂ)^(Fintype.card L)) • DensityTransfer.applyKernel
-        (UnitaryFinite.actionKernel UnitaryFinite.tupleAction
-          (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))) f‖ := by
-      simp only [norm_smul, norm_pow, norm_neg, norm_one, one_pow, one_mul]
-    _ = ‖∑ A : Finset L, ((-1:ℂ)^A.card) • rectOperator (retainedKernel d A) (zeroExtend f)‖ := by
-      rw [alternating_harmonic_action d f hf]
-    _ ≤ ∑ A : Finset L, ‖((-1:ℂ)^A.card) • rectOperator (retainedKernel d A) (zeroExtend f)‖ :=
-      norm_sum_le _ _
-    _ ≤ ∑ A : Finset L, Real.sqrt B * ‖f‖ := by
-      apply Finset.sum_le_sum
-      intro A _
-      simp only [norm_smul, norm_pow, norm_neg, norm_one, one_pow, one_mul]
-      calc _ ≤ ‖rectOperator (retainedKernel d A)‖ * ‖zeroExtend f‖ :=
-              ContinuousLinearMap.le_opNorm _ _
-           _ ≤ Real.sqrt B * ‖zeroExtend f‖ := mul_le_mul_of_nonneg_right (hnorm A) (norm_nonneg _)
-           _ = _ := by rw [norm_zeroExtend]
-    _ = _ := by simp only [Finset.sum_const, Finset.card_univ, Fintype.card_finset, nsmul_eq_mul,
-      Nat.cast_pow, Nat.cast_ofNat]; ring
-
-lemma harmonic_action_zero_one {L : Type*} [Fintype L] (d : ℕ)
-    (hL : Fintype.card L = 1) (f : EuclideanSpace ℂ (L ↪ Card d)) (hf : Harmonic f) :
-    DensityTransfer.applyKernel (UnitaryFinite.actionKernel UnitaryFinite.tupleAction
-      (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))) f = 0 := by
-  classical
-  obtain ⟨i,hi⟩ := Fintype.card_eq_one_iff.mp hL
-  have hz (A : Finset L) : rectOperator (retainedKernel d A) = 0 := by
-    ext v x
-    have hh (y : L → Card d) : ¬ContactEvent d x y := by
-      intro h
-      obtain ⟨j,hj,_⟩ := h i
-      exact hj (hi j).symm
-    simp only [rectOperator_apply, retainedKernel, ite_eq_right (hh _), Complex.ofReal_zero,
-      zero_mul, Finset.sum_const_zero, zero_apply, PiLp.zero_apply]
-  have he := alternating_harmonic_action d f hf
-  simp only [hz, zero_apply, smul_zero, Finset.sum_const_zero, hL, pow_one,
-    neg_smul, one_smul] at he
-  exact neg_eq_zero.mp he.symm
-
-end
-open scoped BigOperators
-
-def MainStatement : Prop :=
-  ∃ Cstar C : ℝ, 0 < Cstar ∧ 0 < C ∧
-    ∀ (d : ℕ), 1 ≤ d → ∀ (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ),
-      let k := μ.card - μ.rowLen 0
-      1 ≤ k → k < 2^d → ∀ reverse : Bool,
-        ‖sweepOperator d μ e reverse‖^2 ≤
-          min 1 ((Cstar * d * k / (2:ℝ)^d) ^ ((k:ℝ)/2)) ∧
-        (k = 1 → sweepOperator d μ e reverse = 0) ∧
-        ‖sweepOperator d μ e reverse‖^2 ≤
-          C^k * (1+(d:ℝ))^(C*k) * ((k:ℝ)/(2:ℝ)^d)^((k:ℝ)/2)
-
-end Thorp.SparseContact
-
-namespace Thorp.SparseContact
-open scoped BigOperators Classical
-
-lemma exists_exact_fixed_unit {α L : Type*} [Fintype α] [Fintype L]
-    (μ : YoungDiagram) (e : α ≃ Specht.Cell μ) (x₀ : L ↪ α)
-    (hL : Fintype.card L = μ.card - μ.rowLen 0) :
-    ∃ w : Specht.hilbertSpace μ,
-      UnitaryFinite.IsFixed UnitaryFinite.tupleAction x₀ (Specht.relabelledUnitary μ e) w ∧ ‖w‖ = 1 := by
-  let K := UnitaryFinite.fixedSpace (V := Specht.hilbertSpace μ) UnitaryFinite.tupleAction x₀ (Specht.relabelledUnitary μ e)
-  have hd : 0 < Module.finrank ℂ (Specht.hilbertSpace μ) := by
-    rw [Specht.finrank_hilbertSpace]
-    exact Module.finrank_pos_iff.mpr (Submodule.nontrivial_iff_ne_bot.mpr (Specht.space_ne_bot μ))
-  have hdim := Specht.relabelled_tuple_dimension μ e x₀
-  rw [hL, Nat.choose_self, mul_one] at hdim
-  have hm : 0 < Module.finrank ℂ K := by
-    by_contra h
-    have hz : Module.finrank ℂ K = 0 := by omega
-    change _ ≤ Module.finrank ℂ K * _ at hdim
-    rw [hz, zero_mul] at hdim
-    omega
-  let B := stdOrthonormalBasis ℂ K
-  let i : Fin (Module.finrank ℂ K) := ⟨0,hm⟩
-  exact ⟨(B i).val, (B i).prop, B.orthonormal.norm_eq_one i⟩
-
-lemma sweep_forward_norm {d : ℕ} (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ) (rev : Bool) :
-    ‖sweepOperator d μ e rev‖ = ‖sweepOperator d μ e false‖ := by
-  cases rev
-  · rfl
-  · simp only [sweepOperator, Bool.false_eq_true, ite_false, ite_true]
-    rw [UnitaryFinite.sampleOperator_inv _ (Specht.relabelledUnitary_unitary μ e),
-      ContinuousLinearMap.adjoint.norm_map]
-
-lemma sweep_norm_le_one (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ) (rev : Bool) :
-    ‖sweepOperator d μ e rev‖ ≤ 1 :=
-  UnitaryFinite.sampleOperator_norm_le _ (Specht.relabelledUnitary_unitary μ e) _
-
-section Generic
-variable {L V : Type*} [Fintype L] [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-  [FiniteDimensional ℂ V]
-
-lemma sample_sweep_bound (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V)
-    [Representation.IsIrreducible ρ] (hρ : UnitaryFinite.IsUnitary ρ)
-    (x₀ : L ↪ Card d) (w : V) (hw : UnitaryFinite.IsFixed UnitaryFinite.tupleAction x₀ ρ w)
-    (hn : ‖w‖ = 1)
-    (hh : ∀ v, Harmonic (UnitaryFinite.normalizedCoefficient UnitaryFinite.tupleAction x₀
-      (UnitaryFinite.tupleAction_transitive x₀) ρ w v))
-    (hδ : 0 < 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)
-    (hδ1 : 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d ≤ 1) :
-    ‖UnitaryFinite.sampleOperator ρ (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))‖ ≤
-      (2:ℝ)^(Fintype.card L) *
-        Real.sqrt ((2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
-          Real.exp (Real.exp 1*(Fintype.card L))) := by
-  rw [← ContinuousLinearMap.adjoint.norm_map (UnitaryFinite.sampleOperator ρ
-    (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))),
-    ← UnitaryFinite.sampleOperator_inv ρ hρ]
-  apply ContinuousLinearMap.opNorm_le_bound _ (by positivity)
-  intro v
-  rw [UnitaryFinite.sampleOperator_inv_apply]
-  have ha := harmonic_action_bound d _ (hh v) hδ hδ1
-  rw [UnitaryFinite.coefficient_random UnitaryFinite.tupleAction x₀
-    (UnitaryFinite.tupleAction_transitive x₀) ρ hρ _ w hw,
-    UnitaryFinite.normalizedCoefficient_norm _ _ _ ρ hρ w hw hn,
-    UnitaryFinite.normalizedCoefficient_norm _ _ _ ρ hρ w hw hn] at ha
-  exact ha
-
-lemma sample_sweep_zero_one (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V)
-    [Representation.IsIrreducible ρ] (hρ : UnitaryFinite.IsUnitary ρ)
-    (x₀ : L ↪ Card d) (w : V) (hw : UnitaryFinite.IsFixed UnitaryFinite.tupleAction x₀ ρ w)
-    (hn : ‖w‖ = 1)
-    (hh : ∀ v, Harmonic (UnitaryFinite.normalizedCoefficient UnitaryFinite.tupleAction x₀
-      (UnitaryFinite.tupleAction_transitive x₀) ρ w v))
-    (hL : Fintype.card L = 1) :
-    UnitaryFinite.sampleOperator ρ (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω)) = 0 := by
-  apply ContinuousLinearMap.adjoint.injective
-  rw [map_zero, ← UnitaryFinite.sampleOperator_inv ρ hρ]
-  apply ContinuousLinearMap.ext
-  intro v
-  rw [UnitaryFinite.sampleOperator_inv_apply]
-  have ha := harmonic_action_zero_one d hL _ (hh v)
-  have he := congrArg norm ha
-  rw [UnitaryFinite.coefficient_random UnitaryFinite.tupleAction x₀
-    (UnitaryFinite.tupleAction_transitive x₀) ρ hρ _ w hw,
-    UnitaryFinite.normalizedCoefficient_norm _ _ _ ρ hρ w hw hn, norm_zero] at he
-  exact norm_eq_zero.mp he
-end Generic
-
-lemma sweep_norm_bound (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ)
-    (hδ : 0 < 2*(d:ℝ)*(μ.card - μ.rowLen 0 : ℕ)/(2:ℝ)^d)
-    (hδ1 : 2*(d:ℝ)*(μ.card - μ.rowLen 0 : ℕ)/(2:ℝ)^d ≤ 1) (rev : Bool) :
-    ‖sweepOperator d μ e rev‖ ≤ (2:ℝ)^(μ.card - μ.rowLen 0) *
-      Real.sqrt ((2*(d:ℝ)*(μ.card - μ.rowLen 0 : ℕ)/(2:ℝ)^d)^(((μ.card - μ.rowLen 0 : ℕ):ℝ)/2) *
-        Real.exp (Real.exp 1*(μ.card - μ.rowLen 0 : ℕ))) := by
-  classical
-  let x₀ : Specht.Tail μ ↪ Card d := (Function.Embedding.subtype _).trans e.symm.toEmbedding
-  obtain ⟨w,hw,hn⟩ := exists_exact_fixed_unit μ e x₀ (Specht.tail_card μ)
-  let := Specht.relabelledUnitary_irreducible μ e
-  rw [sweep_forward_norm μ e rev]
-  have hh (v : Specht.hilbertSpace μ) : Harmonic (UnitaryFinite.normalizedCoefficient
-      (V := Specht.hilbertSpace μ) UnitaryFinite.tupleAction x₀
-      (UnitaryFinite.tupleAction_transitive x₀) (Specht.relabelledUnitary μ e) w v) :=
-    normalizedCoefficient_harmonic μ e (Specht.tail_card μ) x₀ w hw v
-  have he := sample_sweep_bound (V := Specht.hilbertSpace μ) (L := Specht.Tail μ) d (Specht.relabelledUnitary μ e)
-  have he := he (Specht.relabelledUnitary_unitary μ e) x₀ w hw hn hh
-  have he := he (by simpa only [Specht.tail_card] using hδ) (by simpa only [Specht.tail_card] using hδ1)
-  simpa only [sweepOperator, Bool.false_eq_true, ite_false, Specht.tail_card] using he
-
-lemma sweep_zero_one (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ)
-    (hk : μ.card - μ.rowLen 0 = 1) (rev : Bool) : sweepOperator d μ e rev = 0 := by
-  classical
-  let x₀ : Specht.Tail μ ↪ Card d := (Function.Embedding.subtype _).trans e.symm.toEmbedding
-  obtain ⟨w,hw,hn⟩ := exists_exact_fixed_unit μ e x₀ (Specht.tail_card μ)
-  let := Specht.relabelledUnitary_irreducible μ e
-  have hh (v : Specht.hilbertSpace μ) : Harmonic (UnitaryFinite.normalizedCoefficient
-      (V := Specht.hilbertSpace μ) UnitaryFinite.tupleAction x₀
-      (UnitaryFinite.tupleAction_transitive x₀) (Specht.relabelledUnitary μ e) w v) :=
-    normalizedCoefficient_harmonic μ e (Specht.tail_card μ) x₀ w hw v
-  have he := sample_sweep_zero_one (V := Specht.hilbertSpace μ) (L := Specht.Tail μ) d (Specht.relabelledUnitary μ e)
-  have he := he (Specht.relabelledUnitary_unitary μ e) x₀ w hw hn hh
-  have he := he ((Specht.tail_card μ).trans hk)
-  have hz : sweepOperator d μ e false = 0 := by
-    simpa only [sweepOperator, Bool.false_eq_true, ite_false] using he
-  apply (norm_eq_zero (a := sweepOperator d μ e rev)).mp
-  rw [sweep_forward_norm μ e rev, hz]
-  exact @norm_zero (Specht.hilbertSpace μ →L[ℂ] Specht.hilbertSpace μ) _
-
-end Thorp.SparseContact
-
-namespace Thorp.SparseContact
-
-noncomputable def contactConstant : ℝ := 32 * Real.exp (2 * Real.exp 1)
-
-lemma contactConstant_gt_one : 1 < contactConstant := by
-  have h : 1 ≤ Real.exp (2 * Real.exp 1) := Real.one_le_exp (by positivity)
-  unfold contactConstant
-  linarith
-
-lemma contactConstant_gt_two : 2 < contactConstant := by
-  have h : 1 ≤ Real.exp (2 * Real.exp 1) := Real.one_le_exp (by positivity)
-  unfold contactConstant
-  linarith
-
-lemma inflated_contact_bound {δ : ℝ} (hδ : 0 ≤ δ) (k : ℕ) :
-    ((2:ℝ)^k * Real.sqrt (δ^((k:ℝ)/2) * Real.exp (Real.exp 1*k)))^2 =
-      ((contactConstant/2)*δ)^((k:ℝ)/2) := by
-  have hC : 0 ≤ contactConstant/2 := by have := contactConstant_gt_one; positivity
-  rw [mul_pow, Real.sq_sqrt (by positivity), Real.mul_rpow hC hδ]
-  have hc : contactConstant/2 = Real.exp (2*(Real.log 4+Real.exp 1)) := by
-    unfold contactConstant
-    rw [show 2*(Real.log 4+Real.exp 1) = Real.log 4*2+2*Real.exp 1 by ring,
-      Real.exp_add, Real.exp_mul (Real.log 4) 2, Real.exp_log (by norm_num : (0:ℝ)<4)]
-    norm_num
-    ring
-  rw [hc, ←Real.exp_mul]
-  have hpow : ((2:ℝ)^k)^2 = Real.exp (Real.log 4 * k) := by
-    rw [← pow_mul, mul_comm k 2, pow_mul]
-    norm_num only [pow_two]
-    rw [Real.exp_mul (Real.log 4) k, Real.exp_log (by norm_num : (0:ℝ)<4), Real.rpow_natCast]
-  rw [hpow]
-  calc
-    _ = (Real.exp (Real.log 4*k)*Real.exp (Real.exp 1*k))*δ^((k:ℝ)/2) := by ring
-    _ = _ := by rw [←Real.exp_add]; congr 2; ring
-
-lemma contact_crude_arithmetic {C d q : ℝ} (hC : 1 ≤ C) (hd : 1 ≤ d) (hq : 0 ≤ q) (k : ℕ) :
-    (C*d*q)^((k:ℝ)/2) ≤ C^k*(1+d)^(C*k)*q^((k:ℝ)/2) := by
-  rw [Real.mul_rpow (by positivity) hq, Real.mul_rpow (by positivity) (by positivity)]
-  apply mul_le_mul_of_nonneg_right _ (by positivity)
-  apply mul_le_mul
-  · rw [← Real.rpow_natCast]
-    exact Real.rpow_le_rpow_of_exponent_le hC (by linarith [Nat.cast_nonneg (α := ℝ) k])
-  · calc
-      d^((k:ℝ)/2) ≤ (1+d)^((k:ℝ)/2) := Real.rpow_le_rpow (by positivity) (by linarith) (by positivity)
-      _ ≤ (1+d)^(C*k) := Real.rpow_le_rpow_of_exponent_le (by linarith) (by nlinarith [Nat.cast_nonneg (α := ℝ) k])
-  · positivity
-  · positivity
-
-
-lemma sweep_sharp_bound (d : ℕ) (hd : 1 ≤ d) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ)
-    (hk : 1 ≤ μ.card - μ.rowLen 0) (rev : Bool) :
-    ‖sweepOperator d μ e rev‖^2 ≤ min 1
-      ((contactConstant*d*(μ.card - μ.rowLen 0 : ℕ)/(2:ℝ)^d)^(((μ.card - μ.rowLen 0 : ℕ):ℝ)/2)) := by
-  let k := μ.card - μ.rowLen 0
-  have hk' : (0:ℝ) < k := by exact_mod_cast hk
-  have hd' : (0:ℝ) < d := by exact_mod_cast hd
-  have hδ : 0 < 2*(d:ℝ)*k/(2:ℝ)^d := by positivity
-  have hOne : ‖sweepOperator d μ e rev‖^2 ≤ 1 := by
-    have h := sweep_norm_le_one d μ e rev
-    have h0 := norm_nonneg (sweepOperator d μ e rev)
-    nlinarith
-  refine le_min hOne ?_
-  have he : contactConstant * d * (k:ℝ)/(2:ℝ)^d =
-      (contactConstant/2)*(2*(d:ℝ)*k/(2:ℝ)^d) := by ring
-  change _ ≤ (contactConstant * d * (k:ℝ)/(2:ℝ)^d)^((k:ℝ)/2)
-  rw [he]
-  by_cases hs : 2*(d:ℝ)*k/(2:ℝ)^d ≤ 1
-  · have h := sweep_norm_bound d μ e hδ hs rev
-    have hb : 0 ≤ (2:ℝ)^k * Real.sqrt
-        ((2*(d:ℝ)*k/(2:ℝ)^d)^((k:ℝ)/2)*Real.exp (Real.exp 1*k)) := by positivity
-    calc
-      _ ≤ ((2:ℝ)^k * Real.sqrt
-          ((2*(d:ℝ)*k/(2:ℝ)^d)^((k:ℝ)/2)*Real.exp (Real.exp 1*k)))^2 :=
-        (sq_le_sq₀ (norm_nonneg (sweepOperator d μ e rev)) hb).mpr h
-      _ = _ := inflated_contact_bound hδ.le k
-  · apply hOne.trans
-    apply Real.one_le_rpow
-    · have hc : 1 ≤ contactConstant/2 := by have := contactConstant_gt_two; linarith
-      exact one_le_mul_of_one_le_of_one_le hc (le_of_lt (lt_of_not_ge hs))
-    · positivity
-
-theorem uniform_sparse_contact : MainStatement := by
-  refine ⟨contactConstant, contactConstant, (lt_trans (by norm_num) contactConstant_gt_one),
-    (lt_trans (by norm_num) contactConstant_gt_one), ?_⟩
-  intro d hd μ e k hk _ rev
-  have hs := sweep_sharp_bound d hd μ e hk rev
-  refine ⟨hs, ?_, ?_⟩
-  · intro hk1
-    exact sweep_zero_one d μ e hk1 rev
-  · apply (hs.trans (min_le_right _ _)).trans
-    change (contactConstant * d * (k:ℝ)/(2:ℝ)^d)^((k:ℝ)/2) ≤ _
-    rw [mul_div_assoc]
-    exact contact_crude_arithmetic contactConstant_gt_one.le
-      (by exact_mod_cast hd) (by positivity) k
-
-end Thorp.SparseContact
-
-
-namespace Thorp.SparseRegime
-open Filter
-open Thorp.SparseContact
-
-lemma log_contact_eventually : ∀ᶠ d : ℕ in atTop,
-    Real.log (contactConstant*d) ≤ (d:ℝ)^((3:ℝ)/4)/2 := by
-  have hn : Tendsto (fun d : ℕ => (d:ℝ)) atTop atTop := tendsto_natCast_atTop_atTop
-  have hl := ((isLittleO_log_rpow_atTop (by norm_num : (0:ℝ)<3/4)).comp_tendsto hn).def
-    (by norm_num : (0:ℝ)<1/4)
-  have hr := ((tendsto_rpow_atTop (by norm_num : (0:ℝ)<3/4)).comp hn).eventually
-    (eventually_ge_atTop (4*Real.log contactConstant))
-  filter_upwards [hl, hr, eventually_ge_atTop 1] with d hl hr hd
-  have hd' : (0:ℝ)<d := by exact_mod_cast hd
-  have hc : 0<contactConstant := lt_trans (by norm_num) contactConstant_gt_one
-  simp only [Function.comp_apply, Real.norm_eq_abs,
-    abs_of_nonneg (Real.rpow_nonneg (Nat.cast_nonneg d) ((3:ℝ)/4))] at hl
-  simp only [Function.comp_apply] at hr
-  have hl' := (le_abs_self (Real.log (d:ℝ))).trans hl
-  rw [Real.log_mul hc.ne' hd'.ne']
-  linarith
-
-lemma contact_power_saving {d k : ℕ} (hd : 1 ≤ d) (hk : 1 ≤ k)
-    (hlog : Real.log (contactConstant*d) ≤ (d:ℝ)^((3:ℝ)/4)/2)
-    (hs : (d:ℝ)^((3:ℝ)/4) ≤ Real.log ((2:ℝ)^d/k)) :
-    (contactConstant*d*k/(2:ℝ)^d)^((k:ℝ)/2) ≤ ((k:ℝ)/(2:ℝ)^d)^((k:ℝ)/4) := by
-  have hk0 : (0:ℝ)<k := by exact_mod_cast hk
-  have hd0 : (0:ℝ)<d := by exact_mod_cast hd
-  have hc : 0<contactConstant := lt_trans (by norm_num) contactConstant_gt_one
-  have hq : 0<(k:ℝ)/(2:ℝ)^d := by positivity
-  have hmain : 0<contactConstant*d*k/(2:ℝ)^d := by positivity
-  rw [Real.rpow_def_of_pos hmain, Real.rpow_def_of_pos hq]
-  apply Real.exp_le_exp.mpr
-  have hlogq : Real.log ((k:ℝ)/(2:ℝ)^d) = -Real.log ((2:ℝ)^d/k) := by
-    rw [Real.log_div hk0.ne' (by positivity), Real.log_div (by positivity) hk0.ne']
-    ring
-  have he : contactConstant*d*k/(2:ℝ)^d = (contactConstant*d)*((k:ℝ)/(2:ℝ)^d) := by ring
-  rw [he, Real.log_mul (by positivity) hq.ne', hlogq]
-  nlinarith
-
-lemma inverse_dyadic_power (d k : ℕ) (hk : 1 ≤ k) :
-    ((k:ℝ)/(2:ℝ)^d)^((k:ℝ)/4) = Real.exp (-Real.log ((2:ℝ)^d/k)*k/4) := by
-  have hk0 : (0:ℝ)<k := by exact_mod_cast hk
-  rw [Real.rpow_def_of_pos (by positivity)]
-  congr 1
-  rw [Real.log_div hk0.ne' (by positivity), Real.log_div (by positivity) hk0.ne']
-  ring
-
-lemma sparse_finite_correction {d d₀ k : ℕ} (hd : d < d₀) (hk : 1 ≤ k) :
-    1 ≤ Real.exp ((1+(d₀:ℝ)*Real.log 2)*k) * ((k:ℝ)/(2:ℝ)^d)^((k:ℝ)/4) := by
-  have hk0 : (0:ℝ)<k := by exact_mod_cast hk
-  have hk1 : (1:ℝ)≤k := by exact_mod_cast hk
-  have hd' : (d:ℝ) < d₀ := by exact_mod_cast hd
-  have hlogk : 0 ≤ Real.log (k:ℝ) := Real.log_nonneg hk1
-  rw [inverse_dyadic_power d k hk, ←Real.exp_add]
-  apply Real.one_le_exp
-  rw [Real.log_div (by positivity) hk0.ne', Real.log_pow]
-  have hlog2 : 0 ≤ Real.log 2 := (Real.log_pos (by norm_num : (1:ℝ)<2)).le
-  have hdd := mul_le_mul_of_nonneg_right hd'.le hlog2
-  have hs : (d:ℝ)*Real.log 2-Real.log k ≤ 4*(1+(d₀:ℝ)*Real.log 2) := by
-    nlinarith [mul_nonneg (Nat.cast_nonneg (α := ℝ) d₀) hlog2]
-  have hm := mul_le_mul_of_nonneg_right hs hk0.le
-  nlinarith
-
-theorem exponential_sparse_saving : MainStatement := by
-  obtain ⟨d₁, hd₁⟩ := eventually_atTop.mp log_contact_eventually
-  let d₀ := max d₁ 1
-  let C : ℝ := 1+(d₀:ℝ)*Real.log 2
-  have hC : 0<C := by dsimp [C]; positivity
-  refine ⟨C,d₀,hC,by dsimp [d₀]; omega,?_⟩
-  intro d μ e k hk hkn hs rev
-  have hd : 1 ≤ d := by
-    by_contra h
-    have hd0 : d=0 := by omega
-    simp only [hd0, pow_zero] at hkn
-    omega
-  have hz : ‖sweepOperator d μ e rev‖^2 ≤ 1 := by
-    have hn := sweep_norm_le_one d μ e rev
-    have hp := norm_nonneg (sweepOperator d μ e rev)
-    nlinarith
-  have hlarge (hdd : d₀ ≤ d) : ‖sweepOperator d μ e rev‖^2 ≤ ((k:ℝ)/(2:ℝ)^d)^((k:ℝ)/4) := by
-    exact (sweep_sharp_bound d hd μ e hk rev).trans ((min_le_right _ _).trans
-      (contact_power_saving hd hk (hd₁ d (le_trans (le_max_left _ _) hdd)) hs))
-  refine ⟨?_, hlarge⟩
-  by_cases hdd : d₀ ≤ d
-  · apply (hlarge hdd).trans
-    have he : 1 ≤ Real.exp (C*k) := Real.one_le_exp (by positivity)
-    exact le_mul_of_one_le_left (by positivity) he
-  · exact hz.trans (sparse_finite_correction (lt_of_not_ge hdd) hk)
-
-end Thorp.SparseRegime
-
-end ThorpNine.SparseSaving
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.SparseSaving.ModifiedStageEquiv
+
+namespace OAI
+
+namespace ThorpNine.SparseSaving
+
+namespace Thorp.SparseContact
+open scoped BigOperators Classical
+
+noncomputable def typedMarks {N : Type*} [Fintype N] (d : ℕ)
+    (A : N → Finset (Card d)) (ω : N → (SwitchIndex d → Bool)) : Finset (N × Card d) :=
+  typedSet (fun t => butterflyMarks d (A t) (ω t))
+
+lemma typedMarks_card {N : Type*} [Fintype N] (d : ℕ)
+    (A : N → Finset (Card d)) (ω : N → (SwitchIndex d → Bool)) :
+    (typedMarks d A ω).card = ∑ t, (A t).card := by
+  classical
+  rw [typedMarks, typedSet_card]
+  apply Finset.sum_congr rfl
+  intro t _
+  exact Finset.card_image_of_injective _ (butterflyPerm d (decodeButterfly d (ω t))).injective
+
+lemma typedMarks_inclusion {N : Type*} [Fintype N] (d : ℕ)
+    (A : N → Finset (Card d)) (D : Finset (N × Card d)) :
+    inclusionMoment (typedMarks d A) D ≤
+      ∏ v ∈ D, ((A v.1).card:ℝ)/(2:ℝ)^d := by
+  classical
+  have he : inclusionMoment (typedMarks d A) D =
+      ∏ t, inclusionMoment (butterflyMarks d (A t)) (fiber D t) := by
+    rw [inclusionMoment_eq]
+    have hh (ω : N → (SwitchIndex d → Bool)) :
+        (if D ⊆ typedMarks d A ω then (1:ℝ) else 0) =
+        ∏ t, if fiber D t ⊆ butterflyMarks d (A t) (ω t) then (1:ℝ) else 0 := by
+      rw [Fintype.prod_boole]
+      simp only [typedMarks, subset_typedSet]
+    rw [finiteMean_congr hh]
+    simp only [inclusionMoment_eq]
+    exact finiteMean_pi_prod (ι := N) (Ω := fun _ => SwitchIndex d → Bool)
+      (fun t (ω : SwitchIndex d → Bool) =>
+        if fiber D t ⊆ butterflyMarks d (A t) ω then (1:ℝ) else 0)
+  have hprod : (∏ v ∈ D, ((A v.1).card:ℝ)/(2:ℝ)^d) =
+      ∏ t, ∏ x ∈ fiber D t, ((A t).card:ℝ)/(2:ℝ)^d := by
+    conv_lhs => rw [← typedSet_fiber D]
+    exact typedSet_prod _ _
+  rw [he, hprod]
+  apply Finset.prod_le_prod₀
+  · intro t _; exact inclusionMoment_nonneg _ _
+  · intro t _
+    simpa only [Finset.prod_const] using butterflyMarks_inclusion d (A t) (fiber D t)
+
+noncomputable def returnRoute {N : Type*} (d : ℕ) (b : N → Butterfly d)
+    (v : N × Card d) : Finset (SwitchIndex d) :=
+  routeDomain d ((butterflyPerm d (b v.1)).symm v.2) v.2
+
+noncomputable def returnGraph {N : Type*} (d : ℕ) (b : N → Butterfly d) :
+    SimpleGraph (N × Card d) where
+  Adj v w := v ≠ w ∧ ¬Disjoint (returnRoute d b v) (returnRoute d b w)
+  symm := ⟨fun _ _ h => ⟨h.1.symm, fun hh => h.2 hh.symm⟩⟩
+  loopless := ⟨fun _ h => h.1 rfl⟩
+
+lemma returnRoute_card {N : Type*} (d : ℕ) (b : N → Butterfly d) (v : N × Card d) :
+    (returnRoute d b v).card = d := card_routeDomain _ _ _
+
+lemma mem_returnRoute {N : Type*} (d : ℕ) (b : N → Butterfly d)
+    (t : N) (y : Card d) (i : SwitchIndex d) :
+    i ∈ returnRoute d b (t,y) ↔
+      y = butterflyPerm d (b t) (switchUsers d (b t) i).1 ∨
+      y = butterflyPerm d (b t) (switchUsers d (b t) i).2 := by
+  have h := mem_routeDomain_iff_user d (b t) ((butterflyPerm d (b t)).symm y) i
+  simpa only [Equiv.apply_symm_apply, returnRoute, Equiv.eq_symm_apply, eq_comm] using h
+
+lemma returnRoute_incidence {N : Type*} [Fintype N] (d : ℕ) (b : N → Butterfly d)
+    (p : N → ℝ) (i : SwitchIndex d) :
+    (∑ v : N × Card d, if i ∈ returnRoute d b v then p v.1 else 0) =
+      2 * ∑ t, p t := by
+  classical
+  rw [Fintype.sum_prod_type, Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro t _
+  have hne : butterflyPerm d (b t) (switchUsers d (b t) i).1 ≠
+      butterflyPerm d (b t) (switchUsers d (b t) i).2 :=
+    (butterflyPerm d (b t)).injective.ne (switchUsers_ne d (b t) i)
+  simp only [mem_returnRoute]
+  have he (y : Card d) :
+      (if y = butterflyPerm d (b t) (switchUsers d (b t) i).1 ∨
+        y = butterflyPerm d (b t) (switchUsers d (b t) i).2 then p t else 0) =
+      (if y = butterflyPerm d (b t) (switchUsers d (b t) i).1 then p t else 0) +
+      (if y = butterflyPerm d (b t) (switchUsers d (b t) i).2 then p t else 0) := by
+    split_ifs <;> simp_all
+  simp only [he, Finset.sum_add_distrib, Finset.sum_ite_eq', Finset.mem_univ, ite_true]
+  ring
+
+lemma returnGraph_degree {N : Type*} [Fintype N] (d : ℕ) (b : N → Butterfly d)
+    (p : N → ℝ) (hp : ∀ t, 0 ≤ p t) (v : N × Card d) :
+    (∑ w, if (returnGraph d b).Adj v w then p w.1 else 0) ≤
+      2*d*∑ t, p t := by
+  classical
+  calc
+    _ ≤ ∑ w, ∑ i ∈ returnRoute d b v, if i ∈ returnRoute d b w then p w.1 else 0 := by
+      apply Finset.sum_le_sum
+      intro w _
+      split_ifs with h
+      · obtain ⟨i, hi, hw⟩ := Finset.not_disjoint_iff.mp h.2
+        have hh := Finset.single_le_sum
+          (f := fun i => if i ∈ returnRoute d b w then p w.1 else 0)
+          (fun i _ => by split_ifs; exact hp _; exact le_rfl) hi
+        simpa only [ite_eq_left hw] using hh
+      · exact Finset.sum_nonneg (fun i _ => by split_ifs; exact hp _; exact le_rfl)
+    _ = ∑ i ∈ returnRoute d b v, ∑ w, if i ∈ returnRoute d b w then p w.1 else 0 :=
+      Finset.sum_comm
+    _ = _ := by simp only [returnRoute_incidence, Finset.sum_const, returnRoute_card,
+        nsmul_eq_mul]; ring
+
+lemma typed_weight_sum {N : Type*} [Fintype N] (d : ℕ)
+    (A : N → Finset (Card d)) :
+    (∑ v : N × Card d, ((A v.1).card:ℝ)/(2:ℝ)^d) = ∑ t, ((A t).card:ℝ) := by
+  rw [Fintype.sum_prod_type]
+  apply Finset.sum_congr rfl
+  intro t _
+  simp only [Finset.sum_const, Finset.card_univ, card_positions, Nat.cast_pow,
+    Nat.cast_ofNat, nsmul_eq_mul]
+  field_simp
+
+lemma mixed_return_noIsolated {N : Type*} [Fintype N] (d : ℕ)
+    (A : N → Finset (Card d)) (b : N → Butterfly d) (k : ℕ)
+    (hk : ∑ t, (A t).card = k)
+    (hδ : 0 < 2*(d:ℝ)*k/(2:ℝ)^d) (hδ1 : 2*(d:ℝ)*k/(2:ℝ)^d ≤ 1) :
+    finiteMean (fun ω : N → (SwitchIndex d → Bool) =>
+      if NoIsolated (returnGraph d b) (typedMarks d A ω) then (1:ℝ) else 0) ≤
+      (2*(d:ℝ)*k/(2:ℝ)^d)^((k:ℝ)/2) * Real.exp (Real.exp 1*k) := by
+  have hs : (∑ t, ((A t).card:ℝ)) = k := by exact_mod_cast hk
+  have hm := noIsolated_probability_le (returnGraph d b) (typedMarks d A) k
+    (fun ω => (typedMarks_card d A ω).trans hk)
+    (fun v => ((A v.1).card:ℝ)/(2:ℝ)^d) (fun v => by positivity)
+    (typedMarks_inclusion d A) _ hδ hδ1
+    (fun v => (returnGraph_degree d b (fun t => ((A t).card:ℝ)/(2:ℝ)^d)
+      (fun t => by positivity) v).trans_eq (by rw [← Finset.sum_div, hs]; ring))
+  simpa only [typed_weight_sum, hs] using hm
+
+
+lemma mean_fresh_cylinder {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (s : Finset ι) (v : ι → Bool) (f : (ι → Bool) → ℝ)
+    (hf : ∀ i ∈ s, ∀ ω, f (QueryTree.flipCoin i ω) = f ω) :
+    finiteMean (fun ω => if ∀ i ∈ s, ω i = v i then f ω else 0) =
+      finiteMean f / (2:ℝ)^s.card := by
+  induction s using Finset.induction_on with
+  | empty => simp [finiteMean]
+  | @insert i s hi ih =>
+    let g := fun ω => if ∀ j ∈ s, ω j = v j then f ω else 0
+    have hg : ∀ ω, g (QueryTree.flipCoin i ω) = g ω := by
+      intro ω
+      have he : (∀ j ∈ s, QueryTree.flipCoin i ω j = v j) ↔ ∀ j ∈ s, ω j = v j := by
+        apply forall₂_congr
+        intro j hj
+        rw [QueryTree.flipCoin_apply_ne i j (fun h => hi (h ▸ hj))]
+      simp only [g, he, hf i (Finset.mem_insert_self _ _) ω]
+    have he (ω : ι → Bool) :
+        (if ∀ j ∈ insert i s, ω j = v j then f ω else 0) =
+        (if ω i = v i then g ω else 0) := by
+      by_cases h : ω i = v i <;> simp [h,g]
+    rw [finiteMean_congr he, QueryTree.mean_indicator_bit g i (v i) hg]
+    have hs := ih (fun j hj => hf j (Finset.mem_insert_of_mem hj))
+    change finiteMean (fun ω => if ∀ j ∈ s, ω j = v j then f ω else 0) / 2 = _
+    rw [hs, Finset.card_insert_of_notMem hi, pow_succ]
+    ring
+
+noncomputable def endpointProb {L : Type*} (d : ℕ) (A : Finset L)
+    (x y : L → Card d) : ℝ := by
+  classical
+  exact finiteMean (fun ω : SwitchIndex d → Bool =>
+    if ∀ i ∈ A, butterflyPerm d (decodeButterfly d ω) (x i) = y i then (1:ℝ) else 0)
+
+lemma endpointProb_nonneg {L : Type*} (d : ℕ) (A : Finset L)
+    (x y : L → Card d) : 0 ≤ endpointProb d A x y := by
+  classical
+  exact finiteMean_nonneg (fun _ => by positivity)
+
+lemma endpointProb_insert_isolated {L : Type*} (d : ℕ) (A : Finset L)
+    (x y : L → Card d) (i : L)
+    (hiso : ∀ j ∈ A, Disjoint (routeDomain d (x i) (y i)) (routeDomain d (x j) (y j))) :
+    endpointProb d (insert i A) x y = endpointProb d A x y / (2:ℝ)^d := by
+  classical
+  let f := fun ω : SwitchIndex d → Bool =>
+    if ∀ j ∈ A, butterflyPerm d (decodeButterfly d ω) (x j) = y j then (1:ℝ) else 0
+  have hf : ∀ t ∈ routeDomain d (x i) (y i), ∀ ω, f (QueryTree.flipCoin t ω) = f ω := by
+    intro t ht ω
+    have he : (∀ j ∈ A, butterflyPerm d (decodeButterfly d (QueryTree.flipCoin t ω)) (x j) = y j) ↔
+        ∀ j ∈ A, butterflyPerm d (decodeButterfly d ω) (x j) = y j := by
+      apply forall₂_congr
+      intro j hj
+      rw [butterfly_endpoint_iff, butterfly_endpoint_iff]
+      apply forall₂_congr
+      intro s hs
+      rw [QueryTree.flipCoin_apply_ne t s (fun h =>
+        Finset.disjoint_left.mp (hiso j hj) ht (h ▸ hs))]
+    simp only [f, he]
+  have he (ω : SwitchIndex d → Bool) :
+      (if ∀ j ∈ insert i A, butterflyPerm d (decodeButterfly d ω) (x j) = y j then (1:ℝ) else 0) =
+      (if ∀ t ∈ routeDomain d (x i) (y i), ω t = routeValue d (x i) (y i) t then f ω else 0) := by
+    by_cases h : butterflyPerm d (decodeButterfly d ω) (x i) = y i
+    · have hh := (butterfly_endpoint_iff d ω (x i) (y i)).mp h
+      rw [ite_eq_left hh]
+      simp only [Finset.forall_mem_insert, h, true_and, f]
+    · have hh := mt (butterfly_endpoint_iff d ω (x i) (y i)).mpr h
+      simp only [Finset.forall_mem_insert, h, false_and, hh, ite_false]
+  change finiteMean _ = _
+  rw [finiteMean_congr he, mean_fresh_cylinder _ _ f hf, card_routeDomain]
+  rfl
+
+noncomputable def partialKernel {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x y : L → Card d) : ℝ :=
+  ((2:ℝ)^d)^A.card / ((2:ℝ)^d)^(Fintype.card L) * endpointProb d A x y
+
+lemma partialKernel_insert_isolated {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x y : L → Card d) (i : L) (hi : i ∉ A)
+    (hiso : ∀ j ∈ A, Disjoint (routeDomain d (x i) (y i)) (routeDomain d (x j) (y j))) :
+    partialKernel d (insert i A) x y = partialKernel d A x y := by
+  classical
+  rw [partialKernel, endpointProb_insert_isolated d A x y i hiso,
+    Finset.card_insert_of_notMem hi, pow_succ, partialKernel]
+  field_simp
+
+def ContactEvent {L : Type*} (d : ℕ) (x y : L → Card d) : Prop :=
+  ∀ i, ∃ j, i ≠ j ∧ ¬Disjoint (routeDomain d (x i) (y i)) (routeDomain d (x j) (y j))
+
+lemma alternating_cube_cancel {L : Type*} [Fintype L] (f : Finset L → ℝ)
+    (i : L) (hf : ∀ A, i ∉ A → f (insert i A) = f A) :
+    (∑ A : Finset L, (-1:ℝ)^A.card * f A) = 0 := by
+  classical
+  have hi : i ∈ (Finset.univ : Finset L) := Finset.mem_univ _
+  rw [← Finset.powerset_univ, ← Finset.insert_erase hi, Finset.sum_powerset_insert]
+  · rw [← Finset.sum_add_distrib]
+    apply Finset.sum_eq_zero
+    intro A hA
+    have hiA : i ∉ A := fun hh => Finset.notMem_erase i Finset.univ
+      ((Finset.mem_powerset.mp hA) hh)
+    rw [hf A hiA, Finset.card_insert_of_notMem hiA, pow_succ]
+    ring
+  · exact Finset.notMem_erase _ _
+
+lemma alternating_kernel_cancel {L : Type*} [Fintype L] (d : ℕ)
+    (x y : L → Card d) (h : ¬ContactEvent d x y) :
+    (∑ A : Finset L, (-1:ℝ)^A.card * partialKernel d A x y) = 0 := by
+  classical
+  unfold ContactEvent at h
+  push Not at h
+  obtain ⟨i, hi⟩ := h
+  apply alternating_cube_cancel _ i
+  intro A hiA
+  apply partialKernel_insert_isolated d A x y i hiA
+  intro j hj
+  exact hi j (fun hij => hiA (hij ▸ hj))
+
+
+noncomputable def networkOf {L : Type*} (A : Finset L) (i : L) : Option L := by
+  classical
+  exact if i ∈ A then none else some i
+
+noncomputable def mixedTuple {L : Type*} (d : ℕ) (A : Finset L)
+    (ω : Option L → (SwitchIndex d → Bool)) (x : L → Card d) : L → Card d :=
+  fun i => butterflyPerm d (decodeButterfly d (ω (networkOf A i))) (x i)
+
+noncomputable def mixedReturn {L : Type*} (d : ℕ) (A : Finset L)
+    (ω : Option L → (SwitchIndex d → Bool)) (y : L → Card d) : L → Card d :=
+  fun i => (butterflyPerm d (decodeButterfly d (ω (networkOf A i)))).symm (y i)
+
+noncomputable def labelsOf {L : Type*} (A : Finset L) : Option L → Finset L := by
+  classical
+  exact fun t => match t with
+    | none => A
+    | some i => if i ∈ A then ∅ else {i}
+
+lemma mem_labelsOf {L : Type*} (A : Finset L) (i : L) (t : Option L) :
+    i ∈ labelsOf A t ↔ networkOf A i = t := by
+  classical
+  cases t with
+  | none => simp [labelsOf, networkOf]
+  | some j => by_cases hj : j ∈ A <;> by_cases hi : i ∈ A <;> simp_all [labelsOf, networkOf] <;> intro h <;> subst j <;> contradiction
+
+lemma mixedTuple_eq_iff {L : Type*} (d : ℕ) (A : Finset L)
+    (ω : Option L → (SwitchIndex d → Bool)) (x y : L → Card d) :
+    mixedTuple d A ω x = y ↔ ∀ t, ∀ i ∈ labelsOf A t,
+      butterflyPerm d (decodeButterfly d (ω t)) (x i) = y i := by
+  constructor
+  · intro h t i hi
+    have he := congrFun h i
+    simpa only [mixedTuple, mem_labelsOf A i t |>.mp hi] using he
+  · intro h
+    funext i
+    exact h (networkOf A i) i (mem_labelsOf A i _ |>.mpr rfl)
+
+lemma mixedReturn_eq_iff {L : Type*} (d : ℕ) (A : Finset L)
+    (ω : Option L → (SwitchIndex d → Bool)) (y z : L → Card d) :
+    mixedReturn d A ω y = z ↔ mixedTuple d A ω z = y := by
+  constructor
+  · intro h
+    funext i
+    exact (Equiv.symm_apply_eq _).mp (congrFun h i) |>.symm
+  · intro h
+    funext i
+    exact (Equiv.symm_apply_eq _).mpr (congrFun h i).symm
+
+lemma endpointProb_singleton {L : Type*} (d : ℕ) (i : L) (x y : L → Card d) :
+    endpointProb d {i} x y = 1/(2:ℝ)^d := by
+  classical
+  simpa [endpointProb] using butterfly_single_card_uniform d (x i) (y i)
+
+lemma mixedTuple_mass {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x y : L → Card d) :
+    finiteMean (fun ω : Option L → (SwitchIndex d → Bool) =>
+      if mixedTuple d A ω x = y then (1:ℝ) else 0) = partialKernel d A x y := by
+  classical
+  have he (ω : Option L → (SwitchIndex d → Bool)) :
+      (if mixedTuple d A ω x = y then (1:ℝ) else 0) =
+      ∏ t, if ∀ i ∈ labelsOf A t, butterflyPerm d (decodeButterfly d (ω t)) (x i) = y i
+        then (1:ℝ) else 0 := by
+    simp only [Fintype.prod_boole, mixedTuple_eq_iff]
+  rw [finiteMean_congr he]
+  rw [finiteMean_pi_prod (fun t (w : SwitchIndex d → Bool) =>
+    if ∀ i ∈ labelsOf A t, butterflyPerm d (decodeButterfly d w) (x i) = y i then (1:ℝ) else 0)]
+  have hfac (t : Option L) : (finiteMean (fun w : SwitchIndex d → Bool =>
+      if ∀ i ∈ labelsOf A t, butterflyPerm d (decodeButterfly d w) (x i) = y i then (1:ℝ) else 0)) =
+      endpointProb d (labelsOf A t) x y := by
+        unfold endpointProb
+        apply finiteMean_congr
+        intro w
+        split_ifs <;> rfl
+  simp only [hfac]
+  rw [Fintype.prod_option]
+  have hi (i : L) : endpointProb d (labelsOf A (some i)) x y =
+      if i ∈ A then 1 else 1/(2:ℝ)^d := by
+    by_cases h : i ∈ A
+    · simp [labelsOf, h, endpointProb, finiteMean_const]
+    · simp only [labelsOf, ite_eq_right h, endpointProb_singleton]
+  rw [show labelsOf A none = A from rfl]
+  simp only [hi]
+  have hp : (∏ i : L, if i ∈ A then (1:ℝ) else 1/(2:ℝ)^d) =
+      (1/(2:ℝ)^d)^((Fintype.card L) - A.card) := by
+    rw [Finset.prod_ite]
+    simp only [Finset.prod_const_one, one_mul]
+    simp only [Finset.filter_not, Finset.filter_mem_eq_inter, Finset.univ_inter,
+      Finset.prod_const, Finset.card_sdiff, Finset.inter_univ, Finset.card_univ]
+  rw [hp, partialKernel, one_div, inv_pow]
+  have hc : A.card ≤ Fintype.card L := Finset.card_le_univ _
+  have hn : (((2:ℝ)^d)^A.card) * (((2:ℝ)^d)^(Fintype.card L - A.card)) =
+      ((2:ℝ)^d)^(Fintype.card L) := by rw [← pow_add, Nat.add_sub_of_le hc]
+  have hnz : ((2:ℝ)^d)^(Fintype.card L) ≠ 0 := by positivity
+  have hmz : ((2:ℝ)^d)^(Fintype.card L - A.card) ≠ 0 := by positivity
+  field_simp
+  rw [← hn]
+  ring
+
+lemma mixedReturn_mass {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (y z : L → Card d) :
+    finiteMean (fun ω : Option L → (SwitchIndex d → Bool) =>
+      if mixedReturn d A ω y = z then (1:ℝ) else 0) = partialKernel d A z y := by
+  simpa only [mixedReturn_eq_iff] using mixedTuple_mass d A z y
+
+noncomputable def mixedInitial {L : Type*} (d : ℕ) (A : Finset L)
+    (x : L → Card d) (t : Option L) : Finset (Card d) := by
+  classical
+  exact (labelsOf A t).image x
+
+lemma typedMarks_mixedInitial {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x : L → Card d) (ω : Option L → (SwitchIndex d → Bool)) :
+    typedMarks d (mixedInitial d A x) ω = Finset.univ.image
+      (fun i => (networkOf A i, mixedTuple d A ω x i)) := by
+  classical
+  ext v
+  simp only [typedMarks, mem_typedSet, butterflyMarks, mixedInitial,
+    Finset.mem_image, Finset.mem_univ, true_and]
+  constructor
+  · rintro ⟨z, ⟨i,hi,rfl⟩, hv⟩
+    have ht := mem_labelsOf A i v.1 |>.mp hi
+    exact ⟨i, Prod.ext ht (by simpa only [mixedTuple, ht] using hv)⟩
+  · rintro ⟨i, hi⟩
+    have ht := congrArg Prod.fst hi
+    have hv := congrArg Prod.snd hi
+    change networkOf A i = v.1 at ht
+    change mixedTuple d A ω x i = v.2 at hv
+    exact ⟨x i, ⟨i, mem_labelsOf A i v.1 |>.mpr ht, rfl⟩,
+      by simpa only [mixedTuple, ht] using hv⟩
+
+lemma typedTuple_injective {L : Type*} (d : ℕ) (A : Finset L)
+    (x : L → Card d) (hx : Function.Injective x)
+    (ω : Option L → (SwitchIndex d → Bool)) :
+    Function.Injective (fun i => (networkOf A i, mixedTuple d A ω x i)) := by
+  intro i j h
+  have ht := congrArg Prod.fst h
+  have hv := congrArg Prod.snd h
+  change networkOf A i = networkOf A j at ht
+  dsimp only [mixedTuple] at hv
+  rw [ht] at hv
+  exact hx ((butterflyPerm d (decodeButterfly d (ω (networkOf A j)))).injective hv)
+
+lemma mixedInitial_card {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x : L → Card d) (hx : Function.Injective x) :
+    (∑ t, (mixedInitial d A x t).card) = Fintype.card L := by
+  classical
+  let ω : Option L → (SwitchIndex d → Bool) := fun _ _ => false
+  rw [← typedMarks_card d (mixedInitial d A x) ω, typedMarks_mixedInitial,
+    Finset.card_image_of_injective _ (typedTuple_injective d A x hx ω), Finset.card_univ]
+
+lemma contactEvent_mixedReturn_iff {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x : L → Card d) (hx : Function.Injective x)
+    (ω b : Option L → (SwitchIndex d → Bool)) :
+    ContactEvent d (mixedReturn d A b (mixedTuple d A ω x)) (mixedTuple d A ω x) ↔
+      NoIsolated (returnGraph d (fun t => decodeButterfly d (b t)))
+        (typedMarks d (mixedInitial d A x) ω) := by
+  classical
+  rw [typedMarks_mixedInitial]
+  let f := fun i => (networkOf A i, mixedTuple d A ω x i)
+  have hf : Function.Injective f := typedTuple_injective d A x hx ω
+  constructor
+  · intro h v hv
+    obtain ⟨i,_,rfl⟩ := Finset.mem_image.mp hv
+    obtain ⟨j,hij,hc⟩ := h i
+    exact ⟨f j, Finset.mem_image.mpr ⟨j, Finset.mem_univ _,rfl⟩,
+      hf.ne hij, hc⟩
+  · intro h i
+    obtain ⟨v,hv,hij,hc⟩ := h (f i) (Finset.mem_image.mpr ⟨i,Finset.mem_univ _,rfl⟩)
+    obtain ⟨j,_,rfl⟩ := Finset.mem_image.mp hv
+    exact ⟨j, fun he => hij (congrArg f he), hc⟩
+
+lemma probability_mixed_contact {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x : L → Card d) (hx : Function.Injective x)
+    (b : Option L → (SwitchIndex d → Bool))
+    (hδ : 0 < 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)
+    (hδ1 : 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d ≤ 1) :
+    finiteMean (fun ω : Option L → (SwitchIndex d → Bool) =>
+      if ContactEvent d (mixedReturn d A b (mixedTuple d A ω x)) (mixedTuple d A ω x)
+        then (1:ℝ) else 0) ≤
+      (2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
+        Real.exp (Real.exp 1*(Fintype.card L)) := by
+  classical
+  simp only [contactEvent_mixedReturn_iff d A x hx]
+  convert mixed_return_noIsolated d (mixedInitial d A x) (fun t => decodeButterfly d (b t))
+      (Fintype.card L) (mixedInitial_card d A x hx) hδ hδ1 using 1
+  congr 1
+  exact Subsingleton.elim _ _
+
+
+lemma finiteMean_pushforward {Ω X : Type*} [Fintype Ω] [Fintype X]
+    (g : Ω → X) (f : X → ℝ) :
+    finiteMean (fun ω => f (g ω)) =
+      ∑ y, finiteMean (fun ω => if g ω = y then (1:ℝ) else 0) * f y := by
+  classical
+  simp_rw [← finiteMean_mul_const, ← finiteMean_sum]
+  apply finiteMean_congr
+  intro ω
+  simp
+
+lemma mixedTuple_expectation {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x : L → Card d) (f : (L → Card d) → ℝ) :
+    finiteMean (fun ω : Option L → (SwitchIndex d → Bool) => f (mixedTuple d A ω x)) =
+      ∑ y, partialKernel d A x y * f y := by
+  classical
+  rw [finiteMean_pushforward (fun ω : Option L → (SwitchIndex d → Bool) => mixedTuple d A ω x) f]
+  apply Finset.sum_congr rfl
+  intro y _
+  congr 1
+  convert mixedTuple_mass d A x y using 1
+  congr 1
+  funext ω
+  split_ifs <;> rfl
+
+lemma mixedReturn_expectation {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (y : L → Card d) (f : (L → Card d) → ℝ) :
+    finiteMean (fun ω : Option L → (SwitchIndex d → Bool) => f (mixedReturn d A ω y)) =
+      ∑ z, partialKernel d A z y * f z := by
+  classical
+  rw [finiteMean_pushforward (fun ω : Option L → (SwitchIndex d → Bool) => mixedReturn d A ω y) f]
+  apply Finset.sum_congr rfl
+  intro z _
+  congr 1
+  convert mixedReturn_mass d A y z using 1
+  congr 1
+  funext ω
+  split_ifs <;> rfl
+
+lemma partialKernel_nonneg {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x y : L → Card d) : 0 ≤ partialKernel d A x y :=
+  mul_nonneg (by positivity) (endpointProb_nonneg d A x y)
+
+noncomputable def retainedKernel {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x : L ↪ Card d) (y : L → Card d) : ℝ := by
+  classical
+  exact if ContactEvent d x y then partialKernel d A x y else 0
+
+lemma retainedKernel_nonneg {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x : L ↪ Card d) (y : L → Card d) : 0 ≤ retainedKernel d A x y := by
+  classical
+  unfold retainedKernel
+  split_ifs
+  · exact partialKernel_nonneg d A x y
+  · exact le_rfl
+
+lemma retainedKernel_le {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x : L ↪ Card d) (y : L → Card d) : retainedKernel d A x y ≤ partialKernel d A x y := by
+  classical
+  unfold retainedKernel
+  split_ifs
+  · exact le_rfl
+  · exact partialKernel_nonneg d A x y
+
+lemma retained_square_row {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x : L ↪ Card d)
+    (hδ : 0 < 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)
+    (hδ1 : 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d ≤ 1) :
+    (∑ x' : L ↪ Card d, ∑ y : L → Card d, retainedKernel d A x y * retainedKernel d A x' y) ≤
+      (2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
+        Real.exp (Real.exp 1*(Fintype.card L)) := by
+  classical
+  let f := fun y z : L → Card d => if ContactEvent d z y then (1:ℝ) else 0
+  have hsub (y : L → Card d) : (∑ x' : L ↪ Card d, retainedKernel d A x' y) ≤
+      ∑ z : L → Card d, partialKernel d A z y * f y z := by
+    have he (x' : L ↪ Card d) : retainedKernel d A x' y = partialKernel d A x' y * f y x' := by
+      unfold retainedKernel f
+      split_ifs <;> ring
+    simp_rw [he]
+    rw [← Finset.sum_image (f := fun z => partialKernel d A z y * f y z)
+      (Function.Embedding.coe_injective.injOn)]
+    apply Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _)
+    intro z _ _
+    exact mul_nonneg (partialKernel_nonneg d A z y) (by dsimp [f]; positivity)
+  calc
+    _ = ∑ y, retainedKernel d A x y * ∑ x' : L ↪ Card d, retainedKernel d A x' y := by
+      rw [Finset.sum_comm]
+      simp only [Finset.mul_sum]
+    _ ≤ ∑ y, partialKernel d A x y * ∑ z : L → Card d, partialKernel d A z y * f y z := by
+      apply Finset.sum_le_sum
+      intro y _
+      exact mul_le_mul (retainedKernel_le d A x y) (hsub y)
+        (Finset.sum_nonneg (fun x' _ => retainedKernel_nonneg d A x' y)) (partialKernel_nonneg d A x y)
+    _ = finiteMean (fun ω : Option L → (SwitchIndex d → Bool) =>
+        finiteMean (fun b : Option L → (SwitchIndex d → Bool) =>
+          f (mixedTuple d A ω x) (mixedReturn d A b (mixedTuple d A ω x)))) := by
+      rw [mixedTuple_expectation d A x (fun y =>
+        finiteMean (fun b : Option L → (SwitchIndex d → Bool) => f y (mixedReturn d A b y)))]
+      simp only [mixedReturn_expectation]
+    _ = finiteMean (fun b : Option L → (SwitchIndex d → Bool) =>
+        finiteMean (fun ω : Option L → (SwitchIndex d → Bool) =>
+          f (mixedTuple d A ω x) (mixedReturn d A b (mixedTuple d A ω x)))) := finiteMean_comm _
+    _ ≤ _ := by
+      apply (finiteMean_mono (fun b => probability_mixed_contact d A x x.injective b hδ hδ1)).trans_eq
+      exact finiteMean_const _
+
+noncomputable def rectOperator {X Y : Type*} [Fintype X] [Fintype Y]
+    (P : X → Y → ℝ) : EuclideanSpace ℂ Y →L[ℂ] EuclideanSpace ℂ X :=
+  LinearMap.toContinuousLinearMap
+    { toFun := fun v => WithLp.toLp 2 (fun x => ∑ y, (P x y : ℂ)*v y)
+      map_add' := by
+        intro v w
+        ext x
+        simp only [PiLp.add_apply, mul_add, Finset.sum_add_distrib]
+      map_smul' := by
+        intro c v
+        ext x
+        simp only [PiLp.smul_apply, smul_eq_mul, RingHom.id_apply]
+        change (∑ y, (P x y : ℂ)*(c*v y)) = c*∑ y, (P x y : ℂ)*v y
+        rw [Finset.mul_sum]
+        apply Finset.sum_congr rfl
+        intro y _
+        ring }
+
+@[simp] lemma rectOperator_apply {X Y : Type*} [Fintype X] [Fintype Y]
+    (P : X → Y → ℝ) (v : EuclideanSpace ℂ Y) (x : X) :
+    rectOperator P v x = ∑ y, (P x y:ℂ)*v y := rfl
+
+lemma rectOperator_adjoint {X Y : Type*} [Fintype X] [Fintype Y]
+    (P : X → Y → ℝ) : ContinuousLinearMap.adjoint (rectOperator P) =
+      rectOperator (fun y x => P x y) := by
+  apply Eq.symm
+  rw [ContinuousLinearMap.eq_adjoint_iff]
+  intro v w
+  simp only [PiLp.inner_apply, RCLike.inner_apply, rectOperator_apply,
+    map_sum, map_mul, Complex.conj_ofReal, Finset.sum_mul, Finset.mul_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro x _
+  apply Finset.sum_congr rfl
+  intro y _
+  ring
+
+lemma rectOperator_comp_transpose {X Y : Type*} [Fintype X] [Fintype Y]
+    (P : X → Y → ℝ) : (rectOperator P).comp (rectOperator (fun y x => P x y)) =
+      rectOperator (fun x x' => ∑ y, P x y * P x' y) := by
+  ext v x
+  simp only [ContinuousLinearMap.comp_apply, rectOperator_apply, Finset.mul_sum,
+    Complex.ofReal_sum, Complex.ofReal_mul, Finset.sum_mul]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro x' _
+  apply Finset.sum_congr rfl
+  intro y _
+  ring
+
+lemma rectOperator_norm_sq_le {X Y : Type*} [Fintype X] [Fintype Y]
+    (P : X → Y → ℝ) (hP : ∀ x y, 0 ≤ P x y) (B : ℝ) (hB : 0 ≤ B)
+    (hrow : ∀ x, (∑ x', ∑ y, P x y * P x' y) ≤ B) :
+    ‖rectOperator P‖^2 ≤ B := by
+  let S := fun x x' => ∑ y, P x y * P x' y
+  have hs : ∀ x x', S x x' = S x' x := by
+    intro x x'
+    unfold S
+    apply Finset.sum_congr rfl
+    intro y _
+    ring
+  have hbound : ‖rectOperator S‖ ≤ B := by
+    apply ContinuousLinearMap.opNorm_le_bound _ hB
+    intro v
+    exact DensityTransfer.schur_bound S
+      (fun x x' => Finset.sum_nonneg (fun y _ => mul_nonneg (hP x y) (hP x' y))) B hB
+      hrow (fun x' => by
+        calc (∑ x, S x x') = ∑ x, S x' x := Finset.sum_congr rfl (fun x _ => hs x x')
+             _ ≤ B := hrow x') v
+  have hn := ContinuousLinearMap.norm_adjoint_comp_self
+    (ContinuousLinearMap.adjoint (rectOperator P))
+  rw [ContinuousLinearMap.adjoint_adjoint, (ContinuousLinearMap.adjoint.norm_map _),
+    rectOperator_adjoint, rectOperator_comp_transpose] at hn
+  rw [pow_two, ← hn]
+  exact hbound
+
+lemma retained_norm_sq {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (hδ : 0 < 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)
+    (hδ1 : 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d ≤ 1) :
+    ‖rectOperator (retainedKernel d A)‖^2 ≤
+      (2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
+        Real.exp (Real.exp 1*(Fintype.card L)) :=
+  rectOperator_norm_sq_le _ (retainedKernel_nonneg d A) _ (by positivity)
+    (fun x => retained_square_row d A x hδ hδ1)
+
+
+noncomputable def pointStabilizer {α : Type*} (A : Finset α) : Subgroup (Equiv.Perm α) where
+  carrier := {p | ∀ x ∈ A, p x = x}
+  one_mem' := by intro x _; rfl
+  mul_mem' := by intro p q hp hq x hx; change p (q x) = x; rw [hq x hx, hp x hx]
+  inv_mem' := by intro p hp x hx; exact (Equiv.Perm.inv_eq_iff_eq).mpr (hp x hx).symm
+
+lemma swap_mem_pointStabilizer {α : Type*} [DecidableEq α] (A : Finset α)
+    (x y : α) (hx : x ∉ A) (hy : y ∉ A) : Equiv.swap x y ∈ pointStabilizer A := by
+  intro z hz
+  exact Equiv.swap_apply_of_ne_of_ne (fun h => hx (h ▸ hz)) (fun h => hy (h ▸ hz))
+
+lemma point_symmetrizer_zero (μ : YoungDiagram) (A : Finset (Specht.Cell μ))
+    (hA : A.card < μ.card - μ.rowLen 0) (v : Specht.Tabloid μ → ℂ) (hv : v ∈ Specht.space μ) :
+    Specht.symmetrizer (pointStabilizer A) (Specht.tabloidRep μ) v = 0 := by
+  classical
+  induction hv using Submodule.span_induction with
+  | mem v hv =>
+    obtain ⟨p,rfl⟩ := hv
+    let e : {x : Specht.Cell μ // p x ∉ A} ≃ {x : Specht.Cell μ // x ∉ A} :=
+      { toFun := fun x => ⟨p x.1,x.2⟩
+        invFun := fun x => ⟨p.symm x.1, by simpa only [p.apply_symm_apply] using x.2⟩
+        left_inv := fun x => by apply Subtype.ext; exact p.symm_apply_apply _
+        right_inv := fun x => by apply Subtype.ext; exact p.apply_symm_apply _ }
+    have hc : μ.rowLen 0 < Fintype.card {x : Specht.Cell μ // p x ∉ A} := by
+      rw [Fintype.card_congr e, Fintype.card_subtype_compl, Specht.card_cell, Fintype.card_coe]
+      have hr := Specht.rowLen_zero_le_card μ
+      omega
+    obtain ⟨x,y,hne,hcol⟩ := Fintype.exists_ne_map_eq_of_card_lt
+      (fun x : {x : Specht.Cell μ // p x ∉ A} => Specht.colIndex x.1)
+      (by simpa only [Fintype.card_fin] using hc)
+    have hxy : x.1 ≠ y.1 := fun h => hne (Subtype.ext h)
+    let h : pointStabilizer A := ⟨Equiv.swap (p x.1) (p y.1),
+      swap_mem_pointStabilizer A _ _ x.2 y.2⟩
+    apply Specht.symmetrizer_eq_zero_of_neg _ _ h
+    change Specht.tabloidRep μ (Equiv.swap (p x.1) (p y.1))
+      (Specht.tabloidRep μ p (Specht.polytabloid μ)) = -Specht.tabloidRep μ p (Specht.polytabloid μ)
+    rw [←Module.End.mul_apply, ←map_mul, ←Equiv.mul_swap_eq_swap_mul]
+    rw [map_mul, Module.End.mul_apply,
+      Specht.swap_polytabloid x.1 y.1 hxy (congrArg Fin.val hcol), map_neg]
+  | zero => exact map_zero _
+  | add v w hv hw ihv ihw => simp only [map_add,ihv,ihw,add_zero]
+  | smul a v hv ih => simp only [map_smul,ih,smul_zero]
+
+lemma point_fixed_zero (μ : YoungDiagram) (A : Finset (Specht.Cell μ))
+    (hA : A.card < μ.card - μ.rowLen 0) (v : Specht.Tabloid μ → ℂ) (hv : v ∈ Specht.space μ)
+    (hfix : ∀ p, p ∈ pointStabilizer A → Specht.tabloidRep μ p v = v) : v = 0 := by
+  classical
+  have hz := point_symmetrizer_zero μ A hA v hv
+  have he : Specht.symmetrizer (pointStabilizer A) (Specht.tabloidRep μ) v =
+      (Fintype.card (pointStabilizer A) : ℂ) • v := by
+    simp only [Specht.symmetrizer, LinearMap.sum_apply, hfix _ (Subtype.prop _),
+      Finset.sum_const, Finset.card_univ, nsmul_eq_mul, Nat.cast_smul_eq_nsmul]
+  rw [he] at hz
+  exact (smul_eq_zero.mp hz).resolve_left (by exact_mod_cast Fintype.card_ne_zero)
+
+lemma point_unitary_fixed_zero (μ : YoungDiagram) (A : Finset (Specht.Cell μ))
+    (hA : A.card < μ.card - μ.rowLen 0) (v : Specht.hilbertSpace μ)
+    (hfix : ∀ p, p ∈ pointStabilizer A → Specht.unitaryRepresentation μ p v = v) : v = 0 := by
+  let w := (Specht.spaceHilbertEquiv μ).symm v
+  have hw : ∀ p, p ∈ pointStabilizer A → Specht.tabloidRep μ p w.val = w.val := by
+    intro p hp
+    have he := congrArg ((Specht.spaceHilbertEquiv μ).symm) (hfix p hp)
+    change Specht.representation μ p w = w at he
+    exact congrArg Subtype.val he
+  have hz := point_fixed_zero μ A hA w.val w.prop hw
+  have hw0 : w = 0 := Subtype.ext hz
+  have he := congrArg (Specht.spaceHilbertEquiv μ) hw0
+  simpa only [w, LinearEquiv.apply_symm_apply, map_zero] using he
+
+lemma point_relabelled_fixed_zero {α : Type*} [Fintype α] (μ : YoungDiagram)
+    (e : α ≃ Specht.Cell μ) (A : Finset α) (hA : A.card < μ.card - μ.rowLen 0)
+    (v : Specht.hilbertSpace μ)
+    (hfix : ∀ p, p ∈ pointStabilizer A → Specht.relabelledUnitary μ e p v = v) : v = 0 := by
+  classical
+  apply point_unitary_fixed_zero μ (A.image e) (by simpa only [Finset.card_image_of_injective _ e.injective] using hA) v
+  intro p hp
+  have hf : e.symm.permCongr p ∈ pointStabilizer A := by
+    intro x hx
+    have hh := hp (e x) (Finset.mem_image.mpr ⟨x,hx,rfl⟩)
+    simp only [Equiv.permCongr_apply, Equiv.symm_symm, hh, e.symm_apply_apply]
+  have hh := hfix _ hf
+  change Specht.unitaryRepresentation μ (e.permCongr (e.symm.permCongr p)) v = v at hh
+  have he : e.permCongr (e.symm.permCongr p) = p := by
+    apply Equiv.ext
+    intro x
+    simp only [Equiv.permCongr_apply, Equiv.symm_symm, e.apply_symm_apply]
+  simpa only [he] using hh
+
+
+section Orbit
+variable {G X V : Type*} [Group G] [Fintype G] [Fintype X]
+    [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+    (a : G →* Equiv.Perm X) (x₀ : X) (ha : ∀ x, ∃ g, a g x₀ = x)
+    (ρ : Representation ℂ G V) (w : V) (hw : UnitaryFinite.IsFixed a x₀ ρ w)
+
+noncomputable def orbitVector (x : X) : V := ρ (UnitaryFinite.representative a x₀ ha x) w
+
+include hw in
+omit [Fintype G] [Fintype X] [FiniteDimensional ℂ V] in
+lemma orbitVector_at (g : G) : orbitVector a x₀ ha ρ w (a g x₀) = ρ g w := by
+  apply ext_inner_right ℂ
+  intro v
+  exact UnitaryFinite.coefficient_at a x₀ ha ρ w hw v g
+
+include hw in
+omit [Fintype G] [Fintype X] [FiniteDimensional ℂ V] in
+lemma orbitVector_action (g : G) (x : X) :
+    orbitVector a x₀ ha ρ w (a g x) = ρ g (orbitVector a x₀ ha ρ w x) := by
+  obtain ⟨h,rfl⟩ := ha x
+  rw [←Equiv.Perm.mul_apply, ←map_mul, orbitVector_at a x₀ ha ρ w hw,
+    orbitVector_at a x₀ ha ρ w hw, map_mul, Module.End.mul_apply]
+end Orbit
+
+def Harmonic {L α : Type*} [Fintype L] [Fintype α] (f : EuclideanSpace ℂ (L ↪ α)) : Prop :=
+  ∀ (A : Finset L), A ≠ Finset.univ → ∀ z : L → α,
+    (∑ y : L ↪ α, if ∀ i ∈ A, y i = z i then f y else 0) = 0
+
+lemma normalizedCoefficient_harmonic {L α : Type*} [Fintype L] [Fintype α]
+    (μ : YoungDiagram) (e : α ≃ Specht.Cell μ)
+    (hk : Fintype.card L = μ.card - μ.rowLen 0) (x₀ : L ↪ α)
+    (w : Specht.hilbertSpace μ)
+    (hw : UnitaryFinite.IsFixed UnitaryFinite.tupleAction x₀ (Specht.relabelledUnitary μ e) w)
+    (v : Specht.hilbertSpace μ) :
+    Harmonic (UnitaryFinite.normalizedCoefficient UnitaryFinite.tupleAction x₀
+      (UnitaryFinite.tupleAction_transitive x₀) (Specht.relabelledUnitary μ e) w v) := by
+  classical
+  intro A hA z
+  let ρ := Specht.relabelledUnitary μ e
+  let F := orbitVector UnitaryFinite.tupleAction x₀ (UnitaryFinite.tupleAction_transitive x₀) ρ w
+  let S : Specht.hilbertSpace μ := ∑ y : L ↪ α, if ∀ i ∈ A, y i = z i then F y else 0
+  have hcard : (A.image z).card < μ.card - μ.rowLen 0 := by
+    apply (Finset.card_image_le).trans_lt
+    rw [← hk]
+    exact Finset.card_lt_card (Finset.ssubset_univ_iff.mpr hA)
+  have hz : S = 0 := by
+    apply point_relabelled_fixed_zero μ e (A.image z) hcard
+    intro p hp
+    change ρ p S = S
+    simp only [S, map_sum, apply_ite, map_zero]
+    apply Fintype.sum_equiv (UnitaryFinite.tupleAction p)
+    intro y
+    have he : (∀ i ∈ A, UnitaryFinite.tupleAction p y i = z i) ↔ ∀ i ∈ A, y i = z i := by
+      apply forall₂_congr
+      intro i hi
+      have hpi : p (z i) = z i := hp _ (Finset.mem_image.mpr ⟨i, hi,rfl⟩)
+      change p (y i) = z i ↔ y i = z i
+      exact (show (p (y i) = z i) ↔ p (y i) = p (z i) by rw [hpi]).trans p.injective.eq_iff
+    simp only [he]
+    split_ifs
+    · exact (orbitVector_action UnitaryFinite.tupleAction x₀
+        (UnitaryFinite.tupleAction_transitive x₀) ρ w hw p y).symm
+    · rfl
+  let c : ℂ := Real.sqrt ((Module.finrank ℂ (Specht.hilbertSpace μ):ℝ)/(Fintype.card (L ↪ α):ℝ))
+  calc
+    _ = c * inner ℂ S v := by
+      dsimp only [S]
+      rw [sum_inner (𝕜 := ℂ) Finset.univ _ v, Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro y _
+      split_ifs <;> simp_all only [UnitaryFinite.normalizedCoefficient, PiLp.smul_apply,
+        smul_eq_mul, UnitaryFinite.coefficient, F, orbitVector, inner_zero_left (𝕜 := ℂ), mul_zero, c, ρ]
+    _ = 0 := by rw [hz, inner_zero_left (𝕜 := ℂ) v, mul_zero]
+
+end Thorp.SparseContact
+
+namespace Thorp.SparseContact
+
+section
+open scoped BigOperators Classical
+
+lemma sum_injections {L α M : Type*} [Fintype L] [Fintype α] [AddCommMonoid M]
+    (F : (L → α) → M) (hF : ∀ y, ¬Function.Injective y → F y = 0) :
+    (∑ y : L → α, F y) = ∑ y : L ↪ α, F y := by
+  classical
+  rw [← Finset.sum_image Function.Embedding.coe_injective.injOn]
+  apply Eq.symm
+  apply Finset.sum_subset (Finset.subset_univ _)
+  intro y _ hy
+  apply hF y
+  intro h
+  exact hy (Finset.mem_image.mpr ⟨⟨y,h⟩, Finset.mem_univ _, rfl⟩)
+
+noncomputable def zeroExtend {L α : Type*} [Fintype L] [Fintype α]
+    (f : EuclideanSpace ℂ (L ↪ α)) : EuclideanSpace ℂ (L → α) := by
+  classical
+  exact WithLp.toLp 2 (fun y => if h : Function.Injective y then f ⟨y,h⟩ else 0)
+
+@[simp] lemma zeroExtend_inj {L α : Type*} [Fintype L] [Fintype α]
+    (f : EuclideanSpace ℂ (L ↪ α)) (y : L ↪ α) : zeroExtend f y = f y := by
+  classical
+  simp only [zeroExtend, WithLp.ofLp_toLp, dite_eq_left y.injective]
+  congr 1
+
+lemma zeroExtend_noninj {L α : Type*} [Fintype L] [Fintype α]
+    (f : EuclideanSpace ℂ (L ↪ α)) (y : L → α) (hy : ¬Function.Injective y) : zeroExtend f y = 0 := by
+  classical
+  simp only [zeroExtend, WithLp.ofLp_toLp, dite_eq_right hy]
+
+lemma sum_zeroExtend_mul {L α : Type*} [Fintype L] [Fintype α]
+    (f : EuclideanSpace ℂ (L ↪ α)) (g : (L → α) → ℂ) :
+    (∑ y : L → α, g y * zeroExtend f y) = ∑ y : L ↪ α, g y * f y := by
+  rw [sum_injections _ (fun y hy => by rw [zeroExtend_noninj f y hy, mul_zero])]
+  simp only [zeroExtend_inj]
+
+lemma norm_zeroExtend {L α : Type*} [Fintype L] [Fintype α]
+    (f : EuclideanSpace ℂ (L ↪ α)) : ‖zeroExtend f‖ = ‖f‖ := by
+  have hs : ‖zeroExtend f‖^2 = ‖f‖^2 := by
+    rw [EuclideanSpace.norm_sq_eq, EuclideanSpace.norm_sq_eq,
+      sum_injections _ (fun y hy => by rw [zeroExtend_noninj f y hy]; simp)]
+    simp only [zeroExtend_inj]
+  nlinarith [norm_nonneg (zeroExtend f), norm_nonneg f]
+
+lemma partial_proper_zero {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (hA : A ≠ Finset.univ) (f : EuclideanSpace ℂ (L ↪ Card d)) (hf : Harmonic f) :
+    rectOperator (fun (x : L ↪ Card d) (y : L → Card d) => partialKernel d A x y) (zeroExtend f) = 0 := by
+  classical
+  ext x
+  rw [rectOperator_apply, sum_zeroExtend_mul]
+  change (∑ y : L ↪ Card d, (partialKernel d A x y : ℂ) * f y) = 0
+  simp only [partialKernel, endpointProb, finiteMean, Complex.ofReal_mul, Complex.ofReal_div,
+    Complex.ofReal_sum, Complex.ofReal_natCast]
+  simp_rw [div_eq_mul_inv, mul_assoc, ← Finset.mul_sum]
+  rw [mul_eq_zero]
+  right
+  simp only [Finset.sum_mul]
+  rw [Finset.sum_comm]
+  apply mul_eq_zero_of_right
+  apply Finset.sum_eq_zero
+  intro ω _
+  calc
+    _ = ∑ y : L ↪ Card d, (↑(Fintype.card (SwitchIndex d → Bool)) : ℂ)⁻¹ *
+        (if ∀ i ∈ A, y i = butterflyPerm d (decodeButterfly d ω) (x i) then f y else 0) :=
+      by
+        apply Finset.sum_congr rfl
+        intro y _
+        have hh : (∀ i ∈ A, butterflyPerm d (decodeButterfly d ω) (x i) = y i) ↔
+            (∀ i ∈ A, y i = butterflyPerm d (decodeButterfly d ω) (x i)) :=
+          forall₂_congr fun _ _ => eq_comm
+        by_cases h : ∀ i ∈ A, butterflyPerm d (decodeButterfly d ω) (x i) = y i
+        · simp only [ite_eq_left h, ite_eq_left (hh.mp h), Complex.ofReal_one, one_mul]
+        · simp only [ite_eq_right h, ite_eq_right (mt hh.mpr h), Complex.ofReal_zero, zero_mul, mul_zero]
+    _ = 0 := by
+      rw [← Finset.mul_sum]
+      apply mul_eq_zero_of_right
+      convert hf A hA (fun i => butterflyPerm d (decodeButterfly d ω) (x i)) using 1
+      apply Finset.sum_congr rfl
+      intro y _
+      split_ifs <;> rfl
+
+lemma partial_full_action {L : Type*} [Fintype L] (d : ℕ)
+    (f : EuclideanSpace ℂ (L ↪ Card d)) :
+    rectOperator (fun (x : L ↪ Card d) (y : L → Card d) => partialKernel d Finset.univ x y) (zeroExtend f) =
+      DensityTransfer.applyKernel (UnitaryFinite.actionKernel UnitaryFinite.tupleAction
+        (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))) f := by
+  classical
+  ext x
+  rw [rectOperator_apply, sum_zeroExtend_mul]
+  apply Finset.sum_congr rfl
+  intro y _
+  congr 2
+  simp only [partialKernel, Finset.card_univ, div_self (by positivity : ((2:ℝ)^d)^(Fintype.card L) ≠ 0),
+    one_mul, endpointProb, UnitaryFinite.actionKernel]
+  apply finiteMean_congr
+  intro ω
+  congr 1
+  simp only [Finset.mem_univ, forall_const, Function.Embedding.ext_iff]
+  rfl
+
+lemma alternating_retained_eq {L : Type*} [Fintype L] (d : ℕ)
+    (x : L ↪ Card d) (y : L → Card d) :
+    (∑ A : Finset L, (-1:ℝ)^A.card * retainedKernel d A x y) =
+      ∑ A : Finset L, (-1:ℝ)^A.card * partialKernel d A x y := by
+  classical
+  by_cases h : ContactEvent d x y
+  · simp only [retainedKernel, ite_eq_left h]
+  · simp only [retainedKernel, ite_eq_right h, mul_zero, Finset.sum_const_zero]
+    exact (alternating_kernel_cancel d x y h).symm
+
+lemma alternating_harmonic_action {L : Type*} [Fintype L] (d : ℕ)
+    (f : EuclideanSpace ℂ (L ↪ Card d)) (hf : Harmonic f) :
+    (∑ A : Finset L, ((-1:ℂ)^A.card) • rectOperator (retainedKernel d A) (zeroExtend f)) =
+      ((-1:ℂ)^(Fintype.card L)) • DensityTransfer.applyKernel
+        (UnitaryFinite.actionKernel UnitaryFinite.tupleAction
+          (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))) f := by
+  classical
+  calc
+    _ = ∑ A : Finset L, ((-1:ℂ)^A.card) •
+        rectOperator (fun (x : L ↪ Card d) y => partialKernel d A x y) (zeroExtend f) := by
+      ext x
+      simp only [WithLp.ofLp_sum, Finset.sum_apply, PiLp.smul_apply, rectOperator_apply, smul_eq_mul, Finset.mul_sum]
+      rw [Finset.sum_comm, Finset.sum_comm (s := (Finset.univ : Finset (Finset L)))]
+      apply Finset.sum_congr rfl
+      intro y _
+      simp_rw [← mul_assoc, ← Finset.sum_mul]
+      congr 1
+      have he := congrArg (fun t : ℝ => (t:ℂ)) (alternating_retained_eq d x y)
+      simpa only [Complex.ofReal_sum, Complex.ofReal_mul, Complex.ofReal_pow, Complex.ofReal_neg,
+        Complex.ofReal_one] using he
+    _ = ((-1:ℂ)^(Fintype.card L)) •
+        rectOperator (fun (x : L ↪ Card d) y => partialKernel d Finset.univ x y) (zeroExtend f) := by
+      rw [Finset.sum_eq_single (Finset.univ : Finset L)]
+      · simp only [Finset.card_univ]
+      · intro A _ hA
+        rw [partial_proper_zero d A hA f hf, smul_zero]
+      · simp
+    _ = _ := by rw [partial_full_action]
+
+lemma harmonic_action_bound {L : Type*} [Fintype L] (d : ℕ)
+    (f : EuclideanSpace ℂ (L ↪ Card d)) (hf : Harmonic f)
+    (hδ : 0 < 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)
+    (hδ1 : 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d ≤ 1) :
+    ‖DensityTransfer.applyKernel (UnitaryFinite.actionKernel UnitaryFinite.tupleAction
+      (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))) f‖ ≤
+      (2:ℝ)^(Fintype.card L) *
+        Real.sqrt ((2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
+          Real.exp (Real.exp 1*(Fintype.card L))) * ‖f‖ := by
+  classical
+  let B := (2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
+          Real.exp (Real.exp 1*(Fintype.card L))
+  have hB : 0 ≤ B := by dsimp [B]; positivity
+  have hnorm (A : Finset L) : ‖rectOperator (retainedKernel d A)‖ ≤ Real.sqrt B := by
+    have hh := retained_norm_sq d A hδ hδ1
+    have hs := Real.sq_sqrt hB
+    have hn := Real.sqrt_nonneg B
+    nlinarith
+  calc
+    _ = ‖((-1:ℂ)^(Fintype.card L)) • DensityTransfer.applyKernel
+        (UnitaryFinite.actionKernel UnitaryFinite.tupleAction
+          (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))) f‖ := by
+      simp only [norm_smul, norm_pow, norm_neg, norm_one, one_pow, one_mul]
+    _ = ‖∑ A : Finset L, ((-1:ℂ)^A.card) • rectOperator (retainedKernel d A) (zeroExtend f)‖ := by
+      rw [alternating_harmonic_action d f hf]
+    _ ≤ ∑ A : Finset L, ‖((-1:ℂ)^A.card) • rectOperator (retainedKernel d A) (zeroExtend f)‖ :=
+      norm_sum_le _ _
+    _ ≤ ∑ A : Finset L, Real.sqrt B * ‖f‖ := by
+      apply Finset.sum_le_sum
+      intro A _
+      simp only [norm_smul, norm_pow, norm_neg, norm_one, one_pow, one_mul]
+      calc _ ≤ ‖rectOperator (retainedKernel d A)‖ * ‖zeroExtend f‖ :=
+              ContinuousLinearMap.le_opNorm _ _
+           _ ≤ Real.sqrt B * ‖zeroExtend f‖ := mul_le_mul_of_nonneg_right (hnorm A) (norm_nonneg _)
+           _ = _ := by rw [norm_zeroExtend]
+    _ = _ := by simp only [Finset.sum_const, Finset.card_univ, Fintype.card_finset, nsmul_eq_mul,
+      Nat.cast_pow, Nat.cast_ofNat]; ring
+
+lemma harmonic_action_zero_one {L : Type*} [Fintype L] (d : ℕ)
+    (hL : Fintype.card L = 1) (f : EuclideanSpace ℂ (L ↪ Card d)) (hf : Harmonic f) :
+    DensityTransfer.applyKernel (UnitaryFinite.actionKernel UnitaryFinite.tupleAction
+      (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))) f = 0 := by
+  classical
+  obtain ⟨i,hi⟩ := Fintype.card_eq_one_iff.mp hL
+  have hz (A : Finset L) : rectOperator (retainedKernel d A) = 0 := by
+    ext v x
+    have hh (y : L → Card d) : ¬ContactEvent d x y := by
+      intro h
+      obtain ⟨j,hj,_⟩ := h i
+      exact hj (hi j).symm
+    simp only [rectOperator_apply, retainedKernel, ite_eq_right (hh _), Complex.ofReal_zero,
+      zero_mul, Finset.sum_const_zero, zero_apply, PiLp.zero_apply]
+  have he := alternating_harmonic_action d f hf
+  simp only [hz, zero_apply, smul_zero, Finset.sum_const_zero, hL, pow_one,
+    neg_smul, one_smul] at he
+  exact neg_eq_zero.mp he.symm
+
+end
+open scoped BigOperators
+
+def MainStatement : Prop :=
+  ∃ Cstar C : ℝ, 0 < Cstar ∧ 0 < C ∧
+    ∀ (d : ℕ), 1 ≤ d → ∀ (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ),
+      let k := μ.card - μ.rowLen 0
+      1 ≤ k → k < 2^d → ∀ reverse : Bool,
+        ‖sweepOperator d μ e reverse‖^2 ≤
+          min 1 ((Cstar * d * k / (2:ℝ)^d) ^ ((k:ℝ)/2)) ∧
+        (k = 1 → sweepOperator d μ e reverse = 0) ∧
+        ‖sweepOperator d μ e reverse‖^2 ≤
+          C^k * (1+(d:ℝ))^(C*k) * ((k:ℝ)/(2:ℝ)^d)^((k:ℝ)/2)
+
+end Thorp.SparseContact
+
+namespace Thorp.SparseContact
+open scoped BigOperators Classical
+
+lemma exists_exact_fixed_unit {α L : Type*} [Fintype α] [Fintype L]
+    (μ : YoungDiagram) (e : α ≃ Specht.Cell μ) (x₀ : L ↪ α)
+    (hL : Fintype.card L = μ.card - μ.rowLen 0) :
+    ∃ w : Specht.hilbertSpace μ,
+      UnitaryFinite.IsFixed UnitaryFinite.tupleAction x₀ (Specht.relabelledUnitary μ e) w ∧ ‖w‖ = 1 := by
+  let K := UnitaryFinite.fixedSpace (V := Specht.hilbertSpace μ) UnitaryFinite.tupleAction x₀ (Specht.relabelledUnitary μ e)
+  have hd : 0 < Module.finrank ℂ (Specht.hilbertSpace μ) := by
+    rw [Specht.finrank_hilbertSpace]
+    exact Module.finrank_pos_iff.mpr (Submodule.nontrivial_iff_ne_bot.mpr (Specht.space_ne_bot μ))
+  have hdim := Specht.relabelled_tuple_dimension μ e x₀
+  rw [hL, Nat.choose_self, mul_one] at hdim
+  have hm : 0 < Module.finrank ℂ K := by
+    by_contra h
+    have hz : Module.finrank ℂ K = 0 := by omega
+    change _ ≤ Module.finrank ℂ K * _ at hdim
+    rw [hz, zero_mul] at hdim
+    omega
+  let B := stdOrthonormalBasis ℂ K
+  let i : Fin (Module.finrank ℂ K) := ⟨0,hm⟩
+  exact ⟨(B i).val, (B i).prop, B.orthonormal.norm_eq_one i⟩
+
+lemma sweep_forward_norm {d : ℕ} (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ) (rev : Bool) :
+    ‖sweepOperator d μ e rev‖ = ‖sweepOperator d μ e false‖ := by
+  cases rev
+  · rfl
+  · simp only [sweepOperator, Bool.false_eq_true, ite_false, ite_true]
+    rw [UnitaryFinite.sampleOperator_inv _ (Specht.relabelledUnitary_unitary μ e),
+      ContinuousLinearMap.adjoint.norm_map]
+
+lemma sweep_norm_le_one (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ) (rev : Bool) :
+    ‖sweepOperator d μ e rev‖ ≤ 1 :=
+  UnitaryFinite.sampleOperator_norm_le _ (Specht.relabelledUnitary_unitary μ e) _
+
+section Generic
+variable {L V : Type*} [Fintype L] [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+  [FiniteDimensional ℂ V]
+
+lemma sample_sweep_bound (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V)
+    [Representation.IsIrreducible ρ] (hρ : UnitaryFinite.IsUnitary ρ)
+    (x₀ : L ↪ Card d) (w : V) (hw : UnitaryFinite.IsFixed UnitaryFinite.tupleAction x₀ ρ w)
+    (hn : ‖w‖ = 1)
+    (hh : ∀ v, Harmonic (UnitaryFinite.normalizedCoefficient UnitaryFinite.tupleAction x₀
+      (UnitaryFinite.tupleAction_transitive x₀) ρ w v))
+    (hδ : 0 < 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)
+    (hδ1 : 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d ≤ 1) :
+    ‖UnitaryFinite.sampleOperator ρ (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))‖ ≤
+      (2:ℝ)^(Fintype.card L) *
+        Real.sqrt ((2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
+          Real.exp (Real.exp 1*(Fintype.card L))) := by
+  rw [← ContinuousLinearMap.adjoint.norm_map (UnitaryFinite.sampleOperator ρ
+    (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))),
+    ← UnitaryFinite.sampleOperator_inv ρ hρ]
+  apply ContinuousLinearMap.opNorm_le_bound _ (by positivity)
+  intro v
+  rw [UnitaryFinite.sampleOperator_inv_apply]
+  have ha := harmonic_action_bound d _ (hh v) hδ hδ1
+  rw [UnitaryFinite.coefficient_random UnitaryFinite.tupleAction x₀
+    (UnitaryFinite.tupleAction_transitive x₀) ρ hρ _ w hw,
+    UnitaryFinite.normalizedCoefficient_norm _ _ _ ρ hρ w hw hn,
+    UnitaryFinite.normalizedCoefficient_norm _ _ _ ρ hρ w hw hn] at ha
+  exact ha
+
+lemma sample_sweep_zero_one (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V)
+    [Representation.IsIrreducible ρ] (hρ : UnitaryFinite.IsUnitary ρ)
+    (x₀ : L ↪ Card d) (w : V) (hw : UnitaryFinite.IsFixed UnitaryFinite.tupleAction x₀ ρ w)
+    (hn : ‖w‖ = 1)
+    (hh : ∀ v, Harmonic (UnitaryFinite.normalizedCoefficient UnitaryFinite.tupleAction x₀
+      (UnitaryFinite.tupleAction_transitive x₀) ρ w v))
+    (hL : Fintype.card L = 1) :
+    UnitaryFinite.sampleOperator ρ (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω)) = 0 := by
+  apply ContinuousLinearMap.adjoint.injective
+  rw [map_zero, ← UnitaryFinite.sampleOperator_inv ρ hρ]
+  apply ContinuousLinearMap.ext
+  intro v
+  rw [UnitaryFinite.sampleOperator_inv_apply]
+  have ha := harmonic_action_zero_one d hL _ (hh v)
+  have he := congrArg norm ha
+  rw [UnitaryFinite.coefficient_random UnitaryFinite.tupleAction x₀
+    (UnitaryFinite.tupleAction_transitive x₀) ρ hρ _ w hw,
+    UnitaryFinite.normalizedCoefficient_norm _ _ _ ρ hρ w hw hn, norm_zero] at he
+  exact norm_eq_zero.mp he
+end Generic
+
+lemma sweep_norm_bound (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ)
+    (hδ : 0 < 2*(d:ℝ)*(μ.card - μ.rowLen 0 : ℕ)/(2:ℝ)^d)
+    (hδ1 : 2*(d:ℝ)*(μ.card - μ.rowLen 0 : ℕ)/(2:ℝ)^d ≤ 1) (rev : Bool) :
+    ‖sweepOperator d μ e rev‖ ≤ (2:ℝ)^(μ.card - μ.rowLen 0) *
+      Real.sqrt ((2*(d:ℝ)*(μ.card - μ.rowLen 0 : ℕ)/(2:ℝ)^d)^(((μ.card - μ.rowLen 0 : ℕ):ℝ)/2) *
+        Real.exp (Real.exp 1*(μ.card - μ.rowLen 0 : ℕ))) := by
+  classical
+  let x₀ : Specht.Tail μ ↪ Card d := (Function.Embedding.subtype _).trans e.symm.toEmbedding
+  obtain ⟨w,hw,hn⟩ := exists_exact_fixed_unit μ e x₀ (Specht.tail_card μ)
+  let := Specht.relabelledUnitary_irreducible μ e
+  rw [sweep_forward_norm μ e rev]
+  have hh (v : Specht.hilbertSpace μ) : Harmonic (UnitaryFinite.normalizedCoefficient
+      (V := Specht.hilbertSpace μ) UnitaryFinite.tupleAction x₀
+      (UnitaryFinite.tupleAction_transitive x₀) (Specht.relabelledUnitary μ e) w v) :=
+    normalizedCoefficient_harmonic μ e (Specht.tail_card μ) x₀ w hw v
+  have he := sample_sweep_bound (V := Specht.hilbertSpace μ) (L := Specht.Tail μ) d (Specht.relabelledUnitary μ e)
+  have he := he (Specht.relabelledUnitary_unitary μ e) x₀ w hw hn hh
+  have he := he (by simpa only [Specht.tail_card] using hδ) (by simpa only [Specht.tail_card] using hδ1)
+  simpa only [sweepOperator, Bool.false_eq_true, ite_false, Specht.tail_card] using he
+
+lemma sweep_zero_one (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ)
+    (hk : μ.card - μ.rowLen 0 = 1) (rev : Bool) : sweepOperator d μ e rev = 0 := by
+  classical
+  let x₀ : Specht.Tail μ ↪ Card d := (Function.Embedding.subtype _).trans e.symm.toEmbedding
+  obtain ⟨w,hw,hn⟩ := exists_exact_fixed_unit μ e x₀ (Specht.tail_card μ)
+  let := Specht.relabelledUnitary_irreducible μ e
+  have hh (v : Specht.hilbertSpace μ) : Harmonic (UnitaryFinite.normalizedCoefficient
+      (V := Specht.hilbertSpace μ) UnitaryFinite.tupleAction x₀
+      (UnitaryFinite.tupleAction_transitive x₀) (Specht.relabelledUnitary μ e) w v) :=
+    normalizedCoefficient_harmonic μ e (Specht.tail_card μ) x₀ w hw v
+  have he := sample_sweep_zero_one (V := Specht.hilbertSpace μ) (L := Specht.Tail μ) d (Specht.relabelledUnitary μ e)
+  have he := he (Specht.relabelledUnitary_unitary μ e) x₀ w hw hn hh
+  have he := he ((Specht.tail_card μ).trans hk)
+  have hz : sweepOperator d μ e false = 0 := by
+    simpa only [sweepOperator, Bool.false_eq_true, ite_false] using he
+  apply (norm_eq_zero (a := sweepOperator d μ e rev)).mp
+  rw [sweep_forward_norm μ e rev, hz]
+  exact @norm_zero (Specht.hilbertSpace μ →L[ℂ] Specht.hilbertSpace μ) _
+
+end Thorp.SparseContact
+
+namespace Thorp.SparseContact
+
+noncomputable def contactConstant : ℝ := 32 * Real.exp (2 * Real.exp 1)
+
+lemma contactConstant_gt_one : 1 < contactConstant := by
+  have h : 1 ≤ Real.exp (2 * Real.exp 1) := Real.one_le_exp (by positivity)
+  unfold contactConstant
+  linarith
+
+lemma contactConstant_gt_two : 2 < contactConstant := by
+  have h : 1 ≤ Real.exp (2 * Real.exp 1) := Real.one_le_exp (by positivity)
+  unfold contactConstant
+  linarith
+
+lemma inflated_contact_bound {δ : ℝ} (hδ : 0 ≤ δ) (k : ℕ) :
+    ((2:ℝ)^k * Real.sqrt (δ^((k:ℝ)/2) * Real.exp (Real.exp 1*k)))^2 =
+      ((contactConstant/2)*δ)^((k:ℝ)/2) := by
+  have hC : 0 ≤ contactConstant/2 := by have := contactConstant_gt_one; positivity
+  rw [mul_pow, Real.sq_sqrt (by positivity), Real.mul_rpow hC hδ]
+  have hc : contactConstant/2 = Real.exp (2*(Real.log 4+Real.exp 1)) := by
+    unfold contactConstant
+    rw [show 2*(Real.log 4+Real.exp 1) = Real.log 4*2+2*Real.exp 1 by ring,
+      Real.exp_add, Real.exp_mul (Real.log 4) 2, Real.exp_log (by norm_num : (0:ℝ)<4)]
+    norm_num
+    ring
+  rw [hc, ←Real.exp_mul]
+  have hpow : ((2:ℝ)^k)^2 = Real.exp (Real.log 4 * k) := by
+    rw [← pow_mul, mul_comm k 2, pow_mul]
+    norm_num only [pow_two]
+    rw [Real.exp_mul (Real.log 4) k, Real.exp_log (by norm_num : (0:ℝ)<4), Real.rpow_natCast]
+  rw [hpow]
+  calc
+    _ = (Real.exp (Real.log 4*k)*Real.exp (Real.exp 1*k))*δ^((k:ℝ)/2) := by ring
+    _ = _ := by rw [←Real.exp_add]; congr 2; ring
+
+lemma contact_crude_arithmetic {C d q : ℝ} (hC : 1 ≤ C) (hd : 1 ≤ d) (hq : 0 ≤ q) (k : ℕ) :
+    (C*d*q)^((k:ℝ)/2) ≤ C^k*(1+d)^(C*k)*q^((k:ℝ)/2) := by
+  rw [Real.mul_rpow (by positivity) hq, Real.mul_rpow (by positivity) (by positivity)]
+  apply mul_le_mul_of_nonneg_right _ (by positivity)
+  apply mul_le_mul
+  · rw [← Real.rpow_natCast]
+    exact Real.rpow_le_rpow_of_exponent_le hC (by linarith [Nat.cast_nonneg (α := ℝ) k])
+  · calc
+      d^((k:ℝ)/2) ≤ (1+d)^((k:ℝ)/2) := Real.rpow_le_rpow (by positivity) (by linarith) (by positivity)
+      _ ≤ (1+d)^(C*k) := Real.rpow_le_rpow_of_exponent_le (by linarith) (by nlinarith [Nat.cast_nonneg (α := ℝ) k])
+  · positivity
+  · positivity
+
+
+lemma sweep_sharp_bound (d : ℕ) (hd : 1 ≤ d) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ)
+    (hk : 1 ≤ μ.card - μ.rowLen 0) (rev : Bool) :
+    ‖sweepOperator d μ e rev‖^2 ≤ min 1
+      ((contactConstant*d*(μ.card - μ.rowLen 0 : ℕ)/(2:ℝ)^d)^(((μ.card - μ.rowLen 0 : ℕ):ℝ)/2)) := by
+  let k := μ.card - μ.rowLen 0
+  have hk' : (0:ℝ) < k := by exact_mod_cast hk
+  have hd' : (0:ℝ) < d := by exact_mod_cast hd
+  have hδ : 0 < 2*(d:ℝ)*k/(2:ℝ)^d := by positivity
+  have hOne : ‖sweepOperator d μ e rev‖^2 ≤ 1 := by
+    have h := sweep_norm_le_one d μ e rev
+    have h0 := norm_nonneg (sweepOperator d μ e rev)
+    nlinarith
+  refine le_min hOne ?_
+  have he : contactConstant * d * (k:ℝ)/(2:ℝ)^d =
+      (contactConstant/2)*(2*(d:ℝ)*k/(2:ℝ)^d) := by ring
+  change _ ≤ (contactConstant * d * (k:ℝ)/(2:ℝ)^d)^((k:ℝ)/2)
+  rw [he]
+  by_cases hs : 2*(d:ℝ)*k/(2:ℝ)^d ≤ 1
+  · have h := sweep_norm_bound d μ e hδ hs rev
+    have hb : 0 ≤ (2:ℝ)^k * Real.sqrt
+        ((2*(d:ℝ)*k/(2:ℝ)^d)^((k:ℝ)/2)*Real.exp (Real.exp 1*k)) := by positivity
+    calc
+      _ ≤ ((2:ℝ)^k * Real.sqrt
+          ((2*(d:ℝ)*k/(2:ℝ)^d)^((k:ℝ)/2)*Real.exp (Real.exp 1*k)))^2 :=
+        (sq_le_sq₀ (norm_nonneg (sweepOperator d μ e rev)) hb).mpr h
+      _ = _ := inflated_contact_bound hδ.le k
+  · apply hOne.trans
+    apply Real.one_le_rpow
+    · have hc : 1 ≤ contactConstant/2 := by have := contactConstant_gt_two; linarith
+      exact one_le_mul_of_one_le_of_one_le hc (le_of_lt (lt_of_not_ge hs))
+    · positivity
+
+theorem uniform_sparse_contact : MainStatement := by
+  refine ⟨contactConstant, contactConstant, (lt_trans (by norm_num) contactConstant_gt_one),
+    (lt_trans (by norm_num) contactConstant_gt_one), ?_⟩
+  intro d hd μ e k hk _ rev
+  have hs := sweep_sharp_bound d hd μ e hk rev
+  refine ⟨hs, ?_, ?_⟩
+  · intro hk1
+    exact sweep_zero_one d μ e hk1 rev
+  · apply (hs.trans (min_le_right _ _)).trans
+    change (contactConstant * d * (k:ℝ)/(2:ℝ)^d)^((k:ℝ)/2) ≤ _
+    rw [mul_div_assoc]
+    exact contact_crude_arithmetic contactConstant_gt_one.le
+      (by exact_mod_cast hd) (by positivity) k
+
+end Thorp.SparseContact
+
+
+namespace Thorp.SparseRegime
+open Filter
+open Thorp.SparseContact
+
+lemma log_contact_eventually : ∀ᶠ d : ℕ in atTop,
+    Real.log (contactConstant*d) ≤ (d:ℝ)^((3:ℝ)/4)/2 := by
+  have hn : Tendsto (fun d : ℕ => (d:ℝ)) atTop atTop := tendsto_natCast_atTop_atTop
+  have hl := ((isLittleO_log_rpow_atTop (by norm_num : (0:ℝ)<3/4)).comp_tendsto hn).def
+    (by norm_num : (0:ℝ)<1/4)
+  have hr := ((tendsto_rpow_atTop (by norm_num : (0:ℝ)<3/4)).comp hn).eventually
+    (eventually_ge_atTop (4*Real.log contactConstant))
+  filter_upwards [hl, hr, eventually_ge_atTop 1] with d hl hr hd
+  have hd' : (0:ℝ)<d := by exact_mod_cast hd
+  have hc : 0<contactConstant := lt_trans (by norm_num) contactConstant_gt_one
+  simp only [Function.comp_apply, Real.norm_eq_abs,
+    abs_of_nonneg (Real.rpow_nonneg (Nat.cast_nonneg d) ((3:ℝ)/4))] at hl
+  simp only [Function.comp_apply] at hr
+  have hl' := (le_abs_self (Real.log (d:ℝ))).trans hl
+  rw [Real.log_mul hc.ne' hd'.ne']
+  linarith
+
+lemma contact_power_saving {d k : ℕ} (hd : 1 ≤ d) (hk : 1 ≤ k)
+    (hlog : Real.log (contactConstant*d) ≤ (d:ℝ)^((3:ℝ)/4)/2)
+    (hs : (d:ℝ)^((3:ℝ)/4) ≤ Real.log ((2:ℝ)^d/k)) :
+    (contactConstant*d*k/(2:ℝ)^d)^((k:ℝ)/2) ≤ ((k:ℝ)/(2:ℝ)^d)^((k:ℝ)/4) := by
+  have hk0 : (0:ℝ)<k := by exact_mod_cast hk
+  have hd0 : (0:ℝ)<d := by exact_mod_cast hd
+  have hc : 0<contactConstant := lt_trans (by norm_num) contactConstant_gt_one
+  have hq : 0<(k:ℝ)/(2:ℝ)^d := by positivity
+  have hmain : 0<contactConstant*d*k/(2:ℝ)^d := by positivity
+  rw [Real.rpow_def_of_pos hmain, Real.rpow_def_of_pos hq]
+  apply Real.exp_le_exp.mpr
+  have hlogq : Real.log ((k:ℝ)/(2:ℝ)^d) = -Real.log ((2:ℝ)^d/k) := by
+    rw [Real.log_div hk0.ne' (by positivity), Real.log_div (by positivity) hk0.ne']
+    ring
+  have he : contactConstant*d*k/(2:ℝ)^d = (contactConstant*d)*((k:ℝ)/(2:ℝ)^d) := by ring
+  rw [he, Real.log_mul (by positivity) hq.ne', hlogq]
+  nlinarith
+
+lemma inverse_dyadic_power (d k : ℕ) (hk : 1 ≤ k) :
+    ((k:ℝ)/(2:ℝ)^d)^((k:ℝ)/4) = Real.exp (-Real.log ((2:ℝ)^d/k)*k/4) := by
+  have hk0 : (0:ℝ)<k := by exact_mod_cast hk
+  rw [Real.rpow_def_of_pos (by positivity)]
+  congr 1
+  rw [Real.log_div hk0.ne' (by positivity), Real.log_div (by positivity) hk0.ne']
+  ring
+
+lemma sparse_finite_correction {d d₀ k : ℕ} (hd : d < d₀) (hk : 1 ≤ k) :
+    1 ≤ Real.exp ((1+(d₀:ℝ)*Real.log 2)*k) * ((k:ℝ)/(2:ℝ)^d)^((k:ℝ)/4) := by
+  have hk0 : (0:ℝ)<k := by exact_mod_cast hk
+  have hk1 : (1:ℝ)≤k := by exact_mod_cast hk
+  have hd' : (d:ℝ) < d₀ := by exact_mod_cast hd
+  have hlogk : 0 ≤ Real.log (k:ℝ) := Real.log_nonneg hk1
+  rw [inverse_dyadic_power d k hk, ←Real.exp_add]
+  apply Real.one_le_exp
+  rw [Real.log_div (by positivity) hk0.ne', Real.log_pow]
+  have hlog2 : 0 ≤ Real.log 2 := (Real.log_pos (by norm_num : (1:ℝ)<2)).le
+  have hdd := mul_le_mul_of_nonneg_right hd'.le hlog2
+  have hs : (d:ℝ)*Real.log 2-Real.log k ≤ 4*(1+(d₀:ℝ)*Real.log 2) := by
+    nlinarith [mul_nonneg (Nat.cast_nonneg (α := ℝ) d₀) hlog2]
+  have hm := mul_le_mul_of_nonneg_right hs hk0.le
+  nlinarith
+
+theorem exponential_sparse_saving : MainStatement := by
+  obtain ⟨d₁, hd₁⟩ := eventually_atTop.mp log_contact_eventually
+  let d₀ := max d₁ 1
+  let C : ℝ := 1+(d₀:ℝ)*Real.log 2
+  have hC : 0<C := by dsimp [C]; positivity
+  refine ⟨C,d₀,hC,by dsimp [d₀]; omega,?_⟩
+  intro d μ e k hk hkn hs rev
+  have hd : 1 ≤ d := by
+    by_contra h
+    have hd0 : d=0 := by omega
+    simp only [hd0, pow_zero] at hkn
+    omega
+  have hz : ‖sweepOperator d μ e rev‖^2 ≤ 1 := by
+    have hn := sweep_norm_le_one d μ e rev
+    have hp := norm_nonneg (sweepOperator d μ e rev)
+    nlinarith
+  have hlarge (hdd : d₀ ≤ d) : ‖sweepOperator d μ e rev‖^2 ≤ ((k:ℝ)/(2:ℝ)^d)^((k:ℝ)/4) := by
+    exact (sweep_sharp_bound d hd μ e hk rev).trans ((min_le_right _ _).trans
+      (contact_power_saving hd hk (hd₁ d (le_trans (le_max_left _ _) hdd)) hs))
+  refine ⟨?_, hlarge⟩
+  by_cases hdd : d₀ ≤ d
+  · apply (hlarge hdd).trans
+    have he : 1 ≤ Real.exp (C*k) := Real.one_le_exp (by positivity)
+    exact le_mul_of_one_le_left (by positivity) he
+  · exact hz.trans (sparse_finite_correction (lt_of_not_ge hdd) hk)
+
+end Thorp.SparseRegime
+
+end ThorpNine.SparseSaving
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/SparseSaving/Model.lean b/lean/OAI/Probability/ThorpRouting/SparseSaving/Model.lean
index 63018c6852ab142ff8474fd5d35dece3b1ed8fa3..48d298dac76344ad74c9070ef3557a5b5eee43bf 100644
--- a/lean/OAI/Probability/ThorpRouting/SparseSaving/Model.lean
+++ b/lean/OAI/Probability/ThorpRouting/SparseSaving/Model.lean
@@ -1,271 +1,271 @@
-import Mathlib
-
-namespace OAI
-
-namespace ThorpNine.SparseSaving
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-
-abbrev Card (d : ℕ) := Fin d → Bool
-
-def switchFun {d : ℕ} (ξ : Card d → Bool) (x : Card (d + 1)) : Card (d + 1) :=
-  Fin.cons (Bool.xor (x 0) (ξ (Fin.tail x))) (Fin.tail x)
-
-theorem switchFun_involutive {d : ℕ} (ξ : Card d → Bool) :
-    Function.Involutive (switchFun ξ) := by
-  intro x
-  funext i
-  refine Fin.cases ?_ (fun j => ?_) i
-  · simp only [switchFun, Fin.cons_zero, Fin.tail_cons]
-    cases x 0 <;> cases ξ (Fin.tail x) <;> rfl
-  · simp [switchFun, Fin.tail]
-
-def pairSwitch {d : ℕ} (ξ : Card d → Bool) : Equiv.Perm (Card (d + 1)) :=
-  { toFun := switchFun ξ
-    invFun := switchFun ξ
-    left_inv := switchFun_involutive ξ
-    right_inv := switchFun_involutive ξ }
-
-def Butterfly : ℕ → Type
-  | 0 => Unit
-  | d + 1 => (Card d → Bool) × (Bool → Butterfly d)
-
-def childLift {d : ℕ} (p : Bool → Equiv.Perm (Card d)) : Equiv.Perm (Card (d + 1)) where
-  toFun x := Fin.cons (x 0) (p (x 0) (Fin.tail x))
-  invFun x := Fin.cons (x 0) ((p (x 0)).symm (Fin.tail x))
-  left_inv x := by simp only [Fin.cons_zero, Fin.tail_cons, Equiv.symm_apply_apply, Fin.cons_self_tail]
-  right_inv x := by simp only [Fin.cons_zero, Fin.tail_cons, Equiv.apply_symm_apply, Fin.cons_self_tail]
-
-def butterflyPerm : (d : ℕ) → Butterfly d → Equiv.Perm (Card d)
-  | 0, _ => 1
-  | d + 1, b => pairSwitch b.1 * childLift (fun ε => butterflyPerm d (b.2 ε))
-
-def SwitchIndex : ℕ → Type
-  | 0 => Empty
-  | d + 1 => Sum (Card d) (Bool × SwitchIndex d)
-
-instance (d : ℕ) : Fintype (SwitchIndex d) := by
-  induction d with
-  | zero => exact inferInstanceAs (Fintype Empty)
-  | succ d ih => exact inferInstanceAs (Fintype (Sum (Card d) (Bool × SwitchIndex d)))
-
-instance (d : ℕ) : DecidableEq (SwitchIndex d) := by
-  induction d with
-  | zero => exact inferInstanceAs (DecidableEq Empty)
-  | succ d ih => exact inferInstanceAs (DecidableEq (Sum (Card d) (Bool × SwitchIndex d)))
-
-def decodeButterfly : (d : ℕ) → (SwitchIndex d → Bool) → Butterfly d
-  | 0, _ => ()
-  | d + 1, ω => (fun y => ω (Sum.inl y), fun ε =>
-      decodeButterfly d (fun i => ω (Sum.inr (ε,i))))
-
-end Thorp
-
-namespace Thorp.Specht
-open scoped BigOperators Classical
-
-abbrev Cell (μ : YoungDiagram) := {x : ℕ × ℕ // x ∈ μ.cells}
-
-def row {μ : YoungDiagram} (x : Cell μ) : ℕ := x.1.1
-
-def col {μ : YoungDiagram} (x : Cell μ) : ℕ := x.1.2
-
-lemma row_lt_colLen {μ : YoungDiagram} (x : Cell μ) : row x < μ.colLen (col x) :=
-  YoungDiagram.mem_iff_lt_colLen.mp x.2
-
-noncomputable def rowIndex {μ : YoungDiagram} (x : Cell μ) : Fin (μ.colLen 0) :=
-  ⟨row x, (row_lt_colLen x).trans_le (μ.colLen_anti 0 (col x) (Nat.zero_le _))⟩
-
-abbrev Tabloid (μ : YoungDiagram) :=
-  {f : Cell μ → Fin (μ.colLen 0) // ∃ p : Equiv.Perm (Cell μ), f = rowIndex ∘ p}
-
-noncomputable instance (μ : YoungDiagram) : Fintype (Tabloid μ) := Fintype.ofFinite _
-
-noncomputable def baseTabloid (μ : YoungDiagram) : Tabloid μ := ⟨rowIndex, 1, rfl⟩
-
-noncomputable def tabloidAct {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) :
-    Equiv.Perm (Tabloid μ) where
-  toFun f := ⟨fun x => f.1 (p⁻¹ x), by
-    obtain ⟨q,hq⟩ := f.2
-    refine ⟨q*p⁻¹, ?_⟩
-    funext x
-    simp only [hq,Function.comp_apply,Equiv.Perm.mul_apply]⟩
-  invFun f := ⟨fun x => f.1 (p x), by
-    obtain ⟨q,hq⟩ := f.2
-    refine ⟨q*p, ?_⟩
-    funext x
-    simp only [hq,Function.comp_apply,Equiv.Perm.mul_apply]⟩
-  left_inv f := by apply Subtype.ext; funext x; simp
-  right_inv f := by apply Subtype.ext; funext x; simp
-
-@[simp] lemma tabloidAct_apply {μ : YoungDiagram} (p : Equiv.Perm (Cell μ))
-    (f : Tabloid μ) (x : Cell μ) : (tabloidAct p f).1 x = f.1 (p⁻¹ x) := rfl
-
-@[simp] lemma tabloidAct_one (μ : YoungDiagram) : tabloidAct (1 : Equiv.Perm (Cell μ)) = 1 := by
-  ext f x
-  rfl
-
-@[simp] lemma tabloidAct_mul {μ : YoungDiagram} (p q : Equiv.Perm (Cell μ)) :
-    tabloidAct (p*q) = tabloidAct p * tabloidAct q := by
-  ext f x
-  simp only [tabloidAct_apply, mul_inv_rev,Equiv.Perm.mul_apply]
-
-noncomputable def tabloidRep (μ : YoungDiagram) :
-    Representation ℂ (Equiv.Perm (Cell μ)) (Tabloid μ → ℂ) where
-  toFun p :=
-    { toFun := fun v f => v (tabloidAct p⁻¹ f)
-      map_add' := by intros; rfl
-      map_smul' := by intros; rfl }
-  map_one' := by ext v f; simp
-  map_mul' p q := by
-    ext v f
-    simp only [mul_inv_rev,tabloidAct_mul,Equiv.Perm.mul_apply]
-    rfl
-
-noncomputable def delta {μ : YoungDiagram} (f : Tabloid μ) : Tabloid μ → ℂ :=
-  fun g => if g=f then 1 else 0
-
-noncomputable def colGroup (μ : YoungDiagram) : Subgroup (Equiv.Perm (Cell μ)) where
-  carrier := {p | ∀ x, col (p x) = col x}
-  one_mem' := by intro x; rfl
-  mul_mem' := by intro p q hp hq x; exact (hp (q x)).trans (hq x)
-  inv_mem' := by intro p hp x; simpa using (hp (p⁻¹ x)).symm
-
-noncomputable def permSign {α : Type*} [Fintype α] [DecidableEq α] (p : Equiv.Perm α) : ℂ :=
-  ((Equiv.Perm.sign p : ℤ) : ℂ)
-
-
-section
-variable {α V : Type*} [Fintype α] [DecidableEq α] [AddCommGroup V] [Module ℂ V]
-
-noncomputable def alternator (C : Subgroup (Equiv.Perm α))
-    (ρ : Representation ℂ (Equiv.Perm α) V) : Module.End ℂ V := by
-  classical
-  exact ∑ c : C, permSign (c : Equiv.Perm α) • ρ c
-
-end
-
-noncomputable def polytabloid (μ : YoungDiagram) : Tabloid μ → ℂ :=
-  alternator (colGroup μ) (tabloidRep μ) (delta (baseTabloid μ))
-
-noncomputable def space (μ : YoungDiagram) : Submodule ℂ (Tabloid μ → ℂ) :=
-  Submodule.span ℂ (Set.range (fun p : Equiv.Perm (Cell μ) => tabloidRep μ p (polytabloid μ)))
-
-lemma orbit_mem_space (μ : YoungDiagram) (p : Equiv.Perm (Cell μ)) :
-    tabloidRep μ p (polytabloid μ) ∈ space μ := Submodule.subset_span ⟨p,rfl⟩
-
-lemma action_mem_space (μ : YoungDiagram) (p : Equiv.Perm (Cell μ))
-    (v : Tabloid μ → ℂ) (hv : v ∈ space μ) : tabloidRep μ p v ∈ space μ := by
-  induction hv using Submodule.span_induction with
-  | mem v hv =>
-    obtain ⟨q,rfl⟩ := hv
-    change ((tabloidRep μ p)*(tabloidRep μ q)) (polytabloid μ) ∈ space μ
-    rw [←map_mul]
-    exact orbit_mem_space μ (p*q)
-  | zero => simp
-  | add v w hv hw ihv ihw => simpa using (space μ).add_mem ihv ihw
-  | smul a v hv ih => simpa using (space μ).smul_mem a ih
-
-noncomputable def subrepresentation (μ : YoungDiagram) : Subrepresentation (tabloidRep μ) :=
-  ⟨space μ, fun p v hv => action_mem_space μ p v hv⟩
-
-noncomputable def representation (μ : YoungDiagram) :
-    Representation ℂ (Equiv.Perm (Cell μ)) (space μ) :=
-  (subrepresentation μ).toRepresentation
-
-end Thorp.Specht
-
-namespace Thorp.PermutationHilbert
-open scoped BigOperators ComplexConjugate Classical
-open Complex
-variable {X : Type*} [Fintype X]
-
-abbrev H (X : Type*) [Fintype X] := EuclideanSpace ℂ X
-
-end Thorp.PermutationHilbert
-
-namespace Thorp.Specht
-open scoped BigOperators Classical
-
-noncomputable def hilbertEquiv (μ : YoungDiagram) :
-    (Tabloid μ → ℂ) ≃ₗ[ℂ] PermutationHilbert.H (Tabloid μ) :=
-  (WithLp.linearEquiv 2 ℂ (Tabloid μ → ℂ)).symm
-
-noncomputable def hilbertSpace (μ : YoungDiagram) :
-    Submodule ℂ (PermutationHilbert.H (Tabloid μ)) := (space μ).map (hilbertEquiv μ).toLinearMap
-
-end Thorp.Specht
-
-namespace Thorp.UnitaryFinite
-
-section
-open scoped BigOperators ComplexConjugate Classical
-variable {G : Type*} [Group G] [Fintype G]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-noncomputable def continuousRepresentation (ρ : Representation ℂ G V) : G →* (V →L[ℂ] V) where
-  toFun g := (ρ g).toContinuousLinearMap
-  map_one' := by ext v; simp
-  map_mul' g h := by ext v; simp
-
-end
-open scoped BigOperators Classical
-variable {G : Type*} [Group G] [Fintype G]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-variable {Ω Ξ : Type*} [Fintype Ω] [Fintype Ξ]
-
-noncomputable def sampleOperator (ρ : Representation ℂ G V) (P : Ω → G) : V →L[ℂ] V :=
-  (Fintype.card Ω:ℂ)⁻¹ • ∑ ω,continuousRepresentation ρ (P ω)
-
-end Thorp.UnitaryFinite
-
-namespace Thorp.Specht
-open scoped BigOperators Classical
-
-noncomputable def spaceHilbertEquiv (μ : YoungDiagram) : space μ ≃ₗ[ℂ] hilbertSpace μ :=
-  (hilbertEquiv μ).submoduleMap (space μ)
-
-noncomputable def unitaryRepresentation (μ : YoungDiagram) :
-    Representation ℂ (Equiv.Perm (Cell μ)) (hilbertSpace μ) :=
-  ((spaceHilbertEquiv μ).conjAlgEquiv ℂ).toMonoidHom.comp (representation μ)
-
-variable {α ι : Type*} [Fintype α] [Fintype ι]
-
-noncomputable def relabelledUnitary (μ : YoungDiagram) (e : α ≃ Cell μ) :
-    Representation ℂ (Equiv.Perm α) (hilbertSpace μ) :=
-  (unitaryRepresentation μ).comp e.permCongrHom.toMonoidHom
-
-end Thorp.Specht
-
-namespace Thorp.SparseContact
-open scoped BigOperators
-
-noncomputable def sweepOperator (d : ℕ) (μ : YoungDiagram)
-    (e : Card d ≃ Specht.Cell μ) (reverse : Bool) :
-    Specht.hilbertSpace μ →L[ℂ] Specht.hilbertSpace μ :=
-  UnitaryFinite.sampleOperator (V := Specht.hilbertSpace μ) (Specht.relabelledUnitary μ e)
-    (fun ω : SwitchIndex d → Bool => if reverse then
-      (butterflyPerm d (decodeButterfly d ω))⁻¹ else butterflyPerm d (decodeButterfly d ω))
-
-end Thorp.SparseContact
-
-namespace Thorp.SparseRegime
-
-def MainStatement : Prop :=
-  ∃ (C : ℝ) (d₀ : ℕ), 0 < C ∧ 0 < d₀ ∧
-    ∀ (d : ℕ) (μ : YoungDiagram) (e : Thorp.Card d ≃ Thorp.Specht.Cell μ),
-      let k := μ.card - μ.rowLen 0
-      1 ≤ k → k < 2^d → (d:ℝ)^((3:ℝ)/4) ≤ Real.log ((2:ℝ)^d/k) →
-      ∀ rev : Bool,
-        ‖Thorp.SparseContact.sweepOperator d μ e rev‖^2 ≤
-          Real.exp (C*k)*((k:ℝ)/(2:ℝ)^d)^((k:ℝ)/4) ∧
-        (d₀ ≤ d → ‖Thorp.SparseContact.sweepOperator d μ e rev‖^2 ≤
-          ((k:ℝ)/(2:ℝ)^d)^((k:ℝ)/4))
-
-end Thorp.SparseRegime
-
-end ThorpNine.SparseSaving
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+namespace ThorpNine.SparseSaving
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+
+abbrev Card (d : ℕ) := Fin d → Bool
+
+def switchFun {d : ℕ} (ξ : Card d → Bool) (x : Card (d + 1)) : Card (d + 1) :=
+  Fin.cons (Bool.xor (x 0) (ξ (Fin.tail x))) (Fin.tail x)
+
+theorem switchFun_involutive {d : ℕ} (ξ : Card d → Bool) :
+    Function.Involutive (switchFun ξ) := by
+  intro x
+  funext i
+  refine Fin.cases ?_ (fun j => ?_) i
+  · simp only [switchFun, Fin.cons_zero, Fin.tail_cons]
+    cases x 0 <;> cases ξ (Fin.tail x) <;> rfl
+  · simp [switchFun, Fin.tail]
+
+def pairSwitch {d : ℕ} (ξ : Card d → Bool) : Equiv.Perm (Card (d + 1)) :=
+  { toFun := switchFun ξ
+    invFun := switchFun ξ
+    left_inv := switchFun_involutive ξ
+    right_inv := switchFun_involutive ξ }
+
+def Butterfly : ℕ → Type
+  | 0 => Unit
+  | d + 1 => (Card d → Bool) × (Bool → Butterfly d)
+
+def childLift {d : ℕ} (p : Bool → Equiv.Perm (Card d)) : Equiv.Perm (Card (d + 1)) where
+  toFun x := Fin.cons (x 0) (p (x 0) (Fin.tail x))
+  invFun x := Fin.cons (x 0) ((p (x 0)).symm (Fin.tail x))
+  left_inv x := by simp only [Fin.cons_zero, Fin.tail_cons, Equiv.symm_apply_apply, Fin.cons_self_tail]
+  right_inv x := by simp only [Fin.cons_zero, Fin.tail_cons, Equiv.apply_symm_apply, Fin.cons_self_tail]
+
+def butterflyPerm : (d : ℕ) → Butterfly d → Equiv.Perm (Card d)
+  | 0, _ => 1
+  | d + 1, b => pairSwitch b.1 * childLift (fun ε => butterflyPerm d (b.2 ε))
+
+def SwitchIndex : ℕ → Type
+  | 0 => Empty
+  | d + 1 => Sum (Card d) (Bool × SwitchIndex d)
+
+instance (d : ℕ) : Fintype (SwitchIndex d) := by
+  induction d with
+  | zero => exact inferInstanceAs (Fintype Empty)
+  | succ d ih => exact inferInstanceAs (Fintype (Sum (Card d) (Bool × SwitchIndex d)))
+
+instance (d : ℕ) : DecidableEq (SwitchIndex d) := by
+  induction d with
+  | zero => exact inferInstanceAs (DecidableEq Empty)
+  | succ d ih => exact inferInstanceAs (DecidableEq (Sum (Card d) (Bool × SwitchIndex d)))
+
+def decodeButterfly : (d : ℕ) → (SwitchIndex d → Bool) → Butterfly d
+  | 0, _ => ()
+  | d + 1, ω => (fun y => ω (Sum.inl y), fun ε =>
+      decodeButterfly d (fun i => ω (Sum.inr (ε,i))))
+
+end Thorp
+
+namespace Thorp.Specht
+open scoped BigOperators Classical
+
+abbrev Cell (μ : YoungDiagram) := {x : ℕ × ℕ // x ∈ μ.cells}
+
+def row {μ : YoungDiagram} (x : Cell μ) : ℕ := x.1.1
+
+def col {μ : YoungDiagram} (x : Cell μ) : ℕ := x.1.2
+
+lemma row_lt_colLen {μ : YoungDiagram} (x : Cell μ) : row x < μ.colLen (col x) :=
+  YoungDiagram.mem_iff_lt_colLen.mp x.2
+
+noncomputable def rowIndex {μ : YoungDiagram} (x : Cell μ) : Fin (μ.colLen 0) :=
+  ⟨row x, (row_lt_colLen x).trans_le (μ.colLen_anti 0 (col x) (Nat.zero_le _))⟩
+
+abbrev Tabloid (μ : YoungDiagram) :=
+  {f : Cell μ → Fin (μ.colLen 0) // ∃ p : Equiv.Perm (Cell μ), f = rowIndex ∘ p}
+
+noncomputable instance (μ : YoungDiagram) : Fintype (Tabloid μ) := Fintype.ofFinite _
+
+noncomputable def baseTabloid (μ : YoungDiagram) : Tabloid μ := ⟨rowIndex, 1, rfl⟩
+
+noncomputable def tabloidAct {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) :
+    Equiv.Perm (Tabloid μ) where
+  toFun f := ⟨fun x => f.1 (p⁻¹ x), by
+    obtain ⟨q,hq⟩ := f.2
+    refine ⟨q*p⁻¹, ?_⟩
+    funext x
+    simp only [hq,Function.comp_apply,Equiv.Perm.mul_apply]⟩
+  invFun f := ⟨fun x => f.1 (p x), by
+    obtain ⟨q,hq⟩ := f.2
+    refine ⟨q*p, ?_⟩
+    funext x
+    simp only [hq,Function.comp_apply,Equiv.Perm.mul_apply]⟩
+  left_inv f := by apply Subtype.ext; funext x; simp
+  right_inv f := by apply Subtype.ext; funext x; simp
+
+@[simp] lemma tabloidAct_apply {μ : YoungDiagram} (p : Equiv.Perm (Cell μ))
+    (f : Tabloid μ) (x : Cell μ) : (tabloidAct p f).1 x = f.1 (p⁻¹ x) := rfl
+
+@[simp] lemma tabloidAct_one (μ : YoungDiagram) : tabloidAct (1 : Equiv.Perm (Cell μ)) = 1 := by
+  ext f x
+  rfl
+
+@[simp] lemma tabloidAct_mul {μ : YoungDiagram} (p q : Equiv.Perm (Cell μ)) :
+    tabloidAct (p*q) = tabloidAct p * tabloidAct q := by
+  ext f x
+  simp only [tabloidAct_apply, mul_inv_rev,Equiv.Perm.mul_apply]
+
+noncomputable def tabloidRep (μ : YoungDiagram) :
+    Representation ℂ (Equiv.Perm (Cell μ)) (Tabloid μ → ℂ) where
+  toFun p :=
+    { toFun := fun v f => v (tabloidAct p⁻¹ f)
+      map_add' := by intros; rfl
+      map_smul' := by intros; rfl }
+  map_one' := by ext v f; simp
+  map_mul' p q := by
+    ext v f
+    simp only [mul_inv_rev,tabloidAct_mul,Equiv.Perm.mul_apply]
+    rfl
+
+noncomputable def delta {μ : YoungDiagram} (f : Tabloid μ) : Tabloid μ → ℂ :=
+  fun g => if g=f then 1 else 0
+
+noncomputable def colGroup (μ : YoungDiagram) : Subgroup (Equiv.Perm (Cell μ)) where
+  carrier := {p | ∀ x, col (p x) = col x}
+  one_mem' := by intro x; rfl
+  mul_mem' := by intro p q hp hq x; exact (hp (q x)).trans (hq x)
+  inv_mem' := by intro p hp x; simpa using (hp (p⁻¹ x)).symm
+
+noncomputable def permSign {α : Type*} [Fintype α] [DecidableEq α] (p : Equiv.Perm α) : ℂ :=
+  ((Equiv.Perm.sign p : ℤ) : ℂ)
+
+
+section
+variable {α V : Type*} [Fintype α] [DecidableEq α] [AddCommGroup V] [Module ℂ V]
+
+noncomputable def alternator (C : Subgroup (Equiv.Perm α))
+    (ρ : Representation ℂ (Equiv.Perm α) V) : Module.End ℂ V := by
+  classical
+  exact ∑ c : C, permSign (c : Equiv.Perm α) • ρ c
+
+end
+
+noncomputable def polytabloid (μ : YoungDiagram) : Tabloid μ → ℂ :=
+  alternator (colGroup μ) (tabloidRep μ) (delta (baseTabloid μ))
+
+noncomputable def space (μ : YoungDiagram) : Submodule ℂ (Tabloid μ → ℂ) :=
+  Submodule.span ℂ (Set.range (fun p : Equiv.Perm (Cell μ) => tabloidRep μ p (polytabloid μ)))
+
+lemma orbit_mem_space (μ : YoungDiagram) (p : Equiv.Perm (Cell μ)) :
+    tabloidRep μ p (polytabloid μ) ∈ space μ := Submodule.subset_span ⟨p,rfl⟩
+
+lemma action_mem_space (μ : YoungDiagram) (p : Equiv.Perm (Cell μ))
+    (v : Tabloid μ → ℂ) (hv : v ∈ space μ) : tabloidRep μ p v ∈ space μ := by
+  induction hv using Submodule.span_induction with
+  | mem v hv =>
+    obtain ⟨q,rfl⟩ := hv
+    change ((tabloidRep μ p)*(tabloidRep μ q)) (polytabloid μ) ∈ space μ
+    rw [←map_mul]
+    exact orbit_mem_space μ (p*q)
+  | zero => simp
+  | add v w hv hw ihv ihw => simpa using (space μ).add_mem ihv ihw
+  | smul a v hv ih => simpa using (space μ).smul_mem a ih
+
+noncomputable def subrepresentation (μ : YoungDiagram) : Subrepresentation (tabloidRep μ) :=
+  ⟨space μ, fun p v hv => action_mem_space μ p v hv⟩
+
+noncomputable def representation (μ : YoungDiagram) :
+    Representation ℂ (Equiv.Perm (Cell μ)) (space μ) :=
+  (subrepresentation μ).toRepresentation
+
+end Thorp.Specht
+
+namespace Thorp.PermutationHilbert
+open scoped BigOperators ComplexConjugate Classical
+open Complex
+variable {X : Type*} [Fintype X]
+
+abbrev H (X : Type*) [Fintype X] := EuclideanSpace ℂ X
+
+end Thorp.PermutationHilbert
+
+namespace Thorp.Specht
+open scoped BigOperators Classical
+
+noncomputable def hilbertEquiv (μ : YoungDiagram) :
+    (Tabloid μ → ℂ) ≃ₗ[ℂ] PermutationHilbert.H (Tabloid μ) :=
+  (WithLp.linearEquiv 2 ℂ (Tabloid μ → ℂ)).symm
+
+noncomputable def hilbertSpace (μ : YoungDiagram) :
+    Submodule ℂ (PermutationHilbert.H (Tabloid μ)) := (space μ).map (hilbertEquiv μ).toLinearMap
+
+end Thorp.Specht
+
+namespace Thorp.UnitaryFinite
+
+section
+open scoped BigOperators ComplexConjugate Classical
+variable {G : Type*} [Group G] [Fintype G]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+noncomputable def continuousRepresentation (ρ : Representation ℂ G V) : G →* (V →L[ℂ] V) where
+  toFun g := (ρ g).toContinuousLinearMap
+  map_one' := by ext v; simp
+  map_mul' g h := by ext v; simp
+
+end
+open scoped BigOperators Classical
+variable {G : Type*} [Group G] [Fintype G]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+variable {Ω Ξ : Type*} [Fintype Ω] [Fintype Ξ]
+
+noncomputable def sampleOperator (ρ : Representation ℂ G V) (P : Ω → G) : V →L[ℂ] V :=
+  (Fintype.card Ω:ℂ)⁻¹ • ∑ ω,continuousRepresentation ρ (P ω)
+
+end Thorp.UnitaryFinite
+
+namespace Thorp.Specht
+open scoped BigOperators Classical
+
+noncomputable def spaceHilbertEquiv (μ : YoungDiagram) : space μ ≃ₗ[ℂ] hilbertSpace μ :=
+  (hilbertEquiv μ).submoduleMap (space μ)
+
+noncomputable def unitaryRepresentation (μ : YoungDiagram) :
+    Representation ℂ (Equiv.Perm (Cell μ)) (hilbertSpace μ) :=
+  ((spaceHilbertEquiv μ).conjAlgEquiv ℂ).toMonoidHom.comp (representation μ)
+
+variable {α ι : Type*} [Fintype α] [Fintype ι]
+
+noncomputable def relabelledUnitary (μ : YoungDiagram) (e : α ≃ Cell μ) :
+    Representation ℂ (Equiv.Perm α) (hilbertSpace μ) :=
+  (unitaryRepresentation μ).comp e.permCongrHom.toMonoidHom
+
+end Thorp.Specht
+
+namespace Thorp.SparseContact
+open scoped BigOperators
+
+noncomputable def sweepOperator (d : ℕ) (μ : YoungDiagram)
+    (e : Card d ≃ Specht.Cell μ) (reverse : Bool) :
+    Specht.hilbertSpace μ →L[ℂ] Specht.hilbertSpace μ :=
+  UnitaryFinite.sampleOperator (V := Specht.hilbertSpace μ) (Specht.relabelledUnitary μ e)
+    (fun ω : SwitchIndex d → Bool => if reverse then
+      (butterflyPerm d (decodeButterfly d ω))⁻¹ else butterflyPerm d (decodeButterfly d ω))
+
+end Thorp.SparseContact
+
+namespace Thorp.SparseRegime
+
+def MainStatement : Prop :=
+  ∃ (C : ℝ) (d₀ : ℕ), 0 < C ∧ 0 < d₀ ∧
+    ∀ (d : ℕ) (μ : YoungDiagram) (e : Thorp.Card d ≃ Thorp.Specht.Cell μ),
+      let k := μ.card - μ.rowLen 0
+      1 ≤ k → k < 2^d → (d:ℝ)^((3:ℝ)/4) ≤ Real.log ((2:ℝ)^d/k) →
+      ∀ rev : Bool,
+        ‖Thorp.SparseContact.sweepOperator d μ e rev‖^2 ≤
+          Real.exp (C*k)*((k:ℝ)/(2:ℝ)^d)^((k:ℝ)/4) ∧
+        (d₀ ≤ d → ‖Thorp.SparseContact.sweepOperator d μ e rev‖^2 ≤
+          ((k:ℝ)/(2:ℝ)^d)^((k:ℝ)/4))
+
+end Thorp.SparseRegime
+
+end ThorpNine.SparseSaving
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/SparseSaving/ModifiedReverseEquiv.lean b/lean/OAI/Probability/ThorpRouting/SparseSaving/ModifiedReverseEquiv.lean
index 18c26e5d4266b7575f610ae0a65a8408042ae35a..f46f6845b5d1e0f83b1e9f114ad66a26b3a9227d 100644
--- a/lean/OAI/Probability/ThorpRouting/SparseSaving/ModifiedReverseEquiv.lean
+++ b/lean/OAI/Probability/ThorpRouting/SparseSaving/ModifiedReverseEquiv.lean
@@ -1,759 +1,759 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.SparseSaving.ModifiedRoutingMoment
-
-namespace OAI
-
-namespace ThorpNine.SparseSaving
-
-namespace Thorp
-
-noncomputable def modifiedReverseEquiv (L r : ℕ) : Equiv.Perm (ModifiedCoins L r) where
-  toFun := modifiedReverseCoins L r
-  invFun := modifiedReverseCoins L r
-  left_inv := modifiedReverseCoins_involutive L r
-  right_inv := modifiedReverseCoins_involutive L r
-
-lemma modifiedPerm_reverse (L r : ℕ) (E : Finset (Card r)) (ω : ModifiedCoins L r) :
-    modifiedPerm L r E (modifiedReverseCoins L r ω) = (modifiedPerm L r E ω)⁻¹ := by
-  induction r with
-  | zero =>
-      rw [modifiedPerm_zero,modifiedPerm_zero]
-      split_ifs
-      · simp only [inv_one]
-      · simp only [modifiedReverseCoins,mul_inv_rev,inv_inv]
-        simp
-  | succ r ih =>
-      obtain ⟨z,rfl⟩ := (modifiedStepEquiv L r).symm.surjective ω
-      rcases z with ⟨⟨ω₀,ω₁⟩,⟨ξ,η⟩⟩
-      simp only [modifiedReverseCoins,Equiv.apply_symm_apply,modifiedPerm_step,
-        mul_inv_rev,pairSwitch_inv,childLift_inv]
-      rw [mul_assoc]
-      congr 1
-      congr 1
-      apply congrArg childLift
-      funext b
-      cases b <;> simp only [Bool.false_eq_true,↓reduceIte,ih]
-
-lemma modified_kernel_symmetric (L r : ℕ) (E : Finset (Card r))
-    {ι : Type*} [Fintype ι] (e f : ι ↪ Card (L+r)) :
-    PairRouting.tupleProbability (modifiedPerm L r E) e f =
-      PairRouting.tupleProbability (modifiedPerm L r E) f e :=
-  PairRouting.tupleProbability_symmetric _ (modifiedReverseEquiv L r)
-    (modifiedPerm_reverse L r E) e f
-
-end Thorp
-
-namespace Thorp.Specht
-open scoped BigOperators Classical
-
-@[simp] lemma card_cell (μ : YoungDiagram) : Fintype.card (Cell μ) = μ.card := by
-  simp [Cell, YoungDiagram.card]
-
-noncomputable def columnEquiv (μ : YoungDiagram) (j : ℕ) :
-    {x : Cell μ // col x = j} ≃ Fin (μ.colLen j) where
-  toFun x := ⟨row x.1, by simpa [x.2] using row_lt_colLen x.1⟩
-  invFun i := ⟨⟨(i,j), YoungDiagram.mem_iff_lt_colLen.mpr i.2⟩, rfl⟩
-  left_inv x := by apply Subtype.ext; apply Subtype.ext; exact Prod.ext rfl x.2.symm
-  right_inv i := by rfl
-
-noncomputable def columnValues {μ : YoungDiagram} (f : Cell μ → ℕ) (j : ℕ) : Finset ℕ :=
-  (Finset.univ.filter (fun x => col x = j)).image f
-
-lemma mem_columnValues {μ : YoungDiagram} (f : Cell μ → ℕ) (x : Cell μ) :
-    f x ∈ columnValues f (col x) := by
-  classical
-  exact Finset.mem_image.mpr ⟨x, by simp, rfl⟩
-
-lemma card_columnValues {μ : YoungDiagram} (f : Cell μ → ℕ)
-    (hf : ∀ x y, col x = col y → f x = f y → x = y) (j : ℕ) :
-    (columnValues f j).card = μ.colLen j := by
-  classical
-  rw [columnValues, Finset.card_image_iff.mpr]
-  · rw [← Fintype.card_subtype]
-    exact Fintype.card_congr (columnEquiv μ j) |>.trans (Fintype.card_fin _)
-  intro x hx y hy he
-  exact hf x y ((Finset.mem_filter.mp hx).2.trans (Finset.mem_filter.mp hy).2.symm) he
-
-lemma strictMono_fin_nat_ge {n : ℕ} (f : Fin n → ℕ) (hf : StrictMono f) :
-    ∀ i, i.val ≤ f i := by
-  intro i
-  induction h : i.val using Nat.strong_induction_on generalizing i with
-  | h n ih =>
-    by_cases hn : n = 0
-    · omega
-    · have hn' : n-1 < i.val := by omega
-      let j : Fin _ := ⟨n-1, hn'.trans i.isLt⟩
-      have hj := ih (n-1) (by omega) j rfl
-      have hij := hf (show j < i from hn')
-      dsimp [j] at hj hij
-      omega
-
-noncomputable def rankCell {μ : YoungDiagram} (f : Cell μ → ℕ)
-    (hf : ∀ x y, col x = col y → f x = f y → x = y) (x : Cell μ) : Cell μ :=
-  let k := ((columnValues f (col x)).orderIsoOfFin (card_columnValues f hf (col x))).symm
-    ⟨f x, mem_columnValues f x⟩
-  ⟨(k.val,col x), YoungDiagram.mem_iff_lt_colLen.mpr k.isLt⟩
-
-@[simp] lemma rankCell_col {μ : YoungDiagram} (f : Cell μ → ℕ)
-    (hf : ∀ x y, col x = col y → f x = f y → x = y) (x : Cell μ) :
-    col (rankCell f hf x) = col x := rfl
-
-lemma rankCell_orderEmb {μ : YoungDiagram} (f : Cell μ → ℕ)
-    (hf : ∀ x y, col x = col y → f x = f y → x = y) (x : Cell μ) :
-    (columnValues f (col x)).orderEmbOfFin (card_columnValues f hf (col x))
-      ⟨row (rankCell f hf x), row_lt_colLen (rankCell f hf x)⟩ = f x := by
-  change ((columnValues f (col x)).orderIsoOfFin (card_columnValues f hf (col x))
-    (((columnValues f (col x)).orderIsoOfFin (card_columnValues f hf (col x))).symm
-      ⟨f x, mem_columnValues f x⟩)).val = f x
-  simp
-
-lemma rankCell_injective {μ : YoungDiagram} (f : Cell μ → ℕ)
-    (hf : ∀ x y, col x = col y → f x = f y → x = y) :
-    Function.Injective (rankCell f hf) := by
-  intro x y h
-  have hc : col x = col y := by simpa only [rankCell_col] using congrArg (fun z : Cell μ => col z) h
-  apply hf x y hc
-  have hx := rankCell_orderEmb f hf x
-  have hy := rankCell_orderEmb f hf y
-  change (columnValues f (col (rankCell f hf x))).orderEmbOfFin
-    (card_columnValues f hf (col (rankCell f hf x)))
-      ⟨row (rankCell f hf x), row_lt_colLen (rankCell f hf x)⟩ = f x at hx
-  rw [h] at hx
-  exact hx.symm.trans hy
-
-noncomputable def rankPerm {μ : YoungDiagram} (f : Cell μ → ℕ)
-    (hf : ∀ x y, col x = col y → f x = f y → x = y) : Equiv.Perm (Cell μ) :=
-  Equiv.ofBijective (rankCell f hf) ((Finite.injective_iff_bijective).mp (rankCell_injective f hf))
-
-lemma rankCell_row_le {μ : YoungDiagram} (f : Cell μ → ℕ)
-    (hf : ∀ x y, col x = col y → f x = f y → x = y) (x : Cell μ) :
-    row (rankCell f hf x) ≤ f x := by
-  rw [←rankCell_orderEmb f hf x]
-  exact strictMono_fin_nat_ge _ (OrderEmbedding.strictMono _) _
-
-theorem column_transporter (μ : YoungDiagram) (p : Equiv.Perm (Cell μ))
-    (hp : ∀ x y, col x = col y → row (p x) = row (p y) → x = y) :
-    ∃ c : Equiv.Perm (Cell μ), (∀ x, col (c x) = col x) ∧
-      ∀ x, row (c x) = row (p x) := by
-  classical
-  let c := rankPerm (fun x => row (p x)) hp
-  refine ⟨c, fun x => rfl, ?_⟩
-  have hle (x : Cell μ) : row (c x) ≤ row (p x) := rankCell_row_le _ hp x
-  have hsum : ∑ x, row (c x) = ∑ x, row (p x) := by
-    rw [Equiv.sum_comp, Equiv.sum_comp]
-  exact fun x => (Finset.sum_eq_sum_iff_of_le (fun x _ => hle x)).mp hsum x (Finset.mem_univ _)
-
-
-@[simp] lemma tabloidAct_inv {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) :
-    tabloidAct p⁻¹ = (tabloidAct p)⁻¹ := by
-  ext f x
-  simp [tabloidAct,Equiv.Perm.inv_def]
-
-@[simp] lemma tabloidRep_apply {μ : YoungDiagram} (p : Equiv.Perm (Cell μ))
-    (v : Tabloid μ → ℂ) (f : Tabloid μ) : tabloidRep μ p v f = v (tabloidAct p⁻¹ f) := rfl
-
-lemma tabloidRep_delta {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) (f : Tabloid μ) :
-    tabloidRep μ p (delta f) = delta (tabloidAct p f) := by
-  classical
-  ext g
-  simp only [tabloidRep_apply,delta,Equiv.Perm.inv_def]
-  congr 1
-  exact propext (Equiv.symm_apply_eq (tabloidAct p))
-
-@[simp] lemma mem_colGroup {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) :
-    p ∈ colGroup μ ↔ ∀ x, col (p x) = col x := Iff.rfl
-
-lemma colGroup_row_stabilizer {μ : YoungDiagram} (c : colGroup μ)
-    (hc : tabloidAct c (baseTabloid μ) = baseTabloid μ) : (c : Equiv.Perm (Cell μ)) = 1 := by
-  apply Equiv.ext
-  intro x
-  have hr := congrArg (fun f : Tabloid μ => ((f.1 ((c : Equiv.Perm _) x)) : ℕ)) hc
-  simp only [tabloidAct_apply,Equiv.Perm.inv_def,Equiv.symm_apply_apply,baseTabloid,rowIndex] at hr
-  apply Subtype.ext
-  exact Prod.ext hr.symm (c.2 x)
-
-lemma tabloid_column_transporter {μ : YoungDiagram} (f : Tabloid μ)
-    (hf : ∀ x y, col x = col y → f.1 x = f.1 y → x = y) :
-    ∃ c : colGroup μ, f = tabloidAct c (baseTabloid μ) := by
-  obtain ⟨p,hp⟩ := f.2
-  have hp' : ∀ x y, col x = col y → row (p x) = row (p y) → x = y := by
-    intro x y hcol hrow
-    apply hf x y hcol
-    rw [hp]
-    exact Fin.ext hrow
-  obtain ⟨c,hcol,hrow⟩ := column_transporter μ p hp'
-  refine ⟨⟨c⁻¹, (colGroup μ).inv_mem hcol⟩, ?_⟩
-  apply Subtype.ext
-  funext x
-  apply Fin.ext
-  simp only [tabloidAct_apply,baseTabloid,rowIndex,inv_inv,hp,Function.comp_apply]
-  exact (hrow x).symm
-
-
-@[simp] lemma permSign_one {α : Type*} [Fintype α] [DecidableEq α] :
-    permSign (1 : Equiv.Perm α) = 1 := by simp [permSign]
-@[simp] lemma permSign_mul {α : Type*} [Fintype α] [DecidableEq α] (p q : Equiv.Perm α) :
-    permSign (p*q) = permSign p * permSign q := by simp [permSign]
-@[simp] lemma permSign_inv {α : Type*} [Fintype α] [DecidableEq α] (p : Equiv.Perm α) :
-    permSign p⁻¹ = permSign p := by simp [permSign,Equiv.Perm.inv_def]
-lemma permSign_sq {α : Type*} [Fintype α] [DecidableEq α] (p : Equiv.Perm α) :
-    permSign p * permSign p = 1 := by
-  calc
-    permSign p * permSign p = permSign (p*p⁻¹) := by rw [permSign_mul,permSign_inv]
-    _ = 1 := by simp
-@[simp] lemma permSign_swap {α : Type*} [Fintype α] [DecidableEq α] {x y : α} (h : x ≠ y) :
-    permSign (Equiv.swap x y) = -1 := by simp [permSign, h]
-@[simp] lemma conj_permSign {α : Type*} [Fintype α] [DecidableEq α] (p : Equiv.Perm α) :
-    starRingEnd ℂ (permSign p) = permSign p := by simp [permSign]
-
-variable {α V : Type*} [Fintype α] [DecidableEq α] [AddCommGroup V] [Module ℂ V]
-
-lemma alternator_mul (C : Subgroup (Equiv.Perm α))
-    (ρ : Representation ℂ (Equiv.Perm α) V) (c : C) :
-    alternator C ρ * ρ c = permSign (c : Equiv.Perm α) • alternator C ρ := by
-  classical
-  rw [alternator,Finset.sum_mul,Finset.smul_sum]
-  apply Fintype.sum_equiv (Equiv.mulRight c)
-  intro a
-  simp only [smul_mul_assoc,←map_mul,Equiv.coe_mulRight,Subgroup.coe_mul,permSign_mul,smul_smul]
-  congr 1
-  have h := permSign_sq (c : Equiv.Perm α)
-  calc
-    permSign (a : Equiv.Perm α) = permSign (a : Equiv.Perm α)*1 := by ring
-    _ = permSign (a : Equiv.Perm α)*(permSign (c : Equiv.Perm α)*permSign (c : Equiv.Perm α)) := by rw [h]
-    _ = _ := by ring
-
-lemma mul_alternator (C : Subgroup (Equiv.Perm α))
-    (ρ : Representation ℂ (Equiv.Perm α) V) (c : C) :
-    ρ c * alternator C ρ = permSign (c : Equiv.Perm α) • alternator C ρ := by
-  classical
-  rw [alternator,Finset.mul_sum,Finset.smul_sum]
-  apply Fintype.sum_equiv (Equiv.mulLeft c)
-  intro a
-  simp only [mul_smul_comm,←map_mul,Equiv.coe_mulLeft,Subgroup.coe_mul,permSign_mul,smul_smul]
-  congr 1
-  have h := permSign_sq (c : Equiv.Perm α)
-  calc
-    permSign (a : Equiv.Perm α) = 1*permSign (a : Equiv.Perm α) := by ring
-    _ = (permSign (c : Equiv.Perm α)*permSign (c : Equiv.Perm α))*permSign (a : Equiv.Perm α) := by rw [h]
-    _ = _ := by ring
-
-lemma alternator_apply_action (C : Subgroup (Equiv.Perm α))
-    (ρ : Representation ℂ (Equiv.Perm α) V) (c : C) (v : V) :
-    alternator C ρ (ρ c v) = permSign (c : Equiv.Perm α) • alternator C ρ v := by
-  exact congrArg (fun T : Module.End ℂ V => T v) (alternator_mul C ρ c)
-
-lemma action_alternator_apply (C : Subgroup (Equiv.Perm α))
-    (ρ : Representation ℂ (Equiv.Perm α) V) (c : C) (v : V) :
-    ρ c (alternator C ρ v) = permSign (c : Equiv.Perm α) • alternator C ρ v := by
-  exact congrArg (fun T : Module.End ℂ V => T v) (mul_alternator C ρ c)
-
-lemma alternator_eq_zero_of_odd_stabilizer (C : Subgroup (Equiv.Perm α))
-    (ρ : Representation ℂ (Equiv.Perm α) V) (c : C) (v : V)
-    (hc : permSign (c : Equiv.Perm α) = -1) (hv : ρ c v = v) :
-    alternator C ρ v = 0 := by
-  have h := alternator_apply_action C ρ c v
-  rw [hv,hc,neg_one_smul] at h
-  have h2 : (2:ℂ) • alternator C ρ v = 0 := by
-    rw [two_smul]
-    nth_rw 1 [h]
-    exact neg_add_cancel _
-  exact (smul_eq_zero.mp h2).resolve_left (by norm_num)
-
-end Thorp.Specht
-namespace Thorp.Specht
-open scoped BigOperators Classical
-
-lemma swap_mem_colGroup {μ : YoungDiagram} (x y : Cell μ) (hc : col x = col y) :
-    Equiv.swap x y ∈ colGroup μ := by
-  classical
-  intro z
-  by_cases hx : z=x
-  · subst z; simpa using hc.symm
-  by_cases hy : z=y
-  · subst z; simpa using hc
-  simp [Equiv.swap_apply_of_ne_of_ne hx hy]
-
-lemma swap_fixes_tabloid {μ : YoungDiagram} (f : Tabloid μ) (x y : Cell μ)
-    (hxy : f.1 x = f.1 y) : tabloidAct (Equiv.swap x y) f = f := by
-  classical
-  apply Subtype.ext
-  funext z
-  simp only [tabloidAct_apply, Equiv.swap_inv]
-  by_cases hx : z=x
-  · subst z; simpa using hxy.symm
-  by_cases hy : z=y
-  · subst z; simpa using hxy
-  simp [Equiv.swap_apply_of_ne_of_ne hx hy]
-
-lemma alternator_delta_rank_one {μ : YoungDiagram} (f : Tabloid μ) :
-    ∃ z : ℂ, alternator (colGroup μ) (tabloidRep μ) (delta f) = z • polytabloid μ := by
-  classical
-  by_cases hf : ∀ x y, col x = col y → f.1 x = f.1 y → x = y
-  · obtain ⟨c,rfl⟩ := tabloid_column_transporter f hf
-    refine ⟨permSign (c:Equiv.Perm (Cell μ)), ?_⟩
-    rw [←tabloidRep_delta]
-    exact alternator_apply_action _ _ c _
-  · push Not at hf
-    obtain ⟨x,y,hcol,hrow,hne⟩ := hf
-    refine ⟨0, ?_⟩
-    rw [zero_smul]
-    apply alternator_eq_zero_of_odd_stabilizer _ _ ⟨Equiv.swap x y,swap_mem_colGroup x y hcol⟩
-    · exact permSign_swap hne
-    · rw [tabloidRep_delta,swap_fixes_tabloid f x y hrow]
-
-lemma polytabloid_formula (μ : YoungDiagram) :
-    polytabloid μ = ∑ c : colGroup μ, permSign (c:Equiv.Perm (Cell μ)) •
-      delta (tabloidAct c (baseTabloid μ)) := by
-  classical
-  simp only [polytabloid,alternator,LinearMap.sum_apply,LinearMap.smul_apply,tabloidRep_delta]
-
-lemma polytabloid_base (μ : YoungDiagram) : polytabloid μ (baseTabloid μ) = 1 := by
-  classical
-  rw [polytabloid_formula,Finset.sum_apply]
-  rw [Finset.sum_eq_single (1 : colGroup μ)]
-  · simp [delta]
-  · intro c hc hne
-    have hn : baseTabloid μ ≠ tabloidAct c (baseTabloid μ) := by
-      intro h
-      apply hne
-      exact Subtype.ext (colGroup_row_stabilizer c h.symm)
-    simp [Pi.smul_apply,delta,hn]
-  · simp
-
-lemma polytabloid_ne_zero (μ : YoungDiagram) : polytabloid μ ≠ 0 := by
-  intro h
-  have hv := congrArg (fun v : Tabloid μ → ℂ => v (baseTabloid μ)) h
-  simp [polytabloid_base] at hv
-
-lemma delta_decomposition {μ : YoungDiagram} (v : Tabloid μ → ℂ) :
-    v = ∑ f : Tabloid μ, v f • delta f := by
-  classical
-  funext g
-  simp only [Finset.sum_apply,Pi.smul_apply,delta,smul_eq_mul,mul_ite,mul_one,mul_zero]
-  simp
-
-theorem alternator_rank_one (μ : YoungDiagram) (v : Tabloid μ → ℂ) :
-    ∃ z : ℂ, alternator (colGroup μ) (tabloidRep μ) v = z • polytabloid μ := by
-  classical
-  choose z hz using fun f : Tabloid μ => alternator_delta_rank_one f
-  refine ⟨∑ f, v f * z f, ?_⟩
-  conv_lhs => rw [delta_decomposition v]
-  simp only [map_sum,map_smul,hz,smul_smul,Finset.sum_smul]
-
-lemma alternator_rank_one_exact (μ : YoungDiagram) (v : Tabloid μ → ℂ) :
-    alternator (colGroup μ) (tabloidRep μ) v =
-      (alternator (colGroup μ) (tabloidRep μ) v (baseTabloid μ)) • polytabloid μ := by
-  obtain ⟨z,hz⟩ := alternator_rank_one μ v
-  rw [hz]
-  simp only [Pi.smul_apply,polytabloid_base,smul_eq_mul,mul_one]
-
-
-noncomputable def pairing {μ : YoungDiagram} (v w : Tabloid μ → ℂ) : ℂ :=
-  ∑ f, w f * starRingEnd ℂ (v f)
-
-lemma pairing_delta {μ : YoungDiagram} (f : Tabloid μ) (v : Tabloid μ → ℂ) :
-    pairing (delta f) v = v f := by
-  classical
-  simp [pairing,delta]
-
-lemma pairing_sum_left {μ : YoungDiagram} {ι : Type*} [Fintype ι]
-    (v : ι → Tabloid μ → ℂ) (w : Tabloid μ → ℂ) :
-    pairing (∑ i, v i) w = ∑ i, pairing (v i) w := by
-  simp only [pairing,Finset.sum_apply,map_sum,Finset.mul_sum]
-  exact Finset.sum_comm
-
-lemma pairing_smul_left {μ : YoungDiagram} (a : ℂ) (v w : Tabloid μ → ℂ) :
-    pairing (a • v) w = starRingEnd ℂ a * pairing v w := by
-  simp only [pairing,Pi.smul_apply,smul_eq_mul,map_mul,Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro f hf
-  ring
-
-lemma pairing_zero_left {μ : YoungDiagram} (v : Tabloid μ → ℂ) : pairing 0 v = 0 := by
-  simp [pairing]
-
-lemma pairing_add_left {μ : YoungDiagram} (u v w : Tabloid μ → ℂ) :
-    pairing (u+v) w = pairing u w + pairing v w := by
-  simp [pairing,mul_add,Finset.sum_add_distrib]
-
-lemma pairing_self_eq_zero {μ : YoungDiagram} (v : Tabloid μ → ℂ) :
-    pairing v v = 0 ↔ v = 0 := by
-  change inner ℂ (WithLp.toLp 2 v : EuclideanSpace ℂ (Tabloid μ)) (WithLp.toLp 2 v) = 0 ↔ v = 0
-  rw [inner_self_eq_zero]
-  constructor
-  · intro h; exact congrArg WithLp.ofLp h
-  · intro h; subst v; rfl
-
-lemma pairing_action {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) (v w : Tabloid μ → ℂ) :
-    pairing (tabloidRep μ p v) w = pairing v (tabloidRep μ p⁻¹ w) := by
-  classical
-  simp only [pairing,tabloidRep_apply,inv_inv]
-  apply Fintype.sum_equiv (tabloidAct p⁻¹)
-  intro f
-  have hcancel : tabloidAct p (tabloidAct p⁻¹ f) = f := by
-    rw [tabloidAct_inv]
-    exact (tabloidAct p).apply_symm_apply f
-  rw [hcancel]
-
-lemma alternator_base_pairing (μ : YoungDiagram) (v : Tabloid μ → ℂ) :
-    alternator (colGroup μ) (tabloidRep μ) v (baseTabloid μ) = pairing (polytabloid μ) v := by
-  classical
-  simp only [polytabloid_formula,pairing_sum_left,pairing_smul_left,conj_permSign,pairing_delta]
-  simp only [alternator,LinearMap.sum_apply,LinearMap.smul_apply,Finset.sum_apply,Pi.smul_apply,
-    smul_eq_mul,tabloidRep_apply]
-  apply Fintype.sum_equiv (Equiv.inv (colGroup μ))
-  intro c
-  simp only [Equiv.inv_apply,Subgroup.coe_inv,permSign_inv]
-
-theorem alternator_pairing (μ : YoungDiagram) (v : Tabloid μ → ℂ) :
-    alternator (colGroup μ) (tabloidRep μ) v = pairing (polytabloid μ) v • polytabloid μ := by
-  rw [alternator_rank_one_exact,alternator_base_pairing]
-
-
-lemma polytabloid_mem_space (μ : YoungDiagram) : polytabloid μ ∈ space μ := by
-  simpa using orbit_mem_space μ 1
-
-lemma alternator_mem_invariant (μ : YoungDiagram) (W : Submodule ℂ (Tabloid μ → ℂ))
-    (hW : ∀ p v, v ∈ W → tabloidRep μ p v ∈ W) (v : Tabloid μ → ℂ) (hv : v ∈ W) :
-    alternator (colGroup μ) (tabloidRep μ) v ∈ W := by
-  classical
-  simp only [alternator,LinearMap.sum_apply,LinearMap.smul_apply]
-  exact W.sum_mem (fun c _ => W.smul_mem _ (hW c v hv))
-
-theorem submodule_theorem (μ : YoungDiagram) (W : Submodule ℂ (Tabloid μ → ℂ))
-    (hW : ∀ p v, v ∈ W → tabloidRep μ p v ∈ W) :
-    space μ ≤ W ∨ ∀ v ∈ W, ∀ w ∈ space μ, pairing w v = 0 := by
-  classical
-  by_cases hSW : space μ ≤ W
-  · exact Or.inl hSW
-  right
-  have he : polytabloid μ ∉ W := by
-    intro he
-    apply hSW
-    apply Submodule.span_le.mpr
-    rintro _ ⟨p,rfl⟩
-    exact hW p _ he
-  have hpair : ∀ v ∈ W, pairing (polytabloid μ) v = 0 := by
-    intro v hv
-    by_contra hne
-    apply he
-    have hal := alternator_mem_invariant μ W hW v hv
-    rw [alternator_pairing] at hal
-    have hh := W.smul_mem (pairing (polytabloid μ) v)⁻¹ hal
-    simpa only [smul_smul,inv_mul_cancel₀ hne,one_smul] using hh
-  intro v hv w hw
-  induction hw using Submodule.span_induction with
-  | mem w hw =>
-    obtain ⟨p,rfl⟩ := hw
-    rw [pairing_action]
-    exact hpair _ (hW p⁻¹ v hv)
-  | zero => exact pairing_zero_left v
-  | add u w hu hw ihu ihw => rw [pairing_add_left,ihu,ihw,add_zero]
-  | smul a w hw ih => rw [pairing_smul_left,ih,mul_zero]
-
-theorem space_simple (μ : YoungDiagram) (W : Submodule ℂ (Tabloid μ → ℂ))
-    (hW : ∀ p v, v ∈ W → tabloidRep μ p v ∈ W) (hWS : W ≤ space μ) :
-    W = ⊥ ∨ W = space μ := by
-  obtain hSW | horth := submodule_theorem μ W hW
-  · exact Or.inr (le_antisymm hWS hSW)
-  · left
-    apply le_antisymm _ bot_le
-    intro v hv
-    exact (pairing_self_eq_zero v).mp (horth v hv v (hWS hv))
-
-
-lemma space_ne_bot (μ : YoungDiagram) : space μ ≠ ⊥ := by
-  intro h
-  have he := polytabloid_mem_space μ
-  rw [h,Submodule.mem_bot] at he
-  exact polytabloid_ne_zero μ he
-
-theorem representation_irreducible (μ : YoungDiagram) :
-    Representation.IsIrreducible (representation μ) := by
-  have hnt : (⊥ : Subrepresentation (representation μ)) ≠ ⊤ := by
-    intro h
-    have he : (⟨polytabloid μ,polytabloid_mem_space μ⟩ : space μ) ∈
-        (⊥ : Subrepresentation (representation μ)) := by rw [h]; trivial
-    have he' : (⟨polytabloid μ,polytabloid_mem_space μ⟩ : space μ) = 0 := he
-    exact polytabloid_ne_zero μ (congrArg Subtype.val he')
-  refine { exists_pair_ne := ⟨⊥,⊤,hnt⟩, eq_bot_or_eq_top := ?_ }
-  intro U
-  let W : Submodule ℂ (Tabloid μ → ℂ) := U.toSubmodule.map (space μ).subtype
-  have hW : ∀ p v, v ∈ W → tabloidRep μ p v ∈ W := by
-    intro p v hv
-    obtain ⟨u,hu,rfl⟩ := hv
-    exact ⟨representation μ p u,U.apply_mem_toSubmodule p hu,rfl⟩
-  have hWS : W ≤ space μ := by
-    intro v hv
-    obtain ⟨u,hu,rfl⟩ := hv
-    exact u.2
-  obtain hbot | htop := space_simple μ W hW hWS
-  · left
-    apply le_antisymm _ bot_le
-    intro u hu
-    have hmem : (u : Tabloid μ → ℂ) ∈ W := ⟨u,hu,rfl⟩
-    rw [hbot,Submodule.mem_bot] at hmem
-    change u = 0
-    exact Subtype.ext hmem
-  · right
-    apply le_antisymm le_top
-    intro u hu
-    have hmem : (u : Tabloid μ → ℂ) ∈ W := by rw [htop]; exact u.2
-    obtain ⟨v,hv,hvu⟩ := hmem
-    have heq : v = u := Subtype.ext hvu
-    exact heq ▸ hv
-
-
-section Symmetrizer
-variable {α V : Type*} [Fintype α] [DecidableEq α] [AddCommGroup V] [Module ℂ V]
-
-noncomputable def symmetrizer (H : Subgroup (Equiv.Perm α))
-    (ρ : Representation ℂ (Equiv.Perm α) V) : Module.End ℂ V :=
-  ∑ h : H, ρ h
-
-lemma symmetrizer_mul (H : Subgroup (Equiv.Perm α))
-    (ρ : Representation ℂ (Equiv.Perm α) V) (h : H) :
-    symmetrizer H ρ * ρ h = symmetrizer H ρ := by
-  rw [symmetrizer,Finset.sum_mul]
-  exact Fintype.sum_equiv (Equiv.mulRight h) _ _ (fun g => by simp [←map_mul])
-
-lemma symmetrizer_eq_zero_of_neg (H : Subgroup (Equiv.Perm α))
-    (ρ : Representation ℂ (Equiv.Perm α) V) (h : H) (v : V) (hv : ρ h v = -v) :
-    symmetrizer H ρ v = 0 := by
-  have hh := congrArg (fun T : Module.End ℂ V => T v) (symmetrizer_mul H ρ h)
-  change symmetrizer H ρ (ρ h v) = symmetrizer H ρ v at hh
-  rw [hv,map_neg] at hh
-  have h2 : (2:ℂ) • symmetrizer H ρ v = 0 := by
-    rw [two_smul]
-    nth_rw 1 [←hh]
-    exact neg_add_cancel _
-  exact (smul_eq_zero.mp h2).resolve_left (by norm_num)
-
-noncomputable def fiberGroup {β : Type*} (f : α → β) : Subgroup (Equiv.Perm α) where
-  carrier := {p | ∀ x, f (p x) = f x}
-  one_mem' := by intro x; rfl
-  mul_mem' := by intro p q hp hq x; exact (hp (q x)).trans (hq x)
-  inv_mem' := by intro p hp x; simpa using (hp (p⁻¹ x)).symm
-
-omit [Fintype α] in
-lemma swap_mem_fiberGroup {β : Type*} (f : α → β) (x y : α) (hf : f x = f y) :
-    Equiv.swap x y ∈ fiberGroup f := by
-  intro z
-  by_cases hx : z=x
-  · subst z; simpa using hf.symm
-  by_cases hy : z=y
-  · subst z; simpa using hf
-  simp [Equiv.swap_apply_of_ne_of_ne hx hy]
-
-end Symmetrizer
-
-lemma swap_polytabloid {μ : YoungDiagram} (x y : Cell μ) (hne : x ≠ y)
-    (hcol : col x = col y) :
-    tabloidRep μ (Equiv.swap x y) (polytabloid μ) = -polytabloid μ := by
-  have h := action_alternator_apply (colGroup μ) (tabloidRep μ)
-    ⟨Equiv.swap x y,swap_mem_colGroup x y hcol⟩ (delta (baseTabloid μ))
-  simpa only [polytabloid,permSign_swap hne,neg_one_smul] using h
-
-theorem tall_column_average_zero (μ : YoungDiagram) {β : Type*} [Fintype β]
-    (f : Cell μ → β) (j : ℕ) (hj : Fintype.card β < μ.colLen j)
-    (v : Tabloid μ → ℂ) (hv : v ∈ space μ) :
-    symmetrizer (fiberGroup f) (tabloidRep μ) v = 0 := by
-  induction hv using Submodule.span_induction with
-  | mem v hv =>
-    obtain ⟨p,rfl⟩ := hv
-    have hcard : Fintype.card β < Fintype.card {x : Cell μ // col x = j} := by
-      rw [Fintype.card_congr (columnEquiv μ j),Fintype.card_fin]
-      exact hj
-    obtain ⟨x,y,hne,hf⟩ := Fintype.exists_ne_map_eq_of_card_lt
-      (fun x : {x : Cell μ // col x = j} => f (p x.1)) hcard
-    have hxy : x.1 ≠ y.1 := fun h => hne (Subtype.ext h)
-    let h : fiberGroup f := ⟨Equiv.swap (p x.1) (p y.1),swap_mem_fiberGroup f _ _ hf⟩
-    apply symmetrizer_eq_zero_of_neg _ _ h
-    change tabloidRep μ (Equiv.swap (p x.1) (p y.1))
-      (tabloidRep μ p (polytabloid μ)) = -tabloidRep μ p (polytabloid μ)
-    rw [←Module.End.mul_apply,←map_mul,←Equiv.mul_swap_eq_swap_mul]
-    rw [map_mul,Module.End.mul_apply,swap_polytabloid x.1 y.1 hxy (x.2.trans y.2.symm),map_neg]
-  | zero => exact map_zero _
-  | add v w hv hw ihv ihw => simp only [map_add,ihv,ihw,add_zero]
-  | smul a v hv ih => simp only [map_smul,ih,smul_zero]
-
-
-noncomputable def rowEquiv (μ : YoungDiagram) (i : ℕ) :
-    {x : Cell μ // row x = i} ≃ Fin (μ.rowLen i) where
-  toFun x := ⟨col x.1, by
-    have h := YoungDiagram.mem_iff_lt_rowLen.mp x.1.2
-    change col x.1 < μ.rowLen (row x.1) at h
-    simpa only [x.2] using h⟩
-  invFun j := ⟨⟨(i,j),YoungDiagram.mem_iff_lt_rowLen.mpr j.2⟩,rfl⟩
-  left_inv x := by apply Subtype.ext; apply Subtype.ext; exact Prod.ext x.2.symm rfl
-  right_inv j := rfl
-
-def rowCost (μ : YoungDiagram) : ℕ := ∑ x : Cell μ, row x
-
-lemma eq_of_row_preserving (μ ν : YoungDiagram) (e : Cell μ ≃ Cell ν)
-    (he : ∀ x, row (e x) = row x) : μ = ν := by
-  have hlen (i : ℕ) : μ.rowLen i = ν.rowLen i := by
-    let ei : {x : Cell μ // row x = i} ≃ {x : Cell ν // row x = i} :=
-      { toFun := fun x => ⟨e x.1,(he x.1).trans x.2⟩
-        invFun := fun x => ⟨e.symm x.1,by
-          have h := he (e.symm x.1)
-          simp only [Equiv.apply_symm_apply] at h
-          exact h.symm.trans x.2⟩
-        left_inv := fun x => by apply Subtype.ext; exact e.symm_apply_apply _
-        right_inv := fun x => by apply Subtype.ext; exact e.apply_symm_apply _ }
-    simpa only [Fintype.card_fin] using Fintype.card_congr
-      ((rowEquiv μ i).symm.trans (ei.trans (rowEquiv ν i)))
-  apply YoungDiagram.ext
-  ext ⟨i,j⟩
-  simp only [YoungDiagram.mem_cells,YoungDiagram.mem_iff_lt_rowLen,hlen]
-
-lemma column_injective_rowCost (μ ν : YoungDiagram) (e : Cell μ ≃ Cell ν)
-    (h : ∀ x y : Cell μ, col x = col y → row (e x) = row (e y) → x = y) :
-    rowCost μ ≤ rowCost ν ∧ (rowCost μ = rowCost ν → μ = ν) := by
-  let c := rankPerm (fun x => row (e x)) h
-  have hle (x : Cell μ) : row (c x) ≤ row (e x) := rankCell_row_le _ h x
-  have hsumc : ∑ x, row (c x) = rowCost μ := Equiv.sum_comp c _
-  have hsume : ∑ x, row (e x) = rowCost ν := Equiv.sum_comp e _
-  constructor
-  · rw [←hsumc,←hsume]
-    exact Finset.sum_le_sum (fun x _ => hle x)
-  · intro heq
-    have he : ∀ x, row (c x) = row (e x) := by
-      apply fun x => (Finset.sum_eq_sum_iff_of_le (fun x _ => hle x)).mp
-        (hsumc.trans (heq.trans hsume.symm)) x (Finset.mem_univ _)
-    apply eq_of_row_preserving μ ν (c.symm.trans e)
-    intro x
-    simpa using (he (c.symm x)).symm
-
-lemma swap_relabel {α β : Type*} [DecidableEq α] [DecidableEq β]
-    (e : α ≃ β) (x y : α) : e.permCongr (Equiv.swap x y) = Equiv.swap (e x) (e y) := by
-  ext z
-  simp only [Equiv.permCongr_apply]
-  exact e.injective.map_swap x y (e.symm z) |>.trans (by simp)
-
-lemma sign_vector_rowCost (μ ν : YoungDiagram) (e : Cell μ ≃ Cell ν)
-    (v : Tabloid ν → ℂ) (hv : v ≠ 0)
-    (hs : ∀ c : colGroup μ,
-      tabloidRep ν (e.permCongr c) v = permSign (c : Equiv.Perm (Cell μ)) • v) :
-    rowCost μ ≤ rowCost ν ∧ (rowCost μ = rowCost ν → μ = ν) := by
-  obtain ⟨f,hf⟩ : ∃ f, v f ≠ 0 := by
-    by_contra h
-    push Not at h
-    exact hv (funext h)
-  obtain ⟨p,hp⟩ := f.2
-  apply column_injective_rowCost μ ν (e.trans p)
-  intro x y hcol hrow
-  by_contra hne
-  have hfix : tabloidAct (Equiv.swap (e x) (e y)) f = f := by
-    apply swap_fixes_tabloid
-    rw [hp]
-    exact Fin.ext hrow
-  have heq := congrArg (fun w : Tabloid ν → ℂ => w f)
-    (hs ⟨Equiv.swap x y,swap_mem_colGroup x y hcol⟩)
-  rw [swap_relabel,permSign_swap hne] at heq
-  simp only [tabloidRep_apply,Equiv.swap_inv,hfix,Pi.smul_apply,neg_one_smul] at heq
-  apply hf
-  have h2 : (2:ℂ)*(v f) = 0 := by linear_combination heq
-  exact (mul_eq_zero.mp h2).resolve_left (by norm_num)
-
-lemma polytabloid_col_action (μ : YoungDiagram) (c : colGroup μ) :
-    tabloidRep μ c (polytabloid μ) = permSign (c:Equiv.Perm (Cell μ)) • polytabloid μ :=
-  action_alternator_apply _ _ c _
-
-theorem shapes_eq_of_equivariant_equiv (μ ν : YoungDiagram) (e : Cell μ ≃ Cell ν)
-    (T : space μ ≃ₗ[ℂ] space ν)
-    (hT : ∀ p v, T (representation μ p v) = representation ν (e.permCongr p) (T v)) :
-    μ = ν := by
-  let u : space μ := ⟨polytabloid μ,polytabloid_mem_space μ⟩
-  have hu : u ≠ 0 := fun h => polytabloid_ne_zero μ (congrArg Subtype.val h)
-  have hTcol (c : colGroup μ) :
-      tabloidRep ν (e.permCongr c) (T u) = permSign (c:Equiv.Perm (Cell μ)) • (T u : Tabloid ν → ℂ) := by
-    change (representation ν (e.permCongr c) (T u) : Tabloid ν → ℂ) = _
-    rw [←hT]
-    have hh : representation μ c u = permSign (c:Equiv.Perm (Cell μ)) • u :=
-      Subtype.ext (polytabloid_col_action μ c)
-    rw [hh,map_smul]
-    rfl
-  have hfwd := sign_vector_rowCost μ ν e (T u)
-    (fun h => hu (T.injective (by apply Subtype.ext; simpa using h))) hTcol
-  let w : space ν := ⟨polytabloid ν,polytabloid_mem_space ν⟩
-  have hw : w ≠ 0 := fun h => polytabloid_ne_zero ν (congrArg Subtype.val h)
-  have hTi (p : Equiv.Perm (Cell ν)) (v : space ν) :
-      T.symm (representation ν p v) = representation μ (e.symm.permCongr p) (T.symm v) := by
-    apply T.injective
-    rw [T.apply_symm_apply,hT,T.apply_symm_apply]
-    have hp : e.permCongr (e.symm.permCongr p) = p := by
-      apply Equiv.ext
-      intro x
-      simp [Equiv.permCongr_apply]
-    rw [hp]
-  have hTicol (c : colGroup ν) :
-      tabloidRep μ (e.symm.permCongr c) (T.symm w) =
-        permSign (c:Equiv.Perm (Cell ν)) • (T.symm w : Tabloid μ → ℂ) := by
-    change (representation μ (e.symm.permCongr c) (T.symm w) : Tabloid μ → ℂ) = _
-    rw [←hTi]
-    have hh : representation ν c w = permSign (c:Equiv.Perm (Cell ν)) • w :=
-      Subtype.ext (polytabloid_col_action ν c)
-    rw [hh,map_smul]
-    rfl
-  have hbwd := sign_vector_rowCost ν μ e.symm (T.symm w)
-    (fun h => hw (T.symm.injective (by apply Subtype.ext; simpa using h))) hTicol
-  exact hfwd.2 (Nat.le_antisymm hfwd.1 hbwd.1)
-
-
-noncomputable def cellRowsEquiv (μ : YoungDiagram) :
-    Cell μ ≃ (i : Fin (μ.colLen 0)) × Fin (μ.rowLen i) where
-  toFun x := ⟨rowIndex x,⟨col x,YoungDiagram.mem_iff_lt_rowLen.mp x.2⟩⟩
-  invFun p := ⟨(p.1,p.2),YoungDiagram.mem_iff_lt_rowLen.mpr p.2.2⟩
-  left_inv x := by apply Subtype.ext; rfl
-  right_inv p := by rcases p with ⟨⟨i,hi⟩,⟨j,hj⟩⟩; rfl
-
-lemma card_eq_sum_rowLen (μ : YoungDiagram) : μ.card = ∑ i : Fin (μ.colLen 0), μ.rowLen i := by
-  rw [←card_cell,Fintype.card_congr (cellRowsEquiv μ),Fintype.card_sigma]
-  simp only [Fintype.card_fin]
-
-lemma sum_rowLens (μ : YoungDiagram) : μ.rowLens.sum = μ.card := by
-  rw [card_eq_sum_rowLen,YoungDiagram.rowLens]
-  rw [Fin.sum_univ_eq_sum_range]
-  have hh (n : ℕ) : (List.map μ.rowLen (List.range n)).sum = ∑ i ∈ Finset.range n, μ.rowLen i := by
-    induction n with
-    | zero => simp
-    | succ n ih => simp [List.range_succ,List.map_append,Finset.sum_range_succ,ih]
-  exact hh _
-
-noncomputable def diagramPartition (μ : YoungDiagram) : Nat.Partition μ.card where
-  parts := μ.rowLens
-  parts_pos := fun {i} hi => μ.pos_of_mem_rowLens i (by simpa using hi)
-  parts_sum := by simp
-
-noncomputable def partitionDiagram {N : ℕ} (p : Nat.Partition N) : YoungDiagram :=
-  YoungDiagram.ofRowLens (p.parts.sort (· ≥ ·)) (Multiset.pairwise_sort _ _).sortedGE
-
-lemma partitionDiagram_rowLens {N : ℕ} (p : Nat.Partition N) :
-    (partitionDiagram p).rowLens = p.parts.sort (· ≥ ·) := by
-  apply YoungDiagram.rowLens_ofRowLens_eq_self
-  intro x hx
-  exact p.parts_pos ((Multiset.mem_sort _).mp hx)
-
-lemma partitionDiagram_card {N : ℕ} (p : Nat.Partition N) : (partitionDiagram p).card = N := by
-  rw [←sum_rowLens,partitionDiagram_rowLens]
-  change (↑(p.parts.sort (· ≥ ·)) : Multiset ℕ).sum = N
-  rw [Multiset.sort_eq,p.parts_sum]
-
-lemma partitionDiagram_diagramPartition (μ : YoungDiagram) :
-    partitionDiagram (diagramPartition μ) = μ := by
-  apply YoungDiagram.equivListRowLens.injective
-  apply Subtype.ext
-  change (partitionDiagram (diagramPartition μ)).rowLens = μ.rowLens
-  rw [partitionDiagram_rowLens]
-  change (↑μ.rowLens : Multiset ℕ).sort (· ≥ ·) = μ.rowLens
-  rw [Multiset.coe_sort,List.mergeSort_eq_self _ μ.rowLens_sorted.pairwise]
-
-abbrev Shapes (N : ℕ) := {μ : YoungDiagram // μ.card = N}
-
-end Thorp.Specht
-
-end ThorpNine.SparseSaving
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.SparseSaving.ModifiedRoutingMoment
+
+namespace OAI
+
+namespace ThorpNine.SparseSaving
+
+namespace Thorp
+
+noncomputable def modifiedReverseEquiv (L r : ℕ) : Equiv.Perm (ModifiedCoins L r) where
+  toFun := modifiedReverseCoins L r
+  invFun := modifiedReverseCoins L r
+  left_inv := modifiedReverseCoins_involutive L r
+  right_inv := modifiedReverseCoins_involutive L r
+
+lemma modifiedPerm_reverse (L r : ℕ) (E : Finset (Card r)) (ω : ModifiedCoins L r) :
+    modifiedPerm L r E (modifiedReverseCoins L r ω) = (modifiedPerm L r E ω)⁻¹ := by
+  induction r with
+  | zero =>
+      rw [modifiedPerm_zero,modifiedPerm_zero]
+      split_ifs
+      · simp only [inv_one]
+      · simp only [modifiedReverseCoins,mul_inv_rev,inv_inv]
+        simp
+  | succ r ih =>
+      obtain ⟨z,rfl⟩ := (modifiedStepEquiv L r).symm.surjective ω
+      rcases z with ⟨⟨ω₀,ω₁⟩,⟨ξ,η⟩⟩
+      simp only [modifiedReverseCoins,Equiv.apply_symm_apply,modifiedPerm_step,
+        mul_inv_rev,pairSwitch_inv,childLift_inv]
+      rw [mul_assoc]
+      congr 1
+      congr 1
+      apply congrArg childLift
+      funext b
+      cases b <;> simp only [Bool.false_eq_true,↓reduceIte,ih]
+
+lemma modified_kernel_symmetric (L r : ℕ) (E : Finset (Card r))
+    {ι : Type*} [Fintype ι] (e f : ι ↪ Card (L+r)) :
+    PairRouting.tupleProbability (modifiedPerm L r E) e f =
+      PairRouting.tupleProbability (modifiedPerm L r E) f e :=
+  PairRouting.tupleProbability_symmetric _ (modifiedReverseEquiv L r)
+    (modifiedPerm_reverse L r E) e f
+
+end Thorp
+
+namespace Thorp.Specht
+open scoped BigOperators Classical
+
+@[simp] lemma card_cell (μ : YoungDiagram) : Fintype.card (Cell μ) = μ.card := by
+  simp [Cell, YoungDiagram.card]
+
+noncomputable def columnEquiv (μ : YoungDiagram) (j : ℕ) :
+    {x : Cell μ // col x = j} ≃ Fin (μ.colLen j) where
+  toFun x := ⟨row x.1, by simpa [x.2] using row_lt_colLen x.1⟩
+  invFun i := ⟨⟨(i,j), YoungDiagram.mem_iff_lt_colLen.mpr i.2⟩, rfl⟩
+  left_inv x := by apply Subtype.ext; apply Subtype.ext; exact Prod.ext rfl x.2.symm
+  right_inv i := by rfl
+
+noncomputable def columnValues {μ : YoungDiagram} (f : Cell μ → ℕ) (j : ℕ) : Finset ℕ :=
+  (Finset.univ.filter (fun x => col x = j)).image f
+
+lemma mem_columnValues {μ : YoungDiagram} (f : Cell μ → ℕ) (x : Cell μ) :
+    f x ∈ columnValues f (col x) := by
+  classical
+  exact Finset.mem_image.mpr ⟨x, by simp, rfl⟩
+
+lemma card_columnValues {μ : YoungDiagram} (f : Cell μ → ℕ)
+    (hf : ∀ x y, col x = col y → f x = f y → x = y) (j : ℕ) :
+    (columnValues f j).card = μ.colLen j := by
+  classical
+  rw [columnValues, Finset.card_image_iff.mpr]
+  · rw [← Fintype.card_subtype]
+    exact Fintype.card_congr (columnEquiv μ j) |>.trans (Fintype.card_fin _)
+  intro x hx y hy he
+  exact hf x y ((Finset.mem_filter.mp hx).2.trans (Finset.mem_filter.mp hy).2.symm) he
+
+lemma strictMono_fin_nat_ge {n : ℕ} (f : Fin n → ℕ) (hf : StrictMono f) :
+    ∀ i, i.val ≤ f i := by
+  intro i
+  induction h : i.val using Nat.strong_induction_on generalizing i with
+  | h n ih =>
+    by_cases hn : n = 0
+    · omega
+    · have hn' : n-1 < i.val := by omega
+      let j : Fin _ := ⟨n-1, hn'.trans i.isLt⟩
+      have hj := ih (n-1) (by omega) j rfl
+      have hij := hf (show j < i from hn')
+      dsimp [j] at hj hij
+      omega
+
+noncomputable def rankCell {μ : YoungDiagram} (f : Cell μ → ℕ)
+    (hf : ∀ x y, col x = col y → f x = f y → x = y) (x : Cell μ) : Cell μ :=
+  let k := ((columnValues f (col x)).orderIsoOfFin (card_columnValues f hf (col x))).symm
+    ⟨f x, mem_columnValues f x⟩
+  ⟨(k.val,col x), YoungDiagram.mem_iff_lt_colLen.mpr k.isLt⟩
+
+@[simp] lemma rankCell_col {μ : YoungDiagram} (f : Cell μ → ℕ)
+    (hf : ∀ x y, col x = col y → f x = f y → x = y) (x : Cell μ) :
+    col (rankCell f hf x) = col x := rfl
+
+lemma rankCell_orderEmb {μ : YoungDiagram} (f : Cell μ → ℕ)
+    (hf : ∀ x y, col x = col y → f x = f y → x = y) (x : Cell μ) :
+    (columnValues f (col x)).orderEmbOfFin (card_columnValues f hf (col x))
+      ⟨row (rankCell f hf x), row_lt_colLen (rankCell f hf x)⟩ = f x := by
+  change ((columnValues f (col x)).orderIsoOfFin (card_columnValues f hf (col x))
+    (((columnValues f (col x)).orderIsoOfFin (card_columnValues f hf (col x))).symm
+      ⟨f x, mem_columnValues f x⟩)).val = f x
+  simp
+
+lemma rankCell_injective {μ : YoungDiagram} (f : Cell μ → ℕ)
+    (hf : ∀ x y, col x = col y → f x = f y → x = y) :
+    Function.Injective (rankCell f hf) := by
+  intro x y h
+  have hc : col x = col y := by simpa only [rankCell_col] using congrArg (fun z : Cell μ => col z) h
+  apply hf x y hc
+  have hx := rankCell_orderEmb f hf x
+  have hy := rankCell_orderEmb f hf y
+  change (columnValues f (col (rankCell f hf x))).orderEmbOfFin
+    (card_columnValues f hf (col (rankCell f hf x)))
+      ⟨row (rankCell f hf x), row_lt_colLen (rankCell f hf x)⟩ = f x at hx
+  rw [h] at hx
+  exact hx.symm.trans hy
+
+noncomputable def rankPerm {μ : YoungDiagram} (f : Cell μ → ℕ)
+    (hf : ∀ x y, col x = col y → f x = f y → x = y) : Equiv.Perm (Cell μ) :=
+  Equiv.ofBijective (rankCell f hf) ((Finite.injective_iff_bijective).mp (rankCell_injective f hf))
+
+lemma rankCell_row_le {μ : YoungDiagram} (f : Cell μ → ℕ)
+    (hf : ∀ x y, col x = col y → f x = f y → x = y) (x : Cell μ) :
+    row (rankCell f hf x) ≤ f x := by
+  rw [←rankCell_orderEmb f hf x]
+  exact strictMono_fin_nat_ge _ (OrderEmbedding.strictMono _) _
+
+theorem column_transporter (μ : YoungDiagram) (p : Equiv.Perm (Cell μ))
+    (hp : ∀ x y, col x = col y → row (p x) = row (p y) → x = y) :
+    ∃ c : Equiv.Perm (Cell μ), (∀ x, col (c x) = col x) ∧
+      ∀ x, row (c x) = row (p x) := by
+  classical
+  let c := rankPerm (fun x => row (p x)) hp
+  refine ⟨c, fun x => rfl, ?_⟩
+  have hle (x : Cell μ) : row (c x) ≤ row (p x) := rankCell_row_le _ hp x
+  have hsum : ∑ x, row (c x) = ∑ x, row (p x) := by
+    rw [Equiv.sum_comp, Equiv.sum_comp]
+  exact fun x => (Finset.sum_eq_sum_iff_of_le (fun x _ => hle x)).mp hsum x (Finset.mem_univ _)
+
+
+@[simp] lemma tabloidAct_inv {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) :
+    tabloidAct p⁻¹ = (tabloidAct p)⁻¹ := by
+  ext f x
+  simp [tabloidAct,Equiv.Perm.inv_def]
+
+@[simp] lemma tabloidRep_apply {μ : YoungDiagram} (p : Equiv.Perm (Cell μ))
+    (v : Tabloid μ → ℂ) (f : Tabloid μ) : tabloidRep μ p v f = v (tabloidAct p⁻¹ f) := rfl
+
+lemma tabloidRep_delta {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) (f : Tabloid μ) :
+    tabloidRep μ p (delta f) = delta (tabloidAct p f) := by
+  classical
+  ext g
+  simp only [tabloidRep_apply,delta,Equiv.Perm.inv_def]
+  congr 1
+  exact propext (Equiv.symm_apply_eq (tabloidAct p))
+
+@[simp] lemma mem_colGroup {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) :
+    p ∈ colGroup μ ↔ ∀ x, col (p x) = col x := Iff.rfl
+
+lemma colGroup_row_stabilizer {μ : YoungDiagram} (c : colGroup μ)
+    (hc : tabloidAct c (baseTabloid μ) = baseTabloid μ) : (c : Equiv.Perm (Cell μ)) = 1 := by
+  apply Equiv.ext
+  intro x
+  have hr := congrArg (fun f : Tabloid μ => ((f.1 ((c : Equiv.Perm _) x)) : ℕ)) hc
+  simp only [tabloidAct_apply,Equiv.Perm.inv_def,Equiv.symm_apply_apply,baseTabloid,rowIndex] at hr
+  apply Subtype.ext
+  exact Prod.ext hr.symm (c.2 x)
+
+lemma tabloid_column_transporter {μ : YoungDiagram} (f : Tabloid μ)
+    (hf : ∀ x y, col x = col y → f.1 x = f.1 y → x = y) :
+    ∃ c : colGroup μ, f = tabloidAct c (baseTabloid μ) := by
+  obtain ⟨p,hp⟩ := f.2
+  have hp' : ∀ x y, col x = col y → row (p x) = row (p y) → x = y := by
+    intro x y hcol hrow
+    apply hf x y hcol
+    rw [hp]
+    exact Fin.ext hrow
+  obtain ⟨c,hcol,hrow⟩ := column_transporter μ p hp'
+  refine ⟨⟨c⁻¹, (colGroup μ).inv_mem hcol⟩, ?_⟩
+  apply Subtype.ext
+  funext x
+  apply Fin.ext
+  simp only [tabloidAct_apply,baseTabloid,rowIndex,inv_inv,hp,Function.comp_apply]
+  exact (hrow x).symm
+
+
+@[simp] lemma permSign_one {α : Type*} [Fintype α] [DecidableEq α] :
+    permSign (1 : Equiv.Perm α) = 1 := by simp [permSign]
+@[simp] lemma permSign_mul {α : Type*} [Fintype α] [DecidableEq α] (p q : Equiv.Perm α) :
+    permSign (p*q) = permSign p * permSign q := by simp [permSign]
+@[simp] lemma permSign_inv {α : Type*} [Fintype α] [DecidableEq α] (p : Equiv.Perm α) :
+    permSign p⁻¹ = permSign p := by simp [permSign,Equiv.Perm.inv_def]
+lemma permSign_sq {α : Type*} [Fintype α] [DecidableEq α] (p : Equiv.Perm α) :
+    permSign p * permSign p = 1 := by
+  calc
+    permSign p * permSign p = permSign (p*p⁻¹) := by rw [permSign_mul,permSign_inv]
+    _ = 1 := by simp
+@[simp] lemma permSign_swap {α : Type*} [Fintype α] [DecidableEq α] {x y : α} (h : x ≠ y) :
+    permSign (Equiv.swap x y) = -1 := by simp [permSign, h]
+@[simp] lemma conj_permSign {α : Type*} [Fintype α] [DecidableEq α] (p : Equiv.Perm α) :
+    starRingEnd ℂ (permSign p) = permSign p := by simp [permSign]
+
+variable {α V : Type*} [Fintype α] [DecidableEq α] [AddCommGroup V] [Module ℂ V]
+
+lemma alternator_mul (C : Subgroup (Equiv.Perm α))
+    (ρ : Representation ℂ (Equiv.Perm α) V) (c : C) :
+    alternator C ρ * ρ c = permSign (c : Equiv.Perm α) • alternator C ρ := by
+  classical
+  rw [alternator,Finset.sum_mul,Finset.smul_sum]
+  apply Fintype.sum_equiv (Equiv.mulRight c)
+  intro a
+  simp only [smul_mul_assoc,←map_mul,Equiv.coe_mulRight,Subgroup.coe_mul,permSign_mul,smul_smul]
+  congr 1
+  have h := permSign_sq (c : Equiv.Perm α)
+  calc
+    permSign (a : Equiv.Perm α) = permSign (a : Equiv.Perm α)*1 := by ring
+    _ = permSign (a : Equiv.Perm α)*(permSign (c : Equiv.Perm α)*permSign (c : Equiv.Perm α)) := by rw [h]
+    _ = _ := by ring
+
+lemma mul_alternator (C : Subgroup (Equiv.Perm α))
+    (ρ : Representation ℂ (Equiv.Perm α) V) (c : C) :
+    ρ c * alternator C ρ = permSign (c : Equiv.Perm α) • alternator C ρ := by
+  classical
+  rw [alternator,Finset.mul_sum,Finset.smul_sum]
+  apply Fintype.sum_equiv (Equiv.mulLeft c)
+  intro a
+  simp only [mul_smul_comm,←map_mul,Equiv.coe_mulLeft,Subgroup.coe_mul,permSign_mul,smul_smul]
+  congr 1
+  have h := permSign_sq (c : Equiv.Perm α)
+  calc
+    permSign (a : Equiv.Perm α) = 1*permSign (a : Equiv.Perm α) := by ring
+    _ = (permSign (c : Equiv.Perm α)*permSign (c : Equiv.Perm α))*permSign (a : Equiv.Perm α) := by rw [h]
+    _ = _ := by ring
+
+lemma alternator_apply_action (C : Subgroup (Equiv.Perm α))
+    (ρ : Representation ℂ (Equiv.Perm α) V) (c : C) (v : V) :
+    alternator C ρ (ρ c v) = permSign (c : Equiv.Perm α) • alternator C ρ v := by
+  exact congrArg (fun T : Module.End ℂ V => T v) (alternator_mul C ρ c)
+
+lemma action_alternator_apply (C : Subgroup (Equiv.Perm α))
+    (ρ : Representation ℂ (Equiv.Perm α) V) (c : C) (v : V) :
+    ρ c (alternator C ρ v) = permSign (c : Equiv.Perm α) • alternator C ρ v := by
+  exact congrArg (fun T : Module.End ℂ V => T v) (mul_alternator C ρ c)
+
+lemma alternator_eq_zero_of_odd_stabilizer (C : Subgroup (Equiv.Perm α))
+    (ρ : Representation ℂ (Equiv.Perm α) V) (c : C) (v : V)
+    (hc : permSign (c : Equiv.Perm α) = -1) (hv : ρ c v = v) :
+    alternator C ρ v = 0 := by
+  have h := alternator_apply_action C ρ c v
+  rw [hv,hc,neg_one_smul] at h
+  have h2 : (2:ℂ) • alternator C ρ v = 0 := by
+    rw [two_smul]
+    nth_rw 1 [h]
+    exact neg_add_cancel _
+  exact (smul_eq_zero.mp h2).resolve_left (by norm_num)
+
+end Thorp.Specht
+namespace Thorp.Specht
+open scoped BigOperators Classical
+
+lemma swap_mem_colGroup {μ : YoungDiagram} (x y : Cell μ) (hc : col x = col y) :
+    Equiv.swap x y ∈ colGroup μ := by
+  classical
+  intro z
+  by_cases hx : z=x
+  · subst z; simpa using hc.symm
+  by_cases hy : z=y
+  · subst z; simpa using hc
+  simp [Equiv.swap_apply_of_ne_of_ne hx hy]
+
+lemma swap_fixes_tabloid {μ : YoungDiagram} (f : Tabloid μ) (x y : Cell μ)
+    (hxy : f.1 x = f.1 y) : tabloidAct (Equiv.swap x y) f = f := by
+  classical
+  apply Subtype.ext
+  funext z
+  simp only [tabloidAct_apply, Equiv.swap_inv]
+  by_cases hx : z=x
+  · subst z; simpa using hxy.symm
+  by_cases hy : z=y
+  · subst z; simpa using hxy
+  simp [Equiv.swap_apply_of_ne_of_ne hx hy]
+
+lemma alternator_delta_rank_one {μ : YoungDiagram} (f : Tabloid μ) :
+    ∃ z : ℂ, alternator (colGroup μ) (tabloidRep μ) (delta f) = z • polytabloid μ := by
+  classical
+  by_cases hf : ∀ x y, col x = col y → f.1 x = f.1 y → x = y
+  · obtain ⟨c,rfl⟩ := tabloid_column_transporter f hf
+    refine ⟨permSign (c:Equiv.Perm (Cell μ)), ?_⟩
+    rw [←tabloidRep_delta]
+    exact alternator_apply_action _ _ c _
+  · push Not at hf
+    obtain ⟨x,y,hcol,hrow,hne⟩ := hf
+    refine ⟨0, ?_⟩
+    rw [zero_smul]
+    apply alternator_eq_zero_of_odd_stabilizer _ _ ⟨Equiv.swap x y,swap_mem_colGroup x y hcol⟩
+    · exact permSign_swap hne
+    · rw [tabloidRep_delta,swap_fixes_tabloid f x y hrow]
+
+lemma polytabloid_formula (μ : YoungDiagram) :
+    polytabloid μ = ∑ c : colGroup μ, permSign (c:Equiv.Perm (Cell μ)) •
+      delta (tabloidAct c (baseTabloid μ)) := by
+  classical
+  simp only [polytabloid,alternator,LinearMap.sum_apply,LinearMap.smul_apply,tabloidRep_delta]
+
+lemma polytabloid_base (μ : YoungDiagram) : polytabloid μ (baseTabloid μ) = 1 := by
+  classical
+  rw [polytabloid_formula,Finset.sum_apply]
+  rw [Finset.sum_eq_single (1 : colGroup μ)]
+  · simp [delta]
+  · intro c hc hne
+    have hn : baseTabloid μ ≠ tabloidAct c (baseTabloid μ) := by
+      intro h
+      apply hne
+      exact Subtype.ext (colGroup_row_stabilizer c h.symm)
+    simp [Pi.smul_apply,delta,hn]
+  · simp
+
+lemma polytabloid_ne_zero (μ : YoungDiagram) : polytabloid μ ≠ 0 := by
+  intro h
+  have hv := congrArg (fun v : Tabloid μ → ℂ => v (baseTabloid μ)) h
+  simp [polytabloid_base] at hv
+
+lemma delta_decomposition {μ : YoungDiagram} (v : Tabloid μ → ℂ) :
+    v = ∑ f : Tabloid μ, v f • delta f := by
+  classical
+  funext g
+  simp only [Finset.sum_apply,Pi.smul_apply,delta,smul_eq_mul,mul_ite,mul_one,mul_zero]
+  simp
+
+theorem alternator_rank_one (μ : YoungDiagram) (v : Tabloid μ → ℂ) :
+    ∃ z : ℂ, alternator (colGroup μ) (tabloidRep μ) v = z • polytabloid μ := by
+  classical
+  choose z hz using fun f : Tabloid μ => alternator_delta_rank_one f
+  refine ⟨∑ f, v f * z f, ?_⟩
+  conv_lhs => rw [delta_decomposition v]
+  simp only [map_sum,map_smul,hz,smul_smul,Finset.sum_smul]
+
+lemma alternator_rank_one_exact (μ : YoungDiagram) (v : Tabloid μ → ℂ) :
+    alternator (colGroup μ) (tabloidRep μ) v =
+      (alternator (colGroup μ) (tabloidRep μ) v (baseTabloid μ)) • polytabloid μ := by
+  obtain ⟨z,hz⟩ := alternator_rank_one μ v
+  rw [hz]
+  simp only [Pi.smul_apply,polytabloid_base,smul_eq_mul,mul_one]
+
+
+noncomputable def pairing {μ : YoungDiagram} (v w : Tabloid μ → ℂ) : ℂ :=
+  ∑ f, w f * starRingEnd ℂ (v f)
+
+lemma pairing_delta {μ : YoungDiagram} (f : Tabloid μ) (v : Tabloid μ → ℂ) :
+    pairing (delta f) v = v f := by
+  classical
+  simp [pairing,delta]
+
+lemma pairing_sum_left {μ : YoungDiagram} {ι : Type*} [Fintype ι]
+    (v : ι → Tabloid μ → ℂ) (w : Tabloid μ → ℂ) :
+    pairing (∑ i, v i) w = ∑ i, pairing (v i) w := by
+  simp only [pairing,Finset.sum_apply,map_sum,Finset.mul_sum]
+  exact Finset.sum_comm
+
+lemma pairing_smul_left {μ : YoungDiagram} (a : ℂ) (v w : Tabloid μ → ℂ) :
+    pairing (a • v) w = starRingEnd ℂ a * pairing v w := by
+  simp only [pairing,Pi.smul_apply,smul_eq_mul,map_mul,Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro f hf
+  ring
+
+lemma pairing_zero_left {μ : YoungDiagram} (v : Tabloid μ → ℂ) : pairing 0 v = 0 := by
+  simp [pairing]
+
+lemma pairing_add_left {μ : YoungDiagram} (u v w : Tabloid μ → ℂ) :
+    pairing (u+v) w = pairing u w + pairing v w := by
+  simp [pairing,mul_add,Finset.sum_add_distrib]
+
+lemma pairing_self_eq_zero {μ : YoungDiagram} (v : Tabloid μ → ℂ) :
+    pairing v v = 0 ↔ v = 0 := by
+  change inner ℂ (WithLp.toLp 2 v : EuclideanSpace ℂ (Tabloid μ)) (WithLp.toLp 2 v) = 0 ↔ v = 0
+  rw [inner_self_eq_zero]
+  constructor
+  · intro h; exact congrArg WithLp.ofLp h
+  · intro h; subst v; rfl
+
+lemma pairing_action {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) (v w : Tabloid μ → ℂ) :
+    pairing (tabloidRep μ p v) w = pairing v (tabloidRep μ p⁻¹ w) := by
+  classical
+  simp only [pairing,tabloidRep_apply,inv_inv]
+  apply Fintype.sum_equiv (tabloidAct p⁻¹)
+  intro f
+  have hcancel : tabloidAct p (tabloidAct p⁻¹ f) = f := by
+    rw [tabloidAct_inv]
+    exact (tabloidAct p).apply_symm_apply f
+  rw [hcancel]
+
+lemma alternator_base_pairing (μ : YoungDiagram) (v : Tabloid μ → ℂ) :
+    alternator (colGroup μ) (tabloidRep μ) v (baseTabloid μ) = pairing (polytabloid μ) v := by
+  classical
+  simp only [polytabloid_formula,pairing_sum_left,pairing_smul_left,conj_permSign,pairing_delta]
+  simp only [alternator,LinearMap.sum_apply,LinearMap.smul_apply,Finset.sum_apply,Pi.smul_apply,
+    smul_eq_mul,tabloidRep_apply]
+  apply Fintype.sum_equiv (Equiv.inv (colGroup μ))
+  intro c
+  simp only [Equiv.inv_apply,Subgroup.coe_inv,permSign_inv]
+
+theorem alternator_pairing (μ : YoungDiagram) (v : Tabloid μ → ℂ) :
+    alternator (colGroup μ) (tabloidRep μ) v = pairing (polytabloid μ) v • polytabloid μ := by
+  rw [alternator_rank_one_exact,alternator_base_pairing]
+
+
+lemma polytabloid_mem_space (μ : YoungDiagram) : polytabloid μ ∈ space μ := by
+  simpa using orbit_mem_space μ 1
+
+lemma alternator_mem_invariant (μ : YoungDiagram) (W : Submodule ℂ (Tabloid μ → ℂ))
+    (hW : ∀ p v, v ∈ W → tabloidRep μ p v ∈ W) (v : Tabloid μ → ℂ) (hv : v ∈ W) :
+    alternator (colGroup μ) (tabloidRep μ) v ∈ W := by
+  classical
+  simp only [alternator,LinearMap.sum_apply,LinearMap.smul_apply]
+  exact W.sum_mem (fun c _ => W.smul_mem _ (hW c v hv))
+
+theorem submodule_theorem (μ : YoungDiagram) (W : Submodule ℂ (Tabloid μ → ℂ))
+    (hW : ∀ p v, v ∈ W → tabloidRep μ p v ∈ W) :
+    space μ ≤ W ∨ ∀ v ∈ W, ∀ w ∈ space μ, pairing w v = 0 := by
+  classical
+  by_cases hSW : space μ ≤ W
+  · exact Or.inl hSW
+  right
+  have he : polytabloid μ ∉ W := by
+    intro he
+    apply hSW
+    apply Submodule.span_le.mpr
+    rintro _ ⟨p,rfl⟩
+    exact hW p _ he
+  have hpair : ∀ v ∈ W, pairing (polytabloid μ) v = 0 := by
+    intro v hv
+    by_contra hne
+    apply he
+    have hal := alternator_mem_invariant μ W hW v hv
+    rw [alternator_pairing] at hal
+    have hh := W.smul_mem (pairing (polytabloid μ) v)⁻¹ hal
+    simpa only [smul_smul,inv_mul_cancel₀ hne,one_smul] using hh
+  intro v hv w hw
+  induction hw using Submodule.span_induction with
+  | mem w hw =>
+    obtain ⟨p,rfl⟩ := hw
+    rw [pairing_action]
+    exact hpair _ (hW p⁻¹ v hv)
+  | zero => exact pairing_zero_left v
+  | add u w hu hw ihu ihw => rw [pairing_add_left,ihu,ihw,add_zero]
+  | smul a w hw ih => rw [pairing_smul_left,ih,mul_zero]
+
+theorem space_simple (μ : YoungDiagram) (W : Submodule ℂ (Tabloid μ → ℂ))
+    (hW : ∀ p v, v ∈ W → tabloidRep μ p v ∈ W) (hWS : W ≤ space μ) :
+    W = ⊥ ∨ W = space μ := by
+  obtain hSW | horth := submodule_theorem μ W hW
+  · exact Or.inr (le_antisymm hWS hSW)
+  · left
+    apply le_antisymm _ bot_le
+    intro v hv
+    exact (pairing_self_eq_zero v).mp (horth v hv v (hWS hv))
+
+
+lemma space_ne_bot (μ : YoungDiagram) : space μ ≠ ⊥ := by
+  intro h
+  have he := polytabloid_mem_space μ
+  rw [h,Submodule.mem_bot] at he
+  exact polytabloid_ne_zero μ he
+
+theorem representation_irreducible (μ : YoungDiagram) :
+    Representation.IsIrreducible (representation μ) := by
+  have hnt : (⊥ : Subrepresentation (representation μ)) ≠ ⊤ := by
+    intro h
+    have he : (⟨polytabloid μ,polytabloid_mem_space μ⟩ : space μ) ∈
+        (⊥ : Subrepresentation (representation μ)) := by rw [h]; trivial
+    have he' : (⟨polytabloid μ,polytabloid_mem_space μ⟩ : space μ) = 0 := he
+    exact polytabloid_ne_zero μ (congrArg Subtype.val he')
+  refine { exists_pair_ne := ⟨⊥,⊤,hnt⟩, eq_bot_or_eq_top := ?_ }
+  intro U
+  let W : Submodule ℂ (Tabloid μ → ℂ) := U.toSubmodule.map (space μ).subtype
+  have hW : ∀ p v, v ∈ W → tabloidRep μ p v ∈ W := by
+    intro p v hv
+    obtain ⟨u,hu,rfl⟩ := hv
+    exact ⟨representation μ p u,U.apply_mem_toSubmodule p hu,rfl⟩
+  have hWS : W ≤ space μ := by
+    intro v hv
+    obtain ⟨u,hu,rfl⟩ := hv
+    exact u.2
+  obtain hbot | htop := space_simple μ W hW hWS
+  · left
+    apply le_antisymm _ bot_le
+    intro u hu
+    have hmem : (u : Tabloid μ → ℂ) ∈ W := ⟨u,hu,rfl⟩
+    rw [hbot,Submodule.mem_bot] at hmem
+    change u = 0
+    exact Subtype.ext hmem
+  · right
+    apply le_antisymm le_top
+    intro u hu
+    have hmem : (u : Tabloid μ → ℂ) ∈ W := by rw [htop]; exact u.2
+    obtain ⟨v,hv,hvu⟩ := hmem
+    have heq : v = u := Subtype.ext hvu
+    exact heq ▸ hv
+
+
+section Symmetrizer
+variable {α V : Type*} [Fintype α] [DecidableEq α] [AddCommGroup V] [Module ℂ V]
+
+noncomputable def symmetrizer (H : Subgroup (Equiv.Perm α))
+    (ρ : Representation ℂ (Equiv.Perm α) V) : Module.End ℂ V :=
+  ∑ h : H, ρ h
+
+lemma symmetrizer_mul (H : Subgroup (Equiv.Perm α))
+    (ρ : Representation ℂ (Equiv.Perm α) V) (h : H) :
+    symmetrizer H ρ * ρ h = symmetrizer H ρ := by
+  rw [symmetrizer,Finset.sum_mul]
+  exact Fintype.sum_equiv (Equiv.mulRight h) _ _ (fun g => by simp [←map_mul])
+
+lemma symmetrizer_eq_zero_of_neg (H : Subgroup (Equiv.Perm α))
+    (ρ : Representation ℂ (Equiv.Perm α) V) (h : H) (v : V) (hv : ρ h v = -v) :
+    symmetrizer H ρ v = 0 := by
+  have hh := congrArg (fun T : Module.End ℂ V => T v) (symmetrizer_mul H ρ h)
+  change symmetrizer H ρ (ρ h v) = symmetrizer H ρ v at hh
+  rw [hv,map_neg] at hh
+  have h2 : (2:ℂ) • symmetrizer H ρ v = 0 := by
+    rw [two_smul]
+    nth_rw 1 [←hh]
+    exact neg_add_cancel _
+  exact (smul_eq_zero.mp h2).resolve_left (by norm_num)
+
+noncomputable def fiberGroup {β : Type*} (f : α → β) : Subgroup (Equiv.Perm α) where
+  carrier := {p | ∀ x, f (p x) = f x}
+  one_mem' := by intro x; rfl
+  mul_mem' := by intro p q hp hq x; exact (hp (q x)).trans (hq x)
+  inv_mem' := by intro p hp x; simpa using (hp (p⁻¹ x)).symm
+
+omit [Fintype α] in
+lemma swap_mem_fiberGroup {β : Type*} (f : α → β) (x y : α) (hf : f x = f y) :
+    Equiv.swap x y ∈ fiberGroup f := by
+  intro z
+  by_cases hx : z=x
+  · subst z; simpa using hf.symm
+  by_cases hy : z=y
+  · subst z; simpa using hf
+  simp [Equiv.swap_apply_of_ne_of_ne hx hy]
+
+end Symmetrizer
+
+lemma swap_polytabloid {μ : YoungDiagram} (x y : Cell μ) (hne : x ≠ y)
+    (hcol : col x = col y) :
+    tabloidRep μ (Equiv.swap x y) (polytabloid μ) = -polytabloid μ := by
+  have h := action_alternator_apply (colGroup μ) (tabloidRep μ)
+    ⟨Equiv.swap x y,swap_mem_colGroup x y hcol⟩ (delta (baseTabloid μ))
+  simpa only [polytabloid,permSign_swap hne,neg_one_smul] using h
+
+theorem tall_column_average_zero (μ : YoungDiagram) {β : Type*} [Fintype β]
+    (f : Cell μ → β) (j : ℕ) (hj : Fintype.card β < μ.colLen j)
+    (v : Tabloid μ → ℂ) (hv : v ∈ space μ) :
+    symmetrizer (fiberGroup f) (tabloidRep μ) v = 0 := by
+  induction hv using Submodule.span_induction with
+  | mem v hv =>
+    obtain ⟨p,rfl⟩ := hv
+    have hcard : Fintype.card β < Fintype.card {x : Cell μ // col x = j} := by
+      rw [Fintype.card_congr (columnEquiv μ j),Fintype.card_fin]
+      exact hj
+    obtain ⟨x,y,hne,hf⟩ := Fintype.exists_ne_map_eq_of_card_lt
+      (fun x : {x : Cell μ // col x = j} => f (p x.1)) hcard
+    have hxy : x.1 ≠ y.1 := fun h => hne (Subtype.ext h)
+    let h : fiberGroup f := ⟨Equiv.swap (p x.1) (p y.1),swap_mem_fiberGroup f _ _ hf⟩
+    apply symmetrizer_eq_zero_of_neg _ _ h
+    change tabloidRep μ (Equiv.swap (p x.1) (p y.1))
+      (tabloidRep μ p (polytabloid μ)) = -tabloidRep μ p (polytabloid μ)
+    rw [←Module.End.mul_apply,←map_mul,←Equiv.mul_swap_eq_swap_mul]
+    rw [map_mul,Module.End.mul_apply,swap_polytabloid x.1 y.1 hxy (x.2.trans y.2.symm),map_neg]
+  | zero => exact map_zero _
+  | add v w hv hw ihv ihw => simp only [map_add,ihv,ihw,add_zero]
+  | smul a v hv ih => simp only [map_smul,ih,smul_zero]
+
+
+noncomputable def rowEquiv (μ : YoungDiagram) (i : ℕ) :
+    {x : Cell μ // row x = i} ≃ Fin (μ.rowLen i) where
+  toFun x := ⟨col x.1, by
+    have h := YoungDiagram.mem_iff_lt_rowLen.mp x.1.2
+    change col x.1 < μ.rowLen (row x.1) at h
+    simpa only [x.2] using h⟩
+  invFun j := ⟨⟨(i,j),YoungDiagram.mem_iff_lt_rowLen.mpr j.2⟩,rfl⟩
+  left_inv x := by apply Subtype.ext; apply Subtype.ext; exact Prod.ext x.2.symm rfl
+  right_inv j := rfl
+
+def rowCost (μ : YoungDiagram) : ℕ := ∑ x : Cell μ, row x
+
+lemma eq_of_row_preserving (μ ν : YoungDiagram) (e : Cell μ ≃ Cell ν)
+    (he : ∀ x, row (e x) = row x) : μ = ν := by
+  have hlen (i : ℕ) : μ.rowLen i = ν.rowLen i := by
+    let ei : {x : Cell μ // row x = i} ≃ {x : Cell ν // row x = i} :=
+      { toFun := fun x => ⟨e x.1,(he x.1).trans x.2⟩
+        invFun := fun x => ⟨e.symm x.1,by
+          have h := he (e.symm x.1)
+          simp only [Equiv.apply_symm_apply] at h
+          exact h.symm.trans x.2⟩
+        left_inv := fun x => by apply Subtype.ext; exact e.symm_apply_apply _
+        right_inv := fun x => by apply Subtype.ext; exact e.apply_symm_apply _ }
+    simpa only [Fintype.card_fin] using Fintype.card_congr
+      ((rowEquiv μ i).symm.trans (ei.trans (rowEquiv ν i)))
+  apply YoungDiagram.ext
+  ext ⟨i,j⟩
+  simp only [YoungDiagram.mem_cells,YoungDiagram.mem_iff_lt_rowLen,hlen]
+
+lemma column_injective_rowCost (μ ν : YoungDiagram) (e : Cell μ ≃ Cell ν)
+    (h : ∀ x y : Cell μ, col x = col y → row (e x) = row (e y) → x = y) :
+    rowCost μ ≤ rowCost ν ∧ (rowCost μ = rowCost ν → μ = ν) := by
+  let c := rankPerm (fun x => row (e x)) h
+  have hle (x : Cell μ) : row (c x) ≤ row (e x) := rankCell_row_le _ h x
+  have hsumc : ∑ x, row (c x) = rowCost μ := Equiv.sum_comp c _
+  have hsume : ∑ x, row (e x) = rowCost ν := Equiv.sum_comp e _
+  constructor
+  · rw [←hsumc,←hsume]
+    exact Finset.sum_le_sum (fun x _ => hle x)
+  · intro heq
+    have he : ∀ x, row (c x) = row (e x) := by
+      apply fun x => (Finset.sum_eq_sum_iff_of_le (fun x _ => hle x)).mp
+        (hsumc.trans (heq.trans hsume.symm)) x (Finset.mem_univ _)
+    apply eq_of_row_preserving μ ν (c.symm.trans e)
+    intro x
+    simpa using (he (c.symm x)).symm
+
+lemma swap_relabel {α β : Type*} [DecidableEq α] [DecidableEq β]
+    (e : α ≃ β) (x y : α) : e.permCongr (Equiv.swap x y) = Equiv.swap (e x) (e y) := by
+  ext z
+  simp only [Equiv.permCongr_apply]
+  exact e.injective.map_swap x y (e.symm z) |>.trans (by simp)
+
+lemma sign_vector_rowCost (μ ν : YoungDiagram) (e : Cell μ ≃ Cell ν)
+    (v : Tabloid ν → ℂ) (hv : v ≠ 0)
+    (hs : ∀ c : colGroup μ,
+      tabloidRep ν (e.permCongr c) v = permSign (c : Equiv.Perm (Cell μ)) • v) :
+    rowCost μ ≤ rowCost ν ∧ (rowCost μ = rowCost ν → μ = ν) := by
+  obtain ⟨f,hf⟩ : ∃ f, v f ≠ 0 := by
+    by_contra h
+    push Not at h
+    exact hv (funext h)
+  obtain ⟨p,hp⟩ := f.2
+  apply column_injective_rowCost μ ν (e.trans p)
+  intro x y hcol hrow
+  by_contra hne
+  have hfix : tabloidAct (Equiv.swap (e x) (e y)) f = f := by
+    apply swap_fixes_tabloid
+    rw [hp]
+    exact Fin.ext hrow
+  have heq := congrArg (fun w : Tabloid ν → ℂ => w f)
+    (hs ⟨Equiv.swap x y,swap_mem_colGroup x y hcol⟩)
+  rw [swap_relabel,permSign_swap hne] at heq
+  simp only [tabloidRep_apply,Equiv.swap_inv,hfix,Pi.smul_apply,neg_one_smul] at heq
+  apply hf
+  have h2 : (2:ℂ)*(v f) = 0 := by linear_combination heq
+  exact (mul_eq_zero.mp h2).resolve_left (by norm_num)
+
+lemma polytabloid_col_action (μ : YoungDiagram) (c : colGroup μ) :
+    tabloidRep μ c (polytabloid μ) = permSign (c:Equiv.Perm (Cell μ)) • polytabloid μ :=
+  action_alternator_apply _ _ c _
+
+theorem shapes_eq_of_equivariant_equiv (μ ν : YoungDiagram) (e : Cell μ ≃ Cell ν)
+    (T : space μ ≃ₗ[ℂ] space ν)
+    (hT : ∀ p v, T (representation μ p v) = representation ν (e.permCongr p) (T v)) :
+    μ = ν := by
+  let u : space μ := ⟨polytabloid μ,polytabloid_mem_space μ⟩
+  have hu : u ≠ 0 := fun h => polytabloid_ne_zero μ (congrArg Subtype.val h)
+  have hTcol (c : colGroup μ) :
+      tabloidRep ν (e.permCongr c) (T u) = permSign (c:Equiv.Perm (Cell μ)) • (T u : Tabloid ν → ℂ) := by
+    change (representation ν (e.permCongr c) (T u) : Tabloid ν → ℂ) = _
+    rw [←hT]
+    have hh : representation μ c u = permSign (c:Equiv.Perm (Cell μ)) • u :=
+      Subtype.ext (polytabloid_col_action μ c)
+    rw [hh,map_smul]
+    rfl
+  have hfwd := sign_vector_rowCost μ ν e (T u)
+    (fun h => hu (T.injective (by apply Subtype.ext; simpa using h))) hTcol
+  let w : space ν := ⟨polytabloid ν,polytabloid_mem_space ν⟩
+  have hw : w ≠ 0 := fun h => polytabloid_ne_zero ν (congrArg Subtype.val h)
+  have hTi (p : Equiv.Perm (Cell ν)) (v : space ν) :
+      T.symm (representation ν p v) = representation μ (e.symm.permCongr p) (T.symm v) := by
+    apply T.injective
+    rw [T.apply_symm_apply,hT,T.apply_symm_apply]
+    have hp : e.permCongr (e.symm.permCongr p) = p := by
+      apply Equiv.ext
+      intro x
+      simp [Equiv.permCongr_apply]
+    rw [hp]
+  have hTicol (c : colGroup ν) :
+      tabloidRep μ (e.symm.permCongr c) (T.symm w) =
+        permSign (c:Equiv.Perm (Cell ν)) • (T.symm w : Tabloid μ → ℂ) := by
+    change (representation μ (e.symm.permCongr c) (T.symm w) : Tabloid μ → ℂ) = _
+    rw [←hTi]
+    have hh : representation ν c w = permSign (c:Equiv.Perm (Cell ν)) • w :=
+      Subtype.ext (polytabloid_col_action ν c)
+    rw [hh,map_smul]
+    rfl
+  have hbwd := sign_vector_rowCost ν μ e.symm (T.symm w)
+    (fun h => hw (T.symm.injective (by apply Subtype.ext; simpa using h))) hTicol
+  exact hfwd.2 (Nat.le_antisymm hfwd.1 hbwd.1)
+
+
+noncomputable def cellRowsEquiv (μ : YoungDiagram) :
+    Cell μ ≃ (i : Fin (μ.colLen 0)) × Fin (μ.rowLen i) where
+  toFun x := ⟨rowIndex x,⟨col x,YoungDiagram.mem_iff_lt_rowLen.mp x.2⟩⟩
+  invFun p := ⟨(p.1,p.2),YoungDiagram.mem_iff_lt_rowLen.mpr p.2.2⟩
+  left_inv x := by apply Subtype.ext; rfl
+  right_inv p := by rcases p with ⟨⟨i,hi⟩,⟨j,hj⟩⟩; rfl
+
+lemma card_eq_sum_rowLen (μ : YoungDiagram) : μ.card = ∑ i : Fin (μ.colLen 0), μ.rowLen i := by
+  rw [←card_cell,Fintype.card_congr (cellRowsEquiv μ),Fintype.card_sigma]
+  simp only [Fintype.card_fin]
+
+lemma sum_rowLens (μ : YoungDiagram) : μ.rowLens.sum = μ.card := by
+  rw [card_eq_sum_rowLen,YoungDiagram.rowLens]
+  rw [Fin.sum_univ_eq_sum_range]
+  have hh (n : ℕ) : (List.map μ.rowLen (List.range n)).sum = ∑ i ∈ Finset.range n, μ.rowLen i := by
+    induction n with
+    | zero => simp
+    | succ n ih => simp [List.range_succ,List.map_append,Finset.sum_range_succ,ih]
+  exact hh _
+
+noncomputable def diagramPartition (μ : YoungDiagram) : Nat.Partition μ.card where
+  parts := μ.rowLens
+  parts_pos := fun {i} hi => μ.pos_of_mem_rowLens i (by simpa using hi)
+  parts_sum := by simp
+
+noncomputable def partitionDiagram {N : ℕ} (p : Nat.Partition N) : YoungDiagram :=
+  YoungDiagram.ofRowLens (p.parts.sort (· ≥ ·)) (Multiset.pairwise_sort _ _).sortedGE
+
+lemma partitionDiagram_rowLens {N : ℕ} (p : Nat.Partition N) :
+    (partitionDiagram p).rowLens = p.parts.sort (· ≥ ·) := by
+  apply YoungDiagram.rowLens_ofRowLens_eq_self
+  intro x hx
+  exact p.parts_pos ((Multiset.mem_sort _).mp hx)
+
+lemma partitionDiagram_card {N : ℕ} (p : Nat.Partition N) : (partitionDiagram p).card = N := by
+  rw [←sum_rowLens,partitionDiagram_rowLens]
+  change (↑(p.parts.sort (· ≥ ·)) : Multiset ℕ).sum = N
+  rw [Multiset.sort_eq,p.parts_sum]
+
+lemma partitionDiagram_diagramPartition (μ : YoungDiagram) :
+    partitionDiagram (diagramPartition μ) = μ := by
+  apply YoungDiagram.equivListRowLens.injective
+  apply Subtype.ext
+  change (partitionDiagram (diagramPartition μ)).rowLens = μ.rowLens
+  rw [partitionDiagram_rowLens]
+  change (↑μ.rowLens : Multiset ℕ).sort (· ≥ ·) = μ.rowLens
+  rw [Multiset.coe_sort,List.mergeSort_eq_self _ μ.rowLens_sorted.pairwise]
+
+abbrev Shapes (N : ℕ) := {μ : YoungDiagram // μ.card = N}
+
+end Thorp.Specht
+
+end ThorpNine.SparseSaving
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/SparseSaving/ModifiedRoutingMoment.lean b/lean/OAI/Probability/ThorpRouting/SparseSaving/ModifiedRoutingMoment.lean
index 74bea64cc7820a2009ac790ee66641f360ef1054..20afd8bd0576a52fe286009b1ad8cfc75b1e2eb0 100644
--- a/lean/OAI/Probability/ThorpRouting/SparseSaving/ModifiedRoutingMoment.lean
+++ b/lean/OAI/Probability/ThorpRouting/SparseSaving/ModifiedRoutingMoment.lean
@@ -1,858 +1,858 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.SparseSaving.InputTree
-
-namespace OAI
-
-namespace ThorpNine.SparseSaving
-
-namespace Thorp
-
-noncomputable def modifiedRoutingMoment (L r : ℕ) (E : Finset (Card r))
-    {ι : Type*} [Fintype ι] (e f : ι ↪ Card (L+r)) (a : ℝ) : ℝ := by
-  classical
-  exact finiteMean (fun ω : ModifiedCoins L r =>
-    if ∀ i, modifiedPerm L r E ω (e i) = f i then (2:ℝ)^((modifiedCost L r E e ω:ℝ)*a) else 0)
-
-lemma modifiedRoutingMoment_nonneg (L r : ℕ) (E : Finset (Card r))
-    {ι : Type*} [Fintype ι] (e f : ι ↪ Card (L+r)) (a : ℝ) :
-    0 ≤ modifiedRoutingMoment L r E e f a := by
-  classical
-  exact finiteMean_nonneg (fun _ => by split_ifs <;> positivity)
-
-lemma modifiedRoutingMoment_zero (L : ℕ) (E : Finset (Card 0))
-    {ι : Type*} [Fintype ι] (e f : ι ↪ Card L) (a : ℝ) :
-    modifiedRoutingMoment L 0 E e f a = PairRouting.tupleProbability (modifiedPerm L 0 E) e f := by
-  simp only [modifiedRoutingMoment,modifiedCost_zero,Nat.cast_zero,zero_mul,Real.rpow_zero,
-    PairRouting.tupleProbability]
-
-lemma modified_pointwise_zero (L : ℕ) (E : Finset (Card 0))
-    {ι : Type*} [Fintype ι] (e f : ι ↪ Card L) (he : Function.Surjective e) (a : ℝ) (ha : 0 ≤ a) :
-    (PairRouting.tupleProbability (modifiedPerm L 0 E) e f)^(1+a) ≤
-      centralFactor L 0 E a * modifiedRoutingMoment L 0 E e f a := by
-  have hp := PairRouting.tupleProbability_nonneg (modifiedPerm L 0 E) e f
-  rw [modifiedRoutingMoment_zero]
-  have hbound := modified_probability_zero_le L E e f he
-  have hpow := Real.rpow_le_rpow hp hbound ha
-  have hfpos : (0:ℝ) < Nat.factorial (2^L) := Nat.cast_pos.mpr (Nat.factorial_pos _)
-  have heq : (if (fun j => Fin.elim0 j : Card 0) ∈ E then (1:ℝ) else 1/(Nat.factorial (2^L):ℝ))^a =
-      centralFactor L 0 E a := by
-    rw [centralFactor_zero]
-    split_ifs
-    · exact Real.one_rpow _
-    · rw [one_div,Real.inv_rpow (le_of_lt hfpos),Real.rpow_neg (le_of_lt hfpos)]
-  rw [heq] at hpow
-  rw [Real.rpow_add' hp (by linarith : (1+a:ℝ) ≠ 0),Real.rpow_one,mul_comm]
-  exact mul_le_mul_of_nonneg_right hpow hp
-
-
-lemma modified_matches_step (L r : ℕ) (E : Finset (Card (r+1))) {ι : Type*}
-    (e f : ι ↪ Card (L+r+1)) (ω : ModifiedCoins L r × ModifiedCoins L r)
-    (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
-    (∀ i, modifiedPerm L (r+1) E ((modifiedStepEquiv L r).symm (ω,coins)) (e i) = f i) ↔
-    (∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1))
-      ((e.trans (headTailEquiv (L+r)).toEmbedding) i) = ((f.trans (headTailEquiv (L+r)).toEmbedding) i)) := by
-  apply forall_congr'
-  intro i
-  rw [←(headTailEquiv (L+r)).injective.eq_iff,modifiedPerm_coordinates]
-  rfl
-
-lemma modifiedRoutingMoment_step (L r : ℕ) (E : Finset (Card (r+1))) {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (e f : ι ↪ Card (L+r+1)) (a : ℝ) :
-    let x := e.trans (headTailEquiv (L+r)).toEmbedding
-    let y := f.trans (headTailEquiv (L+r)).toEmbedding
-    modifiedRoutingMoment L (r+1) E e f a = PairRouting.outerWeight x y *
-      ∑ c : {c // c ∈ PairRouting.compatibleSet x y},
-        finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
-          if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c.val
-          then (2:ℝ)^((modifiedAssignmentCost L r E x c.val
-            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0) := by
-  classical
-  intro x y
-  let : Fintype (ModifiedCoins L r) := inferInstance
-  rw [modifiedRoutingMoment,finiteMean_equiv (modifiedStepEquiv L r),finiteMean_prod]
-  have he (ω : ModifiedCoins L r × ModifiedCoins L r) :
-      finiteMean (fun coins : (Card (L+r) → Bool) × (Card (L+r) → Bool) =>
-        if ∀ i, modifiedPerm L (r+1) E ((modifiedStepEquiv L r).symm (ω,coins)) (e i) = f i
-        then (2:ℝ)^((modifiedCost L (r+1) E e ((modifiedStepEquiv L r).symm (ω,coins)):ℝ)*a) else 0) =
-      PairRouting.outerWeight x y *
-        ∑ c : {c // c ∈ PairRouting.compatibleSet x y},
-          if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c.val
-          then (2:ℝ)^((modifiedAssignmentCost L r E x c.val
-            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0 := by
-    let W : (ι → Bool) → ℝ := fun c =>
-      if hc : PairRouting.Compatible x c then (2:ℝ)^((modifiedAssignmentCost L r E x c hc ω:ℝ)*a) else 0
-    calc
-      _ = finiteMean (fun coins : (Card (L+r) → Bool) × (Card (L+r) → Bool) =>
-        if ∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1))
-          (x i) = y i then W (PairRouting.colors x coins.1) else 0) := by
-        apply finiteMean_congr
-        intro coins
-        simp only [modified_matches_step,modifiedCost_step]
-        simp only [W,PairRouting.colors_compatible,↓reduceDIte]
-        rfl
-      _ = _ := by
-        rw [PairRouting.sandwich_weighted_probability_subtype]
-        congr 1
-        apply Finset.sum_congr rfl
-        intro c _
-        simp only [W,((PairRouting.mem_compatibleSet x y c.val).mp c.property).1,↓reduceDIte]
-  simp_rw [he]
-  simp_rw [mul_comm (PairRouting.outerWeight x y)]
-  rw [finiteMean_mul_const,finiteMean_sum,mul_comm]
-
-lemma modified_child_moment_product (L r : ℕ) (E : Finset (Card (r+1))) {ι : Type*} [Fintype ι]
-    (x y : ι ↪ Bool × Card (L+r)) (c : ι → Bool)
-    (hcx : PairRouting.Compatible x c) (hcy : PairRouting.Compatible y c) (a : ℝ) :
-    modifiedRoutingMoment L r (exceptionChild E false) (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a *
-      modifiedRoutingMoment L r (exceptionChild E true) (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a =
-    finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
-      if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c
-      then (2:ℝ)^(((modifiedCost L r (exceptionChild E false) (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
-        modifiedCost L r (exceptionChild E true) (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0) := by
-  classical
-  unfold modifiedRoutingMoment
-  rw [←finiteMean_prod_mul]
-  apply finiteMean_congr
-  intro ω
-  let M₀ := ∀ i : {i // c i = false}, modifiedPerm L r (exceptionChild E false) ω.1 (x i.val).2 = (y i.val).2
-  let M₁ := ∀ i : {i // c i = true}, modifiedPerm L r (exceptionChild E true) ω.2 (x i.val).2 = (y i.val).2
-  have hp : (fun b : Bool => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) =
-      (fun b => if b then modifiedPerm L r (exceptionChild E true) ω.2 else modifiedPerm L r (exceptionChild E false) ω.1) := by
-    funext b
-    cases b <;> rfl
-  have hs : PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c ↔
-      M₀ ∧ M₁ := by rw [hp]; exact PairRouting.childMatches_split _ _ _ _ _
-  change (if M₀ then _ else 0) * (if M₁ then _ else 0) = _
-  by_cases h₀ : M₀ <;> by_cases h₁ : M₁
-  · simp only [hs,h₀,h₁,true_and,↓reduceIte]
-    rw [←Real.rpow_add (by norm_num : (0:ℝ)<2)]
-    congr 1
-    ring
-  · simp only [hs,h₀,h₁,and_false,↓reduceIte,mul_zero]
-  · simp only [hs,h₀,h₁,false_and,↓reduceIte,zero_mul]
-  · simp only [hs,h₀,h₁,false_and,↓reduceIte,mul_zero]
-
-lemma modified_probability_step (L r : ℕ) (E : Finset (Card (r+1))) {ι : Type*} [Fintype ι]
-    (e f : ι ↪ Card (L+r+1)) :
-    PairRouting.tupleProbability (modifiedPerm L (r+1) E) e f =
-    finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
-      finiteMean (fun coins : (Card (L+r) → Bool) × (Card (L+r) → Bool) =>
-        if ∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => if b then modifiedPerm L r (exceptionChild E true) ω.2 else modifiedPerm L r (exceptionChild E false) ω.1)
-          ((e.trans (headTailEquiv (L+r)).toEmbedding) i) = ((f.trans (headTailEquiv (L+r)).toEmbedding) i)
-        then (1:ℝ) else 0)) := by
-  classical
-  let : Fintype (ModifiedCoins L r) := inferInstance
-  rw [PairRouting.tupleProbability,finiteMean_equiv (modifiedStepEquiv L r),finiteMean_prod]
-  have hp (b : Bool) (p q : ModifiedCoins L r) :
-      modifiedPerm L r (exceptionChild E b) (if b then p else q) =
-      (if b then modifiedPerm L r (exceptionChild E true) p else modifiedPerm L r (exceptionChild E false) q) := by
-    cases b <;> rfl
-  simp only [modified_matches_step,hp]
-
-
-lemma modified_assignment_moment_bound (L r : ℕ) (E : Finset (Card (r+1))) {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (x y : ι ↪ Bool × Card (L+r)) (c : ι → Bool) (hc : c ∈ PairRouting.compatibleSet x y)
-    (a : ℝ) (ha : 0 ≤ a) :
-    let hcx := ((PairRouting.mem_compatibleSet x y c).mp hc).1
-    PairRouting.outerWeight x y ^ a * ((PairRouting.compatibleSet x y).card:ℝ)^a *
-      ((2:ℝ)^(-(r:ℝ)*Fintype.card ι*a) *
-        finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
-          if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c
-          then (2:ℝ)^(((modifiedCost L r (exceptionChild E false) (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
-            modifiedCost L r (exceptionChild E true) (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0)) ≤
-    (2:ℝ)^(-((r:ℝ)+1)*Fintype.card ι*a) *
-      finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
-        if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c
-        then (2:ℝ)^((modifiedAssignmentCost L r E x c hcx ω:ℝ)*a) else 0) := by
-  classical
-  intro hcx
-  have hscale (q : ℝ) (F : ModifiedCoins L r × ModifiedCoins L r → ℝ) :
-      q * finiteMean F = finiteMean (fun ω => q * F ω) := by
-    simp_rw [mul_comm q,finiteMean_mul_const]
-  rw [←mul_assoc,hscale,hscale]
-  apply finiteMean_mono
-  intro ω
-  split_ifs with hm
-  · have hh := PairRouting.assignment_factor_bound x y c hc
-      (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) hm a ha
-    calc
-      _ ≤ ((2:ℝ)^(((PairRouting.alternatingCycles (PairRouting.coloredEmbedding x c hcx)
-          (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)):ℝ)-Fintype.card ι)*a) *
-          (2:ℝ)^(-(r:ℝ)*Fintype.card ι*a)) *
-          (2:ℝ)^(((modifiedCost L r (exceptionChild E false) (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
-            modifiedCost L r (exceptionChild E true) (PairRouting.childEmbedding x c hcx true) ω.2)*a) :=
-        mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hh (by positivity)) (by positivity)
-      _ = _ := by
-        simp only [←Real.rpow_add (by norm_num : (0:ℝ)<2),modifiedAssignmentCost,Nat.cast_add]
-        congr 1
-        ring
-  · simp only [mul_zero]
-    exact le_rfl
-
-
-lemma modified_child_power_product_bound (L r : ℕ) (E : Finset (Card (r+1)))
-    {ι : Type*} [Fintype ι] (x y : ι ↪ Bool × Card (L+r)) (c : ι → Bool)
-    (hcx : PairRouting.Compatible x c) (hcy : PairRouting.Compatible y c) (a : ℝ)
-    (hfalse : (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E false))
-        (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false))^(1+a) ≤
-      ((2:ℝ)^(-(r:ℝ)*Fintype.card {i // c i = false}*a)*centralFactor L r (exceptionChild E false) a) *
-        modifiedRoutingMoment L r (exceptionChild E false)
-          (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a)
-    (htrue : (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E true))
-        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true))^(1+a) ≤
-      ((2:ℝ)^(-(r:ℝ)*Fintype.card {i // c i = true}*a)*centralFactor L r (exceptionChild E true) a) *
-        modifiedRoutingMoment L r (exceptionChild E true)
-          (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a) :
-    (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E false))
-      (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false))^(1+a) *
-      (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E true))
-        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true))^(1+a) ≤
-    ((2:ℝ)^(-(r:ℝ)*Fintype.card ι*a)*centralFactor L (r+1) E a) *
-      finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
-        if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c
-        then (2:ℝ)^(((modifiedCost L r (exceptionChild E false) (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
-          modifiedCost L r (exceptionChild E true) (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0) := by
-  have hh := mul_le_mul hfalse htrue
-    (Real.rpow_nonneg (PairRouting.tupleProbability_nonneg _ _ _) _)
-    (mul_nonneg (mul_nonneg (by positivity) (centralFactor_nonneg _ _ _ _))
-      (modifiedRoutingMoment_nonneg _ _ _ _ _ _))
-  have he : -(r:ℝ)*Fintype.card {i // c i = false}*a + -(r:ℝ)*Fintype.card {i // c i = true}*a =
-      -(r:ℝ)*Fintype.card ι*a := by
-    rw [←add_mul,←mul_add,←Nat.cast_add,PairRouting.color_card_add]
-  calc
-    _ ≤ _ := hh
-    _ = ((2:ℝ)^(-(r:ℝ)*Fintype.card {i // c i = false}*a)*
-        (2:ℝ)^(-(r:ℝ)*Fintype.card {i // c i = true}*a)) *
-      (centralFactor L r (exceptionChild E false) a * centralFactor L r (exceptionChild E true) a) *
-      (modifiedRoutingMoment L r (exceptionChild E false)
-        (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a *
-      modifiedRoutingMoment L r (exceptionChild E true)
-        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a) := by ring
-    _ = _ := by rw [←Real.rpow_add (by norm_num : (0:ℝ)<2),he,←centralFactor_step,modified_child_moment_product]
-
-
-theorem modified_pointwise_moment (L r : ℕ) (E : Finset (Card r)) {ι : Type*} [Fintype ι]
-    (e f : ι ↪ Card (L+r)) (he : Function.Surjective e) (a : ℝ) (ha : 0 ≤ a) :
-    (PairRouting.tupleProbability (modifiedPerm L r E) e f)^(1+a) ≤
-      ((2:ℝ)^(-(r:ℝ)*Fintype.card ι*a)*centralFactor L r E a) * modifiedRoutingMoment L r E e f a := by
-  classical
-  induction r generalizing ι with
-  | zero =>
-      simpa only [Nat.cast_zero,neg_zero,zero_mul,Real.rpow_zero,one_mul] using
-        modified_pointwise_zero L E e f he a ha
-  | succ r ih =>
-      let x := e.trans (headTailEquiv (L+r)).toEmbedding
-      let y := f.trans (headTailEquiv (L+r)).toEmbedding
-      let W := PairRouting.outerWeight x y
-      let C := PairRouting.compatibleSet x y
-      let p : (ι → Bool) → ℝ := fun c =>
-        (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E false))
-          (fun i : {i // c i = false} => (x i.val).2)
-          (fun i : {i // c i = false} => (y i.val).2))^(1+a) *
-        (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E true))
-          (fun i : {i // c i = true} => (x i.val).2)
-          (fun i : {i // c i = true} => (y i.val).2))^(1+a)
-      let m : {c // c ∈ C} → ℝ := fun c =>
-        finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
-          if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c.val
-          then (2:ℝ)^((modifiedAssignmentCost L r E x c.val
-            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0)
-      have hp (c : {c // c ∈ C}) : W^a * (C.card:ℝ)^a * p c.val ≤
-          ((2:ℝ)^(-((r:ℝ)+1)*Fintype.card ι*a)*centralFactor L (r+1) E a) * m c := by
-        obtain ⟨hcx,hcy⟩ := (PairRouting.mem_compatibleSet x y c.val).mp c.property
-        have h₀ := ih (exceptionChild E false) (PairRouting.childEmbedding x c.val hcx false)
-          (PairRouting.childEmbedding y c.val hcy false)
-          (PairRouting.childEmbedding_surjective x ((headTailEquiv (L+r)).surjective.comp he) c.val hcx false)
-        have h₁ := ih (exceptionChild E true) (PairRouting.childEmbedding x c.val hcx true)
-          (PairRouting.childEmbedding y c.val hcy true)
-          (PairRouting.childEmbedding_surjective x ((headTailEquiv (L+r)).surjective.comp he) c.val hcx true)
-        have hh := modified_child_power_product_bound L r E x y c.val hcx hcy a h₀ h₁
-        have hh' := mul_le_mul_of_nonneg_left hh
-          (mul_nonneg (Real.rpow_nonneg (le_of_lt (PairRouting.outerWeight_pos x y)) a)
-            (Real.rpow_nonneg (Nat.cast_nonneg C.card) a))
-        apply hh'.trans
-        have hm := modified_assignment_moment_bound L r E x y c.val c.property a ha
-        convert mul_le_mul_of_nonneg_left hm (centralFactor_nonneg L (r+1) E a) using 1 <;> ring
-      have hnode := PairRouting.sandwich_power_bound (modifiedPerm L r (exceptionChild E false)) (modifiedPerm L r (exceptionChild E true)) x y a ha
-      rw [←modified_probability_step L r E e f] at hnode
-      change (PairRouting.tupleProbability (modifiedPerm L (r+1) E) e f)^(1+a) ≤
-        W^(1+a) * (C.card:ℝ)^a * ∑ c ∈ C, p c at hnode
-      have hsum : ∑ c ∈ C, p c = ∑ c : {c // c ∈ C}, p c.val := by
-        rw [Finset.sum_subtype C (fun _ => Iff.rfl)]
-      have hW : 0 < W := PairRouting.outerWeight_pos x y
-      calc
-        _ ≤ W^(1+a) * (C.card:ℝ)^a * ∑ c ∈ C, p c := hnode
-        _ = W * ∑ c : {c // c ∈ C}, W^a * (C.card:ℝ)^a * p c.val := by
-          rw [Real.rpow_add hW,Real.rpow_one,hsum,←Finset.mul_sum]
-          ring
-        _ ≤ W * ∑ c : {c // c ∈ C}, ((2:ℝ)^(-((r:ℝ)+1)*Fintype.card ι*a)*centralFactor L (r+1) E a) * m c :=
-          mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun c _ => hp c)) (le_of_lt hW)
-        _ = _ := by
-          rw [←Finset.mul_sum,modifiedRoutingMoment_step]
-          simp only [Nat.cast_add,Nat.cast_one]
-          change W * (((2:ℝ)^(-((r:ℝ)+1)*Fintype.card ι*a)*centralFactor L (r+1) E a) * ∑ c, m c) =
-            ((2:ℝ)^(-((r:ℝ)+1)*Fintype.card ι*a)*centralFactor L (r+1) E a) * (W * ∑ c, m c)
-          ring
-
-
-lemma modifiedRoutingMoment_sum (L r : ℕ) (E : Finset (Card r)) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (L+r)) (a : ℝ) :
-    (∑ f : ι ↪ Card (L+r), modifiedRoutingMoment L r E e f a) =
-      finiteMean (fun ω : ModifiedCoins L r => (2:ℝ)^((modifiedCost L r E e ω:ℝ)*a)) := by
-  classical
-  unfold modifiedRoutingMoment
-  rw [←finiteMean_sum]
-  apply finiteMean_congr
-  intro ω
-  let f₀ : ι ↪ Card (L+r) := e.trans (modifiedPerm L r E ω).toEmbedding
-  rw [Finset.sum_eq_single f₀]
-  · have hm : ∀ i, modifiedPerm L r E ω (e i) = f₀ i := fun _ => rfl
-    simp only [hm,implies_true,↓reduceIte]
-  · intro f _ hf
-    have hm : ¬ ∀ i, modifiedPerm L r E ω (e i) = f i := by
-      intro hh
-      apply hf
-      ext i : 1
-      exact (hh i).symm
-    exact ite_eq_right hm
-  · simp
-
-noncomputable def modifiedRowMoment (L r : ℕ) (E : Finset (Card r)) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (L+r)) (a : ℝ) : ℝ :=
-  finiteMean (fun f : ι ↪ Card (L+r) =>
-    ((Fintype.card (ι ↪ Card (L+r)):ℝ)*PairRouting.tupleProbability (modifiedPerm L r E) e f)^(1+a))
-
-lemma full_tuple_card (d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d)
-    (he : Function.Surjective e) : Fintype.card ι = 2^d := by
-  rw [←card_positions d]
-  exact Fintype.card_congr (Equiv.ofBijective e ⟨e.injective,he⟩)
-
-lemma exception_card_le (r : ℕ) (E : Finset (Card r)) : E.card ≤ 2^r := by
-  simpa only [card_positions] using E.card_le_univ
-
-noncomputable def modifiedNormalization (L r : ℕ) (E : Finset (Card r)) : ℝ :=
-  (Nat.factorial (2^(L+r)):ℝ) /
-    (((2:ℝ)^r)^(2^(L+r)) * (Nat.factorial (2^L):ℝ)^(2^r-E.card))
-
-lemma modifiedNormalization_pos (L r : ℕ) (E : Finset (Card r)) :
-    0 < modifiedNormalization L r E := by unfold modifiedNormalization; positivity
-
-lemma modified_normalization_identity (L r : ℕ) (E : Finset (Card r)) (a : ℝ) :
-    (Nat.factorial (2^(L+r)):ℝ)^a *
-      ((2:ℝ)^(-(r:ℝ)*(2^(L+r):ℕ)*a)*centralFactor L r E a) =
-    modifiedNormalization L r E ^ a := by
-  rw [←mul_assoc,density_normalization,centralFactor,modifiedNormalization,
-    Real.div_rpow (by positivity) (by positivity),Real.div_rpow (by positivity) (by positivity),
-    Real.mul_rpow (by positivity) (by positivity)]
-  have hc : ((2^r-E.card:ℕ):ℝ) = (2:ℝ)^r-E.card := by
-    rw [Nat.cast_sub (exception_card_le r E),Nat.cast_pow,Nat.cast_ofNat]
-  rw [←Real.rpow_natCast (Nat.factorial (2^L):ℝ),←Real.rpow_mul (by positivity),hc,
-    show -((2:ℝ)^r-E.card)*a = -(((2:ℝ)^r-E.card)*a) by ring,
-    Real.rpow_neg (by positivity)]
-  ring
-
-theorem modified_row_density_moment (L r : ℕ) (E : Finset (Card r)) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (L+r)) (he : Function.Surjective e) (a : ℝ) (ha : 0 ≤ a) :
-    modifiedRowMoment L r E e a ≤ modifiedNormalization L r E ^ a *
-      finiteMean (fun ω : ModifiedCoins L r => (2:ℝ)^((modifiedCost L r E e ω:ℝ)*a)) := by
-  classical
-  have : Nonempty (ι ↪ Card (L+r)) := ⟨e⟩
-  rw [modifiedRowMoment,finite_row_power _ (fun f => PairRouting.tupleProbability_nonneg _ _ _)]
-  calc
-    _ ≤ (Fintype.card (ι ↪ Card (L+r)):ℝ)^a *
-      ∑ f : ι ↪ Card (L+r), ((2:ℝ)^(-(r:ℝ)*Fintype.card ι*a)*centralFactor L r E a) *
-        modifiedRoutingMoment L r E e f a :=
-      mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun f _ => modified_pointwise_moment L r E e f he a ha))
-        (Real.rpow_nonneg (Nat.cast_nonneg _) _)
-    _ = _ := by
-      rw [←Finset.mul_sum,modifiedRoutingMoment_sum,←mul_assoc,Fintype.card_embedding_eq,
-        card_positions,full_tuple_card (L+r) e he,Nat.descFactorial_self,modified_normalization_identity]
-
-lemma modified_total_cost_mgf (L r : ℕ) (hL : 0 < L) (E : Finset (Card r))
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) :
-    finiteMean (fun ω : ModifiedCoins L r => momentBase^(modifiedCost L r E e ω)) ≤
-      Real.exp (Fintype.card ι*heightDecay L +
-        3*((L*E.card*2^L:ℕ):ℝ)*Real.log momentBase) := by
-  let B := Real.exp (Fintype.card ι*heightDecay L +
-    3*((L*E.card*2^L:ℕ):ℝ)*Real.log momentBase)
-  rw [finiteMean_prod]
-  apply le_trans (finiteMean_mono (fun U => ?_)) (finiteMean_const B).le
-  rw [finiteMean_prod,finiteMean_comm]
-  apply le_trans (finiteMean_mono (fun X => ?_)) (finiteMean_const B).le
-  exact modified_cost_mgf L r hL E U X e momentBase momentBase_one_le (momentBase_log.trans (by norm_num))
-
-lemma modifiedRowMoment_bound (L r : ℕ) (hL : 0 < L) (E : Finset (Card r))
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (he : Function.Surjective e) :
-    modifiedRowMoment L r E e (1/1000) ≤
-      modifiedNormalization L r E ^ (1/1000:ℝ) *
-        Real.exp ((2^(L+r):ℕ)*heightDecay L +
-          3*((L*E.card*2^L:ℕ):ℝ)*Real.log momentBase) := by
-  have hm := modified_row_density_moment L r E e he (1/1000) (by norm_num)
-  simp_rw [←momentBase_pow] at hm
-  apply hm.trans
-  apply mul_le_mul_of_nonneg_left _ (Real.rpow_nonneg (modifiedNormalization_pos L r E).le _)
-  simpa only [full_tuple_card (L+r) e he] using modified_total_cost_mgf L r hL E e
-
-
-lemma log_factorial_upper {n : ℕ} (hn : 1 ≤ n) :
-    Real.log (Nat.factorial n) ≤ (n:ℝ)*Real.log n-n+Real.log n+1 := by
-  induction n with
-  | zero => omega
-  | succ n ih =>
-      by_cases hz : n = 0
-      · subst n; norm_num
-      have hn0 : (0:ℝ) < n := Nat.cast_pos.mpr (Nat.pos_of_ne_zero hz)
-      have hn1 : (0:ℝ) < n+1 := by positivity
-      have h := Real.one_sub_inv_le_log_of_pos (div_pos hn1 hn0)
-      rw [Real.log_div (ne_of_gt hn1) (ne_of_gt hn0),inv_div] at h
-      have hh := mul_le_mul_of_nonneg_left h hn1.le
-      have hcancel : ((n:ℝ)+1)*(1-(n:ℝ)/(n+1)) = 1 := by field_simp; ring
-      rw [hcancel] at hh
-      have hi := ih (by omega)
-      rw [Nat.factorial_succ,Nat.cast_mul,Real.log_mul (by positivity) (by positivity),Nat.cast_add,Nat.cast_one]
-      nlinarith
-
-lemma log_modifiedNormalization (L r : ℕ) (E : Finset (Card r)) :
-    Real.log (modifiedNormalization L r E) =
-      Real.log (Nat.factorial (2^(L+r))) -
-        ((2^(L+r):ℕ)*((r:ℝ)*Real.log 2)+((2:ℝ)^r-E.card)*Real.log (Nat.factorial (2^L))) := by
-  unfold modifiedNormalization
-  rw [Real.log_div (by positivity) (by positivity),Real.log_mul (by positivity) (by positivity),
-    Real.log_pow,Real.log_pow,Real.log_pow,
-    Nat.cast_sub (exception_card_le r E),Nat.cast_pow,Nat.cast_ofNat]
-  simp only [Nat.cast_pow,Nat.cast_ofNat]
-
-lemma log_modifiedNormalization_le (L r : ℕ) (E : Finset (Card r)) :
-    Real.log (modifiedNormalization L r E) ≤
-      ((L+r:ℕ):ℝ)*Real.log 2+1+E.card*Real.log (Nat.factorial (2^L)) := by
-  have hN : 1 ≤ 2^(L+r) := one_le_pow₀ (by norm_num)
-  have hS : 1 ≤ 2^L := one_le_pow₀ (by norm_num)
-  have hu := log_factorial_upper hN
-  have hl := mul_le_mul_of_nonneg_left (log_factorial_lower hS) (by positivity : (0:ℝ) ≤ (2:ℝ)^r)
-  simp only [Nat.cast_pow,Nat.cast_ofNat,Real.log_pow,Nat.cast_add] at hu hl ⊢
-  rw [log_modifiedNormalization]
-  simp only [pow_add,Nat.cast_mul,Nat.cast_pow,Nat.cast_ofNat] at hu hl ⊢
-  nlinarith
-
-lemma modified_moment_exponential (L r : ℕ) (hL : 0 < L) (E : Finset (Card r))
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (he : Function.Surjective e) :
-    modifiedRowMoment L r E e (1/1000) ≤
-      Real.exp (((L+r:ℕ)*Real.log 2+1+E.card*Real.log (Nat.factorial (2^L)))/1000 +
-        (2^(L+r):ℕ)*heightDecay L+3*((L*E.card*2^L:ℕ):ℝ)*Real.log momentBase) := by
-  apply (modifiedRowMoment_bound L r hL E e he).trans
-  rw [Real.rpow_def_of_pos (modifiedNormalization_pos L r E),←Real.exp_add]
-  apply Real.exp_le_exp.mpr
-  have h := log_modifiedNormalization_le L r E
-  linarith
-
-lemma log_power_factorial_le (L : ℕ) :
-    Real.log (Nat.factorial (2^L)) ≤ (2:ℝ)^L*L := by
-  have hf : (Nat.factorial (2^L):ℝ) ≤ ((2^L:ℕ):ℝ)^(2^L) := by
-    exact_mod_cast Nat.factorial_le_pow (2^L)
-  have hh := Real.log_le_log (by positivity) hf
-  rw [Real.log_pow,Nat.cast_pow,Nat.cast_ofNat,Real.log_pow] at hh
-  apply hh.trans
-  have hl := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
-  have hn : 0 ≤ (2:ℝ)^L*(L:ℝ) := by positivity
-  nlinarith
-
-lemma modified_moment_exception_bound (L r : ℕ) (hL : 0 < L) (E : Finset (Card r))
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (he : Function.Surjective e)
-    (ε : ℝ) (hE : (E.card:ℝ) ≤ ε*(2:ℝ)^r) :
-    modifiedRowMoment L r E e (1/1000) ≤
-      Real.exp (((L+r:ℕ)*Real.log 2+1)/1000 +
-        (2:ℝ)^(L+r)*(heightDecay L+ε*L/250)) := by
-  apply (modified_moment_exponential L r hL E e he).trans
-  apply Real.exp_le_exp.mpr
-  have hs := mul_le_mul_of_nonneg_left (log_power_factorial_le L) (Nat.cast_nonneg E.card)
-  have hb := mul_le_mul_of_nonneg_left momentBase_log
-    (by positivity : (0:ℝ) ≤ 3*((L*E.card*2^L:ℕ):ℝ))
-  have hexc := mul_le_mul_of_nonneg_left hE (by positivity : (0:ℝ) ≤ (2:ℝ)^L*L/250)
-  simp only [Nat.cast_mul,Nat.cast_pow,Nat.cast_ofNat,pow_add] at hs hb hexc ⊢
-  nlinarith
-
-
-lemma heightDecay_tendsto : Filter.Tendsto heightDecay Filter.atTop (nhds 0) := by
-  have h : Filter.Tendsto (fun L : ℕ => (-(1:ℝ)/64)*(L:ℝ)) Filter.atTop Filter.atBot :=
-    (Filter.tendsto_const_mul_atBot_of_neg (by norm_num : -(1:ℝ)/64 < 0)).mpr
-      tendsto_natCast_atTop_atTop
-  have ht := ((Real.tendsto_exp_atBot.comp h).const_mul 68).div_const (1-Real.exp (-(1:ℝ)/64))
-  convert ht using 1
-  · funext L
-    unfold heightDecay
-    dsimp only [Function.comp_apply]
-    rw [show -(L:ℝ)/64 = (-(1:ℝ)/64)*(L:ℝ) by ring]
-  · simp
-
-noncomputable def modifiedOverhead (L r : ℕ) : ℝ :=
-  (((L+r:ℕ):ℝ)*Real.log 2+1)/((2:ℝ)^(L+r)*1000)
-
-lemma modifiedOverhead_tendsto (L : ℕ) :
-    Filter.Tendsto (modifiedOverhead L) Filter.atTop (nhds 0) := by
-  have h₁ := tendsto_pow_const_div_const_pow_of_one_lt 1 (by norm_num : (1:ℝ)<2)
-  have h₀ := tendsto_pow_const_div_const_pow_of_one_lt 0 (by norm_num : (1:ℝ)<2)
-  simp only [pow_one] at h₁
-  simp only [pow_zero] at h₀
-  have ht := ((h₁.const_mul (Real.log 2)).add (h₀.const_mul ((L:ℝ)*Real.log 2+1))).div_const
-    ((2:ℝ)^L*1000)
-  convert ht using 1
-  · funext r
-    simp only [modifiedOverhead,Nat.cast_add,pow_add]
-    field_simp
-    ring
-  · simp
-
-theorem modified_arbitrarily_small_rate (η : ℝ) (hη : 0 < η) :
-    ∃ L : ℕ, 0 < L ∧ ∃ ε : ℝ, 0 < ε ∧ ε < 1/2 ∧
-      ∀ᶠ r : ℕ in Filter.atTop, ∀ (E : Finset (Card r)), (E.card:ℝ) ≤ ε*(2:ℝ)^r →
-        ∀ (e : Card (L+r) ↪ Card (L+r)),
-          modifiedRowMoment L r E e (1/1000) ≤ Real.exp (η*(2:ℝ)^(L+r)) := by
-  have hsmall : ∀ᶠ L : ℕ in Filter.atTop, heightDecay L < η/3 :=
-    heightDecay_tendsto.eventually (gt_mem_nhds (by linarith : (0:ℝ)<η/3))
-  obtain ⟨L,hdec,hL⟩ := (hsmall.and (Filter.eventually_ge_atTop 1)).exists
-  have hL0 : (0:ℝ) < L := Nat.cast_pos.mpr hL
-  let ε : ℝ := min (1/4) (η*250/(3*L))
-  have hε : 0 < ε := lt_min (by norm_num) (by positivity)
-  have hε1 : ε < 1/2 := (min_le_left _ _).trans_lt (by norm_num)
-  have hεL : ε*L/250 ≤ η/3 := by
-    have h := min_le_right (1/4:ℝ) (η*250/(3*L))
-    change ε ≤ η*250/(3*L) at h
-    have hh := mul_le_mul_of_nonneg_right h hL0.le
-    have he : (η*250/(3*(L:ℝ)))*(L:ℝ)/250 = η/3 := by field_simp
-    exact (div_le_div_of_nonneg_right hh (by norm_num)).trans he.le
-  refine ⟨L,hL,ε,hε,hε1,?_⟩
-  have hr : ∀ᶠ r : ℕ in Filter.atTop, modifiedOverhead L r < η/3 :=
-    (modifiedOverhead_tendsto L).eventually (gt_mem_nhds (by linarith : (0:ℝ)<η/3))
-  filter_upwards [hr] with r hr
-  intro E hE e
-  have he := Finite.surjective_of_injective e.injective
-  apply (modified_moment_exception_bound L r hL E e he ε hE).trans
-  apply Real.exp_le_exp.mpr
-  have hn : (0:ℝ) < (2:ℝ)^(L+r) := by positivity
-  have hd : modifiedOverhead L r * (2:ℝ)^(L+r) =
-      (((L+r:ℕ):ℝ)*Real.log 2+1)/1000 := by
-    unfold modifiedOverhead
-    field_simp
-  have hh := mul_le_mul_of_nonneg_right hr.le hn.le
-  rw [hd] at hh
-  have ht := mul_le_mul_of_nonneg_left (show heightDecay L+ε*L/250 ≤ 2*η/3 by linarith) hn.le
-  linarith
-
-
-namespace DensityTransfer
-open scoped BigOperators
-
-noncomputable def applyKernel {X : Type*} [Fintype X] (P : X → X → ℝ)
-    (v : EuclideanSpace ℂ X) : EuclideanSpace ℂ X :=
-  WithLp.toLp 2 (fun x => ∑ y, (P x y : ℂ)*v y)
-
-lemma applyKernel_apply {X : Type*} [Fintype X] (P : X → X → ℝ)
-    (v : EuclideanSpace ℂ X) (x : X) :
-    applyKernel P v x = ∑ y, (P x y : ℂ)*v y := rfl
-
-lemma applyKernel_add {X : Type*} [Fintype X] (P Q : X → X → ℝ)
-    (v : EuclideanSpace ℂ X) :
-    applyKernel (fun x y => P x y+Q x y) v = applyKernel P v+applyKernel Q v := by
-  ext x
-  simp only [applyKernel_apply,Complex.ofReal_add,add_mul,Finset.sum_add_distrib]
-  rfl
-
-lemma kernel_component_sq {X : Type*} [Fintype X] (P : X → X → ℝ)
-    (hP : ∀ x y, 0 ≤ P x y) (v : EuclideanSpace ℂ X) (x : X) :
-    ‖applyKernel P v x‖^2 ≤ (∑ y, P x y) * ∑ y, P x y*‖v y‖^2 := by
-  have hs : ‖applyKernel P v x‖ ≤ ∑ y, P x y*‖v y‖ := by
-    rw [applyKernel_apply]
-    apply (norm_sum_le _ _).trans
-    apply Finset.sum_le_sum
-    intro y _
-    rw [norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (hP x y)]
-  apply (pow_le_pow_left₀ (norm_nonneg _) hs 2).trans
-  exact Finset.sum_sq_le_sum_mul_sum_of_sq_le_mul Finset.univ
-    (fun y _ => hP x y) (fun y _ => mul_nonneg (hP x y) (sq_nonneg _))
-    (fun y _ => by ring_nf; exact le_rfl)
-
-lemma schur_bound {X : Type*} [Fintype X] (P : X → X → ℝ)
-    (hP : ∀ x y, 0 ≤ P x y) (δ : ℝ) (hδ : 0 ≤ δ)
-    (hrow : ∀ x, ∑ y, P x y ≤ δ) (hcol : ∀ y, ∑ x, P x y ≤ δ)
-    (v : EuclideanSpace ℂ X) : ‖applyKernel P v‖ ≤ δ*‖v‖ := by
-  have hsq : ‖applyKernel P v‖^2 ≤ δ^2*‖v‖^2 := by
-    rw [EuclideanSpace.norm_sq_eq,EuclideanSpace.norm_sq_eq]
-    calc
-      _ ≤ ∑ x, δ * ∑ y, P x y*‖v y‖^2 := by
-        apply Finset.sum_le_sum
-        intro x _
-        apply (kernel_component_sq P hP v x).trans
-        exact mul_le_mul_of_nonneg_right (hrow x)
-          (Finset.sum_nonneg (fun y _ => mul_nonneg (hP x y) (sq_nonneg _)))
-      _ = δ * ∑ y, (∑ x, P x y)*‖v y‖^2 := by
-        rw [←Finset.mul_sum,Finset.sum_comm]
-        congr 1
-        apply Finset.sum_congr rfl
-        intro y _
-        exact (Finset.sum_mul Finset.univ _ _).symm
-      _ ≤ δ * ∑ y, δ*‖v y‖^2 := by
-        apply mul_le_mul_of_nonneg_left _ hδ
-        exact Finset.sum_le_sum (fun y _ => mul_le_mul_of_nonneg_right (hcol y) (sq_nonneg _))
-      _ = _ := by rw [←Finset.mul_sum]; ring
-  have hn := norm_nonneg (applyKernel P v)
-  have hr := mul_nonneg hδ (norm_nonneg v)
-  nlinarith
-
-lemma hilbert_schmidt_family {X I : Type*} [Fintype X] [Fintype I]
-    (P : X → X → ℝ) (v : I → EuclideanSpace ℂ X) (hv : Orthonormal ℂ v) :
-    (∑ i, ‖applyKernel P (v i)‖^2) ≤ ∑ x, ∑ y, (P x y)^2 := by
-  simp_rw [EuclideanSpace.norm_sq_eq]
-  rw [Finset.sum_comm]
-  apply Finset.sum_le_sum
-  intro x _
-  let w : EuclideanSpace ℂ X := WithLp.toLp 2 (fun y => (P x y : ℂ))
-  have he (i : I) : ‖applyKernel P (v i) x‖ = ‖inner ℂ (v i) w‖ := by
-    rw [norm_inner_symm]
-    congr 1
-    simp only [applyKernel_apply,PiLp.inner_apply]
-    apply Finset.sum_congr rfl
-    intro y _
-    simp [w,RCLike.inner_apply,mul_comm]
-  simp_rw [he]
-  have h := hv.sum_inner_products_le (s := Finset.univ) w
-  apply h.trans_eq
-  rw [EuclideanSpace.norm_sq_eq]
-  apply Finset.sum_congr rfl
-  intro y _
-  simp [w,Complex.norm_real,Real.norm_eq_abs,sq_abs]
-
-
-lemma tail_density_pointwise (t a z : ℝ) (ht : 0 ≤ t) (ha : 0 ≤ a) (hz : 0 < z) :
-    (if z < t then t else 0) ≤ t^(1+a)*z^(-a) := by
-  split_ifs with h
-  · rw [Real.rpow_neg hz.le,←div_eq_mul_inv]
-    apply (le_div_iff₀ (Real.rpow_pos_of_pos hz a)).mpr
-    rw [Real.rpow_add (lt_trans hz h),Real.rpow_one]
-    exact mul_le_mul_of_nonneg_left (Real.rpow_le_rpow hz.le h.le ha) ht
-  · positivity
-
-lemma tail_density_mean {X : Type*} [Fintype X] (f : X → ℝ) (hf : ∀ x, 0 ≤ f x)
-    (a z : ℝ) (ha : 0 ≤ a) (hz : 0 < z) :
-    finiteMean (fun x => if z < f x then f x else 0) ≤ finiteMean (fun x => f x^(1+a))*z^(-a) := by
-  rw [←finiteMean_mul_const]
-  exact finiteMean_mono (fun x => tail_density_pointwise (f x) a z (hf x) ha hz)
-
-noncomputable def tailKernel {X : Type*} [Fintype X] (P : X → X → ℝ) (z : ℝ) : X → X → ℝ :=
-  fun x y => if z < (Fintype.card X:ℝ)*P x y then P x y else 0
-
-noncomputable def headKernel {X : Type*} [Fintype X] (P : X → X → ℝ) (z : ℝ) : X → X → ℝ :=
-  fun x y => if z < (Fintype.card X:ℝ)*P x y then 0 else P x y
-
-lemma tail_add_head {X : Type*} [Fintype X] (P : X → X → ℝ) (z : ℝ) :
-    (fun x y => tailKernel P z x y+headKernel P z x y) = P := by
-  funext x y
-  unfold tailKernel headKernel
-  split_ifs <;> simp
-
-lemma tailKernel_nonneg {X : Type*} [Fintype X] (P : X → X → ℝ) (z : ℝ)
-    (hP : ∀ x y, 0 ≤ P x y) (x y : X) : 0 ≤ tailKernel P z x y := by
-  unfold tailKernel
-  split_ifs <;> simp only [hP,le_refl]
-
-lemma tailKernel_row {X : Type*} [Fintype X] [Nonempty X] (P : X → X → ℝ)
-    (hP : ∀ x y, 0 ≤ P x y) (a z M : ℝ) (ha : 0 ≤ a) (hz : 0 < z)
-    (hM : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*P x y)^(1+a)) ≤ M) (x : X) :
-    ∑ y, tailKernel P z x y ≤ M*z^(-a) := by
-  have hn : (Fintype.card X:ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
-  have he : finiteMean (fun y => if z < (Fintype.card X:ℝ)*P x y then
-      (Fintype.card X:ℝ)*P x y else 0) = ∑ y, tailKernel P z x y := by
-    unfold finiteMean
-    calc
-      _ = (∑ y, (Fintype.card X:ℝ)*tailKernel P z x y)/(Fintype.card X:ℝ) := by
-        congr 1
-        apply Finset.sum_congr rfl
-        intro y _
-        by_cases hy : z < (Fintype.card X:ℝ)*P x y
-        · simp only [tailKernel,hy,↓reduceIte]
-        · simp only [tailKernel,hy,↓reduceIte,mul_zero]
-      _ = _ := by rw [←Finset.mul_sum]; field_simp
-  rw [←he]
-  exact (tail_density_mean _ (fun y => mul_nonneg (Nat.cast_nonneg _) (hP x y)) a z ha hz).trans
-    (mul_le_mul_of_nonneg_right (hM x) (Real.rpow_nonneg hz.le _))
-
-lemma tailKernel_schur {X : Type*} [Fintype X] [Nonempty X] (P : X → X → ℝ)
-    (hP : ∀ x y, 0 ≤ P x y) (hsym : ∀ x y, P x y = P y x)
-    (a z M : ℝ) (ha : 0 ≤ a) (hz : 0 < z) (hM0 : 0 ≤ M)
-    (hM : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*P x y)^(1+a)) ≤ M)
-    (v : EuclideanSpace ℂ X) :
-    ‖applyKernel (tailKernel P z) v‖ ≤ (M*z^(-a))*‖v‖ := by
-  apply schur_bound _ (tailKernel_nonneg P z hP) _ (by positivity)
-    (tailKernel_row P hP a z M ha hz hM)
-  intro y
-  have hs : (fun x => tailKernel P z x y) = (fun x => tailKernel P z y x) := by
-    funext x
-    simp only [tailKernel,hsym x y]
-  rw [hs]
-  exact tailKernel_row P hP a z M ha hz hM y
-
-lemma headKernel_hs {X : Type*} [Fintype X] [Nonempty X] (P : X → X → ℝ)
-    (hP : ∀ x y, 0 ≤ P x y) (hrow : ∀ x, ∑ y, P x y = 1) (z : ℝ) (hz : 0 ≤ z) :
-    (∑ x, ∑ y, (headKernel P z x y)^2) ≤ z := by
-  have hn : (0:ℝ) < Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
-  have he (x y : X) : (headKernel P z x y)^2 ≤ (z/(Fintype.card X:ℝ))*P x y := by
-    unfold headKernel
-    split_ifs with h
-    · simp only [zero_pow (by omega : (2:ℕ) ≠ 0)]
-      exact mul_nonneg (div_nonneg hz hn.le) (hP x y)
-    · have h' : (Fintype.card X:ℝ)*P x y ≤ z := le_of_not_gt h
-      have hh := mul_le_mul_of_nonneg_right h' (hP x y)
-      rw [show (z/(Fintype.card X:ℝ))*P x y = (z*P x y)/(Fintype.card X:ℝ) by ring]
-      apply (le_div_iff₀ hn).mpr
-      nlinarith
-  calc
-    _ ≤ ∑ x, ∑ y, (z/(Fintype.card X:ℝ))*P x y :=
-      Finset.sum_le_sum (fun x _ => Finset.sum_le_sum (fun y _ => he x y))
-    _ = z := by
-      simp only [←Finset.mul_sum,hrow,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
-      field_simp
-
-
-theorem multiplicity_amplification {X I : Type*} [Fintype X] [Nonempty X]
-    [Fintype I] [Nonempty I] (P : X → X → ℝ)
-    (hP : ∀ x y, 0 ≤ P x y) (hsym : ∀ x y, P x y = P y x)
-    (hrow : ∀ x, ∑ y, P x y = 1) (a z M : ℝ) (ha : 0 ≤ a) (hz : 0 < z) (hM0 : 0 ≤ M)
-    (hM : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*P x y)^(1+a)) ≤ M)
-    (v : I → EuclideanSpace ℂ X) (hv : Orthonormal ℂ v)
-    (μ : ℝ) (hvμ : ∀ i, μ ≤ ‖applyKernel P (v i)‖) :
-    μ ≤ M*z^(-a)+Real.sqrt (z/(Fintype.card I:ℝ)) := by
-  let δ : ℝ := M*z^(-a)
-  have hδ : 0 ≤ δ := by dsimp [δ]; positivity
-  have hm : (0:ℝ) < Fintype.card I := Nat.cast_pos.mpr Fintype.card_pos
-  change μ ≤ δ+_
-  by_cases hμ : μ ≤ δ
-  · exact hμ.trans (le_add_of_nonneg_right (Real.sqrt_nonneg _))
-  have hmd : 0 ≤ μ-δ := by linarith
-  have hsmall (i : I) : μ-δ ≤ ‖applyKernel (headKernel P z) (v i)‖ := by
-    have hr := tailKernel_schur P hP hsym a z M ha hz hM0 hM (v i)
-    rw [hv.norm_eq_one i,mul_one] at hr
-    have he := applyKernel_add (tailKernel P z) (headKernel P z) (v i)
-    rw [tail_add_head] at he
-    have hn := norm_add_le (applyKernel (tailKernel P z) (v i)) (applyKernel (headKernel P z) (v i))
-    rw [←he] at hn
-    have hh := hvμ i
-    change ‖applyKernel (tailKernel P z) (v i)‖ ≤ δ at hr
-    linarith
-  have hsum : (Fintype.card I:ℝ)*(μ-δ)^2 ≤ z := by
-    calc
-      _ = ∑ i : I, (μ-δ)^2 := by simp
-      _ ≤ ∑ i : I, ‖applyKernel (headKernel P z) (v i)‖^2 :=
-        Finset.sum_le_sum (fun i _ => pow_le_pow_left₀ hmd (hsmall i) 2)
-      _ ≤ ∑ x, ∑ y, (headKernel P z x y)^2 := hilbert_schmidt_family _ v hv
-      _ ≤ z := headKernel_hs P hP hrow z hz.le
-  have hsq : (μ-δ)^2 ≤ z/(Fintype.card I:ℝ) := (le_div_iff₀ hm).mpr (by nlinarith)
-  have hs := (Real.le_sqrt hmd (div_nonneg hz.le hm.le)).mpr hsq
-  linarith
-
-end DensityTransfer
-namespace PairRouting
-open scoped BigOperators
-
-lemma tupleProbability_row_sum {ι β Ω : Type*} [Fintype ι] [Fintype β]
-    [DecidableEq β] [Fintype Ω] [Nonempty Ω]
-    (P : Ω → Equiv.Perm β) (e : ι ↪ β) :
-    (∑ f : ι ↪ β, tupleProbability P e f) = 1 := by
-  classical
-  simp only [tupleProbability]
-  rw [←finiteMean_sum]
-  have h (ω : Ω) (f : ι ↪ β) : (∀ i, P ω (e i) = f i) ↔ e.trans (P ω).toEmbedding = f := by
-    constructor
-    · intro h; ext i; exact h i
-    · intro h i; exact congrArg (fun g : ι ↪ β => g i) h
-  simp_rw [h]
-  simp only [Finset.sum_ite_eq,Finset.mem_univ,↓reduceIte,finiteMean_const]
-
-lemma tupleProbability_inverse {ι β Ω : Type*} [Fintype ι] [DecidableEq β]
-    [Fintype Ω] (P : Ω → Equiv.Perm β) (e f : ι → β) :
-    tupleProbability (fun ω => (P ω)⁻¹) e f = tupleProbability P f e := by
-  classical
-  apply finiteMean_congr
-  intro ω
-  congr 1
-  apply propext
-  constructor
-  · intro h i
-    have hh := congrArg (P ω) (h i)
-    simpa only [Equiv.Perm.inv_def,Equiv.apply_symm_apply] using hh.symm
-  · intro h i
-    have hh := congrArg (fun x => (P ω).symm x) (h i)
-    simpa only [Equiv.Perm.inv_def,Equiv.symm_apply_apply] using hh.symm
-
-lemma tupleProbability_equiv {ι β Ω Ω' : Type*} [Fintype ι] [DecidableEq β]
-    [Fintype Ω] [Fintype Ω'] (P : Ω → Equiv.Perm β) (τ : Ω ≃ Ω') (e f : ι → β) :
-    tupleProbability P e f = tupleProbability (fun ω => P (τ.symm ω)) e f :=
-  finiteMean_equiv τ _
-
-lemma tupleProbability_symmetric {ι β Ω : Type*} [Fintype ι] [DecidableEq β]
-    [Fintype Ω] (P : Ω → Equiv.Perm β) (τ : Equiv.Perm Ω)
-    (hτ : ∀ ω, P (τ ω) = (P ω)⁻¹) (e f : ι → β) :
-    tupleProbability P e f = tupleProbability P f e := by
-  rw [tupleProbability_equiv P τ.symm]
-  simp only [Equiv.symm_symm,hτ]
-  exact tupleProbability_inverse P e f
-
-end PairRouting
-
-lemma palindromePerm_reverse (d : ℕ) (ω : BenesCoins d) :
-    palindromePerm d (ω.2,ω.1) = (palindromePerm d ω)⁻¹ := by
-  simp only [palindromePerm,mul_inv_rev]
-  rfl
-
-lemma palindrome_kernel_symmetric (d : ℕ) {ι : Type*} [Fintype ι]
-    (e f : ι ↪ Card d) :
-    PairRouting.tupleProbability (palindromePerm d) e f =
-      PairRouting.tupleProbability (palindromePerm d) f e :=
-  PairRouting.tupleProbability_symmetric _ (Equiv.prodComm _ _) (palindromePerm_reverse d) e f
-
-theorem palindrome_density_transfer (d : ℕ) {ι I : Type*} [Fintype ι] [Fintype I]
-    [Nonempty I] (e₀ : ι ↪ Card d) (hk : 0 < Fintype.card ι)
-    (v : I → EuclideanSpace ℂ (ι ↪ Card d)) (hv : Orthonormal ℂ v)
-    (μ : ℝ) (hvμ : ∀ i, μ ≤ ‖DensityTransfer.applyKernel
-      (fun e f : ι ↪ Card d => PairRouting.tupleProbability (palindromePerm d) e f) (v i)‖)
-    (z : ℝ) (hz : 0 < z) :
-    μ ≤ Real.exp (densityConstant*Fintype.card ι*
-      ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ))*z^(-(1/1000:ℝ))+
-      Real.sqrt (z/(Fintype.card I:ℝ)) := by
-  let : Nonempty (ι ↪ Card d) := ⟨e₀⟩
-  apply DensityTransfer.multiplicity_amplification
-    (fun e f : ι ↪ Card d => PairRouting.tupleProbability (palindromePerm d) e f)
-    (fun e f => PairRouting.tupleProbability_nonneg _ e f)
-    (palindrome_kernel_symmetric d) (PairRouting.tupleProbability_row_sum _)
-    (1/1000) z _ (by norm_num) hz (Real.exp_nonneg _) _ v hv μ hvμ
-  intro e
-  exact palindrome_row_moment_bound d e hk
-
-
-noncomputable def modifiedReverseCoins (L : ℕ) : (r : ℕ) → ModifiedCoins L r → ModifiedCoins L r
-  | 0, ω => ((fun u => butterflyPerm L (decodeButterfly L ω.2.1) * (ω.1 u)⁻¹ *
-      butterflyPerm L (decodeButterfly L ω.2.1)),ω.2)
-  | r+1, ω =>
-      let z := modifiedStepEquiv L r ω
-      (modifiedStepEquiv L r).symm
-        ((modifiedReverseCoins L r z.1.1,modifiedReverseCoins L r z.1.2),(z.2.2,z.2.1))
-
-lemma modifiedReverseCoins_involutive (L r : ℕ) :
-    Function.Involutive (modifiedReverseCoins L r) := by
-  induction r with
-  | zero =>
-      intro ω
-      apply Prod.ext
-      · funext u
-        simp only [modifiedReverseCoins,mul_inv_rev,inv_inv]
-        simp [mul_assoc]
-      · rfl
-  | succ r ih =>
-      change ∀ ω, modifiedReverseCoins L r (modifiedReverseCoins L r ω) = ω at ih
-      intro ω
-      simp only [modifiedReverseCoins,Equiv.apply_symm_apply,ih]
-      exact (modifiedStepEquiv L r).symm_apply_apply ω
-
-end Thorp
-
-end ThorpNine.SparseSaving
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.SparseSaving.InputTree
+
+namespace OAI
+
+namespace ThorpNine.SparseSaving
+
+namespace Thorp
+
+noncomputable def modifiedRoutingMoment (L r : ℕ) (E : Finset (Card r))
+    {ι : Type*} [Fintype ι] (e f : ι ↪ Card (L+r)) (a : ℝ) : ℝ := by
+  classical
+  exact finiteMean (fun ω : ModifiedCoins L r =>
+    if ∀ i, modifiedPerm L r E ω (e i) = f i then (2:ℝ)^((modifiedCost L r E e ω:ℝ)*a) else 0)
+
+lemma modifiedRoutingMoment_nonneg (L r : ℕ) (E : Finset (Card r))
+    {ι : Type*} [Fintype ι] (e f : ι ↪ Card (L+r)) (a : ℝ) :
+    0 ≤ modifiedRoutingMoment L r E e f a := by
+  classical
+  exact finiteMean_nonneg (fun _ => by split_ifs <;> positivity)
+
+lemma modifiedRoutingMoment_zero (L : ℕ) (E : Finset (Card 0))
+    {ι : Type*} [Fintype ι] (e f : ι ↪ Card L) (a : ℝ) :
+    modifiedRoutingMoment L 0 E e f a = PairRouting.tupleProbability (modifiedPerm L 0 E) e f := by
+  simp only [modifiedRoutingMoment,modifiedCost_zero,Nat.cast_zero,zero_mul,Real.rpow_zero,
+    PairRouting.tupleProbability]
+
+lemma modified_pointwise_zero (L : ℕ) (E : Finset (Card 0))
+    {ι : Type*} [Fintype ι] (e f : ι ↪ Card L) (he : Function.Surjective e) (a : ℝ) (ha : 0 ≤ a) :
+    (PairRouting.tupleProbability (modifiedPerm L 0 E) e f)^(1+a) ≤
+      centralFactor L 0 E a * modifiedRoutingMoment L 0 E e f a := by
+  have hp := PairRouting.tupleProbability_nonneg (modifiedPerm L 0 E) e f
+  rw [modifiedRoutingMoment_zero]
+  have hbound := modified_probability_zero_le L E e f he
+  have hpow := Real.rpow_le_rpow hp hbound ha
+  have hfpos : (0:ℝ) < Nat.factorial (2^L) := Nat.cast_pos.mpr (Nat.factorial_pos _)
+  have heq : (if (fun j => Fin.elim0 j : Card 0) ∈ E then (1:ℝ) else 1/(Nat.factorial (2^L):ℝ))^a =
+      centralFactor L 0 E a := by
+    rw [centralFactor_zero]
+    split_ifs
+    · exact Real.one_rpow _
+    · rw [one_div,Real.inv_rpow (le_of_lt hfpos),Real.rpow_neg (le_of_lt hfpos)]
+  rw [heq] at hpow
+  rw [Real.rpow_add' hp (by linarith : (1+a:ℝ) ≠ 0),Real.rpow_one,mul_comm]
+  exact mul_le_mul_of_nonneg_right hpow hp
+
+
+lemma modified_matches_step (L r : ℕ) (E : Finset (Card (r+1))) {ι : Type*}
+    (e f : ι ↪ Card (L+r+1)) (ω : ModifiedCoins L r × ModifiedCoins L r)
+    (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
+    (∀ i, modifiedPerm L (r+1) E ((modifiedStepEquiv L r).symm (ω,coins)) (e i) = f i) ↔
+    (∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1))
+      ((e.trans (headTailEquiv (L+r)).toEmbedding) i) = ((f.trans (headTailEquiv (L+r)).toEmbedding) i)) := by
+  apply forall_congr'
+  intro i
+  rw [←(headTailEquiv (L+r)).injective.eq_iff,modifiedPerm_coordinates]
+  rfl
+
+lemma modifiedRoutingMoment_step (L r : ℕ) (E : Finset (Card (r+1))) {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (e f : ι ↪ Card (L+r+1)) (a : ℝ) :
+    let x := e.trans (headTailEquiv (L+r)).toEmbedding
+    let y := f.trans (headTailEquiv (L+r)).toEmbedding
+    modifiedRoutingMoment L (r+1) E e f a = PairRouting.outerWeight x y *
+      ∑ c : {c // c ∈ PairRouting.compatibleSet x y},
+        finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
+          if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c.val
+          then (2:ℝ)^((modifiedAssignmentCost L r E x c.val
+            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0) := by
+  classical
+  intro x y
+  let : Fintype (ModifiedCoins L r) := inferInstance
+  rw [modifiedRoutingMoment,finiteMean_equiv (modifiedStepEquiv L r),finiteMean_prod]
+  have he (ω : ModifiedCoins L r × ModifiedCoins L r) :
+      finiteMean (fun coins : (Card (L+r) → Bool) × (Card (L+r) → Bool) =>
+        if ∀ i, modifiedPerm L (r+1) E ((modifiedStepEquiv L r).symm (ω,coins)) (e i) = f i
+        then (2:ℝ)^((modifiedCost L (r+1) E e ((modifiedStepEquiv L r).symm (ω,coins)):ℝ)*a) else 0) =
+      PairRouting.outerWeight x y *
+        ∑ c : {c // c ∈ PairRouting.compatibleSet x y},
+          if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c.val
+          then (2:ℝ)^((modifiedAssignmentCost L r E x c.val
+            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0 := by
+    let W : (ι → Bool) → ℝ := fun c =>
+      if hc : PairRouting.Compatible x c then (2:ℝ)^((modifiedAssignmentCost L r E x c hc ω:ℝ)*a) else 0
+    calc
+      _ = finiteMean (fun coins : (Card (L+r) → Bool) × (Card (L+r) → Bool) =>
+        if ∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1))
+          (x i) = y i then W (PairRouting.colors x coins.1) else 0) := by
+        apply finiteMean_congr
+        intro coins
+        simp only [modified_matches_step,modifiedCost_step]
+        simp only [W,PairRouting.colors_compatible,↓reduceDIte]
+        rfl
+      _ = _ := by
+        rw [PairRouting.sandwich_weighted_probability_subtype]
+        congr 1
+        apply Finset.sum_congr rfl
+        intro c _
+        simp only [W,((PairRouting.mem_compatibleSet x y c.val).mp c.property).1,↓reduceDIte]
+  simp_rw [he]
+  simp_rw [mul_comm (PairRouting.outerWeight x y)]
+  rw [finiteMean_mul_const,finiteMean_sum,mul_comm]
+
+lemma modified_child_moment_product (L r : ℕ) (E : Finset (Card (r+1))) {ι : Type*} [Fintype ι]
+    (x y : ι ↪ Bool × Card (L+r)) (c : ι → Bool)
+    (hcx : PairRouting.Compatible x c) (hcy : PairRouting.Compatible y c) (a : ℝ) :
+    modifiedRoutingMoment L r (exceptionChild E false) (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a *
+      modifiedRoutingMoment L r (exceptionChild E true) (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a =
+    finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
+      if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c
+      then (2:ℝ)^(((modifiedCost L r (exceptionChild E false) (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
+        modifiedCost L r (exceptionChild E true) (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0) := by
+  classical
+  unfold modifiedRoutingMoment
+  rw [←finiteMean_prod_mul]
+  apply finiteMean_congr
+  intro ω
+  let M₀ := ∀ i : {i // c i = false}, modifiedPerm L r (exceptionChild E false) ω.1 (x i.val).2 = (y i.val).2
+  let M₁ := ∀ i : {i // c i = true}, modifiedPerm L r (exceptionChild E true) ω.2 (x i.val).2 = (y i.val).2
+  have hp : (fun b : Bool => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) =
+      (fun b => if b then modifiedPerm L r (exceptionChild E true) ω.2 else modifiedPerm L r (exceptionChild E false) ω.1) := by
+    funext b
+    cases b <;> rfl
+  have hs : PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c ↔
+      M₀ ∧ M₁ := by rw [hp]; exact PairRouting.childMatches_split _ _ _ _ _
+  change (if M₀ then _ else 0) * (if M₁ then _ else 0) = _
+  by_cases h₀ : M₀ <;> by_cases h₁ : M₁
+  · simp only [hs,h₀,h₁,true_and,↓reduceIte]
+    rw [←Real.rpow_add (by norm_num : (0:ℝ)<2)]
+    congr 1
+    ring
+  · simp only [hs,h₀,h₁,and_false,↓reduceIte,mul_zero]
+  · simp only [hs,h₀,h₁,false_and,↓reduceIte,zero_mul]
+  · simp only [hs,h₀,h₁,false_and,↓reduceIte,mul_zero]
+
+lemma modified_probability_step (L r : ℕ) (E : Finset (Card (r+1))) {ι : Type*} [Fintype ι]
+    (e f : ι ↪ Card (L+r+1)) :
+    PairRouting.tupleProbability (modifiedPerm L (r+1) E) e f =
+    finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
+      finiteMean (fun coins : (Card (L+r) → Bool) × (Card (L+r) → Bool) =>
+        if ∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => if b then modifiedPerm L r (exceptionChild E true) ω.2 else modifiedPerm L r (exceptionChild E false) ω.1)
+          ((e.trans (headTailEquiv (L+r)).toEmbedding) i) = ((f.trans (headTailEquiv (L+r)).toEmbedding) i)
+        then (1:ℝ) else 0)) := by
+  classical
+  let : Fintype (ModifiedCoins L r) := inferInstance
+  rw [PairRouting.tupleProbability,finiteMean_equiv (modifiedStepEquiv L r),finiteMean_prod]
+  have hp (b : Bool) (p q : ModifiedCoins L r) :
+      modifiedPerm L r (exceptionChild E b) (if b then p else q) =
+      (if b then modifiedPerm L r (exceptionChild E true) p else modifiedPerm L r (exceptionChild E false) q) := by
+    cases b <;> rfl
+  simp only [modified_matches_step,hp]
+
+
+lemma modified_assignment_moment_bound (L r : ℕ) (E : Finset (Card (r+1))) {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (x y : ι ↪ Bool × Card (L+r)) (c : ι → Bool) (hc : c ∈ PairRouting.compatibleSet x y)
+    (a : ℝ) (ha : 0 ≤ a) :
+    let hcx := ((PairRouting.mem_compatibleSet x y c).mp hc).1
+    PairRouting.outerWeight x y ^ a * ((PairRouting.compatibleSet x y).card:ℝ)^a *
+      ((2:ℝ)^(-(r:ℝ)*Fintype.card ι*a) *
+        finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
+          if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c
+          then (2:ℝ)^(((modifiedCost L r (exceptionChild E false) (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
+            modifiedCost L r (exceptionChild E true) (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0)) ≤
+    (2:ℝ)^(-((r:ℝ)+1)*Fintype.card ι*a) *
+      finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
+        if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c
+        then (2:ℝ)^((modifiedAssignmentCost L r E x c hcx ω:ℝ)*a) else 0) := by
+  classical
+  intro hcx
+  have hscale (q : ℝ) (F : ModifiedCoins L r × ModifiedCoins L r → ℝ) :
+      q * finiteMean F = finiteMean (fun ω => q * F ω) := by
+    simp_rw [mul_comm q,finiteMean_mul_const]
+  rw [←mul_assoc,hscale,hscale]
+  apply finiteMean_mono
+  intro ω
+  split_ifs with hm
+  · have hh := PairRouting.assignment_factor_bound x y c hc
+      (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) hm a ha
+    calc
+      _ ≤ ((2:ℝ)^(((PairRouting.alternatingCycles (PairRouting.coloredEmbedding x c hcx)
+          (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)):ℝ)-Fintype.card ι)*a) *
+          (2:ℝ)^(-(r:ℝ)*Fintype.card ι*a)) *
+          (2:ℝ)^(((modifiedCost L r (exceptionChild E false) (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
+            modifiedCost L r (exceptionChild E true) (PairRouting.childEmbedding x c hcx true) ω.2)*a) :=
+        mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hh (by positivity)) (by positivity)
+      _ = _ := by
+        simp only [←Real.rpow_add (by norm_num : (0:ℝ)<2),modifiedAssignmentCost,Nat.cast_add]
+        congr 1
+        ring
+  · simp only [mul_zero]
+    exact le_rfl
+
+
+lemma modified_child_power_product_bound (L r : ℕ) (E : Finset (Card (r+1)))
+    {ι : Type*} [Fintype ι] (x y : ι ↪ Bool × Card (L+r)) (c : ι → Bool)
+    (hcx : PairRouting.Compatible x c) (hcy : PairRouting.Compatible y c) (a : ℝ)
+    (hfalse : (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E false))
+        (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false))^(1+a) ≤
+      ((2:ℝ)^(-(r:ℝ)*Fintype.card {i // c i = false}*a)*centralFactor L r (exceptionChild E false) a) *
+        modifiedRoutingMoment L r (exceptionChild E false)
+          (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a)
+    (htrue : (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E true))
+        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true))^(1+a) ≤
+      ((2:ℝ)^(-(r:ℝ)*Fintype.card {i // c i = true}*a)*centralFactor L r (exceptionChild E true) a) *
+        modifiedRoutingMoment L r (exceptionChild E true)
+          (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a) :
+    (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E false))
+      (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false))^(1+a) *
+      (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E true))
+        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true))^(1+a) ≤
+    ((2:ℝ)^(-(r:ℝ)*Fintype.card ι*a)*centralFactor L (r+1) E a) *
+      finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
+        if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c
+        then (2:ℝ)^(((modifiedCost L r (exceptionChild E false) (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
+          modifiedCost L r (exceptionChild E true) (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0) := by
+  have hh := mul_le_mul hfalse htrue
+    (Real.rpow_nonneg (PairRouting.tupleProbability_nonneg _ _ _) _)
+    (mul_nonneg (mul_nonneg (by positivity) (centralFactor_nonneg _ _ _ _))
+      (modifiedRoutingMoment_nonneg _ _ _ _ _ _))
+  have he : -(r:ℝ)*Fintype.card {i // c i = false}*a + -(r:ℝ)*Fintype.card {i // c i = true}*a =
+      -(r:ℝ)*Fintype.card ι*a := by
+    rw [←add_mul,←mul_add,←Nat.cast_add,PairRouting.color_card_add]
+  calc
+    _ ≤ _ := hh
+    _ = ((2:ℝ)^(-(r:ℝ)*Fintype.card {i // c i = false}*a)*
+        (2:ℝ)^(-(r:ℝ)*Fintype.card {i // c i = true}*a)) *
+      (centralFactor L r (exceptionChild E false) a * centralFactor L r (exceptionChild E true) a) *
+      (modifiedRoutingMoment L r (exceptionChild E false)
+        (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a *
+      modifiedRoutingMoment L r (exceptionChild E true)
+        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a) := by ring
+    _ = _ := by rw [←Real.rpow_add (by norm_num : (0:ℝ)<2),he,←centralFactor_step,modified_child_moment_product]
+
+
+theorem modified_pointwise_moment (L r : ℕ) (E : Finset (Card r)) {ι : Type*} [Fintype ι]
+    (e f : ι ↪ Card (L+r)) (he : Function.Surjective e) (a : ℝ) (ha : 0 ≤ a) :
+    (PairRouting.tupleProbability (modifiedPerm L r E) e f)^(1+a) ≤
+      ((2:ℝ)^(-(r:ℝ)*Fintype.card ι*a)*centralFactor L r E a) * modifiedRoutingMoment L r E e f a := by
+  classical
+  induction r generalizing ι with
+  | zero =>
+      simpa only [Nat.cast_zero,neg_zero,zero_mul,Real.rpow_zero,one_mul] using
+        modified_pointwise_zero L E e f he a ha
+  | succ r ih =>
+      let x := e.trans (headTailEquiv (L+r)).toEmbedding
+      let y := f.trans (headTailEquiv (L+r)).toEmbedding
+      let W := PairRouting.outerWeight x y
+      let C := PairRouting.compatibleSet x y
+      let p : (ι → Bool) → ℝ := fun c =>
+        (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E false))
+          (fun i : {i // c i = false} => (x i.val).2)
+          (fun i : {i // c i = false} => (y i.val).2))^(1+a) *
+        (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E true))
+          (fun i : {i // c i = true} => (x i.val).2)
+          (fun i : {i // c i = true} => (y i.val).2))^(1+a)
+      let m : {c // c ∈ C} → ℝ := fun c =>
+        finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
+          if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c.val
+          then (2:ℝ)^((modifiedAssignmentCost L r E x c.val
+            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0)
+      have hp (c : {c // c ∈ C}) : W^a * (C.card:ℝ)^a * p c.val ≤
+          ((2:ℝ)^(-((r:ℝ)+1)*Fintype.card ι*a)*centralFactor L (r+1) E a) * m c := by
+        obtain ⟨hcx,hcy⟩ := (PairRouting.mem_compatibleSet x y c.val).mp c.property
+        have h₀ := ih (exceptionChild E false) (PairRouting.childEmbedding x c.val hcx false)
+          (PairRouting.childEmbedding y c.val hcy false)
+          (PairRouting.childEmbedding_surjective x ((headTailEquiv (L+r)).surjective.comp he) c.val hcx false)
+        have h₁ := ih (exceptionChild E true) (PairRouting.childEmbedding x c.val hcx true)
+          (PairRouting.childEmbedding y c.val hcy true)
+          (PairRouting.childEmbedding_surjective x ((headTailEquiv (L+r)).surjective.comp he) c.val hcx true)
+        have hh := modified_child_power_product_bound L r E x y c.val hcx hcy a h₀ h₁
+        have hh' := mul_le_mul_of_nonneg_left hh
+          (mul_nonneg (Real.rpow_nonneg (le_of_lt (PairRouting.outerWeight_pos x y)) a)
+            (Real.rpow_nonneg (Nat.cast_nonneg C.card) a))
+        apply hh'.trans
+        have hm := modified_assignment_moment_bound L r E x y c.val c.property a ha
+        convert mul_le_mul_of_nonneg_left hm (centralFactor_nonneg L (r+1) E a) using 1 <;> ring
+      have hnode := PairRouting.sandwich_power_bound (modifiedPerm L r (exceptionChild E false)) (modifiedPerm L r (exceptionChild E true)) x y a ha
+      rw [←modified_probability_step L r E e f] at hnode
+      change (PairRouting.tupleProbability (modifiedPerm L (r+1) E) e f)^(1+a) ≤
+        W^(1+a) * (C.card:ℝ)^a * ∑ c ∈ C, p c at hnode
+      have hsum : ∑ c ∈ C, p c = ∑ c : {c // c ∈ C}, p c.val := by
+        rw [Finset.sum_subtype C (fun _ => Iff.rfl)]
+      have hW : 0 < W := PairRouting.outerWeight_pos x y
+      calc
+        _ ≤ W^(1+a) * (C.card:ℝ)^a * ∑ c ∈ C, p c := hnode
+        _ = W * ∑ c : {c // c ∈ C}, W^a * (C.card:ℝ)^a * p c.val := by
+          rw [Real.rpow_add hW,Real.rpow_one,hsum,←Finset.mul_sum]
+          ring
+        _ ≤ W * ∑ c : {c // c ∈ C}, ((2:ℝ)^(-((r:ℝ)+1)*Fintype.card ι*a)*centralFactor L (r+1) E a) * m c :=
+          mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun c _ => hp c)) (le_of_lt hW)
+        _ = _ := by
+          rw [←Finset.mul_sum,modifiedRoutingMoment_step]
+          simp only [Nat.cast_add,Nat.cast_one]
+          change W * (((2:ℝ)^(-((r:ℝ)+1)*Fintype.card ι*a)*centralFactor L (r+1) E a) * ∑ c, m c) =
+            ((2:ℝ)^(-((r:ℝ)+1)*Fintype.card ι*a)*centralFactor L (r+1) E a) * (W * ∑ c, m c)
+          ring
+
+
+lemma modifiedRoutingMoment_sum (L r : ℕ) (E : Finset (Card r)) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (L+r)) (a : ℝ) :
+    (∑ f : ι ↪ Card (L+r), modifiedRoutingMoment L r E e f a) =
+      finiteMean (fun ω : ModifiedCoins L r => (2:ℝ)^((modifiedCost L r E e ω:ℝ)*a)) := by
+  classical
+  unfold modifiedRoutingMoment
+  rw [←finiteMean_sum]
+  apply finiteMean_congr
+  intro ω
+  let f₀ : ι ↪ Card (L+r) := e.trans (modifiedPerm L r E ω).toEmbedding
+  rw [Finset.sum_eq_single f₀]
+  · have hm : ∀ i, modifiedPerm L r E ω (e i) = f₀ i := fun _ => rfl
+    simp only [hm,implies_true,↓reduceIte]
+  · intro f _ hf
+    have hm : ¬ ∀ i, modifiedPerm L r E ω (e i) = f i := by
+      intro hh
+      apply hf
+      ext i : 1
+      exact (hh i).symm
+    exact ite_eq_right hm
+  · simp
+
+noncomputable def modifiedRowMoment (L r : ℕ) (E : Finset (Card r)) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (L+r)) (a : ℝ) : ℝ :=
+  finiteMean (fun f : ι ↪ Card (L+r) =>
+    ((Fintype.card (ι ↪ Card (L+r)):ℝ)*PairRouting.tupleProbability (modifiedPerm L r E) e f)^(1+a))
+
+lemma full_tuple_card (d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d)
+    (he : Function.Surjective e) : Fintype.card ι = 2^d := by
+  rw [←card_positions d]
+  exact Fintype.card_congr (Equiv.ofBijective e ⟨e.injective,he⟩)
+
+lemma exception_card_le (r : ℕ) (E : Finset (Card r)) : E.card ≤ 2^r := by
+  simpa only [card_positions] using E.card_le_univ
+
+noncomputable def modifiedNormalization (L r : ℕ) (E : Finset (Card r)) : ℝ :=
+  (Nat.factorial (2^(L+r)):ℝ) /
+    (((2:ℝ)^r)^(2^(L+r)) * (Nat.factorial (2^L):ℝ)^(2^r-E.card))
+
+lemma modifiedNormalization_pos (L r : ℕ) (E : Finset (Card r)) :
+    0 < modifiedNormalization L r E := by unfold modifiedNormalization; positivity
+
+lemma modified_normalization_identity (L r : ℕ) (E : Finset (Card r)) (a : ℝ) :
+    (Nat.factorial (2^(L+r)):ℝ)^a *
+      ((2:ℝ)^(-(r:ℝ)*(2^(L+r):ℕ)*a)*centralFactor L r E a) =
+    modifiedNormalization L r E ^ a := by
+  rw [←mul_assoc,density_normalization,centralFactor,modifiedNormalization,
+    Real.div_rpow (by positivity) (by positivity),Real.div_rpow (by positivity) (by positivity),
+    Real.mul_rpow (by positivity) (by positivity)]
+  have hc : ((2^r-E.card:ℕ):ℝ) = (2:ℝ)^r-E.card := by
+    rw [Nat.cast_sub (exception_card_le r E),Nat.cast_pow,Nat.cast_ofNat]
+  rw [←Real.rpow_natCast (Nat.factorial (2^L):ℝ),←Real.rpow_mul (by positivity),hc,
+    show -((2:ℝ)^r-E.card)*a = -(((2:ℝ)^r-E.card)*a) by ring,
+    Real.rpow_neg (by positivity)]
+  ring
+
+theorem modified_row_density_moment (L r : ℕ) (E : Finset (Card r)) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (L+r)) (he : Function.Surjective e) (a : ℝ) (ha : 0 ≤ a) :
+    modifiedRowMoment L r E e a ≤ modifiedNormalization L r E ^ a *
+      finiteMean (fun ω : ModifiedCoins L r => (2:ℝ)^((modifiedCost L r E e ω:ℝ)*a)) := by
+  classical
+  have : Nonempty (ι ↪ Card (L+r)) := ⟨e⟩
+  rw [modifiedRowMoment,finite_row_power _ (fun f => PairRouting.tupleProbability_nonneg _ _ _)]
+  calc
+    _ ≤ (Fintype.card (ι ↪ Card (L+r)):ℝ)^a *
+      ∑ f : ι ↪ Card (L+r), ((2:ℝ)^(-(r:ℝ)*Fintype.card ι*a)*centralFactor L r E a) *
+        modifiedRoutingMoment L r E e f a :=
+      mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun f _ => modified_pointwise_moment L r E e f he a ha))
+        (Real.rpow_nonneg (Nat.cast_nonneg _) _)
+    _ = _ := by
+      rw [←Finset.mul_sum,modifiedRoutingMoment_sum,←mul_assoc,Fintype.card_embedding_eq,
+        card_positions,full_tuple_card (L+r) e he,Nat.descFactorial_self,modified_normalization_identity]
+
+lemma modified_total_cost_mgf (L r : ℕ) (hL : 0 < L) (E : Finset (Card r))
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) :
+    finiteMean (fun ω : ModifiedCoins L r => momentBase^(modifiedCost L r E e ω)) ≤
+      Real.exp (Fintype.card ι*heightDecay L +
+        3*((L*E.card*2^L:ℕ):ℝ)*Real.log momentBase) := by
+  let B := Real.exp (Fintype.card ι*heightDecay L +
+    3*((L*E.card*2^L:ℕ):ℝ)*Real.log momentBase)
+  rw [finiteMean_prod]
+  apply le_trans (finiteMean_mono (fun U => ?_)) (finiteMean_const B).le
+  rw [finiteMean_prod,finiteMean_comm]
+  apply le_trans (finiteMean_mono (fun X => ?_)) (finiteMean_const B).le
+  exact modified_cost_mgf L r hL E U X e momentBase momentBase_one_le (momentBase_log.trans (by norm_num))
+
+lemma modifiedRowMoment_bound (L r : ℕ) (hL : 0 < L) (E : Finset (Card r))
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (he : Function.Surjective e) :
+    modifiedRowMoment L r E e (1/1000) ≤
+      modifiedNormalization L r E ^ (1/1000:ℝ) *
+        Real.exp ((2^(L+r):ℕ)*heightDecay L +
+          3*((L*E.card*2^L:ℕ):ℝ)*Real.log momentBase) := by
+  have hm := modified_row_density_moment L r E e he (1/1000) (by norm_num)
+  simp_rw [←momentBase_pow] at hm
+  apply hm.trans
+  apply mul_le_mul_of_nonneg_left _ (Real.rpow_nonneg (modifiedNormalization_pos L r E).le _)
+  simpa only [full_tuple_card (L+r) e he] using modified_total_cost_mgf L r hL E e
+
+
+lemma log_factorial_upper {n : ℕ} (hn : 1 ≤ n) :
+    Real.log (Nat.factorial n) ≤ (n:ℝ)*Real.log n-n+Real.log n+1 := by
+  induction n with
+  | zero => omega
+  | succ n ih =>
+      by_cases hz : n = 0
+      · subst n; norm_num
+      have hn0 : (0:ℝ) < n := Nat.cast_pos.mpr (Nat.pos_of_ne_zero hz)
+      have hn1 : (0:ℝ) < n+1 := by positivity
+      have h := Real.one_sub_inv_le_log_of_pos (div_pos hn1 hn0)
+      rw [Real.log_div (ne_of_gt hn1) (ne_of_gt hn0),inv_div] at h
+      have hh := mul_le_mul_of_nonneg_left h hn1.le
+      have hcancel : ((n:ℝ)+1)*(1-(n:ℝ)/(n+1)) = 1 := by field_simp; ring
+      rw [hcancel] at hh
+      have hi := ih (by omega)
+      rw [Nat.factorial_succ,Nat.cast_mul,Real.log_mul (by positivity) (by positivity),Nat.cast_add,Nat.cast_one]
+      nlinarith
+
+lemma log_modifiedNormalization (L r : ℕ) (E : Finset (Card r)) :
+    Real.log (modifiedNormalization L r E) =
+      Real.log (Nat.factorial (2^(L+r))) -
+        ((2^(L+r):ℕ)*((r:ℝ)*Real.log 2)+((2:ℝ)^r-E.card)*Real.log (Nat.factorial (2^L))) := by
+  unfold modifiedNormalization
+  rw [Real.log_div (by positivity) (by positivity),Real.log_mul (by positivity) (by positivity),
+    Real.log_pow,Real.log_pow,Real.log_pow,
+    Nat.cast_sub (exception_card_le r E),Nat.cast_pow,Nat.cast_ofNat]
+  simp only [Nat.cast_pow,Nat.cast_ofNat]
+
+lemma log_modifiedNormalization_le (L r : ℕ) (E : Finset (Card r)) :
+    Real.log (modifiedNormalization L r E) ≤
+      ((L+r:ℕ):ℝ)*Real.log 2+1+E.card*Real.log (Nat.factorial (2^L)) := by
+  have hN : 1 ≤ 2^(L+r) := one_le_pow₀ (by norm_num)
+  have hS : 1 ≤ 2^L := one_le_pow₀ (by norm_num)
+  have hu := log_factorial_upper hN
+  have hl := mul_le_mul_of_nonneg_left (log_factorial_lower hS) (by positivity : (0:ℝ) ≤ (2:ℝ)^r)
+  simp only [Nat.cast_pow,Nat.cast_ofNat,Real.log_pow,Nat.cast_add] at hu hl ⊢
+  rw [log_modifiedNormalization]
+  simp only [pow_add,Nat.cast_mul,Nat.cast_pow,Nat.cast_ofNat] at hu hl ⊢
+  nlinarith
+
+lemma modified_moment_exponential (L r : ℕ) (hL : 0 < L) (E : Finset (Card r))
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (he : Function.Surjective e) :
+    modifiedRowMoment L r E e (1/1000) ≤
+      Real.exp (((L+r:ℕ)*Real.log 2+1+E.card*Real.log (Nat.factorial (2^L)))/1000 +
+        (2^(L+r):ℕ)*heightDecay L+3*((L*E.card*2^L:ℕ):ℝ)*Real.log momentBase) := by
+  apply (modifiedRowMoment_bound L r hL E e he).trans
+  rw [Real.rpow_def_of_pos (modifiedNormalization_pos L r E),←Real.exp_add]
+  apply Real.exp_le_exp.mpr
+  have h := log_modifiedNormalization_le L r E
+  linarith
+
+lemma log_power_factorial_le (L : ℕ) :
+    Real.log (Nat.factorial (2^L)) ≤ (2:ℝ)^L*L := by
+  have hf : (Nat.factorial (2^L):ℝ) ≤ ((2^L:ℕ):ℝ)^(2^L) := by
+    exact_mod_cast Nat.factorial_le_pow (2^L)
+  have hh := Real.log_le_log (by positivity) hf
+  rw [Real.log_pow,Nat.cast_pow,Nat.cast_ofNat,Real.log_pow] at hh
+  apply hh.trans
+  have hl := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
+  have hn : 0 ≤ (2:ℝ)^L*(L:ℝ) := by positivity
+  nlinarith
+
+lemma modified_moment_exception_bound (L r : ℕ) (hL : 0 < L) (E : Finset (Card r))
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (he : Function.Surjective e)
+    (ε : ℝ) (hE : (E.card:ℝ) ≤ ε*(2:ℝ)^r) :
+    modifiedRowMoment L r E e (1/1000) ≤
+      Real.exp (((L+r:ℕ)*Real.log 2+1)/1000 +
+        (2:ℝ)^(L+r)*(heightDecay L+ε*L/250)) := by
+  apply (modified_moment_exponential L r hL E e he).trans
+  apply Real.exp_le_exp.mpr
+  have hs := mul_le_mul_of_nonneg_left (log_power_factorial_le L) (Nat.cast_nonneg E.card)
+  have hb := mul_le_mul_of_nonneg_left momentBase_log
+    (by positivity : (0:ℝ) ≤ 3*((L*E.card*2^L:ℕ):ℝ))
+  have hexc := mul_le_mul_of_nonneg_left hE (by positivity : (0:ℝ) ≤ (2:ℝ)^L*L/250)
+  simp only [Nat.cast_mul,Nat.cast_pow,Nat.cast_ofNat,pow_add] at hs hb hexc ⊢
+  nlinarith
+
+
+lemma heightDecay_tendsto : Filter.Tendsto heightDecay Filter.atTop (nhds 0) := by
+  have h : Filter.Tendsto (fun L : ℕ => (-(1:ℝ)/64)*(L:ℝ)) Filter.atTop Filter.atBot :=
+    (Filter.tendsto_const_mul_atBot_of_neg (by norm_num : -(1:ℝ)/64 < 0)).mpr
+      tendsto_natCast_atTop_atTop
+  have ht := ((Real.tendsto_exp_atBot.comp h).const_mul 68).div_const (1-Real.exp (-(1:ℝ)/64))
+  convert ht using 1
+  · funext L
+    unfold heightDecay
+    dsimp only [Function.comp_apply]
+    rw [show -(L:ℝ)/64 = (-(1:ℝ)/64)*(L:ℝ) by ring]
+  · simp
+
+noncomputable def modifiedOverhead (L r : ℕ) : ℝ :=
+  (((L+r:ℕ):ℝ)*Real.log 2+1)/((2:ℝ)^(L+r)*1000)
+
+lemma modifiedOverhead_tendsto (L : ℕ) :
+    Filter.Tendsto (modifiedOverhead L) Filter.atTop (nhds 0) := by
+  have h₁ := tendsto_pow_const_div_const_pow_of_one_lt 1 (by norm_num : (1:ℝ)<2)
+  have h₀ := tendsto_pow_const_div_const_pow_of_one_lt 0 (by norm_num : (1:ℝ)<2)
+  simp only [pow_one] at h₁
+  simp only [pow_zero] at h₀
+  have ht := ((h₁.const_mul (Real.log 2)).add (h₀.const_mul ((L:ℝ)*Real.log 2+1))).div_const
+    ((2:ℝ)^L*1000)
+  convert ht using 1
+  · funext r
+    simp only [modifiedOverhead,Nat.cast_add,pow_add]
+    field_simp
+    ring
+  · simp
+
+theorem modified_arbitrarily_small_rate (η : ℝ) (hη : 0 < η) :
+    ∃ L : ℕ, 0 < L ∧ ∃ ε : ℝ, 0 < ε ∧ ε < 1/2 ∧
+      ∀ᶠ r : ℕ in Filter.atTop, ∀ (E : Finset (Card r)), (E.card:ℝ) ≤ ε*(2:ℝ)^r →
+        ∀ (e : Card (L+r) ↪ Card (L+r)),
+          modifiedRowMoment L r E e (1/1000) ≤ Real.exp (η*(2:ℝ)^(L+r)) := by
+  have hsmall : ∀ᶠ L : ℕ in Filter.atTop, heightDecay L < η/3 :=
+    heightDecay_tendsto.eventually (gt_mem_nhds (by linarith : (0:ℝ)<η/3))
+  obtain ⟨L,hdec,hL⟩ := (hsmall.and (Filter.eventually_ge_atTop 1)).exists
+  have hL0 : (0:ℝ) < L := Nat.cast_pos.mpr hL
+  let ε : ℝ := min (1/4) (η*250/(3*L))
+  have hε : 0 < ε := lt_min (by norm_num) (by positivity)
+  have hε1 : ε < 1/2 := (min_le_left _ _).trans_lt (by norm_num)
+  have hεL : ε*L/250 ≤ η/3 := by
+    have h := min_le_right (1/4:ℝ) (η*250/(3*L))
+    change ε ≤ η*250/(3*L) at h
+    have hh := mul_le_mul_of_nonneg_right h hL0.le
+    have he : (η*250/(3*(L:ℝ)))*(L:ℝ)/250 = η/3 := by field_simp
+    exact (div_le_div_of_nonneg_right hh (by norm_num)).trans he.le
+  refine ⟨L,hL,ε,hε,hε1,?_⟩
+  have hr : ∀ᶠ r : ℕ in Filter.atTop, modifiedOverhead L r < η/3 :=
+    (modifiedOverhead_tendsto L).eventually (gt_mem_nhds (by linarith : (0:ℝ)<η/3))
+  filter_upwards [hr] with r hr
+  intro E hE e
+  have he := Finite.surjective_of_injective e.injective
+  apply (modified_moment_exception_bound L r hL E e he ε hE).trans
+  apply Real.exp_le_exp.mpr
+  have hn : (0:ℝ) < (2:ℝ)^(L+r) := by positivity
+  have hd : modifiedOverhead L r * (2:ℝ)^(L+r) =
+      (((L+r:ℕ):ℝ)*Real.log 2+1)/1000 := by
+    unfold modifiedOverhead
+    field_simp
+  have hh := mul_le_mul_of_nonneg_right hr.le hn.le
+  rw [hd] at hh
+  have ht := mul_le_mul_of_nonneg_left (show heightDecay L+ε*L/250 ≤ 2*η/3 by linarith) hn.le
+  linarith
+
+
+namespace DensityTransfer
+open scoped BigOperators
+
+noncomputable def applyKernel {X : Type*} [Fintype X] (P : X → X → ℝ)
+    (v : EuclideanSpace ℂ X) : EuclideanSpace ℂ X :=
+  WithLp.toLp 2 (fun x => ∑ y, (P x y : ℂ)*v y)
+
+lemma applyKernel_apply {X : Type*} [Fintype X] (P : X → X → ℝ)
+    (v : EuclideanSpace ℂ X) (x : X) :
+    applyKernel P v x = ∑ y, (P x y : ℂ)*v y := rfl
+
+lemma applyKernel_add {X : Type*} [Fintype X] (P Q : X → X → ℝ)
+    (v : EuclideanSpace ℂ X) :
+    applyKernel (fun x y => P x y+Q x y) v = applyKernel P v+applyKernel Q v := by
+  ext x
+  simp only [applyKernel_apply,Complex.ofReal_add,add_mul,Finset.sum_add_distrib]
+  rfl
+
+lemma kernel_component_sq {X : Type*} [Fintype X] (P : X → X → ℝ)
+    (hP : ∀ x y, 0 ≤ P x y) (v : EuclideanSpace ℂ X) (x : X) :
+    ‖applyKernel P v x‖^2 ≤ (∑ y, P x y) * ∑ y, P x y*‖v y‖^2 := by
+  have hs : ‖applyKernel P v x‖ ≤ ∑ y, P x y*‖v y‖ := by
+    rw [applyKernel_apply]
+    apply (norm_sum_le _ _).trans
+    apply Finset.sum_le_sum
+    intro y _
+    rw [norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (hP x y)]
+  apply (pow_le_pow_left₀ (norm_nonneg _) hs 2).trans
+  exact Finset.sum_sq_le_sum_mul_sum_of_sq_le_mul Finset.univ
+    (fun y _ => hP x y) (fun y _ => mul_nonneg (hP x y) (sq_nonneg _))
+    (fun y _ => by ring_nf; exact le_rfl)
+
+lemma schur_bound {X : Type*} [Fintype X] (P : X → X → ℝ)
+    (hP : ∀ x y, 0 ≤ P x y) (δ : ℝ) (hδ : 0 ≤ δ)
+    (hrow : ∀ x, ∑ y, P x y ≤ δ) (hcol : ∀ y, ∑ x, P x y ≤ δ)
+    (v : EuclideanSpace ℂ X) : ‖applyKernel P v‖ ≤ δ*‖v‖ := by
+  have hsq : ‖applyKernel P v‖^2 ≤ δ^2*‖v‖^2 := by
+    rw [EuclideanSpace.norm_sq_eq,EuclideanSpace.norm_sq_eq]
+    calc
+      _ ≤ ∑ x, δ * ∑ y, P x y*‖v y‖^2 := by
+        apply Finset.sum_le_sum
+        intro x _
+        apply (kernel_component_sq P hP v x).trans
+        exact mul_le_mul_of_nonneg_right (hrow x)
+          (Finset.sum_nonneg (fun y _ => mul_nonneg (hP x y) (sq_nonneg _)))
+      _ = δ * ∑ y, (∑ x, P x y)*‖v y‖^2 := by
+        rw [←Finset.mul_sum,Finset.sum_comm]
+        congr 1
+        apply Finset.sum_congr rfl
+        intro y _
+        exact (Finset.sum_mul Finset.univ _ _).symm
+      _ ≤ δ * ∑ y, δ*‖v y‖^2 := by
+        apply mul_le_mul_of_nonneg_left _ hδ
+        exact Finset.sum_le_sum (fun y _ => mul_le_mul_of_nonneg_right (hcol y) (sq_nonneg _))
+      _ = _ := by rw [←Finset.mul_sum]; ring
+  have hn := norm_nonneg (applyKernel P v)
+  have hr := mul_nonneg hδ (norm_nonneg v)
+  nlinarith
+
+lemma hilbert_schmidt_family {X I : Type*} [Fintype X] [Fintype I]
+    (P : X → X → ℝ) (v : I → EuclideanSpace ℂ X) (hv : Orthonormal ℂ v) :
+    (∑ i, ‖applyKernel P (v i)‖^2) ≤ ∑ x, ∑ y, (P x y)^2 := by
+  simp_rw [EuclideanSpace.norm_sq_eq]
+  rw [Finset.sum_comm]
+  apply Finset.sum_le_sum
+  intro x _
+  let w : EuclideanSpace ℂ X := WithLp.toLp 2 (fun y => (P x y : ℂ))
+  have he (i : I) : ‖applyKernel P (v i) x‖ = ‖inner ℂ (v i) w‖ := by
+    rw [norm_inner_symm]
+    congr 1
+    simp only [applyKernel_apply,PiLp.inner_apply]
+    apply Finset.sum_congr rfl
+    intro y _
+    simp [w,RCLike.inner_apply,mul_comm]
+  simp_rw [he]
+  have h := hv.sum_inner_products_le (s := Finset.univ) w
+  apply h.trans_eq
+  rw [EuclideanSpace.norm_sq_eq]
+  apply Finset.sum_congr rfl
+  intro y _
+  simp [w,Complex.norm_real,Real.norm_eq_abs,sq_abs]
+
+
+lemma tail_density_pointwise (t a z : ℝ) (ht : 0 ≤ t) (ha : 0 ≤ a) (hz : 0 < z) :
+    (if z < t then t else 0) ≤ t^(1+a)*z^(-a) := by
+  split_ifs with h
+  · rw [Real.rpow_neg hz.le,←div_eq_mul_inv]
+    apply (le_div_iff₀ (Real.rpow_pos_of_pos hz a)).mpr
+    rw [Real.rpow_add (lt_trans hz h),Real.rpow_one]
+    exact mul_le_mul_of_nonneg_left (Real.rpow_le_rpow hz.le h.le ha) ht
+  · positivity
+
+lemma tail_density_mean {X : Type*} [Fintype X] (f : X → ℝ) (hf : ∀ x, 0 ≤ f x)
+    (a z : ℝ) (ha : 0 ≤ a) (hz : 0 < z) :
+    finiteMean (fun x => if z < f x then f x else 0) ≤ finiteMean (fun x => f x^(1+a))*z^(-a) := by
+  rw [←finiteMean_mul_const]
+  exact finiteMean_mono (fun x => tail_density_pointwise (f x) a z (hf x) ha hz)
+
+noncomputable def tailKernel {X : Type*} [Fintype X] (P : X → X → ℝ) (z : ℝ) : X → X → ℝ :=
+  fun x y => if z < (Fintype.card X:ℝ)*P x y then P x y else 0
+
+noncomputable def headKernel {X : Type*} [Fintype X] (P : X → X → ℝ) (z : ℝ) : X → X → ℝ :=
+  fun x y => if z < (Fintype.card X:ℝ)*P x y then 0 else P x y
+
+lemma tail_add_head {X : Type*} [Fintype X] (P : X → X → ℝ) (z : ℝ) :
+    (fun x y => tailKernel P z x y+headKernel P z x y) = P := by
+  funext x y
+  unfold tailKernel headKernel
+  split_ifs <;> simp
+
+lemma tailKernel_nonneg {X : Type*} [Fintype X] (P : X → X → ℝ) (z : ℝ)
+    (hP : ∀ x y, 0 ≤ P x y) (x y : X) : 0 ≤ tailKernel P z x y := by
+  unfold tailKernel
+  split_ifs <;> simp only [hP,le_refl]
+
+lemma tailKernel_row {X : Type*} [Fintype X] [Nonempty X] (P : X → X → ℝ)
+    (hP : ∀ x y, 0 ≤ P x y) (a z M : ℝ) (ha : 0 ≤ a) (hz : 0 < z)
+    (hM : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*P x y)^(1+a)) ≤ M) (x : X) :
+    ∑ y, tailKernel P z x y ≤ M*z^(-a) := by
+  have hn : (Fintype.card X:ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
+  have he : finiteMean (fun y => if z < (Fintype.card X:ℝ)*P x y then
+      (Fintype.card X:ℝ)*P x y else 0) = ∑ y, tailKernel P z x y := by
+    unfold finiteMean
+    calc
+      _ = (∑ y, (Fintype.card X:ℝ)*tailKernel P z x y)/(Fintype.card X:ℝ) := by
+        congr 1
+        apply Finset.sum_congr rfl
+        intro y _
+        by_cases hy : z < (Fintype.card X:ℝ)*P x y
+        · simp only [tailKernel,hy,↓reduceIte]
+        · simp only [tailKernel,hy,↓reduceIte,mul_zero]
+      _ = _ := by rw [←Finset.mul_sum]; field_simp
+  rw [←he]
+  exact (tail_density_mean _ (fun y => mul_nonneg (Nat.cast_nonneg _) (hP x y)) a z ha hz).trans
+    (mul_le_mul_of_nonneg_right (hM x) (Real.rpow_nonneg hz.le _))
+
+lemma tailKernel_schur {X : Type*} [Fintype X] [Nonempty X] (P : X → X → ℝ)
+    (hP : ∀ x y, 0 ≤ P x y) (hsym : ∀ x y, P x y = P y x)
+    (a z M : ℝ) (ha : 0 ≤ a) (hz : 0 < z) (hM0 : 0 ≤ M)
+    (hM : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*P x y)^(1+a)) ≤ M)
+    (v : EuclideanSpace ℂ X) :
+    ‖applyKernel (tailKernel P z) v‖ ≤ (M*z^(-a))*‖v‖ := by
+  apply schur_bound _ (tailKernel_nonneg P z hP) _ (by positivity)
+    (tailKernel_row P hP a z M ha hz hM)
+  intro y
+  have hs : (fun x => tailKernel P z x y) = (fun x => tailKernel P z y x) := by
+    funext x
+    simp only [tailKernel,hsym x y]
+  rw [hs]
+  exact tailKernel_row P hP a z M ha hz hM y
+
+lemma headKernel_hs {X : Type*} [Fintype X] [Nonempty X] (P : X → X → ℝ)
+    (hP : ∀ x y, 0 ≤ P x y) (hrow : ∀ x, ∑ y, P x y = 1) (z : ℝ) (hz : 0 ≤ z) :
+    (∑ x, ∑ y, (headKernel P z x y)^2) ≤ z := by
+  have hn : (0:ℝ) < Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
+  have he (x y : X) : (headKernel P z x y)^2 ≤ (z/(Fintype.card X:ℝ))*P x y := by
+    unfold headKernel
+    split_ifs with h
+    · simp only [zero_pow (by omega : (2:ℕ) ≠ 0)]
+      exact mul_nonneg (div_nonneg hz hn.le) (hP x y)
+    · have h' : (Fintype.card X:ℝ)*P x y ≤ z := le_of_not_gt h
+      have hh := mul_le_mul_of_nonneg_right h' (hP x y)
+      rw [show (z/(Fintype.card X:ℝ))*P x y = (z*P x y)/(Fintype.card X:ℝ) by ring]
+      apply (le_div_iff₀ hn).mpr
+      nlinarith
+  calc
+    _ ≤ ∑ x, ∑ y, (z/(Fintype.card X:ℝ))*P x y :=
+      Finset.sum_le_sum (fun x _ => Finset.sum_le_sum (fun y _ => he x y))
+    _ = z := by
+      simp only [←Finset.mul_sum,hrow,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
+      field_simp
+
+
+theorem multiplicity_amplification {X I : Type*} [Fintype X] [Nonempty X]
+    [Fintype I] [Nonempty I] (P : X → X → ℝ)
+    (hP : ∀ x y, 0 ≤ P x y) (hsym : ∀ x y, P x y = P y x)
+    (hrow : ∀ x, ∑ y, P x y = 1) (a z M : ℝ) (ha : 0 ≤ a) (hz : 0 < z) (hM0 : 0 ≤ M)
+    (hM : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*P x y)^(1+a)) ≤ M)
+    (v : I → EuclideanSpace ℂ X) (hv : Orthonormal ℂ v)
+    (μ : ℝ) (hvμ : ∀ i, μ ≤ ‖applyKernel P (v i)‖) :
+    μ ≤ M*z^(-a)+Real.sqrt (z/(Fintype.card I:ℝ)) := by
+  let δ : ℝ := M*z^(-a)
+  have hδ : 0 ≤ δ := by dsimp [δ]; positivity
+  have hm : (0:ℝ) < Fintype.card I := Nat.cast_pos.mpr Fintype.card_pos
+  change μ ≤ δ+_
+  by_cases hμ : μ ≤ δ
+  · exact hμ.trans (le_add_of_nonneg_right (Real.sqrt_nonneg _))
+  have hmd : 0 ≤ μ-δ := by linarith
+  have hsmall (i : I) : μ-δ ≤ ‖applyKernel (headKernel P z) (v i)‖ := by
+    have hr := tailKernel_schur P hP hsym a z M ha hz hM0 hM (v i)
+    rw [hv.norm_eq_one i,mul_one] at hr
+    have he := applyKernel_add (tailKernel P z) (headKernel P z) (v i)
+    rw [tail_add_head] at he
+    have hn := norm_add_le (applyKernel (tailKernel P z) (v i)) (applyKernel (headKernel P z) (v i))
+    rw [←he] at hn
+    have hh := hvμ i
+    change ‖applyKernel (tailKernel P z) (v i)‖ ≤ δ at hr
+    linarith
+  have hsum : (Fintype.card I:ℝ)*(μ-δ)^2 ≤ z := by
+    calc
+      _ = ∑ i : I, (μ-δ)^2 := by simp
+      _ ≤ ∑ i : I, ‖applyKernel (headKernel P z) (v i)‖^2 :=
+        Finset.sum_le_sum (fun i _ => pow_le_pow_left₀ hmd (hsmall i) 2)
+      _ ≤ ∑ x, ∑ y, (headKernel P z x y)^2 := hilbert_schmidt_family _ v hv
+      _ ≤ z := headKernel_hs P hP hrow z hz.le
+  have hsq : (μ-δ)^2 ≤ z/(Fintype.card I:ℝ) := (le_div_iff₀ hm).mpr (by nlinarith)
+  have hs := (Real.le_sqrt hmd (div_nonneg hz.le hm.le)).mpr hsq
+  linarith
+
+end DensityTransfer
+namespace PairRouting
+open scoped BigOperators
+
+lemma tupleProbability_row_sum {ι β Ω : Type*} [Fintype ι] [Fintype β]
+    [DecidableEq β] [Fintype Ω] [Nonempty Ω]
+    (P : Ω → Equiv.Perm β) (e : ι ↪ β) :
+    (∑ f : ι ↪ β, tupleProbability P e f) = 1 := by
+  classical
+  simp only [tupleProbability]
+  rw [←finiteMean_sum]
+  have h (ω : Ω) (f : ι ↪ β) : (∀ i, P ω (e i) = f i) ↔ e.trans (P ω).toEmbedding = f := by
+    constructor
+    · intro h; ext i; exact h i
+    · intro h i; exact congrArg (fun g : ι ↪ β => g i) h
+  simp_rw [h]
+  simp only [Finset.sum_ite_eq,Finset.mem_univ,↓reduceIte,finiteMean_const]
+
+lemma tupleProbability_inverse {ι β Ω : Type*} [Fintype ι] [DecidableEq β]
+    [Fintype Ω] (P : Ω → Equiv.Perm β) (e f : ι → β) :
+    tupleProbability (fun ω => (P ω)⁻¹) e f = tupleProbability P f e := by
+  classical
+  apply finiteMean_congr
+  intro ω
+  congr 1
+  apply propext
+  constructor
+  · intro h i
+    have hh := congrArg (P ω) (h i)
+    simpa only [Equiv.Perm.inv_def,Equiv.apply_symm_apply] using hh.symm
+  · intro h i
+    have hh := congrArg (fun x => (P ω).symm x) (h i)
+    simpa only [Equiv.Perm.inv_def,Equiv.symm_apply_apply] using hh.symm
+
+lemma tupleProbability_equiv {ι β Ω Ω' : Type*} [Fintype ι] [DecidableEq β]
+    [Fintype Ω] [Fintype Ω'] (P : Ω → Equiv.Perm β) (τ : Ω ≃ Ω') (e f : ι → β) :
+    tupleProbability P e f = tupleProbability (fun ω => P (τ.symm ω)) e f :=
+  finiteMean_equiv τ _
+
+lemma tupleProbability_symmetric {ι β Ω : Type*} [Fintype ι] [DecidableEq β]
+    [Fintype Ω] (P : Ω → Equiv.Perm β) (τ : Equiv.Perm Ω)
+    (hτ : ∀ ω, P (τ ω) = (P ω)⁻¹) (e f : ι → β) :
+    tupleProbability P e f = tupleProbability P f e := by
+  rw [tupleProbability_equiv P τ.symm]
+  simp only [Equiv.symm_symm,hτ]
+  exact tupleProbability_inverse P e f
+
+end PairRouting
+
+lemma palindromePerm_reverse (d : ℕ) (ω : BenesCoins d) :
+    palindromePerm d (ω.2,ω.1) = (palindromePerm d ω)⁻¹ := by
+  simp only [palindromePerm,mul_inv_rev]
+  rfl
+
+lemma palindrome_kernel_symmetric (d : ℕ) {ι : Type*} [Fintype ι]
+    (e f : ι ↪ Card d) :
+    PairRouting.tupleProbability (palindromePerm d) e f =
+      PairRouting.tupleProbability (palindromePerm d) f e :=
+  PairRouting.tupleProbability_symmetric _ (Equiv.prodComm _ _) (palindromePerm_reverse d) e f
+
+theorem palindrome_density_transfer (d : ℕ) {ι I : Type*} [Fintype ι] [Fintype I]
+    [Nonempty I] (e₀ : ι ↪ Card d) (hk : 0 < Fintype.card ι)
+    (v : I → EuclideanSpace ℂ (ι ↪ Card d)) (hv : Orthonormal ℂ v)
+    (μ : ℝ) (hvμ : ∀ i, μ ≤ ‖DensityTransfer.applyKernel
+      (fun e f : ι ↪ Card d => PairRouting.tupleProbability (palindromePerm d) e f) (v i)‖)
+    (z : ℝ) (hz : 0 < z) :
+    μ ≤ Real.exp (densityConstant*Fintype.card ι*
+      ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ))*z^(-(1/1000:ℝ))+
+      Real.sqrt (z/(Fintype.card I:ℝ)) := by
+  let : Nonempty (ι ↪ Card d) := ⟨e₀⟩
+  apply DensityTransfer.multiplicity_amplification
+    (fun e f : ι ↪ Card d => PairRouting.tupleProbability (palindromePerm d) e f)
+    (fun e f => PairRouting.tupleProbability_nonneg _ e f)
+    (palindrome_kernel_symmetric d) (PairRouting.tupleProbability_row_sum _)
+    (1/1000) z _ (by norm_num) hz (Real.exp_nonneg _) _ v hv μ hvμ
+  intro e
+  exact palindrome_row_moment_bound d e hk
+
+
+noncomputable def modifiedReverseCoins (L : ℕ) : (r : ℕ) → ModifiedCoins L r → ModifiedCoins L r
+  | 0, ω => ((fun u => butterflyPerm L (decodeButterfly L ω.2.1) * (ω.1 u)⁻¹ *
+      butterflyPerm L (decodeButterfly L ω.2.1)),ω.2)
+  | r+1, ω =>
+      let z := modifiedStepEquiv L r ω
+      (modifiedStepEquiv L r).symm
+        ((modifiedReverseCoins L r z.1.1,modifiedReverseCoins L r z.1.2),(z.2.2,z.2.1))
+
+lemma modifiedReverseCoins_involutive (L r : ℕ) :
+    Function.Involutive (modifiedReverseCoins L r) := by
+  induction r with
+  | zero =>
+      intro ω
+      apply Prod.ext
+      · funext u
+        simp only [modifiedReverseCoins,mul_inv_rev,inv_inv]
+        simp [mul_assoc]
+      · rfl
+  | succ r ih =>
+      change ∀ ω, modifiedReverseCoins L r (modifiedReverseCoins L r ω) = ω at ih
+      intro ω
+      simp only [modifiedReverseCoins,Equiv.apply_symm_apply,ih]
+      exact (modifiedStepEquiv L r).symm_apply_apply ω
+
+end Thorp
+
+end ThorpNine.SparseSaving
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/SparseSaving/ModifiedStageEquiv.lean b/lean/OAI/Probability/ThorpRouting/SparseSaving/ModifiedStageEquiv.lean
index 10927a55e5b75a75f7c2bce23f743eca1b1301d0..40a61a294f435e3e0f79786df3f3bd9c910c79f1 100644
--- a/lean/OAI/Probability/ThorpRouting/SparseSaving/ModifiedStageEquiv.lean
+++ b/lean/OAI/Probability/ThorpRouting/SparseSaving/ModifiedStageEquiv.lean
@@ -1,875 +1,875 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.SparseSaving.BlockProduct
-
-namespace OAI
-
-namespace ThorpNine.SparseSaving
-
-namespace Thorp
-open scoped BigOperators Classical
-
-def modifiedStageEquiv (r L : ℕ) :
-    ((OuterCoins r L × (Card r → Equiv.Perm (Card L) × BenesCoins L)) × OuterCoins r L) ≃
-      ModifiedCoins L r :=
-  (modifiedAssembleEquiv _ _ _ _).trans
-    (Equiv.prodCongr (Equiv.refl _) (Equiv.prodCongr (parallelSplitCoinEquiv r L) (parallelSplitCoinEquiv r L)))
-
-lemma modifiedStageEquiv_apply (r L : ℕ)
-    (w : (OuterCoins r L × (Card r → Equiv.Perm (Card L) × BenesCoins L)) × OuterCoins r L) :
-    modifiedStageEquiv r L w=
-      ((fun b => (w.1.2 b).1),
-       (assembleBits r L (fun x => (w.1.2 x.1).2.1 x.2) w.1.1,
-        assembleBits r L (fun x => (w.1.2 x.1).2.2 x.2) w.2)) := by
-  simp only [modifiedStageEquiv,Equiv.trans_apply,Equiv.prodCongr_apply,modifiedAssembleEquiv,
-    Equiv.coe_fn_mk,Prod.map_apply,Equiv.refl_apply,parallelSplitCoinEquiv_apply]
-
-lemma palindrome_stage (r L : ℕ)
-    (w : (OuterCoins r L × (Card r → BenesCoins L)) × OuterCoins r L) :
-    splitPerm r L (palindromePerm (L+r) (palindromeStageEquiv r L w))=
-      outerPerm r L w.1.1*ParallelBlocks.lift (fun b => palindromePerm L (w.1.2 b))*
-        (outerPerm r L w.2)⁻¹ := by
-  rw [palindromeStageEquiv_apply]
-  change splitPerm r L (_*_⁻¹)=_
-  rw [map_mul,map_inv,splitPerm_butterfly,splitPerm_butterfly,mul_inv_rev]
-  rw [mul_assoc,←mul_assoc (ParallelBlocks.lift _),←map_inv,←map_mul,←mul_assoc]
-  rfl
-
-noncomputable def localModifiedPerm (L : ℕ) (e : Bool)
-    (w : Equiv.Perm (Card L) × BenesCoins L) : Equiv.Perm (Card L) :=
-  if e then 1 else butterflyPerm L (decodeButterfly L w.2.1)*w.1⁻¹
-
-lemma modified_stage (r L : ℕ) (E : Finset (Card r))
-    (w : (OuterCoins r L × (Card r → Equiv.Perm (Card L) × BenesCoins L)) × OuterCoins r L) :
-    splitPerm r L (modifiedPerm L r E (modifiedStageEquiv r L w))=
-      outerPerm r L w.1.1*ParallelBlocks.lift (fun b => localModifiedPerm L (decide (b∈E)) (w.1.2 b))*
-        (outerPerm r L w.2)⁻¹ := by
-  rw [modifiedStageEquiv_apply]
-  unfold modifiedPerm modifiedInputTree
-  rw [map_mul,map_inv,splitPerm_butterfly,splitPerm_boundaryTree,mul_inv_rev]
-  rw [mul_assoc,←mul_assoc (ParallelBlocks.lift _),←map_inv,←map_mul,←mul_assoc]
-  congr 2
-  apply congrArg ParallelBlocks.lift
-  funext b
-  simp only [Pi.mul_apply,Pi.inv_apply,modifiedBoundary,boundaryOutput_split,localModifiedPerm]
-  by_cases h : b∈E <;> simp [h]
-
-end Thorp
-namespace Thorp.UnitaryFinite
-open scoped BigOperators Classical
-variable {G : Type*} [Group G]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-variable {Ω Ξ : Type*} [Fintype Ω] [Fintype Ξ]
-
-lemma sampleOperator_sandwich (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
-    (P : Ω → G) (Q : Ξ → G) :
-    sampleOperator ρ (fun w : (Ω × Ξ) × Ω => P w.1.1*Q w.1.2*(P w.2)⁻¹)=
-      sampleOperator ρ P*sampleOperator ρ Q*(sampleOperator ρ P).adjoint := by
-  rw [sampleOperator_prod ρ (fun w : Ω × Ξ => P w.1*Q w.2) (fun w : Ω => (P w)⁻¹),
-    sampleOperator_prod ρ P Q,sampleOperator_inv ρ hρ]
-
-lemma localModified_average (L : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card L)) V) (e : Bool) :
-    sampleOperator ρ (localModifiedPerm L e)=if e then 1 else uniformOperator ρ := by
-  cases e
-  · simp only [Bool.false_eq_true,ite_false]
-    rw [←sampleOperator_equiv ρ _ (Equiv.prodComm _ _)]
-    exact sampleOperator_uniform_mul ρ (fun w : BenesCoins L => butterflyPerm L (decodeButterfly L w.1))
-  · simp only [ite_true]
-    exact sampleOperator_one ρ
-
-end Thorp.UnitaryFinite
-
-namespace Thorp
-open scoped BigOperators Classical
-open UnitaryFinite
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-lemma sampleOperator_palindrome_stage (r L : ℕ)
-    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) (hρ : IsUnitary ρ) :
-    sampleOperator (ρ.comp (splitPerm r L).toMonoidHom) (palindromePerm (L+r))=
-      sampleOperator ρ (outerPerm r L)*
-        sampleOperator ρ (fun w : Card r → BenesCoins L => ParallelBlocks.lift (fun b => palindromePerm L (w b)))*
-          (sampleOperator ρ (outerPerm r L)).adjoint := by
-  have he : ((splitPerm r L).toMonoidHom ∘ palindromePerm (L+r)) ∘ palindromeStageEquiv r L=
-      fun w => outerPerm r L w.1.1*ParallelBlocks.lift (fun b => palindromePerm L (w.1.2 b))*
-        (outerPerm r L w.2)⁻¹ := funext (palindrome_stage r L)
-  calc
-    _ = sampleOperator ρ (((splitPerm r L).toMonoidHom ∘ palindromePerm (L+r)) ∘ palindromeStageEquiv r L) :=
-      (sampleOperator_equiv ρ ((splitPerm r L).toMonoidHom ∘ palindromePerm (L+r)) (palindromeStageEquiv r L)).symm
-    _ = _ := (congrArg (sampleOperator ρ) he).trans
-      (sampleOperator_sandwich ρ hρ (outerPerm r L)
-        (fun w : Card r → BenesCoins L => ParallelBlocks.lift (fun b => palindromePerm L (w b))))
-
-lemma sampleOperator_modified_stage (r L : ℕ) (E : Finset (Card r))
-    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) (hρ : IsUnitary ρ) :
-    sampleOperator (ρ.comp (splitPerm r L).toMonoidHom) (modifiedPerm L r E)=
-      sampleOperator ρ (outerPerm r L)*
-        sampleOperator ρ (fun w : Card r → Equiv.Perm (Card L) × BenesCoins L =>
-          ParallelBlocks.lift (fun b => localModifiedPerm L (decide (b∈E)) (w b)))*
-          (sampleOperator ρ (outerPerm r L)).adjoint := by
-  have he : ((splitPerm r L).toMonoidHom ∘ modifiedPerm L r E) ∘ modifiedStageEquiv r L=
-      fun w => outerPerm r L w.1.1*ParallelBlocks.lift
-        (fun b => localModifiedPerm L (decide (b∈E)) (w.1.2 b))*(outerPerm r L w.2)⁻¹ :=
-    funext (modified_stage r L E)
-  calc
-    _ = sampleOperator ρ (((splitPerm r L).toMonoidHom ∘ modifiedPerm L r E) ∘ modifiedStageEquiv r L) :=
-      (sampleOperator_equiv ρ ((splitPerm r L).toMonoidHom ∘ modifiedPerm L r E) (modifiedStageEquiv r L)).symm
-    _ = _ := (congrArg (sampleOperator ρ) he).trans
-      (sampleOperator_sandwich ρ hρ (outerPerm r L)
-        (fun w : Card r → Equiv.Perm (Card L) × BenesCoins L =>
-          ParallelBlocks.lift (fun b => localModifiedPerm L (decide (b∈E)) (w b))))
-
-end Thorp
-namespace Thorp.CommutingBlocks
-open scoped BigOperators Classical
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
-variable {ι : Type*} [Fintype ι] [DecidableEq ι]
-
-noncomputable def retain (P : ι → V →L[ℂ] V) (E : Finset ι) (i : ι) : V →L[ℂ] V :=
-  if i∈E then 1 else P i
-
-omit [CompleteSpace V] [Fintype ι] in
-lemma retain_commute (P : ι → V →L[ℂ] V) (hP : ∀ i j, Commute (P i) (P j))
-    (E : Finset ι) (i j : ι) : Commute (retain P E i) (retain P E j) := by
-  unfold retain
-  split_ifs <;> first | exact Commute.one_left _ | exact Commute.one_right _ | exact hP _ _
-
-omit [CompleteSpace V] in
-lemma blocks_retain (P : ι → V →L[ℂ] V) (hP : ∀ i j, Commute (P i) (P j)) (E : Finset ι) :
-    blocks (retain P E) (retain_commute P hP E)=coarse P hP E := by
-  have hs : E∪Eᶜ=(Finset.univ : Finset ι) := Finset.union_compl E
-  have hh := Finset.noncommProd_union_of_disjoint (s := E) (t := Eᶜ) disjoint_compl_right
-    (retain P E) (fun _ _ _ _ _ => retain_commute P hP E _ _)
-  simp only [hs] at hh
-  change blocks (retain P E) (retain_commute P hP E)=_ at hh
-  rw [hh]
-  have he : E.noncommProd (retain P E) (fun _ _ _ _ _ => retain_commute P hP E _ _)=1 := by
-    exact (Finset.noncommProd_eq_pow_card E (retain P E)
-      (fun i _ j _ _ => retain_commute P hP E i j) 1 (by intro i hi; exact ite_eq_left hi)).trans (one_pow _)
-  rw [he,one_mul]
-  apply Finset.noncommProd_congr rfl
-  intro i hi
-  exact ite_eq_right (Finset.mem_compl.mp hi)
-
-end Thorp.CommutingBlocks
-
-namespace Thorp.ParallelBlocks
-open scoped BigOperators Classical
-open UnitaryFinite
-variable {B X : Type*} [Fintype B] [Fintype X] [DecidableEq X]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-noncomputable def centralProjection (ρ : Representation ℂ (Equiv.Perm (B × X)) V) (b : B) : V →L[ℂ] V :=
-  uniformOperator (ρ.comp (injection b))
-
-omit [Fintype B] in
-lemma centralProjection_commute (ρ : Representation ℂ (Equiv.Perm (B × X)) V) (b c : B) :
-    Commute (centralProjection ρ b) (centralProjection ρ c) := by
-  simpa only [centralProjection,sampleOperator_uniform] using
-    local_average_commute ρ (fun _ => id : B → Equiv.Perm X → Equiv.Perm X) b c
-
-omit [Fintype B] in
-lemma local_central_commute {Ω : Type*} [Fintype Ω] [Nonempty Ω]
-    (ρ : Representation ℂ (Equiv.Perm (B × X)) V) (P : B → Ω → Equiv.Perm X) (b c : B) :
-    Commute (sampleOperator (ρ.comp (injection b)) (P b)) (centralProjection ρ c) := by
-  by_cases h : b=c
-  · subst c
-    change _*uniformOperator _=uniformOperator _*_
-    rw [sampleOperator_uniform_absorb,sampleOperator_weighted,weighted_uniform_right,sampleLaw_sum,one_smul]
-  · unfold centralProjection
-    rw [←sampleOperator_uniform,sampleOperator_comp,sampleOperator_comp]
-    exact sampleOperator_commute ρ _ _ (fun _ _ => injection_commute h _ _)
-
-end Thorp.ParallelBlocks
-namespace Thorp
-open scoped BigOperators Classical
-open UnitaryFinite ParallelBlocks CommutingBlocks
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-lemma modified_central_average (r L : ℕ) (E : Finset (Card r))
-    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) :
-    sampleOperator ρ (fun w : Card r → Equiv.Perm (Card L) × BenesCoins L =>
-      lift (fun b => localModifiedPerm L (decide (b∈E)) (w b)))=
-      coarse (centralProjection ρ) (centralProjection_commute ρ) E := by
-  rw [sampleOperator_parallel ρ (fun b => localModifiedPerm L (decide (b∈E))),←blocks_retain]
-  apply Finset.noncommProd_congr rfl
-  · intro b _
-    rw [localModified_average]
-    simp only [retain,centralProjection,decide_eq_true_eq]
-  · intro b _ c _ _
-    exact local_average_commute ρ (fun b => localModifiedPerm L (decide (b∈E))) b c
-
-lemma central_local_gap (r L : ℕ)
-    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) (hρ : IsUnitary ρ) (b : Card r) :
-    ‖sampleOperator (ρ.comp (injection b)) (palindromePerm L)*(1-centralProjection ρ b)‖≤blockGap L := by
-  change ‖sampleOperator _ _*(1-uniformOperator _)‖≤_
-  rw [mul_sub,mul_one,sampleOperator_uniform_absorb,sampleOperator_weighted]
-  exact palindrome_off_invariants_gap L _ (unitary_comp ρ hρ _)
-
-lemma central_blocks_exception_bound (r L : ℕ)
-    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) (hρ : IsUnitary ρ)
-    (B : V →L[ℂ] V) (hB0 : ‖B‖≤1) (n : ℕ) :
-    ‖B*blocks (fun b : Card r => sampleOperator (ρ.comp (injection b)) (palindromePerm L))
-      (local_average_commute ρ (fun _ => palindromePerm L))*B.adjoint‖ ≤
-      (blockGap L)^n + ∑ E : Finset (Card r), if E.card<n then
-        ‖B*coarse (centralProjection ρ) (centralProjection_commute ρ) E*B.adjoint‖ else 0 := by
-  have hB : ‖B.adjoint‖≤1 := by
-    rw [ContinuousLinearMap.adjoint.norm_map]
-    exact hB0
-  have h := blocks_exception_bound (centralProjection ρ) (centralProjection_commute ρ)
-    (fun b => uniformOperator_projection (ρ.comp (injection b)) (unitary_comp ρ hρ _))
-    (fun b : Card r => sampleOperator (ρ.comp (injection b)) (palindromePerm L))
-    (local_average_commute ρ (fun _ => palindromePerm L))
-    (local_central_commute ρ (fun _ => palindromePerm L))
-    (fun b => sampleOperator_norm_le (ρ.comp (injection b)) (unitary_comp ρ hρ _) (palindromePerm L))
-    (blockGap L) (blockGap_pos L).le (blockGap_lt_one L).le (central_local_gap r L ρ hρ)
-    B.adjoint hB n
-  simpa only [ContinuousLinearMap.adjoint_adjoint] using h
-
-theorem palindrome_exception_bound (r L : ℕ)
-    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) (hρ : IsUnitary ρ) (n : ℕ) :
-    ‖sampleOperator (ρ.comp (splitPerm r L).toMonoidHom) (palindromePerm (L+r))‖ ≤
-      (blockGap L)^n + ∑ E : Finset (Card r), if E.card<n then
-        ‖sampleOperator (ρ.comp (splitPerm r L).toMonoidHom) (modifiedPerm L r E)‖ else 0 := by
-  rw [sampleOperator_palindrome_stage r L ρ hρ]
-  rw [sampleOperator_parallel ρ (fun _ => palindromePerm L)]
-  apply (central_blocks_exception_bound r L ρ hρ (sampleOperator ρ (outerPerm r L))
-    (sampleOperator_norm_le ρ hρ _) n).trans_eq
-  congr 1
-  apply Finset.sum_congr rfl
-  intro E _
-  split_ifs
-  · rw [sampleOperator_modified_stage r L E ρ hρ,modified_central_average]
-  · rfl
-
-end Thorp
-
-namespace Thorp.UnitaryFinite
-open scoped BigOperators ComplexConjugate Classical
-
-variable {G X Ω : Type*} [Group G] [Fintype G] [Fintype X] [Fintype Ω]
-
-noncomputable def actionKernel (a : G →* Equiv.Perm X) (P : Ω → G) (x y : X) : ℝ :=
-  finiteMean (fun ω => if a (P ω) x=y then 1 else 0)
-
-omit [Fintype G] in
-lemma actionKernel_apply (a : G →* Equiv.Perm X) (P : Ω → G)
-    (v : EuclideanSpace ℂ X) (x : X) :
-    DensityTransfer.applyKernel (actionKernel a P) v x =
-      (Fintype.card Ω:ℂ)⁻¹ * ∑ ω, v (a (P ω) x) := by
-  simp only [DensityTransfer.applyKernel_apply,actionKernel,finiteMean,
-    Complex.ofReal_sum,div_eq_mul_inv,Finset.sum_mul]
-  rw [Finset.sum_comm]
-  calc
-    _ = ∑ ω, (Fintype.card Ω:ℂ)⁻¹ * ∑ y : X, if a (P ω) x=y then v y else 0 := by
-      apply Finset.sum_congr rfl
-      intro ω _
-      rw [Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro y _
-      split_ifs <;> simp_all
-    _ = _ := by simp [Finset.mul_sum]
-
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-noncomputable def randomOperator (ρ : Representation ℂ G V) (P : Ω → G) : Module.End ℂ V :=
-  (Fintype.card Ω:ℂ)⁻¹ • ∑ ω, ρ (P ω)⁻¹
-
-omit [Fintype G] [FiniteDimensional ℂ V] in
-lemma coefficient_random (a : G →* Equiv.Perm X) (x₀ : X)
-    (ha : ∀ x, ∃ g, a g x₀=x) (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
-    (P : Ω → G) (w : V) (hw : IsFixed a x₀ ρ w) (v : V) :
-    DensityTransfer.applyKernel (actionKernel a P) (normalizedCoefficient a x₀ ha ρ w v) =
-      normalizedCoefficient a x₀ ha ρ w (randomOperator ρ P v) := by
-  ext x
-  rw [actionKernel_apply]
-  simp only [normalizedCoefficient,PiLp.smul_apply,smul_eq_mul,Finset.mul_sum,
-    coefficient,randomOperator,LinearMap.smul_apply,LinearMap.sum_apply,
-    inner_smul_right,inner_sum]
-  apply Finset.sum_congr rfl
-  intro ω _
-  have h := coefficient_action a x₀ ha ρ hρ w hw v (P ω) x
-  change inner ℂ (ρ (representative a x₀ ha (a (P ω) x)) w) v =
-    inner ℂ (ρ (representative a x₀ ha x) w) (ρ (P ω)⁻¹ v) at h
-  rw [h]
-  ring
-
-noncomputable def fixedSpace (a : G →* Equiv.Perm X) (x₀ : X) (ρ : Representation ℂ G V) :
-    Submodule ℂ V where
-  carrier := {w | IsFixed a x₀ ρ w}
-  zero_mem' := fun g hg => map_zero _
-  add_mem' := fun hw hz g hg => by rw [map_add,hw g hg,hz g hg]
-  smul_mem' := fun c w hw g hg => by rw [map_smul,hw g hg]
-
-theorem operator_amplification (a : G →* Equiv.Perm X) (x₀ : X)
-    (ha : ∀ x, ∃ g, a g x₀=x) (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
-    (hρ : IsUnitary ρ) (P : Ω → G)
-    (hsym : ∀ x y, actionKernel a P x y=actionKernel a P y x)
-    (hrow : ∀ x, ∑ y, actionKernel a P x y=1)
-    (b z M : ℝ) (hb : 0≤b) (hz : 0<z) (hM0 : 0≤M)
-    (hM : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*actionKernel a P x y)^(1+b)) ≤ M)
-    (hm : 0 < Module.finrank ℂ (fixedSpace a x₀ ρ)) (v : V) (hv : ‖v‖=1) :
-    ‖randomOperator ρ P v‖ ≤ M*z^(-b)+Real.sqrt (z/(Module.finrank ℂ (fixedSpace a x₀ ρ):ℝ)) := by
-  let K := fixedSpace a x₀ ρ
-  let m := Module.finrank ℂ K
-  let : Nonempty X := ⟨x₀⟩
-  let : Nonempty (Fin m) := ⟨⟨0,hm⟩⟩
-  let B := stdOrthonormalBasis ℂ K
-  let w : Fin m → V := fun i => (B i : V)
-  have hfix (i : Fin m) : IsFixed a x₀ ρ (w i) := (B i).2
-  have hw : Orthonormal ℂ w := B.orthonormal.comp_linearIsometry K.subtypeₗᵢ
-  have hc := normalizedCoefficient_orthonormal a x₀ ha ρ hρ w hw hfix v hv
-  have hn (i : Fin m) : ‖DensityTransfer.applyKernel (actionKernel a P)
-      (normalizedCoefficient a x₀ ha ρ (w i) v)‖ = ‖randomOperator ρ P v‖ := by
-    rw [coefficient_random a x₀ ha ρ hρ P (w i) (hfix i),
-      normalizedCoefficient_norm a x₀ ha ρ hρ (w i) (hfix i) (hw.norm_eq_one i)]
-  have h := DensityTransfer.multiplicity_amplification (actionKernel a P)
-    (fun x y => finiteMean_nonneg (fun _ => by split_ifs <;> norm_num)) hsym hrow
-    b z M hb hz hM0 hM _ hc (‖randomOperator ρ P v‖) (fun i => (hn i).ge)
-  simpa only [Fintype.card_fin] using h
-
-end Thorp.UnitaryFinite
-namespace Thorp.UnitaryFinite
-open scoped BigOperators Classical
-
-variable {α ι : Type*} [Fintype α] [Fintype ι]
-
-def tupleAction : Equiv.Perm α →* Equiv.Perm (ι ↪ α) where
-  toFun p :=
-    { toFun := fun e => e.trans p.toEmbedding
-      invFun := fun e => e.trans p.symm.toEmbedding
-      left_inv := fun e => by ext i; exact p.symm_apply_apply (e i)
-      right_inv := fun e => by ext i; exact p.apply_symm_apply (e i) }
-  map_one' := by ext e i; rfl
-  map_mul' p q := by ext e i; rfl
-
-omit [Fintype α] [Fintype ι] in
-@[simp] lemma tupleAction_apply (p : Equiv.Perm α) (e : ι ↪ α) (i : ι) :
-    tupleAction p e i=p (e i) := rfl
-
-omit [Fintype α] in
-lemma tupleAction_transitive (e f : ι ↪ α) : ∃ p, tupleAction p e=f := by
-  obtain ⟨p,hp⟩ := Equiv.Perm.exists_extending_pair e f e.injective f.injective
-  exact ⟨p,Function.Embedding.ext hp⟩
-
-omit [Fintype α] in
-lemma tupleAction_fixed_iff [DecidableEq α] (e : ι ↪ α) (p : Equiv.Perm α) :
-    tupleAction p e=e ↔ ∀ x ∈ Finset.univ.image e, p x=x := by
-  constructor
-  · intro h x hx
-    obtain ⟨i,_,rfl⟩ := Finset.mem_image.mp hx
-    exact congrArg (fun f : ι ↪ α => f i) h
-  · intro h
-    ext i
-    exact h (e i) (Finset.mem_image.mpr ⟨i,Finset.mem_univ _,rfl⟩)
-
-omit [Fintype α] in
-lemma tupleAction_kernel [DecidableEq α] {Ω : Type*} [Fintype Ω] (P : Ω → Equiv.Perm α)
-    (e f : ι ↪ α) : actionKernel tupleAction P e f=PairRouting.tupleProbability P e f := by
-  unfold actionKernel PairRouting.tupleProbability
-  congr 1
-  funext ω
-  congr 1
-  exact propext Function.Embedding.ext_iff
-
-end Thorp.UnitaryFinite
-namespace Thorp.UnitaryFinite
-open scoped BigOperators Classical
-
-lemma irreducible_of_equiv {G V W : Type*} [Group G]
-    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
-    (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
-    (e : Representation.Equiv ρ σ) [Representation.IsIrreducible ρ] :
-    Representation.IsIrreducible σ := by
-  rw [Representation.irreducible_iff_isSimpleModule_asModule]
-  have hi := (Representation.irreducible_iff_isSimpleModule_asModule ρ).mp inferInstance
-  let f : ρ.asModule →ₗ[MonoidAlgebra ℂ G] σ.asModule :=
-    Representation.IntertwiningMap.equivLinearMapAsModule ρ σ e.toIntertwiningMap
-  have hf : Function.Bijective f := e.toLinearEquiv.bijective
-  exact (f.isSimpleModule_iff_of_bijective hf).mp hi
-
-end Thorp.UnitaryFinite
-
-namespace Thorp.Specht
-open scoped BigOperators Classical
-
-lemma unitaryRepresentation_coe (μ : YoungDiagram) (p : Equiv.Perm (Cell μ))
-    (v : hilbertSpace μ) :
-    (unitaryRepresentation μ p v : PermutationHilbert.H (Tabloid μ)) =
-      PermutationHilbert.act (tabloidAct p) v := by
-  obtain ⟨w,rfl⟩ := (spaceHilbertEquiv μ).surjective v
-  change hilbertEquiv μ (tabloidRep μ p w) = _
-  exact hilbert_action μ p w
-
-noncomputable def unitaryEquiv (μ : YoungDiagram) :
-    Representation.Equiv (representation μ) (unitaryRepresentation μ) :=
-  Representation.Equiv.mk (spaceHilbertEquiv μ) (by
-    intro p
-    apply LinearMap.ext
-    intro v
-    change spaceHilbertEquiv μ (representation μ p v) =
-      spaceHilbertEquiv μ (representation μ p ((spaceHilbertEquiv μ).symm (spaceHilbertEquiv μ v)))
-    rw [LinearEquiv.symm_apply_apply])
-
-lemma unitaryRepresentation_irreducible (μ : YoungDiagram) :
-    Representation.IsIrreducible (unitaryRepresentation μ) := by
-  let := representation_irreducible μ
-  exact UnitaryFinite.irreducible_of_equiv _ _ (unitaryEquiv μ)
-
-lemma unitaryRepresentation_unitary (μ : YoungDiagram) :
-    UnitaryFinite.IsUnitary (G := Equiv.Perm (Cell μ)) (V := hilbertSpace μ) (unitaryRepresentation μ) := by
-  intro p v w
-  change inner ℂ (unitaryRepresentation μ p v : PermutationHilbert.H (Tabloid μ))
-      (unitaryRepresentation μ p w : PermutationHilbert.H (Tabloid μ)) = _
-  rw [unitaryRepresentation_coe,unitaryRepresentation_coe]
-  exact (PermutationHilbert.act (tabloidAct p)).inner_map_map v w
-
-
-noncomputable def tupleFixedEquiv (μ : YoungDiagram) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Cell μ) :
-    UnitaryFinite.fixedSpace (V := hilbertSpace μ) UnitaryFinite.tupleAction e (unitaryRepresentation μ) ≃ₗ[ℂ]
-      fixedHilbert μ (Finset.univ.image e) where
-  toFun v := ⟨v.1.1,v.1.2,by
-    intro p hp
-    have h := v.2 p ((UnitaryFinite.tupleAction_fixed_iff e p).mpr hp)
-    exact (unitaryRepresentation_coe μ p v.1).symm.trans (congrArg Subtype.val h)⟩
-  invFun v := ⟨⟨v.1,v.2.1⟩,by
-    intro p hp
-    apply Subtype.ext
-    exact (unitaryRepresentation_coe μ p _).trans
-      (v.2.2 p ((UnitaryFinite.tupleAction_fixed_iff e p).mp hp))⟩
-  left_inv _ := rfl
-  right_inv _ := rfl
-  map_add' _ _ := rfl
-  map_smul' _ _ := rfl
-
-theorem tupleFixed_dimension (μ : YoungDiagram) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Cell μ) :
-    Module.finrank ℂ (hilbertSpace μ) * (Fintype.card ι).choose (μ.card-μ.rowLen 0) ≤
-      Module.finrank ℂ (UnitaryFinite.fixedSpace (V := hilbertSpace μ) UnitaryFinite.tupleAction e (unitaryRepresentation μ)) *
-        μ.card.choose (μ.card-μ.rowLen 0) := by
-  rw [(tupleFixedEquiv μ e).finrank_eq,finrank_hilbertSpace]
-  have h := fixedHilbert_dimension μ (Finset.univ.image e)
-  simpa only [Finset.card_image_of_injective _ e.injective,Finset.card_univ] using h
-
-end Thorp.Specht
-namespace Thorp.UnitaryFinite
-open scoped BigOperators Classical
-
-variable {G H X Ω : Type*} [Group G] [Group H] [Fintype G] [Fintype X] [Fintype Ω]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-omit [Fintype G] in
-lemma sampleOperator_inv_apply (ρ : Representation ℂ G V) (P : Ω → G) (v : V) :
-    sampleOperator ρ (fun ω => (P ω)⁻¹) v=randomOperator ρ P v := by
-  simp only [sampleOperator,randomOperator,smul_apply,
-    sum_apply,continuousRepresentation_apply,LinearMap.smul_apply,LinearMap.sum_apply]
-
-theorem sampleOperator_amplification (a : G →* Equiv.Perm X) (x₀ : X)
-    (ha : ∀ x, ∃ g, a g x₀=x) (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
-    (hρ : IsUnitary ρ) (P : Ω → G)
-    (hsym : ∀ x y, actionKernel a P x y=actionKernel a P y x)
-    (hrow : ∀ x, ∑ y, actionKernel a P x y=1)
-    (b z M : ℝ) (hb : 0≤b) (hz : 0<z) (hM0 : 0≤M)
-    (hM : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*actionKernel a P x y)^(1+b)) ≤ M)
-    (hm : 0 < Module.finrank ℂ (fixedSpace a x₀ ρ)) :
-    ‖sampleOperator ρ P‖ ≤ M*z^(-b)+Real.sqrt (z/(Module.finrank ℂ (fixedSpace a x₀ ρ):ℝ)) := by
-  rw [←ContinuousLinearMap.adjoint.norm_map (sampleOperator ρ P),←sampleOperator_inv ρ hρ]
-  apply ContinuousLinearMap.opNorm_le_of_unit_norm (by positivity)
-  intro v hv
-  rw [sampleOperator_inv_apply]
-  exact operator_amplification a x₀ ha ρ hρ P hsym hrow b z M hb hz hM0 hM hm v hv
-
-omit [Fintype G] [Fintype X] [Fintype Ω] [FiniteDimensional ℂ V] in
-noncomputable def subrepresentationEquiv (ρ : Representation ℂ G V) (e : H ≃* G) :
-    Subrepresentation (ρ.comp e.toMonoidHom) ≃o Subrepresentation ρ where
-  toFun U :=
-    { toSubmodule := U.toSubmodule
-      apply_mem_toSubmodule := fun g v hv => by
-        have h := U.apply_mem_toSubmodule (e.symm g) hv
-        change ρ (e (e.symm g)) v∈U.toSubmodule at h
-        simpa only [e.apply_symm_apply] using h }
-  invFun U :=
-    { toSubmodule := U.toSubmodule
-      apply_mem_toSubmodule := fun h _ hv => U.apply_mem_toSubmodule (e h) hv }
-  left_inv U := by apply Subrepresentation.toSubmodule_injective; rfl
-  right_inv U := by apply Subrepresentation.toSubmodule_injective; rfl
-  map_rel_iff' := Iff.rfl
-
-omit [Fintype G] [Fintype X] [Fintype Ω] [FiniteDimensional ℂ V] in
-lemma irreducible_comp_equiv (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
-    (e : H ≃* G) : Representation.IsIrreducible (ρ.comp e.toMonoidHom) := by
-  exact (subrepresentationEquiv ρ e).isSimpleOrder_iff.mpr inferInstance
-
-end Thorp.UnitaryFinite
-
-namespace Thorp.Specht
-open scoped BigOperators Classical
-
-variable {α ι : Type*} [Fintype α] [Fintype ι]
-
-omit [Fintype α] in
-lemma relabelledUnitary_unitary (μ : YoungDiagram) (e : α ≃ Cell μ) :
-    UnitaryFinite.IsUnitary (V := hilbertSpace μ) (relabelledUnitary μ e) := by
-  intro g v w
-  exact unitaryRepresentation_unitary μ (e.permCongr g) v w
-
-omit [Fintype α] in
-lemma relabelledUnitary_irreducible (μ : YoungDiagram) (e : α ≃ Cell μ) :
-    Representation.IsIrreducible (relabelledUnitary μ e) := by
-  let := unitaryRepresentation_irreducible μ
-  exact UnitaryFinite.irreducible_comp_equiv (V := hilbertSpace μ) (unitaryRepresentation μ) e.permCongrHom
-
-noncomputable def relabelledTupleFixed (μ : YoungDiagram) (e : α ≃ Cell μ) (f : ι ↪ α) :
-    UnitaryFinite.fixedSpace (V := hilbertSpace μ) UnitaryFinite.tupleAction f (relabelledUnitary μ e) ≃ₗ[ℂ]
-    UnitaryFinite.fixedSpace (V := hilbertSpace μ) UnitaryFinite.tupleAction (f.trans e.toEmbedding) (unitaryRepresentation μ) where
-  toFun v := ⟨v.1,by
-    intro g hg
-    have hp : UnitaryFinite.tupleAction (e.symm.permCongr g) f=f := by
-      apply Function.Embedding.ext
-      intro i
-      have hh := congrArg (fun f : ι ↪ Cell μ => f i) hg
-      change g (e (f i))=e (f i) at hh
-      simp only [UnitaryFinite.tupleAction_apply,Equiv.permCongr_apply,Equiv.symm_symm,e.symm_apply_apply,hh]
-    have hh := v.2 (e.symm.permCongr g) hp
-    change unitaryRepresentation μ (e.permCongr (e.symm.permCongr g)) v=v at hh
-    have he : e.permCongr (e.symm.permCongr g)=g := by
-      apply Equiv.ext
-      intro x
-      simp only [Equiv.permCongr_apply,Equiv.symm_symm,e.apply_symm_apply]
-    simpa only [he] using hh⟩
-  invFun v := ⟨v.1,by
-    intro g hg
-    have hp : UnitaryFinite.tupleAction (e.permCongr g) (f.trans e.toEmbedding)=f.trans e.toEmbedding := by
-      apply Function.Embedding.ext
-      intro i
-      have hh := congrArg (fun f : ι ↪ α => f i) hg
-      change g (f i)=f i at hh
-      simp only [UnitaryFinite.tupleAction_apply,Function.Embedding.trans_apply,
-        Equiv.toEmbedding_apply,Equiv.permCongr_apply,e.symm_apply_apply,hh]
-    exact v.2 (e.permCongr g) hp⟩
-  left_inv _ := rfl
-  right_inv _ := rfl
-  map_add' _ _ := rfl
-  map_smul' _ _ := rfl
-
-omit [Fintype α] in
-lemma relabelled_tuple_dimension (μ : YoungDiagram) (e : α ≃ Cell μ) (f : ι ↪ α) :
-    Module.finrank ℂ (hilbertSpace μ) * (Fintype.card ι).choose (μ.card-μ.rowLen 0) ≤
-      Module.finrank ℂ (UnitaryFinite.fixedSpace (V := hilbertSpace μ) UnitaryFinite.tupleAction f (relabelledUnitary μ e)) *
-        μ.card.choose (μ.card-μ.rowLen 0) := by
-  rw [(relabelledTupleFixed μ e f).finrank_eq]
-  exact tupleFixed_dimension μ (f.trans e.toEmbedding)
-
-end Thorp.Specht
-
-namespace Thorp.SparseContact
-open scoped BigOperators Classical
-
-lemma small_dominating_set {V : Type*} [Fintype V] (G : SimpleGraph V)
-    (hG : ∀ v, ∃ w, G.Adj v w) :
-    ∃ R : Finset V, 2 * R.card ≤ Fintype.card V ∧
-      ∀ v, v ∉ R → ∃ w ∈ R, G.Adj v w := by
-  classical
-  obtain ⟨F, hFG, hF, hreach⟩ := G.exists_isAcyclic_reachable_eq_le
-  let c := hF.coloringTwo
-  have hneigh (v : V) : ∃ w, F.Adj v w := by
-    obtain ⟨w, hvw⟩ := hG v
-    have hr : F.Reachable v w := by rw [hreach]; exact hvw.reachable
-    exact hr.nonempty_neighborSet_left hvw.ne
-  let A : Finset V := Finset.univ.filter (fun v => c v = 0)
-  let B : Finset V := Finset.univ.filter (fun v => c v ≠ 0)
-  have hab : A.card + B.card = Fintype.card V := by
-    exact Finset.card_filter_add_card_filter_not _
-  have hcolor {v w : V} (hvw : F.Adj v w) : c v ≠ c w := c.valid hvw
-  by_cases h : A.card ≤ B.card
-  · refine ⟨A, by omega, ?_⟩
-    intro v hv
-    obtain ⟨w, hvw⟩ := hneigh v
-    have hv0 : c v ≠ 0 := by simpa [A] using hv
-    have hw0 : c w = 0 := by
-      have hv1 : c v = 1 := by omega
-      have hn := hcolor hvw
-      omega
-    exact ⟨w, by simp [A, hw0], hFG hvw⟩
-  · refine ⟨B, by omega, ?_⟩
-    intro v hv
-    obtain ⟨w, hvw⟩ := hneigh v
-    have hv0 : c v = 0 := by simpa [B] using hv
-    have hw0 : c w ≠ 0 := by simpa [hv0] using (hcolor hvw).symm
-    exact ⟨w, by simp [B, hw0], hFG hvw⟩
-
-lemma powerset_weight_le_exp {V : Type*} [DecidableEq V] (S : Finset V)
-    (p : V → ℝ) (hp : ∀ v, 0 ≤ p v) :
-    (∑ L ∈ S.powerset, ∏ v ∈ L, p v) ≤ Real.exp (∑ v ∈ S, p v) := by
-  rw [← Finset.prod_one_add, Real.exp_sum]
-  apply Finset.prod_le_prod₀
-  · intro v _; exact add_nonneg (by norm_num) (hp v)
-  · intro v _
-    have h := Real.add_one_le_exp (p v)
-    linarith
-
-lemma fixed_size_weight_le {V : Type*} [DecidableEq V] (S : Finset V)
-    (p : V → ℝ) (hp : ∀ v, 0 ≤ p v) (m : ℕ) (δ : ℝ) (hδ : 0 < δ) :
-    (∑ L ∈ S.powersetCard m, ∏ v ∈ L, p v) ≤
-      δ^m * Real.exp ((∑ v ∈ S, p v)/δ) := by
-  have hsplit (L : Finset V) (hL : L ∈ S.powersetCard m) :
-      (∏ v ∈ L, p v) = δ^m * ∏ v ∈ L, (p v / δ) := by
-    rw [Finset.prod_div_distrib, Finset.prod_const, (Finset.mem_powersetCard.mp hL).2]
-    field_simp
-  calc
-    _ = δ^m * ∑ L ∈ S.powersetCard m, ∏ v ∈ L, (p v/δ) := by
-      rw [Finset.mul_sum]
-      exact Finset.sum_congr rfl hsplit
-    _ ≤ δ^m * ∑ L ∈ S.powerset, ∏ v ∈ L, (p v/δ) := by
-      apply mul_le_mul_of_nonneg_left _ (pow_nonneg hδ.le _)
-      apply Finset.sum_le_sum_of_subset_of_nonneg
-      · intro L hL
-        exact Finset.mem_powerset.mpr (Finset.mem_powersetCard.mp hL).1
-      · intro L _ _; exact Finset.prod_nonneg (fun v _ => div_nonneg (hp v) hδ.le)
-    _ ≤ δ^m * Real.exp (∑ v ∈ S, p v/δ) :=
-      mul_le_mul_of_nonneg_left
-        (powerset_weight_le_exp S (fun v => p v/δ) (fun v => div_nonneg (hp v) hδ.le))
-        (pow_nonneg hδ.le _)
-    _ = _ := by rw [Finset.sum_div]
-
-noncomputable def neighborhood {V : Type*} [Fintype V] (G : SimpleGraph V)
-    (R : Finset V) : Finset V := by
-  classical
-  exact Finset.univ.filter (fun w => ∃ v ∈ R, G.Adj v w)
-
-lemma neighborhood_weight_le {V : Type*} [Fintype V] (G : SimpleGraph V)
-    (p : V → ℝ) (hp : ∀ v, 0 ≤ p v) (δ : ℝ)
-    (hdeg : ∀ v, (∑ w, if G.Adj v w then p w else 0) ≤ δ) (R : Finset V) :
-    (∑ w ∈ neighborhood G R, p w) ≤ R.card * δ := by
-  classical
-  calc
-    _ ≤ ∑ w ∈ neighborhood G R, ∑ v ∈ R, if G.Adj v w then p w else 0 := by
-      apply Finset.sum_le_sum
-      intro w hw
-      obtain ⟨v, hv, hvw⟩ := (Finset.mem_filter.mp hw).2
-      have h : (if G.Adj v w then p w else 0) ≤
-          ∑ u ∈ R, if G.Adj u w then p w else 0 :=
-        Finset.single_le_sum (f := fun u => if G.Adj u w then p w else 0) (fun u _ => by split_ifs; exact hp w; exact le_rfl) hv
-      simpa only [ite_eq_left hvw] using h
-    _ ≤ ∑ w, ∑ v ∈ R, if G.Adj v w then p w else 0 := by
-      apply Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _)
-      intro w _ _
-      exact Finset.sum_nonneg (fun v _ => by split_ifs; exact hp w; exact le_rfl)
-    _ = ∑ v ∈ R, ∑ w, if G.Adj v w then p w else 0 := Finset.sum_comm
-    _ ≤ ∑ v ∈ R, δ := Finset.sum_le_sum (fun v _ => hdeg v)
-    _ = _ := by simp
-
-lemma root_leaves_weight_le {V : Type*} [Fintype V] (G : SimpleGraph V)
-    (p : V → ℝ) (hp : ∀ v, 0 ≤ p v) (k : ℕ) (δ : ℝ)
-    (hδ : 0 < δ) (hδ1 : δ ≤ 1)
-    (hdeg : ∀ v, (∑ w, if G.Adj v w then p w else 0) ≤ δ)
-    (R : Finset V) (hR : 2 * R.card ≤ k) :
-    (∑ L ∈ (neighborhood G R).powersetCard (k-R.card), ∏ v ∈ L, p v) ≤
-      δ^((k:ℝ)/2) * Real.exp R.card := by
-  classical
-  have hmass := neighborhood_weight_le G p hp δ hdeg R
-  have hpow : δ^(k-R.card) ≤ δ^((k:ℝ)/2) := by
-    rw [← Real.rpow_natCast]
-    apply Real.rpow_le_rpow_of_exponent_ge hδ hδ1
-    have hr : R.card ≤ k := by omega
-    rw [Nat.cast_sub hr]
-    have hh : (2:ℝ)*R.card ≤ k := by exact_mod_cast hR
-    linarith
-  apply (fixed_size_weight_le (neighborhood G R) p hp _ δ hδ).trans
-  apply mul_le_mul hpow
-  · apply Real.exp_le_exp.mpr
-    exact (div_le_iff₀ hδ).mpr hmass
-  · positivity
-  · positivity
-
-def NoIsolated {V : Type*} (G : SimpleGraph V) (S : Finset V) : Prop :=
-  ∀ v ∈ S, ∃ w ∈ S, G.Adj v w
-
-lemma small_dominating_subset {V : Type*} [Fintype V] (G : SimpleGraph V)
-    (S : Finset V) (hS : NoIsolated G S) :
-    ∃ R : Finset V, R ⊆ S ∧ 2 * R.card ≤ S.card ∧ S \ R ⊆ neighborhood G R := by
-  classical
-  let H := G.induce (S : Set V)
-  have hH : ∀ v : S, ∃ w : S, H.Adj v w := by
-    intro v
-    obtain ⟨w, hw, hvw⟩ := hS v v.2
-    exact ⟨⟨w, hw⟩, hvw⟩
-  obtain ⟨R, hR, hdom⟩ := small_dominating_set H hH
-  let T := R.image (fun v : S => (v : V))
-  have hc : T.card = R.card := Finset.card_image_of_injective _ Subtype.val_injective
-  refine ⟨T, ?_, ?_, ?_⟩
-  · intro v hv
-    obtain ⟨w, _, rfl⟩ := Finset.mem_image.mp hv
-    exact w.2
-  · simpa [hc, H] using hR
-  · intro v hv
-    have hvS := (Finset.mem_sdiff.mp hv).1
-    have hvR : (⟨v, hvS⟩ : S) ∉ R := by
-      intro hh
-      exact (Finset.mem_sdiff.mp hv).2 (Finset.mem_image.mpr ⟨⟨v,hvS⟩, hh, rfl⟩)
-    obtain ⟨w, hw, hvw⟩ := hdom ⟨v,hvS⟩ hvR
-    exact Finset.mem_filter.mpr ⟨Finset.mem_univ _,
-      ⟨w, Finset.mem_image.mpr ⟨w,hw,rfl⟩, hvw.symm⟩⟩
-
-lemma noIsolated_weight_le {V : Type*} [Fintype V] (G : SimpleGraph V)
-    (p : V → ℝ) (hp : ∀ v, 0 ≤ p v) (k : ℕ) (δ : ℝ)
-    (hδ : 0 < δ) (hδ1 : δ ≤ 1)
-    (hdeg : ∀ v, (∑ w, if G.Adj v w then p w else 0) ≤ δ) :
-    (∑ S : {S : Finset V // S.card = k ∧ NoIsolated G S}, ∏ v ∈ S.val, p v) ≤
-      δ^((k:ℝ)/2) * Real.exp (Real.exp 1 * ∑ v, p v) := by
-  classical
-  let I := {S : Finset V // S.card = k ∧ NoIsolated G S}
-  let root (S : I) := (small_dominating_subset G S.val S.property.2).choose
-  have hroot (S : I) : root S ⊆ S.val ∧ 2*(root S).card ≤ k ∧
-      S.val \ root S ⊆ neighborhood G (root S) := by
-    simpa only [root, S.property.1] using
-      (small_dominating_subset G S.val S.property.2).choose_spec
-  let enc (S : I) : Finset V × Finset V := (root S, S.val \ root S)
-  have hrec (S : I) : (enc S).1 ∪ (enc S).2 = S.val :=
-    Finset.union_sdiff_of_subset (hroot S).1
-  have hinj : Function.Injective enc := by
-    intro S T h
-    apply Subtype.ext
-    rw [← hrec S, ← hrec T, h]
-  let W (z : Finset V × Finset V) : ℝ :=
-    if 2*z.1.card ≤ k ∧ z.2 ∈ (neighborhood G z.1).powersetCard (k-z.1.card)
-    then (∏ v ∈ z.1, p v) * ∏ v ∈ z.2, p v else 0
-  have hW (z : Finset V × Finset V) : 0 ≤ W z := by
-    dsimp [W]
-    split_ifs
-    · exact mul_nonneg (Finset.prod_nonneg (fun v _ => hp v))
-        (Finset.prod_nonneg (fun v _ => hp v))
-    · exact le_rfl
-  have hwenc (S : I) : W (enc S) = ∏ v ∈ S.val, p v := by
-    have hmem : (enc S).2 ∈ (neighborhood G (enc S).1).powersetCard (k-(enc S).1.card) := by
-      refine Finset.mem_powersetCard.mpr ⟨(hroot S).2.2, ?_⟩
-      change (S.val \ root S).card = k-(root S).card
-      rw [Finset.card_sdiff_of_subset (hroot S).1, S.property.1]
-    dsimp only [W]
-    rw [ite_eq_left ⟨(hroot S).2.1,hmem⟩]
-    exact (mul_comm _ _).trans (Finset.prod_sdiff (hroot S).1)
-  have hsum : (∑ S : I, ∏ v ∈ S.val, p v) ≤ ∑ z, W z := by
-    calc
-      _ = ∑ z ∈ Finset.univ.image enc, W z := by
-        rw [Finset.sum_image (fun S _ T _ h => hinj h)]
-        exact Finset.sum_congr rfl (fun S _ => (hwenc S).symm)
-      _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _)
-        (fun z _ _ => hW z)
-  apply hsum.trans
-  rw [Fintype.sum_prod_type]
-  calc
-    (∑ R : Finset V, ∑ L : Finset V, W (R,L)) ≤
-        ∑ R : Finset V, δ^((k:ℝ)/2) * Real.exp R.card * ∏ v ∈ R, p v := by
-      apply Finset.sum_le_sum
-      intro R _
-      by_cases hR : 2*R.card ≤ k
-      · have he : (∑ L : Finset V, W (R,L)) =
-            (∏ v ∈ R, p v) *
-              ∑ L ∈ (neighborhood G R).powersetCard (k-R.card), ∏ v ∈ L, p v := by
-          simp only [W, hR, true_and]
-          rw [← Finset.sum_filter]
-          simp only [Finset.filter_mem_eq_inter, Finset.univ_inter, Finset.mul_sum]
-        rw [he]
-        have h := mul_le_mul_of_nonneg_left (root_leaves_weight_le G p hp k δ hδ hδ1 hdeg R hR)
-          (Finset.prod_nonneg (s := R) (fun v _ => hp v))
-        simpa only [mul_comm, mul_left_comm, mul_assoc] using h
-      · simp only [W, hR, false_and, ite_false, Finset.sum_const_zero]
-        exact mul_nonneg (mul_nonneg (Real.rpow_nonneg hδ.le _) (Real.exp_pos _).le)
-          (Finset.prod_nonneg (fun v _ => hp v))
-    _ = δ^((k:ℝ)/2) * ∑ R : Finset V, ∏ v ∈ R, (Real.exp 1 * p v) := by
-      rw [Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro R _
-      rw [Finset.prod_mul_distrib, Finset.prod_const, ← Real.exp_nat_mul]
-      simp only [mul_one, mul_assoc]
-    _ ≤ δ^((k:ℝ)/2) * Real.exp (∑ v, Real.exp 1 * p v) := by
-      apply mul_le_mul_of_nonneg_left _ (Real.rpow_nonneg hδ.le _)
-      simpa only [Finset.powerset_univ] using
-        powerset_weight_le_exp Finset.univ (fun v => Real.exp 1 * p v)
-          (fun v => mul_nonneg (Real.exp_pos _).le (hp v))
-    _ = _ := by rw [← Finset.mul_sum]
-
-
-lemma noIsolated_probability_le {V Ω : Type*} [Fintype V] [Fintype Ω]
-    (G : SimpleGraph V) (O : Ω → Finset V) (k : ℕ)
-    (hcard : ∀ ω, (O ω).card = k) (p : V → ℝ) (hp : ∀ v, 0 ≤ p v)
-    (hcyl : ∀ D, inclusionMoment O D ≤ ∏ v ∈ D, p v)
-    (δ : ℝ) (hδ : 0 < δ) (hδ1 : δ ≤ 1)
-    (hdeg : ∀ v, (∑ w, if G.Adj v w then p w else 0) ≤ δ) :
-    finiteMean (fun ω => if NoIsolated G (O ω) then (1:ℝ) else 0) ≤
-      δ^((k:ℝ)/2) * Real.exp (Real.exp 1 * ∑ v, p v) := by
-  classical
-  let I := {S : Finset V // S.card = k ∧ NoIsolated G S}
-  calc
-    _ ≤ finiteMean (fun ω => ∑ S : I, if S.val ⊆ O ω then (1:ℝ) else 0) := by
-      apply finiteMean_mono
-      intro ω
-      split_ifs with h
-      · exact (Finset.single_le_sum (f := fun S : I => if S.val ⊆ O ω then (1:ℝ) else 0)
-          (fun S _ => by positivity) (Finset.mem_univ (⟨O ω,hcard ω,h⟩ : I))) |>.trans' (by simp)
-      · positivity
-    _ = ∑ S : I, inclusionMoment O S.val := by
-      rw [finiteMean_sum]
-      rfl
-    _ ≤ ∑ S : I, ∏ v ∈ S.val, p v := Finset.sum_le_sum (fun S _ => hcyl S.val)
-    _ ≤ _ := noIsolated_weight_le G p hp k δ hδ hδ1 hdeg
-
-noncomputable def typedSet {N X : Type*} [Fintype N] [Fintype X]
-    (A : N → Finset X) : Finset (N × X) := by
-  classical
-  exact Finset.univ.filter (fun v => v.2 ∈ A v.1)
-
-@[simp] lemma mem_typedSet {N X : Type*} [Fintype N] [Fintype X]
-    (A : N → Finset X) (v : N × X) : v ∈ typedSet A ↔ v.2 ∈ A v.1 := by
-  classical
-  simp [typedSet]
-
-noncomputable def fiber {N X : Type*} [Fintype X]
-    (S : Finset (N × X)) (t : N) : Finset X := by
-  classical
-  exact Finset.univ.filter (fun x => (t,x) ∈ S)
-
-@[simp] lemma mem_fiber {N X : Type*} [Fintype X]
-    (S : Finset (N × X)) (t : N) (x : X) : x ∈ fiber S t ↔ (t,x) ∈ S := by
-  classical
-  simp [fiber]
-
-lemma typedSet_fiber {N X : Type*} [Fintype N] [Fintype X]
-    (S : Finset (N × X)) : typedSet (fiber S) = S := by
-  classical
-  ext v
-  simp
-
-lemma typedSet_prod {N X : Type*} [Fintype N] [Fintype X]
-    (A : N → Finset X) (p : N × X → ℝ) :
-    (∏ v ∈ typedSet A, p v) = ∏ t, ∏ x ∈ A t, p (t,x) := by
-  classical
-  simp only [typedSet, Finset.prod_filter, Fintype.prod_prod_type]
-  apply Finset.prod_congr rfl
-  intro t _
-  rw [← Finset.prod_filter]
-  simp only [Finset.filter_mem_eq_inter, Finset.univ_inter]
-
-lemma typedSet_card {N X : Type*} [Fintype N] [Fintype X]
-    (A : N → Finset X) : (typedSet A).card = ∑ t, (A t).card := by
-  classical
-  simp only [Finset.card_eq_sum_ones, typedSet, Finset.sum_filter, Fintype.sum_prod_type]
-  apply Finset.sum_congr rfl
-  intro t _
-  rw [← Finset.sum_filter]
-  simp only [Finset.filter_mem_eq_inter, Finset.univ_inter]
-
-lemma subset_typedSet {N X : Type*} [Fintype N] [Fintype X]
-    (A : N → Finset X) (D : Finset (N × X)) :
-    D ⊆ typedSet A ↔ ∀ t, fiber D t ⊆ A t := by
-  constructor
-  · intro h t x hx
-    exact mem_typedSet A (t,x) |>.mp (h (mem_fiber D t x |>.mp hx))
-  · intro h v hv
-    exact mem_typedSet A v |>.mpr (h v.1 (mem_fiber D v.1 v.2 |>.mpr hv))
-
-end Thorp.SparseContact
-
-end ThorpNine.SparseSaving
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.SparseSaving.BlockProduct
+
+namespace OAI
+
+namespace ThorpNine.SparseSaving
+
+namespace Thorp
+open scoped BigOperators Classical
+
+def modifiedStageEquiv (r L : ℕ) :
+    ((OuterCoins r L × (Card r → Equiv.Perm (Card L) × BenesCoins L)) × OuterCoins r L) ≃
+      ModifiedCoins L r :=
+  (modifiedAssembleEquiv _ _ _ _).trans
+    (Equiv.prodCongr (Equiv.refl _) (Equiv.prodCongr (parallelSplitCoinEquiv r L) (parallelSplitCoinEquiv r L)))
+
+lemma modifiedStageEquiv_apply (r L : ℕ)
+    (w : (OuterCoins r L × (Card r → Equiv.Perm (Card L) × BenesCoins L)) × OuterCoins r L) :
+    modifiedStageEquiv r L w=
+      ((fun b => (w.1.2 b).1),
+       (assembleBits r L (fun x => (w.1.2 x.1).2.1 x.2) w.1.1,
+        assembleBits r L (fun x => (w.1.2 x.1).2.2 x.2) w.2)) := by
+  simp only [modifiedStageEquiv,Equiv.trans_apply,Equiv.prodCongr_apply,modifiedAssembleEquiv,
+    Equiv.coe_fn_mk,Prod.map_apply,Equiv.refl_apply,parallelSplitCoinEquiv_apply]
+
+lemma palindrome_stage (r L : ℕ)
+    (w : (OuterCoins r L × (Card r → BenesCoins L)) × OuterCoins r L) :
+    splitPerm r L (palindromePerm (L+r) (palindromeStageEquiv r L w))=
+      outerPerm r L w.1.1*ParallelBlocks.lift (fun b => palindromePerm L (w.1.2 b))*
+        (outerPerm r L w.2)⁻¹ := by
+  rw [palindromeStageEquiv_apply]
+  change splitPerm r L (_*_⁻¹)=_
+  rw [map_mul,map_inv,splitPerm_butterfly,splitPerm_butterfly,mul_inv_rev]
+  rw [mul_assoc,←mul_assoc (ParallelBlocks.lift _),←map_inv,←map_mul,←mul_assoc]
+  rfl
+
+noncomputable def localModifiedPerm (L : ℕ) (e : Bool)
+    (w : Equiv.Perm (Card L) × BenesCoins L) : Equiv.Perm (Card L) :=
+  if e then 1 else butterflyPerm L (decodeButterfly L w.2.1)*w.1⁻¹
+
+lemma modified_stage (r L : ℕ) (E : Finset (Card r))
+    (w : (OuterCoins r L × (Card r → Equiv.Perm (Card L) × BenesCoins L)) × OuterCoins r L) :
+    splitPerm r L (modifiedPerm L r E (modifiedStageEquiv r L w))=
+      outerPerm r L w.1.1*ParallelBlocks.lift (fun b => localModifiedPerm L (decide (b∈E)) (w.1.2 b))*
+        (outerPerm r L w.2)⁻¹ := by
+  rw [modifiedStageEquiv_apply]
+  unfold modifiedPerm modifiedInputTree
+  rw [map_mul,map_inv,splitPerm_butterfly,splitPerm_boundaryTree,mul_inv_rev]
+  rw [mul_assoc,←mul_assoc (ParallelBlocks.lift _),←map_inv,←map_mul,←mul_assoc]
+  congr 2
+  apply congrArg ParallelBlocks.lift
+  funext b
+  simp only [Pi.mul_apply,Pi.inv_apply,modifiedBoundary,boundaryOutput_split,localModifiedPerm]
+  by_cases h : b∈E <;> simp [h]
+
+end Thorp
+namespace Thorp.UnitaryFinite
+open scoped BigOperators Classical
+variable {G : Type*} [Group G]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+variable {Ω Ξ : Type*} [Fintype Ω] [Fintype Ξ]
+
+lemma sampleOperator_sandwich (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
+    (P : Ω → G) (Q : Ξ → G) :
+    sampleOperator ρ (fun w : (Ω × Ξ) × Ω => P w.1.1*Q w.1.2*(P w.2)⁻¹)=
+      sampleOperator ρ P*sampleOperator ρ Q*(sampleOperator ρ P).adjoint := by
+  rw [sampleOperator_prod ρ (fun w : Ω × Ξ => P w.1*Q w.2) (fun w : Ω => (P w)⁻¹),
+    sampleOperator_prod ρ P Q,sampleOperator_inv ρ hρ]
+
+lemma localModified_average (L : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card L)) V) (e : Bool) :
+    sampleOperator ρ (localModifiedPerm L e)=if e then 1 else uniformOperator ρ := by
+  cases e
+  · simp only [Bool.false_eq_true,ite_false]
+    rw [←sampleOperator_equiv ρ _ (Equiv.prodComm _ _)]
+    exact sampleOperator_uniform_mul ρ (fun w : BenesCoins L => butterflyPerm L (decodeButterfly L w.1))
+  · simp only [ite_true]
+    exact sampleOperator_one ρ
+
+end Thorp.UnitaryFinite
+
+namespace Thorp
+open scoped BigOperators Classical
+open UnitaryFinite
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+lemma sampleOperator_palindrome_stage (r L : ℕ)
+    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) (hρ : IsUnitary ρ) :
+    sampleOperator (ρ.comp (splitPerm r L).toMonoidHom) (palindromePerm (L+r))=
+      sampleOperator ρ (outerPerm r L)*
+        sampleOperator ρ (fun w : Card r → BenesCoins L => ParallelBlocks.lift (fun b => palindromePerm L (w b)))*
+          (sampleOperator ρ (outerPerm r L)).adjoint := by
+  have he : ((splitPerm r L).toMonoidHom ∘ palindromePerm (L+r)) ∘ palindromeStageEquiv r L=
+      fun w => outerPerm r L w.1.1*ParallelBlocks.lift (fun b => palindromePerm L (w.1.2 b))*
+        (outerPerm r L w.2)⁻¹ := funext (palindrome_stage r L)
+  calc
+    _ = sampleOperator ρ (((splitPerm r L).toMonoidHom ∘ palindromePerm (L+r)) ∘ palindromeStageEquiv r L) :=
+      (sampleOperator_equiv ρ ((splitPerm r L).toMonoidHom ∘ palindromePerm (L+r)) (palindromeStageEquiv r L)).symm
+    _ = _ := (congrArg (sampleOperator ρ) he).trans
+      (sampleOperator_sandwich ρ hρ (outerPerm r L)
+        (fun w : Card r → BenesCoins L => ParallelBlocks.lift (fun b => palindromePerm L (w b))))
+
+lemma sampleOperator_modified_stage (r L : ℕ) (E : Finset (Card r))
+    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) (hρ : IsUnitary ρ) :
+    sampleOperator (ρ.comp (splitPerm r L).toMonoidHom) (modifiedPerm L r E)=
+      sampleOperator ρ (outerPerm r L)*
+        sampleOperator ρ (fun w : Card r → Equiv.Perm (Card L) × BenesCoins L =>
+          ParallelBlocks.lift (fun b => localModifiedPerm L (decide (b∈E)) (w b)))*
+          (sampleOperator ρ (outerPerm r L)).adjoint := by
+  have he : ((splitPerm r L).toMonoidHom ∘ modifiedPerm L r E) ∘ modifiedStageEquiv r L=
+      fun w => outerPerm r L w.1.1*ParallelBlocks.lift
+        (fun b => localModifiedPerm L (decide (b∈E)) (w.1.2 b))*(outerPerm r L w.2)⁻¹ :=
+    funext (modified_stage r L E)
+  calc
+    _ = sampleOperator ρ (((splitPerm r L).toMonoidHom ∘ modifiedPerm L r E) ∘ modifiedStageEquiv r L) :=
+      (sampleOperator_equiv ρ ((splitPerm r L).toMonoidHom ∘ modifiedPerm L r E) (modifiedStageEquiv r L)).symm
+    _ = _ := (congrArg (sampleOperator ρ) he).trans
+      (sampleOperator_sandwich ρ hρ (outerPerm r L)
+        (fun w : Card r → Equiv.Perm (Card L) × BenesCoins L =>
+          ParallelBlocks.lift (fun b => localModifiedPerm L (decide (b∈E)) (w b))))
+
+end Thorp
+namespace Thorp.CommutingBlocks
+open scoped BigOperators Classical
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
+variable {ι : Type*} [Fintype ι] [DecidableEq ι]
+
+noncomputable def retain (P : ι → V →L[ℂ] V) (E : Finset ι) (i : ι) : V →L[ℂ] V :=
+  if i∈E then 1 else P i
+
+omit [CompleteSpace V] [Fintype ι] in
+lemma retain_commute (P : ι → V →L[ℂ] V) (hP : ∀ i j, Commute (P i) (P j))
+    (E : Finset ι) (i j : ι) : Commute (retain P E i) (retain P E j) := by
+  unfold retain
+  split_ifs <;> first | exact Commute.one_left _ | exact Commute.one_right _ | exact hP _ _
+
+omit [CompleteSpace V] in
+lemma blocks_retain (P : ι → V →L[ℂ] V) (hP : ∀ i j, Commute (P i) (P j)) (E : Finset ι) :
+    blocks (retain P E) (retain_commute P hP E)=coarse P hP E := by
+  have hs : E∪Eᶜ=(Finset.univ : Finset ι) := Finset.union_compl E
+  have hh := Finset.noncommProd_union_of_disjoint (s := E) (t := Eᶜ) disjoint_compl_right
+    (retain P E) (fun _ _ _ _ _ => retain_commute P hP E _ _)
+  simp only [hs] at hh
+  change blocks (retain P E) (retain_commute P hP E)=_ at hh
+  rw [hh]
+  have he : E.noncommProd (retain P E) (fun _ _ _ _ _ => retain_commute P hP E _ _)=1 := by
+    exact (Finset.noncommProd_eq_pow_card E (retain P E)
+      (fun i _ j _ _ => retain_commute P hP E i j) 1 (by intro i hi; exact ite_eq_left hi)).trans (one_pow _)
+  rw [he,one_mul]
+  apply Finset.noncommProd_congr rfl
+  intro i hi
+  exact ite_eq_right (Finset.mem_compl.mp hi)
+
+end Thorp.CommutingBlocks
+
+namespace Thorp.ParallelBlocks
+open scoped BigOperators Classical
+open UnitaryFinite
+variable {B X : Type*} [Fintype B] [Fintype X] [DecidableEq X]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+noncomputable def centralProjection (ρ : Representation ℂ (Equiv.Perm (B × X)) V) (b : B) : V →L[ℂ] V :=
+  uniformOperator (ρ.comp (injection b))
+
+omit [Fintype B] in
+lemma centralProjection_commute (ρ : Representation ℂ (Equiv.Perm (B × X)) V) (b c : B) :
+    Commute (centralProjection ρ b) (centralProjection ρ c) := by
+  simpa only [centralProjection,sampleOperator_uniform] using
+    local_average_commute ρ (fun _ => id : B → Equiv.Perm X → Equiv.Perm X) b c
+
+omit [Fintype B] in
+lemma local_central_commute {Ω : Type*} [Fintype Ω] [Nonempty Ω]
+    (ρ : Representation ℂ (Equiv.Perm (B × X)) V) (P : B → Ω → Equiv.Perm X) (b c : B) :
+    Commute (sampleOperator (ρ.comp (injection b)) (P b)) (centralProjection ρ c) := by
+  by_cases h : b=c
+  · subst c
+    change _*uniformOperator _=uniformOperator _*_
+    rw [sampleOperator_uniform_absorb,sampleOperator_weighted,weighted_uniform_right,sampleLaw_sum,one_smul]
+  · unfold centralProjection
+    rw [←sampleOperator_uniform,sampleOperator_comp,sampleOperator_comp]
+    exact sampleOperator_commute ρ _ _ (fun _ _ => injection_commute h _ _)
+
+end Thorp.ParallelBlocks
+namespace Thorp
+open scoped BigOperators Classical
+open UnitaryFinite ParallelBlocks CommutingBlocks
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+lemma modified_central_average (r L : ℕ) (E : Finset (Card r))
+    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) :
+    sampleOperator ρ (fun w : Card r → Equiv.Perm (Card L) × BenesCoins L =>
+      lift (fun b => localModifiedPerm L (decide (b∈E)) (w b)))=
+      coarse (centralProjection ρ) (centralProjection_commute ρ) E := by
+  rw [sampleOperator_parallel ρ (fun b => localModifiedPerm L (decide (b∈E))),←blocks_retain]
+  apply Finset.noncommProd_congr rfl
+  · intro b _
+    rw [localModified_average]
+    simp only [retain,centralProjection,decide_eq_true_eq]
+  · intro b _ c _ _
+    exact local_average_commute ρ (fun b => localModifiedPerm L (decide (b∈E))) b c
+
+lemma central_local_gap (r L : ℕ)
+    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) (hρ : IsUnitary ρ) (b : Card r) :
+    ‖sampleOperator (ρ.comp (injection b)) (palindromePerm L)*(1-centralProjection ρ b)‖≤blockGap L := by
+  change ‖sampleOperator _ _*(1-uniformOperator _)‖≤_
+  rw [mul_sub,mul_one,sampleOperator_uniform_absorb,sampleOperator_weighted]
+  exact palindrome_off_invariants_gap L _ (unitary_comp ρ hρ _)
+
+lemma central_blocks_exception_bound (r L : ℕ)
+    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) (hρ : IsUnitary ρ)
+    (B : V →L[ℂ] V) (hB0 : ‖B‖≤1) (n : ℕ) :
+    ‖B*blocks (fun b : Card r => sampleOperator (ρ.comp (injection b)) (palindromePerm L))
+      (local_average_commute ρ (fun _ => palindromePerm L))*B.adjoint‖ ≤
+      (blockGap L)^n + ∑ E : Finset (Card r), if E.card<n then
+        ‖B*coarse (centralProjection ρ) (centralProjection_commute ρ) E*B.adjoint‖ else 0 := by
+  have hB : ‖B.adjoint‖≤1 := by
+    rw [ContinuousLinearMap.adjoint.norm_map]
+    exact hB0
+  have h := blocks_exception_bound (centralProjection ρ) (centralProjection_commute ρ)
+    (fun b => uniformOperator_projection (ρ.comp (injection b)) (unitary_comp ρ hρ _))
+    (fun b : Card r => sampleOperator (ρ.comp (injection b)) (palindromePerm L))
+    (local_average_commute ρ (fun _ => palindromePerm L))
+    (local_central_commute ρ (fun _ => palindromePerm L))
+    (fun b => sampleOperator_norm_le (ρ.comp (injection b)) (unitary_comp ρ hρ _) (palindromePerm L))
+    (blockGap L) (blockGap_pos L).le (blockGap_lt_one L).le (central_local_gap r L ρ hρ)
+    B.adjoint hB n
+  simpa only [ContinuousLinearMap.adjoint_adjoint] using h
+
+theorem palindrome_exception_bound (r L : ℕ)
+    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) (hρ : IsUnitary ρ) (n : ℕ) :
+    ‖sampleOperator (ρ.comp (splitPerm r L).toMonoidHom) (palindromePerm (L+r))‖ ≤
+      (blockGap L)^n + ∑ E : Finset (Card r), if E.card<n then
+        ‖sampleOperator (ρ.comp (splitPerm r L).toMonoidHom) (modifiedPerm L r E)‖ else 0 := by
+  rw [sampleOperator_palindrome_stage r L ρ hρ]
+  rw [sampleOperator_parallel ρ (fun _ => palindromePerm L)]
+  apply (central_blocks_exception_bound r L ρ hρ (sampleOperator ρ (outerPerm r L))
+    (sampleOperator_norm_le ρ hρ _) n).trans_eq
+  congr 1
+  apply Finset.sum_congr rfl
+  intro E _
+  split_ifs
+  · rw [sampleOperator_modified_stage r L E ρ hρ,modified_central_average]
+  · rfl
+
+end Thorp
+
+namespace Thorp.UnitaryFinite
+open scoped BigOperators ComplexConjugate Classical
+
+variable {G X Ω : Type*} [Group G] [Fintype G] [Fintype X] [Fintype Ω]
+
+noncomputable def actionKernel (a : G →* Equiv.Perm X) (P : Ω → G) (x y : X) : ℝ :=
+  finiteMean (fun ω => if a (P ω) x=y then 1 else 0)
+
+omit [Fintype G] in
+lemma actionKernel_apply (a : G →* Equiv.Perm X) (P : Ω → G)
+    (v : EuclideanSpace ℂ X) (x : X) :
+    DensityTransfer.applyKernel (actionKernel a P) v x =
+      (Fintype.card Ω:ℂ)⁻¹ * ∑ ω, v (a (P ω) x) := by
+  simp only [DensityTransfer.applyKernel_apply,actionKernel,finiteMean,
+    Complex.ofReal_sum,div_eq_mul_inv,Finset.sum_mul]
+  rw [Finset.sum_comm]
+  calc
+    _ = ∑ ω, (Fintype.card Ω:ℂ)⁻¹ * ∑ y : X, if a (P ω) x=y then v y else 0 := by
+      apply Finset.sum_congr rfl
+      intro ω _
+      rw [Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro y _
+      split_ifs <;> simp_all
+    _ = _ := by simp [Finset.mul_sum]
+
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+noncomputable def randomOperator (ρ : Representation ℂ G V) (P : Ω → G) : Module.End ℂ V :=
+  (Fintype.card Ω:ℂ)⁻¹ • ∑ ω, ρ (P ω)⁻¹
+
+omit [Fintype G] [FiniteDimensional ℂ V] in
+lemma coefficient_random (a : G →* Equiv.Perm X) (x₀ : X)
+    (ha : ∀ x, ∃ g, a g x₀=x) (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
+    (P : Ω → G) (w : V) (hw : IsFixed a x₀ ρ w) (v : V) :
+    DensityTransfer.applyKernel (actionKernel a P) (normalizedCoefficient a x₀ ha ρ w v) =
+      normalizedCoefficient a x₀ ha ρ w (randomOperator ρ P v) := by
+  ext x
+  rw [actionKernel_apply]
+  simp only [normalizedCoefficient,PiLp.smul_apply,smul_eq_mul,Finset.mul_sum,
+    coefficient,randomOperator,LinearMap.smul_apply,LinearMap.sum_apply,
+    inner_smul_right,inner_sum]
+  apply Finset.sum_congr rfl
+  intro ω _
+  have h := coefficient_action a x₀ ha ρ hρ w hw v (P ω) x
+  change inner ℂ (ρ (representative a x₀ ha (a (P ω) x)) w) v =
+    inner ℂ (ρ (representative a x₀ ha x) w) (ρ (P ω)⁻¹ v) at h
+  rw [h]
+  ring
+
+noncomputable def fixedSpace (a : G →* Equiv.Perm X) (x₀ : X) (ρ : Representation ℂ G V) :
+    Submodule ℂ V where
+  carrier := {w | IsFixed a x₀ ρ w}
+  zero_mem' := fun g hg => map_zero _
+  add_mem' := fun hw hz g hg => by rw [map_add,hw g hg,hz g hg]
+  smul_mem' := fun c w hw g hg => by rw [map_smul,hw g hg]
+
+theorem operator_amplification (a : G →* Equiv.Perm X) (x₀ : X)
+    (ha : ∀ x, ∃ g, a g x₀=x) (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
+    (hρ : IsUnitary ρ) (P : Ω → G)
+    (hsym : ∀ x y, actionKernel a P x y=actionKernel a P y x)
+    (hrow : ∀ x, ∑ y, actionKernel a P x y=1)
+    (b z M : ℝ) (hb : 0≤b) (hz : 0<z) (hM0 : 0≤M)
+    (hM : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*actionKernel a P x y)^(1+b)) ≤ M)
+    (hm : 0 < Module.finrank ℂ (fixedSpace a x₀ ρ)) (v : V) (hv : ‖v‖=1) :
+    ‖randomOperator ρ P v‖ ≤ M*z^(-b)+Real.sqrt (z/(Module.finrank ℂ (fixedSpace a x₀ ρ):ℝ)) := by
+  let K := fixedSpace a x₀ ρ
+  let m := Module.finrank ℂ K
+  let : Nonempty X := ⟨x₀⟩
+  let : Nonempty (Fin m) := ⟨⟨0,hm⟩⟩
+  let B := stdOrthonormalBasis ℂ K
+  let w : Fin m → V := fun i => (B i : V)
+  have hfix (i : Fin m) : IsFixed a x₀ ρ (w i) := (B i).2
+  have hw : Orthonormal ℂ w := B.orthonormal.comp_linearIsometry K.subtypeₗᵢ
+  have hc := normalizedCoefficient_orthonormal a x₀ ha ρ hρ w hw hfix v hv
+  have hn (i : Fin m) : ‖DensityTransfer.applyKernel (actionKernel a P)
+      (normalizedCoefficient a x₀ ha ρ (w i) v)‖ = ‖randomOperator ρ P v‖ := by
+    rw [coefficient_random a x₀ ha ρ hρ P (w i) (hfix i),
+      normalizedCoefficient_norm a x₀ ha ρ hρ (w i) (hfix i) (hw.norm_eq_one i)]
+  have h := DensityTransfer.multiplicity_amplification (actionKernel a P)
+    (fun x y => finiteMean_nonneg (fun _ => by split_ifs <;> norm_num)) hsym hrow
+    b z M hb hz hM0 hM _ hc (‖randomOperator ρ P v‖) (fun i => (hn i).ge)
+  simpa only [Fintype.card_fin] using h
+
+end Thorp.UnitaryFinite
+namespace Thorp.UnitaryFinite
+open scoped BigOperators Classical
+
+variable {α ι : Type*} [Fintype α] [Fintype ι]
+
+def tupleAction : Equiv.Perm α →* Equiv.Perm (ι ↪ α) where
+  toFun p :=
+    { toFun := fun e => e.trans p.toEmbedding
+      invFun := fun e => e.trans p.symm.toEmbedding
+      left_inv := fun e => by ext i; exact p.symm_apply_apply (e i)
+      right_inv := fun e => by ext i; exact p.apply_symm_apply (e i) }
+  map_one' := by ext e i; rfl
+  map_mul' p q := by ext e i; rfl
+
+omit [Fintype α] [Fintype ι] in
+@[simp] lemma tupleAction_apply (p : Equiv.Perm α) (e : ι ↪ α) (i : ι) :
+    tupleAction p e i=p (e i) := rfl
+
+omit [Fintype α] in
+lemma tupleAction_transitive (e f : ι ↪ α) : ∃ p, tupleAction p e=f := by
+  obtain ⟨p,hp⟩ := Equiv.Perm.exists_extending_pair e f e.injective f.injective
+  exact ⟨p,Function.Embedding.ext hp⟩
+
+omit [Fintype α] in
+lemma tupleAction_fixed_iff [DecidableEq α] (e : ι ↪ α) (p : Equiv.Perm α) :
+    tupleAction p e=e ↔ ∀ x ∈ Finset.univ.image e, p x=x := by
+  constructor
+  · intro h x hx
+    obtain ⟨i,_,rfl⟩ := Finset.mem_image.mp hx
+    exact congrArg (fun f : ι ↪ α => f i) h
+  · intro h
+    ext i
+    exact h (e i) (Finset.mem_image.mpr ⟨i,Finset.mem_univ _,rfl⟩)
+
+omit [Fintype α] in
+lemma tupleAction_kernel [DecidableEq α] {Ω : Type*} [Fintype Ω] (P : Ω → Equiv.Perm α)
+    (e f : ι ↪ α) : actionKernel tupleAction P e f=PairRouting.tupleProbability P e f := by
+  unfold actionKernel PairRouting.tupleProbability
+  congr 1
+  funext ω
+  congr 1
+  exact propext Function.Embedding.ext_iff
+
+end Thorp.UnitaryFinite
+namespace Thorp.UnitaryFinite
+open scoped BigOperators Classical
+
+lemma irreducible_of_equiv {G V W : Type*} [Group G]
+    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
+    (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
+    (e : Representation.Equiv ρ σ) [Representation.IsIrreducible ρ] :
+    Representation.IsIrreducible σ := by
+  rw [Representation.irreducible_iff_isSimpleModule_asModule]
+  have hi := (Representation.irreducible_iff_isSimpleModule_asModule ρ).mp inferInstance
+  let f : ρ.asModule →ₗ[MonoidAlgebra ℂ G] σ.asModule :=
+    Representation.IntertwiningMap.equivLinearMapAsModule ρ σ e.toIntertwiningMap
+  have hf : Function.Bijective f := e.toLinearEquiv.bijective
+  exact (f.isSimpleModule_iff_of_bijective hf).mp hi
+
+end Thorp.UnitaryFinite
+
+namespace Thorp.Specht
+open scoped BigOperators Classical
+
+lemma unitaryRepresentation_coe (μ : YoungDiagram) (p : Equiv.Perm (Cell μ))
+    (v : hilbertSpace μ) :
+    (unitaryRepresentation μ p v : PermutationHilbert.H (Tabloid μ)) =
+      PermutationHilbert.act (tabloidAct p) v := by
+  obtain ⟨w,rfl⟩ := (spaceHilbertEquiv μ).surjective v
+  change hilbertEquiv μ (tabloidRep μ p w) = _
+  exact hilbert_action μ p w
+
+noncomputable def unitaryEquiv (μ : YoungDiagram) :
+    Representation.Equiv (representation μ) (unitaryRepresentation μ) :=
+  Representation.Equiv.mk (spaceHilbertEquiv μ) (by
+    intro p
+    apply LinearMap.ext
+    intro v
+    change spaceHilbertEquiv μ (representation μ p v) =
+      spaceHilbertEquiv μ (representation μ p ((spaceHilbertEquiv μ).symm (spaceHilbertEquiv μ v)))
+    rw [LinearEquiv.symm_apply_apply])
+
+lemma unitaryRepresentation_irreducible (μ : YoungDiagram) :
+    Representation.IsIrreducible (unitaryRepresentation μ) := by
+  let := representation_irreducible μ
+  exact UnitaryFinite.irreducible_of_equiv _ _ (unitaryEquiv μ)
+
+lemma unitaryRepresentation_unitary (μ : YoungDiagram) :
+    UnitaryFinite.IsUnitary (G := Equiv.Perm (Cell μ)) (V := hilbertSpace μ) (unitaryRepresentation μ) := by
+  intro p v w
+  change inner ℂ (unitaryRepresentation μ p v : PermutationHilbert.H (Tabloid μ))
+      (unitaryRepresentation μ p w : PermutationHilbert.H (Tabloid μ)) = _
+  rw [unitaryRepresentation_coe,unitaryRepresentation_coe]
+  exact (PermutationHilbert.act (tabloidAct p)).inner_map_map v w
+
+
+noncomputable def tupleFixedEquiv (μ : YoungDiagram) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Cell μ) :
+    UnitaryFinite.fixedSpace (V := hilbertSpace μ) UnitaryFinite.tupleAction e (unitaryRepresentation μ) ≃ₗ[ℂ]
+      fixedHilbert μ (Finset.univ.image e) where
+  toFun v := ⟨v.1.1,v.1.2,by
+    intro p hp
+    have h := v.2 p ((UnitaryFinite.tupleAction_fixed_iff e p).mpr hp)
+    exact (unitaryRepresentation_coe μ p v.1).symm.trans (congrArg Subtype.val h)⟩
+  invFun v := ⟨⟨v.1,v.2.1⟩,by
+    intro p hp
+    apply Subtype.ext
+    exact (unitaryRepresentation_coe μ p _).trans
+      (v.2.2 p ((UnitaryFinite.tupleAction_fixed_iff e p).mp hp))⟩
+  left_inv _ := rfl
+  right_inv _ := rfl
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+
+theorem tupleFixed_dimension (μ : YoungDiagram) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Cell μ) :
+    Module.finrank ℂ (hilbertSpace μ) * (Fintype.card ι).choose (μ.card-μ.rowLen 0) ≤
+      Module.finrank ℂ (UnitaryFinite.fixedSpace (V := hilbertSpace μ) UnitaryFinite.tupleAction e (unitaryRepresentation μ)) *
+        μ.card.choose (μ.card-μ.rowLen 0) := by
+  rw [(tupleFixedEquiv μ e).finrank_eq,finrank_hilbertSpace]
+  have h := fixedHilbert_dimension μ (Finset.univ.image e)
+  simpa only [Finset.card_image_of_injective _ e.injective,Finset.card_univ] using h
+
+end Thorp.Specht
+namespace Thorp.UnitaryFinite
+open scoped BigOperators Classical
+
+variable {G H X Ω : Type*} [Group G] [Group H] [Fintype G] [Fintype X] [Fintype Ω]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+omit [Fintype G] in
+lemma sampleOperator_inv_apply (ρ : Representation ℂ G V) (P : Ω → G) (v : V) :
+    sampleOperator ρ (fun ω => (P ω)⁻¹) v=randomOperator ρ P v := by
+  simp only [sampleOperator,randomOperator,smul_apply,
+    sum_apply,continuousRepresentation_apply,LinearMap.smul_apply,LinearMap.sum_apply]
+
+theorem sampleOperator_amplification (a : G →* Equiv.Perm X) (x₀ : X)
+    (ha : ∀ x, ∃ g, a g x₀=x) (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
+    (hρ : IsUnitary ρ) (P : Ω → G)
+    (hsym : ∀ x y, actionKernel a P x y=actionKernel a P y x)
+    (hrow : ∀ x, ∑ y, actionKernel a P x y=1)
+    (b z M : ℝ) (hb : 0≤b) (hz : 0<z) (hM0 : 0≤M)
+    (hM : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*actionKernel a P x y)^(1+b)) ≤ M)
+    (hm : 0 < Module.finrank ℂ (fixedSpace a x₀ ρ)) :
+    ‖sampleOperator ρ P‖ ≤ M*z^(-b)+Real.sqrt (z/(Module.finrank ℂ (fixedSpace a x₀ ρ):ℝ)) := by
+  rw [←ContinuousLinearMap.adjoint.norm_map (sampleOperator ρ P),←sampleOperator_inv ρ hρ]
+  apply ContinuousLinearMap.opNorm_le_of_unit_norm (by positivity)
+  intro v hv
+  rw [sampleOperator_inv_apply]
+  exact operator_amplification a x₀ ha ρ hρ P hsym hrow b z M hb hz hM0 hM hm v hv
+
+omit [Fintype G] [Fintype X] [Fintype Ω] [FiniteDimensional ℂ V] in
+noncomputable def subrepresentationEquiv (ρ : Representation ℂ G V) (e : H ≃* G) :
+    Subrepresentation (ρ.comp e.toMonoidHom) ≃o Subrepresentation ρ where
+  toFun U :=
+    { toSubmodule := U.toSubmodule
+      apply_mem_toSubmodule := fun g v hv => by
+        have h := U.apply_mem_toSubmodule (e.symm g) hv
+        change ρ (e (e.symm g)) v∈U.toSubmodule at h
+        simpa only [e.apply_symm_apply] using h }
+  invFun U :=
+    { toSubmodule := U.toSubmodule
+      apply_mem_toSubmodule := fun h _ hv => U.apply_mem_toSubmodule (e h) hv }
+  left_inv U := by apply Subrepresentation.toSubmodule_injective; rfl
+  right_inv U := by apply Subrepresentation.toSubmodule_injective; rfl
+  map_rel_iff' := Iff.rfl
+
+omit [Fintype G] [Fintype X] [Fintype Ω] [FiniteDimensional ℂ V] in
+lemma irreducible_comp_equiv (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
+    (e : H ≃* G) : Representation.IsIrreducible (ρ.comp e.toMonoidHom) := by
+  exact (subrepresentationEquiv ρ e).isSimpleOrder_iff.mpr inferInstance
+
+end Thorp.UnitaryFinite
+
+namespace Thorp.Specht
+open scoped BigOperators Classical
+
+variable {α ι : Type*} [Fintype α] [Fintype ι]
+
+omit [Fintype α] in
+lemma relabelledUnitary_unitary (μ : YoungDiagram) (e : α ≃ Cell μ) :
+    UnitaryFinite.IsUnitary (V := hilbertSpace μ) (relabelledUnitary μ e) := by
+  intro g v w
+  exact unitaryRepresentation_unitary μ (e.permCongr g) v w
+
+omit [Fintype α] in
+lemma relabelledUnitary_irreducible (μ : YoungDiagram) (e : α ≃ Cell μ) :
+    Representation.IsIrreducible (relabelledUnitary μ e) := by
+  let := unitaryRepresentation_irreducible μ
+  exact UnitaryFinite.irreducible_comp_equiv (V := hilbertSpace μ) (unitaryRepresentation μ) e.permCongrHom
+
+noncomputable def relabelledTupleFixed (μ : YoungDiagram) (e : α ≃ Cell μ) (f : ι ↪ α) :
+    UnitaryFinite.fixedSpace (V := hilbertSpace μ) UnitaryFinite.tupleAction f (relabelledUnitary μ e) ≃ₗ[ℂ]
+    UnitaryFinite.fixedSpace (V := hilbertSpace μ) UnitaryFinite.tupleAction (f.trans e.toEmbedding) (unitaryRepresentation μ) where
+  toFun v := ⟨v.1,by
+    intro g hg
+    have hp : UnitaryFinite.tupleAction (e.symm.permCongr g) f=f := by
+      apply Function.Embedding.ext
+      intro i
+      have hh := congrArg (fun f : ι ↪ Cell μ => f i) hg
+      change g (e (f i))=e (f i) at hh
+      simp only [UnitaryFinite.tupleAction_apply,Equiv.permCongr_apply,Equiv.symm_symm,e.symm_apply_apply,hh]
+    have hh := v.2 (e.symm.permCongr g) hp
+    change unitaryRepresentation μ (e.permCongr (e.symm.permCongr g)) v=v at hh
+    have he : e.permCongr (e.symm.permCongr g)=g := by
+      apply Equiv.ext
+      intro x
+      simp only [Equiv.permCongr_apply,Equiv.symm_symm,e.apply_symm_apply]
+    simpa only [he] using hh⟩
+  invFun v := ⟨v.1,by
+    intro g hg
+    have hp : UnitaryFinite.tupleAction (e.permCongr g) (f.trans e.toEmbedding)=f.trans e.toEmbedding := by
+      apply Function.Embedding.ext
+      intro i
+      have hh := congrArg (fun f : ι ↪ α => f i) hg
+      change g (f i)=f i at hh
+      simp only [UnitaryFinite.tupleAction_apply,Function.Embedding.trans_apply,
+        Equiv.toEmbedding_apply,Equiv.permCongr_apply,e.symm_apply_apply,hh]
+    exact v.2 (e.permCongr g) hp⟩
+  left_inv _ := rfl
+  right_inv _ := rfl
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+
+omit [Fintype α] in
+lemma relabelled_tuple_dimension (μ : YoungDiagram) (e : α ≃ Cell μ) (f : ι ↪ α) :
+    Module.finrank ℂ (hilbertSpace μ) * (Fintype.card ι).choose (μ.card-μ.rowLen 0) ≤
+      Module.finrank ℂ (UnitaryFinite.fixedSpace (V := hilbertSpace μ) UnitaryFinite.tupleAction f (relabelledUnitary μ e)) *
+        μ.card.choose (μ.card-μ.rowLen 0) := by
+  rw [(relabelledTupleFixed μ e f).finrank_eq]
+  exact tupleFixed_dimension μ (f.trans e.toEmbedding)
+
+end Thorp.Specht
+
+namespace Thorp.SparseContact
+open scoped BigOperators Classical
+
+lemma small_dominating_set {V : Type*} [Fintype V] (G : SimpleGraph V)
+    (hG : ∀ v, ∃ w, G.Adj v w) :
+    ∃ R : Finset V, 2 * R.card ≤ Fintype.card V ∧
+      ∀ v, v ∉ R → ∃ w ∈ R, G.Adj v w := by
+  classical
+  obtain ⟨F, hFG, hF, hreach⟩ := G.exists_isAcyclic_reachable_eq_le
+  let c := hF.coloringTwo
+  have hneigh (v : V) : ∃ w, F.Adj v w := by
+    obtain ⟨w, hvw⟩ := hG v
+    have hr : F.Reachable v w := by rw [hreach]; exact hvw.reachable
+    exact hr.nonempty_neighborSet_left hvw.ne
+  let A : Finset V := Finset.univ.filter (fun v => c v = 0)
+  let B : Finset V := Finset.univ.filter (fun v => c v ≠ 0)
+  have hab : A.card + B.card = Fintype.card V := by
+    exact Finset.card_filter_add_card_filter_not _
+  have hcolor {v w : V} (hvw : F.Adj v w) : c v ≠ c w := c.valid hvw
+  by_cases h : A.card ≤ B.card
+  · refine ⟨A, by omega, ?_⟩
+    intro v hv
+    obtain ⟨w, hvw⟩ := hneigh v
+    have hv0 : c v ≠ 0 := by simpa [A] using hv
+    have hw0 : c w = 0 := by
+      have hv1 : c v = 1 := by omega
+      have hn := hcolor hvw
+      omega
+    exact ⟨w, by simp [A, hw0], hFG hvw⟩
+  · refine ⟨B, by omega, ?_⟩
+    intro v hv
+    obtain ⟨w, hvw⟩ := hneigh v
+    have hv0 : c v = 0 := by simpa [B] using hv
+    have hw0 : c w ≠ 0 := by simpa [hv0] using (hcolor hvw).symm
+    exact ⟨w, by simp [B, hw0], hFG hvw⟩
+
+lemma powerset_weight_le_exp {V : Type*} [DecidableEq V] (S : Finset V)
+    (p : V → ℝ) (hp : ∀ v, 0 ≤ p v) :
+    (∑ L ∈ S.powerset, ∏ v ∈ L, p v) ≤ Real.exp (∑ v ∈ S, p v) := by
+  rw [← Finset.prod_one_add, Real.exp_sum]
+  apply Finset.prod_le_prod₀
+  · intro v _; exact add_nonneg (by norm_num) (hp v)
+  · intro v _
+    have h := Real.add_one_le_exp (p v)
+    linarith
+
+lemma fixed_size_weight_le {V : Type*} [DecidableEq V] (S : Finset V)
+    (p : V → ℝ) (hp : ∀ v, 0 ≤ p v) (m : ℕ) (δ : ℝ) (hδ : 0 < δ) :
+    (∑ L ∈ S.powersetCard m, ∏ v ∈ L, p v) ≤
+      δ^m * Real.exp ((∑ v ∈ S, p v)/δ) := by
+  have hsplit (L : Finset V) (hL : L ∈ S.powersetCard m) :
+      (∏ v ∈ L, p v) = δ^m * ∏ v ∈ L, (p v / δ) := by
+    rw [Finset.prod_div_distrib, Finset.prod_const, (Finset.mem_powersetCard.mp hL).2]
+    field_simp
+  calc
+    _ = δ^m * ∑ L ∈ S.powersetCard m, ∏ v ∈ L, (p v/δ) := by
+      rw [Finset.mul_sum]
+      exact Finset.sum_congr rfl hsplit
+    _ ≤ δ^m * ∑ L ∈ S.powerset, ∏ v ∈ L, (p v/δ) := by
+      apply mul_le_mul_of_nonneg_left _ (pow_nonneg hδ.le _)
+      apply Finset.sum_le_sum_of_subset_of_nonneg
+      · intro L hL
+        exact Finset.mem_powerset.mpr (Finset.mem_powersetCard.mp hL).1
+      · intro L _ _; exact Finset.prod_nonneg (fun v _ => div_nonneg (hp v) hδ.le)
+    _ ≤ δ^m * Real.exp (∑ v ∈ S, p v/δ) :=
+      mul_le_mul_of_nonneg_left
+        (powerset_weight_le_exp S (fun v => p v/δ) (fun v => div_nonneg (hp v) hδ.le))
+        (pow_nonneg hδ.le _)
+    _ = _ := by rw [Finset.sum_div]
+
+noncomputable def neighborhood {V : Type*} [Fintype V] (G : SimpleGraph V)
+    (R : Finset V) : Finset V := by
+  classical
+  exact Finset.univ.filter (fun w => ∃ v ∈ R, G.Adj v w)
+
+lemma neighborhood_weight_le {V : Type*} [Fintype V] (G : SimpleGraph V)
+    (p : V → ℝ) (hp : ∀ v, 0 ≤ p v) (δ : ℝ)
+    (hdeg : ∀ v, (∑ w, if G.Adj v w then p w else 0) ≤ δ) (R : Finset V) :
+    (∑ w ∈ neighborhood G R, p w) ≤ R.card * δ := by
+  classical
+  calc
+    _ ≤ ∑ w ∈ neighborhood G R, ∑ v ∈ R, if G.Adj v w then p w else 0 := by
+      apply Finset.sum_le_sum
+      intro w hw
+      obtain ⟨v, hv, hvw⟩ := (Finset.mem_filter.mp hw).2
+      have h : (if G.Adj v w then p w else 0) ≤
+          ∑ u ∈ R, if G.Adj u w then p w else 0 :=
+        Finset.single_le_sum (f := fun u => if G.Adj u w then p w else 0) (fun u _ => by split_ifs; exact hp w; exact le_rfl) hv
+      simpa only [ite_eq_left hvw] using h
+    _ ≤ ∑ w, ∑ v ∈ R, if G.Adj v w then p w else 0 := by
+      apply Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _)
+      intro w _ _
+      exact Finset.sum_nonneg (fun v _ => by split_ifs; exact hp w; exact le_rfl)
+    _ = ∑ v ∈ R, ∑ w, if G.Adj v w then p w else 0 := Finset.sum_comm
+    _ ≤ ∑ v ∈ R, δ := Finset.sum_le_sum (fun v _ => hdeg v)
+    _ = _ := by simp
+
+lemma root_leaves_weight_le {V : Type*} [Fintype V] (G : SimpleGraph V)
+    (p : V → ℝ) (hp : ∀ v, 0 ≤ p v) (k : ℕ) (δ : ℝ)
+    (hδ : 0 < δ) (hδ1 : δ ≤ 1)
+    (hdeg : ∀ v, (∑ w, if G.Adj v w then p w else 0) ≤ δ)
+    (R : Finset V) (hR : 2 * R.card ≤ k) :
+    (∑ L ∈ (neighborhood G R).powersetCard (k-R.card), ∏ v ∈ L, p v) ≤
+      δ^((k:ℝ)/2) * Real.exp R.card := by
+  classical
+  have hmass := neighborhood_weight_le G p hp δ hdeg R
+  have hpow : δ^(k-R.card) ≤ δ^((k:ℝ)/2) := by
+    rw [← Real.rpow_natCast]
+    apply Real.rpow_le_rpow_of_exponent_ge hδ hδ1
+    have hr : R.card ≤ k := by omega
+    rw [Nat.cast_sub hr]
+    have hh : (2:ℝ)*R.card ≤ k := by exact_mod_cast hR
+    linarith
+  apply (fixed_size_weight_le (neighborhood G R) p hp _ δ hδ).trans
+  apply mul_le_mul hpow
+  · apply Real.exp_le_exp.mpr
+    exact (div_le_iff₀ hδ).mpr hmass
+  · positivity
+  · positivity
+
+def NoIsolated {V : Type*} (G : SimpleGraph V) (S : Finset V) : Prop :=
+  ∀ v ∈ S, ∃ w ∈ S, G.Adj v w
+
+lemma small_dominating_subset {V : Type*} [Fintype V] (G : SimpleGraph V)
+    (S : Finset V) (hS : NoIsolated G S) :
+    ∃ R : Finset V, R ⊆ S ∧ 2 * R.card ≤ S.card ∧ S \ R ⊆ neighborhood G R := by
+  classical
+  let H := G.induce (S : Set V)
+  have hH : ∀ v : S, ∃ w : S, H.Adj v w := by
+    intro v
+    obtain ⟨w, hw, hvw⟩ := hS v v.2
+    exact ⟨⟨w, hw⟩, hvw⟩
+  obtain ⟨R, hR, hdom⟩ := small_dominating_set H hH
+  let T := R.image (fun v : S => (v : V))
+  have hc : T.card = R.card := Finset.card_image_of_injective _ Subtype.val_injective
+  refine ⟨T, ?_, ?_, ?_⟩
+  · intro v hv
+    obtain ⟨w, _, rfl⟩ := Finset.mem_image.mp hv
+    exact w.2
+  · simpa [hc, H] using hR
+  · intro v hv
+    have hvS := (Finset.mem_sdiff.mp hv).1
+    have hvR : (⟨v, hvS⟩ : S) ∉ R := by
+      intro hh
+      exact (Finset.mem_sdiff.mp hv).2 (Finset.mem_image.mpr ⟨⟨v,hvS⟩, hh, rfl⟩)
+    obtain ⟨w, hw, hvw⟩ := hdom ⟨v,hvS⟩ hvR
+    exact Finset.mem_filter.mpr ⟨Finset.mem_univ _,
+      ⟨w, Finset.mem_image.mpr ⟨w,hw,rfl⟩, hvw.symm⟩⟩
+
+lemma noIsolated_weight_le {V : Type*} [Fintype V] (G : SimpleGraph V)
+    (p : V → ℝ) (hp : ∀ v, 0 ≤ p v) (k : ℕ) (δ : ℝ)
+    (hδ : 0 < δ) (hδ1 : δ ≤ 1)
+    (hdeg : ∀ v, (∑ w, if G.Adj v w then p w else 0) ≤ δ) :
+    (∑ S : {S : Finset V // S.card = k ∧ NoIsolated G S}, ∏ v ∈ S.val, p v) ≤
+      δ^((k:ℝ)/2) * Real.exp (Real.exp 1 * ∑ v, p v) := by
+  classical
+  let I := {S : Finset V // S.card = k ∧ NoIsolated G S}
+  let root (S : I) := (small_dominating_subset G S.val S.property.2).choose
+  have hroot (S : I) : root S ⊆ S.val ∧ 2*(root S).card ≤ k ∧
+      S.val \ root S ⊆ neighborhood G (root S) := by
+    simpa only [root, S.property.1] using
+      (small_dominating_subset G S.val S.property.2).choose_spec
+  let enc (S : I) : Finset V × Finset V := (root S, S.val \ root S)
+  have hrec (S : I) : (enc S).1 ∪ (enc S).2 = S.val :=
+    Finset.union_sdiff_of_subset (hroot S).1
+  have hinj : Function.Injective enc := by
+    intro S T h
+    apply Subtype.ext
+    rw [← hrec S, ← hrec T, h]
+  let W (z : Finset V × Finset V) : ℝ :=
+    if 2*z.1.card ≤ k ∧ z.2 ∈ (neighborhood G z.1).powersetCard (k-z.1.card)
+    then (∏ v ∈ z.1, p v) * ∏ v ∈ z.2, p v else 0
+  have hW (z : Finset V × Finset V) : 0 ≤ W z := by
+    dsimp [W]
+    split_ifs
+    · exact mul_nonneg (Finset.prod_nonneg (fun v _ => hp v))
+        (Finset.prod_nonneg (fun v _ => hp v))
+    · exact le_rfl
+  have hwenc (S : I) : W (enc S) = ∏ v ∈ S.val, p v := by
+    have hmem : (enc S).2 ∈ (neighborhood G (enc S).1).powersetCard (k-(enc S).1.card) := by
+      refine Finset.mem_powersetCard.mpr ⟨(hroot S).2.2, ?_⟩
+      change (S.val \ root S).card = k-(root S).card
+      rw [Finset.card_sdiff_of_subset (hroot S).1, S.property.1]
+    dsimp only [W]
+    rw [ite_eq_left ⟨(hroot S).2.1,hmem⟩]
+    exact (mul_comm _ _).trans (Finset.prod_sdiff (hroot S).1)
+  have hsum : (∑ S : I, ∏ v ∈ S.val, p v) ≤ ∑ z, W z := by
+    calc
+      _ = ∑ z ∈ Finset.univ.image enc, W z := by
+        rw [Finset.sum_image (fun S _ T _ h => hinj h)]
+        exact Finset.sum_congr rfl (fun S _ => (hwenc S).symm)
+      _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _)
+        (fun z _ _ => hW z)
+  apply hsum.trans
+  rw [Fintype.sum_prod_type]
+  calc
+    (∑ R : Finset V, ∑ L : Finset V, W (R,L)) ≤
+        ∑ R : Finset V, δ^((k:ℝ)/2) * Real.exp R.card * ∏ v ∈ R, p v := by
+      apply Finset.sum_le_sum
+      intro R _
+      by_cases hR : 2*R.card ≤ k
+      · have he : (∑ L : Finset V, W (R,L)) =
+            (∏ v ∈ R, p v) *
+              ∑ L ∈ (neighborhood G R).powersetCard (k-R.card), ∏ v ∈ L, p v := by
+          simp only [W, hR, true_and]
+          rw [← Finset.sum_filter]
+          simp only [Finset.filter_mem_eq_inter, Finset.univ_inter, Finset.mul_sum]
+        rw [he]
+        have h := mul_le_mul_of_nonneg_left (root_leaves_weight_le G p hp k δ hδ hδ1 hdeg R hR)
+          (Finset.prod_nonneg (s := R) (fun v _ => hp v))
+        simpa only [mul_comm, mul_left_comm, mul_assoc] using h
+      · simp only [W, hR, false_and, ite_false, Finset.sum_const_zero]
+        exact mul_nonneg (mul_nonneg (Real.rpow_nonneg hδ.le _) (Real.exp_pos _).le)
+          (Finset.prod_nonneg (fun v _ => hp v))
+    _ = δ^((k:ℝ)/2) * ∑ R : Finset V, ∏ v ∈ R, (Real.exp 1 * p v) := by
+      rw [Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro R _
+      rw [Finset.prod_mul_distrib, Finset.prod_const, ← Real.exp_nat_mul]
+      simp only [mul_one, mul_assoc]
+    _ ≤ δ^((k:ℝ)/2) * Real.exp (∑ v, Real.exp 1 * p v) := by
+      apply mul_le_mul_of_nonneg_left _ (Real.rpow_nonneg hδ.le _)
+      simpa only [Finset.powerset_univ] using
+        powerset_weight_le_exp Finset.univ (fun v => Real.exp 1 * p v)
+          (fun v => mul_nonneg (Real.exp_pos _).le (hp v))
+    _ = _ := by rw [← Finset.mul_sum]
+
+
+lemma noIsolated_probability_le {V Ω : Type*} [Fintype V] [Fintype Ω]
+    (G : SimpleGraph V) (O : Ω → Finset V) (k : ℕ)
+    (hcard : ∀ ω, (O ω).card = k) (p : V → ℝ) (hp : ∀ v, 0 ≤ p v)
+    (hcyl : ∀ D, inclusionMoment O D ≤ ∏ v ∈ D, p v)
+    (δ : ℝ) (hδ : 0 < δ) (hδ1 : δ ≤ 1)
+    (hdeg : ∀ v, (∑ w, if G.Adj v w then p w else 0) ≤ δ) :
+    finiteMean (fun ω => if NoIsolated G (O ω) then (1:ℝ) else 0) ≤
+      δ^((k:ℝ)/2) * Real.exp (Real.exp 1 * ∑ v, p v) := by
+  classical
+  let I := {S : Finset V // S.card = k ∧ NoIsolated G S}
+  calc
+    _ ≤ finiteMean (fun ω => ∑ S : I, if S.val ⊆ O ω then (1:ℝ) else 0) := by
+      apply finiteMean_mono
+      intro ω
+      split_ifs with h
+      · exact (Finset.single_le_sum (f := fun S : I => if S.val ⊆ O ω then (1:ℝ) else 0)
+          (fun S _ => by positivity) (Finset.mem_univ (⟨O ω,hcard ω,h⟩ : I))) |>.trans' (by simp)
+      · positivity
+    _ = ∑ S : I, inclusionMoment O S.val := by
+      rw [finiteMean_sum]
+      rfl
+    _ ≤ ∑ S : I, ∏ v ∈ S.val, p v := Finset.sum_le_sum (fun S _ => hcyl S.val)
+    _ ≤ _ := noIsolated_weight_le G p hp k δ hδ hδ1 hdeg
+
+noncomputable def typedSet {N X : Type*} [Fintype N] [Fintype X]
+    (A : N → Finset X) : Finset (N × X) := by
+  classical
+  exact Finset.univ.filter (fun v => v.2 ∈ A v.1)
+
+@[simp] lemma mem_typedSet {N X : Type*} [Fintype N] [Fintype X]
+    (A : N → Finset X) (v : N × X) : v ∈ typedSet A ↔ v.2 ∈ A v.1 := by
+  classical
+  simp [typedSet]
+
+noncomputable def fiber {N X : Type*} [Fintype X]
+    (S : Finset (N × X)) (t : N) : Finset X := by
+  classical
+  exact Finset.univ.filter (fun x => (t,x) ∈ S)
+
+@[simp] lemma mem_fiber {N X : Type*} [Fintype X]
+    (S : Finset (N × X)) (t : N) (x : X) : x ∈ fiber S t ↔ (t,x) ∈ S := by
+  classical
+  simp [fiber]
+
+lemma typedSet_fiber {N X : Type*} [Fintype N] [Fintype X]
+    (S : Finset (N × X)) : typedSet (fiber S) = S := by
+  classical
+  ext v
+  simp
+
+lemma typedSet_prod {N X : Type*} [Fintype N] [Fintype X]
+    (A : N → Finset X) (p : N × X → ℝ) :
+    (∏ v ∈ typedSet A, p v) = ∏ t, ∏ x ∈ A t, p (t,x) := by
+  classical
+  simp only [typedSet, Finset.prod_filter, Fintype.prod_prod_type]
+  apply Finset.prod_congr rfl
+  intro t _
+  rw [← Finset.prod_filter]
+  simp only [Finset.filter_mem_eq_inter, Finset.univ_inter]
+
+lemma typedSet_card {N X : Type*} [Fintype N] [Fintype X]
+    (A : N → Finset X) : (typedSet A).card = ∑ t, (A t).card := by
+  classical
+  simp only [Finset.card_eq_sum_ones, typedSet, Finset.sum_filter, Fintype.sum_prod_type]
+  apply Finset.sum_congr rfl
+  intro t _
+  rw [← Finset.sum_filter]
+  simp only [Finset.filter_mem_eq_inter, Finset.univ_inter]
+
+lemma subset_typedSet {N X : Type*} [Fintype N] [Fintype X]
+    (A : N → Finset X) (D : Finset (N × X)) :
+    D ⊆ typedSet A ↔ ∀ t, fiber D t ⊆ A t := by
+  constructor
+  · intro h t x hx
+    exact mem_typedSet A (t,x) |>.mp (h (mem_fiber D t x |>.mp hx))
+  · intro h v hv
+    exact mem_typedSet A v |>.mpr (h v.1 (mem_fiber D v.1 v.2 |>.mpr hv))
+
+end Thorp.SparseContact
+
+end ThorpNine.SparseSaving
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/SparseSaving/ParallelFollow.lean b/lean/OAI/Probability/ThorpRouting/SparseSaving/ParallelFollow.lean
index ab46d7064886c3564507e0f2840f9e93024e045a..c05465299846a021d4d3b92e1ffcea2d76b68703 100644
--- a/lean/OAI/Probability/ThorpRouting/SparseSaving/ParallelFollow.lean
+++ b/lean/OAI/Probability/ThorpRouting/SparseSaving/ParallelFollow.lean
@@ -1,883 +1,883 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.SparseSaving.PriorPaths
-
-namespace OAI
-
-namespace ThorpNine.SparseSaving
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-section ParallelButterfly
-variable {α B : Type*} [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] {d : ℕ}
-omit [DecidableEq B] in
-lemma parallel_entropy (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool) (S : Finset α) :
-    (∑ i : B × SwitchIndex d,
-      if (parallelUsers pre ω i).1 ∈ S ∧ (parallelUsers pre ω i).2 ∈ S then (1:ℝ) else 0) ≤
-      (S.card : ℝ) * Real.log S.card / (2 * Real.log 2) := by
-  classical
-  rw [Fintype.sum_prod_type]
-  calc
-    _ ≤ ∑ b : B, (blockMarks pre S b).card * Real.log (blockMarks pre S b).card / (2*Real.log 2) := by
-      apply Finset.sum_le_sum
-      intro b _
-      simpa only [←mem_blockMarks,parallelUsers] using
-        marked_switch_entropy_bound d (decodeButterfly d (fun i => ω (b,i))) (blockMarks pre S b)
-    _ ≤ ∑ b : B, (blockMarks pre S b).card * Real.log S.card / (2*Real.log 2) := by
-      apply Finset.sum_le_sum
-      intro b _
-      by_cases hz : (blockMarks pre S b).card = 0
-      · simp [hz]
-      · apply div_le_div_of_nonneg_right _ (by positivity)
-        apply mul_le_mul_of_nonneg_left _ (by positivity)
-        exact Real.log_le_log (by exact_mod_cast Nat.pos_of_ne_zero hz)
-          (by exact_mod_cast card_blockMarks_le pre S b)
-    _ = _ := by rw [←Finset.sum_div,←Finset.sum_mul,←Nat.cast_sum,sum_card_blockMarks]
-
-noncomputable def parallelFollow (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x : α) (k : α → QueryTree (B × SwitchIndex d)) : QueryTree (B × SwitchIndex d) :=
-  QueryTree.followPath d (fun i => ((pre x).1,i)) (pre x).2 (fun y => k (post ((pre x).1,y)))
-
-omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
-lemma parallelFollow_succeeds (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
-    (parallelFollow pre post x k).succeeds ω = (k (relabeledPerm pre post ω x)).succeeds ω := by
-  exact QueryTree.followPath_succeeds _ _ _ _ _
-
-omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
-lemma parallelFollow_raw_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
-    (parallelFollow pre post x k).charge ω = (k (relabeledPerm pre post ω x)).charge ω := by
-  exact QueryTree.followPath_raw_charge _ _ _ _ _
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma parallelFollow_stale_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
-    (ω : B × SwitchIndex d → Bool) :
-    (parallelFollow pre post x k).staleCharge R ω =
-      (k (relabeledPerm pre post ω x)).staleCharge (R ∪ parallelPath pre ω x) ω := by
-  exact QueryTree.followPath_stale_charge _ _ _ _ _ _
-
-noncomputable def closingDomain (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) : Finset (B × SwitchIndex d) :=
-  (routeDomain d (pre x).2 (post.symm y).2).image (fun i => ((pre x).1,i))
-
-def closingValue (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (i : B × SwitchIndex d) : Bool := routeValue d (pre x).2 (post.symm y).2 i.2
-
-omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
-lemma relabeled_endpoint_pair (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (ω : B × SwitchIndex d → Bool) (x y : α) :
-    relabeledPerm pre post ω x = y ↔
-      (pre x).1 = (post.symm y).1 ∧
-      butterflyPerm d (decodeButterfly d (fun i => ω ((pre x).1,i))) (pre x).2 = (post.symm y).2 := by
-  change post ((pre x).1,_) = y ↔ _
-  rw [←post.eq_symm_apply,Prod.mk.injEq]
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma relabeled_endpoint_iff (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (ω : B × SwitchIndex d → Bool) (x y : α) :
-    relabeledPerm pre post ω x = y ↔
-      (pre x).1 = (post.symm y).1 ∧
-        ∀ i ∈ closingDomain pre post x y, ω i = closingValue pre post x y i := by
-  classical
-  rw [relabeled_endpoint_pair,butterfly_endpoint_iff]
-  simp only [closingDomain,Finset.forall_mem_image,closingValue]
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma closingDomain_eq (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (ω : B × SwitchIndex d → Bool) (x y : α) (h : relabeledPerm pre post ω x = y) :
-    closingDomain pre post x y = parallelPath pre ω x := by
-  rw [closingDomain,parallelPath,(relabeled_endpoint_pair _ _ _ _ _).mp h |>.2]
-
-noncomputable def parallelClose (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (k : QueryTree (B × SwitchIndex d)) : QueryTree (B × SwitchIndex d) :=
-  if (pre x).1 = (post.symm y).1 then
-    QueryTree.forceList (closingDomain pre post x y).toList (closingValue pre post x y) k
-  else QueryTree.reject
-
-omit [Fintype α] in
-lemma parallelClose_succeeds (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (k : QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
-    (parallelClose pre post x y k).succeeds ω =
-      if relabeledPerm pre post ω x = y then k.succeeds ω else false := by
-  classical
-  simp only [relabeled_endpoint_iff,parallelClose]
-  by_cases hb : (pre x).1 = (post.symm y).1
-  · rw [ite_eq_left hb,QueryTree.forceList_succeeds]
-    simp only [Finset.mem_toList,hb,true_and]
-  · simp [hb,QueryTree.succeeds]
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma parallelClose_raw_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (k : QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool)
-    (h : relabeledPerm pre post ω x = y) :
-    (parallelClose pre post x y k).charge ω = k.charge ω + d := by
-  have hh := (relabeled_endpoint_iff _ _ _ _ _).mp h
-  rw [parallelClose,ite_eq_left hh.1,QueryTree.forceList_raw_charge _ _ _ _
-    (by simpa only [Finset.mem_toList] using hh.2),Finset.length_toList,
-    closingDomain_eq _ _ _ _ _ h,card_parallelPath]
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma parallelClose_fresh_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (k : QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
-    (ω : B × SwitchIndex d → Bool) (h : relabeledPerm pre post ω x = y) :
-    (parallelClose pre post x y k).freshCharge R ω =
-      k.freshCharge (R ∪ parallelPath pre ω x) ω + (parallelPath pre ω x \ R).card := by
-  have hh := (relabeled_endpoint_iff _ _ _ _ _).mp h
-  rw [parallelClose,ite_eq_left hh.1,QueryTree.forceList_charge _ _ _ _ _
-    (by simpa only [Finset.mem_toList] using hh.2),Finset.toList_toFinset,
-    closingDomain_eq _ _ _ _ _ h]
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma parallelClose_stale_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (k : QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
-    (ω : B × SwitchIndex d → Bool) (h : relabeledPerm pre post ω x = y) :
-    (parallelClose pre post x y k).staleCharge R ω =
-      k.staleCharge (R ∪ parallelPath pre ω x) ω + (parallelPath pre ω x ∩ R).card := by
-  have h₀ := QueryTree.fresh_add_stale (parallelClose pre post x y k) R ω
-  have h₁ := QueryTree.fresh_add_stale k (R ∪ parallelPath pre ω x) ω
-  rw [parallelClose_raw_charge _ _ _ _ _ _ h,parallelClose_fresh_charge _ _ _ _ _ _ _ h] at h₀
-  have h₂ : (parallelPath pre ω x \ R).card + (parallelPath pre ω x ∩ R).card = d := by
-    rw [Finset.card_sdiff_add_card_inter,card_parallelPath]
-  omega
-
-noncomputable def parallelNetwork (pre : α ≃ B × Card d) (post : B × Card d ≃ α) :
-    RoutingNetwork α (B × SwitchIndex d) d where
-  perm := relabeledPerm pre post
-  path := parallelPath pre
-  users := parallelUsers pre
-  users_ne := parallelUsers_ne pre
-  mem_path := mem_parallelPath pre
-  card_path := card_parallelPath pre
-  entropy := parallel_entropy pre
-  follow := parallelFollow pre post
-  close := parallelClose pre post
-  follow_succeeds := parallelFollow_succeeds pre post
-  follow_raw_charge := parallelFollow_raw_charge pre post
-  follow_stale_charge := parallelFollow_stale_charge pre post
-  close_succeeds := parallelClose_succeeds pre post
-  close_raw_charge := parallelClose_raw_charge pre post
-  close_stale_charge := parallelClose_stale_charge pre post
-
-lemma parallel_cycle_log_mgf (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (S : Finset α) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    Real.log (finiteMean (fun ω : B × SwitchIndex d → Bool =>
-      q^(Fintype.card (RoutingNetwork.SelectedCycle (relabeledPerm pre post ω) S)))) ≤
-      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
-  exact (parallelNetwork pre post).butterfly_cycle_log_mgf S J hJ q hq
-
-end ParallelButterfly
-
-def cardSplit : (r s : ℕ) → Card (s+r) ≃ Card r × Card s
-  | 0, s =>
-    { toFun := fun x => ((fun i => Fin.elim0 i), x)
-      invFun := fun x => x.2
-      left_inv := fun _ => rfl
-      right_inv := fun x => by
-        apply Prod.ext
-        · exact Subsingleton.elim _ _
-        · rfl }
-  | r+1, s =>
-    { toFun := fun x => (Fin.cons (x 0) ((cardSplit r s (Fin.tail x)).1),
-        (cardSplit r s (Fin.tail x)).2)
-      invFun := fun x => Fin.cons (x.1 0) ((cardSplit r s).symm (Fin.tail x.1,x.2))
-      left_inv := by
-        intro x
-        simp only [Fin.cons_zero, Fin.tail_cons, Prod.mk.eta, Equiv.symm_apply_apply, Fin.cons_self_tail]
-      right_inv := by
-        intro x
-        simp only [Fin.cons_zero, Fin.tail_cons, Equiv.apply_symm_apply, Fin.cons_self_tail] }
-
-def assembleBits : (r s : ℕ) → (Card r × SwitchIndex s → Bool) →
-    (Card s × SwitchIndex r → Bool) → SwitchIndex (s+r) → Bool
-  | 0, _, lo, _, i => lo ((fun j => Fin.elim0 j), i)
-  | r+1, s, _, hi, Sum.inl y => hi ((cardSplit r s y).2,Sum.inl (cardSplit r s y).1)
-  | r+1, s, lo, hi, Sum.inr (b,i) => assembleBits r s
-      (fun j => lo (Fin.cons b j.1,j.2)) (fun j => hi (j.1,Sum.inr (b,j.2))) i
-
-lemma butterfly_split (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
-    (hi : Card s × SwitchIndex r → Bool) (x : Card (s+r)) :
-    cardSplit r s (butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi)) x) =
-      let y := cardSplit r s x
-      let z := butterflyPerm s (decodeButterfly s (fun i => lo (y.1,i))) y.2
-      (butterflyPerm r (decodeButterfly r (fun i => hi (z,i))) y.1,z) := by
-  induction r with
-  | zero =>
-    apply Prod.ext
-    · funext i
-      exact Fin.elim0 i
-    · rfl
-  | succ r ih =>
-    let lo' : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons (x 0) j.1,j.2)
-    let hi' : Card s × SwitchIndex r → Bool := fun j => hi (j.1,Sum.inr (x 0,j.2))
-    have hh := ih lo' hi' (Fin.tail x)
-    let u := butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo' hi')) (Fin.tail x)
-    let y := cardSplit r s (Fin.tail x)
-    let z := butterflyPerm s (decodeButterfly s (fun i => lo' (y.1,i))) y.2
-    let w := butterflyPerm r (decodeButterfly r (fun i => hi' (z,i))) y.1
-    change cardSplit r s u = (w,z) at hh
-    change ((Fin.cons (Bool.xor (x 0) (hi ((cardSplit r s u).2,Sum.inl (cardSplit r s u).1)))
-        ((cardSplit r s u).1) : Card (r+1)), (cardSplit r s u).2) =
-      ((Fin.cons (Bool.xor (x 0) (hi (z,Sum.inl w))) w : Card (r+1)),z)
-    rw [hh]
-
-def splitSwitch : (r s : ℕ) → SwitchIndex (s+r) ≃
-    (Card r × SwitchIndex s) ⊕ (Card s × SwitchIndex r)
-  | 0, _ =>
-    { toFun := fun i => Sum.inl ((fun j => Fin.elim0 j),i)
-      invFun := fun i => match i with
-        | Sum.inl (_,j) => j
-        | Sum.inr (_,j) => nomatch j
-      left_inv := fun _ => rfl
-      right_inv := by
-        intro i
-        rcases i with ⟨x,j⟩ | ⟨x,j⟩
-        · exact congrArg Sum.inl (Prod.ext (Subsingleton.elim _ _) rfl)
-        · exact Empty.elim j }
-  | r+1, s =>
-    { toFun := fun i => match i with
-        | Sum.inl y => Sum.inr ((cardSplit r s y).2,Sum.inl (cardSplit r s y).1)
-        | Sum.inr (b,j) => match splitSwitch r s j with
-          | Sum.inl (p,j) => Sum.inl (Fin.cons b p,j)
-          | Sum.inr (q,j) => Sum.inr (q,Sum.inr (b,j))
-      invFun := fun i => match i with
-        | Sum.inl (p,j) => Sum.inr (p 0,(splitSwitch r s).symm (Sum.inl (Fin.tail p,j)))
-        | Sum.inr (q,Sum.inl p) => Sum.inl ((cardSplit r s).symm (p,q))
-        | Sum.inr (q,Sum.inr (b,j)) => Sum.inr (b,(splitSwitch r s).symm (Sum.inr (q,j)))
-      left_inv := by
-        intro i
-        rcases i with y | ⟨b,j⟩
-        · dsimp only
-          rw [Prod.mk.eta, Equiv.symm_apply_apply]
-        · dsimp only
-          cases hz : splitSwitch r s j with
-          | inl z =>
-            rcases z with ⟨p,k⟩
-            simp only [Fin.cons_zero, Fin.tail_cons]
-            rw [←hz,Equiv.symm_apply_apply]
-          | inr z =>
-            rcases z with ⟨q,k⟩
-            dsimp only
-            rw [←hz,Equiv.symm_apply_apply]
-      right_inv := by
-        intro i
-        rcases i with ⟨p,j⟩ | ⟨q,j⟩
-        · simp only [Equiv.apply_symm_apply, Fin.cons_self_tail]
-        · rcases j with p | ⟨b,j⟩ <;> simp only [Equiv.apply_symm_apply] }
-
-lemma assembleBits_eq (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
-    (hi : Card s × SwitchIndex r → Bool) (i : SwitchIndex (s+r)) :
-    assembleBits r s lo hi i = Sum.elim lo hi (splitSwitch r s i) := by
-  induction r with
-  | zero => rfl
-  | succ r ih =>
-    rcases i with y | ⟨b,j⟩
-    · rfl
-    · simp only [assembleBits,splitSwitch,Equiv.coe_fn_mk]
-      rw [ih]
-      cases splitSwitch r s j <;> rfl
-
-def splitCoinEquiv (r s : ℕ) :
-    ((Card r × SwitchIndex s → Bool) × (Card s × SwitchIndex r → Bool)) ≃
-      (SwitchIndex (s+r) → Bool) :=
-  (Equiv.sumArrowEquivProdArrow _ _ Bool).symm.trans
-    (Equiv.arrowCongr (splitSwitch r s).symm (Equiv.refl Bool))
-
-lemma splitCoinEquiv_apply (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
-    (hi : Card s × SwitchIndex r → Bool) :
-    splitCoinEquiv r s (lo,hi) = assembleBits r s lo hi := by
-  funext i
-  exact (assembleBits_eq r s lo hi i).symm
-
-lemma finiteMean_prod {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] (f : Ω × Ω' → ℝ) :
-    finiteMean f = finiteMean (fun x => finiteMean (fun y => f (x,y))) := by
-  simp only [finiteMean, Fintype.sum_prod_type, Fintype.card_prod, Nat.cast_mul,
-    ←Finset.sum_div, div_div, mul_comm]
-
-lemma finiteMean_split (r s : ℕ) (f : (SwitchIndex (s+r) → Bool) → ℝ) :
-    finiteMean f = finiteMean (fun lo : Card r × SwitchIndex s → Bool =>
-      finiteMean (fun hi : Card s × SwitchIndex r → Bool => f (assembleBits r s lo hi))) := by
-  rw [finiteMean_equiv (splitCoinEquiv r s).symm]
-  simp only [Equiv.symm_symm]
-  rw [finiteMean_prod]
-  apply finiteMean_congr
-  intro lo
-  apply finiteMean_congr
-  intro hi
-  rw [splitCoinEquiv_apply]
-
-section PartiallyExposed
-variable {α : Type*} {r s : ℕ}
-
-def freshPre (pre : α ≃ Card (s+r)) (lo : Card r × SwitchIndex s → Bool) :
-    α ≃ Card s × Card r :=
-  ((pre.trans (cardSplit r s)).trans (parallelPerm lo)).trans (Equiv.prodComm _ _)
-
-def freshPost (post : Card (s+r) ≃ α) : Card s × Card r ≃ α :=
-  ((Equiv.prodComm _ _).trans (cardSplit r s).symm).trans post
-
-lemma exposed_perm_eq (pre : α ≃ Card (s+r)) (post : Card (s+r) ≃ α)
-    (lo : Card r × SwitchIndex s → Bool) (hi : Card s × SwitchIndex r → Bool) :
-    (pre.trans (butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi)))).trans post =
-      relabeledPerm (freshPre pre lo) (freshPost post) hi := by
-  ext x
-  simpa only [relabeledPerm, freshPre, freshPost, parallelPerm, Equiv.trans_apply,
-    Equiv.coe_fn_mk, Equiv.prodComm_apply, Prod.swap, Equiv.symm_apply_apply] using
-    congrArg (fun y => post ((cardSplit r s).symm y)) (butterfly_split r s lo hi (pre x))
-
-lemma partially_exposed_cycle_log_mgf [Fintype α] [DecidableEq α]
-    (pre : α ≃ Card (s+r)) (post : Card (s+r) ≃ α)
-    (lo : Card r × SwitchIndex s → Bool) (S : Finset α)
-    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    Real.log (finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(Fintype.card (RoutingNetwork.SelectedCycle
-        ((pre.trans (butterflyPerm (s+r)
-          (decodeButterfly (s+r) (assembleBits r s lo hi)))).trans post) S)))) ≤
-      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^r) := by
-  simp only [exposed_perm_eq]
-  exact parallel_cycle_log_mgf (freshPre pre lo) (freshPost post) S J hJ q hq
-
-end PartiallyExposed
-
-lemma inclusionMoment_eq {α Ω : Type*} [DecidableEq α] [Fintype Ω]
-    (F : Ω → Finset α) (S : Finset α) :
-    inclusionMoment F S = finiteMean (fun ω => if S ⊆ F ω then (1:ℝ) else 0) := by
-  unfold inclusionMoment
-  apply finiteMean_congr
-  intro ω
-  split_ifs <;> rfl
-
-lemma finiteMean_bool (f : Bool → ℝ) : finiteMean f = (f false+f true)/2 := by
-  simp only [finiteMean, Fintype.sum_bool, Fintype.card_bool, Nat.cast_ofNat]
-  ring
-
-lemma finiteMean_const {Ω : Type*} [Fintype Ω] [Nonempty Ω] (c : ℝ) :
-    finiteMean (fun _ : Ω => c) = c := by
-  simp only [finiteMean,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
-  exact mul_div_cancel_left₀ c (Nat.cast_ne_zero.mpr Fintype.card_ne_zero)
-
-lemma finiteMean_pi_prod {ι : Type*} [Fintype ι] [DecidableEq ι] {Ω : ι → Type*}
-    [∀ i, Fintype (Ω i)] (f : (i : ι) → Ω i → ℝ) :
-    finiteMean (fun ω : (i : ι) → Ω i => ∏ i, f i (ω i)) = ∏ i, finiteMean (f i) := by
-  simp only [finiteMean,←Fintype.prod_sum,Fintype.card_pi,Nat.cast_prod,Finset.prod_div_distrib]
-
-section ParallelFairLayer
-variable {β : Type*} [Fintype β] [DecidableEq β]
-
-def pairLayer (ξ : β → Bool) : Equiv.Perm (Bool × β) where
-  toFun x := (Bool.xor x.1 (ξ x.2),x.2)
-  invFun x := (Bool.xor x.1 (ξ x.2),x.2)
-  left_inv x := by
-    rcases x with ⟨b,x⟩
-    change (Bool.xor (Bool.xor b (ξ x)) (ξ x),x) = (b,x)
-    cases b <;> cases ξ x <;> rfl
-  right_inv x := by
-    rcases x with ⟨b,x⟩
-    change (Bool.xor (Bool.xor b (ξ x)) (ξ x),x) = (b,x)
-    cases b <;> cases ξ x <;> rfl
-
-noncomputable def pairFiber (S : Finset (Bool × β)) (b : β) : Finset Bool :=
-  Finset.univ.filter (fun ε => (ε,b) ∈ S)
-
-lemma prod_pairFiber (S : Finset (Bool × β)) (f : Bool × β → ℝ) :
-    (∏ x ∈ S, f x) = ∏ b : β, ∏ ε ∈ pairFiber S b, f (ε,b) := by
-  classical
-  simp only [pairFiber,Finset.prod_filter]
-  rw [Finset.prod_comm,←Fintype.prod_prod_type (fun x : Bool × β => if x ∈ S then f x else 1)]
-  rw [←Finset.prod_filter]
-  congr 1
-  ext x
-  simp
-
-lemma fiber_average_bound (S : Finset Bool) (p : Bool → ℝ) (hp : ∀ b, 0 ≤ p b) :
-    finiteMean (fun c : Bool => ∏ ε ∈ S, p (Bool.xor ε c)) ≤
-      ∏ ε ∈ S, (p ε + p (!ε))/2 := by
-  rw [finiteMean_bool]
-  have hxor : ∀ ε : Bool, Bool.xor ε true = Equiv.swap false true ε := by decide
-  simp only [Bool.xor_false,hxor]
-  calc
-    _ ≤ ∏ ε ∈ S, (p ε+p (Equiv.swap false true ε))/2 :=
-      swap_average_product S p hp false true
-    _ = _ := Finset.prod_congr rfl (fun ε _ => by cases ε <;> rfl)
-
-lemma CylinderBound.fair_parallel {m : Finset (Bool × β) → ℝ} {p : Bool × β → ℝ}
-    (h : CylinderBound m p) (hp : ∀ x, 0 ≤ p x) :
-    CylinderBound (fun S => finiteMean (fun ξ : β → Bool => m (S.image (pairLayer ξ))))
-      (fun x => (p x+p (!x.1,x.2))/2) := by
-  intro S
-  calc
-    _ ≤ finiteMean (fun ξ : β → Bool => ∏ x ∈ S, p (pairLayer ξ x)) := by
-      apply finiteMean_mono
-      intro ξ
-      simpa only [Finset.prod_image (pairLayer ξ).injective.injOn] using h (S.image (pairLayer ξ))
-    _ = ∏ b : β, finiteMean (fun c : Bool => ∏ ε ∈ pairFiber S b, p (Bool.xor ε c,b)) := by
-      calc
-        _ = finiteMean (fun ξ : β → Bool => ∏ b : β,
-            ∏ ε ∈ pairFiber S b, p (Bool.xor ε (ξ b),b)) :=
-          finiteMean_congr (fun ξ => prod_pairFiber S (fun x => p (pairLayer ξ x)))
-        _ = _ := finiteMean_pi_prod (fun b c => ∏ ε ∈ pairFiber S b, p (Bool.xor ε c,b))
-    _ ≤ ∏ b : β, ∏ ε ∈ pairFiber S b, (p (ε,b)+p (!ε,b))/2 := by
-      apply Finset.prod_le_prod₀
-      · intro b _
-        exact div_nonneg (Finset.sum_nonneg (fun c _ =>
-          Finset.prod_nonneg (fun ε _ => hp (Bool.xor ε c,b)))) (Nat.cast_nonneg _)
-      · intro b _
-        exact fiber_average_bound (pairFiber S b) (fun ε => p (ε,b)) (fun ε => hp (ε,b))
-    _ = _ := (prod_pairFiber S (fun x => (p x+p (!x.1,x.2))/2)).symm
-
-omit [Fintype β] in
-lemma subset_pairLayer_image_iff (S A : Finset (Bool × β)) (ξ : β → Bool) :
-    S ⊆ A.image (pairLayer ξ) ↔ S.image (pairLayer ξ) ⊆ A := by
-  have hi (x : Bool × β) : pairLayer ξ (pairLayer ξ x) = x := (pairLayer ξ).left_inv x
-  constructor
-  · intro h
-    have h' := Finset.image_subset_image h (f := pairLayer ξ)
-    simpa only [Finset.image_image,Function.comp_def,hi,Finset.image_id'] using h'
-  · intro h
-    have h' := Finset.image_subset_image h (f := pairLayer ξ)
-    simpa only [Finset.image_image,Function.comp_def,hi,Finset.image_id'] using h'
-
-noncomputable def fairSetLayer {Ω : Type*} (F : Ω → Finset (Bool × β)) :
-    Ω × (β → Bool) → Finset (Bool × β) := fun ω => (F ω.1).image (pairLayer ω.2)
-
-lemma finiteMean_comm {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] (f : Ω → Ω' → ℝ) :
-    finiteMean (fun x => finiteMean (f x)) = finiteMean (fun y => finiteMean (fun x => f x y)) := by
-  simp only [finiteMean,←Finset.sum_div,div_div]
-  rw [Finset.sum_comm,mul_comm]
-
-lemma inclusionMoment_fairSetLayer {Ω : Type*} [Fintype Ω]
-    (F : Ω → Finset (Bool × β)) (S : Finset (Bool × β)) :
-    inclusionMoment (fairSetLayer F) S =
-      finiteMean (fun ξ : β → Bool => inclusionMoment F (S.image (pairLayer ξ))) := by
-  classical
-  simp only [inclusionMoment_eq]
-  have he : (fun ω : Ω × (β → Bool) => if S ⊆ fairSetLayer F ω then (1:ℝ) else 0) =
-      (fun ω : Ω × (β → Bool) => if S.image (pairLayer ω.2) ⊆ F ω.1 then (1:ℝ) else 0) := by
-    funext ω
-    simp only [fairSetLayer,subset_pairLayer_image_iff]
-  apply (congrArg finiteMean he).trans
-  exact (finiteMean_prod (fun ω : Ω × (β → Bool) =>
-    if S.image (pairLayer ω.2) ⊆ F ω.1 then (1:ℝ) else 0)).trans
-      (finiteMean_comm (fun ω ξ => if S.image (pairLayer ξ) ⊆ F ω then (1:ℝ) else 0))
-
-lemma inclusionMoment_fairSetLayer_singleton {Ω : Type*} [Fintype Ω]
-    (F : Ω → Finset (Bool × β)) (x : Bool × β) :
-    inclusionMoment (fairSetLayer F) {x} =
-      (inclusionMoment F {x}+inclusionMoment F {(!x.1,x.2)})/2 := by
-  rw [inclusionMoment_fairSetLayer]
-  simp only [Finset.image_singleton,pairLayer,Equiv.coe_fn_mk]
-  rw [finiteMean_pi_eval (A := fun _ : β => Bool) x.2 (fun c : Bool => inclusionMoment F {(Bool.xor x.1 c,x.2)})]
-  rw [finiteMean_bool]
-  rcases x with ⟨b,x⟩
-  cases b <;> simp only [Bool.xor_false,Bool.false_xor,Bool.true_xor, Bool.not_false,Bool.not_true]
-
-lemma cylinder_fairSetLayer {Ω : Type*} [Fintype Ω]
-    (F : Ω → Finset (Bool × β))
-    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
-    CylinderBound (inclusionMoment (fairSetLayer F))
-      (fun x => inclusionMoment (fairSetLayer F) {x}) := by
-  have hh := h.fair_parallel (fun x => inclusionMoment_nonneg F {x})
-  intro S
-  rw [inclusionMoment_fairSetLayer]
-  simp only [inclusionMoment_fairSetLayer_singleton]
-  exact hh S
-
-end ParallelFairLayer
-
-section RelabelMarks
-variable {α β Ω Ω' : Type*} [DecidableEq α] [DecidableEq β] [Fintype Ω] [Fintype Ω']
-
-omit [Fintype Ω] [Fintype Ω'] in
-lemma subset_image_equiv_iff (e : α ≃ β) (A : Finset α) (S : Finset β) :
-    S ⊆ A.image e ↔ S.image e.symm ⊆ A := by
-  constructor
-  · intro h x hx
-    obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
-    obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp (h hy)
-    simpa only [←hxy,Equiv.symm_apply_apply] using hx
-  · intro h y hy
-    exact Finset.mem_image.mpr ⟨e.symm y,h (Finset.mem_image_of_mem _ hy),e.apply_symm_apply y⟩
-
-omit [Fintype Ω'] in
-lemma inclusionMoment_image_equiv (e : α ≃ β) (F : Ω → Finset α) (S : Finset β) :
-    inclusionMoment (fun ω => (F ω).image e) S = inclusionMoment F (S.image e.symm) := by
-  classical
-  simp only [inclusionMoment_eq]
-  apply finiteMean_congr
-  intro ω
-  simp only [subset_image_equiv_iff]
-
-omit [Fintype Ω'] in
-lemma cylinder_image_equiv (e : α ≃ β) (F : Ω → Finset α)
-    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
-    CylinderBound (inclusionMoment (fun ω => (F ω).image e))
-      (fun x => inclusionMoment (fun ω => (F ω).image e) {x}) := by
-  intro S
-  simp only [inclusionMoment_image_equiv,Finset.image_singleton]
-  have hh := h (S.image e.symm)
-  simpa only [Finset.prod_image e.symm.injective.injOn] using hh
-
-omit [DecidableEq β] in
-lemma inclusionMoment_comp_equiv (e : Ω ≃ Ω') (F : Ω' → Finset α) (S : Finset α) :
-    inclusionMoment (fun ω => F (e ω)) S = inclusionMoment F S := by
-  classical
-  simp only [inclusionMoment_eq]
-  simpa only [Equiv.apply_symm_apply] using
-    finiteMean_equiv e (fun ω => if S ⊆ F (e ω) then (1 : ℝ) else 0)
-
-omit [DecidableEq β] in
-lemma cylinder_comp_equiv (e : Ω ≃ Ω') (F : Ω' → Finset α)
-    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
-    CylinderBound (inclusionMoment (fun ω => F (e ω)))
-      (fun x => inclusionMoment (fun ω => F (e ω)) {x}) := by
-  intro S
-  simpa only [inclusionMoment_comp_equiv] using h S
-end RelabelMarks
-
-lemma finiteMean_prod_mul {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
-    (f : Ω → ℝ) (g : Ω' → ℝ) :
-    finiteMean (fun ω : Ω × Ω' => f ω.1*g ω.2) = finiteMean f * finiteMean g := by
-  simp only [finiteMean,Fintype.sum_prod_type,Fintype.card_prod,Nat.cast_mul,
-    ←Finset.mul_sum,←Finset.sum_mul]
-  ring
-
-section JoinChildren
-variable {β : Type*} [DecidableEq β]
-
-noncomputable def childJoin (A B : Finset β) : Finset (Bool × β) :=
-  A.image (fun x => (false,x)) ∪ B.image (fun x => (true,x))
-
-lemma mem_childJoin (A B : Finset β) (b : Bool) (x : β) :
-    (b,x) ∈ childJoin A B ↔ (if b then x ∈ B else x ∈ A) := by
-  cases b <;> simp [childJoin]
-
-noncomputable def boolFiber (S : Finset (Bool × β)) (b : Bool) : Finset β :=
-  (S.filter (fun x => x.1 = b)).image Prod.snd
-
-lemma mem_boolFiber (S : Finset (Bool × β)) (b : Bool) (x : β) :
-    x ∈ boolFiber S b ↔ (b,x) ∈ S := by
-  constructor
-  · intro h
-    obtain ⟨⟨c,y⟩,hy,hxy⟩ := Finset.mem_image.mp h
-    obtain ⟨hy,hc⟩ := Finset.mem_filter.mp hy
-    change c = b at hc
-    change y = x at hxy
-    simpa only [hc,hxy] using hy
-  · intro h
-    exact Finset.mem_image.mpr ⟨(b,x),Finset.mem_filter.mpr ⟨h,rfl⟩,rfl⟩
-
-lemma childJoin_boolFiber (S : Finset (Bool × β)) :
-    childJoin (boolFiber S false) (boolFiber S true) = S := by
-  ext ⟨b,x⟩
-  rw [mem_childJoin]
-  cases b <;> simp only [Bool.false_eq_true,↓reduceIte,mem_boolFiber]
-
-lemma subset_childJoin_iff (S : Finset (Bool × β)) (A B : Finset β) :
-    S ⊆ childJoin A B ↔ boolFiber S false ⊆ A ∧ boolFiber S true ⊆ B := by
-  constructor
-  · intro h
-    constructor
-    · intro x hx
-      exact (mem_childJoin A B false x).mp (h ((mem_boolFiber S false x).mp hx))
-    · intro x hx
-      exact (mem_childJoin A B true x).mp (h ((mem_boolFiber S true x).mp hx))
-  · rintro ⟨h₀,h₁⟩ ⟨b,x⟩ hx
-    rw [mem_childJoin]
-    cases b
-    · exact h₀ ((mem_boolFiber S false x).mpr hx)
-    · exact h₁ ((mem_boolFiber S true x).mpr hx)
-
-lemma prod_childJoin (A B : Finset β) (f : Bool × β → ℝ) :
-    (∏ x ∈ childJoin A B, f x) = (∏ x ∈ A, f (false,x)) * ∏ x ∈ B, f (true,x) := by
-  classical
-  have hd : Disjoint (A.image (fun x => (false,x))) (B.image (fun x => (true,x))) := by
-    apply Finset.disjoint_left.mpr
-    rintro x hx hy
-    obtain ⟨a,ha,rfl⟩ := Finset.mem_image.mp hx
-    obtain ⟨b,hb,hba⟩ := Finset.mem_image.mp hy
-    exact Bool.false_ne_true (congrArg Prod.fst hba).symm
-  rw [childJoin,Finset.prod_union hd,
-    Finset.prod_image (fun _ _ _ _ h => congrArg Prod.snd h),
-    Finset.prod_image (fun _ _ _ _ h => congrArg Prod.snd h)]
-
-lemma prod_boolFibers (S : Finset (Bool × β)) (f : Bool × β → ℝ) :
-    (∏ x ∈ S, f x) = (∏ x ∈ boolFiber S false, f (false,x)) *
-      ∏ x ∈ boolFiber S true, f (true,x) := by
-  rw [←prod_childJoin,childJoin_boolFiber]
-
-lemma inclusionMoment_childJoin {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
-    (F : Ω → Finset β) (G : Ω' → Finset β) (S : Finset (Bool × β)) :
-    inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) S =
-      inclusionMoment F (boolFiber S false) * inclusionMoment G (boolFiber S true) := by
-  classical
-  simp only [inclusionMoment_eq]
-  rw [←finiteMean_prod_mul]
-  apply finiteMean_congr
-  intro ω
-  simp only [subset_childJoin_iff]
-  split_ifs <;> simp_all
-
-lemma cylinder_childJoin {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] [Nonempty Ω] [Nonempty Ω']
-    (F : Ω → Finset β) (G : Ω' → Finset β)
-    (hF : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x}))
-    (hG : CylinderBound (inclusionMoment G) (fun x => inclusionMoment G {x})) :
-    CylinderBound (inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)))
-      (fun x => inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {x}) := by
-  intro S
-  rw [inclusionMoment_childJoin,prod_boolFibers]
-  have hf (b : Bool) (x : β) : boolFiber {(b,x)} b = {x} := by
-    ext y
-    simp only [mem_boolFiber,Finset.mem_singleton,Prod.mk.injEq,true_and]
-  have hn (b : Bool) (x : β) : boolFiber {(!b,x)} b = ∅ := by
-    ext y
-    cases b <;> simp [mem_boolFiber]
-  have h₀ (x : β) : inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {(false,x)} =
-      inclusionMoment F {x} * inclusionMoment G ∅ := by
-    rw [inclusionMoment_childJoin,hf]
-    have hh := hn true x
-    exact congrArg (fun T => inclusionMoment F {x} * inclusionMoment G T) hh
-  have h₁ (x : β) : inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {(true,x)} =
-      inclusionMoment F ∅ * inclusionMoment G {x} := by
-    rw [inclusionMoment_childJoin,hf]
-    have hh := hn false x
-    exact congrArg (fun T => inclusionMoment F T * inclusionMoment G {x}) hh
-  have emF : inclusionMoment F ∅ = 1 := by simp [inclusionMoment,finiteMean_const]
-  have emG : inclusionMoment G ∅ = 1 := by simp [inclusionMoment,finiteMean_const]
-  simp only [h₀,h₁,emF,emG,mul_one,one_mul]
-  exact mul_le_mul (hF _) (hG _) (inclusionMoment_nonneg _ _)
-    (Finset.prod_nonneg (fun x _ => inclusionMoment_nonneg F {x}))
-end JoinChildren
-
-def coinStepEquiv (d : ℕ) : (SwitchIndex (d+1) → Bool) ≃
-    ((SwitchIndex d → Bool) × (SwitchIndex d → Bool)) × (Card d → Bool) where
-  toFun ω := ((fun i => ω (Sum.inr (false,i)),fun i => ω (Sum.inr (true,i))),
-    fun x => ω (Sum.inl x))
-  invFun ω := Sum.elim ω.2 (fun bi => if bi.1 then ω.1.2 bi.2 else ω.1.1 bi.2)
-  left_inv ω := by
-    funext i
-    cases i with
-    | inl x => rfl
-    | inr bi => cases bi with | mk b i => cases b <;> rfl
-  right_inv ω := by
-    rcases ω with ⟨⟨ω₀,ω₁⟩,ξ⟩
-    rfl
-
-def headTailEquiv (d : ℕ) : Card (d+1) ≃ Bool × Card d :=
-  (Fin.consEquiv (fun _ : Fin (d+1) => Bool)).symm
-
-lemma childMarks_join (d : ℕ) (A : Finset (Card (d+1))) :
-    A = (childMarks A false).image (Fin.cons false) ∪
-      (childMarks A true).image (Fin.cons true) := by
-  classical
-  ext x
-  constructor
-  · intro hx
-    cases hb : x 0
-    · exact Finset.mem_union_left _ (Finset.mem_image.mpr
-        ⟨Fin.tail x,(mem_childMarks _ _ _).mpr (by simpa only [←hb,Fin.cons_self_tail] using hx),
-          by simp only [←hb,Fin.cons_self_tail]⟩)
-    · exact Finset.mem_union_right _ (Finset.mem_image.mpr
-        ⟨Fin.tail x,(mem_childMarks _ _ _).mpr (by simpa only [←hb,Fin.cons_self_tail] using hx),
-          by simp only [←hb,Fin.cons_self_tail]⟩)
-  · intro hx
-    rcases Finset.mem_union.mp hx with hx | hx
-    · obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
-      exact (mem_childMarks _ _ _).mp hy
-    · obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
-      exact (mem_childMarks _ _ _).mp hy
-
-noncomputable def butterflyMarks (d : ℕ) (A : Finset (Card d))
-    (ω : SwitchIndex d → Bool) : Finset (Card d) :=
-  A.image (butterflyPerm d (decodeButterfly d ω))
-
-lemma butterflyMarks_succ (d : ℕ) (A : Finset (Card (d+1))) (ω : SwitchIndex (d+1) → Bool) :
-    butterflyMarks (d+1) A ω =
-      (fairSetLayer (fun p : (SwitchIndex d → Bool) × (SwitchIndex d → Bool) =>
-        childJoin (butterflyMarks d (childMarks A false) p.1)
-          (butterflyMarks d (childMarks A true) p.2)) (coinStepEquiv d ω)).image
-            (headTailEquiv d).symm := by
-  classical
-  unfold butterflyMarks fairSetLayer childJoin
-  rw [Finset.image_union,Finset.image_union]
-  simp only [Finset.image_image]
-  conv_lhs => rw [childMarks_join d A]
-  rw [Finset.image_union]
-  simp only [Finset.image_image]
-  apply congrArg₂ (· ∪ ·)
-  · apply Finset.image_congr
-    intro x _
-    simp only [Function.comp_def,butterflyPerm,decodeButterfly,Equiv.Perm.coe_mul,
-      childLift,pairSwitch,switchFun,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
-      coinStepEquiv,headTailEquiv,Equiv.symm_symm,pairLayer]
-    rfl
-  · apply Finset.image_congr
-    intro x _
-    simp only [Function.comp_def,butterflyPerm,decodeButterfly,Equiv.Perm.coe_mul,
-      childLift,pairSwitch,switchFun,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
-      coinStepEquiv,headTailEquiv,Equiv.symm_symm,pairLayer]
-    rfl
-
-lemma butterflyMarks_cylinder (d : ℕ) (A : Finset (Card d)) :
-    CylinderBound (inclusionMoment (butterflyMarks d A))
-      (fun x => inclusionMoment (butterflyMarks d A) {x}) := by
-  induction d with
-  | zero =>
-    have he : butterflyMarks 0 A = fun _ : SwitchIndex 0 → Bool => A := by
-      funext ω
-      simp [butterflyMarks,butterflyPerm]
-    rw [he]
-    exact cylinder_const A
-  | succ d ih =>
-    have h := cylinder_childJoin (butterflyMarks d (childMarks A false))
-      (butterflyMarks d (childMarks A true)) (ih _) (ih _)
-    have hh := cylinder_image_equiv (headTailEquiv d).symm _ (cylinder_fairSetLayer _ h)
-    have hs := cylinder_comp_equiv (coinStepEquiv d) _ hh
-    simpa only [←butterflyMarks_succ] using hs
-
-lemma indicator_mem_image_perm {α : Type*} [DecidableEq α]
-    (A : Finset α) (p : Equiv.Perm α) (y : α) :
-    (if y ∈ A.image p then (1 : ℝ) else 0) = ∑ x ∈ A, if p x = y then 1 else 0 := by
-  classical
-  by_cases h : y ∈ A.image p
-  · obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp h
-    rw [ite_eq_left (Finset.mem_image.mpr ⟨x,hx,hxy⟩)]
-    symm
-    calc
-      _ = (if p x = y then (1:ℝ) else 0) := Finset.sum_eq_single_of_mem x hx (by
-        intro z _ hzx
-        exact ite_eq_right (fun hz => hzx (p.injective (hz.trans hxy.symm))))
-      _ = 1 := ite_eq_left hxy
-  · rw [ite_eq_right h]
-    symm
-    apply Finset.sum_eq_zero
-    intro x hx
-    exact ite_eq_right (fun hxy => h (Finset.mem_image.mpr ⟨x,hx,hxy⟩))
-
-lemma butterflyMarks_singleton (d : ℕ) (A : Finset (Card d)) (y : Card d) :
-    inclusionMoment (butterflyMarks d A) {y} = (A.card : ℝ)/(2:ℝ)^d := by
-  classical
-  unfold inclusionMoment
-  simp only [Finset.singleton_subset_iff,butterflyMarks,indicator_mem_image_perm]
-  rw [finiteMean_sum]
-  simp only [butterfly_single_card_uniform,Finset.sum_const,nsmul_eq_mul]
-  ring
-
-lemma butterflyMarks_inclusion (d : ℕ) (A S : Finset (Card d)) :
-    inclusionMoment (butterflyMarks d A) S ≤ ((A.card : ℝ)/(2:ℝ)^d)^S.card := by
-  have h := butterflyMarks_cylinder d A S
-  simpa only [butterflyMarks_singleton,Finset.prod_const] using h
-
-namespace CycleColoring
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-abbrev Coloring (p : Equiv.Perm α) (S : Finset α) :=
-  {f : α → Bool // (∀ x, x ∉ S → f x = false) ∧
-    ∀ x ∈ S, p x ∈ S → f (p x) = f x}
-
-noncomputable instance (p : Equiv.Perm α) (S : Finset α) : Fintype (Coloring p S) :=
-  Fintype.ofFinite _
-
-noncomputable def boundary (p : Equiv.Perm α) (S : Finset α) : Finset α :=
-  S.filter (fun x => p.symm x ∉ S)
-
-noncomputable def root {p : Equiv.Perm α} {S : Finset α}
-    (C : RoutingNetwork.SelectedCycle p S) : α := C.nonempty.choose
-
-lemma root_mem {p : Equiv.Perm α} {S : Finset α} (C : RoutingNetwork.SelectedCycle p S) :
-    root C ∈ C.val := C.nonempty.choose_spec
-
-abbrev Seed (p : Equiv.Perm α) (S : Finset α) :=
-  {x // x ∈ boundary p S} ⊕ RoutingNetwork.SelectedCycle p S
-
-noncomputable def record {p : Equiv.Perm α} {S : Finset α}
-    (f : Coloring p S) : Seed p S → Bool :=
-  Sum.elim (fun x => f.val x.val) (fun C => f.val (root C))
-
-lemma orbit_subset_of_predecessor_closed (p : Equiv.Perm α) (D : Finset α)
-    (hD : ∀ x ∈ D, p.symm x ∈ D) {x : α} (hx : x ∈ D) : orbitSet p x ⊆ D := by
-  intro y hy
-  have hh := ((mem_orbitSet p x y).mp hy).inv
-  obtain ⟨n,hn⟩ := hh.exists_nat_pow_eq
-  have hp (k : ℕ) : (p⁻¹ ^ k) x ∈ D := by
-    induction k with
-    | zero => simpa using hx
-    | succ k ih => simpa only [pow_succ',Equiv.Perm.mul_apply,Equiv.Perm.coe_inv] using hD _ ih
-  exact hn ▸ hp n
-
-lemma record_injective (p : Equiv.Perm α) (S : Finset α) :
-    Function.Injective (record (p := p) (S := S)) := by
-  classical
-  intro f g hfg
-  apply Subtype.ext
-  funext x
-  by_contra hne
-  have hx : x ∈ S := by
-    by_contra h
-    exact hne ((f.property.1 x h).trans (g.property.1 x h).symm)
-  let D := S.filter (fun y => f.val y ≠ g.val y)
-  have hxD : x ∈ D := Finset.mem_filter.mpr ⟨hx,hne⟩
-  have hprev (y : α) (hy : y ∈ D) : p.symm y ∈ D := by
-    obtain ⟨hyS,hyne⟩ := Finset.mem_filter.mp hy
-    have hpS : p.symm y ∈ S := by
-      by_contra hn
-      have hseed : y ∈ boundary p S := Finset.mem_filter.mpr ⟨hyS,hn⟩
-      exact hyne (congrFun hfg (Sum.inl ⟨y,hseed⟩))
-    refine Finset.mem_filter.mpr ⟨hpS,?_⟩
-    have hf := f.property.2 (p.symm y) hpS (by simpa only [Equiv.apply_symm_apply] using hyS)
-    have hg := g.property.2 (p.symm y) hpS (by simpa only [Equiv.apply_symm_apply] using hyS)
-    simp only [Equiv.apply_symm_apply] at hf hg
-    intro he
-    exact hyne (hf.trans (he.trans hg.symm))
-  have horb := orbit_subset_of_predecessor_closed p D hprev hxD
-  let C : RoutingNetwork.SelectedCycle p S :=
-    ⟨orbitSet p x,⟨x,rfl⟩,fun y hy => (Finset.mem_filter.mp (horb hy)).1⟩
-  have hrD : root C ∈ D := horb (root_mem C)
-  have hrne := (Finset.mem_filter.mp hrD).2
-  exact hrne (congrFun hfg (Sum.inr C))
-
-lemma card_coloring_le (p : Equiv.Perm α) (S : Finset α) :
-    Fintype.card (Coloring p S) ≤
-      2 ^ ((boundary p S).card + Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
-  have h := Fintype.card_le_of_injective (record (p := p) (S := S)) (record_injective p S)
-  simpa only [Fintype.card_fun,Fintype.card_sum,Fintype.card_coe,Fintype.card_bool] using h
-
-omit [Fintype α] in
-lemma boundary_card (p : Equiv.Perm α) (S : Finset α) :
-    (boundary p S).card = S.card - (S.filter (fun x => p.symm x ∈ S)).card := by
-  classical
-  have h := Finset.card_filter_add_card_filter_not (s := S) (fun x => p.symm x ∈ S)
-  unfold boundary
-  omega
-
-lemma card_coloring_le_edges (p : Equiv.Perm α) (S : Finset α) :
-    Fintype.card (Coloring p S) ≤
-      2 ^ (S.card - (S.filter (fun x => p.symm x ∈ S)).card +
-        Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
-  simpa only [boundary_card] using card_coloring_le p S
-
-
-abbrev OppositeColoring (p : Equiv.Perm α) (S : Finset α) :=
-  {f : α → Bool // (∀ x, x ∉ S → f x = false) ∧
-    ∀ x ∈ S, p x ∈ S → f (p x) = !(f x)}
-
-noncomputable instance (p : Equiv.Perm α) (S : Finset α) : Fintype (OppositeColoring p S) :=
-  Fintype.ofFinite _
-
-noncomputable def bipartitionFlip (p : Equiv.Perm α) (S : Finset α)
-    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x))
-    (f : OppositeColoring p S) : Coloring p S := by
-  classical
-  refine ⟨(fun x => if x ∈ S then Bool.xor (f.val x) (χ x) else false),?_,?_⟩
-  · intro x hx
-    exact ite_eq_right hx
-  · intro x hx hpx
-    simp only [ite_eq_left hx,ite_eq_left hpx,f.property.2 x hx hpx,hχ x]
-    cases f.val x <;> cases χ x <;> rfl
-
-end CycleColoring
-end Thorp
-
-end ThorpNine.SparseSaving
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.SparseSaving.PriorPaths
+
+namespace OAI
+
+namespace ThorpNine.SparseSaving
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+section ParallelButterfly
+variable {α B : Type*} [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] {d : ℕ}
+omit [DecidableEq B] in
+lemma parallel_entropy (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool) (S : Finset α) :
+    (∑ i : B × SwitchIndex d,
+      if (parallelUsers pre ω i).1 ∈ S ∧ (parallelUsers pre ω i).2 ∈ S then (1:ℝ) else 0) ≤
+      (S.card : ℝ) * Real.log S.card / (2 * Real.log 2) := by
+  classical
+  rw [Fintype.sum_prod_type]
+  calc
+    _ ≤ ∑ b : B, (blockMarks pre S b).card * Real.log (blockMarks pre S b).card / (2*Real.log 2) := by
+      apply Finset.sum_le_sum
+      intro b _
+      simpa only [←mem_blockMarks,parallelUsers] using
+        marked_switch_entropy_bound d (decodeButterfly d (fun i => ω (b,i))) (blockMarks pre S b)
+    _ ≤ ∑ b : B, (blockMarks pre S b).card * Real.log S.card / (2*Real.log 2) := by
+      apply Finset.sum_le_sum
+      intro b _
+      by_cases hz : (blockMarks pre S b).card = 0
+      · simp [hz]
+      · apply div_le_div_of_nonneg_right _ (by positivity)
+        apply mul_le_mul_of_nonneg_left _ (by positivity)
+        exact Real.log_le_log (by exact_mod_cast Nat.pos_of_ne_zero hz)
+          (by exact_mod_cast card_blockMarks_le pre S b)
+    _ = _ := by rw [←Finset.sum_div,←Finset.sum_mul,←Nat.cast_sum,sum_card_blockMarks]
+
+noncomputable def parallelFollow (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x : α) (k : α → QueryTree (B × SwitchIndex d)) : QueryTree (B × SwitchIndex d) :=
+  QueryTree.followPath d (fun i => ((pre x).1,i)) (pre x).2 (fun y => k (post ((pre x).1,y)))
+
+omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
+lemma parallelFollow_succeeds (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
+    (parallelFollow pre post x k).succeeds ω = (k (relabeledPerm pre post ω x)).succeeds ω := by
+  exact QueryTree.followPath_succeeds _ _ _ _ _
+
+omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
+lemma parallelFollow_raw_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
+    (parallelFollow pre post x k).charge ω = (k (relabeledPerm pre post ω x)).charge ω := by
+  exact QueryTree.followPath_raw_charge _ _ _ _ _
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma parallelFollow_stale_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
+    (ω : B × SwitchIndex d → Bool) :
+    (parallelFollow pre post x k).staleCharge R ω =
+      (k (relabeledPerm pre post ω x)).staleCharge (R ∪ parallelPath pre ω x) ω := by
+  exact QueryTree.followPath_stale_charge _ _ _ _ _ _
+
+noncomputable def closingDomain (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) : Finset (B × SwitchIndex d) :=
+  (routeDomain d (pre x).2 (post.symm y).2).image (fun i => ((pre x).1,i))
+
+def closingValue (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (i : B × SwitchIndex d) : Bool := routeValue d (pre x).2 (post.symm y).2 i.2
+
+omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
+lemma relabeled_endpoint_pair (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (ω : B × SwitchIndex d → Bool) (x y : α) :
+    relabeledPerm pre post ω x = y ↔
+      (pre x).1 = (post.symm y).1 ∧
+      butterflyPerm d (decodeButterfly d (fun i => ω ((pre x).1,i))) (pre x).2 = (post.symm y).2 := by
+  change post ((pre x).1,_) = y ↔ _
+  rw [←post.eq_symm_apply,Prod.mk.injEq]
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma relabeled_endpoint_iff (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (ω : B × SwitchIndex d → Bool) (x y : α) :
+    relabeledPerm pre post ω x = y ↔
+      (pre x).1 = (post.symm y).1 ∧
+        ∀ i ∈ closingDomain pre post x y, ω i = closingValue pre post x y i := by
+  classical
+  rw [relabeled_endpoint_pair,butterfly_endpoint_iff]
+  simp only [closingDomain,Finset.forall_mem_image,closingValue]
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma closingDomain_eq (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (ω : B × SwitchIndex d → Bool) (x y : α) (h : relabeledPerm pre post ω x = y) :
+    closingDomain pre post x y = parallelPath pre ω x := by
+  rw [closingDomain,parallelPath,(relabeled_endpoint_pair _ _ _ _ _).mp h |>.2]
+
+noncomputable def parallelClose (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (k : QueryTree (B × SwitchIndex d)) : QueryTree (B × SwitchIndex d) :=
+  if (pre x).1 = (post.symm y).1 then
+    QueryTree.forceList (closingDomain pre post x y).toList (closingValue pre post x y) k
+  else QueryTree.reject
+
+omit [Fintype α] in
+lemma parallelClose_succeeds (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (k : QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
+    (parallelClose pre post x y k).succeeds ω =
+      if relabeledPerm pre post ω x = y then k.succeeds ω else false := by
+  classical
+  simp only [relabeled_endpoint_iff,parallelClose]
+  by_cases hb : (pre x).1 = (post.symm y).1
+  · rw [ite_eq_left hb,QueryTree.forceList_succeeds]
+    simp only [Finset.mem_toList,hb,true_and]
+  · simp [hb,QueryTree.succeeds]
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma parallelClose_raw_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (k : QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool)
+    (h : relabeledPerm pre post ω x = y) :
+    (parallelClose pre post x y k).charge ω = k.charge ω + d := by
+  have hh := (relabeled_endpoint_iff _ _ _ _ _).mp h
+  rw [parallelClose,ite_eq_left hh.1,QueryTree.forceList_raw_charge _ _ _ _
+    (by simpa only [Finset.mem_toList] using hh.2),Finset.length_toList,
+    closingDomain_eq _ _ _ _ _ h,card_parallelPath]
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma parallelClose_fresh_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (k : QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
+    (ω : B × SwitchIndex d → Bool) (h : relabeledPerm pre post ω x = y) :
+    (parallelClose pre post x y k).freshCharge R ω =
+      k.freshCharge (R ∪ parallelPath pre ω x) ω + (parallelPath pre ω x \ R).card := by
+  have hh := (relabeled_endpoint_iff _ _ _ _ _).mp h
+  rw [parallelClose,ite_eq_left hh.1,QueryTree.forceList_charge _ _ _ _ _
+    (by simpa only [Finset.mem_toList] using hh.2),Finset.toList_toFinset,
+    closingDomain_eq _ _ _ _ _ h]
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma parallelClose_stale_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (k : QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
+    (ω : B × SwitchIndex d → Bool) (h : relabeledPerm pre post ω x = y) :
+    (parallelClose pre post x y k).staleCharge R ω =
+      k.staleCharge (R ∪ parallelPath pre ω x) ω + (parallelPath pre ω x ∩ R).card := by
+  have h₀ := QueryTree.fresh_add_stale (parallelClose pre post x y k) R ω
+  have h₁ := QueryTree.fresh_add_stale k (R ∪ parallelPath pre ω x) ω
+  rw [parallelClose_raw_charge _ _ _ _ _ _ h,parallelClose_fresh_charge _ _ _ _ _ _ _ h] at h₀
+  have h₂ : (parallelPath pre ω x \ R).card + (parallelPath pre ω x ∩ R).card = d := by
+    rw [Finset.card_sdiff_add_card_inter,card_parallelPath]
+  omega
+
+noncomputable def parallelNetwork (pre : α ≃ B × Card d) (post : B × Card d ≃ α) :
+    RoutingNetwork α (B × SwitchIndex d) d where
+  perm := relabeledPerm pre post
+  path := parallelPath pre
+  users := parallelUsers pre
+  users_ne := parallelUsers_ne pre
+  mem_path := mem_parallelPath pre
+  card_path := card_parallelPath pre
+  entropy := parallel_entropy pre
+  follow := parallelFollow pre post
+  close := parallelClose pre post
+  follow_succeeds := parallelFollow_succeeds pre post
+  follow_raw_charge := parallelFollow_raw_charge pre post
+  follow_stale_charge := parallelFollow_stale_charge pre post
+  close_succeeds := parallelClose_succeeds pre post
+  close_raw_charge := parallelClose_raw_charge pre post
+  close_stale_charge := parallelClose_stale_charge pre post
+
+lemma parallel_cycle_log_mgf (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (S : Finset α) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    Real.log (finiteMean (fun ω : B × SwitchIndex d → Bool =>
+      q^(Fintype.card (RoutingNetwork.SelectedCycle (relabeledPerm pre post ω) S)))) ≤
+      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
+  exact (parallelNetwork pre post).butterfly_cycle_log_mgf S J hJ q hq
+
+end ParallelButterfly
+
+def cardSplit : (r s : ℕ) → Card (s+r) ≃ Card r × Card s
+  | 0, s =>
+    { toFun := fun x => ((fun i => Fin.elim0 i), x)
+      invFun := fun x => x.2
+      left_inv := fun _ => rfl
+      right_inv := fun x => by
+        apply Prod.ext
+        · exact Subsingleton.elim _ _
+        · rfl }
+  | r+1, s =>
+    { toFun := fun x => (Fin.cons (x 0) ((cardSplit r s (Fin.tail x)).1),
+        (cardSplit r s (Fin.tail x)).2)
+      invFun := fun x => Fin.cons (x.1 0) ((cardSplit r s).symm (Fin.tail x.1,x.2))
+      left_inv := by
+        intro x
+        simp only [Fin.cons_zero, Fin.tail_cons, Prod.mk.eta, Equiv.symm_apply_apply, Fin.cons_self_tail]
+      right_inv := by
+        intro x
+        simp only [Fin.cons_zero, Fin.tail_cons, Equiv.apply_symm_apply, Fin.cons_self_tail] }
+
+def assembleBits : (r s : ℕ) → (Card r × SwitchIndex s → Bool) →
+    (Card s × SwitchIndex r → Bool) → SwitchIndex (s+r) → Bool
+  | 0, _, lo, _, i => lo ((fun j => Fin.elim0 j), i)
+  | r+1, s, _, hi, Sum.inl y => hi ((cardSplit r s y).2,Sum.inl (cardSplit r s y).1)
+  | r+1, s, lo, hi, Sum.inr (b,i) => assembleBits r s
+      (fun j => lo (Fin.cons b j.1,j.2)) (fun j => hi (j.1,Sum.inr (b,j.2))) i
+
+lemma butterfly_split (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
+    (hi : Card s × SwitchIndex r → Bool) (x : Card (s+r)) :
+    cardSplit r s (butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi)) x) =
+      let y := cardSplit r s x
+      let z := butterflyPerm s (decodeButterfly s (fun i => lo (y.1,i))) y.2
+      (butterflyPerm r (decodeButterfly r (fun i => hi (z,i))) y.1,z) := by
+  induction r with
+  | zero =>
+    apply Prod.ext
+    · funext i
+      exact Fin.elim0 i
+    · rfl
+  | succ r ih =>
+    let lo' : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons (x 0) j.1,j.2)
+    let hi' : Card s × SwitchIndex r → Bool := fun j => hi (j.1,Sum.inr (x 0,j.2))
+    have hh := ih lo' hi' (Fin.tail x)
+    let u := butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo' hi')) (Fin.tail x)
+    let y := cardSplit r s (Fin.tail x)
+    let z := butterflyPerm s (decodeButterfly s (fun i => lo' (y.1,i))) y.2
+    let w := butterflyPerm r (decodeButterfly r (fun i => hi' (z,i))) y.1
+    change cardSplit r s u = (w,z) at hh
+    change ((Fin.cons (Bool.xor (x 0) (hi ((cardSplit r s u).2,Sum.inl (cardSplit r s u).1)))
+        ((cardSplit r s u).1) : Card (r+1)), (cardSplit r s u).2) =
+      ((Fin.cons (Bool.xor (x 0) (hi (z,Sum.inl w))) w : Card (r+1)),z)
+    rw [hh]
+
+def splitSwitch : (r s : ℕ) → SwitchIndex (s+r) ≃
+    (Card r × SwitchIndex s) ⊕ (Card s × SwitchIndex r)
+  | 0, _ =>
+    { toFun := fun i => Sum.inl ((fun j => Fin.elim0 j),i)
+      invFun := fun i => match i with
+        | Sum.inl (_,j) => j
+        | Sum.inr (_,j) => nomatch j
+      left_inv := fun _ => rfl
+      right_inv := by
+        intro i
+        rcases i with ⟨x,j⟩ | ⟨x,j⟩
+        · exact congrArg Sum.inl (Prod.ext (Subsingleton.elim _ _) rfl)
+        · exact Empty.elim j }
+  | r+1, s =>
+    { toFun := fun i => match i with
+        | Sum.inl y => Sum.inr ((cardSplit r s y).2,Sum.inl (cardSplit r s y).1)
+        | Sum.inr (b,j) => match splitSwitch r s j with
+          | Sum.inl (p,j) => Sum.inl (Fin.cons b p,j)
+          | Sum.inr (q,j) => Sum.inr (q,Sum.inr (b,j))
+      invFun := fun i => match i with
+        | Sum.inl (p,j) => Sum.inr (p 0,(splitSwitch r s).symm (Sum.inl (Fin.tail p,j)))
+        | Sum.inr (q,Sum.inl p) => Sum.inl ((cardSplit r s).symm (p,q))
+        | Sum.inr (q,Sum.inr (b,j)) => Sum.inr (b,(splitSwitch r s).symm (Sum.inr (q,j)))
+      left_inv := by
+        intro i
+        rcases i with y | ⟨b,j⟩
+        · dsimp only
+          rw [Prod.mk.eta, Equiv.symm_apply_apply]
+        · dsimp only
+          cases hz : splitSwitch r s j with
+          | inl z =>
+            rcases z with ⟨p,k⟩
+            simp only [Fin.cons_zero, Fin.tail_cons]
+            rw [←hz,Equiv.symm_apply_apply]
+          | inr z =>
+            rcases z with ⟨q,k⟩
+            dsimp only
+            rw [←hz,Equiv.symm_apply_apply]
+      right_inv := by
+        intro i
+        rcases i with ⟨p,j⟩ | ⟨q,j⟩
+        · simp only [Equiv.apply_symm_apply, Fin.cons_self_tail]
+        · rcases j with p | ⟨b,j⟩ <;> simp only [Equiv.apply_symm_apply] }
+
+lemma assembleBits_eq (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
+    (hi : Card s × SwitchIndex r → Bool) (i : SwitchIndex (s+r)) :
+    assembleBits r s lo hi i = Sum.elim lo hi (splitSwitch r s i) := by
+  induction r with
+  | zero => rfl
+  | succ r ih =>
+    rcases i with y | ⟨b,j⟩
+    · rfl
+    · simp only [assembleBits,splitSwitch,Equiv.coe_fn_mk]
+      rw [ih]
+      cases splitSwitch r s j <;> rfl
+
+def splitCoinEquiv (r s : ℕ) :
+    ((Card r × SwitchIndex s → Bool) × (Card s × SwitchIndex r → Bool)) ≃
+      (SwitchIndex (s+r) → Bool) :=
+  (Equiv.sumArrowEquivProdArrow _ _ Bool).symm.trans
+    (Equiv.arrowCongr (splitSwitch r s).symm (Equiv.refl Bool))
+
+lemma splitCoinEquiv_apply (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
+    (hi : Card s × SwitchIndex r → Bool) :
+    splitCoinEquiv r s (lo,hi) = assembleBits r s lo hi := by
+  funext i
+  exact (assembleBits_eq r s lo hi i).symm
+
+lemma finiteMean_prod {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] (f : Ω × Ω' → ℝ) :
+    finiteMean f = finiteMean (fun x => finiteMean (fun y => f (x,y))) := by
+  simp only [finiteMean, Fintype.sum_prod_type, Fintype.card_prod, Nat.cast_mul,
+    ←Finset.sum_div, div_div, mul_comm]
+
+lemma finiteMean_split (r s : ℕ) (f : (SwitchIndex (s+r) → Bool) → ℝ) :
+    finiteMean f = finiteMean (fun lo : Card r × SwitchIndex s → Bool =>
+      finiteMean (fun hi : Card s × SwitchIndex r → Bool => f (assembleBits r s lo hi))) := by
+  rw [finiteMean_equiv (splitCoinEquiv r s).symm]
+  simp only [Equiv.symm_symm]
+  rw [finiteMean_prod]
+  apply finiteMean_congr
+  intro lo
+  apply finiteMean_congr
+  intro hi
+  rw [splitCoinEquiv_apply]
+
+section PartiallyExposed
+variable {α : Type*} {r s : ℕ}
+
+def freshPre (pre : α ≃ Card (s+r)) (lo : Card r × SwitchIndex s → Bool) :
+    α ≃ Card s × Card r :=
+  ((pre.trans (cardSplit r s)).trans (parallelPerm lo)).trans (Equiv.prodComm _ _)
+
+def freshPost (post : Card (s+r) ≃ α) : Card s × Card r ≃ α :=
+  ((Equiv.prodComm _ _).trans (cardSplit r s).symm).trans post
+
+lemma exposed_perm_eq (pre : α ≃ Card (s+r)) (post : Card (s+r) ≃ α)
+    (lo : Card r × SwitchIndex s → Bool) (hi : Card s × SwitchIndex r → Bool) :
+    (pre.trans (butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi)))).trans post =
+      relabeledPerm (freshPre pre lo) (freshPost post) hi := by
+  ext x
+  simpa only [relabeledPerm, freshPre, freshPost, parallelPerm, Equiv.trans_apply,
+    Equiv.coe_fn_mk, Equiv.prodComm_apply, Prod.swap, Equiv.symm_apply_apply] using
+    congrArg (fun y => post ((cardSplit r s).symm y)) (butterfly_split r s lo hi (pre x))
+
+lemma partially_exposed_cycle_log_mgf [Fintype α] [DecidableEq α]
+    (pre : α ≃ Card (s+r)) (post : Card (s+r) ≃ α)
+    (lo : Card r × SwitchIndex s → Bool) (S : Finset α)
+    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    Real.log (finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(Fintype.card (RoutingNetwork.SelectedCycle
+        ((pre.trans (butterflyPerm (s+r)
+          (decodeButterfly (s+r) (assembleBits r s lo hi)))).trans post) S)))) ≤
+      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^r) := by
+  simp only [exposed_perm_eq]
+  exact parallel_cycle_log_mgf (freshPre pre lo) (freshPost post) S J hJ q hq
+
+end PartiallyExposed
+
+lemma inclusionMoment_eq {α Ω : Type*} [DecidableEq α] [Fintype Ω]
+    (F : Ω → Finset α) (S : Finset α) :
+    inclusionMoment F S = finiteMean (fun ω => if S ⊆ F ω then (1:ℝ) else 0) := by
+  unfold inclusionMoment
+  apply finiteMean_congr
+  intro ω
+  split_ifs <;> rfl
+
+lemma finiteMean_bool (f : Bool → ℝ) : finiteMean f = (f false+f true)/2 := by
+  simp only [finiteMean, Fintype.sum_bool, Fintype.card_bool, Nat.cast_ofNat]
+  ring
+
+lemma finiteMean_const {Ω : Type*} [Fintype Ω] [Nonempty Ω] (c : ℝ) :
+    finiteMean (fun _ : Ω => c) = c := by
+  simp only [finiteMean,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
+  exact mul_div_cancel_left₀ c (Nat.cast_ne_zero.mpr Fintype.card_ne_zero)
+
+lemma finiteMean_pi_prod {ι : Type*} [Fintype ι] [DecidableEq ι] {Ω : ι → Type*}
+    [∀ i, Fintype (Ω i)] (f : (i : ι) → Ω i → ℝ) :
+    finiteMean (fun ω : (i : ι) → Ω i => ∏ i, f i (ω i)) = ∏ i, finiteMean (f i) := by
+  simp only [finiteMean,←Fintype.prod_sum,Fintype.card_pi,Nat.cast_prod,Finset.prod_div_distrib]
+
+section ParallelFairLayer
+variable {β : Type*} [Fintype β] [DecidableEq β]
+
+def pairLayer (ξ : β → Bool) : Equiv.Perm (Bool × β) where
+  toFun x := (Bool.xor x.1 (ξ x.2),x.2)
+  invFun x := (Bool.xor x.1 (ξ x.2),x.2)
+  left_inv x := by
+    rcases x with ⟨b,x⟩
+    change (Bool.xor (Bool.xor b (ξ x)) (ξ x),x) = (b,x)
+    cases b <;> cases ξ x <;> rfl
+  right_inv x := by
+    rcases x with ⟨b,x⟩
+    change (Bool.xor (Bool.xor b (ξ x)) (ξ x),x) = (b,x)
+    cases b <;> cases ξ x <;> rfl
+
+noncomputable def pairFiber (S : Finset (Bool × β)) (b : β) : Finset Bool :=
+  Finset.univ.filter (fun ε => (ε,b) ∈ S)
+
+lemma prod_pairFiber (S : Finset (Bool × β)) (f : Bool × β → ℝ) :
+    (∏ x ∈ S, f x) = ∏ b : β, ∏ ε ∈ pairFiber S b, f (ε,b) := by
+  classical
+  simp only [pairFiber,Finset.prod_filter]
+  rw [Finset.prod_comm,←Fintype.prod_prod_type (fun x : Bool × β => if x ∈ S then f x else 1)]
+  rw [←Finset.prod_filter]
+  congr 1
+  ext x
+  simp
+
+lemma fiber_average_bound (S : Finset Bool) (p : Bool → ℝ) (hp : ∀ b, 0 ≤ p b) :
+    finiteMean (fun c : Bool => ∏ ε ∈ S, p (Bool.xor ε c)) ≤
+      ∏ ε ∈ S, (p ε + p (!ε))/2 := by
+  rw [finiteMean_bool]
+  have hxor : ∀ ε : Bool, Bool.xor ε true = Equiv.swap false true ε := by decide
+  simp only [Bool.xor_false,hxor]
+  calc
+    _ ≤ ∏ ε ∈ S, (p ε+p (Equiv.swap false true ε))/2 :=
+      swap_average_product S p hp false true
+    _ = _ := Finset.prod_congr rfl (fun ε _ => by cases ε <;> rfl)
+
+lemma CylinderBound.fair_parallel {m : Finset (Bool × β) → ℝ} {p : Bool × β → ℝ}
+    (h : CylinderBound m p) (hp : ∀ x, 0 ≤ p x) :
+    CylinderBound (fun S => finiteMean (fun ξ : β → Bool => m (S.image (pairLayer ξ))))
+      (fun x => (p x+p (!x.1,x.2))/2) := by
+  intro S
+  calc
+    _ ≤ finiteMean (fun ξ : β → Bool => ∏ x ∈ S, p (pairLayer ξ x)) := by
+      apply finiteMean_mono
+      intro ξ
+      simpa only [Finset.prod_image (pairLayer ξ).injective.injOn] using h (S.image (pairLayer ξ))
+    _ = ∏ b : β, finiteMean (fun c : Bool => ∏ ε ∈ pairFiber S b, p (Bool.xor ε c,b)) := by
+      calc
+        _ = finiteMean (fun ξ : β → Bool => ∏ b : β,
+            ∏ ε ∈ pairFiber S b, p (Bool.xor ε (ξ b),b)) :=
+          finiteMean_congr (fun ξ => prod_pairFiber S (fun x => p (pairLayer ξ x)))
+        _ = _ := finiteMean_pi_prod (fun b c => ∏ ε ∈ pairFiber S b, p (Bool.xor ε c,b))
+    _ ≤ ∏ b : β, ∏ ε ∈ pairFiber S b, (p (ε,b)+p (!ε,b))/2 := by
+      apply Finset.prod_le_prod₀
+      · intro b _
+        exact div_nonneg (Finset.sum_nonneg (fun c _ =>
+          Finset.prod_nonneg (fun ε _ => hp (Bool.xor ε c,b)))) (Nat.cast_nonneg _)
+      · intro b _
+        exact fiber_average_bound (pairFiber S b) (fun ε => p (ε,b)) (fun ε => hp (ε,b))
+    _ = _ := (prod_pairFiber S (fun x => (p x+p (!x.1,x.2))/2)).symm
+
+omit [Fintype β] in
+lemma subset_pairLayer_image_iff (S A : Finset (Bool × β)) (ξ : β → Bool) :
+    S ⊆ A.image (pairLayer ξ) ↔ S.image (pairLayer ξ) ⊆ A := by
+  have hi (x : Bool × β) : pairLayer ξ (pairLayer ξ x) = x := (pairLayer ξ).left_inv x
+  constructor
+  · intro h
+    have h' := Finset.image_subset_image h (f := pairLayer ξ)
+    simpa only [Finset.image_image,Function.comp_def,hi,Finset.image_id'] using h'
+  · intro h
+    have h' := Finset.image_subset_image h (f := pairLayer ξ)
+    simpa only [Finset.image_image,Function.comp_def,hi,Finset.image_id'] using h'
+
+noncomputable def fairSetLayer {Ω : Type*} (F : Ω → Finset (Bool × β)) :
+    Ω × (β → Bool) → Finset (Bool × β) := fun ω => (F ω.1).image (pairLayer ω.2)
+
+lemma finiteMean_comm {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] (f : Ω → Ω' → ℝ) :
+    finiteMean (fun x => finiteMean (f x)) = finiteMean (fun y => finiteMean (fun x => f x y)) := by
+  simp only [finiteMean,←Finset.sum_div,div_div]
+  rw [Finset.sum_comm,mul_comm]
+
+lemma inclusionMoment_fairSetLayer {Ω : Type*} [Fintype Ω]
+    (F : Ω → Finset (Bool × β)) (S : Finset (Bool × β)) :
+    inclusionMoment (fairSetLayer F) S =
+      finiteMean (fun ξ : β → Bool => inclusionMoment F (S.image (pairLayer ξ))) := by
+  classical
+  simp only [inclusionMoment_eq]
+  have he : (fun ω : Ω × (β → Bool) => if S ⊆ fairSetLayer F ω then (1:ℝ) else 0) =
+      (fun ω : Ω × (β → Bool) => if S.image (pairLayer ω.2) ⊆ F ω.1 then (1:ℝ) else 0) := by
+    funext ω
+    simp only [fairSetLayer,subset_pairLayer_image_iff]
+  apply (congrArg finiteMean he).trans
+  exact (finiteMean_prod (fun ω : Ω × (β → Bool) =>
+    if S.image (pairLayer ω.2) ⊆ F ω.1 then (1:ℝ) else 0)).trans
+      (finiteMean_comm (fun ω ξ => if S.image (pairLayer ξ) ⊆ F ω then (1:ℝ) else 0))
+
+lemma inclusionMoment_fairSetLayer_singleton {Ω : Type*} [Fintype Ω]
+    (F : Ω → Finset (Bool × β)) (x : Bool × β) :
+    inclusionMoment (fairSetLayer F) {x} =
+      (inclusionMoment F {x}+inclusionMoment F {(!x.1,x.2)})/2 := by
+  rw [inclusionMoment_fairSetLayer]
+  simp only [Finset.image_singleton,pairLayer,Equiv.coe_fn_mk]
+  rw [finiteMean_pi_eval (A := fun _ : β => Bool) x.2 (fun c : Bool => inclusionMoment F {(Bool.xor x.1 c,x.2)})]
+  rw [finiteMean_bool]
+  rcases x with ⟨b,x⟩
+  cases b <;> simp only [Bool.xor_false,Bool.false_xor,Bool.true_xor, Bool.not_false,Bool.not_true]
+
+lemma cylinder_fairSetLayer {Ω : Type*} [Fintype Ω]
+    (F : Ω → Finset (Bool × β))
+    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
+    CylinderBound (inclusionMoment (fairSetLayer F))
+      (fun x => inclusionMoment (fairSetLayer F) {x}) := by
+  have hh := h.fair_parallel (fun x => inclusionMoment_nonneg F {x})
+  intro S
+  rw [inclusionMoment_fairSetLayer]
+  simp only [inclusionMoment_fairSetLayer_singleton]
+  exact hh S
+
+end ParallelFairLayer
+
+section RelabelMarks
+variable {α β Ω Ω' : Type*} [DecidableEq α] [DecidableEq β] [Fintype Ω] [Fintype Ω']
+
+omit [Fintype Ω] [Fintype Ω'] in
+lemma subset_image_equiv_iff (e : α ≃ β) (A : Finset α) (S : Finset β) :
+    S ⊆ A.image e ↔ S.image e.symm ⊆ A := by
+  constructor
+  · intro h x hx
+    obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
+    obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp (h hy)
+    simpa only [←hxy,Equiv.symm_apply_apply] using hx
+  · intro h y hy
+    exact Finset.mem_image.mpr ⟨e.symm y,h (Finset.mem_image_of_mem _ hy),e.apply_symm_apply y⟩
+
+omit [Fintype Ω'] in
+lemma inclusionMoment_image_equiv (e : α ≃ β) (F : Ω → Finset α) (S : Finset β) :
+    inclusionMoment (fun ω => (F ω).image e) S = inclusionMoment F (S.image e.symm) := by
+  classical
+  simp only [inclusionMoment_eq]
+  apply finiteMean_congr
+  intro ω
+  simp only [subset_image_equiv_iff]
+
+omit [Fintype Ω'] in
+lemma cylinder_image_equiv (e : α ≃ β) (F : Ω → Finset α)
+    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
+    CylinderBound (inclusionMoment (fun ω => (F ω).image e))
+      (fun x => inclusionMoment (fun ω => (F ω).image e) {x}) := by
+  intro S
+  simp only [inclusionMoment_image_equiv,Finset.image_singleton]
+  have hh := h (S.image e.symm)
+  simpa only [Finset.prod_image e.symm.injective.injOn] using hh
+
+omit [DecidableEq β] in
+lemma inclusionMoment_comp_equiv (e : Ω ≃ Ω') (F : Ω' → Finset α) (S : Finset α) :
+    inclusionMoment (fun ω => F (e ω)) S = inclusionMoment F S := by
+  classical
+  simp only [inclusionMoment_eq]
+  simpa only [Equiv.apply_symm_apply] using
+    finiteMean_equiv e (fun ω => if S ⊆ F (e ω) then (1 : ℝ) else 0)
+
+omit [DecidableEq β] in
+lemma cylinder_comp_equiv (e : Ω ≃ Ω') (F : Ω' → Finset α)
+    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
+    CylinderBound (inclusionMoment (fun ω => F (e ω)))
+      (fun x => inclusionMoment (fun ω => F (e ω)) {x}) := by
+  intro S
+  simpa only [inclusionMoment_comp_equiv] using h S
+end RelabelMarks
+
+lemma finiteMean_prod_mul {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
+    (f : Ω → ℝ) (g : Ω' → ℝ) :
+    finiteMean (fun ω : Ω × Ω' => f ω.1*g ω.2) = finiteMean f * finiteMean g := by
+  simp only [finiteMean,Fintype.sum_prod_type,Fintype.card_prod,Nat.cast_mul,
+    ←Finset.mul_sum,←Finset.sum_mul]
+  ring
+
+section JoinChildren
+variable {β : Type*} [DecidableEq β]
+
+noncomputable def childJoin (A B : Finset β) : Finset (Bool × β) :=
+  A.image (fun x => (false,x)) ∪ B.image (fun x => (true,x))
+
+lemma mem_childJoin (A B : Finset β) (b : Bool) (x : β) :
+    (b,x) ∈ childJoin A B ↔ (if b then x ∈ B else x ∈ A) := by
+  cases b <;> simp [childJoin]
+
+noncomputable def boolFiber (S : Finset (Bool × β)) (b : Bool) : Finset β :=
+  (S.filter (fun x => x.1 = b)).image Prod.snd
+
+lemma mem_boolFiber (S : Finset (Bool × β)) (b : Bool) (x : β) :
+    x ∈ boolFiber S b ↔ (b,x) ∈ S := by
+  constructor
+  · intro h
+    obtain ⟨⟨c,y⟩,hy,hxy⟩ := Finset.mem_image.mp h
+    obtain ⟨hy,hc⟩ := Finset.mem_filter.mp hy
+    change c = b at hc
+    change y = x at hxy
+    simpa only [hc,hxy] using hy
+  · intro h
+    exact Finset.mem_image.mpr ⟨(b,x),Finset.mem_filter.mpr ⟨h,rfl⟩,rfl⟩
+
+lemma childJoin_boolFiber (S : Finset (Bool × β)) :
+    childJoin (boolFiber S false) (boolFiber S true) = S := by
+  ext ⟨b,x⟩
+  rw [mem_childJoin]
+  cases b <;> simp only [Bool.false_eq_true,↓reduceIte,mem_boolFiber]
+
+lemma subset_childJoin_iff (S : Finset (Bool × β)) (A B : Finset β) :
+    S ⊆ childJoin A B ↔ boolFiber S false ⊆ A ∧ boolFiber S true ⊆ B := by
+  constructor
+  · intro h
+    constructor
+    · intro x hx
+      exact (mem_childJoin A B false x).mp (h ((mem_boolFiber S false x).mp hx))
+    · intro x hx
+      exact (mem_childJoin A B true x).mp (h ((mem_boolFiber S true x).mp hx))
+  · rintro ⟨h₀,h₁⟩ ⟨b,x⟩ hx
+    rw [mem_childJoin]
+    cases b
+    · exact h₀ ((mem_boolFiber S false x).mpr hx)
+    · exact h₁ ((mem_boolFiber S true x).mpr hx)
+
+lemma prod_childJoin (A B : Finset β) (f : Bool × β → ℝ) :
+    (∏ x ∈ childJoin A B, f x) = (∏ x ∈ A, f (false,x)) * ∏ x ∈ B, f (true,x) := by
+  classical
+  have hd : Disjoint (A.image (fun x => (false,x))) (B.image (fun x => (true,x))) := by
+    apply Finset.disjoint_left.mpr
+    rintro x hx hy
+    obtain ⟨a,ha,rfl⟩ := Finset.mem_image.mp hx
+    obtain ⟨b,hb,hba⟩ := Finset.mem_image.mp hy
+    exact Bool.false_ne_true (congrArg Prod.fst hba).symm
+  rw [childJoin,Finset.prod_union hd,
+    Finset.prod_image (fun _ _ _ _ h => congrArg Prod.snd h),
+    Finset.prod_image (fun _ _ _ _ h => congrArg Prod.snd h)]
+
+lemma prod_boolFibers (S : Finset (Bool × β)) (f : Bool × β → ℝ) :
+    (∏ x ∈ S, f x) = (∏ x ∈ boolFiber S false, f (false,x)) *
+      ∏ x ∈ boolFiber S true, f (true,x) := by
+  rw [←prod_childJoin,childJoin_boolFiber]
+
+lemma inclusionMoment_childJoin {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
+    (F : Ω → Finset β) (G : Ω' → Finset β) (S : Finset (Bool × β)) :
+    inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) S =
+      inclusionMoment F (boolFiber S false) * inclusionMoment G (boolFiber S true) := by
+  classical
+  simp only [inclusionMoment_eq]
+  rw [←finiteMean_prod_mul]
+  apply finiteMean_congr
+  intro ω
+  simp only [subset_childJoin_iff]
+  split_ifs <;> simp_all
+
+lemma cylinder_childJoin {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] [Nonempty Ω] [Nonempty Ω']
+    (F : Ω → Finset β) (G : Ω' → Finset β)
+    (hF : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x}))
+    (hG : CylinderBound (inclusionMoment G) (fun x => inclusionMoment G {x})) :
+    CylinderBound (inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)))
+      (fun x => inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {x}) := by
+  intro S
+  rw [inclusionMoment_childJoin,prod_boolFibers]
+  have hf (b : Bool) (x : β) : boolFiber {(b,x)} b = {x} := by
+    ext y
+    simp only [mem_boolFiber,Finset.mem_singleton,Prod.mk.injEq,true_and]
+  have hn (b : Bool) (x : β) : boolFiber {(!b,x)} b = ∅ := by
+    ext y
+    cases b <;> simp [mem_boolFiber]
+  have h₀ (x : β) : inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {(false,x)} =
+      inclusionMoment F {x} * inclusionMoment G ∅ := by
+    rw [inclusionMoment_childJoin,hf]
+    have hh := hn true x
+    exact congrArg (fun T => inclusionMoment F {x} * inclusionMoment G T) hh
+  have h₁ (x : β) : inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {(true,x)} =
+      inclusionMoment F ∅ * inclusionMoment G {x} := by
+    rw [inclusionMoment_childJoin,hf]
+    have hh := hn false x
+    exact congrArg (fun T => inclusionMoment F T * inclusionMoment G {x}) hh
+  have emF : inclusionMoment F ∅ = 1 := by simp [inclusionMoment,finiteMean_const]
+  have emG : inclusionMoment G ∅ = 1 := by simp [inclusionMoment,finiteMean_const]
+  simp only [h₀,h₁,emF,emG,mul_one,one_mul]
+  exact mul_le_mul (hF _) (hG _) (inclusionMoment_nonneg _ _)
+    (Finset.prod_nonneg (fun x _ => inclusionMoment_nonneg F {x}))
+end JoinChildren
+
+def coinStepEquiv (d : ℕ) : (SwitchIndex (d+1) → Bool) ≃
+    ((SwitchIndex d → Bool) × (SwitchIndex d → Bool)) × (Card d → Bool) where
+  toFun ω := ((fun i => ω (Sum.inr (false,i)),fun i => ω (Sum.inr (true,i))),
+    fun x => ω (Sum.inl x))
+  invFun ω := Sum.elim ω.2 (fun bi => if bi.1 then ω.1.2 bi.2 else ω.1.1 bi.2)
+  left_inv ω := by
+    funext i
+    cases i with
+    | inl x => rfl
+    | inr bi => cases bi with | mk b i => cases b <;> rfl
+  right_inv ω := by
+    rcases ω with ⟨⟨ω₀,ω₁⟩,ξ⟩
+    rfl
+
+def headTailEquiv (d : ℕ) : Card (d+1) ≃ Bool × Card d :=
+  (Fin.consEquiv (fun _ : Fin (d+1) => Bool)).symm
+
+lemma childMarks_join (d : ℕ) (A : Finset (Card (d+1))) :
+    A = (childMarks A false).image (Fin.cons false) ∪
+      (childMarks A true).image (Fin.cons true) := by
+  classical
+  ext x
+  constructor
+  · intro hx
+    cases hb : x 0
+    · exact Finset.mem_union_left _ (Finset.mem_image.mpr
+        ⟨Fin.tail x,(mem_childMarks _ _ _).mpr (by simpa only [←hb,Fin.cons_self_tail] using hx),
+          by simp only [←hb,Fin.cons_self_tail]⟩)
+    · exact Finset.mem_union_right _ (Finset.mem_image.mpr
+        ⟨Fin.tail x,(mem_childMarks _ _ _).mpr (by simpa only [←hb,Fin.cons_self_tail] using hx),
+          by simp only [←hb,Fin.cons_self_tail]⟩)
+  · intro hx
+    rcases Finset.mem_union.mp hx with hx | hx
+    · obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
+      exact (mem_childMarks _ _ _).mp hy
+    · obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
+      exact (mem_childMarks _ _ _).mp hy
+
+noncomputable def butterflyMarks (d : ℕ) (A : Finset (Card d))
+    (ω : SwitchIndex d → Bool) : Finset (Card d) :=
+  A.image (butterflyPerm d (decodeButterfly d ω))
+
+lemma butterflyMarks_succ (d : ℕ) (A : Finset (Card (d+1))) (ω : SwitchIndex (d+1) → Bool) :
+    butterflyMarks (d+1) A ω =
+      (fairSetLayer (fun p : (SwitchIndex d → Bool) × (SwitchIndex d → Bool) =>
+        childJoin (butterflyMarks d (childMarks A false) p.1)
+          (butterflyMarks d (childMarks A true) p.2)) (coinStepEquiv d ω)).image
+            (headTailEquiv d).symm := by
+  classical
+  unfold butterflyMarks fairSetLayer childJoin
+  rw [Finset.image_union,Finset.image_union]
+  simp only [Finset.image_image]
+  conv_lhs => rw [childMarks_join d A]
+  rw [Finset.image_union]
+  simp only [Finset.image_image]
+  apply congrArg₂ (· ∪ ·)
+  · apply Finset.image_congr
+    intro x _
+    simp only [Function.comp_def,butterflyPerm,decodeButterfly,Equiv.Perm.coe_mul,
+      childLift,pairSwitch,switchFun,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
+      coinStepEquiv,headTailEquiv,Equiv.symm_symm,pairLayer]
+    rfl
+  · apply Finset.image_congr
+    intro x _
+    simp only [Function.comp_def,butterflyPerm,decodeButterfly,Equiv.Perm.coe_mul,
+      childLift,pairSwitch,switchFun,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
+      coinStepEquiv,headTailEquiv,Equiv.symm_symm,pairLayer]
+    rfl
+
+lemma butterflyMarks_cylinder (d : ℕ) (A : Finset (Card d)) :
+    CylinderBound (inclusionMoment (butterflyMarks d A))
+      (fun x => inclusionMoment (butterflyMarks d A) {x}) := by
+  induction d with
+  | zero =>
+    have he : butterflyMarks 0 A = fun _ : SwitchIndex 0 → Bool => A := by
+      funext ω
+      simp [butterflyMarks,butterflyPerm]
+    rw [he]
+    exact cylinder_const A
+  | succ d ih =>
+    have h := cylinder_childJoin (butterflyMarks d (childMarks A false))
+      (butterflyMarks d (childMarks A true)) (ih _) (ih _)
+    have hh := cylinder_image_equiv (headTailEquiv d).symm _ (cylinder_fairSetLayer _ h)
+    have hs := cylinder_comp_equiv (coinStepEquiv d) _ hh
+    simpa only [←butterflyMarks_succ] using hs
+
+lemma indicator_mem_image_perm {α : Type*} [DecidableEq α]
+    (A : Finset α) (p : Equiv.Perm α) (y : α) :
+    (if y ∈ A.image p then (1 : ℝ) else 0) = ∑ x ∈ A, if p x = y then 1 else 0 := by
+  classical
+  by_cases h : y ∈ A.image p
+  · obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp h
+    rw [ite_eq_left (Finset.mem_image.mpr ⟨x,hx,hxy⟩)]
+    symm
+    calc
+      _ = (if p x = y then (1:ℝ) else 0) := Finset.sum_eq_single_of_mem x hx (by
+        intro z _ hzx
+        exact ite_eq_right (fun hz => hzx (p.injective (hz.trans hxy.symm))))
+      _ = 1 := ite_eq_left hxy
+  · rw [ite_eq_right h]
+    symm
+    apply Finset.sum_eq_zero
+    intro x hx
+    exact ite_eq_right (fun hxy => h (Finset.mem_image.mpr ⟨x,hx,hxy⟩))
+
+lemma butterflyMarks_singleton (d : ℕ) (A : Finset (Card d)) (y : Card d) :
+    inclusionMoment (butterflyMarks d A) {y} = (A.card : ℝ)/(2:ℝ)^d := by
+  classical
+  unfold inclusionMoment
+  simp only [Finset.singleton_subset_iff,butterflyMarks,indicator_mem_image_perm]
+  rw [finiteMean_sum]
+  simp only [butterfly_single_card_uniform,Finset.sum_const,nsmul_eq_mul]
+  ring
+
+lemma butterflyMarks_inclusion (d : ℕ) (A S : Finset (Card d)) :
+    inclusionMoment (butterflyMarks d A) S ≤ ((A.card : ℝ)/(2:ℝ)^d)^S.card := by
+  have h := butterflyMarks_cylinder d A S
+  simpa only [butterflyMarks_singleton,Finset.prod_const] using h
+
+namespace CycleColoring
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+abbrev Coloring (p : Equiv.Perm α) (S : Finset α) :=
+  {f : α → Bool // (∀ x, x ∉ S → f x = false) ∧
+    ∀ x ∈ S, p x ∈ S → f (p x) = f x}
+
+noncomputable instance (p : Equiv.Perm α) (S : Finset α) : Fintype (Coloring p S) :=
+  Fintype.ofFinite _
+
+noncomputable def boundary (p : Equiv.Perm α) (S : Finset α) : Finset α :=
+  S.filter (fun x => p.symm x ∉ S)
+
+noncomputable def root {p : Equiv.Perm α} {S : Finset α}
+    (C : RoutingNetwork.SelectedCycle p S) : α := C.nonempty.choose
+
+lemma root_mem {p : Equiv.Perm α} {S : Finset α} (C : RoutingNetwork.SelectedCycle p S) :
+    root C ∈ C.val := C.nonempty.choose_spec
+
+abbrev Seed (p : Equiv.Perm α) (S : Finset α) :=
+  {x // x ∈ boundary p S} ⊕ RoutingNetwork.SelectedCycle p S
+
+noncomputable def record {p : Equiv.Perm α} {S : Finset α}
+    (f : Coloring p S) : Seed p S → Bool :=
+  Sum.elim (fun x => f.val x.val) (fun C => f.val (root C))
+
+lemma orbit_subset_of_predecessor_closed (p : Equiv.Perm α) (D : Finset α)
+    (hD : ∀ x ∈ D, p.symm x ∈ D) {x : α} (hx : x ∈ D) : orbitSet p x ⊆ D := by
+  intro y hy
+  have hh := ((mem_orbitSet p x y).mp hy).inv
+  obtain ⟨n,hn⟩ := hh.exists_nat_pow_eq
+  have hp (k : ℕ) : (p⁻¹ ^ k) x ∈ D := by
+    induction k with
+    | zero => simpa using hx
+    | succ k ih => simpa only [pow_succ',Equiv.Perm.mul_apply,Equiv.Perm.coe_inv] using hD _ ih
+  exact hn ▸ hp n
+
+lemma record_injective (p : Equiv.Perm α) (S : Finset α) :
+    Function.Injective (record (p := p) (S := S)) := by
+  classical
+  intro f g hfg
+  apply Subtype.ext
+  funext x
+  by_contra hne
+  have hx : x ∈ S := by
+    by_contra h
+    exact hne ((f.property.1 x h).trans (g.property.1 x h).symm)
+  let D := S.filter (fun y => f.val y ≠ g.val y)
+  have hxD : x ∈ D := Finset.mem_filter.mpr ⟨hx,hne⟩
+  have hprev (y : α) (hy : y ∈ D) : p.symm y ∈ D := by
+    obtain ⟨hyS,hyne⟩ := Finset.mem_filter.mp hy
+    have hpS : p.symm y ∈ S := by
+      by_contra hn
+      have hseed : y ∈ boundary p S := Finset.mem_filter.mpr ⟨hyS,hn⟩
+      exact hyne (congrFun hfg (Sum.inl ⟨y,hseed⟩))
+    refine Finset.mem_filter.mpr ⟨hpS,?_⟩
+    have hf := f.property.2 (p.symm y) hpS (by simpa only [Equiv.apply_symm_apply] using hyS)
+    have hg := g.property.2 (p.symm y) hpS (by simpa only [Equiv.apply_symm_apply] using hyS)
+    simp only [Equiv.apply_symm_apply] at hf hg
+    intro he
+    exact hyne (hf.trans (he.trans hg.symm))
+  have horb := orbit_subset_of_predecessor_closed p D hprev hxD
+  let C : RoutingNetwork.SelectedCycle p S :=
+    ⟨orbitSet p x,⟨x,rfl⟩,fun y hy => (Finset.mem_filter.mp (horb hy)).1⟩
+  have hrD : root C ∈ D := horb (root_mem C)
+  have hrne := (Finset.mem_filter.mp hrD).2
+  exact hrne (congrFun hfg (Sum.inr C))
+
+lemma card_coloring_le (p : Equiv.Perm α) (S : Finset α) :
+    Fintype.card (Coloring p S) ≤
+      2 ^ ((boundary p S).card + Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
+  have h := Fintype.card_le_of_injective (record (p := p) (S := S)) (record_injective p S)
+  simpa only [Fintype.card_fun,Fintype.card_sum,Fintype.card_coe,Fintype.card_bool] using h
+
+omit [Fintype α] in
+lemma boundary_card (p : Equiv.Perm α) (S : Finset α) :
+    (boundary p S).card = S.card - (S.filter (fun x => p.symm x ∈ S)).card := by
+  classical
+  have h := Finset.card_filter_add_card_filter_not (s := S) (fun x => p.symm x ∈ S)
+  unfold boundary
+  omega
+
+lemma card_coloring_le_edges (p : Equiv.Perm α) (S : Finset α) :
+    Fintype.card (Coloring p S) ≤
+      2 ^ (S.card - (S.filter (fun x => p.symm x ∈ S)).card +
+        Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
+  simpa only [boundary_card] using card_coloring_le p S
+
+
+abbrev OppositeColoring (p : Equiv.Perm α) (S : Finset α) :=
+  {f : α → Bool // (∀ x, x ∉ S → f x = false) ∧
+    ∀ x ∈ S, p x ∈ S → f (p x) = !(f x)}
+
+noncomputable instance (p : Equiv.Perm α) (S : Finset α) : Fintype (OppositeColoring p S) :=
+  Fintype.ofFinite _
+
+noncomputable def bipartitionFlip (p : Equiv.Perm α) (S : Finset α)
+    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x))
+    (f : OppositeColoring p S) : Coloring p S := by
+  classical
+  refine ⟨(fun x => if x ∈ S then Bool.xor (f.val x) (χ x) else false),?_,?_⟩
+  · intro x hx
+    exact ite_eq_right hx
+  · intro x hx hpx
+    simp only [ite_eq_left hx,ite_eq_left hpx,f.property.2 x hx hpx,hχ x]
+    cases f.val x <;> cases χ x <;> rfl
+
+end CycleColoring
+end Thorp
+
+end ThorpNine.SparseSaving
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/SparseSaving/ParallelFreshPre.lean b/lean/OAI/Probability/ThorpRouting/SparseSaving/ParallelFreshPre.lean
index acfe3f3b40dd9fbf26ad0d7ef2499e16b7887936..f41e951969ce79707e4d290dea7ea94dcceafded 100644
--- a/lean/OAI/Probability/ThorpRouting/SparseSaving/ParallelFreshPre.lean
+++ b/lean/OAI/Probability/ThorpRouting/SparseSaving/ParallelFreshPre.lean
@@ -1,848 +1,848 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.SparseSaving.BenesCoins
-
-namespace OAI
-
-namespace ThorpNine.SparseSaving
-
-namespace Thorp
-
-lemma mem_centralBlock (r H : ℕ) (b : Card r) (x : Card (H+r)) :
-    x ∈ centralBlock r H b ↔ (cardSplit r H x).1 = b := by
-  classical
-  constructor
-  · intro h
-    obtain ⟨z,_,rfl⟩ := Finset.mem_image.mp h
-    simp only [Equiv.apply_symm_apply]
-  · intro h
-    apply Finset.mem_image.mpr
-    refine ⟨(cardSplit r H x).2,Finset.mem_univ _,?_⟩
-    rw [←h,Prod.mk.eta,Equiv.symm_apply_apply]
-
-lemma centralBlock_card (r H : ℕ) (b : Card r) : (centralBlock r H b).card = 2^H := by
-  classical
-  rw [centralBlock,Finset.card_image_of_injective]
-  · exact card_positions H
-  · intro x y h
-    exact congrArg Prod.snd ((cardSplit r H).symm.injective h)
-
-lemma centralBlock_disjoint (r H : ℕ) :
-    Pairwise (fun b b' => Disjoint (centralBlock r H b) (centralBlock r H b')) := by
-  classical
-  intro b b' h
-  apply Finset.disjoint_left.mpr
-  intro x hx hy
-  exact h (((mem_centralBlock r H b x).mp hx).symm.trans
-    ((mem_centralBlock r H b' x).mp hy))
-
-lemma centralBlock_zero (H : ℕ) (b : Card 0) : centralBlock 0 H b = Finset.univ := by
-  classical
-  ext x
-  simp only [mem_centralBlock,Finset.mem_univ,iff_true]
-  exact Subsingleton.elim _ _
-
-lemma centralBlock_step (r H : ℕ) (b : Bool) (c : Card r) :
-    centralBlock (r+1) H (Fin.cons b c) = (centralBlock r H c).image (Fin.cons b) := by
-  classical
-  simp only [centralBlock,Finset.image_image,cardSplit,Equiv.coe_fn_symm_mk,
-    Fin.cons_zero,Fin.tail_cons,Function.comp_def]
-
-lemma block_inter_step (r H : ℕ) (b : Bool) (c : Card r) (A : Finset (Card (H+(r+1)))) :
-    centralBlock (r+1) H (Fin.cons b c) ∩ A =
-      (centralBlock r H c ∩ childMarks A b).image (Fin.cons b) := by
-  classical
-  rw [centralBlock_step]
-  ext x
-  constructor
-  · intro hx
-    obtain ⟨hxB,hxA⟩ := Finset.mem_inter.mp hx
-    obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hxB
-    exact Finset.mem_image.mpr ⟨y,Finset.mem_inter.mpr ⟨hy,(mem_childMarks A b y).mpr hxA⟩,rfl⟩
-  · intro hx
-    obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
-    exact Finset.mem_inter.mpr ⟨Finset.mem_image.mpr ⟨y,(Finset.mem_inter.mp hy).1,rfl⟩,
-      (mem_childMarks A b y).mp (Finset.mem_inter.mp hy).2⟩
-
-lemma sparseMass_central_zero (H : ℕ) (A : Finset (Card H)) :
-    sparseMass (centralBlock 0 H) A = crowdedMass A.card := by
-  classical
-  simp only [sparseMass,centralBlock_zero,Finset.univ_inter,crowdedMass,
-    Finset.sum_const,Finset.card_univ,card_positions,pow_zero,one_nsmul]
-
-lemma sparseMass_central_step (r H : ℕ) (A : Finset (Card (H+(r+1)))) :
-    sparseMass (centralBlock (r+1) H) A =
-      sparseMass (centralBlock r H) (childMarks A false) +
-        sparseMass (centralBlock r H) (childMarks A true) := by
-  classical
-  unfold sparseMass
-  rw [←Equiv.sum_comp (headTailEquiv r).symm,Fintype.sum_prod_type]
-  have ht (b : Bool) (c : Card r) : (headTailEquiv r).symm (b,c) = Fin.cons b c := rfl
-  simp only [Fintype.sum_bool,ht]
-  have hc (b : Bool) (c : Card r) :
-      (centralBlock (r+1) H (Fin.cons b c) ∩ A).card =
-      (centralBlock r H c ∩ childMarks A b).card := by
-    rw [block_inter_step]
-    apply Finset.card_image_of_injective
-    intro x y h
-    simpa only [Fin.tail_cons] using congrArg Fin.tail h
-  simp only [hc]
-  exact add_comm _ _
-
-
-lemma inverseButterfly_coordinates_step (d : ℕ)
-    (c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (ξ : Card d → Bool)
-    (x : Card (d+1)) :
-    headTailEquiv d ((butterflyPerm (d+1)
-      (decodeButterfly (d+1) ((coinStepEquiv d).symm (c,ξ)))).symm x) =
-      (Bool.xor (x 0) (ξ (Fin.tail x)),
-        (butterflyPerm d (decodeButterfly d
-          (if Bool.xor (x 0) (ξ (Fin.tail x)) then c.2 else c.1))).symm (Fin.tail x)) := by
-  rw [butterflyPerm_step]
-  rfl
-
-lemma inverseButterflyMarks_child (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (d+1))
-    (c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (ξ : Card d → Bool) (b : Bool) :
-    childMarks (inverseButterflyMarks (d+1) (Finset.univ.image e)
-      ((coinStepEquiv d).symm (c,ξ))) b =
-      inverseButterflyMarks d
-        (Finset.univ.image (PairRouting.childEmbedding (e.trans (headTailEquiv d).toEmbedding)
-          (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) ξ)
-            (PairRouting.colors_compatible _ _) b)) (if b then c.2 else c.1) := by
-  classical
-  let x := e.trans (headTailEquiv d).toEmbedding
-  let col := PairRouting.colors x ξ
-  let child := PairRouting.childEmbedding x col (PairRouting.colors_compatible x ξ) b
-  have hp (i : ι) : headTailEquiv d ((butterflyPerm (d+1)
-      (decodeButterfly (d+1) ((coinStepEquiv d).symm (c,ξ)))).symm (e i)) =
-      (col i,(butterflyPerm d (decodeButterfly d (if col i then c.2 else c.1))).symm (x i).2) :=
-    inverseButterfly_coordinates_step d c ξ (e i)
-  ext y
-  simp only [mem_childMarks,inverseButterflyMarks,Finset.image_image,
-    Finset.mem_image,Finset.mem_univ,true_and,Function.comp_def]
-  constructor
-  · rintro ⟨i,hi⟩
-    have he := congrArg (headTailEquiv d) hi
-    rw [hp i] at he
-    change (col i,_) = (b,y) at he
-    have hb := congrArg Prod.fst he
-    have hy := congrArg Prod.snd he
-    dsimp only at hb hy
-    refine ⟨⟨i,hb⟩,?_⟩
-    change (butterflyPerm d (decodeButterfly d (if b then c.2 else c.1))).symm (x i).2 = y
-    simpa only [hb] using hy
-  · rintro ⟨i,hi⟩
-    refine ⟨i.val,?_⟩
-    apply (headTailEquiv d).injective
-    rw [hp]
-    change (col i.val,_) = (b,y)
-    have hb : col i.val = b := i.property
-    rw [hb]
-    exact Prod.ext rfl hi
-
-lemma inverseButterflyMarks_card (d : ℕ) (A : Finset (Card d)) (ω : SwitchIndex d → Bool) :
-    (inverseButterflyMarks d A ω).card = A.card := by
-  classical
-  exact Finset.card_image_of_injective _ (butterflyPerm d (decodeButterfly d ω)).symm.injective
-
-lemma benes_input_coordinates (d : ℕ) (ω : BenesCoins (d+1)) :
-    ω.2 = (coinStepEquiv d).symm
-      (((benesStepEquiv d ω).1.1.2,(benesStepEquiv d ω).1.2.2),(benesStepEquiv d ω).2.1) := by
-  change ω.2 = (coinStepEquiv d).symm (coinStepEquiv d ω.2)
-  exact ((coinStepEquiv d).symm_apply_apply ω.2).symm
-
-lemma cutoffMass_central (r H : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (H+r)) (ω : BenesCoins (H+r)) :
-    cutoffMass H (H+r) e ω =
-      sparseMass (centralBlock r H) (inverseButterflyMarks (H+r) (Finset.univ.image e) ω.2) := by
-  classical
-  induction r generalizing ι with
-  | zero =>
-    rw [sparseMass_central_zero,inverseButterflyMarks_card,
-      Finset.card_image_of_injective _ e.injective,Finset.card_univ]
-    cases H with
-    | zero =>
-      have hcard : Fintype.card ι ≤ 1 := by
-        have hh := Fintype.card_le_of_injective e e.injective
-        simpa only [Nat.add_zero,card_positions,pow_zero] using hh
-      simp [cutoffMass,crowdedMass,show ¬2≤Fintype.card ι by omega]
-    | succ H => simp only [cutoffMass,le_refl,ite_true]
-  | succ r ih =>
-    simp only [Nat.add_succ] at *
-    rw [cutoffMass,ite_eq_right (by omega : ¬H+r+1≤H)]
-    let σ := benesStepEquiv (H+r) ω
-    let x := e.trans (headTailEquiv (H+r)).toEmbedding
-    let col := PairRouting.colors x σ.2.1
-    let hc := PairRouting.colors_compatible x σ.2.1
-    change cutoffMass H (H+r) (PairRouting.childEmbedding x col hc false) σ.1.1 +
-      cutoffMass H (H+r) (PairRouting.childEmbedding x col hc true) σ.1.2 = _
-    rw [ih,ih,sparseMass_central_step,benes_input_coordinates (H+r) ω]
-    rw [inverseButterflyMarks_child,inverseButterflyMarks_child]
-    rfl
-
-
-theorem palindrome_low_mgf (r H : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (H+r)) (q : ℝ) (hq : 1 ≤ q)
-    (hr : (2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*q^H) ≤ 1/2) :
-    finiteMean (fun ω : BenesCoins (H+r) => q^(palindromeLowCost H (H+r) e ω)) ≤
-      Real.exp ((2:ℝ)^r * (2 * ((2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*q^H))^2)) := by
-  classical
-  have hec : (Finset.univ.image e).card = Fintype.card ι := by
-    rw [Finset.card_image_of_injective _ e.injective,Finset.card_univ]
-  have hm := inverseButterfly_sparse_mgf (H+r) (centralBlock r H)
-    (centralBlock_disjoint r H) (Finset.univ.image e) (q^H)
-    (pow_nonneg (by linarith) _) (fun b => by
-      simpa only [centralBlock_card,hec,Nat.cast_pow,Nat.cast_ofNat] using hr)
-  calc
-    _ ≤ finiteMean (fun ω : BenesCoins (H+r) => (q^H)^(cutoffMass H (H+r) e ω)) := by
-      apply finiteMean_mono
-      intro ω
-      rw [←pow_mul]
-      exact pow_le_pow_right₀ hq (palindromeLowCost_le_cutoff H (H+r) e ω)
-    _ = finiteMean (fun ω : SwitchIndex (H+r) → Bool =>
-        (q^H)^(sparseMass (centralBlock r H)
-          (inverseButterflyMarks (H+r) (Finset.univ.image e) ω))) := by
-      simp_rw [cutoffMass_central]
-      simp only [finiteMean_prod,finiteMean_const]
-    _ ≤ _ := by
-      simpa only [centralBlock_card,hec,Nat.cast_pow,Nat.cast_ofNat,Finset.sum_const,
-        Finset.card_univ,card_positions,nsmul_eq_mul] using hm
-
-variable {α B : Type*} {r s : ℕ}
-
-def parallelFreshPre (pre : α ≃ B × Card (s+r))
-    (lo : B → Card r × SwitchIndex s → Bool) : α ≃ (B × Card s) × Card r :=
-  ((pre.trans (Equiv.prodCongr (Equiv.refl B) (cardSplit r s))).trans
-    (Equiv.prodCongrRight (fun b =>
-      (parallelPerm (lo b)).trans (Equiv.prodComm _ _)))).trans
-    (Equiv.prodAssoc _ _ _).symm
-
-def parallelFreshPost (post : B × Card (s+r) ≃ α) : (B × Card s) × Card r ≃ α :=
-  (((Equiv.prodAssoc _ _ _).trans (Equiv.prodCongr (Equiv.refl B)
-    (Equiv.prodComm _ _))).trans (Equiv.prodCongr (Equiv.refl B) (cardSplit r s).symm)).trans post
-
-def parallelAssembled (lo : B → Card r × SwitchIndex s → Bool)
-    (hi : (B × Card s) × SwitchIndex r → Bool) : B × SwitchIndex (s+r) → Bool :=
-  fun i => assembleBits r s (lo i.1) (fun j => hi ((i.1,j.1),j.2)) i.2
-
-lemma parallel_exposed_perm (pre : α ≃ B × Card (s+r)) (post : B × Card (s+r) ≃ α)
-    (lo : B → Card r × SwitchIndex s → Bool)
-    (hi : (B × Card s) × SwitchIndex r → Bool) :
-    relabeledPerm pre post (parallelAssembled lo hi) =
-      relabeledPerm (parallelFreshPre pre lo) (parallelFreshPost post) hi := by
-  ext x
-  have h := butterfly_split r s (lo (pre x).1)
-    (fun j => hi (((pre x).1,j.1),j.2)) (pre x).2
-  have hh := congrArg (fun y => post ((pre x).1,(cardSplit r s).symm y)) h
-  simpa only [relabeledPerm,parallelAssembled,parallelPerm,parallelFreshPre,parallelFreshPost,
-    Equiv.trans_apply,Equiv.prodCongr_apply,Equiv.prodCongrRight,Equiv.coe_fn_mk,
-    Equiv.refl_apply,Equiv.prodComm_apply,Prod.swap,Equiv.prodAssoc_apply,
-    Equiv.prodAssoc_symm_apply,Prod.map,Equiv.symm_apply_apply] using hh
-
-theorem parallel_partially_exposed_log_mgf [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B]
-    (pre : α ≃ B × Card (s+r)) (post : B × Card (s+r) ≃ α)
-    (lo : B → Card r × SwitchIndex s → Bool) (S : Finset α)
-    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    Real.log (finiteMean (fun hi : (B × Card s) × SwitchIndex r → Bool =>
-      q^(Fintype.card (RoutingNetwork.SelectedCycle
-        (relabeledPerm pre post (parallelAssembled lo hi)) S)))) ≤
-      (S.card:ℝ)/J*Real.log q + 2*q*S.card*Real.sqrt J/Real.sqrt ((2:ℝ)^r) := by
-  simp_rw [parallel_exposed_perm]
-  exact parallel_cycle_log_mgf (parallelFreshPre pre lo) (parallelFreshPost post) S J hJ q hq
-
-end Thorp
-namespace Thorp
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-lemma orbitSet_inverse (p : Equiv.Perm α) (x : α) : orbitSet p⁻¹ x = orbitSet p x := by
-  ext y
-  simp only [mem_orbitSet,Equiv.Perm.sameCycle_inv]
-
-lemma selectedCycle_inverse (p : Equiv.Perm α) (S : Finset α) :
-    Fintype.card (RoutingNetwork.SelectedCycle p⁻¹ S) =
-      Fintype.card (RoutingNetwork.SelectedCycle p S) := by
-  apply Fintype.card_congr
-  refine { toFun := fun C => ⟨C.val,?_,C.property.2⟩
-           invFun := fun C => ⟨C.val,?_,C.property.2⟩
-           left_inv := fun _ => rfl
-           right_inv := fun _ => rfl }
-  · obtain ⟨x,hx⟩ := C.property.1
-    exact ⟨x,hx.trans (orbitSet_inverse p x)⟩
-  · obtain ⟨x,hx⟩ := C.property.1
-    exact ⟨x,hx.trans (orbitSet_inverse p x).symm⟩
-
-lemma orbitSet_conjugate (p g : Equiv.Perm α) (x : α) :
-    orbitSet (g*p*g⁻¹) (g x) = (orbitSet p x).image g := by
-  ext y
-  simp only [mem_orbitSet,Finset.mem_image]
-  rw [Equiv.Perm.sameCycle_conj]
-  change p.SameCycle (g.symm (g x)) (g.symm y) ↔ ∃ z, p.SameCycle x z ∧ g z = y
-  rw [Equiv.symm_apply_apply]
-  constructor
-  · intro h
-    exact ⟨g.symm y,h,g.apply_symm_apply y⟩
-  · rintro ⟨z,hz,rfl⟩
-    simpa only [Equiv.symm_apply_apply] using hz
-
-noncomputable def conjugateCycle (p g : Equiv.Perm α) (S : Finset α)
-    (C : RoutingNetwork.SelectedCycle p S) :
-    RoutingNetwork.SelectedCycle (g*p*g⁻¹) (S.image g) := by
-  refine ⟨C.val.image g,?_,Finset.image_subset_image C.property.2⟩
-  obtain ⟨x,hx⟩ := C.property.1
-  exact ⟨g x,by rw [hx,orbitSet_conjugate]⟩
-
-lemma conjugateCycle_injective (p g : Equiv.Perm α) (S : Finset α) :
-    Function.Injective (conjugateCycle p g S) := by
-  intro C D he
-  apply Subtype.ext
-  exact (Finset.image_injective g.injective) (congrArg Subtype.val he)
-
-lemma selectedCycle_conjugate (p g : Equiv.Perm α) (S : Finset α) :
-    Fintype.card (RoutingNetwork.SelectedCycle (g*p*g⁻¹) (S.image g)) =
-      Fintype.card (RoutingNetwork.SelectedCycle p S) := by
-  apply Nat.le_antisymm
-  · have h := Fintype.card_le_of_injective _
-      (conjugateCycle_injective (g*p*g⁻¹) g⁻¹ (S.image g))
-    have hp : g⁻¹*(g*p*g⁻¹)*g = p := by group
-    rw [inv_inv,hp] at h
-    have hs : (S.image g).image g.symm = S := by
-      ext x
-      simp only [Finset.mem_image]
-      constructor
-      · rintro ⟨y,⟨z,hz,rfl⟩,he⟩
-        have hz' : z = x := by simpa only [Equiv.symm_apply_apply] using he
-        exact hz' ▸ hz
-      · intro hx
-        exact ⟨g x,⟨x,hx,rfl⟩,g.symm_apply_apply x⟩
-    change _ ≤ Fintype.card (RoutingNetwork.SelectedCycle p ((S.image g).image g.symm)) at h
-    rw [hs] at h
-    exact h
-  · exact Fintype.card_le_of_injective _ (conjugateCycle_injective p g S)
-
-lemma alternatingCycles_le_projection {ι : Type*} [Fintype ι]
-    (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    PairRouting.alternatingCycles e p ≤
-      Fintype.card (RoutingNetwork.SelectedCycle (PairRouting.alternatingProjection p)
-        (boolFiber (PairRouting.labelSet e) false)) := by
-  have h := CycleColoring.card_alternating_cycles_le (PairRouting.alternatingProjection p)
-    (boolFiber (PairRouting.labelSet e) false) (boolFiber (PairRouting.labelSet e) true)
-  simpa only [PairRouting.alternatingCycles,childJoin_boolFiber] using h
-
-lemma projectionCycles_forward (X₀ X₁ Y₀ Y₁ : Equiv.Perm α) (S : Finset α) :
-    Fintype.card (RoutingNetwork.SelectedCycle
-      ((Y₀*X₀⁻¹)⁻¹*(Y₁*X₁⁻¹)) S) =
-    Fintype.card (RoutingNetwork.SelectedCycle
-      (X₀⁻¹*X₁*Y₁⁻¹*Y₀) (S.image X₀.symm)) := by
-  let p := (Y₀*X₀⁻¹)⁻¹*(Y₁*X₁⁻¹)
-  calc
-    _ = Fintype.card (RoutingNetwork.SelectedCycle (X₀⁻¹*p*X₀)
-        (S.image X₀.symm)) := by
-      change Fintype.card (RoutingNetwork.SelectedCycle p S) = _
-      have h := (selectedCycle_conjugate p X₀⁻¹ S).symm
-      rw [inv_inv] at h
-      exact h
-    _ = Fintype.card (RoutingNetwork.SelectedCycle (X₀⁻¹*p*X₀)⁻¹
-        (S.image X₀.symm)) := (selectedCycle_inverse _ _).symm
-    _ = _ := by
-      have he : (X₀⁻¹*p*X₀)⁻¹ = X₀⁻¹*X₁*Y₁⁻¹*Y₀ := by dsimp only [p]; group
-      rw [he]
-
-end Thorp
-namespace Thorp
-
-noncomputable def cycleCoefficient (r J : ℕ) (q : ℝ) : ℝ :=
-  Real.log q / J + 2*q*Real.sqrt J/Real.sqrt ((2:ℝ)^r)
-
-lemma cycleCoefficient_nonneg (r J : ℕ) (q : ℝ) (hq : 1 ≤ q) :
-    0 ≤ cycleCoefficient r J q := by
-  unfold cycleCoefficient
-  exact add_nonneg (div_nonneg (Real.log_nonneg hq) (Nat.cast_nonneg _))
-    (div_nonneg (mul_nonneg (by linarith) (Real.sqrt_nonneg _)) (Real.sqrt_nonneg _))
-
-lemma projectionSide_card_le {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι]
-    (e : ι ↪ Bool × α) : (boolFiber (PairRouting.labelSet e) false).card ≤ Fintype.card ι := by
-  have h := card_childJoin (boolFiber (PairRouting.labelSet e) false)
-    (boolFiber (PairRouting.labelSet e) true)
-  rw [childJoin_boolFiber,PairRouting.card_labelSet] at h
-  omega
-
-theorem node_partial_mgf (r s : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Bool × Card (s+r)) (X₀ X₁ Y₁ : Equiv.Perm (Card (s+r)))
-    (lo : Card r × SwitchIndex s → Bool) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(PairRouting.alternatingCycles e (fun b => if b then Y₁*X₁⁻¹ else
-        butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi))*X₀⁻¹))) ≤
-      Real.exp (Fintype.card ι * cycleCoefficient r J q) := by
-  classical
-  let S := boolFiber (PairRouting.labelSet e) false
-  let S' := S.image X₀.symm
-  let post := X₀⁻¹*X₁*Y₁⁻¹
-  let count := fun hi : Card s × SwitchIndex r → Bool =>
-    Fintype.card (RoutingNetwork.SelectedCycle
-      ((Equiv.refl _).trans (butterflyPerm (s+r)
-        (decodeButterfly (s+r) (assembleBits r s lo hi))) |>.trans post) S')
-  have hpoint (hi : Card s × SwitchIndex r → Bool) :
-      PairRouting.alternatingCycles e (fun b => if b then Y₁*X₁⁻¹ else
-        butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi))*X₀⁻¹) ≤ count hi := by
-    refine (alternatingCycles_le_projection e _).trans_eq ?_
-    simp only [PairRouting.alternatingProjection,Bool.false_eq_true,↓reduceIte]
-    change Fintype.card (RoutingNetwork.SelectedCycle
-      ((butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi))*X₀⁻¹)⁻¹ * (Y₁*X₁⁻¹)) S) = count hi
-    exact projectionCycles_forward X₀ X₁ _ Y₁ S
-  have hm := partially_exposed_cycle_log_mgf (Equiv.refl (Card (s+r))) post lo S' J hJ q hq
-  have hp : 0 < finiteMean (fun hi : Card s × SwitchIndex r → Bool => q^(count hi)) :=
-    finiteMean_pos (fun _ => pow_pos (by linarith) _)
-  have hb : finiteMean (fun hi : Card s × SwitchIndex r → Bool => q^(count hi)) ≤
-      Real.exp (S'.card * cycleCoefficient r J q) := by
-    rw [←Real.exp_log hp]
-    apply Real.exp_le_exp.mpr
-    calc
-      _ ≤ _ := hm
-      _ = _ := by unfold cycleCoefficient; ring
-  have hcard : S'.card ≤ Fintype.card ι := by
-    rw [show S'.card = S.card from Finset.card_image_of_injective _ X₀.symm.injective]
-    exact projectionSide_card_le e
-  calc
-    _ ≤ finiteMean (fun hi : Card s × SwitchIndex r → Bool => q^(count hi)) :=
-      finiteMean_mono (fun hi => pow_le_pow_right₀ hq (hpoint hi))
-    _ ≤ Real.exp (S'.card * cycleCoefficient r J q) := hb
-    _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right
-      (by exact_mod_cast hcard) (cycleCoefficient_nonneg r J q hq))
-
-
-def highStepEquiv (r s : ℕ) : (Card s × SwitchIndex (r+1) → Bool) ≃
-    (((Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool)) ×
-      (Card s × Card r → Bool)) where
-  toFun hi := ((fun j => hi (j.1,Sum.inr (false,j.2)),
-    fun j => hi (j.1,Sum.inr (true,j.2))),fun j => hi (j.1,Sum.inl j.2))
-  invFun c := fun j => match j.2 with
-    | Sum.inl y => c.2 (j.1,y)
-    | Sum.inr (b,i) => if b then c.1.2 (j.1,i) else c.1.1 (j.1,i)
-  left_inv hi := by
-    funext j
-    rcases j with ⟨z,y|⟨b,i⟩⟩
-    · rfl
-    · cases b <;> rfl
-  right_inv c := by
-    rcases c with ⟨⟨h₀,h₁⟩,η⟩
-    rfl
-
-lemma assembleBits_step (r s : ℕ) (lo : Card (r+1) × SwitchIndex s → Bool)
-    (c : ((Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool)) ×
-      (Card s × Card r → Bool)) :
-    assembleBits (r+1) s lo ((highStepEquiv r s).symm c) =
-      (coinStepEquiv (s+r)).symm
-        ((assembleBits r s (fun j => lo (Fin.cons false j.1,j.2)) c.1.1,
-          assembleBits r s (fun j => lo (Fin.cons true j.1,j.2)) c.1.2),
-        fun y => c.2 ((cardSplit r s y).2,(cardSplit r s y).1)) := by
-  funext i
-  rcases i with y | ⟨b,i⟩
-  · rfl
-  · cases b <;> rfl
-
-noncomputable def palindromeLevelCost (t : ℕ) : (d : ℕ) → {ι : Type*} → [Fintype ι] →
-    (ι ↪ Card d) → BenesCoins d → ℕ
-  | 0, _, _, _, _ => 0
-  | d+1, _, _, e, ω =>
-      let σ := benesStepEquiv d ω
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x σ.2.1
-      let hc := PairRouting.colors_compatible x σ.2.1
-      if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
-          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) else
-        palindromeLevelCost t d (PairRouting.childEmbedding x c hc false) σ.1.1 +
-          palindromeLevelCost t d (PairRouting.childEmbedding x c hc true) σ.1.2
-
-lemma palindromeLevelCost_step (t d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (d+1)) (X Y : (SwitchIndex d → Bool) × (SwitchIndex d → Bool))
-    (ξ η : Card d → Bool) :
-    palindromeLevelCost t (d+1) e
-      ((coinStepEquiv d).symm (Y,η),(coinStepEquiv d).symm (X,ξ)) =
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x ξ
-      let hc := PairRouting.colors_compatible x ξ
-      if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x ξ)
-        (fun b => palindromePerm d (if b then (Y.2,X.2) else (Y.1,X.1))) else
-        palindromeLevelCost t d (PairRouting.childEmbedding x c hc false) (Y.1,X.1) +
-        palindromeLevelCost t d (PairRouting.childEmbedding x c hc true) (Y.2,X.2) := by
-  rw [palindromeLevelCost]
-  rfl
-
-lemma palindromeLevelCost_gt (t d : ℕ) (h : d < t) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) : palindromeLevelCost t d e ω = 0 := by
-  classical
-  induction d generalizing ι with
-  | zero => rfl
-  | succ d ih =>
-    rw [palindromeLevelCost,ite_eq_right (by omega : t ≠ d+1),ih (by omega),ih (by omega),Nat.zero_add]
-
-
-theorem palindrome_level_partial_mgf (r s u : ℕ) (hu : u < r)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+r)) (X : SwitchIndex (s+r) → Bool)
-    (lo : Card r × SwitchIndex s → Bool) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(palindromeLevelCost (s+u+1) (s+r) e (assembleBits r s lo hi,X))) ≤
-      Real.exp (Fintype.card ι * cycleCoefficient u J q) := by
-  classical
-  induction r generalizing u ι with
-  | zero => omega
-  | succ r ih =>
-    simp only [Nat.add_succ] at *
-    let χ := coinStepEquiv (s+r) X
-    let x := e.trans (headTailEquiv (s+r)).toEmbedding
-    let c := PairRouting.colors x χ.2
-    let hc := PairRouting.colors_compatible x χ.2
-    let e₀ := PairRouting.childEmbedding x c hc false
-    let e₁ := PairRouting.childEmbedding x c hc true
-    let e' := PairRouting.switchedEmbedding x χ.2
-    let lo₀ : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons false j.1,j.2)
-    let lo₁ : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons true j.1,j.2)
-    let Y₀ := fun hi : Card s × SwitchIndex r → Bool => assembleBits r s lo₀ hi
-    let Y₁ := fun hi : Card s × SwitchIndex r → Bool => assembleBits r s lo₁ hi
-    let N := fun hi : (Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool) =>
-      PairRouting.alternatingCycles e' (fun b => if b then palindromePerm (s+r) (Y₁ hi.2,χ.1.2)
-        else palindromePerm (s+r) (Y₀ hi.1,χ.1.1))
-    let L₀ := fun hi : Card s × SwitchIndex r → Bool =>
-      palindromeLevelCost (s+u+1) (s+r) e₀ (Y₀ hi,χ.1.1)
-    let L₁ := fun hi : Card s × SwitchIndex r → Bool =>
-      palindromeLevelCost (s+u+1) (s+r) e₁ (Y₁ hi,χ.1.2)
-    have hs (hi : (Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool))
-        (η : Card s × Card r → Bool) :
-        palindromeLevelCost (s+u+1) (s+r+1) e
-          (assembleBits (r+1) s lo ((highStepEquiv r s).symm (hi,η)),X) =
-          if u = r then N hi else L₀ hi.1 + L₁ hi.2 := by
-      rw [assembleBits_step]
-      conv_lhs => rw [←(coinStepEquiv (s+r)).symm_apply_apply X]
-      rw [palindromeLevelCost_step]
-      simp only [Nat.add_right_cancel_iff,Nat.add_left_cancel_iff]
-      dsimp only [χ,x,c,hc,e₀,e₁,e',lo₀,lo₁,Y₀,Y₁,N,L₀,L₁]
-      congr 2
-      funext b
-      cases b <;> rfl
-    rw [finiteMean_equiv (highStepEquiv r s),finiteMean_prod]
-    simp_rw [hs,finiteMean_const]
-    by_cases hur : u = r
-    · subst u
-      simp only [ite_true]
-      rw [finiteMean_prod,finiteMean_comm]
-      calc
-        _ ≤ finiteMean (fun _hi : Card s × SwitchIndex r → Bool =>
-          Real.exp (Fintype.card ι * cycleCoefficient r J q)) := by
-          apply finiteMean_mono
-          intro hi
-          have hn := node_partial_mgf r s e'
-            (butterflyPerm (s+r) (decodeButterfly (s+r) χ.1.1))
-            (butterflyPerm (s+r) (decodeButterfly (s+r) χ.1.2))
-            (butterflyPerm (s+r) (decodeButterfly (s+r) (Y₁ hi))) lo₀ J hJ q hq
-          exact hn
-        _ = _ := finiteMean_const _
-    · simp only [ite_eq_right hur,pow_add]
-      rw [finiteMean_prod_mul (fun hi => q ^ L₀ hi) (fun hi => q ^ L₁ hi)]
-      have hu' : u < r := by omega
-      have h₀ := ih u hu' e₀ χ.1.1 lo₀
-      have h₁ := ih u hu' e₁ χ.1.2 lo₁
-      have hp := mul_le_mul h₀ h₁ (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
-        (Real.exp_nonneg _)
-      refine hp.trans_eq ?_
-      rw [←Real.exp_add]
-      congr 1
-      have hcount := PairRouting.color_card_add c
-      have hcount' : (Fintype.card {i : ι // c i = false} : ℝ) +
-          Fintype.card {i : ι // c i = true} = Fintype.card ι := by exact_mod_cast hcount
-      change (Fintype.card {i : ι // c i = false} : ℝ) * cycleCoefficient u J q +
-          Fintype.card {i : ι // c i = true} * cycleCoefficient u J q = _
-      rw [←add_mul,hcount']
-
-
-def switchHeight : (d : ℕ) → SwitchIndex d → ℕ
-  | 0, i => nomatch i
-  | d+1, Sum.inl _ => d+1
-  | d+1, Sum.inr (_,i) => switchHeight d i
-
-lemma switchHeight_le (d : ℕ) (i : SwitchIndex d) : switchHeight d i ≤ d := by
-  induction d with
-  | zero => exact Empty.elim i
-  | succ d ih =>
-    rcases i with y | ⟨b,i⟩
-    · exact le_refl _
-    · exact (ih i).trans (Nat.le_succ d)
-
-def AgreeThrough (d s : ℕ) (Y Y' : SwitchIndex d → Bool) : Prop :=
-  ∀ i, switchHeight d i ≤ s → Y i = Y' i
-
-def LayerAdapted (d s : ℕ) {β : Type*} (f : (SwitchIndex d → Bool) → β) : Prop :=
-  ∀ Y Y', AgreeThrough d s Y Y' → f Y = f Y'
-
-lemma agreeThrough_mono {d s t : ℕ} (h : s ≤ t) {Y Y' : SwitchIndex d → Bool}
-    (hh : AgreeThrough d t Y Y') : AgreeThrough d s Y Y' :=
-  fun i hi => hh i (hi.trans h)
-
-lemma layerAdapted_mono {d s t : ℕ} {β : Type*} {f : (SwitchIndex d → Bool) → β}
-    (hf : LayerAdapted d s f) (h : s ≤ t) : LayerAdapted d t f :=
-  fun Y Y' hh => hf Y Y' (agreeThrough_mono h hh)
-
-lemma layerAdapted_comp {d s : ℕ} {α β : Type*} {f : (SwitchIndex d → Bool) → α}
-    (hf : LayerAdapted d s f) (g : α → β) : LayerAdapted d s (g ∘ f) :=
-  fun Y Y' h => congrArg g (hf Y Y' h)
-
-lemma layerAdapted_sum {d s : ℕ} {ι : Type*} (A : Finset ι)
-    (f : ι → (SwitchIndex d → Bool) → ℕ)
-    (hf : ∀ i ∈ A, LayerAdapted d s (f i)) :
-    LayerAdapted d s (fun Y => ∑ i ∈ A, f i Y) := by
-  intro Y Y' h
-  exact Finset.sum_congr rfl (fun i hi => hf i hi Y Y' h)
-
-lemma assembleBits_agree (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
-    (hi hi' : Card s × SwitchIndex r → Bool) :
-    AgreeThrough (s+r) s (assembleBits r s lo hi) (assembleBits r s lo hi') := by
-  induction r with
-  | zero => intro i _; rfl
-  | succ r ih =>
-    intro i h
-    rcases i with y | ⟨b,i⟩
-    · have : s+r+1 ≤ s := h
-      omega
-    · exact ih (fun j => lo (Fin.cons b j.1,j.2))
-        (fun j => hi (j.1,Sum.inr (b,j.2))) (fun j => hi' (j.1,Sum.inr (b,j.2))) i h
-
-lemma palindromeLevelCost_adapted (t d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) :
-    LayerAdapted d (t-1) (fun Y => palindromeLevelCost t d e (Y,X)) := by
-  classical
-  induction d generalizing ι with
-  | zero => intro Y Y' _; rfl
-  | succ d ih =>
-    intro Y Y' h
-    change palindromeLevelCost t (d+1) e (Y,X) = palindromeLevelCost t (d+1) e (Y',X)
-    conv_lhs => rw [←(coinStepEquiv d).symm_apply_apply Y,←(coinStepEquiv d).symm_apply_apply X]
-    conv_rhs => rw [←(coinStepEquiv d).symm_apply_apply Y',←(coinStepEquiv d).symm_apply_apply X]
-    rw [palindromeLevelCost_step,palindromeLevelCost_step]
-    by_cases ht : t = d+1
-    · subst t
-      rw [ite_eq_left rfl,ite_eq_left rfl]
-      have he₀ : (coinStepEquiv d Y).1.1 = (coinStepEquiv d Y').1.1 := by
-        funext i
-        exact h (Sum.inr (false,i)) (by simpa only [switchHeight,Nat.add_sub_cancel] using switchHeight_le d i)
-      have he₁ : (coinStepEquiv d Y).1.2 = (coinStepEquiv d Y').1.2 := by
-        funext i
-        exact h (Sum.inr (true,i)) (by simpa only [switchHeight,Nat.add_sub_cancel] using switchHeight_le d i)
-      rw [he₀,he₁]
-    · rw [ite_eq_right ht,ite_eq_right ht]
-      congr 1
-      · apply ih
-        intro i hi
-        exact h (Sum.inr (false,i)) hi
-      · apply ih
-        intro i hi
-        exact h (Sum.inr (true,i)) hi
-
-lemma adapted_mul_mean_le (r s : ℕ) (F G : (SwitchIndex (s+r) → Bool) → ℝ)
-    (hF : ∀ Y, 0 ≤ F Y) (hA : LayerAdapted (s+r) s F) (B : ℝ)
-    (hG : ∀ lo : Card r × SwitchIndex s → Bool,
-      finiteMean (fun hi : Card s × SwitchIndex r → Bool => G (assembleBits r s lo hi)) ≤ B) :
-    finiteMean (fun Y => F Y*G Y) ≤ finiteMean F * B := by
-  rw [finiteMean_split r s,finiteMean_split r s F,←finiteMean_mul_const]
-  apply finiteMean_mono
-  intro lo
-  let H : Card s × SwitchIndex r → Bool := fun _ => false
-  have hh (hi : Card s × SwitchIndex r → Bool) :
-      F (assembleBits r s lo hi) = F (assembleBits r s lo H) :=
-    hA _ _ (assembleBits_agree r s lo hi H)
-  simp_rw [hh,finiteMean_const]
-  rw [show (fun hi => F (assembleBits r s lo H) * G (assembleBits r s lo hi)) =
-      (fun hi => G (assembleBits r s lo hi) * F (assembleBits r s lo H)) by
-        funext hi; exact mul_comm _ _,finiteMean_mul_const]
-  simpa only [mul_comm] using mul_le_mul_of_nonneg_left (hG lo) (hF _)
-
-
-theorem cycleCoefficient_decay (l r : ℕ) (hr : (l:ℝ)/2 ≤ r)
-    (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ (l:ℝ)/250) :
-    cycleCoefficient r ⌈Real.exp ((l:ℝ)/32)⌉₊ q ≤
-      68 * Real.exp (-(l:ℝ)/64) := by
-  let E := Real.exp ((l:ℝ)/32)
-  let J := ⌈E⌉₊
-  have hl : (0:ℝ) ≤ l := Nat.cast_nonneg _
-  have hE : 1 ≤ E := Real.one_le_exp_iff.mpr (by positivity)
-  have hEpos : 0 < E := Real.exp_pos _
-  have hEJ : E ≤ (J:ℝ) := Nat.le_ceil E
-  have hJ : (1:ℝ) ≤ J := hE.trans hEJ
-  have hJpos : (0:ℝ) < J := lt_of_lt_of_le (by norm_num) hJ
-  have hJE : (J:ℝ) ≤ 2*E := by
-    have h := Nat.ceil_lt_add_one (le_of_lt hEpos)
-    change (J:ℝ) < E+1 at h
-    linarith
-  have hsqJ : Real.sqrt J ≤ (J:ℝ) := by
-    apply Real.sqrt_le_iff.mpr
-    constructor
-    · positivity
-    · nlinarith
-  have hqp : 0 < q := lt_of_lt_of_le (by norm_num) hq
-  have hqE : q ≤ Real.exp ((l:ℝ)/250) := by
-    rw [←Real.exp_log hqp]
-    exact Real.exp_le_exp.mpr hlog
-  have hln : (1/2:ℝ) ≤ Real.log 2 := by
-    have hh := Real.one_sub_inv_le_log_of_pos (by norm_num : (0:ℝ)<2)
-    norm_num at hh ⊢
-    exact hh
-  have hden : Real.exp ((l:ℝ)/8) ≤ Real.sqrt ((2:ℝ)^r) := by
-    have he : (2:ℝ)^r = Real.exp ((r:ℝ)*Real.log 2) := by
-      rw [Real.exp_nat_mul,Real.exp_log (by norm_num)]
-    rw [he,Real.sqrt_eq_rpow,Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp]
-    apply Real.exp_le_exp.mpr
-    have hrn : (0:ℝ) ≤ r := Nat.cast_nonneg _
-    nlinarith
-  have hlog' : Real.log q ≤ (l:ℝ) := by linarith
-  have h₁ : Real.log q / (J:ℝ) ≤ 64*Real.exp (-(l:ℝ)/64) := by
-    calc
-      _ ≤ (l:ℝ)/E := div_le_div₀ (by positivity) hlog' hEpos hEJ
-      _ ≤ (64*Real.exp ((l:ℝ)/64))/E := by
-        apply div_le_div_of_nonneg_right _ (le_of_lt hEpos)
-        have hh := Real.add_one_le_exp ((l:ℝ)/64)
-        linarith
-      _ = _ := by
-        dsimp [E]
-        rw [mul_div_assoc,←Real.exp_sub]
-        congr 2
-        ring
-  have h₂ : 2*q*Real.sqrt J / Real.sqrt ((2:ℝ)^r) ≤
-      4*Real.exp (-(l:ℝ)/64) := by
-    calc
-      _ ≤ (4*Real.exp ((l:ℝ)/250)*E)/Real.exp ((l:ℝ)/8) := by
-        apply div_le_div₀ (by positivity) _ (Real.exp_pos _) hden
-        have hs := hsqJ.trans hJE
-        have hm := mul_le_mul hqE hs (Real.sqrt_nonneg _) (Real.exp_nonneg _)
-        nlinarith
-      _ = 4*Real.exp ((l:ℝ)/250+(l:ℝ)/32-(l:ℝ)/8) := by
-        dsimp [E]
-        rw [mul_assoc,←Real.exp_add,mul_div_assoc,←Real.exp_sub]
-      _ ≤ _ := by
-        apply mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr _) (by norm_num)
-        linarith
-  change Real.log q / (J:ℝ) + 2*q*Real.sqrt J / Real.sqrt ((2:ℝ)^r) ≤ _
-  linarith
-
-lemma cycleCutoff_pos (l : ℕ) : 0 < ⌈Real.exp ((l:ℝ)/32)⌉₊ := by
-  have h := Nat.le_ceil (Real.exp ((l:ℝ)/32))
-  have he := Real.exp_pos ((l:ℝ)/32)
-  have : (0:ℝ) < ⌈Real.exp ((l:ℝ)/32)⌉₊ := lt_of_lt_of_le he h
-  exact_mod_cast this
-
-
-lemma pow_sum_le_mean_powers {ι : Type*} [Fintype ι] [Nonempty ι]
-    (z : ι → ℕ) (q : ℝ) (hq : 0 < q) :
-    q^(∑ i, z i) ≤ finiteMean (fun i => q^(Fintype.card ι*z i)) := by
-  have hn : (Fintype.card ι:ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
-  have hm : finiteMean (fun i => (Fintype.card ι*z i:ℕ)*Real.log q) =
-      (∑ i, z i:ℕ)*Real.log q := by
-    simp only [finiteMean,Nat.cast_mul,Nat.cast_sum,←Finset.mul_sum,←Finset.sum_mul]
-    field_simp
-  have he := exp_finiteMean_le (fun i => (Fintype.card ι*z i:ℕ)*Real.log q)
-  rw [hm,Real.exp_nat_mul,Real.exp_log hq] at he
-  simpa only [Real.exp_nat_mul,Real.exp_log hq] using he
-
-noncomputable def palindromeSlabCost (l d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) : ℕ :=
-  ∑ i : Fin l, palindromeLevelCost (l+i.1+1) d e ω
-
-lemma palindromeSlabCost_adapted (l d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) :
-    LayerAdapted d (2*l) (fun Y => palindromeSlabCost l d e (Y,X)) := by
-  apply layerAdapted_sum
-  intro i _
-  exact layerAdapted_mono (palindromeLevelCost_adapted (l+i.1+1) d e X) (by omega)
-
-theorem palindrome_slab_partial_mgf (r s l : ℕ) (hl : 0 < l) (hs : 2*s ≤ l)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+r)) (X : SwitchIndex (s+r) → Bool)
-    (lo : Card r × SwitchIndex s → Bool) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(palindromeSlabCost l (s+r) e (assembleBits r s lo hi,X))) ≤
-      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
-  classical
-  let : NeZero l := ⟨ne_of_gt hl⟩
-  have hqp : 0 < q := lt_of_lt_of_le (by norm_num) hq
-  calc
-    _ ≤ finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      finiteMean (fun i : Fin l => q^(l*palindromeLevelCost (l+i.1+1) (s+r) e
-        (assembleBits r s lo hi,X)))) := by
-      apply finiteMean_mono
-      intro hi
-      simpa only [palindromeSlabCost,Fintype.card_fin] using
-        pow_sum_le_mean_powers (fun i : Fin l => palindromeLevelCost (l+i.1+1) (s+r) e
-          (assembleBits r s lo hi,X)) q hqp
-    _ = finiteMean (fun i : Fin l => finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(l*palindromeLevelCost (l+i.1+1) (s+r) e (assembleBits r s lo hi,X)))) := finiteMean_comm _
-    _ ≤ finiteMean (fun _i : Fin l =>
-      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64)))) := by
-      apply finiteMean_mono
-      intro i
-      by_cases ht : l+i.1+1 ≤ s+r
-      · have hs' : s ≤ l+i.1 := by omega
-        have he : l+i.1+1 = s+(l+i.1-s)+1 := by omega
-        have hu : l+i.1-s < r := by omega
-        have hm := palindrome_level_partial_mgf r s (l+i.1-s) hu e X lo
-          ⌈Real.exp ((l:ℝ)/32)⌉₊ (cycleCutoff_pos l) (q^l) (one_le_pow₀ hq)
-        have hc := cycleCoefficient_decay l (l+i.1-s) (by
-          rw [Nat.cast_sub hs',Nat.cast_add]
-          have hsr : (2:ℝ)*s ≤ l := by exact_mod_cast hs
-          have hi : (0:ℝ) ≤ i.1 := Nat.cast_nonneg _
-          linarith) (q^l) (one_le_pow₀ hq) (by
-            rw [Real.log_pow]
-            nlinarith [mul_le_mul_of_nonneg_left hlog (Nat.cast_nonneg l)])
-        simp_rw [he,pow_mul]
-        exact hm.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left hc (Nat.cast_nonneg _)))
-      · have hz := palindromeLevelCost_gt (l+i.1+1) (s+r) (by omega) e
-        simp only [hz,Nat.mul_zero,pow_zero,finiteMean_const]
-        exact Real.one_le_exp_iff.mpr (by positivity)
-    _ = _ := finiteMean_const _
-
-
-lemma palindromeSlabCost_gt (l d : ℕ) (h : d ≤ l) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) : palindromeSlabCost l d e ω = 0 := by
-  apply Finset.sum_eq_zero
-  intro i _
-  exact palindromeLevelCost_gt _ _ (by omega) _ _
-
-lemma palindrome_slab_adapted_mul_le (d s l : ℕ) (hsd : s ≤ d) (hl : 0 < l)
-    (hs : 2*s ≤ l) {ι : Type*} [Fintype ι] (e : ι ↪ Card d)
-    (X : SwitchIndex d → Bool) (F : (SwitchIndex d → Bool) → ℝ)
-    (hF : ∀ Y, 0 ≤ F Y) (hA : LayerAdapted d s F) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun Y => F Y*q^(palindromeSlabCost l d e (Y,X))) ≤
-      finiteMean F * Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
-  obtain ⟨r,rfl⟩ := Nat.exists_eq_add_of_le hsd
-  exact adapted_mul_mean_le r s F _ hF hA _
-    (fun lo => palindrome_slab_partial_mgf r s l hl hs e X lo q hq hlog)
-
-def familyEnd (L n : ℕ) : ℕ := if n = 0 then 0 else 2*(L*4^(n-1))
-
-lemma familyEnd_two (L n : ℕ) : 2*familyEnd L n ≤ L*4^n := by
-  cases n with
-  | zero => simp [familyEnd]
-  | succ n => simp only [familyEnd,Nat.succ_ne_zero,ite_false,Nat.add_one_sub_one,pow_succ]; nlinarith
-
-lemma familyEnd_ge (L n j : ℕ) (hj : j < n) : 2*(L*4^j) ≤ familyEnd L n := by
-  cases n with
-  | zero => omega
-  | succ n =>
-    simp only [familyEnd,Nat.succ_ne_zero,ite_false,Nat.add_one_sub_one]
-    apply Nat.mul_le_mul_left
-    exact Nat.mul_le_mul_left L (Nat.pow_le_pow_right (by decide) (by omega : j ≤ n))
-
-noncomputable def palindromeFamilyCost (L n d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) : ℕ :=
-  ∑ j ∈ Finset.range n, palindromeSlabCost (L*4^j) d e ω
-
-lemma palindromeFamilyCost_adapted (L n d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) :
-    LayerAdapted d (familyEnd L n) (fun Y => palindromeFamilyCost L n d e (Y,X)) := by
-  apply layerAdapted_sum
-  intro j hj
-  exact layerAdapted_mono (palindromeSlabCost_adapted (L*4^j) d e X)
-    (familyEnd_ge L n j (Finset.mem_range.mp hj))
-
-end Thorp
-
-end ThorpNine.SparseSaving
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.SparseSaving.BenesCoins
+
+namespace OAI
+
+namespace ThorpNine.SparseSaving
+
+namespace Thorp
+
+lemma mem_centralBlock (r H : ℕ) (b : Card r) (x : Card (H+r)) :
+    x ∈ centralBlock r H b ↔ (cardSplit r H x).1 = b := by
+  classical
+  constructor
+  · intro h
+    obtain ⟨z,_,rfl⟩ := Finset.mem_image.mp h
+    simp only [Equiv.apply_symm_apply]
+  · intro h
+    apply Finset.mem_image.mpr
+    refine ⟨(cardSplit r H x).2,Finset.mem_univ _,?_⟩
+    rw [←h,Prod.mk.eta,Equiv.symm_apply_apply]
+
+lemma centralBlock_card (r H : ℕ) (b : Card r) : (centralBlock r H b).card = 2^H := by
+  classical
+  rw [centralBlock,Finset.card_image_of_injective]
+  · exact card_positions H
+  · intro x y h
+    exact congrArg Prod.snd ((cardSplit r H).symm.injective h)
+
+lemma centralBlock_disjoint (r H : ℕ) :
+    Pairwise (fun b b' => Disjoint (centralBlock r H b) (centralBlock r H b')) := by
+  classical
+  intro b b' h
+  apply Finset.disjoint_left.mpr
+  intro x hx hy
+  exact h (((mem_centralBlock r H b x).mp hx).symm.trans
+    ((mem_centralBlock r H b' x).mp hy))
+
+lemma centralBlock_zero (H : ℕ) (b : Card 0) : centralBlock 0 H b = Finset.univ := by
+  classical
+  ext x
+  simp only [mem_centralBlock,Finset.mem_univ,iff_true]
+  exact Subsingleton.elim _ _
+
+lemma centralBlock_step (r H : ℕ) (b : Bool) (c : Card r) :
+    centralBlock (r+1) H (Fin.cons b c) = (centralBlock r H c).image (Fin.cons b) := by
+  classical
+  simp only [centralBlock,Finset.image_image,cardSplit,Equiv.coe_fn_symm_mk,
+    Fin.cons_zero,Fin.tail_cons,Function.comp_def]
+
+lemma block_inter_step (r H : ℕ) (b : Bool) (c : Card r) (A : Finset (Card (H+(r+1)))) :
+    centralBlock (r+1) H (Fin.cons b c) ∩ A =
+      (centralBlock r H c ∩ childMarks A b).image (Fin.cons b) := by
+  classical
+  rw [centralBlock_step]
+  ext x
+  constructor
+  · intro hx
+    obtain ⟨hxB,hxA⟩ := Finset.mem_inter.mp hx
+    obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hxB
+    exact Finset.mem_image.mpr ⟨y,Finset.mem_inter.mpr ⟨hy,(mem_childMarks A b y).mpr hxA⟩,rfl⟩
+  · intro hx
+    obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
+    exact Finset.mem_inter.mpr ⟨Finset.mem_image.mpr ⟨y,(Finset.mem_inter.mp hy).1,rfl⟩,
+      (mem_childMarks A b y).mp (Finset.mem_inter.mp hy).2⟩
+
+lemma sparseMass_central_zero (H : ℕ) (A : Finset (Card H)) :
+    sparseMass (centralBlock 0 H) A = crowdedMass A.card := by
+  classical
+  simp only [sparseMass,centralBlock_zero,Finset.univ_inter,crowdedMass,
+    Finset.sum_const,Finset.card_univ,card_positions,pow_zero,one_nsmul]
+
+lemma sparseMass_central_step (r H : ℕ) (A : Finset (Card (H+(r+1)))) :
+    sparseMass (centralBlock (r+1) H) A =
+      sparseMass (centralBlock r H) (childMarks A false) +
+        sparseMass (centralBlock r H) (childMarks A true) := by
+  classical
+  unfold sparseMass
+  rw [←Equiv.sum_comp (headTailEquiv r).symm,Fintype.sum_prod_type]
+  have ht (b : Bool) (c : Card r) : (headTailEquiv r).symm (b,c) = Fin.cons b c := rfl
+  simp only [Fintype.sum_bool,ht]
+  have hc (b : Bool) (c : Card r) :
+      (centralBlock (r+1) H (Fin.cons b c) ∩ A).card =
+      (centralBlock r H c ∩ childMarks A b).card := by
+    rw [block_inter_step]
+    apply Finset.card_image_of_injective
+    intro x y h
+    simpa only [Fin.tail_cons] using congrArg Fin.tail h
+  simp only [hc]
+  exact add_comm _ _
+
+
+lemma inverseButterfly_coordinates_step (d : ℕ)
+    (c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (ξ : Card d → Bool)
+    (x : Card (d+1)) :
+    headTailEquiv d ((butterflyPerm (d+1)
+      (decodeButterfly (d+1) ((coinStepEquiv d).symm (c,ξ)))).symm x) =
+      (Bool.xor (x 0) (ξ (Fin.tail x)),
+        (butterflyPerm d (decodeButterfly d
+          (if Bool.xor (x 0) (ξ (Fin.tail x)) then c.2 else c.1))).symm (Fin.tail x)) := by
+  rw [butterflyPerm_step]
+  rfl
+
+lemma inverseButterflyMarks_child (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (d+1))
+    (c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (ξ : Card d → Bool) (b : Bool) :
+    childMarks (inverseButterflyMarks (d+1) (Finset.univ.image e)
+      ((coinStepEquiv d).symm (c,ξ))) b =
+      inverseButterflyMarks d
+        (Finset.univ.image (PairRouting.childEmbedding (e.trans (headTailEquiv d).toEmbedding)
+          (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) ξ)
+            (PairRouting.colors_compatible _ _) b)) (if b then c.2 else c.1) := by
+  classical
+  let x := e.trans (headTailEquiv d).toEmbedding
+  let col := PairRouting.colors x ξ
+  let child := PairRouting.childEmbedding x col (PairRouting.colors_compatible x ξ) b
+  have hp (i : ι) : headTailEquiv d ((butterflyPerm (d+1)
+      (decodeButterfly (d+1) ((coinStepEquiv d).symm (c,ξ)))).symm (e i)) =
+      (col i,(butterflyPerm d (decodeButterfly d (if col i then c.2 else c.1))).symm (x i).2) :=
+    inverseButterfly_coordinates_step d c ξ (e i)
+  ext y
+  simp only [mem_childMarks,inverseButterflyMarks,Finset.image_image,
+    Finset.mem_image,Finset.mem_univ,true_and,Function.comp_def]
+  constructor
+  · rintro ⟨i,hi⟩
+    have he := congrArg (headTailEquiv d) hi
+    rw [hp i] at he
+    change (col i,_) = (b,y) at he
+    have hb := congrArg Prod.fst he
+    have hy := congrArg Prod.snd he
+    dsimp only at hb hy
+    refine ⟨⟨i,hb⟩,?_⟩
+    change (butterflyPerm d (decodeButterfly d (if b then c.2 else c.1))).symm (x i).2 = y
+    simpa only [hb] using hy
+  · rintro ⟨i,hi⟩
+    refine ⟨i.val,?_⟩
+    apply (headTailEquiv d).injective
+    rw [hp]
+    change (col i.val,_) = (b,y)
+    have hb : col i.val = b := i.property
+    rw [hb]
+    exact Prod.ext rfl hi
+
+lemma inverseButterflyMarks_card (d : ℕ) (A : Finset (Card d)) (ω : SwitchIndex d → Bool) :
+    (inverseButterflyMarks d A ω).card = A.card := by
+  classical
+  exact Finset.card_image_of_injective _ (butterflyPerm d (decodeButterfly d ω)).symm.injective
+
+lemma benes_input_coordinates (d : ℕ) (ω : BenesCoins (d+1)) :
+    ω.2 = (coinStepEquiv d).symm
+      (((benesStepEquiv d ω).1.1.2,(benesStepEquiv d ω).1.2.2),(benesStepEquiv d ω).2.1) := by
+  change ω.2 = (coinStepEquiv d).symm (coinStepEquiv d ω.2)
+  exact ((coinStepEquiv d).symm_apply_apply ω.2).symm
+
+lemma cutoffMass_central (r H : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (H+r)) (ω : BenesCoins (H+r)) :
+    cutoffMass H (H+r) e ω =
+      sparseMass (centralBlock r H) (inverseButterflyMarks (H+r) (Finset.univ.image e) ω.2) := by
+  classical
+  induction r generalizing ι with
+  | zero =>
+    rw [sparseMass_central_zero,inverseButterflyMarks_card,
+      Finset.card_image_of_injective _ e.injective,Finset.card_univ]
+    cases H with
+    | zero =>
+      have hcard : Fintype.card ι ≤ 1 := by
+        have hh := Fintype.card_le_of_injective e e.injective
+        simpa only [Nat.add_zero,card_positions,pow_zero] using hh
+      simp [cutoffMass,crowdedMass,show ¬2≤Fintype.card ι by omega]
+    | succ H => simp only [cutoffMass,le_refl,ite_true]
+  | succ r ih =>
+    simp only [Nat.add_succ] at *
+    rw [cutoffMass,ite_eq_right (by omega : ¬H+r+1≤H)]
+    let σ := benesStepEquiv (H+r) ω
+    let x := e.trans (headTailEquiv (H+r)).toEmbedding
+    let col := PairRouting.colors x σ.2.1
+    let hc := PairRouting.colors_compatible x σ.2.1
+    change cutoffMass H (H+r) (PairRouting.childEmbedding x col hc false) σ.1.1 +
+      cutoffMass H (H+r) (PairRouting.childEmbedding x col hc true) σ.1.2 = _
+    rw [ih,ih,sparseMass_central_step,benes_input_coordinates (H+r) ω]
+    rw [inverseButterflyMarks_child,inverseButterflyMarks_child]
+    rfl
+
+
+theorem palindrome_low_mgf (r H : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (H+r)) (q : ℝ) (hq : 1 ≤ q)
+    (hr : (2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*q^H) ≤ 1/2) :
+    finiteMean (fun ω : BenesCoins (H+r) => q^(palindromeLowCost H (H+r) e ω)) ≤
+      Real.exp ((2:ℝ)^r * (2 * ((2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*q^H))^2)) := by
+  classical
+  have hec : (Finset.univ.image e).card = Fintype.card ι := by
+    rw [Finset.card_image_of_injective _ e.injective,Finset.card_univ]
+  have hm := inverseButterfly_sparse_mgf (H+r) (centralBlock r H)
+    (centralBlock_disjoint r H) (Finset.univ.image e) (q^H)
+    (pow_nonneg (by linarith) _) (fun b => by
+      simpa only [centralBlock_card,hec,Nat.cast_pow,Nat.cast_ofNat] using hr)
+  calc
+    _ ≤ finiteMean (fun ω : BenesCoins (H+r) => (q^H)^(cutoffMass H (H+r) e ω)) := by
+      apply finiteMean_mono
+      intro ω
+      rw [←pow_mul]
+      exact pow_le_pow_right₀ hq (palindromeLowCost_le_cutoff H (H+r) e ω)
+    _ = finiteMean (fun ω : SwitchIndex (H+r) → Bool =>
+        (q^H)^(sparseMass (centralBlock r H)
+          (inverseButterflyMarks (H+r) (Finset.univ.image e) ω))) := by
+      simp_rw [cutoffMass_central]
+      simp only [finiteMean_prod,finiteMean_const]
+    _ ≤ _ := by
+      simpa only [centralBlock_card,hec,Nat.cast_pow,Nat.cast_ofNat,Finset.sum_const,
+        Finset.card_univ,card_positions,nsmul_eq_mul] using hm
+
+variable {α B : Type*} {r s : ℕ}
+
+def parallelFreshPre (pre : α ≃ B × Card (s+r))
+    (lo : B → Card r × SwitchIndex s → Bool) : α ≃ (B × Card s) × Card r :=
+  ((pre.trans (Equiv.prodCongr (Equiv.refl B) (cardSplit r s))).trans
+    (Equiv.prodCongrRight (fun b =>
+      (parallelPerm (lo b)).trans (Equiv.prodComm _ _)))).trans
+    (Equiv.prodAssoc _ _ _).symm
+
+def parallelFreshPost (post : B × Card (s+r) ≃ α) : (B × Card s) × Card r ≃ α :=
+  (((Equiv.prodAssoc _ _ _).trans (Equiv.prodCongr (Equiv.refl B)
+    (Equiv.prodComm _ _))).trans (Equiv.prodCongr (Equiv.refl B) (cardSplit r s).symm)).trans post
+
+def parallelAssembled (lo : B → Card r × SwitchIndex s → Bool)
+    (hi : (B × Card s) × SwitchIndex r → Bool) : B × SwitchIndex (s+r) → Bool :=
+  fun i => assembleBits r s (lo i.1) (fun j => hi ((i.1,j.1),j.2)) i.2
+
+lemma parallel_exposed_perm (pre : α ≃ B × Card (s+r)) (post : B × Card (s+r) ≃ α)
+    (lo : B → Card r × SwitchIndex s → Bool)
+    (hi : (B × Card s) × SwitchIndex r → Bool) :
+    relabeledPerm pre post (parallelAssembled lo hi) =
+      relabeledPerm (parallelFreshPre pre lo) (parallelFreshPost post) hi := by
+  ext x
+  have h := butterfly_split r s (lo (pre x).1)
+    (fun j => hi (((pre x).1,j.1),j.2)) (pre x).2
+  have hh := congrArg (fun y => post ((pre x).1,(cardSplit r s).symm y)) h
+  simpa only [relabeledPerm,parallelAssembled,parallelPerm,parallelFreshPre,parallelFreshPost,
+    Equiv.trans_apply,Equiv.prodCongr_apply,Equiv.prodCongrRight,Equiv.coe_fn_mk,
+    Equiv.refl_apply,Equiv.prodComm_apply,Prod.swap,Equiv.prodAssoc_apply,
+    Equiv.prodAssoc_symm_apply,Prod.map,Equiv.symm_apply_apply] using hh
+
+theorem parallel_partially_exposed_log_mgf [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B]
+    (pre : α ≃ B × Card (s+r)) (post : B × Card (s+r) ≃ α)
+    (lo : B → Card r × SwitchIndex s → Bool) (S : Finset α)
+    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    Real.log (finiteMean (fun hi : (B × Card s) × SwitchIndex r → Bool =>
+      q^(Fintype.card (RoutingNetwork.SelectedCycle
+        (relabeledPerm pre post (parallelAssembled lo hi)) S)))) ≤
+      (S.card:ℝ)/J*Real.log q + 2*q*S.card*Real.sqrt J/Real.sqrt ((2:ℝ)^r) := by
+  simp_rw [parallel_exposed_perm]
+  exact parallel_cycle_log_mgf (parallelFreshPre pre lo) (parallelFreshPost post) S J hJ q hq
+
+end Thorp
+namespace Thorp
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+lemma orbitSet_inverse (p : Equiv.Perm α) (x : α) : orbitSet p⁻¹ x = orbitSet p x := by
+  ext y
+  simp only [mem_orbitSet,Equiv.Perm.sameCycle_inv]
+
+lemma selectedCycle_inverse (p : Equiv.Perm α) (S : Finset α) :
+    Fintype.card (RoutingNetwork.SelectedCycle p⁻¹ S) =
+      Fintype.card (RoutingNetwork.SelectedCycle p S) := by
+  apply Fintype.card_congr
+  refine { toFun := fun C => ⟨C.val,?_,C.property.2⟩
+           invFun := fun C => ⟨C.val,?_,C.property.2⟩
+           left_inv := fun _ => rfl
+           right_inv := fun _ => rfl }
+  · obtain ⟨x,hx⟩ := C.property.1
+    exact ⟨x,hx.trans (orbitSet_inverse p x)⟩
+  · obtain ⟨x,hx⟩ := C.property.1
+    exact ⟨x,hx.trans (orbitSet_inverse p x).symm⟩
+
+lemma orbitSet_conjugate (p g : Equiv.Perm α) (x : α) :
+    orbitSet (g*p*g⁻¹) (g x) = (orbitSet p x).image g := by
+  ext y
+  simp only [mem_orbitSet,Finset.mem_image]
+  rw [Equiv.Perm.sameCycle_conj]
+  change p.SameCycle (g.symm (g x)) (g.symm y) ↔ ∃ z, p.SameCycle x z ∧ g z = y
+  rw [Equiv.symm_apply_apply]
+  constructor
+  · intro h
+    exact ⟨g.symm y,h,g.apply_symm_apply y⟩
+  · rintro ⟨z,hz,rfl⟩
+    simpa only [Equiv.symm_apply_apply] using hz
+
+noncomputable def conjugateCycle (p g : Equiv.Perm α) (S : Finset α)
+    (C : RoutingNetwork.SelectedCycle p S) :
+    RoutingNetwork.SelectedCycle (g*p*g⁻¹) (S.image g) := by
+  refine ⟨C.val.image g,?_,Finset.image_subset_image C.property.2⟩
+  obtain ⟨x,hx⟩ := C.property.1
+  exact ⟨g x,by rw [hx,orbitSet_conjugate]⟩
+
+lemma conjugateCycle_injective (p g : Equiv.Perm α) (S : Finset α) :
+    Function.Injective (conjugateCycle p g S) := by
+  intro C D he
+  apply Subtype.ext
+  exact (Finset.image_injective g.injective) (congrArg Subtype.val he)
+
+lemma selectedCycle_conjugate (p g : Equiv.Perm α) (S : Finset α) :
+    Fintype.card (RoutingNetwork.SelectedCycle (g*p*g⁻¹) (S.image g)) =
+      Fintype.card (RoutingNetwork.SelectedCycle p S) := by
+  apply Nat.le_antisymm
+  · have h := Fintype.card_le_of_injective _
+      (conjugateCycle_injective (g*p*g⁻¹) g⁻¹ (S.image g))
+    have hp : g⁻¹*(g*p*g⁻¹)*g = p := by group
+    rw [inv_inv,hp] at h
+    have hs : (S.image g).image g.symm = S := by
+      ext x
+      simp only [Finset.mem_image]
+      constructor
+      · rintro ⟨y,⟨z,hz,rfl⟩,he⟩
+        have hz' : z = x := by simpa only [Equiv.symm_apply_apply] using he
+        exact hz' ▸ hz
+      · intro hx
+        exact ⟨g x,⟨x,hx,rfl⟩,g.symm_apply_apply x⟩
+    change _ ≤ Fintype.card (RoutingNetwork.SelectedCycle p ((S.image g).image g.symm)) at h
+    rw [hs] at h
+    exact h
+  · exact Fintype.card_le_of_injective _ (conjugateCycle_injective p g S)
+
+lemma alternatingCycles_le_projection {ι : Type*} [Fintype ι]
+    (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    PairRouting.alternatingCycles e p ≤
+      Fintype.card (RoutingNetwork.SelectedCycle (PairRouting.alternatingProjection p)
+        (boolFiber (PairRouting.labelSet e) false)) := by
+  have h := CycleColoring.card_alternating_cycles_le (PairRouting.alternatingProjection p)
+    (boolFiber (PairRouting.labelSet e) false) (boolFiber (PairRouting.labelSet e) true)
+  simpa only [PairRouting.alternatingCycles,childJoin_boolFiber] using h
+
+lemma projectionCycles_forward (X₀ X₁ Y₀ Y₁ : Equiv.Perm α) (S : Finset α) :
+    Fintype.card (RoutingNetwork.SelectedCycle
+      ((Y₀*X₀⁻¹)⁻¹*(Y₁*X₁⁻¹)) S) =
+    Fintype.card (RoutingNetwork.SelectedCycle
+      (X₀⁻¹*X₁*Y₁⁻¹*Y₀) (S.image X₀.symm)) := by
+  let p := (Y₀*X₀⁻¹)⁻¹*(Y₁*X₁⁻¹)
+  calc
+    _ = Fintype.card (RoutingNetwork.SelectedCycle (X₀⁻¹*p*X₀)
+        (S.image X₀.symm)) := by
+      change Fintype.card (RoutingNetwork.SelectedCycle p S) = _
+      have h := (selectedCycle_conjugate p X₀⁻¹ S).symm
+      rw [inv_inv] at h
+      exact h
+    _ = Fintype.card (RoutingNetwork.SelectedCycle (X₀⁻¹*p*X₀)⁻¹
+        (S.image X₀.symm)) := (selectedCycle_inverse _ _).symm
+    _ = _ := by
+      have he : (X₀⁻¹*p*X₀)⁻¹ = X₀⁻¹*X₁*Y₁⁻¹*Y₀ := by dsimp only [p]; group
+      rw [he]
+
+end Thorp
+namespace Thorp
+
+noncomputable def cycleCoefficient (r J : ℕ) (q : ℝ) : ℝ :=
+  Real.log q / J + 2*q*Real.sqrt J/Real.sqrt ((2:ℝ)^r)
+
+lemma cycleCoefficient_nonneg (r J : ℕ) (q : ℝ) (hq : 1 ≤ q) :
+    0 ≤ cycleCoefficient r J q := by
+  unfold cycleCoefficient
+  exact add_nonneg (div_nonneg (Real.log_nonneg hq) (Nat.cast_nonneg _))
+    (div_nonneg (mul_nonneg (by linarith) (Real.sqrt_nonneg _)) (Real.sqrt_nonneg _))
+
+lemma projectionSide_card_le {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι]
+    (e : ι ↪ Bool × α) : (boolFiber (PairRouting.labelSet e) false).card ≤ Fintype.card ι := by
+  have h := card_childJoin (boolFiber (PairRouting.labelSet e) false)
+    (boolFiber (PairRouting.labelSet e) true)
+  rw [childJoin_boolFiber,PairRouting.card_labelSet] at h
+  omega
+
+theorem node_partial_mgf (r s : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Bool × Card (s+r)) (X₀ X₁ Y₁ : Equiv.Perm (Card (s+r)))
+    (lo : Card r × SwitchIndex s → Bool) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(PairRouting.alternatingCycles e (fun b => if b then Y₁*X₁⁻¹ else
+        butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi))*X₀⁻¹))) ≤
+      Real.exp (Fintype.card ι * cycleCoefficient r J q) := by
+  classical
+  let S := boolFiber (PairRouting.labelSet e) false
+  let S' := S.image X₀.symm
+  let post := X₀⁻¹*X₁*Y₁⁻¹
+  let count := fun hi : Card s × SwitchIndex r → Bool =>
+    Fintype.card (RoutingNetwork.SelectedCycle
+      ((Equiv.refl _).trans (butterflyPerm (s+r)
+        (decodeButterfly (s+r) (assembleBits r s lo hi))) |>.trans post) S')
+  have hpoint (hi : Card s × SwitchIndex r → Bool) :
+      PairRouting.alternatingCycles e (fun b => if b then Y₁*X₁⁻¹ else
+        butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi))*X₀⁻¹) ≤ count hi := by
+    refine (alternatingCycles_le_projection e _).trans_eq ?_
+    simp only [PairRouting.alternatingProjection,Bool.false_eq_true,↓reduceIte]
+    change Fintype.card (RoutingNetwork.SelectedCycle
+      ((butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi))*X₀⁻¹)⁻¹ * (Y₁*X₁⁻¹)) S) = count hi
+    exact projectionCycles_forward X₀ X₁ _ Y₁ S
+  have hm := partially_exposed_cycle_log_mgf (Equiv.refl (Card (s+r))) post lo S' J hJ q hq
+  have hp : 0 < finiteMean (fun hi : Card s × SwitchIndex r → Bool => q^(count hi)) :=
+    finiteMean_pos (fun _ => pow_pos (by linarith) _)
+  have hb : finiteMean (fun hi : Card s × SwitchIndex r → Bool => q^(count hi)) ≤
+      Real.exp (S'.card * cycleCoefficient r J q) := by
+    rw [←Real.exp_log hp]
+    apply Real.exp_le_exp.mpr
+    calc
+      _ ≤ _ := hm
+      _ = _ := by unfold cycleCoefficient; ring
+  have hcard : S'.card ≤ Fintype.card ι := by
+    rw [show S'.card = S.card from Finset.card_image_of_injective _ X₀.symm.injective]
+    exact projectionSide_card_le e
+  calc
+    _ ≤ finiteMean (fun hi : Card s × SwitchIndex r → Bool => q^(count hi)) :=
+      finiteMean_mono (fun hi => pow_le_pow_right₀ hq (hpoint hi))
+    _ ≤ Real.exp (S'.card * cycleCoefficient r J q) := hb
+    _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right
+      (by exact_mod_cast hcard) (cycleCoefficient_nonneg r J q hq))
+
+
+def highStepEquiv (r s : ℕ) : (Card s × SwitchIndex (r+1) → Bool) ≃
+    (((Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool)) ×
+      (Card s × Card r → Bool)) where
+  toFun hi := ((fun j => hi (j.1,Sum.inr (false,j.2)),
+    fun j => hi (j.1,Sum.inr (true,j.2))),fun j => hi (j.1,Sum.inl j.2))
+  invFun c := fun j => match j.2 with
+    | Sum.inl y => c.2 (j.1,y)
+    | Sum.inr (b,i) => if b then c.1.2 (j.1,i) else c.1.1 (j.1,i)
+  left_inv hi := by
+    funext j
+    rcases j with ⟨z,y|⟨b,i⟩⟩
+    · rfl
+    · cases b <;> rfl
+  right_inv c := by
+    rcases c with ⟨⟨h₀,h₁⟩,η⟩
+    rfl
+
+lemma assembleBits_step (r s : ℕ) (lo : Card (r+1) × SwitchIndex s → Bool)
+    (c : ((Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool)) ×
+      (Card s × Card r → Bool)) :
+    assembleBits (r+1) s lo ((highStepEquiv r s).symm c) =
+      (coinStepEquiv (s+r)).symm
+        ((assembleBits r s (fun j => lo (Fin.cons false j.1,j.2)) c.1.1,
+          assembleBits r s (fun j => lo (Fin.cons true j.1,j.2)) c.1.2),
+        fun y => c.2 ((cardSplit r s y).2,(cardSplit r s y).1)) := by
+  funext i
+  rcases i with y | ⟨b,i⟩
+  · rfl
+  · cases b <;> rfl
+
+noncomputable def palindromeLevelCost (t : ℕ) : (d : ℕ) → {ι : Type*} → [Fintype ι] →
+    (ι ↪ Card d) → BenesCoins d → ℕ
+  | 0, _, _, _, _ => 0
+  | d+1, _, _, e, ω =>
+      let σ := benesStepEquiv d ω
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x σ.2.1
+      let hc := PairRouting.colors_compatible x σ.2.1
+      if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
+          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) else
+        palindromeLevelCost t d (PairRouting.childEmbedding x c hc false) σ.1.1 +
+          palindromeLevelCost t d (PairRouting.childEmbedding x c hc true) σ.1.2
+
+lemma palindromeLevelCost_step (t d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (d+1)) (X Y : (SwitchIndex d → Bool) × (SwitchIndex d → Bool))
+    (ξ η : Card d → Bool) :
+    palindromeLevelCost t (d+1) e
+      ((coinStepEquiv d).symm (Y,η),(coinStepEquiv d).symm (X,ξ)) =
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x ξ
+      let hc := PairRouting.colors_compatible x ξ
+      if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x ξ)
+        (fun b => palindromePerm d (if b then (Y.2,X.2) else (Y.1,X.1))) else
+        palindromeLevelCost t d (PairRouting.childEmbedding x c hc false) (Y.1,X.1) +
+        palindromeLevelCost t d (PairRouting.childEmbedding x c hc true) (Y.2,X.2) := by
+  rw [palindromeLevelCost]
+  rfl
+
+lemma palindromeLevelCost_gt (t d : ℕ) (h : d < t) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) : palindromeLevelCost t d e ω = 0 := by
+  classical
+  induction d generalizing ι with
+  | zero => rfl
+  | succ d ih =>
+    rw [palindromeLevelCost,ite_eq_right (by omega : t ≠ d+1),ih (by omega),ih (by omega),Nat.zero_add]
+
+
+theorem palindrome_level_partial_mgf (r s u : ℕ) (hu : u < r)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+r)) (X : SwitchIndex (s+r) → Bool)
+    (lo : Card r × SwitchIndex s → Bool) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(palindromeLevelCost (s+u+1) (s+r) e (assembleBits r s lo hi,X))) ≤
+      Real.exp (Fintype.card ι * cycleCoefficient u J q) := by
+  classical
+  induction r generalizing u ι with
+  | zero => omega
+  | succ r ih =>
+    simp only [Nat.add_succ] at *
+    let χ := coinStepEquiv (s+r) X
+    let x := e.trans (headTailEquiv (s+r)).toEmbedding
+    let c := PairRouting.colors x χ.2
+    let hc := PairRouting.colors_compatible x χ.2
+    let e₀ := PairRouting.childEmbedding x c hc false
+    let e₁ := PairRouting.childEmbedding x c hc true
+    let e' := PairRouting.switchedEmbedding x χ.2
+    let lo₀ : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons false j.1,j.2)
+    let lo₁ : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons true j.1,j.2)
+    let Y₀ := fun hi : Card s × SwitchIndex r → Bool => assembleBits r s lo₀ hi
+    let Y₁ := fun hi : Card s × SwitchIndex r → Bool => assembleBits r s lo₁ hi
+    let N := fun hi : (Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool) =>
+      PairRouting.alternatingCycles e' (fun b => if b then palindromePerm (s+r) (Y₁ hi.2,χ.1.2)
+        else palindromePerm (s+r) (Y₀ hi.1,χ.1.1))
+    let L₀ := fun hi : Card s × SwitchIndex r → Bool =>
+      palindromeLevelCost (s+u+1) (s+r) e₀ (Y₀ hi,χ.1.1)
+    let L₁ := fun hi : Card s × SwitchIndex r → Bool =>
+      palindromeLevelCost (s+u+1) (s+r) e₁ (Y₁ hi,χ.1.2)
+    have hs (hi : (Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool))
+        (η : Card s × Card r → Bool) :
+        palindromeLevelCost (s+u+1) (s+r+1) e
+          (assembleBits (r+1) s lo ((highStepEquiv r s).symm (hi,η)),X) =
+          if u = r then N hi else L₀ hi.1 + L₁ hi.2 := by
+      rw [assembleBits_step]
+      conv_lhs => rw [←(coinStepEquiv (s+r)).symm_apply_apply X]
+      rw [palindromeLevelCost_step]
+      simp only [Nat.add_right_cancel_iff,Nat.add_left_cancel_iff]
+      dsimp only [χ,x,c,hc,e₀,e₁,e',lo₀,lo₁,Y₀,Y₁,N,L₀,L₁]
+      congr 2
+      funext b
+      cases b <;> rfl
+    rw [finiteMean_equiv (highStepEquiv r s),finiteMean_prod]
+    simp_rw [hs,finiteMean_const]
+    by_cases hur : u = r
+    · subst u
+      simp only [ite_true]
+      rw [finiteMean_prod,finiteMean_comm]
+      calc
+        _ ≤ finiteMean (fun _hi : Card s × SwitchIndex r → Bool =>
+          Real.exp (Fintype.card ι * cycleCoefficient r J q)) := by
+          apply finiteMean_mono
+          intro hi
+          have hn := node_partial_mgf r s e'
+            (butterflyPerm (s+r) (decodeButterfly (s+r) χ.1.1))
+            (butterflyPerm (s+r) (decodeButterfly (s+r) χ.1.2))
+            (butterflyPerm (s+r) (decodeButterfly (s+r) (Y₁ hi))) lo₀ J hJ q hq
+          exact hn
+        _ = _ := finiteMean_const _
+    · simp only [ite_eq_right hur,pow_add]
+      rw [finiteMean_prod_mul (fun hi => q ^ L₀ hi) (fun hi => q ^ L₁ hi)]
+      have hu' : u < r := by omega
+      have h₀ := ih u hu' e₀ χ.1.1 lo₀
+      have h₁ := ih u hu' e₁ χ.1.2 lo₁
+      have hp := mul_le_mul h₀ h₁ (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
+        (Real.exp_nonneg _)
+      refine hp.trans_eq ?_
+      rw [←Real.exp_add]
+      congr 1
+      have hcount := PairRouting.color_card_add c
+      have hcount' : (Fintype.card {i : ι // c i = false} : ℝ) +
+          Fintype.card {i : ι // c i = true} = Fintype.card ι := by exact_mod_cast hcount
+      change (Fintype.card {i : ι // c i = false} : ℝ) * cycleCoefficient u J q +
+          Fintype.card {i : ι // c i = true} * cycleCoefficient u J q = _
+      rw [←add_mul,hcount']
+
+
+def switchHeight : (d : ℕ) → SwitchIndex d → ℕ
+  | 0, i => nomatch i
+  | d+1, Sum.inl _ => d+1
+  | d+1, Sum.inr (_,i) => switchHeight d i
+
+lemma switchHeight_le (d : ℕ) (i : SwitchIndex d) : switchHeight d i ≤ d := by
+  induction d with
+  | zero => exact Empty.elim i
+  | succ d ih =>
+    rcases i with y | ⟨b,i⟩
+    · exact le_refl _
+    · exact (ih i).trans (Nat.le_succ d)
+
+def AgreeThrough (d s : ℕ) (Y Y' : SwitchIndex d → Bool) : Prop :=
+  ∀ i, switchHeight d i ≤ s → Y i = Y' i
+
+def LayerAdapted (d s : ℕ) {β : Type*} (f : (SwitchIndex d → Bool) → β) : Prop :=
+  ∀ Y Y', AgreeThrough d s Y Y' → f Y = f Y'
+
+lemma agreeThrough_mono {d s t : ℕ} (h : s ≤ t) {Y Y' : SwitchIndex d → Bool}
+    (hh : AgreeThrough d t Y Y') : AgreeThrough d s Y Y' :=
+  fun i hi => hh i (hi.trans h)
+
+lemma layerAdapted_mono {d s t : ℕ} {β : Type*} {f : (SwitchIndex d → Bool) → β}
+    (hf : LayerAdapted d s f) (h : s ≤ t) : LayerAdapted d t f :=
+  fun Y Y' hh => hf Y Y' (agreeThrough_mono h hh)
+
+lemma layerAdapted_comp {d s : ℕ} {α β : Type*} {f : (SwitchIndex d → Bool) → α}
+    (hf : LayerAdapted d s f) (g : α → β) : LayerAdapted d s (g ∘ f) :=
+  fun Y Y' h => congrArg g (hf Y Y' h)
+
+lemma layerAdapted_sum {d s : ℕ} {ι : Type*} (A : Finset ι)
+    (f : ι → (SwitchIndex d → Bool) → ℕ)
+    (hf : ∀ i ∈ A, LayerAdapted d s (f i)) :
+    LayerAdapted d s (fun Y => ∑ i ∈ A, f i Y) := by
+  intro Y Y' h
+  exact Finset.sum_congr rfl (fun i hi => hf i hi Y Y' h)
+
+lemma assembleBits_agree (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
+    (hi hi' : Card s × SwitchIndex r → Bool) :
+    AgreeThrough (s+r) s (assembleBits r s lo hi) (assembleBits r s lo hi') := by
+  induction r with
+  | zero => intro i _; rfl
+  | succ r ih =>
+    intro i h
+    rcases i with y | ⟨b,i⟩
+    · have : s+r+1 ≤ s := h
+      omega
+    · exact ih (fun j => lo (Fin.cons b j.1,j.2))
+        (fun j => hi (j.1,Sum.inr (b,j.2))) (fun j => hi' (j.1,Sum.inr (b,j.2))) i h
+
+lemma palindromeLevelCost_adapted (t d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) :
+    LayerAdapted d (t-1) (fun Y => palindromeLevelCost t d e (Y,X)) := by
+  classical
+  induction d generalizing ι with
+  | zero => intro Y Y' _; rfl
+  | succ d ih =>
+    intro Y Y' h
+    change palindromeLevelCost t (d+1) e (Y,X) = palindromeLevelCost t (d+1) e (Y',X)
+    conv_lhs => rw [←(coinStepEquiv d).symm_apply_apply Y,←(coinStepEquiv d).symm_apply_apply X]
+    conv_rhs => rw [←(coinStepEquiv d).symm_apply_apply Y',←(coinStepEquiv d).symm_apply_apply X]
+    rw [palindromeLevelCost_step,palindromeLevelCost_step]
+    by_cases ht : t = d+1
+    · subst t
+      rw [ite_eq_left rfl,ite_eq_left rfl]
+      have he₀ : (coinStepEquiv d Y).1.1 = (coinStepEquiv d Y').1.1 := by
+        funext i
+        exact h (Sum.inr (false,i)) (by simpa only [switchHeight,Nat.add_sub_cancel] using switchHeight_le d i)
+      have he₁ : (coinStepEquiv d Y).1.2 = (coinStepEquiv d Y').1.2 := by
+        funext i
+        exact h (Sum.inr (true,i)) (by simpa only [switchHeight,Nat.add_sub_cancel] using switchHeight_le d i)
+      rw [he₀,he₁]
+    · rw [ite_eq_right ht,ite_eq_right ht]
+      congr 1
+      · apply ih
+        intro i hi
+        exact h (Sum.inr (false,i)) hi
+      · apply ih
+        intro i hi
+        exact h (Sum.inr (true,i)) hi
+
+lemma adapted_mul_mean_le (r s : ℕ) (F G : (SwitchIndex (s+r) → Bool) → ℝ)
+    (hF : ∀ Y, 0 ≤ F Y) (hA : LayerAdapted (s+r) s F) (B : ℝ)
+    (hG : ∀ lo : Card r × SwitchIndex s → Bool,
+      finiteMean (fun hi : Card s × SwitchIndex r → Bool => G (assembleBits r s lo hi)) ≤ B) :
+    finiteMean (fun Y => F Y*G Y) ≤ finiteMean F * B := by
+  rw [finiteMean_split r s,finiteMean_split r s F,←finiteMean_mul_const]
+  apply finiteMean_mono
+  intro lo
+  let H : Card s × SwitchIndex r → Bool := fun _ => false
+  have hh (hi : Card s × SwitchIndex r → Bool) :
+      F (assembleBits r s lo hi) = F (assembleBits r s lo H) :=
+    hA _ _ (assembleBits_agree r s lo hi H)
+  simp_rw [hh,finiteMean_const]
+  rw [show (fun hi => F (assembleBits r s lo H) * G (assembleBits r s lo hi)) =
+      (fun hi => G (assembleBits r s lo hi) * F (assembleBits r s lo H)) by
+        funext hi; exact mul_comm _ _,finiteMean_mul_const]
+  simpa only [mul_comm] using mul_le_mul_of_nonneg_left (hG lo) (hF _)
+
+
+theorem cycleCoefficient_decay (l r : ℕ) (hr : (l:ℝ)/2 ≤ r)
+    (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ (l:ℝ)/250) :
+    cycleCoefficient r ⌈Real.exp ((l:ℝ)/32)⌉₊ q ≤
+      68 * Real.exp (-(l:ℝ)/64) := by
+  let E := Real.exp ((l:ℝ)/32)
+  let J := ⌈E⌉₊
+  have hl : (0:ℝ) ≤ l := Nat.cast_nonneg _
+  have hE : 1 ≤ E := Real.one_le_exp_iff.mpr (by positivity)
+  have hEpos : 0 < E := Real.exp_pos _
+  have hEJ : E ≤ (J:ℝ) := Nat.le_ceil E
+  have hJ : (1:ℝ) ≤ J := hE.trans hEJ
+  have hJpos : (0:ℝ) < J := lt_of_lt_of_le (by norm_num) hJ
+  have hJE : (J:ℝ) ≤ 2*E := by
+    have h := Nat.ceil_lt_add_one (le_of_lt hEpos)
+    change (J:ℝ) < E+1 at h
+    linarith
+  have hsqJ : Real.sqrt J ≤ (J:ℝ) := by
+    apply Real.sqrt_le_iff.mpr
+    constructor
+    · positivity
+    · nlinarith
+  have hqp : 0 < q := lt_of_lt_of_le (by norm_num) hq
+  have hqE : q ≤ Real.exp ((l:ℝ)/250) := by
+    rw [←Real.exp_log hqp]
+    exact Real.exp_le_exp.mpr hlog
+  have hln : (1/2:ℝ) ≤ Real.log 2 := by
+    have hh := Real.one_sub_inv_le_log_of_pos (by norm_num : (0:ℝ)<2)
+    norm_num at hh ⊢
+    exact hh
+  have hden : Real.exp ((l:ℝ)/8) ≤ Real.sqrt ((2:ℝ)^r) := by
+    have he : (2:ℝ)^r = Real.exp ((r:ℝ)*Real.log 2) := by
+      rw [Real.exp_nat_mul,Real.exp_log (by norm_num)]
+    rw [he,Real.sqrt_eq_rpow,Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp]
+    apply Real.exp_le_exp.mpr
+    have hrn : (0:ℝ) ≤ r := Nat.cast_nonneg _
+    nlinarith
+  have hlog' : Real.log q ≤ (l:ℝ) := by linarith
+  have h₁ : Real.log q / (J:ℝ) ≤ 64*Real.exp (-(l:ℝ)/64) := by
+    calc
+      _ ≤ (l:ℝ)/E := div_le_div₀ (by positivity) hlog' hEpos hEJ
+      _ ≤ (64*Real.exp ((l:ℝ)/64))/E := by
+        apply div_le_div_of_nonneg_right _ (le_of_lt hEpos)
+        have hh := Real.add_one_le_exp ((l:ℝ)/64)
+        linarith
+      _ = _ := by
+        dsimp [E]
+        rw [mul_div_assoc,←Real.exp_sub]
+        congr 2
+        ring
+  have h₂ : 2*q*Real.sqrt J / Real.sqrt ((2:ℝ)^r) ≤
+      4*Real.exp (-(l:ℝ)/64) := by
+    calc
+      _ ≤ (4*Real.exp ((l:ℝ)/250)*E)/Real.exp ((l:ℝ)/8) := by
+        apply div_le_div₀ (by positivity) _ (Real.exp_pos _) hden
+        have hs := hsqJ.trans hJE
+        have hm := mul_le_mul hqE hs (Real.sqrt_nonneg _) (Real.exp_nonneg _)
+        nlinarith
+      _ = 4*Real.exp ((l:ℝ)/250+(l:ℝ)/32-(l:ℝ)/8) := by
+        dsimp [E]
+        rw [mul_assoc,←Real.exp_add,mul_div_assoc,←Real.exp_sub]
+      _ ≤ _ := by
+        apply mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr _) (by norm_num)
+        linarith
+  change Real.log q / (J:ℝ) + 2*q*Real.sqrt J / Real.sqrt ((2:ℝ)^r) ≤ _
+  linarith
+
+lemma cycleCutoff_pos (l : ℕ) : 0 < ⌈Real.exp ((l:ℝ)/32)⌉₊ := by
+  have h := Nat.le_ceil (Real.exp ((l:ℝ)/32))
+  have he := Real.exp_pos ((l:ℝ)/32)
+  have : (0:ℝ) < ⌈Real.exp ((l:ℝ)/32)⌉₊ := lt_of_lt_of_le he h
+  exact_mod_cast this
+
+
+lemma pow_sum_le_mean_powers {ι : Type*} [Fintype ι] [Nonempty ι]
+    (z : ι → ℕ) (q : ℝ) (hq : 0 < q) :
+    q^(∑ i, z i) ≤ finiteMean (fun i => q^(Fintype.card ι*z i)) := by
+  have hn : (Fintype.card ι:ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
+  have hm : finiteMean (fun i => (Fintype.card ι*z i:ℕ)*Real.log q) =
+      (∑ i, z i:ℕ)*Real.log q := by
+    simp only [finiteMean,Nat.cast_mul,Nat.cast_sum,←Finset.mul_sum,←Finset.sum_mul]
+    field_simp
+  have he := exp_finiteMean_le (fun i => (Fintype.card ι*z i:ℕ)*Real.log q)
+  rw [hm,Real.exp_nat_mul,Real.exp_log hq] at he
+  simpa only [Real.exp_nat_mul,Real.exp_log hq] using he
+
+noncomputable def palindromeSlabCost (l d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) : ℕ :=
+  ∑ i : Fin l, palindromeLevelCost (l+i.1+1) d e ω
+
+lemma palindromeSlabCost_adapted (l d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) :
+    LayerAdapted d (2*l) (fun Y => palindromeSlabCost l d e (Y,X)) := by
+  apply layerAdapted_sum
+  intro i _
+  exact layerAdapted_mono (palindromeLevelCost_adapted (l+i.1+1) d e X) (by omega)
+
+theorem palindrome_slab_partial_mgf (r s l : ℕ) (hl : 0 < l) (hs : 2*s ≤ l)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+r)) (X : SwitchIndex (s+r) → Bool)
+    (lo : Card r × SwitchIndex s → Bool) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(palindromeSlabCost l (s+r) e (assembleBits r s lo hi,X))) ≤
+      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
+  classical
+  let : NeZero l := ⟨ne_of_gt hl⟩
+  have hqp : 0 < q := lt_of_lt_of_le (by norm_num) hq
+  calc
+    _ ≤ finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      finiteMean (fun i : Fin l => q^(l*palindromeLevelCost (l+i.1+1) (s+r) e
+        (assembleBits r s lo hi,X)))) := by
+      apply finiteMean_mono
+      intro hi
+      simpa only [palindromeSlabCost,Fintype.card_fin] using
+        pow_sum_le_mean_powers (fun i : Fin l => palindromeLevelCost (l+i.1+1) (s+r) e
+          (assembleBits r s lo hi,X)) q hqp
+    _ = finiteMean (fun i : Fin l => finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(l*palindromeLevelCost (l+i.1+1) (s+r) e (assembleBits r s lo hi,X)))) := finiteMean_comm _
+    _ ≤ finiteMean (fun _i : Fin l =>
+      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64)))) := by
+      apply finiteMean_mono
+      intro i
+      by_cases ht : l+i.1+1 ≤ s+r
+      · have hs' : s ≤ l+i.1 := by omega
+        have he : l+i.1+1 = s+(l+i.1-s)+1 := by omega
+        have hu : l+i.1-s < r := by omega
+        have hm := palindrome_level_partial_mgf r s (l+i.1-s) hu e X lo
+          ⌈Real.exp ((l:ℝ)/32)⌉₊ (cycleCutoff_pos l) (q^l) (one_le_pow₀ hq)
+        have hc := cycleCoefficient_decay l (l+i.1-s) (by
+          rw [Nat.cast_sub hs',Nat.cast_add]
+          have hsr : (2:ℝ)*s ≤ l := by exact_mod_cast hs
+          have hi : (0:ℝ) ≤ i.1 := Nat.cast_nonneg _
+          linarith) (q^l) (one_le_pow₀ hq) (by
+            rw [Real.log_pow]
+            nlinarith [mul_le_mul_of_nonneg_left hlog (Nat.cast_nonneg l)])
+        simp_rw [he,pow_mul]
+        exact hm.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left hc (Nat.cast_nonneg _)))
+      · have hz := palindromeLevelCost_gt (l+i.1+1) (s+r) (by omega) e
+        simp only [hz,Nat.mul_zero,pow_zero,finiteMean_const]
+        exact Real.one_le_exp_iff.mpr (by positivity)
+    _ = _ := finiteMean_const _
+
+
+lemma palindromeSlabCost_gt (l d : ℕ) (h : d ≤ l) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) : palindromeSlabCost l d e ω = 0 := by
+  apply Finset.sum_eq_zero
+  intro i _
+  exact palindromeLevelCost_gt _ _ (by omega) _ _
+
+lemma palindrome_slab_adapted_mul_le (d s l : ℕ) (hsd : s ≤ d) (hl : 0 < l)
+    (hs : 2*s ≤ l) {ι : Type*} [Fintype ι] (e : ι ↪ Card d)
+    (X : SwitchIndex d → Bool) (F : (SwitchIndex d → Bool) → ℝ)
+    (hF : ∀ Y, 0 ≤ F Y) (hA : LayerAdapted d s F) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun Y => F Y*q^(palindromeSlabCost l d e (Y,X))) ≤
+      finiteMean F * Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
+  obtain ⟨r,rfl⟩ := Nat.exists_eq_add_of_le hsd
+  exact adapted_mul_mean_le r s F _ hF hA _
+    (fun lo => palindrome_slab_partial_mgf r s l hl hs e X lo q hq hlog)
+
+def familyEnd (L n : ℕ) : ℕ := if n = 0 then 0 else 2*(L*4^(n-1))
+
+lemma familyEnd_two (L n : ℕ) : 2*familyEnd L n ≤ L*4^n := by
+  cases n with
+  | zero => simp [familyEnd]
+  | succ n => simp only [familyEnd,Nat.succ_ne_zero,ite_false,Nat.add_one_sub_one,pow_succ]; nlinarith
+
+lemma familyEnd_ge (L n j : ℕ) (hj : j < n) : 2*(L*4^j) ≤ familyEnd L n := by
+  cases n with
+  | zero => omega
+  | succ n =>
+    simp only [familyEnd,Nat.succ_ne_zero,ite_false,Nat.add_one_sub_one]
+    apply Nat.mul_le_mul_left
+    exact Nat.mul_le_mul_left L (Nat.pow_le_pow_right (by decide) (by omega : j ≤ n))
+
+noncomputable def palindromeFamilyCost (L n d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) : ℕ :=
+  ∑ j ∈ Finset.range n, palindromeSlabCost (L*4^j) d e ω
+
+lemma palindromeFamilyCost_adapted (L n d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) :
+    LayerAdapted d (familyEnd L n) (fun Y => palindromeFamilyCost L n d e (Y,X)) := by
+  apply layerAdapted_sum
+  intro j hj
+  exact layerAdapted_mono (palindromeSlabCost_adapted (L*4^j) d e X)
+    (familyEnd_ge L n j (Finset.mem_range.mp hj))
+
+end Thorp
+
+end ThorpNine.SparseSaving
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/SparseSaving/PriorPaths.lean b/lean/OAI/Probability/ThorpRouting/SparseSaving/PriorPaths.lean
index eaa8b110a2355ffc1afa933c03e181e46d984000..6abafd0c98c6286f1ed6ca30af3156ccd2cf5ae7 100644
--- a/lean/OAI/Probability/ThorpRouting/SparseSaving/PriorPaths.lean
+++ b/lean/OAI/Probability/ThorpRouting/SparseSaving/PriorPaths.lean
@@ -1,882 +1,882 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.SparseSaving.AssignmentSlots
-
-namespace OAI
-
-namespace ThorpNine.SparseSaving
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-namespace RoutingNetwork
-variable {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι] [DecidableEq ι]
-  {d : ℕ} (R : RoutingNetwork α ι d)
-lemma mem_exposedPaths (b : ι → Bool) (V : Finset (α)) (s : ι) :
-    s ∈ exposedPaths R b V ↔ (R.users b s).1 ∈ V ∨ (R.users b s).2 ∈ V := by
-  classical
-  simp only [exposedPaths,Finset.mem_biUnion,R.mem_path]
-  constructor
-  · rintro ⟨x,hx,h | h⟩
-    · exact Or.inl (h ▸ hx)
-    · exact Or.inr (h ▸ hx)
-  · rintro (h | h)
-    · exact ⟨_,h,Or.inl rfl⟩
-    · exact ⟨_,h,Or.inr rfl⟩
-
-lemma terminalCharge_sum (b : ι → Bool) (z : α) (V : Finset (α))
-    (hz : z ∉ V) :
-    (terminalCharge R b z V : ℝ) = ∑ s : ι,
-      ((if (R.users b s).1 = z ∧ (R.users b s).2 ∈ V then (1:ℝ) else 0) +
-      (if (R.users b s).2 = z ∧ (R.users b s).1 ∈ V then (1:ℝ) else 0)) := by
-  classical
-  have he : R.path b z ∩ exposedPaths R b V =
-      Finset.univ.filter (fun s =>
-        ((R.users b s).1 = z ∧ (R.users b s).2 ∈ V) ∨
-        ((R.users b s).2 = z ∧ (R.users b s).1 ∈ V)) := by
-    ext s
-    simp only [Finset.mem_inter,R.mem_path,mem_exposedPaths,
-      Finset.mem_filter,Finset.mem_univ,true_and]
-    by_cases h₁ : (R.users b s).1 = z <;>
-      by_cases h₂ : (R.users b s).2 = z <;> simp [h₁,h₂,hz]
-  rw [terminalCharge,he]
-  simp only [← Finset.sum_boole]
-  apply Finset.sum_congr rfl
-  intro s _
-  by_cases h₁ : (R.users b s).1 = z <;>
-    by_cases h₂ : (R.users b s).2 = z <;> simp [h₁,h₂,hz]
-
-noncomputable def priorPaths {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) : Finset (α) :=
-  Finset.univ.filter (fun x => ∃ i : Fin A, o x = some i ∧ i.val < t)
-
-omit [DecidableEq α] in
-@[simp] lemma mem_priorPaths {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) (x : α) :
-    x ∈ priorPaths o t ↔ ∃ i : Fin A, o x = some i ∧ i.val < t := by
-  classical
-  simp [priorPaths]
-
-lemma terminalCharge_prior_sum {A : ℕ} (o : α → Option (Fin A))
-    (b : ι → Bool) (z : Terminal o) (i : Fin A) :
-    (terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) =
-      ∑ s : ι,
-        ((if (z i).val = (R.users b s).1 ∧ (R.users b s).2 ∈ priorPaths o (i.val+1)
-          then (1:ℝ) else 0) +
-        (if (z i).val = (R.users b s).2 ∧ (R.users b s).1 ∈ priorPaths o (i.val+1)
-          then (1:ℝ) else 0)) := by
-  classical
-  rw [terminalCharge_sum _ _ _ _ (Finset.notMem_erase _ _)]
-  apply Finset.sum_congr rfl
-  intro s _
-  have hn := R.users_ne b s
-  simp only [Finset.mem_erase]
-  by_cases h₁ : (z i).val = (R.users b s).1 <;>
-    by_cases h₂ : (z i).val = (R.users b s).2 <;> simp_all [eq_comm]
-
-lemma rotationMeeting_eq_slot_sum {A : ℕ} (o : α → Option (Fin A))
-    (z : Terminal o) (u v : α) :
-    rotationMeeting o z u v = ∑ i : Fin A,
-      ((if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) +
-      (if (z i).val = v ∧ u ∈ priorPaths o (i.val+1) then (1:ℝ) else 0)) := by
-  classical
-  have only_owner (u : α) (a : Fin A) (hou : o u = some a)
-      (i : Fin A) (hia : i ≠ a) : (z i).val ≠ u := by
-    intro h
-    have hi := (z i).property
-    rw [h,hou] at hi
-    exact hia (Option.some.inj hi).symm
-  have sum_one (u v : α) (a : Fin A) (hou : o u = some a) :
-      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) =
-      if (z a).val = u ∧ v ∈ priorPaths o (a.val+1) then 1 else 0 := by
-    apply Finset.sum_eq_single a
-    · intro i _ hia
-      simp [only_owner u a hou i hia]
-    · simp
-  have sum_none (u v : α) (hou : o u = none) :
-      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) = 0 := by
-    apply Finset.sum_eq_zero
-    intro i _
-    have hn : (z i).val ≠ u := by
-      intro h
-      have hi := (z i).property
-      rw [h,hou] at hi
-      contradiction
-    simp [hn]
-  rw [Finset.sum_add_distrib]
-  cases hou : o u with
-  | none =>
-    rw [sum_none u v hou]
-    simp [rotationMeeting,hou,mem_priorPaths]
-  | some a =>
-    rw [sum_one u v a hou]
-    cases hov : o v with
-    | none =>
-      rw [sum_none v u hov]
-      simp [rotationMeeting,hou,hov,mem_priorPaths]
-    | some b =>
-      rw [sum_one v u b hov]
-      simp only [rotationMeeting,hou,hov,mem_priorPaths,Option.some.injEq]
-      rcases lt_trichotomy a b with hab | rfl | hba
-      · have hn : ¬ b.val < a.val+1 := by omega
-        have hy : a.val < b.val+1 := by omega
-        simp [hab,hn,hy]
-      · simp
-      · have hn : ¬ a.val < b.val+1 := by omega
-        have hy : b.val < a.val+1 := by omega
-        simp [hba,not_lt_of_ge (le_of_lt hba),hn,hy]
-
-lemma rotationCharge_eq_terminal_sum {A : ℕ} (o : α → Option (Fin A))
-    (b : ι → Bool) (z : Terminal o) :
-    rotationCharge R o b z = ∑ i : Fin A,
-      (terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) := by
-  simp only [rotationCharge,rotationMeeting_eq_slot_sum,terminalCharge_prior_sum]
-  exact Finset.sum_comm
-
-omit [DecidableEq α] in
-@[simp] lemma priorPaths_zero {A : ℕ} (o : α → Option (Fin A)) : priorPaths o 0 = ∅ := by
-  classical
-  ext x
-  simp
-
-lemma priorPaths_succ {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) (ht : t < A) :
-    priorPaths o (t+1) = priorPaths o t ∪ cycleFiber o ⟨t,ht⟩ := by
-  classical
-  ext x
-  simp only [mem_priorPaths,Finset.mem_union,mem_cycleFiber]
-  constructor
-  · rintro ⟨i,hi,hit⟩
-    by_cases h : i.val < t
-    · exact Or.inl ⟨i,hi,h⟩
-    · have he : i = ⟨t,ht⟩ := Fin.ext (by change i.val = t; omega)
-      exact Or.inr (he ▸ hi)
-  · rintro (⟨i,hi,hit⟩ | hi)
-    · exact ⟨i,hi,by omega⟩
-    · exact ⟨⟨t,ht⟩,hi,by change t < t+1; omega⟩
-
-omit [DecidableEq α] in
-lemma priorPaths_disjoint {A : ℕ} (o : α → Option (Fin A)) (i : Fin A) :
-    Disjoint (priorPaths o i.val) (cycleFiber o i) := by
-  classical
-  apply Finset.disjoint_left.mpr
-  intro x hx hx'
-  obtain ⟨j,hj,hji⟩ := (mem_priorPaths _ _ _).mp hx
-  have he : j = i := Option.some.inj (hj.symm.trans ((mem_cycleFiber _ _ _).mp hx'))
-  subst j
-  exact (lt_irrefl _ hji)
-
-namespace QueryTree
-open Thorp.QueryTree
-
-lemma seekCycles_owned {A : ℕ} (S : Finset (α)) (r : Fin A → ℕ)
-    (o : α → Option (Fin A)) (z : Terminal o) (ω : ι → Bool)
-    (hS : ∀ i, cycleFiber o i ⊆ S)
-    (hO : ∀ i, orbitSet (R.perm ω) (z i).val = cycleFiber o i)
-    (hC : ∀ i, (cycleFiber o i).card = r i+1) :
-    let p := R.perm ω
-    let L := List.ofFn (fun i : Fin A => (p (z i).val,r i))
-    (seekCycles R S L ∅).succeeds ω = true ∧
-      ((seekCycles R S L ∅).staleCharge ∅ ω : ℝ) = rotationCharge R o (ω) z := by
-  classical
-  let b := ω
-  let p := R.perm b
-  let f (i : Fin A) := (p (z i).val,r i)
-  let c (i : Fin A) : ℝ := terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val)
-  have hc (i : Fin A) : Function.minimalPeriod p (z i).val - 1 = r i := by
-    have he := orbitSet_card p (z i).val
-    rw [hO,hC] at he
-    omega
-  have hs (t : ℕ) (ht : t ≤ A) :
-      (seekCycles R S ((List.ofFn f).drop t) (priorPaths o t)).succeeds ω = true ∧
-        ((seekCycles R S ((List.ofFn f).drop t) (priorPaths o t)).staleCharge (exposedPaths R b (priorPaths o t)) ω : ℝ) =
-          ((List.ofFn c).drop t).sum := by
-    induction ht using Nat.decreasingInduction with
-    | self => simp only [List.length_ofFn,le_refl,List.drop_eq_nil_of_le,List.sum_nil,seekCycles,Thorp.QueryTree.succeeds,staleCharge,Nat.cast_zero,and_self]
-    | of_succ t ht ih =>
-      have htf : t < (List.ofFn f).length := by simpa using ht
-      have htc : t < (List.ofFn c).length := by simpa using ht
-      rw [List.drop_eq_getElem_cons htf,List.drop_eq_getElem_cons htc]
-      simp only [List.getElem_ofFn,List.sum_cons,seekCycles]
-      have h := seekCycle_orbit R S (priorPaths o t) (z ⟨t,ht⟩).val
-        (fun V => seekCycles R S ((List.ofFn f).drop (t+1)) V) ω
-        (by rw [hO]; exact hS _) (by rw [hO]; exact priorPaths_disjoint o ⟨t,ht⟩)
-      dsimp only at h
-      rw [hc,hO,←priorPaths_succ o t ht] at h
-      dsimp only [f]
-      constructor
-      · exact h.1.trans ih.1
-      · rw [h.2,Nat.cast_add,ih.2]
-        exact add_comm _ _
-  have h := hs 0 (Nat.zero_le _)
-  simp only [List.drop_zero,priorPaths_zero,exposedPaths_empty,List.sum_ofFn] at h
-  dsimp only
-  exact ⟨h.1,h.2.trans (rotationCharge_eq_terminal_sum R o b z).symm⟩
-
-end QueryTree
-section Collections
-variable {A : ℕ}
-
-def ValidCycleSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ)
-    (o : α → Option (Fin A)) : Prop :=
-  (∀ i, (cycleFiber o i).card = r i + 1) ∧
-  (∀ i, cycleFiber o i ⊆ S) ∧
-  (∀ i x, o x = some i → orbitSet p x = cycleFiber o i)
-
-abbrev CycleSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
-  {o : α → Option (Fin A) // ValidCycleSlots p S r o}
-
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
-    Fintype (CycleSlots p S r) := Fintype.ofFinite _
-
-lemma CycleSlots.terminal_nonempty {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (c : CycleSlots p S r) (i : Fin A) : Nonempty {x : α // c.val x = some i} := by
-  have hn : (cycleFiber c.val i).Nonempty := Finset.card_pos.mp (by rw [c.property.1]; omega)
-  obtain ⟨x,hx⟩ := hn
-  exact ⟨x,(mem_cycleFiber _ _ _).mp hx⟩
-
-omit [DecidableEq α] in
-lemma owner_ext {o o' : α → Option (Fin A)}
-    (h : ∀ i, cycleFiber o i = cycleFiber o' i) : o = o' := by
-  funext x
-  have he (i : Fin A) : o x = some i ↔ o' x = some i := by
-    simpa only [mem_cycleFiber] using (Finset.ext_iff.mp (h i) x)
-  cases ho : o x with
-  | some i => exact ((he i).mp ho).symm
-  | none =>
-    cases ho' : o' x with
-    | none => rfl
-    | some i => have hn := (he i).mpr ho'; rw [ho] at hn; contradiction
-
-abbrev RootedSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
-  Σ c : CycleSlots p S r, Terminal c.val
-
-noncomputable def rootedStart {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (c : RootedSlots p S r) : Fin A → S := fun i =>
-  ⟨p (c.2 i).val,c.1.property.2.1 i (by
-    rw [←c.1.property.2.2 i (c.2 i).val (c.2 i).property]
-    exact (mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.apply_right (Equiv.Perm.SameCycle.refl p _)))⟩
-
-lemma rootedStart_injective {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ} :
-    Function.Injective (rootedStart (p:=p) (S:=S) (r:=r)) := by
-  rintro ⟨c,z⟩ ⟨c',z'⟩ h
-  have hz (i : Fin A) : (z i).val = (z' i).val :=
-    p.injective (congrArg Subtype.val (congrFun h i))
-  have hc : c = c' := Subtype.ext (owner_ext (fun i => by
-    rw [←c.property.2.2 i (z i).val (z i).property,
-      ←c'.property.2.2 i (z' i).val (z' i).property,hz]))
-  subst c'
-  have hzz : z = z' := funext (fun i => Subtype.ext (hz i))
-  subst z'
-  rfl
-
-lemma rooted_weight_sum_le (S : Finset (α)) (r : Fin A → ℕ) (ω : ι → Bool) :
-    (∑ c : RootedSlots (R.perm ω) S r,
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2)) ≤
-    ∑ a : Fin A → S,
-      if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-        ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-      else 0 := by
-  classical
-  let p := R.perm ω
-  let g (a : Fin A → S) : ℝ :=
-    if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-    else 0
-  have he (c : RootedSlots p S r) :
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2) =
-      g (rootedStart c) := by
-    have hs := QueryTree.seekCycles_owned R S r c.1.val c.2 ω c.1.property.2.1
-      (fun i => c.1.property.2.2 i (c.2 i).val (c.2 i).property) c.1.property.1
-    dsimp only at hs
-    unfold g
-    rw [ite_eq_left (show (QueryTree.seekCycles R S (List.ofFn (fun i => ((rootedStart c i).val,r i))) ∅).succeeds ω = true from hs.1)]
-    rw [← hs.2,neg_mul,Real.exp_neg,mul_comm (Real.log 2),Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
-    rfl
-  simp_rw [he]
-  exact sum_comp_injective_le rootedStart rootedStart_injective g (fun a => by dsimp [g]; split_ifs <;> positivity)
-
-lemma collection_rotation_lower (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r)
-    (ω : ι → Bool) (c : CycleSlots (R.perm ω) S r) :
-    collectionWeight d r ≤ ∑ z : Terminal c.val,
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z) := by
-  classical
-  let (i : Fin A) : Nonempty {x : α // c.val x = some i} := c.terminal_nonempty i
-  have hm : Monotone (fun i => (cycleFiber c.val i).card) := by
-    intro i j hij
-    dsimp only
-    rw [c.property.1,c.property.1]
-    exact Nat.add_le_add_right (hr hij) 1
-  have hb := rotation_exp_bound R hm (ω)
-  have hcard : (Fintype.card (Terminal c.val) : ℝ) = ∏ i : Fin A, (r i+1 : ℝ) := by
-    rw [Fintype.card_pi,Nat.cast_prod]
-    apply Finset.prod_congr rfl
-    intro i _
-    rw [card_cycleFiber,c.property.1,Nat.cast_add,Nat.cast_one]
-  have hp : (0:ℝ) < Fintype.card (Terminal c.val) := Nat.cast_pos.mpr Fintype.card_pos
-  have hl : rotationLower d r ≤ finiteMean (fun z : Terminal c.val => Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z)) := by
-    simpa only [rotationLower,c.property.1,Nat.cast_add,Nat.cast_one] using hb
-  have hh := (le_div_iff₀ hp).mp hl
-  rw [hcard] at hh
-  unfold collectionWeight
-  rw [← Finset.mul_sum]
-  calc
-    _ = ((2:ℝ)^d)^A * (rotationLower d r * ∏ i : Fin A, (r i+1 : ℝ)) := by ring
-    _ ≤ _ := mul_le_mul_of_nonneg_left hh (by positivity)
-
-lemma cycleSlots_weighted_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : ι → Bool =>
-      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) *
-      collectionWeight d r ≤ (S.card : ℝ)^A := by
-  classical
-  let g (a : Fin A → S) (ω : ι → Bool) : ℝ :=
-    if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-    else 0
-  have hω (ω : ι → Bool) :
-      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ) * collectionWeight d r ≤
-        ∑ a : Fin A → S, g a ω := by
-    calc
-      _ = ∑ _c : CycleSlots (R.perm ω) S r, collectionWeight d r := by simp
-      _ ≤ ∑ c : CycleSlots (R.perm ω) S r,
-          ∑ z : Terminal c.val, ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z) :=
-        Finset.sum_le_sum (fun c _ => collection_rotation_lower R S r hr ω c)
-      _ = ∑ c : RootedSlots (R.perm ω) S r,
-          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2) :=
-        (Fintype.sum_sigma (fun c : RootedSlots (R.perm ω) S r =>
-          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2))).symm
-      _ ≤ _ := rooted_weight_sum_le R S r ω
-  have hb := finiteMean_mono hω
-  rw [finiteMean_mul_const,finiteMean_sum] at hb
-  calc
-    _ ≤ ∑ a : Fin A → S, finiteMean (g a) := hb
-    _ ≤ ∑ _a : Fin A → S, (1:ℝ) := by
-      apply Finset.sum_le_sum
-      intro a _
-      have hh := QueryTree.cycles_weighted_success R S (List.ofFn (fun i : Fin A => ((a i).val,r i)))
-      simpa only [List.length_ofFn] using hh
-    _ = _ := by simp
-
-lemma cycleSlots_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : ι → Bool =>
-      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
-        (S.card : ℝ)^A / collectionWeight d r := by
-  exact (le_div_iff₀ (collectionWeight_pos d r)).mpr (cycleSlots_weighted_moment R S r hr)
-
-lemma butterfly_cycleSlots_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : ι → Bool =>
-      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
-        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
-  have h := cycleSlots_moment R S r hr
-  rw [collectionWeight_eq_sqrt,div_mul_eq_div_mul_one_div] at h
-  rw [div_pow]
-  simpa only [one_div,Finset.prod_inv_distrib] using h
-
-end Collections
-
-section CycleCounting
-variable {A : ℕ}
-
-abbrev SelectedCycle (p : Equiv.Perm (α)) (S : Finset (α)) :=
-  {C : Finset (α) // (∃ x, C = orbitSet p x) ∧ C ⊆ S}
-
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) :
-    Fintype (SelectedCycle p S) := Fintype.ofFinite _
-
-lemma SelectedCycle.nonempty {p : Equiv.Perm (α)} {S : Finset (α)}
-    (C : SelectedCycle p S) : C.val.Nonempty := by
-  obtain ⟨x,hx⟩ := C.property.1
-  rw [hx]
-  exact orbitSet_nonempty p x
-
-lemma SelectedCycle.orbit_eq {p : Equiv.Perm (α)} {S : Finset (α)}
-    (C : SelectedCycle p S) {x : α} (hx : x ∈ C.val) : orbitSet p x = C.val := by
-  obtain ⟨y,hy⟩ := C.property.1
-  rw [hy] at hx ⊢
-  exact (orbitSet_eq p ((mem_orbitSet _ _ _).mp hx)).symm
-
-lemma SelectedCycle.eq_of_mem {p : Equiv.Perm (α)} {S : Finset (α)}
-    (C D : SelectedCycle p S) {x : α} (hC : x ∈ C.val) (hD : x ∈ D.val) : C = D := by
-  apply Subtype.ext
-  exact (C.orbit_eq hC).symm.trans (D.orbit_eq hD)
-
-abbrev SlotAssignment (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
-  {f : Fin A → SelectedCycle p S // Function.Injective f ∧ ∀ i, (f i).val.card = r i+1}
-
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
-    Fintype (SlotAssignment p S r) := Fintype.ofFinite _
-
-noncomputable def slotsToAssignment {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (c : CycleSlots p S r) : SlotAssignment p S r := by
-  classical
-  let f (i : Fin A) : SelectedCycle p S := ⟨cycleFiber c.val i,by
-    obtain ⟨x,hx⟩ := c.terminal_nonempty i
-    exact ⟨⟨x,(c.property.2.2 i x hx).symm⟩,c.property.2.1 i⟩⟩
-  refine ⟨f,?_,fun i => c.property.1 i⟩
-  intro i j hij
-  have he : cycleFiber c.val i = cycleFiber c.val j := congrArg Subtype.val hij
-  obtain ⟨x,hx⟩ := c.terminal_nonempty i
-  have hxi : x ∈ cycleFiber c.val i := (mem_cycleFiber _ _ _).mpr hx
-  rw [he] at hxi
-  exact Option.some.inj (hx.symm.trans ((mem_cycleFiber _ _ _).mp hxi))
-
-noncomputable def assignmentOwner {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) (x : α) : Option (Fin A) := by
-  classical
-  exact if h : ∃ i, x ∈ (f.val i).val then some (Classical.choose h) else none
-
-lemma assignmentOwner_eq_some {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) (x : α) (i : Fin A) :
-    assignmentOwner f x = some i ↔ x ∈ (f.val i).val := by
-  classical
-  unfold assignmentOwner
-  split_ifs with h
-  · constructor
-    · intro hi
-      have hx := Classical.choose_spec h
-      rw [Option.some.inj hi] at hx
-      exact hx
-    · intro hx
-      have he := (f.val (Classical.choose h)).eq_of_mem (f.val i) (Classical.choose_spec h) hx
-      exact congrArg some (f.property.1 he)
-  · constructor
-    · intro hi; contradiction
-    · intro hx; exact (h ⟨i,hx⟩).elim
-
-lemma assignmentOwner_fiber {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) (i : Fin A) : cycleFiber (assignmentOwner f) i = (f.val i).val := by
-  ext x
-  simp only [mem_cycleFiber,assignmentOwner_eq_some]
-
-noncomputable def assignmentToSlots {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) : CycleSlots p S r :=
-  ⟨assignmentOwner f,by
-    refine ⟨?_,?_,?_⟩
-    · intro i
-      rw [assignmentOwner_fiber]
-      exact f.property.2 i
-    · intro i
-      rw [assignmentOwner_fiber]
-      exact (f.val i).property.2
-    · intro i x hx
-      rw [assignmentOwner_fiber]
-      exact (f.val i).orbit_eq ((assignmentOwner_eq_some f x i).mp hx)⟩
-
-noncomputable def cycleSlotsEquivAssignments (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
-    CycleSlots p S r ≃ SlotAssignment p S r where
-  toFun := slotsToAssignment
-  invFun := assignmentToSlots
-  left_inv c := by
-    apply Subtype.ext
-    apply owner_ext
-    intro i
-    rw [assignmentToSlots,assignmentOwner_fiber]
-    rfl
-  right_inv f := by
-    apply Subtype.ext
-    funext i
-    apply Subtype.ext
-    exact assignmentOwner_fiber f i
-
-end CycleCounting
-
-section GroupedCounting
-variable {A : ℕ}
-
-abbrev LengthClass (r : Fin A → ℕ) := {j : ℕ // j ∈ Finset.univ.image (fun i => r i + 1)}
-
-def slotClass (r : Fin A → ℕ) (i : Fin A) : LengthClass r :=
-  ⟨r i+1, Finset.mem_image.mpr ⟨i, Finset.mem_univ _, rfl⟩⟩
-
-abbrev SlotsOfLength (r : Fin A → ℕ) (j : LengthClass r) := {i : Fin A // slotClass r i = j}
-abbrev CyclesOfLength (p : Equiv.Perm (α)) (S : Finset (α)) (j : ℕ) :=
-  {C : SelectedCycle p S // C.val.card = j}
-
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (j : ℕ) :
-    Fintype (CyclesOfLength p S j) := Fintype.ofFinite _
-
-abbrev GroupedAssignments (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
-  ∀ j : LengthClass r, SlotsOfLength r j ↪ CyclesOfLength p S j.val
-
-noncomputable def assignmentToGroups {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) : GroupedAssignments p S r := fun _j =>
-  { toFun := fun i => ⟨f.val i.val, (f.property.2 i.val).trans (congrArg Subtype.val i.property)⟩
-    inj' := fun _i _k h => Subtype.ext (f.property.1 (congrArg Subtype.val h)) }
-
-noncomputable def groupsEmbedding {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (F : GroupedAssignments p S r) : Fin A ↪ SelectedCycle p S :=
-  ((Equiv.sigmaFiberEquiv (slotClass r)).symm.toEmbedding.trans
-    (Function.Embedding.sigmaMap (Function.Embedding.refl _) F)).trans
-    ((Function.Embedding.sigmaMap (Function.Embedding.subtype _) (fun _ => Function.Embedding.refl _)).trans
-      (Equiv.sigmaFiberEquiv (fun C : SelectedCycle p S => C.val.card)).toEmbedding)
-
-noncomputable def groupsToAssignment {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (F : GroupedAssignments p S r) : SlotAssignment p S r :=
-  ⟨groupsEmbedding F, (groupsEmbedding F).injective,
-    fun i => (F (slotClass r i) ⟨i,rfl⟩).property⟩
-
-noncomputable def assignmentEquivGroups (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
-    SlotAssignment p S r ≃ GroupedAssignments p S r where
-  toFun := assignmentToGroups
-  invFun := groupsToAssignment
-  left_inv f := by
-    apply Subtype.ext
-    rfl
-  right_inv F := by
-    funext j
-    apply Function.Embedding.ext
-    intro i
-    rcases i with ⟨i,hi⟩
-    subst j
-    rfl
-
-lemma card_cycleSlots_factorial (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
-    Fintype.card (CycleSlots p S r) =
-      ∏ j : LengthClass r, (Fintype.card (CyclesOfLength p S j.val)).descFactorial
-        (Fintype.card (SlotsOfLength r j)) := by
-  classical
-  rw [Fintype.card_congr (cycleSlotsEquivAssignments p S r),
-    Fintype.card_congr (assignmentEquivGroups p S r), Fintype.card_pi]
-  simp only [Fintype.card_embedding_eq]
-
-lemma butterfly_factorial_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : ι → Bool =>
-       (∏ j : LengthClass r,
-         (Fintype.card (CyclesOfLength (R.perm ω) S j.val)).descFactorial
-           (Fintype.card (SlotsOfLength r j)) : ℕ)) ≤
-      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-        ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
-  simpa only [←card_cycleSlots_factorial] using butterfly_cycleSlots_moment R S r hr
-
-end GroupedCounting
-
-section AllLengths
-variable {A : ℕ}
-
-noncomputable def assignmentRelabel (p : Equiv.Perm (α)) (S : Finset (α))
-    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
-    SlotAssignment p S r ≃ SlotAssignment p S (r ∘ e) where
-  toFun f := ⟨f.val ∘ e, f.property.1.comp e.injective, fun i => f.property.2 (e i)⟩
-  invFun f := ⟨f.val ∘ e.symm, f.property.1.comp e.symm.injective, fun i => by
-    simpa only [Function.comp_apply,Equiv.apply_symm_apply] using f.property.2 (e.symm i)⟩
-  left_inv f := by apply Subtype.ext; funext i; simp
-  right_inv f := by apply Subtype.ext; funext i; simp
-
-lemma card_cycleSlots_relabel (p : Equiv.Perm (α)) (S : Finset (α))
-    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
-    Fintype.card (CycleSlots p S r) = Fintype.card (CycleSlots p S (r ∘ e)) := by
-  exact Fintype.card_congr ((cycleSlotsEquivAssignments p S r).trans
-    ((assignmentRelabel p S r e).trans (cycleSlotsEquivAssignments p S (r ∘ e)).symm))
-
-lemma butterfly_cycleSlots_moment_all (S : Finset (α)) (r : Fin A → ℕ) :
-    finiteMean (fun ω : ι → Bool =>
-      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
-        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
-  have h := butterfly_cycleSlots_moment R S (r ∘ Tuple.sort r) (Tuple.monotone_sort r)
-  simp_rw [←card_cycleSlots_relabel] at h
-  have hp := Equiv.prod_comp (Tuple.sort r) (fun i : Fin A => (1 / Real.sqrt (r i+1) : ℝ))
-  simpa only [Function.comp_apply,hp] using h
-
-abbrev ShortCycle (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :=
-  {C : SelectedCycle p S // C.val.card ≤ J}
-
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    Fintype (ShortCycle p S J) := Fintype.ofFinite _
-
-noncomputable def shortCycleLength {p : Equiv.Perm (α)} {S : Finset (α)} {J : ℕ}
-    (C : ShortCycle p S J) : Fin J :=
-  ⟨C.val.val.card-1, by have := C.val.nonempty.card_pos; have := C.property; omega⟩
-
-lemma shortCycleLength_add_one {p : Equiv.Perm (α)} {S : Finset (α)} {J : ℕ}
-    (C : ShortCycle p S J) : (shortCycleLength C : ℕ)+1 = C.val.val.card := by
-  have := C.val.nonempty.card_pos
-  simp only [shortCycleLength]
-  omega
-
-noncomputable def shortEmbeddingEquiv (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    (Fin A ↪ ShortCycle p S J) ≃
-      Σ r : Fin A → Fin J, SlotAssignment p S (fun i => (r i : ℕ)) where
-  toFun f := ⟨fun i => shortCycleLength (f i),
-    ⟨fun i => (f i).val, fun i j h => f.injective (Subtype.ext h),
-      fun i => (shortCycleLength_add_one (f i)).symm⟩⟩
-  invFun c :=
-    { toFun := fun i => ⟨c.2.val i, by rw [c.2.property.2]; exact (c.1 i).isLt⟩
-      inj' := fun _i _j h => c.2.property.1 (congrArg Subtype.val h) }
-  left_inv f := by
-    apply Function.Embedding.ext
-    intro i
-    rfl
-  right_inv c := by
-    rcases c with ⟨r,f⟩
-    have he : (fun i => shortCycleLength (⟨f.val i, by rw [f.property.2]; exact (r i).isLt⟩ : ShortCycle p S J)) = r := by
-      funext i
-      apply Fin.ext
-      simp only [shortCycleLength,f.property.2,Nat.add_sub_cancel]
-    apply Sigma.ext he
-    apply (Subtype.heq_iff_coe_eq (by
-      intro x
-      change (Function.Injective x ∧ ∀ i, (x i).val.card = (f.val i).val.card - 1 + 1) ↔ _
-      simp only [f.property.2,Nat.add_sub_cancel])).mpr
-    rfl
-
-lemma shortCycle_factorial_eq (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    (Fintype.card (ShortCycle p S J)).descFactorial A =
-      ∑ r : Fin A → Fin J, Fintype.card (CycleSlots p S (fun i => (r i : ℕ))) := by
-  classical
-  have hc := Fintype.card_embedding_eq (α := Fin A) (β := ShortCycle p S J)
-  simp only [Fintype.card_fin] at hc
-  rw [←hc, Fintype.card_congr (shortEmbeddingEquiv p S J),Fintype.card_sigma]
-  apply Finset.sum_congr rfl
-  intro r _
-  exact (Fintype.card_congr (cycleSlotsEquivAssignments p S (fun i => (r i : ℕ)))).symm
-
-end AllLengths
-
-section CycleMGF
-
-lemma butterfly_shortCycle_moment (S : Finset (α)) (J A : ℕ) :
-    finiteMean (fun ω : ι → Bool =>
-      ((Fintype.card (ShortCycle (R.perm ω) S J)).descFactorial A : ℝ)) ≤
-      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))^A := by
-  classical
-  simp_rw [shortCycle_factorial_eq,Nat.cast_sum,finiteMean_sum]
-  calc
-    _ ≤ ∑ r : Fin A → Fin J,
-        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-          ∏ i : Fin A, (1 / Real.sqrt ((r i : ℕ)+1)) := by
-      apply Finset.sum_le_sum
-      intro r _
-      exact butterfly_cycleSlots_moment_all R S (fun i => (r i : ℕ))
-    _ = _ := by
-      rw [←Finset.mul_sum,mul_pow]
-      congr 1
-      rw [←Fintype.prod_sum (fun (_ : Fin A) (j : Fin J) => (1 / Real.sqrt ((j:ℕ)+1) : ℝ))]
-      simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fin]
-
-noncomputable def cycleMass (p : Equiv.Perm (α)) (S : Finset (α)) : ℕ :=
-  ∑ C : SelectedCycle p S, C.val.card
-
-lemma cycleMass_le (p : Equiv.Perm (α)) (S : Finset (α)) : cycleMass p S ≤ S.card := by
-  classical
-  let f : (Σ C : SelectedCycle p S, {x : α // x ∈ C.val}) → S :=
-    fun c => ⟨c.2.val,c.1.property.2 c.2.property⟩
-  have hi : Function.Injective f := by
-    intro c e h
-    have hx : c.2.val = e.2.val := congrArg Subtype.val h
-    have hC : c.1 = e.1 := c.1.eq_of_mem e.1 c.2.property (hx ▸ e.2.property)
-    rcases c with ⟨c,x⟩
-    rcases e with ⟨e,y⟩
-    dsimp only at hC
-    subst e
-    have he : x=y := Subtype.ext hx
-    cases he
-    rfl
-  have hh := Fintype.card_le_of_injective f hi
-  simpa only [Fintype.card_sigma,Fintype.card_coe,cycleMass] using hh
-
-lemma shortCycle_card_le (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    Fintype.card (ShortCycle p S J) ≤ S.card := by
-  classical
-  calc
-    _ ≤ Fintype.card (SelectedCycle p S) := Fintype.card_subtype_le _
-    _ = ∑ _ : SelectedCycle p S, 1 := by simp
-    _ ≤ cycleMass p S := Finset.sum_le_sum (fun C _ => C.nonempty.card_pos)
-    _ ≤ _ := cycleMass_le p S
-
-lemma butterfly_shortCycle_mgf (S : Finset (α)) (J : ℕ) (q : ℝ) (hq : 1 ≤ q) :
-    finiteMean (fun ω : ι → Bool =>
-      q^(Fintype.card (ShortCycle (R.perm ω) S J))) ≤
-      Real.exp ((q-1)*((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))) := by
-  apply factorial_mgf _ S.card _ q (fun ω => shortCycle_card_le _ _ _)
-  · positivity
-  · exact hq
-  · exact butterfly_shortCycle_moment R S J
-
-end CycleMGF
-
-
-abbrev LongCycle (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :=
-  {C : SelectedCycle p S // ¬ C.val.card ≤ J}
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    Fintype (LongCycle p S J) := Fintype.ofFinite _
-
-lemma selectedCycle_card_split (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    Fintype.card (SelectedCycle p S) =
-      Fintype.card (ShortCycle p S J) + Fintype.card (LongCycle p S J) := by
-  classical
-  rw [←Fintype.card_congr (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J)),Fintype.card_sum]
-
-lemma longCycle_card_bound (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    J * Fintype.card (LongCycle p S J) ≤ S.card := by
-  classical
-  have hs := Equiv.sum_comp (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J))
-    (fun C : SelectedCycle p S => C.val.card)
-  rw [Fintype.sum_sum_type] at hs
-  have h0 : J * Fintype.card (LongCycle p S J) ≤
-      ∑ C : LongCycle p S J, C.val.val.card := by
-    calc
-      _ = ∑ _C : LongCycle p S J, J := by simp [mul_comm]
-      _ ≤ _ := Finset.sum_le_sum (fun C _ => Nat.le_of_lt (Nat.lt_of_not_ge C.property))
-  have hh := cycleMass_le p S
-  unfold cycleMass at hh
-  rw [←hs] at hh
-  exact h0.trans ((Nat.le_add_left _ _).trans hh)
-
-lemma longCycle_pow_bound (p : Equiv.Perm (α)) (S : Finset (α))
-    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    q^(Fintype.card (LongCycle p S J)) ≤ Real.exp ((S.card : ℝ)/J * Real.log q) := by
-  have hq0 : 0 < q := lt_of_lt_of_le zero_lt_one hq
-  have hc : (Fintype.card (LongCycle p S J) : ℝ) ≤ (S.card : ℝ)/J := by
-    apply (le_div_iff₀ (by exact_mod_cast hJ)).mpr
-    have hh := longCycle_card_bound p S J
-    exact_mod_cast (by simpa only [mul_comm] using hh)
-  calc
-    _ = Real.exp ((Fintype.card (LongCycle p S J) : ℝ)*Real.log q) := by
-      rw [Real.exp_nat_mul,Real.exp_log hq0]
-    _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right hc (Real.log_nonneg hq))
-
-lemma butterfly_cycle_log_mgf (S : Finset (α)) (J : ℕ) (hJ : 0 < J)
-    (q : ℝ) (hq : 1 ≤ q) :
-    Real.log (finiteMean (fun ω : ι → Bool =>
-      q^(Fintype.card (SelectedCycle (R.perm ω) S)))) ≤
-      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
-  let lam : ℝ := (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1)
-  let c : ℝ := (S.card : ℝ)/J * Real.log q
-  have hnum : (q-1)*lam ≤ 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
-    have hsum := sum_inv_sqrt J
-    have hp : 0 ≤ (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) := by positivity
-    have h0 : 0 ≤ 2*Real.sqrt J := by positivity
-    have hh := mul_le_mul_of_nonneg_left hsum hp
-    have hq1 : 0 ≤ q-1 := sub_nonneg.mpr hq
-    calc
-      _ ≤ (q-1)*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
-        mul_le_mul_of_nonneg_left hh hq1
-      _ ≤ q*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
-        mul_le_mul_of_nonneg_right (by linarith) (mul_nonneg hp h0)
-      _ = _ := by ring
-  have hm : finiteMean (fun ω : ι → Bool =>
-      q^(Fintype.card (SelectedCycle (R.perm ω) S))) ≤
-      Real.exp (c+(q-1)*lam) := by
-    calc
-      _ ≤ finiteMean (fun ω : ι → Bool =>
-          q^(Fintype.card (ShortCycle (R.perm ω) S J))*Real.exp c) := by
-        apply finiteMean_mono
-        intro ω
-        rw [selectedCycle_card_split _ _ J,pow_add]
-        exact mul_le_mul_of_nonneg_left (longCycle_pow_bound _ S J hJ q hq) (by positivity)
-      _ = finiteMean (fun ω : ι → Bool =>
-          q^(Fintype.card (ShortCycle (R.perm ω) S J))) * Real.exp c :=
-        finiteMean_mul_const _ _
-      _ ≤ Real.exp ((q-1)*lam) * Real.exp c :=
-        mul_le_mul_of_nonneg_right (butterfly_shortCycle_mgf R S J q hq) (Real.exp_nonneg _)
-      _ = _ := by rw [←Real.exp_add,add_comm]
-  have hpos : 0 < finiteMean (fun ω : ι → Bool =>
-      q^(Fintype.card (SelectedCycle (R.perm ω) S))) := by
-    apply finiteMean_pos
-    intro ω
-    positivity
-  have hlog := Real.log_le_log hpos hm
-  rw [Real.log_exp] at hlog
-  dsimp [c] at hlog
-  linarith
-
-
-end RoutingNetwork
-
-variable {α B : Type*} [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] {d : ℕ}
-
-def parallelPerm (ω : B × SwitchIndex d → Bool) : Equiv.Perm (B × Card d) where
-  toFun x := (x.1,butterflyPerm d (decodeButterfly d (fun i => ω (x.1,i))) x.2)
-  invFun y := (y.1,(butterflyPerm d (decodeButterfly d (fun i => ω (y.1,i)))).symm y.2)
-  left_inv x := by simp
-  right_inv y := by simp
-
-def relabeledPerm (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (ω : B × SwitchIndex d → Bool) : Equiv.Perm α :=
-  (pre.trans (parallelPerm ω)).trans post
-
-noncomputable def parallelPath (pre : α ≃ B × Card d)
-    (ω : B × SwitchIndex d → Bool) (x : α) : Finset (B × SwitchIndex d) :=
-  (routeDomain d (pre x).2
-    (butterflyPerm d (decodeButterfly d (fun i => ω ((pre x).1,i))) (pre x).2)).image
-      (fun i => ((pre x).1,i))
-
-def parallelUsers (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
-    (i : B × SwitchIndex d) : α × α :=
-  let u := switchUsers d (decodeButterfly d (fun j => ω (i.1,j))) i.2
-  (pre.symm (i.1,u.1),pre.symm (i.1,u.2))
-
-omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
-lemma parallelUsers_ne (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
-    (i : B × SwitchIndex d) : (parallelUsers pre ω i).1 ≠ (parallelUsers pre ω i).2 := by
-  intro h
-  exact switchUsers_ne d _ _ (congrArg Prod.snd (pre.symm.injective h))
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma mem_parallelPath (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
-    (x : α) (i : B × SwitchIndex d) :
-    i ∈ parallelPath pre ω x ↔ (parallelUsers pre ω i).1 = x ∨ (parallelUsers pre ω i).2 = x := by
-  classical
-  obtain ⟨b,j⟩ := i
-  simp only [parallelPath,Finset.mem_image,Prod.mk.injEq]
-  constructor
-  · rintro ⟨k,hk,hb,rfl⟩
-    subst b
-    rcases (mem_routeDomain_iff_user d _ _ _).mp hk with h | h
-    · left
-      apply pre.injective
-      change pre (pre.symm ((pre x).1,_)) = pre x
-      rw [Equiv.apply_symm_apply]
-      exact Prod.ext rfl h
-    · right
-      apply pre.injective
-      change pre (pre.symm ((pre x).1,_)) = pre x
-      rw [Equiv.apply_symm_apply]
-      exact Prod.ext rfl h
-  · intro h
-    have hh : (b,(switchUsers d (decodeButterfly d (fun j => ω (b,j))) j).1) = pre x ∨
-        (b,(switchUsers d (decodeButterfly d (fun j => ω (b,j))) j).2) = pre x := by
-      rcases h with h | h
-      · exact Or.inl (by simpa only [parallelUsers,Equiv.apply_symm_apply] using congrArg pre h)
-      · exact Or.inr (by simpa only [parallelUsers,Equiv.apply_symm_apply] using congrArg pre h)
-    have hb : b = (pre x).1 := hh.elim (congrArg Prod.fst) (congrArg Prod.fst)
-    subst b
-    refine ⟨j,?_,rfl,rfl⟩
-    apply (mem_routeDomain_iff_user d _ _ _).mpr
-    exact hh.elim (fun h => Or.inl (congrArg Prod.snd h)) (fun h => Or.inr (congrArg Prod.snd h))
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma card_parallelPath (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool) (x : α) :
-    (parallelPath pre ω x).card = d := by
-  classical
-  rw [parallelPath,Finset.card_image_of_injective _ (fun _ _ h => Prod.mk.inj h |>.2),card_routeDomain]
-
-noncomputable def blockMarks (pre : α ≃ B × Card d) (S : Finset α) (b : B) : Finset (Card d) :=
-  Finset.univ.filter (fun x => pre.symm (b,x) ∈ S)
-
-omit [Fintype α] [Fintype B] [DecidableEq B] in
-@[simp] lemma mem_blockMarks (pre : α ≃ B × Card d) (S : Finset α) (b : B) (x : Card d) :
-    x ∈ blockMarks pre S b ↔ pre.symm (b,x) ∈ S := by
-  classical
-  simp [blockMarks]
-
-omit [DecidableEq B] in
-lemma sum_card_blockMarks (pre : α ≃ B × Card d) (S : Finset α) :
-    ∑ b : B, (blockMarks pre S b).card = S.card := by
-  classical
-  calc
-    _ = ∑ b : B, ∑ x : Card d, if pre.symm (b,x) ∈ S then 1 else 0 := by
-      simp only [blockMarks,Finset.card_filter]
-    _ = ∑ u : B × Card d, if pre.symm u ∈ S then 1 else 0 := (Fintype.sum_prod_type (fun u : B × Card d => if pre.symm u ∈ S then (1:ℕ) else 0)).symm
-    _ = ∑ x : α, if x ∈ S then 1 else 0 := Equiv.sum_comp pre.symm (fun x : α => if x ∈ S then (1:ℕ) else 0)
-    _ = _ := by simp
-
-omit [DecidableEq B] in
-lemma card_blockMarks_le (pre : α ≃ B × Card d) (S : Finset α) (b : B) :
-    (blockMarks pre S b).card ≤ S.card := by
-  rw [←sum_card_blockMarks pre S]
-  exact Finset.single_le_sum (f := fun b : B => (blockMarks pre S b).card) (fun _ _ => Nat.zero_le _) (Finset.mem_univ b)
-
-end Thorp
-
-end ThorpNine.SparseSaving
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.SparseSaving.AssignmentSlots
+
+namespace OAI
+
+namespace ThorpNine.SparseSaving
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+namespace RoutingNetwork
+variable {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι] [DecidableEq ι]
+  {d : ℕ} (R : RoutingNetwork α ι d)
+lemma mem_exposedPaths (b : ι → Bool) (V : Finset (α)) (s : ι) :
+    s ∈ exposedPaths R b V ↔ (R.users b s).1 ∈ V ∨ (R.users b s).2 ∈ V := by
+  classical
+  simp only [exposedPaths,Finset.mem_biUnion,R.mem_path]
+  constructor
+  · rintro ⟨x,hx,h | h⟩
+    · exact Or.inl (h ▸ hx)
+    · exact Or.inr (h ▸ hx)
+  · rintro (h | h)
+    · exact ⟨_,h,Or.inl rfl⟩
+    · exact ⟨_,h,Or.inr rfl⟩
+
+lemma terminalCharge_sum (b : ι → Bool) (z : α) (V : Finset (α))
+    (hz : z ∉ V) :
+    (terminalCharge R b z V : ℝ) = ∑ s : ι,
+      ((if (R.users b s).1 = z ∧ (R.users b s).2 ∈ V then (1:ℝ) else 0) +
+      (if (R.users b s).2 = z ∧ (R.users b s).1 ∈ V then (1:ℝ) else 0)) := by
+  classical
+  have he : R.path b z ∩ exposedPaths R b V =
+      Finset.univ.filter (fun s =>
+        ((R.users b s).1 = z ∧ (R.users b s).2 ∈ V) ∨
+        ((R.users b s).2 = z ∧ (R.users b s).1 ∈ V)) := by
+    ext s
+    simp only [Finset.mem_inter,R.mem_path,mem_exposedPaths,
+      Finset.mem_filter,Finset.mem_univ,true_and]
+    by_cases h₁ : (R.users b s).1 = z <;>
+      by_cases h₂ : (R.users b s).2 = z <;> simp [h₁,h₂,hz]
+  rw [terminalCharge,he]
+  simp only [← Finset.sum_boole]
+  apply Finset.sum_congr rfl
+  intro s _
+  by_cases h₁ : (R.users b s).1 = z <;>
+    by_cases h₂ : (R.users b s).2 = z <;> simp [h₁,h₂,hz]
+
+noncomputable def priorPaths {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) : Finset (α) :=
+  Finset.univ.filter (fun x => ∃ i : Fin A, o x = some i ∧ i.val < t)
+
+omit [DecidableEq α] in
+@[simp] lemma mem_priorPaths {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) (x : α) :
+    x ∈ priorPaths o t ↔ ∃ i : Fin A, o x = some i ∧ i.val < t := by
+  classical
+  simp [priorPaths]
+
+lemma terminalCharge_prior_sum {A : ℕ} (o : α → Option (Fin A))
+    (b : ι → Bool) (z : Terminal o) (i : Fin A) :
+    (terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) =
+      ∑ s : ι,
+        ((if (z i).val = (R.users b s).1 ∧ (R.users b s).2 ∈ priorPaths o (i.val+1)
+          then (1:ℝ) else 0) +
+        (if (z i).val = (R.users b s).2 ∧ (R.users b s).1 ∈ priorPaths o (i.val+1)
+          then (1:ℝ) else 0)) := by
+  classical
+  rw [terminalCharge_sum _ _ _ _ (Finset.notMem_erase _ _)]
+  apply Finset.sum_congr rfl
+  intro s _
+  have hn := R.users_ne b s
+  simp only [Finset.mem_erase]
+  by_cases h₁ : (z i).val = (R.users b s).1 <;>
+    by_cases h₂ : (z i).val = (R.users b s).2 <;> simp_all [eq_comm]
+
+lemma rotationMeeting_eq_slot_sum {A : ℕ} (o : α → Option (Fin A))
+    (z : Terminal o) (u v : α) :
+    rotationMeeting o z u v = ∑ i : Fin A,
+      ((if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) +
+      (if (z i).val = v ∧ u ∈ priorPaths o (i.val+1) then (1:ℝ) else 0)) := by
+  classical
+  have only_owner (u : α) (a : Fin A) (hou : o u = some a)
+      (i : Fin A) (hia : i ≠ a) : (z i).val ≠ u := by
+    intro h
+    have hi := (z i).property
+    rw [h,hou] at hi
+    exact hia (Option.some.inj hi).symm
+  have sum_one (u v : α) (a : Fin A) (hou : o u = some a) :
+      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) =
+      if (z a).val = u ∧ v ∈ priorPaths o (a.val+1) then 1 else 0 := by
+    apply Finset.sum_eq_single a
+    · intro i _ hia
+      simp [only_owner u a hou i hia]
+    · simp
+  have sum_none (u v : α) (hou : o u = none) :
+      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) = 0 := by
+    apply Finset.sum_eq_zero
+    intro i _
+    have hn : (z i).val ≠ u := by
+      intro h
+      have hi := (z i).property
+      rw [h,hou] at hi
+      contradiction
+    simp [hn]
+  rw [Finset.sum_add_distrib]
+  cases hou : o u with
+  | none =>
+    rw [sum_none u v hou]
+    simp [rotationMeeting,hou,mem_priorPaths]
+  | some a =>
+    rw [sum_one u v a hou]
+    cases hov : o v with
+    | none =>
+      rw [sum_none v u hov]
+      simp [rotationMeeting,hou,hov,mem_priorPaths]
+    | some b =>
+      rw [sum_one v u b hov]
+      simp only [rotationMeeting,hou,hov,mem_priorPaths,Option.some.injEq]
+      rcases lt_trichotomy a b with hab | rfl | hba
+      · have hn : ¬ b.val < a.val+1 := by omega
+        have hy : a.val < b.val+1 := by omega
+        simp [hab,hn,hy]
+      · simp
+      · have hn : ¬ a.val < b.val+1 := by omega
+        have hy : b.val < a.val+1 := by omega
+        simp [hba,not_lt_of_ge (le_of_lt hba),hn,hy]
+
+lemma rotationCharge_eq_terminal_sum {A : ℕ} (o : α → Option (Fin A))
+    (b : ι → Bool) (z : Terminal o) :
+    rotationCharge R o b z = ∑ i : Fin A,
+      (terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) := by
+  simp only [rotationCharge,rotationMeeting_eq_slot_sum,terminalCharge_prior_sum]
+  exact Finset.sum_comm
+
+omit [DecidableEq α] in
+@[simp] lemma priorPaths_zero {A : ℕ} (o : α → Option (Fin A)) : priorPaths o 0 = ∅ := by
+  classical
+  ext x
+  simp
+
+lemma priorPaths_succ {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) (ht : t < A) :
+    priorPaths o (t+1) = priorPaths o t ∪ cycleFiber o ⟨t,ht⟩ := by
+  classical
+  ext x
+  simp only [mem_priorPaths,Finset.mem_union,mem_cycleFiber]
+  constructor
+  · rintro ⟨i,hi,hit⟩
+    by_cases h : i.val < t
+    · exact Or.inl ⟨i,hi,h⟩
+    · have he : i = ⟨t,ht⟩ := Fin.ext (by change i.val = t; omega)
+      exact Or.inr (he ▸ hi)
+  · rintro (⟨i,hi,hit⟩ | hi)
+    · exact ⟨i,hi,by omega⟩
+    · exact ⟨⟨t,ht⟩,hi,by change t < t+1; omega⟩
+
+omit [DecidableEq α] in
+lemma priorPaths_disjoint {A : ℕ} (o : α → Option (Fin A)) (i : Fin A) :
+    Disjoint (priorPaths o i.val) (cycleFiber o i) := by
+  classical
+  apply Finset.disjoint_left.mpr
+  intro x hx hx'
+  obtain ⟨j,hj,hji⟩ := (mem_priorPaths _ _ _).mp hx
+  have he : j = i := Option.some.inj (hj.symm.trans ((mem_cycleFiber _ _ _).mp hx'))
+  subst j
+  exact (lt_irrefl _ hji)
+
+namespace QueryTree
+open Thorp.QueryTree
+
+lemma seekCycles_owned {A : ℕ} (S : Finset (α)) (r : Fin A → ℕ)
+    (o : α → Option (Fin A)) (z : Terminal o) (ω : ι → Bool)
+    (hS : ∀ i, cycleFiber o i ⊆ S)
+    (hO : ∀ i, orbitSet (R.perm ω) (z i).val = cycleFiber o i)
+    (hC : ∀ i, (cycleFiber o i).card = r i+1) :
+    let p := R.perm ω
+    let L := List.ofFn (fun i : Fin A => (p (z i).val,r i))
+    (seekCycles R S L ∅).succeeds ω = true ∧
+      ((seekCycles R S L ∅).staleCharge ∅ ω : ℝ) = rotationCharge R o (ω) z := by
+  classical
+  let b := ω
+  let p := R.perm b
+  let f (i : Fin A) := (p (z i).val,r i)
+  let c (i : Fin A) : ℝ := terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val)
+  have hc (i : Fin A) : Function.minimalPeriod p (z i).val - 1 = r i := by
+    have he := orbitSet_card p (z i).val
+    rw [hO,hC] at he
+    omega
+  have hs (t : ℕ) (ht : t ≤ A) :
+      (seekCycles R S ((List.ofFn f).drop t) (priorPaths o t)).succeeds ω = true ∧
+        ((seekCycles R S ((List.ofFn f).drop t) (priorPaths o t)).staleCharge (exposedPaths R b (priorPaths o t)) ω : ℝ) =
+          ((List.ofFn c).drop t).sum := by
+    induction ht using Nat.decreasingInduction with
+    | self => simp only [List.length_ofFn,le_refl,List.drop_eq_nil_of_le,List.sum_nil,seekCycles,Thorp.QueryTree.succeeds,staleCharge,Nat.cast_zero,and_self]
+    | of_succ t ht ih =>
+      have htf : t < (List.ofFn f).length := by simpa using ht
+      have htc : t < (List.ofFn c).length := by simpa using ht
+      rw [List.drop_eq_getElem_cons htf,List.drop_eq_getElem_cons htc]
+      simp only [List.getElem_ofFn,List.sum_cons,seekCycles]
+      have h := seekCycle_orbit R S (priorPaths o t) (z ⟨t,ht⟩).val
+        (fun V => seekCycles R S ((List.ofFn f).drop (t+1)) V) ω
+        (by rw [hO]; exact hS _) (by rw [hO]; exact priorPaths_disjoint o ⟨t,ht⟩)
+      dsimp only at h
+      rw [hc,hO,←priorPaths_succ o t ht] at h
+      dsimp only [f]
+      constructor
+      · exact h.1.trans ih.1
+      · rw [h.2,Nat.cast_add,ih.2]
+        exact add_comm _ _
+  have h := hs 0 (Nat.zero_le _)
+  simp only [List.drop_zero,priorPaths_zero,exposedPaths_empty,List.sum_ofFn] at h
+  dsimp only
+  exact ⟨h.1,h.2.trans (rotationCharge_eq_terminal_sum R o b z).symm⟩
+
+end QueryTree
+section Collections
+variable {A : ℕ}
+
+def ValidCycleSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ)
+    (o : α → Option (Fin A)) : Prop :=
+  (∀ i, (cycleFiber o i).card = r i + 1) ∧
+  (∀ i, cycleFiber o i ⊆ S) ∧
+  (∀ i x, o x = some i → orbitSet p x = cycleFiber o i)
+
+abbrev CycleSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
+  {o : α → Option (Fin A) // ValidCycleSlots p S r o}
+
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
+    Fintype (CycleSlots p S r) := Fintype.ofFinite _
+
+lemma CycleSlots.terminal_nonempty {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (c : CycleSlots p S r) (i : Fin A) : Nonempty {x : α // c.val x = some i} := by
+  have hn : (cycleFiber c.val i).Nonempty := Finset.card_pos.mp (by rw [c.property.1]; omega)
+  obtain ⟨x,hx⟩ := hn
+  exact ⟨x,(mem_cycleFiber _ _ _).mp hx⟩
+
+omit [DecidableEq α] in
+lemma owner_ext {o o' : α → Option (Fin A)}
+    (h : ∀ i, cycleFiber o i = cycleFiber o' i) : o = o' := by
+  funext x
+  have he (i : Fin A) : o x = some i ↔ o' x = some i := by
+    simpa only [mem_cycleFiber] using (Finset.ext_iff.mp (h i) x)
+  cases ho : o x with
+  | some i => exact ((he i).mp ho).symm
+  | none =>
+    cases ho' : o' x with
+    | none => rfl
+    | some i => have hn := (he i).mpr ho'; rw [ho] at hn; contradiction
+
+abbrev RootedSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
+  Σ c : CycleSlots p S r, Terminal c.val
+
+noncomputable def rootedStart {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (c : RootedSlots p S r) : Fin A → S := fun i =>
+  ⟨p (c.2 i).val,c.1.property.2.1 i (by
+    rw [←c.1.property.2.2 i (c.2 i).val (c.2 i).property]
+    exact (mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.apply_right (Equiv.Perm.SameCycle.refl p _)))⟩
+
+lemma rootedStart_injective {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ} :
+    Function.Injective (rootedStart (p:=p) (S:=S) (r:=r)) := by
+  rintro ⟨c,z⟩ ⟨c',z'⟩ h
+  have hz (i : Fin A) : (z i).val = (z' i).val :=
+    p.injective (congrArg Subtype.val (congrFun h i))
+  have hc : c = c' := Subtype.ext (owner_ext (fun i => by
+    rw [←c.property.2.2 i (z i).val (z i).property,
+      ←c'.property.2.2 i (z' i).val (z' i).property,hz]))
+  subst c'
+  have hzz : z = z' := funext (fun i => Subtype.ext (hz i))
+  subst z'
+  rfl
+
+lemma rooted_weight_sum_le (S : Finset (α)) (r : Fin A → ℕ) (ω : ι → Bool) :
+    (∑ c : RootedSlots (R.perm ω) S r,
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2)) ≤
+    ∑ a : Fin A → S,
+      if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+        ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+      else 0 := by
+  classical
+  let p := R.perm ω
+  let g (a : Fin A → S) : ℝ :=
+    if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+    else 0
+  have he (c : RootedSlots p S r) :
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2) =
+      g (rootedStart c) := by
+    have hs := QueryTree.seekCycles_owned R S r c.1.val c.2 ω c.1.property.2.1
+      (fun i => c.1.property.2.2 i (c.2 i).val (c.2 i).property) c.1.property.1
+    dsimp only at hs
+    unfold g
+    rw [ite_eq_left (show (QueryTree.seekCycles R S (List.ofFn (fun i => ((rootedStart c i).val,r i))) ∅).succeeds ω = true from hs.1)]
+    rw [← hs.2,neg_mul,Real.exp_neg,mul_comm (Real.log 2),Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
+    rfl
+  simp_rw [he]
+  exact sum_comp_injective_le rootedStart rootedStart_injective g (fun a => by dsimp [g]; split_ifs <;> positivity)
+
+lemma collection_rotation_lower (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r)
+    (ω : ι → Bool) (c : CycleSlots (R.perm ω) S r) :
+    collectionWeight d r ≤ ∑ z : Terminal c.val,
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z) := by
+  classical
+  let (i : Fin A) : Nonempty {x : α // c.val x = some i} := c.terminal_nonempty i
+  have hm : Monotone (fun i => (cycleFiber c.val i).card) := by
+    intro i j hij
+    dsimp only
+    rw [c.property.1,c.property.1]
+    exact Nat.add_le_add_right (hr hij) 1
+  have hb := rotation_exp_bound R hm (ω)
+  have hcard : (Fintype.card (Terminal c.val) : ℝ) = ∏ i : Fin A, (r i+1 : ℝ) := by
+    rw [Fintype.card_pi,Nat.cast_prod]
+    apply Finset.prod_congr rfl
+    intro i _
+    rw [card_cycleFiber,c.property.1,Nat.cast_add,Nat.cast_one]
+  have hp : (0:ℝ) < Fintype.card (Terminal c.val) := Nat.cast_pos.mpr Fintype.card_pos
+  have hl : rotationLower d r ≤ finiteMean (fun z : Terminal c.val => Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z)) := by
+    simpa only [rotationLower,c.property.1,Nat.cast_add,Nat.cast_one] using hb
+  have hh := (le_div_iff₀ hp).mp hl
+  rw [hcard] at hh
+  unfold collectionWeight
+  rw [← Finset.mul_sum]
+  calc
+    _ = ((2:ℝ)^d)^A * (rotationLower d r * ∏ i : Fin A, (r i+1 : ℝ)) := by ring
+    _ ≤ _ := mul_le_mul_of_nonneg_left hh (by positivity)
+
+lemma cycleSlots_weighted_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : ι → Bool =>
+      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) *
+      collectionWeight d r ≤ (S.card : ℝ)^A := by
+  classical
+  let g (a : Fin A → S) (ω : ι → Bool) : ℝ :=
+    if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+    else 0
+  have hω (ω : ι → Bool) :
+      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ) * collectionWeight d r ≤
+        ∑ a : Fin A → S, g a ω := by
+    calc
+      _ = ∑ _c : CycleSlots (R.perm ω) S r, collectionWeight d r := by simp
+      _ ≤ ∑ c : CycleSlots (R.perm ω) S r,
+          ∑ z : Terminal c.val, ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z) :=
+        Finset.sum_le_sum (fun c _ => collection_rotation_lower R S r hr ω c)
+      _ = ∑ c : RootedSlots (R.perm ω) S r,
+          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2) :=
+        (Fintype.sum_sigma (fun c : RootedSlots (R.perm ω) S r =>
+          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2))).symm
+      _ ≤ _ := rooted_weight_sum_le R S r ω
+  have hb := finiteMean_mono hω
+  rw [finiteMean_mul_const,finiteMean_sum] at hb
+  calc
+    _ ≤ ∑ a : Fin A → S, finiteMean (g a) := hb
+    _ ≤ ∑ _a : Fin A → S, (1:ℝ) := by
+      apply Finset.sum_le_sum
+      intro a _
+      have hh := QueryTree.cycles_weighted_success R S (List.ofFn (fun i : Fin A => ((a i).val,r i)))
+      simpa only [List.length_ofFn] using hh
+    _ = _ := by simp
+
+lemma cycleSlots_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : ι → Bool =>
+      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
+        (S.card : ℝ)^A / collectionWeight d r := by
+  exact (le_div_iff₀ (collectionWeight_pos d r)).mpr (cycleSlots_weighted_moment R S r hr)
+
+lemma butterfly_cycleSlots_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : ι → Bool =>
+      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
+        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
+  have h := cycleSlots_moment R S r hr
+  rw [collectionWeight_eq_sqrt,div_mul_eq_div_mul_one_div] at h
+  rw [div_pow]
+  simpa only [one_div,Finset.prod_inv_distrib] using h
+
+end Collections
+
+section CycleCounting
+variable {A : ℕ}
+
+abbrev SelectedCycle (p : Equiv.Perm (α)) (S : Finset (α)) :=
+  {C : Finset (α) // (∃ x, C = orbitSet p x) ∧ C ⊆ S}
+
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) :
+    Fintype (SelectedCycle p S) := Fintype.ofFinite _
+
+lemma SelectedCycle.nonempty {p : Equiv.Perm (α)} {S : Finset (α)}
+    (C : SelectedCycle p S) : C.val.Nonempty := by
+  obtain ⟨x,hx⟩ := C.property.1
+  rw [hx]
+  exact orbitSet_nonempty p x
+
+lemma SelectedCycle.orbit_eq {p : Equiv.Perm (α)} {S : Finset (α)}
+    (C : SelectedCycle p S) {x : α} (hx : x ∈ C.val) : orbitSet p x = C.val := by
+  obtain ⟨y,hy⟩ := C.property.1
+  rw [hy] at hx ⊢
+  exact (orbitSet_eq p ((mem_orbitSet _ _ _).mp hx)).symm
+
+lemma SelectedCycle.eq_of_mem {p : Equiv.Perm (α)} {S : Finset (α)}
+    (C D : SelectedCycle p S) {x : α} (hC : x ∈ C.val) (hD : x ∈ D.val) : C = D := by
+  apply Subtype.ext
+  exact (C.orbit_eq hC).symm.trans (D.orbit_eq hD)
+
+abbrev SlotAssignment (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
+  {f : Fin A → SelectedCycle p S // Function.Injective f ∧ ∀ i, (f i).val.card = r i+1}
+
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
+    Fintype (SlotAssignment p S r) := Fintype.ofFinite _
+
+noncomputable def slotsToAssignment {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (c : CycleSlots p S r) : SlotAssignment p S r := by
+  classical
+  let f (i : Fin A) : SelectedCycle p S := ⟨cycleFiber c.val i,by
+    obtain ⟨x,hx⟩ := c.terminal_nonempty i
+    exact ⟨⟨x,(c.property.2.2 i x hx).symm⟩,c.property.2.1 i⟩⟩
+  refine ⟨f,?_,fun i => c.property.1 i⟩
+  intro i j hij
+  have he : cycleFiber c.val i = cycleFiber c.val j := congrArg Subtype.val hij
+  obtain ⟨x,hx⟩ := c.terminal_nonempty i
+  have hxi : x ∈ cycleFiber c.val i := (mem_cycleFiber _ _ _).mpr hx
+  rw [he] at hxi
+  exact Option.some.inj (hx.symm.trans ((mem_cycleFiber _ _ _).mp hxi))
+
+noncomputable def assignmentOwner {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) (x : α) : Option (Fin A) := by
+  classical
+  exact if h : ∃ i, x ∈ (f.val i).val then some (Classical.choose h) else none
+
+lemma assignmentOwner_eq_some {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) (x : α) (i : Fin A) :
+    assignmentOwner f x = some i ↔ x ∈ (f.val i).val := by
+  classical
+  unfold assignmentOwner
+  split_ifs with h
+  · constructor
+    · intro hi
+      have hx := Classical.choose_spec h
+      rw [Option.some.inj hi] at hx
+      exact hx
+    · intro hx
+      have he := (f.val (Classical.choose h)).eq_of_mem (f.val i) (Classical.choose_spec h) hx
+      exact congrArg some (f.property.1 he)
+  · constructor
+    · intro hi; contradiction
+    · intro hx; exact (h ⟨i,hx⟩).elim
+
+lemma assignmentOwner_fiber {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) (i : Fin A) : cycleFiber (assignmentOwner f) i = (f.val i).val := by
+  ext x
+  simp only [mem_cycleFiber,assignmentOwner_eq_some]
+
+noncomputable def assignmentToSlots {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) : CycleSlots p S r :=
+  ⟨assignmentOwner f,by
+    refine ⟨?_,?_,?_⟩
+    · intro i
+      rw [assignmentOwner_fiber]
+      exact f.property.2 i
+    · intro i
+      rw [assignmentOwner_fiber]
+      exact (f.val i).property.2
+    · intro i x hx
+      rw [assignmentOwner_fiber]
+      exact (f.val i).orbit_eq ((assignmentOwner_eq_some f x i).mp hx)⟩
+
+noncomputable def cycleSlotsEquivAssignments (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
+    CycleSlots p S r ≃ SlotAssignment p S r where
+  toFun := slotsToAssignment
+  invFun := assignmentToSlots
+  left_inv c := by
+    apply Subtype.ext
+    apply owner_ext
+    intro i
+    rw [assignmentToSlots,assignmentOwner_fiber]
+    rfl
+  right_inv f := by
+    apply Subtype.ext
+    funext i
+    apply Subtype.ext
+    exact assignmentOwner_fiber f i
+
+end CycleCounting
+
+section GroupedCounting
+variable {A : ℕ}
+
+abbrev LengthClass (r : Fin A → ℕ) := {j : ℕ // j ∈ Finset.univ.image (fun i => r i + 1)}
+
+def slotClass (r : Fin A → ℕ) (i : Fin A) : LengthClass r :=
+  ⟨r i+1, Finset.mem_image.mpr ⟨i, Finset.mem_univ _, rfl⟩⟩
+
+abbrev SlotsOfLength (r : Fin A → ℕ) (j : LengthClass r) := {i : Fin A // slotClass r i = j}
+abbrev CyclesOfLength (p : Equiv.Perm (α)) (S : Finset (α)) (j : ℕ) :=
+  {C : SelectedCycle p S // C.val.card = j}
+
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (j : ℕ) :
+    Fintype (CyclesOfLength p S j) := Fintype.ofFinite _
+
+abbrev GroupedAssignments (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
+  ∀ j : LengthClass r, SlotsOfLength r j ↪ CyclesOfLength p S j.val
+
+noncomputable def assignmentToGroups {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) : GroupedAssignments p S r := fun _j =>
+  { toFun := fun i => ⟨f.val i.val, (f.property.2 i.val).trans (congrArg Subtype.val i.property)⟩
+    inj' := fun _i _k h => Subtype.ext (f.property.1 (congrArg Subtype.val h)) }
+
+noncomputable def groupsEmbedding {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (F : GroupedAssignments p S r) : Fin A ↪ SelectedCycle p S :=
+  ((Equiv.sigmaFiberEquiv (slotClass r)).symm.toEmbedding.trans
+    (Function.Embedding.sigmaMap (Function.Embedding.refl _) F)).trans
+    ((Function.Embedding.sigmaMap (Function.Embedding.subtype _) (fun _ => Function.Embedding.refl _)).trans
+      (Equiv.sigmaFiberEquiv (fun C : SelectedCycle p S => C.val.card)).toEmbedding)
+
+noncomputable def groupsToAssignment {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (F : GroupedAssignments p S r) : SlotAssignment p S r :=
+  ⟨groupsEmbedding F, (groupsEmbedding F).injective,
+    fun i => (F (slotClass r i) ⟨i,rfl⟩).property⟩
+
+noncomputable def assignmentEquivGroups (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
+    SlotAssignment p S r ≃ GroupedAssignments p S r where
+  toFun := assignmentToGroups
+  invFun := groupsToAssignment
+  left_inv f := by
+    apply Subtype.ext
+    rfl
+  right_inv F := by
+    funext j
+    apply Function.Embedding.ext
+    intro i
+    rcases i with ⟨i,hi⟩
+    subst j
+    rfl
+
+lemma card_cycleSlots_factorial (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
+    Fintype.card (CycleSlots p S r) =
+      ∏ j : LengthClass r, (Fintype.card (CyclesOfLength p S j.val)).descFactorial
+        (Fintype.card (SlotsOfLength r j)) := by
+  classical
+  rw [Fintype.card_congr (cycleSlotsEquivAssignments p S r),
+    Fintype.card_congr (assignmentEquivGroups p S r), Fintype.card_pi]
+  simp only [Fintype.card_embedding_eq]
+
+lemma butterfly_factorial_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : ι → Bool =>
+       (∏ j : LengthClass r,
+         (Fintype.card (CyclesOfLength (R.perm ω) S j.val)).descFactorial
+           (Fintype.card (SlotsOfLength r j)) : ℕ)) ≤
+      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+        ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
+  simpa only [←card_cycleSlots_factorial] using butterfly_cycleSlots_moment R S r hr
+
+end GroupedCounting
+
+section AllLengths
+variable {A : ℕ}
+
+noncomputable def assignmentRelabel (p : Equiv.Perm (α)) (S : Finset (α))
+    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
+    SlotAssignment p S r ≃ SlotAssignment p S (r ∘ e) where
+  toFun f := ⟨f.val ∘ e, f.property.1.comp e.injective, fun i => f.property.2 (e i)⟩
+  invFun f := ⟨f.val ∘ e.symm, f.property.1.comp e.symm.injective, fun i => by
+    simpa only [Function.comp_apply,Equiv.apply_symm_apply] using f.property.2 (e.symm i)⟩
+  left_inv f := by apply Subtype.ext; funext i; simp
+  right_inv f := by apply Subtype.ext; funext i; simp
+
+lemma card_cycleSlots_relabel (p : Equiv.Perm (α)) (S : Finset (α))
+    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
+    Fintype.card (CycleSlots p S r) = Fintype.card (CycleSlots p S (r ∘ e)) := by
+  exact Fintype.card_congr ((cycleSlotsEquivAssignments p S r).trans
+    ((assignmentRelabel p S r e).trans (cycleSlotsEquivAssignments p S (r ∘ e)).symm))
+
+lemma butterfly_cycleSlots_moment_all (S : Finset (α)) (r : Fin A → ℕ) :
+    finiteMean (fun ω : ι → Bool =>
+      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
+        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
+  have h := butterfly_cycleSlots_moment R S (r ∘ Tuple.sort r) (Tuple.monotone_sort r)
+  simp_rw [←card_cycleSlots_relabel] at h
+  have hp := Equiv.prod_comp (Tuple.sort r) (fun i : Fin A => (1 / Real.sqrt (r i+1) : ℝ))
+  simpa only [Function.comp_apply,hp] using h
+
+abbrev ShortCycle (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :=
+  {C : SelectedCycle p S // C.val.card ≤ J}
+
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    Fintype (ShortCycle p S J) := Fintype.ofFinite _
+
+noncomputable def shortCycleLength {p : Equiv.Perm (α)} {S : Finset (α)} {J : ℕ}
+    (C : ShortCycle p S J) : Fin J :=
+  ⟨C.val.val.card-1, by have := C.val.nonempty.card_pos; have := C.property; omega⟩
+
+lemma shortCycleLength_add_one {p : Equiv.Perm (α)} {S : Finset (α)} {J : ℕ}
+    (C : ShortCycle p S J) : (shortCycleLength C : ℕ)+1 = C.val.val.card := by
+  have := C.val.nonempty.card_pos
+  simp only [shortCycleLength]
+  omega
+
+noncomputable def shortEmbeddingEquiv (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    (Fin A ↪ ShortCycle p S J) ≃
+      Σ r : Fin A → Fin J, SlotAssignment p S (fun i => (r i : ℕ)) where
+  toFun f := ⟨fun i => shortCycleLength (f i),
+    ⟨fun i => (f i).val, fun i j h => f.injective (Subtype.ext h),
+      fun i => (shortCycleLength_add_one (f i)).symm⟩⟩
+  invFun c :=
+    { toFun := fun i => ⟨c.2.val i, by rw [c.2.property.2]; exact (c.1 i).isLt⟩
+      inj' := fun _i _j h => c.2.property.1 (congrArg Subtype.val h) }
+  left_inv f := by
+    apply Function.Embedding.ext
+    intro i
+    rfl
+  right_inv c := by
+    rcases c with ⟨r,f⟩
+    have he : (fun i => shortCycleLength (⟨f.val i, by rw [f.property.2]; exact (r i).isLt⟩ : ShortCycle p S J)) = r := by
+      funext i
+      apply Fin.ext
+      simp only [shortCycleLength,f.property.2,Nat.add_sub_cancel]
+    apply Sigma.ext he
+    apply (Subtype.heq_iff_coe_eq (by
+      intro x
+      change (Function.Injective x ∧ ∀ i, (x i).val.card = (f.val i).val.card - 1 + 1) ↔ _
+      simp only [f.property.2,Nat.add_sub_cancel])).mpr
+    rfl
+
+lemma shortCycle_factorial_eq (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    (Fintype.card (ShortCycle p S J)).descFactorial A =
+      ∑ r : Fin A → Fin J, Fintype.card (CycleSlots p S (fun i => (r i : ℕ))) := by
+  classical
+  have hc := Fintype.card_embedding_eq (α := Fin A) (β := ShortCycle p S J)
+  simp only [Fintype.card_fin] at hc
+  rw [←hc, Fintype.card_congr (shortEmbeddingEquiv p S J),Fintype.card_sigma]
+  apply Finset.sum_congr rfl
+  intro r _
+  exact (Fintype.card_congr (cycleSlotsEquivAssignments p S (fun i => (r i : ℕ)))).symm
+
+end AllLengths
+
+section CycleMGF
+
+lemma butterfly_shortCycle_moment (S : Finset (α)) (J A : ℕ) :
+    finiteMean (fun ω : ι → Bool =>
+      ((Fintype.card (ShortCycle (R.perm ω) S J)).descFactorial A : ℝ)) ≤
+      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))^A := by
+  classical
+  simp_rw [shortCycle_factorial_eq,Nat.cast_sum,finiteMean_sum]
+  calc
+    _ ≤ ∑ r : Fin A → Fin J,
+        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+          ∏ i : Fin A, (1 / Real.sqrt ((r i : ℕ)+1)) := by
+      apply Finset.sum_le_sum
+      intro r _
+      exact butterfly_cycleSlots_moment_all R S (fun i => (r i : ℕ))
+    _ = _ := by
+      rw [←Finset.mul_sum,mul_pow]
+      congr 1
+      rw [←Fintype.prod_sum (fun (_ : Fin A) (j : Fin J) => (1 / Real.sqrt ((j:ℕ)+1) : ℝ))]
+      simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fin]
+
+noncomputable def cycleMass (p : Equiv.Perm (α)) (S : Finset (α)) : ℕ :=
+  ∑ C : SelectedCycle p S, C.val.card
+
+lemma cycleMass_le (p : Equiv.Perm (α)) (S : Finset (α)) : cycleMass p S ≤ S.card := by
+  classical
+  let f : (Σ C : SelectedCycle p S, {x : α // x ∈ C.val}) → S :=
+    fun c => ⟨c.2.val,c.1.property.2 c.2.property⟩
+  have hi : Function.Injective f := by
+    intro c e h
+    have hx : c.2.val = e.2.val := congrArg Subtype.val h
+    have hC : c.1 = e.1 := c.1.eq_of_mem e.1 c.2.property (hx ▸ e.2.property)
+    rcases c with ⟨c,x⟩
+    rcases e with ⟨e,y⟩
+    dsimp only at hC
+    subst e
+    have he : x=y := Subtype.ext hx
+    cases he
+    rfl
+  have hh := Fintype.card_le_of_injective f hi
+  simpa only [Fintype.card_sigma,Fintype.card_coe,cycleMass] using hh
+
+lemma shortCycle_card_le (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    Fintype.card (ShortCycle p S J) ≤ S.card := by
+  classical
+  calc
+    _ ≤ Fintype.card (SelectedCycle p S) := Fintype.card_subtype_le _
+    _ = ∑ _ : SelectedCycle p S, 1 := by simp
+    _ ≤ cycleMass p S := Finset.sum_le_sum (fun C _ => C.nonempty.card_pos)
+    _ ≤ _ := cycleMass_le p S
+
+lemma butterfly_shortCycle_mgf (S : Finset (α)) (J : ℕ) (q : ℝ) (hq : 1 ≤ q) :
+    finiteMean (fun ω : ι → Bool =>
+      q^(Fintype.card (ShortCycle (R.perm ω) S J))) ≤
+      Real.exp ((q-1)*((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))) := by
+  apply factorial_mgf _ S.card _ q (fun ω => shortCycle_card_le _ _ _)
+  · positivity
+  · exact hq
+  · exact butterfly_shortCycle_moment R S J
+
+end CycleMGF
+
+
+abbrev LongCycle (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :=
+  {C : SelectedCycle p S // ¬ C.val.card ≤ J}
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    Fintype (LongCycle p S J) := Fintype.ofFinite _
+
+lemma selectedCycle_card_split (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    Fintype.card (SelectedCycle p S) =
+      Fintype.card (ShortCycle p S J) + Fintype.card (LongCycle p S J) := by
+  classical
+  rw [←Fintype.card_congr (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J)),Fintype.card_sum]
+
+lemma longCycle_card_bound (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    J * Fintype.card (LongCycle p S J) ≤ S.card := by
+  classical
+  have hs := Equiv.sum_comp (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J))
+    (fun C : SelectedCycle p S => C.val.card)
+  rw [Fintype.sum_sum_type] at hs
+  have h0 : J * Fintype.card (LongCycle p S J) ≤
+      ∑ C : LongCycle p S J, C.val.val.card := by
+    calc
+      _ = ∑ _C : LongCycle p S J, J := by simp [mul_comm]
+      _ ≤ _ := Finset.sum_le_sum (fun C _ => Nat.le_of_lt (Nat.lt_of_not_ge C.property))
+  have hh := cycleMass_le p S
+  unfold cycleMass at hh
+  rw [←hs] at hh
+  exact h0.trans ((Nat.le_add_left _ _).trans hh)
+
+lemma longCycle_pow_bound (p : Equiv.Perm (α)) (S : Finset (α))
+    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    q^(Fintype.card (LongCycle p S J)) ≤ Real.exp ((S.card : ℝ)/J * Real.log q) := by
+  have hq0 : 0 < q := lt_of_lt_of_le zero_lt_one hq
+  have hc : (Fintype.card (LongCycle p S J) : ℝ) ≤ (S.card : ℝ)/J := by
+    apply (le_div_iff₀ (by exact_mod_cast hJ)).mpr
+    have hh := longCycle_card_bound p S J
+    exact_mod_cast (by simpa only [mul_comm] using hh)
+  calc
+    _ = Real.exp ((Fintype.card (LongCycle p S J) : ℝ)*Real.log q) := by
+      rw [Real.exp_nat_mul,Real.exp_log hq0]
+    _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right hc (Real.log_nonneg hq))
+
+lemma butterfly_cycle_log_mgf (S : Finset (α)) (J : ℕ) (hJ : 0 < J)
+    (q : ℝ) (hq : 1 ≤ q) :
+    Real.log (finiteMean (fun ω : ι → Bool =>
+      q^(Fintype.card (SelectedCycle (R.perm ω) S)))) ≤
+      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
+  let lam : ℝ := (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1)
+  let c : ℝ := (S.card : ℝ)/J * Real.log q
+  have hnum : (q-1)*lam ≤ 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
+    have hsum := sum_inv_sqrt J
+    have hp : 0 ≤ (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) := by positivity
+    have h0 : 0 ≤ 2*Real.sqrt J := by positivity
+    have hh := mul_le_mul_of_nonneg_left hsum hp
+    have hq1 : 0 ≤ q-1 := sub_nonneg.mpr hq
+    calc
+      _ ≤ (q-1)*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
+        mul_le_mul_of_nonneg_left hh hq1
+      _ ≤ q*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
+        mul_le_mul_of_nonneg_right (by linarith) (mul_nonneg hp h0)
+      _ = _ := by ring
+  have hm : finiteMean (fun ω : ι → Bool =>
+      q^(Fintype.card (SelectedCycle (R.perm ω) S))) ≤
+      Real.exp (c+(q-1)*lam) := by
+    calc
+      _ ≤ finiteMean (fun ω : ι → Bool =>
+          q^(Fintype.card (ShortCycle (R.perm ω) S J))*Real.exp c) := by
+        apply finiteMean_mono
+        intro ω
+        rw [selectedCycle_card_split _ _ J,pow_add]
+        exact mul_le_mul_of_nonneg_left (longCycle_pow_bound _ S J hJ q hq) (by positivity)
+      _ = finiteMean (fun ω : ι → Bool =>
+          q^(Fintype.card (ShortCycle (R.perm ω) S J))) * Real.exp c :=
+        finiteMean_mul_const _ _
+      _ ≤ Real.exp ((q-1)*lam) * Real.exp c :=
+        mul_le_mul_of_nonneg_right (butterfly_shortCycle_mgf R S J q hq) (Real.exp_nonneg _)
+      _ = _ := by rw [←Real.exp_add,add_comm]
+  have hpos : 0 < finiteMean (fun ω : ι → Bool =>
+      q^(Fintype.card (SelectedCycle (R.perm ω) S))) := by
+    apply finiteMean_pos
+    intro ω
+    positivity
+  have hlog := Real.log_le_log hpos hm
+  rw [Real.log_exp] at hlog
+  dsimp [c] at hlog
+  linarith
+
+
+end RoutingNetwork
+
+variable {α B : Type*} [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] {d : ℕ}
+
+def parallelPerm (ω : B × SwitchIndex d → Bool) : Equiv.Perm (B × Card d) where
+  toFun x := (x.1,butterflyPerm d (decodeButterfly d (fun i => ω (x.1,i))) x.2)
+  invFun y := (y.1,(butterflyPerm d (decodeButterfly d (fun i => ω (y.1,i)))).symm y.2)
+  left_inv x := by simp
+  right_inv y := by simp
+
+def relabeledPerm (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (ω : B × SwitchIndex d → Bool) : Equiv.Perm α :=
+  (pre.trans (parallelPerm ω)).trans post
+
+noncomputable def parallelPath (pre : α ≃ B × Card d)
+    (ω : B × SwitchIndex d → Bool) (x : α) : Finset (B × SwitchIndex d) :=
+  (routeDomain d (pre x).2
+    (butterflyPerm d (decodeButterfly d (fun i => ω ((pre x).1,i))) (pre x).2)).image
+      (fun i => ((pre x).1,i))
+
+def parallelUsers (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
+    (i : B × SwitchIndex d) : α × α :=
+  let u := switchUsers d (decodeButterfly d (fun j => ω (i.1,j))) i.2
+  (pre.symm (i.1,u.1),pre.symm (i.1,u.2))
+
+omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
+lemma parallelUsers_ne (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
+    (i : B × SwitchIndex d) : (parallelUsers pre ω i).1 ≠ (parallelUsers pre ω i).2 := by
+  intro h
+  exact switchUsers_ne d _ _ (congrArg Prod.snd (pre.symm.injective h))
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma mem_parallelPath (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
+    (x : α) (i : B × SwitchIndex d) :
+    i ∈ parallelPath pre ω x ↔ (parallelUsers pre ω i).1 = x ∨ (parallelUsers pre ω i).2 = x := by
+  classical
+  obtain ⟨b,j⟩ := i
+  simp only [parallelPath,Finset.mem_image,Prod.mk.injEq]
+  constructor
+  · rintro ⟨k,hk,hb,rfl⟩
+    subst b
+    rcases (mem_routeDomain_iff_user d _ _ _).mp hk with h | h
+    · left
+      apply pre.injective
+      change pre (pre.symm ((pre x).1,_)) = pre x
+      rw [Equiv.apply_symm_apply]
+      exact Prod.ext rfl h
+    · right
+      apply pre.injective
+      change pre (pre.symm ((pre x).1,_)) = pre x
+      rw [Equiv.apply_symm_apply]
+      exact Prod.ext rfl h
+  · intro h
+    have hh : (b,(switchUsers d (decodeButterfly d (fun j => ω (b,j))) j).1) = pre x ∨
+        (b,(switchUsers d (decodeButterfly d (fun j => ω (b,j))) j).2) = pre x := by
+      rcases h with h | h
+      · exact Or.inl (by simpa only [parallelUsers,Equiv.apply_symm_apply] using congrArg pre h)
+      · exact Or.inr (by simpa only [parallelUsers,Equiv.apply_symm_apply] using congrArg pre h)
+    have hb : b = (pre x).1 := hh.elim (congrArg Prod.fst) (congrArg Prod.fst)
+    subst b
+    refine ⟨j,?_,rfl,rfl⟩
+    apply (mem_routeDomain_iff_user d _ _ _).mpr
+    exact hh.elim (fun h => Or.inl (congrArg Prod.snd h)) (fun h => Or.inr (congrArg Prod.snd h))
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma card_parallelPath (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool) (x : α) :
+    (parallelPath pre ω x).card = d := by
+  classical
+  rw [parallelPath,Finset.card_image_of_injective _ (fun _ _ h => Prod.mk.inj h |>.2),card_routeDomain]
+
+noncomputable def blockMarks (pre : α ≃ B × Card d) (S : Finset α) (b : B) : Finset (Card d) :=
+  Finset.univ.filter (fun x => pre.symm (b,x) ∈ S)
+
+omit [Fintype α] [Fintype B] [DecidableEq B] in
+@[simp] lemma mem_blockMarks (pre : α ≃ B × Card d) (S : Finset α) (b : B) (x : Card d) :
+    x ∈ blockMarks pre S b ↔ pre.symm (b,x) ∈ S := by
+  classical
+  simp [blockMarks]
+
+omit [DecidableEq B] in
+lemma sum_card_blockMarks (pre : α ≃ B × Card d) (S : Finset α) :
+    ∑ b : B, (blockMarks pre S b).card = S.card := by
+  classical
+  calc
+    _ = ∑ b : B, ∑ x : Card d, if pre.symm (b,x) ∈ S then 1 else 0 := by
+      simp only [blockMarks,Finset.card_filter]
+    _ = ∑ u : B × Card d, if pre.symm u ∈ S then 1 else 0 := (Fintype.sum_prod_type (fun u : B × Card d => if pre.symm u ∈ S then (1:ℕ) else 0)).symm
+    _ = ∑ x : α, if x ∈ S then 1 else 0 := Equiv.sum_comp pre.symm (fun x : α => if x ∈ S then (1:ℕ) else 0)
+    _ = _ := by simp
+
+omit [DecidableEq B] in
+lemma card_blockMarks_le (pre : α ≃ B × Card d) (S : Finset α) (b : B) :
+    (blockMarks pre S b).card ≤ S.card := by
+  rw [←sum_card_blockMarks pre S]
+  exact Finset.single_le_sum (f := fun b : B => (blockMarks pre S b).card) (fun _ _ => Nat.zero_le _) (Finset.mem_univ b)
+
+end Thorp
+
+end ThorpNine.SparseSaving
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/SparseSaving/Representative.lean b/lean/OAI/Probability/ThorpRouting/SparseSaving/Representative.lean
index efdb126563ceb26e4e27fc6a55d89d42882f48b2..d09582b80efffb74e632364171a4126e5a099e2b 100644
--- a/lean/OAI/Probability/ThorpRouting/SparseSaving/Representative.lean
+++ b/lean/OAI/Probability/ThorpRouting/SparseSaving/Representative.lean
@@ -1,893 +1,893 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.SparseSaving.ShapePartitionEquiv
-
-namespace OAI
-
-namespace ThorpNine.SparseSaving
-
-namespace Thorp.UnitaryFinite
-open scoped BigOperators ComplexConjugate Classical
-variable {G : Type*} [Group G] [Fintype G]
-variable {V W : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-  [NormedAddCommGroup W] [InnerProductSpace ℂ W] [FiniteDimensional ℂ W]
-omit [FiniteDimensional ℂ V] in
-lemma averageMap_self_trace (ρ : Representation ℂ G V) (A : Module.End ℂ V) :
-    LinearMap.trace ℂ V (averageMap ρ ρ A).toLinearMap =
-      (Fintype.card G:ℂ) * LinearMap.trace ℂ V A := by
-  change LinearMap.trace ℂ V (∑ g, ρ g * A * ρ g⁻¹) = _
-  rw [map_sum]
-  have hj (g : G) : LinearMap.trace ℂ V (ρ g*A*ρ g⁻¹) = LinearMap.trace ℂ V A := by
-    rw [LinearMap.trace_mul_cycle,←map_mul,inv_mul_cancel,map_one,one_mul]
-  simp only [hj,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
-
-lemma averageMap_scalar (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
-    (A : Module.End ℂ V) :
-    ∃ c : ℂ, (averageMap ρ ρ A).toLinearMap = c • LinearMap.id ∧
-      c * (Module.finrank ℂ V:ℂ) = (Fintype.card G:ℂ)*LinearMap.trace ℂ V A := by
-  obtain ⟨c,hc⟩ := (Representation.IsIrreducible.algebraMap_intertwiningMap_bijective_of_isAlgClosed (ρ := ρ)).surjective (averageMap ρ ρ A)
-  have he : (averageMap ρ ρ A).toLinearMap = c • LinearMap.id := by
-    rw [←hc]
-    rfl
-  refine ⟨c,he,?_⟩
-  rw [←averageMap_self_trace ρ A,he,map_smul,LinearMap.trace_id,smul_eq_mul]
-
-theorem coefficient_orthogonality (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
-    (hρ : IsUnitary ρ) (u v w z : V) :
-    (Module.finrank ℂ V:ℂ) * ∑ g, inner ℂ u (ρ g v) * inner ℂ (ρ g w) z =
-      (Fintype.card G:ℂ) * inner ℂ w v * inner ℂ u z := by
-  obtain ⟨c,hc,htrace⟩ := averageMap_scalar ρ (InnerProductSpace.rankOne ℂ v w).toLinearMap
-  rw [InnerProductSpace.trace_rankOne] at htrace
-  have happ := congrArg (fun A : Module.End ℂ V => inner ℂ u (A z)) hc
-  change inner ℂ u ((∑ g, ρ g * (InnerProductSpace.rankOne ℂ v w).toLinearMap * ρ g⁻¹) z) = _ at happ
-  simp only [LinearMap.sum_apply,Module.End.mul_apply,ContinuousLinearMap.coe_coe,
-    InnerProductSpace.rankOne_apply,map_smul,unitary_inner_inv ρ hρ,inner_sum,
-    inner_smul_right,LinearMap.smul_apply,LinearMap.id_apply] at happ
-  have hsum : ∑ g, inner ℂ u (ρ g v) * inner ℂ (ρ g w) z = c * inner ℂ u z := by
-    simpa only [mul_comm] using happ
-  rw [hsum]
-  calc
-    _ = (c*(Module.finrank ℂ V:ℂ))*inner ℂ u z := by ring
-    _ = _ := by rw [htrace]
-
-end Thorp.UnitaryFinite
-namespace Thorp.UnitaryFinite
-open scoped BigOperators ComplexConjugate Classical
-
-variable {G X : Type*} [Group G] [Fintype G] [Fintype X]
-  (a : G →* Equiv.Perm X) (x₀ : X) (ha : ∀ x, ∃ g, a g x₀=x)
-
-noncomputable def representative (x : X) : G := Classical.choose (ha x)
-
-omit [Fintype G] [Fintype X] in
-lemma representative_apply (x : X) : a (representative a x₀ ha x) x₀=x :=
-  Classical.choose_spec (ha x)
-
-include ha in
-lemma orbit_sum (F : X → ℂ) :
-    (Fintype.card X:ℂ)*∑ g, F (a g x₀) = (Fintype.card G:ℂ)*∑ x, F x := by
-  have ht (x : X) : ∑ g, F (a g x)=∑ g, F (a g x₀) := by
-    obtain ⟨t,rfl⟩ := ha x
-    symm
-    apply Fintype.sum_equiv (Equiv.mulRight t⁻¹)
-    intro g
-    simp
-  calc
-    _ = ∑ x : X, ∑ g : G, F (a g x) := by simp only [ht,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
-    _ = ∑ g : G, ∑ x : X, F (a g x) := Finset.sum_comm
-    _ = ∑ g : G, ∑ x : X, F x := by
-      apply Finset.sum_congr rfl
-      intro g _
-      exact Equiv.sum_comp (a g) F
-    _ = _ := by simp
-
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V)
-
-def IsFixed (w : V) : Prop := ∀ g, a g x₀=x₀ → ρ g w=w
-
-noncomputable def coefficient (w v : V) : EuclideanSpace ℂ X :=
-  WithLp.toLp 2 (fun x => inner ℂ (ρ (representative a x₀ ha x) w) v)
-
-omit [Fintype G] [Fintype X] [FiniteDimensional ℂ V] in
-lemma coefficient_at (w : V) (hw : IsFixed a x₀ ρ w) (v : V) (g : G) :
-    coefficient a x₀ ha ρ w v (a g x₀) = inner ℂ (ρ g w) v := by
-  let r := representative a x₀ ha (a g x₀)
-  have hr : a r x₀=a g x₀ := representative_apply a x₀ ha _
-  have hs : a (g⁻¹*r) x₀=x₀ := by
-    simp only [map_mul,Equiv.Perm.mul_apply]
-    rw [hr,←Equiv.Perm.mul_apply,←map_mul,inv_mul_cancel,map_one]
-    rfl
-  have he := congrArg (ρ g) (hw (g⁻¹*r) hs)
-  have hh : ρ r w=ρ g w := by
-    simpa only [map_mul,Module.End.mul_apply,←Module.End.mul_apply,←map_mul,
-      mul_inv_cancel_left] using he
-  change inner ℂ (ρ r w) v= _
-  rw [hh]
-
-lemma coefficient_inner [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
-    (w z : V) (hw : IsFixed a x₀ ρ w) (hz : IsFixed a x₀ ρ z) (v t : V) :
-    (Module.finrank ℂ V:ℂ)*inner ℂ (coefficient a x₀ ha ρ w v) (coefficient a x₀ ha ρ z t) =
-      (Fintype.card X:ℂ)*inner ℂ z w*inner ℂ v t := by
-  have hs := orbit_sum a x₀ ha (fun x => star (coefficient a x₀ ha ρ w v x)*coefficient a x₀ ha ρ z t x)
-  simp only [coefficient_at a x₀ ha ρ w hw,coefficient_at a x₀ ha ρ z hz] at hs
-  simp only [←starRingEnd_apply,inner_conj_symm] at hs
-  simp only [starRingEnd_apply] at hs
-  have ho := coefficient_orthogonality ρ hρ v w z t
-  have hcard : (Fintype.card G:ℂ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
-  apply mul_left_cancel₀ hcard
-  rw [PiLp.inner_apply]
-  simp only [RCLike.inner_apply,starRingEnd_apply]
-  have hsum : ∑ x, coefficient a x₀ ha ρ z t x * star (coefficient a x₀ ha ρ w v x) =
-      ∑ x, star (coefficient a x₀ ha ρ w v x)*coefficient a x₀ ha ρ z t x := by
-    apply Finset.sum_congr rfl
-    intro x _
-    ring
-  rw [hsum]
-  calc
-    _ = (Module.finrank ℂ V:ℂ)*((Fintype.card G:ℂ)*∑ x,
-        star (coefficient a x₀ ha ρ w v x)*coefficient a x₀ ha ρ z t x) := by ring
-    _ = (Fintype.card X:ℂ)*((Module.finrank ℂ V:ℂ)*∑ g,
-        inner ℂ v (ρ g w)*inner ℂ (ρ g z) t) := by rw [←hs]; ring
-    _ = _ := by rw [ho]; ring
-
-omit [Fintype G] [Fintype X] [FiniteDimensional ℂ V] in
-lemma coefficient_action (hρ : IsUnitary ρ) (w : V) (hw : IsFixed a x₀ ρ w)
-    (v : V) (g : G) (x : X) :
-    coefficient a x₀ ha ρ w v (a g x) = coefficient a x₀ ha ρ w (ρ g⁻¹ v) x := by
-  obtain ⟨r,rfl⟩ := ha x
-  rw [←Equiv.Perm.mul_apply,←map_mul,coefficient_at a x₀ ha ρ w hw,
-    coefficient_at a x₀ ha ρ w hw]
-  simp only [map_mul,Module.End.mul_apply,unitary_inner_inv ρ hρ]
-
-noncomputable def normalizedCoefficient (w v : V) : EuclideanSpace ℂ X :=
-  (Real.sqrt ((Module.finrank ℂ V:ℝ)/(Fintype.card X:ℝ)) : ℂ) • coefficient a x₀ ha ρ w v
-
-lemma normalizedCoefficient_inner [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
-    (w z : V) (hw : IsFixed a x₀ ρ w) (hz : IsFixed a x₀ ρ z) (v t : V) :
-    inner ℂ (normalizedCoefficient a x₀ ha ρ w v) (normalizedCoefficient a x₀ ha ρ z t) =
-      inner ℂ z w*inner ℂ v t := by
-  let : Nonempty X := ⟨x₀⟩
-  have hX : (Fintype.card X:ℂ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
-  have hs : (Real.sqrt ((Module.finrank ℂ V:ℝ)/(Fintype.card X:ℝ)) : ℂ)^2 =
-      (Module.finrank ℂ V:ℂ)/(Fintype.card X:ℂ) := by
-    rw [←Complex.ofReal_pow,Real.sq_sqrt (by positivity)]
-    simp
-  simp only [normalizedCoefficient,inner_smul_left,inner_smul_right]
-  simp only [Complex.conj_ofReal]
-  rw [←mul_assoc,←pow_two,hs]
-  have hi := coefficient_inner a x₀ ha ρ hρ w z hw hz v t
-  field_simp
-  simpa only [mul_assoc] using hi
-
-lemma normalizedCoefficient_orthonormal [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
-    {I : Type*} (w : I → V) (hw : Orthonormal ℂ w)
-    (hfix : ∀ i, IsFixed a x₀ ρ (w i)) (v : V) (hv : ‖v‖=1) :
-    Orthonormal ℂ (fun i => normalizedCoefficient a x₀ ha ρ (w i) v) := by
-  rw [orthonormal_iff_ite] at hw ⊢
-  intro i j
-  rw [normalizedCoefficient_inner a x₀ ha ρ hρ _ _ (hfix i) (hfix j),hw,
-    inner_self_eq_norm_sq_to_K,hv]
-  simp [eq_comm]
-
-lemma normalizedCoefficient_norm [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
-    (w : V) (hw : IsFixed a x₀ ρ w) (hn : ‖w‖=1) (v : V) :
-    ‖normalizedCoefficient a x₀ ha ρ w v‖=‖v‖ := by
-  have he := normalizedCoefficient_inner a x₀ ha ρ hρ w w hw hw v v
-  rw [inner_self_eq_norm_sq_to_K,inner_self_eq_norm_sq_to_K,hn] at he
-  simp only [inner_self_eq_norm_sq_to_K] at he
-  norm_num at he
-  have hs : ‖normalizedCoefficient a x₀ ha ρ w v‖^2=‖v‖^2 := by exact_mod_cast he
-  nlinarith [norm_nonneg (normalizedCoefficient a x₀ ha ρ w v),norm_nonneg v]
-
-end Thorp.UnitaryFinite
-
-namespace Thorp.FiniteConvolution
-open scoped BigOperators Classical
-variable {G : Type*} [Group G] [Fintype G]
-
-noncomputable def conv (p q : G → ℝ) (g : G) : ℝ := ∑ h, p h*q (h⁻¹*g)
-noncomputable def power (p : G → ℝ) : ℕ → G → ℝ
-  | 0 => fun g => if g=1 then 1 else 0
-  | n+1 => conv p (power p n)
-
-lemma conv_nonneg {p q : G → ℝ} (hp : ∀ g, 0 ≤ p g) (hq : ∀ g, 0 ≤ q g) :
-    ∀ g, 0 ≤ conv p q g := fun _ => Finset.sum_nonneg (fun h _ => mul_nonneg (hp h) (hq _))
-
-lemma conv_pos_of {p q : G → ℝ} (hp : ∀ g, 0 ≤ p g) (hq : ∀ g, 0 ≤ q g)
-    {a b : G} (ha : 0 < p a) (hb : 0 < q b) : 0 < conv p q (a*b) := by
-  apply lt_of_lt_of_le (mul_pos ha hb)
-  have h := Finset.single_le_sum (fun g _ => mul_nonneg (hp g) (hq (g⁻¹*(a*b))))
-    (Finset.mem_univ a)
-  simpa [conv] using h
-
-lemma sum_conv (p q : G → ℝ) : ∑ g, conv p q g=(∑ g,p g)*(∑ g,q g) := by
-  unfold conv
-  rw [Finset.sum_comm,Finset.sum_mul]
-  apply Finset.sum_congr rfl
-  intro h _
-  rw [←Finset.mul_sum]
-  congr 1
-  exact Fintype.sum_equiv (Equiv.mulLeft h⁻¹) _ _ (fun _ => rfl)
-
-lemma power_nonneg {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) : ∀ n g, 0 ≤ power p n g := by
-  intro n
-  induction n with
-  | zero => intro g; simp only [power]; split_ifs  <;> norm_num
-  | succ n ih => exact conv_nonneg hp ih
-
-lemma sum_power {p : G → ℝ} (hp : ∑ g,p g=1) : ∀ n, ∑ g,power p n g=1 := by
-  intro n
-  induction n with
-  | zero => simp [power]
-  | succ n ih => rw [power,sum_conv,hp,ih,one_mul]
-
-lemma power_list_pos {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) (l : List G)
-    (hl : ∀ g ∈ l, 0 < p g) : 0 < power p l.length l.prod := by
-  induction l with
-  | nil => simp [power]
-  | cons a l ih =>
-    simpa only [List.length_cons,List.prod_cons,power] using
-      conv_pos_of hp (power_nonneg hp l.length) (hl a (by simp))
-        (ih (fun g hg => hl g (by simp [hg])))
-
-lemma power_pos_mono {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) (h1 : 0 < p 1)
-    {n m : ℕ} (hnm : n ≤ m) {g : G} (hg : 0 < power p n g) : 0 < power p m g := by
-  induction m, hnm using Nat.le_induction with
-  | base => exact hg
-  | succ m _ ih =>
-    simpa only [one_mul,power] using conv_pos_of hp (power_nonneg hp m) h1 ih
-
-theorem exists_positive_power {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) (h1 : 0 < p 1)
-    (hgen : Subgroup.closure {g | 0 < p g}=⊤) :
-    ∃ n : ℕ, ∀ g, 0 < power p (2^n) g := by
-  have hx (g : G) : g ∈ Submonoid.closure {g | 0 < p g} := by
-    rw [←Subgroup.closure_toSubmonoid_of_finite,hgen]
-    trivial
-  choose l hl hprod using fun g => Submonoid.exists_list_of_mem_closure (hx g)
-  let n := Finset.univ.sup (fun g => (l g).length)
-  refine ⟨n,fun g => ?_⟩
-  have hlen : (l g).length ≤ n := Finset.le_sup (f := fun g => (l g).length) (Finset.mem_univ g)
-  have hpow : n ≤ 2^n := Nat.le_of_lt (Nat.lt_two_pow_self)
-  have hpos := power_list_pos hp (l g) (hl g)
-  rw [hprod g] at hpos
-  exact power_pos_mono hp h1 (hlen.trans hpow) hpos
-
-structure Minorant (p : G → ℝ) where
-  exponent : ℕ
-  mass : ℝ
-  mass_pos : 0 < mass
-  mass_le_one : mass ≤ 1
-  lower : ∀ g, mass/(Fintype.card G:ℝ) ≤ power p (2^exponent) g
-
-noncomputable def minorant {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) (hsum : ∑ g,p g=1)
-    (h1 : 0 < p 1) (hgen : Subgroup.closure {g | 0 < p g}=⊤) : Minorant p := by
-  let n := Classical.choose (exists_positive_power hp h1 hgen)
-  have hn := Classical.choose_spec (exists_positive_power hp h1 hgen)
-  let a := Finset.univ.inf' Finset.univ_nonempty (power p (2^n))
-  have ha : 0 < a := (Finset.lt_inf'_iff _).mpr (fun g _ => hn g)
-  have hag (g : G) : a ≤ power p (2^n) g := Finset.inf'_le _ (Finset.mem_univ g)
-  have hN : (0:ℝ) < Fintype.card G := Nat.cast_pos.mpr Fintype.card_pos
-  have hsum' := Finset.sum_le_sum (s := Finset.univ) (fun g _ => hag g)
-  simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,sum_power hsum] at hsum'
-  refine ⟨n,(Fintype.card G:ℝ)*a,mul_pos hN ha,hsum',fun g => ?_⟩
-  have he : (Fintype.card G:ℝ)*a/(Fintype.card G:ℝ)=a := by field_simp
-  rw [he]
-  exact hag g
-
-end Thorp.FiniteConvolution
-namespace Thorp.UnitaryFinite
-
-section
-open scoped BigOperators ComplexConjugate Classical
-
-variable {G : Type*} [Group G] [Fintype G]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-omit [Fintype G] in
-@[simp] lemma continuousRepresentation_apply (ρ : Representation ℂ G V) (g : G) (v : V) :
-    continuousRepresentation ρ g v=ρ g v := rfl
-
-omit [Fintype G] [FiniteDimensional ℂ V] in
-lemma unitary_norm (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (g : G) (v : V) :
-    ‖ρ g v‖=‖v‖ := by
-  rw [norm_eq_sqrt_re_inner (𝕜 := ℂ),hρ,norm_eq_sqrt_re_inner (𝕜 := ℂ)]
-
-omit [Fintype G] in
-lemma continuousRepresentation_norm_le (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (g : G) :
-    ‖continuousRepresentation ρ g‖≤1 := by
-  apply ContinuousLinearMap.opNorm_le_bound _ zero_le_one
-  intro v
-  simp only [continuousRepresentation_apply,unitary_norm ρ hρ,one_mul,le_refl]
-
-omit [Fintype G] in
-lemma continuousRepresentation_adjoint (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (g : G) :
-    (continuousRepresentation ρ g).adjoint=continuousRepresentation ρ g⁻¹ := by
-  ext w
-  apply ext_inner_left ℂ
-  intro v
-  rw [ContinuousLinearMap.adjoint_inner_right]
-  exact (unitary_inner_inv ρ hρ g v w).symm
-
-noncomputable def weightedOperator (ρ : Representation ℂ G V) (p : G → ℝ) : V →L[ℂ] V :=
-  ∑ g, (p g : ℂ) • continuousRepresentation ρ g
-
-lemma weightedOperator_norm_le (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
-    (p : G → ℝ) (hp : ∀ g, 0≤p g) : ‖weightedOperator ρ p‖ ≤ ∑ g, p g := by
-  apply (norm_sum_le _ _).trans
-  apply Finset.sum_le_sum
-  intro g _
-  rw [norm_smul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (hp g)]
-  exact mul_le_of_le_one_right (hp g) (continuousRepresentation_norm_le ρ hρ g)
-
-lemma weightedOperator_symmetric (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
-    (p : G → ℝ) (hp : ∀ g, p g⁻¹=p g) : IsSelfAdjoint (weightedOperator ρ p) := by
-  change (weightedOperator ρ p).adjoint=weightedOperator ρ p
-  unfold weightedOperator
-  rw [map_sum]
-  simp only [map_smulₛₗ,Complex.conj_ofReal,continuousRepresentation_adjoint ρ hρ]
-  apply Fintype.sum_equiv (Equiv.inv G)
-  intro g
-  simp [hp]
-
-noncomputable def uniformOperator (ρ : Representation ℂ G V) : V →L[ℂ] V :=
-  weightedOperator ρ (fun _ => (Fintype.card G:ℝ)⁻¹)
-
-lemma uniformOperator_symmetric (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) :
-    IsSelfAdjoint (uniformOperator ρ) := weightedOperator_symmetric ρ hρ _ (fun _ => rfl)
-
-lemma uniformOperator_left (ρ : Representation ℂ G V) (g : G) :
-    continuousRepresentation ρ g * uniformOperator ρ=uniformOperator ρ := by
-  unfold uniformOperator weightedOperator
-  rw [Finset.mul_sum]
-  simp only [mul_smul_comm,←map_mul]
-  exact Fintype.sum_equiv (Equiv.mulLeft g) _ _ (fun _ => rfl)
-
-lemma uniformOperator_right (ρ : Representation ℂ G V) (g : G) :
-    uniformOperator ρ * continuousRepresentation ρ g=uniformOperator ρ := by
-  unfold uniformOperator weightedOperator
-  rw [Finset.sum_mul]
-  simp only [smul_mul_assoc,←map_mul]
-  exact Fintype.sum_equiv (Equiv.mulRight g) _ _ (fun _ => rfl)
-
-lemma uniformOperator_idempotent (ρ : Representation ℂ G V) :
-    IsIdempotentElem (uniformOperator ρ) := by
-  change uniformOperator ρ * uniformOperator ρ=uniformOperator ρ
-  nth_rw 1 [uniformOperator,weightedOperator]
-  rw [Finset.sum_mul]
-  simp only [smul_mul_assoc,uniformOperator_left,Finset.sum_const,Finset.card_univ,
-    ←Nat.cast_smul_eq_nsmul ℂ,smul_smul,Complex.ofReal_inv,Complex.ofReal_natCast]
-  rw [mul_inv_cancel₀ (by exact_mod_cast Fintype.card_ne_zero),one_smul]
-
-lemma uniformOperator_projection (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) :
-    IsStarProjection (uniformOperator ρ) :=
-  ⟨uniformOperator_idempotent ρ,uniformOperator_symmetric ρ hρ⟩
-
-lemma weighted_uniform_left (ρ : Representation ℂ G V) (p : G → ℝ) :
-    weightedOperator ρ p * uniformOperator ρ=(∑ g,p g : ℝ) • uniformOperator ρ := by
-  unfold weightedOperator
-  rw [Finset.sum_mul]
-  simp only [smul_mul_assoc,uniformOperator_left,←Finset.sum_smul,←Complex.ofReal_sum]
-  rfl
-
-lemma weighted_uniform_right (ρ : Representation ℂ G V) (p : G → ℝ) :
-    uniformOperator ρ * weightedOperator ρ p=(∑ g,p g : ℝ) • uniformOperator ρ := by
-  unfold weightedOperator
-  rw [Finset.mul_sum]
-  simp only [mul_smul_comm,uniformOperator_right,←Finset.sum_smul,←Complex.ofReal_sum]
-  rfl
-
-theorem doeblin_off_invariants (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
-    (p : G → ℝ) (hp : ∑ g,p g=1) (ε : ℝ)
-    (hminor : ∀ g, ε/(Fintype.card G:ℝ)≤p g) :
-    ‖weightedOperator ρ p * (1-uniformOperator ρ)‖≤1-ε := by
-  let q : G → ℝ := fun g => p g-ε/(Fintype.card G:ℝ)
-  have hq : ∀ g, 0≤q g := fun g => sub_nonneg.mpr (hminor g)
-  have hcard : (Fintype.card G:ℝ)≠0 := by exact_mod_cast Fintype.card_ne_zero
-  have hsum : ∑ g,q g=1-ε := by
-    simp only [q,Finset.sum_sub_distrib,hp,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
-    field_simp
-  have he : weightedOperator ρ p=weightedOperator ρ q+(ε:ℂ) • uniformOperator ρ := by
-    simp only [uniformOperator,weightedOperator]
-    rw [Finset.smul_sum,←Finset.sum_add_distrib]
-    apply Finset.sum_congr rfl
-    intro g _
-    simp only [q,Complex.ofReal_sub,Complex.ofReal_div,smul_smul,←add_smul,
-      Complex.ofReal_inv,Complex.ofReal_natCast]
-    congr 1
-    field_simp
-    ring
-  have hz : uniformOperator ρ*(1-uniformOperator ρ)=0 := by
-    rw [mul_sub,mul_one,(uniformOperator_idempotent ρ).eq,sub_self]
-  rw [he,add_mul,smul_mul_assoc,hz,smul_zero,add_zero]
-  calc
-    _ ≤ ‖weightedOperator ρ q‖*‖1-uniformOperator ρ‖ := norm_mul_le _ _
-    _ ≤ (1-ε)*1 := mul_le_mul (by rw [←hsum]; exact weightedOperator_norm_le ρ hρ q hq)
-      ((uniformOperator_projection ρ hρ).one_sub.norm_le _) (norm_nonneg _)
-      (by rw [←hsum]; exact Finset.sum_nonneg (fun g _ => hq g))
-    _ = _ := mul_one _
-
-end
-open scoped BigOperators Classical
-variable {G : Type*} [Group G] [Fintype G]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-lemma weightedOperator_conv (ρ : Representation ℂ G V) (p q : G → ℝ) :
-    weightedOperator ρ (FiniteConvolution.conv p q)=weightedOperator ρ p*weightedOperator ρ q := by
-  unfold weightedOperator FiniteConvolution.conv
-  simp only [Complex.ofReal_sum,Finset.sum_smul]
-  rw [Finset.sum_comm,Finset.sum_mul]
-  apply Finset.sum_congr rfl
-  intro h _
-  rw [Finset.mul_sum]
-  simp only [smul_mul_assoc,mul_smul_comm,smul_smul,←map_mul]
-  apply Fintype.sum_equiv (Equiv.mulLeft h⁻¹)
-  intro g
-  simp [Complex.ofReal_mul,mul_comm]
-
-lemma weightedOperator_power (ρ : Representation ℂ G V) (p : G → ℝ) (n : ℕ) :
-    weightedOperator ρ (FiniteConvolution.power p n)=(weightedOperator ρ p)^n := by
-  induction n with
-  | zero => simp [FiniteConvolution.power,weightedOperator]
-  | succ n ih => rw [FiniteConvolution.power,weightedOperator_conv,ih,pow_succ']
-
-lemma off_invariants_pow (ρ : Representation ℂ G V) (p : G → ℝ)
-    (hp : ∑ g,p g=1) (n : ℕ) (hn : n≠0) :
-    (weightedOperator ρ p-uniformOperator ρ)^n=
-      (weightedOperator ρ p)^n*(1-uniformOperator ρ) := by
-  have hAP := weighted_uniform_left ρ p
-  have hPA := weighted_uniform_right ρ p
-  rw [hp,one_smul] at hAP hPA
-  have hc : Commute (weightedOperator ρ p) (1-uniformOperator ρ) := by
-    show _*_= _*_
-    rw [mul_sub,sub_mul,mul_one,one_mul,hAP,hPA]
-  have he : weightedOperator ρ p-uniformOperator ρ=
-      weightedOperator ρ p*(1-uniformOperator ρ) := by rw [mul_sub,mul_one,hAP]
-  rw [he,hc.mul_pow,((uniformOperator_idempotent ρ).one_sub).pow_eq hn]
-
-noncomputable def minorantGap {p : G → ℝ} (c : FiniteConvolution.Minorant p) : ℝ :=
-  1-c.mass/(2*(2:ℝ)^c.exponent)
-
-omit [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V] in
-lemma minorantGap_pos {p : G → ℝ} (c : FiniteConvolution.Minorant p) : 0 < minorantGap c := by
-  have hn : (1:ℝ) ≤ 2^c.exponent := one_le_pow₀ (by norm_num)
-  have hq : c.mass/(2*(2:ℝ)^c.exponent) ≤ 1/2 := by
-    apply (div_le_iff₀ (by positivity)).mpr
-    nlinarith [c.mass_le_one]
-  unfold minorantGap
-  linarith
-
-omit [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V] in
-lemma minorantGap_lt_one {p : G → ℝ} (c : FiniteConvolution.Minorant p) : minorantGap c < 1 := by
-  unfold minorantGap
-  exact sub_lt_self _ (div_pos c.mass_pos (by positivity))
-
-theorem norm_off_invariants_le_gap (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
-    (p : G → ℝ) (hp : ∑ g,p g=1) (hsym : ∀ g,p g⁻¹=p g)
-    (c : FiniteConvolution.Minorant p) :
-    ‖weightedOperator ρ p-uniformOperator ρ‖ ≤ minorantGap c := by
-  have hself := (weightedOperator_symmetric ρ hρ p hsym).sub (uniformOperator_symmetric ρ hρ)
-  have hbound := doeblin_off_invariants ρ hρ (FiniteConvolution.power p (2^c.exponent))
-    (FiniteConvolution.sum_power hp _) c.mass c.lower
-  rw [weightedOperator_power,←off_invariants_pow ρ p hp _ (by positivity),
-    hself.norm_pow_two_pow] at hbound
-  have hn : (1:ℝ) ≤ 2^c.exponent := one_le_pow₀ (by norm_num)
-  have hq : c.mass/(2*(2:ℝ)^c.exponent) ≤ 1/2 := by
-    apply (div_le_iff₀ (by positivity)).mpr
-    nlinarith [c.mass_le_one]
-  have hbern := one_add_mul_le_pow
-    (a := -(c.mass/(2*(2:ℝ)^c.exponent))) (by linarith : (-2:ℝ) ≤ -(c.mass/(2*(2:ℝ)^c.exponent)))
-    (2^c.exponent)
-  have he : 1+(2^c.exponent:ℕ)*(-(c.mass/(2*(2:ℝ)^c.exponent)))=1-c.mass/2 := by
-    push_cast
-    field_simp
-    ring
-  rw [he,show 1+ -(c.mass/(2*(2:ℝ)^c.exponent))=minorantGap c by rfl] at hbern
-  by_contra h
-  have hlt : minorantGap c < ‖weightedOperator ρ p-uniformOperator ρ‖ := lt_of_not_ge h
-  have hpows := pow_lt_pow_left₀ hlt (minorantGap_pos c).le (by positivity : (2^c.exponent:ℕ)≠0)
-  linarith [c.mass_pos]
-
-end Thorp.UnitaryFinite
-namespace Thorp
-open scoped BigOperators Classical
-
-def butterflyIdentity : (d : ℕ) → Butterfly d
-  | 0 => ()
-  | d+1 => (fun _ => false,fun _ => butterflyIdentity d)
-
-lemma pairSwitch_false (d : ℕ) : pairSwitch (d := d) (fun _ => false)=1 := by
-  apply Equiv.ext
-  intro x
-  change switchFun (fun _ => false) x=x
-  simp [switchFun]
-
-lemma childLift_one (d : ℕ) : childLift (d := d) (fun _ => 1)=1 := by
-  ext x i
-  simp [childLift]
-
-lemma butterflyIdentity_perm (d : ℕ) : butterflyPerm d (butterflyIdentity d)=1 := by
-  induction d with
-  | zero => rfl
-  | succ d ih => simp only [butterflyIdentity,butterflyPerm,ih,pairSwitch_false,childLift_one,mul_one]
-
-noncomputable def childInjection (d : ℕ) (b : Bool) : Equiv.Perm (Card d) →* Equiv.Perm (Card (d+1)) where
-  toFun p := childLift (fun c => if c=b then p else 1)
-  map_one' := by simpa only [ite_self] using childLift_one d
-  map_mul' p q := by
-    rw [childLift_mul]
-    congr 1
-    funext c
-    split_ifs <;> simp
-
-lemma childInjection_swap (d : ℕ) (b : Bool) (u v : Card d) :
-    childInjection d b (Equiv.swap u v)=Equiv.swap (Fin.cons b u) (Fin.cons b v) := by
-  apply Equiv.ext
-  intro x
-  obtain ⟨c,w,rfl⟩ : ∃ c w, x=Fin.cons c w := ⟨x 0,Fin.tail x,(Fin.cons_self_tail x).symm⟩
-  change (Fin.cons c ((if c=b then Equiv.swap u v else 1) w) : Card (d+1))=_
-  by_cases hc : c=b
-  · subst c
-    simp only [↓reduceIte]
-    rw [Equiv.swap_apply_def]
-    split_ifs with h₁ h₂
-    · subst w; simp
-    · subst w; simp
-    · rw [Equiv.swap_apply_of_ne_of_ne]
-      · exact fun h => h₁ (by simpa using congrArg Fin.tail h)
-      · exact fun h => h₂ (by simpa using congrArg Fin.tail h)
-  · rw [ite_eq_right hc,Equiv.Perm.one_apply,Equiv.swap_apply_of_ne_of_ne]
-    · intro h; exact hc (by simpa using congrArg (fun z => z 0) h)
-    · intro h; exact hc (by simpa using congrArg (fun z => z 0) h)
-
-lemma pairSwitch_single (d : ℕ) (u : Card d) :
-    pairSwitch (fun v => decide (v=u))=Equiv.swap (Fin.cons false u) (Fin.cons true u) := by
-  apply Equiv.ext
-  intro x
-  obtain ⟨c,w,rfl⟩ : ∃ c w, x=Fin.cons c w := ⟨x 0,Fin.tail x,(Fin.cons_self_tail x).symm⟩
-  change switchFun (fun v => decide (v=u)) (Fin.cons c w)=_
-  by_cases h : w=u
-  · subst w
-    cases c <;> simp [switchFun]
-  · have ha : (Fin.cons c w : Card (d+1))≠Fin.cons false u := fun he => h (by simpa using congrArg Fin.tail he)
-    have hb : (Fin.cons c w : Card (d+1))≠Fin.cons true u := fun he => h (by simpa using congrArg Fin.tail he)
-    simp [switchFun,h,Equiv.swap_apply_of_ne_of_ne ha hb]
-
-theorem butterfly_generates (d : ℕ) (H : Subgroup (Equiv.Perm (Card d)))
-    (hH : ∀ ω : Butterfly d, butterflyPerm d ω∈H) : H=⊤ := by
-  induction d with
-  | zero =>
-    apply top_unique
-    intro p _
-    have hp : p=1 := Subsingleton.elim _ _
-    rw [hp]
-    exact H.one_mem
-  | succ d ih =>
-    have hchild (b : Bool) (p : Equiv.Perm (Card d)) : childInjection d b p∈H := by
-      have hc : H.comap (childInjection d b)=⊤ := by
-        apply ih
-        intro ω
-        change childLift (fun c => if c=b then butterflyPerm d ω else 1)∈H
-        have h := hH ((fun _ => false),fun c => if c=b then ω else butterflyIdentity d)
-        simpa only [butterflyPerm,pairSwitch_false,one_mul,apply_ite,butterflyIdentity_perm] using h
-      have hp : p∈H.comap (childInjection d b) := by rw [hc]; trivial
-      exact hp
-    have hpair (u : Card d) : Equiv.swap (Fin.cons false u) (Fin.cons true u)∈H := by
-      rw [←pairSwitch_single]
-      have h := hH ((fun v => decide (v=u)),fun _ => butterflyIdentity d)
-      simpa only [butterflyPerm,butterflyIdentity_perm,childLift_one,mul_one] using h
-    have hswap (x y : Card (d+1)) : Equiv.swap x y∈H := by
-      have ht (a b : Bool) (u : Card d) : Equiv.swap (Fin.cons a u) (Fin.cons b u)∈H := by
-        cases a <;> cases b
-        · rw [Equiv.swap_self]; exact H.one_mem
-        · exact hpair _
-        · rw [Equiv.swap_comm]; exact hpair _
-        · rw [Equiv.swap_self]; exact H.one_mem
-      have h₁ : Equiv.swap x (Fin.cons (y 0) (Fin.tail x))∈H := by
-        convert ht (x 0) (y 0) (Fin.tail x) using 1
-        rw [Fin.cons_self_tail]
-      have h₂ := hchild (y 0) (Equiv.swap (Fin.tail x) (Fin.tail y))
-      rw [childInjection_swap,Fin.cons_self_tail] at h₂
-      exact SubmonoidClass.swap_mem_trans H h₁ h₂
-    rw [eq_top_iff,←Equiv.Perm.closure_isSwap,Subgroup.closure_le]
-    rintro _ ⟨x,y,_,rfl⟩
-    exact hswap x y
-
-lemma butterfly_closure (d : ℕ) : Subgroup.closure (Set.range (butterflyPerm d))=⊤ :=
-  butterfly_generates d _ (fun ω => Subgroup.subset_closure ⟨ω,rfl⟩)
-
-variable {Ω G : Type*} [Fintype Ω] [Fintype G]
-
-noncomputable def sampleLaw (P : Ω → G) (g : G) : ℝ :=
-  finiteMean (fun ω => if P ω=g then 1 else 0)
-
-omit [Fintype G] in
-lemma sampleLaw_nonneg (P : Ω → G) (g : G) : 0 ≤ sampleLaw P g :=
-  finiteMean_nonneg (fun _ => by split_ifs <;> norm_num)
-
-lemma sampleLaw_sum [Nonempty Ω] (P : Ω → G) : ∑ g,sampleLaw P g=1 := by
-  unfold sampleLaw
-  rw [←finiteMean_sum]
-  simp only [Finset.sum_ite_eq,Finset.mem_univ,↓reduceIte,finiteMean_const]
-
-omit [Fintype G] in
-lemma sampleLaw_pos_iff [Nonempty Ω] (P : Ω → G) (g : G) :
-    0<sampleLaw P g ↔ ∃ ω,P ω=g := by
-  unfold sampleLaw finiteMean
-  rw [div_pos_iff_of_pos_right (Nat.cast_pos.mpr Fintype.card_pos)]
-  rw [Finset.sum_pos_iff_of_nonneg (fun _ _ => by split_ifs <;> norm_num)]
-  simp only [Finset.mem_univ,true_and]
-  apply exists_congr
-  intro ω
-  split_ifs <;> simp_all
-
-omit [Fintype G] in
-lemma sampleLaw_equiv {Ω' : Type*} [Fintype Ω'] (P : Ω → G) (e : Ω' ≃ Ω) :
-    sampleLaw (P ∘ e)=sampleLaw P := by
-  funext g
-  simpa only [sampleLaw,Function.comp_apply,Equiv.apply_symm_apply] using
-    finiteMean_equiv e (fun ω => if P (e ω)=g then (1:ℝ) else 0)
-
-variable [Group G]
-omit [Fintype G] in
-lemma sampleLaw_inverse (P : Ω → G) (g : G) :
-    sampleLaw (fun ω => (P ω)⁻¹) g=sampleLaw P g⁻¹ := by
-  apply finiteMean_congr
-  intro ω
-  rw [inv_eq_iff_eq_inv]
-
-omit [Fintype G] in
-lemma sampleLaw_symmetric (P : Ω → G) (e : Equiv.Perm Ω)
-    (he : ∀ ω,P (e ω)=(P ω)⁻¹) (g : G) : sampleLaw P g⁻¹=sampleLaw P g := by
-  rw [←sampleLaw_inverse]
-  have hh : (fun ω => (P ω)⁻¹)=P ∘ e := funext (fun ω => (he ω).symm)
-  rw [hh,sampleLaw_equiv]
-
-end Thorp
-
-namespace Thorp
-open scoped BigOperators Classical
-
-lemma palindrome_identity (d : ℕ) : ∃ ω,palindromePerm d ω=1 := by
-  refine ⟨((butterflyCoinEquiv d).symm (butterflyIdentity d),
-    (butterflyCoinEquiv d).symm (butterflyIdentity d)),?_⟩
-  exact mul_inv_cancel _
-
-lemma palindrome_range_contains_butterfly (d : ℕ) (ω : Butterfly d) :
-    ∃ ξ,palindromePerm d ξ=butterflyPerm d ω := by
-  refine ⟨((butterflyCoinEquiv d).symm ω,(butterflyCoinEquiv d).symm (butterflyIdentity d)),?_⟩
-  unfold palindromePerm
-  have h (β : Butterfly d) : decodeButterfly d ((butterflyCoinEquiv d).symm β)=β :=
-    (butterflyCoinEquiv d).apply_symm_apply β
-  rw [h,h,butterflyIdentity_perm,show (1 : Equiv.Perm (Card d)).symm=1 from rfl,mul_one]
-
-lemma palindrome_support_generates (d : ℕ) :
-    Subgroup.closure {g | 0<sampleLaw (palindromePerm d) g}=⊤ := by
-  apply butterfly_generates
-  intro ω
-  apply Subgroup.subset_closure
-  exact (sampleLaw_pos_iff _ _).mpr (palindrome_range_contains_butterfly d ω)
-
-noncomputable def palindromeMinorant (d : ℕ) :
-    FiniteConvolution.Minorant (sampleLaw (palindromePerm d)) :=
-  FiniteConvolution.minorant (sampleLaw_nonneg _) (sampleLaw_sum _)
-    ((sampleLaw_pos_iff _ _).mpr (palindrome_identity d)) (palindrome_support_generates d)
-
-noncomputable def blockGap (d : ℕ) : ℝ := UnitaryFinite.minorantGap (palindromeMinorant d)
-
-lemma blockGap_pos (d : ℕ) : 0<blockGap d := UnitaryFinite.minorantGap_pos _
-lemma blockGap_lt_one (d : ℕ) : blockGap d<1 := UnitaryFinite.minorantGap_lt_one _
-
-theorem palindrome_off_invariants_gap (d : ℕ) {V : Type*}
-    [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-    (ρ : Representation ℂ (Equiv.Perm (Card d)) V) (hρ : UnitaryFinite.IsUnitary ρ) :
-    ‖UnitaryFinite.weightedOperator ρ (sampleLaw (palindromePerm d))-
-      UnitaryFinite.uniformOperator ρ‖≤blockGap d := by
-  apply UnitaryFinite.norm_off_invariants_le_gap ρ hρ _ (sampleLaw_sum _)
-  exact sampleLaw_symmetric _ (Equiv.prodComm _ _) (palindromePerm_reverse d)
-
-end Thorp
-
-namespace Thorp.CommutingBlocks
-open scoped BigOperators Classical
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
-variable {ι : Type*} [Fintype ι]
-
-omit [CompleteSpace V] in
-lemma orthogonal_sum_norm_sq (v : ι → V) (hv : ∀ i j, i≠j → inner ℂ (v i) (v j)=0) :
-    ‖∑ i,v i‖^2=∑ i,‖v i‖^2 := by
-  rw [norm_sq_eq_re_inner (𝕜 := ℂ),sum_inner]
-  simp only [inner_sum,map_sum]
-  apply Finset.sum_congr rfl
-  intro i _
-  rw [Finset.sum_eq_single i]
-  · exact (norm_sq_eq_re_inner (𝕜 := ℂ) (v i)).symm
-  · intro j _ hji
-    rw [hv i j hji.symm]
-    rfl
-  · simp
-
-lemma projection_inner_zero (P Q : V →L[ℂ] V) (hP : IsStarProjection P)
-    (hPQ : P*Q=0) (v w : V) : inner ℂ (P v) (Q w)=0 := by
-  rw [←ContinuousLinearMap.adjoint_inner_right]
-  change inner ℂ v ((star P*Q) w)=0
-  rw [hP.isSelfAdjoint.star_eq,hPQ]
-  simp
-
-lemma projection_sandwich_norm (P B : V →L[ℂ] V) (hP : IsStarProjection P) :
-    ‖B.adjoint*P*B‖=‖P*B‖^2 := by
-  have he : star (P*B)*(P*B)=B.adjoint*P*B := by
-    rw [star_mul,hP.isSelfAdjoint.star_eq]
-    change B.adjoint*P*(P*B)=_
-    rw [←mul_assoc,mul_assoc B.adjoint P P,hP.isIdempotentElem.eq]
-  rw [←he,CStarRing.norm_star_mul_self]
-  ring
-
-theorem resolution_norm_bound (P : ι → V →L[ℂ] V)
-    (hP : ∀ i, IsStarProjection (P i))
-    (horth : ∀ i j, i≠j → P i*P j=0) (hsum : ∑ i,P i=1)
-    (A : V →L[ℂ] V) (hcomm : ∀ i, Commute A (P i))
-    (c : ℝ) (hc : 0≤c) (hbound : ∀ i, ‖A*P i‖≤c) : ‖A‖≤c := by
-  apply ContinuousLinearMap.opNorm_le_bound _ hc
-  intro v
-  have hsplit (w : V) : ∑ i,P i w=w := by
-    rw [←sum_apply,hsum]
-    rfl
-  have hp := orthogonal_sum_norm_sq (fun i => P i v)
-    (fun i j hij => projection_inner_zero _ _ (hP i) (horth i j hij) v v)
-  rw [hsplit] at hp
-  have ha := orthogonal_sum_norm_sq (fun i => P i (A v))
-    (fun i j hij => projection_inner_zero _ _ (hP i) (horth i j hij) (A v) (A v))
-  rw [hsplit] at ha
-  have he (i : ι) : P i (A v)=(A*P i) (P i v) := by
-    change (P i*A) v=(A*P i*P i) v
-    rw [mul_assoc,(hP i).isIdempotentElem.eq,(hcomm i).eq]
-  have hh (i : ι) : ‖P i (A v)‖≤c*‖P i v‖ := by
-    rw [he]
-    exact ((A*P i).le_opNorm _).trans (mul_le_mul_of_nonneg_right (hbound i) (norm_nonneg _))
-  have hsq := Finset.sum_le_sum (s := Finset.univ) (fun i _ =>
-    pow_le_pow_left₀ (norm_nonneg _) (hh i) 2)
-  simp_rw [mul_pow] at hsq
-  rw [←Finset.mul_sum,←hp,←ha,←mul_pow] at hsq
-  exact (sq_le_sq₀ (norm_nonneg _) (mul_nonneg hc (norm_nonneg _))).mp hsq
-
-theorem sector_sandwich_le (T P A B : V →L[ℂ] V)
-    (hT : IsStarProjection T) (hP : IsStarProjection P)
-    (hTP : T*P=T) (hcomm : Commute A T) (hA : ‖A‖≤1) :
-    ‖B.adjoint*A*T*B‖≤‖B.adjoint*P*B‖ := by
-  have he : B.adjoint*A*T*B=(T*B).adjoint*A*(T*B) := by
-    change _=star (T*B)*A*(T*B)
-    rw [star_mul,hT.isSelfAdjoint.star_eq]
-    change _=B.adjoint*T*A*(T*B)
-    rw [mul_assoc B.adjoint T A,←hcomm.eq]
-    simp only [mul_assoc]
-    rw [←mul_assoc T T B,hT.isIdempotentElem.eq]
-  have hTB : ‖T*B‖≤‖P*B‖ := by
-    have he : T*B=T*(P*B) := by rw [←mul_assoc,hTP]
-    rw [he]
-    exact (norm_mul_le _ _).trans ((mul_le_mul_of_nonneg_right (hT.norm_le _) (norm_nonneg _)).trans_eq (one_mul _))
-  rw [he,projection_sandwich_norm P B hP]
-  calc
-    _ ≤ ‖(T*B).adjoint‖*‖A‖*‖T*B‖ := (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right (norm_mul_le _ _) (norm_nonneg _))
-    _ ≤ ‖T*B‖^2 := by rw [ContinuousLinearMap.adjoint.norm_map]; nlinarith [norm_nonneg (T*B)]
-    _ ≤ _ := pow_le_pow_left₀ (norm_nonneg _) hTB 2
-
-omit [CompleteSpace V] [Fintype ι] in
-lemma resolution_commute (P : ι → V →L[ℂ] V)
-    (horth : ∀ i j, i≠j → P i*P j=0) (i j : ι) : Commute (P i) (P j) := by
-  by_cases hij : i=j
-  · subst j; exact Commute.refl _
-  · exact (horth i j hij).trans (horth j i (Ne.symm hij)).symm
-
-omit [Fintype ι] in
-lemma resolution_sum_mul (P : ι → V →L[ℂ] V) (hP : ∀ i, IsStarProjection (P i))
-    (horth : ∀ i j, i≠j → P i*P j=0) (S : Finset ι) (j : ι) :
-    (∑ i∈S,P i)*P j=if j∈S then P j else 0 := by
-  rw [Finset.sum_mul]
-  have he (i : ι) : P i*P j=if i=j then P j else 0 := by
-    split_ifs with h
-    · subst i; exact (hP j).isIdempotentElem.eq
-    · exact horth i j h
-  simp only [he,Finset.sum_ite_eq']
-
-lemma resolution_truncated_norm (P : ι → V →L[ℂ] V)
-    (hP : ∀ i, IsStarProjection (P i))
-    (horth : ∀ i j, i≠j → P i*P j=0) (hsum : ∑ i,P i=1)
-    (A : V →L[ℂ] V) (hcomm : ∀ i, Commute A (P i))
-    (S : Finset ι) (c : ℝ) (hc : 0≤c) (hbound : ∀ i∈S, ‖A*P i‖≤c) :
-    ‖A*(∑ i∈S,P i)‖≤c := by
-  apply resolution_norm_bound P hP horth hsum _ _ c hc
-  · intro i
-    rw [mul_assoc,resolution_sum_mul P hP horth]
-    split_ifs with hi
-    · exact hbound i hi
-    · simp [hc]
-  · intro i
-    exact (hcomm i).mul_left (Commute.sum_left S P (P i) fun j _ => resolution_commute P horth j i)
-
-theorem resolution_exception_bound (T Q : ι → V →L[ℂ] V)
-    (hT : ∀ i, IsStarProjection (T i))
-    (horth : ∀ i j, i≠j → T i*T j=0) (hsum : ∑ i,T i=1)
-    (hQ : ∀ i, IsStarProjection (Q i)) (hTQ : ∀ i,T i*Q i=T i)
-    (A B : V →L[ℂ] V) (hcomm : ∀ i, Commute A (T i))
-    (hA : ‖A‖≤1) (hB : ‖B‖≤1)
-    (S : Finset ι) (c : ℝ) (hc : 0≤c) (hbound : ∀ i∉S, ‖A*T i‖≤c) :
-    ‖B.adjoint*A*B‖≤c+∑ i∈S,‖B.adjoint*Q i*B‖ := by
-  have hhigh : ‖A*(∑ i∈Sᶜ,T i)‖≤c := resolution_truncated_norm T hT horth hsum A hcomm
-    Sᶜ c hc (fun i hi => hbound i (Finset.mem_compl.mp hi))
-  have hsplit : (∑ i∈Sᶜ,T i)+(∑ i∈S,T i)=1 := by
-    rw [Finset.sum_compl_add_sum,hsum]
-  have he : B.adjoint*A*B=B.adjoint*(A*(∑ i∈Sᶜ,T i))*B+
-      ∑ i∈S,B.adjoint*A*T i*B := by
-    rw [←Finset.sum_mul,←Finset.mul_sum]
-    rw [←add_mul]
-    simp only [mul_assoc]
-    rw [←mul_add,←mul_add,hsplit]
-    simp only [mul_one,mul_assoc]
-  rw [he]
-  apply (norm_add_le _ _).trans
-  apply add_le_add
-  · calc
-      _ ≤ ‖B.adjoint‖*‖A*(∑ i∈Sᶜ,T i)‖*‖B‖ :=
-        (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right (norm_mul_le _ _) (norm_nonneg _))
-      _ ≤ 1*c*1 := mul_le_mul (mul_le_mul (by simpa only [ContinuousLinearMap.adjoint.norm_map] using hB)
-        hhigh (norm_nonneg _) zero_le_one) hB (norm_nonneg _) (by simpa using hc)
-      _ = c := by ring
-  · apply (norm_sum_le _ _).trans
-    exact Finset.sum_le_sum fun i _ => sector_sandwich_le _ _ _ _ (hT i) (hQ i) (hTQ i) (hcomm i) hA
-
-variable [DecidableEq ι]
-variable (P : ι → V →L[ℂ] V) (hcomm : ∀ i j, Commute (P i) (P j))
-
-noncomputable def choice (E : Finset ι) (i : ι) : V →L[ℂ] V :=
-  if i∈E then 1-P i else P i
-
-include hcomm
-omit [CompleteSpace V] [Fintype ι] in
-lemma choice_commute (E F : Finset ι) (i j : ι) :
-    Commute (choice P E i) (choice P F j) := by
-  unfold choice
-  split_ifs
-  · exact (Commute.one_left _).sub_left ((Commute.one_right _).sub_right (hcomm i j))
-  · exact (Commute.one_left _).sub_left (hcomm i j)
-  · exact (Commute.one_right _).sub_right (hcomm i j)
-  · exact hcomm i j
-
-noncomputable def sector (s E : Finset ι) : V →L[ℂ] V :=
-  s.noncommProd (choice P E) (fun i _ j _ _ => choice_commute P hcomm E E i j)
-
-omit [CompleteSpace V] [Fintype ι] in
-lemma sector_insert (s E : Finset ι) (a : ι) (ha : a∉s) :
-    sector P hcomm (insert a s) E=choice P E a*sector P hcomm s E := by
-  exact Finset.noncommProd_insert_of_notMem _ _ _ _ ha
-
-omit [CompleteSpace V] [Fintype ι] in
-lemma sector_congr (s E F : Finset ι) (h : ∀ i∈s, (i∈E ↔ i∈F)) :
-    sector P hcomm s E=sector P hcomm s F := by
-  apply Finset.noncommProd_congr rfl
-  intro i hi
-  simp only [choice,h i hi]
-
-omit [CompleteSpace V] [Fintype ι] in
-lemma sector_factor_commute (s E F : Finset ι) (i : ι) :
-    Commute (choice P E i) (sector P hcomm s F) :=
-  Finset.noncommProd_commute _ _ _ _ (fun j _ => choice_commute P hcomm E F i j)
-
-omit [Fintype ι] in
-lemma sector_projection (hP : ∀ i, IsStarProjection (P i)) (s E : Finset ι) :
-    IsStarProjection (sector P hcomm s E) := by
-  induction s using Finset.induction_on with
-  | empty => exact IsStarProjection.one _
-  | @insert a s ha ih =>
-    rw [sector_insert P hcomm s E a ha]
-    have hc : IsStarProjection (choice P E a) := by
-      unfold choice
-      split_ifs
-      · exact (hP a).one_sub
-      · exact hP a
-    exact hc.mul ih (sector_factor_commute P hcomm s E E a)
-
-end Thorp.CommutingBlocks
-
-end ThorpNine.SparseSaving
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.SparseSaving.ShapePartitionEquiv
+
+namespace OAI
+
+namespace ThorpNine.SparseSaving
+
+namespace Thorp.UnitaryFinite
+open scoped BigOperators ComplexConjugate Classical
+variable {G : Type*} [Group G] [Fintype G]
+variable {V W : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+  [NormedAddCommGroup W] [InnerProductSpace ℂ W] [FiniteDimensional ℂ W]
+omit [FiniteDimensional ℂ V] in
+lemma averageMap_self_trace (ρ : Representation ℂ G V) (A : Module.End ℂ V) :
+    LinearMap.trace ℂ V (averageMap ρ ρ A).toLinearMap =
+      (Fintype.card G:ℂ) * LinearMap.trace ℂ V A := by
+  change LinearMap.trace ℂ V (∑ g, ρ g * A * ρ g⁻¹) = _
+  rw [map_sum]
+  have hj (g : G) : LinearMap.trace ℂ V (ρ g*A*ρ g⁻¹) = LinearMap.trace ℂ V A := by
+    rw [LinearMap.trace_mul_cycle,←map_mul,inv_mul_cancel,map_one,one_mul]
+  simp only [hj,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
+
+lemma averageMap_scalar (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
+    (A : Module.End ℂ V) :
+    ∃ c : ℂ, (averageMap ρ ρ A).toLinearMap = c • LinearMap.id ∧
+      c * (Module.finrank ℂ V:ℂ) = (Fintype.card G:ℂ)*LinearMap.trace ℂ V A := by
+  obtain ⟨c,hc⟩ := (Representation.IsIrreducible.algebraMap_intertwiningMap_bijective_of_isAlgClosed (ρ := ρ)).surjective (averageMap ρ ρ A)
+  have he : (averageMap ρ ρ A).toLinearMap = c • LinearMap.id := by
+    rw [←hc]
+    rfl
+  refine ⟨c,he,?_⟩
+  rw [←averageMap_self_trace ρ A,he,map_smul,LinearMap.trace_id,smul_eq_mul]
+
+theorem coefficient_orthogonality (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
+    (hρ : IsUnitary ρ) (u v w z : V) :
+    (Module.finrank ℂ V:ℂ) * ∑ g, inner ℂ u (ρ g v) * inner ℂ (ρ g w) z =
+      (Fintype.card G:ℂ) * inner ℂ w v * inner ℂ u z := by
+  obtain ⟨c,hc,htrace⟩ := averageMap_scalar ρ (InnerProductSpace.rankOne ℂ v w).toLinearMap
+  rw [InnerProductSpace.trace_rankOne] at htrace
+  have happ := congrArg (fun A : Module.End ℂ V => inner ℂ u (A z)) hc
+  change inner ℂ u ((∑ g, ρ g * (InnerProductSpace.rankOne ℂ v w).toLinearMap * ρ g⁻¹) z) = _ at happ
+  simp only [LinearMap.sum_apply,Module.End.mul_apply,ContinuousLinearMap.coe_coe,
+    InnerProductSpace.rankOne_apply,map_smul,unitary_inner_inv ρ hρ,inner_sum,
+    inner_smul_right,LinearMap.smul_apply,LinearMap.id_apply] at happ
+  have hsum : ∑ g, inner ℂ u (ρ g v) * inner ℂ (ρ g w) z = c * inner ℂ u z := by
+    simpa only [mul_comm] using happ
+  rw [hsum]
+  calc
+    _ = (c*(Module.finrank ℂ V:ℂ))*inner ℂ u z := by ring
+    _ = _ := by rw [htrace]
+
+end Thorp.UnitaryFinite
+namespace Thorp.UnitaryFinite
+open scoped BigOperators ComplexConjugate Classical
+
+variable {G X : Type*} [Group G] [Fintype G] [Fintype X]
+  (a : G →* Equiv.Perm X) (x₀ : X) (ha : ∀ x, ∃ g, a g x₀=x)
+
+noncomputable def representative (x : X) : G := Classical.choose (ha x)
+
+omit [Fintype G] [Fintype X] in
+lemma representative_apply (x : X) : a (representative a x₀ ha x) x₀=x :=
+  Classical.choose_spec (ha x)
+
+include ha in
+lemma orbit_sum (F : X → ℂ) :
+    (Fintype.card X:ℂ)*∑ g, F (a g x₀) = (Fintype.card G:ℂ)*∑ x, F x := by
+  have ht (x : X) : ∑ g, F (a g x)=∑ g, F (a g x₀) := by
+    obtain ⟨t,rfl⟩ := ha x
+    symm
+    apply Fintype.sum_equiv (Equiv.mulRight t⁻¹)
+    intro g
+    simp
+  calc
+    _ = ∑ x : X, ∑ g : G, F (a g x) := by simp only [ht,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
+    _ = ∑ g : G, ∑ x : X, F (a g x) := Finset.sum_comm
+    _ = ∑ g : G, ∑ x : X, F x := by
+      apply Finset.sum_congr rfl
+      intro g _
+      exact Equiv.sum_comp (a g) F
+    _ = _ := by simp
+
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V)
+
+def IsFixed (w : V) : Prop := ∀ g, a g x₀=x₀ → ρ g w=w
+
+noncomputable def coefficient (w v : V) : EuclideanSpace ℂ X :=
+  WithLp.toLp 2 (fun x => inner ℂ (ρ (representative a x₀ ha x) w) v)
+
+omit [Fintype G] [Fintype X] [FiniteDimensional ℂ V] in
+lemma coefficient_at (w : V) (hw : IsFixed a x₀ ρ w) (v : V) (g : G) :
+    coefficient a x₀ ha ρ w v (a g x₀) = inner ℂ (ρ g w) v := by
+  let r := representative a x₀ ha (a g x₀)
+  have hr : a r x₀=a g x₀ := representative_apply a x₀ ha _
+  have hs : a (g⁻¹*r) x₀=x₀ := by
+    simp only [map_mul,Equiv.Perm.mul_apply]
+    rw [hr,←Equiv.Perm.mul_apply,←map_mul,inv_mul_cancel,map_one]
+    rfl
+  have he := congrArg (ρ g) (hw (g⁻¹*r) hs)
+  have hh : ρ r w=ρ g w := by
+    simpa only [map_mul,Module.End.mul_apply,←Module.End.mul_apply,←map_mul,
+      mul_inv_cancel_left] using he
+  change inner ℂ (ρ r w) v= _
+  rw [hh]
+
+lemma coefficient_inner [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
+    (w z : V) (hw : IsFixed a x₀ ρ w) (hz : IsFixed a x₀ ρ z) (v t : V) :
+    (Module.finrank ℂ V:ℂ)*inner ℂ (coefficient a x₀ ha ρ w v) (coefficient a x₀ ha ρ z t) =
+      (Fintype.card X:ℂ)*inner ℂ z w*inner ℂ v t := by
+  have hs := orbit_sum a x₀ ha (fun x => star (coefficient a x₀ ha ρ w v x)*coefficient a x₀ ha ρ z t x)
+  simp only [coefficient_at a x₀ ha ρ w hw,coefficient_at a x₀ ha ρ z hz] at hs
+  simp only [←starRingEnd_apply,inner_conj_symm] at hs
+  simp only [starRingEnd_apply] at hs
+  have ho := coefficient_orthogonality ρ hρ v w z t
+  have hcard : (Fintype.card G:ℂ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
+  apply mul_left_cancel₀ hcard
+  rw [PiLp.inner_apply]
+  simp only [RCLike.inner_apply,starRingEnd_apply]
+  have hsum : ∑ x, coefficient a x₀ ha ρ z t x * star (coefficient a x₀ ha ρ w v x) =
+      ∑ x, star (coefficient a x₀ ha ρ w v x)*coefficient a x₀ ha ρ z t x := by
+    apply Finset.sum_congr rfl
+    intro x _
+    ring
+  rw [hsum]
+  calc
+    _ = (Module.finrank ℂ V:ℂ)*((Fintype.card G:ℂ)*∑ x,
+        star (coefficient a x₀ ha ρ w v x)*coefficient a x₀ ha ρ z t x) := by ring
+    _ = (Fintype.card X:ℂ)*((Module.finrank ℂ V:ℂ)*∑ g,
+        inner ℂ v (ρ g w)*inner ℂ (ρ g z) t) := by rw [←hs]; ring
+    _ = _ := by rw [ho]; ring
+
+omit [Fintype G] [Fintype X] [FiniteDimensional ℂ V] in
+lemma coefficient_action (hρ : IsUnitary ρ) (w : V) (hw : IsFixed a x₀ ρ w)
+    (v : V) (g : G) (x : X) :
+    coefficient a x₀ ha ρ w v (a g x) = coefficient a x₀ ha ρ w (ρ g⁻¹ v) x := by
+  obtain ⟨r,rfl⟩ := ha x
+  rw [←Equiv.Perm.mul_apply,←map_mul,coefficient_at a x₀ ha ρ w hw,
+    coefficient_at a x₀ ha ρ w hw]
+  simp only [map_mul,Module.End.mul_apply,unitary_inner_inv ρ hρ]
+
+noncomputable def normalizedCoefficient (w v : V) : EuclideanSpace ℂ X :=
+  (Real.sqrt ((Module.finrank ℂ V:ℝ)/(Fintype.card X:ℝ)) : ℂ) • coefficient a x₀ ha ρ w v
+
+lemma normalizedCoefficient_inner [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
+    (w z : V) (hw : IsFixed a x₀ ρ w) (hz : IsFixed a x₀ ρ z) (v t : V) :
+    inner ℂ (normalizedCoefficient a x₀ ha ρ w v) (normalizedCoefficient a x₀ ha ρ z t) =
+      inner ℂ z w*inner ℂ v t := by
+  let : Nonempty X := ⟨x₀⟩
+  have hX : (Fintype.card X:ℂ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
+  have hs : (Real.sqrt ((Module.finrank ℂ V:ℝ)/(Fintype.card X:ℝ)) : ℂ)^2 =
+      (Module.finrank ℂ V:ℂ)/(Fintype.card X:ℂ) := by
+    rw [←Complex.ofReal_pow,Real.sq_sqrt (by positivity)]
+    simp
+  simp only [normalizedCoefficient,inner_smul_left,inner_smul_right]
+  simp only [Complex.conj_ofReal]
+  rw [←mul_assoc,←pow_two,hs]
+  have hi := coefficient_inner a x₀ ha ρ hρ w z hw hz v t
+  field_simp
+  simpa only [mul_assoc] using hi
+
+lemma normalizedCoefficient_orthonormal [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
+    {I : Type*} (w : I → V) (hw : Orthonormal ℂ w)
+    (hfix : ∀ i, IsFixed a x₀ ρ (w i)) (v : V) (hv : ‖v‖=1) :
+    Orthonormal ℂ (fun i => normalizedCoefficient a x₀ ha ρ (w i) v) := by
+  rw [orthonormal_iff_ite] at hw ⊢
+  intro i j
+  rw [normalizedCoefficient_inner a x₀ ha ρ hρ _ _ (hfix i) (hfix j),hw,
+    inner_self_eq_norm_sq_to_K,hv]
+  simp [eq_comm]
+
+lemma normalizedCoefficient_norm [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
+    (w : V) (hw : IsFixed a x₀ ρ w) (hn : ‖w‖=1) (v : V) :
+    ‖normalizedCoefficient a x₀ ha ρ w v‖=‖v‖ := by
+  have he := normalizedCoefficient_inner a x₀ ha ρ hρ w w hw hw v v
+  rw [inner_self_eq_norm_sq_to_K,inner_self_eq_norm_sq_to_K,hn] at he
+  simp only [inner_self_eq_norm_sq_to_K] at he
+  norm_num at he
+  have hs : ‖normalizedCoefficient a x₀ ha ρ w v‖^2=‖v‖^2 := by exact_mod_cast he
+  nlinarith [norm_nonneg (normalizedCoefficient a x₀ ha ρ w v),norm_nonneg v]
+
+end Thorp.UnitaryFinite
+
+namespace Thorp.FiniteConvolution
+open scoped BigOperators Classical
+variable {G : Type*} [Group G] [Fintype G]
+
+noncomputable def conv (p q : G → ℝ) (g : G) : ℝ := ∑ h, p h*q (h⁻¹*g)
+noncomputable def power (p : G → ℝ) : ℕ → G → ℝ
+  | 0 => fun g => if g=1 then 1 else 0
+  | n+1 => conv p (power p n)
+
+lemma conv_nonneg {p q : G → ℝ} (hp : ∀ g, 0 ≤ p g) (hq : ∀ g, 0 ≤ q g) :
+    ∀ g, 0 ≤ conv p q g := fun _ => Finset.sum_nonneg (fun h _ => mul_nonneg (hp h) (hq _))
+
+lemma conv_pos_of {p q : G → ℝ} (hp : ∀ g, 0 ≤ p g) (hq : ∀ g, 0 ≤ q g)
+    {a b : G} (ha : 0 < p a) (hb : 0 < q b) : 0 < conv p q (a*b) := by
+  apply lt_of_lt_of_le (mul_pos ha hb)
+  have h := Finset.single_le_sum (fun g _ => mul_nonneg (hp g) (hq (g⁻¹*(a*b))))
+    (Finset.mem_univ a)
+  simpa [conv] using h
+
+lemma sum_conv (p q : G → ℝ) : ∑ g, conv p q g=(∑ g,p g)*(∑ g,q g) := by
+  unfold conv
+  rw [Finset.sum_comm,Finset.sum_mul]
+  apply Finset.sum_congr rfl
+  intro h _
+  rw [←Finset.mul_sum]
+  congr 1
+  exact Fintype.sum_equiv (Equiv.mulLeft h⁻¹) _ _ (fun _ => rfl)
+
+lemma power_nonneg {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) : ∀ n g, 0 ≤ power p n g := by
+  intro n
+  induction n with
+  | zero => intro g; simp only [power]; split_ifs  <;> norm_num
+  | succ n ih => exact conv_nonneg hp ih
+
+lemma sum_power {p : G → ℝ} (hp : ∑ g,p g=1) : ∀ n, ∑ g,power p n g=1 := by
+  intro n
+  induction n with
+  | zero => simp [power]
+  | succ n ih => rw [power,sum_conv,hp,ih,one_mul]
+
+lemma power_list_pos {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) (l : List G)
+    (hl : ∀ g ∈ l, 0 < p g) : 0 < power p l.length l.prod := by
+  induction l with
+  | nil => simp [power]
+  | cons a l ih =>
+    simpa only [List.length_cons,List.prod_cons,power] using
+      conv_pos_of hp (power_nonneg hp l.length) (hl a (by simp))
+        (ih (fun g hg => hl g (by simp [hg])))
+
+lemma power_pos_mono {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) (h1 : 0 < p 1)
+    {n m : ℕ} (hnm : n ≤ m) {g : G} (hg : 0 < power p n g) : 0 < power p m g := by
+  induction m, hnm using Nat.le_induction with
+  | base => exact hg
+  | succ m _ ih =>
+    simpa only [one_mul,power] using conv_pos_of hp (power_nonneg hp m) h1 ih
+
+theorem exists_positive_power {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) (h1 : 0 < p 1)
+    (hgen : Subgroup.closure {g | 0 < p g}=⊤) :
+    ∃ n : ℕ, ∀ g, 0 < power p (2^n) g := by
+  have hx (g : G) : g ∈ Submonoid.closure {g | 0 < p g} := by
+    rw [←Subgroup.closure_toSubmonoid_of_finite,hgen]
+    trivial
+  choose l hl hprod using fun g => Submonoid.exists_list_of_mem_closure (hx g)
+  let n := Finset.univ.sup (fun g => (l g).length)
+  refine ⟨n,fun g => ?_⟩
+  have hlen : (l g).length ≤ n := Finset.le_sup (f := fun g => (l g).length) (Finset.mem_univ g)
+  have hpow : n ≤ 2^n := Nat.le_of_lt (Nat.lt_two_pow_self)
+  have hpos := power_list_pos hp (l g) (hl g)
+  rw [hprod g] at hpos
+  exact power_pos_mono hp h1 (hlen.trans hpow) hpos
+
+structure Minorant (p : G → ℝ) where
+  exponent : ℕ
+  mass : ℝ
+  mass_pos : 0 < mass
+  mass_le_one : mass ≤ 1
+  lower : ∀ g, mass/(Fintype.card G:ℝ) ≤ power p (2^exponent) g
+
+noncomputable def minorant {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) (hsum : ∑ g,p g=1)
+    (h1 : 0 < p 1) (hgen : Subgroup.closure {g | 0 < p g}=⊤) : Minorant p := by
+  let n := Classical.choose (exists_positive_power hp h1 hgen)
+  have hn := Classical.choose_spec (exists_positive_power hp h1 hgen)
+  let a := Finset.univ.inf' Finset.univ_nonempty (power p (2^n))
+  have ha : 0 < a := (Finset.lt_inf'_iff _).mpr (fun g _ => hn g)
+  have hag (g : G) : a ≤ power p (2^n) g := Finset.inf'_le _ (Finset.mem_univ g)
+  have hN : (0:ℝ) < Fintype.card G := Nat.cast_pos.mpr Fintype.card_pos
+  have hsum' := Finset.sum_le_sum (s := Finset.univ) (fun g _ => hag g)
+  simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,sum_power hsum] at hsum'
+  refine ⟨n,(Fintype.card G:ℝ)*a,mul_pos hN ha,hsum',fun g => ?_⟩
+  have he : (Fintype.card G:ℝ)*a/(Fintype.card G:ℝ)=a := by field_simp
+  rw [he]
+  exact hag g
+
+end Thorp.FiniteConvolution
+namespace Thorp.UnitaryFinite
+
+section
+open scoped BigOperators ComplexConjugate Classical
+
+variable {G : Type*} [Group G] [Fintype G]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+omit [Fintype G] in
+@[simp] lemma continuousRepresentation_apply (ρ : Representation ℂ G V) (g : G) (v : V) :
+    continuousRepresentation ρ g v=ρ g v := rfl
+
+omit [Fintype G] [FiniteDimensional ℂ V] in
+lemma unitary_norm (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (g : G) (v : V) :
+    ‖ρ g v‖=‖v‖ := by
+  rw [norm_eq_sqrt_re_inner (𝕜 := ℂ),hρ,norm_eq_sqrt_re_inner (𝕜 := ℂ)]
+
+omit [Fintype G] in
+lemma continuousRepresentation_norm_le (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (g : G) :
+    ‖continuousRepresentation ρ g‖≤1 := by
+  apply ContinuousLinearMap.opNorm_le_bound _ zero_le_one
+  intro v
+  simp only [continuousRepresentation_apply,unitary_norm ρ hρ,one_mul,le_refl]
+
+omit [Fintype G] in
+lemma continuousRepresentation_adjoint (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (g : G) :
+    (continuousRepresentation ρ g).adjoint=continuousRepresentation ρ g⁻¹ := by
+  ext w
+  apply ext_inner_left ℂ
+  intro v
+  rw [ContinuousLinearMap.adjoint_inner_right]
+  exact (unitary_inner_inv ρ hρ g v w).symm
+
+noncomputable def weightedOperator (ρ : Representation ℂ G V) (p : G → ℝ) : V →L[ℂ] V :=
+  ∑ g, (p g : ℂ) • continuousRepresentation ρ g
+
+lemma weightedOperator_norm_le (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
+    (p : G → ℝ) (hp : ∀ g, 0≤p g) : ‖weightedOperator ρ p‖ ≤ ∑ g, p g := by
+  apply (norm_sum_le _ _).trans
+  apply Finset.sum_le_sum
+  intro g _
+  rw [norm_smul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (hp g)]
+  exact mul_le_of_le_one_right (hp g) (continuousRepresentation_norm_le ρ hρ g)
+
+lemma weightedOperator_symmetric (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
+    (p : G → ℝ) (hp : ∀ g, p g⁻¹=p g) : IsSelfAdjoint (weightedOperator ρ p) := by
+  change (weightedOperator ρ p).adjoint=weightedOperator ρ p
+  unfold weightedOperator
+  rw [map_sum]
+  simp only [map_smulₛₗ,Complex.conj_ofReal,continuousRepresentation_adjoint ρ hρ]
+  apply Fintype.sum_equiv (Equiv.inv G)
+  intro g
+  simp [hp]
+
+noncomputable def uniformOperator (ρ : Representation ℂ G V) : V →L[ℂ] V :=
+  weightedOperator ρ (fun _ => (Fintype.card G:ℝ)⁻¹)
+
+lemma uniformOperator_symmetric (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) :
+    IsSelfAdjoint (uniformOperator ρ) := weightedOperator_symmetric ρ hρ _ (fun _ => rfl)
+
+lemma uniformOperator_left (ρ : Representation ℂ G V) (g : G) :
+    continuousRepresentation ρ g * uniformOperator ρ=uniformOperator ρ := by
+  unfold uniformOperator weightedOperator
+  rw [Finset.mul_sum]
+  simp only [mul_smul_comm,←map_mul]
+  exact Fintype.sum_equiv (Equiv.mulLeft g) _ _ (fun _ => rfl)
+
+lemma uniformOperator_right (ρ : Representation ℂ G V) (g : G) :
+    uniformOperator ρ * continuousRepresentation ρ g=uniformOperator ρ := by
+  unfold uniformOperator weightedOperator
+  rw [Finset.sum_mul]
+  simp only [smul_mul_assoc,←map_mul]
+  exact Fintype.sum_equiv (Equiv.mulRight g) _ _ (fun _ => rfl)
+
+lemma uniformOperator_idempotent (ρ : Representation ℂ G V) :
+    IsIdempotentElem (uniformOperator ρ) := by
+  change uniformOperator ρ * uniformOperator ρ=uniformOperator ρ
+  nth_rw 1 [uniformOperator,weightedOperator]
+  rw [Finset.sum_mul]
+  simp only [smul_mul_assoc,uniformOperator_left,Finset.sum_const,Finset.card_univ,
+    ←Nat.cast_smul_eq_nsmul ℂ,smul_smul,Complex.ofReal_inv,Complex.ofReal_natCast]
+  rw [mul_inv_cancel₀ (by exact_mod_cast Fintype.card_ne_zero),one_smul]
+
+lemma uniformOperator_projection (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) :
+    IsStarProjection (uniformOperator ρ) :=
+  ⟨uniformOperator_idempotent ρ,uniformOperator_symmetric ρ hρ⟩
+
+lemma weighted_uniform_left (ρ : Representation ℂ G V) (p : G → ℝ) :
+    weightedOperator ρ p * uniformOperator ρ=(∑ g,p g : ℝ) • uniformOperator ρ := by
+  unfold weightedOperator
+  rw [Finset.sum_mul]
+  simp only [smul_mul_assoc,uniformOperator_left,←Finset.sum_smul,←Complex.ofReal_sum]
+  rfl
+
+lemma weighted_uniform_right (ρ : Representation ℂ G V) (p : G → ℝ) :
+    uniformOperator ρ * weightedOperator ρ p=(∑ g,p g : ℝ) • uniformOperator ρ := by
+  unfold weightedOperator
+  rw [Finset.mul_sum]
+  simp only [mul_smul_comm,uniformOperator_right,←Finset.sum_smul,←Complex.ofReal_sum]
+  rfl
+
+theorem doeblin_off_invariants (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
+    (p : G → ℝ) (hp : ∑ g,p g=1) (ε : ℝ)
+    (hminor : ∀ g, ε/(Fintype.card G:ℝ)≤p g) :
+    ‖weightedOperator ρ p * (1-uniformOperator ρ)‖≤1-ε := by
+  let q : G → ℝ := fun g => p g-ε/(Fintype.card G:ℝ)
+  have hq : ∀ g, 0≤q g := fun g => sub_nonneg.mpr (hminor g)
+  have hcard : (Fintype.card G:ℝ)≠0 := by exact_mod_cast Fintype.card_ne_zero
+  have hsum : ∑ g,q g=1-ε := by
+    simp only [q,Finset.sum_sub_distrib,hp,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
+    field_simp
+  have he : weightedOperator ρ p=weightedOperator ρ q+(ε:ℂ) • uniformOperator ρ := by
+    simp only [uniformOperator,weightedOperator]
+    rw [Finset.smul_sum,←Finset.sum_add_distrib]
+    apply Finset.sum_congr rfl
+    intro g _
+    simp only [q,Complex.ofReal_sub,Complex.ofReal_div,smul_smul,←add_smul,
+      Complex.ofReal_inv,Complex.ofReal_natCast]
+    congr 1
+    field_simp
+    ring
+  have hz : uniformOperator ρ*(1-uniformOperator ρ)=0 := by
+    rw [mul_sub,mul_one,(uniformOperator_idempotent ρ).eq,sub_self]
+  rw [he,add_mul,smul_mul_assoc,hz,smul_zero,add_zero]
+  calc
+    _ ≤ ‖weightedOperator ρ q‖*‖1-uniformOperator ρ‖ := norm_mul_le _ _
+    _ ≤ (1-ε)*1 := mul_le_mul (by rw [←hsum]; exact weightedOperator_norm_le ρ hρ q hq)
+      ((uniformOperator_projection ρ hρ).one_sub.norm_le _) (norm_nonneg _)
+      (by rw [←hsum]; exact Finset.sum_nonneg (fun g _ => hq g))
+    _ = _ := mul_one _
+
+end
+open scoped BigOperators Classical
+variable {G : Type*} [Group G] [Fintype G]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+lemma weightedOperator_conv (ρ : Representation ℂ G V) (p q : G → ℝ) :
+    weightedOperator ρ (FiniteConvolution.conv p q)=weightedOperator ρ p*weightedOperator ρ q := by
+  unfold weightedOperator FiniteConvolution.conv
+  simp only [Complex.ofReal_sum,Finset.sum_smul]
+  rw [Finset.sum_comm,Finset.sum_mul]
+  apply Finset.sum_congr rfl
+  intro h _
+  rw [Finset.mul_sum]
+  simp only [smul_mul_assoc,mul_smul_comm,smul_smul,←map_mul]
+  apply Fintype.sum_equiv (Equiv.mulLeft h⁻¹)
+  intro g
+  simp [Complex.ofReal_mul,mul_comm]
+
+lemma weightedOperator_power (ρ : Representation ℂ G V) (p : G → ℝ) (n : ℕ) :
+    weightedOperator ρ (FiniteConvolution.power p n)=(weightedOperator ρ p)^n := by
+  induction n with
+  | zero => simp [FiniteConvolution.power,weightedOperator]
+  | succ n ih => rw [FiniteConvolution.power,weightedOperator_conv,ih,pow_succ']
+
+lemma off_invariants_pow (ρ : Representation ℂ G V) (p : G → ℝ)
+    (hp : ∑ g,p g=1) (n : ℕ) (hn : n≠0) :
+    (weightedOperator ρ p-uniformOperator ρ)^n=
+      (weightedOperator ρ p)^n*(1-uniformOperator ρ) := by
+  have hAP := weighted_uniform_left ρ p
+  have hPA := weighted_uniform_right ρ p
+  rw [hp,one_smul] at hAP hPA
+  have hc : Commute (weightedOperator ρ p) (1-uniformOperator ρ) := by
+    show _*_= _*_
+    rw [mul_sub,sub_mul,mul_one,one_mul,hAP,hPA]
+  have he : weightedOperator ρ p-uniformOperator ρ=
+      weightedOperator ρ p*(1-uniformOperator ρ) := by rw [mul_sub,mul_one,hAP]
+  rw [he,hc.mul_pow,((uniformOperator_idempotent ρ).one_sub).pow_eq hn]
+
+noncomputable def minorantGap {p : G → ℝ} (c : FiniteConvolution.Minorant p) : ℝ :=
+  1-c.mass/(2*(2:ℝ)^c.exponent)
+
+omit [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V] in
+lemma minorantGap_pos {p : G → ℝ} (c : FiniteConvolution.Minorant p) : 0 < minorantGap c := by
+  have hn : (1:ℝ) ≤ 2^c.exponent := one_le_pow₀ (by norm_num)
+  have hq : c.mass/(2*(2:ℝ)^c.exponent) ≤ 1/2 := by
+    apply (div_le_iff₀ (by positivity)).mpr
+    nlinarith [c.mass_le_one]
+  unfold minorantGap
+  linarith
+
+omit [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V] in
+lemma minorantGap_lt_one {p : G → ℝ} (c : FiniteConvolution.Minorant p) : minorantGap c < 1 := by
+  unfold minorantGap
+  exact sub_lt_self _ (div_pos c.mass_pos (by positivity))
+
+theorem norm_off_invariants_le_gap (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
+    (p : G → ℝ) (hp : ∑ g,p g=1) (hsym : ∀ g,p g⁻¹=p g)
+    (c : FiniteConvolution.Minorant p) :
+    ‖weightedOperator ρ p-uniformOperator ρ‖ ≤ minorantGap c := by
+  have hself := (weightedOperator_symmetric ρ hρ p hsym).sub (uniformOperator_symmetric ρ hρ)
+  have hbound := doeblin_off_invariants ρ hρ (FiniteConvolution.power p (2^c.exponent))
+    (FiniteConvolution.sum_power hp _) c.mass c.lower
+  rw [weightedOperator_power,←off_invariants_pow ρ p hp _ (by positivity),
+    hself.norm_pow_two_pow] at hbound
+  have hn : (1:ℝ) ≤ 2^c.exponent := one_le_pow₀ (by norm_num)
+  have hq : c.mass/(2*(2:ℝ)^c.exponent) ≤ 1/2 := by
+    apply (div_le_iff₀ (by positivity)).mpr
+    nlinarith [c.mass_le_one]
+  have hbern := one_add_mul_le_pow
+    (a := -(c.mass/(2*(2:ℝ)^c.exponent))) (by linarith : (-2:ℝ) ≤ -(c.mass/(2*(2:ℝ)^c.exponent)))
+    (2^c.exponent)
+  have he : 1+(2^c.exponent:ℕ)*(-(c.mass/(2*(2:ℝ)^c.exponent)))=1-c.mass/2 := by
+    push_cast
+    field_simp
+    ring
+  rw [he,show 1+ -(c.mass/(2*(2:ℝ)^c.exponent))=minorantGap c by rfl] at hbern
+  by_contra h
+  have hlt : minorantGap c < ‖weightedOperator ρ p-uniformOperator ρ‖ := lt_of_not_ge h
+  have hpows := pow_lt_pow_left₀ hlt (minorantGap_pos c).le (by positivity : (2^c.exponent:ℕ)≠0)
+  linarith [c.mass_pos]
+
+end Thorp.UnitaryFinite
+namespace Thorp
+open scoped BigOperators Classical
+
+def butterflyIdentity : (d : ℕ) → Butterfly d
+  | 0 => ()
+  | d+1 => (fun _ => false,fun _ => butterflyIdentity d)
+
+lemma pairSwitch_false (d : ℕ) : pairSwitch (d := d) (fun _ => false)=1 := by
+  apply Equiv.ext
+  intro x
+  change switchFun (fun _ => false) x=x
+  simp [switchFun]
+
+lemma childLift_one (d : ℕ) : childLift (d := d) (fun _ => 1)=1 := by
+  ext x i
+  simp [childLift]
+
+lemma butterflyIdentity_perm (d : ℕ) : butterflyPerm d (butterflyIdentity d)=1 := by
+  induction d with
+  | zero => rfl
+  | succ d ih => simp only [butterflyIdentity,butterflyPerm,ih,pairSwitch_false,childLift_one,mul_one]
+
+noncomputable def childInjection (d : ℕ) (b : Bool) : Equiv.Perm (Card d) →* Equiv.Perm (Card (d+1)) where
+  toFun p := childLift (fun c => if c=b then p else 1)
+  map_one' := by simpa only [ite_self] using childLift_one d
+  map_mul' p q := by
+    rw [childLift_mul]
+    congr 1
+    funext c
+    split_ifs <;> simp
+
+lemma childInjection_swap (d : ℕ) (b : Bool) (u v : Card d) :
+    childInjection d b (Equiv.swap u v)=Equiv.swap (Fin.cons b u) (Fin.cons b v) := by
+  apply Equiv.ext
+  intro x
+  obtain ⟨c,w,rfl⟩ : ∃ c w, x=Fin.cons c w := ⟨x 0,Fin.tail x,(Fin.cons_self_tail x).symm⟩
+  change (Fin.cons c ((if c=b then Equiv.swap u v else 1) w) : Card (d+1))=_
+  by_cases hc : c=b
+  · subst c
+    simp only [↓reduceIte]
+    rw [Equiv.swap_apply_def]
+    split_ifs with h₁ h₂
+    · subst w; simp
+    · subst w; simp
+    · rw [Equiv.swap_apply_of_ne_of_ne]
+      · exact fun h => h₁ (by simpa using congrArg Fin.tail h)
+      · exact fun h => h₂ (by simpa using congrArg Fin.tail h)
+  · rw [ite_eq_right hc,Equiv.Perm.one_apply,Equiv.swap_apply_of_ne_of_ne]
+    · intro h; exact hc (by simpa using congrArg (fun z => z 0) h)
+    · intro h; exact hc (by simpa using congrArg (fun z => z 0) h)
+
+lemma pairSwitch_single (d : ℕ) (u : Card d) :
+    pairSwitch (fun v => decide (v=u))=Equiv.swap (Fin.cons false u) (Fin.cons true u) := by
+  apply Equiv.ext
+  intro x
+  obtain ⟨c,w,rfl⟩ : ∃ c w, x=Fin.cons c w := ⟨x 0,Fin.tail x,(Fin.cons_self_tail x).symm⟩
+  change switchFun (fun v => decide (v=u)) (Fin.cons c w)=_
+  by_cases h : w=u
+  · subst w
+    cases c <;> simp [switchFun]
+  · have ha : (Fin.cons c w : Card (d+1))≠Fin.cons false u := fun he => h (by simpa using congrArg Fin.tail he)
+    have hb : (Fin.cons c w : Card (d+1))≠Fin.cons true u := fun he => h (by simpa using congrArg Fin.tail he)
+    simp [switchFun,h,Equiv.swap_apply_of_ne_of_ne ha hb]
+
+theorem butterfly_generates (d : ℕ) (H : Subgroup (Equiv.Perm (Card d)))
+    (hH : ∀ ω : Butterfly d, butterflyPerm d ω∈H) : H=⊤ := by
+  induction d with
+  | zero =>
+    apply top_unique
+    intro p _
+    have hp : p=1 := Subsingleton.elim _ _
+    rw [hp]
+    exact H.one_mem
+  | succ d ih =>
+    have hchild (b : Bool) (p : Equiv.Perm (Card d)) : childInjection d b p∈H := by
+      have hc : H.comap (childInjection d b)=⊤ := by
+        apply ih
+        intro ω
+        change childLift (fun c => if c=b then butterflyPerm d ω else 1)∈H
+        have h := hH ((fun _ => false),fun c => if c=b then ω else butterflyIdentity d)
+        simpa only [butterflyPerm,pairSwitch_false,one_mul,apply_ite,butterflyIdentity_perm] using h
+      have hp : p∈H.comap (childInjection d b) := by rw [hc]; trivial
+      exact hp
+    have hpair (u : Card d) : Equiv.swap (Fin.cons false u) (Fin.cons true u)∈H := by
+      rw [←pairSwitch_single]
+      have h := hH ((fun v => decide (v=u)),fun _ => butterflyIdentity d)
+      simpa only [butterflyPerm,butterflyIdentity_perm,childLift_one,mul_one] using h
+    have hswap (x y : Card (d+1)) : Equiv.swap x y∈H := by
+      have ht (a b : Bool) (u : Card d) : Equiv.swap (Fin.cons a u) (Fin.cons b u)∈H := by
+        cases a <;> cases b
+        · rw [Equiv.swap_self]; exact H.one_mem
+        · exact hpair _
+        · rw [Equiv.swap_comm]; exact hpair _
+        · rw [Equiv.swap_self]; exact H.one_mem
+      have h₁ : Equiv.swap x (Fin.cons (y 0) (Fin.tail x))∈H := by
+        convert ht (x 0) (y 0) (Fin.tail x) using 1
+        rw [Fin.cons_self_tail]
+      have h₂ := hchild (y 0) (Equiv.swap (Fin.tail x) (Fin.tail y))
+      rw [childInjection_swap,Fin.cons_self_tail] at h₂
+      exact SubmonoidClass.swap_mem_trans H h₁ h₂
+    rw [eq_top_iff,←Equiv.Perm.closure_isSwap,Subgroup.closure_le]
+    rintro _ ⟨x,y,_,rfl⟩
+    exact hswap x y
+
+lemma butterfly_closure (d : ℕ) : Subgroup.closure (Set.range (butterflyPerm d))=⊤ :=
+  butterfly_generates d _ (fun ω => Subgroup.subset_closure ⟨ω,rfl⟩)
+
+variable {Ω G : Type*} [Fintype Ω] [Fintype G]
+
+noncomputable def sampleLaw (P : Ω → G) (g : G) : ℝ :=
+  finiteMean (fun ω => if P ω=g then 1 else 0)
+
+omit [Fintype G] in
+lemma sampleLaw_nonneg (P : Ω → G) (g : G) : 0 ≤ sampleLaw P g :=
+  finiteMean_nonneg (fun _ => by split_ifs <;> norm_num)
+
+lemma sampleLaw_sum [Nonempty Ω] (P : Ω → G) : ∑ g,sampleLaw P g=1 := by
+  unfold sampleLaw
+  rw [←finiteMean_sum]
+  simp only [Finset.sum_ite_eq,Finset.mem_univ,↓reduceIte,finiteMean_const]
+
+omit [Fintype G] in
+lemma sampleLaw_pos_iff [Nonempty Ω] (P : Ω → G) (g : G) :
+    0<sampleLaw P g ↔ ∃ ω,P ω=g := by
+  unfold sampleLaw finiteMean
+  rw [div_pos_iff_of_pos_right (Nat.cast_pos.mpr Fintype.card_pos)]
+  rw [Finset.sum_pos_iff_of_nonneg (fun _ _ => by split_ifs <;> norm_num)]
+  simp only [Finset.mem_univ,true_and]
+  apply exists_congr
+  intro ω
+  split_ifs <;> simp_all
+
+omit [Fintype G] in
+lemma sampleLaw_equiv {Ω' : Type*} [Fintype Ω'] (P : Ω → G) (e : Ω' ≃ Ω) :
+    sampleLaw (P ∘ e)=sampleLaw P := by
+  funext g
+  simpa only [sampleLaw,Function.comp_apply,Equiv.apply_symm_apply] using
+    finiteMean_equiv e (fun ω => if P (e ω)=g then (1:ℝ) else 0)
+
+variable [Group G]
+omit [Fintype G] in
+lemma sampleLaw_inverse (P : Ω → G) (g : G) :
+    sampleLaw (fun ω => (P ω)⁻¹) g=sampleLaw P g⁻¹ := by
+  apply finiteMean_congr
+  intro ω
+  rw [inv_eq_iff_eq_inv]
+
+omit [Fintype G] in
+lemma sampleLaw_symmetric (P : Ω → G) (e : Equiv.Perm Ω)
+    (he : ∀ ω,P (e ω)=(P ω)⁻¹) (g : G) : sampleLaw P g⁻¹=sampleLaw P g := by
+  rw [←sampleLaw_inverse]
+  have hh : (fun ω => (P ω)⁻¹)=P ∘ e := funext (fun ω => (he ω).symm)
+  rw [hh,sampleLaw_equiv]
+
+end Thorp
+
+namespace Thorp
+open scoped BigOperators Classical
+
+lemma palindrome_identity (d : ℕ) : ∃ ω,palindromePerm d ω=1 := by
+  refine ⟨((butterflyCoinEquiv d).symm (butterflyIdentity d),
+    (butterflyCoinEquiv d).symm (butterflyIdentity d)),?_⟩
+  exact mul_inv_cancel _
+
+lemma palindrome_range_contains_butterfly (d : ℕ) (ω : Butterfly d) :
+    ∃ ξ,palindromePerm d ξ=butterflyPerm d ω := by
+  refine ⟨((butterflyCoinEquiv d).symm ω,(butterflyCoinEquiv d).symm (butterflyIdentity d)),?_⟩
+  unfold palindromePerm
+  have h (β : Butterfly d) : decodeButterfly d ((butterflyCoinEquiv d).symm β)=β :=
+    (butterflyCoinEquiv d).apply_symm_apply β
+  rw [h,h,butterflyIdentity_perm,show (1 : Equiv.Perm (Card d)).symm=1 from rfl,mul_one]
+
+lemma palindrome_support_generates (d : ℕ) :
+    Subgroup.closure {g | 0<sampleLaw (palindromePerm d) g}=⊤ := by
+  apply butterfly_generates
+  intro ω
+  apply Subgroup.subset_closure
+  exact (sampleLaw_pos_iff _ _).mpr (palindrome_range_contains_butterfly d ω)
+
+noncomputable def palindromeMinorant (d : ℕ) :
+    FiniteConvolution.Minorant (sampleLaw (palindromePerm d)) :=
+  FiniteConvolution.minorant (sampleLaw_nonneg _) (sampleLaw_sum _)
+    ((sampleLaw_pos_iff _ _).mpr (palindrome_identity d)) (palindrome_support_generates d)
+
+noncomputable def blockGap (d : ℕ) : ℝ := UnitaryFinite.minorantGap (palindromeMinorant d)
+
+lemma blockGap_pos (d : ℕ) : 0<blockGap d := UnitaryFinite.minorantGap_pos _
+lemma blockGap_lt_one (d : ℕ) : blockGap d<1 := UnitaryFinite.minorantGap_lt_one _
+
+theorem palindrome_off_invariants_gap (d : ℕ) {V : Type*}
+    [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+    (ρ : Representation ℂ (Equiv.Perm (Card d)) V) (hρ : UnitaryFinite.IsUnitary ρ) :
+    ‖UnitaryFinite.weightedOperator ρ (sampleLaw (palindromePerm d))-
+      UnitaryFinite.uniformOperator ρ‖≤blockGap d := by
+  apply UnitaryFinite.norm_off_invariants_le_gap ρ hρ _ (sampleLaw_sum _)
+  exact sampleLaw_symmetric _ (Equiv.prodComm _ _) (palindromePerm_reverse d)
+
+end Thorp
+
+namespace Thorp.CommutingBlocks
+open scoped BigOperators Classical
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
+variable {ι : Type*} [Fintype ι]
+
+omit [CompleteSpace V] in
+lemma orthogonal_sum_norm_sq (v : ι → V) (hv : ∀ i j, i≠j → inner ℂ (v i) (v j)=0) :
+    ‖∑ i,v i‖^2=∑ i,‖v i‖^2 := by
+  rw [norm_sq_eq_re_inner (𝕜 := ℂ),sum_inner]
+  simp only [inner_sum,map_sum]
+  apply Finset.sum_congr rfl
+  intro i _
+  rw [Finset.sum_eq_single i]
+  · exact (norm_sq_eq_re_inner (𝕜 := ℂ) (v i)).symm
+  · intro j _ hji
+    rw [hv i j hji.symm]
+    rfl
+  · simp
+
+lemma projection_inner_zero (P Q : V →L[ℂ] V) (hP : IsStarProjection P)
+    (hPQ : P*Q=0) (v w : V) : inner ℂ (P v) (Q w)=0 := by
+  rw [←ContinuousLinearMap.adjoint_inner_right]
+  change inner ℂ v ((star P*Q) w)=0
+  rw [hP.isSelfAdjoint.star_eq,hPQ]
+  simp
+
+lemma projection_sandwich_norm (P B : V →L[ℂ] V) (hP : IsStarProjection P) :
+    ‖B.adjoint*P*B‖=‖P*B‖^2 := by
+  have he : star (P*B)*(P*B)=B.adjoint*P*B := by
+    rw [star_mul,hP.isSelfAdjoint.star_eq]
+    change B.adjoint*P*(P*B)=_
+    rw [←mul_assoc,mul_assoc B.adjoint P P,hP.isIdempotentElem.eq]
+  rw [←he,CStarRing.norm_star_mul_self]
+  ring
+
+theorem resolution_norm_bound (P : ι → V →L[ℂ] V)
+    (hP : ∀ i, IsStarProjection (P i))
+    (horth : ∀ i j, i≠j → P i*P j=0) (hsum : ∑ i,P i=1)
+    (A : V →L[ℂ] V) (hcomm : ∀ i, Commute A (P i))
+    (c : ℝ) (hc : 0≤c) (hbound : ∀ i, ‖A*P i‖≤c) : ‖A‖≤c := by
+  apply ContinuousLinearMap.opNorm_le_bound _ hc
+  intro v
+  have hsplit (w : V) : ∑ i,P i w=w := by
+    rw [←sum_apply,hsum]
+    rfl
+  have hp := orthogonal_sum_norm_sq (fun i => P i v)
+    (fun i j hij => projection_inner_zero _ _ (hP i) (horth i j hij) v v)
+  rw [hsplit] at hp
+  have ha := orthogonal_sum_norm_sq (fun i => P i (A v))
+    (fun i j hij => projection_inner_zero _ _ (hP i) (horth i j hij) (A v) (A v))
+  rw [hsplit] at ha
+  have he (i : ι) : P i (A v)=(A*P i) (P i v) := by
+    change (P i*A) v=(A*P i*P i) v
+    rw [mul_assoc,(hP i).isIdempotentElem.eq,(hcomm i).eq]
+  have hh (i : ι) : ‖P i (A v)‖≤c*‖P i v‖ := by
+    rw [he]
+    exact ((A*P i).le_opNorm _).trans (mul_le_mul_of_nonneg_right (hbound i) (norm_nonneg _))
+  have hsq := Finset.sum_le_sum (s := Finset.univ) (fun i _ =>
+    pow_le_pow_left₀ (norm_nonneg _) (hh i) 2)
+  simp_rw [mul_pow] at hsq
+  rw [←Finset.mul_sum,←hp,←ha,←mul_pow] at hsq
+  exact (sq_le_sq₀ (norm_nonneg _) (mul_nonneg hc (norm_nonneg _))).mp hsq
+
+theorem sector_sandwich_le (T P A B : V →L[ℂ] V)
+    (hT : IsStarProjection T) (hP : IsStarProjection P)
+    (hTP : T*P=T) (hcomm : Commute A T) (hA : ‖A‖≤1) :
+    ‖B.adjoint*A*T*B‖≤‖B.adjoint*P*B‖ := by
+  have he : B.adjoint*A*T*B=(T*B).adjoint*A*(T*B) := by
+    change _=star (T*B)*A*(T*B)
+    rw [star_mul,hT.isSelfAdjoint.star_eq]
+    change _=B.adjoint*T*A*(T*B)
+    rw [mul_assoc B.adjoint T A,←hcomm.eq]
+    simp only [mul_assoc]
+    rw [←mul_assoc T T B,hT.isIdempotentElem.eq]
+  have hTB : ‖T*B‖≤‖P*B‖ := by
+    have he : T*B=T*(P*B) := by rw [←mul_assoc,hTP]
+    rw [he]
+    exact (norm_mul_le _ _).trans ((mul_le_mul_of_nonneg_right (hT.norm_le _) (norm_nonneg _)).trans_eq (one_mul _))
+  rw [he,projection_sandwich_norm P B hP]
+  calc
+    _ ≤ ‖(T*B).adjoint‖*‖A‖*‖T*B‖ := (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right (norm_mul_le _ _) (norm_nonneg _))
+    _ ≤ ‖T*B‖^2 := by rw [ContinuousLinearMap.adjoint.norm_map]; nlinarith [norm_nonneg (T*B)]
+    _ ≤ _ := pow_le_pow_left₀ (norm_nonneg _) hTB 2
+
+omit [CompleteSpace V] [Fintype ι] in
+lemma resolution_commute (P : ι → V →L[ℂ] V)
+    (horth : ∀ i j, i≠j → P i*P j=0) (i j : ι) : Commute (P i) (P j) := by
+  by_cases hij : i=j
+  · subst j; exact Commute.refl _
+  · exact (horth i j hij).trans (horth j i (Ne.symm hij)).symm
+
+omit [Fintype ι] in
+lemma resolution_sum_mul (P : ι → V →L[ℂ] V) (hP : ∀ i, IsStarProjection (P i))
+    (horth : ∀ i j, i≠j → P i*P j=0) (S : Finset ι) (j : ι) :
+    (∑ i∈S,P i)*P j=if j∈S then P j else 0 := by
+  rw [Finset.sum_mul]
+  have he (i : ι) : P i*P j=if i=j then P j else 0 := by
+    split_ifs with h
+    · subst i; exact (hP j).isIdempotentElem.eq
+    · exact horth i j h
+  simp only [he,Finset.sum_ite_eq']
+
+lemma resolution_truncated_norm (P : ι → V →L[ℂ] V)
+    (hP : ∀ i, IsStarProjection (P i))
+    (horth : ∀ i j, i≠j → P i*P j=0) (hsum : ∑ i,P i=1)
+    (A : V →L[ℂ] V) (hcomm : ∀ i, Commute A (P i))
+    (S : Finset ι) (c : ℝ) (hc : 0≤c) (hbound : ∀ i∈S, ‖A*P i‖≤c) :
+    ‖A*(∑ i∈S,P i)‖≤c := by
+  apply resolution_norm_bound P hP horth hsum _ _ c hc
+  · intro i
+    rw [mul_assoc,resolution_sum_mul P hP horth]
+    split_ifs with hi
+    · exact hbound i hi
+    · simp [hc]
+  · intro i
+    exact (hcomm i).mul_left (Commute.sum_left S P (P i) fun j _ => resolution_commute P horth j i)
+
+theorem resolution_exception_bound (T Q : ι → V →L[ℂ] V)
+    (hT : ∀ i, IsStarProjection (T i))
+    (horth : ∀ i j, i≠j → T i*T j=0) (hsum : ∑ i,T i=1)
+    (hQ : ∀ i, IsStarProjection (Q i)) (hTQ : ∀ i,T i*Q i=T i)
+    (A B : V →L[ℂ] V) (hcomm : ∀ i, Commute A (T i))
+    (hA : ‖A‖≤1) (hB : ‖B‖≤1)
+    (S : Finset ι) (c : ℝ) (hc : 0≤c) (hbound : ∀ i∉S, ‖A*T i‖≤c) :
+    ‖B.adjoint*A*B‖≤c+∑ i∈S,‖B.adjoint*Q i*B‖ := by
+  have hhigh : ‖A*(∑ i∈Sᶜ,T i)‖≤c := resolution_truncated_norm T hT horth hsum A hcomm
+    Sᶜ c hc (fun i hi => hbound i (Finset.mem_compl.mp hi))
+  have hsplit : (∑ i∈Sᶜ,T i)+(∑ i∈S,T i)=1 := by
+    rw [Finset.sum_compl_add_sum,hsum]
+  have he : B.adjoint*A*B=B.adjoint*(A*(∑ i∈Sᶜ,T i))*B+
+      ∑ i∈S,B.adjoint*A*T i*B := by
+    rw [←Finset.sum_mul,←Finset.mul_sum]
+    rw [←add_mul]
+    simp only [mul_assoc]
+    rw [←mul_add,←mul_add,hsplit]
+    simp only [mul_one,mul_assoc]
+  rw [he]
+  apply (norm_add_le _ _).trans
+  apply add_le_add
+  · calc
+      _ ≤ ‖B.adjoint‖*‖A*(∑ i∈Sᶜ,T i)‖*‖B‖ :=
+        (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right (norm_mul_le _ _) (norm_nonneg _))
+      _ ≤ 1*c*1 := mul_le_mul (mul_le_mul (by simpa only [ContinuousLinearMap.adjoint.norm_map] using hB)
+        hhigh (norm_nonneg _) zero_le_one) hB (norm_nonneg _) (by simpa using hc)
+      _ = c := by ring
+  · apply (norm_sum_le _ _).trans
+    exact Finset.sum_le_sum fun i _ => sector_sandwich_le _ _ _ _ (hT i) (hQ i) (hTQ i) (hcomm i) hA
+
+variable [DecidableEq ι]
+variable (P : ι → V →L[ℂ] V) (hcomm : ∀ i j, Commute (P i) (P j))
+
+noncomputable def choice (E : Finset ι) (i : ι) : V →L[ℂ] V :=
+  if i∈E then 1-P i else P i
+
+include hcomm
+omit [CompleteSpace V] [Fintype ι] in
+lemma choice_commute (E F : Finset ι) (i j : ι) :
+    Commute (choice P E i) (choice P F j) := by
+  unfold choice
+  split_ifs
+  · exact (Commute.one_left _).sub_left ((Commute.one_right _).sub_right (hcomm i j))
+  · exact (Commute.one_left _).sub_left (hcomm i j)
+  · exact (Commute.one_right _).sub_right (hcomm i j)
+  · exact hcomm i j
+
+noncomputable def sector (s E : Finset ι) : V →L[ℂ] V :=
+  s.noncommProd (choice P E) (fun i _ j _ _ => choice_commute P hcomm E E i j)
+
+omit [CompleteSpace V] [Fintype ι] in
+lemma sector_insert (s E : Finset ι) (a : ι) (ha : a∉s) :
+    sector P hcomm (insert a s) E=choice P E a*sector P hcomm s E := by
+  exact Finset.noncommProd_insert_of_notMem _ _ _ _ ha
+
+omit [CompleteSpace V] [Fintype ι] in
+lemma sector_congr (s E F : Finset ι) (h : ∀ i∈s, (i∈E ↔ i∈F)) :
+    sector P hcomm s E=sector P hcomm s F := by
+  apply Finset.noncommProd_congr rfl
+  intro i hi
+  simp only [choice,h i hi]
+
+omit [CompleteSpace V] [Fintype ι] in
+lemma sector_factor_commute (s E F : Finset ι) (i : ι) :
+    Commute (choice P E i) (sector P hcomm s F) :=
+  Finset.noncommProd_commute _ _ _ _ (fun j _ => choice_commute P hcomm E F i j)
+
+omit [Fintype ι] in
+lemma sector_projection (hP : ∀ i, IsStarProjection (P i)) (s E : Finset ι) :
+    IsStarProjection (sector P hcomm s E) := by
+  induction s using Finset.induction_on with
+  | empty => exact IsStarProjection.one _
+  | @insert a s ha ih =>
+    rw [sector_insert P hcomm s E a ha]
+    have hc : IsStarProjection (choice P E a) := by
+      unfold choice
+      split_ifs
+      · exact (hP a).one_sub
+      · exact hP a
+    exact hc.mul ih (sector_factor_commute P hcomm s E E a)
+
+end Thorp.CommutingBlocks
+
+end ThorpNine.SparseSaving
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/SparseSaving/RotationMeeting.lean b/lean/OAI/Probability/ThorpRouting/SparseSaving/RotationMeeting.lean
index 5e9006b90fa97478b2a158790d87b5ccf357b3b7..d67d3d6c82fc18d34b58b2312bb4ef3e4dc37ac8 100644
--- a/lean/OAI/Probability/ThorpRouting/SparseSaving/RotationMeeting.lean
+++ b/lean/OAI/Probability/ThorpRouting/SparseSaving/RotationMeeting.lean
@@ -1,881 +1,881 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.SparseSaving.ExposedCoins
-
-namespace OAI
-
-namespace ThorpNine.SparseSaving
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-section Rotation
-variable {d A : ℕ}
-
-noncomputable def rotationMeeting (o : Card d → Option (Fin A)) (z : Terminal o)
-    (u v : Card d) : ℝ :=
-  match o u, o v with
-  | some i, some j => if i < j then (if (z j).val = v then 1 else 0)
-      else if j < i then (if (z i).val = u then 1 else 0)
-      else (if (z i).val = u then 1 else 0) + (if (z i).val = v then 1 else 0)
-  | _, _ => 0
-
-lemma card_cycleFiber (o : Card d → Option (Fin A)) (i : Fin A) :
-    Fintype.card {x : Card d // o x = some i} = (cycleFiber o i).card := by
-  classical
-  simp [cycleFiber,Fintype.card_subtype]
-
-lemma terminal_mean {o : Card d → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
-    (i : Fin A) (x : Card d) (hx : o x = some i) :
-    finiteMean (fun z : Terminal o => if (z i).val = x then (1:ℝ) else 0) = slotWeight o i := by
-  classical
-  have hev := finiteMean_pi_eval (A := fun i : Fin A => {x : Card d // o x = some i})
-    i (fun z => if z.val = x then (1:ℝ) else 0)
-  rw [hev]
-  have he : (fun z : {x : Card d // o x = some i} => if z.val = x then (1:ℝ) else 0) =
-      (fun z => if z = ⟨x,hx⟩ then 1 else 0) := by
-    funext z
-    simp only [Subtype.ext_iff]
-  rw [he,finiteMean_eq_indicator,card_cycleFiber,slotWeight]
-
-lemma slotWeight_antitone (o : Card d → Option (Fin A))
-    (hn : ∀ i, (cycleFiber o i).Nonempty)
-    (hm : Monotone (fun i => (cycleFiber o i).card)) : Antitone (slotWeight o) := by
-  intro i j hij
-  unfold slotWeight
-  apply one_div_le_one_div_of_le
-  · exact_mod_cast Finset.card_pos.mpr (hn i)
-  · exact_mod_cast hm hij
-
-lemma rotationMeeting_mean {o : Card d → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
-    (hm : Monotone (fun i => (cycleFiber o i).card)) (u v : Card d) :
-    finiteMean (fun z : Terminal o => rotationMeeting o z u v) =
-      min (pathWeight o u) (pathWeight o v) + withinWeight o u v := by
-  classical
-  have hn : ∀ i, (cycleFiber o i).Nonempty := by
-    intro i
-    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : Card d // o x = some i})
-    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
-  have hw := slotWeight_antitone o hn hm
-  cases hu : o u with
-  | none =>
-    cases hv : o v with
-    | none => simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean]
-    | some j =>
-      have hj : 0 ≤ slotWeight o j := by unfold slotWeight; positivity
-      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_left hj]
-  | some i =>
-    cases hv : o v with
-    | none =>
-      have hi : 0 ≤ slotWeight o i := by unfold slotWeight; positivity
-      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_right hi]
-    | some j =>
-      simp only [rotationMeeting,pathWeight,withinWeight,hu,hv]
-      rcases lt_trichotomy i j with hij | rfl | hji
-      · simp only [hij,↓reduceIte,ne_of_lt hij]
-        rw [terminal_mean j v hv,min_eq_right (hw (le_of_lt hij)),add_zero]
-      · simp only [lt_self_iff_false,↓reduceIte,min_self]
-        rw [finiteMean_add,terminal_mean i u hu,terminal_mean i v hv]
-      · simp only [not_lt_of_ge (le_of_lt hji),hji,↓reduceIte,ne_of_gt hji]
-        rw [terminal_mean i u hu,min_eq_left (hw (le_of_lt hji)),add_zero]
-
-lemma pathWeight_sum (o : Card d → Option (Fin A)) (hn : ∀ i, (cycleFiber o i).Nonempty) :
-    ∑ x : Card d, pathWeight o x = (A : ℝ) := by
-  classical
-  have he (x : Card d) : pathWeight o x =
-      ∑ i : Fin A, if o x = some i then slotWeight o i else 0 := by
-    cases hx : o x with
-    | none => simp [pathWeight,hx]
-    | some j => simp [pathWeight,hx]
-  simp only [he]
-  rw [Finset.sum_comm]
-  have hf (i : Fin A) : (∑ x : Card d, if o x = some i then slotWeight o i else 0) = 1 := by
-    rw [← Finset.sum_filter]
-    change (∑ _x ∈ cycleFiber o i, slotWeight o i) = 1
-    simp only [Finset.sum_const,nsmul_eq_mul,slotWeight]
-    exact mul_one_div_cancel (by exact_mod_cast (Finset.card_pos.mpr (hn i)).ne')
-  simp only [hf,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,mul_one]
-
-lemma withinWeight_sum_bound (o : Card d → Option (Fin A))
-    (hn : ∀ i, (cycleFiber o i).Nonempty) (b : Butterfly d) :
-    (∑ s : SwitchIndex d, withinWeight o (switchUsers d b s).1 (switchUsers d b s).2) ≤
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
-  classical
-  have he (u v : Card d) : withinWeight o u v =
-      ∑ i : Fin A, (if u ∈ cycleFiber o i ∧ v ∈ cycleFiber o i then (1:ℝ) else 0) * slotWeight o i := by
-    simp only [mem_cycleFiber]
-    cases hu : o u <;> cases hv : o v <;> simp [withinWeight,hu,hv,ite_and,eq_comm]
-  simp only [he]
-  rw [Finset.sum_comm]
-  apply Finset.sum_le_sum
-  intro i _
-  rw [← Finset.sum_mul]
-  have hi := marked_switch_entropy_bound d b (cycleFiber o i)
-  have hpos : (0:ℝ) < (cycleFiber o i).card := by exact_mod_cast Finset.card_pos.mpr (hn i)
-  have hmul := mul_le_mul_of_nonneg_right hi (show 0 ≤ slotWeight o i by unfold slotWeight; positivity)
-  calc
-    _ ≤ ((cycleFiber o i).card : ℝ) * Real.log (cycleFiber o i).card /
-        (2 * Real.log 2) * slotWeight o i := hmul
-    _ = _ := by unfold slotWeight; field_simp
-
-noncomputable def rotationCharge (o : Card d → Option (Fin A)) (b : Butterfly d)
-    (z : Terminal o) : ℝ :=
-  ∑ s : SwitchIndex d, rotationMeeting o z (switchUsers d b s).1 (switchUsers d b s).2
-
-lemma rotationCharge_mean_bound {o : Card d → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
-    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : Butterfly d) :
-    finiteMean (rotationCharge o b) ≤ (A:ℝ) * d / 2 +
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
-  classical
-  have hn : ∀ i, (cycleFiber o i).Nonempty := by
-    intro i
-    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : Card d // o x = some i})
-    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
-  unfold rotationCharge
-  rw [finiteMean_sum]
-  simp only [rotationMeeting_mean hm,Finset.sum_add_distrib]
-  have h₁ := weighted_switch_min_bound d b (pathWeight o)
-  rw [pathWeight_sum o hn] at h₁
-  have h₂ := withinWeight_sum_bound o hn b
-  linarith
-
-lemma rotation_exp_bound {o : Card d → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
-    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : Butterfly d) :
-    Real.exp (-Real.log 2 * ((A:ℝ)*d/2 +
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2))) ≤
-      finiteMean (fun z : Terminal o => Real.exp (-Real.log 2 * rotationCharge o b z)) := by
-  have hb := rotationCharge_mean_bound hm b
-  have hlog : 0 ≤ Real.log 2 := Real.log_nonneg (by norm_num)
-  have hm' : -Real.log 2 * ((A:ℝ)*d/2 +
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2)) ≤
-      finiteMean (fun z : Terminal o => -Real.log 2 * rotationCharge o b z) := by
-    have he := finiteMean_mul_const (rotationCharge o b) (-Real.log 2)
-    simp only [mul_comm] at he
-    rw [he]
-    exact mul_le_mul_of_nonpos_left hb (neg_nonpos.mpr hlog)
-  exact (Real.exp_le_exp.mpr hm').trans (exp_finiteMean_le _)
-
-end Rotation
-
-section OrbitCore
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-noncomputable def orbitSet (p : Equiv.Perm α) (x : α) : Finset α :=
-  Finset.univ.filter (p.SameCycle x)
-
-@[simp] lemma mem_orbitSet (p : Equiv.Perm α) (x y : α) :
-    y ∈ orbitSet p x ↔ p.SameCycle x y := by
-  classical
-  simp [orbitSet]
-
-lemma orbitSet_nonempty (p : Equiv.Perm α) (x : α) : (orbitSet p x).Nonempty :=
-  ⟨x,(mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.refl _ _)⟩
-
-lemma orbitSet_eq (p : Equiv.Perm α) {x y : α} (h : p.SameCycle x y) :
-    orbitSet p x = orbitSet p y := by
-  ext z
-  simp only [mem_orbitSet]
-  exact ⟨fun hz => h.symm.trans hz,fun hz => h.trans hz⟩
-
-@[simp] lemma orbitSet_apply (p : Equiv.Perm α) (x : α) : orbitSet p (p x) = orbitSet p x := by
-  ext y
-  simp only [mem_orbitSet,Equiv.Perm.sameCycle_apply_left]
-
-lemma orbitSet_eq_or_disjoint (p : Equiv.Perm α) (x y : α) :
-    orbitSet p x = orbitSet p y ∨ Disjoint (orbitSet p x) (orbitSet p y) := by
-  classical
-  by_cases h : p.SameCycle x y
-  · exact Or.inl (orbitSet_eq p h)
-  · right
-    apply Finset.disjoint_left.mpr
-    intro z hz hz'
-    exact h (((mem_orbitSet _ _ _).mp hz).trans ((mem_orbitSet _ _ _).mp hz').symm)
-
-omit [DecidableEq α] in
-lemma perm_minimalPeriod_pos (p : Equiv.Perm α) (x : α) :
-    0 < Function.minimalPeriod p x :=
-  Function.minimalPeriod_pos_of_mem_periodicPts (p.injective.mem_periodicPts x)
-
-lemma orbitSet_card (p : Equiv.Perm α) (x : α) :
-    (orbitSet p x).card = Function.minimalPeriod p x := by
-  classical
-  let f : Fin (Function.minimalPeriod p x) → {y // p.SameCycle x y} := fun n =>
-    ⟨p^[n.val] x,by rw [Equiv.Perm.iterate_eq_pow]; exact ⟨(n.val:ℤ),by simp⟩⟩
-  have hf : Function.Bijective f := by
-    constructor
-    · intro i j hij
-      apply Fin.ext
-      exact Function.iterate_injOn_Iio_minimalPeriod i.isLt j.isLt (congrArg Subtype.val hij)
-    · intro y
-      obtain ⟨n,hn⟩ := y.property.exists_nat_pow_eq
-      refine ⟨⟨n % Function.minimalPeriod p x, Nat.mod_lt _ (perm_minimalPeriod_pos p x)⟩,?_⟩
-      apply Subtype.ext
-      change p^[n % Function.minimalPeriod p x] x = y.val
-      rw [Function.iterate_mod_minimalPeriod_eq,Equiv.Perm.iterate_eq_pow,hn]
-  have hc := Fintype.card_congr (Equiv.ofBijective f hf)
-  simpa [orbitSet,Fintype.card_subtype] using hc.symm
-
-def walkSet (p : Equiv.Perm α) (r : ℕ) (x : α) : Finset α :=
-  (Finset.range r).image (fun n => p^[n] x)
-
-omit [Fintype α] in
-@[simp] lemma mem_walkSet (p : Equiv.Perm α) (r : ℕ) (x y : α) :
-    y ∈ walkSet p r x ↔ ∃ n < r, p^[n] x = y := by
-  simp [walkSet]
-
-omit [Fintype α] in
-@[simp] lemma walkSet_zero (p : Equiv.Perm α) (x : α) : walkSet p 0 x = ∅ := by
-  simp [walkSet]
-
-omit [Fintype α] in
-lemma walkSet_succ_last (p : Equiv.Perm α) (r : ℕ) (x : α) :
-    walkSet p (r+1) x = insert (p^[r] x) (walkSet p r x) := by
-  simp [walkSet,Finset.range_add_one]
-
-omit [Fintype α] in
-lemma walkSet_succ_first (p : Equiv.Perm α) (r : ℕ) (x : α) :
-    walkSet p (r+1) x = insert x (walkSet p r (p x)) := by
-  ext y
-  simp only [mem_walkSet,Finset.mem_insert]
-  constructor
-  · rintro ⟨n,hn,hny⟩
-    cases n with
-    | zero => exact Or.inl (by simpa using hny.symm)
-    | succ n => exact Or.inr ⟨n,by omega,by simpa only [Function.iterate_succ_apply] using hny⟩
-  · rintro (rfl | ⟨n,hn,hny⟩)
-    · exact ⟨0,by omega,rfl⟩
-    · exact ⟨n+1,by omega,by simpa only [Function.iterate_succ_apply] using hny⟩
-
-omit [Fintype α] in
-lemma not_mem_short_walk (p : Equiv.Perm α) (x : α) {r : ℕ}
-    (hr : r < Function.minimalPeriod p x) : x ∉ walkSet p r (p x) := by
-  rintro hx
-  obtain ⟨n,hn,hnx⟩ := (mem_walkSet _ _ _ _).mp hx
-  have hperiod : Function.IsPeriodicPt p (n+1) x := by
-    simpa only [Function.IsPeriodicPt,Function.IsFixedPt,Function.iterate_succ_apply] using hnx
-  have hp := hperiod.minimalPeriod_le (by omega : 0 < n+1)
-  omega
-
-lemma walkSet_period (p : Equiv.Perm α) (x : α) :
-    walkSet p (Function.minimalPeriod p x) x = orbitSet p x := by
-  ext y
-  simp only [mem_walkSet,mem_orbitSet]
-  constructor
-  · rintro ⟨n,_,rfl⟩
-    rw [Equiv.Perm.iterate_eq_pow]
-    exact ⟨(n:ℤ),by simp⟩
-  · intro h
-    obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
-    refine ⟨n % Function.minimalPeriod p x,Nat.mod_lt _ (perm_minimalPeriod_pos p x),?_⟩
-    rw [Function.iterate_mod_minimalPeriod_eq,Equiv.Perm.iterate_eq_pow,hn]
-
-omit [Fintype α] in
-lemma walkSet_card (p : Equiv.Perm α) (x : α) {r : ℕ}
-    (hr : r ≤ Function.minimalPeriod p x) : (walkSet p r x).card = r := by
-  rw [walkSet,Finset.card_image_of_injOn]
-  · exact Finset.card_range _
-  · intro i hi j hj hij
-    exact Function.iterate_injOn_Iio_minimalPeriod
-      (lt_of_lt_of_le (Finset.mem_range.mp hi) hr)
-      (lt_of_lt_of_le (Finset.mem_range.mp hj) hr) hij
-
-lemma walkSet_subset_orbit (p : Equiv.Perm α) (r : ℕ) (x : α) :
-    walkSet p r x ⊆ orbitSet p x := by
-  intro y hy
-  obtain ⟨n,_,rfl⟩ := (mem_walkSet _ _ _ _).mp hy
-  rw [mem_orbitSet,Equiv.Perm.iterate_eq_pow]
-  exact ⟨(n:ℤ),by simp⟩
-
-def cycleTrace (p : Equiv.Perm α) (S : Finset α) (a : α) :
-    ℕ → α → Finset α → Option (Finset α × α)
-  | 0, x, V => if x ∈ S ∧ x ∉ V ∧ p x = a then some (insert x V,x) else none
-  | r+1, x, V => if x ∈ S ∧ x ∉ V then cycleTrace p S a r (p x) (insert x V) else none
-
-lemma cycleTrace_walk (p : Equiv.Perm α) (S : Finset α) (r : ℕ) (x : α) (V : Finset α)
-    (hr : r < Function.minimalPeriod p x) (hS : walkSet p (r+1) x ⊆ S)
-    (hV : Disjoint V (walkSet p (r+1) x)) :
-    cycleTrace p S (p^[r+1] x) r x V = some (V ∪ walkSet p (r+1) x,p^[r] x) := by
-  induction r generalizing x V with
-  | zero =>
-    have hxS : x ∈ S := hS (by simp [walkSet_succ_first])
-    have hxV : x ∉ V := fun h => Finset.disjoint_left.mp hV h (by simp [walkSet_succ_first])
-    simp [cycleTrace,hxS,hxV,walkSet_succ_first]
-  | succ r ih =>
-    have hxS : x ∈ S := hS (by rw [walkSet_succ_first]; exact Finset.mem_insert_self _ _)
-    have hxV : x ∉ V := fun h => Finset.disjoint_left.mp hV h
-      (by rw [walkSet_succ_first]; exact Finset.mem_insert_self _ _)
-    have hr' : r < Function.minimalPeriod p (p x) := by
-      rw [Function.minimalPeriod_apply (p.injective.mem_periodicPts x)]
-      omega
-    have hS' : walkSet p (r+1) (p x) ⊆ S := by
-      intro y hy
-      apply hS
-      rw [walkSet_succ_first]
-      exact Finset.mem_insert_of_mem hy
-    have hV' : Disjoint (insert x V) (walkSet p (r+1) (p x)) := by
-      apply Finset.disjoint_left.mpr
-      intro y hy hy'
-      rcases Finset.mem_insert.mp hy with hy | hy
-      · subst y
-        exact not_mem_short_walk p x hr hy'
-      · apply Finset.disjoint_left.mp hV hy
-        rw [walkSet_succ_first]
-        exact Finset.mem_insert_of_mem hy'
-    rw [cycleTrace,ite_eq_left ⟨hxS,hxV⟩,Function.iterate_succ_apply p (r+1) x,
-      ih _ _ hr' hS' hV',walkSet_succ_first p (r+1) x,Function.iterate_succ_apply]
-    simp only [Finset.insert_union,Finset.union_insert]
-
-lemma cycleTrace_orbit (p : Equiv.Perm α) (S V : Finset α) (z : α)
-    (hS : orbitSet p z ⊆ S) (hV : Disjoint V (orbitSet p z)) :
-    cycleTrace p S (p z) (Function.minimalPeriod p z - 1) (p z) V =
-      some (V ∪ orbitSet p z,z) := by
-  have hpos := perm_minimalPeriod_pos p z
-  have he : Function.minimalPeriod p (p z) = Function.minimalPeriod p z :=
-    Function.minimalPeriod_apply (p.injective.mem_periodicPts z)
-  have hp : Function.minimalPeriod p z - 1 + 1 = Function.minimalPeriod p z := by omega
-  have hw : walkSet p (Function.minimalPeriod p z - 1 + 1) (p z) = orbitSet p z := by
-    rw [hp,←he,walkSet_period,orbitSet_apply]
-  have hr : Function.minimalPeriod p z - 1 < Function.minimalPeriod p (p z) := by rw [he]; omega
-  have h := cycleTrace_walk p S (Function.minimalPeriod p z-1) (p z) V hr
-    (by simpa only [hw] using hS) (by simpa only [hw] using hV)
-  have ht : p^[Function.minimalPeriod p z-1] (p z) = z := by
-    rw [← Function.iterate_succ_apply,show (Function.minimalPeriod p z-1).succ =
-      Function.minimalPeriod p z by omega,Function.iterate_minimalPeriod]
-  have ha : p^[Function.minimalPeriod p z-1+1] (p z) = p z := by
-    rw [hp,←he,Function.iterate_minimalPeriod]
-  simpa only [ha,ht,hw] using h
-
-end OrbitCore
-
-noncomputable def exposedPaths (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) :
-    Finset (SwitchIndex d) :=
-  V.biUnion (fun x => routeDomain d x (butterflyPerm d b x))
-
-@[simp] lemma exposedPaths_empty (d : ℕ) (b : Butterfly d) : exposedPaths d b ∅ = ∅ := by
-  classical
-  simp [exposedPaths]
-
-lemma exposedPaths_insert (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) (x : Card d) :
-    exposedPaths d b (insert x V) = exposedPaths d b V ∪ routeDomain d x (butterflyPerm d b x) := by
-  classical
-  simp [exposedPaths,Finset.union_comm]
-
-noncomputable def terminalCharge (d : ℕ) (b : Butterfly d) (z : Card d) (V : Finset (Card d)) : ℕ :=
-  (routeDomain d z (butterflyPerm d b z) ∩ exposedPaths d b V).card
-
-namespace QueryTree
-
-lemma followPath_stale_charge {ι : Type*} [DecidableEq ι]
-    (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
-    (k : Card d → QueryTree ι) (R : Finset ι) (ω : ι → Bool) :
-    staleCharge (followPath d e x k) R ω =
-      staleCharge (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x))
-        (R ∪ (routeDomain d x (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).image e) ω := by
-  have h₀ := fresh_add_stale (followPath d e x k) R ω
-  have h₁ := fresh_add_stale (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x))
-    (R ∪ (routeDomain d x (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).image e) ω
-  rw [followPath_charge,followPath_raw_charge] at h₀
-  omega
-
-lemma seekCycle_semantics (d : ℕ) (S : Finset (Card d)) (a : Card d) (r : ℕ)
-    (x : Card d) (V W : Finset (Card d)) (z : Card d)
-    (k : Finset (Card d) → QueryTree (SwitchIndex d)) (ω : SwitchIndex d → Bool)
-    (h : cycleTrace (butterflyPerm d (decodeButterfly d ω)) S a r x V = some (W,z)) :
-    (seekCycle d S a r x V k).succeeds ω = (k W).succeeds ω ∧
-      (seekCycle d S a r x V k).staleCharge (exposedPaths d (decodeButterfly d ω) V) ω =
-        (k W).staleCharge (exposedPaths d (decodeButterfly d ω) W) ω +
-          terminalCharge d (decodeButterfly d ω) z (W.erase z) := by
-  classical
-  induction r generalizing x V with
-  | zero =>
-    simp only [cycleTrace] at h
-    split_ifs at h with hv
-    obtain ⟨hxS,hxV,hxa⟩ := hv
-    obtain ⟨rfl,rfl⟩ := Prod.mk.inj (Option.some.inj h)
-    simp only [seekCycle,ite_eq_left (show x ∈ S ∧ x ∉ V from ⟨hxS,hxV⟩),closePath_succeeds,ite_eq_left hxa]
-    constructor
-    · trivial
-    · rw [closePath_stale_charge _ _ _ _ _ _ hxa,exposedPaths_insert,
-        Finset.erase_insert hxV]
-      simp only [terminalCharge,hxa]
-  | succ r ih =>
-    simp only [cycleTrace] at h
-    split_ifs at h with hv
-    obtain ⟨hxS,hxV⟩ := hv
-    rw [seekCycle,ite_eq_left ⟨hxS,hxV⟩,followPath_succeeds,followPath_stale_charge]
-    simp only [Function.comp_id,Finset.image_id]
-    rw [← exposedPaths_insert]
-    exact ih _ _ h
-
-lemma seekCycle_orbit (d : ℕ) (S V : Finset (Card d)) (z : Card d)
-    (k : Finset (Card d) → QueryTree (SwitchIndex d)) (ω : SwitchIndex d → Bool)
-    (hS : orbitSet (butterflyPerm d (decodeButterfly d ω)) z ⊆ S)
-    (hV : Disjoint V (orbitSet (butterflyPerm d (decodeButterfly d ω)) z)) :
-    let p := butterflyPerm d (decodeButterfly d ω)
-    let W := V ∪ orbitSet p z
-    (seekCycle d S (p z) (Function.minimalPeriod p z-1) (p z) V k).succeeds ω = (k W).succeeds ω ∧
-      (seekCycle d S (p z) (Function.minimalPeriod p z-1) (p z) V k).staleCharge
-          (exposedPaths d (decodeButterfly d ω) V) ω =
-        (k W).staleCharge (exposedPaths d (decodeButterfly d ω) W) ω +
-          terminalCharge d (decodeButterfly d ω) z (W.erase z) := by
-  exact seekCycle_semantics _ _ _ _ _ _ _ _ _ _ (cycleTrace_orbit _ _ _ _ hS hV)
-
-end QueryTree
-
-lemma mem_exposedPaths (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) (s : SwitchIndex d) :
-    s ∈ exposedPaths d b V ↔ (switchUsers d b s).1 ∈ V ∨ (switchUsers d b s).2 ∈ V := by
-  classical
-  simp only [exposedPaths,Finset.mem_biUnion,mem_routeDomain_iff_user]
-  constructor
-  · rintro ⟨x,hx,h | h⟩
-    · exact Or.inl (h ▸ hx)
-    · exact Or.inr (h ▸ hx)
-  · rintro (h | h)
-    · exact ⟨_,h,Or.inl rfl⟩
-    · exact ⟨_,h,Or.inr rfl⟩
-
-lemma terminalCharge_sum (d : ℕ) (b : Butterfly d) (z : Card d) (V : Finset (Card d))
-    (hz : z ∉ V) :
-    (terminalCharge d b z V : ℝ) = ∑ s : SwitchIndex d,
-      ((if (switchUsers d b s).1 = z ∧ (switchUsers d b s).2 ∈ V then (1:ℝ) else 0) +
-      (if (switchUsers d b s).2 = z ∧ (switchUsers d b s).1 ∈ V then (1:ℝ) else 0)) := by
-  classical
-  have he : routeDomain d z (butterflyPerm d b z) ∩ exposedPaths d b V =
-      Finset.univ.filter (fun s =>
-        ((switchUsers d b s).1 = z ∧ (switchUsers d b s).2 ∈ V) ∨
-        ((switchUsers d b s).2 = z ∧ (switchUsers d b s).1 ∈ V)) := by
-    ext s
-    simp only [Finset.mem_inter,mem_routeDomain_iff_user,mem_exposedPaths,
-      Finset.mem_filter,Finset.mem_univ,true_and]
-    by_cases h₁ : (switchUsers d b s).1 = z <;>
-      by_cases h₂ : (switchUsers d b s).2 = z <;> simp [h₁,h₂,hz]
-  rw [terminalCharge,he]
-  simp only [← Finset.sum_boole]
-  apply Finset.sum_congr rfl
-  intro s _
-  by_cases h₁ : (switchUsers d b s).1 = z <;>
-    by_cases h₂ : (switchUsers d b s).2 = z <;> simp [h₁,h₂,hz]
-
-noncomputable def priorPaths {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) : Finset (Card d) :=
-  Finset.univ.filter (fun x => ∃ i : Fin A, o x = some i ∧ i.val < t)
-
-@[simp] lemma mem_priorPaths {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) (x : Card d) :
-    x ∈ priorPaths o t ↔ ∃ i : Fin A, o x = some i ∧ i.val < t := by
-  classical
-  simp [priorPaths]
-
-lemma terminalCharge_prior_sum {d A : ℕ} (o : Card d → Option (Fin A))
-    (b : Butterfly d) (z : Terminal o) (i : Fin A) :
-    (terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) =
-      ∑ s : SwitchIndex d,
-        ((if (z i).val = (switchUsers d b s).1 ∧ (switchUsers d b s).2 ∈ priorPaths o (i.val+1)
-          then (1:ℝ) else 0) +
-        (if (z i).val = (switchUsers d b s).2 ∧ (switchUsers d b s).1 ∈ priorPaths o (i.val+1)
-          then (1:ℝ) else 0)) := by
-  classical
-  rw [terminalCharge_sum _ _ _ _ (Finset.notMem_erase _ _)]
-  apply Finset.sum_congr rfl
-  intro s _
-  have hn := switchUsers_ne d b s
-  simp only [Finset.mem_erase]
-  by_cases h₁ : (z i).val = (switchUsers d b s).1 <;>
-    by_cases h₂ : (z i).val = (switchUsers d b s).2 <;> simp_all [eq_comm]
-
-lemma rotationMeeting_eq_slot_sum {d A : ℕ} (o : Card d → Option (Fin A))
-    (z : Terminal o) (u v : Card d) :
-    rotationMeeting o z u v = ∑ i : Fin A,
-      ((if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) +
-      (if (z i).val = v ∧ u ∈ priorPaths o (i.val+1) then (1:ℝ) else 0)) := by
-  classical
-  have only_owner (u : Card d) (a : Fin A) (hou : o u = some a)
-      (i : Fin A) (hia : i ≠ a) : (z i).val ≠ u := by
-    intro h
-    have hi := (z i).property
-    rw [h,hou] at hi
-    exact hia (Option.some.inj hi).symm
-  have sum_one (u v : Card d) (a : Fin A) (hou : o u = some a) :
-      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) =
-      if (z a).val = u ∧ v ∈ priorPaths o (a.val+1) then 1 else 0 := by
-    apply Finset.sum_eq_single a
-    · intro i _ hia
-      simp [only_owner u a hou i hia]
-    · simp
-  have sum_none (u v : Card d) (hou : o u = none) :
-      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) = 0 := by
-    apply Finset.sum_eq_zero
-    intro i _
-    have hn : (z i).val ≠ u := by
-      intro h
-      have hi := (z i).property
-      rw [h,hou] at hi
-      contradiction
-    simp [hn]
-  rw [Finset.sum_add_distrib]
-  cases hou : o u with
-  | none =>
-    rw [sum_none u v hou]
-    simp [rotationMeeting,hou,mem_priorPaths]
-  | some a =>
-    rw [sum_one u v a hou]
-    cases hov : o v with
-    | none =>
-      rw [sum_none v u hov]
-      simp [rotationMeeting,hou,hov,mem_priorPaths]
-    | some b =>
-      rw [sum_one v u b hov]
-      simp only [rotationMeeting,hou,hov,mem_priorPaths,Option.some.injEq]
-      rcases lt_trichotomy a b with hab | rfl | hba
-      · have hn : ¬ b.val < a.val+1 := by omega
-        have hy : a.val < b.val+1 := by omega
-        simp [hab,hn,hy]
-      · simp
-      · have hn : ¬ a.val < b.val+1 := by omega
-        have hy : b.val < a.val+1 := by omega
-        simp [hba,not_lt_of_ge (le_of_lt hba),hn,hy]
-
-lemma rotationCharge_eq_terminal_sum {d A : ℕ} (o : Card d → Option (Fin A))
-    (b : Butterfly d) (z : Terminal o) :
-    rotationCharge o b z = ∑ i : Fin A,
-      (terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) := by
-  simp only [rotationCharge,rotationMeeting_eq_slot_sum,terminalCharge_prior_sum]
-  exact Finset.sum_comm
-
-@[simp] lemma priorPaths_zero {d A : ℕ} (o : Card d → Option (Fin A)) : priorPaths o 0 = ∅ := by
-  classical
-  ext x
-  simp
-
-lemma priorPaths_succ {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) (ht : t < A) :
-    priorPaths o (t+1) = priorPaths o t ∪ cycleFiber o ⟨t,ht⟩ := by
-  classical
-  ext x
-  simp only [mem_priorPaths,Finset.mem_union,mem_cycleFiber]
-  constructor
-  · rintro ⟨i,hi,hit⟩
-    by_cases h : i.val < t
-    · exact Or.inl ⟨i,hi,h⟩
-    · have he : i = ⟨t,ht⟩ := Fin.ext (by change i.val = t; omega)
-      exact Or.inr (he ▸ hi)
-  · rintro (⟨i,hi,hit⟩ | hi)
-    · exact ⟨i,hi,by omega⟩
-    · exact ⟨⟨t,ht⟩,hi,by change t < t+1; omega⟩
-
-lemma priorPaths_disjoint {d A : ℕ} (o : Card d → Option (Fin A)) (i : Fin A) :
-    Disjoint (priorPaths o i.val) (cycleFiber o i) := by
-  classical
-  apply Finset.disjoint_left.mpr
-  intro x hx hx'
-  obtain ⟨j,hj,hji⟩ := (mem_priorPaths _ _ _).mp hx
-  have he : j = i := Option.some.inj (hj.symm.trans ((mem_cycleFiber _ _ _).mp hx'))
-  subst j
-  exact (lt_irrefl _ hji)
-
-namespace QueryTree
-
-lemma seekCycles_owned {d A : ℕ} (S : Finset (Card d)) (r : Fin A → ℕ)
-    (o : Card d → Option (Fin A)) (z : Terminal o) (ω : SwitchIndex d → Bool)
-    (hS : ∀ i, cycleFiber o i ⊆ S)
-    (hO : ∀ i, orbitSet (butterflyPerm d (decodeButterfly d ω)) (z i).val = cycleFiber o i)
-    (hC : ∀ i, (cycleFiber o i).card = r i+1) :
-    let p := butterflyPerm d (decodeButterfly d ω)
-    let L := List.ofFn (fun i : Fin A => (p (z i).val,r i))
-    (seekCycles d S L ∅).succeeds ω = true ∧
-      ((seekCycles d S L ∅).staleCharge ∅ ω : ℝ) = rotationCharge o (decodeButterfly d ω) z := by
-  classical
-  let b := decodeButterfly d ω
-  let p := butterflyPerm d b
-  let f (i : Fin A) := (p (z i).val,r i)
-  let c (i : Fin A) : ℝ := terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val)
-  have hc (i : Fin A) : Function.minimalPeriod p (z i).val - 1 = r i := by
-    have he := orbitSet_card p (z i).val
-    rw [hO,hC] at he
-    omega
-  have hs (t : ℕ) (ht : t ≤ A) :
-      (seekCycles d S ((List.ofFn f).drop t) (priorPaths o t)).succeeds ω = true ∧
-        ((seekCycles d S ((List.ofFn f).drop t) (priorPaths o t)).staleCharge (exposedPaths d b (priorPaths o t)) ω : ℝ) =
-          ((List.ofFn c).drop t).sum := by
-    induction ht using Nat.decreasingInduction with
-    | self => simp only [List.length_ofFn,le_refl,List.drop_eq_nil_of_le,List.sum_nil,seekCycles,succeeds,staleCharge,Nat.cast_zero,and_self]
-    | of_succ t ht ih =>
-      have htf : t < (List.ofFn f).length := by simpa using ht
-      have htc : t < (List.ofFn c).length := by simpa using ht
-      rw [List.drop_eq_getElem_cons htf,List.drop_eq_getElem_cons htc]
-      simp only [List.getElem_ofFn,List.sum_cons,seekCycles]
-      have h := seekCycle_orbit d S (priorPaths o t) (z ⟨t,ht⟩).val
-        (fun V => seekCycles d S ((List.ofFn f).drop (t+1)) V) ω
-        (by rw [hO]; exact hS _) (by rw [hO]; exact priorPaths_disjoint o ⟨t,ht⟩)
-      dsimp only at h
-      rw [hc,hO,←priorPaths_succ o t ht] at h
-      dsimp only [f]
-      constructor
-      · exact h.1.trans ih.1
-      · rw [h.2,Nat.cast_add,ih.2]
-        exact add_comm _ _
-  have h := hs 0 (Nat.zero_le _)
-  simp only [List.drop_zero,priorPaths_zero,exposedPaths_empty,List.sum_ofFn] at h
-  dsimp only
-  exact ⟨h.1,h.2.trans (rotationCharge_eq_terminal_sum o b z).symm⟩
-
-end QueryTree
-
-section Collections
-variable {d A : ℕ}
-
-def ValidCycleSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ)
-    (o : Card d → Option (Fin A)) : Prop :=
-  (∀ i, (cycleFiber o i).card = r i + 1) ∧
-  (∀ i, cycleFiber o i ⊆ S) ∧
-  (∀ i x, o x = some i → orbitSet p x = cycleFiber o i)
-
-abbrev CycleSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
-  {o : Card d → Option (Fin A) // ValidCycleSlots p S r o}
-
-noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
-    Fintype (CycleSlots p S r) := Fintype.ofFinite _
-
-lemma CycleSlots.terminal_nonempty {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (c : CycleSlots p S r) (i : Fin A) : Nonempty {x : Card d // c.val x = some i} := by
-  have hn : (cycleFiber c.val i).Nonempty := Finset.card_pos.mp (by rw [c.property.1]; omega)
-  obtain ⟨x,hx⟩ := hn
-  exact ⟨x,(mem_cycleFiber _ _ _).mp hx⟩
-
-lemma owner_ext {o o' : Card d → Option (Fin A)}
-    (h : ∀ i, cycleFiber o i = cycleFiber o' i) : o = o' := by
-  funext x
-  have he (i : Fin A) : o x = some i ↔ o' x = some i := by
-    simpa only [mem_cycleFiber] using (Finset.ext_iff.mp (h i) x)
-  cases ho : o x with
-  | some i => exact ((he i).mp ho).symm
-  | none =>
-    cases ho' : o' x with
-    | none => rfl
-    | some i => have hn := (he i).mpr ho'; rw [ho] at hn; contradiction
-
-abbrev RootedSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
-  Σ c : CycleSlots p S r, Terminal c.val
-
-noncomputable def rootedStart {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (c : RootedSlots p S r) : Fin A → S := fun i =>
-  ⟨p (c.2 i).val,c.1.property.2.1 i (by
-    rw [←c.1.property.2.2 i (c.2 i).val (c.2 i).property]
-    exact (mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.apply_right (Equiv.Perm.SameCycle.refl p _)))⟩
-
-lemma rootedStart_injective {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ} :
-    Function.Injective (rootedStart (p:=p) (S:=S) (r:=r)) := by
-  rintro ⟨c,z⟩ ⟨c',z'⟩ h
-  have hz (i : Fin A) : (z i).val = (z' i).val :=
-    p.injective (congrArg Subtype.val (congrFun h i))
-  have hc : c = c' := Subtype.ext (owner_ext (fun i => by
-    rw [←c.property.2.2 i (z i).val (z i).property,
-      ←c'.property.2.2 i (z' i).val (z' i).property,hz]))
-  subst c'
-  have hzz : z = z' := funext (fun i => Subtype.ext (hz i))
-  subst z'
-  rfl
-
-lemma sum_comp_injective_le {α β : Type*} [Fintype α] [Fintype β]
-    (e : α → β) (he : Function.Injective e) (f : β → ℝ) (hf : ∀ b, 0 ≤ f b) :
-    ∑ a, f (e a) ≤ ∑ b, f b := by
-  classical
-  calc
-    _ = ∑ b ∈ Finset.univ.image e, f b := (Finset.sum_image (fun _ _ _ _ h => he h)).symm
-    _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _) (fun b _ _ => hf b)
-
-lemma rooted_weight_sum_le (S : Finset (Card d)) (r : Fin A → ℕ) (ω : SwitchIndex d → Bool) :
-    (∑ c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r,
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2)) ≤
-    ∑ a : Fin A → S,
-      if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-        ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-      else 0 := by
-  classical
-  let p := butterflyPerm d (decodeButterfly d ω)
-  let g (a : Fin A → S) : ℝ :=
-    if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-    else 0
-  have he (c : RootedSlots p S r) :
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2) =
-      g (rootedStart c) := by
-    have hs := QueryTree.seekCycles_owned S r c.1.val c.2 ω c.1.property.2.1
-      (fun i => c.1.property.2.2 i (c.2 i).val (c.2 i).property) c.1.property.1
-    dsimp only at hs
-    unfold g
-    rw [ite_eq_left (show (QueryTree.seekCycles d S (List.ofFn (fun i => ((rootedStart c i).val,r i))) ∅).succeeds ω = true from hs.1)]
-    rw [← hs.2,neg_mul,Real.exp_neg,mul_comm (Real.log 2),Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
-    rfl
-  simp_rw [he]
-  exact sum_comp_injective_le rootedStart rootedStart_injective g (fun a => by dsimp [g]; split_ifs <;> positivity)
-
-noncomputable def rotationLower (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : ℝ :=
-  Real.exp (-Real.log 2 * ((A:ℝ)*d/2 + ∑ i : Fin A, Real.log (r i+1) / (2 * Real.log 2)))
-
-noncomputable def collectionWeight (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : ℝ :=
-  ((2:ℝ)^d)^A * (∏ i : Fin A, (r i+1 : ℝ)) * rotationLower d r
-
-lemma collectionWeight_pos (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : 0 < collectionWeight d r := by
-  unfold collectionWeight rotationLower
-  positivity
-
-lemma collection_rotation_lower (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r)
-    (ω : SwitchIndex d → Bool) (c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) :
-    collectionWeight d r ≤ ∑ z : Terminal c.val,
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z) := by
-  classical
-  let (i : Fin A) : Nonempty {x : Card d // c.val x = some i} := c.terminal_nonempty i
-  have hm : Monotone (fun i => (cycleFiber c.val i).card) := by
-    intro i j hij
-    dsimp only
-    rw [c.property.1,c.property.1]
-    exact Nat.add_le_add_right (hr hij) 1
-  have hb := rotation_exp_bound hm (decodeButterfly d ω)
-  have hcard : (Fintype.card (Terminal c.val) : ℝ) = ∏ i : Fin A, (r i+1 : ℝ) := by
-    rw [Fintype.card_pi,Nat.cast_prod]
-    apply Finset.prod_congr rfl
-    intro i _
-    rw [card_cycleFiber,c.property.1,Nat.cast_add,Nat.cast_one]
-  have hp : (0:ℝ) < Fintype.card (Terminal c.val) := Nat.cast_pos.mpr Fintype.card_pos
-  have hl : rotationLower d r ≤ finiteMean (fun z : Terminal c.val => Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z)) := by
-    simpa only [rotationLower,c.property.1,Nat.cast_add,Nat.cast_one] using hb
-  have hh := (le_div_iff₀ hp).mp hl
-  rw [hcard] at hh
-  unfold collectionWeight
-  rw [← Finset.mul_sum]
-  calc
-    _ = ((2:ℝ)^d)^A * (rotationLower d r * ∏ i : Fin A, (r i+1 : ℝ)) := by ring
-    _ ≤ _ := mul_le_mul_of_nonneg_left hh (by positivity)
-
-lemma cycleSlots_weighted_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) *
-      collectionWeight d r ≤ (S.card : ℝ)^A := by
-  classical
-  let g (a : Fin A → S) (ω : SwitchIndex d → Bool) : ℝ :=
-    if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-    else 0
-  have hω (ω : SwitchIndex d → Bool) :
-      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ) * collectionWeight d r ≤
-        ∑ a : Fin A → S, g a ω := by
-    calc
-      _ = ∑ _c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r, collectionWeight d r := by simp
-      _ ≤ ∑ c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r,
-          ∑ z : Terminal c.val, ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z) :=
-        Finset.sum_le_sum (fun c _ => collection_rotation_lower S r hr ω c)
-      _ = ∑ c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r,
-          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2) :=
-        (Fintype.sum_sigma (fun c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r =>
-          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2))).symm
-      _ ≤ _ := rooted_weight_sum_le S r ω
-  have hb := finiteMean_mono hω
-  rw [finiteMean_mul_const,finiteMean_sum] at hb
-  calc
-    _ ≤ ∑ a : Fin A → S, finiteMean (g a) := hb
-    _ ≤ ∑ _a : Fin A → S, (1:ℝ) := by
-      apply Finset.sum_le_sum
-      intro a _
-      have hh := QueryTree.cycles_weighted_success d S (List.ofFn (fun i : Fin A => ((a i).val,r i)))
-      simpa only [List.length_ofFn] using hh
-    _ = _ := by simp
-
-lemma cycleSlots_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) ≤
-        (S.card : ℝ)^A / collectionWeight d r := by
-  exact (le_div_iff₀ (collectionWeight_pos d r)).mpr (cycleSlots_weighted_moment S r hr)
-
-lemma collectionWeight_eq_sqrt (d : ℕ) {A : ℕ} (r : Fin A → ℕ) :
-    collectionWeight d r = (Real.sqrt ((2:ℝ)^d))^A * ∏ i : Fin A, Real.sqrt (r i+1) := by
-  have hn : ((2:ℝ)^d)^A = Real.exp ((A:ℝ)*d*Real.log 2) := by
-    rw [mul_assoc,Real.exp_nat_mul,Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
-  have hp : (∏ i : Fin A, (r i+1 : ℝ)) = Real.exp (∑ i : Fin A, Real.log (r i+1)) := by
-    rw [Real.exp_sum]
-    apply Finset.prod_congr rfl
-    intro i _
-    exact (Real.exp_log (by positivity)).symm
-  have hs : Real.sqrt ((2:ℝ)^d) = Real.exp ((d:ℝ)*Real.log 2/2) := by
-    rw [Real.sqrt_eq_rpow,Real.rpow_def_of_pos (by positivity),Real.log_pow]
-    congr 1
-    ring
-  have hj (i : Fin A) : Real.sqrt (r i+1) = Real.exp (Real.log (r i+1)/2) := by
-    rw [Real.sqrt_eq_rpow,Real.rpow_def_of_pos (by positivity)]
-    congr 1
-    ring
-  simp_rw [collectionWeight,hn,hp,rotationLower,hs,hj]
-  rw [←Real.exp_add,←Real.exp_add,←Real.exp_nat_mul,←Real.exp_sum,←Real.exp_add]
-  congr 1
-  rw [←Finset.sum_div,←Finset.sum_div]
-  have hl : Real.log 2 ≠ 0 := ne_of_gt (Real.log_pos (by norm_num))
-  field_simp
-  ring
-
-lemma butterfly_cycleSlots_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) ≤
-        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
-  have h := cycleSlots_moment S r hr
-  rw [collectionWeight_eq_sqrt,div_mul_eq_div_mul_one_div] at h
-  rw [div_pow]
-  simpa only [one_div,Finset.prod_inv_distrib] using h
-
-end Collections
-
-variable {d A : ℕ}
-
-abbrev SelectedCycle (p : Equiv.Perm (Card d)) (S : Finset (Card d)) :=
-  {C : Finset (Card d) // (∃ x, C = orbitSet p x) ∧ C ⊆ S}
-
-noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) :
-    Fintype (SelectedCycle p S) := Fintype.ofFinite _
-
-lemma SelectedCycle.nonempty {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
-    (C : SelectedCycle p S) : C.val.Nonempty := by
-  obtain ⟨x,hx⟩ := C.property.1
-  rw [hx]
-  exact orbitSet_nonempty p x
-
-lemma SelectedCycle.orbit_eq {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
-    (C : SelectedCycle p S) {x : Card d} (hx : x ∈ C.val) : orbitSet p x = C.val := by
-  obtain ⟨y,hy⟩ := C.property.1
-  rw [hy] at hx ⊢
-  exact (orbitSet_eq p ((mem_orbitSet _ _ _).mp hx)).symm
-
-lemma SelectedCycle.eq_of_mem {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
-    (C D : SelectedCycle p S) {x : Card d} (hC : x ∈ C.val) (hD : x ∈ D.val) : C = D := by
-  apply Subtype.ext
-  exact (C.orbit_eq hC).symm.trans (D.orbit_eq hD)
-
-abbrev SlotAssignment (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
-  {f : Fin A → SelectedCycle p S // Function.Injective f ∧ ∀ i, (f i).val.card = r i+1}
-
-noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
-    Fintype (SlotAssignment p S r) := Fintype.ofFinite _
-
-noncomputable def slotsToAssignment {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (c : CycleSlots p S r) : SlotAssignment p S r := by
-  classical
-  let f (i : Fin A) : SelectedCycle p S := ⟨cycleFiber c.val i,by
-    obtain ⟨x,hx⟩ := c.terminal_nonempty i
-    exact ⟨⟨x,(c.property.2.2 i x hx).symm⟩,c.property.2.1 i⟩⟩
-  refine ⟨f,?_,fun i => c.property.1 i⟩
-  intro i j hij
-  have he : cycleFiber c.val i = cycleFiber c.val j := congrArg Subtype.val hij
-  obtain ⟨x,hx⟩ := c.terminal_nonempty i
-  have hxi : x ∈ cycleFiber c.val i := (mem_cycleFiber _ _ _).mpr hx
-  rw [he] at hxi
-  exact Option.some.inj (hx.symm.trans ((mem_cycleFiber _ _ _).mp hxi))
-
-noncomputable def assignmentOwner {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) (x : Card d) : Option (Fin A) := by
-  classical
-  exact if h : ∃ i, x ∈ (f.val i).val then some (Classical.choose h) else none
-
-end Thorp
-
-end ThorpNine.SparseSaving
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.SparseSaving.ExposedCoins
+
+namespace OAI
+
+namespace ThorpNine.SparseSaving
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+section Rotation
+variable {d A : ℕ}
+
+noncomputable def rotationMeeting (o : Card d → Option (Fin A)) (z : Terminal o)
+    (u v : Card d) : ℝ :=
+  match o u, o v with
+  | some i, some j => if i < j then (if (z j).val = v then 1 else 0)
+      else if j < i then (if (z i).val = u then 1 else 0)
+      else (if (z i).val = u then 1 else 0) + (if (z i).val = v then 1 else 0)
+  | _, _ => 0
+
+lemma card_cycleFiber (o : Card d → Option (Fin A)) (i : Fin A) :
+    Fintype.card {x : Card d // o x = some i} = (cycleFiber o i).card := by
+  classical
+  simp [cycleFiber,Fintype.card_subtype]
+
+lemma terminal_mean {o : Card d → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
+    (i : Fin A) (x : Card d) (hx : o x = some i) :
+    finiteMean (fun z : Terminal o => if (z i).val = x then (1:ℝ) else 0) = slotWeight o i := by
+  classical
+  have hev := finiteMean_pi_eval (A := fun i : Fin A => {x : Card d // o x = some i})
+    i (fun z => if z.val = x then (1:ℝ) else 0)
+  rw [hev]
+  have he : (fun z : {x : Card d // o x = some i} => if z.val = x then (1:ℝ) else 0) =
+      (fun z => if z = ⟨x,hx⟩ then 1 else 0) := by
+    funext z
+    simp only [Subtype.ext_iff]
+  rw [he,finiteMean_eq_indicator,card_cycleFiber,slotWeight]
+
+lemma slotWeight_antitone (o : Card d → Option (Fin A))
+    (hn : ∀ i, (cycleFiber o i).Nonempty)
+    (hm : Monotone (fun i => (cycleFiber o i).card)) : Antitone (slotWeight o) := by
+  intro i j hij
+  unfold slotWeight
+  apply one_div_le_one_div_of_le
+  · exact_mod_cast Finset.card_pos.mpr (hn i)
+  · exact_mod_cast hm hij
+
+lemma rotationMeeting_mean {o : Card d → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
+    (hm : Monotone (fun i => (cycleFiber o i).card)) (u v : Card d) :
+    finiteMean (fun z : Terminal o => rotationMeeting o z u v) =
+      min (pathWeight o u) (pathWeight o v) + withinWeight o u v := by
+  classical
+  have hn : ∀ i, (cycleFiber o i).Nonempty := by
+    intro i
+    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : Card d // o x = some i})
+    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
+  have hw := slotWeight_antitone o hn hm
+  cases hu : o u with
+  | none =>
+    cases hv : o v with
+    | none => simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean]
+    | some j =>
+      have hj : 0 ≤ slotWeight o j := by unfold slotWeight; positivity
+      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_left hj]
+  | some i =>
+    cases hv : o v with
+    | none =>
+      have hi : 0 ≤ slotWeight o i := by unfold slotWeight; positivity
+      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_right hi]
+    | some j =>
+      simp only [rotationMeeting,pathWeight,withinWeight,hu,hv]
+      rcases lt_trichotomy i j with hij | rfl | hji
+      · simp only [hij,↓reduceIte,ne_of_lt hij]
+        rw [terminal_mean j v hv,min_eq_right (hw (le_of_lt hij)),add_zero]
+      · simp only [lt_self_iff_false,↓reduceIte,min_self]
+        rw [finiteMean_add,terminal_mean i u hu,terminal_mean i v hv]
+      · simp only [not_lt_of_ge (le_of_lt hji),hji,↓reduceIte,ne_of_gt hji]
+        rw [terminal_mean i u hu,min_eq_left (hw (le_of_lt hji)),add_zero]
+
+lemma pathWeight_sum (o : Card d → Option (Fin A)) (hn : ∀ i, (cycleFiber o i).Nonempty) :
+    ∑ x : Card d, pathWeight o x = (A : ℝ) := by
+  classical
+  have he (x : Card d) : pathWeight o x =
+      ∑ i : Fin A, if o x = some i then slotWeight o i else 0 := by
+    cases hx : o x with
+    | none => simp [pathWeight,hx]
+    | some j => simp [pathWeight,hx]
+  simp only [he]
+  rw [Finset.sum_comm]
+  have hf (i : Fin A) : (∑ x : Card d, if o x = some i then slotWeight o i else 0) = 1 := by
+    rw [← Finset.sum_filter]
+    change (∑ _x ∈ cycleFiber o i, slotWeight o i) = 1
+    simp only [Finset.sum_const,nsmul_eq_mul,slotWeight]
+    exact mul_one_div_cancel (by exact_mod_cast (Finset.card_pos.mpr (hn i)).ne')
+  simp only [hf,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,mul_one]
+
+lemma withinWeight_sum_bound (o : Card d → Option (Fin A))
+    (hn : ∀ i, (cycleFiber o i).Nonempty) (b : Butterfly d) :
+    (∑ s : SwitchIndex d, withinWeight o (switchUsers d b s).1 (switchUsers d b s).2) ≤
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
+  classical
+  have he (u v : Card d) : withinWeight o u v =
+      ∑ i : Fin A, (if u ∈ cycleFiber o i ∧ v ∈ cycleFiber o i then (1:ℝ) else 0) * slotWeight o i := by
+    simp only [mem_cycleFiber]
+    cases hu : o u <;> cases hv : o v <;> simp [withinWeight,hu,hv,ite_and,eq_comm]
+  simp only [he]
+  rw [Finset.sum_comm]
+  apply Finset.sum_le_sum
+  intro i _
+  rw [← Finset.sum_mul]
+  have hi := marked_switch_entropy_bound d b (cycleFiber o i)
+  have hpos : (0:ℝ) < (cycleFiber o i).card := by exact_mod_cast Finset.card_pos.mpr (hn i)
+  have hmul := mul_le_mul_of_nonneg_right hi (show 0 ≤ slotWeight o i by unfold slotWeight; positivity)
+  calc
+    _ ≤ ((cycleFiber o i).card : ℝ) * Real.log (cycleFiber o i).card /
+        (2 * Real.log 2) * slotWeight o i := hmul
+    _ = _ := by unfold slotWeight; field_simp
+
+noncomputable def rotationCharge (o : Card d → Option (Fin A)) (b : Butterfly d)
+    (z : Terminal o) : ℝ :=
+  ∑ s : SwitchIndex d, rotationMeeting o z (switchUsers d b s).1 (switchUsers d b s).2
+
+lemma rotationCharge_mean_bound {o : Card d → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
+    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : Butterfly d) :
+    finiteMean (rotationCharge o b) ≤ (A:ℝ) * d / 2 +
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
+  classical
+  have hn : ∀ i, (cycleFiber o i).Nonempty := by
+    intro i
+    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : Card d // o x = some i})
+    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
+  unfold rotationCharge
+  rw [finiteMean_sum]
+  simp only [rotationMeeting_mean hm,Finset.sum_add_distrib]
+  have h₁ := weighted_switch_min_bound d b (pathWeight o)
+  rw [pathWeight_sum o hn] at h₁
+  have h₂ := withinWeight_sum_bound o hn b
+  linarith
+
+lemma rotation_exp_bound {o : Card d → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
+    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : Butterfly d) :
+    Real.exp (-Real.log 2 * ((A:ℝ)*d/2 +
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2))) ≤
+      finiteMean (fun z : Terminal o => Real.exp (-Real.log 2 * rotationCharge o b z)) := by
+  have hb := rotationCharge_mean_bound hm b
+  have hlog : 0 ≤ Real.log 2 := Real.log_nonneg (by norm_num)
+  have hm' : -Real.log 2 * ((A:ℝ)*d/2 +
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2)) ≤
+      finiteMean (fun z : Terminal o => -Real.log 2 * rotationCharge o b z) := by
+    have he := finiteMean_mul_const (rotationCharge o b) (-Real.log 2)
+    simp only [mul_comm] at he
+    rw [he]
+    exact mul_le_mul_of_nonpos_left hb (neg_nonpos.mpr hlog)
+  exact (Real.exp_le_exp.mpr hm').trans (exp_finiteMean_le _)
+
+end Rotation
+
+section OrbitCore
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+noncomputable def orbitSet (p : Equiv.Perm α) (x : α) : Finset α :=
+  Finset.univ.filter (p.SameCycle x)
+
+@[simp] lemma mem_orbitSet (p : Equiv.Perm α) (x y : α) :
+    y ∈ orbitSet p x ↔ p.SameCycle x y := by
+  classical
+  simp [orbitSet]
+
+lemma orbitSet_nonempty (p : Equiv.Perm α) (x : α) : (orbitSet p x).Nonempty :=
+  ⟨x,(mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.refl _ _)⟩
+
+lemma orbitSet_eq (p : Equiv.Perm α) {x y : α} (h : p.SameCycle x y) :
+    orbitSet p x = orbitSet p y := by
+  ext z
+  simp only [mem_orbitSet]
+  exact ⟨fun hz => h.symm.trans hz,fun hz => h.trans hz⟩
+
+@[simp] lemma orbitSet_apply (p : Equiv.Perm α) (x : α) : orbitSet p (p x) = orbitSet p x := by
+  ext y
+  simp only [mem_orbitSet,Equiv.Perm.sameCycle_apply_left]
+
+lemma orbitSet_eq_or_disjoint (p : Equiv.Perm α) (x y : α) :
+    orbitSet p x = orbitSet p y ∨ Disjoint (orbitSet p x) (orbitSet p y) := by
+  classical
+  by_cases h : p.SameCycle x y
+  · exact Or.inl (orbitSet_eq p h)
+  · right
+    apply Finset.disjoint_left.mpr
+    intro z hz hz'
+    exact h (((mem_orbitSet _ _ _).mp hz).trans ((mem_orbitSet _ _ _).mp hz').symm)
+
+omit [DecidableEq α] in
+lemma perm_minimalPeriod_pos (p : Equiv.Perm α) (x : α) :
+    0 < Function.minimalPeriod p x :=
+  Function.minimalPeriod_pos_of_mem_periodicPts (p.injective.mem_periodicPts x)
+
+lemma orbitSet_card (p : Equiv.Perm α) (x : α) :
+    (orbitSet p x).card = Function.minimalPeriod p x := by
+  classical
+  let f : Fin (Function.minimalPeriod p x) → {y // p.SameCycle x y} := fun n =>
+    ⟨p^[n.val] x,by rw [Equiv.Perm.iterate_eq_pow]; exact ⟨(n.val:ℤ),by simp⟩⟩
+  have hf : Function.Bijective f := by
+    constructor
+    · intro i j hij
+      apply Fin.ext
+      exact Function.iterate_injOn_Iio_minimalPeriod i.isLt j.isLt (congrArg Subtype.val hij)
+    · intro y
+      obtain ⟨n,hn⟩ := y.property.exists_nat_pow_eq
+      refine ⟨⟨n % Function.minimalPeriod p x, Nat.mod_lt _ (perm_minimalPeriod_pos p x)⟩,?_⟩
+      apply Subtype.ext
+      change p^[n % Function.minimalPeriod p x] x = y.val
+      rw [Function.iterate_mod_minimalPeriod_eq,Equiv.Perm.iterate_eq_pow,hn]
+  have hc := Fintype.card_congr (Equiv.ofBijective f hf)
+  simpa [orbitSet,Fintype.card_subtype] using hc.symm
+
+def walkSet (p : Equiv.Perm α) (r : ℕ) (x : α) : Finset α :=
+  (Finset.range r).image (fun n => p^[n] x)
+
+omit [Fintype α] in
+@[simp] lemma mem_walkSet (p : Equiv.Perm α) (r : ℕ) (x y : α) :
+    y ∈ walkSet p r x ↔ ∃ n < r, p^[n] x = y := by
+  simp [walkSet]
+
+omit [Fintype α] in
+@[simp] lemma walkSet_zero (p : Equiv.Perm α) (x : α) : walkSet p 0 x = ∅ := by
+  simp [walkSet]
+
+omit [Fintype α] in
+lemma walkSet_succ_last (p : Equiv.Perm α) (r : ℕ) (x : α) :
+    walkSet p (r+1) x = insert (p^[r] x) (walkSet p r x) := by
+  simp [walkSet,Finset.range_add_one]
+
+omit [Fintype α] in
+lemma walkSet_succ_first (p : Equiv.Perm α) (r : ℕ) (x : α) :
+    walkSet p (r+1) x = insert x (walkSet p r (p x)) := by
+  ext y
+  simp only [mem_walkSet,Finset.mem_insert]
+  constructor
+  · rintro ⟨n,hn,hny⟩
+    cases n with
+    | zero => exact Or.inl (by simpa using hny.symm)
+    | succ n => exact Or.inr ⟨n,by omega,by simpa only [Function.iterate_succ_apply] using hny⟩
+  · rintro (rfl | ⟨n,hn,hny⟩)
+    · exact ⟨0,by omega,rfl⟩
+    · exact ⟨n+1,by omega,by simpa only [Function.iterate_succ_apply] using hny⟩
+
+omit [Fintype α] in
+lemma not_mem_short_walk (p : Equiv.Perm α) (x : α) {r : ℕ}
+    (hr : r < Function.minimalPeriod p x) : x ∉ walkSet p r (p x) := by
+  rintro hx
+  obtain ⟨n,hn,hnx⟩ := (mem_walkSet _ _ _ _).mp hx
+  have hperiod : Function.IsPeriodicPt p (n+1) x := by
+    simpa only [Function.IsPeriodicPt,Function.IsFixedPt,Function.iterate_succ_apply] using hnx
+  have hp := hperiod.minimalPeriod_le (by omega : 0 < n+1)
+  omega
+
+lemma walkSet_period (p : Equiv.Perm α) (x : α) :
+    walkSet p (Function.minimalPeriod p x) x = orbitSet p x := by
+  ext y
+  simp only [mem_walkSet,mem_orbitSet]
+  constructor
+  · rintro ⟨n,_,rfl⟩
+    rw [Equiv.Perm.iterate_eq_pow]
+    exact ⟨(n:ℤ),by simp⟩
+  · intro h
+    obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
+    refine ⟨n % Function.minimalPeriod p x,Nat.mod_lt _ (perm_minimalPeriod_pos p x),?_⟩
+    rw [Function.iterate_mod_minimalPeriod_eq,Equiv.Perm.iterate_eq_pow,hn]
+
+omit [Fintype α] in
+lemma walkSet_card (p : Equiv.Perm α) (x : α) {r : ℕ}
+    (hr : r ≤ Function.minimalPeriod p x) : (walkSet p r x).card = r := by
+  rw [walkSet,Finset.card_image_of_injOn]
+  · exact Finset.card_range _
+  · intro i hi j hj hij
+    exact Function.iterate_injOn_Iio_minimalPeriod
+      (lt_of_lt_of_le (Finset.mem_range.mp hi) hr)
+      (lt_of_lt_of_le (Finset.mem_range.mp hj) hr) hij
+
+lemma walkSet_subset_orbit (p : Equiv.Perm α) (r : ℕ) (x : α) :
+    walkSet p r x ⊆ orbitSet p x := by
+  intro y hy
+  obtain ⟨n,_,rfl⟩ := (mem_walkSet _ _ _ _).mp hy
+  rw [mem_orbitSet,Equiv.Perm.iterate_eq_pow]
+  exact ⟨(n:ℤ),by simp⟩
+
+def cycleTrace (p : Equiv.Perm α) (S : Finset α) (a : α) :
+    ℕ → α → Finset α → Option (Finset α × α)
+  | 0, x, V => if x ∈ S ∧ x ∉ V ∧ p x = a then some (insert x V,x) else none
+  | r+1, x, V => if x ∈ S ∧ x ∉ V then cycleTrace p S a r (p x) (insert x V) else none
+
+lemma cycleTrace_walk (p : Equiv.Perm α) (S : Finset α) (r : ℕ) (x : α) (V : Finset α)
+    (hr : r < Function.minimalPeriod p x) (hS : walkSet p (r+1) x ⊆ S)
+    (hV : Disjoint V (walkSet p (r+1) x)) :
+    cycleTrace p S (p^[r+1] x) r x V = some (V ∪ walkSet p (r+1) x,p^[r] x) := by
+  induction r generalizing x V with
+  | zero =>
+    have hxS : x ∈ S := hS (by simp [walkSet_succ_first])
+    have hxV : x ∉ V := fun h => Finset.disjoint_left.mp hV h (by simp [walkSet_succ_first])
+    simp [cycleTrace,hxS,hxV,walkSet_succ_first]
+  | succ r ih =>
+    have hxS : x ∈ S := hS (by rw [walkSet_succ_first]; exact Finset.mem_insert_self _ _)
+    have hxV : x ∉ V := fun h => Finset.disjoint_left.mp hV h
+      (by rw [walkSet_succ_first]; exact Finset.mem_insert_self _ _)
+    have hr' : r < Function.minimalPeriod p (p x) := by
+      rw [Function.minimalPeriod_apply (p.injective.mem_periodicPts x)]
+      omega
+    have hS' : walkSet p (r+1) (p x) ⊆ S := by
+      intro y hy
+      apply hS
+      rw [walkSet_succ_first]
+      exact Finset.mem_insert_of_mem hy
+    have hV' : Disjoint (insert x V) (walkSet p (r+1) (p x)) := by
+      apply Finset.disjoint_left.mpr
+      intro y hy hy'
+      rcases Finset.mem_insert.mp hy with hy | hy
+      · subst y
+        exact not_mem_short_walk p x hr hy'
+      · apply Finset.disjoint_left.mp hV hy
+        rw [walkSet_succ_first]
+        exact Finset.mem_insert_of_mem hy'
+    rw [cycleTrace,ite_eq_left ⟨hxS,hxV⟩,Function.iterate_succ_apply p (r+1) x,
+      ih _ _ hr' hS' hV',walkSet_succ_first p (r+1) x,Function.iterate_succ_apply]
+    simp only [Finset.insert_union,Finset.union_insert]
+
+lemma cycleTrace_orbit (p : Equiv.Perm α) (S V : Finset α) (z : α)
+    (hS : orbitSet p z ⊆ S) (hV : Disjoint V (orbitSet p z)) :
+    cycleTrace p S (p z) (Function.minimalPeriod p z - 1) (p z) V =
+      some (V ∪ orbitSet p z,z) := by
+  have hpos := perm_minimalPeriod_pos p z
+  have he : Function.minimalPeriod p (p z) = Function.minimalPeriod p z :=
+    Function.minimalPeriod_apply (p.injective.mem_periodicPts z)
+  have hp : Function.minimalPeriod p z - 1 + 1 = Function.minimalPeriod p z := by omega
+  have hw : walkSet p (Function.minimalPeriod p z - 1 + 1) (p z) = orbitSet p z := by
+    rw [hp,←he,walkSet_period,orbitSet_apply]
+  have hr : Function.minimalPeriod p z - 1 < Function.minimalPeriod p (p z) := by rw [he]; omega
+  have h := cycleTrace_walk p S (Function.minimalPeriod p z-1) (p z) V hr
+    (by simpa only [hw] using hS) (by simpa only [hw] using hV)
+  have ht : p^[Function.minimalPeriod p z-1] (p z) = z := by
+    rw [← Function.iterate_succ_apply,show (Function.minimalPeriod p z-1).succ =
+      Function.minimalPeriod p z by omega,Function.iterate_minimalPeriod]
+  have ha : p^[Function.minimalPeriod p z-1+1] (p z) = p z := by
+    rw [hp,←he,Function.iterate_minimalPeriod]
+  simpa only [ha,ht,hw] using h
+
+end OrbitCore
+
+noncomputable def exposedPaths (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) :
+    Finset (SwitchIndex d) :=
+  V.biUnion (fun x => routeDomain d x (butterflyPerm d b x))
+
+@[simp] lemma exposedPaths_empty (d : ℕ) (b : Butterfly d) : exposedPaths d b ∅ = ∅ := by
+  classical
+  simp [exposedPaths]
+
+lemma exposedPaths_insert (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) (x : Card d) :
+    exposedPaths d b (insert x V) = exposedPaths d b V ∪ routeDomain d x (butterflyPerm d b x) := by
+  classical
+  simp [exposedPaths,Finset.union_comm]
+
+noncomputable def terminalCharge (d : ℕ) (b : Butterfly d) (z : Card d) (V : Finset (Card d)) : ℕ :=
+  (routeDomain d z (butterflyPerm d b z) ∩ exposedPaths d b V).card
+
+namespace QueryTree
+
+lemma followPath_stale_charge {ι : Type*} [DecidableEq ι]
+    (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
+    (k : Card d → QueryTree ι) (R : Finset ι) (ω : ι → Bool) :
+    staleCharge (followPath d e x k) R ω =
+      staleCharge (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x))
+        (R ∪ (routeDomain d x (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).image e) ω := by
+  have h₀ := fresh_add_stale (followPath d e x k) R ω
+  have h₁ := fresh_add_stale (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x))
+    (R ∪ (routeDomain d x (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).image e) ω
+  rw [followPath_charge,followPath_raw_charge] at h₀
+  omega
+
+lemma seekCycle_semantics (d : ℕ) (S : Finset (Card d)) (a : Card d) (r : ℕ)
+    (x : Card d) (V W : Finset (Card d)) (z : Card d)
+    (k : Finset (Card d) → QueryTree (SwitchIndex d)) (ω : SwitchIndex d → Bool)
+    (h : cycleTrace (butterflyPerm d (decodeButterfly d ω)) S a r x V = some (W,z)) :
+    (seekCycle d S a r x V k).succeeds ω = (k W).succeeds ω ∧
+      (seekCycle d S a r x V k).staleCharge (exposedPaths d (decodeButterfly d ω) V) ω =
+        (k W).staleCharge (exposedPaths d (decodeButterfly d ω) W) ω +
+          terminalCharge d (decodeButterfly d ω) z (W.erase z) := by
+  classical
+  induction r generalizing x V with
+  | zero =>
+    simp only [cycleTrace] at h
+    split_ifs at h with hv
+    obtain ⟨hxS,hxV,hxa⟩ := hv
+    obtain ⟨rfl,rfl⟩ := Prod.mk.inj (Option.some.inj h)
+    simp only [seekCycle,ite_eq_left (show x ∈ S ∧ x ∉ V from ⟨hxS,hxV⟩),closePath_succeeds,ite_eq_left hxa]
+    constructor
+    · trivial
+    · rw [closePath_stale_charge _ _ _ _ _ _ hxa,exposedPaths_insert,
+        Finset.erase_insert hxV]
+      simp only [terminalCharge,hxa]
+  | succ r ih =>
+    simp only [cycleTrace] at h
+    split_ifs at h with hv
+    obtain ⟨hxS,hxV⟩ := hv
+    rw [seekCycle,ite_eq_left ⟨hxS,hxV⟩,followPath_succeeds,followPath_stale_charge]
+    simp only [Function.comp_id,Finset.image_id]
+    rw [← exposedPaths_insert]
+    exact ih _ _ h
+
+lemma seekCycle_orbit (d : ℕ) (S V : Finset (Card d)) (z : Card d)
+    (k : Finset (Card d) → QueryTree (SwitchIndex d)) (ω : SwitchIndex d → Bool)
+    (hS : orbitSet (butterflyPerm d (decodeButterfly d ω)) z ⊆ S)
+    (hV : Disjoint V (orbitSet (butterflyPerm d (decodeButterfly d ω)) z)) :
+    let p := butterflyPerm d (decodeButterfly d ω)
+    let W := V ∪ orbitSet p z
+    (seekCycle d S (p z) (Function.minimalPeriod p z-1) (p z) V k).succeeds ω = (k W).succeeds ω ∧
+      (seekCycle d S (p z) (Function.minimalPeriod p z-1) (p z) V k).staleCharge
+          (exposedPaths d (decodeButterfly d ω) V) ω =
+        (k W).staleCharge (exposedPaths d (decodeButterfly d ω) W) ω +
+          terminalCharge d (decodeButterfly d ω) z (W.erase z) := by
+  exact seekCycle_semantics _ _ _ _ _ _ _ _ _ _ (cycleTrace_orbit _ _ _ _ hS hV)
+
+end QueryTree
+
+lemma mem_exposedPaths (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) (s : SwitchIndex d) :
+    s ∈ exposedPaths d b V ↔ (switchUsers d b s).1 ∈ V ∨ (switchUsers d b s).2 ∈ V := by
+  classical
+  simp only [exposedPaths,Finset.mem_biUnion,mem_routeDomain_iff_user]
+  constructor
+  · rintro ⟨x,hx,h | h⟩
+    · exact Or.inl (h ▸ hx)
+    · exact Or.inr (h ▸ hx)
+  · rintro (h | h)
+    · exact ⟨_,h,Or.inl rfl⟩
+    · exact ⟨_,h,Or.inr rfl⟩
+
+lemma terminalCharge_sum (d : ℕ) (b : Butterfly d) (z : Card d) (V : Finset (Card d))
+    (hz : z ∉ V) :
+    (terminalCharge d b z V : ℝ) = ∑ s : SwitchIndex d,
+      ((if (switchUsers d b s).1 = z ∧ (switchUsers d b s).2 ∈ V then (1:ℝ) else 0) +
+      (if (switchUsers d b s).2 = z ∧ (switchUsers d b s).1 ∈ V then (1:ℝ) else 0)) := by
+  classical
+  have he : routeDomain d z (butterflyPerm d b z) ∩ exposedPaths d b V =
+      Finset.univ.filter (fun s =>
+        ((switchUsers d b s).1 = z ∧ (switchUsers d b s).2 ∈ V) ∨
+        ((switchUsers d b s).2 = z ∧ (switchUsers d b s).1 ∈ V)) := by
+    ext s
+    simp only [Finset.mem_inter,mem_routeDomain_iff_user,mem_exposedPaths,
+      Finset.mem_filter,Finset.mem_univ,true_and]
+    by_cases h₁ : (switchUsers d b s).1 = z <;>
+      by_cases h₂ : (switchUsers d b s).2 = z <;> simp [h₁,h₂,hz]
+  rw [terminalCharge,he]
+  simp only [← Finset.sum_boole]
+  apply Finset.sum_congr rfl
+  intro s _
+  by_cases h₁ : (switchUsers d b s).1 = z <;>
+    by_cases h₂ : (switchUsers d b s).2 = z <;> simp [h₁,h₂,hz]
+
+noncomputable def priorPaths {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) : Finset (Card d) :=
+  Finset.univ.filter (fun x => ∃ i : Fin A, o x = some i ∧ i.val < t)
+
+@[simp] lemma mem_priorPaths {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) (x : Card d) :
+    x ∈ priorPaths o t ↔ ∃ i : Fin A, o x = some i ∧ i.val < t := by
+  classical
+  simp [priorPaths]
+
+lemma terminalCharge_prior_sum {d A : ℕ} (o : Card d → Option (Fin A))
+    (b : Butterfly d) (z : Terminal o) (i : Fin A) :
+    (terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) =
+      ∑ s : SwitchIndex d,
+        ((if (z i).val = (switchUsers d b s).1 ∧ (switchUsers d b s).2 ∈ priorPaths o (i.val+1)
+          then (1:ℝ) else 0) +
+        (if (z i).val = (switchUsers d b s).2 ∧ (switchUsers d b s).1 ∈ priorPaths o (i.val+1)
+          then (1:ℝ) else 0)) := by
+  classical
+  rw [terminalCharge_sum _ _ _ _ (Finset.notMem_erase _ _)]
+  apply Finset.sum_congr rfl
+  intro s _
+  have hn := switchUsers_ne d b s
+  simp only [Finset.mem_erase]
+  by_cases h₁ : (z i).val = (switchUsers d b s).1 <;>
+    by_cases h₂ : (z i).val = (switchUsers d b s).2 <;> simp_all [eq_comm]
+
+lemma rotationMeeting_eq_slot_sum {d A : ℕ} (o : Card d → Option (Fin A))
+    (z : Terminal o) (u v : Card d) :
+    rotationMeeting o z u v = ∑ i : Fin A,
+      ((if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) +
+      (if (z i).val = v ∧ u ∈ priorPaths o (i.val+1) then (1:ℝ) else 0)) := by
+  classical
+  have only_owner (u : Card d) (a : Fin A) (hou : o u = some a)
+      (i : Fin A) (hia : i ≠ a) : (z i).val ≠ u := by
+    intro h
+    have hi := (z i).property
+    rw [h,hou] at hi
+    exact hia (Option.some.inj hi).symm
+  have sum_one (u v : Card d) (a : Fin A) (hou : o u = some a) :
+      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) =
+      if (z a).val = u ∧ v ∈ priorPaths o (a.val+1) then 1 else 0 := by
+    apply Finset.sum_eq_single a
+    · intro i _ hia
+      simp [only_owner u a hou i hia]
+    · simp
+  have sum_none (u v : Card d) (hou : o u = none) :
+      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) = 0 := by
+    apply Finset.sum_eq_zero
+    intro i _
+    have hn : (z i).val ≠ u := by
+      intro h
+      have hi := (z i).property
+      rw [h,hou] at hi
+      contradiction
+    simp [hn]
+  rw [Finset.sum_add_distrib]
+  cases hou : o u with
+  | none =>
+    rw [sum_none u v hou]
+    simp [rotationMeeting,hou,mem_priorPaths]
+  | some a =>
+    rw [sum_one u v a hou]
+    cases hov : o v with
+    | none =>
+      rw [sum_none v u hov]
+      simp [rotationMeeting,hou,hov,mem_priorPaths]
+    | some b =>
+      rw [sum_one v u b hov]
+      simp only [rotationMeeting,hou,hov,mem_priorPaths,Option.some.injEq]
+      rcases lt_trichotomy a b with hab | rfl | hba
+      · have hn : ¬ b.val < a.val+1 := by omega
+        have hy : a.val < b.val+1 := by omega
+        simp [hab,hn,hy]
+      · simp
+      · have hn : ¬ a.val < b.val+1 := by omega
+        have hy : b.val < a.val+1 := by omega
+        simp [hba,not_lt_of_ge (le_of_lt hba),hn,hy]
+
+lemma rotationCharge_eq_terminal_sum {d A : ℕ} (o : Card d → Option (Fin A))
+    (b : Butterfly d) (z : Terminal o) :
+    rotationCharge o b z = ∑ i : Fin A,
+      (terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) := by
+  simp only [rotationCharge,rotationMeeting_eq_slot_sum,terminalCharge_prior_sum]
+  exact Finset.sum_comm
+
+@[simp] lemma priorPaths_zero {d A : ℕ} (o : Card d → Option (Fin A)) : priorPaths o 0 = ∅ := by
+  classical
+  ext x
+  simp
+
+lemma priorPaths_succ {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) (ht : t < A) :
+    priorPaths o (t+1) = priorPaths o t ∪ cycleFiber o ⟨t,ht⟩ := by
+  classical
+  ext x
+  simp only [mem_priorPaths,Finset.mem_union,mem_cycleFiber]
+  constructor
+  · rintro ⟨i,hi,hit⟩
+    by_cases h : i.val < t
+    · exact Or.inl ⟨i,hi,h⟩
+    · have he : i = ⟨t,ht⟩ := Fin.ext (by change i.val = t; omega)
+      exact Or.inr (he ▸ hi)
+  · rintro (⟨i,hi,hit⟩ | hi)
+    · exact ⟨i,hi,by omega⟩
+    · exact ⟨⟨t,ht⟩,hi,by change t < t+1; omega⟩
+
+lemma priorPaths_disjoint {d A : ℕ} (o : Card d → Option (Fin A)) (i : Fin A) :
+    Disjoint (priorPaths o i.val) (cycleFiber o i) := by
+  classical
+  apply Finset.disjoint_left.mpr
+  intro x hx hx'
+  obtain ⟨j,hj,hji⟩ := (mem_priorPaths _ _ _).mp hx
+  have he : j = i := Option.some.inj (hj.symm.trans ((mem_cycleFiber _ _ _).mp hx'))
+  subst j
+  exact (lt_irrefl _ hji)
+
+namespace QueryTree
+
+lemma seekCycles_owned {d A : ℕ} (S : Finset (Card d)) (r : Fin A → ℕ)
+    (o : Card d → Option (Fin A)) (z : Terminal o) (ω : SwitchIndex d → Bool)
+    (hS : ∀ i, cycleFiber o i ⊆ S)
+    (hO : ∀ i, orbitSet (butterflyPerm d (decodeButterfly d ω)) (z i).val = cycleFiber o i)
+    (hC : ∀ i, (cycleFiber o i).card = r i+1) :
+    let p := butterflyPerm d (decodeButterfly d ω)
+    let L := List.ofFn (fun i : Fin A => (p (z i).val,r i))
+    (seekCycles d S L ∅).succeeds ω = true ∧
+      ((seekCycles d S L ∅).staleCharge ∅ ω : ℝ) = rotationCharge o (decodeButterfly d ω) z := by
+  classical
+  let b := decodeButterfly d ω
+  let p := butterflyPerm d b
+  let f (i : Fin A) := (p (z i).val,r i)
+  let c (i : Fin A) : ℝ := terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val)
+  have hc (i : Fin A) : Function.minimalPeriod p (z i).val - 1 = r i := by
+    have he := orbitSet_card p (z i).val
+    rw [hO,hC] at he
+    omega
+  have hs (t : ℕ) (ht : t ≤ A) :
+      (seekCycles d S ((List.ofFn f).drop t) (priorPaths o t)).succeeds ω = true ∧
+        ((seekCycles d S ((List.ofFn f).drop t) (priorPaths o t)).staleCharge (exposedPaths d b (priorPaths o t)) ω : ℝ) =
+          ((List.ofFn c).drop t).sum := by
+    induction ht using Nat.decreasingInduction with
+    | self => simp only [List.length_ofFn,le_refl,List.drop_eq_nil_of_le,List.sum_nil,seekCycles,succeeds,staleCharge,Nat.cast_zero,and_self]
+    | of_succ t ht ih =>
+      have htf : t < (List.ofFn f).length := by simpa using ht
+      have htc : t < (List.ofFn c).length := by simpa using ht
+      rw [List.drop_eq_getElem_cons htf,List.drop_eq_getElem_cons htc]
+      simp only [List.getElem_ofFn,List.sum_cons,seekCycles]
+      have h := seekCycle_orbit d S (priorPaths o t) (z ⟨t,ht⟩).val
+        (fun V => seekCycles d S ((List.ofFn f).drop (t+1)) V) ω
+        (by rw [hO]; exact hS _) (by rw [hO]; exact priorPaths_disjoint o ⟨t,ht⟩)
+      dsimp only at h
+      rw [hc,hO,←priorPaths_succ o t ht] at h
+      dsimp only [f]
+      constructor
+      · exact h.1.trans ih.1
+      · rw [h.2,Nat.cast_add,ih.2]
+        exact add_comm _ _
+  have h := hs 0 (Nat.zero_le _)
+  simp only [List.drop_zero,priorPaths_zero,exposedPaths_empty,List.sum_ofFn] at h
+  dsimp only
+  exact ⟨h.1,h.2.trans (rotationCharge_eq_terminal_sum o b z).symm⟩
+
+end QueryTree
+
+section Collections
+variable {d A : ℕ}
+
+def ValidCycleSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ)
+    (o : Card d → Option (Fin A)) : Prop :=
+  (∀ i, (cycleFiber o i).card = r i + 1) ∧
+  (∀ i, cycleFiber o i ⊆ S) ∧
+  (∀ i x, o x = some i → orbitSet p x = cycleFiber o i)
+
+abbrev CycleSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
+  {o : Card d → Option (Fin A) // ValidCycleSlots p S r o}
+
+noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
+    Fintype (CycleSlots p S r) := Fintype.ofFinite _
+
+lemma CycleSlots.terminal_nonempty {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (c : CycleSlots p S r) (i : Fin A) : Nonempty {x : Card d // c.val x = some i} := by
+  have hn : (cycleFiber c.val i).Nonempty := Finset.card_pos.mp (by rw [c.property.1]; omega)
+  obtain ⟨x,hx⟩ := hn
+  exact ⟨x,(mem_cycleFiber _ _ _).mp hx⟩
+
+lemma owner_ext {o o' : Card d → Option (Fin A)}
+    (h : ∀ i, cycleFiber o i = cycleFiber o' i) : o = o' := by
+  funext x
+  have he (i : Fin A) : o x = some i ↔ o' x = some i := by
+    simpa only [mem_cycleFiber] using (Finset.ext_iff.mp (h i) x)
+  cases ho : o x with
+  | some i => exact ((he i).mp ho).symm
+  | none =>
+    cases ho' : o' x with
+    | none => rfl
+    | some i => have hn := (he i).mpr ho'; rw [ho] at hn; contradiction
+
+abbrev RootedSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
+  Σ c : CycleSlots p S r, Terminal c.val
+
+noncomputable def rootedStart {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (c : RootedSlots p S r) : Fin A → S := fun i =>
+  ⟨p (c.2 i).val,c.1.property.2.1 i (by
+    rw [←c.1.property.2.2 i (c.2 i).val (c.2 i).property]
+    exact (mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.apply_right (Equiv.Perm.SameCycle.refl p _)))⟩
+
+lemma rootedStart_injective {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ} :
+    Function.Injective (rootedStart (p:=p) (S:=S) (r:=r)) := by
+  rintro ⟨c,z⟩ ⟨c',z'⟩ h
+  have hz (i : Fin A) : (z i).val = (z' i).val :=
+    p.injective (congrArg Subtype.val (congrFun h i))
+  have hc : c = c' := Subtype.ext (owner_ext (fun i => by
+    rw [←c.property.2.2 i (z i).val (z i).property,
+      ←c'.property.2.2 i (z' i).val (z' i).property,hz]))
+  subst c'
+  have hzz : z = z' := funext (fun i => Subtype.ext (hz i))
+  subst z'
+  rfl
+
+lemma sum_comp_injective_le {α β : Type*} [Fintype α] [Fintype β]
+    (e : α → β) (he : Function.Injective e) (f : β → ℝ) (hf : ∀ b, 0 ≤ f b) :
+    ∑ a, f (e a) ≤ ∑ b, f b := by
+  classical
+  calc
+    _ = ∑ b ∈ Finset.univ.image e, f b := (Finset.sum_image (fun _ _ _ _ h => he h)).symm
+    _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _) (fun b _ _ => hf b)
+
+lemma rooted_weight_sum_le (S : Finset (Card d)) (r : Fin A → ℕ) (ω : SwitchIndex d → Bool) :
+    (∑ c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r,
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2)) ≤
+    ∑ a : Fin A → S,
+      if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+        ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+      else 0 := by
+  classical
+  let p := butterflyPerm d (decodeButterfly d ω)
+  let g (a : Fin A → S) : ℝ :=
+    if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+    else 0
+  have he (c : RootedSlots p S r) :
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2) =
+      g (rootedStart c) := by
+    have hs := QueryTree.seekCycles_owned S r c.1.val c.2 ω c.1.property.2.1
+      (fun i => c.1.property.2.2 i (c.2 i).val (c.2 i).property) c.1.property.1
+    dsimp only at hs
+    unfold g
+    rw [ite_eq_left (show (QueryTree.seekCycles d S (List.ofFn (fun i => ((rootedStart c i).val,r i))) ∅).succeeds ω = true from hs.1)]
+    rw [← hs.2,neg_mul,Real.exp_neg,mul_comm (Real.log 2),Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
+    rfl
+  simp_rw [he]
+  exact sum_comp_injective_le rootedStart rootedStart_injective g (fun a => by dsimp [g]; split_ifs <;> positivity)
+
+noncomputable def rotationLower (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : ℝ :=
+  Real.exp (-Real.log 2 * ((A:ℝ)*d/2 + ∑ i : Fin A, Real.log (r i+1) / (2 * Real.log 2)))
+
+noncomputable def collectionWeight (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : ℝ :=
+  ((2:ℝ)^d)^A * (∏ i : Fin A, (r i+1 : ℝ)) * rotationLower d r
+
+lemma collectionWeight_pos (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : 0 < collectionWeight d r := by
+  unfold collectionWeight rotationLower
+  positivity
+
+lemma collection_rotation_lower (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r)
+    (ω : SwitchIndex d → Bool) (c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) :
+    collectionWeight d r ≤ ∑ z : Terminal c.val,
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z) := by
+  classical
+  let (i : Fin A) : Nonempty {x : Card d // c.val x = some i} := c.terminal_nonempty i
+  have hm : Monotone (fun i => (cycleFiber c.val i).card) := by
+    intro i j hij
+    dsimp only
+    rw [c.property.1,c.property.1]
+    exact Nat.add_le_add_right (hr hij) 1
+  have hb := rotation_exp_bound hm (decodeButterfly d ω)
+  have hcard : (Fintype.card (Terminal c.val) : ℝ) = ∏ i : Fin A, (r i+1 : ℝ) := by
+    rw [Fintype.card_pi,Nat.cast_prod]
+    apply Finset.prod_congr rfl
+    intro i _
+    rw [card_cycleFiber,c.property.1,Nat.cast_add,Nat.cast_one]
+  have hp : (0:ℝ) < Fintype.card (Terminal c.val) := Nat.cast_pos.mpr Fintype.card_pos
+  have hl : rotationLower d r ≤ finiteMean (fun z : Terminal c.val => Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z)) := by
+    simpa only [rotationLower,c.property.1,Nat.cast_add,Nat.cast_one] using hb
+  have hh := (le_div_iff₀ hp).mp hl
+  rw [hcard] at hh
+  unfold collectionWeight
+  rw [← Finset.mul_sum]
+  calc
+    _ = ((2:ℝ)^d)^A * (rotationLower d r * ∏ i : Fin A, (r i+1 : ℝ)) := by ring
+    _ ≤ _ := mul_le_mul_of_nonneg_left hh (by positivity)
+
+lemma cycleSlots_weighted_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) *
+      collectionWeight d r ≤ (S.card : ℝ)^A := by
+  classical
+  let g (a : Fin A → S) (ω : SwitchIndex d → Bool) : ℝ :=
+    if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+    else 0
+  have hω (ω : SwitchIndex d → Bool) :
+      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ) * collectionWeight d r ≤
+        ∑ a : Fin A → S, g a ω := by
+    calc
+      _ = ∑ _c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r, collectionWeight d r := by simp
+      _ ≤ ∑ c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r,
+          ∑ z : Terminal c.val, ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z) :=
+        Finset.sum_le_sum (fun c _ => collection_rotation_lower S r hr ω c)
+      _ = ∑ c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r,
+          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2) :=
+        (Fintype.sum_sigma (fun c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r =>
+          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2))).symm
+      _ ≤ _ := rooted_weight_sum_le S r ω
+  have hb := finiteMean_mono hω
+  rw [finiteMean_mul_const,finiteMean_sum] at hb
+  calc
+    _ ≤ ∑ a : Fin A → S, finiteMean (g a) := hb
+    _ ≤ ∑ _a : Fin A → S, (1:ℝ) := by
+      apply Finset.sum_le_sum
+      intro a _
+      have hh := QueryTree.cycles_weighted_success d S (List.ofFn (fun i : Fin A => ((a i).val,r i)))
+      simpa only [List.length_ofFn] using hh
+    _ = _ := by simp
+
+lemma cycleSlots_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) ≤
+        (S.card : ℝ)^A / collectionWeight d r := by
+  exact (le_div_iff₀ (collectionWeight_pos d r)).mpr (cycleSlots_weighted_moment S r hr)
+
+lemma collectionWeight_eq_sqrt (d : ℕ) {A : ℕ} (r : Fin A → ℕ) :
+    collectionWeight d r = (Real.sqrt ((2:ℝ)^d))^A * ∏ i : Fin A, Real.sqrt (r i+1) := by
+  have hn : ((2:ℝ)^d)^A = Real.exp ((A:ℝ)*d*Real.log 2) := by
+    rw [mul_assoc,Real.exp_nat_mul,Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
+  have hp : (∏ i : Fin A, (r i+1 : ℝ)) = Real.exp (∑ i : Fin A, Real.log (r i+1)) := by
+    rw [Real.exp_sum]
+    apply Finset.prod_congr rfl
+    intro i _
+    exact (Real.exp_log (by positivity)).symm
+  have hs : Real.sqrt ((2:ℝ)^d) = Real.exp ((d:ℝ)*Real.log 2/2) := by
+    rw [Real.sqrt_eq_rpow,Real.rpow_def_of_pos (by positivity),Real.log_pow]
+    congr 1
+    ring
+  have hj (i : Fin A) : Real.sqrt (r i+1) = Real.exp (Real.log (r i+1)/2) := by
+    rw [Real.sqrt_eq_rpow,Real.rpow_def_of_pos (by positivity)]
+    congr 1
+    ring
+  simp_rw [collectionWeight,hn,hp,rotationLower,hs,hj]
+  rw [←Real.exp_add,←Real.exp_add,←Real.exp_nat_mul,←Real.exp_sum,←Real.exp_add]
+  congr 1
+  rw [←Finset.sum_div,←Finset.sum_div]
+  have hl : Real.log 2 ≠ 0 := ne_of_gt (Real.log_pos (by norm_num))
+  field_simp
+  ring
+
+lemma butterfly_cycleSlots_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) ≤
+        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
+  have h := cycleSlots_moment S r hr
+  rw [collectionWeight_eq_sqrt,div_mul_eq_div_mul_one_div] at h
+  rw [div_pow]
+  simpa only [one_div,Finset.prod_inv_distrib] using h
+
+end Collections
+
+variable {d A : ℕ}
+
+abbrev SelectedCycle (p : Equiv.Perm (Card d)) (S : Finset (Card d)) :=
+  {C : Finset (Card d) // (∃ x, C = orbitSet p x) ∧ C ⊆ S}
+
+noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) :
+    Fintype (SelectedCycle p S) := Fintype.ofFinite _
+
+lemma SelectedCycle.nonempty {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
+    (C : SelectedCycle p S) : C.val.Nonempty := by
+  obtain ⟨x,hx⟩ := C.property.1
+  rw [hx]
+  exact orbitSet_nonempty p x
+
+lemma SelectedCycle.orbit_eq {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
+    (C : SelectedCycle p S) {x : Card d} (hx : x ∈ C.val) : orbitSet p x = C.val := by
+  obtain ⟨y,hy⟩ := C.property.1
+  rw [hy] at hx ⊢
+  exact (orbitSet_eq p ((mem_orbitSet _ _ _).mp hx)).symm
+
+lemma SelectedCycle.eq_of_mem {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
+    (C D : SelectedCycle p S) {x : Card d} (hC : x ∈ C.val) (hD : x ∈ D.val) : C = D := by
+  apply Subtype.ext
+  exact (C.orbit_eq hC).symm.trans (D.orbit_eq hD)
+
+abbrev SlotAssignment (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
+  {f : Fin A → SelectedCycle p S // Function.Injective f ∧ ∀ i, (f i).val.card = r i+1}
+
+noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
+    Fintype (SlotAssignment p S r) := Fintype.ofFinite _
+
+noncomputable def slotsToAssignment {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (c : CycleSlots p S r) : SlotAssignment p S r := by
+  classical
+  let f (i : Fin A) : SelectedCycle p S := ⟨cycleFiber c.val i,by
+    obtain ⟨x,hx⟩ := c.terminal_nonempty i
+    exact ⟨⟨x,(c.property.2.2 i x hx).symm⟩,c.property.2.1 i⟩⟩
+  refine ⟨f,?_,fun i => c.property.1 i⟩
+  intro i j hij
+  have he : cycleFiber c.val i = cycleFiber c.val j := congrArg Subtype.val hij
+  obtain ⟨x,hx⟩ := c.terminal_nonempty i
+  have hxi : x ∈ cycleFiber c.val i := (mem_cycleFiber _ _ _).mpr hx
+  rw [he] at hxi
+  exact Option.some.inj (hx.symm.trans ((mem_cycleFiber _ _ _).mp hxi))
+
+noncomputable def assignmentOwner {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) (x : Card d) : Option (Fin A) := by
+  classical
+  exact if h : ∃ i, x ∈ (f.val i).val then some (Classical.choose h) else none
+
+end Thorp
+
+end ThorpNine.SparseSaving
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/SparseSaving/ShapePartitionEquiv.lean b/lean/OAI/Probability/ThorpRouting/SparseSaving/ShapePartitionEquiv.lean
index 9bb08fdadbe57b18324fa03a5b4e36135a34bbea..fe114e6bb8f940a6996b13df34afd82c295f365b 100644
--- a/lean/OAI/Probability/ThorpRouting/SparseSaving/ShapePartitionEquiv.lean
+++ b/lean/OAI/Probability/ThorpRouting/SparseSaving/ShapePartitionEquiv.lean
@@ -1,875 +1,875 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.SparseSaving.ModifiedReverseEquiv
-
-namespace OAI
-
-namespace ThorpNine.SparseSaving
-
-namespace Thorp.Specht
-open scoped BigOperators Classical
-
-noncomputable def shapePartitionEquiv (N : ℕ) : Shapes N ≃ Nat.Partition N where
-  toFun μ := { parts := μ.1.rowLens
-               parts_pos := fun {i} hi => μ.1.pos_of_mem_rowLens i (by simpa using hi)
-               parts_sum := by simpa using (sum_rowLens μ.1).trans μ.2 }
-  invFun p := ⟨partitionDiagram p,partitionDiagram_card p⟩
-  left_inv μ := by
-    apply Subtype.ext
-    obtain ⟨μ,rfl⟩ := μ
-    exact partitionDiagram_diagramPartition μ
-  right_inv p := by
-    apply Nat.Partition.ext
-    change ((partitionDiagram p).rowLens : Multiset ℕ) = p.parts
-    rw [partitionDiagram_rowLens,Multiset.sort_eq]
-
-noncomputable instance (N : ℕ) : Fintype (Shapes N) := Fintype.ofEquiv _ (shapePartitionEquiv N).symm
-
-noncomputable def conjugacyPartition {α : Type*} [Fintype α] [DecidableEq α] :
-    ConjClasses (Equiv.Perm α) → Nat.Partition (Fintype.card α) :=
-  Quotient.lift Equiv.Perm.partition (fun _ _ hpq => (Equiv.Perm.partition_eq_of_isConj).mp hpq)
-
-lemma conjugacyPartition_injective {α : Type*} [Fintype α] [DecidableEq α] :
-    Function.Injective (conjugacyPartition (α := α)) := by
-  intro p q
-  induction p using Quotient.inductionOn with | _ p =>
-    induction q using Quotient.inductionOn with | _ q =>
-      intro h
-      exact Quotient.sound ((Equiv.Perm.partition_eq_of_isConj).mpr h)
-
-lemma card_conjugacy_le_shapes (N : ℕ) :
-    Nat.card (ConjClasses (Equiv.Perm (Fin N))) ≤ Fintype.card (Shapes N) := by
-  have h := Nat.card_le_card_of_injective _ (conjugacyPartition_injective (α := Fin N))
-  simpa only [Fintype.card_fin, Nat.card_eq_fintype_card, Fintype.card_congr (shapePartitionEquiv N)] using h
-
-noncomputable def labels {N : ℕ} (μ : Shapes N) : Cell μ.1 ≃ Fin N :=
-  (Fintype.equivFin (Cell μ.1)).trans (finCongr ((card_cell μ.1).trans μ.2))
-
-noncomputable def labelledRepresentation {N : ℕ} (μ : Shapes N) :
-    Representation ℂ (Equiv.Perm (Fin N)) (space μ.1) :=
-  (representation μ.1).comp (labels μ).symm.permCongrHom.toMonoidHom
-
-lemma labelledRepresentation_irreducible {N : ℕ} (μ : Shapes N) :
-    Representation.IsIrreducible (labelledRepresentation μ) := by
-  let e := (labels μ).symm.permCongrHom
-  let ρ := representation μ.1
-  have hn : (⊥ : Subrepresentation (labelledRepresentation μ)) ≠ ⊤ := by
-    intro h
-    have he : (⟨polytabloid μ.1,polytabloid_mem_space μ.1⟩ : space μ.1) ∈
-        (⊥ : Subrepresentation (labelledRepresentation μ)) := by rw [h]; trivial
-    exact polytabloid_ne_zero μ.1 (congrArg Subtype.val he)
-  refine { exists_pair_ne := ⟨⊥,⊤,hn⟩,eq_bot_or_eq_top := ?_ }
-  intro U
-  let U' : Subrepresentation ρ :=
-    { toSubmodule := U.toSubmodule
-      apply_mem_toSubmodule := fun p v hv => by
-        have hh := U.apply_mem_toSubmodule (e.symm p) hv
-        change ρ (e (e.symm p)) v ∈ U.toSubmodule at hh
-        simpa using hh }
-  have := representation_irreducible μ.1
-  obtain h | h := IsSimpleOrder.eq_bot_or_eq_top U'
-  · left
-    apply Subrepresentation.toSubmodule_injective
-    exact congrArg (fun R : Subrepresentation ρ => R.toSubmodule) h
-  · right
-    apply Subrepresentation.toSubmodule_injective
-    exact congrArg (fun R : Subrepresentation ρ => R.toSubmodule) h
-
-lemma labelledRepresentation_equiv_shapes {N : ℕ} (μ ν : Shapes N)
-    (T : Representation.Equiv (labelledRepresentation μ) (labelledRepresentation ν)) : μ = ν := by
-  apply Subtype.ext
-  apply shapes_eq_of_equivariant_equiv μ.1 ν.1 ((labels μ).trans (labels ν).symm) T.toLinearEquiv
-  intro p v
-  have hh := Representation.IntertwiningMap.isIntertwining _ _ T.toIntertwiningMap ((labels μ).permCongr p) v
-  change T.toLinearEquiv
-    (representation μ.1 ((labels μ).symm.permCongr ((labels μ).permCongr p)) v) =
-      representation ν.1 ((labels ν).symm.permCongr ((labels μ).permCongr p)) (T.toLinearEquiv v) at hh
-  have hp : (labels μ).symm.permCongr ((labels μ).permCongr p) = p := by
-    apply Equiv.ext
-    intro x
-    simp [Equiv.permCongr_apply]
-  rw [hp] at hh
-  convert hh using 2
-  congr 1
-
-end Thorp.Specht
-
-namespace Thorp.FiniteCharacters
-open scoped BigOperators Classical
-
-section Group
-variable {G : Type*} [Group G] [Fintype G]
-
-noncomputable def classRep (c : ConjClasses G) : G := (ConjClasses.exists_rep c).choose
-omit [Fintype G] in
-lemma classRep_spec (c : ConjClasses G) : ConjClasses.mk (classRep c) = c :=
-  (ConjClasses.exists_rep c).choose_spec
-
-variable {V : Type*} [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
-
-noncomputable def characterClass (ρ : Representation ℂ G V) : ConjClasses G → ℂ :=
-  fun c => ρ.character (classRep c)
-
-omit [Fintype G] [FiniteDimensional ℂ V] in
-lemma characterClass_mk (ρ : Representation ℂ G V) (g : G) :
-    characterClass ρ (ConjClasses.mk g) = ρ.character g := by
-  have h := ConjClasses.mk_eq_mk_iff_isConj.mp (classRep_spec (ConjClasses.mk g))
-  obtain ⟨p,hp⟩ := isConj_iff.mp h
-  change ρ.character (classRep (ConjClasses.mk g)) = ρ.character g
-  exact (Representation.char_conj ρ _ p).symm.trans (congrArg ρ.character hp)
-
-noncomputable def characterDual (ρ : Representation ℂ G V) :
-    (ConjClasses G → ℂ) →ₗ[ℂ] ℂ where
-  toFun f := (Nat.card G : ℂ)⁻¹ * ∑ g, f (ConjClasses.mk g) * ρ.character g⁻¹
-  map_add' f h := by simp [Finset.sum_add_distrib,add_mul,mul_add]
-  map_smul' a f := by
-    simp only [Pi.smul_apply,smul_eq_mul,RingHom.id_apply]
-    simp_rw [mul_assoc]
-    rw [←Finset.mul_sum]
-    ring
-
-lemma characterDual_characterClass (ρ : Representation ℂ G V)
-    {W : Type*} [AddCommGroup W] [Module ℂ W] [FiniteDimensional ℂ W]
-    (σ : Representation ℂ G W) [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ] :
-    characterDual ρ (characterClass σ) = if Nonempty (Representation.Equiv ρ σ) then 1 else 0 := by
-  let : Invertible (Nat.card G : ℂ) := invertibleOfNonzero (by
-    exact_mod_cast (Nat.card_pos (α := G)).ne')
-  change (Nat.card G : ℂ)⁻¹ * ∑ g, characterClass σ (ConjClasses.mk g) * ρ.character g⁻¹ = _
-  simp only [characterClass_mk]
-  exact Representation.char_orthonormal σ ρ
-
-end Group
-
-theorem complete_of_card_conjugacy_le
-    {G ι : Type*} [Group G] [Fintype G] [Fintype ι]
-    (V : ι → Type*) [∀ i, AddCommGroup (V i)] [∀ i, Module ℂ (V i)]
-    [∀ i, FiniteDimensional ℂ (V i)]
-    (ρ : ∀ i, Representation ℂ G (V i)) [∀ i, Representation.IsIrreducible (ρ i)]
-    (hneq : ∀ i j, i ≠ j → ¬Nonempty (Representation.Equiv (ρ i) (ρ j)))
-    (hcard : Nat.card (ConjClasses G) ≤ Fintype.card ι)
-    {W : Type*} [AddCommGroup W] [Module ℂ W] [FiniteDimensional ℂ W]
-    (σ : Representation ℂ G W) [Representation.IsIrreducible σ] :
-    ∃ i, Nonempty (Representation.Equiv (ρ i) σ) := by
-  let : Fintype (ConjClasses G) := Fintype.ofFinite _
-  have horth (i j : ι) : characterDual (ρ i) (characterClass (ρ j)) = if i=j then 1 else 0 := by
-    rw [characterDual_characterClass]
-    by_cases hij : i=j
-    · subst j
-      simp only [ite_true]
-      exact ite_eq_left ⟨Representation.Equiv.refl _⟩
-    · rw [ite_eq_right hij,ite_eq_right (hneq i j hij)]
-  have hLI : LinearIndependent ℂ (fun i => characterClass (ρ i)) := by
-    apply LinearIndependent.of_pairwise_dual_eq_zero_one _ (fun i => characterDual (ρ i))
-    · intro i j hij
-      exact (horth i j).trans (ite_eq_right hij)
-    · intro i
-      simpa using horth i i
-  have heq : Fintype.card ι = Module.finrank ℂ (ConjClasses G → ℂ) := by
-    apply Nat.le_antisymm hLI.fintype_card_le_finrank
-    simpa only [Module.finrank_pi,Module.finrank_self,Finset.sum_const,Finset.card_univ,
-      smul_eq_mul,mul_one,Nat.card_eq_fintype_card] using hcard
-  have hspan := hLI.span_eq_top_of_card_eq_finrank' heq
-  by_contra h
-  push Not at h
-  have hker : Submodule.span ℂ (Set.range (fun i => characterClass (ρ i))) ≤
-      (characterDual σ).ker := by
-    apply Submodule.span_le.mpr
-    rintro f ⟨i,rfl⟩
-    change characterDual σ (characterClass (ρ i)) = 0
-    rw [characterDual_characterClass,ite_eq_right]
-    rintro ⟨T⟩
-    exact (h i).false T.symm
-  rw [hspan] at hker
-  have hz : characterDual σ (characterClass σ) = 0 := hker (show characterClass σ ∈ ⊤ from trivial)
-  rw [characterDual_characterClass,ite_eq_left ⟨Representation.Equiv.refl σ⟩] at hz
-  exact one_ne_zero hz
-
-end Thorp.FiniteCharacters
-
-namespace Thorp.Specht
-open scoped Classical
-
-theorem labelled_complete {N : ℕ}
-    {V : Type*} [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
-    (ρ : Representation ℂ (Equiv.Perm (Fin N)) V) [Representation.IsIrreducible ρ] :
-    ∃ μ : Shapes N, Nonempty (Representation.Equiv (labelledRepresentation μ) ρ) := by
-  let (μ : Shapes N) : Representation.IsIrreducible (labelledRepresentation μ) :=
-    labelledRepresentation_irreducible μ
-  apply FiniteCharacters.complete_of_card_conjugacy_le
-    (fun μ : Shapes N => space μ.1) labelledRepresentation _ (card_conjugacy_le_shapes N) ρ
-  intro μ ν h ⟨T⟩
-  exact h (labelledRepresentation_equiv_shapes μ ν T)
-
-end Thorp.Specht
-
-namespace Thorp.PermutationHilbert
-
-section
-open scoped BigOperators ComplexConjugate Classical
-open Complex
-
-variable {X : Type*} [Fintype X]
-
-noncomputable def act (p : Equiv.Perm X) : H X ≃ₗᵢ[ℂ] H X :=
-  LinearIsometryEquiv.piLpCongrLeft 2 ℂ ℂ p
-
-@[simp] lemma act_apply (p : Equiv.Perm X) (v : H X) (x : X) :
-    act p v x = v (p⁻¹ x) := rfl
-
-lemma act_single [DecidableEq X] (p : Equiv.Perm X) (x : X) :
-    act p (EuclideanSpace.single x (1:ℂ)) = EuclideanSpace.single (p x) 1 := by
-  exact EuclideanSpace.piLpCongrLeft_single p x 1
-
-lemma trace_projection (W : Submodule ℂ (H X)) :
-    ∑ x, (W.starProjection (EuclideanSpace.single x (1:ℂ))) x = Module.finrank ℂ W := by
-  classical
-  have hp : LinearMap.IsProj W W.starProjection.toLinearMap := by
-    constructor
-    · intro x; exact (W.orthogonalProjectionOnto x).2
-    · intro x hx; exact W.starProjection_eq_self_iff.mpr hx
-  have ht := hp.trace
-  rw [LinearMap.trace_eq_matrix_trace ℂ (EuclideanSpace.basisFun X ℂ).toBasis] at ht
-  simpa only [Matrix.trace,Matrix.diag_apply,LinearMap.toMatrix_apply,EuclideanSpace.basisFun_repr,
-    EuclideanSpace.basisFun_apply,OrthonormalBasis.coe_toBasis,
-    OrthonormalBasis.coe_toBasis_repr_apply,ContinuousLinearMap.coe_coe] using ht
-
-lemma projection_diagonal (W : Submodule ℂ (H X)) (x : X) :
-    (W.starProjection (EuclideanSpace.single x (1:ℂ))) x =
-      (‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 : ℝ) := by
-  classical
-  have hh := W.inner_starProjection_left_eq_right
-    (EuclideanSpace.single x (1:ℂ)) (W.starProjection (EuclideanSpace.single x (1:ℂ)))
-  have hf : W.starProjection (W.starProjection (EuclideanSpace.single x (1:ℂ))) =
-      W.starProjection (EuclideanSpace.single x (1:ℂ)) :=
-    W.starProjection_eq_self_iff.mpr (W.orthogonalProjectionOnto _).2
-  rw [hf] at hh
-  rw [inner_self_eq_norm_sq_to_K,EuclideanSpace.inner_single_left] at hh
-  simpa using hh.symm
-
-lemma projection_comm (W : Submodule ℂ (H X)) (p : Equiv.Perm X)
-    (hW : W.map (act p).toLinearEquiv.toLinearMap = W) (v : H X) :
-    act p (W.starProjection v) = W.starProjection (act p v) := by
-  have hh := (act p).toLinearIsometry.map_starProjection W v
-  change act p (W.starProjection v) =
-    (W.map (act p).toLinearEquiv.toLinearMap).starProjection (act p v) at hh
-  simpa only [hW] using hh
-
-lemma uniform_diagonal (W : Submodule ℂ (H X))
-    (htrans : ∀ x y : X, ∃ p : Equiv.Perm X, p x = y ∧
-      W.map (act p).toLinearEquiv.toLinearMap = W) (x : X) :
-    (Fintype.card X:ℝ) * ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 =
-      Module.finrank ℂ W := by
-  classical
-  have hh (y : X) : (W.starProjection (EuclideanSpace.single y (1:ℂ))) y =
-      (‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 : ℝ) := by
-    obtain ⟨p,hp,hW⟩ := htrans x y
-    subst y
-    rw [projection_diagonal,←act_single p x,←projection_comm W p hW,(act p).norm_map]
-  have ht := trace_projection W
-  simp_rw [hh] at ht
-  have hhreal : (∑ _y : X, ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2) =
-      (Module.finrank ℂ W:ℝ) := by exact_mod_cast ht
-  simpa using hhreal
-
-theorem uncertainty (W : Submodule ℂ (H X))
-    (htrans : ∀ x y : X, ∃ p : Equiv.Perm X, p x = y ∧
-      W.map (act p).toLinearEquiv.toLinearMap = W)
-    (v : H X) (hv : v ∈ W) (x : X) :
-    (Fintype.card X:ℝ) * ‖v x‖^2 ≤ (Module.finrank ℂ W:ℝ) * ‖v‖^2 := by
-  classical
-  have hh : inner ℂ (W.starProjection (EuclideanSpace.single x (1:ℂ))) v = v x := by
-    rw [W.inner_starProjection_left_eq_right,W.starProjection_eq_self_iff.mpr hv,
-      EuclideanSpace.inner_single_left]
-    simp
-  have hc := norm_inner_le_norm (𝕜 := ℂ) (W.starProjection (EuclideanSpace.single x (1:ℂ))) v
-  rw [hh] at hc
-  have hs : ‖v x‖^2 ≤ ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 * ‖v‖^2 := by
-    nlinarith [norm_nonneg (v x),norm_nonneg v,
-      norm_nonneg (W.starProjection (EuclideanSpace.single x (1:ℂ)))]
-  calc
-    (Fintype.card X:ℝ) * ‖v x‖^2 ≤
-      (Fintype.card X:ℝ) * (‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 * ‖v‖^2) :=
-      mul_le_mul_of_nonneg_left hs (Nat.cast_nonneg _)
-    _ = _ := by rw [←mul_assoc,uniform_diagonal W htrans x]
-
-end
-open scoped BigOperators Classical
-
-variable {X : Type*} [Fintype X]
-
-lemma trace_projection_real (W : Submodule ℂ (H X)) :
-    ∑ x, ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 = (Module.finrank ℂ W:ℝ) := by
-  have hh := trace_projection W
-  simp_rw [projection_diagonal] at hh
-  exact_mod_cast hh
-
-theorem footprint_dimension (W K : Submodule ℂ (H X)) (hKW : K ≤ W)
-    (htrans : ∀ x y : X, ∃ p : Equiv.Perm X, p x = y ∧
-      W.map (act p).toLinearEquiv.toLinearMap = W)
-    (S : Finset X) (hS : ∀ x ∈ S, W.starProjection (EuclideanSpace.single x (1:ℂ)) ∈ K) :
-    (S.card:ℝ) * (Module.finrank ℂ W:ℝ) ≤
-      (Fintype.card X:ℝ) * (Module.finrank ℂ K:ℝ) := by
-  have he (x : X) (hx : x ∈ S) :
-      K.starProjection (EuclideanSpace.single x (1:ℂ)) =
-        W.starProjection (EuclideanSpace.single x (1:ℂ)) := by
-    apply K.eq_starProjection_of_mem_of_inner_eq_zero (hS x hx)
-    intro v hv
-    exact W.starProjection_inner_eq_zero _ v (hKW hv)
-  have hs : ∑ x ∈ S, ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 ≤
-      (Module.finrank ℂ K:ℝ) := by
-    rw [←trace_projection_real K]
-    calc
-      _ = ∑ x ∈ S, ‖K.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 := by
-        apply Finset.sum_congr rfl
-        intro x hx
-        rw [he x hx]
-      _ ≤ _ := Finset.sum_le_univ_sum_of_nonneg (fun x => sq_nonneg _)
-  have hmul := mul_le_mul_of_nonneg_left hs (Nat.cast_nonneg (Fintype.card X): (0:ℝ) ≤ _)
-  rw [Finset.mul_sum] at hmul
-  simp_rw [uniform_diagonal W htrans] at hmul
-  simpa using hmul
-
-end Thorp.PermutationHilbert
-namespace Thorp.Specht
-open scoped BigOperators Classical
-
-lemma finrank_hilbertSpace (μ : YoungDiagram) :
-    Module.finrank ℂ (hilbertSpace μ) = Module.finrank ℂ (space μ) :=
-  (hilbertEquiv μ).finrank_map_eq (space μ)
-
-lemma hilbert_action (μ : YoungDiagram) (p : Equiv.Perm (Cell μ)) (v : Tabloid μ → ℂ) :
-    hilbertEquiv μ (tabloidRep μ p v) =
-      PermutationHilbert.act (tabloidAct p) (hilbertEquiv μ v) := by
-  ext f
-  simp [hilbertEquiv,PermutationHilbert.act_apply]
-
-lemma hilbertSpace_invariant (μ : YoungDiagram) (p : Equiv.Perm (Cell μ)) :
-    (hilbertSpace μ).map (PermutationHilbert.act (tabloidAct p)).toLinearEquiv.toLinearMap =
-      hilbertSpace μ := by
-  apply le_antisymm
-  · rintro _ ⟨v,⟨w,hw,rfl⟩,rfl⟩
-    exact ⟨tabloidRep μ p w,action_mem_space μ p w hw,hilbert_action μ p w⟩
-  · rintro _ ⟨w,hw,rfl⟩
-    refine ⟨hilbertEquiv μ (tabloidRep μ p⁻¹ w),
-      ⟨tabloidRep μ p⁻¹ w,action_mem_space μ p⁻¹ w hw,rfl⟩, ?_⟩
-    change PermutationHilbert.act (tabloidAct p) (hilbertEquiv μ (tabloidRep μ p⁻¹ w)) = hilbertEquiv μ w
-    rw [←hilbert_action]
-    change hilbertEquiv μ ((tabloidRep μ p * tabloidRep μ p⁻¹) w) = _
-    rw [←map_mul,mul_inv_cancel,map_one]
-    rfl
-
-lemma tabloidAct_transitive (μ : YoungDiagram) (f g : Tabloid μ) :
-    ∃ p : Equiv.Perm (Cell μ), tabloidAct p f = g := by
-  obtain ⟨q,hq⟩ := f.2
-  obtain ⟨r,hr⟩ := g.2
-  refine ⟨r⁻¹*q, ?_⟩
-  apply Subtype.ext
-  funext x
-  simp only [tabloidAct_apply,mul_inv_rev,Equiv.Perm.mul_apply,hq,hr,
-    Function.comp_apply,Equiv.Perm.inv_def,Equiv.apply_symm_apply,Equiv.symm_symm]
-
-lemma polytabloid_pairing (μ : YoungDiagram) :
-    pairing (polytabloid μ) (polytabloid μ) = Fintype.card (colGroup μ) := by
-  have h := alternator_pairing μ (polytabloid μ)
-  have he (c : colGroup μ) : tabloidRep μ c (polytabloid μ) =
-      permSign (c:Equiv.Perm (Cell μ)) • polytabloid μ :=
-    action_alternator_apply (colGroup μ) (tabloidRep μ) c _
-  have hb := congrArg (fun v : Tabloid μ → ℂ => v (baseTabloid μ)) h
-  simp only [alternator,LinearMap.sum_apply,LinearMap.smul_apply,he,
-    Pi.smul_apply,smul_eq_mul,Finset.sum_apply,polytabloid_base,mul_one] at hb
-  simp only [permSign_sq,Finset.sum_const,Finset.card_univ,nsmul_eq_mul,mul_one] at hb
-  exact hb.symm
-
-lemma polytabloid_norm_sq (μ : YoungDiagram) :
-    ‖hilbertEquiv μ (polytabloid μ)‖^2 = (Fintype.card (colGroup μ):ℝ) := by
-  have hp := polytabloid_pairing μ
-  change inner ℂ (hilbertEquiv μ (polytabloid μ)) (hilbertEquiv μ (polytabloid μ)) = _ at hp
-  rw [inner_self_eq_norm_sq_to_K] at hp
-  have hre := congrArg Complex.re hp
-  simpa [pow_two] using hre
-
-theorem dimension_times_column_ge_tabloids (μ : YoungDiagram) :
-    Fintype.card (Tabloid μ) ≤ Module.finrank ℂ (space μ) * Fintype.card (colGroup μ) := by
-  have ht (f g : Tabloid μ) : ∃ p : Equiv.Perm (Tabloid μ), p f = g ∧
-      (hilbertSpace μ).map (PermutationHilbert.act p).toLinearEquiv.toLinearMap = hilbertSpace μ := by
-    obtain ⟨p,hp⟩ := tabloidAct_transitive μ f g
-    exact ⟨tabloidAct p,hp,hilbertSpace_invariant μ p⟩
-  have hu := PermutationHilbert.uncertainty (hilbertSpace μ) ht
-    (hilbertEquiv μ (polytabloid μ)) ⟨polytabloid μ,polytabloid_mem_space μ,rfl⟩ (baseTabloid μ)
-  have he : hilbertEquiv μ (polytabloid μ) (baseTabloid μ) = 1 := polytabloid_base μ
-  rw [he,norm_one,one_pow,mul_one,finrank_hilbertSpace,polytabloid_norm_sq] at hu
-  exact_mod_cast hu
-
-
-section FiberGroup
-variable {α β γ : Type*} [Fintype α] [Fintype β] [Fintype γ]
-
-noncomputable def fiberGroupEquiv (f : α → β) :
-    fiberGroup f ≃ (∀ b, Equiv.Perm {x // f x = b}) where
-  toFun p b :=
-    { toFun x := ⟨p.1 x.1,(p.2 x.1).trans x.2⟩
-      invFun x := ⟨p.1⁻¹ x.1, by
-        have hh := p.2 (p.1⁻¹ x.1)
-        simp only [Equiv.Perm.inv_def,Equiv.apply_symm_apply] at hh
-        exact hh.symm.trans x.2⟩
-      left_inv x := by apply Subtype.ext; simp
-      right_inv x := by apply Subtype.ext; simp }
-  invFun q :=
-    ⟨(Equiv.sigmaFiberEquiv f).permCongr (Equiv.Perm.sigmaCongrRight q),
-      fun x => (q (f x) ⟨x,rfl⟩).2⟩
-  left_inv p := by
-    apply Subtype.ext
-    apply Equiv.ext
-    intro x
-    rfl
-  right_inv q := by
-    funext b
-    apply Equiv.ext
-    intro x
-    apply Subtype.ext
-    change (q (f x.1) ⟨x.1,rfl⟩).1 = (q b x).1
-    cases x with
-    | mk x hx => dsimp only at *; subst b; rfl
-
-lemma nat_card_fiberGroup (f : α → β) :
-    Nat.card (fiberGroup f) = ∏ b, (Nat.card {x // f x=b}).factorial := by
-  rw [Nat.card_congr (fiberGroupEquiv f),Nat.card_pi]
-  simp only [Nat.card_eq_fintype_card,Fintype.card_perm]
-
-omit [Fintype β] [Fintype γ] in
-
-lemma card_fiberGroups_mul_le (f : α → β) (g : α → γ)
-    (hinj : ∀ x y, f x = f y → g x = g y → x=y) :
-    Fintype.card (fiberGroup f) * Fintype.card (fiberGroup g) ≤ (Fintype.card α).factorial := by
-  have hi : Function.Injective (fun p : fiberGroup f × fiberGroup g => p.1.1*p.2.1) := by
-    intro p q hpq
-    dsimp only at hpq
-    have he : p.1.1⁻¹*q.1.1 = p.2.1*q.2.1⁻¹ := by
-      apply inv_mul_eq_iff_eq_mul.mpr
-      rw [←mul_assoc,hpq,mul_assoc,mul_inv_cancel,mul_one]
-    have hf : p.1.1⁻¹*q.1.1 ∈ fiberGroup f := (fiberGroup f).mul_mem ((fiberGroup f).inv_mem p.1.2) q.1.2
-    have hg : p.1.1⁻¹*q.1.1 ∈ fiberGroup g := he ▸ (fiberGroup g).mul_mem p.2.2 ((fiberGroup g).inv_mem q.2.2)
-    have h1 : p.1.1⁻¹*q.1.1 = 1 := by
-      apply Equiv.ext
-      intro x
-      exact hinj _ x (hf x) (hg x)
-    have hpf : p.1=q.1 := Subtype.ext (inv_mul_eq_one.mp h1)
-    have hpg : p.2=q.2 := by
-      apply Subtype.ext
-      have hh := hpq
-      rw [show p.1.1=q.1.1 from congrArg Subtype.val hpf] at hh
-      exact mul_left_cancel hh
-    exact Prod.ext hpf hpg
-  have hc := Fintype.card_le_of_injective _ hi
-  simpa only [Fintype.card_prod,Fintype.card_perm] using hc
-
-end FiberGroup
-
-noncomputable def rowGroup (μ : YoungDiagram) : Subgroup (Equiv.Perm (Cell μ)) :=
-  fiberGroup (rowIndex (μ := μ))
-
-noncomputable def colIndex {μ : YoungDiagram} (x : Cell μ) : Fin (μ.rowLen 0) :=
-  ⟨col x, (YoungDiagram.mem_iff_lt_rowLen.mp x.2).trans_le (μ.rowLen_anti 0 (row x) (Nat.zero_le _))⟩
-
-lemma colGroup_eq (μ : YoungDiagram) : colGroup μ = fiberGroup (colIndex (μ := μ)) := by
-  ext p
-  constructor
-  · intro hp x; exact Fin.ext (hp x)
-  · intro hp x; exact congrArg Fin.val (hp x)
-
-noncomputable instance tabloidMulAction (μ : YoungDiagram) : MulAction (Equiv.Perm (Cell μ)) (Tabloid μ) where
-  smul p f := tabloidAct p f
-  one_smul f := by change tabloidAct 1 f = f; rw [tabloidAct_one]; rfl
-  mul_smul p q f := by change tabloidAct (p*q) f = tabloidAct p (tabloidAct q f); rw [tabloidAct_mul]; rfl
-
-lemma tabloid_orbit_all (μ : YoungDiagram) :
-    MulAction.orbit (Equiv.Perm (Cell μ)) (baseTabloid μ) = Set.univ := by
-  ext f
-  simp only [Set.mem_univ,iff_true]
-  obtain ⟨p,hp⟩ := tabloidAct_transitive μ (baseTabloid μ) f
-  exact ⟨p,hp⟩
-
-lemma tabloid_stabilizer (μ : YoungDiagram) :
-    MulAction.stabilizer (Equiv.Perm (Cell μ)) (baseTabloid μ) = rowGroup μ := by
-  ext p
-  change tabloidAct p (baseTabloid μ) = baseTabloid μ ↔ ∀ x, rowIndex (p x) = rowIndex x
-  constructor
-  · intro h x
-    have hh := congrArg (fun f : Tabloid μ => f.1 (p x)) h
-    simpa only [tabloidAct_apply,baseTabloid,Equiv.Perm.inv_def,Equiv.symm_apply_apply] using hh.symm
-  · intro h
-    apply Subtype.ext
-    funext x
-    exact (h (p⁻¹ x)).symm.trans (by simp [baseTabloid])
-
-lemma card_tabloid_mul_row (μ : YoungDiagram) :
-    Fintype.card (Tabloid μ)*Fintype.card (rowGroup μ) = μ.card.factorial := by
-  have hh := MulAction.card_orbit_mul_card_stabilizer_eq_card_group
-    (Equiv.Perm (Cell μ)) (baseTabloid μ)
-  have hc : Fintype.card (MulAction.orbit (Equiv.Perm (Cell μ)) (baseTabloid μ)) =
-      Fintype.card (Tabloid μ) := by
-    rw [tabloid_orbit_all,Fintype.card_setUniv]
-  simpa only [hc,tabloid_stabilizer,Fintype.card_perm,card_cell] using hh
-
-theorem factorial_le_dimension_row_column (μ : YoungDiagram) :
-    μ.card.factorial ≤ Module.finrank ℂ (space μ)*Fintype.card (rowGroup μ)*Fintype.card (colGroup μ) := by
-  rw [←card_tabloid_mul_row]
-  nlinarith [dimension_times_column_ge_tabloids μ]
-
-
-lemma card_row_fiber (μ : YoungDiagram) (i : Fin (μ.colLen 0)) :
-    Fintype.card {x : Cell μ // rowIndex x = i} = μ.rowLen i := by
-  let e : {x : Cell μ // rowIndex x = i} ≃ {x : Cell μ // row x = i} :=
-    Equiv.subtypeEquivRight (fun x => Fin.ext_iff)
-  rw [Fintype.card_congr e,Fintype.card_congr (rowEquiv μ i),Fintype.card_fin]
-
-lemma card_col_fiber (μ : YoungDiagram) (j : Fin (μ.rowLen 0)) :
-    Fintype.card {x : Cell μ // colIndex x = j} = μ.colLen j := by
-  let e : {x : Cell μ // colIndex x = j} ≃ {x : Cell μ // col x = j} :=
-    Equiv.subtypeEquivRight (fun x => Fin.ext_iff)
-  rw [Fintype.card_congr e,Fintype.card_congr (columnEquiv μ j),Fintype.card_fin]
-
-lemma card_rowGroup (μ : YoungDiagram) :
-    Fintype.card (rowGroup μ) = ∏ i : Fin (μ.colLen 0), (μ.rowLen i).factorial := by
-  rw [←Nat.card_eq_fintype_card,rowGroup,nat_card_fiberGroup]
-  simp only [Nat.card_eq_fintype_card]
-  simp only [card_row_fiber]
-
-lemma card_colGroup (μ : YoungDiagram) :
-    Fintype.card (colGroup μ) = ∏ j : Fin (μ.rowLen 0), (μ.colLen j).factorial := by
-  rw [←Nat.card_eq_fintype_card,colGroup_eq,nat_card_fiberGroup]
-  simp only [Nat.card_eq_fintype_card,card_col_fiber]
-
-abbrev Tail (μ : YoungDiagram) := {x : Cell μ // row x ≠ 0}
-
-lemma tail_card (μ : YoungDiagram) : Fintype.card (Tail μ) = μ.card - μ.rowLen 0 := by
-  have hh := Fintype.card_subtype_compl (p := fun x : Cell μ => row x=0)
-  rw [Fintype.card_congr (rowEquiv μ 0),Fintype.card_fin,card_cell] at hh
-  exact hh
-
-lemma rowLen_zero_le_card (μ : YoungDiagram) : μ.rowLen 0 ≤ μ.card := by
-  have h := Fintype.card_subtype_le (fun x : Cell μ => row x=0)
-  rwa [Fintype.card_congr (rowEquiv μ 0),Fintype.card_fin,card_cell] at h
-
-noncomputable def tailColumnEquiv (μ : YoungDiagram) (j : Fin (μ.rowLen 0)) :
-    {x : Tail μ // colIndex x.1 = j} ≃ Fin (μ.colLen j-1) where
-  toFun x := ⟨row x.1.1-1, by
-    have hx := row_lt_colLen x.1.1
-    have hc : col x.1.1 = j := congrArg Fin.val x.2
-    rw [hc] at hx
-    have hz := x.1.2
-    omega⟩
-  invFun i := ⟨⟨⟨(i+1,j),YoungDiagram.mem_iff_lt_colLen.mpr (by omega)⟩,by simp [row]⟩,rfl⟩
-  left_inv x := by
-    apply Subtype.ext
-    apply Subtype.ext
-    apply Subtype.ext
-    apply Prod.ext
-    · change row x.1.1 - 1 + 1 = row x.1.1
-      have := x.1.2
-      omega
-    · exact (congrArg Fin.val x.2).symm
-  right_inv i := by apply Fin.ext; simp [row]
-
-lemma card_tail_col (μ : YoungDiagram) (j : Fin (μ.rowLen 0)) :
-    Fintype.card {x : Tail μ // colIndex x.1 = j} = μ.colLen j-1 := by
-  rw [Fintype.card_congr (tailColumnEquiv μ j),Fintype.card_fin]
-
-lemma card_tail_row (μ : YoungDiagram) (i : Fin (μ.colLen 0)) :
-    Fintype.card {x : Tail μ // rowIndex x.1 = i} = if i.val=0 then 0 else μ.rowLen i := by
-  by_cases hi : i.val=0
-  · rw [ite_eq_left hi]
-    apply Fintype.card_eq_zero_iff.mpr
-    refine ⟨fun x => x.1.2 ?_⟩
-    exact (congrArg Fin.val x.2).trans hi
-  · rw [ite_eq_right hi]
-    let e : {x : Tail μ // rowIndex x.1 = i} ≃ {x : Cell μ // rowIndex x=i} :=
-      { toFun := fun x => ⟨x.1.1,x.2⟩
-        invFun := fun x => ⟨⟨x.1,fun hz => hi ((congrArg Fin.val x.2).symm.trans hz)⟩,x.2⟩
-        left_inv := fun x => rfl
-        right_inv := fun x => rfl }
-    rw [Fintype.card_congr e,card_row_fiber]
-
-lemma tail_row_group (μ : YoungDiagram) :
-    (μ.rowLen 0).factorial * Fintype.card (fiberGroup (fun x : Tail μ => rowIndex x.1)) =
-      Fintype.card (rowGroup μ) := by
-  rw [←Nat.card_eq_fintype_card (α := fiberGroup (fun x : Tail μ => rowIndex x.1)),nat_card_fiberGroup,card_rowGroup]
-  simp only [Nat.card_eq_fintype_card]
-  simp only [card_tail_row]
-  by_cases hn : μ.colLen 0=0
-  · have hr : μ.rowLen 0=0 := by
-      by_contra h
-      have hmem : (0,0) ∈ μ := YoungDiagram.mem_iff_lt_rowLen.mpr (Nat.pos_of_ne_zero h)
-      have hh := YoungDiagram.mem_iff_lt_colLen.mp hmem
-      omega
-    let : IsEmpty (Fin (μ.colLen 0)) := ⟨fun i => by have := i.isLt; omega⟩
-    simp [hr]
-  · let z : Fin (μ.colLen 0) := ⟨0,Nat.pos_of_ne_zero hn⟩
-    have hz : ∀ i : Fin (μ.colLen 0), i.val=0 ↔ i=z := fun i => by simp [z,Fin.ext_iff]
-    simp only [hz]
-    have hprod : (∏ x : Fin (μ.colLen 0), (if x=z then 0 else μ.rowLen x).factorial) =
-        ∏ x ∈ Finset.univ.erase z, (μ.rowLen x).factorial := by
-      rw [←Finset.prod_erase_mul Finset.univ (fun x : Fin (μ.colLen 0) => (if x=z then 0 else μ.rowLen x).factorial) (Finset.mem_univ z)]
-      simp only [ite_true,Nat.factorial_zero,mul_one]
-      apply Finset.prod_congr rfl
-      intro x hx
-      rw [ite_eq_right (Finset.mem_erase.mp hx).1]
-    rw [hprod]
-    exact Finset.mul_prod_erase Finset.univ (fun i : Fin (μ.colLen 0) => (μ.rowLen i).factorial) (Finset.mem_univ z)
-
-lemma sum_tail_columns (μ : YoungDiagram) :
-    ∑ j : Fin (μ.rowLen 0), (μ.colLen j-1) = μ.card-μ.rowLen 0 := by
-  simp_rw [←card_tail_col]
-  rw [←Fintype.card_sigma,Fintype.card_congr (Equiv.sigmaFiberEquiv (fun x : Tail μ => colIndex x.1)),tail_card]
-
-lemma column_pos (μ : YoungDiagram) (j : Fin (μ.rowLen 0)) : 0 < μ.colLen j :=
-  YoungDiagram.mem_iff_lt_colLen.mp (YoungDiagram.mem_iff_lt_rowLen.mpr j.isLt)
-
-lemma nat_le_two_pow_pred (n : ℕ) (hn : 0<n) : n ≤ 2^(n-1) := by
-  obtain ⟨n,rfl⟩ := Nat.exists_eq_succ_of_ne_zero hn.ne'
-  simp only [Nat.add_one_sub_one]
-  induction n with
-  | zero => norm_num
-  | succ n ih => rw [pow_succ]; omega
-
-lemma column_group_le_tail (μ : YoungDiagram) :
-    Fintype.card (colGroup μ) ≤ 2^(μ.card-μ.rowLen 0) *
-      Fintype.card (fiberGroup (fun x : Tail μ => colIndex x.1)) := by
-  rw [card_colGroup,←Nat.card_eq_fintype_card (α := fiberGroup (fun x : Tail μ => colIndex x.1)),nat_card_fiberGroup]
-  simp only [Nat.card_eq_fintype_card]
-  simp only [card_tail_col]
-  calc
-    _ = ∏ j : Fin (μ.rowLen 0), μ.colLen j * (μ.colLen j-1).factorial := by
-      apply Finset.prod_congr rfl
-      intro j hj
-      exact (Nat.mul_factorial_pred (column_pos μ j).ne').symm
-    _ ≤ ∏ j : Fin (μ.rowLen 0), 2^(μ.colLen j-1) * (μ.colLen j-1).factorial := by
-      apply Finset.prod_le_prod
-      intro j hj
-      exact Nat.mul_le_mul_right _ (nat_le_two_pow_pred _ (column_pos μ j))
-    _ = _ := by rw [Finset.prod_mul_distrib,Finset.prod_pow_eq_pow_sum,sum_tail_columns]
-
-theorem choose_tail_le_dimension (μ : YoungDiagram) :
-    μ.card.choose (μ.card-μ.rowLen 0) ≤ Module.finrank ℂ (space μ) * 2^(μ.card-μ.rowLen 0) := by
-  let j := μ.card-μ.rowLen 0
-  have htail := card_fiberGroups_mul_le
-    (fun x : Tail μ => rowIndex x.1) (fun x : Tail μ => colIndex x.1)
-    (fun x y hr hc => Subtype.ext (Subtype.ext (Prod.ext (congrArg Fin.val hr) (congrArg Fin.val hc))))
-  rw [tail_card] at htail
-  have hc := column_group_le_tail μ
-  have hr := tail_row_group μ
-  have hd := factorial_le_dimension_row_column μ
-  simp only [←Nat.card_eq_fintype_card] at htail hc hr hd
-  have hh : μ.card.factorial ≤ Module.finrank ℂ (space μ) * 2^j *
-      ((μ.rowLen 0).factorial * j.factorial) := by
-    calc
-      _ ≤ _ := hd
-      _ ≤ Module.finrank ℂ (space μ) * Nat.card (rowGroup μ) *
-          (2^j * Nat.card (fiberGroup (fun x : Tail μ => colIndex x.1))) := Nat.mul_le_mul_left _ hc
-      _ = Module.finrank ℂ (space μ) * 2^j * (μ.rowLen 0).factorial *
-          (Nat.card (fiberGroup (fun x : Tail μ => rowIndex x.1)) *
-           Nat.card (fiberGroup (fun x : Tail μ => colIndex x.1))) := by rw [←hr]; ring
-      _ ≤ _ := by
-        calc
-          _ ≤ Module.finrank ℂ (space μ) * 2^j * (μ.rowLen 0).factorial * j.factorial :=
-            Nat.mul_le_mul_left _ htail
-          _ = _ := by ring
-  have hj : j ≤ μ.card := Nat.sub_le _ _
-  have hs : μ.card-j = μ.rowLen 0 := by dsimp [j]; have := rowLen_zero_le_card μ; omega
-  have hf := Nat.choose_mul_factorial_mul_factorial hj
-  rw [hs] at hf
-  rw [←hf] at hh
-  have hpos : 0 < (μ.rowLen 0).factorial * j.factorial := Nat.mul_pos (Nat.factorial_pos _) (Nat.factorial_pos _)
-  nlinarith
-
-
-lemma exists_perm_mem_iff {α : Type*} [Fintype α] (s t : Finset α) (hc : s.card=t.card) :
-    ∃ p : Equiv.Perm α, ∀ x, p x ∈ t ↔ x ∈ s := by
-  let e : s ≃ t := Finset.equivOfCardEq hc
-  let ec : {x : α // x ∉ s} ≃ {x : α // x ∉ t} := Fintype.equivOfCardEq (by
-    simp only [Fintype.card_subtype_compl,Fintype.card_coe,hc])
-  let p := (Equiv.sumCompl (· ∈ s)).symm.trans ((Equiv.sumCongr e ec).trans (Equiv.sumCompl (· ∈ t)))
-  refine ⟨p,fun x => ?_⟩
-  by_cases hx : x ∈ s
-  · have hp : p x ∈ t := by
-      change ((Equiv.sumCompl (· ∈ t)) ((Equiv.sumCongr e ec) ((Equiv.sumCompl (· ∈ s)).symm x))) ∈ t
-      rw [Equiv.sumCompl_symm_apply_of_pos hx]
-      exact (e ⟨x,hx⟩).2
-    exact iff_of_true hp hx
-  · have hp : p x ∉ t := by
-      change ((Equiv.sumCompl (· ∈ t)) ((Equiv.sumCongr e ec) ((Equiv.sumCompl (· ∈ s)).symm x))) ∉ t
-      rw [Equiv.sumCompl_symm_apply_of_neg hx]
-      exact (ec ⟨x,hx⟩).2
-    exact iff_of_false hp hx
-
-noncomputable def tailSet {μ : YoungDiagram} (f : Tabloid μ) : Finset (Cell μ) :=
-  Finset.univ.filter (fun x => (f.1 x).val ≠ 0)
-
-@[simp] lemma mem_tailSet {μ : YoungDiagram} (f : Tabloid μ) (x : Cell μ) :
-    x ∈ tailSet f ↔ (f.1 x).val ≠ 0 := by simp [tailSet]
-
-lemma tailSet_act {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) (f : Tabloid μ) :
-    tailSet (tabloidAct p f) = (tailSet f).image p := by
-  ext x
-  simp only [mem_tailSet,tabloidAct_apply,Finset.mem_image]
-  constructor
-  · intro h; exact ⟨p⁻¹ x,h,by simp⟩
-  · rintro ⟨y,hy,rfl⟩; simpa using hy
-
-lemma card_tailSet {μ : YoungDiagram} (f : Tabloid μ) :
-    (tailSet f).card = μ.card-μ.rowLen 0 := by
-  obtain ⟨p,rfl⟩ := tabloidAct_transitive μ (baseTabloid μ) f
-  rw [tailSet_act,Finset.card_image_of_injective _ p.injective]
-  change (Finset.univ.filter (fun x : Cell μ => row x ≠ 0)).card = _
-  rw [←Fintype.card_subtype]
-  exact tail_card μ
-
-noncomputable def footprint (μ : YoungDiagram) (A : Finset (Cell μ)) : Finset (Tabloid μ) :=
-  Finset.univ.filter (fun f => tailSet f ⊆ A)
-
-lemma tail_fiber_card_eq (μ : YoungDiagram) (s t : Finset (Cell μ)) (hst : s.card=t.card) :
-    (Finset.univ.filter (fun f : Tabloid μ => tailSet f=s)).card =
-      (Finset.univ.filter (fun f : Tabloid μ => tailSet f=t)).card := by
-  obtain ⟨p,hp⟩ := exists_perm_mem_iff s t hst
-  have he : s.image p=t := by
-    ext x
-    constructor
-    · intro hx
-      obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
-      exact (hp y).mpr hy
-    · intro hx
-      exact Finset.mem_image.mpr ⟨p⁻¹ x,(hp (p⁻¹ x)).mp (by simpa using hx),by simp⟩
-  apply Finset.card_equiv (tabloidAct p)
-  intro f
-  simp only [Finset.mem_filter,Finset.mem_univ,true_and,tailSet_act,←he]
-  exact (Finset.image_inj p.injective).symm
-
-lemma footprint_card_product (μ : YoungDiagram) (A : Finset (Cell μ)) :
-    (footprint μ A).card * μ.card.choose (μ.card-μ.rowLen 0) =
-      Fintype.card (Tabloid μ) * A.card.choose (μ.card-μ.rowLen 0) := by
-  let j := μ.card-μ.rowLen 0
-  let s := tailSet (baseTabloid μ)
-  let c := (Finset.univ.filter (fun f : Tabloid μ => tailSet f=s)).card
-  have hconstant (t : Finset (Cell μ)) (ht : t.card=j) :
-      (Finset.univ.filter (fun f : Tabloid μ => tailSet f=t)).card = c :=
-    tail_fiber_card_eq μ t s (ht.trans (card_tailSet _).symm)
-  have hcount (A : Finset (Cell μ)) : (footprint μ A).card = A.card.choose j*c := by
-    have h := Finset.sum_card_fiberwise_eq_card_filter
-      (Finset.univ : Finset (Tabloid μ)) (A.powersetCard j) tailSet
-    have he : (Finset.univ.filter (fun f : Tabloid μ => tailSet f ∈ A.powersetCard j)) = footprint μ A := by
-      ext f
-      simp only [Finset.mem_filter,Finset.mem_univ,true_and,Finset.mem_powersetCard,
-        card_tailSet,footprint]
-      simp [j]
-    rw [he] at h
-    rw [←h]
-    calc
-      _ = ∑ _t ∈ A.powersetCard j, c := by
-        apply Finset.sum_congr rfl
-        intro t ht
-        exact hconstant t (Finset.mem_powersetCard.mp ht).2
-      _ = _ := by simp [Finset.card_powersetCard]
-  have hfull : (footprint μ Finset.univ).card = Fintype.card (Tabloid μ) := by
-    simp only [footprint,Finset.subset_univ,Finset.filter_true,Finset.card_univ]
-  have hN := hcount Finset.univ
-  rw [hfull,Finset.card_univ,card_cell] at hN
-  rw [hcount,hN]
-  ring
-
-lemma footprint_fixed {μ : YoungDiagram} (A : Finset (Cell μ))
-    (f : Tabloid μ) (hf : f ∈ footprint μ A) (p : Equiv.Perm (Cell μ))
-    (hp : ∀ x ∈ A, p x=x) : tabloidAct p f=f := by
-  have htail : tailSet f ⊆ A := (Finset.mem_filter.mp hf).2
-  apply Subtype.ext
-  funext x
-  change f.1 (p⁻¹ x) = f.1 x
-  by_cases hx : x ∈ A
-  · rw [show p⁻¹ x=x from p.injective (by simpa using (hp x hx).symm)]
-  · have hy : p⁻¹ x ∉ A := by
-      intro h
-      have hh := hp (p⁻¹ x) h
-      have he : x=p⁻¹ x := by simpa using hh
-      exact hx (he ▸ h)
-    apply Fin.ext
-    have hx0 : (f.1 x).val=0 := by by_contra h; exact hx (htail (mem_tailSet f x |>.mpr h))
-    have hy0 : (f.1 (p⁻¹ x)).val=0 := by by_contra h; exact hy (htail (mem_tailSet f _ |>.mpr h))
-    exact hy0.trans hx0.symm
-
-noncomputable def fixedHilbert (μ : YoungDiagram) (A : Finset (Cell μ)) :
-    Submodule ℂ (PermutationHilbert.H (Tabloid μ)) where
-  carrier := {v | v ∈ hilbertSpace μ ∧ ∀ p : Equiv.Perm (Cell μ),
-    (∀ x ∈ A, p x=x) → PermutationHilbert.act (tabloidAct p) v=v}
-  zero_mem' := ⟨Submodule.zero_mem _,fun p hp => map_zero _⟩
-  add_mem' := fun hv hw => ⟨Submodule.add_mem _ hv.1 hw.1,fun p hp => by
-    rw [map_add,hv.2 p hp,hw.2 p hp]⟩
-  smul_mem' := fun c v hv => ⟨Submodule.smul_mem _ c hv.1,fun p hp => by
-    rw [map_smul,hv.2 p hp]⟩
-
-lemma fixedHilbert_le (μ : YoungDiagram) (A : Finset (Cell μ)) : fixedHilbert μ A ≤ hilbertSpace μ :=
-  fun _ hv => hv.1
-
-theorem fixedHilbert_dimension (μ : YoungDiagram) (A : Finset (Cell μ)) :
-    Module.finrank ℂ (space μ) * A.card.choose (μ.card-μ.rowLen 0) ≤
-      Module.finrank ℂ (fixedHilbert μ A) * μ.card.choose (μ.card-μ.rowLen 0) := by
-  let : DecidableEq (Tabloid μ) := Classical.decEq _
-  have ht (f g : Tabloid μ) : ∃ p : Equiv.Perm (Tabloid μ), p f=g ∧
-      (hilbertSpace μ).map (PermutationHilbert.act p).toLinearEquiv.toLinearMap = hilbertSpace μ := by
-    obtain ⟨p,hp⟩ := tabloidAct_transitive μ f g
-    exact ⟨tabloidAct p,hp,hilbertSpace_invariant μ p⟩
-  have hproj (f : Tabloid μ) (hf : f ∈ footprint μ A) :
-      (hilbertSpace μ).starProjection (EuclideanSpace.single f (1:ℂ)) ∈ fixedHilbert μ A := by
-    refine ⟨((hilbertSpace μ).orthogonalProjectionOnto _).2,fun p hp => ?_⟩
-    rw [PermutationHilbert.projection_comm _ _ (hilbertSpace_invariant μ p),
-      PermutationHilbert.act_single,footprint_fixed A f hf p hp]
-  have h := PermutationHilbert.footprint_dimension (hilbertSpace μ) (fixedHilbert μ A)
-    (fixedHilbert_le μ A) ht (footprint μ A) hproj
-  rw [finrank_hilbertSpace] at h
-  have hn : (footprint μ A).card * Module.finrank ℂ (space μ) ≤
-      Fintype.card (Tabloid μ)*Module.finrank ℂ (fixedHilbert μ A) := by exact_mod_cast h
-  have hc := footprint_card_product μ A
-  have hpos : 0 < Fintype.card (Tabloid μ) := Fintype.card_pos_iff.mpr ⟨baseTabloid μ⟩
-  have hh := Nat.mul_le_mul_right (μ.card.choose (μ.card-μ.rowLen 0)) hn
-  nlinarith
-
-end Thorp.Specht
-
-namespace Thorp.UnitaryFinite
-open scoped BigOperators ComplexConjugate Classical
-
-variable {G : Type*} [Group G] [Fintype G]
-variable {V W : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-  [NormedAddCommGroup W] [InnerProductSpace ℂ W] [FiniteDimensional ℂ W]
-
-def IsUnitary (ρ : Representation ℂ G V) : Prop :=
-  ∀ g v w, inner ℂ (ρ g v) (ρ g w) = inner ℂ v w
-
-omit [Fintype G] [FiniteDimensional ℂ V] in
-lemma unitary_inner_inv (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (g : G) (v w : V) :
-    inner ℂ v (ρ g⁻¹ w) = inner ℂ (ρ g v) w := by
-  have h := hρ g v (ρ g⁻¹ w)
-  have he : ρ g (ρ g⁻¹ w) = w := by
-    change (ρ g * ρ g⁻¹) w = w
-    rw [←map_mul,mul_inv_cancel,map_one]
-    rfl
-  rw [he] at h
-  exact h.symm
-
-noncomputable def averageMap (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
-    (A : V →ₗ[ℂ] W) : Representation.IntertwiningMap ρ σ where
-  toLinearMap := ∑ g, (σ g).comp (A.comp (ρ g⁻¹))
-  isIntertwining' h := by
-    ext v
-    simp only [LinearMap.comp_apply,LinearMap.sum_apply,map_sum]
-    apply Fintype.sum_equiv (Equiv.mulLeft h⁻¹)
-    intro g
-    simp only [Equiv.coe_mulLeft,mul_inv_rev,inv_inv,map_mul,Module.End.mul_apply]
-    have hh (w : W) : σ h (σ h⁻¹ w)=w := by
-      change (σ h*σ h⁻¹) w=w
-      rw [←map_mul,mul_inv_cancel,map_one]
-      rfl
-    rw [hh]
-
-end Thorp.UnitaryFinite
-
-end ThorpNine.SparseSaving
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.SparseSaving.ModifiedReverseEquiv
+
+namespace OAI
+
+namespace ThorpNine.SparseSaving
+
+namespace Thorp.Specht
+open scoped BigOperators Classical
+
+noncomputable def shapePartitionEquiv (N : ℕ) : Shapes N ≃ Nat.Partition N where
+  toFun μ := { parts := μ.1.rowLens
+               parts_pos := fun {i} hi => μ.1.pos_of_mem_rowLens i (by simpa using hi)
+               parts_sum := by simpa using (sum_rowLens μ.1).trans μ.2 }
+  invFun p := ⟨partitionDiagram p,partitionDiagram_card p⟩
+  left_inv μ := by
+    apply Subtype.ext
+    obtain ⟨μ,rfl⟩ := μ
+    exact partitionDiagram_diagramPartition μ
+  right_inv p := by
+    apply Nat.Partition.ext
+    change ((partitionDiagram p).rowLens : Multiset ℕ) = p.parts
+    rw [partitionDiagram_rowLens,Multiset.sort_eq]
+
+noncomputable instance (N : ℕ) : Fintype (Shapes N) := Fintype.ofEquiv _ (shapePartitionEquiv N).symm
+
+noncomputable def conjugacyPartition {α : Type*} [Fintype α] [DecidableEq α] :
+    ConjClasses (Equiv.Perm α) → Nat.Partition (Fintype.card α) :=
+  Quotient.lift Equiv.Perm.partition (fun _ _ hpq => (Equiv.Perm.partition_eq_of_isConj).mp hpq)
+
+lemma conjugacyPartition_injective {α : Type*} [Fintype α] [DecidableEq α] :
+    Function.Injective (conjugacyPartition (α := α)) := by
+  intro p q
+  induction p using Quotient.inductionOn with | _ p =>
+    induction q using Quotient.inductionOn with | _ q =>
+      intro h
+      exact Quotient.sound ((Equiv.Perm.partition_eq_of_isConj).mpr h)
+
+lemma card_conjugacy_le_shapes (N : ℕ) :
+    Nat.card (ConjClasses (Equiv.Perm (Fin N))) ≤ Fintype.card (Shapes N) := by
+  have h := Nat.card_le_card_of_injective _ (conjugacyPartition_injective (α := Fin N))
+  simpa only [Fintype.card_fin, Nat.card_eq_fintype_card, Fintype.card_congr (shapePartitionEquiv N)] using h
+
+noncomputable def labels {N : ℕ} (μ : Shapes N) : Cell μ.1 ≃ Fin N :=
+  (Fintype.equivFin (Cell μ.1)).trans (finCongr ((card_cell μ.1).trans μ.2))
+
+noncomputable def labelledRepresentation {N : ℕ} (μ : Shapes N) :
+    Representation ℂ (Equiv.Perm (Fin N)) (space μ.1) :=
+  (representation μ.1).comp (labels μ).symm.permCongrHom.toMonoidHom
+
+lemma labelledRepresentation_irreducible {N : ℕ} (μ : Shapes N) :
+    Representation.IsIrreducible (labelledRepresentation μ) := by
+  let e := (labels μ).symm.permCongrHom
+  let ρ := representation μ.1
+  have hn : (⊥ : Subrepresentation (labelledRepresentation μ)) ≠ ⊤ := by
+    intro h
+    have he : (⟨polytabloid μ.1,polytabloid_mem_space μ.1⟩ : space μ.1) ∈
+        (⊥ : Subrepresentation (labelledRepresentation μ)) := by rw [h]; trivial
+    exact polytabloid_ne_zero μ.1 (congrArg Subtype.val he)
+  refine { exists_pair_ne := ⟨⊥,⊤,hn⟩,eq_bot_or_eq_top := ?_ }
+  intro U
+  let U' : Subrepresentation ρ :=
+    { toSubmodule := U.toSubmodule
+      apply_mem_toSubmodule := fun p v hv => by
+        have hh := U.apply_mem_toSubmodule (e.symm p) hv
+        change ρ (e (e.symm p)) v ∈ U.toSubmodule at hh
+        simpa using hh }
+  have := representation_irreducible μ.1
+  obtain h | h := IsSimpleOrder.eq_bot_or_eq_top U'
+  · left
+    apply Subrepresentation.toSubmodule_injective
+    exact congrArg (fun R : Subrepresentation ρ => R.toSubmodule) h
+  · right
+    apply Subrepresentation.toSubmodule_injective
+    exact congrArg (fun R : Subrepresentation ρ => R.toSubmodule) h
+
+lemma labelledRepresentation_equiv_shapes {N : ℕ} (μ ν : Shapes N)
+    (T : Representation.Equiv (labelledRepresentation μ) (labelledRepresentation ν)) : μ = ν := by
+  apply Subtype.ext
+  apply shapes_eq_of_equivariant_equiv μ.1 ν.1 ((labels μ).trans (labels ν).symm) T.toLinearEquiv
+  intro p v
+  have hh := Representation.IntertwiningMap.isIntertwining _ _ T.toIntertwiningMap ((labels μ).permCongr p) v
+  change T.toLinearEquiv
+    (representation μ.1 ((labels μ).symm.permCongr ((labels μ).permCongr p)) v) =
+      representation ν.1 ((labels ν).symm.permCongr ((labels μ).permCongr p)) (T.toLinearEquiv v) at hh
+  have hp : (labels μ).symm.permCongr ((labels μ).permCongr p) = p := by
+    apply Equiv.ext
+    intro x
+    simp [Equiv.permCongr_apply]
+  rw [hp] at hh
+  convert hh using 2
+  congr 1
+
+end Thorp.Specht
+
+namespace Thorp.FiniteCharacters
+open scoped BigOperators Classical
+
+section Group
+variable {G : Type*} [Group G] [Fintype G]
+
+noncomputable def classRep (c : ConjClasses G) : G := (ConjClasses.exists_rep c).choose
+omit [Fintype G] in
+lemma classRep_spec (c : ConjClasses G) : ConjClasses.mk (classRep c) = c :=
+  (ConjClasses.exists_rep c).choose_spec
+
+variable {V : Type*} [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
+
+noncomputable def characterClass (ρ : Representation ℂ G V) : ConjClasses G → ℂ :=
+  fun c => ρ.character (classRep c)
+
+omit [Fintype G] [FiniteDimensional ℂ V] in
+lemma characterClass_mk (ρ : Representation ℂ G V) (g : G) :
+    characterClass ρ (ConjClasses.mk g) = ρ.character g := by
+  have h := ConjClasses.mk_eq_mk_iff_isConj.mp (classRep_spec (ConjClasses.mk g))
+  obtain ⟨p,hp⟩ := isConj_iff.mp h
+  change ρ.character (classRep (ConjClasses.mk g)) = ρ.character g
+  exact (Representation.char_conj ρ _ p).symm.trans (congrArg ρ.character hp)
+
+noncomputable def characterDual (ρ : Representation ℂ G V) :
+    (ConjClasses G → ℂ) →ₗ[ℂ] ℂ where
+  toFun f := (Nat.card G : ℂ)⁻¹ * ∑ g, f (ConjClasses.mk g) * ρ.character g⁻¹
+  map_add' f h := by simp [Finset.sum_add_distrib,add_mul,mul_add]
+  map_smul' a f := by
+    simp only [Pi.smul_apply,smul_eq_mul,RingHom.id_apply]
+    simp_rw [mul_assoc]
+    rw [←Finset.mul_sum]
+    ring
+
+lemma characterDual_characterClass (ρ : Representation ℂ G V)
+    {W : Type*} [AddCommGroup W] [Module ℂ W] [FiniteDimensional ℂ W]
+    (σ : Representation ℂ G W) [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ] :
+    characterDual ρ (characterClass σ) = if Nonempty (Representation.Equiv ρ σ) then 1 else 0 := by
+  let : Invertible (Nat.card G : ℂ) := invertibleOfNonzero (by
+    exact_mod_cast (Nat.card_pos (α := G)).ne')
+  change (Nat.card G : ℂ)⁻¹ * ∑ g, characterClass σ (ConjClasses.mk g) * ρ.character g⁻¹ = _
+  simp only [characterClass_mk]
+  exact Representation.char_orthonormal σ ρ
+
+end Group
+
+theorem complete_of_card_conjugacy_le
+    {G ι : Type*} [Group G] [Fintype G] [Fintype ι]
+    (V : ι → Type*) [∀ i, AddCommGroup (V i)] [∀ i, Module ℂ (V i)]
+    [∀ i, FiniteDimensional ℂ (V i)]
+    (ρ : ∀ i, Representation ℂ G (V i)) [∀ i, Representation.IsIrreducible (ρ i)]
+    (hneq : ∀ i j, i ≠ j → ¬Nonempty (Representation.Equiv (ρ i) (ρ j)))
+    (hcard : Nat.card (ConjClasses G) ≤ Fintype.card ι)
+    {W : Type*} [AddCommGroup W] [Module ℂ W] [FiniteDimensional ℂ W]
+    (σ : Representation ℂ G W) [Representation.IsIrreducible σ] :
+    ∃ i, Nonempty (Representation.Equiv (ρ i) σ) := by
+  let : Fintype (ConjClasses G) := Fintype.ofFinite _
+  have horth (i j : ι) : characterDual (ρ i) (characterClass (ρ j)) = if i=j then 1 else 0 := by
+    rw [characterDual_characterClass]
+    by_cases hij : i=j
+    · subst j
+      simp only [ite_true]
+      exact ite_eq_left ⟨Representation.Equiv.refl _⟩
+    · rw [ite_eq_right hij,ite_eq_right (hneq i j hij)]
+  have hLI : LinearIndependent ℂ (fun i => characterClass (ρ i)) := by
+    apply LinearIndependent.of_pairwise_dual_eq_zero_one _ (fun i => characterDual (ρ i))
+    · intro i j hij
+      exact (horth i j).trans (ite_eq_right hij)
+    · intro i
+      simpa using horth i i
+  have heq : Fintype.card ι = Module.finrank ℂ (ConjClasses G → ℂ) := by
+    apply Nat.le_antisymm hLI.fintype_card_le_finrank
+    simpa only [Module.finrank_pi,Module.finrank_self,Finset.sum_const,Finset.card_univ,
+      smul_eq_mul,mul_one,Nat.card_eq_fintype_card] using hcard
+  have hspan := hLI.span_eq_top_of_card_eq_finrank' heq
+  by_contra h
+  push Not at h
+  have hker : Submodule.span ℂ (Set.range (fun i => characterClass (ρ i))) ≤
+      (characterDual σ).ker := by
+    apply Submodule.span_le.mpr
+    rintro f ⟨i,rfl⟩
+    change characterDual σ (characterClass (ρ i)) = 0
+    rw [characterDual_characterClass,ite_eq_right]
+    rintro ⟨T⟩
+    exact (h i).false T.symm
+  rw [hspan] at hker
+  have hz : characterDual σ (characterClass σ) = 0 := hker (show characterClass σ ∈ ⊤ from trivial)
+  rw [characterDual_characterClass,ite_eq_left ⟨Representation.Equiv.refl σ⟩] at hz
+  exact one_ne_zero hz
+
+end Thorp.FiniteCharacters
+
+namespace Thorp.Specht
+open scoped Classical
+
+theorem labelled_complete {N : ℕ}
+    {V : Type*} [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
+    (ρ : Representation ℂ (Equiv.Perm (Fin N)) V) [Representation.IsIrreducible ρ] :
+    ∃ μ : Shapes N, Nonempty (Representation.Equiv (labelledRepresentation μ) ρ) := by
+  let (μ : Shapes N) : Representation.IsIrreducible (labelledRepresentation μ) :=
+    labelledRepresentation_irreducible μ
+  apply FiniteCharacters.complete_of_card_conjugacy_le
+    (fun μ : Shapes N => space μ.1) labelledRepresentation _ (card_conjugacy_le_shapes N) ρ
+  intro μ ν h ⟨T⟩
+  exact h (labelledRepresentation_equiv_shapes μ ν T)
+
+end Thorp.Specht
+
+namespace Thorp.PermutationHilbert
+
+section
+open scoped BigOperators ComplexConjugate Classical
+open Complex
+
+variable {X : Type*} [Fintype X]
+
+noncomputable def act (p : Equiv.Perm X) : H X ≃ₗᵢ[ℂ] H X :=
+  LinearIsometryEquiv.piLpCongrLeft 2 ℂ ℂ p
+
+@[simp] lemma act_apply (p : Equiv.Perm X) (v : H X) (x : X) :
+    act p v x = v (p⁻¹ x) := rfl
+
+lemma act_single [DecidableEq X] (p : Equiv.Perm X) (x : X) :
+    act p (EuclideanSpace.single x (1:ℂ)) = EuclideanSpace.single (p x) 1 := by
+  exact EuclideanSpace.piLpCongrLeft_single p x 1
+
+lemma trace_projection (W : Submodule ℂ (H X)) :
+    ∑ x, (W.starProjection (EuclideanSpace.single x (1:ℂ))) x = Module.finrank ℂ W := by
+  classical
+  have hp : LinearMap.IsProj W W.starProjection.toLinearMap := by
+    constructor
+    · intro x; exact (W.orthogonalProjectionOnto x).2
+    · intro x hx; exact W.starProjection_eq_self_iff.mpr hx
+  have ht := hp.trace
+  rw [LinearMap.trace_eq_matrix_trace ℂ (EuclideanSpace.basisFun X ℂ).toBasis] at ht
+  simpa only [Matrix.trace,Matrix.diag_apply,LinearMap.toMatrix_apply,EuclideanSpace.basisFun_repr,
+    EuclideanSpace.basisFun_apply,OrthonormalBasis.coe_toBasis,
+    OrthonormalBasis.coe_toBasis_repr_apply,ContinuousLinearMap.coe_coe] using ht
+
+lemma projection_diagonal (W : Submodule ℂ (H X)) (x : X) :
+    (W.starProjection (EuclideanSpace.single x (1:ℂ))) x =
+      (‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 : ℝ) := by
+  classical
+  have hh := W.inner_starProjection_left_eq_right
+    (EuclideanSpace.single x (1:ℂ)) (W.starProjection (EuclideanSpace.single x (1:ℂ)))
+  have hf : W.starProjection (W.starProjection (EuclideanSpace.single x (1:ℂ))) =
+      W.starProjection (EuclideanSpace.single x (1:ℂ)) :=
+    W.starProjection_eq_self_iff.mpr (W.orthogonalProjectionOnto _).2
+  rw [hf] at hh
+  rw [inner_self_eq_norm_sq_to_K,EuclideanSpace.inner_single_left] at hh
+  simpa using hh.symm
+
+lemma projection_comm (W : Submodule ℂ (H X)) (p : Equiv.Perm X)
+    (hW : W.map (act p).toLinearEquiv.toLinearMap = W) (v : H X) :
+    act p (W.starProjection v) = W.starProjection (act p v) := by
+  have hh := (act p).toLinearIsometry.map_starProjection W v
+  change act p (W.starProjection v) =
+    (W.map (act p).toLinearEquiv.toLinearMap).starProjection (act p v) at hh
+  simpa only [hW] using hh
+
+lemma uniform_diagonal (W : Submodule ℂ (H X))
+    (htrans : ∀ x y : X, ∃ p : Equiv.Perm X, p x = y ∧
+      W.map (act p).toLinearEquiv.toLinearMap = W) (x : X) :
+    (Fintype.card X:ℝ) * ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 =
+      Module.finrank ℂ W := by
+  classical
+  have hh (y : X) : (W.starProjection (EuclideanSpace.single y (1:ℂ))) y =
+      (‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 : ℝ) := by
+    obtain ⟨p,hp,hW⟩ := htrans x y
+    subst y
+    rw [projection_diagonal,←act_single p x,←projection_comm W p hW,(act p).norm_map]
+  have ht := trace_projection W
+  simp_rw [hh] at ht
+  have hhreal : (∑ _y : X, ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2) =
+      (Module.finrank ℂ W:ℝ) := by exact_mod_cast ht
+  simpa using hhreal
+
+theorem uncertainty (W : Submodule ℂ (H X))
+    (htrans : ∀ x y : X, ∃ p : Equiv.Perm X, p x = y ∧
+      W.map (act p).toLinearEquiv.toLinearMap = W)
+    (v : H X) (hv : v ∈ W) (x : X) :
+    (Fintype.card X:ℝ) * ‖v x‖^2 ≤ (Module.finrank ℂ W:ℝ) * ‖v‖^2 := by
+  classical
+  have hh : inner ℂ (W.starProjection (EuclideanSpace.single x (1:ℂ))) v = v x := by
+    rw [W.inner_starProjection_left_eq_right,W.starProjection_eq_self_iff.mpr hv,
+      EuclideanSpace.inner_single_left]
+    simp
+  have hc := norm_inner_le_norm (𝕜 := ℂ) (W.starProjection (EuclideanSpace.single x (1:ℂ))) v
+  rw [hh] at hc
+  have hs : ‖v x‖^2 ≤ ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 * ‖v‖^2 := by
+    nlinarith [norm_nonneg (v x),norm_nonneg v,
+      norm_nonneg (W.starProjection (EuclideanSpace.single x (1:ℂ)))]
+  calc
+    (Fintype.card X:ℝ) * ‖v x‖^2 ≤
+      (Fintype.card X:ℝ) * (‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 * ‖v‖^2) :=
+      mul_le_mul_of_nonneg_left hs (Nat.cast_nonneg _)
+    _ = _ := by rw [←mul_assoc,uniform_diagonal W htrans x]
+
+end
+open scoped BigOperators Classical
+
+variable {X : Type*} [Fintype X]
+
+lemma trace_projection_real (W : Submodule ℂ (H X)) :
+    ∑ x, ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 = (Module.finrank ℂ W:ℝ) := by
+  have hh := trace_projection W
+  simp_rw [projection_diagonal] at hh
+  exact_mod_cast hh
+
+theorem footprint_dimension (W K : Submodule ℂ (H X)) (hKW : K ≤ W)
+    (htrans : ∀ x y : X, ∃ p : Equiv.Perm X, p x = y ∧
+      W.map (act p).toLinearEquiv.toLinearMap = W)
+    (S : Finset X) (hS : ∀ x ∈ S, W.starProjection (EuclideanSpace.single x (1:ℂ)) ∈ K) :
+    (S.card:ℝ) * (Module.finrank ℂ W:ℝ) ≤
+      (Fintype.card X:ℝ) * (Module.finrank ℂ K:ℝ) := by
+  have he (x : X) (hx : x ∈ S) :
+      K.starProjection (EuclideanSpace.single x (1:ℂ)) =
+        W.starProjection (EuclideanSpace.single x (1:ℂ)) := by
+    apply K.eq_starProjection_of_mem_of_inner_eq_zero (hS x hx)
+    intro v hv
+    exact W.starProjection_inner_eq_zero _ v (hKW hv)
+  have hs : ∑ x ∈ S, ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 ≤
+      (Module.finrank ℂ K:ℝ) := by
+    rw [←trace_projection_real K]
+    calc
+      _ = ∑ x ∈ S, ‖K.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 := by
+        apply Finset.sum_congr rfl
+        intro x hx
+        rw [he x hx]
+      _ ≤ _ := Finset.sum_le_univ_sum_of_nonneg (fun x => sq_nonneg _)
+  have hmul := mul_le_mul_of_nonneg_left hs (Nat.cast_nonneg (Fintype.card X): (0:ℝ) ≤ _)
+  rw [Finset.mul_sum] at hmul
+  simp_rw [uniform_diagonal W htrans] at hmul
+  simpa using hmul
+
+end Thorp.PermutationHilbert
+namespace Thorp.Specht
+open scoped BigOperators Classical
+
+lemma finrank_hilbertSpace (μ : YoungDiagram) :
+    Module.finrank ℂ (hilbertSpace μ) = Module.finrank ℂ (space μ) :=
+  (hilbertEquiv μ).finrank_map_eq (space μ)
+
+lemma hilbert_action (μ : YoungDiagram) (p : Equiv.Perm (Cell μ)) (v : Tabloid μ → ℂ) :
+    hilbertEquiv μ (tabloidRep μ p v) =
+      PermutationHilbert.act (tabloidAct p) (hilbertEquiv μ v) := by
+  ext f
+  simp [hilbertEquiv,PermutationHilbert.act_apply]
+
+lemma hilbertSpace_invariant (μ : YoungDiagram) (p : Equiv.Perm (Cell μ)) :
+    (hilbertSpace μ).map (PermutationHilbert.act (tabloidAct p)).toLinearEquiv.toLinearMap =
+      hilbertSpace μ := by
+  apply le_antisymm
+  · rintro _ ⟨v,⟨w,hw,rfl⟩,rfl⟩
+    exact ⟨tabloidRep μ p w,action_mem_space μ p w hw,hilbert_action μ p w⟩
+  · rintro _ ⟨w,hw,rfl⟩
+    refine ⟨hilbertEquiv μ (tabloidRep μ p⁻¹ w),
+      ⟨tabloidRep μ p⁻¹ w,action_mem_space μ p⁻¹ w hw,rfl⟩, ?_⟩
+    change PermutationHilbert.act (tabloidAct p) (hilbertEquiv μ (tabloidRep μ p⁻¹ w)) = hilbertEquiv μ w
+    rw [←hilbert_action]
+    change hilbertEquiv μ ((tabloidRep μ p * tabloidRep μ p⁻¹) w) = _
+    rw [←map_mul,mul_inv_cancel,map_one]
+    rfl
+
+lemma tabloidAct_transitive (μ : YoungDiagram) (f g : Tabloid μ) :
+    ∃ p : Equiv.Perm (Cell μ), tabloidAct p f = g := by
+  obtain ⟨q,hq⟩ := f.2
+  obtain ⟨r,hr⟩ := g.2
+  refine ⟨r⁻¹*q, ?_⟩
+  apply Subtype.ext
+  funext x
+  simp only [tabloidAct_apply,mul_inv_rev,Equiv.Perm.mul_apply,hq,hr,
+    Function.comp_apply,Equiv.Perm.inv_def,Equiv.apply_symm_apply,Equiv.symm_symm]
+
+lemma polytabloid_pairing (μ : YoungDiagram) :
+    pairing (polytabloid μ) (polytabloid μ) = Fintype.card (colGroup μ) := by
+  have h := alternator_pairing μ (polytabloid μ)
+  have he (c : colGroup μ) : tabloidRep μ c (polytabloid μ) =
+      permSign (c:Equiv.Perm (Cell μ)) • polytabloid μ :=
+    action_alternator_apply (colGroup μ) (tabloidRep μ) c _
+  have hb := congrArg (fun v : Tabloid μ → ℂ => v (baseTabloid μ)) h
+  simp only [alternator,LinearMap.sum_apply,LinearMap.smul_apply,he,
+    Pi.smul_apply,smul_eq_mul,Finset.sum_apply,polytabloid_base,mul_one] at hb
+  simp only [permSign_sq,Finset.sum_const,Finset.card_univ,nsmul_eq_mul,mul_one] at hb
+  exact hb.symm
+
+lemma polytabloid_norm_sq (μ : YoungDiagram) :
+    ‖hilbertEquiv μ (polytabloid μ)‖^2 = (Fintype.card (colGroup μ):ℝ) := by
+  have hp := polytabloid_pairing μ
+  change inner ℂ (hilbertEquiv μ (polytabloid μ)) (hilbertEquiv μ (polytabloid μ)) = _ at hp
+  rw [inner_self_eq_norm_sq_to_K] at hp
+  have hre := congrArg Complex.re hp
+  simpa [pow_two] using hre
+
+theorem dimension_times_column_ge_tabloids (μ : YoungDiagram) :
+    Fintype.card (Tabloid μ) ≤ Module.finrank ℂ (space μ) * Fintype.card (colGroup μ) := by
+  have ht (f g : Tabloid μ) : ∃ p : Equiv.Perm (Tabloid μ), p f = g ∧
+      (hilbertSpace μ).map (PermutationHilbert.act p).toLinearEquiv.toLinearMap = hilbertSpace μ := by
+    obtain ⟨p,hp⟩ := tabloidAct_transitive μ f g
+    exact ⟨tabloidAct p,hp,hilbertSpace_invariant μ p⟩
+  have hu := PermutationHilbert.uncertainty (hilbertSpace μ) ht
+    (hilbertEquiv μ (polytabloid μ)) ⟨polytabloid μ,polytabloid_mem_space μ,rfl⟩ (baseTabloid μ)
+  have he : hilbertEquiv μ (polytabloid μ) (baseTabloid μ) = 1 := polytabloid_base μ
+  rw [he,norm_one,one_pow,mul_one,finrank_hilbertSpace,polytabloid_norm_sq] at hu
+  exact_mod_cast hu
+
+
+section FiberGroup
+variable {α β γ : Type*} [Fintype α] [Fintype β] [Fintype γ]
+
+noncomputable def fiberGroupEquiv (f : α → β) :
+    fiberGroup f ≃ (∀ b, Equiv.Perm {x // f x = b}) where
+  toFun p b :=
+    { toFun x := ⟨p.1 x.1,(p.2 x.1).trans x.2⟩
+      invFun x := ⟨p.1⁻¹ x.1, by
+        have hh := p.2 (p.1⁻¹ x.1)
+        simp only [Equiv.Perm.inv_def,Equiv.apply_symm_apply] at hh
+        exact hh.symm.trans x.2⟩
+      left_inv x := by apply Subtype.ext; simp
+      right_inv x := by apply Subtype.ext; simp }
+  invFun q :=
+    ⟨(Equiv.sigmaFiberEquiv f).permCongr (Equiv.Perm.sigmaCongrRight q),
+      fun x => (q (f x) ⟨x,rfl⟩).2⟩
+  left_inv p := by
+    apply Subtype.ext
+    apply Equiv.ext
+    intro x
+    rfl
+  right_inv q := by
+    funext b
+    apply Equiv.ext
+    intro x
+    apply Subtype.ext
+    change (q (f x.1) ⟨x.1,rfl⟩).1 = (q b x).1
+    cases x with
+    | mk x hx => dsimp only at *; subst b; rfl
+
+lemma nat_card_fiberGroup (f : α → β) :
+    Nat.card (fiberGroup f) = ∏ b, (Nat.card {x // f x=b}).factorial := by
+  rw [Nat.card_congr (fiberGroupEquiv f),Nat.card_pi]
+  simp only [Nat.card_eq_fintype_card,Fintype.card_perm]
+
+omit [Fintype β] [Fintype γ] in
+
+lemma card_fiberGroups_mul_le (f : α → β) (g : α → γ)
+    (hinj : ∀ x y, f x = f y → g x = g y → x=y) :
+    Fintype.card (fiberGroup f) * Fintype.card (fiberGroup g) ≤ (Fintype.card α).factorial := by
+  have hi : Function.Injective (fun p : fiberGroup f × fiberGroup g => p.1.1*p.2.1) := by
+    intro p q hpq
+    dsimp only at hpq
+    have he : p.1.1⁻¹*q.1.1 = p.2.1*q.2.1⁻¹ := by
+      apply inv_mul_eq_iff_eq_mul.mpr
+      rw [←mul_assoc,hpq,mul_assoc,mul_inv_cancel,mul_one]
+    have hf : p.1.1⁻¹*q.1.1 ∈ fiberGroup f := (fiberGroup f).mul_mem ((fiberGroup f).inv_mem p.1.2) q.1.2
+    have hg : p.1.1⁻¹*q.1.1 ∈ fiberGroup g := he ▸ (fiberGroup g).mul_mem p.2.2 ((fiberGroup g).inv_mem q.2.2)
+    have h1 : p.1.1⁻¹*q.1.1 = 1 := by
+      apply Equiv.ext
+      intro x
+      exact hinj _ x (hf x) (hg x)
+    have hpf : p.1=q.1 := Subtype.ext (inv_mul_eq_one.mp h1)
+    have hpg : p.2=q.2 := by
+      apply Subtype.ext
+      have hh := hpq
+      rw [show p.1.1=q.1.1 from congrArg Subtype.val hpf] at hh
+      exact mul_left_cancel hh
+    exact Prod.ext hpf hpg
+  have hc := Fintype.card_le_of_injective _ hi
+  simpa only [Fintype.card_prod,Fintype.card_perm] using hc
+
+end FiberGroup
+
+noncomputable def rowGroup (μ : YoungDiagram) : Subgroup (Equiv.Perm (Cell μ)) :=
+  fiberGroup (rowIndex (μ := μ))
+
+noncomputable def colIndex {μ : YoungDiagram} (x : Cell μ) : Fin (μ.rowLen 0) :=
+  ⟨col x, (YoungDiagram.mem_iff_lt_rowLen.mp x.2).trans_le (μ.rowLen_anti 0 (row x) (Nat.zero_le _))⟩
+
+lemma colGroup_eq (μ : YoungDiagram) : colGroup μ = fiberGroup (colIndex (μ := μ)) := by
+  ext p
+  constructor
+  · intro hp x; exact Fin.ext (hp x)
+  · intro hp x; exact congrArg Fin.val (hp x)
+
+noncomputable instance tabloidMulAction (μ : YoungDiagram) : MulAction (Equiv.Perm (Cell μ)) (Tabloid μ) where
+  smul p f := tabloidAct p f
+  one_smul f := by change tabloidAct 1 f = f; rw [tabloidAct_one]; rfl
+  mul_smul p q f := by change tabloidAct (p*q) f = tabloidAct p (tabloidAct q f); rw [tabloidAct_mul]; rfl
+
+lemma tabloid_orbit_all (μ : YoungDiagram) :
+    MulAction.orbit (Equiv.Perm (Cell μ)) (baseTabloid μ) = Set.univ := by
+  ext f
+  simp only [Set.mem_univ,iff_true]
+  obtain ⟨p,hp⟩ := tabloidAct_transitive μ (baseTabloid μ) f
+  exact ⟨p,hp⟩
+
+lemma tabloid_stabilizer (μ : YoungDiagram) :
+    MulAction.stabilizer (Equiv.Perm (Cell μ)) (baseTabloid μ) = rowGroup μ := by
+  ext p
+  change tabloidAct p (baseTabloid μ) = baseTabloid μ ↔ ∀ x, rowIndex (p x) = rowIndex x
+  constructor
+  · intro h x
+    have hh := congrArg (fun f : Tabloid μ => f.1 (p x)) h
+    simpa only [tabloidAct_apply,baseTabloid,Equiv.Perm.inv_def,Equiv.symm_apply_apply] using hh.symm
+  · intro h
+    apply Subtype.ext
+    funext x
+    exact (h (p⁻¹ x)).symm.trans (by simp [baseTabloid])
+
+lemma card_tabloid_mul_row (μ : YoungDiagram) :
+    Fintype.card (Tabloid μ)*Fintype.card (rowGroup μ) = μ.card.factorial := by
+  have hh := MulAction.card_orbit_mul_card_stabilizer_eq_card_group
+    (Equiv.Perm (Cell μ)) (baseTabloid μ)
+  have hc : Fintype.card (MulAction.orbit (Equiv.Perm (Cell μ)) (baseTabloid μ)) =
+      Fintype.card (Tabloid μ) := by
+    rw [tabloid_orbit_all,Fintype.card_setUniv]
+  simpa only [hc,tabloid_stabilizer,Fintype.card_perm,card_cell] using hh
+
+theorem factorial_le_dimension_row_column (μ : YoungDiagram) :
+    μ.card.factorial ≤ Module.finrank ℂ (space μ)*Fintype.card (rowGroup μ)*Fintype.card (colGroup μ) := by
+  rw [←card_tabloid_mul_row]
+  nlinarith [dimension_times_column_ge_tabloids μ]
+
+
+lemma card_row_fiber (μ : YoungDiagram) (i : Fin (μ.colLen 0)) :
+    Fintype.card {x : Cell μ // rowIndex x = i} = μ.rowLen i := by
+  let e : {x : Cell μ // rowIndex x = i} ≃ {x : Cell μ // row x = i} :=
+    Equiv.subtypeEquivRight (fun x => Fin.ext_iff)
+  rw [Fintype.card_congr e,Fintype.card_congr (rowEquiv μ i),Fintype.card_fin]
+
+lemma card_col_fiber (μ : YoungDiagram) (j : Fin (μ.rowLen 0)) :
+    Fintype.card {x : Cell μ // colIndex x = j} = μ.colLen j := by
+  let e : {x : Cell μ // colIndex x = j} ≃ {x : Cell μ // col x = j} :=
+    Equiv.subtypeEquivRight (fun x => Fin.ext_iff)
+  rw [Fintype.card_congr e,Fintype.card_congr (columnEquiv μ j),Fintype.card_fin]
+
+lemma card_rowGroup (μ : YoungDiagram) :
+    Fintype.card (rowGroup μ) = ∏ i : Fin (μ.colLen 0), (μ.rowLen i).factorial := by
+  rw [←Nat.card_eq_fintype_card,rowGroup,nat_card_fiberGroup]
+  simp only [Nat.card_eq_fintype_card]
+  simp only [card_row_fiber]
+
+lemma card_colGroup (μ : YoungDiagram) :
+    Fintype.card (colGroup μ) = ∏ j : Fin (μ.rowLen 0), (μ.colLen j).factorial := by
+  rw [←Nat.card_eq_fintype_card,colGroup_eq,nat_card_fiberGroup]
+  simp only [Nat.card_eq_fintype_card,card_col_fiber]
+
+abbrev Tail (μ : YoungDiagram) := {x : Cell μ // row x ≠ 0}
+
+lemma tail_card (μ : YoungDiagram) : Fintype.card (Tail μ) = μ.card - μ.rowLen 0 := by
+  have hh := Fintype.card_subtype_compl (p := fun x : Cell μ => row x=0)
+  rw [Fintype.card_congr (rowEquiv μ 0),Fintype.card_fin,card_cell] at hh
+  exact hh
+
+lemma rowLen_zero_le_card (μ : YoungDiagram) : μ.rowLen 0 ≤ μ.card := by
+  have h := Fintype.card_subtype_le (fun x : Cell μ => row x=0)
+  rwa [Fintype.card_congr (rowEquiv μ 0),Fintype.card_fin,card_cell] at h
+
+noncomputable def tailColumnEquiv (μ : YoungDiagram) (j : Fin (μ.rowLen 0)) :
+    {x : Tail μ // colIndex x.1 = j} ≃ Fin (μ.colLen j-1) where
+  toFun x := ⟨row x.1.1-1, by
+    have hx := row_lt_colLen x.1.1
+    have hc : col x.1.1 = j := congrArg Fin.val x.2
+    rw [hc] at hx
+    have hz := x.1.2
+    omega⟩
+  invFun i := ⟨⟨⟨(i+1,j),YoungDiagram.mem_iff_lt_colLen.mpr (by omega)⟩,by simp [row]⟩,rfl⟩
+  left_inv x := by
+    apply Subtype.ext
+    apply Subtype.ext
+    apply Subtype.ext
+    apply Prod.ext
+    · change row x.1.1 - 1 + 1 = row x.1.1
+      have := x.1.2
+      omega
+    · exact (congrArg Fin.val x.2).symm
+  right_inv i := by apply Fin.ext; simp [row]
+
+lemma card_tail_col (μ : YoungDiagram) (j : Fin (μ.rowLen 0)) :
+    Fintype.card {x : Tail μ // colIndex x.1 = j} = μ.colLen j-1 := by
+  rw [Fintype.card_congr (tailColumnEquiv μ j),Fintype.card_fin]
+
+lemma card_tail_row (μ : YoungDiagram) (i : Fin (μ.colLen 0)) :
+    Fintype.card {x : Tail μ // rowIndex x.1 = i} = if i.val=0 then 0 else μ.rowLen i := by
+  by_cases hi : i.val=0
+  · rw [ite_eq_left hi]
+    apply Fintype.card_eq_zero_iff.mpr
+    refine ⟨fun x => x.1.2 ?_⟩
+    exact (congrArg Fin.val x.2).trans hi
+  · rw [ite_eq_right hi]
+    let e : {x : Tail μ // rowIndex x.1 = i} ≃ {x : Cell μ // rowIndex x=i} :=
+      { toFun := fun x => ⟨x.1.1,x.2⟩
+        invFun := fun x => ⟨⟨x.1,fun hz => hi ((congrArg Fin.val x.2).symm.trans hz)⟩,x.2⟩
+        left_inv := fun x => rfl
+        right_inv := fun x => rfl }
+    rw [Fintype.card_congr e,card_row_fiber]
+
+lemma tail_row_group (μ : YoungDiagram) :
+    (μ.rowLen 0).factorial * Fintype.card (fiberGroup (fun x : Tail μ => rowIndex x.1)) =
+      Fintype.card (rowGroup μ) := by
+  rw [←Nat.card_eq_fintype_card (α := fiberGroup (fun x : Tail μ => rowIndex x.1)),nat_card_fiberGroup,card_rowGroup]
+  simp only [Nat.card_eq_fintype_card]
+  simp only [card_tail_row]
+  by_cases hn : μ.colLen 0=0
+  · have hr : μ.rowLen 0=0 := by
+      by_contra h
+      have hmem : (0,0) ∈ μ := YoungDiagram.mem_iff_lt_rowLen.mpr (Nat.pos_of_ne_zero h)
+      have hh := YoungDiagram.mem_iff_lt_colLen.mp hmem
+      omega
+    let : IsEmpty (Fin (μ.colLen 0)) := ⟨fun i => by have := i.isLt; omega⟩
+    simp [hr]
+  · let z : Fin (μ.colLen 0) := ⟨0,Nat.pos_of_ne_zero hn⟩
+    have hz : ∀ i : Fin (μ.colLen 0), i.val=0 ↔ i=z := fun i => by simp [z,Fin.ext_iff]
+    simp only [hz]
+    have hprod : (∏ x : Fin (μ.colLen 0), (if x=z then 0 else μ.rowLen x).factorial) =
+        ∏ x ∈ Finset.univ.erase z, (μ.rowLen x).factorial := by
+      rw [←Finset.prod_erase_mul Finset.univ (fun x : Fin (μ.colLen 0) => (if x=z then 0 else μ.rowLen x).factorial) (Finset.mem_univ z)]
+      simp only [ite_true,Nat.factorial_zero,mul_one]
+      apply Finset.prod_congr rfl
+      intro x hx
+      rw [ite_eq_right (Finset.mem_erase.mp hx).1]
+    rw [hprod]
+    exact Finset.mul_prod_erase Finset.univ (fun i : Fin (μ.colLen 0) => (μ.rowLen i).factorial) (Finset.mem_univ z)
+
+lemma sum_tail_columns (μ : YoungDiagram) :
+    ∑ j : Fin (μ.rowLen 0), (μ.colLen j-1) = μ.card-μ.rowLen 0 := by
+  simp_rw [←card_tail_col]
+  rw [←Fintype.card_sigma,Fintype.card_congr (Equiv.sigmaFiberEquiv (fun x : Tail μ => colIndex x.1)),tail_card]
+
+lemma column_pos (μ : YoungDiagram) (j : Fin (μ.rowLen 0)) : 0 < μ.colLen j :=
+  YoungDiagram.mem_iff_lt_colLen.mp (YoungDiagram.mem_iff_lt_rowLen.mpr j.isLt)
+
+lemma nat_le_two_pow_pred (n : ℕ) (hn : 0<n) : n ≤ 2^(n-1) := by
+  obtain ⟨n,rfl⟩ := Nat.exists_eq_succ_of_ne_zero hn.ne'
+  simp only [Nat.add_one_sub_one]
+  induction n with
+  | zero => norm_num
+  | succ n ih => rw [pow_succ]; omega
+
+lemma column_group_le_tail (μ : YoungDiagram) :
+    Fintype.card (colGroup μ) ≤ 2^(μ.card-μ.rowLen 0) *
+      Fintype.card (fiberGroup (fun x : Tail μ => colIndex x.1)) := by
+  rw [card_colGroup,←Nat.card_eq_fintype_card (α := fiberGroup (fun x : Tail μ => colIndex x.1)),nat_card_fiberGroup]
+  simp only [Nat.card_eq_fintype_card]
+  simp only [card_tail_col]
+  calc
+    _ = ∏ j : Fin (μ.rowLen 0), μ.colLen j * (μ.colLen j-1).factorial := by
+      apply Finset.prod_congr rfl
+      intro j hj
+      exact (Nat.mul_factorial_pred (column_pos μ j).ne').symm
+    _ ≤ ∏ j : Fin (μ.rowLen 0), 2^(μ.colLen j-1) * (μ.colLen j-1).factorial := by
+      apply Finset.prod_le_prod
+      intro j hj
+      exact Nat.mul_le_mul_right _ (nat_le_two_pow_pred _ (column_pos μ j))
+    _ = _ := by rw [Finset.prod_mul_distrib,Finset.prod_pow_eq_pow_sum,sum_tail_columns]
+
+theorem choose_tail_le_dimension (μ : YoungDiagram) :
+    μ.card.choose (μ.card-μ.rowLen 0) ≤ Module.finrank ℂ (space μ) * 2^(μ.card-μ.rowLen 0) := by
+  let j := μ.card-μ.rowLen 0
+  have htail := card_fiberGroups_mul_le
+    (fun x : Tail μ => rowIndex x.1) (fun x : Tail μ => colIndex x.1)
+    (fun x y hr hc => Subtype.ext (Subtype.ext (Prod.ext (congrArg Fin.val hr) (congrArg Fin.val hc))))
+  rw [tail_card] at htail
+  have hc := column_group_le_tail μ
+  have hr := tail_row_group μ
+  have hd := factorial_le_dimension_row_column μ
+  simp only [←Nat.card_eq_fintype_card] at htail hc hr hd
+  have hh : μ.card.factorial ≤ Module.finrank ℂ (space μ) * 2^j *
+      ((μ.rowLen 0).factorial * j.factorial) := by
+    calc
+      _ ≤ _ := hd
+      _ ≤ Module.finrank ℂ (space μ) * Nat.card (rowGroup μ) *
+          (2^j * Nat.card (fiberGroup (fun x : Tail μ => colIndex x.1))) := Nat.mul_le_mul_left _ hc
+      _ = Module.finrank ℂ (space μ) * 2^j * (μ.rowLen 0).factorial *
+          (Nat.card (fiberGroup (fun x : Tail μ => rowIndex x.1)) *
+           Nat.card (fiberGroup (fun x : Tail μ => colIndex x.1))) := by rw [←hr]; ring
+      _ ≤ _ := by
+        calc
+          _ ≤ Module.finrank ℂ (space μ) * 2^j * (μ.rowLen 0).factorial * j.factorial :=
+            Nat.mul_le_mul_left _ htail
+          _ = _ := by ring
+  have hj : j ≤ μ.card := Nat.sub_le _ _
+  have hs : μ.card-j = μ.rowLen 0 := by dsimp [j]; have := rowLen_zero_le_card μ; omega
+  have hf := Nat.choose_mul_factorial_mul_factorial hj
+  rw [hs] at hf
+  rw [←hf] at hh
+  have hpos : 0 < (μ.rowLen 0).factorial * j.factorial := Nat.mul_pos (Nat.factorial_pos _) (Nat.factorial_pos _)
+  nlinarith
+
+
+lemma exists_perm_mem_iff {α : Type*} [Fintype α] (s t : Finset α) (hc : s.card=t.card) :
+    ∃ p : Equiv.Perm α, ∀ x, p x ∈ t ↔ x ∈ s := by
+  let e : s ≃ t := Finset.equivOfCardEq hc
+  let ec : {x : α // x ∉ s} ≃ {x : α // x ∉ t} := Fintype.equivOfCardEq (by
+    simp only [Fintype.card_subtype_compl,Fintype.card_coe,hc])
+  let p := (Equiv.sumCompl (· ∈ s)).symm.trans ((Equiv.sumCongr e ec).trans (Equiv.sumCompl (· ∈ t)))
+  refine ⟨p,fun x => ?_⟩
+  by_cases hx : x ∈ s
+  · have hp : p x ∈ t := by
+      change ((Equiv.sumCompl (· ∈ t)) ((Equiv.sumCongr e ec) ((Equiv.sumCompl (· ∈ s)).symm x))) ∈ t
+      rw [Equiv.sumCompl_symm_apply_of_pos hx]
+      exact (e ⟨x,hx⟩).2
+    exact iff_of_true hp hx
+  · have hp : p x ∉ t := by
+      change ((Equiv.sumCompl (· ∈ t)) ((Equiv.sumCongr e ec) ((Equiv.sumCompl (· ∈ s)).symm x))) ∉ t
+      rw [Equiv.sumCompl_symm_apply_of_neg hx]
+      exact (ec ⟨x,hx⟩).2
+    exact iff_of_false hp hx
+
+noncomputable def tailSet {μ : YoungDiagram} (f : Tabloid μ) : Finset (Cell μ) :=
+  Finset.univ.filter (fun x => (f.1 x).val ≠ 0)
+
+@[simp] lemma mem_tailSet {μ : YoungDiagram} (f : Tabloid μ) (x : Cell μ) :
+    x ∈ tailSet f ↔ (f.1 x).val ≠ 0 := by simp [tailSet]
+
+lemma tailSet_act {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) (f : Tabloid μ) :
+    tailSet (tabloidAct p f) = (tailSet f).image p := by
+  ext x
+  simp only [mem_tailSet,tabloidAct_apply,Finset.mem_image]
+  constructor
+  · intro h; exact ⟨p⁻¹ x,h,by simp⟩
+  · rintro ⟨y,hy,rfl⟩; simpa using hy
+
+lemma card_tailSet {μ : YoungDiagram} (f : Tabloid μ) :
+    (tailSet f).card = μ.card-μ.rowLen 0 := by
+  obtain ⟨p,rfl⟩ := tabloidAct_transitive μ (baseTabloid μ) f
+  rw [tailSet_act,Finset.card_image_of_injective _ p.injective]
+  change (Finset.univ.filter (fun x : Cell μ => row x ≠ 0)).card = _
+  rw [←Fintype.card_subtype]
+  exact tail_card μ
+
+noncomputable def footprint (μ : YoungDiagram) (A : Finset (Cell μ)) : Finset (Tabloid μ) :=
+  Finset.univ.filter (fun f => tailSet f ⊆ A)
+
+lemma tail_fiber_card_eq (μ : YoungDiagram) (s t : Finset (Cell μ)) (hst : s.card=t.card) :
+    (Finset.univ.filter (fun f : Tabloid μ => tailSet f=s)).card =
+      (Finset.univ.filter (fun f : Tabloid μ => tailSet f=t)).card := by
+  obtain ⟨p,hp⟩ := exists_perm_mem_iff s t hst
+  have he : s.image p=t := by
+    ext x
+    constructor
+    · intro hx
+      obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
+      exact (hp y).mpr hy
+    · intro hx
+      exact Finset.mem_image.mpr ⟨p⁻¹ x,(hp (p⁻¹ x)).mp (by simpa using hx),by simp⟩
+  apply Finset.card_equiv (tabloidAct p)
+  intro f
+  simp only [Finset.mem_filter,Finset.mem_univ,true_and,tailSet_act,←he]
+  exact (Finset.image_inj p.injective).symm
+
+lemma footprint_card_product (μ : YoungDiagram) (A : Finset (Cell μ)) :
+    (footprint μ A).card * μ.card.choose (μ.card-μ.rowLen 0) =
+      Fintype.card (Tabloid μ) * A.card.choose (μ.card-μ.rowLen 0) := by
+  let j := μ.card-μ.rowLen 0
+  let s := tailSet (baseTabloid μ)
+  let c := (Finset.univ.filter (fun f : Tabloid μ => tailSet f=s)).card
+  have hconstant (t : Finset (Cell μ)) (ht : t.card=j) :
+      (Finset.univ.filter (fun f : Tabloid μ => tailSet f=t)).card = c :=
+    tail_fiber_card_eq μ t s (ht.trans (card_tailSet _).symm)
+  have hcount (A : Finset (Cell μ)) : (footprint μ A).card = A.card.choose j*c := by
+    have h := Finset.sum_card_fiberwise_eq_card_filter
+      (Finset.univ : Finset (Tabloid μ)) (A.powersetCard j) tailSet
+    have he : (Finset.univ.filter (fun f : Tabloid μ => tailSet f ∈ A.powersetCard j)) = footprint μ A := by
+      ext f
+      simp only [Finset.mem_filter,Finset.mem_univ,true_and,Finset.mem_powersetCard,
+        card_tailSet,footprint]
+      simp [j]
+    rw [he] at h
+    rw [←h]
+    calc
+      _ = ∑ _t ∈ A.powersetCard j, c := by
+        apply Finset.sum_congr rfl
+        intro t ht
+        exact hconstant t (Finset.mem_powersetCard.mp ht).2
+      _ = _ := by simp [Finset.card_powersetCard]
+  have hfull : (footprint μ Finset.univ).card = Fintype.card (Tabloid μ) := by
+    simp only [footprint,Finset.subset_univ,Finset.filter_true,Finset.card_univ]
+  have hN := hcount Finset.univ
+  rw [hfull,Finset.card_univ,card_cell] at hN
+  rw [hcount,hN]
+  ring
+
+lemma footprint_fixed {μ : YoungDiagram} (A : Finset (Cell μ))
+    (f : Tabloid μ) (hf : f ∈ footprint μ A) (p : Equiv.Perm (Cell μ))
+    (hp : ∀ x ∈ A, p x=x) : tabloidAct p f=f := by
+  have htail : tailSet f ⊆ A := (Finset.mem_filter.mp hf).2
+  apply Subtype.ext
+  funext x
+  change f.1 (p⁻¹ x) = f.1 x
+  by_cases hx : x ∈ A
+  · rw [show p⁻¹ x=x from p.injective (by simpa using (hp x hx).symm)]
+  · have hy : p⁻¹ x ∉ A := by
+      intro h
+      have hh := hp (p⁻¹ x) h
+      have he : x=p⁻¹ x := by simpa using hh
+      exact hx (he ▸ h)
+    apply Fin.ext
+    have hx0 : (f.1 x).val=0 := by by_contra h; exact hx (htail (mem_tailSet f x |>.mpr h))
+    have hy0 : (f.1 (p⁻¹ x)).val=0 := by by_contra h; exact hy (htail (mem_tailSet f _ |>.mpr h))
+    exact hy0.trans hx0.symm
+
+noncomputable def fixedHilbert (μ : YoungDiagram) (A : Finset (Cell μ)) :
+    Submodule ℂ (PermutationHilbert.H (Tabloid μ)) where
+  carrier := {v | v ∈ hilbertSpace μ ∧ ∀ p : Equiv.Perm (Cell μ),
+    (∀ x ∈ A, p x=x) → PermutationHilbert.act (tabloidAct p) v=v}
+  zero_mem' := ⟨Submodule.zero_mem _,fun p hp => map_zero _⟩
+  add_mem' := fun hv hw => ⟨Submodule.add_mem _ hv.1 hw.1,fun p hp => by
+    rw [map_add,hv.2 p hp,hw.2 p hp]⟩
+  smul_mem' := fun c v hv => ⟨Submodule.smul_mem _ c hv.1,fun p hp => by
+    rw [map_smul,hv.2 p hp]⟩
+
+lemma fixedHilbert_le (μ : YoungDiagram) (A : Finset (Cell μ)) : fixedHilbert μ A ≤ hilbertSpace μ :=
+  fun _ hv => hv.1
+
+theorem fixedHilbert_dimension (μ : YoungDiagram) (A : Finset (Cell μ)) :
+    Module.finrank ℂ (space μ) * A.card.choose (μ.card-μ.rowLen 0) ≤
+      Module.finrank ℂ (fixedHilbert μ A) * μ.card.choose (μ.card-μ.rowLen 0) := by
+  let : DecidableEq (Tabloid μ) := Classical.decEq _
+  have ht (f g : Tabloid μ) : ∃ p : Equiv.Perm (Tabloid μ), p f=g ∧
+      (hilbertSpace μ).map (PermutationHilbert.act p).toLinearEquiv.toLinearMap = hilbertSpace μ := by
+    obtain ⟨p,hp⟩ := tabloidAct_transitive μ f g
+    exact ⟨tabloidAct p,hp,hilbertSpace_invariant μ p⟩
+  have hproj (f : Tabloid μ) (hf : f ∈ footprint μ A) :
+      (hilbertSpace μ).starProjection (EuclideanSpace.single f (1:ℂ)) ∈ fixedHilbert μ A := by
+    refine ⟨((hilbertSpace μ).orthogonalProjectionOnto _).2,fun p hp => ?_⟩
+    rw [PermutationHilbert.projection_comm _ _ (hilbertSpace_invariant μ p),
+      PermutationHilbert.act_single,footprint_fixed A f hf p hp]
+  have h := PermutationHilbert.footprint_dimension (hilbertSpace μ) (fixedHilbert μ A)
+    (fixedHilbert_le μ A) ht (footprint μ A) hproj
+  rw [finrank_hilbertSpace] at h
+  have hn : (footprint μ A).card * Module.finrank ℂ (space μ) ≤
+      Fintype.card (Tabloid μ)*Module.finrank ℂ (fixedHilbert μ A) := by exact_mod_cast h
+  have hc := footprint_card_product μ A
+  have hpos : 0 < Fintype.card (Tabloid μ) := Fintype.card_pos_iff.mpr ⟨baseTabloid μ⟩
+  have hh := Nat.mul_le_mul_right (μ.card.choose (μ.card-μ.rowLen 0)) hn
+  nlinarith
+
+end Thorp.Specht
+
+namespace Thorp.UnitaryFinite
+open scoped BigOperators ComplexConjugate Classical
+
+variable {G : Type*} [Group G] [Fintype G]
+variable {V W : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+  [NormedAddCommGroup W] [InnerProductSpace ℂ W] [FiniteDimensional ℂ W]
+
+def IsUnitary (ρ : Representation ℂ G V) : Prop :=
+  ∀ g v w, inner ℂ (ρ g v) (ρ g w) = inner ℂ v w
+
+omit [Fintype G] [FiniteDimensional ℂ V] in
+lemma unitary_inner_inv (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (g : G) (v w : V) :
+    inner ℂ v (ρ g⁻¹ w) = inner ℂ (ρ g v) w := by
+  have h := hρ g v (ρ g⁻¹ w)
+  have he : ρ g (ρ g⁻¹ w) = w := by
+    change (ρ g * ρ g⁻¹) w = w
+    rw [←map_mul,mul_inv_cancel,map_one]
+    rfl
+  rw [he] at h
+  exact h.symm
+
+noncomputable def averageMap (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
+    (A : V →ₗ[ℂ] W) : Representation.IntertwiningMap ρ σ where
+  toLinearMap := ∑ g, (σ g).comp (A.comp (ρ g⁻¹))
+  isIntertwining' h := by
+    ext v
+    simp only [LinearMap.comp_apply,LinearMap.sum_apply,map_sum]
+    apply Fintype.sum_equiv (Equiv.mulLeft h⁻¹)
+    intro g
+    simp only [Equiv.coe_mulLeft,mul_inv_rev,inv_inv,map_mul,Module.End.mul_apply]
+    have hh (w : W) : σ h (σ h⁻¹ w)=w := by
+      change (σ h*σ h⁻¹) w=w
+      rw [←map_mul,mul_inv_cancel,map_one]
+      rfl
+    rw [hh]
+
+end Thorp.UnitaryFinite
+
+end ThorpNine.SparseSaving
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Tail/AlternatingPerm.lean b/lean/OAI/Probability/ThorpRouting/Tail/AlternatingPerm.lean
index bf8fa9db14025f4415850faedc05e1562f138c5f..b9baf5d656b154f43fe629b27a642feeef80f6e4 100644
--- a/lean/OAI/Probability/ThorpRouting/Tail/AlternatingPerm.lean
+++ b/lean/OAI/Probability/ThorpRouting/Tail/AlternatingPerm.lean
@@ -1,895 +1,895 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Tail.ParallelFollow
-
-namespace OAI
-
-namespace ThorpNine.Tail
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-namespace CycleColoring
-variable {α : Type*} [Fintype α] [DecidableEq α]
-omit [Fintype α] in
-lemma bipartitionFlip_injective (p : Equiv.Perm α) (S : Finset α)
-    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x)) :
-    Function.Injective (bipartitionFlip p S χ hχ) := by
-  classical
-  intro f g he
-  apply Subtype.ext
-  funext x
-  by_cases hx : x ∈ S
-  · have hh := congrFun (congrArg Subtype.val he) x
-    change (if x ∈ S then Bool.xor (f.val x) (χ x) else false) =
-      (if x ∈ S then Bool.xor (g.val x) (χ x) else false) at hh
-    simp only [ite_eq_left hx] at hh
-    cases hfx : f.val x <;> cases hgx : g.val x <;> cases hcx : χ x <;> simp_all
-  · exact (f.property.1 x hx).trans (g.property.1 x hx).symm
-
-lemma card_opposite_le (p : Equiv.Perm α) (S : Finset α)
-    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x)) :
-    Fintype.card (OppositeColoring p S) ≤
-      2 ^ ((boundary p S).card + Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
-  exact (Fintype.card_le_of_injective _ (bipartitionFlip_injective p S χ hχ)).trans
-    (card_coloring_le p S)
-
-def alternatingPerm (p : Equiv.Perm α) : Equiv.Perm (Bool × α) where
-  toFun x := if x.1 then (false,p x.2) else (true,x.2)
-  invFun x := if x.1 then (false,x.2) else (true,p.symm x.2)
-  left_inv x := by rcases x with ⟨b,x⟩; cases b <;> simp
-  right_inv x := by rcases x with ⟨b,x⟩; cases b <;> simp
-
-omit [Fintype α] [DecidableEq α] in
-lemma alternatingPerm_flip (p : Equiv.Perm α) (x : Bool × α) :
-    (alternatingPerm p x).1 = !x.1 := by
-  rcases x with ⟨b,x⟩
-  cases b <;> rfl
-
-omit [Fintype α] [DecidableEq α] in
-lemma alternating_step_sameCycle (p : Equiv.Perm α) (x : Bool × α) :
-    p.SameCycle x.2 (alternatingPerm p x).2 := by
-  rcases x with ⟨b,x⟩
-  cases b
-  · exact Equiv.Perm.SameCycle.refl p x
-  · exact (Equiv.Perm.SameCycle.refl p x).apply_right
-
-omit [DecidableEq α] in
-lemma alternating_sameCycle_snd (p : Equiv.Perm α) {x y : Bool × α}
-    (h : (alternatingPerm p).SameCycle x y) : p.SameCycle x.2 y.2 := by
-  obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
-  have hp (k : ℕ) : p.SameCycle x.2 (((alternatingPerm p)^k) x).2 := by
-    induction k with
-    | zero => exact Equiv.Perm.SameCycle.refl p x.2
-    | succ k ih =>
-      rw [pow_succ',Equiv.Perm.mul_apply]
-      exact ih.trans (alternating_step_sameCycle p _)
-  exact hn ▸ hp n
-
-omit [DecidableEq α] in
-lemma alternating_zero_sameCycle (p : Equiv.Perm α) {x y : α}
-    (h : p.SameCycle x y) : (alternatingPerm p).SameCycle (false,x) (false,y) := by
-  obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
-  have hp (k : ℕ) : (alternatingPerm p).SameCycle (false,x) (false,(p^k) x) := by
-    induction k with
-    | zero => exact Equiv.Perm.SameCycle.refl _ _
-    | succ k ih =>
-      have hs := ih.apply_right.apply_right
-      simpa only [pow_succ',Equiv.Perm.mul_apply,alternatingPerm,Equiv.coe_fn_mk,
-        Bool.false_eq_true,↓reduceIte] using hs
-  simpa only [hn] using hp n
-
-omit [Fintype α] [DecidableEq α] in
-lemma alternating_sameCycle_zero (p : Equiv.Perm α) (x : Bool × α) :
-    (alternatingPerm p).SameCycle x (false,x.2) := by
-  rcases x with ⟨b,x⟩
-  cases b
-  · exact Equiv.Perm.SameCycle.refl _ _
-  · exact ((Equiv.Perm.SameCycle.refl (alternatingPerm p) (false,x)).apply_right).symm
-
-omit [DecidableEq α] in
-lemma alternating_sameCycle_iff (p : Equiv.Perm α) (x y : Bool × α) :
-    (alternatingPerm p).SameCycle x y ↔ p.SameCycle x.2 y.2 := by
-  constructor
-  · exact alternating_sameCycle_snd p
-  · intro h
-    exact (alternating_sameCycle_zero p x).trans
-      ((alternating_zero_sameCycle p h).trans (alternating_sameCycle_zero p y).symm)
-
-lemma orbit_alternating (p : Equiv.Perm α) (x : Bool × α) :
-    orbitSet (alternatingPerm p) x = childJoin (orbitSet p x.2) (orbitSet p x.2) := by
-  ext ⟨b,y⟩
-  simp only [mem_orbitSet,alternating_sameCycle_iff,mem_childJoin,ite_self]
-
-omit [Fintype α] in
-lemma boolFiber_childJoin (A B : Finset α) (b : Bool) :
-    boolFiber (childJoin A B) b = if b then B else A := by
-  ext x
-  cases b <;> simp only [mem_boolFiber,mem_childJoin,Bool.false_eq_true,↓reduceIte]
-
-noncomputable def projectCycle (p : Equiv.Perm α) (A B : Finset α)
-    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) :
-    RoutingNetwork.SelectedCycle p A := by
-  refine ⟨boolFiber C.val false,?_,?_⟩
-  · obtain ⟨x,hx⟩ := C.property.1
-    refine ⟨x.2,?_⟩
-    rw [hx,orbit_alternating,boolFiber_childJoin]
-    rfl
-  · intro x hx
-    exact (mem_childJoin A B false x).mp (C.property.2 ((mem_boolFiber _ _ _).mp hx))
-
-lemma selected_alternating_eq (p : Equiv.Perm α) (A B : Finset α)
-    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) :
-    C.val = childJoin (projectCycle p A B C).val (projectCycle p A B C).val := by
-  change C.val = childJoin (boolFiber C.val false) (boolFiber C.val false)
-  obtain ⟨x,hx⟩ := C.property.1
-  rw [hx,orbit_alternating,boolFiber_childJoin]
-  rfl
-
-lemma projectCycle_injective (p : Equiv.Perm α) (A B : Finset α) :
-    Function.Injective (projectCycle p A B) := by
-  intro C D he
-  apply Subtype.ext
-  rw [selected_alternating_eq p A B C,selected_alternating_eq p A B D,he]
-
-lemma card_alternating_cycles_le (p : Equiv.Perm α) (A B : Finset α) :
-    Fintype.card (RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) ≤
-      Fintype.card (RoutingNetwork.SelectedCycle p A) :=
-  Fintype.card_le_of_injective _ (projectCycle_injective p A B)
-
-end CycleColoring
-
-section PairAssignmentCount
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-omit [Fintype α] in
-lemma mem_image_perm_iff (p : Equiv.Perm α) (A : Finset α) (y : α) :
-    y ∈ A.image p ↔ p.symm y ∈ A := by
-  constructor
-  · intro h
-    obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp h
-    simpa only [←hxy,Equiv.symm_apply_apply] using hx
-  · intro h
-    exact Finset.mem_image.mpr ⟨p.symm y,h,p.apply_symm_apply y⟩
-
-omit [Fintype α] in
-lemma card_childJoin (A B : Finset α) : (childJoin A B).card = A.card+B.card := by
-  have hd : Disjoint (A.image (fun x => (false,x))) (B.image (fun x => (true,x))) := by
-    apply Finset.disjoint_left.mpr
-    rintro x hx hy
-    obtain ⟨a,ha,rfl⟩ := Finset.mem_image.mp hx
-    obtain ⟨b,hb,hba⟩ := Finset.mem_image.mp hy
-    exact Bool.false_ne_true (congrArg Prod.fst hba).symm
-  have hi (b : Bool) : Function.Injective (fun x : α => (b,x)) :=
-    fun _ _ h => congrArg Prod.snd h
-  rw [childJoin,Finset.card_union_of_disjoint hd,
-    Finset.card_image_of_injective A (hi false),Finset.card_image_of_injective B (hi true)]
-
-omit [Fintype α] in
-lemma alternating_internal (p : Equiv.Perm α) (A B : Finset α) :
-    ((childJoin A B).filter (fun x => (CycleColoring.alternatingPerm p).symm x ∈ childJoin A B)) =
-      childJoin (A ∩ B.image p) (B ∩ A) := by
-  ext ⟨b,x⟩
-  rw [Finset.mem_filter]
-  cases b <;> simp only [mem_childJoin,CycleColoring.alternatingPerm,
-    Equiv.symm_mk,Equiv.coe_fn_mk,Bool.false_eq_true,↓reduceIte,Finset.mem_inter,mem_image_perm_iff]
-
-omit [Fintype α] in
-lemma alternating_boundary_card (p : Equiv.Perm α) (A B : Finset α) :
-    (CycleColoring.boundary (CycleColoring.alternatingPerm p) (childJoin A B)).card =
-      (A ∪ B).card + (A ∪ B.image p).card - (A.card+B.card) := by
-  rw [CycleColoring.boundary_card,alternating_internal,card_childJoin,card_childJoin]
-  have h₀ := Finset.card_union_add_card_inter A B
-  have h₁ := Finset.card_union_add_card_inter A (B.image p)
-  rw [Finset.card_image_of_injective _ p.injective] at h₁
-  have h₂ := Finset.inter_comm B A
-  rw [h₂]
-  omega
-
-lemma compatible_assignment_bound (p : Equiv.Perm α) (A B : Finset α) :
-    Fintype.card (CycleColoring.OppositeColoring (CycleColoring.alternatingPerm p) (childJoin A B)) ≤
-      2 ^ ((A ∪ B).card + (A ∪ B.image p).card - (A.card+B.card) +
-        Fintype.card (RoutingNetwork.SelectedCycle p A)) := by
-  have h := CycleColoring.card_opposite_le (CycleColoring.alternatingPerm p) (childJoin A B)
-    Prod.fst (CycleColoring.alternatingPerm_flip p)
-  rw [alternating_boundary_card] at h
-  exact h.trans (Nat.pow_le_pow_right (by norm_num)
-    (Nat.add_le_add_left (CycleColoring.card_alternating_cycles_le p A B) _))
-
-end PairAssignmentCount
-
-lemma finiteMean_nonneg {Ω : Type*} [Fintype Ω] {f : Ω → ℝ}
-    (h : ∀ ω, 0 ≤ f ω) : 0 ≤ finiteMean f := by
-  exact div_nonneg (Finset.sum_nonneg (fun ω _ => h ω)) (Nat.cast_nonneg _)
-
-lemma boolFiber_headTail (d : ℕ) (S : Finset (Card (d+1))) (b : Bool) :
-    boolFiber (S.image (headTailEquiv d)) b = childMarks S b := by
-  classical
-  ext x
-  rw [mem_boolFiber,mem_childMarks]
-  constructor
-  · intro h
-    obtain ⟨y,hy,he⟩ := Finset.mem_image.mp h
-    have hh := congrArg (headTailEquiv d).symm he
-    rw [Equiv.symm_apply_apply] at hh
-    change y = Fin.cons b x at hh
-    exact hh ▸ hy
-  · intro h
-    exact Finset.mem_image.mpr ⟨Fin.cons b x,h,by rfl⟩
-
-lemma prod_childMarks (d : ℕ) (S : Finset (Card (d+1))) (f : Card (d+1) → ℝ) :
-    (∏ x ∈ S, f x) =
-      (∏ x ∈ childMarks S false, f (Fin.cons false x)) *
-      ∏ x ∈ childMarks S true, f (Fin.cons true x) := by
-  classical
-  have h := prod_boolFibers (S.image (headTailEquiv d)) (fun x => f ((headTailEquiv d).symm x))
-  rw [Finset.prod_image (headTailEquiv d).injective.injOn,
-    boolFiber_headTail,boolFiber_headTail] at h
-  simp only [Equiv.symm_apply_apply] at h
-  exact h
-
-noncomputable def headAverage {d : ℕ} (p : Card (d+1) → ℝ) (y : Card d) : ℝ :=
-  (p (Fin.cons false y)+p (Fin.cons true y))/2
-
-lemma finiteMean_headAverage (d : ℕ) (p : Card (d+1) → ℝ) :
-    finiteMean (headAverage p) = finiteMean p := by
-  rw [finiteMean_equiv (headTailEquiv d),finiteMean_prod,finiteMean_comm]
-  apply finiteMean_congr
-  intro y
-  rw [finiteMean_bool]
-  rfl
-
-lemma pairSwitch_product_bound (d : ℕ) (S : Finset (Card (d+1)))
-    (p : Card (d+1) → ℝ) (hp : ∀ x, 0 ≤ p x) :
-    finiteMean (fun ξ : Card d → Bool => ∏ x ∈ S, p (pairSwitch ξ x)) ≤
-      ∏ x ∈ S, headAverage p (Fin.tail x) := by
-  classical
-  have h : CylinderBound (fun T => ∏ x ∈ T, p ((headTailEquiv d).symm x))
-      (fun x => p ((headTailEquiv d).symm x)) := fun _ => le_rfl
-  have hh := h.fair_parallel (fun x => hp _) (S.image (headTailEquiv d))
-  simp only [Finset.prod_image (pairLayer _).injective.injOn,
-    Finset.prod_image (headTailEquiv d).injective.injOn] at hh
-  convert hh using 1
-  · apply finiteMean_congr
-    intro ξ
-    apply Finset.prod_congr rfl
-    intro x _
-    rfl
-  · apply Finset.prod_congr rfl
-    intro x _
-    change (p (Fin.cons false (Fin.tail x))+p (Fin.cons true (Fin.tail x)))/2 =
-      (p (Fin.cons (x 0) (Fin.tail x))+p (Fin.cons (!(x 0)) (Fin.tail x)))/2
-    cases x 0
-    · rfl
-    · simp only [Bool.not_true]
-      ring
-
-lemma butterflyPerm_step (d : ℕ)
-    (c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (ξ : Card d → Bool) :
-    butterflyPerm (d+1) (decodeButterfly (d+1) ((coinStepEquiv d).symm (c,ξ))) =
-    pairSwitch ξ * childLift (fun b => butterflyPerm d (decodeButterfly d (if b then c.2 else c.1))) := by
-  change pairSwitch (fun y => (coinStepEquiv d).symm (c,ξ) (Sum.inl y)) *
-    childLift (fun b => butterflyPerm d (decodeButterfly d
-      (fun i => (coinStepEquiv d).symm (c,ξ) (Sum.inr (b,i))))) = _
-  apply congrArg₂ (· * ·)
-  · rfl
-  · apply congrArg childLift
-    funext b
-    apply congrArg (fun ω => butterflyPerm d (decodeButterfly d ω))
-    funext i
-    cases b <;> rfl
-
-lemma butterfly_product_bound (d : ℕ) (S : Finset (Card d))
-    (p : Card d → ℝ) (hp : ∀ x, 0 ≤ p x) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      ∏ x ∈ S, p (butterflyPerm d (decodeButterfly d ω) x)) ≤ (finiteMean p)^S.card := by
-  classical
-  induction d with
-  | zero =>
-    have he : p = fun _ => p (fun i => Fin.elim0 i) := by
-      funext x
-      exact congrArg p (Subsingleton.elim _ _)
-    rw [he]
-    simp only [Finset.prod_const,finiteMean_const,le_refl]
-  | succ d ih =>
-    let g : Card d → ℝ := headAverage p
-    have hg : ∀ x, 0 ≤ g x := fun x => div_nonneg (add_nonneg (hp _) (hp _)) (by norm_num)
-    rw [finiteMean_equiv (coinStepEquiv d),finiteMean_prod]
-    calc
-      _ ≤ finiteMean (fun c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool) =>
-        (∏ x ∈ childMarks S false, g (butterflyPerm d (decodeButterfly d c.1) x)) *
-        ∏ x ∈ childMarks S true, g (butterflyPerm d (decodeButterfly d c.2) x)) := by
-        apply finiteMean_mono
-        intro c
-        let f : Equiv.Perm (Card (d+1)) := childLift (fun b =>
-          butterflyPerm d (decodeButterfly d (if b then c.2 else c.1)))
-        have hh := pairSwitch_product_bound d (S.image f) p hp
-        simp only [Finset.prod_image f.injective.injOn] at hh
-        convert hh using 1
-        · apply finiteMean_congr
-          intro ξ
-          apply Finset.prod_congr rfl
-          intro x _
-          rw [butterflyPerm_step]
-          rfl
-        · rw [prod_childMarks]
-          simp only [f,childLift,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
-            Bool.false_eq_true,↓reduceIte]
-          rfl
-      _ = (finiteMean (fun c : SwitchIndex d → Bool =>
-        ∏ x ∈ childMarks S false, g (butterflyPerm d (decodeButterfly d c) x))) *
-        finiteMean (fun c : SwitchIndex d → Bool =>
-        ∏ x ∈ childMarks S true, g (butterflyPerm d (decodeButterfly d c) x)) :=
-          finiteMean_prod_mul
-            (fun c : SwitchIndex d → Bool => ∏ x ∈ childMarks S false,
-              g (butterflyPerm d (decodeButterfly d c) x))
-            (fun c : SwitchIndex d → Bool => ∏ x ∈ childMarks S true,
-              g (butterflyPerm d (decodeButterfly d c) x))
-      _ ≤ (finiteMean g)^(childMarks S false).card * (finiteMean g)^(childMarks S true).card :=
-        mul_le_mul (ih (childMarks S false) g hg) (ih (childMarks S true) g hg)
-          (finiteMean_nonneg fun c => Finset.prod_nonneg (fun x _ => hg _))
-          (pow_nonneg (finiteMean_nonneg hg) _)
-      _ = (finiteMean p)^S.card := by
-        rw [←pow_add,card_childMarks_add,finiteMean_headAverage]
-
-noncomputable def inverseButterflyMarks (d : ℕ) (A : Finset (Card d))
-    (ω : SwitchIndex d → Bool) : Finset (Card d) :=
-  A.image (butterflyPerm d (decodeButterfly d ω)).symm
-
-lemma inverseButterflyMarks_inclusion (d : ℕ) (A S : Finset (Card d)) :
-    inclusionMoment (inverseButterflyMarks d A) S ≤
-      ((A.card : ℝ)/(2:ℝ)^d)^S.card := by
-  classical
-  let p : Card d → ℝ := fun x => if x ∈ A then 1 else 0
-  have hm : finiteMean p = (A.card : ℝ)/(2:ℝ)^d := by
-    simp only [finiteMean,p,card_positions,Nat.cast_pow,Nat.cast_ofNat]
-    congr 1
-    simp
-  have he : inclusionMoment (inverseButterflyMarks d A) S =
-      finiteMean (fun ω : SwitchIndex d → Bool =>
-        ∏ x ∈ S, p (butterflyPerm d (decodeButterfly d ω) x)) := by
-    rw [inclusionMoment_eq]
-    apply finiteMean_congr
-    intro ω
-    simp only [inverseButterflyMarks,subset_image_equiv_iff,Equiv.symm_symm,indicator_subset_prod,
-      Finset.prod_image (butterflyPerm d (decodeButterfly d ω)).injective.injOn,p]
-  rw [he]
-  simpa only [hm] using butterfly_product_bound d S p (fun x => by dsimp [p]; split_ifs <;> norm_num)
-
-namespace PairRouting
-
-section
-variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
-
-noncomputable def occupied (x : ι → Bool × α) : Finset α :=
-  Finset.univ.image (fun i => (x i).2)
-
-omit [Fintype α] in
-lemma mem_occupied (x : ι → Bool × α) (y : α) : y ∈ occupied x ↔ ∃ i, (x i).2 = y := by
-  simp [occupied]
-
-def required (x : ι → Bool × α) (c : ι → Bool) (i : ι) : Bool := Bool.xor (x i).1 (c i)
-
-def Compatible (x : ι → Bool × α) (c : ι → Bool) : Prop :=
-  ∀ i j, (x i).2 = (x j).2 → required x c i = required x c j
-
-def colors (x : ι → Bool × α) (ξ : α → Bool) (i : ι) : Bool :=
-  Bool.xor (x i).1 (ξ (x i).2)
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma required_colors (x : ι → Bool × α) (ξ : α → Bool) (i : ι) :
-    required x (colors x ξ) i = ξ (x i).2 := by
-  simp only [required,colors]
-  cases (x i).1 <;> cases ξ (x i).2 <;> rfl
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma colors_compatible (x : ι → Bool × α) (ξ : α → Bool) : Compatible x (colors x ξ) := by
-  intro i j he
-  simp only [required_colors,he]
-
-noncomputable def chosenCoin (x : ι → Bool × α) (c : ι → Bool) (y : α) : Bool := by
-  classical
-  exact if h : ∃ i, (x i).2 = y then required x c (Classical.choose h) else false
-
-omit [Fintype α] in
-lemma chosenCoin_apply (x : ι → Bool × α) (c : ι → Bool) (hc : Compatible x c) (i : ι) :
-    chosenCoin x c (x i).2 = required x c i := by
-  classical
-  simp only [chosenCoin,dite_eq_left (show ∃ j, (x j).2 = (x i).2 from ⟨i,rfl⟩)]
-  exact hc _ i (Classical.choose_spec (show ∃ j, (x j).2 = (x i).2 from ⟨i,rfl⟩))
-
-omit [Fintype α] in
-lemma colors_eq_iff (x : ι → Bool × α) (c : ι → Bool) (hc : Compatible x c) (ξ : α → Bool) :
-    colors x ξ = c ↔ ∀ y ∈ occupied x, ξ y = chosenCoin x c y := by
-  constructor
-  · intro he y hy
-    obtain ⟨i,rfl⟩ := (mem_occupied x y).mp hy
-    rw [chosenCoin_apply x c hc i]
-    rw [←he,required_colors]
-  · intro he
-    funext i
-    apply (xor_eq_iff _ _ _).mpr
-    exact (he _ ((mem_occupied x _).mpr ⟨i,rfl⟩)).trans (chosenCoin_apply x c hc i)
-
-noncomputable instance (x : ι → Bool × α) (c : ι → Bool) : Decidable (Compatible x c) :=
-  Classical.propDecidable _
-
-lemma colors_probability (x : ι → Bool × α) (c : ι → Bool) :
-    finiteMean (fun ξ : α → Bool => if colors x ξ = c then (1 : ℝ) else 0) =
-      if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0 := by
-  classical
-  by_cases hc : Compatible x c
-  · rw [ite_eq_left hc]
-    have he := fun ξ => colors_eq_iff x c hc ξ
-    simp only [he]
-    exact mean_coinCylinder (occupied x) (chosenCoin x c)
-  · rw [ite_eq_right hc]
-    have he : ∀ ξ, colors x ξ ≠ c := fun ξ hh => hc (hh ▸ colors_compatible x ξ)
-    simp only [he,ite_false,finiteMean_const]
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma Compatible.tail_injective {x : ι ↪ Bool × α} {c : ι → Bool} (hc : Compatible x c) (b : Bool) :
-    Function.Injective (fun i : {i : ι // c i = b} => (x i.val).2) := by
-  intro i j he
-  apply Subtype.ext
-  apply x.injective
-  apply Prod.ext
-  · have hh := hc i j he
-    simp only [required,i.property,j.property] at hh
-    cases h₀ : (x i.val).1 <;> cases h₁ : (x j.val).1 <;> cases b <;> simp_all
-  · exact he
-
-def childEmbedding (x : ι ↪ Bool × α) (c : ι → Bool) (hc : Compatible x c) (b : Bool) :
-    {i : ι // c i = b} ↪ α := ⟨fun i => (x i.val).2,hc.tail_injective b⟩
-
-
-def lift (p : Bool → Equiv.Perm α) : Equiv.Perm (Bool × α) where
-  toFun x := (x.1,p x.1 x.2)
-  invFun x := (x.1,(p x.1).symm x.2)
-  left_inv x := by simp
-  right_inv x := by simp
-
-def sandwich (ξ η : α → Bool) (p : Bool → Equiv.Perm α) : Equiv.Perm (Bool × α) :=
-  pairLayer η * lift p * pairLayer ξ
-
-def ChildMatches (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) : Prop :=
-  ∀ i, p (c i) (x i).2 = (y i).2
-
-noncomputable instance (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
-    Decidable (ChildMatches p x y c) := Classical.propDecidable _
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma sandwich_matches_iff (ξ η : α → Bool) (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) :
-    (∀ i, sandwich ξ η p (x i) = y i) ↔
-      colors x ξ = colors y η ∧ ChildMatches p x y (colors x ξ) := by
-  have he (i : ι) : sandwich ξ η p (x i) = y i ↔
-      colors x ξ i = colors y η i ∧ p (colors x ξ i) (x i).2 = (y i).2 := by
-    change pairLayer η (lift p (pairLayer ξ (x i))) = y i ↔ _
-    rw [←Equiv.eq_symm_apply]
-    change (colors x ξ i,p (colors x ξ i) (x i).2) = (colors y η i,(y i).2) ↔ _
-    exact ⟨Prod.mk.inj,fun h => Prod.ext h.1 h.2⟩
-  simp only [he,forall_and,←funext_iff,ChildMatches]
-
-lemma sandwich_probability [DecidableEq ι] (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) :
-    finiteMean (fun coins : (α → Bool) × (α → Bool) =>
-      if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then (1:ℝ) else 0) =
-    ∑ c : ι → Bool,
-      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
-      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) *
-      (if ChildMatches p x y c then 1 else 0) := by
-  classical
-  have he (coins : (α → Bool) × (α → Bool)) :
-      (if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then (1:ℝ) else 0) =
-      ∑ c : ι → Bool, (if colors x coins.1 = c then (1:ℝ) else 0) *
-        (if colors y coins.2 = c then 1 else 0) * (if ChildMatches p x y c then 1 else 0) := by
-    rw [Finset.sum_eq_single (colors x coins.1)]
-    · by_cases hc : colors x coins.1 = colors y coins.2
-      · simp only [sandwich_matches_iff,hc,true_and,↓reduceIte,one_mul]
-      · simp only [sandwich_matches_iff,hc,false_and,↓reduceIte,one_mul,Ne.symm hc,zero_mul]
-    · intro c _ hc
-      simp only [Ne.symm hc,ite_false,zero_mul]
-    · simp
-  simp_rw [he]
-  rw [finiteMean_sum]
-  apply Finset.sum_congr rfl
-  intro c _
-  rw [finiteMean_mul_const]
-  rw [finiteMean_prod_mul
-    (fun ξ : α → Bool => if colors x ξ = c then (1:ℝ) else 0)
-    (fun η : α → Bool => if colors y η = c then (1:ℝ) else 0),
-    colors_probability,colors_probability]
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma childMatches_split (p₀ p₁ : Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
-    ChildMatches (fun b => if b then p₁ else p₀) x y c ↔
-      (∀ i : {i // c i = false}, p₀ (x i.val).2 = (y i.val).2) ∧
-      (∀ i : {i // c i = true}, p₁ (x i.val).2 = (y i.val).2) := by
-  constructor
-  · intro h
-    constructor
-    · intro i
-      simpa only [i.property,Bool.false_eq_true,↓reduceIte] using h i.val
-    · intro i
-      simpa only [i.property,↓reduceIte] using h i.val
-  · rintro ⟨h₀,h₁⟩ i
-    cases hi : c i
-    · simpa only [hi,Bool.false_eq_true,↓reduceIte] using h₀ ⟨i,hi⟩
-    · simpa only [hi,↓reduceIte] using h₁ ⟨i,hi⟩
-
-omit [Fintype α] in
-lemma childMatches_mean {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
-    (P : Ω → Equiv.Perm α) (Q : Ω' → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
-    finiteMean (fun ω : Ω × Ω' =>
-      if ChildMatches (fun b => if b then Q ω.2 else P ω.1) x y c then (1:ℝ) else 0) =
-      tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
-        (fun i : {i // c i = false} => (y i.val).2) *
-      tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
-        (fun i : {i // c i = true} => (y i.val).2) := by
-  classical
-  let f : Ω → ℝ := fun ω => if ∀ i : {i // c i = false}, P ω (x i.val).2 = (y i.val).2 then 1 else 0
-  let g : Ω' → ℝ := fun ω => if ∀ i : {i // c i = true}, Q ω (x i.val).2 = (y i.val).2 then 1 else 0
-  have he (ω : Ω × Ω') :
-      (if ChildMatches (fun b => if b then Q ω.2 else P ω.1) x y c then (1:ℝ) else 0) = f ω.1*g ω.2 := by
-    simp only [childMatches_split,f,g]
-    split_ifs <;> simp_all
-  simp_rw [he]
-  exact finiteMean_prod_mul f g
-
-lemma sandwich_mean [DecidableEq ι] {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
-    (P : Ω → Equiv.Perm α) (Q : Ω' → Equiv.Perm α) (x y : ι → Bool × α) :
-    finiteMean (fun ω : Ω × Ω' => finiteMean (fun coins : (α → Bool) × (α → Bool) =>
-      if ∀ i, sandwich coins.1 coins.2 (fun b => if b then Q ω.2 else P ω.1) (x i) = y i
-      then (1:ℝ) else 0)) =
-    ∑ c : ι → Bool,
-      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
-      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) *
-      (tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
-        (fun i : {i // c i = false} => (y i.val).2) *
-      tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
-        (fun i : {i // c i = true} => (y i.val).2)) := by
-  classical
-  simp_rw [sandwich_probability]
-  rw [finiteMean_sum]
-  apply Finset.sum_congr rfl
-  intro c _
-  simp only [mul_comm ((if Compatible x c then _ else _) * _),finiteMean_mul_const,
-    childMatches_mean]
-
-
-abbrev BothAssignments (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :=
-  {c : ι → Bool // Compatible e c ∧ Compatible (fun i => lift p (e i)) c}
-
-noncomputable instance (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    Fintype (BothAssignments e p) := Fintype.ofFinite _
-
-noncomputable def labelSet (e : ι ↪ Bool × α) : Finset (Bool × α) := Finset.univ.image e
-
-def alternatingProjection (p : Bool → Equiv.Perm α) : Equiv.Perm α := (p false).symm * p true
-
-noncomputable def assignmentColoring (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α)
-    (c : BothAssignments e p) : CycleColoring.OppositeColoring
-      (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e) := by
-  classical
-  refine ⟨Function.extend e c.val (fun _ => false),?_,?_⟩
-  · intro z hz
-    apply Function.extend_apply'
-    rintro ⟨i,rfl⟩
-    exact hz (Finset.mem_image.mpr ⟨i,Finset.mem_univ _,rfl⟩)
-  · intro z hz hnext
-    obtain ⟨i,_,hi⟩ := Finset.mem_image.mp hz
-    obtain ⟨j,_,hj⟩ := Finset.mem_image.mp hnext
-    have hfi := e.injective.extend_apply c.val (fun _ => false) i
-    have hfj := e.injective.extend_apply c.val (fun _ => false) j
-    rw [hi] at hfi
-    rw [hj] at hfj
-    rw [hfi,hfj]
-    rcases z with ⟨b,a⟩
-    cases b
-    · have hj' : e j = (true,a) := hj
-      have ht : (e i).2 = (e j).2 := by rw [hi,hj']
-      have hh := c.property.1 i j ht
-      simp only [required,hi,hj'] at hh
-      revert hh
-      generalize c.val i = bi
-      generalize c.val j = bj
-      cases bi <;> cases bj <;> decide
-    · have hj' : e j = (false,alternatingProjection p a) := hj
-      have ht : (lift p (e i)).2 = (lift p (e j)).2 := by
-        simp only [hi,hj',lift,Equiv.coe_fn_mk,alternatingProjection,Equiv.Perm.coe_mul,
-          Function.comp_apply,Equiv.apply_symm_apply]
-      have hh := c.property.2 i j ht
-      simp only [required,hi,hj',lift,Equiv.coe_fn_mk] at hh
-      revert hh
-      generalize c.val i = bi
-      generalize c.val j = bj
-      cases bi <;> cases bj <;> decide
-
-omit [Fintype α] in
-lemma assignmentColoring_injective (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    Function.Injective (assignmentColoring e p) := by
-  intro c c' he
-  apply Subtype.ext
-  funext i
-  have hh := congrFun (congrArg Subtype.val he) (e i)
-  change Function.extend e c.val (fun _ => false) (e i) =
-    Function.extend e c'.val (fun _ => false) (e i) at hh
-  simpa only [e.injective.extend_apply] using hh
-
-lemma bothAssignments_card (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    Fintype.card (BothAssignments e p) ≤
-      2 ^ ((boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true).card +
-        (boolFiber (labelSet e) false ∪ (boolFiber (labelSet e) true).image
-          (alternatingProjection p)).card -
-        ((boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card) +
-        Fintype.card (RoutingNetwork.SelectedCycle (alternatingProjection p) (boolFiber (labelSet e) false))) := by
-  have h := compatible_assignment_bound (alternatingProjection p)
-    (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
-  rw [childJoin_boolFiber] at h
-  exact (Fintype.card_le_of_injective _ (assignmentColoring_injective e p)).trans h
-
-omit [Fintype α] in
-lemma occupied_labelSet (e : ι ↪ Bool × α) : occupied e = (labelSet e).image Prod.snd := by
-  classical
-  simp only [occupied,labelSet,Finset.image_image,Function.comp_def]
-
-omit [Fintype ι] [Fintype α] in
-lemma image_snd_childJoin (A B : Finset α) : (childJoin A B).image Prod.snd = A ∪ B := by
-  classical
-  simp only [childJoin,Finset.image_union,Finset.image_image,Function.comp_def,Finset.image_id']
-
-omit [Fintype ι] [Fintype α] in
-lemma image_lift_childJoin (p : Bool → Equiv.Perm α) (A B : Finset α) :
-    (childJoin A B).image (fun x => (lift p x).2) = A.image (p false) ∪ B.image (p true) := by
-  classical
-  simp only [childJoin,Finset.image_union,Finset.image_image,Function.comp_def,lift,Equiv.coe_fn_mk]
-
-omit [Fintype α] in
-lemma occupied_lift_card (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    (occupied (fun i => lift p (e i))).card =
-      (boolFiber (labelSet e) false ∪ (boolFiber (labelSet e) true).image (alternatingProjection p)).card := by
-  classical
-  have he : occupied (fun i => lift p (e i)) = (labelSet e).image (fun x => (lift p x).2) := by
-    simp only [occupied,labelSet,Finset.image_image,Function.comp_def]
-  rw [he,←childJoin_boolFiber (labelSet e),image_lift_childJoin]
-  simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
-  let A := boolFiber (labelSet e) false
-  let B := boolFiber (labelSet e) true
-  change (A.image (p false) ∪ B.image (p true)).card = (A ∪ B.image (alternatingProjection p)).card
-  have hh : A.image (p false) ∪ B.image (p true) =
-      (A ∪ B.image (alternatingProjection p)).image (p false) := by
-    simp only [Finset.image_union,Finset.image_image,Function.comp_def,
-      alternatingProjection,Equiv.Perm.coe_mul,Equiv.apply_symm_apply]
-  rw [hh,Finset.card_image_of_injective _ (p false).injective]
-
-lemma bothAssignments_occupied_bound (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    Fintype.card (BothAssignments e p) ≤
-      2 ^ ((occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι +
-        Fintype.card (RoutingNetwork.SelectedCycle (alternatingProjection p) (boolFiber (labelSet e) false))) := by
-  have he : (occupied e).card =
-      (boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true).card := by
-    rw [occupied_labelSet,←childJoin_boolFiber (labelSet e),image_snd_childJoin]
-    simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
-  have hc : (boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card = Fintype.card ι := by
-    rw [←card_childJoin,childJoin_boolFiber,labelSet,Finset.card_image_of_injective _ e.injective,
-      Finset.card_univ]
-  simpa only [he,occupied_lift_card,←hc] using bothAssignments_card e p
-
-omit [Fintype ι] [Fintype α] [DecidableEq α] in
-lemma compatible_pairLayer_iff (x : ι → Bool × α) (c : ι → Bool) (ξ : α → Bool) :
-    Compatible (fun i => pairLayer ξ (x i)) c ↔ Compatible x c := by
-  have he (i j : ι) (ht : (x i).2 = (x j).2) :
-      required (fun i => pairLayer ξ (x i)) c i = required (fun i => pairLayer ξ (x i)) c j ↔
-      required x c i = required x c j := by
-    simp only [required,pairLayer,Equiv.coe_fn_mk,ht]
-    cases (x i).1 <;> cases (x j).1 <;> cases c i <;> cases c j <;> cases ξ (x j).2 <;> decide
-  constructor
-  · intro h i j ht
-    exact (he i j ht).mp (h i j ht)
-  · intro h i j ht
-    exact (he i j ht).mpr (h i j ht)
-
-
-noncomputable def alternatingCycles (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) : ℕ :=
-  Fintype.card (RoutingNetwork.SelectedCycle
-    (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e))
-
-omit [Fintype α] in
-lemma occupied_eq_fibers (e : ι ↪ Bool × α) :
-    occupied e = boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true := by
-  rw [occupied_labelSet,←childJoin_boolFiber (labelSet e),image_snd_childJoin]
-  simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
-
-omit [Fintype α] in
-lemma card_labelSet (e : ι ↪ Bool × α) : (labelSet e).card = Fintype.card ι := by
-  classical
-  exact (Finset.card_image_of_injective _ e.injective).trans (Finset.card_univ)
-
-omit [Fintype α] in
-lemma occupied_card_double (e : ι ↪ Bool × α) : Fintype.card ι ≤ 2*(occupied e).card := by
-  have h₀ : (boolFiber (labelSet e) false).card ≤ (occupied e).card := by
-    rw [occupied_eq_fibers]
-    exact Finset.card_le_card Finset.subset_union_left
-  have h₁ : (boolFiber (labelSet e) true).card ≤ (occupied e).card := by
-    rw [occupied_eq_fibers]
-    exact Finset.card_le_card Finset.subset_union_right
-  have he : (boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card = Fintype.card ι := by
-    rw [←card_childJoin,childJoin_boolFiber,card_labelSet]
-  omega
-
-lemma bothAssignments_exact_bound (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    Fintype.card (BothAssignments e p) ≤
-      2 ^ ((occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι +
-        alternatingCycles e p) := by
-  have h := CycleColoring.card_opposite_le
-    (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e)
-    Prod.fst (CycleColoring.alternatingPerm_flip (alternatingProjection p))
-  have hb : (CycleColoring.boundary (CycleColoring.alternatingPerm (alternatingProjection p))
-      (labelSet e)).card =
-      (occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι := by
-    have hh := alternating_boundary_card (alternatingProjection p)
-      (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
-    rw [childJoin_boolFiber] at hh
-    rw [occupied_eq_fibers,occupied_lift_card]
-    have he := card_childJoin (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
-    rw [childJoin_boolFiber,card_labelSet] at he
-    simpa only [he] using hh
-  rw [hb] at h
-  exact (Fintype.card_le_of_injective _ (assignmentColoring_injective e p)).trans h
-
-abbrev Assignments (x y : ι → Bool × α) :=
-  {c : ι → Bool // Compatible x c ∧ Compatible y c}
-
-noncomputable instance (x y : ι → Bool × α) : Fintype (Assignments x y) := Fintype.ofFinite _
-
-def switchedEmbedding (e : ι ↪ Bool × α) (ξ : α → Bool) : ι ↪ Bool × α :=
-  e.trans (pairLayer ξ).toEmbedding
-
-def routedEmbedding (e : ι ↪ Bool × α) (ξ η : α → Bool) (p : Bool → Equiv.Perm α) :
-    ι ↪ Bool × α := e.trans (sandwich ξ η p).toEmbedding
-
-omit [Fintype α] in
-lemma occupied_pairLayer (x : ι → Bool × α) (ξ : α → Bool) :
-    occupied (fun i => pairLayer ξ (x i)) = occupied x := rfl
-
-noncomputable def assignmentsEquiv (e : ι ↪ Bool × α) (ξ η : α → Bool)
-    (p : Bool → Equiv.Perm α) :
-    Assignments e (routedEmbedding e ξ η p) ≃ BothAssignments (switchedEmbedding e ξ) p := by
-  apply Equiv.subtypeEquivRight
-  intro c
-  change Compatible e c ∧ Compatible (fun i => pairLayer η (lift p (pairLayer ξ (e i)))) c ↔ _
-  rw [compatible_pairLayer_iff]
-  exact and_congr (compatible_pairLayer_iff e c ξ).symm Iff.rfl
-
-lemma assignments_actual_bound (e : ι ↪ Bool × α) (ξ η : α → Bool) (p : Bool → Equiv.Perm α) :
-    Fintype.card (Assignments e (routedEmbedding e ξ η p)) ≤
-      2 ^ ((occupied e).card + (occupied (routedEmbedding e ξ η p)).card - Fintype.card ι +
-        alternatingCycles (switchedEmbedding e ξ) p) := by
-  rw [Fintype.card_congr (assignmentsEquiv e ξ η p)]
-  exact bothAssignments_exact_bound (switchedEmbedding e ξ) p
-
-omit [Fintype α] in
-lemma assignments_actual_exponent_nonneg (e : ι ↪ Bool × α) (ξ η : α → Bool)
-    (p : Bool → Equiv.Perm α) :
-    Fintype.card ι ≤ (occupied e).card + (occupied (routedEmbedding e ξ η p)).card := by
-  have h₀ := occupied_card_double e
-  have h₁ := occupied_card_double (routedEmbedding e ξ η p)
-  omega
-
-end
-variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α] [DecidableEq ι]
-
-noncomputable def compatibleSet (x y : ι → Bool × α) : Finset (ι → Bool) := by
-  classical
-  exact Finset.univ.filter (fun c => Compatible x c ∧ Compatible y c)
-
-omit [Fintype α] [DecidableEq α] in
-lemma mem_compatibleSet (x y : ι → Bool × α) (c : ι → Bool) :
-    c ∈ compatibleSet x y ↔ Compatible x c ∧ Compatible y c := by
-  classical
-  simp only [compatibleSet,Finset.mem_filter,Finset.mem_univ,true_and]
-
-omit [Fintype α] [DecidableEq α] in
-lemma card_compatibleSet (x y : ι → Bool × α) :
-    (compatibleSet x y).card = Fintype.card (Assignments x y) := by
-  classical
-  exact (Fintype.card_subtype _).symm
-
-noncomputable def outerWeight (x y : ι → Bool × α) : ℝ :=
-  1/(2:ℝ)^((occupied x).card+(occupied y).card)
-
-omit [Fintype α] [DecidableEq ι] in
-lemma outerWeight_pos (x y : ι → Bool × α) : 0 < outerWeight x y := by
-  unfold outerWeight
-  positivity
-
-omit [Fintype α] in
-lemma sum_colors_factor (x y : ι → Bool × α) (f : (ι → Bool) → ℝ) :
-    (∑ c : ι → Bool,
-      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
-      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) * f c) =
-      outerWeight x y * ∑ c ∈ compatibleSet x y, f c := by
-  classical
-  rw [compatibleSet,Finset.sum_filter,Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro c _
-  by_cases hx : Compatible x c <;> by_cases hy : Compatible y c
-  · simp only [hx,hy,true_and,↓reduceIte,outerWeight,pow_add,one_div_mul_one_div]
-  · simp only [hx,hy,and_false,↓reduceIte,mul_zero,zero_mul]
-  · simp only [hx,hy,false_and,↓reduceIte,mul_zero,zero_mul]
-  · simp only [hx,hy,false_and,↓reduceIte,mul_zero,zero_mul]
-
-lemma sandwich_weighted_probability (p : Bool → Equiv.Perm α) (x y : ι → Bool × α)
-    (W : (ι → Bool) → ℝ) :
-    finiteMean (fun coins : (α → Bool) × (α → Bool) =>
-      if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then W (colors x coins.1) else 0) =
-    outerWeight x y * ∑ c ∈ compatibleSet x y, if ChildMatches p x y c then W c else 0 := by
-  classical
-  have he (coins : (α → Bool) × (α → Bool)) :
-      (if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then W (colors x coins.1) else 0) =
-      ∑ c : ι → Bool, (if colors x coins.1 = c then (1:ℝ) else 0) *
-        (if colors y coins.2 = c then 1 else 0) * (if ChildMatches p x y c then W c else 0) := by
-    rw [Finset.sum_eq_single (colors x coins.1)]
-    · by_cases hc : colors x coins.1 = colors y coins.2
-      · simp only [sandwich_matches_iff,hc,true_and,↓reduceIte,one_mul]
-      · simp only [sandwich_matches_iff,hc,false_and,↓reduceIte,one_mul,Ne.symm hc,zero_mul]
-    · intro c _ hc
-      simp only [Ne.symm hc,ite_false,zero_mul]
-    · simp
-  simp_rw [he]
-  rw [finiteMean_sum,←sum_colors_factor]
-  apply Finset.sum_congr rfl
-  intro c _
-  rw [finiteMean_mul_const,finiteMean_prod_mul
-    (fun ξ : α → Bool => if colors x ξ = c then (1:ℝ) else 0)
-    (fun η : α → Bool => if colors y η = c then (1:ℝ) else 0),
-    colors_probability,colors_probability]
-
-omit [Fintype α] [DecidableEq ι] in
-lemma tupleProbability_nonneg {Ω β : Type*} [Fintype Ω] [DecidableEq β]
-    (P : Ω → Equiv.Perm β) (x y : ι → β) : 0 ≤ tupleProbability P x y := by
-  classical
-  exact finiteMean_nonneg (fun ω => by split_ifs <;> norm_num)
-
-lemma sandwich_power_bound {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
-    (P : Ω → Equiv.Perm α) (Q : Ω' → Equiv.Perm α) (x y : ι → Bool × α)
-    (a : ℝ) (ha : 0 ≤ a) :
-    (finiteMean (fun ω : Ω × Ω' => finiteMean (fun coins : (α → Bool) × (α → Bool) =>
-      if ∀ i, sandwich coins.1 coins.2 (fun b => if b then Q ω.2 else P ω.1) (x i) = y i
-      then (1:ℝ) else 0)))^(1+a) ≤
-    (outerWeight x y)^(1+a) * ((compatibleSet x y).card:ℝ)^a *
-      ∑ c ∈ compatibleSet x y,
-      (tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
-        (fun i : {i // c i = false} => (y i.val).2))^(1+a) *
-      (tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
-        (fun i : {i // c i = true} => (y i.val).2))^(1+a) := by
-  classical
-  rw [sandwich_mean,sum_colors_factor]
-  let f (c : ι → Bool) : ℝ :=
-      tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
-        (fun i : {i // c i = false} => (y i.val).2) *
-      tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
-        (fun i : {i // c i = true} => (y i.val).2)
-  have hf (c : ι → Bool) : 0 ≤ f c := mul_nonneg (tupleProbability_nonneg _ _ _)
-    (tupleProbability_nonneg _ _ _)
-  rw [Real.mul_rpow (le_of_lt (outerWeight_pos _ _)) (Finset.sum_nonneg (fun c _ => hf c))]
-  have hh := Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg (compatibleSet x y)
-    (show 1 ≤ 1+a by linarith) (fun c _ => hf c)
-  have he : 1+a-1=a := by ring
-  rw [he] at hh
-  calc
-    _ ≤ (outerWeight x y)^(1+a) *
-        (((compatibleSet x y).card:ℝ)^a * ∑ c ∈ compatibleSet x y, (f c)^(1+a)) :=
-      mul_le_mul_of_nonneg_left hh (Real.rpow_nonneg (le_of_lt (outerWeight_pos _ _)) _)
-    _ = _ := by
-      rw [←mul_assoc]
-      congr 1
-      apply Finset.sum_congr rfl
-      intro c _
-      exact Real.mul_rpow (tupleProbability_nonneg _ _ _) (tupleProbability_nonneg _ _ _)
-
-end PairRouting
-
-end Thorp
-
-end ThorpNine.Tail
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Tail.ParallelFollow
+
+namespace OAI
+
+namespace ThorpNine.Tail
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+namespace CycleColoring
+variable {α : Type*} [Fintype α] [DecidableEq α]
+omit [Fintype α] in
+lemma bipartitionFlip_injective (p : Equiv.Perm α) (S : Finset α)
+    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x)) :
+    Function.Injective (bipartitionFlip p S χ hχ) := by
+  classical
+  intro f g he
+  apply Subtype.ext
+  funext x
+  by_cases hx : x ∈ S
+  · have hh := congrFun (congrArg Subtype.val he) x
+    change (if x ∈ S then Bool.xor (f.val x) (χ x) else false) =
+      (if x ∈ S then Bool.xor (g.val x) (χ x) else false) at hh
+    simp only [ite_eq_left hx] at hh
+    cases hfx : f.val x <;> cases hgx : g.val x <;> cases hcx : χ x <;> simp_all
+  · exact (f.property.1 x hx).trans (g.property.1 x hx).symm
+
+lemma card_opposite_le (p : Equiv.Perm α) (S : Finset α)
+    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x)) :
+    Fintype.card (OppositeColoring p S) ≤
+      2 ^ ((boundary p S).card + Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
+  exact (Fintype.card_le_of_injective _ (bipartitionFlip_injective p S χ hχ)).trans
+    (card_coloring_le p S)
+
+def alternatingPerm (p : Equiv.Perm α) : Equiv.Perm (Bool × α) where
+  toFun x := if x.1 then (false,p x.2) else (true,x.2)
+  invFun x := if x.1 then (false,x.2) else (true,p.symm x.2)
+  left_inv x := by rcases x with ⟨b,x⟩; cases b <;> simp
+  right_inv x := by rcases x with ⟨b,x⟩; cases b <;> simp
+
+omit [Fintype α] [DecidableEq α] in
+lemma alternatingPerm_flip (p : Equiv.Perm α) (x : Bool × α) :
+    (alternatingPerm p x).1 = !x.1 := by
+  rcases x with ⟨b,x⟩
+  cases b <;> rfl
+
+omit [Fintype α] [DecidableEq α] in
+lemma alternating_step_sameCycle (p : Equiv.Perm α) (x : Bool × α) :
+    p.SameCycle x.2 (alternatingPerm p x).2 := by
+  rcases x with ⟨b,x⟩
+  cases b
+  · exact Equiv.Perm.SameCycle.refl p x
+  · exact (Equiv.Perm.SameCycle.refl p x).apply_right
+
+omit [DecidableEq α] in
+lemma alternating_sameCycle_snd (p : Equiv.Perm α) {x y : Bool × α}
+    (h : (alternatingPerm p).SameCycle x y) : p.SameCycle x.2 y.2 := by
+  obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
+  have hp (k : ℕ) : p.SameCycle x.2 (((alternatingPerm p)^k) x).2 := by
+    induction k with
+    | zero => exact Equiv.Perm.SameCycle.refl p x.2
+    | succ k ih =>
+      rw [pow_succ',Equiv.Perm.mul_apply]
+      exact ih.trans (alternating_step_sameCycle p _)
+  exact hn ▸ hp n
+
+omit [DecidableEq α] in
+lemma alternating_zero_sameCycle (p : Equiv.Perm α) {x y : α}
+    (h : p.SameCycle x y) : (alternatingPerm p).SameCycle (false,x) (false,y) := by
+  obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
+  have hp (k : ℕ) : (alternatingPerm p).SameCycle (false,x) (false,(p^k) x) := by
+    induction k with
+    | zero => exact Equiv.Perm.SameCycle.refl _ _
+    | succ k ih =>
+      have hs := ih.apply_right.apply_right
+      simpa only [pow_succ',Equiv.Perm.mul_apply,alternatingPerm,Equiv.coe_fn_mk,
+        Bool.false_eq_true,↓reduceIte] using hs
+  simpa only [hn] using hp n
+
+omit [Fintype α] [DecidableEq α] in
+lemma alternating_sameCycle_zero (p : Equiv.Perm α) (x : Bool × α) :
+    (alternatingPerm p).SameCycle x (false,x.2) := by
+  rcases x with ⟨b,x⟩
+  cases b
+  · exact Equiv.Perm.SameCycle.refl _ _
+  · exact ((Equiv.Perm.SameCycle.refl (alternatingPerm p) (false,x)).apply_right).symm
+
+omit [DecidableEq α] in
+lemma alternating_sameCycle_iff (p : Equiv.Perm α) (x y : Bool × α) :
+    (alternatingPerm p).SameCycle x y ↔ p.SameCycle x.2 y.2 := by
+  constructor
+  · exact alternating_sameCycle_snd p
+  · intro h
+    exact (alternating_sameCycle_zero p x).trans
+      ((alternating_zero_sameCycle p h).trans (alternating_sameCycle_zero p y).symm)
+
+lemma orbit_alternating (p : Equiv.Perm α) (x : Bool × α) :
+    orbitSet (alternatingPerm p) x = childJoin (orbitSet p x.2) (orbitSet p x.2) := by
+  ext ⟨b,y⟩
+  simp only [mem_orbitSet,alternating_sameCycle_iff,mem_childJoin,ite_self]
+
+omit [Fintype α] in
+lemma boolFiber_childJoin (A B : Finset α) (b : Bool) :
+    boolFiber (childJoin A B) b = if b then B else A := by
+  ext x
+  cases b <;> simp only [mem_boolFiber,mem_childJoin,Bool.false_eq_true,↓reduceIte]
+
+noncomputable def projectCycle (p : Equiv.Perm α) (A B : Finset α)
+    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) :
+    RoutingNetwork.SelectedCycle p A := by
+  refine ⟨boolFiber C.val false,?_,?_⟩
+  · obtain ⟨x,hx⟩ := C.property.1
+    refine ⟨x.2,?_⟩
+    rw [hx,orbit_alternating,boolFiber_childJoin]
+    rfl
+  · intro x hx
+    exact (mem_childJoin A B false x).mp (C.property.2 ((mem_boolFiber _ _ _).mp hx))
+
+lemma selected_alternating_eq (p : Equiv.Perm α) (A B : Finset α)
+    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) :
+    C.val = childJoin (projectCycle p A B C).val (projectCycle p A B C).val := by
+  change C.val = childJoin (boolFiber C.val false) (boolFiber C.val false)
+  obtain ⟨x,hx⟩ := C.property.1
+  rw [hx,orbit_alternating,boolFiber_childJoin]
+  rfl
+
+lemma projectCycle_injective (p : Equiv.Perm α) (A B : Finset α) :
+    Function.Injective (projectCycle p A B) := by
+  intro C D he
+  apply Subtype.ext
+  rw [selected_alternating_eq p A B C,selected_alternating_eq p A B D,he]
+
+lemma card_alternating_cycles_le (p : Equiv.Perm α) (A B : Finset α) :
+    Fintype.card (RoutingNetwork.SelectedCycle (alternatingPerm p) (childJoin A B)) ≤
+      Fintype.card (RoutingNetwork.SelectedCycle p A) :=
+  Fintype.card_le_of_injective _ (projectCycle_injective p A B)
+
+end CycleColoring
+
+section PairAssignmentCount
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+omit [Fintype α] in
+lemma mem_image_perm_iff (p : Equiv.Perm α) (A : Finset α) (y : α) :
+    y ∈ A.image p ↔ p.symm y ∈ A := by
+  constructor
+  · intro h
+    obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp h
+    simpa only [←hxy,Equiv.symm_apply_apply] using hx
+  · intro h
+    exact Finset.mem_image.mpr ⟨p.symm y,h,p.apply_symm_apply y⟩
+
+omit [Fintype α] in
+lemma card_childJoin (A B : Finset α) : (childJoin A B).card = A.card+B.card := by
+  have hd : Disjoint (A.image (fun x => (false,x))) (B.image (fun x => (true,x))) := by
+    apply Finset.disjoint_left.mpr
+    rintro x hx hy
+    obtain ⟨a,ha,rfl⟩ := Finset.mem_image.mp hx
+    obtain ⟨b,hb,hba⟩ := Finset.mem_image.mp hy
+    exact Bool.false_ne_true (congrArg Prod.fst hba).symm
+  have hi (b : Bool) : Function.Injective (fun x : α => (b,x)) :=
+    fun _ _ h => congrArg Prod.snd h
+  rw [childJoin,Finset.card_union_of_disjoint hd,
+    Finset.card_image_of_injective A (hi false),Finset.card_image_of_injective B (hi true)]
+
+omit [Fintype α] in
+lemma alternating_internal (p : Equiv.Perm α) (A B : Finset α) :
+    ((childJoin A B).filter (fun x => (CycleColoring.alternatingPerm p).symm x ∈ childJoin A B)) =
+      childJoin (A ∩ B.image p) (B ∩ A) := by
+  ext ⟨b,x⟩
+  rw [Finset.mem_filter]
+  cases b <;> simp only [mem_childJoin,CycleColoring.alternatingPerm,
+    Equiv.symm_mk,Equiv.coe_fn_mk,Bool.false_eq_true,↓reduceIte,Finset.mem_inter,mem_image_perm_iff]
+
+omit [Fintype α] in
+lemma alternating_boundary_card (p : Equiv.Perm α) (A B : Finset α) :
+    (CycleColoring.boundary (CycleColoring.alternatingPerm p) (childJoin A B)).card =
+      (A ∪ B).card + (A ∪ B.image p).card - (A.card+B.card) := by
+  rw [CycleColoring.boundary_card,alternating_internal,card_childJoin,card_childJoin]
+  have h₀ := Finset.card_union_add_card_inter A B
+  have h₁ := Finset.card_union_add_card_inter A (B.image p)
+  rw [Finset.card_image_of_injective _ p.injective] at h₁
+  have h₂ := Finset.inter_comm B A
+  rw [h₂]
+  omega
+
+lemma compatible_assignment_bound (p : Equiv.Perm α) (A B : Finset α) :
+    Fintype.card (CycleColoring.OppositeColoring (CycleColoring.alternatingPerm p) (childJoin A B)) ≤
+      2 ^ ((A ∪ B).card + (A ∪ B.image p).card - (A.card+B.card) +
+        Fintype.card (RoutingNetwork.SelectedCycle p A)) := by
+  have h := CycleColoring.card_opposite_le (CycleColoring.alternatingPerm p) (childJoin A B)
+    Prod.fst (CycleColoring.alternatingPerm_flip p)
+  rw [alternating_boundary_card] at h
+  exact h.trans (Nat.pow_le_pow_right (by norm_num)
+    (Nat.add_le_add_left (CycleColoring.card_alternating_cycles_le p A B) _))
+
+end PairAssignmentCount
+
+lemma finiteMean_nonneg {Ω : Type*} [Fintype Ω] {f : Ω → ℝ}
+    (h : ∀ ω, 0 ≤ f ω) : 0 ≤ finiteMean f := by
+  exact div_nonneg (Finset.sum_nonneg (fun ω _ => h ω)) (Nat.cast_nonneg _)
+
+lemma boolFiber_headTail (d : ℕ) (S : Finset (Card (d+1))) (b : Bool) :
+    boolFiber (S.image (headTailEquiv d)) b = childMarks S b := by
+  classical
+  ext x
+  rw [mem_boolFiber,mem_childMarks]
+  constructor
+  · intro h
+    obtain ⟨y,hy,he⟩ := Finset.mem_image.mp h
+    have hh := congrArg (headTailEquiv d).symm he
+    rw [Equiv.symm_apply_apply] at hh
+    change y = Fin.cons b x at hh
+    exact hh ▸ hy
+  · intro h
+    exact Finset.mem_image.mpr ⟨Fin.cons b x,h,by rfl⟩
+
+lemma prod_childMarks (d : ℕ) (S : Finset (Card (d+1))) (f : Card (d+1) → ℝ) :
+    (∏ x ∈ S, f x) =
+      (∏ x ∈ childMarks S false, f (Fin.cons false x)) *
+      ∏ x ∈ childMarks S true, f (Fin.cons true x) := by
+  classical
+  have h := prod_boolFibers (S.image (headTailEquiv d)) (fun x => f ((headTailEquiv d).symm x))
+  rw [Finset.prod_image (headTailEquiv d).injective.injOn,
+    boolFiber_headTail,boolFiber_headTail] at h
+  simp only [Equiv.symm_apply_apply] at h
+  exact h
+
+noncomputable def headAverage {d : ℕ} (p : Card (d+1) → ℝ) (y : Card d) : ℝ :=
+  (p (Fin.cons false y)+p (Fin.cons true y))/2
+
+lemma finiteMean_headAverage (d : ℕ) (p : Card (d+1) → ℝ) :
+    finiteMean (headAverage p) = finiteMean p := by
+  rw [finiteMean_equiv (headTailEquiv d),finiteMean_prod,finiteMean_comm]
+  apply finiteMean_congr
+  intro y
+  rw [finiteMean_bool]
+  rfl
+
+lemma pairSwitch_product_bound (d : ℕ) (S : Finset (Card (d+1)))
+    (p : Card (d+1) → ℝ) (hp : ∀ x, 0 ≤ p x) :
+    finiteMean (fun ξ : Card d → Bool => ∏ x ∈ S, p (pairSwitch ξ x)) ≤
+      ∏ x ∈ S, headAverage p (Fin.tail x) := by
+  classical
+  have h : CylinderBound (fun T => ∏ x ∈ T, p ((headTailEquiv d).symm x))
+      (fun x => p ((headTailEquiv d).symm x)) := fun _ => le_rfl
+  have hh := h.fair_parallel (fun x => hp _) (S.image (headTailEquiv d))
+  simp only [Finset.prod_image (pairLayer _).injective.injOn,
+    Finset.prod_image (headTailEquiv d).injective.injOn] at hh
+  convert hh using 1
+  · apply finiteMean_congr
+    intro ξ
+    apply Finset.prod_congr rfl
+    intro x _
+    rfl
+  · apply Finset.prod_congr rfl
+    intro x _
+    change (p (Fin.cons false (Fin.tail x))+p (Fin.cons true (Fin.tail x)))/2 =
+      (p (Fin.cons (x 0) (Fin.tail x))+p (Fin.cons (!(x 0)) (Fin.tail x)))/2
+    cases x 0
+    · rfl
+    · simp only [Bool.not_true]
+      ring
+
+lemma butterflyPerm_step (d : ℕ)
+    (c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (ξ : Card d → Bool) :
+    butterflyPerm (d+1) (decodeButterfly (d+1) ((coinStepEquiv d).symm (c,ξ))) =
+    pairSwitch ξ * childLift (fun b => butterflyPerm d (decodeButterfly d (if b then c.2 else c.1))) := by
+  change pairSwitch (fun y => (coinStepEquiv d).symm (c,ξ) (Sum.inl y)) *
+    childLift (fun b => butterflyPerm d (decodeButterfly d
+      (fun i => (coinStepEquiv d).symm (c,ξ) (Sum.inr (b,i))))) = _
+  apply congrArg₂ (· * ·)
+  · rfl
+  · apply congrArg childLift
+    funext b
+    apply congrArg (fun ω => butterflyPerm d (decodeButterfly d ω))
+    funext i
+    cases b <;> rfl
+
+lemma butterfly_product_bound (d : ℕ) (S : Finset (Card d))
+    (p : Card d → ℝ) (hp : ∀ x, 0 ≤ p x) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      ∏ x ∈ S, p (butterflyPerm d (decodeButterfly d ω) x)) ≤ (finiteMean p)^S.card := by
+  classical
+  induction d with
+  | zero =>
+    have he : p = fun _ => p (fun i => Fin.elim0 i) := by
+      funext x
+      exact congrArg p (Subsingleton.elim _ _)
+    rw [he]
+    simp only [Finset.prod_const,finiteMean_const,le_refl]
+  | succ d ih =>
+    let g : Card d → ℝ := headAverage p
+    have hg : ∀ x, 0 ≤ g x := fun x => div_nonneg (add_nonneg (hp _) (hp _)) (by norm_num)
+    rw [finiteMean_equiv (coinStepEquiv d),finiteMean_prod]
+    calc
+      _ ≤ finiteMean (fun c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool) =>
+        (∏ x ∈ childMarks S false, g (butterflyPerm d (decodeButterfly d c.1) x)) *
+        ∏ x ∈ childMarks S true, g (butterflyPerm d (decodeButterfly d c.2) x)) := by
+        apply finiteMean_mono
+        intro c
+        let f : Equiv.Perm (Card (d+1)) := childLift (fun b =>
+          butterflyPerm d (decodeButterfly d (if b then c.2 else c.1)))
+        have hh := pairSwitch_product_bound d (S.image f) p hp
+        simp only [Finset.prod_image f.injective.injOn] at hh
+        convert hh using 1
+        · apply finiteMean_congr
+          intro ξ
+          apply Finset.prod_congr rfl
+          intro x _
+          rw [butterflyPerm_step]
+          rfl
+        · rw [prod_childMarks]
+          simp only [f,childLift,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
+            Bool.false_eq_true,↓reduceIte]
+          rfl
+      _ = (finiteMean (fun c : SwitchIndex d → Bool =>
+        ∏ x ∈ childMarks S false, g (butterflyPerm d (decodeButterfly d c) x))) *
+        finiteMean (fun c : SwitchIndex d → Bool =>
+        ∏ x ∈ childMarks S true, g (butterflyPerm d (decodeButterfly d c) x)) :=
+          finiteMean_prod_mul
+            (fun c : SwitchIndex d → Bool => ∏ x ∈ childMarks S false,
+              g (butterflyPerm d (decodeButterfly d c) x))
+            (fun c : SwitchIndex d → Bool => ∏ x ∈ childMarks S true,
+              g (butterflyPerm d (decodeButterfly d c) x))
+      _ ≤ (finiteMean g)^(childMarks S false).card * (finiteMean g)^(childMarks S true).card :=
+        mul_le_mul (ih (childMarks S false) g hg) (ih (childMarks S true) g hg)
+          (finiteMean_nonneg fun c => Finset.prod_nonneg (fun x _ => hg _))
+          (pow_nonneg (finiteMean_nonneg hg) _)
+      _ = (finiteMean p)^S.card := by
+        rw [←pow_add,card_childMarks_add,finiteMean_headAverage]
+
+noncomputable def inverseButterflyMarks (d : ℕ) (A : Finset (Card d))
+    (ω : SwitchIndex d → Bool) : Finset (Card d) :=
+  A.image (butterflyPerm d (decodeButterfly d ω)).symm
+
+lemma inverseButterflyMarks_inclusion (d : ℕ) (A S : Finset (Card d)) :
+    inclusionMoment (inverseButterflyMarks d A) S ≤
+      ((A.card : ℝ)/(2:ℝ)^d)^S.card := by
+  classical
+  let p : Card d → ℝ := fun x => if x ∈ A then 1 else 0
+  have hm : finiteMean p = (A.card : ℝ)/(2:ℝ)^d := by
+    simp only [finiteMean,p,card_positions,Nat.cast_pow,Nat.cast_ofNat]
+    congr 1
+    simp
+  have he : inclusionMoment (inverseButterflyMarks d A) S =
+      finiteMean (fun ω : SwitchIndex d → Bool =>
+        ∏ x ∈ S, p (butterflyPerm d (decodeButterfly d ω) x)) := by
+    rw [inclusionMoment_eq]
+    apply finiteMean_congr
+    intro ω
+    simp only [inverseButterflyMarks,subset_image_equiv_iff,Equiv.symm_symm,indicator_subset_prod,
+      Finset.prod_image (butterflyPerm d (decodeButterfly d ω)).injective.injOn,p]
+  rw [he]
+  simpa only [hm] using butterfly_product_bound d S p (fun x => by dsimp [p]; split_ifs <;> norm_num)
+
+namespace PairRouting
+
+section
+variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
+
+noncomputable def occupied (x : ι → Bool × α) : Finset α :=
+  Finset.univ.image (fun i => (x i).2)
+
+omit [Fintype α] in
+lemma mem_occupied (x : ι → Bool × α) (y : α) : y ∈ occupied x ↔ ∃ i, (x i).2 = y := by
+  simp [occupied]
+
+def required (x : ι → Bool × α) (c : ι → Bool) (i : ι) : Bool := Bool.xor (x i).1 (c i)
+
+def Compatible (x : ι → Bool × α) (c : ι → Bool) : Prop :=
+  ∀ i j, (x i).2 = (x j).2 → required x c i = required x c j
+
+def colors (x : ι → Bool × α) (ξ : α → Bool) (i : ι) : Bool :=
+  Bool.xor (x i).1 (ξ (x i).2)
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma required_colors (x : ι → Bool × α) (ξ : α → Bool) (i : ι) :
+    required x (colors x ξ) i = ξ (x i).2 := by
+  simp only [required,colors]
+  cases (x i).1 <;> cases ξ (x i).2 <;> rfl
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma colors_compatible (x : ι → Bool × α) (ξ : α → Bool) : Compatible x (colors x ξ) := by
+  intro i j he
+  simp only [required_colors,he]
+
+noncomputable def chosenCoin (x : ι → Bool × α) (c : ι → Bool) (y : α) : Bool := by
+  classical
+  exact if h : ∃ i, (x i).2 = y then required x c (Classical.choose h) else false
+
+omit [Fintype α] in
+lemma chosenCoin_apply (x : ι → Bool × α) (c : ι → Bool) (hc : Compatible x c) (i : ι) :
+    chosenCoin x c (x i).2 = required x c i := by
+  classical
+  simp only [chosenCoin,dite_eq_left (show ∃ j, (x j).2 = (x i).2 from ⟨i,rfl⟩)]
+  exact hc _ i (Classical.choose_spec (show ∃ j, (x j).2 = (x i).2 from ⟨i,rfl⟩))
+
+omit [Fintype α] in
+lemma colors_eq_iff (x : ι → Bool × α) (c : ι → Bool) (hc : Compatible x c) (ξ : α → Bool) :
+    colors x ξ = c ↔ ∀ y ∈ occupied x, ξ y = chosenCoin x c y := by
+  constructor
+  · intro he y hy
+    obtain ⟨i,rfl⟩ := (mem_occupied x y).mp hy
+    rw [chosenCoin_apply x c hc i]
+    rw [←he,required_colors]
+  · intro he
+    funext i
+    apply (xor_eq_iff _ _ _).mpr
+    exact (he _ ((mem_occupied x _).mpr ⟨i,rfl⟩)).trans (chosenCoin_apply x c hc i)
+
+noncomputable instance (x : ι → Bool × α) (c : ι → Bool) : Decidable (Compatible x c) :=
+  Classical.propDecidable _
+
+lemma colors_probability (x : ι → Bool × α) (c : ι → Bool) :
+    finiteMean (fun ξ : α → Bool => if colors x ξ = c then (1 : ℝ) else 0) =
+      if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0 := by
+  classical
+  by_cases hc : Compatible x c
+  · rw [ite_eq_left hc]
+    have he := fun ξ => colors_eq_iff x c hc ξ
+    simp only [he]
+    exact mean_coinCylinder (occupied x) (chosenCoin x c)
+  · rw [ite_eq_right hc]
+    have he : ∀ ξ, colors x ξ ≠ c := fun ξ hh => hc (hh ▸ colors_compatible x ξ)
+    simp only [he,ite_false,finiteMean_const]
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma Compatible.tail_injective {x : ι ↪ Bool × α} {c : ι → Bool} (hc : Compatible x c) (b : Bool) :
+    Function.Injective (fun i : {i : ι // c i = b} => (x i.val).2) := by
+  intro i j he
+  apply Subtype.ext
+  apply x.injective
+  apply Prod.ext
+  · have hh := hc i j he
+    simp only [required,i.property,j.property] at hh
+    cases h₀ : (x i.val).1 <;> cases h₁ : (x j.val).1 <;> cases b <;> simp_all
+  · exact he
+
+def childEmbedding (x : ι ↪ Bool × α) (c : ι → Bool) (hc : Compatible x c) (b : Bool) :
+    {i : ι // c i = b} ↪ α := ⟨fun i => (x i.val).2,hc.tail_injective b⟩
+
+
+def lift (p : Bool → Equiv.Perm α) : Equiv.Perm (Bool × α) where
+  toFun x := (x.1,p x.1 x.2)
+  invFun x := (x.1,(p x.1).symm x.2)
+  left_inv x := by simp
+  right_inv x := by simp
+
+def sandwich (ξ η : α → Bool) (p : Bool → Equiv.Perm α) : Equiv.Perm (Bool × α) :=
+  pairLayer η * lift p * pairLayer ξ
+
+def ChildMatches (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) : Prop :=
+  ∀ i, p (c i) (x i).2 = (y i).2
+
+noncomputable instance (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
+    Decidable (ChildMatches p x y c) := Classical.propDecidable _
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma sandwich_matches_iff (ξ η : α → Bool) (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) :
+    (∀ i, sandwich ξ η p (x i) = y i) ↔
+      colors x ξ = colors y η ∧ ChildMatches p x y (colors x ξ) := by
+  have he (i : ι) : sandwich ξ η p (x i) = y i ↔
+      colors x ξ i = colors y η i ∧ p (colors x ξ i) (x i).2 = (y i).2 := by
+    change pairLayer η (lift p (pairLayer ξ (x i))) = y i ↔ _
+    rw [←Equiv.eq_symm_apply]
+    change (colors x ξ i,p (colors x ξ i) (x i).2) = (colors y η i,(y i).2) ↔ _
+    exact ⟨Prod.mk.inj,fun h => Prod.ext h.1 h.2⟩
+  simp only [he,forall_and,←funext_iff,ChildMatches]
+
+lemma sandwich_probability [DecidableEq ι] (p : Bool → Equiv.Perm α) (x y : ι → Bool × α) :
+    finiteMean (fun coins : (α → Bool) × (α → Bool) =>
+      if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then (1:ℝ) else 0) =
+    ∑ c : ι → Bool,
+      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
+      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) *
+      (if ChildMatches p x y c then 1 else 0) := by
+  classical
+  have he (coins : (α → Bool) × (α → Bool)) :
+      (if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then (1:ℝ) else 0) =
+      ∑ c : ι → Bool, (if colors x coins.1 = c then (1:ℝ) else 0) *
+        (if colors y coins.2 = c then 1 else 0) * (if ChildMatches p x y c then 1 else 0) := by
+    rw [Finset.sum_eq_single (colors x coins.1)]
+    · by_cases hc : colors x coins.1 = colors y coins.2
+      · simp only [sandwich_matches_iff,hc,true_and,↓reduceIte,one_mul]
+      · simp only [sandwich_matches_iff,hc,false_and,↓reduceIte,one_mul,Ne.symm hc,zero_mul]
+    · intro c _ hc
+      simp only [Ne.symm hc,ite_false,zero_mul]
+    · simp
+  simp_rw [he]
+  rw [finiteMean_sum]
+  apply Finset.sum_congr rfl
+  intro c _
+  rw [finiteMean_mul_const]
+  rw [finiteMean_prod_mul
+    (fun ξ : α → Bool => if colors x ξ = c then (1:ℝ) else 0)
+    (fun η : α → Bool => if colors y η = c then (1:ℝ) else 0),
+    colors_probability,colors_probability]
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma childMatches_split (p₀ p₁ : Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
+    ChildMatches (fun b => if b then p₁ else p₀) x y c ↔
+      (∀ i : {i // c i = false}, p₀ (x i.val).2 = (y i.val).2) ∧
+      (∀ i : {i // c i = true}, p₁ (x i.val).2 = (y i.val).2) := by
+  constructor
+  · intro h
+    constructor
+    · intro i
+      simpa only [i.property,Bool.false_eq_true,↓reduceIte] using h i.val
+    · intro i
+      simpa only [i.property,↓reduceIte] using h i.val
+  · rintro ⟨h₀,h₁⟩ i
+    cases hi : c i
+    · simpa only [hi,Bool.false_eq_true,↓reduceIte] using h₀ ⟨i,hi⟩
+    · simpa only [hi,↓reduceIte] using h₁ ⟨i,hi⟩
+
+omit [Fintype α] in
+lemma childMatches_mean {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
+    (P : Ω → Equiv.Perm α) (Q : Ω' → Equiv.Perm α) (x y : ι → Bool × α) (c : ι → Bool) :
+    finiteMean (fun ω : Ω × Ω' =>
+      if ChildMatches (fun b => if b then Q ω.2 else P ω.1) x y c then (1:ℝ) else 0) =
+      tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
+        (fun i : {i // c i = false} => (y i.val).2) *
+      tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
+        (fun i : {i // c i = true} => (y i.val).2) := by
+  classical
+  let f : Ω → ℝ := fun ω => if ∀ i : {i // c i = false}, P ω (x i.val).2 = (y i.val).2 then 1 else 0
+  let g : Ω' → ℝ := fun ω => if ∀ i : {i // c i = true}, Q ω (x i.val).2 = (y i.val).2 then 1 else 0
+  have he (ω : Ω × Ω') :
+      (if ChildMatches (fun b => if b then Q ω.2 else P ω.1) x y c then (1:ℝ) else 0) = f ω.1*g ω.2 := by
+    simp only [childMatches_split,f,g]
+    split_ifs <;> simp_all
+  simp_rw [he]
+  exact finiteMean_prod_mul f g
+
+lemma sandwich_mean [DecidableEq ι] {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
+    (P : Ω → Equiv.Perm α) (Q : Ω' → Equiv.Perm α) (x y : ι → Bool × α) :
+    finiteMean (fun ω : Ω × Ω' => finiteMean (fun coins : (α → Bool) × (α → Bool) =>
+      if ∀ i, sandwich coins.1 coins.2 (fun b => if b then Q ω.2 else P ω.1) (x i) = y i
+      then (1:ℝ) else 0)) =
+    ∑ c : ι → Bool,
+      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
+      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) *
+      (tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
+        (fun i : {i // c i = false} => (y i.val).2) *
+      tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
+        (fun i : {i // c i = true} => (y i.val).2)) := by
+  classical
+  simp_rw [sandwich_probability]
+  rw [finiteMean_sum]
+  apply Finset.sum_congr rfl
+  intro c _
+  simp only [mul_comm ((if Compatible x c then _ else _) * _),finiteMean_mul_const,
+    childMatches_mean]
+
+
+abbrev BothAssignments (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :=
+  {c : ι → Bool // Compatible e c ∧ Compatible (fun i => lift p (e i)) c}
+
+noncomputable instance (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    Fintype (BothAssignments e p) := Fintype.ofFinite _
+
+noncomputable def labelSet (e : ι ↪ Bool × α) : Finset (Bool × α) := Finset.univ.image e
+
+def alternatingProjection (p : Bool → Equiv.Perm α) : Equiv.Perm α := (p false).symm * p true
+
+noncomputable def assignmentColoring (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α)
+    (c : BothAssignments e p) : CycleColoring.OppositeColoring
+      (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e) := by
+  classical
+  refine ⟨Function.extend e c.val (fun _ => false),?_,?_⟩
+  · intro z hz
+    apply Function.extend_apply'
+    rintro ⟨i,rfl⟩
+    exact hz (Finset.mem_image.mpr ⟨i,Finset.mem_univ _,rfl⟩)
+  · intro z hz hnext
+    obtain ⟨i,_,hi⟩ := Finset.mem_image.mp hz
+    obtain ⟨j,_,hj⟩ := Finset.mem_image.mp hnext
+    have hfi := e.injective.extend_apply c.val (fun _ => false) i
+    have hfj := e.injective.extend_apply c.val (fun _ => false) j
+    rw [hi] at hfi
+    rw [hj] at hfj
+    rw [hfi,hfj]
+    rcases z with ⟨b,a⟩
+    cases b
+    · have hj' : e j = (true,a) := hj
+      have ht : (e i).2 = (e j).2 := by rw [hi,hj']
+      have hh := c.property.1 i j ht
+      simp only [required,hi,hj'] at hh
+      revert hh
+      generalize c.val i = bi
+      generalize c.val j = bj
+      cases bi <;> cases bj <;> decide
+    · have hj' : e j = (false,alternatingProjection p a) := hj
+      have ht : (lift p (e i)).2 = (lift p (e j)).2 := by
+        simp only [hi,hj',lift,Equiv.coe_fn_mk,alternatingProjection,Equiv.Perm.coe_mul,
+          Function.comp_apply,Equiv.apply_symm_apply]
+      have hh := c.property.2 i j ht
+      simp only [required,hi,hj',lift,Equiv.coe_fn_mk] at hh
+      revert hh
+      generalize c.val i = bi
+      generalize c.val j = bj
+      cases bi <;> cases bj <;> decide
+
+omit [Fintype α] in
+lemma assignmentColoring_injective (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    Function.Injective (assignmentColoring e p) := by
+  intro c c' he
+  apply Subtype.ext
+  funext i
+  have hh := congrFun (congrArg Subtype.val he) (e i)
+  change Function.extend e c.val (fun _ => false) (e i) =
+    Function.extend e c'.val (fun _ => false) (e i) at hh
+  simpa only [e.injective.extend_apply] using hh
+
+lemma bothAssignments_card (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    Fintype.card (BothAssignments e p) ≤
+      2 ^ ((boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true).card +
+        (boolFiber (labelSet e) false ∪ (boolFiber (labelSet e) true).image
+          (alternatingProjection p)).card -
+        ((boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card) +
+        Fintype.card (RoutingNetwork.SelectedCycle (alternatingProjection p) (boolFiber (labelSet e) false))) := by
+  have h := compatible_assignment_bound (alternatingProjection p)
+    (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
+  rw [childJoin_boolFiber] at h
+  exact (Fintype.card_le_of_injective _ (assignmentColoring_injective e p)).trans h
+
+omit [Fintype α] in
+lemma occupied_labelSet (e : ι ↪ Bool × α) : occupied e = (labelSet e).image Prod.snd := by
+  classical
+  simp only [occupied,labelSet,Finset.image_image,Function.comp_def]
+
+omit [Fintype ι] [Fintype α] in
+lemma image_snd_childJoin (A B : Finset α) : (childJoin A B).image Prod.snd = A ∪ B := by
+  classical
+  simp only [childJoin,Finset.image_union,Finset.image_image,Function.comp_def,Finset.image_id']
+
+omit [Fintype ι] [Fintype α] in
+lemma image_lift_childJoin (p : Bool → Equiv.Perm α) (A B : Finset α) :
+    (childJoin A B).image (fun x => (lift p x).2) = A.image (p false) ∪ B.image (p true) := by
+  classical
+  simp only [childJoin,Finset.image_union,Finset.image_image,Function.comp_def,lift,Equiv.coe_fn_mk]
+
+omit [Fintype α] in
+lemma occupied_lift_card (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    (occupied (fun i => lift p (e i))).card =
+      (boolFiber (labelSet e) false ∪ (boolFiber (labelSet e) true).image (alternatingProjection p)).card := by
+  classical
+  have he : occupied (fun i => lift p (e i)) = (labelSet e).image (fun x => (lift p x).2) := by
+    simp only [occupied,labelSet,Finset.image_image,Function.comp_def]
+  rw [he,←childJoin_boolFiber (labelSet e),image_lift_childJoin]
+  simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
+  let A := boolFiber (labelSet e) false
+  let B := boolFiber (labelSet e) true
+  change (A.image (p false) ∪ B.image (p true)).card = (A ∪ B.image (alternatingProjection p)).card
+  have hh : A.image (p false) ∪ B.image (p true) =
+      (A ∪ B.image (alternatingProjection p)).image (p false) := by
+    simp only [Finset.image_union,Finset.image_image,Function.comp_def,
+      alternatingProjection,Equiv.Perm.coe_mul,Equiv.apply_symm_apply]
+  rw [hh,Finset.card_image_of_injective _ (p false).injective]
+
+lemma bothAssignments_occupied_bound (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    Fintype.card (BothAssignments e p) ≤
+      2 ^ ((occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι +
+        Fintype.card (RoutingNetwork.SelectedCycle (alternatingProjection p) (boolFiber (labelSet e) false))) := by
+  have he : (occupied e).card =
+      (boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true).card := by
+    rw [occupied_labelSet,←childJoin_boolFiber (labelSet e),image_snd_childJoin]
+    simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
+  have hc : (boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card = Fintype.card ι := by
+    rw [←card_childJoin,childJoin_boolFiber,labelSet,Finset.card_image_of_injective _ e.injective,
+      Finset.card_univ]
+  simpa only [he,occupied_lift_card,←hc] using bothAssignments_card e p
+
+omit [Fintype ι] [Fintype α] [DecidableEq α] in
+lemma compatible_pairLayer_iff (x : ι → Bool × α) (c : ι → Bool) (ξ : α → Bool) :
+    Compatible (fun i => pairLayer ξ (x i)) c ↔ Compatible x c := by
+  have he (i j : ι) (ht : (x i).2 = (x j).2) :
+      required (fun i => pairLayer ξ (x i)) c i = required (fun i => pairLayer ξ (x i)) c j ↔
+      required x c i = required x c j := by
+    simp only [required,pairLayer,Equiv.coe_fn_mk,ht]
+    cases (x i).1 <;> cases (x j).1 <;> cases c i <;> cases c j <;> cases ξ (x j).2 <;> decide
+  constructor
+  · intro h i j ht
+    exact (he i j ht).mp (h i j ht)
+  · intro h i j ht
+    exact (he i j ht).mpr (h i j ht)
+
+
+noncomputable def alternatingCycles (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) : ℕ :=
+  Fintype.card (RoutingNetwork.SelectedCycle
+    (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e))
+
+omit [Fintype α] in
+lemma occupied_eq_fibers (e : ι ↪ Bool × α) :
+    occupied e = boolFiber (labelSet e) false ∪ boolFiber (labelSet e) true := by
+  rw [occupied_labelSet,←childJoin_boolFiber (labelSet e),image_snd_childJoin]
+  simp only [CycleColoring.boolFiber_childJoin,Bool.false_eq_true,↓reduceIte]
+
+omit [Fintype α] in
+lemma card_labelSet (e : ι ↪ Bool × α) : (labelSet e).card = Fintype.card ι := by
+  classical
+  exact (Finset.card_image_of_injective _ e.injective).trans (Finset.card_univ)
+
+omit [Fintype α] in
+lemma occupied_card_double (e : ι ↪ Bool × α) : Fintype.card ι ≤ 2*(occupied e).card := by
+  have h₀ : (boolFiber (labelSet e) false).card ≤ (occupied e).card := by
+    rw [occupied_eq_fibers]
+    exact Finset.card_le_card Finset.subset_union_left
+  have h₁ : (boolFiber (labelSet e) true).card ≤ (occupied e).card := by
+    rw [occupied_eq_fibers]
+    exact Finset.card_le_card Finset.subset_union_right
+  have he : (boolFiber (labelSet e) false).card+(boolFiber (labelSet e) true).card = Fintype.card ι := by
+    rw [←card_childJoin,childJoin_boolFiber,card_labelSet]
+  omega
+
+lemma bothAssignments_exact_bound (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    Fintype.card (BothAssignments e p) ≤
+      2 ^ ((occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι +
+        alternatingCycles e p) := by
+  have h := CycleColoring.card_opposite_le
+    (CycleColoring.alternatingPerm (alternatingProjection p)) (labelSet e)
+    Prod.fst (CycleColoring.alternatingPerm_flip (alternatingProjection p))
+  have hb : (CycleColoring.boundary (CycleColoring.alternatingPerm (alternatingProjection p))
+      (labelSet e)).card =
+      (occupied e).card + (occupied (fun i => lift p (e i))).card - Fintype.card ι := by
+    have hh := alternating_boundary_card (alternatingProjection p)
+      (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
+    rw [childJoin_boolFiber] at hh
+    rw [occupied_eq_fibers,occupied_lift_card]
+    have he := card_childJoin (boolFiber (labelSet e) false) (boolFiber (labelSet e) true)
+    rw [childJoin_boolFiber,card_labelSet] at he
+    simpa only [he] using hh
+  rw [hb] at h
+  exact (Fintype.card_le_of_injective _ (assignmentColoring_injective e p)).trans h
+
+abbrev Assignments (x y : ι → Bool × α) :=
+  {c : ι → Bool // Compatible x c ∧ Compatible y c}
+
+noncomputable instance (x y : ι → Bool × α) : Fintype (Assignments x y) := Fintype.ofFinite _
+
+def switchedEmbedding (e : ι ↪ Bool × α) (ξ : α → Bool) : ι ↪ Bool × α :=
+  e.trans (pairLayer ξ).toEmbedding
+
+def routedEmbedding (e : ι ↪ Bool × α) (ξ η : α → Bool) (p : Bool → Equiv.Perm α) :
+    ι ↪ Bool × α := e.trans (sandwich ξ η p).toEmbedding
+
+omit [Fintype α] in
+lemma occupied_pairLayer (x : ι → Bool × α) (ξ : α → Bool) :
+    occupied (fun i => pairLayer ξ (x i)) = occupied x := rfl
+
+noncomputable def assignmentsEquiv (e : ι ↪ Bool × α) (ξ η : α → Bool)
+    (p : Bool → Equiv.Perm α) :
+    Assignments e (routedEmbedding e ξ η p) ≃ BothAssignments (switchedEmbedding e ξ) p := by
+  apply Equiv.subtypeEquivRight
+  intro c
+  change Compatible e c ∧ Compatible (fun i => pairLayer η (lift p (pairLayer ξ (e i)))) c ↔ _
+  rw [compatible_pairLayer_iff]
+  exact and_congr (compatible_pairLayer_iff e c ξ).symm Iff.rfl
+
+lemma assignments_actual_bound (e : ι ↪ Bool × α) (ξ η : α → Bool) (p : Bool → Equiv.Perm α) :
+    Fintype.card (Assignments e (routedEmbedding e ξ η p)) ≤
+      2 ^ ((occupied e).card + (occupied (routedEmbedding e ξ η p)).card - Fintype.card ι +
+        alternatingCycles (switchedEmbedding e ξ) p) := by
+  rw [Fintype.card_congr (assignmentsEquiv e ξ η p)]
+  exact bothAssignments_exact_bound (switchedEmbedding e ξ) p
+
+omit [Fintype α] in
+lemma assignments_actual_exponent_nonneg (e : ι ↪ Bool × α) (ξ η : α → Bool)
+    (p : Bool → Equiv.Perm α) :
+    Fintype.card ι ≤ (occupied e).card + (occupied (routedEmbedding e ξ η p)).card := by
+  have h₀ := occupied_card_double e
+  have h₁ := occupied_card_double (routedEmbedding e ξ η p)
+  omega
+
+end
+variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α] [DecidableEq ι]
+
+noncomputable def compatibleSet (x y : ι → Bool × α) : Finset (ι → Bool) := by
+  classical
+  exact Finset.univ.filter (fun c => Compatible x c ∧ Compatible y c)
+
+omit [Fintype α] [DecidableEq α] in
+lemma mem_compatibleSet (x y : ι → Bool × α) (c : ι → Bool) :
+    c ∈ compatibleSet x y ↔ Compatible x c ∧ Compatible y c := by
+  classical
+  simp only [compatibleSet,Finset.mem_filter,Finset.mem_univ,true_and]
+
+omit [Fintype α] [DecidableEq α] in
+lemma card_compatibleSet (x y : ι → Bool × α) :
+    (compatibleSet x y).card = Fintype.card (Assignments x y) := by
+  classical
+  exact (Fintype.card_subtype _).symm
+
+noncomputable def outerWeight (x y : ι → Bool × α) : ℝ :=
+  1/(2:ℝ)^((occupied x).card+(occupied y).card)
+
+omit [Fintype α] [DecidableEq ι] in
+lemma outerWeight_pos (x y : ι → Bool × α) : 0 < outerWeight x y := by
+  unfold outerWeight
+  positivity
+
+omit [Fintype α] in
+lemma sum_colors_factor (x y : ι → Bool × α) (f : (ι → Bool) → ℝ) :
+    (∑ c : ι → Bool,
+      (if Compatible x c then 1/(2:ℝ)^(occupied x).card else 0) *
+      (if Compatible y c then 1/(2:ℝ)^(occupied y).card else 0) * f c) =
+      outerWeight x y * ∑ c ∈ compatibleSet x y, f c := by
+  classical
+  rw [compatibleSet,Finset.sum_filter,Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro c _
+  by_cases hx : Compatible x c <;> by_cases hy : Compatible y c
+  · simp only [hx,hy,true_and,↓reduceIte,outerWeight,pow_add,one_div_mul_one_div]
+  · simp only [hx,hy,and_false,↓reduceIte,mul_zero,zero_mul]
+  · simp only [hx,hy,false_and,↓reduceIte,mul_zero,zero_mul]
+  · simp only [hx,hy,false_and,↓reduceIte,mul_zero,zero_mul]
+
+lemma sandwich_weighted_probability (p : Bool → Equiv.Perm α) (x y : ι → Bool × α)
+    (W : (ι → Bool) → ℝ) :
+    finiteMean (fun coins : (α → Bool) × (α → Bool) =>
+      if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then W (colors x coins.1) else 0) =
+    outerWeight x y * ∑ c ∈ compatibleSet x y, if ChildMatches p x y c then W c else 0 := by
+  classical
+  have he (coins : (α → Bool) × (α → Bool)) :
+      (if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then W (colors x coins.1) else 0) =
+      ∑ c : ι → Bool, (if colors x coins.1 = c then (1:ℝ) else 0) *
+        (if colors y coins.2 = c then 1 else 0) * (if ChildMatches p x y c then W c else 0) := by
+    rw [Finset.sum_eq_single (colors x coins.1)]
+    · by_cases hc : colors x coins.1 = colors y coins.2
+      · simp only [sandwich_matches_iff,hc,true_and,↓reduceIte,one_mul]
+      · simp only [sandwich_matches_iff,hc,false_and,↓reduceIte,one_mul,Ne.symm hc,zero_mul]
+    · intro c _ hc
+      simp only [Ne.symm hc,ite_false,zero_mul]
+    · simp
+  simp_rw [he]
+  rw [finiteMean_sum,←sum_colors_factor]
+  apply Finset.sum_congr rfl
+  intro c _
+  rw [finiteMean_mul_const,finiteMean_prod_mul
+    (fun ξ : α → Bool => if colors x ξ = c then (1:ℝ) else 0)
+    (fun η : α → Bool => if colors y η = c then (1:ℝ) else 0),
+    colors_probability,colors_probability]
+
+omit [Fintype α] [DecidableEq ι] in
+lemma tupleProbability_nonneg {Ω β : Type*} [Fintype Ω] [DecidableEq β]
+    (P : Ω → Equiv.Perm β) (x y : ι → β) : 0 ≤ tupleProbability P x y := by
+  classical
+  exact finiteMean_nonneg (fun ω => by split_ifs <;> norm_num)
+
+lemma sandwich_power_bound {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
+    (P : Ω → Equiv.Perm α) (Q : Ω' → Equiv.Perm α) (x y : ι → Bool × α)
+    (a : ℝ) (ha : 0 ≤ a) :
+    (finiteMean (fun ω : Ω × Ω' => finiteMean (fun coins : (α → Bool) × (α → Bool) =>
+      if ∀ i, sandwich coins.1 coins.2 (fun b => if b then Q ω.2 else P ω.1) (x i) = y i
+      then (1:ℝ) else 0)))^(1+a) ≤
+    (outerWeight x y)^(1+a) * ((compatibleSet x y).card:ℝ)^a *
+      ∑ c ∈ compatibleSet x y,
+      (tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
+        (fun i : {i // c i = false} => (y i.val).2))^(1+a) *
+      (tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
+        (fun i : {i // c i = true} => (y i.val).2))^(1+a) := by
+  classical
+  rw [sandwich_mean,sum_colors_factor]
+  let f (c : ι → Bool) : ℝ :=
+      tupleProbability P (fun i : {i // c i = false} => (x i.val).2)
+        (fun i : {i // c i = false} => (y i.val).2) *
+      tupleProbability Q (fun i : {i // c i = true} => (x i.val).2)
+        (fun i : {i // c i = true} => (y i.val).2)
+  have hf (c : ι → Bool) : 0 ≤ f c := mul_nonneg (tupleProbability_nonneg _ _ _)
+    (tupleProbability_nonneg _ _ _)
+  rw [Real.mul_rpow (le_of_lt (outerWeight_pos _ _)) (Finset.sum_nonneg (fun c _ => hf c))]
+  have hh := Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg (compatibleSet x y)
+    (show 1 ≤ 1+a by linarith) (fun c _ => hf c)
+  have he : 1+a-1=a := by ring
+  rw [he] at hh
+  calc
+    _ ≤ (outerWeight x y)^(1+a) *
+        (((compatibleSet x y).card:ℝ)^a * ∑ c ∈ compatibleSet x y, (f c)^(1+a)) :=
+      mul_le_mul_of_nonneg_left hh (Real.rpow_nonneg (le_of_lt (outerWeight_pos _ _)) _)
+    _ = _ := by
+      rw [←mul_assoc]
+      congr 1
+      apply Finset.sum_congr rfl
+      intro c _
+      exact Real.mul_rpow (tupleProbability_nonneg _ _ _) (tupleProbability_nonneg _ _ _)
+
+end PairRouting
+
+end Thorp
+
+end ThorpNine.Tail
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Tail/AssignmentSlots.lean b/lean/OAI/Probability/ThorpRouting/Tail/AssignmentSlots.lean
index 46616eb1781026f7e5461c839715510bb5a3fe61..9962cc730866365c6273a9685f46bf91e9289d1e 100644
--- a/lean/OAI/Probability/ThorpRouting/Tail/AssignmentSlots.lean
+++ b/lean/OAI/Probability/ThorpRouting/Tail/AssignmentSlots.lean
@@ -1,834 +1,834 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Tail.RotationMeeting
-
-namespace OAI
-
-namespace ThorpNine.Tail
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-section CycleCounting
-variable {d A : ℕ}
-
-lemma assignmentOwner_eq_some {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) (x : Card d) (i : Fin A) :
-    assignmentOwner f x = some i ↔ x ∈ (f.val i).val := by
-  classical
-  unfold assignmentOwner
-  split_ifs with h
-  · constructor
-    · intro hi
-      have hx := Classical.choose_spec h
-      rw [Option.some.inj hi] at hx
-      exact hx
-    · intro hx
-      have he := (f.val (Classical.choose h)).eq_of_mem (f.val i) (Classical.choose_spec h) hx
-      exact congrArg some (f.property.1 he)
-  · constructor
-    · intro hi; contradiction
-    · intro hx; exact (h ⟨i,hx⟩).elim
-
-lemma assignmentOwner_fiber {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) (i : Fin A) : cycleFiber (assignmentOwner f) i = (f.val i).val := by
-  ext x
-  simp only [mem_cycleFiber,assignmentOwner_eq_some]
-
-noncomputable def assignmentToSlots {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) : CycleSlots p S r :=
-  ⟨assignmentOwner f,by
-    refine ⟨?_,?_,?_⟩
-    · intro i
-      rw [assignmentOwner_fiber]
-      exact f.property.2 i
-    · intro i
-      rw [assignmentOwner_fiber]
-      exact (f.val i).property.2
-    · intro i x hx
-      rw [assignmentOwner_fiber]
-      exact (f.val i).orbit_eq ((assignmentOwner_eq_some f x i).mp hx)⟩
-
-noncomputable def cycleSlotsEquivAssignments (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
-    CycleSlots p S r ≃ SlotAssignment p S r where
-  toFun := slotsToAssignment
-  invFun := assignmentToSlots
-  left_inv c := by
-    apply Subtype.ext
-    apply owner_ext
-    intro i
-    rw [assignmentToSlots,assignmentOwner_fiber]
-    rfl
-  right_inv f := by
-    apply Subtype.ext
-    funext i
-    apply Subtype.ext
-    exact assignmentOwner_fiber f i
-
-end CycleCounting
-
-section GroupedCounting
-variable {d A : ℕ}
-
-abbrev LengthClass (r : Fin A → ℕ) := {j : ℕ // j ∈ Finset.univ.image (fun i => r i + 1)}
-
-def slotClass (r : Fin A → ℕ) (i : Fin A) : LengthClass r :=
-  ⟨r i+1, Finset.mem_image.mpr ⟨i, Finset.mem_univ _, rfl⟩⟩
-
-abbrev SlotsOfLength (r : Fin A → ℕ) (j : LengthClass r) := {i : Fin A // slotClass r i = j}
-abbrev CyclesOfLength (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (j : ℕ) :=
-  {C : SelectedCycle p S // C.val.card = j}
-
-noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (j : ℕ) :
-    Fintype (CyclesOfLength p S j) := Fintype.ofFinite _
-
-abbrev GroupedAssignments (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
-  ∀ j : LengthClass r, SlotsOfLength r j ↪ CyclesOfLength p S j.val
-
-noncomputable def assignmentToGroups {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) : GroupedAssignments p S r := fun _j =>
-  { toFun := fun i => ⟨f.val i.val, (f.property.2 i.val).trans (congrArg Subtype.val i.property)⟩
-    inj' := fun _i _k h => Subtype.ext (f.property.1 (congrArg Subtype.val h)) }
-
-noncomputable def groupsEmbedding {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (F : GroupedAssignments p S r) : Fin A ↪ SelectedCycle p S :=
-  ((Equiv.sigmaFiberEquiv (slotClass r)).symm.toEmbedding.trans
-    (Function.Embedding.sigmaMap (Function.Embedding.refl _) F)).trans
-    ((Function.Embedding.sigmaMap (Function.Embedding.subtype _) (fun _ => Function.Embedding.refl _)).trans
-      (Equiv.sigmaFiberEquiv (fun C : SelectedCycle p S => C.val.card)).toEmbedding)
-
-noncomputable def groupsToAssignment {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (F : GroupedAssignments p S r) : SlotAssignment p S r :=
-  ⟨groupsEmbedding F, (groupsEmbedding F).injective,
-    fun i => (F (slotClass r i) ⟨i,rfl⟩).property⟩
-
-noncomputable def assignmentEquivGroups (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
-    SlotAssignment p S r ≃ GroupedAssignments p S r where
-  toFun := assignmentToGroups
-  invFun := groupsToAssignment
-  left_inv f := by
-    apply Subtype.ext
-    rfl
-  right_inv F := by
-    funext j
-    apply Function.Embedding.ext
-    intro i
-    rcases i with ⟨i,hi⟩
-    subst j
-    rfl
-
-lemma card_cycleSlots_factorial (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
-    Fintype.card (CycleSlots p S r) =
-      ∏ j : LengthClass r, (Fintype.card (CyclesOfLength p S j.val)).descFactorial
-        (Fintype.card (SlotsOfLength r j)) := by
-  classical
-  rw [Fintype.card_congr (cycleSlotsEquivAssignments p S r),
-    Fintype.card_congr (assignmentEquivGroups p S r), Fintype.card_pi]
-  simp only [Fintype.card_embedding_eq]
-
-lemma butterfly_factorial_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-       (∏ j : LengthClass r,
-         (Fintype.card (CyclesOfLength (butterflyPerm d (decodeButterfly d ω)) S j.val)).descFactorial
-           (Fintype.card (SlotsOfLength r j)) : ℕ)) ≤
-      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-        ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
-  simpa only [←card_cycleSlots_factorial] using butterfly_cycleSlots_moment S r hr
-
-end GroupedCounting
-
-section AllLengths
-variable {d A : ℕ}
-
-noncomputable def assignmentRelabel (p : Equiv.Perm (Card d)) (S : Finset (Card d))
-    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
-    SlotAssignment p S r ≃ SlotAssignment p S (r ∘ e) where
-  toFun f := ⟨f.val ∘ e, f.property.1.comp e.injective, fun i => f.property.2 (e i)⟩
-  invFun f := ⟨f.val ∘ e.symm, f.property.1.comp e.symm.injective, fun i => by
-    simpa only [Function.comp_apply,Equiv.apply_symm_apply] using f.property.2 (e.symm i)⟩
-  left_inv f := by apply Subtype.ext; funext i; simp
-  right_inv f := by apply Subtype.ext; funext i; simp
-
-lemma card_cycleSlots_relabel (p : Equiv.Perm (Card d)) (S : Finset (Card d))
-    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
-    Fintype.card (CycleSlots p S r) = Fintype.card (CycleSlots p S (r ∘ e)) := by
-  exact Fintype.card_congr ((cycleSlotsEquivAssignments p S r).trans
-    ((assignmentRelabel p S r e).trans (cycleSlotsEquivAssignments p S (r ∘ e)).symm))
-
-lemma butterfly_cycleSlots_moment_all (S : Finset (Card d)) (r : Fin A → ℕ) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) ≤
-        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
-  have h := butterfly_cycleSlots_moment S (r ∘ Tuple.sort r) (Tuple.monotone_sort r)
-  simp_rw [←card_cycleSlots_relabel] at h
-  have hp := Equiv.prod_comp (Tuple.sort r) (fun i : Fin A => (1 / Real.sqrt (r i+1) : ℝ))
-  simpa only [Function.comp_apply,hp] using h
-
-abbrev ShortCycle (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :=
-  {C : SelectedCycle p S // C.val.card ≤ J}
-
-noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
-    Fintype (ShortCycle p S J) := Fintype.ofFinite _
-
-noncomputable def shortCycleLength {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {J : ℕ}
-    (C : ShortCycle p S J) : Fin J :=
-  ⟨C.val.val.card-1, by have := C.val.nonempty.card_pos; have := C.property; omega⟩
-
-lemma shortCycleLength_add_one {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {J : ℕ}
-    (C : ShortCycle p S J) : (shortCycleLength C : ℕ)+1 = C.val.val.card := by
-  have := C.val.nonempty.card_pos
-  simp only [shortCycleLength]
-  omega
-
-noncomputable def shortEmbeddingEquiv (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
-    (Fin A ↪ ShortCycle p S J) ≃
-      Σ r : Fin A → Fin J, SlotAssignment p S (fun i => (r i : ℕ)) where
-  toFun f := ⟨fun i => shortCycleLength (f i),
-    ⟨fun i => (f i).val, fun i j h => f.injective (Subtype.ext h),
-      fun i => (shortCycleLength_add_one (f i)).symm⟩⟩
-  invFun c :=
-    { toFun := fun i => ⟨c.2.val i, by rw [c.2.property.2]; exact (c.1 i).isLt⟩
-      inj' := fun _i _j h => c.2.property.1 (congrArg Subtype.val h) }
-  left_inv f := by
-    apply Function.Embedding.ext
-    intro i
-    rfl
-  right_inv c := by
-    rcases c with ⟨r,f⟩
-    have he : (fun i => shortCycleLength (⟨f.val i, by rw [f.property.2]; exact (r i).isLt⟩ : ShortCycle p S J)) = r := by
-      funext i
-      apply Fin.ext
-      simp only [shortCycleLength,f.property.2,Nat.add_sub_cancel]
-    apply Sigma.ext he
-    apply (Subtype.heq_iff_coe_eq (by
-      intro x
-      change (Function.Injective x ∧ ∀ i, (x i).val.card = (f.val i).val.card - 1 + 1) ↔ _
-      simp only [f.property.2,Nat.add_sub_cancel])).mpr
-    rfl
-
-lemma shortCycle_factorial_eq (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
-    (Fintype.card (ShortCycle p S J)).descFactorial A =
-      ∑ r : Fin A → Fin J, Fintype.card (CycleSlots p S (fun i => (r i : ℕ))) := by
-  classical
-  have hc := Fintype.card_embedding_eq (α := Fin A) (β := ShortCycle p S J)
-  simp only [Fintype.card_fin] at hc
-  rw [←hc, Fintype.card_congr (shortEmbeddingEquiv p S J),Fintype.card_sigma]
-  apply Finset.sum_congr rfl
-  intro r _
-  exact (Fintype.card_congr (cycleSlotsEquivAssignments p S (fun i => (r i : ℕ)))).symm
-
-end AllLengths
-
-lemma factorial_mgf {Ω : Type*} [Fintype Ω] (F : Ω → ℕ) (k : ℕ) (Λ q : ℝ)
-    (hF : ∀ ω, F ω ≤ k) (hΛ : 0 ≤ Λ) (hq : 1 ≤ q)
-    (hm : ∀ a, finiteMean (fun ω => ((F ω).descFactorial a : ℝ)) ≤ Λ^a) :
-    finiteMean (fun ω => q^(F ω)) ≤ Real.exp ((q-1)*Λ) := by
-  have hc (n a : ℕ) : (n.choose a : ℝ) = (n.descFactorial a : ℝ) / (a.factorial : ℝ) := by
-    rw [Nat.descFactorial_eq_factorial_mul_choose,Nat.cast_mul]
-    exact (mul_div_cancel_left₀ _ (by exact_mod_cast Nat.factorial_ne_zero a)).symm
-  have hex (ω : Ω) : q^(F ω) = ∑ a ∈ Finset.range (k+1),
-      ((F ω).descFactorial a : ℝ) * ((q-1)^a / (a.factorial : ℝ)) := by
-    calc
-      _ = ((q-1)+1)^(F ω) := by congr 1; ring
-      _ = ∑ a ∈ Finset.range (F ω+1), (q-1)^a * (F ω).choose a := by
-        rw [add_pow]
-        simp only [one_pow,mul_one]
-      _ = ∑ a ∈ Finset.range (k+1), (q-1)^a * (F ω).choose a := by
-        apply Finset.sum_subset (Finset.range_mono (Nat.add_le_add_right (hF ω) _))
-        intro a _ ha
-        have hn : F ω < a := by have := (Finset.mem_range.not.mp ha); omega
-        rw [Nat.choose_eq_zero_of_lt hn,Nat.cast_zero,mul_zero]
-      _ = _ := by
-        apply Finset.sum_congr rfl
-        intro a _
-        rw [hc]
-        ring
-  simp_rw [hex,finiteMean_sum,finiteMean_mul_const]
-  calc
-    _ ≤ ∑ a ∈ Finset.range (k+1), Λ^a * ((q-1)^a / (a.factorial : ℝ)) := by
-      apply Finset.sum_le_sum
-      intro a _
-      exact mul_le_mul_of_nonneg_right (hm a) (by positivity)
-    _ = ∑ a ∈ Finset.range (k+1), ((q-1)*Λ)^a / (a.factorial : ℝ) := by
-      apply Finset.sum_congr rfl
-      intro a _
-      rw [mul_pow]
-      ring
-    _ ≤ _ := Real.sum_le_exp_of_nonneg (mul_nonneg (sub_nonneg.mpr hq) hΛ) _
-
-lemma inv_sqrt_step (j : ℕ) :
-    1 / Real.sqrt (j+1) ≤ 2*(Real.sqrt (j+1)-Real.sqrt j) := by
-  have hp : 0 < Real.sqrt (j+1) := Real.sqrt_pos.mpr (by positivity)
-  apply (div_le_iff₀ hp).mpr
-  have h1 := Real.sq_sqrt (show (0:ℝ) ≤ j+1 by positivity)
-  have h0 := Real.sq_sqrt (show (0:ℝ) ≤ j by positivity)
-  nlinarith [sq_nonneg (Real.sqrt (j+1)-Real.sqrt j)]
-
-lemma sum_inv_sqrt (J : ℕ) :
-    (∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1)) ≤ 2*Real.sqrt J := by
-  have h (n : ℕ) : (∑ j ∈ Finset.range n, 1 / Real.sqrt (j+1)) ≤ 2*Real.sqrt n := by
-    induction n with
-    | zero => simp
-    | succ n ih =>
-      rw [Finset.sum_range_succ]
-      have hs := inv_sqrt_step n
-      push_cast
-      linarith
-  rw [Fin.sum_univ_eq_sum_range (fun j : ℕ => (1 / Real.sqrt (j+1) : ℝ)) J]
-  exact h J
-section CycleMGF
-variable {d : ℕ}
-
-lemma butterfly_shortCycle_moment (S : Finset (Card d)) (J A : ℕ) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      ((Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J)).descFactorial A : ℝ)) ≤
-      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))^A := by
-  classical
-  simp_rw [shortCycle_factorial_eq,Nat.cast_sum,finiteMean_sum]
-  calc
-    _ ≤ ∑ r : Fin A → Fin J,
-        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-          ∏ i : Fin A, (1 / Real.sqrt ((r i : ℕ)+1)) := by
-      apply Finset.sum_le_sum
-      intro r _
-      exact butterfly_cycleSlots_moment_all S (fun i => (r i : ℕ))
-    _ = _ := by
-      rw [←Finset.mul_sum,mul_pow]
-      congr 1
-      rw [←Fintype.prod_sum (fun (_ : Fin A) (j : Fin J) => (1 / Real.sqrt ((j:ℕ)+1) : ℝ))]
-      simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fin]
-
-noncomputable def cycleMass (p : Equiv.Perm (Card d)) (S : Finset (Card d)) : ℕ :=
-  ∑ C : SelectedCycle p S, C.val.card
-
-lemma cycleMass_le (p : Equiv.Perm (Card d)) (S : Finset (Card d)) : cycleMass p S ≤ S.card := by
-  classical
-  let f : (Σ C : SelectedCycle p S, {x : Card d // x ∈ C.val}) → S :=
-    fun c => ⟨c.2.val,c.1.property.2 c.2.property⟩
-  have hi : Function.Injective f := by
-    intro c e h
-    have hx : c.2.val = e.2.val := congrArg Subtype.val h
-    have hC : c.1 = e.1 := c.1.eq_of_mem e.1 c.2.property (hx ▸ e.2.property)
-    rcases c with ⟨c,x⟩
-    rcases e with ⟨e,y⟩
-    dsimp only at hC
-    subst e
-    have he : x=y := Subtype.ext hx
-    cases he
-    rfl
-  have hh := Fintype.card_le_of_injective f hi
-  simpa only [Fintype.card_sigma,Fintype.card_coe,cycleMass] using hh
-
-lemma shortCycle_card_le (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
-    Fintype.card (ShortCycle p S J) ≤ S.card := by
-  classical
-  calc
-    _ ≤ Fintype.card (SelectedCycle p S) := Fintype.card_subtype_le _
-    _ = ∑ _ : SelectedCycle p S, 1 := by simp
-    _ ≤ cycleMass p S := Finset.sum_le_sum (fun C _ => C.nonempty.card_pos)
-    _ ≤ _ := cycleMass_le p S
-
-lemma butterfly_shortCycle_mgf (S : Finset (Card d)) (J : ℕ) (q : ℝ) (hq : 1 ≤ q) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      q^(Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J))) ≤
-      Real.exp ((q-1)*((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))) := by
-  apply factorial_mgf _ S.card _ q (fun ω => shortCycle_card_le _ _ _)
-  · positivity
-  · exact hq
-  · exact butterfly_shortCycle_moment S J
-
-end CycleMGF
-
-section FullCycleMGF
-variable {d : ℕ}
-
-abbrev LongCycle (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :=
-  {C : SelectedCycle p S // ¬ C.val.card ≤ J}
-noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
-    Fintype (LongCycle p S J) := Fintype.ofFinite _
-
-lemma selectedCycle_card_split (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
-    Fintype.card (SelectedCycle p S) =
-      Fintype.card (ShortCycle p S J) + Fintype.card (LongCycle p S J) := by
-  classical
-  rw [←Fintype.card_congr (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J)),Fintype.card_sum]
-
-lemma longCycle_card_bound (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
-    J * Fintype.card (LongCycle p S J) ≤ S.card := by
-  classical
-  have hs := Equiv.sum_comp (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J))
-    (fun C : SelectedCycle p S => C.val.card)
-  rw [Fintype.sum_sum_type] at hs
-  have h0 : J * Fintype.card (LongCycle p S J) ≤
-      ∑ C : LongCycle p S J, C.val.val.card := by
-    calc
-      _ = ∑ _C : LongCycle p S J, J := by simp [mul_comm]
-      _ ≤ _ := Finset.sum_le_sum (fun C _ => Nat.le_of_lt (Nat.lt_of_not_ge C.property))
-  have hh := cycleMass_le p S
-  unfold cycleMass at hh
-  rw [←hs] at hh
-  exact h0.trans ((Nat.le_add_left _ _).trans hh)
-
-lemma finiteMean_pos {Ω : Type*} [Fintype Ω] [Nonempty Ω] {f : Ω → ℝ}
-    (h : ∀ ω, 0 < f ω) : 0 < finiteMean f := by
-  apply div_pos
-  · exact Finset.sum_pos (fun ω _ => h ω) Finset.univ_nonempty
-  · exact Nat.cast_pos.mpr Fintype.card_pos
-
-lemma longCycle_pow_bound (p : Equiv.Perm (Card d)) (S : Finset (Card d))
-    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    q^(Fintype.card (LongCycle p S J)) ≤ Real.exp ((S.card : ℝ)/J * Real.log q) := by
-  have hq0 : 0 < q := lt_of_lt_of_le zero_lt_one hq
-  have hc : (Fintype.card (LongCycle p S J) : ℝ) ≤ (S.card : ℝ)/J := by
-    apply (le_div_iff₀ (by exact_mod_cast hJ)).mpr
-    have hh := longCycle_card_bound p S J
-    exact_mod_cast (by simpa only [mul_comm] using hh)
-  calc
-    _ = Real.exp ((Fintype.card (LongCycle p S J) : ℝ)*Real.log q) := by
-      rw [Real.exp_nat_mul,Real.exp_log hq0]
-    _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right hc (Real.log_nonneg hq))
-
-lemma butterfly_cycle_log_mgf (S : Finset (Card d)) (J : ℕ) (hJ : 0 < J)
-    (q : ℝ) (hq : 1 ≤ q) :
-    Real.log (finiteMean (fun ω : SwitchIndex d → Bool =>
-      q^(Fintype.card (SelectedCycle (butterflyPerm d (decodeButterfly d ω)) S)))) ≤
-      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
-  let lam : ℝ := (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1)
-  let c : ℝ := (S.card : ℝ)/J * Real.log q
-  have hnum : (q-1)*lam ≤ 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
-    have hsum := sum_inv_sqrt J
-    have hp : 0 ≤ (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) := by positivity
-    have h0 : 0 ≤ 2*Real.sqrt J := by positivity
-    have hh := mul_le_mul_of_nonneg_left hsum hp
-    have hq1 : 0 ≤ q-1 := sub_nonneg.mpr hq
-    calc
-      _ ≤ (q-1)*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
-        mul_le_mul_of_nonneg_left hh hq1
-      _ ≤ q*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
-        mul_le_mul_of_nonneg_right (by linarith) (mul_nonneg hp h0)
-      _ = _ := by ring
-  have hm : finiteMean (fun ω : SwitchIndex d → Bool =>
-      q^(Fintype.card (SelectedCycle (butterflyPerm d (decodeButterfly d ω)) S))) ≤
-      Real.exp (c+(q-1)*lam) := by
-    calc
-      _ ≤ finiteMean (fun ω : SwitchIndex d → Bool =>
-          q^(Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J))*Real.exp c) := by
-        apply finiteMean_mono
-        intro ω
-        rw [selectedCycle_card_split _ _ J,pow_add]
-        exact mul_le_mul_of_nonneg_left (longCycle_pow_bound _ S J hJ q hq) (by positivity)
-      _ = finiteMean (fun ω : SwitchIndex d → Bool =>
-          q^(Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J))) * Real.exp c :=
-        finiteMean_mul_const _ _
-      _ ≤ Real.exp ((q-1)*lam) * Real.exp c :=
-        mul_le_mul_of_nonneg_right (butterfly_shortCycle_mgf S J q hq) (Real.exp_nonneg _)
-      _ = _ := by rw [←Real.exp_add,add_comm]
-  have hpos : 0 < finiteMean (fun ω : SwitchIndex d → Bool =>
-      q^(Fintype.card (SelectedCycle (butterflyPerm d (decodeButterfly d ω)) S))) := by
-    apply finiteMean_pos
-    intro ω
-    positivity
-  have hlog := Real.log_le_log hpos hm
-  rw [Real.log_exp] at hlog
-  dsimp [c] at hlog
-  linarith
-
-end FullCycleMGF
-
-structure RoutingNetwork (α ι : Type*) [Fintype α] [DecidableEq α]
-    [Fintype ι] [DecidableEq ι] (d : ℕ) where
-  perm : (ι → Bool) → Equiv.Perm α
-  path : (ι → Bool) → α → Finset ι
-  users : (ι → Bool) → ι → α × α
-  users_ne : ∀ ω i, (users ω i).1 ≠ (users ω i).2
-  mem_path : ∀ ω x i, i ∈ path ω x ↔ (users ω i).1 = x ∨ (users ω i).2 = x
-  card_path : ∀ ω x, (path ω x).card = d
-  entropy : ∀ ω (S : Finset α),
-    (∑ i : ι, if (users ω i).1 ∈ S ∧ (users ω i).2 ∈ S then (1:ℝ) else 0) ≤
-      (S.card : ℝ) * Real.log S.card / (2 * Real.log 2)
-  follow : α → (α → QueryTree ι) → QueryTree ι
-  close : α → α → QueryTree ι → QueryTree ι
-  follow_succeeds : ∀ x k ω, (follow x k).succeeds ω = (k (perm ω x)).succeeds ω
-  follow_raw_charge : ∀ x k ω, (follow x k).charge ω = (k (perm ω x)).charge ω
-  follow_stale_charge : ∀ x k R ω,
-    (follow x k).staleCharge R ω = (k (perm ω x)).staleCharge (R ∪ path ω x) ω
-  close_succeeds : ∀ x y k ω,
-    (close x y k).succeeds ω = if perm ω x = y then k.succeeds ω else false
-  close_raw_charge : ∀ x y k ω, perm ω x = y → (close x y k).charge ω = k.charge ω + d
-  close_stale_charge : ∀ x y k R ω, perm ω x = y →
-    (close x y k).staleCharge R ω = k.staleCharge (R ∪ path ω x) ω + (path ω x ∩ R).card
-
-namespace RoutingNetwork
-variable {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι] [DecidableEq ι]
-  {d : ℕ} (R : RoutingNetwork α ι d)
-
-lemma card_switches_using (ω : ι → Bool) (x : α) :
-    (Finset.univ.filter (fun i : ι =>
-      (R.users ω i).1 = x ∨ (R.users ω i).2 = x)).card = d := by
-  classical
-  have he : Finset.univ.filter (fun i : ι =>
-      (R.users ω i).1 = x ∨ (R.users ω i).2 = x) = R.path ω x := by
-    ext i
-    simp only [Finset.mem_filter,Finset.mem_univ,true_and,R.mem_path]
-  rw [he,R.card_path]
-lemma switch_incidence_sum (b : ι → Bool) (w : α → ℝ) :
-    ∑ i : ι, (w (R.users b i).1 + w (R.users b i).2) =
-      (d : ℝ) * ∑ x : α, w x := by
-  classical
-  calc
-    _ = ∑ i : ι, ∑ x : α,
-        if (R.users b i).1 = x ∨ (R.users b i).2 = x then w x else 0 := by
-      apply Finset.sum_congr rfl
-      intro i _
-      rw [Finset.sum_ite]
-      simp only [Finset.sum_const_zero, add_zero]
-      have he : Finset.univ.filter (fun x : α =>
-          (R.users b i).1 = x ∨ (R.users b i).2 = x) =
-          {(R.users b i).1, (R.users b i).2} := by
-        ext x; simp [eq_comm]
-      rw [he, Finset.sum_pair (R.users_ne b i)]
-    _ = ∑ x : α, ∑ i : ι,
-        if (R.users b i).1 = x ∨ (R.users b i).2 = x then w x else 0 :=
-      Finset.sum_comm
-    _ = ∑ x : α, (d : ℝ) * w x := by
-      apply Finset.sum_congr rfl
-      intro x _
-      rw [Finset.sum_ite]
-      simp only [Finset.sum_const_zero, add_zero, Finset.sum_const, nsmul_eq_mul,
-        card_switches_using]
-    _ = _ := by rw [Finset.mul_sum]
-
-lemma weighted_switch_min_bound (b : ι → Bool) (w : α → ℝ) :
-    ∑ i : ι, min (w (R.users b i).1) (w (R.users b i).2) ≤
-      (d : ℝ) / 2 * ∑ x : α, w x := by
-  have h : 2 * (∑ i : ι,
-      min (w (R.users b i).1) (w (R.users b i).2)) ≤
-      ∑ i : ι, (w (R.users b i).1 + w (R.users b i).2) := by
-    rw [Finset.mul_sum]
-    apply Finset.sum_le_sum
-    intro i _
-    linarith [min_le_left (w (R.users b i).1) (w (R.users b i).2),
-      min_le_right (w (R.users b i).1) (w (R.users b i).2)]
-  rw [switch_incidence_sum] at h
-  linarith
-section Rotation
-variable {A : ℕ}
-
-def cycleFiber (o : α → Option (Fin A)) (i : Fin A) : Finset (α) := by
-  classical
-  exact Finset.univ.filter (fun x => o x = some i)
-
-omit [DecidableEq α] in
-@[simp] lemma mem_cycleFiber (o : α → Option (Fin A)) (i : Fin A) (x : α) :
-    x ∈ cycleFiber o i ↔ o x = some i := by
-  classical
-  simp [cycleFiber]
-
-abbrev Terminal (o : α → Option (Fin A)) := ∀ i : Fin A, {x : α // o x = some i}
-
-noncomputable def slotWeight (o : α → Option (Fin A)) (i : Fin A) : ℝ :=
-  1 / (cycleFiber o i).card
-
-noncomputable def pathWeight (o : α → Option (Fin A)) (x : α) : ℝ :=
-  match o x with
-  | none => 0
-  | some i => slotWeight o i
-
-noncomputable def withinWeight (o : α → Option (Fin A)) (u v : α) : ℝ :=
-  match o u, o v with
-  | some i, some j => if i = j then slotWeight o i else 0
-  | _, _ => 0
-
-noncomputable def rotationMeeting (o : α → Option (Fin A)) (z : Terminal o)
-    (u v : α) : ℝ :=
-  match o u, o v with
-  | some i, some j => if i < j then (if (z j).val = v then 1 else 0)
-      else if j < i then (if (z i).val = u then 1 else 0)
-      else (if (z i).val = u then 1 else 0) + (if (z i).val = v then 1 else 0)
-  | _, _ => 0
-
-omit [DecidableEq α] in
-lemma card_cycleFiber (o : α → Option (Fin A)) (i : Fin A) :
-    Fintype.card {x : α // o x = some i} = (cycleFiber o i).card := by
-  classical
-  simp [cycleFiber,Fintype.card_subtype]
-
-lemma terminal_mean {o : α → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
-    (i : Fin A) (x : α) (hx : o x = some i) :
-    finiteMean (fun z : Terminal o => if (z i).val = x then (1:ℝ) else 0) = slotWeight o i := by
-  classical
-  have hev := finiteMean_pi_eval (A := fun i : Fin A => {x : α // o x = some i})
-    i (fun z => if z.val = x then (1:ℝ) else 0)
-  rw [hev]
-  have he : (fun z : {x : α // o x = some i} => if z.val = x then (1:ℝ) else 0) =
-      (fun z => if z = ⟨x,hx⟩ then 1 else 0) := by
-    funext z
-    simp only [Subtype.ext_iff]
-  rw [he,finiteMean_eq_indicator,card_cycleFiber,slotWeight]
-
-omit [DecidableEq α] in
-lemma slotWeight_antitone (o : α → Option (Fin A))
-    (hn : ∀ i, (cycleFiber o i).Nonempty)
-    (hm : Monotone (fun i => (cycleFiber o i).card)) : Antitone (slotWeight o) := by
-  intro i j hij
-  unfold slotWeight
-  apply one_div_le_one_div_of_le
-  · exact_mod_cast Finset.card_pos.mpr (hn i)
-  · exact_mod_cast hm hij
-
-lemma rotationMeeting_mean {o : α → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
-    (hm : Monotone (fun i => (cycleFiber o i).card)) (u v : α) :
-    finiteMean (fun z : Terminal o => rotationMeeting o z u v) =
-      min (pathWeight o u) (pathWeight o v) + withinWeight o u v := by
-  classical
-  have hn : ∀ i, (cycleFiber o i).Nonempty := by
-    intro i
-    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : α // o x = some i})
-    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
-  have hw := slotWeight_antitone o hn hm
-  cases hu : o u with
-  | none =>
-    cases hv : o v with
-    | none => simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean]
-    | some j =>
-      have hj : 0 ≤ slotWeight o j := by unfold slotWeight; positivity
-      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_left hj]
-  | some i =>
-    cases hv : o v with
-    | none =>
-      have hi : 0 ≤ slotWeight o i := by unfold slotWeight; positivity
-      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_right hi]
-    | some j =>
-      simp only [rotationMeeting,pathWeight,withinWeight,hu,hv]
-      rcases lt_trichotomy i j with hij | rfl | hji
-      · simp only [hij,↓reduceIte,ne_of_lt hij]
-        rw [terminal_mean j v hv,min_eq_right (hw (le_of_lt hij)),add_zero]
-      · simp only [lt_self_iff_false,↓reduceIte,min_self]
-        rw [finiteMean_add,terminal_mean i u hu,terminal_mean i v hv]
-      · simp only [not_lt_of_ge (le_of_lt hji),hji,↓reduceIte,ne_of_gt hji]
-        rw [terminal_mean i u hu,min_eq_left (hw (le_of_lt hji)),add_zero]
-
-omit [DecidableEq α] in
-lemma pathWeight_sum (o : α → Option (Fin A)) (hn : ∀ i, (cycleFiber o i).Nonempty) :
-    ∑ x : α, pathWeight o x = (A : ℝ) := by
-  classical
-  have he (x : α) : pathWeight o x =
-      ∑ i : Fin A, if o x = some i then slotWeight o i else 0 := by
-    cases hx : o x with
-    | none => simp [pathWeight,hx]
-    | some j => simp [pathWeight,hx]
-  simp only [he]
-  rw [Finset.sum_comm]
-  have hf (i : Fin A) : (∑ x : α, if o x = some i then slotWeight o i else 0) = 1 := by
-    rw [← Finset.sum_filter]
-    change (∑ _x ∈ cycleFiber o i, slotWeight o i) = 1
-    simp only [Finset.sum_const,nsmul_eq_mul,slotWeight]
-    exact mul_one_div_cancel (by exact_mod_cast (Finset.card_pos.mpr (hn i)).ne')
-  simp only [hf,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,mul_one]
-
-lemma withinWeight_sum_bound (o : α → Option (Fin A))
-    (hn : ∀ i, (cycleFiber o i).Nonempty) (b : ι → Bool) :
-    (∑ s : ι, withinWeight o (R.users b s).1 (R.users b s).2) ≤
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
-  classical
-  have he (u v : α) : withinWeight o u v =
-      ∑ i : Fin A, (if u ∈ cycleFiber o i ∧ v ∈ cycleFiber o i then (1:ℝ) else 0) * slotWeight o i := by
-    simp only [mem_cycleFiber]
-    cases hu : o u <;> cases hv : o v <;> simp [withinWeight,hu,hv,ite_and,eq_comm]
-  simp only [he]
-  rw [Finset.sum_comm]
-  apply Finset.sum_le_sum
-  intro i _
-  rw [← Finset.sum_mul]
-  have hi := R.entropy b (cycleFiber o i)
-  have hpos : (0:ℝ) < (cycleFiber o i).card := by exact_mod_cast Finset.card_pos.mpr (hn i)
-  have hmul := mul_le_mul_of_nonneg_right hi (show 0 ≤ slotWeight o i by unfold slotWeight; positivity)
-  calc
-    _ ≤ ((cycleFiber o i).card : ℝ) * Real.log (cycleFiber o i).card /
-        (2 * Real.log 2) * slotWeight o i := hmul
-    _ = _ := by unfold slotWeight; field_simp
-
-noncomputable def rotationCharge (o : α → Option (Fin A)) (b : ι → Bool)
-    (z : Terminal o) : ℝ :=
-  ∑ s : ι, rotationMeeting o z (R.users b s).1 (R.users b s).2
-
-lemma rotationCharge_mean_bound {o : α → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
-    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : ι → Bool) :
-    finiteMean (rotationCharge R o b) ≤ (A:ℝ) * d / 2 +
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
-  classical
-  have hn : ∀ i, (cycleFiber o i).Nonempty := by
-    intro i
-    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : α // o x = some i})
-    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
-  unfold rotationCharge
-  rw [finiteMean_sum]
-  simp only [rotationMeeting_mean hm,Finset.sum_add_distrib]
-  have h₁ := weighted_switch_min_bound R b (pathWeight o)
-  rw [pathWeight_sum o hn] at h₁
-  have h₂ := withinWeight_sum_bound R o hn b
-  linarith
-
-lemma rotation_exp_bound {o : α → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
-    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : ι → Bool) :
-    Real.exp (-Real.log 2 * ((A:ℝ)*d/2 +
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2))) ≤
-      finiteMean (fun z : Terminal o => Real.exp (-Real.log 2 * rotationCharge R o b z)) := by
-  have hb := rotationCharge_mean_bound R hm b
-  have hlog : 0 ≤ Real.log 2 := Real.log_nonneg (by norm_num)
-  have hm' : -Real.log 2 * ((A:ℝ)*d/2 +
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2)) ≤
-      finiteMean (fun z : Terminal o => -Real.log 2 * rotationCharge R o b z) := by
-    have he := finiteMean_mul_const (rotationCharge R o b) (-Real.log 2)
-    simp only [mul_comm] at he
-    rw [he]
-    exact mul_le_mul_of_nonpos_left hb (neg_nonpos.mpr hlog)
-  exact (Real.exp_le_exp.mpr hm').trans (exp_finiteMean_le _)
-
-end Rotation
-
-namespace QueryTree
-open Thorp.QueryTree
-
-noncomputable def seekCycle (R : RoutingNetwork α ι d) (S : Finset (α)) (a : α) :
-    ℕ → α → Finset (α) → (Finset (α) → QueryTree (ι)) →
-      QueryTree (ι)
-  | 0, x, V, k => if x ∈ S ∧ x ∉ V then R.close x a (k (insert x V)) else reject
-  | r + 1, x, V, k => if x ∈ S ∧ x ∉ V then R.follow x
-      (fun y => seekCycle R S a r y (insert x V) k) else reject
-
-lemma seekCycle_raw_charge (S : Finset (α)) (a : α) (r : ℕ)
-    (x : α) (V : Finset (α)) (k : Finset (α) → QueryTree (ι))
-    (ω : ι → Bool) (K : ℕ)
-    (hK : ∀ V, (k V).succeeds ω = true → (k V).charge ω = K)
-    (hs : (seekCycle R S a r x V k).succeeds ω = true) :
-    (seekCycle R S a r x V k).charge ω = K + d := by
-  induction r generalizing x V with
-  | zero =>
-    by_cases hv : x ∈ S ∧ x ∉ V
-    · rw [seekCycle,ite_eq_left hv,R.close_succeeds] at hs
-      split_ifs at hs with he
-      · rw [seekCycle,ite_eq_left hv,R.close_raw_charge _ _ _ _ he,hK _ hs]
-    · simp [seekCycle,hv,Thorp.QueryTree.succeeds] at hs
-  | succ r ih =>
-    by_cases hv : x ∈ S ∧ x ∉ V
-    · rw [seekCycle,ite_eq_left hv,R.follow_succeeds] at hs
-      rw [seekCycle,ite_eq_left hv,R.follow_raw_charge]
-      exact ih _ _ hs
-    · simp [seekCycle,hv,Thorp.QueryTree.succeeds] at hs
-
-noncomputable def seekCycles (R : RoutingNetwork α ι d) (S : Finset (α)) :
-    List (α × ℕ) → Finset (α) → QueryTree (ι)
-  | [], _ => accept
-  | (a,r) :: L, V => seekCycle R S a r a V (fun V => seekCycles R S L V)
-
-lemma seekCycles_raw_charge (S : Finset (α)) (L : List (α × ℕ))
-    (V : Finset (α)) (ω : ι → Bool)
-    (hs : (seekCycles R S L V).succeeds ω = true) :
-    (seekCycles R S L V).charge ω = L.length * d := by
-  induction L generalizing V with
-  | nil => simp [seekCycles,charge]
-  | cons ar L ih =>
-    obtain ⟨a,r⟩ := ar
-    rw [seekCycles]
-    rw [seekCycle_raw_charge R S a r a V (fun V => seekCycles R S L V) ω (L.length*d)
-      (fun V hs => ih V hs) hs]
-    simp [Nat.add_mul]
-
-lemma cycles_weighted_success (S : Finset (α)) (L : List (α × ℕ)) :
-    finiteMean (fun ω : ι → Bool =>
-      if (seekCycles R S L ∅).succeeds ω then
-        ((2:ℝ)^d)^L.length / (2:ℝ)^(seekCycles R S L ∅).staleCharge ∅ ω else 0) ≤ 1 := by
-  let T := seekCycles R S L ∅
-  have hh := mean_freshCharge_le_one T
-  have he : finiteMean (fun ω : ι → Bool =>
-      if T.succeeds ω then (2:ℝ)^T.freshCharge ∅ ω else 0) =
-      finiteMean (fun ω : ι → Bool =>
-        if T.succeeds ω then ((2:ℝ)^d)^L.length / (2:ℝ)^T.staleCharge ∅ ω else 0) := by
-    apply finiteMean_congr
-    intro ω
-    cases hs : T.succeeds ω
-    · simp
-    · simp only [↓reduceIte]
-      have hc := fresh_add_stale T ∅ ω
-      rw [seekCycles_raw_charge R S L ∅ ω hs] at hc
-      have hp : ((2:ℝ)^d)^L.length = (2:ℝ)^T.freshCharge ∅ ω * (2:ℝ)^T.staleCharge ∅ ω := by
-        rw [←pow_add,hc,←pow_mul,Nat.mul_comm]
-      rw [hp,mul_div_cancel_right₀ _ (pow_ne_zero _ (by norm_num))]
-  rw [← he]
-  exact hh
-
-end QueryTree
-noncomputable def exposedPaths (b : ι → Bool) (V : Finset (α)) :
-    Finset (ι) :=
-  V.biUnion (fun x => R.path b x)
-
-@[simp] lemma exposedPaths_empty (b : ι → Bool) : exposedPaths R b ∅ = ∅ := by
-  classical
-  simp [exposedPaths]
-
-lemma exposedPaths_insert (b : ι → Bool) (V : Finset (α)) (x : α) :
-    exposedPaths R b (insert x V) = exposedPaths R b V ∪ R.path b x := by
-  classical
-  simp [exposedPaths,Finset.union_comm]
-
-noncomputable def terminalCharge (b : ι → Bool) (z : α) (V : Finset (α)) : ℕ :=
-  (R.path b z ∩ exposedPaths R b V).card
-
-namespace QueryTree
-open Thorp.QueryTree
-
-lemma seekCycle_semantics (S : Finset (α)) (a : α) (r : ℕ)
-    (x : α) (V W : Finset (α)) (z : α)
-    (k : Finset (α) → QueryTree (ι)) (ω : ι → Bool)
-    (h : cycleTrace (R.perm ω) S a r x V = some (W,z)) :
-    (seekCycle R S a r x V k).succeeds ω = (k W).succeeds ω ∧
-      (seekCycle R S a r x V k).staleCharge (exposedPaths R (ω) V) ω =
-        (k W).staleCharge (exposedPaths R (ω) W) ω +
-          terminalCharge R (ω) z (W.erase z) := by
-  classical
-  induction r generalizing x V with
-  | zero =>
-    simp only [cycleTrace] at h
-    split_ifs at h with hv
-    obtain ⟨hxS,hxV,hxa⟩ := hv
-    obtain ⟨rfl,rfl⟩ := Prod.mk.inj (Option.some.inj h)
-    simp only [seekCycle,ite_eq_left (show x ∈ S ∧ x ∉ V from ⟨hxS,hxV⟩),R.close_succeeds,ite_eq_left hxa]
-    constructor
-    · trivial
-    · rw [R.close_stale_charge _ _ _ _ _ hxa,exposedPaths_insert,
-        Finset.erase_insert hxV]
-      simp only [terminalCharge]
-  | succ r ih =>
-    simp only [cycleTrace] at h
-    split_ifs at h with hv
-    obtain ⟨hxS,hxV⟩ := hv
-    rw [seekCycle,ite_eq_left ⟨hxS,hxV⟩,R.follow_succeeds,R.follow_stale_charge]
-    rw [← exposedPaths_insert]
-    exact ih _ _ h
-
-lemma seekCycle_orbit (S V : Finset (α)) (z : α)
-    (k : Finset (α) → QueryTree (ι)) (ω : ι → Bool)
-    (hS : orbitSet (R.perm ω) z ⊆ S)
-    (hV : Disjoint V (orbitSet (R.perm ω) z)) :
-    let p := R.perm ω
-    let W := V ∪ orbitSet p z
-    (seekCycle R S (p z) (Function.minimalPeriod p z-1) (p z) V k).succeeds ω = (k W).succeeds ω ∧
-      (seekCycle R S (p z) (Function.minimalPeriod p z-1) (p z) V k).staleCharge
-          (exposedPaths R (ω) V) ω =
-        (k W).staleCharge (exposedPaths R (ω) W) ω +
-          terminalCharge R (ω) z (W.erase z) := by
-  exact seekCycle_semantics _ _ _ _ _ _ _ _ _ _ (cycleTrace_orbit _ _ _ _ hS hV)
-
-end QueryTree
-
-end RoutingNetwork
-end Thorp
-
-end ThorpNine.Tail
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Tail.RotationMeeting
+
+namespace OAI
+
+namespace ThorpNine.Tail
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+section CycleCounting
+variable {d A : ℕ}
+
+lemma assignmentOwner_eq_some {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) (x : Card d) (i : Fin A) :
+    assignmentOwner f x = some i ↔ x ∈ (f.val i).val := by
+  classical
+  unfold assignmentOwner
+  split_ifs with h
+  · constructor
+    · intro hi
+      have hx := Classical.choose_spec h
+      rw [Option.some.inj hi] at hx
+      exact hx
+    · intro hx
+      have he := (f.val (Classical.choose h)).eq_of_mem (f.val i) (Classical.choose_spec h) hx
+      exact congrArg some (f.property.1 he)
+  · constructor
+    · intro hi; contradiction
+    · intro hx; exact (h ⟨i,hx⟩).elim
+
+lemma assignmentOwner_fiber {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) (i : Fin A) : cycleFiber (assignmentOwner f) i = (f.val i).val := by
+  ext x
+  simp only [mem_cycleFiber,assignmentOwner_eq_some]
+
+noncomputable def assignmentToSlots {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) : CycleSlots p S r :=
+  ⟨assignmentOwner f,by
+    refine ⟨?_,?_,?_⟩
+    · intro i
+      rw [assignmentOwner_fiber]
+      exact f.property.2 i
+    · intro i
+      rw [assignmentOwner_fiber]
+      exact (f.val i).property.2
+    · intro i x hx
+      rw [assignmentOwner_fiber]
+      exact (f.val i).orbit_eq ((assignmentOwner_eq_some f x i).mp hx)⟩
+
+noncomputable def cycleSlotsEquivAssignments (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
+    CycleSlots p S r ≃ SlotAssignment p S r where
+  toFun := slotsToAssignment
+  invFun := assignmentToSlots
+  left_inv c := by
+    apply Subtype.ext
+    apply owner_ext
+    intro i
+    rw [assignmentToSlots,assignmentOwner_fiber]
+    rfl
+  right_inv f := by
+    apply Subtype.ext
+    funext i
+    apply Subtype.ext
+    exact assignmentOwner_fiber f i
+
+end CycleCounting
+
+section GroupedCounting
+variable {d A : ℕ}
+
+abbrev LengthClass (r : Fin A → ℕ) := {j : ℕ // j ∈ Finset.univ.image (fun i => r i + 1)}
+
+def slotClass (r : Fin A → ℕ) (i : Fin A) : LengthClass r :=
+  ⟨r i+1, Finset.mem_image.mpr ⟨i, Finset.mem_univ _, rfl⟩⟩
+
+abbrev SlotsOfLength (r : Fin A → ℕ) (j : LengthClass r) := {i : Fin A // slotClass r i = j}
+abbrev CyclesOfLength (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (j : ℕ) :=
+  {C : SelectedCycle p S // C.val.card = j}
+
+noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (j : ℕ) :
+    Fintype (CyclesOfLength p S j) := Fintype.ofFinite _
+
+abbrev GroupedAssignments (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
+  ∀ j : LengthClass r, SlotsOfLength r j ↪ CyclesOfLength p S j.val
+
+noncomputable def assignmentToGroups {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) : GroupedAssignments p S r := fun _j =>
+  { toFun := fun i => ⟨f.val i.val, (f.property.2 i.val).trans (congrArg Subtype.val i.property)⟩
+    inj' := fun _i _k h => Subtype.ext (f.property.1 (congrArg Subtype.val h)) }
+
+noncomputable def groupsEmbedding {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (F : GroupedAssignments p S r) : Fin A ↪ SelectedCycle p S :=
+  ((Equiv.sigmaFiberEquiv (slotClass r)).symm.toEmbedding.trans
+    (Function.Embedding.sigmaMap (Function.Embedding.refl _) F)).trans
+    ((Function.Embedding.sigmaMap (Function.Embedding.subtype _) (fun _ => Function.Embedding.refl _)).trans
+      (Equiv.sigmaFiberEquiv (fun C : SelectedCycle p S => C.val.card)).toEmbedding)
+
+noncomputable def groupsToAssignment {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (F : GroupedAssignments p S r) : SlotAssignment p S r :=
+  ⟨groupsEmbedding F, (groupsEmbedding F).injective,
+    fun i => (F (slotClass r i) ⟨i,rfl⟩).property⟩
+
+noncomputable def assignmentEquivGroups (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
+    SlotAssignment p S r ≃ GroupedAssignments p S r where
+  toFun := assignmentToGroups
+  invFun := groupsToAssignment
+  left_inv f := by
+    apply Subtype.ext
+    rfl
+  right_inv F := by
+    funext j
+    apply Function.Embedding.ext
+    intro i
+    rcases i with ⟨i,hi⟩
+    subst j
+    rfl
+
+lemma card_cycleSlots_factorial (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
+    Fintype.card (CycleSlots p S r) =
+      ∏ j : LengthClass r, (Fintype.card (CyclesOfLength p S j.val)).descFactorial
+        (Fintype.card (SlotsOfLength r j)) := by
+  classical
+  rw [Fintype.card_congr (cycleSlotsEquivAssignments p S r),
+    Fintype.card_congr (assignmentEquivGroups p S r), Fintype.card_pi]
+  simp only [Fintype.card_embedding_eq]
+
+lemma butterfly_factorial_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+       (∏ j : LengthClass r,
+         (Fintype.card (CyclesOfLength (butterflyPerm d (decodeButterfly d ω)) S j.val)).descFactorial
+           (Fintype.card (SlotsOfLength r j)) : ℕ)) ≤
+      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+        ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
+  simpa only [←card_cycleSlots_factorial] using butterfly_cycleSlots_moment S r hr
+
+end GroupedCounting
+
+section AllLengths
+variable {d A : ℕ}
+
+noncomputable def assignmentRelabel (p : Equiv.Perm (Card d)) (S : Finset (Card d))
+    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
+    SlotAssignment p S r ≃ SlotAssignment p S (r ∘ e) where
+  toFun f := ⟨f.val ∘ e, f.property.1.comp e.injective, fun i => f.property.2 (e i)⟩
+  invFun f := ⟨f.val ∘ e.symm, f.property.1.comp e.symm.injective, fun i => by
+    simpa only [Function.comp_apply,Equiv.apply_symm_apply] using f.property.2 (e.symm i)⟩
+  left_inv f := by apply Subtype.ext; funext i; simp
+  right_inv f := by apply Subtype.ext; funext i; simp
+
+lemma card_cycleSlots_relabel (p : Equiv.Perm (Card d)) (S : Finset (Card d))
+    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
+    Fintype.card (CycleSlots p S r) = Fintype.card (CycleSlots p S (r ∘ e)) := by
+  exact Fintype.card_congr ((cycleSlotsEquivAssignments p S r).trans
+    ((assignmentRelabel p S r e).trans (cycleSlotsEquivAssignments p S (r ∘ e)).symm))
+
+lemma butterfly_cycleSlots_moment_all (S : Finset (Card d)) (r : Fin A → ℕ) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) ≤
+        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
+  have h := butterfly_cycleSlots_moment S (r ∘ Tuple.sort r) (Tuple.monotone_sort r)
+  simp_rw [←card_cycleSlots_relabel] at h
+  have hp := Equiv.prod_comp (Tuple.sort r) (fun i : Fin A => (1 / Real.sqrt (r i+1) : ℝ))
+  simpa only [Function.comp_apply,hp] using h
+
+abbrev ShortCycle (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :=
+  {C : SelectedCycle p S // C.val.card ≤ J}
+
+noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
+    Fintype (ShortCycle p S J) := Fintype.ofFinite _
+
+noncomputable def shortCycleLength {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {J : ℕ}
+    (C : ShortCycle p S J) : Fin J :=
+  ⟨C.val.val.card-1, by have := C.val.nonempty.card_pos; have := C.property; omega⟩
+
+lemma shortCycleLength_add_one {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {J : ℕ}
+    (C : ShortCycle p S J) : (shortCycleLength C : ℕ)+1 = C.val.val.card := by
+  have := C.val.nonempty.card_pos
+  simp only [shortCycleLength]
+  omega
+
+noncomputable def shortEmbeddingEquiv (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
+    (Fin A ↪ ShortCycle p S J) ≃
+      Σ r : Fin A → Fin J, SlotAssignment p S (fun i => (r i : ℕ)) where
+  toFun f := ⟨fun i => shortCycleLength (f i),
+    ⟨fun i => (f i).val, fun i j h => f.injective (Subtype.ext h),
+      fun i => (shortCycleLength_add_one (f i)).symm⟩⟩
+  invFun c :=
+    { toFun := fun i => ⟨c.2.val i, by rw [c.2.property.2]; exact (c.1 i).isLt⟩
+      inj' := fun _i _j h => c.2.property.1 (congrArg Subtype.val h) }
+  left_inv f := by
+    apply Function.Embedding.ext
+    intro i
+    rfl
+  right_inv c := by
+    rcases c with ⟨r,f⟩
+    have he : (fun i => shortCycleLength (⟨f.val i, by rw [f.property.2]; exact (r i).isLt⟩ : ShortCycle p S J)) = r := by
+      funext i
+      apply Fin.ext
+      simp only [shortCycleLength,f.property.2,Nat.add_sub_cancel]
+    apply Sigma.ext he
+    apply (Subtype.heq_iff_coe_eq (by
+      intro x
+      change (Function.Injective x ∧ ∀ i, (x i).val.card = (f.val i).val.card - 1 + 1) ↔ _
+      simp only [f.property.2,Nat.add_sub_cancel])).mpr
+    rfl
+
+lemma shortCycle_factorial_eq (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
+    (Fintype.card (ShortCycle p S J)).descFactorial A =
+      ∑ r : Fin A → Fin J, Fintype.card (CycleSlots p S (fun i => (r i : ℕ))) := by
+  classical
+  have hc := Fintype.card_embedding_eq (α := Fin A) (β := ShortCycle p S J)
+  simp only [Fintype.card_fin] at hc
+  rw [←hc, Fintype.card_congr (shortEmbeddingEquiv p S J),Fintype.card_sigma]
+  apply Finset.sum_congr rfl
+  intro r _
+  exact (Fintype.card_congr (cycleSlotsEquivAssignments p S (fun i => (r i : ℕ)))).symm
+
+end AllLengths
+
+lemma factorial_mgf {Ω : Type*} [Fintype Ω] (F : Ω → ℕ) (k : ℕ) (Λ q : ℝ)
+    (hF : ∀ ω, F ω ≤ k) (hΛ : 0 ≤ Λ) (hq : 1 ≤ q)
+    (hm : ∀ a, finiteMean (fun ω => ((F ω).descFactorial a : ℝ)) ≤ Λ^a) :
+    finiteMean (fun ω => q^(F ω)) ≤ Real.exp ((q-1)*Λ) := by
+  have hc (n a : ℕ) : (n.choose a : ℝ) = (n.descFactorial a : ℝ) / (a.factorial : ℝ) := by
+    rw [Nat.descFactorial_eq_factorial_mul_choose,Nat.cast_mul]
+    exact (mul_div_cancel_left₀ _ (by exact_mod_cast Nat.factorial_ne_zero a)).symm
+  have hex (ω : Ω) : q^(F ω) = ∑ a ∈ Finset.range (k+1),
+      ((F ω).descFactorial a : ℝ) * ((q-1)^a / (a.factorial : ℝ)) := by
+    calc
+      _ = ((q-1)+1)^(F ω) := by congr 1; ring
+      _ = ∑ a ∈ Finset.range (F ω+1), (q-1)^a * (F ω).choose a := by
+        rw [add_pow]
+        simp only [one_pow,mul_one]
+      _ = ∑ a ∈ Finset.range (k+1), (q-1)^a * (F ω).choose a := by
+        apply Finset.sum_subset (Finset.range_mono (Nat.add_le_add_right (hF ω) _))
+        intro a _ ha
+        have hn : F ω < a := by have := (Finset.mem_range.not.mp ha); omega
+        rw [Nat.choose_eq_zero_of_lt hn,Nat.cast_zero,mul_zero]
+      _ = _ := by
+        apply Finset.sum_congr rfl
+        intro a _
+        rw [hc]
+        ring
+  simp_rw [hex,finiteMean_sum,finiteMean_mul_const]
+  calc
+    _ ≤ ∑ a ∈ Finset.range (k+1), Λ^a * ((q-1)^a / (a.factorial : ℝ)) := by
+      apply Finset.sum_le_sum
+      intro a _
+      exact mul_le_mul_of_nonneg_right (hm a) (by positivity)
+    _ = ∑ a ∈ Finset.range (k+1), ((q-1)*Λ)^a / (a.factorial : ℝ) := by
+      apply Finset.sum_congr rfl
+      intro a _
+      rw [mul_pow]
+      ring
+    _ ≤ _ := Real.sum_le_exp_of_nonneg (mul_nonneg (sub_nonneg.mpr hq) hΛ) _
+
+lemma inv_sqrt_step (j : ℕ) :
+    1 / Real.sqrt (j+1) ≤ 2*(Real.sqrt (j+1)-Real.sqrt j) := by
+  have hp : 0 < Real.sqrt (j+1) := Real.sqrt_pos.mpr (by positivity)
+  apply (div_le_iff₀ hp).mpr
+  have h1 := Real.sq_sqrt (show (0:ℝ) ≤ j+1 by positivity)
+  have h0 := Real.sq_sqrt (show (0:ℝ) ≤ j by positivity)
+  nlinarith [sq_nonneg (Real.sqrt (j+1)-Real.sqrt j)]
+
+lemma sum_inv_sqrt (J : ℕ) :
+    (∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1)) ≤ 2*Real.sqrt J := by
+  have h (n : ℕ) : (∑ j ∈ Finset.range n, 1 / Real.sqrt (j+1)) ≤ 2*Real.sqrt n := by
+    induction n with
+    | zero => simp
+    | succ n ih =>
+      rw [Finset.sum_range_succ]
+      have hs := inv_sqrt_step n
+      push_cast
+      linarith
+  rw [Fin.sum_univ_eq_sum_range (fun j : ℕ => (1 / Real.sqrt (j+1) : ℝ)) J]
+  exact h J
+section CycleMGF
+variable {d : ℕ}
+
+lemma butterfly_shortCycle_moment (S : Finset (Card d)) (J A : ℕ) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      ((Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J)).descFactorial A : ℝ)) ≤
+      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))^A := by
+  classical
+  simp_rw [shortCycle_factorial_eq,Nat.cast_sum,finiteMean_sum]
+  calc
+    _ ≤ ∑ r : Fin A → Fin J,
+        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+          ∏ i : Fin A, (1 / Real.sqrt ((r i : ℕ)+1)) := by
+      apply Finset.sum_le_sum
+      intro r _
+      exact butterfly_cycleSlots_moment_all S (fun i => (r i : ℕ))
+    _ = _ := by
+      rw [←Finset.mul_sum,mul_pow]
+      congr 1
+      rw [←Fintype.prod_sum (fun (_ : Fin A) (j : Fin J) => (1 / Real.sqrt ((j:ℕ)+1) : ℝ))]
+      simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fin]
+
+noncomputable def cycleMass (p : Equiv.Perm (Card d)) (S : Finset (Card d)) : ℕ :=
+  ∑ C : SelectedCycle p S, C.val.card
+
+lemma cycleMass_le (p : Equiv.Perm (Card d)) (S : Finset (Card d)) : cycleMass p S ≤ S.card := by
+  classical
+  let f : (Σ C : SelectedCycle p S, {x : Card d // x ∈ C.val}) → S :=
+    fun c => ⟨c.2.val,c.1.property.2 c.2.property⟩
+  have hi : Function.Injective f := by
+    intro c e h
+    have hx : c.2.val = e.2.val := congrArg Subtype.val h
+    have hC : c.1 = e.1 := c.1.eq_of_mem e.1 c.2.property (hx ▸ e.2.property)
+    rcases c with ⟨c,x⟩
+    rcases e with ⟨e,y⟩
+    dsimp only at hC
+    subst e
+    have he : x=y := Subtype.ext hx
+    cases he
+    rfl
+  have hh := Fintype.card_le_of_injective f hi
+  simpa only [Fintype.card_sigma,Fintype.card_coe,cycleMass] using hh
+
+lemma shortCycle_card_le (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
+    Fintype.card (ShortCycle p S J) ≤ S.card := by
+  classical
+  calc
+    _ ≤ Fintype.card (SelectedCycle p S) := Fintype.card_subtype_le _
+    _ = ∑ _ : SelectedCycle p S, 1 := by simp
+    _ ≤ cycleMass p S := Finset.sum_le_sum (fun C _ => C.nonempty.card_pos)
+    _ ≤ _ := cycleMass_le p S
+
+lemma butterfly_shortCycle_mgf (S : Finset (Card d)) (J : ℕ) (q : ℝ) (hq : 1 ≤ q) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      q^(Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J))) ≤
+      Real.exp ((q-1)*((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))) := by
+  apply factorial_mgf _ S.card _ q (fun ω => shortCycle_card_le _ _ _)
+  · positivity
+  · exact hq
+  · exact butterfly_shortCycle_moment S J
+
+end CycleMGF
+
+section FullCycleMGF
+variable {d : ℕ}
+
+abbrev LongCycle (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :=
+  {C : SelectedCycle p S // ¬ C.val.card ≤ J}
+noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
+    Fintype (LongCycle p S J) := Fintype.ofFinite _
+
+lemma selectedCycle_card_split (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
+    Fintype.card (SelectedCycle p S) =
+      Fintype.card (ShortCycle p S J) + Fintype.card (LongCycle p S J) := by
+  classical
+  rw [←Fintype.card_congr (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J)),Fintype.card_sum]
+
+lemma longCycle_card_bound (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (J : ℕ) :
+    J * Fintype.card (LongCycle p S J) ≤ S.card := by
+  classical
+  have hs := Equiv.sum_comp (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J))
+    (fun C : SelectedCycle p S => C.val.card)
+  rw [Fintype.sum_sum_type] at hs
+  have h0 : J * Fintype.card (LongCycle p S J) ≤
+      ∑ C : LongCycle p S J, C.val.val.card := by
+    calc
+      _ = ∑ _C : LongCycle p S J, J := by simp [mul_comm]
+      _ ≤ _ := Finset.sum_le_sum (fun C _ => Nat.le_of_lt (Nat.lt_of_not_ge C.property))
+  have hh := cycleMass_le p S
+  unfold cycleMass at hh
+  rw [←hs] at hh
+  exact h0.trans ((Nat.le_add_left _ _).trans hh)
+
+lemma finiteMean_pos {Ω : Type*} [Fintype Ω] [Nonempty Ω] {f : Ω → ℝ}
+    (h : ∀ ω, 0 < f ω) : 0 < finiteMean f := by
+  apply div_pos
+  · exact Finset.sum_pos (fun ω _ => h ω) Finset.univ_nonempty
+  · exact Nat.cast_pos.mpr Fintype.card_pos
+
+lemma longCycle_pow_bound (p : Equiv.Perm (Card d)) (S : Finset (Card d))
+    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    q^(Fintype.card (LongCycle p S J)) ≤ Real.exp ((S.card : ℝ)/J * Real.log q) := by
+  have hq0 : 0 < q := lt_of_lt_of_le zero_lt_one hq
+  have hc : (Fintype.card (LongCycle p S J) : ℝ) ≤ (S.card : ℝ)/J := by
+    apply (le_div_iff₀ (by exact_mod_cast hJ)).mpr
+    have hh := longCycle_card_bound p S J
+    exact_mod_cast (by simpa only [mul_comm] using hh)
+  calc
+    _ = Real.exp ((Fintype.card (LongCycle p S J) : ℝ)*Real.log q) := by
+      rw [Real.exp_nat_mul,Real.exp_log hq0]
+    _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right hc (Real.log_nonneg hq))
+
+lemma butterfly_cycle_log_mgf (S : Finset (Card d)) (J : ℕ) (hJ : 0 < J)
+    (q : ℝ) (hq : 1 ≤ q) :
+    Real.log (finiteMean (fun ω : SwitchIndex d → Bool =>
+      q^(Fintype.card (SelectedCycle (butterflyPerm d (decodeButterfly d ω)) S)))) ≤
+      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
+  let lam : ℝ := (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1)
+  let c : ℝ := (S.card : ℝ)/J * Real.log q
+  have hnum : (q-1)*lam ≤ 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
+    have hsum := sum_inv_sqrt J
+    have hp : 0 ≤ (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) := by positivity
+    have h0 : 0 ≤ 2*Real.sqrt J := by positivity
+    have hh := mul_le_mul_of_nonneg_left hsum hp
+    have hq1 : 0 ≤ q-1 := sub_nonneg.mpr hq
+    calc
+      _ ≤ (q-1)*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
+        mul_le_mul_of_nonneg_left hh hq1
+      _ ≤ q*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
+        mul_le_mul_of_nonneg_right (by linarith) (mul_nonneg hp h0)
+      _ = _ := by ring
+  have hm : finiteMean (fun ω : SwitchIndex d → Bool =>
+      q^(Fintype.card (SelectedCycle (butterflyPerm d (decodeButterfly d ω)) S))) ≤
+      Real.exp (c+(q-1)*lam) := by
+    calc
+      _ ≤ finiteMean (fun ω : SwitchIndex d → Bool =>
+          q^(Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J))*Real.exp c) := by
+        apply finiteMean_mono
+        intro ω
+        rw [selectedCycle_card_split _ _ J,pow_add]
+        exact mul_le_mul_of_nonneg_left (longCycle_pow_bound _ S J hJ q hq) (by positivity)
+      _ = finiteMean (fun ω : SwitchIndex d → Bool =>
+          q^(Fintype.card (ShortCycle (butterflyPerm d (decodeButterfly d ω)) S J))) * Real.exp c :=
+        finiteMean_mul_const _ _
+      _ ≤ Real.exp ((q-1)*lam) * Real.exp c :=
+        mul_le_mul_of_nonneg_right (butterfly_shortCycle_mgf S J q hq) (Real.exp_nonneg _)
+      _ = _ := by rw [←Real.exp_add,add_comm]
+  have hpos : 0 < finiteMean (fun ω : SwitchIndex d → Bool =>
+      q^(Fintype.card (SelectedCycle (butterflyPerm d (decodeButterfly d ω)) S))) := by
+    apply finiteMean_pos
+    intro ω
+    positivity
+  have hlog := Real.log_le_log hpos hm
+  rw [Real.log_exp] at hlog
+  dsimp [c] at hlog
+  linarith
+
+end FullCycleMGF
+
+structure RoutingNetwork (α ι : Type*) [Fintype α] [DecidableEq α]
+    [Fintype ι] [DecidableEq ι] (d : ℕ) where
+  perm : (ι → Bool) → Equiv.Perm α
+  path : (ι → Bool) → α → Finset ι
+  users : (ι → Bool) → ι → α × α
+  users_ne : ∀ ω i, (users ω i).1 ≠ (users ω i).2
+  mem_path : ∀ ω x i, i ∈ path ω x ↔ (users ω i).1 = x ∨ (users ω i).2 = x
+  card_path : ∀ ω x, (path ω x).card = d
+  entropy : ∀ ω (S : Finset α),
+    (∑ i : ι, if (users ω i).1 ∈ S ∧ (users ω i).2 ∈ S then (1:ℝ) else 0) ≤
+      (S.card : ℝ) * Real.log S.card / (2 * Real.log 2)
+  follow : α → (α → QueryTree ι) → QueryTree ι
+  close : α → α → QueryTree ι → QueryTree ι
+  follow_succeeds : ∀ x k ω, (follow x k).succeeds ω = (k (perm ω x)).succeeds ω
+  follow_raw_charge : ∀ x k ω, (follow x k).charge ω = (k (perm ω x)).charge ω
+  follow_stale_charge : ∀ x k R ω,
+    (follow x k).staleCharge R ω = (k (perm ω x)).staleCharge (R ∪ path ω x) ω
+  close_succeeds : ∀ x y k ω,
+    (close x y k).succeeds ω = if perm ω x = y then k.succeeds ω else false
+  close_raw_charge : ∀ x y k ω, perm ω x = y → (close x y k).charge ω = k.charge ω + d
+  close_stale_charge : ∀ x y k R ω, perm ω x = y →
+    (close x y k).staleCharge R ω = k.staleCharge (R ∪ path ω x) ω + (path ω x ∩ R).card
+
+namespace RoutingNetwork
+variable {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι] [DecidableEq ι]
+  {d : ℕ} (R : RoutingNetwork α ι d)
+
+lemma card_switches_using (ω : ι → Bool) (x : α) :
+    (Finset.univ.filter (fun i : ι =>
+      (R.users ω i).1 = x ∨ (R.users ω i).2 = x)).card = d := by
+  classical
+  have he : Finset.univ.filter (fun i : ι =>
+      (R.users ω i).1 = x ∨ (R.users ω i).2 = x) = R.path ω x := by
+    ext i
+    simp only [Finset.mem_filter,Finset.mem_univ,true_and,R.mem_path]
+  rw [he,R.card_path]
+lemma switch_incidence_sum (b : ι → Bool) (w : α → ℝ) :
+    ∑ i : ι, (w (R.users b i).1 + w (R.users b i).2) =
+      (d : ℝ) * ∑ x : α, w x := by
+  classical
+  calc
+    _ = ∑ i : ι, ∑ x : α,
+        if (R.users b i).1 = x ∨ (R.users b i).2 = x then w x else 0 := by
+      apply Finset.sum_congr rfl
+      intro i _
+      rw [Finset.sum_ite]
+      simp only [Finset.sum_const_zero, add_zero]
+      have he : Finset.univ.filter (fun x : α =>
+          (R.users b i).1 = x ∨ (R.users b i).2 = x) =
+          {(R.users b i).1, (R.users b i).2} := by
+        ext x; simp [eq_comm]
+      rw [he, Finset.sum_pair (R.users_ne b i)]
+    _ = ∑ x : α, ∑ i : ι,
+        if (R.users b i).1 = x ∨ (R.users b i).2 = x then w x else 0 :=
+      Finset.sum_comm
+    _ = ∑ x : α, (d : ℝ) * w x := by
+      apply Finset.sum_congr rfl
+      intro x _
+      rw [Finset.sum_ite]
+      simp only [Finset.sum_const_zero, add_zero, Finset.sum_const, nsmul_eq_mul,
+        card_switches_using]
+    _ = _ := by rw [Finset.mul_sum]
+
+lemma weighted_switch_min_bound (b : ι → Bool) (w : α → ℝ) :
+    ∑ i : ι, min (w (R.users b i).1) (w (R.users b i).2) ≤
+      (d : ℝ) / 2 * ∑ x : α, w x := by
+  have h : 2 * (∑ i : ι,
+      min (w (R.users b i).1) (w (R.users b i).2)) ≤
+      ∑ i : ι, (w (R.users b i).1 + w (R.users b i).2) := by
+    rw [Finset.mul_sum]
+    apply Finset.sum_le_sum
+    intro i _
+    linarith [min_le_left (w (R.users b i).1) (w (R.users b i).2),
+      min_le_right (w (R.users b i).1) (w (R.users b i).2)]
+  rw [switch_incidence_sum] at h
+  linarith
+section Rotation
+variable {A : ℕ}
+
+def cycleFiber (o : α → Option (Fin A)) (i : Fin A) : Finset (α) := by
+  classical
+  exact Finset.univ.filter (fun x => o x = some i)
+
+omit [DecidableEq α] in
+@[simp] lemma mem_cycleFiber (o : α → Option (Fin A)) (i : Fin A) (x : α) :
+    x ∈ cycleFiber o i ↔ o x = some i := by
+  classical
+  simp [cycleFiber]
+
+abbrev Terminal (o : α → Option (Fin A)) := ∀ i : Fin A, {x : α // o x = some i}
+
+noncomputable def slotWeight (o : α → Option (Fin A)) (i : Fin A) : ℝ :=
+  1 / (cycleFiber o i).card
+
+noncomputable def pathWeight (o : α → Option (Fin A)) (x : α) : ℝ :=
+  match o x with
+  | none => 0
+  | some i => slotWeight o i
+
+noncomputable def withinWeight (o : α → Option (Fin A)) (u v : α) : ℝ :=
+  match o u, o v with
+  | some i, some j => if i = j then slotWeight o i else 0
+  | _, _ => 0
+
+noncomputable def rotationMeeting (o : α → Option (Fin A)) (z : Terminal o)
+    (u v : α) : ℝ :=
+  match o u, o v with
+  | some i, some j => if i < j then (if (z j).val = v then 1 else 0)
+      else if j < i then (if (z i).val = u then 1 else 0)
+      else (if (z i).val = u then 1 else 0) + (if (z i).val = v then 1 else 0)
+  | _, _ => 0
+
+omit [DecidableEq α] in
+lemma card_cycleFiber (o : α → Option (Fin A)) (i : Fin A) :
+    Fintype.card {x : α // o x = some i} = (cycleFiber o i).card := by
+  classical
+  simp [cycleFiber,Fintype.card_subtype]
+
+lemma terminal_mean {o : α → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
+    (i : Fin A) (x : α) (hx : o x = some i) :
+    finiteMean (fun z : Terminal o => if (z i).val = x then (1:ℝ) else 0) = slotWeight o i := by
+  classical
+  have hev := finiteMean_pi_eval (A := fun i : Fin A => {x : α // o x = some i})
+    i (fun z => if z.val = x then (1:ℝ) else 0)
+  rw [hev]
+  have he : (fun z : {x : α // o x = some i} => if z.val = x then (1:ℝ) else 0) =
+      (fun z => if z = ⟨x,hx⟩ then 1 else 0) := by
+    funext z
+    simp only [Subtype.ext_iff]
+  rw [he,finiteMean_eq_indicator,card_cycleFiber,slotWeight]
+
+omit [DecidableEq α] in
+lemma slotWeight_antitone (o : α → Option (Fin A))
+    (hn : ∀ i, (cycleFiber o i).Nonempty)
+    (hm : Monotone (fun i => (cycleFiber o i).card)) : Antitone (slotWeight o) := by
+  intro i j hij
+  unfold slotWeight
+  apply one_div_le_one_div_of_le
+  · exact_mod_cast Finset.card_pos.mpr (hn i)
+  · exact_mod_cast hm hij
+
+lemma rotationMeeting_mean {o : α → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
+    (hm : Monotone (fun i => (cycleFiber o i).card)) (u v : α) :
+    finiteMean (fun z : Terminal o => rotationMeeting o z u v) =
+      min (pathWeight o u) (pathWeight o v) + withinWeight o u v := by
+  classical
+  have hn : ∀ i, (cycleFiber o i).Nonempty := by
+    intro i
+    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : α // o x = some i})
+    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
+  have hw := slotWeight_antitone o hn hm
+  cases hu : o u with
+  | none =>
+    cases hv : o v with
+    | none => simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean]
+    | some j =>
+      have hj : 0 ≤ slotWeight o j := by unfold slotWeight; positivity
+      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_left hj]
+  | some i =>
+    cases hv : o v with
+    | none =>
+      have hi : 0 ≤ slotWeight o i := by unfold slotWeight; positivity
+      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_right hi]
+    | some j =>
+      simp only [rotationMeeting,pathWeight,withinWeight,hu,hv]
+      rcases lt_trichotomy i j with hij | rfl | hji
+      · simp only [hij,↓reduceIte,ne_of_lt hij]
+        rw [terminal_mean j v hv,min_eq_right (hw (le_of_lt hij)),add_zero]
+      · simp only [lt_self_iff_false,↓reduceIte,min_self]
+        rw [finiteMean_add,terminal_mean i u hu,terminal_mean i v hv]
+      · simp only [not_lt_of_ge (le_of_lt hji),hji,↓reduceIte,ne_of_gt hji]
+        rw [terminal_mean i u hu,min_eq_left (hw (le_of_lt hji)),add_zero]
+
+omit [DecidableEq α] in
+lemma pathWeight_sum (o : α → Option (Fin A)) (hn : ∀ i, (cycleFiber o i).Nonempty) :
+    ∑ x : α, pathWeight o x = (A : ℝ) := by
+  classical
+  have he (x : α) : pathWeight o x =
+      ∑ i : Fin A, if o x = some i then slotWeight o i else 0 := by
+    cases hx : o x with
+    | none => simp [pathWeight,hx]
+    | some j => simp [pathWeight,hx]
+  simp only [he]
+  rw [Finset.sum_comm]
+  have hf (i : Fin A) : (∑ x : α, if o x = some i then slotWeight o i else 0) = 1 := by
+    rw [← Finset.sum_filter]
+    change (∑ _x ∈ cycleFiber o i, slotWeight o i) = 1
+    simp only [Finset.sum_const,nsmul_eq_mul,slotWeight]
+    exact mul_one_div_cancel (by exact_mod_cast (Finset.card_pos.mpr (hn i)).ne')
+  simp only [hf,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,mul_one]
+
+lemma withinWeight_sum_bound (o : α → Option (Fin A))
+    (hn : ∀ i, (cycleFiber o i).Nonempty) (b : ι → Bool) :
+    (∑ s : ι, withinWeight o (R.users b s).1 (R.users b s).2) ≤
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
+  classical
+  have he (u v : α) : withinWeight o u v =
+      ∑ i : Fin A, (if u ∈ cycleFiber o i ∧ v ∈ cycleFiber o i then (1:ℝ) else 0) * slotWeight o i := by
+    simp only [mem_cycleFiber]
+    cases hu : o u <;> cases hv : o v <;> simp [withinWeight,hu,hv,ite_and,eq_comm]
+  simp only [he]
+  rw [Finset.sum_comm]
+  apply Finset.sum_le_sum
+  intro i _
+  rw [← Finset.sum_mul]
+  have hi := R.entropy b (cycleFiber o i)
+  have hpos : (0:ℝ) < (cycleFiber o i).card := by exact_mod_cast Finset.card_pos.mpr (hn i)
+  have hmul := mul_le_mul_of_nonneg_right hi (show 0 ≤ slotWeight o i by unfold slotWeight; positivity)
+  calc
+    _ ≤ ((cycleFiber o i).card : ℝ) * Real.log (cycleFiber o i).card /
+        (2 * Real.log 2) * slotWeight o i := hmul
+    _ = _ := by unfold slotWeight; field_simp
+
+noncomputable def rotationCharge (o : α → Option (Fin A)) (b : ι → Bool)
+    (z : Terminal o) : ℝ :=
+  ∑ s : ι, rotationMeeting o z (R.users b s).1 (R.users b s).2
+
+lemma rotationCharge_mean_bound {o : α → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
+    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : ι → Bool) :
+    finiteMean (rotationCharge R o b) ≤ (A:ℝ) * d / 2 +
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
+  classical
+  have hn : ∀ i, (cycleFiber o i).Nonempty := by
+    intro i
+    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : α // o x = some i})
+    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
+  unfold rotationCharge
+  rw [finiteMean_sum]
+  simp only [rotationMeeting_mean hm,Finset.sum_add_distrib]
+  have h₁ := weighted_switch_min_bound R b (pathWeight o)
+  rw [pathWeight_sum o hn] at h₁
+  have h₂ := withinWeight_sum_bound R o hn b
+  linarith
+
+lemma rotation_exp_bound {o : α → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : α // o x = some i}]
+    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : ι → Bool) :
+    Real.exp (-Real.log 2 * ((A:ℝ)*d/2 +
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2))) ≤
+      finiteMean (fun z : Terminal o => Real.exp (-Real.log 2 * rotationCharge R o b z)) := by
+  have hb := rotationCharge_mean_bound R hm b
+  have hlog : 0 ≤ Real.log 2 := Real.log_nonneg (by norm_num)
+  have hm' : -Real.log 2 * ((A:ℝ)*d/2 +
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2)) ≤
+      finiteMean (fun z : Terminal o => -Real.log 2 * rotationCharge R o b z) := by
+    have he := finiteMean_mul_const (rotationCharge R o b) (-Real.log 2)
+    simp only [mul_comm] at he
+    rw [he]
+    exact mul_le_mul_of_nonpos_left hb (neg_nonpos.mpr hlog)
+  exact (Real.exp_le_exp.mpr hm').trans (exp_finiteMean_le _)
+
+end Rotation
+
+namespace QueryTree
+open Thorp.QueryTree
+
+noncomputable def seekCycle (R : RoutingNetwork α ι d) (S : Finset (α)) (a : α) :
+    ℕ → α → Finset (α) → (Finset (α) → QueryTree (ι)) →
+      QueryTree (ι)
+  | 0, x, V, k => if x ∈ S ∧ x ∉ V then R.close x a (k (insert x V)) else reject
+  | r + 1, x, V, k => if x ∈ S ∧ x ∉ V then R.follow x
+      (fun y => seekCycle R S a r y (insert x V) k) else reject
+
+lemma seekCycle_raw_charge (S : Finset (α)) (a : α) (r : ℕ)
+    (x : α) (V : Finset (α)) (k : Finset (α) → QueryTree (ι))
+    (ω : ι → Bool) (K : ℕ)
+    (hK : ∀ V, (k V).succeeds ω = true → (k V).charge ω = K)
+    (hs : (seekCycle R S a r x V k).succeeds ω = true) :
+    (seekCycle R S a r x V k).charge ω = K + d := by
+  induction r generalizing x V with
+  | zero =>
+    by_cases hv : x ∈ S ∧ x ∉ V
+    · rw [seekCycle,ite_eq_left hv,R.close_succeeds] at hs
+      split_ifs at hs with he
+      · rw [seekCycle,ite_eq_left hv,R.close_raw_charge _ _ _ _ he,hK _ hs]
+    · simp [seekCycle,hv,Thorp.QueryTree.succeeds] at hs
+  | succ r ih =>
+    by_cases hv : x ∈ S ∧ x ∉ V
+    · rw [seekCycle,ite_eq_left hv,R.follow_succeeds] at hs
+      rw [seekCycle,ite_eq_left hv,R.follow_raw_charge]
+      exact ih _ _ hs
+    · simp [seekCycle,hv,Thorp.QueryTree.succeeds] at hs
+
+noncomputable def seekCycles (R : RoutingNetwork α ι d) (S : Finset (α)) :
+    List (α × ℕ) → Finset (α) → QueryTree (ι)
+  | [], _ => accept
+  | (a,r) :: L, V => seekCycle R S a r a V (fun V => seekCycles R S L V)
+
+lemma seekCycles_raw_charge (S : Finset (α)) (L : List (α × ℕ))
+    (V : Finset (α)) (ω : ι → Bool)
+    (hs : (seekCycles R S L V).succeeds ω = true) :
+    (seekCycles R S L V).charge ω = L.length * d := by
+  induction L generalizing V with
+  | nil => simp [seekCycles,charge]
+  | cons ar L ih =>
+    obtain ⟨a,r⟩ := ar
+    rw [seekCycles]
+    rw [seekCycle_raw_charge R S a r a V (fun V => seekCycles R S L V) ω (L.length*d)
+      (fun V hs => ih V hs) hs]
+    simp [Nat.add_mul]
+
+lemma cycles_weighted_success (S : Finset (α)) (L : List (α × ℕ)) :
+    finiteMean (fun ω : ι → Bool =>
+      if (seekCycles R S L ∅).succeeds ω then
+        ((2:ℝ)^d)^L.length / (2:ℝ)^(seekCycles R S L ∅).staleCharge ∅ ω else 0) ≤ 1 := by
+  let T := seekCycles R S L ∅
+  have hh := mean_freshCharge_le_one T
+  have he : finiteMean (fun ω : ι → Bool =>
+      if T.succeeds ω then (2:ℝ)^T.freshCharge ∅ ω else 0) =
+      finiteMean (fun ω : ι → Bool =>
+        if T.succeeds ω then ((2:ℝ)^d)^L.length / (2:ℝ)^T.staleCharge ∅ ω else 0) := by
+    apply finiteMean_congr
+    intro ω
+    cases hs : T.succeeds ω
+    · simp
+    · simp only [↓reduceIte]
+      have hc := fresh_add_stale T ∅ ω
+      rw [seekCycles_raw_charge R S L ∅ ω hs] at hc
+      have hp : ((2:ℝ)^d)^L.length = (2:ℝ)^T.freshCharge ∅ ω * (2:ℝ)^T.staleCharge ∅ ω := by
+        rw [←pow_add,hc,←pow_mul,Nat.mul_comm]
+      rw [hp,mul_div_cancel_right₀ _ (pow_ne_zero _ (by norm_num))]
+  rw [← he]
+  exact hh
+
+end QueryTree
+noncomputable def exposedPaths (b : ι → Bool) (V : Finset (α)) :
+    Finset (ι) :=
+  V.biUnion (fun x => R.path b x)
+
+@[simp] lemma exposedPaths_empty (b : ι → Bool) : exposedPaths R b ∅ = ∅ := by
+  classical
+  simp [exposedPaths]
+
+lemma exposedPaths_insert (b : ι → Bool) (V : Finset (α)) (x : α) :
+    exposedPaths R b (insert x V) = exposedPaths R b V ∪ R.path b x := by
+  classical
+  simp [exposedPaths,Finset.union_comm]
+
+noncomputable def terminalCharge (b : ι → Bool) (z : α) (V : Finset (α)) : ℕ :=
+  (R.path b z ∩ exposedPaths R b V).card
+
+namespace QueryTree
+open Thorp.QueryTree
+
+lemma seekCycle_semantics (S : Finset (α)) (a : α) (r : ℕ)
+    (x : α) (V W : Finset (α)) (z : α)
+    (k : Finset (α) → QueryTree (ι)) (ω : ι → Bool)
+    (h : cycleTrace (R.perm ω) S a r x V = some (W,z)) :
+    (seekCycle R S a r x V k).succeeds ω = (k W).succeeds ω ∧
+      (seekCycle R S a r x V k).staleCharge (exposedPaths R (ω) V) ω =
+        (k W).staleCharge (exposedPaths R (ω) W) ω +
+          terminalCharge R (ω) z (W.erase z) := by
+  classical
+  induction r generalizing x V with
+  | zero =>
+    simp only [cycleTrace] at h
+    split_ifs at h with hv
+    obtain ⟨hxS,hxV,hxa⟩ := hv
+    obtain ⟨rfl,rfl⟩ := Prod.mk.inj (Option.some.inj h)
+    simp only [seekCycle,ite_eq_left (show x ∈ S ∧ x ∉ V from ⟨hxS,hxV⟩),R.close_succeeds,ite_eq_left hxa]
+    constructor
+    · trivial
+    · rw [R.close_stale_charge _ _ _ _ _ hxa,exposedPaths_insert,
+        Finset.erase_insert hxV]
+      simp only [terminalCharge]
+  | succ r ih =>
+    simp only [cycleTrace] at h
+    split_ifs at h with hv
+    obtain ⟨hxS,hxV⟩ := hv
+    rw [seekCycle,ite_eq_left ⟨hxS,hxV⟩,R.follow_succeeds,R.follow_stale_charge]
+    rw [← exposedPaths_insert]
+    exact ih _ _ h
+
+lemma seekCycle_orbit (S V : Finset (α)) (z : α)
+    (k : Finset (α) → QueryTree (ι)) (ω : ι → Bool)
+    (hS : orbitSet (R.perm ω) z ⊆ S)
+    (hV : Disjoint V (orbitSet (R.perm ω) z)) :
+    let p := R.perm ω
+    let W := V ∪ orbitSet p z
+    (seekCycle R S (p z) (Function.minimalPeriod p z-1) (p z) V k).succeeds ω = (k W).succeeds ω ∧
+      (seekCycle R S (p z) (Function.minimalPeriod p z-1) (p z) V k).staleCharge
+          (exposedPaths R (ω) V) ω =
+        (k W).staleCharge (exposedPaths R (ω) W) ω +
+          terminalCharge R (ω) z (W.erase z) := by
+  exact seekCycle_semantics _ _ _ _ _ _ _ _ _ _ (cycleTrace_orbit _ _ _ _ hS hV)
+
+end QueryTree
+
+end RoutingNetwork
+end Thorp
+
+end ThorpNine.Tail
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Tail/BenesCoins.lean b/lean/OAI/Probability/ThorpRouting/Tail/BenesCoins.lean
index 558b73dc83c5ee60c6cbbccdb59f80eec7619d41..d3ae2994a5b3fda121a4f4112f1e4007d8404428 100644
--- a/lean/OAI/Probability/ThorpRouting/Tail/BenesCoins.lean
+++ b/lean/OAI/Probability/ThorpRouting/Tail/BenesCoins.lean
@@ -1,882 +1,882 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Tail.AlternatingPerm
-
-namespace OAI
-
-namespace ThorpNine.Tail
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-
-abbrev BenesCoins (d : ℕ) := (SwitchIndex d → Bool) × (SwitchIndex d → Bool)
-
-def palindromePerm (d : ℕ) (ω : BenesCoins d) : Equiv.Perm (Card d) :=
-  butterflyPerm d (decodeButterfly d ω.1) * (butterflyPerm d (decodeButterfly d ω.2)).symm
-
-def sandwichShuffle {A B C D E F : Type*} :
-    ((A × B) × C) × ((D × E) × F) ≃ ((A × D) × (B × E)) × (F × C) where
-  toFun x := (((x.1.1.1,x.2.1.1),(x.1.1.2,x.2.1.2)),(x.2.2,x.1.2))
-  invFun x := (((x.1.1.1,x.1.2.1),x.2.2),((x.1.1.2,x.1.2.2),x.2.1))
-  left_inv _ := rfl
-  right_inv _ := rfl
-
-def benesStepEquiv (d : ℕ) : BenesCoins (d+1) ≃
-    (BenesCoins d × BenesCoins d) × ((Card d → Bool) × (Card d → Bool)) :=
-  (Equiv.prodCongr (coinStepEquiv d) (coinStepEquiv d)).trans sandwichShuffle
-
-lemma childLift_mul (d : ℕ) (p q : Bool → Equiv.Perm (Card d)) :
-    childLift p * childLift q = childLift (fun b => p b * q b) := by
-  ext x i
-  simp only [Equiv.Perm.mul_apply,childLift,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons]
-
-lemma childLift_inv (d : ℕ) (p : Bool → Equiv.Perm (Card d)) :
-    (childLift p)⁻¹ = childLift (fun b => (p b)⁻¹) := rfl
-
-lemma pairSwitch_inv (d : ℕ) (ξ : Card d → Bool) : (pairSwitch ξ)⁻¹ = pairSwitch ξ := rfl
-
-lemma palindromePerm_step (d : ℕ) (c : BenesCoins d × BenesCoins d)
-    (coins : (Card d → Bool) × (Card d → Bool)) :
-    palindromePerm (d+1) ((benesStepEquiv d).symm (c,coins)) =
-      pairSwitch coins.2 * childLift (fun b => palindromePerm d (if b then c.2 else c.1)) *
-        pairSwitch coins.1 := by
-  change (butterflyPerm (d+1) (decodeButterfly (d+1)
-      ((coinStepEquiv d).symm ((c.1.1,c.2.1),coins.2)))) *
-    (butterflyPerm (d+1) (decodeButterfly (d+1)
-      ((coinStepEquiv d).symm ((c.1.2,c.2.2),coins.1))))⁻¹ = _
-  rw [butterflyPerm_step,butterflyPerm_step,mul_inv_rev,childLift_inv,pairSwitch_inv]
-  rw [mul_assoc (pairSwitch coins.2),←mul_assoc (childLift _),childLift_mul,←mul_assoc]
-  congr 2
-  apply congrArg childLift
-  funext b
-  cases b <;> rfl
-
-lemma headTail_switch (d : ℕ) (ξ : Card d → Bool) (x : Card (d+1)) :
-    headTailEquiv d (pairSwitch ξ x) = pairLayer ξ (headTailEquiv d x) := rfl
-
-lemma headTail_child (d : ℕ) (p : Bool → Equiv.Perm (Card d)) (x : Card (d+1)) :
-    headTailEquiv d (childLift p x) = PairRouting.lift p (headTailEquiv d x) := rfl
-
-lemma palindromePerm_coordinates (d : ℕ) (c : BenesCoins d × BenesCoins d)
-    (coins : (Card d → Bool) × (Card d → Bool)) (x : Card (d+1)) :
-    headTailEquiv d (palindromePerm (d+1) ((benesStepEquiv d).symm (c,coins)) x) =
-      PairRouting.sandwich coins.1 coins.2 (fun b => palindromePerm d (if b then c.2 else c.1))
-        (headTailEquiv d x) := by
-  rw [palindromePerm_step]
-  simp only [Equiv.Perm.mul_apply,headTail_switch,headTail_child]
-  rfl
-
-noncomputable def palindromeCost : (d : ℕ) → {ι : Type*} → [Fintype ι] →
-    (ι ↪ Card d) → BenesCoins d → ℕ
-  | 0, _, _, _, _ => 0
-  | d+1, _, _, e, ω =>
-      let σ := benesStepEquiv d ω
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x σ.2.1
-      let hc := PairRouting.colors_compatible x σ.2.1
-      PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
-          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) +
-        palindromeCost d (PairRouting.childEmbedding x c hc false) σ.1.1 +
-        palindromeCost d (PairRouting.childEmbedding x c hc true) σ.1.2
-
-namespace CycleColoring
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-lemma alternating_cycle_two (p : Equiv.Perm α) (S : Finset (Bool × α))
-    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) S) : 2 ≤ C.val.card := by
-  obtain ⟨x,hx⟩ := C.nonempty
-  have hpx : alternatingPerm p x ∈ C.val := by
-    rw [←C.orbit_eq hx,mem_orbitSet]
-    exact (Equiv.Perm.SameCycle.refl _ _).apply_right
-  have hne : x ≠ alternatingPerm p x := by
-    intro he
-    have hh := congrArg Prod.fst he
-    rw [alternatingPerm_flip] at hh
-    cases hb : x.1 <;> simp only [hb, Bool.not_false, Bool.not_true] at hh <;> cases hh
-  have hs : {x,alternatingPerm p x} ⊆ C.val := by
-    intro y hy
-    simp only [Finset.mem_insert,Finset.mem_singleton] at hy
-    rcases hy with rfl | rfl <;> assumption
-  have hh := Finset.card_le_card hs
-  simpa only [Finset.card_pair hne] using hh
-
-lemma alternating_cycle_count_double (p : Equiv.Perm α) (S : Finset (Bool × α)) :
-    2 * Fintype.card (RoutingNetwork.SelectedCycle (alternatingPerm p) S) ≤ S.card := by
-  classical
-  calc
-    _ = ∑ _C : RoutingNetwork.SelectedCycle (alternatingPerm p) S, 2 := by simp [Nat.mul_comm]
-    _ ≤ RoutingNetwork.cycleMass (alternatingPerm p) S :=
-      Finset.sum_le_sum (fun C _ => alternating_cycle_two p S C)
-    _ ≤ _ := RoutingNetwork.cycleMass_le _ _
-
-lemma alternating_cycle_count_zero (p : Equiv.Perm α) (S : Finset (Bool × α))
-    (hS : S.card ≤ 1) : Fintype.card (RoutingNetwork.SelectedCycle (alternatingPerm p) S) = 0 := by
-  have h := alternating_cycle_count_double p S
-  omega
-
-end CycleColoring
-
-namespace PairRouting
-
-section
-variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
-
-lemma alternatingCycles_double (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    2 * alternatingCycles e p ≤ Fintype.card ι := by
-  have hh := CycleColoring.alternating_cycle_count_double (alternatingProjection p) (labelSet e)
-  simpa only [alternatingCycles, card_labelSet] using hh
-
-lemma alternatingCycles_zero (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α)
-    (hι : Fintype.card ι ≤ 1) : alternatingCycles e p = 0 := by
-  have hh := alternatingCycles_double e p
-  omega
-
-end
-variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α] [DecidableEq ι]
-
-omit [Fintype α] [DecidableEq ι] in
-lemma colors_chosenCoin (x : ι → Bool × α) (c : ι → Bool) (hc : Compatible x c) :
-    colors x (chosenCoin x c) = c := by
-  funext i
-  simp only [colors,chosenCoin_apply x c hc i,required]
-  cases (x i).1 <;> cases c i <;> rfl
-
-noncomputable def coloredEmbedding (x : ι ↪ Bool × α) (c : ι → Bool) (_hc : Compatible x c) :
-    ι ↪ Bool × α := switchedEmbedding x (chosenCoin x c)
-
-omit [Fintype α] [DecidableEq ι] in
-lemma coloredEmbedding_apply (x : ι ↪ Bool × α) (c : ι → Bool) (hc : Compatible x c) (i : ι) :
-    coloredEmbedding x c hc i = (c i,(x i).2) := by
-  apply Prod.ext
-  · exact congrFun (colors_chosenCoin x c hc) i
-  · rfl
-
-omit [Fintype α] [DecidableEq ι] in
-lemma switchedEmbedding_eq_colored (x : ι ↪ Bool × α) (c : ι → Bool) (hc : Compatible x c)
-    (ξ : α → Bool) (he : colors x ξ = c) :
-    switchedEmbedding x ξ = coloredEmbedding x c hc := by
-  ext i : 1
-  rw [coloredEmbedding_apply]
-  apply Prod.ext
-  · exact congrFun he i
-  · rfl
-
-lemma assignments_match_bound (x y : ι ↪ Bool × α) (c : ι → Bool)
-    (hc : c ∈ compatibleSet x y) (p : Bool → Equiv.Perm α) (hm : ChildMatches p x y c) :
-    (compatibleSet x y).card ≤
-      2 ^ ((occupied x).card + (occupied y).card - Fintype.card ι +
-        alternatingCycles (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p) := by
-  obtain ⟨hcx,hcy⟩ := (mem_compatibleSet x y c).mp hc
-  let ξ := chosenCoin x c
-  let η := chosenCoin y c
-  have he : routedEmbedding x ξ η p = y := by
-    ext i : 1
-    have h := (sandwich_matches_iff ξ η p x y).mpr
-      ⟨(colors_chosenCoin x c hcx).trans (colors_chosenCoin y c hcy).symm,by
-        simpa only [ξ,colors_chosenCoin x c hcx] using hm⟩
-    exact h i
-  have hh := assignments_actual_bound x ξ η p
-  rw [he] at hh
-  rw [card_compatibleSet]
-  exact hh
-
-omit [Fintype α] [DecidableEq ι] in
-lemma occupied_sum_ge (x y : ι ↪ Bool × α) :
-    Fintype.card ι ≤ (occupied x).card + (occupied y).card := by
-  have h₀ := occupied_card_double x
-  have h₁ := occupied_card_double y
-  omega
-
-lemma assignment_weight_bound (x y : ι ↪ Bool × α) (c : ι → Bool)
-    (hc : c ∈ compatibleSet x y) (p : Bool → Equiv.Perm α) (hm : ChildMatches p x y c) :
-    ((compatibleSet x y).card:ℝ) * outerWeight x y ≤
-      (2:ℝ)^((alternatingCycles
-        (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p : ℝ) - Fintype.card ι) := by
-  let z := alternatingCycles (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p
-  let b := (occupied x).card+(occupied y).card
-  let h := Fintype.card ι
-  have hb : h ≤ b := occupied_sum_ge x y
-  have hn : ((compatibleSet x y).card:ℝ) ≤ (2:ℝ)^(b-h+z) := by
-    exact_mod_cast assignments_match_bound x y c hc p hm
-  change ((compatibleSet x y).card:ℝ) * (1/(2:ℝ)^b) ≤ (2:ℝ)^((z:ℝ)-h)
-  rw [mul_one_div]
-  calc
-    _ ≤ (2:ℝ)^(b-h+z)/(2:ℝ)^b := div_le_div_of_nonneg_right hn (by positivity)
-    _ = _ := by
-      rw [←Real.rpow_natCast,←Real.rpow_natCast,←Real.rpow_sub (by norm_num : (0:ℝ)<2)]
-      congr 1
-      rw [Nat.cast_add,Nat.cast_sub hb]
-      ring
-
-lemma assignment_factor_bound (x y : ι ↪ Bool × α) (c : ι → Bool)
-    (hc : c ∈ compatibleSet x y) (p : Bool → Equiv.Perm α) (hm : ChildMatches p x y c)
-    (a : ℝ) (ha : 0 ≤ a) :
-    (outerWeight x y)^a * ((compatibleSet x y).card:ℝ)^a ≤
-      (2:ℝ)^(((alternatingCycles
-        (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p : ℝ) - Fintype.card ι)*a) := by
-  have h := Real.rpow_le_rpow (mul_nonneg (Nat.cast_nonneg _) (le_of_lt (outerWeight_pos x y)))
-    (assignment_weight_bound x y c hc p hm) ha
-  rw [Real.mul_rpow (Nat.cast_nonneg _) (le_of_lt (outerWeight_pos x y)),
-    ←Real.rpow_mul (by norm_num : (0:ℝ)≤2)] at h
-  simpa only [mul_comm] using h
-
-
-lemma sandwich_weighted_probability_subtype (p : Bool → Equiv.Perm α) (x y : ι → Bool × α)
-    (W : (ι → Bool) → ℝ) :
-    finiteMean (fun coins : (α → Bool) × (α → Bool) =>
-      if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then W (colors x coins.1) else 0) =
-    outerWeight x y * ∑ c : {c // c ∈ compatibleSet x y},
-      if ChildMatches p x y c.val then W c.val else 0 := by
-  rw [sandwich_weighted_probability]
-  congr 1
-  rw [Finset.sum_subtype (compatibleSet x y) (fun _ => Iff.rfl)]
-
-omit [Fintype α] [DecidableEq α] [DecidableEq ι] in
-lemma color_card_add (c : ι → Bool) :
-    Fintype.card {i // c i = false} + Fintype.card {i // c i = true} = Fintype.card ι := by
-  have he : {i // c i = true} ≃ {i // ¬c i = false} := Equiv.subtypeEquivRight fun i => by
-    cases c i <;> simp
-  rw [Fintype.card_congr he]
-  rw [Fintype.card_subtype_compl]
-  exact Nat.add_sub_of_le (Fintype.card_subtype_le _)
-
-end PairRouting
-
-noncomputable def assignmentCost (d : ℕ) {ι : Type*} [Fintype ι]
-    (x : ι ↪ Bool × Card d) (c : ι → Bool) (hc : PairRouting.Compatible x c)
-    (ω : BenesCoins d × BenesCoins d) : ℕ :=
-  PairRouting.alternatingCycles (PairRouting.coloredEmbedding x c hc)
-      (fun b => palindromePerm d (if b then ω.2 else ω.1)) +
-    palindromeCost d (PairRouting.childEmbedding x c hc false) ω.1 +
-    palindromeCost d (PairRouting.childEmbedding x c hc true) ω.2
-
-lemma palindromeCost_eq (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (d+1)) (σ : BenesCoins (d+1)) :
-    palindromeCost (d+1) e σ =
-      assignmentCost d (e.trans (headTailEquiv d).toEmbedding)
-        (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) (benesStepEquiv d σ).2.1)
-        (PairRouting.colors_compatible _ _) (benesStepEquiv d σ).1 := by
-  unfold palindromeCost assignmentCost
-  dsimp only
-  rw [PairRouting.switchedEmbedding_eq_colored _ _ (PairRouting.colors_compatible _ _) _ rfl]
-
-lemma palindromeCost_step (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (d+1)) (ω : BenesCoins d × BenesCoins d)
-    (coins : (Card d → Bool) × (Card d → Bool)) :
-    palindromeCost (d+1) e ((benesStepEquiv d).symm (ω,coins)) =
-      assignmentCost d (e.trans (headTailEquiv d).toEmbedding)
-        (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) coins.1)
-        (PairRouting.colors_compatible _ _) ω := by
-  rw [palindromeCost_eq]
-  simp only [Equiv.apply_symm_apply]
-
-noncomputable def routingMoment (d : ℕ) {ι : Type*} [Fintype ι]
-    (e f : ι ↪ Card d) (a : ℝ) : ℝ := by
-  classical
-  exact finiteMean (fun ω : BenesCoins d =>
-    if ∀ i, palindromePerm d ω (e i) = f i then (2:ℝ)^((palindromeCost d e ω:ℝ)*a) else 0)
-
-lemma routingMoment_nonneg (d : ℕ) {ι : Type*} [Fintype ι]
-    (e f : ι ↪ Card d) (a : ℝ) : 0 ≤ routingMoment d e f a := by
-  classical
-  exact finiteMean_nonneg (fun _ => by split_ifs <;> positivity)
-
-lemma palindrome_matches_step (d : ℕ) {ι : Type*}
-    (e f : ι ↪ Card (d+1)) (ω : BenesCoins d × BenesCoins d)
-    (coins : (Card d → Bool) × (Card d → Bool)) :
-    (∀ i, palindromePerm (d+1) ((benesStepEquiv d).symm (ω,coins)) (e i) = f i) ↔
-    (∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => palindromePerm d (if b then ω.2 else ω.1))
-      ((e.trans (headTailEquiv d).toEmbedding) i) = ((f.trans (headTailEquiv d).toEmbedding) i)) := by
-  apply forall_congr'
-  intro i
-  rw [←(headTailEquiv d).injective.eq_iff,palindromePerm_coordinates]
-  rfl
-
-lemma routingMoment_step (d : ℕ) {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (e f : ι ↪ Card (d+1)) (a : ℝ) :
-    let x := e.trans (headTailEquiv d).toEmbedding
-    let y := f.trans (headTailEquiv d).toEmbedding
-    routingMoment (d+1) e f a = PairRouting.outerWeight x y *
-      ∑ c : {c // c ∈ PairRouting.compatibleSet x y},
-        finiteMean (fun ω : BenesCoins d × BenesCoins d =>
-          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c.val
-          then (2:ℝ)^((assignmentCost d x c.val
-            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0) := by
-  classical
-  intro x y
-  rw [routingMoment,finiteMean_equiv (benesStepEquiv d),finiteMean_prod]
-  have he (ω : BenesCoins d × BenesCoins d) :
-      finiteMean (fun coins : (Card d → Bool) × (Card d → Bool) =>
-        if ∀ i, palindromePerm (d+1) ((benesStepEquiv d).symm (ω,coins)) (e i) = f i
-        then (2:ℝ)^((palindromeCost (d+1) e ((benesStepEquiv d).symm (ω,coins)):ℝ)*a) else 0) =
-      PairRouting.outerWeight x y *
-        ∑ c : {c // c ∈ PairRouting.compatibleSet x y},
-          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c.val
-          then (2:ℝ)^((assignmentCost d x c.val
-            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0 := by
-    let W : (ι → Bool) → ℝ := fun c =>
-      if hc : PairRouting.Compatible x c then (2:ℝ)^((assignmentCost d x c hc ω:ℝ)*a) else 0
-    calc
-      _ = finiteMean (fun coins : (Card d → Bool) × (Card d → Bool) =>
-        if ∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => palindromePerm d (if b then ω.2 else ω.1))
-          (x i) = y i then W (PairRouting.colors x coins.1) else 0) := by
-        apply finiteMean_congr
-        intro coins
-        simp only [palindrome_matches_step,palindromeCost_step]
-        simp only [W,PairRouting.colors_compatible,↓reduceDIte]
-        rfl
-      _ = _ := by
-        rw [PairRouting.sandwich_weighted_probability_subtype]
-        congr 1
-        apply Finset.sum_congr rfl
-        intro c _
-        simp only [W,((PairRouting.mem_compatibleSet x y c.val).mp c.property).1,↓reduceDIte]
-  simp_rw [he]
-  simp_rw [mul_comm (PairRouting.outerWeight x y)]
-  rw [finiteMean_mul_const,finiteMean_sum,mul_comm]
-
-lemma child_moment_product (d : ℕ) {ι : Type*} [Fintype ι]
-    (x y : ι ↪ Bool × Card d) (c : ι → Bool)
-    (hcx : PairRouting.Compatible x c) (hcy : PairRouting.Compatible y c) (a : ℝ) :
-    routingMoment d (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a *
-      routingMoment d (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a =
-    finiteMean (fun ω : BenesCoins d × BenesCoins d =>
-      if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
-      then (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
-        palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0) := by
-  classical
-  unfold routingMoment
-  rw [←finiteMean_prod_mul]
-  apply finiteMean_congr
-  intro ω
-  let M₀ := ∀ i : {i // c i = false}, palindromePerm d ω.1 (x i.val).2 = (y i.val).2
-  let M₁ := ∀ i : {i // c i = true}, palindromePerm d ω.2 (x i.val).2 = (y i.val).2
-  have hp : (fun b : Bool => palindromePerm d (if b then ω.2 else ω.1)) =
-      (fun b => if b then palindromePerm d ω.2 else palindromePerm d ω.1) := by
-    funext b
-    cases b <;> rfl
-  have hs : PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c ↔
-      M₀ ∧ M₁ := by rw [hp]; exact PairRouting.childMatches_split _ _ _ _ _
-  change (if M₀ then _ else 0) * (if M₁ then _ else 0) = _
-  by_cases h₀ : M₀ <;> by_cases h₁ : M₁
-  · simp only [hs,h₀,h₁,true_and,↓reduceIte]
-    rw [←Real.rpow_add (by norm_num : (0:ℝ)<2)]
-    congr 1
-    ring
-  · simp only [hs,h₀,h₁,and_false,↓reduceIte,mul_zero]
-  · simp only [hs,h₀,h₁,false_and,↓reduceIte,zero_mul]
-  · simp only [hs,h₀,h₁,false_and,↓reduceIte,mul_zero]
-
-lemma palindrome_probability_step (d : ℕ) {ι : Type*} [Fintype ι]
-    (e f : ι ↪ Card (d+1)) :
-    PairRouting.tupleProbability (palindromePerm (d+1)) e f =
-    finiteMean (fun ω : BenesCoins d × BenesCoins d =>
-      finiteMean (fun coins : (Card d → Bool) × (Card d → Bool) =>
-        if ∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => if b then palindromePerm d ω.2 else palindromePerm d ω.1)
-          ((e.trans (headTailEquiv d).toEmbedding) i) = ((f.trans (headTailEquiv d).toEmbedding) i)
-        then (1:ℝ) else 0)) := by
-  classical
-  rw [PairRouting.tupleProbability,finiteMean_equiv (benesStepEquiv d),finiteMean_prod]
-  have hp (b : Bool) (p q : BenesCoins d) :
-      palindromePerm d (if b then p else q) = (if b then palindromePerm d p else palindromePerm d q) := by
-    cases b <;> rfl
-  simp only [palindrome_matches_step,hp]
-
-lemma child_power_product_bound (d : ℕ) {ι : Type*} [Fintype ι]
-    (x y : ι ↪ Bool × Card d) (c : ι → Bool)
-    (hcx : PairRouting.Compatible x c) (hcy : PairRouting.Compatible y c) (a : ℝ)
-    (hfalse : (PairRouting.tupleProbability (palindromePerm d)
-        (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false))^(1+a) ≤
-      (2:ℝ)^(-(d:ℝ)*Fintype.card {i // c i = false}*a) *
-        routingMoment d (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a)
-    (htrue : (PairRouting.tupleProbability (palindromePerm d)
-        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true))^(1+a) ≤
-      (2:ℝ)^(-(d:ℝ)*Fintype.card {i // c i = true}*a) *
-        routingMoment d (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a) :
-    (PairRouting.tupleProbability (palindromePerm d)
-      (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false))^(1+a) *
-      (PairRouting.tupleProbability (palindromePerm d)
-        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true))^(1+a) ≤
-    (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) *
-      finiteMean (fun ω : BenesCoins d × BenesCoins d =>
-        if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
-        then (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
-          palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0) := by
-  have hh := mul_le_mul hfalse htrue
-    (Real.rpow_nonneg (PairRouting.tupleProbability_nonneg _ _ _) _)
-    (mul_nonneg (by positivity) (routingMoment_nonneg _ _ _ _))
-  rw [mul_mul_mul_comm,child_moment_product,←Real.rpow_add (by norm_num : (0:ℝ)<2)] at hh
-  have he : -(d:ℝ)*Fintype.card {i // c i = false}*a + -(d:ℝ)*Fintype.card {i // c i = true}*a =
-      -(d:ℝ)*Fintype.card ι*a := by
-    rw [←add_mul,←mul_add,←Nat.cast_add,PairRouting.color_card_add]
-  rwa [he] at hh
-
-lemma assignment_moment_bound (d : ℕ) {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (x y : ι ↪ Bool × Card d) (c : ι → Bool) (hc : c ∈ PairRouting.compatibleSet x y)
-    (a : ℝ) (ha : 0 ≤ a) :
-    let hcx := ((PairRouting.mem_compatibleSet x y c).mp hc).1
-    PairRouting.outerWeight x y ^ a * ((PairRouting.compatibleSet x y).card:ℝ)^a *
-      ((2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) *
-        finiteMean (fun ω : BenesCoins d × BenesCoins d =>
-          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
-          then (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
-            palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0)) ≤
-    (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) *
-      finiteMean (fun ω : BenesCoins d × BenesCoins d =>
-        if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
-        then (2:ℝ)^((assignmentCost d x c hcx ω:ℝ)*a) else 0) := by
-  classical
-  intro hcx
-  have hscale (q : ℝ) (F : BenesCoins d × BenesCoins d → ℝ) :
-      q * finiteMean F = finiteMean (fun ω => q * F ω) := by
-    simp_rw [mul_comm q,finiteMean_mul_const]
-  rw [←mul_assoc,hscale,hscale]
-  apply finiteMean_mono
-  intro ω
-  split_ifs with hm
-  · have hh := PairRouting.assignment_factor_bound x y c hc
-      (fun b => palindromePerm d (if b then ω.2 else ω.1)) hm a ha
-    calc
-      _ ≤ ((2:ℝ)^(((PairRouting.alternatingCycles (PairRouting.coloredEmbedding x c hcx)
-          (fun b => palindromePerm d (if b then ω.2 else ω.1)):ℝ)-Fintype.card ι)*a) *
-          (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a)) *
-          (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
-            palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) :=
-        mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hh (by positivity)) (by positivity)
-      _ = _ := by
-        simp only [←Real.rpow_add (by norm_num : (0:ℝ)<2),assignmentCost,Nat.cast_add]
-        congr 1
-        ring
-  · simp only [mul_zero]
-    exact le_rfl
-
-theorem palindrome_pointwise_moment (d : ℕ) {ι : Type*} [Fintype ι]
-    (e f : ι ↪ Card d) (a : ℝ) (ha : 0 ≤ a) :
-    (PairRouting.tupleProbability (palindromePerm d) e f)^(1+a) ≤
-      (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) * routingMoment d e f a := by
-  classical
-  induction d generalizing ι with
-  | zero =>
-      have he : ∀ (ω : BenesCoins 0) (i : ι), palindromePerm 0 ω (e i) = f i :=
-        fun _ _ => Subsingleton.elim _ _
-      simp [PairRouting.tupleProbability,routingMoment,he,palindromeCost,finiteMean_const]
-  | succ d ih =>
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let y := f.trans (headTailEquiv d).toEmbedding
-      let W := PairRouting.outerWeight x y
-      let C := PairRouting.compatibleSet x y
-      let p : (ι → Bool) → ℝ := fun c =>
-        (PairRouting.tupleProbability (palindromePerm d)
-          (fun i : {i // c i = false} => (x i.val).2)
-          (fun i : {i // c i = false} => (y i.val).2))^(1+a) *
-        (PairRouting.tupleProbability (palindromePerm d)
-          (fun i : {i // c i = true} => (x i.val).2)
-          (fun i : {i // c i = true} => (y i.val).2))^(1+a)
-      let m : {c // c ∈ C} → ℝ := fun c =>
-        finiteMean (fun ω : BenesCoins d × BenesCoins d =>
-          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c.val
-          then (2:ℝ)^((assignmentCost d x c.val
-            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0)
-      have hp (c : {c // c ∈ C}) : W^a * (C.card:ℝ)^a * p c.val ≤
-          (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * m c := by
-        obtain ⟨hcx,hcy⟩ := (PairRouting.mem_compatibleSet x y c.val).mp c.property
-        have h₀ := ih (PairRouting.childEmbedding x c.val hcx false)
-          (PairRouting.childEmbedding y c.val hcy false)
-        have h₁ := ih (PairRouting.childEmbedding x c.val hcx true)
-          (PairRouting.childEmbedding y c.val hcy true)
-        have hh := child_power_product_bound d x y c.val hcx hcy a h₀ h₁
-        have hh' := mul_le_mul_of_nonneg_left hh
-          (mul_nonneg (Real.rpow_nonneg (le_of_lt (PairRouting.outerWeight_pos x y)) a)
-            (Real.rpow_nonneg (Nat.cast_nonneg C.card) a))
-        exact hh'.trans (assignment_moment_bound d x y c.val c.property a ha)
-      have hnode := PairRouting.sandwich_power_bound (palindromePerm d) (palindromePerm d) x y a ha
-      rw [←palindrome_probability_step d e f] at hnode
-      change (PairRouting.tupleProbability (palindromePerm (d+1)) e f)^(1+a) ≤
-        W^(1+a) * (C.card:ℝ)^a * ∑ c ∈ C, p c at hnode
-      have hsum : ∑ c ∈ C, p c = ∑ c : {c // c ∈ C}, p c.val := by
-        rw [Finset.sum_subtype C (fun _ => Iff.rfl)]
-      have hW : 0 < W := PairRouting.outerWeight_pos x y
-      calc
-        _ ≤ W^(1+a) * (C.card:ℝ)^a * ∑ c ∈ C, p c := hnode
-        _ = W * ∑ c : {c // c ∈ C}, W^a * (C.card:ℝ)^a * p c.val := by
-          rw [Real.rpow_add hW,Real.rpow_one,hsum,←Finset.mul_sum]
-          ring
-        _ ≤ W * ∑ c : {c // c ∈ C}, (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * m c :=
-          mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun c _ => hp c)) (le_of_lt hW)
-        _ = _ := by
-          rw [←Finset.mul_sum,routingMoment_step]
-          simp only [Nat.cast_add,Nat.cast_one]
-          change W * ((2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * ∑ c, m c) =
-            (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * (W * ∑ c, m c)
-          ring
-
-lemma routingMoment_sum (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (a : ℝ) :
-    (∑ f : ι ↪ Card d, routingMoment d e f a) =
-      finiteMean (fun ω : BenesCoins d => (2:ℝ)^((palindromeCost d e ω:ℝ)*a)) := by
-  classical
-  unfold routingMoment
-  rw [←finiteMean_sum]
-  apply finiteMean_congr
-  intro ω
-  let f₀ : ι ↪ Card d := e.trans (palindromePerm d ω).toEmbedding
-  rw [Finset.sum_eq_single f₀]
-  · have hm : ∀ i, palindromePerm d ω (e i) = f₀ i := fun _ => rfl
-    simp only [hm,implies_true,↓reduceIte]
-  · intro f _ hf
-    have hm : ¬ ∀ i, palindromePerm d ω (e i) = f i := by
-      intro hh
-      apply hf
-      ext i : 1
-      exact (hh i).symm
-    exact ite_eq_right hm
-  · simp
-
-lemma finite_row_power {X : Type*} [Fintype X] [Nonempty X]
-    (p : X → ℝ) (hp : ∀ x, 0 ≤ p x) (a : ℝ) :
-    finiteMean (fun x => ((Fintype.card X:ℝ)*p x)^(1+a)) =
-      (Fintype.card X:ℝ)^a * ∑ x, (p x)^(1+a) := by
-  have hn : (0:ℝ) < Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
-  simp only [finiteMean,Real.mul_rpow (le_of_lt hn) (hp _),←Finset.mul_sum]
-  rw [Real.rpow_add hn,Real.rpow_one]
-  field_simp
-
-noncomputable def palindromeRowMoment (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (a : ℝ) : ℝ :=
-  finiteMean (fun f : ι ↪ Card d =>
-    ((Fintype.card (ι ↪ Card d):ℝ)*PairRouting.tupleProbability (palindromePerm d) e f)^(1+a))
-
-lemma density_normalization (d h n : ℕ) (a : ℝ) :
-    (n:ℝ)^a * (2:ℝ)^(-(d:ℝ)*h*a) = ((n:ℝ)/((2:ℝ)^d)^h)^a := by
-  rw [Real.div_rpow (Nat.cast_nonneg _) (by positivity),←pow_mul,
-    ←Real.rpow_natCast,←Real.rpow_mul (by norm_num : (0:ℝ) ≤ 2),Nat.cast_mul]
-  rw [show -(d:ℝ)*h*a = -((d:ℝ)*h*a) by ring,
-    Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2),div_eq_mul_inv]
-
-theorem palindrome_row_density_moment (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (a : ℝ) (ha : 0 ≤ a) :
-    palindromeRowMoment d e a ≤
-      (((2^d).descFactorial (Fintype.card ι):ℝ)/((2^d:ℕ):ℝ)^(Fintype.card ι))^a *
-        finiteMean (fun ω : BenesCoins d => (2:ℝ)^((palindromeCost d e ω:ℝ)*a)) := by
-  classical
-  have : Nonempty (ι ↪ Card d) := ⟨e⟩
-  rw [palindromeRowMoment,finite_row_power _ (fun f => PairRouting.tupleProbability_nonneg _ _ _)]
-  calc
-    _ ≤ (Fintype.card (ι ↪ Card d):ℝ)^a *
-      ∑ f : ι ↪ Card d, (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) * routingMoment d e f a :=
-      mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun f _ => palindrome_pointwise_moment d e f a ha))
-        (Real.rpow_nonneg (Nat.cast_nonneg _) _)
-    _ = _ := by
-      rw [←Finset.mul_sum,routingMoment_sum,←mul_assoc,density_normalization,
-        Fintype.card_embedding_eq,card_positions]
-      norm_cast
-
-end Thorp
-namespace Thorp
-
-lemma subset_biUnion_indicator {α β : Type*} [DecidableEq α] [Fintype β]
-    (A : Finset α) (T : β → Finset α) :
-    (if Finset.univ.biUnion T ⊆ A then (1:ℝ) else 0) =
-      ∏ b, if T b ⊆ A then (1:ℝ) else 0 := by
-  classical
-  by_cases h : Finset.univ.biUnion T ⊆ A
-  · rw [ite_eq_left h]
-    symm
-    apply Finset.prod_eq_one
-    intro b _
-    rw [ite_eq_left]
-    exact (Finset.subset_biUnion_of_mem T (Finset.mem_univ b)).trans h
-  · rw [ite_eq_right h]
-    obtain ⟨x,hx,hxA⟩ := Finset.not_subset.mp h
-    obtain ⟨b,_,hxb⟩ := Finset.mem_biUnion.mp hx
-    symm
-    apply Finset.prod_eq_zero (Finset.mem_univ b)
-    rw [ite_eq_right]
-    exact fun hb => hxA (hb hxb)
-
-lemma cylinder_block_polynomial {α β Ω : Type*} [DecidableEq α] [Fintype β]
-    [Fintype Ω] (τ : β → Type*) [∀ b, Fintype (τ b)]
-    (F : Ω → Finset α) (p : α → ℝ)
-    (hF : CylinderBound (inclusionMoment F) p)
-    (T : ∀ b, τ b → Finset α) (c : ∀ b, τ b → ℝ)
-    (hc : ∀ b s, 0 ≤ c b s)
-    (hd : ∀ b b', b ≠ b' → ∀ s s', Disjoint (T b s) (T b' s')) :
-    finiteMean (fun ω => ∏ b, ∑ s, c b s * if T b s ⊆ F ω then 1 else 0) ≤
-      ∏ b, ∑ s, c b s * ∏ x ∈ T b s, p x := by
-  classical
-  simp_rw [Fintype.prod_sum]
-  rw [finiteMean_sum]
-  apply Finset.sum_le_sum
-  intro s _
-  let U := Finset.univ.biUnion (fun b => T b (s b))
-  have he : (fun ω => ∏ b, c b (s b) * if T b (s b) ⊆ F ω then 1 else 0) =
-      fun ω => (if U ⊆ F ω then 1 else 0) * ∏ b, c b (s b) := by
-    funext ω
-    rw [Finset.prod_mul_distrib,←subset_biUnion_indicator]
-    exact mul_comm _ _
-  rw [he,finiteMean_mul_const]
-  have hdis : ((Finset.univ : Finset β) : Set β).PairwiseDisjoint (fun b => T b (s b)) := by
-    intro b _ b' _ hbb
-    exact hd b b' hbb (s b) (s b')
-  have hmean : finiteMean (fun ω => if U ⊆ F ω then 1 else 0) = inclusionMoment F U := by
-    unfold inclusionMoment
-    apply finiteMean_congr
-    intro ω
-    by_cases hh : U ⊆ F ω <;> simp [hh]
-  calc
-    _ ≤ (∏ x ∈ U, p x) * ∏ b, c b (s b) :=
-      mul_le_mul_of_nonneg_right (hmean.le.trans (hF U)) (Finset.prod_nonneg (fun b _ => hc b (s b)))
-    _ = _ := by
-      dsimp only [U]
-      rw [Finset.prod_biUnion hdis,←Finset.prod_mul_distrib]
-      apply Finset.prod_congr rfl
-      intro b _
-      exact mul_comm _ _
-
-
-noncomputable def sparseCoefficient {α : Type*} (w : ℝ) (T : Finset α) : ℝ :=
-  if T.card = 1 then 0 else w^T.card
-
-lemma sparseCoefficient_nonneg {α : Type*} (w : ℝ) (hw : 0 ≤ w) (T : Finset α) :
-    0 ≤ sparseCoefficient w T := by
-  unfold sparseCoefficient
-  split_ifs <;> positivity
-
-lemma sparse_block_pointwise {α : Type*} [DecidableEq α] (A D : Finset α)
-    (w : ℝ) (hw : 0 ≤ w) :
-    w^(if 2 ≤ (D ∩ A).card then (D ∩ A).card else 0) ≤
-      ∑ T : ↑D.powerset, sparseCoefficient w T.val * if T.val ⊆ A then 1 else 0 := by
-  classical
-  let V := D ∩ A
-  let T : ↑D.powerset := if 2 ≤ V.card then
-    ⟨V,Finset.mem_powerset.mpr Finset.inter_subset_left⟩ else
-    ⟨∅,Finset.mem_powerset.mpr (Finset.empty_subset D)⟩
-  calc
-    _ = sparseCoefficient w T.val * (if T.val ⊆ A then 1 else 0) := by
-      by_cases h : 2 ≤ V.card
-      · have hn : V.card ≠ 1 := by omega
-        simp [T,ite_eq_left h,sparseCoefficient,hn,show V ⊆ A from Finset.inter_subset_right,V]
-      · simp [T,ite_eq_right h,sparseCoefficient,V] at *
-    _ ≤ _ := Finset.single_le_sum (f := fun S : ↑D.powerset =>
-      sparseCoefficient w S.val * (if S.val ⊆ A then 1 else 0))
-      (fun S _ => mul_nonneg (sparseCoefficient_nonneg w hw S.val)
-        (by split_ifs <;> norm_num)) (Finset.mem_univ T)
-
-lemma geometric_sum_half (n : ℕ) {x : ℝ} (hx : 0 ≤ x) (hhalf : x ≤ 1/2) :
-    ∑ i ∈ Finset.range n, x^i ≤ 2 := by
-  induction n with
-  | zero => simp
-  | succ n ih =>
-    rw [Finset.sum_range_succ',pow_zero]
-    simp only [pow_succ',←Finset.mul_sum]
-    nlinarith
-
-lemma sparse_binomial_bound (R : ℕ) (z : ℝ) (hz : 0 ≤ z) (hr : (R:ℝ)*z ≤ 1/2) :
-    ∑ i ∈ Finset.range (R+1), (R.choose i:ℝ) * (if i = 1 then 0 else z^i) ≤
-      1 + 2*((R:ℝ)*z)^2 := by
-  classical
-  let x := (R:ℝ)*z
-  have hx : 0 ≤ x := mul_nonneg (Nat.cast_nonneg _) hz
-  have hbound (i : ℕ) : (R.choose i:ℝ)*(if i=1 then 0 else z^i) ≤
-      if i=1 then 0 else x^i := by
-    by_cases h : i=1
-    · simp [h]
-    · simp only [ite_eq_right h]
-      dsimp [x]
-      rw [mul_pow]
-      apply mul_le_mul_of_nonneg_right _ (pow_nonneg hz i)
-      exact_mod_cast Nat.choose_le_pow R i
-  calc
-    _ ≤ ∑ i ∈ Finset.range (R+1), if i=1 then 0 else x^i :=
-      Finset.sum_le_sum (fun i _ => hbound i)
-    _ ≤ 1+2*x^2 := by
-      rcases R with _ | R
-      · simp [x]
-      rcases R with _ | R
-      · norm_num [Finset.sum_range_succ]
-        positivity
-      have he : ∑ i ∈ Finset.range (R+2+1), (if i=1 then 0 else x^i) =
-          1 + x^2*∑ i ∈ Finset.range (R+1), x^i := by
-        rw [show R+2+1=2+(R+1) by omega,Finset.sum_range_add]
-        have hn (i : ℕ) : 2+i ≠ 1 := by omega
-        simp only [hn,ite_false,pow_add,←Finset.mul_sum]
-        norm_num [Finset.sum_range_succ]
-      rw [he]
-      have hg := geometric_sum_half (R+1) hx hr
-      nlinarith [sq_nonneg x]
-
-lemma sparse_block_product_bound {α : Type*} [DecidableEq α]
-    (D : Finset α) (p w : ℝ) (hp : 0 ≤ p) (hw : 0 ≤ w)
-    (hr : (D.card:ℝ)*(p*w) ≤ 1/2) :
-    ∑ T : ↑D.powerset, sparseCoefficient w T.val * p^T.val.card ≤
-      Real.exp (2*((D.card:ℝ)*(p*w))^2) := by
-  classical
-  have he : ∑ T : ↑D.powerset, sparseCoefficient w T.val * p^T.val.card =
-      ∑ i ∈ Finset.range (D.card+1), (D.card.choose i:ℝ)*(if i=1 then 0 else (p*w)^i) := by
-    rw [Finset.sum_coe_sort D.powerset (fun T => sparseCoefficient w T*p^T.card)]
-    have hh : (fun T : Finset α => sparseCoefficient w T*p^T.card) =
-        fun T => if T.card=1 then 0 else (p*w)^T.card := by
-      funext T
-      unfold sparseCoefficient
-      split_ifs <;> simp [mul_pow,mul_comm]
-    change (∑ T ∈ D.powerset, (fun T : Finset α => sparseCoefficient w T*p^T.card) T) = _
-    rw [hh,Finset.sum_powerset_apply_card (fun i => if i=1 then (0:ℝ) else (p*w)^i)]
-    simp only [nsmul_eq_mul]
-  rw [he]
-  exact (sparse_binomial_bound D.card (p*w) (mul_nonneg hp hw) hr).trans
-    (by simpa only [add_comm] using Real.add_one_le_exp (2*((D.card:ℝ)*(p*w))^2))
-
-
-noncomputable def sparseMass {α β : Type*} [DecidableEq α] [Fintype β]
-    (D : β → Finset α) (A : Finset α) : ℕ :=
-  ∑ b, if 2 ≤ (D b ∩ A).card then (D b ∩ A).card else 0
-
-theorem sparse_block_mgf {α β Ω : Type*} [DecidableEq α] [Fintype β] [Fintype Ω]
-    (D : β → Finset α) (hd : Pairwise (fun b b' => Disjoint (D b) (D b')))
-    (F : Ω → Finset α) (p w : ℝ) (hp : 0 ≤ p) (hw : 0 ≤ w)
-    (hF : ∀ T, inclusionMoment F T ≤ p^T.card)
-    (hr : ∀ b, ((D b).card:ℝ)*(p*w) ≤ 1/2) :
-    finiteMean (fun ω => w^(sparseMass D (F ω))) ≤
-      Real.exp (∑ b, 2*(((D b).card:ℝ)*(p*w))^2) := by
-  classical
-  have hc : CylinderBound (inclusionMoment F) (fun _ => p) := by
-    intro T
-    simpa only [Finset.prod_const] using hF T
-  have hb := cylinder_block_polynomial (fun b => ↑(D b).powerset) F (fun _ => p) hc
-    (fun _ S => S.val) (fun _ S => sparseCoefficient w S.val)
-    (fun _ S => sparseCoefficient_nonneg w hw S.val)
-    (fun b b' h S S' => (hd h).mono
-      (Finset.mem_powerset.mp S.property) (Finset.mem_powerset.mp S'.property))
-  calc
-    _ ≤ finiteMean (fun ω => ∏ b, ∑ S : ↑(D b).powerset,
-        sparseCoefficient w S.val * if S.val ⊆ F ω then 1 else 0) := by
-      apply finiteMean_mono
-      intro ω
-      rw [sparseMass,←Finset.prod_pow_eq_pow_sum]
-      apply Finset.prod_le_prod₀ (fun _ _ => pow_nonneg hw _)
-      intro b _
-      exact sparse_block_pointwise (F ω) (D b) w hw
-    _ ≤ ∏ b, ∑ S : ↑(D b).powerset, sparseCoefficient w S.val * p^S.val.card := by
-      simpa only [Finset.prod_const] using hb
-    _ ≤ _ := by
-      rw [Real.exp_sum]
-      apply Finset.prod_le_prod₀
-      · intro b _
-        exact Finset.sum_nonneg (fun S _ => mul_nonneg
-          (sparseCoefficient_nonneg w hw S.val) (pow_nonneg hp _))
-      · intro b _
-        exact sparse_block_product_bound (D b) p w hp hw (hr b)
-
-theorem inverseButterfly_sparse_mgf (d : ℕ) {β : Type*} [Fintype β]
-    (D : β → Finset (Card d)) (hd : Pairwise (fun b b' => Disjoint (D b) (D b')))
-    (A : Finset (Card d)) (w : ℝ) (hw : 0 ≤ w)
-    (hr : ∀ b, ((D b).card:ℝ)*((A.card/(2:ℝ)^d)*w) ≤ 1/2) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      w^(sparseMass D (inverseButterflyMarks d A ω))) ≤
-      Real.exp (∑ b, 2*(((D b).card:ℝ)*((A.card/(2:ℝ)^d)*w))^2) := by
-  apply sparse_block_mgf D hd (inverseButterflyMarks d A) (A.card/(2:ℝ)^d) w
-    (by positivity) hw
-  · exact inverseButterflyMarks_inclusion d A
-  · exact hr
-
-
-def crowdedMass (h : ℕ) : ℕ := if 2 ≤ h then h else 0
-
-lemma crowdedMass_le (h : ℕ) : crowdedMass h ≤ h := by
-  unfold crowdedMass
-  split_ifs <;> omega
-
-lemma crowdedMass_add (u v : ℕ) : crowdedMass u + crowdedMass v ≤ crowdedMass (u+v) := by
-  unfold crowdedMass
-  split_ifs <;> omega
-
-lemma alternatingCycles_le_crowded {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
-    (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    PairRouting.alternatingCycles e p ≤ crowdedMass (Fintype.card ι) := by
-  unfold crowdedMass
-  split_ifs with h
-  · have hh := PairRouting.alternatingCycles_double e p
-    omega
-  · rw [PairRouting.alternatingCycles_zero e p (by omega)]
-
-lemma palindromeCost_le_crowded (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) :
-    palindromeCost d e ω ≤ d*crowdedMass (Fintype.card ι) := by
-  classical
-  induction d generalizing ι with
-  | zero => simp [palindromeCost]
-  | succ d ih =>
-    let σ := benesStepEquiv d ω
-    let x := e.trans (headTailEquiv d).toEmbedding
-    let c := PairRouting.colors x σ.2.1
-    let hc := PairRouting.colors_compatible x σ.2.1
-    have h₀ := ih (PairRouting.childEmbedding x c hc false) σ.1.1
-    have h₁ := ih (PairRouting.childEmbedding x c hc true) σ.1.2
-    have hz := alternatingCycles_le_crowded (PairRouting.switchedEmbedding x σ.2.1)
-      (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1))
-    have hs := crowdedMass_add (Fintype.card {i // c i = false})
-      (Fintype.card {i // c i = true})
-    rw [PairRouting.color_card_add] at hs
-    change PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
-        (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) +
-        palindromeCost d (PairRouting.childEmbedding x c hc false) σ.1.1 +
-        palindromeCost d (PairRouting.childEmbedding x c hc true) σ.1.2 ≤ _
-    nlinarith
-
-noncomputable def palindromeLowCost (H : ℕ) : (d : ℕ) → {ι : Type*} → [Fintype ι] →
-    (ι ↪ Card d) → BenesCoins d → ℕ
-  | 0, _, _, _, _ => 0
-  | d+1, _, _, e, ω =>
-      let σ := benesStepEquiv d ω
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x σ.2.1
-      let hc := PairRouting.colors_compatible x σ.2.1
-      (if d+1 ≤ H then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
-          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) else 0) +
-        palindromeLowCost H d (PairRouting.childEmbedding x c hc false) σ.1.1 +
-        palindromeLowCost H d (PairRouting.childEmbedding x c hc true) σ.1.2
-
-noncomputable def cutoffMass (H : ℕ) : (d : ℕ) → {ι : Type*} → [Fintype ι] →
-    (ι ↪ Card d) → BenesCoins d → ℕ
-  | 0, _, _, _, _ => 0
-  | d+1, ι, _, e, ω =>
-      if d+1 ≤ H then crowdedMass (Fintype.card ι) else
-      let σ := benesStepEquiv d ω
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x σ.2.1
-      let hc := PairRouting.colors_compatible x σ.2.1
-      cutoffMass H d (PairRouting.childEmbedding x c hc false) σ.1.1 +
-        cutoffMass H d (PairRouting.childEmbedding x c hc true) σ.1.2
-
-lemma palindromeLowCost_eq (H d : ℕ) (h : d ≤ H) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) :
-    palindromeLowCost H d e ω = palindromeCost d e ω := by
-  classical
-  induction d generalizing ι with
-  | zero => rfl
-  | succ d ih =>
-    rw [palindromeLowCost,palindromeCost,ite_eq_left h]
-    congr 2 <;> exact ih (by omega) _ _
-
-lemma palindromeLowCost_le_cutoff (H d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) :
-    palindromeLowCost H d e ω ≤ H*cutoffMass H d e ω := by
-  classical
-  induction d generalizing ι with
-  | zero => simp [palindromeLowCost,cutoffMass]
-  | succ d ih =>
-    by_cases h : d+1 ≤ H
-    · rw [palindromeLowCost_eq H (d+1) h,cutoffMass,ite_eq_left h]
-      exact (palindromeCost_le_crowded (d+1) e ω).trans (Nat.mul_le_mul_right _ h)
-    · rw [palindromeLowCost,cutoffMass,ite_eq_right h,ite_eq_right h,zero_add,Nat.mul_add]
-      exact Nat.add_le_add (ih _ _) (ih _ _)
-
-
-noncomputable def centralBlock (r H : ℕ) (b : Card r) : Finset (Card (H+r)) :=
-  Finset.univ.image (fun z : Card H => (cardSplit r H).symm (b,z))
-
-end Thorp
-
-end ThorpNine.Tail
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Tail.AlternatingPerm
+
+namespace OAI
+
+namespace ThorpNine.Tail
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+
+abbrev BenesCoins (d : ℕ) := (SwitchIndex d → Bool) × (SwitchIndex d → Bool)
+
+def palindromePerm (d : ℕ) (ω : BenesCoins d) : Equiv.Perm (Card d) :=
+  butterflyPerm d (decodeButterfly d ω.1) * (butterflyPerm d (decodeButterfly d ω.2)).symm
+
+def sandwichShuffle {A B C D E F : Type*} :
+    ((A × B) × C) × ((D × E) × F) ≃ ((A × D) × (B × E)) × (F × C) where
+  toFun x := (((x.1.1.1,x.2.1.1),(x.1.1.2,x.2.1.2)),(x.2.2,x.1.2))
+  invFun x := (((x.1.1.1,x.1.2.1),x.2.2),((x.1.1.2,x.1.2.2),x.2.1))
+  left_inv _ := rfl
+  right_inv _ := rfl
+
+def benesStepEquiv (d : ℕ) : BenesCoins (d+1) ≃
+    (BenesCoins d × BenesCoins d) × ((Card d → Bool) × (Card d → Bool)) :=
+  (Equiv.prodCongr (coinStepEquiv d) (coinStepEquiv d)).trans sandwichShuffle
+
+lemma childLift_mul (d : ℕ) (p q : Bool → Equiv.Perm (Card d)) :
+    childLift p * childLift q = childLift (fun b => p b * q b) := by
+  ext x i
+  simp only [Equiv.Perm.mul_apply,childLift,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons]
+
+lemma childLift_inv (d : ℕ) (p : Bool → Equiv.Perm (Card d)) :
+    (childLift p)⁻¹ = childLift (fun b => (p b)⁻¹) := rfl
+
+lemma pairSwitch_inv (d : ℕ) (ξ : Card d → Bool) : (pairSwitch ξ)⁻¹ = pairSwitch ξ := rfl
+
+lemma palindromePerm_step (d : ℕ) (c : BenesCoins d × BenesCoins d)
+    (coins : (Card d → Bool) × (Card d → Bool)) :
+    palindromePerm (d+1) ((benesStepEquiv d).symm (c,coins)) =
+      pairSwitch coins.2 * childLift (fun b => palindromePerm d (if b then c.2 else c.1)) *
+        pairSwitch coins.1 := by
+  change (butterflyPerm (d+1) (decodeButterfly (d+1)
+      ((coinStepEquiv d).symm ((c.1.1,c.2.1),coins.2)))) *
+    (butterflyPerm (d+1) (decodeButterfly (d+1)
+      ((coinStepEquiv d).symm ((c.1.2,c.2.2),coins.1))))⁻¹ = _
+  rw [butterflyPerm_step,butterflyPerm_step,mul_inv_rev,childLift_inv,pairSwitch_inv]
+  rw [mul_assoc (pairSwitch coins.2),←mul_assoc (childLift _),childLift_mul,←mul_assoc]
+  congr 2
+  apply congrArg childLift
+  funext b
+  cases b <;> rfl
+
+lemma headTail_switch (d : ℕ) (ξ : Card d → Bool) (x : Card (d+1)) :
+    headTailEquiv d (pairSwitch ξ x) = pairLayer ξ (headTailEquiv d x) := rfl
+
+lemma headTail_child (d : ℕ) (p : Bool → Equiv.Perm (Card d)) (x : Card (d+1)) :
+    headTailEquiv d (childLift p x) = PairRouting.lift p (headTailEquiv d x) := rfl
+
+lemma palindromePerm_coordinates (d : ℕ) (c : BenesCoins d × BenesCoins d)
+    (coins : (Card d → Bool) × (Card d → Bool)) (x : Card (d+1)) :
+    headTailEquiv d (palindromePerm (d+1) ((benesStepEquiv d).symm (c,coins)) x) =
+      PairRouting.sandwich coins.1 coins.2 (fun b => palindromePerm d (if b then c.2 else c.1))
+        (headTailEquiv d x) := by
+  rw [palindromePerm_step]
+  simp only [Equiv.Perm.mul_apply,headTail_switch,headTail_child]
+  rfl
+
+noncomputable def palindromeCost : (d : ℕ) → {ι : Type*} → [Fintype ι] →
+    (ι ↪ Card d) → BenesCoins d → ℕ
+  | 0, _, _, _, _ => 0
+  | d+1, _, _, e, ω =>
+      let σ := benesStepEquiv d ω
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x σ.2.1
+      let hc := PairRouting.colors_compatible x σ.2.1
+      PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
+          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) +
+        palindromeCost d (PairRouting.childEmbedding x c hc false) σ.1.1 +
+        palindromeCost d (PairRouting.childEmbedding x c hc true) σ.1.2
+
+namespace CycleColoring
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+lemma alternating_cycle_two (p : Equiv.Perm α) (S : Finset (Bool × α))
+    (C : RoutingNetwork.SelectedCycle (alternatingPerm p) S) : 2 ≤ C.val.card := by
+  obtain ⟨x,hx⟩ := C.nonempty
+  have hpx : alternatingPerm p x ∈ C.val := by
+    rw [←C.orbit_eq hx,mem_orbitSet]
+    exact (Equiv.Perm.SameCycle.refl _ _).apply_right
+  have hne : x ≠ alternatingPerm p x := by
+    intro he
+    have hh := congrArg Prod.fst he
+    rw [alternatingPerm_flip] at hh
+    cases hb : x.1 <;> simp only [hb, Bool.not_false, Bool.not_true] at hh <;> cases hh
+  have hs : {x,alternatingPerm p x} ⊆ C.val := by
+    intro y hy
+    simp only [Finset.mem_insert,Finset.mem_singleton] at hy
+    rcases hy with rfl | rfl <;> assumption
+  have hh := Finset.card_le_card hs
+  simpa only [Finset.card_pair hne] using hh
+
+lemma alternating_cycle_count_double (p : Equiv.Perm α) (S : Finset (Bool × α)) :
+    2 * Fintype.card (RoutingNetwork.SelectedCycle (alternatingPerm p) S) ≤ S.card := by
+  classical
+  calc
+    _ = ∑ _C : RoutingNetwork.SelectedCycle (alternatingPerm p) S, 2 := by simp [Nat.mul_comm]
+    _ ≤ RoutingNetwork.cycleMass (alternatingPerm p) S :=
+      Finset.sum_le_sum (fun C _ => alternating_cycle_two p S C)
+    _ ≤ _ := RoutingNetwork.cycleMass_le _ _
+
+lemma alternating_cycle_count_zero (p : Equiv.Perm α) (S : Finset (Bool × α))
+    (hS : S.card ≤ 1) : Fintype.card (RoutingNetwork.SelectedCycle (alternatingPerm p) S) = 0 := by
+  have h := alternating_cycle_count_double p S
+  omega
+
+end CycleColoring
+
+namespace PairRouting
+
+section
+variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
+
+lemma alternatingCycles_double (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    2 * alternatingCycles e p ≤ Fintype.card ι := by
+  have hh := CycleColoring.alternating_cycle_count_double (alternatingProjection p) (labelSet e)
+  simpa only [alternatingCycles, card_labelSet] using hh
+
+lemma alternatingCycles_zero (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α)
+    (hι : Fintype.card ι ≤ 1) : alternatingCycles e p = 0 := by
+  have hh := alternatingCycles_double e p
+  omega
+
+end
+variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α] [DecidableEq ι]
+
+omit [Fintype α] [DecidableEq ι] in
+lemma colors_chosenCoin (x : ι → Bool × α) (c : ι → Bool) (hc : Compatible x c) :
+    colors x (chosenCoin x c) = c := by
+  funext i
+  simp only [colors,chosenCoin_apply x c hc i,required]
+  cases (x i).1 <;> cases c i <;> rfl
+
+noncomputable def coloredEmbedding (x : ι ↪ Bool × α) (c : ι → Bool) (_hc : Compatible x c) :
+    ι ↪ Bool × α := switchedEmbedding x (chosenCoin x c)
+
+omit [Fintype α] [DecidableEq ι] in
+lemma coloredEmbedding_apply (x : ι ↪ Bool × α) (c : ι → Bool) (hc : Compatible x c) (i : ι) :
+    coloredEmbedding x c hc i = (c i,(x i).2) := by
+  apply Prod.ext
+  · exact congrFun (colors_chosenCoin x c hc) i
+  · rfl
+
+omit [Fintype α] [DecidableEq ι] in
+lemma switchedEmbedding_eq_colored (x : ι ↪ Bool × α) (c : ι → Bool) (hc : Compatible x c)
+    (ξ : α → Bool) (he : colors x ξ = c) :
+    switchedEmbedding x ξ = coloredEmbedding x c hc := by
+  ext i : 1
+  rw [coloredEmbedding_apply]
+  apply Prod.ext
+  · exact congrFun he i
+  · rfl
+
+lemma assignments_match_bound (x y : ι ↪ Bool × α) (c : ι → Bool)
+    (hc : c ∈ compatibleSet x y) (p : Bool → Equiv.Perm α) (hm : ChildMatches p x y c) :
+    (compatibleSet x y).card ≤
+      2 ^ ((occupied x).card + (occupied y).card - Fintype.card ι +
+        alternatingCycles (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p) := by
+  obtain ⟨hcx,hcy⟩ := (mem_compatibleSet x y c).mp hc
+  let ξ := chosenCoin x c
+  let η := chosenCoin y c
+  have he : routedEmbedding x ξ η p = y := by
+    ext i : 1
+    have h := (sandwich_matches_iff ξ η p x y).mpr
+      ⟨(colors_chosenCoin x c hcx).trans (colors_chosenCoin y c hcy).symm,by
+        simpa only [ξ,colors_chosenCoin x c hcx] using hm⟩
+    exact h i
+  have hh := assignments_actual_bound x ξ η p
+  rw [he] at hh
+  rw [card_compatibleSet]
+  exact hh
+
+omit [Fintype α] [DecidableEq ι] in
+lemma occupied_sum_ge (x y : ι ↪ Bool × α) :
+    Fintype.card ι ≤ (occupied x).card + (occupied y).card := by
+  have h₀ := occupied_card_double x
+  have h₁ := occupied_card_double y
+  omega
+
+lemma assignment_weight_bound (x y : ι ↪ Bool × α) (c : ι → Bool)
+    (hc : c ∈ compatibleSet x y) (p : Bool → Equiv.Perm α) (hm : ChildMatches p x y c) :
+    ((compatibleSet x y).card:ℝ) * outerWeight x y ≤
+      (2:ℝ)^((alternatingCycles
+        (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p : ℝ) - Fintype.card ι) := by
+  let z := alternatingCycles (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p
+  let b := (occupied x).card+(occupied y).card
+  let h := Fintype.card ι
+  have hb : h ≤ b := occupied_sum_ge x y
+  have hn : ((compatibleSet x y).card:ℝ) ≤ (2:ℝ)^(b-h+z) := by
+    exact_mod_cast assignments_match_bound x y c hc p hm
+  change ((compatibleSet x y).card:ℝ) * (1/(2:ℝ)^b) ≤ (2:ℝ)^((z:ℝ)-h)
+  rw [mul_one_div]
+  calc
+    _ ≤ (2:ℝ)^(b-h+z)/(2:ℝ)^b := div_le_div_of_nonneg_right hn (by positivity)
+    _ = _ := by
+      rw [←Real.rpow_natCast,←Real.rpow_natCast,←Real.rpow_sub (by norm_num : (0:ℝ)<2)]
+      congr 1
+      rw [Nat.cast_add,Nat.cast_sub hb]
+      ring
+
+lemma assignment_factor_bound (x y : ι ↪ Bool × α) (c : ι → Bool)
+    (hc : c ∈ compatibleSet x y) (p : Bool → Equiv.Perm α) (hm : ChildMatches p x y c)
+    (a : ℝ) (ha : 0 ≤ a) :
+    (outerWeight x y)^a * ((compatibleSet x y).card:ℝ)^a ≤
+      (2:ℝ)^(((alternatingCycles
+        (coloredEmbedding x c (mem_compatibleSet x y c |>.mp hc).1) p : ℝ) - Fintype.card ι)*a) := by
+  have h := Real.rpow_le_rpow (mul_nonneg (Nat.cast_nonneg _) (le_of_lt (outerWeight_pos x y)))
+    (assignment_weight_bound x y c hc p hm) ha
+  rw [Real.mul_rpow (Nat.cast_nonneg _) (le_of_lt (outerWeight_pos x y)),
+    ←Real.rpow_mul (by norm_num : (0:ℝ)≤2)] at h
+  simpa only [mul_comm] using h
+
+
+lemma sandwich_weighted_probability_subtype (p : Bool → Equiv.Perm α) (x y : ι → Bool × α)
+    (W : (ι → Bool) → ℝ) :
+    finiteMean (fun coins : (α → Bool) × (α → Bool) =>
+      if ∀ i, sandwich coins.1 coins.2 p (x i) = y i then W (colors x coins.1) else 0) =
+    outerWeight x y * ∑ c : {c // c ∈ compatibleSet x y},
+      if ChildMatches p x y c.val then W c.val else 0 := by
+  rw [sandwich_weighted_probability]
+  congr 1
+  rw [Finset.sum_subtype (compatibleSet x y) (fun _ => Iff.rfl)]
+
+omit [Fintype α] [DecidableEq α] [DecidableEq ι] in
+lemma color_card_add (c : ι → Bool) :
+    Fintype.card {i // c i = false} + Fintype.card {i // c i = true} = Fintype.card ι := by
+  have he : {i // c i = true} ≃ {i // ¬c i = false} := Equiv.subtypeEquivRight fun i => by
+    cases c i <;> simp
+  rw [Fintype.card_congr he]
+  rw [Fintype.card_subtype_compl]
+  exact Nat.add_sub_of_le (Fintype.card_subtype_le _)
+
+end PairRouting
+
+noncomputable def assignmentCost (d : ℕ) {ι : Type*} [Fintype ι]
+    (x : ι ↪ Bool × Card d) (c : ι → Bool) (hc : PairRouting.Compatible x c)
+    (ω : BenesCoins d × BenesCoins d) : ℕ :=
+  PairRouting.alternatingCycles (PairRouting.coloredEmbedding x c hc)
+      (fun b => palindromePerm d (if b then ω.2 else ω.1)) +
+    palindromeCost d (PairRouting.childEmbedding x c hc false) ω.1 +
+    palindromeCost d (PairRouting.childEmbedding x c hc true) ω.2
+
+lemma palindromeCost_eq (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (d+1)) (σ : BenesCoins (d+1)) :
+    palindromeCost (d+1) e σ =
+      assignmentCost d (e.trans (headTailEquiv d).toEmbedding)
+        (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) (benesStepEquiv d σ).2.1)
+        (PairRouting.colors_compatible _ _) (benesStepEquiv d σ).1 := by
+  unfold palindromeCost assignmentCost
+  dsimp only
+  rw [PairRouting.switchedEmbedding_eq_colored _ _ (PairRouting.colors_compatible _ _) _ rfl]
+
+lemma palindromeCost_step (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (d+1)) (ω : BenesCoins d × BenesCoins d)
+    (coins : (Card d → Bool) × (Card d → Bool)) :
+    palindromeCost (d+1) e ((benesStepEquiv d).symm (ω,coins)) =
+      assignmentCost d (e.trans (headTailEquiv d).toEmbedding)
+        (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) coins.1)
+        (PairRouting.colors_compatible _ _) ω := by
+  rw [palindromeCost_eq]
+  simp only [Equiv.apply_symm_apply]
+
+noncomputable def routingMoment (d : ℕ) {ι : Type*} [Fintype ι]
+    (e f : ι ↪ Card d) (a : ℝ) : ℝ := by
+  classical
+  exact finiteMean (fun ω : BenesCoins d =>
+    if ∀ i, palindromePerm d ω (e i) = f i then (2:ℝ)^((palindromeCost d e ω:ℝ)*a) else 0)
+
+lemma routingMoment_nonneg (d : ℕ) {ι : Type*} [Fintype ι]
+    (e f : ι ↪ Card d) (a : ℝ) : 0 ≤ routingMoment d e f a := by
+  classical
+  exact finiteMean_nonneg (fun _ => by split_ifs <;> positivity)
+
+lemma palindrome_matches_step (d : ℕ) {ι : Type*}
+    (e f : ι ↪ Card (d+1)) (ω : BenesCoins d × BenesCoins d)
+    (coins : (Card d → Bool) × (Card d → Bool)) :
+    (∀ i, palindromePerm (d+1) ((benesStepEquiv d).symm (ω,coins)) (e i) = f i) ↔
+    (∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => palindromePerm d (if b then ω.2 else ω.1))
+      ((e.trans (headTailEquiv d).toEmbedding) i) = ((f.trans (headTailEquiv d).toEmbedding) i)) := by
+  apply forall_congr'
+  intro i
+  rw [←(headTailEquiv d).injective.eq_iff,palindromePerm_coordinates]
+  rfl
+
+lemma routingMoment_step (d : ℕ) {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (e f : ι ↪ Card (d+1)) (a : ℝ) :
+    let x := e.trans (headTailEquiv d).toEmbedding
+    let y := f.trans (headTailEquiv d).toEmbedding
+    routingMoment (d+1) e f a = PairRouting.outerWeight x y *
+      ∑ c : {c // c ∈ PairRouting.compatibleSet x y},
+        finiteMean (fun ω : BenesCoins d × BenesCoins d =>
+          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c.val
+          then (2:ℝ)^((assignmentCost d x c.val
+            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0) := by
+  classical
+  intro x y
+  rw [routingMoment,finiteMean_equiv (benesStepEquiv d),finiteMean_prod]
+  have he (ω : BenesCoins d × BenesCoins d) :
+      finiteMean (fun coins : (Card d → Bool) × (Card d → Bool) =>
+        if ∀ i, palindromePerm (d+1) ((benesStepEquiv d).symm (ω,coins)) (e i) = f i
+        then (2:ℝ)^((palindromeCost (d+1) e ((benesStepEquiv d).symm (ω,coins)):ℝ)*a) else 0) =
+      PairRouting.outerWeight x y *
+        ∑ c : {c // c ∈ PairRouting.compatibleSet x y},
+          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c.val
+          then (2:ℝ)^((assignmentCost d x c.val
+            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0 := by
+    let W : (ι → Bool) → ℝ := fun c =>
+      if hc : PairRouting.Compatible x c then (2:ℝ)^((assignmentCost d x c hc ω:ℝ)*a) else 0
+    calc
+      _ = finiteMean (fun coins : (Card d → Bool) × (Card d → Bool) =>
+        if ∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => palindromePerm d (if b then ω.2 else ω.1))
+          (x i) = y i then W (PairRouting.colors x coins.1) else 0) := by
+        apply finiteMean_congr
+        intro coins
+        simp only [palindrome_matches_step,palindromeCost_step]
+        simp only [W,PairRouting.colors_compatible,↓reduceDIte]
+        rfl
+      _ = _ := by
+        rw [PairRouting.sandwich_weighted_probability_subtype]
+        congr 1
+        apply Finset.sum_congr rfl
+        intro c _
+        simp only [W,((PairRouting.mem_compatibleSet x y c.val).mp c.property).1,↓reduceDIte]
+  simp_rw [he]
+  simp_rw [mul_comm (PairRouting.outerWeight x y)]
+  rw [finiteMean_mul_const,finiteMean_sum,mul_comm]
+
+lemma child_moment_product (d : ℕ) {ι : Type*} [Fintype ι]
+    (x y : ι ↪ Bool × Card d) (c : ι → Bool)
+    (hcx : PairRouting.Compatible x c) (hcy : PairRouting.Compatible y c) (a : ℝ) :
+    routingMoment d (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a *
+      routingMoment d (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a =
+    finiteMean (fun ω : BenesCoins d × BenesCoins d =>
+      if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
+      then (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
+        palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0) := by
+  classical
+  unfold routingMoment
+  rw [←finiteMean_prod_mul]
+  apply finiteMean_congr
+  intro ω
+  let M₀ := ∀ i : {i // c i = false}, palindromePerm d ω.1 (x i.val).2 = (y i.val).2
+  let M₁ := ∀ i : {i // c i = true}, palindromePerm d ω.2 (x i.val).2 = (y i.val).2
+  have hp : (fun b : Bool => palindromePerm d (if b then ω.2 else ω.1)) =
+      (fun b => if b then palindromePerm d ω.2 else palindromePerm d ω.1) := by
+    funext b
+    cases b <;> rfl
+  have hs : PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c ↔
+      M₀ ∧ M₁ := by rw [hp]; exact PairRouting.childMatches_split _ _ _ _ _
+  change (if M₀ then _ else 0) * (if M₁ then _ else 0) = _
+  by_cases h₀ : M₀ <;> by_cases h₁ : M₁
+  · simp only [hs,h₀,h₁,true_and,↓reduceIte]
+    rw [←Real.rpow_add (by norm_num : (0:ℝ)<2)]
+    congr 1
+    ring
+  · simp only [hs,h₀,h₁,and_false,↓reduceIte,mul_zero]
+  · simp only [hs,h₀,h₁,false_and,↓reduceIte,zero_mul]
+  · simp only [hs,h₀,h₁,false_and,↓reduceIte,mul_zero]
+
+lemma palindrome_probability_step (d : ℕ) {ι : Type*} [Fintype ι]
+    (e f : ι ↪ Card (d+1)) :
+    PairRouting.tupleProbability (palindromePerm (d+1)) e f =
+    finiteMean (fun ω : BenesCoins d × BenesCoins d =>
+      finiteMean (fun coins : (Card d → Bool) × (Card d → Bool) =>
+        if ∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => if b then palindromePerm d ω.2 else palindromePerm d ω.1)
+          ((e.trans (headTailEquiv d).toEmbedding) i) = ((f.trans (headTailEquiv d).toEmbedding) i)
+        then (1:ℝ) else 0)) := by
+  classical
+  rw [PairRouting.tupleProbability,finiteMean_equiv (benesStepEquiv d),finiteMean_prod]
+  have hp (b : Bool) (p q : BenesCoins d) :
+      palindromePerm d (if b then p else q) = (if b then palindromePerm d p else palindromePerm d q) := by
+    cases b <;> rfl
+  simp only [palindrome_matches_step,hp]
+
+lemma child_power_product_bound (d : ℕ) {ι : Type*} [Fintype ι]
+    (x y : ι ↪ Bool × Card d) (c : ι → Bool)
+    (hcx : PairRouting.Compatible x c) (hcy : PairRouting.Compatible y c) (a : ℝ)
+    (hfalse : (PairRouting.tupleProbability (palindromePerm d)
+        (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false))^(1+a) ≤
+      (2:ℝ)^(-(d:ℝ)*Fintype.card {i // c i = false}*a) *
+        routingMoment d (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a)
+    (htrue : (PairRouting.tupleProbability (palindromePerm d)
+        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true))^(1+a) ≤
+      (2:ℝ)^(-(d:ℝ)*Fintype.card {i // c i = true}*a) *
+        routingMoment d (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a) :
+    (PairRouting.tupleProbability (palindromePerm d)
+      (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false))^(1+a) *
+      (PairRouting.tupleProbability (palindromePerm d)
+        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true))^(1+a) ≤
+    (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) *
+      finiteMean (fun ω : BenesCoins d × BenesCoins d =>
+        if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
+        then (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
+          palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0) := by
+  have hh := mul_le_mul hfalse htrue
+    (Real.rpow_nonneg (PairRouting.tupleProbability_nonneg _ _ _) _)
+    (mul_nonneg (by positivity) (routingMoment_nonneg _ _ _ _))
+  rw [mul_mul_mul_comm,child_moment_product,←Real.rpow_add (by norm_num : (0:ℝ)<2)] at hh
+  have he : -(d:ℝ)*Fintype.card {i // c i = false}*a + -(d:ℝ)*Fintype.card {i // c i = true}*a =
+      -(d:ℝ)*Fintype.card ι*a := by
+    rw [←add_mul,←mul_add,←Nat.cast_add,PairRouting.color_card_add]
+  rwa [he] at hh
+
+lemma assignment_moment_bound (d : ℕ) {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (x y : ι ↪ Bool × Card d) (c : ι → Bool) (hc : c ∈ PairRouting.compatibleSet x y)
+    (a : ℝ) (ha : 0 ≤ a) :
+    let hcx := ((PairRouting.mem_compatibleSet x y c).mp hc).1
+    PairRouting.outerWeight x y ^ a * ((PairRouting.compatibleSet x y).card:ℝ)^a *
+      ((2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) *
+        finiteMean (fun ω : BenesCoins d × BenesCoins d =>
+          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
+          then (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
+            palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0)) ≤
+    (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) *
+      finiteMean (fun ω : BenesCoins d × BenesCoins d =>
+        if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c
+        then (2:ℝ)^((assignmentCost d x c hcx ω:ℝ)*a) else 0) := by
+  classical
+  intro hcx
+  have hscale (q : ℝ) (F : BenesCoins d × BenesCoins d → ℝ) :
+      q * finiteMean F = finiteMean (fun ω => q * F ω) := by
+    simp_rw [mul_comm q,finiteMean_mul_const]
+  rw [←mul_assoc,hscale,hscale]
+  apply finiteMean_mono
+  intro ω
+  split_ifs with hm
+  · have hh := PairRouting.assignment_factor_bound x y c hc
+      (fun b => palindromePerm d (if b then ω.2 else ω.1)) hm a ha
+    calc
+      _ ≤ ((2:ℝ)^(((PairRouting.alternatingCycles (PairRouting.coloredEmbedding x c hcx)
+          (fun b => palindromePerm d (if b then ω.2 else ω.1)):ℝ)-Fintype.card ι)*a) *
+          (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a)) *
+          (2:ℝ)^(((palindromeCost d (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
+            palindromeCost d (PairRouting.childEmbedding x c hcx true) ω.2)*a) :=
+        mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hh (by positivity)) (by positivity)
+      _ = _ := by
+        simp only [←Real.rpow_add (by norm_num : (0:ℝ)<2),assignmentCost,Nat.cast_add]
+        congr 1
+        ring
+  · simp only [mul_zero]
+    exact le_rfl
+
+theorem palindrome_pointwise_moment (d : ℕ) {ι : Type*} [Fintype ι]
+    (e f : ι ↪ Card d) (a : ℝ) (ha : 0 ≤ a) :
+    (PairRouting.tupleProbability (palindromePerm d) e f)^(1+a) ≤
+      (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) * routingMoment d e f a := by
+  classical
+  induction d generalizing ι with
+  | zero =>
+      have he : ∀ (ω : BenesCoins 0) (i : ι), palindromePerm 0 ω (e i) = f i :=
+        fun _ _ => Subsingleton.elim _ _
+      simp [PairRouting.tupleProbability,routingMoment,he,palindromeCost,finiteMean_const]
+  | succ d ih =>
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let y := f.trans (headTailEquiv d).toEmbedding
+      let W := PairRouting.outerWeight x y
+      let C := PairRouting.compatibleSet x y
+      let p : (ι → Bool) → ℝ := fun c =>
+        (PairRouting.tupleProbability (palindromePerm d)
+          (fun i : {i // c i = false} => (x i.val).2)
+          (fun i : {i // c i = false} => (y i.val).2))^(1+a) *
+        (PairRouting.tupleProbability (palindromePerm d)
+          (fun i : {i // c i = true} => (x i.val).2)
+          (fun i : {i // c i = true} => (y i.val).2))^(1+a)
+      let m : {c // c ∈ C} → ℝ := fun c =>
+        finiteMean (fun ω : BenesCoins d × BenesCoins d =>
+          if PairRouting.ChildMatches (fun b => palindromePerm d (if b then ω.2 else ω.1)) x y c.val
+          then (2:ℝ)^((assignmentCost d x c.val
+            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0)
+      have hp (c : {c // c ∈ C}) : W^a * (C.card:ℝ)^a * p c.val ≤
+          (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * m c := by
+        obtain ⟨hcx,hcy⟩ := (PairRouting.mem_compatibleSet x y c.val).mp c.property
+        have h₀ := ih (PairRouting.childEmbedding x c.val hcx false)
+          (PairRouting.childEmbedding y c.val hcy false)
+        have h₁ := ih (PairRouting.childEmbedding x c.val hcx true)
+          (PairRouting.childEmbedding y c.val hcy true)
+        have hh := child_power_product_bound d x y c.val hcx hcy a h₀ h₁
+        have hh' := mul_le_mul_of_nonneg_left hh
+          (mul_nonneg (Real.rpow_nonneg (le_of_lt (PairRouting.outerWeight_pos x y)) a)
+            (Real.rpow_nonneg (Nat.cast_nonneg C.card) a))
+        exact hh'.trans (assignment_moment_bound d x y c.val c.property a ha)
+      have hnode := PairRouting.sandwich_power_bound (palindromePerm d) (palindromePerm d) x y a ha
+      rw [←palindrome_probability_step d e f] at hnode
+      change (PairRouting.tupleProbability (palindromePerm (d+1)) e f)^(1+a) ≤
+        W^(1+a) * (C.card:ℝ)^a * ∑ c ∈ C, p c at hnode
+      have hsum : ∑ c ∈ C, p c = ∑ c : {c // c ∈ C}, p c.val := by
+        rw [Finset.sum_subtype C (fun _ => Iff.rfl)]
+      have hW : 0 < W := PairRouting.outerWeight_pos x y
+      calc
+        _ ≤ W^(1+a) * (C.card:ℝ)^a * ∑ c ∈ C, p c := hnode
+        _ = W * ∑ c : {c // c ∈ C}, W^a * (C.card:ℝ)^a * p c.val := by
+          rw [Real.rpow_add hW,Real.rpow_one,hsum,←Finset.mul_sum]
+          ring
+        _ ≤ W * ∑ c : {c // c ∈ C}, (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * m c :=
+          mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun c _ => hp c)) (le_of_lt hW)
+        _ = _ := by
+          rw [←Finset.mul_sum,routingMoment_step]
+          simp only [Nat.cast_add,Nat.cast_one]
+          change W * ((2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * ∑ c, m c) =
+            (2:ℝ)^(-((d:ℝ)+1)*Fintype.card ι*a) * (W * ∑ c, m c)
+          ring
+
+lemma routingMoment_sum (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (a : ℝ) :
+    (∑ f : ι ↪ Card d, routingMoment d e f a) =
+      finiteMean (fun ω : BenesCoins d => (2:ℝ)^((palindromeCost d e ω:ℝ)*a)) := by
+  classical
+  unfold routingMoment
+  rw [←finiteMean_sum]
+  apply finiteMean_congr
+  intro ω
+  let f₀ : ι ↪ Card d := e.trans (palindromePerm d ω).toEmbedding
+  rw [Finset.sum_eq_single f₀]
+  · have hm : ∀ i, palindromePerm d ω (e i) = f₀ i := fun _ => rfl
+    simp only [hm,implies_true,↓reduceIte]
+  · intro f _ hf
+    have hm : ¬ ∀ i, palindromePerm d ω (e i) = f i := by
+      intro hh
+      apply hf
+      ext i : 1
+      exact (hh i).symm
+    exact ite_eq_right hm
+  · simp
+
+lemma finite_row_power {X : Type*} [Fintype X] [Nonempty X]
+    (p : X → ℝ) (hp : ∀ x, 0 ≤ p x) (a : ℝ) :
+    finiteMean (fun x => ((Fintype.card X:ℝ)*p x)^(1+a)) =
+      (Fintype.card X:ℝ)^a * ∑ x, (p x)^(1+a) := by
+  have hn : (0:ℝ) < Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
+  simp only [finiteMean,Real.mul_rpow (le_of_lt hn) (hp _),←Finset.mul_sum]
+  rw [Real.rpow_add hn,Real.rpow_one]
+  field_simp
+
+noncomputable def palindromeRowMoment (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (a : ℝ) : ℝ :=
+  finiteMean (fun f : ι ↪ Card d =>
+    ((Fintype.card (ι ↪ Card d):ℝ)*PairRouting.tupleProbability (palindromePerm d) e f)^(1+a))
+
+lemma density_normalization (d h n : ℕ) (a : ℝ) :
+    (n:ℝ)^a * (2:ℝ)^(-(d:ℝ)*h*a) = ((n:ℝ)/((2:ℝ)^d)^h)^a := by
+  rw [Real.div_rpow (Nat.cast_nonneg _) (by positivity),←pow_mul,
+    ←Real.rpow_natCast,←Real.rpow_mul (by norm_num : (0:ℝ) ≤ 2),Nat.cast_mul]
+  rw [show -(d:ℝ)*h*a = -((d:ℝ)*h*a) by ring,
+    Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2),div_eq_mul_inv]
+
+theorem palindrome_row_density_moment (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (a : ℝ) (ha : 0 ≤ a) :
+    palindromeRowMoment d e a ≤
+      (((2^d).descFactorial (Fintype.card ι):ℝ)/((2^d:ℕ):ℝ)^(Fintype.card ι))^a *
+        finiteMean (fun ω : BenesCoins d => (2:ℝ)^((palindromeCost d e ω:ℝ)*a)) := by
+  classical
+  have : Nonempty (ι ↪ Card d) := ⟨e⟩
+  rw [palindromeRowMoment,finite_row_power _ (fun f => PairRouting.tupleProbability_nonneg _ _ _)]
+  calc
+    _ ≤ (Fintype.card (ι ↪ Card d):ℝ)^a *
+      ∑ f : ι ↪ Card d, (2:ℝ)^(-(d:ℝ)*Fintype.card ι*a) * routingMoment d e f a :=
+      mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun f _ => palindrome_pointwise_moment d e f a ha))
+        (Real.rpow_nonneg (Nat.cast_nonneg _) _)
+    _ = _ := by
+      rw [←Finset.mul_sum,routingMoment_sum,←mul_assoc,density_normalization,
+        Fintype.card_embedding_eq,card_positions]
+      norm_cast
+
+end Thorp
+namespace Thorp
+
+lemma subset_biUnion_indicator {α β : Type*} [DecidableEq α] [Fintype β]
+    (A : Finset α) (T : β → Finset α) :
+    (if Finset.univ.biUnion T ⊆ A then (1:ℝ) else 0) =
+      ∏ b, if T b ⊆ A then (1:ℝ) else 0 := by
+  classical
+  by_cases h : Finset.univ.biUnion T ⊆ A
+  · rw [ite_eq_left h]
+    symm
+    apply Finset.prod_eq_one
+    intro b _
+    rw [ite_eq_left]
+    exact (Finset.subset_biUnion_of_mem T (Finset.mem_univ b)).trans h
+  · rw [ite_eq_right h]
+    obtain ⟨x,hx,hxA⟩ := Finset.not_subset.mp h
+    obtain ⟨b,_,hxb⟩ := Finset.mem_biUnion.mp hx
+    symm
+    apply Finset.prod_eq_zero (Finset.mem_univ b)
+    rw [ite_eq_right]
+    exact fun hb => hxA (hb hxb)
+
+lemma cylinder_block_polynomial {α β Ω : Type*} [DecidableEq α] [Fintype β]
+    [Fintype Ω] (τ : β → Type*) [∀ b, Fintype (τ b)]
+    (F : Ω → Finset α) (p : α → ℝ)
+    (hF : CylinderBound (inclusionMoment F) p)
+    (T : ∀ b, τ b → Finset α) (c : ∀ b, τ b → ℝ)
+    (hc : ∀ b s, 0 ≤ c b s)
+    (hd : ∀ b b', b ≠ b' → ∀ s s', Disjoint (T b s) (T b' s')) :
+    finiteMean (fun ω => ∏ b, ∑ s, c b s * if T b s ⊆ F ω then 1 else 0) ≤
+      ∏ b, ∑ s, c b s * ∏ x ∈ T b s, p x := by
+  classical
+  simp_rw [Fintype.prod_sum]
+  rw [finiteMean_sum]
+  apply Finset.sum_le_sum
+  intro s _
+  let U := Finset.univ.biUnion (fun b => T b (s b))
+  have he : (fun ω => ∏ b, c b (s b) * if T b (s b) ⊆ F ω then 1 else 0) =
+      fun ω => (if U ⊆ F ω then 1 else 0) * ∏ b, c b (s b) := by
+    funext ω
+    rw [Finset.prod_mul_distrib,←subset_biUnion_indicator]
+    exact mul_comm _ _
+  rw [he,finiteMean_mul_const]
+  have hdis : ((Finset.univ : Finset β) : Set β).PairwiseDisjoint (fun b => T b (s b)) := by
+    intro b _ b' _ hbb
+    exact hd b b' hbb (s b) (s b')
+  have hmean : finiteMean (fun ω => if U ⊆ F ω then 1 else 0) = inclusionMoment F U := by
+    unfold inclusionMoment
+    apply finiteMean_congr
+    intro ω
+    by_cases hh : U ⊆ F ω <;> simp [hh]
+  calc
+    _ ≤ (∏ x ∈ U, p x) * ∏ b, c b (s b) :=
+      mul_le_mul_of_nonneg_right (hmean.le.trans (hF U)) (Finset.prod_nonneg (fun b _ => hc b (s b)))
+    _ = _ := by
+      dsimp only [U]
+      rw [Finset.prod_biUnion hdis,←Finset.prod_mul_distrib]
+      apply Finset.prod_congr rfl
+      intro b _
+      exact mul_comm _ _
+
+
+noncomputable def sparseCoefficient {α : Type*} (w : ℝ) (T : Finset α) : ℝ :=
+  if T.card = 1 then 0 else w^T.card
+
+lemma sparseCoefficient_nonneg {α : Type*} (w : ℝ) (hw : 0 ≤ w) (T : Finset α) :
+    0 ≤ sparseCoefficient w T := by
+  unfold sparseCoefficient
+  split_ifs <;> positivity
+
+lemma sparse_block_pointwise {α : Type*} [DecidableEq α] (A D : Finset α)
+    (w : ℝ) (hw : 0 ≤ w) :
+    w^(if 2 ≤ (D ∩ A).card then (D ∩ A).card else 0) ≤
+      ∑ T : ↑D.powerset, sparseCoefficient w T.val * if T.val ⊆ A then 1 else 0 := by
+  classical
+  let V := D ∩ A
+  let T : ↑D.powerset := if 2 ≤ V.card then
+    ⟨V,Finset.mem_powerset.mpr Finset.inter_subset_left⟩ else
+    ⟨∅,Finset.mem_powerset.mpr (Finset.empty_subset D)⟩
+  calc
+    _ = sparseCoefficient w T.val * (if T.val ⊆ A then 1 else 0) := by
+      by_cases h : 2 ≤ V.card
+      · have hn : V.card ≠ 1 := by omega
+        simp [T,ite_eq_left h,sparseCoefficient,hn,show V ⊆ A from Finset.inter_subset_right,V]
+      · simp [T,ite_eq_right h,sparseCoefficient,V] at *
+    _ ≤ _ := Finset.single_le_sum (f := fun S : ↑D.powerset =>
+      sparseCoefficient w S.val * (if S.val ⊆ A then 1 else 0))
+      (fun S _ => mul_nonneg (sparseCoefficient_nonneg w hw S.val)
+        (by split_ifs <;> norm_num)) (Finset.mem_univ T)
+
+lemma geometric_sum_half (n : ℕ) {x : ℝ} (hx : 0 ≤ x) (hhalf : x ≤ 1/2) :
+    ∑ i ∈ Finset.range n, x^i ≤ 2 := by
+  induction n with
+  | zero => simp
+  | succ n ih =>
+    rw [Finset.sum_range_succ',pow_zero]
+    simp only [pow_succ',←Finset.mul_sum]
+    nlinarith
+
+lemma sparse_binomial_bound (R : ℕ) (z : ℝ) (hz : 0 ≤ z) (hr : (R:ℝ)*z ≤ 1/2) :
+    ∑ i ∈ Finset.range (R+1), (R.choose i:ℝ) * (if i = 1 then 0 else z^i) ≤
+      1 + 2*((R:ℝ)*z)^2 := by
+  classical
+  let x := (R:ℝ)*z
+  have hx : 0 ≤ x := mul_nonneg (Nat.cast_nonneg _) hz
+  have hbound (i : ℕ) : (R.choose i:ℝ)*(if i=1 then 0 else z^i) ≤
+      if i=1 then 0 else x^i := by
+    by_cases h : i=1
+    · simp [h]
+    · simp only [ite_eq_right h]
+      dsimp [x]
+      rw [mul_pow]
+      apply mul_le_mul_of_nonneg_right _ (pow_nonneg hz i)
+      exact_mod_cast Nat.choose_le_pow R i
+  calc
+    _ ≤ ∑ i ∈ Finset.range (R+1), if i=1 then 0 else x^i :=
+      Finset.sum_le_sum (fun i _ => hbound i)
+    _ ≤ 1+2*x^2 := by
+      rcases R with _ | R
+      · simp [x]
+      rcases R with _ | R
+      · norm_num [Finset.sum_range_succ]
+        positivity
+      have he : ∑ i ∈ Finset.range (R+2+1), (if i=1 then 0 else x^i) =
+          1 + x^2*∑ i ∈ Finset.range (R+1), x^i := by
+        rw [show R+2+1=2+(R+1) by omega,Finset.sum_range_add]
+        have hn (i : ℕ) : 2+i ≠ 1 := by omega
+        simp only [hn,ite_false,pow_add,←Finset.mul_sum]
+        norm_num [Finset.sum_range_succ]
+      rw [he]
+      have hg := geometric_sum_half (R+1) hx hr
+      nlinarith [sq_nonneg x]
+
+lemma sparse_block_product_bound {α : Type*} [DecidableEq α]
+    (D : Finset α) (p w : ℝ) (hp : 0 ≤ p) (hw : 0 ≤ w)
+    (hr : (D.card:ℝ)*(p*w) ≤ 1/2) :
+    ∑ T : ↑D.powerset, sparseCoefficient w T.val * p^T.val.card ≤
+      Real.exp (2*((D.card:ℝ)*(p*w))^2) := by
+  classical
+  have he : ∑ T : ↑D.powerset, sparseCoefficient w T.val * p^T.val.card =
+      ∑ i ∈ Finset.range (D.card+1), (D.card.choose i:ℝ)*(if i=1 then 0 else (p*w)^i) := by
+    rw [Finset.sum_coe_sort D.powerset (fun T => sparseCoefficient w T*p^T.card)]
+    have hh : (fun T : Finset α => sparseCoefficient w T*p^T.card) =
+        fun T => if T.card=1 then 0 else (p*w)^T.card := by
+      funext T
+      unfold sparseCoefficient
+      split_ifs <;> simp [mul_pow,mul_comm]
+    change (∑ T ∈ D.powerset, (fun T : Finset α => sparseCoefficient w T*p^T.card) T) = _
+    rw [hh,Finset.sum_powerset_apply_card (fun i => if i=1 then (0:ℝ) else (p*w)^i)]
+    simp only [nsmul_eq_mul]
+  rw [he]
+  exact (sparse_binomial_bound D.card (p*w) (mul_nonneg hp hw) hr).trans
+    (by simpa only [add_comm] using Real.add_one_le_exp (2*((D.card:ℝ)*(p*w))^2))
+
+
+noncomputable def sparseMass {α β : Type*} [DecidableEq α] [Fintype β]
+    (D : β → Finset α) (A : Finset α) : ℕ :=
+  ∑ b, if 2 ≤ (D b ∩ A).card then (D b ∩ A).card else 0
+
+theorem sparse_block_mgf {α β Ω : Type*} [DecidableEq α] [Fintype β] [Fintype Ω]
+    (D : β → Finset α) (hd : Pairwise (fun b b' => Disjoint (D b) (D b')))
+    (F : Ω → Finset α) (p w : ℝ) (hp : 0 ≤ p) (hw : 0 ≤ w)
+    (hF : ∀ T, inclusionMoment F T ≤ p^T.card)
+    (hr : ∀ b, ((D b).card:ℝ)*(p*w) ≤ 1/2) :
+    finiteMean (fun ω => w^(sparseMass D (F ω))) ≤
+      Real.exp (∑ b, 2*(((D b).card:ℝ)*(p*w))^2) := by
+  classical
+  have hc : CylinderBound (inclusionMoment F) (fun _ => p) := by
+    intro T
+    simpa only [Finset.prod_const] using hF T
+  have hb := cylinder_block_polynomial (fun b => ↑(D b).powerset) F (fun _ => p) hc
+    (fun _ S => S.val) (fun _ S => sparseCoefficient w S.val)
+    (fun _ S => sparseCoefficient_nonneg w hw S.val)
+    (fun b b' h S S' => (hd h).mono
+      (Finset.mem_powerset.mp S.property) (Finset.mem_powerset.mp S'.property))
+  calc
+    _ ≤ finiteMean (fun ω => ∏ b, ∑ S : ↑(D b).powerset,
+        sparseCoefficient w S.val * if S.val ⊆ F ω then 1 else 0) := by
+      apply finiteMean_mono
+      intro ω
+      rw [sparseMass,←Finset.prod_pow_eq_pow_sum]
+      apply Finset.prod_le_prod₀ (fun _ _ => pow_nonneg hw _)
+      intro b _
+      exact sparse_block_pointwise (F ω) (D b) w hw
+    _ ≤ ∏ b, ∑ S : ↑(D b).powerset, sparseCoefficient w S.val * p^S.val.card := by
+      simpa only [Finset.prod_const] using hb
+    _ ≤ _ := by
+      rw [Real.exp_sum]
+      apply Finset.prod_le_prod₀
+      · intro b _
+        exact Finset.sum_nonneg (fun S _ => mul_nonneg
+          (sparseCoefficient_nonneg w hw S.val) (pow_nonneg hp _))
+      · intro b _
+        exact sparse_block_product_bound (D b) p w hp hw (hr b)
+
+theorem inverseButterfly_sparse_mgf (d : ℕ) {β : Type*} [Fintype β]
+    (D : β → Finset (Card d)) (hd : Pairwise (fun b b' => Disjoint (D b) (D b')))
+    (A : Finset (Card d)) (w : ℝ) (hw : 0 ≤ w)
+    (hr : ∀ b, ((D b).card:ℝ)*((A.card/(2:ℝ)^d)*w) ≤ 1/2) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      w^(sparseMass D (inverseButterflyMarks d A ω))) ≤
+      Real.exp (∑ b, 2*(((D b).card:ℝ)*((A.card/(2:ℝ)^d)*w))^2) := by
+  apply sparse_block_mgf D hd (inverseButterflyMarks d A) (A.card/(2:ℝ)^d) w
+    (by positivity) hw
+  · exact inverseButterflyMarks_inclusion d A
+  · exact hr
+
+
+def crowdedMass (h : ℕ) : ℕ := if 2 ≤ h then h else 0
+
+lemma crowdedMass_le (h : ℕ) : crowdedMass h ≤ h := by
+  unfold crowdedMass
+  split_ifs <;> omega
+
+lemma crowdedMass_add (u v : ℕ) : crowdedMass u + crowdedMass v ≤ crowdedMass (u+v) := by
+  unfold crowdedMass
+  split_ifs <;> omega
+
+lemma alternatingCycles_le_crowded {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
+    (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    PairRouting.alternatingCycles e p ≤ crowdedMass (Fintype.card ι) := by
+  unfold crowdedMass
+  split_ifs with h
+  · have hh := PairRouting.alternatingCycles_double e p
+    omega
+  · rw [PairRouting.alternatingCycles_zero e p (by omega)]
+
+lemma palindromeCost_le_crowded (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) :
+    palindromeCost d e ω ≤ d*crowdedMass (Fintype.card ι) := by
+  classical
+  induction d generalizing ι with
+  | zero => simp [palindromeCost]
+  | succ d ih =>
+    let σ := benesStepEquiv d ω
+    let x := e.trans (headTailEquiv d).toEmbedding
+    let c := PairRouting.colors x σ.2.1
+    let hc := PairRouting.colors_compatible x σ.2.1
+    have h₀ := ih (PairRouting.childEmbedding x c hc false) σ.1.1
+    have h₁ := ih (PairRouting.childEmbedding x c hc true) σ.1.2
+    have hz := alternatingCycles_le_crowded (PairRouting.switchedEmbedding x σ.2.1)
+      (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1))
+    have hs := crowdedMass_add (Fintype.card {i // c i = false})
+      (Fintype.card {i // c i = true})
+    rw [PairRouting.color_card_add] at hs
+    change PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
+        (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) +
+        palindromeCost d (PairRouting.childEmbedding x c hc false) σ.1.1 +
+        palindromeCost d (PairRouting.childEmbedding x c hc true) σ.1.2 ≤ _
+    nlinarith
+
+noncomputable def palindromeLowCost (H : ℕ) : (d : ℕ) → {ι : Type*} → [Fintype ι] →
+    (ι ↪ Card d) → BenesCoins d → ℕ
+  | 0, _, _, _, _ => 0
+  | d+1, _, _, e, ω =>
+      let σ := benesStepEquiv d ω
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x σ.2.1
+      let hc := PairRouting.colors_compatible x σ.2.1
+      (if d+1 ≤ H then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
+          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) else 0) +
+        palindromeLowCost H d (PairRouting.childEmbedding x c hc false) σ.1.1 +
+        palindromeLowCost H d (PairRouting.childEmbedding x c hc true) σ.1.2
+
+noncomputable def cutoffMass (H : ℕ) : (d : ℕ) → {ι : Type*} → [Fintype ι] →
+    (ι ↪ Card d) → BenesCoins d → ℕ
+  | 0, _, _, _, _ => 0
+  | d+1, ι, _, e, ω =>
+      if d+1 ≤ H then crowdedMass (Fintype.card ι) else
+      let σ := benesStepEquiv d ω
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x σ.2.1
+      let hc := PairRouting.colors_compatible x σ.2.1
+      cutoffMass H d (PairRouting.childEmbedding x c hc false) σ.1.1 +
+        cutoffMass H d (PairRouting.childEmbedding x c hc true) σ.1.2
+
+lemma palindromeLowCost_eq (H d : ℕ) (h : d ≤ H) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) :
+    palindromeLowCost H d e ω = palindromeCost d e ω := by
+  classical
+  induction d generalizing ι with
+  | zero => rfl
+  | succ d ih =>
+    rw [palindromeLowCost,palindromeCost,ite_eq_left h]
+    congr 2 <;> exact ih (by omega) _ _
+
+lemma palindromeLowCost_le_cutoff (H d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) :
+    palindromeLowCost H d e ω ≤ H*cutoffMass H d e ω := by
+  classical
+  induction d generalizing ι with
+  | zero => simp [palindromeLowCost,cutoffMass]
+  | succ d ih =>
+    by_cases h : d+1 ≤ H
+    · rw [palindromeLowCost_eq H (d+1) h,cutoffMass,ite_eq_left h]
+      exact (palindromeCost_le_crowded (d+1) e ω).trans (Nat.mul_le_mul_right _ h)
+    · rw [palindromeLowCost,cutoffMass,ite_eq_right h,ite_eq_right h,zero_add,Nat.mul_add]
+      exact Nat.add_le_add (ih _ _) (ih _ _)
+
+
+noncomputable def centralBlock (r H : ℕ) (b : Card r) : Finset (Card (H+r)) :=
+  Finset.univ.image (fun z : Card H => (cardSplit r H).symm (b,z))
+
+end Thorp
+
+end ThorpNine.Tail
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Tail/BlockProduct.lean b/lean/OAI/Probability/ThorpRouting/Tail/BlockProduct.lean
index d1e0e59e196042fb39ebf75aaf74823ef9bb9a8c..8ec586b2d81b7193ef52579686688dae2aa35f43 100644
--- a/lean/OAI/Probability/ThorpRouting/Tail/BlockProduct.lean
+++ b/lean/OAI/Probability/ThorpRouting/Tail/BlockProduct.lean
@@ -1,894 +1,894 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Tail.Representative
-
-namespace OAI
-
-namespace ThorpNine.Tail
-
-namespace Thorp.CommutingBlocks
-open scoped BigOperators Classical
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
-variable {ι : Type*} [Fintype ι]
-variable [DecidableEq ι]
-variable (P : ι → V →L[ℂ] V) (hcomm : ∀ i j, Commute (P i) (P j))
-include hcomm
-omit [CompleteSpace V] [Fintype ι] in
-lemma sector_sum (s : Finset ι) : ∑ E∈s.powerset, sector P hcomm s E=1 := by
-  induction s using Finset.induction_on with
-  | empty =>
-    simp only [Finset.powerset_empty,Finset.sum_singleton]
-    exact Finset.noncommProd_empty _ _
-  | @insert a s ha ih =>
-    rw [Finset.sum_powerset_insert ha]
-    have h₁ (E : Finset ι) (hE : E∈s.powerset) :
-        sector P hcomm (insert a s) E=P a*sector P hcomm s E := by
-      rw [sector_insert P hcomm s E a ha]
-      congr 1
-      exact ite_eq_right (fun h => ha ((Finset.mem_powerset.mp hE) h))
-    have h₂ (E : Finset ι) (_hE : E∈s.powerset) :
-        sector P hcomm (insert a s) (insert a E)=(1-P a)*sector P hcomm s E := by
-      rw [sector_insert P hcomm s (insert a E) a ha]
-      simp only [choice,Finset.mem_insert_self,↓reduceIte]
-      congr 1
-      apply sector_congr
-      intro i hi
-      simp [ne_of_mem_of_not_mem hi ha]
-    rw [Finset.sum_congr rfl h₁,Finset.sum_congr rfl h₂,
-      ←Finset.mul_sum,←Finset.mul_sum,ih]
-    simp
-
-omit [Fintype ι] in
-lemma sector_absorb_factor (s E : Finset ι) (i : ι) (hi : i∈s)
-    (hP : ∀ i, IsStarProjection (P i)) :
-    choice P E i*sector P hcomm s E=sector P hcomm s E := by
-  have hc : IsIdempotentElem (choice P E i) := by
-    unfold choice
-    split_ifs
-    · exact (hP i).one_sub.isIdempotentElem
-    · exact (hP i).isIdempotentElem
-  have hsplit := Finset.mul_noncommProd_erase s hi (choice P E)
-    (fun i _ j _ _ => choice_commute P hcomm E E i j)
-  change choice P E i*sector P hcomm s E=sector P hcomm s E
-  change _=sector P hcomm s E at hsplit
-  nth_rw 1 [←hsplit]
-  rw [←mul_assoc,hc.eq,hsplit]
-
-omit [Fintype ι] in
-lemma sector_orthogonal (s E F : Finset ι) (hE : E⊆s) (hF : F⊆s) (hEF : E≠F)
-    (hP : ∀ i, IsStarProjection (P i)) :
-    sector P hcomm s E*sector P hcomm s F=0 := by
-  have hdiff : ∃ i∈s, ¬(i∈E ↔ i∈F) := by
-    by_contra h
-    push Not at h
-    exact hEF (Finset.ext fun i => by
-      by_cases hi : i∈s
-      · exact h i hi
-      · exact iff_of_false (fun h => hi (hE h)) (fun h => hi (hF h)))
-  obtain ⟨i,hi,hdiff⟩ := hdiff
-  have hz : choice P E i*choice P F i=0 := by
-    by_cases he : i∈E <;> by_cases hf : i∈F
-    · exact (hdiff (iff_of_true he hf)).elim
-    · simpa only [choice,ite_eq_left he,ite_eq_right hf] using (hP i).one_sub_mul_self
-    · simpa only [choice,ite_eq_right he,ite_eq_left hf] using (hP i).mul_one_sub_self
-    · exact (hdiff (iff_of_false he hf)).elim
-  have hEi := sector_absorb_factor P hcomm s E i hi hP
-  have hFi := sector_absorb_factor P hcomm s F i hi hP
-  calc
-    _ = sector P hcomm s E*(choice P E i*choice P F i)*sector P hcomm s F := by
-      simp only [mul_assoc]
-      rw [hFi]
-      rw [←mul_assoc,(sector_factor_commute P hcomm s E E i).eq.symm,hEi]
-    _ = 0 := by rw [hz,mul_zero,zero_mul]
-
-noncomputable def blockProduct (A : ι → V →L[ℂ] V)
-    (hA : ∀ i j, Commute (A i) (A j)) (s : Finset ι) : V →L[ℂ] V :=
-  s.noncommProd A (fun i _ j _ _ => hA i j)
-
-omit hcomm [Fintype ι] in
-lemma norm_noncommProd_le (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
-    (s : Finset ι) : ‖blockProduct A hA s‖≤∏ i∈s,‖A i‖ := by
-  induction s using Finset.induction_on with
-  | empty => exact ContinuousLinearMap.norm_id_le
-  | @insert a s ha ih =>
-    have he : blockProduct A hA (insert a s)=A a*blockProduct A hA s :=
-      Finset.noncommProd_insert_of_notMem _ _ _ _ ha
-    rw [he,Finset.prod_insert ha]
-    exact (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_left ih (norm_nonneg _))
-
-noncomputable def blocks (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j)) : V →L[ℂ] V :=
-  blockProduct A hA Finset.univ
-
-omit hcomm in
-lemma blocks_norm_le_one (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
-    (hn : ∀ i,‖A i‖≤1) : ‖blocks A hA‖≤1 :=
-  (norm_noncommProd_le A hA Finset.univ).trans (Finset.prod_le_one₀ (fun _ _ => norm_nonneg _) (fun i _ => hn i))
-
-omit [CompleteSpace V] in
-lemma blocks_sector_commute (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
-    (hAP : ∀ i j, Commute (A i) (P j)) (E : Finset ι) :
-    Commute (blocks A hA) (sector P hcomm Finset.univ E) := by
-  unfold blocks blockProduct
-  apply (Finset.noncommProd_commute _ _ (fun i _ j _ _ => hA i j) _ _).symm
-  intro i _
-  unfold sector
-  apply (Finset.noncommProd_commute _ _ (fun i _ j _ _ => choice_commute P hcomm E E i j) _ _).symm
-  intro j _
-  unfold choice
-  split_ifs
-  · exact (Commute.one_right _).sub_right (hAP i j)
-  · exact hAP i j
-
-lemma blocks_sector_norm (hP : ∀ i, IsStarProjection (P i))
-    (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
-    (hAP : ∀ i j, Commute (A i) (P j)) (hn : ∀ i,‖A i‖≤1)
-    (θ : ℝ) (_hθ : 0≤θ) (hgap : ∀ i,‖A i*(1-P i)‖≤θ) (E : Finset ι) :
-    ‖blocks A hA*sector P hcomm Finset.univ E‖≤θ^E.card := by
-  have hAC (i j : ι) : Commute (A i) (choice P E j) := by
-    unfold choice
-    split_ifs
-    · exact (Commute.one_right _).sub_right (hAP i j)
-    · exact hAP i j
-  have hprod := Finset.noncommProd_mul_distrib (s := Finset.univ) A (choice P E)
-    (fun i _ j _ _ => hA i j) (fun i _ j _ _ => choice_commute P hcomm E E i j)
-    (fun i _ j _ _ => (hAC j i).symm)
-  change _=blocks A hA*sector P hcomm Finset.univ E at hprod
-  rw [←hprod]
-  have hc (i j : ι) : Commute ((A*choice P E) i) ((A*choice P E) j) :=
-    ((hA i j).mul_right (hAC i j)).mul_left ((hAC j i).symm.mul_right (choice_commute P hcomm E E i j))
-  apply (norm_noncommProd_le _ hc Finset.univ).trans
-  calc
-    _ ≤ ∏ i : ι, if i∈E then θ else 1 := by
-      apply Finset.prod_le_prod₀ (fun i _ => norm_nonneg _)
-      intro i _
-      dsimp only [Pi.mul_apply]
-      unfold choice
-      split_ifs with hi
-      · exact hgap i
-      · exact (norm_mul_le _ _).trans (by nlinarith [hn i,(hP i).norm_le (P i),norm_nonneg (A i),norm_nonneg (P i)])
-    _ = _ := by rw [Fintype.prod_ite_mem,Finset.prod_const]
-
-noncomputable def coarse (E : Finset ι) : V →L[ℂ] V :=
-  Eᶜ.noncommProd P (fun i _ j _ _ => hcomm i j)
-
-lemma coarse_projection (hP : ∀ i, IsStarProjection (P i)) (E : Finset ι) :
-    IsStarProjection (coarse P hcomm E) := by
-  have he : coarse P hcomm E=sector P hcomm Eᶜ ∅ := by
-    apply Finset.noncommProd_congr rfl
-    intro i _
-    simp [choice]
-  rw [he]
-  exact sector_projection P hcomm hP _ _
-
-lemma sector_coarse (hP : ∀ i, IsStarProjection (P i)) (E : Finset ι) :
-    sector P hcomm Finset.univ E*coarse P hcomm E=sector P hcomm Finset.univ E := by
-  have hpi (i : ι) (hi : i∈Eᶜ) : sector P hcomm Finset.univ E*P i=sector P hcomm Finset.univ E := by
-    have h := sector_absorb_factor P hcomm Finset.univ E i (Finset.mem_univ i) hP
-    rw [(sector_factor_commute P hcomm Finset.univ E E i).eq] at h
-    simpa only [choice,ite_eq_right (Finset.mem_compl.mp hi)] using h
-  have hp (s : Finset ι) (hs : ∀ i∈s,
-      sector P hcomm Finset.univ E*P i=sector P hcomm Finset.univ E) :
-      sector P hcomm Finset.univ E*blockProduct P hcomm s=sector P hcomm Finset.univ E := by
-    induction s using Finset.induction_on with
-    | empty => exact mul_one _
-    | @insert a s ha ih =>
-      have he : blockProduct P hcomm (insert a s)=P a*blockProduct P hcomm s :=
-        Finset.noncommProd_insert_of_notMem _ _ _ _ ha
-      rw [he,←mul_assoc,hs a (Finset.mem_insert_self _ _)]
-      exact ih (fun i hi => hs i (Finset.mem_insert_of_mem hi))
-  exact hp Eᶜ hpi
-
-theorem blocks_exception_bound (hP : ∀ i, IsStarProjection (P i))
-    (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
-    (hAP : ∀ i j, Commute (A i) (P j)) (hn : ∀ i,‖A i‖≤1)
-    (θ : ℝ) (hθ : 0≤θ) (hθ1 : θ≤1) (hgap : ∀ i,‖A i*(1-P i)‖≤θ)
-    (B : V →L[ℂ] V) (hB : ‖B‖≤1) (n : ℕ) :
-    ‖B.adjoint*blocks A hA*B‖≤θ^n+
-      ∑ E : Finset ι, if E.card<n then ‖B.adjoint*coarse P hcomm E*B‖ else 0 := by
-  let T : Finset ι → V →L[ℂ] V := sector P hcomm Finset.univ
-  have hsum : ∑ E,T E=1 := by
-    simpa only [Finset.powerset_univ] using sector_sum P hcomm Finset.univ
-  have hbound := resolution_exception_bound T (coarse P hcomm)
-    (fun E => sector_projection P hcomm hP _ E)
-    (fun E F hEF => sector_orthogonal P hcomm _ E F (Finset.subset_univ _) (Finset.subset_univ _) hEF hP)
-    hsum (coarse_projection P hcomm hP) (sector_coarse P hcomm hP)
-    (blocks A hA) B (blocks_sector_commute P hcomm A hA hAP)
-    (blocks_norm_le_one A hA hn) hB (Finset.univ.filter fun E : Finset ι => E.card<n)
-    (θ^n) (pow_nonneg hθ _) (by
-      intro E hE
-      have hEn : n≤E.card := by simpa using hE
-      exact (blocks_sector_norm P hcomm hP A hA hAP hn θ hθ hgap E).trans
-        (pow_le_pow_of_le_one hθ hθ1 hEn))
-  simpa only [Finset.sum_filter] using hbound
-
-
-end Thorp.CommutingBlocks
-
-namespace Thorp.UnitaryFinite
-open scoped BigOperators Classical
-
-variable {G : Type*} [Group G] [Fintype G]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-variable {Ω Ξ : Type*} [Fintype Ω] [Fintype Ξ]
-
-lemma sampleOperator_weighted (ρ : Representation ℂ G V) (P : Ω → G) :
-    sampleOperator ρ P=weightedOperator ρ (sampleLaw P) := by
-  unfold sampleOperator weightedOperator sampleLaw finiteMean
-  simp only [Complex.ofReal_sum,div_eq_mul_inv,Finset.sum_mul,Finset.sum_smul]
-  rw [Finset.sum_comm]
-  simp only [Finset.smul_sum]
-  apply Finset.sum_congr rfl
-  intro ω _
-  calc
-    _ = ∑ g, if P ω=g then (Fintype.card Ω:ℂ)⁻¹ • continuousRepresentation ρ g else 0 := by
-      symm
-      simp only [Finset.sum_ite_eq,Finset.mem_univ,↓reduceIte]
-    _ = _ := by
-      apply Finset.sum_congr rfl
-      intro g _
-      split_ifs <;> simp_all
-
-lemma sampleOperator_norm_le [Nonempty Ω] (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (P : Ω → G) :
-    ‖sampleOperator ρ P‖≤1 := by
-  rw [sampleOperator_weighted]
-  exact (weightedOperator_norm_le ρ hρ _ (sampleLaw_nonneg _)).trans_eq (sampleLaw_sum P)
-
-omit [Fintype G] in
-lemma sampleOperator_equiv (ρ : Representation ℂ G V) (P : Ω → G) (e : Ξ ≃ Ω) :
-    sampleOperator ρ (P ∘ e)=sampleOperator ρ P := by
-  simp only [sampleOperator,Fintype.card_congr e,Function.comp_apply]
-  rw [Equiv.sum_comp e (fun ω => continuousRepresentation ρ (P ω))]
-
-omit [Fintype G] in
-lemma sampleOperator_prod (ρ : Representation ℂ G V) (P : Ω → G) (Q : Ξ → G) :
-    sampleOperator ρ (fun ω : Ω × Ξ => P ω.1*Q ω.2)=sampleOperator ρ P*sampleOperator ρ Q := by
-  simp only [sampleOperator,Fintype.card_prod,Nat.cast_mul,mul_inv_rev,map_mul,Fintype.sum_prod_type,
-    Finset.sum_mul,Finset.mul_sum,smul_mul_smul,Finset.smul_sum]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro ω _
-  apply Finset.sum_congr rfl
-  intro ξ _
-  rw [mul_comm]
-
-omit [Fintype G] in
-lemma sampleOperator_inv (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (P : Ω → G) :
-    sampleOperator ρ (fun ω => (P ω)⁻¹)=(sampleOperator ρ P).adjoint := by
-  simp only [sampleOperator,map_smulₛₗ,map_sum,map_inv₀,map_natCast,
-    continuousRepresentation_adjoint ρ hρ]
-
-lemma sampleOperator_palindrome (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V)
-    (hρ : IsUnitary ρ) :
-    sampleOperator ρ (palindromePerm d)=
-      sampleOperator ρ (fun ω => butterflyPerm d (decodeButterfly d ω))*
-      (sampleOperator ρ (fun ω => butterflyPerm d (decodeButterfly d ω))).adjoint := by
-  rw [←sampleOperator_inv ρ hρ,←sampleOperator_prod]
-  rfl
-
-omit [Fintype G] in
-lemma sampleOperator_one [Nonempty Ω] (ρ : Representation ℂ G V) :
-    sampleOperator ρ (fun _ : Ω => 1)=1 := by
-  simp only [sampleOperator,map_one,Finset.sum_const,Finset.card_univ,←Nat.cast_smul_eq_nsmul ℂ,
-    smul_smul]
-  rw [inv_mul_cancel₀ (by exact_mod_cast Fintype.card_ne_zero),one_smul]
-
-lemma sampleOperator_uniform (ρ : Representation ℂ G V) : sampleOperator ρ (id : G → G)=uniformOperator ρ := by
-  simp only [sampleOperator,uniformOperator,weightedOperator,Complex.ofReal_inv,Complex.ofReal_natCast,
-    Finset.smul_sum,id_eq]
-
-lemma sampleOperator_uniform_mul [Nonempty Ω] (ρ : Representation ℂ G V) (P : Ω → G) :
-    sampleOperator ρ (fun ω : Ω × G => P ω.1*ω.2⁻¹)=uniformOperator ρ := by
-  rw [sampleOperator_prod]
-  have he : sampleOperator ρ (fun g : G => g⁻¹)=uniformOperator ρ := by
-    rw [←sampleOperator_uniform]
-    exact sampleOperator_equiv ρ id (Equiv.inv G)
-  rw [he,sampleOperator_weighted,weighted_uniform_left,sampleLaw_sum,one_smul]
-
-end Thorp.UnitaryFinite
-
-namespace Thorp.FiniteAverage
-open scoped BigOperators Classical
-variable {I Ω A : Type*} [Fintype I] [DecidableEq I] [Fintype Ω]
-    [Ring A] [Algebra ℂ A]
-
-noncomputable def orderedProduct : MultilinearMap ℂ (fun _ : I => A) A :=
-  MultilinearMap.domDomCongr (Fintype.equivFin I).symm
-    (MultilinearMap.mkPiAlgebraFin ℂ (Fintype.card I) A)
-
-lemma orderedProduct_eq (f : I → A) (hc : Pairwise fun i j => Commute (f i) (f j)) :
-    orderedProduct f=Finset.univ.noncommProd f (fun _ _ _ _ hij => hc hij) := by
-  unfold orderedProduct
-  rw [MultilinearMap.domDomCongr_apply,MultilinearMap.mkPiAlgebraFin_apply]
-  let l := List.ofFn (Fintype.equivFin I).symm
-  have hl : l.Nodup := List.nodup_ofFn.mpr (Fintype.equivFin I).symm.injective
-  have hu : l.toFinset=Finset.univ := by
-    ext i
-    simp only [List.mem_toFinset,List.mem_ofFn,Finset.mem_univ,iff_true,l]
-    exact (Fintype.equivFin I).symm.surjective i
-  have he := Finset.noncommProd_toFinset l f (by intro i hi j hj hij; exact hc hij) hl
-  rw [hu] at he
-  rw [he]
-  congr 1
-  exact List.map_ofFn.symm
-
-lemma multilinear_average (F : MultilinearMap ℂ (fun _ : I => A) A) (x : I → Ω → A) :
-    F (fun i => (Fintype.card Ω:ℂ)⁻¹ • ∑ ω,x i ω)=
-      (Fintype.card (I → Ω):ℂ)⁻¹ • ∑ ω : I → Ω,F (fun i => x i (ω i)) := by
-  rw [F.map_smul_univ,F.map_sum]
-  simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fun,Nat.cast_pow,inv_pow]
-
-lemma ordered_average (x : I → Ω → A)
-    (hc : ∀ ω, Pairwise fun i j => Commute (x i (ω i)) (x j (ω j)))
-    (havg : Pairwise fun i j => Commute ((Fintype.card Ω:ℂ)⁻¹ • ∑ ω,x i ω)
-      ((Fintype.card Ω:ℂ)⁻¹ • ∑ ω,x j ω)) :
-    (Finset.univ.noncommProd (fun i => (Fintype.card Ω:ℂ)⁻¹ • ∑ ω,x i ω)
-      (fun _ _ _ _ hij => havg hij))=
-    (Fintype.card (I → Ω):ℂ)⁻¹ • ∑ ω : I → Ω,
-      Finset.univ.noncommProd (fun i => x i (ω i)) (fun _ _ _ _ hij => hc ω hij) := by
-  rw [←orderedProduct_eq _ havg,multilinear_average]
-  simp only [orderedProduct_eq _ (hc _)]
-
-end Thorp.FiniteAverage
-namespace Thorp.FiniteCharacters
-open Module
-open scoped BigOperators Classical
-
-variable {G ι : Type*} [Group G] [Fintype G] [Fintype ι]
-variable (V : ι → Type*) [∀ i, AddCommGroup (V i)] [∀ i, Module ℂ (V i)]
-  [∀ i, FiniteDimensional ℂ (V i)]
-variable (ρ : ∀ i, Representation ℂ G (V i)) [∀ i, Representation.IsIrreducible (ρ i)]
-variable (hneq : ∀ i j, i≠j → ¬Nonempty (Representation.Equiv (ρ i) (ρ j)))
-  (hcard : Nat.card (ConjClasses G)≤Fintype.card ι)
-include hneq hcard
-
-lemma character_expansion (f : ConjClasses G → ℂ) :
-    f=∑ i, characterDual (ρ i) f • characterClass (ρ i) := by
-  let : Fintype (ConjClasses G) := Fintype.ofFinite _
-  have horth (i j : ι) : characterDual (ρ i) (characterClass (ρ j))=if i=j then 1 else 0 := by
-    rw [characterDual_characterClass]
-    by_cases h : i=j
-    · subst j; rw [ite_eq_left ⟨Representation.Equiv.refl _⟩,ite_eq_left rfl]
-    · rw [ite_eq_right h,ite_eq_right (hneq i j h)]
-  have hLI : LinearIndependent ℂ (fun i => characterClass (ρ i)) := by
-    apply LinearIndependent.of_pairwise_dual_eq_zero_one _ (fun i => characterDual (ρ i))
-    · intro i j hij; exact (horth i j).trans (ite_eq_right hij)
-    · intro i; simpa using horth i i
-  have heq : Fintype.card ι=Module.finrank ℂ (ConjClasses G → ℂ) := by
-    apply Nat.le_antisymm hLI.fintype_card_le_finrank
-    simpa only [Module.finrank_pi,Module.finrank_self,Finset.sum_const,Finset.card_univ,
-      smul_eq_mul,mul_one,Nat.card_eq_fintype_card] using hcard
-  let b := Basis.mk hLI (by rw [hLI.span_eq_top_of_card_eq_finrank' heq])
-  have hb (i : ι) : b i=characterClass (ρ i) := Basis.mk_apply _ _ _
-  have hd (i : ι) : b.coord i=characterDual (ρ i) := by
-    apply b.ext
-    intro j
-    rw [hb,horth]
-    simp only [←hb,Basis.coord_apply,Basis.repr_self,Finsupp.single_apply]
-    split_ifs <;> simp_all
-  have h := b.sum_repr f
-  simpa only [hb,show ∀ i,b.repr f i=characterDual (ρ i) f from fun i => congrArg (fun l => l f) (hd i)] using h.symm
-
-theorem sum_dimension_sq : ∑ i, (Module.finrank ℂ (V i))^2=Fintype.card G := by
-  let f : ConjClasses G → ℂ := fun c => if c=ConjClasses.mk (1:G) then 1 else 0
-  have hf (g : G) : f (ConjClasses.mk g)=if g=1 then 1 else 0 := by
-    simp only [f,ConjClasses.mk_eq_mk_iff_isConj,isConj_one_left]
-  have hd (i : ι) : characterDual (ρ i) f=(Fintype.card G:ℂ)⁻¹*(Module.finrank ℂ (V i):ℂ) := by
-    change (Nat.card G:ℂ)⁻¹ * ∑ g, f (ConjClasses.mk g)*(ρ i).character g⁻¹=_
-    simp only [hf,ite_mul,one_mul,zero_mul,Finset.sum_ite_eq',Finset.mem_univ,↓reduceIte,inv_one,
-      Representation.char_one,Nat.card_eq_fintype_card]
-  have he := congrFun (character_expansion V ρ hneq hcard f) (ConjClasses.mk (1:G))
-  rw [hf,ite_eq_left rfl] at he
-  simp only [Finset.sum_apply,Pi.smul_apply,smul_eq_mul,hd,characterClass_mk,Representation.char_one] at he
-  have hN : (Fintype.card G:ℂ)≠0 := by exact_mod_cast Fintype.card_ne_zero
-  have hs : (∑ i,(Module.finrank ℂ (V i):ℂ)^2)=(Fintype.card G:ℂ) := by
-    calc
-      _ = (Fintype.card G:ℂ) * ∑ i,(Fintype.card G:ℂ)⁻¹ *
-          (Module.finrank ℂ (V i):ℂ)*(Module.finrank ℂ (V i):ℂ) := by
-        rw [Finset.mul_sum]
-        apply Finset.sum_congr rfl
-        intro i _
-        field_simp
-      _ = _ := by rw [←he,mul_one]
-  exact_mod_cast hs
-
-end Thorp.FiniteCharacters
-
-namespace Thorp.UnitaryFinite
-open scoped BigOperators ComplexConjugate Classical
-
-variable {G : Type*} [Group G] [Fintype G]
-variable {V W : Type*} [NormedAddCommGroup V] [NormedAddCommGroup W]
-  [InnerProductSpace ℂ V] [InnerProductSpace ℂ W]
-  [FiniteDimensional ℂ V] [FiniteDimensional ℂ W]
-
-omit [FiniteDimensional ℂ V] [FiniteDimensional ℂ W] in
-lemma averageMap_zero (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
-    [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ]
-    (hne : ¬Nonempty (Representation.Equiv ρ σ)) (A : V →ₗ[ℂ] W) :
-    averageMap ρ σ A=0 := by
-  rcases Representation.IsIrreducible.bijective_or_eq_zero (averageMap ρ σ A) with h|h
-  · exact (hne ⟨Representation.IntertwiningMap.ofBijective _ h⟩).elim
-  · exact h
-
-omit [FiniteDimensional ℂ V] [FiniteDimensional ℂ W] in
-lemma coefficient_cross_orthogonality (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
-    [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ]
-    (hρ : IsUnitary ρ) (hne : ¬Nonempty (Representation.Equiv ρ σ))
-    (u z : W) (v w : V) :
-    ∑ g, inner ℂ u (σ g z)*inner ℂ (ρ g w) v=0 := by
-  have he := averageMap_zero ρ σ hne (InnerProductSpace.rankOne ℂ z w).toLinearMap
-  have hv := congrArg (fun T : Representation.IntertwiningMap ρ σ => inner ℂ u (T v)) he
-  change inner ℂ u ((∑ g, (σ g).comp ((InnerProductSpace.rankOne ℂ z w).toLinearMap.comp (ρ g⁻¹))) v)=_ at hv
-  simpa only [LinearMap.sum_apply,LinearMap.comp_apply,ContinuousLinearMap.coe_coe,
-    InnerProductSpace.rankOne_apply,map_smul,unitary_inner_inv ρ hρ,inner_sum,
-    inner_smul_right,Representation.IntertwiningMap.coe_zero,Pi.zero_apply,inner_zero_right,mul_comm] using hv
-
-end Thorp.UnitaryFinite
-namespace Thorp.UnitaryFinite
-open scoped BigOperators ComplexConjugate Classical
-
-variable {G : Type*} [Group G] [Fintype G]
-
-def regularAction : G →* Equiv.Perm G where
-  toFun := Equiv.mulLeft
-  map_one' := by ext; simp
-  map_mul' _ _ := by ext; simp
-
-omit [Fintype G] in
-lemma regular_transitive (x : G) : ∃ g,regularAction g (1:G)=x := ⟨x,mul_one x⟩
-
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-omit [Fintype G] [FiniteDimensional ℂ V] in
-lemma regular_fixed (ρ : Representation ℂ G V) (w : V) : IsFixed regularAction (1:G) ρ w := by
-  intro g hg
-  have he : g=1 := by simpa only [regularAction,MonoidHom.coe_mk,OneHom.coe_mk,Equiv.coe_mulLeft,mul_one] using hg
-  subst g
-  simp
-
-noncomputable def regularCoefficient (ρ : Representation ℂ G V) (w v : V) : EuclideanSpace ℂ G :=
-  normalizedCoefficient regularAction 1 regular_transitive ρ w v
-
-omit [FiniteDimensional ℂ V] in
-lemma regularCoefficient_apply (ρ : Representation ℂ G V) (w v : V) (g : G) :
-    regularCoefficient ρ w v g = (Real.sqrt ((Module.finrank ℂ V:ℝ)/(Fintype.card G:ℝ)):ℂ)*inner ℂ (ρ g w) v := by
-  unfold regularCoefficient normalizedCoefficient
-  simp only [PiLp.smul_apply,smul_eq_mul]
-  congr 1
-  have he := coefficient_at regularAction 1 regular_transitive ρ w (regular_fixed ρ w) v g
-  simpa only [regularAction,MonoidHom.coe_mk,OneHom.coe_mk,Equiv.coe_mulLeft,mul_one] using he
-
-lemma regularCoefficient_inner (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
-    (hρ : IsUnitary ρ) (w z v t : V) :
-    inner ℂ (regularCoefficient ρ w v) (regularCoefficient ρ z t)=inner ℂ z w*inner ℂ v t :=
-  normalizedCoefficient_inner regularAction 1 regular_transitive ρ hρ w z
-    (regular_fixed ρ w) (regular_fixed ρ z) v t
-
-variable {W : Type*} [NormedAddCommGroup W] [InnerProductSpace ℂ W] [FiniteDimensional ℂ W]
-
-omit [FiniteDimensional ℂ V] [FiniteDimensional ℂ W] in
-lemma regularCoefficient_cross (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
-    [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ]
-    (hσ : IsUnitary σ) (hne : ¬Nonempty (Representation.Equiv σ ρ))
-    (w v : V) (z t : W) : inner ℂ (regularCoefficient ρ w v) (regularCoefficient σ z t)=0 := by
-  rw [PiLp.inner_apply]
-  simp only [RCLike.inner_apply,regularCoefficient_apply,map_mul,Complex.conj_ofReal,
-    inner_conj_symm]
-  have he := coefficient_cross_orthogonality σ ρ hσ hne v w t z
-  calc
-    _ = ((Real.sqrt ((Module.finrank ℂ V:ℝ)/(Fintype.card G:ℝ)):ℂ)*
-        (Real.sqrt ((Module.finrank ℂ W:ℝ)/(Fintype.card G:ℝ)):ℂ)) *
-        ∑ g,inner ℂ v (ρ g w)*inner ℂ (σ g z) t := by
-      rw [Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro g _
-      ring
-    _ = 0 := by rw [he,mul_zero]
-
-end Thorp.UnitaryFinite
-
-namespace Thorp.FiniteFourier
-open scoped BigOperators ComplexConjugate Classical
-open UnitaryFinite
-variable {G ι : Type*} [Group G] [Fintype G] [Fintype ι]
-variable (V : ι → Type*) [∀ i, NormedAddCommGroup (V i)] [∀ i, InnerProductSpace ℂ (V i)]
-  [∀ i, FiniteDimensional ℂ (V i)]
-variable (ρ : ∀ i, Representation ℂ G (V i)) [∀ i, Representation.IsIrreducible (ρ i)]
-variable (hρ : ∀ i, IsUnitary (ρ i))
-variable (hneq : ∀ i j, i≠j → ¬Nonempty (Representation.Equiv (ρ i) (ρ j)))
-  (hcard : Nat.card (ConjClasses G)≤Fintype.card ι)
-
-
-abbrev CoefficientIndex := Σ i,Fin (Module.finrank ℂ (V i)) × Fin (Module.finrank ℂ (V i))
-
-noncomputable def coefficientFamily (a : CoefficientIndex V) : EuclideanSpace ℂ G :=
-  regularCoefficient (ρ a.1) (stdOrthonormalBasis ℂ (V a.1) a.2.1) (stdOrthonormalBasis ℂ (V a.1) a.2.2)
-
-omit [Fintype ι] in
-include hρ hneq in
-lemma coefficientFamily_orthonormal : Orthonormal ℂ (coefficientFamily V ρ) := by
-  rw [orthonormal_iff_ite]
-  rintro ⟨i,a,b⟩ ⟨j,c,d⟩
-  dsimp only [coefficientFamily]
-  by_cases hij : i=j
-  · subst j
-    rw [regularCoefficient_inner _ (hρ i)]
-    simp only [(stdOrthonormalBasis ℂ (V i)).inner_eq_ite]
-    by_cases hac : a=c <;> by_cases hbd : b=d <;> simp [hac,hbd,eq_comm]
-  · rw [regularCoefficient_cross _ _ (hρ j) (hneq j i (Ne.symm hij))]
-    rw [ite_eq_right (by intro h; exact hij (congrArg Sigma.fst h))]
-
-include ρ hneq hcard in
-lemma coefficientIndex_card : Fintype.card (CoefficientIndex V)=Fintype.card G := by
-  simp only [CoefficientIndex,Fintype.card_sigma,Fintype.card_prod,Fintype.card_fin,←pow_two]
-  exact FiniteCharacters.sum_dimension_sq V ρ hneq hcard
-
-noncomputable def coefficientBasis : OrthonormalBasis (CoefficientIndex V) ℂ (EuclideanSpace ℂ G) :=
-  OrthonormalBasis.mk (coefficientFamily_orthonormal V ρ hρ hneq)
-    (by rw [(coefficientFamily_orthonormal V ρ hρ hneq).linearIndependent.span_eq_top_of_card_eq_finrank'
-          (by rw [coefficientIndex_card V ρ hneq hcard,finrank_euclideanSpace])])
-
-lemma coefficientBasis_apply (a : CoefficientIndex V) :
-    coefficientBasis V ρ hρ hneq hcard a=coefficientFamily V ρ a := congrFun (OrthonormalBasis.coe_mk _ _) a
-
-include hρ hneq hcard in
-
-theorem parseval (f : EuclideanSpace ℂ G) :
-    ∑ a : CoefficientIndex V, ‖inner ℂ (coefficientFamily V ρ a) f‖^2=‖f‖^2 := by
-  simpa only [coefficientBasis_apply] using
-    (coefficientBasis V ρ hρ hneq hcard).sum_sq_norm_inner_right f
-
-
-noncomputable def realVector (p : G → ℝ) : EuclideanSpace ℂ G := WithLp.toLp 2 (fun g => (p g:ℂ))
-
-omit [Fintype ι] [∀ i, Representation.IsIrreducible (ρ i)] in
-lemma coefficient_realVector (p : G → ℝ) (a : CoefficientIndex V) :
-    inner ℂ (coefficientFamily V ρ a) (realVector p)=
-      (Real.sqrt ((Module.finrank ℂ (V a.1):ℝ)/(Fintype.card G:ℝ)):ℂ)*
-      inner ℂ (stdOrthonormalBasis ℂ (V a.1) a.2.2)
-        (weightedOperator (ρ a.1) p (stdOrthonormalBasis ℂ (V a.1) a.2.1)) := by
-  simp only [PiLp.inner_apply,coefficientFamily,regularCoefficient_apply,realVector,
-    RCLike.inner_apply,map_mul,Complex.conj_ofReal,inner_conj_symm,
-    weightedOperator,sum_apply,smul_apply,
-    continuousRepresentation_apply,inner_sum,inner_smul_right,Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro g _
-  ring
-
-omit [Fintype ι] [∀ i, Representation.IsIrreducible (ρ i)] in
-lemma coefficient_realVector_sq (p : G → ℝ) (a : CoefficientIndex V) :
-    ‖inner ℂ (coefficientFamily V ρ a) (realVector p)‖^2=
-      ((Module.finrank ℂ (V a.1):ℝ)/(Fintype.card G:ℝ))*
-      ‖inner ℂ (stdOrthonormalBasis ℂ (V a.1) a.2.2)
-        (weightedOperator (ρ a.1) p (stdOrthonormalBasis ℂ (V a.1) a.2.1))‖^2 := by
-  rw [coefficient_realVector,norm_mul,Complex.norm_real,Real.norm_eq_abs,
-    abs_of_nonneg (Real.sqrt_nonneg _),mul_pow,Real.sq_sqrt (div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _))]
-
-include hρ hneq hcard in
-lemma parseval_operator (p : G → ℝ) :
-    (Fintype.card G:ℝ)*‖realVector p‖^2=
-      ∑ i,(Module.finrank ℂ (V i):ℝ)*∑ a : Fin (Module.finrank ℂ (V i)),
-        ‖weightedOperator (ρ i) p (stdOrthonormalBasis ℂ (V i) a)‖^2 := by
-  rw [←parseval V ρ hρ hneq hcard (realVector p)]
-  simp only [Fintype.sum_sigma,Fintype.sum_prod_type,coefficient_realVector_sq]
-  have hN : (Fintype.card G:ℝ)≠0 := by exact_mod_cast Fintype.card_ne_zero
-  rw [Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro i _
-  have he (a : Fin (Module.finrank ℂ (V i))) :
-      (∑ b : Fin (Module.finrank ℂ (V i)),
-        (Module.finrank ℂ (V i):ℝ)/(Fintype.card G:ℝ)*
-          ‖inner ℂ (stdOrthonormalBasis ℂ (V i) b)
-            (weightedOperator (ρ i) p (stdOrthonormalBasis ℂ (V i) a))‖^2)=
-        (Module.finrank ℂ (V i):ℝ)/(Fintype.card G:ℝ)*
-          ‖weightedOperator (ρ i) p (stdOrthonormalBasis ℂ (V i) a)‖^2 := by
-    rw [←Finset.mul_sum,(stdOrthonormalBasis ℂ (V i)).sum_sq_norm_inner_right]
-  simp_rw [he]
-  rw [←Finset.mul_sum]
-  field_simp
-
-include hρ hneq hcard in
-lemma realVector_bound (p : G → ℝ) :
-    (Fintype.card G:ℝ)*‖realVector p‖^2≤
-      ∑ i,(Module.finrank ℂ (V i):ℝ)^2*‖weightedOperator (ρ i) p‖^2 := by
-  rw [parseval_operator V ρ hρ hneq hcard]
-  apply Finset.sum_le_sum
-  intro i _
-  calc
-    _ ≤ (Module.finrank ℂ (V i):ℝ)*∑ _a : Fin (Module.finrank ℂ (V i)),‖weightedOperator (ρ i) p‖^2 := by
-      apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
-      apply Finset.sum_le_sum
-      intro a _
-      apply pow_le_pow_left₀ (norm_nonneg _)
-      simpa only [(stdOrthonormalBasis ℂ (V i)).norm_eq_one,mul_one] using
-        (weightedOperator (ρ i) p).le_opNorm (stdOrthonormalBasis ℂ (V i) a)
-    _ = _ := by simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]; ring
-
-include hρ hneq hcard in
-
-theorem l1_fourier_bound (p : G → ℝ) :
-    (∑ g,|p g|)^2≤∑ i,(Module.finrank ℂ (V i):ℝ)^2*‖weightedOperator (ρ i) p‖^2 := by
-  apply le_trans _ (realVector_bound V ρ hρ hneq hcard p)
-  have he : ‖realVector p‖^2=∑ g,|p g|^2 := by
-    rw [EuclideanSpace.norm_sq_eq]
-    simp only [realVector,PiLp.toLp_apply,Complex.norm_real,Real.norm_eq_abs]
-  rw [he]
-  simpa only [Finset.card_univ] using sq_sum_le_card_mul_sum_sq (s := Finset.univ) (f := fun g => |p g|)
-
-end Thorp.FiniteFourier
-
-namespace Thorp.ParallelBlocks
-open scoped BigOperators Classical
-variable {B X : Type*}
-
-def lift : (B → Equiv.Perm X) →* Equiv.Perm (B × X) where
-  toFun p :=
-    { toFun := fun x => (x.1,p x.1 x.2)
-      invFun := fun x => (x.1,(p x.1).symm x.2)
-      left_inv := fun x => by simp
-      right_inv := fun x => by simp }
-  map_one' := rfl
-  map_mul' _ _ := rfl
-
-@[simp] lemma lift_apply (p : B → Equiv.Perm X) (x : B × X) : lift p x=(x.1,p x.1 x.2) := rfl
-
-noncomputable def injection (b : B) : Equiv.Perm X →* Equiv.Perm (B × X) :=
-  lift.comp
-    { toFun := fun p c => if c=b then p else 1
-      map_one' := by funext c; simp
-      map_mul' := by intro p q; funext c; by_cases h : c=b <;> simp [h] }
-
-@[simp] lemma injection_apply (b : B) (p : Equiv.Perm X) (x : B × X) :
-    injection b p x=(x.1,if x.1=b then p x.2 else x.2) := by
-  simp only [injection,MonoidHom.comp_apply,lift_apply]
-  by_cases h : x.1=b <;> simp [h]
-
-lemma injection_commute {b c : B} (h : b≠c) (p q : Equiv.Perm X) :
-    Commute (injection b p) (injection c q) := by
-  apply Equiv.ext
-  intro x
-  simp only [Equiv.Perm.mul_apply,injection_apply]
-  by_cases hb : x.1=b <;> by_cases hc : x.1=c <;> simp_all
-
-variable [Fintype B]
-
-lemma lift_product (p : B → Equiv.Perm X) :
-    lift p=Finset.univ.noncommProd (fun b => injection b (p b))
-      (fun _ _ _ _ h => injection_commute h _ _) := by
-  have he (s : Finset B) (x : B × X) :
-      s.noncommProd (fun b => injection b (p b))
-        (fun _ _ _ _ h => injection_commute h _ _) x=
-          (x.1,if x.1∈s then p x.1 x.2 else x.2) := by
-    induction s using Finset.induction_on with
-    | empty =>
-      have hz : Finset.noncommProd (∅ : Finset B) (fun b => injection b (p b))
-          (fun _ _ _ _ h => injection_commute h _ _)=1 := Finset.noncommProd_empty _ _
-      rw [hz]; simp
-    | @insert a s ha ih =>
-      have hz : (insert a s).noncommProd (fun b => injection b (p b))
-          (fun _ _ _ _ h => injection_commute h _ _)=injection a (p a)*
-          s.noncommProd (fun b => injection b (p b))
-            (fun _ _ _ _ h => injection_commute h _ _) :=
-        Finset.noncommProd_insert_of_notMem _ _ _ _ ha
-      rw [hz,Equiv.Perm.mul_apply,ih,injection_apply]
-      by_cases h : x.1=a
-      · subst a; simp [ha]
-      · simp [h]
-  apply Equiv.ext
-  intro x
-  rw [he]
-  simp
-
-end Thorp.ParallelBlocks
-namespace Thorp.UnitaryFinite
-open scoped BigOperators Classical
-
-variable {G H : Type*} [Group G] [Group H]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-variable {Ω Ξ : Type*} [Fintype Ω] [Fintype Ξ]
-
-lemma continuousRepresentation_comp (ρ : Representation ℂ G V) (φ : H →* G) (h : H) :
-    continuousRepresentation (ρ.comp φ) h=continuousRepresentation ρ (φ h) := rfl
-
-lemma sampleOperator_comp (ρ : Representation ℂ G V) (φ : H →* G) (P : Ω → H) :
-    sampleOperator (ρ.comp φ) P=sampleOperator ρ (φ ∘ P) := rfl
-
-omit [FiniteDimensional ℂ V] in
-lemma unitary_comp (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (φ : H →* G) :
-    IsUnitary (ρ.comp φ) := fun h v w => hρ (φ h) v w
-
-lemma sampleOperator_commute (ρ : Representation ℂ G V) (P : Ω → G) (Q : Ξ → G)
-    (h : ∀ ω ξ, Commute (P ω) (Q ξ)) :
-    Commute (sampleOperator ρ P) (sampleOperator ρ Q) := by
-  unfold sampleOperator
-  apply Commute.smul_left
-  apply Commute.smul_right
-  apply Commute.sum_left
-  intro ω _
-  apply Commute.sum_right
-  intro ξ _
-  exact (h ω ξ).map (continuousRepresentation ρ)
-
-lemma sampleOperator_uniform_absorb [Fintype G] [Nonempty Ω]
-    (ρ : Representation ℂ G V) (P : Ω → G) :
-    sampleOperator ρ P*uniformOperator ρ=uniformOperator ρ := by
-  rw [sampleOperator_weighted,weighted_uniform_left,sampleLaw_sum,one_smul]
-
-end Thorp.UnitaryFinite
-namespace Thorp.ParallelBlocks
-open scoped BigOperators Classical
-open UnitaryFinite
-variable {B X Ω : Type*} [Fintype B] [DecidableEq B] [Fintype Ω]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-omit [Fintype B] [DecidableEq B] in
-lemma local_average_commute (ρ : Representation ℂ (Equiv.Perm (B × X)) V)
-    (P : B → Ω → Equiv.Perm X) (b c : B) :
-    Commute (sampleOperator (ρ.comp (injection b)) (P b))
-      (sampleOperator (ρ.comp (injection c)) (P c)) := by
-  by_cases h : b=c
-  · subst c; exact Commute.refl _
-  · rw [sampleOperator_comp,sampleOperator_comp]
-    exact sampleOperator_commute ρ _ _ (fun _ _ => injection_commute h _ _)
-
-lemma sampleOperator_parallel (ρ : Representation ℂ (Equiv.Perm (B × X)) V)
-    (P : B → Ω → Equiv.Perm X) :
-    sampleOperator ρ (fun ω : B → Ω => lift (fun b => P b (ω b)))=
-      CommutingBlocks.blocks (fun b => sampleOperator (ρ.comp (injection b)) (P b))
-        (local_average_commute ρ P) := by
-  let A (b : B) (w : Ω) := continuousRepresentation ρ (injection b (P b w))
-  have hc : ∀ ω : B → Ω, Pairwise fun b c => Commute (A b (ω b)) (A c (ω c)) := by
-    intro ω b c h
-    exact (injection_commute h _ _).map (continuousRepresentation ρ)
-  have havg : Pairwise fun b c => Commute ((Fintype.card Ω:ℂ)⁻¹ • ∑ w,A b w)
-      ((Fintype.card Ω:ℂ)⁻¹ • ∑ w,A c w) := by
-    intro b c _
-    exact local_average_commute ρ P b c
-  have he (ω : B → Ω) : continuousRepresentation ρ (lift (fun b => P b (ω b)))=
-      Finset.univ.noncommProd (fun b => A b (ω b)) (fun _ _ _ _ h => hc ω h) := by
-    rw [lift_product]
-    exact Finset.map_noncommProd _ _ _ (continuousRepresentation ρ)
-  unfold sampleOperator
-  simp_rw [he]
-  exact (FiniteAverage.ordered_average A hc havg).symm
-
-end Thorp.ParallelBlocks
-namespace Thorp
-open scoped BigOperators Classical
-
-noncomputable def splitPerm (r L : ℕ) : Equiv.Perm (Card (L+r)) ≃* Equiv.Perm (Card r × Card L) :=
-  (cardSplit r L).permCongrHom
-
-noncomputable def outerPerm (r L : ℕ) (hi : Card L × SwitchIndex r → Bool) :
-    Equiv.Perm (Card r × Card L) :=
-  (Equiv.prodComm (Card L) (Card r)).permCongrHom
-    (ParallelBlocks.lift (fun x => butterflyPerm r (decodeButterfly r (fun i => hi (x,i)))))
-
-lemma outerPerm_apply (r L : ℕ) (hi : Card L × SwitchIndex r → Bool) (x : Card r × Card L) :
-    outerPerm r L hi x=(butterflyPerm r (decodeButterfly r (fun i => hi (x.2,i))) x.1,x.2) := rfl
-
-lemma splitPerm_butterfly (r L : ℕ) (lo : Card r × SwitchIndex L → Bool)
-    (hi : Card L × SwitchIndex r → Bool) :
-    splitPerm r L (butterflyPerm (L+r) (decodeButterfly (L+r) (assembleBits r L lo hi)))=
-      outerPerm r L hi*ParallelBlocks.lift
-        (fun b => butterflyPerm L (decodeButterfly L (fun i => lo (b,i)))) := by
-  apply Equiv.ext
-  intro x
-  change cardSplit r L (butterflyPerm (L+r) (decodeButterfly (L+r) (assembleBits r L lo hi))
-    ((cardSplit r L).symm x))=_
-  rw [butterfly_split,Equiv.apply_symm_apply]
-  rfl
-
-lemma boundaryTree_node_raw (L r : ℕ) (U : Card (r+1) → Equiv.Perm (Card L))
-    (X : SwitchIndex (L+r+1) → Bool) :
-    boundaryTree L (r+1) U X=InputTree.node (fun y => X (Sum.inl y))
-      (fun b => boundaryTree L r (fun u => U (Fin.cons b u)) (fun i => X (Sum.inr (b,i)))) := by
-  conv_lhs => unfold boundaryTree
-  congr 1
-  funext b
-  cases b <;> simp only [Bool.false_eq_true,ite_false,ite_true] <;> rfl
-
-lemma boundaryOutput_node_raw (L r : ℕ) (Y : SwitchIndex (L+r+1) → Bool) (b : Card (r+1)) :
-    boundaryOutput L (r+1) Y b=
-      boundaryOutput L r (fun i => Y (Sum.inr (b 0,i))) (Fin.tail b) := by
-  conv_lhs => unfold boundaryOutput
-  cases b 0 <;> simp only [Bool.false_eq_true,ite_false,ite_true] <;> rfl
-
-lemma boundaryTree_split (L r : ℕ) (U : Card r → Equiv.Perm (Card L))
-    (lo : Card r × SwitchIndex L → Bool) (hi : Card L × SwitchIndex r → Bool) (x : Card (L+r)) :
-    cardSplit r L ((boundaryTree L r U (assembleBits r L lo hi)).perm x)=
-      let y := cardSplit r L x
-      let z := U y.1 y.2
-      (butterflyPerm r (decodeButterfly r (fun i => hi (z,i))) y.1,z) := by
-  induction r with
-  | zero =>
-    apply Prod.ext
-    · exact Subsingleton.elim _ _
-    · rfl
-  | succ r ih =>
-    let lo' : Card r × SwitchIndex L → Bool := fun j => lo (Fin.cons (x 0) j.1,j.2)
-    let hi' : Card L × SwitchIndex r → Bool := fun j => hi (j.1,Sum.inr (x 0,j.2))
-    let U' : Card r → Equiv.Perm (Card L) := fun b => U (Fin.cons (x 0) b)
-    have hh := ih U' lo' hi' (Fin.tail x)
-    let u := (boundaryTree L r U' (assembleBits r L lo' hi')).perm (Fin.tail x)
-    let y := cardSplit r L (Fin.tail x)
-    let z := U' y.1 y.2
-    let w := butterflyPerm r (decodeButterfly r (fun i => hi' (z,i))) y.1
-    change cardSplit r L u=(w,z) at hh
-    rw [boundaryTree_node_raw]
-    change ((Fin.cons (Bool.xor (x 0) (hi ((cardSplit r L u).2,Sum.inl (cardSplit r L u).1)))
-      ((cardSplit r L u).1) : Card (r+1)),(cardSplit r L u).2)=
-        ((Fin.cons (Bool.xor (x 0) (hi (z,Sum.inl w))) w : Card (r+1)),z)
-    rw [hh]
-
-lemma splitPerm_boundaryTree (L r : ℕ) (U : Card r → Equiv.Perm (Card L))
-    (lo : Card r × SwitchIndex L → Bool) (hi : Card L × SwitchIndex r → Bool) :
-    splitPerm r L (boundaryTree L r U (assembleBits r L lo hi)).perm=
-      outerPerm r L hi*ParallelBlocks.lift U := by
-  apply Equiv.ext
-  intro x
-  change cardSplit r L ((boundaryTree L r U (assembleBits r L lo hi)).perm
-    ((cardSplit r L).symm x))=_
-  rw [boundaryTree_split,Equiv.apply_symm_apply]
-  rfl
-
-lemma boundaryOutput_split (L r : ℕ) (lo : Card r × SwitchIndex L → Bool)
-    (hi : Card L × SwitchIndex r → Bool) (b : Card r) :
-    boundaryOutput L r (assembleBits r L lo hi) b=
-      butterflyPerm L (decodeButterfly L (fun i => lo (b,i))) := by
-  induction r with
-  | zero =>
-    rw [Subsingleton.elim b (fun j => Fin.elim0 j)]
-    rfl
-  | succ r ih =>
-    rw [boundaryOutput_node_raw]
-    change boundaryOutput L r (assembleBits r L (fun j => lo (Fin.cons (b 0) j.1,j.2))
-      (fun j => hi (j.1,Sum.inr (b 0,j.2)))) (Fin.tail b)=_
-    rw [ih]
-    simp only [Fin.cons_self_tail]
-
-
-abbrev OuterCoins (r L : ℕ) := Card L × SwitchIndex r → Bool
-abbrev LocalCoins (L : ℕ) := SwitchIndex L → Bool
-
-def parallelSplitCoinEquiv (r L : ℕ) :
-    ((Card r → LocalCoins L) × OuterCoins r L) ≃ LocalCoins (L+r) :=
-  (Equiv.prodCongr (Equiv.curry (Card r) (SwitchIndex L) Bool).symm (Equiv.refl _)).trans
-    (splitCoinEquiv r L)
-
-lemma parallelSplitCoinEquiv_apply (r L : ℕ) (c : Card r → LocalCoins L) (h : OuterCoins r L) :
-    parallelSplitCoinEquiv r L (c,h)=assembleBits r L (fun x => c x.1 x.2) h := by
-  exact splitCoinEquiv_apply _ _ _ _
-
-def pairAssembleEquiv (B C H : Type*) :
-    ((H × (B → C × C)) × H) ≃ (((B → C) × H) × ((B → C) × H)) where
-  toFun w := (((fun b => (w.1.2 b).1),w.1.1),((fun b => (w.1.2 b).2),w.2))
-  invFun w := ((w.1.2,fun b => (w.1.1 b,w.2.1 b)),w.2.2)
-  left_inv _w := rfl
-  right_inv _w := rfl
-
-def palindromeStageEquiv (r L : ℕ) :
-    ((OuterCoins r L × (Card r → BenesCoins L)) × OuterCoins r L) ≃ BenesCoins (L+r) :=
-  (pairAssembleEquiv _ _ _).trans
-    (Equiv.prodCongr (parallelSplitCoinEquiv r L) (parallelSplitCoinEquiv r L))
-
-lemma palindromeStageEquiv_apply (r L : ℕ)
-    (w : (OuterCoins r L × (Card r → BenesCoins L)) × OuterCoins r L) :
-    palindromeStageEquiv r L w=
-      (assembleBits r L (fun x => (w.1.2 x.1).1 x.2) w.1.1,
-       assembleBits r L (fun x => (w.1.2 x.1).2 x.2) w.2) := by
-  simp only [palindromeStageEquiv,Equiv.trans_apply,Equiv.prodCongr_apply,pairAssembleEquiv,
-    Equiv.coe_fn_mk,Prod.map_apply,parallelSplitCoinEquiv_apply]
-
-def modifiedAssembleEquiv (B C H U : Type*) :
-    ((H × (B → U × (C × C))) × H) ≃ (B → U) × (((B → C) × H) × ((B → C) × H)) where
-  toFun w := ((fun b => (w.1.2 b).1),
-    (((fun b => (w.1.2 b).2.1),w.1.1),((fun b => (w.1.2 b).2.2),w.2)))
-  invFun w := ((w.2.1.2,fun b => (w.1 b,(w.2.1.1 b,w.2.2.1 b))),w.2.2.2)
-  left_inv _w := rfl
-  right_inv _w := rfl
-
-end Thorp
-
-end ThorpNine.Tail
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Tail.Representative
+
+namespace OAI
+
+namespace ThorpNine.Tail
+
+namespace Thorp.CommutingBlocks
+open scoped BigOperators Classical
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
+variable {ι : Type*} [Fintype ι]
+variable [DecidableEq ι]
+variable (P : ι → V →L[ℂ] V) (hcomm : ∀ i j, Commute (P i) (P j))
+include hcomm
+omit [CompleteSpace V] [Fintype ι] in
+lemma sector_sum (s : Finset ι) : ∑ E∈s.powerset, sector P hcomm s E=1 := by
+  induction s using Finset.induction_on with
+  | empty =>
+    simp only [Finset.powerset_empty,Finset.sum_singleton]
+    exact Finset.noncommProd_empty _ _
+  | @insert a s ha ih =>
+    rw [Finset.sum_powerset_insert ha]
+    have h₁ (E : Finset ι) (hE : E∈s.powerset) :
+        sector P hcomm (insert a s) E=P a*sector P hcomm s E := by
+      rw [sector_insert P hcomm s E a ha]
+      congr 1
+      exact ite_eq_right (fun h => ha ((Finset.mem_powerset.mp hE) h))
+    have h₂ (E : Finset ι) (_hE : E∈s.powerset) :
+        sector P hcomm (insert a s) (insert a E)=(1-P a)*sector P hcomm s E := by
+      rw [sector_insert P hcomm s (insert a E) a ha]
+      simp only [choice,Finset.mem_insert_self,↓reduceIte]
+      congr 1
+      apply sector_congr
+      intro i hi
+      simp [ne_of_mem_of_not_mem hi ha]
+    rw [Finset.sum_congr rfl h₁,Finset.sum_congr rfl h₂,
+      ←Finset.mul_sum,←Finset.mul_sum,ih]
+    simp
+
+omit [Fintype ι] in
+lemma sector_absorb_factor (s E : Finset ι) (i : ι) (hi : i∈s)
+    (hP : ∀ i, IsStarProjection (P i)) :
+    choice P E i*sector P hcomm s E=sector P hcomm s E := by
+  have hc : IsIdempotentElem (choice P E i) := by
+    unfold choice
+    split_ifs
+    · exact (hP i).one_sub.isIdempotentElem
+    · exact (hP i).isIdempotentElem
+  have hsplit := Finset.mul_noncommProd_erase s hi (choice P E)
+    (fun i _ j _ _ => choice_commute P hcomm E E i j)
+  change choice P E i*sector P hcomm s E=sector P hcomm s E
+  change _=sector P hcomm s E at hsplit
+  nth_rw 1 [←hsplit]
+  rw [←mul_assoc,hc.eq,hsplit]
+
+omit [Fintype ι] in
+lemma sector_orthogonal (s E F : Finset ι) (hE : E⊆s) (hF : F⊆s) (hEF : E≠F)
+    (hP : ∀ i, IsStarProjection (P i)) :
+    sector P hcomm s E*sector P hcomm s F=0 := by
+  have hdiff : ∃ i∈s, ¬(i∈E ↔ i∈F) := by
+    by_contra h
+    push Not at h
+    exact hEF (Finset.ext fun i => by
+      by_cases hi : i∈s
+      · exact h i hi
+      · exact iff_of_false (fun h => hi (hE h)) (fun h => hi (hF h)))
+  obtain ⟨i,hi,hdiff⟩ := hdiff
+  have hz : choice P E i*choice P F i=0 := by
+    by_cases he : i∈E <;> by_cases hf : i∈F
+    · exact (hdiff (iff_of_true he hf)).elim
+    · simpa only [choice,ite_eq_left he,ite_eq_right hf] using (hP i).one_sub_mul_self
+    · simpa only [choice,ite_eq_right he,ite_eq_left hf] using (hP i).mul_one_sub_self
+    · exact (hdiff (iff_of_false he hf)).elim
+  have hEi := sector_absorb_factor P hcomm s E i hi hP
+  have hFi := sector_absorb_factor P hcomm s F i hi hP
+  calc
+    _ = sector P hcomm s E*(choice P E i*choice P F i)*sector P hcomm s F := by
+      simp only [mul_assoc]
+      rw [hFi]
+      rw [←mul_assoc,(sector_factor_commute P hcomm s E E i).eq.symm,hEi]
+    _ = 0 := by rw [hz,mul_zero,zero_mul]
+
+noncomputable def blockProduct (A : ι → V →L[ℂ] V)
+    (hA : ∀ i j, Commute (A i) (A j)) (s : Finset ι) : V →L[ℂ] V :=
+  s.noncommProd A (fun i _ j _ _ => hA i j)
+
+omit hcomm [Fintype ι] in
+lemma norm_noncommProd_le (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
+    (s : Finset ι) : ‖blockProduct A hA s‖≤∏ i∈s,‖A i‖ := by
+  induction s using Finset.induction_on with
+  | empty => exact ContinuousLinearMap.norm_id_le
+  | @insert a s ha ih =>
+    have he : blockProduct A hA (insert a s)=A a*blockProduct A hA s :=
+      Finset.noncommProd_insert_of_notMem _ _ _ _ ha
+    rw [he,Finset.prod_insert ha]
+    exact (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_left ih (norm_nonneg _))
+
+noncomputable def blocks (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j)) : V →L[ℂ] V :=
+  blockProduct A hA Finset.univ
+
+omit hcomm in
+lemma blocks_norm_le_one (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
+    (hn : ∀ i,‖A i‖≤1) : ‖blocks A hA‖≤1 :=
+  (norm_noncommProd_le A hA Finset.univ).trans (Finset.prod_le_one₀ (fun _ _ => norm_nonneg _) (fun i _ => hn i))
+
+omit [CompleteSpace V] in
+lemma blocks_sector_commute (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
+    (hAP : ∀ i j, Commute (A i) (P j)) (E : Finset ι) :
+    Commute (blocks A hA) (sector P hcomm Finset.univ E) := by
+  unfold blocks blockProduct
+  apply (Finset.noncommProd_commute _ _ (fun i _ j _ _ => hA i j) _ _).symm
+  intro i _
+  unfold sector
+  apply (Finset.noncommProd_commute _ _ (fun i _ j _ _ => choice_commute P hcomm E E i j) _ _).symm
+  intro j _
+  unfold choice
+  split_ifs
+  · exact (Commute.one_right _).sub_right (hAP i j)
+  · exact hAP i j
+
+lemma blocks_sector_norm (hP : ∀ i, IsStarProjection (P i))
+    (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
+    (hAP : ∀ i j, Commute (A i) (P j)) (hn : ∀ i,‖A i‖≤1)
+    (θ : ℝ) (_hθ : 0≤θ) (hgap : ∀ i,‖A i*(1-P i)‖≤θ) (E : Finset ι) :
+    ‖blocks A hA*sector P hcomm Finset.univ E‖≤θ^E.card := by
+  have hAC (i j : ι) : Commute (A i) (choice P E j) := by
+    unfold choice
+    split_ifs
+    · exact (Commute.one_right _).sub_right (hAP i j)
+    · exact hAP i j
+  have hprod := Finset.noncommProd_mul_distrib (s := Finset.univ) A (choice P E)
+    (fun i _ j _ _ => hA i j) (fun i _ j _ _ => choice_commute P hcomm E E i j)
+    (fun i _ j _ _ => (hAC j i).symm)
+  change _=blocks A hA*sector P hcomm Finset.univ E at hprod
+  rw [←hprod]
+  have hc (i j : ι) : Commute ((A*choice P E) i) ((A*choice P E) j) :=
+    ((hA i j).mul_right (hAC i j)).mul_left ((hAC j i).symm.mul_right (choice_commute P hcomm E E i j))
+  apply (norm_noncommProd_le _ hc Finset.univ).trans
+  calc
+    _ ≤ ∏ i : ι, if i∈E then θ else 1 := by
+      apply Finset.prod_le_prod₀ (fun i _ => norm_nonneg _)
+      intro i _
+      dsimp only [Pi.mul_apply]
+      unfold choice
+      split_ifs with hi
+      · exact hgap i
+      · exact (norm_mul_le _ _).trans (by nlinarith [hn i,(hP i).norm_le (P i),norm_nonneg (A i),norm_nonneg (P i)])
+    _ = _ := by rw [Fintype.prod_ite_mem,Finset.prod_const]
+
+noncomputable def coarse (E : Finset ι) : V →L[ℂ] V :=
+  Eᶜ.noncommProd P (fun i _ j _ _ => hcomm i j)
+
+lemma coarse_projection (hP : ∀ i, IsStarProjection (P i)) (E : Finset ι) :
+    IsStarProjection (coarse P hcomm E) := by
+  have he : coarse P hcomm E=sector P hcomm Eᶜ ∅ := by
+    apply Finset.noncommProd_congr rfl
+    intro i _
+    simp [choice]
+  rw [he]
+  exact sector_projection P hcomm hP _ _
+
+lemma sector_coarse (hP : ∀ i, IsStarProjection (P i)) (E : Finset ι) :
+    sector P hcomm Finset.univ E*coarse P hcomm E=sector P hcomm Finset.univ E := by
+  have hpi (i : ι) (hi : i∈Eᶜ) : sector P hcomm Finset.univ E*P i=sector P hcomm Finset.univ E := by
+    have h := sector_absorb_factor P hcomm Finset.univ E i (Finset.mem_univ i) hP
+    rw [(sector_factor_commute P hcomm Finset.univ E E i).eq] at h
+    simpa only [choice,ite_eq_right (Finset.mem_compl.mp hi)] using h
+  have hp (s : Finset ι) (hs : ∀ i∈s,
+      sector P hcomm Finset.univ E*P i=sector P hcomm Finset.univ E) :
+      sector P hcomm Finset.univ E*blockProduct P hcomm s=sector P hcomm Finset.univ E := by
+    induction s using Finset.induction_on with
+    | empty => exact mul_one _
+    | @insert a s ha ih =>
+      have he : blockProduct P hcomm (insert a s)=P a*blockProduct P hcomm s :=
+        Finset.noncommProd_insert_of_notMem _ _ _ _ ha
+      rw [he,←mul_assoc,hs a (Finset.mem_insert_self _ _)]
+      exact ih (fun i hi => hs i (Finset.mem_insert_of_mem hi))
+  exact hp Eᶜ hpi
+
+theorem blocks_exception_bound (hP : ∀ i, IsStarProjection (P i))
+    (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
+    (hAP : ∀ i j, Commute (A i) (P j)) (hn : ∀ i,‖A i‖≤1)
+    (θ : ℝ) (hθ : 0≤θ) (hθ1 : θ≤1) (hgap : ∀ i,‖A i*(1-P i)‖≤θ)
+    (B : V →L[ℂ] V) (hB : ‖B‖≤1) (n : ℕ) :
+    ‖B.adjoint*blocks A hA*B‖≤θ^n+
+      ∑ E : Finset ι, if E.card<n then ‖B.adjoint*coarse P hcomm E*B‖ else 0 := by
+  let T : Finset ι → V →L[ℂ] V := sector P hcomm Finset.univ
+  have hsum : ∑ E,T E=1 := by
+    simpa only [Finset.powerset_univ] using sector_sum P hcomm Finset.univ
+  have hbound := resolution_exception_bound T (coarse P hcomm)
+    (fun E => sector_projection P hcomm hP _ E)
+    (fun E F hEF => sector_orthogonal P hcomm _ E F (Finset.subset_univ _) (Finset.subset_univ _) hEF hP)
+    hsum (coarse_projection P hcomm hP) (sector_coarse P hcomm hP)
+    (blocks A hA) B (blocks_sector_commute P hcomm A hA hAP)
+    (blocks_norm_le_one A hA hn) hB (Finset.univ.filter fun E : Finset ι => E.card<n)
+    (θ^n) (pow_nonneg hθ _) (by
+      intro E hE
+      have hEn : n≤E.card := by simpa using hE
+      exact (blocks_sector_norm P hcomm hP A hA hAP hn θ hθ hgap E).trans
+        (pow_le_pow_of_le_one hθ hθ1 hEn))
+  simpa only [Finset.sum_filter] using hbound
+
+
+end Thorp.CommutingBlocks
+
+namespace Thorp.UnitaryFinite
+open scoped BigOperators Classical
+
+variable {G : Type*} [Group G] [Fintype G]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+variable {Ω Ξ : Type*} [Fintype Ω] [Fintype Ξ]
+
+lemma sampleOperator_weighted (ρ : Representation ℂ G V) (P : Ω → G) :
+    sampleOperator ρ P=weightedOperator ρ (sampleLaw P) := by
+  unfold sampleOperator weightedOperator sampleLaw finiteMean
+  simp only [Complex.ofReal_sum,div_eq_mul_inv,Finset.sum_mul,Finset.sum_smul]
+  rw [Finset.sum_comm]
+  simp only [Finset.smul_sum]
+  apply Finset.sum_congr rfl
+  intro ω _
+  calc
+    _ = ∑ g, if P ω=g then (Fintype.card Ω:ℂ)⁻¹ • continuousRepresentation ρ g else 0 := by
+      symm
+      simp only [Finset.sum_ite_eq,Finset.mem_univ,↓reduceIte]
+    _ = _ := by
+      apply Finset.sum_congr rfl
+      intro g _
+      split_ifs <;> simp_all
+
+lemma sampleOperator_norm_le [Nonempty Ω] (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (P : Ω → G) :
+    ‖sampleOperator ρ P‖≤1 := by
+  rw [sampleOperator_weighted]
+  exact (weightedOperator_norm_le ρ hρ _ (sampleLaw_nonneg _)).trans_eq (sampleLaw_sum P)
+
+omit [Fintype G] in
+lemma sampleOperator_equiv (ρ : Representation ℂ G V) (P : Ω → G) (e : Ξ ≃ Ω) :
+    sampleOperator ρ (P ∘ e)=sampleOperator ρ P := by
+  simp only [sampleOperator,Fintype.card_congr e,Function.comp_apply]
+  rw [Equiv.sum_comp e (fun ω => continuousRepresentation ρ (P ω))]
+
+omit [Fintype G] in
+lemma sampleOperator_prod (ρ : Representation ℂ G V) (P : Ω → G) (Q : Ξ → G) :
+    sampleOperator ρ (fun ω : Ω × Ξ => P ω.1*Q ω.2)=sampleOperator ρ P*sampleOperator ρ Q := by
+  simp only [sampleOperator,Fintype.card_prod,Nat.cast_mul,mul_inv_rev,map_mul,Fintype.sum_prod_type,
+    Finset.sum_mul,Finset.mul_sum,smul_mul_smul,Finset.smul_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro ω _
+  apply Finset.sum_congr rfl
+  intro ξ _
+  rw [mul_comm]
+
+omit [Fintype G] in
+lemma sampleOperator_inv (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (P : Ω → G) :
+    sampleOperator ρ (fun ω => (P ω)⁻¹)=(sampleOperator ρ P).adjoint := by
+  simp only [sampleOperator,map_smulₛₗ,map_sum,map_inv₀,map_natCast,
+    continuousRepresentation_adjoint ρ hρ]
+
+lemma sampleOperator_palindrome (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V)
+    (hρ : IsUnitary ρ) :
+    sampleOperator ρ (palindromePerm d)=
+      sampleOperator ρ (fun ω => butterflyPerm d (decodeButterfly d ω))*
+      (sampleOperator ρ (fun ω => butterflyPerm d (decodeButterfly d ω))).adjoint := by
+  rw [←sampleOperator_inv ρ hρ,←sampleOperator_prod]
+  rfl
+
+omit [Fintype G] in
+lemma sampleOperator_one [Nonempty Ω] (ρ : Representation ℂ G V) :
+    sampleOperator ρ (fun _ : Ω => 1)=1 := by
+  simp only [sampleOperator,map_one,Finset.sum_const,Finset.card_univ,←Nat.cast_smul_eq_nsmul ℂ,
+    smul_smul]
+  rw [inv_mul_cancel₀ (by exact_mod_cast Fintype.card_ne_zero),one_smul]
+
+lemma sampleOperator_uniform (ρ : Representation ℂ G V) : sampleOperator ρ (id : G → G)=uniformOperator ρ := by
+  simp only [sampleOperator,uniformOperator,weightedOperator,Complex.ofReal_inv,Complex.ofReal_natCast,
+    Finset.smul_sum,id_eq]
+
+lemma sampleOperator_uniform_mul [Nonempty Ω] (ρ : Representation ℂ G V) (P : Ω → G) :
+    sampleOperator ρ (fun ω : Ω × G => P ω.1*ω.2⁻¹)=uniformOperator ρ := by
+  rw [sampleOperator_prod]
+  have he : sampleOperator ρ (fun g : G => g⁻¹)=uniformOperator ρ := by
+    rw [←sampleOperator_uniform]
+    exact sampleOperator_equiv ρ id (Equiv.inv G)
+  rw [he,sampleOperator_weighted,weighted_uniform_left,sampleLaw_sum,one_smul]
+
+end Thorp.UnitaryFinite
+
+namespace Thorp.FiniteAverage
+open scoped BigOperators Classical
+variable {I Ω A : Type*} [Fintype I] [DecidableEq I] [Fintype Ω]
+    [Ring A] [Algebra ℂ A]
+
+noncomputable def orderedProduct : MultilinearMap ℂ (fun _ : I => A) A :=
+  MultilinearMap.domDomCongr (Fintype.equivFin I).symm
+    (MultilinearMap.mkPiAlgebraFin ℂ (Fintype.card I) A)
+
+lemma orderedProduct_eq (f : I → A) (hc : Pairwise fun i j => Commute (f i) (f j)) :
+    orderedProduct f=Finset.univ.noncommProd f (fun _ _ _ _ hij => hc hij) := by
+  unfold orderedProduct
+  rw [MultilinearMap.domDomCongr_apply,MultilinearMap.mkPiAlgebraFin_apply]
+  let l := List.ofFn (Fintype.equivFin I).symm
+  have hl : l.Nodup := List.nodup_ofFn.mpr (Fintype.equivFin I).symm.injective
+  have hu : l.toFinset=Finset.univ := by
+    ext i
+    simp only [List.mem_toFinset,List.mem_ofFn,Finset.mem_univ,iff_true,l]
+    exact (Fintype.equivFin I).symm.surjective i
+  have he := Finset.noncommProd_toFinset l f (by intro i hi j hj hij; exact hc hij) hl
+  rw [hu] at he
+  rw [he]
+  congr 1
+  exact List.map_ofFn.symm
+
+lemma multilinear_average (F : MultilinearMap ℂ (fun _ : I => A) A) (x : I → Ω → A) :
+    F (fun i => (Fintype.card Ω:ℂ)⁻¹ • ∑ ω,x i ω)=
+      (Fintype.card (I → Ω):ℂ)⁻¹ • ∑ ω : I → Ω,F (fun i => x i (ω i)) := by
+  rw [F.map_smul_univ,F.map_sum]
+  simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fun,Nat.cast_pow,inv_pow]
+
+lemma ordered_average (x : I → Ω → A)
+    (hc : ∀ ω, Pairwise fun i j => Commute (x i (ω i)) (x j (ω j)))
+    (havg : Pairwise fun i j => Commute ((Fintype.card Ω:ℂ)⁻¹ • ∑ ω,x i ω)
+      ((Fintype.card Ω:ℂ)⁻¹ • ∑ ω,x j ω)) :
+    (Finset.univ.noncommProd (fun i => (Fintype.card Ω:ℂ)⁻¹ • ∑ ω,x i ω)
+      (fun _ _ _ _ hij => havg hij))=
+    (Fintype.card (I → Ω):ℂ)⁻¹ • ∑ ω : I → Ω,
+      Finset.univ.noncommProd (fun i => x i (ω i)) (fun _ _ _ _ hij => hc ω hij) := by
+  rw [←orderedProduct_eq _ havg,multilinear_average]
+  simp only [orderedProduct_eq _ (hc _)]
+
+end Thorp.FiniteAverage
+namespace Thorp.FiniteCharacters
+open Module
+open scoped BigOperators Classical
+
+variable {G ι : Type*} [Group G] [Fintype G] [Fintype ι]
+variable (V : ι → Type*) [∀ i, AddCommGroup (V i)] [∀ i, Module ℂ (V i)]
+  [∀ i, FiniteDimensional ℂ (V i)]
+variable (ρ : ∀ i, Representation ℂ G (V i)) [∀ i, Representation.IsIrreducible (ρ i)]
+variable (hneq : ∀ i j, i≠j → ¬Nonempty (Representation.Equiv (ρ i) (ρ j)))
+  (hcard : Nat.card (ConjClasses G)≤Fintype.card ι)
+include hneq hcard
+
+lemma character_expansion (f : ConjClasses G → ℂ) :
+    f=∑ i, characterDual (ρ i) f • characterClass (ρ i) := by
+  let : Fintype (ConjClasses G) := Fintype.ofFinite _
+  have horth (i j : ι) : characterDual (ρ i) (characterClass (ρ j))=if i=j then 1 else 0 := by
+    rw [characterDual_characterClass]
+    by_cases h : i=j
+    · subst j; rw [ite_eq_left ⟨Representation.Equiv.refl _⟩,ite_eq_left rfl]
+    · rw [ite_eq_right h,ite_eq_right (hneq i j h)]
+  have hLI : LinearIndependent ℂ (fun i => characterClass (ρ i)) := by
+    apply LinearIndependent.of_pairwise_dual_eq_zero_one _ (fun i => characterDual (ρ i))
+    · intro i j hij; exact (horth i j).trans (ite_eq_right hij)
+    · intro i; simpa using horth i i
+  have heq : Fintype.card ι=Module.finrank ℂ (ConjClasses G → ℂ) := by
+    apply Nat.le_antisymm hLI.fintype_card_le_finrank
+    simpa only [Module.finrank_pi,Module.finrank_self,Finset.sum_const,Finset.card_univ,
+      smul_eq_mul,mul_one,Nat.card_eq_fintype_card] using hcard
+  let b := Basis.mk hLI (by rw [hLI.span_eq_top_of_card_eq_finrank' heq])
+  have hb (i : ι) : b i=characterClass (ρ i) := Basis.mk_apply _ _ _
+  have hd (i : ι) : b.coord i=characterDual (ρ i) := by
+    apply b.ext
+    intro j
+    rw [hb,horth]
+    simp only [←hb,Basis.coord_apply,Basis.repr_self,Finsupp.single_apply]
+    split_ifs <;> simp_all
+  have h := b.sum_repr f
+  simpa only [hb,show ∀ i,b.repr f i=characterDual (ρ i) f from fun i => congrArg (fun l => l f) (hd i)] using h.symm
+
+theorem sum_dimension_sq : ∑ i, (Module.finrank ℂ (V i))^2=Fintype.card G := by
+  let f : ConjClasses G → ℂ := fun c => if c=ConjClasses.mk (1:G) then 1 else 0
+  have hf (g : G) : f (ConjClasses.mk g)=if g=1 then 1 else 0 := by
+    simp only [f,ConjClasses.mk_eq_mk_iff_isConj,isConj_one_left]
+  have hd (i : ι) : characterDual (ρ i) f=(Fintype.card G:ℂ)⁻¹*(Module.finrank ℂ (V i):ℂ) := by
+    change (Nat.card G:ℂ)⁻¹ * ∑ g, f (ConjClasses.mk g)*(ρ i).character g⁻¹=_
+    simp only [hf,ite_mul,one_mul,zero_mul,Finset.sum_ite_eq',Finset.mem_univ,↓reduceIte,inv_one,
+      Representation.char_one,Nat.card_eq_fintype_card]
+  have he := congrFun (character_expansion V ρ hneq hcard f) (ConjClasses.mk (1:G))
+  rw [hf,ite_eq_left rfl] at he
+  simp only [Finset.sum_apply,Pi.smul_apply,smul_eq_mul,hd,characterClass_mk,Representation.char_one] at he
+  have hN : (Fintype.card G:ℂ)≠0 := by exact_mod_cast Fintype.card_ne_zero
+  have hs : (∑ i,(Module.finrank ℂ (V i):ℂ)^2)=(Fintype.card G:ℂ) := by
+    calc
+      _ = (Fintype.card G:ℂ) * ∑ i,(Fintype.card G:ℂ)⁻¹ *
+          (Module.finrank ℂ (V i):ℂ)*(Module.finrank ℂ (V i):ℂ) := by
+        rw [Finset.mul_sum]
+        apply Finset.sum_congr rfl
+        intro i _
+        field_simp
+      _ = _ := by rw [←he,mul_one]
+  exact_mod_cast hs
+
+end Thorp.FiniteCharacters
+
+namespace Thorp.UnitaryFinite
+open scoped BigOperators ComplexConjugate Classical
+
+variable {G : Type*} [Group G] [Fintype G]
+variable {V W : Type*} [NormedAddCommGroup V] [NormedAddCommGroup W]
+  [InnerProductSpace ℂ V] [InnerProductSpace ℂ W]
+  [FiniteDimensional ℂ V] [FiniteDimensional ℂ W]
+
+omit [FiniteDimensional ℂ V] [FiniteDimensional ℂ W] in
+lemma averageMap_zero (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
+    [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ]
+    (hne : ¬Nonempty (Representation.Equiv ρ σ)) (A : V →ₗ[ℂ] W) :
+    averageMap ρ σ A=0 := by
+  rcases Representation.IsIrreducible.bijective_or_eq_zero (averageMap ρ σ A) with h|h
+  · exact (hne ⟨Representation.IntertwiningMap.ofBijective _ h⟩).elim
+  · exact h
+
+omit [FiniteDimensional ℂ V] [FiniteDimensional ℂ W] in
+lemma coefficient_cross_orthogonality (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
+    [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ]
+    (hρ : IsUnitary ρ) (hne : ¬Nonempty (Representation.Equiv ρ σ))
+    (u z : W) (v w : V) :
+    ∑ g, inner ℂ u (σ g z)*inner ℂ (ρ g w) v=0 := by
+  have he := averageMap_zero ρ σ hne (InnerProductSpace.rankOne ℂ z w).toLinearMap
+  have hv := congrArg (fun T : Representation.IntertwiningMap ρ σ => inner ℂ u (T v)) he
+  change inner ℂ u ((∑ g, (σ g).comp ((InnerProductSpace.rankOne ℂ z w).toLinearMap.comp (ρ g⁻¹))) v)=_ at hv
+  simpa only [LinearMap.sum_apply,LinearMap.comp_apply,ContinuousLinearMap.coe_coe,
+    InnerProductSpace.rankOne_apply,map_smul,unitary_inner_inv ρ hρ,inner_sum,
+    inner_smul_right,Representation.IntertwiningMap.coe_zero,Pi.zero_apply,inner_zero_right,mul_comm] using hv
+
+end Thorp.UnitaryFinite
+namespace Thorp.UnitaryFinite
+open scoped BigOperators ComplexConjugate Classical
+
+variable {G : Type*} [Group G] [Fintype G]
+
+def regularAction : G →* Equiv.Perm G where
+  toFun := Equiv.mulLeft
+  map_one' := by ext; simp
+  map_mul' _ _ := by ext; simp
+
+omit [Fintype G] in
+lemma regular_transitive (x : G) : ∃ g,regularAction g (1:G)=x := ⟨x,mul_one x⟩
+
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+omit [Fintype G] [FiniteDimensional ℂ V] in
+lemma regular_fixed (ρ : Representation ℂ G V) (w : V) : IsFixed regularAction (1:G) ρ w := by
+  intro g hg
+  have he : g=1 := by simpa only [regularAction,MonoidHom.coe_mk,OneHom.coe_mk,Equiv.coe_mulLeft,mul_one] using hg
+  subst g
+  simp
+
+noncomputable def regularCoefficient (ρ : Representation ℂ G V) (w v : V) : EuclideanSpace ℂ G :=
+  normalizedCoefficient regularAction 1 regular_transitive ρ w v
+
+omit [FiniteDimensional ℂ V] in
+lemma regularCoefficient_apply (ρ : Representation ℂ G V) (w v : V) (g : G) :
+    regularCoefficient ρ w v g = (Real.sqrt ((Module.finrank ℂ V:ℝ)/(Fintype.card G:ℝ)):ℂ)*inner ℂ (ρ g w) v := by
+  unfold regularCoefficient normalizedCoefficient
+  simp only [PiLp.smul_apply,smul_eq_mul]
+  congr 1
+  have he := coefficient_at regularAction 1 regular_transitive ρ w (regular_fixed ρ w) v g
+  simpa only [regularAction,MonoidHom.coe_mk,OneHom.coe_mk,Equiv.coe_mulLeft,mul_one] using he
+
+lemma regularCoefficient_inner (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
+    (hρ : IsUnitary ρ) (w z v t : V) :
+    inner ℂ (regularCoefficient ρ w v) (regularCoefficient ρ z t)=inner ℂ z w*inner ℂ v t :=
+  normalizedCoefficient_inner regularAction 1 regular_transitive ρ hρ w z
+    (regular_fixed ρ w) (regular_fixed ρ z) v t
+
+variable {W : Type*} [NormedAddCommGroup W] [InnerProductSpace ℂ W] [FiniteDimensional ℂ W]
+
+omit [FiniteDimensional ℂ V] [FiniteDimensional ℂ W] in
+lemma regularCoefficient_cross (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
+    [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ]
+    (hσ : IsUnitary σ) (hne : ¬Nonempty (Representation.Equiv σ ρ))
+    (w v : V) (z t : W) : inner ℂ (regularCoefficient ρ w v) (regularCoefficient σ z t)=0 := by
+  rw [PiLp.inner_apply]
+  simp only [RCLike.inner_apply,regularCoefficient_apply,map_mul,Complex.conj_ofReal,
+    inner_conj_symm]
+  have he := coefficient_cross_orthogonality σ ρ hσ hne v w t z
+  calc
+    _ = ((Real.sqrt ((Module.finrank ℂ V:ℝ)/(Fintype.card G:ℝ)):ℂ)*
+        (Real.sqrt ((Module.finrank ℂ W:ℝ)/(Fintype.card G:ℝ)):ℂ)) *
+        ∑ g,inner ℂ v (ρ g w)*inner ℂ (σ g z) t := by
+      rw [Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro g _
+      ring
+    _ = 0 := by rw [he,mul_zero]
+
+end Thorp.UnitaryFinite
+
+namespace Thorp.FiniteFourier
+open scoped BigOperators ComplexConjugate Classical
+open UnitaryFinite
+variable {G ι : Type*} [Group G] [Fintype G] [Fintype ι]
+variable (V : ι → Type*) [∀ i, NormedAddCommGroup (V i)] [∀ i, InnerProductSpace ℂ (V i)]
+  [∀ i, FiniteDimensional ℂ (V i)]
+variable (ρ : ∀ i, Representation ℂ G (V i)) [∀ i, Representation.IsIrreducible (ρ i)]
+variable (hρ : ∀ i, IsUnitary (ρ i))
+variable (hneq : ∀ i j, i≠j → ¬Nonempty (Representation.Equiv (ρ i) (ρ j)))
+  (hcard : Nat.card (ConjClasses G)≤Fintype.card ι)
+
+
+abbrev CoefficientIndex := Σ i,Fin (Module.finrank ℂ (V i)) × Fin (Module.finrank ℂ (V i))
+
+noncomputable def coefficientFamily (a : CoefficientIndex V) : EuclideanSpace ℂ G :=
+  regularCoefficient (ρ a.1) (stdOrthonormalBasis ℂ (V a.1) a.2.1) (stdOrthonormalBasis ℂ (V a.1) a.2.2)
+
+omit [Fintype ι] in
+include hρ hneq in
+lemma coefficientFamily_orthonormal : Orthonormal ℂ (coefficientFamily V ρ) := by
+  rw [orthonormal_iff_ite]
+  rintro ⟨i,a,b⟩ ⟨j,c,d⟩
+  dsimp only [coefficientFamily]
+  by_cases hij : i=j
+  · subst j
+    rw [regularCoefficient_inner _ (hρ i)]
+    simp only [(stdOrthonormalBasis ℂ (V i)).inner_eq_ite]
+    by_cases hac : a=c <;> by_cases hbd : b=d <;> simp [hac,hbd,eq_comm]
+  · rw [regularCoefficient_cross _ _ (hρ j) (hneq j i (Ne.symm hij))]
+    rw [ite_eq_right (by intro h; exact hij (congrArg Sigma.fst h))]
+
+include ρ hneq hcard in
+lemma coefficientIndex_card : Fintype.card (CoefficientIndex V)=Fintype.card G := by
+  simp only [CoefficientIndex,Fintype.card_sigma,Fintype.card_prod,Fintype.card_fin,←pow_two]
+  exact FiniteCharacters.sum_dimension_sq V ρ hneq hcard
+
+noncomputable def coefficientBasis : OrthonormalBasis (CoefficientIndex V) ℂ (EuclideanSpace ℂ G) :=
+  OrthonormalBasis.mk (coefficientFamily_orthonormal V ρ hρ hneq)
+    (by rw [(coefficientFamily_orthonormal V ρ hρ hneq).linearIndependent.span_eq_top_of_card_eq_finrank'
+          (by rw [coefficientIndex_card V ρ hneq hcard,finrank_euclideanSpace])])
+
+lemma coefficientBasis_apply (a : CoefficientIndex V) :
+    coefficientBasis V ρ hρ hneq hcard a=coefficientFamily V ρ a := congrFun (OrthonormalBasis.coe_mk _ _) a
+
+include hρ hneq hcard in
+
+theorem parseval (f : EuclideanSpace ℂ G) :
+    ∑ a : CoefficientIndex V, ‖inner ℂ (coefficientFamily V ρ a) f‖^2=‖f‖^2 := by
+  simpa only [coefficientBasis_apply] using
+    (coefficientBasis V ρ hρ hneq hcard).sum_sq_norm_inner_right f
+
+
+noncomputable def realVector (p : G → ℝ) : EuclideanSpace ℂ G := WithLp.toLp 2 (fun g => (p g:ℂ))
+
+omit [Fintype ι] [∀ i, Representation.IsIrreducible (ρ i)] in
+lemma coefficient_realVector (p : G → ℝ) (a : CoefficientIndex V) :
+    inner ℂ (coefficientFamily V ρ a) (realVector p)=
+      (Real.sqrt ((Module.finrank ℂ (V a.1):ℝ)/(Fintype.card G:ℝ)):ℂ)*
+      inner ℂ (stdOrthonormalBasis ℂ (V a.1) a.2.2)
+        (weightedOperator (ρ a.1) p (stdOrthonormalBasis ℂ (V a.1) a.2.1)) := by
+  simp only [PiLp.inner_apply,coefficientFamily,regularCoefficient_apply,realVector,
+    RCLike.inner_apply,map_mul,Complex.conj_ofReal,inner_conj_symm,
+    weightedOperator,sum_apply,smul_apply,
+    continuousRepresentation_apply,inner_sum,inner_smul_right,Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro g _
+  ring
+
+omit [Fintype ι] [∀ i, Representation.IsIrreducible (ρ i)] in
+lemma coefficient_realVector_sq (p : G → ℝ) (a : CoefficientIndex V) :
+    ‖inner ℂ (coefficientFamily V ρ a) (realVector p)‖^2=
+      ((Module.finrank ℂ (V a.1):ℝ)/(Fintype.card G:ℝ))*
+      ‖inner ℂ (stdOrthonormalBasis ℂ (V a.1) a.2.2)
+        (weightedOperator (ρ a.1) p (stdOrthonormalBasis ℂ (V a.1) a.2.1))‖^2 := by
+  rw [coefficient_realVector,norm_mul,Complex.norm_real,Real.norm_eq_abs,
+    abs_of_nonneg (Real.sqrt_nonneg _),mul_pow,Real.sq_sqrt (div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _))]
+
+include hρ hneq hcard in
+lemma parseval_operator (p : G → ℝ) :
+    (Fintype.card G:ℝ)*‖realVector p‖^2=
+      ∑ i,(Module.finrank ℂ (V i):ℝ)*∑ a : Fin (Module.finrank ℂ (V i)),
+        ‖weightedOperator (ρ i) p (stdOrthonormalBasis ℂ (V i) a)‖^2 := by
+  rw [←parseval V ρ hρ hneq hcard (realVector p)]
+  simp only [Fintype.sum_sigma,Fintype.sum_prod_type,coefficient_realVector_sq]
+  have hN : (Fintype.card G:ℝ)≠0 := by exact_mod_cast Fintype.card_ne_zero
+  rw [Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro i _
+  have he (a : Fin (Module.finrank ℂ (V i))) :
+      (∑ b : Fin (Module.finrank ℂ (V i)),
+        (Module.finrank ℂ (V i):ℝ)/(Fintype.card G:ℝ)*
+          ‖inner ℂ (stdOrthonormalBasis ℂ (V i) b)
+            (weightedOperator (ρ i) p (stdOrthonormalBasis ℂ (V i) a))‖^2)=
+        (Module.finrank ℂ (V i):ℝ)/(Fintype.card G:ℝ)*
+          ‖weightedOperator (ρ i) p (stdOrthonormalBasis ℂ (V i) a)‖^2 := by
+    rw [←Finset.mul_sum,(stdOrthonormalBasis ℂ (V i)).sum_sq_norm_inner_right]
+  simp_rw [he]
+  rw [←Finset.mul_sum]
+  field_simp
+
+include hρ hneq hcard in
+lemma realVector_bound (p : G → ℝ) :
+    (Fintype.card G:ℝ)*‖realVector p‖^2≤
+      ∑ i,(Module.finrank ℂ (V i):ℝ)^2*‖weightedOperator (ρ i) p‖^2 := by
+  rw [parseval_operator V ρ hρ hneq hcard]
+  apply Finset.sum_le_sum
+  intro i _
+  calc
+    _ ≤ (Module.finrank ℂ (V i):ℝ)*∑ _a : Fin (Module.finrank ℂ (V i)),‖weightedOperator (ρ i) p‖^2 := by
+      apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
+      apply Finset.sum_le_sum
+      intro a _
+      apply pow_le_pow_left₀ (norm_nonneg _)
+      simpa only [(stdOrthonormalBasis ℂ (V i)).norm_eq_one,mul_one] using
+        (weightedOperator (ρ i) p).le_opNorm (stdOrthonormalBasis ℂ (V i) a)
+    _ = _ := by simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]; ring
+
+include hρ hneq hcard in
+
+theorem l1_fourier_bound (p : G → ℝ) :
+    (∑ g,|p g|)^2≤∑ i,(Module.finrank ℂ (V i):ℝ)^2*‖weightedOperator (ρ i) p‖^2 := by
+  apply le_trans _ (realVector_bound V ρ hρ hneq hcard p)
+  have he : ‖realVector p‖^2=∑ g,|p g|^2 := by
+    rw [EuclideanSpace.norm_sq_eq]
+    simp only [realVector,PiLp.toLp_apply,Complex.norm_real,Real.norm_eq_abs]
+  rw [he]
+  simpa only [Finset.card_univ] using sq_sum_le_card_mul_sum_sq (s := Finset.univ) (f := fun g => |p g|)
+
+end Thorp.FiniteFourier
+
+namespace Thorp.ParallelBlocks
+open scoped BigOperators Classical
+variable {B X : Type*}
+
+def lift : (B → Equiv.Perm X) →* Equiv.Perm (B × X) where
+  toFun p :=
+    { toFun := fun x => (x.1,p x.1 x.2)
+      invFun := fun x => (x.1,(p x.1).symm x.2)
+      left_inv := fun x => by simp
+      right_inv := fun x => by simp }
+  map_one' := rfl
+  map_mul' _ _ := rfl
+
+@[simp] lemma lift_apply (p : B → Equiv.Perm X) (x : B × X) : lift p x=(x.1,p x.1 x.2) := rfl
+
+noncomputable def injection (b : B) : Equiv.Perm X →* Equiv.Perm (B × X) :=
+  lift.comp
+    { toFun := fun p c => if c=b then p else 1
+      map_one' := by funext c; simp
+      map_mul' := by intro p q; funext c; by_cases h : c=b <;> simp [h] }
+
+@[simp] lemma injection_apply (b : B) (p : Equiv.Perm X) (x : B × X) :
+    injection b p x=(x.1,if x.1=b then p x.2 else x.2) := by
+  simp only [injection,MonoidHom.comp_apply,lift_apply]
+  by_cases h : x.1=b <;> simp [h]
+
+lemma injection_commute {b c : B} (h : b≠c) (p q : Equiv.Perm X) :
+    Commute (injection b p) (injection c q) := by
+  apply Equiv.ext
+  intro x
+  simp only [Equiv.Perm.mul_apply,injection_apply]
+  by_cases hb : x.1=b <;> by_cases hc : x.1=c <;> simp_all
+
+variable [Fintype B]
+
+lemma lift_product (p : B → Equiv.Perm X) :
+    lift p=Finset.univ.noncommProd (fun b => injection b (p b))
+      (fun _ _ _ _ h => injection_commute h _ _) := by
+  have he (s : Finset B) (x : B × X) :
+      s.noncommProd (fun b => injection b (p b))
+        (fun _ _ _ _ h => injection_commute h _ _) x=
+          (x.1,if x.1∈s then p x.1 x.2 else x.2) := by
+    induction s using Finset.induction_on with
+    | empty =>
+      have hz : Finset.noncommProd (∅ : Finset B) (fun b => injection b (p b))
+          (fun _ _ _ _ h => injection_commute h _ _)=1 := Finset.noncommProd_empty _ _
+      rw [hz]; simp
+    | @insert a s ha ih =>
+      have hz : (insert a s).noncommProd (fun b => injection b (p b))
+          (fun _ _ _ _ h => injection_commute h _ _)=injection a (p a)*
+          s.noncommProd (fun b => injection b (p b))
+            (fun _ _ _ _ h => injection_commute h _ _) :=
+        Finset.noncommProd_insert_of_notMem _ _ _ _ ha
+      rw [hz,Equiv.Perm.mul_apply,ih,injection_apply]
+      by_cases h : x.1=a
+      · subst a; simp [ha]
+      · simp [h]
+  apply Equiv.ext
+  intro x
+  rw [he]
+  simp
+
+end Thorp.ParallelBlocks
+namespace Thorp.UnitaryFinite
+open scoped BigOperators Classical
+
+variable {G H : Type*} [Group G] [Group H]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+variable {Ω Ξ : Type*} [Fintype Ω] [Fintype Ξ]
+
+lemma continuousRepresentation_comp (ρ : Representation ℂ G V) (φ : H →* G) (h : H) :
+    continuousRepresentation (ρ.comp φ) h=continuousRepresentation ρ (φ h) := rfl
+
+lemma sampleOperator_comp (ρ : Representation ℂ G V) (φ : H →* G) (P : Ω → H) :
+    sampleOperator (ρ.comp φ) P=sampleOperator ρ (φ ∘ P) := rfl
+
+omit [FiniteDimensional ℂ V] in
+lemma unitary_comp (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (φ : H →* G) :
+    IsUnitary (ρ.comp φ) := fun h v w => hρ (φ h) v w
+
+lemma sampleOperator_commute (ρ : Representation ℂ G V) (P : Ω → G) (Q : Ξ → G)
+    (h : ∀ ω ξ, Commute (P ω) (Q ξ)) :
+    Commute (sampleOperator ρ P) (sampleOperator ρ Q) := by
+  unfold sampleOperator
+  apply Commute.smul_left
+  apply Commute.smul_right
+  apply Commute.sum_left
+  intro ω _
+  apply Commute.sum_right
+  intro ξ _
+  exact (h ω ξ).map (continuousRepresentation ρ)
+
+lemma sampleOperator_uniform_absorb [Fintype G] [Nonempty Ω]
+    (ρ : Representation ℂ G V) (P : Ω → G) :
+    sampleOperator ρ P*uniformOperator ρ=uniformOperator ρ := by
+  rw [sampleOperator_weighted,weighted_uniform_left,sampleLaw_sum,one_smul]
+
+end Thorp.UnitaryFinite
+namespace Thorp.ParallelBlocks
+open scoped BigOperators Classical
+open UnitaryFinite
+variable {B X Ω : Type*} [Fintype B] [DecidableEq B] [Fintype Ω]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+omit [Fintype B] [DecidableEq B] in
+lemma local_average_commute (ρ : Representation ℂ (Equiv.Perm (B × X)) V)
+    (P : B → Ω → Equiv.Perm X) (b c : B) :
+    Commute (sampleOperator (ρ.comp (injection b)) (P b))
+      (sampleOperator (ρ.comp (injection c)) (P c)) := by
+  by_cases h : b=c
+  · subst c; exact Commute.refl _
+  · rw [sampleOperator_comp,sampleOperator_comp]
+    exact sampleOperator_commute ρ _ _ (fun _ _ => injection_commute h _ _)
+
+lemma sampleOperator_parallel (ρ : Representation ℂ (Equiv.Perm (B × X)) V)
+    (P : B → Ω → Equiv.Perm X) :
+    sampleOperator ρ (fun ω : B → Ω => lift (fun b => P b (ω b)))=
+      CommutingBlocks.blocks (fun b => sampleOperator (ρ.comp (injection b)) (P b))
+        (local_average_commute ρ P) := by
+  let A (b : B) (w : Ω) := continuousRepresentation ρ (injection b (P b w))
+  have hc : ∀ ω : B → Ω, Pairwise fun b c => Commute (A b (ω b)) (A c (ω c)) := by
+    intro ω b c h
+    exact (injection_commute h _ _).map (continuousRepresentation ρ)
+  have havg : Pairwise fun b c => Commute ((Fintype.card Ω:ℂ)⁻¹ • ∑ w,A b w)
+      ((Fintype.card Ω:ℂ)⁻¹ • ∑ w,A c w) := by
+    intro b c _
+    exact local_average_commute ρ P b c
+  have he (ω : B → Ω) : continuousRepresentation ρ (lift (fun b => P b (ω b)))=
+      Finset.univ.noncommProd (fun b => A b (ω b)) (fun _ _ _ _ h => hc ω h) := by
+    rw [lift_product]
+    exact Finset.map_noncommProd _ _ _ (continuousRepresentation ρ)
+  unfold sampleOperator
+  simp_rw [he]
+  exact (FiniteAverage.ordered_average A hc havg).symm
+
+end Thorp.ParallelBlocks
+namespace Thorp
+open scoped BigOperators Classical
+
+noncomputable def splitPerm (r L : ℕ) : Equiv.Perm (Card (L+r)) ≃* Equiv.Perm (Card r × Card L) :=
+  (cardSplit r L).permCongrHom
+
+noncomputable def outerPerm (r L : ℕ) (hi : Card L × SwitchIndex r → Bool) :
+    Equiv.Perm (Card r × Card L) :=
+  (Equiv.prodComm (Card L) (Card r)).permCongrHom
+    (ParallelBlocks.lift (fun x => butterflyPerm r (decodeButterfly r (fun i => hi (x,i)))))
+
+lemma outerPerm_apply (r L : ℕ) (hi : Card L × SwitchIndex r → Bool) (x : Card r × Card L) :
+    outerPerm r L hi x=(butterflyPerm r (decodeButterfly r (fun i => hi (x.2,i))) x.1,x.2) := rfl
+
+lemma splitPerm_butterfly (r L : ℕ) (lo : Card r × SwitchIndex L → Bool)
+    (hi : Card L × SwitchIndex r → Bool) :
+    splitPerm r L (butterflyPerm (L+r) (decodeButterfly (L+r) (assembleBits r L lo hi)))=
+      outerPerm r L hi*ParallelBlocks.lift
+        (fun b => butterflyPerm L (decodeButterfly L (fun i => lo (b,i)))) := by
+  apply Equiv.ext
+  intro x
+  change cardSplit r L (butterflyPerm (L+r) (decodeButterfly (L+r) (assembleBits r L lo hi))
+    ((cardSplit r L).symm x))=_
+  rw [butterfly_split,Equiv.apply_symm_apply]
+  rfl
+
+lemma boundaryTree_node_raw (L r : ℕ) (U : Card (r+1) → Equiv.Perm (Card L))
+    (X : SwitchIndex (L+r+1) → Bool) :
+    boundaryTree L (r+1) U X=InputTree.node (fun y => X (Sum.inl y))
+      (fun b => boundaryTree L r (fun u => U (Fin.cons b u)) (fun i => X (Sum.inr (b,i)))) := by
+  conv_lhs => unfold boundaryTree
+  congr 1
+  funext b
+  cases b <;> simp only [Bool.false_eq_true,ite_false,ite_true] <;> rfl
+
+lemma boundaryOutput_node_raw (L r : ℕ) (Y : SwitchIndex (L+r+1) → Bool) (b : Card (r+1)) :
+    boundaryOutput L (r+1) Y b=
+      boundaryOutput L r (fun i => Y (Sum.inr (b 0,i))) (Fin.tail b) := by
+  conv_lhs => unfold boundaryOutput
+  cases b 0 <;> simp only [Bool.false_eq_true,ite_false,ite_true] <;> rfl
+
+lemma boundaryTree_split (L r : ℕ) (U : Card r → Equiv.Perm (Card L))
+    (lo : Card r × SwitchIndex L → Bool) (hi : Card L × SwitchIndex r → Bool) (x : Card (L+r)) :
+    cardSplit r L ((boundaryTree L r U (assembleBits r L lo hi)).perm x)=
+      let y := cardSplit r L x
+      let z := U y.1 y.2
+      (butterflyPerm r (decodeButterfly r (fun i => hi (z,i))) y.1,z) := by
+  induction r with
+  | zero =>
+    apply Prod.ext
+    · exact Subsingleton.elim _ _
+    · rfl
+  | succ r ih =>
+    let lo' : Card r × SwitchIndex L → Bool := fun j => lo (Fin.cons (x 0) j.1,j.2)
+    let hi' : Card L × SwitchIndex r → Bool := fun j => hi (j.1,Sum.inr (x 0,j.2))
+    let U' : Card r → Equiv.Perm (Card L) := fun b => U (Fin.cons (x 0) b)
+    have hh := ih U' lo' hi' (Fin.tail x)
+    let u := (boundaryTree L r U' (assembleBits r L lo' hi')).perm (Fin.tail x)
+    let y := cardSplit r L (Fin.tail x)
+    let z := U' y.1 y.2
+    let w := butterflyPerm r (decodeButterfly r (fun i => hi' (z,i))) y.1
+    change cardSplit r L u=(w,z) at hh
+    rw [boundaryTree_node_raw]
+    change ((Fin.cons (Bool.xor (x 0) (hi ((cardSplit r L u).2,Sum.inl (cardSplit r L u).1)))
+      ((cardSplit r L u).1) : Card (r+1)),(cardSplit r L u).2)=
+        ((Fin.cons (Bool.xor (x 0) (hi (z,Sum.inl w))) w : Card (r+1)),z)
+    rw [hh]
+
+lemma splitPerm_boundaryTree (L r : ℕ) (U : Card r → Equiv.Perm (Card L))
+    (lo : Card r × SwitchIndex L → Bool) (hi : Card L × SwitchIndex r → Bool) :
+    splitPerm r L (boundaryTree L r U (assembleBits r L lo hi)).perm=
+      outerPerm r L hi*ParallelBlocks.lift U := by
+  apply Equiv.ext
+  intro x
+  change cardSplit r L ((boundaryTree L r U (assembleBits r L lo hi)).perm
+    ((cardSplit r L).symm x))=_
+  rw [boundaryTree_split,Equiv.apply_symm_apply]
+  rfl
+
+lemma boundaryOutput_split (L r : ℕ) (lo : Card r × SwitchIndex L → Bool)
+    (hi : Card L × SwitchIndex r → Bool) (b : Card r) :
+    boundaryOutput L r (assembleBits r L lo hi) b=
+      butterflyPerm L (decodeButterfly L (fun i => lo (b,i))) := by
+  induction r with
+  | zero =>
+    rw [Subsingleton.elim b (fun j => Fin.elim0 j)]
+    rfl
+  | succ r ih =>
+    rw [boundaryOutput_node_raw]
+    change boundaryOutput L r (assembleBits r L (fun j => lo (Fin.cons (b 0) j.1,j.2))
+      (fun j => hi (j.1,Sum.inr (b 0,j.2)))) (Fin.tail b)=_
+    rw [ih]
+    simp only [Fin.cons_self_tail]
+
+
+abbrev OuterCoins (r L : ℕ) := Card L × SwitchIndex r → Bool
+abbrev LocalCoins (L : ℕ) := SwitchIndex L → Bool
+
+def parallelSplitCoinEquiv (r L : ℕ) :
+    ((Card r → LocalCoins L) × OuterCoins r L) ≃ LocalCoins (L+r) :=
+  (Equiv.prodCongr (Equiv.curry (Card r) (SwitchIndex L) Bool).symm (Equiv.refl _)).trans
+    (splitCoinEquiv r L)
+
+lemma parallelSplitCoinEquiv_apply (r L : ℕ) (c : Card r → LocalCoins L) (h : OuterCoins r L) :
+    parallelSplitCoinEquiv r L (c,h)=assembleBits r L (fun x => c x.1 x.2) h := by
+  exact splitCoinEquiv_apply _ _ _ _
+
+def pairAssembleEquiv (B C H : Type*) :
+    ((H × (B → C × C)) × H) ≃ (((B → C) × H) × ((B → C) × H)) where
+  toFun w := (((fun b => (w.1.2 b).1),w.1.1),((fun b => (w.1.2 b).2),w.2))
+  invFun w := ((w.1.2,fun b => (w.1.1 b,w.2.1 b)),w.2.2)
+  left_inv _w := rfl
+  right_inv _w := rfl
+
+def palindromeStageEquiv (r L : ℕ) :
+    ((OuterCoins r L × (Card r → BenesCoins L)) × OuterCoins r L) ≃ BenesCoins (L+r) :=
+  (pairAssembleEquiv _ _ _).trans
+    (Equiv.prodCongr (parallelSplitCoinEquiv r L) (parallelSplitCoinEquiv r L))
+
+lemma palindromeStageEquiv_apply (r L : ℕ)
+    (w : (OuterCoins r L × (Card r → BenesCoins L)) × OuterCoins r L) :
+    palindromeStageEquiv r L w=
+      (assembleBits r L (fun x => (w.1.2 x.1).1 x.2) w.1.1,
+       assembleBits r L (fun x => (w.1.2 x.1).2 x.2) w.2) := by
+  simp only [palindromeStageEquiv,Equiv.trans_apply,Equiv.prodCongr_apply,pairAssembleEquiv,
+    Equiv.coe_fn_mk,Prod.map_apply,parallelSplitCoinEquiv_apply]
+
+def modifiedAssembleEquiv (B C H U : Type*) :
+    ((H × (B → U × (C × C))) × H) ≃ (B → U) × (((B → C) × H) × ((B → C) × H)) where
+  toFun w := ((fun b => (w.1.2 b).1),
+    (((fun b => (w.1.2 b).2.1),w.1.1),((fun b => (w.1.2 b).2.2),w.2)))
+  invFun w := ((w.2.1.2,fun b => (w.1 b,(w.2.1.1 b,w.2.2.1 b))),w.2.2.2)
+  left_inv _w := rfl
+  right_inv _w := rfl
+
+end Thorp
+
+end ThorpNine.Tail
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Tail/Coins.lean b/lean/OAI/Probability/ThorpRouting/Tail/Coins.lean
index 41235ab3ac26e9e0e0e023da7a0b9ffb678cf7a6..1bd8eb4dfa14a8f08441e8f93081675ff49f34d0 100644
--- a/lean/OAI/Probability/ThorpRouting/Tail/Coins.lean
+++ b/lean/OAI/Probability/ThorpRouting/Tail/Coins.lean
@@ -1,764 +1,764 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Tail.Model
-
-namespace OAI
-
-namespace ThorpNine.Tail
-
-namespace Thorp
-
-open scoped BigOperators
-open Filter
-
-def Coins : ℕ → Type
-  | 0 => Unit
-  | d + 1 => Card d → Bool
-
-instance (d : ℕ) : Fintype (Coins d) := by
-  cases d <;> dsimp [Coins] <;> infer_instance
-
-instance (d : ℕ) : Nonempty (Coins d) := by
-  cases d <;> dsimp [Coins] <;> infer_instance
-
-def rotate (d : ℕ) : Equiv.Perm (Card d) where
-  toFun x i := x (finRotate d i)
-  invFun x i := x ((finRotate d).symm i)
-  left_inv x := by funext i; simp only [Equiv.apply_symm_apply]
-  right_inv x := by funext i; simp only [Equiv.symm_apply_apply]
-
-def step : (d : ℕ) → Coins d → Equiv.Perm (Card d)
-  | 0, _ => 1
-  | d + 1, ξ => rotate (d + 1) * pairSwitch ξ
-
-def run (d : ℕ) : (t : ℕ) → (Fin t → Coins d) → Equiv.Perm (Card d)
-  | 0, _ => 1
-  | t + 1, ω => step d (ω (Fin.last t)) * run d t (fun i => ω i.castSucc)
-
-noncomputable def law (d t : ℕ) (g : Equiv.Perm (Card d)) : ℝ :=
-  (Fintype.card {ω : Fin t → Coins d // run d t ω = g} : ℝ) /
-    (Fintype.card (Fin t → Coins d) : ℝ)
-
-noncomputable def distance (d t : ℕ) : ℝ :=
-  (1 / 2 : ℝ) * ∑ g : Equiv.Perm (Card d),
-    |law d t g - 1 / (Nat.factorial (2 ^ d) : ℝ)|
-
-noncomputable def mixingTime (d : ℕ) : ℕ :=
-  sInf {t : ℕ | distance d t ≤ (1 / 4 : ℝ)}
-
-noncomputable def supportTimeBound (d : ℕ) : ℤ :=
-  ⌈(2 / (2 : ℝ) ^ d) *
-    (Real.log ((3 / 4 : ℝ) * (Nat.factorial (2 ^ d) : ℝ)) / Real.log 2)⌉
-
-def OptimalOrder : Prop :=
-  ∃ C : ℕ, 0 < C ∧
-    Tendsto (fun d : ℕ => distance d (C * d)) atTop (nhds 0) ∧
-    (∀ d t : ℕ, 1 - (2 : ℝ) ^ ((t : ℝ) * (2 : ℝ) ^ d / 2) /
-      (Nat.factorial (2 ^ d) : ℝ) ≤ distance d t) ∧
-    (∀ᶠ d : ℕ in atTop,
-      supportTimeBound d ≤ (mixingTime d : ℤ) ∧ mixingTime d ≤ C * d) ∧
-    (∃ K : ℝ, ∀ᶠ d : ℕ in atTop, 2 * (d : ℝ) - K ≤ (mixingTime d : ℝ)) ∧
-    Asymptotics.IsTheta atTop (fun d : ℕ => (mixingTime d : ℝ)) (fun d : ℕ => (d : ℝ))
-
-noncomputable def finiteLaw {α β : Type*} [Fintype α] (f : α → β) (b : β) : ℝ := by
-  classical
-  exact (Fintype.card {a : α // f a = b} : ℝ) / (Fintype.card α : ℝ)
-
-lemma sum_card_fibers {α β : Type*} [Fintype α] [Fintype β] [DecidableEq β] (f : α → β) :
-    ∑ b, Fintype.card {a : α // f a = b} = Fintype.card α := by
-  classical
-  rw [← Fintype.card_sigma]
-  exact Fintype.card_congr (Equiv.sigmaFiberEquiv f)
-
-lemma finiteLaw_nonneg {α β : Type*} [Fintype α] (f : α → β) (b : β) :
-    0 ≤ finiteLaw f b := by
-  exact div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _)
-
-lemma finiteLaw_sum {α β : Type*} [Fintype α] [Nonempty α] [Fintype β] (f : α → β) :
-    ∑ b, finiteLaw f b = 1 := by
-  classical
-  simp only [finiteLaw, ← Finset.sum_div]
-  have hc : (∑ b, (Fintype.card {a : α // f a = b} : ℝ)) = (Fintype.card α : ℝ) := by
-    exact_mod_cast sum_card_fibers f
-  rw [hc]
-  exact div_self (Nat.cast_ne_zero.mpr Fintype.card_ne_zero)
-
-lemma finiteLaw_eq_zero_of_not_mem {α β : Type*} [Fintype α] [DecidableEq β]
-    (f : α → β) {b : β} (hb : b ∉ Finset.univ.image f) : finiteLaw f b = 0 := by
-  classical
-  have : IsEmpty {a : α // f a = b} := ⟨fun a => hb (Finset.mem_image.mpr
-    ⟨a.val, Finset.mem_univ _, a.property⟩)⟩
-  simp [finiteLaw]
-
-lemma event_le_totalVariation {β : Type*} [Fintype β] [DecidableEq β]
-    (p q : β → ℝ) (hpq : ∑ b, (p b - q b) = 0) (s : Finset β) :
-    (∑ b ∈ s, p b) - (∑ b ∈ s, q b) ≤ (1 / 2 : ℝ) * ∑ b, |p b - q b| := by
-  have h₁ : (∑ b ∈ s, (p b - q b)) ≤ ∑ b ∈ s, |p b - q b| :=
-    Finset.sum_le_sum (fun b _ => le_abs_self (p b - q b))
-  have h₂ : -(∑ b ∈ sᶜ, (p b - q b)) ≤ ∑ b ∈ sᶜ, |p b - q b| := by
-    rw [← Finset.sum_neg_distrib]
-    exact Finset.sum_le_sum (fun b _ => neg_le_abs (p b - q b))
-  have hz := Finset.sum_add_sum_compl s (fun b => p b - q b)
-  have ha := Finset.sum_add_sum_compl s (fun b => |p b - q b|)
-  rw [hpq] at hz
-  rw [← Finset.sum_sub_distrib]
-  linarith
-
-lemma finiteLaw_support_bound {α β : Type*} [Fintype α] [Nonempty α]
-    [Fintype β] [Nonempty β] (f : α → β) :
-    1 - (Fintype.card α : ℝ) / (Fintype.card β : ℝ) ≤
-      (1 / 2 : ℝ) * ∑ b, |finiteLaw f b - 1 / (Fintype.card β : ℝ)| := by
-  classical
-  let s := Finset.univ.image f
-  have hs : ∑ b ∈ s, finiteLaw f b = 1 := by
-    have h := Finset.sum_subset (Finset.subset_univ s)
-      (fun b _ hb => finiteLaw_eq_zero_of_not_mem f hb)
-    simpa only [finiteLaw_sum] using h
-  have hβ : (Fintype.card β : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
-  have hq : (∑ _b : β, 1 / (Fintype.card β : ℝ)) = 1 := by simp [hβ]
-  have hd : (∑ b, (finiteLaw f b - 1 / (Fintype.card β : ℝ))) = 0 := by
-    rw [Finset.sum_sub_distrib, finiteLaw_sum, hq, sub_self]
-  have he := event_le_totalVariation (finiteLaw f) (fun _ => 1 / (Fintype.card β : ℝ)) hd s
-  rw [hs] at he
-  have hsq : (∑ _b ∈ s, 1 / (Fintype.card β : ℝ)) = (s.card : ℝ) / Fintype.card β := by
-    simp [div_eq_mul_inv]
-  rw [hsq] at he
-  have hcard : (s.card : ℝ) ≤ Fintype.card α := by
-    exact_mod_cast (Finset.card_image_le (s := (Finset.univ : Finset α)) (f := f))
-  have hdiv := div_le_div_of_nonneg_right hcard (Nat.cast_nonneg (Fintype.card β) : (0 : ℝ) ≤ _)
-  linarith
-
-lemma card_positions (d : ℕ) : Fintype.card (Card d) = 2 ^ d := by
-  simp [Card]
-
-lemma card_permutations (d : ℕ) : Fintype.card (Equiv.Perm (Card d)) = Nat.factorial (2 ^ d) := by
-  rw [Fintype.card_perm, card_positions]
-
-lemma card_coins_succ (d : ℕ) : Fintype.card (Coins (d + 1)) = 2 ^ (2 ^ d) := by
-  simp [Coins]
-
-lemma card_tapes_succ (d t : ℕ) : Fintype.card (Fin t → Coins (d + 1)) = 2 ^ (t * 2 ^ d) := by
-  rw [Fintype.card_fun, card_coins_succ, Fintype.card_fin, ← pow_mul, Nat.mul_comm]
-
-lemma law_eq_finiteLaw (d t : ℕ) (g : Equiv.Perm (Card d)) :
-    law d t g = finiteLaw (run d t) g := by
-  unfold law finiteLaw
-  apply congrArg (fun n : ℕ => (n : ℝ) / (Fintype.card (Fin t → Coins d) : ℝ))
-  exact congrArg (@Fintype.card _) (Subsingleton.elim _ _)
-
-lemma law_nonneg (d t : ℕ) (g : Equiv.Perm (Card d)) : 0 ≤ law d t g := by
-  rw [law_eq_finiteLaw]
-  exact finiteLaw_nonneg (run d t) g
-
-lemma law_sum (d t : ℕ) : ∑ g, law d t g = 1 := by
-  simp only [law_eq_finiteLaw]
-  exact finiteLaw_sum (run d t)
-
-lemma physical_support_bound_succ (d t : ℕ) :
-    1 - (2 : ℝ) ^ (t * 2 ^ d) / (Nat.factorial (2 ^ (d + 1)) : ℝ) ≤ distance (d + 1) t := by
-  have : Nonempty (Equiv.Perm (Card (d + 1))) := ⟨1⟩
-  have h := finiteLaw_support_bound (run (d + 1) t)
-  rw [card_tapes_succ, card_permutations] at h
-  simpa only [Nat.cast_pow, Nat.cast_ofNat, distance, law_eq_finiteLaw] using h
-
-lemma step_succ_formula (d : ℕ) (ξ : Coins (d + 1)) (x : Card (d + 1)) :
-    step (d + 1) ξ x = Fin.snoc (Fin.tail x) (Bool.xor (x 0) (ξ (Fin.tail x))) := by
-  change (fun i => switchFun ξ x (finRotate (d + 1) i)) = _
-  rw [Fin.snoc_eq_cons_rotate]
-  rfl
-
-lemma distance_nonneg (d t : ℕ) : 0 ≤ distance d t := by
-  unfold distance
-  exact mul_nonneg (by norm_num) (Finset.sum_nonneg (fun _ _ => abs_nonneg _))
-
-lemma physical_support_bound (d t : ℕ) :
-    1 - (2 : ℝ) ^ ((t : ℝ) * (2 : ℝ) ^ d / 2) /
-      (Nat.factorial (2 ^ d) : ℝ) ≤ distance d t := by
-  cases d with
-  | zero =>
-    have h : 1 ≤ (2 : ℝ) ^ ((t : ℝ) / 2) := Real.one_le_rpow (by norm_num) (by positivity)
-    simpa only [pow_zero, Nat.factorial_one, Nat.cast_one, div_one, mul_one] using
-      le_trans (sub_nonpos.mpr h) (distance_nonneg 0 t)
-  | succ d =>
-    have he : (t : ℝ) * (2 : ℝ) ^ (d + 1) / 2 = (t * 2 ^ d : ℕ) := by
-      push_cast
-      rw [pow_succ]
-      ring
-    rw [he, Real.rpow_natCast]
-    exact physical_support_bound_succ d t
-
-lemma supportTimeBound_le_of_distance_le (d t : ℕ) (ht : distance d t ≤ (1 / 4 : ℝ)) :
-    supportTimeBound d ≤ (t : ℤ) := by
-  have hF : (0 : ℝ) < Nat.factorial (2 ^ d) := by positivity
-  have hN : (0 : ℝ) < (2 : ℝ) ^ d := by positivity
-  have hL : (0 : ℝ) < Real.log 2 := Real.log_pos (by norm_num)
-  have hb := physical_support_bound d t
-  have hp : (3 / 4 : ℝ) ≤
-      (2 : ℝ) ^ ((t : ℝ) * (2 : ℝ) ^ d / 2) / (Nat.factorial (2 ^ d) : ℝ) := by
-    linarith
-  have hr := (le_div_iff₀ hF).mp hp
-  have hlog := Real.log_le_log (show (0 : ℝ) < (3 / 4 : ℝ) * Nat.factorial (2 ^ d) by positivity) hr
-  rw [Real.log_rpow (by norm_num : (0 : ℝ) < 2)] at hlog
-  unfold supportTimeBound
-  apply Int.ceil_le.mpr
-  simp only [Int.cast_natCast]
-  calc
-    2 / (2 : ℝ) ^ d * (Real.log ((3 / 4 : ℝ) * Nat.factorial (2 ^ d)) / Real.log 2) =
-        Real.log ((3 / 4 : ℝ) * Nat.factorial (2 ^ d)) / ((2 : ℝ) ^ d / 2 * Real.log 2) := by
-      field_simp
-    _ ≤ (t : ℝ) := (div_le_iff₀ (mul_pos (div_pos hN (by norm_num)) hL)).mpr (by
-      nlinarith [hlog])
-
-lemma supportTimeBound_le_mixingTime {d : ℕ}
-    (h : ({t : ℕ | distance d t ≤ (1 / 4 : ℝ)} : Set ℕ).Nonempty) :
-    supportTimeBound d ≤ (mixingTime d : ℤ) := by
-  exact supportTimeBound_le_of_distance_le d (mixingTime d) (Nat.sInf_mem h)
-
-lemma log_factorial_lower {n : ℕ} (hn : 1 ≤ n) :
-    (n : ℝ) * Real.log n - n ≤ Real.log (Nat.factorial n) := by
-  have h := Stirling.le_log_factorial_stirling (show n ≠ 0 by omega)
-  have hln : 0 ≤ Real.log (n : ℝ) := Real.log_nonneg (by exact_mod_cast hn)
-  have hlp : 0 ≤ Real.log (2 * Real.pi) := Real.log_nonneg (by
-    have := Real.pi_gt_three
-    linarith)
-  linarith
-
-lemma log_three_quarters_lower : -(1 : ℝ) ≤ Real.log (3 / 4 : ℝ) := by
-  have h := Real.log_le_sub_one_of_pos (show (0 : ℝ) < 4 / 3 by norm_num)
-  have he : Real.log (4 / 3 : ℝ) = - Real.log (3 / 4 : ℝ) := by
-    rw [← Real.log_inv]
-    congr 1
-    norm_num
-  rw [he] at h
-  linarith
-
-lemma linear_le_supportTimeBound (d : ℕ) :
-    2 * (d : ℝ) - 4 / Real.log 2 ≤ (supportTimeBound d : ℝ) := by
-  have hN : 1 ≤ (2 : ℝ) ^ d := one_le_pow₀ (by norm_num)
-  have hNp : 0 < (2 : ℝ) ^ d := by positivity
-  have hL : 0 < Real.log 2 := Real.log_pos (by norm_num)
-  have hf := log_factorial_lower (show 1 ≤ 2 ^ d by exact one_le_pow₀ (by norm_num))
-  have hlog : (2 : ℝ) ^ d * ((d : ℝ) * Real.log 2) - (2 : ℝ) ^ d - 1 ≤
-      Real.log ((3 / 4 : ℝ) * (Nat.factorial (2 ^ d) : ℝ)) := by
-    rw [Real.log_mul (by norm_num) (by positivity)]
-    push_cast at hf
-    rw [Real.log_pow] at hf
-    linarith [log_three_quarters_lower]
-  apply le_trans _ (Int.le_ceil _)
-  have hmul : (2 * (d : ℝ) - 4 / Real.log 2) *
-      ((2 : ℝ) ^ d / 2 * Real.log 2) ≤
-      Real.log ((3 / 4 : ℝ) * (Nat.factorial (2 ^ d) : ℝ)) := by
-    have hcancel : (2 * (d : ℝ) - 4 / Real.log 2) *
-        ((2 : ℝ) ^ d / 2 * Real.log 2) =
-        (2 : ℝ) ^ d * ((d : ℝ) * Real.log 2) - 2 * (2 : ℝ) ^ d := by
-      field_simp
-      ring
-    rw [hcancel]
-    linarith
-  convert (le_div_iff₀ (mul_pos (div_pos hNp (by norm_num)) hL)).mpr hmul using 1
-  field_simp
-
-lemma linear_le_time_of_distance_le (d t : ℕ)
-    (ht : distance d t ≤ (1 / 4 : ℝ)) :
-    2 * (d : ℝ) - 4 / Real.log 2 ≤ (t : ℝ) := by
-  have h := supportTimeBound_le_of_distance_le d t ht
-  have hr : (supportTimeBound d : ℝ) ≤ (t : ℝ) := by exact_mod_cast h
-  exact (linear_le_supportTimeBound d).trans hr
-
-lemma entropy_chord (p : ℝ) (hp : 0 ≤ p) (hq : p ≤ 1 / 2) :
-    2 * p * Real.log 2 ≤ Real.binEntropy p := by
-  have hc := Real.strictConcave_binEntropy.concaveOn.2
-    (show (0 : ℝ) ∈ Set.Icc (0 : ℝ) 1 by norm_num)
-    (show (2 : ℝ)⁻¹ ∈ Set.Icc (0 : ℝ) 1 by norm_num)
-    (show 0 ≤ 1 - 2 * p by linarith)
-    (show 0 ≤ 2 * p by positivity)
-    (show (1 - 2 * p) + 2 * p = 1 by ring)
-  simpa only [Real.binEntropy_zero, Real.binEntropy_two_inv,
-    smul_eq_mul, mul_zero, zero_add, show 2 * p * (2 : ℝ)⁻¹ = p by ring] using hc
-
-lemma split_entropy_lower (a b : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) :
-    2 * min a b * Real.log 2 ≤
-      (a + b) * Real.log (a + b) - a * Real.log a - b * Real.log b := by
-  wlog hab : a ≤ b generalizing a b
-  · have hba := this b a hb ha (le_of_not_ge hab)
-    simpa only [min_comm, add_comm b a] using hba.trans_eq (show (b + a) * Real.log (b + a) - b * Real.log b - a * Real.log a =
-      (a + b) * Real.log (a + b) - a * Real.log a - b * Real.log b by rw [add_comm b a]; ring)
-  rw [min_eq_left hab]
-  by_cases hz : a = 0
-  · subst a
-    simp
-  have hap : 0 < a := lt_of_le_of_ne ha (Ne.symm hz)
-  have hbp : 0 < b := hap.trans_le hab
-  have hs : 0 < a + b := add_pos hap hbp
-  have hp : 0 ≤ a / (a + b) := div_nonneg ha hs.le
-  have hq : a / (a + b) ≤ 1 / 2 := (div_le_iff₀ hs).mpr (by linarith)
-  have hc := mul_le_mul_of_nonneg_left (entropy_chord (a / (a + b)) hp hq) hs.le
-  have he : (a + b) * Real.binEntropy (a / (a + b)) =
-      (a + b) * Real.log (a + b) - a * Real.log a - b * Real.log b := by
-    unfold Real.binEntropy
-    rw [show 1 - a / (a + b) = b / (a + b) by field_simp; ring]
-    simp only [Real.log_inv]
-    rw [Real.log_div (ne_of_gt hap) (ne_of_gt hs),
-      Real.log_div (ne_of_gt hbp) (ne_of_gt hs)]
-    field_simp
-    ring
-  rw [he] at hc
-  calc
-    2 * a * Real.log 2 = (a + b) * (2 * (a / (a + b)) * Real.log 2) := by
-      field_simp
-    _ ≤ _ := hc
-
-noncomputable def childMarks {d : ℕ} (s : Finset (Card (d + 1))) (ε : Bool) :
-    Finset (Card d) := (s.filter (fun x => x 0 = ε)).image Fin.tail
-
-lemma card_childMarks {d : ℕ} (s : Finset (Card (d + 1))) (ε : Bool) :
-    (childMarks s ε).card = (s.filter (fun x => x 0 = ε)).card := by
-  classical
-  unfold childMarks
-  apply Finset.card_image_iff.mpr
-  intro x hx y hy hxy
-  have hx0 := (Finset.mem_filter.mp hx).2
-  have hy0 := (Finset.mem_filter.mp hy).2
-  rw [← Fin.cons_self_tail x, ← Fin.cons_self_tail y, hx0, hy0, hxy]
-
-lemma card_childMarks_add {d : ℕ} (s : Finset (Card (d + 1))) :
-    (childMarks s false).card + (childMarks s true).card = s.card := by
-  classical
-  rw [card_childMarks, card_childMarks]
-  simpa only [Bool.not_eq_false] using
-    (Finset.card_filter_add_card_filter_not (s := s) (fun x => x 0 = false))
-
-noncomputable def topMeetings {d : ℕ} (b : Butterfly (d + 1))
-    (s : Finset (Card (d + 1))) : Finset (Card d) :=
-  ((childMarks s false).image (butterflyPerm d (b.2 false))) ∩
-    ((childMarks s true).image (butterflyPerm d (b.2 true)))
-
-lemma card_topMeetings_le {d : ℕ} (b : Butterfly (d + 1)) (s : Finset (Card (d + 1))) :
-    (topMeetings b s).card ≤ min (childMarks s false).card (childMarks s true).card := by
-  classical
-  apply le_min
-  · exact (Finset.card_le_card Finset.inter_subset_left).trans_eq
-      (Finset.card_image_of_injective _ (butterflyPerm d (b.2 false)).injective)
-  · exact (Finset.card_le_card Finset.inter_subset_right).trans_eq
-      (Finset.card_image_of_injective _ (butterflyPerm d (b.2 true)).injective)
-
-noncomputable def encounters : (d : ℕ) → Butterfly d → Finset (Card d) → ℕ
-  | 0, _, _ => 0
-  | d + 1, b, s => encounters d (b.2 false) (childMarks s false) +
-      encounters d (b.2 true) (childMarks s true) + (topMeetings b s).card
-
-lemma butterfly_encounters (d : ℕ) (b : Butterfly d) (s : Finset (Card d)) :
-    (encounters d b s : ℝ) ≤ (s.card : ℝ) * Real.log s.card / (2 * Real.log 2) := by
-  induction d with
-  | zero =>
-    simp only [encounters, Nat.cast_zero]
-    have hs : s.card ≤ 1 := by
-      simpa [Card] using Finset.card_le_univ s
-    interval_cases s.card <;> simp
-  | succ d ih =>
-    have h₀ := ih (b.2 false) (childMarks s false)
-    have h₁ := ih (b.2 true) (childMarks s true)
-    have ht : ((topMeetings b s).card : ℝ) ≤
-        min ((childMarks s false).card : ℝ) ((childMarks s true).card : ℝ) := by
-      exact_mod_cast card_topMeetings_le b s
-    have he := split_entropy_lower ((childMarks s false).card : ℝ)
-      ((childMarks s true).card : ℝ) (by positivity) (by positivity)
-    have hs : ((childMarks s false).card : ℝ) + (childMarks s true).card = s.card := by
-      exact_mod_cast card_childMarks_add s
-    rw [hs] at he
-    have hL : 0 < 2 * Real.log 2 := by positivity
-    rw [le_div_iff₀ hL] at h₀ h₁ ⊢
-    simp only [encounters, Nat.cast_add]
-    nlinarith [mul_le_mul_of_nonneg_right ht hL.le]
-
-lemma prod_split_at {α : Type*} [DecidableEq α] (s : Finset α) (f : α → ℝ) (a : α) :
-    ∏ i ∈ s, f i = (if a ∈ s then f a else 1) * ∏ i ∈ s.erase a, f i := by
-  by_cases ha : a ∈ s
-  · rw [ite_eq_left ha]
-    exact (Finset.mul_prod_erase s f ha).symm
-  · simp [ha]
-
-lemma prod_split_two {α : Type*} [DecidableEq α] (s : Finset α) (f : α → ℝ)
-    (a b : α) (hab : a ≠ b) :
-    ∏ i ∈ s, f i = (if a ∈ s then f a else 1) * (if b ∈ s then f b else 1) *
-      ∏ i ∈ (s.erase a).erase b, f i := by
-  calc
-    ∏ i ∈ s, f i = (if a ∈ s then f a else 1) * ∏ i ∈ s.erase a, f i :=
-      prod_split_at s f a
-    _ = _ := by
-      rw [prod_split_at (s.erase a) f b]
-      have hm : (b ∈ s.erase a) ↔ b ∈ s :=
-        ⟨fun h => (Finset.mem_erase.mp h).2, fun h => Finset.mem_erase.mpr ⟨Ne.symm hab, h⟩⟩
-      simp only [hm, mul_assoc]
-
-lemma swap_average_product {α : Type*} [DecidableEq α] (s : Finset α) (p : α → ℝ)
-    (hp : ∀ i, 0 ≤ p i) (a b : α) :
-    ((∏ i ∈ s, p i) + ∏ i ∈ s, p (Equiv.swap a b i)) / 2 ≤
-      ∏ i ∈ s, (p i + p (Equiv.swap a b i)) / 2 := by
-  by_cases hab : a = b
-  · subst b
-    simp only [Equiv.swap_self, Equiv.refl_apply]
-    have h : (fun i => (p i + p i) / 2) = p := by funext i; ring
-    rw [h]
-    linarith
-  let t := (s.erase a).erase b
-  have hfix : ∀ i ∈ t, Equiv.swap a b i = i := by
-    intro i hi
-    exact Equiv.swap_apply_of_ne_of_ne (Finset.mem_erase.mp (Finset.mem_erase.mp hi).2).1
-      (Finset.mem_erase.mp hi).1
-  have hprod : (∏ i ∈ t, p (Equiv.swap a b i)) = ∏ i ∈ t, p i :=
-    Finset.prod_congr rfl (fun i hi => by rw [hfix i hi])
-  have havg : (∏ i ∈ t, (p i + p (Equiv.swap a b i)) / 2) = ∏ i ∈ t, p i :=
-    Finset.prod_congr rfl (fun i hi => by rw [hfix i hi]; ring)
-  rw [prod_split_two s p a b hab,
-    prod_split_two s (fun i => p (Equiv.swap a b i)) a b hab,
-    prod_split_two s (fun i => (p i + p (Equiv.swap a b i)) / 2) a b hab]
-  change ((if a ∈ s then p a else 1) * (if b ∈ s then p b else 1) * (∏ i ∈ t, p i) +
-    (if a ∈ s then p (Equiv.swap a b a) else 1) *
-      (if b ∈ s then p (Equiv.swap a b b) else 1) * (∏ i ∈ t, p (Equiv.swap a b i))) / 2 ≤ _
-  rw [hprod]
-  change _ ≤ (if a ∈ s then (p a + p (Equiv.swap a b a)) / 2 else 1) *
-    (if b ∈ s then (p b + p (Equiv.swap a b b)) / 2 else 1) *
-      (∏ i ∈ t, (p i + p (Equiv.swap a b i)) / 2)
-  rw [havg, Equiv.swap_apply_left, Equiv.swap_apply_right]
-  have ht : 0 ≤ ∏ i ∈ t, p i := Finset.prod_nonneg (fun i _ => hp i)
-  by_cases ha : a ∈ s <;> by_cases hb : b ∈ s <;> simp only [ha, hb, ↓reduceIte]
-  · have hsq := mul_nonneg (sq_nonneg (p a - p b)) ht
-    nlinarith
-  all_goals ring_nf; rfl
-
-def CylinderBound {α : Type*} (m : Finset α → ℝ) (p : α → ℝ) : Prop :=
-  ∀ s, m s ≤ ∏ i ∈ s, p i
-
-lemma CylinderBound.fair_swap {α : Type*} [DecidableEq α]
-    {m : Finset α → ℝ} {p : α → ℝ} (h : CylinderBound m p)
-    (hp : ∀ i, 0 ≤ p i) (a b : α) :
-    CylinderBound (fun s => (m s + m (s.image (Equiv.swap a b))) / 2)
-      (fun i => (p i + p (Equiv.swap a b i)) / 2) := by
-  intro s
-  have h₀ := h s
-  have h₁ := h (s.image (Equiv.swap a b))
-  rw [Finset.prod_image (Equiv.swap a b).injective.injOn] at h₁
-  exact (show (m s + m (s.image (Equiv.swap a b))) / 2 ≤
-      ((∏ i ∈ s, p i) + ∏ i ∈ s, p (Equiv.swap a b i)) / 2 by linarith).trans
-    (swap_average_product s p hp a b)
-
-noncomputable def inclusionMoment {α Ω : Type*} [Fintype Ω]
-    (F : Ω → Finset α) (s : Finset α) : ℝ := by
-  classical
-  exact finiteMean (fun ω => if s ⊆ F ω then 1 else 0)
-
-lemma inclusionMoment_nonneg {α Ω : Type*} [Fintype Ω]
-    (F : Ω → Finset α) (s : Finset α) : 0 ≤ inclusionMoment F s := by
-  classical
-  unfold inclusionMoment finiteMean
-  positivity
-
-lemma indicator_subset_prod {α : Type*} [DecidableEq α] (A s : Finset α) :
-    (if s ⊆ A then (1 : ℝ) else 0) = ∏ i ∈ s, (if i ∈ A then 1 else 0) := by
-  by_cases h : s ⊆ A
-  · rw [ite_eq_left h]
-    exact (Finset.prod_eq_one (fun i hi => ite_eq_left (h hi))).symm
-  · rw [ite_eq_right h]
-    obtain ⟨i, hi, hiA⟩ := Finset.not_subset.mp h
-    exact (Finset.prod_eq_zero hi (ite_eq_right hiA)).symm
-
-lemma inclusionMoment_const {α Ω : Type*} [DecidableEq α] [Fintype Ω] [Nonempty Ω]
-    (A s : Finset α) : inclusionMoment (fun _ : Ω => A) s =
-      if s ⊆ A then 1 else 0 := by
-  classical
-  have hn : (Fintype.card Ω : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
-  by_cases h : s ⊆ A <;> simp [inclusionMoment, finiteMean, h, hn]
-
-lemma cylinder_const {α Ω : Type*} [Fintype Ω] [Nonempty Ω] (A : Finset α) :
-    CylinderBound (inclusionMoment (fun _ : Ω => A))
-      (fun i => inclusionMoment (fun _ : Ω => A) {i}) := by
-  classical
-  intro s
-  simp only [inclusionMoment_const, Finset.singleton_subset_iff]
-  exact (indicator_subset_prod A s).le
-
-noncomputable def fairSetSwap {α Ω : Type*} [DecidableEq α]
-    (F : Ω → Finset α) (a b : α) : Ω × Bool → Finset α :=
-  fun ω => if ω.2 then (F ω.1).image (Equiv.swap a b) else F ω.1
-
-lemma subset_swap_image_iff {α : Type*} [DecidableEq α]
-    (A s : Finset α) (a b : α) :
-    s ⊆ A.image (Equiv.swap a b) ↔ s.image (Equiv.swap a b) ⊆ A := by
-  constructor
-  · intro h
-    have h' := Finset.image_subset_image h (f := Equiv.swap a b)
-    simpa only [Finset.image_image, Function.comp_def, Equiv.swap_apply_self, Finset.image_id'] using h'
-  · intro h
-    have h' := Finset.image_subset_image h (f := Equiv.swap a b)
-    simpa only [Finset.image_image, Function.comp_def, Equiv.swap_apply_self, Finset.image_id'] using h'
-
-lemma inclusionMoment_fairSetSwap {α Ω : Type*} [DecidableEq α] [Fintype Ω]
-    (F : Ω → Finset α) (a b : α) (s : Finset α) :
-    inclusionMoment (fairSetSwap F a b) s =
-      (inclusionMoment F s + inclusionMoment F (s.image (Equiv.swap a b))) / 2 := by
-  classical
-  simp only [inclusionMoment, finiteMean, Fintype.sum_prod_type, Fintype.sum_bool,
-    fairSetSwap, Bool.false_eq_true, ↓reduceIte, Fintype.card_prod, Fintype.card_bool,
-    Nat.cast_mul, Nat.cast_ofNat, subset_swap_image_iff, Finset.sum_add_distrib]
-  rw [div_mul_eq_div_div, add_div]
-  ring_nf
-
-lemma cylinder_fairSetSwap {α Ω : Type*} [DecidableEq α] [Fintype Ω]
-    (F : Ω → Finset α) (a b : α)
-    (h : CylinderBound (inclusionMoment F) (fun i => inclusionMoment F {i})) :
-    CylinderBound (inclusionMoment (fairSetSwap F a b))
-      (fun i => inclusionMoment (fairSetSwap F a b) {i}) := by
-  classical
-  have h' := h.fair_swap (fun i => inclusionMoment_nonneg F {i}) a b
-  intro s
-  simpa only [inclusionMoment_fairSetSwap, Finset.image_singleton] using h' s
-
-def SwitchSamples : ℕ → Type
-  | 0 => Unit
-  | n + 1 => SwitchSamples n × Bool
-
-instance (n : ℕ) : Fintype (SwitchSamples n) := by
-  induction n with
-  | zero => exact inferInstanceAs (Fintype Unit)
-  | succ n ih => exact inferInstanceAs (Fintype (SwitchSamples n × Bool))
-
-instance (n : ℕ) : Nonempty (SwitchSamples n) := by
-  induction n with
-  | zero => exact inferInstanceAs (Nonempty Unit)
-  | succ n ih => exact inferInstanceAs (Nonempty (SwitchSamples n × Bool))
-
-noncomputable def switchedSet {α : Type*} [DecidableEq α] (A : Finset α) :
-    (L : List (α × α)) → SwitchSamples L.length → Finset α
-  | [], _ => A
-  | (a,b) :: L, ω => fairSetSwap (switchedSet A L) a b ω
-
-lemma switchedSet_cylinder {α : Type*} [DecidableEq α] (A : Finset α)
-    (L : List (α × α)) :
-    CylinderBound (inclusionMoment (switchedSet A L))
-      (fun i => inclusionMoment (switchedSet A L) {i}) := by
-  induction L with
-  | nil => exact cylinder_const A
-  | cons e L ih => exact cylinder_fairSetSwap (switchedSet A L) e.1 e.2 ih
-
-inductive QueryTree (ι : Type*)
-  | reject : QueryTree ι
-  | accept : QueryTree ι
-  | branch (i : ι) (zero one : QueryTree ι) : QueryTree ι
-  | force (i : ι) (b : Bool) (next : QueryTree ι) : QueryTree ι
-
-namespace QueryTree
-
-variable {ι : Type*} [DecidableEq ι]
-
-def queries : QueryTree ι → Finset ι
-  | reject => ∅
-  | accept => ∅
-  | branch i zero one => insert i (queries zero ∪ queries one)
-  | force i _ next => insert i (queries next)
-
-def Fresh : QueryTree ι → Prop
-  | reject => True
-  | accept => True
-  | branch i zero one => i ∉ queries zero ∧ i ∉ queries one ∧ Fresh zero ∧ Fresh one
-  | force i _ next => i ∉ queries next ∧ Fresh next
-
-def weighted : QueryTree ι → (ι → Bool) → ℝ
-  | reject, _ => 0
-  | accept, _ => 1
-  | branch i zero one, ω => if ω i then weighted one ω else weighted zero ω
-  | force i b next, ω => if ω i = b then 2 * weighted next ω else 0
-
-omit [DecidableEq ι] in
-lemma weighted_nonneg (T : QueryTree ι) (ω : ι → Bool) : 0 ≤ T.weighted ω := by
-  induction T with
-  | reject => exact le_rfl
-  | accept => exact zero_le_one
-  | branch i zero one ih₀ ih₁ => simpa only [weighted] using ite_nonneg ih₁ ih₀
-  | force i b next ih => simpa only [weighted] using ite_nonneg (mul_nonneg (by norm_num) ih) le_rfl
-
-lemma weighted_congr (T : QueryTree ι) {ω η : ι → Bool}
-    (h : ∀ i ∈ T.queries, ω i = η i) : T.weighted ω = T.weighted η := by
-  induction T with
-  | reject => rfl
-  | accept => rfl
-  | branch i zero one ih₀ ih₁ =>
-    have hi := h i (Finset.mem_insert_self _ _)
-    simp only [weighted, hi]
-    rw [ih₀ (fun j hj => h j (Finset.mem_insert_of_mem (Finset.mem_union_left _ hj))),
-      ih₁ (fun j hj => h j (Finset.mem_insert_of_mem (Finset.mem_union_right _ hj)))]
-  | force i b next ih =>
-    have hi := h i (Finset.mem_insert_self _ _)
-    simp only [weighted, hi]
-    rw [ih (fun j hj => h j (Finset.mem_insert_of_mem hj))]
-
-def flipCoin (i : ι) : Equiv.Perm (ι → Bool) :=
-  (show Function.Involutive (fun ω : ι → Bool => Function.update ω i (!(ω i))) from by
-    intro ω
-    funext j
-    by_cases h : j = i
-    · subst j; simp
-    · simp [Function.update_of_ne h]).toPerm
-
-lemma flipCoin_apply_eq (i : ι) (ω : ι → Bool) : flipCoin i ω i = !(ω i) := by
-  change Function.update ω i (!(ω i)) i = _
-  exact Function.update_self _ _ _
-
-lemma flipCoin_apply_ne (i j : ι) (h : j ≠ i) (ω : ι → Bool) :
-    flipCoin i ω j = ω j := by
-  change Function.update ω i (!(ω i)) j = _
-  exact Function.update_of_ne h _ _
-
-lemma weighted_flip_of_not_mem (T : QueryTree ι) (i : ι) (hi : i ∉ T.queries)
-    (ω : ι → Bool) : T.weighted (flipCoin i ω) = T.weighted ω := by
-  apply weighted_congr
-  intro j hj
-  exact flipCoin_apply_ne i j (fun h => hi (h ▸ hj)) ω
-
-lemma mean_indicator_bit [Fintype ι] (f : (ι → Bool) → ℝ) (i : ι) (b : Bool)
-    (hf : ∀ ω, f (flipCoin i ω) = f ω) :
-    finiteMean (fun ω => if ω i = b then f ω else 0) = finiteMean f / 2 := by
-  let g := fun ω : ι → Bool => if ω i = b then f ω else 0
-  have hp (ω : ι → Bool) : g ω + g (flipCoin i ω) = f ω := by
-    dsimp [g]
-    rw [flipCoin_apply_eq, hf]
-    cases hω : ω i <;> cases b <;> simp
-  have hs := Equiv.sum_comp (flipCoin i) g
-  have hadd : (∑ ω, g ω) + (∑ ω, g (flipCoin i ω)) = ∑ ω, f ω := by
-    rw [← Finset.sum_add_distrib]
-    exact Finset.sum_congr rfl (fun ω _ => hp ω)
-  unfold finiteMean
-  change (∑ ω, g ω) / _ = _
-  rw [hs] at hadd
-  have hh : (∑ ω, g ω) = (∑ ω, f ω) / 2 := by linarith
-  rw [hh]
-  ring
-
-lemma mean_weighted_le_one [Fintype ι] (T : QueryTree ι) (hT : T.Fresh) :
-    finiteMean T.weighted ≤ 1 := by
-  induction T with
-  | reject => simp [weighted, finiteMean]
-  | accept => simp [weighted, finiteMean]
-  | branch i zero one ih₀ ih₁ =>
-    obtain ⟨hz, ho, hfz, hfo⟩ := hT
-    have h₀ := mean_indicator_bit zero.weighted i false (zero.weighted_flip_of_not_mem i hz)
-    have h₁ := mean_indicator_bit one.weighted i true (one.weighted_flip_of_not_mem i ho)
-    have he : finiteMean (branch i zero one).weighted =
-        finiteMean (fun ω => if ω i = false then zero.weighted ω else 0) +
-        finiteMean (fun ω => if ω i = true then one.weighted ω else 0) := by
-      unfold finiteMean
-      rw [← add_div, ← Finset.sum_add_distrib]
-      congr 1
-      apply Finset.sum_congr rfl
-      intro ω _
-      cases hω : ω i <;> simp [weighted, hω]
-    rw [he, h₀, h₁]
-    linarith [ih₀ hfz, ih₁ hfo]
-  | force i b next ih =>
-    obtain ⟨hi, hn⟩ := hT
-    have hm := mean_indicator_bit next.weighted i b (next.weighted_flip_of_not_mem i hi)
-    have he : finiteMean (force i b next).weighted =
-        2 * finiteMean (fun ω => if ω i = b then next.weighted ω else 0) := by
-      unfold finiteMean
-      rw [← mul_div_assoc, Finset.mul_sum]
-      congr 1
-      apply Finset.sum_congr rfl
-      intro ω _
-      by_cases h : ω i = b <;> simp [weighted, h]
-    rw [he, hm]
-    linarith [ih hn]
-
-def succeeds : QueryTree ι → (ι → Bool) → Bool
-  | reject, _ => false
-  | accept, _ => true
-  | branch i zero one, ω => if ω i then succeeds one ω else succeeds zero ω
-  | force i b next, ω => if ω i = b then succeeds next ω else false
-
-def charge : QueryTree ι → (ι → Bool) → ℕ
-  | reject, _ => 0
-  | accept, _ => 0
-  | branch i zero one, ω => if ω i then charge one ω else charge zero ω
-  | force i b next, ω => if ω i = b then charge next ω + 1 else 0
-
-omit [DecidableEq ι] in
-lemma weighted_eq (T : QueryTree ι) (ω : ι → Bool) :
-    T.weighted ω = if T.succeeds ω then (2 : ℝ) ^ T.charge ω else 0 := by
-  induction T with
-  | reject => rfl
-  | accept => norm_num [weighted, succeeds, charge]
-  | branch i zero one ih₀ ih₁ =>
-    cases h : ω i <;> simp [weighted, succeeds, charge, h, ih₀, ih₁]
-  | force i b next ih =>
-    by_cases h : ω i = b
-    · simp only [weighted, succeeds, charge, h, ↓reduceIte, ih]
-      cases hs : next.succeeds ω <;> simp [pow_succ, mul_comm]
-    · simp [weighted, succeeds, h]
-
-lemma success_probability_le [Fintype ι] (T : QueryTree ι) (hT : T.Fresh)
-    (K : ℕ) (hK : ∀ ω, T.succeeds ω = true → K ≤ T.charge ω) :
-    finiteMean (fun ω => if T.succeeds ω then (1 : ℝ) else 0) ≤ 1 / (2 : ℝ) ^ K := by
-  have hw (ω : ι → Bool) :
-      (2 : ℝ) ^ K * (if T.succeeds ω then (1 : ℝ) else 0) ≤ T.weighted ω := by
-    rw [weighted_eq]
-    cases hs : T.succeeds ω
-    · simp
-    · simpa using pow_le_pow_right₀ (show (1 : ℝ) ≤ 2 by norm_num) (hK ω hs)
-  have hn : 0 ≤ (Fintype.card (ι → Bool) : ℝ) := Nat.cast_nonneg _
-  have hm := div_le_div_of_nonneg_right
-    (Finset.sum_le_sum (fun ω (_ : ω ∈ Finset.univ) => hw ω)) hn
-  rw [← Finset.mul_sum, mul_div_assoc] at hm
-  change (2 : ℝ) ^ K * finiteMean (fun ω => if T.succeeds ω then 1 else 0) ≤
-    finiteMean T.weighted at hm
-  exact (le_div_iff₀ (pow_pos (by norm_num) K)).mpr (by
-    rw [mul_comm]
-    exact hm.trans (mean_weighted_le_one T hT))
-
-end QueryTree
-
-def encodeButterfly : (d : ℕ) → Butterfly d → SwitchIndex d → Bool
-  | 0, _, i => nomatch i
-  | _d + 1, b, Sum.inl y => b.1 y
-  | d + 1, b, Sum.inr (ε,i) => encodeButterfly d (b.2 ε) i
-
-lemma decode_encodeButterfly (d : ℕ) (b : Butterfly d) :
-    decodeButterfly d (encodeButterfly d b) = b := by
-  induction d with
-  | zero => exact Unit.ext _ _
-  | succ d ih =>
-    apply Prod.ext
-    · rfl
-    · funext ε
-      exact ih (b.2 ε)
-
-lemma encode_decodeButterfly (d : ℕ) (ω : SwitchIndex d → Bool) :
-    encodeButterfly d (decodeButterfly d ω) = ω := by
-  induction d with
-  | zero => funext i; exact Empty.elim i
-  | succ d ih =>
-    funext i
-    cases i with
-    | inl y => rfl
-    | inr z => exact congrFun (ih (fun i => ω (Sum.inr (z.1,i)))) z.2
-
-def butterflyCoinEquiv (d : ℕ) : (SwitchIndex d → Bool) ≃ Butterfly d where
-  toFun := decodeButterfly d
-  invFun := encodeButterfly d
-  left_inv := encode_decodeButterfly d
-  right_inv := decode_encodeButterfly d
-
-noncomputable def routeDomain : (d : ℕ) → Card d → Card d → Finset (SwitchIndex d)
-  | 0, _, _ => ∅
-  | d + 1, x, y => by
-      classical
-      exact insert (Sum.inl (Fin.tail y))
-        ((routeDomain d (Fin.tail x) (Fin.tail y)).image (fun i => Sum.inr (x 0,i)))
-
-def routeValue : (d : ℕ) → Card d → Card d → SwitchIndex d → Bool
-  | 0, _, _, i => nomatch i
-  | _d + 1, x, y, Sum.inl _ => Bool.xor (x 0) (y 0)
-  | d + 1, x, y, Sum.inr (_,i) => routeValue d (Fin.tail x) (Fin.tail y) i
-
-end Thorp
-
-end ThorpNine.Tail
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Tail.Model
+
+namespace OAI
+
+namespace ThorpNine.Tail
+
+namespace Thorp
+
+open scoped BigOperators
+open Filter
+
+def Coins : ℕ → Type
+  | 0 => Unit
+  | d + 1 => Card d → Bool
+
+instance (d : ℕ) : Fintype (Coins d) := by
+  cases d <;> dsimp [Coins] <;> infer_instance
+
+instance (d : ℕ) : Nonempty (Coins d) := by
+  cases d <;> dsimp [Coins] <;> infer_instance
+
+def rotate (d : ℕ) : Equiv.Perm (Card d) where
+  toFun x i := x (finRotate d i)
+  invFun x i := x ((finRotate d).symm i)
+  left_inv x := by funext i; simp only [Equiv.apply_symm_apply]
+  right_inv x := by funext i; simp only [Equiv.symm_apply_apply]
+
+def step : (d : ℕ) → Coins d → Equiv.Perm (Card d)
+  | 0, _ => 1
+  | d + 1, ξ => rotate (d + 1) * pairSwitch ξ
+
+def run (d : ℕ) : (t : ℕ) → (Fin t → Coins d) → Equiv.Perm (Card d)
+  | 0, _ => 1
+  | t + 1, ω => step d (ω (Fin.last t)) * run d t (fun i => ω i.castSucc)
+
+noncomputable def law (d t : ℕ) (g : Equiv.Perm (Card d)) : ℝ :=
+  (Fintype.card {ω : Fin t → Coins d // run d t ω = g} : ℝ) /
+    (Fintype.card (Fin t → Coins d) : ℝ)
+
+noncomputable def distance (d t : ℕ) : ℝ :=
+  (1 / 2 : ℝ) * ∑ g : Equiv.Perm (Card d),
+    |law d t g - 1 / (Nat.factorial (2 ^ d) : ℝ)|
+
+noncomputable def mixingTime (d : ℕ) : ℕ :=
+  sInf {t : ℕ | distance d t ≤ (1 / 4 : ℝ)}
+
+noncomputable def supportTimeBound (d : ℕ) : ℤ :=
+  ⌈(2 / (2 : ℝ) ^ d) *
+    (Real.log ((3 / 4 : ℝ) * (Nat.factorial (2 ^ d) : ℝ)) / Real.log 2)⌉
+
+def OptimalOrder : Prop :=
+  ∃ C : ℕ, 0 < C ∧
+    Tendsto (fun d : ℕ => distance d (C * d)) atTop (nhds 0) ∧
+    (∀ d t : ℕ, 1 - (2 : ℝ) ^ ((t : ℝ) * (2 : ℝ) ^ d / 2) /
+      (Nat.factorial (2 ^ d) : ℝ) ≤ distance d t) ∧
+    (∀ᶠ d : ℕ in atTop,
+      supportTimeBound d ≤ (mixingTime d : ℤ) ∧ mixingTime d ≤ C * d) ∧
+    (∃ K : ℝ, ∀ᶠ d : ℕ in atTop, 2 * (d : ℝ) - K ≤ (mixingTime d : ℝ)) ∧
+    Asymptotics.IsTheta atTop (fun d : ℕ => (mixingTime d : ℝ)) (fun d : ℕ => (d : ℝ))
+
+noncomputable def finiteLaw {α β : Type*} [Fintype α] (f : α → β) (b : β) : ℝ := by
+  classical
+  exact (Fintype.card {a : α // f a = b} : ℝ) / (Fintype.card α : ℝ)
+
+lemma sum_card_fibers {α β : Type*} [Fintype α] [Fintype β] [DecidableEq β] (f : α → β) :
+    ∑ b, Fintype.card {a : α // f a = b} = Fintype.card α := by
+  classical
+  rw [← Fintype.card_sigma]
+  exact Fintype.card_congr (Equiv.sigmaFiberEquiv f)
+
+lemma finiteLaw_nonneg {α β : Type*} [Fintype α] (f : α → β) (b : β) :
+    0 ≤ finiteLaw f b := by
+  exact div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _)
+
+lemma finiteLaw_sum {α β : Type*} [Fintype α] [Nonempty α] [Fintype β] (f : α → β) :
+    ∑ b, finiteLaw f b = 1 := by
+  classical
+  simp only [finiteLaw, ← Finset.sum_div]
+  have hc : (∑ b, (Fintype.card {a : α // f a = b} : ℝ)) = (Fintype.card α : ℝ) := by
+    exact_mod_cast sum_card_fibers f
+  rw [hc]
+  exact div_self (Nat.cast_ne_zero.mpr Fintype.card_ne_zero)
+
+lemma finiteLaw_eq_zero_of_not_mem {α β : Type*} [Fintype α] [DecidableEq β]
+    (f : α → β) {b : β} (hb : b ∉ Finset.univ.image f) : finiteLaw f b = 0 := by
+  classical
+  have : IsEmpty {a : α // f a = b} := ⟨fun a => hb (Finset.mem_image.mpr
+    ⟨a.val, Finset.mem_univ _, a.property⟩)⟩
+  simp [finiteLaw]
+
+lemma event_le_totalVariation {β : Type*} [Fintype β] [DecidableEq β]
+    (p q : β → ℝ) (hpq : ∑ b, (p b - q b) = 0) (s : Finset β) :
+    (∑ b ∈ s, p b) - (∑ b ∈ s, q b) ≤ (1 / 2 : ℝ) * ∑ b, |p b - q b| := by
+  have h₁ : (∑ b ∈ s, (p b - q b)) ≤ ∑ b ∈ s, |p b - q b| :=
+    Finset.sum_le_sum (fun b _ => le_abs_self (p b - q b))
+  have h₂ : -(∑ b ∈ sᶜ, (p b - q b)) ≤ ∑ b ∈ sᶜ, |p b - q b| := by
+    rw [← Finset.sum_neg_distrib]
+    exact Finset.sum_le_sum (fun b _ => neg_le_abs (p b - q b))
+  have hz := Finset.sum_add_sum_compl s (fun b => p b - q b)
+  have ha := Finset.sum_add_sum_compl s (fun b => |p b - q b|)
+  rw [hpq] at hz
+  rw [← Finset.sum_sub_distrib]
+  linarith
+
+lemma finiteLaw_support_bound {α β : Type*} [Fintype α] [Nonempty α]
+    [Fintype β] [Nonempty β] (f : α → β) :
+    1 - (Fintype.card α : ℝ) / (Fintype.card β : ℝ) ≤
+      (1 / 2 : ℝ) * ∑ b, |finiteLaw f b - 1 / (Fintype.card β : ℝ)| := by
+  classical
+  let s := Finset.univ.image f
+  have hs : ∑ b ∈ s, finiteLaw f b = 1 := by
+    have h := Finset.sum_subset (Finset.subset_univ s)
+      (fun b _ hb => finiteLaw_eq_zero_of_not_mem f hb)
+    simpa only [finiteLaw_sum] using h
+  have hβ : (Fintype.card β : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
+  have hq : (∑ _b : β, 1 / (Fintype.card β : ℝ)) = 1 := by simp [hβ]
+  have hd : (∑ b, (finiteLaw f b - 1 / (Fintype.card β : ℝ))) = 0 := by
+    rw [Finset.sum_sub_distrib, finiteLaw_sum, hq, sub_self]
+  have he := event_le_totalVariation (finiteLaw f) (fun _ => 1 / (Fintype.card β : ℝ)) hd s
+  rw [hs] at he
+  have hsq : (∑ _b ∈ s, 1 / (Fintype.card β : ℝ)) = (s.card : ℝ) / Fintype.card β := by
+    simp [div_eq_mul_inv]
+  rw [hsq] at he
+  have hcard : (s.card : ℝ) ≤ Fintype.card α := by
+    exact_mod_cast (Finset.card_image_le (s := (Finset.univ : Finset α)) (f := f))
+  have hdiv := div_le_div_of_nonneg_right hcard (Nat.cast_nonneg (Fintype.card β) : (0 : ℝ) ≤ _)
+  linarith
+
+lemma card_positions (d : ℕ) : Fintype.card (Card d) = 2 ^ d := by
+  simp [Card]
+
+lemma card_permutations (d : ℕ) : Fintype.card (Equiv.Perm (Card d)) = Nat.factorial (2 ^ d) := by
+  rw [Fintype.card_perm, card_positions]
+
+lemma card_coins_succ (d : ℕ) : Fintype.card (Coins (d + 1)) = 2 ^ (2 ^ d) := by
+  simp [Coins]
+
+lemma card_tapes_succ (d t : ℕ) : Fintype.card (Fin t → Coins (d + 1)) = 2 ^ (t * 2 ^ d) := by
+  rw [Fintype.card_fun, card_coins_succ, Fintype.card_fin, ← pow_mul, Nat.mul_comm]
+
+lemma law_eq_finiteLaw (d t : ℕ) (g : Equiv.Perm (Card d)) :
+    law d t g = finiteLaw (run d t) g := by
+  unfold law finiteLaw
+  apply congrArg (fun n : ℕ => (n : ℝ) / (Fintype.card (Fin t → Coins d) : ℝ))
+  exact congrArg (@Fintype.card _) (Subsingleton.elim _ _)
+
+lemma law_nonneg (d t : ℕ) (g : Equiv.Perm (Card d)) : 0 ≤ law d t g := by
+  rw [law_eq_finiteLaw]
+  exact finiteLaw_nonneg (run d t) g
+
+lemma law_sum (d t : ℕ) : ∑ g, law d t g = 1 := by
+  simp only [law_eq_finiteLaw]
+  exact finiteLaw_sum (run d t)
+
+lemma physical_support_bound_succ (d t : ℕ) :
+    1 - (2 : ℝ) ^ (t * 2 ^ d) / (Nat.factorial (2 ^ (d + 1)) : ℝ) ≤ distance (d + 1) t := by
+  have : Nonempty (Equiv.Perm (Card (d + 1))) := ⟨1⟩
+  have h := finiteLaw_support_bound (run (d + 1) t)
+  rw [card_tapes_succ, card_permutations] at h
+  simpa only [Nat.cast_pow, Nat.cast_ofNat, distance, law_eq_finiteLaw] using h
+
+lemma step_succ_formula (d : ℕ) (ξ : Coins (d + 1)) (x : Card (d + 1)) :
+    step (d + 1) ξ x = Fin.snoc (Fin.tail x) (Bool.xor (x 0) (ξ (Fin.tail x))) := by
+  change (fun i => switchFun ξ x (finRotate (d + 1) i)) = _
+  rw [Fin.snoc_eq_cons_rotate]
+  rfl
+
+lemma distance_nonneg (d t : ℕ) : 0 ≤ distance d t := by
+  unfold distance
+  exact mul_nonneg (by norm_num) (Finset.sum_nonneg (fun _ _ => abs_nonneg _))
+
+lemma physical_support_bound (d t : ℕ) :
+    1 - (2 : ℝ) ^ ((t : ℝ) * (2 : ℝ) ^ d / 2) /
+      (Nat.factorial (2 ^ d) : ℝ) ≤ distance d t := by
+  cases d with
+  | zero =>
+    have h : 1 ≤ (2 : ℝ) ^ ((t : ℝ) / 2) := Real.one_le_rpow (by norm_num) (by positivity)
+    simpa only [pow_zero, Nat.factorial_one, Nat.cast_one, div_one, mul_one] using
+      le_trans (sub_nonpos.mpr h) (distance_nonneg 0 t)
+  | succ d =>
+    have he : (t : ℝ) * (2 : ℝ) ^ (d + 1) / 2 = (t * 2 ^ d : ℕ) := by
+      push_cast
+      rw [pow_succ]
+      ring
+    rw [he, Real.rpow_natCast]
+    exact physical_support_bound_succ d t
+
+lemma supportTimeBound_le_of_distance_le (d t : ℕ) (ht : distance d t ≤ (1 / 4 : ℝ)) :
+    supportTimeBound d ≤ (t : ℤ) := by
+  have hF : (0 : ℝ) < Nat.factorial (2 ^ d) := by positivity
+  have hN : (0 : ℝ) < (2 : ℝ) ^ d := by positivity
+  have hL : (0 : ℝ) < Real.log 2 := Real.log_pos (by norm_num)
+  have hb := physical_support_bound d t
+  have hp : (3 / 4 : ℝ) ≤
+      (2 : ℝ) ^ ((t : ℝ) * (2 : ℝ) ^ d / 2) / (Nat.factorial (2 ^ d) : ℝ) := by
+    linarith
+  have hr := (le_div_iff₀ hF).mp hp
+  have hlog := Real.log_le_log (show (0 : ℝ) < (3 / 4 : ℝ) * Nat.factorial (2 ^ d) by positivity) hr
+  rw [Real.log_rpow (by norm_num : (0 : ℝ) < 2)] at hlog
+  unfold supportTimeBound
+  apply Int.ceil_le.mpr
+  simp only [Int.cast_natCast]
+  calc
+    2 / (2 : ℝ) ^ d * (Real.log ((3 / 4 : ℝ) * Nat.factorial (2 ^ d)) / Real.log 2) =
+        Real.log ((3 / 4 : ℝ) * Nat.factorial (2 ^ d)) / ((2 : ℝ) ^ d / 2 * Real.log 2) := by
+      field_simp
+    _ ≤ (t : ℝ) := (div_le_iff₀ (mul_pos (div_pos hN (by norm_num)) hL)).mpr (by
+      nlinarith [hlog])
+
+lemma supportTimeBound_le_mixingTime {d : ℕ}
+    (h : ({t : ℕ | distance d t ≤ (1 / 4 : ℝ)} : Set ℕ).Nonempty) :
+    supportTimeBound d ≤ (mixingTime d : ℤ) := by
+  exact supportTimeBound_le_of_distance_le d (mixingTime d) (Nat.sInf_mem h)
+
+lemma log_factorial_lower {n : ℕ} (hn : 1 ≤ n) :
+    (n : ℝ) * Real.log n - n ≤ Real.log (Nat.factorial n) := by
+  have h := Stirling.le_log_factorial_stirling (show n ≠ 0 by omega)
+  have hln : 0 ≤ Real.log (n : ℝ) := Real.log_nonneg (by exact_mod_cast hn)
+  have hlp : 0 ≤ Real.log (2 * Real.pi) := Real.log_nonneg (by
+    have := Real.pi_gt_three
+    linarith)
+  linarith
+
+lemma log_three_quarters_lower : -(1 : ℝ) ≤ Real.log (3 / 4 : ℝ) := by
+  have h := Real.log_le_sub_one_of_pos (show (0 : ℝ) < 4 / 3 by norm_num)
+  have he : Real.log (4 / 3 : ℝ) = - Real.log (3 / 4 : ℝ) := by
+    rw [← Real.log_inv]
+    congr 1
+    norm_num
+  rw [he] at h
+  linarith
+
+lemma linear_le_supportTimeBound (d : ℕ) :
+    2 * (d : ℝ) - 4 / Real.log 2 ≤ (supportTimeBound d : ℝ) := by
+  have hN : 1 ≤ (2 : ℝ) ^ d := one_le_pow₀ (by norm_num)
+  have hNp : 0 < (2 : ℝ) ^ d := by positivity
+  have hL : 0 < Real.log 2 := Real.log_pos (by norm_num)
+  have hf := log_factorial_lower (show 1 ≤ 2 ^ d by exact one_le_pow₀ (by norm_num))
+  have hlog : (2 : ℝ) ^ d * ((d : ℝ) * Real.log 2) - (2 : ℝ) ^ d - 1 ≤
+      Real.log ((3 / 4 : ℝ) * (Nat.factorial (2 ^ d) : ℝ)) := by
+    rw [Real.log_mul (by norm_num) (by positivity)]
+    push_cast at hf
+    rw [Real.log_pow] at hf
+    linarith [log_three_quarters_lower]
+  apply le_trans _ (Int.le_ceil _)
+  have hmul : (2 * (d : ℝ) - 4 / Real.log 2) *
+      ((2 : ℝ) ^ d / 2 * Real.log 2) ≤
+      Real.log ((3 / 4 : ℝ) * (Nat.factorial (2 ^ d) : ℝ)) := by
+    have hcancel : (2 * (d : ℝ) - 4 / Real.log 2) *
+        ((2 : ℝ) ^ d / 2 * Real.log 2) =
+        (2 : ℝ) ^ d * ((d : ℝ) * Real.log 2) - 2 * (2 : ℝ) ^ d := by
+      field_simp
+      ring
+    rw [hcancel]
+    linarith
+  convert (le_div_iff₀ (mul_pos (div_pos hNp (by norm_num)) hL)).mpr hmul using 1
+  field_simp
+
+lemma linear_le_time_of_distance_le (d t : ℕ)
+    (ht : distance d t ≤ (1 / 4 : ℝ)) :
+    2 * (d : ℝ) - 4 / Real.log 2 ≤ (t : ℝ) := by
+  have h := supportTimeBound_le_of_distance_le d t ht
+  have hr : (supportTimeBound d : ℝ) ≤ (t : ℝ) := by exact_mod_cast h
+  exact (linear_le_supportTimeBound d).trans hr
+
+lemma entropy_chord (p : ℝ) (hp : 0 ≤ p) (hq : p ≤ 1 / 2) :
+    2 * p * Real.log 2 ≤ Real.binEntropy p := by
+  have hc := Real.strictConcave_binEntropy.concaveOn.2
+    (show (0 : ℝ) ∈ Set.Icc (0 : ℝ) 1 by norm_num)
+    (show (2 : ℝ)⁻¹ ∈ Set.Icc (0 : ℝ) 1 by norm_num)
+    (show 0 ≤ 1 - 2 * p by linarith)
+    (show 0 ≤ 2 * p by positivity)
+    (show (1 - 2 * p) + 2 * p = 1 by ring)
+  simpa only [Real.binEntropy_zero, Real.binEntropy_two_inv,
+    smul_eq_mul, mul_zero, zero_add, show 2 * p * (2 : ℝ)⁻¹ = p by ring] using hc
+
+lemma split_entropy_lower (a b : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) :
+    2 * min a b * Real.log 2 ≤
+      (a + b) * Real.log (a + b) - a * Real.log a - b * Real.log b := by
+  wlog hab : a ≤ b generalizing a b
+  · have hba := this b a hb ha (le_of_not_ge hab)
+    simpa only [min_comm, add_comm b a] using hba.trans_eq (show (b + a) * Real.log (b + a) - b * Real.log b - a * Real.log a =
+      (a + b) * Real.log (a + b) - a * Real.log a - b * Real.log b by rw [add_comm b a]; ring)
+  rw [min_eq_left hab]
+  by_cases hz : a = 0
+  · subst a
+    simp
+  have hap : 0 < a := lt_of_le_of_ne ha (Ne.symm hz)
+  have hbp : 0 < b := hap.trans_le hab
+  have hs : 0 < a + b := add_pos hap hbp
+  have hp : 0 ≤ a / (a + b) := div_nonneg ha hs.le
+  have hq : a / (a + b) ≤ 1 / 2 := (div_le_iff₀ hs).mpr (by linarith)
+  have hc := mul_le_mul_of_nonneg_left (entropy_chord (a / (a + b)) hp hq) hs.le
+  have he : (a + b) * Real.binEntropy (a / (a + b)) =
+      (a + b) * Real.log (a + b) - a * Real.log a - b * Real.log b := by
+    unfold Real.binEntropy
+    rw [show 1 - a / (a + b) = b / (a + b) by field_simp; ring]
+    simp only [Real.log_inv]
+    rw [Real.log_div (ne_of_gt hap) (ne_of_gt hs),
+      Real.log_div (ne_of_gt hbp) (ne_of_gt hs)]
+    field_simp
+    ring
+  rw [he] at hc
+  calc
+    2 * a * Real.log 2 = (a + b) * (2 * (a / (a + b)) * Real.log 2) := by
+      field_simp
+    _ ≤ _ := hc
+
+noncomputable def childMarks {d : ℕ} (s : Finset (Card (d + 1))) (ε : Bool) :
+    Finset (Card d) := (s.filter (fun x => x 0 = ε)).image Fin.tail
+
+lemma card_childMarks {d : ℕ} (s : Finset (Card (d + 1))) (ε : Bool) :
+    (childMarks s ε).card = (s.filter (fun x => x 0 = ε)).card := by
+  classical
+  unfold childMarks
+  apply Finset.card_image_iff.mpr
+  intro x hx y hy hxy
+  have hx0 := (Finset.mem_filter.mp hx).2
+  have hy0 := (Finset.mem_filter.mp hy).2
+  rw [← Fin.cons_self_tail x, ← Fin.cons_self_tail y, hx0, hy0, hxy]
+
+lemma card_childMarks_add {d : ℕ} (s : Finset (Card (d + 1))) :
+    (childMarks s false).card + (childMarks s true).card = s.card := by
+  classical
+  rw [card_childMarks, card_childMarks]
+  simpa only [Bool.not_eq_false] using
+    (Finset.card_filter_add_card_filter_not (s := s) (fun x => x 0 = false))
+
+noncomputable def topMeetings {d : ℕ} (b : Butterfly (d + 1))
+    (s : Finset (Card (d + 1))) : Finset (Card d) :=
+  ((childMarks s false).image (butterflyPerm d (b.2 false))) ∩
+    ((childMarks s true).image (butterflyPerm d (b.2 true)))
+
+lemma card_topMeetings_le {d : ℕ} (b : Butterfly (d + 1)) (s : Finset (Card (d + 1))) :
+    (topMeetings b s).card ≤ min (childMarks s false).card (childMarks s true).card := by
+  classical
+  apply le_min
+  · exact (Finset.card_le_card Finset.inter_subset_left).trans_eq
+      (Finset.card_image_of_injective _ (butterflyPerm d (b.2 false)).injective)
+  · exact (Finset.card_le_card Finset.inter_subset_right).trans_eq
+      (Finset.card_image_of_injective _ (butterflyPerm d (b.2 true)).injective)
+
+noncomputable def encounters : (d : ℕ) → Butterfly d → Finset (Card d) → ℕ
+  | 0, _, _ => 0
+  | d + 1, b, s => encounters d (b.2 false) (childMarks s false) +
+      encounters d (b.2 true) (childMarks s true) + (topMeetings b s).card
+
+lemma butterfly_encounters (d : ℕ) (b : Butterfly d) (s : Finset (Card d)) :
+    (encounters d b s : ℝ) ≤ (s.card : ℝ) * Real.log s.card / (2 * Real.log 2) := by
+  induction d with
+  | zero =>
+    simp only [encounters, Nat.cast_zero]
+    have hs : s.card ≤ 1 := by
+      simpa [Card] using Finset.card_le_univ s
+    interval_cases s.card <;> simp
+  | succ d ih =>
+    have h₀ := ih (b.2 false) (childMarks s false)
+    have h₁ := ih (b.2 true) (childMarks s true)
+    have ht : ((topMeetings b s).card : ℝ) ≤
+        min ((childMarks s false).card : ℝ) ((childMarks s true).card : ℝ) := by
+      exact_mod_cast card_topMeetings_le b s
+    have he := split_entropy_lower ((childMarks s false).card : ℝ)
+      ((childMarks s true).card : ℝ) (by positivity) (by positivity)
+    have hs : ((childMarks s false).card : ℝ) + (childMarks s true).card = s.card := by
+      exact_mod_cast card_childMarks_add s
+    rw [hs] at he
+    have hL : 0 < 2 * Real.log 2 := by positivity
+    rw [le_div_iff₀ hL] at h₀ h₁ ⊢
+    simp only [encounters, Nat.cast_add]
+    nlinarith [mul_le_mul_of_nonneg_right ht hL.le]
+
+lemma prod_split_at {α : Type*} [DecidableEq α] (s : Finset α) (f : α → ℝ) (a : α) :
+    ∏ i ∈ s, f i = (if a ∈ s then f a else 1) * ∏ i ∈ s.erase a, f i := by
+  by_cases ha : a ∈ s
+  · rw [ite_eq_left ha]
+    exact (Finset.mul_prod_erase s f ha).symm
+  · simp [ha]
+
+lemma prod_split_two {α : Type*} [DecidableEq α] (s : Finset α) (f : α → ℝ)
+    (a b : α) (hab : a ≠ b) :
+    ∏ i ∈ s, f i = (if a ∈ s then f a else 1) * (if b ∈ s then f b else 1) *
+      ∏ i ∈ (s.erase a).erase b, f i := by
+  calc
+    ∏ i ∈ s, f i = (if a ∈ s then f a else 1) * ∏ i ∈ s.erase a, f i :=
+      prod_split_at s f a
+    _ = _ := by
+      rw [prod_split_at (s.erase a) f b]
+      have hm : (b ∈ s.erase a) ↔ b ∈ s :=
+        ⟨fun h => (Finset.mem_erase.mp h).2, fun h => Finset.mem_erase.mpr ⟨Ne.symm hab, h⟩⟩
+      simp only [hm, mul_assoc]
+
+lemma swap_average_product {α : Type*} [DecidableEq α] (s : Finset α) (p : α → ℝ)
+    (hp : ∀ i, 0 ≤ p i) (a b : α) :
+    ((∏ i ∈ s, p i) + ∏ i ∈ s, p (Equiv.swap a b i)) / 2 ≤
+      ∏ i ∈ s, (p i + p (Equiv.swap a b i)) / 2 := by
+  by_cases hab : a = b
+  · subst b
+    simp only [Equiv.swap_self, Equiv.refl_apply]
+    have h : (fun i => (p i + p i) / 2) = p := by funext i; ring
+    rw [h]
+    linarith
+  let t := (s.erase a).erase b
+  have hfix : ∀ i ∈ t, Equiv.swap a b i = i := by
+    intro i hi
+    exact Equiv.swap_apply_of_ne_of_ne (Finset.mem_erase.mp (Finset.mem_erase.mp hi).2).1
+      (Finset.mem_erase.mp hi).1
+  have hprod : (∏ i ∈ t, p (Equiv.swap a b i)) = ∏ i ∈ t, p i :=
+    Finset.prod_congr rfl (fun i hi => by rw [hfix i hi])
+  have havg : (∏ i ∈ t, (p i + p (Equiv.swap a b i)) / 2) = ∏ i ∈ t, p i :=
+    Finset.prod_congr rfl (fun i hi => by rw [hfix i hi]; ring)
+  rw [prod_split_two s p a b hab,
+    prod_split_two s (fun i => p (Equiv.swap a b i)) a b hab,
+    prod_split_two s (fun i => (p i + p (Equiv.swap a b i)) / 2) a b hab]
+  change ((if a ∈ s then p a else 1) * (if b ∈ s then p b else 1) * (∏ i ∈ t, p i) +
+    (if a ∈ s then p (Equiv.swap a b a) else 1) *
+      (if b ∈ s then p (Equiv.swap a b b) else 1) * (∏ i ∈ t, p (Equiv.swap a b i))) / 2 ≤ _
+  rw [hprod]
+  change _ ≤ (if a ∈ s then (p a + p (Equiv.swap a b a)) / 2 else 1) *
+    (if b ∈ s then (p b + p (Equiv.swap a b b)) / 2 else 1) *
+      (∏ i ∈ t, (p i + p (Equiv.swap a b i)) / 2)
+  rw [havg, Equiv.swap_apply_left, Equiv.swap_apply_right]
+  have ht : 0 ≤ ∏ i ∈ t, p i := Finset.prod_nonneg (fun i _ => hp i)
+  by_cases ha : a ∈ s <;> by_cases hb : b ∈ s <;> simp only [ha, hb, ↓reduceIte]
+  · have hsq := mul_nonneg (sq_nonneg (p a - p b)) ht
+    nlinarith
+  all_goals ring_nf; rfl
+
+def CylinderBound {α : Type*} (m : Finset α → ℝ) (p : α → ℝ) : Prop :=
+  ∀ s, m s ≤ ∏ i ∈ s, p i
+
+lemma CylinderBound.fair_swap {α : Type*} [DecidableEq α]
+    {m : Finset α → ℝ} {p : α → ℝ} (h : CylinderBound m p)
+    (hp : ∀ i, 0 ≤ p i) (a b : α) :
+    CylinderBound (fun s => (m s + m (s.image (Equiv.swap a b))) / 2)
+      (fun i => (p i + p (Equiv.swap a b i)) / 2) := by
+  intro s
+  have h₀ := h s
+  have h₁ := h (s.image (Equiv.swap a b))
+  rw [Finset.prod_image (Equiv.swap a b).injective.injOn] at h₁
+  exact (show (m s + m (s.image (Equiv.swap a b))) / 2 ≤
+      ((∏ i ∈ s, p i) + ∏ i ∈ s, p (Equiv.swap a b i)) / 2 by linarith).trans
+    (swap_average_product s p hp a b)
+
+noncomputable def inclusionMoment {α Ω : Type*} [Fintype Ω]
+    (F : Ω → Finset α) (s : Finset α) : ℝ := by
+  classical
+  exact finiteMean (fun ω => if s ⊆ F ω then 1 else 0)
+
+lemma inclusionMoment_nonneg {α Ω : Type*} [Fintype Ω]
+    (F : Ω → Finset α) (s : Finset α) : 0 ≤ inclusionMoment F s := by
+  classical
+  unfold inclusionMoment finiteMean
+  positivity
+
+lemma indicator_subset_prod {α : Type*} [DecidableEq α] (A s : Finset α) :
+    (if s ⊆ A then (1 : ℝ) else 0) = ∏ i ∈ s, (if i ∈ A then 1 else 0) := by
+  by_cases h : s ⊆ A
+  · rw [ite_eq_left h]
+    exact (Finset.prod_eq_one (fun i hi => ite_eq_left (h hi))).symm
+  · rw [ite_eq_right h]
+    obtain ⟨i, hi, hiA⟩ := Finset.not_subset.mp h
+    exact (Finset.prod_eq_zero hi (ite_eq_right hiA)).symm
+
+lemma inclusionMoment_const {α Ω : Type*} [DecidableEq α] [Fintype Ω] [Nonempty Ω]
+    (A s : Finset α) : inclusionMoment (fun _ : Ω => A) s =
+      if s ⊆ A then 1 else 0 := by
+  classical
+  have hn : (Fintype.card Ω : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
+  by_cases h : s ⊆ A <;> simp [inclusionMoment, finiteMean, h, hn]
+
+lemma cylinder_const {α Ω : Type*} [Fintype Ω] [Nonempty Ω] (A : Finset α) :
+    CylinderBound (inclusionMoment (fun _ : Ω => A))
+      (fun i => inclusionMoment (fun _ : Ω => A) {i}) := by
+  classical
+  intro s
+  simp only [inclusionMoment_const, Finset.singleton_subset_iff]
+  exact (indicator_subset_prod A s).le
+
+noncomputable def fairSetSwap {α Ω : Type*} [DecidableEq α]
+    (F : Ω → Finset α) (a b : α) : Ω × Bool → Finset α :=
+  fun ω => if ω.2 then (F ω.1).image (Equiv.swap a b) else F ω.1
+
+lemma subset_swap_image_iff {α : Type*} [DecidableEq α]
+    (A s : Finset α) (a b : α) :
+    s ⊆ A.image (Equiv.swap a b) ↔ s.image (Equiv.swap a b) ⊆ A := by
+  constructor
+  · intro h
+    have h' := Finset.image_subset_image h (f := Equiv.swap a b)
+    simpa only [Finset.image_image, Function.comp_def, Equiv.swap_apply_self, Finset.image_id'] using h'
+  · intro h
+    have h' := Finset.image_subset_image h (f := Equiv.swap a b)
+    simpa only [Finset.image_image, Function.comp_def, Equiv.swap_apply_self, Finset.image_id'] using h'
+
+lemma inclusionMoment_fairSetSwap {α Ω : Type*} [DecidableEq α] [Fintype Ω]
+    (F : Ω → Finset α) (a b : α) (s : Finset α) :
+    inclusionMoment (fairSetSwap F a b) s =
+      (inclusionMoment F s + inclusionMoment F (s.image (Equiv.swap a b))) / 2 := by
+  classical
+  simp only [inclusionMoment, finiteMean, Fintype.sum_prod_type, Fintype.sum_bool,
+    fairSetSwap, Bool.false_eq_true, ↓reduceIte, Fintype.card_prod, Fintype.card_bool,
+    Nat.cast_mul, Nat.cast_ofNat, subset_swap_image_iff, Finset.sum_add_distrib]
+  rw [div_mul_eq_div_div, add_div]
+  ring_nf
+
+lemma cylinder_fairSetSwap {α Ω : Type*} [DecidableEq α] [Fintype Ω]
+    (F : Ω → Finset α) (a b : α)
+    (h : CylinderBound (inclusionMoment F) (fun i => inclusionMoment F {i})) :
+    CylinderBound (inclusionMoment (fairSetSwap F a b))
+      (fun i => inclusionMoment (fairSetSwap F a b) {i}) := by
+  classical
+  have h' := h.fair_swap (fun i => inclusionMoment_nonneg F {i}) a b
+  intro s
+  simpa only [inclusionMoment_fairSetSwap, Finset.image_singleton] using h' s
+
+def SwitchSamples : ℕ → Type
+  | 0 => Unit
+  | n + 1 => SwitchSamples n × Bool
+
+instance (n : ℕ) : Fintype (SwitchSamples n) := by
+  induction n with
+  | zero => exact inferInstanceAs (Fintype Unit)
+  | succ n ih => exact inferInstanceAs (Fintype (SwitchSamples n × Bool))
+
+instance (n : ℕ) : Nonempty (SwitchSamples n) := by
+  induction n with
+  | zero => exact inferInstanceAs (Nonempty Unit)
+  | succ n ih => exact inferInstanceAs (Nonempty (SwitchSamples n × Bool))
+
+noncomputable def switchedSet {α : Type*} [DecidableEq α] (A : Finset α) :
+    (L : List (α × α)) → SwitchSamples L.length → Finset α
+  | [], _ => A
+  | (a,b) :: L, ω => fairSetSwap (switchedSet A L) a b ω
+
+lemma switchedSet_cylinder {α : Type*} [DecidableEq α] (A : Finset α)
+    (L : List (α × α)) :
+    CylinderBound (inclusionMoment (switchedSet A L))
+      (fun i => inclusionMoment (switchedSet A L) {i}) := by
+  induction L with
+  | nil => exact cylinder_const A
+  | cons e L ih => exact cylinder_fairSetSwap (switchedSet A L) e.1 e.2 ih
+
+inductive QueryTree (ι : Type*)
+  | reject : QueryTree ι
+  | accept : QueryTree ι
+  | branch (i : ι) (zero one : QueryTree ι) : QueryTree ι
+  | force (i : ι) (b : Bool) (next : QueryTree ι) : QueryTree ι
+
+namespace QueryTree
+
+variable {ι : Type*} [DecidableEq ι]
+
+def queries : QueryTree ι → Finset ι
+  | reject => ∅
+  | accept => ∅
+  | branch i zero one => insert i (queries zero ∪ queries one)
+  | force i _ next => insert i (queries next)
+
+def Fresh : QueryTree ι → Prop
+  | reject => True
+  | accept => True
+  | branch i zero one => i ∉ queries zero ∧ i ∉ queries one ∧ Fresh zero ∧ Fresh one
+  | force i _ next => i ∉ queries next ∧ Fresh next
+
+def weighted : QueryTree ι → (ι → Bool) → ℝ
+  | reject, _ => 0
+  | accept, _ => 1
+  | branch i zero one, ω => if ω i then weighted one ω else weighted zero ω
+  | force i b next, ω => if ω i = b then 2 * weighted next ω else 0
+
+omit [DecidableEq ι] in
+lemma weighted_nonneg (T : QueryTree ι) (ω : ι → Bool) : 0 ≤ T.weighted ω := by
+  induction T with
+  | reject => exact le_rfl
+  | accept => exact zero_le_one
+  | branch i zero one ih₀ ih₁ => simpa only [weighted] using ite_nonneg ih₁ ih₀
+  | force i b next ih => simpa only [weighted] using ite_nonneg (mul_nonneg (by norm_num) ih) le_rfl
+
+lemma weighted_congr (T : QueryTree ι) {ω η : ι → Bool}
+    (h : ∀ i ∈ T.queries, ω i = η i) : T.weighted ω = T.weighted η := by
+  induction T with
+  | reject => rfl
+  | accept => rfl
+  | branch i zero one ih₀ ih₁ =>
+    have hi := h i (Finset.mem_insert_self _ _)
+    simp only [weighted, hi]
+    rw [ih₀ (fun j hj => h j (Finset.mem_insert_of_mem (Finset.mem_union_left _ hj))),
+      ih₁ (fun j hj => h j (Finset.mem_insert_of_mem (Finset.mem_union_right _ hj)))]
+  | force i b next ih =>
+    have hi := h i (Finset.mem_insert_self _ _)
+    simp only [weighted, hi]
+    rw [ih (fun j hj => h j (Finset.mem_insert_of_mem hj))]
+
+def flipCoin (i : ι) : Equiv.Perm (ι → Bool) :=
+  (show Function.Involutive (fun ω : ι → Bool => Function.update ω i (!(ω i))) from by
+    intro ω
+    funext j
+    by_cases h : j = i
+    · subst j; simp
+    · simp [Function.update_of_ne h]).toPerm
+
+lemma flipCoin_apply_eq (i : ι) (ω : ι → Bool) : flipCoin i ω i = !(ω i) := by
+  change Function.update ω i (!(ω i)) i = _
+  exact Function.update_self _ _ _
+
+lemma flipCoin_apply_ne (i j : ι) (h : j ≠ i) (ω : ι → Bool) :
+    flipCoin i ω j = ω j := by
+  change Function.update ω i (!(ω i)) j = _
+  exact Function.update_of_ne h _ _
+
+lemma weighted_flip_of_not_mem (T : QueryTree ι) (i : ι) (hi : i ∉ T.queries)
+    (ω : ι → Bool) : T.weighted (flipCoin i ω) = T.weighted ω := by
+  apply weighted_congr
+  intro j hj
+  exact flipCoin_apply_ne i j (fun h => hi (h ▸ hj)) ω
+
+lemma mean_indicator_bit [Fintype ι] (f : (ι → Bool) → ℝ) (i : ι) (b : Bool)
+    (hf : ∀ ω, f (flipCoin i ω) = f ω) :
+    finiteMean (fun ω => if ω i = b then f ω else 0) = finiteMean f / 2 := by
+  let g := fun ω : ι → Bool => if ω i = b then f ω else 0
+  have hp (ω : ι → Bool) : g ω + g (flipCoin i ω) = f ω := by
+    dsimp [g]
+    rw [flipCoin_apply_eq, hf]
+    cases hω : ω i <;> cases b <;> simp
+  have hs := Equiv.sum_comp (flipCoin i) g
+  have hadd : (∑ ω, g ω) + (∑ ω, g (flipCoin i ω)) = ∑ ω, f ω := by
+    rw [← Finset.sum_add_distrib]
+    exact Finset.sum_congr rfl (fun ω _ => hp ω)
+  unfold finiteMean
+  change (∑ ω, g ω) / _ = _
+  rw [hs] at hadd
+  have hh : (∑ ω, g ω) = (∑ ω, f ω) / 2 := by linarith
+  rw [hh]
+  ring
+
+lemma mean_weighted_le_one [Fintype ι] (T : QueryTree ι) (hT : T.Fresh) :
+    finiteMean T.weighted ≤ 1 := by
+  induction T with
+  | reject => simp [weighted, finiteMean]
+  | accept => simp [weighted, finiteMean]
+  | branch i zero one ih₀ ih₁ =>
+    obtain ⟨hz, ho, hfz, hfo⟩ := hT
+    have h₀ := mean_indicator_bit zero.weighted i false (zero.weighted_flip_of_not_mem i hz)
+    have h₁ := mean_indicator_bit one.weighted i true (one.weighted_flip_of_not_mem i ho)
+    have he : finiteMean (branch i zero one).weighted =
+        finiteMean (fun ω => if ω i = false then zero.weighted ω else 0) +
+        finiteMean (fun ω => if ω i = true then one.weighted ω else 0) := by
+      unfold finiteMean
+      rw [← add_div, ← Finset.sum_add_distrib]
+      congr 1
+      apply Finset.sum_congr rfl
+      intro ω _
+      cases hω : ω i <;> simp [weighted, hω]
+    rw [he, h₀, h₁]
+    linarith [ih₀ hfz, ih₁ hfo]
+  | force i b next ih =>
+    obtain ⟨hi, hn⟩ := hT
+    have hm := mean_indicator_bit next.weighted i b (next.weighted_flip_of_not_mem i hi)
+    have he : finiteMean (force i b next).weighted =
+        2 * finiteMean (fun ω => if ω i = b then next.weighted ω else 0) := by
+      unfold finiteMean
+      rw [← mul_div_assoc, Finset.mul_sum]
+      congr 1
+      apply Finset.sum_congr rfl
+      intro ω _
+      by_cases h : ω i = b <;> simp [weighted, h]
+    rw [he, hm]
+    linarith [ih hn]
+
+def succeeds : QueryTree ι → (ι → Bool) → Bool
+  | reject, _ => false
+  | accept, _ => true
+  | branch i zero one, ω => if ω i then succeeds one ω else succeeds zero ω
+  | force i b next, ω => if ω i = b then succeeds next ω else false
+
+def charge : QueryTree ι → (ι → Bool) → ℕ
+  | reject, _ => 0
+  | accept, _ => 0
+  | branch i zero one, ω => if ω i then charge one ω else charge zero ω
+  | force i b next, ω => if ω i = b then charge next ω + 1 else 0
+
+omit [DecidableEq ι] in
+lemma weighted_eq (T : QueryTree ι) (ω : ι → Bool) :
+    T.weighted ω = if T.succeeds ω then (2 : ℝ) ^ T.charge ω else 0 := by
+  induction T with
+  | reject => rfl
+  | accept => norm_num [weighted, succeeds, charge]
+  | branch i zero one ih₀ ih₁ =>
+    cases h : ω i <;> simp [weighted, succeeds, charge, h, ih₀, ih₁]
+  | force i b next ih =>
+    by_cases h : ω i = b
+    · simp only [weighted, succeeds, charge, h, ↓reduceIte, ih]
+      cases hs : next.succeeds ω <;> simp [pow_succ, mul_comm]
+    · simp [weighted, succeeds, h]
+
+lemma success_probability_le [Fintype ι] (T : QueryTree ι) (hT : T.Fresh)
+    (K : ℕ) (hK : ∀ ω, T.succeeds ω = true → K ≤ T.charge ω) :
+    finiteMean (fun ω => if T.succeeds ω then (1 : ℝ) else 0) ≤ 1 / (2 : ℝ) ^ K := by
+  have hw (ω : ι → Bool) :
+      (2 : ℝ) ^ K * (if T.succeeds ω then (1 : ℝ) else 0) ≤ T.weighted ω := by
+    rw [weighted_eq]
+    cases hs : T.succeeds ω
+    · simp
+    · simpa using pow_le_pow_right₀ (show (1 : ℝ) ≤ 2 by norm_num) (hK ω hs)
+  have hn : 0 ≤ (Fintype.card (ι → Bool) : ℝ) := Nat.cast_nonneg _
+  have hm := div_le_div_of_nonneg_right
+    (Finset.sum_le_sum (fun ω (_ : ω ∈ Finset.univ) => hw ω)) hn
+  rw [← Finset.mul_sum, mul_div_assoc] at hm
+  change (2 : ℝ) ^ K * finiteMean (fun ω => if T.succeeds ω then 1 else 0) ≤
+    finiteMean T.weighted at hm
+  exact (le_div_iff₀ (pow_pos (by norm_num) K)).mpr (by
+    rw [mul_comm]
+    exact hm.trans (mean_weighted_le_one T hT))
+
+end QueryTree
+
+def encodeButterfly : (d : ℕ) → Butterfly d → SwitchIndex d → Bool
+  | 0, _, i => nomatch i
+  | _d + 1, b, Sum.inl y => b.1 y
+  | d + 1, b, Sum.inr (ε,i) => encodeButterfly d (b.2 ε) i
+
+lemma decode_encodeButterfly (d : ℕ) (b : Butterfly d) :
+    decodeButterfly d (encodeButterfly d b) = b := by
+  induction d with
+  | zero => exact Unit.ext _ _
+  | succ d ih =>
+    apply Prod.ext
+    · rfl
+    · funext ε
+      exact ih (b.2 ε)
+
+lemma encode_decodeButterfly (d : ℕ) (ω : SwitchIndex d → Bool) :
+    encodeButterfly d (decodeButterfly d ω) = ω := by
+  induction d with
+  | zero => funext i; exact Empty.elim i
+  | succ d ih =>
+    funext i
+    cases i with
+    | inl y => rfl
+    | inr z => exact congrFun (ih (fun i => ω (Sum.inr (z.1,i)))) z.2
+
+def butterflyCoinEquiv (d : ℕ) : (SwitchIndex d → Bool) ≃ Butterfly d where
+  toFun := decodeButterfly d
+  invFun := encodeButterfly d
+  left_inv := encode_decodeButterfly d
+  right_inv := decode_encodeButterfly d
+
+noncomputable def routeDomain : (d : ℕ) → Card d → Card d → Finset (SwitchIndex d)
+  | 0, _, _ => ∅
+  | d + 1, x, y => by
+      classical
+      exact insert (Sum.inl (Fin.tail y))
+        ((routeDomain d (Fin.tail x) (Fin.tail y)).image (fun i => Sum.inr (x 0,i)))
+
+def routeValue : (d : ℕ) → Card d → Card d → SwitchIndex d → Bool
+  | 0, _, _, i => nomatch i
+  | _d + 1, x, y, Sum.inl _ => Bool.xor (x 0) (y 0)
+  | d + 1, x, y, Sum.inr (_,i) => routeValue d (Fin.tail x) (Fin.tail y) i
+
+end Thorp
+
+end ThorpNine.Tail
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Tail/ExposedCoins.lean b/lean/OAI/Probability/ThorpRouting/Tail/ExposedCoins.lean
index 6ff3cfeeac4f5cecd3517a414799559106006a0c..3503b48e92c3773f3cdaac85342621cb1ab8364e 100644
--- a/lean/OAI/Probability/ThorpRouting/Tail/ExposedCoins.lean
+++ b/lean/OAI/Probability/ThorpRouting/Tail/ExposedCoins.lean
@@ -1,860 +1,860 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Tail.Coins
-
-namespace OAI
-
-namespace ThorpNine.Tail
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-
-lemma card_routeDomain (d : ℕ) (x y : Card d) : (routeDomain d x y).card = d := by
-  classical
-  induction d with
-  | zero => rfl
-  | succ d ih =>
-    change (insert (Sum.inl (Fin.tail y))
-      ((routeDomain d (Fin.tail x) (Fin.tail y)).image
-        (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) :
-      Finset (Sum (Card d) (Bool × SwitchIndex d))).card = d + 1
-    rw [Finset.card_insert_of_notMem (by simp)]
-    rw [Finset.card_image_of_injective _ (by intro i j h; cases h; rfl), ih]
-
-lemma xor_eq_iff (a b c : Bool) : Bool.xor a b = c ↔ b = Bool.xor a c := by
-  cases a <;> cases b <;> cases c <;> decide
-
-lemma butterfly_endpoint_iff (d : ℕ) (ω : SwitchIndex d → Bool) (x y : Card d) :
-    butterflyPerm d (decodeButterfly d ω) x = y ↔
-      ∀ i ∈ routeDomain d x y, ω i = routeValue d x y i := by
-  classical
-  induction d with
-  | zero =>
-    have he : butterflyPerm 0 (decodeButterfly 0 ω) x = y := Subsingleton.elim _ _
-    simp [he, routeDomain]
-  | succ d ih =>
-    let v := butterflyPerm d (decodeButterfly d (fun i => ω (Sum.inr (x 0,i)))) (Fin.tail x)
-    change Fin.cons (Bool.xor (x 0) (ω (Sum.inl v))) v = y ↔ _
-    have hv : v = Fin.tail y ↔
-        ∀ i ∈ routeDomain d (Fin.tail x) (Fin.tail y),
-          ω (Sum.inr (x 0,i)) = routeValue d (Fin.tail x) (Fin.tail y) i :=
-      ih _ _ _
-    constructor
-    · intro h i hi
-      have ht : v = Fin.tail y := by simpa using congrArg Fin.tail h
-      have hh : ω (Sum.inl (Fin.tail y)) = Bool.xor (x 0) (y 0) := by
-        have he := congrFun h 0
-        simp only [Fin.cons_zero, ht] at he
-        exact (xor_eq_iff _ _ _).mp he
-      rcases Finset.mem_insert.mp hi with htop | hchild
-      · subst i
-        exact hh
-      · obtain ⟨j, hj, rfl⟩ := Finset.mem_image.mp hchild
-        exact hv.mp ht j hj
-    · intro h
-      have hh := h (Sum.inl (Fin.tail y)) (Finset.mem_insert_self _ _)
-      have ht : v = Fin.tail y := hv.mpr (fun i hi =>
-        h (Sum.inr (x 0,i)) (Finset.mem_insert_of_mem (Finset.mem_image_of_mem _ hi)))
-      have hh' : Bool.xor (x 0) (ω (Sum.inl (Fin.tail y))) = y 0 :=
-        (xor_eq_iff _ _ _).mpr hh
-      rw [ht, hh', Fin.cons_self_tail]
-
-lemma finiteMean_congr {Ω : Type*} [Fintype Ω] {f g : Ω → ℝ}
-    (h : ∀ ω, f ω = g ω) : finiteMean f = finiteMean g := by
-  unfold finiteMean
-  congr 1
-  exact Finset.sum_congr rfl (fun ω _ => h ω)
-
-lemma mean_coinCylinder {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (s : Finset ι) (v : ι → Bool) :
-    finiteMean (fun ω : ι → Bool => if ∀ i ∈ s, ω i = v i then (1 : ℝ) else 0) =
-      1 / (2 : ℝ) ^ s.card := by
-  induction s using Finset.induction_on with
-  | empty => simp [finiteMean]
-  | @insert i s hi ih =>
-    let f := fun ω : ι → Bool => if ∀ j ∈ s, ω j = v j then (1 : ℝ) else 0
-    have hf : ∀ ω, f (QueryTree.flipCoin i ω) = f ω := by
-      intro ω
-      have he : (∀ j ∈ s, QueryTree.flipCoin i ω j = v j) ↔ ∀ j ∈ s, ω j = v j := by
-        apply forall₂_congr
-        intro j hj
-        rw [QueryTree.flipCoin_apply_ne i j (fun h => hi (h ▸ hj))]
-      dsimp [f]
-      simp only [he]
-    have hm := QueryTree.mean_indicator_bit f i (v i) hf
-    have he : (fun ω : ι → Bool => if ∀ j ∈ insert i s, ω j = v j then (1 : ℝ) else 0) =
-        (fun ω => if ω i = v i then f ω else 0) := by
-      funext ω
-      by_cases h : ω i = v i <;> simp [f, h]
-    rw [he, hm]
-    change finiteMean (fun ω : ι → Bool => if ∀ j ∈ s, ω j = v j then (1 : ℝ) else 0) / 2 = _
-    rw [ih, Finset.card_insert_of_notMem hi, pow_succ]
-    ring
-
-def exposedCoins {ι : Type*} [DecidableEq ι] (R : Finset ι) (η ω : ι → Bool) : ι → Bool :=
-  fun i => if i ∈ R then η i else ω i
-
-lemma mean_coinCylinder_exposed {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (s R : Finset ι) (v η : ι → Bool) :
-    finiteMean (fun ω : ι → Bool =>
-      if ∀ i ∈ s, exposedCoins R η ω i = v i then (1 : ℝ) else 0) =
-    if (∀ i ∈ s ∩ R, η i = v i) then 1 / (2 : ℝ) ^ (s \ R).card else 0 := by
-  have he (ω : ι → Bool) : (∀ i ∈ s, exposedCoins R η ω i = v i) ↔
-      (∀ i ∈ s ∩ R, η i = v i) ∧ (∀ i ∈ s \ R, ω i = v i) := by
-    constructor
-    · intro h
-      constructor
-      · intro i hi
-        simpa [exposedCoins, (Finset.mem_inter.mp hi).2] using h i (Finset.mem_inter.mp hi).1
-      · intro i hi
-        simpa [exposedCoins, (Finset.mem_sdiff.mp hi).2] using h i (Finset.mem_sdiff.mp hi).1
-    · rintro ⟨ho, hf⟩ i hi
-      by_cases hR : i ∈ R
-      · simpa [exposedCoins, hR] using ho i (Finset.mem_inter.mpr ⟨hi,hR⟩)
-      · simpa [exposedCoins, hR] using hf i (Finset.mem_sdiff.mpr ⟨hi,hR⟩)
-  have hm : finiteMean (fun ω : ι → Bool =>
-      if ∀ i ∈ s, exposedCoins R η ω i = v i then (1 : ℝ) else 0) =
-      finiteMean (fun ω : ι → Bool =>
-        if (∀ i ∈ s ∩ R, η i = v i) ∧ (∀ i ∈ s \ R, ω i = v i) then (1 : ℝ) else 0) :=
-    finiteMean_congr (fun ω => by simp only [he ω])
-  rw [hm]
-  by_cases h : ∀ i ∈ s ∩ R, η i = v i
-  · have hp : (∀ i ∈ s ∩ R, η i = v i) ↔ True := iff_true_intro h
-    simp only [hp, true_and, ↓reduceIte]
-    exact mean_coinCylinder (s \ R) v
-  · have hp : (∀ i ∈ s ∩ R, η i = v i) ↔ False := iff_false_intro h
-    simp only [hp, false_and, ↓reduceIte]
-    simp [finiteMean]
-
-lemma butterfly_closure_probability (d : ℕ) (x y : Card d)
-    (R : Finset (SwitchIndex d)) (η : SwitchIndex d → Bool) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      if butterflyPerm d (decodeButterfly d (exposedCoins R η ω)) x = y then (1 : ℝ) else 0) =
-      if (∀ i ∈ routeDomain d x y ∩ R, η i = routeValue d x y i) then
-        (2 : ℝ) ^ (routeDomain d x y ∩ R).card / (2 : ℝ) ^ d else 0 := by
-  have he : finiteMean (fun ω : SwitchIndex d → Bool =>
-      if butterflyPerm d (decodeButterfly d (exposedCoins R η ω)) x = y then (1 : ℝ) else 0) =
-      finiteMean (fun ω : SwitchIndex d → Bool =>
-        if ∀ i ∈ routeDomain d x y, exposedCoins R η ω i = routeValue d x y i then (1 : ℝ) else 0) :=
-    finiteMean_congr (fun ω => by simp only [butterfly_endpoint_iff])
-  rw [he, mean_coinCylinder_exposed]
-  congr 1
-  have hc : (routeDomain d x y \ R).card + (routeDomain d x y ∩ R).card = d := by
-    rw [Finset.card_sdiff_add_card_inter, card_routeDomain]
-  have hp : (2 : ℝ) ^ d = (2 : ℝ) ^ (routeDomain d x y \ R).card *
-      (2 : ℝ) ^ (routeDomain d x y ∩ R).card := by
-    rw [← pow_add]
-    exact congrArg (fun n => (2 : ℝ) ^ n) hc.symm
-  rw [hp]
-  have hn : (2 : ℝ) ^ (routeDomain d x y ∩ R).card ≠ 0 := pow_ne_zero _ (by norm_num)
-  field_simp
-
-lemma butterfly_single_card_uniform (d : ℕ) (x y : Card d) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      if butterflyPerm d (decodeButterfly d ω) x = y then (1 : ℝ) else 0) = 1 / (2 : ℝ) ^ d := by
-  have he : finiteMean (fun ω : SwitchIndex d → Bool =>
-      if butterflyPerm d (decodeButterfly d ω) x = y then (1 : ℝ) else 0) =
-      finiteMean (fun ω : SwitchIndex d → Bool =>
-        if ∀ i ∈ routeDomain d x y, ω i = routeValue d x y i then (1 : ℝ) else 0) :=
-    finiteMean_congr (fun ω => by simp only [butterfly_endpoint_iff])
-  rw [he, mean_coinCylinder, card_routeDomain]
-
-def switchUsers : (d : ℕ) → Butterfly d → SwitchIndex d → Card d × Card d
-  | 0, _, i => nomatch i
-  | d + 1, b, Sum.inl y =>
-      (Fin.cons false ((butterflyPerm d (b.2 false)).symm y),
-       Fin.cons true ((butterflyPerm d (b.2 true)).symm y))
-  | d + 1, b, Sum.inr (ε,i) =>
-      (Fin.cons ε (switchUsers d (b.2 ε) i).1,
-       Fin.cons ε (switchUsers d (b.2 ε) i).2)
-
-lemma switchUsers_ne (d : ℕ) (b : Butterfly d) (i : SwitchIndex d) :
-    (switchUsers d b i).1 ≠ (switchUsers d b i).2 := by
-  induction d with
-  | zero => exact Empty.elim i
-  | succ d ih =>
-    cases i with
-    | inl y =>
-      intro h
-      have := congrFun h 0
-      simp only [switchUsers, Fin.cons_zero] at this
-      exact Bool.false_ne_true this
-    | inr z =>
-      intro h
-      exact ih (b.2 z.1) z.2 (by simpa only [switchUsers, Fin.tail_cons] using congrArg Fin.tail h)
-
-lemma cons_eq_iff {d : ℕ} (ε : Bool) (a : Card d) (x : Card (d + 1)) :
-    Fin.cons ε a = x ↔ ε = x 0 ∧ a = Fin.tail x := by
-  constructor
-  · intro h
-    exact ⟨by simpa only [Fin.cons_zero] using congrFun h 0,
-      by simpa only [Fin.tail_cons] using congrArg Fin.tail h⟩
-  · rintro ⟨rfl,rfl⟩
-    exact Fin.cons_self_tail x
-
-lemma mem_routeDomain_iff_user (d : ℕ) (b : Butterfly d) (x : Card d) (i : SwitchIndex d) :
-    i ∈ routeDomain d x (butterflyPerm d b x) ↔
-      (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x := by
-  classical
-  induction d with
-  | zero => exact Empty.elim i
-  | succ d ih =>
-    cases i with
-    | inl y =>
-      change Sum.inl y ∈ insert (Sum.inl (Fin.tail (butterflyPerm (d+1) b x)))
-        ((routeDomain d (Fin.tail x) (Fin.tail (butterflyPerm (d+1) b x))).image
-          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) ↔ _
-      simp only [Finset.mem_insert, Sum.inl.injEq, Finset.mem_image,
-        Sum.inr_ne_inl, and_false, exists_false, or_false]
-      change y = butterflyPerm d (b.2 (x 0)) (Fin.tail x) ↔ _
-      simp only [switchUsers, cons_eq_iff]
-      cases x 0 <;> simp [Equiv.symm_apply_eq]
-    | inr z =>
-      obtain ⟨ε,i⟩ := z
-      change Sum.inr (ε,i) ∈ insert (Sum.inl (Fin.tail (butterflyPerm (d+1) b x)))
-        ((routeDomain d (Fin.tail x) (Fin.tail (butterflyPerm (d+1) b x))).image
-          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) ↔ _
-      simp only [Finset.mem_insert, Sum.inr_ne_inl, false_or, Finset.mem_image,
-        Sum.inr.injEq, Prod.mk.injEq]
-      change (∃ j, j ∈ routeDomain d (Fin.tail x) (butterflyPerm d (b.2 (x 0)) (Fin.tail x)) ∧
-        x 0 = ε ∧ j = i) ↔ _
-      simp only [switchUsers, cons_eq_iff]
-      constructor
-      · rintro ⟨j,hj,hε,rfl⟩
-        subst ε
-        rcases (ih (b.2 (x 0)) (Fin.tail x) j).mp hj with h | h
-        · exact Or.inl ⟨rfl,h⟩
-        · exact Or.inr ⟨rfl,h⟩
-      · rintro (⟨hε,h⟩ | ⟨hε,h⟩)
-        · subst ε
-          exact ⟨i,(ih (b.2 (x 0)) (Fin.tail x) i).mpr (Or.inl h),rfl,rfl⟩
-        · subst ε
-          exact ⟨i,(ih (b.2 (x 0)) (Fin.tail x) i).mpr (Or.inr h),rfl,rfl⟩
-
-lemma card_switches_using (d : ℕ) (b : Butterfly d) (x : Card d) :
-    (Finset.univ.filter (fun i : SwitchIndex d =>
-      (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x)).card = d := by
-  classical
-  have he : Finset.univ.filter (fun i : SwitchIndex d =>
-      (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x) =
-      routeDomain d x (butterflyPerm d b x) := by
-    ext i
-    simp only [Finset.mem_filter, Finset.mem_univ, true_and, mem_routeDomain_iff_user]
-  rw [he,card_routeDomain]
-
-lemma switch_incidence_sum (d : ℕ) (b : Butterfly d) (w : Card d → ℝ) :
-    ∑ i : SwitchIndex d, (w (switchUsers d b i).1 + w (switchUsers d b i).2) =
-      (d : ℝ) * ∑ x : Card d, w x := by
-  classical
-  calc
-    _ = ∑ i : SwitchIndex d, ∑ x : Card d,
-        if (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x then w x else 0 := by
-      apply Finset.sum_congr rfl
-      intro i _
-      rw [Finset.sum_ite]
-      simp only [Finset.sum_const_zero, add_zero]
-      have he : Finset.univ.filter (fun x : Card d =>
-          (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x) =
-          {(switchUsers d b i).1, (switchUsers d b i).2} := by
-        ext x; simp [eq_comm]
-      rw [he, Finset.sum_pair (switchUsers_ne d b i)]
-    _ = ∑ x : Card d, ∑ i : SwitchIndex d,
-        if (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x then w x else 0 :=
-      Finset.sum_comm
-    _ = ∑ x : Card d, (d : ℝ) * w x := by
-      apply Finset.sum_congr rfl
-      intro x _
-      rw [Finset.sum_ite]
-      simp only [Finset.sum_const_zero, add_zero, Finset.sum_const, nsmul_eq_mul,
-        card_switches_using]
-    _ = _ := by rw [Finset.mul_sum]
-
-lemma weighted_switch_min_bound (d : ℕ) (b : Butterfly d) (w : Card d → ℝ) :
-    ∑ i : SwitchIndex d, min (w (switchUsers d b i).1) (w (switchUsers d b i).2) ≤
-      (d : ℝ) / 2 * ∑ x : Card d, w x := by
-  have h : 2 * (∑ i : SwitchIndex d,
-      min (w (switchUsers d b i).1) (w (switchUsers d b i).2)) ≤
-      ∑ i : SwitchIndex d, (w (switchUsers d b i).1 + w (switchUsers d b i).2) := by
-    rw [Finset.mul_sum]
-    apply Finset.sum_le_sum
-    intro i _
-    linarith [min_le_left (w (switchUsers d b i).1) (w (switchUsers d b i).2),
-      min_le_right (w (switchUsers d b i).1) (w (switchUsers d b i).2)]
-  rw [switch_incidence_sum] at h
-  linarith
-
-lemma mem_childMarks {d : ℕ} (s : Finset (Card (d + 1))) (ε : Bool) (x : Card d) :
-    x ∈ childMarks s ε ↔ Fin.cons ε x ∈ s := by
-  classical
-  constructor
-  · intro h
-    obtain ⟨y,hy,hxy⟩ := Finset.mem_image.mp h
-    obtain ⟨hys,hyε⟩ := Finset.mem_filter.mp hy
-    rw [← hxy, ← hyε, Fin.cons_self_tail]
-    exact hys
-  · intro h
-    exact Finset.mem_image.mpr ⟨Fin.cons ε x,
-      Finset.mem_filter.mpr ⟨h,by simp⟩,by simp⟩
-
-lemma mem_topMeetings {d : ℕ} (b : Butterfly (d + 1))
-    (s : Finset (Card (d + 1))) (y : Card d) :
-    y ∈ topMeetings b s ↔
-      (switchUsers (d+1) b (Sum.inl y)).1 ∈ s ∧
-      (switchUsers (d+1) b (Sum.inl y)).2 ∈ s := by
-  classical
-  have he (ε : Bool) : y ∈ (childMarks s ε).image (butterflyPerm d (b.2 ε)) ↔
-      (butterflyPerm d (b.2 ε)).symm y ∈ childMarks s ε := by
-    constructor
-    · intro h
-      obtain ⟨x,hx,rfl⟩ := Finset.mem_image.mp h
-      simpa only [Equiv.symm_apply_apply] using hx
-    · intro h
-      exact Finset.mem_image.mpr ⟨_,h,Equiv.apply_symm_apply _ _⟩
-  simp only [topMeetings, Finset.mem_inter, he, mem_childMarks, switchUsers]
-
-lemma encounters_eq_switch_sum (d : ℕ) (b : Butterfly d) (s : Finset (Card d)) :
-    encounters d b s = ∑ i : SwitchIndex d,
-      if (switchUsers d b i).1 ∈ s ∧ (switchUsers d b i).2 ∈ s then 1 else 0 := by
-  classical
-  induction d with
-  | zero =>
-    simp only [encounters]
-    symm
-    apply Finset.sum_eq_zero
-    intro i _
-    exact Empty.elim i
-  | succ d ih =>
-    change encounters d (b.2 false) (childMarks s false) +
-      encounters d (b.2 true) (childMarks s true) + (topMeetings b s).card =
-        ∑ i : Sum (Card d) (Bool × SwitchIndex d), _
-    rw [Fintype.sum_sum_type, Fintype.sum_prod_type]
-    simp only [Fintype.sum_bool]
-    rw [ih,ih]
-    have ht : (∑ y : Card d,
-        if (switchUsers (d+1) b (Sum.inl y)).1 ∈ s ∧
-          (switchUsers (d+1) b (Sum.inl y)).2 ∈ s then 1 else 0) =
-        (topMeetings b s).card := by
-      simp only [← mem_topMeetings]
-      simp
-    rw [ht]
-    simp only [mem_childMarks, switchUsers]
-    ac_rfl
-
-lemma marked_switch_entropy_bound (d : ℕ) (b : Butterfly d) (s : Finset (Card d)) :
-    (∑ i : SwitchIndex d,
-      if (switchUsers d b i).1 ∈ s ∧ (switchUsers d b i).2 ∈ s then (1:ℝ) else 0) ≤
-      (s.card : ℝ) * Real.log s.card / (2 * Real.log 2) := by
-  classical
-  have he : (encounters d b s : ℝ) =
-      ∑ i : SwitchIndex d,
-        if (switchUsers d b i).1 ∈ s ∧ (switchUsers d b i).2 ∈ s then (1:ℝ) else 0 := by
-    rw [encounters_eq_switch_sum, Nat.cast_sum]
-    apply Finset.sum_congr rfl
-    intro i _
-    split_ifs <;> simp
-  rw [← he]
-  exact butterfly_encounters d b s
-
-namespace QueryTree
-variable {ι : Type*} [DecidableEq ι]
-
-def memo : QueryTree ι → Finset ι → (ι → Bool) → QueryTree ι
-  | reject, _, _ => reject
-  | accept, _, _ => accept
-  | branch i zero one, R, η =>
-      if i ∈ R then (if η i then memo one R η else memo zero R η) else
-      branch i (memo zero (insert i R) (Function.update η i false))
-        (memo one (insert i R) (Function.update η i true))
-  | force i b next, R, η =>
-      if i ∈ R then (if η i = b then memo next R η else reject) else
-      force i b (memo next (insert i R) (Function.update η i b))
-
-lemma memo_queries_disjoint (T : QueryTree ι) (R : Finset ι) (η : ι → Bool) :
-    Disjoint (memo T R η).queries R := by
-  induction T generalizing R η with
-  | reject => simp [memo,queries]
-  | accept => simp [memo,queries]
-  | branch i zero one ih₀ ih₁ =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte]
-      cases η i <;> simp only [Bool.false_eq_true,↓reduceIte]
-      · exact ih₀ R η
-      · exact ih₁ R η
-    · simp only [memo,hi,↓reduceIte,queries,Finset.disjoint_insert_left,Finset.disjoint_union_left]
-      exact ⟨by simp, (ih₀ (insert i R) _).mono_right (Finset.subset_insert _ _),
-        (ih₁ (insert i R) _).mono_right (Finset.subset_insert _ _)⟩
-  | force i b next ih =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte]
-      by_cases hb : η i = b
-      · simpa only [hb,↓reduceIte] using ih R η
-      · simp [hb,queries]
-    · simp only [memo,hi,↓reduceIte,queries,Finset.disjoint_insert_left]
-      exact ⟨by simp,(ih (insert i R) _).mono_right (Finset.subset_insert _ _)⟩
-
-lemma memo_fresh (T : QueryTree ι) (R : Finset ι) (η : ι → Bool) :
-    (memo T R η).Fresh := by
-  induction T generalizing R η with
-  | reject => trivial
-  | accept => trivial
-  | branch i zero one ih₀ ih₁ =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte]
-      cases η i <;> simp only [Bool.false_eq_true,↓reduceIte]
-      · exact ih₀ R η
-      · exact ih₁ R η
-    · simp only [memo,hi,↓reduceIte,Fresh]
-      exact ⟨fun hm => Finset.disjoint_left.mp (memo_queries_disjoint zero (insert i R) _) hm
-          (Finset.mem_insert_self _ _),
-        fun hm => Finset.disjoint_left.mp (memo_queries_disjoint one (insert i R) _) hm
-          (Finset.mem_insert_self _ _), ih₀ _ _,ih₁ _ _⟩
-  | force i b next ih =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte]
-      by_cases hb : η i = b
-      · simpa only [hb,↓reduceIte] using ih R η
-      · simp [hb,Fresh]
-    · simp only [memo,hi,↓reduceIte,Fresh]
-      exact ⟨fun hm => Finset.disjoint_left.mp (memo_queries_disjoint next (insert i R) _) hm
-        (Finset.mem_insert_self _ _),ih _ _⟩
-
-lemma update_agrees {R : Finset ι} {η ω : ι → Bool}
-    (h : ∀ i ∈ R, η i = ω i) (i : ι) :
-    ∀ j ∈ insert i R, Function.update η i (ω i) j = ω j := by
-  intro j hj
-  rcases Finset.mem_insert.mp hj with rfl | hj
-  · simp
-  · by_cases hji : j = i
-    · subst j; simp
-    · rw [Function.update_of_ne hji]
-      exact h j hj
-
-lemma memo_succeeds (T : QueryTree ι) (R : Finset ι) (η ω : ι → Bool)
-    (h : ∀ i ∈ R, η i = ω i) : (memo T R η).succeeds ω = T.succeeds ω := by
-  induction T generalizing R η with
-  | reject => rfl
-  | accept => rfl
-  | branch i zero one ih₀ ih₁ =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte,succeeds,h i hi]
-      cases ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
-      · exact ih₀ R η h
-      · exact ih₁ R η h
-    · simp only [memo,hi,↓reduceIte,succeeds]
-      cases hb : ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
-      · exact ih₀ _ _ (by simpa only [hb] using update_agrees h i)
-      · exact ih₁ _ _ (by simpa only [hb] using update_agrees h i)
-  | force i b next ih =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte,h i hi]
-      by_cases hb : ω i = b
-      · simp only [hb,↓reduceIte,succeeds]
-        exact ih R η h
-      · simp [hb,succeeds]
-    · simp only [memo,hi,↓reduceIte,succeeds]
-      by_cases hb : ω i = b
-      · simp only [hb,↓reduceIte]
-        exact ih _ _ (by simpa only [hb] using update_agrees h i)
-      · simp [hb]
-
-def freshCharge : QueryTree ι → Finset ι → (ι → Bool) → ℕ
-  | reject, _, _ => 0
-  | accept, _, _ => 0
-  | branch i zero one, R, ω => if ω i then freshCharge one (insert i R) ω else freshCharge zero (insert i R) ω
-  | force i b next, R, ω => if ω i = b then
-      freshCharge next (insert i R) ω + (if i ∈ R then 0 else 1) else 0
-
-lemma memo_charge (T : QueryTree ι) (R : Finset ι) (η ω : ι → Bool)
-    (h : ∀ i ∈ R, η i = ω i) : (memo T R η).charge ω = freshCharge T R ω := by
-  induction T generalizing R η with
-  | reject => rfl
-  | accept => rfl
-  | branch i zero one ih₀ ih₁ =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte,freshCharge,h i hi,Finset.insert_eq_of_mem hi]
-      cases ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
-      · exact ih₀ R η h
-      · exact ih₁ R η h
-    · simp only [memo,hi,↓reduceIte,charge,freshCharge]
-      cases hb : ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
-      · exact ih₀ _ _ (by simpa only [hb] using update_agrees h i)
-      · exact ih₁ _ _ (by simpa only [hb] using update_agrees h i)
-  | force i b next ih =>
-    by_cases hi : i ∈ R
-    · simp only [memo,hi,↓reduceIte,h i hi,freshCharge,Finset.insert_eq_of_mem hi,Nat.add_zero]
-      by_cases hb : ω i = b
-      · simp only [hb,↓reduceIte]
-        exact ih R η h
-      · simp [hb,charge]
-    · simp only [memo,hi,↓reduceIte,charge,freshCharge]
-      by_cases hb : ω i = b
-      · simp only [hb,↓reduceIte]
-        rw [ih _ _ (by simpa only [hb] using update_agrees h i)]
-      · simp [hb]
-
-lemma mean_freshCharge_le_one [Fintype ι] (T : QueryTree ι) :
-    finiteMean (fun ω : ι → Bool =>
-      if T.succeeds ω then (2:ℝ) ^ T.freshCharge ∅ ω else 0) ≤ 1 := by
-  let η : ι → Bool := fun _ => false
-  have hm := mean_weighted_le_one (memo T ∅ η) (memo_fresh T ∅ η)
-  have he : finiteMean (memo T ∅ η).weighted =
-      finiteMean (fun ω : ι → Bool =>
-        if T.succeeds ω then (2:ℝ) ^ T.freshCharge ∅ ω else 0) := by
-    apply finiteMean_congr
-    intro ω
-    rw [weighted_eq, memo_succeeds _ _ _ _ (by simp), memo_charge _ _ _ _ (by simp)]
-  rw [he] at hm
-  exact hm
-
-
-def followPath : (d : ℕ) → (SwitchIndex d → ι) → Card d →
-    (Card d → QueryTree ι) → QueryTree ι
-  | 0, _, x, k => k x
-  | d + 1, e, x, k => followPath d (fun i => e (Sum.inr (x 0,i))) (Fin.tail x)
-      (fun y => branch (e (Sum.inl y))
-        (k (Fin.cons (Bool.xor (x 0) false) y))
-        (k (Fin.cons (Bool.xor (x 0) true) y)))
-
-omit [DecidableEq ι] in
-lemma followPath_succeeds (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
-    (k : Card d → QueryTree ι) (ω : ι → Bool) :
-    (followPath d e x k).succeeds ω =
-      (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).succeeds ω := by
-  induction d with
-  | zero => rfl
-  | succ d ih =>
-    rw [followPath, ih]
-    let y := butterflyPerm d
-      (decodeButterfly d (fun i => ω (e (Sum.inr (x 0,i))))) (Fin.tail x)
-    change (if ω (e (Sum.inl y)) then
-      (k (Fin.cons (Bool.xor (x 0) true) y)).succeeds ω else
-      (k (Fin.cons (Bool.xor (x 0) false) y)).succeeds ω) =
-      (k (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).succeeds ω
-    cases ω (e (Sum.inl y)) <;> rfl
-
-lemma followPath_charge (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
-    (k : Card d → QueryTree ι) (R : Finset ι) (ω : ι → Bool) :
-    freshCharge (followPath d e x k) R ω =
-      freshCharge (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x))
-        (R ∪ (routeDomain d x (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).image e) ω := by
-  classical
-  induction d generalizing R with
-  | zero => simp only [followPath, routeDomain, Finset.image_empty, Finset.union_empty]; rfl
-  | succ d ih =>
-    rw [followPath, ih]
-    let y := butterflyPerm d
-      (decodeButterfly d (fun i => ω (e (Sum.inr (x 0,i))))) (Fin.tail x)
-    let S := (routeDomain d (Fin.tail x) y).image (fun i => e (Sum.inr (x 0,i)))
-    change (if ω (e (Sum.inl y)) then
-      freshCharge (k (Fin.cons (Bool.xor (x 0) true) y)) (insert (e (Sum.inl y)) (R ∪ S)) ω else
-      freshCharge (k (Fin.cons (Bool.xor (x 0) false) y)) (insert (e (Sum.inl y)) (R ∪ S)) ω) =
-      freshCharge (k (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y))
-        (R ∪ (routeDomain (d+1) x
-          (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).image e) ω
-    have hs : (routeDomain (d+1) x
-        (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).image e =
-        insert (e (Sum.inl y)) S := by
-      let e' : Sum (Card d) (Bool × SwitchIndex d) → ι := e
-      change (insert (Sum.inl y)
-        ((routeDomain d (Fin.tail x) y).image
-          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) :
-          Finset (Sum (Card d) (Bool × SwitchIndex d))).image e' = _
-      rw [Finset.image_insert, Finset.image_image]
-      rfl
-    rw [hs,Finset.union_insert]
-    cases ω (e (Sum.inl y)) <;> rfl
-
-def forceList : List ι → (ι → Bool) → QueryTree ι → QueryTree ι
-  | [], _, k => k
-  | i :: L, v, k => force i (v i) (forceList L v k)
-
-omit [DecidableEq ι] in
-lemma forceList_succeeds (L : List ι) (v : ι → Bool) (k : QueryTree ι) (ω : ι → Bool) :
-    (forceList L v k).succeeds ω =
-      if ∀ i ∈ L, ω i = v i then k.succeeds ω else false := by
-  induction L with
-  | nil => simp [forceList]
-  | cons i L ih =>
-    simp only [forceList,succeeds,ih,List.mem_cons,forall_eq_or_imp]
-    by_cases h : ω i = v i <;> simp [h]
-
-lemma forceList_charge (L : List ι) (v : ι → Bool) (k : QueryTree ι)
-    (R : Finset ι) (ω : ι → Bool) (h : ∀ i ∈ L, ω i = v i) :
-    freshCharge (forceList L v k) R ω =
-      freshCharge k (R ∪ L.toFinset) ω + (L.toFinset \ R).card := by
-  induction L generalizing R with
-  | nil => simp [forceList]
-  | cons i L ih =>
-    rw [forceList,freshCharge,ite_eq_left (h i List.mem_cons_self),
-      ih (insert i R) (fun j hj => h j (List.mem_cons_of_mem _ hj))]
-    simp only [List.toFinset_cons]
-    have hs : insert i R ∪ L.toFinset = R ∪ insert i L.toFinset := by ext j; simp
-    rw [hs]
-    by_cases hi : i ∈ R
-    · rw [ite_eq_left hi]
-      have he : L.toFinset \ insert i R = insert i L.toFinset \ R := by
-        ext j
-        by_cases hji : j = i
-        · subst j; simp [hi]
-        · simp [hji]
-      rw [he,Nat.add_zero]
-    · rw [ite_eq_right hi]
-      have he : insert i L.toFinset \ R = insert i (L.toFinset \ insert i R) := by
-        ext j
-        by_cases hji : j = i
-        · subst j; simp [hi]
-        · simp [hji]
-      rw [he, Finset.card_insert_of_notMem (by simp)]
-      omega
-
-
-def staleCharge : QueryTree ι → Finset ι → (ι → Bool) → ℕ
-  | reject, _, _ => 0
-  | accept, _, _ => 0
-  | branch i zero one, R, ω => if ω i then staleCharge one (insert i R) ω
-      else staleCharge zero (insert i R) ω
-  | force i b next, R, ω => if ω i = b then
-      staleCharge next (insert i R) ω + (if i ∈ R then 1 else 0) else 0
-
-lemma fresh_add_stale (T : QueryTree ι) (R : Finset ι) (ω : ι → Bool) :
-    T.freshCharge R ω + T.staleCharge R ω = T.charge ω := by
-  induction T generalizing R with
-  | reject => rfl
-  | accept => rfl
-  | branch i zero one ih₀ ih₁ =>
-    cases hb : ω i <;> simp [freshCharge,staleCharge,charge,hb,ih₀,ih₁]
-  | force i b next ih =>
-    by_cases hb : ω i = b
-    · simp only [freshCharge,staleCharge,charge,hb,↓reduceIte]
-      have he := ih (insert i R)
-      by_cases hi : i ∈ R <;> simp only [hi,↓reduceIte] <;> omega
-    · simp [freshCharge,staleCharge,charge,hb]
-
-omit [DecidableEq ι] in
-lemma followPath_raw_charge (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
-    (k : Card d → QueryTree ι) (ω : ι → Bool) :
-    (followPath d e x k).charge ω =
-      (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).charge ω := by
-  induction d with
-  | zero => rfl
-  | succ d ih =>
-    rw [followPath, ih]
-    let y := butterflyPerm d
-      (decodeButterfly d (fun i => ω (e (Sum.inr (x 0,i))))) (Fin.tail x)
-    change (if ω (e (Sum.inl y)) then
-      (k (Fin.cons (Bool.xor (x 0) true) y)).charge ω else
-      (k (Fin.cons (Bool.xor (x 0) false) y)).charge ω) =
-      (k (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).charge ω
-    cases ω (e (Sum.inl y)) <;> rfl
-
-omit [DecidableEq ι] in
-lemma forceList_raw_charge (L : List ι) (v : ι → Bool) (k : QueryTree ι) (ω : ι → Bool)
-    (h : ∀ i ∈ L, ω i = v i) :
-    (forceList L v k).charge ω = k.charge ω + L.length := by
-  induction L with
-  | nil => simp [forceList]
-  | cons i L ih =>
-    rw [forceList,charge,ite_eq_left (h i List.mem_cons_self),
-      ih (fun j hj => h j (List.mem_cons_of_mem _ hj))]
-    simp only [List.length_cons,Nat.add_assoc]
-
-noncomputable def closePath (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d)) :
-    QueryTree (SwitchIndex d) :=
-  forceList (routeDomain d x y).toList (routeValue d x y) k
-
-lemma closePath_succeeds (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
-    (ω : SwitchIndex d → Bool) :
-    (closePath d x y k).succeeds ω =
-      if butterflyPerm d (decodeButterfly d ω) x = y then k.succeeds ω else false := by
-  unfold closePath
-  rw [forceList_succeeds]
-  simp only [Finset.mem_toList, ← butterfly_endpoint_iff]
-
-lemma closePath_raw_charge (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
-    (ω : SwitchIndex d → Bool) (h : butterflyPerm d (decodeButterfly d ω) x = y) :
-    (closePath d x y k).charge ω = k.charge ω + d := by
-  unfold closePath
-  rw [forceList_raw_charge _ _ _ _ (by simpa only [Finset.mem_toList] using
-    (butterfly_endpoint_iff d ω x y).mp h), Finset.length_toList, card_routeDomain]
-
-lemma closePath_fresh_charge (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
-    (R : Finset (SwitchIndex d)) (ω : SwitchIndex d → Bool)
-    (h : butterflyPerm d (decodeButterfly d ω) x = y) :
-    (closePath d x y k).freshCharge R ω =
-      k.freshCharge (R ∪ routeDomain d x y) ω + (routeDomain d x y \ R).card := by
-  unfold closePath
-  rw [forceList_charge _ _ _ _ _ (by simpa only [Finset.mem_toList] using
-    (butterfly_endpoint_iff d ω x y).mp h), Finset.toList_toFinset]
-
-lemma closePath_stale_charge (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
-    (R : Finset (SwitchIndex d)) (ω : SwitchIndex d → Bool)
-    (h : butterflyPerm d (decodeButterfly d ω) x = y) :
-    (closePath d x y k).staleCharge R ω =
-      k.staleCharge (R ∪ routeDomain d x y) ω + (routeDomain d x y ∩ R).card := by
-  have h₀ := fresh_add_stale (closePath d x y k) R ω
-  have h₁ := fresh_add_stale k (R ∪ routeDomain d x y) ω
-  rw [closePath_raw_charge _ _ _ _ _ h,closePath_fresh_charge _ _ _ _ _ _ h] at h₀
-  have h₂ : (routeDomain d x y \ R).card + (routeDomain d x y ∩ R).card = d := by
-    rw [Finset.card_sdiff_add_card_inter,card_routeDomain]
-  omega
-
-noncomputable def seekCycle (d : ℕ) (S : Finset (Card d)) (a : Card d) :
-    ℕ → Card d → Finset (Card d) → (Finset (Card d) → QueryTree (SwitchIndex d)) →
-      QueryTree (SwitchIndex d)
-  | 0, x, V, k => if x ∈ S ∧ x ∉ V then closePath d x a (k (insert x V)) else reject
-  | r + 1, x, V, k => if x ∈ S ∧ x ∉ V then followPath d id x
-      (fun y => seekCycle d S a r y (insert x V) k) else reject
-
-lemma seekCycle_raw_charge (d : ℕ) (S : Finset (Card d)) (a : Card d) (r : ℕ)
-    (x : Card d) (V : Finset (Card d)) (k : Finset (Card d) → QueryTree (SwitchIndex d))
-    (ω : SwitchIndex d → Bool) (K : ℕ)
-    (hK : ∀ V, (k V).succeeds ω = true → (k V).charge ω = K)
-    (hs : (seekCycle d S a r x V k).succeeds ω = true) :
-    (seekCycle d S a r x V k).charge ω = K + d := by
-  induction r generalizing x V with
-  | zero =>
-    by_cases hv : x ∈ S ∧ x ∉ V
-    · rw [seekCycle,ite_eq_left hv,closePath_succeeds] at hs
-      split_ifs at hs with he
-      · rw [seekCycle,ite_eq_left hv,closePath_raw_charge _ _ _ _ _ he,hK _ hs]
-    · simp [seekCycle,hv,succeeds] at hs
-  | succ r ih =>
-    by_cases hv : x ∈ S ∧ x ∉ V
-    · rw [seekCycle,ite_eq_left hv,followPath_succeeds] at hs
-      rw [seekCycle,ite_eq_left hv,followPath_raw_charge]
-      exact ih _ _ hs
-    · simp [seekCycle,hv,succeeds] at hs
-
-noncomputable def seekCycles (d : ℕ) (S : Finset (Card d)) :
-    List (Card d × ℕ) → Finset (Card d) → QueryTree (SwitchIndex d)
-  | [], _ => accept
-  | (a,r) :: L, V => seekCycle d S a r a V (fun V => seekCycles d S L V)
-
-lemma seekCycles_raw_charge (d : ℕ) (S : Finset (Card d)) (L : List (Card d × ℕ))
-    (V : Finset (Card d)) (ω : SwitchIndex d → Bool)
-    (hs : (seekCycles d S L V).succeeds ω = true) :
-    (seekCycles d S L V).charge ω = L.length * d := by
-  induction L generalizing V with
-  | nil => simp [seekCycles,charge]
-  | cons ar L ih =>
-    obtain ⟨a,r⟩ := ar
-    rw [seekCycles]
-    rw [seekCycle_raw_charge d S a r a V (fun V => seekCycles d S L V) ω (L.length*d)
-      (fun V hs => ih V hs) hs]
-    simp [Nat.add_mul]
-
-lemma cycles_weighted_success (d : ℕ) (S : Finset (Card d)) (L : List (Card d × ℕ)) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      if (seekCycles d S L ∅).succeeds ω then
-        ((2:ℝ)^d)^L.length / (2:ℝ)^(seekCycles d S L ∅).staleCharge ∅ ω else 0) ≤ 1 := by
-  let T := seekCycles d S L ∅
-  have hh := mean_freshCharge_le_one T
-  have he : finiteMean (fun ω : SwitchIndex d → Bool =>
-      if T.succeeds ω then (2:ℝ)^T.freshCharge ∅ ω else 0) =
-      finiteMean (fun ω : SwitchIndex d → Bool =>
-        if T.succeeds ω then ((2:ℝ)^d)^L.length / (2:ℝ)^T.staleCharge ∅ ω else 0) := by
-    apply finiteMean_congr
-    intro ω
-    cases hs : T.succeeds ω
-    · simp
-    · simp only [↓reduceIte]
-      have hc := fresh_add_stale T ∅ ω
-      rw [seekCycles_raw_charge d S L ∅ ω hs] at hc
-      have hp : ((2:ℝ)^d)^L.length = (2:ℝ)^T.freshCharge ∅ ω * (2:ℝ)^T.staleCharge ∅ ω := by
-        rw [←pow_add,hc,←pow_mul,Nat.mul_comm]
-      rw [hp,mul_div_cancel_right₀ _ (pow_ne_zero _ (by norm_num))]
-  rw [← he]
-  exact hh
-
-end QueryTree
-
-lemma finiteMean_add {Ω : Type*} [Fintype Ω] (f g : Ω → ℝ) :
-    finiteMean (fun ω => f ω + g ω) = finiteMean f + finiteMean g := by
-  simp only [finiteMean,Finset.sum_add_distrib,add_div]
-
-lemma finiteMean_mul_const {Ω : Type*} [Fintype Ω] (f : Ω → ℝ) (c : ℝ) :
-    finiteMean (fun ω => f ω * c) = finiteMean f * c := by
-  unfold finiteMean
-  rw [← Finset.sum_mul]
-  ring
-
-lemma finiteMean_sum {Ω α : Type*} [Fintype Ω] (s : Finset α) (f : α → Ω → ℝ) :
-    finiteMean (fun ω => ∑ i ∈ s, f i ω) = ∑ i ∈ s, finiteMean (f i) := by
-  simp only [finiteMean,Finset.sum_div]
-  exact Finset.sum_comm
-
-lemma finiteMean_mono {Ω : Type*} [Fintype Ω] {f g : Ω → ℝ}
-    (h : ∀ ω, f ω ≤ g ω) : finiteMean f ≤ finiteMean g := by
-  exact div_le_div_of_nonneg_right (Finset.sum_le_sum (fun ω _ => h ω)) (Nat.cast_nonneg _)
-
-lemma finiteMean_equiv {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
-    (e : Ω ≃ Ω') (f : Ω → ℝ) :
-    finiteMean f = finiteMean (fun ω => f (e.symm ω)) := by
-  unfold finiteMean
-  rw [Fintype.card_congr e]
-  congr 1
-  exact Fintype.sum_equiv e _ _ (fun ω => by simp)
-
-lemma finiteMean_prod_left {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] [Nonempty Ω']
-    (f : Ω → ℝ) : finiteMean (fun ω : Ω × Ω' => f ω.1) = finiteMean f := by
-  simp only [finiteMean,Fintype.sum_prod_type,Finset.sum_const,nsmul_eq_mul,
-    ← Finset.mul_sum,Fintype.card_prod,Nat.cast_mul,Finset.card_univ]
-  have hn : (Fintype.card Ω' : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
-  field_simp
-
-lemma finiteMean_pi_eval {C : Type*} [Fintype C] [DecidableEq C]
-    {A : C → Type*} [∀ c, Fintype (A c)] [∀ c, Nonempty (A c)]
-    (c : C) (g : A c → ℝ) :
-    finiteMean (fun z : ∀ c, A c => g (z c)) = finiteMean g := by
-  rw [finiteMean_equiv (Equiv.piSplitAt c A)]
-  have he : (fun z : A c × ((j : {j // j ≠ c}) → A j) =>
-      g ((Equiv.piSplitAt c A).symm z c)) = (fun z => g z.1) := by
-    funext z
-    simp [Equiv.piSplitAt]
-  rw [he,finiteMean_prod_left]
-
-lemma finiteMean_eq_indicator {Ω : Type*} [Fintype Ω] [DecidableEq Ω] (a : Ω) :
-    finiteMean (fun ω : Ω => if ω = a then (1:ℝ) else 0) = 1 / Fintype.card Ω := by
-  simp [finiteMean]
-
-lemma exp_finiteMean_le {Ω : Type*} [Fintype Ω] [Nonempty Ω] (f : Ω → ℝ) :
-    Real.exp (finiteMean f) ≤ finiteMean (fun ω => Real.exp (f ω)) := by
-  have hn : (Fintype.card Ω : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
-  have hw : ∑ ω : Ω, (1 / (Fintype.card Ω : ℝ)) = 1 := by simp [hn]
-  have hj := convexOn_exp.map_sum_le (t := Finset.univ)
-    (w := fun _ : Ω => 1 / (Fintype.card Ω : ℝ)) (p := f)
-    (fun _ _ => by positivity) hw (fun _ _ => Set.mem_univ _)
-  simp only [smul_eq_mul, ← Finset.mul_sum] at hj
-  simpa only [finiteMean, div_eq_mul_inv, one_div, mul_comm, mul_one, one_mul] using hj
-
-variable {d A : ℕ}
-
-def cycleFiber (o : Card d → Option (Fin A)) (i : Fin A) : Finset (Card d) := by
-  classical
-  exact Finset.univ.filter (fun x => o x = some i)
-
-@[simp] lemma mem_cycleFiber (o : Card d → Option (Fin A)) (i : Fin A) (x : Card d) :
-    x ∈ cycleFiber o i ↔ o x = some i := by
-  classical
-  simp [cycleFiber]
-
-abbrev Terminal (o : Card d → Option (Fin A)) := ∀ i : Fin A, {x : Card d // o x = some i}
-
-noncomputable def slotWeight (o : Card d → Option (Fin A)) (i : Fin A) : ℝ :=
-  1 / (cycleFiber o i).card
-
-noncomputable def pathWeight (o : Card d → Option (Fin A)) (x : Card d) : ℝ :=
-  match o x with
-  | none => 0
-  | some i => slotWeight o i
-
-noncomputable def withinWeight (o : Card d → Option (Fin A)) (u v : Card d) : ℝ :=
-  match o u, o v with
-  | some i, some j => if i = j then slotWeight o i else 0
-  | _, _ => 0
-
-end Thorp
-
-end ThorpNine.Tail
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Tail.Coins
+
+namespace OAI
+
+namespace ThorpNine.Tail
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+
+lemma card_routeDomain (d : ℕ) (x y : Card d) : (routeDomain d x y).card = d := by
+  classical
+  induction d with
+  | zero => rfl
+  | succ d ih =>
+    change (insert (Sum.inl (Fin.tail y))
+      ((routeDomain d (Fin.tail x) (Fin.tail y)).image
+        (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) :
+      Finset (Sum (Card d) (Bool × SwitchIndex d))).card = d + 1
+    rw [Finset.card_insert_of_notMem (by simp)]
+    rw [Finset.card_image_of_injective _ (by intro i j h; cases h; rfl), ih]
+
+lemma xor_eq_iff (a b c : Bool) : Bool.xor a b = c ↔ b = Bool.xor a c := by
+  cases a <;> cases b <;> cases c <;> decide
+
+lemma butterfly_endpoint_iff (d : ℕ) (ω : SwitchIndex d → Bool) (x y : Card d) :
+    butterflyPerm d (decodeButterfly d ω) x = y ↔
+      ∀ i ∈ routeDomain d x y, ω i = routeValue d x y i := by
+  classical
+  induction d with
+  | zero =>
+    have he : butterflyPerm 0 (decodeButterfly 0 ω) x = y := Subsingleton.elim _ _
+    simp [he, routeDomain]
+  | succ d ih =>
+    let v := butterflyPerm d (decodeButterfly d (fun i => ω (Sum.inr (x 0,i)))) (Fin.tail x)
+    change Fin.cons (Bool.xor (x 0) (ω (Sum.inl v))) v = y ↔ _
+    have hv : v = Fin.tail y ↔
+        ∀ i ∈ routeDomain d (Fin.tail x) (Fin.tail y),
+          ω (Sum.inr (x 0,i)) = routeValue d (Fin.tail x) (Fin.tail y) i :=
+      ih _ _ _
+    constructor
+    · intro h i hi
+      have ht : v = Fin.tail y := by simpa using congrArg Fin.tail h
+      have hh : ω (Sum.inl (Fin.tail y)) = Bool.xor (x 0) (y 0) := by
+        have he := congrFun h 0
+        simp only [Fin.cons_zero, ht] at he
+        exact (xor_eq_iff _ _ _).mp he
+      rcases Finset.mem_insert.mp hi with htop | hchild
+      · subst i
+        exact hh
+      · obtain ⟨j, hj, rfl⟩ := Finset.mem_image.mp hchild
+        exact hv.mp ht j hj
+    · intro h
+      have hh := h (Sum.inl (Fin.tail y)) (Finset.mem_insert_self _ _)
+      have ht : v = Fin.tail y := hv.mpr (fun i hi =>
+        h (Sum.inr (x 0,i)) (Finset.mem_insert_of_mem (Finset.mem_image_of_mem _ hi)))
+      have hh' : Bool.xor (x 0) (ω (Sum.inl (Fin.tail y))) = y 0 :=
+        (xor_eq_iff _ _ _).mpr hh
+      rw [ht, hh', Fin.cons_self_tail]
+
+lemma finiteMean_congr {Ω : Type*} [Fintype Ω] {f g : Ω → ℝ}
+    (h : ∀ ω, f ω = g ω) : finiteMean f = finiteMean g := by
+  unfold finiteMean
+  congr 1
+  exact Finset.sum_congr rfl (fun ω _ => h ω)
+
+lemma mean_coinCylinder {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (s : Finset ι) (v : ι → Bool) :
+    finiteMean (fun ω : ι → Bool => if ∀ i ∈ s, ω i = v i then (1 : ℝ) else 0) =
+      1 / (2 : ℝ) ^ s.card := by
+  induction s using Finset.induction_on with
+  | empty => simp [finiteMean]
+  | @insert i s hi ih =>
+    let f := fun ω : ι → Bool => if ∀ j ∈ s, ω j = v j then (1 : ℝ) else 0
+    have hf : ∀ ω, f (QueryTree.flipCoin i ω) = f ω := by
+      intro ω
+      have he : (∀ j ∈ s, QueryTree.flipCoin i ω j = v j) ↔ ∀ j ∈ s, ω j = v j := by
+        apply forall₂_congr
+        intro j hj
+        rw [QueryTree.flipCoin_apply_ne i j (fun h => hi (h ▸ hj))]
+      dsimp [f]
+      simp only [he]
+    have hm := QueryTree.mean_indicator_bit f i (v i) hf
+    have he : (fun ω : ι → Bool => if ∀ j ∈ insert i s, ω j = v j then (1 : ℝ) else 0) =
+        (fun ω => if ω i = v i then f ω else 0) := by
+      funext ω
+      by_cases h : ω i = v i <;> simp [f, h]
+    rw [he, hm]
+    change finiteMean (fun ω : ι → Bool => if ∀ j ∈ s, ω j = v j then (1 : ℝ) else 0) / 2 = _
+    rw [ih, Finset.card_insert_of_notMem hi, pow_succ]
+    ring
+
+def exposedCoins {ι : Type*} [DecidableEq ι] (R : Finset ι) (η ω : ι → Bool) : ι → Bool :=
+  fun i => if i ∈ R then η i else ω i
+
+lemma mean_coinCylinder_exposed {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (s R : Finset ι) (v η : ι → Bool) :
+    finiteMean (fun ω : ι → Bool =>
+      if ∀ i ∈ s, exposedCoins R η ω i = v i then (1 : ℝ) else 0) =
+    if (∀ i ∈ s ∩ R, η i = v i) then 1 / (2 : ℝ) ^ (s \ R).card else 0 := by
+  have he (ω : ι → Bool) : (∀ i ∈ s, exposedCoins R η ω i = v i) ↔
+      (∀ i ∈ s ∩ R, η i = v i) ∧ (∀ i ∈ s \ R, ω i = v i) := by
+    constructor
+    · intro h
+      constructor
+      · intro i hi
+        simpa [exposedCoins, (Finset.mem_inter.mp hi).2] using h i (Finset.mem_inter.mp hi).1
+      · intro i hi
+        simpa [exposedCoins, (Finset.mem_sdiff.mp hi).2] using h i (Finset.mem_sdiff.mp hi).1
+    · rintro ⟨ho, hf⟩ i hi
+      by_cases hR : i ∈ R
+      · simpa [exposedCoins, hR] using ho i (Finset.mem_inter.mpr ⟨hi,hR⟩)
+      · simpa [exposedCoins, hR] using hf i (Finset.mem_sdiff.mpr ⟨hi,hR⟩)
+  have hm : finiteMean (fun ω : ι → Bool =>
+      if ∀ i ∈ s, exposedCoins R η ω i = v i then (1 : ℝ) else 0) =
+      finiteMean (fun ω : ι → Bool =>
+        if (∀ i ∈ s ∩ R, η i = v i) ∧ (∀ i ∈ s \ R, ω i = v i) then (1 : ℝ) else 0) :=
+    finiteMean_congr (fun ω => by simp only [he ω])
+  rw [hm]
+  by_cases h : ∀ i ∈ s ∩ R, η i = v i
+  · have hp : (∀ i ∈ s ∩ R, η i = v i) ↔ True := iff_true_intro h
+    simp only [hp, true_and, ↓reduceIte]
+    exact mean_coinCylinder (s \ R) v
+  · have hp : (∀ i ∈ s ∩ R, η i = v i) ↔ False := iff_false_intro h
+    simp only [hp, false_and, ↓reduceIte]
+    simp [finiteMean]
+
+lemma butterfly_closure_probability (d : ℕ) (x y : Card d)
+    (R : Finset (SwitchIndex d)) (η : SwitchIndex d → Bool) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      if butterflyPerm d (decodeButterfly d (exposedCoins R η ω)) x = y then (1 : ℝ) else 0) =
+      if (∀ i ∈ routeDomain d x y ∩ R, η i = routeValue d x y i) then
+        (2 : ℝ) ^ (routeDomain d x y ∩ R).card / (2 : ℝ) ^ d else 0 := by
+  have he : finiteMean (fun ω : SwitchIndex d → Bool =>
+      if butterflyPerm d (decodeButterfly d (exposedCoins R η ω)) x = y then (1 : ℝ) else 0) =
+      finiteMean (fun ω : SwitchIndex d → Bool =>
+        if ∀ i ∈ routeDomain d x y, exposedCoins R η ω i = routeValue d x y i then (1 : ℝ) else 0) :=
+    finiteMean_congr (fun ω => by simp only [butterfly_endpoint_iff])
+  rw [he, mean_coinCylinder_exposed]
+  congr 1
+  have hc : (routeDomain d x y \ R).card + (routeDomain d x y ∩ R).card = d := by
+    rw [Finset.card_sdiff_add_card_inter, card_routeDomain]
+  have hp : (2 : ℝ) ^ d = (2 : ℝ) ^ (routeDomain d x y \ R).card *
+      (2 : ℝ) ^ (routeDomain d x y ∩ R).card := by
+    rw [← pow_add]
+    exact congrArg (fun n => (2 : ℝ) ^ n) hc.symm
+  rw [hp]
+  have hn : (2 : ℝ) ^ (routeDomain d x y ∩ R).card ≠ 0 := pow_ne_zero _ (by norm_num)
+  field_simp
+
+lemma butterfly_single_card_uniform (d : ℕ) (x y : Card d) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      if butterflyPerm d (decodeButterfly d ω) x = y then (1 : ℝ) else 0) = 1 / (2 : ℝ) ^ d := by
+  have he : finiteMean (fun ω : SwitchIndex d → Bool =>
+      if butterflyPerm d (decodeButterfly d ω) x = y then (1 : ℝ) else 0) =
+      finiteMean (fun ω : SwitchIndex d → Bool =>
+        if ∀ i ∈ routeDomain d x y, ω i = routeValue d x y i then (1 : ℝ) else 0) :=
+    finiteMean_congr (fun ω => by simp only [butterfly_endpoint_iff])
+  rw [he, mean_coinCylinder, card_routeDomain]
+
+def switchUsers : (d : ℕ) → Butterfly d → SwitchIndex d → Card d × Card d
+  | 0, _, i => nomatch i
+  | d + 1, b, Sum.inl y =>
+      (Fin.cons false ((butterflyPerm d (b.2 false)).symm y),
+       Fin.cons true ((butterflyPerm d (b.2 true)).symm y))
+  | d + 1, b, Sum.inr (ε,i) =>
+      (Fin.cons ε (switchUsers d (b.2 ε) i).1,
+       Fin.cons ε (switchUsers d (b.2 ε) i).2)
+
+lemma switchUsers_ne (d : ℕ) (b : Butterfly d) (i : SwitchIndex d) :
+    (switchUsers d b i).1 ≠ (switchUsers d b i).2 := by
+  induction d with
+  | zero => exact Empty.elim i
+  | succ d ih =>
+    cases i with
+    | inl y =>
+      intro h
+      have := congrFun h 0
+      simp only [switchUsers, Fin.cons_zero] at this
+      exact Bool.false_ne_true this
+    | inr z =>
+      intro h
+      exact ih (b.2 z.1) z.2 (by simpa only [switchUsers, Fin.tail_cons] using congrArg Fin.tail h)
+
+lemma cons_eq_iff {d : ℕ} (ε : Bool) (a : Card d) (x : Card (d + 1)) :
+    Fin.cons ε a = x ↔ ε = x 0 ∧ a = Fin.tail x := by
+  constructor
+  · intro h
+    exact ⟨by simpa only [Fin.cons_zero] using congrFun h 0,
+      by simpa only [Fin.tail_cons] using congrArg Fin.tail h⟩
+  · rintro ⟨rfl,rfl⟩
+    exact Fin.cons_self_tail x
+
+lemma mem_routeDomain_iff_user (d : ℕ) (b : Butterfly d) (x : Card d) (i : SwitchIndex d) :
+    i ∈ routeDomain d x (butterflyPerm d b x) ↔
+      (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x := by
+  classical
+  induction d with
+  | zero => exact Empty.elim i
+  | succ d ih =>
+    cases i with
+    | inl y =>
+      change Sum.inl y ∈ insert (Sum.inl (Fin.tail (butterflyPerm (d+1) b x)))
+        ((routeDomain d (Fin.tail x) (Fin.tail (butterflyPerm (d+1) b x))).image
+          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) ↔ _
+      simp only [Finset.mem_insert, Sum.inl.injEq, Finset.mem_image,
+        Sum.inr_ne_inl, and_false, exists_false, or_false]
+      change y = butterflyPerm d (b.2 (x 0)) (Fin.tail x) ↔ _
+      simp only [switchUsers, cons_eq_iff]
+      cases x 0 <;> simp [Equiv.symm_apply_eq]
+    | inr z =>
+      obtain ⟨ε,i⟩ := z
+      change Sum.inr (ε,i) ∈ insert (Sum.inl (Fin.tail (butterflyPerm (d+1) b x)))
+        ((routeDomain d (Fin.tail x) (Fin.tail (butterflyPerm (d+1) b x))).image
+          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) ↔ _
+      simp only [Finset.mem_insert, Sum.inr_ne_inl, false_or, Finset.mem_image,
+        Sum.inr.injEq, Prod.mk.injEq]
+      change (∃ j, j ∈ routeDomain d (Fin.tail x) (butterflyPerm d (b.2 (x 0)) (Fin.tail x)) ∧
+        x 0 = ε ∧ j = i) ↔ _
+      simp only [switchUsers, cons_eq_iff]
+      constructor
+      · rintro ⟨j,hj,hε,rfl⟩
+        subst ε
+        rcases (ih (b.2 (x 0)) (Fin.tail x) j).mp hj with h | h
+        · exact Or.inl ⟨rfl,h⟩
+        · exact Or.inr ⟨rfl,h⟩
+      · rintro (⟨hε,h⟩ | ⟨hε,h⟩)
+        · subst ε
+          exact ⟨i,(ih (b.2 (x 0)) (Fin.tail x) i).mpr (Or.inl h),rfl,rfl⟩
+        · subst ε
+          exact ⟨i,(ih (b.2 (x 0)) (Fin.tail x) i).mpr (Or.inr h),rfl,rfl⟩
+
+lemma card_switches_using (d : ℕ) (b : Butterfly d) (x : Card d) :
+    (Finset.univ.filter (fun i : SwitchIndex d =>
+      (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x)).card = d := by
+  classical
+  have he : Finset.univ.filter (fun i : SwitchIndex d =>
+      (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x) =
+      routeDomain d x (butterflyPerm d b x) := by
+    ext i
+    simp only [Finset.mem_filter, Finset.mem_univ, true_and, mem_routeDomain_iff_user]
+  rw [he,card_routeDomain]
+
+lemma switch_incidence_sum (d : ℕ) (b : Butterfly d) (w : Card d → ℝ) :
+    ∑ i : SwitchIndex d, (w (switchUsers d b i).1 + w (switchUsers d b i).2) =
+      (d : ℝ) * ∑ x : Card d, w x := by
+  classical
+  calc
+    _ = ∑ i : SwitchIndex d, ∑ x : Card d,
+        if (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x then w x else 0 := by
+      apply Finset.sum_congr rfl
+      intro i _
+      rw [Finset.sum_ite]
+      simp only [Finset.sum_const_zero, add_zero]
+      have he : Finset.univ.filter (fun x : Card d =>
+          (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x) =
+          {(switchUsers d b i).1, (switchUsers d b i).2} := by
+        ext x; simp [eq_comm]
+      rw [he, Finset.sum_pair (switchUsers_ne d b i)]
+    _ = ∑ x : Card d, ∑ i : SwitchIndex d,
+        if (switchUsers d b i).1 = x ∨ (switchUsers d b i).2 = x then w x else 0 :=
+      Finset.sum_comm
+    _ = ∑ x : Card d, (d : ℝ) * w x := by
+      apply Finset.sum_congr rfl
+      intro x _
+      rw [Finset.sum_ite]
+      simp only [Finset.sum_const_zero, add_zero, Finset.sum_const, nsmul_eq_mul,
+        card_switches_using]
+    _ = _ := by rw [Finset.mul_sum]
+
+lemma weighted_switch_min_bound (d : ℕ) (b : Butterfly d) (w : Card d → ℝ) :
+    ∑ i : SwitchIndex d, min (w (switchUsers d b i).1) (w (switchUsers d b i).2) ≤
+      (d : ℝ) / 2 * ∑ x : Card d, w x := by
+  have h : 2 * (∑ i : SwitchIndex d,
+      min (w (switchUsers d b i).1) (w (switchUsers d b i).2)) ≤
+      ∑ i : SwitchIndex d, (w (switchUsers d b i).1 + w (switchUsers d b i).2) := by
+    rw [Finset.mul_sum]
+    apply Finset.sum_le_sum
+    intro i _
+    linarith [min_le_left (w (switchUsers d b i).1) (w (switchUsers d b i).2),
+      min_le_right (w (switchUsers d b i).1) (w (switchUsers d b i).2)]
+  rw [switch_incidence_sum] at h
+  linarith
+
+lemma mem_childMarks {d : ℕ} (s : Finset (Card (d + 1))) (ε : Bool) (x : Card d) :
+    x ∈ childMarks s ε ↔ Fin.cons ε x ∈ s := by
+  classical
+  constructor
+  · intro h
+    obtain ⟨y,hy,hxy⟩ := Finset.mem_image.mp h
+    obtain ⟨hys,hyε⟩ := Finset.mem_filter.mp hy
+    rw [← hxy, ← hyε, Fin.cons_self_tail]
+    exact hys
+  · intro h
+    exact Finset.mem_image.mpr ⟨Fin.cons ε x,
+      Finset.mem_filter.mpr ⟨h,by simp⟩,by simp⟩
+
+lemma mem_topMeetings {d : ℕ} (b : Butterfly (d + 1))
+    (s : Finset (Card (d + 1))) (y : Card d) :
+    y ∈ topMeetings b s ↔
+      (switchUsers (d+1) b (Sum.inl y)).1 ∈ s ∧
+      (switchUsers (d+1) b (Sum.inl y)).2 ∈ s := by
+  classical
+  have he (ε : Bool) : y ∈ (childMarks s ε).image (butterflyPerm d (b.2 ε)) ↔
+      (butterflyPerm d (b.2 ε)).symm y ∈ childMarks s ε := by
+    constructor
+    · intro h
+      obtain ⟨x,hx,rfl⟩ := Finset.mem_image.mp h
+      simpa only [Equiv.symm_apply_apply] using hx
+    · intro h
+      exact Finset.mem_image.mpr ⟨_,h,Equiv.apply_symm_apply _ _⟩
+  simp only [topMeetings, Finset.mem_inter, he, mem_childMarks, switchUsers]
+
+lemma encounters_eq_switch_sum (d : ℕ) (b : Butterfly d) (s : Finset (Card d)) :
+    encounters d b s = ∑ i : SwitchIndex d,
+      if (switchUsers d b i).1 ∈ s ∧ (switchUsers d b i).2 ∈ s then 1 else 0 := by
+  classical
+  induction d with
+  | zero =>
+    simp only [encounters]
+    symm
+    apply Finset.sum_eq_zero
+    intro i _
+    exact Empty.elim i
+  | succ d ih =>
+    change encounters d (b.2 false) (childMarks s false) +
+      encounters d (b.2 true) (childMarks s true) + (topMeetings b s).card =
+        ∑ i : Sum (Card d) (Bool × SwitchIndex d), _
+    rw [Fintype.sum_sum_type, Fintype.sum_prod_type]
+    simp only [Fintype.sum_bool]
+    rw [ih,ih]
+    have ht : (∑ y : Card d,
+        if (switchUsers (d+1) b (Sum.inl y)).1 ∈ s ∧
+          (switchUsers (d+1) b (Sum.inl y)).2 ∈ s then 1 else 0) =
+        (topMeetings b s).card := by
+      simp only [← mem_topMeetings]
+      simp
+    rw [ht]
+    simp only [mem_childMarks, switchUsers]
+    ac_rfl
+
+lemma marked_switch_entropy_bound (d : ℕ) (b : Butterfly d) (s : Finset (Card d)) :
+    (∑ i : SwitchIndex d,
+      if (switchUsers d b i).1 ∈ s ∧ (switchUsers d b i).2 ∈ s then (1:ℝ) else 0) ≤
+      (s.card : ℝ) * Real.log s.card / (2 * Real.log 2) := by
+  classical
+  have he : (encounters d b s : ℝ) =
+      ∑ i : SwitchIndex d,
+        if (switchUsers d b i).1 ∈ s ∧ (switchUsers d b i).2 ∈ s then (1:ℝ) else 0 := by
+    rw [encounters_eq_switch_sum, Nat.cast_sum]
+    apply Finset.sum_congr rfl
+    intro i _
+    split_ifs <;> simp
+  rw [← he]
+  exact butterfly_encounters d b s
+
+namespace QueryTree
+variable {ι : Type*} [DecidableEq ι]
+
+def memo : QueryTree ι → Finset ι → (ι → Bool) → QueryTree ι
+  | reject, _, _ => reject
+  | accept, _, _ => accept
+  | branch i zero one, R, η =>
+      if i ∈ R then (if η i then memo one R η else memo zero R η) else
+      branch i (memo zero (insert i R) (Function.update η i false))
+        (memo one (insert i R) (Function.update η i true))
+  | force i b next, R, η =>
+      if i ∈ R then (if η i = b then memo next R η else reject) else
+      force i b (memo next (insert i R) (Function.update η i b))
+
+lemma memo_queries_disjoint (T : QueryTree ι) (R : Finset ι) (η : ι → Bool) :
+    Disjoint (memo T R η).queries R := by
+  induction T generalizing R η with
+  | reject => simp [memo,queries]
+  | accept => simp [memo,queries]
+  | branch i zero one ih₀ ih₁ =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte]
+      cases η i <;> simp only [Bool.false_eq_true,↓reduceIte]
+      · exact ih₀ R η
+      · exact ih₁ R η
+    · simp only [memo,hi,↓reduceIte,queries,Finset.disjoint_insert_left,Finset.disjoint_union_left]
+      exact ⟨by simp, (ih₀ (insert i R) _).mono_right (Finset.subset_insert _ _),
+        (ih₁ (insert i R) _).mono_right (Finset.subset_insert _ _)⟩
+  | force i b next ih =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte]
+      by_cases hb : η i = b
+      · simpa only [hb,↓reduceIte] using ih R η
+      · simp [hb,queries]
+    · simp only [memo,hi,↓reduceIte,queries,Finset.disjoint_insert_left]
+      exact ⟨by simp,(ih (insert i R) _).mono_right (Finset.subset_insert _ _)⟩
+
+lemma memo_fresh (T : QueryTree ι) (R : Finset ι) (η : ι → Bool) :
+    (memo T R η).Fresh := by
+  induction T generalizing R η with
+  | reject => trivial
+  | accept => trivial
+  | branch i zero one ih₀ ih₁ =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte]
+      cases η i <;> simp only [Bool.false_eq_true,↓reduceIte]
+      · exact ih₀ R η
+      · exact ih₁ R η
+    · simp only [memo,hi,↓reduceIte,Fresh]
+      exact ⟨fun hm => Finset.disjoint_left.mp (memo_queries_disjoint zero (insert i R) _) hm
+          (Finset.mem_insert_self _ _),
+        fun hm => Finset.disjoint_left.mp (memo_queries_disjoint one (insert i R) _) hm
+          (Finset.mem_insert_self _ _), ih₀ _ _,ih₁ _ _⟩
+  | force i b next ih =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte]
+      by_cases hb : η i = b
+      · simpa only [hb,↓reduceIte] using ih R η
+      · simp [hb,Fresh]
+    · simp only [memo,hi,↓reduceIte,Fresh]
+      exact ⟨fun hm => Finset.disjoint_left.mp (memo_queries_disjoint next (insert i R) _) hm
+        (Finset.mem_insert_self _ _),ih _ _⟩
+
+lemma update_agrees {R : Finset ι} {η ω : ι → Bool}
+    (h : ∀ i ∈ R, η i = ω i) (i : ι) :
+    ∀ j ∈ insert i R, Function.update η i (ω i) j = ω j := by
+  intro j hj
+  rcases Finset.mem_insert.mp hj with rfl | hj
+  · simp
+  · by_cases hji : j = i
+    · subst j; simp
+    · rw [Function.update_of_ne hji]
+      exact h j hj
+
+lemma memo_succeeds (T : QueryTree ι) (R : Finset ι) (η ω : ι → Bool)
+    (h : ∀ i ∈ R, η i = ω i) : (memo T R η).succeeds ω = T.succeeds ω := by
+  induction T generalizing R η with
+  | reject => rfl
+  | accept => rfl
+  | branch i zero one ih₀ ih₁ =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte,succeeds,h i hi]
+      cases ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
+      · exact ih₀ R η h
+      · exact ih₁ R η h
+    · simp only [memo,hi,↓reduceIte,succeeds]
+      cases hb : ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
+      · exact ih₀ _ _ (by simpa only [hb] using update_agrees h i)
+      · exact ih₁ _ _ (by simpa only [hb] using update_agrees h i)
+  | force i b next ih =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte,h i hi]
+      by_cases hb : ω i = b
+      · simp only [hb,↓reduceIte,succeeds]
+        exact ih R η h
+      · simp [hb,succeeds]
+    · simp only [memo,hi,↓reduceIte,succeeds]
+      by_cases hb : ω i = b
+      · simp only [hb,↓reduceIte]
+        exact ih _ _ (by simpa only [hb] using update_agrees h i)
+      · simp [hb]
+
+def freshCharge : QueryTree ι → Finset ι → (ι → Bool) → ℕ
+  | reject, _, _ => 0
+  | accept, _, _ => 0
+  | branch i zero one, R, ω => if ω i then freshCharge one (insert i R) ω else freshCharge zero (insert i R) ω
+  | force i b next, R, ω => if ω i = b then
+      freshCharge next (insert i R) ω + (if i ∈ R then 0 else 1) else 0
+
+lemma memo_charge (T : QueryTree ι) (R : Finset ι) (η ω : ι → Bool)
+    (h : ∀ i ∈ R, η i = ω i) : (memo T R η).charge ω = freshCharge T R ω := by
+  induction T generalizing R η with
+  | reject => rfl
+  | accept => rfl
+  | branch i zero one ih₀ ih₁ =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte,freshCharge,h i hi,Finset.insert_eq_of_mem hi]
+      cases ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
+      · exact ih₀ R η h
+      · exact ih₁ R η h
+    · simp only [memo,hi,↓reduceIte,charge,freshCharge]
+      cases hb : ω i <;> simp only [Bool.false_eq_true,↓reduceIte]
+      · exact ih₀ _ _ (by simpa only [hb] using update_agrees h i)
+      · exact ih₁ _ _ (by simpa only [hb] using update_agrees h i)
+  | force i b next ih =>
+    by_cases hi : i ∈ R
+    · simp only [memo,hi,↓reduceIte,h i hi,freshCharge,Finset.insert_eq_of_mem hi,Nat.add_zero]
+      by_cases hb : ω i = b
+      · simp only [hb,↓reduceIte]
+        exact ih R η h
+      · simp [hb,charge]
+    · simp only [memo,hi,↓reduceIte,charge,freshCharge]
+      by_cases hb : ω i = b
+      · simp only [hb,↓reduceIte]
+        rw [ih _ _ (by simpa only [hb] using update_agrees h i)]
+      · simp [hb]
+
+lemma mean_freshCharge_le_one [Fintype ι] (T : QueryTree ι) :
+    finiteMean (fun ω : ι → Bool =>
+      if T.succeeds ω then (2:ℝ) ^ T.freshCharge ∅ ω else 0) ≤ 1 := by
+  let η : ι → Bool := fun _ => false
+  have hm := mean_weighted_le_one (memo T ∅ η) (memo_fresh T ∅ η)
+  have he : finiteMean (memo T ∅ η).weighted =
+      finiteMean (fun ω : ι → Bool =>
+        if T.succeeds ω then (2:ℝ) ^ T.freshCharge ∅ ω else 0) := by
+    apply finiteMean_congr
+    intro ω
+    rw [weighted_eq, memo_succeeds _ _ _ _ (by simp), memo_charge _ _ _ _ (by simp)]
+  rw [he] at hm
+  exact hm
+
+
+def followPath : (d : ℕ) → (SwitchIndex d → ι) → Card d →
+    (Card d → QueryTree ι) → QueryTree ι
+  | 0, _, x, k => k x
+  | d + 1, e, x, k => followPath d (fun i => e (Sum.inr (x 0,i))) (Fin.tail x)
+      (fun y => branch (e (Sum.inl y))
+        (k (Fin.cons (Bool.xor (x 0) false) y))
+        (k (Fin.cons (Bool.xor (x 0) true) y)))
+
+omit [DecidableEq ι] in
+lemma followPath_succeeds (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
+    (k : Card d → QueryTree ι) (ω : ι → Bool) :
+    (followPath d e x k).succeeds ω =
+      (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).succeeds ω := by
+  induction d with
+  | zero => rfl
+  | succ d ih =>
+    rw [followPath, ih]
+    let y := butterflyPerm d
+      (decodeButterfly d (fun i => ω (e (Sum.inr (x 0,i))))) (Fin.tail x)
+    change (if ω (e (Sum.inl y)) then
+      (k (Fin.cons (Bool.xor (x 0) true) y)).succeeds ω else
+      (k (Fin.cons (Bool.xor (x 0) false) y)).succeeds ω) =
+      (k (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).succeeds ω
+    cases ω (e (Sum.inl y)) <;> rfl
+
+lemma followPath_charge (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
+    (k : Card d → QueryTree ι) (R : Finset ι) (ω : ι → Bool) :
+    freshCharge (followPath d e x k) R ω =
+      freshCharge (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x))
+        (R ∪ (routeDomain d x (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).image e) ω := by
+  classical
+  induction d generalizing R with
+  | zero => simp only [followPath, routeDomain, Finset.image_empty, Finset.union_empty]; rfl
+  | succ d ih =>
+    rw [followPath, ih]
+    let y := butterflyPerm d
+      (decodeButterfly d (fun i => ω (e (Sum.inr (x 0,i))))) (Fin.tail x)
+    let S := (routeDomain d (Fin.tail x) y).image (fun i => e (Sum.inr (x 0,i)))
+    change (if ω (e (Sum.inl y)) then
+      freshCharge (k (Fin.cons (Bool.xor (x 0) true) y)) (insert (e (Sum.inl y)) (R ∪ S)) ω else
+      freshCharge (k (Fin.cons (Bool.xor (x 0) false) y)) (insert (e (Sum.inl y)) (R ∪ S)) ω) =
+      freshCharge (k (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y))
+        (R ∪ (routeDomain (d+1) x
+          (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).image e) ω
+    have hs : (routeDomain (d+1) x
+        (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).image e =
+        insert (e (Sum.inl y)) S := by
+      let e' : Sum (Card d) (Bool × SwitchIndex d) → ι := e
+      change (insert (Sum.inl y)
+        ((routeDomain d (Fin.tail x) y).image
+          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) :
+          Finset (Sum (Card d) (Bool × SwitchIndex d))).image e' = _
+      rw [Finset.image_insert, Finset.image_image]
+      rfl
+    rw [hs,Finset.union_insert]
+    cases ω (e (Sum.inl y)) <;> rfl
+
+def forceList : List ι → (ι → Bool) → QueryTree ι → QueryTree ι
+  | [], _, k => k
+  | i :: L, v, k => force i (v i) (forceList L v k)
+
+omit [DecidableEq ι] in
+lemma forceList_succeeds (L : List ι) (v : ι → Bool) (k : QueryTree ι) (ω : ι → Bool) :
+    (forceList L v k).succeeds ω =
+      if ∀ i ∈ L, ω i = v i then k.succeeds ω else false := by
+  induction L with
+  | nil => simp [forceList]
+  | cons i L ih =>
+    simp only [forceList,succeeds,ih,List.mem_cons,forall_eq_or_imp]
+    by_cases h : ω i = v i <;> simp [h]
+
+lemma forceList_charge (L : List ι) (v : ι → Bool) (k : QueryTree ι)
+    (R : Finset ι) (ω : ι → Bool) (h : ∀ i ∈ L, ω i = v i) :
+    freshCharge (forceList L v k) R ω =
+      freshCharge k (R ∪ L.toFinset) ω + (L.toFinset \ R).card := by
+  induction L generalizing R with
+  | nil => simp [forceList]
+  | cons i L ih =>
+    rw [forceList,freshCharge,ite_eq_left (h i List.mem_cons_self),
+      ih (insert i R) (fun j hj => h j (List.mem_cons_of_mem _ hj))]
+    simp only [List.toFinset_cons]
+    have hs : insert i R ∪ L.toFinset = R ∪ insert i L.toFinset := by ext j; simp
+    rw [hs]
+    by_cases hi : i ∈ R
+    · rw [ite_eq_left hi]
+      have he : L.toFinset \ insert i R = insert i L.toFinset \ R := by
+        ext j
+        by_cases hji : j = i
+        · subst j; simp [hi]
+        · simp [hji]
+      rw [he,Nat.add_zero]
+    · rw [ite_eq_right hi]
+      have he : insert i L.toFinset \ R = insert i (L.toFinset \ insert i R) := by
+        ext j
+        by_cases hji : j = i
+        · subst j; simp [hi]
+        · simp [hji]
+      rw [he, Finset.card_insert_of_notMem (by simp)]
+      omega
+
+
+def staleCharge : QueryTree ι → Finset ι → (ι → Bool) → ℕ
+  | reject, _, _ => 0
+  | accept, _, _ => 0
+  | branch i zero one, R, ω => if ω i then staleCharge one (insert i R) ω
+      else staleCharge zero (insert i R) ω
+  | force i b next, R, ω => if ω i = b then
+      staleCharge next (insert i R) ω + (if i ∈ R then 1 else 0) else 0
+
+lemma fresh_add_stale (T : QueryTree ι) (R : Finset ι) (ω : ι → Bool) :
+    T.freshCharge R ω + T.staleCharge R ω = T.charge ω := by
+  induction T generalizing R with
+  | reject => rfl
+  | accept => rfl
+  | branch i zero one ih₀ ih₁ =>
+    cases hb : ω i <;> simp [freshCharge,staleCharge,charge,hb,ih₀,ih₁]
+  | force i b next ih =>
+    by_cases hb : ω i = b
+    · simp only [freshCharge,staleCharge,charge,hb,↓reduceIte]
+      have he := ih (insert i R)
+      by_cases hi : i ∈ R <;> simp only [hi,↓reduceIte] <;> omega
+    · simp [freshCharge,staleCharge,charge,hb]
+
+omit [DecidableEq ι] in
+lemma followPath_raw_charge (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
+    (k : Card d → QueryTree ι) (ω : ι → Bool) :
+    (followPath d e x k).charge ω =
+      (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).charge ω := by
+  induction d with
+  | zero => rfl
+  | succ d ih =>
+    rw [followPath, ih]
+    let y := butterflyPerm d
+      (decodeButterfly d (fun i => ω (e (Sum.inr (x 0,i))))) (Fin.tail x)
+    change (if ω (e (Sum.inl y)) then
+      (k (Fin.cons (Bool.xor (x 0) true) y)).charge ω else
+      (k (Fin.cons (Bool.xor (x 0) false) y)).charge ω) =
+      (k (Fin.cons (Bool.xor (x 0) (ω (e (Sum.inl y)))) y)).charge ω
+    cases ω (e (Sum.inl y)) <;> rfl
+
+omit [DecidableEq ι] in
+lemma forceList_raw_charge (L : List ι) (v : ι → Bool) (k : QueryTree ι) (ω : ι → Bool)
+    (h : ∀ i ∈ L, ω i = v i) :
+    (forceList L v k).charge ω = k.charge ω + L.length := by
+  induction L with
+  | nil => simp [forceList]
+  | cons i L ih =>
+    rw [forceList,charge,ite_eq_left (h i List.mem_cons_self),
+      ih (fun j hj => h j (List.mem_cons_of_mem _ hj))]
+    simp only [List.length_cons,Nat.add_assoc]
+
+noncomputable def closePath (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d)) :
+    QueryTree (SwitchIndex d) :=
+  forceList (routeDomain d x y).toList (routeValue d x y) k
+
+lemma closePath_succeeds (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
+    (ω : SwitchIndex d → Bool) :
+    (closePath d x y k).succeeds ω =
+      if butterflyPerm d (decodeButterfly d ω) x = y then k.succeeds ω else false := by
+  unfold closePath
+  rw [forceList_succeeds]
+  simp only [Finset.mem_toList, ← butterfly_endpoint_iff]
+
+lemma closePath_raw_charge (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
+    (ω : SwitchIndex d → Bool) (h : butterflyPerm d (decodeButterfly d ω) x = y) :
+    (closePath d x y k).charge ω = k.charge ω + d := by
+  unfold closePath
+  rw [forceList_raw_charge _ _ _ _ (by simpa only [Finset.mem_toList] using
+    (butterfly_endpoint_iff d ω x y).mp h), Finset.length_toList, card_routeDomain]
+
+lemma closePath_fresh_charge (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
+    (R : Finset (SwitchIndex d)) (ω : SwitchIndex d → Bool)
+    (h : butterflyPerm d (decodeButterfly d ω) x = y) :
+    (closePath d x y k).freshCharge R ω =
+      k.freshCharge (R ∪ routeDomain d x y) ω + (routeDomain d x y \ R).card := by
+  unfold closePath
+  rw [forceList_charge _ _ _ _ _ (by simpa only [Finset.mem_toList] using
+    (butterfly_endpoint_iff d ω x y).mp h), Finset.toList_toFinset]
+
+lemma closePath_stale_charge (d : ℕ) (x y : Card d) (k : QueryTree (SwitchIndex d))
+    (R : Finset (SwitchIndex d)) (ω : SwitchIndex d → Bool)
+    (h : butterflyPerm d (decodeButterfly d ω) x = y) :
+    (closePath d x y k).staleCharge R ω =
+      k.staleCharge (R ∪ routeDomain d x y) ω + (routeDomain d x y ∩ R).card := by
+  have h₀ := fresh_add_stale (closePath d x y k) R ω
+  have h₁ := fresh_add_stale k (R ∪ routeDomain d x y) ω
+  rw [closePath_raw_charge _ _ _ _ _ h,closePath_fresh_charge _ _ _ _ _ _ h] at h₀
+  have h₂ : (routeDomain d x y \ R).card + (routeDomain d x y ∩ R).card = d := by
+    rw [Finset.card_sdiff_add_card_inter,card_routeDomain]
+  omega
+
+noncomputable def seekCycle (d : ℕ) (S : Finset (Card d)) (a : Card d) :
+    ℕ → Card d → Finset (Card d) → (Finset (Card d) → QueryTree (SwitchIndex d)) →
+      QueryTree (SwitchIndex d)
+  | 0, x, V, k => if x ∈ S ∧ x ∉ V then closePath d x a (k (insert x V)) else reject
+  | r + 1, x, V, k => if x ∈ S ∧ x ∉ V then followPath d id x
+      (fun y => seekCycle d S a r y (insert x V) k) else reject
+
+lemma seekCycle_raw_charge (d : ℕ) (S : Finset (Card d)) (a : Card d) (r : ℕ)
+    (x : Card d) (V : Finset (Card d)) (k : Finset (Card d) → QueryTree (SwitchIndex d))
+    (ω : SwitchIndex d → Bool) (K : ℕ)
+    (hK : ∀ V, (k V).succeeds ω = true → (k V).charge ω = K)
+    (hs : (seekCycle d S a r x V k).succeeds ω = true) :
+    (seekCycle d S a r x V k).charge ω = K + d := by
+  induction r generalizing x V with
+  | zero =>
+    by_cases hv : x ∈ S ∧ x ∉ V
+    · rw [seekCycle,ite_eq_left hv,closePath_succeeds] at hs
+      split_ifs at hs with he
+      · rw [seekCycle,ite_eq_left hv,closePath_raw_charge _ _ _ _ _ he,hK _ hs]
+    · simp [seekCycle,hv,succeeds] at hs
+  | succ r ih =>
+    by_cases hv : x ∈ S ∧ x ∉ V
+    · rw [seekCycle,ite_eq_left hv,followPath_succeeds] at hs
+      rw [seekCycle,ite_eq_left hv,followPath_raw_charge]
+      exact ih _ _ hs
+    · simp [seekCycle,hv,succeeds] at hs
+
+noncomputable def seekCycles (d : ℕ) (S : Finset (Card d)) :
+    List (Card d × ℕ) → Finset (Card d) → QueryTree (SwitchIndex d)
+  | [], _ => accept
+  | (a,r) :: L, V => seekCycle d S a r a V (fun V => seekCycles d S L V)
+
+lemma seekCycles_raw_charge (d : ℕ) (S : Finset (Card d)) (L : List (Card d × ℕ))
+    (V : Finset (Card d)) (ω : SwitchIndex d → Bool)
+    (hs : (seekCycles d S L V).succeeds ω = true) :
+    (seekCycles d S L V).charge ω = L.length * d := by
+  induction L generalizing V with
+  | nil => simp [seekCycles,charge]
+  | cons ar L ih =>
+    obtain ⟨a,r⟩ := ar
+    rw [seekCycles]
+    rw [seekCycle_raw_charge d S a r a V (fun V => seekCycles d S L V) ω (L.length*d)
+      (fun V hs => ih V hs) hs]
+    simp [Nat.add_mul]
+
+lemma cycles_weighted_success (d : ℕ) (S : Finset (Card d)) (L : List (Card d × ℕ)) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      if (seekCycles d S L ∅).succeeds ω then
+        ((2:ℝ)^d)^L.length / (2:ℝ)^(seekCycles d S L ∅).staleCharge ∅ ω else 0) ≤ 1 := by
+  let T := seekCycles d S L ∅
+  have hh := mean_freshCharge_le_one T
+  have he : finiteMean (fun ω : SwitchIndex d → Bool =>
+      if T.succeeds ω then (2:ℝ)^T.freshCharge ∅ ω else 0) =
+      finiteMean (fun ω : SwitchIndex d → Bool =>
+        if T.succeeds ω then ((2:ℝ)^d)^L.length / (2:ℝ)^T.staleCharge ∅ ω else 0) := by
+    apply finiteMean_congr
+    intro ω
+    cases hs : T.succeeds ω
+    · simp
+    · simp only [↓reduceIte]
+      have hc := fresh_add_stale T ∅ ω
+      rw [seekCycles_raw_charge d S L ∅ ω hs] at hc
+      have hp : ((2:ℝ)^d)^L.length = (2:ℝ)^T.freshCharge ∅ ω * (2:ℝ)^T.staleCharge ∅ ω := by
+        rw [←pow_add,hc,←pow_mul,Nat.mul_comm]
+      rw [hp,mul_div_cancel_right₀ _ (pow_ne_zero _ (by norm_num))]
+  rw [← he]
+  exact hh
+
+end QueryTree
+
+lemma finiteMean_add {Ω : Type*} [Fintype Ω] (f g : Ω → ℝ) :
+    finiteMean (fun ω => f ω + g ω) = finiteMean f + finiteMean g := by
+  simp only [finiteMean,Finset.sum_add_distrib,add_div]
+
+lemma finiteMean_mul_const {Ω : Type*} [Fintype Ω] (f : Ω → ℝ) (c : ℝ) :
+    finiteMean (fun ω => f ω * c) = finiteMean f * c := by
+  unfold finiteMean
+  rw [← Finset.sum_mul]
+  ring
+
+lemma finiteMean_sum {Ω α : Type*} [Fintype Ω] (s : Finset α) (f : α → Ω → ℝ) :
+    finiteMean (fun ω => ∑ i ∈ s, f i ω) = ∑ i ∈ s, finiteMean (f i) := by
+  simp only [finiteMean,Finset.sum_div]
+  exact Finset.sum_comm
+
+lemma finiteMean_mono {Ω : Type*} [Fintype Ω] {f g : Ω → ℝ}
+    (h : ∀ ω, f ω ≤ g ω) : finiteMean f ≤ finiteMean g := by
+  exact div_le_div_of_nonneg_right (Finset.sum_le_sum (fun ω _ => h ω)) (Nat.cast_nonneg _)
+
+lemma finiteMean_equiv {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
+    (e : Ω ≃ Ω') (f : Ω → ℝ) :
+    finiteMean f = finiteMean (fun ω => f (e.symm ω)) := by
+  unfold finiteMean
+  rw [Fintype.card_congr e]
+  congr 1
+  exact Fintype.sum_equiv e _ _ (fun ω => by simp)
+
+lemma finiteMean_prod_left {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] [Nonempty Ω']
+    (f : Ω → ℝ) : finiteMean (fun ω : Ω × Ω' => f ω.1) = finiteMean f := by
+  simp only [finiteMean,Fintype.sum_prod_type,Finset.sum_const,nsmul_eq_mul,
+    ← Finset.mul_sum,Fintype.card_prod,Nat.cast_mul,Finset.card_univ]
+  have hn : (Fintype.card Ω' : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
+  field_simp
+
+lemma finiteMean_pi_eval {C : Type*} [Fintype C] [DecidableEq C]
+    {A : C → Type*} [∀ c, Fintype (A c)] [∀ c, Nonempty (A c)]
+    (c : C) (g : A c → ℝ) :
+    finiteMean (fun z : ∀ c, A c => g (z c)) = finiteMean g := by
+  rw [finiteMean_equiv (Equiv.piSplitAt c A)]
+  have he : (fun z : A c × ((j : {j // j ≠ c}) → A j) =>
+      g ((Equiv.piSplitAt c A).symm z c)) = (fun z => g z.1) := by
+    funext z
+    simp [Equiv.piSplitAt]
+  rw [he,finiteMean_prod_left]
+
+lemma finiteMean_eq_indicator {Ω : Type*} [Fintype Ω] [DecidableEq Ω] (a : Ω) :
+    finiteMean (fun ω : Ω => if ω = a then (1:ℝ) else 0) = 1 / Fintype.card Ω := by
+  simp [finiteMean]
+
+lemma exp_finiteMean_le {Ω : Type*} [Fintype Ω] [Nonempty Ω] (f : Ω → ℝ) :
+    Real.exp (finiteMean f) ≤ finiteMean (fun ω => Real.exp (f ω)) := by
+  have hn : (Fintype.card Ω : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
+  have hw : ∑ ω : Ω, (1 / (Fintype.card Ω : ℝ)) = 1 := by simp [hn]
+  have hj := convexOn_exp.map_sum_le (t := Finset.univ)
+    (w := fun _ : Ω => 1 / (Fintype.card Ω : ℝ)) (p := f)
+    (fun _ _ => by positivity) hw (fun _ _ => Set.mem_univ _)
+  simp only [smul_eq_mul, ← Finset.mul_sum] at hj
+  simpa only [finiteMean, div_eq_mul_inv, one_div, mul_comm, mul_one, one_mul] using hj
+
+variable {d A : ℕ}
+
+def cycleFiber (o : Card d → Option (Fin A)) (i : Fin A) : Finset (Card d) := by
+  classical
+  exact Finset.univ.filter (fun x => o x = some i)
+
+@[simp] lemma mem_cycleFiber (o : Card d → Option (Fin A)) (i : Fin A) (x : Card d) :
+    x ∈ cycleFiber o i ↔ o x = some i := by
+  classical
+  simp [cycleFiber]
+
+abbrev Terminal (o : Card d → Option (Fin A)) := ∀ i : Fin A, {x : Card d // o x = some i}
+
+noncomputable def slotWeight (o : Card d → Option (Fin A)) (i : Fin A) : ℝ :=
+  1 / (cycleFiber o i).card
+
+noncomputable def pathWeight (o : Card d → Option (Fin A)) (x : Card d) : ℝ :=
+  match o x with
+  | none => 0
+  | some i => slotWeight o i
+
+noncomputable def withinWeight (o : Card d → Option (Fin A)) (u v : Card d) : ℝ :=
+  match o u, o v with
+  | some i, some j => if i = j then slotWeight o i else 0
+  | _, _ => 0
+
+end Thorp
+
+end ThorpNine.Tail
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Tail/HeightDecay.lean b/lean/OAI/Probability/ThorpRouting/Tail/HeightDecay.lean
index a92bc3ae098080a8ff1f1a970e04b17c1ad099f0..246c76f1ca260e4a5493bfec3978efb922a1c8da 100644
--- a/lean/OAI/Probability/ThorpRouting/Tail/HeightDecay.lean
+++ b/lean/OAI/Probability/ThorpRouting/Tail/HeightDecay.lean
@@ -1,863 +1,863 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Tail.ParallelFreshPre
-
-namespace OAI
-
-namespace ThorpNine.Tail
-
-namespace Thorp
-
-theorem palindrome_family_mgf (L n d : ℕ) (hL : 0 < L) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) ≤
-      Real.exp (Fintype.card ι * (68*∑ j ∈ Finset.range n,
-        Real.exp (-((L*4^j:ℕ):ℝ)/64))) := by
-  induction n with
-  | zero => simp only [palindromeFamilyCost,Finset.range_zero,Finset.sum_empty,pow_zero,
-      finiteMean_const,mul_zero,Real.exp_zero,le_refl]
-  | succ n ih =>
-    let l := L*4^n
-    have hl : 0 < l := Nat.mul_pos hL (pow_pos (by decide) _)
-    have hs : 2*familyEnd L n ≤ l := familyEnd_two L n
-    have hc : ∀ Y, palindromeFamilyCost L (n+1) d e (Y,X) =
-        palindromeFamilyCost L n d e (Y,X)+palindromeSlabCost l d e (Y,X) := by
-      intro Y
-      exact Finset.sum_range_succ _ _
-    have hm : finiteMean (fun Y => q^(palindromeFamilyCost L (n+1) d e (Y,X))) ≤
-        finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) *
-          Real.exp (Fintype.card ι*(68*Real.exp (-(l:ℝ)/64))) := by
-      by_cases hld : l ≤ d
-      · simp_rw [hc,pow_add]
-        exact palindrome_slab_adapted_mul_le d (familyEnd L n) l (by omega) hl hs e X _
-          (fun _ => pow_nonneg (by linarith) _)
-          (layerAdapted_comp (palindromeFamilyCost_adapted L n d e X) (fun c => q^c))
-          q hq hlog
-      · simp_rw [hc,palindromeSlabCost_gt l d (by omega) e,Nat.add_zero]
-        exact le_mul_of_one_le_right
-          (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
-          (Real.one_le_exp_iff.mpr (by positivity))
-    refine (hm.trans (mul_le_mul_of_nonneg_right ih (Real.exp_nonneg _))).trans_eq ?_
-    rw [←Real.exp_add,Finset.sum_range_succ]
-    congr 1
-    dsimp [l]
-    ring
-
-lemma pow_four_ge (j : ℕ) : j+1 ≤ 4^j := by
-  induction j with
-  | zero => norm_num
-  | succ j ih => rw [pow_succ]; omega
-
-lemma sum_family_decay (L n : ℕ) (hL : 0 < L) :
-    (∑ j ∈ Finset.range n, Real.exp (-((L*4^j:ℕ):ℝ)/64)) ≤
-      Real.exp (-(L:ℝ)/64)/(1-Real.exp (-(1:ℝ)/64)) := by
-  let r := Real.exp (-(1:ℝ)/64)
-  have hr : 0 < r := Real.exp_pos _
-  have hr1 : r < 1 := Real.exp_lt_one_iff.mpr (by norm_num)
-  have hg : (∑ j ∈ Finset.range n, r^j) ≤ 1/(1-r) := by
-    apply (le_div_iff₀ (by linarith : (0:ℝ)<1-r)).mpr
-    have hh := geom_sum_mul r n
-    nlinarith [pow_nonneg (le_of_lt hr) n]
-  calc
-    _ ≤ ∑ j ∈ Finset.range n, Real.exp (-(L:ℝ)/64)*r^j := by
-      apply Finset.sum_le_sum
-      intro j _
-      have hj := pow_four_ge j
-      have hlj : L+j ≤ L*4^j := by nlinarith
-      have hlj' : (L:ℝ)+j ≤ (L*4^j:ℕ) := by exact_mod_cast hlj
-      rw [←Real.exp_nat_mul,←Real.exp_add]
-      exact Real.exp_le_exp.mpr (by linarith)
-    _ = Real.exp (-(L:ℝ)/64)*(∑ j ∈ Finset.range n, r^j) := by rw [Finset.mul_sum]
-    _ ≤ Real.exp (-(L:ℝ)/64)*(1/(1-r)) :=
-      mul_le_mul_of_nonneg_left hg (Real.exp_nonneg _)
-    _ = _ := by ring
-
-
-lemma palindromeLevelCost_step_add (t d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (d+1)) (ω : BenesCoins (d+1)) :
-    palindromeLevelCost t (d+1) e ω =
-      let σ := benesStepEquiv d ω
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x σ.2.1
-      let hc := PairRouting.colors_compatible x σ.2.1
-      (if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
-          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) else 0) +
-        palindromeLevelCost t d (PairRouting.childEmbedding x c hc false) σ.1.1 +
-        palindromeLevelCost t d (PairRouting.childEmbedding x c hc true) σ.1.2 := by
-  classical
-  rw [palindromeLevelCost]
-  dsimp only
-  split_ifs with h
-  · simp only [palindromeLevelCost_gt t d (by omega),Nat.add_zero]
-  · simp only [Nat.zero_add]
-
-lemma palindromeLowCost_sum (H d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) :
-    palindromeLowCost H d e ω = ∑ i ∈ Finset.range H, palindromeLevelCost (i+1) d e ω := by
-  classical
-  induction d generalizing ι with
-  | zero => simp only [palindromeLowCost,palindromeLevelCost,Finset.sum_const_zero]
-  | succ d ih =>
-    rw [palindromeLowCost]
-    simp only [palindromeLevelCost_step_add,Finset.sum_add_distrib]
-    rw [←ih,←ih]
-    congr 2
-    simp only [Nat.add_right_cancel_iff,Finset.sum_ite_eq',Finset.mem_range,Nat.succ_le_iff]
-
-lemma palindromeLowCost_slab (L d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) :
-    palindromeLowCost L d e ω + palindromeSlabCost L d e ω = palindromeLowCost (2*L) d e ω := by
-  rw [palindromeLowCost_sum,palindromeLowCost_sum,show 2*L=L+L by omega,Finset.sum_range_add]
-  congr 1
-  exact Fin.sum_univ_eq_sum_range (fun i => palindromeLevelCost (L+i+1) d e ω) L
-
-lemma palindromeCost_family (L n d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) :
-    palindromeLowCost L d e ω + palindromeFamilyCost L n d e ω +
-      palindromeFamilyCost (2*L) n d e ω = palindromeLowCost (L*4^n) d e ω := by
-  induction n with
-  | zero => simp only [palindromeFamilyCost,Finset.range_zero,Finset.sum_empty,Nat.add_zero,
-      pow_zero,mul_one]
-  | succ n ih =>
-    have he : (2*L)*4^n = 2*(L*4^n) := by ring
-    have he' : L*4^(n+1) = 2*(2*(L*4^n)) := by rw [pow_succ]; ring
-    simp only [palindromeFamilyCost,Finset.sum_range_succ] at ih ⊢
-    rw [he,he',←palindromeLowCost_slab,←palindromeLowCost_slab,←ih]
-    omega
-
-
-lemma finiteMean_mul_sq_le {Ω : Type*} [Fintype Ω] (f g : Ω → ℝ) :
-    (finiteMean (fun ω => f ω*g ω))^2 ≤ finiteMean (fun ω => (f ω)^2)*finiteMean (fun ω => (g ω)^2) := by
-  have h := Finset.sum_mul_sq_le_sq_mul_sq Finset.univ f g
-  simp only [finiteMean,div_pow,div_mul_div_comm,←pow_two]
-  exact div_le_div_of_nonneg_right h (sq_nonneg _)
-
-lemma finiteMean_mul_exp_le {Ω : Type*} [Fintype Ω] (f g : Ω → ℝ) (A B : ℝ)
-    (hf : finiteMean (fun ω => (f ω)^2) ≤ Real.exp A)
-    (hg : finiteMean (fun ω => (g ω)^2) ≤ Real.exp B) :
-    finiteMean (fun ω => f ω*g ω) ≤ Real.exp ((A+B)/2) := by
-  have hc := finiteMean_mul_sq_le f g
-  have hn : 0 ≤ finiteMean (fun ω => (g ω)^2) := finiteMean_nonneg (fun _ => sq_nonneg _)
-  have hu := mul_le_mul hf hg hn (Real.exp_nonneg _)
-  have he : Real.exp A*Real.exp B = (Real.exp ((A+B)/2))^2 := by
-    rw [←Real.exp_add,←Real.exp_nat_mul]
-    congr 1
-    push_cast
-    ring
-  rw [he] at hu
-  have hh := hc.trans hu
-  have hep := Real.exp_pos ((A+B)/2)
-  nlinarith
-
-
-noncomputable def heightDecay (H : ℕ) : ℝ :=
-  68*Real.exp (-(H:ℝ)/64)/(1-Real.exp (-(1:ℝ)/64))
-
-lemma heightDecay_nonneg (H : ℕ) : 0 ≤ heightDecay H := by
-  have h : Real.exp (-(1:ℝ)/64) < 1 := Real.exp_lt_one_iff.mpr (by norm_num)
-  apply div_nonneg (by positivity) (by linarith)
-
-lemma heightDecay_antitone : Antitone heightDecay := by
-  intro L H h
-  have hden : 0 ≤ 1-Real.exp (-(1:ℝ)/64) := by
-    have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
-    linarith
-  apply div_le_div_of_nonneg_right _ hden
-  apply mul_le_mul_of_nonneg_left _ (by norm_num)
-  apply Real.exp_le_exp.mpr
-  have h' : (L:ℝ) ≤ H := by exact_mod_cast h
-  linarith
-
-lemma palindrome_family_mgf_decay (L n d : ℕ) (hL : 0 < L) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) ≤
-      Real.exp (Fintype.card ι*heightDecay L) := by
-  apply (palindrome_family_mgf L n d hL e X q hq hlog).trans
-  apply Real.exp_le_exp.mpr
-  apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
-  have h := mul_le_mul_of_nonneg_left (sum_family_decay L n hL) (by norm_num : (0:ℝ)≤68)
-  simpa only [heightDecay,mul_div_assoc] using h
-
-theorem palindrome_high_mgf (H n d : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/500) :
-    finiteMean (fun Y => q^(palindromeFamilyCost H n d e (Y,X)+
-      palindromeFamilyCost (2*H) n d e (Y,X))) ≤ Real.exp (Fintype.card ι*heightDecay H) := by
-  have hq2 := one_le_pow₀ (n:=2) hq
-  have hlog2 : Real.log (q^2) ≤ 1/250 := by rw [Real.log_pow]; norm_num; linarith
-  have h₀ := palindrome_family_mgf_decay H n d hH e X (q^2) hq2 hlog2
-  have h₁ := palindrome_family_mgf_decay (2*H) n d (by omega) e X (q^2) hq2 hlog2
-  have h₁' := h₁.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left
-    (heightDecay_antitone (show H ≤ 2*H by omega)) (Nat.cast_nonneg _)))
-  have hh := finiteMean_mul_exp_le (fun Y => q^(palindromeFamilyCost H n d e (Y,X)))
-    (fun Y => q^(palindromeFamilyCost (2*H) n d e (Y,X)))
-    (Fintype.card ι*heightDecay H) (Fintype.card ι*heightDecay H)
-    (by simpa only [pow_right_comm] using h₀) (by simpa only [pow_right_comm] using h₁')
-  simpa only [←pow_add,add_self_div_two] using hh
-
-lemma palindrome_high_full_mgf (H n d : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/500) :
-    finiteMean (fun ω : BenesCoins d => q^(palindromeFamilyCost H n d e ω+
-      palindromeFamilyCost (2*H) n d e ω)) ≤ Real.exp (Fintype.card ι*heightDecay H) := by
-  rw [finiteMean_prod,finiteMean_comm]
-  calc
-    _ ≤ finiteMean (fun _X : SwitchIndex d → Bool => Real.exp (Fintype.card ι*heightDecay H)) := by
-      apply finiteMean_mono
-      intro X
-      exact palindrome_high_mgf H n d hH e X q hq hlog
-    _ = _ := finiteMean_const _
-
-theorem palindrome_cost_mgf_cutoff (r H : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (H+r)) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/1000)
-    (hr : (2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*(q^2)^H) ≤ 1/2) :
-    finiteMean (fun ω : BenesCoins (H+r) => q^(palindromeCost (H+r) e ω)) ≤
-      Real.exp (((2:ℝ)^r*(2*((2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*(q^2)^H))^2)+
-        Fintype.card ι*heightDecay H)/2) := by
-  let d := H+r
-  have hd : d ≤ H*4^d := by
-    have hh := pow_four_ge d
-    nlinarith
-  have hcost (ω : BenesCoins d) : palindromeCost d e ω = palindromeLowCost H d e ω+
-      (palindromeFamilyCost H d d e ω+palindromeFamilyCost (2*H) d d e ω) := by
-    have hh := palindromeCost_family H d d e ω
-    rw [palindromeLowCost_eq _ _ hd] at hh
-    simpa only [Nat.add_assoc] using hh.symm
-  have hq2 := one_le_pow₀ (n:=2) hq
-  have hlog2 : Real.log (q^2) ≤ 1/500 := by rw [Real.log_pow]; norm_num; linarith
-  have hlow := palindrome_low_mgf r H e (q^2) hq2 hr
-  have hhigh := palindrome_high_full_mgf H d d hH e (q^2) hq2 hlog2
-  have hh := finiteMean_mul_exp_le (fun ω : BenesCoins d => q^(palindromeLowCost H d e ω))
-    (fun ω : BenesCoins d => q^(palindromeFamilyCost H d d e ω+palindromeFamilyCost (2*H) d d e ω))
-    _ _ (by simpa only [pow_right_comm] using hlow) (by simpa only [pow_right_comm] using hhigh)
-  change finiteMean (fun ω : BenesCoins d => q^(palindromeCost d e ω)) ≤ _
-  simpa only [hcost,pow_add,pow_right_comm] using hh
-
-
-lemma cutoffMass_le (H d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (ω : BenesCoins d) :
-    cutoffMass H d e ω ≤ Fintype.card ι := by
-  classical
-  induction d generalizing ι with
-  | zero => exact Nat.zero_le _
-  | succ d ih =>
-    rw [cutoffMass]
-    split_ifs
-    · exact crowdedMass_le _
-    · exact (Nat.add_le_add (ih _ _) (ih _ _)).trans_eq (PairRouting.color_card_add _)
-
-lemma palindromeLowCost_le (H d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) : palindromeLowCost H d e ω ≤ H*Fintype.card ι :=
-  (palindromeLowCost_le_cutoff H d e ω).trans (Nat.mul_le_mul_left H (cutoffMass_le H d e ω))
-
-theorem palindrome_cost_mgf_coarse (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/1000) :
-    finiteMean (fun ω : BenesCoins d => q^(palindromeCost d e ω)) ≤
-      Real.exp (Fintype.card ι*(1+heightDecay 1)) := by
-  have hd : d ≤ 1*4^d := by have hh := pow_four_ge d; omega
-  have hcost (ω : BenesCoins d) : palindromeCost d e ω = palindromeLowCost 1 d e ω+
-      (palindromeFamilyCost 1 d d e ω+palindromeFamilyCost 2 d d e ω) := by
-    have hh := palindromeCost_family 1 d d e ω
-    rw [palindromeLowCost_eq _ _ hd] at hh
-    simpa only [Nat.add_assoc] using hh.symm
-  have hq2 := one_le_pow₀ (n:=2) hq
-  have hlog2 : Real.log (q^2) ≤ 1/500 := by rw [Real.log_pow]; norm_num; linarith
-  have hlow : finiteMean (fun ω : BenesCoins d => (q^2)^(palindromeLowCost 1 d e ω)) ≤
-      Real.exp (Fintype.card ι) := by
-    calc
-      _ ≤ finiteMean (fun _ω : BenesCoins d => (q^2)^(Fintype.card ι)) := by
-        apply finiteMean_mono
-        intro ω
-        apply pow_le_pow_right₀ hq2
-        simpa only [one_mul] using palindromeLowCost_le 1 d e ω
-      _ = (q^2)^(Fintype.card ι) := finiteMean_const _
-      _ = Real.exp (Fintype.card ι * Real.log (q^2)) := by
-        rw [Real.exp_nat_mul,Real.exp_log (by positivity : 0<q^2)]
-      _ ≤ _ := Real.exp_le_exp.mpr (by
-        have hh := mul_le_mul_of_nonneg_left hlog2 (Nat.cast_nonneg (Fintype.card ι) : (0:ℝ)≤_)
-        nlinarith [Nat.cast_nonneg (α:=ℝ) (Fintype.card ι)])
-  have hhigh := palindrome_high_full_mgf 1 d d (by decide) e (q^2) hq2 hlog2
-  have hh := finiteMean_mul_exp_le (fun ω : BenesCoins d => q^(palindromeLowCost 1 d e ω))
-    (fun ω : BenesCoins d => q^(palindromeFamilyCost 1 d d e ω+palindromeFamilyCost 2 d d e ω))
-    (Fintype.card ι) (Fintype.card ι*heightDecay 1)
-    (by simpa only [pow_right_comm] using hlow) (by simpa only [pow_right_comm] using hhigh)
-  simp only [←pow_add,←hcost] at hh
-  apply hh.trans (Real.exp_le_exp.mpr _)
-  have hk : (0:ℝ) ≤ Fintype.card ι := Nat.cast_nonneg _
-  have hD := heightDecay_nonneg 1
-  nlinarith
-
-noncomputable def momentBase : ℝ := (2:ℝ)^(1/1000:ℝ)
-
-lemma momentBase_one_le : 1 ≤ momentBase := by
-  exact Real.one_le_rpow (by norm_num) (by norm_num)
-
-lemma momentBase_log : Real.log momentBase ≤ 1/1000 := by
-  rw [momentBase,Real.log_rpow (by norm_num)]
-  have hh := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
-  norm_num at hh ⊢
-  linarith
-
-lemma momentBase_pow (c : ℕ) : momentBase^c = (2:ℝ)^((c:ℝ)*(1/1000:ℝ)) := by
-  rw [mul_comm,Real.rpow_mul_natCast (by norm_num)]
-  rfl
-
-lemma palindromeRowMoment_le_cost (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) :
-    palindromeRowMoment d e (1/1000) ≤
-      finiteMean (fun ω : BenesCoins d => momentBase^(palindromeCost d e ω)) := by
-  have hn : (0:ℝ) < ((2^d:ℕ):ℝ)^(Fintype.card ι) := by positivity
-  have hn0 : (0:ℝ) ≤ ((2^d).descFactorial (Fintype.card ι):ℝ) /
-      ((2^d:ℕ):ℝ)^(Fintype.card ι) := by positivity
-  have hn1 : ((2^d).descFactorial (Fintype.card ι):ℝ) /
-      ((2^d:ℕ):ℝ)^(Fintype.card ι) ≤ 1 := by
-    apply (div_le_one hn).mpr
-    exact_mod_cast Nat.descFactorial_le_pow (2^d) (Fintype.card ι)
-  have hpow := Real.rpow_le_one hn0 hn1 (by norm_num : (0:ℝ)≤1/1000)
-  have hm := palindrome_row_density_moment d e (1/1000) (by norm_num)
-  simp_rw [←momentBase_pow] at hm
-  exact hm.trans (mul_le_of_le_one_left (finiteMean_nonneg (fun _ =>
-    pow_nonneg (le_trans (by norm_num) momentBase_one_le) _)) hpow)
-
-theorem palindrome_row_moment_coarse (d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) :
-    palindromeRowMoment d e (1/1000) ≤ Real.exp (Fintype.card ι*(1+heightDecay 1)) :=
-  (palindromeRowMoment_le_cost d e).trans
-    (palindrome_cost_mgf_coarse d e momentBase momentBase_one_le momentBase_log)
-
-
-noncomputable def densityCutoff (p : ℝ) : ℕ := ⌊-Real.log p/(4*Real.log 2)⌋₊
-
-lemma log_small_density {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
-    Real.log p ≤ -8*Real.log 2 := by
-  have he : Real.log (1/256:ℝ) = -8*Real.log 2 := by
-    rw [show (1/256:ℝ) = 1/2^8 by norm_num,
-      Real.log_div (by norm_num) (by positivity),Real.log_one,Real.log_pow]
-    ring
-  exact (Real.log_le_log hp hs).trans_eq he
-
-lemma densityCutoff_bounds {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
-    0 < densityCutoff p ∧
-    (densityCutoff p:ℝ)*(4*Real.log 2) ≤ -Real.log p ∧
-    -Real.log p < ((densityCutoff p:ℝ)+1)*(4*Real.log 2) := by
-  have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
-  have hd : 0 < 4*Real.log 2 := by positivity
-  have hh := log_small_density hp hs
-  have hx : 1 ≤ -Real.log p/(4*Real.log 2) := by
-    apply (le_div_iff₀ hd).mpr
-    nlinarith
-  refine ⟨Nat.floor_pos.mpr hx,?_,?_⟩
-  · exact (le_div_iff₀ hd).mp (Nat.floor_le (by linarith : 0 ≤ -Real.log p/(4*Real.log 2)))
-  · exact (div_lt_iff₀ hd).mp (Nat.lt_floor_add_one (-Real.log p/(4*Real.log 2)))
-
-lemma densityCutoff_decay {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
-    Real.exp (-(densityCutoff p:ℝ)/64) ≤ Real.exp (1/64) * p^(1/512:ℝ) := by
-  have hh := (densityCutoff_bounds hp hs).2.2
-  have hb : Real.log 2 ≤ 2 := by
-    have := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
-    linarith
-  have hH : 0 ≤ (densityCutoff p:ℝ)+1 := by positivity
-  have hu := mul_le_mul_of_nonneg_left hb hH
-  rw [Real.rpow_def_of_pos hp,←Real.exp_add]
-  apply Real.exp_le_exp.mpr
-  nlinarith
-
-lemma densityCutoff_power {p q : ℝ} (hp : 0 < p) (hs : p ≤ 1/256)
-    (hq : 0 < q) (hlog : Real.log q ≤ Real.log 2/4) (n : ℕ) (hn : n ≤ 4)
-    (hq1 : 1 ≤ q) :
-    p*(2:ℝ)^(densityCutoff p)*q^(n*densityCutoff p) ≤ p^(1/2:ℝ) := by
-  have hcut := (densityCutoff_bounds hp hs).2.1
-  have hH : 0 ≤ (densityCutoff p:ℝ) := Nat.cast_nonneg _
-  have hn' : (n:ℝ) ≤ 4 := by exact_mod_cast hn
-  have hln := Real.log_nonneg hq1
-  have hl := mul_le_mul_of_nonneg_right hn' hln
-  have hnl : (n:ℝ)*Real.log q ≤ Real.log 2 := by linarith
-  have hm := mul_le_mul_of_nonneg_left hnl hH
-  have heq : p*(2:ℝ)^(densityCutoff p)*q^(n*densityCutoff p) =
-      Real.exp (Real.log p+(densityCutoff p:ℝ)*Real.log 2+
-        (n*densityCutoff p:ℕ)*Real.log q) := by
-    rw [Real.exp_add,Real.exp_add,Real.exp_nat_mul,Real.exp_nat_mul,
-      Real.exp_log hp,Real.exp_log hq,Real.exp_log (by norm_num : (0:ℝ)<2)]
-  rw [heq,Real.rpow_def_of_pos hp]
-  apply Real.exp_le_exp.mpr
-  push_cast
-  nlinarith
-
-lemma densityCutoff_sparse {p q : ℝ} (hp : 0 < p) (hs : p ≤ 1/256)
-    (hq : 0 < q) (hlog : Real.log q ≤ Real.log 2/4) (hq1 : 1 ≤ q) :
-    (2:ℝ)^(densityCutoff p)*(p*(q^2)^(densityCutoff p)) ≤ 1/2 := by
-  have hh := densityCutoff_power hp hs hq hlog 2 (by decide) hq1
-  rw [←pow_mul] at ⊢
-  have hle : p^(1/2:ℝ) ≤ 1/2 := by
-    have hlogp := log_small_density hp hs
-    have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
-    rw [Real.rpow_def_of_pos hp]
-    calc
-      _ ≤ Real.exp (-Real.log 2) := by apply Real.exp_le_exp.mpr; nlinarith
-      _ = 1/2 := by rw [Real.exp_neg,Real.exp_log (by norm_num : (0:ℝ)<2)]; norm_num
-  nlinarith [hh.trans hle]
-
-
-lemma momentBase_log_exact : Real.log momentBase = Real.log 2/1000 := by
-  rw [momentBase,Real.log_rpow (by norm_num)]
-  ring
-
-lemma densityCutoff_le_height (d k : ℕ) (hk : 0 < k) :
-    densityCutoff ((k:ℝ)/(2:ℝ)^d) ≤ d := by
-  have hk' : (1:ℝ) ≤ k := by exact_mod_cast hk
-  have hp : 0 < (k:ℝ)/(2:ℝ)^d := by positivity
-  have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
-  have hlogk := Real.log_nonneg hk'
-  by_cases hx : 0 ≤ -Real.log ((k:ℝ)/(2:ℝ)^d)/(4*Real.log 2)
-  · have hh := (le_div_iff₀ (by positivity : 0 < 4*Real.log 2)).mp (Nat.floor_le hx)
-    have hd : (0:ℝ) ≤ d := Nat.cast_nonneg _
-    have hH : (0:ℝ) ≤ densityCutoff ((k:ℝ)/(2:ℝ)^d) := Nat.cast_nonneg _
-    change (densityCutoff ((k:ℝ)/(2:ℝ)^d):ℝ)*(4*Real.log 2) ≤ -Real.log ((k:ℝ)/(2:ℝ)^d) at hh
-    rw [Real.log_div (by positivity) (by positivity),Real.log_pow] at hh
-    have hle : (densityCutoff ((k:ℝ)/(2:ℝ)^d):ℝ) ≤ d := by nlinarith
-    exact_mod_cast hle
-  · have hz := Nat.floor_of_nonpos (le_of_not_ge hx)
-    simp only [densityCutoff,hz,Nat.zero_le]
-
-noncomputable def densityTailConstant : ℝ :=
-  68*Real.exp (1/64)/(1-Real.exp (-(1:ℝ)/64))
-
-lemma densityTailConstant_nonneg : 0 ≤ densityTailConstant := by
-  have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
-  apply div_nonneg (by positivity) (by linarith)
-
-lemma heightDecay_cutoff {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
-    heightDecay (densityCutoff p) ≤ densityTailConstant*p^(1/512:ℝ) := by
-  have hden : 0 ≤ 1-Real.exp (-(1:ℝ)/64) := by
-    have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
-    linarith
-  have hh := div_le_div_of_nonneg_right
-    (mul_le_mul_of_nonneg_left (densityCutoff_decay hp hs) (by norm_num : (0:ℝ)≤68)) hden
-  simpa only [heightDecay,densityTailConstant,mul_assoc,div_mul_eq_mul_div] using hh
-
-lemma sparse_low_algebra (r H k : ℕ) (q : ℝ) :
-    (2:ℝ)^r*(2*((2:ℝ)^H*(((k:ℝ)/(2:ℝ)^(H+r))*(q^2)^H))^2) =
-    2*k*(((k:ℝ)/(2:ℝ)^(H+r))*(2:ℝ)^H*q^(4*H)) := by
-  rw [pow_add,←pow_mul,show 4*H = (2*H)*2 by omega,pow_mul]
-  have hH : (2:ℝ)^H ≠ 0 := by positivity
-  have hr : (2:ℝ)^r ≠ 0 := by positivity
-  field_simp
-  ring
-
-lemma palindrome_row_moment_small (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι)
-    (hs : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1/256) :
-    palindromeRowMoment d e (1/1000) ≤
-      Real.exp (Fintype.card ι*(1+densityTailConstant)*
-        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ)) := by
-  generalize hpdef : ((Fintype.card ι:ℝ)/(2:ℝ)^d) = p at hs ⊢
-  have hp : 0 < p := by rw [←hpdef]; positivity
-  have hle := densityCutoff_le_height d (Fintype.card ι) hk
-  rw [hpdef] at hle
-  obtain ⟨r,hr⟩ := Nat.exists_eq_add_of_le hle
-  subst d
-  have hH := (densityCutoff_bounds hp hs).1
-  have hq := momentBase_one_le
-  have hqp : 0 < momentBase := by linarith
-  have hlog : Real.log momentBase ≤ Real.log 2/4 := by
-    rw [momentBase_log_exact]
-    have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
-    linarith
-  have hsp := densityCutoff_sparse hp hs hqp hlog hq
-  rw [←hpdef] at hsp
-  have hh := palindrome_cost_mgf_cutoff r (densityCutoff p) hH e momentBase hq momentBase_log
-    (by simpa only [hpdef] using hsp)
-  apply (palindromeRowMoment_le_cost _ e).trans (hh.trans (Real.exp_le_exp.mpr _))
-  rw [sparse_low_algebra,hpdef]
-  have hlow := densityCutoff_power hp hs hqp hlog 4 (by decide) hq
-  have hp1 : p ≤ 1 := by linarith
-  have he := Real.rpow_le_rpow_of_exponent_ge hp hp1 (by norm_num : (1:ℝ)/512 ≤ 1/2)
-  have hlow' := mul_le_mul_of_nonneg_left (hlow.trans he)
-    (by positivity : (0:ℝ)≤2*Fintype.card ι)
-  have hhigh := mul_le_mul_of_nonneg_left (heightDecay_cutoff hp hs)
-    (Nat.cast_nonneg (Fintype.card ι): (0:ℝ)≤_)
-  have hnon : 0 ≤ Fintype.card ι*densityTailConstant*p^(1/512:ℝ) :=
-    mul_nonneg (mul_nonneg (Nat.cast_nonneg _) densityTailConstant_nonneg)
-      (Real.rpow_nonneg (le_of_lt hp) _)
-  nlinarith
-
-noncomputable def densityConstant : ℝ :=
-  256*(1+heightDecay 1)+1+densityTailConstant
-
-lemma densityConstant_pos : 0 < densityConstant := by
-  dsimp [densityConstant]
-  have hh := heightDecay_nonneg 1
-  have ht := densityTailConstant_nonneg
-  linarith
-
-theorem palindrome_row_moment_bound (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι) :
-    palindromeRowMoment d e (1/1000) ≤ Real.exp (densityConstant*Fintype.card ι*
-        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ)) := by
-  have hp : 0 < (Fintype.card ι:ℝ)/(2:ℝ)^d := by positivity
-  have hp1 : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1 := by
-    apply (div_le_one (by positivity)).mpr
-    have h := Fintype.card_le_of_injective e e.injective
-    rw [card_positions] at h
-    exact_mod_cast h
-  have hk0 : (0:ℝ) ≤ Fintype.card ι := Nat.cast_nonneg _
-  have hpow := Real.rpow_nonneg (le_of_lt hp) (1/512:ℝ)
-  by_cases hs : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1/256
-  · apply (palindrome_row_moment_small d e hk hs).trans (Real.exp_le_exp.mpr _)
-    have hD : 1+densityTailConstant ≤ densityConstant := by
-      dsimp [densityConstant]
-      have := heightDecay_nonneg 1
-      linarith
-    have hh := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hD hk0) hpow
-    nlinarith
-  · have hlow : (1:ℝ)/256 ≤ ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ) := by
-      have he := Real.rpow_le_rpow_of_exponent_ge hp hp1 (by norm_num : (1:ℝ)/512 ≤ 1)
-      rw [Real.rpow_one] at he
-      exact (le_of_not_ge hs).trans he
-    have hD : 256*(1+heightDecay 1) ≤ densityConstant := by
-      dsimp [densityConstant]
-      have := densityTailConstant_nonneg
-      linarith
-    apply (palindrome_row_moment_coarse d e).trans (Real.exp_le_exp.mpr _)
-    have hD0 : 0 ≤ 1+heightDecay 1 := by have := heightDecay_nonneg 1; linarith
-    have ha := mul_le_mul_of_nonneg_left hlow (mul_nonneg (by norm_num : (0:ℝ)≤256) hD0)
-    have hb := mul_le_mul_of_nonneg_right hD hpow
-    have hc := mul_le_mul_of_nonneg_right (ha.trans hb) hk0
-    nlinarith
-
-lemma palindromeRowMoment_nonneg (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (a : ℝ) : 0 ≤ palindromeRowMoment d e a := by
-  apply finiteMean_nonneg
-  intro f
-  exact Real.rpow_nonneg (mul_nonneg (Nat.cast_nonneg _)
-    (PairRouting.tupleProbability_nonneg _ _ _)) _
-
-theorem palindrome_log_row_moment (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι) :
-    Real.log (palindromeRowMoment d e (1/1000)) ≤ densityConstant*Fintype.card ι*
-        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ) := by
-  by_cases hz : palindromeRowMoment d e (1/1000) = 0
-  · rw [hz,Real.log_zero]
-    exact mul_nonneg (mul_nonneg (le_of_lt densityConstant_pos) (Nat.cast_nonneg _))
-      (Real.rpow_nonneg (by positivity) _)
-  · exact (Real.log_le_iff_le_exp (lt_of_le_of_ne (palindromeRowMoment_nonneg _ _ _) (Ne.symm hz))).mpr
-      (palindrome_row_moment_bound d e hk)
-
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-lemma orbitSet_eq_of_agree (p q : Equiv.Perm α) (x : α)
-    (h : ∀ z ∈ orbitSet p x, p z = q z) : orbitSet p x = orbitSet q x := by
-  have hp (n : ℕ) : (p^n) x ∈ orbitSet p x := by
-    apply (mem_orbitSet _ _ _).mpr
-    exact Equiv.Perm.sameCycle_pow_right.mpr (Equiv.Perm.SameCycle.refl _ _)
-  have he (n : ℕ) : (p^n) x = (q^n) x := by
-    induction n with
-    | zero => rfl
-    | succ n ih =>
-      rw [pow_succ',pow_succ',Equiv.Perm.mul_apply,Equiv.Perm.mul_apply,←ih]
-      exact h _ (hp n)
-  ext y
-  simp only [mem_orbitSet]
-  constructor
-  · intro hy
-    obtain ⟨n,hn⟩ := hy.exists_nat_pow_eq
-    rw [←hn,he]
-    exact Equiv.Perm.sameCycle_pow_right.mpr (Equiv.Perm.SameCycle.refl _ _)
-  · intro hy
-    obtain ⟨n,hn⟩ := hy.exists_nat_pow_eq
-    rw [←hn,←he]
-    exact Equiv.Perm.sameCycle_pow_right.mpr (Equiv.Perm.SameCycle.refl _ _)
-
-noncomputable def survivingCycle (p q : Equiv.Perm α) (S B : Finset α)
-    (h : ∀ x ∉ B, p x = q x) (C : RoutingNetwork.SelectedCycle p S)
-    (hC : ∀ x ∈ C.val, x ∉ B) : RoutingNetwork.SelectedCycle q S := by
-  refine ⟨C.val,?_,C.property.2⟩
-  obtain ⟨x,hx⟩ := C.property.1
-  refine ⟨x,hx.trans (orbitSet_eq_of_agree p q x ?_)⟩
-  intro z hz
-  exact h z (hC z (hx ▸ hz))
-
-noncomputable def perturbCycleMap (p q : Equiv.Perm α) (S B : Finset α)
-    (h : ∀ x ∉ B, p x = q x) (C : RoutingNetwork.SelectedCycle p S) :
-    RoutingNetwork.SelectedCycle q S ⊕ {x // x ∈ B} := by
-  classical
-  by_cases hb : ∃ x ∈ C.val, x ∈ B
-  · exact Sum.inr ⟨hb.choose,hb.choose_spec.2⟩
-  · exact Sum.inl (survivingCycle p q S B h C (by simpa only [not_exists,not_and] using hb))
-
-lemma perturbCycleMap_injective (p q : Equiv.Perm α) (S B : Finset α)
-    (h : ∀ x ∉ B, p x = q x) : Function.Injective (perturbCycleMap p q S B h) := by
-  classical
-  intro C D he
-  by_cases hC : ∃ x ∈ C.val, x ∈ B <;>
-    by_cases hD : ∃ x ∈ D.val, x ∈ B
-  · simp only [perturbCycleMap,dite_eq_left hC,dite_eq_left hD,Sum.inr.injEq] at he
-    have hx : hC.choose = hD.choose := congrArg Subtype.val he
-    exact RoutingNetwork.SelectedCycle.eq_of_mem C D hC.choose_spec.1
-      (hx.symm ▸ hD.choose_spec.1)
-  · simp only [perturbCycleMap,dite_eq_left hC,dite_eq_right hD] at he
-    cases he
-  · simp only [perturbCycleMap,dite_eq_right hC,dite_eq_left hD] at he
-    cases he
-  · simp only [perturbCycleMap,dite_eq_right hC,dite_eq_right hD,Sum.inl.injEq] at he
-    have hval : C.val = D.val := congrArg (fun Z : RoutingNetwork.SelectedCycle q S => Z.val) he
-    exact Subtype.ext hval
-
-theorem selectedCycle_perturb_le (p q : Equiv.Perm α) (S B : Finset α)
-    (h : ∀ x ∉ B, p x = q x) :
-    Fintype.card (RoutingNetwork.SelectedCycle p S) ≤
-      Fintype.card (RoutingNetwork.SelectedCycle q S)+B.card := by
-  have hh := Fintype.card_le_of_injective _ (perturbCycleMap_injective p q S B h)
-  simpa only [Fintype.card_sum,Fintype.card_coe] using hh
-
-
-noncomputable def permDistance (p q : Equiv.Perm α) : ℕ :=
-  (Finset.univ.filter (fun x => p x ≠ q x)).card
-
-lemma permDistance_sum (p q : Equiv.Perm α) :
-    permDistance p q = ∑ x : α, if p x = q x then 0 else 1 := by
-  classical
-  simp [permDistance,Finset.card_filter]
-
-lemma permDistance_refl (p : Equiv.Perm α) : permDistance p p = 0 := by
-  simp [permDistance]
-
-lemma permDistance_le (p q : Equiv.Perm α) : permDistance p q ≤ Fintype.card α :=
-  (Finset.card_filter_le _ _).trans_eq (Finset.card_univ)
-
-lemma permDistance_symm (p q : Equiv.Perm α) : permDistance p q = permDistance q p := by
-  simp only [permDistance,ne_comm]
-
-lemma permDistance_triangle (p q r : Equiv.Perm α) :
-    permDistance p r ≤ permDistance p q+permDistance q r := by
-  classical
-  simp only [permDistance_sum,←Finset.sum_add_distrib]
-  apply Finset.sum_le_sum
-  intro x _
-  by_cases hpq : p x = q x <;> by_cases hqr : q x = r x <;>
-    by_cases hpr : p x = r x <;> simp only [hpq,hqr,hpr,↓reduceIte] <;>
-    omega
-
-lemma permDistance_mul_left (p q g : Equiv.Perm α) :
-    permDistance (g*p) (g*q) = permDistance p q := by
-  classical
-  unfold permDistance
-  congr 1
-  ext x
-  simp only [Finset.mem_filter,Finset.mem_univ,true_and]
-  exact not_congr g.injective.eq_iff
-
-lemma permDistance_mul_right (p q g : Equiv.Perm α) :
-    permDistance (p*g) (q*g) = permDistance p q := by
-  classical
-  simp only [permDistance_sum,Equiv.Perm.mul_apply]
-  exact Fintype.sum_equiv g _ _ (fun _ => rfl)
-
-lemma permDistance_inv (p q : Equiv.Perm α) : permDistance p⁻¹ q⁻¹ = permDistance p q := by
-  have hh := permDistance_mul_left p⁻¹ q⁻¹ q
-  have hhh := permDistance_mul_right (q*p⁻¹) 1 p
-  simp only [mul_inv_cancel,one_mul,inv_mul_cancel_right] at hh hhh
-  rw [←hhh] at hh
-  rw [←hh,permDistance_symm]
-
-lemma permDistance_mul_le (p q g h : Equiv.Perm α) :
-    permDistance (p*g) (q*h) ≤ permDistance p q+permDistance g h := by
-  have hh := permDistance_triangle (p*g) (q*g) (q*h)
-  simpa only [permDistance_mul_left,permDistance_mul_right] using hh
-
-lemma selectedCycle_distance_le (p q : Equiv.Perm α) (S : Finset α) :
-    Fintype.card (RoutingNetwork.SelectedCycle p S) ≤
-      Fintype.card (RoutingNetwork.SelectedCycle q S)+permDistance p q := by
-  apply selectedCycle_perturb_le
-  intro x hx
-  simpa only [Finset.mem_filter,Finset.mem_univ,true_and,not_not] using hx
-
-lemma permDistance_alternating (p q : Equiv.Perm α) :
-    permDistance (CycleColoring.alternatingPerm p) (CycleColoring.alternatingPerm q) = permDistance p q := by
-  classical
-  simp only [permDistance_sum,Fintype.sum_prod_type,Fintype.sum_bool]
-  simp [CycleColoring.alternatingPerm]
-
-lemma alternatingCycles_distance_le {ι : Type*} [Fintype ι]
-    (e : ι ↪ Bool × α) (p q : Bool → Equiv.Perm α) :
-    PairRouting.alternatingCycles e p ≤ PairRouting.alternatingCycles e q+
-      (permDistance (p false) (q false)+permDistance (p true) (q true)) := by
-  have hh := selectedCycle_distance_le
-    (CycleColoring.alternatingPerm (PairRouting.alternatingProjection p))
-    (CycleColoring.alternatingPerm (PairRouting.alternatingProjection q)) (PairRouting.labelSet e)
-  rw [permDistance_alternating] at hh
-  apply hh.trans
-  apply Nat.add_le_add_left
-  dsimp only [PairRouting.alternatingProjection]
-  exact (permDistance_mul_le _ _ _ _).trans_eq (by
-    change permDistance (p false)⁻¹ (q false)⁻¹ + _ = _
-    rw [permDistance_inv])
-
-end Thorp
-namespace Thorp
-
-def centralBlockPerm (L : ℕ) : (r : ℕ) → (Card r → Equiv.Perm (Card L)) →
-    Equiv.Perm (Card (L+r))
-  | 0, C => C (fun j => Fin.elim0 j)
-  | r+1, C => childLift (fun b => centralBlockPerm L r (fun u => C (Fin.cons b u)))
-
-def centralPalindromePerm (L r : ℕ) (C : Card r → Equiv.Perm (Card L))
-    (ω : BenesCoins (L+r)) : Equiv.Perm (Card (L+r)) :=
-  butterflyPerm (L+r) (decodeButterfly (L+r) ω.1) * centralBlockPerm L r C *
-    (butterflyPerm (L+r) (decodeButterfly (L+r) ω.2))⁻¹
-
-lemma centralPalindromePerm_step (L r : ℕ) (C : Card (r+1) → Equiv.Perm (Card L))
-    (c : BenesCoins (L+r) × BenesCoins (L+r))
-    (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
-    centralPalindromePerm L (r+1) C ((benesStepEquiv (L+r)).symm (c,coins)) =
-      pairSwitch coins.2 * childLift (fun b => centralPalindromePerm L r
-        (fun u => C (Fin.cons b u)) (if b then c.2 else c.1)) * pairSwitch coins.1 := by
-  change ((butterflyPerm (L+r+1) (decodeButterfly (L+r+1)
-      ((coinStepEquiv (L+r)).symm ((c.1.1,c.2.1),coins.2))))) *
-    childLift (fun b => centralBlockPerm L r (fun u => C (Fin.cons b u))) *
-    (butterflyPerm (L+r+1) (decodeButterfly (L+r+1)
-      ((coinStepEquiv (L+r)).symm ((c.1.2,c.2.2),coins.1))))⁻¹ = _
-  rw [butterflyPerm_step,butterflyPerm_step,mul_inv_rev,childLift_inv,pairSwitch_inv]
-  simp only [mul_assoc]
-  rw [←mul_assoc (childLift _),childLift_mul,←mul_assoc (childLift _),childLift_mul]
-  congr 2
-  apply congrArg childLift
-  funext b
-  cases b <;> rfl
-
-noncomputable def centralLevelCost (t L : ℕ) : (r : ℕ) →
-    (Card r → Equiv.Perm (Card L)) → {ι : Type*} → [Fintype ι] →
-    (ι ↪ Card (L+r)) → BenesCoins (L+r) → ℕ
-  | 0, _, _, _, _, _ => 0
-  | r+1, C, _, _, e, ω =>
-    let σ := benesStepEquiv (L+r) ω
-    let x := e.trans (headTailEquiv (L+r)).toEmbedding
-    let c := PairRouting.colors x σ.2.1
-    let hc := PairRouting.colors_compatible x σ.2.1
-    if t = L+r+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
-      (fun b => centralPalindromePerm L r (fun u => C (Fin.cons b u)) (if b then σ.1.2 else σ.1.1))
-    else centralLevelCost t L r (fun u => C (Fin.cons false u))
-      (PairRouting.childEmbedding x c hc false) σ.1.1 +
-      centralLevelCost t L r (fun u => C (Fin.cons true u))
-      (PairRouting.childEmbedding x c hc true) σ.1.2
-
-noncomputable def centralSlabCost (l L r : ℕ) (C : Card r → Equiv.Perm (Card L))
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (ω : BenesCoins (L+r)) : ℕ :=
-  ∑ i : Fin l, centralLevelCost (l+i.val+1) L r C e ω
-
-lemma permDistance_childLift (d : ℕ) (p q : Bool → Equiv.Perm (Card d)) :
-    permDistance (childLift p) (childLift q) =
-      permDistance (p false) (q false)+permDistance (p true) (q true) := by
-  classical
-  rw [permDistance_sum]
-  have hh : (∑ x : Card (d+1), if childLift p x = childLift q x then 0 else 1) =
-      ∑ x : Bool × Card d, if p x.1 x.2 = q x.1 x.2 then 0 else 1 := by
-    apply Fintype.sum_equiv (headTailEquiv d)
-    intro x
-    have hx : (headTailEquiv d) x = (x 0,Fin.tail x) := rfl
-    rw [hx]
-    congr 1
-    apply propext
-    constructor
-    · intro h
-      have ht := congrArg Fin.tail h
-      simpa only [childLift,Equiv.coe_fn_mk,Fin.tail_cons] using ht
-    · intro h
-      dsimp only [childLift,Equiv.coe_fn_mk]
-      exact congrArg (fun z : Card d => (Fin.cons (x 0) z : Card (d+1))) h
-  rw [hh]
-  simp only [Fintype.sum_prod_type,Fintype.sum_bool,permDistance_sum]
-  omega
-
-noncomputable def centralDistance (L r : ℕ) (C D : Card r → Equiv.Perm (Card L)) : ℕ :=
-  ∑ u : Card r, permDistance (C u) (D u)
-
-lemma centralDistance_step (L r : ℕ) (C D : Card (r+1) → Equiv.Perm (Card L)) :
-    centralDistance L (r+1) C D =
-      centralDistance L r (fun u => C (Fin.cons false u)) (fun u => D (Fin.cons false u))+
-      centralDistance L r (fun u => C (Fin.cons true u)) (fun u => D (Fin.cons true u)) := by
-  classical
-  unfold centralDistance
-  rw [Fintype.sum_equiv (headTailEquiv r) _
-    (fun x : Bool × Card r => permDistance (C (Fin.cons x.1 x.2)) (D (Fin.cons x.1 x.2)))
-      (fun x => by
-        change permDistance (C x) (D x) =
-          permDistance (C (Fin.cons (x 0) (Fin.tail x))) (D (Fin.cons (x 0) (Fin.tail x)))
-        rw [Fin.cons_self_tail])]
-  simp only [Fintype.sum_prod_type,Fintype.sum_bool]
-  omega
-
-lemma permDistance_centralBlock (L r : ℕ) (C D : Card r → Equiv.Perm (Card L)) :
-    permDistance (centralBlockPerm L r C) (centralBlockPerm L r D) = centralDistance L r C D := by
-  induction r with
-  | zero =>
-    simp only [centralBlockPerm,centralDistance,Fintype.sum_unique]
-    congr 1
-  | succ r ih =>
-    rw [centralBlockPerm,centralBlockPerm,permDistance_childLift,centralDistance_step,ih,ih]
-
-lemma permDistance_centralPalindrome (L r : ℕ) (C D : Card r → Equiv.Perm (Card L))
-    (ω : BenesCoins (L+r)) :
-    permDistance (centralPalindromePerm L r C ω) (centralPalindromePerm L r D ω) =
-      centralDistance L r C D := by
-  rw [centralPalindromePerm,centralPalindromePerm,permDistance_mul_right,
-    permDistance_mul_left,permDistance_centralBlock]
-
-lemma centralLevelCost_step (t L r : ℕ) (C : Card (r+1) → Equiv.Perm (Card L))
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r+1))
-    (ω : BenesCoins (L+r) × BenesCoins (L+r))
-    (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
-    centralLevelCost t L (r+1) C e ((benesStepEquiv (L+r)).symm (ω,coins)) =
-      let x := e.trans (headTailEquiv (L+r)).toEmbedding
-      let c := PairRouting.colors x coins.1
-      let hc := PairRouting.colors_compatible x coins.1
-      if t = L+r+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x coins.1)
-        (fun b => centralPalindromePerm L r (fun u => C (Fin.cons b u)) (if b then ω.2 else ω.1))
-      else centralLevelCost t L r (fun u => C (Fin.cons false u))
-        (PairRouting.childEmbedding x c hc false) ω.1 +
-        centralLevelCost t L r (fun u => C (Fin.cons true u))
-        (PairRouting.childEmbedding x c hc true) ω.2 := by
-  rw [centralLevelCost]
-  rfl
-
-lemma centralLevelCost_perturb (t L r : ℕ) (C D : Card r → Equiv.Perm (Card L))
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (ω : BenesCoins (L+r)) :
-    centralLevelCost t L r C e ω ≤ centralLevelCost t L r D e ω+centralDistance L r C D := by
-  induction r generalizing ι with
-  | zero => simp only [centralLevelCost,Nat.zero_add,Nat.zero_le]
-  | succ r ih =>
-    rw [←(benesStepEquiv (L+r)).symm_apply_apply ω,centralLevelCost_step,centralLevelCost_step]
-    dsimp only
-    split_ifs with ht
-    · have hh := alternatingCycles_distance_le
-        (PairRouting.switchedEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
-          (benesStepEquiv (L+r) ω).2.1)
-        (fun b => centralPalindromePerm L r (fun u => C (Fin.cons b u))
-          (if b then (benesStepEquiv (L+r) ω).1.2 else (benesStepEquiv (L+r) ω).1.1))
-        (fun b => centralPalindromePerm L r (fun u => D (Fin.cons b u))
-          (if b then (benesStepEquiv (L+r) ω).1.2 else (benesStepEquiv (L+r) ω).1.1))
-      simpa only [Bool.false_eq_true,↓reduceIte,permDistance_centralPalindrome,←centralDistance_step] using hh
-    · rw [centralDistance_step]
-      have h₀ := ih (fun u => C (Fin.cons false u)) (fun u => D (Fin.cons false u))
-        (PairRouting.childEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
-          (PairRouting.colors (e.trans (headTailEquiv (L+r)).toEmbedding) (benesStepEquiv (L+r) ω).2.1)
-          (PairRouting.colors_compatible _ _) false)
-        (benesStepEquiv (L+r) ω).1.1
-      have h₁ := ih (fun u => C (Fin.cons true u)) (fun u => D (Fin.cons true u))
-        (PairRouting.childEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
-          (PairRouting.colors (e.trans (headTailEquiv (L+r)).toEmbedding) (benesStepEquiv (L+r) ω).2.1)
-          (PairRouting.colors_compatible _ _) true)
-        (benesStepEquiv (L+r) ω).1.2
-      omega
-
-lemma centralSlabCost_perturb (l L r : ℕ) (C D : Card r → Equiv.Perm (Card L))
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (ω : BenesCoins (L+r)) :
-    centralSlabCost l L r C e ω ≤ centralSlabCost l L r D e ω+l*centralDistance L r C D := by
-  have hh := Finset.sum_le_sum (fun (i : Fin l) (_ : i ∈ Finset.univ) =>
-    centralLevelCost_perturb (l+i.val+1) L r C D e ω)
-  simpa only [Finset.sum_add_distrib,Finset.sum_const,Finset.card_univ,Fintype.card_fin,
-    nsmul_eq_mul,Nat.cast_id,centralSlabCost] using hh
-
-end Thorp
-
-end ThorpNine.Tail
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Tail.ParallelFreshPre
+
+namespace OAI
+
+namespace ThorpNine.Tail
+
+namespace Thorp
+
+theorem palindrome_family_mgf (L n d : ℕ) (hL : 0 < L) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) ≤
+      Real.exp (Fintype.card ι * (68*∑ j ∈ Finset.range n,
+        Real.exp (-((L*4^j:ℕ):ℝ)/64))) := by
+  induction n with
+  | zero => simp only [palindromeFamilyCost,Finset.range_zero,Finset.sum_empty,pow_zero,
+      finiteMean_const,mul_zero,Real.exp_zero,le_refl]
+  | succ n ih =>
+    let l := L*4^n
+    have hl : 0 < l := Nat.mul_pos hL (pow_pos (by decide) _)
+    have hs : 2*familyEnd L n ≤ l := familyEnd_two L n
+    have hc : ∀ Y, palindromeFamilyCost L (n+1) d e (Y,X) =
+        palindromeFamilyCost L n d e (Y,X)+palindromeSlabCost l d e (Y,X) := by
+      intro Y
+      exact Finset.sum_range_succ _ _
+    have hm : finiteMean (fun Y => q^(palindromeFamilyCost L (n+1) d e (Y,X))) ≤
+        finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) *
+          Real.exp (Fintype.card ι*(68*Real.exp (-(l:ℝ)/64))) := by
+      by_cases hld : l ≤ d
+      · simp_rw [hc,pow_add]
+        exact palindrome_slab_adapted_mul_le d (familyEnd L n) l (by omega) hl hs e X _
+          (fun _ => pow_nonneg (by linarith) _)
+          (layerAdapted_comp (palindromeFamilyCost_adapted L n d e X) (fun c => q^c))
+          q hq hlog
+      · simp_rw [hc,palindromeSlabCost_gt l d (by omega) e,Nat.add_zero]
+        exact le_mul_of_one_le_right
+          (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
+          (Real.one_le_exp_iff.mpr (by positivity))
+    refine (hm.trans (mul_le_mul_of_nonneg_right ih (Real.exp_nonneg _))).trans_eq ?_
+    rw [←Real.exp_add,Finset.sum_range_succ]
+    congr 1
+    dsimp [l]
+    ring
+
+lemma pow_four_ge (j : ℕ) : j+1 ≤ 4^j := by
+  induction j with
+  | zero => norm_num
+  | succ j ih => rw [pow_succ]; omega
+
+lemma sum_family_decay (L n : ℕ) (hL : 0 < L) :
+    (∑ j ∈ Finset.range n, Real.exp (-((L*4^j:ℕ):ℝ)/64)) ≤
+      Real.exp (-(L:ℝ)/64)/(1-Real.exp (-(1:ℝ)/64)) := by
+  let r := Real.exp (-(1:ℝ)/64)
+  have hr : 0 < r := Real.exp_pos _
+  have hr1 : r < 1 := Real.exp_lt_one_iff.mpr (by norm_num)
+  have hg : (∑ j ∈ Finset.range n, r^j) ≤ 1/(1-r) := by
+    apply (le_div_iff₀ (by linarith : (0:ℝ)<1-r)).mpr
+    have hh := geom_sum_mul r n
+    nlinarith [pow_nonneg (le_of_lt hr) n]
+  calc
+    _ ≤ ∑ j ∈ Finset.range n, Real.exp (-(L:ℝ)/64)*r^j := by
+      apply Finset.sum_le_sum
+      intro j _
+      have hj := pow_four_ge j
+      have hlj : L+j ≤ L*4^j := by nlinarith
+      have hlj' : (L:ℝ)+j ≤ (L*4^j:ℕ) := by exact_mod_cast hlj
+      rw [←Real.exp_nat_mul,←Real.exp_add]
+      exact Real.exp_le_exp.mpr (by linarith)
+    _ = Real.exp (-(L:ℝ)/64)*(∑ j ∈ Finset.range n, r^j) := by rw [Finset.mul_sum]
+    _ ≤ Real.exp (-(L:ℝ)/64)*(1/(1-r)) :=
+      mul_le_mul_of_nonneg_left hg (Real.exp_nonneg _)
+    _ = _ := by ring
+
+
+lemma palindromeLevelCost_step_add (t d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (d+1)) (ω : BenesCoins (d+1)) :
+    palindromeLevelCost t (d+1) e ω =
+      let σ := benesStepEquiv d ω
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x σ.2.1
+      let hc := PairRouting.colors_compatible x σ.2.1
+      (if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
+          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) else 0) +
+        palindromeLevelCost t d (PairRouting.childEmbedding x c hc false) σ.1.1 +
+        palindromeLevelCost t d (PairRouting.childEmbedding x c hc true) σ.1.2 := by
+  classical
+  rw [palindromeLevelCost]
+  dsimp only
+  split_ifs with h
+  · simp only [palindromeLevelCost_gt t d (by omega),Nat.add_zero]
+  · simp only [Nat.zero_add]
+
+lemma palindromeLowCost_sum (H d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) :
+    palindromeLowCost H d e ω = ∑ i ∈ Finset.range H, palindromeLevelCost (i+1) d e ω := by
+  classical
+  induction d generalizing ι with
+  | zero => simp only [palindromeLowCost,palindromeLevelCost,Finset.sum_const_zero]
+  | succ d ih =>
+    rw [palindromeLowCost]
+    simp only [palindromeLevelCost_step_add,Finset.sum_add_distrib]
+    rw [←ih,←ih]
+    congr 2
+    simp only [Nat.add_right_cancel_iff,Finset.sum_ite_eq',Finset.mem_range,Nat.succ_le_iff]
+
+lemma palindromeLowCost_slab (L d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) :
+    palindromeLowCost L d e ω + palindromeSlabCost L d e ω = palindromeLowCost (2*L) d e ω := by
+  rw [palindromeLowCost_sum,palindromeLowCost_sum,show 2*L=L+L by omega,Finset.sum_range_add]
+  congr 1
+  exact Fin.sum_univ_eq_sum_range (fun i => palindromeLevelCost (L+i+1) d e ω) L
+
+lemma palindromeCost_family (L n d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) :
+    palindromeLowCost L d e ω + palindromeFamilyCost L n d e ω +
+      palindromeFamilyCost (2*L) n d e ω = palindromeLowCost (L*4^n) d e ω := by
+  induction n with
+  | zero => simp only [palindromeFamilyCost,Finset.range_zero,Finset.sum_empty,Nat.add_zero,
+      pow_zero,mul_one]
+  | succ n ih =>
+    have he : (2*L)*4^n = 2*(L*4^n) := by ring
+    have he' : L*4^(n+1) = 2*(2*(L*4^n)) := by rw [pow_succ]; ring
+    simp only [palindromeFamilyCost,Finset.sum_range_succ] at ih ⊢
+    rw [he,he',←palindromeLowCost_slab,←palindromeLowCost_slab,←ih]
+    omega
+
+
+lemma finiteMean_mul_sq_le {Ω : Type*} [Fintype Ω] (f g : Ω → ℝ) :
+    (finiteMean (fun ω => f ω*g ω))^2 ≤ finiteMean (fun ω => (f ω)^2)*finiteMean (fun ω => (g ω)^2) := by
+  have h := Finset.sum_mul_sq_le_sq_mul_sq Finset.univ f g
+  simp only [finiteMean,div_pow,div_mul_div_comm,←pow_two]
+  exact div_le_div_of_nonneg_right h (sq_nonneg _)
+
+lemma finiteMean_mul_exp_le {Ω : Type*} [Fintype Ω] (f g : Ω → ℝ) (A B : ℝ)
+    (hf : finiteMean (fun ω => (f ω)^2) ≤ Real.exp A)
+    (hg : finiteMean (fun ω => (g ω)^2) ≤ Real.exp B) :
+    finiteMean (fun ω => f ω*g ω) ≤ Real.exp ((A+B)/2) := by
+  have hc := finiteMean_mul_sq_le f g
+  have hn : 0 ≤ finiteMean (fun ω => (g ω)^2) := finiteMean_nonneg (fun _ => sq_nonneg _)
+  have hu := mul_le_mul hf hg hn (Real.exp_nonneg _)
+  have he : Real.exp A*Real.exp B = (Real.exp ((A+B)/2))^2 := by
+    rw [←Real.exp_add,←Real.exp_nat_mul]
+    congr 1
+    push_cast
+    ring
+  rw [he] at hu
+  have hh := hc.trans hu
+  have hep := Real.exp_pos ((A+B)/2)
+  nlinarith
+
+
+noncomputable def heightDecay (H : ℕ) : ℝ :=
+  68*Real.exp (-(H:ℝ)/64)/(1-Real.exp (-(1:ℝ)/64))
+
+lemma heightDecay_nonneg (H : ℕ) : 0 ≤ heightDecay H := by
+  have h : Real.exp (-(1:ℝ)/64) < 1 := Real.exp_lt_one_iff.mpr (by norm_num)
+  apply div_nonneg (by positivity) (by linarith)
+
+lemma heightDecay_antitone : Antitone heightDecay := by
+  intro L H h
+  have hden : 0 ≤ 1-Real.exp (-(1:ℝ)/64) := by
+    have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
+    linarith
+  apply div_le_div_of_nonneg_right _ hden
+  apply mul_le_mul_of_nonneg_left _ (by norm_num)
+  apply Real.exp_le_exp.mpr
+  have h' : (L:ℝ) ≤ H := by exact_mod_cast h
+  linarith
+
+lemma palindrome_family_mgf_decay (L n d : ℕ) (hL : 0 < L) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun Y => q^(palindromeFamilyCost L n d e (Y,X))) ≤
+      Real.exp (Fintype.card ι*heightDecay L) := by
+  apply (palindrome_family_mgf L n d hL e X q hq hlog).trans
+  apply Real.exp_le_exp.mpr
+  apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
+  have h := mul_le_mul_of_nonneg_left (sum_family_decay L n hL) (by norm_num : (0:ℝ)≤68)
+  simpa only [heightDecay,mul_div_assoc] using h
+
+theorem palindrome_high_mgf (H n d : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/500) :
+    finiteMean (fun Y => q^(palindromeFamilyCost H n d e (Y,X)+
+      palindromeFamilyCost (2*H) n d e (Y,X))) ≤ Real.exp (Fintype.card ι*heightDecay H) := by
+  have hq2 := one_le_pow₀ (n:=2) hq
+  have hlog2 : Real.log (q^2) ≤ 1/250 := by rw [Real.log_pow]; norm_num; linarith
+  have h₀ := palindrome_family_mgf_decay H n d hH e X (q^2) hq2 hlog2
+  have h₁ := palindrome_family_mgf_decay (2*H) n d (by omega) e X (q^2) hq2 hlog2
+  have h₁' := h₁.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left
+    (heightDecay_antitone (show H ≤ 2*H by omega)) (Nat.cast_nonneg _)))
+  have hh := finiteMean_mul_exp_le (fun Y => q^(palindromeFamilyCost H n d e (Y,X)))
+    (fun Y => q^(palindromeFamilyCost (2*H) n d e (Y,X)))
+    (Fintype.card ι*heightDecay H) (Fintype.card ι*heightDecay H)
+    (by simpa only [pow_right_comm] using h₀) (by simpa only [pow_right_comm] using h₁')
+  simpa only [←pow_add,add_self_div_two] using hh
+
+lemma palindrome_high_full_mgf (H n d : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/500) :
+    finiteMean (fun ω : BenesCoins d => q^(palindromeFamilyCost H n d e ω+
+      palindromeFamilyCost (2*H) n d e ω)) ≤ Real.exp (Fintype.card ι*heightDecay H) := by
+  rw [finiteMean_prod,finiteMean_comm]
+  calc
+    _ ≤ finiteMean (fun _X : SwitchIndex d → Bool => Real.exp (Fintype.card ι*heightDecay H)) := by
+      apply finiteMean_mono
+      intro X
+      exact palindrome_high_mgf H n d hH e X q hq hlog
+    _ = _ := finiteMean_const _
+
+theorem palindrome_cost_mgf_cutoff (r H : ℕ) (hH : 0 < H) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (H+r)) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/1000)
+    (hr : (2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*(q^2)^H) ≤ 1/2) :
+    finiteMean (fun ω : BenesCoins (H+r) => q^(palindromeCost (H+r) e ω)) ≤
+      Real.exp (((2:ℝ)^r*(2*((2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*(q^2)^H))^2)+
+        Fintype.card ι*heightDecay H)/2) := by
+  let d := H+r
+  have hd : d ≤ H*4^d := by
+    have hh := pow_four_ge d
+    nlinarith
+  have hcost (ω : BenesCoins d) : palindromeCost d e ω = palindromeLowCost H d e ω+
+      (palindromeFamilyCost H d d e ω+palindromeFamilyCost (2*H) d d e ω) := by
+    have hh := palindromeCost_family H d d e ω
+    rw [palindromeLowCost_eq _ _ hd] at hh
+    simpa only [Nat.add_assoc] using hh.symm
+  have hq2 := one_le_pow₀ (n:=2) hq
+  have hlog2 : Real.log (q^2) ≤ 1/500 := by rw [Real.log_pow]; norm_num; linarith
+  have hlow := palindrome_low_mgf r H e (q^2) hq2 hr
+  have hhigh := palindrome_high_full_mgf H d d hH e (q^2) hq2 hlog2
+  have hh := finiteMean_mul_exp_le (fun ω : BenesCoins d => q^(palindromeLowCost H d e ω))
+    (fun ω : BenesCoins d => q^(palindromeFamilyCost H d d e ω+palindromeFamilyCost (2*H) d d e ω))
+    _ _ (by simpa only [pow_right_comm] using hlow) (by simpa only [pow_right_comm] using hhigh)
+  change finiteMean (fun ω : BenesCoins d => q^(palindromeCost d e ω)) ≤ _
+  simpa only [hcost,pow_add,pow_right_comm] using hh
+
+
+lemma cutoffMass_le (H d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (ω : BenesCoins d) :
+    cutoffMass H d e ω ≤ Fintype.card ι := by
+  classical
+  induction d generalizing ι with
+  | zero => exact Nat.zero_le _
+  | succ d ih =>
+    rw [cutoffMass]
+    split_ifs
+    · exact crowdedMass_le _
+    · exact (Nat.add_le_add (ih _ _) (ih _ _)).trans_eq (PairRouting.color_card_add _)
+
+lemma palindromeLowCost_le (H d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) : palindromeLowCost H d e ω ≤ H*Fintype.card ι :=
+  (palindromeLowCost_le_cutoff H d e ω).trans (Nat.mul_le_mul_left H (cutoffMass_le H d e ω))
+
+theorem palindrome_cost_mgf_coarse (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/1000) :
+    finiteMean (fun ω : BenesCoins d => q^(palindromeCost d e ω)) ≤
+      Real.exp (Fintype.card ι*(1+heightDecay 1)) := by
+  have hd : d ≤ 1*4^d := by have hh := pow_four_ge d; omega
+  have hcost (ω : BenesCoins d) : palindromeCost d e ω = palindromeLowCost 1 d e ω+
+      (palindromeFamilyCost 1 d d e ω+palindromeFamilyCost 2 d d e ω) := by
+    have hh := palindromeCost_family 1 d d e ω
+    rw [palindromeLowCost_eq _ _ hd] at hh
+    simpa only [Nat.add_assoc] using hh.symm
+  have hq2 := one_le_pow₀ (n:=2) hq
+  have hlog2 : Real.log (q^2) ≤ 1/500 := by rw [Real.log_pow]; norm_num; linarith
+  have hlow : finiteMean (fun ω : BenesCoins d => (q^2)^(palindromeLowCost 1 d e ω)) ≤
+      Real.exp (Fintype.card ι) := by
+    calc
+      _ ≤ finiteMean (fun _ω : BenesCoins d => (q^2)^(Fintype.card ι)) := by
+        apply finiteMean_mono
+        intro ω
+        apply pow_le_pow_right₀ hq2
+        simpa only [one_mul] using palindromeLowCost_le 1 d e ω
+      _ = (q^2)^(Fintype.card ι) := finiteMean_const _
+      _ = Real.exp (Fintype.card ι * Real.log (q^2)) := by
+        rw [Real.exp_nat_mul,Real.exp_log (by positivity : 0<q^2)]
+      _ ≤ _ := Real.exp_le_exp.mpr (by
+        have hh := mul_le_mul_of_nonneg_left hlog2 (Nat.cast_nonneg (Fintype.card ι) : (0:ℝ)≤_)
+        nlinarith [Nat.cast_nonneg (α:=ℝ) (Fintype.card ι)])
+  have hhigh := palindrome_high_full_mgf 1 d d (by decide) e (q^2) hq2 hlog2
+  have hh := finiteMean_mul_exp_le (fun ω : BenesCoins d => q^(palindromeLowCost 1 d e ω))
+    (fun ω : BenesCoins d => q^(palindromeFamilyCost 1 d d e ω+palindromeFamilyCost 2 d d e ω))
+    (Fintype.card ι) (Fintype.card ι*heightDecay 1)
+    (by simpa only [pow_right_comm] using hlow) (by simpa only [pow_right_comm] using hhigh)
+  simp only [←pow_add,←hcost] at hh
+  apply hh.trans (Real.exp_le_exp.mpr _)
+  have hk : (0:ℝ) ≤ Fintype.card ι := Nat.cast_nonneg _
+  have hD := heightDecay_nonneg 1
+  nlinarith
+
+noncomputable def momentBase : ℝ := (2:ℝ)^(1/1000:ℝ)
+
+lemma momentBase_one_le : 1 ≤ momentBase := by
+  exact Real.one_le_rpow (by norm_num) (by norm_num)
+
+lemma momentBase_log : Real.log momentBase ≤ 1/1000 := by
+  rw [momentBase,Real.log_rpow (by norm_num)]
+  have hh := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
+  norm_num at hh ⊢
+  linarith
+
+lemma momentBase_pow (c : ℕ) : momentBase^c = (2:ℝ)^((c:ℝ)*(1/1000:ℝ)) := by
+  rw [mul_comm,Real.rpow_mul_natCast (by norm_num)]
+  rfl
+
+lemma palindromeRowMoment_le_cost (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) :
+    palindromeRowMoment d e (1/1000) ≤
+      finiteMean (fun ω : BenesCoins d => momentBase^(palindromeCost d e ω)) := by
+  have hn : (0:ℝ) < ((2^d:ℕ):ℝ)^(Fintype.card ι) := by positivity
+  have hn0 : (0:ℝ) ≤ ((2^d).descFactorial (Fintype.card ι):ℝ) /
+      ((2^d:ℕ):ℝ)^(Fintype.card ι) := by positivity
+  have hn1 : ((2^d).descFactorial (Fintype.card ι):ℝ) /
+      ((2^d:ℕ):ℝ)^(Fintype.card ι) ≤ 1 := by
+    apply (div_le_one hn).mpr
+    exact_mod_cast Nat.descFactorial_le_pow (2^d) (Fintype.card ι)
+  have hpow := Real.rpow_le_one hn0 hn1 (by norm_num : (0:ℝ)≤1/1000)
+  have hm := palindrome_row_density_moment d e (1/1000) (by norm_num)
+  simp_rw [←momentBase_pow] at hm
+  exact hm.trans (mul_le_of_le_one_left (finiteMean_nonneg (fun _ =>
+    pow_nonneg (le_trans (by norm_num) momentBase_one_le) _)) hpow)
+
+theorem palindrome_row_moment_coarse (d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) :
+    palindromeRowMoment d e (1/1000) ≤ Real.exp (Fintype.card ι*(1+heightDecay 1)) :=
+  (palindromeRowMoment_le_cost d e).trans
+    (palindrome_cost_mgf_coarse d e momentBase momentBase_one_le momentBase_log)
+
+
+noncomputable def densityCutoff (p : ℝ) : ℕ := ⌊-Real.log p/(4*Real.log 2)⌋₊
+
+lemma log_small_density {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
+    Real.log p ≤ -8*Real.log 2 := by
+  have he : Real.log (1/256:ℝ) = -8*Real.log 2 := by
+    rw [show (1/256:ℝ) = 1/2^8 by norm_num,
+      Real.log_div (by norm_num) (by positivity),Real.log_one,Real.log_pow]
+    ring
+  exact (Real.log_le_log hp hs).trans_eq he
+
+lemma densityCutoff_bounds {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
+    0 < densityCutoff p ∧
+    (densityCutoff p:ℝ)*(4*Real.log 2) ≤ -Real.log p ∧
+    -Real.log p < ((densityCutoff p:ℝ)+1)*(4*Real.log 2) := by
+  have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
+  have hd : 0 < 4*Real.log 2 := by positivity
+  have hh := log_small_density hp hs
+  have hx : 1 ≤ -Real.log p/(4*Real.log 2) := by
+    apply (le_div_iff₀ hd).mpr
+    nlinarith
+  refine ⟨Nat.floor_pos.mpr hx,?_,?_⟩
+  · exact (le_div_iff₀ hd).mp (Nat.floor_le (by linarith : 0 ≤ -Real.log p/(4*Real.log 2)))
+  · exact (div_lt_iff₀ hd).mp (Nat.lt_floor_add_one (-Real.log p/(4*Real.log 2)))
+
+lemma densityCutoff_decay {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
+    Real.exp (-(densityCutoff p:ℝ)/64) ≤ Real.exp (1/64) * p^(1/512:ℝ) := by
+  have hh := (densityCutoff_bounds hp hs).2.2
+  have hb : Real.log 2 ≤ 2 := by
+    have := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
+    linarith
+  have hH : 0 ≤ (densityCutoff p:ℝ)+1 := by positivity
+  have hu := mul_le_mul_of_nonneg_left hb hH
+  rw [Real.rpow_def_of_pos hp,←Real.exp_add]
+  apply Real.exp_le_exp.mpr
+  nlinarith
+
+lemma densityCutoff_power {p q : ℝ} (hp : 0 < p) (hs : p ≤ 1/256)
+    (hq : 0 < q) (hlog : Real.log q ≤ Real.log 2/4) (n : ℕ) (hn : n ≤ 4)
+    (hq1 : 1 ≤ q) :
+    p*(2:ℝ)^(densityCutoff p)*q^(n*densityCutoff p) ≤ p^(1/2:ℝ) := by
+  have hcut := (densityCutoff_bounds hp hs).2.1
+  have hH : 0 ≤ (densityCutoff p:ℝ) := Nat.cast_nonneg _
+  have hn' : (n:ℝ) ≤ 4 := by exact_mod_cast hn
+  have hln := Real.log_nonneg hq1
+  have hl := mul_le_mul_of_nonneg_right hn' hln
+  have hnl : (n:ℝ)*Real.log q ≤ Real.log 2 := by linarith
+  have hm := mul_le_mul_of_nonneg_left hnl hH
+  have heq : p*(2:ℝ)^(densityCutoff p)*q^(n*densityCutoff p) =
+      Real.exp (Real.log p+(densityCutoff p:ℝ)*Real.log 2+
+        (n*densityCutoff p:ℕ)*Real.log q) := by
+    rw [Real.exp_add,Real.exp_add,Real.exp_nat_mul,Real.exp_nat_mul,
+      Real.exp_log hp,Real.exp_log hq,Real.exp_log (by norm_num : (0:ℝ)<2)]
+  rw [heq,Real.rpow_def_of_pos hp]
+  apply Real.exp_le_exp.mpr
+  push_cast
+  nlinarith
+
+lemma densityCutoff_sparse {p q : ℝ} (hp : 0 < p) (hs : p ≤ 1/256)
+    (hq : 0 < q) (hlog : Real.log q ≤ Real.log 2/4) (hq1 : 1 ≤ q) :
+    (2:ℝ)^(densityCutoff p)*(p*(q^2)^(densityCutoff p)) ≤ 1/2 := by
+  have hh := densityCutoff_power hp hs hq hlog 2 (by decide) hq1
+  rw [←pow_mul] at ⊢
+  have hle : p^(1/2:ℝ) ≤ 1/2 := by
+    have hlogp := log_small_density hp hs
+    have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
+    rw [Real.rpow_def_of_pos hp]
+    calc
+      _ ≤ Real.exp (-Real.log 2) := by apply Real.exp_le_exp.mpr; nlinarith
+      _ = 1/2 := by rw [Real.exp_neg,Real.exp_log (by norm_num : (0:ℝ)<2)]; norm_num
+  nlinarith [hh.trans hle]
+
+
+lemma momentBase_log_exact : Real.log momentBase = Real.log 2/1000 := by
+  rw [momentBase,Real.log_rpow (by norm_num)]
+  ring
+
+lemma densityCutoff_le_height (d k : ℕ) (hk : 0 < k) :
+    densityCutoff ((k:ℝ)/(2:ℝ)^d) ≤ d := by
+  have hk' : (1:ℝ) ≤ k := by exact_mod_cast hk
+  have hp : 0 < (k:ℝ)/(2:ℝ)^d := by positivity
+  have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
+  have hlogk := Real.log_nonneg hk'
+  by_cases hx : 0 ≤ -Real.log ((k:ℝ)/(2:ℝ)^d)/(4*Real.log 2)
+  · have hh := (le_div_iff₀ (by positivity : 0 < 4*Real.log 2)).mp (Nat.floor_le hx)
+    have hd : (0:ℝ) ≤ d := Nat.cast_nonneg _
+    have hH : (0:ℝ) ≤ densityCutoff ((k:ℝ)/(2:ℝ)^d) := Nat.cast_nonneg _
+    change (densityCutoff ((k:ℝ)/(2:ℝ)^d):ℝ)*(4*Real.log 2) ≤ -Real.log ((k:ℝ)/(2:ℝ)^d) at hh
+    rw [Real.log_div (by positivity) (by positivity),Real.log_pow] at hh
+    have hle : (densityCutoff ((k:ℝ)/(2:ℝ)^d):ℝ) ≤ d := by nlinarith
+    exact_mod_cast hle
+  · have hz := Nat.floor_of_nonpos (le_of_not_ge hx)
+    simp only [densityCutoff,hz,Nat.zero_le]
+
+noncomputable def densityTailConstant : ℝ :=
+  68*Real.exp (1/64)/(1-Real.exp (-(1:ℝ)/64))
+
+lemma densityTailConstant_nonneg : 0 ≤ densityTailConstant := by
+  have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
+  apply div_nonneg (by positivity) (by linarith)
+
+lemma heightDecay_cutoff {p : ℝ} (hp : 0 < p) (hs : p ≤ 1/256) :
+    heightDecay (densityCutoff p) ≤ densityTailConstant*p^(1/512:ℝ) := by
+  have hden : 0 ≤ 1-Real.exp (-(1:ℝ)/64) := by
+    have h := Real.exp_lt_one_iff.mpr (by norm_num : -(1:ℝ)/64 < 0)
+    linarith
+  have hh := div_le_div_of_nonneg_right
+    (mul_le_mul_of_nonneg_left (densityCutoff_decay hp hs) (by norm_num : (0:ℝ)≤68)) hden
+  simpa only [heightDecay,densityTailConstant,mul_assoc,div_mul_eq_mul_div] using hh
+
+lemma sparse_low_algebra (r H k : ℕ) (q : ℝ) :
+    (2:ℝ)^r*(2*((2:ℝ)^H*(((k:ℝ)/(2:ℝ)^(H+r))*(q^2)^H))^2) =
+    2*k*(((k:ℝ)/(2:ℝ)^(H+r))*(2:ℝ)^H*q^(4*H)) := by
+  rw [pow_add,←pow_mul,show 4*H = (2*H)*2 by omega,pow_mul]
+  have hH : (2:ℝ)^H ≠ 0 := by positivity
+  have hr : (2:ℝ)^r ≠ 0 := by positivity
+  field_simp
+  ring
+
+lemma palindrome_row_moment_small (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι)
+    (hs : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1/256) :
+    palindromeRowMoment d e (1/1000) ≤
+      Real.exp (Fintype.card ι*(1+densityTailConstant)*
+        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ)) := by
+  generalize hpdef : ((Fintype.card ι:ℝ)/(2:ℝ)^d) = p at hs ⊢
+  have hp : 0 < p := by rw [←hpdef]; positivity
+  have hle := densityCutoff_le_height d (Fintype.card ι) hk
+  rw [hpdef] at hle
+  obtain ⟨r,hr⟩ := Nat.exists_eq_add_of_le hle
+  subst d
+  have hH := (densityCutoff_bounds hp hs).1
+  have hq := momentBase_one_le
+  have hqp : 0 < momentBase := by linarith
+  have hlog : Real.log momentBase ≤ Real.log 2/4 := by
+    rw [momentBase_log_exact]
+    have h2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
+    linarith
+  have hsp := densityCutoff_sparse hp hs hqp hlog hq
+  rw [←hpdef] at hsp
+  have hh := palindrome_cost_mgf_cutoff r (densityCutoff p) hH e momentBase hq momentBase_log
+    (by simpa only [hpdef] using hsp)
+  apply (palindromeRowMoment_le_cost _ e).trans (hh.trans (Real.exp_le_exp.mpr _))
+  rw [sparse_low_algebra,hpdef]
+  have hlow := densityCutoff_power hp hs hqp hlog 4 (by decide) hq
+  have hp1 : p ≤ 1 := by linarith
+  have he := Real.rpow_le_rpow_of_exponent_ge hp hp1 (by norm_num : (1:ℝ)/512 ≤ 1/2)
+  have hlow' := mul_le_mul_of_nonneg_left (hlow.trans he)
+    (by positivity : (0:ℝ)≤2*Fintype.card ι)
+  have hhigh := mul_le_mul_of_nonneg_left (heightDecay_cutoff hp hs)
+    (Nat.cast_nonneg (Fintype.card ι): (0:ℝ)≤_)
+  have hnon : 0 ≤ Fintype.card ι*densityTailConstant*p^(1/512:ℝ) :=
+    mul_nonneg (mul_nonneg (Nat.cast_nonneg _) densityTailConstant_nonneg)
+      (Real.rpow_nonneg (le_of_lt hp) _)
+  nlinarith
+
+noncomputable def densityConstant : ℝ :=
+  256*(1+heightDecay 1)+1+densityTailConstant
+
+lemma densityConstant_pos : 0 < densityConstant := by
+  dsimp [densityConstant]
+  have hh := heightDecay_nonneg 1
+  have ht := densityTailConstant_nonneg
+  linarith
+
+theorem palindrome_row_moment_bound (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι) :
+    palindromeRowMoment d e (1/1000) ≤ Real.exp (densityConstant*Fintype.card ι*
+        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ)) := by
+  have hp : 0 < (Fintype.card ι:ℝ)/(2:ℝ)^d := by positivity
+  have hp1 : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1 := by
+    apply (div_le_one (by positivity)).mpr
+    have h := Fintype.card_le_of_injective e e.injective
+    rw [card_positions] at h
+    exact_mod_cast h
+  have hk0 : (0:ℝ) ≤ Fintype.card ι := Nat.cast_nonneg _
+  have hpow := Real.rpow_nonneg (le_of_lt hp) (1/512:ℝ)
+  by_cases hs : (Fintype.card ι:ℝ)/(2:ℝ)^d ≤ 1/256
+  · apply (palindrome_row_moment_small d e hk hs).trans (Real.exp_le_exp.mpr _)
+    have hD : 1+densityTailConstant ≤ densityConstant := by
+      dsimp [densityConstant]
+      have := heightDecay_nonneg 1
+      linarith
+    have hh := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hD hk0) hpow
+    nlinarith
+  · have hlow : (1:ℝ)/256 ≤ ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ) := by
+      have he := Real.rpow_le_rpow_of_exponent_ge hp hp1 (by norm_num : (1:ℝ)/512 ≤ 1)
+      rw [Real.rpow_one] at he
+      exact (le_of_not_ge hs).trans he
+    have hD : 256*(1+heightDecay 1) ≤ densityConstant := by
+      dsimp [densityConstant]
+      have := densityTailConstant_nonneg
+      linarith
+    apply (palindrome_row_moment_coarse d e).trans (Real.exp_le_exp.mpr _)
+    have hD0 : 0 ≤ 1+heightDecay 1 := by have := heightDecay_nonneg 1; linarith
+    have ha := mul_le_mul_of_nonneg_left hlow (mul_nonneg (by norm_num : (0:ℝ)≤256) hD0)
+    have hb := mul_le_mul_of_nonneg_right hD hpow
+    have hc := mul_le_mul_of_nonneg_right (ha.trans hb) hk0
+    nlinarith
+
+lemma palindromeRowMoment_nonneg (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (a : ℝ) : 0 ≤ palindromeRowMoment d e a := by
+  apply finiteMean_nonneg
+  intro f
+  exact Real.rpow_nonneg (mul_nonneg (Nat.cast_nonneg _)
+    (PairRouting.tupleProbability_nonneg _ _ _)) _
+
+theorem palindrome_log_row_moment (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (hk : 0 < Fintype.card ι) :
+    Real.log (palindromeRowMoment d e (1/1000)) ≤ densityConstant*Fintype.card ι*
+        ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ) := by
+  by_cases hz : palindromeRowMoment d e (1/1000) = 0
+  · rw [hz,Real.log_zero]
+    exact mul_nonneg (mul_nonneg (le_of_lt densityConstant_pos) (Nat.cast_nonneg _))
+      (Real.rpow_nonneg (by positivity) _)
+  · exact (Real.log_le_iff_le_exp (lt_of_le_of_ne (palindromeRowMoment_nonneg _ _ _) (Ne.symm hz))).mpr
+      (palindrome_row_moment_bound d e hk)
+
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+lemma orbitSet_eq_of_agree (p q : Equiv.Perm α) (x : α)
+    (h : ∀ z ∈ orbitSet p x, p z = q z) : orbitSet p x = orbitSet q x := by
+  have hp (n : ℕ) : (p^n) x ∈ orbitSet p x := by
+    apply (mem_orbitSet _ _ _).mpr
+    exact Equiv.Perm.sameCycle_pow_right.mpr (Equiv.Perm.SameCycle.refl _ _)
+  have he (n : ℕ) : (p^n) x = (q^n) x := by
+    induction n with
+    | zero => rfl
+    | succ n ih =>
+      rw [pow_succ',pow_succ',Equiv.Perm.mul_apply,Equiv.Perm.mul_apply,←ih]
+      exact h _ (hp n)
+  ext y
+  simp only [mem_orbitSet]
+  constructor
+  · intro hy
+    obtain ⟨n,hn⟩ := hy.exists_nat_pow_eq
+    rw [←hn,he]
+    exact Equiv.Perm.sameCycle_pow_right.mpr (Equiv.Perm.SameCycle.refl _ _)
+  · intro hy
+    obtain ⟨n,hn⟩ := hy.exists_nat_pow_eq
+    rw [←hn,←he]
+    exact Equiv.Perm.sameCycle_pow_right.mpr (Equiv.Perm.SameCycle.refl _ _)
+
+noncomputable def survivingCycle (p q : Equiv.Perm α) (S B : Finset α)
+    (h : ∀ x ∉ B, p x = q x) (C : RoutingNetwork.SelectedCycle p S)
+    (hC : ∀ x ∈ C.val, x ∉ B) : RoutingNetwork.SelectedCycle q S := by
+  refine ⟨C.val,?_,C.property.2⟩
+  obtain ⟨x,hx⟩ := C.property.1
+  refine ⟨x,hx.trans (orbitSet_eq_of_agree p q x ?_)⟩
+  intro z hz
+  exact h z (hC z (hx ▸ hz))
+
+noncomputable def perturbCycleMap (p q : Equiv.Perm α) (S B : Finset α)
+    (h : ∀ x ∉ B, p x = q x) (C : RoutingNetwork.SelectedCycle p S) :
+    RoutingNetwork.SelectedCycle q S ⊕ {x // x ∈ B} := by
+  classical
+  by_cases hb : ∃ x ∈ C.val, x ∈ B
+  · exact Sum.inr ⟨hb.choose,hb.choose_spec.2⟩
+  · exact Sum.inl (survivingCycle p q S B h C (by simpa only [not_exists,not_and] using hb))
+
+lemma perturbCycleMap_injective (p q : Equiv.Perm α) (S B : Finset α)
+    (h : ∀ x ∉ B, p x = q x) : Function.Injective (perturbCycleMap p q S B h) := by
+  classical
+  intro C D he
+  by_cases hC : ∃ x ∈ C.val, x ∈ B <;>
+    by_cases hD : ∃ x ∈ D.val, x ∈ B
+  · simp only [perturbCycleMap,dite_eq_left hC,dite_eq_left hD,Sum.inr.injEq] at he
+    have hx : hC.choose = hD.choose := congrArg Subtype.val he
+    exact RoutingNetwork.SelectedCycle.eq_of_mem C D hC.choose_spec.1
+      (hx.symm ▸ hD.choose_spec.1)
+  · simp only [perturbCycleMap,dite_eq_left hC,dite_eq_right hD] at he
+    cases he
+  · simp only [perturbCycleMap,dite_eq_right hC,dite_eq_left hD] at he
+    cases he
+  · simp only [perturbCycleMap,dite_eq_right hC,dite_eq_right hD,Sum.inl.injEq] at he
+    have hval : C.val = D.val := congrArg (fun Z : RoutingNetwork.SelectedCycle q S => Z.val) he
+    exact Subtype.ext hval
+
+theorem selectedCycle_perturb_le (p q : Equiv.Perm α) (S B : Finset α)
+    (h : ∀ x ∉ B, p x = q x) :
+    Fintype.card (RoutingNetwork.SelectedCycle p S) ≤
+      Fintype.card (RoutingNetwork.SelectedCycle q S)+B.card := by
+  have hh := Fintype.card_le_of_injective _ (perturbCycleMap_injective p q S B h)
+  simpa only [Fintype.card_sum,Fintype.card_coe] using hh
+
+
+noncomputable def permDistance (p q : Equiv.Perm α) : ℕ :=
+  (Finset.univ.filter (fun x => p x ≠ q x)).card
+
+lemma permDistance_sum (p q : Equiv.Perm α) :
+    permDistance p q = ∑ x : α, if p x = q x then 0 else 1 := by
+  classical
+  simp [permDistance,Finset.card_filter]
+
+lemma permDistance_refl (p : Equiv.Perm α) : permDistance p p = 0 := by
+  simp [permDistance]
+
+lemma permDistance_le (p q : Equiv.Perm α) : permDistance p q ≤ Fintype.card α :=
+  (Finset.card_filter_le _ _).trans_eq (Finset.card_univ)
+
+lemma permDistance_symm (p q : Equiv.Perm α) : permDistance p q = permDistance q p := by
+  simp only [permDistance,ne_comm]
+
+lemma permDistance_triangle (p q r : Equiv.Perm α) :
+    permDistance p r ≤ permDistance p q+permDistance q r := by
+  classical
+  simp only [permDistance_sum,←Finset.sum_add_distrib]
+  apply Finset.sum_le_sum
+  intro x _
+  by_cases hpq : p x = q x <;> by_cases hqr : q x = r x <;>
+    by_cases hpr : p x = r x <;> simp only [hpq,hqr,hpr,↓reduceIte] <;>
+    omega
+
+lemma permDistance_mul_left (p q g : Equiv.Perm α) :
+    permDistance (g*p) (g*q) = permDistance p q := by
+  classical
+  unfold permDistance
+  congr 1
+  ext x
+  simp only [Finset.mem_filter,Finset.mem_univ,true_and]
+  exact not_congr g.injective.eq_iff
+
+lemma permDistance_mul_right (p q g : Equiv.Perm α) :
+    permDistance (p*g) (q*g) = permDistance p q := by
+  classical
+  simp only [permDistance_sum,Equiv.Perm.mul_apply]
+  exact Fintype.sum_equiv g _ _ (fun _ => rfl)
+
+lemma permDistance_inv (p q : Equiv.Perm α) : permDistance p⁻¹ q⁻¹ = permDistance p q := by
+  have hh := permDistance_mul_left p⁻¹ q⁻¹ q
+  have hhh := permDistance_mul_right (q*p⁻¹) 1 p
+  simp only [mul_inv_cancel,one_mul,inv_mul_cancel_right] at hh hhh
+  rw [←hhh] at hh
+  rw [←hh,permDistance_symm]
+
+lemma permDistance_mul_le (p q g h : Equiv.Perm α) :
+    permDistance (p*g) (q*h) ≤ permDistance p q+permDistance g h := by
+  have hh := permDistance_triangle (p*g) (q*g) (q*h)
+  simpa only [permDistance_mul_left,permDistance_mul_right] using hh
+
+lemma selectedCycle_distance_le (p q : Equiv.Perm α) (S : Finset α) :
+    Fintype.card (RoutingNetwork.SelectedCycle p S) ≤
+      Fintype.card (RoutingNetwork.SelectedCycle q S)+permDistance p q := by
+  apply selectedCycle_perturb_le
+  intro x hx
+  simpa only [Finset.mem_filter,Finset.mem_univ,true_and,not_not] using hx
+
+lemma permDistance_alternating (p q : Equiv.Perm α) :
+    permDistance (CycleColoring.alternatingPerm p) (CycleColoring.alternatingPerm q) = permDistance p q := by
+  classical
+  simp only [permDistance_sum,Fintype.sum_prod_type,Fintype.sum_bool]
+  simp [CycleColoring.alternatingPerm]
+
+lemma alternatingCycles_distance_le {ι : Type*} [Fintype ι]
+    (e : ι ↪ Bool × α) (p q : Bool → Equiv.Perm α) :
+    PairRouting.alternatingCycles e p ≤ PairRouting.alternatingCycles e q+
+      (permDistance (p false) (q false)+permDistance (p true) (q true)) := by
+  have hh := selectedCycle_distance_le
+    (CycleColoring.alternatingPerm (PairRouting.alternatingProjection p))
+    (CycleColoring.alternatingPerm (PairRouting.alternatingProjection q)) (PairRouting.labelSet e)
+  rw [permDistance_alternating] at hh
+  apply hh.trans
+  apply Nat.add_le_add_left
+  dsimp only [PairRouting.alternatingProjection]
+  exact (permDistance_mul_le _ _ _ _).trans_eq (by
+    change permDistance (p false)⁻¹ (q false)⁻¹ + _ = _
+    rw [permDistance_inv])
+
+end Thorp
+namespace Thorp
+
+def centralBlockPerm (L : ℕ) : (r : ℕ) → (Card r → Equiv.Perm (Card L)) →
+    Equiv.Perm (Card (L+r))
+  | 0, C => C (fun j => Fin.elim0 j)
+  | r+1, C => childLift (fun b => centralBlockPerm L r (fun u => C (Fin.cons b u)))
+
+def centralPalindromePerm (L r : ℕ) (C : Card r → Equiv.Perm (Card L))
+    (ω : BenesCoins (L+r)) : Equiv.Perm (Card (L+r)) :=
+  butterflyPerm (L+r) (decodeButterfly (L+r) ω.1) * centralBlockPerm L r C *
+    (butterflyPerm (L+r) (decodeButterfly (L+r) ω.2))⁻¹
+
+lemma centralPalindromePerm_step (L r : ℕ) (C : Card (r+1) → Equiv.Perm (Card L))
+    (c : BenesCoins (L+r) × BenesCoins (L+r))
+    (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
+    centralPalindromePerm L (r+1) C ((benesStepEquiv (L+r)).symm (c,coins)) =
+      pairSwitch coins.2 * childLift (fun b => centralPalindromePerm L r
+        (fun u => C (Fin.cons b u)) (if b then c.2 else c.1)) * pairSwitch coins.1 := by
+  change ((butterflyPerm (L+r+1) (decodeButterfly (L+r+1)
+      ((coinStepEquiv (L+r)).symm ((c.1.1,c.2.1),coins.2))))) *
+    childLift (fun b => centralBlockPerm L r (fun u => C (Fin.cons b u))) *
+    (butterflyPerm (L+r+1) (decodeButterfly (L+r+1)
+      ((coinStepEquiv (L+r)).symm ((c.1.2,c.2.2),coins.1))))⁻¹ = _
+  rw [butterflyPerm_step,butterflyPerm_step,mul_inv_rev,childLift_inv,pairSwitch_inv]
+  simp only [mul_assoc]
+  rw [←mul_assoc (childLift _),childLift_mul,←mul_assoc (childLift _),childLift_mul]
+  congr 2
+  apply congrArg childLift
+  funext b
+  cases b <;> rfl
+
+noncomputable def centralLevelCost (t L : ℕ) : (r : ℕ) →
+    (Card r → Equiv.Perm (Card L)) → {ι : Type*} → [Fintype ι] →
+    (ι ↪ Card (L+r)) → BenesCoins (L+r) → ℕ
+  | 0, _, _, _, _, _ => 0
+  | r+1, C, _, _, e, ω =>
+    let σ := benesStepEquiv (L+r) ω
+    let x := e.trans (headTailEquiv (L+r)).toEmbedding
+    let c := PairRouting.colors x σ.2.1
+    let hc := PairRouting.colors_compatible x σ.2.1
+    if t = L+r+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
+      (fun b => centralPalindromePerm L r (fun u => C (Fin.cons b u)) (if b then σ.1.2 else σ.1.1))
+    else centralLevelCost t L r (fun u => C (Fin.cons false u))
+      (PairRouting.childEmbedding x c hc false) σ.1.1 +
+      centralLevelCost t L r (fun u => C (Fin.cons true u))
+      (PairRouting.childEmbedding x c hc true) σ.1.2
+
+noncomputable def centralSlabCost (l L r : ℕ) (C : Card r → Equiv.Perm (Card L))
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (ω : BenesCoins (L+r)) : ℕ :=
+  ∑ i : Fin l, centralLevelCost (l+i.val+1) L r C e ω
+
+lemma permDistance_childLift (d : ℕ) (p q : Bool → Equiv.Perm (Card d)) :
+    permDistance (childLift p) (childLift q) =
+      permDistance (p false) (q false)+permDistance (p true) (q true) := by
+  classical
+  rw [permDistance_sum]
+  have hh : (∑ x : Card (d+1), if childLift p x = childLift q x then 0 else 1) =
+      ∑ x : Bool × Card d, if p x.1 x.2 = q x.1 x.2 then 0 else 1 := by
+    apply Fintype.sum_equiv (headTailEquiv d)
+    intro x
+    have hx : (headTailEquiv d) x = (x 0,Fin.tail x) := rfl
+    rw [hx]
+    congr 1
+    apply propext
+    constructor
+    · intro h
+      have ht := congrArg Fin.tail h
+      simpa only [childLift,Equiv.coe_fn_mk,Fin.tail_cons] using ht
+    · intro h
+      dsimp only [childLift,Equiv.coe_fn_mk]
+      exact congrArg (fun z : Card d => (Fin.cons (x 0) z : Card (d+1))) h
+  rw [hh]
+  simp only [Fintype.sum_prod_type,Fintype.sum_bool,permDistance_sum]
+  omega
+
+noncomputable def centralDistance (L r : ℕ) (C D : Card r → Equiv.Perm (Card L)) : ℕ :=
+  ∑ u : Card r, permDistance (C u) (D u)
+
+lemma centralDistance_step (L r : ℕ) (C D : Card (r+1) → Equiv.Perm (Card L)) :
+    centralDistance L (r+1) C D =
+      centralDistance L r (fun u => C (Fin.cons false u)) (fun u => D (Fin.cons false u))+
+      centralDistance L r (fun u => C (Fin.cons true u)) (fun u => D (Fin.cons true u)) := by
+  classical
+  unfold centralDistance
+  rw [Fintype.sum_equiv (headTailEquiv r) _
+    (fun x : Bool × Card r => permDistance (C (Fin.cons x.1 x.2)) (D (Fin.cons x.1 x.2)))
+      (fun x => by
+        change permDistance (C x) (D x) =
+          permDistance (C (Fin.cons (x 0) (Fin.tail x))) (D (Fin.cons (x 0) (Fin.tail x)))
+        rw [Fin.cons_self_tail])]
+  simp only [Fintype.sum_prod_type,Fintype.sum_bool]
+  omega
+
+lemma permDistance_centralBlock (L r : ℕ) (C D : Card r → Equiv.Perm (Card L)) :
+    permDistance (centralBlockPerm L r C) (centralBlockPerm L r D) = centralDistance L r C D := by
+  induction r with
+  | zero =>
+    simp only [centralBlockPerm,centralDistance,Fintype.sum_unique]
+    congr 1
+  | succ r ih =>
+    rw [centralBlockPerm,centralBlockPerm,permDistance_childLift,centralDistance_step,ih,ih]
+
+lemma permDistance_centralPalindrome (L r : ℕ) (C D : Card r → Equiv.Perm (Card L))
+    (ω : BenesCoins (L+r)) :
+    permDistance (centralPalindromePerm L r C ω) (centralPalindromePerm L r D ω) =
+      centralDistance L r C D := by
+  rw [centralPalindromePerm,centralPalindromePerm,permDistance_mul_right,
+    permDistance_mul_left,permDistance_centralBlock]
+
+lemma centralLevelCost_step (t L r : ℕ) (C : Card (r+1) → Equiv.Perm (Card L))
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r+1))
+    (ω : BenesCoins (L+r) × BenesCoins (L+r))
+    (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
+    centralLevelCost t L (r+1) C e ((benesStepEquiv (L+r)).symm (ω,coins)) =
+      let x := e.trans (headTailEquiv (L+r)).toEmbedding
+      let c := PairRouting.colors x coins.1
+      let hc := PairRouting.colors_compatible x coins.1
+      if t = L+r+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x coins.1)
+        (fun b => centralPalindromePerm L r (fun u => C (Fin.cons b u)) (if b then ω.2 else ω.1))
+      else centralLevelCost t L r (fun u => C (Fin.cons false u))
+        (PairRouting.childEmbedding x c hc false) ω.1 +
+        centralLevelCost t L r (fun u => C (Fin.cons true u))
+        (PairRouting.childEmbedding x c hc true) ω.2 := by
+  rw [centralLevelCost]
+  rfl
+
+lemma centralLevelCost_perturb (t L r : ℕ) (C D : Card r → Equiv.Perm (Card L))
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (ω : BenesCoins (L+r)) :
+    centralLevelCost t L r C e ω ≤ centralLevelCost t L r D e ω+centralDistance L r C D := by
+  induction r generalizing ι with
+  | zero => simp only [centralLevelCost,Nat.zero_add,Nat.zero_le]
+  | succ r ih =>
+    rw [←(benesStepEquiv (L+r)).symm_apply_apply ω,centralLevelCost_step,centralLevelCost_step]
+    dsimp only
+    split_ifs with ht
+    · have hh := alternatingCycles_distance_le
+        (PairRouting.switchedEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
+          (benesStepEquiv (L+r) ω).2.1)
+        (fun b => centralPalindromePerm L r (fun u => C (Fin.cons b u))
+          (if b then (benesStepEquiv (L+r) ω).1.2 else (benesStepEquiv (L+r) ω).1.1))
+        (fun b => centralPalindromePerm L r (fun u => D (Fin.cons b u))
+          (if b then (benesStepEquiv (L+r) ω).1.2 else (benesStepEquiv (L+r) ω).1.1))
+      simpa only [Bool.false_eq_true,↓reduceIte,permDistance_centralPalindrome,←centralDistance_step] using hh
+    · rw [centralDistance_step]
+      have h₀ := ih (fun u => C (Fin.cons false u)) (fun u => D (Fin.cons false u))
+        (PairRouting.childEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
+          (PairRouting.colors (e.trans (headTailEquiv (L+r)).toEmbedding) (benesStepEquiv (L+r) ω).2.1)
+          (PairRouting.colors_compatible _ _) false)
+        (benesStepEquiv (L+r) ω).1.1
+      have h₁ := ih (fun u => C (Fin.cons true u)) (fun u => D (Fin.cons true u))
+        (PairRouting.childEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
+          (PairRouting.colors (e.trans (headTailEquiv (L+r)).toEmbedding) (benesStepEquiv (L+r) ω).2.1)
+          (PairRouting.colors_compatible _ _) true)
+        (benesStepEquiv (L+r) ω).1.2
+      omega
+
+lemma centralSlabCost_perturb (l L r : ℕ) (C D : Card r → Equiv.Perm (Card L))
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (ω : BenesCoins (L+r)) :
+    centralSlabCost l L r C e ω ≤ centralSlabCost l L r D e ω+l*centralDistance L r C D := by
+  have hh := Finset.sum_le_sum (fun (i : Fin l) (_ : i ∈ Finset.univ) =>
+    centralLevelCost_perturb (l+i.val+1) L r C D e ω)
+  simpa only [Finset.sum_add_distrib,Finset.sum_const,Finset.card_univ,Fintype.card_fin,
+    nsmul_eq_mul,Nat.cast_id,centralSlabCost] using hh
+
+end Thorp
+
+end ThorpNine.Tail
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Tail/InputTree.lean b/lean/OAI/Probability/ThorpRouting/Tail/InputTree.lean
index 07b425a7131ce490660d31c903ad590aab9bf91e..0ef5fe2863f0ae3e568f0b018087afaf3ce84690 100644
--- a/lean/OAI/Probability/ThorpRouting/Tail/InputTree.lean
+++ b/lean/OAI/Probability/ThorpRouting/Tail/InputTree.lean
@@ -1,884 +1,884 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Tail.HeightDecay
-
-namespace OAI
-
-namespace ThorpNine.Tail
-
-namespace Thorp
-
-lemma centralDistance_exceptional (L r : ℕ) (C D : Card r → Equiv.Perm (Card L))
-    (E : Finset (Card r)) (h : ∀ u ∉ E, C u = D u) :
-    centralDistance L r C D ≤ E.card*2^L := by
-  classical
-  calc
-    _ ≤ ∑ u : Card r, if u ∈ E then 2^L else 0 := by
-      apply Finset.sum_le_sum
-      intro u _
-      split_ifs with hu
-      · simpa only [card_positions] using permDistance_le (C u) (D u)
-      · rw [h u hu,permDistance_refl]
-    _ = _ := by simp
-
-
-inductive InputTree : ℕ → Type
-  | tip {d : ℕ} (p : Equiv.Perm (Card d)) : InputTree d
-  | node {d : ℕ} (ξ : Card d → Bool) (T : Bool → InputTree d) : InputTree (d+1)
-
-def InputTree.perm : {d : ℕ} → InputTree d → Equiv.Perm (Card d)
-  | _, .tip p => p
-  | _, .node ξ T => pairSwitch ξ * childLift (fun b => (T b).perm)
-
-noncomputable def InputTree.levelCost (t : ℕ) : {d : ℕ} → InputTree d →
-    {ι : Type*} → [Fintype ι] → (ι ↪ Card d) → (SwitchIndex d → Bool) → ℕ
-  | _, .tip _, _, _, _, _ => 0
-  | d+1, .node ξ T, _, _, e, Y =>
-    let x := e.trans (headTailEquiv d).toEmbedding
-    let c := PairRouting.colors x ξ
-    let hc := PairRouting.colors_compatible x ξ
-    let σ := coinStepEquiv d Y
-    if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x ξ)
-      (fun b => if b then butterflyPerm d (decodeButterfly d σ.1.2) * (T true).perm⁻¹
-        else butterflyPerm d (decodeButterfly d σ.1.1) * (T false).perm⁻¹)
-    else (T false).levelCost t (PairRouting.childEmbedding x c hc false) σ.1.1 +
-      (T true).levelCost t (PairRouting.childEmbedding x c hc true) σ.1.2
-
-lemma InputTree.levelCost_node (t d : ℕ) (ξ : Card d → Bool) (T : Bool → InputTree d)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (d+1))
-    (Y : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (η : Card d → Bool) :
-    (InputTree.node ξ T).levelCost t e ((coinStepEquiv d).symm (Y,η)) =
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x ξ
-      let hc := PairRouting.colors_compatible x ξ
-      if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x ξ)
-        (fun b => if b then butterflyPerm d (decodeButterfly d Y.2) * (T true).perm⁻¹
-          else butterflyPerm d (decodeButterfly d Y.1) * (T false).perm⁻¹)
-      else (T false).levelCost t (PairRouting.childEmbedding x c hc false) Y.1 +
-        (T true).levelCost t (PairRouting.childEmbedding x c hc true) Y.2 := by
-  rw [InputTree.levelCost]
-  rfl
-
-lemma InputTree.levelCost_gt (t d : ℕ) (T : InputTree d) (ht : d < t)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) :
-    T.levelCost t e Y = 0 := by
-  induction T generalizing ι with
-  | tip p => rfl
-  | @node d ξ T ih =>
-    rw [InputTree.levelCost,ite_eq_right (by omega : t ≠ d+1),ih false (by omega),ih true (by omega)]
-
-lemma InputTree.levelCost_adapted (t d : ℕ) (T : InputTree d)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) :
-    LayerAdapted d (t-1) (fun Y => T.levelCost t e Y) := by
-  induction T generalizing ι with
-  | tip p => intro Y Y' _; rfl
-  | @node d ξ T ih =>
-    intro Y Y' h
-    dsimp only
-    conv_lhs => rw [←(coinStepEquiv d).symm_apply_apply Y]
-    conv_rhs => rw [←(coinStepEquiv d).symm_apply_apply Y']
-    rw [InputTree.levelCost_node t d ξ T e ((coinStepEquiv d Y).1) ((coinStepEquiv d Y).2),
-      InputTree.levelCost_node t d ξ T e ((coinStepEquiv d Y').1) ((coinStepEquiv d Y').2)]
-    by_cases ht : t = d+1
-    · subst t
-      rw [ite_eq_left rfl,ite_eq_left rfl]
-      have he₀ : (coinStepEquiv d Y).1.1 = (coinStepEquiv d Y').1.1 := by
-        funext i
-        exact h (Sum.inr (false,i)) (by simpa only [switchHeight,Nat.add_sub_cancel] using switchHeight_le d i)
-      have he₁ : (coinStepEquiv d Y).1.2 = (coinStepEquiv d Y').1.2 := by
-        funext i
-        exact h (Sum.inr (true,i)) (by simpa only [switchHeight,Nat.add_sub_cancel] using switchHeight_le d i)
-      rw [he₀,he₁]
-    · rw [ite_eq_right ht,ite_eq_right ht]
-      congr 1
-      · apply ih false
-        intro i hi
-        exact h (Sum.inr (false,i)) hi
-      · apply ih true
-        intro i hi
-        exact h (Sum.inr (true,i)) hi
-
-theorem InputTree.level_partial_mgf (r s u : ℕ) (hu : u < r)
-    (T : InputTree (s+r)) {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+r))
-    (lo : Card r × SwitchIndex s → Bool) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(T.levelCost (s+u+1) e (assembleBits r s lo hi))) ≤
-      Real.exp (Fintype.card ι * cycleCoefficient u J q) := by
-  classical
-  induction r generalizing u ι with
-  | zero => omega
-  | succ r ih =>
-    simp only [Nat.add_succ] at *
-    cases T with
-    | tip p =>
-      simp only [InputTree.levelCost,pow_zero,finiteMean_const]
-      exact Real.one_le_exp (mul_nonneg (Nat.cast_nonneg _) (cycleCoefficient_nonneg u J q hq))
-    | node ξ T =>
-      let x := e.trans (headTailEquiv (s+r)).toEmbedding
-      let c := PairRouting.colors x ξ
-      let hc := PairRouting.colors_compatible x ξ
-      let e₀ := PairRouting.childEmbedding x c hc false
-      let e₁ := PairRouting.childEmbedding x c hc true
-      let e' := PairRouting.switchedEmbedding x ξ
-      let lo₀ : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons false j.1,j.2)
-      let lo₁ : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons true j.1,j.2)
-      let Y₀ := fun hi : Card s × SwitchIndex r → Bool => assembleBits r s lo₀ hi
-      let Y₁ := fun hi : Card s × SwitchIndex r → Bool => assembleBits r s lo₁ hi
-      let N := fun hi : (Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool) =>
-        PairRouting.alternatingCycles e' (fun b =>
-          if b then butterflyPerm (s+r) (decodeButterfly (s+r) (Y₁ hi.2)) * (T true).perm⁻¹
-          else butterflyPerm (s+r) (decodeButterfly (s+r) (Y₀ hi.1)) * (T false).perm⁻¹)
-      let L₀ := fun hi : Card s × SwitchIndex r → Bool =>
-        (T false).levelCost (s+u+1) e₀ (Y₀ hi)
-      let L₁ := fun hi : Card s × SwitchIndex r → Bool =>
-        (T true).levelCost (s+u+1) e₁ (Y₁ hi)
-      have hs (hi : (Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool))
-          (η : Card s × Card r → Bool) :
-          (InputTree.node ξ T).levelCost (s+u+1) e
-            (assembleBits (r+1) s lo ((highStepEquiv r s).symm (hi,η))) =
-            if u = r then N hi else L₀ hi.1 + L₁ hi.2 := by
-        rw [assembleBits_step]
-        rw [InputTree.levelCost_node (s+u+1) (s+r) ξ T e (Y₀ hi.1,Y₁ hi.2) (fun y => η (((cardSplit r s) y).2,((cardSplit r s) y).1))]
-        simp only [Nat.add_right_cancel_iff,Nat.add_left_cancel_iff]
-        rfl
-      rw [finiteMean_equiv (highStepEquiv r s),finiteMean_prod]
-      simp_rw [hs,finiteMean_const]
-      by_cases hur : u = r
-      · subst u
-        simp only [ite_true]
-        rw [finiteMean_prod,finiteMean_comm]
-        calc
-          _ ≤ finiteMean (fun _hi : Card s × SwitchIndex r → Bool =>
-            Real.exp (Fintype.card ι * cycleCoefficient r J q)) := by
-            apply finiteMean_mono
-            intro hi
-            have hn := node_partial_mgf r s e' (T false).perm (T true).perm
-              (butterflyPerm (s+r) (decodeButterfly (s+r) (Y₁ hi))) lo₀ J hJ q hq
-            exact hn
-          _ = _ := finiteMean_const _
-      · simp only [ite_eq_right hur,pow_add]
-        rw [finiteMean_prod_mul (fun hi => q ^ L₀ hi) (fun hi => q ^ L₁ hi)]
-        have hu' : u < r := by omega
-        have h₀ := ih u hu' (T false) e₀ lo₀
-        have h₁ := ih u hu' (T true) e₁ lo₁
-        have hp := mul_le_mul h₀ h₁ (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
-          (Real.exp_nonneg _)
-        refine hp.trans_eq ?_
-        rw [←Real.exp_add]
-        congr 1
-        have hcount := PairRouting.color_card_add c
-        have hcount' : (Fintype.card {i : ι // c i = false} : ℝ) +
-            Fintype.card {i : ι // c i = true} = Fintype.card ι := by exact_mod_cast hcount
-        change (Fintype.card {i : ι // c i = false} : ℝ) * cycleCoefficient u J q +
-            Fintype.card {i : ι // c i = true} * cycleCoefficient u J q = _
-        rw [←add_mul,hcount']
-
-
-noncomputable def InputTree.slabCost (l d : ℕ) (T : InputTree d) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) : ℕ :=
-  ∑ i : Fin l, T.levelCost (l+i.1+1) e Y
-
-lemma InputTree.slabCost_adapted (l d : ℕ) (T : InputTree d) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) :
-    LayerAdapted d (2*l) (fun Y => InputTree.slabCost l d T e Y) := by
-  apply layerAdapted_sum
-  intro i _
-  exact layerAdapted_mono (InputTree.levelCost_adapted (l+i.1+1) d T e) (by omega)
-
-theorem InputTree.slab_partial_mgf (r s l : ℕ) (hl : 0 < l) (hs : 2*s ≤ l) (T : InputTree (s+r))
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+r))
-    (lo : Card r × SwitchIndex s → Bool) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(InputTree.slabCost l (s+r) T e (assembleBits r s lo hi))) ≤
-      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
-  classical
-  let : NeZero l := ⟨ne_of_gt hl⟩
-  have hqp : 0 < q := lt_of_lt_of_le (by norm_num) hq
-  calc
-    _ ≤ finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      finiteMean (fun i : Fin l => q^(l*T.levelCost (l+i.1+1) e
-        (assembleBits r s lo hi)))) := by
-      apply finiteMean_mono
-      intro hi
-      simpa only [InputTree.slabCost,Fintype.card_fin] using
-        pow_sum_le_mean_powers (fun i : Fin l => T.levelCost (l+i.1+1) e
-          (assembleBits r s lo hi)) q hqp
-    _ = finiteMean (fun i : Fin l => finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(l*T.levelCost (l+i.1+1) e (assembleBits r s lo hi)))) := finiteMean_comm _
-    _ ≤ finiteMean (fun _i : Fin l =>
-      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64)))) := by
-      apply finiteMean_mono
-      intro i
-      by_cases ht : l+i.1+1 ≤ s+r
-      · have hs' : s ≤ l+i.1 := by omega
-        have he : l+i.1+1 = s+(l+i.1-s)+1 := by omega
-        have hu : l+i.1-s < r := by omega
-        have hm := InputTree.level_partial_mgf r s (l+i.1-s) hu T e lo
-          ⌈Real.exp ((l:ℝ)/32)⌉₊ (cycleCutoff_pos l) (q^l) (one_le_pow₀ hq)
-        have hc := cycleCoefficient_decay l (l+i.1-s) (by
-          rw [Nat.cast_sub hs',Nat.cast_add]
-          have hsr : (2:ℝ)*s ≤ l := by exact_mod_cast hs
-          have hi : (0:ℝ) ≤ i.1 := Nat.cast_nonneg _
-          linarith) (q^l) (one_le_pow₀ hq) (by
-            rw [Real.log_pow]
-            nlinarith [mul_le_mul_of_nonneg_left hlog (Nat.cast_nonneg l)])
-        simp_rw [he,pow_mul]
-        exact hm.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left hc (Nat.cast_nonneg _)))
-      · have hz := InputTree.levelCost_gt (l+i.1+1) (s+r) T (by omega) e
-        simp only [hz,Nat.mul_zero,pow_zero,finiteMean_const]
-        exact Real.one_le_exp_iff.mpr (by positivity)
-    _ = _ := finiteMean_const _
-
-
-def boundaryTree (L : ℕ) : (r : ℕ) → (Card r → Equiv.Perm (Card L)) →
-    (SwitchIndex (L+r) → Bool) → InputTree (L+r)
-  | 0, U, _ => .tip (U (fun j => Fin.elim0 j))
-  | r+1, U, X => .node (coinStepEquiv (L+r) X).2 (fun b =>
-      boundaryTree L r (fun u => U (Fin.cons b u))
-        (if b then (coinStepEquiv (L+r) X).1.2 else (coinStepEquiv (L+r) X).1.1))
-
-def boundaryOutput (L : ℕ) : (r : ℕ) → (SwitchIndex (L+r) → Bool) →
-    Card r → Equiv.Perm (Card L)
-  | 0, Y, _ => butterflyPerm L (decodeButterfly L Y)
-  | r+1, Y, u => boundaryOutput L r
-      (if u 0 then (coinStepEquiv (L+r) Y).1.2 else (coinStepEquiv (L+r) Y).1.1) (Fin.tail u)
-
-lemma boundaryOutput_adapted (L r : ℕ) :
-    LayerAdapted (L+r) L (boundaryOutput L r) := by
-  induction r with
-  | zero =>
-    intro Y Y' h
-    have he : Y = Y' := funext (fun i => h i (by simpa only [Nat.add_zero] using switchHeight_le L i))
-    rw [he]
-  | succ r ih =>
-    intro Y Y' h
-    funext u
-    dsimp only [boundaryOutput]
-    cases hb : u 0 <;> simp only [Bool.false_eq_true,↓reduceIte]
-    · exact congrFun (ih _ _ (fun i hi => h (Sum.inr (false,i)) hi)) (Fin.tail u)
-    · exact congrFun (ih _ _ (fun i hi => h (Sum.inr (true,i)) hi)) (Fin.tail u)
-
-noncomputable def modifiedBoundary (L r : ℕ) (E : Finset (Card r))
-    (U : Card r → Equiv.Perm (Card L)) (Y : SwitchIndex (L+r) → Bool) :
-    Card r → Equiv.Perm (Card L) := fun u => if u ∈ E then boundaryOutput L r Y u else U u
-
-lemma modifiedBoundary_adapted (L r : ℕ) (E : Finset (Card r))
-    (U : Card r → Equiv.Perm (Card L)) :
-    LayerAdapted (L+r) L (modifiedBoundary L r E U) := by
-  intro Y Y' h
-  unfold modifiedBoundary
-  rw [boundaryOutput_adapted L r Y Y' h]
-
-lemma permDistance_boundaryTree (L r : ℕ) (U V : Card r → Equiv.Perm (Card L))
-    (X : SwitchIndex (L+r) → Bool) :
-    permDistance (boundaryTree L r U X).perm (boundaryTree L r V X).perm =
-      centralDistance L r U V := by
-  induction r with
-  | zero =>
-    simp only [boundaryTree,InputTree.perm,centralDistance,Fintype.sum_unique]
-    congr 1
-  | succ r ih =>
-    simp only [boundaryTree,InputTree.perm,permDistance_mul_left]
-    rw [permDistance_childLift,centralDistance_step,ih,ih]
-
-lemma boundaryTree_levelCost_perturb (t L r : ℕ) (U V : Card r → Equiv.Perm (Card L))
-    (X : SwitchIndex (L+r) → Bool) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (L+r)) (Y : SwitchIndex (L+r) → Bool) :
-    (boundaryTree L r U X).levelCost t e Y ≤
-      (boundaryTree L r V X).levelCost t e Y + centralDistance L r U V := by
-  induction r generalizing ι with
-  | zero => simp only [boundaryTree,InputTree.levelCost,Nat.zero_add,Nat.zero_le]
-  | succ r ih =>
-    rw [boundaryTree,boundaryTree,InputTree.levelCost,InputTree.levelCost]
-    by_cases ht : t = L+r+1
-    · simp only [ite_eq_left ht]
-      have hh := alternatingCycles_distance_le
-        (PairRouting.switchedEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
-          (coinStepEquiv (L+r) X).2)
-        (fun b => if b then butterflyPerm (L+r) (decodeButterfly (L+r) (coinStepEquiv (L+r) Y).1.2) *
-            (boundaryTree L r (fun u => U (Fin.cons true u)) (coinStepEquiv (L+r) X).1.2).perm⁻¹
-          else butterflyPerm (L+r) (decodeButterfly (L+r) (coinStepEquiv (L+r) Y).1.1) *
-            (boundaryTree L r (fun u => U (Fin.cons false u)) (coinStepEquiv (L+r) X).1.1).perm⁻¹)
-        (fun b => if b then butterflyPerm (L+r) (decodeButterfly (L+r) (coinStepEquiv (L+r) Y).1.2) *
-            (boundaryTree L r (fun u => V (Fin.cons true u)) (coinStepEquiv (L+r) X).1.2).perm⁻¹
-          else butterflyPerm (L+r) (decodeButterfly (L+r) (coinStepEquiv (L+r) Y).1.1) *
-            (boundaryTree L r (fun u => V (Fin.cons false u)) (coinStepEquiv (L+r) X).1.1).perm⁻¹)
-      simpa only [Bool.false_eq_true,↓reduceIte,permDistance_mul_left,permDistance_inv,
-        permDistance_boundaryTree,←centralDistance_step] using hh
-    · simp only [ite_eq_right ht,Bool.false_eq_true,↓reduceIte]
-      rw [centralDistance_step]
-      have h₀ := ih (fun u => U (Fin.cons false u)) (fun u => V (Fin.cons false u))
-        (coinStepEquiv (L+r) X).1.1
-        (PairRouting.childEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
-          (PairRouting.colors (e.trans (headTailEquiv (L+r)).toEmbedding) (coinStepEquiv (L+r) X).2)
-          (PairRouting.colors_compatible _ _) false) (coinStepEquiv (L+r) Y).1.1
-      have h₁ := ih (fun u => U (Fin.cons true u)) (fun u => V (Fin.cons true u))
-        (coinStepEquiv (L+r) X).1.2
-        (PairRouting.childEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
-          (PairRouting.colors (e.trans (headTailEquiv (L+r)).toEmbedding) (coinStepEquiv (L+r) X).2)
-          (PairRouting.colors_compatible _ _) true) (coinStepEquiv (L+r) Y).1.2
-      omega
-
-lemma boundaryTree_slabCost_perturb (l L r : ℕ) (U V : Card r → Equiv.Perm (Card L))
-    (X : SwitchIndex (L+r) → Bool) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (L+r)) (Y : SwitchIndex (L+r) → Bool) :
-    InputTree.slabCost l (L+r) (boundaryTree L r U X) e Y ≤
-      InputTree.slabCost l (L+r) (boundaryTree L r V X) e Y + l*centralDistance L r U V := by
-  have hh := Finset.sum_le_sum (fun (i : Fin l) (_ : i ∈ Finset.univ) =>
-    boundaryTree_levelCost_perturb (l+i.val+1) L r U V X e Y)
-  simpa only [Finset.sum_add_distrib,Finset.sum_const,Finset.card_univ,Fintype.card_fin,
-    nsmul_eq_mul,Nat.cast_id,InputTree.slabCost] using hh
-
-lemma modifiedBoundary_distance_le (L r : ℕ) (E : Finset (Card r))
-    (U : Card r → Equiv.Perm (Card L)) (Y : SwitchIndex (L+r) → Bool) :
-    centralDistance L r (modifiedBoundary L r E U Y) U ≤ E.card*2^L := by
-  apply centralDistance_exceptional
-  intro u hu
-  exact ite_eq_right hu
-
-
-lemma InputTree.slabCost_gt (l d : ℕ) (T : InputTree d) (hd : d ≤ l)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) :
-    InputTree.slabCost l d T e Y = 0 := by
-  apply Finset.sum_eq_zero
-  intro i _
-  exact InputTree.levelCost_gt _ _ T (by omega) _ _
-
-lemma InputTree.random_slab_adapted_mul_le (d s l : ℕ) (hsd : s ≤ d) (hl : 0 < l)
-    (hs : 2*s ≤ l) (T : (SwitchIndex d → Bool) → InputTree d)
-    (hT : LayerAdapted d s T) {ι : Type*} [Fintype ι] (e : ι ↪ Card d)
-    (F : (SwitchIndex d → Bool) → ℝ) (hF : ∀ Y, 0 ≤ F Y) (hA : LayerAdapted d s F)
-    (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun Y => F Y*q^(InputTree.slabCost l d (T Y) e Y)) ≤
-      finiteMean F * Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
-  obtain ⟨r,rfl⟩ := Nat.exists_eq_add_of_le hsd
-  apply adapted_mul_mean_le r s F _ hF hA
-  intro lo
-  let hi₀ : Card s × SwitchIndex r → Bool := fun _ => false
-  have he (hi : Card s × SwitchIndex r → Bool) :
-      T (assembleBits r s lo hi) = T (assembleBits r s lo hi₀) :=
-    hT _ _ (assembleBits_agree r s lo hi hi₀)
-  simp_rw [he]
-  exact InputTree.slab_partial_mgf r s l hl hs _ e lo q hq hlog
-
-lemma InputTree.slab_mgf (l d : ℕ) (hl : 0 < l) (T : InputTree d)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun Y => q^(InputTree.slabCost l d T e Y)) ≤
-      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
-  have hh := InputTree.random_slab_adapted_mul_le d 0 l (Nat.zero_le _) hl (Nat.zero_le _)
-    (fun _ => T) (fun _ _ _ => rfl) e (fun _ => 1) (fun _ => by norm_num)
-    (fun _ _ _ => rfl) q hq hlog
-  simpa only [one_mul,finiteMean_const] using hh
-
-noncomputable def InputTree.familyCost (L n d : ℕ)
-    (T : (SwitchIndex d → Bool) → InputTree d) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) : ℕ :=
-  ∑ j ∈ Finset.range n, InputTree.slabCost (L*4^j) d (T Y) e Y
-
-lemma InputTree.random_slabCost_adapted (l d s : ℕ) (T : (SwitchIndex d → Bool) → InputTree d)
-    (hT : LayerAdapted d s T) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) :
-    LayerAdapted d (max s (2*l)) (fun Y => InputTree.slabCost l d (T Y) e Y) := by
-  intro Y Y' h
-  dsimp only
-  rw [hT _ _ (agreeThrough_mono (le_max_left _ _) h)]
-  exact InputTree.slabCost_adapted l d (T Y') e Y Y'
-    (agreeThrough_mono (le_max_right _ _) h)
-
-lemma InputTree.familyCost_adapted (L n d s : ℕ) (T : (SwitchIndex d → Bool) → InputTree d)
-    (hT : LayerAdapted d s T) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) :
-    LayerAdapted d (max s (familyEnd L n)) (fun Y => InputTree.familyCost L n d T e Y) := by
-  apply layerAdapted_sum
-  intro j hj
-  exact layerAdapted_mono (InputTree.random_slabCost_adapted (L*4^j) d s T hT e)
-    (max_le_max_left s (familyEnd_ge L n j (Finset.mem_range.mp hj)))
-
-theorem InputTree.family_mgf_first (L n d s : ℕ) (hL : 0 < L) (hs : s ≤ L)
-    (T : (SwitchIndex d → Bool) → InputTree d) (hT : LayerAdapted d s T)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) (p : ℝ) (hp : 0 ≤ p)
-    (hfirst : finiteMean (fun Y => q^(InputTree.slabCost L d (T Y) e Y)) ≤
-      Real.exp (p+Fintype.card ι*(68*Real.exp (-(L:ℝ)/64)))) :
-    finiteMean (fun Y => q^(InputTree.familyCost L n d T e Y)) ≤
-      Real.exp (p+Fintype.card ι*(68*∑ j ∈ Finset.range n,
-        Real.exp (-((L*4^j:ℕ):ℝ)/64))) := by
-  induction n with
-  | zero =>
-    simp only [InputTree.familyCost,Finset.range_zero,Finset.sum_empty,pow_zero,
-      finiteMean_const,mul_zero,add_zero]
-    exact Real.one_le_exp hp
-  | succ n ih =>
-    by_cases hn : n = 0
-    · subst n
-      simpa only [InputTree.familyCost,Nat.zero_add,Finset.sum_range_one,pow_zero,mul_one] using hfirst
-    · let l := L*4^n
-      have hl : 0 < l := Nat.mul_pos hL (pow_pos (by decide) _)
-      have hsl : 2*familyEnd L n ≤ l := familyEnd_two L n
-      have hs' : s ≤ familyEnd L n := by
-        have hh := familyEnd_ge L n 0 (Nat.pos_of_ne_zero hn)
-        simp only [pow_zero,mul_one] at hh
-        omega
-      have hc : ∀ Y, InputTree.familyCost L (n+1) d T e Y =
-          InputTree.familyCost L n d T e Y+InputTree.slabCost l d (T Y) e Y := by
-        intro Y
-        exact Finset.sum_range_succ _ _
-      have hm : finiteMean (fun Y => q^(InputTree.familyCost L (n+1) d T e Y)) ≤
-          finiteMean (fun Y => q^(InputTree.familyCost L n d T e Y)) *
-            Real.exp (Fintype.card ι*(68*Real.exp (-(l:ℝ)/64))) := by
-        by_cases hld : l ≤ d
-        · simp_rw [hc,pow_add]
-          apply InputTree.random_slab_adapted_mul_le d (familyEnd L n) l (by omega) hl hsl T
-            (layerAdapted_mono hT hs') e _ (fun _ => pow_nonneg (by linarith) _)
-          · apply layerAdapted_comp
-            exact layerAdapted_mono (InputTree.familyCost_adapted L n d s T hT e) (max_le hs' le_rfl)
-          · exact hq
-          · exact hlog
-        · simp_rw [hc,InputTree.slabCost_gt l d _ (by omega),Nat.add_zero]
-          exact le_mul_of_one_le_right
-            (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
-            (Real.one_le_exp_iff.mpr (by positivity))
-      refine (hm.trans (mul_le_mul_of_nonneg_right ih (Real.exp_nonneg _))).trans_eq ?_
-      rw [←Real.exp_add,Finset.sum_range_succ]
-      congr 1
-      dsimp only [l]
-      ring
-
-lemma InputTree.family_mgf_first_decay (L n d s : ℕ) (hL : 0 < L) (hs : s ≤ L)
-    (T : (SwitchIndex d → Bool) → InputTree d) (hT : LayerAdapted d s T)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) (p : ℝ) (hp : 0 ≤ p)
-    (hfirst : finiteMean (fun Y => q^(InputTree.slabCost L d (T Y) e Y)) ≤
-      Real.exp (p+Fintype.card ι*(68*Real.exp (-(L:ℝ)/64)))) :
-    finiteMean (fun Y => q^(InputTree.familyCost L n d T e Y)) ≤
-      Real.exp (p+Fintype.card ι*heightDecay L) := by
-  apply (InputTree.family_mgf_first L n d s hL hs T hT e q hq hlog p hp hfirst).trans
-  apply Real.exp_le_exp.mpr
-  apply add_le_add_right
-  apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
-  have hh := mul_le_mul_of_nonneg_left (sum_family_decay L n hL) (by norm_num : (0:ℝ)≤68)
-  simpa only [heightDecay,mul_div_assoc] using hh
-
-
-noncomputable def modifiedInputTree (L r : ℕ) (E : Finset (Card r))
-    (U : Card r → Equiv.Perm (Card L)) (X Y : SwitchIndex (L+r) → Bool) : InputTree (L+r) :=
-  boundaryTree L r (modifiedBoundary L r E U Y) X
-
-lemma modifiedInputTree_adapted (L r : ℕ) (E : Finset (Card r))
-    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool) :
-    LayerAdapted (L+r) L (modifiedInputTree L r E U X) := by
-  intro Y Y' h
-  unfold modifiedInputTree
-  rw [modifiedBoundary_adapted L r E U Y Y' h]
-
-lemma modified_slab_mgf (l L r : ℕ) (hl : 0 < l) (E : Finset (Card r))
-    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun Y => q^(InputTree.slabCost l (L+r) (modifiedInputTree L r E U X Y) e Y)) ≤
-      Real.exp (((l*E.card*2^L:ℕ):ℝ)*Real.log q+
-        Fintype.card ι*(68*Real.exp (-(l:ℝ)/64))) := by
-  have hqp : 0 < q := by linarith
-  have hpoint (Y : SwitchIndex (L+r) → Bool) :
-      InputTree.slabCost l (L+r) (modifiedInputTree L r E U X Y) e Y ≤
-        InputTree.slabCost l (L+r) (boundaryTree L r U X) e Y+l*E.card*2^L := by
-    apply (boundaryTree_slabCost_perturb l L r (modifiedBoundary L r E U Y) U X e Y).trans
-    apply Nat.add_le_add_left
-    simpa only [Nat.mul_assoc] using Nat.mul_le_mul_left l (modifiedBoundary_distance_le L r E U Y)
-  calc
-    _ ≤ finiteMean (fun Y => q^(InputTree.slabCost l (L+r) (boundaryTree L r U X) e Y+l*E.card*2^L)) :=
-      finiteMean_mono (fun Y => pow_le_pow_right₀ hq (hpoint Y))
-    _ = finiteMean (fun Y => q^(InputTree.slabCost l (L+r) (boundaryTree L r U X) e Y)) *
-        q^(l*E.card*2^L) := by simp_rw [pow_add]; exact finiteMean_mul_const _ _
-    _ ≤ Real.exp (Fintype.card ι*(68*Real.exp (-(l:ℝ)/64))) * q^(l*E.card*2^L) :=
-      mul_le_mul_of_nonneg_right (InputTree.slab_mgf l (L+r) hl _ e q hq hlog) (pow_nonneg (by linarith) _)
-    _ = _ := by
-      have he : Real.exp (((l*E.card*2^L:ℕ):ℝ)*Real.log q) = q^(l*E.card*2^L) := by
-        rw [Real.exp_nat_mul,Real.exp_log hqp]
-      rw [Real.exp_add,he,mul_comm]
-
-lemma modified_family_mgf (l n L r : ℕ) (hl : 0 < l) (hLl : L ≤ l) (E : Finset (Card r))
-    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun Y => q^(InputTree.familyCost l n (L+r) (modifiedInputTree L r E U X) e Y)) ≤
-      Real.exp (((l*E.card*2^L:ℕ):ℝ)*Real.log q+Fintype.card ι*heightDecay l) := by
-  exact InputTree.family_mgf_first_decay l n (L+r) L hl hLl _
-    (modifiedInputTree_adapted L r E U X) e q hq hlog _
-    (mul_nonneg (Nat.cast_nonneg _) (Real.log_nonneg hq))
-    (modified_slab_mgf l L r hl E U X e q hq hlog)
-
-theorem modified_high_mgf (L n r : ℕ) (hL : 0 < L) (E : Finset (Card r))
-    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/500) :
-    finiteMean (fun Y => q^(InputTree.familyCost L n (L+r) (modifiedInputTree L r E U X) e Y+
-      InputTree.familyCost (2*L) n (L+r) (modifiedInputTree L r E U X) e Y)) ≤
-      Real.exp (Fintype.card ι*heightDecay L +
-        3*((L*E.card*2^L:ℕ):ℝ)*Real.log q) := by
-  have hq2 := one_le_pow₀ (n:=2) hq
-  have hlog2 : Real.log (q^2) ≤ 1/250 := by rw [Real.log_pow]; norm_num; linarith
-  have h₀ := modified_family_mgf L n L r hL le_rfl E U X e (q^2) hq2 hlog2
-  have h₁ := modified_family_mgf (2*L) n L r (by omega) (by omega) E U X e (q^2) hq2 hlog2
-  have h₁' : finiteMean (fun Y => (q^2)^(InputTree.familyCost (2*L) n (L+r)
-      (modifiedInputTree L r E U X) e Y)) ≤
-      Real.exp ((((2*L)*E.card*2^L:ℕ):ℝ)*Real.log (q^2)+Fintype.card ι*heightDecay L) := by
-    apply h₁.trans
-    apply Real.exp_le_exp.mpr
-    exact add_le_add_right (mul_le_mul_of_nonneg_left
-      (heightDecay_antitone (show L ≤ 2*L by omega)) (Nat.cast_nonneg _)) _
-  have hh := finiteMean_mul_exp_le
-    (fun Y => q^(InputTree.familyCost L n (L+r) (modifiedInputTree L r E U X) e Y))
-    (fun Y => q^(InputTree.familyCost (2*L) n (L+r) (modifiedInputTree L r E U X) e Y))
-    _ _ (by simpa only [pow_right_comm] using h₀) (by simpa only [pow_right_comm] using h₁')
-  simp only [←pow_add] at hh
-  refine hh.trans_eq ?_
-  congr 1
-  simp only [Real.log_pow,Nat.cast_mul,Nat.cast_ofNat]
-  ring
-
-noncomputable def InputTree.lowCost (H d : ℕ) (T : InputTree d)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) : ℕ :=
-  ∑ i ∈ Finset.range H, T.levelCost (i+1) e Y
-
-lemma InputTree.lowCost_slab (L d : ℕ) (T : InputTree d)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) :
-    InputTree.lowCost L d T e Y+InputTree.slabCost L d T e Y = InputTree.lowCost (2*L) d T e Y := by
-  rw [InputTree.lowCost,InputTree.lowCost,show 2*L=L+L by omega,Finset.sum_range_add]
-  congr 1
-  exact Fin.sum_univ_eq_sum_range (fun i => T.levelCost (L+i+1) e Y) L
-
-lemma InputTree.cost_family (L n d : ℕ) (T : (SwitchIndex d → Bool) → InputTree d)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) :
-    InputTree.lowCost L d (T Y) e Y + InputTree.familyCost L n d T e Y +
-      InputTree.familyCost (2*L) n d T e Y = InputTree.lowCost (L*4^n) d (T Y) e Y := by
-  induction n with
-  | zero => simp only [InputTree.familyCost,Finset.range_zero,Finset.sum_empty,Nat.add_zero,
-      pow_zero,mul_one]
-  | succ n ih =>
-    have he : (2*L)*4^n = 2*(L*4^n) := by ring
-    have he' : L*4^(n+1) = 2*(2*(L*4^n)) := by rw [pow_succ]; ring
-    simp only [InputTree.familyCost,Finset.sum_range_succ] at ih ⊢
-    rw [he,he',←InputTree.lowCost_slab,←InputTree.lowCost_slab,←ih]
-    omega
-
-lemma InputTree.lowCost_ge (H d : ℕ) (hd : d ≤ H) (T : InputTree d)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) :
-    InputTree.lowCost H d T e Y = InputTree.lowCost d d T e Y := by
-  symm
-  apply Finset.sum_subset (Finset.range_mono hd)
-  intro i _ hi
-  have hi' : d ≤ i := by simpa only [Finset.mem_range,not_lt] using hi
-  exact InputTree.levelCost_gt (i+1) d T (by omega) e Y
-
-lemma boundaryTree_levelCost_low (t L r : ℕ) (ht : t ≤ L)
-    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (Y : SwitchIndex (L+r) → Bool) :
-    (boundaryTree L r U X).levelCost t e Y = 0 := by
-  induction r generalizing ι with
-  | zero => rfl
-  | succ r ih =>
-    rw [boundaryTree,InputTree.levelCost,ite_eq_right (by omega : t ≠ L+r+1),ih,ih]
-
-lemma boundaryTree_lowCost_zero (L r : ℕ) (U : Card r → Equiv.Perm (Card L))
-    (X : SwitchIndex (L+r) → Bool) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (L+r)) (Y : SwitchIndex (L+r) → Bool) :
-    InputTree.lowCost L (L+r) (boundaryTree L r U X) e Y = 0 := by
-  apply Finset.sum_eq_zero
-  intro i hi
-  exact boundaryTree_levelCost_low (i+1) L r (by simpa only [Finset.mem_range,Nat.lt_iff_add_one_le] using hi) U X e Y
-
-theorem modified_cost_mgf (L r : ℕ) (hL : 0 < L) (E : Finset (Card r))
-    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/500) :
-    finiteMean (fun Y => q^(InputTree.lowCost (L+r) (L+r) (modifiedInputTree L r E U X Y) e Y)) ≤
-      Real.exp (Fintype.card ι*heightDecay L +
-        3*((L*E.card*2^L:ℕ):ℝ)*Real.log q) := by
-  have hbound : L+r ≤ L*4^(L+r) := by
-    have h₀ := pow_four_ge (L+r)
-    have h₁ := Nat.mul_le_mul_right (4^(L+r)) hL
-    nlinarith
-  have he (Y : SwitchIndex (L+r) → Bool) :=
-    InputTree.cost_family L (L+r) (L+r) (modifiedInputTree L r E U X) e Y
-  have hz (Y : SwitchIndex (L+r) → Bool) :
-      InputTree.lowCost L (L+r) (modifiedInputTree L r E U X Y) e Y = 0 :=
-    boundaryTree_lowCost_zero L r _ X e Y
-  simp only [hz,Nat.zero_add,InputTree.lowCost_ge _ _ hbound] at he
-  simp_rw [←he]
-  exact modified_high_mgf L (L+r) r hL E U X e q hq hlog
-
-
-noncomputable def exceptionChild {r : ℕ} (E : Finset (Card (r+1))) (b : Bool) : Finset (Card r) :=
-  Finset.univ.filter (fun u => Fin.cons b u ∈ E)
-
-lemma mem_exceptionChild {r : ℕ} (E : Finset (Card (r+1))) (b : Bool) (u : Card r) :
-    u ∈ exceptionChild E b ↔ Fin.cons b u ∈ E := by
-  classical
-  simp only [exceptionChild,Finset.mem_filter,Finset.mem_univ,true_and]
-
-lemma exceptionChild_card_add {r : ℕ} (E : Finset (Card (r+1))) :
-    (exceptionChild E false).card+(exceptionChild E true).card = E.card := by
-  classical
-  have he : (∑ u : Card (r+1), if u ∈ E then 1 else 0) = E.card := by simp
-  rw [Fintype.sum_equiv (headTailEquiv r) _
-    (fun x : Bool × Card r => if Fin.cons x.1 x.2 ∈ E then 1 else 0)
-    (fun x => by change (if x ∈ E then 1 else 0) = (if Fin.cons (x 0) (Fin.tail x) ∈ E then 1 else 0); rw [Fin.cons_self_tail])] at he
-  simp only [Fintype.sum_prod_type,Fintype.sum_bool] at he
-  simpa only [exceptionChild,Finset.card_filter,add_comm] using he
-
-def seedSplitEquiv (r : ℕ) (A : Type*) : (Card (r+1) → A) ≃ (Card r → A) × (Card r → A) where
-  toFun U := (fun u => U (Fin.cons false u),fun u => U (Fin.cons true u))
-  invFun U := fun u => if u 0 then U.2 (Fin.tail u) else U.1 (Fin.tail u)
-  left_inv U := by
-    funext u
-    dsimp only
-    split_ifs with h
-    · rw [←h,Fin.cons_self_tail]
-    · have h' : u 0 = false := by cases hh : u 0 <;> simp_all
-      rw [←h',Fin.cons_self_tail]
-  right_inv U := by simp only [Fin.cons_zero,Fin.tail_cons,Bool.false_eq_true,↓reduceIte]
-
-def seedSandwichShuffle {A B C D E : Type*} :
-    (A × B) × ((C × D) × E) ≃ ((A × C) × (B × D)) × E where
-  toFun x := (((x.1.1,x.2.1.1),(x.1.2,x.2.1.2)),x.2.2)
-  invFun x := ((x.1.1.1,x.1.2.1),((x.1.1.2,x.1.2.2),x.2))
-  left_inv _ := rfl
-  right_inv _ := rfl
-
-abbrev ModifiedCoins (L r : ℕ) := (Card r → Equiv.Perm (Card L)) × BenesCoins (L+r)
-
-def modifiedStepEquiv (L r : ℕ) : ModifiedCoins L (r+1) ≃
-    (ModifiedCoins L r × ModifiedCoins L r) × ((Card (L+r) → Bool) × (Card (L+r) → Bool)) :=
-  (Equiv.prodCongr (seedSplitEquiv r (Equiv.Perm (Card L))) (benesStepEquiv (L+r))).trans
-    seedSandwichShuffle
-
-noncomputable def modifiedPerm (L r : ℕ) (E : Finset (Card r)) (ω : ModifiedCoins L r) :
-    Equiv.Perm (Card (L+r)) :=
-  butterflyPerm (L+r) (decodeButterfly (L+r) ω.2.1) *
-    (modifiedInputTree L r E ω.1 ω.2.2 ω.2.1).perm⁻¹
-
-noncomputable def modifiedCost (L r : ℕ) (E : Finset (Card r))
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (ω : ModifiedCoins L r) : ℕ :=
-  InputTree.lowCost (L+r) (L+r) (modifiedInputTree L r E ω.1 ω.2.2 ω.2.1) e ω.2.1
-
-lemma modifiedBoundary_cons (L r : ℕ) (E : Finset (Card (r+1)))
-    (U : Card (r+1) → Equiv.Perm (Card L)) (Y : SwitchIndex (L+r+1) → Bool) (b : Bool) :
-    (fun u => modifiedBoundary L (r+1) E U Y (Fin.cons b u)) =
-    modifiedBoundary L r (exceptionChild E b) (fun u => U (Fin.cons b u))
-      (if b then (coinStepEquiv (L+r) Y).1.2 else (coinStepEquiv (L+r) Y).1.1) := by
-  funext u
-  simp only [modifiedBoundary,mem_exceptionChild,boundaryOutput,Fin.cons_zero,Fin.tail_cons]
-
-lemma modifiedInputTree_node (L r : ℕ) (E : Finset (Card (r+1)))
-    (ω : ModifiedCoins L r × ModifiedCoins L r) (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
-    let z := (modifiedStepEquiv L r).symm (ω,coins)
-    modifiedInputTree L (r+1) E z.1 z.2.2 z.2.1 =
-      InputTree.node coins.1 (fun b =>
-        let w := if b then ω.2 else ω.1
-        modifiedInputTree L r (exceptionChild E b) w.1 w.2.2 w.2.1) := by
-  dsimp only
-  unfold modifiedInputTree
-  rw [boundaryTree]
-  change InputTree.node coins.1 (fun b => boundaryTree L r _ _) = _
-  congr 1
-  funext b
-  rw [modifiedBoundary_cons]
-  cases b <;> rfl
-
-lemma modifiedPerm_step (L r : ℕ) (E : Finset (Card (r+1)))
-    (ω : ModifiedCoins L r × ModifiedCoins L r) (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
-    modifiedPerm L (r+1) E ((modifiedStepEquiv L r).symm (ω,coins)) =
-      pairSwitch coins.2 * childLift (fun b => modifiedPerm L r (exceptionChild E b)
-        (if b then ω.2 else ω.1)) * pairSwitch coins.1 := by
-  rw [modifiedPerm,modifiedInputTree_node,InputTree.perm,mul_inv_rev,pairSwitch_inv,childLift_inv]
-  change (butterflyPerm (L+r+1) (decodeButterfly (L+r+1)
-    ((coinStepEquiv (L+r)).symm ((ω.1.2.1,ω.2.2.1),coins.2)))) * _ = _
-  rw [butterflyPerm_step,mul_assoc,←mul_assoc (childLift _),childLift_mul,←mul_assoc]
-  congr 2
-  apply congrArg childLift
-  funext b
-  cases b <;> rfl
-
-lemma modifiedPerm_coordinates (L r : ℕ) (E : Finset (Card (r+1)))
-    (ω : ModifiedCoins L r × ModifiedCoins L r) (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool))
-    (x : Card (L+r+1)) :
-    headTailEquiv (L+r) (modifiedPerm L (r+1) E ((modifiedStepEquiv L r).symm (ω,coins)) x) =
-      PairRouting.sandwich coins.1 coins.2 (fun b => modifiedPerm L r (exceptionChild E b)
-        (if b then ω.2 else ω.1)) (headTailEquiv (L+r) x) := by
-  rw [modifiedPerm_step]
-  simp only [Equiv.Perm.mul_apply,headTail_switch,headTail_child]
-  rfl
-
-lemma InputTree.lowCost_node (d : ℕ) (ξ : Card d → Bool) (T : Bool → InputTree d)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (d+1)) (Y : (SwitchIndex d → Bool) × (SwitchIndex d → Bool))
-    (η : Card d → Bool) :
-    InputTree.lowCost (d+1) (d+1) (.node ξ T) e ((coinStepEquiv d).symm (Y,η)) =
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x ξ
-      let hc := PairRouting.colors_compatible x ξ
-      PairRouting.alternatingCycles (PairRouting.switchedEmbedding x ξ)
-        (fun b => if b then butterflyPerm d (decodeButterfly d Y.2) * (T true).perm⁻¹
-          else butterflyPerm d (decodeButterfly d Y.1) * (T false).perm⁻¹) +
-        InputTree.lowCost d d (T false) (PairRouting.childEmbedding x c hc false) Y.1 +
-        InputTree.lowCost d d (T true) (PairRouting.childEmbedding x c hc true) Y.2 := by
-  dsimp only
-  rw [InputTree.lowCost,Finset.sum_range_succ,InputTree.levelCost_node,ite_eq_left rfl]
-  have he : (∑ i ∈ Finset.range d,
-      (.node ξ T : InputTree (d+1)).levelCost (i+1) e ((coinStepEquiv d).symm (Y,η))) =
-      InputTree.lowCost d d (T false)
-        (PairRouting.childEmbedding (e.trans (headTailEquiv d).toEmbedding)
-          (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) ξ) (PairRouting.colors_compatible _ _) false) Y.1 +
-      InputTree.lowCost d d (T true)
-        (PairRouting.childEmbedding (e.trans (headTailEquiv d).toEmbedding)
-          (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) ξ) (PairRouting.colors_compatible _ _) true) Y.2 := by
-    rw [InputTree.lowCost,InputTree.lowCost,←Finset.sum_add_distrib]
-    apply Finset.sum_congr rfl
-    intro i hi
-    rw [InputTree.levelCost_node,ite_eq_right (by simp only [Finset.mem_range] at hi; omega : i+1 ≠ d+1)]
-  rw [he]
-  omega
-
-noncomputable def modifiedAssignmentCost (L r : ℕ) (E : Finset (Card (r+1)))
-    {ι : Type*} [Fintype ι] (x : ι ↪ Bool × Card (L+r)) (c : ι → Bool)
-    (hc : PairRouting.Compatible x c) (ω : ModifiedCoins L r × ModifiedCoins L r) : ℕ :=
-  PairRouting.alternatingCycles (PairRouting.coloredEmbedding x c hc)
-    (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) +
-  modifiedCost L r (exceptionChild E false) (PairRouting.childEmbedding x c hc false) ω.1 +
-  modifiedCost L r (exceptionChild E true) (PairRouting.childEmbedding x c hc true) ω.2
-
-lemma modifiedCost_step (L r : ℕ) (E : Finset (Card (r+1)))
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r+1))
-    (ω : ModifiedCoins L r × ModifiedCoins L r) (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
-    modifiedCost L (r+1) E e ((modifiedStepEquiv L r).symm (ω,coins)) =
-      modifiedAssignmentCost L r E (e.trans (headTailEquiv (L+r)).toEmbedding)
-        (PairRouting.colors (e.trans (headTailEquiv (L+r)).toEmbedding) coins.1)
-        (PairRouting.colors_compatible _ _) ω := by
-  rw [modifiedCost,modifiedInputTree_node]
-  change InputTree.lowCost (L+r+1) (L+r+1) _ e
-    ((coinStepEquiv (L+r)).symm ((ω.1.2.1,ω.2.2.1),coins.2)) = _
-  rw [InputTree.lowCost_node]
-  unfold modifiedAssignmentCost
-  dsimp only
-  simp only [Bool.false_eq_true,↓reduceIte]
-  rw [PairRouting.switchedEmbedding_eq_colored _ _ (PairRouting.colors_compatible _ _) _ rfl]
-  congr 2
-
-
-namespace PairRouting
-variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α] [DecidableEq ι]
-
-omit [Fintype α] [DecidableEq ι] in
-lemma childEmbedding_surjective (x : ι ↪ Bool × α) (hx : Function.Surjective x)
-    (c : ι → Bool) (hc : Compatible x c) (b : Bool) :
-    Function.Surjective (childEmbedding x c hc b) := by
-  intro y
-  have hs : Function.Surjective (coloredEmbedding x c hc) :=
-    (pairLayer (chosenCoin x c)).surjective.comp hx
-  obtain ⟨i,hi⟩ := hs (b,y)
-  rw [coloredEmbedding_apply] at hi
-  exact ⟨⟨i,congrArg Prod.fst hi⟩,congrArg Prod.snd hi⟩
-
-omit [DecidableEq ι] in
-lemma uniformPerm_tuple_upper (e f : ι ↪ α) (he : Function.Surjective e) (p : Equiv.Perm α) :
-    finiteMean (fun u : Equiv.Perm α => if ∀ i, (p*u⁻¹) (e i) = f i then (1:ℝ) else 0) ≤
-      1/(Fintype.card (Equiv.Perm α):ℝ) := by
-  classical
-  let M : Equiv.Perm α → Prop := fun u => ∀ i, (p*u⁻¹) (e i) = f i
-  have huniq (u v : Equiv.Perm α) (hu : M u) (hv : M v) : u = v := by
-    have hh : p*u⁻¹ = p*v⁻¹ := by
-      ext x
-      obtain ⟨i,rfl⟩ := he x
-      exact (hu i).trans (hv i).symm
-    exact inv_injective (mul_left_cancel hh)
-  unfold finiteMean
-  apply div_le_div_of_nonneg_right _ (Nat.cast_nonneg _)
-  change (∑ u, if M u then (1:ℝ) else 0) ≤ 1
-  by_cases h : ∃ u, M u
-  · obtain ⟨u,hu⟩ := h
-    rw [Finset.sum_eq_single u]
-    · simp only [hu,↓reduceIte,le_refl]
-    · intro v _ hv
-      exact ite_eq_right (fun hh => hv (huniq v u hh hu))
-    · simp
-  · push Not at h
-    simp only [h,ite_false,Finset.sum_const_zero,zero_le_one]
-
-end PairRouting
-
-noncomputable def centralFactor (L r : ℕ) (E : Finset (Card r)) (a : ℝ) : ℝ :=
-  (Nat.factorial (2^L):ℝ)^(-((2:ℝ)^r-E.card)*a)
-
-lemma centralFactor_nonneg (L r : ℕ) (E : Finset (Card r)) (a : ℝ) :
-    0 ≤ centralFactor L r E a := Real.rpow_nonneg (Nat.cast_nonneg _) _
-
-lemma centralFactor_step (L r : ℕ) (E : Finset (Card (r+1))) (a : ℝ) :
-    centralFactor L (r+1) E a =
-      centralFactor L r (exceptionChild E false) a * centralFactor L r (exceptionChild E true) a := by
-  unfold centralFactor
-  rw [←Real.rpow_add (Nat.cast_pos.mpr (Nat.factorial_pos _))]
-  congr 1
-  have he : ((exceptionChild E false).card:ℝ)+(exceptionChild E true).card = E.card := by
-    exact_mod_cast exceptionChild_card_add E
-  rw [pow_succ,←he]
-  ring
-
-lemma modifiedPerm_zero (L : ℕ) (E : Finset (Card 0)) (ω : ModifiedCoins L 0) :
-    modifiedPerm L 0 E ω =
-      if (fun j => Fin.elim0 j : Card 0) ∈ E then 1
-      else butterflyPerm L (decodeButterfly L ω.2.1)*(ω.1 (fun j => Fin.elim0 j))⁻¹ := by
-  unfold modifiedPerm modifiedInputTree boundaryTree InputTree.perm modifiedBoundary boundaryOutput
-  split_ifs
-  · exact mul_inv_cancel _
-  · rfl
-
-lemma modifiedCost_zero (L : ℕ) (E : Finset (Card 0)) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card L) (ω : ModifiedCoins L 0) : modifiedCost L 0 E e ω = 0 := by
-  simp only [modifiedCost,modifiedInputTree,boundaryTree,InputTree.lowCost,
-    InputTree.levelCost,Finset.sum_const_zero]
-
-lemma centralFactor_zero (L : ℕ) (E : Finset (Card 0)) (a : ℝ) :
-    centralFactor L 0 E a =
-      if (fun j => Fin.elim0 j : Card 0) ∈ E then 1 else (Nat.factorial (2^L):ℝ)^(-a) := by
-  classical
-  let z : Card 0 := fun j => Fin.elim0 j
-  have he : (E.card:ℝ) = if z ∈ E then 1 else 0 := by
-    by_cases hz : z ∈ E
-    · have he : E = {z} := by
-        ext u
-        rw [Subsingleton.elim u z]
-        simp only [hz,Finset.mem_singleton]
-      rw [he,ite_eq_left (by simpa only [he] using hz),Finset.card_singleton,Nat.cast_one]
-    · have he : E = ∅ := Finset.eq_empty_iff_forall_notMem.mpr (fun u hu => hz (Subsingleton.elim u z ▸ hu))
-      rw [he,Finset.card_empty,Nat.cast_zero,ite_eq_right (by simp)]
-  unfold centralFactor
-  rw [pow_zero,he]
-  split_ifs <;> simp
-
-lemma modified_probability_zero_le (L : ℕ) (E : Finset (Card 0))
-    {ι : Type*} [Fintype ι] (e f : ι ↪ Card L) (he : Function.Surjective e) :
-    PairRouting.tupleProbability (modifiedPerm L 0 E) e f ≤
-      if (fun j => Fin.elim0 j : Card 0) ∈ E then 1 else 1/(Nat.factorial (2^L):ℝ) := by
-  classical
-  rw [PairRouting.tupleProbability,finiteMean_prod,finiteMean_comm]
-  let B : ℝ := if (fun j => Fin.elim0 j : Card 0) ∈ E then 1 else 1/(Nat.factorial (2^L):ℝ)
-  change _ ≤ B
-  calc
-    _ ≤ finiteMean (fun _ω : BenesCoins L => B) := by
-      apply finiteMean_mono
-      intro ω
-      simp only [modifiedPerm_zero]
-      by_cases h : (fun j => Fin.elim0 j : Card 0) ∈ E
-      · simp only [h,↓reduceIte,Equiv.Perm.one_apply,finiteMean_const,B]
-        split_ifs <;> norm_num
-      · simp only [h,↓reduceIte,B]
-        rw [finiteMean_pi_eval (A := fun _ : Card 0 => Equiv.Perm (Card L)) (fun j => Fin.elim0 j) (fun u : Equiv.Perm (Card L) =>
-          if ∀ i, (butterflyPerm L (decodeButterfly L ω.1)*u⁻¹) (e i) = f i then (1:ℝ) else 0)]
-        convert PairRouting.uniformPerm_tuple_upper e f he (butterflyPerm L (decodeButterfly L ω.1)) using 1
-        rw [Fintype.card_perm,card_positions]
-    _ = _ := finiteMean_const B
-
-end Thorp
-
-end ThorpNine.Tail
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Tail.HeightDecay
+
+namespace OAI
+
+namespace ThorpNine.Tail
+
+namespace Thorp
+
+lemma centralDistance_exceptional (L r : ℕ) (C D : Card r → Equiv.Perm (Card L))
+    (E : Finset (Card r)) (h : ∀ u ∉ E, C u = D u) :
+    centralDistance L r C D ≤ E.card*2^L := by
+  classical
+  calc
+    _ ≤ ∑ u : Card r, if u ∈ E then 2^L else 0 := by
+      apply Finset.sum_le_sum
+      intro u _
+      split_ifs with hu
+      · simpa only [card_positions] using permDistance_le (C u) (D u)
+      · rw [h u hu,permDistance_refl]
+    _ = _ := by simp
+
+
+inductive InputTree : ℕ → Type
+  | tip {d : ℕ} (p : Equiv.Perm (Card d)) : InputTree d
+  | node {d : ℕ} (ξ : Card d → Bool) (T : Bool → InputTree d) : InputTree (d+1)
+
+def InputTree.perm : {d : ℕ} → InputTree d → Equiv.Perm (Card d)
+  | _, .tip p => p
+  | _, .node ξ T => pairSwitch ξ * childLift (fun b => (T b).perm)
+
+noncomputable def InputTree.levelCost (t : ℕ) : {d : ℕ} → InputTree d →
+    {ι : Type*} → [Fintype ι] → (ι ↪ Card d) → (SwitchIndex d → Bool) → ℕ
+  | _, .tip _, _, _, _, _ => 0
+  | d+1, .node ξ T, _, _, e, Y =>
+    let x := e.trans (headTailEquiv d).toEmbedding
+    let c := PairRouting.colors x ξ
+    let hc := PairRouting.colors_compatible x ξ
+    let σ := coinStepEquiv d Y
+    if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x ξ)
+      (fun b => if b then butterflyPerm d (decodeButterfly d σ.1.2) * (T true).perm⁻¹
+        else butterflyPerm d (decodeButterfly d σ.1.1) * (T false).perm⁻¹)
+    else (T false).levelCost t (PairRouting.childEmbedding x c hc false) σ.1.1 +
+      (T true).levelCost t (PairRouting.childEmbedding x c hc true) σ.1.2
+
+lemma InputTree.levelCost_node (t d : ℕ) (ξ : Card d → Bool) (T : Bool → InputTree d)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (d+1))
+    (Y : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (η : Card d → Bool) :
+    (InputTree.node ξ T).levelCost t e ((coinStepEquiv d).symm (Y,η)) =
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x ξ
+      let hc := PairRouting.colors_compatible x ξ
+      if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x ξ)
+        (fun b => if b then butterflyPerm d (decodeButterfly d Y.2) * (T true).perm⁻¹
+          else butterflyPerm d (decodeButterfly d Y.1) * (T false).perm⁻¹)
+      else (T false).levelCost t (PairRouting.childEmbedding x c hc false) Y.1 +
+        (T true).levelCost t (PairRouting.childEmbedding x c hc true) Y.2 := by
+  rw [InputTree.levelCost]
+  rfl
+
+lemma InputTree.levelCost_gt (t d : ℕ) (T : InputTree d) (ht : d < t)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) :
+    T.levelCost t e Y = 0 := by
+  induction T generalizing ι with
+  | tip p => rfl
+  | @node d ξ T ih =>
+    rw [InputTree.levelCost,ite_eq_right (by omega : t ≠ d+1),ih false (by omega),ih true (by omega)]
+
+lemma InputTree.levelCost_adapted (t d : ℕ) (T : InputTree d)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) :
+    LayerAdapted d (t-1) (fun Y => T.levelCost t e Y) := by
+  induction T generalizing ι with
+  | tip p => intro Y Y' _; rfl
+  | @node d ξ T ih =>
+    intro Y Y' h
+    dsimp only
+    conv_lhs => rw [←(coinStepEquiv d).symm_apply_apply Y]
+    conv_rhs => rw [←(coinStepEquiv d).symm_apply_apply Y']
+    rw [InputTree.levelCost_node t d ξ T e ((coinStepEquiv d Y).1) ((coinStepEquiv d Y).2),
+      InputTree.levelCost_node t d ξ T e ((coinStepEquiv d Y').1) ((coinStepEquiv d Y').2)]
+    by_cases ht : t = d+1
+    · subst t
+      rw [ite_eq_left rfl,ite_eq_left rfl]
+      have he₀ : (coinStepEquiv d Y).1.1 = (coinStepEquiv d Y').1.1 := by
+        funext i
+        exact h (Sum.inr (false,i)) (by simpa only [switchHeight,Nat.add_sub_cancel] using switchHeight_le d i)
+      have he₁ : (coinStepEquiv d Y).1.2 = (coinStepEquiv d Y').1.2 := by
+        funext i
+        exact h (Sum.inr (true,i)) (by simpa only [switchHeight,Nat.add_sub_cancel] using switchHeight_le d i)
+      rw [he₀,he₁]
+    · rw [ite_eq_right ht,ite_eq_right ht]
+      congr 1
+      · apply ih false
+        intro i hi
+        exact h (Sum.inr (false,i)) hi
+      · apply ih true
+        intro i hi
+        exact h (Sum.inr (true,i)) hi
+
+theorem InputTree.level_partial_mgf (r s u : ℕ) (hu : u < r)
+    (T : InputTree (s+r)) {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+r))
+    (lo : Card r × SwitchIndex s → Bool) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(T.levelCost (s+u+1) e (assembleBits r s lo hi))) ≤
+      Real.exp (Fintype.card ι * cycleCoefficient u J q) := by
+  classical
+  induction r generalizing u ι with
+  | zero => omega
+  | succ r ih =>
+    simp only [Nat.add_succ] at *
+    cases T with
+    | tip p =>
+      simp only [InputTree.levelCost,pow_zero,finiteMean_const]
+      exact Real.one_le_exp (mul_nonneg (Nat.cast_nonneg _) (cycleCoefficient_nonneg u J q hq))
+    | node ξ T =>
+      let x := e.trans (headTailEquiv (s+r)).toEmbedding
+      let c := PairRouting.colors x ξ
+      let hc := PairRouting.colors_compatible x ξ
+      let e₀ := PairRouting.childEmbedding x c hc false
+      let e₁ := PairRouting.childEmbedding x c hc true
+      let e' := PairRouting.switchedEmbedding x ξ
+      let lo₀ : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons false j.1,j.2)
+      let lo₁ : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons true j.1,j.2)
+      let Y₀ := fun hi : Card s × SwitchIndex r → Bool => assembleBits r s lo₀ hi
+      let Y₁ := fun hi : Card s × SwitchIndex r → Bool => assembleBits r s lo₁ hi
+      let N := fun hi : (Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool) =>
+        PairRouting.alternatingCycles e' (fun b =>
+          if b then butterflyPerm (s+r) (decodeButterfly (s+r) (Y₁ hi.2)) * (T true).perm⁻¹
+          else butterflyPerm (s+r) (decodeButterfly (s+r) (Y₀ hi.1)) * (T false).perm⁻¹)
+      let L₀ := fun hi : Card s × SwitchIndex r → Bool =>
+        (T false).levelCost (s+u+1) e₀ (Y₀ hi)
+      let L₁ := fun hi : Card s × SwitchIndex r → Bool =>
+        (T true).levelCost (s+u+1) e₁ (Y₁ hi)
+      have hs (hi : (Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool))
+          (η : Card s × Card r → Bool) :
+          (InputTree.node ξ T).levelCost (s+u+1) e
+            (assembleBits (r+1) s lo ((highStepEquiv r s).symm (hi,η))) =
+            if u = r then N hi else L₀ hi.1 + L₁ hi.2 := by
+        rw [assembleBits_step]
+        rw [InputTree.levelCost_node (s+u+1) (s+r) ξ T e (Y₀ hi.1,Y₁ hi.2) (fun y => η (((cardSplit r s) y).2,((cardSplit r s) y).1))]
+        simp only [Nat.add_right_cancel_iff,Nat.add_left_cancel_iff]
+        rfl
+      rw [finiteMean_equiv (highStepEquiv r s),finiteMean_prod]
+      simp_rw [hs,finiteMean_const]
+      by_cases hur : u = r
+      · subst u
+        simp only [ite_true]
+        rw [finiteMean_prod,finiteMean_comm]
+        calc
+          _ ≤ finiteMean (fun _hi : Card s × SwitchIndex r → Bool =>
+            Real.exp (Fintype.card ι * cycleCoefficient r J q)) := by
+            apply finiteMean_mono
+            intro hi
+            have hn := node_partial_mgf r s e' (T false).perm (T true).perm
+              (butterflyPerm (s+r) (decodeButterfly (s+r) (Y₁ hi))) lo₀ J hJ q hq
+            exact hn
+          _ = _ := finiteMean_const _
+      · simp only [ite_eq_right hur,pow_add]
+        rw [finiteMean_prod_mul (fun hi => q ^ L₀ hi) (fun hi => q ^ L₁ hi)]
+        have hu' : u < r := by omega
+        have h₀ := ih u hu' (T false) e₀ lo₀
+        have h₁ := ih u hu' (T true) e₁ lo₁
+        have hp := mul_le_mul h₀ h₁ (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
+          (Real.exp_nonneg _)
+        refine hp.trans_eq ?_
+        rw [←Real.exp_add]
+        congr 1
+        have hcount := PairRouting.color_card_add c
+        have hcount' : (Fintype.card {i : ι // c i = false} : ℝ) +
+            Fintype.card {i : ι // c i = true} = Fintype.card ι := by exact_mod_cast hcount
+        change (Fintype.card {i : ι // c i = false} : ℝ) * cycleCoefficient u J q +
+            Fintype.card {i : ι // c i = true} * cycleCoefficient u J q = _
+        rw [←add_mul,hcount']
+
+
+noncomputable def InputTree.slabCost (l d : ℕ) (T : InputTree d) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) : ℕ :=
+  ∑ i : Fin l, T.levelCost (l+i.1+1) e Y
+
+lemma InputTree.slabCost_adapted (l d : ℕ) (T : InputTree d) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) :
+    LayerAdapted d (2*l) (fun Y => InputTree.slabCost l d T e Y) := by
+  apply layerAdapted_sum
+  intro i _
+  exact layerAdapted_mono (InputTree.levelCost_adapted (l+i.1+1) d T e) (by omega)
+
+theorem InputTree.slab_partial_mgf (r s l : ℕ) (hl : 0 < l) (hs : 2*s ≤ l) (T : InputTree (s+r))
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+r))
+    (lo : Card r × SwitchIndex s → Bool) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(InputTree.slabCost l (s+r) T e (assembleBits r s lo hi))) ≤
+      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
+  classical
+  let : NeZero l := ⟨ne_of_gt hl⟩
+  have hqp : 0 < q := lt_of_lt_of_le (by norm_num) hq
+  calc
+    _ ≤ finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      finiteMean (fun i : Fin l => q^(l*T.levelCost (l+i.1+1) e
+        (assembleBits r s lo hi)))) := by
+      apply finiteMean_mono
+      intro hi
+      simpa only [InputTree.slabCost,Fintype.card_fin] using
+        pow_sum_le_mean_powers (fun i : Fin l => T.levelCost (l+i.1+1) e
+          (assembleBits r s lo hi)) q hqp
+    _ = finiteMean (fun i : Fin l => finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(l*T.levelCost (l+i.1+1) e (assembleBits r s lo hi)))) := finiteMean_comm _
+    _ ≤ finiteMean (fun _i : Fin l =>
+      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64)))) := by
+      apply finiteMean_mono
+      intro i
+      by_cases ht : l+i.1+1 ≤ s+r
+      · have hs' : s ≤ l+i.1 := by omega
+        have he : l+i.1+1 = s+(l+i.1-s)+1 := by omega
+        have hu : l+i.1-s < r := by omega
+        have hm := InputTree.level_partial_mgf r s (l+i.1-s) hu T e lo
+          ⌈Real.exp ((l:ℝ)/32)⌉₊ (cycleCutoff_pos l) (q^l) (one_le_pow₀ hq)
+        have hc := cycleCoefficient_decay l (l+i.1-s) (by
+          rw [Nat.cast_sub hs',Nat.cast_add]
+          have hsr : (2:ℝ)*s ≤ l := by exact_mod_cast hs
+          have hi : (0:ℝ) ≤ i.1 := Nat.cast_nonneg _
+          linarith) (q^l) (one_le_pow₀ hq) (by
+            rw [Real.log_pow]
+            nlinarith [mul_le_mul_of_nonneg_left hlog (Nat.cast_nonneg l)])
+        simp_rw [he,pow_mul]
+        exact hm.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left hc (Nat.cast_nonneg _)))
+      · have hz := InputTree.levelCost_gt (l+i.1+1) (s+r) T (by omega) e
+        simp only [hz,Nat.mul_zero,pow_zero,finiteMean_const]
+        exact Real.one_le_exp_iff.mpr (by positivity)
+    _ = _ := finiteMean_const _
+
+
+def boundaryTree (L : ℕ) : (r : ℕ) → (Card r → Equiv.Perm (Card L)) →
+    (SwitchIndex (L+r) → Bool) → InputTree (L+r)
+  | 0, U, _ => .tip (U (fun j => Fin.elim0 j))
+  | r+1, U, X => .node (coinStepEquiv (L+r) X).2 (fun b =>
+      boundaryTree L r (fun u => U (Fin.cons b u))
+        (if b then (coinStepEquiv (L+r) X).1.2 else (coinStepEquiv (L+r) X).1.1))
+
+def boundaryOutput (L : ℕ) : (r : ℕ) → (SwitchIndex (L+r) → Bool) →
+    Card r → Equiv.Perm (Card L)
+  | 0, Y, _ => butterflyPerm L (decodeButterfly L Y)
+  | r+1, Y, u => boundaryOutput L r
+      (if u 0 then (coinStepEquiv (L+r) Y).1.2 else (coinStepEquiv (L+r) Y).1.1) (Fin.tail u)
+
+lemma boundaryOutput_adapted (L r : ℕ) :
+    LayerAdapted (L+r) L (boundaryOutput L r) := by
+  induction r with
+  | zero =>
+    intro Y Y' h
+    have he : Y = Y' := funext (fun i => h i (by simpa only [Nat.add_zero] using switchHeight_le L i))
+    rw [he]
+  | succ r ih =>
+    intro Y Y' h
+    funext u
+    dsimp only [boundaryOutput]
+    cases hb : u 0 <;> simp only [Bool.false_eq_true,↓reduceIte]
+    · exact congrFun (ih _ _ (fun i hi => h (Sum.inr (false,i)) hi)) (Fin.tail u)
+    · exact congrFun (ih _ _ (fun i hi => h (Sum.inr (true,i)) hi)) (Fin.tail u)
+
+noncomputable def modifiedBoundary (L r : ℕ) (E : Finset (Card r))
+    (U : Card r → Equiv.Perm (Card L)) (Y : SwitchIndex (L+r) → Bool) :
+    Card r → Equiv.Perm (Card L) := fun u => if u ∈ E then boundaryOutput L r Y u else U u
+
+lemma modifiedBoundary_adapted (L r : ℕ) (E : Finset (Card r))
+    (U : Card r → Equiv.Perm (Card L)) :
+    LayerAdapted (L+r) L (modifiedBoundary L r E U) := by
+  intro Y Y' h
+  unfold modifiedBoundary
+  rw [boundaryOutput_adapted L r Y Y' h]
+
+lemma permDistance_boundaryTree (L r : ℕ) (U V : Card r → Equiv.Perm (Card L))
+    (X : SwitchIndex (L+r) → Bool) :
+    permDistance (boundaryTree L r U X).perm (boundaryTree L r V X).perm =
+      centralDistance L r U V := by
+  induction r with
+  | zero =>
+    simp only [boundaryTree,InputTree.perm,centralDistance,Fintype.sum_unique]
+    congr 1
+  | succ r ih =>
+    simp only [boundaryTree,InputTree.perm,permDistance_mul_left]
+    rw [permDistance_childLift,centralDistance_step,ih,ih]
+
+lemma boundaryTree_levelCost_perturb (t L r : ℕ) (U V : Card r → Equiv.Perm (Card L))
+    (X : SwitchIndex (L+r) → Bool) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (L+r)) (Y : SwitchIndex (L+r) → Bool) :
+    (boundaryTree L r U X).levelCost t e Y ≤
+      (boundaryTree L r V X).levelCost t e Y + centralDistance L r U V := by
+  induction r generalizing ι with
+  | zero => simp only [boundaryTree,InputTree.levelCost,Nat.zero_add,Nat.zero_le]
+  | succ r ih =>
+    rw [boundaryTree,boundaryTree,InputTree.levelCost,InputTree.levelCost]
+    by_cases ht : t = L+r+1
+    · simp only [ite_eq_left ht]
+      have hh := alternatingCycles_distance_le
+        (PairRouting.switchedEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
+          (coinStepEquiv (L+r) X).2)
+        (fun b => if b then butterflyPerm (L+r) (decodeButterfly (L+r) (coinStepEquiv (L+r) Y).1.2) *
+            (boundaryTree L r (fun u => U (Fin.cons true u)) (coinStepEquiv (L+r) X).1.2).perm⁻¹
+          else butterflyPerm (L+r) (decodeButterfly (L+r) (coinStepEquiv (L+r) Y).1.1) *
+            (boundaryTree L r (fun u => U (Fin.cons false u)) (coinStepEquiv (L+r) X).1.1).perm⁻¹)
+        (fun b => if b then butterflyPerm (L+r) (decodeButterfly (L+r) (coinStepEquiv (L+r) Y).1.2) *
+            (boundaryTree L r (fun u => V (Fin.cons true u)) (coinStepEquiv (L+r) X).1.2).perm⁻¹
+          else butterflyPerm (L+r) (decodeButterfly (L+r) (coinStepEquiv (L+r) Y).1.1) *
+            (boundaryTree L r (fun u => V (Fin.cons false u)) (coinStepEquiv (L+r) X).1.1).perm⁻¹)
+      simpa only [Bool.false_eq_true,↓reduceIte,permDistance_mul_left,permDistance_inv,
+        permDistance_boundaryTree,←centralDistance_step] using hh
+    · simp only [ite_eq_right ht,Bool.false_eq_true,↓reduceIte]
+      rw [centralDistance_step]
+      have h₀ := ih (fun u => U (Fin.cons false u)) (fun u => V (Fin.cons false u))
+        (coinStepEquiv (L+r) X).1.1
+        (PairRouting.childEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
+          (PairRouting.colors (e.trans (headTailEquiv (L+r)).toEmbedding) (coinStepEquiv (L+r) X).2)
+          (PairRouting.colors_compatible _ _) false) (coinStepEquiv (L+r) Y).1.1
+      have h₁ := ih (fun u => U (Fin.cons true u)) (fun u => V (Fin.cons true u))
+        (coinStepEquiv (L+r) X).1.2
+        (PairRouting.childEmbedding (e.trans (headTailEquiv (L+r)).toEmbedding)
+          (PairRouting.colors (e.trans (headTailEquiv (L+r)).toEmbedding) (coinStepEquiv (L+r) X).2)
+          (PairRouting.colors_compatible _ _) true) (coinStepEquiv (L+r) Y).1.2
+      omega
+
+lemma boundaryTree_slabCost_perturb (l L r : ℕ) (U V : Card r → Equiv.Perm (Card L))
+    (X : SwitchIndex (L+r) → Bool) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (L+r)) (Y : SwitchIndex (L+r) → Bool) :
+    InputTree.slabCost l (L+r) (boundaryTree L r U X) e Y ≤
+      InputTree.slabCost l (L+r) (boundaryTree L r V X) e Y + l*centralDistance L r U V := by
+  have hh := Finset.sum_le_sum (fun (i : Fin l) (_ : i ∈ Finset.univ) =>
+    boundaryTree_levelCost_perturb (l+i.val+1) L r U V X e Y)
+  simpa only [Finset.sum_add_distrib,Finset.sum_const,Finset.card_univ,Fintype.card_fin,
+    nsmul_eq_mul,Nat.cast_id,InputTree.slabCost] using hh
+
+lemma modifiedBoundary_distance_le (L r : ℕ) (E : Finset (Card r))
+    (U : Card r → Equiv.Perm (Card L)) (Y : SwitchIndex (L+r) → Bool) :
+    centralDistance L r (modifiedBoundary L r E U Y) U ≤ E.card*2^L := by
+  apply centralDistance_exceptional
+  intro u hu
+  exact ite_eq_right hu
+
+
+lemma InputTree.slabCost_gt (l d : ℕ) (T : InputTree d) (hd : d ≤ l)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) :
+    InputTree.slabCost l d T e Y = 0 := by
+  apply Finset.sum_eq_zero
+  intro i _
+  exact InputTree.levelCost_gt _ _ T (by omega) _ _
+
+lemma InputTree.random_slab_adapted_mul_le (d s l : ℕ) (hsd : s ≤ d) (hl : 0 < l)
+    (hs : 2*s ≤ l) (T : (SwitchIndex d → Bool) → InputTree d)
+    (hT : LayerAdapted d s T) {ι : Type*} [Fintype ι] (e : ι ↪ Card d)
+    (F : (SwitchIndex d → Bool) → ℝ) (hF : ∀ Y, 0 ≤ F Y) (hA : LayerAdapted d s F)
+    (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun Y => F Y*q^(InputTree.slabCost l d (T Y) e Y)) ≤
+      finiteMean F * Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
+  obtain ⟨r,rfl⟩ := Nat.exists_eq_add_of_le hsd
+  apply adapted_mul_mean_le r s F _ hF hA
+  intro lo
+  let hi₀ : Card s × SwitchIndex r → Bool := fun _ => false
+  have he (hi : Card s × SwitchIndex r → Bool) :
+      T (assembleBits r s lo hi) = T (assembleBits r s lo hi₀) :=
+    hT _ _ (assembleBits_agree r s lo hi hi₀)
+  simp_rw [he]
+  exact InputTree.slab_partial_mgf r s l hl hs _ e lo q hq hlog
+
+lemma InputTree.slab_mgf (l d : ℕ) (hl : 0 < l) (T : InputTree d)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun Y => q^(InputTree.slabCost l d T e Y)) ≤
+      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
+  have hh := InputTree.random_slab_adapted_mul_le d 0 l (Nat.zero_le _) hl (Nat.zero_le _)
+    (fun _ => T) (fun _ _ _ => rfl) e (fun _ => 1) (fun _ => by norm_num)
+    (fun _ _ _ => rfl) q hq hlog
+  simpa only [one_mul,finiteMean_const] using hh
+
+noncomputable def InputTree.familyCost (L n d : ℕ)
+    (T : (SwitchIndex d → Bool) → InputTree d) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) : ℕ :=
+  ∑ j ∈ Finset.range n, InputTree.slabCost (L*4^j) d (T Y) e Y
+
+lemma InputTree.random_slabCost_adapted (l d s : ℕ) (T : (SwitchIndex d → Bool) → InputTree d)
+    (hT : LayerAdapted d s T) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) :
+    LayerAdapted d (max s (2*l)) (fun Y => InputTree.slabCost l d (T Y) e Y) := by
+  intro Y Y' h
+  dsimp only
+  rw [hT _ _ (agreeThrough_mono (le_max_left _ _) h)]
+  exact InputTree.slabCost_adapted l d (T Y') e Y Y'
+    (agreeThrough_mono (le_max_right _ _) h)
+
+lemma InputTree.familyCost_adapted (L n d s : ℕ) (T : (SwitchIndex d → Bool) → InputTree d)
+    (hT : LayerAdapted d s T) {ι : Type*} [Fintype ι] (e : ι ↪ Card d) :
+    LayerAdapted d (max s (familyEnd L n)) (fun Y => InputTree.familyCost L n d T e Y) := by
+  apply layerAdapted_sum
+  intro j hj
+  exact layerAdapted_mono (InputTree.random_slabCost_adapted (L*4^j) d s T hT e)
+    (max_le_max_left s (familyEnd_ge L n j (Finset.mem_range.mp hj)))
+
+theorem InputTree.family_mgf_first (L n d s : ℕ) (hL : 0 < L) (hs : s ≤ L)
+    (T : (SwitchIndex d → Bool) → InputTree d) (hT : LayerAdapted d s T)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) (p : ℝ) (hp : 0 ≤ p)
+    (hfirst : finiteMean (fun Y => q^(InputTree.slabCost L d (T Y) e Y)) ≤
+      Real.exp (p+Fintype.card ι*(68*Real.exp (-(L:ℝ)/64)))) :
+    finiteMean (fun Y => q^(InputTree.familyCost L n d T e Y)) ≤
+      Real.exp (p+Fintype.card ι*(68*∑ j ∈ Finset.range n,
+        Real.exp (-((L*4^j:ℕ):ℝ)/64))) := by
+  induction n with
+  | zero =>
+    simp only [InputTree.familyCost,Finset.range_zero,Finset.sum_empty,pow_zero,
+      finiteMean_const,mul_zero,add_zero]
+    exact Real.one_le_exp hp
+  | succ n ih =>
+    by_cases hn : n = 0
+    · subst n
+      simpa only [InputTree.familyCost,Nat.zero_add,Finset.sum_range_one,pow_zero,mul_one] using hfirst
+    · let l := L*4^n
+      have hl : 0 < l := Nat.mul_pos hL (pow_pos (by decide) _)
+      have hsl : 2*familyEnd L n ≤ l := familyEnd_two L n
+      have hs' : s ≤ familyEnd L n := by
+        have hh := familyEnd_ge L n 0 (Nat.pos_of_ne_zero hn)
+        simp only [pow_zero,mul_one] at hh
+        omega
+      have hc : ∀ Y, InputTree.familyCost L (n+1) d T e Y =
+          InputTree.familyCost L n d T e Y+InputTree.slabCost l d (T Y) e Y := by
+        intro Y
+        exact Finset.sum_range_succ _ _
+      have hm : finiteMean (fun Y => q^(InputTree.familyCost L (n+1) d T e Y)) ≤
+          finiteMean (fun Y => q^(InputTree.familyCost L n d T e Y)) *
+            Real.exp (Fintype.card ι*(68*Real.exp (-(l:ℝ)/64))) := by
+        by_cases hld : l ≤ d
+        · simp_rw [hc,pow_add]
+          apply InputTree.random_slab_adapted_mul_le d (familyEnd L n) l (by omega) hl hsl T
+            (layerAdapted_mono hT hs') e _ (fun _ => pow_nonneg (by linarith) _)
+          · apply layerAdapted_comp
+            exact layerAdapted_mono (InputTree.familyCost_adapted L n d s T hT e) (max_le hs' le_rfl)
+          · exact hq
+          · exact hlog
+        · simp_rw [hc,InputTree.slabCost_gt l d _ (by omega),Nat.add_zero]
+          exact le_mul_of_one_le_right
+            (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
+            (Real.one_le_exp_iff.mpr (by positivity))
+      refine (hm.trans (mul_le_mul_of_nonneg_right ih (Real.exp_nonneg _))).trans_eq ?_
+      rw [←Real.exp_add,Finset.sum_range_succ]
+      congr 1
+      dsimp only [l]
+      ring
+
+lemma InputTree.family_mgf_first_decay (L n d s : ℕ) (hL : 0 < L) (hs : s ≤ L)
+    (T : (SwitchIndex d → Bool) → InputTree d) (hT : LayerAdapted d s T)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) (p : ℝ) (hp : 0 ≤ p)
+    (hfirst : finiteMean (fun Y => q^(InputTree.slabCost L d (T Y) e Y)) ≤
+      Real.exp (p+Fintype.card ι*(68*Real.exp (-(L:ℝ)/64)))) :
+    finiteMean (fun Y => q^(InputTree.familyCost L n d T e Y)) ≤
+      Real.exp (p+Fintype.card ι*heightDecay L) := by
+  apply (InputTree.family_mgf_first L n d s hL hs T hT e q hq hlog p hp hfirst).trans
+  apply Real.exp_le_exp.mpr
+  apply add_le_add_right
+  apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
+  have hh := mul_le_mul_of_nonneg_left (sum_family_decay L n hL) (by norm_num : (0:ℝ)≤68)
+  simpa only [heightDecay,mul_div_assoc] using hh
+
+
+noncomputable def modifiedInputTree (L r : ℕ) (E : Finset (Card r))
+    (U : Card r → Equiv.Perm (Card L)) (X Y : SwitchIndex (L+r) → Bool) : InputTree (L+r) :=
+  boundaryTree L r (modifiedBoundary L r E U Y) X
+
+lemma modifiedInputTree_adapted (L r : ℕ) (E : Finset (Card r))
+    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool) :
+    LayerAdapted (L+r) L (modifiedInputTree L r E U X) := by
+  intro Y Y' h
+  unfold modifiedInputTree
+  rw [modifiedBoundary_adapted L r E U Y Y' h]
+
+lemma modified_slab_mgf (l L r : ℕ) (hl : 0 < l) (E : Finset (Card r))
+    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun Y => q^(InputTree.slabCost l (L+r) (modifiedInputTree L r E U X Y) e Y)) ≤
+      Real.exp (((l*E.card*2^L:ℕ):ℝ)*Real.log q+
+        Fintype.card ι*(68*Real.exp (-(l:ℝ)/64))) := by
+  have hqp : 0 < q := by linarith
+  have hpoint (Y : SwitchIndex (L+r) → Bool) :
+      InputTree.slabCost l (L+r) (modifiedInputTree L r E U X Y) e Y ≤
+        InputTree.slabCost l (L+r) (boundaryTree L r U X) e Y+l*E.card*2^L := by
+    apply (boundaryTree_slabCost_perturb l L r (modifiedBoundary L r E U Y) U X e Y).trans
+    apply Nat.add_le_add_left
+    simpa only [Nat.mul_assoc] using Nat.mul_le_mul_left l (modifiedBoundary_distance_le L r E U Y)
+  calc
+    _ ≤ finiteMean (fun Y => q^(InputTree.slabCost l (L+r) (boundaryTree L r U X) e Y+l*E.card*2^L)) :=
+      finiteMean_mono (fun Y => pow_le_pow_right₀ hq (hpoint Y))
+    _ = finiteMean (fun Y => q^(InputTree.slabCost l (L+r) (boundaryTree L r U X) e Y)) *
+        q^(l*E.card*2^L) := by simp_rw [pow_add]; exact finiteMean_mul_const _ _
+    _ ≤ Real.exp (Fintype.card ι*(68*Real.exp (-(l:ℝ)/64))) * q^(l*E.card*2^L) :=
+      mul_le_mul_of_nonneg_right (InputTree.slab_mgf l (L+r) hl _ e q hq hlog) (pow_nonneg (by linarith) _)
+    _ = _ := by
+      have he : Real.exp (((l*E.card*2^L:ℕ):ℝ)*Real.log q) = q^(l*E.card*2^L) := by
+        rw [Real.exp_nat_mul,Real.exp_log hqp]
+      rw [Real.exp_add,he,mul_comm]
+
+lemma modified_family_mgf (l n L r : ℕ) (hl : 0 < l) (hLl : L ≤ l) (E : Finset (Card r))
+    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun Y => q^(InputTree.familyCost l n (L+r) (modifiedInputTree L r E U X) e Y)) ≤
+      Real.exp (((l*E.card*2^L:ℕ):ℝ)*Real.log q+Fintype.card ι*heightDecay l) := by
+  exact InputTree.family_mgf_first_decay l n (L+r) L hl hLl _
+    (modifiedInputTree_adapted L r E U X) e q hq hlog _
+    (mul_nonneg (Nat.cast_nonneg _) (Real.log_nonneg hq))
+    (modified_slab_mgf l L r hl E U X e q hq hlog)
+
+theorem modified_high_mgf (L n r : ℕ) (hL : 0 < L) (E : Finset (Card r))
+    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/500) :
+    finiteMean (fun Y => q^(InputTree.familyCost L n (L+r) (modifiedInputTree L r E U X) e Y+
+      InputTree.familyCost (2*L) n (L+r) (modifiedInputTree L r E U X) e Y)) ≤
+      Real.exp (Fintype.card ι*heightDecay L +
+        3*((L*E.card*2^L:ℕ):ℝ)*Real.log q) := by
+  have hq2 := one_le_pow₀ (n:=2) hq
+  have hlog2 : Real.log (q^2) ≤ 1/250 := by rw [Real.log_pow]; norm_num; linarith
+  have h₀ := modified_family_mgf L n L r hL le_rfl E U X e (q^2) hq2 hlog2
+  have h₁ := modified_family_mgf (2*L) n L r (by omega) (by omega) E U X e (q^2) hq2 hlog2
+  have h₁' : finiteMean (fun Y => (q^2)^(InputTree.familyCost (2*L) n (L+r)
+      (modifiedInputTree L r E U X) e Y)) ≤
+      Real.exp ((((2*L)*E.card*2^L:ℕ):ℝ)*Real.log (q^2)+Fintype.card ι*heightDecay L) := by
+    apply h₁.trans
+    apply Real.exp_le_exp.mpr
+    exact add_le_add_right (mul_le_mul_of_nonneg_left
+      (heightDecay_antitone (show L ≤ 2*L by omega)) (Nat.cast_nonneg _)) _
+  have hh := finiteMean_mul_exp_le
+    (fun Y => q^(InputTree.familyCost L n (L+r) (modifiedInputTree L r E U X) e Y))
+    (fun Y => q^(InputTree.familyCost (2*L) n (L+r) (modifiedInputTree L r E U X) e Y))
+    _ _ (by simpa only [pow_right_comm] using h₀) (by simpa only [pow_right_comm] using h₁')
+  simp only [←pow_add] at hh
+  refine hh.trans_eq ?_
+  congr 1
+  simp only [Real.log_pow,Nat.cast_mul,Nat.cast_ofNat]
+  ring
+
+noncomputable def InputTree.lowCost (H d : ℕ) (T : InputTree d)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) : ℕ :=
+  ∑ i ∈ Finset.range H, T.levelCost (i+1) e Y
+
+lemma InputTree.lowCost_slab (L d : ℕ) (T : InputTree d)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) :
+    InputTree.lowCost L d T e Y+InputTree.slabCost L d T e Y = InputTree.lowCost (2*L) d T e Y := by
+  rw [InputTree.lowCost,InputTree.lowCost,show 2*L=L+L by omega,Finset.sum_range_add]
+  congr 1
+  exact Fin.sum_univ_eq_sum_range (fun i => T.levelCost (L+i+1) e Y) L
+
+lemma InputTree.cost_family (L n d : ℕ) (T : (SwitchIndex d → Bool) → InputTree d)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) :
+    InputTree.lowCost L d (T Y) e Y + InputTree.familyCost L n d T e Y +
+      InputTree.familyCost (2*L) n d T e Y = InputTree.lowCost (L*4^n) d (T Y) e Y := by
+  induction n with
+  | zero => simp only [InputTree.familyCost,Finset.range_zero,Finset.sum_empty,Nat.add_zero,
+      pow_zero,mul_one]
+  | succ n ih =>
+    have he : (2*L)*4^n = 2*(L*4^n) := by ring
+    have he' : L*4^(n+1) = 2*(2*(L*4^n)) := by rw [pow_succ]; ring
+    simp only [InputTree.familyCost,Finset.sum_range_succ] at ih ⊢
+    rw [he,he',←InputTree.lowCost_slab,←InputTree.lowCost_slab,←ih]
+    omega
+
+lemma InputTree.lowCost_ge (H d : ℕ) (hd : d ≤ H) (T : InputTree d)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card d) (Y : SwitchIndex d → Bool) :
+    InputTree.lowCost H d T e Y = InputTree.lowCost d d T e Y := by
+  symm
+  apply Finset.sum_subset (Finset.range_mono hd)
+  intro i _ hi
+  have hi' : d ≤ i := by simpa only [Finset.mem_range,not_lt] using hi
+  exact InputTree.levelCost_gt (i+1) d T (by omega) e Y
+
+lemma boundaryTree_levelCost_low (t L r : ℕ) (ht : t ≤ L)
+    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (Y : SwitchIndex (L+r) → Bool) :
+    (boundaryTree L r U X).levelCost t e Y = 0 := by
+  induction r generalizing ι with
+  | zero => rfl
+  | succ r ih =>
+    rw [boundaryTree,InputTree.levelCost,ite_eq_right (by omega : t ≠ L+r+1),ih,ih]
+
+lemma boundaryTree_lowCost_zero (L r : ℕ) (U : Card r → Equiv.Perm (Card L))
+    (X : SwitchIndex (L+r) → Bool) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (L+r)) (Y : SwitchIndex (L+r) → Bool) :
+    InputTree.lowCost L (L+r) (boundaryTree L r U X) e Y = 0 := by
+  apply Finset.sum_eq_zero
+  intro i hi
+  exact boundaryTree_levelCost_low (i+1) L r (by simpa only [Finset.mem_range,Nat.lt_iff_add_one_le] using hi) U X e Y
+
+theorem modified_cost_mgf (L r : ℕ) (hL : 0 < L) (E : Finset (Card r))
+    (U : Card r → Equiv.Perm (Card L)) (X : SwitchIndex (L+r) → Bool)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/500) :
+    finiteMean (fun Y => q^(InputTree.lowCost (L+r) (L+r) (modifiedInputTree L r E U X Y) e Y)) ≤
+      Real.exp (Fintype.card ι*heightDecay L +
+        3*((L*E.card*2^L:ℕ):ℝ)*Real.log q) := by
+  have hbound : L+r ≤ L*4^(L+r) := by
+    have h₀ := pow_four_ge (L+r)
+    have h₁ := Nat.mul_le_mul_right (4^(L+r)) hL
+    nlinarith
+  have he (Y : SwitchIndex (L+r) → Bool) :=
+    InputTree.cost_family L (L+r) (L+r) (modifiedInputTree L r E U X) e Y
+  have hz (Y : SwitchIndex (L+r) → Bool) :
+      InputTree.lowCost L (L+r) (modifiedInputTree L r E U X Y) e Y = 0 :=
+    boundaryTree_lowCost_zero L r _ X e Y
+  simp only [hz,Nat.zero_add,InputTree.lowCost_ge _ _ hbound] at he
+  simp_rw [←he]
+  exact modified_high_mgf L (L+r) r hL E U X e q hq hlog
+
+
+noncomputable def exceptionChild {r : ℕ} (E : Finset (Card (r+1))) (b : Bool) : Finset (Card r) :=
+  Finset.univ.filter (fun u => Fin.cons b u ∈ E)
+
+lemma mem_exceptionChild {r : ℕ} (E : Finset (Card (r+1))) (b : Bool) (u : Card r) :
+    u ∈ exceptionChild E b ↔ Fin.cons b u ∈ E := by
+  classical
+  simp only [exceptionChild,Finset.mem_filter,Finset.mem_univ,true_and]
+
+lemma exceptionChild_card_add {r : ℕ} (E : Finset (Card (r+1))) :
+    (exceptionChild E false).card+(exceptionChild E true).card = E.card := by
+  classical
+  have he : (∑ u : Card (r+1), if u ∈ E then 1 else 0) = E.card := by simp
+  rw [Fintype.sum_equiv (headTailEquiv r) _
+    (fun x : Bool × Card r => if Fin.cons x.1 x.2 ∈ E then 1 else 0)
+    (fun x => by change (if x ∈ E then 1 else 0) = (if Fin.cons (x 0) (Fin.tail x) ∈ E then 1 else 0); rw [Fin.cons_self_tail])] at he
+  simp only [Fintype.sum_prod_type,Fintype.sum_bool] at he
+  simpa only [exceptionChild,Finset.card_filter,add_comm] using he
+
+def seedSplitEquiv (r : ℕ) (A : Type*) : (Card (r+1) → A) ≃ (Card r → A) × (Card r → A) where
+  toFun U := (fun u => U (Fin.cons false u),fun u => U (Fin.cons true u))
+  invFun U := fun u => if u 0 then U.2 (Fin.tail u) else U.1 (Fin.tail u)
+  left_inv U := by
+    funext u
+    dsimp only
+    split_ifs with h
+    · rw [←h,Fin.cons_self_tail]
+    · have h' : u 0 = false := by cases hh : u 0 <;> simp_all
+      rw [←h',Fin.cons_self_tail]
+  right_inv U := by simp only [Fin.cons_zero,Fin.tail_cons,Bool.false_eq_true,↓reduceIte]
+
+def seedSandwichShuffle {A B C D E : Type*} :
+    (A × B) × ((C × D) × E) ≃ ((A × C) × (B × D)) × E where
+  toFun x := (((x.1.1,x.2.1.1),(x.1.2,x.2.1.2)),x.2.2)
+  invFun x := ((x.1.1.1,x.1.2.1),((x.1.1.2,x.1.2.2),x.2))
+  left_inv _ := rfl
+  right_inv _ := rfl
+
+abbrev ModifiedCoins (L r : ℕ) := (Card r → Equiv.Perm (Card L)) × BenesCoins (L+r)
+
+def modifiedStepEquiv (L r : ℕ) : ModifiedCoins L (r+1) ≃
+    (ModifiedCoins L r × ModifiedCoins L r) × ((Card (L+r) → Bool) × (Card (L+r) → Bool)) :=
+  (Equiv.prodCongr (seedSplitEquiv r (Equiv.Perm (Card L))) (benesStepEquiv (L+r))).trans
+    seedSandwichShuffle
+
+noncomputable def modifiedPerm (L r : ℕ) (E : Finset (Card r)) (ω : ModifiedCoins L r) :
+    Equiv.Perm (Card (L+r)) :=
+  butterflyPerm (L+r) (decodeButterfly (L+r) ω.2.1) *
+    (modifiedInputTree L r E ω.1 ω.2.2 ω.2.1).perm⁻¹
+
+noncomputable def modifiedCost (L r : ℕ) (E : Finset (Card r))
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (ω : ModifiedCoins L r) : ℕ :=
+  InputTree.lowCost (L+r) (L+r) (modifiedInputTree L r E ω.1 ω.2.2 ω.2.1) e ω.2.1
+
+lemma modifiedBoundary_cons (L r : ℕ) (E : Finset (Card (r+1)))
+    (U : Card (r+1) → Equiv.Perm (Card L)) (Y : SwitchIndex (L+r+1) → Bool) (b : Bool) :
+    (fun u => modifiedBoundary L (r+1) E U Y (Fin.cons b u)) =
+    modifiedBoundary L r (exceptionChild E b) (fun u => U (Fin.cons b u))
+      (if b then (coinStepEquiv (L+r) Y).1.2 else (coinStepEquiv (L+r) Y).1.1) := by
+  funext u
+  simp only [modifiedBoundary,mem_exceptionChild,boundaryOutput,Fin.cons_zero,Fin.tail_cons]
+
+lemma modifiedInputTree_node (L r : ℕ) (E : Finset (Card (r+1)))
+    (ω : ModifiedCoins L r × ModifiedCoins L r) (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
+    let z := (modifiedStepEquiv L r).symm (ω,coins)
+    modifiedInputTree L (r+1) E z.1 z.2.2 z.2.1 =
+      InputTree.node coins.1 (fun b =>
+        let w := if b then ω.2 else ω.1
+        modifiedInputTree L r (exceptionChild E b) w.1 w.2.2 w.2.1) := by
+  dsimp only
+  unfold modifiedInputTree
+  rw [boundaryTree]
+  change InputTree.node coins.1 (fun b => boundaryTree L r _ _) = _
+  congr 1
+  funext b
+  rw [modifiedBoundary_cons]
+  cases b <;> rfl
+
+lemma modifiedPerm_step (L r : ℕ) (E : Finset (Card (r+1)))
+    (ω : ModifiedCoins L r × ModifiedCoins L r) (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
+    modifiedPerm L (r+1) E ((modifiedStepEquiv L r).symm (ω,coins)) =
+      pairSwitch coins.2 * childLift (fun b => modifiedPerm L r (exceptionChild E b)
+        (if b then ω.2 else ω.1)) * pairSwitch coins.1 := by
+  rw [modifiedPerm,modifiedInputTree_node,InputTree.perm,mul_inv_rev,pairSwitch_inv,childLift_inv]
+  change (butterflyPerm (L+r+1) (decodeButterfly (L+r+1)
+    ((coinStepEquiv (L+r)).symm ((ω.1.2.1,ω.2.2.1),coins.2)))) * _ = _
+  rw [butterflyPerm_step,mul_assoc,←mul_assoc (childLift _),childLift_mul,←mul_assoc]
+  congr 2
+  apply congrArg childLift
+  funext b
+  cases b <;> rfl
+
+lemma modifiedPerm_coordinates (L r : ℕ) (E : Finset (Card (r+1)))
+    (ω : ModifiedCoins L r × ModifiedCoins L r) (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool))
+    (x : Card (L+r+1)) :
+    headTailEquiv (L+r) (modifiedPerm L (r+1) E ((modifiedStepEquiv L r).symm (ω,coins)) x) =
+      PairRouting.sandwich coins.1 coins.2 (fun b => modifiedPerm L r (exceptionChild E b)
+        (if b then ω.2 else ω.1)) (headTailEquiv (L+r) x) := by
+  rw [modifiedPerm_step]
+  simp only [Equiv.Perm.mul_apply,headTail_switch,headTail_child]
+  rfl
+
+lemma InputTree.lowCost_node (d : ℕ) (ξ : Card d → Bool) (T : Bool → InputTree d)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (d+1)) (Y : (SwitchIndex d → Bool) × (SwitchIndex d → Bool))
+    (η : Card d → Bool) :
+    InputTree.lowCost (d+1) (d+1) (.node ξ T) e ((coinStepEquiv d).symm (Y,η)) =
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x ξ
+      let hc := PairRouting.colors_compatible x ξ
+      PairRouting.alternatingCycles (PairRouting.switchedEmbedding x ξ)
+        (fun b => if b then butterflyPerm d (decodeButterfly d Y.2) * (T true).perm⁻¹
+          else butterflyPerm d (decodeButterfly d Y.1) * (T false).perm⁻¹) +
+        InputTree.lowCost d d (T false) (PairRouting.childEmbedding x c hc false) Y.1 +
+        InputTree.lowCost d d (T true) (PairRouting.childEmbedding x c hc true) Y.2 := by
+  dsimp only
+  rw [InputTree.lowCost,Finset.sum_range_succ,InputTree.levelCost_node,ite_eq_left rfl]
+  have he : (∑ i ∈ Finset.range d,
+      (.node ξ T : InputTree (d+1)).levelCost (i+1) e ((coinStepEquiv d).symm (Y,η))) =
+      InputTree.lowCost d d (T false)
+        (PairRouting.childEmbedding (e.trans (headTailEquiv d).toEmbedding)
+          (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) ξ) (PairRouting.colors_compatible _ _) false) Y.1 +
+      InputTree.lowCost d d (T true)
+        (PairRouting.childEmbedding (e.trans (headTailEquiv d).toEmbedding)
+          (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) ξ) (PairRouting.colors_compatible _ _) true) Y.2 := by
+    rw [InputTree.lowCost,InputTree.lowCost,←Finset.sum_add_distrib]
+    apply Finset.sum_congr rfl
+    intro i hi
+    rw [InputTree.levelCost_node,ite_eq_right (by simp only [Finset.mem_range] at hi; omega : i+1 ≠ d+1)]
+  rw [he]
+  omega
+
+noncomputable def modifiedAssignmentCost (L r : ℕ) (E : Finset (Card (r+1)))
+    {ι : Type*} [Fintype ι] (x : ι ↪ Bool × Card (L+r)) (c : ι → Bool)
+    (hc : PairRouting.Compatible x c) (ω : ModifiedCoins L r × ModifiedCoins L r) : ℕ :=
+  PairRouting.alternatingCycles (PairRouting.coloredEmbedding x c hc)
+    (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) +
+  modifiedCost L r (exceptionChild E false) (PairRouting.childEmbedding x c hc false) ω.1 +
+  modifiedCost L r (exceptionChild E true) (PairRouting.childEmbedding x c hc true) ω.2
+
+lemma modifiedCost_step (L r : ℕ) (E : Finset (Card (r+1)))
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r+1))
+    (ω : ModifiedCoins L r × ModifiedCoins L r) (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
+    modifiedCost L (r+1) E e ((modifiedStepEquiv L r).symm (ω,coins)) =
+      modifiedAssignmentCost L r E (e.trans (headTailEquiv (L+r)).toEmbedding)
+        (PairRouting.colors (e.trans (headTailEquiv (L+r)).toEmbedding) coins.1)
+        (PairRouting.colors_compatible _ _) ω := by
+  rw [modifiedCost,modifiedInputTree_node]
+  change InputTree.lowCost (L+r+1) (L+r+1) _ e
+    ((coinStepEquiv (L+r)).symm ((ω.1.2.1,ω.2.2.1),coins.2)) = _
+  rw [InputTree.lowCost_node]
+  unfold modifiedAssignmentCost
+  dsimp only
+  simp only [Bool.false_eq_true,↓reduceIte]
+  rw [PairRouting.switchedEmbedding_eq_colored _ _ (PairRouting.colors_compatible _ _) _ rfl]
+  congr 2
+
+
+namespace PairRouting
+variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α] [DecidableEq ι]
+
+omit [Fintype α] [DecidableEq ι] in
+lemma childEmbedding_surjective (x : ι ↪ Bool × α) (hx : Function.Surjective x)
+    (c : ι → Bool) (hc : Compatible x c) (b : Bool) :
+    Function.Surjective (childEmbedding x c hc b) := by
+  intro y
+  have hs : Function.Surjective (coloredEmbedding x c hc) :=
+    (pairLayer (chosenCoin x c)).surjective.comp hx
+  obtain ⟨i,hi⟩ := hs (b,y)
+  rw [coloredEmbedding_apply] at hi
+  exact ⟨⟨i,congrArg Prod.fst hi⟩,congrArg Prod.snd hi⟩
+
+omit [DecidableEq ι] in
+lemma uniformPerm_tuple_upper (e f : ι ↪ α) (he : Function.Surjective e) (p : Equiv.Perm α) :
+    finiteMean (fun u : Equiv.Perm α => if ∀ i, (p*u⁻¹) (e i) = f i then (1:ℝ) else 0) ≤
+      1/(Fintype.card (Equiv.Perm α):ℝ) := by
+  classical
+  let M : Equiv.Perm α → Prop := fun u => ∀ i, (p*u⁻¹) (e i) = f i
+  have huniq (u v : Equiv.Perm α) (hu : M u) (hv : M v) : u = v := by
+    have hh : p*u⁻¹ = p*v⁻¹ := by
+      ext x
+      obtain ⟨i,rfl⟩ := he x
+      exact (hu i).trans (hv i).symm
+    exact inv_injective (mul_left_cancel hh)
+  unfold finiteMean
+  apply div_le_div_of_nonneg_right _ (Nat.cast_nonneg _)
+  change (∑ u, if M u then (1:ℝ) else 0) ≤ 1
+  by_cases h : ∃ u, M u
+  · obtain ⟨u,hu⟩ := h
+    rw [Finset.sum_eq_single u]
+    · simp only [hu,↓reduceIte,le_refl]
+    · intro v _ hv
+      exact ite_eq_right (fun hh => hv (huniq v u hh hu))
+    · simp
+  · push Not at h
+    simp only [h,ite_false,Finset.sum_const_zero,zero_le_one]
+
+end PairRouting
+
+noncomputable def centralFactor (L r : ℕ) (E : Finset (Card r)) (a : ℝ) : ℝ :=
+  (Nat.factorial (2^L):ℝ)^(-((2:ℝ)^r-E.card)*a)
+
+lemma centralFactor_nonneg (L r : ℕ) (E : Finset (Card r)) (a : ℝ) :
+    0 ≤ centralFactor L r E a := Real.rpow_nonneg (Nat.cast_nonneg _) _
+
+lemma centralFactor_step (L r : ℕ) (E : Finset (Card (r+1))) (a : ℝ) :
+    centralFactor L (r+1) E a =
+      centralFactor L r (exceptionChild E false) a * centralFactor L r (exceptionChild E true) a := by
+  unfold centralFactor
+  rw [←Real.rpow_add (Nat.cast_pos.mpr (Nat.factorial_pos _))]
+  congr 1
+  have he : ((exceptionChild E false).card:ℝ)+(exceptionChild E true).card = E.card := by
+    exact_mod_cast exceptionChild_card_add E
+  rw [pow_succ,←he]
+  ring
+
+lemma modifiedPerm_zero (L : ℕ) (E : Finset (Card 0)) (ω : ModifiedCoins L 0) :
+    modifiedPerm L 0 E ω =
+      if (fun j => Fin.elim0 j : Card 0) ∈ E then 1
+      else butterflyPerm L (decodeButterfly L ω.2.1)*(ω.1 (fun j => Fin.elim0 j))⁻¹ := by
+  unfold modifiedPerm modifiedInputTree boundaryTree InputTree.perm modifiedBoundary boundaryOutput
+  split_ifs
+  · exact mul_inv_cancel _
+  · rfl
+
+lemma modifiedCost_zero (L : ℕ) (E : Finset (Card 0)) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card L) (ω : ModifiedCoins L 0) : modifiedCost L 0 E e ω = 0 := by
+  simp only [modifiedCost,modifiedInputTree,boundaryTree,InputTree.lowCost,
+    InputTree.levelCost,Finset.sum_const_zero]
+
+lemma centralFactor_zero (L : ℕ) (E : Finset (Card 0)) (a : ℝ) :
+    centralFactor L 0 E a =
+      if (fun j => Fin.elim0 j : Card 0) ∈ E then 1 else (Nat.factorial (2^L):ℝ)^(-a) := by
+  classical
+  let z : Card 0 := fun j => Fin.elim0 j
+  have he : (E.card:ℝ) = if z ∈ E then 1 else 0 := by
+    by_cases hz : z ∈ E
+    · have he : E = {z} := by
+        ext u
+        rw [Subsingleton.elim u z]
+        simp only [hz,Finset.mem_singleton]
+      rw [he,ite_eq_left (by simpa only [he] using hz),Finset.card_singleton,Nat.cast_one]
+    · have he : E = ∅ := Finset.eq_empty_iff_forall_notMem.mpr (fun u hu => hz (Subsingleton.elim u z ▸ hu))
+      rw [he,Finset.card_empty,Nat.cast_zero,ite_eq_right (by simp)]
+  unfold centralFactor
+  rw [pow_zero,he]
+  split_ifs <;> simp
+
+lemma modified_probability_zero_le (L : ℕ) (E : Finset (Card 0))
+    {ι : Type*} [Fintype ι] (e f : ι ↪ Card L) (he : Function.Surjective e) :
+    PairRouting.tupleProbability (modifiedPerm L 0 E) e f ≤
+      if (fun j => Fin.elim0 j : Card 0) ∈ E then 1 else 1/(Nat.factorial (2^L):ℝ) := by
+  classical
+  rw [PairRouting.tupleProbability,finiteMean_prod,finiteMean_comm]
+  let B : ℝ := if (fun j => Fin.elim0 j : Card 0) ∈ E then 1 else 1/(Nat.factorial (2^L):ℝ)
+  change _ ≤ B
+  calc
+    _ ≤ finiteMean (fun _ω : BenesCoins L => B) := by
+      apply finiteMean_mono
+      intro ω
+      simp only [modifiedPerm_zero]
+      by_cases h : (fun j => Fin.elim0 j : Card 0) ∈ E
+      · simp only [h,↓reduceIte,Equiv.Perm.one_apply,finiteMean_const,B]
+        split_ifs <;> norm_num
+      · simp only [h,↓reduceIte,B]
+        rw [finiteMean_pi_eval (A := fun _ : Card 0 => Equiv.Perm (Card L)) (fun j => Fin.elim0 j) (fun u : Equiv.Perm (Card L) =>
+          if ∀ i, (butterflyPerm L (decodeButterfly L ω.1)*u⁻¹) (e i) = f i then (1:ℝ) else 0)]
+        convert PairRouting.uniformPerm_tuple_upper e f he (butterflyPerm L (decodeButterfly L ω.1)) using 1
+        rw [Fintype.card_perm,card_positions]
+    _ = _ := finiteMean_const B
+
+end Thorp
+
+end ThorpNine.Tail
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Tail/InsertionPerm.lean b/lean/OAI/Probability/ThorpRouting/Tail/InsertionPerm.lean
index b2c8df089170b57838cf6f80ad9f0ccbe9a9ada6..ba165f51950572e1326be362c1415780596feadf 100644
--- a/lean/OAI/Probability/ThorpRouting/Tail/InsertionPerm.lean
+++ b/lean/OAI/Probability/ThorpRouting/Tail/InsertionPerm.lean
@@ -1,897 +1,897 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Tail.MainStatement
-
-namespace OAI
-
-namespace ThorpNine.Tail
-
-namespace Thorp.DimensionEstimates
-open scoped BigOperators Classical
-
-section
-open Filter
-
-theorem exists_near_cutoff (C : ℝ) (hC : 0≤C) :
-    ∃ X : ℝ, 2≤X ∧ ∀ (N j D m : ℕ), 0<j → X≤(N:ℝ)/j →
-      N.choose j≤D*2^j →
-      D * (chosenTuple N j).choose j ≤ m * N.choose j →
-      j≤chosenTuple N j ∧ chosenTuple N j≤N ∧
-      nearFraction*Real.log D≤Real.log m ∧
-      C*(chosenTuple N j:ℝ)*((chosenTuple N j:ℝ)/N)^(1/512:ℝ)≤Real.log m/4000 ∧
-      (1/2:ℝ)*Real.log N≤Real.log D ∧ (j:ℝ)≤Real.log D := by
-  obtain ⟨X,hX⟩ := eventually_atTop.mp (near_conditions_eventually C)
-  refine ⟨max 2 X,le_max_left _ _,?_⟩
-  intro N j D m hj hNx hD hfoot
-  have hc := hX ((N:ℝ)/j) ((le_max_right _ _).trans hNx)
-  have hjp : (0:ℝ)<j := by exact_mod_cast hj
-  have hN : 2*j≤N := by exact_mod_cast (le_div_iff₀ hjp).mp hc.1
-  have hk := chosenTuple_bounds N j hj (by linarith [hc.1]) hc.2.2.1
-  have hm := chosenTuple_log_footprint N j D m hj hN hD hc.2.1 hc.2.2.2.1 hfoot
-  have hd := near_row_log_uniform N j D hj hN hD hc.2.2.2.2.1
-  refine ⟨hk.1,hk.2.1,hm,?_,hd⟩
-  apply (chosenTuple_density_bound N j hj (by linarith [hc.1]) hc.2.2.1 C hC).trans
-  have hh := mul_le_mul_of_nonneg_right hc.2.2.2.2.2 hjp.le
-  dsimp [nearFraction] at hm ⊢
-  dsimp [nearFraction] at hh
-  nlinarith [hd.2]
-
-end
-
-lemma amplification_numerical (m u : ℝ) (hm : 1 ≤ m) (hu : u ≤ Real.log m/4000) :
-    Real.exp u*(Real.exp (Real.log m/2))^(-(1/1000:ℝ))+
-      Real.sqrt (Real.exp (Real.log m/2)/m) ≤ 2*Real.exp (-Real.log m/4000) := by
-  have hmp : 0 < m := lt_of_lt_of_le zero_lt_one hm
-  have hl : 0≤Real.log m := Real.log_nonneg hm
-  have he : Real.exp (Real.log m/2)/m=Real.exp (-Real.log m/2) := by
-    calc
-      _ = Real.exp (Real.log m/2)/Real.exp (Real.log m) := by rw [Real.exp_log hmp]
-      _ = _ := by rw [←Real.exp_sub]; congr 1; ring
-  have hs : Real.sqrt (Real.exp (-Real.log m/2))=Real.exp (-Real.log m/4) := by
-    rw [Real.sqrt_eq_rpow,Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp]
-    congr 1
-    ring
-  rw [Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp,←Real.exp_add,he,hs]
-  have h₁ : u+(Real.log m/2)*(-(1/1000:ℝ))≤ -Real.log m/4000 := by linarith
-  have h₂ : -Real.log m/4≤ -Real.log m/4000 := by linarith
-  linarith [Real.exp_le_exp.mpr h₁,Real.exp_le_exp.mpr h₂]
-
-lemma amplification_absorb (x : ℝ) (hx : 8000*Real.log 2≤x) :
-    2*Real.exp (-x/4000)≤Real.exp (-x/8000) := by
-  rw [show (2:ℝ)=Real.exp (Real.log 2) by rw [Real.exp_log (by norm_num)],←Real.exp_add]
-  exact Real.exp_le_exp.mpr (by linarith)
-
-lemma amplification_decay (m u : ℝ) (hm : 1 ≤ m) (hu : u ≤ Real.log m/4000)
-    (hl : 8000*Real.log 2≤Real.log m) :
-    Real.exp u*(Real.exp (Real.log m/2))^(-(1/1000:ℝ))+
-      Real.sqrt (Real.exp (Real.log m/2)/m) ≤ Real.exp (-Real.log m/8000) :=
-  (amplification_numerical m u hm hu).trans (amplification_absorb _ hl)
-
-end Thorp.DimensionEstimates
-
-namespace Thorp.UnitaryFinite
-open scoped BigOperators Classical
-
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-omit [FiniteDimensional ℂ V] in
-lemma full_tuple_fixed (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V) :
-    fixedSpace tupleAction (Function.Embedding.refl (Card d)) ρ=⊤ := by
-  apply Submodule.eq_top_iff'.mpr
-  intro v g hg
-  have he : g=1 := by
-    apply Equiv.ext
-    intro x
-    exact congrArg (fun f : Card d ↪ Card d => f x) hg
-  rw [he,map_one]
-  rfl
-
-omit [FiniteDimensional ℂ V] in
-lemma full_tuple_finrank (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V) :
-    Module.finrank ℂ (fixedSpace tupleAction (Function.Embedding.refl (Card d)) ρ)=Module.finrank ℂ V := by
-  rw [full_tuple_fixed,finrank_top]
-
-theorem palindrome_irrep_amplification (d : ℕ) {ι : Type*} [Fintype ι]
-    (f : ι ↪ Card d) (hk : 0<Fintype.card ι)
-    (ρ : Representation ℂ (Equiv.Perm (Card d)) V) [Representation.IsIrreducible ρ]
-    (hρ : IsUnitary ρ) (hm : 0<Module.finrank ℂ (fixedSpace tupleAction f ρ))
-    (z : ℝ) (hz : 0<z) :
-    ‖sampleOperator ρ (palindromePerm d)‖ ≤
-      Real.exp (densityConstant*Fintype.card ι*((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ))*
-        z^(-(1/1000:ℝ))+Real.sqrt (z/(Module.finrank ℂ (fixedSpace tupleAction f ρ):ℝ)) := by
-  apply sampleOperator_amplification tupleAction f (tupleAction_transitive f) ρ hρ
-    (palindromePerm d) _ _ (1/1000) z _ (by norm_num) hz (by positivity) _ hm
-  · intro e f
-    simp only [tupleAction_kernel]
-    exact palindrome_kernel_symmetric d e f
-  · intro e
-    simp only [tupleAction_kernel]
-    exact PairRouting.tupleProbability_row_sum _ e
-  · intro e
-    simp only [tupleAction_kernel]
-    exact palindrome_row_moment_bound d e hk
-
-theorem modified_irrep_amplification (L r : ℕ) (E : Finset (Card r))
-    (ρ : Representation ℂ (Equiv.Perm (Card (L+r))) V) [Representation.IsIrreducible ρ]
-    (hρ : IsUnitary ρ) (hm : 0<Module.finrank ℂ V)
-    (u z : ℝ) (hz : 0<z)
-    (hM : ∀ e : Card (L+r) ↪ Card (L+r), modifiedRowMoment L r E e (1/1000)≤Real.exp u) :
-    ‖sampleOperator ρ (modifiedPerm L r E)‖ ≤
-      Real.exp u*z^(-(1/1000:ℝ))+Real.sqrt (z/(Module.finrank ℂ V:ℝ)) := by
-  let f := Function.Embedding.refl (Card (L+r))
-  have h := sampleOperator_amplification tupleAction f (tupleAction_transitive f) ρ hρ
-    (modifiedPerm L r E) (fun e f => by simpa only [tupleAction_kernel] using modified_kernel_symmetric L r E e f)
-    (fun e => by simpa only [tupleAction_kernel] using PairRouting.tupleProbability_row_sum (modifiedPerm L r E) e)
-    (1/1000) z (Real.exp u) (by norm_num) hz (by positivity)
-    (fun e => by simpa only [tupleAction_kernel,modifiedRowMoment] using hM e)
-    (by simpa only [f,full_tuple_finrank] using hm)
-  simpa only [f,full_tuple_finrank] using h
-
-end Thorp.UnitaryFinite
-
-namespace Thorp.Specht
-open scoped BigOperators Classical
-
-theorem choose_tail_le_of_factorial (μ : YoungDiagram) (D : ℕ)
-    (hd : μ.card.factorial ≤ D * Fintype.card (rowGroup μ) * Fintype.card (colGroup μ)) :
-    μ.card.choose (μ.card-μ.rowLen 0) ≤ D * 2^(μ.card-μ.rowLen 0) := by
-  let j := μ.card-μ.rowLen 0
-  have htail := card_fiberGroups_mul_le
-    (fun x : TailCell μ => rowIndex x.1) (fun x : TailCell μ => colIndex x.1)
-    (fun x y hr hc => Subtype.ext (Subtype.ext (Prod.ext (congrArg Fin.val hr) (congrArg Fin.val hc))))
-  rw [tail_card] at htail
-  have hc := column_group_le_tail μ
-  have hr := tail_row_group μ
-  simp only [←Nat.card_eq_fintype_card] at htail hc hr hd
-  have hh : μ.card.factorial ≤ D * 2^j *
-      ((μ.rowLen 0).factorial * j.factorial) := by
-    calc
-      _ ≤ _ := hd
-      _ ≤ D * Nat.card (rowGroup μ) *
-          (2^j * Nat.card (fiberGroup (fun x : TailCell μ => colIndex x.1))) := Nat.mul_le_mul_left _ hc
-      _ = D * 2^j * (μ.rowLen 0).factorial *
-          (Nat.card (fiberGroup (fun x : TailCell μ => rowIndex x.1)) *
-           Nat.card (fiberGroup (fun x : TailCell μ => colIndex x.1))) := by rw [←hr]; ring
-      _ ≤ _ := by
-        calc
-          _ ≤ D * 2^j * (μ.rowLen 0).factorial * j.factorial :=
-            Nat.mul_le_mul_left _ htail
-          _ = _ := by ring
-  have hj : j ≤ μ.card := Nat.sub_le _ _
-  have hs : μ.card-j = μ.rowLen 0 := by dsimp [j]; have := rowLen_zero_le_card μ; omega
-  have hf := Nat.choose_mul_factorial_mul_factorial hj
-  rw [hs] at hf
-  rw [←hf] at hh
-  have hpos : 0 < (μ.rowLen 0).factorial * j.factorial := Nat.mul_pos (Nat.factorial_pos _) (Nat.factorial_pos _)
-  nlinarith
-
-lemma card_rowGroup_transpose (μ : YoungDiagram) :
-    Fintype.card (rowGroup μ.transpose) = Fintype.card (colGroup μ) := by
-  rw [card_rowGroup,card_colGroup]
-  apply Fintype.prod_equiv (finCongr (μ.colLen_transpose 0))
-  intro i
-  change (μ.transpose.rowLen i).factorial = (μ.colLen i).factorial
-  rw [YoungDiagram.rowLen_transpose]
-
-lemma card_colGroup_transpose (μ : YoungDiagram) :
-    Fintype.card (colGroup μ.transpose) = Fintype.card (rowGroup μ) := by
-  rw [card_colGroup,card_rowGroup]
-  apply Fintype.prod_equiv (finCongr (μ.rowLen_transpose 0))
-  intro i
-  change (μ.transpose.colLen i).factorial = (μ.rowLen i).factorial
-  rw [YoungDiagram.colLen_transpose]
-
-lemma card_transpose (μ : YoungDiagram) : μ.transpose.card=μ.card := by
-  unfold YoungDiagram.card YoungDiagram.transpose
-  exact Finset.card_map _
-
-theorem choose_column_tail_le_dimension (μ : YoungDiagram) :
-    μ.card.choose (μ.card-μ.colLen 0) ≤ Module.finrank ℂ (space μ) * 2^(μ.card-μ.colLen 0) := by
-  have hd := factorial_le_dimension_row_column μ
-  have ht := choose_tail_le_of_factorial μ.transpose (Module.finrank ℂ (space μ))
-  simp only [card_transpose,card_rowGroup_transpose,card_colGroup_transpose,YoungDiagram.rowLen_transpose] at ht
-  apply ht
-  nlinarith
-
-open DimensionEstimates Filter
-
-lemma dimension_log_incidence (μ : YoungDiagram) (hμ : 0<μ.card) :
-    (2*Real.log 2-1)*(μ.card:ℝ)-Real.log 2*((μ.rowLen 0:ℝ)+μ.colLen 0)+1≤
-      Real.log (Module.finrank ℂ (space μ)) := by
-  let : Nonempty (Cell μ) := Fintype.card_pos_iff.mp (by simpa only [card_cell] using hμ)
-  have hd := factorial_le_dimension_row_column μ
-  rw [card_rowGroup,card_colGroup] at hd
-  have hr (i : Fin (μ.colLen 0)) : Nat.card {x : Cell μ // rowIndex x=i}=μ.rowLen i := by
-    let e : {x : Cell μ // rowIndex x=i} ≃ {x : Cell μ // row x=i} :=
-      Equiv.subtypeEquivRight (fun x => Fin.ext_iff)
-    rw [Nat.card_congr e,Nat.card_congr (rowEquiv μ i),Nat.card_fin]
-  have hc (i : Fin (μ.rowLen 0)) : Nat.card {x : Cell μ // colIndex x=i}=μ.colLen i := by
-    let e : {x : Cell μ // colIndex x=i} ≃ {x : Cell μ // col x=i} :=
-      Equiv.subtypeEquivRight (fun x => Fin.ext_iff)
-    rw [Nat.card_congr e,Nat.card_congr (columnEquiv μ i),Nat.card_fin]
-  have hcell : Nat.card (Cell μ)=μ.card := by rw [Nat.card_eq_fintype_card,card_cell]
-  have h := dimension_log_from_incidence (rowIndex (μ := μ)) (colIndex (μ := μ))
-    (fun x y h => Subtype.ext (Prod.ext (congrArg (fun z => (z.1:ℕ)) h)
-      (congrArg (fun z => (z.2:ℕ)) h))) (Module.finrank ℂ (space μ))
-    (by simpa only [←Nat.card_eq_fintype_card,hr,hc,hcell] using hd)
-  simpa only [card_cell,Fintype.card_fin,add_comm (μ.colLen 0:ℝ)] using h
-
-lemma medium_dimension_raw (μ : YoungDiagram) (hμ : 0<μ.card) (δ : ℝ) (hδ : 0<δ)
-    (hδ2 : δ≤1/2) (hrow : δ*(μ.card:ℝ)≤μ.card-μ.rowLen 0)
-    (hcol : (μ.colLen 0:ℝ)≤(μ.card:ℝ)/2) :
-    min ((3/2:ℝ)*Real.log 2-1) (entropyGap*δ)*(μ.card:ℝ)-2*Real.log μ.card-1≤
-      Real.log (Module.finrank ℂ (space μ)) := by
-  let c := (3/2:ℝ)*Real.log 2-1
-  have hN : (0:ℝ)<μ.card := by exact_mod_cast hμ
-  have hlog : 0≤Real.log μ.card := Real.log_nonneg (by exact_mod_cast hμ)
-  have hc : 0≤Real.log 2 := Real.log_nonneg (by norm_num)
-  have hr := rowLen_zero_le_card μ
-  have hcl : μ.colLen 0≤μ.card := by
-    simpa only [YoungDiagram.rowLen_transpose,card_transpose] using rowLen_zero_le_card μ.transpose
-  by_cases hsum : (μ.rowLen 0:ℝ)+μ.colLen 0≤(μ.card:ℝ)/2
-  · have hi := dimension_log_incidence μ hμ
-    have hm := mul_le_mul_of_nonneg_right (min_le_left c (entropyGap*δ)) hN.le
-    dsimp [c] at hm
-    nlinarith
-  · have hbig : (μ.card:ℝ)/4≤μ.rowLen 0 ∨ (μ.card:ℝ)/4≤μ.colLen 0 := by
-      by_contra h
-      push Not at h
-      linarith
-    rcases hbig with hrb | hcb
-    · have hj : 0<μ.card-μ.rowLen 0 := by
-        rw [←Nat.cast_pos (α := ℝ),Nat.cast_sub hr]
-        exact lt_of_lt_of_le (mul_pos hδ hN) hrow
-      have hjN : μ.card-μ.rowLen 0<μ.card := by
-        have : 0<μ.rowLen 0 := by exact_mod_cast (by linarith : (0:ℝ)<μ.rowLen 0)
-        omega
-      have hi := log_dimension_from_binomial μ.card (μ.card-μ.rowLen 0)
-        (Module.finrank ℂ (space μ)) hj hjN
-        (by rw [Nat.cast_sub hr]; linarith) (choose_tail_le_dimension μ)
-      rw [Nat.cast_sub hr] at hi
-      have hm := mul_le_mul_of_nonneg_right (min_le_right c (entropyGap*δ)) hN.le
-      nlinarith [entropyGap_pos]
-    · have hj : 0<μ.card-μ.colLen 0 := by
-        rw [←Nat.cast_pos (α := ℝ),Nat.cast_sub hcl]
-        linarith
-      have hjN : μ.card-μ.colLen 0<μ.card := by
-        have : 0<μ.colLen 0 := by exact_mod_cast (by linarith : (0:ℝ)<μ.colLen 0)
-        omega
-      have hi := log_dimension_from_binomial μ.card (μ.card-μ.colLen 0)
-        (Module.finrank ℂ (space μ)) hj hjN
-        (by rw [Nat.cast_sub hcl]; linarith) (choose_column_tail_le_dimension μ)
-      rw [Nat.cast_sub hcl] at hi
-      have hm := mul_le_mul_of_nonneg_right (min_le_right c (entropyGap*δ)) hN.le
-      have ht := mul_le_mul_of_nonneg_left (show δ*(μ.card:ℝ)≤(μ.card:ℝ)-μ.colLen 0 by nlinarith) entropyGap_pos.le
-      linarith
-
-end Thorp.Specht
-
-namespace Thorp.DimensionEstimates
-
-section
-open scoped BigOperators Classical
-open Filter
-
-lemma finset_power_sum {ι : Type*} [Fintype ι] (x : ℝ) :
-    ∑ E : Finset ι, x^E.card=(1+x)^Fintype.card ι := by
-  have h := Finset.prod_one_add (f := fun _ : ι => x) Finset.univ
-  simpa only [Finset.prod_const,Finset.card_univ,Finset.powerset_univ] using h.symm
-
-lemma small_sets_count {ι : Type*} [Fintype ι] (ε t : ℝ) (ht : 0≤t) :
-    (∑ E : Finset ι, if (E.card:ℝ)<ε*Fintype.card ι then (1:ℝ) else 0) ≤
-      Real.exp ((t*ε+Real.log (1+Real.exp (-t)))*Fintype.card ι) := by
-  have hpow (E : Finset ι) : (Real.exp (-t))^E.card=Real.exp (-t*(E.card:ℝ)) := by
-    rw [←Real.exp_nat_mul]
-    congr 1
-    ring
-  calc
-    _ ≤ ∑ E : Finset ι, Real.exp (t*ε*Fintype.card ι)*(Real.exp (-t))^E.card := by
-      apply Finset.sum_le_sum
-      intro E _
-      split_ifs with hE
-      · rw [hpow,←Real.exp_add,←Real.exp_zero]
-        apply Real.exp_le_exp.mpr
-        nlinarith [mul_le_mul_of_nonneg_left hE.le ht]
-      · positivity
-    _ = Real.exp (t*ε*Fintype.card ι)*(1+Real.exp (-t))^Fintype.card ι := by
-      rw [←Finset.mul_sum,finset_power_sum]
-    _ = _ := by
-      have he : (1+Real.exp (-t))^Fintype.card ι=
-          Real.exp (Real.log (1+Real.exp (-t))*(Fintype.card ι:ℝ)) := by
-        rw [mul_comm,Real.exp_nat_mul,Real.exp_log (by positivity)]
-      rw [he,←Real.exp_add]
-      congr 1
-      ring
-
-lemma small_sets_ceil_bound {ι : Type*} [Fintype ι] (ε t : ℝ) (ht : 0≤t) :
-    (∑ E : Finset ι, if E.card<⌈ε*Fintype.card ι⌉₊ then (1:ℝ) else 0) ≤
-      Real.exp ((t*ε+Real.log (1+Real.exp (-t)))*Fintype.card ι) := by
-  have he (E : Finset ι) : E.card<⌈ε*Fintype.card ι⌉₊ ↔ (E.card:ℝ)<ε*Fintype.card ι := Nat.lt_ceil
-  simp only [he]
-  exact small_sets_count ε t ht
-
-lemma exists_small_set_rate (κ ε₀ : ℝ) (hκ : 0<κ) (hε₀ : 0<ε₀) :
-    ∃ ε : ℝ, 0<ε ∧ ε≤ε₀ ∧ ε<1/2 ∧ ∃ t : ℝ, 0≤t ∧
-      t*ε+Real.log (1+Real.exp (-t))≤κ := by
-  have ht : Tendsto (fun t : ℝ => Real.log (1+Real.exp (-t))) atTop (nhds 0) := by
-    have he : Tendsto (fun t : ℝ => Real.exp (-t)) atTop (nhds 0) :=
-      Real.tendsto_exp_atBot.comp tendsto_neg_atTop_atBot
-    have hl := (tendsto_const_nhds.add he).log (by norm_num : (1:ℝ)+0≠0)
-    simpa only [add_zero,Real.log_one] using hl
-  obtain ⟨t,ht,hlog⟩ := ((eventually_ge_atTop (1:ℝ)).and
-    (ht.eventually (gt_mem_nhds (by linarith : (0:ℝ)<κ/2)))).exists
-  let ε := min ε₀ (min (1/4) (κ/(2*t)))
-  have htp : 0<t := by linarith
-  have hε : 0<ε := lt_min hε₀ (lt_min (by norm_num) (by positivity))
-  refine ⟨ε,hε,min_le_left _ _,(min_le_right _ _).trans_lt ((min_le_left _ _).trans_lt (by norm_num)),t,htp.le,?_⟩
-  have hsmall : ε≤κ/(2*t) := (min_le_right _ _).trans (min_le_right _ _)
-  have hm := (le_div_iff₀ (by positivity : (0:ℝ)<2*t)).mp hsmall
-  linarith
-
-end
-open Filter
-
-lemma log_error_eventually (a : ℝ) (ha : 0<a) :
-    ∀ᶠ N : ℕ in atTop, 2*Real.log N+1≤(a/2)*(N:ℝ) := by
-  have h := (tendsto_natCast_atTop_atTop (R := ℝ)).eventually
-    (Real.isLittleO_log_id_atTop.bound (by linarith : 0<a/8))
-  have hg := (tendsto_natCast_atTop_atTop (R := ℝ)).eventually (eventually_ge_atTop (4/a))
-  filter_upwards [h,hg,eventually_ge_atTop 1] with N h hg hN
-  have hn : (1:ℝ)≤N := by exact_mod_cast hN
-  simp only [Real.norm_eq_abs,abs_of_nonneg (Real.log_nonneg hn),id_eq,abs_of_nonneg (by positivity : (0:ℝ)≤N)] at h
-  have hh : 4≤(N:ℝ)*a := (div_le_iff₀ ha).mp hg
-  linarith
-
-end Thorp.DimensionEstimates
-
-namespace Thorp.Specht
-open scoped BigOperators Classical
-open DimensionEstimates Filter
-
-theorem medium_dimension (δ : ℝ) (hδ : 0<δ) (hδ2 : δ≤1/2) :
-    ∃ h : ℝ, 0<h ∧ ∀ᶠ N : ℕ in atTop, ∀ μ : YoungDiagram, μ.card=N →
-      δ*(N:ℝ)≤N-μ.rowLen 0 → (μ.colLen 0:ℝ)≤(N:ℝ)/2 →
-      h*(N:ℝ)≤Real.log (Module.finrank ℂ (space μ)) := by
-  let a := min ((3/2:ℝ)*Real.log 2-1) (entropyGap*δ)
-  have ha : 0<a := by
-    apply lt_min
-    · nlinarith [Real.log_two_gt_d9]
-    · exact mul_pos entropyGap_pos hδ
-  refine ⟨a/2,by positivity,?_⟩
-  filter_upwards [log_error_eventually a ha,eventually_ge_atTop 1] with N hN hn
-  intro μ hμ hr hc
-  have hp : 0<μ.card := by omega
-  have hh := medium_dimension_raw μ hp δ hδ hδ2 (by simpa only [hμ] using hr) (by simpa only [hμ] using hc)
-  rw [hμ] at hh
-  change a*(N:ℝ)-2*Real.log N-1≤_ at hh
-  linarith
-
-end Thorp.Specht
-
-namespace Thorp
-open scoped BigOperators Classical
-
-lemma insertNth_castSucc_snoc {α : Type*} {d : ℕ} (i : Fin (d+1)) (b a : α) (y : Fin d → α) :
-    Fin.insertNth i.castSucc b (Fin.snoc (α := fun _ => α) y a)=Fin.snoc (α := fun _ => α) (Fin.insertNth i b y) a := by
-  apply Fin.lastCases ?_ (fun j => ?_) |> fun h => funext h
-  · rw [←show i.castSucc.succAbove (Fin.last d)=Fin.last (d+1) from
-        Fin.succAbove_castSucc_of_le i (Fin.last d) (Fin.le_last i),Fin.insertNth_apply_succAbove]
-    simp
-  · cases j using Fin.succAboveCases i with
-    | x => simp
-    | p j =>
-      have he : i.castSucc.succAbove j.castSucc=(i.succAbove j).castSucc := by
-        simp only [Fin.succAbove,Fin.castSucc_lt_castSucc_iff]
-        split_ifs <;> rfl
-      rw [←he,Fin.insertNth_apply_succAbove]
-      simp
-
-lemma rotate_cons (d : ℕ) (b : Bool) (y : Card d) :
-    rotate (d+1) (Fin.cons b y)=Fin.snoc y b := (Fin.snoc_eq_cons_rotate y b).symm
-
-lemma rotate_insert (d : ℕ) (i : Fin (d+1)) (b : Bool) (y : Card (d+1)) :
-    rotate (d+2) (Fin.insertNth i.succ b y)=Fin.insertNth i.castSucc b (rotate (d+1) y) := by
-  rw [←Fin.cons_self_tail y,Fin.insertNth_succ_cons,rotate_cons,rotate_cons,insertNth_castSucc_snoc]
-
-def insertionPerm (d : ℕ) (i : Fin (d+1)) : Equiv.Perm (Card (d+1)) :=
-  (Fin.consEquiv (fun _ : Fin (d+1) => Bool)).symm.trans
-    (Fin.insertNthEquiv (fun _ : Fin (d+1) => Bool) i)
-
-lemma insertionPerm_apply (d : ℕ) (i : Fin (d+1)) (x : Card (d+1)) :
-    insertionPerm d i x=Fin.insertNth i (x 0) (Fin.tail x) := rfl
-
-lemma insertionPerm_zero (d : ℕ) : insertionPerm d 0=1 := by
-  ext x j
-  simp [insertionPerm_apply]
-
-lemma insertionPerm_last (d : ℕ) : insertionPerm d (Fin.last d)=rotate (d+1) := by
-  apply Equiv.ext
-  intro x
-  rw [insertionPerm_apply,Fin.insertNth_last',←rotate_cons,Fin.cons_self_tail]
-
-lemma physical_carry_step (d : ℕ) (i : Fin (d+1)) (ξ : Bool → Coins (d+1)) :
-    step (d+2) (fun y => ξ (y i) (Fin.removeNth i y))*insertionPerm (d+1) i.succ=
-      insertionPerm (d+1) i.castSucc*childLift (fun b => step (d+1) (ξ b)) := by
-  apply Equiv.ext
-  intro x
-  have he : Fin.cons (Bool.xor ((Fin.tail x) 0) (ξ (x 0) (Fin.tail (Fin.tail x))))
-      (Fin.insertNth i (x 0) (Fin.tail (Fin.tail x))) =
-      (Fin.insertNth i.succ (x 0) (switchFun (ξ (x 0)) (Fin.tail x)) : Card (d+2)) := by
-    simp only [switchFun,Fin.insertNth_succ_cons]
-  have hs : switchFun (fun y => ξ (y i) (Fin.removeNth i y))
-      (Fin.insertNth i.succ (x 0) (Fin.tail x)) =
-      (Fin.insertNth i.succ (x 0) (switchFun (ξ (x 0)) (Fin.tail x)) : Card (d+2)) := by
-    conv_lhs => rw [←Fin.cons_self_tail (Fin.tail x),Fin.insertNth_succ_cons]
-    simp only [switchFun,Fin.cons_zero,Fin.tail_cons,Fin.insertNth_apply_same,Fin.removeNth_insertNth]
-    exact he
-  change rotate (d+2) (switchFun (fun y => ξ (y i) (Fin.removeNth i y))
-      (Fin.insertNth i.succ (x 0) (Fin.tail x))) = _
-  rw [hs,rotate_insert]
-  rfl
-
-end Thorp
-namespace Thorp.UnitaryFinite
-open scoped BigOperators Classical
-variable {G : Type*} [Group G]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-variable {Ω : Type*} [Fintype Ω]
-
-lemma sampleOperator_mul_const (ρ : Representation ℂ G V) (P : Ω → G) (g : G) :
-    sampleOperator ρ (fun ω => P ω*g)=sampleOperator ρ P*continuousRepresentation ρ g := by
-  simp only [sampleOperator,map_mul,Finset.sum_mul,smul_mul_assoc]
-
-lemma sampleOperator_const_mul (ρ : Representation ℂ G V) (P : Ω → G) (g : G) :
-    sampleOperator ρ (fun ω => g*P ω)=continuousRepresentation ρ g*sampleOperator ρ P := by
-  simp only [sampleOperator,map_mul,Finset.mul_sum,mul_smul_comm]
-
-lemma sampleOperator_run (d t : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V) :
-    sampleOperator ρ (run d t)=(sampleOperator ρ (step d))^t := by
-  induction t with
-  | zero => exact sampleOperator_one ρ
-  | succ t ih =>
-    let e : Coins d × (Fin t → Coins d) ≃ (Fin (t+1) → Coins d) :=
-      Fin.snocEquiv (fun _ => Coins d)
-    have he : run d (t+1) ∘ e=fun w => step d w.1*run d t w.2 := by
-      funext w
-      simp only [Function.comp_apply,e,Fin.snocEquiv_apply,
-        run,Fin.snoc_last,Fin.snoc_castSucc]
-    rw [←sampleOperator_equiv ρ (run d (t+1)) e,he,sampleOperator_prod,ih,pow_succ']
-
-end Thorp.UnitaryFinite
-
-namespace Thorp
-open scoped BigOperators Classical
-open UnitaryFinite
-
-def carryCoins (d : ℕ) (i : Fin (d+1)) : (Bool → Coins (d+1)) ≃ Coins (d+2) where
-  toFun ξ y := ξ (y i) (Fin.removeNth i y)
-  invFun ξ b y := ξ (Fin.insertNth i b y)
-  left_inv ξ := by funext b y; simp
-  right_inv ξ := by funext y; simp
-
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-lemma physical_carry_average (d : ℕ) (i : Fin d)
-    (ρ : Representation ℂ (Equiv.Perm (Card (d+1))) V) :
-    sampleOperator ρ (step (d+1))*continuousRepresentation ρ (insertionPerm d i.succ)=
-      continuousRepresentation ρ (insertionPerm d i.castSucc)*
-        sampleOperator ρ (fun ξ : Bool → Coins d => childLift (fun b => step d (ξ b))) := by
-  cases d with
-  | zero => exact Fin.elim0 i
-  | succ d =>
-    have he : (fun ξ : Bool → Coins (d+1) =>
-        step (d+2) (carryCoins d i ξ)*insertionPerm (d+1) i.succ)=
-      (fun ξ => insertionPerm (d+1) i.castSucc*childLift (fun b => step (d+1) (ξ b))) := by
-      funext ξ
-      exact physical_carry_step d i ξ
-    have h := congrArg (sampleOperator ρ) he
-    rw [sampleOperator_mul_const,sampleOperator_const_mul] at h
-    change sampleOperator ρ (step (d+2) ∘ carryCoins d i)*_=_ at h
-    rw [sampleOperator_equiv] at h
-    exact h
-
-theorem physical_sweep_recursion (d : ℕ)
-    (ρ : Representation ℂ (Equiv.Perm (Card (d+1))) V) :
-    (sampleOperator ρ (step (d+1)))^(d+1)=
-      (sampleOperator ρ (fun ξ : Bool → Coins d => childLift (fun b => step d (ξ b))))^d*
-        sampleOperator ρ (pairSwitch (d := d)) := by
-  let A := sampleOperator ρ (step (d+1))
-  let K := sampleOperator ρ (fun ξ : Bool → Coins d => childLift (fun b => step d (ξ b)))
-  have h (t : ℕ) (ht : t≤d) : A^t*continuousRepresentation ρ (insertionPerm d ⟨t,by omega⟩)=K^t := by
-    induction t with
-    | zero =>
-      change 1*continuousRepresentation ρ (insertionPerm d 0)=1
-      rw [insertionPerm_zero,map_one,one_mul]
-    | succ t ih =>
-      have hc := physical_carry_average d (⟨t,by omega⟩ : Fin d) ρ
-      change A*continuousRepresentation ρ (insertionPerm d ⟨t+1,by omega⟩)=
-        continuousRepresentation ρ (insertionPerm d ⟨t,by omega⟩)*K at hc
-      rw [pow_succ,mul_assoc,hc,←mul_assoc,ih (by omega),←pow_succ]
-  have he : sampleOperator ρ (step (d+1))=
-      continuousRepresentation ρ (rotate (d+1))*sampleOperator ρ (pairSwitch (d := d)) :=
-    sampleOperator_const_mul ρ (pairSwitch (d := d)) (rotate (d+1))
-  calc
-    _ = A^d*A := pow_succ A d
-    _ = A^d*(continuousRepresentation ρ (insertionPerm d (Fin.last d))*
-        sampleOperator ρ (pairSwitch (d := d))) := by rw [insertionPerm_last]; rw [←he]
-    _ = _ := by
-      rw [←mul_assoc]
-      exact congrArg (fun z => z*sampleOperator ρ (pairSwitch (d := d))) (h d le_rfl)
-
-end Thorp
-namespace Thorp
-open scoped BigOperators Classical
-open UnitaryFinite
-
-lemma childInjection_product (d : ℕ) (p : Bool → Equiv.Perm (Card d)) :
-    childInjection d false (p false)*childInjection d true (p true)=childLift p := by
-  change childLift _*childLift _=_
-  rw [childLift_mul]
-  congr 1
-  funext b
-  cases b <;> simp
-
-lemma childInjection_commute (d : ℕ) (p q : Equiv.Perm (Card d)) :
-    Commute (childInjection d false p) (childInjection d true q) := by
-  change childLift _*childLift _=childLift _*childLift _
-  rw [childLift_mul,childLift_mul]
-  congr 1
-  funext b
-  cases b <;> simp
-
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-variable {Ω : Type*} [Fintype Ω]
-
-lemma sampleOperator_child (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card (d+1))) V)
-    (P : Ω → Equiv.Perm (Card d)) :
-    sampleOperator ρ (fun ω : Bool → Ω => childLift (fun b => P (ω b)))=
-      sampleOperator (ρ.comp (childInjection d false)) P*
-        sampleOperator (ρ.comp (childInjection d true)) P := by
-  let e := (Equiv.boolArrowEquivProd Ω).symm
-  have he : (fun ω : Bool → Ω => childLift (fun b => P (ω b))) ∘ e=
-      fun w : Ω × Ω => childInjection d false (P w.1)*childInjection d true (P w.2) := by
-    funext w
-    change childLift (fun b => P (e w b))=_
-    rw [←childInjection_product]
-    rfl
-  rw [←sampleOperator_equiv ρ _ e,he]
-  exact sampleOperator_prod ρ (fun w => childInjection d false (P w))
-    (fun w => childInjection d true (P w))
-
-lemma sampleOperator_child_commute (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card (d+1))) V)
-    (P : Ω → Equiv.Perm (Card d)) :
-    Commute (sampleOperator (ρ.comp (childInjection d false)) P)
-      (sampleOperator (ρ.comp (childInjection d true)) P) := by
-  rw [sampleOperator_comp,sampleOperator_comp]
-  exact sampleOperator_commute ρ _ _ (fun _ _ => childInjection_commute d _ _)
-
-lemma inverse_butterfly_average_step (d : ℕ)
-    (ρ : Representation ℂ (Equiv.Perm (Card (d+1))) V) :
-    sampleOperator ρ (fun ω => (butterflyPerm (d+1) (decodeButterfly (d+1) ω))⁻¹)=
-      sampleOperator ρ (fun ω : Bool → (SwitchIndex d → Bool) =>
-        childLift (fun b => (butterflyPerm d (decodeButterfly d (ω b)))⁻¹))*
-      sampleOperator ρ (pairSwitch (d := d)) := by
-  let e := ((Equiv.prodCongr (Equiv.boolArrowEquivProd (SwitchIndex d → Bool)) (Equiv.refl (Card d → Bool))).trans
-    (coinStepEquiv d).symm)
-  have he : (fun ω => (butterflyPerm (d+1) (decodeButterfly (d+1) ω))⁻¹) ∘ e=
-      fun w : (Bool → (SwitchIndex d → Bool)) × (Card d → Bool) =>
-        childLift (fun b => (butterflyPerm d (decodeButterfly d (w.1 b)))⁻¹)*pairSwitch w.2 := by
-    funext w
-    change (butterflyPerm (d+1) (decodeButterfly (d+1)
-      ((coinStepEquiv d).symm ((w.1 false,w.1 true),w.2))))⁻¹=_
-    rw [butterflyPerm_step,mul_inv_rev,childLift_inv,pairSwitch_inv]
-    congr 2
-    funext b
-    cases b <;> rfl
-  rw [←sampleOperator_equiv ρ _ e,he]
-  exact sampleOperator_prod ρ
-    (fun ω : Bool → (SwitchIndex d → Bool) => childLift (fun b => (butterflyPerm d (decodeButterfly d (ω b)))⁻¹))
-    (pairSwitch (d := d))
-
-theorem physical_sweep_inverse_butterfly (d : ℕ)
-    (ρ : Representation ℂ (Equiv.Perm (Card d)) V) :
-    (sampleOperator ρ (step d))^d=
-      sampleOperator ρ (fun ω => (butterflyPerm d (decodeButterfly d ω))⁻¹) := by
-  induction d with
-  | zero =>
-    symm
-    exact sampleOperator_one ρ
-  | succ d ih =>
-    rw [physical_sweep_recursion,sampleOperator_child,
-      (sampleOperator_child_commute d ρ (step d)).mul_pow,ih,ih,
-      inverse_butterfly_average_step]
-    rw [sampleOperator_child d ρ (fun ω => (butterflyPerm d (decodeButterfly d ω))⁻¹)]
-
-lemma physical_sweep_adjoint (d : ℕ)
-    (ρ : Representation ℂ (Equiv.Perm (Card d)) V) (hρ : IsUnitary ρ) :
-    sampleOperator ρ (run d d)=
-      (sampleOperator ρ (fun ω => butterflyPerm d (decodeButterfly d ω))).adjoint := by
-  rw [sampleOperator_run,physical_sweep_inverse_butterfly,sampleOperator_inv ρ hρ]
-
-end Thorp
-
-namespace Thorp.Specht
-open scoped BigOperators Classical
-open UnitaryFinite DimensionEstimates Filter
-
-lemma dimension_pos (μ : YoungDiagram) : 0 < Module.finrank ℂ (space μ) := by
-  have h := factorial_le_dimension_row_column μ
-  by_contra hn
-  have hz : Module.finrank ℂ (space μ)=0 := by omega
-  rw [hz,zero_mul,zero_mul] at h
-  exact (Nat.factorial_pos μ.card).not_ge h
-
-lemma near_numerical_cutoff :
-    ∃ X : ℝ, 2 ≤ X ∧ ∀ (N j D : ℕ), 0 < j → X ≤ (N:ℝ)/j → N.choose j ≤ D * 2^j →
-      ∃ k : ℕ, j ≤ k ∧ k ≤ N ∧ ∀ m : ℕ, D * k.choose j ≤ m * N.choose j →
-        nearFraction * Real.log D ≤ Real.log m ∧
-        densityConstant * (k:ℝ) * ((k:ℝ)/N)^(1/512:ℝ) ≤ Real.log m/4000 ∧
-        (1/2:ℝ) * Real.log N ≤ Real.log D ∧ (j:ℝ) ≤ Real.log D := by
-  obtain ⟨X,hX,h⟩ := exists_near_cutoff densityConstant densityConstant_pos.le
-  refine ⟨X,hX,?_⟩
-  intro N j D hj hx hd
-  have hjN : j ≤ N := by
-    have hjp : (0:ℝ) < j := by exact_mod_cast hj
-    have hh : (j:ℝ) ≤ N := by
-      have := (le_div_iff₀ hjp).mp (hX.trans hx)
-      linarith
-    exact_mod_cast hh
-  have hc : 1 ≤ N.choose j := Nat.choose_pos hjN
-  have hfoot : D * (chosenTuple N j).choose j ≤ (D * (chosenTuple N j).choose j) * N.choose j :=
-    Nat.le_mul_of_pos_right _ hc
-  have hh := h N j D _ hj hx hd hfoot
-  refine ⟨chosenTuple N j,hh.1,hh.2.1,?_⟩
-  intro m hm
-  exact (h N j D m hj hx hd hm).2.2
-
-lemma near_row_amplification (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Cell μ)
-    {k : ℕ} (f : Fin k ↪ Card d) (hk : 0<k)
-    (hm : 0<Module.finrank ℂ (fixedSpace (V := hilbertSpace μ) tupleAction f (relabelledUnitary μ e))) :
-    ‖sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e) (palindromePerm d)‖ ≤
-      Real.exp (densityConstant*(k:ℝ)*((k:ℝ)/(2:ℝ)^d)^(1/512:ℝ))*
-        (Real.exp (Real.log (Module.finrank ℂ (fixedSpace (V := hilbertSpace μ) tupleAction f (relabelledUnitary μ e)))/2))^(-(1/1000:ℝ))+
-      Real.sqrt ((Real.exp (Real.log (Module.finrank ℂ (fixedSpace (V := hilbertSpace μ) tupleAction f (relabelledUnitary μ e)))/2))/
-        (Module.finrank ℂ (fixedSpace (V := hilbertSpace μ) tupleAction f (relabelledUnitary μ e)):ℝ)) := by
-  let := relabelledUnitary_irreducible μ e
-  let : NormedAddCommGroup (hilbertSpace μ) := inferInstance
-  let : InnerProductSpace ℂ (hilbertSpace μ) := inferInstance
-  let : FiniteDimensional ℂ (hilbertSpace μ) := inferInstance
-  have aux := @palindrome_irrep_amplification (hilbertSpace μ) _ _ _ d (Fin k) _ f
-    (by simpa only [Fintype.card_fin] using hk) (relabelledUnitary μ e) inferInstance
-    (relabelledUnitary_unitary μ e) hm
-    (Real.exp (Real.log (Module.finrank ℂ (fixedSpace (V := hilbertSpace μ) tupleAction f (relabelledUnitary μ e)))/2))
-    (Real.exp_pos _)
-  simpa only [Fintype.card_fin] using aux
-
-theorem near_row_spectral :
-    ∃ X : ℝ, 2 ≤ X ∧ ∀ (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Cell μ),
-      0 < μ.card-μ.rowLen 0 → X ≤ (μ.card:ℝ)/(μ.card-μ.rowLen 0:ℕ) →
-      32768000 * Real.log 2 ≤ (1/2:ℝ) * Real.log μ.card →
-      ‖sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e) (palindromePerm d)‖ ≤
-        Real.exp (-(nearFraction/8000) * Real.log (Module.finrank ℂ (space μ))) ∧
-      (1/2:ℝ) * Real.log μ.card ≤ Real.log (Module.finrank ℂ (space μ)) ∧
-      (μ.card-μ.rowLen 0:ℕ) ≤ Real.log (Module.finrank ℂ (space μ)) := by
-  obtain ⟨X,hX,h⟩ := near_numerical_cutoff
-  refine ⟨X,hX,?_⟩
-  intro d μ e hj hx hlarge
-  let j := μ.card-μ.rowLen 0
-  let D := Module.finrank ℂ (space μ)
-  have hcard : μ.card=2^d := by rw [←card_cell,←Fintype.card_congr e,card_positions]
-  obtain ⟨k,hjk,hk,hnum⟩ := h μ.card j D hj hx (choose_tail_le_dimension μ)
-  obtain ⟨f⟩ := Function.Embedding.nonempty_of_card_le
-    (show Fintype.card (Fin k) ≤ Fintype.card (Card d) by simpa only [Fintype.card_fin,card_positions,←hcard] using hk)
-  let m := Module.finrank ℂ (fixedSpace (V := hilbertSpace μ) tupleAction f (relabelledUnitary μ e))
-  have hfoot : D * k.choose j ≤ m * μ.card.choose j := by
-    simpa only [Fintype.card_fin,finrank_hilbertSpace] using relabelled_tuple_dimension μ e f
-  have hm : 0 < m := by
-    have hc := Nat.choose_pos hjk
-    have hd := dimension_pos μ
-    by_contra hn
-    have hz : m=0 := by omega
-    rw [hz,zero_mul] at hfoot
-    exact (Nat.mul_pos hd hc).not_ge hfoot
-  obtain ⟨hlm,hden,hlog,hjlog⟩ := hnum m hfoot
-  have ha := near_row_amplification d μ e f (hj.trans_le hjk) hm
-  have hcardR : (μ.card:ℝ)=(2:ℝ)^d := by exact_mod_cast hcard
-  rw [←hcardR] at ha
-  have hl : 8000 * Real.log 2 ≤ Real.log m := by
-    dsimp [nearFraction] at hlm
-    linarith
-  have hn := amplification_decay (m:ℝ) (densityConstant * k * ((k:ℝ)/μ.card)^(1/512:ℝ))
-    (by exact_mod_cast hm) hden hl
-  refine ⟨ha.trans (hn.trans ?_),hlog,hjlog⟩
-  apply Real.exp_le_exp.mpr
-  dsimp [D] at hlm
-  linarith
-
-end Thorp.Specht
-
-namespace Thorp.UnitaryFinite
-open scoped BigOperators Classical
-open DimensionEstimates
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-theorem full_palindrome_decay (d : ℕ)
-    (ρ : Representation ℂ (Equiv.Perm (Card d)) V) [Representation.IsIrreducible ρ]
-    (hρ : IsUnitary ρ) (hm : 0<Module.finrank ℂ V)
-    (hden : densityConstant*(2:ℝ)^d≤Real.log (Module.finrank ℂ V)/4000)
-    (hl : 8000*Real.log 2≤Real.log (Module.finrank ℂ V)) :
-    ‖sampleOperator ρ (palindromePerm d)‖≤Real.exp (-Real.log (Module.finrank ℂ V)/8000) := by
-  have h := palindrome_irrep_amplification d (Function.Embedding.refl (Card d))
-    (Fintype.card_pos) ρ hρ (by simpa only [full_tuple_finrank] using hm)
-    (Real.exp (Real.log (Module.finrank ℂ V)/2)) (Real.exp_pos _)
-  simp only [full_tuple_finrank,card_positions,Nat.cast_pow,Nat.cast_ofNat,
-    div_self (by positivity : (2:ℝ)^d≠0),Real.one_rpow,mul_one] at h
-  exact h.trans (amplification_decay _ _ (by exact_mod_cast hm) hden hl)
-
-theorem full_modified_decay (L r : ℕ) (E : Finset (Card r))
-    (ρ : Representation ℂ (Equiv.Perm (Card (L+r))) V) [Representation.IsIrreducible ρ]
-    (hρ : IsUnitary ρ) (hm : 0<Module.finrank ℂ V) (u : ℝ)
-    (hM : ∀ e : Card (L+r) ↪ Card (L+r), modifiedRowMoment L r E e (1/1000)≤Real.exp u)
-    (hden : u≤Real.log (Module.finrank ℂ V)/4000)
-    (hl : 8000*Real.log 2≤Real.log (Module.finrank ℂ V)) :
-    ‖sampleOperator ρ (modifiedPerm L r E)‖≤Real.exp (-Real.log (Module.finrank ℂ V)/8000) := by
-  exact (modified_irrep_amplification L r E ρ hρ hm u
-    (Real.exp (Real.log (Module.finrank ℂ V)/2)) (Real.exp_pos _) hM).trans
-    (amplification_decay _ _ (by exact_mod_cast hm) hden hl)
-
-lemma palindrome_exception_cards (r L : ℕ)
-    (ρ : Representation ℂ (Equiv.Perm (Card (L+r))) V) (hρ : IsUnitary ρ) (n : ℕ) :
-    ‖sampleOperator ρ (palindromePerm (L+r))‖ ≤
-      (blockGap L)^n+∑ E : Finset (Card r), if E.card<n then
-        ‖sampleOperator ρ (modifiedPerm L r E)‖ else 0 := by
-  let σ := ρ.comp (splitPerm r L).symm.toMonoidHom
-  have he : σ.comp (splitPerm r L).toMonoidHom=ρ := by
-    ext g v
-    change ρ ((splitPerm r L).symm ((splitPerm r L) g)) v=ρ g v
-    rw [MulEquiv.symm_apply_apply]
-  have h := palindrome_exception_bound r L σ (unitary_comp ρ hρ _) n
-  simpa only [he] using h
-
-end Thorp.UnitaryFinite
-
-namespace Thorp.DimensionEstimates
-open scoped BigOperators Classical
-open Filter
-
-lemma medium_parameters (h : ℝ) (hh : 0<h) :
-    ∃ L : ℕ, 0<L ∧ ∃ ε t : ℝ, 0<ε ∧ ε<1/2 ∧ 0≤t ∧
-      t*ε+Real.log (1+Real.exp (-t))≤h*(2:ℝ)^L/16000 ∧
-      ∀ᶠ r : ℕ in atTop, ∀ (E : Finset (Card r)), (E.card:ℝ)<ε*(2:ℝ)^r →
-        ∀ e : Card (L+r) ↪ Card (L+r),
-          modifiedRowMoment L r E e (1/1000)≤Real.exp ((h/4000)*(2:ℝ)^(L+r)) := by
-  obtain ⟨L,hL,ε₀,hε₀,_,hM⟩ := modified_arbitrarily_small_rate (h/4000) (by positivity)
-  obtain ⟨ε,hε,hεle,hεhalf,t,ht,hcount⟩ := exists_small_set_rate
-    (h*(2:ℝ)^L/16000) ε₀ (by positivity) hε₀
-  refine ⟨L,hL,ε,t,hε,hεhalf,ht,hcount,?_⟩
-  filter_upwards [hM] with r hM
-  intro E hE e
-  exact hM E (hE.le.trans (mul_le_mul_of_nonneg_right hεle (by positivity))) e
-
-lemma geometric_ceil_bound (θ ε : ℝ) (hθ : 0<θ) (hθ1 : θ<1) (M : ℕ) :
-    θ^⌈ε*M⌉₊≤Real.exp (Real.log θ*ε*M) := by
-  conv_lhs => rw [←Real.exp_log hθ,←Real.exp_nat_mul]
-  apply Real.exp_le_exp.mpr
-  have hg := mul_le_mul_of_nonpos_left (Nat.le_ceil (ε*M)) (Real.log_nonpos hθ.le hθ1.le)
-  nlinarith
-
-end Thorp.DimensionEstimates
-
-namespace Thorp.UnitaryFinite
-open scoped BigOperators Classical
-open DimensionEstimates
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-theorem medium_exception_decay (L r : ℕ) (h ε t : ℝ) (ht : 0≤t)
-    (hcount : t*ε+Real.log (1+Real.exp (-t))≤h*(2:ℝ)^L/16000)
-    (ρ : Representation ℂ (Equiv.Perm (Card (L+r))) V) [Representation.IsIrreducible ρ]
-    (hρ : IsUnitary ρ) (hm : 0<Module.finrank ℂ V)
-    (hd : h*(2:ℝ)^(L+r)≤Real.log (Module.finrank ℂ V))
-    (hl : 8000*Real.log 2≤Real.log (Module.finrank ℂ V))
-    (hM : ∀ (E : Finset (Card r)), (E.card:ℝ)<ε*(2:ℝ)^r →
-      ∀ e : Card (L+r) ↪ Card (L+r),
-        modifiedRowMoment L r E e (1/1000)≤Real.exp ((h/4000)*(2:ℝ)^(L+r))) :
-    ‖sampleOperator ρ (palindromePerm (L+r))‖≤
-      Real.exp (Real.log (blockGap L)*ε*(2:ℝ)^r)+Real.exp (-h*(2:ℝ)^(L+r)/16000) := by
-  have hmE (E : Finset (Card r)) (hE : E.card<⌈ε*(2:ℝ)^r⌉₊) :
-      ‖sampleOperator ρ (modifiedPerm L r E)‖≤Real.exp (-h*(2:ℝ)^(L+r)/8000) := by
-    apply (full_modified_decay L r E ρ hρ hm ((h/4000)*(2:ℝ)^(L+r))
-      (hM E (Nat.lt_ceil.mp hE)) (by linarith) hl).trans
-    apply Real.exp_le_exp.mpr
-    linarith
-  have hs : (∑ E : Finset (Card r), if E.card<⌈ε*(2:ℝ)^r⌉₊ then
-      ‖sampleOperator ρ (modifiedPerm L r E)‖ else 0) ≤
-      Real.exp (-h*(2:ℝ)^(L+r)/16000) := by
-    calc
-      _ ≤ (∑ E : Finset (Card r), if E.card<⌈ε*(2:ℝ)^r⌉₊ then (1:ℝ) else 0)*
-          Real.exp (-h*(2:ℝ)^(L+r)/8000) := by
-        rw [Finset.sum_mul]
-        apply Finset.sum_le_sum
-        intro E _
-        split_ifs with hE
-        · simpa only [one_mul] using hmE E hE
-        · simp
-      _ ≤ Real.exp ((t*ε+Real.log (1+Real.exp (-t)))*(2:ℝ)^r)*
-          Real.exp (-h*(2:ℝ)^(L+r)/8000) := by
-        apply mul_le_mul_of_nonneg_right _ (Real.exp_pos _).le
-        simpa only [card_positions,Nat.cast_pow,Nat.cast_ofNat] using
-          small_sets_ceil_bound (ι := Card r) ε t ht
-      _ ≤ _ := by
-        rw [←Real.exp_add]
-        apply Real.exp_le_exp.mpr
-        have hh := mul_le_mul_of_nonneg_right hcount (by positivity : (0:ℝ)≤(2:ℝ)^r)
-        rw [pow_add]
-        nlinarith
-  apply (palindrome_exception_cards r L ρ hρ ⌈ε*(2:ℝ)^r⌉₊).trans
-  apply add_le_add _ hs
-  simpa only [Nat.cast_pow,Nat.cast_ofNat] using
-    geometric_ceil_bound (blockGap L) ε (blockGap_pos L) (blockGap_lt_one L) (2^r)
-
-end Thorp.UnitaryFinite
-
-namespace Thorp.Specht
-open scoped BigOperators Classical
-
-lemma partition_count (N : ℕ) : Fintype.card (Nat.Partition N)≤2^N := by
-  have h := Fintype.card_le_of_surjective (Nat.Partition.ofComposition N) Nat.Partition.ofComposition_surj
-  rw [composition_card] at h
-  exact h.trans (Nat.pow_le_pow_right (by norm_num) (Nat.sub_le N 1))
-
-lemma shapes_count (N : ℕ) : Fintype.card (Shapes N)≤2^N := by
-  rw [Fintype.card_congr (shapePartitionEquiv N)]
-  exact partition_count N
-
-lemma rowLens_headD (μ : YoungDiagram) : μ.rowLens.headD 0=μ.rowLen 0 := by
-  cases he : μ.rowLens with
-  | nil =>
-    have hc := sum_rowLens μ
-    rw [he,List.sum_nil] at hc
-    have hr := rowLen_zero_le_card μ
-    simp only [List.headD_nil]
-    omega
-  | cons a l =>
-    have hh := YoungDiagram.get_rowLens (μ := μ) (i := 0) (h := by simp [he])
-    simpa only [he,List.getElem_cons_zero,List.headD_cons] using hh
-
-lemma rowLens_tail_sum (μ : YoungDiagram) : μ.rowLens.tail.sum=μ.card-μ.rowLen 0 := by
-  have hh := rowLens_headD μ
-  have hc := sum_rowLens μ
-  cases he : μ.rowLens with
-  | nil => simp only [he,List.headD_nil] at hh; simp only [he,List.sum_nil] at hc; simp [←hc]
-  | cons a l =>
-    simp only [he,List.headD_cons] at hh
-    simp only [he,List.sum_cons] at hc
-    simp only [List.tail_cons]
-    omega
-
-lemma positive_list_eq_of_tail_sum {a b : List ℕ}
-    (ha : ∀ x∈a, 0<x) (hb : ∀ x∈b, 0<x) (ht : a.tail=b.tail) (hs : a.sum=b.sum) : a=b := by
-  cases a with
-  | nil =>
-    cases b with
-    | nil => rfl
-    | cons x l =>
-      simp only [List.tail_nil,List.tail_cons] at ht
-      subst l
-      have hx := hb x (by simp)
-      simp only [List.sum_nil,List.sum_cons,add_zero] at hs
-      omega
-  | cons x l =>
-    cases b with
-    | nil =>
-      simp only [List.tail_nil,List.tail_cons] at ht
-      subst l
-      have hx := ha x (by simp)
-      simp only [List.sum_nil,List.sum_cons,add_zero] at hs
-      omega
-    | cons y m =>
-      simp only [List.tail_cons] at ht
-      subst m
-      simp only [List.sum_cons] at hs
-      congr 1
-      omega
-
-end Thorp.Specht
-
-end ThorpNine.Tail
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Tail.MainStatement
+
+namespace OAI
+
+namespace ThorpNine.Tail
+
+namespace Thorp.DimensionEstimates
+open scoped BigOperators Classical
+
+section
+open Filter
+
+theorem exists_near_cutoff (C : ℝ) (hC : 0≤C) :
+    ∃ X : ℝ, 2≤X ∧ ∀ (N j D m : ℕ), 0<j → X≤(N:ℝ)/j →
+      N.choose j≤D*2^j →
+      D * (chosenTuple N j).choose j ≤ m * N.choose j →
+      j≤chosenTuple N j ∧ chosenTuple N j≤N ∧
+      nearFraction*Real.log D≤Real.log m ∧
+      C*(chosenTuple N j:ℝ)*((chosenTuple N j:ℝ)/N)^(1/512:ℝ)≤Real.log m/4000 ∧
+      (1/2:ℝ)*Real.log N≤Real.log D ∧ (j:ℝ)≤Real.log D := by
+  obtain ⟨X,hX⟩ := eventually_atTop.mp (near_conditions_eventually C)
+  refine ⟨max 2 X,le_max_left _ _,?_⟩
+  intro N j D m hj hNx hD hfoot
+  have hc := hX ((N:ℝ)/j) ((le_max_right _ _).trans hNx)
+  have hjp : (0:ℝ)<j := by exact_mod_cast hj
+  have hN : 2*j≤N := by exact_mod_cast (le_div_iff₀ hjp).mp hc.1
+  have hk := chosenTuple_bounds N j hj (by linarith [hc.1]) hc.2.2.1
+  have hm := chosenTuple_log_footprint N j D m hj hN hD hc.2.1 hc.2.2.2.1 hfoot
+  have hd := near_row_log_uniform N j D hj hN hD hc.2.2.2.2.1
+  refine ⟨hk.1,hk.2.1,hm,?_,hd⟩
+  apply (chosenTuple_density_bound N j hj (by linarith [hc.1]) hc.2.2.1 C hC).trans
+  have hh := mul_le_mul_of_nonneg_right hc.2.2.2.2.2 hjp.le
+  dsimp [nearFraction] at hm ⊢
+  dsimp [nearFraction] at hh
+  nlinarith [hd.2]
+
+end
+
+lemma amplification_numerical (m u : ℝ) (hm : 1 ≤ m) (hu : u ≤ Real.log m/4000) :
+    Real.exp u*(Real.exp (Real.log m/2))^(-(1/1000:ℝ))+
+      Real.sqrt (Real.exp (Real.log m/2)/m) ≤ 2*Real.exp (-Real.log m/4000) := by
+  have hmp : 0 < m := lt_of_lt_of_le zero_lt_one hm
+  have hl : 0≤Real.log m := Real.log_nonneg hm
+  have he : Real.exp (Real.log m/2)/m=Real.exp (-Real.log m/2) := by
+    calc
+      _ = Real.exp (Real.log m/2)/Real.exp (Real.log m) := by rw [Real.exp_log hmp]
+      _ = _ := by rw [←Real.exp_sub]; congr 1; ring
+  have hs : Real.sqrt (Real.exp (-Real.log m/2))=Real.exp (-Real.log m/4) := by
+    rw [Real.sqrt_eq_rpow,Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp]
+    congr 1
+    ring
+  rw [Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp,←Real.exp_add,he,hs]
+  have h₁ : u+(Real.log m/2)*(-(1/1000:ℝ))≤ -Real.log m/4000 := by linarith
+  have h₂ : -Real.log m/4≤ -Real.log m/4000 := by linarith
+  linarith [Real.exp_le_exp.mpr h₁,Real.exp_le_exp.mpr h₂]
+
+lemma amplification_absorb (x : ℝ) (hx : 8000*Real.log 2≤x) :
+    2*Real.exp (-x/4000)≤Real.exp (-x/8000) := by
+  rw [show (2:ℝ)=Real.exp (Real.log 2) by rw [Real.exp_log (by norm_num)],←Real.exp_add]
+  exact Real.exp_le_exp.mpr (by linarith)
+
+lemma amplification_decay (m u : ℝ) (hm : 1 ≤ m) (hu : u ≤ Real.log m/4000)
+    (hl : 8000*Real.log 2≤Real.log m) :
+    Real.exp u*(Real.exp (Real.log m/2))^(-(1/1000:ℝ))+
+      Real.sqrt (Real.exp (Real.log m/2)/m) ≤ Real.exp (-Real.log m/8000) :=
+  (amplification_numerical m u hm hu).trans (amplification_absorb _ hl)
+
+end Thorp.DimensionEstimates
+
+namespace Thorp.UnitaryFinite
+open scoped BigOperators Classical
+
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+omit [FiniteDimensional ℂ V] in
+lemma full_tuple_fixed (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V) :
+    fixedSpace tupleAction (Function.Embedding.refl (Card d)) ρ=⊤ := by
+  apply Submodule.eq_top_iff'.mpr
+  intro v g hg
+  have he : g=1 := by
+    apply Equiv.ext
+    intro x
+    exact congrArg (fun f : Card d ↪ Card d => f x) hg
+  rw [he,map_one]
+  rfl
+
+omit [FiniteDimensional ℂ V] in
+lemma full_tuple_finrank (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V) :
+    Module.finrank ℂ (fixedSpace tupleAction (Function.Embedding.refl (Card d)) ρ)=Module.finrank ℂ V := by
+  rw [full_tuple_fixed,finrank_top]
+
+theorem palindrome_irrep_amplification (d : ℕ) {ι : Type*} [Fintype ι]
+    (f : ι ↪ Card d) (hk : 0<Fintype.card ι)
+    (ρ : Representation ℂ (Equiv.Perm (Card d)) V) [Representation.IsIrreducible ρ]
+    (hρ : IsUnitary ρ) (hm : 0<Module.finrank ℂ (fixedSpace tupleAction f ρ))
+    (z : ℝ) (hz : 0<z) :
+    ‖sampleOperator ρ (palindromePerm d)‖ ≤
+      Real.exp (densityConstant*Fintype.card ι*((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ))*
+        z^(-(1/1000:ℝ))+Real.sqrt (z/(Module.finrank ℂ (fixedSpace tupleAction f ρ):ℝ)) := by
+  apply sampleOperator_amplification tupleAction f (tupleAction_transitive f) ρ hρ
+    (palindromePerm d) _ _ (1/1000) z _ (by norm_num) hz (by positivity) _ hm
+  · intro e f
+    simp only [tupleAction_kernel]
+    exact palindrome_kernel_symmetric d e f
+  · intro e
+    simp only [tupleAction_kernel]
+    exact PairRouting.tupleProbability_row_sum _ e
+  · intro e
+    simp only [tupleAction_kernel]
+    exact palindrome_row_moment_bound d e hk
+
+theorem modified_irrep_amplification (L r : ℕ) (E : Finset (Card r))
+    (ρ : Representation ℂ (Equiv.Perm (Card (L+r))) V) [Representation.IsIrreducible ρ]
+    (hρ : IsUnitary ρ) (hm : 0<Module.finrank ℂ V)
+    (u z : ℝ) (hz : 0<z)
+    (hM : ∀ e : Card (L+r) ↪ Card (L+r), modifiedRowMoment L r E e (1/1000)≤Real.exp u) :
+    ‖sampleOperator ρ (modifiedPerm L r E)‖ ≤
+      Real.exp u*z^(-(1/1000:ℝ))+Real.sqrt (z/(Module.finrank ℂ V:ℝ)) := by
+  let f := Function.Embedding.refl (Card (L+r))
+  have h := sampleOperator_amplification tupleAction f (tupleAction_transitive f) ρ hρ
+    (modifiedPerm L r E) (fun e f => by simpa only [tupleAction_kernel] using modified_kernel_symmetric L r E e f)
+    (fun e => by simpa only [tupleAction_kernel] using PairRouting.tupleProbability_row_sum (modifiedPerm L r E) e)
+    (1/1000) z (Real.exp u) (by norm_num) hz (by positivity)
+    (fun e => by simpa only [tupleAction_kernel,modifiedRowMoment] using hM e)
+    (by simpa only [f,full_tuple_finrank] using hm)
+  simpa only [f,full_tuple_finrank] using h
+
+end Thorp.UnitaryFinite
+
+namespace Thorp.Specht
+open scoped BigOperators Classical
+
+theorem choose_tail_le_of_factorial (μ : YoungDiagram) (D : ℕ)
+    (hd : μ.card.factorial ≤ D * Fintype.card (rowGroup μ) * Fintype.card (colGroup μ)) :
+    μ.card.choose (μ.card-μ.rowLen 0) ≤ D * 2^(μ.card-μ.rowLen 0) := by
+  let j := μ.card-μ.rowLen 0
+  have htail := card_fiberGroups_mul_le
+    (fun x : TailCell μ => rowIndex x.1) (fun x : TailCell μ => colIndex x.1)
+    (fun x y hr hc => Subtype.ext (Subtype.ext (Prod.ext (congrArg Fin.val hr) (congrArg Fin.val hc))))
+  rw [tail_card] at htail
+  have hc := column_group_le_tail μ
+  have hr := tail_row_group μ
+  simp only [←Nat.card_eq_fintype_card] at htail hc hr hd
+  have hh : μ.card.factorial ≤ D * 2^j *
+      ((μ.rowLen 0).factorial * j.factorial) := by
+    calc
+      _ ≤ _ := hd
+      _ ≤ D * Nat.card (rowGroup μ) *
+          (2^j * Nat.card (fiberGroup (fun x : TailCell μ => colIndex x.1))) := Nat.mul_le_mul_left _ hc
+      _ = D * 2^j * (μ.rowLen 0).factorial *
+          (Nat.card (fiberGroup (fun x : TailCell μ => rowIndex x.1)) *
+           Nat.card (fiberGroup (fun x : TailCell μ => colIndex x.1))) := by rw [←hr]; ring
+      _ ≤ _ := by
+        calc
+          _ ≤ D * 2^j * (μ.rowLen 0).factorial * j.factorial :=
+            Nat.mul_le_mul_left _ htail
+          _ = _ := by ring
+  have hj : j ≤ μ.card := Nat.sub_le _ _
+  have hs : μ.card-j = μ.rowLen 0 := by dsimp [j]; have := rowLen_zero_le_card μ; omega
+  have hf := Nat.choose_mul_factorial_mul_factorial hj
+  rw [hs] at hf
+  rw [←hf] at hh
+  have hpos : 0 < (μ.rowLen 0).factorial * j.factorial := Nat.mul_pos (Nat.factorial_pos _) (Nat.factorial_pos _)
+  nlinarith
+
+lemma card_rowGroup_transpose (μ : YoungDiagram) :
+    Fintype.card (rowGroup μ.transpose) = Fintype.card (colGroup μ) := by
+  rw [card_rowGroup,card_colGroup]
+  apply Fintype.prod_equiv (finCongr (μ.colLen_transpose 0))
+  intro i
+  change (μ.transpose.rowLen i).factorial = (μ.colLen i).factorial
+  rw [YoungDiagram.rowLen_transpose]
+
+lemma card_colGroup_transpose (μ : YoungDiagram) :
+    Fintype.card (colGroup μ.transpose) = Fintype.card (rowGroup μ) := by
+  rw [card_colGroup,card_rowGroup]
+  apply Fintype.prod_equiv (finCongr (μ.rowLen_transpose 0))
+  intro i
+  change (μ.transpose.colLen i).factorial = (μ.rowLen i).factorial
+  rw [YoungDiagram.colLen_transpose]
+
+lemma card_transpose (μ : YoungDiagram) : μ.transpose.card=μ.card := by
+  unfold YoungDiagram.card YoungDiagram.transpose
+  exact Finset.card_map _
+
+theorem choose_column_tail_le_dimension (μ : YoungDiagram) :
+    μ.card.choose (μ.card-μ.colLen 0) ≤ Module.finrank ℂ (space μ) * 2^(μ.card-μ.colLen 0) := by
+  have hd := factorial_le_dimension_row_column μ
+  have ht := choose_tail_le_of_factorial μ.transpose (Module.finrank ℂ (space μ))
+  simp only [card_transpose,card_rowGroup_transpose,card_colGroup_transpose,YoungDiagram.rowLen_transpose] at ht
+  apply ht
+  nlinarith
+
+open DimensionEstimates Filter
+
+lemma dimension_log_incidence (μ : YoungDiagram) (hμ : 0<μ.card) :
+    (2*Real.log 2-1)*(μ.card:ℝ)-Real.log 2*((μ.rowLen 0:ℝ)+μ.colLen 0)+1≤
+      Real.log (Module.finrank ℂ (space μ)) := by
+  let : Nonempty (Cell μ) := Fintype.card_pos_iff.mp (by simpa only [card_cell] using hμ)
+  have hd := factorial_le_dimension_row_column μ
+  rw [card_rowGroup,card_colGroup] at hd
+  have hr (i : Fin (μ.colLen 0)) : Nat.card {x : Cell μ // rowIndex x=i}=μ.rowLen i := by
+    let e : {x : Cell μ // rowIndex x=i} ≃ {x : Cell μ // row x=i} :=
+      Equiv.subtypeEquivRight (fun x => Fin.ext_iff)
+    rw [Nat.card_congr e,Nat.card_congr (rowEquiv μ i),Nat.card_fin]
+  have hc (i : Fin (μ.rowLen 0)) : Nat.card {x : Cell μ // colIndex x=i}=μ.colLen i := by
+    let e : {x : Cell μ // colIndex x=i} ≃ {x : Cell μ // col x=i} :=
+      Equiv.subtypeEquivRight (fun x => Fin.ext_iff)
+    rw [Nat.card_congr e,Nat.card_congr (columnEquiv μ i),Nat.card_fin]
+  have hcell : Nat.card (Cell μ)=μ.card := by rw [Nat.card_eq_fintype_card,card_cell]
+  have h := dimension_log_from_incidence (rowIndex (μ := μ)) (colIndex (μ := μ))
+    (fun x y h => Subtype.ext (Prod.ext (congrArg (fun z => (z.1:ℕ)) h)
+      (congrArg (fun z => (z.2:ℕ)) h))) (Module.finrank ℂ (space μ))
+    (by simpa only [←Nat.card_eq_fintype_card,hr,hc,hcell] using hd)
+  simpa only [card_cell,Fintype.card_fin,add_comm (μ.colLen 0:ℝ)] using h
+
+lemma medium_dimension_raw (μ : YoungDiagram) (hμ : 0<μ.card) (δ : ℝ) (hδ : 0<δ)
+    (hδ2 : δ≤1/2) (hrow : δ*(μ.card:ℝ)≤μ.card-μ.rowLen 0)
+    (hcol : (μ.colLen 0:ℝ)≤(μ.card:ℝ)/2) :
+    min ((3/2:ℝ)*Real.log 2-1) (entropyGap*δ)*(μ.card:ℝ)-2*Real.log μ.card-1≤
+      Real.log (Module.finrank ℂ (space μ)) := by
+  let c := (3/2:ℝ)*Real.log 2-1
+  have hN : (0:ℝ)<μ.card := by exact_mod_cast hμ
+  have hlog : 0≤Real.log μ.card := Real.log_nonneg (by exact_mod_cast hμ)
+  have hc : 0≤Real.log 2 := Real.log_nonneg (by norm_num)
+  have hr := rowLen_zero_le_card μ
+  have hcl : μ.colLen 0≤μ.card := by
+    simpa only [YoungDiagram.rowLen_transpose,card_transpose] using rowLen_zero_le_card μ.transpose
+  by_cases hsum : (μ.rowLen 0:ℝ)+μ.colLen 0≤(μ.card:ℝ)/2
+  · have hi := dimension_log_incidence μ hμ
+    have hm := mul_le_mul_of_nonneg_right (min_le_left c (entropyGap*δ)) hN.le
+    dsimp [c] at hm
+    nlinarith
+  · have hbig : (μ.card:ℝ)/4≤μ.rowLen 0 ∨ (μ.card:ℝ)/4≤μ.colLen 0 := by
+      by_contra h
+      push Not at h
+      linarith
+    rcases hbig with hrb | hcb
+    · have hj : 0<μ.card-μ.rowLen 0 := by
+        rw [←Nat.cast_pos (α := ℝ),Nat.cast_sub hr]
+        exact lt_of_lt_of_le (mul_pos hδ hN) hrow
+      have hjN : μ.card-μ.rowLen 0<μ.card := by
+        have : 0<μ.rowLen 0 := by exact_mod_cast (by linarith : (0:ℝ)<μ.rowLen 0)
+        omega
+      have hi := log_dimension_from_binomial μ.card (μ.card-μ.rowLen 0)
+        (Module.finrank ℂ (space μ)) hj hjN
+        (by rw [Nat.cast_sub hr]; linarith) (choose_tail_le_dimension μ)
+      rw [Nat.cast_sub hr] at hi
+      have hm := mul_le_mul_of_nonneg_right (min_le_right c (entropyGap*δ)) hN.le
+      nlinarith [entropyGap_pos]
+    · have hj : 0<μ.card-μ.colLen 0 := by
+        rw [←Nat.cast_pos (α := ℝ),Nat.cast_sub hcl]
+        linarith
+      have hjN : μ.card-μ.colLen 0<μ.card := by
+        have : 0<μ.colLen 0 := by exact_mod_cast (by linarith : (0:ℝ)<μ.colLen 0)
+        omega
+      have hi := log_dimension_from_binomial μ.card (μ.card-μ.colLen 0)
+        (Module.finrank ℂ (space μ)) hj hjN
+        (by rw [Nat.cast_sub hcl]; linarith) (choose_column_tail_le_dimension μ)
+      rw [Nat.cast_sub hcl] at hi
+      have hm := mul_le_mul_of_nonneg_right (min_le_right c (entropyGap*δ)) hN.le
+      have ht := mul_le_mul_of_nonneg_left (show δ*(μ.card:ℝ)≤(μ.card:ℝ)-μ.colLen 0 by nlinarith) entropyGap_pos.le
+      linarith
+
+end Thorp.Specht
+
+namespace Thorp.DimensionEstimates
+
+section
+open scoped BigOperators Classical
+open Filter
+
+lemma finset_power_sum {ι : Type*} [Fintype ι] (x : ℝ) :
+    ∑ E : Finset ι, x^E.card=(1+x)^Fintype.card ι := by
+  have h := Finset.prod_one_add (f := fun _ : ι => x) Finset.univ
+  simpa only [Finset.prod_const,Finset.card_univ,Finset.powerset_univ] using h.symm
+
+lemma small_sets_count {ι : Type*} [Fintype ι] (ε t : ℝ) (ht : 0≤t) :
+    (∑ E : Finset ι, if (E.card:ℝ)<ε*Fintype.card ι then (1:ℝ) else 0) ≤
+      Real.exp ((t*ε+Real.log (1+Real.exp (-t)))*Fintype.card ι) := by
+  have hpow (E : Finset ι) : (Real.exp (-t))^E.card=Real.exp (-t*(E.card:ℝ)) := by
+    rw [←Real.exp_nat_mul]
+    congr 1
+    ring
+  calc
+    _ ≤ ∑ E : Finset ι, Real.exp (t*ε*Fintype.card ι)*(Real.exp (-t))^E.card := by
+      apply Finset.sum_le_sum
+      intro E _
+      split_ifs with hE
+      · rw [hpow,←Real.exp_add,←Real.exp_zero]
+        apply Real.exp_le_exp.mpr
+        nlinarith [mul_le_mul_of_nonneg_left hE.le ht]
+      · positivity
+    _ = Real.exp (t*ε*Fintype.card ι)*(1+Real.exp (-t))^Fintype.card ι := by
+      rw [←Finset.mul_sum,finset_power_sum]
+    _ = _ := by
+      have he : (1+Real.exp (-t))^Fintype.card ι=
+          Real.exp (Real.log (1+Real.exp (-t))*(Fintype.card ι:ℝ)) := by
+        rw [mul_comm,Real.exp_nat_mul,Real.exp_log (by positivity)]
+      rw [he,←Real.exp_add]
+      congr 1
+      ring
+
+lemma small_sets_ceil_bound {ι : Type*} [Fintype ι] (ε t : ℝ) (ht : 0≤t) :
+    (∑ E : Finset ι, if E.card<⌈ε*Fintype.card ι⌉₊ then (1:ℝ) else 0) ≤
+      Real.exp ((t*ε+Real.log (1+Real.exp (-t)))*Fintype.card ι) := by
+  have he (E : Finset ι) : E.card<⌈ε*Fintype.card ι⌉₊ ↔ (E.card:ℝ)<ε*Fintype.card ι := Nat.lt_ceil
+  simp only [he]
+  exact small_sets_count ε t ht
+
+lemma exists_small_set_rate (κ ε₀ : ℝ) (hκ : 0<κ) (hε₀ : 0<ε₀) :
+    ∃ ε : ℝ, 0<ε ∧ ε≤ε₀ ∧ ε<1/2 ∧ ∃ t : ℝ, 0≤t ∧
+      t*ε+Real.log (1+Real.exp (-t))≤κ := by
+  have ht : Tendsto (fun t : ℝ => Real.log (1+Real.exp (-t))) atTop (nhds 0) := by
+    have he : Tendsto (fun t : ℝ => Real.exp (-t)) atTop (nhds 0) :=
+      Real.tendsto_exp_atBot.comp tendsto_neg_atTop_atBot
+    have hl := (tendsto_const_nhds.add he).log (by norm_num : (1:ℝ)+0≠0)
+    simpa only [add_zero,Real.log_one] using hl
+  obtain ⟨t,ht,hlog⟩ := ((eventually_ge_atTop (1:ℝ)).and
+    (ht.eventually (gt_mem_nhds (by linarith : (0:ℝ)<κ/2)))).exists
+  let ε := min ε₀ (min (1/4) (κ/(2*t)))
+  have htp : 0<t := by linarith
+  have hε : 0<ε := lt_min hε₀ (lt_min (by norm_num) (by positivity))
+  refine ⟨ε,hε,min_le_left _ _,(min_le_right _ _).trans_lt ((min_le_left _ _).trans_lt (by norm_num)),t,htp.le,?_⟩
+  have hsmall : ε≤κ/(2*t) := (min_le_right _ _).trans (min_le_right _ _)
+  have hm := (le_div_iff₀ (by positivity : (0:ℝ)<2*t)).mp hsmall
+  linarith
+
+end
+open Filter
+
+lemma log_error_eventually (a : ℝ) (ha : 0<a) :
+    ∀ᶠ N : ℕ in atTop, 2*Real.log N+1≤(a/2)*(N:ℝ) := by
+  have h := (tendsto_natCast_atTop_atTop (R := ℝ)).eventually
+    (Real.isLittleO_log_id_atTop.bound (by linarith : 0<a/8))
+  have hg := (tendsto_natCast_atTop_atTop (R := ℝ)).eventually (eventually_ge_atTop (4/a))
+  filter_upwards [h,hg,eventually_ge_atTop 1] with N h hg hN
+  have hn : (1:ℝ)≤N := by exact_mod_cast hN
+  simp only [Real.norm_eq_abs,abs_of_nonneg (Real.log_nonneg hn),id_eq,abs_of_nonneg (by positivity : (0:ℝ)≤N)] at h
+  have hh : 4≤(N:ℝ)*a := (div_le_iff₀ ha).mp hg
+  linarith
+
+end Thorp.DimensionEstimates
+
+namespace Thorp.Specht
+open scoped BigOperators Classical
+open DimensionEstimates Filter
+
+theorem medium_dimension (δ : ℝ) (hδ : 0<δ) (hδ2 : δ≤1/2) :
+    ∃ h : ℝ, 0<h ∧ ∀ᶠ N : ℕ in atTop, ∀ μ : YoungDiagram, μ.card=N →
+      δ*(N:ℝ)≤N-μ.rowLen 0 → (μ.colLen 0:ℝ)≤(N:ℝ)/2 →
+      h*(N:ℝ)≤Real.log (Module.finrank ℂ (space μ)) := by
+  let a := min ((3/2:ℝ)*Real.log 2-1) (entropyGap*δ)
+  have ha : 0<a := by
+    apply lt_min
+    · nlinarith [Real.log_two_gt_d9]
+    · exact mul_pos entropyGap_pos hδ
+  refine ⟨a/2,by positivity,?_⟩
+  filter_upwards [log_error_eventually a ha,eventually_ge_atTop 1] with N hN hn
+  intro μ hμ hr hc
+  have hp : 0<μ.card := by omega
+  have hh := medium_dimension_raw μ hp δ hδ hδ2 (by simpa only [hμ] using hr) (by simpa only [hμ] using hc)
+  rw [hμ] at hh
+  change a*(N:ℝ)-2*Real.log N-1≤_ at hh
+  linarith
+
+end Thorp.Specht
+
+namespace Thorp
+open scoped BigOperators Classical
+
+lemma insertNth_castSucc_snoc {α : Type*} {d : ℕ} (i : Fin (d+1)) (b a : α) (y : Fin d → α) :
+    Fin.insertNth i.castSucc b (Fin.snoc (α := fun _ => α) y a)=Fin.snoc (α := fun _ => α) (Fin.insertNth i b y) a := by
+  apply Fin.lastCases ?_ (fun j => ?_) |> fun h => funext h
+  · rw [←show i.castSucc.succAbove (Fin.last d)=Fin.last (d+1) from
+        Fin.succAbove_castSucc_of_le i (Fin.last d) (Fin.le_last i),Fin.insertNth_apply_succAbove]
+    simp
+  · cases j using Fin.succAboveCases i with
+    | x => simp
+    | p j =>
+      have he : i.castSucc.succAbove j.castSucc=(i.succAbove j).castSucc := by
+        simp only [Fin.succAbove,Fin.castSucc_lt_castSucc_iff]
+        split_ifs <;> rfl
+      rw [←he,Fin.insertNth_apply_succAbove]
+      simp
+
+lemma rotate_cons (d : ℕ) (b : Bool) (y : Card d) :
+    rotate (d+1) (Fin.cons b y)=Fin.snoc y b := (Fin.snoc_eq_cons_rotate y b).symm
+
+lemma rotate_insert (d : ℕ) (i : Fin (d+1)) (b : Bool) (y : Card (d+1)) :
+    rotate (d+2) (Fin.insertNth i.succ b y)=Fin.insertNth i.castSucc b (rotate (d+1) y) := by
+  rw [←Fin.cons_self_tail y,Fin.insertNth_succ_cons,rotate_cons,rotate_cons,insertNth_castSucc_snoc]
+
+def insertionPerm (d : ℕ) (i : Fin (d+1)) : Equiv.Perm (Card (d+1)) :=
+  (Fin.consEquiv (fun _ : Fin (d+1) => Bool)).symm.trans
+    (Fin.insertNthEquiv (fun _ : Fin (d+1) => Bool) i)
+
+lemma insertionPerm_apply (d : ℕ) (i : Fin (d+1)) (x : Card (d+1)) :
+    insertionPerm d i x=Fin.insertNth i (x 0) (Fin.tail x) := rfl
+
+lemma insertionPerm_zero (d : ℕ) : insertionPerm d 0=1 := by
+  ext x j
+  simp [insertionPerm_apply]
+
+lemma insertionPerm_last (d : ℕ) : insertionPerm d (Fin.last d)=rotate (d+1) := by
+  apply Equiv.ext
+  intro x
+  rw [insertionPerm_apply,Fin.insertNth_last',←rotate_cons,Fin.cons_self_tail]
+
+lemma physical_carry_step (d : ℕ) (i : Fin (d+1)) (ξ : Bool → Coins (d+1)) :
+    step (d+2) (fun y => ξ (y i) (Fin.removeNth i y))*insertionPerm (d+1) i.succ=
+      insertionPerm (d+1) i.castSucc*childLift (fun b => step (d+1) (ξ b)) := by
+  apply Equiv.ext
+  intro x
+  have he : Fin.cons (Bool.xor ((Fin.tail x) 0) (ξ (x 0) (Fin.tail (Fin.tail x))))
+      (Fin.insertNth i (x 0) (Fin.tail (Fin.tail x))) =
+      (Fin.insertNth i.succ (x 0) (switchFun (ξ (x 0)) (Fin.tail x)) : Card (d+2)) := by
+    simp only [switchFun,Fin.insertNth_succ_cons]
+  have hs : switchFun (fun y => ξ (y i) (Fin.removeNth i y))
+      (Fin.insertNth i.succ (x 0) (Fin.tail x)) =
+      (Fin.insertNth i.succ (x 0) (switchFun (ξ (x 0)) (Fin.tail x)) : Card (d+2)) := by
+    conv_lhs => rw [←Fin.cons_self_tail (Fin.tail x),Fin.insertNth_succ_cons]
+    simp only [switchFun,Fin.cons_zero,Fin.tail_cons,Fin.insertNth_apply_same,Fin.removeNth_insertNth]
+    exact he
+  change rotate (d+2) (switchFun (fun y => ξ (y i) (Fin.removeNth i y))
+      (Fin.insertNth i.succ (x 0) (Fin.tail x))) = _
+  rw [hs,rotate_insert]
+  rfl
+
+end Thorp
+namespace Thorp.UnitaryFinite
+open scoped BigOperators Classical
+variable {G : Type*} [Group G]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+variable {Ω : Type*} [Fintype Ω]
+
+lemma sampleOperator_mul_const (ρ : Representation ℂ G V) (P : Ω → G) (g : G) :
+    sampleOperator ρ (fun ω => P ω*g)=sampleOperator ρ P*continuousRepresentation ρ g := by
+  simp only [sampleOperator,map_mul,Finset.sum_mul,smul_mul_assoc]
+
+lemma sampleOperator_const_mul (ρ : Representation ℂ G V) (P : Ω → G) (g : G) :
+    sampleOperator ρ (fun ω => g*P ω)=continuousRepresentation ρ g*sampleOperator ρ P := by
+  simp only [sampleOperator,map_mul,Finset.mul_sum,mul_smul_comm]
+
+lemma sampleOperator_run (d t : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V) :
+    sampleOperator ρ (run d t)=(sampleOperator ρ (step d))^t := by
+  induction t with
+  | zero => exact sampleOperator_one ρ
+  | succ t ih =>
+    let e : Coins d × (Fin t → Coins d) ≃ (Fin (t+1) → Coins d) :=
+      Fin.snocEquiv (fun _ => Coins d)
+    have he : run d (t+1) ∘ e=fun w => step d w.1*run d t w.2 := by
+      funext w
+      simp only [Function.comp_apply,e,Fin.snocEquiv_apply,
+        run,Fin.snoc_last,Fin.snoc_castSucc]
+    rw [←sampleOperator_equiv ρ (run d (t+1)) e,he,sampleOperator_prod,ih,pow_succ']
+
+end Thorp.UnitaryFinite
+
+namespace Thorp
+open scoped BigOperators Classical
+open UnitaryFinite
+
+def carryCoins (d : ℕ) (i : Fin (d+1)) : (Bool → Coins (d+1)) ≃ Coins (d+2) where
+  toFun ξ y := ξ (y i) (Fin.removeNth i y)
+  invFun ξ b y := ξ (Fin.insertNth i b y)
+  left_inv ξ := by funext b y; simp
+  right_inv ξ := by funext y; simp
+
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+lemma physical_carry_average (d : ℕ) (i : Fin d)
+    (ρ : Representation ℂ (Equiv.Perm (Card (d+1))) V) :
+    sampleOperator ρ (step (d+1))*continuousRepresentation ρ (insertionPerm d i.succ)=
+      continuousRepresentation ρ (insertionPerm d i.castSucc)*
+        sampleOperator ρ (fun ξ : Bool → Coins d => childLift (fun b => step d (ξ b))) := by
+  cases d with
+  | zero => exact Fin.elim0 i
+  | succ d =>
+    have he : (fun ξ : Bool → Coins (d+1) =>
+        step (d+2) (carryCoins d i ξ)*insertionPerm (d+1) i.succ)=
+      (fun ξ => insertionPerm (d+1) i.castSucc*childLift (fun b => step (d+1) (ξ b))) := by
+      funext ξ
+      exact physical_carry_step d i ξ
+    have h := congrArg (sampleOperator ρ) he
+    rw [sampleOperator_mul_const,sampleOperator_const_mul] at h
+    change sampleOperator ρ (step (d+2) ∘ carryCoins d i)*_=_ at h
+    rw [sampleOperator_equiv] at h
+    exact h
+
+theorem physical_sweep_recursion (d : ℕ)
+    (ρ : Representation ℂ (Equiv.Perm (Card (d+1))) V) :
+    (sampleOperator ρ (step (d+1)))^(d+1)=
+      (sampleOperator ρ (fun ξ : Bool → Coins d => childLift (fun b => step d (ξ b))))^d*
+        sampleOperator ρ (pairSwitch (d := d)) := by
+  let A := sampleOperator ρ (step (d+1))
+  let K := sampleOperator ρ (fun ξ : Bool → Coins d => childLift (fun b => step d (ξ b)))
+  have h (t : ℕ) (ht : t≤d) : A^t*continuousRepresentation ρ (insertionPerm d ⟨t,by omega⟩)=K^t := by
+    induction t with
+    | zero =>
+      change 1*continuousRepresentation ρ (insertionPerm d 0)=1
+      rw [insertionPerm_zero,map_one,one_mul]
+    | succ t ih =>
+      have hc := physical_carry_average d (⟨t,by omega⟩ : Fin d) ρ
+      change A*continuousRepresentation ρ (insertionPerm d ⟨t+1,by omega⟩)=
+        continuousRepresentation ρ (insertionPerm d ⟨t,by omega⟩)*K at hc
+      rw [pow_succ,mul_assoc,hc,←mul_assoc,ih (by omega),←pow_succ]
+  have he : sampleOperator ρ (step (d+1))=
+      continuousRepresentation ρ (rotate (d+1))*sampleOperator ρ (pairSwitch (d := d)) :=
+    sampleOperator_const_mul ρ (pairSwitch (d := d)) (rotate (d+1))
+  calc
+    _ = A^d*A := pow_succ A d
+    _ = A^d*(continuousRepresentation ρ (insertionPerm d (Fin.last d))*
+        sampleOperator ρ (pairSwitch (d := d))) := by rw [insertionPerm_last]; rw [←he]
+    _ = _ := by
+      rw [←mul_assoc]
+      exact congrArg (fun z => z*sampleOperator ρ (pairSwitch (d := d))) (h d le_rfl)
+
+end Thorp
+namespace Thorp
+open scoped BigOperators Classical
+open UnitaryFinite
+
+lemma childInjection_product (d : ℕ) (p : Bool → Equiv.Perm (Card d)) :
+    childInjection d false (p false)*childInjection d true (p true)=childLift p := by
+  change childLift _*childLift _=_
+  rw [childLift_mul]
+  congr 1
+  funext b
+  cases b <;> simp
+
+lemma childInjection_commute (d : ℕ) (p q : Equiv.Perm (Card d)) :
+    Commute (childInjection d false p) (childInjection d true q) := by
+  change childLift _*childLift _=childLift _*childLift _
+  rw [childLift_mul,childLift_mul]
+  congr 1
+  funext b
+  cases b <;> simp
+
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+variable {Ω : Type*} [Fintype Ω]
+
+lemma sampleOperator_child (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card (d+1))) V)
+    (P : Ω → Equiv.Perm (Card d)) :
+    sampleOperator ρ (fun ω : Bool → Ω => childLift (fun b => P (ω b)))=
+      sampleOperator (ρ.comp (childInjection d false)) P*
+        sampleOperator (ρ.comp (childInjection d true)) P := by
+  let e := (Equiv.boolArrowEquivProd Ω).symm
+  have he : (fun ω : Bool → Ω => childLift (fun b => P (ω b))) ∘ e=
+      fun w : Ω × Ω => childInjection d false (P w.1)*childInjection d true (P w.2) := by
+    funext w
+    change childLift (fun b => P (e w b))=_
+    rw [←childInjection_product]
+    rfl
+  rw [←sampleOperator_equiv ρ _ e,he]
+  exact sampleOperator_prod ρ (fun w => childInjection d false (P w))
+    (fun w => childInjection d true (P w))
+
+lemma sampleOperator_child_commute (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card (d+1))) V)
+    (P : Ω → Equiv.Perm (Card d)) :
+    Commute (sampleOperator (ρ.comp (childInjection d false)) P)
+      (sampleOperator (ρ.comp (childInjection d true)) P) := by
+  rw [sampleOperator_comp,sampleOperator_comp]
+  exact sampleOperator_commute ρ _ _ (fun _ _ => childInjection_commute d _ _)
+
+lemma inverse_butterfly_average_step (d : ℕ)
+    (ρ : Representation ℂ (Equiv.Perm (Card (d+1))) V) :
+    sampleOperator ρ (fun ω => (butterflyPerm (d+1) (decodeButterfly (d+1) ω))⁻¹)=
+      sampleOperator ρ (fun ω : Bool → (SwitchIndex d → Bool) =>
+        childLift (fun b => (butterflyPerm d (decodeButterfly d (ω b)))⁻¹))*
+      sampleOperator ρ (pairSwitch (d := d)) := by
+  let e := ((Equiv.prodCongr (Equiv.boolArrowEquivProd (SwitchIndex d → Bool)) (Equiv.refl (Card d → Bool))).trans
+    (coinStepEquiv d).symm)
+  have he : (fun ω => (butterflyPerm (d+1) (decodeButterfly (d+1) ω))⁻¹) ∘ e=
+      fun w : (Bool → (SwitchIndex d → Bool)) × (Card d → Bool) =>
+        childLift (fun b => (butterflyPerm d (decodeButterfly d (w.1 b)))⁻¹)*pairSwitch w.2 := by
+    funext w
+    change (butterflyPerm (d+1) (decodeButterfly (d+1)
+      ((coinStepEquiv d).symm ((w.1 false,w.1 true),w.2))))⁻¹=_
+    rw [butterflyPerm_step,mul_inv_rev,childLift_inv,pairSwitch_inv]
+    congr 2
+    funext b
+    cases b <;> rfl
+  rw [←sampleOperator_equiv ρ _ e,he]
+  exact sampleOperator_prod ρ
+    (fun ω : Bool → (SwitchIndex d → Bool) => childLift (fun b => (butterflyPerm d (decodeButterfly d (ω b)))⁻¹))
+    (pairSwitch (d := d))
+
+theorem physical_sweep_inverse_butterfly (d : ℕ)
+    (ρ : Representation ℂ (Equiv.Perm (Card d)) V) :
+    (sampleOperator ρ (step d))^d=
+      sampleOperator ρ (fun ω => (butterflyPerm d (decodeButterfly d ω))⁻¹) := by
+  induction d with
+  | zero =>
+    symm
+    exact sampleOperator_one ρ
+  | succ d ih =>
+    rw [physical_sweep_recursion,sampleOperator_child,
+      (sampleOperator_child_commute d ρ (step d)).mul_pow,ih,ih,
+      inverse_butterfly_average_step]
+    rw [sampleOperator_child d ρ (fun ω => (butterflyPerm d (decodeButterfly d ω))⁻¹)]
+
+lemma physical_sweep_adjoint (d : ℕ)
+    (ρ : Representation ℂ (Equiv.Perm (Card d)) V) (hρ : IsUnitary ρ) :
+    sampleOperator ρ (run d d)=
+      (sampleOperator ρ (fun ω => butterflyPerm d (decodeButterfly d ω))).adjoint := by
+  rw [sampleOperator_run,physical_sweep_inverse_butterfly,sampleOperator_inv ρ hρ]
+
+end Thorp
+
+namespace Thorp.Specht
+open scoped BigOperators Classical
+open UnitaryFinite DimensionEstimates Filter
+
+lemma dimension_pos (μ : YoungDiagram) : 0 < Module.finrank ℂ (space μ) := by
+  have h := factorial_le_dimension_row_column μ
+  by_contra hn
+  have hz : Module.finrank ℂ (space μ)=0 := by omega
+  rw [hz,zero_mul,zero_mul] at h
+  exact (Nat.factorial_pos μ.card).not_ge h
+
+lemma near_numerical_cutoff :
+    ∃ X : ℝ, 2 ≤ X ∧ ∀ (N j D : ℕ), 0 < j → X ≤ (N:ℝ)/j → N.choose j ≤ D * 2^j →
+      ∃ k : ℕ, j ≤ k ∧ k ≤ N ∧ ∀ m : ℕ, D * k.choose j ≤ m * N.choose j →
+        nearFraction * Real.log D ≤ Real.log m ∧
+        densityConstant * (k:ℝ) * ((k:ℝ)/N)^(1/512:ℝ) ≤ Real.log m/4000 ∧
+        (1/2:ℝ) * Real.log N ≤ Real.log D ∧ (j:ℝ) ≤ Real.log D := by
+  obtain ⟨X,hX,h⟩ := exists_near_cutoff densityConstant densityConstant_pos.le
+  refine ⟨X,hX,?_⟩
+  intro N j D hj hx hd
+  have hjN : j ≤ N := by
+    have hjp : (0:ℝ) < j := by exact_mod_cast hj
+    have hh : (j:ℝ) ≤ N := by
+      have := (le_div_iff₀ hjp).mp (hX.trans hx)
+      linarith
+    exact_mod_cast hh
+  have hc : 1 ≤ N.choose j := Nat.choose_pos hjN
+  have hfoot : D * (chosenTuple N j).choose j ≤ (D * (chosenTuple N j).choose j) * N.choose j :=
+    Nat.le_mul_of_pos_right _ hc
+  have hh := h N j D _ hj hx hd hfoot
+  refine ⟨chosenTuple N j,hh.1,hh.2.1,?_⟩
+  intro m hm
+  exact (h N j D m hj hx hd hm).2.2
+
+lemma near_row_amplification (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Cell μ)
+    {k : ℕ} (f : Fin k ↪ Card d) (hk : 0<k)
+    (hm : 0<Module.finrank ℂ (fixedSpace (V := hilbertSpace μ) tupleAction f (relabelledUnitary μ e))) :
+    ‖sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e) (palindromePerm d)‖ ≤
+      Real.exp (densityConstant*(k:ℝ)*((k:ℝ)/(2:ℝ)^d)^(1/512:ℝ))*
+        (Real.exp (Real.log (Module.finrank ℂ (fixedSpace (V := hilbertSpace μ) tupleAction f (relabelledUnitary μ e)))/2))^(-(1/1000:ℝ))+
+      Real.sqrt ((Real.exp (Real.log (Module.finrank ℂ (fixedSpace (V := hilbertSpace μ) tupleAction f (relabelledUnitary μ e)))/2))/
+        (Module.finrank ℂ (fixedSpace (V := hilbertSpace μ) tupleAction f (relabelledUnitary μ e)):ℝ)) := by
+  let := relabelledUnitary_irreducible μ e
+  let : NormedAddCommGroup (hilbertSpace μ) := inferInstance
+  let : InnerProductSpace ℂ (hilbertSpace μ) := inferInstance
+  let : FiniteDimensional ℂ (hilbertSpace μ) := inferInstance
+  have aux := @palindrome_irrep_amplification (hilbertSpace μ) _ _ _ d (Fin k) _ f
+    (by simpa only [Fintype.card_fin] using hk) (relabelledUnitary μ e) inferInstance
+    (relabelledUnitary_unitary μ e) hm
+    (Real.exp (Real.log (Module.finrank ℂ (fixedSpace (V := hilbertSpace μ) tupleAction f (relabelledUnitary μ e)))/2))
+    (Real.exp_pos _)
+  simpa only [Fintype.card_fin] using aux
+
+theorem near_row_spectral :
+    ∃ X : ℝ, 2 ≤ X ∧ ∀ (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Cell μ),
+      0 < μ.card-μ.rowLen 0 → X ≤ (μ.card:ℝ)/(μ.card-μ.rowLen 0:ℕ) →
+      32768000 * Real.log 2 ≤ (1/2:ℝ) * Real.log μ.card →
+      ‖sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e) (palindromePerm d)‖ ≤
+        Real.exp (-(nearFraction/8000) * Real.log (Module.finrank ℂ (space μ))) ∧
+      (1/2:ℝ) * Real.log μ.card ≤ Real.log (Module.finrank ℂ (space μ)) ∧
+      (μ.card-μ.rowLen 0:ℕ) ≤ Real.log (Module.finrank ℂ (space μ)) := by
+  obtain ⟨X,hX,h⟩ := near_numerical_cutoff
+  refine ⟨X,hX,?_⟩
+  intro d μ e hj hx hlarge
+  let j := μ.card-μ.rowLen 0
+  let D := Module.finrank ℂ (space μ)
+  have hcard : μ.card=2^d := by rw [←card_cell,←Fintype.card_congr e,card_positions]
+  obtain ⟨k,hjk,hk,hnum⟩ := h μ.card j D hj hx (choose_tail_le_dimension μ)
+  obtain ⟨f⟩ := Function.Embedding.nonempty_of_card_le
+    (show Fintype.card (Fin k) ≤ Fintype.card (Card d) by simpa only [Fintype.card_fin,card_positions,←hcard] using hk)
+  let m := Module.finrank ℂ (fixedSpace (V := hilbertSpace μ) tupleAction f (relabelledUnitary μ e))
+  have hfoot : D * k.choose j ≤ m * μ.card.choose j := by
+    simpa only [Fintype.card_fin,finrank_hilbertSpace] using relabelled_tuple_dimension μ e f
+  have hm : 0 < m := by
+    have hc := Nat.choose_pos hjk
+    have hd := dimension_pos μ
+    by_contra hn
+    have hz : m=0 := by omega
+    rw [hz,zero_mul] at hfoot
+    exact (Nat.mul_pos hd hc).not_ge hfoot
+  obtain ⟨hlm,hden,hlog,hjlog⟩ := hnum m hfoot
+  have ha := near_row_amplification d μ e f (hj.trans_le hjk) hm
+  have hcardR : (μ.card:ℝ)=(2:ℝ)^d := by exact_mod_cast hcard
+  rw [←hcardR] at ha
+  have hl : 8000 * Real.log 2 ≤ Real.log m := by
+    dsimp [nearFraction] at hlm
+    linarith
+  have hn := amplification_decay (m:ℝ) (densityConstant * k * ((k:ℝ)/μ.card)^(1/512:ℝ))
+    (by exact_mod_cast hm) hden hl
+  refine ⟨ha.trans (hn.trans ?_),hlog,hjlog⟩
+  apply Real.exp_le_exp.mpr
+  dsimp [D] at hlm
+  linarith
+
+end Thorp.Specht
+
+namespace Thorp.UnitaryFinite
+open scoped BigOperators Classical
+open DimensionEstimates
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+theorem full_palindrome_decay (d : ℕ)
+    (ρ : Representation ℂ (Equiv.Perm (Card d)) V) [Representation.IsIrreducible ρ]
+    (hρ : IsUnitary ρ) (hm : 0<Module.finrank ℂ V)
+    (hden : densityConstant*(2:ℝ)^d≤Real.log (Module.finrank ℂ V)/4000)
+    (hl : 8000*Real.log 2≤Real.log (Module.finrank ℂ V)) :
+    ‖sampleOperator ρ (palindromePerm d)‖≤Real.exp (-Real.log (Module.finrank ℂ V)/8000) := by
+  have h := palindrome_irrep_amplification d (Function.Embedding.refl (Card d))
+    (Fintype.card_pos) ρ hρ (by simpa only [full_tuple_finrank] using hm)
+    (Real.exp (Real.log (Module.finrank ℂ V)/2)) (Real.exp_pos _)
+  simp only [full_tuple_finrank,card_positions,Nat.cast_pow,Nat.cast_ofNat,
+    div_self (by positivity : (2:ℝ)^d≠0),Real.one_rpow,mul_one] at h
+  exact h.trans (amplification_decay _ _ (by exact_mod_cast hm) hden hl)
+
+theorem full_modified_decay (L r : ℕ) (E : Finset (Card r))
+    (ρ : Representation ℂ (Equiv.Perm (Card (L+r))) V) [Representation.IsIrreducible ρ]
+    (hρ : IsUnitary ρ) (hm : 0<Module.finrank ℂ V) (u : ℝ)
+    (hM : ∀ e : Card (L+r) ↪ Card (L+r), modifiedRowMoment L r E e (1/1000)≤Real.exp u)
+    (hden : u≤Real.log (Module.finrank ℂ V)/4000)
+    (hl : 8000*Real.log 2≤Real.log (Module.finrank ℂ V)) :
+    ‖sampleOperator ρ (modifiedPerm L r E)‖≤Real.exp (-Real.log (Module.finrank ℂ V)/8000) := by
+  exact (modified_irrep_amplification L r E ρ hρ hm u
+    (Real.exp (Real.log (Module.finrank ℂ V)/2)) (Real.exp_pos _) hM).trans
+    (amplification_decay _ _ (by exact_mod_cast hm) hden hl)
+
+lemma palindrome_exception_cards (r L : ℕ)
+    (ρ : Representation ℂ (Equiv.Perm (Card (L+r))) V) (hρ : IsUnitary ρ) (n : ℕ) :
+    ‖sampleOperator ρ (palindromePerm (L+r))‖ ≤
+      (blockGap L)^n+∑ E : Finset (Card r), if E.card<n then
+        ‖sampleOperator ρ (modifiedPerm L r E)‖ else 0 := by
+  let σ := ρ.comp (splitPerm r L).symm.toMonoidHom
+  have he : σ.comp (splitPerm r L).toMonoidHom=ρ := by
+    ext g v
+    change ρ ((splitPerm r L).symm ((splitPerm r L) g)) v=ρ g v
+    rw [MulEquiv.symm_apply_apply]
+  have h := palindrome_exception_bound r L σ (unitary_comp ρ hρ _) n
+  simpa only [he] using h
+
+end Thorp.UnitaryFinite
+
+namespace Thorp.DimensionEstimates
+open scoped BigOperators Classical
+open Filter
+
+lemma medium_parameters (h : ℝ) (hh : 0<h) :
+    ∃ L : ℕ, 0<L ∧ ∃ ε t : ℝ, 0<ε ∧ ε<1/2 ∧ 0≤t ∧
+      t*ε+Real.log (1+Real.exp (-t))≤h*(2:ℝ)^L/16000 ∧
+      ∀ᶠ r : ℕ in atTop, ∀ (E : Finset (Card r)), (E.card:ℝ)<ε*(2:ℝ)^r →
+        ∀ e : Card (L+r) ↪ Card (L+r),
+          modifiedRowMoment L r E e (1/1000)≤Real.exp ((h/4000)*(2:ℝ)^(L+r)) := by
+  obtain ⟨L,hL,ε₀,hε₀,_,hM⟩ := modified_arbitrarily_small_rate (h/4000) (by positivity)
+  obtain ⟨ε,hε,hεle,hεhalf,t,ht,hcount⟩ := exists_small_set_rate
+    (h*(2:ℝ)^L/16000) ε₀ (by positivity) hε₀
+  refine ⟨L,hL,ε,t,hε,hεhalf,ht,hcount,?_⟩
+  filter_upwards [hM] with r hM
+  intro E hE e
+  exact hM E (hE.le.trans (mul_le_mul_of_nonneg_right hεle (by positivity))) e
+
+lemma geometric_ceil_bound (θ ε : ℝ) (hθ : 0<θ) (hθ1 : θ<1) (M : ℕ) :
+    θ^⌈ε*M⌉₊≤Real.exp (Real.log θ*ε*M) := by
+  conv_lhs => rw [←Real.exp_log hθ,←Real.exp_nat_mul]
+  apply Real.exp_le_exp.mpr
+  have hg := mul_le_mul_of_nonpos_left (Nat.le_ceil (ε*M)) (Real.log_nonpos hθ.le hθ1.le)
+  nlinarith
+
+end Thorp.DimensionEstimates
+
+namespace Thorp.UnitaryFinite
+open scoped BigOperators Classical
+open DimensionEstimates
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+theorem medium_exception_decay (L r : ℕ) (h ε t : ℝ) (ht : 0≤t)
+    (hcount : t*ε+Real.log (1+Real.exp (-t))≤h*(2:ℝ)^L/16000)
+    (ρ : Representation ℂ (Equiv.Perm (Card (L+r))) V) [Representation.IsIrreducible ρ]
+    (hρ : IsUnitary ρ) (hm : 0<Module.finrank ℂ V)
+    (hd : h*(2:ℝ)^(L+r)≤Real.log (Module.finrank ℂ V))
+    (hl : 8000*Real.log 2≤Real.log (Module.finrank ℂ V))
+    (hM : ∀ (E : Finset (Card r)), (E.card:ℝ)<ε*(2:ℝ)^r →
+      ∀ e : Card (L+r) ↪ Card (L+r),
+        modifiedRowMoment L r E e (1/1000)≤Real.exp ((h/4000)*(2:ℝ)^(L+r))) :
+    ‖sampleOperator ρ (palindromePerm (L+r))‖≤
+      Real.exp (Real.log (blockGap L)*ε*(2:ℝ)^r)+Real.exp (-h*(2:ℝ)^(L+r)/16000) := by
+  have hmE (E : Finset (Card r)) (hE : E.card<⌈ε*(2:ℝ)^r⌉₊) :
+      ‖sampleOperator ρ (modifiedPerm L r E)‖≤Real.exp (-h*(2:ℝ)^(L+r)/8000) := by
+    apply (full_modified_decay L r E ρ hρ hm ((h/4000)*(2:ℝ)^(L+r))
+      (hM E (Nat.lt_ceil.mp hE)) (by linarith) hl).trans
+    apply Real.exp_le_exp.mpr
+    linarith
+  have hs : (∑ E : Finset (Card r), if E.card<⌈ε*(2:ℝ)^r⌉₊ then
+      ‖sampleOperator ρ (modifiedPerm L r E)‖ else 0) ≤
+      Real.exp (-h*(2:ℝ)^(L+r)/16000) := by
+    calc
+      _ ≤ (∑ E : Finset (Card r), if E.card<⌈ε*(2:ℝ)^r⌉₊ then (1:ℝ) else 0)*
+          Real.exp (-h*(2:ℝ)^(L+r)/8000) := by
+        rw [Finset.sum_mul]
+        apply Finset.sum_le_sum
+        intro E _
+        split_ifs with hE
+        · simpa only [one_mul] using hmE E hE
+        · simp
+      _ ≤ Real.exp ((t*ε+Real.log (1+Real.exp (-t)))*(2:ℝ)^r)*
+          Real.exp (-h*(2:ℝ)^(L+r)/8000) := by
+        apply mul_le_mul_of_nonneg_right _ (Real.exp_pos _).le
+        simpa only [card_positions,Nat.cast_pow,Nat.cast_ofNat] using
+          small_sets_ceil_bound (ι := Card r) ε t ht
+      _ ≤ _ := by
+        rw [←Real.exp_add]
+        apply Real.exp_le_exp.mpr
+        have hh := mul_le_mul_of_nonneg_right hcount (by positivity : (0:ℝ)≤(2:ℝ)^r)
+        rw [pow_add]
+        nlinarith
+  apply (palindrome_exception_cards r L ρ hρ ⌈ε*(2:ℝ)^r⌉₊).trans
+  apply add_le_add _ hs
+  simpa only [Nat.cast_pow,Nat.cast_ofNat] using
+    geometric_ceil_bound (blockGap L) ε (blockGap_pos L) (blockGap_lt_one L) (2^r)
+
+end Thorp.UnitaryFinite
+
+namespace Thorp.Specht
+open scoped BigOperators Classical
+
+lemma partition_count (N : ℕ) : Fintype.card (Nat.Partition N)≤2^N := by
+  have h := Fintype.card_le_of_surjective (Nat.Partition.ofComposition N) Nat.Partition.ofComposition_surj
+  rw [composition_card] at h
+  exact h.trans (Nat.pow_le_pow_right (by norm_num) (Nat.sub_le N 1))
+
+lemma shapes_count (N : ℕ) : Fintype.card (Shapes N)≤2^N := by
+  rw [Fintype.card_congr (shapePartitionEquiv N)]
+  exact partition_count N
+
+lemma rowLens_headD (μ : YoungDiagram) : μ.rowLens.headD 0=μ.rowLen 0 := by
+  cases he : μ.rowLens with
+  | nil =>
+    have hc := sum_rowLens μ
+    rw [he,List.sum_nil] at hc
+    have hr := rowLen_zero_le_card μ
+    simp only [List.headD_nil]
+    omega
+  | cons a l =>
+    have hh := YoungDiagram.get_rowLens (μ := μ) (i := 0) (h := by simp [he])
+    simpa only [he,List.getElem_cons_zero,List.headD_cons] using hh
+
+lemma rowLens_tail_sum (μ : YoungDiagram) : μ.rowLens.tail.sum=μ.card-μ.rowLen 0 := by
+  have hh := rowLens_headD μ
+  have hc := sum_rowLens μ
+  cases he : μ.rowLens with
+  | nil => simp only [he,List.headD_nil] at hh; simp only [he,List.sum_nil] at hc; simp [←hc]
+  | cons a l =>
+    simp only [he,List.headD_cons] at hh
+    simp only [he,List.sum_cons] at hc
+    simp only [List.tail_cons]
+    omega
+
+lemma positive_list_eq_of_tail_sum {a b : List ℕ}
+    (ha : ∀ x∈a, 0<x) (hb : ∀ x∈b, 0<x) (ht : a.tail=b.tail) (hs : a.sum=b.sum) : a=b := by
+  cases a with
+  | nil =>
+    cases b with
+    | nil => rfl
+    | cons x l =>
+      simp only [List.tail_nil,List.tail_cons] at ht
+      subst l
+      have hx := hb x (by simp)
+      simp only [List.sum_nil,List.sum_cons,add_zero] at hs
+      omega
+  | cons x l =>
+    cases b with
+    | nil =>
+      simp only [List.tail_nil,List.tail_cons] at ht
+      subst l
+      have hx := ha x (by simp)
+      simp only [List.sum_nil,List.sum_cons,add_zero] at hs
+      omega
+    | cons y m =>
+      simp only [List.tail_cons] at ht
+      subst m
+      simp only [List.sum_cons] at hs
+      congr 1
+      omega
+
+end Thorp.Specht
+
+end ThorpNine.Tail
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Tail/Main.lean b/lean/OAI/Probability/ThorpRouting/Tail/Main.lean
index fed929d1dc2c75860ecf29f5eae2f6f00f7268aa..ee13edc8b1a0ae16e775f9442886af9157bed854 100644
--- a/lean/OAI/Probability/ThorpRouting/Tail/Main.lean
+++ b/lean/OAI/Probability/ThorpRouting/Tail/Main.lean
@@ -1,693 +1,693 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Tail.TailShapes
-
-namespace OAI
-
-namespace ThorpNine.Tail
-
-namespace Thorp.StrongTail
-open scoped BigOperators Classical
-open Specht
-
-noncomputable def tailCellEquiv (μ : YoungDiagram) : Cell (tailDiagram μ) ≃ TailCell μ where
-  toFun x := ⟨⟨(x.1.1+1,x.1.2),(mem_tailDiagram μ _ _).mp x.2⟩,by simp [row]⟩
-  invFun x := ⟨(row x.1-1,col x.1),(mem_tailDiagram μ _ _).mpr (by
-    have he : row x.1-1+1 = row x.1 := by have := x.2; omega
-    change (row x.1-1+1,col x.1) ∈ μ
-    rw [he]
-    exact x.1.2)⟩
-  left_inv x := by apply Subtype.ext; simp [row,col]
-  right_inv x := by
-    apply Subtype.ext; apply Subtype.ext; apply Prod.ext
-    · change row x.1-1+1 = row x.1
-      have := x.2; omega
-    · rfl
-
-noncomputable abbrev upCell (μ : YoungDiagram) (x : Cell (tailDiagram μ)) : Cell μ := (tailCellEquiv μ x).1
-
-@[simp] lemma upCell_row (μ : YoungDiagram) (x : Cell (tailDiagram μ)) : row (upCell μ x) = row x+1 := rfl
-@[simp] lemma upCell_col (μ : YoungDiagram) (x : Cell (tailDiagram μ)) : col (upCell μ x) = col x := rfl
-
-lemma tail_card_eq (μ : YoungDiagram) : (tailDiagram μ).card = μ.card-μ.rowLen 0 := by
-  rw [←card_cell, Fintype.card_congr (tailCellEquiv μ), tail_card]
-
-noncomputable def extendPerm (μ : YoungDiagram) (p : Equiv.Perm (Cell (tailDiagram μ))) :
-    Equiv.Perm (Cell μ) := p.extendDomain (tailCellEquiv μ)
-
-@[simp] lemma extendPerm_up (μ : YoungDiagram) (p : Equiv.Perm (Cell (tailDiagram μ)))
-    (x : Cell (tailDiagram μ)) : extendPerm μ p (upCell μ x) = upCell μ (p x) :=
-  Equiv.Perm.extendDomain_apply_image p (tailCellEquiv μ) x
-
-lemma extendPerm_top (μ : YoungDiagram) (p : Equiv.Perm (Cell (tailDiagram μ)))
-    (x : Cell μ) (hx : row x=0) : extendPerm μ p x = x :=
-  Equiv.Perm.extendDomain_apply_not_subtype p (tailCellEquiv μ) (not_not.mpr hx)
-
-@[simp] lemma extendPerm_inv (μ : YoungDiagram) (p : Equiv.Perm (Cell (tailDiagram μ))) :
-    (extendPerm μ p)⁻¹ = extendPerm μ p⁻¹ := Equiv.Perm.extendDomain_symm _ _
-
-@[simp] lemma extendPerm_sign (μ : YoungDiagram) (p : Equiv.Perm (Cell (tailDiagram μ))) :
-    permSign (extendPerm μ p) = permSign p := by
-  simp [extendPerm,permSign,Equiv.Perm.sign_extendDomain]
-
-noncomputable def liftRows (μ : YoungDiagram) (f : Tabloid (tailDiagram μ)) (x : Cell μ) : Fin (μ.colLen 0) :=
-  if hx : row x ≠ 0 then
-    ⟨(f.1 ((tailCellEquiv μ).symm ⟨x,hx⟩)).val+1,by
-      have h := (f.1 ((tailCellEquiv μ).symm ⟨x,hx⟩)).isLt
-      have hh := tail_height μ
-      omega⟩
-  else rowIndex x
-
-@[simp] lemma liftRows_up (μ : YoungDiagram) (f : Tabloid (tailDiagram μ))
-    (x : Cell (tailDiagram μ)) : (liftRows μ f (upCell μ x)).val = (f.1 x).val+1 := by
-  simp [liftRows,upCell]
-
-lemma liftRows_top (μ : YoungDiagram) (f : Tabloid (tailDiagram μ)) (x : Cell μ) (hx : row x=0) :
-    liftRows μ f x = rowIndex x := by simp [liftRows,hx]
-
-lemma liftRows_perm (μ : YoungDiagram) (f : Tabloid (tailDiagram μ))
-    (p : Equiv.Perm (Cell (tailDiagram μ))) (hf : f.1 = rowIndex ∘ p) :
-    liftRows μ f = rowIndex ∘ extendPerm μ p := by
-  funext x
-  by_cases hx : row x = 0
-  · simp [liftRows_top μ f x hx,extendPerm_top μ p x hx]
-  · let y := (tailCellEquiv μ).symm ⟨x,hx⟩
-    have he : upCell μ y = x := congrArg Subtype.val ((tailCellEquiv μ).apply_symm_apply ⟨x,hx⟩)
-    rw [←he]
-    apply Fin.ext
-    simp [hf,Function.comp_apply,rowIndex]
-
-noncomputable def liftTabloid (μ : YoungDiagram) (f : Tabloid (tailDiagram μ)) : Tabloid μ :=
-  ⟨liftRows μ f, by obtain ⟨p,hp⟩ := f.2; exact ⟨extendPerm μ p,liftRows_perm μ f p hp⟩⟩
-
-@[simp] lemma liftTabloid_base (μ : YoungDiagram) :
-    liftTabloid μ (baseTabloid (tailDiagram μ)) = baseTabloid μ := by
-  apply Subtype.ext
-  funext x
-  by_cases hx : row x=0
-  · exact liftRows_top μ _ x hx
-  · let y := (tailCellEquiv μ).symm ⟨x,hx⟩
-    have he : upCell μ y = x := congrArg Subtype.val ((tailCellEquiv μ).apply_symm_apply ⟨x,hx⟩)
-    rw [←he]
-    apply Fin.ext
-    exact liftRows_up μ _ y
-
-lemma liftTabloid_action (μ : YoungDiagram) (p : Equiv.Perm (Cell (tailDiagram μ)))
-    (f : Tabloid (tailDiagram μ)) :
-    liftTabloid μ (tabloidAct p f) = tabloidAct (extendPerm μ p) (liftTabloid μ f) := by
-  apply Subtype.ext
-  funext x
-  change liftRows μ (tabloidAct p f) x = liftRows μ f ((extendPerm μ p)⁻¹ x)
-  rw [extendPerm_inv]
-  by_cases hx : row x=0
-  · rw [extendPerm_top μ p⁻¹ x hx,liftRows_top μ _ x hx,liftRows_top μ _ x hx]
-  · let y := (tailCellEquiv μ).symm ⟨x,hx⟩
-    have he : upCell μ y = x := congrArg Subtype.val ((tailCellEquiv μ).apply_symm_apply ⟨x,hx⟩)
-    rw [←he,extendPerm_up]
-    apply Fin.ext
-    simp only [liftRows_up,tabloidAct_apply]
-
-noncomputable def restrictTabloids (μ : YoungDiagram) :
-    (Tabloid μ → ℂ) →ₗ[ℂ] (Tabloid (tailDiagram μ) → ℂ) where
-  toFun v := fun f => v (liftTabloid μ f)
-  map_add' := by intros; rfl
-  map_smul' := by intros; rfl
-
-lemma restrict_action (μ : YoungDiagram) (p : Equiv.Perm (Cell (tailDiagram μ)))
-    (v : Tabloid μ → ℂ) :
-    restrictTabloids μ (tabloidRep μ (extendPerm μ p) v) =
-      tabloidRep (tailDiagram μ) p (restrictTabloids μ v) := by
-  ext f
-  change v (tabloidAct (extendPerm μ p)⁻¹ (liftTabloid μ f)) = v (liftTabloid μ (tabloidAct p⁻¹ f))
-  rw [extendPerm_inv,liftTabloid_action]
-
-lemma extend_colGroup (μ : YoungDiagram) (c : colGroup (tailDiagram μ)) :
-    extendPerm μ c ∈ colGroup μ := by
-  intro x
-  by_cases hx : row x=0
-  · rw [extendPerm_top μ c x hx]
-  · let y := (tailCellEquiv μ).symm ⟨x,hx⟩
-    have he : upCell μ y = x := congrArg Subtype.val ((tailCellEquiv μ).apply_symm_apply ⟨x,hx⟩)
-    rw [←he,extendPerm_up,upCell_col,upCell_col]
-    exact c.2 y
-
-lemma restrict_polytabloid (μ : YoungDiagram) :
-    restrictTabloids μ (polytabloid μ) = polytabloid (tailDiagram μ) := by
-  let v := restrictTabloids μ (polytabloid μ)
-  have hv (c : colGroup (tailDiagram μ)) :
-      tabloidRep (tailDiagram μ) c v = permSign (c:Equiv.Perm (Cell (tailDiagram μ))) • v := by
-    rw [←restrict_action]
-    rw [polytabloid_col_action μ ⟨extendPerm μ c,extend_colGroup μ c⟩]
-    simp only [map_smul,extendPerm_sign]
-    rfl
-  have ha : alternator (colGroup (tailDiagram μ)) (tabloidRep (tailDiagram μ)) v =
-      (Fintype.card (colGroup (tailDiagram μ)):ℂ) • v := by
-    simp only [alternator,LinearMap.sum_apply,LinearMap.smul_apply,hv,smul_smul,permSign_sq,one_smul]
-    ext f
-    simp
-  have hb : v (baseTabloid (tailDiagram μ)) = 1 := by
-    change polytabloid μ (liftTabloid μ (baseTabloid (tailDiagram μ))) = 1
-    rw [liftTabloid_base, polytabloid_base]
-  have he := alternator_rank_one_exact (tailDiagram μ) v
-  rw [ha] at he
-  simp only [Pi.smul_apply,hb,smul_eq_mul,mul_one] at he
-  have hc : (Fintype.card (colGroup (tailDiagram μ)):ℂ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
-  exact (smul_right_injective _ hc) he
-
-lemma tail_space_le_image (μ : YoungDiagram) :
-    space (tailDiagram μ) ≤ (space μ).map (restrictTabloids μ) := by
-  apply Submodule.span_le.mpr
-  rintro _ ⟨p,rfl⟩
-  refine ⟨tabloidRep μ (extendPerm μ p) (polytabloid μ), orbit_mem_space μ _, ?_⟩
-  rw [restrict_action,restrict_polytabloid]
-
-theorem tail_dimension_le (μ : YoungDiagram) :
-    Module.finrank ℂ (space (tailDiagram μ)) ≤ Module.finrank ℂ (space μ) :=
-  (Submodule.finrank_mono (tail_space_le_image μ)).trans (Submodule.finrank_map_le _ _)
-
-end Thorp.StrongTail
-
-namespace Thorp.ButterflyReverse
-open scoped BigOperators
-
-abbrev Site (d : ℕ) := {t : Fin d × Card d // t.2 t.1 = false}
-
-def site : (d : ℕ) → SwitchIndex d → Site d
-  | 0, t => nomatch t
-  | _+1, Sum.inl y => ⟨(0, Fin.cons false y), rfl⟩
-  | d+1, Sum.inr (b,t) =>
-      ⟨((site d t).1.1.succ, Fin.cons b (site d t).1.2), (site d t).2⟩
-
-lemma site_injective (d : ℕ) : Function.Injective (site d) := by
-  induction d with
-  | zero => intro t; exact Empty.elim t
-  | succ d ih =>
-    intro s t h
-    cases s with
-    | inl y =>
-      cases t with
-      | inl z =>
-        have he := congrArg (fun a : Site (d+1) => Fin.tail a.1.2) h
-        exact congrArg Sum.inl he
-      | inr t =>
-        have he := congrArg (fun a : Site (d+1) => a.1.1.val) h
-        simp [site] at he
-    | inr s =>
-      cases t with
-      | inl y =>
-        have he := congrArg (fun a : Site (d+1) => a.1.1.val) h
-        simp [site] at he
-      | inr t =>
-        obtain ⟨b,s⟩ := s
-        obtain ⟨c,t⟩ := t
-        have hbc := congrArg (fun a : Site (d+1) => a.1.2 0) h
-        simp only [site, Fin.cons_zero] at hbc
-        have hst : site d s = site d t := by
-          apply Subtype.ext
-          apply Prod.ext
-          · exact Fin.succ_inj.mp (congrArg (fun a : Site (d+1) => a.1.1) h)
-          · exact congrArg (fun a : Site (d+1) => Fin.tail a.1.2) h
-        cases hbc
-        cases ih hst
-        rfl
-
-lemma site_surjective (d : ℕ) : Function.Surjective (site d) := by
-  induction d with
-  | zero => intro t; exact Fin.elim0 t.1.1
-  | succ d ih =>
-    rintro ⟨⟨i,x⟩,hx⟩
-    refine Fin.cases ?_ (fun j => ?_) i hx
-    · intro hx0
-      refine ⟨Sum.inl (Fin.tail x), ?_⟩
-      apply Subtype.ext
-      simp only [site, Prod.mk.injEq, true_and]
-      rw [←hx0,Fin.cons_self_tail]
-    · intro hxj
-      obtain ⟨t,ht⟩ := ih ⟨(j,Fin.tail x),hxj⟩
-      refine ⟨Sum.inr (x 0,t), ?_⟩
-      apply Subtype.ext
-      simp only [site,ht,Fin.cons_self_tail]
-
-noncomputable def siteEquiv (d : ℕ) : SwitchIndex d ≃ Site d :=
-  Equiv.ofBijective (site d) ⟨site_injective d,site_surjective d⟩
-
-def reverseCard (d : ℕ) : Equiv.Perm (Card d) where
-  toFun x := fun i => x i.rev
-  invFun x := fun i => x i.rev
-  left_inv x := by funext i; simp
-  right_inv x := by funext i; simp
-
-@[simp] lemma reverseCard_apply (d : ℕ) (x : Card d) (i : Fin d) :
-    reverseCard d x i = x i.rev := rfl
-
-@[simp] lemma reverseCard_involutive (d : ℕ) (x : Card d) :
-    reverseCard d (reverseCard d x) = x := by funext i; simp
-
-def mirrorSite (d : ℕ) : Equiv.Perm (Site d) where
-  toFun s := ⟨(s.1.1.rev,reverseCard d s.1.2),by simpa using s.2⟩
-  invFun s := ⟨(s.1.1.rev,reverseCard d s.1.2),by simpa using s.2⟩
-  left_inv s := by apply Subtype.ext; simp
-  right_inv s := by apply Subtype.ext; simp
-
-noncomputable def mirrorSwitch (d : ℕ) : Equiv.Perm (SwitchIndex d) :=
-  (siteEquiv d).trans ((mirrorSite d).trans (siteEquiv d).symm)
-
-lemma site_mirrorSwitch (d : ℕ) (t : SwitchIndex d) :
-    site d (mirrorSwitch d t) = mirrorSite d (site d t) := by
-  exact (siteEquiv d).apply_symm_apply _
-
-lemma routeDomain_site (d : ℕ) (x y : Card d) (t : SwitchIndex d) :
-    t ∈ routeDomain d x y ↔
-      ∀ j : Fin d, j ≠ (site d t).1.1 →
-      (site d t).1.2 j = if j < (site d t).1.1 then x j else y j := by
-  classical
-  induction d with
-  | zero => exact Empty.elim t
-  | succ d ih =>
-    cases t with
-    | inl z =>
-      change Sum.inl z ∈ (insert (Sum.inl (Fin.tail y))
-        ((routeDomain d (Fin.tail x) (Fin.tail y)).image
-          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) :
-        Finset (Sum (Card d) (Bool × SwitchIndex d))) ↔ _
-      simp only [Finset.mem_insert, Sum.inl.injEq,Finset.mem_image,
-        reduceCtorEq,and_false,exists_false,or_false]
-      constructor
-      · intro hz j hj
-        subst z
-        have hj0 : j ≠ 0 := hj
-        obtain ⟨i,rfl⟩ := Fin.eq_succ_of_ne_zero hj0
-        simp [site,Fin.tail]
-      · intro h
-        funext j
-        have he := h j.succ (by simp [site])
-        simpa [site,Fin.tail] using he
-    | inr z =>
-      obtain ⟨b,t⟩ := z
-      change Sum.inr (b,t) ∈ (insert (Sum.inl (Fin.tail y))
-        ((routeDomain d (Fin.tail x) (Fin.tail y)).image
-          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) :
-        Finset (Sum (Card d) (Bool × SwitchIndex d))) ↔ _
-      simp only [Finset.mem_insert,reduceCtorEq,false_or,Finset.mem_image,Sum.inr.injEq,Prod.mk.injEq]
-      constructor
-      · rintro ⟨s,hs,hb,ht⟩
-        subst s
-        have hst := (ih (Fin.tail x) (Fin.tail y) t).mp hs
-        intro j hj
-        refine Fin.cases ?_ (fun i => ?_) j hj
-        · intro _
-          simp [site,hb]
-        · intro hi
-          have hi' : i ≠ (site d t).1.1 := by simpa [site] using hi
-          simpa [site,Fin.tail] using hst i hi'
-      · intro h
-        have hb := h 0 (by simpa only [site] using (Fin.succ_ne_zero (site d t).1.1).symm)
-        have hb' : x 0 = b := by simpa [site] using hb.symm
-        refine ⟨t,?_,hb',rfl⟩
-        apply (ih (Fin.tail x) (Fin.tail y) t).mpr
-        intro j hj
-        have he := h j.succ (by simpa [site] using hj)
-        simpa [site,Fin.tail] using he
-
-lemma routeValue_site (d : ℕ) (x y : Card d) (t : SwitchIndex d) :
-    routeValue d x y t = Bool.xor (x (site d t).1.1) (y (site d t).1.1) := by
-  induction d with
-  | zero => exact Empty.elim t
-  | succ d ih =>
-    cases t with
-    | inl z => rfl
-    | inr z => exact ih (Fin.tail x) (Fin.tail y) z.2
-
-lemma mirrorSwitch_involutive (d : ℕ) (t : SwitchIndex d) :
-    mirrorSwitch d (mirrorSwitch d t) = t := by
-  apply site_injective d
-  rw [site_mirrorSwitch,site_mirrorSwitch]
-  apply Subtype.ext
-  simp [mirrorSite]
-
-lemma mirror_routeDomain (d : ℕ) (x y : Card d) (t : SwitchIndex d)
-    (ht : t ∈ routeDomain d x y) :
-    mirrorSwitch d t ∈ routeDomain d (reverseCard d y) (reverseCard d x) := by
-  rw [routeDomain_site,site_mirrorSwitch]
-  intro j hj
-  have hj' : j.rev ≠ (site d t).1.1 := by
-    intro h
-    apply hj
-    change j = (site d t).1.1.rev
-    rw [←h,Fin.rev_rev]
-  have he := (routeDomain_site d x y t).mp ht j.rev hj'
-  change (site d t).1.2 j.rev =
-    if j < (site d t).1.1.rev then y j.rev else x j.rev
-  rw [he]
-  have hord : j < (site d t).1.1.rev ↔ ¬j.rev < (site d t).1.1 := by
-    rw [Fin.lt_rev_iff]
-    constructor
-    · intro h
-      exact not_lt_of_ge h.le
-    · intro h
-      exact lt_of_le_of_ne (le_of_not_gt h) (Ne.symm hj')
-  by_cases h : j.rev < (site d t).1.1 <;> simp [h,hord]
-
-lemma mirror_routeValue (d : ℕ) (x y : Card d) (t : SwitchIndex d) :
-    routeValue d (reverseCard d y) (reverseCard d x) (mirrorSwitch d t) =
-      routeValue d x y t := by
-  rw [routeValue_site,routeValue_site,site_mirrorSwitch]
-  simp only [mirrorSite,Equiv.coe_fn_mk,reverseCard_apply,Fin.rev_rev]
-  exact Bool.xor_comm _ _
-
-noncomputable def mirrorCoins (d : ℕ) : Equiv.Perm (SwitchIndex d → Bool) where
-  toFun ω := fun t => ω (mirrorSwitch d t)
-  invFun ω := fun t => ω (mirrorSwitch d t)
-  left_inv ω := by funext t; change ω (mirrorSwitch d (mirrorSwitch d t)) = ω t; rw [mirrorSwitch_involutive]
-  right_inv ω := by funext t; change ω (mirrorSwitch d (mirrorSwitch d t)) = ω t; rw [mirrorSwitch_involutive]
-
-lemma mirror_endpoint (d : ℕ) (ω : SwitchIndex d → Bool) (x y : Card d)
-    (hxy : butterflyPerm d (decodeButterfly d ω) x = y) :
-    butterflyPerm d (decodeButterfly d (mirrorCoins d ω)) (reverseCard d y) =
-      reverseCard d x := by
-  apply (butterfly_endpoint_iff d _ _ _).mpr
-  intro t ht
-  have ht' := mirror_routeDomain d (reverseCard d y) (reverseCard d x) t ht
-  simp only [reverseCard_involutive] at ht'
-  have he := (butterfly_endpoint_iff d ω x y).mp hxy (mirrorSwitch d t) ht'
-  change ω (mirrorSwitch d t) = _
-  rw [he]
-  have hv := mirror_routeValue d x y (mirrorSwitch d t)
-  rw [mirrorSwitch_involutive] at hv
-  exact hv.symm
-
-lemma butterfly_mirror (d : ℕ) (ω : SwitchIndex d → Bool) :
-    butterflyPerm d (decodeButterfly d (mirrorCoins d ω)) =
-      reverseCard d * (butterflyPerm d (decodeButterfly d ω))⁻¹ * reverseCard d := by
-  apply Equiv.ext
-  intro x
-  have he := mirror_endpoint d ω
-    ((butterflyPerm d (decodeButterfly d ω))⁻¹ (reverseCard d x)) (reverseCard d x)
-    (Equiv.apply_symm_apply _ _)
-  simpa only [reverseCard_involutive,Equiv.Perm.mul_apply] using he
-
-def reverseOrder (d : ℕ) : Equiv.Perm (Fin d) where
-  toFun := Fin.rev
-  invFun := Fin.rev
-  left_inv := Fin.rev_rev
-  right_inv := Fin.rev_rev
-
-lemma relabel_mul (d : ℕ) (a b : Equiv.Perm (Fin d)) :
-    StrongSmoothing.coordinateRelabel d (a*b) =
-      StrongSmoothing.coordinateRelabel d a * StrongSmoothing.coordinateRelabel d b := rfl
-
-lemma relabel_reverse (d : ℕ) :
-    StrongSmoothing.coordinateRelabel d (reverseOrder d) = reverseCard d := rfl
-
-lemma reverseCard_sq (d : ℕ) : reverseCard d * reverseCard d = 1 := by
-  ext x i
-  simp
-
-lemma sweep_mirror (d : ℕ) (order : Equiv.Perm (Fin d)) (ω : SwitchIndex d → Bool) :
-    StrongSmoothing.sweep d (order*reverseOrder d) (mirrorCoins d ω) =
-      (StrongSmoothing.sweep d order ω)⁻¹ := by
-  simp only [StrongSmoothing.sweep,relabel_mul,relabel_reverse,butterfly_mirror,mul_inv_rev]
-  have hR : (reverseCard d)⁻¹ = reverseCard d := by
-    exact inv_eq_of_mul_eq_one_right (reverseCard_sq d)
-  rw [hR]
-  calc
-    _ = StrongSmoothing.coordinateRelabel d order *
-        (reverseCard d*reverseCard d) *
-        (butterflyPerm d (decodeButterfly d ω))⁻¹ *
-        (reverseCard d*reverseCard d) * (StrongSmoothing.coordinateRelabel d order)⁻¹ := by group
-    _ = _ := by rw [reverseCard_sq]; group
-
-lemma sweepKernel_reverse (d l : ℕ) (order : Equiv.Perm (Fin d)) :
-    StrongSmoothing.sweepKernel d l (order*reverseOrder d) =
-      (StrongSmoothing.sweepKernel d l order).transpose := by
-  classical
-  rw [StrongSmoothing.sweepKernel_law,StrongSmoothing.sweepKernel_law,
-    ←StrongSmoothing.PermLaw.inverse]
-  ext x y
-  unfold StrongSmoothing.PermLaw.kernel
-  rw [finiteMean_equiv (mirrorCoins d)]
-  apply finiteMean_congr
-  intro ω
-  change (if StrongSmoothing.tuplePerm
-      (StrongSmoothing.sweep d (order*reverseOrder d) (mirrorCoins d ω)) x = y then (1:ℝ) else 0) = _
-  rw [sweep_mirror,StrongSmoothing.tuplePerm_inverse]
-
-lemma reflectedKernel_reverse (d l : ℕ) (order : Equiv.Perm (Fin d)) :
-    StrongSmoothing.reflectedKernel d l order true =
-      StrongSmoothing.reflectedKernel d l (order*reverseOrder d) false := by
-  simp only [StrongSmoothing.reflectedKernel,Bool.false_eq_true,
-    ↓reduceIte,sweepKernel_reverse,Matrix.transpose_transpose]
-
-end Thorp.ButterflyReverse
-
-namespace Thorp.StrongTail
-open scoped BigOperators Classical
-
-section
-open StrongSmoothing
-
-lemma reflected_nonneg_both (d l : ℕ) (order : Equiv.Perm (Fin d)) (rev : Bool)
-    (x y : Tuples d l) : 0 ≤ reflectedKernel d l order rev x y := by
-  cases rev
-  · exact reflected_nonneg _ _ _ _ _
-  · rw [ButterflyReverse.reflectedKernel_reverse]
-    exact reflected_nonneg _ _ _ _ _
-
-lemma reflected_symm_both (d l : ℕ) (order : Equiv.Perm (Fin d)) (rev : Bool) :
-    (reflectedKernel d l order rev).IsSymm := by
-  cases rev
-  · exact reflected_symmetric _ _ _
-  · rw [ButterflyReverse.reflectedKernel_reverse]
-    exact reflected_symmetric _ _ _
-
-lemma reflected_moment_both (d l : ℕ) (order : Equiv.Perm (Fin d)) (rev : Bool)
-    (x : Tuples d l) :
-    finiteMean (fun y : Tuples d l =>
-      ((Fintype.card (Tuples d l):ℝ)*reflectedKernel d l order rev x y)^(1+(1/1000:ℝ))) ≤
-      Real.exp (l*(1+heightDecay 1)) := by
-  cases rev
-  · exact reflected_moment _ _ _ _
-  · rw [ButterflyReverse.reflectedKernel_reverse]
-    exact reflected_moment _ _ _ _
-
-lemma reflected_moment_small (d l : ℕ) (order : Equiv.Perm (Fin d)) (rev : Bool)
-    (x : Tuples d l) {p : ℝ} (hp : 0 < p) (hpq : p ≤ 1+(1/1000:ℝ)) :
-    finiteMean (fun y : Tuples d l =>
-      ((Fintype.card (Tuples d l):ℝ)*reflectedKernel d l order rev x y)^p) ≤
-      Real.exp (l*(1+heightDecay 1)) := by
-  let : Nonempty (Tuples d l) := ⟨x⟩
-  let q : ℝ := 1+1/1000
-  let f := fun y : Tuples d l => (Fintype.card (Tuples d l):ℝ)*reflectedKernel d l order rev x y
-  have hf : ∀ y, 0 ≤ f y := fun y => mul_nonneg (Nat.cast_nonneg _) (reflected_nonneg_both _ _ _ _ _ _)
-  have hq : 0 < q := by norm_num [q]
-  have hr : 0 ≤ p/q := (div_pos hp hq).le
-  have hr1 : p/q ≤ 1 := (div_le_one hq).mpr hpq
-  have he (y) : (f y)^p = ((f y)^q)^(p/q) := by
-    rw [←Real.rpow_mul (hf y)]
-    congr 1
-    field_simp
-  simp_rw [show (fun y : Tuples d l => ((Fintype.card (Tuples d l):ℝ)*reflectedKernel d l order rev x y)^p)
-    = fun y => ((f y)^q)^(p/q) from funext he]
-  apply (mean_rpow_concave _ (fun y => Real.rpow_nonneg (hf y) _) hr hr1).trans
-  apply (Real.rpow_le_rpow (finiteMean_nonneg (fun y => Real.rpow_nonneg (hf y) _))
-    (reflected_moment_both d l order rev x) hr).trans
-  rw [Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp]
-  apply Real.exp_le_exp.mpr
-  have hA : 0 ≤ (l:ℝ)*(1+heightDecay 1) := by
-    have hh := heightDecay_nonneg 1
-    positivity
-  nlinarith
-
-lemma log_factorial_le_power (n N : ℕ) (hn : n ≤ N) (_hN : 0 < N) :
-    Real.log (n.factorial:ℝ) ≤ (n:ℝ)*Real.log N := by
-  have hb : (n.factorial:ℝ) ≤ (N:ℝ)^n := by
-    exact_mod_cast (Nat.factorial_le_pow n).trans (Nat.pow_le_pow_left hn n)
-  simpa only [Real.log_pow] using Real.log_le_log (by positivity : (0:ℝ)<n.factorial) hb
-
-lemma log_normalization_le (N l : ℕ) (hN : 0 < N) (hl : l ≤ N) :
-    Real.log ((N:ℝ)^l/(N.descFactorial l:ℝ)) ≤ 2*l := by
-  have hNp : (0:ℝ)<N := by exact_mod_cast hN
-  have hMp : (0:ℝ)<N.descFactorial l := by exact_mod_cast Nat.descFactorial_pos.mpr hl
-  rw [Real.log_div (by positivity) hMp.ne',Real.log_pow]
-  by_cases hlow : 2*l ≤ N
-  · have hhalf : (N:ℝ)/2 ≤ (N+1-l:ℕ) := by
-      have hNat : N ≤ 2*(N+1-l) := by omega
-      exact (div_le_iff₀ (by norm_num : (0:ℝ)<2)).mpr (by simpa only [mul_comm] using (show (N:ℝ) ≤ 2*(N+1-l:ℕ) by exact_mod_cast hNat))
-    have hb : ((N:ℝ)/2)^l ≤ (N.descFactorial l:ℝ) := by
-      apply (pow_le_pow_left₀ (by positivity) hhalf l).trans
-      exact_mod_cast Nat.pow_sub_le_descFactorial N l
-    have hh := Real.log_le_log (pow_pos (by positivity : (0:ℝ)<(N:ℝ)/2) l) hb
-    rw [Real.log_pow,Real.log_div hNp.ne' (by norm_num)] at hh
-    have ht : Real.log (2:ℝ) ≤ 1 := by simpa only [show (2:ℝ)-1=1 by norm_num] using Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
-    nlinarith [(show (0:ℝ) ≤ l by positivity)]
-  · have he : (N.descFactorial l:ℝ)*(Nat.factorial (N-l):ℝ) = N.factorial := by
-      have he := Nat.factorial_mul_descFactorial hl
-      exact_mod_cast (by simpa only [mul_comm] using he : N.descFactorial l * (N-l).factorial = N.factorial)
-    have hh := congrArg Real.log he
-    rw [Real.log_mul hMp.ne' (by positivity)] at hh
-    have hfull := log_factorial_lower (show 1 ≤ N by omega)
-    have hrem := log_factorial_le_power (N-l) N (Nat.sub_le _ _) hN
-    rw [Nat.cast_sub hl] at hrem
-    have hNle : (N:ℝ) ≤ 2*l := by exact_mod_cast (by omega : N ≤ 2*l)
-    linarith
-
-lemma normalized_moment_identity {X : Type*} [Fintype X] [Nonempty X]
-    (f : X → ℝ) (hf : ∀ x, 0 ≤ f x) {N : ℝ} (hN : 0 < N) (p : ℝ) :
-    N⁻¹*(∑ x, (N*f x)^p) =
-     (N/(Fintype.card X:ℝ))^(p-1) * finiteMean (fun x => ((Fintype.card X:ℝ)*f x)^p) := by
-  let M : ℝ := Fintype.card X
-  have hM : 0 < M := Nat.cast_pos.mpr Fintype.card_pos
-  have he (x) : (N*f x)^p = (N/M)^p*(M*f x)^p := by
-    rw [←Real.mul_rpow (div_nonneg hN.le hM.le) (mul_nonneg hM.le (hf x))]
-    congr 1
-    field_simp
-  simp_rw [he]
-  rw [←Finset.mul_sum,Real.rpow_sub (div_pos hN hM),Real.rpow_one]
-  unfold finiteMean
-  dsimp [M] at hM ⊢
-  field_simp
-
-lemma density_at_small_exponent {p : ℝ} (hp : 1 < p) (hpq : p ≤ 1+(1/1000:ℝ)) :
-    DensityAt p (3+heightDecay 1) := by
-  intro d l hl order rev x
-  dsimp only
-  let : Nonempty (Tuples d l) := ⟨x⟩
-  let H := reflectedKernel d l order rev
-  have hm := reflected_moment_small d l order rev x (by linarith : 0<p) hpq
-  have hb : Real.exp ((l:ℝ)*(1+heightDecay 1)) ≤ Real.exp ((3+heightDecay 1)*l) := by
-    apply Real.exp_le_exp.mpr
-    nlinarith [(show (0:ℝ) ≤ l by positivity)]
-  have hs (y) : reflectedKernel d l order rev y x = reflectedKernel d l order rev x y := (reflected_symm_both d l order rev).apply _ _
-  have hpow : (((2:ℝ)^d)^l)⁻¹ * (∑ y, ((((2:ℝ)^d)^l)*H x y)^p) ≤
-      Real.exp ((3+heightDecay 1)*l) := by
-    rw [normalized_moment_identity (fun y => H x y) (reflected_nonneg_both d l order rev x)
-      (by positivity) p]
-    have hrat : ((2:ℝ)^d)^l/(Fintype.card (Tuples d l):ℝ) ≤ Real.exp (2*l) := by
-      apply (Real.log_le_iff_le_exp (by positivity)).mp
-      rw [tuples_card]
-      exact_mod_cast log_normalization_le (2^d) l (by positivity) hl
-    have hr : 0 ≤ ((2:ℝ)^d)^l/(Fintype.card (Tuples d l):ℝ) := by positivity
-    have hrp : (((2:ℝ)^d)^l/(Fintype.card (Tuples d l):ℝ))^(p-1) ≤ Real.exp (2*l) := by
-      apply (Real.rpow_le_rpow hr hrat (by linarith)).trans
-      rw [Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp]
-      apply Real.exp_le_exp.mpr
-      have hpl : p-1 ≤ 1 := by linarith
-      nlinarith [(show (0:ℝ) ≤ l by positivity)]
-    apply (mul_le_mul hrp hm (finiteMean_nonneg (fun y => Real.rpow_nonneg
-      (mul_nonneg (Nat.cast_nonneg _) (reflected_nonneg_both d l order rev x y)) _)) (Real.exp_pos _).le).trans
-    rw [←Real.exp_add]
-    apply Real.exp_le_exp.mpr
-    ring_nf
-    rfl
-  dsimp [H] at hpow
-  refine ⟨hm.trans hb,?_,hpow,?_⟩
-  · simpa only [hs] using hm.trans hb
-  · simpa only [hs] using hpow
-
-end
-open Specht UnitaryFinite
-
-lemma dimension_le_factorial (μ : YoungDiagram) :
-    Module.finrank ℂ (space μ) ≤ μ.card.factorial := by
-  have hdim : Module.finrank ℂ (space μ) ≤ Fintype.card (Tabloid μ) := by
-    simpa using Submodule.finrank_le (space μ)
-  have hrow : 1 ≤ Fintype.card (rowGroup μ) := Fintype.card_pos
-  have hc := card_tabloid_mul_row μ
-  nlinarith
-
-lemma tail_card_exact (μ : YoungDiagram) :
-    (tailDiagram μ).card = μ.card - μ.rowLen 0 := by
-  rw [←card_cell,Fintype.card_congr (tailCellEquiv μ),tail_card]
-
-lemma tail_dimension_le_factorial (μ : YoungDiagram) :
-    Module.finrank ℂ (space (tailDiagram μ)) ≤ (μ.card - μ.rowLen 0).factorial := by
-  simpa only [tail_card_exact] using dimension_le_factorial (tailDiagram μ)
-
-lemma sweep_sq_palindrome (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Cell μ) (rev : Bool) :
-    ‖SparseContact.sweepOperator d μ e rev‖^2 =
-      ‖sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e) (palindromePerm d)‖ := by
-  rw [SparseContact.sweep_forward_norm, sampleOperator_palindrome _ _ (relabelledUnitary_unitary μ e)]
-  change ‖sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e)
-    (fun ω => butterflyPerm d (decodeButterfly d ω))‖^2 = _
-  rw [show (sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e)
-    (fun ω => butterflyPerm d (decodeButterfly d ω))).adjoint =
-    star (sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e)
-    (fun ω => butterflyPerm d (decodeButterfly d ω))) from rfl]
-  simpa only [pow_two] using (CStarRing.norm_self_mul_star (x := sampleOperator (V := hilbertSpace μ)
-    (relabelledUnitary μ e) (fun ω => butterflyPerm d (decodeButterfly d ω)))).symm
-
-lemma sweep_sq_le_one (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Cell μ) (rev : Bool) :
-    ‖SparseContact.sweepOperator d μ e rev‖^2 ≤ 1 := by
-  have h := SparseContact.sweep_norm_le_one d μ e rev
-  nlinarith [norm_nonneg (SparseContact.sweepOperator d μ e rev)]
-
-theorem strong_tail : MainStatement := by
-  obtain ⟨a,h,ha,hh,dE⟩ := Specht.spectral_regimes
-  obtain ⟨D0,hD0⟩ := Filter.eventually_atTop.mp dE
-  let d0 := max D0 1
-  let t : ℝ := min a (1/1000)
-  have ht : 0 < t := lt_min ha (by norm_num)
-  have hta : t ≤ a := min_le_left _ _
-  have htq : t ≤ 1/1000 := min_le_right _ _
-  let b := t/(1+t)
-  have hden : 0 < 1+t := by linarith
-  have hb : 0 < b := div_pos ht hden
-  have hbt : b ≤ t := (div_le_iff₀ hden).mpr (by nlinarith [sq_nonneg t])
-  have hba : b ≤ a := hbt.trans hta
-  have hb1 : b ≤ 1 := (div_le_one hden).mpr (by linarith)
-  let C := Real.log (((2^d0).factorial:ℕ):ℝ)
-  have hC : 0 ≤ C := Real.log_nonneg (by exact_mod_cast Nat.factorial_pos (2^d0))
-  refine ⟨1+t,3+heightDecay 1,C,by linarith,by linarith,
-    density_at_small_exponent (by linarith) (by linarith),?_⟩
-  intro d μ e rev
-  let k := μ.card - μ.rowLen 0
-  let D : ℝ := Module.finrank ℂ (space (tailDiagram μ))
-  have hD : 1 ≤ D := by dsimp [D]; exact_mod_cast dimension_pos (tailDiagram μ)
-  have hDp : 0 < D := by linarith
-  have hlog : 0 ≤ Real.log D := Real.log_nonneg hD
-  have hμ : μ.card=2^d := by rw [←card_cell,←Fintype.card_congr e,card_positions]
-  have hform : -(((1+t)-1)/(1+t)) = -b := by dsimp [b]; congr 1; ring
-  rw [hform]
-  change ‖SparseContact.sweepOperator d μ e rev‖^2 ≤ Real.exp (C*k)*D^(-b)
-  rw [Real.rpow_def_of_pos hDp,←Real.exp_add]
-  by_cases hk : k=0
-  · have hDone : D=1 := by
-      have hn := tail_dimension_le_factorial μ
-      change Module.finrank ℂ (space (tailDiagram μ)) ≤ k.factorial at hn
-      rw [hk,Nat.factorial_zero] at hn
-      have hu : D ≤ 1 := by dsimp [D]; exact_mod_cast hn
-      linarith
-    rw [hk,Nat.cast_zero,mul_zero,hDone,Real.log_one,zero_mul,add_zero,Real.exp_zero]
-    exact sweep_sq_le_one d μ e rev
-  · have hkpos : 0<k := by omega
-    by_cases hd : d0 ≤ d
-    · have hdpos : 0<d := by have := le_max_right D0 1; omega
-      by_cases htall : (μ.card:ℝ)/2 < μ.colLen 0
-      · rw [sweep_sq_palindrome,Specht.palindrome_tall_zero_general d hdpos μ e htall,
-          @norm_zero (hilbertSpace μ →L[ℂ] hilbertSpace μ) _]
-        exact (Real.exp_pos _).le
-      · have hg := (hD0 d ((le_max_left _ _).trans hd) μ e hkpos (le_of_not_gt htall)).1
-        have hrootpos : (0:ℝ)<Module.finrank ℂ (space μ) := by exact_mod_cast dimension_pos μ
-        have hlogle : Real.log D ≤ Real.log (Module.finrank ℂ (space μ):ℝ) :=
-          Real.log_le_log hDp (by dsimp [D]; exact_mod_cast tail_dimension_le μ)
-        rw [sweep_sq_palindrome]
-        apply hg.trans
-        apply Real.exp_le_exp.mpr
-        have hbound := mul_le_mul hba hlogle hlog ha.le
-        have hck : 0 ≤ C*(k:ℝ) := mul_nonneg hC (Nat.cast_nonneg _)
-        nlinarith
-    · have hpow : 2^d ≤ 2^d0 := Nat.pow_le_pow_right (by decide) (by omega)
-      have hklarge : k ≤ 2^d0 := (by simp [k,hμ] : k ≤ 2^d).trans hpow
-      have hdim : D ≤ ((2^d0).factorial:ℝ) := by
-        dsimp [D]
-        exact_mod_cast (tail_dimension_le_factorial μ).trans (Nat.factorial_le hklarge)
-      have hlC : Real.log D ≤ C := Real.log_le_log hDp hdim
-      have hk1 : (1:ℝ) ≤ k := by exact_mod_cast hkpos
-      have hprod : b*Real.log D ≤ C*(k:ℝ) := by nlinarith
-      apply (sweep_sq_le_one d μ e rev).trans
-      exact Real.one_le_exp_iff.mpr (by nlinarith)
-
-end Thorp.StrongTail
-
-end ThorpNine.Tail
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Tail.TailShapes
+
+namespace OAI
+
+namespace ThorpNine.Tail
+
+namespace Thorp.StrongTail
+open scoped BigOperators Classical
+open Specht
+
+noncomputable def tailCellEquiv (μ : YoungDiagram) : Cell (tailDiagram μ) ≃ TailCell μ where
+  toFun x := ⟨⟨(x.1.1+1,x.1.2),(mem_tailDiagram μ _ _).mp x.2⟩,by simp [row]⟩
+  invFun x := ⟨(row x.1-1,col x.1),(mem_tailDiagram μ _ _).mpr (by
+    have he : row x.1-1+1 = row x.1 := by have := x.2; omega
+    change (row x.1-1+1,col x.1) ∈ μ
+    rw [he]
+    exact x.1.2)⟩
+  left_inv x := by apply Subtype.ext; simp [row,col]
+  right_inv x := by
+    apply Subtype.ext; apply Subtype.ext; apply Prod.ext
+    · change row x.1-1+1 = row x.1
+      have := x.2; omega
+    · rfl
+
+noncomputable abbrev upCell (μ : YoungDiagram) (x : Cell (tailDiagram μ)) : Cell μ := (tailCellEquiv μ x).1
+
+@[simp] lemma upCell_row (μ : YoungDiagram) (x : Cell (tailDiagram μ)) : row (upCell μ x) = row x+1 := rfl
+@[simp] lemma upCell_col (μ : YoungDiagram) (x : Cell (tailDiagram μ)) : col (upCell μ x) = col x := rfl
+
+lemma tail_card_eq (μ : YoungDiagram) : (tailDiagram μ).card = μ.card-μ.rowLen 0 := by
+  rw [←card_cell, Fintype.card_congr (tailCellEquiv μ), tail_card]
+
+noncomputable def extendPerm (μ : YoungDiagram) (p : Equiv.Perm (Cell (tailDiagram μ))) :
+    Equiv.Perm (Cell μ) := p.extendDomain (tailCellEquiv μ)
+
+@[simp] lemma extendPerm_up (μ : YoungDiagram) (p : Equiv.Perm (Cell (tailDiagram μ)))
+    (x : Cell (tailDiagram μ)) : extendPerm μ p (upCell μ x) = upCell μ (p x) :=
+  Equiv.Perm.extendDomain_apply_image p (tailCellEquiv μ) x
+
+lemma extendPerm_top (μ : YoungDiagram) (p : Equiv.Perm (Cell (tailDiagram μ)))
+    (x : Cell μ) (hx : row x=0) : extendPerm μ p x = x :=
+  Equiv.Perm.extendDomain_apply_not_subtype p (tailCellEquiv μ) (not_not.mpr hx)
+
+@[simp] lemma extendPerm_inv (μ : YoungDiagram) (p : Equiv.Perm (Cell (tailDiagram μ))) :
+    (extendPerm μ p)⁻¹ = extendPerm μ p⁻¹ := Equiv.Perm.extendDomain_symm _ _
+
+@[simp] lemma extendPerm_sign (μ : YoungDiagram) (p : Equiv.Perm (Cell (tailDiagram μ))) :
+    permSign (extendPerm μ p) = permSign p := by
+  simp [extendPerm,permSign,Equiv.Perm.sign_extendDomain]
+
+noncomputable def liftRows (μ : YoungDiagram) (f : Tabloid (tailDiagram μ)) (x : Cell μ) : Fin (μ.colLen 0) :=
+  if hx : row x ≠ 0 then
+    ⟨(f.1 ((tailCellEquiv μ).symm ⟨x,hx⟩)).val+1,by
+      have h := (f.1 ((tailCellEquiv μ).symm ⟨x,hx⟩)).isLt
+      have hh := tail_height μ
+      omega⟩
+  else rowIndex x
+
+@[simp] lemma liftRows_up (μ : YoungDiagram) (f : Tabloid (tailDiagram μ))
+    (x : Cell (tailDiagram μ)) : (liftRows μ f (upCell μ x)).val = (f.1 x).val+1 := by
+  simp [liftRows,upCell]
+
+lemma liftRows_top (μ : YoungDiagram) (f : Tabloid (tailDiagram μ)) (x : Cell μ) (hx : row x=0) :
+    liftRows μ f x = rowIndex x := by simp [liftRows,hx]
+
+lemma liftRows_perm (μ : YoungDiagram) (f : Tabloid (tailDiagram μ))
+    (p : Equiv.Perm (Cell (tailDiagram μ))) (hf : f.1 = rowIndex ∘ p) :
+    liftRows μ f = rowIndex ∘ extendPerm μ p := by
+  funext x
+  by_cases hx : row x = 0
+  · simp [liftRows_top μ f x hx,extendPerm_top μ p x hx]
+  · let y := (tailCellEquiv μ).symm ⟨x,hx⟩
+    have he : upCell μ y = x := congrArg Subtype.val ((tailCellEquiv μ).apply_symm_apply ⟨x,hx⟩)
+    rw [←he]
+    apply Fin.ext
+    simp [hf,Function.comp_apply,rowIndex]
+
+noncomputable def liftTabloid (μ : YoungDiagram) (f : Tabloid (tailDiagram μ)) : Tabloid μ :=
+  ⟨liftRows μ f, by obtain ⟨p,hp⟩ := f.2; exact ⟨extendPerm μ p,liftRows_perm μ f p hp⟩⟩
+
+@[simp] lemma liftTabloid_base (μ : YoungDiagram) :
+    liftTabloid μ (baseTabloid (tailDiagram μ)) = baseTabloid μ := by
+  apply Subtype.ext
+  funext x
+  by_cases hx : row x=0
+  · exact liftRows_top μ _ x hx
+  · let y := (tailCellEquiv μ).symm ⟨x,hx⟩
+    have he : upCell μ y = x := congrArg Subtype.val ((tailCellEquiv μ).apply_symm_apply ⟨x,hx⟩)
+    rw [←he]
+    apply Fin.ext
+    exact liftRows_up μ _ y
+
+lemma liftTabloid_action (μ : YoungDiagram) (p : Equiv.Perm (Cell (tailDiagram μ)))
+    (f : Tabloid (tailDiagram μ)) :
+    liftTabloid μ (tabloidAct p f) = tabloidAct (extendPerm μ p) (liftTabloid μ f) := by
+  apply Subtype.ext
+  funext x
+  change liftRows μ (tabloidAct p f) x = liftRows μ f ((extendPerm μ p)⁻¹ x)
+  rw [extendPerm_inv]
+  by_cases hx : row x=0
+  · rw [extendPerm_top μ p⁻¹ x hx,liftRows_top μ _ x hx,liftRows_top μ _ x hx]
+  · let y := (tailCellEquiv μ).symm ⟨x,hx⟩
+    have he : upCell μ y = x := congrArg Subtype.val ((tailCellEquiv μ).apply_symm_apply ⟨x,hx⟩)
+    rw [←he,extendPerm_up]
+    apply Fin.ext
+    simp only [liftRows_up,tabloidAct_apply]
+
+noncomputable def restrictTabloids (μ : YoungDiagram) :
+    (Tabloid μ → ℂ) →ₗ[ℂ] (Tabloid (tailDiagram μ) → ℂ) where
+  toFun v := fun f => v (liftTabloid μ f)
+  map_add' := by intros; rfl
+  map_smul' := by intros; rfl
+
+lemma restrict_action (μ : YoungDiagram) (p : Equiv.Perm (Cell (tailDiagram μ)))
+    (v : Tabloid μ → ℂ) :
+    restrictTabloids μ (tabloidRep μ (extendPerm μ p) v) =
+      tabloidRep (tailDiagram μ) p (restrictTabloids μ v) := by
+  ext f
+  change v (tabloidAct (extendPerm μ p)⁻¹ (liftTabloid μ f)) = v (liftTabloid μ (tabloidAct p⁻¹ f))
+  rw [extendPerm_inv,liftTabloid_action]
+
+lemma extend_colGroup (μ : YoungDiagram) (c : colGroup (tailDiagram μ)) :
+    extendPerm μ c ∈ colGroup μ := by
+  intro x
+  by_cases hx : row x=0
+  · rw [extendPerm_top μ c x hx]
+  · let y := (tailCellEquiv μ).symm ⟨x,hx⟩
+    have he : upCell μ y = x := congrArg Subtype.val ((tailCellEquiv μ).apply_symm_apply ⟨x,hx⟩)
+    rw [←he,extendPerm_up,upCell_col,upCell_col]
+    exact c.2 y
+
+lemma restrict_polytabloid (μ : YoungDiagram) :
+    restrictTabloids μ (polytabloid μ) = polytabloid (tailDiagram μ) := by
+  let v := restrictTabloids μ (polytabloid μ)
+  have hv (c : colGroup (tailDiagram μ)) :
+      tabloidRep (tailDiagram μ) c v = permSign (c:Equiv.Perm (Cell (tailDiagram μ))) • v := by
+    rw [←restrict_action]
+    rw [polytabloid_col_action μ ⟨extendPerm μ c,extend_colGroup μ c⟩]
+    simp only [map_smul,extendPerm_sign]
+    rfl
+  have ha : alternator (colGroup (tailDiagram μ)) (tabloidRep (tailDiagram μ)) v =
+      (Fintype.card (colGroup (tailDiagram μ)):ℂ) • v := by
+    simp only [alternator,LinearMap.sum_apply,LinearMap.smul_apply,hv,smul_smul,permSign_sq,one_smul]
+    ext f
+    simp
+  have hb : v (baseTabloid (tailDiagram μ)) = 1 := by
+    change polytabloid μ (liftTabloid μ (baseTabloid (tailDiagram μ))) = 1
+    rw [liftTabloid_base, polytabloid_base]
+  have he := alternator_rank_one_exact (tailDiagram μ) v
+  rw [ha] at he
+  simp only [Pi.smul_apply,hb,smul_eq_mul,mul_one] at he
+  have hc : (Fintype.card (colGroup (tailDiagram μ)):ℂ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
+  exact (smul_right_injective _ hc) he
+
+lemma tail_space_le_image (μ : YoungDiagram) :
+    space (tailDiagram μ) ≤ (space μ).map (restrictTabloids μ) := by
+  apply Submodule.span_le.mpr
+  rintro _ ⟨p,rfl⟩
+  refine ⟨tabloidRep μ (extendPerm μ p) (polytabloid μ), orbit_mem_space μ _, ?_⟩
+  rw [restrict_action,restrict_polytabloid]
+
+theorem tail_dimension_le (μ : YoungDiagram) :
+    Module.finrank ℂ (space (tailDiagram μ)) ≤ Module.finrank ℂ (space μ) :=
+  (Submodule.finrank_mono (tail_space_le_image μ)).trans (Submodule.finrank_map_le _ _)
+
+end Thorp.StrongTail
+
+namespace Thorp.ButterflyReverse
+open scoped BigOperators
+
+abbrev Site (d : ℕ) := {t : Fin d × Card d // t.2 t.1 = false}
+
+def site : (d : ℕ) → SwitchIndex d → Site d
+  | 0, t => nomatch t
+  | _+1, Sum.inl y => ⟨(0, Fin.cons false y), rfl⟩
+  | d+1, Sum.inr (b,t) =>
+      ⟨((site d t).1.1.succ, Fin.cons b (site d t).1.2), (site d t).2⟩
+
+lemma site_injective (d : ℕ) : Function.Injective (site d) := by
+  induction d with
+  | zero => intro t; exact Empty.elim t
+  | succ d ih =>
+    intro s t h
+    cases s with
+    | inl y =>
+      cases t with
+      | inl z =>
+        have he := congrArg (fun a : Site (d+1) => Fin.tail a.1.2) h
+        exact congrArg Sum.inl he
+      | inr t =>
+        have he := congrArg (fun a : Site (d+1) => a.1.1.val) h
+        simp [site] at he
+    | inr s =>
+      cases t with
+      | inl y =>
+        have he := congrArg (fun a : Site (d+1) => a.1.1.val) h
+        simp [site] at he
+      | inr t =>
+        obtain ⟨b,s⟩ := s
+        obtain ⟨c,t⟩ := t
+        have hbc := congrArg (fun a : Site (d+1) => a.1.2 0) h
+        simp only [site, Fin.cons_zero] at hbc
+        have hst : site d s = site d t := by
+          apply Subtype.ext
+          apply Prod.ext
+          · exact Fin.succ_inj.mp (congrArg (fun a : Site (d+1) => a.1.1) h)
+          · exact congrArg (fun a : Site (d+1) => Fin.tail a.1.2) h
+        cases hbc
+        cases ih hst
+        rfl
+
+lemma site_surjective (d : ℕ) : Function.Surjective (site d) := by
+  induction d with
+  | zero => intro t; exact Fin.elim0 t.1.1
+  | succ d ih =>
+    rintro ⟨⟨i,x⟩,hx⟩
+    refine Fin.cases ?_ (fun j => ?_) i hx
+    · intro hx0
+      refine ⟨Sum.inl (Fin.tail x), ?_⟩
+      apply Subtype.ext
+      simp only [site, Prod.mk.injEq, true_and]
+      rw [←hx0,Fin.cons_self_tail]
+    · intro hxj
+      obtain ⟨t,ht⟩ := ih ⟨(j,Fin.tail x),hxj⟩
+      refine ⟨Sum.inr (x 0,t), ?_⟩
+      apply Subtype.ext
+      simp only [site,ht,Fin.cons_self_tail]
+
+noncomputable def siteEquiv (d : ℕ) : SwitchIndex d ≃ Site d :=
+  Equiv.ofBijective (site d) ⟨site_injective d,site_surjective d⟩
+
+def reverseCard (d : ℕ) : Equiv.Perm (Card d) where
+  toFun x := fun i => x i.rev
+  invFun x := fun i => x i.rev
+  left_inv x := by funext i; simp
+  right_inv x := by funext i; simp
+
+@[simp] lemma reverseCard_apply (d : ℕ) (x : Card d) (i : Fin d) :
+    reverseCard d x i = x i.rev := rfl
+
+@[simp] lemma reverseCard_involutive (d : ℕ) (x : Card d) :
+    reverseCard d (reverseCard d x) = x := by funext i; simp
+
+def mirrorSite (d : ℕ) : Equiv.Perm (Site d) where
+  toFun s := ⟨(s.1.1.rev,reverseCard d s.1.2),by simpa using s.2⟩
+  invFun s := ⟨(s.1.1.rev,reverseCard d s.1.2),by simpa using s.2⟩
+  left_inv s := by apply Subtype.ext; simp
+  right_inv s := by apply Subtype.ext; simp
+
+noncomputable def mirrorSwitch (d : ℕ) : Equiv.Perm (SwitchIndex d) :=
+  (siteEquiv d).trans ((mirrorSite d).trans (siteEquiv d).symm)
+
+lemma site_mirrorSwitch (d : ℕ) (t : SwitchIndex d) :
+    site d (mirrorSwitch d t) = mirrorSite d (site d t) := by
+  exact (siteEquiv d).apply_symm_apply _
+
+lemma routeDomain_site (d : ℕ) (x y : Card d) (t : SwitchIndex d) :
+    t ∈ routeDomain d x y ↔
+      ∀ j : Fin d, j ≠ (site d t).1.1 →
+      (site d t).1.2 j = if j < (site d t).1.1 then x j else y j := by
+  classical
+  induction d with
+  | zero => exact Empty.elim t
+  | succ d ih =>
+    cases t with
+    | inl z =>
+      change Sum.inl z ∈ (insert (Sum.inl (Fin.tail y))
+        ((routeDomain d (Fin.tail x) (Fin.tail y)).image
+          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) :
+        Finset (Sum (Card d) (Bool × SwitchIndex d))) ↔ _
+      simp only [Finset.mem_insert, Sum.inl.injEq,Finset.mem_image,
+        reduceCtorEq,and_false,exists_false,or_false]
+      constructor
+      · intro hz j hj
+        subst z
+        have hj0 : j ≠ 0 := hj
+        obtain ⟨i,rfl⟩ := Fin.eq_succ_of_ne_zero hj0
+        simp [site,Fin.tail]
+      · intro h
+        funext j
+        have he := h j.succ (by simp [site])
+        simpa [site,Fin.tail] using he
+    | inr z =>
+      obtain ⟨b,t⟩ := z
+      change Sum.inr (b,t) ∈ (insert (Sum.inl (Fin.tail y))
+        ((routeDomain d (Fin.tail x) (Fin.tail y)).image
+          (fun i => (Sum.inr (x 0,i) : Sum (Card d) (Bool × SwitchIndex d)))) :
+        Finset (Sum (Card d) (Bool × SwitchIndex d))) ↔ _
+      simp only [Finset.mem_insert,reduceCtorEq,false_or,Finset.mem_image,Sum.inr.injEq,Prod.mk.injEq]
+      constructor
+      · rintro ⟨s,hs,hb,ht⟩
+        subst s
+        have hst := (ih (Fin.tail x) (Fin.tail y) t).mp hs
+        intro j hj
+        refine Fin.cases ?_ (fun i => ?_) j hj
+        · intro _
+          simp [site,hb]
+        · intro hi
+          have hi' : i ≠ (site d t).1.1 := by simpa [site] using hi
+          simpa [site,Fin.tail] using hst i hi'
+      · intro h
+        have hb := h 0 (by simpa only [site] using (Fin.succ_ne_zero (site d t).1.1).symm)
+        have hb' : x 0 = b := by simpa [site] using hb.symm
+        refine ⟨t,?_,hb',rfl⟩
+        apply (ih (Fin.tail x) (Fin.tail y) t).mpr
+        intro j hj
+        have he := h j.succ (by simpa [site] using hj)
+        simpa [site,Fin.tail] using he
+
+lemma routeValue_site (d : ℕ) (x y : Card d) (t : SwitchIndex d) :
+    routeValue d x y t = Bool.xor (x (site d t).1.1) (y (site d t).1.1) := by
+  induction d with
+  | zero => exact Empty.elim t
+  | succ d ih =>
+    cases t with
+    | inl z => rfl
+    | inr z => exact ih (Fin.tail x) (Fin.tail y) z.2
+
+lemma mirrorSwitch_involutive (d : ℕ) (t : SwitchIndex d) :
+    mirrorSwitch d (mirrorSwitch d t) = t := by
+  apply site_injective d
+  rw [site_mirrorSwitch,site_mirrorSwitch]
+  apply Subtype.ext
+  simp [mirrorSite]
+
+lemma mirror_routeDomain (d : ℕ) (x y : Card d) (t : SwitchIndex d)
+    (ht : t ∈ routeDomain d x y) :
+    mirrorSwitch d t ∈ routeDomain d (reverseCard d y) (reverseCard d x) := by
+  rw [routeDomain_site,site_mirrorSwitch]
+  intro j hj
+  have hj' : j.rev ≠ (site d t).1.1 := by
+    intro h
+    apply hj
+    change j = (site d t).1.1.rev
+    rw [←h,Fin.rev_rev]
+  have he := (routeDomain_site d x y t).mp ht j.rev hj'
+  change (site d t).1.2 j.rev =
+    if j < (site d t).1.1.rev then y j.rev else x j.rev
+  rw [he]
+  have hord : j < (site d t).1.1.rev ↔ ¬j.rev < (site d t).1.1 := by
+    rw [Fin.lt_rev_iff]
+    constructor
+    · intro h
+      exact not_lt_of_ge h.le
+    · intro h
+      exact lt_of_le_of_ne (le_of_not_gt h) (Ne.symm hj')
+  by_cases h : j.rev < (site d t).1.1 <;> simp [h,hord]
+
+lemma mirror_routeValue (d : ℕ) (x y : Card d) (t : SwitchIndex d) :
+    routeValue d (reverseCard d y) (reverseCard d x) (mirrorSwitch d t) =
+      routeValue d x y t := by
+  rw [routeValue_site,routeValue_site,site_mirrorSwitch]
+  simp only [mirrorSite,Equiv.coe_fn_mk,reverseCard_apply,Fin.rev_rev]
+  exact Bool.xor_comm _ _
+
+noncomputable def mirrorCoins (d : ℕ) : Equiv.Perm (SwitchIndex d → Bool) where
+  toFun ω := fun t => ω (mirrorSwitch d t)
+  invFun ω := fun t => ω (mirrorSwitch d t)
+  left_inv ω := by funext t; change ω (mirrorSwitch d (mirrorSwitch d t)) = ω t; rw [mirrorSwitch_involutive]
+  right_inv ω := by funext t; change ω (mirrorSwitch d (mirrorSwitch d t)) = ω t; rw [mirrorSwitch_involutive]
+
+lemma mirror_endpoint (d : ℕ) (ω : SwitchIndex d → Bool) (x y : Card d)
+    (hxy : butterflyPerm d (decodeButterfly d ω) x = y) :
+    butterflyPerm d (decodeButterfly d (mirrorCoins d ω)) (reverseCard d y) =
+      reverseCard d x := by
+  apply (butterfly_endpoint_iff d _ _ _).mpr
+  intro t ht
+  have ht' := mirror_routeDomain d (reverseCard d y) (reverseCard d x) t ht
+  simp only [reverseCard_involutive] at ht'
+  have he := (butterfly_endpoint_iff d ω x y).mp hxy (mirrorSwitch d t) ht'
+  change ω (mirrorSwitch d t) = _
+  rw [he]
+  have hv := mirror_routeValue d x y (mirrorSwitch d t)
+  rw [mirrorSwitch_involutive] at hv
+  exact hv.symm
+
+lemma butterfly_mirror (d : ℕ) (ω : SwitchIndex d → Bool) :
+    butterflyPerm d (decodeButterfly d (mirrorCoins d ω)) =
+      reverseCard d * (butterflyPerm d (decodeButterfly d ω))⁻¹ * reverseCard d := by
+  apply Equiv.ext
+  intro x
+  have he := mirror_endpoint d ω
+    ((butterflyPerm d (decodeButterfly d ω))⁻¹ (reverseCard d x)) (reverseCard d x)
+    (Equiv.apply_symm_apply _ _)
+  simpa only [reverseCard_involutive,Equiv.Perm.mul_apply] using he
+
+def reverseOrder (d : ℕ) : Equiv.Perm (Fin d) where
+  toFun := Fin.rev
+  invFun := Fin.rev
+  left_inv := Fin.rev_rev
+  right_inv := Fin.rev_rev
+
+lemma relabel_mul (d : ℕ) (a b : Equiv.Perm (Fin d)) :
+    StrongSmoothing.coordinateRelabel d (a*b) =
+      StrongSmoothing.coordinateRelabel d a * StrongSmoothing.coordinateRelabel d b := rfl
+
+lemma relabel_reverse (d : ℕ) :
+    StrongSmoothing.coordinateRelabel d (reverseOrder d) = reverseCard d := rfl
+
+lemma reverseCard_sq (d : ℕ) : reverseCard d * reverseCard d = 1 := by
+  ext x i
+  simp
+
+lemma sweep_mirror (d : ℕ) (order : Equiv.Perm (Fin d)) (ω : SwitchIndex d → Bool) :
+    StrongSmoothing.sweep d (order*reverseOrder d) (mirrorCoins d ω) =
+      (StrongSmoothing.sweep d order ω)⁻¹ := by
+  simp only [StrongSmoothing.sweep,relabel_mul,relabel_reverse,butterfly_mirror,mul_inv_rev]
+  have hR : (reverseCard d)⁻¹ = reverseCard d := by
+    exact inv_eq_of_mul_eq_one_right (reverseCard_sq d)
+  rw [hR]
+  calc
+    _ = StrongSmoothing.coordinateRelabel d order *
+        (reverseCard d*reverseCard d) *
+        (butterflyPerm d (decodeButterfly d ω))⁻¹ *
+        (reverseCard d*reverseCard d) * (StrongSmoothing.coordinateRelabel d order)⁻¹ := by group
+    _ = _ := by rw [reverseCard_sq]; group
+
+lemma sweepKernel_reverse (d l : ℕ) (order : Equiv.Perm (Fin d)) :
+    StrongSmoothing.sweepKernel d l (order*reverseOrder d) =
+      (StrongSmoothing.sweepKernel d l order).transpose := by
+  classical
+  rw [StrongSmoothing.sweepKernel_law,StrongSmoothing.sweepKernel_law,
+    ←StrongSmoothing.PermLaw.inverse]
+  ext x y
+  unfold StrongSmoothing.PermLaw.kernel
+  rw [finiteMean_equiv (mirrorCoins d)]
+  apply finiteMean_congr
+  intro ω
+  change (if StrongSmoothing.tuplePerm
+      (StrongSmoothing.sweep d (order*reverseOrder d) (mirrorCoins d ω)) x = y then (1:ℝ) else 0) = _
+  rw [sweep_mirror,StrongSmoothing.tuplePerm_inverse]
+
+lemma reflectedKernel_reverse (d l : ℕ) (order : Equiv.Perm (Fin d)) :
+    StrongSmoothing.reflectedKernel d l order true =
+      StrongSmoothing.reflectedKernel d l (order*reverseOrder d) false := by
+  simp only [StrongSmoothing.reflectedKernel,Bool.false_eq_true,
+    ↓reduceIte,sweepKernel_reverse,Matrix.transpose_transpose]
+
+end Thorp.ButterflyReverse
+
+namespace Thorp.StrongTail
+open scoped BigOperators Classical
+
+section
+open StrongSmoothing
+
+lemma reflected_nonneg_both (d l : ℕ) (order : Equiv.Perm (Fin d)) (rev : Bool)
+    (x y : Tuples d l) : 0 ≤ reflectedKernel d l order rev x y := by
+  cases rev
+  · exact reflected_nonneg _ _ _ _ _
+  · rw [ButterflyReverse.reflectedKernel_reverse]
+    exact reflected_nonneg _ _ _ _ _
+
+lemma reflected_symm_both (d l : ℕ) (order : Equiv.Perm (Fin d)) (rev : Bool) :
+    (reflectedKernel d l order rev).IsSymm := by
+  cases rev
+  · exact reflected_symmetric _ _ _
+  · rw [ButterflyReverse.reflectedKernel_reverse]
+    exact reflected_symmetric _ _ _
+
+lemma reflected_moment_both (d l : ℕ) (order : Equiv.Perm (Fin d)) (rev : Bool)
+    (x : Tuples d l) :
+    finiteMean (fun y : Tuples d l =>
+      ((Fintype.card (Tuples d l):ℝ)*reflectedKernel d l order rev x y)^(1+(1/1000:ℝ))) ≤
+      Real.exp (l*(1+heightDecay 1)) := by
+  cases rev
+  · exact reflected_moment _ _ _ _
+  · rw [ButterflyReverse.reflectedKernel_reverse]
+    exact reflected_moment _ _ _ _
+
+lemma reflected_moment_small (d l : ℕ) (order : Equiv.Perm (Fin d)) (rev : Bool)
+    (x : Tuples d l) {p : ℝ} (hp : 0 < p) (hpq : p ≤ 1+(1/1000:ℝ)) :
+    finiteMean (fun y : Tuples d l =>
+      ((Fintype.card (Tuples d l):ℝ)*reflectedKernel d l order rev x y)^p) ≤
+      Real.exp (l*(1+heightDecay 1)) := by
+  let : Nonempty (Tuples d l) := ⟨x⟩
+  let q : ℝ := 1+1/1000
+  let f := fun y : Tuples d l => (Fintype.card (Tuples d l):ℝ)*reflectedKernel d l order rev x y
+  have hf : ∀ y, 0 ≤ f y := fun y => mul_nonneg (Nat.cast_nonneg _) (reflected_nonneg_both _ _ _ _ _ _)
+  have hq : 0 < q := by norm_num [q]
+  have hr : 0 ≤ p/q := (div_pos hp hq).le
+  have hr1 : p/q ≤ 1 := (div_le_one hq).mpr hpq
+  have he (y) : (f y)^p = ((f y)^q)^(p/q) := by
+    rw [←Real.rpow_mul (hf y)]
+    congr 1
+    field_simp
+  simp_rw [show (fun y : Tuples d l => ((Fintype.card (Tuples d l):ℝ)*reflectedKernel d l order rev x y)^p)
+    = fun y => ((f y)^q)^(p/q) from funext he]
+  apply (mean_rpow_concave _ (fun y => Real.rpow_nonneg (hf y) _) hr hr1).trans
+  apply (Real.rpow_le_rpow (finiteMean_nonneg (fun y => Real.rpow_nonneg (hf y) _))
+    (reflected_moment_both d l order rev x) hr).trans
+  rw [Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp]
+  apply Real.exp_le_exp.mpr
+  have hA : 0 ≤ (l:ℝ)*(1+heightDecay 1) := by
+    have hh := heightDecay_nonneg 1
+    positivity
+  nlinarith
+
+lemma log_factorial_le_power (n N : ℕ) (hn : n ≤ N) (_hN : 0 < N) :
+    Real.log (n.factorial:ℝ) ≤ (n:ℝ)*Real.log N := by
+  have hb : (n.factorial:ℝ) ≤ (N:ℝ)^n := by
+    exact_mod_cast (Nat.factorial_le_pow n).trans (Nat.pow_le_pow_left hn n)
+  simpa only [Real.log_pow] using Real.log_le_log (by positivity : (0:ℝ)<n.factorial) hb
+
+lemma log_normalization_le (N l : ℕ) (hN : 0 < N) (hl : l ≤ N) :
+    Real.log ((N:ℝ)^l/(N.descFactorial l:ℝ)) ≤ 2*l := by
+  have hNp : (0:ℝ)<N := by exact_mod_cast hN
+  have hMp : (0:ℝ)<N.descFactorial l := by exact_mod_cast Nat.descFactorial_pos.mpr hl
+  rw [Real.log_div (by positivity) hMp.ne',Real.log_pow]
+  by_cases hlow : 2*l ≤ N
+  · have hhalf : (N:ℝ)/2 ≤ (N+1-l:ℕ) := by
+      have hNat : N ≤ 2*(N+1-l) := by omega
+      exact (div_le_iff₀ (by norm_num : (0:ℝ)<2)).mpr (by simpa only [mul_comm] using (show (N:ℝ) ≤ 2*(N+1-l:ℕ) by exact_mod_cast hNat))
+    have hb : ((N:ℝ)/2)^l ≤ (N.descFactorial l:ℝ) := by
+      apply (pow_le_pow_left₀ (by positivity) hhalf l).trans
+      exact_mod_cast Nat.pow_sub_le_descFactorial N l
+    have hh := Real.log_le_log (pow_pos (by positivity : (0:ℝ)<(N:ℝ)/2) l) hb
+    rw [Real.log_pow,Real.log_div hNp.ne' (by norm_num)] at hh
+    have ht : Real.log (2:ℝ) ≤ 1 := by simpa only [show (2:ℝ)-1=1 by norm_num] using Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
+    nlinarith [(show (0:ℝ) ≤ l by positivity)]
+  · have he : (N.descFactorial l:ℝ)*(Nat.factorial (N-l):ℝ) = N.factorial := by
+      have he := Nat.factorial_mul_descFactorial hl
+      exact_mod_cast (by simpa only [mul_comm] using he : N.descFactorial l * (N-l).factorial = N.factorial)
+    have hh := congrArg Real.log he
+    rw [Real.log_mul hMp.ne' (by positivity)] at hh
+    have hfull := log_factorial_lower (show 1 ≤ N by omega)
+    have hrem := log_factorial_le_power (N-l) N (Nat.sub_le _ _) hN
+    rw [Nat.cast_sub hl] at hrem
+    have hNle : (N:ℝ) ≤ 2*l := by exact_mod_cast (by omega : N ≤ 2*l)
+    linarith
+
+lemma normalized_moment_identity {X : Type*} [Fintype X] [Nonempty X]
+    (f : X → ℝ) (hf : ∀ x, 0 ≤ f x) {N : ℝ} (hN : 0 < N) (p : ℝ) :
+    N⁻¹*(∑ x, (N*f x)^p) =
+     (N/(Fintype.card X:ℝ))^(p-1) * finiteMean (fun x => ((Fintype.card X:ℝ)*f x)^p) := by
+  let M : ℝ := Fintype.card X
+  have hM : 0 < M := Nat.cast_pos.mpr Fintype.card_pos
+  have he (x) : (N*f x)^p = (N/M)^p*(M*f x)^p := by
+    rw [←Real.mul_rpow (div_nonneg hN.le hM.le) (mul_nonneg hM.le (hf x))]
+    congr 1
+    field_simp
+  simp_rw [he]
+  rw [←Finset.mul_sum,Real.rpow_sub (div_pos hN hM),Real.rpow_one]
+  unfold finiteMean
+  dsimp [M] at hM ⊢
+  field_simp
+
+lemma density_at_small_exponent {p : ℝ} (hp : 1 < p) (hpq : p ≤ 1+(1/1000:ℝ)) :
+    DensityAt p (3+heightDecay 1) := by
+  intro d l hl order rev x
+  dsimp only
+  let : Nonempty (Tuples d l) := ⟨x⟩
+  let H := reflectedKernel d l order rev
+  have hm := reflected_moment_small d l order rev x (by linarith : 0<p) hpq
+  have hb : Real.exp ((l:ℝ)*(1+heightDecay 1)) ≤ Real.exp ((3+heightDecay 1)*l) := by
+    apply Real.exp_le_exp.mpr
+    nlinarith [(show (0:ℝ) ≤ l by positivity)]
+  have hs (y) : reflectedKernel d l order rev y x = reflectedKernel d l order rev x y := (reflected_symm_both d l order rev).apply _ _
+  have hpow : (((2:ℝ)^d)^l)⁻¹ * (∑ y, ((((2:ℝ)^d)^l)*H x y)^p) ≤
+      Real.exp ((3+heightDecay 1)*l) := by
+    rw [normalized_moment_identity (fun y => H x y) (reflected_nonneg_both d l order rev x)
+      (by positivity) p]
+    have hrat : ((2:ℝ)^d)^l/(Fintype.card (Tuples d l):ℝ) ≤ Real.exp (2*l) := by
+      apply (Real.log_le_iff_le_exp (by positivity)).mp
+      rw [tuples_card]
+      exact_mod_cast log_normalization_le (2^d) l (by positivity) hl
+    have hr : 0 ≤ ((2:ℝ)^d)^l/(Fintype.card (Tuples d l):ℝ) := by positivity
+    have hrp : (((2:ℝ)^d)^l/(Fintype.card (Tuples d l):ℝ))^(p-1) ≤ Real.exp (2*l) := by
+      apply (Real.rpow_le_rpow hr hrat (by linarith)).trans
+      rw [Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp]
+      apply Real.exp_le_exp.mpr
+      have hpl : p-1 ≤ 1 := by linarith
+      nlinarith [(show (0:ℝ) ≤ l by positivity)]
+    apply (mul_le_mul hrp hm (finiteMean_nonneg (fun y => Real.rpow_nonneg
+      (mul_nonneg (Nat.cast_nonneg _) (reflected_nonneg_both d l order rev x y)) _)) (Real.exp_pos _).le).trans
+    rw [←Real.exp_add]
+    apply Real.exp_le_exp.mpr
+    ring_nf
+    rfl
+  dsimp [H] at hpow
+  refine ⟨hm.trans hb,?_,hpow,?_⟩
+  · simpa only [hs] using hm.trans hb
+  · simpa only [hs] using hpow
+
+end
+open Specht UnitaryFinite
+
+lemma dimension_le_factorial (μ : YoungDiagram) :
+    Module.finrank ℂ (space μ) ≤ μ.card.factorial := by
+  have hdim : Module.finrank ℂ (space μ) ≤ Fintype.card (Tabloid μ) := by
+    simpa using Submodule.finrank_le (space μ)
+  have hrow : 1 ≤ Fintype.card (rowGroup μ) := Fintype.card_pos
+  have hc := card_tabloid_mul_row μ
+  nlinarith
+
+lemma tail_card_exact (μ : YoungDiagram) :
+    (tailDiagram μ).card = μ.card - μ.rowLen 0 := by
+  rw [←card_cell,Fintype.card_congr (tailCellEquiv μ),tail_card]
+
+lemma tail_dimension_le_factorial (μ : YoungDiagram) :
+    Module.finrank ℂ (space (tailDiagram μ)) ≤ (μ.card - μ.rowLen 0).factorial := by
+  simpa only [tail_card_exact] using dimension_le_factorial (tailDiagram μ)
+
+lemma sweep_sq_palindrome (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Cell μ) (rev : Bool) :
+    ‖SparseContact.sweepOperator d μ e rev‖^2 =
+      ‖sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e) (palindromePerm d)‖ := by
+  rw [SparseContact.sweep_forward_norm, sampleOperator_palindrome _ _ (relabelledUnitary_unitary μ e)]
+  change ‖sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e)
+    (fun ω => butterflyPerm d (decodeButterfly d ω))‖^2 = _
+  rw [show (sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e)
+    (fun ω => butterflyPerm d (decodeButterfly d ω))).adjoint =
+    star (sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e)
+    (fun ω => butterflyPerm d (decodeButterfly d ω))) from rfl]
+  simpa only [pow_two] using (CStarRing.norm_self_mul_star (x := sampleOperator (V := hilbertSpace μ)
+    (relabelledUnitary μ e) (fun ω => butterflyPerm d (decodeButterfly d ω)))).symm
+
+lemma sweep_sq_le_one (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Cell μ) (rev : Bool) :
+    ‖SparseContact.sweepOperator d μ e rev‖^2 ≤ 1 := by
+  have h := SparseContact.sweep_norm_le_one d μ e rev
+  nlinarith [norm_nonneg (SparseContact.sweepOperator d μ e rev)]
+
+theorem strong_tail : MainStatement := by
+  obtain ⟨a,h,ha,hh,dE⟩ := Specht.spectral_regimes
+  obtain ⟨D0,hD0⟩ := Filter.eventually_atTop.mp dE
+  let d0 := max D0 1
+  let t : ℝ := min a (1/1000)
+  have ht : 0 < t := lt_min ha (by norm_num)
+  have hta : t ≤ a := min_le_left _ _
+  have htq : t ≤ 1/1000 := min_le_right _ _
+  let b := t/(1+t)
+  have hden : 0 < 1+t := by linarith
+  have hb : 0 < b := div_pos ht hden
+  have hbt : b ≤ t := (div_le_iff₀ hden).mpr (by nlinarith [sq_nonneg t])
+  have hba : b ≤ a := hbt.trans hta
+  have hb1 : b ≤ 1 := (div_le_one hden).mpr (by linarith)
+  let C := Real.log (((2^d0).factorial:ℕ):ℝ)
+  have hC : 0 ≤ C := Real.log_nonneg (by exact_mod_cast Nat.factorial_pos (2^d0))
+  refine ⟨1+t,3+heightDecay 1,C,by linarith,by linarith,
+    density_at_small_exponent (by linarith) (by linarith),?_⟩
+  intro d μ e rev
+  let k := μ.card - μ.rowLen 0
+  let D : ℝ := Module.finrank ℂ (space (tailDiagram μ))
+  have hD : 1 ≤ D := by dsimp [D]; exact_mod_cast dimension_pos (tailDiagram μ)
+  have hDp : 0 < D := by linarith
+  have hlog : 0 ≤ Real.log D := Real.log_nonneg hD
+  have hμ : μ.card=2^d := by rw [←card_cell,←Fintype.card_congr e,card_positions]
+  have hform : -(((1+t)-1)/(1+t)) = -b := by dsimp [b]; congr 1; ring
+  rw [hform]
+  change ‖SparseContact.sweepOperator d μ e rev‖^2 ≤ Real.exp (C*k)*D^(-b)
+  rw [Real.rpow_def_of_pos hDp,←Real.exp_add]
+  by_cases hk : k=0
+  · have hDone : D=1 := by
+      have hn := tail_dimension_le_factorial μ
+      change Module.finrank ℂ (space (tailDiagram μ)) ≤ k.factorial at hn
+      rw [hk,Nat.factorial_zero] at hn
+      have hu : D ≤ 1 := by dsimp [D]; exact_mod_cast hn
+      linarith
+    rw [hk,Nat.cast_zero,mul_zero,hDone,Real.log_one,zero_mul,add_zero,Real.exp_zero]
+    exact sweep_sq_le_one d μ e rev
+  · have hkpos : 0<k := by omega
+    by_cases hd : d0 ≤ d
+    · have hdpos : 0<d := by have := le_max_right D0 1; omega
+      by_cases htall : (μ.card:ℝ)/2 < μ.colLen 0
+      · rw [sweep_sq_palindrome,Specht.palindrome_tall_zero_general d hdpos μ e htall,
+          @norm_zero (hilbertSpace μ →L[ℂ] hilbertSpace μ) _]
+        exact (Real.exp_pos _).le
+      · have hg := (hD0 d ((le_max_left _ _).trans hd) μ e hkpos (le_of_not_gt htall)).1
+        have hrootpos : (0:ℝ)<Module.finrank ℂ (space μ) := by exact_mod_cast dimension_pos μ
+        have hlogle : Real.log D ≤ Real.log (Module.finrank ℂ (space μ):ℝ) :=
+          Real.log_le_log hDp (by dsimp [D]; exact_mod_cast tail_dimension_le μ)
+        rw [sweep_sq_palindrome]
+        apply hg.trans
+        apply Real.exp_le_exp.mpr
+        have hbound := mul_le_mul hba hlogle hlog ha.le
+        have hck : 0 ≤ C*(k:ℝ) := mul_nonneg hC (Nat.cast_nonneg _)
+        nlinarith
+    · have hpow : 2^d ≤ 2^d0 := Nat.pow_le_pow_right (by decide) (by omega)
+      have hklarge : k ≤ 2^d0 := (by simp [k,hμ] : k ≤ 2^d).trans hpow
+      have hdim : D ≤ ((2^d0).factorial:ℝ) := by
+        dsimp [D]
+        exact_mod_cast (tail_dimension_le_factorial μ).trans (Nat.factorial_le hklarge)
+      have hlC : Real.log D ≤ C := Real.log_le_log hDp hdim
+      have hk1 : (1:ℝ) ≤ k := by exact_mod_cast hkpos
+      have hprod : b*Real.log D ≤ C*(k:ℝ) := by nlinarith
+      apply (sweep_sq_le_one d μ e rev).trans
+      exact Real.one_le_exp_iff.mpr (by nlinarith)
+
+end Thorp.StrongTail
+
+end ThorpNine.Tail
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Tail/MainStatement.lean b/lean/OAI/Probability/ThorpRouting/Tail/MainStatement.lean
index 68e914bf76c5000d83d518e0db0f424ef878232c..d8c8ea1a2db883130b1e183c9d0df99947793663 100644
--- a/lean/OAI/Probability/ThorpRouting/Tail/MainStatement.lean
+++ b/lean/OAI/Probability/ThorpRouting/Tail/MainStatement.lean
@@ -1,850 +1,850 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Tail.TypedMarks
-
-namespace OAI
-
-namespace ThorpNine.Tail
-
-namespace Thorp.SparseContact
-
-section
-open scoped BigOperators Classical
-
-lemma partial_full_action {L : Type*} [Fintype L] (d : ℕ)
-    (f : EuclideanSpace ℂ (L ↪ Card d)) :
-    rectOperator (fun (x : L ↪ Card d) (y : L → Card d) => partialKernel d Finset.univ x y) (zeroExtend f) =
-      DensityTransfer.applyKernel (UnitaryFinite.actionKernel UnitaryFinite.tupleAction
-        (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))) f := by
-  classical
-  ext x
-  rw [rectOperator_apply, sum_zeroExtend_mul]
-  apply Finset.sum_congr rfl
-  intro y _
-  congr 2
-  simp only [partialKernel, Finset.card_univ, div_self (by positivity : ((2:ℝ)^d)^(Fintype.card L) ≠ 0),
-    one_mul, endpointProb, UnitaryFinite.actionKernel]
-  apply finiteMean_congr
-  intro ω
-  congr 1
-  simp only [Finset.mem_univ, forall_const, Function.Embedding.ext_iff]
-  rfl
-
-lemma alternating_retained_eq {L : Type*} [Fintype L] (d : ℕ)
-    (x : L ↪ Card d) (y : L → Card d) :
-    (∑ A : Finset L, (-1:ℝ)^A.card * retainedKernel d A x y) =
-      ∑ A : Finset L, (-1:ℝ)^A.card * partialKernel d A x y := by
-  classical
-  by_cases h : ContactEvent d x y
-  · simp only [retainedKernel, ite_eq_left h]
-  · simp only [retainedKernel, ite_eq_right h, mul_zero, Finset.sum_const_zero]
-    exact (alternating_kernel_cancel d x y h).symm
-
-lemma alternating_harmonic_action {L : Type*} [Fintype L] (d : ℕ)
-    (f : EuclideanSpace ℂ (L ↪ Card d)) (hf : Harmonic f) :
-    (∑ A : Finset L, ((-1:ℂ)^A.card) • rectOperator (retainedKernel d A) (zeroExtend f)) =
-      ((-1:ℂ)^(Fintype.card L)) • DensityTransfer.applyKernel
-        (UnitaryFinite.actionKernel UnitaryFinite.tupleAction
-          (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))) f := by
-  classical
-  calc
-    _ = ∑ A : Finset L, ((-1:ℂ)^A.card) •
-        rectOperator (fun (x : L ↪ Card d) y => partialKernel d A x y) (zeroExtend f) := by
-      ext x
-      simp only [WithLp.ofLp_sum, Finset.sum_apply, PiLp.smul_apply, rectOperator_apply, smul_eq_mul, Finset.mul_sum]
-      rw [Finset.sum_comm, Finset.sum_comm (s := (Finset.univ : Finset (Finset L)))]
-      apply Finset.sum_congr rfl
-      intro y _
-      simp_rw [← mul_assoc, ← Finset.sum_mul]
-      congr 1
-      have he := congrArg (fun t : ℝ => (t:ℂ)) (alternating_retained_eq d x y)
-      simpa only [Complex.ofReal_sum, Complex.ofReal_mul, Complex.ofReal_pow, Complex.ofReal_neg,
-        Complex.ofReal_one] using he
-    _ = ((-1:ℂ)^(Fintype.card L)) •
-        rectOperator (fun (x : L ↪ Card d) y => partialKernel d Finset.univ x y) (zeroExtend f) := by
-      rw [Finset.sum_eq_single (Finset.univ : Finset L)]
-      · simp only [Finset.card_univ]
-      · intro A _ hA
-        rw [partial_proper_zero d A hA f hf, smul_zero]
-      · simp
-    _ = _ := by rw [partial_full_action]
-
-lemma harmonic_action_bound {L : Type*} [Fintype L] (d : ℕ)
-    (f : EuclideanSpace ℂ (L ↪ Card d)) (hf : Harmonic f)
-    (hδ : 0 < 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)
-    (hδ1 : 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d ≤ 1) :
-    ‖DensityTransfer.applyKernel (UnitaryFinite.actionKernel UnitaryFinite.tupleAction
-      (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))) f‖ ≤
-      (2:ℝ)^(Fintype.card L) *
-        Real.sqrt ((2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
-          Real.exp (Real.exp 1*(Fintype.card L))) * ‖f‖ := by
-  classical
-  let B := (2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
-          Real.exp (Real.exp 1*(Fintype.card L))
-  have hB : 0 ≤ B := by dsimp [B]; positivity
-  have hnorm (A : Finset L) : ‖rectOperator (retainedKernel d A)‖ ≤ Real.sqrt B := by
-    have hh := retained_norm_sq d A hδ hδ1
-    have hs := Real.sq_sqrt hB
-    have hn := Real.sqrt_nonneg B
-    nlinarith
-  calc
-    _ = ‖((-1:ℂ)^(Fintype.card L)) • DensityTransfer.applyKernel
-        (UnitaryFinite.actionKernel UnitaryFinite.tupleAction
-          (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))) f‖ := by
-      simp only [norm_smul, norm_pow, norm_neg, norm_one, one_pow, one_mul]
-    _ = ‖∑ A : Finset L, ((-1:ℂ)^A.card) • rectOperator (retainedKernel d A) (zeroExtend f)‖ := by
-      rw [alternating_harmonic_action d f hf]
-    _ ≤ ∑ A : Finset L, ‖((-1:ℂ)^A.card) • rectOperator (retainedKernel d A) (zeroExtend f)‖ :=
-      norm_sum_le _ _
-    _ ≤ ∑ A : Finset L, Real.sqrt B * ‖f‖ := by
-      apply Finset.sum_le_sum
-      intro A _
-      simp only [norm_smul, norm_pow, norm_neg, norm_one, one_pow, one_mul]
-      calc _ ≤ ‖rectOperator (retainedKernel d A)‖ * ‖zeroExtend f‖ :=
-              ContinuousLinearMap.le_opNorm _ _
-           _ ≤ Real.sqrt B * ‖zeroExtend f‖ := mul_le_mul_of_nonneg_right (hnorm A) (norm_nonneg _)
-           _ = _ := by rw [norm_zeroExtend]
-    _ = _ := by simp only [Finset.sum_const, Finset.card_univ, Fintype.card_finset, nsmul_eq_mul,
-      Nat.cast_pow, Nat.cast_ofNat]; ring
-
-lemma harmonic_action_zero_one {L : Type*} [Fintype L] (d : ℕ)
-    (hL : Fintype.card L = 1) (f : EuclideanSpace ℂ (L ↪ Card d)) (hf : Harmonic f) :
-    DensityTransfer.applyKernel (UnitaryFinite.actionKernel UnitaryFinite.tupleAction
-      (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))) f = 0 := by
-  classical
-  obtain ⟨i,hi⟩ := Fintype.card_eq_one_iff.mp hL
-  have hz (A : Finset L) : rectOperator (retainedKernel d A) = 0 := by
-    ext v x
-    have hh (y : L → Card d) : ¬ContactEvent d x y := by
-      intro h
-      obtain ⟨j,hj,_⟩ := h i
-      exact hj (hi j).symm
-    simp only [rectOperator_apply, retainedKernel, ite_eq_right (hh _), Complex.ofReal_zero,
-      zero_mul, Finset.sum_const_zero, zero_apply, PiLp.zero_apply]
-  have he := alternating_harmonic_action d f hf
-  simp only [hz, zero_apply, smul_zero, Finset.sum_const_zero, hL, pow_one,
-    neg_smul, one_smul] at he
-  exact neg_eq_zero.mp he.symm
-
-end
-open scoped BigOperators
-
-def MainStatement : Prop :=
-  ∃ Cstar C : ℝ, 0 < Cstar ∧ 0 < C ∧
-    ∀ (d : ℕ), 1 ≤ d → ∀ (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ),
-      let k := μ.card - μ.rowLen 0
-      1 ≤ k → k < 2^d → ∀ reverse : Bool,
-        ‖sweepOperator d μ e reverse‖^2 ≤
-          min 1 ((Cstar * d * k / (2:ℝ)^d) ^ ((k:ℝ)/2)) ∧
-        (k = 1 → sweepOperator d μ e reverse = 0) ∧
-        ‖sweepOperator d μ e reverse‖^2 ≤
-          C^k * (1+(d:ℝ))^(C*k) * ((k:ℝ)/(2:ℝ)^d)^((k:ℝ)/2)
-
-end Thorp.SparseContact
-
-namespace Thorp.SparseContact
-open scoped BigOperators Classical
-
-lemma exists_exact_fixed_unit {α L : Type*} [Fintype α] [Fintype L]
-    (μ : YoungDiagram) (e : α ≃ Specht.Cell μ) (x₀ : L ↪ α)
-    (hL : Fintype.card L = μ.card - μ.rowLen 0) :
-    ∃ w : Specht.hilbertSpace μ,
-      UnitaryFinite.IsFixed UnitaryFinite.tupleAction x₀ (Specht.relabelledUnitary μ e) w ∧ ‖w‖ = 1 := by
-  let K := UnitaryFinite.fixedSpace (V := Specht.hilbertSpace μ) UnitaryFinite.tupleAction x₀ (Specht.relabelledUnitary μ e)
-  have hd : 0 < Module.finrank ℂ (Specht.hilbertSpace μ) := by
-    rw [Specht.finrank_hilbertSpace]
-    exact Module.finrank_pos_iff.mpr (Submodule.nontrivial_iff_ne_bot.mpr (Specht.space_ne_bot μ))
-  have hdim := Specht.relabelled_tuple_dimension μ e x₀
-  rw [hL, Nat.choose_self, mul_one] at hdim
-  have hm : 0 < Module.finrank ℂ K := by
-    by_contra h
-    have hz : Module.finrank ℂ K = 0 := by omega
-    change _ ≤ Module.finrank ℂ K * _ at hdim
-    rw [hz, zero_mul] at hdim
-    omega
-  let B := stdOrthonormalBasis ℂ K
-  let i : Fin (Module.finrank ℂ K) := ⟨0,hm⟩
-  exact ⟨(B i).val, (B i).prop, B.orthonormal.norm_eq_one i⟩
-
-lemma sweep_forward_norm {d : ℕ} (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ) (rev : Bool) :
-    ‖sweepOperator d μ e rev‖ = ‖sweepOperator d μ e false‖ := by
-  cases rev
-  · rfl
-  · simp only [sweepOperator, Bool.false_eq_true, ite_false, ite_true]
-    rw [UnitaryFinite.sampleOperator_inv _ (Specht.relabelledUnitary_unitary μ e),
-      ContinuousLinearMap.adjoint.norm_map]
-
-lemma sweep_norm_le_one (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ) (rev : Bool) :
-    ‖sweepOperator d μ e rev‖ ≤ 1 :=
-  UnitaryFinite.sampleOperator_norm_le _ (Specht.relabelledUnitary_unitary μ e) _
-
-section Generic
-variable {L V : Type*} [Fintype L] [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-  [FiniteDimensional ℂ V]
-
-lemma sample_sweep_bound (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V)
-    [Representation.IsIrreducible ρ] (hρ : UnitaryFinite.IsUnitary ρ)
-    (x₀ : L ↪ Card d) (w : V) (hw : UnitaryFinite.IsFixed UnitaryFinite.tupleAction x₀ ρ w)
-    (hn : ‖w‖ = 1)
-    (hh : ∀ v, Harmonic (UnitaryFinite.normalizedCoefficient UnitaryFinite.tupleAction x₀
-      (UnitaryFinite.tupleAction_transitive x₀) ρ w v))
-    (hδ : 0 < 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)
-    (hδ1 : 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d ≤ 1) :
-    ‖UnitaryFinite.sampleOperator ρ (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))‖ ≤
-      (2:ℝ)^(Fintype.card L) *
-        Real.sqrt ((2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
-          Real.exp (Real.exp 1*(Fintype.card L))) := by
-  rw [← ContinuousLinearMap.adjoint.norm_map (UnitaryFinite.sampleOperator ρ
-    (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))),
-    ← UnitaryFinite.sampleOperator_inv ρ hρ]
-  apply ContinuousLinearMap.opNorm_le_bound _ (by positivity)
-  intro v
-  rw [UnitaryFinite.sampleOperator_inv_apply]
-  have ha := harmonic_action_bound d _ (hh v) hδ hδ1
-  rw [UnitaryFinite.coefficient_random UnitaryFinite.tupleAction x₀
-    (UnitaryFinite.tupleAction_transitive x₀) ρ hρ _ w hw,
-    UnitaryFinite.normalizedCoefficient_norm _ _ _ ρ hρ w hw hn,
-    UnitaryFinite.normalizedCoefficient_norm _ _ _ ρ hρ w hw hn] at ha
-  exact ha
-
-lemma sample_sweep_zero_one (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V)
-    [Representation.IsIrreducible ρ] (hρ : UnitaryFinite.IsUnitary ρ)
-    (x₀ : L ↪ Card d) (w : V) (hw : UnitaryFinite.IsFixed UnitaryFinite.tupleAction x₀ ρ w)
-    (hn : ‖w‖ = 1)
-    (hh : ∀ v, Harmonic (UnitaryFinite.normalizedCoefficient UnitaryFinite.tupleAction x₀
-      (UnitaryFinite.tupleAction_transitive x₀) ρ w v))
-    (hL : Fintype.card L = 1) :
-    UnitaryFinite.sampleOperator ρ (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω)) = 0 := by
-  apply ContinuousLinearMap.adjoint.injective
-  rw [map_zero, ← UnitaryFinite.sampleOperator_inv ρ hρ]
-  apply ContinuousLinearMap.ext
-  intro v
-  rw [UnitaryFinite.sampleOperator_inv_apply]
-  have ha := harmonic_action_zero_one d hL _ (hh v)
-  have he := congrArg norm ha
-  rw [UnitaryFinite.coefficient_random UnitaryFinite.tupleAction x₀
-    (UnitaryFinite.tupleAction_transitive x₀) ρ hρ _ w hw,
-    UnitaryFinite.normalizedCoefficient_norm _ _ _ ρ hρ w hw hn, norm_zero] at he
-  exact norm_eq_zero.mp he
-end Generic
-
-lemma sweep_norm_bound (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ)
-    (hδ : 0 < 2*(d:ℝ)*(μ.card - μ.rowLen 0 : ℕ)/(2:ℝ)^d)
-    (hδ1 : 2*(d:ℝ)*(μ.card - μ.rowLen 0 : ℕ)/(2:ℝ)^d ≤ 1) (rev : Bool) :
-    ‖sweepOperator d μ e rev‖ ≤ (2:ℝ)^(μ.card - μ.rowLen 0) *
-      Real.sqrt ((2*(d:ℝ)*(μ.card - μ.rowLen 0 : ℕ)/(2:ℝ)^d)^(((μ.card - μ.rowLen 0 : ℕ):ℝ)/2) *
-        Real.exp (Real.exp 1*(μ.card - μ.rowLen 0 : ℕ))) := by
-  classical
-  let x₀ : Specht.TailCell μ ↪ Card d := (Function.Embedding.subtype _).trans e.symm.toEmbedding
-  obtain ⟨w,hw,hn⟩ := exists_exact_fixed_unit μ e x₀ (Specht.tail_card μ)
-  let := Specht.relabelledUnitary_irreducible μ e
-  rw [sweep_forward_norm μ e rev]
-  have hh (v : Specht.hilbertSpace μ) : Harmonic (UnitaryFinite.normalizedCoefficient
-      (V := Specht.hilbertSpace μ) UnitaryFinite.tupleAction x₀
-      (UnitaryFinite.tupleAction_transitive x₀) (Specht.relabelledUnitary μ e) w v) :=
-    normalizedCoefficient_harmonic μ e (Specht.tail_card μ) x₀ w hw v
-  have he := sample_sweep_bound (V := Specht.hilbertSpace μ) (L := Specht.TailCell μ) d (Specht.relabelledUnitary μ e)
-  have he := he (Specht.relabelledUnitary_unitary μ e) x₀ w hw hn hh
-  have he := he (by simpa only [Specht.tail_card] using hδ) (by simpa only [Specht.tail_card] using hδ1)
-  simpa only [sweepOperator, Bool.false_eq_true, ite_false, Specht.tail_card] using he
-
-lemma sweep_zero_one (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ)
-    (hk : μ.card - μ.rowLen 0 = 1) (rev : Bool) : sweepOperator d μ e rev = 0 := by
-  classical
-  let x₀ : Specht.TailCell μ ↪ Card d := (Function.Embedding.subtype _).trans e.symm.toEmbedding
-  obtain ⟨w,hw,hn⟩ := exists_exact_fixed_unit μ e x₀ (Specht.tail_card μ)
-  let := Specht.relabelledUnitary_irreducible μ e
-  have hh (v : Specht.hilbertSpace μ) : Harmonic (UnitaryFinite.normalizedCoefficient
-      (V := Specht.hilbertSpace μ) UnitaryFinite.tupleAction x₀
-      (UnitaryFinite.tupleAction_transitive x₀) (Specht.relabelledUnitary μ e) w v) :=
-    normalizedCoefficient_harmonic μ e (Specht.tail_card μ) x₀ w hw v
-  have he := sample_sweep_zero_one (V := Specht.hilbertSpace μ) (L := Specht.TailCell μ) d (Specht.relabelledUnitary μ e)
-  have he := he (Specht.relabelledUnitary_unitary μ e) x₀ w hw hn hh
-  have he := he ((Specht.tail_card μ).trans hk)
-  have hz : sweepOperator d μ e false = 0 := by
-    simpa only [sweepOperator, Bool.false_eq_true, ite_false] using he
-  apply (norm_eq_zero (a := sweepOperator d μ e rev)).mp
-  rw [sweep_forward_norm μ e rev, hz]
-  exact @norm_zero (Specht.hilbertSpace μ →L[ℂ] Specht.hilbertSpace μ) _
-
-end Thorp.SparseContact
-
-namespace Thorp.SparseContact
-
-noncomputable def contactConstant : ℝ := 32 * Real.exp (2 * Real.exp 1)
-
-lemma contactConstant_gt_one : 1 < contactConstant := by
-  have h : 1 ≤ Real.exp (2 * Real.exp 1) := Real.one_le_exp (by positivity)
-  unfold contactConstant
-  linarith
-
-lemma contactConstant_gt_two : 2 < contactConstant := by
-  have h : 1 ≤ Real.exp (2 * Real.exp 1) := Real.one_le_exp (by positivity)
-  unfold contactConstant
-  linarith
-
-lemma inflated_contact_bound {δ : ℝ} (hδ : 0 ≤ δ) (k : ℕ) :
-    ((2:ℝ)^k * Real.sqrt (δ^((k:ℝ)/2) * Real.exp (Real.exp 1*k)))^2 =
-      ((contactConstant/2)*δ)^((k:ℝ)/2) := by
-  have hC : 0 ≤ contactConstant/2 := by have := contactConstant_gt_one; positivity
-  rw [mul_pow, Real.sq_sqrt (by positivity), Real.mul_rpow hC hδ]
-  have hc : contactConstant/2 = Real.exp (2*(Real.log 4+Real.exp 1)) := by
-    unfold contactConstant
-    rw [show 2*(Real.log 4+Real.exp 1) = Real.log 4*2+2*Real.exp 1 by ring,
-      Real.exp_add, Real.exp_mul (Real.log 4) 2, Real.exp_log (by norm_num : (0:ℝ)<4)]
-    norm_num
-    ring
-  rw [hc, ←Real.exp_mul]
-  have hpow : ((2:ℝ)^k)^2 = Real.exp (Real.log 4 * k) := by
-    rw [← pow_mul, mul_comm k 2, pow_mul]
-    norm_num only [pow_two]
-    rw [Real.exp_mul (Real.log 4) k, Real.exp_log (by norm_num : (0:ℝ)<4), Real.rpow_natCast]
-  rw [hpow]
-  calc
-    _ = (Real.exp (Real.log 4*k)*Real.exp (Real.exp 1*k))*δ^((k:ℝ)/2) := by ring
-    _ = _ := by rw [←Real.exp_add]; congr 2; ring
-
-lemma contact_crude_arithmetic {C d q : ℝ} (hC : 1 ≤ C) (hd : 1 ≤ d) (hq : 0 ≤ q) (k : ℕ) :
-    (C*d*q)^((k:ℝ)/2) ≤ C^k*(1+d)^(C*k)*q^((k:ℝ)/2) := by
-  rw [Real.mul_rpow (by positivity) hq, Real.mul_rpow (by positivity) (by positivity)]
-  apply mul_le_mul_of_nonneg_right _ (by positivity)
-  apply mul_le_mul
-  · rw [← Real.rpow_natCast]
-    exact Real.rpow_le_rpow_of_exponent_le hC (by linarith [Nat.cast_nonneg (α := ℝ) k])
-  · calc
-      d^((k:ℝ)/2) ≤ (1+d)^((k:ℝ)/2) := Real.rpow_le_rpow (by positivity) (by linarith) (by positivity)
-      _ ≤ (1+d)^(C*k) := Real.rpow_le_rpow_of_exponent_le (by linarith) (by nlinarith [Nat.cast_nonneg (α := ℝ) k])
-  · positivity
-  · positivity
-
-
-lemma sweep_sharp_bound (d : ℕ) (hd : 1 ≤ d) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ)
-    (hk : 1 ≤ μ.card - μ.rowLen 0) (rev : Bool) :
-    ‖sweepOperator d μ e rev‖^2 ≤ min 1
-      ((contactConstant*d*(μ.card - μ.rowLen 0 : ℕ)/(2:ℝ)^d)^(((μ.card - μ.rowLen 0 : ℕ):ℝ)/2)) := by
-  let k := μ.card - μ.rowLen 0
-  have hk' : (0:ℝ) < k := by exact_mod_cast hk
-  have hd' : (0:ℝ) < d := by exact_mod_cast hd
-  have hδ : 0 < 2*(d:ℝ)*k/(2:ℝ)^d := by positivity
-  have hOne : ‖sweepOperator d μ e rev‖^2 ≤ 1 := by
-    have h := sweep_norm_le_one d μ e rev
-    have h0 := norm_nonneg (sweepOperator d μ e rev)
-    nlinarith
-  refine le_min hOne ?_
-  have he : contactConstant * d * (k:ℝ)/(2:ℝ)^d =
-      (contactConstant/2)*(2*(d:ℝ)*k/(2:ℝ)^d) := by ring
-  change _ ≤ (contactConstant * d * (k:ℝ)/(2:ℝ)^d)^((k:ℝ)/2)
-  rw [he]
-  by_cases hs : 2*(d:ℝ)*k/(2:ℝ)^d ≤ 1
-  · have h := sweep_norm_bound d μ e hδ hs rev
-    have hb : 0 ≤ (2:ℝ)^k * Real.sqrt
-        ((2*(d:ℝ)*k/(2:ℝ)^d)^((k:ℝ)/2)*Real.exp (Real.exp 1*k)) := by positivity
-    calc
-      _ ≤ ((2:ℝ)^k * Real.sqrt
-          ((2*(d:ℝ)*k/(2:ℝ)^d)^((k:ℝ)/2)*Real.exp (Real.exp 1*k)))^2 :=
-        (sq_le_sq₀ (norm_nonneg (sweepOperator d μ e rev)) hb).mpr h
-      _ = _ := inflated_contact_bound hδ.le k
-  · apply hOne.trans
-    apply Real.one_le_rpow
-    · have hc : 1 ≤ contactConstant/2 := by have := contactConstant_gt_two; linarith
-      exact one_le_mul_of_one_le_of_one_le hc (le_of_lt (lt_of_not_ge hs))
-    · positivity
-
-theorem uniform_sparse_contact : MainStatement := by
-  refine ⟨contactConstant, contactConstant, (lt_trans (by norm_num) contactConstant_gt_one),
-    (lt_trans (by norm_num) contactConstant_gt_one), ?_⟩
-  intro d hd μ e k hk _ rev
-  have hs := sweep_sharp_bound d hd μ e hk rev
-  refine ⟨hs, ?_, ?_⟩
-  · intro hk1
-    exact sweep_zero_one d μ e hk1 rev
-  · apply (hs.trans (min_le_right _ _)).trans
-    change (contactConstant * d * (k:ℝ)/(2:ℝ)^d)^((k:ℝ)/2) ≤ _
-    rw [mul_div_assoc]
-    exact contact_crude_arithmetic contactConstant_gt_one.le
-      (by exact_mod_cast hd) (by positivity) k
-
-end Thorp.SparseContact
-
-namespace Thorp.DimensionEstimates
-open scoped BigOperators
-
-lemma log_factorial_upper_binary (n : ℕ) (hn : 0<n) :
-    Real.log (n.factorial:ℝ) ≤ (n:ℝ)*Real.log n-Real.log 2*((n:ℝ)-1) := by
-  induction n, hn using Nat.le_induction with
-  | base => norm_num
-  | succ n hn ih =>
-    have hn0 : (0:ℝ)<n := by exact_mod_cast hn
-    have hb : (2:ℝ) ≤ (1+(n:ℝ)⁻¹)^n := by
-      have h := one_add_mul_le_pow (by have := inv_nonneg.mpr hn0.le; linarith : (-2:ℝ)≤(n:ℝ)⁻¹) n
-      norm_num [mul_inv_cancel₀ hn0.ne'] at h ⊢
-      exact h
-    have hl := Real.log_le_log (by norm_num : (0:ℝ)<2) hb
-    rw [Real.log_pow] at hl
-    have hs : Real.log (1+(n:ℝ)⁻¹)=Real.log ((n:ℝ)+1)-Real.log n := by
-      rw [←Real.log_div (by positivity) hn0.ne']
-      congr 1
-      field_simp
-    rw [hs] at hl
-    rw [Nat.factorial_succ,Nat.cast_mul,Real.log_mul (by positivity) (by positivity),Nat.cast_add,Nat.cast_one]
-    nlinarith
-
-lemma log_factorial_lower (n : ℕ) (hn : 0<n) :
-    (n:ℝ)*Real.log n-n+1 ≤ Real.log (n.factorial:ℝ) := by
-  induction n, hn using Nat.le_induction with
-  | base => norm_num
-  | succ n hn ih =>
-    have hn0 : (0:ℝ)<n := by exact_mod_cast hn
-    have hl := Real.log_le_sub_one_of_pos (show 0<((n:ℝ)+1)/n by positivity)
-    rw [Real.log_div (by positivity) hn0.ne'] at hl
-    have hh := mul_le_mul_of_nonneg_left hl hn0.le
-    have he : (n:ℝ)*(((n:ℝ)+1)/n-1)=1 := by field_simp; ring
-    rw [he] at hh
-    rw [Nat.factorial_succ,Nat.cast_mul,Real.log_mul (by positivity) (by positivity),Nat.cast_add,Nat.cast_one]
-    nlinarith
-
-lemma log_factorial_upper (n : ℕ) (hn : 0<n) :
-    Real.log (n.factorial:ℝ) ≤ (n:ℝ)*Real.log n-n+Real.log n+1 := by
-  induction n, hn using Nat.le_induction with
-  | base => norm_num
-  | succ n hn ih =>
-    have hn0 : (0:ℝ)<n := by exact_mod_cast hn
-    have hl := Real.one_sub_inv_le_log_of_pos (show 0<((n:ℝ)+1)/n by positivity)
-    rw [Real.log_div (by positivity) hn0.ne'] at hl
-    have hh := mul_le_mul_of_nonneg_left hl (show 0≤(n:ℝ)+1 by positivity)
-    have he : ((n:ℝ)+1)*(1-(((n:ℝ)+1)/n)⁻¹)=1 := by field_simp; ring
-    rw [he] at hh
-    rw [Nat.factorial_succ,Nat.cast_mul,Real.log_mul (by positivity) (by positivity),Nat.cast_add,Nat.cast_one]
-    nlinarith
-
-variable {X I J : Type*} [Fintype X] [Fintype I] [Fintype J]
-open scoped Classical
-
-noncomputable def fiberWeight (f : X → I) (i : I) : ℝ := Fintype.card {x // f x=i}
-
-omit [Fintype I] in
-lemma fiberWeight_nonneg (f : X → I) (i : I) : 0≤fiberWeight f i := Nat.cast_nonneg _
-
-omit [Fintype I] in
-lemma fiberWeight_pos (f : X → I) (x : X) : 0<fiberWeight f (f x) := by
-  let : Nonempty {y // f y=f x} := ⟨⟨x,rfl⟩⟩
-  exact Nat.cast_pos.mpr Fintype.card_pos
-
-lemma fiber_sum (f : X → I) (u : I → ℝ) :
-    ∑ x, u (f x)=∑ i, fiberWeight f i*u i := by
-  rw [←Equiv.sum_comp (Equiv.sigmaFiberEquiv f)]
-  simp only [Equiv.sigmaFiberEquiv_apply,Fintype.sum_sigma]
-  apply Finset.sum_congr rfl
-  intro i _
-  calc
-    (∑ x : {x // f x=i}, u (f x)) = ∑ _x : {x // f x=i}, u i :=
-      Finset.sum_congr rfl (fun x _ => congrArg u x.property)
-    _ = _ := by simp [fiberWeight]
-
-lemma sum_fiberWeight (f : X → I) : ∑ i, fiberWeight f i=Fintype.card X := by
-  have h := fiber_sum f (fun _ => 1)
-  simpa using h.symm
-
-theorem incidence_entropy (f : X → I) (g : X → J)
-    (hinj : Function.Injective (fun x => (f x,g x))) :
-    (∑ i, fiberWeight f i*Real.log (fiberWeight f i))+
-      (∑ j, fiberWeight g j*Real.log (fiberWeight g j)) ≤
-        (Fintype.card X:ℝ)*Real.log (Fintype.card X) := by
-  by_cases hX : Nonempty X
-  · let : Nonempty X := hX
-    have hN : (0:ℝ)<Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
-    have hprod : ∑ x, fiberWeight f (f x)*fiberWeight g (g x) ≤ (Fintype.card X:ℝ)^2 := by
-      calc
-        _ ≤ ∑ p : I × J, fiberWeight f p.1*fiberWeight g p.2 := by
-          calc
-            _ = ∑ p ∈ Finset.univ.image (fun x => (f x,g x)), fiberWeight f p.1*fiberWeight g p.2 := by
-              rw [Finset.sum_image (fun x _ y _ h => hinj h)]
-            _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _)
-              (fun p _ _ => mul_nonneg (fiberWeight_nonneg f p.1) (fiberWeight_nonneg g p.2))
-        _ = _ := by
-          rw [Fintype.sum_prod_type]
-          simp_rw [←Finset.mul_sum]
-          rw [←Finset.sum_mul,sum_fiberWeight,sum_fiberWeight,pow_two]
-    have hdiv : (∑ x, fiberWeight f (f x)*fiberWeight g (g x))/(Fintype.card X:ℝ) ≤ Fintype.card X := by
-      apply (div_le_iff₀ hN).mpr
-      simpa only [pow_two] using hprod
-    have hl (x : X) := Real.log_le_sub_one_of_pos
-      (show 0<fiberWeight f (f x)*fiberWeight g (g x)/(Fintype.card X:ℝ) from
-        div_pos (mul_pos (fiberWeight_pos f x) (fiberWeight_pos g x)) hN)
-    have hs := Finset.sum_le_sum (s := Finset.univ) (fun x _ => hl x)
-    simp_rw [Real.log_div (mul_pos (fiberWeight_pos f _) (fiberWeight_pos g _)).ne' hN.ne',
-      Real.log_mul (fiberWeight_pos f _).ne' (fiberWeight_pos g _).ne'] at hs
-    rw [Finset.sum_sub_distrib,Finset.sum_add_distrib,
-      fiber_sum f (fun i => Real.log (fiberWeight f i)),
-      fiber_sum g (fun j => Real.log (fiberWeight g j)),
-      Finset.sum_sub_distrib,←Finset.sum_div] at hs
-    simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,mul_one] at hs
-    linarith
-  · let : IsEmpty X := not_nonempty_iff.mp hX
-    have hf (i : I) : fiberWeight f i=0 := by simp [fiberWeight]
-    have hg (j : J) : fiberWeight g j=0 := by simp [fiberWeight]
-    simp [hf,hg]
-
-end Thorp.DimensionEstimates
-
-namespace Thorp.DimensionEstimates
-open scoped BigOperators Classical
-
-lemma log_factorial_upper_binary_all (n : ℕ) :
-    Real.log (n.factorial:ℝ) ≤ (n:ℝ)*Real.log n-Real.log 2*((n:ℝ)-1) := by
-  by_cases hn : n=0
-  · subst n
-    simpa using (Real.log_nonneg (by norm_num : (1:ℝ)≤2))
-  · exact log_factorial_upper_binary n (Nat.pos_of_ne_zero hn)
-
-lemma log_fiber_factorials {X I : Type*} [Fintype X] [Fintype I] (f : X → I) :
-    Real.log ((∏ i, (Fintype.card {x // f x=i}).factorial : ℕ):ℝ) ≤
-      ∑ i, fiberWeight f i*Real.log (fiberWeight f i) -
-        Real.log 2*((Fintype.card X:ℝ)-Fintype.card I) := by
-  rw [Nat.cast_prod,Real.log_prod (fun _ _ => by positivity)]
-  have h := Finset.sum_le_sum (s := Finset.univ) (fun i _ =>
-    log_factorial_upper_binary_all (Fintype.card {x // f x=i}))
-  change ∑ i, Real.log ((Fintype.card {x // f x=i}).factorial:ℝ) ≤
-    ∑ i, (fiberWeight f i*Real.log (fiberWeight f i)-Real.log 2*(fiberWeight f i-1)) at h
-  rw [Finset.sum_sub_distrib,←Finset.mul_sum,Finset.sum_sub_distrib,sum_fiberWeight] at h
-  simpa using h
-
-theorem dimension_log_from_incidence {X I J : Type*} [Fintype X] [Fintype I] [Fintype J]
-    [Nonempty X] (f : X → I) (g : X → J)
-    (hinj : Function.Injective (fun x => (f x,g x))) (D : ℕ)
-    (hD : (Fintype.card X).factorial ≤ D *
-      (∏ i, (Fintype.card {x // f x=i}).factorial) *
-      (∏ j, (Fintype.card {x // g x=j}).factorial)) :
-    ((2*Real.log 2-1)*(Fintype.card X:ℝ)-
-      Real.log 2*((Fintype.card I:ℝ)+Fintype.card J)+1) ≤ Real.log D := by
-  have hd : 0<D := by
-    by_contra h
-    have : D=0 := by omega
-    simp [this] at hD
-    exact (Nat.factorial_pos _).ne' hD
-  have hf := log_fiber_factorials f
-  have hg := log_fiber_factorials g
-  have hi := incidence_entropy f g hinj
-  have hlo := log_factorial_lower (Fintype.card X) Fintype.card_pos
-  have hlog := Real.log_le_log (by positivity : (0:ℝ)<(Fintype.card X).factorial)
-    (show ((Fintype.card X).factorial:ℝ) ≤ D *
-      (∏ i, (Fintype.card {x // f x=i}).factorial) *
-      (∏ j, (Fintype.card {x // g x=j}).factorial) by exact_mod_cast hD)
-  rw [Real.log_mul (by positivity) (by positivity),
-    Real.log_mul (by positivity) (by positivity)] at hlog
-  nlinarith
-
-
-lemma scaled_binEntropy (N j : ℝ) (hN : 0<N) (hj : 0<j) (hjN : j<N) :
-    N*Real.binEntropy (j/N)=N*Real.log N-j*Real.log j-(N-j)*Real.log (N-j) := by
-  have hsub : 0<N-j := sub_pos.mpr hjN
-  have he : 1-j/N=(N-j)/N := by field_simp
-  unfold Real.binEntropy
-  rw [he,Real.log_inv,Real.log_inv,Real.log_div hj.ne' hN.ne',
-    Real.log_div hsub.ne' hN.ne']
-  field_simp
-  ring
-
-lemma log_choose_lower_entropy (N j : ℕ) (hj : 0<j) (hjN : j<N) :
-    (N:ℝ)*Real.binEntropy ((j:ℝ)/N)-2*Real.log N-1 ≤ Real.log (N.choose j:ℝ) := by
-  have hN : 0<N := lt_trans hj hjN
-  have ht : 0<N-j := Nat.sub_pos_of_lt hjN
-  have hj0 : (0:ℝ)<j := by exact_mod_cast hj
-  have hN0 : (0:ℝ)<N := by exact_mod_cast hN
-  have hjN0 : (j:ℝ)<N := by exact_mod_cast hjN
-  have hchoose : (0:ℝ)<N.choose j := by exact_mod_cast Nat.choose_pos hjN.le
-  have hfac := congrArg (fun n : ℕ => Real.log (n:ℝ)) (Nat.choose_mul_factorial_mul_factorial hjN.le)
-  simp only [Nat.cast_mul] at hfac
-  rw [Real.log_mul (mul_pos hchoose (by positivity)).ne' (by positivity),
-    Real.log_mul hchoose.ne' (by positivity)] at hfac
-  have hlo := log_factorial_lower N hN
-  have hji := log_factorial_upper j hj
-  have hti := log_factorial_upper (N-j) ht
-  have hjlog := Real.log_le_log hj0 hjN0.le
-  have htlog := Real.log_le_log (by exact_mod_cast ht : (0:ℝ)<(N-j:ℕ))
-    (by exact_mod_cast Nat.sub_le N j : ((N-j:ℕ):ℝ)≤N)
-  rw [Nat.cast_sub hjN.le] at hti htlog
-  rw [scaled_binEntropy N j hN0 hj0 hjN0]
-  linarith
-
-lemma binEntropy_three_quarters :
-    Real.binEntropy (3/4:ℝ)=2*Real.log 2-(3/4:ℝ)*Real.log 3 := by
-  have h4 : Real.log (4:ℝ)=2*Real.log 2 := by
-    rw [show (4:ℝ)=2^2 by norm_num,Real.log_pow]; norm_num
-  unfold Real.binEntropy
-  norm_num only [show (1-(3/4:ℝ))=1/4 by norm_num]
-  rw [Real.log_div (by norm_num) (by norm_num),h4]
-  ring
-
-noncomputable def entropyGap : ℝ := Real.log (32/27:ℝ)/3
-
-lemma entropyGap_pos : 0<entropyGap := by
-  unfold entropyGap
-  exact div_pos (Real.log_pos (by norm_num)) (by norm_num)
-
-lemma entropyGap_eq : entropyGap=(5*Real.log 2-3*Real.log 3)/3 := by
-  unfold entropyGap
-  rw [Real.log_div (by norm_num) (by norm_num),show (32:ℝ)=2^5 by norm_num,
-    show (27:ℝ)=3^3 by norm_num,Real.log_pow,Real.log_pow]
-  norm_num
-
-lemma binEntropy_tail_lower (q : ℝ) (hq : 0≤q) (hq' : q≤3/4) :
-    q*Real.log 2+entropyGap*q ≤ Real.binEntropy q := by
-  have hc := Real.strictConcave_binEntropy.concaveOn.2
-    (show (0:ℝ)∈Set.Icc 0 1 by constructor <;> norm_num)
-    (show (3/4:ℝ)∈Set.Icc 0 1 by constructor <;> norm_num)
-    (show 0≤1-4*q/3 by linarith) (show 0≤4*q/3 by positivity)
-    (show (1-4*q/3)+(4*q/3)=1 by ring)
-  simp only [Real.binEntropy_zero,binEntropy_three_quarters,smul_eq_mul,mul_zero,zero_add] at hc
-  have he : 4*q/3*(3/4:ℝ)=q := by ring
-  rw [he] at hc
-  rw [entropyGap_eq]
-  linarith
-
-lemma log_dimension_from_binomial (N j D : ℕ) (hj : 0<j) (hjN : j<N)
-    (hfrac : (j:ℝ)≤(3/4:ℝ)*N) (hD : N.choose j≤D*2^j) :
-    entropyGap*(j:ℝ)-2*Real.log N-1 ≤ Real.log D := by
-  have hpos : 0<N.choose j := Nat.choose_pos hjN.le
-  have hd : 0<D := by nlinarith [Nat.zero_le (2^j)]
-  have hN : (0:ℝ)<N := by exact_mod_cast lt_trans hj hjN
-  have hm := Real.log_le_log (by exact_mod_cast hpos : (0:ℝ)<N.choose j)
-    (show (N.choose j:ℝ)≤D*2^j by exact_mod_cast hD)
-  rw [Real.log_mul (by positivity) (by positivity),Real.log_pow] at hm
-  have he := binEntropy_tail_lower ((j:ℝ)/N) (by positivity)
-    ((div_le_iff₀ hN).mpr hfrac)
-  have hel := mul_le_mul_of_nonneg_left he hN.le
-  have hbin := log_choose_lower_entropy N j hj hjN
-  have heq : (N:ℝ)*((j:ℝ)/N*Real.log 2+entropyGap*((j:ℝ)/N))=
-      (j:ℝ)*Real.log 2+entropyGap*j := by field_simp
-  rw [heq] at hel
-  linarith
-
-lemma balanced_dimension_coefficient_pos : 0<(3/2:ℝ)*Real.log 2-1 := by
-  linarith [Real.log_two_gt_d9]
-
-
-lemma log_choose_upper (N j : ℕ) (hN : 0<N) (hj : j≤N) :
-    Real.log (N.choose j:ℝ) ≤ (j:ℝ)*Real.log N-Real.log (j.factorial:ℝ) := by
-  have hh := Real.log_le_log (by exact_mod_cast Nat.choose_pos hj : (0:ℝ)<N.choose j)
-    (Nat.choose_le_pow_div j N : (N.choose j:ℝ)≤(N:ℝ)^j/(j.factorial:ℝ))
-  rw [Real.log_div (by positivity : (N:ℝ)^j≠0) (by positivity : (j.factorial:ℝ)≠0),Real.log_pow] at hh
-  exact hh
-
-lemma log_choose_lower (N j : ℕ) (hj : j≤N) :
-    (j:ℝ)*Real.log (N+1-j:ℕ)-Real.log (j.factorial:ℝ) ≤ Real.log (N.choose j:ℝ) := by
-  have hp : 0<N+1-j := by omega
-  have hh := Real.log_le_log (by positivity : (0:ℝ)<((N+1-j:ℕ):ℝ)^j/(j.factorial:ℝ))
-    (Nat.pow_le_choose j N : (((N+1-j:ℕ):ℝ)^j)/(j.factorial:ℝ)≤N.choose j)
-  rw [Real.log_div (by positivity : (((N+1-j:ℕ):ℝ)^j)≠0) (by positivity : (j.factorial:ℝ)≠0),Real.log_pow] at hh
-  exact hh
-
-lemma footprint_log_bound (N k j D m : ℕ) (hN : 0<N) (hk : j≤k) (hj : j≤N)
-    (hD : 0<D) (hfoot : D * k.choose j ≤ m * N.choose j) :
-    Real.log D-(j:ℝ)*(Real.log N-Real.log (k+1-j:ℕ)) ≤ Real.log m := by
-  have hkpos := Nat.choose_pos hk
-  have hprod : 0<D*k.choose j := Nat.mul_pos hD hkpos
-  have hm : 0 < m := by
-    by_contra h
-    have he : m=0 := by omega
-    simp only [he,zero_mul] at hfoot
-    omega
-  have hh := Real.log_le_log (by exact_mod_cast hprod : (0:ℝ)<D*k.choose j)
-    (show (D:ℝ)*k.choose j≤ m*N.choose j by exact_mod_cast hfoot)
-  rw [Real.log_mul (by positivity) (by exact_mod_cast hkpos.ne'),
-    Real.log_mul (by positivity) (by exact_mod_cast (Nat.choose_pos hj).ne')] at hh
-  have hlo := log_choose_lower k j hk
-  have hup := log_choose_upper N j hN hj
-  nlinarith
-
-lemma log_factorial_le_mul_log (j : ℕ) (_hj : 0<j) :
-    Real.log (j.factorial:ℝ)≤(j:ℝ)*Real.log j := by
-  have h := Real.log_le_log (by positivity : (0:ℝ)<j.factorial)
-    (show (j.factorial:ℝ)≤(j:ℝ)^j by exact_mod_cast Nat.factorial_le_pow j)
-  simpa only [Real.log_pow] using h
-
-lemma near_row_log_dimension (N j D : ℕ) (hj : 0<j) (hsmall : 2*j≤N)
-    (hD : N.choose j≤D*2^j) :
-    (j:ℝ)*(Real.log ((N:ℝ)/j)-2*Real.log 2) ≤ Real.log D := by
-  have hjN : j≤N := by omega
-  have hN : 0<N := by omega
-  have hNp : (0:ℝ)<N := by exact_mod_cast hN
-  have hjp : (0:ℝ)<j := by exact_mod_cast hj
-  have hn : 0<N+1-j := by omega
-  have hnreal : (N:ℝ)/2≤(N+1-j:ℕ) := by
-    rw [Nat.cast_sub (by omega : j≤N+1),Nat.cast_add,Nat.cast_one]
-    have hc : 2*(j:ℝ)≤N := by exact_mod_cast hsmall
-    linarith
-  have hDpos : 0<D := by
-    have hc := Nat.choose_pos hjN
-    by_contra h
-    have he : D=0 := by omega
-    simp only [he,zero_mul] at hD
-    omega
-  have hm := Real.log_le_log (by exact_mod_cast Nat.choose_pos hjN : (0:ℝ)<N.choose j)
-    (show (N.choose j:ℝ)≤D*2^j by exact_mod_cast hD)
-  rw [Real.log_mul (by positivity) (by positivity),Real.log_pow] at hm
-  have hl := Real.log_le_log (by positivity : (0:ℝ)<(N:ℝ)/2) hnreal
-  rw [Real.log_div hNp.ne' (by norm_num)] at hl
-  have hh := log_choose_lower N j hjN
-  have hf := log_factorial_le_mul_log j hj
-  rw [Real.log_div hNp.ne' hjp.ne']
-  nlinarith
-
-open Filter
-
-noncomputable def nearPower : ℝ := 1/2048
-noncomputable def nearFraction : ℝ := 1/4096
-noncomputable def tailPower : ℝ := 1535/1048576
-
-noncomputable def chosenTuple (N j : ℕ) : ℕ :=
-  ⌈(j:ℝ)*((N:ℝ)/j)^nearPower⌉₊
-
-lemma chosenTuple_bounds (N j : ℕ) (hj : 0<j) (hx : 1≤(N:ℝ)/j)
-    (hhalf : ((N:ℝ)/j)^(nearPower-1)≤1/2) :
-    j≤chosenTuple N j ∧ chosenTuple N j≤N ∧
-      (chosenTuple N j:ℝ)≤2*j*((N:ℝ)/j)^nearPower := by
-  let x : ℝ := (N:ℝ)/j
-  have hjp : (0:ℝ)<j := by exact_mod_cast hj
-  have hj1 : (1:ℝ)≤j := by exact_mod_cast hj
-  have hxp : 0<x := lt_of_lt_of_le zero_lt_one hx
-  have hpow : 1≤x^nearPower := Real.one_le_rpow hx (by norm_num [nearPower])
-  have hceil : (j:ℝ)*x^nearPower≤chosenTuple N j := Nat.le_ceil _
-  have hceil' : (chosenTuple N j:ℝ)<(j:ℝ)*x^nearPower+1 := Nat.ceil_lt_add_one (by positivity)
-  have he : (j:ℝ)*x=N := by dsimp [x]; field_simp
-  have hh : x^nearPower≤x/2 := by
-    rw [Real.rpow_sub hxp,Real.rpow_one] at hhalf
-    nlinarith [(div_le_iff₀ hxp).mp hhalf]
-  constructor
-  · exact_mod_cast (le_mul_of_one_le_right hjp.le hpow).trans hceil
-  constructor
-  · apply Nat.ceil_le.mpr
-    change (j:ℝ)*x^nearPower≤N
-    nlinarith
-  · change (chosenTuple N j:ℝ)≤2*(j:ℝ)*x^nearPower
-    nlinarith
-
-lemma chosenTuple_tail_lower (N j : ℕ) (hj : 0<j)
-    (hx : 2≤((N:ℝ)/j)^nearPower) :
-    (j:ℝ)*((N:ℝ)/j)^nearPower/2 ≤ (chosenTuple N j+1-j:ℕ) := by
-  have hjp : (0:ℝ)<j := by exact_mod_cast hj
-  have hc : (j:ℝ)*((N:ℝ)/j)^nearPower≤chosenTuple N j := Nat.le_ceil _
-  have hjk : j≤chosenTuple N j := by exact_mod_cast (by nlinarith : (j:ℝ)≤chosenTuple N j)
-  rw [Nat.cast_sub (by omega : j≤chosenTuple N j+1),Nat.cast_add,Nat.cast_one]
-  nlinarith
-
-lemma chosenTuple_log_footprint (N j D m : ℕ) (hj : 0<j)
-    (hN : 2*j≤N) (hD : N.choose j≤D*2^j)
-    (hx : 2≤((N:ℝ)/j)^nearPower)
-    (hlog : 8*Real.log 2/nearPower≤Real.log ((N:ℝ)/j))
-    (hfoot : D * (chosenTuple N j).choose j ≤ m * N.choose j) :
-    nearFraction*Real.log D≤Real.log m := by
-  let x : ℝ := (N:ℝ)/j
-  have hjp : (0:ℝ)<j := by exact_mod_cast hj
-  have hNp : (0:ℝ)<N := by exact_mod_cast (by omega : 0<N)
-  have hxp : 0<x := by dsimp [x]; positivity
-  have hdpos : 0<D := by
-    have hc := Nat.choose_pos (show j≤N by omega)
-    by_contra h
-    have hz : D=0 := by omega
-    simp only [hz,zero_mul] at hD
-    omega
-  have htail := chosenTuple_tail_lower N j hj hx
-  have hjk : j≤chosenTuple N j := by
-    have hc : (j:ℝ)*x^nearPower≤chosenTuple N j := Nat.le_ceil _
-    exact_mod_cast (by nlinarith : (j:ℝ)≤chosenTuple N j)
-  have hh := footprint_log_bound N (chosenTuple N j) j D m (by omega) hjk (by omega) hdpos hfoot
-  have htlog := Real.log_le_log (by positivity : (0:ℝ)<(j:ℝ)*x^nearPower/2) htail
-  rw [Real.log_div (by positivity) (by norm_num),Real.log_mul hjp.ne' (by positivity),Real.log_rpow hxp] at htlog
-  have hxlog : Real.log x=Real.log N-Real.log j := Real.log_div hNp.ne' hjp.ne'
-  have hdim := near_row_log_dimension N j D hj hN hD
-  change (j:ℝ)*(Real.log x-2*Real.log 2)≤Real.log D at hdim
-  have hh' : Real.log D-(j:ℝ)*((1-nearPower)*Real.log x+Real.log 2)≤Real.log m := by
-    have hm := mul_le_mul_of_nonneg_left htlog hjp.le
-    rw [hxlog] at hm
-    nlinarith
-  have hlg : (3-nearPower)*Real.log 2≤nearFraction*Real.log x := by
-    dsimp [nearPower,nearFraction] at hlog ⊢
-    have := Real.log_pos (by norm_num : (1:ℝ)<2)
-    linarith
-  have hlg' := mul_le_mul_of_nonneg_left hlg hjp.le
-  dsimp [nearPower,nearFraction] at hh' hlg' ⊢
-  nlinarith
-
-lemma near_row_log_uniform (N j D : ℕ) (hj : 0<j) (hN : 2*j≤N)
-    (hD : N.choose j≤D*2^j) (hlog : 4*Real.log 2+2≤Real.log ((N:ℝ)/j)) :
-    (1/2:ℝ)*Real.log N≤Real.log D ∧ (j:ℝ)≤Real.log D := by
-  have hjp : (0:ℝ)<j := by exact_mod_cast hj
-  have hj1 : (1:ℝ)≤j := by exact_mod_cast hj
-  have hNp : (0:ℝ)<N := by exact_mod_cast (by omega : 0<N)
-  have hd := near_row_log_dimension N j D hj hN hD
-  have hjlog := Real.log_le_sub_one_of_pos hjp
-  have hl2 : 0≤Real.log 2 := Real.log_nonneg (by norm_num)
-  have hi : 1≤Real.log ((N:ℝ)/j) := by linarith
-  have he : Real.log N=Real.log j+Real.log ((N:ℝ)/j) := by rw [Real.log_div hNp.ne' hjp.ne']; ring
-  constructor
-  · rw [he]
-    nlinarith
-  · nlinarith
-
-
-lemma chosenTuple_density_bound (N j : ℕ) (hj : 0<j) (hx : 1≤(N:ℝ)/j)
-    (hhalf : ((N:ℝ)/j)^(nearPower-1)≤1/2) (C : ℝ) (hC : 0≤C) :
-    C*(chosenTuple N j:ℝ)*((chosenTuple N j:ℝ)/N)^(1/512:ℝ) ≤
-      C*2^(1+(1/512:ℝ))*(j:ℝ)*((N:ℝ)/j)^(-tailPower) := by
-  let x : ℝ := (N:ℝ)/j
-  have hjp : (0:ℝ)<j := by exact_mod_cast hj
-  have hxp : 0<x := lt_of_lt_of_le zero_lt_one hx
-  have hNp : (0:ℝ)<N := by
-    have : (0:ℝ)<(N:ℝ)/j := hxp
-    exact (div_pos_iff.mp this).resolve_right (by intro h; exact (not_lt_of_ge hjp.le) h.2) |>.1
-  have he : (j:ℝ)*x=N := by dsimp [x]; field_simp
-  have hk := (chosenTuple_bounds N j hj hx hhalf).2.2
-  change (chosenTuple N j:ℝ)≤2*(j:ℝ)*x^nearPower at hk
-  have hratio : (chosenTuple N j:ℝ)/N≤2*x^(nearPower-1) := by
-    rw [Real.rpow_sub hxp,Real.rpow_one]
-    apply (div_le_iff₀ hNp).mpr
-    rw [←he]
-    have : 2*(x^nearPower/x)*((j:ℝ)*x)=2*(j:ℝ)*x^nearPower := by field_simp
-    rw [this]
-    exact hk
-  have hpow := Real.rpow_le_rpow (by positivity : (0:ℝ)≤(chosenTuple N j:ℝ)/N) hratio
-    (by norm_num : (0:ℝ)≤1/512)
-  have hmain := mul_le_mul hk hpow (by positivity) (by positivity)
-  have hrpow : (2*x^(nearPower-1))^(1/512:ℝ)=
-      (2:ℝ)^(1/512:ℝ)*x^((nearPower-1)*(1/512)) := by
-    rw [Real.mul_rpow (by norm_num) (by positivity),←Real.rpow_mul hxp.le]
-  rw [hrpow] at hmain
-  have heq : 2*(j:ℝ)*x^nearPower*((2:ℝ)^(1/512:ℝ)*x^((nearPower-1)*(1/512)))=
-      (2:ℝ)^(1+(1/512:ℝ))*(j:ℝ)*x^(-tailPower) := by
-    rw [Real.rpow_add (by norm_num : (0:ℝ)<2),Real.rpow_one]
-    have hp : nearPower+(nearPower-1)*(1/512) = -tailPower := by norm_num [nearPower,tailPower]
-    rw [←hp,Real.rpow_add hxp]
-    ring
-  rw [heq] at hmain
-  have h := mul_le_mul_of_nonneg_left hmain hC
-  simpa only [mul_assoc] using h
-
-lemma near_conditions_eventually (C : ℝ) :
-    ∀ᶠ x : ℝ in atTop, 2≤x ∧ 2≤x^nearPower ∧ x^(nearPower-1)≤1/2 ∧
-      8*Real.log 2/nearPower≤Real.log x ∧ 4*Real.log 2+2≤Real.log x ∧
-      C*2^(1+(1/512:ℝ))*x^(-tailPower) ≤ nearFraction/4000 := by
-  have hpow := (tendsto_rpow_atTop (by norm_num [nearPower] : 0<nearPower)).eventually
-    (eventually_ge_atTop (2:ℝ))
-  have hhalf : ∀ᶠ x : ℝ in atTop, x^(nearPower-1)≤1/2 := by
-    have h := (tendsto_rpow_neg_atTop (by norm_num [nearPower] : 0<1-nearPower)).eventually
-      (gt_mem_nhds (by norm_num : (0:ℝ)<1/2))
-    filter_upwards [h] with x hx
-    simpa only [neg_sub] using hx.le
-  have hdec : Tendsto (fun x : ℝ => C*2^(1+(1/512:ℝ))*x^(-tailPower)) atTop (nhds 0) := by
-    simpa only [mul_zero] using
-      (tendsto_const_nhds.mul (tendsto_rpow_neg_atTop (by norm_num [tailPower] : 0<tailPower)))
-  have hd := hdec.eventually (gt_mem_nhds (by norm_num [nearFraction] : (0:ℝ)<nearFraction/4000))
-  filter_upwards [eventually_ge_atTop (2:ℝ),hpow,hhalf,
-    Real.tendsto_log_atTop.eventually (eventually_ge_atTop (8*Real.log 2/nearPower)),
-    Real.tendsto_log_atTop.eventually (eventually_ge_atTop (4*Real.log 2+2)),hd] with x hx hp hh hl hl' hd
-  exact ⟨hx,hp,hh,hl,hl',hd.le⟩
-
-end Thorp.DimensionEstimates
-
-end ThorpNine.Tail
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Tail.TypedMarks
+
+namespace OAI
+
+namespace ThorpNine.Tail
+
+namespace Thorp.SparseContact
+
+section
+open scoped BigOperators Classical
+
+lemma partial_full_action {L : Type*} [Fintype L] (d : ℕ)
+    (f : EuclideanSpace ℂ (L ↪ Card d)) :
+    rectOperator (fun (x : L ↪ Card d) (y : L → Card d) => partialKernel d Finset.univ x y) (zeroExtend f) =
+      DensityTransfer.applyKernel (UnitaryFinite.actionKernel UnitaryFinite.tupleAction
+        (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))) f := by
+  classical
+  ext x
+  rw [rectOperator_apply, sum_zeroExtend_mul]
+  apply Finset.sum_congr rfl
+  intro y _
+  congr 2
+  simp only [partialKernel, Finset.card_univ, div_self (by positivity : ((2:ℝ)^d)^(Fintype.card L) ≠ 0),
+    one_mul, endpointProb, UnitaryFinite.actionKernel]
+  apply finiteMean_congr
+  intro ω
+  congr 1
+  simp only [Finset.mem_univ, forall_const, Function.Embedding.ext_iff]
+  rfl
+
+lemma alternating_retained_eq {L : Type*} [Fintype L] (d : ℕ)
+    (x : L ↪ Card d) (y : L → Card d) :
+    (∑ A : Finset L, (-1:ℝ)^A.card * retainedKernel d A x y) =
+      ∑ A : Finset L, (-1:ℝ)^A.card * partialKernel d A x y := by
+  classical
+  by_cases h : ContactEvent d x y
+  · simp only [retainedKernel, ite_eq_left h]
+  · simp only [retainedKernel, ite_eq_right h, mul_zero, Finset.sum_const_zero]
+    exact (alternating_kernel_cancel d x y h).symm
+
+lemma alternating_harmonic_action {L : Type*} [Fintype L] (d : ℕ)
+    (f : EuclideanSpace ℂ (L ↪ Card d)) (hf : Harmonic f) :
+    (∑ A : Finset L, ((-1:ℂ)^A.card) • rectOperator (retainedKernel d A) (zeroExtend f)) =
+      ((-1:ℂ)^(Fintype.card L)) • DensityTransfer.applyKernel
+        (UnitaryFinite.actionKernel UnitaryFinite.tupleAction
+          (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))) f := by
+  classical
+  calc
+    _ = ∑ A : Finset L, ((-1:ℂ)^A.card) •
+        rectOperator (fun (x : L ↪ Card d) y => partialKernel d A x y) (zeroExtend f) := by
+      ext x
+      simp only [WithLp.ofLp_sum, Finset.sum_apply, PiLp.smul_apply, rectOperator_apply, smul_eq_mul, Finset.mul_sum]
+      rw [Finset.sum_comm, Finset.sum_comm (s := (Finset.univ : Finset (Finset L)))]
+      apply Finset.sum_congr rfl
+      intro y _
+      simp_rw [← mul_assoc, ← Finset.sum_mul]
+      congr 1
+      have he := congrArg (fun t : ℝ => (t:ℂ)) (alternating_retained_eq d x y)
+      simpa only [Complex.ofReal_sum, Complex.ofReal_mul, Complex.ofReal_pow, Complex.ofReal_neg,
+        Complex.ofReal_one] using he
+    _ = ((-1:ℂ)^(Fintype.card L)) •
+        rectOperator (fun (x : L ↪ Card d) y => partialKernel d Finset.univ x y) (zeroExtend f) := by
+      rw [Finset.sum_eq_single (Finset.univ : Finset L)]
+      · simp only [Finset.card_univ]
+      · intro A _ hA
+        rw [partial_proper_zero d A hA f hf, smul_zero]
+      · simp
+    _ = _ := by rw [partial_full_action]
+
+lemma harmonic_action_bound {L : Type*} [Fintype L] (d : ℕ)
+    (f : EuclideanSpace ℂ (L ↪ Card d)) (hf : Harmonic f)
+    (hδ : 0 < 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)
+    (hδ1 : 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d ≤ 1) :
+    ‖DensityTransfer.applyKernel (UnitaryFinite.actionKernel UnitaryFinite.tupleAction
+      (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))) f‖ ≤
+      (2:ℝ)^(Fintype.card L) *
+        Real.sqrt ((2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
+          Real.exp (Real.exp 1*(Fintype.card L))) * ‖f‖ := by
+  classical
+  let B := (2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
+          Real.exp (Real.exp 1*(Fintype.card L))
+  have hB : 0 ≤ B := by dsimp [B]; positivity
+  have hnorm (A : Finset L) : ‖rectOperator (retainedKernel d A)‖ ≤ Real.sqrt B := by
+    have hh := retained_norm_sq d A hδ hδ1
+    have hs := Real.sq_sqrt hB
+    have hn := Real.sqrt_nonneg B
+    nlinarith
+  calc
+    _ = ‖((-1:ℂ)^(Fintype.card L)) • DensityTransfer.applyKernel
+        (UnitaryFinite.actionKernel UnitaryFinite.tupleAction
+          (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))) f‖ := by
+      simp only [norm_smul, norm_pow, norm_neg, norm_one, one_pow, one_mul]
+    _ = ‖∑ A : Finset L, ((-1:ℂ)^A.card) • rectOperator (retainedKernel d A) (zeroExtend f)‖ := by
+      rw [alternating_harmonic_action d f hf]
+    _ ≤ ∑ A : Finset L, ‖((-1:ℂ)^A.card) • rectOperator (retainedKernel d A) (zeroExtend f)‖ :=
+      norm_sum_le _ _
+    _ ≤ ∑ A : Finset L, Real.sqrt B * ‖f‖ := by
+      apply Finset.sum_le_sum
+      intro A _
+      simp only [norm_smul, norm_pow, norm_neg, norm_one, one_pow, one_mul]
+      calc _ ≤ ‖rectOperator (retainedKernel d A)‖ * ‖zeroExtend f‖ :=
+              ContinuousLinearMap.le_opNorm _ _
+           _ ≤ Real.sqrt B * ‖zeroExtend f‖ := mul_le_mul_of_nonneg_right (hnorm A) (norm_nonneg _)
+           _ = _ := by rw [norm_zeroExtend]
+    _ = _ := by simp only [Finset.sum_const, Finset.card_univ, Fintype.card_finset, nsmul_eq_mul,
+      Nat.cast_pow, Nat.cast_ofNat]; ring
+
+lemma harmonic_action_zero_one {L : Type*} [Fintype L] (d : ℕ)
+    (hL : Fintype.card L = 1) (f : EuclideanSpace ℂ (L ↪ Card d)) (hf : Harmonic f) :
+    DensityTransfer.applyKernel (UnitaryFinite.actionKernel UnitaryFinite.tupleAction
+      (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))) f = 0 := by
+  classical
+  obtain ⟨i,hi⟩ := Fintype.card_eq_one_iff.mp hL
+  have hz (A : Finset L) : rectOperator (retainedKernel d A) = 0 := by
+    ext v x
+    have hh (y : L → Card d) : ¬ContactEvent d x y := by
+      intro h
+      obtain ⟨j,hj,_⟩ := h i
+      exact hj (hi j).symm
+    simp only [rectOperator_apply, retainedKernel, ite_eq_right (hh _), Complex.ofReal_zero,
+      zero_mul, Finset.sum_const_zero, zero_apply, PiLp.zero_apply]
+  have he := alternating_harmonic_action d f hf
+  simp only [hz, zero_apply, smul_zero, Finset.sum_const_zero, hL, pow_one,
+    neg_smul, one_smul] at he
+  exact neg_eq_zero.mp he.symm
+
+end
+open scoped BigOperators
+
+def MainStatement : Prop :=
+  ∃ Cstar C : ℝ, 0 < Cstar ∧ 0 < C ∧
+    ∀ (d : ℕ), 1 ≤ d → ∀ (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ),
+      let k := μ.card - μ.rowLen 0
+      1 ≤ k → k < 2^d → ∀ reverse : Bool,
+        ‖sweepOperator d μ e reverse‖^2 ≤
+          min 1 ((Cstar * d * k / (2:ℝ)^d) ^ ((k:ℝ)/2)) ∧
+        (k = 1 → sweepOperator d μ e reverse = 0) ∧
+        ‖sweepOperator d μ e reverse‖^2 ≤
+          C^k * (1+(d:ℝ))^(C*k) * ((k:ℝ)/(2:ℝ)^d)^((k:ℝ)/2)
+
+end Thorp.SparseContact
+
+namespace Thorp.SparseContact
+open scoped BigOperators Classical
+
+lemma exists_exact_fixed_unit {α L : Type*} [Fintype α] [Fintype L]
+    (μ : YoungDiagram) (e : α ≃ Specht.Cell μ) (x₀ : L ↪ α)
+    (hL : Fintype.card L = μ.card - μ.rowLen 0) :
+    ∃ w : Specht.hilbertSpace μ,
+      UnitaryFinite.IsFixed UnitaryFinite.tupleAction x₀ (Specht.relabelledUnitary μ e) w ∧ ‖w‖ = 1 := by
+  let K := UnitaryFinite.fixedSpace (V := Specht.hilbertSpace μ) UnitaryFinite.tupleAction x₀ (Specht.relabelledUnitary μ e)
+  have hd : 0 < Module.finrank ℂ (Specht.hilbertSpace μ) := by
+    rw [Specht.finrank_hilbertSpace]
+    exact Module.finrank_pos_iff.mpr (Submodule.nontrivial_iff_ne_bot.mpr (Specht.space_ne_bot μ))
+  have hdim := Specht.relabelled_tuple_dimension μ e x₀
+  rw [hL, Nat.choose_self, mul_one] at hdim
+  have hm : 0 < Module.finrank ℂ K := by
+    by_contra h
+    have hz : Module.finrank ℂ K = 0 := by omega
+    change _ ≤ Module.finrank ℂ K * _ at hdim
+    rw [hz, zero_mul] at hdim
+    omega
+  let B := stdOrthonormalBasis ℂ K
+  let i : Fin (Module.finrank ℂ K) := ⟨0,hm⟩
+  exact ⟨(B i).val, (B i).prop, B.orthonormal.norm_eq_one i⟩
+
+lemma sweep_forward_norm {d : ℕ} (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ) (rev : Bool) :
+    ‖sweepOperator d μ e rev‖ = ‖sweepOperator d μ e false‖ := by
+  cases rev
+  · rfl
+  · simp only [sweepOperator, Bool.false_eq_true, ite_false, ite_true]
+    rw [UnitaryFinite.sampleOperator_inv _ (Specht.relabelledUnitary_unitary μ e),
+      ContinuousLinearMap.adjoint.norm_map]
+
+lemma sweep_norm_le_one (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ) (rev : Bool) :
+    ‖sweepOperator d μ e rev‖ ≤ 1 :=
+  UnitaryFinite.sampleOperator_norm_le _ (Specht.relabelledUnitary_unitary μ e) _
+
+section Generic
+variable {L V : Type*} [Fintype L] [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+  [FiniteDimensional ℂ V]
+
+lemma sample_sweep_bound (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V)
+    [Representation.IsIrreducible ρ] (hρ : UnitaryFinite.IsUnitary ρ)
+    (x₀ : L ↪ Card d) (w : V) (hw : UnitaryFinite.IsFixed UnitaryFinite.tupleAction x₀ ρ w)
+    (hn : ‖w‖ = 1)
+    (hh : ∀ v, Harmonic (UnitaryFinite.normalizedCoefficient UnitaryFinite.tupleAction x₀
+      (UnitaryFinite.tupleAction_transitive x₀) ρ w v))
+    (hδ : 0 < 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)
+    (hδ1 : 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d ≤ 1) :
+    ‖UnitaryFinite.sampleOperator ρ (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))‖ ≤
+      (2:ℝ)^(Fintype.card L) *
+        Real.sqrt ((2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
+          Real.exp (Real.exp 1*(Fintype.card L))) := by
+  rw [← ContinuousLinearMap.adjoint.norm_map (UnitaryFinite.sampleOperator ρ
+    (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω))),
+    ← UnitaryFinite.sampleOperator_inv ρ hρ]
+  apply ContinuousLinearMap.opNorm_le_bound _ (by positivity)
+  intro v
+  rw [UnitaryFinite.sampleOperator_inv_apply]
+  have ha := harmonic_action_bound d _ (hh v) hδ hδ1
+  rw [UnitaryFinite.coefficient_random UnitaryFinite.tupleAction x₀
+    (UnitaryFinite.tupleAction_transitive x₀) ρ hρ _ w hw,
+    UnitaryFinite.normalizedCoefficient_norm _ _ _ ρ hρ w hw hn,
+    UnitaryFinite.normalizedCoefficient_norm _ _ _ ρ hρ w hw hn] at ha
+  exact ha
+
+lemma sample_sweep_zero_one (d : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card d)) V)
+    [Representation.IsIrreducible ρ] (hρ : UnitaryFinite.IsUnitary ρ)
+    (x₀ : L ↪ Card d) (w : V) (hw : UnitaryFinite.IsFixed UnitaryFinite.tupleAction x₀ ρ w)
+    (hn : ‖w‖ = 1)
+    (hh : ∀ v, Harmonic (UnitaryFinite.normalizedCoefficient UnitaryFinite.tupleAction x₀
+      (UnitaryFinite.tupleAction_transitive x₀) ρ w v))
+    (hL : Fintype.card L = 1) :
+    UnitaryFinite.sampleOperator ρ (fun ω : SwitchIndex d → Bool => butterflyPerm d (decodeButterfly d ω)) = 0 := by
+  apply ContinuousLinearMap.adjoint.injective
+  rw [map_zero, ← UnitaryFinite.sampleOperator_inv ρ hρ]
+  apply ContinuousLinearMap.ext
+  intro v
+  rw [UnitaryFinite.sampleOperator_inv_apply]
+  have ha := harmonic_action_zero_one d hL _ (hh v)
+  have he := congrArg norm ha
+  rw [UnitaryFinite.coefficient_random UnitaryFinite.tupleAction x₀
+    (UnitaryFinite.tupleAction_transitive x₀) ρ hρ _ w hw,
+    UnitaryFinite.normalizedCoefficient_norm _ _ _ ρ hρ w hw hn, norm_zero] at he
+  exact norm_eq_zero.mp he
+end Generic
+
+lemma sweep_norm_bound (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ)
+    (hδ : 0 < 2*(d:ℝ)*(μ.card - μ.rowLen 0 : ℕ)/(2:ℝ)^d)
+    (hδ1 : 2*(d:ℝ)*(μ.card - μ.rowLen 0 : ℕ)/(2:ℝ)^d ≤ 1) (rev : Bool) :
+    ‖sweepOperator d μ e rev‖ ≤ (2:ℝ)^(μ.card - μ.rowLen 0) *
+      Real.sqrt ((2*(d:ℝ)*(μ.card - μ.rowLen 0 : ℕ)/(2:ℝ)^d)^(((μ.card - μ.rowLen 0 : ℕ):ℝ)/2) *
+        Real.exp (Real.exp 1*(μ.card - μ.rowLen 0 : ℕ))) := by
+  classical
+  let x₀ : Specht.TailCell μ ↪ Card d := (Function.Embedding.subtype _).trans e.symm.toEmbedding
+  obtain ⟨w,hw,hn⟩ := exists_exact_fixed_unit μ e x₀ (Specht.tail_card μ)
+  let := Specht.relabelledUnitary_irreducible μ e
+  rw [sweep_forward_norm μ e rev]
+  have hh (v : Specht.hilbertSpace μ) : Harmonic (UnitaryFinite.normalizedCoefficient
+      (V := Specht.hilbertSpace μ) UnitaryFinite.tupleAction x₀
+      (UnitaryFinite.tupleAction_transitive x₀) (Specht.relabelledUnitary μ e) w v) :=
+    normalizedCoefficient_harmonic μ e (Specht.tail_card μ) x₀ w hw v
+  have he := sample_sweep_bound (V := Specht.hilbertSpace μ) (L := Specht.TailCell μ) d (Specht.relabelledUnitary μ e)
+  have he := he (Specht.relabelledUnitary_unitary μ e) x₀ w hw hn hh
+  have he := he (by simpa only [Specht.tail_card] using hδ) (by simpa only [Specht.tail_card] using hδ1)
+  simpa only [sweepOperator, Bool.false_eq_true, ite_false, Specht.tail_card] using he
+
+lemma sweep_zero_one (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ)
+    (hk : μ.card - μ.rowLen 0 = 1) (rev : Bool) : sweepOperator d μ e rev = 0 := by
+  classical
+  let x₀ : Specht.TailCell μ ↪ Card d := (Function.Embedding.subtype _).trans e.symm.toEmbedding
+  obtain ⟨w,hw,hn⟩ := exists_exact_fixed_unit μ e x₀ (Specht.tail_card μ)
+  let := Specht.relabelledUnitary_irreducible μ e
+  have hh (v : Specht.hilbertSpace μ) : Harmonic (UnitaryFinite.normalizedCoefficient
+      (V := Specht.hilbertSpace μ) UnitaryFinite.tupleAction x₀
+      (UnitaryFinite.tupleAction_transitive x₀) (Specht.relabelledUnitary μ e) w v) :=
+    normalizedCoefficient_harmonic μ e (Specht.tail_card μ) x₀ w hw v
+  have he := sample_sweep_zero_one (V := Specht.hilbertSpace μ) (L := Specht.TailCell μ) d (Specht.relabelledUnitary μ e)
+  have he := he (Specht.relabelledUnitary_unitary μ e) x₀ w hw hn hh
+  have he := he ((Specht.tail_card μ).trans hk)
+  have hz : sweepOperator d μ e false = 0 := by
+    simpa only [sweepOperator, Bool.false_eq_true, ite_false] using he
+  apply (norm_eq_zero (a := sweepOperator d μ e rev)).mp
+  rw [sweep_forward_norm μ e rev, hz]
+  exact @norm_zero (Specht.hilbertSpace μ →L[ℂ] Specht.hilbertSpace μ) _
+
+end Thorp.SparseContact
+
+namespace Thorp.SparseContact
+
+noncomputable def contactConstant : ℝ := 32 * Real.exp (2 * Real.exp 1)
+
+lemma contactConstant_gt_one : 1 < contactConstant := by
+  have h : 1 ≤ Real.exp (2 * Real.exp 1) := Real.one_le_exp (by positivity)
+  unfold contactConstant
+  linarith
+
+lemma contactConstant_gt_two : 2 < contactConstant := by
+  have h : 1 ≤ Real.exp (2 * Real.exp 1) := Real.one_le_exp (by positivity)
+  unfold contactConstant
+  linarith
+
+lemma inflated_contact_bound {δ : ℝ} (hδ : 0 ≤ δ) (k : ℕ) :
+    ((2:ℝ)^k * Real.sqrt (δ^((k:ℝ)/2) * Real.exp (Real.exp 1*k)))^2 =
+      ((contactConstant/2)*δ)^((k:ℝ)/2) := by
+  have hC : 0 ≤ contactConstant/2 := by have := contactConstant_gt_one; positivity
+  rw [mul_pow, Real.sq_sqrt (by positivity), Real.mul_rpow hC hδ]
+  have hc : contactConstant/2 = Real.exp (2*(Real.log 4+Real.exp 1)) := by
+    unfold contactConstant
+    rw [show 2*(Real.log 4+Real.exp 1) = Real.log 4*2+2*Real.exp 1 by ring,
+      Real.exp_add, Real.exp_mul (Real.log 4) 2, Real.exp_log (by norm_num : (0:ℝ)<4)]
+    norm_num
+    ring
+  rw [hc, ←Real.exp_mul]
+  have hpow : ((2:ℝ)^k)^2 = Real.exp (Real.log 4 * k) := by
+    rw [← pow_mul, mul_comm k 2, pow_mul]
+    norm_num only [pow_two]
+    rw [Real.exp_mul (Real.log 4) k, Real.exp_log (by norm_num : (0:ℝ)<4), Real.rpow_natCast]
+  rw [hpow]
+  calc
+    _ = (Real.exp (Real.log 4*k)*Real.exp (Real.exp 1*k))*δ^((k:ℝ)/2) := by ring
+    _ = _ := by rw [←Real.exp_add]; congr 2; ring
+
+lemma contact_crude_arithmetic {C d q : ℝ} (hC : 1 ≤ C) (hd : 1 ≤ d) (hq : 0 ≤ q) (k : ℕ) :
+    (C*d*q)^((k:ℝ)/2) ≤ C^k*(1+d)^(C*k)*q^((k:ℝ)/2) := by
+  rw [Real.mul_rpow (by positivity) hq, Real.mul_rpow (by positivity) (by positivity)]
+  apply mul_le_mul_of_nonneg_right _ (by positivity)
+  apply mul_le_mul
+  · rw [← Real.rpow_natCast]
+    exact Real.rpow_le_rpow_of_exponent_le hC (by linarith [Nat.cast_nonneg (α := ℝ) k])
+  · calc
+      d^((k:ℝ)/2) ≤ (1+d)^((k:ℝ)/2) := Real.rpow_le_rpow (by positivity) (by linarith) (by positivity)
+      _ ≤ (1+d)^(C*k) := Real.rpow_le_rpow_of_exponent_le (by linarith) (by nlinarith [Nat.cast_nonneg (α := ℝ) k])
+  · positivity
+  · positivity
+
+
+lemma sweep_sharp_bound (d : ℕ) (hd : 1 ≤ d) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ)
+    (hk : 1 ≤ μ.card - μ.rowLen 0) (rev : Bool) :
+    ‖sweepOperator d μ e rev‖^2 ≤ min 1
+      ((contactConstant*d*(μ.card - μ.rowLen 0 : ℕ)/(2:ℝ)^d)^(((μ.card - μ.rowLen 0 : ℕ):ℝ)/2)) := by
+  let k := μ.card - μ.rowLen 0
+  have hk' : (0:ℝ) < k := by exact_mod_cast hk
+  have hd' : (0:ℝ) < d := by exact_mod_cast hd
+  have hδ : 0 < 2*(d:ℝ)*k/(2:ℝ)^d := by positivity
+  have hOne : ‖sweepOperator d μ e rev‖^2 ≤ 1 := by
+    have h := sweep_norm_le_one d μ e rev
+    have h0 := norm_nonneg (sweepOperator d μ e rev)
+    nlinarith
+  refine le_min hOne ?_
+  have he : contactConstant * d * (k:ℝ)/(2:ℝ)^d =
+      (contactConstant/2)*(2*(d:ℝ)*k/(2:ℝ)^d) := by ring
+  change _ ≤ (contactConstant * d * (k:ℝ)/(2:ℝ)^d)^((k:ℝ)/2)
+  rw [he]
+  by_cases hs : 2*(d:ℝ)*k/(2:ℝ)^d ≤ 1
+  · have h := sweep_norm_bound d μ e hδ hs rev
+    have hb : 0 ≤ (2:ℝ)^k * Real.sqrt
+        ((2*(d:ℝ)*k/(2:ℝ)^d)^((k:ℝ)/2)*Real.exp (Real.exp 1*k)) := by positivity
+    calc
+      _ ≤ ((2:ℝ)^k * Real.sqrt
+          ((2*(d:ℝ)*k/(2:ℝ)^d)^((k:ℝ)/2)*Real.exp (Real.exp 1*k)))^2 :=
+        (sq_le_sq₀ (norm_nonneg (sweepOperator d μ e rev)) hb).mpr h
+      _ = _ := inflated_contact_bound hδ.le k
+  · apply hOne.trans
+    apply Real.one_le_rpow
+    · have hc : 1 ≤ contactConstant/2 := by have := contactConstant_gt_two; linarith
+      exact one_le_mul_of_one_le_of_one_le hc (le_of_lt (lt_of_not_ge hs))
+    · positivity
+
+theorem uniform_sparse_contact : MainStatement := by
+  refine ⟨contactConstant, contactConstant, (lt_trans (by norm_num) contactConstant_gt_one),
+    (lt_trans (by norm_num) contactConstant_gt_one), ?_⟩
+  intro d hd μ e k hk _ rev
+  have hs := sweep_sharp_bound d hd μ e hk rev
+  refine ⟨hs, ?_, ?_⟩
+  · intro hk1
+    exact sweep_zero_one d μ e hk1 rev
+  · apply (hs.trans (min_le_right _ _)).trans
+    change (contactConstant * d * (k:ℝ)/(2:ℝ)^d)^((k:ℝ)/2) ≤ _
+    rw [mul_div_assoc]
+    exact contact_crude_arithmetic contactConstant_gt_one.le
+      (by exact_mod_cast hd) (by positivity) k
+
+end Thorp.SparseContact
+
+namespace Thorp.DimensionEstimates
+open scoped BigOperators
+
+lemma log_factorial_upper_binary (n : ℕ) (hn : 0<n) :
+    Real.log (n.factorial:ℝ) ≤ (n:ℝ)*Real.log n-Real.log 2*((n:ℝ)-1) := by
+  induction n, hn using Nat.le_induction with
+  | base => norm_num
+  | succ n hn ih =>
+    have hn0 : (0:ℝ)<n := by exact_mod_cast hn
+    have hb : (2:ℝ) ≤ (1+(n:ℝ)⁻¹)^n := by
+      have h := one_add_mul_le_pow (by have := inv_nonneg.mpr hn0.le; linarith : (-2:ℝ)≤(n:ℝ)⁻¹) n
+      norm_num [mul_inv_cancel₀ hn0.ne'] at h ⊢
+      exact h
+    have hl := Real.log_le_log (by norm_num : (0:ℝ)<2) hb
+    rw [Real.log_pow] at hl
+    have hs : Real.log (1+(n:ℝ)⁻¹)=Real.log ((n:ℝ)+1)-Real.log n := by
+      rw [←Real.log_div (by positivity) hn0.ne']
+      congr 1
+      field_simp
+    rw [hs] at hl
+    rw [Nat.factorial_succ,Nat.cast_mul,Real.log_mul (by positivity) (by positivity),Nat.cast_add,Nat.cast_one]
+    nlinarith
+
+lemma log_factorial_lower (n : ℕ) (hn : 0<n) :
+    (n:ℝ)*Real.log n-n+1 ≤ Real.log (n.factorial:ℝ) := by
+  induction n, hn using Nat.le_induction with
+  | base => norm_num
+  | succ n hn ih =>
+    have hn0 : (0:ℝ)<n := by exact_mod_cast hn
+    have hl := Real.log_le_sub_one_of_pos (show 0<((n:ℝ)+1)/n by positivity)
+    rw [Real.log_div (by positivity) hn0.ne'] at hl
+    have hh := mul_le_mul_of_nonneg_left hl hn0.le
+    have he : (n:ℝ)*(((n:ℝ)+1)/n-1)=1 := by field_simp; ring
+    rw [he] at hh
+    rw [Nat.factorial_succ,Nat.cast_mul,Real.log_mul (by positivity) (by positivity),Nat.cast_add,Nat.cast_one]
+    nlinarith
+
+lemma log_factorial_upper (n : ℕ) (hn : 0<n) :
+    Real.log (n.factorial:ℝ) ≤ (n:ℝ)*Real.log n-n+Real.log n+1 := by
+  induction n, hn using Nat.le_induction with
+  | base => norm_num
+  | succ n hn ih =>
+    have hn0 : (0:ℝ)<n := by exact_mod_cast hn
+    have hl := Real.one_sub_inv_le_log_of_pos (show 0<((n:ℝ)+1)/n by positivity)
+    rw [Real.log_div (by positivity) hn0.ne'] at hl
+    have hh := mul_le_mul_of_nonneg_left hl (show 0≤(n:ℝ)+1 by positivity)
+    have he : ((n:ℝ)+1)*(1-(((n:ℝ)+1)/n)⁻¹)=1 := by field_simp; ring
+    rw [he] at hh
+    rw [Nat.factorial_succ,Nat.cast_mul,Real.log_mul (by positivity) (by positivity),Nat.cast_add,Nat.cast_one]
+    nlinarith
+
+variable {X I J : Type*} [Fintype X] [Fintype I] [Fintype J]
+open scoped Classical
+
+noncomputable def fiberWeight (f : X → I) (i : I) : ℝ := Fintype.card {x // f x=i}
+
+omit [Fintype I] in
+lemma fiberWeight_nonneg (f : X → I) (i : I) : 0≤fiberWeight f i := Nat.cast_nonneg _
+
+omit [Fintype I] in
+lemma fiberWeight_pos (f : X → I) (x : X) : 0<fiberWeight f (f x) := by
+  let : Nonempty {y // f y=f x} := ⟨⟨x,rfl⟩⟩
+  exact Nat.cast_pos.mpr Fintype.card_pos
+
+lemma fiber_sum (f : X → I) (u : I → ℝ) :
+    ∑ x, u (f x)=∑ i, fiberWeight f i*u i := by
+  rw [←Equiv.sum_comp (Equiv.sigmaFiberEquiv f)]
+  simp only [Equiv.sigmaFiberEquiv_apply,Fintype.sum_sigma]
+  apply Finset.sum_congr rfl
+  intro i _
+  calc
+    (∑ x : {x // f x=i}, u (f x)) = ∑ _x : {x // f x=i}, u i :=
+      Finset.sum_congr rfl (fun x _ => congrArg u x.property)
+    _ = _ := by simp [fiberWeight]
+
+lemma sum_fiberWeight (f : X → I) : ∑ i, fiberWeight f i=Fintype.card X := by
+  have h := fiber_sum f (fun _ => 1)
+  simpa using h.symm
+
+theorem incidence_entropy (f : X → I) (g : X → J)
+    (hinj : Function.Injective (fun x => (f x,g x))) :
+    (∑ i, fiberWeight f i*Real.log (fiberWeight f i))+
+      (∑ j, fiberWeight g j*Real.log (fiberWeight g j)) ≤
+        (Fintype.card X:ℝ)*Real.log (Fintype.card X) := by
+  by_cases hX : Nonempty X
+  · let : Nonempty X := hX
+    have hN : (0:ℝ)<Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
+    have hprod : ∑ x, fiberWeight f (f x)*fiberWeight g (g x) ≤ (Fintype.card X:ℝ)^2 := by
+      calc
+        _ ≤ ∑ p : I × J, fiberWeight f p.1*fiberWeight g p.2 := by
+          calc
+            _ = ∑ p ∈ Finset.univ.image (fun x => (f x,g x)), fiberWeight f p.1*fiberWeight g p.2 := by
+              rw [Finset.sum_image (fun x _ y _ h => hinj h)]
+            _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _)
+              (fun p _ _ => mul_nonneg (fiberWeight_nonneg f p.1) (fiberWeight_nonneg g p.2))
+        _ = _ := by
+          rw [Fintype.sum_prod_type]
+          simp_rw [←Finset.mul_sum]
+          rw [←Finset.sum_mul,sum_fiberWeight,sum_fiberWeight,pow_two]
+    have hdiv : (∑ x, fiberWeight f (f x)*fiberWeight g (g x))/(Fintype.card X:ℝ) ≤ Fintype.card X := by
+      apply (div_le_iff₀ hN).mpr
+      simpa only [pow_two] using hprod
+    have hl (x : X) := Real.log_le_sub_one_of_pos
+      (show 0<fiberWeight f (f x)*fiberWeight g (g x)/(Fintype.card X:ℝ) from
+        div_pos (mul_pos (fiberWeight_pos f x) (fiberWeight_pos g x)) hN)
+    have hs := Finset.sum_le_sum (s := Finset.univ) (fun x _ => hl x)
+    simp_rw [Real.log_div (mul_pos (fiberWeight_pos f _) (fiberWeight_pos g _)).ne' hN.ne',
+      Real.log_mul (fiberWeight_pos f _).ne' (fiberWeight_pos g _).ne'] at hs
+    rw [Finset.sum_sub_distrib,Finset.sum_add_distrib,
+      fiber_sum f (fun i => Real.log (fiberWeight f i)),
+      fiber_sum g (fun j => Real.log (fiberWeight g j)),
+      Finset.sum_sub_distrib,←Finset.sum_div] at hs
+    simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,mul_one] at hs
+    linarith
+  · let : IsEmpty X := not_nonempty_iff.mp hX
+    have hf (i : I) : fiberWeight f i=0 := by simp [fiberWeight]
+    have hg (j : J) : fiberWeight g j=0 := by simp [fiberWeight]
+    simp [hf,hg]
+
+end Thorp.DimensionEstimates
+
+namespace Thorp.DimensionEstimates
+open scoped BigOperators Classical
+
+lemma log_factorial_upper_binary_all (n : ℕ) :
+    Real.log (n.factorial:ℝ) ≤ (n:ℝ)*Real.log n-Real.log 2*((n:ℝ)-1) := by
+  by_cases hn : n=0
+  · subst n
+    simpa using (Real.log_nonneg (by norm_num : (1:ℝ)≤2))
+  · exact log_factorial_upper_binary n (Nat.pos_of_ne_zero hn)
+
+lemma log_fiber_factorials {X I : Type*} [Fintype X] [Fintype I] (f : X → I) :
+    Real.log ((∏ i, (Fintype.card {x // f x=i}).factorial : ℕ):ℝ) ≤
+      ∑ i, fiberWeight f i*Real.log (fiberWeight f i) -
+        Real.log 2*((Fintype.card X:ℝ)-Fintype.card I) := by
+  rw [Nat.cast_prod,Real.log_prod (fun _ _ => by positivity)]
+  have h := Finset.sum_le_sum (s := Finset.univ) (fun i _ =>
+    log_factorial_upper_binary_all (Fintype.card {x // f x=i}))
+  change ∑ i, Real.log ((Fintype.card {x // f x=i}).factorial:ℝ) ≤
+    ∑ i, (fiberWeight f i*Real.log (fiberWeight f i)-Real.log 2*(fiberWeight f i-1)) at h
+  rw [Finset.sum_sub_distrib,←Finset.mul_sum,Finset.sum_sub_distrib,sum_fiberWeight] at h
+  simpa using h
+
+theorem dimension_log_from_incidence {X I J : Type*} [Fintype X] [Fintype I] [Fintype J]
+    [Nonempty X] (f : X → I) (g : X → J)
+    (hinj : Function.Injective (fun x => (f x,g x))) (D : ℕ)
+    (hD : (Fintype.card X).factorial ≤ D *
+      (∏ i, (Fintype.card {x // f x=i}).factorial) *
+      (∏ j, (Fintype.card {x // g x=j}).factorial)) :
+    ((2*Real.log 2-1)*(Fintype.card X:ℝ)-
+      Real.log 2*((Fintype.card I:ℝ)+Fintype.card J)+1) ≤ Real.log D := by
+  have hd : 0<D := by
+    by_contra h
+    have : D=0 := by omega
+    simp [this] at hD
+    exact (Nat.factorial_pos _).ne' hD
+  have hf := log_fiber_factorials f
+  have hg := log_fiber_factorials g
+  have hi := incidence_entropy f g hinj
+  have hlo := log_factorial_lower (Fintype.card X) Fintype.card_pos
+  have hlog := Real.log_le_log (by positivity : (0:ℝ)<(Fintype.card X).factorial)
+    (show ((Fintype.card X).factorial:ℝ) ≤ D *
+      (∏ i, (Fintype.card {x // f x=i}).factorial) *
+      (∏ j, (Fintype.card {x // g x=j}).factorial) by exact_mod_cast hD)
+  rw [Real.log_mul (by positivity) (by positivity),
+    Real.log_mul (by positivity) (by positivity)] at hlog
+  nlinarith
+
+
+lemma scaled_binEntropy (N j : ℝ) (hN : 0<N) (hj : 0<j) (hjN : j<N) :
+    N*Real.binEntropy (j/N)=N*Real.log N-j*Real.log j-(N-j)*Real.log (N-j) := by
+  have hsub : 0<N-j := sub_pos.mpr hjN
+  have he : 1-j/N=(N-j)/N := by field_simp
+  unfold Real.binEntropy
+  rw [he,Real.log_inv,Real.log_inv,Real.log_div hj.ne' hN.ne',
+    Real.log_div hsub.ne' hN.ne']
+  field_simp
+  ring
+
+lemma log_choose_lower_entropy (N j : ℕ) (hj : 0<j) (hjN : j<N) :
+    (N:ℝ)*Real.binEntropy ((j:ℝ)/N)-2*Real.log N-1 ≤ Real.log (N.choose j:ℝ) := by
+  have hN : 0<N := lt_trans hj hjN
+  have ht : 0<N-j := Nat.sub_pos_of_lt hjN
+  have hj0 : (0:ℝ)<j := by exact_mod_cast hj
+  have hN0 : (0:ℝ)<N := by exact_mod_cast hN
+  have hjN0 : (j:ℝ)<N := by exact_mod_cast hjN
+  have hchoose : (0:ℝ)<N.choose j := by exact_mod_cast Nat.choose_pos hjN.le
+  have hfac := congrArg (fun n : ℕ => Real.log (n:ℝ)) (Nat.choose_mul_factorial_mul_factorial hjN.le)
+  simp only [Nat.cast_mul] at hfac
+  rw [Real.log_mul (mul_pos hchoose (by positivity)).ne' (by positivity),
+    Real.log_mul hchoose.ne' (by positivity)] at hfac
+  have hlo := log_factorial_lower N hN
+  have hji := log_factorial_upper j hj
+  have hti := log_factorial_upper (N-j) ht
+  have hjlog := Real.log_le_log hj0 hjN0.le
+  have htlog := Real.log_le_log (by exact_mod_cast ht : (0:ℝ)<(N-j:ℕ))
+    (by exact_mod_cast Nat.sub_le N j : ((N-j:ℕ):ℝ)≤N)
+  rw [Nat.cast_sub hjN.le] at hti htlog
+  rw [scaled_binEntropy N j hN0 hj0 hjN0]
+  linarith
+
+lemma binEntropy_three_quarters :
+    Real.binEntropy (3/4:ℝ)=2*Real.log 2-(3/4:ℝ)*Real.log 3 := by
+  have h4 : Real.log (4:ℝ)=2*Real.log 2 := by
+    rw [show (4:ℝ)=2^2 by norm_num,Real.log_pow]; norm_num
+  unfold Real.binEntropy
+  norm_num only [show (1-(3/4:ℝ))=1/4 by norm_num]
+  rw [Real.log_div (by norm_num) (by norm_num),h4]
+  ring
+
+noncomputable def entropyGap : ℝ := Real.log (32/27:ℝ)/3
+
+lemma entropyGap_pos : 0<entropyGap := by
+  unfold entropyGap
+  exact div_pos (Real.log_pos (by norm_num)) (by norm_num)
+
+lemma entropyGap_eq : entropyGap=(5*Real.log 2-3*Real.log 3)/3 := by
+  unfold entropyGap
+  rw [Real.log_div (by norm_num) (by norm_num),show (32:ℝ)=2^5 by norm_num,
+    show (27:ℝ)=3^3 by norm_num,Real.log_pow,Real.log_pow]
+  norm_num
+
+lemma binEntropy_tail_lower (q : ℝ) (hq : 0≤q) (hq' : q≤3/4) :
+    q*Real.log 2+entropyGap*q ≤ Real.binEntropy q := by
+  have hc := Real.strictConcave_binEntropy.concaveOn.2
+    (show (0:ℝ)∈Set.Icc 0 1 by constructor <;> norm_num)
+    (show (3/4:ℝ)∈Set.Icc 0 1 by constructor <;> norm_num)
+    (show 0≤1-4*q/3 by linarith) (show 0≤4*q/3 by positivity)
+    (show (1-4*q/3)+(4*q/3)=1 by ring)
+  simp only [Real.binEntropy_zero,binEntropy_three_quarters,smul_eq_mul,mul_zero,zero_add] at hc
+  have he : 4*q/3*(3/4:ℝ)=q := by ring
+  rw [he] at hc
+  rw [entropyGap_eq]
+  linarith
+
+lemma log_dimension_from_binomial (N j D : ℕ) (hj : 0<j) (hjN : j<N)
+    (hfrac : (j:ℝ)≤(3/4:ℝ)*N) (hD : N.choose j≤D*2^j) :
+    entropyGap*(j:ℝ)-2*Real.log N-1 ≤ Real.log D := by
+  have hpos : 0<N.choose j := Nat.choose_pos hjN.le
+  have hd : 0<D := by nlinarith [Nat.zero_le (2^j)]
+  have hN : (0:ℝ)<N := by exact_mod_cast lt_trans hj hjN
+  have hm := Real.log_le_log (by exact_mod_cast hpos : (0:ℝ)<N.choose j)
+    (show (N.choose j:ℝ)≤D*2^j by exact_mod_cast hD)
+  rw [Real.log_mul (by positivity) (by positivity),Real.log_pow] at hm
+  have he := binEntropy_tail_lower ((j:ℝ)/N) (by positivity)
+    ((div_le_iff₀ hN).mpr hfrac)
+  have hel := mul_le_mul_of_nonneg_left he hN.le
+  have hbin := log_choose_lower_entropy N j hj hjN
+  have heq : (N:ℝ)*((j:ℝ)/N*Real.log 2+entropyGap*((j:ℝ)/N))=
+      (j:ℝ)*Real.log 2+entropyGap*j := by field_simp
+  rw [heq] at hel
+  linarith
+
+lemma balanced_dimension_coefficient_pos : 0<(3/2:ℝ)*Real.log 2-1 := by
+  linarith [Real.log_two_gt_d9]
+
+
+lemma log_choose_upper (N j : ℕ) (hN : 0<N) (hj : j≤N) :
+    Real.log (N.choose j:ℝ) ≤ (j:ℝ)*Real.log N-Real.log (j.factorial:ℝ) := by
+  have hh := Real.log_le_log (by exact_mod_cast Nat.choose_pos hj : (0:ℝ)<N.choose j)
+    (Nat.choose_le_pow_div j N : (N.choose j:ℝ)≤(N:ℝ)^j/(j.factorial:ℝ))
+  rw [Real.log_div (by positivity : (N:ℝ)^j≠0) (by positivity : (j.factorial:ℝ)≠0),Real.log_pow] at hh
+  exact hh
+
+lemma log_choose_lower (N j : ℕ) (hj : j≤N) :
+    (j:ℝ)*Real.log (N+1-j:ℕ)-Real.log (j.factorial:ℝ) ≤ Real.log (N.choose j:ℝ) := by
+  have hp : 0<N+1-j := by omega
+  have hh := Real.log_le_log (by positivity : (0:ℝ)<((N+1-j:ℕ):ℝ)^j/(j.factorial:ℝ))
+    (Nat.pow_le_choose j N : (((N+1-j:ℕ):ℝ)^j)/(j.factorial:ℝ)≤N.choose j)
+  rw [Real.log_div (by positivity : (((N+1-j:ℕ):ℝ)^j)≠0) (by positivity : (j.factorial:ℝ)≠0),Real.log_pow] at hh
+  exact hh
+
+lemma footprint_log_bound (N k j D m : ℕ) (hN : 0<N) (hk : j≤k) (hj : j≤N)
+    (hD : 0<D) (hfoot : D * k.choose j ≤ m * N.choose j) :
+    Real.log D-(j:ℝ)*(Real.log N-Real.log (k+1-j:ℕ)) ≤ Real.log m := by
+  have hkpos := Nat.choose_pos hk
+  have hprod : 0<D*k.choose j := Nat.mul_pos hD hkpos
+  have hm : 0 < m := by
+    by_contra h
+    have he : m=0 := by omega
+    simp only [he,zero_mul] at hfoot
+    omega
+  have hh := Real.log_le_log (by exact_mod_cast hprod : (0:ℝ)<D*k.choose j)
+    (show (D:ℝ)*k.choose j≤ m*N.choose j by exact_mod_cast hfoot)
+  rw [Real.log_mul (by positivity) (by exact_mod_cast hkpos.ne'),
+    Real.log_mul (by positivity) (by exact_mod_cast (Nat.choose_pos hj).ne')] at hh
+  have hlo := log_choose_lower k j hk
+  have hup := log_choose_upper N j hN hj
+  nlinarith
+
+lemma log_factorial_le_mul_log (j : ℕ) (_hj : 0<j) :
+    Real.log (j.factorial:ℝ)≤(j:ℝ)*Real.log j := by
+  have h := Real.log_le_log (by positivity : (0:ℝ)<j.factorial)
+    (show (j.factorial:ℝ)≤(j:ℝ)^j by exact_mod_cast Nat.factorial_le_pow j)
+  simpa only [Real.log_pow] using h
+
+lemma near_row_log_dimension (N j D : ℕ) (hj : 0<j) (hsmall : 2*j≤N)
+    (hD : N.choose j≤D*2^j) :
+    (j:ℝ)*(Real.log ((N:ℝ)/j)-2*Real.log 2) ≤ Real.log D := by
+  have hjN : j≤N := by omega
+  have hN : 0<N := by omega
+  have hNp : (0:ℝ)<N := by exact_mod_cast hN
+  have hjp : (0:ℝ)<j := by exact_mod_cast hj
+  have hn : 0<N+1-j := by omega
+  have hnreal : (N:ℝ)/2≤(N+1-j:ℕ) := by
+    rw [Nat.cast_sub (by omega : j≤N+1),Nat.cast_add,Nat.cast_one]
+    have hc : 2*(j:ℝ)≤N := by exact_mod_cast hsmall
+    linarith
+  have hDpos : 0<D := by
+    have hc := Nat.choose_pos hjN
+    by_contra h
+    have he : D=0 := by omega
+    simp only [he,zero_mul] at hD
+    omega
+  have hm := Real.log_le_log (by exact_mod_cast Nat.choose_pos hjN : (0:ℝ)<N.choose j)
+    (show (N.choose j:ℝ)≤D*2^j by exact_mod_cast hD)
+  rw [Real.log_mul (by positivity) (by positivity),Real.log_pow] at hm
+  have hl := Real.log_le_log (by positivity : (0:ℝ)<(N:ℝ)/2) hnreal
+  rw [Real.log_div hNp.ne' (by norm_num)] at hl
+  have hh := log_choose_lower N j hjN
+  have hf := log_factorial_le_mul_log j hj
+  rw [Real.log_div hNp.ne' hjp.ne']
+  nlinarith
+
+open Filter
+
+noncomputable def nearPower : ℝ := 1/2048
+noncomputable def nearFraction : ℝ := 1/4096
+noncomputable def tailPower : ℝ := 1535/1048576
+
+noncomputable def chosenTuple (N j : ℕ) : ℕ :=
+  ⌈(j:ℝ)*((N:ℝ)/j)^nearPower⌉₊
+
+lemma chosenTuple_bounds (N j : ℕ) (hj : 0<j) (hx : 1≤(N:ℝ)/j)
+    (hhalf : ((N:ℝ)/j)^(nearPower-1)≤1/2) :
+    j≤chosenTuple N j ∧ chosenTuple N j≤N ∧
+      (chosenTuple N j:ℝ)≤2*j*((N:ℝ)/j)^nearPower := by
+  let x : ℝ := (N:ℝ)/j
+  have hjp : (0:ℝ)<j := by exact_mod_cast hj
+  have hj1 : (1:ℝ)≤j := by exact_mod_cast hj
+  have hxp : 0<x := lt_of_lt_of_le zero_lt_one hx
+  have hpow : 1≤x^nearPower := Real.one_le_rpow hx (by norm_num [nearPower])
+  have hceil : (j:ℝ)*x^nearPower≤chosenTuple N j := Nat.le_ceil _
+  have hceil' : (chosenTuple N j:ℝ)<(j:ℝ)*x^nearPower+1 := Nat.ceil_lt_add_one (by positivity)
+  have he : (j:ℝ)*x=N := by dsimp [x]; field_simp
+  have hh : x^nearPower≤x/2 := by
+    rw [Real.rpow_sub hxp,Real.rpow_one] at hhalf
+    nlinarith [(div_le_iff₀ hxp).mp hhalf]
+  constructor
+  · exact_mod_cast (le_mul_of_one_le_right hjp.le hpow).trans hceil
+  constructor
+  · apply Nat.ceil_le.mpr
+    change (j:ℝ)*x^nearPower≤N
+    nlinarith
+  · change (chosenTuple N j:ℝ)≤2*(j:ℝ)*x^nearPower
+    nlinarith
+
+lemma chosenTuple_tail_lower (N j : ℕ) (hj : 0<j)
+    (hx : 2≤((N:ℝ)/j)^nearPower) :
+    (j:ℝ)*((N:ℝ)/j)^nearPower/2 ≤ (chosenTuple N j+1-j:ℕ) := by
+  have hjp : (0:ℝ)<j := by exact_mod_cast hj
+  have hc : (j:ℝ)*((N:ℝ)/j)^nearPower≤chosenTuple N j := Nat.le_ceil _
+  have hjk : j≤chosenTuple N j := by exact_mod_cast (by nlinarith : (j:ℝ)≤chosenTuple N j)
+  rw [Nat.cast_sub (by omega : j≤chosenTuple N j+1),Nat.cast_add,Nat.cast_one]
+  nlinarith
+
+lemma chosenTuple_log_footprint (N j D m : ℕ) (hj : 0<j)
+    (hN : 2*j≤N) (hD : N.choose j≤D*2^j)
+    (hx : 2≤((N:ℝ)/j)^nearPower)
+    (hlog : 8*Real.log 2/nearPower≤Real.log ((N:ℝ)/j))
+    (hfoot : D * (chosenTuple N j).choose j ≤ m * N.choose j) :
+    nearFraction*Real.log D≤Real.log m := by
+  let x : ℝ := (N:ℝ)/j
+  have hjp : (0:ℝ)<j := by exact_mod_cast hj
+  have hNp : (0:ℝ)<N := by exact_mod_cast (by omega : 0<N)
+  have hxp : 0<x := by dsimp [x]; positivity
+  have hdpos : 0<D := by
+    have hc := Nat.choose_pos (show j≤N by omega)
+    by_contra h
+    have hz : D=0 := by omega
+    simp only [hz,zero_mul] at hD
+    omega
+  have htail := chosenTuple_tail_lower N j hj hx
+  have hjk : j≤chosenTuple N j := by
+    have hc : (j:ℝ)*x^nearPower≤chosenTuple N j := Nat.le_ceil _
+    exact_mod_cast (by nlinarith : (j:ℝ)≤chosenTuple N j)
+  have hh := footprint_log_bound N (chosenTuple N j) j D m (by omega) hjk (by omega) hdpos hfoot
+  have htlog := Real.log_le_log (by positivity : (0:ℝ)<(j:ℝ)*x^nearPower/2) htail
+  rw [Real.log_div (by positivity) (by norm_num),Real.log_mul hjp.ne' (by positivity),Real.log_rpow hxp] at htlog
+  have hxlog : Real.log x=Real.log N-Real.log j := Real.log_div hNp.ne' hjp.ne'
+  have hdim := near_row_log_dimension N j D hj hN hD
+  change (j:ℝ)*(Real.log x-2*Real.log 2)≤Real.log D at hdim
+  have hh' : Real.log D-(j:ℝ)*((1-nearPower)*Real.log x+Real.log 2)≤Real.log m := by
+    have hm := mul_le_mul_of_nonneg_left htlog hjp.le
+    rw [hxlog] at hm
+    nlinarith
+  have hlg : (3-nearPower)*Real.log 2≤nearFraction*Real.log x := by
+    dsimp [nearPower,nearFraction] at hlog ⊢
+    have := Real.log_pos (by norm_num : (1:ℝ)<2)
+    linarith
+  have hlg' := mul_le_mul_of_nonneg_left hlg hjp.le
+  dsimp [nearPower,nearFraction] at hh' hlg' ⊢
+  nlinarith
+
+lemma near_row_log_uniform (N j D : ℕ) (hj : 0<j) (hN : 2*j≤N)
+    (hD : N.choose j≤D*2^j) (hlog : 4*Real.log 2+2≤Real.log ((N:ℝ)/j)) :
+    (1/2:ℝ)*Real.log N≤Real.log D ∧ (j:ℝ)≤Real.log D := by
+  have hjp : (0:ℝ)<j := by exact_mod_cast hj
+  have hj1 : (1:ℝ)≤j := by exact_mod_cast hj
+  have hNp : (0:ℝ)<N := by exact_mod_cast (by omega : 0<N)
+  have hd := near_row_log_dimension N j D hj hN hD
+  have hjlog := Real.log_le_sub_one_of_pos hjp
+  have hl2 : 0≤Real.log 2 := Real.log_nonneg (by norm_num)
+  have hi : 1≤Real.log ((N:ℝ)/j) := by linarith
+  have he : Real.log N=Real.log j+Real.log ((N:ℝ)/j) := by rw [Real.log_div hNp.ne' hjp.ne']; ring
+  constructor
+  · rw [he]
+    nlinarith
+  · nlinarith
+
+
+lemma chosenTuple_density_bound (N j : ℕ) (hj : 0<j) (hx : 1≤(N:ℝ)/j)
+    (hhalf : ((N:ℝ)/j)^(nearPower-1)≤1/2) (C : ℝ) (hC : 0≤C) :
+    C*(chosenTuple N j:ℝ)*((chosenTuple N j:ℝ)/N)^(1/512:ℝ) ≤
+      C*2^(1+(1/512:ℝ))*(j:ℝ)*((N:ℝ)/j)^(-tailPower) := by
+  let x : ℝ := (N:ℝ)/j
+  have hjp : (0:ℝ)<j := by exact_mod_cast hj
+  have hxp : 0<x := lt_of_lt_of_le zero_lt_one hx
+  have hNp : (0:ℝ)<N := by
+    have : (0:ℝ)<(N:ℝ)/j := hxp
+    exact (div_pos_iff.mp this).resolve_right (by intro h; exact (not_lt_of_ge hjp.le) h.2) |>.1
+  have he : (j:ℝ)*x=N := by dsimp [x]; field_simp
+  have hk := (chosenTuple_bounds N j hj hx hhalf).2.2
+  change (chosenTuple N j:ℝ)≤2*(j:ℝ)*x^nearPower at hk
+  have hratio : (chosenTuple N j:ℝ)/N≤2*x^(nearPower-1) := by
+    rw [Real.rpow_sub hxp,Real.rpow_one]
+    apply (div_le_iff₀ hNp).mpr
+    rw [←he]
+    have : 2*(x^nearPower/x)*((j:ℝ)*x)=2*(j:ℝ)*x^nearPower := by field_simp
+    rw [this]
+    exact hk
+  have hpow := Real.rpow_le_rpow (by positivity : (0:ℝ)≤(chosenTuple N j:ℝ)/N) hratio
+    (by norm_num : (0:ℝ)≤1/512)
+  have hmain := mul_le_mul hk hpow (by positivity) (by positivity)
+  have hrpow : (2*x^(nearPower-1))^(1/512:ℝ)=
+      (2:ℝ)^(1/512:ℝ)*x^((nearPower-1)*(1/512)) := by
+    rw [Real.mul_rpow (by norm_num) (by positivity),←Real.rpow_mul hxp.le]
+  rw [hrpow] at hmain
+  have heq : 2*(j:ℝ)*x^nearPower*((2:ℝ)^(1/512:ℝ)*x^((nearPower-1)*(1/512)))=
+      (2:ℝ)^(1+(1/512:ℝ))*(j:ℝ)*x^(-tailPower) := by
+    rw [Real.rpow_add (by norm_num : (0:ℝ)<2),Real.rpow_one]
+    have hp : nearPower+(nearPower-1)*(1/512) = -tailPower := by norm_num [nearPower,tailPower]
+    rw [←hp,Real.rpow_add hxp]
+    ring
+  rw [heq] at hmain
+  have h := mul_le_mul_of_nonneg_left hmain hC
+  simpa only [mul_assoc] using h
+
+lemma near_conditions_eventually (C : ℝ) :
+    ∀ᶠ x : ℝ in atTop, 2≤x ∧ 2≤x^nearPower ∧ x^(nearPower-1)≤1/2 ∧
+      8*Real.log 2/nearPower≤Real.log x ∧ 4*Real.log 2+2≤Real.log x ∧
+      C*2^(1+(1/512:ℝ))*x^(-tailPower) ≤ nearFraction/4000 := by
+  have hpow := (tendsto_rpow_atTop (by norm_num [nearPower] : 0<nearPower)).eventually
+    (eventually_ge_atTop (2:ℝ))
+  have hhalf : ∀ᶠ x : ℝ in atTop, x^(nearPower-1)≤1/2 := by
+    have h := (tendsto_rpow_neg_atTop (by norm_num [nearPower] : 0<1-nearPower)).eventually
+      (gt_mem_nhds (by norm_num : (0:ℝ)<1/2))
+    filter_upwards [h] with x hx
+    simpa only [neg_sub] using hx.le
+  have hdec : Tendsto (fun x : ℝ => C*2^(1+(1/512:ℝ))*x^(-tailPower)) atTop (nhds 0) := by
+    simpa only [mul_zero] using
+      (tendsto_const_nhds.mul (tendsto_rpow_neg_atTop (by norm_num [tailPower] : 0<tailPower)))
+  have hd := hdec.eventually (gt_mem_nhds (by norm_num [nearFraction] : (0:ℝ)<nearFraction/4000))
+  filter_upwards [eventually_ge_atTop (2:ℝ),hpow,hhalf,
+    Real.tendsto_log_atTop.eventually (eventually_ge_atTop (8*Real.log 2/nearPower)),
+    Real.tendsto_log_atTop.eventually (eventually_ge_atTop (4*Real.log 2+2)),hd] with x hx hp hh hl hl' hd
+  exact ⟨hx,hp,hh,hl,hl',hd.le⟩
+
+end Thorp.DimensionEstimates
+
+end ThorpNine.Tail
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Tail/Model.lean b/lean/OAI/Probability/ThorpRouting/Tail/Model.lean
index 0365246b60a635c0100fa7931911946f783065d7..7dd680f1530cc1d1da2f56c2e37260e699ec34dc 100644
--- a/lean/OAI/Probability/ThorpRouting/Tail/Model.lean
+++ b/lean/OAI/Probability/ThorpRouting/Tail/Model.lean
@@ -1,335 +1,335 @@
-import Mathlib
-
-namespace OAI
-
-namespace ThorpNine.Tail
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-
-abbrev Card (d : ℕ) := Fin d → Bool
-
-def switchFun {d : ℕ} (ξ : Card d → Bool) (x : Card (d + 1)) : Card (d + 1) :=
-  Fin.cons (Bool.xor (x 0) (ξ (Fin.tail x))) (Fin.tail x)
-
-theorem switchFun_involutive {d : ℕ} (ξ : Card d → Bool) :
-    Function.Involutive (switchFun ξ) := by
-  intro x
-  funext i
-  refine Fin.cases ?_ (fun j => ?_) i
-  · simp only [switchFun, Fin.cons_zero, Fin.tail_cons]
-    cases x 0 <;> cases ξ (Fin.tail x) <;> rfl
-  · simp [switchFun, Fin.tail]
-
-def pairSwitch {d : ℕ} (ξ : Card d → Bool) : Equiv.Perm (Card (d + 1)) :=
-  { toFun := switchFun ξ
-    invFun := switchFun ξ
-    left_inv := switchFun_involutive ξ
-    right_inv := switchFun_involutive ξ }
-
-def Butterfly : ℕ → Type
-  | 0 => Unit
-  | d + 1 => (Card d → Bool) × (Bool → Butterfly d)
-
-def childLift {d : ℕ} (p : Bool → Equiv.Perm (Card d)) : Equiv.Perm (Card (d + 1)) where
-  toFun x := Fin.cons (x 0) (p (x 0) (Fin.tail x))
-  invFun x := Fin.cons (x 0) ((p (x 0)).symm (Fin.tail x))
-  left_inv x := by simp only [Fin.cons_zero, Fin.tail_cons, Equiv.symm_apply_apply, Fin.cons_self_tail]
-  right_inv x := by simp only [Fin.cons_zero, Fin.tail_cons, Equiv.apply_symm_apply, Fin.cons_self_tail]
-
-def butterflyPerm : (d : ℕ) → Butterfly d → Equiv.Perm (Card d)
-  | 0, _ => 1
-  | d + 1, b => pairSwitch b.1 * childLift (fun ε => butterflyPerm d (b.2 ε))
-
-noncomputable def finiteMean {Ω : Type*} [Fintype Ω] (f : Ω → ℝ) : ℝ :=
-  (∑ ω, f ω) / Fintype.card Ω
-
-def SwitchIndex : ℕ → Type
-  | 0 => Empty
-  | d + 1 => Sum (Card d) (Bool × SwitchIndex d)
-
-instance (d : ℕ) : Fintype (SwitchIndex d) := by
-  induction d with
-  | zero => exact inferInstanceAs (Fintype Empty)
-  | succ d ih => exact inferInstanceAs (Fintype (Sum (Card d) (Bool × SwitchIndex d)))
-
-instance (d : ℕ) : DecidableEq (SwitchIndex d) := by
-  induction d with
-  | zero => exact inferInstanceAs (DecidableEq Empty)
-  | succ d ih => exact inferInstanceAs (DecidableEq (Sum (Card d) (Bool × SwitchIndex d)))
-
-def decodeButterfly : (d : ℕ) → (SwitchIndex d → Bool) → Butterfly d
-  | 0, _ => ()
-  | d + 1, ω => (fun y => ω (Sum.inl y), fun ε =>
-      decodeButterfly d (fun i => ω (Sum.inr (ε,i))))
-
-end Thorp
-
-namespace Thorp.PairRouting
-open scoped BigOperators
-open Filter
-variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
-
-noncomputable def tupleProbability {Ω β : Type*} [Fintype Ω] [DecidableEq β]
-    (P : Ω → Equiv.Perm β) (x y : ι → β) : ℝ := by
-  classical
-  exact finiteMean (fun ω => if ∀ i, P ω (x i) = y i then 1 else 0)
-
-end Thorp.PairRouting
-
-namespace Thorp.Specht
-open scoped BigOperators Classical
-
-abbrev Cell (μ : YoungDiagram) := {x : ℕ × ℕ // x ∈ μ.cells}
-
-def row {μ : YoungDiagram} (x : Cell μ) : ℕ := x.1.1
-
-def col {μ : YoungDiagram} (x : Cell μ) : ℕ := x.1.2
-
-lemma row_lt_colLen {μ : YoungDiagram} (x : Cell μ) : row x < μ.colLen (col x) :=
-  YoungDiagram.mem_iff_lt_colLen.mp x.2
-
-noncomputable def rowIndex {μ : YoungDiagram} (x : Cell μ) : Fin (μ.colLen 0) :=
-  ⟨row x, (row_lt_colLen x).trans_le (μ.colLen_anti 0 (col x) (Nat.zero_le _))⟩
-
-abbrev Tabloid (μ : YoungDiagram) :=
-  {f : Cell μ → Fin (μ.colLen 0) // ∃ p : Equiv.Perm (Cell μ), f = rowIndex ∘ p}
-
-noncomputable instance (μ : YoungDiagram) : Fintype (Tabloid μ) := Fintype.ofFinite _
-
-noncomputable def baseTabloid (μ : YoungDiagram) : Tabloid μ := ⟨rowIndex, 1, rfl⟩
-
-noncomputable def tabloidAct {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) :
-    Equiv.Perm (Tabloid μ) where
-  toFun f := ⟨fun x => f.1 (p⁻¹ x), by
-    obtain ⟨q,hq⟩ := f.2
-    refine ⟨q*p⁻¹, ?_⟩
-    funext x
-    simp only [hq,Function.comp_apply,Equiv.Perm.mul_apply]⟩
-  invFun f := ⟨fun x => f.1 (p x), by
-    obtain ⟨q,hq⟩ := f.2
-    refine ⟨q*p, ?_⟩
-    funext x
-    simp only [hq,Function.comp_apply,Equiv.Perm.mul_apply]⟩
-  left_inv f := by apply Subtype.ext; funext x; simp
-  right_inv f := by apply Subtype.ext; funext x; simp
-
-@[simp] lemma tabloidAct_apply {μ : YoungDiagram} (p : Equiv.Perm (Cell μ))
-    (f : Tabloid μ) (x : Cell μ) : (tabloidAct p f).1 x = f.1 (p⁻¹ x) := rfl
-
-@[simp] lemma tabloidAct_one (μ : YoungDiagram) : tabloidAct (1 : Equiv.Perm (Cell μ)) = 1 := by
-  ext f x
-  rfl
-
-@[simp] lemma tabloidAct_mul {μ : YoungDiagram} (p q : Equiv.Perm (Cell μ)) :
-    tabloidAct (p*q) = tabloidAct p * tabloidAct q := by
-  ext f x
-  simp only [tabloidAct_apply, mul_inv_rev,Equiv.Perm.mul_apply]
-
-noncomputable def tabloidRep (μ : YoungDiagram) :
-    Representation ℂ (Equiv.Perm (Cell μ)) (Tabloid μ → ℂ) where
-  toFun p :=
-    { toFun := fun v f => v (tabloidAct p⁻¹ f)
-      map_add' := by intros; rfl
-      map_smul' := by intros; rfl }
-  map_one' := by ext v f; simp
-  map_mul' p q := by
-    ext v f
-    simp only [mul_inv_rev,tabloidAct_mul,Equiv.Perm.mul_apply]
-    rfl
-
-noncomputable def delta {μ : YoungDiagram} (f : Tabloid μ) : Tabloid μ → ℂ :=
-  fun g => if g=f then 1 else 0
-
-noncomputable def colGroup (μ : YoungDiagram) : Subgroup (Equiv.Perm (Cell μ)) where
-  carrier := {p | ∀ x, col (p x) = col x}
-  one_mem' := by intro x; rfl
-  mul_mem' := by intro p q hp hq x; exact (hp (q x)).trans (hq x)
-  inv_mem' := by intro p hp x; simpa using (hp (p⁻¹ x)).symm
-
-noncomputable def permSign {α : Type*} [Fintype α] [DecidableEq α] (p : Equiv.Perm α) : ℂ :=
-  ((Equiv.Perm.sign p : ℤ) : ℂ)
-
-
-section
-variable {α V : Type*} [Fintype α] [DecidableEq α] [AddCommGroup V] [Module ℂ V]
-
-noncomputable def alternator (C : Subgroup (Equiv.Perm α))
-    (ρ : Representation ℂ (Equiv.Perm α) V) : Module.End ℂ V := by
-  classical
-  exact ∑ c : C, permSign (c : Equiv.Perm α) • ρ c
-
-end
-
-noncomputable def polytabloid (μ : YoungDiagram) : Tabloid μ → ℂ :=
-  alternator (colGroup μ) (tabloidRep μ) (delta (baseTabloid μ))
-
-noncomputable def space (μ : YoungDiagram) : Submodule ℂ (Tabloid μ → ℂ) :=
-  Submodule.span ℂ (Set.range (fun p : Equiv.Perm (Cell μ) => tabloidRep μ p (polytabloid μ)))
-
-lemma orbit_mem_space (μ : YoungDiagram) (p : Equiv.Perm (Cell μ)) :
-    tabloidRep μ p (polytabloid μ) ∈ space μ := Submodule.subset_span ⟨p,rfl⟩
-
-lemma action_mem_space (μ : YoungDiagram) (p : Equiv.Perm (Cell μ))
-    (v : Tabloid μ → ℂ) (hv : v ∈ space μ) : tabloidRep μ p v ∈ space μ := by
-  induction hv using Submodule.span_induction with
-  | mem v hv =>
-    obtain ⟨q,rfl⟩ := hv
-    change ((tabloidRep μ p)*(tabloidRep μ q)) (polytabloid μ) ∈ space μ
-    rw [←map_mul]
-    exact orbit_mem_space μ (p*q)
-  | zero => simp
-  | add v w hv hw ihv ihw => simpa using (space μ).add_mem ihv ihw
-  | smul a v hv ih => simpa using (space μ).smul_mem a ih
-
-noncomputable def subrepresentation (μ : YoungDiagram) : Subrepresentation (tabloidRep μ) :=
-  ⟨space μ, fun p v hv => action_mem_space μ p v hv⟩
-
-noncomputable def representation (μ : YoungDiagram) :
-    Representation ℂ (Equiv.Perm (Cell μ)) (space μ) :=
-  (subrepresentation μ).toRepresentation
-
-end Thorp.Specht
-
-namespace Thorp.PermutationHilbert
-open scoped BigOperators ComplexConjugate Classical
-open Complex
-variable {X : Type*} [Fintype X]
-
-abbrev H (X : Type*) [Fintype X] := EuclideanSpace ℂ X
-
-end Thorp.PermutationHilbert
-
-namespace Thorp.Specht
-open scoped BigOperators Classical
-
-noncomputable def hilbertEquiv (μ : YoungDiagram) :
-    (Tabloid μ → ℂ) ≃ₗ[ℂ] PermutationHilbert.H (Tabloid μ) :=
-  (WithLp.linearEquiv 2 ℂ (Tabloid μ → ℂ)).symm
-
-noncomputable def hilbertSpace (μ : YoungDiagram) :
-    Submodule ℂ (PermutationHilbert.H (Tabloid μ)) := (space μ).map (hilbertEquiv μ).toLinearMap
-
-end Thorp.Specht
-
-namespace Thorp.UnitaryFinite
-
-section
-open scoped BigOperators ComplexConjugate Classical
-variable {G : Type*} [Group G] [Fintype G]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-noncomputable def continuousRepresentation (ρ : Representation ℂ G V) : G →* (V →L[ℂ] V) where
-  toFun g := (ρ g).toContinuousLinearMap
-  map_one' := by ext v; simp
-  map_mul' g h := by ext v; simp
-
-end
-open scoped BigOperators Classical
-variable {G : Type*} [Group G] [Fintype G]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-variable {Ω Ξ : Type*} [Fintype Ω] [Fintype Ξ]
-
-noncomputable def sampleOperator (ρ : Representation ℂ G V) (P : Ω → G) : V →L[ℂ] V :=
-  (Fintype.card Ω:ℂ)⁻¹ • ∑ ω,continuousRepresentation ρ (P ω)
-
-end Thorp.UnitaryFinite
-
-namespace Thorp.Specht
-open scoped BigOperators Classical
-
-noncomputable def spaceHilbertEquiv (μ : YoungDiagram) : space μ ≃ₗ[ℂ] hilbertSpace μ :=
-  (hilbertEquiv μ).submoduleMap (space μ)
-
-noncomputable def unitaryRepresentation (μ : YoungDiagram) :
-    Representation ℂ (Equiv.Perm (Cell μ)) (hilbertSpace μ) :=
-  ((spaceHilbertEquiv μ).conjAlgEquiv ℂ).toMonoidHom.comp (representation μ)
-
-variable {α ι : Type*} [Fintype α] [Fintype ι]
-
-noncomputable def relabelledUnitary (μ : YoungDiagram) (e : α ≃ Cell μ) :
-    Representation ℂ (Equiv.Perm α) (hilbertSpace μ) :=
-  (unitaryRepresentation μ).comp e.permCongrHom.toMonoidHom
-
-end Thorp.Specht
-
-namespace Thorp.SparseContact
-open scoped BigOperators
-
-noncomputable def sweepOperator (d : ℕ) (μ : YoungDiagram)
-    (e : Card d ≃ Specht.Cell μ) (reverse : Bool) :
-    Specht.hilbertSpace μ →L[ℂ] Specht.hilbertSpace μ :=
-  UnitaryFinite.sampleOperator (V := Specht.hilbertSpace μ) (Specht.relabelledUnitary μ e)
-    (fun ω : SwitchIndex d → Bool => if reverse then
-      (butterflyPerm d (decodeButterfly d ω))⁻¹ else butterflyPerm d (decodeButterfly d ω))
-
-end Thorp.SparseContact
-
-namespace Thorp.StrongSmoothing
-
-def coordinateRelabel (d : ℕ) (order : Equiv.Perm (Fin d)) : Equiv.Perm (Card d) where
-  toFun x := fun i => x (order.symm i)
-  invFun x := fun i => x (order i)
-  left_inv x := by funext i; simp
-  right_inv x := by funext i; simp
-
-def sweep (d : ℕ) (order : Equiv.Perm (Fin d)) (ω : SwitchIndex d → Bool) :
-    Equiv.Perm (Card d) :=
-  coordinateRelabel d order * butterflyPerm d (decodeButterfly d ω) *
-    (coordinateRelabel d order)⁻¹
-
-abbrev Tuples (d l : ℕ) := Fin l ↪ Card d
-
-noncomputable def sweepKernel (d l : ℕ) (order : Equiv.Perm (Fin d)) :
-    Matrix (Tuples d l) (Tuples d l) ℝ :=
-  fun x y => PairRouting.tupleProbability (sweep d order) x y
-
-noncomputable def reflectedKernel (d l : ℕ) (order : Equiv.Perm (Fin d)) (orientation : Bool) :
-    Matrix (Tuples d l) (Tuples d l) ℝ :=
-  let A := sweepKernel d l order
-  if orientation then A * A.transpose else A.transpose * A
-
-end Thorp.StrongSmoothing
-
-namespace Thorp.StrongTail
-open scoped BigOperators Classical
-
-section
-open Specht
-
-def tailDiagram (μ : YoungDiagram) : YoungDiagram :=
-  YoungDiagram.ofRowLens μ.rowLens.tail μ.rowLens_sorted.pairwise.tail.sortedGE
-
-end
-open StrongSmoothing
-
-def DensityAt (p C : ℝ) : Prop :=
-  ∀ (d l : ℕ), l ≤ 2^d → ∀ (order : Equiv.Perm (Fin d)) (rev : Bool)
-    (x : StrongSmoothing.Tuples d l),
-    let H := StrongSmoothing.reflectedKernel d l order rev
-    (Thorp.finiteMean (fun y => ((Fintype.card (StrongSmoothing.Tuples d l):ℝ)*H x y)^p)
-      ≤ Real.exp (C*l)) ∧
-    (Thorp.finiteMean (fun y => ((Fintype.card (StrongSmoothing.Tuples d l):ℝ)*H y x)^p)
-      ≤ Real.exp (C*l)) ∧
-    (((2:ℝ)^d)^l)⁻¹ * (∑ y, ((((2:ℝ)^d)^l)*H x y)^p) ≤ Real.exp (C*l) ∧
-    (((2:ℝ)^d)^l)⁻¹ * (∑ y, ((((2:ℝ)^d)^l)*H y x)^p) ≤ Real.exp (C*l)
-
-end Thorp.StrongTail
-
-namespace Thorp.StrongTail
-open scoped BigOperators Classical
-open Specht UnitaryFinite
-
-def MainStatement : Prop :=
-  ∃ p Cdensity C : ℝ, 1 < p ∧ p ≤ 2 ∧ DensityAt p Cdensity ∧
-    ∀ (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ) (rev : Bool),
-      ‖SparseContact.sweepOperator d μ e rev‖^2 ≤
-       Real.exp (C*(μ.card-μ.rowLen 0 : ℕ)) *
-       (Module.finrank ℂ (Specht.space (tailDiagram μ)):ℝ)^(-((p-1)/p))
-
-end Thorp.StrongTail
-
-end ThorpNine.Tail
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+namespace ThorpNine.Tail
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+
+abbrev Card (d : ℕ) := Fin d → Bool
+
+def switchFun {d : ℕ} (ξ : Card d → Bool) (x : Card (d + 1)) : Card (d + 1) :=
+  Fin.cons (Bool.xor (x 0) (ξ (Fin.tail x))) (Fin.tail x)
+
+theorem switchFun_involutive {d : ℕ} (ξ : Card d → Bool) :
+    Function.Involutive (switchFun ξ) := by
+  intro x
+  funext i
+  refine Fin.cases ?_ (fun j => ?_) i
+  · simp only [switchFun, Fin.cons_zero, Fin.tail_cons]
+    cases x 0 <;> cases ξ (Fin.tail x) <;> rfl
+  · simp [switchFun, Fin.tail]
+
+def pairSwitch {d : ℕ} (ξ : Card d → Bool) : Equiv.Perm (Card (d + 1)) :=
+  { toFun := switchFun ξ
+    invFun := switchFun ξ
+    left_inv := switchFun_involutive ξ
+    right_inv := switchFun_involutive ξ }
+
+def Butterfly : ℕ → Type
+  | 0 => Unit
+  | d + 1 => (Card d → Bool) × (Bool → Butterfly d)
+
+def childLift {d : ℕ} (p : Bool → Equiv.Perm (Card d)) : Equiv.Perm (Card (d + 1)) where
+  toFun x := Fin.cons (x 0) (p (x 0) (Fin.tail x))
+  invFun x := Fin.cons (x 0) ((p (x 0)).symm (Fin.tail x))
+  left_inv x := by simp only [Fin.cons_zero, Fin.tail_cons, Equiv.symm_apply_apply, Fin.cons_self_tail]
+  right_inv x := by simp only [Fin.cons_zero, Fin.tail_cons, Equiv.apply_symm_apply, Fin.cons_self_tail]
+
+def butterflyPerm : (d : ℕ) → Butterfly d → Equiv.Perm (Card d)
+  | 0, _ => 1
+  | d + 1, b => pairSwitch b.1 * childLift (fun ε => butterflyPerm d (b.2 ε))
+
+noncomputable def finiteMean {Ω : Type*} [Fintype Ω] (f : Ω → ℝ) : ℝ :=
+  (∑ ω, f ω) / Fintype.card Ω
+
+def SwitchIndex : ℕ → Type
+  | 0 => Empty
+  | d + 1 => Sum (Card d) (Bool × SwitchIndex d)
+
+instance (d : ℕ) : Fintype (SwitchIndex d) := by
+  induction d with
+  | zero => exact inferInstanceAs (Fintype Empty)
+  | succ d ih => exact inferInstanceAs (Fintype (Sum (Card d) (Bool × SwitchIndex d)))
+
+instance (d : ℕ) : DecidableEq (SwitchIndex d) := by
+  induction d with
+  | zero => exact inferInstanceAs (DecidableEq Empty)
+  | succ d ih => exact inferInstanceAs (DecidableEq (Sum (Card d) (Bool × SwitchIndex d)))
+
+def decodeButterfly : (d : ℕ) → (SwitchIndex d → Bool) → Butterfly d
+  | 0, _ => ()
+  | d + 1, ω => (fun y => ω (Sum.inl y), fun ε =>
+      decodeButterfly d (fun i => ω (Sum.inr (ε,i))))
+
+end Thorp
+
+namespace Thorp.PairRouting
+open scoped BigOperators
+open Filter
+variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
+
+noncomputable def tupleProbability {Ω β : Type*} [Fintype Ω] [DecidableEq β]
+    (P : Ω → Equiv.Perm β) (x y : ι → β) : ℝ := by
+  classical
+  exact finiteMean (fun ω => if ∀ i, P ω (x i) = y i then 1 else 0)
+
+end Thorp.PairRouting
+
+namespace Thorp.Specht
+open scoped BigOperators Classical
+
+abbrev Cell (μ : YoungDiagram) := {x : ℕ × ℕ // x ∈ μ.cells}
+
+def row {μ : YoungDiagram} (x : Cell μ) : ℕ := x.1.1
+
+def col {μ : YoungDiagram} (x : Cell μ) : ℕ := x.1.2
+
+lemma row_lt_colLen {μ : YoungDiagram} (x : Cell μ) : row x < μ.colLen (col x) :=
+  YoungDiagram.mem_iff_lt_colLen.mp x.2
+
+noncomputable def rowIndex {μ : YoungDiagram} (x : Cell μ) : Fin (μ.colLen 0) :=
+  ⟨row x, (row_lt_colLen x).trans_le (μ.colLen_anti 0 (col x) (Nat.zero_le _))⟩
+
+abbrev Tabloid (μ : YoungDiagram) :=
+  {f : Cell μ → Fin (μ.colLen 0) // ∃ p : Equiv.Perm (Cell μ), f = rowIndex ∘ p}
+
+noncomputable instance (μ : YoungDiagram) : Fintype (Tabloid μ) := Fintype.ofFinite _
+
+noncomputable def baseTabloid (μ : YoungDiagram) : Tabloid μ := ⟨rowIndex, 1, rfl⟩
+
+noncomputable def tabloidAct {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) :
+    Equiv.Perm (Tabloid μ) where
+  toFun f := ⟨fun x => f.1 (p⁻¹ x), by
+    obtain ⟨q,hq⟩ := f.2
+    refine ⟨q*p⁻¹, ?_⟩
+    funext x
+    simp only [hq,Function.comp_apply,Equiv.Perm.mul_apply]⟩
+  invFun f := ⟨fun x => f.1 (p x), by
+    obtain ⟨q,hq⟩ := f.2
+    refine ⟨q*p, ?_⟩
+    funext x
+    simp only [hq,Function.comp_apply,Equiv.Perm.mul_apply]⟩
+  left_inv f := by apply Subtype.ext; funext x; simp
+  right_inv f := by apply Subtype.ext; funext x; simp
+
+@[simp] lemma tabloidAct_apply {μ : YoungDiagram} (p : Equiv.Perm (Cell μ))
+    (f : Tabloid μ) (x : Cell μ) : (tabloidAct p f).1 x = f.1 (p⁻¹ x) := rfl
+
+@[simp] lemma tabloidAct_one (μ : YoungDiagram) : tabloidAct (1 : Equiv.Perm (Cell μ)) = 1 := by
+  ext f x
+  rfl
+
+@[simp] lemma tabloidAct_mul {μ : YoungDiagram} (p q : Equiv.Perm (Cell μ)) :
+    tabloidAct (p*q) = tabloidAct p * tabloidAct q := by
+  ext f x
+  simp only [tabloidAct_apply, mul_inv_rev,Equiv.Perm.mul_apply]
+
+noncomputable def tabloidRep (μ : YoungDiagram) :
+    Representation ℂ (Equiv.Perm (Cell μ)) (Tabloid μ → ℂ) where
+  toFun p :=
+    { toFun := fun v f => v (tabloidAct p⁻¹ f)
+      map_add' := by intros; rfl
+      map_smul' := by intros; rfl }
+  map_one' := by ext v f; simp
+  map_mul' p q := by
+    ext v f
+    simp only [mul_inv_rev,tabloidAct_mul,Equiv.Perm.mul_apply]
+    rfl
+
+noncomputable def delta {μ : YoungDiagram} (f : Tabloid μ) : Tabloid μ → ℂ :=
+  fun g => if g=f then 1 else 0
+
+noncomputable def colGroup (μ : YoungDiagram) : Subgroup (Equiv.Perm (Cell μ)) where
+  carrier := {p | ∀ x, col (p x) = col x}
+  one_mem' := by intro x; rfl
+  mul_mem' := by intro p q hp hq x; exact (hp (q x)).trans (hq x)
+  inv_mem' := by intro p hp x; simpa using (hp (p⁻¹ x)).symm
+
+noncomputable def permSign {α : Type*} [Fintype α] [DecidableEq α] (p : Equiv.Perm α) : ℂ :=
+  ((Equiv.Perm.sign p : ℤ) : ℂ)
+
+
+section
+variable {α V : Type*} [Fintype α] [DecidableEq α] [AddCommGroup V] [Module ℂ V]
+
+noncomputable def alternator (C : Subgroup (Equiv.Perm α))
+    (ρ : Representation ℂ (Equiv.Perm α) V) : Module.End ℂ V := by
+  classical
+  exact ∑ c : C, permSign (c : Equiv.Perm α) • ρ c
+
+end
+
+noncomputable def polytabloid (μ : YoungDiagram) : Tabloid μ → ℂ :=
+  alternator (colGroup μ) (tabloidRep μ) (delta (baseTabloid μ))
+
+noncomputable def space (μ : YoungDiagram) : Submodule ℂ (Tabloid μ → ℂ) :=
+  Submodule.span ℂ (Set.range (fun p : Equiv.Perm (Cell μ) => tabloidRep μ p (polytabloid μ)))
+
+lemma orbit_mem_space (μ : YoungDiagram) (p : Equiv.Perm (Cell μ)) :
+    tabloidRep μ p (polytabloid μ) ∈ space μ := Submodule.subset_span ⟨p,rfl⟩
+
+lemma action_mem_space (μ : YoungDiagram) (p : Equiv.Perm (Cell μ))
+    (v : Tabloid μ → ℂ) (hv : v ∈ space μ) : tabloidRep μ p v ∈ space μ := by
+  induction hv using Submodule.span_induction with
+  | mem v hv =>
+    obtain ⟨q,rfl⟩ := hv
+    change ((tabloidRep μ p)*(tabloidRep μ q)) (polytabloid μ) ∈ space μ
+    rw [←map_mul]
+    exact orbit_mem_space μ (p*q)
+  | zero => simp
+  | add v w hv hw ihv ihw => simpa using (space μ).add_mem ihv ihw
+  | smul a v hv ih => simpa using (space μ).smul_mem a ih
+
+noncomputable def subrepresentation (μ : YoungDiagram) : Subrepresentation (tabloidRep μ) :=
+  ⟨space μ, fun p v hv => action_mem_space μ p v hv⟩
+
+noncomputable def representation (μ : YoungDiagram) :
+    Representation ℂ (Equiv.Perm (Cell μ)) (space μ) :=
+  (subrepresentation μ).toRepresentation
+
+end Thorp.Specht
+
+namespace Thorp.PermutationHilbert
+open scoped BigOperators ComplexConjugate Classical
+open Complex
+variable {X : Type*} [Fintype X]
+
+abbrev H (X : Type*) [Fintype X] := EuclideanSpace ℂ X
+
+end Thorp.PermutationHilbert
+
+namespace Thorp.Specht
+open scoped BigOperators Classical
+
+noncomputable def hilbertEquiv (μ : YoungDiagram) :
+    (Tabloid μ → ℂ) ≃ₗ[ℂ] PermutationHilbert.H (Tabloid μ) :=
+  (WithLp.linearEquiv 2 ℂ (Tabloid μ → ℂ)).symm
+
+noncomputable def hilbertSpace (μ : YoungDiagram) :
+    Submodule ℂ (PermutationHilbert.H (Tabloid μ)) := (space μ).map (hilbertEquiv μ).toLinearMap
+
+end Thorp.Specht
+
+namespace Thorp.UnitaryFinite
+
+section
+open scoped BigOperators ComplexConjugate Classical
+variable {G : Type*} [Group G] [Fintype G]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+noncomputable def continuousRepresentation (ρ : Representation ℂ G V) : G →* (V →L[ℂ] V) where
+  toFun g := (ρ g).toContinuousLinearMap
+  map_one' := by ext v; simp
+  map_mul' g h := by ext v; simp
+
+end
+open scoped BigOperators Classical
+variable {G : Type*} [Group G] [Fintype G]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+variable {Ω Ξ : Type*} [Fintype Ω] [Fintype Ξ]
+
+noncomputable def sampleOperator (ρ : Representation ℂ G V) (P : Ω → G) : V →L[ℂ] V :=
+  (Fintype.card Ω:ℂ)⁻¹ • ∑ ω,continuousRepresentation ρ (P ω)
+
+end Thorp.UnitaryFinite
+
+namespace Thorp.Specht
+open scoped BigOperators Classical
+
+noncomputable def spaceHilbertEquiv (μ : YoungDiagram) : space μ ≃ₗ[ℂ] hilbertSpace μ :=
+  (hilbertEquiv μ).submoduleMap (space μ)
+
+noncomputable def unitaryRepresentation (μ : YoungDiagram) :
+    Representation ℂ (Equiv.Perm (Cell μ)) (hilbertSpace μ) :=
+  ((spaceHilbertEquiv μ).conjAlgEquiv ℂ).toMonoidHom.comp (representation μ)
+
+variable {α ι : Type*} [Fintype α] [Fintype ι]
+
+noncomputable def relabelledUnitary (μ : YoungDiagram) (e : α ≃ Cell μ) :
+    Representation ℂ (Equiv.Perm α) (hilbertSpace μ) :=
+  (unitaryRepresentation μ).comp e.permCongrHom.toMonoidHom
+
+end Thorp.Specht
+
+namespace Thorp.SparseContact
+open scoped BigOperators
+
+noncomputable def sweepOperator (d : ℕ) (μ : YoungDiagram)
+    (e : Card d ≃ Specht.Cell μ) (reverse : Bool) :
+    Specht.hilbertSpace μ →L[ℂ] Specht.hilbertSpace μ :=
+  UnitaryFinite.sampleOperator (V := Specht.hilbertSpace μ) (Specht.relabelledUnitary μ e)
+    (fun ω : SwitchIndex d → Bool => if reverse then
+      (butterflyPerm d (decodeButterfly d ω))⁻¹ else butterflyPerm d (decodeButterfly d ω))
+
+end Thorp.SparseContact
+
+namespace Thorp.StrongSmoothing
+
+def coordinateRelabel (d : ℕ) (order : Equiv.Perm (Fin d)) : Equiv.Perm (Card d) where
+  toFun x := fun i => x (order.symm i)
+  invFun x := fun i => x (order i)
+  left_inv x := by funext i; simp
+  right_inv x := by funext i; simp
+
+def sweep (d : ℕ) (order : Equiv.Perm (Fin d)) (ω : SwitchIndex d → Bool) :
+    Equiv.Perm (Card d) :=
+  coordinateRelabel d order * butterflyPerm d (decodeButterfly d ω) *
+    (coordinateRelabel d order)⁻¹
+
+abbrev Tuples (d l : ℕ) := Fin l ↪ Card d
+
+noncomputable def sweepKernel (d l : ℕ) (order : Equiv.Perm (Fin d)) :
+    Matrix (Tuples d l) (Tuples d l) ℝ :=
+  fun x y => PairRouting.tupleProbability (sweep d order) x y
+
+noncomputable def reflectedKernel (d l : ℕ) (order : Equiv.Perm (Fin d)) (orientation : Bool) :
+    Matrix (Tuples d l) (Tuples d l) ℝ :=
+  let A := sweepKernel d l order
+  if orientation then A * A.transpose else A.transpose * A
+
+end Thorp.StrongSmoothing
+
+namespace Thorp.StrongTail
+open scoped BigOperators Classical
+
+section
+open Specht
+
+def tailDiagram (μ : YoungDiagram) : YoungDiagram :=
+  YoungDiagram.ofRowLens μ.rowLens.tail μ.rowLens_sorted.pairwise.tail.sortedGE
+
+end
+open StrongSmoothing
+
+def DensityAt (p C : ℝ) : Prop :=
+  ∀ (d l : ℕ), l ≤ 2^d → ∀ (order : Equiv.Perm (Fin d)) (rev : Bool)
+    (x : StrongSmoothing.Tuples d l),
+    let H := StrongSmoothing.reflectedKernel d l order rev
+    (Thorp.finiteMean (fun y => ((Fintype.card (StrongSmoothing.Tuples d l):ℝ)*H x y)^p)
+      ≤ Real.exp (C*l)) ∧
+    (Thorp.finiteMean (fun y => ((Fintype.card (StrongSmoothing.Tuples d l):ℝ)*H y x)^p)
+      ≤ Real.exp (C*l)) ∧
+    (((2:ℝ)^d)^l)⁻¹ * (∑ y, ((((2:ℝ)^d)^l)*H x y)^p) ≤ Real.exp (C*l) ∧
+    (((2:ℝ)^d)^l)⁻¹ * (∑ y, ((((2:ℝ)^d)^l)*H y x)^p) ≤ Real.exp (C*l)
+
+end Thorp.StrongTail
+
+namespace Thorp.StrongTail
+open scoped BigOperators Classical
+open Specht UnitaryFinite
+
+def MainStatement : Prop :=
+  ∃ p Cdensity C : ℝ, 1 < p ∧ p ≤ 2 ∧ DensityAt p Cdensity ∧
+    ∀ (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Specht.Cell μ) (rev : Bool),
+      ‖SparseContact.sweepOperator d μ e rev‖^2 ≤
+       Real.exp (C*(μ.card-μ.rowLen 0 : ℕ)) *
+       (Module.finrank ℂ (Specht.space (tailDiagram μ)):ℝ)^(-((p-1)/p))
+
+end Thorp.StrongTail
+
+end ThorpNine.Tail
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Tail/ModifiedReverseEquiv.lean b/lean/OAI/Probability/ThorpRouting/Tail/ModifiedReverseEquiv.lean
index dff6e00a2aaed8360aa6c1da1d0a1a4992e4a7ae..fa1f4c5243702e29b44a40d97d98f9122a566ee5 100644
--- a/lean/OAI/Probability/ThorpRouting/Tail/ModifiedReverseEquiv.lean
+++ b/lean/OAI/Probability/ThorpRouting/Tail/ModifiedReverseEquiv.lean
@@ -1,759 +1,759 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Tail.ModifiedRoutingMoment
-
-namespace OAI
-
-namespace ThorpNine.Tail
-
-namespace Thorp
-
-noncomputable def modifiedReverseEquiv (L r : ℕ) : Equiv.Perm (ModifiedCoins L r) where
-  toFun := modifiedReverseCoins L r
-  invFun := modifiedReverseCoins L r
-  left_inv := modifiedReverseCoins_involutive L r
-  right_inv := modifiedReverseCoins_involutive L r
-
-lemma modifiedPerm_reverse (L r : ℕ) (E : Finset (Card r)) (ω : ModifiedCoins L r) :
-    modifiedPerm L r E (modifiedReverseCoins L r ω) = (modifiedPerm L r E ω)⁻¹ := by
-  induction r with
-  | zero =>
-      rw [modifiedPerm_zero,modifiedPerm_zero]
-      split_ifs
-      · simp only [inv_one]
-      · simp only [modifiedReverseCoins,mul_inv_rev,inv_inv]
-        simp
-  | succ r ih =>
-      obtain ⟨z,rfl⟩ := (modifiedStepEquiv L r).symm.surjective ω
-      rcases z with ⟨⟨ω₀,ω₁⟩,⟨ξ,η⟩⟩
-      simp only [modifiedReverseCoins,Equiv.apply_symm_apply,modifiedPerm_step,
-        mul_inv_rev,pairSwitch_inv,childLift_inv]
-      rw [mul_assoc]
-      congr 1
-      congr 1
-      apply congrArg childLift
-      funext b
-      cases b <;> simp only [Bool.false_eq_true,↓reduceIte,ih]
-
-lemma modified_kernel_symmetric (L r : ℕ) (E : Finset (Card r))
-    {ι : Type*} [Fintype ι] (e f : ι ↪ Card (L+r)) :
-    PairRouting.tupleProbability (modifiedPerm L r E) e f =
-      PairRouting.tupleProbability (modifiedPerm L r E) f e :=
-  PairRouting.tupleProbability_symmetric _ (modifiedReverseEquiv L r)
-    (modifiedPerm_reverse L r E) e f
-
-end Thorp
-
-namespace Thorp.Specht
-open scoped BigOperators Classical
-
-@[simp] lemma card_cell (μ : YoungDiagram) : Fintype.card (Cell μ) = μ.card := by
-  simp [Cell, YoungDiagram.card]
-
-noncomputable def columnEquiv (μ : YoungDiagram) (j : ℕ) :
-    {x : Cell μ // col x = j} ≃ Fin (μ.colLen j) where
-  toFun x := ⟨row x.1, by simpa [x.2] using row_lt_colLen x.1⟩
-  invFun i := ⟨⟨(i,j), YoungDiagram.mem_iff_lt_colLen.mpr i.2⟩, rfl⟩
-  left_inv x := by apply Subtype.ext; apply Subtype.ext; exact Prod.ext rfl x.2.symm
-  right_inv i := by rfl
-
-noncomputable def columnValues {μ : YoungDiagram} (f : Cell μ → ℕ) (j : ℕ) : Finset ℕ :=
-  (Finset.univ.filter (fun x => col x = j)).image f
-
-lemma mem_columnValues {μ : YoungDiagram} (f : Cell μ → ℕ) (x : Cell μ) :
-    f x ∈ columnValues f (col x) := by
-  classical
-  exact Finset.mem_image.mpr ⟨x, by simp, rfl⟩
-
-lemma card_columnValues {μ : YoungDiagram} (f : Cell μ → ℕ)
-    (hf : ∀ x y, col x = col y → f x = f y → x = y) (j : ℕ) :
-    (columnValues f j).card = μ.colLen j := by
-  classical
-  rw [columnValues, Finset.card_image_iff.mpr]
-  · rw [← Fintype.card_subtype]
-    exact Fintype.card_congr (columnEquiv μ j) |>.trans (Fintype.card_fin _)
-  intro x hx y hy he
-  exact hf x y ((Finset.mem_filter.mp hx).2.trans (Finset.mem_filter.mp hy).2.symm) he
-
-lemma strictMono_fin_nat_ge {n : ℕ} (f : Fin n → ℕ) (hf : StrictMono f) :
-    ∀ i, i.val ≤ f i := by
-  intro i
-  induction h : i.val using Nat.strong_induction_on generalizing i with
-  | h n ih =>
-    by_cases hn : n = 0
-    · omega
-    · have hn' : n-1 < i.val := by omega
-      let j : Fin _ := ⟨n-1, hn'.trans i.isLt⟩
-      have hj := ih (n-1) (by omega) j rfl
-      have hij := hf (show j < i from hn')
-      dsimp [j] at hj hij
-      omega
-
-noncomputable def rankCell {μ : YoungDiagram} (f : Cell μ → ℕ)
-    (hf : ∀ x y, col x = col y → f x = f y → x = y) (x : Cell μ) : Cell μ :=
-  let k := ((columnValues f (col x)).orderIsoOfFin (card_columnValues f hf (col x))).symm
-    ⟨f x, mem_columnValues f x⟩
-  ⟨(k.val,col x), YoungDiagram.mem_iff_lt_colLen.mpr k.isLt⟩
-
-@[simp] lemma rankCell_col {μ : YoungDiagram} (f : Cell μ → ℕ)
-    (hf : ∀ x y, col x = col y → f x = f y → x = y) (x : Cell μ) :
-    col (rankCell f hf x) = col x := rfl
-
-lemma rankCell_orderEmb {μ : YoungDiagram} (f : Cell μ → ℕ)
-    (hf : ∀ x y, col x = col y → f x = f y → x = y) (x : Cell μ) :
-    (columnValues f (col x)).orderEmbOfFin (card_columnValues f hf (col x))
-      ⟨row (rankCell f hf x), row_lt_colLen (rankCell f hf x)⟩ = f x := by
-  change ((columnValues f (col x)).orderIsoOfFin (card_columnValues f hf (col x))
-    (((columnValues f (col x)).orderIsoOfFin (card_columnValues f hf (col x))).symm
-      ⟨f x, mem_columnValues f x⟩)).val = f x
-  simp
-
-lemma rankCell_injective {μ : YoungDiagram} (f : Cell μ → ℕ)
-    (hf : ∀ x y, col x = col y → f x = f y → x = y) :
-    Function.Injective (rankCell f hf) := by
-  intro x y h
-  have hc : col x = col y := by simpa only [rankCell_col] using congrArg (fun z : Cell μ => col z) h
-  apply hf x y hc
-  have hx := rankCell_orderEmb f hf x
-  have hy := rankCell_orderEmb f hf y
-  change (columnValues f (col (rankCell f hf x))).orderEmbOfFin
-    (card_columnValues f hf (col (rankCell f hf x)))
-      ⟨row (rankCell f hf x), row_lt_colLen (rankCell f hf x)⟩ = f x at hx
-  rw [h] at hx
-  exact hx.symm.trans hy
-
-noncomputable def rankPerm {μ : YoungDiagram} (f : Cell μ → ℕ)
-    (hf : ∀ x y, col x = col y → f x = f y → x = y) : Equiv.Perm (Cell μ) :=
-  Equiv.ofBijective (rankCell f hf) ((Finite.injective_iff_bijective).mp (rankCell_injective f hf))
-
-lemma rankCell_row_le {μ : YoungDiagram} (f : Cell μ → ℕ)
-    (hf : ∀ x y, col x = col y → f x = f y → x = y) (x : Cell μ) :
-    row (rankCell f hf x) ≤ f x := by
-  rw [←rankCell_orderEmb f hf x]
-  exact strictMono_fin_nat_ge _ (OrderEmbedding.strictMono _) _
-
-theorem column_transporter (μ : YoungDiagram) (p : Equiv.Perm (Cell μ))
-    (hp : ∀ x y, col x = col y → row (p x) = row (p y) → x = y) :
-    ∃ c : Equiv.Perm (Cell μ), (∀ x, col (c x) = col x) ∧
-      ∀ x, row (c x) = row (p x) := by
-  classical
-  let c := rankPerm (fun x => row (p x)) hp
-  refine ⟨c, fun x => rfl, ?_⟩
-  have hle (x : Cell μ) : row (c x) ≤ row (p x) := rankCell_row_le _ hp x
-  have hsum : ∑ x, row (c x) = ∑ x, row (p x) := by
-    rw [Equiv.sum_comp, Equiv.sum_comp]
-  exact fun x => (Finset.sum_eq_sum_iff_of_le (fun x _ => hle x)).mp hsum x (Finset.mem_univ _)
-
-
-@[simp] lemma tabloidAct_inv {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) :
-    tabloidAct p⁻¹ = (tabloidAct p)⁻¹ := by
-  ext f x
-  simp [tabloidAct,Equiv.Perm.inv_def]
-
-@[simp] lemma tabloidRep_apply {μ : YoungDiagram} (p : Equiv.Perm (Cell μ))
-    (v : Tabloid μ → ℂ) (f : Tabloid μ) : tabloidRep μ p v f = v (tabloidAct p⁻¹ f) := rfl
-
-lemma tabloidRep_delta {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) (f : Tabloid μ) :
-    tabloidRep μ p (delta f) = delta (tabloidAct p f) := by
-  classical
-  ext g
-  simp only [tabloidRep_apply,delta,Equiv.Perm.inv_def]
-  congr 1
-  exact propext (Equiv.symm_apply_eq (tabloidAct p))
-
-@[simp] lemma mem_colGroup {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) :
-    p ∈ colGroup μ ↔ ∀ x, col (p x) = col x := Iff.rfl
-
-lemma colGroup_row_stabilizer {μ : YoungDiagram} (c : colGroup μ)
-    (hc : tabloidAct c (baseTabloid μ) = baseTabloid μ) : (c : Equiv.Perm (Cell μ)) = 1 := by
-  apply Equiv.ext
-  intro x
-  have hr := congrArg (fun f : Tabloid μ => ((f.1 ((c : Equiv.Perm _) x)) : ℕ)) hc
-  simp only [tabloidAct_apply,Equiv.Perm.inv_def,Equiv.symm_apply_apply,baseTabloid,rowIndex] at hr
-  apply Subtype.ext
-  exact Prod.ext hr.symm (c.2 x)
-
-lemma tabloid_column_transporter {μ : YoungDiagram} (f : Tabloid μ)
-    (hf : ∀ x y, col x = col y → f.1 x = f.1 y → x = y) :
-    ∃ c : colGroup μ, f = tabloidAct c (baseTabloid μ) := by
-  obtain ⟨p,hp⟩ := f.2
-  have hp' : ∀ x y, col x = col y → row (p x) = row (p y) → x = y := by
-    intro x y hcol hrow
-    apply hf x y hcol
-    rw [hp]
-    exact Fin.ext hrow
-  obtain ⟨c,hcol,hrow⟩ := column_transporter μ p hp'
-  refine ⟨⟨c⁻¹, (colGroup μ).inv_mem hcol⟩, ?_⟩
-  apply Subtype.ext
-  funext x
-  apply Fin.ext
-  simp only [tabloidAct_apply,baseTabloid,rowIndex,inv_inv,hp,Function.comp_apply]
-  exact (hrow x).symm
-
-
-@[simp] lemma permSign_one {α : Type*} [Fintype α] [DecidableEq α] :
-    permSign (1 : Equiv.Perm α) = 1 := by simp [permSign]
-@[simp] lemma permSign_mul {α : Type*} [Fintype α] [DecidableEq α] (p q : Equiv.Perm α) :
-    permSign (p*q) = permSign p * permSign q := by simp [permSign]
-@[simp] lemma permSign_inv {α : Type*} [Fintype α] [DecidableEq α] (p : Equiv.Perm α) :
-    permSign p⁻¹ = permSign p := by simp [permSign,Equiv.Perm.inv_def]
-lemma permSign_sq {α : Type*} [Fintype α] [DecidableEq α] (p : Equiv.Perm α) :
-    permSign p * permSign p = 1 := by
-  calc
-    permSign p * permSign p = permSign (p*p⁻¹) := by rw [permSign_mul,permSign_inv]
-    _ = 1 := by simp
-@[simp] lemma permSign_swap {α : Type*} [Fintype α] [DecidableEq α] {x y : α} (h : x ≠ y) :
-    permSign (Equiv.swap x y) = -1 := by simp [permSign, h]
-@[simp] lemma conj_permSign {α : Type*} [Fintype α] [DecidableEq α] (p : Equiv.Perm α) :
-    starRingEnd ℂ (permSign p) = permSign p := by simp [permSign]
-
-variable {α V : Type*} [Fintype α] [DecidableEq α] [AddCommGroup V] [Module ℂ V]
-
-lemma alternator_mul (C : Subgroup (Equiv.Perm α))
-    (ρ : Representation ℂ (Equiv.Perm α) V) (c : C) :
-    alternator C ρ * ρ c = permSign (c : Equiv.Perm α) • alternator C ρ := by
-  classical
-  rw [alternator,Finset.sum_mul,Finset.smul_sum]
-  apply Fintype.sum_equiv (Equiv.mulRight c)
-  intro a
-  simp only [smul_mul_assoc,←map_mul,Equiv.coe_mulRight,Subgroup.coe_mul,permSign_mul,smul_smul]
-  congr 1
-  have h := permSign_sq (c : Equiv.Perm α)
-  calc
-    permSign (a : Equiv.Perm α) = permSign (a : Equiv.Perm α)*1 := by ring
-    _ = permSign (a : Equiv.Perm α)*(permSign (c : Equiv.Perm α)*permSign (c : Equiv.Perm α)) := by rw [h]
-    _ = _ := by ring
-
-lemma mul_alternator (C : Subgroup (Equiv.Perm α))
-    (ρ : Representation ℂ (Equiv.Perm α) V) (c : C) :
-    ρ c * alternator C ρ = permSign (c : Equiv.Perm α) • alternator C ρ := by
-  classical
-  rw [alternator,Finset.mul_sum,Finset.smul_sum]
-  apply Fintype.sum_equiv (Equiv.mulLeft c)
-  intro a
-  simp only [mul_smul_comm,←map_mul,Equiv.coe_mulLeft,Subgroup.coe_mul,permSign_mul,smul_smul]
-  congr 1
-  have h := permSign_sq (c : Equiv.Perm α)
-  calc
-    permSign (a : Equiv.Perm α) = 1*permSign (a : Equiv.Perm α) := by ring
-    _ = (permSign (c : Equiv.Perm α)*permSign (c : Equiv.Perm α))*permSign (a : Equiv.Perm α) := by rw [h]
-    _ = _ := by ring
-
-lemma alternator_apply_action (C : Subgroup (Equiv.Perm α))
-    (ρ : Representation ℂ (Equiv.Perm α) V) (c : C) (v : V) :
-    alternator C ρ (ρ c v) = permSign (c : Equiv.Perm α) • alternator C ρ v := by
-  exact congrArg (fun T : Module.End ℂ V => T v) (alternator_mul C ρ c)
-
-lemma action_alternator_apply (C : Subgroup (Equiv.Perm α))
-    (ρ : Representation ℂ (Equiv.Perm α) V) (c : C) (v : V) :
-    ρ c (alternator C ρ v) = permSign (c : Equiv.Perm α) • alternator C ρ v := by
-  exact congrArg (fun T : Module.End ℂ V => T v) (mul_alternator C ρ c)
-
-lemma alternator_eq_zero_of_odd_stabilizer (C : Subgroup (Equiv.Perm α))
-    (ρ : Representation ℂ (Equiv.Perm α) V) (c : C) (v : V)
-    (hc : permSign (c : Equiv.Perm α) = -1) (hv : ρ c v = v) :
-    alternator C ρ v = 0 := by
-  have h := alternator_apply_action C ρ c v
-  rw [hv,hc,neg_one_smul] at h
-  have h2 : (2:ℂ) • alternator C ρ v = 0 := by
-    rw [two_smul]
-    nth_rw 1 [h]
-    exact neg_add_cancel _
-  exact (smul_eq_zero.mp h2).resolve_left (by norm_num)
-
-end Thorp.Specht
-namespace Thorp.Specht
-open scoped BigOperators Classical
-
-lemma swap_mem_colGroup {μ : YoungDiagram} (x y : Cell μ) (hc : col x = col y) :
-    Equiv.swap x y ∈ colGroup μ := by
-  classical
-  intro z
-  by_cases hx : z=x
-  · subst z; simpa using hc.symm
-  by_cases hy : z=y
-  · subst z; simpa using hc
-  simp [Equiv.swap_apply_of_ne_of_ne hx hy]
-
-lemma swap_fixes_tabloid {μ : YoungDiagram} (f : Tabloid μ) (x y : Cell μ)
-    (hxy : f.1 x = f.1 y) : tabloidAct (Equiv.swap x y) f = f := by
-  classical
-  apply Subtype.ext
-  funext z
-  simp only [tabloidAct_apply, Equiv.swap_inv]
-  by_cases hx : z=x
-  · subst z; simpa using hxy.symm
-  by_cases hy : z=y
-  · subst z; simpa using hxy
-  simp [Equiv.swap_apply_of_ne_of_ne hx hy]
-
-lemma alternator_delta_rank_one {μ : YoungDiagram} (f : Tabloid μ) :
-    ∃ z : ℂ, alternator (colGroup μ) (tabloidRep μ) (delta f) = z • polytabloid μ := by
-  classical
-  by_cases hf : ∀ x y, col x = col y → f.1 x = f.1 y → x = y
-  · obtain ⟨c,rfl⟩ := tabloid_column_transporter f hf
-    refine ⟨permSign (c:Equiv.Perm (Cell μ)), ?_⟩
-    rw [←tabloidRep_delta]
-    exact alternator_apply_action _ _ c _
-  · push Not at hf
-    obtain ⟨x,y,hcol,hrow,hne⟩ := hf
-    refine ⟨0, ?_⟩
-    rw [zero_smul]
-    apply alternator_eq_zero_of_odd_stabilizer _ _ ⟨Equiv.swap x y,swap_mem_colGroup x y hcol⟩
-    · exact permSign_swap hne
-    · rw [tabloidRep_delta,swap_fixes_tabloid f x y hrow]
-
-lemma polytabloid_formula (μ : YoungDiagram) :
-    polytabloid μ = ∑ c : colGroup μ, permSign (c:Equiv.Perm (Cell μ)) •
-      delta (tabloidAct c (baseTabloid μ)) := by
-  classical
-  simp only [polytabloid,alternator,LinearMap.sum_apply,LinearMap.smul_apply,tabloidRep_delta]
-
-lemma polytabloid_base (μ : YoungDiagram) : polytabloid μ (baseTabloid μ) = 1 := by
-  classical
-  rw [polytabloid_formula,Finset.sum_apply]
-  rw [Finset.sum_eq_single (1 : colGroup μ)]
-  · simp [delta]
-  · intro c hc hne
-    have hn : baseTabloid μ ≠ tabloidAct c (baseTabloid μ) := by
-      intro h
-      apply hne
-      exact Subtype.ext (colGroup_row_stabilizer c h.symm)
-    simp [Pi.smul_apply,delta,hn]
-  · simp
-
-lemma polytabloid_ne_zero (μ : YoungDiagram) : polytabloid μ ≠ 0 := by
-  intro h
-  have hv := congrArg (fun v : Tabloid μ → ℂ => v (baseTabloid μ)) h
-  simp [polytabloid_base] at hv
-
-lemma delta_decomposition {μ : YoungDiagram} (v : Tabloid μ → ℂ) :
-    v = ∑ f : Tabloid μ, v f • delta f := by
-  classical
-  funext g
-  simp only [Finset.sum_apply,Pi.smul_apply,delta,smul_eq_mul,mul_ite,mul_one,mul_zero]
-  simp
-
-theorem alternator_rank_one (μ : YoungDiagram) (v : Tabloid μ → ℂ) :
-    ∃ z : ℂ, alternator (colGroup μ) (tabloidRep μ) v = z • polytabloid μ := by
-  classical
-  choose z hz using fun f : Tabloid μ => alternator_delta_rank_one f
-  refine ⟨∑ f, v f * z f, ?_⟩
-  conv_lhs => rw [delta_decomposition v]
-  simp only [map_sum,map_smul,hz,smul_smul,Finset.sum_smul]
-
-lemma alternator_rank_one_exact (μ : YoungDiagram) (v : Tabloid μ → ℂ) :
-    alternator (colGroup μ) (tabloidRep μ) v =
-      (alternator (colGroup μ) (tabloidRep μ) v (baseTabloid μ)) • polytabloid μ := by
-  obtain ⟨z,hz⟩ := alternator_rank_one μ v
-  rw [hz]
-  simp only [Pi.smul_apply,polytabloid_base,smul_eq_mul,mul_one]
-
-
-noncomputable def pairing {μ : YoungDiagram} (v w : Tabloid μ → ℂ) : ℂ :=
-  ∑ f, w f * starRingEnd ℂ (v f)
-
-lemma pairing_delta {μ : YoungDiagram} (f : Tabloid μ) (v : Tabloid μ → ℂ) :
-    pairing (delta f) v = v f := by
-  classical
-  simp [pairing,delta]
-
-lemma pairing_sum_left {μ : YoungDiagram} {ι : Type*} [Fintype ι]
-    (v : ι → Tabloid μ → ℂ) (w : Tabloid μ → ℂ) :
-    pairing (∑ i, v i) w = ∑ i, pairing (v i) w := by
-  simp only [pairing,Finset.sum_apply,map_sum,Finset.mul_sum]
-  exact Finset.sum_comm
-
-lemma pairing_smul_left {μ : YoungDiagram} (a : ℂ) (v w : Tabloid μ → ℂ) :
-    pairing (a • v) w = starRingEnd ℂ a * pairing v w := by
-  simp only [pairing,Pi.smul_apply,smul_eq_mul,map_mul,Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro f hf
-  ring
-
-lemma pairing_zero_left {μ : YoungDiagram} (v : Tabloid μ → ℂ) : pairing 0 v = 0 := by
-  simp [pairing]
-
-lemma pairing_add_left {μ : YoungDiagram} (u v w : Tabloid μ → ℂ) :
-    pairing (u+v) w = pairing u w + pairing v w := by
-  simp [pairing,mul_add,Finset.sum_add_distrib]
-
-lemma pairing_self_eq_zero {μ : YoungDiagram} (v : Tabloid μ → ℂ) :
-    pairing v v = 0 ↔ v = 0 := by
-  change inner ℂ (WithLp.toLp 2 v : EuclideanSpace ℂ (Tabloid μ)) (WithLp.toLp 2 v) = 0 ↔ v = 0
-  rw [inner_self_eq_zero]
-  constructor
-  · intro h; exact congrArg WithLp.ofLp h
-  · intro h; subst v; rfl
-
-lemma pairing_action {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) (v w : Tabloid μ → ℂ) :
-    pairing (tabloidRep μ p v) w = pairing v (tabloidRep μ p⁻¹ w) := by
-  classical
-  simp only [pairing,tabloidRep_apply,inv_inv]
-  apply Fintype.sum_equiv (tabloidAct p⁻¹)
-  intro f
-  have hcancel : tabloidAct p (tabloidAct p⁻¹ f) = f := by
-    rw [tabloidAct_inv]
-    exact (tabloidAct p).apply_symm_apply f
-  rw [hcancel]
-
-lemma alternator_base_pairing (μ : YoungDiagram) (v : Tabloid μ → ℂ) :
-    alternator (colGroup μ) (tabloidRep μ) v (baseTabloid μ) = pairing (polytabloid μ) v := by
-  classical
-  simp only [polytabloid_formula,pairing_sum_left,pairing_smul_left,conj_permSign,pairing_delta]
-  simp only [alternator,LinearMap.sum_apply,LinearMap.smul_apply,Finset.sum_apply,Pi.smul_apply,
-    smul_eq_mul,tabloidRep_apply]
-  apply Fintype.sum_equiv (Equiv.inv (colGroup μ))
-  intro c
-  simp only [Equiv.inv_apply,Subgroup.coe_inv,permSign_inv]
-
-theorem alternator_pairing (μ : YoungDiagram) (v : Tabloid μ → ℂ) :
-    alternator (colGroup μ) (tabloidRep μ) v = pairing (polytabloid μ) v • polytabloid μ := by
-  rw [alternator_rank_one_exact,alternator_base_pairing]
-
-
-lemma polytabloid_mem_space (μ : YoungDiagram) : polytabloid μ ∈ space μ := by
-  simpa using orbit_mem_space μ 1
-
-lemma alternator_mem_invariant (μ : YoungDiagram) (W : Submodule ℂ (Tabloid μ → ℂ))
-    (hW : ∀ p v, v ∈ W → tabloidRep μ p v ∈ W) (v : Tabloid μ → ℂ) (hv : v ∈ W) :
-    alternator (colGroup μ) (tabloidRep μ) v ∈ W := by
-  classical
-  simp only [alternator,LinearMap.sum_apply,LinearMap.smul_apply]
-  exact W.sum_mem (fun c _ => W.smul_mem _ (hW c v hv))
-
-theorem submodule_theorem (μ : YoungDiagram) (W : Submodule ℂ (Tabloid μ → ℂ))
-    (hW : ∀ p v, v ∈ W → tabloidRep μ p v ∈ W) :
-    space μ ≤ W ∨ ∀ v ∈ W, ∀ w ∈ space μ, pairing w v = 0 := by
-  classical
-  by_cases hSW : space μ ≤ W
-  · exact Or.inl hSW
-  right
-  have he : polytabloid μ ∉ W := by
-    intro he
-    apply hSW
-    apply Submodule.span_le.mpr
-    rintro _ ⟨p,rfl⟩
-    exact hW p _ he
-  have hpair : ∀ v ∈ W, pairing (polytabloid μ) v = 0 := by
-    intro v hv
-    by_contra hne
-    apply he
-    have hal := alternator_mem_invariant μ W hW v hv
-    rw [alternator_pairing] at hal
-    have hh := W.smul_mem (pairing (polytabloid μ) v)⁻¹ hal
-    simpa only [smul_smul,inv_mul_cancel₀ hne,one_smul] using hh
-  intro v hv w hw
-  induction hw using Submodule.span_induction with
-  | mem w hw =>
-    obtain ⟨p,rfl⟩ := hw
-    rw [pairing_action]
-    exact hpair _ (hW p⁻¹ v hv)
-  | zero => exact pairing_zero_left v
-  | add u w hu hw ihu ihw => rw [pairing_add_left,ihu,ihw,add_zero]
-  | smul a w hw ih => rw [pairing_smul_left,ih,mul_zero]
-
-theorem space_simple (μ : YoungDiagram) (W : Submodule ℂ (Tabloid μ → ℂ))
-    (hW : ∀ p v, v ∈ W → tabloidRep μ p v ∈ W) (hWS : W ≤ space μ) :
-    W = ⊥ ∨ W = space μ := by
-  obtain hSW | horth := submodule_theorem μ W hW
-  · exact Or.inr (le_antisymm hWS hSW)
-  · left
-    apply le_antisymm _ bot_le
-    intro v hv
-    exact (pairing_self_eq_zero v).mp (horth v hv v (hWS hv))
-
-
-lemma space_ne_bot (μ : YoungDiagram) : space μ ≠ ⊥ := by
-  intro h
-  have he := polytabloid_mem_space μ
-  rw [h,Submodule.mem_bot] at he
-  exact polytabloid_ne_zero μ he
-
-theorem representation_irreducible (μ : YoungDiagram) :
-    Representation.IsIrreducible (representation μ) := by
-  have hnt : (⊥ : Subrepresentation (representation μ)) ≠ ⊤ := by
-    intro h
-    have he : (⟨polytabloid μ,polytabloid_mem_space μ⟩ : space μ) ∈
-        (⊥ : Subrepresentation (representation μ)) := by rw [h]; trivial
-    have he' : (⟨polytabloid μ,polytabloid_mem_space μ⟩ : space μ) = 0 := he
-    exact polytabloid_ne_zero μ (congrArg Subtype.val he')
-  refine { exists_pair_ne := ⟨⊥,⊤,hnt⟩, eq_bot_or_eq_top := ?_ }
-  intro U
-  let W : Submodule ℂ (Tabloid μ → ℂ) := U.toSubmodule.map (space μ).subtype
-  have hW : ∀ p v, v ∈ W → tabloidRep μ p v ∈ W := by
-    intro p v hv
-    obtain ⟨u,hu,rfl⟩ := hv
-    exact ⟨representation μ p u,U.apply_mem_toSubmodule p hu,rfl⟩
-  have hWS : W ≤ space μ := by
-    intro v hv
-    obtain ⟨u,hu,rfl⟩ := hv
-    exact u.2
-  obtain hbot | htop := space_simple μ W hW hWS
-  · left
-    apply le_antisymm _ bot_le
-    intro u hu
-    have hmem : (u : Tabloid μ → ℂ) ∈ W := ⟨u,hu,rfl⟩
-    rw [hbot,Submodule.mem_bot] at hmem
-    change u = 0
-    exact Subtype.ext hmem
-  · right
-    apply le_antisymm le_top
-    intro u hu
-    have hmem : (u : Tabloid μ → ℂ) ∈ W := by rw [htop]; exact u.2
-    obtain ⟨v,hv,hvu⟩ := hmem
-    have heq : v = u := Subtype.ext hvu
-    exact heq ▸ hv
-
-
-section Symmetrizer
-variable {α V : Type*} [Fintype α] [DecidableEq α] [AddCommGroup V] [Module ℂ V]
-
-noncomputable def symmetrizer (H : Subgroup (Equiv.Perm α))
-    (ρ : Representation ℂ (Equiv.Perm α) V) : Module.End ℂ V :=
-  ∑ h : H, ρ h
-
-lemma symmetrizer_mul (H : Subgroup (Equiv.Perm α))
-    (ρ : Representation ℂ (Equiv.Perm α) V) (h : H) :
-    symmetrizer H ρ * ρ h = symmetrizer H ρ := by
-  rw [symmetrizer,Finset.sum_mul]
-  exact Fintype.sum_equiv (Equiv.mulRight h) _ _ (fun g => by simp [←map_mul])
-
-lemma symmetrizer_eq_zero_of_neg (H : Subgroup (Equiv.Perm α))
-    (ρ : Representation ℂ (Equiv.Perm α) V) (h : H) (v : V) (hv : ρ h v = -v) :
-    symmetrizer H ρ v = 0 := by
-  have hh := congrArg (fun T : Module.End ℂ V => T v) (symmetrizer_mul H ρ h)
-  change symmetrizer H ρ (ρ h v) = symmetrizer H ρ v at hh
-  rw [hv,map_neg] at hh
-  have h2 : (2:ℂ) • symmetrizer H ρ v = 0 := by
-    rw [two_smul]
-    nth_rw 1 [←hh]
-    exact neg_add_cancel _
-  exact (smul_eq_zero.mp h2).resolve_left (by norm_num)
-
-noncomputable def fiberGroup {β : Type*} (f : α → β) : Subgroup (Equiv.Perm α) where
-  carrier := {p | ∀ x, f (p x) = f x}
-  one_mem' := by intro x; rfl
-  mul_mem' := by intro p q hp hq x; exact (hp (q x)).trans (hq x)
-  inv_mem' := by intro p hp x; simpa using (hp (p⁻¹ x)).symm
-
-omit [Fintype α] in
-lemma swap_mem_fiberGroup {β : Type*} (f : α → β) (x y : α) (hf : f x = f y) :
-    Equiv.swap x y ∈ fiberGroup f := by
-  intro z
-  by_cases hx : z=x
-  · subst z; simpa using hf.symm
-  by_cases hy : z=y
-  · subst z; simpa using hf
-  simp [Equiv.swap_apply_of_ne_of_ne hx hy]
-
-end Symmetrizer
-
-lemma swap_polytabloid {μ : YoungDiagram} (x y : Cell μ) (hne : x ≠ y)
-    (hcol : col x = col y) :
-    tabloidRep μ (Equiv.swap x y) (polytabloid μ) = -polytabloid μ := by
-  have h := action_alternator_apply (colGroup μ) (tabloidRep μ)
-    ⟨Equiv.swap x y,swap_mem_colGroup x y hcol⟩ (delta (baseTabloid μ))
-  simpa only [polytabloid,permSign_swap hne,neg_one_smul] using h
-
-theorem tall_column_average_zero (μ : YoungDiagram) {β : Type*} [Fintype β]
-    (f : Cell μ → β) (j : ℕ) (hj : Fintype.card β < μ.colLen j)
-    (v : Tabloid μ → ℂ) (hv : v ∈ space μ) :
-    symmetrizer (fiberGroup f) (tabloidRep μ) v = 0 := by
-  induction hv using Submodule.span_induction with
-  | mem v hv =>
-    obtain ⟨p,rfl⟩ := hv
-    have hcard : Fintype.card β < Fintype.card {x : Cell μ // col x = j} := by
-      rw [Fintype.card_congr (columnEquiv μ j),Fintype.card_fin]
-      exact hj
-    obtain ⟨x,y,hne,hf⟩ := Fintype.exists_ne_map_eq_of_card_lt
-      (fun x : {x : Cell μ // col x = j} => f (p x.1)) hcard
-    have hxy : x.1 ≠ y.1 := fun h => hne (Subtype.ext h)
-    let h : fiberGroup f := ⟨Equiv.swap (p x.1) (p y.1),swap_mem_fiberGroup f _ _ hf⟩
-    apply symmetrizer_eq_zero_of_neg _ _ h
-    change tabloidRep μ (Equiv.swap (p x.1) (p y.1))
-      (tabloidRep μ p (polytabloid μ)) = -tabloidRep μ p (polytabloid μ)
-    rw [←Module.End.mul_apply,←map_mul,←Equiv.mul_swap_eq_swap_mul]
-    rw [map_mul,Module.End.mul_apply,swap_polytabloid x.1 y.1 hxy (x.2.trans y.2.symm),map_neg]
-  | zero => exact map_zero _
-  | add v w hv hw ihv ihw => simp only [map_add,ihv,ihw,add_zero]
-  | smul a v hv ih => simp only [map_smul,ih,smul_zero]
-
-
-noncomputable def rowEquiv (μ : YoungDiagram) (i : ℕ) :
-    {x : Cell μ // row x = i} ≃ Fin (μ.rowLen i) where
-  toFun x := ⟨col x.1, by
-    have h := YoungDiagram.mem_iff_lt_rowLen.mp x.1.2
-    change col x.1 < μ.rowLen (row x.1) at h
-    simpa only [x.2] using h⟩
-  invFun j := ⟨⟨(i,j),YoungDiagram.mem_iff_lt_rowLen.mpr j.2⟩,rfl⟩
-  left_inv x := by apply Subtype.ext; apply Subtype.ext; exact Prod.ext x.2.symm rfl
-  right_inv j := rfl
-
-def rowCost (μ : YoungDiagram) : ℕ := ∑ x : Cell μ, row x
-
-lemma eq_of_row_preserving (μ ν : YoungDiagram) (e : Cell μ ≃ Cell ν)
-    (he : ∀ x, row (e x) = row x) : μ = ν := by
-  have hlen (i : ℕ) : μ.rowLen i = ν.rowLen i := by
-    let ei : {x : Cell μ // row x = i} ≃ {x : Cell ν // row x = i} :=
-      { toFun := fun x => ⟨e x.1,(he x.1).trans x.2⟩
-        invFun := fun x => ⟨e.symm x.1,by
-          have h := he (e.symm x.1)
-          simp only [Equiv.apply_symm_apply] at h
-          exact h.symm.trans x.2⟩
-        left_inv := fun x => by apply Subtype.ext; exact e.symm_apply_apply _
-        right_inv := fun x => by apply Subtype.ext; exact e.apply_symm_apply _ }
-    simpa only [Fintype.card_fin] using Fintype.card_congr
-      ((rowEquiv μ i).symm.trans (ei.trans (rowEquiv ν i)))
-  apply YoungDiagram.ext
-  ext ⟨i,j⟩
-  simp only [YoungDiagram.mem_cells,YoungDiagram.mem_iff_lt_rowLen,hlen]
-
-lemma column_injective_rowCost (μ ν : YoungDiagram) (e : Cell μ ≃ Cell ν)
-    (h : ∀ x y : Cell μ, col x = col y → row (e x) = row (e y) → x = y) :
-    rowCost μ ≤ rowCost ν ∧ (rowCost μ = rowCost ν → μ = ν) := by
-  let c := rankPerm (fun x => row (e x)) h
-  have hle (x : Cell μ) : row (c x) ≤ row (e x) := rankCell_row_le _ h x
-  have hsumc : ∑ x, row (c x) = rowCost μ := Equiv.sum_comp c _
-  have hsume : ∑ x, row (e x) = rowCost ν := Equiv.sum_comp e _
-  constructor
-  · rw [←hsumc,←hsume]
-    exact Finset.sum_le_sum (fun x _ => hle x)
-  · intro heq
-    have he : ∀ x, row (c x) = row (e x) := by
-      apply fun x => (Finset.sum_eq_sum_iff_of_le (fun x _ => hle x)).mp
-        (hsumc.trans (heq.trans hsume.symm)) x (Finset.mem_univ _)
-    apply eq_of_row_preserving μ ν (c.symm.trans e)
-    intro x
-    simpa using (he (c.symm x)).symm
-
-lemma swap_relabel {α β : Type*} [DecidableEq α] [DecidableEq β]
-    (e : α ≃ β) (x y : α) : e.permCongr (Equiv.swap x y) = Equiv.swap (e x) (e y) := by
-  ext z
-  simp only [Equiv.permCongr_apply]
-  exact e.injective.map_swap x y (e.symm z) |>.trans (by simp)
-
-lemma sign_vector_rowCost (μ ν : YoungDiagram) (e : Cell μ ≃ Cell ν)
-    (v : Tabloid ν → ℂ) (hv : v ≠ 0)
-    (hs : ∀ c : colGroup μ,
-      tabloidRep ν (e.permCongr c) v = permSign (c : Equiv.Perm (Cell μ)) • v) :
-    rowCost μ ≤ rowCost ν ∧ (rowCost μ = rowCost ν → μ = ν) := by
-  obtain ⟨f,hf⟩ : ∃ f, v f ≠ 0 := by
-    by_contra h
-    push Not at h
-    exact hv (funext h)
-  obtain ⟨p,hp⟩ := f.2
-  apply column_injective_rowCost μ ν (e.trans p)
-  intro x y hcol hrow
-  by_contra hne
-  have hfix : tabloidAct (Equiv.swap (e x) (e y)) f = f := by
-    apply swap_fixes_tabloid
-    rw [hp]
-    exact Fin.ext hrow
-  have heq := congrArg (fun w : Tabloid ν → ℂ => w f)
-    (hs ⟨Equiv.swap x y,swap_mem_colGroup x y hcol⟩)
-  rw [swap_relabel,permSign_swap hne] at heq
-  simp only [tabloidRep_apply,Equiv.swap_inv,hfix,Pi.smul_apply,neg_one_smul] at heq
-  apply hf
-  have h2 : (2:ℂ)*(v f) = 0 := by linear_combination heq
-  exact (mul_eq_zero.mp h2).resolve_left (by norm_num)
-
-lemma polytabloid_col_action (μ : YoungDiagram) (c : colGroup μ) :
-    tabloidRep μ c (polytabloid μ) = permSign (c:Equiv.Perm (Cell μ)) • polytabloid μ :=
-  action_alternator_apply _ _ c _
-
-theorem shapes_eq_of_equivariant_equiv (μ ν : YoungDiagram) (e : Cell μ ≃ Cell ν)
-    (T : space μ ≃ₗ[ℂ] space ν)
-    (hT : ∀ p v, T (representation μ p v) = representation ν (e.permCongr p) (T v)) :
-    μ = ν := by
-  let u : space μ := ⟨polytabloid μ,polytabloid_mem_space μ⟩
-  have hu : u ≠ 0 := fun h => polytabloid_ne_zero μ (congrArg Subtype.val h)
-  have hTcol (c : colGroup μ) :
-      tabloidRep ν (e.permCongr c) (T u) = permSign (c:Equiv.Perm (Cell μ)) • (T u : Tabloid ν → ℂ) := by
-    change (representation ν (e.permCongr c) (T u) : Tabloid ν → ℂ) = _
-    rw [←hT]
-    have hh : representation μ c u = permSign (c:Equiv.Perm (Cell μ)) • u :=
-      Subtype.ext (polytabloid_col_action μ c)
-    rw [hh,map_smul]
-    rfl
-  have hfwd := sign_vector_rowCost μ ν e (T u)
-    (fun h => hu (T.injective (by apply Subtype.ext; simpa using h))) hTcol
-  let w : space ν := ⟨polytabloid ν,polytabloid_mem_space ν⟩
-  have hw : w ≠ 0 := fun h => polytabloid_ne_zero ν (congrArg Subtype.val h)
-  have hTi (p : Equiv.Perm (Cell ν)) (v : space ν) :
-      T.symm (representation ν p v) = representation μ (e.symm.permCongr p) (T.symm v) := by
-    apply T.injective
-    rw [T.apply_symm_apply,hT,T.apply_symm_apply]
-    have hp : e.permCongr (e.symm.permCongr p) = p := by
-      apply Equiv.ext
-      intro x
-      simp [Equiv.permCongr_apply]
-    rw [hp]
-  have hTicol (c : colGroup ν) :
-      tabloidRep μ (e.symm.permCongr c) (T.symm w) =
-        permSign (c:Equiv.Perm (Cell ν)) • (T.symm w : Tabloid μ → ℂ) := by
-    change (representation μ (e.symm.permCongr c) (T.symm w) : Tabloid μ → ℂ) = _
-    rw [←hTi]
-    have hh : representation ν c w = permSign (c:Equiv.Perm (Cell ν)) • w :=
-      Subtype.ext (polytabloid_col_action ν c)
-    rw [hh,map_smul]
-    rfl
-  have hbwd := sign_vector_rowCost ν μ e.symm (T.symm w)
-    (fun h => hw (T.symm.injective (by apply Subtype.ext; simpa using h))) hTicol
-  exact hfwd.2 (Nat.le_antisymm hfwd.1 hbwd.1)
-
-
-noncomputable def cellRowsEquiv (μ : YoungDiagram) :
-    Cell μ ≃ (i : Fin (μ.colLen 0)) × Fin (μ.rowLen i) where
-  toFun x := ⟨rowIndex x,⟨col x,YoungDiagram.mem_iff_lt_rowLen.mp x.2⟩⟩
-  invFun p := ⟨(p.1,p.2),YoungDiagram.mem_iff_lt_rowLen.mpr p.2.2⟩
-  left_inv x := by apply Subtype.ext; rfl
-  right_inv p := by rcases p with ⟨⟨i,hi⟩,⟨j,hj⟩⟩; rfl
-
-lemma card_eq_sum_rowLen (μ : YoungDiagram) : μ.card = ∑ i : Fin (μ.colLen 0), μ.rowLen i := by
-  rw [←card_cell,Fintype.card_congr (cellRowsEquiv μ),Fintype.card_sigma]
-  simp only [Fintype.card_fin]
-
-lemma sum_rowLens (μ : YoungDiagram) : μ.rowLens.sum = μ.card := by
-  rw [card_eq_sum_rowLen,YoungDiagram.rowLens]
-  rw [Fin.sum_univ_eq_sum_range]
-  have hh (n : ℕ) : (List.map μ.rowLen (List.range n)).sum = ∑ i ∈ Finset.range n, μ.rowLen i := by
-    induction n with
-    | zero => simp
-    | succ n ih => simp [List.range_succ,List.map_append,Finset.sum_range_succ,ih]
-  exact hh _
-
-noncomputable def diagramPartition (μ : YoungDiagram) : Nat.Partition μ.card where
-  parts := μ.rowLens
-  parts_pos := fun {i} hi => μ.pos_of_mem_rowLens i (by simpa using hi)
-  parts_sum := by simp
-
-noncomputable def partitionDiagram {N : ℕ} (p : Nat.Partition N) : YoungDiagram :=
-  YoungDiagram.ofRowLens (p.parts.sort (· ≥ ·)) (Multiset.pairwise_sort _ _).sortedGE
-
-lemma partitionDiagram_rowLens {N : ℕ} (p : Nat.Partition N) :
-    (partitionDiagram p).rowLens = p.parts.sort (· ≥ ·) := by
-  apply YoungDiagram.rowLens_ofRowLens_eq_self
-  intro x hx
-  exact p.parts_pos ((Multiset.mem_sort _).mp hx)
-
-lemma partitionDiagram_card {N : ℕ} (p : Nat.Partition N) : (partitionDiagram p).card = N := by
-  rw [←sum_rowLens,partitionDiagram_rowLens]
-  change (↑(p.parts.sort (· ≥ ·)) : Multiset ℕ).sum = N
-  rw [Multiset.sort_eq,p.parts_sum]
-
-lemma partitionDiagram_diagramPartition (μ : YoungDiagram) :
-    partitionDiagram (diagramPartition μ) = μ := by
-  apply YoungDiagram.equivListRowLens.injective
-  apply Subtype.ext
-  change (partitionDiagram (diagramPartition μ)).rowLens = μ.rowLens
-  rw [partitionDiagram_rowLens]
-  change (↑μ.rowLens : Multiset ℕ).sort (· ≥ ·) = μ.rowLens
-  rw [Multiset.coe_sort,List.mergeSort_eq_self _ μ.rowLens_sorted.pairwise]
-
-abbrev Shapes (N : ℕ) := {μ : YoungDiagram // μ.card = N}
-
-end Thorp.Specht
-
-end ThorpNine.Tail
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Tail.ModifiedRoutingMoment
+
+namespace OAI
+
+namespace ThorpNine.Tail
+
+namespace Thorp
+
+noncomputable def modifiedReverseEquiv (L r : ℕ) : Equiv.Perm (ModifiedCoins L r) where
+  toFun := modifiedReverseCoins L r
+  invFun := modifiedReverseCoins L r
+  left_inv := modifiedReverseCoins_involutive L r
+  right_inv := modifiedReverseCoins_involutive L r
+
+lemma modifiedPerm_reverse (L r : ℕ) (E : Finset (Card r)) (ω : ModifiedCoins L r) :
+    modifiedPerm L r E (modifiedReverseCoins L r ω) = (modifiedPerm L r E ω)⁻¹ := by
+  induction r with
+  | zero =>
+      rw [modifiedPerm_zero,modifiedPerm_zero]
+      split_ifs
+      · simp only [inv_one]
+      · simp only [modifiedReverseCoins,mul_inv_rev,inv_inv]
+        simp
+  | succ r ih =>
+      obtain ⟨z,rfl⟩ := (modifiedStepEquiv L r).symm.surjective ω
+      rcases z with ⟨⟨ω₀,ω₁⟩,⟨ξ,η⟩⟩
+      simp only [modifiedReverseCoins,Equiv.apply_symm_apply,modifiedPerm_step,
+        mul_inv_rev,pairSwitch_inv,childLift_inv]
+      rw [mul_assoc]
+      congr 1
+      congr 1
+      apply congrArg childLift
+      funext b
+      cases b <;> simp only [Bool.false_eq_true,↓reduceIte,ih]
+
+lemma modified_kernel_symmetric (L r : ℕ) (E : Finset (Card r))
+    {ι : Type*} [Fintype ι] (e f : ι ↪ Card (L+r)) :
+    PairRouting.tupleProbability (modifiedPerm L r E) e f =
+      PairRouting.tupleProbability (modifiedPerm L r E) f e :=
+  PairRouting.tupleProbability_symmetric _ (modifiedReverseEquiv L r)
+    (modifiedPerm_reverse L r E) e f
+
+end Thorp
+
+namespace Thorp.Specht
+open scoped BigOperators Classical
+
+@[simp] lemma card_cell (μ : YoungDiagram) : Fintype.card (Cell μ) = μ.card := by
+  simp [Cell, YoungDiagram.card]
+
+noncomputable def columnEquiv (μ : YoungDiagram) (j : ℕ) :
+    {x : Cell μ // col x = j} ≃ Fin (μ.colLen j) where
+  toFun x := ⟨row x.1, by simpa [x.2] using row_lt_colLen x.1⟩
+  invFun i := ⟨⟨(i,j), YoungDiagram.mem_iff_lt_colLen.mpr i.2⟩, rfl⟩
+  left_inv x := by apply Subtype.ext; apply Subtype.ext; exact Prod.ext rfl x.2.symm
+  right_inv i := by rfl
+
+noncomputable def columnValues {μ : YoungDiagram} (f : Cell μ → ℕ) (j : ℕ) : Finset ℕ :=
+  (Finset.univ.filter (fun x => col x = j)).image f
+
+lemma mem_columnValues {μ : YoungDiagram} (f : Cell μ → ℕ) (x : Cell μ) :
+    f x ∈ columnValues f (col x) := by
+  classical
+  exact Finset.mem_image.mpr ⟨x, by simp, rfl⟩
+
+lemma card_columnValues {μ : YoungDiagram} (f : Cell μ → ℕ)
+    (hf : ∀ x y, col x = col y → f x = f y → x = y) (j : ℕ) :
+    (columnValues f j).card = μ.colLen j := by
+  classical
+  rw [columnValues, Finset.card_image_iff.mpr]
+  · rw [← Fintype.card_subtype]
+    exact Fintype.card_congr (columnEquiv μ j) |>.trans (Fintype.card_fin _)
+  intro x hx y hy he
+  exact hf x y ((Finset.mem_filter.mp hx).2.trans (Finset.mem_filter.mp hy).2.symm) he
+
+lemma strictMono_fin_nat_ge {n : ℕ} (f : Fin n → ℕ) (hf : StrictMono f) :
+    ∀ i, i.val ≤ f i := by
+  intro i
+  induction h : i.val using Nat.strong_induction_on generalizing i with
+  | h n ih =>
+    by_cases hn : n = 0
+    · omega
+    · have hn' : n-1 < i.val := by omega
+      let j : Fin _ := ⟨n-1, hn'.trans i.isLt⟩
+      have hj := ih (n-1) (by omega) j rfl
+      have hij := hf (show j < i from hn')
+      dsimp [j] at hj hij
+      omega
+
+noncomputable def rankCell {μ : YoungDiagram} (f : Cell μ → ℕ)
+    (hf : ∀ x y, col x = col y → f x = f y → x = y) (x : Cell μ) : Cell μ :=
+  let k := ((columnValues f (col x)).orderIsoOfFin (card_columnValues f hf (col x))).symm
+    ⟨f x, mem_columnValues f x⟩
+  ⟨(k.val,col x), YoungDiagram.mem_iff_lt_colLen.mpr k.isLt⟩
+
+@[simp] lemma rankCell_col {μ : YoungDiagram} (f : Cell μ → ℕ)
+    (hf : ∀ x y, col x = col y → f x = f y → x = y) (x : Cell μ) :
+    col (rankCell f hf x) = col x := rfl
+
+lemma rankCell_orderEmb {μ : YoungDiagram} (f : Cell μ → ℕ)
+    (hf : ∀ x y, col x = col y → f x = f y → x = y) (x : Cell μ) :
+    (columnValues f (col x)).orderEmbOfFin (card_columnValues f hf (col x))
+      ⟨row (rankCell f hf x), row_lt_colLen (rankCell f hf x)⟩ = f x := by
+  change ((columnValues f (col x)).orderIsoOfFin (card_columnValues f hf (col x))
+    (((columnValues f (col x)).orderIsoOfFin (card_columnValues f hf (col x))).symm
+      ⟨f x, mem_columnValues f x⟩)).val = f x
+  simp
+
+lemma rankCell_injective {μ : YoungDiagram} (f : Cell μ → ℕ)
+    (hf : ∀ x y, col x = col y → f x = f y → x = y) :
+    Function.Injective (rankCell f hf) := by
+  intro x y h
+  have hc : col x = col y := by simpa only [rankCell_col] using congrArg (fun z : Cell μ => col z) h
+  apply hf x y hc
+  have hx := rankCell_orderEmb f hf x
+  have hy := rankCell_orderEmb f hf y
+  change (columnValues f (col (rankCell f hf x))).orderEmbOfFin
+    (card_columnValues f hf (col (rankCell f hf x)))
+      ⟨row (rankCell f hf x), row_lt_colLen (rankCell f hf x)⟩ = f x at hx
+  rw [h] at hx
+  exact hx.symm.trans hy
+
+noncomputable def rankPerm {μ : YoungDiagram} (f : Cell μ → ℕ)
+    (hf : ∀ x y, col x = col y → f x = f y → x = y) : Equiv.Perm (Cell μ) :=
+  Equiv.ofBijective (rankCell f hf) ((Finite.injective_iff_bijective).mp (rankCell_injective f hf))
+
+lemma rankCell_row_le {μ : YoungDiagram} (f : Cell μ → ℕ)
+    (hf : ∀ x y, col x = col y → f x = f y → x = y) (x : Cell μ) :
+    row (rankCell f hf x) ≤ f x := by
+  rw [←rankCell_orderEmb f hf x]
+  exact strictMono_fin_nat_ge _ (OrderEmbedding.strictMono _) _
+
+theorem column_transporter (μ : YoungDiagram) (p : Equiv.Perm (Cell μ))
+    (hp : ∀ x y, col x = col y → row (p x) = row (p y) → x = y) :
+    ∃ c : Equiv.Perm (Cell μ), (∀ x, col (c x) = col x) ∧
+      ∀ x, row (c x) = row (p x) := by
+  classical
+  let c := rankPerm (fun x => row (p x)) hp
+  refine ⟨c, fun x => rfl, ?_⟩
+  have hle (x : Cell μ) : row (c x) ≤ row (p x) := rankCell_row_le _ hp x
+  have hsum : ∑ x, row (c x) = ∑ x, row (p x) := by
+    rw [Equiv.sum_comp, Equiv.sum_comp]
+  exact fun x => (Finset.sum_eq_sum_iff_of_le (fun x _ => hle x)).mp hsum x (Finset.mem_univ _)
+
+
+@[simp] lemma tabloidAct_inv {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) :
+    tabloidAct p⁻¹ = (tabloidAct p)⁻¹ := by
+  ext f x
+  simp [tabloidAct,Equiv.Perm.inv_def]
+
+@[simp] lemma tabloidRep_apply {μ : YoungDiagram} (p : Equiv.Perm (Cell μ))
+    (v : Tabloid μ → ℂ) (f : Tabloid μ) : tabloidRep μ p v f = v (tabloidAct p⁻¹ f) := rfl
+
+lemma tabloidRep_delta {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) (f : Tabloid μ) :
+    tabloidRep μ p (delta f) = delta (tabloidAct p f) := by
+  classical
+  ext g
+  simp only [tabloidRep_apply,delta,Equiv.Perm.inv_def]
+  congr 1
+  exact propext (Equiv.symm_apply_eq (tabloidAct p))
+
+@[simp] lemma mem_colGroup {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) :
+    p ∈ colGroup μ ↔ ∀ x, col (p x) = col x := Iff.rfl
+
+lemma colGroup_row_stabilizer {μ : YoungDiagram} (c : colGroup μ)
+    (hc : tabloidAct c (baseTabloid μ) = baseTabloid μ) : (c : Equiv.Perm (Cell μ)) = 1 := by
+  apply Equiv.ext
+  intro x
+  have hr := congrArg (fun f : Tabloid μ => ((f.1 ((c : Equiv.Perm _) x)) : ℕ)) hc
+  simp only [tabloidAct_apply,Equiv.Perm.inv_def,Equiv.symm_apply_apply,baseTabloid,rowIndex] at hr
+  apply Subtype.ext
+  exact Prod.ext hr.symm (c.2 x)
+
+lemma tabloid_column_transporter {μ : YoungDiagram} (f : Tabloid μ)
+    (hf : ∀ x y, col x = col y → f.1 x = f.1 y → x = y) :
+    ∃ c : colGroup μ, f = tabloidAct c (baseTabloid μ) := by
+  obtain ⟨p,hp⟩ := f.2
+  have hp' : ∀ x y, col x = col y → row (p x) = row (p y) → x = y := by
+    intro x y hcol hrow
+    apply hf x y hcol
+    rw [hp]
+    exact Fin.ext hrow
+  obtain ⟨c,hcol,hrow⟩ := column_transporter μ p hp'
+  refine ⟨⟨c⁻¹, (colGroup μ).inv_mem hcol⟩, ?_⟩
+  apply Subtype.ext
+  funext x
+  apply Fin.ext
+  simp only [tabloidAct_apply,baseTabloid,rowIndex,inv_inv,hp,Function.comp_apply]
+  exact (hrow x).symm
+
+
+@[simp] lemma permSign_one {α : Type*} [Fintype α] [DecidableEq α] :
+    permSign (1 : Equiv.Perm α) = 1 := by simp [permSign]
+@[simp] lemma permSign_mul {α : Type*} [Fintype α] [DecidableEq α] (p q : Equiv.Perm α) :
+    permSign (p*q) = permSign p * permSign q := by simp [permSign]
+@[simp] lemma permSign_inv {α : Type*} [Fintype α] [DecidableEq α] (p : Equiv.Perm α) :
+    permSign p⁻¹ = permSign p := by simp [permSign,Equiv.Perm.inv_def]
+lemma permSign_sq {α : Type*} [Fintype α] [DecidableEq α] (p : Equiv.Perm α) :
+    permSign p * permSign p = 1 := by
+  calc
+    permSign p * permSign p = permSign (p*p⁻¹) := by rw [permSign_mul,permSign_inv]
+    _ = 1 := by simp
+@[simp] lemma permSign_swap {α : Type*} [Fintype α] [DecidableEq α] {x y : α} (h : x ≠ y) :
+    permSign (Equiv.swap x y) = -1 := by simp [permSign, h]
+@[simp] lemma conj_permSign {α : Type*} [Fintype α] [DecidableEq α] (p : Equiv.Perm α) :
+    starRingEnd ℂ (permSign p) = permSign p := by simp [permSign]
+
+variable {α V : Type*} [Fintype α] [DecidableEq α] [AddCommGroup V] [Module ℂ V]
+
+lemma alternator_mul (C : Subgroup (Equiv.Perm α))
+    (ρ : Representation ℂ (Equiv.Perm α) V) (c : C) :
+    alternator C ρ * ρ c = permSign (c : Equiv.Perm α) • alternator C ρ := by
+  classical
+  rw [alternator,Finset.sum_mul,Finset.smul_sum]
+  apply Fintype.sum_equiv (Equiv.mulRight c)
+  intro a
+  simp only [smul_mul_assoc,←map_mul,Equiv.coe_mulRight,Subgroup.coe_mul,permSign_mul,smul_smul]
+  congr 1
+  have h := permSign_sq (c : Equiv.Perm α)
+  calc
+    permSign (a : Equiv.Perm α) = permSign (a : Equiv.Perm α)*1 := by ring
+    _ = permSign (a : Equiv.Perm α)*(permSign (c : Equiv.Perm α)*permSign (c : Equiv.Perm α)) := by rw [h]
+    _ = _ := by ring
+
+lemma mul_alternator (C : Subgroup (Equiv.Perm α))
+    (ρ : Representation ℂ (Equiv.Perm α) V) (c : C) :
+    ρ c * alternator C ρ = permSign (c : Equiv.Perm α) • alternator C ρ := by
+  classical
+  rw [alternator,Finset.mul_sum,Finset.smul_sum]
+  apply Fintype.sum_equiv (Equiv.mulLeft c)
+  intro a
+  simp only [mul_smul_comm,←map_mul,Equiv.coe_mulLeft,Subgroup.coe_mul,permSign_mul,smul_smul]
+  congr 1
+  have h := permSign_sq (c : Equiv.Perm α)
+  calc
+    permSign (a : Equiv.Perm α) = 1*permSign (a : Equiv.Perm α) := by ring
+    _ = (permSign (c : Equiv.Perm α)*permSign (c : Equiv.Perm α))*permSign (a : Equiv.Perm α) := by rw [h]
+    _ = _ := by ring
+
+lemma alternator_apply_action (C : Subgroup (Equiv.Perm α))
+    (ρ : Representation ℂ (Equiv.Perm α) V) (c : C) (v : V) :
+    alternator C ρ (ρ c v) = permSign (c : Equiv.Perm α) • alternator C ρ v := by
+  exact congrArg (fun T : Module.End ℂ V => T v) (alternator_mul C ρ c)
+
+lemma action_alternator_apply (C : Subgroup (Equiv.Perm α))
+    (ρ : Representation ℂ (Equiv.Perm α) V) (c : C) (v : V) :
+    ρ c (alternator C ρ v) = permSign (c : Equiv.Perm α) • alternator C ρ v := by
+  exact congrArg (fun T : Module.End ℂ V => T v) (mul_alternator C ρ c)
+
+lemma alternator_eq_zero_of_odd_stabilizer (C : Subgroup (Equiv.Perm α))
+    (ρ : Representation ℂ (Equiv.Perm α) V) (c : C) (v : V)
+    (hc : permSign (c : Equiv.Perm α) = -1) (hv : ρ c v = v) :
+    alternator C ρ v = 0 := by
+  have h := alternator_apply_action C ρ c v
+  rw [hv,hc,neg_one_smul] at h
+  have h2 : (2:ℂ) • alternator C ρ v = 0 := by
+    rw [two_smul]
+    nth_rw 1 [h]
+    exact neg_add_cancel _
+  exact (smul_eq_zero.mp h2).resolve_left (by norm_num)
+
+end Thorp.Specht
+namespace Thorp.Specht
+open scoped BigOperators Classical
+
+lemma swap_mem_colGroup {μ : YoungDiagram} (x y : Cell μ) (hc : col x = col y) :
+    Equiv.swap x y ∈ colGroup μ := by
+  classical
+  intro z
+  by_cases hx : z=x
+  · subst z; simpa using hc.symm
+  by_cases hy : z=y
+  · subst z; simpa using hc
+  simp [Equiv.swap_apply_of_ne_of_ne hx hy]
+
+lemma swap_fixes_tabloid {μ : YoungDiagram} (f : Tabloid μ) (x y : Cell μ)
+    (hxy : f.1 x = f.1 y) : tabloidAct (Equiv.swap x y) f = f := by
+  classical
+  apply Subtype.ext
+  funext z
+  simp only [tabloidAct_apply, Equiv.swap_inv]
+  by_cases hx : z=x
+  · subst z; simpa using hxy.symm
+  by_cases hy : z=y
+  · subst z; simpa using hxy
+  simp [Equiv.swap_apply_of_ne_of_ne hx hy]
+
+lemma alternator_delta_rank_one {μ : YoungDiagram} (f : Tabloid μ) :
+    ∃ z : ℂ, alternator (colGroup μ) (tabloidRep μ) (delta f) = z • polytabloid μ := by
+  classical
+  by_cases hf : ∀ x y, col x = col y → f.1 x = f.1 y → x = y
+  · obtain ⟨c,rfl⟩ := tabloid_column_transporter f hf
+    refine ⟨permSign (c:Equiv.Perm (Cell μ)), ?_⟩
+    rw [←tabloidRep_delta]
+    exact alternator_apply_action _ _ c _
+  · push Not at hf
+    obtain ⟨x,y,hcol,hrow,hne⟩ := hf
+    refine ⟨0, ?_⟩
+    rw [zero_smul]
+    apply alternator_eq_zero_of_odd_stabilizer _ _ ⟨Equiv.swap x y,swap_mem_colGroup x y hcol⟩
+    · exact permSign_swap hne
+    · rw [tabloidRep_delta,swap_fixes_tabloid f x y hrow]
+
+lemma polytabloid_formula (μ : YoungDiagram) :
+    polytabloid μ = ∑ c : colGroup μ, permSign (c:Equiv.Perm (Cell μ)) •
+      delta (tabloidAct c (baseTabloid μ)) := by
+  classical
+  simp only [polytabloid,alternator,LinearMap.sum_apply,LinearMap.smul_apply,tabloidRep_delta]
+
+lemma polytabloid_base (μ : YoungDiagram) : polytabloid μ (baseTabloid μ) = 1 := by
+  classical
+  rw [polytabloid_formula,Finset.sum_apply]
+  rw [Finset.sum_eq_single (1 : colGroup μ)]
+  · simp [delta]
+  · intro c hc hne
+    have hn : baseTabloid μ ≠ tabloidAct c (baseTabloid μ) := by
+      intro h
+      apply hne
+      exact Subtype.ext (colGroup_row_stabilizer c h.symm)
+    simp [Pi.smul_apply,delta,hn]
+  · simp
+
+lemma polytabloid_ne_zero (μ : YoungDiagram) : polytabloid μ ≠ 0 := by
+  intro h
+  have hv := congrArg (fun v : Tabloid μ → ℂ => v (baseTabloid μ)) h
+  simp [polytabloid_base] at hv
+
+lemma delta_decomposition {μ : YoungDiagram} (v : Tabloid μ → ℂ) :
+    v = ∑ f : Tabloid μ, v f • delta f := by
+  classical
+  funext g
+  simp only [Finset.sum_apply,Pi.smul_apply,delta,smul_eq_mul,mul_ite,mul_one,mul_zero]
+  simp
+
+theorem alternator_rank_one (μ : YoungDiagram) (v : Tabloid μ → ℂ) :
+    ∃ z : ℂ, alternator (colGroup μ) (tabloidRep μ) v = z • polytabloid μ := by
+  classical
+  choose z hz using fun f : Tabloid μ => alternator_delta_rank_one f
+  refine ⟨∑ f, v f * z f, ?_⟩
+  conv_lhs => rw [delta_decomposition v]
+  simp only [map_sum,map_smul,hz,smul_smul,Finset.sum_smul]
+
+lemma alternator_rank_one_exact (μ : YoungDiagram) (v : Tabloid μ → ℂ) :
+    alternator (colGroup μ) (tabloidRep μ) v =
+      (alternator (colGroup μ) (tabloidRep μ) v (baseTabloid μ)) • polytabloid μ := by
+  obtain ⟨z,hz⟩ := alternator_rank_one μ v
+  rw [hz]
+  simp only [Pi.smul_apply,polytabloid_base,smul_eq_mul,mul_one]
+
+
+noncomputable def pairing {μ : YoungDiagram} (v w : Tabloid μ → ℂ) : ℂ :=
+  ∑ f, w f * starRingEnd ℂ (v f)
+
+lemma pairing_delta {μ : YoungDiagram} (f : Tabloid μ) (v : Tabloid μ → ℂ) :
+    pairing (delta f) v = v f := by
+  classical
+  simp [pairing,delta]
+
+lemma pairing_sum_left {μ : YoungDiagram} {ι : Type*} [Fintype ι]
+    (v : ι → Tabloid μ → ℂ) (w : Tabloid μ → ℂ) :
+    pairing (∑ i, v i) w = ∑ i, pairing (v i) w := by
+  simp only [pairing,Finset.sum_apply,map_sum,Finset.mul_sum]
+  exact Finset.sum_comm
+
+lemma pairing_smul_left {μ : YoungDiagram} (a : ℂ) (v w : Tabloid μ → ℂ) :
+    pairing (a • v) w = starRingEnd ℂ a * pairing v w := by
+  simp only [pairing,Pi.smul_apply,smul_eq_mul,map_mul,Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro f hf
+  ring
+
+lemma pairing_zero_left {μ : YoungDiagram} (v : Tabloid μ → ℂ) : pairing 0 v = 0 := by
+  simp [pairing]
+
+lemma pairing_add_left {μ : YoungDiagram} (u v w : Tabloid μ → ℂ) :
+    pairing (u+v) w = pairing u w + pairing v w := by
+  simp [pairing,mul_add,Finset.sum_add_distrib]
+
+lemma pairing_self_eq_zero {μ : YoungDiagram} (v : Tabloid μ → ℂ) :
+    pairing v v = 0 ↔ v = 0 := by
+  change inner ℂ (WithLp.toLp 2 v : EuclideanSpace ℂ (Tabloid μ)) (WithLp.toLp 2 v) = 0 ↔ v = 0
+  rw [inner_self_eq_zero]
+  constructor
+  · intro h; exact congrArg WithLp.ofLp h
+  · intro h; subst v; rfl
+
+lemma pairing_action {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) (v w : Tabloid μ → ℂ) :
+    pairing (tabloidRep μ p v) w = pairing v (tabloidRep μ p⁻¹ w) := by
+  classical
+  simp only [pairing,tabloidRep_apply,inv_inv]
+  apply Fintype.sum_equiv (tabloidAct p⁻¹)
+  intro f
+  have hcancel : tabloidAct p (tabloidAct p⁻¹ f) = f := by
+    rw [tabloidAct_inv]
+    exact (tabloidAct p).apply_symm_apply f
+  rw [hcancel]
+
+lemma alternator_base_pairing (μ : YoungDiagram) (v : Tabloid μ → ℂ) :
+    alternator (colGroup μ) (tabloidRep μ) v (baseTabloid μ) = pairing (polytabloid μ) v := by
+  classical
+  simp only [polytabloid_formula,pairing_sum_left,pairing_smul_left,conj_permSign,pairing_delta]
+  simp only [alternator,LinearMap.sum_apply,LinearMap.smul_apply,Finset.sum_apply,Pi.smul_apply,
+    smul_eq_mul,tabloidRep_apply]
+  apply Fintype.sum_equiv (Equiv.inv (colGroup μ))
+  intro c
+  simp only [Equiv.inv_apply,Subgroup.coe_inv,permSign_inv]
+
+theorem alternator_pairing (μ : YoungDiagram) (v : Tabloid μ → ℂ) :
+    alternator (colGroup μ) (tabloidRep μ) v = pairing (polytabloid μ) v • polytabloid μ := by
+  rw [alternator_rank_one_exact,alternator_base_pairing]
+
+
+lemma polytabloid_mem_space (μ : YoungDiagram) : polytabloid μ ∈ space μ := by
+  simpa using orbit_mem_space μ 1
+
+lemma alternator_mem_invariant (μ : YoungDiagram) (W : Submodule ℂ (Tabloid μ → ℂ))
+    (hW : ∀ p v, v ∈ W → tabloidRep μ p v ∈ W) (v : Tabloid μ → ℂ) (hv : v ∈ W) :
+    alternator (colGroup μ) (tabloidRep μ) v ∈ W := by
+  classical
+  simp only [alternator,LinearMap.sum_apply,LinearMap.smul_apply]
+  exact W.sum_mem (fun c _ => W.smul_mem _ (hW c v hv))
+
+theorem submodule_theorem (μ : YoungDiagram) (W : Submodule ℂ (Tabloid μ → ℂ))
+    (hW : ∀ p v, v ∈ W → tabloidRep μ p v ∈ W) :
+    space μ ≤ W ∨ ∀ v ∈ W, ∀ w ∈ space μ, pairing w v = 0 := by
+  classical
+  by_cases hSW : space μ ≤ W
+  · exact Or.inl hSW
+  right
+  have he : polytabloid μ ∉ W := by
+    intro he
+    apply hSW
+    apply Submodule.span_le.mpr
+    rintro _ ⟨p,rfl⟩
+    exact hW p _ he
+  have hpair : ∀ v ∈ W, pairing (polytabloid μ) v = 0 := by
+    intro v hv
+    by_contra hne
+    apply he
+    have hal := alternator_mem_invariant μ W hW v hv
+    rw [alternator_pairing] at hal
+    have hh := W.smul_mem (pairing (polytabloid μ) v)⁻¹ hal
+    simpa only [smul_smul,inv_mul_cancel₀ hne,one_smul] using hh
+  intro v hv w hw
+  induction hw using Submodule.span_induction with
+  | mem w hw =>
+    obtain ⟨p,rfl⟩ := hw
+    rw [pairing_action]
+    exact hpair _ (hW p⁻¹ v hv)
+  | zero => exact pairing_zero_left v
+  | add u w hu hw ihu ihw => rw [pairing_add_left,ihu,ihw,add_zero]
+  | smul a w hw ih => rw [pairing_smul_left,ih,mul_zero]
+
+theorem space_simple (μ : YoungDiagram) (W : Submodule ℂ (Tabloid μ → ℂ))
+    (hW : ∀ p v, v ∈ W → tabloidRep μ p v ∈ W) (hWS : W ≤ space μ) :
+    W = ⊥ ∨ W = space μ := by
+  obtain hSW | horth := submodule_theorem μ W hW
+  · exact Or.inr (le_antisymm hWS hSW)
+  · left
+    apply le_antisymm _ bot_le
+    intro v hv
+    exact (pairing_self_eq_zero v).mp (horth v hv v (hWS hv))
+
+
+lemma space_ne_bot (μ : YoungDiagram) : space μ ≠ ⊥ := by
+  intro h
+  have he := polytabloid_mem_space μ
+  rw [h,Submodule.mem_bot] at he
+  exact polytabloid_ne_zero μ he
+
+theorem representation_irreducible (μ : YoungDiagram) :
+    Representation.IsIrreducible (representation μ) := by
+  have hnt : (⊥ : Subrepresentation (representation μ)) ≠ ⊤ := by
+    intro h
+    have he : (⟨polytabloid μ,polytabloid_mem_space μ⟩ : space μ) ∈
+        (⊥ : Subrepresentation (representation μ)) := by rw [h]; trivial
+    have he' : (⟨polytabloid μ,polytabloid_mem_space μ⟩ : space μ) = 0 := he
+    exact polytabloid_ne_zero μ (congrArg Subtype.val he')
+  refine { exists_pair_ne := ⟨⊥,⊤,hnt⟩, eq_bot_or_eq_top := ?_ }
+  intro U
+  let W : Submodule ℂ (Tabloid μ → ℂ) := U.toSubmodule.map (space μ).subtype
+  have hW : ∀ p v, v ∈ W → tabloidRep μ p v ∈ W := by
+    intro p v hv
+    obtain ⟨u,hu,rfl⟩ := hv
+    exact ⟨representation μ p u,U.apply_mem_toSubmodule p hu,rfl⟩
+  have hWS : W ≤ space μ := by
+    intro v hv
+    obtain ⟨u,hu,rfl⟩ := hv
+    exact u.2
+  obtain hbot | htop := space_simple μ W hW hWS
+  · left
+    apply le_antisymm _ bot_le
+    intro u hu
+    have hmem : (u : Tabloid μ → ℂ) ∈ W := ⟨u,hu,rfl⟩
+    rw [hbot,Submodule.mem_bot] at hmem
+    change u = 0
+    exact Subtype.ext hmem
+  · right
+    apply le_antisymm le_top
+    intro u hu
+    have hmem : (u : Tabloid μ → ℂ) ∈ W := by rw [htop]; exact u.2
+    obtain ⟨v,hv,hvu⟩ := hmem
+    have heq : v = u := Subtype.ext hvu
+    exact heq ▸ hv
+
+
+section Symmetrizer
+variable {α V : Type*} [Fintype α] [DecidableEq α] [AddCommGroup V] [Module ℂ V]
+
+noncomputable def symmetrizer (H : Subgroup (Equiv.Perm α))
+    (ρ : Representation ℂ (Equiv.Perm α) V) : Module.End ℂ V :=
+  ∑ h : H, ρ h
+
+lemma symmetrizer_mul (H : Subgroup (Equiv.Perm α))
+    (ρ : Representation ℂ (Equiv.Perm α) V) (h : H) :
+    symmetrizer H ρ * ρ h = symmetrizer H ρ := by
+  rw [symmetrizer,Finset.sum_mul]
+  exact Fintype.sum_equiv (Equiv.mulRight h) _ _ (fun g => by simp [←map_mul])
+
+lemma symmetrizer_eq_zero_of_neg (H : Subgroup (Equiv.Perm α))
+    (ρ : Representation ℂ (Equiv.Perm α) V) (h : H) (v : V) (hv : ρ h v = -v) :
+    symmetrizer H ρ v = 0 := by
+  have hh := congrArg (fun T : Module.End ℂ V => T v) (symmetrizer_mul H ρ h)
+  change symmetrizer H ρ (ρ h v) = symmetrizer H ρ v at hh
+  rw [hv,map_neg] at hh
+  have h2 : (2:ℂ) • symmetrizer H ρ v = 0 := by
+    rw [two_smul]
+    nth_rw 1 [←hh]
+    exact neg_add_cancel _
+  exact (smul_eq_zero.mp h2).resolve_left (by norm_num)
+
+noncomputable def fiberGroup {β : Type*} (f : α → β) : Subgroup (Equiv.Perm α) where
+  carrier := {p | ∀ x, f (p x) = f x}
+  one_mem' := by intro x; rfl
+  mul_mem' := by intro p q hp hq x; exact (hp (q x)).trans (hq x)
+  inv_mem' := by intro p hp x; simpa using (hp (p⁻¹ x)).symm
+
+omit [Fintype α] in
+lemma swap_mem_fiberGroup {β : Type*} (f : α → β) (x y : α) (hf : f x = f y) :
+    Equiv.swap x y ∈ fiberGroup f := by
+  intro z
+  by_cases hx : z=x
+  · subst z; simpa using hf.symm
+  by_cases hy : z=y
+  · subst z; simpa using hf
+  simp [Equiv.swap_apply_of_ne_of_ne hx hy]
+
+end Symmetrizer
+
+lemma swap_polytabloid {μ : YoungDiagram} (x y : Cell μ) (hne : x ≠ y)
+    (hcol : col x = col y) :
+    tabloidRep μ (Equiv.swap x y) (polytabloid μ) = -polytabloid μ := by
+  have h := action_alternator_apply (colGroup μ) (tabloidRep μ)
+    ⟨Equiv.swap x y,swap_mem_colGroup x y hcol⟩ (delta (baseTabloid μ))
+  simpa only [polytabloid,permSign_swap hne,neg_one_smul] using h
+
+theorem tall_column_average_zero (μ : YoungDiagram) {β : Type*} [Fintype β]
+    (f : Cell μ → β) (j : ℕ) (hj : Fintype.card β < μ.colLen j)
+    (v : Tabloid μ → ℂ) (hv : v ∈ space μ) :
+    symmetrizer (fiberGroup f) (tabloidRep μ) v = 0 := by
+  induction hv using Submodule.span_induction with
+  | mem v hv =>
+    obtain ⟨p,rfl⟩ := hv
+    have hcard : Fintype.card β < Fintype.card {x : Cell μ // col x = j} := by
+      rw [Fintype.card_congr (columnEquiv μ j),Fintype.card_fin]
+      exact hj
+    obtain ⟨x,y,hne,hf⟩ := Fintype.exists_ne_map_eq_of_card_lt
+      (fun x : {x : Cell μ // col x = j} => f (p x.1)) hcard
+    have hxy : x.1 ≠ y.1 := fun h => hne (Subtype.ext h)
+    let h : fiberGroup f := ⟨Equiv.swap (p x.1) (p y.1),swap_mem_fiberGroup f _ _ hf⟩
+    apply symmetrizer_eq_zero_of_neg _ _ h
+    change tabloidRep μ (Equiv.swap (p x.1) (p y.1))
+      (tabloidRep μ p (polytabloid μ)) = -tabloidRep μ p (polytabloid μ)
+    rw [←Module.End.mul_apply,←map_mul,←Equiv.mul_swap_eq_swap_mul]
+    rw [map_mul,Module.End.mul_apply,swap_polytabloid x.1 y.1 hxy (x.2.trans y.2.symm),map_neg]
+  | zero => exact map_zero _
+  | add v w hv hw ihv ihw => simp only [map_add,ihv,ihw,add_zero]
+  | smul a v hv ih => simp only [map_smul,ih,smul_zero]
+
+
+noncomputable def rowEquiv (μ : YoungDiagram) (i : ℕ) :
+    {x : Cell μ // row x = i} ≃ Fin (μ.rowLen i) where
+  toFun x := ⟨col x.1, by
+    have h := YoungDiagram.mem_iff_lt_rowLen.mp x.1.2
+    change col x.1 < μ.rowLen (row x.1) at h
+    simpa only [x.2] using h⟩
+  invFun j := ⟨⟨(i,j),YoungDiagram.mem_iff_lt_rowLen.mpr j.2⟩,rfl⟩
+  left_inv x := by apply Subtype.ext; apply Subtype.ext; exact Prod.ext x.2.symm rfl
+  right_inv j := rfl
+
+def rowCost (μ : YoungDiagram) : ℕ := ∑ x : Cell μ, row x
+
+lemma eq_of_row_preserving (μ ν : YoungDiagram) (e : Cell μ ≃ Cell ν)
+    (he : ∀ x, row (e x) = row x) : μ = ν := by
+  have hlen (i : ℕ) : μ.rowLen i = ν.rowLen i := by
+    let ei : {x : Cell μ // row x = i} ≃ {x : Cell ν // row x = i} :=
+      { toFun := fun x => ⟨e x.1,(he x.1).trans x.2⟩
+        invFun := fun x => ⟨e.symm x.1,by
+          have h := he (e.symm x.1)
+          simp only [Equiv.apply_symm_apply] at h
+          exact h.symm.trans x.2⟩
+        left_inv := fun x => by apply Subtype.ext; exact e.symm_apply_apply _
+        right_inv := fun x => by apply Subtype.ext; exact e.apply_symm_apply _ }
+    simpa only [Fintype.card_fin] using Fintype.card_congr
+      ((rowEquiv μ i).symm.trans (ei.trans (rowEquiv ν i)))
+  apply YoungDiagram.ext
+  ext ⟨i,j⟩
+  simp only [YoungDiagram.mem_cells,YoungDiagram.mem_iff_lt_rowLen,hlen]
+
+lemma column_injective_rowCost (μ ν : YoungDiagram) (e : Cell μ ≃ Cell ν)
+    (h : ∀ x y : Cell μ, col x = col y → row (e x) = row (e y) → x = y) :
+    rowCost μ ≤ rowCost ν ∧ (rowCost μ = rowCost ν → μ = ν) := by
+  let c := rankPerm (fun x => row (e x)) h
+  have hle (x : Cell μ) : row (c x) ≤ row (e x) := rankCell_row_le _ h x
+  have hsumc : ∑ x, row (c x) = rowCost μ := Equiv.sum_comp c _
+  have hsume : ∑ x, row (e x) = rowCost ν := Equiv.sum_comp e _
+  constructor
+  · rw [←hsumc,←hsume]
+    exact Finset.sum_le_sum (fun x _ => hle x)
+  · intro heq
+    have he : ∀ x, row (c x) = row (e x) := by
+      apply fun x => (Finset.sum_eq_sum_iff_of_le (fun x _ => hle x)).mp
+        (hsumc.trans (heq.trans hsume.symm)) x (Finset.mem_univ _)
+    apply eq_of_row_preserving μ ν (c.symm.trans e)
+    intro x
+    simpa using (he (c.symm x)).symm
+
+lemma swap_relabel {α β : Type*} [DecidableEq α] [DecidableEq β]
+    (e : α ≃ β) (x y : α) : e.permCongr (Equiv.swap x y) = Equiv.swap (e x) (e y) := by
+  ext z
+  simp only [Equiv.permCongr_apply]
+  exact e.injective.map_swap x y (e.symm z) |>.trans (by simp)
+
+lemma sign_vector_rowCost (μ ν : YoungDiagram) (e : Cell μ ≃ Cell ν)
+    (v : Tabloid ν → ℂ) (hv : v ≠ 0)
+    (hs : ∀ c : colGroup μ,
+      tabloidRep ν (e.permCongr c) v = permSign (c : Equiv.Perm (Cell μ)) • v) :
+    rowCost μ ≤ rowCost ν ∧ (rowCost μ = rowCost ν → μ = ν) := by
+  obtain ⟨f,hf⟩ : ∃ f, v f ≠ 0 := by
+    by_contra h
+    push Not at h
+    exact hv (funext h)
+  obtain ⟨p,hp⟩ := f.2
+  apply column_injective_rowCost μ ν (e.trans p)
+  intro x y hcol hrow
+  by_contra hne
+  have hfix : tabloidAct (Equiv.swap (e x) (e y)) f = f := by
+    apply swap_fixes_tabloid
+    rw [hp]
+    exact Fin.ext hrow
+  have heq := congrArg (fun w : Tabloid ν → ℂ => w f)
+    (hs ⟨Equiv.swap x y,swap_mem_colGroup x y hcol⟩)
+  rw [swap_relabel,permSign_swap hne] at heq
+  simp only [tabloidRep_apply,Equiv.swap_inv,hfix,Pi.smul_apply,neg_one_smul] at heq
+  apply hf
+  have h2 : (2:ℂ)*(v f) = 0 := by linear_combination heq
+  exact (mul_eq_zero.mp h2).resolve_left (by norm_num)
+
+lemma polytabloid_col_action (μ : YoungDiagram) (c : colGroup μ) :
+    tabloidRep μ c (polytabloid μ) = permSign (c:Equiv.Perm (Cell μ)) • polytabloid μ :=
+  action_alternator_apply _ _ c _
+
+theorem shapes_eq_of_equivariant_equiv (μ ν : YoungDiagram) (e : Cell μ ≃ Cell ν)
+    (T : space μ ≃ₗ[ℂ] space ν)
+    (hT : ∀ p v, T (representation μ p v) = representation ν (e.permCongr p) (T v)) :
+    μ = ν := by
+  let u : space μ := ⟨polytabloid μ,polytabloid_mem_space μ⟩
+  have hu : u ≠ 0 := fun h => polytabloid_ne_zero μ (congrArg Subtype.val h)
+  have hTcol (c : colGroup μ) :
+      tabloidRep ν (e.permCongr c) (T u) = permSign (c:Equiv.Perm (Cell μ)) • (T u : Tabloid ν → ℂ) := by
+    change (representation ν (e.permCongr c) (T u) : Tabloid ν → ℂ) = _
+    rw [←hT]
+    have hh : representation μ c u = permSign (c:Equiv.Perm (Cell μ)) • u :=
+      Subtype.ext (polytabloid_col_action μ c)
+    rw [hh,map_smul]
+    rfl
+  have hfwd := sign_vector_rowCost μ ν e (T u)
+    (fun h => hu (T.injective (by apply Subtype.ext; simpa using h))) hTcol
+  let w : space ν := ⟨polytabloid ν,polytabloid_mem_space ν⟩
+  have hw : w ≠ 0 := fun h => polytabloid_ne_zero ν (congrArg Subtype.val h)
+  have hTi (p : Equiv.Perm (Cell ν)) (v : space ν) :
+      T.symm (representation ν p v) = representation μ (e.symm.permCongr p) (T.symm v) := by
+    apply T.injective
+    rw [T.apply_symm_apply,hT,T.apply_symm_apply]
+    have hp : e.permCongr (e.symm.permCongr p) = p := by
+      apply Equiv.ext
+      intro x
+      simp [Equiv.permCongr_apply]
+    rw [hp]
+  have hTicol (c : colGroup ν) :
+      tabloidRep μ (e.symm.permCongr c) (T.symm w) =
+        permSign (c:Equiv.Perm (Cell ν)) • (T.symm w : Tabloid μ → ℂ) := by
+    change (representation μ (e.symm.permCongr c) (T.symm w) : Tabloid μ → ℂ) = _
+    rw [←hTi]
+    have hh : representation ν c w = permSign (c:Equiv.Perm (Cell ν)) • w :=
+      Subtype.ext (polytabloid_col_action ν c)
+    rw [hh,map_smul]
+    rfl
+  have hbwd := sign_vector_rowCost ν μ e.symm (T.symm w)
+    (fun h => hw (T.symm.injective (by apply Subtype.ext; simpa using h))) hTicol
+  exact hfwd.2 (Nat.le_antisymm hfwd.1 hbwd.1)
+
+
+noncomputable def cellRowsEquiv (μ : YoungDiagram) :
+    Cell μ ≃ (i : Fin (μ.colLen 0)) × Fin (μ.rowLen i) where
+  toFun x := ⟨rowIndex x,⟨col x,YoungDiagram.mem_iff_lt_rowLen.mp x.2⟩⟩
+  invFun p := ⟨(p.1,p.2),YoungDiagram.mem_iff_lt_rowLen.mpr p.2.2⟩
+  left_inv x := by apply Subtype.ext; rfl
+  right_inv p := by rcases p with ⟨⟨i,hi⟩,⟨j,hj⟩⟩; rfl
+
+lemma card_eq_sum_rowLen (μ : YoungDiagram) : μ.card = ∑ i : Fin (μ.colLen 0), μ.rowLen i := by
+  rw [←card_cell,Fintype.card_congr (cellRowsEquiv μ),Fintype.card_sigma]
+  simp only [Fintype.card_fin]
+
+lemma sum_rowLens (μ : YoungDiagram) : μ.rowLens.sum = μ.card := by
+  rw [card_eq_sum_rowLen,YoungDiagram.rowLens]
+  rw [Fin.sum_univ_eq_sum_range]
+  have hh (n : ℕ) : (List.map μ.rowLen (List.range n)).sum = ∑ i ∈ Finset.range n, μ.rowLen i := by
+    induction n with
+    | zero => simp
+    | succ n ih => simp [List.range_succ,List.map_append,Finset.sum_range_succ,ih]
+  exact hh _
+
+noncomputable def diagramPartition (μ : YoungDiagram) : Nat.Partition μ.card where
+  parts := μ.rowLens
+  parts_pos := fun {i} hi => μ.pos_of_mem_rowLens i (by simpa using hi)
+  parts_sum := by simp
+
+noncomputable def partitionDiagram {N : ℕ} (p : Nat.Partition N) : YoungDiagram :=
+  YoungDiagram.ofRowLens (p.parts.sort (· ≥ ·)) (Multiset.pairwise_sort _ _).sortedGE
+
+lemma partitionDiagram_rowLens {N : ℕ} (p : Nat.Partition N) :
+    (partitionDiagram p).rowLens = p.parts.sort (· ≥ ·) := by
+  apply YoungDiagram.rowLens_ofRowLens_eq_self
+  intro x hx
+  exact p.parts_pos ((Multiset.mem_sort _).mp hx)
+
+lemma partitionDiagram_card {N : ℕ} (p : Nat.Partition N) : (partitionDiagram p).card = N := by
+  rw [←sum_rowLens,partitionDiagram_rowLens]
+  change (↑(p.parts.sort (· ≥ ·)) : Multiset ℕ).sum = N
+  rw [Multiset.sort_eq,p.parts_sum]
+
+lemma partitionDiagram_diagramPartition (μ : YoungDiagram) :
+    partitionDiagram (diagramPartition μ) = μ := by
+  apply YoungDiagram.equivListRowLens.injective
+  apply Subtype.ext
+  change (partitionDiagram (diagramPartition μ)).rowLens = μ.rowLens
+  rw [partitionDiagram_rowLens]
+  change (↑μ.rowLens : Multiset ℕ).sort (· ≥ ·) = μ.rowLens
+  rw [Multiset.coe_sort,List.mergeSort_eq_self _ μ.rowLens_sorted.pairwise]
+
+abbrev Shapes (N : ℕ) := {μ : YoungDiagram // μ.card = N}
+
+end Thorp.Specht
+
+end ThorpNine.Tail
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Tail/ModifiedRoutingMoment.lean b/lean/OAI/Probability/ThorpRouting/Tail/ModifiedRoutingMoment.lean
index 06b176f506ba1afe4751ca32cfe226ba234ceca4..01160500b6142ccad89b18a0182dcbde158349e2 100644
--- a/lean/OAI/Probability/ThorpRouting/Tail/ModifiedRoutingMoment.lean
+++ b/lean/OAI/Probability/ThorpRouting/Tail/ModifiedRoutingMoment.lean
@@ -1,858 +1,858 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Tail.InputTree
-
-namespace OAI
-
-namespace ThorpNine.Tail
-
-namespace Thorp
-
-noncomputable def modifiedRoutingMoment (L r : ℕ) (E : Finset (Card r))
-    {ι : Type*} [Fintype ι] (e f : ι ↪ Card (L+r)) (a : ℝ) : ℝ := by
-  classical
-  exact finiteMean (fun ω : ModifiedCoins L r =>
-    if ∀ i, modifiedPerm L r E ω (e i) = f i then (2:ℝ)^((modifiedCost L r E e ω:ℝ)*a) else 0)
-
-lemma modifiedRoutingMoment_nonneg (L r : ℕ) (E : Finset (Card r))
-    {ι : Type*} [Fintype ι] (e f : ι ↪ Card (L+r)) (a : ℝ) :
-    0 ≤ modifiedRoutingMoment L r E e f a := by
-  classical
-  exact finiteMean_nonneg (fun _ => by split_ifs <;> positivity)
-
-lemma modifiedRoutingMoment_zero (L : ℕ) (E : Finset (Card 0))
-    {ι : Type*} [Fintype ι] (e f : ι ↪ Card L) (a : ℝ) :
-    modifiedRoutingMoment L 0 E e f a = PairRouting.tupleProbability (modifiedPerm L 0 E) e f := by
-  simp only [modifiedRoutingMoment,modifiedCost_zero,Nat.cast_zero,zero_mul,Real.rpow_zero,
-    PairRouting.tupleProbability]
-
-lemma modified_pointwise_zero (L : ℕ) (E : Finset (Card 0))
-    {ι : Type*} [Fintype ι] (e f : ι ↪ Card L) (he : Function.Surjective e) (a : ℝ) (ha : 0 ≤ a) :
-    (PairRouting.tupleProbability (modifiedPerm L 0 E) e f)^(1+a) ≤
-      centralFactor L 0 E a * modifiedRoutingMoment L 0 E e f a := by
-  have hp := PairRouting.tupleProbability_nonneg (modifiedPerm L 0 E) e f
-  rw [modifiedRoutingMoment_zero]
-  have hbound := modified_probability_zero_le L E e f he
-  have hpow := Real.rpow_le_rpow hp hbound ha
-  have hfpos : (0:ℝ) < Nat.factorial (2^L) := Nat.cast_pos.mpr (Nat.factorial_pos _)
-  have heq : (if (fun j => Fin.elim0 j : Card 0) ∈ E then (1:ℝ) else 1/(Nat.factorial (2^L):ℝ))^a =
-      centralFactor L 0 E a := by
-    rw [centralFactor_zero]
-    split_ifs
-    · exact Real.one_rpow _
-    · rw [one_div,Real.inv_rpow (le_of_lt hfpos),Real.rpow_neg (le_of_lt hfpos)]
-  rw [heq] at hpow
-  rw [Real.rpow_add' hp (by linarith : (1+a:ℝ) ≠ 0),Real.rpow_one,mul_comm]
-  exact mul_le_mul_of_nonneg_right hpow hp
-
-
-lemma modified_matches_step (L r : ℕ) (E : Finset (Card (r+1))) {ι : Type*}
-    (e f : ι ↪ Card (L+r+1)) (ω : ModifiedCoins L r × ModifiedCoins L r)
-    (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
-    (∀ i, modifiedPerm L (r+1) E ((modifiedStepEquiv L r).symm (ω,coins)) (e i) = f i) ↔
-    (∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1))
-      ((e.trans (headTailEquiv (L+r)).toEmbedding) i) = ((f.trans (headTailEquiv (L+r)).toEmbedding) i)) := by
-  apply forall_congr'
-  intro i
-  rw [←(headTailEquiv (L+r)).injective.eq_iff,modifiedPerm_coordinates]
-  rfl
-
-lemma modifiedRoutingMoment_step (L r : ℕ) (E : Finset (Card (r+1))) {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (e f : ι ↪ Card (L+r+1)) (a : ℝ) :
-    let x := e.trans (headTailEquiv (L+r)).toEmbedding
-    let y := f.trans (headTailEquiv (L+r)).toEmbedding
-    modifiedRoutingMoment L (r+1) E e f a = PairRouting.outerWeight x y *
-      ∑ c : {c // c ∈ PairRouting.compatibleSet x y},
-        finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
-          if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c.val
-          then (2:ℝ)^((modifiedAssignmentCost L r E x c.val
-            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0) := by
-  classical
-  intro x y
-  let : Fintype (ModifiedCoins L r) := inferInstance
-  rw [modifiedRoutingMoment,finiteMean_equiv (modifiedStepEquiv L r),finiteMean_prod]
-  have he (ω : ModifiedCoins L r × ModifiedCoins L r) :
-      finiteMean (fun coins : (Card (L+r) → Bool) × (Card (L+r) → Bool) =>
-        if ∀ i, modifiedPerm L (r+1) E ((modifiedStepEquiv L r).symm (ω,coins)) (e i) = f i
-        then (2:ℝ)^((modifiedCost L (r+1) E e ((modifiedStepEquiv L r).symm (ω,coins)):ℝ)*a) else 0) =
-      PairRouting.outerWeight x y *
-        ∑ c : {c // c ∈ PairRouting.compatibleSet x y},
-          if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c.val
-          then (2:ℝ)^((modifiedAssignmentCost L r E x c.val
-            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0 := by
-    let W : (ι → Bool) → ℝ := fun c =>
-      if hc : PairRouting.Compatible x c then (2:ℝ)^((modifiedAssignmentCost L r E x c hc ω:ℝ)*a) else 0
-    calc
-      _ = finiteMean (fun coins : (Card (L+r) → Bool) × (Card (L+r) → Bool) =>
-        if ∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1))
-          (x i) = y i then W (PairRouting.colors x coins.1) else 0) := by
-        apply finiteMean_congr
-        intro coins
-        simp only [modified_matches_step,modifiedCost_step]
-        simp only [W,PairRouting.colors_compatible,↓reduceDIte]
-        rfl
-      _ = _ := by
-        rw [PairRouting.sandwich_weighted_probability_subtype]
-        congr 1
-        apply Finset.sum_congr rfl
-        intro c _
-        simp only [W,((PairRouting.mem_compatibleSet x y c.val).mp c.property).1,↓reduceDIte]
-  simp_rw [he]
-  simp_rw [mul_comm (PairRouting.outerWeight x y)]
-  rw [finiteMean_mul_const,finiteMean_sum,mul_comm]
-
-lemma modified_child_moment_product (L r : ℕ) (E : Finset (Card (r+1))) {ι : Type*} [Fintype ι]
-    (x y : ι ↪ Bool × Card (L+r)) (c : ι → Bool)
-    (hcx : PairRouting.Compatible x c) (hcy : PairRouting.Compatible y c) (a : ℝ) :
-    modifiedRoutingMoment L r (exceptionChild E false) (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a *
-      modifiedRoutingMoment L r (exceptionChild E true) (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a =
-    finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
-      if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c
-      then (2:ℝ)^(((modifiedCost L r (exceptionChild E false) (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
-        modifiedCost L r (exceptionChild E true) (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0) := by
-  classical
-  unfold modifiedRoutingMoment
-  rw [←finiteMean_prod_mul]
-  apply finiteMean_congr
-  intro ω
-  let M₀ := ∀ i : {i // c i = false}, modifiedPerm L r (exceptionChild E false) ω.1 (x i.val).2 = (y i.val).2
-  let M₁ := ∀ i : {i // c i = true}, modifiedPerm L r (exceptionChild E true) ω.2 (x i.val).2 = (y i.val).2
-  have hp : (fun b : Bool => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) =
-      (fun b => if b then modifiedPerm L r (exceptionChild E true) ω.2 else modifiedPerm L r (exceptionChild E false) ω.1) := by
-    funext b
-    cases b <;> rfl
-  have hs : PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c ↔
-      M₀ ∧ M₁ := by rw [hp]; exact PairRouting.childMatches_split _ _ _ _ _
-  change (if M₀ then _ else 0) * (if M₁ then _ else 0) = _
-  by_cases h₀ : M₀ <;> by_cases h₁ : M₁
-  · simp only [hs,h₀,h₁,true_and,↓reduceIte]
-    rw [←Real.rpow_add (by norm_num : (0:ℝ)<2)]
-    congr 1
-    ring
-  · simp only [hs,h₀,h₁,and_false,↓reduceIte,mul_zero]
-  · simp only [hs,h₀,h₁,false_and,↓reduceIte,zero_mul]
-  · simp only [hs,h₀,h₁,false_and,↓reduceIte,mul_zero]
-
-lemma modified_probability_step (L r : ℕ) (E : Finset (Card (r+1))) {ι : Type*} [Fintype ι]
-    (e f : ι ↪ Card (L+r+1)) :
-    PairRouting.tupleProbability (modifiedPerm L (r+1) E) e f =
-    finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
-      finiteMean (fun coins : (Card (L+r) → Bool) × (Card (L+r) → Bool) =>
-        if ∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => if b then modifiedPerm L r (exceptionChild E true) ω.2 else modifiedPerm L r (exceptionChild E false) ω.1)
-          ((e.trans (headTailEquiv (L+r)).toEmbedding) i) = ((f.trans (headTailEquiv (L+r)).toEmbedding) i)
-        then (1:ℝ) else 0)) := by
-  classical
-  let : Fintype (ModifiedCoins L r) := inferInstance
-  rw [PairRouting.tupleProbability,finiteMean_equiv (modifiedStepEquiv L r),finiteMean_prod]
-  have hp (b : Bool) (p q : ModifiedCoins L r) :
-      modifiedPerm L r (exceptionChild E b) (if b then p else q) =
-      (if b then modifiedPerm L r (exceptionChild E true) p else modifiedPerm L r (exceptionChild E false) q) := by
-    cases b <;> rfl
-  simp only [modified_matches_step,hp]
-
-
-lemma modified_assignment_moment_bound (L r : ℕ) (E : Finset (Card (r+1))) {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (x y : ι ↪ Bool × Card (L+r)) (c : ι → Bool) (hc : c ∈ PairRouting.compatibleSet x y)
-    (a : ℝ) (ha : 0 ≤ a) :
-    let hcx := ((PairRouting.mem_compatibleSet x y c).mp hc).1
-    PairRouting.outerWeight x y ^ a * ((PairRouting.compatibleSet x y).card:ℝ)^a *
-      ((2:ℝ)^(-(r:ℝ)*Fintype.card ι*a) *
-        finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
-          if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c
-          then (2:ℝ)^(((modifiedCost L r (exceptionChild E false) (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
-            modifiedCost L r (exceptionChild E true) (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0)) ≤
-    (2:ℝ)^(-((r:ℝ)+1)*Fintype.card ι*a) *
-      finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
-        if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c
-        then (2:ℝ)^((modifiedAssignmentCost L r E x c hcx ω:ℝ)*a) else 0) := by
-  classical
-  intro hcx
-  have hscale (q : ℝ) (F : ModifiedCoins L r × ModifiedCoins L r → ℝ) :
-      q * finiteMean F = finiteMean (fun ω => q * F ω) := by
-    simp_rw [mul_comm q,finiteMean_mul_const]
-  rw [←mul_assoc,hscale,hscale]
-  apply finiteMean_mono
-  intro ω
-  split_ifs with hm
-  · have hh := PairRouting.assignment_factor_bound x y c hc
-      (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) hm a ha
-    calc
-      _ ≤ ((2:ℝ)^(((PairRouting.alternatingCycles (PairRouting.coloredEmbedding x c hcx)
-          (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)):ℝ)-Fintype.card ι)*a) *
-          (2:ℝ)^(-(r:ℝ)*Fintype.card ι*a)) *
-          (2:ℝ)^(((modifiedCost L r (exceptionChild E false) (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
-            modifiedCost L r (exceptionChild E true) (PairRouting.childEmbedding x c hcx true) ω.2)*a) :=
-        mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hh (by positivity)) (by positivity)
-      _ = _ := by
-        simp only [←Real.rpow_add (by norm_num : (0:ℝ)<2),modifiedAssignmentCost,Nat.cast_add]
-        congr 1
-        ring
-  · simp only [mul_zero]
-    exact le_rfl
-
-
-lemma modified_child_power_product_bound (L r : ℕ) (E : Finset (Card (r+1)))
-    {ι : Type*} [Fintype ι] (x y : ι ↪ Bool × Card (L+r)) (c : ι → Bool)
-    (hcx : PairRouting.Compatible x c) (hcy : PairRouting.Compatible y c) (a : ℝ)
-    (hfalse : (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E false))
-        (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false))^(1+a) ≤
-      ((2:ℝ)^(-(r:ℝ)*Fintype.card {i // c i = false}*a)*centralFactor L r (exceptionChild E false) a) *
-        modifiedRoutingMoment L r (exceptionChild E false)
-          (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a)
-    (htrue : (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E true))
-        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true))^(1+a) ≤
-      ((2:ℝ)^(-(r:ℝ)*Fintype.card {i // c i = true}*a)*centralFactor L r (exceptionChild E true) a) *
-        modifiedRoutingMoment L r (exceptionChild E true)
-          (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a) :
-    (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E false))
-      (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false))^(1+a) *
-      (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E true))
-        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true))^(1+a) ≤
-    ((2:ℝ)^(-(r:ℝ)*Fintype.card ι*a)*centralFactor L (r+1) E a) *
-      finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
-        if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c
-        then (2:ℝ)^(((modifiedCost L r (exceptionChild E false) (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
-          modifiedCost L r (exceptionChild E true) (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0) := by
-  have hh := mul_le_mul hfalse htrue
-    (Real.rpow_nonneg (PairRouting.tupleProbability_nonneg _ _ _) _)
-    (mul_nonneg (mul_nonneg (by positivity) (centralFactor_nonneg _ _ _ _))
-      (modifiedRoutingMoment_nonneg _ _ _ _ _ _))
-  have he : -(r:ℝ)*Fintype.card {i // c i = false}*a + -(r:ℝ)*Fintype.card {i // c i = true}*a =
-      -(r:ℝ)*Fintype.card ι*a := by
-    rw [←add_mul,←mul_add,←Nat.cast_add,PairRouting.color_card_add]
-  calc
-    _ ≤ _ := hh
-    _ = ((2:ℝ)^(-(r:ℝ)*Fintype.card {i // c i = false}*a)*
-        (2:ℝ)^(-(r:ℝ)*Fintype.card {i // c i = true}*a)) *
-      (centralFactor L r (exceptionChild E false) a * centralFactor L r (exceptionChild E true) a) *
-      (modifiedRoutingMoment L r (exceptionChild E false)
-        (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a *
-      modifiedRoutingMoment L r (exceptionChild E true)
-        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a) := by ring
-    _ = _ := by rw [←Real.rpow_add (by norm_num : (0:ℝ)<2),he,←centralFactor_step,modified_child_moment_product]
-
-
-theorem modified_pointwise_moment (L r : ℕ) (E : Finset (Card r)) {ι : Type*} [Fintype ι]
-    (e f : ι ↪ Card (L+r)) (he : Function.Surjective e) (a : ℝ) (ha : 0 ≤ a) :
-    (PairRouting.tupleProbability (modifiedPerm L r E) e f)^(1+a) ≤
-      ((2:ℝ)^(-(r:ℝ)*Fintype.card ι*a)*centralFactor L r E a) * modifiedRoutingMoment L r E e f a := by
-  classical
-  induction r generalizing ι with
-  | zero =>
-      simpa only [Nat.cast_zero,neg_zero,zero_mul,Real.rpow_zero,one_mul] using
-        modified_pointwise_zero L E e f he a ha
-  | succ r ih =>
-      let x := e.trans (headTailEquiv (L+r)).toEmbedding
-      let y := f.trans (headTailEquiv (L+r)).toEmbedding
-      let W := PairRouting.outerWeight x y
-      let C := PairRouting.compatibleSet x y
-      let p : (ι → Bool) → ℝ := fun c =>
-        (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E false))
-          (fun i : {i // c i = false} => (x i.val).2)
-          (fun i : {i // c i = false} => (y i.val).2))^(1+a) *
-        (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E true))
-          (fun i : {i // c i = true} => (x i.val).2)
-          (fun i : {i // c i = true} => (y i.val).2))^(1+a)
-      let m : {c // c ∈ C} → ℝ := fun c =>
-        finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
-          if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c.val
-          then (2:ℝ)^((modifiedAssignmentCost L r E x c.val
-            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0)
-      have hp (c : {c // c ∈ C}) : W^a * (C.card:ℝ)^a * p c.val ≤
-          ((2:ℝ)^(-((r:ℝ)+1)*Fintype.card ι*a)*centralFactor L (r+1) E a) * m c := by
-        obtain ⟨hcx,hcy⟩ := (PairRouting.mem_compatibleSet x y c.val).mp c.property
-        have h₀ := ih (exceptionChild E false) (PairRouting.childEmbedding x c.val hcx false)
-          (PairRouting.childEmbedding y c.val hcy false)
-          (PairRouting.childEmbedding_surjective x ((headTailEquiv (L+r)).surjective.comp he) c.val hcx false)
-        have h₁ := ih (exceptionChild E true) (PairRouting.childEmbedding x c.val hcx true)
-          (PairRouting.childEmbedding y c.val hcy true)
-          (PairRouting.childEmbedding_surjective x ((headTailEquiv (L+r)).surjective.comp he) c.val hcx true)
-        have hh := modified_child_power_product_bound L r E x y c.val hcx hcy a h₀ h₁
-        have hh' := mul_le_mul_of_nonneg_left hh
-          (mul_nonneg (Real.rpow_nonneg (le_of_lt (PairRouting.outerWeight_pos x y)) a)
-            (Real.rpow_nonneg (Nat.cast_nonneg C.card) a))
-        apply hh'.trans
-        have hm := modified_assignment_moment_bound L r E x y c.val c.property a ha
-        convert mul_le_mul_of_nonneg_left hm (centralFactor_nonneg L (r+1) E a) using 1 <;> ring
-      have hnode := PairRouting.sandwich_power_bound (modifiedPerm L r (exceptionChild E false)) (modifiedPerm L r (exceptionChild E true)) x y a ha
-      rw [←modified_probability_step L r E e f] at hnode
-      change (PairRouting.tupleProbability (modifiedPerm L (r+1) E) e f)^(1+a) ≤
-        W^(1+a) * (C.card:ℝ)^a * ∑ c ∈ C, p c at hnode
-      have hsum : ∑ c ∈ C, p c = ∑ c : {c // c ∈ C}, p c.val := by
-        rw [Finset.sum_subtype C (fun _ => Iff.rfl)]
-      have hW : 0 < W := PairRouting.outerWeight_pos x y
-      calc
-        _ ≤ W^(1+a) * (C.card:ℝ)^a * ∑ c ∈ C, p c := hnode
-        _ = W * ∑ c : {c // c ∈ C}, W^a * (C.card:ℝ)^a * p c.val := by
-          rw [Real.rpow_add hW,Real.rpow_one,hsum,←Finset.mul_sum]
-          ring
-        _ ≤ W * ∑ c : {c // c ∈ C}, ((2:ℝ)^(-((r:ℝ)+1)*Fintype.card ι*a)*centralFactor L (r+1) E a) * m c :=
-          mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun c _ => hp c)) (le_of_lt hW)
-        _ = _ := by
-          rw [←Finset.mul_sum,modifiedRoutingMoment_step]
-          simp only [Nat.cast_add,Nat.cast_one]
-          change W * (((2:ℝ)^(-((r:ℝ)+1)*Fintype.card ι*a)*centralFactor L (r+1) E a) * ∑ c, m c) =
-            ((2:ℝ)^(-((r:ℝ)+1)*Fintype.card ι*a)*centralFactor L (r+1) E a) * (W * ∑ c, m c)
-          ring
-
-
-lemma modifiedRoutingMoment_sum (L r : ℕ) (E : Finset (Card r)) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (L+r)) (a : ℝ) :
-    (∑ f : ι ↪ Card (L+r), modifiedRoutingMoment L r E e f a) =
-      finiteMean (fun ω : ModifiedCoins L r => (2:ℝ)^((modifiedCost L r E e ω:ℝ)*a)) := by
-  classical
-  unfold modifiedRoutingMoment
-  rw [←finiteMean_sum]
-  apply finiteMean_congr
-  intro ω
-  let f₀ : ι ↪ Card (L+r) := e.trans (modifiedPerm L r E ω).toEmbedding
-  rw [Finset.sum_eq_single f₀]
-  · have hm : ∀ i, modifiedPerm L r E ω (e i) = f₀ i := fun _ => rfl
-    simp only [hm,implies_true,↓reduceIte]
-  · intro f _ hf
-    have hm : ¬ ∀ i, modifiedPerm L r E ω (e i) = f i := by
-      intro hh
-      apply hf
-      ext i : 1
-      exact (hh i).symm
-    exact ite_eq_right hm
-  · simp
-
-noncomputable def modifiedRowMoment (L r : ℕ) (E : Finset (Card r)) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (L+r)) (a : ℝ) : ℝ :=
-  finiteMean (fun f : ι ↪ Card (L+r) =>
-    ((Fintype.card (ι ↪ Card (L+r)):ℝ)*PairRouting.tupleProbability (modifiedPerm L r E) e f)^(1+a))
-
-lemma full_tuple_card (d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d)
-    (he : Function.Surjective e) : Fintype.card ι = 2^d := by
-  rw [←card_positions d]
-  exact Fintype.card_congr (Equiv.ofBijective e ⟨e.injective,he⟩)
-
-lemma exception_card_le (r : ℕ) (E : Finset (Card r)) : E.card ≤ 2^r := by
-  simpa only [card_positions] using E.card_le_univ
-
-noncomputable def modifiedNormalization (L r : ℕ) (E : Finset (Card r)) : ℝ :=
-  (Nat.factorial (2^(L+r)):ℝ) /
-    (((2:ℝ)^r)^(2^(L+r)) * (Nat.factorial (2^L):ℝ)^(2^r-E.card))
-
-lemma modifiedNormalization_pos (L r : ℕ) (E : Finset (Card r)) :
-    0 < modifiedNormalization L r E := by unfold modifiedNormalization; positivity
-
-lemma modified_normalization_identity (L r : ℕ) (E : Finset (Card r)) (a : ℝ) :
-    (Nat.factorial (2^(L+r)):ℝ)^a *
-      ((2:ℝ)^(-(r:ℝ)*(2^(L+r):ℕ)*a)*centralFactor L r E a) =
-    modifiedNormalization L r E ^ a := by
-  rw [←mul_assoc,density_normalization,centralFactor,modifiedNormalization,
-    Real.div_rpow (by positivity) (by positivity),Real.div_rpow (by positivity) (by positivity),
-    Real.mul_rpow (by positivity) (by positivity)]
-  have hc : ((2^r-E.card:ℕ):ℝ) = (2:ℝ)^r-E.card := by
-    rw [Nat.cast_sub (exception_card_le r E),Nat.cast_pow,Nat.cast_ofNat]
-  rw [←Real.rpow_natCast (Nat.factorial (2^L):ℝ),←Real.rpow_mul (by positivity),hc,
-    show -((2:ℝ)^r-E.card)*a = -(((2:ℝ)^r-E.card)*a) by ring,
-    Real.rpow_neg (by positivity)]
-  ring
-
-theorem modified_row_density_moment (L r : ℕ) (E : Finset (Card r)) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (L+r)) (he : Function.Surjective e) (a : ℝ) (ha : 0 ≤ a) :
-    modifiedRowMoment L r E e a ≤ modifiedNormalization L r E ^ a *
-      finiteMean (fun ω : ModifiedCoins L r => (2:ℝ)^((modifiedCost L r E e ω:ℝ)*a)) := by
-  classical
-  have : Nonempty (ι ↪ Card (L+r)) := ⟨e⟩
-  rw [modifiedRowMoment,finite_row_power _ (fun f => PairRouting.tupleProbability_nonneg _ _ _)]
-  calc
-    _ ≤ (Fintype.card (ι ↪ Card (L+r)):ℝ)^a *
-      ∑ f : ι ↪ Card (L+r), ((2:ℝ)^(-(r:ℝ)*Fintype.card ι*a)*centralFactor L r E a) *
-        modifiedRoutingMoment L r E e f a :=
-      mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun f _ => modified_pointwise_moment L r E e f he a ha))
-        (Real.rpow_nonneg (Nat.cast_nonneg _) _)
-    _ = _ := by
-      rw [←Finset.mul_sum,modifiedRoutingMoment_sum,←mul_assoc,Fintype.card_embedding_eq,
-        card_positions,full_tuple_card (L+r) e he,Nat.descFactorial_self,modified_normalization_identity]
-
-lemma modified_total_cost_mgf (L r : ℕ) (hL : 0 < L) (E : Finset (Card r))
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) :
-    finiteMean (fun ω : ModifiedCoins L r => momentBase^(modifiedCost L r E e ω)) ≤
-      Real.exp (Fintype.card ι*heightDecay L +
-        3*((L*E.card*2^L:ℕ):ℝ)*Real.log momentBase) := by
-  let B := Real.exp (Fintype.card ι*heightDecay L +
-    3*((L*E.card*2^L:ℕ):ℝ)*Real.log momentBase)
-  rw [finiteMean_prod]
-  apply le_trans (finiteMean_mono (fun U => ?_)) (finiteMean_const B).le
-  rw [finiteMean_prod,finiteMean_comm]
-  apply le_trans (finiteMean_mono (fun X => ?_)) (finiteMean_const B).le
-  exact modified_cost_mgf L r hL E U X e momentBase momentBase_one_le (momentBase_log.trans (by norm_num))
-
-lemma modifiedRowMoment_bound (L r : ℕ) (hL : 0 < L) (E : Finset (Card r))
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (he : Function.Surjective e) :
-    modifiedRowMoment L r E e (1/1000) ≤
-      modifiedNormalization L r E ^ (1/1000:ℝ) *
-        Real.exp ((2^(L+r):ℕ)*heightDecay L +
-          3*((L*E.card*2^L:ℕ):ℝ)*Real.log momentBase) := by
-  have hm := modified_row_density_moment L r E e he (1/1000) (by norm_num)
-  simp_rw [←momentBase_pow] at hm
-  apply hm.trans
-  apply mul_le_mul_of_nonneg_left _ (Real.rpow_nonneg (modifiedNormalization_pos L r E).le _)
-  simpa only [full_tuple_card (L+r) e he] using modified_total_cost_mgf L r hL E e
-
-
-lemma log_factorial_upper {n : ℕ} (hn : 1 ≤ n) :
-    Real.log (Nat.factorial n) ≤ (n:ℝ)*Real.log n-n+Real.log n+1 := by
-  induction n with
-  | zero => omega
-  | succ n ih =>
-      by_cases hz : n = 0
-      · subst n; norm_num
-      have hn0 : (0:ℝ) < n := Nat.cast_pos.mpr (Nat.pos_of_ne_zero hz)
-      have hn1 : (0:ℝ) < n+1 := by positivity
-      have h := Real.one_sub_inv_le_log_of_pos (div_pos hn1 hn0)
-      rw [Real.log_div (ne_of_gt hn1) (ne_of_gt hn0),inv_div] at h
-      have hh := mul_le_mul_of_nonneg_left h hn1.le
-      have hcancel : ((n:ℝ)+1)*(1-(n:ℝ)/(n+1)) = 1 := by field_simp; ring
-      rw [hcancel] at hh
-      have hi := ih (by omega)
-      rw [Nat.factorial_succ,Nat.cast_mul,Real.log_mul (by positivity) (by positivity),Nat.cast_add,Nat.cast_one]
-      nlinarith
-
-lemma log_modifiedNormalization (L r : ℕ) (E : Finset (Card r)) :
-    Real.log (modifiedNormalization L r E) =
-      Real.log (Nat.factorial (2^(L+r))) -
-        ((2^(L+r):ℕ)*((r:ℝ)*Real.log 2)+((2:ℝ)^r-E.card)*Real.log (Nat.factorial (2^L))) := by
-  unfold modifiedNormalization
-  rw [Real.log_div (by positivity) (by positivity),Real.log_mul (by positivity) (by positivity),
-    Real.log_pow,Real.log_pow,Real.log_pow,
-    Nat.cast_sub (exception_card_le r E),Nat.cast_pow,Nat.cast_ofNat]
-  simp only [Nat.cast_pow,Nat.cast_ofNat]
-
-lemma log_modifiedNormalization_le (L r : ℕ) (E : Finset (Card r)) :
-    Real.log (modifiedNormalization L r E) ≤
-      ((L+r:ℕ):ℝ)*Real.log 2+1+E.card*Real.log (Nat.factorial (2^L)) := by
-  have hN : 1 ≤ 2^(L+r) := one_le_pow₀ (by norm_num)
-  have hS : 1 ≤ 2^L := one_le_pow₀ (by norm_num)
-  have hu := log_factorial_upper hN
-  have hl := mul_le_mul_of_nonneg_left (log_factorial_lower hS) (by positivity : (0:ℝ) ≤ (2:ℝ)^r)
-  simp only [Nat.cast_pow,Nat.cast_ofNat,Real.log_pow,Nat.cast_add] at hu hl ⊢
-  rw [log_modifiedNormalization]
-  simp only [pow_add,Nat.cast_mul,Nat.cast_pow,Nat.cast_ofNat] at hu hl ⊢
-  nlinarith
-
-lemma modified_moment_exponential (L r : ℕ) (hL : 0 < L) (E : Finset (Card r))
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (he : Function.Surjective e) :
-    modifiedRowMoment L r E e (1/1000) ≤
-      Real.exp (((L+r:ℕ)*Real.log 2+1+E.card*Real.log (Nat.factorial (2^L)))/1000 +
-        (2^(L+r):ℕ)*heightDecay L+3*((L*E.card*2^L:ℕ):ℝ)*Real.log momentBase) := by
-  apply (modifiedRowMoment_bound L r hL E e he).trans
-  rw [Real.rpow_def_of_pos (modifiedNormalization_pos L r E),←Real.exp_add]
-  apply Real.exp_le_exp.mpr
-  have h := log_modifiedNormalization_le L r E
-  linarith
-
-lemma log_power_factorial_le (L : ℕ) :
-    Real.log (Nat.factorial (2^L)) ≤ (2:ℝ)^L*L := by
-  have hf : (Nat.factorial (2^L):ℝ) ≤ ((2^L:ℕ):ℝ)^(2^L) := by
-    exact_mod_cast Nat.factorial_le_pow (2^L)
-  have hh := Real.log_le_log (by positivity) hf
-  rw [Real.log_pow,Nat.cast_pow,Nat.cast_ofNat,Real.log_pow] at hh
-  apply hh.trans
-  have hl := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
-  have hn : 0 ≤ (2:ℝ)^L*(L:ℝ) := by positivity
-  nlinarith
-
-lemma modified_moment_exception_bound (L r : ℕ) (hL : 0 < L) (E : Finset (Card r))
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (he : Function.Surjective e)
-    (ε : ℝ) (hE : (E.card:ℝ) ≤ ε*(2:ℝ)^r) :
-    modifiedRowMoment L r E e (1/1000) ≤
-      Real.exp (((L+r:ℕ)*Real.log 2+1)/1000 +
-        (2:ℝ)^(L+r)*(heightDecay L+ε*L/250)) := by
-  apply (modified_moment_exponential L r hL E e he).trans
-  apply Real.exp_le_exp.mpr
-  have hs := mul_le_mul_of_nonneg_left (log_power_factorial_le L) (Nat.cast_nonneg E.card)
-  have hb := mul_le_mul_of_nonneg_left momentBase_log
-    (by positivity : (0:ℝ) ≤ 3*((L*E.card*2^L:ℕ):ℝ))
-  have hexc := mul_le_mul_of_nonneg_left hE (by positivity : (0:ℝ) ≤ (2:ℝ)^L*L/250)
-  simp only [Nat.cast_mul,Nat.cast_pow,Nat.cast_ofNat,pow_add] at hs hb hexc ⊢
-  nlinarith
-
-
-lemma heightDecay_tendsto : Filter.Tendsto heightDecay Filter.atTop (nhds 0) := by
-  have h : Filter.Tendsto (fun L : ℕ => (-(1:ℝ)/64)*(L:ℝ)) Filter.atTop Filter.atBot :=
-    (Filter.tendsto_const_mul_atBot_of_neg (by norm_num : -(1:ℝ)/64 < 0)).mpr
-      tendsto_natCast_atTop_atTop
-  have ht := ((Real.tendsto_exp_atBot.comp h).const_mul 68).div_const (1-Real.exp (-(1:ℝ)/64))
-  convert ht using 1
-  · funext L
-    unfold heightDecay
-    dsimp only [Function.comp_apply]
-    rw [show -(L:ℝ)/64 = (-(1:ℝ)/64)*(L:ℝ) by ring]
-  · simp
-
-noncomputable def modifiedOverhead (L r : ℕ) : ℝ :=
-  (((L+r:ℕ):ℝ)*Real.log 2+1)/((2:ℝ)^(L+r)*1000)
-
-lemma modifiedOverhead_tendsto (L : ℕ) :
-    Filter.Tendsto (modifiedOverhead L) Filter.atTop (nhds 0) := by
-  have h₁ := tendsto_pow_const_div_const_pow_of_one_lt 1 (by norm_num : (1:ℝ)<2)
-  have h₀ := tendsto_pow_const_div_const_pow_of_one_lt 0 (by norm_num : (1:ℝ)<2)
-  simp only [pow_one] at h₁
-  simp only [pow_zero] at h₀
-  have ht := ((h₁.const_mul (Real.log 2)).add (h₀.const_mul ((L:ℝ)*Real.log 2+1))).div_const
-    ((2:ℝ)^L*1000)
-  convert ht using 1
-  · funext r
-    simp only [modifiedOverhead,Nat.cast_add,pow_add]
-    field_simp
-    ring
-  · simp
-
-theorem modified_arbitrarily_small_rate (η : ℝ) (hη : 0 < η) :
-    ∃ L : ℕ, 0 < L ∧ ∃ ε : ℝ, 0 < ε ∧ ε < 1/2 ∧
-      ∀ᶠ r : ℕ in Filter.atTop, ∀ (E : Finset (Card r)), (E.card:ℝ) ≤ ε*(2:ℝ)^r →
-        ∀ (e : Card (L+r) ↪ Card (L+r)),
-          modifiedRowMoment L r E e (1/1000) ≤ Real.exp (η*(2:ℝ)^(L+r)) := by
-  have hsmall : ∀ᶠ L : ℕ in Filter.atTop, heightDecay L < η/3 :=
-    heightDecay_tendsto.eventually (gt_mem_nhds (by linarith : (0:ℝ)<η/3))
-  obtain ⟨L,hdec,hL⟩ := (hsmall.and (Filter.eventually_ge_atTop 1)).exists
-  have hL0 : (0:ℝ) < L := Nat.cast_pos.mpr hL
-  let ε : ℝ := min (1/4) (η*250/(3*L))
-  have hε : 0 < ε := lt_min (by norm_num) (by positivity)
-  have hε1 : ε < 1/2 := (min_le_left _ _).trans_lt (by norm_num)
-  have hεL : ε*L/250 ≤ η/3 := by
-    have h := min_le_right (1/4:ℝ) (η*250/(3*L))
-    change ε ≤ η*250/(3*L) at h
-    have hh := mul_le_mul_of_nonneg_right h hL0.le
-    have he : (η*250/(3*(L:ℝ)))*(L:ℝ)/250 = η/3 := by field_simp
-    exact (div_le_div_of_nonneg_right hh (by norm_num)).trans he.le
-  refine ⟨L,hL,ε,hε,hε1,?_⟩
-  have hr : ∀ᶠ r : ℕ in Filter.atTop, modifiedOverhead L r < η/3 :=
-    (modifiedOverhead_tendsto L).eventually (gt_mem_nhds (by linarith : (0:ℝ)<η/3))
-  filter_upwards [hr] with r hr
-  intro E hE e
-  have he := Finite.surjective_of_injective e.injective
-  apply (modified_moment_exception_bound L r hL E e he ε hE).trans
-  apply Real.exp_le_exp.mpr
-  have hn : (0:ℝ) < (2:ℝ)^(L+r) := by positivity
-  have hd : modifiedOverhead L r * (2:ℝ)^(L+r) =
-      (((L+r:ℕ):ℝ)*Real.log 2+1)/1000 := by
-    unfold modifiedOverhead
-    field_simp
-  have hh := mul_le_mul_of_nonneg_right hr.le hn.le
-  rw [hd] at hh
-  have ht := mul_le_mul_of_nonneg_left (show heightDecay L+ε*L/250 ≤ 2*η/3 by linarith) hn.le
-  linarith
-
-
-namespace DensityTransfer
-open scoped BigOperators
-
-noncomputable def applyKernel {X : Type*} [Fintype X] (P : X → X → ℝ)
-    (v : EuclideanSpace ℂ X) : EuclideanSpace ℂ X :=
-  WithLp.toLp 2 (fun x => ∑ y, (P x y : ℂ)*v y)
-
-lemma applyKernel_apply {X : Type*} [Fintype X] (P : X → X → ℝ)
-    (v : EuclideanSpace ℂ X) (x : X) :
-    applyKernel P v x = ∑ y, (P x y : ℂ)*v y := rfl
-
-lemma applyKernel_add {X : Type*} [Fintype X] (P Q : X → X → ℝ)
-    (v : EuclideanSpace ℂ X) :
-    applyKernel (fun x y => P x y+Q x y) v = applyKernel P v+applyKernel Q v := by
-  ext x
-  simp only [applyKernel_apply,Complex.ofReal_add,add_mul,Finset.sum_add_distrib]
-  rfl
-
-lemma kernel_component_sq {X : Type*} [Fintype X] (P : X → X → ℝ)
-    (hP : ∀ x y, 0 ≤ P x y) (v : EuclideanSpace ℂ X) (x : X) :
-    ‖applyKernel P v x‖^2 ≤ (∑ y, P x y) * ∑ y, P x y*‖v y‖^2 := by
-  have hs : ‖applyKernel P v x‖ ≤ ∑ y, P x y*‖v y‖ := by
-    rw [applyKernel_apply]
-    apply (norm_sum_le _ _).trans
-    apply Finset.sum_le_sum
-    intro y _
-    rw [norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (hP x y)]
-  apply (pow_le_pow_left₀ (norm_nonneg _) hs 2).trans
-  exact Finset.sum_sq_le_sum_mul_sum_of_sq_le_mul Finset.univ
-    (fun y _ => hP x y) (fun y _ => mul_nonneg (hP x y) (sq_nonneg _))
-    (fun y _ => by ring_nf; exact le_rfl)
-
-lemma schur_bound {X : Type*} [Fintype X] (P : X → X → ℝ)
-    (hP : ∀ x y, 0 ≤ P x y) (δ : ℝ) (hδ : 0 ≤ δ)
-    (hrow : ∀ x, ∑ y, P x y ≤ δ) (hcol : ∀ y, ∑ x, P x y ≤ δ)
-    (v : EuclideanSpace ℂ X) : ‖applyKernel P v‖ ≤ δ*‖v‖ := by
-  have hsq : ‖applyKernel P v‖^2 ≤ δ^2*‖v‖^2 := by
-    rw [EuclideanSpace.norm_sq_eq,EuclideanSpace.norm_sq_eq]
-    calc
-      _ ≤ ∑ x, δ * ∑ y, P x y*‖v y‖^2 := by
-        apply Finset.sum_le_sum
-        intro x _
-        apply (kernel_component_sq P hP v x).trans
-        exact mul_le_mul_of_nonneg_right (hrow x)
-          (Finset.sum_nonneg (fun y _ => mul_nonneg (hP x y) (sq_nonneg _)))
-      _ = δ * ∑ y, (∑ x, P x y)*‖v y‖^2 := by
-        rw [←Finset.mul_sum,Finset.sum_comm]
-        congr 1
-        apply Finset.sum_congr rfl
-        intro y _
-        exact (Finset.sum_mul Finset.univ _ _).symm
-      _ ≤ δ * ∑ y, δ*‖v y‖^2 := by
-        apply mul_le_mul_of_nonneg_left _ hδ
-        exact Finset.sum_le_sum (fun y _ => mul_le_mul_of_nonneg_right (hcol y) (sq_nonneg _))
-      _ = _ := by rw [←Finset.mul_sum]; ring
-  have hn := norm_nonneg (applyKernel P v)
-  have hr := mul_nonneg hδ (norm_nonneg v)
-  nlinarith
-
-lemma hilbert_schmidt_family {X I : Type*} [Fintype X] [Fintype I]
-    (P : X → X → ℝ) (v : I → EuclideanSpace ℂ X) (hv : Orthonormal ℂ v) :
-    (∑ i, ‖applyKernel P (v i)‖^2) ≤ ∑ x, ∑ y, (P x y)^2 := by
-  simp_rw [EuclideanSpace.norm_sq_eq]
-  rw [Finset.sum_comm]
-  apply Finset.sum_le_sum
-  intro x _
-  let w : EuclideanSpace ℂ X := WithLp.toLp 2 (fun y => (P x y : ℂ))
-  have he (i : I) : ‖applyKernel P (v i) x‖ = ‖inner ℂ (v i) w‖ := by
-    rw [norm_inner_symm]
-    congr 1
-    simp only [applyKernel_apply,PiLp.inner_apply]
-    apply Finset.sum_congr rfl
-    intro y _
-    simp [w,RCLike.inner_apply,mul_comm]
-  simp_rw [he]
-  have h := hv.sum_inner_products_le (s := Finset.univ) w
-  apply h.trans_eq
-  rw [EuclideanSpace.norm_sq_eq]
-  apply Finset.sum_congr rfl
-  intro y _
-  simp [w,Complex.norm_real,Real.norm_eq_abs,sq_abs]
-
-
-lemma tail_density_pointwise (t a z : ℝ) (ht : 0 ≤ t) (ha : 0 ≤ a) (hz : 0 < z) :
-    (if z < t then t else 0) ≤ t^(1+a)*z^(-a) := by
-  split_ifs with h
-  · rw [Real.rpow_neg hz.le,←div_eq_mul_inv]
-    apply (le_div_iff₀ (Real.rpow_pos_of_pos hz a)).mpr
-    rw [Real.rpow_add (lt_trans hz h),Real.rpow_one]
-    exact mul_le_mul_of_nonneg_left (Real.rpow_le_rpow hz.le h.le ha) ht
-  · positivity
-
-lemma tail_density_mean {X : Type*} [Fintype X] (f : X → ℝ) (hf : ∀ x, 0 ≤ f x)
-    (a z : ℝ) (ha : 0 ≤ a) (hz : 0 < z) :
-    finiteMean (fun x => if z < f x then f x else 0) ≤ finiteMean (fun x => f x^(1+a))*z^(-a) := by
-  rw [←finiteMean_mul_const]
-  exact finiteMean_mono (fun x => tail_density_pointwise (f x) a z (hf x) ha hz)
-
-noncomputable def tailKernel {X : Type*} [Fintype X] (P : X → X → ℝ) (z : ℝ) : X → X → ℝ :=
-  fun x y => if z < (Fintype.card X:ℝ)*P x y then P x y else 0
-
-noncomputable def headKernel {X : Type*} [Fintype X] (P : X → X → ℝ) (z : ℝ) : X → X → ℝ :=
-  fun x y => if z < (Fintype.card X:ℝ)*P x y then 0 else P x y
-
-lemma tail_add_head {X : Type*} [Fintype X] (P : X → X → ℝ) (z : ℝ) :
-    (fun x y => tailKernel P z x y+headKernel P z x y) = P := by
-  funext x y
-  unfold tailKernel headKernel
-  split_ifs <;> simp
-
-lemma tailKernel_nonneg {X : Type*} [Fintype X] (P : X → X → ℝ) (z : ℝ)
-    (hP : ∀ x y, 0 ≤ P x y) (x y : X) : 0 ≤ tailKernel P z x y := by
-  unfold tailKernel
-  split_ifs <;> simp only [hP,le_refl]
-
-lemma tailKernel_row {X : Type*} [Fintype X] [Nonempty X] (P : X → X → ℝ)
-    (hP : ∀ x y, 0 ≤ P x y) (a z M : ℝ) (ha : 0 ≤ a) (hz : 0 < z)
-    (hM : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*P x y)^(1+a)) ≤ M) (x : X) :
-    ∑ y, tailKernel P z x y ≤ M*z^(-a) := by
-  have hn : (Fintype.card X:ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
-  have he : finiteMean (fun y => if z < (Fintype.card X:ℝ)*P x y then
-      (Fintype.card X:ℝ)*P x y else 0) = ∑ y, tailKernel P z x y := by
-    unfold finiteMean
-    calc
-      _ = (∑ y, (Fintype.card X:ℝ)*tailKernel P z x y)/(Fintype.card X:ℝ) := by
-        congr 1
-        apply Finset.sum_congr rfl
-        intro y _
-        by_cases hy : z < (Fintype.card X:ℝ)*P x y
-        · simp only [tailKernel,hy,↓reduceIte]
-        · simp only [tailKernel,hy,↓reduceIte,mul_zero]
-      _ = _ := by rw [←Finset.mul_sum]; field_simp
-  rw [←he]
-  exact (tail_density_mean _ (fun y => mul_nonneg (Nat.cast_nonneg _) (hP x y)) a z ha hz).trans
-    (mul_le_mul_of_nonneg_right (hM x) (Real.rpow_nonneg hz.le _))
-
-lemma tailKernel_schur {X : Type*} [Fintype X] [Nonempty X] (P : X → X → ℝ)
-    (hP : ∀ x y, 0 ≤ P x y) (hsym : ∀ x y, P x y = P y x)
-    (a z M : ℝ) (ha : 0 ≤ a) (hz : 0 < z) (hM0 : 0 ≤ M)
-    (hM : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*P x y)^(1+a)) ≤ M)
-    (v : EuclideanSpace ℂ X) :
-    ‖applyKernel (tailKernel P z) v‖ ≤ (M*z^(-a))*‖v‖ := by
-  apply schur_bound _ (tailKernel_nonneg P z hP) _ (by positivity)
-    (tailKernel_row P hP a z M ha hz hM)
-  intro y
-  have hs : (fun x => tailKernel P z x y) = (fun x => tailKernel P z y x) := by
-    funext x
-    simp only [tailKernel,hsym x y]
-  rw [hs]
-  exact tailKernel_row P hP a z M ha hz hM y
-
-lemma headKernel_hs {X : Type*} [Fintype X] [Nonempty X] (P : X → X → ℝ)
-    (hP : ∀ x y, 0 ≤ P x y) (hrow : ∀ x, ∑ y, P x y = 1) (z : ℝ) (hz : 0 ≤ z) :
-    (∑ x, ∑ y, (headKernel P z x y)^2) ≤ z := by
-  have hn : (0:ℝ) < Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
-  have he (x y : X) : (headKernel P z x y)^2 ≤ (z/(Fintype.card X:ℝ))*P x y := by
-    unfold headKernel
-    split_ifs with h
-    · simp only [zero_pow (by omega : (2:ℕ) ≠ 0)]
-      exact mul_nonneg (div_nonneg hz hn.le) (hP x y)
-    · have h' : (Fintype.card X:ℝ)*P x y ≤ z := le_of_not_gt h
-      have hh := mul_le_mul_of_nonneg_right h' (hP x y)
-      rw [show (z/(Fintype.card X:ℝ))*P x y = (z*P x y)/(Fintype.card X:ℝ) by ring]
-      apply (le_div_iff₀ hn).mpr
-      nlinarith
-  calc
-    _ ≤ ∑ x, ∑ y, (z/(Fintype.card X:ℝ))*P x y :=
-      Finset.sum_le_sum (fun x _ => Finset.sum_le_sum (fun y _ => he x y))
-    _ = z := by
-      simp only [←Finset.mul_sum,hrow,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
-      field_simp
-
-
-theorem multiplicity_amplification {X I : Type*} [Fintype X] [Nonempty X]
-    [Fintype I] [Nonempty I] (P : X → X → ℝ)
-    (hP : ∀ x y, 0 ≤ P x y) (hsym : ∀ x y, P x y = P y x)
-    (hrow : ∀ x, ∑ y, P x y = 1) (a z M : ℝ) (ha : 0 ≤ a) (hz : 0 < z) (hM0 : 0 ≤ M)
-    (hM : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*P x y)^(1+a)) ≤ M)
-    (v : I → EuclideanSpace ℂ X) (hv : Orthonormal ℂ v)
-    (μ : ℝ) (hvμ : ∀ i, μ ≤ ‖applyKernel P (v i)‖) :
-    μ ≤ M*z^(-a)+Real.sqrt (z/(Fintype.card I:ℝ)) := by
-  let δ : ℝ := M*z^(-a)
-  have hδ : 0 ≤ δ := by dsimp [δ]; positivity
-  have hm : (0:ℝ) < Fintype.card I := Nat.cast_pos.mpr Fintype.card_pos
-  change μ ≤ δ+_
-  by_cases hμ : μ ≤ δ
-  · exact hμ.trans (le_add_of_nonneg_right (Real.sqrt_nonneg _))
-  have hmd : 0 ≤ μ-δ := by linarith
-  have hsmall (i : I) : μ-δ ≤ ‖applyKernel (headKernel P z) (v i)‖ := by
-    have hr := tailKernel_schur P hP hsym a z M ha hz hM0 hM (v i)
-    rw [hv.norm_eq_one i,mul_one] at hr
-    have he := applyKernel_add (tailKernel P z) (headKernel P z) (v i)
-    rw [tail_add_head] at he
-    have hn := norm_add_le (applyKernel (tailKernel P z) (v i)) (applyKernel (headKernel P z) (v i))
-    rw [←he] at hn
-    have hh := hvμ i
-    change ‖applyKernel (tailKernel P z) (v i)‖ ≤ δ at hr
-    linarith
-  have hsum : (Fintype.card I:ℝ)*(μ-δ)^2 ≤ z := by
-    calc
-      _ = ∑ i : I, (μ-δ)^2 := by simp
-      _ ≤ ∑ i : I, ‖applyKernel (headKernel P z) (v i)‖^2 :=
-        Finset.sum_le_sum (fun i _ => pow_le_pow_left₀ hmd (hsmall i) 2)
-      _ ≤ ∑ x, ∑ y, (headKernel P z x y)^2 := hilbert_schmidt_family _ v hv
-      _ ≤ z := headKernel_hs P hP hrow z hz.le
-  have hsq : (μ-δ)^2 ≤ z/(Fintype.card I:ℝ) := (le_div_iff₀ hm).mpr (by nlinarith)
-  have hs := (Real.le_sqrt hmd (div_nonneg hz.le hm.le)).mpr hsq
-  linarith
-
-end DensityTransfer
-namespace PairRouting
-open scoped BigOperators
-
-lemma tupleProbability_row_sum {ι β Ω : Type*} [Fintype ι] [Fintype β]
-    [DecidableEq β] [Fintype Ω] [Nonempty Ω]
-    (P : Ω → Equiv.Perm β) (e : ι ↪ β) :
-    (∑ f : ι ↪ β, tupleProbability P e f) = 1 := by
-  classical
-  simp only [tupleProbability]
-  rw [←finiteMean_sum]
-  have h (ω : Ω) (f : ι ↪ β) : (∀ i, P ω (e i) = f i) ↔ e.trans (P ω).toEmbedding = f := by
-    constructor
-    · intro h; ext i; exact h i
-    · intro h i; exact congrArg (fun g : ι ↪ β => g i) h
-  simp_rw [h]
-  simp only [Finset.sum_ite_eq,Finset.mem_univ,↓reduceIte,finiteMean_const]
-
-lemma tupleProbability_inverse {ι β Ω : Type*} [Fintype ι] [DecidableEq β]
-    [Fintype Ω] (P : Ω → Equiv.Perm β) (e f : ι → β) :
-    tupleProbability (fun ω => (P ω)⁻¹) e f = tupleProbability P f e := by
-  classical
-  apply finiteMean_congr
-  intro ω
-  congr 1
-  apply propext
-  constructor
-  · intro h i
-    have hh := congrArg (P ω) (h i)
-    simpa only [Equiv.Perm.inv_def,Equiv.apply_symm_apply] using hh.symm
-  · intro h i
-    have hh := congrArg (fun x => (P ω).symm x) (h i)
-    simpa only [Equiv.Perm.inv_def,Equiv.symm_apply_apply] using hh.symm
-
-lemma tupleProbability_equiv {ι β Ω Ω' : Type*} [Fintype ι] [DecidableEq β]
-    [Fintype Ω] [Fintype Ω'] (P : Ω → Equiv.Perm β) (τ : Ω ≃ Ω') (e f : ι → β) :
-    tupleProbability P e f = tupleProbability (fun ω => P (τ.symm ω)) e f :=
-  finiteMean_equiv τ _
-
-lemma tupleProbability_symmetric {ι β Ω : Type*} [Fintype ι] [DecidableEq β]
-    [Fintype Ω] (P : Ω → Equiv.Perm β) (τ : Equiv.Perm Ω)
-    (hτ : ∀ ω, P (τ ω) = (P ω)⁻¹) (e f : ι → β) :
-    tupleProbability P e f = tupleProbability P f e := by
-  rw [tupleProbability_equiv P τ.symm]
-  simp only [Equiv.symm_symm,hτ]
-  exact tupleProbability_inverse P e f
-
-end PairRouting
-
-lemma palindromePerm_reverse (d : ℕ) (ω : BenesCoins d) :
-    palindromePerm d (ω.2,ω.1) = (palindromePerm d ω)⁻¹ := by
-  simp only [palindromePerm,mul_inv_rev]
-  rfl
-
-lemma palindrome_kernel_symmetric (d : ℕ) {ι : Type*} [Fintype ι]
-    (e f : ι ↪ Card d) :
-    PairRouting.tupleProbability (palindromePerm d) e f =
-      PairRouting.tupleProbability (palindromePerm d) f e :=
-  PairRouting.tupleProbability_symmetric _ (Equiv.prodComm _ _) (palindromePerm_reverse d) e f
-
-theorem palindrome_density_transfer (d : ℕ) {ι I : Type*} [Fintype ι] [Fintype I]
-    [Nonempty I] (e₀ : ι ↪ Card d) (hk : 0 < Fintype.card ι)
-    (v : I → EuclideanSpace ℂ (ι ↪ Card d)) (hv : Orthonormal ℂ v)
-    (μ : ℝ) (hvμ : ∀ i, μ ≤ ‖DensityTransfer.applyKernel
-      (fun e f : ι ↪ Card d => PairRouting.tupleProbability (palindromePerm d) e f) (v i)‖)
-    (z : ℝ) (hz : 0 < z) :
-    μ ≤ Real.exp (densityConstant*Fintype.card ι*
-      ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ))*z^(-(1/1000:ℝ))+
-      Real.sqrt (z/(Fintype.card I:ℝ)) := by
-  let : Nonempty (ι ↪ Card d) := ⟨e₀⟩
-  apply DensityTransfer.multiplicity_amplification
-    (fun e f : ι ↪ Card d => PairRouting.tupleProbability (palindromePerm d) e f)
-    (fun e f => PairRouting.tupleProbability_nonneg _ e f)
-    (palindrome_kernel_symmetric d) (PairRouting.tupleProbability_row_sum _)
-    (1/1000) z _ (by norm_num) hz (Real.exp_nonneg _) _ v hv μ hvμ
-  intro e
-  exact palindrome_row_moment_bound d e hk
-
-
-noncomputable def modifiedReverseCoins (L : ℕ) : (r : ℕ) → ModifiedCoins L r → ModifiedCoins L r
-  | 0, ω => ((fun u => butterflyPerm L (decodeButterfly L ω.2.1) * (ω.1 u)⁻¹ *
-      butterflyPerm L (decodeButterfly L ω.2.1)),ω.2)
-  | r+1, ω =>
-      let z := modifiedStepEquiv L r ω
-      (modifiedStepEquiv L r).symm
-        ((modifiedReverseCoins L r z.1.1,modifiedReverseCoins L r z.1.2),(z.2.2,z.2.1))
-
-lemma modifiedReverseCoins_involutive (L r : ℕ) :
-    Function.Involutive (modifiedReverseCoins L r) := by
-  induction r with
-  | zero =>
-      intro ω
-      apply Prod.ext
-      · funext u
-        simp only [modifiedReverseCoins,mul_inv_rev,inv_inv]
-        simp [mul_assoc]
-      · rfl
-  | succ r ih =>
-      change ∀ ω, modifiedReverseCoins L r (modifiedReverseCoins L r ω) = ω at ih
-      intro ω
-      simp only [modifiedReverseCoins,Equiv.apply_symm_apply,ih]
-      exact (modifiedStepEquiv L r).symm_apply_apply ω
-
-end Thorp
-
-end ThorpNine.Tail
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Tail.InputTree
+
+namespace OAI
+
+namespace ThorpNine.Tail
+
+namespace Thorp
+
+noncomputable def modifiedRoutingMoment (L r : ℕ) (E : Finset (Card r))
+    {ι : Type*} [Fintype ι] (e f : ι ↪ Card (L+r)) (a : ℝ) : ℝ := by
+  classical
+  exact finiteMean (fun ω : ModifiedCoins L r =>
+    if ∀ i, modifiedPerm L r E ω (e i) = f i then (2:ℝ)^((modifiedCost L r E e ω:ℝ)*a) else 0)
+
+lemma modifiedRoutingMoment_nonneg (L r : ℕ) (E : Finset (Card r))
+    {ι : Type*} [Fintype ι] (e f : ι ↪ Card (L+r)) (a : ℝ) :
+    0 ≤ modifiedRoutingMoment L r E e f a := by
+  classical
+  exact finiteMean_nonneg (fun _ => by split_ifs <;> positivity)
+
+lemma modifiedRoutingMoment_zero (L : ℕ) (E : Finset (Card 0))
+    {ι : Type*} [Fintype ι] (e f : ι ↪ Card L) (a : ℝ) :
+    modifiedRoutingMoment L 0 E e f a = PairRouting.tupleProbability (modifiedPerm L 0 E) e f := by
+  simp only [modifiedRoutingMoment,modifiedCost_zero,Nat.cast_zero,zero_mul,Real.rpow_zero,
+    PairRouting.tupleProbability]
+
+lemma modified_pointwise_zero (L : ℕ) (E : Finset (Card 0))
+    {ι : Type*} [Fintype ι] (e f : ι ↪ Card L) (he : Function.Surjective e) (a : ℝ) (ha : 0 ≤ a) :
+    (PairRouting.tupleProbability (modifiedPerm L 0 E) e f)^(1+a) ≤
+      centralFactor L 0 E a * modifiedRoutingMoment L 0 E e f a := by
+  have hp := PairRouting.tupleProbability_nonneg (modifiedPerm L 0 E) e f
+  rw [modifiedRoutingMoment_zero]
+  have hbound := modified_probability_zero_le L E e f he
+  have hpow := Real.rpow_le_rpow hp hbound ha
+  have hfpos : (0:ℝ) < Nat.factorial (2^L) := Nat.cast_pos.mpr (Nat.factorial_pos _)
+  have heq : (if (fun j => Fin.elim0 j : Card 0) ∈ E then (1:ℝ) else 1/(Nat.factorial (2^L):ℝ))^a =
+      centralFactor L 0 E a := by
+    rw [centralFactor_zero]
+    split_ifs
+    · exact Real.one_rpow _
+    · rw [one_div,Real.inv_rpow (le_of_lt hfpos),Real.rpow_neg (le_of_lt hfpos)]
+  rw [heq] at hpow
+  rw [Real.rpow_add' hp (by linarith : (1+a:ℝ) ≠ 0),Real.rpow_one,mul_comm]
+  exact mul_le_mul_of_nonneg_right hpow hp
+
+
+lemma modified_matches_step (L r : ℕ) (E : Finset (Card (r+1))) {ι : Type*}
+    (e f : ι ↪ Card (L+r+1)) (ω : ModifiedCoins L r × ModifiedCoins L r)
+    (coins : (Card (L+r) → Bool) × (Card (L+r) → Bool)) :
+    (∀ i, modifiedPerm L (r+1) E ((modifiedStepEquiv L r).symm (ω,coins)) (e i) = f i) ↔
+    (∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1))
+      ((e.trans (headTailEquiv (L+r)).toEmbedding) i) = ((f.trans (headTailEquiv (L+r)).toEmbedding) i)) := by
+  apply forall_congr'
+  intro i
+  rw [←(headTailEquiv (L+r)).injective.eq_iff,modifiedPerm_coordinates]
+  rfl
+
+lemma modifiedRoutingMoment_step (L r : ℕ) (E : Finset (Card (r+1))) {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (e f : ι ↪ Card (L+r+1)) (a : ℝ) :
+    let x := e.trans (headTailEquiv (L+r)).toEmbedding
+    let y := f.trans (headTailEquiv (L+r)).toEmbedding
+    modifiedRoutingMoment L (r+1) E e f a = PairRouting.outerWeight x y *
+      ∑ c : {c // c ∈ PairRouting.compatibleSet x y},
+        finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
+          if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c.val
+          then (2:ℝ)^((modifiedAssignmentCost L r E x c.val
+            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0) := by
+  classical
+  intro x y
+  let : Fintype (ModifiedCoins L r) := inferInstance
+  rw [modifiedRoutingMoment,finiteMean_equiv (modifiedStepEquiv L r),finiteMean_prod]
+  have he (ω : ModifiedCoins L r × ModifiedCoins L r) :
+      finiteMean (fun coins : (Card (L+r) → Bool) × (Card (L+r) → Bool) =>
+        if ∀ i, modifiedPerm L (r+1) E ((modifiedStepEquiv L r).symm (ω,coins)) (e i) = f i
+        then (2:ℝ)^((modifiedCost L (r+1) E e ((modifiedStepEquiv L r).symm (ω,coins)):ℝ)*a) else 0) =
+      PairRouting.outerWeight x y *
+        ∑ c : {c // c ∈ PairRouting.compatibleSet x y},
+          if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c.val
+          then (2:ℝ)^((modifiedAssignmentCost L r E x c.val
+            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0 := by
+    let W : (ι → Bool) → ℝ := fun c =>
+      if hc : PairRouting.Compatible x c then (2:ℝ)^((modifiedAssignmentCost L r E x c hc ω:ℝ)*a) else 0
+    calc
+      _ = finiteMean (fun coins : (Card (L+r) → Bool) × (Card (L+r) → Bool) =>
+        if ∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1))
+          (x i) = y i then W (PairRouting.colors x coins.1) else 0) := by
+        apply finiteMean_congr
+        intro coins
+        simp only [modified_matches_step,modifiedCost_step]
+        simp only [W,PairRouting.colors_compatible,↓reduceDIte]
+        rfl
+      _ = _ := by
+        rw [PairRouting.sandwich_weighted_probability_subtype]
+        congr 1
+        apply Finset.sum_congr rfl
+        intro c _
+        simp only [W,((PairRouting.mem_compatibleSet x y c.val).mp c.property).1,↓reduceDIte]
+  simp_rw [he]
+  simp_rw [mul_comm (PairRouting.outerWeight x y)]
+  rw [finiteMean_mul_const,finiteMean_sum,mul_comm]
+
+lemma modified_child_moment_product (L r : ℕ) (E : Finset (Card (r+1))) {ι : Type*} [Fintype ι]
+    (x y : ι ↪ Bool × Card (L+r)) (c : ι → Bool)
+    (hcx : PairRouting.Compatible x c) (hcy : PairRouting.Compatible y c) (a : ℝ) :
+    modifiedRoutingMoment L r (exceptionChild E false) (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a *
+      modifiedRoutingMoment L r (exceptionChild E true) (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a =
+    finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
+      if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c
+      then (2:ℝ)^(((modifiedCost L r (exceptionChild E false) (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
+        modifiedCost L r (exceptionChild E true) (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0) := by
+  classical
+  unfold modifiedRoutingMoment
+  rw [←finiteMean_prod_mul]
+  apply finiteMean_congr
+  intro ω
+  let M₀ := ∀ i : {i // c i = false}, modifiedPerm L r (exceptionChild E false) ω.1 (x i.val).2 = (y i.val).2
+  let M₁ := ∀ i : {i // c i = true}, modifiedPerm L r (exceptionChild E true) ω.2 (x i.val).2 = (y i.val).2
+  have hp : (fun b : Bool => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) =
+      (fun b => if b then modifiedPerm L r (exceptionChild E true) ω.2 else modifiedPerm L r (exceptionChild E false) ω.1) := by
+    funext b
+    cases b <;> rfl
+  have hs : PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c ↔
+      M₀ ∧ M₁ := by rw [hp]; exact PairRouting.childMatches_split _ _ _ _ _
+  change (if M₀ then _ else 0) * (if M₁ then _ else 0) = _
+  by_cases h₀ : M₀ <;> by_cases h₁ : M₁
+  · simp only [hs,h₀,h₁,true_and,↓reduceIte]
+    rw [←Real.rpow_add (by norm_num : (0:ℝ)<2)]
+    congr 1
+    ring
+  · simp only [hs,h₀,h₁,and_false,↓reduceIte,mul_zero]
+  · simp only [hs,h₀,h₁,false_and,↓reduceIte,zero_mul]
+  · simp only [hs,h₀,h₁,false_and,↓reduceIte,mul_zero]
+
+lemma modified_probability_step (L r : ℕ) (E : Finset (Card (r+1))) {ι : Type*} [Fintype ι]
+    (e f : ι ↪ Card (L+r+1)) :
+    PairRouting.tupleProbability (modifiedPerm L (r+1) E) e f =
+    finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
+      finiteMean (fun coins : (Card (L+r) → Bool) × (Card (L+r) → Bool) =>
+        if ∀ i, PairRouting.sandwich coins.1 coins.2 (fun b => if b then modifiedPerm L r (exceptionChild E true) ω.2 else modifiedPerm L r (exceptionChild E false) ω.1)
+          ((e.trans (headTailEquiv (L+r)).toEmbedding) i) = ((f.trans (headTailEquiv (L+r)).toEmbedding) i)
+        then (1:ℝ) else 0)) := by
+  classical
+  let : Fintype (ModifiedCoins L r) := inferInstance
+  rw [PairRouting.tupleProbability,finiteMean_equiv (modifiedStepEquiv L r),finiteMean_prod]
+  have hp (b : Bool) (p q : ModifiedCoins L r) :
+      modifiedPerm L r (exceptionChild E b) (if b then p else q) =
+      (if b then modifiedPerm L r (exceptionChild E true) p else modifiedPerm L r (exceptionChild E false) q) := by
+    cases b <;> rfl
+  simp only [modified_matches_step,hp]
+
+
+lemma modified_assignment_moment_bound (L r : ℕ) (E : Finset (Card (r+1))) {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (x y : ι ↪ Bool × Card (L+r)) (c : ι → Bool) (hc : c ∈ PairRouting.compatibleSet x y)
+    (a : ℝ) (ha : 0 ≤ a) :
+    let hcx := ((PairRouting.mem_compatibleSet x y c).mp hc).1
+    PairRouting.outerWeight x y ^ a * ((PairRouting.compatibleSet x y).card:ℝ)^a *
+      ((2:ℝ)^(-(r:ℝ)*Fintype.card ι*a) *
+        finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
+          if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c
+          then (2:ℝ)^(((modifiedCost L r (exceptionChild E false) (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
+            modifiedCost L r (exceptionChild E true) (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0)) ≤
+    (2:ℝ)^(-((r:ℝ)+1)*Fintype.card ι*a) *
+      finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
+        if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c
+        then (2:ℝ)^((modifiedAssignmentCost L r E x c hcx ω:ℝ)*a) else 0) := by
+  classical
+  intro hcx
+  have hscale (q : ℝ) (F : ModifiedCoins L r × ModifiedCoins L r → ℝ) :
+      q * finiteMean F = finiteMean (fun ω => q * F ω) := by
+    simp_rw [mul_comm q,finiteMean_mul_const]
+  rw [←mul_assoc,hscale,hscale]
+  apply finiteMean_mono
+  intro ω
+  split_ifs with hm
+  · have hh := PairRouting.assignment_factor_bound x y c hc
+      (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) hm a ha
+    calc
+      _ ≤ ((2:ℝ)^(((PairRouting.alternatingCycles (PairRouting.coloredEmbedding x c hcx)
+          (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)):ℝ)-Fintype.card ι)*a) *
+          (2:ℝ)^(-(r:ℝ)*Fintype.card ι*a)) *
+          (2:ℝ)^(((modifiedCost L r (exceptionChild E false) (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
+            modifiedCost L r (exceptionChild E true) (PairRouting.childEmbedding x c hcx true) ω.2)*a) :=
+        mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hh (by positivity)) (by positivity)
+      _ = _ := by
+        simp only [←Real.rpow_add (by norm_num : (0:ℝ)<2),modifiedAssignmentCost,Nat.cast_add]
+        congr 1
+        ring
+  · simp only [mul_zero]
+    exact le_rfl
+
+
+lemma modified_child_power_product_bound (L r : ℕ) (E : Finset (Card (r+1)))
+    {ι : Type*} [Fintype ι] (x y : ι ↪ Bool × Card (L+r)) (c : ι → Bool)
+    (hcx : PairRouting.Compatible x c) (hcy : PairRouting.Compatible y c) (a : ℝ)
+    (hfalse : (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E false))
+        (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false))^(1+a) ≤
+      ((2:ℝ)^(-(r:ℝ)*Fintype.card {i // c i = false}*a)*centralFactor L r (exceptionChild E false) a) *
+        modifiedRoutingMoment L r (exceptionChild E false)
+          (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a)
+    (htrue : (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E true))
+        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true))^(1+a) ≤
+      ((2:ℝ)^(-(r:ℝ)*Fintype.card {i // c i = true}*a)*centralFactor L r (exceptionChild E true) a) *
+        modifiedRoutingMoment L r (exceptionChild E true)
+          (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a) :
+    (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E false))
+      (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false))^(1+a) *
+      (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E true))
+        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true))^(1+a) ≤
+    ((2:ℝ)^(-(r:ℝ)*Fintype.card ι*a)*centralFactor L (r+1) E a) *
+      finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
+        if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c
+        then (2:ℝ)^(((modifiedCost L r (exceptionChild E false) (PairRouting.childEmbedding x c hcx false) ω.1 : ℝ) +
+          modifiedCost L r (exceptionChild E true) (PairRouting.childEmbedding x c hcx true) ω.2)*a) else 0) := by
+  have hh := mul_le_mul hfalse htrue
+    (Real.rpow_nonneg (PairRouting.tupleProbability_nonneg _ _ _) _)
+    (mul_nonneg (mul_nonneg (by positivity) (centralFactor_nonneg _ _ _ _))
+      (modifiedRoutingMoment_nonneg _ _ _ _ _ _))
+  have he : -(r:ℝ)*Fintype.card {i // c i = false}*a + -(r:ℝ)*Fintype.card {i // c i = true}*a =
+      -(r:ℝ)*Fintype.card ι*a := by
+    rw [←add_mul,←mul_add,←Nat.cast_add,PairRouting.color_card_add]
+  calc
+    _ ≤ _ := hh
+    _ = ((2:ℝ)^(-(r:ℝ)*Fintype.card {i // c i = false}*a)*
+        (2:ℝ)^(-(r:ℝ)*Fintype.card {i // c i = true}*a)) *
+      (centralFactor L r (exceptionChild E false) a * centralFactor L r (exceptionChild E true) a) *
+      (modifiedRoutingMoment L r (exceptionChild E false)
+        (PairRouting.childEmbedding x c hcx false) (PairRouting.childEmbedding y c hcy false) a *
+      modifiedRoutingMoment L r (exceptionChild E true)
+        (PairRouting.childEmbedding x c hcx true) (PairRouting.childEmbedding y c hcy true) a) := by ring
+    _ = _ := by rw [←Real.rpow_add (by norm_num : (0:ℝ)<2),he,←centralFactor_step,modified_child_moment_product]
+
+
+theorem modified_pointwise_moment (L r : ℕ) (E : Finset (Card r)) {ι : Type*} [Fintype ι]
+    (e f : ι ↪ Card (L+r)) (he : Function.Surjective e) (a : ℝ) (ha : 0 ≤ a) :
+    (PairRouting.tupleProbability (modifiedPerm L r E) e f)^(1+a) ≤
+      ((2:ℝ)^(-(r:ℝ)*Fintype.card ι*a)*centralFactor L r E a) * modifiedRoutingMoment L r E e f a := by
+  classical
+  induction r generalizing ι with
+  | zero =>
+      simpa only [Nat.cast_zero,neg_zero,zero_mul,Real.rpow_zero,one_mul] using
+        modified_pointwise_zero L E e f he a ha
+  | succ r ih =>
+      let x := e.trans (headTailEquiv (L+r)).toEmbedding
+      let y := f.trans (headTailEquiv (L+r)).toEmbedding
+      let W := PairRouting.outerWeight x y
+      let C := PairRouting.compatibleSet x y
+      let p : (ι → Bool) → ℝ := fun c =>
+        (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E false))
+          (fun i : {i // c i = false} => (x i.val).2)
+          (fun i : {i // c i = false} => (y i.val).2))^(1+a) *
+        (PairRouting.tupleProbability (modifiedPerm L r (exceptionChild E true))
+          (fun i : {i // c i = true} => (x i.val).2)
+          (fun i : {i // c i = true} => (y i.val).2))^(1+a)
+      let m : {c // c ∈ C} → ℝ := fun c =>
+        finiteMean (fun ω : ModifiedCoins L r × ModifiedCoins L r =>
+          if PairRouting.ChildMatches (fun b => modifiedPerm L r (exceptionChild E b) (if b then ω.2 else ω.1)) x y c.val
+          then (2:ℝ)^((modifiedAssignmentCost L r E x c.val
+            ((PairRouting.mem_compatibleSet x y c.val).mp c.property).1 ω:ℝ)*a) else 0)
+      have hp (c : {c // c ∈ C}) : W^a * (C.card:ℝ)^a * p c.val ≤
+          ((2:ℝ)^(-((r:ℝ)+1)*Fintype.card ι*a)*centralFactor L (r+1) E a) * m c := by
+        obtain ⟨hcx,hcy⟩ := (PairRouting.mem_compatibleSet x y c.val).mp c.property
+        have h₀ := ih (exceptionChild E false) (PairRouting.childEmbedding x c.val hcx false)
+          (PairRouting.childEmbedding y c.val hcy false)
+          (PairRouting.childEmbedding_surjective x ((headTailEquiv (L+r)).surjective.comp he) c.val hcx false)
+        have h₁ := ih (exceptionChild E true) (PairRouting.childEmbedding x c.val hcx true)
+          (PairRouting.childEmbedding y c.val hcy true)
+          (PairRouting.childEmbedding_surjective x ((headTailEquiv (L+r)).surjective.comp he) c.val hcx true)
+        have hh := modified_child_power_product_bound L r E x y c.val hcx hcy a h₀ h₁
+        have hh' := mul_le_mul_of_nonneg_left hh
+          (mul_nonneg (Real.rpow_nonneg (le_of_lt (PairRouting.outerWeight_pos x y)) a)
+            (Real.rpow_nonneg (Nat.cast_nonneg C.card) a))
+        apply hh'.trans
+        have hm := modified_assignment_moment_bound L r E x y c.val c.property a ha
+        convert mul_le_mul_of_nonneg_left hm (centralFactor_nonneg L (r+1) E a) using 1 <;> ring
+      have hnode := PairRouting.sandwich_power_bound (modifiedPerm L r (exceptionChild E false)) (modifiedPerm L r (exceptionChild E true)) x y a ha
+      rw [←modified_probability_step L r E e f] at hnode
+      change (PairRouting.tupleProbability (modifiedPerm L (r+1) E) e f)^(1+a) ≤
+        W^(1+a) * (C.card:ℝ)^a * ∑ c ∈ C, p c at hnode
+      have hsum : ∑ c ∈ C, p c = ∑ c : {c // c ∈ C}, p c.val := by
+        rw [Finset.sum_subtype C (fun _ => Iff.rfl)]
+      have hW : 0 < W := PairRouting.outerWeight_pos x y
+      calc
+        _ ≤ W^(1+a) * (C.card:ℝ)^a * ∑ c ∈ C, p c := hnode
+        _ = W * ∑ c : {c // c ∈ C}, W^a * (C.card:ℝ)^a * p c.val := by
+          rw [Real.rpow_add hW,Real.rpow_one,hsum,←Finset.mul_sum]
+          ring
+        _ ≤ W * ∑ c : {c // c ∈ C}, ((2:ℝ)^(-((r:ℝ)+1)*Fintype.card ι*a)*centralFactor L (r+1) E a) * m c :=
+          mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun c _ => hp c)) (le_of_lt hW)
+        _ = _ := by
+          rw [←Finset.mul_sum,modifiedRoutingMoment_step]
+          simp only [Nat.cast_add,Nat.cast_one]
+          change W * (((2:ℝ)^(-((r:ℝ)+1)*Fintype.card ι*a)*centralFactor L (r+1) E a) * ∑ c, m c) =
+            ((2:ℝ)^(-((r:ℝ)+1)*Fintype.card ι*a)*centralFactor L (r+1) E a) * (W * ∑ c, m c)
+          ring
+
+
+lemma modifiedRoutingMoment_sum (L r : ℕ) (E : Finset (Card r)) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (L+r)) (a : ℝ) :
+    (∑ f : ι ↪ Card (L+r), modifiedRoutingMoment L r E e f a) =
+      finiteMean (fun ω : ModifiedCoins L r => (2:ℝ)^((modifiedCost L r E e ω:ℝ)*a)) := by
+  classical
+  unfold modifiedRoutingMoment
+  rw [←finiteMean_sum]
+  apply finiteMean_congr
+  intro ω
+  let f₀ : ι ↪ Card (L+r) := e.trans (modifiedPerm L r E ω).toEmbedding
+  rw [Finset.sum_eq_single f₀]
+  · have hm : ∀ i, modifiedPerm L r E ω (e i) = f₀ i := fun _ => rfl
+    simp only [hm,implies_true,↓reduceIte]
+  · intro f _ hf
+    have hm : ¬ ∀ i, modifiedPerm L r E ω (e i) = f i := by
+      intro hh
+      apply hf
+      ext i : 1
+      exact (hh i).symm
+    exact ite_eq_right hm
+  · simp
+
+noncomputable def modifiedRowMoment (L r : ℕ) (E : Finset (Card r)) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (L+r)) (a : ℝ) : ℝ :=
+  finiteMean (fun f : ι ↪ Card (L+r) =>
+    ((Fintype.card (ι ↪ Card (L+r)):ℝ)*PairRouting.tupleProbability (modifiedPerm L r E) e f)^(1+a))
+
+lemma full_tuple_card (d : ℕ) {ι : Type*} [Fintype ι] (e : ι ↪ Card d)
+    (he : Function.Surjective e) : Fintype.card ι = 2^d := by
+  rw [←card_positions d]
+  exact Fintype.card_congr (Equiv.ofBijective e ⟨e.injective,he⟩)
+
+lemma exception_card_le (r : ℕ) (E : Finset (Card r)) : E.card ≤ 2^r := by
+  simpa only [card_positions] using E.card_le_univ
+
+noncomputable def modifiedNormalization (L r : ℕ) (E : Finset (Card r)) : ℝ :=
+  (Nat.factorial (2^(L+r)):ℝ) /
+    (((2:ℝ)^r)^(2^(L+r)) * (Nat.factorial (2^L):ℝ)^(2^r-E.card))
+
+lemma modifiedNormalization_pos (L r : ℕ) (E : Finset (Card r)) :
+    0 < modifiedNormalization L r E := by unfold modifiedNormalization; positivity
+
+lemma modified_normalization_identity (L r : ℕ) (E : Finset (Card r)) (a : ℝ) :
+    (Nat.factorial (2^(L+r)):ℝ)^a *
+      ((2:ℝ)^(-(r:ℝ)*(2^(L+r):ℕ)*a)*centralFactor L r E a) =
+    modifiedNormalization L r E ^ a := by
+  rw [←mul_assoc,density_normalization,centralFactor,modifiedNormalization,
+    Real.div_rpow (by positivity) (by positivity),Real.div_rpow (by positivity) (by positivity),
+    Real.mul_rpow (by positivity) (by positivity)]
+  have hc : ((2^r-E.card:ℕ):ℝ) = (2:ℝ)^r-E.card := by
+    rw [Nat.cast_sub (exception_card_le r E),Nat.cast_pow,Nat.cast_ofNat]
+  rw [←Real.rpow_natCast (Nat.factorial (2^L):ℝ),←Real.rpow_mul (by positivity),hc,
+    show -((2:ℝ)^r-E.card)*a = -(((2:ℝ)^r-E.card)*a) by ring,
+    Real.rpow_neg (by positivity)]
+  ring
+
+theorem modified_row_density_moment (L r : ℕ) (E : Finset (Card r)) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (L+r)) (he : Function.Surjective e) (a : ℝ) (ha : 0 ≤ a) :
+    modifiedRowMoment L r E e a ≤ modifiedNormalization L r E ^ a *
+      finiteMean (fun ω : ModifiedCoins L r => (2:ℝ)^((modifiedCost L r E e ω:ℝ)*a)) := by
+  classical
+  have : Nonempty (ι ↪ Card (L+r)) := ⟨e⟩
+  rw [modifiedRowMoment,finite_row_power _ (fun f => PairRouting.tupleProbability_nonneg _ _ _)]
+  calc
+    _ ≤ (Fintype.card (ι ↪ Card (L+r)):ℝ)^a *
+      ∑ f : ι ↪ Card (L+r), ((2:ℝ)^(-(r:ℝ)*Fintype.card ι*a)*centralFactor L r E a) *
+        modifiedRoutingMoment L r E e f a :=
+      mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun f _ => modified_pointwise_moment L r E e f he a ha))
+        (Real.rpow_nonneg (Nat.cast_nonneg _) _)
+    _ = _ := by
+      rw [←Finset.mul_sum,modifiedRoutingMoment_sum,←mul_assoc,Fintype.card_embedding_eq,
+        card_positions,full_tuple_card (L+r) e he,Nat.descFactorial_self,modified_normalization_identity]
+
+lemma modified_total_cost_mgf (L r : ℕ) (hL : 0 < L) (E : Finset (Card r))
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) :
+    finiteMean (fun ω : ModifiedCoins L r => momentBase^(modifiedCost L r E e ω)) ≤
+      Real.exp (Fintype.card ι*heightDecay L +
+        3*((L*E.card*2^L:ℕ):ℝ)*Real.log momentBase) := by
+  let B := Real.exp (Fintype.card ι*heightDecay L +
+    3*((L*E.card*2^L:ℕ):ℝ)*Real.log momentBase)
+  rw [finiteMean_prod]
+  apply le_trans (finiteMean_mono (fun U => ?_)) (finiteMean_const B).le
+  rw [finiteMean_prod,finiteMean_comm]
+  apply le_trans (finiteMean_mono (fun X => ?_)) (finiteMean_const B).le
+  exact modified_cost_mgf L r hL E U X e momentBase momentBase_one_le (momentBase_log.trans (by norm_num))
+
+lemma modifiedRowMoment_bound (L r : ℕ) (hL : 0 < L) (E : Finset (Card r))
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (he : Function.Surjective e) :
+    modifiedRowMoment L r E e (1/1000) ≤
+      modifiedNormalization L r E ^ (1/1000:ℝ) *
+        Real.exp ((2^(L+r):ℕ)*heightDecay L +
+          3*((L*E.card*2^L:ℕ):ℝ)*Real.log momentBase) := by
+  have hm := modified_row_density_moment L r E e he (1/1000) (by norm_num)
+  simp_rw [←momentBase_pow] at hm
+  apply hm.trans
+  apply mul_le_mul_of_nonneg_left _ (Real.rpow_nonneg (modifiedNormalization_pos L r E).le _)
+  simpa only [full_tuple_card (L+r) e he] using modified_total_cost_mgf L r hL E e
+
+
+lemma log_factorial_upper {n : ℕ} (hn : 1 ≤ n) :
+    Real.log (Nat.factorial n) ≤ (n:ℝ)*Real.log n-n+Real.log n+1 := by
+  induction n with
+  | zero => omega
+  | succ n ih =>
+      by_cases hz : n = 0
+      · subst n; norm_num
+      have hn0 : (0:ℝ) < n := Nat.cast_pos.mpr (Nat.pos_of_ne_zero hz)
+      have hn1 : (0:ℝ) < n+1 := by positivity
+      have h := Real.one_sub_inv_le_log_of_pos (div_pos hn1 hn0)
+      rw [Real.log_div (ne_of_gt hn1) (ne_of_gt hn0),inv_div] at h
+      have hh := mul_le_mul_of_nonneg_left h hn1.le
+      have hcancel : ((n:ℝ)+1)*(1-(n:ℝ)/(n+1)) = 1 := by field_simp; ring
+      rw [hcancel] at hh
+      have hi := ih (by omega)
+      rw [Nat.factorial_succ,Nat.cast_mul,Real.log_mul (by positivity) (by positivity),Nat.cast_add,Nat.cast_one]
+      nlinarith
+
+lemma log_modifiedNormalization (L r : ℕ) (E : Finset (Card r)) :
+    Real.log (modifiedNormalization L r E) =
+      Real.log (Nat.factorial (2^(L+r))) -
+        ((2^(L+r):ℕ)*((r:ℝ)*Real.log 2)+((2:ℝ)^r-E.card)*Real.log (Nat.factorial (2^L))) := by
+  unfold modifiedNormalization
+  rw [Real.log_div (by positivity) (by positivity),Real.log_mul (by positivity) (by positivity),
+    Real.log_pow,Real.log_pow,Real.log_pow,
+    Nat.cast_sub (exception_card_le r E),Nat.cast_pow,Nat.cast_ofNat]
+  simp only [Nat.cast_pow,Nat.cast_ofNat]
+
+lemma log_modifiedNormalization_le (L r : ℕ) (E : Finset (Card r)) :
+    Real.log (modifiedNormalization L r E) ≤
+      ((L+r:ℕ):ℝ)*Real.log 2+1+E.card*Real.log (Nat.factorial (2^L)) := by
+  have hN : 1 ≤ 2^(L+r) := one_le_pow₀ (by norm_num)
+  have hS : 1 ≤ 2^L := one_le_pow₀ (by norm_num)
+  have hu := log_factorial_upper hN
+  have hl := mul_le_mul_of_nonneg_left (log_factorial_lower hS) (by positivity : (0:ℝ) ≤ (2:ℝ)^r)
+  simp only [Nat.cast_pow,Nat.cast_ofNat,Real.log_pow,Nat.cast_add] at hu hl ⊢
+  rw [log_modifiedNormalization]
+  simp only [pow_add,Nat.cast_mul,Nat.cast_pow,Nat.cast_ofNat] at hu hl ⊢
+  nlinarith
+
+lemma modified_moment_exponential (L r : ℕ) (hL : 0 < L) (E : Finset (Card r))
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (he : Function.Surjective e) :
+    modifiedRowMoment L r E e (1/1000) ≤
+      Real.exp (((L+r:ℕ)*Real.log 2+1+E.card*Real.log (Nat.factorial (2^L)))/1000 +
+        (2^(L+r):ℕ)*heightDecay L+3*((L*E.card*2^L:ℕ):ℝ)*Real.log momentBase) := by
+  apply (modifiedRowMoment_bound L r hL E e he).trans
+  rw [Real.rpow_def_of_pos (modifiedNormalization_pos L r E),←Real.exp_add]
+  apply Real.exp_le_exp.mpr
+  have h := log_modifiedNormalization_le L r E
+  linarith
+
+lemma log_power_factorial_le (L : ℕ) :
+    Real.log (Nat.factorial (2^L)) ≤ (2:ℝ)^L*L := by
+  have hf : (Nat.factorial (2^L):ℝ) ≤ ((2^L:ℕ):ℝ)^(2^L) := by
+    exact_mod_cast Nat.factorial_le_pow (2^L)
+  have hh := Real.log_le_log (by positivity) hf
+  rw [Real.log_pow,Nat.cast_pow,Nat.cast_ofNat,Real.log_pow] at hh
+  apply hh.trans
+  have hl := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
+  have hn : 0 ≤ (2:ℝ)^L*(L:ℝ) := by positivity
+  nlinarith
+
+lemma modified_moment_exception_bound (L r : ℕ) (hL : 0 < L) (E : Finset (Card r))
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (L+r)) (he : Function.Surjective e)
+    (ε : ℝ) (hE : (E.card:ℝ) ≤ ε*(2:ℝ)^r) :
+    modifiedRowMoment L r E e (1/1000) ≤
+      Real.exp (((L+r:ℕ)*Real.log 2+1)/1000 +
+        (2:ℝ)^(L+r)*(heightDecay L+ε*L/250)) := by
+  apply (modified_moment_exponential L r hL E e he).trans
+  apply Real.exp_le_exp.mpr
+  have hs := mul_le_mul_of_nonneg_left (log_power_factorial_le L) (Nat.cast_nonneg E.card)
+  have hb := mul_le_mul_of_nonneg_left momentBase_log
+    (by positivity : (0:ℝ) ≤ 3*((L*E.card*2^L:ℕ):ℝ))
+  have hexc := mul_le_mul_of_nonneg_left hE (by positivity : (0:ℝ) ≤ (2:ℝ)^L*L/250)
+  simp only [Nat.cast_mul,Nat.cast_pow,Nat.cast_ofNat,pow_add] at hs hb hexc ⊢
+  nlinarith
+
+
+lemma heightDecay_tendsto : Filter.Tendsto heightDecay Filter.atTop (nhds 0) := by
+  have h : Filter.Tendsto (fun L : ℕ => (-(1:ℝ)/64)*(L:ℝ)) Filter.atTop Filter.atBot :=
+    (Filter.tendsto_const_mul_atBot_of_neg (by norm_num : -(1:ℝ)/64 < 0)).mpr
+      tendsto_natCast_atTop_atTop
+  have ht := ((Real.tendsto_exp_atBot.comp h).const_mul 68).div_const (1-Real.exp (-(1:ℝ)/64))
+  convert ht using 1
+  · funext L
+    unfold heightDecay
+    dsimp only [Function.comp_apply]
+    rw [show -(L:ℝ)/64 = (-(1:ℝ)/64)*(L:ℝ) by ring]
+  · simp
+
+noncomputable def modifiedOverhead (L r : ℕ) : ℝ :=
+  (((L+r:ℕ):ℝ)*Real.log 2+1)/((2:ℝ)^(L+r)*1000)
+
+lemma modifiedOverhead_tendsto (L : ℕ) :
+    Filter.Tendsto (modifiedOverhead L) Filter.atTop (nhds 0) := by
+  have h₁ := tendsto_pow_const_div_const_pow_of_one_lt 1 (by norm_num : (1:ℝ)<2)
+  have h₀ := tendsto_pow_const_div_const_pow_of_one_lt 0 (by norm_num : (1:ℝ)<2)
+  simp only [pow_one] at h₁
+  simp only [pow_zero] at h₀
+  have ht := ((h₁.const_mul (Real.log 2)).add (h₀.const_mul ((L:ℝ)*Real.log 2+1))).div_const
+    ((2:ℝ)^L*1000)
+  convert ht using 1
+  · funext r
+    simp only [modifiedOverhead,Nat.cast_add,pow_add]
+    field_simp
+    ring
+  · simp
+
+theorem modified_arbitrarily_small_rate (η : ℝ) (hη : 0 < η) :
+    ∃ L : ℕ, 0 < L ∧ ∃ ε : ℝ, 0 < ε ∧ ε < 1/2 ∧
+      ∀ᶠ r : ℕ in Filter.atTop, ∀ (E : Finset (Card r)), (E.card:ℝ) ≤ ε*(2:ℝ)^r →
+        ∀ (e : Card (L+r) ↪ Card (L+r)),
+          modifiedRowMoment L r E e (1/1000) ≤ Real.exp (η*(2:ℝ)^(L+r)) := by
+  have hsmall : ∀ᶠ L : ℕ in Filter.atTop, heightDecay L < η/3 :=
+    heightDecay_tendsto.eventually (gt_mem_nhds (by linarith : (0:ℝ)<η/3))
+  obtain ⟨L,hdec,hL⟩ := (hsmall.and (Filter.eventually_ge_atTop 1)).exists
+  have hL0 : (0:ℝ) < L := Nat.cast_pos.mpr hL
+  let ε : ℝ := min (1/4) (η*250/(3*L))
+  have hε : 0 < ε := lt_min (by norm_num) (by positivity)
+  have hε1 : ε < 1/2 := (min_le_left _ _).trans_lt (by norm_num)
+  have hεL : ε*L/250 ≤ η/3 := by
+    have h := min_le_right (1/4:ℝ) (η*250/(3*L))
+    change ε ≤ η*250/(3*L) at h
+    have hh := mul_le_mul_of_nonneg_right h hL0.le
+    have he : (η*250/(3*(L:ℝ)))*(L:ℝ)/250 = η/3 := by field_simp
+    exact (div_le_div_of_nonneg_right hh (by norm_num)).trans he.le
+  refine ⟨L,hL,ε,hε,hε1,?_⟩
+  have hr : ∀ᶠ r : ℕ in Filter.atTop, modifiedOverhead L r < η/3 :=
+    (modifiedOverhead_tendsto L).eventually (gt_mem_nhds (by linarith : (0:ℝ)<η/3))
+  filter_upwards [hr] with r hr
+  intro E hE e
+  have he := Finite.surjective_of_injective e.injective
+  apply (modified_moment_exception_bound L r hL E e he ε hE).trans
+  apply Real.exp_le_exp.mpr
+  have hn : (0:ℝ) < (2:ℝ)^(L+r) := by positivity
+  have hd : modifiedOverhead L r * (2:ℝ)^(L+r) =
+      (((L+r:ℕ):ℝ)*Real.log 2+1)/1000 := by
+    unfold modifiedOverhead
+    field_simp
+  have hh := mul_le_mul_of_nonneg_right hr.le hn.le
+  rw [hd] at hh
+  have ht := mul_le_mul_of_nonneg_left (show heightDecay L+ε*L/250 ≤ 2*η/3 by linarith) hn.le
+  linarith
+
+
+namespace DensityTransfer
+open scoped BigOperators
+
+noncomputable def applyKernel {X : Type*} [Fintype X] (P : X → X → ℝ)
+    (v : EuclideanSpace ℂ X) : EuclideanSpace ℂ X :=
+  WithLp.toLp 2 (fun x => ∑ y, (P x y : ℂ)*v y)
+
+lemma applyKernel_apply {X : Type*} [Fintype X] (P : X → X → ℝ)
+    (v : EuclideanSpace ℂ X) (x : X) :
+    applyKernel P v x = ∑ y, (P x y : ℂ)*v y := rfl
+
+lemma applyKernel_add {X : Type*} [Fintype X] (P Q : X → X → ℝ)
+    (v : EuclideanSpace ℂ X) :
+    applyKernel (fun x y => P x y+Q x y) v = applyKernel P v+applyKernel Q v := by
+  ext x
+  simp only [applyKernel_apply,Complex.ofReal_add,add_mul,Finset.sum_add_distrib]
+  rfl
+
+lemma kernel_component_sq {X : Type*} [Fintype X] (P : X → X → ℝ)
+    (hP : ∀ x y, 0 ≤ P x y) (v : EuclideanSpace ℂ X) (x : X) :
+    ‖applyKernel P v x‖^2 ≤ (∑ y, P x y) * ∑ y, P x y*‖v y‖^2 := by
+  have hs : ‖applyKernel P v x‖ ≤ ∑ y, P x y*‖v y‖ := by
+    rw [applyKernel_apply]
+    apply (norm_sum_le _ _).trans
+    apply Finset.sum_le_sum
+    intro y _
+    rw [norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (hP x y)]
+  apply (pow_le_pow_left₀ (norm_nonneg _) hs 2).trans
+  exact Finset.sum_sq_le_sum_mul_sum_of_sq_le_mul Finset.univ
+    (fun y _ => hP x y) (fun y _ => mul_nonneg (hP x y) (sq_nonneg _))
+    (fun y _ => by ring_nf; exact le_rfl)
+
+lemma schur_bound {X : Type*} [Fintype X] (P : X → X → ℝ)
+    (hP : ∀ x y, 0 ≤ P x y) (δ : ℝ) (hδ : 0 ≤ δ)
+    (hrow : ∀ x, ∑ y, P x y ≤ δ) (hcol : ∀ y, ∑ x, P x y ≤ δ)
+    (v : EuclideanSpace ℂ X) : ‖applyKernel P v‖ ≤ δ*‖v‖ := by
+  have hsq : ‖applyKernel P v‖^2 ≤ δ^2*‖v‖^2 := by
+    rw [EuclideanSpace.norm_sq_eq,EuclideanSpace.norm_sq_eq]
+    calc
+      _ ≤ ∑ x, δ * ∑ y, P x y*‖v y‖^2 := by
+        apply Finset.sum_le_sum
+        intro x _
+        apply (kernel_component_sq P hP v x).trans
+        exact mul_le_mul_of_nonneg_right (hrow x)
+          (Finset.sum_nonneg (fun y _ => mul_nonneg (hP x y) (sq_nonneg _)))
+      _ = δ * ∑ y, (∑ x, P x y)*‖v y‖^2 := by
+        rw [←Finset.mul_sum,Finset.sum_comm]
+        congr 1
+        apply Finset.sum_congr rfl
+        intro y _
+        exact (Finset.sum_mul Finset.univ _ _).symm
+      _ ≤ δ * ∑ y, δ*‖v y‖^2 := by
+        apply mul_le_mul_of_nonneg_left _ hδ
+        exact Finset.sum_le_sum (fun y _ => mul_le_mul_of_nonneg_right (hcol y) (sq_nonneg _))
+      _ = _ := by rw [←Finset.mul_sum]; ring
+  have hn := norm_nonneg (applyKernel P v)
+  have hr := mul_nonneg hδ (norm_nonneg v)
+  nlinarith
+
+lemma hilbert_schmidt_family {X I : Type*} [Fintype X] [Fintype I]
+    (P : X → X → ℝ) (v : I → EuclideanSpace ℂ X) (hv : Orthonormal ℂ v) :
+    (∑ i, ‖applyKernel P (v i)‖^2) ≤ ∑ x, ∑ y, (P x y)^2 := by
+  simp_rw [EuclideanSpace.norm_sq_eq]
+  rw [Finset.sum_comm]
+  apply Finset.sum_le_sum
+  intro x _
+  let w : EuclideanSpace ℂ X := WithLp.toLp 2 (fun y => (P x y : ℂ))
+  have he (i : I) : ‖applyKernel P (v i) x‖ = ‖inner ℂ (v i) w‖ := by
+    rw [norm_inner_symm]
+    congr 1
+    simp only [applyKernel_apply,PiLp.inner_apply]
+    apply Finset.sum_congr rfl
+    intro y _
+    simp [w,RCLike.inner_apply,mul_comm]
+  simp_rw [he]
+  have h := hv.sum_inner_products_le (s := Finset.univ) w
+  apply h.trans_eq
+  rw [EuclideanSpace.norm_sq_eq]
+  apply Finset.sum_congr rfl
+  intro y _
+  simp [w,Complex.norm_real,Real.norm_eq_abs,sq_abs]
+
+
+lemma tail_density_pointwise (t a z : ℝ) (ht : 0 ≤ t) (ha : 0 ≤ a) (hz : 0 < z) :
+    (if z < t then t else 0) ≤ t^(1+a)*z^(-a) := by
+  split_ifs with h
+  · rw [Real.rpow_neg hz.le,←div_eq_mul_inv]
+    apply (le_div_iff₀ (Real.rpow_pos_of_pos hz a)).mpr
+    rw [Real.rpow_add (lt_trans hz h),Real.rpow_one]
+    exact mul_le_mul_of_nonneg_left (Real.rpow_le_rpow hz.le h.le ha) ht
+  · positivity
+
+lemma tail_density_mean {X : Type*} [Fintype X] (f : X → ℝ) (hf : ∀ x, 0 ≤ f x)
+    (a z : ℝ) (ha : 0 ≤ a) (hz : 0 < z) :
+    finiteMean (fun x => if z < f x then f x else 0) ≤ finiteMean (fun x => f x^(1+a))*z^(-a) := by
+  rw [←finiteMean_mul_const]
+  exact finiteMean_mono (fun x => tail_density_pointwise (f x) a z (hf x) ha hz)
+
+noncomputable def tailKernel {X : Type*} [Fintype X] (P : X → X → ℝ) (z : ℝ) : X → X → ℝ :=
+  fun x y => if z < (Fintype.card X:ℝ)*P x y then P x y else 0
+
+noncomputable def headKernel {X : Type*} [Fintype X] (P : X → X → ℝ) (z : ℝ) : X → X → ℝ :=
+  fun x y => if z < (Fintype.card X:ℝ)*P x y then 0 else P x y
+
+lemma tail_add_head {X : Type*} [Fintype X] (P : X → X → ℝ) (z : ℝ) :
+    (fun x y => tailKernel P z x y+headKernel P z x y) = P := by
+  funext x y
+  unfold tailKernel headKernel
+  split_ifs <;> simp
+
+lemma tailKernel_nonneg {X : Type*} [Fintype X] (P : X → X → ℝ) (z : ℝ)
+    (hP : ∀ x y, 0 ≤ P x y) (x y : X) : 0 ≤ tailKernel P z x y := by
+  unfold tailKernel
+  split_ifs <;> simp only [hP,le_refl]
+
+lemma tailKernel_row {X : Type*} [Fintype X] [Nonempty X] (P : X → X → ℝ)
+    (hP : ∀ x y, 0 ≤ P x y) (a z M : ℝ) (ha : 0 ≤ a) (hz : 0 < z)
+    (hM : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*P x y)^(1+a)) ≤ M) (x : X) :
+    ∑ y, tailKernel P z x y ≤ M*z^(-a) := by
+  have hn : (Fintype.card X:ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
+  have he : finiteMean (fun y => if z < (Fintype.card X:ℝ)*P x y then
+      (Fintype.card X:ℝ)*P x y else 0) = ∑ y, tailKernel P z x y := by
+    unfold finiteMean
+    calc
+      _ = (∑ y, (Fintype.card X:ℝ)*tailKernel P z x y)/(Fintype.card X:ℝ) := by
+        congr 1
+        apply Finset.sum_congr rfl
+        intro y _
+        by_cases hy : z < (Fintype.card X:ℝ)*P x y
+        · simp only [tailKernel,hy,↓reduceIte]
+        · simp only [tailKernel,hy,↓reduceIte,mul_zero]
+      _ = _ := by rw [←Finset.mul_sum]; field_simp
+  rw [←he]
+  exact (tail_density_mean _ (fun y => mul_nonneg (Nat.cast_nonneg _) (hP x y)) a z ha hz).trans
+    (mul_le_mul_of_nonneg_right (hM x) (Real.rpow_nonneg hz.le _))
+
+lemma tailKernel_schur {X : Type*} [Fintype X] [Nonempty X] (P : X → X → ℝ)
+    (hP : ∀ x y, 0 ≤ P x y) (hsym : ∀ x y, P x y = P y x)
+    (a z M : ℝ) (ha : 0 ≤ a) (hz : 0 < z) (hM0 : 0 ≤ M)
+    (hM : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*P x y)^(1+a)) ≤ M)
+    (v : EuclideanSpace ℂ X) :
+    ‖applyKernel (tailKernel P z) v‖ ≤ (M*z^(-a))*‖v‖ := by
+  apply schur_bound _ (tailKernel_nonneg P z hP) _ (by positivity)
+    (tailKernel_row P hP a z M ha hz hM)
+  intro y
+  have hs : (fun x => tailKernel P z x y) = (fun x => tailKernel P z y x) := by
+    funext x
+    simp only [tailKernel,hsym x y]
+  rw [hs]
+  exact tailKernel_row P hP a z M ha hz hM y
+
+lemma headKernel_hs {X : Type*} [Fintype X] [Nonempty X] (P : X → X → ℝ)
+    (hP : ∀ x y, 0 ≤ P x y) (hrow : ∀ x, ∑ y, P x y = 1) (z : ℝ) (hz : 0 ≤ z) :
+    (∑ x, ∑ y, (headKernel P z x y)^2) ≤ z := by
+  have hn : (0:ℝ) < Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
+  have he (x y : X) : (headKernel P z x y)^2 ≤ (z/(Fintype.card X:ℝ))*P x y := by
+    unfold headKernel
+    split_ifs with h
+    · simp only [zero_pow (by omega : (2:ℕ) ≠ 0)]
+      exact mul_nonneg (div_nonneg hz hn.le) (hP x y)
+    · have h' : (Fintype.card X:ℝ)*P x y ≤ z := le_of_not_gt h
+      have hh := mul_le_mul_of_nonneg_right h' (hP x y)
+      rw [show (z/(Fintype.card X:ℝ))*P x y = (z*P x y)/(Fintype.card X:ℝ) by ring]
+      apply (le_div_iff₀ hn).mpr
+      nlinarith
+  calc
+    _ ≤ ∑ x, ∑ y, (z/(Fintype.card X:ℝ))*P x y :=
+      Finset.sum_le_sum (fun x _ => Finset.sum_le_sum (fun y _ => he x y))
+    _ = z := by
+      simp only [←Finset.mul_sum,hrow,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
+      field_simp
+
+
+theorem multiplicity_amplification {X I : Type*} [Fintype X] [Nonempty X]
+    [Fintype I] [Nonempty I] (P : X → X → ℝ)
+    (hP : ∀ x y, 0 ≤ P x y) (hsym : ∀ x y, P x y = P y x)
+    (hrow : ∀ x, ∑ y, P x y = 1) (a z M : ℝ) (ha : 0 ≤ a) (hz : 0 < z) (hM0 : 0 ≤ M)
+    (hM : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*P x y)^(1+a)) ≤ M)
+    (v : I → EuclideanSpace ℂ X) (hv : Orthonormal ℂ v)
+    (μ : ℝ) (hvμ : ∀ i, μ ≤ ‖applyKernel P (v i)‖) :
+    μ ≤ M*z^(-a)+Real.sqrt (z/(Fintype.card I:ℝ)) := by
+  let δ : ℝ := M*z^(-a)
+  have hδ : 0 ≤ δ := by dsimp [δ]; positivity
+  have hm : (0:ℝ) < Fintype.card I := Nat.cast_pos.mpr Fintype.card_pos
+  change μ ≤ δ+_
+  by_cases hμ : μ ≤ δ
+  · exact hμ.trans (le_add_of_nonneg_right (Real.sqrt_nonneg _))
+  have hmd : 0 ≤ μ-δ := by linarith
+  have hsmall (i : I) : μ-δ ≤ ‖applyKernel (headKernel P z) (v i)‖ := by
+    have hr := tailKernel_schur P hP hsym a z M ha hz hM0 hM (v i)
+    rw [hv.norm_eq_one i,mul_one] at hr
+    have he := applyKernel_add (tailKernel P z) (headKernel P z) (v i)
+    rw [tail_add_head] at he
+    have hn := norm_add_le (applyKernel (tailKernel P z) (v i)) (applyKernel (headKernel P z) (v i))
+    rw [←he] at hn
+    have hh := hvμ i
+    change ‖applyKernel (tailKernel P z) (v i)‖ ≤ δ at hr
+    linarith
+  have hsum : (Fintype.card I:ℝ)*(μ-δ)^2 ≤ z := by
+    calc
+      _ = ∑ i : I, (μ-δ)^2 := by simp
+      _ ≤ ∑ i : I, ‖applyKernel (headKernel P z) (v i)‖^2 :=
+        Finset.sum_le_sum (fun i _ => pow_le_pow_left₀ hmd (hsmall i) 2)
+      _ ≤ ∑ x, ∑ y, (headKernel P z x y)^2 := hilbert_schmidt_family _ v hv
+      _ ≤ z := headKernel_hs P hP hrow z hz.le
+  have hsq : (μ-δ)^2 ≤ z/(Fintype.card I:ℝ) := (le_div_iff₀ hm).mpr (by nlinarith)
+  have hs := (Real.le_sqrt hmd (div_nonneg hz.le hm.le)).mpr hsq
+  linarith
+
+end DensityTransfer
+namespace PairRouting
+open scoped BigOperators
+
+lemma tupleProbability_row_sum {ι β Ω : Type*} [Fintype ι] [Fintype β]
+    [DecidableEq β] [Fintype Ω] [Nonempty Ω]
+    (P : Ω → Equiv.Perm β) (e : ι ↪ β) :
+    (∑ f : ι ↪ β, tupleProbability P e f) = 1 := by
+  classical
+  simp only [tupleProbability]
+  rw [←finiteMean_sum]
+  have h (ω : Ω) (f : ι ↪ β) : (∀ i, P ω (e i) = f i) ↔ e.trans (P ω).toEmbedding = f := by
+    constructor
+    · intro h; ext i; exact h i
+    · intro h i; exact congrArg (fun g : ι ↪ β => g i) h
+  simp_rw [h]
+  simp only [Finset.sum_ite_eq,Finset.mem_univ,↓reduceIte,finiteMean_const]
+
+lemma tupleProbability_inverse {ι β Ω : Type*} [Fintype ι] [DecidableEq β]
+    [Fintype Ω] (P : Ω → Equiv.Perm β) (e f : ι → β) :
+    tupleProbability (fun ω => (P ω)⁻¹) e f = tupleProbability P f e := by
+  classical
+  apply finiteMean_congr
+  intro ω
+  congr 1
+  apply propext
+  constructor
+  · intro h i
+    have hh := congrArg (P ω) (h i)
+    simpa only [Equiv.Perm.inv_def,Equiv.apply_symm_apply] using hh.symm
+  · intro h i
+    have hh := congrArg (fun x => (P ω).symm x) (h i)
+    simpa only [Equiv.Perm.inv_def,Equiv.symm_apply_apply] using hh.symm
+
+lemma tupleProbability_equiv {ι β Ω Ω' : Type*} [Fintype ι] [DecidableEq β]
+    [Fintype Ω] [Fintype Ω'] (P : Ω → Equiv.Perm β) (τ : Ω ≃ Ω') (e f : ι → β) :
+    tupleProbability P e f = tupleProbability (fun ω => P (τ.symm ω)) e f :=
+  finiteMean_equiv τ _
+
+lemma tupleProbability_symmetric {ι β Ω : Type*} [Fintype ι] [DecidableEq β]
+    [Fintype Ω] (P : Ω → Equiv.Perm β) (τ : Equiv.Perm Ω)
+    (hτ : ∀ ω, P (τ ω) = (P ω)⁻¹) (e f : ι → β) :
+    tupleProbability P e f = tupleProbability P f e := by
+  rw [tupleProbability_equiv P τ.symm]
+  simp only [Equiv.symm_symm,hτ]
+  exact tupleProbability_inverse P e f
+
+end PairRouting
+
+lemma palindromePerm_reverse (d : ℕ) (ω : BenesCoins d) :
+    palindromePerm d (ω.2,ω.1) = (palindromePerm d ω)⁻¹ := by
+  simp only [palindromePerm,mul_inv_rev]
+  rfl
+
+lemma palindrome_kernel_symmetric (d : ℕ) {ι : Type*} [Fintype ι]
+    (e f : ι ↪ Card d) :
+    PairRouting.tupleProbability (palindromePerm d) e f =
+      PairRouting.tupleProbability (palindromePerm d) f e :=
+  PairRouting.tupleProbability_symmetric _ (Equiv.prodComm _ _) (palindromePerm_reverse d) e f
+
+theorem palindrome_density_transfer (d : ℕ) {ι I : Type*} [Fintype ι] [Fintype I]
+    [Nonempty I] (e₀ : ι ↪ Card d) (hk : 0 < Fintype.card ι)
+    (v : I → EuclideanSpace ℂ (ι ↪ Card d)) (hv : Orthonormal ℂ v)
+    (μ : ℝ) (hvμ : ∀ i, μ ≤ ‖DensityTransfer.applyKernel
+      (fun e f : ι ↪ Card d => PairRouting.tupleProbability (palindromePerm d) e f) (v i)‖)
+    (z : ℝ) (hz : 0 < z) :
+    μ ≤ Real.exp (densityConstant*Fintype.card ι*
+      ((Fintype.card ι:ℝ)/(2:ℝ)^d)^(1/512:ℝ))*z^(-(1/1000:ℝ))+
+      Real.sqrt (z/(Fintype.card I:ℝ)) := by
+  let : Nonempty (ι ↪ Card d) := ⟨e₀⟩
+  apply DensityTransfer.multiplicity_amplification
+    (fun e f : ι ↪ Card d => PairRouting.tupleProbability (palindromePerm d) e f)
+    (fun e f => PairRouting.tupleProbability_nonneg _ e f)
+    (palindrome_kernel_symmetric d) (PairRouting.tupleProbability_row_sum _)
+    (1/1000) z _ (by norm_num) hz (Real.exp_nonneg _) _ v hv μ hvμ
+  intro e
+  exact palindrome_row_moment_bound d e hk
+
+
+noncomputable def modifiedReverseCoins (L : ℕ) : (r : ℕ) → ModifiedCoins L r → ModifiedCoins L r
+  | 0, ω => ((fun u => butterflyPerm L (decodeButterfly L ω.2.1) * (ω.1 u)⁻¹ *
+      butterflyPerm L (decodeButterfly L ω.2.1)),ω.2)
+  | r+1, ω =>
+      let z := modifiedStepEquiv L r ω
+      (modifiedStepEquiv L r).symm
+        ((modifiedReverseCoins L r z.1.1,modifiedReverseCoins L r z.1.2),(z.2.2,z.2.1))
+
+lemma modifiedReverseCoins_involutive (L r : ℕ) :
+    Function.Involutive (modifiedReverseCoins L r) := by
+  induction r with
+  | zero =>
+      intro ω
+      apply Prod.ext
+      · funext u
+        simp only [modifiedReverseCoins,mul_inv_rev,inv_inv]
+        simp [mul_assoc]
+      · rfl
+  | succ r ih =>
+      change ∀ ω, modifiedReverseCoins L r (modifiedReverseCoins L r ω) = ω at ih
+      intro ω
+      simp only [modifiedReverseCoins,Equiv.apply_symm_apply,ih]
+      exact (modifiedStepEquiv L r).symm_apply_apply ω
+
+end Thorp
+
+end ThorpNine.Tail
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Tail/ModifiedStageEquiv.lean b/lean/OAI/Probability/ThorpRouting/Tail/ModifiedStageEquiv.lean
index 13ad57c4526c43799816178fd6435d60f169afe4..59d7098af0723459ab6ed5ac4c2ff73d00e9e302 100644
--- a/lean/OAI/Probability/ThorpRouting/Tail/ModifiedStageEquiv.lean
+++ b/lean/OAI/Probability/ThorpRouting/Tail/ModifiedStageEquiv.lean
@@ -1,875 +1,875 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Tail.BlockProduct
-
-namespace OAI
-
-namespace ThorpNine.Tail
-
-namespace Thorp
-open scoped BigOperators Classical
-
-def modifiedStageEquiv (r L : ℕ) :
-    ((OuterCoins r L × (Card r → Equiv.Perm (Card L) × BenesCoins L)) × OuterCoins r L) ≃
-      ModifiedCoins L r :=
-  (modifiedAssembleEquiv _ _ _ _).trans
-    (Equiv.prodCongr (Equiv.refl _) (Equiv.prodCongr (parallelSplitCoinEquiv r L) (parallelSplitCoinEquiv r L)))
-
-lemma modifiedStageEquiv_apply (r L : ℕ)
-    (w : (OuterCoins r L × (Card r → Equiv.Perm (Card L) × BenesCoins L)) × OuterCoins r L) :
-    modifiedStageEquiv r L w=
-      ((fun b => (w.1.2 b).1),
-       (assembleBits r L (fun x => (w.1.2 x.1).2.1 x.2) w.1.1,
-        assembleBits r L (fun x => (w.1.2 x.1).2.2 x.2) w.2)) := by
-  simp only [modifiedStageEquiv,Equiv.trans_apply,Equiv.prodCongr_apply,modifiedAssembleEquiv,
-    Equiv.coe_fn_mk,Prod.map_apply,Equiv.refl_apply,parallelSplitCoinEquiv_apply]
-
-lemma palindrome_stage (r L : ℕ)
-    (w : (OuterCoins r L × (Card r → BenesCoins L)) × OuterCoins r L) :
-    splitPerm r L (palindromePerm (L+r) (palindromeStageEquiv r L w))=
-      outerPerm r L w.1.1*ParallelBlocks.lift (fun b => palindromePerm L (w.1.2 b))*
-        (outerPerm r L w.2)⁻¹ := by
-  rw [palindromeStageEquiv_apply]
-  change splitPerm r L (_*_⁻¹)=_
-  rw [map_mul,map_inv,splitPerm_butterfly,splitPerm_butterfly,mul_inv_rev]
-  rw [mul_assoc,←mul_assoc (ParallelBlocks.lift _),←map_inv,←map_mul,←mul_assoc]
-  rfl
-
-noncomputable def localModifiedPerm (L : ℕ) (e : Bool)
-    (w : Equiv.Perm (Card L) × BenesCoins L) : Equiv.Perm (Card L) :=
-  if e then 1 else butterflyPerm L (decodeButterfly L w.2.1)*w.1⁻¹
-
-lemma modified_stage (r L : ℕ) (E : Finset (Card r))
-    (w : (OuterCoins r L × (Card r → Equiv.Perm (Card L) × BenesCoins L)) × OuterCoins r L) :
-    splitPerm r L (modifiedPerm L r E (modifiedStageEquiv r L w))=
-      outerPerm r L w.1.1*ParallelBlocks.lift (fun b => localModifiedPerm L (decide (b∈E)) (w.1.2 b))*
-        (outerPerm r L w.2)⁻¹ := by
-  rw [modifiedStageEquiv_apply]
-  unfold modifiedPerm modifiedInputTree
-  rw [map_mul,map_inv,splitPerm_butterfly,splitPerm_boundaryTree,mul_inv_rev]
-  rw [mul_assoc,←mul_assoc (ParallelBlocks.lift _),←map_inv,←map_mul,←mul_assoc]
-  congr 2
-  apply congrArg ParallelBlocks.lift
-  funext b
-  simp only [Pi.mul_apply,Pi.inv_apply,modifiedBoundary,boundaryOutput_split,localModifiedPerm]
-  by_cases h : b∈E <;> simp [h]
-
-end Thorp
-namespace Thorp.UnitaryFinite
-open scoped BigOperators Classical
-variable {G : Type*} [Group G]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-variable {Ω Ξ : Type*} [Fintype Ω] [Fintype Ξ]
-
-lemma sampleOperator_sandwich (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
-    (P : Ω → G) (Q : Ξ → G) :
-    sampleOperator ρ (fun w : (Ω × Ξ) × Ω => P w.1.1*Q w.1.2*(P w.2)⁻¹)=
-      sampleOperator ρ P*sampleOperator ρ Q*(sampleOperator ρ P).adjoint := by
-  rw [sampleOperator_prod ρ (fun w : Ω × Ξ => P w.1*Q w.2) (fun w : Ω => (P w)⁻¹),
-    sampleOperator_prod ρ P Q,sampleOperator_inv ρ hρ]
-
-lemma localModified_average (L : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card L)) V) (e : Bool) :
-    sampleOperator ρ (localModifiedPerm L e)=if e then 1 else uniformOperator ρ := by
-  cases e
-  · simp only [Bool.false_eq_true,ite_false]
-    rw [←sampleOperator_equiv ρ _ (Equiv.prodComm _ _)]
-    exact sampleOperator_uniform_mul ρ (fun w : BenesCoins L => butterflyPerm L (decodeButterfly L w.1))
-  · simp only [ite_true]
-    exact sampleOperator_one ρ
-
-end Thorp.UnitaryFinite
-
-namespace Thorp
-open scoped BigOperators Classical
-open UnitaryFinite
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-lemma sampleOperator_palindrome_stage (r L : ℕ)
-    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) (hρ : IsUnitary ρ) :
-    sampleOperator (ρ.comp (splitPerm r L).toMonoidHom) (palindromePerm (L+r))=
-      sampleOperator ρ (outerPerm r L)*
-        sampleOperator ρ (fun w : Card r → BenesCoins L => ParallelBlocks.lift (fun b => palindromePerm L (w b)))*
-          (sampleOperator ρ (outerPerm r L)).adjoint := by
-  have he : ((splitPerm r L).toMonoidHom ∘ palindromePerm (L+r)) ∘ palindromeStageEquiv r L=
-      fun w => outerPerm r L w.1.1*ParallelBlocks.lift (fun b => palindromePerm L (w.1.2 b))*
-        (outerPerm r L w.2)⁻¹ := funext (palindrome_stage r L)
-  calc
-    _ = sampleOperator ρ (((splitPerm r L).toMonoidHom ∘ palindromePerm (L+r)) ∘ palindromeStageEquiv r L) :=
-      (sampleOperator_equiv ρ ((splitPerm r L).toMonoidHom ∘ palindromePerm (L+r)) (palindromeStageEquiv r L)).symm
-    _ = _ := (congrArg (sampleOperator ρ) he).trans
-      (sampleOperator_sandwich ρ hρ (outerPerm r L)
-        (fun w : Card r → BenesCoins L => ParallelBlocks.lift (fun b => palindromePerm L (w b))))
-
-lemma sampleOperator_modified_stage (r L : ℕ) (E : Finset (Card r))
-    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) (hρ : IsUnitary ρ) :
-    sampleOperator (ρ.comp (splitPerm r L).toMonoidHom) (modifiedPerm L r E)=
-      sampleOperator ρ (outerPerm r L)*
-        sampleOperator ρ (fun w : Card r → Equiv.Perm (Card L) × BenesCoins L =>
-          ParallelBlocks.lift (fun b => localModifiedPerm L (decide (b∈E)) (w b)))*
-          (sampleOperator ρ (outerPerm r L)).adjoint := by
-  have he : ((splitPerm r L).toMonoidHom ∘ modifiedPerm L r E) ∘ modifiedStageEquiv r L=
-      fun w => outerPerm r L w.1.1*ParallelBlocks.lift
-        (fun b => localModifiedPerm L (decide (b∈E)) (w.1.2 b))*(outerPerm r L w.2)⁻¹ :=
-    funext (modified_stage r L E)
-  calc
-    _ = sampleOperator ρ (((splitPerm r L).toMonoidHom ∘ modifiedPerm L r E) ∘ modifiedStageEquiv r L) :=
-      (sampleOperator_equiv ρ ((splitPerm r L).toMonoidHom ∘ modifiedPerm L r E) (modifiedStageEquiv r L)).symm
-    _ = _ := (congrArg (sampleOperator ρ) he).trans
-      (sampleOperator_sandwich ρ hρ (outerPerm r L)
-        (fun w : Card r → Equiv.Perm (Card L) × BenesCoins L =>
-          ParallelBlocks.lift (fun b => localModifiedPerm L (decide (b∈E)) (w b))))
-
-end Thorp
-namespace Thorp.CommutingBlocks
-open scoped BigOperators Classical
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
-variable {ι : Type*} [Fintype ι] [DecidableEq ι]
-
-noncomputable def retain (P : ι → V →L[ℂ] V) (E : Finset ι) (i : ι) : V →L[ℂ] V :=
-  if i∈E then 1 else P i
-
-omit [CompleteSpace V] [Fintype ι] in
-lemma retain_commute (P : ι → V →L[ℂ] V) (hP : ∀ i j, Commute (P i) (P j))
-    (E : Finset ι) (i j : ι) : Commute (retain P E i) (retain P E j) := by
-  unfold retain
-  split_ifs <;> first | exact Commute.one_left _ | exact Commute.one_right _ | exact hP _ _
-
-omit [CompleteSpace V] in
-lemma blocks_retain (P : ι → V →L[ℂ] V) (hP : ∀ i j, Commute (P i) (P j)) (E : Finset ι) :
-    blocks (retain P E) (retain_commute P hP E)=coarse P hP E := by
-  have hs : E∪Eᶜ=(Finset.univ : Finset ι) := Finset.union_compl E
-  have hh := Finset.noncommProd_union_of_disjoint (s := E) (t := Eᶜ) disjoint_compl_right
-    (retain P E) (fun _ _ _ _ _ => retain_commute P hP E _ _)
-  simp only [hs] at hh
-  change blocks (retain P E) (retain_commute P hP E)=_ at hh
-  rw [hh]
-  have he : E.noncommProd (retain P E) (fun _ _ _ _ _ => retain_commute P hP E _ _)=1 := by
-    exact (Finset.noncommProd_eq_pow_card E (retain P E)
-      (fun i _ j _ _ => retain_commute P hP E i j) 1 (by intro i hi; exact ite_eq_left hi)).trans (one_pow _)
-  rw [he,one_mul]
-  apply Finset.noncommProd_congr rfl
-  intro i hi
-  exact ite_eq_right (Finset.mem_compl.mp hi)
-
-end Thorp.CommutingBlocks
-
-namespace Thorp.ParallelBlocks
-open scoped BigOperators Classical
-open UnitaryFinite
-variable {B X : Type*} [Fintype B] [Fintype X] [DecidableEq X]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-noncomputable def centralProjection (ρ : Representation ℂ (Equiv.Perm (B × X)) V) (b : B) : V →L[ℂ] V :=
-  uniformOperator (ρ.comp (injection b))
-
-omit [Fintype B] in
-lemma centralProjection_commute (ρ : Representation ℂ (Equiv.Perm (B × X)) V) (b c : B) :
-    Commute (centralProjection ρ b) (centralProjection ρ c) := by
-  simpa only [centralProjection,sampleOperator_uniform] using
-    local_average_commute ρ (fun _ => id : B → Equiv.Perm X → Equiv.Perm X) b c
-
-omit [Fintype B] in
-lemma local_central_commute {Ω : Type*} [Fintype Ω] [Nonempty Ω]
-    (ρ : Representation ℂ (Equiv.Perm (B × X)) V) (P : B → Ω → Equiv.Perm X) (b c : B) :
-    Commute (sampleOperator (ρ.comp (injection b)) (P b)) (centralProjection ρ c) := by
-  by_cases h : b=c
-  · subst c
-    change _*uniformOperator _=uniformOperator _*_
-    rw [sampleOperator_uniform_absorb,sampleOperator_weighted,weighted_uniform_right,sampleLaw_sum,one_smul]
-  · unfold centralProjection
-    rw [←sampleOperator_uniform,sampleOperator_comp,sampleOperator_comp]
-    exact sampleOperator_commute ρ _ _ (fun _ _ => injection_commute h _ _)
-
-end Thorp.ParallelBlocks
-namespace Thorp
-open scoped BigOperators Classical
-open UnitaryFinite ParallelBlocks CommutingBlocks
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-lemma modified_central_average (r L : ℕ) (E : Finset (Card r))
-    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) :
-    sampleOperator ρ (fun w : Card r → Equiv.Perm (Card L) × BenesCoins L =>
-      lift (fun b => localModifiedPerm L (decide (b∈E)) (w b)))=
-      coarse (centralProjection ρ) (centralProjection_commute ρ) E := by
-  rw [sampleOperator_parallel ρ (fun b => localModifiedPerm L (decide (b∈E))),←blocks_retain]
-  apply Finset.noncommProd_congr rfl
-  · intro b _
-    rw [localModified_average]
-    simp only [retain,centralProjection,decide_eq_true_eq]
-  · intro b _ c _ _
-    exact local_average_commute ρ (fun b => localModifiedPerm L (decide (b∈E))) b c
-
-lemma central_local_gap (r L : ℕ)
-    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) (hρ : IsUnitary ρ) (b : Card r) :
-    ‖sampleOperator (ρ.comp (injection b)) (palindromePerm L)*(1-centralProjection ρ b)‖≤blockGap L := by
-  change ‖sampleOperator _ _*(1-uniformOperator _)‖≤_
-  rw [mul_sub,mul_one,sampleOperator_uniform_absorb,sampleOperator_weighted]
-  exact palindrome_off_invariants_gap L _ (unitary_comp ρ hρ _)
-
-lemma central_blocks_exception_bound (r L : ℕ)
-    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) (hρ : IsUnitary ρ)
-    (B : V →L[ℂ] V) (hB0 : ‖B‖≤1) (n : ℕ) :
-    ‖B*blocks (fun b : Card r => sampleOperator (ρ.comp (injection b)) (palindromePerm L))
-      (local_average_commute ρ (fun _ => palindromePerm L))*B.adjoint‖ ≤
-      (blockGap L)^n + ∑ E : Finset (Card r), if E.card<n then
-        ‖B*coarse (centralProjection ρ) (centralProjection_commute ρ) E*B.adjoint‖ else 0 := by
-  have hB : ‖B.adjoint‖≤1 := by
-    rw [ContinuousLinearMap.adjoint.norm_map]
-    exact hB0
-  have h := blocks_exception_bound (centralProjection ρ) (centralProjection_commute ρ)
-    (fun b => uniformOperator_projection (ρ.comp (injection b)) (unitary_comp ρ hρ _))
-    (fun b : Card r => sampleOperator (ρ.comp (injection b)) (palindromePerm L))
-    (local_average_commute ρ (fun _ => palindromePerm L))
-    (local_central_commute ρ (fun _ => palindromePerm L))
-    (fun b => sampleOperator_norm_le (ρ.comp (injection b)) (unitary_comp ρ hρ _) (palindromePerm L))
-    (blockGap L) (blockGap_pos L).le (blockGap_lt_one L).le (central_local_gap r L ρ hρ)
-    B.adjoint hB n
-  simpa only [ContinuousLinearMap.adjoint_adjoint] using h
-
-theorem palindrome_exception_bound (r L : ℕ)
-    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) (hρ : IsUnitary ρ) (n : ℕ) :
-    ‖sampleOperator (ρ.comp (splitPerm r L).toMonoidHom) (palindromePerm (L+r))‖ ≤
-      (blockGap L)^n + ∑ E : Finset (Card r), if E.card<n then
-        ‖sampleOperator (ρ.comp (splitPerm r L).toMonoidHom) (modifiedPerm L r E)‖ else 0 := by
-  rw [sampleOperator_palindrome_stage r L ρ hρ]
-  rw [sampleOperator_parallel ρ (fun _ => palindromePerm L)]
-  apply (central_blocks_exception_bound r L ρ hρ (sampleOperator ρ (outerPerm r L))
-    (sampleOperator_norm_le ρ hρ _) n).trans_eq
-  congr 1
-  apply Finset.sum_congr rfl
-  intro E _
-  split_ifs
-  · rw [sampleOperator_modified_stage r L E ρ hρ,modified_central_average]
-  · rfl
-
-end Thorp
-
-namespace Thorp.UnitaryFinite
-open scoped BigOperators ComplexConjugate Classical
-
-variable {G X Ω : Type*} [Group G] [Fintype G] [Fintype X] [Fintype Ω]
-
-noncomputable def actionKernel (a : G →* Equiv.Perm X) (P : Ω → G) (x y : X) : ℝ :=
-  finiteMean (fun ω => if a (P ω) x=y then 1 else 0)
-
-omit [Fintype G] in
-lemma actionKernel_apply (a : G →* Equiv.Perm X) (P : Ω → G)
-    (v : EuclideanSpace ℂ X) (x : X) :
-    DensityTransfer.applyKernel (actionKernel a P) v x =
-      (Fintype.card Ω:ℂ)⁻¹ * ∑ ω, v (a (P ω) x) := by
-  simp only [DensityTransfer.applyKernel_apply,actionKernel,finiteMean,
-    Complex.ofReal_sum,div_eq_mul_inv,Finset.sum_mul]
-  rw [Finset.sum_comm]
-  calc
-    _ = ∑ ω, (Fintype.card Ω:ℂ)⁻¹ * ∑ y : X, if a (P ω) x=y then v y else 0 := by
-      apply Finset.sum_congr rfl
-      intro ω _
-      rw [Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro y _
-      split_ifs <;> simp_all
-    _ = _ := by simp [Finset.mul_sum]
-
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-noncomputable def randomOperator (ρ : Representation ℂ G V) (P : Ω → G) : Module.End ℂ V :=
-  (Fintype.card Ω:ℂ)⁻¹ • ∑ ω, ρ (P ω)⁻¹
-
-omit [Fintype G] [FiniteDimensional ℂ V] in
-lemma coefficient_random (a : G →* Equiv.Perm X) (x₀ : X)
-    (ha : ∀ x, ∃ g, a g x₀=x) (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
-    (P : Ω → G) (w : V) (hw : IsFixed a x₀ ρ w) (v : V) :
-    DensityTransfer.applyKernel (actionKernel a P) (normalizedCoefficient a x₀ ha ρ w v) =
-      normalizedCoefficient a x₀ ha ρ w (randomOperator ρ P v) := by
-  ext x
-  rw [actionKernel_apply]
-  simp only [normalizedCoefficient,PiLp.smul_apply,smul_eq_mul,Finset.mul_sum,
-    coefficient,randomOperator,LinearMap.smul_apply,LinearMap.sum_apply,
-    inner_smul_right,inner_sum]
-  apply Finset.sum_congr rfl
-  intro ω _
-  have h := coefficient_action a x₀ ha ρ hρ w hw v (P ω) x
-  change inner ℂ (ρ (representative a x₀ ha (a (P ω) x)) w) v =
-    inner ℂ (ρ (representative a x₀ ha x) w) (ρ (P ω)⁻¹ v) at h
-  rw [h]
-  ring
-
-noncomputable def fixedSpace (a : G →* Equiv.Perm X) (x₀ : X) (ρ : Representation ℂ G V) :
-    Submodule ℂ V where
-  carrier := {w | IsFixed a x₀ ρ w}
-  zero_mem' := fun g hg => map_zero _
-  add_mem' := fun hw hz g hg => by rw [map_add,hw g hg,hz g hg]
-  smul_mem' := fun c w hw g hg => by rw [map_smul,hw g hg]
-
-theorem operator_amplification (a : G →* Equiv.Perm X) (x₀ : X)
-    (ha : ∀ x, ∃ g, a g x₀=x) (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
-    (hρ : IsUnitary ρ) (P : Ω → G)
-    (hsym : ∀ x y, actionKernel a P x y=actionKernel a P y x)
-    (hrow : ∀ x, ∑ y, actionKernel a P x y=1)
-    (b z M : ℝ) (hb : 0≤b) (hz : 0<z) (hM0 : 0≤M)
-    (hM : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*actionKernel a P x y)^(1+b)) ≤ M)
-    (hm : 0 < Module.finrank ℂ (fixedSpace a x₀ ρ)) (v : V) (hv : ‖v‖=1) :
-    ‖randomOperator ρ P v‖ ≤ M*z^(-b)+Real.sqrt (z/(Module.finrank ℂ (fixedSpace a x₀ ρ):ℝ)) := by
-  let K := fixedSpace a x₀ ρ
-  let m := Module.finrank ℂ K
-  let : Nonempty X := ⟨x₀⟩
-  let : Nonempty (Fin m) := ⟨⟨0,hm⟩⟩
-  let B := stdOrthonormalBasis ℂ K
-  let w : Fin m → V := fun i => (B i : V)
-  have hfix (i : Fin m) : IsFixed a x₀ ρ (w i) := (B i).2
-  have hw : Orthonormal ℂ w := B.orthonormal.comp_linearIsometry K.subtypeₗᵢ
-  have hc := normalizedCoefficient_orthonormal a x₀ ha ρ hρ w hw hfix v hv
-  have hn (i : Fin m) : ‖DensityTransfer.applyKernel (actionKernel a P)
-      (normalizedCoefficient a x₀ ha ρ (w i) v)‖ = ‖randomOperator ρ P v‖ := by
-    rw [coefficient_random a x₀ ha ρ hρ P (w i) (hfix i),
-      normalizedCoefficient_norm a x₀ ha ρ hρ (w i) (hfix i) (hw.norm_eq_one i)]
-  have h := DensityTransfer.multiplicity_amplification (actionKernel a P)
-    (fun x y => finiteMean_nonneg (fun _ => by split_ifs <;> norm_num)) hsym hrow
-    b z M hb hz hM0 hM _ hc (‖randomOperator ρ P v‖) (fun i => (hn i).ge)
-  simpa only [Fintype.card_fin] using h
-
-end Thorp.UnitaryFinite
-namespace Thorp.UnitaryFinite
-open scoped BigOperators Classical
-
-variable {α ι : Type*} [Fintype α] [Fintype ι]
-
-def tupleAction : Equiv.Perm α →* Equiv.Perm (ι ↪ α) where
-  toFun p :=
-    { toFun := fun e => e.trans p.toEmbedding
-      invFun := fun e => e.trans p.symm.toEmbedding
-      left_inv := fun e => by ext i; exact p.symm_apply_apply (e i)
-      right_inv := fun e => by ext i; exact p.apply_symm_apply (e i) }
-  map_one' := by ext e i; rfl
-  map_mul' p q := by ext e i; rfl
-
-omit [Fintype α] [Fintype ι] in
-@[simp] lemma tupleAction_apply (p : Equiv.Perm α) (e : ι ↪ α) (i : ι) :
-    tupleAction p e i=p (e i) := rfl
-
-omit [Fintype α] in
-lemma tupleAction_transitive (e f : ι ↪ α) : ∃ p, tupleAction p e=f := by
-  obtain ⟨p,hp⟩ := Equiv.Perm.exists_extending_pair e f e.injective f.injective
-  exact ⟨p,Function.Embedding.ext hp⟩
-
-omit [Fintype α] in
-lemma tupleAction_fixed_iff [DecidableEq α] (e : ι ↪ α) (p : Equiv.Perm α) :
-    tupleAction p e=e ↔ ∀ x ∈ Finset.univ.image e, p x=x := by
-  constructor
-  · intro h x hx
-    obtain ⟨i,_,rfl⟩ := Finset.mem_image.mp hx
-    exact congrArg (fun f : ι ↪ α => f i) h
-  · intro h
-    ext i
-    exact h (e i) (Finset.mem_image.mpr ⟨i,Finset.mem_univ _,rfl⟩)
-
-omit [Fintype α] in
-lemma tupleAction_kernel [DecidableEq α] {Ω : Type*} [Fintype Ω] (P : Ω → Equiv.Perm α)
-    (e f : ι ↪ α) : actionKernel tupleAction P e f=PairRouting.tupleProbability P e f := by
-  unfold actionKernel PairRouting.tupleProbability
-  congr 1
-  funext ω
-  congr 1
-  exact propext Function.Embedding.ext_iff
-
-end Thorp.UnitaryFinite
-namespace Thorp.UnitaryFinite
-open scoped BigOperators Classical
-
-lemma irreducible_of_equiv {G V W : Type*} [Group G]
-    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
-    (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
-    (e : Representation.Equiv ρ σ) [Representation.IsIrreducible ρ] :
-    Representation.IsIrreducible σ := by
-  rw [Representation.irreducible_iff_isSimpleModule_asModule]
-  have hi := (Representation.irreducible_iff_isSimpleModule_asModule ρ).mp inferInstance
-  let f : ρ.asModule →ₗ[MonoidAlgebra ℂ G] σ.asModule :=
-    Representation.IntertwiningMap.equivLinearMapAsModule ρ σ e.toIntertwiningMap
-  have hf : Function.Bijective f := e.toLinearEquiv.bijective
-  exact (f.isSimpleModule_iff_of_bijective hf).mp hi
-
-end Thorp.UnitaryFinite
-
-namespace Thorp.Specht
-open scoped BigOperators Classical
-
-lemma unitaryRepresentation_coe (μ : YoungDiagram) (p : Equiv.Perm (Cell μ))
-    (v : hilbertSpace μ) :
-    (unitaryRepresentation μ p v : PermutationHilbert.H (Tabloid μ)) =
-      PermutationHilbert.act (tabloidAct p) v := by
-  obtain ⟨w,rfl⟩ := (spaceHilbertEquiv μ).surjective v
-  change hilbertEquiv μ (tabloidRep μ p w) = _
-  exact hilbert_action μ p w
-
-noncomputable def unitaryEquiv (μ : YoungDiagram) :
-    Representation.Equiv (representation μ) (unitaryRepresentation μ) :=
-  Representation.Equiv.mk (spaceHilbertEquiv μ) (by
-    intro p
-    apply LinearMap.ext
-    intro v
-    change spaceHilbertEquiv μ (representation μ p v) =
-      spaceHilbertEquiv μ (representation μ p ((spaceHilbertEquiv μ).symm (spaceHilbertEquiv μ v)))
-    rw [LinearEquiv.symm_apply_apply])
-
-lemma unitaryRepresentation_irreducible (μ : YoungDiagram) :
-    Representation.IsIrreducible (unitaryRepresentation μ) := by
-  let := representation_irreducible μ
-  exact UnitaryFinite.irreducible_of_equiv _ _ (unitaryEquiv μ)
-
-lemma unitaryRepresentation_unitary (μ : YoungDiagram) :
-    UnitaryFinite.IsUnitary (G := Equiv.Perm (Cell μ)) (V := hilbertSpace μ) (unitaryRepresentation μ) := by
-  intro p v w
-  change inner ℂ (unitaryRepresentation μ p v : PermutationHilbert.H (Tabloid μ))
-      (unitaryRepresentation μ p w : PermutationHilbert.H (Tabloid μ)) = _
-  rw [unitaryRepresentation_coe,unitaryRepresentation_coe]
-  exact (PermutationHilbert.act (tabloidAct p)).inner_map_map v w
-
-
-noncomputable def tupleFixedEquiv (μ : YoungDiagram) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Cell μ) :
-    UnitaryFinite.fixedSpace (V := hilbertSpace μ) UnitaryFinite.tupleAction e (unitaryRepresentation μ) ≃ₗ[ℂ]
-      fixedHilbert μ (Finset.univ.image e) where
-  toFun v := ⟨v.1.1,v.1.2,by
-    intro p hp
-    have h := v.2 p ((UnitaryFinite.tupleAction_fixed_iff e p).mpr hp)
-    exact (unitaryRepresentation_coe μ p v.1).symm.trans (congrArg Subtype.val h)⟩
-  invFun v := ⟨⟨v.1,v.2.1⟩,by
-    intro p hp
-    apply Subtype.ext
-    exact (unitaryRepresentation_coe μ p _).trans
-      (v.2.2 p ((UnitaryFinite.tupleAction_fixed_iff e p).mp hp))⟩
-  left_inv _ := rfl
-  right_inv _ := rfl
-  map_add' _ _ := rfl
-  map_smul' _ _ := rfl
-
-theorem tupleFixed_dimension (μ : YoungDiagram) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Cell μ) :
-    Module.finrank ℂ (hilbertSpace μ) * (Fintype.card ι).choose (μ.card-μ.rowLen 0) ≤
-      Module.finrank ℂ (UnitaryFinite.fixedSpace (V := hilbertSpace μ) UnitaryFinite.tupleAction e (unitaryRepresentation μ)) *
-        μ.card.choose (μ.card-μ.rowLen 0) := by
-  rw [(tupleFixedEquiv μ e).finrank_eq,finrank_hilbertSpace]
-  have h := fixedHilbert_dimension μ (Finset.univ.image e)
-  simpa only [Finset.card_image_of_injective _ e.injective,Finset.card_univ] using h
-
-end Thorp.Specht
-namespace Thorp.UnitaryFinite
-open scoped BigOperators Classical
-
-variable {G H X Ω : Type*} [Group G] [Group H] [Fintype G] [Fintype X] [Fintype Ω]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-omit [Fintype G] in
-lemma sampleOperator_inv_apply (ρ : Representation ℂ G V) (P : Ω → G) (v : V) :
-    sampleOperator ρ (fun ω => (P ω)⁻¹) v=randomOperator ρ P v := by
-  simp only [sampleOperator,randomOperator,smul_apply,
-    sum_apply,continuousRepresentation_apply,LinearMap.smul_apply,LinearMap.sum_apply]
-
-theorem sampleOperator_amplification (a : G →* Equiv.Perm X) (x₀ : X)
-    (ha : ∀ x, ∃ g, a g x₀=x) (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
-    (hρ : IsUnitary ρ) (P : Ω → G)
-    (hsym : ∀ x y, actionKernel a P x y=actionKernel a P y x)
-    (hrow : ∀ x, ∑ y, actionKernel a P x y=1)
-    (b z M : ℝ) (hb : 0≤b) (hz : 0<z) (hM0 : 0≤M)
-    (hM : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*actionKernel a P x y)^(1+b)) ≤ M)
-    (hm : 0 < Module.finrank ℂ (fixedSpace a x₀ ρ)) :
-    ‖sampleOperator ρ P‖ ≤ M*z^(-b)+Real.sqrt (z/(Module.finrank ℂ (fixedSpace a x₀ ρ):ℝ)) := by
-  rw [←ContinuousLinearMap.adjoint.norm_map (sampleOperator ρ P),←sampleOperator_inv ρ hρ]
-  apply ContinuousLinearMap.opNorm_le_of_unit_norm (by positivity)
-  intro v hv
-  rw [sampleOperator_inv_apply]
-  exact operator_amplification a x₀ ha ρ hρ P hsym hrow b z M hb hz hM0 hM hm v hv
-
-omit [Fintype G] [Fintype X] [Fintype Ω] [FiniteDimensional ℂ V] in
-noncomputable def subrepresentationEquiv (ρ : Representation ℂ G V) (e : H ≃* G) :
-    Subrepresentation (ρ.comp e.toMonoidHom) ≃o Subrepresentation ρ where
-  toFun U :=
-    { toSubmodule := U.toSubmodule
-      apply_mem_toSubmodule := fun g v hv => by
-        have h := U.apply_mem_toSubmodule (e.symm g) hv
-        change ρ (e (e.symm g)) v∈U.toSubmodule at h
-        simpa only [e.apply_symm_apply] using h }
-  invFun U :=
-    { toSubmodule := U.toSubmodule
-      apply_mem_toSubmodule := fun h _ hv => U.apply_mem_toSubmodule (e h) hv }
-  left_inv U := by apply Subrepresentation.toSubmodule_injective; rfl
-  right_inv U := by apply Subrepresentation.toSubmodule_injective; rfl
-  map_rel_iff' := Iff.rfl
-
-omit [Fintype G] [Fintype X] [Fintype Ω] [FiniteDimensional ℂ V] in
-lemma irreducible_comp_equiv (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
-    (e : H ≃* G) : Representation.IsIrreducible (ρ.comp e.toMonoidHom) := by
-  exact (subrepresentationEquiv ρ e).isSimpleOrder_iff.mpr inferInstance
-
-end Thorp.UnitaryFinite
-
-namespace Thorp.Specht
-open scoped BigOperators Classical
-
-variable {α ι : Type*} [Fintype α] [Fintype ι]
-
-omit [Fintype α] in
-lemma relabelledUnitary_unitary (μ : YoungDiagram) (e : α ≃ Cell μ) :
-    UnitaryFinite.IsUnitary (V := hilbertSpace μ) (relabelledUnitary μ e) := by
-  intro g v w
-  exact unitaryRepresentation_unitary μ (e.permCongr g) v w
-
-omit [Fintype α] in
-lemma relabelledUnitary_irreducible (μ : YoungDiagram) (e : α ≃ Cell μ) :
-    Representation.IsIrreducible (relabelledUnitary μ e) := by
-  let := unitaryRepresentation_irreducible μ
-  exact UnitaryFinite.irreducible_comp_equiv (V := hilbertSpace μ) (unitaryRepresentation μ) e.permCongrHom
-
-noncomputable def relabelledTupleFixed (μ : YoungDiagram) (e : α ≃ Cell μ) (f : ι ↪ α) :
-    UnitaryFinite.fixedSpace (V := hilbertSpace μ) UnitaryFinite.tupleAction f (relabelledUnitary μ e) ≃ₗ[ℂ]
-    UnitaryFinite.fixedSpace (V := hilbertSpace μ) UnitaryFinite.tupleAction (f.trans e.toEmbedding) (unitaryRepresentation μ) where
-  toFun v := ⟨v.1,by
-    intro g hg
-    have hp : UnitaryFinite.tupleAction (e.symm.permCongr g) f=f := by
-      apply Function.Embedding.ext
-      intro i
-      have hh := congrArg (fun f : ι ↪ Cell μ => f i) hg
-      change g (e (f i))=e (f i) at hh
-      simp only [UnitaryFinite.tupleAction_apply,Equiv.permCongr_apply,Equiv.symm_symm,e.symm_apply_apply,hh]
-    have hh := v.2 (e.symm.permCongr g) hp
-    change unitaryRepresentation μ (e.permCongr (e.symm.permCongr g)) v=v at hh
-    have he : e.permCongr (e.symm.permCongr g)=g := by
-      apply Equiv.ext
-      intro x
-      simp only [Equiv.permCongr_apply,Equiv.symm_symm,e.apply_symm_apply]
-    simpa only [he] using hh⟩
-  invFun v := ⟨v.1,by
-    intro g hg
-    have hp : UnitaryFinite.tupleAction (e.permCongr g) (f.trans e.toEmbedding)=f.trans e.toEmbedding := by
-      apply Function.Embedding.ext
-      intro i
-      have hh := congrArg (fun f : ι ↪ α => f i) hg
-      change g (f i)=f i at hh
-      simp only [UnitaryFinite.tupleAction_apply,Function.Embedding.trans_apply,
-        Equiv.toEmbedding_apply,Equiv.permCongr_apply,e.symm_apply_apply,hh]
-    exact v.2 (e.permCongr g) hp⟩
-  left_inv _ := rfl
-  right_inv _ := rfl
-  map_add' _ _ := rfl
-  map_smul' _ _ := rfl
-
-omit [Fintype α] in
-lemma relabelled_tuple_dimension (μ : YoungDiagram) (e : α ≃ Cell μ) (f : ι ↪ α) :
-    Module.finrank ℂ (hilbertSpace μ) * (Fintype.card ι).choose (μ.card-μ.rowLen 0) ≤
-      Module.finrank ℂ (UnitaryFinite.fixedSpace (V := hilbertSpace μ) UnitaryFinite.tupleAction f (relabelledUnitary μ e)) *
-        μ.card.choose (μ.card-μ.rowLen 0) := by
-  rw [(relabelledTupleFixed μ e f).finrank_eq]
-  exact tupleFixed_dimension μ (f.trans e.toEmbedding)
-
-end Thorp.Specht
-
-namespace Thorp.SparseContact
-open scoped BigOperators Classical
-
-lemma small_dominating_set {V : Type*} [Fintype V] (G : SimpleGraph V)
-    (hG : ∀ v, ∃ w, G.Adj v w) :
-    ∃ R : Finset V, 2 * R.card ≤ Fintype.card V ∧
-      ∀ v, v ∉ R → ∃ w ∈ R, G.Adj v w := by
-  classical
-  obtain ⟨F, hFG, hF, hreach⟩ := G.exists_isAcyclic_reachable_eq_le
-  let c := hF.coloringTwo
-  have hneigh (v : V) : ∃ w, F.Adj v w := by
-    obtain ⟨w, hvw⟩ := hG v
-    have hr : F.Reachable v w := by rw [hreach]; exact hvw.reachable
-    exact hr.nonempty_neighborSet_left hvw.ne
-  let A : Finset V := Finset.univ.filter (fun v => c v = 0)
-  let B : Finset V := Finset.univ.filter (fun v => c v ≠ 0)
-  have hab : A.card + B.card = Fintype.card V := by
-    exact Finset.card_filter_add_card_filter_not _
-  have hcolor {v w : V} (hvw : F.Adj v w) : c v ≠ c w := c.valid hvw
-  by_cases h : A.card ≤ B.card
-  · refine ⟨A, by omega, ?_⟩
-    intro v hv
-    obtain ⟨w, hvw⟩ := hneigh v
-    have hv0 : c v ≠ 0 := by simpa [A] using hv
-    have hw0 : c w = 0 := by
-      have hv1 : c v = 1 := by omega
-      have hn := hcolor hvw
-      omega
-    exact ⟨w, by simp [A, hw0], hFG hvw⟩
-  · refine ⟨B, by omega, ?_⟩
-    intro v hv
-    obtain ⟨w, hvw⟩ := hneigh v
-    have hv0 : c v = 0 := by simpa [B] using hv
-    have hw0 : c w ≠ 0 := by simpa [hv0] using (hcolor hvw).symm
-    exact ⟨w, by simp [B, hw0], hFG hvw⟩
-
-lemma powerset_weight_le_exp {V : Type*} [DecidableEq V] (S : Finset V)
-    (p : V → ℝ) (hp : ∀ v, 0 ≤ p v) :
-    (∑ L ∈ S.powerset, ∏ v ∈ L, p v) ≤ Real.exp (∑ v ∈ S, p v) := by
-  rw [← Finset.prod_one_add, Real.exp_sum]
-  apply Finset.prod_le_prod₀
-  · intro v _; exact add_nonneg (by norm_num) (hp v)
-  · intro v _
-    have h := Real.add_one_le_exp (p v)
-    linarith
-
-lemma fixed_size_weight_le {V : Type*} [DecidableEq V] (S : Finset V)
-    (p : V → ℝ) (hp : ∀ v, 0 ≤ p v) (m : ℕ) (δ : ℝ) (hδ : 0 < δ) :
-    (∑ L ∈ S.powersetCard m, ∏ v ∈ L, p v) ≤
-      δ^m * Real.exp ((∑ v ∈ S, p v)/δ) := by
-  have hsplit (L : Finset V) (hL : L ∈ S.powersetCard m) :
-      (∏ v ∈ L, p v) = δ^m * ∏ v ∈ L, (p v / δ) := by
-    rw [Finset.prod_div_distrib, Finset.prod_const, (Finset.mem_powersetCard.mp hL).2]
-    field_simp
-  calc
-    _ = δ^m * ∑ L ∈ S.powersetCard m, ∏ v ∈ L, (p v/δ) := by
-      rw [Finset.mul_sum]
-      exact Finset.sum_congr rfl hsplit
-    _ ≤ δ^m * ∑ L ∈ S.powerset, ∏ v ∈ L, (p v/δ) := by
-      apply mul_le_mul_of_nonneg_left _ (pow_nonneg hδ.le _)
-      apply Finset.sum_le_sum_of_subset_of_nonneg
-      · intro L hL
-        exact Finset.mem_powerset.mpr (Finset.mem_powersetCard.mp hL).1
-      · intro L _ _; exact Finset.prod_nonneg (fun v _ => div_nonneg (hp v) hδ.le)
-    _ ≤ δ^m * Real.exp (∑ v ∈ S, p v/δ) :=
-      mul_le_mul_of_nonneg_left
-        (powerset_weight_le_exp S (fun v => p v/δ) (fun v => div_nonneg (hp v) hδ.le))
-        (pow_nonneg hδ.le _)
-    _ = _ := by rw [Finset.sum_div]
-
-noncomputable def neighborhood {V : Type*} [Fintype V] (G : SimpleGraph V)
-    (R : Finset V) : Finset V := by
-  classical
-  exact Finset.univ.filter (fun w => ∃ v ∈ R, G.Adj v w)
-
-lemma neighborhood_weight_le {V : Type*} [Fintype V] (G : SimpleGraph V)
-    (p : V → ℝ) (hp : ∀ v, 0 ≤ p v) (δ : ℝ)
-    (hdeg : ∀ v, (∑ w, if G.Adj v w then p w else 0) ≤ δ) (R : Finset V) :
-    (∑ w ∈ neighborhood G R, p w) ≤ R.card * δ := by
-  classical
-  calc
-    _ ≤ ∑ w ∈ neighborhood G R, ∑ v ∈ R, if G.Adj v w then p w else 0 := by
-      apply Finset.sum_le_sum
-      intro w hw
-      obtain ⟨v, hv, hvw⟩ := (Finset.mem_filter.mp hw).2
-      have h : (if G.Adj v w then p w else 0) ≤
-          ∑ u ∈ R, if G.Adj u w then p w else 0 :=
-        Finset.single_le_sum (f := fun u => if G.Adj u w then p w else 0) (fun u _ => by split_ifs; exact hp w; exact le_rfl) hv
-      simpa only [ite_eq_left hvw] using h
-    _ ≤ ∑ w, ∑ v ∈ R, if G.Adj v w then p w else 0 := by
-      apply Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _)
-      intro w _ _
-      exact Finset.sum_nonneg (fun v _ => by split_ifs; exact hp w; exact le_rfl)
-    _ = ∑ v ∈ R, ∑ w, if G.Adj v w then p w else 0 := Finset.sum_comm
-    _ ≤ ∑ v ∈ R, δ := Finset.sum_le_sum (fun v _ => hdeg v)
-    _ = _ := by simp
-
-lemma root_leaves_weight_le {V : Type*} [Fintype V] (G : SimpleGraph V)
-    (p : V → ℝ) (hp : ∀ v, 0 ≤ p v) (k : ℕ) (δ : ℝ)
-    (hδ : 0 < δ) (hδ1 : δ ≤ 1)
-    (hdeg : ∀ v, (∑ w, if G.Adj v w then p w else 0) ≤ δ)
-    (R : Finset V) (hR : 2 * R.card ≤ k) :
-    (∑ L ∈ (neighborhood G R).powersetCard (k-R.card), ∏ v ∈ L, p v) ≤
-      δ^((k:ℝ)/2) * Real.exp R.card := by
-  classical
-  have hmass := neighborhood_weight_le G p hp δ hdeg R
-  have hpow : δ^(k-R.card) ≤ δ^((k:ℝ)/2) := by
-    rw [← Real.rpow_natCast]
-    apply Real.rpow_le_rpow_of_exponent_ge hδ hδ1
-    have hr : R.card ≤ k := by omega
-    rw [Nat.cast_sub hr]
-    have hh : (2:ℝ)*R.card ≤ k := by exact_mod_cast hR
-    linarith
-  apply (fixed_size_weight_le (neighborhood G R) p hp _ δ hδ).trans
-  apply mul_le_mul hpow
-  · apply Real.exp_le_exp.mpr
-    exact (div_le_iff₀ hδ).mpr hmass
-  · positivity
-  · positivity
-
-def NoIsolated {V : Type*} (G : SimpleGraph V) (S : Finset V) : Prop :=
-  ∀ v ∈ S, ∃ w ∈ S, G.Adj v w
-
-lemma small_dominating_subset {V : Type*} [Fintype V] (G : SimpleGraph V)
-    (S : Finset V) (hS : NoIsolated G S) :
-    ∃ R : Finset V, R ⊆ S ∧ 2 * R.card ≤ S.card ∧ S \ R ⊆ neighborhood G R := by
-  classical
-  let H := G.induce (S : Set V)
-  have hH : ∀ v : S, ∃ w : S, H.Adj v w := by
-    intro v
-    obtain ⟨w, hw, hvw⟩ := hS v v.2
-    exact ⟨⟨w, hw⟩, hvw⟩
-  obtain ⟨R, hR, hdom⟩ := small_dominating_set H hH
-  let T := R.image (fun v : S => (v : V))
-  have hc : T.card = R.card := Finset.card_image_of_injective _ Subtype.val_injective
-  refine ⟨T, ?_, ?_, ?_⟩
-  · intro v hv
-    obtain ⟨w, _, rfl⟩ := Finset.mem_image.mp hv
-    exact w.2
-  · simpa [hc, H] using hR
-  · intro v hv
-    have hvS := (Finset.mem_sdiff.mp hv).1
-    have hvR : (⟨v, hvS⟩ : S) ∉ R := by
-      intro hh
-      exact (Finset.mem_sdiff.mp hv).2 (Finset.mem_image.mpr ⟨⟨v,hvS⟩, hh, rfl⟩)
-    obtain ⟨w, hw, hvw⟩ := hdom ⟨v,hvS⟩ hvR
-    exact Finset.mem_filter.mpr ⟨Finset.mem_univ _,
-      ⟨w, Finset.mem_image.mpr ⟨w,hw,rfl⟩, hvw.symm⟩⟩
-
-lemma noIsolated_weight_le {V : Type*} [Fintype V] (G : SimpleGraph V)
-    (p : V → ℝ) (hp : ∀ v, 0 ≤ p v) (k : ℕ) (δ : ℝ)
-    (hδ : 0 < δ) (hδ1 : δ ≤ 1)
-    (hdeg : ∀ v, (∑ w, if G.Adj v w then p w else 0) ≤ δ) :
-    (∑ S : {S : Finset V // S.card = k ∧ NoIsolated G S}, ∏ v ∈ S.val, p v) ≤
-      δ^((k:ℝ)/2) * Real.exp (Real.exp 1 * ∑ v, p v) := by
-  classical
-  let I := {S : Finset V // S.card = k ∧ NoIsolated G S}
-  let root (S : I) := (small_dominating_subset G S.val S.property.2).choose
-  have hroot (S : I) : root S ⊆ S.val ∧ 2*(root S).card ≤ k ∧
-      S.val \ root S ⊆ neighborhood G (root S) := by
-    simpa only [root, S.property.1] using
-      (small_dominating_subset G S.val S.property.2).choose_spec
-  let enc (S : I) : Finset V × Finset V := (root S, S.val \ root S)
-  have hrec (S : I) : (enc S).1 ∪ (enc S).2 = S.val :=
-    Finset.union_sdiff_of_subset (hroot S).1
-  have hinj : Function.Injective enc := by
-    intro S T h
-    apply Subtype.ext
-    rw [← hrec S, ← hrec T, h]
-  let W (z : Finset V × Finset V) : ℝ :=
-    if 2*z.1.card ≤ k ∧ z.2 ∈ (neighborhood G z.1).powersetCard (k-z.1.card)
-    then (∏ v ∈ z.1, p v) * ∏ v ∈ z.2, p v else 0
-  have hW (z : Finset V × Finset V) : 0 ≤ W z := by
-    dsimp [W]
-    split_ifs
-    · exact mul_nonneg (Finset.prod_nonneg (fun v _ => hp v))
-        (Finset.prod_nonneg (fun v _ => hp v))
-    · exact le_rfl
-  have hwenc (S : I) : W (enc S) = ∏ v ∈ S.val, p v := by
-    have hmem : (enc S).2 ∈ (neighborhood G (enc S).1).powersetCard (k-(enc S).1.card) := by
-      refine Finset.mem_powersetCard.mpr ⟨(hroot S).2.2, ?_⟩
-      change (S.val \ root S).card = k-(root S).card
-      rw [Finset.card_sdiff_of_subset (hroot S).1, S.property.1]
-    dsimp only [W]
-    rw [ite_eq_left ⟨(hroot S).2.1,hmem⟩]
-    exact (mul_comm _ _).trans (Finset.prod_sdiff (hroot S).1)
-  have hsum : (∑ S : I, ∏ v ∈ S.val, p v) ≤ ∑ z, W z := by
-    calc
-      _ = ∑ z ∈ Finset.univ.image enc, W z := by
-        rw [Finset.sum_image (fun S _ T _ h => hinj h)]
-        exact Finset.sum_congr rfl (fun S _ => (hwenc S).symm)
-      _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _)
-        (fun z _ _ => hW z)
-  apply hsum.trans
-  rw [Fintype.sum_prod_type]
-  calc
-    (∑ R : Finset V, ∑ L : Finset V, W (R,L)) ≤
-        ∑ R : Finset V, δ^((k:ℝ)/2) * Real.exp R.card * ∏ v ∈ R, p v := by
-      apply Finset.sum_le_sum
-      intro R _
-      by_cases hR : 2*R.card ≤ k
-      · have he : (∑ L : Finset V, W (R,L)) =
-            (∏ v ∈ R, p v) *
-              ∑ L ∈ (neighborhood G R).powersetCard (k-R.card), ∏ v ∈ L, p v := by
-          simp only [W, hR, true_and]
-          rw [← Finset.sum_filter]
-          simp only [Finset.filter_mem_eq_inter, Finset.univ_inter, Finset.mul_sum]
-        rw [he]
-        have h := mul_le_mul_of_nonneg_left (root_leaves_weight_le G p hp k δ hδ hδ1 hdeg R hR)
-          (Finset.prod_nonneg (s := R) (fun v _ => hp v))
-        simpa only [mul_comm, mul_left_comm, mul_assoc] using h
-      · simp only [W, hR, false_and, ite_false, Finset.sum_const_zero]
-        exact mul_nonneg (mul_nonneg (Real.rpow_nonneg hδ.le _) (Real.exp_pos _).le)
-          (Finset.prod_nonneg (fun v _ => hp v))
-    _ = δ^((k:ℝ)/2) * ∑ R : Finset V, ∏ v ∈ R, (Real.exp 1 * p v) := by
-      rw [Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro R _
-      rw [Finset.prod_mul_distrib, Finset.prod_const, ← Real.exp_nat_mul]
-      simp only [mul_one, mul_assoc]
-    _ ≤ δ^((k:ℝ)/2) * Real.exp (∑ v, Real.exp 1 * p v) := by
-      apply mul_le_mul_of_nonneg_left _ (Real.rpow_nonneg hδ.le _)
-      simpa only [Finset.powerset_univ] using
-        powerset_weight_le_exp Finset.univ (fun v => Real.exp 1 * p v)
-          (fun v => mul_nonneg (Real.exp_pos _).le (hp v))
-    _ = _ := by rw [← Finset.mul_sum]
-
-
-lemma noIsolated_probability_le {V Ω : Type*} [Fintype V] [Fintype Ω]
-    (G : SimpleGraph V) (O : Ω → Finset V) (k : ℕ)
-    (hcard : ∀ ω, (O ω).card = k) (p : V → ℝ) (hp : ∀ v, 0 ≤ p v)
-    (hcyl : ∀ D, inclusionMoment O D ≤ ∏ v ∈ D, p v)
-    (δ : ℝ) (hδ : 0 < δ) (hδ1 : δ ≤ 1)
-    (hdeg : ∀ v, (∑ w, if G.Adj v w then p w else 0) ≤ δ) :
-    finiteMean (fun ω => if NoIsolated G (O ω) then (1:ℝ) else 0) ≤
-      δ^((k:ℝ)/2) * Real.exp (Real.exp 1 * ∑ v, p v) := by
-  classical
-  let I := {S : Finset V // S.card = k ∧ NoIsolated G S}
-  calc
-    _ ≤ finiteMean (fun ω => ∑ S : I, if S.val ⊆ O ω then (1:ℝ) else 0) := by
-      apply finiteMean_mono
-      intro ω
-      split_ifs with h
-      · exact (Finset.single_le_sum (f := fun S : I => if S.val ⊆ O ω then (1:ℝ) else 0)
-          (fun S _ => by positivity) (Finset.mem_univ (⟨O ω,hcard ω,h⟩ : I))) |>.trans' (by simp)
-      · positivity
-    _ = ∑ S : I, inclusionMoment O S.val := by
-      rw [finiteMean_sum]
-      rfl
-    _ ≤ ∑ S : I, ∏ v ∈ S.val, p v := Finset.sum_le_sum (fun S _ => hcyl S.val)
-    _ ≤ _ := noIsolated_weight_le G p hp k δ hδ hδ1 hdeg
-
-noncomputable def typedSet {N X : Type*} [Fintype N] [Fintype X]
-    (A : N → Finset X) : Finset (N × X) := by
-  classical
-  exact Finset.univ.filter (fun v => v.2 ∈ A v.1)
-
-@[simp] lemma mem_typedSet {N X : Type*} [Fintype N] [Fintype X]
-    (A : N → Finset X) (v : N × X) : v ∈ typedSet A ↔ v.2 ∈ A v.1 := by
-  classical
-  simp [typedSet]
-
-noncomputable def fiber {N X : Type*} [Fintype X]
-    (S : Finset (N × X)) (t : N) : Finset X := by
-  classical
-  exact Finset.univ.filter (fun x => (t,x) ∈ S)
-
-@[simp] lemma mem_fiber {N X : Type*} [Fintype X]
-    (S : Finset (N × X)) (t : N) (x : X) : x ∈ fiber S t ↔ (t,x) ∈ S := by
-  classical
-  simp [fiber]
-
-lemma typedSet_fiber {N X : Type*} [Fintype N] [Fintype X]
-    (S : Finset (N × X)) : typedSet (fiber S) = S := by
-  classical
-  ext v
-  simp
-
-lemma typedSet_prod {N X : Type*} [Fintype N] [Fintype X]
-    (A : N → Finset X) (p : N × X → ℝ) :
-    (∏ v ∈ typedSet A, p v) = ∏ t, ∏ x ∈ A t, p (t,x) := by
-  classical
-  simp only [typedSet, Finset.prod_filter, Fintype.prod_prod_type]
-  apply Finset.prod_congr rfl
-  intro t _
-  rw [← Finset.prod_filter]
-  simp only [Finset.filter_mem_eq_inter, Finset.univ_inter]
-
-lemma typedSet_card {N X : Type*} [Fintype N] [Fintype X]
-    (A : N → Finset X) : (typedSet A).card = ∑ t, (A t).card := by
-  classical
-  simp only [Finset.card_eq_sum_ones, typedSet, Finset.sum_filter, Fintype.sum_prod_type]
-  apply Finset.sum_congr rfl
-  intro t _
-  rw [← Finset.sum_filter]
-  simp only [Finset.filter_mem_eq_inter, Finset.univ_inter]
-
-lemma subset_typedSet {N X : Type*} [Fintype N] [Fintype X]
-    (A : N → Finset X) (D : Finset (N × X)) :
-    D ⊆ typedSet A ↔ ∀ t, fiber D t ⊆ A t := by
-  constructor
-  · intro h t x hx
-    exact mem_typedSet A (t,x) |>.mp (h (mem_fiber D t x |>.mp hx))
-  · intro h v hv
-    exact mem_typedSet A v |>.mpr (h v.1 (mem_fiber D v.1 v.2 |>.mpr hv))
-
-end Thorp.SparseContact
-
-end ThorpNine.Tail
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Tail.BlockProduct
+
+namespace OAI
+
+namespace ThorpNine.Tail
+
+namespace Thorp
+open scoped BigOperators Classical
+
+def modifiedStageEquiv (r L : ℕ) :
+    ((OuterCoins r L × (Card r → Equiv.Perm (Card L) × BenesCoins L)) × OuterCoins r L) ≃
+      ModifiedCoins L r :=
+  (modifiedAssembleEquiv _ _ _ _).trans
+    (Equiv.prodCongr (Equiv.refl _) (Equiv.prodCongr (parallelSplitCoinEquiv r L) (parallelSplitCoinEquiv r L)))
+
+lemma modifiedStageEquiv_apply (r L : ℕ)
+    (w : (OuterCoins r L × (Card r → Equiv.Perm (Card L) × BenesCoins L)) × OuterCoins r L) :
+    modifiedStageEquiv r L w=
+      ((fun b => (w.1.2 b).1),
+       (assembleBits r L (fun x => (w.1.2 x.1).2.1 x.2) w.1.1,
+        assembleBits r L (fun x => (w.1.2 x.1).2.2 x.2) w.2)) := by
+  simp only [modifiedStageEquiv,Equiv.trans_apply,Equiv.prodCongr_apply,modifiedAssembleEquiv,
+    Equiv.coe_fn_mk,Prod.map_apply,Equiv.refl_apply,parallelSplitCoinEquiv_apply]
+
+lemma palindrome_stage (r L : ℕ)
+    (w : (OuterCoins r L × (Card r → BenesCoins L)) × OuterCoins r L) :
+    splitPerm r L (palindromePerm (L+r) (palindromeStageEquiv r L w))=
+      outerPerm r L w.1.1*ParallelBlocks.lift (fun b => palindromePerm L (w.1.2 b))*
+        (outerPerm r L w.2)⁻¹ := by
+  rw [palindromeStageEquiv_apply]
+  change splitPerm r L (_*_⁻¹)=_
+  rw [map_mul,map_inv,splitPerm_butterfly,splitPerm_butterfly,mul_inv_rev]
+  rw [mul_assoc,←mul_assoc (ParallelBlocks.lift _),←map_inv,←map_mul,←mul_assoc]
+  rfl
+
+noncomputable def localModifiedPerm (L : ℕ) (e : Bool)
+    (w : Equiv.Perm (Card L) × BenesCoins L) : Equiv.Perm (Card L) :=
+  if e then 1 else butterflyPerm L (decodeButterfly L w.2.1)*w.1⁻¹
+
+lemma modified_stage (r L : ℕ) (E : Finset (Card r))
+    (w : (OuterCoins r L × (Card r → Equiv.Perm (Card L) × BenesCoins L)) × OuterCoins r L) :
+    splitPerm r L (modifiedPerm L r E (modifiedStageEquiv r L w))=
+      outerPerm r L w.1.1*ParallelBlocks.lift (fun b => localModifiedPerm L (decide (b∈E)) (w.1.2 b))*
+        (outerPerm r L w.2)⁻¹ := by
+  rw [modifiedStageEquiv_apply]
+  unfold modifiedPerm modifiedInputTree
+  rw [map_mul,map_inv,splitPerm_butterfly,splitPerm_boundaryTree,mul_inv_rev]
+  rw [mul_assoc,←mul_assoc (ParallelBlocks.lift _),←map_inv,←map_mul,←mul_assoc]
+  congr 2
+  apply congrArg ParallelBlocks.lift
+  funext b
+  simp only [Pi.mul_apply,Pi.inv_apply,modifiedBoundary,boundaryOutput_split,localModifiedPerm]
+  by_cases h : b∈E <;> simp [h]
+
+end Thorp
+namespace Thorp.UnitaryFinite
+open scoped BigOperators Classical
+variable {G : Type*} [Group G]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+variable {Ω Ξ : Type*} [Fintype Ω] [Fintype Ξ]
+
+lemma sampleOperator_sandwich (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
+    (P : Ω → G) (Q : Ξ → G) :
+    sampleOperator ρ (fun w : (Ω × Ξ) × Ω => P w.1.1*Q w.1.2*(P w.2)⁻¹)=
+      sampleOperator ρ P*sampleOperator ρ Q*(sampleOperator ρ P).adjoint := by
+  rw [sampleOperator_prod ρ (fun w : Ω × Ξ => P w.1*Q w.2) (fun w : Ω => (P w)⁻¹),
+    sampleOperator_prod ρ P Q,sampleOperator_inv ρ hρ]
+
+lemma localModified_average (L : ℕ) (ρ : Representation ℂ (Equiv.Perm (Card L)) V) (e : Bool) :
+    sampleOperator ρ (localModifiedPerm L e)=if e then 1 else uniformOperator ρ := by
+  cases e
+  · simp only [Bool.false_eq_true,ite_false]
+    rw [←sampleOperator_equiv ρ _ (Equiv.prodComm _ _)]
+    exact sampleOperator_uniform_mul ρ (fun w : BenesCoins L => butterflyPerm L (decodeButterfly L w.1))
+  · simp only [ite_true]
+    exact sampleOperator_one ρ
+
+end Thorp.UnitaryFinite
+
+namespace Thorp
+open scoped BigOperators Classical
+open UnitaryFinite
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+lemma sampleOperator_palindrome_stage (r L : ℕ)
+    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) (hρ : IsUnitary ρ) :
+    sampleOperator (ρ.comp (splitPerm r L).toMonoidHom) (palindromePerm (L+r))=
+      sampleOperator ρ (outerPerm r L)*
+        sampleOperator ρ (fun w : Card r → BenesCoins L => ParallelBlocks.lift (fun b => palindromePerm L (w b)))*
+          (sampleOperator ρ (outerPerm r L)).adjoint := by
+  have he : ((splitPerm r L).toMonoidHom ∘ palindromePerm (L+r)) ∘ palindromeStageEquiv r L=
+      fun w => outerPerm r L w.1.1*ParallelBlocks.lift (fun b => palindromePerm L (w.1.2 b))*
+        (outerPerm r L w.2)⁻¹ := funext (palindrome_stage r L)
+  calc
+    _ = sampleOperator ρ (((splitPerm r L).toMonoidHom ∘ palindromePerm (L+r)) ∘ palindromeStageEquiv r L) :=
+      (sampleOperator_equiv ρ ((splitPerm r L).toMonoidHom ∘ palindromePerm (L+r)) (palindromeStageEquiv r L)).symm
+    _ = _ := (congrArg (sampleOperator ρ) he).trans
+      (sampleOperator_sandwich ρ hρ (outerPerm r L)
+        (fun w : Card r → BenesCoins L => ParallelBlocks.lift (fun b => palindromePerm L (w b))))
+
+lemma sampleOperator_modified_stage (r L : ℕ) (E : Finset (Card r))
+    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) (hρ : IsUnitary ρ) :
+    sampleOperator (ρ.comp (splitPerm r L).toMonoidHom) (modifiedPerm L r E)=
+      sampleOperator ρ (outerPerm r L)*
+        sampleOperator ρ (fun w : Card r → Equiv.Perm (Card L) × BenesCoins L =>
+          ParallelBlocks.lift (fun b => localModifiedPerm L (decide (b∈E)) (w b)))*
+          (sampleOperator ρ (outerPerm r L)).adjoint := by
+  have he : ((splitPerm r L).toMonoidHom ∘ modifiedPerm L r E) ∘ modifiedStageEquiv r L=
+      fun w => outerPerm r L w.1.1*ParallelBlocks.lift
+        (fun b => localModifiedPerm L (decide (b∈E)) (w.1.2 b))*(outerPerm r L w.2)⁻¹ :=
+    funext (modified_stage r L E)
+  calc
+    _ = sampleOperator ρ (((splitPerm r L).toMonoidHom ∘ modifiedPerm L r E) ∘ modifiedStageEquiv r L) :=
+      (sampleOperator_equiv ρ ((splitPerm r L).toMonoidHom ∘ modifiedPerm L r E) (modifiedStageEquiv r L)).symm
+    _ = _ := (congrArg (sampleOperator ρ) he).trans
+      (sampleOperator_sandwich ρ hρ (outerPerm r L)
+        (fun w : Card r → Equiv.Perm (Card L) × BenesCoins L =>
+          ParallelBlocks.lift (fun b => localModifiedPerm L (decide (b∈E)) (w b))))
+
+end Thorp
+namespace Thorp.CommutingBlocks
+open scoped BigOperators Classical
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
+variable {ι : Type*} [Fintype ι] [DecidableEq ι]
+
+noncomputable def retain (P : ι → V →L[ℂ] V) (E : Finset ι) (i : ι) : V →L[ℂ] V :=
+  if i∈E then 1 else P i
+
+omit [CompleteSpace V] [Fintype ι] in
+lemma retain_commute (P : ι → V →L[ℂ] V) (hP : ∀ i j, Commute (P i) (P j))
+    (E : Finset ι) (i j : ι) : Commute (retain P E i) (retain P E j) := by
+  unfold retain
+  split_ifs <;> first | exact Commute.one_left _ | exact Commute.one_right _ | exact hP _ _
+
+omit [CompleteSpace V] in
+lemma blocks_retain (P : ι → V →L[ℂ] V) (hP : ∀ i j, Commute (P i) (P j)) (E : Finset ι) :
+    blocks (retain P E) (retain_commute P hP E)=coarse P hP E := by
+  have hs : E∪Eᶜ=(Finset.univ : Finset ι) := Finset.union_compl E
+  have hh := Finset.noncommProd_union_of_disjoint (s := E) (t := Eᶜ) disjoint_compl_right
+    (retain P E) (fun _ _ _ _ _ => retain_commute P hP E _ _)
+  simp only [hs] at hh
+  change blocks (retain P E) (retain_commute P hP E)=_ at hh
+  rw [hh]
+  have he : E.noncommProd (retain P E) (fun _ _ _ _ _ => retain_commute P hP E _ _)=1 := by
+    exact (Finset.noncommProd_eq_pow_card E (retain P E)
+      (fun i _ j _ _ => retain_commute P hP E i j) 1 (by intro i hi; exact ite_eq_left hi)).trans (one_pow _)
+  rw [he,one_mul]
+  apply Finset.noncommProd_congr rfl
+  intro i hi
+  exact ite_eq_right (Finset.mem_compl.mp hi)
+
+end Thorp.CommutingBlocks
+
+namespace Thorp.ParallelBlocks
+open scoped BigOperators Classical
+open UnitaryFinite
+variable {B X : Type*} [Fintype B] [Fintype X] [DecidableEq X]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+noncomputable def centralProjection (ρ : Representation ℂ (Equiv.Perm (B × X)) V) (b : B) : V →L[ℂ] V :=
+  uniformOperator (ρ.comp (injection b))
+
+omit [Fintype B] in
+lemma centralProjection_commute (ρ : Representation ℂ (Equiv.Perm (B × X)) V) (b c : B) :
+    Commute (centralProjection ρ b) (centralProjection ρ c) := by
+  simpa only [centralProjection,sampleOperator_uniform] using
+    local_average_commute ρ (fun _ => id : B → Equiv.Perm X → Equiv.Perm X) b c
+
+omit [Fintype B] in
+lemma local_central_commute {Ω : Type*} [Fintype Ω] [Nonempty Ω]
+    (ρ : Representation ℂ (Equiv.Perm (B × X)) V) (P : B → Ω → Equiv.Perm X) (b c : B) :
+    Commute (sampleOperator (ρ.comp (injection b)) (P b)) (centralProjection ρ c) := by
+  by_cases h : b=c
+  · subst c
+    change _*uniformOperator _=uniformOperator _*_
+    rw [sampleOperator_uniform_absorb,sampleOperator_weighted,weighted_uniform_right,sampleLaw_sum,one_smul]
+  · unfold centralProjection
+    rw [←sampleOperator_uniform,sampleOperator_comp,sampleOperator_comp]
+    exact sampleOperator_commute ρ _ _ (fun _ _ => injection_commute h _ _)
+
+end Thorp.ParallelBlocks
+namespace Thorp
+open scoped BigOperators Classical
+open UnitaryFinite ParallelBlocks CommutingBlocks
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+lemma modified_central_average (r L : ℕ) (E : Finset (Card r))
+    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) :
+    sampleOperator ρ (fun w : Card r → Equiv.Perm (Card L) × BenesCoins L =>
+      lift (fun b => localModifiedPerm L (decide (b∈E)) (w b)))=
+      coarse (centralProjection ρ) (centralProjection_commute ρ) E := by
+  rw [sampleOperator_parallel ρ (fun b => localModifiedPerm L (decide (b∈E))),←blocks_retain]
+  apply Finset.noncommProd_congr rfl
+  · intro b _
+    rw [localModified_average]
+    simp only [retain,centralProjection,decide_eq_true_eq]
+  · intro b _ c _ _
+    exact local_average_commute ρ (fun b => localModifiedPerm L (decide (b∈E))) b c
+
+lemma central_local_gap (r L : ℕ)
+    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) (hρ : IsUnitary ρ) (b : Card r) :
+    ‖sampleOperator (ρ.comp (injection b)) (palindromePerm L)*(1-centralProjection ρ b)‖≤blockGap L := by
+  change ‖sampleOperator _ _*(1-uniformOperator _)‖≤_
+  rw [mul_sub,mul_one,sampleOperator_uniform_absorb,sampleOperator_weighted]
+  exact palindrome_off_invariants_gap L _ (unitary_comp ρ hρ _)
+
+lemma central_blocks_exception_bound (r L : ℕ)
+    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) (hρ : IsUnitary ρ)
+    (B : V →L[ℂ] V) (hB0 : ‖B‖≤1) (n : ℕ) :
+    ‖B*blocks (fun b : Card r => sampleOperator (ρ.comp (injection b)) (palindromePerm L))
+      (local_average_commute ρ (fun _ => palindromePerm L))*B.adjoint‖ ≤
+      (blockGap L)^n + ∑ E : Finset (Card r), if E.card<n then
+        ‖B*coarse (centralProjection ρ) (centralProjection_commute ρ) E*B.adjoint‖ else 0 := by
+  have hB : ‖B.adjoint‖≤1 := by
+    rw [ContinuousLinearMap.adjoint.norm_map]
+    exact hB0
+  have h := blocks_exception_bound (centralProjection ρ) (centralProjection_commute ρ)
+    (fun b => uniformOperator_projection (ρ.comp (injection b)) (unitary_comp ρ hρ _))
+    (fun b : Card r => sampleOperator (ρ.comp (injection b)) (palindromePerm L))
+    (local_average_commute ρ (fun _ => palindromePerm L))
+    (local_central_commute ρ (fun _ => palindromePerm L))
+    (fun b => sampleOperator_norm_le (ρ.comp (injection b)) (unitary_comp ρ hρ _) (palindromePerm L))
+    (blockGap L) (blockGap_pos L).le (blockGap_lt_one L).le (central_local_gap r L ρ hρ)
+    B.adjoint hB n
+  simpa only [ContinuousLinearMap.adjoint_adjoint] using h
+
+theorem palindrome_exception_bound (r L : ℕ)
+    (ρ : Representation ℂ (Equiv.Perm (Card r × Card L)) V) (hρ : IsUnitary ρ) (n : ℕ) :
+    ‖sampleOperator (ρ.comp (splitPerm r L).toMonoidHom) (palindromePerm (L+r))‖ ≤
+      (blockGap L)^n + ∑ E : Finset (Card r), if E.card<n then
+        ‖sampleOperator (ρ.comp (splitPerm r L).toMonoidHom) (modifiedPerm L r E)‖ else 0 := by
+  rw [sampleOperator_palindrome_stage r L ρ hρ]
+  rw [sampleOperator_parallel ρ (fun _ => palindromePerm L)]
+  apply (central_blocks_exception_bound r L ρ hρ (sampleOperator ρ (outerPerm r L))
+    (sampleOperator_norm_le ρ hρ _) n).trans_eq
+  congr 1
+  apply Finset.sum_congr rfl
+  intro E _
+  split_ifs
+  · rw [sampleOperator_modified_stage r L E ρ hρ,modified_central_average]
+  · rfl
+
+end Thorp
+
+namespace Thorp.UnitaryFinite
+open scoped BigOperators ComplexConjugate Classical
+
+variable {G X Ω : Type*} [Group G] [Fintype G] [Fintype X] [Fintype Ω]
+
+noncomputable def actionKernel (a : G →* Equiv.Perm X) (P : Ω → G) (x y : X) : ℝ :=
+  finiteMean (fun ω => if a (P ω) x=y then 1 else 0)
+
+omit [Fintype G] in
+lemma actionKernel_apply (a : G →* Equiv.Perm X) (P : Ω → G)
+    (v : EuclideanSpace ℂ X) (x : X) :
+    DensityTransfer.applyKernel (actionKernel a P) v x =
+      (Fintype.card Ω:ℂ)⁻¹ * ∑ ω, v (a (P ω) x) := by
+  simp only [DensityTransfer.applyKernel_apply,actionKernel,finiteMean,
+    Complex.ofReal_sum,div_eq_mul_inv,Finset.sum_mul]
+  rw [Finset.sum_comm]
+  calc
+    _ = ∑ ω, (Fintype.card Ω:ℂ)⁻¹ * ∑ y : X, if a (P ω) x=y then v y else 0 := by
+      apply Finset.sum_congr rfl
+      intro ω _
+      rw [Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro y _
+      split_ifs <;> simp_all
+    _ = _ := by simp [Finset.mul_sum]
+
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+noncomputable def randomOperator (ρ : Representation ℂ G V) (P : Ω → G) : Module.End ℂ V :=
+  (Fintype.card Ω:ℂ)⁻¹ • ∑ ω, ρ (P ω)⁻¹
+
+omit [Fintype G] [FiniteDimensional ℂ V] in
+lemma coefficient_random (a : G →* Equiv.Perm X) (x₀ : X)
+    (ha : ∀ x, ∃ g, a g x₀=x) (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
+    (P : Ω → G) (w : V) (hw : IsFixed a x₀ ρ w) (v : V) :
+    DensityTransfer.applyKernel (actionKernel a P) (normalizedCoefficient a x₀ ha ρ w v) =
+      normalizedCoefficient a x₀ ha ρ w (randomOperator ρ P v) := by
+  ext x
+  rw [actionKernel_apply]
+  simp only [normalizedCoefficient,PiLp.smul_apply,smul_eq_mul,Finset.mul_sum,
+    coefficient,randomOperator,LinearMap.smul_apply,LinearMap.sum_apply,
+    inner_smul_right,inner_sum]
+  apply Finset.sum_congr rfl
+  intro ω _
+  have h := coefficient_action a x₀ ha ρ hρ w hw v (P ω) x
+  change inner ℂ (ρ (representative a x₀ ha (a (P ω) x)) w) v =
+    inner ℂ (ρ (representative a x₀ ha x) w) (ρ (P ω)⁻¹ v) at h
+  rw [h]
+  ring
+
+noncomputable def fixedSpace (a : G →* Equiv.Perm X) (x₀ : X) (ρ : Representation ℂ G V) :
+    Submodule ℂ V where
+  carrier := {w | IsFixed a x₀ ρ w}
+  zero_mem' := fun g hg => map_zero _
+  add_mem' := fun hw hz g hg => by rw [map_add,hw g hg,hz g hg]
+  smul_mem' := fun c w hw g hg => by rw [map_smul,hw g hg]
+
+theorem operator_amplification (a : G →* Equiv.Perm X) (x₀ : X)
+    (ha : ∀ x, ∃ g, a g x₀=x) (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
+    (hρ : IsUnitary ρ) (P : Ω → G)
+    (hsym : ∀ x y, actionKernel a P x y=actionKernel a P y x)
+    (hrow : ∀ x, ∑ y, actionKernel a P x y=1)
+    (b z M : ℝ) (hb : 0≤b) (hz : 0<z) (hM0 : 0≤M)
+    (hM : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*actionKernel a P x y)^(1+b)) ≤ M)
+    (hm : 0 < Module.finrank ℂ (fixedSpace a x₀ ρ)) (v : V) (hv : ‖v‖=1) :
+    ‖randomOperator ρ P v‖ ≤ M*z^(-b)+Real.sqrt (z/(Module.finrank ℂ (fixedSpace a x₀ ρ):ℝ)) := by
+  let K := fixedSpace a x₀ ρ
+  let m := Module.finrank ℂ K
+  let : Nonempty X := ⟨x₀⟩
+  let : Nonempty (Fin m) := ⟨⟨0,hm⟩⟩
+  let B := stdOrthonormalBasis ℂ K
+  let w : Fin m → V := fun i => (B i : V)
+  have hfix (i : Fin m) : IsFixed a x₀ ρ (w i) := (B i).2
+  have hw : Orthonormal ℂ w := B.orthonormal.comp_linearIsometry K.subtypeₗᵢ
+  have hc := normalizedCoefficient_orthonormal a x₀ ha ρ hρ w hw hfix v hv
+  have hn (i : Fin m) : ‖DensityTransfer.applyKernel (actionKernel a P)
+      (normalizedCoefficient a x₀ ha ρ (w i) v)‖ = ‖randomOperator ρ P v‖ := by
+    rw [coefficient_random a x₀ ha ρ hρ P (w i) (hfix i),
+      normalizedCoefficient_norm a x₀ ha ρ hρ (w i) (hfix i) (hw.norm_eq_one i)]
+  have h := DensityTransfer.multiplicity_amplification (actionKernel a P)
+    (fun x y => finiteMean_nonneg (fun _ => by split_ifs <;> norm_num)) hsym hrow
+    b z M hb hz hM0 hM _ hc (‖randomOperator ρ P v‖) (fun i => (hn i).ge)
+  simpa only [Fintype.card_fin] using h
+
+end Thorp.UnitaryFinite
+namespace Thorp.UnitaryFinite
+open scoped BigOperators Classical
+
+variable {α ι : Type*} [Fintype α] [Fintype ι]
+
+def tupleAction : Equiv.Perm α →* Equiv.Perm (ι ↪ α) where
+  toFun p :=
+    { toFun := fun e => e.trans p.toEmbedding
+      invFun := fun e => e.trans p.symm.toEmbedding
+      left_inv := fun e => by ext i; exact p.symm_apply_apply (e i)
+      right_inv := fun e => by ext i; exact p.apply_symm_apply (e i) }
+  map_one' := by ext e i; rfl
+  map_mul' p q := by ext e i; rfl
+
+omit [Fintype α] [Fintype ι] in
+@[simp] lemma tupleAction_apply (p : Equiv.Perm α) (e : ι ↪ α) (i : ι) :
+    tupleAction p e i=p (e i) := rfl
+
+omit [Fintype α] in
+lemma tupleAction_transitive (e f : ι ↪ α) : ∃ p, tupleAction p e=f := by
+  obtain ⟨p,hp⟩ := Equiv.Perm.exists_extending_pair e f e.injective f.injective
+  exact ⟨p,Function.Embedding.ext hp⟩
+
+omit [Fintype α] in
+lemma tupleAction_fixed_iff [DecidableEq α] (e : ι ↪ α) (p : Equiv.Perm α) :
+    tupleAction p e=e ↔ ∀ x ∈ Finset.univ.image e, p x=x := by
+  constructor
+  · intro h x hx
+    obtain ⟨i,_,rfl⟩ := Finset.mem_image.mp hx
+    exact congrArg (fun f : ι ↪ α => f i) h
+  · intro h
+    ext i
+    exact h (e i) (Finset.mem_image.mpr ⟨i,Finset.mem_univ _,rfl⟩)
+
+omit [Fintype α] in
+lemma tupleAction_kernel [DecidableEq α] {Ω : Type*} [Fintype Ω] (P : Ω → Equiv.Perm α)
+    (e f : ι ↪ α) : actionKernel tupleAction P e f=PairRouting.tupleProbability P e f := by
+  unfold actionKernel PairRouting.tupleProbability
+  congr 1
+  funext ω
+  congr 1
+  exact propext Function.Embedding.ext_iff
+
+end Thorp.UnitaryFinite
+namespace Thorp.UnitaryFinite
+open scoped BigOperators Classical
+
+lemma irreducible_of_equiv {G V W : Type*} [Group G]
+    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
+    (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
+    (e : Representation.Equiv ρ σ) [Representation.IsIrreducible ρ] :
+    Representation.IsIrreducible σ := by
+  rw [Representation.irreducible_iff_isSimpleModule_asModule]
+  have hi := (Representation.irreducible_iff_isSimpleModule_asModule ρ).mp inferInstance
+  let f : ρ.asModule →ₗ[MonoidAlgebra ℂ G] σ.asModule :=
+    Representation.IntertwiningMap.equivLinearMapAsModule ρ σ e.toIntertwiningMap
+  have hf : Function.Bijective f := e.toLinearEquiv.bijective
+  exact (f.isSimpleModule_iff_of_bijective hf).mp hi
+
+end Thorp.UnitaryFinite
+
+namespace Thorp.Specht
+open scoped BigOperators Classical
+
+lemma unitaryRepresentation_coe (μ : YoungDiagram) (p : Equiv.Perm (Cell μ))
+    (v : hilbertSpace μ) :
+    (unitaryRepresentation μ p v : PermutationHilbert.H (Tabloid μ)) =
+      PermutationHilbert.act (tabloidAct p) v := by
+  obtain ⟨w,rfl⟩ := (spaceHilbertEquiv μ).surjective v
+  change hilbertEquiv μ (tabloidRep μ p w) = _
+  exact hilbert_action μ p w
+
+noncomputable def unitaryEquiv (μ : YoungDiagram) :
+    Representation.Equiv (representation μ) (unitaryRepresentation μ) :=
+  Representation.Equiv.mk (spaceHilbertEquiv μ) (by
+    intro p
+    apply LinearMap.ext
+    intro v
+    change spaceHilbertEquiv μ (representation μ p v) =
+      spaceHilbertEquiv μ (representation μ p ((spaceHilbertEquiv μ).symm (spaceHilbertEquiv μ v)))
+    rw [LinearEquiv.symm_apply_apply])
+
+lemma unitaryRepresentation_irreducible (μ : YoungDiagram) :
+    Representation.IsIrreducible (unitaryRepresentation μ) := by
+  let := representation_irreducible μ
+  exact UnitaryFinite.irreducible_of_equiv _ _ (unitaryEquiv μ)
+
+lemma unitaryRepresentation_unitary (μ : YoungDiagram) :
+    UnitaryFinite.IsUnitary (G := Equiv.Perm (Cell μ)) (V := hilbertSpace μ) (unitaryRepresentation μ) := by
+  intro p v w
+  change inner ℂ (unitaryRepresentation μ p v : PermutationHilbert.H (Tabloid μ))
+      (unitaryRepresentation μ p w : PermutationHilbert.H (Tabloid μ)) = _
+  rw [unitaryRepresentation_coe,unitaryRepresentation_coe]
+  exact (PermutationHilbert.act (tabloidAct p)).inner_map_map v w
+
+
+noncomputable def tupleFixedEquiv (μ : YoungDiagram) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Cell μ) :
+    UnitaryFinite.fixedSpace (V := hilbertSpace μ) UnitaryFinite.tupleAction e (unitaryRepresentation μ) ≃ₗ[ℂ]
+      fixedHilbert μ (Finset.univ.image e) where
+  toFun v := ⟨v.1.1,v.1.2,by
+    intro p hp
+    have h := v.2 p ((UnitaryFinite.tupleAction_fixed_iff e p).mpr hp)
+    exact (unitaryRepresentation_coe μ p v.1).symm.trans (congrArg Subtype.val h)⟩
+  invFun v := ⟨⟨v.1,v.2.1⟩,by
+    intro p hp
+    apply Subtype.ext
+    exact (unitaryRepresentation_coe μ p _).trans
+      (v.2.2 p ((UnitaryFinite.tupleAction_fixed_iff e p).mp hp))⟩
+  left_inv _ := rfl
+  right_inv _ := rfl
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+
+theorem tupleFixed_dimension (μ : YoungDiagram) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Cell μ) :
+    Module.finrank ℂ (hilbertSpace μ) * (Fintype.card ι).choose (μ.card-μ.rowLen 0) ≤
+      Module.finrank ℂ (UnitaryFinite.fixedSpace (V := hilbertSpace μ) UnitaryFinite.tupleAction e (unitaryRepresentation μ)) *
+        μ.card.choose (μ.card-μ.rowLen 0) := by
+  rw [(tupleFixedEquiv μ e).finrank_eq,finrank_hilbertSpace]
+  have h := fixedHilbert_dimension μ (Finset.univ.image e)
+  simpa only [Finset.card_image_of_injective _ e.injective,Finset.card_univ] using h
+
+end Thorp.Specht
+namespace Thorp.UnitaryFinite
+open scoped BigOperators Classical
+
+variable {G H X Ω : Type*} [Group G] [Group H] [Fintype G] [Fintype X] [Fintype Ω]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+omit [Fintype G] in
+lemma sampleOperator_inv_apply (ρ : Representation ℂ G V) (P : Ω → G) (v : V) :
+    sampleOperator ρ (fun ω => (P ω)⁻¹) v=randomOperator ρ P v := by
+  simp only [sampleOperator,randomOperator,smul_apply,
+    sum_apply,continuousRepresentation_apply,LinearMap.smul_apply,LinearMap.sum_apply]
+
+theorem sampleOperator_amplification (a : G →* Equiv.Perm X) (x₀ : X)
+    (ha : ∀ x, ∃ g, a g x₀=x) (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
+    (hρ : IsUnitary ρ) (P : Ω → G)
+    (hsym : ∀ x y, actionKernel a P x y=actionKernel a P y x)
+    (hrow : ∀ x, ∑ y, actionKernel a P x y=1)
+    (b z M : ℝ) (hb : 0≤b) (hz : 0<z) (hM0 : 0≤M)
+    (hM : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*actionKernel a P x y)^(1+b)) ≤ M)
+    (hm : 0 < Module.finrank ℂ (fixedSpace a x₀ ρ)) :
+    ‖sampleOperator ρ P‖ ≤ M*z^(-b)+Real.sqrt (z/(Module.finrank ℂ (fixedSpace a x₀ ρ):ℝ)) := by
+  rw [←ContinuousLinearMap.adjoint.norm_map (sampleOperator ρ P),←sampleOperator_inv ρ hρ]
+  apply ContinuousLinearMap.opNorm_le_of_unit_norm (by positivity)
+  intro v hv
+  rw [sampleOperator_inv_apply]
+  exact operator_amplification a x₀ ha ρ hρ P hsym hrow b z M hb hz hM0 hM hm v hv
+
+omit [Fintype G] [Fintype X] [Fintype Ω] [FiniteDimensional ℂ V] in
+noncomputable def subrepresentationEquiv (ρ : Representation ℂ G V) (e : H ≃* G) :
+    Subrepresentation (ρ.comp e.toMonoidHom) ≃o Subrepresentation ρ where
+  toFun U :=
+    { toSubmodule := U.toSubmodule
+      apply_mem_toSubmodule := fun g v hv => by
+        have h := U.apply_mem_toSubmodule (e.symm g) hv
+        change ρ (e (e.symm g)) v∈U.toSubmodule at h
+        simpa only [e.apply_symm_apply] using h }
+  invFun U :=
+    { toSubmodule := U.toSubmodule
+      apply_mem_toSubmodule := fun h _ hv => U.apply_mem_toSubmodule (e h) hv }
+  left_inv U := by apply Subrepresentation.toSubmodule_injective; rfl
+  right_inv U := by apply Subrepresentation.toSubmodule_injective; rfl
+  map_rel_iff' := Iff.rfl
+
+omit [Fintype G] [Fintype X] [Fintype Ω] [FiniteDimensional ℂ V] in
+lemma irreducible_comp_equiv (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
+    (e : H ≃* G) : Representation.IsIrreducible (ρ.comp e.toMonoidHom) := by
+  exact (subrepresentationEquiv ρ e).isSimpleOrder_iff.mpr inferInstance
+
+end Thorp.UnitaryFinite
+
+namespace Thorp.Specht
+open scoped BigOperators Classical
+
+variable {α ι : Type*} [Fintype α] [Fintype ι]
+
+omit [Fintype α] in
+lemma relabelledUnitary_unitary (μ : YoungDiagram) (e : α ≃ Cell μ) :
+    UnitaryFinite.IsUnitary (V := hilbertSpace μ) (relabelledUnitary μ e) := by
+  intro g v w
+  exact unitaryRepresentation_unitary μ (e.permCongr g) v w
+
+omit [Fintype α] in
+lemma relabelledUnitary_irreducible (μ : YoungDiagram) (e : α ≃ Cell μ) :
+    Representation.IsIrreducible (relabelledUnitary μ e) := by
+  let := unitaryRepresentation_irreducible μ
+  exact UnitaryFinite.irreducible_comp_equiv (V := hilbertSpace μ) (unitaryRepresentation μ) e.permCongrHom
+
+noncomputable def relabelledTupleFixed (μ : YoungDiagram) (e : α ≃ Cell μ) (f : ι ↪ α) :
+    UnitaryFinite.fixedSpace (V := hilbertSpace μ) UnitaryFinite.tupleAction f (relabelledUnitary μ e) ≃ₗ[ℂ]
+    UnitaryFinite.fixedSpace (V := hilbertSpace μ) UnitaryFinite.tupleAction (f.trans e.toEmbedding) (unitaryRepresentation μ) where
+  toFun v := ⟨v.1,by
+    intro g hg
+    have hp : UnitaryFinite.tupleAction (e.symm.permCongr g) f=f := by
+      apply Function.Embedding.ext
+      intro i
+      have hh := congrArg (fun f : ι ↪ Cell μ => f i) hg
+      change g (e (f i))=e (f i) at hh
+      simp only [UnitaryFinite.tupleAction_apply,Equiv.permCongr_apply,Equiv.symm_symm,e.symm_apply_apply,hh]
+    have hh := v.2 (e.symm.permCongr g) hp
+    change unitaryRepresentation μ (e.permCongr (e.symm.permCongr g)) v=v at hh
+    have he : e.permCongr (e.symm.permCongr g)=g := by
+      apply Equiv.ext
+      intro x
+      simp only [Equiv.permCongr_apply,Equiv.symm_symm,e.apply_symm_apply]
+    simpa only [he] using hh⟩
+  invFun v := ⟨v.1,by
+    intro g hg
+    have hp : UnitaryFinite.tupleAction (e.permCongr g) (f.trans e.toEmbedding)=f.trans e.toEmbedding := by
+      apply Function.Embedding.ext
+      intro i
+      have hh := congrArg (fun f : ι ↪ α => f i) hg
+      change g (f i)=f i at hh
+      simp only [UnitaryFinite.tupleAction_apply,Function.Embedding.trans_apply,
+        Equiv.toEmbedding_apply,Equiv.permCongr_apply,e.symm_apply_apply,hh]
+    exact v.2 (e.permCongr g) hp⟩
+  left_inv _ := rfl
+  right_inv _ := rfl
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+
+omit [Fintype α] in
+lemma relabelled_tuple_dimension (μ : YoungDiagram) (e : α ≃ Cell μ) (f : ι ↪ α) :
+    Module.finrank ℂ (hilbertSpace μ) * (Fintype.card ι).choose (μ.card-μ.rowLen 0) ≤
+      Module.finrank ℂ (UnitaryFinite.fixedSpace (V := hilbertSpace μ) UnitaryFinite.tupleAction f (relabelledUnitary μ e)) *
+        μ.card.choose (μ.card-μ.rowLen 0) := by
+  rw [(relabelledTupleFixed μ e f).finrank_eq]
+  exact tupleFixed_dimension μ (f.trans e.toEmbedding)
+
+end Thorp.Specht
+
+namespace Thorp.SparseContact
+open scoped BigOperators Classical
+
+lemma small_dominating_set {V : Type*} [Fintype V] (G : SimpleGraph V)
+    (hG : ∀ v, ∃ w, G.Adj v w) :
+    ∃ R : Finset V, 2 * R.card ≤ Fintype.card V ∧
+      ∀ v, v ∉ R → ∃ w ∈ R, G.Adj v w := by
+  classical
+  obtain ⟨F, hFG, hF, hreach⟩ := G.exists_isAcyclic_reachable_eq_le
+  let c := hF.coloringTwo
+  have hneigh (v : V) : ∃ w, F.Adj v w := by
+    obtain ⟨w, hvw⟩ := hG v
+    have hr : F.Reachable v w := by rw [hreach]; exact hvw.reachable
+    exact hr.nonempty_neighborSet_left hvw.ne
+  let A : Finset V := Finset.univ.filter (fun v => c v = 0)
+  let B : Finset V := Finset.univ.filter (fun v => c v ≠ 0)
+  have hab : A.card + B.card = Fintype.card V := by
+    exact Finset.card_filter_add_card_filter_not _
+  have hcolor {v w : V} (hvw : F.Adj v w) : c v ≠ c w := c.valid hvw
+  by_cases h : A.card ≤ B.card
+  · refine ⟨A, by omega, ?_⟩
+    intro v hv
+    obtain ⟨w, hvw⟩ := hneigh v
+    have hv0 : c v ≠ 0 := by simpa [A] using hv
+    have hw0 : c w = 0 := by
+      have hv1 : c v = 1 := by omega
+      have hn := hcolor hvw
+      omega
+    exact ⟨w, by simp [A, hw0], hFG hvw⟩
+  · refine ⟨B, by omega, ?_⟩
+    intro v hv
+    obtain ⟨w, hvw⟩ := hneigh v
+    have hv0 : c v = 0 := by simpa [B] using hv
+    have hw0 : c w ≠ 0 := by simpa [hv0] using (hcolor hvw).symm
+    exact ⟨w, by simp [B, hw0], hFG hvw⟩
+
+lemma powerset_weight_le_exp {V : Type*} [DecidableEq V] (S : Finset V)
+    (p : V → ℝ) (hp : ∀ v, 0 ≤ p v) :
+    (∑ L ∈ S.powerset, ∏ v ∈ L, p v) ≤ Real.exp (∑ v ∈ S, p v) := by
+  rw [← Finset.prod_one_add, Real.exp_sum]
+  apply Finset.prod_le_prod₀
+  · intro v _; exact add_nonneg (by norm_num) (hp v)
+  · intro v _
+    have h := Real.add_one_le_exp (p v)
+    linarith
+
+lemma fixed_size_weight_le {V : Type*} [DecidableEq V] (S : Finset V)
+    (p : V → ℝ) (hp : ∀ v, 0 ≤ p v) (m : ℕ) (δ : ℝ) (hδ : 0 < δ) :
+    (∑ L ∈ S.powersetCard m, ∏ v ∈ L, p v) ≤
+      δ^m * Real.exp ((∑ v ∈ S, p v)/δ) := by
+  have hsplit (L : Finset V) (hL : L ∈ S.powersetCard m) :
+      (∏ v ∈ L, p v) = δ^m * ∏ v ∈ L, (p v / δ) := by
+    rw [Finset.prod_div_distrib, Finset.prod_const, (Finset.mem_powersetCard.mp hL).2]
+    field_simp
+  calc
+    _ = δ^m * ∑ L ∈ S.powersetCard m, ∏ v ∈ L, (p v/δ) := by
+      rw [Finset.mul_sum]
+      exact Finset.sum_congr rfl hsplit
+    _ ≤ δ^m * ∑ L ∈ S.powerset, ∏ v ∈ L, (p v/δ) := by
+      apply mul_le_mul_of_nonneg_left _ (pow_nonneg hδ.le _)
+      apply Finset.sum_le_sum_of_subset_of_nonneg
+      · intro L hL
+        exact Finset.mem_powerset.mpr (Finset.mem_powersetCard.mp hL).1
+      · intro L _ _; exact Finset.prod_nonneg (fun v _ => div_nonneg (hp v) hδ.le)
+    _ ≤ δ^m * Real.exp (∑ v ∈ S, p v/δ) :=
+      mul_le_mul_of_nonneg_left
+        (powerset_weight_le_exp S (fun v => p v/δ) (fun v => div_nonneg (hp v) hδ.le))
+        (pow_nonneg hδ.le _)
+    _ = _ := by rw [Finset.sum_div]
+
+noncomputable def neighborhood {V : Type*} [Fintype V] (G : SimpleGraph V)
+    (R : Finset V) : Finset V := by
+  classical
+  exact Finset.univ.filter (fun w => ∃ v ∈ R, G.Adj v w)
+
+lemma neighborhood_weight_le {V : Type*} [Fintype V] (G : SimpleGraph V)
+    (p : V → ℝ) (hp : ∀ v, 0 ≤ p v) (δ : ℝ)
+    (hdeg : ∀ v, (∑ w, if G.Adj v w then p w else 0) ≤ δ) (R : Finset V) :
+    (∑ w ∈ neighborhood G R, p w) ≤ R.card * δ := by
+  classical
+  calc
+    _ ≤ ∑ w ∈ neighborhood G R, ∑ v ∈ R, if G.Adj v w then p w else 0 := by
+      apply Finset.sum_le_sum
+      intro w hw
+      obtain ⟨v, hv, hvw⟩ := (Finset.mem_filter.mp hw).2
+      have h : (if G.Adj v w then p w else 0) ≤
+          ∑ u ∈ R, if G.Adj u w then p w else 0 :=
+        Finset.single_le_sum (f := fun u => if G.Adj u w then p w else 0) (fun u _ => by split_ifs; exact hp w; exact le_rfl) hv
+      simpa only [ite_eq_left hvw] using h
+    _ ≤ ∑ w, ∑ v ∈ R, if G.Adj v w then p w else 0 := by
+      apply Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _)
+      intro w _ _
+      exact Finset.sum_nonneg (fun v _ => by split_ifs; exact hp w; exact le_rfl)
+    _ = ∑ v ∈ R, ∑ w, if G.Adj v w then p w else 0 := Finset.sum_comm
+    _ ≤ ∑ v ∈ R, δ := Finset.sum_le_sum (fun v _ => hdeg v)
+    _ = _ := by simp
+
+lemma root_leaves_weight_le {V : Type*} [Fintype V] (G : SimpleGraph V)
+    (p : V → ℝ) (hp : ∀ v, 0 ≤ p v) (k : ℕ) (δ : ℝ)
+    (hδ : 0 < δ) (hδ1 : δ ≤ 1)
+    (hdeg : ∀ v, (∑ w, if G.Adj v w then p w else 0) ≤ δ)
+    (R : Finset V) (hR : 2 * R.card ≤ k) :
+    (∑ L ∈ (neighborhood G R).powersetCard (k-R.card), ∏ v ∈ L, p v) ≤
+      δ^((k:ℝ)/2) * Real.exp R.card := by
+  classical
+  have hmass := neighborhood_weight_le G p hp δ hdeg R
+  have hpow : δ^(k-R.card) ≤ δ^((k:ℝ)/2) := by
+    rw [← Real.rpow_natCast]
+    apply Real.rpow_le_rpow_of_exponent_ge hδ hδ1
+    have hr : R.card ≤ k := by omega
+    rw [Nat.cast_sub hr]
+    have hh : (2:ℝ)*R.card ≤ k := by exact_mod_cast hR
+    linarith
+  apply (fixed_size_weight_le (neighborhood G R) p hp _ δ hδ).trans
+  apply mul_le_mul hpow
+  · apply Real.exp_le_exp.mpr
+    exact (div_le_iff₀ hδ).mpr hmass
+  · positivity
+  · positivity
+
+def NoIsolated {V : Type*} (G : SimpleGraph V) (S : Finset V) : Prop :=
+  ∀ v ∈ S, ∃ w ∈ S, G.Adj v w
+
+lemma small_dominating_subset {V : Type*} [Fintype V] (G : SimpleGraph V)
+    (S : Finset V) (hS : NoIsolated G S) :
+    ∃ R : Finset V, R ⊆ S ∧ 2 * R.card ≤ S.card ∧ S \ R ⊆ neighborhood G R := by
+  classical
+  let H := G.induce (S : Set V)
+  have hH : ∀ v : S, ∃ w : S, H.Adj v w := by
+    intro v
+    obtain ⟨w, hw, hvw⟩ := hS v v.2
+    exact ⟨⟨w, hw⟩, hvw⟩
+  obtain ⟨R, hR, hdom⟩ := small_dominating_set H hH
+  let T := R.image (fun v : S => (v : V))
+  have hc : T.card = R.card := Finset.card_image_of_injective _ Subtype.val_injective
+  refine ⟨T, ?_, ?_, ?_⟩
+  · intro v hv
+    obtain ⟨w, _, rfl⟩ := Finset.mem_image.mp hv
+    exact w.2
+  · simpa [hc, H] using hR
+  · intro v hv
+    have hvS := (Finset.mem_sdiff.mp hv).1
+    have hvR : (⟨v, hvS⟩ : S) ∉ R := by
+      intro hh
+      exact (Finset.mem_sdiff.mp hv).2 (Finset.mem_image.mpr ⟨⟨v,hvS⟩, hh, rfl⟩)
+    obtain ⟨w, hw, hvw⟩ := hdom ⟨v,hvS⟩ hvR
+    exact Finset.mem_filter.mpr ⟨Finset.mem_univ _,
+      ⟨w, Finset.mem_image.mpr ⟨w,hw,rfl⟩, hvw.symm⟩⟩
+
+lemma noIsolated_weight_le {V : Type*} [Fintype V] (G : SimpleGraph V)
+    (p : V → ℝ) (hp : ∀ v, 0 ≤ p v) (k : ℕ) (δ : ℝ)
+    (hδ : 0 < δ) (hδ1 : δ ≤ 1)
+    (hdeg : ∀ v, (∑ w, if G.Adj v w then p w else 0) ≤ δ) :
+    (∑ S : {S : Finset V // S.card = k ∧ NoIsolated G S}, ∏ v ∈ S.val, p v) ≤
+      δ^((k:ℝ)/2) * Real.exp (Real.exp 1 * ∑ v, p v) := by
+  classical
+  let I := {S : Finset V // S.card = k ∧ NoIsolated G S}
+  let root (S : I) := (small_dominating_subset G S.val S.property.2).choose
+  have hroot (S : I) : root S ⊆ S.val ∧ 2*(root S).card ≤ k ∧
+      S.val \ root S ⊆ neighborhood G (root S) := by
+    simpa only [root, S.property.1] using
+      (small_dominating_subset G S.val S.property.2).choose_spec
+  let enc (S : I) : Finset V × Finset V := (root S, S.val \ root S)
+  have hrec (S : I) : (enc S).1 ∪ (enc S).2 = S.val :=
+    Finset.union_sdiff_of_subset (hroot S).1
+  have hinj : Function.Injective enc := by
+    intro S T h
+    apply Subtype.ext
+    rw [← hrec S, ← hrec T, h]
+  let W (z : Finset V × Finset V) : ℝ :=
+    if 2*z.1.card ≤ k ∧ z.2 ∈ (neighborhood G z.1).powersetCard (k-z.1.card)
+    then (∏ v ∈ z.1, p v) * ∏ v ∈ z.2, p v else 0
+  have hW (z : Finset V × Finset V) : 0 ≤ W z := by
+    dsimp [W]
+    split_ifs
+    · exact mul_nonneg (Finset.prod_nonneg (fun v _ => hp v))
+        (Finset.prod_nonneg (fun v _ => hp v))
+    · exact le_rfl
+  have hwenc (S : I) : W (enc S) = ∏ v ∈ S.val, p v := by
+    have hmem : (enc S).2 ∈ (neighborhood G (enc S).1).powersetCard (k-(enc S).1.card) := by
+      refine Finset.mem_powersetCard.mpr ⟨(hroot S).2.2, ?_⟩
+      change (S.val \ root S).card = k-(root S).card
+      rw [Finset.card_sdiff_of_subset (hroot S).1, S.property.1]
+    dsimp only [W]
+    rw [ite_eq_left ⟨(hroot S).2.1,hmem⟩]
+    exact (mul_comm _ _).trans (Finset.prod_sdiff (hroot S).1)
+  have hsum : (∑ S : I, ∏ v ∈ S.val, p v) ≤ ∑ z, W z := by
+    calc
+      _ = ∑ z ∈ Finset.univ.image enc, W z := by
+        rw [Finset.sum_image (fun S _ T _ h => hinj h)]
+        exact Finset.sum_congr rfl (fun S _ => (hwenc S).symm)
+      _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _)
+        (fun z _ _ => hW z)
+  apply hsum.trans
+  rw [Fintype.sum_prod_type]
+  calc
+    (∑ R : Finset V, ∑ L : Finset V, W (R,L)) ≤
+        ∑ R : Finset V, δ^((k:ℝ)/2) * Real.exp R.card * ∏ v ∈ R, p v := by
+      apply Finset.sum_le_sum
+      intro R _
+      by_cases hR : 2*R.card ≤ k
+      · have he : (∑ L : Finset V, W (R,L)) =
+            (∏ v ∈ R, p v) *
+              ∑ L ∈ (neighborhood G R).powersetCard (k-R.card), ∏ v ∈ L, p v := by
+          simp only [W, hR, true_and]
+          rw [← Finset.sum_filter]
+          simp only [Finset.filter_mem_eq_inter, Finset.univ_inter, Finset.mul_sum]
+        rw [he]
+        have h := mul_le_mul_of_nonneg_left (root_leaves_weight_le G p hp k δ hδ hδ1 hdeg R hR)
+          (Finset.prod_nonneg (s := R) (fun v _ => hp v))
+        simpa only [mul_comm, mul_left_comm, mul_assoc] using h
+      · simp only [W, hR, false_and, ite_false, Finset.sum_const_zero]
+        exact mul_nonneg (mul_nonneg (Real.rpow_nonneg hδ.le _) (Real.exp_pos _).le)
+          (Finset.prod_nonneg (fun v _ => hp v))
+    _ = δ^((k:ℝ)/2) * ∑ R : Finset V, ∏ v ∈ R, (Real.exp 1 * p v) := by
+      rw [Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro R _
+      rw [Finset.prod_mul_distrib, Finset.prod_const, ← Real.exp_nat_mul]
+      simp only [mul_one, mul_assoc]
+    _ ≤ δ^((k:ℝ)/2) * Real.exp (∑ v, Real.exp 1 * p v) := by
+      apply mul_le_mul_of_nonneg_left _ (Real.rpow_nonneg hδ.le _)
+      simpa only [Finset.powerset_univ] using
+        powerset_weight_le_exp Finset.univ (fun v => Real.exp 1 * p v)
+          (fun v => mul_nonneg (Real.exp_pos _).le (hp v))
+    _ = _ := by rw [← Finset.mul_sum]
+
+
+lemma noIsolated_probability_le {V Ω : Type*} [Fintype V] [Fintype Ω]
+    (G : SimpleGraph V) (O : Ω → Finset V) (k : ℕ)
+    (hcard : ∀ ω, (O ω).card = k) (p : V → ℝ) (hp : ∀ v, 0 ≤ p v)
+    (hcyl : ∀ D, inclusionMoment O D ≤ ∏ v ∈ D, p v)
+    (δ : ℝ) (hδ : 0 < δ) (hδ1 : δ ≤ 1)
+    (hdeg : ∀ v, (∑ w, if G.Adj v w then p w else 0) ≤ δ) :
+    finiteMean (fun ω => if NoIsolated G (O ω) then (1:ℝ) else 0) ≤
+      δ^((k:ℝ)/2) * Real.exp (Real.exp 1 * ∑ v, p v) := by
+  classical
+  let I := {S : Finset V // S.card = k ∧ NoIsolated G S}
+  calc
+    _ ≤ finiteMean (fun ω => ∑ S : I, if S.val ⊆ O ω then (1:ℝ) else 0) := by
+      apply finiteMean_mono
+      intro ω
+      split_ifs with h
+      · exact (Finset.single_le_sum (f := fun S : I => if S.val ⊆ O ω then (1:ℝ) else 0)
+          (fun S _ => by positivity) (Finset.mem_univ (⟨O ω,hcard ω,h⟩ : I))) |>.trans' (by simp)
+      · positivity
+    _ = ∑ S : I, inclusionMoment O S.val := by
+      rw [finiteMean_sum]
+      rfl
+    _ ≤ ∑ S : I, ∏ v ∈ S.val, p v := Finset.sum_le_sum (fun S _ => hcyl S.val)
+    _ ≤ _ := noIsolated_weight_le G p hp k δ hδ hδ1 hdeg
+
+noncomputable def typedSet {N X : Type*} [Fintype N] [Fintype X]
+    (A : N → Finset X) : Finset (N × X) := by
+  classical
+  exact Finset.univ.filter (fun v => v.2 ∈ A v.1)
+
+@[simp] lemma mem_typedSet {N X : Type*} [Fintype N] [Fintype X]
+    (A : N → Finset X) (v : N × X) : v ∈ typedSet A ↔ v.2 ∈ A v.1 := by
+  classical
+  simp [typedSet]
+
+noncomputable def fiber {N X : Type*} [Fintype X]
+    (S : Finset (N × X)) (t : N) : Finset X := by
+  classical
+  exact Finset.univ.filter (fun x => (t,x) ∈ S)
+
+@[simp] lemma mem_fiber {N X : Type*} [Fintype X]
+    (S : Finset (N × X)) (t : N) (x : X) : x ∈ fiber S t ↔ (t,x) ∈ S := by
+  classical
+  simp [fiber]
+
+lemma typedSet_fiber {N X : Type*} [Fintype N] [Fintype X]
+    (S : Finset (N × X)) : typedSet (fiber S) = S := by
+  classical
+  ext v
+  simp
+
+lemma typedSet_prod {N X : Type*} [Fintype N] [Fintype X]
+    (A : N → Finset X) (p : N × X → ℝ) :
+    (∏ v ∈ typedSet A, p v) = ∏ t, ∏ x ∈ A t, p (t,x) := by
+  classical
+  simp only [typedSet, Finset.prod_filter, Fintype.prod_prod_type]
+  apply Finset.prod_congr rfl
+  intro t _
+  rw [← Finset.prod_filter]
+  simp only [Finset.filter_mem_eq_inter, Finset.univ_inter]
+
+lemma typedSet_card {N X : Type*} [Fintype N] [Fintype X]
+    (A : N → Finset X) : (typedSet A).card = ∑ t, (A t).card := by
+  classical
+  simp only [Finset.card_eq_sum_ones, typedSet, Finset.sum_filter, Fintype.sum_prod_type]
+  apply Finset.sum_congr rfl
+  intro t _
+  rw [← Finset.sum_filter]
+  simp only [Finset.filter_mem_eq_inter, Finset.univ_inter]
+
+lemma subset_typedSet {N X : Type*} [Fintype N] [Fintype X]
+    (A : N → Finset X) (D : Finset (N × X)) :
+    D ⊆ typedSet A ↔ ∀ t, fiber D t ⊆ A t := by
+  constructor
+  · intro h t x hx
+    exact mem_typedSet A (t,x) |>.mp (h (mem_fiber D t x |>.mp hx))
+  · intro h v hv
+    exact mem_typedSet A v |>.mpr (h v.1 (mem_fiber D v.1 v.2 |>.mpr hv))
+
+end Thorp.SparseContact
+
+end ThorpNine.Tail
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Tail/ParallelFollow.lean b/lean/OAI/Probability/ThorpRouting/Tail/ParallelFollow.lean
index 181b8568fb5dc308c24575c5ab31bd23bd3c7a96..6671f9d03082cbe2ee2b28a71e78c2665169ca79 100644
--- a/lean/OAI/Probability/ThorpRouting/Tail/ParallelFollow.lean
+++ b/lean/OAI/Probability/ThorpRouting/Tail/ParallelFollow.lean
@@ -1,883 +1,883 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Tail.PriorPaths
-
-namespace OAI
-
-namespace ThorpNine.Tail
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-section ParallelButterfly
-variable {α B : Type*} [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] {d : ℕ}
-omit [DecidableEq B] in
-lemma parallel_entropy (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool) (S : Finset α) :
-    (∑ i : B × SwitchIndex d,
-      if (parallelUsers pre ω i).1 ∈ S ∧ (parallelUsers pre ω i).2 ∈ S then (1:ℝ) else 0) ≤
-      (S.card : ℝ) * Real.log S.card / (2 * Real.log 2) := by
-  classical
-  rw [Fintype.sum_prod_type]
-  calc
-    _ ≤ ∑ b : B, (blockMarks pre S b).card * Real.log (blockMarks pre S b).card / (2*Real.log 2) := by
-      apply Finset.sum_le_sum
-      intro b _
-      simpa only [←mem_blockMarks,parallelUsers] using
-        marked_switch_entropy_bound d (decodeButterfly d (fun i => ω (b,i))) (blockMarks pre S b)
-    _ ≤ ∑ b : B, (blockMarks pre S b).card * Real.log S.card / (2*Real.log 2) := by
-      apply Finset.sum_le_sum
-      intro b _
-      by_cases hz : (blockMarks pre S b).card = 0
-      · simp [hz]
-      · apply div_le_div_of_nonneg_right _ (by positivity)
-        apply mul_le_mul_of_nonneg_left _ (by positivity)
-        exact Real.log_le_log (by exact_mod_cast Nat.pos_of_ne_zero hz)
-          (by exact_mod_cast card_blockMarks_le pre S b)
-    _ = _ := by rw [←Finset.sum_div,←Finset.sum_mul,←Nat.cast_sum,sum_card_blockMarks]
-
-noncomputable def parallelFollow (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x : α) (k : α → QueryTree (B × SwitchIndex d)) : QueryTree (B × SwitchIndex d) :=
-  QueryTree.followPath d (fun i => ((pre x).1,i)) (pre x).2 (fun y => k (post ((pre x).1,y)))
-
-omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
-lemma parallelFollow_succeeds (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
-    (parallelFollow pre post x k).succeeds ω = (k (relabeledPerm pre post ω x)).succeeds ω := by
-  exact QueryTree.followPath_succeeds _ _ _ _ _
-
-omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
-lemma parallelFollow_raw_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
-    (parallelFollow pre post x k).charge ω = (k (relabeledPerm pre post ω x)).charge ω := by
-  exact QueryTree.followPath_raw_charge _ _ _ _ _
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma parallelFollow_stale_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
-    (ω : B × SwitchIndex d → Bool) :
-    (parallelFollow pre post x k).staleCharge R ω =
-      (k (relabeledPerm pre post ω x)).staleCharge (R ∪ parallelPath pre ω x) ω := by
-  exact QueryTree.followPath_stale_charge _ _ _ _ _ _
-
-noncomputable def closingDomain (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) : Finset (B × SwitchIndex d) :=
-  (routeDomain d (pre x).2 (post.symm y).2).image (fun i => ((pre x).1,i))
-
-def closingValue (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (i : B × SwitchIndex d) : Bool := routeValue d (pre x).2 (post.symm y).2 i.2
-
-omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
-lemma relabeled_endpoint_pair (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (ω : B × SwitchIndex d → Bool) (x y : α) :
-    relabeledPerm pre post ω x = y ↔
-      (pre x).1 = (post.symm y).1 ∧
-      butterflyPerm d (decodeButterfly d (fun i => ω ((pre x).1,i))) (pre x).2 = (post.symm y).2 := by
-  change post ((pre x).1,_) = y ↔ _
-  rw [←post.eq_symm_apply,Prod.mk.injEq]
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma relabeled_endpoint_iff (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (ω : B × SwitchIndex d → Bool) (x y : α) :
-    relabeledPerm pre post ω x = y ↔
-      (pre x).1 = (post.symm y).1 ∧
-        ∀ i ∈ closingDomain pre post x y, ω i = closingValue pre post x y i := by
-  classical
-  rw [relabeled_endpoint_pair,butterfly_endpoint_iff]
-  simp only [closingDomain,Finset.forall_mem_image,closingValue]
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma closingDomain_eq (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (ω : B × SwitchIndex d → Bool) (x y : α) (h : relabeledPerm pre post ω x = y) :
-    closingDomain pre post x y = parallelPath pre ω x := by
-  rw [closingDomain,parallelPath,(relabeled_endpoint_pair _ _ _ _ _).mp h |>.2]
-
-noncomputable def parallelClose (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (k : QueryTree (B × SwitchIndex d)) : QueryTree (B × SwitchIndex d) :=
-  if (pre x).1 = (post.symm y).1 then
-    QueryTree.forceList (closingDomain pre post x y).toList (closingValue pre post x y) k
-  else QueryTree.reject
-
-omit [Fintype α] in
-lemma parallelClose_succeeds (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (k : QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
-    (parallelClose pre post x y k).succeeds ω =
-      if relabeledPerm pre post ω x = y then k.succeeds ω else false := by
-  classical
-  simp only [relabeled_endpoint_iff,parallelClose]
-  by_cases hb : (pre x).1 = (post.symm y).1
-  · rw [ite_eq_left hb,QueryTree.forceList_succeeds]
-    simp only [Finset.mem_toList,hb,true_and]
-  · simp [hb,QueryTree.succeeds]
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma parallelClose_raw_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (k : QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool)
-    (h : relabeledPerm pre post ω x = y) :
-    (parallelClose pre post x y k).charge ω = k.charge ω + d := by
-  have hh := (relabeled_endpoint_iff _ _ _ _ _).mp h
-  rw [parallelClose,ite_eq_left hh.1,QueryTree.forceList_raw_charge _ _ _ _
-    (by simpa only [Finset.mem_toList] using hh.2),Finset.length_toList,
-    closingDomain_eq _ _ _ _ _ h,card_parallelPath]
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma parallelClose_fresh_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (k : QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
-    (ω : B × SwitchIndex d → Bool) (h : relabeledPerm pre post ω x = y) :
-    (parallelClose pre post x y k).freshCharge R ω =
-      k.freshCharge (R ∪ parallelPath pre ω x) ω + (parallelPath pre ω x \ R).card := by
-  have hh := (relabeled_endpoint_iff _ _ _ _ _).mp h
-  rw [parallelClose,ite_eq_left hh.1,QueryTree.forceList_charge _ _ _ _ _
-    (by simpa only [Finset.mem_toList] using hh.2),Finset.toList_toFinset,
-    closingDomain_eq _ _ _ _ _ h]
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma parallelClose_stale_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (x y : α) (k : QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
-    (ω : B × SwitchIndex d → Bool) (h : relabeledPerm pre post ω x = y) :
-    (parallelClose pre post x y k).staleCharge R ω =
-      k.staleCharge (R ∪ parallelPath pre ω x) ω + (parallelPath pre ω x ∩ R).card := by
-  have h₀ := QueryTree.fresh_add_stale (parallelClose pre post x y k) R ω
-  have h₁ := QueryTree.fresh_add_stale k (R ∪ parallelPath pre ω x) ω
-  rw [parallelClose_raw_charge _ _ _ _ _ _ h,parallelClose_fresh_charge _ _ _ _ _ _ _ h] at h₀
-  have h₂ : (parallelPath pre ω x \ R).card + (parallelPath pre ω x ∩ R).card = d := by
-    rw [Finset.card_sdiff_add_card_inter,card_parallelPath]
-  omega
-
-noncomputable def parallelNetwork (pre : α ≃ B × Card d) (post : B × Card d ≃ α) :
-    RoutingNetwork α (B × SwitchIndex d) d where
-  perm := relabeledPerm pre post
-  path := parallelPath pre
-  users := parallelUsers pre
-  users_ne := parallelUsers_ne pre
-  mem_path := mem_parallelPath pre
-  card_path := card_parallelPath pre
-  entropy := parallel_entropy pre
-  follow := parallelFollow pre post
-  close := parallelClose pre post
-  follow_succeeds := parallelFollow_succeeds pre post
-  follow_raw_charge := parallelFollow_raw_charge pre post
-  follow_stale_charge := parallelFollow_stale_charge pre post
-  close_succeeds := parallelClose_succeeds pre post
-  close_raw_charge := parallelClose_raw_charge pre post
-  close_stale_charge := parallelClose_stale_charge pre post
-
-lemma parallel_cycle_log_mgf (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (S : Finset α) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    Real.log (finiteMean (fun ω : B × SwitchIndex d → Bool =>
-      q^(Fintype.card (RoutingNetwork.SelectedCycle (relabeledPerm pre post ω) S)))) ≤
-      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
-  exact (parallelNetwork pre post).butterfly_cycle_log_mgf S J hJ q hq
-
-end ParallelButterfly
-
-def cardSplit : (r s : ℕ) → Card (s+r) ≃ Card r × Card s
-  | 0, s =>
-    { toFun := fun x => ((fun i => Fin.elim0 i), x)
-      invFun := fun x => x.2
-      left_inv := fun _ => rfl
-      right_inv := fun x => by
-        apply Prod.ext
-        · exact Subsingleton.elim _ _
-        · rfl }
-  | r+1, s =>
-    { toFun := fun x => (Fin.cons (x 0) ((cardSplit r s (Fin.tail x)).1),
-        (cardSplit r s (Fin.tail x)).2)
-      invFun := fun x => Fin.cons (x.1 0) ((cardSplit r s).symm (Fin.tail x.1,x.2))
-      left_inv := by
-        intro x
-        simp only [Fin.cons_zero, Fin.tail_cons, Prod.mk.eta, Equiv.symm_apply_apply, Fin.cons_self_tail]
-      right_inv := by
-        intro x
-        simp only [Fin.cons_zero, Fin.tail_cons, Equiv.apply_symm_apply, Fin.cons_self_tail] }
-
-def assembleBits : (r s : ℕ) → (Card r × SwitchIndex s → Bool) →
-    (Card s × SwitchIndex r → Bool) → SwitchIndex (s+r) → Bool
-  | 0, _, lo, _, i => lo ((fun j => Fin.elim0 j), i)
-  | r+1, s, _, hi, Sum.inl y => hi ((cardSplit r s y).2,Sum.inl (cardSplit r s y).1)
-  | r+1, s, lo, hi, Sum.inr (b,i) => assembleBits r s
-      (fun j => lo (Fin.cons b j.1,j.2)) (fun j => hi (j.1,Sum.inr (b,j.2))) i
-
-lemma butterfly_split (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
-    (hi : Card s × SwitchIndex r → Bool) (x : Card (s+r)) :
-    cardSplit r s (butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi)) x) =
-      let y := cardSplit r s x
-      let z := butterflyPerm s (decodeButterfly s (fun i => lo (y.1,i))) y.2
-      (butterflyPerm r (decodeButterfly r (fun i => hi (z,i))) y.1,z) := by
-  induction r with
-  | zero =>
-    apply Prod.ext
-    · funext i
-      exact Fin.elim0 i
-    · rfl
-  | succ r ih =>
-    let lo' : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons (x 0) j.1,j.2)
-    let hi' : Card s × SwitchIndex r → Bool := fun j => hi (j.1,Sum.inr (x 0,j.2))
-    have hh := ih lo' hi' (Fin.tail x)
-    let u := butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo' hi')) (Fin.tail x)
-    let y := cardSplit r s (Fin.tail x)
-    let z := butterflyPerm s (decodeButterfly s (fun i => lo' (y.1,i))) y.2
-    let w := butterflyPerm r (decodeButterfly r (fun i => hi' (z,i))) y.1
-    change cardSplit r s u = (w,z) at hh
-    change ((Fin.cons (Bool.xor (x 0) (hi ((cardSplit r s u).2,Sum.inl (cardSplit r s u).1)))
-        ((cardSplit r s u).1) : Card (r+1)), (cardSplit r s u).2) =
-      ((Fin.cons (Bool.xor (x 0) (hi (z,Sum.inl w))) w : Card (r+1)),z)
-    rw [hh]
-
-def splitSwitch : (r s : ℕ) → SwitchIndex (s+r) ≃
-    (Card r × SwitchIndex s) ⊕ (Card s × SwitchIndex r)
-  | 0, _ =>
-    { toFun := fun i => Sum.inl ((fun j => Fin.elim0 j),i)
-      invFun := fun i => match i with
-        | Sum.inl (_,j) => j
-        | Sum.inr (_,j) => nomatch j
-      left_inv := fun _ => rfl
-      right_inv := by
-        intro i
-        rcases i with ⟨x,j⟩ | ⟨x,j⟩
-        · exact congrArg Sum.inl (Prod.ext (Subsingleton.elim _ _) rfl)
-        · exact Empty.elim j }
-  | r+1, s =>
-    { toFun := fun i => match i with
-        | Sum.inl y => Sum.inr ((cardSplit r s y).2,Sum.inl (cardSplit r s y).1)
-        | Sum.inr (b,j) => match splitSwitch r s j with
-          | Sum.inl (p,j) => Sum.inl (Fin.cons b p,j)
-          | Sum.inr (q,j) => Sum.inr (q,Sum.inr (b,j))
-      invFun := fun i => match i with
-        | Sum.inl (p,j) => Sum.inr (p 0,(splitSwitch r s).symm (Sum.inl (Fin.tail p,j)))
-        | Sum.inr (q,Sum.inl p) => Sum.inl ((cardSplit r s).symm (p,q))
-        | Sum.inr (q,Sum.inr (b,j)) => Sum.inr (b,(splitSwitch r s).symm (Sum.inr (q,j)))
-      left_inv := by
-        intro i
-        rcases i with y | ⟨b,j⟩
-        · dsimp only
-          rw [Prod.mk.eta, Equiv.symm_apply_apply]
-        · dsimp only
-          cases hz : splitSwitch r s j with
-          | inl z =>
-            rcases z with ⟨p,k⟩
-            simp only [Fin.cons_zero, Fin.tail_cons]
-            rw [←hz,Equiv.symm_apply_apply]
-          | inr z =>
-            rcases z with ⟨q,k⟩
-            dsimp only
-            rw [←hz,Equiv.symm_apply_apply]
-      right_inv := by
-        intro i
-        rcases i with ⟨p,j⟩ | ⟨q,j⟩
-        · simp only [Equiv.apply_symm_apply, Fin.cons_self_tail]
-        · rcases j with p | ⟨b,j⟩ <;> simp only [Equiv.apply_symm_apply] }
-
-lemma assembleBits_eq (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
-    (hi : Card s × SwitchIndex r → Bool) (i : SwitchIndex (s+r)) :
-    assembleBits r s lo hi i = Sum.elim lo hi (splitSwitch r s i) := by
-  induction r with
-  | zero => rfl
-  | succ r ih =>
-    rcases i with y | ⟨b,j⟩
-    · rfl
-    · simp only [assembleBits,splitSwitch,Equiv.coe_fn_mk]
-      rw [ih]
-      cases splitSwitch r s j <;> rfl
-
-def splitCoinEquiv (r s : ℕ) :
-    ((Card r × SwitchIndex s → Bool) × (Card s × SwitchIndex r → Bool)) ≃
-      (SwitchIndex (s+r) → Bool) :=
-  (Equiv.sumArrowEquivProdArrow _ _ Bool).symm.trans
-    (Equiv.arrowCongr (splitSwitch r s).symm (Equiv.refl Bool))
-
-lemma splitCoinEquiv_apply (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
-    (hi : Card s × SwitchIndex r → Bool) :
-    splitCoinEquiv r s (lo,hi) = assembleBits r s lo hi := by
-  funext i
-  exact (assembleBits_eq r s lo hi i).symm
-
-lemma finiteMean_prod {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] (f : Ω × Ω' → ℝ) :
-    finiteMean f = finiteMean (fun x => finiteMean (fun y => f (x,y))) := by
-  simp only [finiteMean, Fintype.sum_prod_type, Fintype.card_prod, Nat.cast_mul,
-    ←Finset.sum_div, div_div, mul_comm]
-
-lemma finiteMean_split (r s : ℕ) (f : (SwitchIndex (s+r) → Bool) → ℝ) :
-    finiteMean f = finiteMean (fun lo : Card r × SwitchIndex s → Bool =>
-      finiteMean (fun hi : Card s × SwitchIndex r → Bool => f (assembleBits r s lo hi))) := by
-  rw [finiteMean_equiv (splitCoinEquiv r s).symm]
-  simp only [Equiv.symm_symm]
-  rw [finiteMean_prod]
-  apply finiteMean_congr
-  intro lo
-  apply finiteMean_congr
-  intro hi
-  rw [splitCoinEquiv_apply]
-
-section PartiallyExposed
-variable {α : Type*} {r s : ℕ}
-
-def freshPre (pre : α ≃ Card (s+r)) (lo : Card r × SwitchIndex s → Bool) :
-    α ≃ Card s × Card r :=
-  ((pre.trans (cardSplit r s)).trans (parallelPerm lo)).trans (Equiv.prodComm _ _)
-
-def freshPost (post : Card (s+r) ≃ α) : Card s × Card r ≃ α :=
-  ((Equiv.prodComm _ _).trans (cardSplit r s).symm).trans post
-
-lemma exposed_perm_eq (pre : α ≃ Card (s+r)) (post : Card (s+r) ≃ α)
-    (lo : Card r × SwitchIndex s → Bool) (hi : Card s × SwitchIndex r → Bool) :
-    (pre.trans (butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi)))).trans post =
-      relabeledPerm (freshPre pre lo) (freshPost post) hi := by
-  ext x
-  simpa only [relabeledPerm, freshPre, freshPost, parallelPerm, Equiv.trans_apply,
-    Equiv.coe_fn_mk, Equiv.prodComm_apply, Prod.swap, Equiv.symm_apply_apply] using
-    congrArg (fun y => post ((cardSplit r s).symm y)) (butterfly_split r s lo hi (pre x))
-
-lemma partially_exposed_cycle_log_mgf [Fintype α] [DecidableEq α]
-    (pre : α ≃ Card (s+r)) (post : Card (s+r) ≃ α)
-    (lo : Card r × SwitchIndex s → Bool) (S : Finset α)
-    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    Real.log (finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(Fintype.card (RoutingNetwork.SelectedCycle
-        ((pre.trans (butterflyPerm (s+r)
-          (decodeButterfly (s+r) (assembleBits r s lo hi)))).trans post) S)))) ≤
-      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^r) := by
-  simp only [exposed_perm_eq]
-  exact parallel_cycle_log_mgf (freshPre pre lo) (freshPost post) S J hJ q hq
-
-end PartiallyExposed
-
-lemma inclusionMoment_eq {α Ω : Type*} [DecidableEq α] [Fintype Ω]
-    (F : Ω → Finset α) (S : Finset α) :
-    inclusionMoment F S = finiteMean (fun ω => if S ⊆ F ω then (1:ℝ) else 0) := by
-  unfold inclusionMoment
-  apply finiteMean_congr
-  intro ω
-  split_ifs <;> rfl
-
-lemma finiteMean_bool (f : Bool → ℝ) : finiteMean f = (f false+f true)/2 := by
-  simp only [finiteMean, Fintype.sum_bool, Fintype.card_bool, Nat.cast_ofNat]
-  ring
-
-lemma finiteMean_const {Ω : Type*} [Fintype Ω] [Nonempty Ω] (c : ℝ) :
-    finiteMean (fun _ : Ω => c) = c := by
-  simp only [finiteMean,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
-  exact mul_div_cancel_left₀ c (Nat.cast_ne_zero.mpr Fintype.card_ne_zero)
-
-lemma finiteMean_pi_prod {ι : Type*} [Fintype ι] [DecidableEq ι] {Ω : ι → Type*}
-    [∀ i, Fintype (Ω i)] (f : (i : ι) → Ω i → ℝ) :
-    finiteMean (fun ω : (i : ι) → Ω i => ∏ i, f i (ω i)) = ∏ i, finiteMean (f i) := by
-  simp only [finiteMean,←Fintype.prod_sum,Fintype.card_pi,Nat.cast_prod,Finset.prod_div_distrib]
-
-section ParallelFairLayer
-variable {β : Type*} [Fintype β] [DecidableEq β]
-
-def pairLayer (ξ : β → Bool) : Equiv.Perm (Bool × β) where
-  toFun x := (Bool.xor x.1 (ξ x.2),x.2)
-  invFun x := (Bool.xor x.1 (ξ x.2),x.2)
-  left_inv x := by
-    rcases x with ⟨b,x⟩
-    change (Bool.xor (Bool.xor b (ξ x)) (ξ x),x) = (b,x)
-    cases b <;> cases ξ x <;> rfl
-  right_inv x := by
-    rcases x with ⟨b,x⟩
-    change (Bool.xor (Bool.xor b (ξ x)) (ξ x),x) = (b,x)
-    cases b <;> cases ξ x <;> rfl
-
-noncomputable def pairFiber (S : Finset (Bool × β)) (b : β) : Finset Bool :=
-  Finset.univ.filter (fun ε => (ε,b) ∈ S)
-
-lemma prod_pairFiber (S : Finset (Bool × β)) (f : Bool × β → ℝ) :
-    (∏ x ∈ S, f x) = ∏ b : β, ∏ ε ∈ pairFiber S b, f (ε,b) := by
-  classical
-  simp only [pairFiber,Finset.prod_filter]
-  rw [Finset.prod_comm,←Fintype.prod_prod_type (fun x : Bool × β => if x ∈ S then f x else 1)]
-  rw [←Finset.prod_filter]
-  congr 1
-  ext x
-  simp
-
-lemma fiber_average_bound (S : Finset Bool) (p : Bool → ℝ) (hp : ∀ b, 0 ≤ p b) :
-    finiteMean (fun c : Bool => ∏ ε ∈ S, p (Bool.xor ε c)) ≤
-      ∏ ε ∈ S, (p ε + p (!ε))/2 := by
-  rw [finiteMean_bool]
-  have hxor : ∀ ε : Bool, Bool.xor ε true = Equiv.swap false true ε := by decide
-  simp only [Bool.xor_false,hxor]
-  calc
-    _ ≤ ∏ ε ∈ S, (p ε+p (Equiv.swap false true ε))/2 :=
-      swap_average_product S p hp false true
-    _ = _ := Finset.prod_congr rfl (fun ε _ => by cases ε <;> rfl)
-
-lemma CylinderBound.fair_parallel {m : Finset (Bool × β) → ℝ} {p : Bool × β → ℝ}
-    (h : CylinderBound m p) (hp : ∀ x, 0 ≤ p x) :
-    CylinderBound (fun S => finiteMean (fun ξ : β → Bool => m (S.image (pairLayer ξ))))
-      (fun x => (p x+p (!x.1,x.2))/2) := by
-  intro S
-  calc
-    _ ≤ finiteMean (fun ξ : β → Bool => ∏ x ∈ S, p (pairLayer ξ x)) := by
-      apply finiteMean_mono
-      intro ξ
-      simpa only [Finset.prod_image (pairLayer ξ).injective.injOn] using h (S.image (pairLayer ξ))
-    _ = ∏ b : β, finiteMean (fun c : Bool => ∏ ε ∈ pairFiber S b, p (Bool.xor ε c,b)) := by
-      calc
-        _ = finiteMean (fun ξ : β → Bool => ∏ b : β,
-            ∏ ε ∈ pairFiber S b, p (Bool.xor ε (ξ b),b)) :=
-          finiteMean_congr (fun ξ => prod_pairFiber S (fun x => p (pairLayer ξ x)))
-        _ = _ := finiteMean_pi_prod (fun b c => ∏ ε ∈ pairFiber S b, p (Bool.xor ε c,b))
-    _ ≤ ∏ b : β, ∏ ε ∈ pairFiber S b, (p (ε,b)+p (!ε,b))/2 := by
-      apply Finset.prod_le_prod₀
-      · intro b _
-        exact div_nonneg (Finset.sum_nonneg (fun c _ =>
-          Finset.prod_nonneg (fun ε _ => hp (Bool.xor ε c,b)))) (Nat.cast_nonneg _)
-      · intro b _
-        exact fiber_average_bound (pairFiber S b) (fun ε => p (ε,b)) (fun ε => hp (ε,b))
-    _ = _ := (prod_pairFiber S (fun x => (p x+p (!x.1,x.2))/2)).symm
-
-omit [Fintype β] in
-lemma subset_pairLayer_image_iff (S A : Finset (Bool × β)) (ξ : β → Bool) :
-    S ⊆ A.image (pairLayer ξ) ↔ S.image (pairLayer ξ) ⊆ A := by
-  have hi (x : Bool × β) : pairLayer ξ (pairLayer ξ x) = x := (pairLayer ξ).left_inv x
-  constructor
-  · intro h
-    have h' := Finset.image_subset_image h (f := pairLayer ξ)
-    simpa only [Finset.image_image,Function.comp_def,hi,Finset.image_id'] using h'
-  · intro h
-    have h' := Finset.image_subset_image h (f := pairLayer ξ)
-    simpa only [Finset.image_image,Function.comp_def,hi,Finset.image_id'] using h'
-
-noncomputable def fairSetLayer {Ω : Type*} (F : Ω → Finset (Bool × β)) :
-    Ω × (β → Bool) → Finset (Bool × β) := fun ω => (F ω.1).image (pairLayer ω.2)
-
-lemma finiteMean_comm {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] (f : Ω → Ω' → ℝ) :
-    finiteMean (fun x => finiteMean (f x)) = finiteMean (fun y => finiteMean (fun x => f x y)) := by
-  simp only [finiteMean,←Finset.sum_div,div_div]
-  rw [Finset.sum_comm,mul_comm]
-
-lemma inclusionMoment_fairSetLayer {Ω : Type*} [Fintype Ω]
-    (F : Ω → Finset (Bool × β)) (S : Finset (Bool × β)) :
-    inclusionMoment (fairSetLayer F) S =
-      finiteMean (fun ξ : β → Bool => inclusionMoment F (S.image (pairLayer ξ))) := by
-  classical
-  simp only [inclusionMoment_eq]
-  have he : (fun ω : Ω × (β → Bool) => if S ⊆ fairSetLayer F ω then (1:ℝ) else 0) =
-      (fun ω : Ω × (β → Bool) => if S.image (pairLayer ω.2) ⊆ F ω.1 then (1:ℝ) else 0) := by
-    funext ω
-    simp only [fairSetLayer,subset_pairLayer_image_iff]
-  apply (congrArg finiteMean he).trans
-  exact (finiteMean_prod (fun ω : Ω × (β → Bool) =>
-    if S.image (pairLayer ω.2) ⊆ F ω.1 then (1:ℝ) else 0)).trans
-      (finiteMean_comm (fun ω ξ => if S.image (pairLayer ξ) ⊆ F ω then (1:ℝ) else 0))
-
-lemma inclusionMoment_fairSetLayer_singleton {Ω : Type*} [Fintype Ω]
-    (F : Ω → Finset (Bool × β)) (x : Bool × β) :
-    inclusionMoment (fairSetLayer F) {x} =
-      (inclusionMoment F {x}+inclusionMoment F {(!x.1,x.2)})/2 := by
-  rw [inclusionMoment_fairSetLayer]
-  simp only [Finset.image_singleton,pairLayer,Equiv.coe_fn_mk]
-  rw [finiteMean_pi_eval (A := fun _ : β => Bool) x.2 (fun c : Bool => inclusionMoment F {(Bool.xor x.1 c,x.2)})]
-  rw [finiteMean_bool]
-  rcases x with ⟨b,x⟩
-  cases b <;> simp only [Bool.xor_false,Bool.false_xor,Bool.true_xor, Bool.not_false,Bool.not_true]
-
-lemma cylinder_fairSetLayer {Ω : Type*} [Fintype Ω]
-    (F : Ω → Finset (Bool × β))
-    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
-    CylinderBound (inclusionMoment (fairSetLayer F))
-      (fun x => inclusionMoment (fairSetLayer F) {x}) := by
-  have hh := h.fair_parallel (fun x => inclusionMoment_nonneg F {x})
-  intro S
-  rw [inclusionMoment_fairSetLayer]
-  simp only [inclusionMoment_fairSetLayer_singleton]
-  exact hh S
-
-end ParallelFairLayer
-
-section RelabelMarks
-variable {α β Ω Ω' : Type*} [DecidableEq α] [DecidableEq β] [Fintype Ω] [Fintype Ω']
-
-omit [Fintype Ω] [Fintype Ω'] in
-lemma subset_image_equiv_iff (e : α ≃ β) (A : Finset α) (S : Finset β) :
-    S ⊆ A.image e ↔ S.image e.symm ⊆ A := by
-  constructor
-  · intro h x hx
-    obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
-    obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp (h hy)
-    simpa only [←hxy,Equiv.symm_apply_apply] using hx
-  · intro h y hy
-    exact Finset.mem_image.mpr ⟨e.symm y,h (Finset.mem_image_of_mem _ hy),e.apply_symm_apply y⟩
-
-omit [Fintype Ω'] in
-lemma inclusionMoment_image_equiv (e : α ≃ β) (F : Ω → Finset α) (S : Finset β) :
-    inclusionMoment (fun ω => (F ω).image e) S = inclusionMoment F (S.image e.symm) := by
-  classical
-  simp only [inclusionMoment_eq]
-  apply finiteMean_congr
-  intro ω
-  simp only [subset_image_equiv_iff]
-
-omit [Fintype Ω'] in
-lemma cylinder_image_equiv (e : α ≃ β) (F : Ω → Finset α)
-    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
-    CylinderBound (inclusionMoment (fun ω => (F ω).image e))
-      (fun x => inclusionMoment (fun ω => (F ω).image e) {x}) := by
-  intro S
-  simp only [inclusionMoment_image_equiv,Finset.image_singleton]
-  have hh := h (S.image e.symm)
-  simpa only [Finset.prod_image e.symm.injective.injOn] using hh
-
-omit [DecidableEq β] in
-lemma inclusionMoment_comp_equiv (e : Ω ≃ Ω') (F : Ω' → Finset α) (S : Finset α) :
-    inclusionMoment (fun ω => F (e ω)) S = inclusionMoment F S := by
-  classical
-  simp only [inclusionMoment_eq]
-  simpa only [Equiv.apply_symm_apply] using
-    finiteMean_equiv e (fun ω => if S ⊆ F (e ω) then (1 : ℝ) else 0)
-
-omit [DecidableEq β] in
-lemma cylinder_comp_equiv (e : Ω ≃ Ω') (F : Ω' → Finset α)
-    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
-    CylinderBound (inclusionMoment (fun ω => F (e ω)))
-      (fun x => inclusionMoment (fun ω => F (e ω)) {x}) := by
-  intro S
-  simpa only [inclusionMoment_comp_equiv] using h S
-end RelabelMarks
-
-lemma finiteMean_prod_mul {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
-    (f : Ω → ℝ) (g : Ω' → ℝ) :
-    finiteMean (fun ω : Ω × Ω' => f ω.1*g ω.2) = finiteMean f * finiteMean g := by
-  simp only [finiteMean,Fintype.sum_prod_type,Fintype.card_prod,Nat.cast_mul,
-    ←Finset.mul_sum,←Finset.sum_mul]
-  ring
-
-section JoinChildren
-variable {β : Type*} [DecidableEq β]
-
-noncomputable def childJoin (A B : Finset β) : Finset (Bool × β) :=
-  A.image (fun x => (false,x)) ∪ B.image (fun x => (true,x))
-
-lemma mem_childJoin (A B : Finset β) (b : Bool) (x : β) :
-    (b,x) ∈ childJoin A B ↔ (if b then x ∈ B else x ∈ A) := by
-  cases b <;> simp [childJoin]
-
-noncomputable def boolFiber (S : Finset (Bool × β)) (b : Bool) : Finset β :=
-  (S.filter (fun x => x.1 = b)).image Prod.snd
-
-lemma mem_boolFiber (S : Finset (Bool × β)) (b : Bool) (x : β) :
-    x ∈ boolFiber S b ↔ (b,x) ∈ S := by
-  constructor
-  · intro h
-    obtain ⟨⟨c,y⟩,hy,hxy⟩ := Finset.mem_image.mp h
-    obtain ⟨hy,hc⟩ := Finset.mem_filter.mp hy
-    change c = b at hc
-    change y = x at hxy
-    simpa only [hc,hxy] using hy
-  · intro h
-    exact Finset.mem_image.mpr ⟨(b,x),Finset.mem_filter.mpr ⟨h,rfl⟩,rfl⟩
-
-lemma childJoin_boolFiber (S : Finset (Bool × β)) :
-    childJoin (boolFiber S false) (boolFiber S true) = S := by
-  ext ⟨b,x⟩
-  rw [mem_childJoin]
-  cases b <;> simp only [Bool.false_eq_true,↓reduceIte,mem_boolFiber]
-
-lemma subset_childJoin_iff (S : Finset (Bool × β)) (A B : Finset β) :
-    S ⊆ childJoin A B ↔ boolFiber S false ⊆ A ∧ boolFiber S true ⊆ B := by
-  constructor
-  · intro h
-    constructor
-    · intro x hx
-      exact (mem_childJoin A B false x).mp (h ((mem_boolFiber S false x).mp hx))
-    · intro x hx
-      exact (mem_childJoin A B true x).mp (h ((mem_boolFiber S true x).mp hx))
-  · rintro ⟨h₀,h₁⟩ ⟨b,x⟩ hx
-    rw [mem_childJoin]
-    cases b
-    · exact h₀ ((mem_boolFiber S false x).mpr hx)
-    · exact h₁ ((mem_boolFiber S true x).mpr hx)
-
-lemma prod_childJoin (A B : Finset β) (f : Bool × β → ℝ) :
-    (∏ x ∈ childJoin A B, f x) = (∏ x ∈ A, f (false,x)) * ∏ x ∈ B, f (true,x) := by
-  classical
-  have hd : Disjoint (A.image (fun x => (false,x))) (B.image (fun x => (true,x))) := by
-    apply Finset.disjoint_left.mpr
-    rintro x hx hy
-    obtain ⟨a,ha,rfl⟩ := Finset.mem_image.mp hx
-    obtain ⟨b,hb,hba⟩ := Finset.mem_image.mp hy
-    exact Bool.false_ne_true (congrArg Prod.fst hba).symm
-  rw [childJoin,Finset.prod_union hd,
-    Finset.prod_image (fun _ _ _ _ h => congrArg Prod.snd h),
-    Finset.prod_image (fun _ _ _ _ h => congrArg Prod.snd h)]
-
-lemma prod_boolFibers (S : Finset (Bool × β)) (f : Bool × β → ℝ) :
-    (∏ x ∈ S, f x) = (∏ x ∈ boolFiber S false, f (false,x)) *
-      ∏ x ∈ boolFiber S true, f (true,x) := by
-  rw [←prod_childJoin,childJoin_boolFiber]
-
-lemma inclusionMoment_childJoin {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
-    (F : Ω → Finset β) (G : Ω' → Finset β) (S : Finset (Bool × β)) :
-    inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) S =
-      inclusionMoment F (boolFiber S false) * inclusionMoment G (boolFiber S true) := by
-  classical
-  simp only [inclusionMoment_eq]
-  rw [←finiteMean_prod_mul]
-  apply finiteMean_congr
-  intro ω
-  simp only [subset_childJoin_iff]
-  split_ifs <;> simp_all
-
-lemma cylinder_childJoin {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] [Nonempty Ω] [Nonempty Ω']
-    (F : Ω → Finset β) (G : Ω' → Finset β)
-    (hF : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x}))
-    (hG : CylinderBound (inclusionMoment G) (fun x => inclusionMoment G {x})) :
-    CylinderBound (inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)))
-      (fun x => inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {x}) := by
-  intro S
-  rw [inclusionMoment_childJoin,prod_boolFibers]
-  have hf (b : Bool) (x : β) : boolFiber {(b,x)} b = {x} := by
-    ext y
-    simp only [mem_boolFiber,Finset.mem_singleton,Prod.mk.injEq,true_and]
-  have hn (b : Bool) (x : β) : boolFiber {(!b,x)} b = ∅ := by
-    ext y
-    cases b <;> simp [mem_boolFiber]
-  have h₀ (x : β) : inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {(false,x)} =
-      inclusionMoment F {x} * inclusionMoment G ∅ := by
-    rw [inclusionMoment_childJoin,hf]
-    have hh := hn true x
-    exact congrArg (fun T => inclusionMoment F {x} * inclusionMoment G T) hh
-  have h₁ (x : β) : inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {(true,x)} =
-      inclusionMoment F ∅ * inclusionMoment G {x} := by
-    rw [inclusionMoment_childJoin,hf]
-    have hh := hn false x
-    exact congrArg (fun T => inclusionMoment F T * inclusionMoment G {x}) hh
-  have emF : inclusionMoment F ∅ = 1 := by simp [inclusionMoment,finiteMean_const]
-  have emG : inclusionMoment G ∅ = 1 := by simp [inclusionMoment,finiteMean_const]
-  simp only [h₀,h₁,emF,emG,mul_one,one_mul]
-  exact mul_le_mul (hF _) (hG _) (inclusionMoment_nonneg _ _)
-    (Finset.prod_nonneg (fun x _ => inclusionMoment_nonneg F {x}))
-end JoinChildren
-
-def coinStepEquiv (d : ℕ) : (SwitchIndex (d+1) → Bool) ≃
-    ((SwitchIndex d → Bool) × (SwitchIndex d → Bool)) × (Card d → Bool) where
-  toFun ω := ((fun i => ω (Sum.inr (false,i)),fun i => ω (Sum.inr (true,i))),
-    fun x => ω (Sum.inl x))
-  invFun ω := Sum.elim ω.2 (fun bi => if bi.1 then ω.1.2 bi.2 else ω.1.1 bi.2)
-  left_inv ω := by
-    funext i
-    cases i with
-    | inl x => rfl
-    | inr bi => cases bi with | mk b i => cases b <;> rfl
-  right_inv ω := by
-    rcases ω with ⟨⟨ω₀,ω₁⟩,ξ⟩
-    rfl
-
-def headTailEquiv (d : ℕ) : Card (d+1) ≃ Bool × Card d :=
-  (Fin.consEquiv (fun _ : Fin (d+1) => Bool)).symm
-
-lemma childMarks_join (d : ℕ) (A : Finset (Card (d+1))) :
-    A = (childMarks A false).image (Fin.cons false) ∪
-      (childMarks A true).image (Fin.cons true) := by
-  classical
-  ext x
-  constructor
-  · intro hx
-    cases hb : x 0
-    · exact Finset.mem_union_left _ (Finset.mem_image.mpr
-        ⟨Fin.tail x,(mem_childMarks _ _ _).mpr (by simpa only [←hb,Fin.cons_self_tail] using hx),
-          by simp only [←hb,Fin.cons_self_tail]⟩)
-    · exact Finset.mem_union_right _ (Finset.mem_image.mpr
-        ⟨Fin.tail x,(mem_childMarks _ _ _).mpr (by simpa only [←hb,Fin.cons_self_tail] using hx),
-          by simp only [←hb,Fin.cons_self_tail]⟩)
-  · intro hx
-    rcases Finset.mem_union.mp hx with hx | hx
-    · obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
-      exact (mem_childMarks _ _ _).mp hy
-    · obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
-      exact (mem_childMarks _ _ _).mp hy
-
-noncomputable def butterflyMarks (d : ℕ) (A : Finset (Card d))
-    (ω : SwitchIndex d → Bool) : Finset (Card d) :=
-  A.image (butterflyPerm d (decodeButterfly d ω))
-
-lemma butterflyMarks_succ (d : ℕ) (A : Finset (Card (d+1))) (ω : SwitchIndex (d+1) → Bool) :
-    butterflyMarks (d+1) A ω =
-      (fairSetLayer (fun p : (SwitchIndex d → Bool) × (SwitchIndex d → Bool) =>
-        childJoin (butterflyMarks d (childMarks A false) p.1)
-          (butterflyMarks d (childMarks A true) p.2)) (coinStepEquiv d ω)).image
-            (headTailEquiv d).symm := by
-  classical
-  unfold butterflyMarks fairSetLayer childJoin
-  rw [Finset.image_union,Finset.image_union]
-  simp only [Finset.image_image]
-  conv_lhs => rw [childMarks_join d A]
-  rw [Finset.image_union]
-  simp only [Finset.image_image]
-  apply congrArg₂ (· ∪ ·)
-  · apply Finset.image_congr
-    intro x _
-    simp only [Function.comp_def,butterflyPerm,decodeButterfly,Equiv.Perm.coe_mul,
-      childLift,pairSwitch,switchFun,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
-      coinStepEquiv,headTailEquiv,Equiv.symm_symm,pairLayer]
-    rfl
-  · apply Finset.image_congr
-    intro x _
-    simp only [Function.comp_def,butterflyPerm,decodeButterfly,Equiv.Perm.coe_mul,
-      childLift,pairSwitch,switchFun,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
-      coinStepEquiv,headTailEquiv,Equiv.symm_symm,pairLayer]
-    rfl
-
-lemma butterflyMarks_cylinder (d : ℕ) (A : Finset (Card d)) :
-    CylinderBound (inclusionMoment (butterflyMarks d A))
-      (fun x => inclusionMoment (butterflyMarks d A) {x}) := by
-  induction d with
-  | zero =>
-    have he : butterflyMarks 0 A = fun _ : SwitchIndex 0 → Bool => A := by
-      funext ω
-      simp [butterflyMarks,butterflyPerm]
-    rw [he]
-    exact cylinder_const A
-  | succ d ih =>
-    have h := cylinder_childJoin (butterflyMarks d (childMarks A false))
-      (butterflyMarks d (childMarks A true)) (ih _) (ih _)
-    have hh := cylinder_image_equiv (headTailEquiv d).symm _ (cylinder_fairSetLayer _ h)
-    have hs := cylinder_comp_equiv (coinStepEquiv d) _ hh
-    simpa only [←butterflyMarks_succ] using hs
-
-lemma indicator_mem_image_perm {α : Type*} [DecidableEq α]
-    (A : Finset α) (p : Equiv.Perm α) (y : α) :
-    (if y ∈ A.image p then (1 : ℝ) else 0) = ∑ x ∈ A, if p x = y then 1 else 0 := by
-  classical
-  by_cases h : y ∈ A.image p
-  · obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp h
-    rw [ite_eq_left (Finset.mem_image.mpr ⟨x,hx,hxy⟩)]
-    symm
-    calc
-      _ = (if p x = y then (1:ℝ) else 0) := Finset.sum_eq_single_of_mem x hx (by
-        intro z _ hzx
-        exact ite_eq_right (fun hz => hzx (p.injective (hz.trans hxy.symm))))
-      _ = 1 := ite_eq_left hxy
-  · rw [ite_eq_right h]
-    symm
-    apply Finset.sum_eq_zero
-    intro x hx
-    exact ite_eq_right (fun hxy => h (Finset.mem_image.mpr ⟨x,hx,hxy⟩))
-
-lemma butterflyMarks_singleton (d : ℕ) (A : Finset (Card d)) (y : Card d) :
-    inclusionMoment (butterflyMarks d A) {y} = (A.card : ℝ)/(2:ℝ)^d := by
-  classical
-  unfold inclusionMoment
-  simp only [Finset.singleton_subset_iff,butterflyMarks,indicator_mem_image_perm]
-  rw [finiteMean_sum]
-  simp only [butterfly_single_card_uniform,Finset.sum_const,nsmul_eq_mul]
-  ring
-
-lemma butterflyMarks_inclusion (d : ℕ) (A S : Finset (Card d)) :
-    inclusionMoment (butterflyMarks d A) S ≤ ((A.card : ℝ)/(2:ℝ)^d)^S.card := by
-  have h := butterflyMarks_cylinder d A S
-  simpa only [butterflyMarks_singleton,Finset.prod_const] using h
-
-namespace CycleColoring
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-abbrev Coloring (p : Equiv.Perm α) (S : Finset α) :=
-  {f : α → Bool // (∀ x, x ∉ S → f x = false) ∧
-    ∀ x ∈ S, p x ∈ S → f (p x) = f x}
-
-noncomputable instance (p : Equiv.Perm α) (S : Finset α) : Fintype (Coloring p S) :=
-  Fintype.ofFinite _
-
-noncomputable def boundary (p : Equiv.Perm α) (S : Finset α) : Finset α :=
-  S.filter (fun x => p.symm x ∉ S)
-
-noncomputable def root {p : Equiv.Perm α} {S : Finset α}
-    (C : RoutingNetwork.SelectedCycle p S) : α := C.nonempty.choose
-
-lemma root_mem {p : Equiv.Perm α} {S : Finset α} (C : RoutingNetwork.SelectedCycle p S) :
-    root C ∈ C.val := C.nonempty.choose_spec
-
-abbrev Seed (p : Equiv.Perm α) (S : Finset α) :=
-  {x // x ∈ boundary p S} ⊕ RoutingNetwork.SelectedCycle p S
-
-noncomputable def record {p : Equiv.Perm α} {S : Finset α}
-    (f : Coloring p S) : Seed p S → Bool :=
-  Sum.elim (fun x => f.val x.val) (fun C => f.val (root C))
-
-lemma orbit_subset_of_predecessor_closed (p : Equiv.Perm α) (D : Finset α)
-    (hD : ∀ x ∈ D, p.symm x ∈ D) {x : α} (hx : x ∈ D) : orbitSet p x ⊆ D := by
-  intro y hy
-  have hh := ((mem_orbitSet p x y).mp hy).inv
-  obtain ⟨n,hn⟩ := hh.exists_nat_pow_eq
-  have hp (k : ℕ) : (p⁻¹ ^ k) x ∈ D := by
-    induction k with
-    | zero => simpa using hx
-    | succ k ih => simpa only [pow_succ',Equiv.Perm.mul_apply,Equiv.Perm.coe_inv] using hD _ ih
-  exact hn ▸ hp n
-
-lemma record_injective (p : Equiv.Perm α) (S : Finset α) :
-    Function.Injective (record (p := p) (S := S)) := by
-  classical
-  intro f g hfg
-  apply Subtype.ext
-  funext x
-  by_contra hne
-  have hx : x ∈ S := by
-    by_contra h
-    exact hne ((f.property.1 x h).trans (g.property.1 x h).symm)
-  let D := S.filter (fun y => f.val y ≠ g.val y)
-  have hxD : x ∈ D := Finset.mem_filter.mpr ⟨hx,hne⟩
-  have hprev (y : α) (hy : y ∈ D) : p.symm y ∈ D := by
-    obtain ⟨hyS,hyne⟩ := Finset.mem_filter.mp hy
-    have hpS : p.symm y ∈ S := by
-      by_contra hn
-      have hseed : y ∈ boundary p S := Finset.mem_filter.mpr ⟨hyS,hn⟩
-      exact hyne (congrFun hfg (Sum.inl ⟨y,hseed⟩))
-    refine Finset.mem_filter.mpr ⟨hpS,?_⟩
-    have hf := f.property.2 (p.symm y) hpS (by simpa only [Equiv.apply_symm_apply] using hyS)
-    have hg := g.property.2 (p.symm y) hpS (by simpa only [Equiv.apply_symm_apply] using hyS)
-    simp only [Equiv.apply_symm_apply] at hf hg
-    intro he
-    exact hyne (hf.trans (he.trans hg.symm))
-  have horb := orbit_subset_of_predecessor_closed p D hprev hxD
-  let C : RoutingNetwork.SelectedCycle p S :=
-    ⟨orbitSet p x,⟨x,rfl⟩,fun y hy => (Finset.mem_filter.mp (horb hy)).1⟩
-  have hrD : root C ∈ D := horb (root_mem C)
-  have hrne := (Finset.mem_filter.mp hrD).2
-  exact hrne (congrFun hfg (Sum.inr C))
-
-lemma card_coloring_le (p : Equiv.Perm α) (S : Finset α) :
-    Fintype.card (Coloring p S) ≤
-      2 ^ ((boundary p S).card + Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
-  have h := Fintype.card_le_of_injective (record (p := p) (S := S)) (record_injective p S)
-  simpa only [Fintype.card_fun,Fintype.card_sum,Fintype.card_coe,Fintype.card_bool] using h
-
-omit [Fintype α] in
-lemma boundary_card (p : Equiv.Perm α) (S : Finset α) :
-    (boundary p S).card = S.card - (S.filter (fun x => p.symm x ∈ S)).card := by
-  classical
-  have h := Finset.card_filter_add_card_filter_not (s := S) (fun x => p.symm x ∈ S)
-  unfold boundary
-  omega
-
-lemma card_coloring_le_edges (p : Equiv.Perm α) (S : Finset α) :
-    Fintype.card (Coloring p S) ≤
-      2 ^ (S.card - (S.filter (fun x => p.symm x ∈ S)).card +
-        Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
-  simpa only [boundary_card] using card_coloring_le p S
-
-
-abbrev OppositeColoring (p : Equiv.Perm α) (S : Finset α) :=
-  {f : α → Bool // (∀ x, x ∉ S → f x = false) ∧
-    ∀ x ∈ S, p x ∈ S → f (p x) = !(f x)}
-
-noncomputable instance (p : Equiv.Perm α) (S : Finset α) : Fintype (OppositeColoring p S) :=
-  Fintype.ofFinite _
-
-noncomputable def bipartitionFlip (p : Equiv.Perm α) (S : Finset α)
-    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x))
-    (f : OppositeColoring p S) : Coloring p S := by
-  classical
-  refine ⟨(fun x => if x ∈ S then Bool.xor (f.val x) (χ x) else false),?_,?_⟩
-  · intro x hx
-    exact ite_eq_right hx
-  · intro x hx hpx
-    simp only [ite_eq_left hx,ite_eq_left hpx,f.property.2 x hx hpx,hχ x]
-    cases f.val x <;> cases χ x <;> rfl
-
-end CycleColoring
-end Thorp
-
-end ThorpNine.Tail
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Tail.PriorPaths
+
+namespace OAI
+
+namespace ThorpNine.Tail
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+section ParallelButterfly
+variable {α B : Type*} [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] {d : ℕ}
+omit [DecidableEq B] in
+lemma parallel_entropy (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool) (S : Finset α) :
+    (∑ i : B × SwitchIndex d,
+      if (parallelUsers pre ω i).1 ∈ S ∧ (parallelUsers pre ω i).2 ∈ S then (1:ℝ) else 0) ≤
+      (S.card : ℝ) * Real.log S.card / (2 * Real.log 2) := by
+  classical
+  rw [Fintype.sum_prod_type]
+  calc
+    _ ≤ ∑ b : B, (blockMarks pre S b).card * Real.log (blockMarks pre S b).card / (2*Real.log 2) := by
+      apply Finset.sum_le_sum
+      intro b _
+      simpa only [←mem_blockMarks,parallelUsers] using
+        marked_switch_entropy_bound d (decodeButterfly d (fun i => ω (b,i))) (blockMarks pre S b)
+    _ ≤ ∑ b : B, (blockMarks pre S b).card * Real.log S.card / (2*Real.log 2) := by
+      apply Finset.sum_le_sum
+      intro b _
+      by_cases hz : (blockMarks pre S b).card = 0
+      · simp [hz]
+      · apply div_le_div_of_nonneg_right _ (by positivity)
+        apply mul_le_mul_of_nonneg_left _ (by positivity)
+        exact Real.log_le_log (by exact_mod_cast Nat.pos_of_ne_zero hz)
+          (by exact_mod_cast card_blockMarks_le pre S b)
+    _ = _ := by rw [←Finset.sum_div,←Finset.sum_mul,←Nat.cast_sum,sum_card_blockMarks]
+
+noncomputable def parallelFollow (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x : α) (k : α → QueryTree (B × SwitchIndex d)) : QueryTree (B × SwitchIndex d) :=
+  QueryTree.followPath d (fun i => ((pre x).1,i)) (pre x).2 (fun y => k (post ((pre x).1,y)))
+
+omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
+lemma parallelFollow_succeeds (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
+    (parallelFollow pre post x k).succeeds ω = (k (relabeledPerm pre post ω x)).succeeds ω := by
+  exact QueryTree.followPath_succeeds _ _ _ _ _
+
+omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
+lemma parallelFollow_raw_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
+    (parallelFollow pre post x k).charge ω = (k (relabeledPerm pre post ω x)).charge ω := by
+  exact QueryTree.followPath_raw_charge _ _ _ _ _
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma parallelFollow_stale_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x : α) (k : α → QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
+    (ω : B × SwitchIndex d → Bool) :
+    (parallelFollow pre post x k).staleCharge R ω =
+      (k (relabeledPerm pre post ω x)).staleCharge (R ∪ parallelPath pre ω x) ω := by
+  exact QueryTree.followPath_stale_charge _ _ _ _ _ _
+
+noncomputable def closingDomain (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) : Finset (B × SwitchIndex d) :=
+  (routeDomain d (pre x).2 (post.symm y).2).image (fun i => ((pre x).1,i))
+
+def closingValue (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (i : B × SwitchIndex d) : Bool := routeValue d (pre x).2 (post.symm y).2 i.2
+
+omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
+lemma relabeled_endpoint_pair (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (ω : B × SwitchIndex d → Bool) (x y : α) :
+    relabeledPerm pre post ω x = y ↔
+      (pre x).1 = (post.symm y).1 ∧
+      butterflyPerm d (decodeButterfly d (fun i => ω ((pre x).1,i))) (pre x).2 = (post.symm y).2 := by
+  change post ((pre x).1,_) = y ↔ _
+  rw [←post.eq_symm_apply,Prod.mk.injEq]
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma relabeled_endpoint_iff (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (ω : B × SwitchIndex d → Bool) (x y : α) :
+    relabeledPerm pre post ω x = y ↔
+      (pre x).1 = (post.symm y).1 ∧
+        ∀ i ∈ closingDomain pre post x y, ω i = closingValue pre post x y i := by
+  classical
+  rw [relabeled_endpoint_pair,butterfly_endpoint_iff]
+  simp only [closingDomain,Finset.forall_mem_image,closingValue]
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma closingDomain_eq (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (ω : B × SwitchIndex d → Bool) (x y : α) (h : relabeledPerm pre post ω x = y) :
+    closingDomain pre post x y = parallelPath pre ω x := by
+  rw [closingDomain,parallelPath,(relabeled_endpoint_pair _ _ _ _ _).mp h |>.2]
+
+noncomputable def parallelClose (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (k : QueryTree (B × SwitchIndex d)) : QueryTree (B × SwitchIndex d) :=
+  if (pre x).1 = (post.symm y).1 then
+    QueryTree.forceList (closingDomain pre post x y).toList (closingValue pre post x y) k
+  else QueryTree.reject
+
+omit [Fintype α] in
+lemma parallelClose_succeeds (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (k : QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool) :
+    (parallelClose pre post x y k).succeeds ω =
+      if relabeledPerm pre post ω x = y then k.succeeds ω else false := by
+  classical
+  simp only [relabeled_endpoint_iff,parallelClose]
+  by_cases hb : (pre x).1 = (post.symm y).1
+  · rw [ite_eq_left hb,QueryTree.forceList_succeeds]
+    simp only [Finset.mem_toList,hb,true_and]
+  · simp [hb,QueryTree.succeeds]
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma parallelClose_raw_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (k : QueryTree (B × SwitchIndex d)) (ω : B × SwitchIndex d → Bool)
+    (h : relabeledPerm pre post ω x = y) :
+    (parallelClose pre post x y k).charge ω = k.charge ω + d := by
+  have hh := (relabeled_endpoint_iff _ _ _ _ _).mp h
+  rw [parallelClose,ite_eq_left hh.1,QueryTree.forceList_raw_charge _ _ _ _
+    (by simpa only [Finset.mem_toList] using hh.2),Finset.length_toList,
+    closingDomain_eq _ _ _ _ _ h,card_parallelPath]
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma parallelClose_fresh_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (k : QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
+    (ω : B × SwitchIndex d → Bool) (h : relabeledPerm pre post ω x = y) :
+    (parallelClose pre post x y k).freshCharge R ω =
+      k.freshCharge (R ∪ parallelPath pre ω x) ω + (parallelPath pre ω x \ R).card := by
+  have hh := (relabeled_endpoint_iff _ _ _ _ _).mp h
+  rw [parallelClose,ite_eq_left hh.1,QueryTree.forceList_charge _ _ _ _ _
+    (by simpa only [Finset.mem_toList] using hh.2),Finset.toList_toFinset,
+    closingDomain_eq _ _ _ _ _ h]
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma parallelClose_stale_charge (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (x y : α) (k : QueryTree (B × SwitchIndex d)) (R : Finset (B × SwitchIndex d))
+    (ω : B × SwitchIndex d → Bool) (h : relabeledPerm pre post ω x = y) :
+    (parallelClose pre post x y k).staleCharge R ω =
+      k.staleCharge (R ∪ parallelPath pre ω x) ω + (parallelPath pre ω x ∩ R).card := by
+  have h₀ := QueryTree.fresh_add_stale (parallelClose pre post x y k) R ω
+  have h₁ := QueryTree.fresh_add_stale k (R ∪ parallelPath pre ω x) ω
+  rw [parallelClose_raw_charge _ _ _ _ _ _ h,parallelClose_fresh_charge _ _ _ _ _ _ _ h] at h₀
+  have h₂ : (parallelPath pre ω x \ R).card + (parallelPath pre ω x ∩ R).card = d := by
+    rw [Finset.card_sdiff_add_card_inter,card_parallelPath]
+  omega
+
+noncomputable def parallelNetwork (pre : α ≃ B × Card d) (post : B × Card d ≃ α) :
+    RoutingNetwork α (B × SwitchIndex d) d where
+  perm := relabeledPerm pre post
+  path := parallelPath pre
+  users := parallelUsers pre
+  users_ne := parallelUsers_ne pre
+  mem_path := mem_parallelPath pre
+  card_path := card_parallelPath pre
+  entropy := parallel_entropy pre
+  follow := parallelFollow pre post
+  close := parallelClose pre post
+  follow_succeeds := parallelFollow_succeeds pre post
+  follow_raw_charge := parallelFollow_raw_charge pre post
+  follow_stale_charge := parallelFollow_stale_charge pre post
+  close_succeeds := parallelClose_succeeds pre post
+  close_raw_charge := parallelClose_raw_charge pre post
+  close_stale_charge := parallelClose_stale_charge pre post
+
+lemma parallel_cycle_log_mgf (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (S : Finset α) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    Real.log (finiteMean (fun ω : B × SwitchIndex d → Bool =>
+      q^(Fintype.card (RoutingNetwork.SelectedCycle (relabeledPerm pre post ω) S)))) ≤
+      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
+  exact (parallelNetwork pre post).butterfly_cycle_log_mgf S J hJ q hq
+
+end ParallelButterfly
+
+def cardSplit : (r s : ℕ) → Card (s+r) ≃ Card r × Card s
+  | 0, s =>
+    { toFun := fun x => ((fun i => Fin.elim0 i), x)
+      invFun := fun x => x.2
+      left_inv := fun _ => rfl
+      right_inv := fun x => by
+        apply Prod.ext
+        · exact Subsingleton.elim _ _
+        · rfl }
+  | r+1, s =>
+    { toFun := fun x => (Fin.cons (x 0) ((cardSplit r s (Fin.tail x)).1),
+        (cardSplit r s (Fin.tail x)).2)
+      invFun := fun x => Fin.cons (x.1 0) ((cardSplit r s).symm (Fin.tail x.1,x.2))
+      left_inv := by
+        intro x
+        simp only [Fin.cons_zero, Fin.tail_cons, Prod.mk.eta, Equiv.symm_apply_apply, Fin.cons_self_tail]
+      right_inv := by
+        intro x
+        simp only [Fin.cons_zero, Fin.tail_cons, Equiv.apply_symm_apply, Fin.cons_self_tail] }
+
+def assembleBits : (r s : ℕ) → (Card r × SwitchIndex s → Bool) →
+    (Card s × SwitchIndex r → Bool) → SwitchIndex (s+r) → Bool
+  | 0, _, lo, _, i => lo ((fun j => Fin.elim0 j), i)
+  | r+1, s, _, hi, Sum.inl y => hi ((cardSplit r s y).2,Sum.inl (cardSplit r s y).1)
+  | r+1, s, lo, hi, Sum.inr (b,i) => assembleBits r s
+      (fun j => lo (Fin.cons b j.1,j.2)) (fun j => hi (j.1,Sum.inr (b,j.2))) i
+
+lemma butterfly_split (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
+    (hi : Card s × SwitchIndex r → Bool) (x : Card (s+r)) :
+    cardSplit r s (butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi)) x) =
+      let y := cardSplit r s x
+      let z := butterflyPerm s (decodeButterfly s (fun i => lo (y.1,i))) y.2
+      (butterflyPerm r (decodeButterfly r (fun i => hi (z,i))) y.1,z) := by
+  induction r with
+  | zero =>
+    apply Prod.ext
+    · funext i
+      exact Fin.elim0 i
+    · rfl
+  | succ r ih =>
+    let lo' : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons (x 0) j.1,j.2)
+    let hi' : Card s × SwitchIndex r → Bool := fun j => hi (j.1,Sum.inr (x 0,j.2))
+    have hh := ih lo' hi' (Fin.tail x)
+    let u := butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo' hi')) (Fin.tail x)
+    let y := cardSplit r s (Fin.tail x)
+    let z := butterflyPerm s (decodeButterfly s (fun i => lo' (y.1,i))) y.2
+    let w := butterflyPerm r (decodeButterfly r (fun i => hi' (z,i))) y.1
+    change cardSplit r s u = (w,z) at hh
+    change ((Fin.cons (Bool.xor (x 0) (hi ((cardSplit r s u).2,Sum.inl (cardSplit r s u).1)))
+        ((cardSplit r s u).1) : Card (r+1)), (cardSplit r s u).2) =
+      ((Fin.cons (Bool.xor (x 0) (hi (z,Sum.inl w))) w : Card (r+1)),z)
+    rw [hh]
+
+def splitSwitch : (r s : ℕ) → SwitchIndex (s+r) ≃
+    (Card r × SwitchIndex s) ⊕ (Card s × SwitchIndex r)
+  | 0, _ =>
+    { toFun := fun i => Sum.inl ((fun j => Fin.elim0 j),i)
+      invFun := fun i => match i with
+        | Sum.inl (_,j) => j
+        | Sum.inr (_,j) => nomatch j
+      left_inv := fun _ => rfl
+      right_inv := by
+        intro i
+        rcases i with ⟨x,j⟩ | ⟨x,j⟩
+        · exact congrArg Sum.inl (Prod.ext (Subsingleton.elim _ _) rfl)
+        · exact Empty.elim j }
+  | r+1, s =>
+    { toFun := fun i => match i with
+        | Sum.inl y => Sum.inr ((cardSplit r s y).2,Sum.inl (cardSplit r s y).1)
+        | Sum.inr (b,j) => match splitSwitch r s j with
+          | Sum.inl (p,j) => Sum.inl (Fin.cons b p,j)
+          | Sum.inr (q,j) => Sum.inr (q,Sum.inr (b,j))
+      invFun := fun i => match i with
+        | Sum.inl (p,j) => Sum.inr (p 0,(splitSwitch r s).symm (Sum.inl (Fin.tail p,j)))
+        | Sum.inr (q,Sum.inl p) => Sum.inl ((cardSplit r s).symm (p,q))
+        | Sum.inr (q,Sum.inr (b,j)) => Sum.inr (b,(splitSwitch r s).symm (Sum.inr (q,j)))
+      left_inv := by
+        intro i
+        rcases i with y | ⟨b,j⟩
+        · dsimp only
+          rw [Prod.mk.eta, Equiv.symm_apply_apply]
+        · dsimp only
+          cases hz : splitSwitch r s j with
+          | inl z =>
+            rcases z with ⟨p,k⟩
+            simp only [Fin.cons_zero, Fin.tail_cons]
+            rw [←hz,Equiv.symm_apply_apply]
+          | inr z =>
+            rcases z with ⟨q,k⟩
+            dsimp only
+            rw [←hz,Equiv.symm_apply_apply]
+      right_inv := by
+        intro i
+        rcases i with ⟨p,j⟩ | ⟨q,j⟩
+        · simp only [Equiv.apply_symm_apply, Fin.cons_self_tail]
+        · rcases j with p | ⟨b,j⟩ <;> simp only [Equiv.apply_symm_apply] }
+
+lemma assembleBits_eq (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
+    (hi : Card s × SwitchIndex r → Bool) (i : SwitchIndex (s+r)) :
+    assembleBits r s lo hi i = Sum.elim lo hi (splitSwitch r s i) := by
+  induction r with
+  | zero => rfl
+  | succ r ih =>
+    rcases i with y | ⟨b,j⟩
+    · rfl
+    · simp only [assembleBits,splitSwitch,Equiv.coe_fn_mk]
+      rw [ih]
+      cases splitSwitch r s j <;> rfl
+
+def splitCoinEquiv (r s : ℕ) :
+    ((Card r × SwitchIndex s → Bool) × (Card s × SwitchIndex r → Bool)) ≃
+      (SwitchIndex (s+r) → Bool) :=
+  (Equiv.sumArrowEquivProdArrow _ _ Bool).symm.trans
+    (Equiv.arrowCongr (splitSwitch r s).symm (Equiv.refl Bool))
+
+lemma splitCoinEquiv_apply (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
+    (hi : Card s × SwitchIndex r → Bool) :
+    splitCoinEquiv r s (lo,hi) = assembleBits r s lo hi := by
+  funext i
+  exact (assembleBits_eq r s lo hi i).symm
+
+lemma finiteMean_prod {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] (f : Ω × Ω' → ℝ) :
+    finiteMean f = finiteMean (fun x => finiteMean (fun y => f (x,y))) := by
+  simp only [finiteMean, Fintype.sum_prod_type, Fintype.card_prod, Nat.cast_mul,
+    ←Finset.sum_div, div_div, mul_comm]
+
+lemma finiteMean_split (r s : ℕ) (f : (SwitchIndex (s+r) → Bool) → ℝ) :
+    finiteMean f = finiteMean (fun lo : Card r × SwitchIndex s → Bool =>
+      finiteMean (fun hi : Card s × SwitchIndex r → Bool => f (assembleBits r s lo hi))) := by
+  rw [finiteMean_equiv (splitCoinEquiv r s).symm]
+  simp only [Equiv.symm_symm]
+  rw [finiteMean_prod]
+  apply finiteMean_congr
+  intro lo
+  apply finiteMean_congr
+  intro hi
+  rw [splitCoinEquiv_apply]
+
+section PartiallyExposed
+variable {α : Type*} {r s : ℕ}
+
+def freshPre (pre : α ≃ Card (s+r)) (lo : Card r × SwitchIndex s → Bool) :
+    α ≃ Card s × Card r :=
+  ((pre.trans (cardSplit r s)).trans (parallelPerm lo)).trans (Equiv.prodComm _ _)
+
+def freshPost (post : Card (s+r) ≃ α) : Card s × Card r ≃ α :=
+  ((Equiv.prodComm _ _).trans (cardSplit r s).symm).trans post
+
+lemma exposed_perm_eq (pre : α ≃ Card (s+r)) (post : Card (s+r) ≃ α)
+    (lo : Card r × SwitchIndex s → Bool) (hi : Card s × SwitchIndex r → Bool) :
+    (pre.trans (butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi)))).trans post =
+      relabeledPerm (freshPre pre lo) (freshPost post) hi := by
+  ext x
+  simpa only [relabeledPerm, freshPre, freshPost, parallelPerm, Equiv.trans_apply,
+    Equiv.coe_fn_mk, Equiv.prodComm_apply, Prod.swap, Equiv.symm_apply_apply] using
+    congrArg (fun y => post ((cardSplit r s).symm y)) (butterfly_split r s lo hi (pre x))
+
+lemma partially_exposed_cycle_log_mgf [Fintype α] [DecidableEq α]
+    (pre : α ≃ Card (s+r)) (post : Card (s+r) ≃ α)
+    (lo : Card r × SwitchIndex s → Bool) (S : Finset α)
+    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    Real.log (finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(Fintype.card (RoutingNetwork.SelectedCycle
+        ((pre.trans (butterflyPerm (s+r)
+          (decodeButterfly (s+r) (assembleBits r s lo hi)))).trans post) S)))) ≤
+      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^r) := by
+  simp only [exposed_perm_eq]
+  exact parallel_cycle_log_mgf (freshPre pre lo) (freshPost post) S J hJ q hq
+
+end PartiallyExposed
+
+lemma inclusionMoment_eq {α Ω : Type*} [DecidableEq α] [Fintype Ω]
+    (F : Ω → Finset α) (S : Finset α) :
+    inclusionMoment F S = finiteMean (fun ω => if S ⊆ F ω then (1:ℝ) else 0) := by
+  unfold inclusionMoment
+  apply finiteMean_congr
+  intro ω
+  split_ifs <;> rfl
+
+lemma finiteMean_bool (f : Bool → ℝ) : finiteMean f = (f false+f true)/2 := by
+  simp only [finiteMean, Fintype.sum_bool, Fintype.card_bool, Nat.cast_ofNat]
+  ring
+
+lemma finiteMean_const {Ω : Type*} [Fintype Ω] [Nonempty Ω] (c : ℝ) :
+    finiteMean (fun _ : Ω => c) = c := by
+  simp only [finiteMean,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
+  exact mul_div_cancel_left₀ c (Nat.cast_ne_zero.mpr Fintype.card_ne_zero)
+
+lemma finiteMean_pi_prod {ι : Type*} [Fintype ι] [DecidableEq ι] {Ω : ι → Type*}
+    [∀ i, Fintype (Ω i)] (f : (i : ι) → Ω i → ℝ) :
+    finiteMean (fun ω : (i : ι) → Ω i => ∏ i, f i (ω i)) = ∏ i, finiteMean (f i) := by
+  simp only [finiteMean,←Fintype.prod_sum,Fintype.card_pi,Nat.cast_prod,Finset.prod_div_distrib]
+
+section ParallelFairLayer
+variable {β : Type*} [Fintype β] [DecidableEq β]
+
+def pairLayer (ξ : β → Bool) : Equiv.Perm (Bool × β) where
+  toFun x := (Bool.xor x.1 (ξ x.2),x.2)
+  invFun x := (Bool.xor x.1 (ξ x.2),x.2)
+  left_inv x := by
+    rcases x with ⟨b,x⟩
+    change (Bool.xor (Bool.xor b (ξ x)) (ξ x),x) = (b,x)
+    cases b <;> cases ξ x <;> rfl
+  right_inv x := by
+    rcases x with ⟨b,x⟩
+    change (Bool.xor (Bool.xor b (ξ x)) (ξ x),x) = (b,x)
+    cases b <;> cases ξ x <;> rfl
+
+noncomputable def pairFiber (S : Finset (Bool × β)) (b : β) : Finset Bool :=
+  Finset.univ.filter (fun ε => (ε,b) ∈ S)
+
+lemma prod_pairFiber (S : Finset (Bool × β)) (f : Bool × β → ℝ) :
+    (∏ x ∈ S, f x) = ∏ b : β, ∏ ε ∈ pairFiber S b, f (ε,b) := by
+  classical
+  simp only [pairFiber,Finset.prod_filter]
+  rw [Finset.prod_comm,←Fintype.prod_prod_type (fun x : Bool × β => if x ∈ S then f x else 1)]
+  rw [←Finset.prod_filter]
+  congr 1
+  ext x
+  simp
+
+lemma fiber_average_bound (S : Finset Bool) (p : Bool → ℝ) (hp : ∀ b, 0 ≤ p b) :
+    finiteMean (fun c : Bool => ∏ ε ∈ S, p (Bool.xor ε c)) ≤
+      ∏ ε ∈ S, (p ε + p (!ε))/2 := by
+  rw [finiteMean_bool]
+  have hxor : ∀ ε : Bool, Bool.xor ε true = Equiv.swap false true ε := by decide
+  simp only [Bool.xor_false,hxor]
+  calc
+    _ ≤ ∏ ε ∈ S, (p ε+p (Equiv.swap false true ε))/2 :=
+      swap_average_product S p hp false true
+    _ = _ := Finset.prod_congr rfl (fun ε _ => by cases ε <;> rfl)
+
+lemma CylinderBound.fair_parallel {m : Finset (Bool × β) → ℝ} {p : Bool × β → ℝ}
+    (h : CylinderBound m p) (hp : ∀ x, 0 ≤ p x) :
+    CylinderBound (fun S => finiteMean (fun ξ : β → Bool => m (S.image (pairLayer ξ))))
+      (fun x => (p x+p (!x.1,x.2))/2) := by
+  intro S
+  calc
+    _ ≤ finiteMean (fun ξ : β → Bool => ∏ x ∈ S, p (pairLayer ξ x)) := by
+      apply finiteMean_mono
+      intro ξ
+      simpa only [Finset.prod_image (pairLayer ξ).injective.injOn] using h (S.image (pairLayer ξ))
+    _ = ∏ b : β, finiteMean (fun c : Bool => ∏ ε ∈ pairFiber S b, p (Bool.xor ε c,b)) := by
+      calc
+        _ = finiteMean (fun ξ : β → Bool => ∏ b : β,
+            ∏ ε ∈ pairFiber S b, p (Bool.xor ε (ξ b),b)) :=
+          finiteMean_congr (fun ξ => prod_pairFiber S (fun x => p (pairLayer ξ x)))
+        _ = _ := finiteMean_pi_prod (fun b c => ∏ ε ∈ pairFiber S b, p (Bool.xor ε c,b))
+    _ ≤ ∏ b : β, ∏ ε ∈ pairFiber S b, (p (ε,b)+p (!ε,b))/2 := by
+      apply Finset.prod_le_prod₀
+      · intro b _
+        exact div_nonneg (Finset.sum_nonneg (fun c _ =>
+          Finset.prod_nonneg (fun ε _ => hp (Bool.xor ε c,b)))) (Nat.cast_nonneg _)
+      · intro b _
+        exact fiber_average_bound (pairFiber S b) (fun ε => p (ε,b)) (fun ε => hp (ε,b))
+    _ = _ := (prod_pairFiber S (fun x => (p x+p (!x.1,x.2))/2)).symm
+
+omit [Fintype β] in
+lemma subset_pairLayer_image_iff (S A : Finset (Bool × β)) (ξ : β → Bool) :
+    S ⊆ A.image (pairLayer ξ) ↔ S.image (pairLayer ξ) ⊆ A := by
+  have hi (x : Bool × β) : pairLayer ξ (pairLayer ξ x) = x := (pairLayer ξ).left_inv x
+  constructor
+  · intro h
+    have h' := Finset.image_subset_image h (f := pairLayer ξ)
+    simpa only [Finset.image_image,Function.comp_def,hi,Finset.image_id'] using h'
+  · intro h
+    have h' := Finset.image_subset_image h (f := pairLayer ξ)
+    simpa only [Finset.image_image,Function.comp_def,hi,Finset.image_id'] using h'
+
+noncomputable def fairSetLayer {Ω : Type*} (F : Ω → Finset (Bool × β)) :
+    Ω × (β → Bool) → Finset (Bool × β) := fun ω => (F ω.1).image (pairLayer ω.2)
+
+lemma finiteMean_comm {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] (f : Ω → Ω' → ℝ) :
+    finiteMean (fun x => finiteMean (f x)) = finiteMean (fun y => finiteMean (fun x => f x y)) := by
+  simp only [finiteMean,←Finset.sum_div,div_div]
+  rw [Finset.sum_comm,mul_comm]
+
+lemma inclusionMoment_fairSetLayer {Ω : Type*} [Fintype Ω]
+    (F : Ω → Finset (Bool × β)) (S : Finset (Bool × β)) :
+    inclusionMoment (fairSetLayer F) S =
+      finiteMean (fun ξ : β → Bool => inclusionMoment F (S.image (pairLayer ξ))) := by
+  classical
+  simp only [inclusionMoment_eq]
+  have he : (fun ω : Ω × (β → Bool) => if S ⊆ fairSetLayer F ω then (1:ℝ) else 0) =
+      (fun ω : Ω × (β → Bool) => if S.image (pairLayer ω.2) ⊆ F ω.1 then (1:ℝ) else 0) := by
+    funext ω
+    simp only [fairSetLayer,subset_pairLayer_image_iff]
+  apply (congrArg finiteMean he).trans
+  exact (finiteMean_prod (fun ω : Ω × (β → Bool) =>
+    if S.image (pairLayer ω.2) ⊆ F ω.1 then (1:ℝ) else 0)).trans
+      (finiteMean_comm (fun ω ξ => if S.image (pairLayer ξ) ⊆ F ω then (1:ℝ) else 0))
+
+lemma inclusionMoment_fairSetLayer_singleton {Ω : Type*} [Fintype Ω]
+    (F : Ω → Finset (Bool × β)) (x : Bool × β) :
+    inclusionMoment (fairSetLayer F) {x} =
+      (inclusionMoment F {x}+inclusionMoment F {(!x.1,x.2)})/2 := by
+  rw [inclusionMoment_fairSetLayer]
+  simp only [Finset.image_singleton,pairLayer,Equiv.coe_fn_mk]
+  rw [finiteMean_pi_eval (A := fun _ : β => Bool) x.2 (fun c : Bool => inclusionMoment F {(Bool.xor x.1 c,x.2)})]
+  rw [finiteMean_bool]
+  rcases x with ⟨b,x⟩
+  cases b <;> simp only [Bool.xor_false,Bool.false_xor,Bool.true_xor, Bool.not_false,Bool.not_true]
+
+lemma cylinder_fairSetLayer {Ω : Type*} [Fintype Ω]
+    (F : Ω → Finset (Bool × β))
+    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
+    CylinderBound (inclusionMoment (fairSetLayer F))
+      (fun x => inclusionMoment (fairSetLayer F) {x}) := by
+  have hh := h.fair_parallel (fun x => inclusionMoment_nonneg F {x})
+  intro S
+  rw [inclusionMoment_fairSetLayer]
+  simp only [inclusionMoment_fairSetLayer_singleton]
+  exact hh S
+
+end ParallelFairLayer
+
+section RelabelMarks
+variable {α β Ω Ω' : Type*} [DecidableEq α] [DecidableEq β] [Fintype Ω] [Fintype Ω']
+
+omit [Fintype Ω] [Fintype Ω'] in
+lemma subset_image_equiv_iff (e : α ≃ β) (A : Finset α) (S : Finset β) :
+    S ⊆ A.image e ↔ S.image e.symm ⊆ A := by
+  constructor
+  · intro h x hx
+    obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
+    obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp (h hy)
+    simpa only [←hxy,Equiv.symm_apply_apply] using hx
+  · intro h y hy
+    exact Finset.mem_image.mpr ⟨e.symm y,h (Finset.mem_image_of_mem _ hy),e.apply_symm_apply y⟩
+
+omit [Fintype Ω'] in
+lemma inclusionMoment_image_equiv (e : α ≃ β) (F : Ω → Finset α) (S : Finset β) :
+    inclusionMoment (fun ω => (F ω).image e) S = inclusionMoment F (S.image e.symm) := by
+  classical
+  simp only [inclusionMoment_eq]
+  apply finiteMean_congr
+  intro ω
+  simp only [subset_image_equiv_iff]
+
+omit [Fintype Ω'] in
+lemma cylinder_image_equiv (e : α ≃ β) (F : Ω → Finset α)
+    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
+    CylinderBound (inclusionMoment (fun ω => (F ω).image e))
+      (fun x => inclusionMoment (fun ω => (F ω).image e) {x}) := by
+  intro S
+  simp only [inclusionMoment_image_equiv,Finset.image_singleton]
+  have hh := h (S.image e.symm)
+  simpa only [Finset.prod_image e.symm.injective.injOn] using hh
+
+omit [DecidableEq β] in
+lemma inclusionMoment_comp_equiv (e : Ω ≃ Ω') (F : Ω' → Finset α) (S : Finset α) :
+    inclusionMoment (fun ω => F (e ω)) S = inclusionMoment F S := by
+  classical
+  simp only [inclusionMoment_eq]
+  simpa only [Equiv.apply_symm_apply] using
+    finiteMean_equiv e (fun ω => if S ⊆ F (e ω) then (1 : ℝ) else 0)
+
+omit [DecidableEq β] in
+lemma cylinder_comp_equiv (e : Ω ≃ Ω') (F : Ω' → Finset α)
+    (h : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x})) :
+    CylinderBound (inclusionMoment (fun ω => F (e ω)))
+      (fun x => inclusionMoment (fun ω => F (e ω)) {x}) := by
+  intro S
+  simpa only [inclusionMoment_comp_equiv] using h S
+end RelabelMarks
+
+lemma finiteMean_prod_mul {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
+    (f : Ω → ℝ) (g : Ω' → ℝ) :
+    finiteMean (fun ω : Ω × Ω' => f ω.1*g ω.2) = finiteMean f * finiteMean g := by
+  simp only [finiteMean,Fintype.sum_prod_type,Fintype.card_prod,Nat.cast_mul,
+    ←Finset.mul_sum,←Finset.sum_mul]
+  ring
+
+section JoinChildren
+variable {β : Type*} [DecidableEq β]
+
+noncomputable def childJoin (A B : Finset β) : Finset (Bool × β) :=
+  A.image (fun x => (false,x)) ∪ B.image (fun x => (true,x))
+
+lemma mem_childJoin (A B : Finset β) (b : Bool) (x : β) :
+    (b,x) ∈ childJoin A B ↔ (if b then x ∈ B else x ∈ A) := by
+  cases b <;> simp [childJoin]
+
+noncomputable def boolFiber (S : Finset (Bool × β)) (b : Bool) : Finset β :=
+  (S.filter (fun x => x.1 = b)).image Prod.snd
+
+lemma mem_boolFiber (S : Finset (Bool × β)) (b : Bool) (x : β) :
+    x ∈ boolFiber S b ↔ (b,x) ∈ S := by
+  constructor
+  · intro h
+    obtain ⟨⟨c,y⟩,hy,hxy⟩ := Finset.mem_image.mp h
+    obtain ⟨hy,hc⟩ := Finset.mem_filter.mp hy
+    change c = b at hc
+    change y = x at hxy
+    simpa only [hc,hxy] using hy
+  · intro h
+    exact Finset.mem_image.mpr ⟨(b,x),Finset.mem_filter.mpr ⟨h,rfl⟩,rfl⟩
+
+lemma childJoin_boolFiber (S : Finset (Bool × β)) :
+    childJoin (boolFiber S false) (boolFiber S true) = S := by
+  ext ⟨b,x⟩
+  rw [mem_childJoin]
+  cases b <;> simp only [Bool.false_eq_true,↓reduceIte,mem_boolFiber]
+
+lemma subset_childJoin_iff (S : Finset (Bool × β)) (A B : Finset β) :
+    S ⊆ childJoin A B ↔ boolFiber S false ⊆ A ∧ boolFiber S true ⊆ B := by
+  constructor
+  · intro h
+    constructor
+    · intro x hx
+      exact (mem_childJoin A B false x).mp (h ((mem_boolFiber S false x).mp hx))
+    · intro x hx
+      exact (mem_childJoin A B true x).mp (h ((mem_boolFiber S true x).mp hx))
+  · rintro ⟨h₀,h₁⟩ ⟨b,x⟩ hx
+    rw [mem_childJoin]
+    cases b
+    · exact h₀ ((mem_boolFiber S false x).mpr hx)
+    · exact h₁ ((mem_boolFiber S true x).mpr hx)
+
+lemma prod_childJoin (A B : Finset β) (f : Bool × β → ℝ) :
+    (∏ x ∈ childJoin A B, f x) = (∏ x ∈ A, f (false,x)) * ∏ x ∈ B, f (true,x) := by
+  classical
+  have hd : Disjoint (A.image (fun x => (false,x))) (B.image (fun x => (true,x))) := by
+    apply Finset.disjoint_left.mpr
+    rintro x hx hy
+    obtain ⟨a,ha,rfl⟩ := Finset.mem_image.mp hx
+    obtain ⟨b,hb,hba⟩ := Finset.mem_image.mp hy
+    exact Bool.false_ne_true (congrArg Prod.fst hba).symm
+  rw [childJoin,Finset.prod_union hd,
+    Finset.prod_image (fun _ _ _ _ h => congrArg Prod.snd h),
+    Finset.prod_image (fun _ _ _ _ h => congrArg Prod.snd h)]
+
+lemma prod_boolFibers (S : Finset (Bool × β)) (f : Bool × β → ℝ) :
+    (∏ x ∈ S, f x) = (∏ x ∈ boolFiber S false, f (false,x)) *
+      ∏ x ∈ boolFiber S true, f (true,x) := by
+  rw [←prod_childJoin,childJoin_boolFiber]
+
+lemma inclusionMoment_childJoin {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω']
+    (F : Ω → Finset β) (G : Ω' → Finset β) (S : Finset (Bool × β)) :
+    inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) S =
+      inclusionMoment F (boolFiber S false) * inclusionMoment G (boolFiber S true) := by
+  classical
+  simp only [inclusionMoment_eq]
+  rw [←finiteMean_prod_mul]
+  apply finiteMean_congr
+  intro ω
+  simp only [subset_childJoin_iff]
+  split_ifs <;> simp_all
+
+lemma cylinder_childJoin {Ω Ω' : Type*} [Fintype Ω] [Fintype Ω'] [Nonempty Ω] [Nonempty Ω']
+    (F : Ω → Finset β) (G : Ω' → Finset β)
+    (hF : CylinderBound (inclusionMoment F) (fun x => inclusionMoment F {x}))
+    (hG : CylinderBound (inclusionMoment G) (fun x => inclusionMoment G {x})) :
+    CylinderBound (inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)))
+      (fun x => inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {x}) := by
+  intro S
+  rw [inclusionMoment_childJoin,prod_boolFibers]
+  have hf (b : Bool) (x : β) : boolFiber {(b,x)} b = {x} := by
+    ext y
+    simp only [mem_boolFiber,Finset.mem_singleton,Prod.mk.injEq,true_and]
+  have hn (b : Bool) (x : β) : boolFiber {(!b,x)} b = ∅ := by
+    ext y
+    cases b <;> simp [mem_boolFiber]
+  have h₀ (x : β) : inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {(false,x)} =
+      inclusionMoment F {x} * inclusionMoment G ∅ := by
+    rw [inclusionMoment_childJoin,hf]
+    have hh := hn true x
+    exact congrArg (fun T => inclusionMoment F {x} * inclusionMoment G T) hh
+  have h₁ (x : β) : inclusionMoment (fun ω : Ω × Ω' => childJoin (F ω.1) (G ω.2)) {(true,x)} =
+      inclusionMoment F ∅ * inclusionMoment G {x} := by
+    rw [inclusionMoment_childJoin,hf]
+    have hh := hn false x
+    exact congrArg (fun T => inclusionMoment F T * inclusionMoment G {x}) hh
+  have emF : inclusionMoment F ∅ = 1 := by simp [inclusionMoment,finiteMean_const]
+  have emG : inclusionMoment G ∅ = 1 := by simp [inclusionMoment,finiteMean_const]
+  simp only [h₀,h₁,emF,emG,mul_one,one_mul]
+  exact mul_le_mul (hF _) (hG _) (inclusionMoment_nonneg _ _)
+    (Finset.prod_nonneg (fun x _ => inclusionMoment_nonneg F {x}))
+end JoinChildren
+
+def coinStepEquiv (d : ℕ) : (SwitchIndex (d+1) → Bool) ≃
+    ((SwitchIndex d → Bool) × (SwitchIndex d → Bool)) × (Card d → Bool) where
+  toFun ω := ((fun i => ω (Sum.inr (false,i)),fun i => ω (Sum.inr (true,i))),
+    fun x => ω (Sum.inl x))
+  invFun ω := Sum.elim ω.2 (fun bi => if bi.1 then ω.1.2 bi.2 else ω.1.1 bi.2)
+  left_inv ω := by
+    funext i
+    cases i with
+    | inl x => rfl
+    | inr bi => cases bi with | mk b i => cases b <;> rfl
+  right_inv ω := by
+    rcases ω with ⟨⟨ω₀,ω₁⟩,ξ⟩
+    rfl
+
+def headTailEquiv (d : ℕ) : Card (d+1) ≃ Bool × Card d :=
+  (Fin.consEquiv (fun _ : Fin (d+1) => Bool)).symm
+
+lemma childMarks_join (d : ℕ) (A : Finset (Card (d+1))) :
+    A = (childMarks A false).image (Fin.cons false) ∪
+      (childMarks A true).image (Fin.cons true) := by
+  classical
+  ext x
+  constructor
+  · intro hx
+    cases hb : x 0
+    · exact Finset.mem_union_left _ (Finset.mem_image.mpr
+        ⟨Fin.tail x,(mem_childMarks _ _ _).mpr (by simpa only [←hb,Fin.cons_self_tail] using hx),
+          by simp only [←hb,Fin.cons_self_tail]⟩)
+    · exact Finset.mem_union_right _ (Finset.mem_image.mpr
+        ⟨Fin.tail x,(mem_childMarks _ _ _).mpr (by simpa only [←hb,Fin.cons_self_tail] using hx),
+          by simp only [←hb,Fin.cons_self_tail]⟩)
+  · intro hx
+    rcases Finset.mem_union.mp hx with hx | hx
+    · obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
+      exact (mem_childMarks _ _ _).mp hy
+    · obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
+      exact (mem_childMarks _ _ _).mp hy
+
+noncomputable def butterflyMarks (d : ℕ) (A : Finset (Card d))
+    (ω : SwitchIndex d → Bool) : Finset (Card d) :=
+  A.image (butterflyPerm d (decodeButterfly d ω))
+
+lemma butterflyMarks_succ (d : ℕ) (A : Finset (Card (d+1))) (ω : SwitchIndex (d+1) → Bool) :
+    butterflyMarks (d+1) A ω =
+      (fairSetLayer (fun p : (SwitchIndex d → Bool) × (SwitchIndex d → Bool) =>
+        childJoin (butterflyMarks d (childMarks A false) p.1)
+          (butterflyMarks d (childMarks A true) p.2)) (coinStepEquiv d ω)).image
+            (headTailEquiv d).symm := by
+  classical
+  unfold butterflyMarks fairSetLayer childJoin
+  rw [Finset.image_union,Finset.image_union]
+  simp only [Finset.image_image]
+  conv_lhs => rw [childMarks_join d A]
+  rw [Finset.image_union]
+  simp only [Finset.image_image]
+  apply congrArg₂ (· ∪ ·)
+  · apply Finset.image_congr
+    intro x _
+    simp only [Function.comp_def,butterflyPerm,decodeButterfly,Equiv.Perm.coe_mul,
+      childLift,pairSwitch,switchFun,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
+      coinStepEquiv,headTailEquiv,Equiv.symm_symm,pairLayer]
+    rfl
+  · apply Finset.image_congr
+    intro x _
+    simp only [Function.comp_def,butterflyPerm,decodeButterfly,Equiv.Perm.coe_mul,
+      childLift,pairSwitch,switchFun,Equiv.coe_fn_mk,Fin.cons_zero,Fin.tail_cons,
+      coinStepEquiv,headTailEquiv,Equiv.symm_symm,pairLayer]
+    rfl
+
+lemma butterflyMarks_cylinder (d : ℕ) (A : Finset (Card d)) :
+    CylinderBound (inclusionMoment (butterflyMarks d A))
+      (fun x => inclusionMoment (butterflyMarks d A) {x}) := by
+  induction d with
+  | zero =>
+    have he : butterflyMarks 0 A = fun _ : SwitchIndex 0 → Bool => A := by
+      funext ω
+      simp [butterflyMarks,butterflyPerm]
+    rw [he]
+    exact cylinder_const A
+  | succ d ih =>
+    have h := cylinder_childJoin (butterflyMarks d (childMarks A false))
+      (butterflyMarks d (childMarks A true)) (ih _) (ih _)
+    have hh := cylinder_image_equiv (headTailEquiv d).symm _ (cylinder_fairSetLayer _ h)
+    have hs := cylinder_comp_equiv (coinStepEquiv d) _ hh
+    simpa only [←butterflyMarks_succ] using hs
+
+lemma indicator_mem_image_perm {α : Type*} [DecidableEq α]
+    (A : Finset α) (p : Equiv.Perm α) (y : α) :
+    (if y ∈ A.image p then (1 : ℝ) else 0) = ∑ x ∈ A, if p x = y then 1 else 0 := by
+  classical
+  by_cases h : y ∈ A.image p
+  · obtain ⟨x,hx,hxy⟩ := Finset.mem_image.mp h
+    rw [ite_eq_left (Finset.mem_image.mpr ⟨x,hx,hxy⟩)]
+    symm
+    calc
+      _ = (if p x = y then (1:ℝ) else 0) := Finset.sum_eq_single_of_mem x hx (by
+        intro z _ hzx
+        exact ite_eq_right (fun hz => hzx (p.injective (hz.trans hxy.symm))))
+      _ = 1 := ite_eq_left hxy
+  · rw [ite_eq_right h]
+    symm
+    apply Finset.sum_eq_zero
+    intro x hx
+    exact ite_eq_right (fun hxy => h (Finset.mem_image.mpr ⟨x,hx,hxy⟩))
+
+lemma butterflyMarks_singleton (d : ℕ) (A : Finset (Card d)) (y : Card d) :
+    inclusionMoment (butterflyMarks d A) {y} = (A.card : ℝ)/(2:ℝ)^d := by
+  classical
+  unfold inclusionMoment
+  simp only [Finset.singleton_subset_iff,butterflyMarks,indicator_mem_image_perm]
+  rw [finiteMean_sum]
+  simp only [butterfly_single_card_uniform,Finset.sum_const,nsmul_eq_mul]
+  ring
+
+lemma butterflyMarks_inclusion (d : ℕ) (A S : Finset (Card d)) :
+    inclusionMoment (butterflyMarks d A) S ≤ ((A.card : ℝ)/(2:ℝ)^d)^S.card := by
+  have h := butterflyMarks_cylinder d A S
+  simpa only [butterflyMarks_singleton,Finset.prod_const] using h
+
+namespace CycleColoring
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+abbrev Coloring (p : Equiv.Perm α) (S : Finset α) :=
+  {f : α → Bool // (∀ x, x ∉ S → f x = false) ∧
+    ∀ x ∈ S, p x ∈ S → f (p x) = f x}
+
+noncomputable instance (p : Equiv.Perm α) (S : Finset α) : Fintype (Coloring p S) :=
+  Fintype.ofFinite _
+
+noncomputable def boundary (p : Equiv.Perm α) (S : Finset α) : Finset α :=
+  S.filter (fun x => p.symm x ∉ S)
+
+noncomputable def root {p : Equiv.Perm α} {S : Finset α}
+    (C : RoutingNetwork.SelectedCycle p S) : α := C.nonempty.choose
+
+lemma root_mem {p : Equiv.Perm α} {S : Finset α} (C : RoutingNetwork.SelectedCycle p S) :
+    root C ∈ C.val := C.nonempty.choose_spec
+
+abbrev Seed (p : Equiv.Perm α) (S : Finset α) :=
+  {x // x ∈ boundary p S} ⊕ RoutingNetwork.SelectedCycle p S
+
+noncomputable def record {p : Equiv.Perm α} {S : Finset α}
+    (f : Coloring p S) : Seed p S → Bool :=
+  Sum.elim (fun x => f.val x.val) (fun C => f.val (root C))
+
+lemma orbit_subset_of_predecessor_closed (p : Equiv.Perm α) (D : Finset α)
+    (hD : ∀ x ∈ D, p.symm x ∈ D) {x : α} (hx : x ∈ D) : orbitSet p x ⊆ D := by
+  intro y hy
+  have hh := ((mem_orbitSet p x y).mp hy).inv
+  obtain ⟨n,hn⟩ := hh.exists_nat_pow_eq
+  have hp (k : ℕ) : (p⁻¹ ^ k) x ∈ D := by
+    induction k with
+    | zero => simpa using hx
+    | succ k ih => simpa only [pow_succ',Equiv.Perm.mul_apply,Equiv.Perm.coe_inv] using hD _ ih
+  exact hn ▸ hp n
+
+lemma record_injective (p : Equiv.Perm α) (S : Finset α) :
+    Function.Injective (record (p := p) (S := S)) := by
+  classical
+  intro f g hfg
+  apply Subtype.ext
+  funext x
+  by_contra hne
+  have hx : x ∈ S := by
+    by_contra h
+    exact hne ((f.property.1 x h).trans (g.property.1 x h).symm)
+  let D := S.filter (fun y => f.val y ≠ g.val y)
+  have hxD : x ∈ D := Finset.mem_filter.mpr ⟨hx,hne⟩
+  have hprev (y : α) (hy : y ∈ D) : p.symm y ∈ D := by
+    obtain ⟨hyS,hyne⟩ := Finset.mem_filter.mp hy
+    have hpS : p.symm y ∈ S := by
+      by_contra hn
+      have hseed : y ∈ boundary p S := Finset.mem_filter.mpr ⟨hyS,hn⟩
+      exact hyne (congrFun hfg (Sum.inl ⟨y,hseed⟩))
+    refine Finset.mem_filter.mpr ⟨hpS,?_⟩
+    have hf := f.property.2 (p.symm y) hpS (by simpa only [Equiv.apply_symm_apply] using hyS)
+    have hg := g.property.2 (p.symm y) hpS (by simpa only [Equiv.apply_symm_apply] using hyS)
+    simp only [Equiv.apply_symm_apply] at hf hg
+    intro he
+    exact hyne (hf.trans (he.trans hg.symm))
+  have horb := orbit_subset_of_predecessor_closed p D hprev hxD
+  let C : RoutingNetwork.SelectedCycle p S :=
+    ⟨orbitSet p x,⟨x,rfl⟩,fun y hy => (Finset.mem_filter.mp (horb hy)).1⟩
+  have hrD : root C ∈ D := horb (root_mem C)
+  have hrne := (Finset.mem_filter.mp hrD).2
+  exact hrne (congrFun hfg (Sum.inr C))
+
+lemma card_coloring_le (p : Equiv.Perm α) (S : Finset α) :
+    Fintype.card (Coloring p S) ≤
+      2 ^ ((boundary p S).card + Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
+  have h := Fintype.card_le_of_injective (record (p := p) (S := S)) (record_injective p S)
+  simpa only [Fintype.card_fun,Fintype.card_sum,Fintype.card_coe,Fintype.card_bool] using h
+
+omit [Fintype α] in
+lemma boundary_card (p : Equiv.Perm α) (S : Finset α) :
+    (boundary p S).card = S.card - (S.filter (fun x => p.symm x ∈ S)).card := by
+  classical
+  have h := Finset.card_filter_add_card_filter_not (s := S) (fun x => p.symm x ∈ S)
+  unfold boundary
+  omega
+
+lemma card_coloring_le_edges (p : Equiv.Perm α) (S : Finset α) :
+    Fintype.card (Coloring p S) ≤
+      2 ^ (S.card - (S.filter (fun x => p.symm x ∈ S)).card +
+        Fintype.card (RoutingNetwork.SelectedCycle p S)) := by
+  simpa only [boundary_card] using card_coloring_le p S
+
+
+abbrev OppositeColoring (p : Equiv.Perm α) (S : Finset α) :=
+  {f : α → Bool // (∀ x, x ∉ S → f x = false) ∧
+    ∀ x ∈ S, p x ∈ S → f (p x) = !(f x)}
+
+noncomputable instance (p : Equiv.Perm α) (S : Finset α) : Fintype (OppositeColoring p S) :=
+  Fintype.ofFinite _
+
+noncomputable def bipartitionFlip (p : Equiv.Perm α) (S : Finset α)
+    (χ : α → Bool) (hχ : ∀ x, χ (p x) = !(χ x))
+    (f : OppositeColoring p S) : Coloring p S := by
+  classical
+  refine ⟨(fun x => if x ∈ S then Bool.xor (f.val x) (χ x) else false),?_,?_⟩
+  · intro x hx
+    exact ite_eq_right hx
+  · intro x hx hpx
+    simp only [ite_eq_left hx,ite_eq_left hpx,f.property.2 x hx hpx,hχ x]
+    cases f.val x <;> cases χ x <;> rfl
+
+end CycleColoring
+end Thorp
+
+end ThorpNine.Tail
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Tail/ParallelFreshPre.lean b/lean/OAI/Probability/ThorpRouting/Tail/ParallelFreshPre.lean
index ba43c4a96ac00c01c8b119495bda9c7f702646e2..16d21a5d265fd87c35dbde327db432d3fd76e696 100644
--- a/lean/OAI/Probability/ThorpRouting/Tail/ParallelFreshPre.lean
+++ b/lean/OAI/Probability/ThorpRouting/Tail/ParallelFreshPre.lean
@@ -1,848 +1,848 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Tail.BenesCoins
-
-namespace OAI
-
-namespace ThorpNine.Tail
-
-namespace Thorp
-
-lemma mem_centralBlock (r H : ℕ) (b : Card r) (x : Card (H+r)) :
-    x ∈ centralBlock r H b ↔ (cardSplit r H x).1 = b := by
-  classical
-  constructor
-  · intro h
-    obtain ⟨z,_,rfl⟩ := Finset.mem_image.mp h
-    simp only [Equiv.apply_symm_apply]
-  · intro h
-    apply Finset.mem_image.mpr
-    refine ⟨(cardSplit r H x).2,Finset.mem_univ _,?_⟩
-    rw [←h,Prod.mk.eta,Equiv.symm_apply_apply]
-
-lemma centralBlock_card (r H : ℕ) (b : Card r) : (centralBlock r H b).card = 2^H := by
-  classical
-  rw [centralBlock,Finset.card_image_of_injective]
-  · exact card_positions H
-  · intro x y h
-    exact congrArg Prod.snd ((cardSplit r H).symm.injective h)
-
-lemma centralBlock_disjoint (r H : ℕ) :
-    Pairwise (fun b b' => Disjoint (centralBlock r H b) (centralBlock r H b')) := by
-  classical
-  intro b b' h
-  apply Finset.disjoint_left.mpr
-  intro x hx hy
-  exact h (((mem_centralBlock r H b x).mp hx).symm.trans
-    ((mem_centralBlock r H b' x).mp hy))
-
-lemma centralBlock_zero (H : ℕ) (b : Card 0) : centralBlock 0 H b = Finset.univ := by
-  classical
-  ext x
-  simp only [mem_centralBlock,Finset.mem_univ,iff_true]
-  exact Subsingleton.elim _ _
-
-lemma centralBlock_step (r H : ℕ) (b : Bool) (c : Card r) :
-    centralBlock (r+1) H (Fin.cons b c) = (centralBlock r H c).image (Fin.cons b) := by
-  classical
-  simp only [centralBlock,Finset.image_image,cardSplit,Equiv.coe_fn_symm_mk,
-    Fin.cons_zero,Fin.tail_cons,Function.comp_def]
-
-lemma block_inter_step (r H : ℕ) (b : Bool) (c : Card r) (A : Finset (Card (H+(r+1)))) :
-    centralBlock (r+1) H (Fin.cons b c) ∩ A =
-      (centralBlock r H c ∩ childMarks A b).image (Fin.cons b) := by
-  classical
-  rw [centralBlock_step]
-  ext x
-  constructor
-  · intro hx
-    obtain ⟨hxB,hxA⟩ := Finset.mem_inter.mp hx
-    obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hxB
-    exact Finset.mem_image.mpr ⟨y,Finset.mem_inter.mpr ⟨hy,(mem_childMarks A b y).mpr hxA⟩,rfl⟩
-  · intro hx
-    obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
-    exact Finset.mem_inter.mpr ⟨Finset.mem_image.mpr ⟨y,(Finset.mem_inter.mp hy).1,rfl⟩,
-      (mem_childMarks A b y).mp (Finset.mem_inter.mp hy).2⟩
-
-lemma sparseMass_central_zero (H : ℕ) (A : Finset (Card H)) :
-    sparseMass (centralBlock 0 H) A = crowdedMass A.card := by
-  classical
-  simp only [sparseMass,centralBlock_zero,Finset.univ_inter,crowdedMass,
-    Finset.sum_const,Finset.card_univ,card_positions,pow_zero,one_nsmul]
-
-lemma sparseMass_central_step (r H : ℕ) (A : Finset (Card (H+(r+1)))) :
-    sparseMass (centralBlock (r+1) H) A =
-      sparseMass (centralBlock r H) (childMarks A false) +
-        sparseMass (centralBlock r H) (childMarks A true) := by
-  classical
-  unfold sparseMass
-  rw [←Equiv.sum_comp (headTailEquiv r).symm,Fintype.sum_prod_type]
-  have ht (b : Bool) (c : Card r) : (headTailEquiv r).symm (b,c) = Fin.cons b c := rfl
-  simp only [Fintype.sum_bool,ht]
-  have hc (b : Bool) (c : Card r) :
-      (centralBlock (r+1) H (Fin.cons b c) ∩ A).card =
-      (centralBlock r H c ∩ childMarks A b).card := by
-    rw [block_inter_step]
-    apply Finset.card_image_of_injective
-    intro x y h
-    simpa only [Fin.tail_cons] using congrArg Fin.tail h
-  simp only [hc]
-  exact add_comm _ _
-
-
-lemma inverseButterfly_coordinates_step (d : ℕ)
-    (c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (ξ : Card d → Bool)
-    (x : Card (d+1)) :
-    headTailEquiv d ((butterflyPerm (d+1)
-      (decodeButterfly (d+1) ((coinStepEquiv d).symm (c,ξ)))).symm x) =
-      (Bool.xor (x 0) (ξ (Fin.tail x)),
-        (butterflyPerm d (decodeButterfly d
-          (if Bool.xor (x 0) (ξ (Fin.tail x)) then c.2 else c.1))).symm (Fin.tail x)) := by
-  rw [butterflyPerm_step]
-  rfl
-
-lemma inverseButterflyMarks_child (d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (d+1))
-    (c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (ξ : Card d → Bool) (b : Bool) :
-    childMarks (inverseButterflyMarks (d+1) (Finset.univ.image e)
-      ((coinStepEquiv d).symm (c,ξ))) b =
-      inverseButterflyMarks d
-        (Finset.univ.image (PairRouting.childEmbedding (e.trans (headTailEquiv d).toEmbedding)
-          (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) ξ)
-            (PairRouting.colors_compatible _ _) b)) (if b then c.2 else c.1) := by
-  classical
-  let x := e.trans (headTailEquiv d).toEmbedding
-  let col := PairRouting.colors x ξ
-  let child := PairRouting.childEmbedding x col (PairRouting.colors_compatible x ξ) b
-  have hp (i : ι) : headTailEquiv d ((butterflyPerm (d+1)
-      (decodeButterfly (d+1) ((coinStepEquiv d).symm (c,ξ)))).symm (e i)) =
-      (col i,(butterflyPerm d (decodeButterfly d (if col i then c.2 else c.1))).symm (x i).2) :=
-    inverseButterfly_coordinates_step d c ξ (e i)
-  ext y
-  simp only [mem_childMarks,inverseButterflyMarks,Finset.image_image,
-    Finset.mem_image,Finset.mem_univ,true_and,Function.comp_def]
-  constructor
-  · rintro ⟨i,hi⟩
-    have he := congrArg (headTailEquiv d) hi
-    rw [hp i] at he
-    change (col i,_) = (b,y) at he
-    have hb := congrArg Prod.fst he
-    have hy := congrArg Prod.snd he
-    dsimp only at hb hy
-    refine ⟨⟨i,hb⟩,?_⟩
-    change (butterflyPerm d (decodeButterfly d (if b then c.2 else c.1))).symm (x i).2 = y
-    simpa only [hb] using hy
-  · rintro ⟨i,hi⟩
-    refine ⟨i.val,?_⟩
-    apply (headTailEquiv d).injective
-    rw [hp]
-    change (col i.val,_) = (b,y)
-    have hb : col i.val = b := i.property
-    rw [hb]
-    exact Prod.ext rfl hi
-
-lemma inverseButterflyMarks_card (d : ℕ) (A : Finset (Card d)) (ω : SwitchIndex d → Bool) :
-    (inverseButterflyMarks d A ω).card = A.card := by
-  classical
-  exact Finset.card_image_of_injective _ (butterflyPerm d (decodeButterfly d ω)).symm.injective
-
-lemma benes_input_coordinates (d : ℕ) (ω : BenesCoins (d+1)) :
-    ω.2 = (coinStepEquiv d).symm
-      (((benesStepEquiv d ω).1.1.2,(benesStepEquiv d ω).1.2.2),(benesStepEquiv d ω).2.1) := by
-  change ω.2 = (coinStepEquiv d).symm (coinStepEquiv d ω.2)
-  exact ((coinStepEquiv d).symm_apply_apply ω.2).symm
-
-lemma cutoffMass_central (r H : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (H+r)) (ω : BenesCoins (H+r)) :
-    cutoffMass H (H+r) e ω =
-      sparseMass (centralBlock r H) (inverseButterflyMarks (H+r) (Finset.univ.image e) ω.2) := by
-  classical
-  induction r generalizing ι with
-  | zero =>
-    rw [sparseMass_central_zero,inverseButterflyMarks_card,
-      Finset.card_image_of_injective _ e.injective,Finset.card_univ]
-    cases H with
-    | zero =>
-      have hcard : Fintype.card ι ≤ 1 := by
-        have hh := Fintype.card_le_of_injective e e.injective
-        simpa only [Nat.add_zero,card_positions,pow_zero] using hh
-      simp [cutoffMass,crowdedMass,show ¬2≤Fintype.card ι by omega]
-    | succ H => simp only [cutoffMass,le_refl,ite_true]
-  | succ r ih =>
-    simp only [Nat.add_succ] at *
-    rw [cutoffMass,ite_eq_right (by omega : ¬H+r+1≤H)]
-    let σ := benesStepEquiv (H+r) ω
-    let x := e.trans (headTailEquiv (H+r)).toEmbedding
-    let col := PairRouting.colors x σ.2.1
-    let hc := PairRouting.colors_compatible x σ.2.1
-    change cutoffMass H (H+r) (PairRouting.childEmbedding x col hc false) σ.1.1 +
-      cutoffMass H (H+r) (PairRouting.childEmbedding x col hc true) σ.1.2 = _
-    rw [ih,ih,sparseMass_central_step,benes_input_coordinates (H+r) ω]
-    rw [inverseButterflyMarks_child,inverseButterflyMarks_child]
-    rfl
-
-
-theorem palindrome_low_mgf (r H : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (H+r)) (q : ℝ) (hq : 1 ≤ q)
-    (hr : (2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*q^H) ≤ 1/2) :
-    finiteMean (fun ω : BenesCoins (H+r) => q^(palindromeLowCost H (H+r) e ω)) ≤
-      Real.exp ((2:ℝ)^r * (2 * ((2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*q^H))^2)) := by
-  classical
-  have hec : (Finset.univ.image e).card = Fintype.card ι := by
-    rw [Finset.card_image_of_injective _ e.injective,Finset.card_univ]
-  have hm := inverseButterfly_sparse_mgf (H+r) (centralBlock r H)
-    (centralBlock_disjoint r H) (Finset.univ.image e) (q^H)
-    (pow_nonneg (by linarith) _) (fun b => by
-      simpa only [centralBlock_card,hec,Nat.cast_pow,Nat.cast_ofNat] using hr)
-  calc
-    _ ≤ finiteMean (fun ω : BenesCoins (H+r) => (q^H)^(cutoffMass H (H+r) e ω)) := by
-      apply finiteMean_mono
-      intro ω
-      rw [←pow_mul]
-      exact pow_le_pow_right₀ hq (palindromeLowCost_le_cutoff H (H+r) e ω)
-    _ = finiteMean (fun ω : SwitchIndex (H+r) → Bool =>
-        (q^H)^(sparseMass (centralBlock r H)
-          (inverseButterflyMarks (H+r) (Finset.univ.image e) ω))) := by
-      simp_rw [cutoffMass_central]
-      simp only [finiteMean_prod,finiteMean_const]
-    _ ≤ _ := by
-      simpa only [centralBlock_card,hec,Nat.cast_pow,Nat.cast_ofNat,Finset.sum_const,
-        Finset.card_univ,card_positions,nsmul_eq_mul] using hm
-
-variable {α B : Type*} {r s : ℕ}
-
-def parallelFreshPre (pre : α ≃ B × Card (s+r))
-    (lo : B → Card r × SwitchIndex s → Bool) : α ≃ (B × Card s) × Card r :=
-  ((pre.trans (Equiv.prodCongr (Equiv.refl B) (cardSplit r s))).trans
-    (Equiv.prodCongrRight (fun b =>
-      (parallelPerm (lo b)).trans (Equiv.prodComm _ _)))).trans
-    (Equiv.prodAssoc _ _ _).symm
-
-def parallelFreshPost (post : B × Card (s+r) ≃ α) : (B × Card s) × Card r ≃ α :=
-  (((Equiv.prodAssoc _ _ _).trans (Equiv.prodCongr (Equiv.refl B)
-    (Equiv.prodComm _ _))).trans (Equiv.prodCongr (Equiv.refl B) (cardSplit r s).symm)).trans post
-
-def parallelAssembled (lo : B → Card r × SwitchIndex s → Bool)
-    (hi : (B × Card s) × SwitchIndex r → Bool) : B × SwitchIndex (s+r) → Bool :=
-  fun i => assembleBits r s (lo i.1) (fun j => hi ((i.1,j.1),j.2)) i.2
-
-lemma parallel_exposed_perm (pre : α ≃ B × Card (s+r)) (post : B × Card (s+r) ≃ α)
-    (lo : B → Card r × SwitchIndex s → Bool)
-    (hi : (B × Card s) × SwitchIndex r → Bool) :
-    relabeledPerm pre post (parallelAssembled lo hi) =
-      relabeledPerm (parallelFreshPre pre lo) (parallelFreshPost post) hi := by
-  ext x
-  have h := butterfly_split r s (lo (pre x).1)
-    (fun j => hi (((pre x).1,j.1),j.2)) (pre x).2
-  have hh := congrArg (fun y => post ((pre x).1,(cardSplit r s).symm y)) h
-  simpa only [relabeledPerm,parallelAssembled,parallelPerm,parallelFreshPre,parallelFreshPost,
-    Equiv.trans_apply,Equiv.prodCongr_apply,Equiv.prodCongrRight,Equiv.coe_fn_mk,
-    Equiv.refl_apply,Equiv.prodComm_apply,Prod.swap,Equiv.prodAssoc_apply,
-    Equiv.prodAssoc_symm_apply,Prod.map,Equiv.symm_apply_apply] using hh
-
-theorem parallel_partially_exposed_log_mgf [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B]
-    (pre : α ≃ B × Card (s+r)) (post : B × Card (s+r) ≃ α)
-    (lo : B → Card r × SwitchIndex s → Bool) (S : Finset α)
-    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    Real.log (finiteMean (fun hi : (B × Card s) × SwitchIndex r → Bool =>
-      q^(Fintype.card (RoutingNetwork.SelectedCycle
-        (relabeledPerm pre post (parallelAssembled lo hi)) S)))) ≤
-      (S.card:ℝ)/J*Real.log q + 2*q*S.card*Real.sqrt J/Real.sqrt ((2:ℝ)^r) := by
-  simp_rw [parallel_exposed_perm]
-  exact parallel_cycle_log_mgf (parallelFreshPre pre lo) (parallelFreshPost post) S J hJ q hq
-
-end Thorp
-namespace Thorp
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-lemma orbitSet_inverse (p : Equiv.Perm α) (x : α) : orbitSet p⁻¹ x = orbitSet p x := by
-  ext y
-  simp only [mem_orbitSet,Equiv.Perm.sameCycle_inv]
-
-lemma selectedCycle_inverse (p : Equiv.Perm α) (S : Finset α) :
-    Fintype.card (RoutingNetwork.SelectedCycle p⁻¹ S) =
-      Fintype.card (RoutingNetwork.SelectedCycle p S) := by
-  apply Fintype.card_congr
-  refine { toFun := fun C => ⟨C.val,?_,C.property.2⟩
-           invFun := fun C => ⟨C.val,?_,C.property.2⟩
-           left_inv := fun _ => rfl
-           right_inv := fun _ => rfl }
-  · obtain ⟨x,hx⟩ := C.property.1
-    exact ⟨x,hx.trans (orbitSet_inverse p x)⟩
-  · obtain ⟨x,hx⟩ := C.property.1
-    exact ⟨x,hx.trans (orbitSet_inverse p x).symm⟩
-
-lemma orbitSet_conjugate (p g : Equiv.Perm α) (x : α) :
-    orbitSet (g*p*g⁻¹) (g x) = (orbitSet p x).image g := by
-  ext y
-  simp only [mem_orbitSet,Finset.mem_image]
-  rw [Equiv.Perm.sameCycle_conj]
-  change p.SameCycle (g.symm (g x)) (g.symm y) ↔ ∃ z, p.SameCycle x z ∧ g z = y
-  rw [Equiv.symm_apply_apply]
-  constructor
-  · intro h
-    exact ⟨g.symm y,h,g.apply_symm_apply y⟩
-  · rintro ⟨z,hz,rfl⟩
-    simpa only [Equiv.symm_apply_apply] using hz
-
-noncomputable def conjugateCycle (p g : Equiv.Perm α) (S : Finset α)
-    (C : RoutingNetwork.SelectedCycle p S) :
-    RoutingNetwork.SelectedCycle (g*p*g⁻¹) (S.image g) := by
-  refine ⟨C.val.image g,?_,Finset.image_subset_image C.property.2⟩
-  obtain ⟨x,hx⟩ := C.property.1
-  exact ⟨g x,by rw [hx,orbitSet_conjugate]⟩
-
-lemma conjugateCycle_injective (p g : Equiv.Perm α) (S : Finset α) :
-    Function.Injective (conjugateCycle p g S) := by
-  intro C D he
-  apply Subtype.ext
-  exact (Finset.image_injective g.injective) (congrArg Subtype.val he)
-
-lemma selectedCycle_conjugate (p g : Equiv.Perm α) (S : Finset α) :
-    Fintype.card (RoutingNetwork.SelectedCycle (g*p*g⁻¹) (S.image g)) =
-      Fintype.card (RoutingNetwork.SelectedCycle p S) := by
-  apply Nat.le_antisymm
-  · have h := Fintype.card_le_of_injective _
-      (conjugateCycle_injective (g*p*g⁻¹) g⁻¹ (S.image g))
-    have hp : g⁻¹*(g*p*g⁻¹)*g = p := by group
-    rw [inv_inv,hp] at h
-    have hs : (S.image g).image g.symm = S := by
-      ext x
-      simp only [Finset.mem_image]
-      constructor
-      · rintro ⟨y,⟨z,hz,rfl⟩,he⟩
-        have hz' : z = x := by simpa only [Equiv.symm_apply_apply] using he
-        exact hz' ▸ hz
-      · intro hx
-        exact ⟨g x,⟨x,hx,rfl⟩,g.symm_apply_apply x⟩
-    change _ ≤ Fintype.card (RoutingNetwork.SelectedCycle p ((S.image g).image g.symm)) at h
-    rw [hs] at h
-    exact h
-  · exact Fintype.card_le_of_injective _ (conjugateCycle_injective p g S)
-
-lemma alternatingCycles_le_projection {ι : Type*} [Fintype ι]
-    (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
-    PairRouting.alternatingCycles e p ≤
-      Fintype.card (RoutingNetwork.SelectedCycle (PairRouting.alternatingProjection p)
-        (boolFiber (PairRouting.labelSet e) false)) := by
-  have h := CycleColoring.card_alternating_cycles_le (PairRouting.alternatingProjection p)
-    (boolFiber (PairRouting.labelSet e) false) (boolFiber (PairRouting.labelSet e) true)
-  simpa only [PairRouting.alternatingCycles,childJoin_boolFiber] using h
-
-lemma projectionCycles_forward (X₀ X₁ Y₀ Y₁ : Equiv.Perm α) (S : Finset α) :
-    Fintype.card (RoutingNetwork.SelectedCycle
-      ((Y₀*X₀⁻¹)⁻¹*(Y₁*X₁⁻¹)) S) =
-    Fintype.card (RoutingNetwork.SelectedCycle
-      (X₀⁻¹*X₁*Y₁⁻¹*Y₀) (S.image X₀.symm)) := by
-  let p := (Y₀*X₀⁻¹)⁻¹*(Y₁*X₁⁻¹)
-  calc
-    _ = Fintype.card (RoutingNetwork.SelectedCycle (X₀⁻¹*p*X₀)
-        (S.image X₀.symm)) := by
-      change Fintype.card (RoutingNetwork.SelectedCycle p S) = _
-      have h := (selectedCycle_conjugate p X₀⁻¹ S).symm
-      rw [inv_inv] at h
-      exact h
-    _ = Fintype.card (RoutingNetwork.SelectedCycle (X₀⁻¹*p*X₀)⁻¹
-        (S.image X₀.symm)) := (selectedCycle_inverse _ _).symm
-    _ = _ := by
-      have he : (X₀⁻¹*p*X₀)⁻¹ = X₀⁻¹*X₁*Y₁⁻¹*Y₀ := by dsimp only [p]; group
-      rw [he]
-
-end Thorp
-namespace Thorp
-
-noncomputable def cycleCoefficient (r J : ℕ) (q : ℝ) : ℝ :=
-  Real.log q / J + 2*q*Real.sqrt J/Real.sqrt ((2:ℝ)^r)
-
-lemma cycleCoefficient_nonneg (r J : ℕ) (q : ℝ) (hq : 1 ≤ q) :
-    0 ≤ cycleCoefficient r J q := by
-  unfold cycleCoefficient
-  exact add_nonneg (div_nonneg (Real.log_nonneg hq) (Nat.cast_nonneg _))
-    (div_nonneg (mul_nonneg (by linarith) (Real.sqrt_nonneg _)) (Real.sqrt_nonneg _))
-
-lemma projectionSide_card_le {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι]
-    (e : ι ↪ Bool × α) : (boolFiber (PairRouting.labelSet e) false).card ≤ Fintype.card ι := by
-  have h := card_childJoin (boolFiber (PairRouting.labelSet e) false)
-    (boolFiber (PairRouting.labelSet e) true)
-  rw [childJoin_boolFiber,PairRouting.card_labelSet] at h
-  omega
-
-theorem node_partial_mgf (r s : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Bool × Card (s+r)) (X₀ X₁ Y₁ : Equiv.Perm (Card (s+r)))
-    (lo : Card r × SwitchIndex s → Bool) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(PairRouting.alternatingCycles e (fun b => if b then Y₁*X₁⁻¹ else
-        butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi))*X₀⁻¹))) ≤
-      Real.exp (Fintype.card ι * cycleCoefficient r J q) := by
-  classical
-  let S := boolFiber (PairRouting.labelSet e) false
-  let S' := S.image X₀.symm
-  let post := X₀⁻¹*X₁*Y₁⁻¹
-  let count := fun hi : Card s × SwitchIndex r → Bool =>
-    Fintype.card (RoutingNetwork.SelectedCycle
-      ((Equiv.refl _).trans (butterflyPerm (s+r)
-        (decodeButterfly (s+r) (assembleBits r s lo hi))) |>.trans post) S')
-  have hpoint (hi : Card s × SwitchIndex r → Bool) :
-      PairRouting.alternatingCycles e (fun b => if b then Y₁*X₁⁻¹ else
-        butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi))*X₀⁻¹) ≤ count hi := by
-    refine (alternatingCycles_le_projection e _).trans_eq ?_
-    simp only [PairRouting.alternatingProjection,Bool.false_eq_true,↓reduceIte]
-    change Fintype.card (RoutingNetwork.SelectedCycle
-      ((butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi))*X₀⁻¹)⁻¹ * (Y₁*X₁⁻¹)) S) = count hi
-    exact projectionCycles_forward X₀ X₁ _ Y₁ S
-  have hm := partially_exposed_cycle_log_mgf (Equiv.refl (Card (s+r))) post lo S' J hJ q hq
-  have hp : 0 < finiteMean (fun hi : Card s × SwitchIndex r → Bool => q^(count hi)) :=
-    finiteMean_pos (fun _ => pow_pos (by linarith) _)
-  have hb : finiteMean (fun hi : Card s × SwitchIndex r → Bool => q^(count hi)) ≤
-      Real.exp (S'.card * cycleCoefficient r J q) := by
-    rw [←Real.exp_log hp]
-    apply Real.exp_le_exp.mpr
-    calc
-      _ ≤ _ := hm
-      _ = _ := by unfold cycleCoefficient; ring
-  have hcard : S'.card ≤ Fintype.card ι := by
-    rw [show S'.card = S.card from Finset.card_image_of_injective _ X₀.symm.injective]
-    exact projectionSide_card_le e
-  calc
-    _ ≤ finiteMean (fun hi : Card s × SwitchIndex r → Bool => q^(count hi)) :=
-      finiteMean_mono (fun hi => pow_le_pow_right₀ hq (hpoint hi))
-    _ ≤ Real.exp (S'.card * cycleCoefficient r J q) := hb
-    _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right
-      (by exact_mod_cast hcard) (cycleCoefficient_nonneg r J q hq))
-
-
-def highStepEquiv (r s : ℕ) : (Card s × SwitchIndex (r+1) → Bool) ≃
-    (((Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool)) ×
-      (Card s × Card r → Bool)) where
-  toFun hi := ((fun j => hi (j.1,Sum.inr (false,j.2)),
-    fun j => hi (j.1,Sum.inr (true,j.2))),fun j => hi (j.1,Sum.inl j.2))
-  invFun c := fun j => match j.2 with
-    | Sum.inl y => c.2 (j.1,y)
-    | Sum.inr (b,i) => if b then c.1.2 (j.1,i) else c.1.1 (j.1,i)
-  left_inv hi := by
-    funext j
-    rcases j with ⟨z,y|⟨b,i⟩⟩
-    · rfl
-    · cases b <;> rfl
-  right_inv c := by
-    rcases c with ⟨⟨h₀,h₁⟩,η⟩
-    rfl
-
-lemma assembleBits_step (r s : ℕ) (lo : Card (r+1) × SwitchIndex s → Bool)
-    (c : ((Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool)) ×
-      (Card s × Card r → Bool)) :
-    assembleBits (r+1) s lo ((highStepEquiv r s).symm c) =
-      (coinStepEquiv (s+r)).symm
-        ((assembleBits r s (fun j => lo (Fin.cons false j.1,j.2)) c.1.1,
-          assembleBits r s (fun j => lo (Fin.cons true j.1,j.2)) c.1.2),
-        fun y => c.2 ((cardSplit r s y).2,(cardSplit r s y).1)) := by
-  funext i
-  rcases i with y | ⟨b,i⟩
-  · rfl
-  · cases b <;> rfl
-
-noncomputable def palindromeLevelCost (t : ℕ) : (d : ℕ) → {ι : Type*} → [Fintype ι] →
-    (ι ↪ Card d) → BenesCoins d → ℕ
-  | 0, _, _, _, _ => 0
-  | d+1, _, _, e, ω =>
-      let σ := benesStepEquiv d ω
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x σ.2.1
-      let hc := PairRouting.colors_compatible x σ.2.1
-      if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
-          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) else
-        palindromeLevelCost t d (PairRouting.childEmbedding x c hc false) σ.1.1 +
-          palindromeLevelCost t d (PairRouting.childEmbedding x c hc true) σ.1.2
-
-lemma palindromeLevelCost_step (t d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card (d+1)) (X Y : (SwitchIndex d → Bool) × (SwitchIndex d → Bool))
-    (ξ η : Card d → Bool) :
-    palindromeLevelCost t (d+1) e
-      ((coinStepEquiv d).symm (Y,η),(coinStepEquiv d).symm (X,ξ)) =
-      let x := e.trans (headTailEquiv d).toEmbedding
-      let c := PairRouting.colors x ξ
-      let hc := PairRouting.colors_compatible x ξ
-      if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x ξ)
-        (fun b => palindromePerm d (if b then (Y.2,X.2) else (Y.1,X.1))) else
-        palindromeLevelCost t d (PairRouting.childEmbedding x c hc false) (Y.1,X.1) +
-        palindromeLevelCost t d (PairRouting.childEmbedding x c hc true) (Y.2,X.2) := by
-  rw [palindromeLevelCost]
-  rfl
-
-lemma palindromeLevelCost_gt (t d : ℕ) (h : d < t) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) : palindromeLevelCost t d e ω = 0 := by
-  classical
-  induction d generalizing ι with
-  | zero => rfl
-  | succ d ih =>
-    rw [palindromeLevelCost,ite_eq_right (by omega : t ≠ d+1),ih (by omega),ih (by omega),Nat.zero_add]
-
-
-theorem palindrome_level_partial_mgf (r s u : ℕ) (hu : u < r)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+r)) (X : SwitchIndex (s+r) → Bool)
-    (lo : Card r × SwitchIndex s → Bool) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(palindromeLevelCost (s+u+1) (s+r) e (assembleBits r s lo hi,X))) ≤
-      Real.exp (Fintype.card ι * cycleCoefficient u J q) := by
-  classical
-  induction r generalizing u ι with
-  | zero => omega
-  | succ r ih =>
-    simp only [Nat.add_succ] at *
-    let χ := coinStepEquiv (s+r) X
-    let x := e.trans (headTailEquiv (s+r)).toEmbedding
-    let c := PairRouting.colors x χ.2
-    let hc := PairRouting.colors_compatible x χ.2
-    let e₀ := PairRouting.childEmbedding x c hc false
-    let e₁ := PairRouting.childEmbedding x c hc true
-    let e' := PairRouting.switchedEmbedding x χ.2
-    let lo₀ : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons false j.1,j.2)
-    let lo₁ : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons true j.1,j.2)
-    let Y₀ := fun hi : Card s × SwitchIndex r → Bool => assembleBits r s lo₀ hi
-    let Y₁ := fun hi : Card s × SwitchIndex r → Bool => assembleBits r s lo₁ hi
-    let N := fun hi : (Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool) =>
-      PairRouting.alternatingCycles e' (fun b => if b then palindromePerm (s+r) (Y₁ hi.2,χ.1.2)
-        else palindromePerm (s+r) (Y₀ hi.1,χ.1.1))
-    let L₀ := fun hi : Card s × SwitchIndex r → Bool =>
-      palindromeLevelCost (s+u+1) (s+r) e₀ (Y₀ hi,χ.1.1)
-    let L₁ := fun hi : Card s × SwitchIndex r → Bool =>
-      palindromeLevelCost (s+u+1) (s+r) e₁ (Y₁ hi,χ.1.2)
-    have hs (hi : (Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool))
-        (η : Card s × Card r → Bool) :
-        palindromeLevelCost (s+u+1) (s+r+1) e
-          (assembleBits (r+1) s lo ((highStepEquiv r s).symm (hi,η)),X) =
-          if u = r then N hi else L₀ hi.1 + L₁ hi.2 := by
-      rw [assembleBits_step]
-      conv_lhs => rw [←(coinStepEquiv (s+r)).symm_apply_apply X]
-      rw [palindromeLevelCost_step]
-      simp only [Nat.add_right_cancel_iff,Nat.add_left_cancel_iff]
-      dsimp only [χ,x,c,hc,e₀,e₁,e',lo₀,lo₁,Y₀,Y₁,N,L₀,L₁]
-      congr 2
-      funext b
-      cases b <;> rfl
-    rw [finiteMean_equiv (highStepEquiv r s),finiteMean_prod]
-    simp_rw [hs,finiteMean_const]
-    by_cases hur : u = r
-    · subst u
-      simp only [ite_true]
-      rw [finiteMean_prod,finiteMean_comm]
-      calc
-        _ ≤ finiteMean (fun _hi : Card s × SwitchIndex r → Bool =>
-          Real.exp (Fintype.card ι * cycleCoefficient r J q)) := by
-          apply finiteMean_mono
-          intro hi
-          have hn := node_partial_mgf r s e'
-            (butterflyPerm (s+r) (decodeButterfly (s+r) χ.1.1))
-            (butterflyPerm (s+r) (decodeButterfly (s+r) χ.1.2))
-            (butterflyPerm (s+r) (decodeButterfly (s+r) (Y₁ hi))) lo₀ J hJ q hq
-          exact hn
-        _ = _ := finiteMean_const _
-    · simp only [ite_eq_right hur,pow_add]
-      rw [finiteMean_prod_mul (fun hi => q ^ L₀ hi) (fun hi => q ^ L₁ hi)]
-      have hu' : u < r := by omega
-      have h₀ := ih u hu' e₀ χ.1.1 lo₀
-      have h₁ := ih u hu' e₁ χ.1.2 lo₁
-      have hp := mul_le_mul h₀ h₁ (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
-        (Real.exp_nonneg _)
-      refine hp.trans_eq ?_
-      rw [←Real.exp_add]
-      congr 1
-      have hcount := PairRouting.color_card_add c
-      have hcount' : (Fintype.card {i : ι // c i = false} : ℝ) +
-          Fintype.card {i : ι // c i = true} = Fintype.card ι := by exact_mod_cast hcount
-      change (Fintype.card {i : ι // c i = false} : ℝ) * cycleCoefficient u J q +
-          Fintype.card {i : ι // c i = true} * cycleCoefficient u J q = _
-      rw [←add_mul,hcount']
-
-
-def switchHeight : (d : ℕ) → SwitchIndex d → ℕ
-  | 0, i => nomatch i
-  | d+1, Sum.inl _ => d+1
-  | d+1, Sum.inr (_,i) => switchHeight d i
-
-lemma switchHeight_le (d : ℕ) (i : SwitchIndex d) : switchHeight d i ≤ d := by
-  induction d with
-  | zero => exact Empty.elim i
-  | succ d ih =>
-    rcases i with y | ⟨b,i⟩
-    · exact le_refl _
-    · exact (ih i).trans (Nat.le_succ d)
-
-def AgreeThrough (d s : ℕ) (Y Y' : SwitchIndex d → Bool) : Prop :=
-  ∀ i, switchHeight d i ≤ s → Y i = Y' i
-
-def LayerAdapted (d s : ℕ) {β : Type*} (f : (SwitchIndex d → Bool) → β) : Prop :=
-  ∀ Y Y', AgreeThrough d s Y Y' → f Y = f Y'
-
-lemma agreeThrough_mono {d s t : ℕ} (h : s ≤ t) {Y Y' : SwitchIndex d → Bool}
-    (hh : AgreeThrough d t Y Y') : AgreeThrough d s Y Y' :=
-  fun i hi => hh i (hi.trans h)
-
-lemma layerAdapted_mono {d s t : ℕ} {β : Type*} {f : (SwitchIndex d → Bool) → β}
-    (hf : LayerAdapted d s f) (h : s ≤ t) : LayerAdapted d t f :=
-  fun Y Y' hh => hf Y Y' (agreeThrough_mono h hh)
-
-lemma layerAdapted_comp {d s : ℕ} {α β : Type*} {f : (SwitchIndex d → Bool) → α}
-    (hf : LayerAdapted d s f) (g : α → β) : LayerAdapted d s (g ∘ f) :=
-  fun Y Y' h => congrArg g (hf Y Y' h)
-
-lemma layerAdapted_sum {d s : ℕ} {ι : Type*} (A : Finset ι)
-    (f : ι → (SwitchIndex d → Bool) → ℕ)
-    (hf : ∀ i ∈ A, LayerAdapted d s (f i)) :
-    LayerAdapted d s (fun Y => ∑ i ∈ A, f i Y) := by
-  intro Y Y' h
-  exact Finset.sum_congr rfl (fun i hi => hf i hi Y Y' h)
-
-lemma assembleBits_agree (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
-    (hi hi' : Card s × SwitchIndex r → Bool) :
-    AgreeThrough (s+r) s (assembleBits r s lo hi) (assembleBits r s lo hi') := by
-  induction r with
-  | zero => intro i _; rfl
-  | succ r ih =>
-    intro i h
-    rcases i with y | ⟨b,i⟩
-    · have : s+r+1 ≤ s := h
-      omega
-    · exact ih (fun j => lo (Fin.cons b j.1,j.2))
-        (fun j => hi (j.1,Sum.inr (b,j.2))) (fun j => hi' (j.1,Sum.inr (b,j.2))) i h
-
-lemma palindromeLevelCost_adapted (t d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) :
-    LayerAdapted d (t-1) (fun Y => palindromeLevelCost t d e (Y,X)) := by
-  classical
-  induction d generalizing ι with
-  | zero => intro Y Y' _; rfl
-  | succ d ih =>
-    intro Y Y' h
-    change palindromeLevelCost t (d+1) e (Y,X) = palindromeLevelCost t (d+1) e (Y',X)
-    conv_lhs => rw [←(coinStepEquiv d).symm_apply_apply Y,←(coinStepEquiv d).symm_apply_apply X]
-    conv_rhs => rw [←(coinStepEquiv d).symm_apply_apply Y',←(coinStepEquiv d).symm_apply_apply X]
-    rw [palindromeLevelCost_step,palindromeLevelCost_step]
-    by_cases ht : t = d+1
-    · subst t
-      rw [ite_eq_left rfl,ite_eq_left rfl]
-      have he₀ : (coinStepEquiv d Y).1.1 = (coinStepEquiv d Y').1.1 := by
-        funext i
-        exact h (Sum.inr (false,i)) (by simpa only [switchHeight,Nat.add_sub_cancel] using switchHeight_le d i)
-      have he₁ : (coinStepEquiv d Y).1.2 = (coinStepEquiv d Y').1.2 := by
-        funext i
-        exact h (Sum.inr (true,i)) (by simpa only [switchHeight,Nat.add_sub_cancel] using switchHeight_le d i)
-      rw [he₀,he₁]
-    · rw [ite_eq_right ht,ite_eq_right ht]
-      congr 1
-      · apply ih
-        intro i hi
-        exact h (Sum.inr (false,i)) hi
-      · apply ih
-        intro i hi
-        exact h (Sum.inr (true,i)) hi
-
-lemma adapted_mul_mean_le (r s : ℕ) (F G : (SwitchIndex (s+r) → Bool) → ℝ)
-    (hF : ∀ Y, 0 ≤ F Y) (hA : LayerAdapted (s+r) s F) (B : ℝ)
-    (hG : ∀ lo : Card r × SwitchIndex s → Bool,
-      finiteMean (fun hi : Card s × SwitchIndex r → Bool => G (assembleBits r s lo hi)) ≤ B) :
-    finiteMean (fun Y => F Y*G Y) ≤ finiteMean F * B := by
-  rw [finiteMean_split r s,finiteMean_split r s F,←finiteMean_mul_const]
-  apply finiteMean_mono
-  intro lo
-  let H : Card s × SwitchIndex r → Bool := fun _ => false
-  have hh (hi : Card s × SwitchIndex r → Bool) :
-      F (assembleBits r s lo hi) = F (assembleBits r s lo H) :=
-    hA _ _ (assembleBits_agree r s lo hi H)
-  simp_rw [hh,finiteMean_const]
-  rw [show (fun hi => F (assembleBits r s lo H) * G (assembleBits r s lo hi)) =
-      (fun hi => G (assembleBits r s lo hi) * F (assembleBits r s lo H)) by
-        funext hi; exact mul_comm _ _,finiteMean_mul_const]
-  simpa only [mul_comm] using mul_le_mul_of_nonneg_left (hG lo) (hF _)
-
-
-theorem cycleCoefficient_decay (l r : ℕ) (hr : (l:ℝ)/2 ≤ r)
-    (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ (l:ℝ)/250) :
-    cycleCoefficient r ⌈Real.exp ((l:ℝ)/32)⌉₊ q ≤
-      68 * Real.exp (-(l:ℝ)/64) := by
-  let E := Real.exp ((l:ℝ)/32)
-  let J := ⌈E⌉₊
-  have hl : (0:ℝ) ≤ l := Nat.cast_nonneg _
-  have hE : 1 ≤ E := Real.one_le_exp_iff.mpr (by positivity)
-  have hEpos : 0 < E := Real.exp_pos _
-  have hEJ : E ≤ (J:ℝ) := Nat.le_ceil E
-  have hJ : (1:ℝ) ≤ J := hE.trans hEJ
-  have hJpos : (0:ℝ) < J := lt_of_lt_of_le (by norm_num) hJ
-  have hJE : (J:ℝ) ≤ 2*E := by
-    have h := Nat.ceil_lt_add_one (le_of_lt hEpos)
-    change (J:ℝ) < E+1 at h
-    linarith
-  have hsqJ : Real.sqrt J ≤ (J:ℝ) := by
-    apply Real.sqrt_le_iff.mpr
-    constructor
-    · positivity
-    · nlinarith
-  have hqp : 0 < q := lt_of_lt_of_le (by norm_num) hq
-  have hqE : q ≤ Real.exp ((l:ℝ)/250) := by
-    rw [←Real.exp_log hqp]
-    exact Real.exp_le_exp.mpr hlog
-  have hln : (1/2:ℝ) ≤ Real.log 2 := by
-    have hh := Real.one_sub_inv_le_log_of_pos (by norm_num : (0:ℝ)<2)
-    norm_num at hh ⊢
-    exact hh
-  have hden : Real.exp ((l:ℝ)/8) ≤ Real.sqrt ((2:ℝ)^r) := by
-    have he : (2:ℝ)^r = Real.exp ((r:ℝ)*Real.log 2) := by
-      rw [Real.exp_nat_mul,Real.exp_log (by norm_num)]
-    rw [he,Real.sqrt_eq_rpow,Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp]
-    apply Real.exp_le_exp.mpr
-    have hrn : (0:ℝ) ≤ r := Nat.cast_nonneg _
-    nlinarith
-  have hlog' : Real.log q ≤ (l:ℝ) := by linarith
-  have h₁ : Real.log q / (J:ℝ) ≤ 64*Real.exp (-(l:ℝ)/64) := by
-    calc
-      _ ≤ (l:ℝ)/E := div_le_div₀ (by positivity) hlog' hEpos hEJ
-      _ ≤ (64*Real.exp ((l:ℝ)/64))/E := by
-        apply div_le_div_of_nonneg_right _ (le_of_lt hEpos)
-        have hh := Real.add_one_le_exp ((l:ℝ)/64)
-        linarith
-      _ = _ := by
-        dsimp [E]
-        rw [mul_div_assoc,←Real.exp_sub]
-        congr 2
-        ring
-  have h₂ : 2*q*Real.sqrt J / Real.sqrt ((2:ℝ)^r) ≤
-      4*Real.exp (-(l:ℝ)/64) := by
-    calc
-      _ ≤ (4*Real.exp ((l:ℝ)/250)*E)/Real.exp ((l:ℝ)/8) := by
-        apply div_le_div₀ (by positivity) _ (Real.exp_pos _) hden
-        have hs := hsqJ.trans hJE
-        have hm := mul_le_mul hqE hs (Real.sqrt_nonneg _) (Real.exp_nonneg _)
-        nlinarith
-      _ = 4*Real.exp ((l:ℝ)/250+(l:ℝ)/32-(l:ℝ)/8) := by
-        dsimp [E]
-        rw [mul_assoc,←Real.exp_add,mul_div_assoc,←Real.exp_sub]
-      _ ≤ _ := by
-        apply mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr _) (by norm_num)
-        linarith
-  change Real.log q / (J:ℝ) + 2*q*Real.sqrt J / Real.sqrt ((2:ℝ)^r) ≤ _
-  linarith
-
-lemma cycleCutoff_pos (l : ℕ) : 0 < ⌈Real.exp ((l:ℝ)/32)⌉₊ := by
-  have h := Nat.le_ceil (Real.exp ((l:ℝ)/32))
-  have he := Real.exp_pos ((l:ℝ)/32)
-  have : (0:ℝ) < ⌈Real.exp ((l:ℝ)/32)⌉₊ := lt_of_lt_of_le he h
-  exact_mod_cast this
-
-
-lemma pow_sum_le_mean_powers {ι : Type*} [Fintype ι] [Nonempty ι]
-    (z : ι → ℕ) (q : ℝ) (hq : 0 < q) :
-    q^(∑ i, z i) ≤ finiteMean (fun i => q^(Fintype.card ι*z i)) := by
-  have hn : (Fintype.card ι:ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
-  have hm : finiteMean (fun i => (Fintype.card ι*z i:ℕ)*Real.log q) =
-      (∑ i, z i:ℕ)*Real.log q := by
-    simp only [finiteMean,Nat.cast_mul,Nat.cast_sum,←Finset.mul_sum,←Finset.sum_mul]
-    field_simp
-  have he := exp_finiteMean_le (fun i => (Fintype.card ι*z i:ℕ)*Real.log q)
-  rw [hm,Real.exp_nat_mul,Real.exp_log hq] at he
-  simpa only [Real.exp_nat_mul,Real.exp_log hq] using he
-
-noncomputable def palindromeSlabCost (l d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) : ℕ :=
-  ∑ i : Fin l, palindromeLevelCost (l+i.1+1) d e ω
-
-lemma palindromeSlabCost_adapted (l d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) :
-    LayerAdapted d (2*l) (fun Y => palindromeSlabCost l d e (Y,X)) := by
-  apply layerAdapted_sum
-  intro i _
-  exact layerAdapted_mono (palindromeLevelCost_adapted (l+i.1+1) d e X) (by omega)
-
-theorem palindrome_slab_partial_mgf (r s l : ℕ) (hl : 0 < l) (hs : 2*s ≤ l)
-    {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+r)) (X : SwitchIndex (s+r) → Bool)
-    (lo : Card r × SwitchIndex s → Bool) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(palindromeSlabCost l (s+r) e (assembleBits r s lo hi,X))) ≤
-      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
-  classical
-  let : NeZero l := ⟨ne_of_gt hl⟩
-  have hqp : 0 < q := lt_of_lt_of_le (by norm_num) hq
-  calc
-    _ ≤ finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      finiteMean (fun i : Fin l => q^(l*palindromeLevelCost (l+i.1+1) (s+r) e
-        (assembleBits r s lo hi,X)))) := by
-      apply finiteMean_mono
-      intro hi
-      simpa only [palindromeSlabCost,Fintype.card_fin] using
-        pow_sum_le_mean_powers (fun i : Fin l => palindromeLevelCost (l+i.1+1) (s+r) e
-          (assembleBits r s lo hi,X)) q hqp
-    _ = finiteMean (fun i : Fin l => finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
-      q^(l*palindromeLevelCost (l+i.1+1) (s+r) e (assembleBits r s lo hi,X)))) := finiteMean_comm _
-    _ ≤ finiteMean (fun _i : Fin l =>
-      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64)))) := by
-      apply finiteMean_mono
-      intro i
-      by_cases ht : l+i.1+1 ≤ s+r
-      · have hs' : s ≤ l+i.1 := by omega
-        have he : l+i.1+1 = s+(l+i.1-s)+1 := by omega
-        have hu : l+i.1-s < r := by omega
-        have hm := palindrome_level_partial_mgf r s (l+i.1-s) hu e X lo
-          ⌈Real.exp ((l:ℝ)/32)⌉₊ (cycleCutoff_pos l) (q^l) (one_le_pow₀ hq)
-        have hc := cycleCoefficient_decay l (l+i.1-s) (by
-          rw [Nat.cast_sub hs',Nat.cast_add]
-          have hsr : (2:ℝ)*s ≤ l := by exact_mod_cast hs
-          have hi : (0:ℝ) ≤ i.1 := Nat.cast_nonneg _
-          linarith) (q^l) (one_le_pow₀ hq) (by
-            rw [Real.log_pow]
-            nlinarith [mul_le_mul_of_nonneg_left hlog (Nat.cast_nonneg l)])
-        simp_rw [he,pow_mul]
-        exact hm.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left hc (Nat.cast_nonneg _)))
-      · have hz := palindromeLevelCost_gt (l+i.1+1) (s+r) (by omega) e
-        simp only [hz,Nat.mul_zero,pow_zero,finiteMean_const]
-        exact Real.one_le_exp_iff.mpr (by positivity)
-    _ = _ := finiteMean_const _
-
-
-lemma palindromeSlabCost_gt (l d : ℕ) (h : d ≤ l) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) : palindromeSlabCost l d e ω = 0 := by
-  apply Finset.sum_eq_zero
-  intro i _
-  exact palindromeLevelCost_gt _ _ (by omega) _ _
-
-lemma palindrome_slab_adapted_mul_le (d s l : ℕ) (hsd : s ≤ d) (hl : 0 < l)
-    (hs : 2*s ≤ l) {ι : Type*} [Fintype ι] (e : ι ↪ Card d)
-    (X : SwitchIndex d → Bool) (F : (SwitchIndex d → Bool) → ℝ)
-    (hF : ∀ Y, 0 ≤ F Y) (hA : LayerAdapted d s F) (q : ℝ) (hq : 1 ≤ q)
-    (hlog : Real.log q ≤ 1/250) :
-    finiteMean (fun Y => F Y*q^(palindromeSlabCost l d e (Y,X))) ≤
-      finiteMean F * Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
-  obtain ⟨r,rfl⟩ := Nat.exists_eq_add_of_le hsd
-  exact adapted_mul_mean_le r s F _ hF hA _
-    (fun lo => palindrome_slab_partial_mgf r s l hl hs e X lo q hq hlog)
-
-def familyEnd (L n : ℕ) : ℕ := if n = 0 then 0 else 2*(L*4^(n-1))
-
-lemma familyEnd_two (L n : ℕ) : 2*familyEnd L n ≤ L*4^n := by
-  cases n with
-  | zero => simp [familyEnd]
-  | succ n => simp only [familyEnd,Nat.succ_ne_zero,ite_false,Nat.add_one_sub_one,pow_succ]; nlinarith
-
-lemma familyEnd_ge (L n j : ℕ) (hj : j < n) : 2*(L*4^j) ≤ familyEnd L n := by
-  cases n with
-  | zero => omega
-  | succ n =>
-    simp only [familyEnd,Nat.succ_ne_zero,ite_false,Nat.add_one_sub_one]
-    apply Nat.mul_le_mul_left
-    exact Nat.mul_le_mul_left L (Nat.pow_le_pow_right (by decide) (by omega : j ≤ n))
-
-noncomputable def palindromeFamilyCost (L n d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (ω : BenesCoins d) : ℕ :=
-  ∑ j ∈ Finset.range n, palindromeSlabCost (L*4^j) d e ω
-
-lemma palindromeFamilyCost_adapted (L n d : ℕ) {ι : Type*} [Fintype ι]
-    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) :
-    LayerAdapted d (familyEnd L n) (fun Y => palindromeFamilyCost L n d e (Y,X)) := by
-  apply layerAdapted_sum
-  intro j hj
-  exact layerAdapted_mono (palindromeSlabCost_adapted (L*4^j) d e X)
-    (familyEnd_ge L n j (Finset.mem_range.mp hj))
-
-end Thorp
-
-end ThorpNine.Tail
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Tail.BenesCoins
+
+namespace OAI
+
+namespace ThorpNine.Tail
+
+namespace Thorp
+
+lemma mem_centralBlock (r H : ℕ) (b : Card r) (x : Card (H+r)) :
+    x ∈ centralBlock r H b ↔ (cardSplit r H x).1 = b := by
+  classical
+  constructor
+  · intro h
+    obtain ⟨z,_,rfl⟩ := Finset.mem_image.mp h
+    simp only [Equiv.apply_symm_apply]
+  · intro h
+    apply Finset.mem_image.mpr
+    refine ⟨(cardSplit r H x).2,Finset.mem_univ _,?_⟩
+    rw [←h,Prod.mk.eta,Equiv.symm_apply_apply]
+
+lemma centralBlock_card (r H : ℕ) (b : Card r) : (centralBlock r H b).card = 2^H := by
+  classical
+  rw [centralBlock,Finset.card_image_of_injective]
+  · exact card_positions H
+  · intro x y h
+    exact congrArg Prod.snd ((cardSplit r H).symm.injective h)
+
+lemma centralBlock_disjoint (r H : ℕ) :
+    Pairwise (fun b b' => Disjoint (centralBlock r H b) (centralBlock r H b')) := by
+  classical
+  intro b b' h
+  apply Finset.disjoint_left.mpr
+  intro x hx hy
+  exact h (((mem_centralBlock r H b x).mp hx).symm.trans
+    ((mem_centralBlock r H b' x).mp hy))
+
+lemma centralBlock_zero (H : ℕ) (b : Card 0) : centralBlock 0 H b = Finset.univ := by
+  classical
+  ext x
+  simp only [mem_centralBlock,Finset.mem_univ,iff_true]
+  exact Subsingleton.elim _ _
+
+lemma centralBlock_step (r H : ℕ) (b : Bool) (c : Card r) :
+    centralBlock (r+1) H (Fin.cons b c) = (centralBlock r H c).image (Fin.cons b) := by
+  classical
+  simp only [centralBlock,Finset.image_image,cardSplit,Equiv.coe_fn_symm_mk,
+    Fin.cons_zero,Fin.tail_cons,Function.comp_def]
+
+lemma block_inter_step (r H : ℕ) (b : Bool) (c : Card r) (A : Finset (Card (H+(r+1)))) :
+    centralBlock (r+1) H (Fin.cons b c) ∩ A =
+      (centralBlock r H c ∩ childMarks A b).image (Fin.cons b) := by
+  classical
+  rw [centralBlock_step]
+  ext x
+  constructor
+  · intro hx
+    obtain ⟨hxB,hxA⟩ := Finset.mem_inter.mp hx
+    obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hxB
+    exact Finset.mem_image.mpr ⟨y,Finset.mem_inter.mpr ⟨hy,(mem_childMarks A b y).mpr hxA⟩,rfl⟩
+  · intro hx
+    obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
+    exact Finset.mem_inter.mpr ⟨Finset.mem_image.mpr ⟨y,(Finset.mem_inter.mp hy).1,rfl⟩,
+      (mem_childMarks A b y).mp (Finset.mem_inter.mp hy).2⟩
+
+lemma sparseMass_central_zero (H : ℕ) (A : Finset (Card H)) :
+    sparseMass (centralBlock 0 H) A = crowdedMass A.card := by
+  classical
+  simp only [sparseMass,centralBlock_zero,Finset.univ_inter,crowdedMass,
+    Finset.sum_const,Finset.card_univ,card_positions,pow_zero,one_nsmul]
+
+lemma sparseMass_central_step (r H : ℕ) (A : Finset (Card (H+(r+1)))) :
+    sparseMass (centralBlock (r+1) H) A =
+      sparseMass (centralBlock r H) (childMarks A false) +
+        sparseMass (centralBlock r H) (childMarks A true) := by
+  classical
+  unfold sparseMass
+  rw [←Equiv.sum_comp (headTailEquiv r).symm,Fintype.sum_prod_type]
+  have ht (b : Bool) (c : Card r) : (headTailEquiv r).symm (b,c) = Fin.cons b c := rfl
+  simp only [Fintype.sum_bool,ht]
+  have hc (b : Bool) (c : Card r) :
+      (centralBlock (r+1) H (Fin.cons b c) ∩ A).card =
+      (centralBlock r H c ∩ childMarks A b).card := by
+    rw [block_inter_step]
+    apply Finset.card_image_of_injective
+    intro x y h
+    simpa only [Fin.tail_cons] using congrArg Fin.tail h
+  simp only [hc]
+  exact add_comm _ _
+
+
+lemma inverseButterfly_coordinates_step (d : ℕ)
+    (c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (ξ : Card d → Bool)
+    (x : Card (d+1)) :
+    headTailEquiv d ((butterflyPerm (d+1)
+      (decodeButterfly (d+1) ((coinStepEquiv d).symm (c,ξ)))).symm x) =
+      (Bool.xor (x 0) (ξ (Fin.tail x)),
+        (butterflyPerm d (decodeButterfly d
+          (if Bool.xor (x 0) (ξ (Fin.tail x)) then c.2 else c.1))).symm (Fin.tail x)) := by
+  rw [butterflyPerm_step]
+  rfl
+
+lemma inverseButterflyMarks_child (d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (d+1))
+    (c : (SwitchIndex d → Bool) × (SwitchIndex d → Bool)) (ξ : Card d → Bool) (b : Bool) :
+    childMarks (inverseButterflyMarks (d+1) (Finset.univ.image e)
+      ((coinStepEquiv d).symm (c,ξ))) b =
+      inverseButterflyMarks d
+        (Finset.univ.image (PairRouting.childEmbedding (e.trans (headTailEquiv d).toEmbedding)
+          (PairRouting.colors (e.trans (headTailEquiv d).toEmbedding) ξ)
+            (PairRouting.colors_compatible _ _) b)) (if b then c.2 else c.1) := by
+  classical
+  let x := e.trans (headTailEquiv d).toEmbedding
+  let col := PairRouting.colors x ξ
+  let child := PairRouting.childEmbedding x col (PairRouting.colors_compatible x ξ) b
+  have hp (i : ι) : headTailEquiv d ((butterflyPerm (d+1)
+      (decodeButterfly (d+1) ((coinStepEquiv d).symm (c,ξ)))).symm (e i)) =
+      (col i,(butterflyPerm d (decodeButterfly d (if col i then c.2 else c.1))).symm (x i).2) :=
+    inverseButterfly_coordinates_step d c ξ (e i)
+  ext y
+  simp only [mem_childMarks,inverseButterflyMarks,Finset.image_image,
+    Finset.mem_image,Finset.mem_univ,true_and,Function.comp_def]
+  constructor
+  · rintro ⟨i,hi⟩
+    have he := congrArg (headTailEquiv d) hi
+    rw [hp i] at he
+    change (col i,_) = (b,y) at he
+    have hb := congrArg Prod.fst he
+    have hy := congrArg Prod.snd he
+    dsimp only at hb hy
+    refine ⟨⟨i,hb⟩,?_⟩
+    change (butterflyPerm d (decodeButterfly d (if b then c.2 else c.1))).symm (x i).2 = y
+    simpa only [hb] using hy
+  · rintro ⟨i,hi⟩
+    refine ⟨i.val,?_⟩
+    apply (headTailEquiv d).injective
+    rw [hp]
+    change (col i.val,_) = (b,y)
+    have hb : col i.val = b := i.property
+    rw [hb]
+    exact Prod.ext rfl hi
+
+lemma inverseButterflyMarks_card (d : ℕ) (A : Finset (Card d)) (ω : SwitchIndex d → Bool) :
+    (inverseButterflyMarks d A ω).card = A.card := by
+  classical
+  exact Finset.card_image_of_injective _ (butterflyPerm d (decodeButterfly d ω)).symm.injective
+
+lemma benes_input_coordinates (d : ℕ) (ω : BenesCoins (d+1)) :
+    ω.2 = (coinStepEquiv d).symm
+      (((benesStepEquiv d ω).1.1.2,(benesStepEquiv d ω).1.2.2),(benesStepEquiv d ω).2.1) := by
+  change ω.2 = (coinStepEquiv d).symm (coinStepEquiv d ω.2)
+  exact ((coinStepEquiv d).symm_apply_apply ω.2).symm
+
+lemma cutoffMass_central (r H : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (H+r)) (ω : BenesCoins (H+r)) :
+    cutoffMass H (H+r) e ω =
+      sparseMass (centralBlock r H) (inverseButterflyMarks (H+r) (Finset.univ.image e) ω.2) := by
+  classical
+  induction r generalizing ι with
+  | zero =>
+    rw [sparseMass_central_zero,inverseButterflyMarks_card,
+      Finset.card_image_of_injective _ e.injective,Finset.card_univ]
+    cases H with
+    | zero =>
+      have hcard : Fintype.card ι ≤ 1 := by
+        have hh := Fintype.card_le_of_injective e e.injective
+        simpa only [Nat.add_zero,card_positions,pow_zero] using hh
+      simp [cutoffMass,crowdedMass,show ¬2≤Fintype.card ι by omega]
+    | succ H => simp only [cutoffMass,le_refl,ite_true]
+  | succ r ih =>
+    simp only [Nat.add_succ] at *
+    rw [cutoffMass,ite_eq_right (by omega : ¬H+r+1≤H)]
+    let σ := benesStepEquiv (H+r) ω
+    let x := e.trans (headTailEquiv (H+r)).toEmbedding
+    let col := PairRouting.colors x σ.2.1
+    let hc := PairRouting.colors_compatible x σ.2.1
+    change cutoffMass H (H+r) (PairRouting.childEmbedding x col hc false) σ.1.1 +
+      cutoffMass H (H+r) (PairRouting.childEmbedding x col hc true) σ.1.2 = _
+    rw [ih,ih,sparseMass_central_step,benes_input_coordinates (H+r) ω]
+    rw [inverseButterflyMarks_child,inverseButterflyMarks_child]
+    rfl
+
+
+theorem palindrome_low_mgf (r H : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (H+r)) (q : ℝ) (hq : 1 ≤ q)
+    (hr : (2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*q^H) ≤ 1/2) :
+    finiteMean (fun ω : BenesCoins (H+r) => q^(palindromeLowCost H (H+r) e ω)) ≤
+      Real.exp ((2:ℝ)^r * (2 * ((2:ℝ)^H*((Fintype.card ι/(2:ℝ)^(H+r))*q^H))^2)) := by
+  classical
+  have hec : (Finset.univ.image e).card = Fintype.card ι := by
+    rw [Finset.card_image_of_injective _ e.injective,Finset.card_univ]
+  have hm := inverseButterfly_sparse_mgf (H+r) (centralBlock r H)
+    (centralBlock_disjoint r H) (Finset.univ.image e) (q^H)
+    (pow_nonneg (by linarith) _) (fun b => by
+      simpa only [centralBlock_card,hec,Nat.cast_pow,Nat.cast_ofNat] using hr)
+  calc
+    _ ≤ finiteMean (fun ω : BenesCoins (H+r) => (q^H)^(cutoffMass H (H+r) e ω)) := by
+      apply finiteMean_mono
+      intro ω
+      rw [←pow_mul]
+      exact pow_le_pow_right₀ hq (palindromeLowCost_le_cutoff H (H+r) e ω)
+    _ = finiteMean (fun ω : SwitchIndex (H+r) → Bool =>
+        (q^H)^(sparseMass (centralBlock r H)
+          (inverseButterflyMarks (H+r) (Finset.univ.image e) ω))) := by
+      simp_rw [cutoffMass_central]
+      simp only [finiteMean_prod,finiteMean_const]
+    _ ≤ _ := by
+      simpa only [centralBlock_card,hec,Nat.cast_pow,Nat.cast_ofNat,Finset.sum_const,
+        Finset.card_univ,card_positions,nsmul_eq_mul] using hm
+
+variable {α B : Type*} {r s : ℕ}
+
+def parallelFreshPre (pre : α ≃ B × Card (s+r))
+    (lo : B → Card r × SwitchIndex s → Bool) : α ≃ (B × Card s) × Card r :=
+  ((pre.trans (Equiv.prodCongr (Equiv.refl B) (cardSplit r s))).trans
+    (Equiv.prodCongrRight (fun b =>
+      (parallelPerm (lo b)).trans (Equiv.prodComm _ _)))).trans
+    (Equiv.prodAssoc _ _ _).symm
+
+def parallelFreshPost (post : B × Card (s+r) ≃ α) : (B × Card s) × Card r ≃ α :=
+  (((Equiv.prodAssoc _ _ _).trans (Equiv.prodCongr (Equiv.refl B)
+    (Equiv.prodComm _ _))).trans (Equiv.prodCongr (Equiv.refl B) (cardSplit r s).symm)).trans post
+
+def parallelAssembled (lo : B → Card r × SwitchIndex s → Bool)
+    (hi : (B × Card s) × SwitchIndex r → Bool) : B × SwitchIndex (s+r) → Bool :=
+  fun i => assembleBits r s (lo i.1) (fun j => hi ((i.1,j.1),j.2)) i.2
+
+lemma parallel_exposed_perm (pre : α ≃ B × Card (s+r)) (post : B × Card (s+r) ≃ α)
+    (lo : B → Card r × SwitchIndex s → Bool)
+    (hi : (B × Card s) × SwitchIndex r → Bool) :
+    relabeledPerm pre post (parallelAssembled lo hi) =
+      relabeledPerm (parallelFreshPre pre lo) (parallelFreshPost post) hi := by
+  ext x
+  have h := butterfly_split r s (lo (pre x).1)
+    (fun j => hi (((pre x).1,j.1),j.2)) (pre x).2
+  have hh := congrArg (fun y => post ((pre x).1,(cardSplit r s).symm y)) h
+  simpa only [relabeledPerm,parallelAssembled,parallelPerm,parallelFreshPre,parallelFreshPost,
+    Equiv.trans_apply,Equiv.prodCongr_apply,Equiv.prodCongrRight,Equiv.coe_fn_mk,
+    Equiv.refl_apply,Equiv.prodComm_apply,Prod.swap,Equiv.prodAssoc_apply,
+    Equiv.prodAssoc_symm_apply,Prod.map,Equiv.symm_apply_apply] using hh
+
+theorem parallel_partially_exposed_log_mgf [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B]
+    (pre : α ≃ B × Card (s+r)) (post : B × Card (s+r) ≃ α)
+    (lo : B → Card r × SwitchIndex s → Bool) (S : Finset α)
+    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    Real.log (finiteMean (fun hi : (B × Card s) × SwitchIndex r → Bool =>
+      q^(Fintype.card (RoutingNetwork.SelectedCycle
+        (relabeledPerm pre post (parallelAssembled lo hi)) S)))) ≤
+      (S.card:ℝ)/J*Real.log q + 2*q*S.card*Real.sqrt J/Real.sqrt ((2:ℝ)^r) := by
+  simp_rw [parallel_exposed_perm]
+  exact parallel_cycle_log_mgf (parallelFreshPre pre lo) (parallelFreshPost post) S J hJ q hq
+
+end Thorp
+namespace Thorp
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+lemma orbitSet_inverse (p : Equiv.Perm α) (x : α) : orbitSet p⁻¹ x = orbitSet p x := by
+  ext y
+  simp only [mem_orbitSet,Equiv.Perm.sameCycle_inv]
+
+lemma selectedCycle_inverse (p : Equiv.Perm α) (S : Finset α) :
+    Fintype.card (RoutingNetwork.SelectedCycle p⁻¹ S) =
+      Fintype.card (RoutingNetwork.SelectedCycle p S) := by
+  apply Fintype.card_congr
+  refine { toFun := fun C => ⟨C.val,?_,C.property.2⟩
+           invFun := fun C => ⟨C.val,?_,C.property.2⟩
+           left_inv := fun _ => rfl
+           right_inv := fun _ => rfl }
+  · obtain ⟨x,hx⟩ := C.property.1
+    exact ⟨x,hx.trans (orbitSet_inverse p x)⟩
+  · obtain ⟨x,hx⟩ := C.property.1
+    exact ⟨x,hx.trans (orbitSet_inverse p x).symm⟩
+
+lemma orbitSet_conjugate (p g : Equiv.Perm α) (x : α) :
+    orbitSet (g*p*g⁻¹) (g x) = (orbitSet p x).image g := by
+  ext y
+  simp only [mem_orbitSet,Finset.mem_image]
+  rw [Equiv.Perm.sameCycle_conj]
+  change p.SameCycle (g.symm (g x)) (g.symm y) ↔ ∃ z, p.SameCycle x z ∧ g z = y
+  rw [Equiv.symm_apply_apply]
+  constructor
+  · intro h
+    exact ⟨g.symm y,h,g.apply_symm_apply y⟩
+  · rintro ⟨z,hz,rfl⟩
+    simpa only [Equiv.symm_apply_apply] using hz
+
+noncomputable def conjugateCycle (p g : Equiv.Perm α) (S : Finset α)
+    (C : RoutingNetwork.SelectedCycle p S) :
+    RoutingNetwork.SelectedCycle (g*p*g⁻¹) (S.image g) := by
+  refine ⟨C.val.image g,?_,Finset.image_subset_image C.property.2⟩
+  obtain ⟨x,hx⟩ := C.property.1
+  exact ⟨g x,by rw [hx,orbitSet_conjugate]⟩
+
+lemma conjugateCycle_injective (p g : Equiv.Perm α) (S : Finset α) :
+    Function.Injective (conjugateCycle p g S) := by
+  intro C D he
+  apply Subtype.ext
+  exact (Finset.image_injective g.injective) (congrArg Subtype.val he)
+
+lemma selectedCycle_conjugate (p g : Equiv.Perm α) (S : Finset α) :
+    Fintype.card (RoutingNetwork.SelectedCycle (g*p*g⁻¹) (S.image g)) =
+      Fintype.card (RoutingNetwork.SelectedCycle p S) := by
+  apply Nat.le_antisymm
+  · have h := Fintype.card_le_of_injective _
+      (conjugateCycle_injective (g*p*g⁻¹) g⁻¹ (S.image g))
+    have hp : g⁻¹*(g*p*g⁻¹)*g = p := by group
+    rw [inv_inv,hp] at h
+    have hs : (S.image g).image g.symm = S := by
+      ext x
+      simp only [Finset.mem_image]
+      constructor
+      · rintro ⟨y,⟨z,hz,rfl⟩,he⟩
+        have hz' : z = x := by simpa only [Equiv.symm_apply_apply] using he
+        exact hz' ▸ hz
+      · intro hx
+        exact ⟨g x,⟨x,hx,rfl⟩,g.symm_apply_apply x⟩
+    change _ ≤ Fintype.card (RoutingNetwork.SelectedCycle p ((S.image g).image g.symm)) at h
+    rw [hs] at h
+    exact h
+  · exact Fintype.card_le_of_injective _ (conjugateCycle_injective p g S)
+
+lemma alternatingCycles_le_projection {ι : Type*} [Fintype ι]
+    (e : ι ↪ Bool × α) (p : Bool → Equiv.Perm α) :
+    PairRouting.alternatingCycles e p ≤
+      Fintype.card (RoutingNetwork.SelectedCycle (PairRouting.alternatingProjection p)
+        (boolFiber (PairRouting.labelSet e) false)) := by
+  have h := CycleColoring.card_alternating_cycles_le (PairRouting.alternatingProjection p)
+    (boolFiber (PairRouting.labelSet e) false) (boolFiber (PairRouting.labelSet e) true)
+  simpa only [PairRouting.alternatingCycles,childJoin_boolFiber] using h
+
+lemma projectionCycles_forward (X₀ X₁ Y₀ Y₁ : Equiv.Perm α) (S : Finset α) :
+    Fintype.card (RoutingNetwork.SelectedCycle
+      ((Y₀*X₀⁻¹)⁻¹*(Y₁*X₁⁻¹)) S) =
+    Fintype.card (RoutingNetwork.SelectedCycle
+      (X₀⁻¹*X₁*Y₁⁻¹*Y₀) (S.image X₀.symm)) := by
+  let p := (Y₀*X₀⁻¹)⁻¹*(Y₁*X₁⁻¹)
+  calc
+    _ = Fintype.card (RoutingNetwork.SelectedCycle (X₀⁻¹*p*X₀)
+        (S.image X₀.symm)) := by
+      change Fintype.card (RoutingNetwork.SelectedCycle p S) = _
+      have h := (selectedCycle_conjugate p X₀⁻¹ S).symm
+      rw [inv_inv] at h
+      exact h
+    _ = Fintype.card (RoutingNetwork.SelectedCycle (X₀⁻¹*p*X₀)⁻¹
+        (S.image X₀.symm)) := (selectedCycle_inverse _ _).symm
+    _ = _ := by
+      have he : (X₀⁻¹*p*X₀)⁻¹ = X₀⁻¹*X₁*Y₁⁻¹*Y₀ := by dsimp only [p]; group
+      rw [he]
+
+end Thorp
+namespace Thorp
+
+noncomputable def cycleCoefficient (r J : ℕ) (q : ℝ) : ℝ :=
+  Real.log q / J + 2*q*Real.sqrt J/Real.sqrt ((2:ℝ)^r)
+
+lemma cycleCoefficient_nonneg (r J : ℕ) (q : ℝ) (hq : 1 ≤ q) :
+    0 ≤ cycleCoefficient r J q := by
+  unfold cycleCoefficient
+  exact add_nonneg (div_nonneg (Real.log_nonneg hq) (Nat.cast_nonneg _))
+    (div_nonneg (mul_nonneg (by linarith) (Real.sqrt_nonneg _)) (Real.sqrt_nonneg _))
+
+lemma projectionSide_card_le {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι]
+    (e : ι ↪ Bool × α) : (boolFiber (PairRouting.labelSet e) false).card ≤ Fintype.card ι := by
+  have h := card_childJoin (boolFiber (PairRouting.labelSet e) false)
+    (boolFiber (PairRouting.labelSet e) true)
+  rw [childJoin_boolFiber,PairRouting.card_labelSet] at h
+  omega
+
+theorem node_partial_mgf (r s : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Bool × Card (s+r)) (X₀ X₁ Y₁ : Equiv.Perm (Card (s+r)))
+    (lo : Card r × SwitchIndex s → Bool) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(PairRouting.alternatingCycles e (fun b => if b then Y₁*X₁⁻¹ else
+        butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi))*X₀⁻¹))) ≤
+      Real.exp (Fintype.card ι * cycleCoefficient r J q) := by
+  classical
+  let S := boolFiber (PairRouting.labelSet e) false
+  let S' := S.image X₀.symm
+  let post := X₀⁻¹*X₁*Y₁⁻¹
+  let count := fun hi : Card s × SwitchIndex r → Bool =>
+    Fintype.card (RoutingNetwork.SelectedCycle
+      ((Equiv.refl _).trans (butterflyPerm (s+r)
+        (decodeButterfly (s+r) (assembleBits r s lo hi))) |>.trans post) S')
+  have hpoint (hi : Card s × SwitchIndex r → Bool) :
+      PairRouting.alternatingCycles e (fun b => if b then Y₁*X₁⁻¹ else
+        butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi))*X₀⁻¹) ≤ count hi := by
+    refine (alternatingCycles_le_projection e _).trans_eq ?_
+    simp only [PairRouting.alternatingProjection,Bool.false_eq_true,↓reduceIte]
+    change Fintype.card (RoutingNetwork.SelectedCycle
+      ((butterflyPerm (s+r) (decodeButterfly (s+r) (assembleBits r s lo hi))*X₀⁻¹)⁻¹ * (Y₁*X₁⁻¹)) S) = count hi
+    exact projectionCycles_forward X₀ X₁ _ Y₁ S
+  have hm := partially_exposed_cycle_log_mgf (Equiv.refl (Card (s+r))) post lo S' J hJ q hq
+  have hp : 0 < finiteMean (fun hi : Card s × SwitchIndex r → Bool => q^(count hi)) :=
+    finiteMean_pos (fun _ => pow_pos (by linarith) _)
+  have hb : finiteMean (fun hi : Card s × SwitchIndex r → Bool => q^(count hi)) ≤
+      Real.exp (S'.card * cycleCoefficient r J q) := by
+    rw [←Real.exp_log hp]
+    apply Real.exp_le_exp.mpr
+    calc
+      _ ≤ _ := hm
+      _ = _ := by unfold cycleCoefficient; ring
+  have hcard : S'.card ≤ Fintype.card ι := by
+    rw [show S'.card = S.card from Finset.card_image_of_injective _ X₀.symm.injective]
+    exact projectionSide_card_le e
+  calc
+    _ ≤ finiteMean (fun hi : Card s × SwitchIndex r → Bool => q^(count hi)) :=
+      finiteMean_mono (fun hi => pow_le_pow_right₀ hq (hpoint hi))
+    _ ≤ Real.exp (S'.card * cycleCoefficient r J q) := hb
+    _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right
+      (by exact_mod_cast hcard) (cycleCoefficient_nonneg r J q hq))
+
+
+def highStepEquiv (r s : ℕ) : (Card s × SwitchIndex (r+1) → Bool) ≃
+    (((Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool)) ×
+      (Card s × Card r → Bool)) where
+  toFun hi := ((fun j => hi (j.1,Sum.inr (false,j.2)),
+    fun j => hi (j.1,Sum.inr (true,j.2))),fun j => hi (j.1,Sum.inl j.2))
+  invFun c := fun j => match j.2 with
+    | Sum.inl y => c.2 (j.1,y)
+    | Sum.inr (b,i) => if b then c.1.2 (j.1,i) else c.1.1 (j.1,i)
+  left_inv hi := by
+    funext j
+    rcases j with ⟨z,y|⟨b,i⟩⟩
+    · rfl
+    · cases b <;> rfl
+  right_inv c := by
+    rcases c with ⟨⟨h₀,h₁⟩,η⟩
+    rfl
+
+lemma assembleBits_step (r s : ℕ) (lo : Card (r+1) × SwitchIndex s → Bool)
+    (c : ((Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool)) ×
+      (Card s × Card r → Bool)) :
+    assembleBits (r+1) s lo ((highStepEquiv r s).symm c) =
+      (coinStepEquiv (s+r)).symm
+        ((assembleBits r s (fun j => lo (Fin.cons false j.1,j.2)) c.1.1,
+          assembleBits r s (fun j => lo (Fin.cons true j.1,j.2)) c.1.2),
+        fun y => c.2 ((cardSplit r s y).2,(cardSplit r s y).1)) := by
+  funext i
+  rcases i with y | ⟨b,i⟩
+  · rfl
+  · cases b <;> rfl
+
+noncomputable def palindromeLevelCost (t : ℕ) : (d : ℕ) → {ι : Type*} → [Fintype ι] →
+    (ι ↪ Card d) → BenesCoins d → ℕ
+  | 0, _, _, _, _ => 0
+  | d+1, _, _, e, ω =>
+      let σ := benesStepEquiv d ω
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x σ.2.1
+      let hc := PairRouting.colors_compatible x σ.2.1
+      if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x σ.2.1)
+          (fun b => palindromePerm d (if b then σ.1.2 else σ.1.1)) else
+        palindromeLevelCost t d (PairRouting.childEmbedding x c hc false) σ.1.1 +
+          palindromeLevelCost t d (PairRouting.childEmbedding x c hc true) σ.1.2
+
+lemma palindromeLevelCost_step (t d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card (d+1)) (X Y : (SwitchIndex d → Bool) × (SwitchIndex d → Bool))
+    (ξ η : Card d → Bool) :
+    palindromeLevelCost t (d+1) e
+      ((coinStepEquiv d).symm (Y,η),(coinStepEquiv d).symm (X,ξ)) =
+      let x := e.trans (headTailEquiv d).toEmbedding
+      let c := PairRouting.colors x ξ
+      let hc := PairRouting.colors_compatible x ξ
+      if t = d+1 then PairRouting.alternatingCycles (PairRouting.switchedEmbedding x ξ)
+        (fun b => palindromePerm d (if b then (Y.2,X.2) else (Y.1,X.1))) else
+        palindromeLevelCost t d (PairRouting.childEmbedding x c hc false) (Y.1,X.1) +
+        palindromeLevelCost t d (PairRouting.childEmbedding x c hc true) (Y.2,X.2) := by
+  rw [palindromeLevelCost]
+  rfl
+
+lemma palindromeLevelCost_gt (t d : ℕ) (h : d < t) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) : palindromeLevelCost t d e ω = 0 := by
+  classical
+  induction d generalizing ι with
+  | zero => rfl
+  | succ d ih =>
+    rw [palindromeLevelCost,ite_eq_right (by omega : t ≠ d+1),ih (by omega),ih (by omega),Nat.zero_add]
+
+
+theorem palindrome_level_partial_mgf (r s u : ℕ) (hu : u < r)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+r)) (X : SwitchIndex (s+r) → Bool)
+    (lo : Card r × SwitchIndex s → Bool) (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(palindromeLevelCost (s+u+1) (s+r) e (assembleBits r s lo hi,X))) ≤
+      Real.exp (Fintype.card ι * cycleCoefficient u J q) := by
+  classical
+  induction r generalizing u ι with
+  | zero => omega
+  | succ r ih =>
+    simp only [Nat.add_succ] at *
+    let χ := coinStepEquiv (s+r) X
+    let x := e.trans (headTailEquiv (s+r)).toEmbedding
+    let c := PairRouting.colors x χ.2
+    let hc := PairRouting.colors_compatible x χ.2
+    let e₀ := PairRouting.childEmbedding x c hc false
+    let e₁ := PairRouting.childEmbedding x c hc true
+    let e' := PairRouting.switchedEmbedding x χ.2
+    let lo₀ : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons false j.1,j.2)
+    let lo₁ : Card r × SwitchIndex s → Bool := fun j => lo (Fin.cons true j.1,j.2)
+    let Y₀ := fun hi : Card s × SwitchIndex r → Bool => assembleBits r s lo₀ hi
+    let Y₁ := fun hi : Card s × SwitchIndex r → Bool => assembleBits r s lo₁ hi
+    let N := fun hi : (Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool) =>
+      PairRouting.alternatingCycles e' (fun b => if b then palindromePerm (s+r) (Y₁ hi.2,χ.1.2)
+        else palindromePerm (s+r) (Y₀ hi.1,χ.1.1))
+    let L₀ := fun hi : Card s × SwitchIndex r → Bool =>
+      palindromeLevelCost (s+u+1) (s+r) e₀ (Y₀ hi,χ.1.1)
+    let L₁ := fun hi : Card s × SwitchIndex r → Bool =>
+      palindromeLevelCost (s+u+1) (s+r) e₁ (Y₁ hi,χ.1.2)
+    have hs (hi : (Card s × SwitchIndex r → Bool) × (Card s × SwitchIndex r → Bool))
+        (η : Card s × Card r → Bool) :
+        palindromeLevelCost (s+u+1) (s+r+1) e
+          (assembleBits (r+1) s lo ((highStepEquiv r s).symm (hi,η)),X) =
+          if u = r then N hi else L₀ hi.1 + L₁ hi.2 := by
+      rw [assembleBits_step]
+      conv_lhs => rw [←(coinStepEquiv (s+r)).symm_apply_apply X]
+      rw [palindromeLevelCost_step]
+      simp only [Nat.add_right_cancel_iff,Nat.add_left_cancel_iff]
+      dsimp only [χ,x,c,hc,e₀,e₁,e',lo₀,lo₁,Y₀,Y₁,N,L₀,L₁]
+      congr 2
+      funext b
+      cases b <;> rfl
+    rw [finiteMean_equiv (highStepEquiv r s),finiteMean_prod]
+    simp_rw [hs,finiteMean_const]
+    by_cases hur : u = r
+    · subst u
+      simp only [ite_true]
+      rw [finiteMean_prod,finiteMean_comm]
+      calc
+        _ ≤ finiteMean (fun _hi : Card s × SwitchIndex r → Bool =>
+          Real.exp (Fintype.card ι * cycleCoefficient r J q)) := by
+          apply finiteMean_mono
+          intro hi
+          have hn := node_partial_mgf r s e'
+            (butterflyPerm (s+r) (decodeButterfly (s+r) χ.1.1))
+            (butterflyPerm (s+r) (decodeButterfly (s+r) χ.1.2))
+            (butterflyPerm (s+r) (decodeButterfly (s+r) (Y₁ hi))) lo₀ J hJ q hq
+          exact hn
+        _ = _ := finiteMean_const _
+    · simp only [ite_eq_right hur,pow_add]
+      rw [finiteMean_prod_mul (fun hi => q ^ L₀ hi) (fun hi => q ^ L₁ hi)]
+      have hu' : u < r := by omega
+      have h₀ := ih u hu' e₀ χ.1.1 lo₀
+      have h₁ := ih u hu' e₁ χ.1.2 lo₁
+      have hp := mul_le_mul h₀ h₁ (finiteMean_nonneg (fun _ => pow_nonneg (by linarith) _))
+        (Real.exp_nonneg _)
+      refine hp.trans_eq ?_
+      rw [←Real.exp_add]
+      congr 1
+      have hcount := PairRouting.color_card_add c
+      have hcount' : (Fintype.card {i : ι // c i = false} : ℝ) +
+          Fintype.card {i : ι // c i = true} = Fintype.card ι := by exact_mod_cast hcount
+      change (Fintype.card {i : ι // c i = false} : ℝ) * cycleCoefficient u J q +
+          Fintype.card {i : ι // c i = true} * cycleCoefficient u J q = _
+      rw [←add_mul,hcount']
+
+
+def switchHeight : (d : ℕ) → SwitchIndex d → ℕ
+  | 0, i => nomatch i
+  | d+1, Sum.inl _ => d+1
+  | d+1, Sum.inr (_,i) => switchHeight d i
+
+lemma switchHeight_le (d : ℕ) (i : SwitchIndex d) : switchHeight d i ≤ d := by
+  induction d with
+  | zero => exact Empty.elim i
+  | succ d ih =>
+    rcases i with y | ⟨b,i⟩
+    · exact le_refl _
+    · exact (ih i).trans (Nat.le_succ d)
+
+def AgreeThrough (d s : ℕ) (Y Y' : SwitchIndex d → Bool) : Prop :=
+  ∀ i, switchHeight d i ≤ s → Y i = Y' i
+
+def LayerAdapted (d s : ℕ) {β : Type*} (f : (SwitchIndex d → Bool) → β) : Prop :=
+  ∀ Y Y', AgreeThrough d s Y Y' → f Y = f Y'
+
+lemma agreeThrough_mono {d s t : ℕ} (h : s ≤ t) {Y Y' : SwitchIndex d → Bool}
+    (hh : AgreeThrough d t Y Y') : AgreeThrough d s Y Y' :=
+  fun i hi => hh i (hi.trans h)
+
+lemma layerAdapted_mono {d s t : ℕ} {β : Type*} {f : (SwitchIndex d → Bool) → β}
+    (hf : LayerAdapted d s f) (h : s ≤ t) : LayerAdapted d t f :=
+  fun Y Y' hh => hf Y Y' (agreeThrough_mono h hh)
+
+lemma layerAdapted_comp {d s : ℕ} {α β : Type*} {f : (SwitchIndex d → Bool) → α}
+    (hf : LayerAdapted d s f) (g : α → β) : LayerAdapted d s (g ∘ f) :=
+  fun Y Y' h => congrArg g (hf Y Y' h)
+
+lemma layerAdapted_sum {d s : ℕ} {ι : Type*} (A : Finset ι)
+    (f : ι → (SwitchIndex d → Bool) → ℕ)
+    (hf : ∀ i ∈ A, LayerAdapted d s (f i)) :
+    LayerAdapted d s (fun Y => ∑ i ∈ A, f i Y) := by
+  intro Y Y' h
+  exact Finset.sum_congr rfl (fun i hi => hf i hi Y Y' h)
+
+lemma assembleBits_agree (r s : ℕ) (lo : Card r × SwitchIndex s → Bool)
+    (hi hi' : Card s × SwitchIndex r → Bool) :
+    AgreeThrough (s+r) s (assembleBits r s lo hi) (assembleBits r s lo hi') := by
+  induction r with
+  | zero => intro i _; rfl
+  | succ r ih =>
+    intro i h
+    rcases i with y | ⟨b,i⟩
+    · have : s+r+1 ≤ s := h
+      omega
+    · exact ih (fun j => lo (Fin.cons b j.1,j.2))
+        (fun j => hi (j.1,Sum.inr (b,j.2))) (fun j => hi' (j.1,Sum.inr (b,j.2))) i h
+
+lemma palindromeLevelCost_adapted (t d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) :
+    LayerAdapted d (t-1) (fun Y => palindromeLevelCost t d e (Y,X)) := by
+  classical
+  induction d generalizing ι with
+  | zero => intro Y Y' _; rfl
+  | succ d ih =>
+    intro Y Y' h
+    change palindromeLevelCost t (d+1) e (Y,X) = palindromeLevelCost t (d+1) e (Y',X)
+    conv_lhs => rw [←(coinStepEquiv d).symm_apply_apply Y,←(coinStepEquiv d).symm_apply_apply X]
+    conv_rhs => rw [←(coinStepEquiv d).symm_apply_apply Y',←(coinStepEquiv d).symm_apply_apply X]
+    rw [palindromeLevelCost_step,palindromeLevelCost_step]
+    by_cases ht : t = d+1
+    · subst t
+      rw [ite_eq_left rfl,ite_eq_left rfl]
+      have he₀ : (coinStepEquiv d Y).1.1 = (coinStepEquiv d Y').1.1 := by
+        funext i
+        exact h (Sum.inr (false,i)) (by simpa only [switchHeight,Nat.add_sub_cancel] using switchHeight_le d i)
+      have he₁ : (coinStepEquiv d Y).1.2 = (coinStepEquiv d Y').1.2 := by
+        funext i
+        exact h (Sum.inr (true,i)) (by simpa only [switchHeight,Nat.add_sub_cancel] using switchHeight_le d i)
+      rw [he₀,he₁]
+    · rw [ite_eq_right ht,ite_eq_right ht]
+      congr 1
+      · apply ih
+        intro i hi
+        exact h (Sum.inr (false,i)) hi
+      · apply ih
+        intro i hi
+        exact h (Sum.inr (true,i)) hi
+
+lemma adapted_mul_mean_le (r s : ℕ) (F G : (SwitchIndex (s+r) → Bool) → ℝ)
+    (hF : ∀ Y, 0 ≤ F Y) (hA : LayerAdapted (s+r) s F) (B : ℝ)
+    (hG : ∀ lo : Card r × SwitchIndex s → Bool,
+      finiteMean (fun hi : Card s × SwitchIndex r → Bool => G (assembleBits r s lo hi)) ≤ B) :
+    finiteMean (fun Y => F Y*G Y) ≤ finiteMean F * B := by
+  rw [finiteMean_split r s,finiteMean_split r s F,←finiteMean_mul_const]
+  apply finiteMean_mono
+  intro lo
+  let H : Card s × SwitchIndex r → Bool := fun _ => false
+  have hh (hi : Card s × SwitchIndex r → Bool) :
+      F (assembleBits r s lo hi) = F (assembleBits r s lo H) :=
+    hA _ _ (assembleBits_agree r s lo hi H)
+  simp_rw [hh,finiteMean_const]
+  rw [show (fun hi => F (assembleBits r s lo H) * G (assembleBits r s lo hi)) =
+      (fun hi => G (assembleBits r s lo hi) * F (assembleBits r s lo H)) by
+        funext hi; exact mul_comm _ _,finiteMean_mul_const]
+  simpa only [mul_comm] using mul_le_mul_of_nonneg_left (hG lo) (hF _)
+
+
+theorem cycleCoefficient_decay (l r : ℕ) (hr : (l:ℝ)/2 ≤ r)
+    (q : ℝ) (hq : 1 ≤ q) (hlog : Real.log q ≤ (l:ℝ)/250) :
+    cycleCoefficient r ⌈Real.exp ((l:ℝ)/32)⌉₊ q ≤
+      68 * Real.exp (-(l:ℝ)/64) := by
+  let E := Real.exp ((l:ℝ)/32)
+  let J := ⌈E⌉₊
+  have hl : (0:ℝ) ≤ l := Nat.cast_nonneg _
+  have hE : 1 ≤ E := Real.one_le_exp_iff.mpr (by positivity)
+  have hEpos : 0 < E := Real.exp_pos _
+  have hEJ : E ≤ (J:ℝ) := Nat.le_ceil E
+  have hJ : (1:ℝ) ≤ J := hE.trans hEJ
+  have hJpos : (0:ℝ) < J := lt_of_lt_of_le (by norm_num) hJ
+  have hJE : (J:ℝ) ≤ 2*E := by
+    have h := Nat.ceil_lt_add_one (le_of_lt hEpos)
+    change (J:ℝ) < E+1 at h
+    linarith
+  have hsqJ : Real.sqrt J ≤ (J:ℝ) := by
+    apply Real.sqrt_le_iff.mpr
+    constructor
+    · positivity
+    · nlinarith
+  have hqp : 0 < q := lt_of_lt_of_le (by norm_num) hq
+  have hqE : q ≤ Real.exp ((l:ℝ)/250) := by
+    rw [←Real.exp_log hqp]
+    exact Real.exp_le_exp.mpr hlog
+  have hln : (1/2:ℝ) ≤ Real.log 2 := by
+    have hh := Real.one_sub_inv_le_log_of_pos (by norm_num : (0:ℝ)<2)
+    norm_num at hh ⊢
+    exact hh
+  have hden : Real.exp ((l:ℝ)/8) ≤ Real.sqrt ((2:ℝ)^r) := by
+    have he : (2:ℝ)^r = Real.exp ((r:ℝ)*Real.log 2) := by
+      rw [Real.exp_nat_mul,Real.exp_log (by norm_num)]
+    rw [he,Real.sqrt_eq_rpow,Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp]
+    apply Real.exp_le_exp.mpr
+    have hrn : (0:ℝ) ≤ r := Nat.cast_nonneg _
+    nlinarith
+  have hlog' : Real.log q ≤ (l:ℝ) := by linarith
+  have h₁ : Real.log q / (J:ℝ) ≤ 64*Real.exp (-(l:ℝ)/64) := by
+    calc
+      _ ≤ (l:ℝ)/E := div_le_div₀ (by positivity) hlog' hEpos hEJ
+      _ ≤ (64*Real.exp ((l:ℝ)/64))/E := by
+        apply div_le_div_of_nonneg_right _ (le_of_lt hEpos)
+        have hh := Real.add_one_le_exp ((l:ℝ)/64)
+        linarith
+      _ = _ := by
+        dsimp [E]
+        rw [mul_div_assoc,←Real.exp_sub]
+        congr 2
+        ring
+  have h₂ : 2*q*Real.sqrt J / Real.sqrt ((2:ℝ)^r) ≤
+      4*Real.exp (-(l:ℝ)/64) := by
+    calc
+      _ ≤ (4*Real.exp ((l:ℝ)/250)*E)/Real.exp ((l:ℝ)/8) := by
+        apply div_le_div₀ (by positivity) _ (Real.exp_pos _) hden
+        have hs := hsqJ.trans hJE
+        have hm := mul_le_mul hqE hs (Real.sqrt_nonneg _) (Real.exp_nonneg _)
+        nlinarith
+      _ = 4*Real.exp ((l:ℝ)/250+(l:ℝ)/32-(l:ℝ)/8) := by
+        dsimp [E]
+        rw [mul_assoc,←Real.exp_add,mul_div_assoc,←Real.exp_sub]
+      _ ≤ _ := by
+        apply mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr _) (by norm_num)
+        linarith
+  change Real.log q / (J:ℝ) + 2*q*Real.sqrt J / Real.sqrt ((2:ℝ)^r) ≤ _
+  linarith
+
+lemma cycleCutoff_pos (l : ℕ) : 0 < ⌈Real.exp ((l:ℝ)/32)⌉₊ := by
+  have h := Nat.le_ceil (Real.exp ((l:ℝ)/32))
+  have he := Real.exp_pos ((l:ℝ)/32)
+  have : (0:ℝ) < ⌈Real.exp ((l:ℝ)/32)⌉₊ := lt_of_lt_of_le he h
+  exact_mod_cast this
+
+
+lemma pow_sum_le_mean_powers {ι : Type*} [Fintype ι] [Nonempty ι]
+    (z : ι → ℕ) (q : ℝ) (hq : 0 < q) :
+    q^(∑ i, z i) ≤ finiteMean (fun i => q^(Fintype.card ι*z i)) := by
+  have hn : (Fintype.card ι:ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
+  have hm : finiteMean (fun i => (Fintype.card ι*z i:ℕ)*Real.log q) =
+      (∑ i, z i:ℕ)*Real.log q := by
+    simp only [finiteMean,Nat.cast_mul,Nat.cast_sum,←Finset.mul_sum,←Finset.sum_mul]
+    field_simp
+  have he := exp_finiteMean_le (fun i => (Fintype.card ι*z i:ℕ)*Real.log q)
+  rw [hm,Real.exp_nat_mul,Real.exp_log hq] at he
+  simpa only [Real.exp_nat_mul,Real.exp_log hq] using he
+
+noncomputable def palindromeSlabCost (l d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) : ℕ :=
+  ∑ i : Fin l, palindromeLevelCost (l+i.1+1) d e ω
+
+lemma palindromeSlabCost_adapted (l d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) :
+    LayerAdapted d (2*l) (fun Y => palindromeSlabCost l d e (Y,X)) := by
+  apply layerAdapted_sum
+  intro i _
+  exact layerAdapted_mono (palindromeLevelCost_adapted (l+i.1+1) d e X) (by omega)
+
+theorem palindrome_slab_partial_mgf (r s l : ℕ) (hl : 0 < l) (hs : 2*s ≤ l)
+    {ι : Type*} [Fintype ι] (e : ι ↪ Card (s+r)) (X : SwitchIndex (s+r) → Bool)
+    (lo : Card r × SwitchIndex s → Bool) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(palindromeSlabCost l (s+r) e (assembleBits r s lo hi,X))) ≤
+      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
+  classical
+  let : NeZero l := ⟨ne_of_gt hl⟩
+  have hqp : 0 < q := lt_of_lt_of_le (by norm_num) hq
+  calc
+    _ ≤ finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      finiteMean (fun i : Fin l => q^(l*palindromeLevelCost (l+i.1+1) (s+r) e
+        (assembleBits r s lo hi,X)))) := by
+      apply finiteMean_mono
+      intro hi
+      simpa only [palindromeSlabCost,Fintype.card_fin] using
+        pow_sum_le_mean_powers (fun i : Fin l => palindromeLevelCost (l+i.1+1) (s+r) e
+          (assembleBits r s lo hi,X)) q hqp
+    _ = finiteMean (fun i : Fin l => finiteMean (fun hi : Card s × SwitchIndex r → Bool =>
+      q^(l*palindromeLevelCost (l+i.1+1) (s+r) e (assembleBits r s lo hi,X)))) := finiteMean_comm _
+    _ ≤ finiteMean (fun _i : Fin l =>
+      Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64)))) := by
+      apply finiteMean_mono
+      intro i
+      by_cases ht : l+i.1+1 ≤ s+r
+      · have hs' : s ≤ l+i.1 := by omega
+        have he : l+i.1+1 = s+(l+i.1-s)+1 := by omega
+        have hu : l+i.1-s < r := by omega
+        have hm := palindrome_level_partial_mgf r s (l+i.1-s) hu e X lo
+          ⌈Real.exp ((l:ℝ)/32)⌉₊ (cycleCutoff_pos l) (q^l) (one_le_pow₀ hq)
+        have hc := cycleCoefficient_decay l (l+i.1-s) (by
+          rw [Nat.cast_sub hs',Nat.cast_add]
+          have hsr : (2:ℝ)*s ≤ l := by exact_mod_cast hs
+          have hi : (0:ℝ) ≤ i.1 := Nat.cast_nonneg _
+          linarith) (q^l) (one_le_pow₀ hq) (by
+            rw [Real.log_pow]
+            nlinarith [mul_le_mul_of_nonneg_left hlog (Nat.cast_nonneg l)])
+        simp_rw [he,pow_mul]
+        exact hm.trans (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left hc (Nat.cast_nonneg _)))
+      · have hz := palindromeLevelCost_gt (l+i.1+1) (s+r) (by omega) e
+        simp only [hz,Nat.mul_zero,pow_zero,finiteMean_const]
+        exact Real.one_le_exp_iff.mpr (by positivity)
+    _ = _ := finiteMean_const _
+
+
+lemma palindromeSlabCost_gt (l d : ℕ) (h : d ≤ l) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) : palindromeSlabCost l d e ω = 0 := by
+  apply Finset.sum_eq_zero
+  intro i _
+  exact palindromeLevelCost_gt _ _ (by omega) _ _
+
+lemma palindrome_slab_adapted_mul_le (d s l : ℕ) (hsd : s ≤ d) (hl : 0 < l)
+    (hs : 2*s ≤ l) {ι : Type*} [Fintype ι] (e : ι ↪ Card d)
+    (X : SwitchIndex d → Bool) (F : (SwitchIndex d → Bool) → ℝ)
+    (hF : ∀ Y, 0 ≤ F Y) (hA : LayerAdapted d s F) (q : ℝ) (hq : 1 ≤ q)
+    (hlog : Real.log q ≤ 1/250) :
+    finiteMean (fun Y => F Y*q^(palindromeSlabCost l d e (Y,X))) ≤
+      finiteMean F * Real.exp (Fintype.card ι * (68*Real.exp (-(l:ℝ)/64))) := by
+  obtain ⟨r,rfl⟩ := Nat.exists_eq_add_of_le hsd
+  exact adapted_mul_mean_le r s F _ hF hA _
+    (fun lo => palindrome_slab_partial_mgf r s l hl hs e X lo q hq hlog)
+
+def familyEnd (L n : ℕ) : ℕ := if n = 0 then 0 else 2*(L*4^(n-1))
+
+lemma familyEnd_two (L n : ℕ) : 2*familyEnd L n ≤ L*4^n := by
+  cases n with
+  | zero => simp [familyEnd]
+  | succ n => simp only [familyEnd,Nat.succ_ne_zero,ite_false,Nat.add_one_sub_one,pow_succ]; nlinarith
+
+lemma familyEnd_ge (L n j : ℕ) (hj : j < n) : 2*(L*4^j) ≤ familyEnd L n := by
+  cases n with
+  | zero => omega
+  | succ n =>
+    simp only [familyEnd,Nat.succ_ne_zero,ite_false,Nat.add_one_sub_one]
+    apply Nat.mul_le_mul_left
+    exact Nat.mul_le_mul_left L (Nat.pow_le_pow_right (by decide) (by omega : j ≤ n))
+
+noncomputable def palindromeFamilyCost (L n d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (ω : BenesCoins d) : ℕ :=
+  ∑ j ∈ Finset.range n, palindromeSlabCost (L*4^j) d e ω
+
+lemma palindromeFamilyCost_adapted (L n d : ℕ) {ι : Type*} [Fintype ι]
+    (e : ι ↪ Card d) (X : SwitchIndex d → Bool) :
+    LayerAdapted d (familyEnd L n) (fun Y => palindromeFamilyCost L n d e (Y,X)) := by
+  apply layerAdapted_sum
+  intro j hj
+  exact layerAdapted_mono (palindromeSlabCost_adapted (L*4^j) d e X)
+    (familyEnd_ge L n j (Finset.mem_range.mp hj))
+
+end Thorp
+
+end ThorpNine.Tail
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Tail/PriorPaths.lean b/lean/OAI/Probability/ThorpRouting/Tail/PriorPaths.lean
index ddfcaeec5cc0a335cf99cbce628913b9f9264f0b..52f76f1bf375e197116a89f843094972fcef4070 100644
--- a/lean/OAI/Probability/ThorpRouting/Tail/PriorPaths.lean
+++ b/lean/OAI/Probability/ThorpRouting/Tail/PriorPaths.lean
@@ -1,882 +1,882 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Tail.AssignmentSlots
-
-namespace OAI
-
-namespace ThorpNine.Tail
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-namespace RoutingNetwork
-variable {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι] [DecidableEq ι]
-  {d : ℕ} (R : RoutingNetwork α ι d)
-lemma mem_exposedPaths (b : ι → Bool) (V : Finset (α)) (s : ι) :
-    s ∈ exposedPaths R b V ↔ (R.users b s).1 ∈ V ∨ (R.users b s).2 ∈ V := by
-  classical
-  simp only [exposedPaths,Finset.mem_biUnion,R.mem_path]
-  constructor
-  · rintro ⟨x,hx,h | h⟩
-    · exact Or.inl (h ▸ hx)
-    · exact Or.inr (h ▸ hx)
-  · rintro (h | h)
-    · exact ⟨_,h,Or.inl rfl⟩
-    · exact ⟨_,h,Or.inr rfl⟩
-
-lemma terminalCharge_sum (b : ι → Bool) (z : α) (V : Finset (α))
-    (hz : z ∉ V) :
-    (terminalCharge R b z V : ℝ) = ∑ s : ι,
-      ((if (R.users b s).1 = z ∧ (R.users b s).2 ∈ V then (1:ℝ) else 0) +
-      (if (R.users b s).2 = z ∧ (R.users b s).1 ∈ V then (1:ℝ) else 0)) := by
-  classical
-  have he : R.path b z ∩ exposedPaths R b V =
-      Finset.univ.filter (fun s =>
-        ((R.users b s).1 = z ∧ (R.users b s).2 ∈ V) ∨
-        ((R.users b s).2 = z ∧ (R.users b s).1 ∈ V)) := by
-    ext s
-    simp only [Finset.mem_inter,R.mem_path,mem_exposedPaths,
-      Finset.mem_filter,Finset.mem_univ,true_and]
-    by_cases h₁ : (R.users b s).1 = z <;>
-      by_cases h₂ : (R.users b s).2 = z <;> simp [h₁,h₂,hz]
-  rw [terminalCharge,he]
-  simp only [← Finset.sum_boole]
-  apply Finset.sum_congr rfl
-  intro s _
-  by_cases h₁ : (R.users b s).1 = z <;>
-    by_cases h₂ : (R.users b s).2 = z <;> simp [h₁,h₂,hz]
-
-noncomputable def priorPaths {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) : Finset (α) :=
-  Finset.univ.filter (fun x => ∃ i : Fin A, o x = some i ∧ i.val < t)
-
-omit [DecidableEq α] in
-@[simp] lemma mem_priorPaths {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) (x : α) :
-    x ∈ priorPaths o t ↔ ∃ i : Fin A, o x = some i ∧ i.val < t := by
-  classical
-  simp [priorPaths]
-
-lemma terminalCharge_prior_sum {A : ℕ} (o : α → Option (Fin A))
-    (b : ι → Bool) (z : Terminal o) (i : Fin A) :
-    (terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) =
-      ∑ s : ι,
-        ((if (z i).val = (R.users b s).1 ∧ (R.users b s).2 ∈ priorPaths o (i.val+1)
-          then (1:ℝ) else 0) +
-        (if (z i).val = (R.users b s).2 ∧ (R.users b s).1 ∈ priorPaths o (i.val+1)
-          then (1:ℝ) else 0)) := by
-  classical
-  rw [terminalCharge_sum _ _ _ _ (Finset.notMem_erase _ _)]
-  apply Finset.sum_congr rfl
-  intro s _
-  have hn := R.users_ne b s
-  simp only [Finset.mem_erase]
-  by_cases h₁ : (z i).val = (R.users b s).1 <;>
-    by_cases h₂ : (z i).val = (R.users b s).2 <;> simp_all [eq_comm]
-
-lemma rotationMeeting_eq_slot_sum {A : ℕ} (o : α → Option (Fin A))
-    (z : Terminal o) (u v : α) :
-    rotationMeeting o z u v = ∑ i : Fin A,
-      ((if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) +
-      (if (z i).val = v ∧ u ∈ priorPaths o (i.val+1) then (1:ℝ) else 0)) := by
-  classical
-  have only_owner (u : α) (a : Fin A) (hou : o u = some a)
-      (i : Fin A) (hia : i ≠ a) : (z i).val ≠ u := by
-    intro h
-    have hi := (z i).property
-    rw [h,hou] at hi
-    exact hia (Option.some.inj hi).symm
-  have sum_one (u v : α) (a : Fin A) (hou : o u = some a) :
-      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) =
-      if (z a).val = u ∧ v ∈ priorPaths o (a.val+1) then 1 else 0 := by
-    apply Finset.sum_eq_single a
-    · intro i _ hia
-      simp [only_owner u a hou i hia]
-    · simp
-  have sum_none (u v : α) (hou : o u = none) :
-      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) = 0 := by
-    apply Finset.sum_eq_zero
-    intro i _
-    have hn : (z i).val ≠ u := by
-      intro h
-      have hi := (z i).property
-      rw [h,hou] at hi
-      contradiction
-    simp [hn]
-  rw [Finset.sum_add_distrib]
-  cases hou : o u with
-  | none =>
-    rw [sum_none u v hou]
-    simp [rotationMeeting,hou,mem_priorPaths]
-  | some a =>
-    rw [sum_one u v a hou]
-    cases hov : o v with
-    | none =>
-      rw [sum_none v u hov]
-      simp [rotationMeeting,hou,hov,mem_priorPaths]
-    | some b =>
-      rw [sum_one v u b hov]
-      simp only [rotationMeeting,hou,hov,mem_priorPaths,Option.some.injEq]
-      rcases lt_trichotomy a b with hab | rfl | hba
-      · have hn : ¬ b.val < a.val+1 := by omega
-        have hy : a.val < b.val+1 := by omega
-        simp [hab,hn,hy]
-      · simp
-      · have hn : ¬ a.val < b.val+1 := by omega
-        have hy : b.val < a.val+1 := by omega
-        simp [hba,not_lt_of_ge (le_of_lt hba),hn,hy]
-
-lemma rotationCharge_eq_terminal_sum {A : ℕ} (o : α → Option (Fin A))
-    (b : ι → Bool) (z : Terminal o) :
-    rotationCharge R o b z = ∑ i : Fin A,
-      (terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) := by
-  simp only [rotationCharge,rotationMeeting_eq_slot_sum,terminalCharge_prior_sum]
-  exact Finset.sum_comm
-
-omit [DecidableEq α] in
-@[simp] lemma priorPaths_zero {A : ℕ} (o : α → Option (Fin A)) : priorPaths o 0 = ∅ := by
-  classical
-  ext x
-  simp
-
-lemma priorPaths_succ {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) (ht : t < A) :
-    priorPaths o (t+1) = priorPaths o t ∪ cycleFiber o ⟨t,ht⟩ := by
-  classical
-  ext x
-  simp only [mem_priorPaths,Finset.mem_union,mem_cycleFiber]
-  constructor
-  · rintro ⟨i,hi,hit⟩
-    by_cases h : i.val < t
-    · exact Or.inl ⟨i,hi,h⟩
-    · have he : i = ⟨t,ht⟩ := Fin.ext (by change i.val = t; omega)
-      exact Or.inr (he ▸ hi)
-  · rintro (⟨i,hi,hit⟩ | hi)
-    · exact ⟨i,hi,by omega⟩
-    · exact ⟨⟨t,ht⟩,hi,by change t < t+1; omega⟩
-
-omit [DecidableEq α] in
-lemma priorPaths_disjoint {A : ℕ} (o : α → Option (Fin A)) (i : Fin A) :
-    Disjoint (priorPaths o i.val) (cycleFiber o i) := by
-  classical
-  apply Finset.disjoint_left.mpr
-  intro x hx hx'
-  obtain ⟨j,hj,hji⟩ := (mem_priorPaths _ _ _).mp hx
-  have he : j = i := Option.some.inj (hj.symm.trans ((mem_cycleFiber _ _ _).mp hx'))
-  subst j
-  exact (lt_irrefl _ hji)
-
-namespace QueryTree
-open Thorp.QueryTree
-
-lemma seekCycles_owned {A : ℕ} (S : Finset (α)) (r : Fin A → ℕ)
-    (o : α → Option (Fin A)) (z : Terminal o) (ω : ι → Bool)
-    (hS : ∀ i, cycleFiber o i ⊆ S)
-    (hO : ∀ i, orbitSet (R.perm ω) (z i).val = cycleFiber o i)
-    (hC : ∀ i, (cycleFiber o i).card = r i+1) :
-    let p := R.perm ω
-    let L := List.ofFn (fun i : Fin A => (p (z i).val,r i))
-    (seekCycles R S L ∅).succeeds ω = true ∧
-      ((seekCycles R S L ∅).staleCharge ∅ ω : ℝ) = rotationCharge R o (ω) z := by
-  classical
-  let b := ω
-  let p := R.perm b
-  let f (i : Fin A) := (p (z i).val,r i)
-  let c (i : Fin A) : ℝ := terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val)
-  have hc (i : Fin A) : Function.minimalPeriod p (z i).val - 1 = r i := by
-    have he := orbitSet_card p (z i).val
-    rw [hO,hC] at he
-    omega
-  have hs (t : ℕ) (ht : t ≤ A) :
-      (seekCycles R S ((List.ofFn f).drop t) (priorPaths o t)).succeeds ω = true ∧
-        ((seekCycles R S ((List.ofFn f).drop t) (priorPaths o t)).staleCharge (exposedPaths R b (priorPaths o t)) ω : ℝ) =
-          ((List.ofFn c).drop t).sum := by
-    induction ht using Nat.decreasingInduction with
-    | self => simp only [List.length_ofFn,le_refl,List.drop_eq_nil_of_le,List.sum_nil,seekCycles,Thorp.QueryTree.succeeds,staleCharge,Nat.cast_zero,and_self]
-    | of_succ t ht ih =>
-      have htf : t < (List.ofFn f).length := by simpa using ht
-      have htc : t < (List.ofFn c).length := by simpa using ht
-      rw [List.drop_eq_getElem_cons htf,List.drop_eq_getElem_cons htc]
-      simp only [List.getElem_ofFn,List.sum_cons,seekCycles]
-      have h := seekCycle_orbit R S (priorPaths o t) (z ⟨t,ht⟩).val
-        (fun V => seekCycles R S ((List.ofFn f).drop (t+1)) V) ω
-        (by rw [hO]; exact hS _) (by rw [hO]; exact priorPaths_disjoint o ⟨t,ht⟩)
-      dsimp only at h
-      rw [hc,hO,←priorPaths_succ o t ht] at h
-      dsimp only [f]
-      constructor
-      · exact h.1.trans ih.1
-      · rw [h.2,Nat.cast_add,ih.2]
-        exact add_comm _ _
-  have h := hs 0 (Nat.zero_le _)
-  simp only [List.drop_zero,priorPaths_zero,exposedPaths_empty,List.sum_ofFn] at h
-  dsimp only
-  exact ⟨h.1,h.2.trans (rotationCharge_eq_terminal_sum R o b z).symm⟩
-
-end QueryTree
-section Collections
-variable {A : ℕ}
-
-def ValidCycleSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ)
-    (o : α → Option (Fin A)) : Prop :=
-  (∀ i, (cycleFiber o i).card = r i + 1) ∧
-  (∀ i, cycleFiber o i ⊆ S) ∧
-  (∀ i x, o x = some i → orbitSet p x = cycleFiber o i)
-
-abbrev CycleSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
-  {o : α → Option (Fin A) // ValidCycleSlots p S r o}
-
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
-    Fintype (CycleSlots p S r) := Fintype.ofFinite _
-
-lemma CycleSlots.terminal_nonempty {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (c : CycleSlots p S r) (i : Fin A) : Nonempty {x : α // c.val x = some i} := by
-  have hn : (cycleFiber c.val i).Nonempty := Finset.card_pos.mp (by rw [c.property.1]; omega)
-  obtain ⟨x,hx⟩ := hn
-  exact ⟨x,(mem_cycleFiber _ _ _).mp hx⟩
-
-omit [DecidableEq α] in
-lemma owner_ext {o o' : α → Option (Fin A)}
-    (h : ∀ i, cycleFiber o i = cycleFiber o' i) : o = o' := by
-  funext x
-  have he (i : Fin A) : o x = some i ↔ o' x = some i := by
-    simpa only [mem_cycleFiber] using (Finset.ext_iff.mp (h i) x)
-  cases ho : o x with
-  | some i => exact ((he i).mp ho).symm
-  | none =>
-    cases ho' : o' x with
-    | none => rfl
-    | some i => have hn := (he i).mpr ho'; rw [ho] at hn; contradiction
-
-abbrev RootedSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
-  Σ c : CycleSlots p S r, Terminal c.val
-
-noncomputable def rootedStart {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (c : RootedSlots p S r) : Fin A → S := fun i =>
-  ⟨p (c.2 i).val,c.1.property.2.1 i (by
-    rw [←c.1.property.2.2 i (c.2 i).val (c.2 i).property]
-    exact (mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.apply_right (Equiv.Perm.SameCycle.refl p _)))⟩
-
-lemma rootedStart_injective {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ} :
-    Function.Injective (rootedStart (p:=p) (S:=S) (r:=r)) := by
-  rintro ⟨c,z⟩ ⟨c',z'⟩ h
-  have hz (i : Fin A) : (z i).val = (z' i).val :=
-    p.injective (congrArg Subtype.val (congrFun h i))
-  have hc : c = c' := Subtype.ext (owner_ext (fun i => by
-    rw [←c.property.2.2 i (z i).val (z i).property,
-      ←c'.property.2.2 i (z' i).val (z' i).property,hz]))
-  subst c'
-  have hzz : z = z' := funext (fun i => Subtype.ext (hz i))
-  subst z'
-  rfl
-
-lemma rooted_weight_sum_le (S : Finset (α)) (r : Fin A → ℕ) (ω : ι → Bool) :
-    (∑ c : RootedSlots (R.perm ω) S r,
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2)) ≤
-    ∑ a : Fin A → S,
-      if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-        ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-      else 0 := by
-  classical
-  let p := R.perm ω
-  let g (a : Fin A → S) : ℝ :=
-    if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-    else 0
-  have he (c : RootedSlots p S r) :
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2) =
-      g (rootedStart c) := by
-    have hs := QueryTree.seekCycles_owned R S r c.1.val c.2 ω c.1.property.2.1
-      (fun i => c.1.property.2.2 i (c.2 i).val (c.2 i).property) c.1.property.1
-    dsimp only at hs
-    unfold g
-    rw [ite_eq_left (show (QueryTree.seekCycles R S (List.ofFn (fun i => ((rootedStart c i).val,r i))) ∅).succeeds ω = true from hs.1)]
-    rw [← hs.2,neg_mul,Real.exp_neg,mul_comm (Real.log 2),Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
-    rfl
-  simp_rw [he]
-  exact sum_comp_injective_le rootedStart rootedStart_injective g (fun a => by dsimp [g]; split_ifs <;> positivity)
-
-lemma collection_rotation_lower (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r)
-    (ω : ι → Bool) (c : CycleSlots (R.perm ω) S r) :
-    collectionWeight d r ≤ ∑ z : Terminal c.val,
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z) := by
-  classical
-  let (i : Fin A) : Nonempty {x : α // c.val x = some i} := c.terminal_nonempty i
-  have hm : Monotone (fun i => (cycleFiber c.val i).card) := by
-    intro i j hij
-    dsimp only
-    rw [c.property.1,c.property.1]
-    exact Nat.add_le_add_right (hr hij) 1
-  have hb := rotation_exp_bound R hm (ω)
-  have hcard : (Fintype.card (Terminal c.val) : ℝ) = ∏ i : Fin A, (r i+1 : ℝ) := by
-    rw [Fintype.card_pi,Nat.cast_prod]
-    apply Finset.prod_congr rfl
-    intro i _
-    rw [card_cycleFiber,c.property.1,Nat.cast_add,Nat.cast_one]
-  have hp : (0:ℝ) < Fintype.card (Terminal c.val) := Nat.cast_pos.mpr Fintype.card_pos
-  have hl : rotationLower d r ≤ finiteMean (fun z : Terminal c.val => Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z)) := by
-    simpa only [rotationLower,c.property.1,Nat.cast_add,Nat.cast_one] using hb
-  have hh := (le_div_iff₀ hp).mp hl
-  rw [hcard] at hh
-  unfold collectionWeight
-  rw [← Finset.mul_sum]
-  calc
-    _ = ((2:ℝ)^d)^A * (rotationLower d r * ∏ i : Fin A, (r i+1 : ℝ)) := by ring
-    _ ≤ _ := mul_le_mul_of_nonneg_left hh (by positivity)
-
-lemma cycleSlots_weighted_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : ι → Bool =>
-      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) *
-      collectionWeight d r ≤ (S.card : ℝ)^A := by
-  classical
-  let g (a : Fin A → S) (ω : ι → Bool) : ℝ :=
-    if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-    else 0
-  have hω (ω : ι → Bool) :
-      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ) * collectionWeight d r ≤
-        ∑ a : Fin A → S, g a ω := by
-    calc
-      _ = ∑ _c : CycleSlots (R.perm ω) S r, collectionWeight d r := by simp
-      _ ≤ ∑ c : CycleSlots (R.perm ω) S r,
-          ∑ z : Terminal c.val, ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z) :=
-        Finset.sum_le_sum (fun c _ => collection_rotation_lower R S r hr ω c)
-      _ = ∑ c : RootedSlots (R.perm ω) S r,
-          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2) :=
-        (Fintype.sum_sigma (fun c : RootedSlots (R.perm ω) S r =>
-          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2))).symm
-      _ ≤ _ := rooted_weight_sum_le R S r ω
-  have hb := finiteMean_mono hω
-  rw [finiteMean_mul_const,finiteMean_sum] at hb
-  calc
-    _ ≤ ∑ a : Fin A → S, finiteMean (g a) := hb
-    _ ≤ ∑ _a : Fin A → S, (1:ℝ) := by
-      apply Finset.sum_le_sum
-      intro a _
-      have hh := QueryTree.cycles_weighted_success R S (List.ofFn (fun i : Fin A => ((a i).val,r i)))
-      simpa only [List.length_ofFn] using hh
-    _ = _ := by simp
-
-lemma cycleSlots_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : ι → Bool =>
-      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
-        (S.card : ℝ)^A / collectionWeight d r := by
-  exact (le_div_iff₀ (collectionWeight_pos d r)).mpr (cycleSlots_weighted_moment R S r hr)
-
-lemma butterfly_cycleSlots_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : ι → Bool =>
-      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
-        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
-  have h := cycleSlots_moment R S r hr
-  rw [collectionWeight_eq_sqrt,div_mul_eq_div_mul_one_div] at h
-  rw [div_pow]
-  simpa only [one_div,Finset.prod_inv_distrib] using h
-
-end Collections
-
-section CycleCounting
-variable {A : ℕ}
-
-abbrev SelectedCycle (p : Equiv.Perm (α)) (S : Finset (α)) :=
-  {C : Finset (α) // (∃ x, C = orbitSet p x) ∧ C ⊆ S}
-
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) :
-    Fintype (SelectedCycle p S) := Fintype.ofFinite _
-
-lemma SelectedCycle.nonempty {p : Equiv.Perm (α)} {S : Finset (α)}
-    (C : SelectedCycle p S) : C.val.Nonempty := by
-  obtain ⟨x,hx⟩ := C.property.1
-  rw [hx]
-  exact orbitSet_nonempty p x
-
-lemma SelectedCycle.orbit_eq {p : Equiv.Perm (α)} {S : Finset (α)}
-    (C : SelectedCycle p S) {x : α} (hx : x ∈ C.val) : orbitSet p x = C.val := by
-  obtain ⟨y,hy⟩ := C.property.1
-  rw [hy] at hx ⊢
-  exact (orbitSet_eq p ((mem_orbitSet _ _ _).mp hx)).symm
-
-lemma SelectedCycle.eq_of_mem {p : Equiv.Perm (α)} {S : Finset (α)}
-    (C D : SelectedCycle p S) {x : α} (hC : x ∈ C.val) (hD : x ∈ D.val) : C = D := by
-  apply Subtype.ext
-  exact (C.orbit_eq hC).symm.trans (D.orbit_eq hD)
-
-abbrev SlotAssignment (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
-  {f : Fin A → SelectedCycle p S // Function.Injective f ∧ ∀ i, (f i).val.card = r i+1}
-
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
-    Fintype (SlotAssignment p S r) := Fintype.ofFinite _
-
-noncomputable def slotsToAssignment {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (c : CycleSlots p S r) : SlotAssignment p S r := by
-  classical
-  let f (i : Fin A) : SelectedCycle p S := ⟨cycleFiber c.val i,by
-    obtain ⟨x,hx⟩ := c.terminal_nonempty i
-    exact ⟨⟨x,(c.property.2.2 i x hx).symm⟩,c.property.2.1 i⟩⟩
-  refine ⟨f,?_,fun i => c.property.1 i⟩
-  intro i j hij
-  have he : cycleFiber c.val i = cycleFiber c.val j := congrArg Subtype.val hij
-  obtain ⟨x,hx⟩ := c.terminal_nonempty i
-  have hxi : x ∈ cycleFiber c.val i := (mem_cycleFiber _ _ _).mpr hx
-  rw [he] at hxi
-  exact Option.some.inj (hx.symm.trans ((mem_cycleFiber _ _ _).mp hxi))
-
-noncomputable def assignmentOwner {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) (x : α) : Option (Fin A) := by
-  classical
-  exact if h : ∃ i, x ∈ (f.val i).val then some (Classical.choose h) else none
-
-lemma assignmentOwner_eq_some {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) (x : α) (i : Fin A) :
-    assignmentOwner f x = some i ↔ x ∈ (f.val i).val := by
-  classical
-  unfold assignmentOwner
-  split_ifs with h
-  · constructor
-    · intro hi
-      have hx := Classical.choose_spec h
-      rw [Option.some.inj hi] at hx
-      exact hx
-    · intro hx
-      have he := (f.val (Classical.choose h)).eq_of_mem (f.val i) (Classical.choose_spec h) hx
-      exact congrArg some (f.property.1 he)
-  · constructor
-    · intro hi; contradiction
-    · intro hx; exact (h ⟨i,hx⟩).elim
-
-lemma assignmentOwner_fiber {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) (i : Fin A) : cycleFiber (assignmentOwner f) i = (f.val i).val := by
-  ext x
-  simp only [mem_cycleFiber,assignmentOwner_eq_some]
-
-noncomputable def assignmentToSlots {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) : CycleSlots p S r :=
-  ⟨assignmentOwner f,by
-    refine ⟨?_,?_,?_⟩
-    · intro i
-      rw [assignmentOwner_fiber]
-      exact f.property.2 i
-    · intro i
-      rw [assignmentOwner_fiber]
-      exact (f.val i).property.2
-    · intro i x hx
-      rw [assignmentOwner_fiber]
-      exact (f.val i).orbit_eq ((assignmentOwner_eq_some f x i).mp hx)⟩
-
-noncomputable def cycleSlotsEquivAssignments (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
-    CycleSlots p S r ≃ SlotAssignment p S r where
-  toFun := slotsToAssignment
-  invFun := assignmentToSlots
-  left_inv c := by
-    apply Subtype.ext
-    apply owner_ext
-    intro i
-    rw [assignmentToSlots,assignmentOwner_fiber]
-    rfl
-  right_inv f := by
-    apply Subtype.ext
-    funext i
-    apply Subtype.ext
-    exact assignmentOwner_fiber f i
-
-end CycleCounting
-
-section GroupedCounting
-variable {A : ℕ}
-
-abbrev LengthClass (r : Fin A → ℕ) := {j : ℕ // j ∈ Finset.univ.image (fun i => r i + 1)}
-
-def slotClass (r : Fin A → ℕ) (i : Fin A) : LengthClass r :=
-  ⟨r i+1, Finset.mem_image.mpr ⟨i, Finset.mem_univ _, rfl⟩⟩
-
-abbrev SlotsOfLength (r : Fin A → ℕ) (j : LengthClass r) := {i : Fin A // slotClass r i = j}
-abbrev CyclesOfLength (p : Equiv.Perm (α)) (S : Finset (α)) (j : ℕ) :=
-  {C : SelectedCycle p S // C.val.card = j}
-
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (j : ℕ) :
-    Fintype (CyclesOfLength p S j) := Fintype.ofFinite _
-
-abbrev GroupedAssignments (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
-  ∀ j : LengthClass r, SlotsOfLength r j ↪ CyclesOfLength p S j.val
-
-noncomputable def assignmentToGroups {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) : GroupedAssignments p S r := fun _j =>
-  { toFun := fun i => ⟨f.val i.val, (f.property.2 i.val).trans (congrArg Subtype.val i.property)⟩
-    inj' := fun _i _k h => Subtype.ext (f.property.1 (congrArg Subtype.val h)) }
-
-noncomputable def groupsEmbedding {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (F : GroupedAssignments p S r) : Fin A ↪ SelectedCycle p S :=
-  ((Equiv.sigmaFiberEquiv (slotClass r)).symm.toEmbedding.trans
-    (Function.Embedding.sigmaMap (Function.Embedding.refl _) F)).trans
-    ((Function.Embedding.sigmaMap (Function.Embedding.subtype _) (fun _ => Function.Embedding.refl _)).trans
-      (Equiv.sigmaFiberEquiv (fun C : SelectedCycle p S => C.val.card)).toEmbedding)
-
-noncomputable def groupsToAssignment {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
-    (F : GroupedAssignments p S r) : SlotAssignment p S r :=
-  ⟨groupsEmbedding F, (groupsEmbedding F).injective,
-    fun i => (F (slotClass r i) ⟨i,rfl⟩).property⟩
-
-noncomputable def assignmentEquivGroups (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
-    SlotAssignment p S r ≃ GroupedAssignments p S r where
-  toFun := assignmentToGroups
-  invFun := groupsToAssignment
-  left_inv f := by
-    apply Subtype.ext
-    rfl
-  right_inv F := by
-    funext j
-    apply Function.Embedding.ext
-    intro i
-    rcases i with ⟨i,hi⟩
-    subst j
-    rfl
-
-lemma card_cycleSlots_factorial (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
-    Fintype.card (CycleSlots p S r) =
-      ∏ j : LengthClass r, (Fintype.card (CyclesOfLength p S j.val)).descFactorial
-        (Fintype.card (SlotsOfLength r j)) := by
-  classical
-  rw [Fintype.card_congr (cycleSlotsEquivAssignments p S r),
-    Fintype.card_congr (assignmentEquivGroups p S r), Fintype.card_pi]
-  simp only [Fintype.card_embedding_eq]
-
-lemma butterfly_factorial_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : ι → Bool =>
-       (∏ j : LengthClass r,
-         (Fintype.card (CyclesOfLength (R.perm ω) S j.val)).descFactorial
-           (Fintype.card (SlotsOfLength r j)) : ℕ)) ≤
-      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-        ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
-  simpa only [←card_cycleSlots_factorial] using butterfly_cycleSlots_moment R S r hr
-
-end GroupedCounting
-
-section AllLengths
-variable {A : ℕ}
-
-noncomputable def assignmentRelabel (p : Equiv.Perm (α)) (S : Finset (α))
-    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
-    SlotAssignment p S r ≃ SlotAssignment p S (r ∘ e) where
-  toFun f := ⟨f.val ∘ e, f.property.1.comp e.injective, fun i => f.property.2 (e i)⟩
-  invFun f := ⟨f.val ∘ e.symm, f.property.1.comp e.symm.injective, fun i => by
-    simpa only [Function.comp_apply,Equiv.apply_symm_apply] using f.property.2 (e.symm i)⟩
-  left_inv f := by apply Subtype.ext; funext i; simp
-  right_inv f := by apply Subtype.ext; funext i; simp
-
-lemma card_cycleSlots_relabel (p : Equiv.Perm (α)) (S : Finset (α))
-    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
-    Fintype.card (CycleSlots p S r) = Fintype.card (CycleSlots p S (r ∘ e)) := by
-  exact Fintype.card_congr ((cycleSlotsEquivAssignments p S r).trans
-    ((assignmentRelabel p S r e).trans (cycleSlotsEquivAssignments p S (r ∘ e)).symm))
-
-lemma butterfly_cycleSlots_moment_all (S : Finset (α)) (r : Fin A → ℕ) :
-    finiteMean (fun ω : ι → Bool =>
-      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
-        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
-  have h := butterfly_cycleSlots_moment R S (r ∘ Tuple.sort r) (Tuple.monotone_sort r)
-  simp_rw [←card_cycleSlots_relabel] at h
-  have hp := Equiv.prod_comp (Tuple.sort r) (fun i : Fin A => (1 / Real.sqrt (r i+1) : ℝ))
-  simpa only [Function.comp_apply,hp] using h
-
-abbrev ShortCycle (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :=
-  {C : SelectedCycle p S // C.val.card ≤ J}
-
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    Fintype (ShortCycle p S J) := Fintype.ofFinite _
-
-noncomputable def shortCycleLength {p : Equiv.Perm (α)} {S : Finset (α)} {J : ℕ}
-    (C : ShortCycle p S J) : Fin J :=
-  ⟨C.val.val.card-1, by have := C.val.nonempty.card_pos; have := C.property; omega⟩
-
-lemma shortCycleLength_add_one {p : Equiv.Perm (α)} {S : Finset (α)} {J : ℕ}
-    (C : ShortCycle p S J) : (shortCycleLength C : ℕ)+1 = C.val.val.card := by
-  have := C.val.nonempty.card_pos
-  simp only [shortCycleLength]
-  omega
-
-noncomputable def shortEmbeddingEquiv (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    (Fin A ↪ ShortCycle p S J) ≃
-      Σ r : Fin A → Fin J, SlotAssignment p S (fun i => (r i : ℕ)) where
-  toFun f := ⟨fun i => shortCycleLength (f i),
-    ⟨fun i => (f i).val, fun i j h => f.injective (Subtype.ext h),
-      fun i => (shortCycleLength_add_one (f i)).symm⟩⟩
-  invFun c :=
-    { toFun := fun i => ⟨c.2.val i, by rw [c.2.property.2]; exact (c.1 i).isLt⟩
-      inj' := fun _i _j h => c.2.property.1 (congrArg Subtype.val h) }
-  left_inv f := by
-    apply Function.Embedding.ext
-    intro i
-    rfl
-  right_inv c := by
-    rcases c with ⟨r,f⟩
-    have he : (fun i => shortCycleLength (⟨f.val i, by rw [f.property.2]; exact (r i).isLt⟩ : ShortCycle p S J)) = r := by
-      funext i
-      apply Fin.ext
-      simp only [shortCycleLength,f.property.2,Nat.add_sub_cancel]
-    apply Sigma.ext he
-    apply (Subtype.heq_iff_coe_eq (by
-      intro x
-      change (Function.Injective x ∧ ∀ i, (x i).val.card = (f.val i).val.card - 1 + 1) ↔ _
-      simp only [f.property.2,Nat.add_sub_cancel])).mpr
-    rfl
-
-lemma shortCycle_factorial_eq (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    (Fintype.card (ShortCycle p S J)).descFactorial A =
-      ∑ r : Fin A → Fin J, Fintype.card (CycleSlots p S (fun i => (r i : ℕ))) := by
-  classical
-  have hc := Fintype.card_embedding_eq (α := Fin A) (β := ShortCycle p S J)
-  simp only [Fintype.card_fin] at hc
-  rw [←hc, Fintype.card_congr (shortEmbeddingEquiv p S J),Fintype.card_sigma]
-  apply Finset.sum_congr rfl
-  intro r _
-  exact (Fintype.card_congr (cycleSlotsEquivAssignments p S (fun i => (r i : ℕ)))).symm
-
-end AllLengths
-
-section CycleMGF
-
-lemma butterfly_shortCycle_moment (S : Finset (α)) (J A : ℕ) :
-    finiteMean (fun ω : ι → Bool =>
-      ((Fintype.card (ShortCycle (R.perm ω) S J)).descFactorial A : ℝ)) ≤
-      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))^A := by
-  classical
-  simp_rw [shortCycle_factorial_eq,Nat.cast_sum,finiteMean_sum]
-  calc
-    _ ≤ ∑ r : Fin A → Fin J,
-        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-          ∏ i : Fin A, (1 / Real.sqrt ((r i : ℕ)+1)) := by
-      apply Finset.sum_le_sum
-      intro r _
-      exact butterfly_cycleSlots_moment_all R S (fun i => (r i : ℕ))
-    _ = _ := by
-      rw [←Finset.mul_sum,mul_pow]
-      congr 1
-      rw [←Fintype.prod_sum (fun (_ : Fin A) (j : Fin J) => (1 / Real.sqrt ((j:ℕ)+1) : ℝ))]
-      simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fin]
-
-noncomputable def cycleMass (p : Equiv.Perm (α)) (S : Finset (α)) : ℕ :=
-  ∑ C : SelectedCycle p S, C.val.card
-
-lemma cycleMass_le (p : Equiv.Perm (α)) (S : Finset (α)) : cycleMass p S ≤ S.card := by
-  classical
-  let f : (Σ C : SelectedCycle p S, {x : α // x ∈ C.val}) → S :=
-    fun c => ⟨c.2.val,c.1.property.2 c.2.property⟩
-  have hi : Function.Injective f := by
-    intro c e h
-    have hx : c.2.val = e.2.val := congrArg Subtype.val h
-    have hC : c.1 = e.1 := c.1.eq_of_mem e.1 c.2.property (hx ▸ e.2.property)
-    rcases c with ⟨c,x⟩
-    rcases e with ⟨e,y⟩
-    dsimp only at hC
-    subst e
-    have he : x=y := Subtype.ext hx
-    cases he
-    rfl
-  have hh := Fintype.card_le_of_injective f hi
-  simpa only [Fintype.card_sigma,Fintype.card_coe,cycleMass] using hh
-
-lemma shortCycle_card_le (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    Fintype.card (ShortCycle p S J) ≤ S.card := by
-  classical
-  calc
-    _ ≤ Fintype.card (SelectedCycle p S) := Fintype.card_subtype_le _
-    _ = ∑ _ : SelectedCycle p S, 1 := by simp
-    _ ≤ cycleMass p S := Finset.sum_le_sum (fun C _ => C.nonempty.card_pos)
-    _ ≤ _ := cycleMass_le p S
-
-lemma butterfly_shortCycle_mgf (S : Finset (α)) (J : ℕ) (q : ℝ) (hq : 1 ≤ q) :
-    finiteMean (fun ω : ι → Bool =>
-      q^(Fintype.card (ShortCycle (R.perm ω) S J))) ≤
-      Real.exp ((q-1)*((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))) := by
-  apply factorial_mgf _ S.card _ q (fun ω => shortCycle_card_le _ _ _)
-  · positivity
-  · exact hq
-  · exact butterfly_shortCycle_moment R S J
-
-end CycleMGF
-
-
-abbrev LongCycle (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :=
-  {C : SelectedCycle p S // ¬ C.val.card ≤ J}
-noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    Fintype (LongCycle p S J) := Fintype.ofFinite _
-
-lemma selectedCycle_card_split (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    Fintype.card (SelectedCycle p S) =
-      Fintype.card (ShortCycle p S J) + Fintype.card (LongCycle p S J) := by
-  classical
-  rw [←Fintype.card_congr (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J)),Fintype.card_sum]
-
-lemma longCycle_card_bound (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
-    J * Fintype.card (LongCycle p S J) ≤ S.card := by
-  classical
-  have hs := Equiv.sum_comp (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J))
-    (fun C : SelectedCycle p S => C.val.card)
-  rw [Fintype.sum_sum_type] at hs
-  have h0 : J * Fintype.card (LongCycle p S J) ≤
-      ∑ C : LongCycle p S J, C.val.val.card := by
-    calc
-      _ = ∑ _C : LongCycle p S J, J := by simp [mul_comm]
-      _ ≤ _ := Finset.sum_le_sum (fun C _ => Nat.le_of_lt (Nat.lt_of_not_ge C.property))
-  have hh := cycleMass_le p S
-  unfold cycleMass at hh
-  rw [←hs] at hh
-  exact h0.trans ((Nat.le_add_left _ _).trans hh)
-
-lemma longCycle_pow_bound (p : Equiv.Perm (α)) (S : Finset (α))
-    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
-    q^(Fintype.card (LongCycle p S J)) ≤ Real.exp ((S.card : ℝ)/J * Real.log q) := by
-  have hq0 : 0 < q := lt_of_lt_of_le zero_lt_one hq
-  have hc : (Fintype.card (LongCycle p S J) : ℝ) ≤ (S.card : ℝ)/J := by
-    apply (le_div_iff₀ (by exact_mod_cast hJ)).mpr
-    have hh := longCycle_card_bound p S J
-    exact_mod_cast (by simpa only [mul_comm] using hh)
-  calc
-    _ = Real.exp ((Fintype.card (LongCycle p S J) : ℝ)*Real.log q) := by
-      rw [Real.exp_nat_mul,Real.exp_log hq0]
-    _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right hc (Real.log_nonneg hq))
-
-lemma butterfly_cycle_log_mgf (S : Finset (α)) (J : ℕ) (hJ : 0 < J)
-    (q : ℝ) (hq : 1 ≤ q) :
-    Real.log (finiteMean (fun ω : ι → Bool =>
-      q^(Fintype.card (SelectedCycle (R.perm ω) S)))) ≤
-      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
-  let lam : ℝ := (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1)
-  let c : ℝ := (S.card : ℝ)/J * Real.log q
-  have hnum : (q-1)*lam ≤ 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
-    have hsum := sum_inv_sqrt J
-    have hp : 0 ≤ (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) := by positivity
-    have h0 : 0 ≤ 2*Real.sqrt J := by positivity
-    have hh := mul_le_mul_of_nonneg_left hsum hp
-    have hq1 : 0 ≤ q-1 := sub_nonneg.mpr hq
-    calc
-      _ ≤ (q-1)*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
-        mul_le_mul_of_nonneg_left hh hq1
-      _ ≤ q*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
-        mul_le_mul_of_nonneg_right (by linarith) (mul_nonneg hp h0)
-      _ = _ := by ring
-  have hm : finiteMean (fun ω : ι → Bool =>
-      q^(Fintype.card (SelectedCycle (R.perm ω) S))) ≤
-      Real.exp (c+(q-1)*lam) := by
-    calc
-      _ ≤ finiteMean (fun ω : ι → Bool =>
-          q^(Fintype.card (ShortCycle (R.perm ω) S J))*Real.exp c) := by
-        apply finiteMean_mono
-        intro ω
-        rw [selectedCycle_card_split _ _ J,pow_add]
-        exact mul_le_mul_of_nonneg_left (longCycle_pow_bound _ S J hJ q hq) (by positivity)
-      _ = finiteMean (fun ω : ι → Bool =>
-          q^(Fintype.card (ShortCycle (R.perm ω) S J))) * Real.exp c :=
-        finiteMean_mul_const _ _
-      _ ≤ Real.exp ((q-1)*lam) * Real.exp c :=
-        mul_le_mul_of_nonneg_right (butterfly_shortCycle_mgf R S J q hq) (Real.exp_nonneg _)
-      _ = _ := by rw [←Real.exp_add,add_comm]
-  have hpos : 0 < finiteMean (fun ω : ι → Bool =>
-      q^(Fintype.card (SelectedCycle (R.perm ω) S))) := by
-    apply finiteMean_pos
-    intro ω
-    positivity
-  have hlog := Real.log_le_log hpos hm
-  rw [Real.log_exp] at hlog
-  dsimp [c] at hlog
-  linarith
-
-
-end RoutingNetwork
-
-variable {α B : Type*} [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] {d : ℕ}
-
-def parallelPerm (ω : B × SwitchIndex d → Bool) : Equiv.Perm (B × Card d) where
-  toFun x := (x.1,butterflyPerm d (decodeButterfly d (fun i => ω (x.1,i))) x.2)
-  invFun y := (y.1,(butterflyPerm d (decodeButterfly d (fun i => ω (y.1,i)))).symm y.2)
-  left_inv x := by simp
-  right_inv y := by simp
-
-def relabeledPerm (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
-    (ω : B × SwitchIndex d → Bool) : Equiv.Perm α :=
-  (pre.trans (parallelPerm ω)).trans post
-
-noncomputable def parallelPath (pre : α ≃ B × Card d)
-    (ω : B × SwitchIndex d → Bool) (x : α) : Finset (B × SwitchIndex d) :=
-  (routeDomain d (pre x).2
-    (butterflyPerm d (decodeButterfly d (fun i => ω ((pre x).1,i))) (pre x).2)).image
-      (fun i => ((pre x).1,i))
-
-def parallelUsers (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
-    (i : B × SwitchIndex d) : α × α :=
-  let u := switchUsers d (decodeButterfly d (fun j => ω (i.1,j))) i.2
-  (pre.symm (i.1,u.1),pre.symm (i.1,u.2))
-
-omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
-lemma parallelUsers_ne (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
-    (i : B × SwitchIndex d) : (parallelUsers pre ω i).1 ≠ (parallelUsers pre ω i).2 := by
-  intro h
-  exact switchUsers_ne d _ _ (congrArg Prod.snd (pre.symm.injective h))
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma mem_parallelPath (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
-    (x : α) (i : B × SwitchIndex d) :
-    i ∈ parallelPath pre ω x ↔ (parallelUsers pre ω i).1 = x ∨ (parallelUsers pre ω i).2 = x := by
-  classical
-  obtain ⟨b,j⟩ := i
-  simp only [parallelPath,Finset.mem_image,Prod.mk.injEq]
-  constructor
-  · rintro ⟨k,hk,hb,rfl⟩
-    subst b
-    rcases (mem_routeDomain_iff_user d _ _ _).mp hk with h | h
-    · left
-      apply pre.injective
-      change pre (pre.symm ((pre x).1,_)) = pre x
-      rw [Equiv.apply_symm_apply]
-      exact Prod.ext rfl h
-    · right
-      apply pre.injective
-      change pre (pre.symm ((pre x).1,_)) = pre x
-      rw [Equiv.apply_symm_apply]
-      exact Prod.ext rfl h
-  · intro h
-    have hh : (b,(switchUsers d (decodeButterfly d (fun j => ω (b,j))) j).1) = pre x ∨
-        (b,(switchUsers d (decodeButterfly d (fun j => ω (b,j))) j).2) = pre x := by
-      rcases h with h | h
-      · exact Or.inl (by simpa only [parallelUsers,Equiv.apply_symm_apply] using congrArg pre h)
-      · exact Or.inr (by simpa only [parallelUsers,Equiv.apply_symm_apply] using congrArg pre h)
-    have hb : b = (pre x).1 := hh.elim (congrArg Prod.fst) (congrArg Prod.fst)
-    subst b
-    refine ⟨j,?_,rfl,rfl⟩
-    apply (mem_routeDomain_iff_user d _ _ _).mpr
-    exact hh.elim (fun h => Or.inl (congrArg Prod.snd h)) (fun h => Or.inr (congrArg Prod.snd h))
-
-omit [Fintype α] [DecidableEq α] [Fintype B] in
-lemma card_parallelPath (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool) (x : α) :
-    (parallelPath pre ω x).card = d := by
-  classical
-  rw [parallelPath,Finset.card_image_of_injective _ (fun _ _ h => Prod.mk.inj h |>.2),card_routeDomain]
-
-noncomputable def blockMarks (pre : α ≃ B × Card d) (S : Finset α) (b : B) : Finset (Card d) :=
-  Finset.univ.filter (fun x => pre.symm (b,x) ∈ S)
-
-omit [Fintype α] [Fintype B] [DecidableEq B] in
-@[simp] lemma mem_blockMarks (pre : α ≃ B × Card d) (S : Finset α) (b : B) (x : Card d) :
-    x ∈ blockMarks pre S b ↔ pre.symm (b,x) ∈ S := by
-  classical
-  simp [blockMarks]
-
-omit [DecidableEq B] in
-lemma sum_card_blockMarks (pre : α ≃ B × Card d) (S : Finset α) :
-    ∑ b : B, (blockMarks pre S b).card = S.card := by
-  classical
-  calc
-    _ = ∑ b : B, ∑ x : Card d, if pre.symm (b,x) ∈ S then 1 else 0 := by
-      simp only [blockMarks,Finset.card_filter]
-    _ = ∑ u : B × Card d, if pre.symm u ∈ S then 1 else 0 := (Fintype.sum_prod_type (fun u : B × Card d => if pre.symm u ∈ S then (1:ℕ) else 0)).symm
-    _ = ∑ x : α, if x ∈ S then 1 else 0 := Equiv.sum_comp pre.symm (fun x : α => if x ∈ S then (1:ℕ) else 0)
-    _ = _ := by simp
-
-omit [DecidableEq B] in
-lemma card_blockMarks_le (pre : α ≃ B × Card d) (S : Finset α) (b : B) :
-    (blockMarks pre S b).card ≤ S.card := by
-  rw [←sum_card_blockMarks pre S]
-  exact Finset.single_le_sum (f := fun b : B => (blockMarks pre S b).card) (fun _ _ => Nat.zero_le _) (Finset.mem_univ b)
-
-end Thorp
-
-end ThorpNine.Tail
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Tail.AssignmentSlots
+
+namespace OAI
+
+namespace ThorpNine.Tail
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+namespace RoutingNetwork
+variable {α ι : Type*} [Fintype α] [DecidableEq α] [Fintype ι] [DecidableEq ι]
+  {d : ℕ} (R : RoutingNetwork α ι d)
+lemma mem_exposedPaths (b : ι → Bool) (V : Finset (α)) (s : ι) :
+    s ∈ exposedPaths R b V ↔ (R.users b s).1 ∈ V ∨ (R.users b s).2 ∈ V := by
+  classical
+  simp only [exposedPaths,Finset.mem_biUnion,R.mem_path]
+  constructor
+  · rintro ⟨x,hx,h | h⟩
+    · exact Or.inl (h ▸ hx)
+    · exact Or.inr (h ▸ hx)
+  · rintro (h | h)
+    · exact ⟨_,h,Or.inl rfl⟩
+    · exact ⟨_,h,Or.inr rfl⟩
+
+lemma terminalCharge_sum (b : ι → Bool) (z : α) (V : Finset (α))
+    (hz : z ∉ V) :
+    (terminalCharge R b z V : ℝ) = ∑ s : ι,
+      ((if (R.users b s).1 = z ∧ (R.users b s).2 ∈ V then (1:ℝ) else 0) +
+      (if (R.users b s).2 = z ∧ (R.users b s).1 ∈ V then (1:ℝ) else 0)) := by
+  classical
+  have he : R.path b z ∩ exposedPaths R b V =
+      Finset.univ.filter (fun s =>
+        ((R.users b s).1 = z ∧ (R.users b s).2 ∈ V) ∨
+        ((R.users b s).2 = z ∧ (R.users b s).1 ∈ V)) := by
+    ext s
+    simp only [Finset.mem_inter,R.mem_path,mem_exposedPaths,
+      Finset.mem_filter,Finset.mem_univ,true_and]
+    by_cases h₁ : (R.users b s).1 = z <;>
+      by_cases h₂ : (R.users b s).2 = z <;> simp [h₁,h₂,hz]
+  rw [terminalCharge,he]
+  simp only [← Finset.sum_boole]
+  apply Finset.sum_congr rfl
+  intro s _
+  by_cases h₁ : (R.users b s).1 = z <;>
+    by_cases h₂ : (R.users b s).2 = z <;> simp [h₁,h₂,hz]
+
+noncomputable def priorPaths {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) : Finset (α) :=
+  Finset.univ.filter (fun x => ∃ i : Fin A, o x = some i ∧ i.val < t)
+
+omit [DecidableEq α] in
+@[simp] lemma mem_priorPaths {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) (x : α) :
+    x ∈ priorPaths o t ↔ ∃ i : Fin A, o x = some i ∧ i.val < t := by
+  classical
+  simp [priorPaths]
+
+lemma terminalCharge_prior_sum {A : ℕ} (o : α → Option (Fin A))
+    (b : ι → Bool) (z : Terminal o) (i : Fin A) :
+    (terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) =
+      ∑ s : ι,
+        ((if (z i).val = (R.users b s).1 ∧ (R.users b s).2 ∈ priorPaths o (i.val+1)
+          then (1:ℝ) else 0) +
+        (if (z i).val = (R.users b s).2 ∧ (R.users b s).1 ∈ priorPaths o (i.val+1)
+          then (1:ℝ) else 0)) := by
+  classical
+  rw [terminalCharge_sum _ _ _ _ (Finset.notMem_erase _ _)]
+  apply Finset.sum_congr rfl
+  intro s _
+  have hn := R.users_ne b s
+  simp only [Finset.mem_erase]
+  by_cases h₁ : (z i).val = (R.users b s).1 <;>
+    by_cases h₂ : (z i).val = (R.users b s).2 <;> simp_all [eq_comm]
+
+lemma rotationMeeting_eq_slot_sum {A : ℕ} (o : α → Option (Fin A))
+    (z : Terminal o) (u v : α) :
+    rotationMeeting o z u v = ∑ i : Fin A,
+      ((if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) +
+      (if (z i).val = v ∧ u ∈ priorPaths o (i.val+1) then (1:ℝ) else 0)) := by
+  classical
+  have only_owner (u : α) (a : Fin A) (hou : o u = some a)
+      (i : Fin A) (hia : i ≠ a) : (z i).val ≠ u := by
+    intro h
+    have hi := (z i).property
+    rw [h,hou] at hi
+    exact hia (Option.some.inj hi).symm
+  have sum_one (u v : α) (a : Fin A) (hou : o u = some a) :
+      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) =
+      if (z a).val = u ∧ v ∈ priorPaths o (a.val+1) then 1 else 0 := by
+    apply Finset.sum_eq_single a
+    · intro i _ hia
+      simp [only_owner u a hou i hia]
+    · simp
+  have sum_none (u v : α) (hou : o u = none) :
+      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) = 0 := by
+    apply Finset.sum_eq_zero
+    intro i _
+    have hn : (z i).val ≠ u := by
+      intro h
+      have hi := (z i).property
+      rw [h,hou] at hi
+      contradiction
+    simp [hn]
+  rw [Finset.sum_add_distrib]
+  cases hou : o u with
+  | none =>
+    rw [sum_none u v hou]
+    simp [rotationMeeting,hou,mem_priorPaths]
+  | some a =>
+    rw [sum_one u v a hou]
+    cases hov : o v with
+    | none =>
+      rw [sum_none v u hov]
+      simp [rotationMeeting,hou,hov,mem_priorPaths]
+    | some b =>
+      rw [sum_one v u b hov]
+      simp only [rotationMeeting,hou,hov,mem_priorPaths,Option.some.injEq]
+      rcases lt_trichotomy a b with hab | rfl | hba
+      · have hn : ¬ b.val < a.val+1 := by omega
+        have hy : a.val < b.val+1 := by omega
+        simp [hab,hn,hy]
+      · simp
+      · have hn : ¬ a.val < b.val+1 := by omega
+        have hy : b.val < a.val+1 := by omega
+        simp [hba,not_lt_of_ge (le_of_lt hba),hn,hy]
+
+lemma rotationCharge_eq_terminal_sum {A : ℕ} (o : α → Option (Fin A))
+    (b : ι → Bool) (z : Terminal o) :
+    rotationCharge R o b z = ∑ i : Fin A,
+      (terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) := by
+  simp only [rotationCharge,rotationMeeting_eq_slot_sum,terminalCharge_prior_sum]
+  exact Finset.sum_comm
+
+omit [DecidableEq α] in
+@[simp] lemma priorPaths_zero {A : ℕ} (o : α → Option (Fin A)) : priorPaths o 0 = ∅ := by
+  classical
+  ext x
+  simp
+
+lemma priorPaths_succ {A : ℕ} (o : α → Option (Fin A)) (t : ℕ) (ht : t < A) :
+    priorPaths o (t+1) = priorPaths o t ∪ cycleFiber o ⟨t,ht⟩ := by
+  classical
+  ext x
+  simp only [mem_priorPaths,Finset.mem_union,mem_cycleFiber]
+  constructor
+  · rintro ⟨i,hi,hit⟩
+    by_cases h : i.val < t
+    · exact Or.inl ⟨i,hi,h⟩
+    · have he : i = ⟨t,ht⟩ := Fin.ext (by change i.val = t; omega)
+      exact Or.inr (he ▸ hi)
+  · rintro (⟨i,hi,hit⟩ | hi)
+    · exact ⟨i,hi,by omega⟩
+    · exact ⟨⟨t,ht⟩,hi,by change t < t+1; omega⟩
+
+omit [DecidableEq α] in
+lemma priorPaths_disjoint {A : ℕ} (o : α → Option (Fin A)) (i : Fin A) :
+    Disjoint (priorPaths o i.val) (cycleFiber o i) := by
+  classical
+  apply Finset.disjoint_left.mpr
+  intro x hx hx'
+  obtain ⟨j,hj,hji⟩ := (mem_priorPaths _ _ _).mp hx
+  have he : j = i := Option.some.inj (hj.symm.trans ((mem_cycleFiber _ _ _).mp hx'))
+  subst j
+  exact (lt_irrefl _ hji)
+
+namespace QueryTree
+open Thorp.QueryTree
+
+lemma seekCycles_owned {A : ℕ} (S : Finset (α)) (r : Fin A → ℕ)
+    (o : α → Option (Fin A)) (z : Terminal o) (ω : ι → Bool)
+    (hS : ∀ i, cycleFiber o i ⊆ S)
+    (hO : ∀ i, orbitSet (R.perm ω) (z i).val = cycleFiber o i)
+    (hC : ∀ i, (cycleFiber o i).card = r i+1) :
+    let p := R.perm ω
+    let L := List.ofFn (fun i : Fin A => (p (z i).val,r i))
+    (seekCycles R S L ∅).succeeds ω = true ∧
+      ((seekCycles R S L ∅).staleCharge ∅ ω : ℝ) = rotationCharge R o (ω) z := by
+  classical
+  let b := ω
+  let p := R.perm b
+  let f (i : Fin A) := (p (z i).val,r i)
+  let c (i : Fin A) : ℝ := terminalCharge R b (z i).val ((priorPaths o (i.val+1)).erase (z i).val)
+  have hc (i : Fin A) : Function.minimalPeriod p (z i).val - 1 = r i := by
+    have he := orbitSet_card p (z i).val
+    rw [hO,hC] at he
+    omega
+  have hs (t : ℕ) (ht : t ≤ A) :
+      (seekCycles R S ((List.ofFn f).drop t) (priorPaths o t)).succeeds ω = true ∧
+        ((seekCycles R S ((List.ofFn f).drop t) (priorPaths o t)).staleCharge (exposedPaths R b (priorPaths o t)) ω : ℝ) =
+          ((List.ofFn c).drop t).sum := by
+    induction ht using Nat.decreasingInduction with
+    | self => simp only [List.length_ofFn,le_refl,List.drop_eq_nil_of_le,List.sum_nil,seekCycles,Thorp.QueryTree.succeeds,staleCharge,Nat.cast_zero,and_self]
+    | of_succ t ht ih =>
+      have htf : t < (List.ofFn f).length := by simpa using ht
+      have htc : t < (List.ofFn c).length := by simpa using ht
+      rw [List.drop_eq_getElem_cons htf,List.drop_eq_getElem_cons htc]
+      simp only [List.getElem_ofFn,List.sum_cons,seekCycles]
+      have h := seekCycle_orbit R S (priorPaths o t) (z ⟨t,ht⟩).val
+        (fun V => seekCycles R S ((List.ofFn f).drop (t+1)) V) ω
+        (by rw [hO]; exact hS _) (by rw [hO]; exact priorPaths_disjoint o ⟨t,ht⟩)
+      dsimp only at h
+      rw [hc,hO,←priorPaths_succ o t ht] at h
+      dsimp only [f]
+      constructor
+      · exact h.1.trans ih.1
+      · rw [h.2,Nat.cast_add,ih.2]
+        exact add_comm _ _
+  have h := hs 0 (Nat.zero_le _)
+  simp only [List.drop_zero,priorPaths_zero,exposedPaths_empty,List.sum_ofFn] at h
+  dsimp only
+  exact ⟨h.1,h.2.trans (rotationCharge_eq_terminal_sum R o b z).symm⟩
+
+end QueryTree
+section Collections
+variable {A : ℕ}
+
+def ValidCycleSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ)
+    (o : α → Option (Fin A)) : Prop :=
+  (∀ i, (cycleFiber o i).card = r i + 1) ∧
+  (∀ i, cycleFiber o i ⊆ S) ∧
+  (∀ i x, o x = some i → orbitSet p x = cycleFiber o i)
+
+abbrev CycleSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
+  {o : α → Option (Fin A) // ValidCycleSlots p S r o}
+
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
+    Fintype (CycleSlots p S r) := Fintype.ofFinite _
+
+lemma CycleSlots.terminal_nonempty {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (c : CycleSlots p S r) (i : Fin A) : Nonempty {x : α // c.val x = some i} := by
+  have hn : (cycleFiber c.val i).Nonempty := Finset.card_pos.mp (by rw [c.property.1]; omega)
+  obtain ⟨x,hx⟩ := hn
+  exact ⟨x,(mem_cycleFiber _ _ _).mp hx⟩
+
+omit [DecidableEq α] in
+lemma owner_ext {o o' : α → Option (Fin A)}
+    (h : ∀ i, cycleFiber o i = cycleFiber o' i) : o = o' := by
+  funext x
+  have he (i : Fin A) : o x = some i ↔ o' x = some i := by
+    simpa only [mem_cycleFiber] using (Finset.ext_iff.mp (h i) x)
+  cases ho : o x with
+  | some i => exact ((he i).mp ho).symm
+  | none =>
+    cases ho' : o' x with
+    | none => rfl
+    | some i => have hn := (he i).mpr ho'; rw [ho] at hn; contradiction
+
+abbrev RootedSlots (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
+  Σ c : CycleSlots p S r, Terminal c.val
+
+noncomputable def rootedStart {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (c : RootedSlots p S r) : Fin A → S := fun i =>
+  ⟨p (c.2 i).val,c.1.property.2.1 i (by
+    rw [←c.1.property.2.2 i (c.2 i).val (c.2 i).property]
+    exact (mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.apply_right (Equiv.Perm.SameCycle.refl p _)))⟩
+
+lemma rootedStart_injective {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ} :
+    Function.Injective (rootedStart (p:=p) (S:=S) (r:=r)) := by
+  rintro ⟨c,z⟩ ⟨c',z'⟩ h
+  have hz (i : Fin A) : (z i).val = (z' i).val :=
+    p.injective (congrArg Subtype.val (congrFun h i))
+  have hc : c = c' := Subtype.ext (owner_ext (fun i => by
+    rw [←c.property.2.2 i (z i).val (z i).property,
+      ←c'.property.2.2 i (z' i).val (z' i).property,hz]))
+  subst c'
+  have hzz : z = z' := funext (fun i => Subtype.ext (hz i))
+  subst z'
+  rfl
+
+lemma rooted_weight_sum_le (S : Finset (α)) (r : Fin A → ℕ) (ω : ι → Bool) :
+    (∑ c : RootedSlots (R.perm ω) S r,
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2)) ≤
+    ∑ a : Fin A → S,
+      if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+        ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+      else 0 := by
+  classical
+  let p := R.perm ω
+  let g (a : Fin A → S) : ℝ :=
+    if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+    else 0
+  have he (c : RootedSlots p S r) :
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2) =
+      g (rootedStart c) := by
+    have hs := QueryTree.seekCycles_owned R S r c.1.val c.2 ω c.1.property.2.1
+      (fun i => c.1.property.2.2 i (c.2 i).val (c.2 i).property) c.1.property.1
+    dsimp only at hs
+    unfold g
+    rw [ite_eq_left (show (QueryTree.seekCycles R S (List.ofFn (fun i => ((rootedStart c i).val,r i))) ∅).succeeds ω = true from hs.1)]
+    rw [← hs.2,neg_mul,Real.exp_neg,mul_comm (Real.log 2),Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
+    rfl
+  simp_rw [he]
+  exact sum_comp_injective_le rootedStart rootedStart_injective g (fun a => by dsimp [g]; split_ifs <;> positivity)
+
+lemma collection_rotation_lower (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r)
+    (ω : ι → Bool) (c : CycleSlots (R.perm ω) S r) :
+    collectionWeight d r ≤ ∑ z : Terminal c.val,
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z) := by
+  classical
+  let (i : Fin A) : Nonempty {x : α // c.val x = some i} := c.terminal_nonempty i
+  have hm : Monotone (fun i => (cycleFiber c.val i).card) := by
+    intro i j hij
+    dsimp only
+    rw [c.property.1,c.property.1]
+    exact Nat.add_le_add_right (hr hij) 1
+  have hb := rotation_exp_bound R hm (ω)
+  have hcard : (Fintype.card (Terminal c.val) : ℝ) = ∏ i : Fin A, (r i+1 : ℝ) := by
+    rw [Fintype.card_pi,Nat.cast_prod]
+    apply Finset.prod_congr rfl
+    intro i _
+    rw [card_cycleFiber,c.property.1,Nat.cast_add,Nat.cast_one]
+  have hp : (0:ℝ) < Fintype.card (Terminal c.val) := Nat.cast_pos.mpr Fintype.card_pos
+  have hl : rotationLower d r ≤ finiteMean (fun z : Terminal c.val => Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z)) := by
+    simpa only [rotationLower,c.property.1,Nat.cast_add,Nat.cast_one] using hb
+  have hh := (le_div_iff₀ hp).mp hl
+  rw [hcard] at hh
+  unfold collectionWeight
+  rw [← Finset.mul_sum]
+  calc
+    _ = ((2:ℝ)^d)^A * (rotationLower d r * ∏ i : Fin A, (r i+1 : ℝ)) := by ring
+    _ ≤ _ := mul_le_mul_of_nonneg_left hh (by positivity)
+
+lemma cycleSlots_weighted_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : ι → Bool =>
+      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) *
+      collectionWeight d r ≤ (S.card : ℝ)^A := by
+  classical
+  let g (a : Fin A → S) (ω : ι → Bool) : ℝ :=
+    if (QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles R S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+    else 0
+  have hω (ω : ι → Bool) :
+      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ) * collectionWeight d r ≤
+        ∑ a : Fin A → S, g a ω := by
+    calc
+      _ = ∑ _c : CycleSlots (R.perm ω) S r, collectionWeight d r := by simp
+      _ ≤ ∑ c : CycleSlots (R.perm ω) S r,
+          ∑ z : Terminal c.val, ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.val (ω) z) :=
+        Finset.sum_le_sum (fun c _ => collection_rotation_lower R S r hr ω c)
+      _ = ∑ c : RootedSlots (R.perm ω) S r,
+          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2) :=
+        (Fintype.sum_sigma (fun c : RootedSlots (R.perm ω) S r =>
+          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge R c.1.val (ω) c.2))).symm
+      _ ≤ _ := rooted_weight_sum_le R S r ω
+  have hb := finiteMean_mono hω
+  rw [finiteMean_mul_const,finiteMean_sum] at hb
+  calc
+    _ ≤ ∑ a : Fin A → S, finiteMean (g a) := hb
+    _ ≤ ∑ _a : Fin A → S, (1:ℝ) := by
+      apply Finset.sum_le_sum
+      intro a _
+      have hh := QueryTree.cycles_weighted_success R S (List.ofFn (fun i : Fin A => ((a i).val,r i)))
+      simpa only [List.length_ofFn] using hh
+    _ = _ := by simp
+
+lemma cycleSlots_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : ι → Bool =>
+      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
+        (S.card : ℝ)^A / collectionWeight d r := by
+  exact (le_div_iff₀ (collectionWeight_pos d r)).mpr (cycleSlots_weighted_moment R S r hr)
+
+lemma butterfly_cycleSlots_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : ι → Bool =>
+      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
+        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
+  have h := cycleSlots_moment R S r hr
+  rw [collectionWeight_eq_sqrt,div_mul_eq_div_mul_one_div] at h
+  rw [div_pow]
+  simpa only [one_div,Finset.prod_inv_distrib] using h
+
+end Collections
+
+section CycleCounting
+variable {A : ℕ}
+
+abbrev SelectedCycle (p : Equiv.Perm (α)) (S : Finset (α)) :=
+  {C : Finset (α) // (∃ x, C = orbitSet p x) ∧ C ⊆ S}
+
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) :
+    Fintype (SelectedCycle p S) := Fintype.ofFinite _
+
+lemma SelectedCycle.nonempty {p : Equiv.Perm (α)} {S : Finset (α)}
+    (C : SelectedCycle p S) : C.val.Nonempty := by
+  obtain ⟨x,hx⟩ := C.property.1
+  rw [hx]
+  exact orbitSet_nonempty p x
+
+lemma SelectedCycle.orbit_eq {p : Equiv.Perm (α)} {S : Finset (α)}
+    (C : SelectedCycle p S) {x : α} (hx : x ∈ C.val) : orbitSet p x = C.val := by
+  obtain ⟨y,hy⟩ := C.property.1
+  rw [hy] at hx ⊢
+  exact (orbitSet_eq p ((mem_orbitSet _ _ _).mp hx)).symm
+
+lemma SelectedCycle.eq_of_mem {p : Equiv.Perm (α)} {S : Finset (α)}
+    (C D : SelectedCycle p S) {x : α} (hC : x ∈ C.val) (hD : x ∈ D.val) : C = D := by
+  apply Subtype.ext
+  exact (C.orbit_eq hC).symm.trans (D.orbit_eq hD)
+
+abbrev SlotAssignment (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
+  {f : Fin A → SelectedCycle p S // Function.Injective f ∧ ∀ i, (f i).val.card = r i+1}
+
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
+    Fintype (SlotAssignment p S r) := Fintype.ofFinite _
+
+noncomputable def slotsToAssignment {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (c : CycleSlots p S r) : SlotAssignment p S r := by
+  classical
+  let f (i : Fin A) : SelectedCycle p S := ⟨cycleFiber c.val i,by
+    obtain ⟨x,hx⟩ := c.terminal_nonempty i
+    exact ⟨⟨x,(c.property.2.2 i x hx).symm⟩,c.property.2.1 i⟩⟩
+  refine ⟨f,?_,fun i => c.property.1 i⟩
+  intro i j hij
+  have he : cycleFiber c.val i = cycleFiber c.val j := congrArg Subtype.val hij
+  obtain ⟨x,hx⟩ := c.terminal_nonempty i
+  have hxi : x ∈ cycleFiber c.val i := (mem_cycleFiber _ _ _).mpr hx
+  rw [he] at hxi
+  exact Option.some.inj (hx.symm.trans ((mem_cycleFiber _ _ _).mp hxi))
+
+noncomputable def assignmentOwner {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) (x : α) : Option (Fin A) := by
+  classical
+  exact if h : ∃ i, x ∈ (f.val i).val then some (Classical.choose h) else none
+
+lemma assignmentOwner_eq_some {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) (x : α) (i : Fin A) :
+    assignmentOwner f x = some i ↔ x ∈ (f.val i).val := by
+  classical
+  unfold assignmentOwner
+  split_ifs with h
+  · constructor
+    · intro hi
+      have hx := Classical.choose_spec h
+      rw [Option.some.inj hi] at hx
+      exact hx
+    · intro hx
+      have he := (f.val (Classical.choose h)).eq_of_mem (f.val i) (Classical.choose_spec h) hx
+      exact congrArg some (f.property.1 he)
+  · constructor
+    · intro hi; contradiction
+    · intro hx; exact (h ⟨i,hx⟩).elim
+
+lemma assignmentOwner_fiber {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) (i : Fin A) : cycleFiber (assignmentOwner f) i = (f.val i).val := by
+  ext x
+  simp only [mem_cycleFiber,assignmentOwner_eq_some]
+
+noncomputable def assignmentToSlots {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) : CycleSlots p S r :=
+  ⟨assignmentOwner f,by
+    refine ⟨?_,?_,?_⟩
+    · intro i
+      rw [assignmentOwner_fiber]
+      exact f.property.2 i
+    · intro i
+      rw [assignmentOwner_fiber]
+      exact (f.val i).property.2
+    · intro i x hx
+      rw [assignmentOwner_fiber]
+      exact (f.val i).orbit_eq ((assignmentOwner_eq_some f x i).mp hx)⟩
+
+noncomputable def cycleSlotsEquivAssignments (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
+    CycleSlots p S r ≃ SlotAssignment p S r where
+  toFun := slotsToAssignment
+  invFun := assignmentToSlots
+  left_inv c := by
+    apply Subtype.ext
+    apply owner_ext
+    intro i
+    rw [assignmentToSlots,assignmentOwner_fiber]
+    rfl
+  right_inv f := by
+    apply Subtype.ext
+    funext i
+    apply Subtype.ext
+    exact assignmentOwner_fiber f i
+
+end CycleCounting
+
+section GroupedCounting
+variable {A : ℕ}
+
+abbrev LengthClass (r : Fin A → ℕ) := {j : ℕ // j ∈ Finset.univ.image (fun i => r i + 1)}
+
+def slotClass (r : Fin A → ℕ) (i : Fin A) : LengthClass r :=
+  ⟨r i+1, Finset.mem_image.mpr ⟨i, Finset.mem_univ _, rfl⟩⟩
+
+abbrev SlotsOfLength (r : Fin A → ℕ) (j : LengthClass r) := {i : Fin A // slotClass r i = j}
+abbrev CyclesOfLength (p : Equiv.Perm (α)) (S : Finset (α)) (j : ℕ) :=
+  {C : SelectedCycle p S // C.val.card = j}
+
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (j : ℕ) :
+    Fintype (CyclesOfLength p S j) := Fintype.ofFinite _
+
+abbrev GroupedAssignments (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :=
+  ∀ j : LengthClass r, SlotsOfLength r j ↪ CyclesOfLength p S j.val
+
+noncomputable def assignmentToGroups {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) : GroupedAssignments p S r := fun _j =>
+  { toFun := fun i => ⟨f.val i.val, (f.property.2 i.val).trans (congrArg Subtype.val i.property)⟩
+    inj' := fun _i _k h => Subtype.ext (f.property.1 (congrArg Subtype.val h)) }
+
+noncomputable def groupsEmbedding {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (F : GroupedAssignments p S r) : Fin A ↪ SelectedCycle p S :=
+  ((Equiv.sigmaFiberEquiv (slotClass r)).symm.toEmbedding.trans
+    (Function.Embedding.sigmaMap (Function.Embedding.refl _) F)).trans
+    ((Function.Embedding.sigmaMap (Function.Embedding.subtype _) (fun _ => Function.Embedding.refl _)).trans
+      (Equiv.sigmaFiberEquiv (fun C : SelectedCycle p S => C.val.card)).toEmbedding)
+
+noncomputable def groupsToAssignment {p : Equiv.Perm (α)} {S : Finset (α)} {r : Fin A → ℕ}
+    (F : GroupedAssignments p S r) : SlotAssignment p S r :=
+  ⟨groupsEmbedding F, (groupsEmbedding F).injective,
+    fun i => (F (slotClass r i) ⟨i,rfl⟩).property⟩
+
+noncomputable def assignmentEquivGroups (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
+    SlotAssignment p S r ≃ GroupedAssignments p S r where
+  toFun := assignmentToGroups
+  invFun := groupsToAssignment
+  left_inv f := by
+    apply Subtype.ext
+    rfl
+  right_inv F := by
+    funext j
+    apply Function.Embedding.ext
+    intro i
+    rcases i with ⟨i,hi⟩
+    subst j
+    rfl
+
+lemma card_cycleSlots_factorial (p : Equiv.Perm (α)) (S : Finset (α)) (r : Fin A → ℕ) :
+    Fintype.card (CycleSlots p S r) =
+      ∏ j : LengthClass r, (Fintype.card (CyclesOfLength p S j.val)).descFactorial
+        (Fintype.card (SlotsOfLength r j)) := by
+  classical
+  rw [Fintype.card_congr (cycleSlotsEquivAssignments p S r),
+    Fintype.card_congr (assignmentEquivGroups p S r), Fintype.card_pi]
+  simp only [Fintype.card_embedding_eq]
+
+lemma butterfly_factorial_moment (S : Finset (α)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : ι → Bool =>
+       (∏ j : LengthClass r,
+         (Fintype.card (CyclesOfLength (R.perm ω) S j.val)).descFactorial
+           (Fintype.card (SlotsOfLength r j)) : ℕ)) ≤
+      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+        ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
+  simpa only [←card_cycleSlots_factorial] using butterfly_cycleSlots_moment R S r hr
+
+end GroupedCounting
+
+section AllLengths
+variable {A : ℕ}
+
+noncomputable def assignmentRelabel (p : Equiv.Perm (α)) (S : Finset (α))
+    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
+    SlotAssignment p S r ≃ SlotAssignment p S (r ∘ e) where
+  toFun f := ⟨f.val ∘ e, f.property.1.comp e.injective, fun i => f.property.2 (e i)⟩
+  invFun f := ⟨f.val ∘ e.symm, f.property.1.comp e.symm.injective, fun i => by
+    simpa only [Function.comp_apply,Equiv.apply_symm_apply] using f.property.2 (e.symm i)⟩
+  left_inv f := by apply Subtype.ext; funext i; simp
+  right_inv f := by apply Subtype.ext; funext i; simp
+
+lemma card_cycleSlots_relabel (p : Equiv.Perm (α)) (S : Finset (α))
+    (r : Fin A → ℕ) (e : Equiv.Perm (Fin A)) :
+    Fintype.card (CycleSlots p S r) = Fintype.card (CycleSlots p S (r ∘ e)) := by
+  exact Fintype.card_congr ((cycleSlotsEquivAssignments p S r).trans
+    ((assignmentRelabel p S r e).trans (cycleSlotsEquivAssignments p S (r ∘ e)).symm))
+
+lemma butterfly_cycleSlots_moment_all (S : Finset (α)) (r : Fin A → ℕ) :
+    finiteMean (fun ω : ι → Bool =>
+      (Fintype.card (CycleSlots (R.perm ω) S r) : ℝ)) ≤
+        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
+  have h := butterfly_cycleSlots_moment R S (r ∘ Tuple.sort r) (Tuple.monotone_sort r)
+  simp_rw [←card_cycleSlots_relabel] at h
+  have hp := Equiv.prod_comp (Tuple.sort r) (fun i : Fin A => (1 / Real.sqrt (r i+1) : ℝ))
+  simpa only [Function.comp_apply,hp] using h
+
+abbrev ShortCycle (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :=
+  {C : SelectedCycle p S // C.val.card ≤ J}
+
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    Fintype (ShortCycle p S J) := Fintype.ofFinite _
+
+noncomputable def shortCycleLength {p : Equiv.Perm (α)} {S : Finset (α)} {J : ℕ}
+    (C : ShortCycle p S J) : Fin J :=
+  ⟨C.val.val.card-1, by have := C.val.nonempty.card_pos; have := C.property; omega⟩
+
+lemma shortCycleLength_add_one {p : Equiv.Perm (α)} {S : Finset (α)} {J : ℕ}
+    (C : ShortCycle p S J) : (shortCycleLength C : ℕ)+1 = C.val.val.card := by
+  have := C.val.nonempty.card_pos
+  simp only [shortCycleLength]
+  omega
+
+noncomputable def shortEmbeddingEquiv (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    (Fin A ↪ ShortCycle p S J) ≃
+      Σ r : Fin A → Fin J, SlotAssignment p S (fun i => (r i : ℕ)) where
+  toFun f := ⟨fun i => shortCycleLength (f i),
+    ⟨fun i => (f i).val, fun i j h => f.injective (Subtype.ext h),
+      fun i => (shortCycleLength_add_one (f i)).symm⟩⟩
+  invFun c :=
+    { toFun := fun i => ⟨c.2.val i, by rw [c.2.property.2]; exact (c.1 i).isLt⟩
+      inj' := fun _i _j h => c.2.property.1 (congrArg Subtype.val h) }
+  left_inv f := by
+    apply Function.Embedding.ext
+    intro i
+    rfl
+  right_inv c := by
+    rcases c with ⟨r,f⟩
+    have he : (fun i => shortCycleLength (⟨f.val i, by rw [f.property.2]; exact (r i).isLt⟩ : ShortCycle p S J)) = r := by
+      funext i
+      apply Fin.ext
+      simp only [shortCycleLength,f.property.2,Nat.add_sub_cancel]
+    apply Sigma.ext he
+    apply (Subtype.heq_iff_coe_eq (by
+      intro x
+      change (Function.Injective x ∧ ∀ i, (x i).val.card = (f.val i).val.card - 1 + 1) ↔ _
+      simp only [f.property.2,Nat.add_sub_cancel])).mpr
+    rfl
+
+lemma shortCycle_factorial_eq (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    (Fintype.card (ShortCycle p S J)).descFactorial A =
+      ∑ r : Fin A → Fin J, Fintype.card (CycleSlots p S (fun i => (r i : ℕ))) := by
+  classical
+  have hc := Fintype.card_embedding_eq (α := Fin A) (β := ShortCycle p S J)
+  simp only [Fintype.card_fin] at hc
+  rw [←hc, Fintype.card_congr (shortEmbeddingEquiv p S J),Fintype.card_sigma]
+  apply Finset.sum_congr rfl
+  intro r _
+  exact (Fintype.card_congr (cycleSlotsEquivAssignments p S (fun i => (r i : ℕ)))).symm
+
+end AllLengths
+
+section CycleMGF
+
+lemma butterfly_shortCycle_moment (S : Finset (α)) (J A : ℕ) :
+    finiteMean (fun ω : ι → Bool =>
+      ((Fintype.card (ShortCycle (R.perm ω) S J)).descFactorial A : ℝ)) ≤
+      ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))^A := by
+  classical
+  simp_rw [shortCycle_factorial_eq,Nat.cast_sum,finiteMean_sum]
+  calc
+    _ ≤ ∑ r : Fin A → Fin J,
+        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+          ∏ i : Fin A, (1 / Real.sqrt ((r i : ℕ)+1)) := by
+      apply Finset.sum_le_sum
+      intro r _
+      exact butterfly_cycleSlots_moment_all R S (fun i => (r i : ℕ))
+    _ = _ := by
+      rw [←Finset.mul_sum,mul_pow]
+      congr 1
+      rw [←Fintype.prod_sum (fun (_ : Fin A) (j : Fin J) => (1 / Real.sqrt ((j:ℕ)+1) : ℝ))]
+      simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fin]
+
+noncomputable def cycleMass (p : Equiv.Perm (α)) (S : Finset (α)) : ℕ :=
+  ∑ C : SelectedCycle p S, C.val.card
+
+lemma cycleMass_le (p : Equiv.Perm (α)) (S : Finset (α)) : cycleMass p S ≤ S.card := by
+  classical
+  let f : (Σ C : SelectedCycle p S, {x : α // x ∈ C.val}) → S :=
+    fun c => ⟨c.2.val,c.1.property.2 c.2.property⟩
+  have hi : Function.Injective f := by
+    intro c e h
+    have hx : c.2.val = e.2.val := congrArg Subtype.val h
+    have hC : c.1 = e.1 := c.1.eq_of_mem e.1 c.2.property (hx ▸ e.2.property)
+    rcases c with ⟨c,x⟩
+    rcases e with ⟨e,y⟩
+    dsimp only at hC
+    subst e
+    have he : x=y := Subtype.ext hx
+    cases he
+    rfl
+  have hh := Fintype.card_le_of_injective f hi
+  simpa only [Fintype.card_sigma,Fintype.card_coe,cycleMass] using hh
+
+lemma shortCycle_card_le (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    Fintype.card (ShortCycle p S J) ≤ S.card := by
+  classical
+  calc
+    _ ≤ Fintype.card (SelectedCycle p S) := Fintype.card_subtype_le _
+    _ = ∑ _ : SelectedCycle p S, 1 := by simp
+    _ ≤ cycleMass p S := Finset.sum_le_sum (fun C _ => C.nonempty.card_pos)
+    _ ≤ _ := cycleMass_le p S
+
+lemma butterfly_shortCycle_mgf (S : Finset (α)) (J : ℕ) (q : ℝ) (hq : 1 ≤ q) :
+    finiteMean (fun ω : ι → Bool =>
+      q^(Fintype.card (ShortCycle (R.perm ω) S J))) ≤
+      Real.exp ((q-1)*((S.card : ℝ) / Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1))) := by
+  apply factorial_mgf _ S.card _ q (fun ω => shortCycle_card_le _ _ _)
+  · positivity
+  · exact hq
+  · exact butterfly_shortCycle_moment R S J
+
+end CycleMGF
+
+
+abbrev LongCycle (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :=
+  {C : SelectedCycle p S // ¬ C.val.card ≤ J}
+noncomputable instance (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    Fintype (LongCycle p S J) := Fintype.ofFinite _
+
+lemma selectedCycle_card_split (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    Fintype.card (SelectedCycle p S) =
+      Fintype.card (ShortCycle p S J) + Fintype.card (LongCycle p S J) := by
+  classical
+  rw [←Fintype.card_congr (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J)),Fintype.card_sum]
+
+lemma longCycle_card_bound (p : Equiv.Perm (α)) (S : Finset (α)) (J : ℕ) :
+    J * Fintype.card (LongCycle p S J) ≤ S.card := by
+  classical
+  have hs := Equiv.sum_comp (Equiv.sumCompl (fun C : SelectedCycle p S => C.val.card ≤ J))
+    (fun C : SelectedCycle p S => C.val.card)
+  rw [Fintype.sum_sum_type] at hs
+  have h0 : J * Fintype.card (LongCycle p S J) ≤
+      ∑ C : LongCycle p S J, C.val.val.card := by
+    calc
+      _ = ∑ _C : LongCycle p S J, J := by simp [mul_comm]
+      _ ≤ _ := Finset.sum_le_sum (fun C _ => Nat.le_of_lt (Nat.lt_of_not_ge C.property))
+  have hh := cycleMass_le p S
+  unfold cycleMass at hh
+  rw [←hs] at hh
+  exact h0.trans ((Nat.le_add_left _ _).trans hh)
+
+lemma longCycle_pow_bound (p : Equiv.Perm (α)) (S : Finset (α))
+    (J : ℕ) (hJ : 0 < J) (q : ℝ) (hq : 1 ≤ q) :
+    q^(Fintype.card (LongCycle p S J)) ≤ Real.exp ((S.card : ℝ)/J * Real.log q) := by
+  have hq0 : 0 < q := lt_of_lt_of_le zero_lt_one hq
+  have hc : (Fintype.card (LongCycle p S J) : ℝ) ≤ (S.card : ℝ)/J := by
+    apply (le_div_iff₀ (by exact_mod_cast hJ)).mpr
+    have hh := longCycle_card_bound p S J
+    exact_mod_cast (by simpa only [mul_comm] using hh)
+  calc
+    _ = Real.exp ((Fintype.card (LongCycle p S J) : ℝ)*Real.log q) := by
+      rw [Real.exp_nat_mul,Real.exp_log hq0]
+    _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right hc (Real.log_nonneg hq))
+
+lemma butterfly_cycle_log_mgf (S : Finset (α)) (J : ℕ) (hJ : 0 < J)
+    (q : ℝ) (hq : 1 ≤ q) :
+    Real.log (finiteMean (fun ω : ι → Bool =>
+      q^(Fintype.card (SelectedCycle (R.perm ω) S)))) ≤
+      (S.card : ℝ)/J * Real.log q + 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
+  let lam : ℝ := (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) * ∑ j : Fin J, 1 / Real.sqrt ((j:ℕ)+1)
+  let c : ℝ := (S.card : ℝ)/J * Real.log q
+  have hnum : (q-1)*lam ≤ 2*q*S.card*Real.sqrt J / Real.sqrt ((2:ℝ)^d) := by
+    have hsum := sum_inv_sqrt J
+    have hp : 0 ≤ (S.card : ℝ)/Real.sqrt ((2:ℝ)^d) := by positivity
+    have h0 : 0 ≤ 2*Real.sqrt J := by positivity
+    have hh := mul_le_mul_of_nonneg_left hsum hp
+    have hq1 : 0 ≤ q-1 := sub_nonneg.mpr hq
+    calc
+      _ ≤ (q-1)*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
+        mul_le_mul_of_nonneg_left hh hq1
+      _ ≤ q*((S.card : ℝ)/Real.sqrt ((2:ℝ)^d)*(2*Real.sqrt J)) :=
+        mul_le_mul_of_nonneg_right (by linarith) (mul_nonneg hp h0)
+      _ = _ := by ring
+  have hm : finiteMean (fun ω : ι → Bool =>
+      q^(Fintype.card (SelectedCycle (R.perm ω) S))) ≤
+      Real.exp (c+(q-1)*lam) := by
+    calc
+      _ ≤ finiteMean (fun ω : ι → Bool =>
+          q^(Fintype.card (ShortCycle (R.perm ω) S J))*Real.exp c) := by
+        apply finiteMean_mono
+        intro ω
+        rw [selectedCycle_card_split _ _ J,pow_add]
+        exact mul_le_mul_of_nonneg_left (longCycle_pow_bound _ S J hJ q hq) (by positivity)
+      _ = finiteMean (fun ω : ι → Bool =>
+          q^(Fintype.card (ShortCycle (R.perm ω) S J))) * Real.exp c :=
+        finiteMean_mul_const _ _
+      _ ≤ Real.exp ((q-1)*lam) * Real.exp c :=
+        mul_le_mul_of_nonneg_right (butterfly_shortCycle_mgf R S J q hq) (Real.exp_nonneg _)
+      _ = _ := by rw [←Real.exp_add,add_comm]
+  have hpos : 0 < finiteMean (fun ω : ι → Bool =>
+      q^(Fintype.card (SelectedCycle (R.perm ω) S))) := by
+    apply finiteMean_pos
+    intro ω
+    positivity
+  have hlog := Real.log_le_log hpos hm
+  rw [Real.log_exp] at hlog
+  dsimp [c] at hlog
+  linarith
+
+
+end RoutingNetwork
+
+variable {α B : Type*} [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] {d : ℕ}
+
+def parallelPerm (ω : B × SwitchIndex d → Bool) : Equiv.Perm (B × Card d) where
+  toFun x := (x.1,butterflyPerm d (decodeButterfly d (fun i => ω (x.1,i))) x.2)
+  invFun y := (y.1,(butterflyPerm d (decodeButterfly d (fun i => ω (y.1,i)))).symm y.2)
+  left_inv x := by simp
+  right_inv y := by simp
+
+def relabeledPerm (pre : α ≃ B × Card d) (post : B × Card d ≃ α)
+    (ω : B × SwitchIndex d → Bool) : Equiv.Perm α :=
+  (pre.trans (parallelPerm ω)).trans post
+
+noncomputable def parallelPath (pre : α ≃ B × Card d)
+    (ω : B × SwitchIndex d → Bool) (x : α) : Finset (B × SwitchIndex d) :=
+  (routeDomain d (pre x).2
+    (butterflyPerm d (decodeButterfly d (fun i => ω ((pre x).1,i))) (pre x).2)).image
+      (fun i => ((pre x).1,i))
+
+def parallelUsers (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
+    (i : B × SwitchIndex d) : α × α :=
+  let u := switchUsers d (decodeButterfly d (fun j => ω (i.1,j))) i.2
+  (pre.symm (i.1,u.1),pre.symm (i.1,u.2))
+
+omit [Fintype α] [DecidableEq α] [Fintype B] [DecidableEq B] in
+lemma parallelUsers_ne (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
+    (i : B × SwitchIndex d) : (parallelUsers pre ω i).1 ≠ (parallelUsers pre ω i).2 := by
+  intro h
+  exact switchUsers_ne d _ _ (congrArg Prod.snd (pre.symm.injective h))
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma mem_parallelPath (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool)
+    (x : α) (i : B × SwitchIndex d) :
+    i ∈ parallelPath pre ω x ↔ (parallelUsers pre ω i).1 = x ∨ (parallelUsers pre ω i).2 = x := by
+  classical
+  obtain ⟨b,j⟩ := i
+  simp only [parallelPath,Finset.mem_image,Prod.mk.injEq]
+  constructor
+  · rintro ⟨k,hk,hb,rfl⟩
+    subst b
+    rcases (mem_routeDomain_iff_user d _ _ _).mp hk with h | h
+    · left
+      apply pre.injective
+      change pre (pre.symm ((pre x).1,_)) = pre x
+      rw [Equiv.apply_symm_apply]
+      exact Prod.ext rfl h
+    · right
+      apply pre.injective
+      change pre (pre.symm ((pre x).1,_)) = pre x
+      rw [Equiv.apply_symm_apply]
+      exact Prod.ext rfl h
+  · intro h
+    have hh : (b,(switchUsers d (decodeButterfly d (fun j => ω (b,j))) j).1) = pre x ∨
+        (b,(switchUsers d (decodeButterfly d (fun j => ω (b,j))) j).2) = pre x := by
+      rcases h with h | h
+      · exact Or.inl (by simpa only [parallelUsers,Equiv.apply_symm_apply] using congrArg pre h)
+      · exact Or.inr (by simpa only [parallelUsers,Equiv.apply_symm_apply] using congrArg pre h)
+    have hb : b = (pre x).1 := hh.elim (congrArg Prod.fst) (congrArg Prod.fst)
+    subst b
+    refine ⟨j,?_,rfl,rfl⟩
+    apply (mem_routeDomain_iff_user d _ _ _).mpr
+    exact hh.elim (fun h => Or.inl (congrArg Prod.snd h)) (fun h => Or.inr (congrArg Prod.snd h))
+
+omit [Fintype α] [DecidableEq α] [Fintype B] in
+lemma card_parallelPath (pre : α ≃ B × Card d) (ω : B × SwitchIndex d → Bool) (x : α) :
+    (parallelPath pre ω x).card = d := by
+  classical
+  rw [parallelPath,Finset.card_image_of_injective _ (fun _ _ h => Prod.mk.inj h |>.2),card_routeDomain]
+
+noncomputable def blockMarks (pre : α ≃ B × Card d) (S : Finset α) (b : B) : Finset (Card d) :=
+  Finset.univ.filter (fun x => pre.symm (b,x) ∈ S)
+
+omit [Fintype α] [Fintype B] [DecidableEq B] in
+@[simp] lemma mem_blockMarks (pre : α ≃ B × Card d) (S : Finset α) (b : B) (x : Card d) :
+    x ∈ blockMarks pre S b ↔ pre.symm (b,x) ∈ S := by
+  classical
+  simp [blockMarks]
+
+omit [DecidableEq B] in
+lemma sum_card_blockMarks (pre : α ≃ B × Card d) (S : Finset α) :
+    ∑ b : B, (blockMarks pre S b).card = S.card := by
+  classical
+  calc
+    _ = ∑ b : B, ∑ x : Card d, if pre.symm (b,x) ∈ S then 1 else 0 := by
+      simp only [blockMarks,Finset.card_filter]
+    _ = ∑ u : B × Card d, if pre.symm u ∈ S then 1 else 0 := (Fintype.sum_prod_type (fun u : B × Card d => if pre.symm u ∈ S then (1:ℕ) else 0)).symm
+    _ = ∑ x : α, if x ∈ S then 1 else 0 := Equiv.sum_comp pre.symm (fun x : α => if x ∈ S then (1:ℕ) else 0)
+    _ = _ := by simp
+
+omit [DecidableEq B] in
+lemma card_blockMarks_le (pre : α ≃ B × Card d) (S : Finset α) (b : B) :
+    (blockMarks pre S b).card ≤ S.card := by
+  rw [←sum_card_blockMarks pre S]
+  exact Finset.single_le_sum (f := fun b : B => (blockMarks pre S b).card) (fun _ _ => Nat.zero_le _) (Finset.mem_univ b)
+
+end Thorp
+
+end ThorpNine.Tail
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Tail/Representative.lean b/lean/OAI/Probability/ThorpRouting/Tail/Representative.lean
index 6bef312e9bd6b4a09ae9c810e06e97c0dc59a898..c83efd8041159acf3daf609ae2cd01dab4773ad7 100644
--- a/lean/OAI/Probability/ThorpRouting/Tail/Representative.lean
+++ b/lean/OAI/Probability/ThorpRouting/Tail/Representative.lean
@@ -1,893 +1,893 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Tail.ShapePartitionEquiv
-
-namespace OAI
-
-namespace ThorpNine.Tail
-
-namespace Thorp.UnitaryFinite
-open scoped BigOperators ComplexConjugate Classical
-variable {G : Type*} [Group G] [Fintype G]
-variable {V W : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-  [NormedAddCommGroup W] [InnerProductSpace ℂ W] [FiniteDimensional ℂ W]
-omit [FiniteDimensional ℂ V] in
-lemma averageMap_self_trace (ρ : Representation ℂ G V) (A : Module.End ℂ V) :
-    LinearMap.trace ℂ V (averageMap ρ ρ A).toLinearMap =
-      (Fintype.card G:ℂ) * LinearMap.trace ℂ V A := by
-  change LinearMap.trace ℂ V (∑ g, ρ g * A * ρ g⁻¹) = _
-  rw [map_sum]
-  have hj (g : G) : LinearMap.trace ℂ V (ρ g*A*ρ g⁻¹) = LinearMap.trace ℂ V A := by
-    rw [LinearMap.trace_mul_cycle,←map_mul,inv_mul_cancel,map_one,one_mul]
-  simp only [hj,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
-
-lemma averageMap_scalar (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
-    (A : Module.End ℂ V) :
-    ∃ c : ℂ, (averageMap ρ ρ A).toLinearMap = c • LinearMap.id ∧
-      c * (Module.finrank ℂ V:ℂ) = (Fintype.card G:ℂ)*LinearMap.trace ℂ V A := by
-  obtain ⟨c,hc⟩ := (Representation.IsIrreducible.algebraMap_intertwiningMap_bijective_of_isAlgClosed (ρ := ρ)).surjective (averageMap ρ ρ A)
-  have he : (averageMap ρ ρ A).toLinearMap = c • LinearMap.id := by
-    rw [←hc]
-    rfl
-  refine ⟨c,he,?_⟩
-  rw [←averageMap_self_trace ρ A,he,map_smul,LinearMap.trace_id,smul_eq_mul]
-
-theorem coefficient_orthogonality (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
-    (hρ : IsUnitary ρ) (u v w z : V) :
-    (Module.finrank ℂ V:ℂ) * ∑ g, inner ℂ u (ρ g v) * inner ℂ (ρ g w) z =
-      (Fintype.card G:ℂ) * inner ℂ w v * inner ℂ u z := by
-  obtain ⟨c,hc,htrace⟩ := averageMap_scalar ρ (InnerProductSpace.rankOne ℂ v w).toLinearMap
-  rw [InnerProductSpace.trace_rankOne] at htrace
-  have happ := congrArg (fun A : Module.End ℂ V => inner ℂ u (A z)) hc
-  change inner ℂ u ((∑ g, ρ g * (InnerProductSpace.rankOne ℂ v w).toLinearMap * ρ g⁻¹) z) = _ at happ
-  simp only [LinearMap.sum_apply,Module.End.mul_apply,ContinuousLinearMap.coe_coe,
-    InnerProductSpace.rankOne_apply,map_smul,unitary_inner_inv ρ hρ,inner_sum,
-    inner_smul_right,LinearMap.smul_apply,LinearMap.id_apply] at happ
-  have hsum : ∑ g, inner ℂ u (ρ g v) * inner ℂ (ρ g w) z = c * inner ℂ u z := by
-    simpa only [mul_comm] using happ
-  rw [hsum]
-  calc
-    _ = (c*(Module.finrank ℂ V:ℂ))*inner ℂ u z := by ring
-    _ = _ := by rw [htrace]
-
-end Thorp.UnitaryFinite
-namespace Thorp.UnitaryFinite
-open scoped BigOperators ComplexConjugate Classical
-
-variable {G X : Type*} [Group G] [Fintype G] [Fintype X]
-  (a : G →* Equiv.Perm X) (x₀ : X) (ha : ∀ x, ∃ g, a g x₀=x)
-
-noncomputable def representative (x : X) : G := Classical.choose (ha x)
-
-omit [Fintype G] [Fintype X] in
-lemma representative_apply (x : X) : a (representative a x₀ ha x) x₀=x :=
-  Classical.choose_spec (ha x)
-
-include ha in
-lemma orbit_sum (F : X → ℂ) :
-    (Fintype.card X:ℂ)*∑ g, F (a g x₀) = (Fintype.card G:ℂ)*∑ x, F x := by
-  have ht (x : X) : ∑ g, F (a g x)=∑ g, F (a g x₀) := by
-    obtain ⟨t,rfl⟩ := ha x
-    symm
-    apply Fintype.sum_equiv (Equiv.mulRight t⁻¹)
-    intro g
-    simp
-  calc
-    _ = ∑ x : X, ∑ g : G, F (a g x) := by simp only [ht,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
-    _ = ∑ g : G, ∑ x : X, F (a g x) := Finset.sum_comm
-    _ = ∑ g : G, ∑ x : X, F x := by
-      apply Finset.sum_congr rfl
-      intro g _
-      exact Equiv.sum_comp (a g) F
-    _ = _ := by simp
-
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V)
-
-def IsFixed (w : V) : Prop := ∀ g, a g x₀=x₀ → ρ g w=w
-
-noncomputable def coefficient (w v : V) : EuclideanSpace ℂ X :=
-  WithLp.toLp 2 (fun x => inner ℂ (ρ (representative a x₀ ha x) w) v)
-
-omit [Fintype G] [Fintype X] [FiniteDimensional ℂ V] in
-lemma coefficient_at (w : V) (hw : IsFixed a x₀ ρ w) (v : V) (g : G) :
-    coefficient a x₀ ha ρ w v (a g x₀) = inner ℂ (ρ g w) v := by
-  let r := representative a x₀ ha (a g x₀)
-  have hr : a r x₀=a g x₀ := representative_apply a x₀ ha _
-  have hs : a (g⁻¹*r) x₀=x₀ := by
-    simp only [map_mul,Equiv.Perm.mul_apply]
-    rw [hr,←Equiv.Perm.mul_apply,←map_mul,inv_mul_cancel,map_one]
-    rfl
-  have he := congrArg (ρ g) (hw (g⁻¹*r) hs)
-  have hh : ρ r w=ρ g w := by
-    simpa only [map_mul,Module.End.mul_apply,←Module.End.mul_apply,←map_mul,
-      mul_inv_cancel_left] using he
-  change inner ℂ (ρ r w) v= _
-  rw [hh]
-
-lemma coefficient_inner [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
-    (w z : V) (hw : IsFixed a x₀ ρ w) (hz : IsFixed a x₀ ρ z) (v t : V) :
-    (Module.finrank ℂ V:ℂ)*inner ℂ (coefficient a x₀ ha ρ w v) (coefficient a x₀ ha ρ z t) =
-      (Fintype.card X:ℂ)*inner ℂ z w*inner ℂ v t := by
-  have hs := orbit_sum a x₀ ha (fun x => star (coefficient a x₀ ha ρ w v x)*coefficient a x₀ ha ρ z t x)
-  simp only [coefficient_at a x₀ ha ρ w hw,coefficient_at a x₀ ha ρ z hz] at hs
-  simp only [←starRingEnd_apply,inner_conj_symm] at hs
-  simp only [starRingEnd_apply] at hs
-  have ho := coefficient_orthogonality ρ hρ v w z t
-  have hcard : (Fintype.card G:ℂ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
-  apply mul_left_cancel₀ hcard
-  rw [PiLp.inner_apply]
-  simp only [RCLike.inner_apply,starRingEnd_apply]
-  have hsum : ∑ x, coefficient a x₀ ha ρ z t x * star (coefficient a x₀ ha ρ w v x) =
-      ∑ x, star (coefficient a x₀ ha ρ w v x)*coefficient a x₀ ha ρ z t x := by
-    apply Finset.sum_congr rfl
-    intro x _
-    ring
-  rw [hsum]
-  calc
-    _ = (Module.finrank ℂ V:ℂ)*((Fintype.card G:ℂ)*∑ x,
-        star (coefficient a x₀ ha ρ w v x)*coefficient a x₀ ha ρ z t x) := by ring
-    _ = (Fintype.card X:ℂ)*((Module.finrank ℂ V:ℂ)*∑ g,
-        inner ℂ v (ρ g w)*inner ℂ (ρ g z) t) := by rw [←hs]; ring
-    _ = _ := by rw [ho]; ring
-
-omit [Fintype G] [Fintype X] [FiniteDimensional ℂ V] in
-lemma coefficient_action (hρ : IsUnitary ρ) (w : V) (hw : IsFixed a x₀ ρ w)
-    (v : V) (g : G) (x : X) :
-    coefficient a x₀ ha ρ w v (a g x) = coefficient a x₀ ha ρ w (ρ g⁻¹ v) x := by
-  obtain ⟨r,rfl⟩ := ha x
-  rw [←Equiv.Perm.mul_apply,←map_mul,coefficient_at a x₀ ha ρ w hw,
-    coefficient_at a x₀ ha ρ w hw]
-  simp only [map_mul,Module.End.mul_apply,unitary_inner_inv ρ hρ]
-
-noncomputable def normalizedCoefficient (w v : V) : EuclideanSpace ℂ X :=
-  (Real.sqrt ((Module.finrank ℂ V:ℝ)/(Fintype.card X:ℝ)) : ℂ) • coefficient a x₀ ha ρ w v
-
-lemma normalizedCoefficient_inner [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
-    (w z : V) (hw : IsFixed a x₀ ρ w) (hz : IsFixed a x₀ ρ z) (v t : V) :
-    inner ℂ (normalizedCoefficient a x₀ ha ρ w v) (normalizedCoefficient a x₀ ha ρ z t) =
-      inner ℂ z w*inner ℂ v t := by
-  let : Nonempty X := ⟨x₀⟩
-  have hX : (Fintype.card X:ℂ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
-  have hs : (Real.sqrt ((Module.finrank ℂ V:ℝ)/(Fintype.card X:ℝ)) : ℂ)^2 =
-      (Module.finrank ℂ V:ℂ)/(Fintype.card X:ℂ) := by
-    rw [←Complex.ofReal_pow,Real.sq_sqrt (by positivity)]
-    simp
-  simp only [normalizedCoefficient,inner_smul_left,inner_smul_right]
-  simp only [Complex.conj_ofReal]
-  rw [←mul_assoc,←pow_two,hs]
-  have hi := coefficient_inner a x₀ ha ρ hρ w z hw hz v t
-  field_simp
-  simpa only [mul_assoc] using hi
-
-lemma normalizedCoefficient_orthonormal [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
-    {I : Type*} (w : I → V) (hw : Orthonormal ℂ w)
-    (hfix : ∀ i, IsFixed a x₀ ρ (w i)) (v : V) (hv : ‖v‖=1) :
-    Orthonormal ℂ (fun i => normalizedCoefficient a x₀ ha ρ (w i) v) := by
-  rw [orthonormal_iff_ite] at hw ⊢
-  intro i j
-  rw [normalizedCoefficient_inner a x₀ ha ρ hρ _ _ (hfix i) (hfix j),hw,
-    inner_self_eq_norm_sq_to_K,hv]
-  simp [eq_comm]
-
-lemma normalizedCoefficient_norm [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
-    (w : V) (hw : IsFixed a x₀ ρ w) (hn : ‖w‖=1) (v : V) :
-    ‖normalizedCoefficient a x₀ ha ρ w v‖=‖v‖ := by
-  have he := normalizedCoefficient_inner a x₀ ha ρ hρ w w hw hw v v
-  rw [inner_self_eq_norm_sq_to_K,inner_self_eq_norm_sq_to_K,hn] at he
-  simp only [inner_self_eq_norm_sq_to_K] at he
-  norm_num at he
-  have hs : ‖normalizedCoefficient a x₀ ha ρ w v‖^2=‖v‖^2 := by exact_mod_cast he
-  nlinarith [norm_nonneg (normalizedCoefficient a x₀ ha ρ w v),norm_nonneg v]
-
-end Thorp.UnitaryFinite
-
-namespace Thorp.FiniteConvolution
-open scoped BigOperators Classical
-variable {G : Type*} [Group G] [Fintype G]
-
-noncomputable def conv (p q : G → ℝ) (g : G) : ℝ := ∑ h, p h*q (h⁻¹*g)
-noncomputable def power (p : G → ℝ) : ℕ → G → ℝ
-  | 0 => fun g => if g=1 then 1 else 0
-  | n+1 => conv p (power p n)
-
-lemma conv_nonneg {p q : G → ℝ} (hp : ∀ g, 0 ≤ p g) (hq : ∀ g, 0 ≤ q g) :
-    ∀ g, 0 ≤ conv p q g := fun _ => Finset.sum_nonneg (fun h _ => mul_nonneg (hp h) (hq _))
-
-lemma conv_pos_of {p q : G → ℝ} (hp : ∀ g, 0 ≤ p g) (hq : ∀ g, 0 ≤ q g)
-    {a b : G} (ha : 0 < p a) (hb : 0 < q b) : 0 < conv p q (a*b) := by
-  apply lt_of_lt_of_le (mul_pos ha hb)
-  have h := Finset.single_le_sum (fun g _ => mul_nonneg (hp g) (hq (g⁻¹*(a*b))))
-    (Finset.mem_univ a)
-  simpa [conv] using h
-
-lemma sum_conv (p q : G → ℝ) : ∑ g, conv p q g=(∑ g,p g)*(∑ g,q g) := by
-  unfold conv
-  rw [Finset.sum_comm,Finset.sum_mul]
-  apply Finset.sum_congr rfl
-  intro h _
-  rw [←Finset.mul_sum]
-  congr 1
-  exact Fintype.sum_equiv (Equiv.mulLeft h⁻¹) _ _ (fun _ => rfl)
-
-lemma power_nonneg {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) : ∀ n g, 0 ≤ power p n g := by
-  intro n
-  induction n with
-  | zero => intro g; simp only [power]; split_ifs  <;> norm_num
-  | succ n ih => exact conv_nonneg hp ih
-
-lemma sum_power {p : G → ℝ} (hp : ∑ g,p g=1) : ∀ n, ∑ g,power p n g=1 := by
-  intro n
-  induction n with
-  | zero => simp [power]
-  | succ n ih => rw [power,sum_conv,hp,ih,one_mul]
-
-lemma power_list_pos {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) (l : List G)
-    (hl : ∀ g ∈ l, 0 < p g) : 0 < power p l.length l.prod := by
-  induction l with
-  | nil => simp [power]
-  | cons a l ih =>
-    simpa only [List.length_cons,List.prod_cons,power] using
-      conv_pos_of hp (power_nonneg hp l.length) (hl a (by simp))
-        (ih (fun g hg => hl g (by simp [hg])))
-
-lemma power_pos_mono {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) (h1 : 0 < p 1)
-    {n m : ℕ} (hnm : n ≤ m) {g : G} (hg : 0 < power p n g) : 0 < power p m g := by
-  induction m, hnm using Nat.le_induction with
-  | base => exact hg
-  | succ m _ ih =>
-    simpa only [one_mul,power] using conv_pos_of hp (power_nonneg hp m) h1 ih
-
-theorem exists_positive_power {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) (h1 : 0 < p 1)
-    (hgen : Subgroup.closure {g | 0 < p g}=⊤) :
-    ∃ n : ℕ, ∀ g, 0 < power p (2^n) g := by
-  have hx (g : G) : g ∈ Submonoid.closure {g | 0 < p g} := by
-    rw [←Subgroup.closure_toSubmonoid_of_finite,hgen]
-    trivial
-  choose l hl hprod using fun g => Submonoid.exists_list_of_mem_closure (hx g)
-  let n := Finset.univ.sup (fun g => (l g).length)
-  refine ⟨n,fun g => ?_⟩
-  have hlen : (l g).length ≤ n := Finset.le_sup (f := fun g => (l g).length) (Finset.mem_univ g)
-  have hpow : n ≤ 2^n := Nat.le_of_lt (Nat.lt_two_pow_self)
-  have hpos := power_list_pos hp (l g) (hl g)
-  rw [hprod g] at hpos
-  exact power_pos_mono hp h1 (hlen.trans hpow) hpos
-
-structure Minorant (p : G → ℝ) where
-  exponent : ℕ
-  mass : ℝ
-  mass_pos : 0 < mass
-  mass_le_one : mass ≤ 1
-  lower : ∀ g, mass/(Fintype.card G:ℝ) ≤ power p (2^exponent) g
-
-noncomputable def minorant {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) (hsum : ∑ g,p g=1)
-    (h1 : 0 < p 1) (hgen : Subgroup.closure {g | 0 < p g}=⊤) : Minorant p := by
-  let n := Classical.choose (exists_positive_power hp h1 hgen)
-  have hn := Classical.choose_spec (exists_positive_power hp h1 hgen)
-  let a := Finset.univ.inf' Finset.univ_nonempty (power p (2^n))
-  have ha : 0 < a := (Finset.lt_inf'_iff _).mpr (fun g _ => hn g)
-  have hag (g : G) : a ≤ power p (2^n) g := Finset.inf'_le _ (Finset.mem_univ g)
-  have hN : (0:ℝ) < Fintype.card G := Nat.cast_pos.mpr Fintype.card_pos
-  have hsum' := Finset.sum_le_sum (s := Finset.univ) (fun g _ => hag g)
-  simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,sum_power hsum] at hsum'
-  refine ⟨n,(Fintype.card G:ℝ)*a,mul_pos hN ha,hsum',fun g => ?_⟩
-  have he : (Fintype.card G:ℝ)*a/(Fintype.card G:ℝ)=a := by field_simp
-  rw [he]
-  exact hag g
-
-end Thorp.FiniteConvolution
-namespace Thorp.UnitaryFinite
-
-section
-open scoped BigOperators ComplexConjugate Classical
-
-variable {G : Type*} [Group G] [Fintype G]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-omit [Fintype G] in
-@[simp] lemma continuousRepresentation_apply (ρ : Representation ℂ G V) (g : G) (v : V) :
-    continuousRepresentation ρ g v=ρ g v := rfl
-
-omit [Fintype G] [FiniteDimensional ℂ V] in
-lemma unitary_norm (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (g : G) (v : V) :
-    ‖ρ g v‖=‖v‖ := by
-  rw [norm_eq_sqrt_re_inner (𝕜 := ℂ),hρ,norm_eq_sqrt_re_inner (𝕜 := ℂ)]
-
-omit [Fintype G] in
-lemma continuousRepresentation_norm_le (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (g : G) :
-    ‖continuousRepresentation ρ g‖≤1 := by
-  apply ContinuousLinearMap.opNorm_le_bound _ zero_le_one
-  intro v
-  simp only [continuousRepresentation_apply,unitary_norm ρ hρ,one_mul,le_refl]
-
-omit [Fintype G] in
-lemma continuousRepresentation_adjoint (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (g : G) :
-    (continuousRepresentation ρ g).adjoint=continuousRepresentation ρ g⁻¹ := by
-  ext w
-  apply ext_inner_left ℂ
-  intro v
-  rw [ContinuousLinearMap.adjoint_inner_right]
-  exact (unitary_inner_inv ρ hρ g v w).symm
-
-noncomputable def weightedOperator (ρ : Representation ℂ G V) (p : G → ℝ) : V →L[ℂ] V :=
-  ∑ g, (p g : ℂ) • continuousRepresentation ρ g
-
-lemma weightedOperator_norm_le (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
-    (p : G → ℝ) (hp : ∀ g, 0≤p g) : ‖weightedOperator ρ p‖ ≤ ∑ g, p g := by
-  apply (norm_sum_le _ _).trans
-  apply Finset.sum_le_sum
-  intro g _
-  rw [norm_smul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (hp g)]
-  exact mul_le_of_le_one_right (hp g) (continuousRepresentation_norm_le ρ hρ g)
-
-lemma weightedOperator_symmetric (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
-    (p : G → ℝ) (hp : ∀ g, p g⁻¹=p g) : IsSelfAdjoint (weightedOperator ρ p) := by
-  change (weightedOperator ρ p).adjoint=weightedOperator ρ p
-  unfold weightedOperator
-  rw [map_sum]
-  simp only [map_smulₛₗ,Complex.conj_ofReal,continuousRepresentation_adjoint ρ hρ]
-  apply Fintype.sum_equiv (Equiv.inv G)
-  intro g
-  simp [hp]
-
-noncomputable def uniformOperator (ρ : Representation ℂ G V) : V →L[ℂ] V :=
-  weightedOperator ρ (fun _ => (Fintype.card G:ℝ)⁻¹)
-
-lemma uniformOperator_symmetric (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) :
-    IsSelfAdjoint (uniformOperator ρ) := weightedOperator_symmetric ρ hρ _ (fun _ => rfl)
-
-lemma uniformOperator_left (ρ : Representation ℂ G V) (g : G) :
-    continuousRepresentation ρ g * uniformOperator ρ=uniformOperator ρ := by
-  unfold uniformOperator weightedOperator
-  rw [Finset.mul_sum]
-  simp only [mul_smul_comm,←map_mul]
-  exact Fintype.sum_equiv (Equiv.mulLeft g) _ _ (fun _ => rfl)
-
-lemma uniformOperator_right (ρ : Representation ℂ G V) (g : G) :
-    uniformOperator ρ * continuousRepresentation ρ g=uniformOperator ρ := by
-  unfold uniformOperator weightedOperator
-  rw [Finset.sum_mul]
-  simp only [smul_mul_assoc,←map_mul]
-  exact Fintype.sum_equiv (Equiv.mulRight g) _ _ (fun _ => rfl)
-
-lemma uniformOperator_idempotent (ρ : Representation ℂ G V) :
-    IsIdempotentElem (uniformOperator ρ) := by
-  change uniformOperator ρ * uniformOperator ρ=uniformOperator ρ
-  nth_rw 1 [uniformOperator,weightedOperator]
-  rw [Finset.sum_mul]
-  simp only [smul_mul_assoc,uniformOperator_left,Finset.sum_const,Finset.card_univ,
-    ←Nat.cast_smul_eq_nsmul ℂ,smul_smul,Complex.ofReal_inv,Complex.ofReal_natCast]
-  rw [mul_inv_cancel₀ (by exact_mod_cast Fintype.card_ne_zero),one_smul]
-
-lemma uniformOperator_projection (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) :
-    IsStarProjection (uniformOperator ρ) :=
-  ⟨uniformOperator_idempotent ρ,uniformOperator_symmetric ρ hρ⟩
-
-lemma weighted_uniform_left (ρ : Representation ℂ G V) (p : G → ℝ) :
-    weightedOperator ρ p * uniformOperator ρ=(∑ g,p g : ℝ) • uniformOperator ρ := by
-  unfold weightedOperator
-  rw [Finset.sum_mul]
-  simp only [smul_mul_assoc,uniformOperator_left,←Finset.sum_smul,←Complex.ofReal_sum]
-  rfl
-
-lemma weighted_uniform_right (ρ : Representation ℂ G V) (p : G → ℝ) :
-    uniformOperator ρ * weightedOperator ρ p=(∑ g,p g : ℝ) • uniformOperator ρ := by
-  unfold weightedOperator
-  rw [Finset.mul_sum]
-  simp only [mul_smul_comm,uniformOperator_right,←Finset.sum_smul,←Complex.ofReal_sum]
-  rfl
-
-theorem doeblin_off_invariants (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
-    (p : G → ℝ) (hp : ∑ g,p g=1) (ε : ℝ)
-    (hminor : ∀ g, ε/(Fintype.card G:ℝ)≤p g) :
-    ‖weightedOperator ρ p * (1-uniformOperator ρ)‖≤1-ε := by
-  let q : G → ℝ := fun g => p g-ε/(Fintype.card G:ℝ)
-  have hq : ∀ g, 0≤q g := fun g => sub_nonneg.mpr (hminor g)
-  have hcard : (Fintype.card G:ℝ)≠0 := by exact_mod_cast Fintype.card_ne_zero
-  have hsum : ∑ g,q g=1-ε := by
-    simp only [q,Finset.sum_sub_distrib,hp,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
-    field_simp
-  have he : weightedOperator ρ p=weightedOperator ρ q+(ε:ℂ) • uniformOperator ρ := by
-    simp only [uniformOperator,weightedOperator]
-    rw [Finset.smul_sum,←Finset.sum_add_distrib]
-    apply Finset.sum_congr rfl
-    intro g _
-    simp only [q,Complex.ofReal_sub,Complex.ofReal_div,smul_smul,←add_smul,
-      Complex.ofReal_inv,Complex.ofReal_natCast]
-    congr 1
-    field_simp
-    ring
-  have hz : uniformOperator ρ*(1-uniformOperator ρ)=0 := by
-    rw [mul_sub,mul_one,(uniformOperator_idempotent ρ).eq,sub_self]
-  rw [he,add_mul,smul_mul_assoc,hz,smul_zero,add_zero]
-  calc
-    _ ≤ ‖weightedOperator ρ q‖*‖1-uniformOperator ρ‖ := norm_mul_le _ _
-    _ ≤ (1-ε)*1 := mul_le_mul (by rw [←hsum]; exact weightedOperator_norm_le ρ hρ q hq)
-      ((uniformOperator_projection ρ hρ).one_sub.norm_le _) (norm_nonneg _)
-      (by rw [←hsum]; exact Finset.sum_nonneg (fun g _ => hq g))
-    _ = _ := mul_one _
-
-end
-open scoped BigOperators Classical
-variable {G : Type*} [Group G] [Fintype G]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-lemma weightedOperator_conv (ρ : Representation ℂ G V) (p q : G → ℝ) :
-    weightedOperator ρ (FiniteConvolution.conv p q)=weightedOperator ρ p*weightedOperator ρ q := by
-  unfold weightedOperator FiniteConvolution.conv
-  simp only [Complex.ofReal_sum,Finset.sum_smul]
-  rw [Finset.sum_comm,Finset.sum_mul]
-  apply Finset.sum_congr rfl
-  intro h _
-  rw [Finset.mul_sum]
-  simp only [smul_mul_assoc,mul_smul_comm,smul_smul,←map_mul]
-  apply Fintype.sum_equiv (Equiv.mulLeft h⁻¹)
-  intro g
-  simp [Complex.ofReal_mul,mul_comm]
-
-lemma weightedOperator_power (ρ : Representation ℂ G V) (p : G → ℝ) (n : ℕ) :
-    weightedOperator ρ (FiniteConvolution.power p n)=(weightedOperator ρ p)^n := by
-  induction n with
-  | zero => simp [FiniteConvolution.power,weightedOperator]
-  | succ n ih => rw [FiniteConvolution.power,weightedOperator_conv,ih,pow_succ']
-
-lemma off_invariants_pow (ρ : Representation ℂ G V) (p : G → ℝ)
-    (hp : ∑ g,p g=1) (n : ℕ) (hn : n≠0) :
-    (weightedOperator ρ p-uniformOperator ρ)^n=
-      (weightedOperator ρ p)^n*(1-uniformOperator ρ) := by
-  have hAP := weighted_uniform_left ρ p
-  have hPA := weighted_uniform_right ρ p
-  rw [hp,one_smul] at hAP hPA
-  have hc : Commute (weightedOperator ρ p) (1-uniformOperator ρ) := by
-    show _*_= _*_
-    rw [mul_sub,sub_mul,mul_one,one_mul,hAP,hPA]
-  have he : weightedOperator ρ p-uniformOperator ρ=
-      weightedOperator ρ p*(1-uniformOperator ρ) := by rw [mul_sub,mul_one,hAP]
-  rw [he,hc.mul_pow,((uniformOperator_idempotent ρ).one_sub).pow_eq hn]
-
-noncomputable def minorantGap {p : G → ℝ} (c : FiniteConvolution.Minorant p) : ℝ :=
-  1-c.mass/(2*(2:ℝ)^c.exponent)
-
-omit [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V] in
-lemma minorantGap_pos {p : G → ℝ} (c : FiniteConvolution.Minorant p) : 0 < minorantGap c := by
-  have hn : (1:ℝ) ≤ 2^c.exponent := one_le_pow₀ (by norm_num)
-  have hq : c.mass/(2*(2:ℝ)^c.exponent) ≤ 1/2 := by
-    apply (div_le_iff₀ (by positivity)).mpr
-    nlinarith [c.mass_le_one]
-  unfold minorantGap
-  linarith
-
-omit [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V] in
-lemma minorantGap_lt_one {p : G → ℝ} (c : FiniteConvolution.Minorant p) : minorantGap c < 1 := by
-  unfold minorantGap
-  exact sub_lt_self _ (div_pos c.mass_pos (by positivity))
-
-theorem norm_off_invariants_le_gap (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
-    (p : G → ℝ) (hp : ∑ g,p g=1) (hsym : ∀ g,p g⁻¹=p g)
-    (c : FiniteConvolution.Minorant p) :
-    ‖weightedOperator ρ p-uniformOperator ρ‖ ≤ minorantGap c := by
-  have hself := (weightedOperator_symmetric ρ hρ p hsym).sub (uniformOperator_symmetric ρ hρ)
-  have hbound := doeblin_off_invariants ρ hρ (FiniteConvolution.power p (2^c.exponent))
-    (FiniteConvolution.sum_power hp _) c.mass c.lower
-  rw [weightedOperator_power,←off_invariants_pow ρ p hp _ (by positivity),
-    hself.norm_pow_two_pow] at hbound
-  have hn : (1:ℝ) ≤ 2^c.exponent := one_le_pow₀ (by norm_num)
-  have hq : c.mass/(2*(2:ℝ)^c.exponent) ≤ 1/2 := by
-    apply (div_le_iff₀ (by positivity)).mpr
-    nlinarith [c.mass_le_one]
-  have hbern := one_add_mul_le_pow
-    (a := -(c.mass/(2*(2:ℝ)^c.exponent))) (by linarith : (-2:ℝ) ≤ -(c.mass/(2*(2:ℝ)^c.exponent)))
-    (2^c.exponent)
-  have he : 1+(2^c.exponent:ℕ)*(-(c.mass/(2*(2:ℝ)^c.exponent)))=1-c.mass/2 := by
-    push_cast
-    field_simp
-    ring
-  rw [he,show 1+ -(c.mass/(2*(2:ℝ)^c.exponent))=minorantGap c by rfl] at hbern
-  by_contra h
-  have hlt : minorantGap c < ‖weightedOperator ρ p-uniformOperator ρ‖ := lt_of_not_ge h
-  have hpows := pow_lt_pow_left₀ hlt (minorantGap_pos c).le (by positivity : (2^c.exponent:ℕ)≠0)
-  linarith [c.mass_pos]
-
-end Thorp.UnitaryFinite
-namespace Thorp
-open scoped BigOperators Classical
-
-def butterflyIdentity : (d : ℕ) → Butterfly d
-  | 0 => ()
-  | d+1 => (fun _ => false,fun _ => butterflyIdentity d)
-
-lemma pairSwitch_false (d : ℕ) : pairSwitch (d := d) (fun _ => false)=1 := by
-  apply Equiv.ext
-  intro x
-  change switchFun (fun _ => false) x=x
-  simp [switchFun]
-
-lemma childLift_one (d : ℕ) : childLift (d := d) (fun _ => 1)=1 := by
-  ext x i
-  simp [childLift]
-
-lemma butterflyIdentity_perm (d : ℕ) : butterflyPerm d (butterflyIdentity d)=1 := by
-  induction d with
-  | zero => rfl
-  | succ d ih => simp only [butterflyIdentity,butterflyPerm,ih,pairSwitch_false,childLift_one,mul_one]
-
-noncomputable def childInjection (d : ℕ) (b : Bool) : Equiv.Perm (Card d) →* Equiv.Perm (Card (d+1)) where
-  toFun p := childLift (fun c => if c=b then p else 1)
-  map_one' := by simpa only [ite_self] using childLift_one d
-  map_mul' p q := by
-    rw [childLift_mul]
-    congr 1
-    funext c
-    split_ifs <;> simp
-
-lemma childInjection_swap (d : ℕ) (b : Bool) (u v : Card d) :
-    childInjection d b (Equiv.swap u v)=Equiv.swap (Fin.cons b u) (Fin.cons b v) := by
-  apply Equiv.ext
-  intro x
-  obtain ⟨c,w,rfl⟩ : ∃ c w, x=Fin.cons c w := ⟨x 0,Fin.tail x,(Fin.cons_self_tail x).symm⟩
-  change (Fin.cons c ((if c=b then Equiv.swap u v else 1) w) : Card (d+1))=_
-  by_cases hc : c=b
-  · subst c
-    simp only [↓reduceIte]
-    rw [Equiv.swap_apply_def]
-    split_ifs with h₁ h₂
-    · subst w; simp
-    · subst w; simp
-    · rw [Equiv.swap_apply_of_ne_of_ne]
-      · exact fun h => h₁ (by simpa using congrArg Fin.tail h)
-      · exact fun h => h₂ (by simpa using congrArg Fin.tail h)
-  · rw [ite_eq_right hc,Equiv.Perm.one_apply,Equiv.swap_apply_of_ne_of_ne]
-    · intro h; exact hc (by simpa using congrArg (fun z => z 0) h)
-    · intro h; exact hc (by simpa using congrArg (fun z => z 0) h)
-
-lemma pairSwitch_single (d : ℕ) (u : Card d) :
-    pairSwitch (fun v => decide (v=u))=Equiv.swap (Fin.cons false u) (Fin.cons true u) := by
-  apply Equiv.ext
-  intro x
-  obtain ⟨c,w,rfl⟩ : ∃ c w, x=Fin.cons c w := ⟨x 0,Fin.tail x,(Fin.cons_self_tail x).symm⟩
-  change switchFun (fun v => decide (v=u)) (Fin.cons c w)=_
-  by_cases h : w=u
-  · subst w
-    cases c <;> simp [switchFun]
-  · have ha : (Fin.cons c w : Card (d+1))≠Fin.cons false u := fun he => h (by simpa using congrArg Fin.tail he)
-    have hb : (Fin.cons c w : Card (d+1))≠Fin.cons true u := fun he => h (by simpa using congrArg Fin.tail he)
-    simp [switchFun,h,Equiv.swap_apply_of_ne_of_ne ha hb]
-
-theorem butterfly_generates (d : ℕ) (H : Subgroup (Equiv.Perm (Card d)))
-    (hH : ∀ ω : Butterfly d, butterflyPerm d ω∈H) : H=⊤ := by
-  induction d with
-  | zero =>
-    apply top_unique
-    intro p _
-    have hp : p=1 := Subsingleton.elim _ _
-    rw [hp]
-    exact H.one_mem
-  | succ d ih =>
-    have hchild (b : Bool) (p : Equiv.Perm (Card d)) : childInjection d b p∈H := by
-      have hc : H.comap (childInjection d b)=⊤ := by
-        apply ih
-        intro ω
-        change childLift (fun c => if c=b then butterflyPerm d ω else 1)∈H
-        have h := hH ((fun _ => false),fun c => if c=b then ω else butterflyIdentity d)
-        simpa only [butterflyPerm,pairSwitch_false,one_mul,apply_ite,butterflyIdentity_perm] using h
-      have hp : p∈H.comap (childInjection d b) := by rw [hc]; trivial
-      exact hp
-    have hpair (u : Card d) : Equiv.swap (Fin.cons false u) (Fin.cons true u)∈H := by
-      rw [←pairSwitch_single]
-      have h := hH ((fun v => decide (v=u)),fun _ => butterflyIdentity d)
-      simpa only [butterflyPerm,butterflyIdentity_perm,childLift_one,mul_one] using h
-    have hswap (x y : Card (d+1)) : Equiv.swap x y∈H := by
-      have ht (a b : Bool) (u : Card d) : Equiv.swap (Fin.cons a u) (Fin.cons b u)∈H := by
-        cases a <;> cases b
-        · rw [Equiv.swap_self]; exact H.one_mem
-        · exact hpair _
-        · rw [Equiv.swap_comm]; exact hpair _
-        · rw [Equiv.swap_self]; exact H.one_mem
-      have h₁ : Equiv.swap x (Fin.cons (y 0) (Fin.tail x))∈H := by
-        convert ht (x 0) (y 0) (Fin.tail x) using 1
-        rw [Fin.cons_self_tail]
-      have h₂ := hchild (y 0) (Equiv.swap (Fin.tail x) (Fin.tail y))
-      rw [childInjection_swap,Fin.cons_self_tail] at h₂
-      exact SubmonoidClass.swap_mem_trans H h₁ h₂
-    rw [eq_top_iff,←Equiv.Perm.closure_isSwap,Subgroup.closure_le]
-    rintro _ ⟨x,y,_,rfl⟩
-    exact hswap x y
-
-lemma butterfly_closure (d : ℕ) : Subgroup.closure (Set.range (butterflyPerm d))=⊤ :=
-  butterfly_generates d _ (fun ω => Subgroup.subset_closure ⟨ω,rfl⟩)
-
-variable {Ω G : Type*} [Fintype Ω] [Fintype G]
-
-noncomputable def sampleLaw (P : Ω → G) (g : G) : ℝ :=
-  finiteMean (fun ω => if P ω=g then 1 else 0)
-
-omit [Fintype G] in
-lemma sampleLaw_nonneg (P : Ω → G) (g : G) : 0 ≤ sampleLaw P g :=
-  finiteMean_nonneg (fun _ => by split_ifs <;> norm_num)
-
-lemma sampleLaw_sum [Nonempty Ω] (P : Ω → G) : ∑ g,sampleLaw P g=1 := by
-  unfold sampleLaw
-  rw [←finiteMean_sum]
-  simp only [Finset.sum_ite_eq,Finset.mem_univ,↓reduceIte,finiteMean_const]
-
-omit [Fintype G] in
-lemma sampleLaw_pos_iff [Nonempty Ω] (P : Ω → G) (g : G) :
-    0<sampleLaw P g ↔ ∃ ω,P ω=g := by
-  unfold sampleLaw finiteMean
-  rw [div_pos_iff_of_pos_right (Nat.cast_pos.mpr Fintype.card_pos)]
-  rw [Finset.sum_pos_iff_of_nonneg (fun _ _ => by split_ifs <;> norm_num)]
-  simp only [Finset.mem_univ,true_and]
-  apply exists_congr
-  intro ω
-  split_ifs <;> simp_all
-
-omit [Fintype G] in
-lemma sampleLaw_equiv {Ω' : Type*} [Fintype Ω'] (P : Ω → G) (e : Ω' ≃ Ω) :
-    sampleLaw (P ∘ e)=sampleLaw P := by
-  funext g
-  simpa only [sampleLaw,Function.comp_apply,Equiv.apply_symm_apply] using
-    finiteMean_equiv e (fun ω => if P (e ω)=g then (1:ℝ) else 0)
-
-variable [Group G]
-omit [Fintype G] in
-lemma sampleLaw_inverse (P : Ω → G) (g : G) :
-    sampleLaw (fun ω => (P ω)⁻¹) g=sampleLaw P g⁻¹ := by
-  apply finiteMean_congr
-  intro ω
-  rw [inv_eq_iff_eq_inv]
-
-omit [Fintype G] in
-lemma sampleLaw_symmetric (P : Ω → G) (e : Equiv.Perm Ω)
-    (he : ∀ ω,P (e ω)=(P ω)⁻¹) (g : G) : sampleLaw P g⁻¹=sampleLaw P g := by
-  rw [←sampleLaw_inverse]
-  have hh : (fun ω => (P ω)⁻¹)=P ∘ e := funext (fun ω => (he ω).symm)
-  rw [hh,sampleLaw_equiv]
-
-end Thorp
-
-namespace Thorp
-open scoped BigOperators Classical
-
-lemma palindrome_identity (d : ℕ) : ∃ ω,palindromePerm d ω=1 := by
-  refine ⟨((butterflyCoinEquiv d).symm (butterflyIdentity d),
-    (butterflyCoinEquiv d).symm (butterflyIdentity d)),?_⟩
-  exact mul_inv_cancel _
-
-lemma palindrome_range_contains_butterfly (d : ℕ) (ω : Butterfly d) :
-    ∃ ξ,palindromePerm d ξ=butterflyPerm d ω := by
-  refine ⟨((butterflyCoinEquiv d).symm ω,(butterflyCoinEquiv d).symm (butterflyIdentity d)),?_⟩
-  unfold palindromePerm
-  have h (β : Butterfly d) : decodeButterfly d ((butterflyCoinEquiv d).symm β)=β :=
-    (butterflyCoinEquiv d).apply_symm_apply β
-  rw [h,h,butterflyIdentity_perm,show (1 : Equiv.Perm (Card d)).symm=1 from rfl,mul_one]
-
-lemma palindrome_support_generates (d : ℕ) :
-    Subgroup.closure {g | 0<sampleLaw (palindromePerm d) g}=⊤ := by
-  apply butterfly_generates
-  intro ω
-  apply Subgroup.subset_closure
-  exact (sampleLaw_pos_iff _ _).mpr (palindrome_range_contains_butterfly d ω)
-
-noncomputable def palindromeMinorant (d : ℕ) :
-    FiniteConvolution.Minorant (sampleLaw (palindromePerm d)) :=
-  FiniteConvolution.minorant (sampleLaw_nonneg _) (sampleLaw_sum _)
-    ((sampleLaw_pos_iff _ _).mpr (palindrome_identity d)) (palindrome_support_generates d)
-
-noncomputable def blockGap (d : ℕ) : ℝ := UnitaryFinite.minorantGap (palindromeMinorant d)
-
-lemma blockGap_pos (d : ℕ) : 0<blockGap d := UnitaryFinite.minorantGap_pos _
-lemma blockGap_lt_one (d : ℕ) : blockGap d<1 := UnitaryFinite.minorantGap_lt_one _
-
-theorem palindrome_off_invariants_gap (d : ℕ) {V : Type*}
-    [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-    (ρ : Representation ℂ (Equiv.Perm (Card d)) V) (hρ : UnitaryFinite.IsUnitary ρ) :
-    ‖UnitaryFinite.weightedOperator ρ (sampleLaw (palindromePerm d))-
-      UnitaryFinite.uniformOperator ρ‖≤blockGap d := by
-  apply UnitaryFinite.norm_off_invariants_le_gap ρ hρ _ (sampleLaw_sum _)
-  exact sampleLaw_symmetric _ (Equiv.prodComm _ _) (palindromePerm_reverse d)
-
-end Thorp
-
-namespace Thorp.CommutingBlocks
-open scoped BigOperators Classical
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
-variable {ι : Type*} [Fintype ι]
-
-omit [CompleteSpace V] in
-lemma orthogonal_sum_norm_sq (v : ι → V) (hv : ∀ i j, i≠j → inner ℂ (v i) (v j)=0) :
-    ‖∑ i,v i‖^2=∑ i,‖v i‖^2 := by
-  rw [norm_sq_eq_re_inner (𝕜 := ℂ),sum_inner]
-  simp only [inner_sum,map_sum]
-  apply Finset.sum_congr rfl
-  intro i _
-  rw [Finset.sum_eq_single i]
-  · exact (norm_sq_eq_re_inner (𝕜 := ℂ) (v i)).symm
-  · intro j _ hji
-    rw [hv i j hji.symm]
-    rfl
-  · simp
-
-lemma projection_inner_zero (P Q : V →L[ℂ] V) (hP : IsStarProjection P)
-    (hPQ : P*Q=0) (v w : V) : inner ℂ (P v) (Q w)=0 := by
-  rw [←ContinuousLinearMap.adjoint_inner_right]
-  change inner ℂ v ((star P*Q) w)=0
-  rw [hP.isSelfAdjoint.star_eq,hPQ]
-  simp
-
-lemma projection_sandwich_norm (P B : V →L[ℂ] V) (hP : IsStarProjection P) :
-    ‖B.adjoint*P*B‖=‖P*B‖^2 := by
-  have he : star (P*B)*(P*B)=B.adjoint*P*B := by
-    rw [star_mul,hP.isSelfAdjoint.star_eq]
-    change B.adjoint*P*(P*B)=_
-    rw [←mul_assoc,mul_assoc B.adjoint P P,hP.isIdempotentElem.eq]
-  rw [←he,CStarRing.norm_star_mul_self]
-  ring
-
-theorem resolution_norm_bound (P : ι → V →L[ℂ] V)
-    (hP : ∀ i, IsStarProjection (P i))
-    (horth : ∀ i j, i≠j → P i*P j=0) (hsum : ∑ i,P i=1)
-    (A : V →L[ℂ] V) (hcomm : ∀ i, Commute A (P i))
-    (c : ℝ) (hc : 0≤c) (hbound : ∀ i, ‖A*P i‖≤c) : ‖A‖≤c := by
-  apply ContinuousLinearMap.opNorm_le_bound _ hc
-  intro v
-  have hsplit (w : V) : ∑ i,P i w=w := by
-    rw [←sum_apply,hsum]
-    rfl
-  have hp := orthogonal_sum_norm_sq (fun i => P i v)
-    (fun i j hij => projection_inner_zero _ _ (hP i) (horth i j hij) v v)
-  rw [hsplit] at hp
-  have ha := orthogonal_sum_norm_sq (fun i => P i (A v))
-    (fun i j hij => projection_inner_zero _ _ (hP i) (horth i j hij) (A v) (A v))
-  rw [hsplit] at ha
-  have he (i : ι) : P i (A v)=(A*P i) (P i v) := by
-    change (P i*A) v=(A*P i*P i) v
-    rw [mul_assoc,(hP i).isIdempotentElem.eq,(hcomm i).eq]
-  have hh (i : ι) : ‖P i (A v)‖≤c*‖P i v‖ := by
-    rw [he]
-    exact ((A*P i).le_opNorm _).trans (mul_le_mul_of_nonneg_right (hbound i) (norm_nonneg _))
-  have hsq := Finset.sum_le_sum (s := Finset.univ) (fun i _ =>
-    pow_le_pow_left₀ (norm_nonneg _) (hh i) 2)
-  simp_rw [mul_pow] at hsq
-  rw [←Finset.mul_sum,←hp,←ha,←mul_pow] at hsq
-  exact (sq_le_sq₀ (norm_nonneg _) (mul_nonneg hc (norm_nonneg _))).mp hsq
-
-theorem sector_sandwich_le (T P A B : V →L[ℂ] V)
-    (hT : IsStarProjection T) (hP : IsStarProjection P)
-    (hTP : T*P=T) (hcomm : Commute A T) (hA : ‖A‖≤1) :
-    ‖B.adjoint*A*T*B‖≤‖B.adjoint*P*B‖ := by
-  have he : B.adjoint*A*T*B=(T*B).adjoint*A*(T*B) := by
-    change _=star (T*B)*A*(T*B)
-    rw [star_mul,hT.isSelfAdjoint.star_eq]
-    change _=B.adjoint*T*A*(T*B)
-    rw [mul_assoc B.adjoint T A,←hcomm.eq]
-    simp only [mul_assoc]
-    rw [←mul_assoc T T B,hT.isIdempotentElem.eq]
-  have hTB : ‖T*B‖≤‖P*B‖ := by
-    have he : T*B=T*(P*B) := by rw [←mul_assoc,hTP]
-    rw [he]
-    exact (norm_mul_le _ _).trans ((mul_le_mul_of_nonneg_right (hT.norm_le _) (norm_nonneg _)).trans_eq (one_mul _))
-  rw [he,projection_sandwich_norm P B hP]
-  calc
-    _ ≤ ‖(T*B).adjoint‖*‖A‖*‖T*B‖ := (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right (norm_mul_le _ _) (norm_nonneg _))
-    _ ≤ ‖T*B‖^2 := by rw [ContinuousLinearMap.adjoint.norm_map]; nlinarith [norm_nonneg (T*B)]
-    _ ≤ _ := pow_le_pow_left₀ (norm_nonneg _) hTB 2
-
-omit [CompleteSpace V] [Fintype ι] in
-lemma resolution_commute (P : ι → V →L[ℂ] V)
-    (horth : ∀ i j, i≠j → P i*P j=0) (i j : ι) : Commute (P i) (P j) := by
-  by_cases hij : i=j
-  · subst j; exact Commute.refl _
-  · exact (horth i j hij).trans (horth j i (Ne.symm hij)).symm
-
-omit [Fintype ι] in
-lemma resolution_sum_mul (P : ι → V →L[ℂ] V) (hP : ∀ i, IsStarProjection (P i))
-    (horth : ∀ i j, i≠j → P i*P j=0) (S : Finset ι) (j : ι) :
-    (∑ i∈S,P i)*P j=if j∈S then P j else 0 := by
-  rw [Finset.sum_mul]
-  have he (i : ι) : P i*P j=if i=j then P j else 0 := by
-    split_ifs with h
-    · subst i; exact (hP j).isIdempotentElem.eq
-    · exact horth i j h
-  simp only [he,Finset.sum_ite_eq']
-
-lemma resolution_truncated_norm (P : ι → V →L[ℂ] V)
-    (hP : ∀ i, IsStarProjection (P i))
-    (horth : ∀ i j, i≠j → P i*P j=0) (hsum : ∑ i,P i=1)
-    (A : V →L[ℂ] V) (hcomm : ∀ i, Commute A (P i))
-    (S : Finset ι) (c : ℝ) (hc : 0≤c) (hbound : ∀ i∈S, ‖A*P i‖≤c) :
-    ‖A*(∑ i∈S,P i)‖≤c := by
-  apply resolution_norm_bound P hP horth hsum _ _ c hc
-  · intro i
-    rw [mul_assoc,resolution_sum_mul P hP horth]
-    split_ifs with hi
-    · exact hbound i hi
-    · simp [hc]
-  · intro i
-    exact (hcomm i).mul_left (Commute.sum_left S P (P i) fun j _ => resolution_commute P horth j i)
-
-theorem resolution_exception_bound (T Q : ι → V →L[ℂ] V)
-    (hT : ∀ i, IsStarProjection (T i))
-    (horth : ∀ i j, i≠j → T i*T j=0) (hsum : ∑ i,T i=1)
-    (hQ : ∀ i, IsStarProjection (Q i)) (hTQ : ∀ i,T i*Q i=T i)
-    (A B : V →L[ℂ] V) (hcomm : ∀ i, Commute A (T i))
-    (hA : ‖A‖≤1) (hB : ‖B‖≤1)
-    (S : Finset ι) (c : ℝ) (hc : 0≤c) (hbound : ∀ i∉S, ‖A*T i‖≤c) :
-    ‖B.adjoint*A*B‖≤c+∑ i∈S,‖B.adjoint*Q i*B‖ := by
-  have hhigh : ‖A*(∑ i∈Sᶜ,T i)‖≤c := resolution_truncated_norm T hT horth hsum A hcomm
-    Sᶜ c hc (fun i hi => hbound i (Finset.mem_compl.mp hi))
-  have hsplit : (∑ i∈Sᶜ,T i)+(∑ i∈S,T i)=1 := by
-    rw [Finset.sum_compl_add_sum,hsum]
-  have he : B.adjoint*A*B=B.adjoint*(A*(∑ i∈Sᶜ,T i))*B+
-      ∑ i∈S,B.adjoint*A*T i*B := by
-    rw [←Finset.sum_mul,←Finset.mul_sum]
-    rw [←add_mul]
-    simp only [mul_assoc]
-    rw [←mul_add,←mul_add,hsplit]
-    simp only [mul_one,mul_assoc]
-  rw [he]
-  apply (norm_add_le _ _).trans
-  apply add_le_add
-  · calc
-      _ ≤ ‖B.adjoint‖*‖A*(∑ i∈Sᶜ,T i)‖*‖B‖ :=
-        (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right (norm_mul_le _ _) (norm_nonneg _))
-      _ ≤ 1*c*1 := mul_le_mul (mul_le_mul (by simpa only [ContinuousLinearMap.adjoint.norm_map] using hB)
-        hhigh (norm_nonneg _) zero_le_one) hB (norm_nonneg _) (by simpa using hc)
-      _ = c := by ring
-  · apply (norm_sum_le _ _).trans
-    exact Finset.sum_le_sum fun i _ => sector_sandwich_le _ _ _ _ (hT i) (hQ i) (hTQ i) (hcomm i) hA
-
-variable [DecidableEq ι]
-variable (P : ι → V →L[ℂ] V) (hcomm : ∀ i j, Commute (P i) (P j))
-
-noncomputable def choice (E : Finset ι) (i : ι) : V →L[ℂ] V :=
-  if i∈E then 1-P i else P i
-
-include hcomm
-omit [CompleteSpace V] [Fintype ι] in
-lemma choice_commute (E F : Finset ι) (i j : ι) :
-    Commute (choice P E i) (choice P F j) := by
-  unfold choice
-  split_ifs
-  · exact (Commute.one_left _).sub_left ((Commute.one_right _).sub_right (hcomm i j))
-  · exact (Commute.one_left _).sub_left (hcomm i j)
-  · exact (Commute.one_right _).sub_right (hcomm i j)
-  · exact hcomm i j
-
-noncomputable def sector (s E : Finset ι) : V →L[ℂ] V :=
-  s.noncommProd (choice P E) (fun i _ j _ _ => choice_commute P hcomm E E i j)
-
-omit [CompleteSpace V] [Fintype ι] in
-lemma sector_insert (s E : Finset ι) (a : ι) (ha : a∉s) :
-    sector P hcomm (insert a s) E=choice P E a*sector P hcomm s E := by
-  exact Finset.noncommProd_insert_of_notMem _ _ _ _ ha
-
-omit [CompleteSpace V] [Fintype ι] in
-lemma sector_congr (s E F : Finset ι) (h : ∀ i∈s, (i∈E ↔ i∈F)) :
-    sector P hcomm s E=sector P hcomm s F := by
-  apply Finset.noncommProd_congr rfl
-  intro i hi
-  simp only [choice,h i hi]
-
-omit [CompleteSpace V] [Fintype ι] in
-lemma sector_factor_commute (s E F : Finset ι) (i : ι) :
-    Commute (choice P E i) (sector P hcomm s F) :=
-  Finset.noncommProd_commute _ _ _ _ (fun j _ => choice_commute P hcomm E F i j)
-
-omit [Fintype ι] in
-lemma sector_projection (hP : ∀ i, IsStarProjection (P i)) (s E : Finset ι) :
-    IsStarProjection (sector P hcomm s E) := by
-  induction s using Finset.induction_on with
-  | empty => exact IsStarProjection.one _
-  | @insert a s ha ih =>
-    rw [sector_insert P hcomm s E a ha]
-    have hc : IsStarProjection (choice P E a) := by
-      unfold choice
-      split_ifs
-      · exact (hP a).one_sub
-      · exact hP a
-    exact hc.mul ih (sector_factor_commute P hcomm s E E a)
-
-end Thorp.CommutingBlocks
-
-end ThorpNine.Tail
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Tail.ShapePartitionEquiv
+
+namespace OAI
+
+namespace ThorpNine.Tail
+
+namespace Thorp.UnitaryFinite
+open scoped BigOperators ComplexConjugate Classical
+variable {G : Type*} [Group G] [Fintype G]
+variable {V W : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+  [NormedAddCommGroup W] [InnerProductSpace ℂ W] [FiniteDimensional ℂ W]
+omit [FiniteDimensional ℂ V] in
+lemma averageMap_self_trace (ρ : Representation ℂ G V) (A : Module.End ℂ V) :
+    LinearMap.trace ℂ V (averageMap ρ ρ A).toLinearMap =
+      (Fintype.card G:ℂ) * LinearMap.trace ℂ V A := by
+  change LinearMap.trace ℂ V (∑ g, ρ g * A * ρ g⁻¹) = _
+  rw [map_sum]
+  have hj (g : G) : LinearMap.trace ℂ V (ρ g*A*ρ g⁻¹) = LinearMap.trace ℂ V A := by
+    rw [LinearMap.trace_mul_cycle,←map_mul,inv_mul_cancel,map_one,one_mul]
+  simp only [hj,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
+
+lemma averageMap_scalar (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
+    (A : Module.End ℂ V) :
+    ∃ c : ℂ, (averageMap ρ ρ A).toLinearMap = c • LinearMap.id ∧
+      c * (Module.finrank ℂ V:ℂ) = (Fintype.card G:ℂ)*LinearMap.trace ℂ V A := by
+  obtain ⟨c,hc⟩ := (Representation.IsIrreducible.algebraMap_intertwiningMap_bijective_of_isAlgClosed (ρ := ρ)).surjective (averageMap ρ ρ A)
+  have he : (averageMap ρ ρ A).toLinearMap = c • LinearMap.id := by
+    rw [←hc]
+    rfl
+  refine ⟨c,he,?_⟩
+  rw [←averageMap_self_trace ρ A,he,map_smul,LinearMap.trace_id,smul_eq_mul]
+
+theorem coefficient_orthogonality (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
+    (hρ : IsUnitary ρ) (u v w z : V) :
+    (Module.finrank ℂ V:ℂ) * ∑ g, inner ℂ u (ρ g v) * inner ℂ (ρ g w) z =
+      (Fintype.card G:ℂ) * inner ℂ w v * inner ℂ u z := by
+  obtain ⟨c,hc,htrace⟩ := averageMap_scalar ρ (InnerProductSpace.rankOne ℂ v w).toLinearMap
+  rw [InnerProductSpace.trace_rankOne] at htrace
+  have happ := congrArg (fun A : Module.End ℂ V => inner ℂ u (A z)) hc
+  change inner ℂ u ((∑ g, ρ g * (InnerProductSpace.rankOne ℂ v w).toLinearMap * ρ g⁻¹) z) = _ at happ
+  simp only [LinearMap.sum_apply,Module.End.mul_apply,ContinuousLinearMap.coe_coe,
+    InnerProductSpace.rankOne_apply,map_smul,unitary_inner_inv ρ hρ,inner_sum,
+    inner_smul_right,LinearMap.smul_apply,LinearMap.id_apply] at happ
+  have hsum : ∑ g, inner ℂ u (ρ g v) * inner ℂ (ρ g w) z = c * inner ℂ u z := by
+    simpa only [mul_comm] using happ
+  rw [hsum]
+  calc
+    _ = (c*(Module.finrank ℂ V:ℂ))*inner ℂ u z := by ring
+    _ = _ := by rw [htrace]
+
+end Thorp.UnitaryFinite
+namespace Thorp.UnitaryFinite
+open scoped BigOperators ComplexConjugate Classical
+
+variable {G X : Type*} [Group G] [Fintype G] [Fintype X]
+  (a : G →* Equiv.Perm X) (x₀ : X) (ha : ∀ x, ∃ g, a g x₀=x)
+
+noncomputable def representative (x : X) : G := Classical.choose (ha x)
+
+omit [Fintype G] [Fintype X] in
+lemma representative_apply (x : X) : a (representative a x₀ ha x) x₀=x :=
+  Classical.choose_spec (ha x)
+
+include ha in
+lemma orbit_sum (F : X → ℂ) :
+    (Fintype.card X:ℂ)*∑ g, F (a g x₀) = (Fintype.card G:ℂ)*∑ x, F x := by
+  have ht (x : X) : ∑ g, F (a g x)=∑ g, F (a g x₀) := by
+    obtain ⟨t,rfl⟩ := ha x
+    symm
+    apply Fintype.sum_equiv (Equiv.mulRight t⁻¹)
+    intro g
+    simp
+  calc
+    _ = ∑ x : X, ∑ g : G, F (a g x) := by simp only [ht,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
+    _ = ∑ g : G, ∑ x : X, F (a g x) := Finset.sum_comm
+    _ = ∑ g : G, ∑ x : X, F x := by
+      apply Finset.sum_congr rfl
+      intro g _
+      exact Equiv.sum_comp (a g) F
+    _ = _ := by simp
+
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V)
+
+def IsFixed (w : V) : Prop := ∀ g, a g x₀=x₀ → ρ g w=w
+
+noncomputable def coefficient (w v : V) : EuclideanSpace ℂ X :=
+  WithLp.toLp 2 (fun x => inner ℂ (ρ (representative a x₀ ha x) w) v)
+
+omit [Fintype G] [Fintype X] [FiniteDimensional ℂ V] in
+lemma coefficient_at (w : V) (hw : IsFixed a x₀ ρ w) (v : V) (g : G) :
+    coefficient a x₀ ha ρ w v (a g x₀) = inner ℂ (ρ g w) v := by
+  let r := representative a x₀ ha (a g x₀)
+  have hr : a r x₀=a g x₀ := representative_apply a x₀ ha _
+  have hs : a (g⁻¹*r) x₀=x₀ := by
+    simp only [map_mul,Equiv.Perm.mul_apply]
+    rw [hr,←Equiv.Perm.mul_apply,←map_mul,inv_mul_cancel,map_one]
+    rfl
+  have he := congrArg (ρ g) (hw (g⁻¹*r) hs)
+  have hh : ρ r w=ρ g w := by
+    simpa only [map_mul,Module.End.mul_apply,←Module.End.mul_apply,←map_mul,
+      mul_inv_cancel_left] using he
+  change inner ℂ (ρ r w) v= _
+  rw [hh]
+
+lemma coefficient_inner [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
+    (w z : V) (hw : IsFixed a x₀ ρ w) (hz : IsFixed a x₀ ρ z) (v t : V) :
+    (Module.finrank ℂ V:ℂ)*inner ℂ (coefficient a x₀ ha ρ w v) (coefficient a x₀ ha ρ z t) =
+      (Fintype.card X:ℂ)*inner ℂ z w*inner ℂ v t := by
+  have hs := orbit_sum a x₀ ha (fun x => star (coefficient a x₀ ha ρ w v x)*coefficient a x₀ ha ρ z t x)
+  simp only [coefficient_at a x₀ ha ρ w hw,coefficient_at a x₀ ha ρ z hz] at hs
+  simp only [←starRingEnd_apply,inner_conj_symm] at hs
+  simp only [starRingEnd_apply] at hs
+  have ho := coefficient_orthogonality ρ hρ v w z t
+  have hcard : (Fintype.card G:ℂ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
+  apply mul_left_cancel₀ hcard
+  rw [PiLp.inner_apply]
+  simp only [RCLike.inner_apply,starRingEnd_apply]
+  have hsum : ∑ x, coefficient a x₀ ha ρ z t x * star (coefficient a x₀ ha ρ w v x) =
+      ∑ x, star (coefficient a x₀ ha ρ w v x)*coefficient a x₀ ha ρ z t x := by
+    apply Finset.sum_congr rfl
+    intro x _
+    ring
+  rw [hsum]
+  calc
+    _ = (Module.finrank ℂ V:ℂ)*((Fintype.card G:ℂ)*∑ x,
+        star (coefficient a x₀ ha ρ w v x)*coefficient a x₀ ha ρ z t x) := by ring
+    _ = (Fintype.card X:ℂ)*((Module.finrank ℂ V:ℂ)*∑ g,
+        inner ℂ v (ρ g w)*inner ℂ (ρ g z) t) := by rw [←hs]; ring
+    _ = _ := by rw [ho]; ring
+
+omit [Fintype G] [Fintype X] [FiniteDimensional ℂ V] in
+lemma coefficient_action (hρ : IsUnitary ρ) (w : V) (hw : IsFixed a x₀ ρ w)
+    (v : V) (g : G) (x : X) :
+    coefficient a x₀ ha ρ w v (a g x) = coefficient a x₀ ha ρ w (ρ g⁻¹ v) x := by
+  obtain ⟨r,rfl⟩ := ha x
+  rw [←Equiv.Perm.mul_apply,←map_mul,coefficient_at a x₀ ha ρ w hw,
+    coefficient_at a x₀ ha ρ w hw]
+  simp only [map_mul,Module.End.mul_apply,unitary_inner_inv ρ hρ]
+
+noncomputable def normalizedCoefficient (w v : V) : EuclideanSpace ℂ X :=
+  (Real.sqrt ((Module.finrank ℂ V:ℝ)/(Fintype.card X:ℝ)) : ℂ) • coefficient a x₀ ha ρ w v
+
+lemma normalizedCoefficient_inner [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
+    (w z : V) (hw : IsFixed a x₀ ρ w) (hz : IsFixed a x₀ ρ z) (v t : V) :
+    inner ℂ (normalizedCoefficient a x₀ ha ρ w v) (normalizedCoefficient a x₀ ha ρ z t) =
+      inner ℂ z w*inner ℂ v t := by
+  let : Nonempty X := ⟨x₀⟩
+  have hX : (Fintype.card X:ℂ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
+  have hs : (Real.sqrt ((Module.finrank ℂ V:ℝ)/(Fintype.card X:ℝ)) : ℂ)^2 =
+      (Module.finrank ℂ V:ℂ)/(Fintype.card X:ℂ) := by
+    rw [←Complex.ofReal_pow,Real.sq_sqrt (by positivity)]
+    simp
+  simp only [normalizedCoefficient,inner_smul_left,inner_smul_right]
+  simp only [Complex.conj_ofReal]
+  rw [←mul_assoc,←pow_two,hs]
+  have hi := coefficient_inner a x₀ ha ρ hρ w z hw hz v t
+  field_simp
+  simpa only [mul_assoc] using hi
+
+lemma normalizedCoefficient_orthonormal [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
+    {I : Type*} (w : I → V) (hw : Orthonormal ℂ w)
+    (hfix : ∀ i, IsFixed a x₀ ρ (w i)) (v : V) (hv : ‖v‖=1) :
+    Orthonormal ℂ (fun i => normalizedCoefficient a x₀ ha ρ (w i) v) := by
+  rw [orthonormal_iff_ite] at hw ⊢
+  intro i j
+  rw [normalizedCoefficient_inner a x₀ ha ρ hρ _ _ (hfix i) (hfix j),hw,
+    inner_self_eq_norm_sq_to_K,hv]
+  simp [eq_comm]
+
+lemma normalizedCoefficient_norm [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
+    (w : V) (hw : IsFixed a x₀ ρ w) (hn : ‖w‖=1) (v : V) :
+    ‖normalizedCoefficient a x₀ ha ρ w v‖=‖v‖ := by
+  have he := normalizedCoefficient_inner a x₀ ha ρ hρ w w hw hw v v
+  rw [inner_self_eq_norm_sq_to_K,inner_self_eq_norm_sq_to_K,hn] at he
+  simp only [inner_self_eq_norm_sq_to_K] at he
+  norm_num at he
+  have hs : ‖normalizedCoefficient a x₀ ha ρ w v‖^2=‖v‖^2 := by exact_mod_cast he
+  nlinarith [norm_nonneg (normalizedCoefficient a x₀ ha ρ w v),norm_nonneg v]
+
+end Thorp.UnitaryFinite
+
+namespace Thorp.FiniteConvolution
+open scoped BigOperators Classical
+variable {G : Type*} [Group G] [Fintype G]
+
+noncomputable def conv (p q : G → ℝ) (g : G) : ℝ := ∑ h, p h*q (h⁻¹*g)
+noncomputable def power (p : G → ℝ) : ℕ → G → ℝ
+  | 0 => fun g => if g=1 then 1 else 0
+  | n+1 => conv p (power p n)
+
+lemma conv_nonneg {p q : G → ℝ} (hp : ∀ g, 0 ≤ p g) (hq : ∀ g, 0 ≤ q g) :
+    ∀ g, 0 ≤ conv p q g := fun _ => Finset.sum_nonneg (fun h _ => mul_nonneg (hp h) (hq _))
+
+lemma conv_pos_of {p q : G → ℝ} (hp : ∀ g, 0 ≤ p g) (hq : ∀ g, 0 ≤ q g)
+    {a b : G} (ha : 0 < p a) (hb : 0 < q b) : 0 < conv p q (a*b) := by
+  apply lt_of_lt_of_le (mul_pos ha hb)
+  have h := Finset.single_le_sum (fun g _ => mul_nonneg (hp g) (hq (g⁻¹*(a*b))))
+    (Finset.mem_univ a)
+  simpa [conv] using h
+
+lemma sum_conv (p q : G → ℝ) : ∑ g, conv p q g=(∑ g,p g)*(∑ g,q g) := by
+  unfold conv
+  rw [Finset.sum_comm,Finset.sum_mul]
+  apply Finset.sum_congr rfl
+  intro h _
+  rw [←Finset.mul_sum]
+  congr 1
+  exact Fintype.sum_equiv (Equiv.mulLeft h⁻¹) _ _ (fun _ => rfl)
+
+lemma power_nonneg {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) : ∀ n g, 0 ≤ power p n g := by
+  intro n
+  induction n with
+  | zero => intro g; simp only [power]; split_ifs  <;> norm_num
+  | succ n ih => exact conv_nonneg hp ih
+
+lemma sum_power {p : G → ℝ} (hp : ∑ g,p g=1) : ∀ n, ∑ g,power p n g=1 := by
+  intro n
+  induction n with
+  | zero => simp [power]
+  | succ n ih => rw [power,sum_conv,hp,ih,one_mul]
+
+lemma power_list_pos {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) (l : List G)
+    (hl : ∀ g ∈ l, 0 < p g) : 0 < power p l.length l.prod := by
+  induction l with
+  | nil => simp [power]
+  | cons a l ih =>
+    simpa only [List.length_cons,List.prod_cons,power] using
+      conv_pos_of hp (power_nonneg hp l.length) (hl a (by simp))
+        (ih (fun g hg => hl g (by simp [hg])))
+
+lemma power_pos_mono {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) (h1 : 0 < p 1)
+    {n m : ℕ} (hnm : n ≤ m) {g : G} (hg : 0 < power p n g) : 0 < power p m g := by
+  induction m, hnm using Nat.le_induction with
+  | base => exact hg
+  | succ m _ ih =>
+    simpa only [one_mul,power] using conv_pos_of hp (power_nonneg hp m) h1 ih
+
+theorem exists_positive_power {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) (h1 : 0 < p 1)
+    (hgen : Subgroup.closure {g | 0 < p g}=⊤) :
+    ∃ n : ℕ, ∀ g, 0 < power p (2^n) g := by
+  have hx (g : G) : g ∈ Submonoid.closure {g | 0 < p g} := by
+    rw [←Subgroup.closure_toSubmonoid_of_finite,hgen]
+    trivial
+  choose l hl hprod using fun g => Submonoid.exists_list_of_mem_closure (hx g)
+  let n := Finset.univ.sup (fun g => (l g).length)
+  refine ⟨n,fun g => ?_⟩
+  have hlen : (l g).length ≤ n := Finset.le_sup (f := fun g => (l g).length) (Finset.mem_univ g)
+  have hpow : n ≤ 2^n := Nat.le_of_lt (Nat.lt_two_pow_self)
+  have hpos := power_list_pos hp (l g) (hl g)
+  rw [hprod g] at hpos
+  exact power_pos_mono hp h1 (hlen.trans hpow) hpos
+
+structure Minorant (p : G → ℝ) where
+  exponent : ℕ
+  mass : ℝ
+  mass_pos : 0 < mass
+  mass_le_one : mass ≤ 1
+  lower : ∀ g, mass/(Fintype.card G:ℝ) ≤ power p (2^exponent) g
+
+noncomputable def minorant {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) (hsum : ∑ g,p g=1)
+    (h1 : 0 < p 1) (hgen : Subgroup.closure {g | 0 < p g}=⊤) : Minorant p := by
+  let n := Classical.choose (exists_positive_power hp h1 hgen)
+  have hn := Classical.choose_spec (exists_positive_power hp h1 hgen)
+  let a := Finset.univ.inf' Finset.univ_nonempty (power p (2^n))
+  have ha : 0 < a := (Finset.lt_inf'_iff _).mpr (fun g _ => hn g)
+  have hag (g : G) : a ≤ power p (2^n) g := Finset.inf'_le _ (Finset.mem_univ g)
+  have hN : (0:ℝ) < Fintype.card G := Nat.cast_pos.mpr Fintype.card_pos
+  have hsum' := Finset.sum_le_sum (s := Finset.univ) (fun g _ => hag g)
+  simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,sum_power hsum] at hsum'
+  refine ⟨n,(Fintype.card G:ℝ)*a,mul_pos hN ha,hsum',fun g => ?_⟩
+  have he : (Fintype.card G:ℝ)*a/(Fintype.card G:ℝ)=a := by field_simp
+  rw [he]
+  exact hag g
+
+end Thorp.FiniteConvolution
+namespace Thorp.UnitaryFinite
+
+section
+open scoped BigOperators ComplexConjugate Classical
+
+variable {G : Type*} [Group G] [Fintype G]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+omit [Fintype G] in
+@[simp] lemma continuousRepresentation_apply (ρ : Representation ℂ G V) (g : G) (v : V) :
+    continuousRepresentation ρ g v=ρ g v := rfl
+
+omit [Fintype G] [FiniteDimensional ℂ V] in
+lemma unitary_norm (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (g : G) (v : V) :
+    ‖ρ g v‖=‖v‖ := by
+  rw [norm_eq_sqrt_re_inner (𝕜 := ℂ),hρ,norm_eq_sqrt_re_inner (𝕜 := ℂ)]
+
+omit [Fintype G] in
+lemma continuousRepresentation_norm_le (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (g : G) :
+    ‖continuousRepresentation ρ g‖≤1 := by
+  apply ContinuousLinearMap.opNorm_le_bound _ zero_le_one
+  intro v
+  simp only [continuousRepresentation_apply,unitary_norm ρ hρ,one_mul,le_refl]
+
+omit [Fintype G] in
+lemma continuousRepresentation_adjoint (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (g : G) :
+    (continuousRepresentation ρ g).adjoint=continuousRepresentation ρ g⁻¹ := by
+  ext w
+  apply ext_inner_left ℂ
+  intro v
+  rw [ContinuousLinearMap.adjoint_inner_right]
+  exact (unitary_inner_inv ρ hρ g v w).symm
+
+noncomputable def weightedOperator (ρ : Representation ℂ G V) (p : G → ℝ) : V →L[ℂ] V :=
+  ∑ g, (p g : ℂ) • continuousRepresentation ρ g
+
+lemma weightedOperator_norm_le (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
+    (p : G → ℝ) (hp : ∀ g, 0≤p g) : ‖weightedOperator ρ p‖ ≤ ∑ g, p g := by
+  apply (norm_sum_le _ _).trans
+  apply Finset.sum_le_sum
+  intro g _
+  rw [norm_smul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (hp g)]
+  exact mul_le_of_le_one_right (hp g) (continuousRepresentation_norm_le ρ hρ g)
+
+lemma weightedOperator_symmetric (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
+    (p : G → ℝ) (hp : ∀ g, p g⁻¹=p g) : IsSelfAdjoint (weightedOperator ρ p) := by
+  change (weightedOperator ρ p).adjoint=weightedOperator ρ p
+  unfold weightedOperator
+  rw [map_sum]
+  simp only [map_smulₛₗ,Complex.conj_ofReal,continuousRepresentation_adjoint ρ hρ]
+  apply Fintype.sum_equiv (Equiv.inv G)
+  intro g
+  simp [hp]
+
+noncomputable def uniformOperator (ρ : Representation ℂ G V) : V →L[ℂ] V :=
+  weightedOperator ρ (fun _ => (Fintype.card G:ℝ)⁻¹)
+
+lemma uniformOperator_symmetric (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) :
+    IsSelfAdjoint (uniformOperator ρ) := weightedOperator_symmetric ρ hρ _ (fun _ => rfl)
+
+lemma uniformOperator_left (ρ : Representation ℂ G V) (g : G) :
+    continuousRepresentation ρ g * uniformOperator ρ=uniformOperator ρ := by
+  unfold uniformOperator weightedOperator
+  rw [Finset.mul_sum]
+  simp only [mul_smul_comm,←map_mul]
+  exact Fintype.sum_equiv (Equiv.mulLeft g) _ _ (fun _ => rfl)
+
+lemma uniformOperator_right (ρ : Representation ℂ G V) (g : G) :
+    uniformOperator ρ * continuousRepresentation ρ g=uniformOperator ρ := by
+  unfold uniformOperator weightedOperator
+  rw [Finset.sum_mul]
+  simp only [smul_mul_assoc,←map_mul]
+  exact Fintype.sum_equiv (Equiv.mulRight g) _ _ (fun _ => rfl)
+
+lemma uniformOperator_idempotent (ρ : Representation ℂ G V) :
+    IsIdempotentElem (uniformOperator ρ) := by
+  change uniformOperator ρ * uniformOperator ρ=uniformOperator ρ
+  nth_rw 1 [uniformOperator,weightedOperator]
+  rw [Finset.sum_mul]
+  simp only [smul_mul_assoc,uniformOperator_left,Finset.sum_const,Finset.card_univ,
+    ←Nat.cast_smul_eq_nsmul ℂ,smul_smul,Complex.ofReal_inv,Complex.ofReal_natCast]
+  rw [mul_inv_cancel₀ (by exact_mod_cast Fintype.card_ne_zero),one_smul]
+
+lemma uniformOperator_projection (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) :
+    IsStarProjection (uniformOperator ρ) :=
+  ⟨uniformOperator_idempotent ρ,uniformOperator_symmetric ρ hρ⟩
+
+lemma weighted_uniform_left (ρ : Representation ℂ G V) (p : G → ℝ) :
+    weightedOperator ρ p * uniformOperator ρ=(∑ g,p g : ℝ) • uniformOperator ρ := by
+  unfold weightedOperator
+  rw [Finset.sum_mul]
+  simp only [smul_mul_assoc,uniformOperator_left,←Finset.sum_smul,←Complex.ofReal_sum]
+  rfl
+
+lemma weighted_uniform_right (ρ : Representation ℂ G V) (p : G → ℝ) :
+    uniformOperator ρ * weightedOperator ρ p=(∑ g,p g : ℝ) • uniformOperator ρ := by
+  unfold weightedOperator
+  rw [Finset.mul_sum]
+  simp only [mul_smul_comm,uniformOperator_right,←Finset.sum_smul,←Complex.ofReal_sum]
+  rfl
+
+theorem doeblin_off_invariants (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
+    (p : G → ℝ) (hp : ∑ g,p g=1) (ε : ℝ)
+    (hminor : ∀ g, ε/(Fintype.card G:ℝ)≤p g) :
+    ‖weightedOperator ρ p * (1-uniformOperator ρ)‖≤1-ε := by
+  let q : G → ℝ := fun g => p g-ε/(Fintype.card G:ℝ)
+  have hq : ∀ g, 0≤q g := fun g => sub_nonneg.mpr (hminor g)
+  have hcard : (Fintype.card G:ℝ)≠0 := by exact_mod_cast Fintype.card_ne_zero
+  have hsum : ∑ g,q g=1-ε := by
+    simp only [q,Finset.sum_sub_distrib,hp,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
+    field_simp
+  have he : weightedOperator ρ p=weightedOperator ρ q+(ε:ℂ) • uniformOperator ρ := by
+    simp only [uniformOperator,weightedOperator]
+    rw [Finset.smul_sum,←Finset.sum_add_distrib]
+    apply Finset.sum_congr rfl
+    intro g _
+    simp only [q,Complex.ofReal_sub,Complex.ofReal_div,smul_smul,←add_smul,
+      Complex.ofReal_inv,Complex.ofReal_natCast]
+    congr 1
+    field_simp
+    ring
+  have hz : uniformOperator ρ*(1-uniformOperator ρ)=0 := by
+    rw [mul_sub,mul_one,(uniformOperator_idempotent ρ).eq,sub_self]
+  rw [he,add_mul,smul_mul_assoc,hz,smul_zero,add_zero]
+  calc
+    _ ≤ ‖weightedOperator ρ q‖*‖1-uniformOperator ρ‖ := norm_mul_le _ _
+    _ ≤ (1-ε)*1 := mul_le_mul (by rw [←hsum]; exact weightedOperator_norm_le ρ hρ q hq)
+      ((uniformOperator_projection ρ hρ).one_sub.norm_le _) (norm_nonneg _)
+      (by rw [←hsum]; exact Finset.sum_nonneg (fun g _ => hq g))
+    _ = _ := mul_one _
+
+end
+open scoped BigOperators Classical
+variable {G : Type*} [Group G] [Fintype G]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+lemma weightedOperator_conv (ρ : Representation ℂ G V) (p q : G → ℝ) :
+    weightedOperator ρ (FiniteConvolution.conv p q)=weightedOperator ρ p*weightedOperator ρ q := by
+  unfold weightedOperator FiniteConvolution.conv
+  simp only [Complex.ofReal_sum,Finset.sum_smul]
+  rw [Finset.sum_comm,Finset.sum_mul]
+  apply Finset.sum_congr rfl
+  intro h _
+  rw [Finset.mul_sum]
+  simp only [smul_mul_assoc,mul_smul_comm,smul_smul,←map_mul]
+  apply Fintype.sum_equiv (Equiv.mulLeft h⁻¹)
+  intro g
+  simp [Complex.ofReal_mul,mul_comm]
+
+lemma weightedOperator_power (ρ : Representation ℂ G V) (p : G → ℝ) (n : ℕ) :
+    weightedOperator ρ (FiniteConvolution.power p n)=(weightedOperator ρ p)^n := by
+  induction n with
+  | zero => simp [FiniteConvolution.power,weightedOperator]
+  | succ n ih => rw [FiniteConvolution.power,weightedOperator_conv,ih,pow_succ']
+
+lemma off_invariants_pow (ρ : Representation ℂ G V) (p : G → ℝ)
+    (hp : ∑ g,p g=1) (n : ℕ) (hn : n≠0) :
+    (weightedOperator ρ p-uniformOperator ρ)^n=
+      (weightedOperator ρ p)^n*(1-uniformOperator ρ) := by
+  have hAP := weighted_uniform_left ρ p
+  have hPA := weighted_uniform_right ρ p
+  rw [hp,one_smul] at hAP hPA
+  have hc : Commute (weightedOperator ρ p) (1-uniformOperator ρ) := by
+    show _*_= _*_
+    rw [mul_sub,sub_mul,mul_one,one_mul,hAP,hPA]
+  have he : weightedOperator ρ p-uniformOperator ρ=
+      weightedOperator ρ p*(1-uniformOperator ρ) := by rw [mul_sub,mul_one,hAP]
+  rw [he,hc.mul_pow,((uniformOperator_idempotent ρ).one_sub).pow_eq hn]
+
+noncomputable def minorantGap {p : G → ℝ} (c : FiniteConvolution.Minorant p) : ℝ :=
+  1-c.mass/(2*(2:ℝ)^c.exponent)
+
+omit [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V] in
+lemma minorantGap_pos {p : G → ℝ} (c : FiniteConvolution.Minorant p) : 0 < minorantGap c := by
+  have hn : (1:ℝ) ≤ 2^c.exponent := one_le_pow₀ (by norm_num)
+  have hq : c.mass/(2*(2:ℝ)^c.exponent) ≤ 1/2 := by
+    apply (div_le_iff₀ (by positivity)).mpr
+    nlinarith [c.mass_le_one]
+  unfold minorantGap
+  linarith
+
+omit [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V] in
+lemma minorantGap_lt_one {p : G → ℝ} (c : FiniteConvolution.Minorant p) : minorantGap c < 1 := by
+  unfold minorantGap
+  exact sub_lt_self _ (div_pos c.mass_pos (by positivity))
+
+theorem norm_off_invariants_le_gap (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
+    (p : G → ℝ) (hp : ∑ g,p g=1) (hsym : ∀ g,p g⁻¹=p g)
+    (c : FiniteConvolution.Minorant p) :
+    ‖weightedOperator ρ p-uniformOperator ρ‖ ≤ minorantGap c := by
+  have hself := (weightedOperator_symmetric ρ hρ p hsym).sub (uniformOperator_symmetric ρ hρ)
+  have hbound := doeblin_off_invariants ρ hρ (FiniteConvolution.power p (2^c.exponent))
+    (FiniteConvolution.sum_power hp _) c.mass c.lower
+  rw [weightedOperator_power,←off_invariants_pow ρ p hp _ (by positivity),
+    hself.norm_pow_two_pow] at hbound
+  have hn : (1:ℝ) ≤ 2^c.exponent := one_le_pow₀ (by norm_num)
+  have hq : c.mass/(2*(2:ℝ)^c.exponent) ≤ 1/2 := by
+    apply (div_le_iff₀ (by positivity)).mpr
+    nlinarith [c.mass_le_one]
+  have hbern := one_add_mul_le_pow
+    (a := -(c.mass/(2*(2:ℝ)^c.exponent))) (by linarith : (-2:ℝ) ≤ -(c.mass/(2*(2:ℝ)^c.exponent)))
+    (2^c.exponent)
+  have he : 1+(2^c.exponent:ℕ)*(-(c.mass/(2*(2:ℝ)^c.exponent)))=1-c.mass/2 := by
+    push_cast
+    field_simp
+    ring
+  rw [he,show 1+ -(c.mass/(2*(2:ℝ)^c.exponent))=minorantGap c by rfl] at hbern
+  by_contra h
+  have hlt : minorantGap c < ‖weightedOperator ρ p-uniformOperator ρ‖ := lt_of_not_ge h
+  have hpows := pow_lt_pow_left₀ hlt (minorantGap_pos c).le (by positivity : (2^c.exponent:ℕ)≠0)
+  linarith [c.mass_pos]
+
+end Thorp.UnitaryFinite
+namespace Thorp
+open scoped BigOperators Classical
+
+def butterflyIdentity : (d : ℕ) → Butterfly d
+  | 0 => ()
+  | d+1 => (fun _ => false,fun _ => butterflyIdentity d)
+
+lemma pairSwitch_false (d : ℕ) : pairSwitch (d := d) (fun _ => false)=1 := by
+  apply Equiv.ext
+  intro x
+  change switchFun (fun _ => false) x=x
+  simp [switchFun]
+
+lemma childLift_one (d : ℕ) : childLift (d := d) (fun _ => 1)=1 := by
+  ext x i
+  simp [childLift]
+
+lemma butterflyIdentity_perm (d : ℕ) : butterflyPerm d (butterflyIdentity d)=1 := by
+  induction d with
+  | zero => rfl
+  | succ d ih => simp only [butterflyIdentity,butterflyPerm,ih,pairSwitch_false,childLift_one,mul_one]
+
+noncomputable def childInjection (d : ℕ) (b : Bool) : Equiv.Perm (Card d) →* Equiv.Perm (Card (d+1)) where
+  toFun p := childLift (fun c => if c=b then p else 1)
+  map_one' := by simpa only [ite_self] using childLift_one d
+  map_mul' p q := by
+    rw [childLift_mul]
+    congr 1
+    funext c
+    split_ifs <;> simp
+
+lemma childInjection_swap (d : ℕ) (b : Bool) (u v : Card d) :
+    childInjection d b (Equiv.swap u v)=Equiv.swap (Fin.cons b u) (Fin.cons b v) := by
+  apply Equiv.ext
+  intro x
+  obtain ⟨c,w,rfl⟩ : ∃ c w, x=Fin.cons c w := ⟨x 0,Fin.tail x,(Fin.cons_self_tail x).symm⟩
+  change (Fin.cons c ((if c=b then Equiv.swap u v else 1) w) : Card (d+1))=_
+  by_cases hc : c=b
+  · subst c
+    simp only [↓reduceIte]
+    rw [Equiv.swap_apply_def]
+    split_ifs with h₁ h₂
+    · subst w; simp
+    · subst w; simp
+    · rw [Equiv.swap_apply_of_ne_of_ne]
+      · exact fun h => h₁ (by simpa using congrArg Fin.tail h)
+      · exact fun h => h₂ (by simpa using congrArg Fin.tail h)
+  · rw [ite_eq_right hc,Equiv.Perm.one_apply,Equiv.swap_apply_of_ne_of_ne]
+    · intro h; exact hc (by simpa using congrArg (fun z => z 0) h)
+    · intro h; exact hc (by simpa using congrArg (fun z => z 0) h)
+
+lemma pairSwitch_single (d : ℕ) (u : Card d) :
+    pairSwitch (fun v => decide (v=u))=Equiv.swap (Fin.cons false u) (Fin.cons true u) := by
+  apply Equiv.ext
+  intro x
+  obtain ⟨c,w,rfl⟩ : ∃ c w, x=Fin.cons c w := ⟨x 0,Fin.tail x,(Fin.cons_self_tail x).symm⟩
+  change switchFun (fun v => decide (v=u)) (Fin.cons c w)=_
+  by_cases h : w=u
+  · subst w
+    cases c <;> simp [switchFun]
+  · have ha : (Fin.cons c w : Card (d+1))≠Fin.cons false u := fun he => h (by simpa using congrArg Fin.tail he)
+    have hb : (Fin.cons c w : Card (d+1))≠Fin.cons true u := fun he => h (by simpa using congrArg Fin.tail he)
+    simp [switchFun,h,Equiv.swap_apply_of_ne_of_ne ha hb]
+
+theorem butterfly_generates (d : ℕ) (H : Subgroup (Equiv.Perm (Card d)))
+    (hH : ∀ ω : Butterfly d, butterflyPerm d ω∈H) : H=⊤ := by
+  induction d with
+  | zero =>
+    apply top_unique
+    intro p _
+    have hp : p=1 := Subsingleton.elim _ _
+    rw [hp]
+    exact H.one_mem
+  | succ d ih =>
+    have hchild (b : Bool) (p : Equiv.Perm (Card d)) : childInjection d b p∈H := by
+      have hc : H.comap (childInjection d b)=⊤ := by
+        apply ih
+        intro ω
+        change childLift (fun c => if c=b then butterflyPerm d ω else 1)∈H
+        have h := hH ((fun _ => false),fun c => if c=b then ω else butterflyIdentity d)
+        simpa only [butterflyPerm,pairSwitch_false,one_mul,apply_ite,butterflyIdentity_perm] using h
+      have hp : p∈H.comap (childInjection d b) := by rw [hc]; trivial
+      exact hp
+    have hpair (u : Card d) : Equiv.swap (Fin.cons false u) (Fin.cons true u)∈H := by
+      rw [←pairSwitch_single]
+      have h := hH ((fun v => decide (v=u)),fun _ => butterflyIdentity d)
+      simpa only [butterflyPerm,butterflyIdentity_perm,childLift_one,mul_one] using h
+    have hswap (x y : Card (d+1)) : Equiv.swap x y∈H := by
+      have ht (a b : Bool) (u : Card d) : Equiv.swap (Fin.cons a u) (Fin.cons b u)∈H := by
+        cases a <;> cases b
+        · rw [Equiv.swap_self]; exact H.one_mem
+        · exact hpair _
+        · rw [Equiv.swap_comm]; exact hpair _
+        · rw [Equiv.swap_self]; exact H.one_mem
+      have h₁ : Equiv.swap x (Fin.cons (y 0) (Fin.tail x))∈H := by
+        convert ht (x 0) (y 0) (Fin.tail x) using 1
+        rw [Fin.cons_self_tail]
+      have h₂ := hchild (y 0) (Equiv.swap (Fin.tail x) (Fin.tail y))
+      rw [childInjection_swap,Fin.cons_self_tail] at h₂
+      exact SubmonoidClass.swap_mem_trans H h₁ h₂
+    rw [eq_top_iff,←Equiv.Perm.closure_isSwap,Subgroup.closure_le]
+    rintro _ ⟨x,y,_,rfl⟩
+    exact hswap x y
+
+lemma butterfly_closure (d : ℕ) : Subgroup.closure (Set.range (butterflyPerm d))=⊤ :=
+  butterfly_generates d _ (fun ω => Subgroup.subset_closure ⟨ω,rfl⟩)
+
+variable {Ω G : Type*} [Fintype Ω] [Fintype G]
+
+noncomputable def sampleLaw (P : Ω → G) (g : G) : ℝ :=
+  finiteMean (fun ω => if P ω=g then 1 else 0)
+
+omit [Fintype G] in
+lemma sampleLaw_nonneg (P : Ω → G) (g : G) : 0 ≤ sampleLaw P g :=
+  finiteMean_nonneg (fun _ => by split_ifs <;> norm_num)
+
+lemma sampleLaw_sum [Nonempty Ω] (P : Ω → G) : ∑ g,sampleLaw P g=1 := by
+  unfold sampleLaw
+  rw [←finiteMean_sum]
+  simp only [Finset.sum_ite_eq,Finset.mem_univ,↓reduceIte,finiteMean_const]
+
+omit [Fintype G] in
+lemma sampleLaw_pos_iff [Nonempty Ω] (P : Ω → G) (g : G) :
+    0<sampleLaw P g ↔ ∃ ω,P ω=g := by
+  unfold sampleLaw finiteMean
+  rw [div_pos_iff_of_pos_right (Nat.cast_pos.mpr Fintype.card_pos)]
+  rw [Finset.sum_pos_iff_of_nonneg (fun _ _ => by split_ifs <;> norm_num)]
+  simp only [Finset.mem_univ,true_and]
+  apply exists_congr
+  intro ω
+  split_ifs <;> simp_all
+
+omit [Fintype G] in
+lemma sampleLaw_equiv {Ω' : Type*} [Fintype Ω'] (P : Ω → G) (e : Ω' ≃ Ω) :
+    sampleLaw (P ∘ e)=sampleLaw P := by
+  funext g
+  simpa only [sampleLaw,Function.comp_apply,Equiv.apply_symm_apply] using
+    finiteMean_equiv e (fun ω => if P (e ω)=g then (1:ℝ) else 0)
+
+variable [Group G]
+omit [Fintype G] in
+lemma sampleLaw_inverse (P : Ω → G) (g : G) :
+    sampleLaw (fun ω => (P ω)⁻¹) g=sampleLaw P g⁻¹ := by
+  apply finiteMean_congr
+  intro ω
+  rw [inv_eq_iff_eq_inv]
+
+omit [Fintype G] in
+lemma sampleLaw_symmetric (P : Ω → G) (e : Equiv.Perm Ω)
+    (he : ∀ ω,P (e ω)=(P ω)⁻¹) (g : G) : sampleLaw P g⁻¹=sampleLaw P g := by
+  rw [←sampleLaw_inverse]
+  have hh : (fun ω => (P ω)⁻¹)=P ∘ e := funext (fun ω => (he ω).symm)
+  rw [hh,sampleLaw_equiv]
+
+end Thorp
+
+namespace Thorp
+open scoped BigOperators Classical
+
+lemma palindrome_identity (d : ℕ) : ∃ ω,palindromePerm d ω=1 := by
+  refine ⟨((butterflyCoinEquiv d).symm (butterflyIdentity d),
+    (butterflyCoinEquiv d).symm (butterflyIdentity d)),?_⟩
+  exact mul_inv_cancel _
+
+lemma palindrome_range_contains_butterfly (d : ℕ) (ω : Butterfly d) :
+    ∃ ξ,palindromePerm d ξ=butterflyPerm d ω := by
+  refine ⟨((butterflyCoinEquiv d).symm ω,(butterflyCoinEquiv d).symm (butterflyIdentity d)),?_⟩
+  unfold palindromePerm
+  have h (β : Butterfly d) : decodeButterfly d ((butterflyCoinEquiv d).symm β)=β :=
+    (butterflyCoinEquiv d).apply_symm_apply β
+  rw [h,h,butterflyIdentity_perm,show (1 : Equiv.Perm (Card d)).symm=1 from rfl,mul_one]
+
+lemma palindrome_support_generates (d : ℕ) :
+    Subgroup.closure {g | 0<sampleLaw (palindromePerm d) g}=⊤ := by
+  apply butterfly_generates
+  intro ω
+  apply Subgroup.subset_closure
+  exact (sampleLaw_pos_iff _ _).mpr (palindrome_range_contains_butterfly d ω)
+
+noncomputable def palindromeMinorant (d : ℕ) :
+    FiniteConvolution.Minorant (sampleLaw (palindromePerm d)) :=
+  FiniteConvolution.minorant (sampleLaw_nonneg _) (sampleLaw_sum _)
+    ((sampleLaw_pos_iff _ _).mpr (palindrome_identity d)) (palindrome_support_generates d)
+
+noncomputable def blockGap (d : ℕ) : ℝ := UnitaryFinite.minorantGap (palindromeMinorant d)
+
+lemma blockGap_pos (d : ℕ) : 0<blockGap d := UnitaryFinite.minorantGap_pos _
+lemma blockGap_lt_one (d : ℕ) : blockGap d<1 := UnitaryFinite.minorantGap_lt_one _
+
+theorem palindrome_off_invariants_gap (d : ℕ) {V : Type*}
+    [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+    (ρ : Representation ℂ (Equiv.Perm (Card d)) V) (hρ : UnitaryFinite.IsUnitary ρ) :
+    ‖UnitaryFinite.weightedOperator ρ (sampleLaw (palindromePerm d))-
+      UnitaryFinite.uniformOperator ρ‖≤blockGap d := by
+  apply UnitaryFinite.norm_off_invariants_le_gap ρ hρ _ (sampleLaw_sum _)
+  exact sampleLaw_symmetric _ (Equiv.prodComm _ _) (palindromePerm_reverse d)
+
+end Thorp
+
+namespace Thorp.CommutingBlocks
+open scoped BigOperators Classical
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
+variable {ι : Type*} [Fintype ι]
+
+omit [CompleteSpace V] in
+lemma orthogonal_sum_norm_sq (v : ι → V) (hv : ∀ i j, i≠j → inner ℂ (v i) (v j)=0) :
+    ‖∑ i,v i‖^2=∑ i,‖v i‖^2 := by
+  rw [norm_sq_eq_re_inner (𝕜 := ℂ),sum_inner]
+  simp only [inner_sum,map_sum]
+  apply Finset.sum_congr rfl
+  intro i _
+  rw [Finset.sum_eq_single i]
+  · exact (norm_sq_eq_re_inner (𝕜 := ℂ) (v i)).symm
+  · intro j _ hji
+    rw [hv i j hji.symm]
+    rfl
+  · simp
+
+lemma projection_inner_zero (P Q : V →L[ℂ] V) (hP : IsStarProjection P)
+    (hPQ : P*Q=0) (v w : V) : inner ℂ (P v) (Q w)=0 := by
+  rw [←ContinuousLinearMap.adjoint_inner_right]
+  change inner ℂ v ((star P*Q) w)=0
+  rw [hP.isSelfAdjoint.star_eq,hPQ]
+  simp
+
+lemma projection_sandwich_norm (P B : V →L[ℂ] V) (hP : IsStarProjection P) :
+    ‖B.adjoint*P*B‖=‖P*B‖^2 := by
+  have he : star (P*B)*(P*B)=B.adjoint*P*B := by
+    rw [star_mul,hP.isSelfAdjoint.star_eq]
+    change B.adjoint*P*(P*B)=_
+    rw [←mul_assoc,mul_assoc B.adjoint P P,hP.isIdempotentElem.eq]
+  rw [←he,CStarRing.norm_star_mul_self]
+  ring
+
+theorem resolution_norm_bound (P : ι → V →L[ℂ] V)
+    (hP : ∀ i, IsStarProjection (P i))
+    (horth : ∀ i j, i≠j → P i*P j=0) (hsum : ∑ i,P i=1)
+    (A : V →L[ℂ] V) (hcomm : ∀ i, Commute A (P i))
+    (c : ℝ) (hc : 0≤c) (hbound : ∀ i, ‖A*P i‖≤c) : ‖A‖≤c := by
+  apply ContinuousLinearMap.opNorm_le_bound _ hc
+  intro v
+  have hsplit (w : V) : ∑ i,P i w=w := by
+    rw [←sum_apply,hsum]
+    rfl
+  have hp := orthogonal_sum_norm_sq (fun i => P i v)
+    (fun i j hij => projection_inner_zero _ _ (hP i) (horth i j hij) v v)
+  rw [hsplit] at hp
+  have ha := orthogonal_sum_norm_sq (fun i => P i (A v))
+    (fun i j hij => projection_inner_zero _ _ (hP i) (horth i j hij) (A v) (A v))
+  rw [hsplit] at ha
+  have he (i : ι) : P i (A v)=(A*P i) (P i v) := by
+    change (P i*A) v=(A*P i*P i) v
+    rw [mul_assoc,(hP i).isIdempotentElem.eq,(hcomm i).eq]
+  have hh (i : ι) : ‖P i (A v)‖≤c*‖P i v‖ := by
+    rw [he]
+    exact ((A*P i).le_opNorm _).trans (mul_le_mul_of_nonneg_right (hbound i) (norm_nonneg _))
+  have hsq := Finset.sum_le_sum (s := Finset.univ) (fun i _ =>
+    pow_le_pow_left₀ (norm_nonneg _) (hh i) 2)
+  simp_rw [mul_pow] at hsq
+  rw [←Finset.mul_sum,←hp,←ha,←mul_pow] at hsq
+  exact (sq_le_sq₀ (norm_nonneg _) (mul_nonneg hc (norm_nonneg _))).mp hsq
+
+theorem sector_sandwich_le (T P A B : V →L[ℂ] V)
+    (hT : IsStarProjection T) (hP : IsStarProjection P)
+    (hTP : T*P=T) (hcomm : Commute A T) (hA : ‖A‖≤1) :
+    ‖B.adjoint*A*T*B‖≤‖B.adjoint*P*B‖ := by
+  have he : B.adjoint*A*T*B=(T*B).adjoint*A*(T*B) := by
+    change _=star (T*B)*A*(T*B)
+    rw [star_mul,hT.isSelfAdjoint.star_eq]
+    change _=B.adjoint*T*A*(T*B)
+    rw [mul_assoc B.adjoint T A,←hcomm.eq]
+    simp only [mul_assoc]
+    rw [←mul_assoc T T B,hT.isIdempotentElem.eq]
+  have hTB : ‖T*B‖≤‖P*B‖ := by
+    have he : T*B=T*(P*B) := by rw [←mul_assoc,hTP]
+    rw [he]
+    exact (norm_mul_le _ _).trans ((mul_le_mul_of_nonneg_right (hT.norm_le _) (norm_nonneg _)).trans_eq (one_mul _))
+  rw [he,projection_sandwich_norm P B hP]
+  calc
+    _ ≤ ‖(T*B).adjoint‖*‖A‖*‖T*B‖ := (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right (norm_mul_le _ _) (norm_nonneg _))
+    _ ≤ ‖T*B‖^2 := by rw [ContinuousLinearMap.adjoint.norm_map]; nlinarith [norm_nonneg (T*B)]
+    _ ≤ _ := pow_le_pow_left₀ (norm_nonneg _) hTB 2
+
+omit [CompleteSpace V] [Fintype ι] in
+lemma resolution_commute (P : ι → V →L[ℂ] V)
+    (horth : ∀ i j, i≠j → P i*P j=0) (i j : ι) : Commute (P i) (P j) := by
+  by_cases hij : i=j
+  · subst j; exact Commute.refl _
+  · exact (horth i j hij).trans (horth j i (Ne.symm hij)).symm
+
+omit [Fintype ι] in
+lemma resolution_sum_mul (P : ι → V →L[ℂ] V) (hP : ∀ i, IsStarProjection (P i))
+    (horth : ∀ i j, i≠j → P i*P j=0) (S : Finset ι) (j : ι) :
+    (∑ i∈S,P i)*P j=if j∈S then P j else 0 := by
+  rw [Finset.sum_mul]
+  have he (i : ι) : P i*P j=if i=j then P j else 0 := by
+    split_ifs with h
+    · subst i; exact (hP j).isIdempotentElem.eq
+    · exact horth i j h
+  simp only [he,Finset.sum_ite_eq']
+
+lemma resolution_truncated_norm (P : ι → V →L[ℂ] V)
+    (hP : ∀ i, IsStarProjection (P i))
+    (horth : ∀ i j, i≠j → P i*P j=0) (hsum : ∑ i,P i=1)
+    (A : V →L[ℂ] V) (hcomm : ∀ i, Commute A (P i))
+    (S : Finset ι) (c : ℝ) (hc : 0≤c) (hbound : ∀ i∈S, ‖A*P i‖≤c) :
+    ‖A*(∑ i∈S,P i)‖≤c := by
+  apply resolution_norm_bound P hP horth hsum _ _ c hc
+  · intro i
+    rw [mul_assoc,resolution_sum_mul P hP horth]
+    split_ifs with hi
+    · exact hbound i hi
+    · simp [hc]
+  · intro i
+    exact (hcomm i).mul_left (Commute.sum_left S P (P i) fun j _ => resolution_commute P horth j i)
+
+theorem resolution_exception_bound (T Q : ι → V →L[ℂ] V)
+    (hT : ∀ i, IsStarProjection (T i))
+    (horth : ∀ i j, i≠j → T i*T j=0) (hsum : ∑ i,T i=1)
+    (hQ : ∀ i, IsStarProjection (Q i)) (hTQ : ∀ i,T i*Q i=T i)
+    (A B : V →L[ℂ] V) (hcomm : ∀ i, Commute A (T i))
+    (hA : ‖A‖≤1) (hB : ‖B‖≤1)
+    (S : Finset ι) (c : ℝ) (hc : 0≤c) (hbound : ∀ i∉S, ‖A*T i‖≤c) :
+    ‖B.adjoint*A*B‖≤c+∑ i∈S,‖B.adjoint*Q i*B‖ := by
+  have hhigh : ‖A*(∑ i∈Sᶜ,T i)‖≤c := resolution_truncated_norm T hT horth hsum A hcomm
+    Sᶜ c hc (fun i hi => hbound i (Finset.mem_compl.mp hi))
+  have hsplit : (∑ i∈Sᶜ,T i)+(∑ i∈S,T i)=1 := by
+    rw [Finset.sum_compl_add_sum,hsum]
+  have he : B.adjoint*A*B=B.adjoint*(A*(∑ i∈Sᶜ,T i))*B+
+      ∑ i∈S,B.adjoint*A*T i*B := by
+    rw [←Finset.sum_mul,←Finset.mul_sum]
+    rw [←add_mul]
+    simp only [mul_assoc]
+    rw [←mul_add,←mul_add,hsplit]
+    simp only [mul_one,mul_assoc]
+  rw [he]
+  apply (norm_add_le _ _).trans
+  apply add_le_add
+  · calc
+      _ ≤ ‖B.adjoint‖*‖A*(∑ i∈Sᶜ,T i)‖*‖B‖ :=
+        (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right (norm_mul_le _ _) (norm_nonneg _))
+      _ ≤ 1*c*1 := mul_le_mul (mul_le_mul (by simpa only [ContinuousLinearMap.adjoint.norm_map] using hB)
+        hhigh (norm_nonneg _) zero_le_one) hB (norm_nonneg _) (by simpa using hc)
+      _ = c := by ring
+  · apply (norm_sum_le _ _).trans
+    exact Finset.sum_le_sum fun i _ => sector_sandwich_le _ _ _ _ (hT i) (hQ i) (hTQ i) (hcomm i) hA
+
+variable [DecidableEq ι]
+variable (P : ι → V →L[ℂ] V) (hcomm : ∀ i j, Commute (P i) (P j))
+
+noncomputable def choice (E : Finset ι) (i : ι) : V →L[ℂ] V :=
+  if i∈E then 1-P i else P i
+
+include hcomm
+omit [CompleteSpace V] [Fintype ι] in
+lemma choice_commute (E F : Finset ι) (i j : ι) :
+    Commute (choice P E i) (choice P F j) := by
+  unfold choice
+  split_ifs
+  · exact (Commute.one_left _).sub_left ((Commute.one_right _).sub_right (hcomm i j))
+  · exact (Commute.one_left _).sub_left (hcomm i j)
+  · exact (Commute.one_right _).sub_right (hcomm i j)
+  · exact hcomm i j
+
+noncomputable def sector (s E : Finset ι) : V →L[ℂ] V :=
+  s.noncommProd (choice P E) (fun i _ j _ _ => choice_commute P hcomm E E i j)
+
+omit [CompleteSpace V] [Fintype ι] in
+lemma sector_insert (s E : Finset ι) (a : ι) (ha : a∉s) :
+    sector P hcomm (insert a s) E=choice P E a*sector P hcomm s E := by
+  exact Finset.noncommProd_insert_of_notMem _ _ _ _ ha
+
+omit [CompleteSpace V] [Fintype ι] in
+lemma sector_congr (s E F : Finset ι) (h : ∀ i∈s, (i∈E ↔ i∈F)) :
+    sector P hcomm s E=sector P hcomm s F := by
+  apply Finset.noncommProd_congr rfl
+  intro i hi
+  simp only [choice,h i hi]
+
+omit [CompleteSpace V] [Fintype ι] in
+lemma sector_factor_commute (s E F : Finset ι) (i : ι) :
+    Commute (choice P E i) (sector P hcomm s F) :=
+  Finset.noncommProd_commute _ _ _ _ (fun j _ => choice_commute P hcomm E F i j)
+
+omit [Fintype ι] in
+lemma sector_projection (hP : ∀ i, IsStarProjection (P i)) (s E : Finset ι) :
+    IsStarProjection (sector P hcomm s E) := by
+  induction s using Finset.induction_on with
+  | empty => exact IsStarProjection.one _
+  | @insert a s ha ih =>
+    rw [sector_insert P hcomm s E a ha]
+    have hc : IsStarProjection (choice P E a) := by
+      unfold choice
+      split_ifs
+      · exact (hP a).one_sub
+      · exact hP a
+    exact hc.mul ih (sector_factor_commute P hcomm s E E a)
+
+end Thorp.CommutingBlocks
+
+end ThorpNine.Tail
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Tail/RotationMeeting.lean b/lean/OAI/Probability/ThorpRouting/Tail/RotationMeeting.lean
index 89c647041ba230346896ee180ea4ea8c8e31cf6b..e9f2de5b6b1d7762caff841bf2a89c050f4ae057 100644
--- a/lean/OAI/Probability/ThorpRouting/Tail/RotationMeeting.lean
+++ b/lean/OAI/Probability/ThorpRouting/Tail/RotationMeeting.lean
@@ -1,881 +1,881 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Tail.ExposedCoins
-
-namespace OAI
-
-namespace ThorpNine.Tail
-
-namespace Thorp
-open scoped BigOperators
-open Filter
-section Rotation
-variable {d A : ℕ}
-
-noncomputable def rotationMeeting (o : Card d → Option (Fin A)) (z : Terminal o)
-    (u v : Card d) : ℝ :=
-  match o u, o v with
-  | some i, some j => if i < j then (if (z j).val = v then 1 else 0)
-      else if j < i then (if (z i).val = u then 1 else 0)
-      else (if (z i).val = u then 1 else 0) + (if (z i).val = v then 1 else 0)
-  | _, _ => 0
-
-lemma card_cycleFiber (o : Card d → Option (Fin A)) (i : Fin A) :
-    Fintype.card {x : Card d // o x = some i} = (cycleFiber o i).card := by
-  classical
-  simp [cycleFiber,Fintype.card_subtype]
-
-lemma terminal_mean {o : Card d → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
-    (i : Fin A) (x : Card d) (hx : o x = some i) :
-    finiteMean (fun z : Terminal o => if (z i).val = x then (1:ℝ) else 0) = slotWeight o i := by
-  classical
-  have hev := finiteMean_pi_eval (A := fun i : Fin A => {x : Card d // o x = some i})
-    i (fun z => if z.val = x then (1:ℝ) else 0)
-  rw [hev]
-  have he : (fun z : {x : Card d // o x = some i} => if z.val = x then (1:ℝ) else 0) =
-      (fun z => if z = ⟨x,hx⟩ then 1 else 0) := by
-    funext z
-    simp only [Subtype.ext_iff]
-  rw [he,finiteMean_eq_indicator,card_cycleFiber,slotWeight]
-
-lemma slotWeight_antitone (o : Card d → Option (Fin A))
-    (hn : ∀ i, (cycleFiber o i).Nonempty)
-    (hm : Monotone (fun i => (cycleFiber o i).card)) : Antitone (slotWeight o) := by
-  intro i j hij
-  unfold slotWeight
-  apply one_div_le_one_div_of_le
-  · exact_mod_cast Finset.card_pos.mpr (hn i)
-  · exact_mod_cast hm hij
-
-lemma rotationMeeting_mean {o : Card d → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
-    (hm : Monotone (fun i => (cycleFiber o i).card)) (u v : Card d) :
-    finiteMean (fun z : Terminal o => rotationMeeting o z u v) =
-      min (pathWeight o u) (pathWeight o v) + withinWeight o u v := by
-  classical
-  have hn : ∀ i, (cycleFiber o i).Nonempty := by
-    intro i
-    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : Card d // o x = some i})
-    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
-  have hw := slotWeight_antitone o hn hm
-  cases hu : o u with
-  | none =>
-    cases hv : o v with
-    | none => simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean]
-    | some j =>
-      have hj : 0 ≤ slotWeight o j := by unfold slotWeight; positivity
-      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_left hj]
-  | some i =>
-    cases hv : o v with
-    | none =>
-      have hi : 0 ≤ slotWeight o i := by unfold slotWeight; positivity
-      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_right hi]
-    | some j =>
-      simp only [rotationMeeting,pathWeight,withinWeight,hu,hv]
-      rcases lt_trichotomy i j with hij | rfl | hji
-      · simp only [hij,↓reduceIte,ne_of_lt hij]
-        rw [terminal_mean j v hv,min_eq_right (hw (le_of_lt hij)),add_zero]
-      · simp only [lt_self_iff_false,↓reduceIte,min_self]
-        rw [finiteMean_add,terminal_mean i u hu,terminal_mean i v hv]
-      · simp only [not_lt_of_ge (le_of_lt hji),hji,↓reduceIte,ne_of_gt hji]
-        rw [terminal_mean i u hu,min_eq_left (hw (le_of_lt hji)),add_zero]
-
-lemma pathWeight_sum (o : Card d → Option (Fin A)) (hn : ∀ i, (cycleFiber o i).Nonempty) :
-    ∑ x : Card d, pathWeight o x = (A : ℝ) := by
-  classical
-  have he (x : Card d) : pathWeight o x =
-      ∑ i : Fin A, if o x = some i then slotWeight o i else 0 := by
-    cases hx : o x with
-    | none => simp [pathWeight,hx]
-    | some j => simp [pathWeight,hx]
-  simp only [he]
-  rw [Finset.sum_comm]
-  have hf (i : Fin A) : (∑ x : Card d, if o x = some i then slotWeight o i else 0) = 1 := by
-    rw [← Finset.sum_filter]
-    change (∑ _x ∈ cycleFiber o i, slotWeight o i) = 1
-    simp only [Finset.sum_const,nsmul_eq_mul,slotWeight]
-    exact mul_one_div_cancel (by exact_mod_cast (Finset.card_pos.mpr (hn i)).ne')
-  simp only [hf,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,mul_one]
-
-lemma withinWeight_sum_bound (o : Card d → Option (Fin A))
-    (hn : ∀ i, (cycleFiber o i).Nonempty) (b : Butterfly d) :
-    (∑ s : SwitchIndex d, withinWeight o (switchUsers d b s).1 (switchUsers d b s).2) ≤
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
-  classical
-  have he (u v : Card d) : withinWeight o u v =
-      ∑ i : Fin A, (if u ∈ cycleFiber o i ∧ v ∈ cycleFiber o i then (1:ℝ) else 0) * slotWeight o i := by
-    simp only [mem_cycleFiber]
-    cases hu : o u <;> cases hv : o v <;> simp [withinWeight,hu,hv,ite_and,eq_comm]
-  simp only [he]
-  rw [Finset.sum_comm]
-  apply Finset.sum_le_sum
-  intro i _
-  rw [← Finset.sum_mul]
-  have hi := marked_switch_entropy_bound d b (cycleFiber o i)
-  have hpos : (0:ℝ) < (cycleFiber o i).card := by exact_mod_cast Finset.card_pos.mpr (hn i)
-  have hmul := mul_le_mul_of_nonneg_right hi (show 0 ≤ slotWeight o i by unfold slotWeight; positivity)
-  calc
-    _ ≤ ((cycleFiber o i).card : ℝ) * Real.log (cycleFiber o i).card /
-        (2 * Real.log 2) * slotWeight o i := hmul
-    _ = _ := by unfold slotWeight; field_simp
-
-noncomputable def rotationCharge (o : Card d → Option (Fin A)) (b : Butterfly d)
-    (z : Terminal o) : ℝ :=
-  ∑ s : SwitchIndex d, rotationMeeting o z (switchUsers d b s).1 (switchUsers d b s).2
-
-lemma rotationCharge_mean_bound {o : Card d → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
-    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : Butterfly d) :
-    finiteMean (rotationCharge o b) ≤ (A:ℝ) * d / 2 +
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
-  classical
-  have hn : ∀ i, (cycleFiber o i).Nonempty := by
-    intro i
-    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : Card d // o x = some i})
-    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
-  unfold rotationCharge
-  rw [finiteMean_sum]
-  simp only [rotationMeeting_mean hm,Finset.sum_add_distrib]
-  have h₁ := weighted_switch_min_bound d b (pathWeight o)
-  rw [pathWeight_sum o hn] at h₁
-  have h₂ := withinWeight_sum_bound o hn b
-  linarith
-
-lemma rotation_exp_bound {o : Card d → Option (Fin A)}
-    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
-    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : Butterfly d) :
-    Real.exp (-Real.log 2 * ((A:ℝ)*d/2 +
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2))) ≤
-      finiteMean (fun z : Terminal o => Real.exp (-Real.log 2 * rotationCharge o b z)) := by
-  have hb := rotationCharge_mean_bound hm b
-  have hlog : 0 ≤ Real.log 2 := Real.log_nonneg (by norm_num)
-  have hm' : -Real.log 2 * ((A:ℝ)*d/2 +
-      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2)) ≤
-      finiteMean (fun z : Terminal o => -Real.log 2 * rotationCharge o b z) := by
-    have he := finiteMean_mul_const (rotationCharge o b) (-Real.log 2)
-    simp only [mul_comm] at he
-    rw [he]
-    exact mul_le_mul_of_nonpos_left hb (neg_nonpos.mpr hlog)
-  exact (Real.exp_le_exp.mpr hm').trans (exp_finiteMean_le _)
-
-end Rotation
-
-section OrbitCore
-variable {α : Type*} [Fintype α] [DecidableEq α]
-
-noncomputable def orbitSet (p : Equiv.Perm α) (x : α) : Finset α :=
-  Finset.univ.filter (p.SameCycle x)
-
-@[simp] lemma mem_orbitSet (p : Equiv.Perm α) (x y : α) :
-    y ∈ orbitSet p x ↔ p.SameCycle x y := by
-  classical
-  simp [orbitSet]
-
-lemma orbitSet_nonempty (p : Equiv.Perm α) (x : α) : (orbitSet p x).Nonempty :=
-  ⟨x,(mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.refl _ _)⟩
-
-lemma orbitSet_eq (p : Equiv.Perm α) {x y : α} (h : p.SameCycle x y) :
-    orbitSet p x = orbitSet p y := by
-  ext z
-  simp only [mem_orbitSet]
-  exact ⟨fun hz => h.symm.trans hz,fun hz => h.trans hz⟩
-
-@[simp] lemma orbitSet_apply (p : Equiv.Perm α) (x : α) : orbitSet p (p x) = orbitSet p x := by
-  ext y
-  simp only [mem_orbitSet,Equiv.Perm.sameCycle_apply_left]
-
-lemma orbitSet_eq_or_disjoint (p : Equiv.Perm α) (x y : α) :
-    orbitSet p x = orbitSet p y ∨ Disjoint (orbitSet p x) (orbitSet p y) := by
-  classical
-  by_cases h : p.SameCycle x y
-  · exact Or.inl (orbitSet_eq p h)
-  · right
-    apply Finset.disjoint_left.mpr
-    intro z hz hz'
-    exact h (((mem_orbitSet _ _ _).mp hz).trans ((mem_orbitSet _ _ _).mp hz').symm)
-
-omit [DecidableEq α] in
-lemma perm_minimalPeriod_pos (p : Equiv.Perm α) (x : α) :
-    0 < Function.minimalPeriod p x :=
-  Function.minimalPeriod_pos_of_mem_periodicPts (p.injective.mem_periodicPts x)
-
-lemma orbitSet_card (p : Equiv.Perm α) (x : α) :
-    (orbitSet p x).card = Function.minimalPeriod p x := by
-  classical
-  let f : Fin (Function.minimalPeriod p x) → {y // p.SameCycle x y} := fun n =>
-    ⟨p^[n.val] x,by rw [Equiv.Perm.iterate_eq_pow]; exact ⟨(n.val:ℤ),by simp⟩⟩
-  have hf : Function.Bijective f := by
-    constructor
-    · intro i j hij
-      apply Fin.ext
-      exact Function.iterate_injOn_Iio_minimalPeriod i.isLt j.isLt (congrArg Subtype.val hij)
-    · intro y
-      obtain ⟨n,hn⟩ := y.property.exists_nat_pow_eq
-      refine ⟨⟨n % Function.minimalPeriod p x, Nat.mod_lt _ (perm_minimalPeriod_pos p x)⟩,?_⟩
-      apply Subtype.ext
-      change p^[n % Function.minimalPeriod p x] x = y.val
-      rw [Function.iterate_mod_minimalPeriod_eq,Equiv.Perm.iterate_eq_pow,hn]
-  have hc := Fintype.card_congr (Equiv.ofBijective f hf)
-  simpa [orbitSet,Fintype.card_subtype] using hc.symm
-
-def walkSet (p : Equiv.Perm α) (r : ℕ) (x : α) : Finset α :=
-  (Finset.range r).image (fun n => p^[n] x)
-
-omit [Fintype α] in
-@[simp] lemma mem_walkSet (p : Equiv.Perm α) (r : ℕ) (x y : α) :
-    y ∈ walkSet p r x ↔ ∃ n < r, p^[n] x = y := by
-  simp [walkSet]
-
-omit [Fintype α] in
-@[simp] lemma walkSet_zero (p : Equiv.Perm α) (x : α) : walkSet p 0 x = ∅ := by
-  simp [walkSet]
-
-omit [Fintype α] in
-lemma walkSet_succ_last (p : Equiv.Perm α) (r : ℕ) (x : α) :
-    walkSet p (r+1) x = insert (p^[r] x) (walkSet p r x) := by
-  simp [walkSet,Finset.range_add_one]
-
-omit [Fintype α] in
-lemma walkSet_succ_first (p : Equiv.Perm α) (r : ℕ) (x : α) :
-    walkSet p (r+1) x = insert x (walkSet p r (p x)) := by
-  ext y
-  simp only [mem_walkSet,Finset.mem_insert]
-  constructor
-  · rintro ⟨n,hn,hny⟩
-    cases n with
-    | zero => exact Or.inl (by simpa using hny.symm)
-    | succ n => exact Or.inr ⟨n,by omega,by simpa only [Function.iterate_succ_apply] using hny⟩
-  · rintro (rfl | ⟨n,hn,hny⟩)
-    · exact ⟨0,by omega,rfl⟩
-    · exact ⟨n+1,by omega,by simpa only [Function.iterate_succ_apply] using hny⟩
-
-omit [Fintype α] in
-lemma not_mem_short_walk (p : Equiv.Perm α) (x : α) {r : ℕ}
-    (hr : r < Function.minimalPeriod p x) : x ∉ walkSet p r (p x) := by
-  rintro hx
-  obtain ⟨n,hn,hnx⟩ := (mem_walkSet _ _ _ _).mp hx
-  have hperiod : Function.IsPeriodicPt p (n+1) x := by
-    simpa only [Function.IsPeriodicPt,Function.IsFixedPt,Function.iterate_succ_apply] using hnx
-  have hp := hperiod.minimalPeriod_le (by omega : 0 < n+1)
-  omega
-
-lemma walkSet_period (p : Equiv.Perm α) (x : α) :
-    walkSet p (Function.minimalPeriod p x) x = orbitSet p x := by
-  ext y
-  simp only [mem_walkSet,mem_orbitSet]
-  constructor
-  · rintro ⟨n,_,rfl⟩
-    rw [Equiv.Perm.iterate_eq_pow]
-    exact ⟨(n:ℤ),by simp⟩
-  · intro h
-    obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
-    refine ⟨n % Function.minimalPeriod p x,Nat.mod_lt _ (perm_minimalPeriod_pos p x),?_⟩
-    rw [Function.iterate_mod_minimalPeriod_eq,Equiv.Perm.iterate_eq_pow,hn]
-
-omit [Fintype α] in
-lemma walkSet_card (p : Equiv.Perm α) (x : α) {r : ℕ}
-    (hr : r ≤ Function.minimalPeriod p x) : (walkSet p r x).card = r := by
-  rw [walkSet,Finset.card_image_of_injOn]
-  · exact Finset.card_range _
-  · intro i hi j hj hij
-    exact Function.iterate_injOn_Iio_minimalPeriod
-      (lt_of_lt_of_le (Finset.mem_range.mp hi) hr)
-      (lt_of_lt_of_le (Finset.mem_range.mp hj) hr) hij
-
-lemma walkSet_subset_orbit (p : Equiv.Perm α) (r : ℕ) (x : α) :
-    walkSet p r x ⊆ orbitSet p x := by
-  intro y hy
-  obtain ⟨n,_,rfl⟩ := (mem_walkSet _ _ _ _).mp hy
-  rw [mem_orbitSet,Equiv.Perm.iterate_eq_pow]
-  exact ⟨(n:ℤ),by simp⟩
-
-def cycleTrace (p : Equiv.Perm α) (S : Finset α) (a : α) :
-    ℕ → α → Finset α → Option (Finset α × α)
-  | 0, x, V => if x ∈ S ∧ x ∉ V ∧ p x = a then some (insert x V,x) else none
-  | r+1, x, V => if x ∈ S ∧ x ∉ V then cycleTrace p S a r (p x) (insert x V) else none
-
-lemma cycleTrace_walk (p : Equiv.Perm α) (S : Finset α) (r : ℕ) (x : α) (V : Finset α)
-    (hr : r < Function.minimalPeriod p x) (hS : walkSet p (r+1) x ⊆ S)
-    (hV : Disjoint V (walkSet p (r+1) x)) :
-    cycleTrace p S (p^[r+1] x) r x V = some (V ∪ walkSet p (r+1) x,p^[r] x) := by
-  induction r generalizing x V with
-  | zero =>
-    have hxS : x ∈ S := hS (by simp [walkSet_succ_first])
-    have hxV : x ∉ V := fun h => Finset.disjoint_left.mp hV h (by simp [walkSet_succ_first])
-    simp [cycleTrace,hxS,hxV,walkSet_succ_first]
-  | succ r ih =>
-    have hxS : x ∈ S := hS (by rw [walkSet_succ_first]; exact Finset.mem_insert_self _ _)
-    have hxV : x ∉ V := fun h => Finset.disjoint_left.mp hV h
-      (by rw [walkSet_succ_first]; exact Finset.mem_insert_self _ _)
-    have hr' : r < Function.minimalPeriod p (p x) := by
-      rw [Function.minimalPeriod_apply (p.injective.mem_periodicPts x)]
-      omega
-    have hS' : walkSet p (r+1) (p x) ⊆ S := by
-      intro y hy
-      apply hS
-      rw [walkSet_succ_first]
-      exact Finset.mem_insert_of_mem hy
-    have hV' : Disjoint (insert x V) (walkSet p (r+1) (p x)) := by
-      apply Finset.disjoint_left.mpr
-      intro y hy hy'
-      rcases Finset.mem_insert.mp hy with hy | hy
-      · subst y
-        exact not_mem_short_walk p x hr hy'
-      · apply Finset.disjoint_left.mp hV hy
-        rw [walkSet_succ_first]
-        exact Finset.mem_insert_of_mem hy'
-    rw [cycleTrace,ite_eq_left ⟨hxS,hxV⟩,Function.iterate_succ_apply p (r+1) x,
-      ih _ _ hr' hS' hV',walkSet_succ_first p (r+1) x,Function.iterate_succ_apply]
-    simp only [Finset.insert_union,Finset.union_insert]
-
-lemma cycleTrace_orbit (p : Equiv.Perm α) (S V : Finset α) (z : α)
-    (hS : orbitSet p z ⊆ S) (hV : Disjoint V (orbitSet p z)) :
-    cycleTrace p S (p z) (Function.minimalPeriod p z - 1) (p z) V =
-      some (V ∪ orbitSet p z,z) := by
-  have hpos := perm_minimalPeriod_pos p z
-  have he : Function.minimalPeriod p (p z) = Function.minimalPeriod p z :=
-    Function.minimalPeriod_apply (p.injective.mem_periodicPts z)
-  have hp : Function.minimalPeriod p z - 1 + 1 = Function.minimalPeriod p z := by omega
-  have hw : walkSet p (Function.minimalPeriod p z - 1 + 1) (p z) = orbitSet p z := by
-    rw [hp,←he,walkSet_period,orbitSet_apply]
-  have hr : Function.minimalPeriod p z - 1 < Function.minimalPeriod p (p z) := by rw [he]; omega
-  have h := cycleTrace_walk p S (Function.minimalPeriod p z-1) (p z) V hr
-    (by simpa only [hw] using hS) (by simpa only [hw] using hV)
-  have ht : p^[Function.minimalPeriod p z-1] (p z) = z := by
-    rw [← Function.iterate_succ_apply,show (Function.minimalPeriod p z-1).succ =
-      Function.minimalPeriod p z by omega,Function.iterate_minimalPeriod]
-  have ha : p^[Function.minimalPeriod p z-1+1] (p z) = p z := by
-    rw [hp,←he,Function.iterate_minimalPeriod]
-  simpa only [ha,ht,hw] using h
-
-end OrbitCore
-
-noncomputable def exposedPaths (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) :
-    Finset (SwitchIndex d) :=
-  V.biUnion (fun x => routeDomain d x (butterflyPerm d b x))
-
-@[simp] lemma exposedPaths_empty (d : ℕ) (b : Butterfly d) : exposedPaths d b ∅ = ∅ := by
-  classical
-  simp [exposedPaths]
-
-lemma exposedPaths_insert (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) (x : Card d) :
-    exposedPaths d b (insert x V) = exposedPaths d b V ∪ routeDomain d x (butterflyPerm d b x) := by
-  classical
-  simp [exposedPaths,Finset.union_comm]
-
-noncomputable def terminalCharge (d : ℕ) (b : Butterfly d) (z : Card d) (V : Finset (Card d)) : ℕ :=
-  (routeDomain d z (butterflyPerm d b z) ∩ exposedPaths d b V).card
-
-namespace QueryTree
-
-lemma followPath_stale_charge {ι : Type*} [DecidableEq ι]
-    (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
-    (k : Card d → QueryTree ι) (R : Finset ι) (ω : ι → Bool) :
-    staleCharge (followPath d e x k) R ω =
-      staleCharge (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x))
-        (R ∪ (routeDomain d x (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).image e) ω := by
-  have h₀ := fresh_add_stale (followPath d e x k) R ω
-  have h₁ := fresh_add_stale (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x))
-    (R ∪ (routeDomain d x (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).image e) ω
-  rw [followPath_charge,followPath_raw_charge] at h₀
-  omega
-
-lemma seekCycle_semantics (d : ℕ) (S : Finset (Card d)) (a : Card d) (r : ℕ)
-    (x : Card d) (V W : Finset (Card d)) (z : Card d)
-    (k : Finset (Card d) → QueryTree (SwitchIndex d)) (ω : SwitchIndex d → Bool)
-    (h : cycleTrace (butterflyPerm d (decodeButterfly d ω)) S a r x V = some (W,z)) :
-    (seekCycle d S a r x V k).succeeds ω = (k W).succeeds ω ∧
-      (seekCycle d S a r x V k).staleCharge (exposedPaths d (decodeButterfly d ω) V) ω =
-        (k W).staleCharge (exposedPaths d (decodeButterfly d ω) W) ω +
-          terminalCharge d (decodeButterfly d ω) z (W.erase z) := by
-  classical
-  induction r generalizing x V with
-  | zero =>
-    simp only [cycleTrace] at h
-    split_ifs at h with hv
-    obtain ⟨hxS,hxV,hxa⟩ := hv
-    obtain ⟨rfl,rfl⟩ := Prod.mk.inj (Option.some.inj h)
-    simp only [seekCycle,ite_eq_left (show x ∈ S ∧ x ∉ V from ⟨hxS,hxV⟩),closePath_succeeds,ite_eq_left hxa]
-    constructor
-    · trivial
-    · rw [closePath_stale_charge _ _ _ _ _ _ hxa,exposedPaths_insert,
-        Finset.erase_insert hxV]
-      simp only [terminalCharge,hxa]
-  | succ r ih =>
-    simp only [cycleTrace] at h
-    split_ifs at h with hv
-    obtain ⟨hxS,hxV⟩ := hv
-    rw [seekCycle,ite_eq_left ⟨hxS,hxV⟩,followPath_succeeds,followPath_stale_charge]
-    simp only [Function.comp_id,Finset.image_id]
-    rw [← exposedPaths_insert]
-    exact ih _ _ h
-
-lemma seekCycle_orbit (d : ℕ) (S V : Finset (Card d)) (z : Card d)
-    (k : Finset (Card d) → QueryTree (SwitchIndex d)) (ω : SwitchIndex d → Bool)
-    (hS : orbitSet (butterflyPerm d (decodeButterfly d ω)) z ⊆ S)
-    (hV : Disjoint V (orbitSet (butterflyPerm d (decodeButterfly d ω)) z)) :
-    let p := butterflyPerm d (decodeButterfly d ω)
-    let W := V ∪ orbitSet p z
-    (seekCycle d S (p z) (Function.minimalPeriod p z-1) (p z) V k).succeeds ω = (k W).succeeds ω ∧
-      (seekCycle d S (p z) (Function.minimalPeriod p z-1) (p z) V k).staleCharge
-          (exposedPaths d (decodeButterfly d ω) V) ω =
-        (k W).staleCharge (exposedPaths d (decodeButterfly d ω) W) ω +
-          terminalCharge d (decodeButterfly d ω) z (W.erase z) := by
-  exact seekCycle_semantics _ _ _ _ _ _ _ _ _ _ (cycleTrace_orbit _ _ _ _ hS hV)
-
-end QueryTree
-
-lemma mem_exposedPaths (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) (s : SwitchIndex d) :
-    s ∈ exposedPaths d b V ↔ (switchUsers d b s).1 ∈ V ∨ (switchUsers d b s).2 ∈ V := by
-  classical
-  simp only [exposedPaths,Finset.mem_biUnion,mem_routeDomain_iff_user]
-  constructor
-  · rintro ⟨x,hx,h | h⟩
-    · exact Or.inl (h ▸ hx)
-    · exact Or.inr (h ▸ hx)
-  · rintro (h | h)
-    · exact ⟨_,h,Or.inl rfl⟩
-    · exact ⟨_,h,Or.inr rfl⟩
-
-lemma terminalCharge_sum (d : ℕ) (b : Butterfly d) (z : Card d) (V : Finset (Card d))
-    (hz : z ∉ V) :
-    (terminalCharge d b z V : ℝ) = ∑ s : SwitchIndex d,
-      ((if (switchUsers d b s).1 = z ∧ (switchUsers d b s).2 ∈ V then (1:ℝ) else 0) +
-      (if (switchUsers d b s).2 = z ∧ (switchUsers d b s).1 ∈ V then (1:ℝ) else 0)) := by
-  classical
-  have he : routeDomain d z (butterflyPerm d b z) ∩ exposedPaths d b V =
-      Finset.univ.filter (fun s =>
-        ((switchUsers d b s).1 = z ∧ (switchUsers d b s).2 ∈ V) ∨
-        ((switchUsers d b s).2 = z ∧ (switchUsers d b s).1 ∈ V)) := by
-    ext s
-    simp only [Finset.mem_inter,mem_routeDomain_iff_user,mem_exposedPaths,
-      Finset.mem_filter,Finset.mem_univ,true_and]
-    by_cases h₁ : (switchUsers d b s).1 = z <;>
-      by_cases h₂ : (switchUsers d b s).2 = z <;> simp [h₁,h₂,hz]
-  rw [terminalCharge,he]
-  simp only [← Finset.sum_boole]
-  apply Finset.sum_congr rfl
-  intro s _
-  by_cases h₁ : (switchUsers d b s).1 = z <;>
-    by_cases h₂ : (switchUsers d b s).2 = z <;> simp [h₁,h₂,hz]
-
-noncomputable def priorPaths {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) : Finset (Card d) :=
-  Finset.univ.filter (fun x => ∃ i : Fin A, o x = some i ∧ i.val < t)
-
-@[simp] lemma mem_priorPaths {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) (x : Card d) :
-    x ∈ priorPaths o t ↔ ∃ i : Fin A, o x = some i ∧ i.val < t := by
-  classical
-  simp [priorPaths]
-
-lemma terminalCharge_prior_sum {d A : ℕ} (o : Card d → Option (Fin A))
-    (b : Butterfly d) (z : Terminal o) (i : Fin A) :
-    (terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) =
-      ∑ s : SwitchIndex d,
-        ((if (z i).val = (switchUsers d b s).1 ∧ (switchUsers d b s).2 ∈ priorPaths o (i.val+1)
-          then (1:ℝ) else 0) +
-        (if (z i).val = (switchUsers d b s).2 ∧ (switchUsers d b s).1 ∈ priorPaths o (i.val+1)
-          then (1:ℝ) else 0)) := by
-  classical
-  rw [terminalCharge_sum _ _ _ _ (Finset.notMem_erase _ _)]
-  apply Finset.sum_congr rfl
-  intro s _
-  have hn := switchUsers_ne d b s
-  simp only [Finset.mem_erase]
-  by_cases h₁ : (z i).val = (switchUsers d b s).1 <;>
-    by_cases h₂ : (z i).val = (switchUsers d b s).2 <;> simp_all [eq_comm]
-
-lemma rotationMeeting_eq_slot_sum {d A : ℕ} (o : Card d → Option (Fin A))
-    (z : Terminal o) (u v : Card d) :
-    rotationMeeting o z u v = ∑ i : Fin A,
-      ((if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) +
-      (if (z i).val = v ∧ u ∈ priorPaths o (i.val+1) then (1:ℝ) else 0)) := by
-  classical
-  have only_owner (u : Card d) (a : Fin A) (hou : o u = some a)
-      (i : Fin A) (hia : i ≠ a) : (z i).val ≠ u := by
-    intro h
-    have hi := (z i).property
-    rw [h,hou] at hi
-    exact hia (Option.some.inj hi).symm
-  have sum_one (u v : Card d) (a : Fin A) (hou : o u = some a) :
-      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) =
-      if (z a).val = u ∧ v ∈ priorPaths o (a.val+1) then 1 else 0 := by
-    apply Finset.sum_eq_single a
-    · intro i _ hia
-      simp [only_owner u a hou i hia]
-    · simp
-  have sum_none (u v : Card d) (hou : o u = none) :
-      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) = 0 := by
-    apply Finset.sum_eq_zero
-    intro i _
-    have hn : (z i).val ≠ u := by
-      intro h
-      have hi := (z i).property
-      rw [h,hou] at hi
-      contradiction
-    simp [hn]
-  rw [Finset.sum_add_distrib]
-  cases hou : o u with
-  | none =>
-    rw [sum_none u v hou]
-    simp [rotationMeeting,hou,mem_priorPaths]
-  | some a =>
-    rw [sum_one u v a hou]
-    cases hov : o v with
-    | none =>
-      rw [sum_none v u hov]
-      simp [rotationMeeting,hou,hov,mem_priorPaths]
-    | some b =>
-      rw [sum_one v u b hov]
-      simp only [rotationMeeting,hou,hov,mem_priorPaths,Option.some.injEq]
-      rcases lt_trichotomy a b with hab | rfl | hba
-      · have hn : ¬ b.val < a.val+1 := by omega
-        have hy : a.val < b.val+1 := by omega
-        simp [hab,hn,hy]
-      · simp
-      · have hn : ¬ a.val < b.val+1 := by omega
-        have hy : b.val < a.val+1 := by omega
-        simp [hba,not_lt_of_ge (le_of_lt hba),hn,hy]
-
-lemma rotationCharge_eq_terminal_sum {d A : ℕ} (o : Card d → Option (Fin A))
-    (b : Butterfly d) (z : Terminal o) :
-    rotationCharge o b z = ∑ i : Fin A,
-      (terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) := by
-  simp only [rotationCharge,rotationMeeting_eq_slot_sum,terminalCharge_prior_sum]
-  exact Finset.sum_comm
-
-@[simp] lemma priorPaths_zero {d A : ℕ} (o : Card d → Option (Fin A)) : priorPaths o 0 = ∅ := by
-  classical
-  ext x
-  simp
-
-lemma priorPaths_succ {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) (ht : t < A) :
-    priorPaths o (t+1) = priorPaths o t ∪ cycleFiber o ⟨t,ht⟩ := by
-  classical
-  ext x
-  simp only [mem_priorPaths,Finset.mem_union,mem_cycleFiber]
-  constructor
-  · rintro ⟨i,hi,hit⟩
-    by_cases h : i.val < t
-    · exact Or.inl ⟨i,hi,h⟩
-    · have he : i = ⟨t,ht⟩ := Fin.ext (by change i.val = t; omega)
-      exact Or.inr (he ▸ hi)
-  · rintro (⟨i,hi,hit⟩ | hi)
-    · exact ⟨i,hi,by omega⟩
-    · exact ⟨⟨t,ht⟩,hi,by change t < t+1; omega⟩
-
-lemma priorPaths_disjoint {d A : ℕ} (o : Card d → Option (Fin A)) (i : Fin A) :
-    Disjoint (priorPaths o i.val) (cycleFiber o i) := by
-  classical
-  apply Finset.disjoint_left.mpr
-  intro x hx hx'
-  obtain ⟨j,hj,hji⟩ := (mem_priorPaths _ _ _).mp hx
-  have he : j = i := Option.some.inj (hj.symm.trans ((mem_cycleFiber _ _ _).mp hx'))
-  subst j
-  exact (lt_irrefl _ hji)
-
-namespace QueryTree
-
-lemma seekCycles_owned {d A : ℕ} (S : Finset (Card d)) (r : Fin A → ℕ)
-    (o : Card d → Option (Fin A)) (z : Terminal o) (ω : SwitchIndex d → Bool)
-    (hS : ∀ i, cycleFiber o i ⊆ S)
-    (hO : ∀ i, orbitSet (butterflyPerm d (decodeButterfly d ω)) (z i).val = cycleFiber o i)
-    (hC : ∀ i, (cycleFiber o i).card = r i+1) :
-    let p := butterflyPerm d (decodeButterfly d ω)
-    let L := List.ofFn (fun i : Fin A => (p (z i).val,r i))
-    (seekCycles d S L ∅).succeeds ω = true ∧
-      ((seekCycles d S L ∅).staleCharge ∅ ω : ℝ) = rotationCharge o (decodeButterfly d ω) z := by
-  classical
-  let b := decodeButterfly d ω
-  let p := butterflyPerm d b
-  let f (i : Fin A) := (p (z i).val,r i)
-  let c (i : Fin A) : ℝ := terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val)
-  have hc (i : Fin A) : Function.minimalPeriod p (z i).val - 1 = r i := by
-    have he := orbitSet_card p (z i).val
-    rw [hO,hC] at he
-    omega
-  have hs (t : ℕ) (ht : t ≤ A) :
-      (seekCycles d S ((List.ofFn f).drop t) (priorPaths o t)).succeeds ω = true ∧
-        ((seekCycles d S ((List.ofFn f).drop t) (priorPaths o t)).staleCharge (exposedPaths d b (priorPaths o t)) ω : ℝ) =
-          ((List.ofFn c).drop t).sum := by
-    induction ht using Nat.decreasingInduction with
-    | self => simp only [List.length_ofFn,le_refl,List.drop_eq_nil_of_le,List.sum_nil,seekCycles,succeeds,staleCharge,Nat.cast_zero,and_self]
-    | of_succ t ht ih =>
-      have htf : t < (List.ofFn f).length := by simpa using ht
-      have htc : t < (List.ofFn c).length := by simpa using ht
-      rw [List.drop_eq_getElem_cons htf,List.drop_eq_getElem_cons htc]
-      simp only [List.getElem_ofFn,List.sum_cons,seekCycles]
-      have h := seekCycle_orbit d S (priorPaths o t) (z ⟨t,ht⟩).val
-        (fun V => seekCycles d S ((List.ofFn f).drop (t+1)) V) ω
-        (by rw [hO]; exact hS _) (by rw [hO]; exact priorPaths_disjoint o ⟨t,ht⟩)
-      dsimp only at h
-      rw [hc,hO,←priorPaths_succ o t ht] at h
-      dsimp only [f]
-      constructor
-      · exact h.1.trans ih.1
-      · rw [h.2,Nat.cast_add,ih.2]
-        exact add_comm _ _
-  have h := hs 0 (Nat.zero_le _)
-  simp only [List.drop_zero,priorPaths_zero,exposedPaths_empty,List.sum_ofFn] at h
-  dsimp only
-  exact ⟨h.1,h.2.trans (rotationCharge_eq_terminal_sum o b z).symm⟩
-
-end QueryTree
-
-section Collections
-variable {d A : ℕ}
-
-def ValidCycleSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ)
-    (o : Card d → Option (Fin A)) : Prop :=
-  (∀ i, (cycleFiber o i).card = r i + 1) ∧
-  (∀ i, cycleFiber o i ⊆ S) ∧
-  (∀ i x, o x = some i → orbitSet p x = cycleFiber o i)
-
-abbrev CycleSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
-  {o : Card d → Option (Fin A) // ValidCycleSlots p S r o}
-
-noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
-    Fintype (CycleSlots p S r) := Fintype.ofFinite _
-
-lemma CycleSlots.terminal_nonempty {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (c : CycleSlots p S r) (i : Fin A) : Nonempty {x : Card d // c.val x = some i} := by
-  have hn : (cycleFiber c.val i).Nonempty := Finset.card_pos.mp (by rw [c.property.1]; omega)
-  obtain ⟨x,hx⟩ := hn
-  exact ⟨x,(mem_cycleFiber _ _ _).mp hx⟩
-
-lemma owner_ext {o o' : Card d → Option (Fin A)}
-    (h : ∀ i, cycleFiber o i = cycleFiber o' i) : o = o' := by
-  funext x
-  have he (i : Fin A) : o x = some i ↔ o' x = some i := by
-    simpa only [mem_cycleFiber] using (Finset.ext_iff.mp (h i) x)
-  cases ho : o x with
-  | some i => exact ((he i).mp ho).symm
-  | none =>
-    cases ho' : o' x with
-    | none => rfl
-    | some i => have hn := (he i).mpr ho'; rw [ho] at hn; contradiction
-
-abbrev RootedSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
-  Σ c : CycleSlots p S r, Terminal c.val
-
-noncomputable def rootedStart {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (c : RootedSlots p S r) : Fin A → S := fun i =>
-  ⟨p (c.2 i).val,c.1.property.2.1 i (by
-    rw [←c.1.property.2.2 i (c.2 i).val (c.2 i).property]
-    exact (mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.apply_right (Equiv.Perm.SameCycle.refl p _)))⟩
-
-lemma rootedStart_injective {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ} :
-    Function.Injective (rootedStart (p:=p) (S:=S) (r:=r)) := by
-  rintro ⟨c,z⟩ ⟨c',z'⟩ h
-  have hz (i : Fin A) : (z i).val = (z' i).val :=
-    p.injective (congrArg Subtype.val (congrFun h i))
-  have hc : c = c' := Subtype.ext (owner_ext (fun i => by
-    rw [←c.property.2.2 i (z i).val (z i).property,
-      ←c'.property.2.2 i (z' i).val (z' i).property,hz]))
-  subst c'
-  have hzz : z = z' := funext (fun i => Subtype.ext (hz i))
-  subst z'
-  rfl
-
-lemma sum_comp_injective_le {α β : Type*} [Fintype α] [Fintype β]
-    (e : α → β) (he : Function.Injective e) (f : β → ℝ) (hf : ∀ b, 0 ≤ f b) :
-    ∑ a, f (e a) ≤ ∑ b, f b := by
-  classical
-  calc
-    _ = ∑ b ∈ Finset.univ.image e, f b := (Finset.sum_image (fun _ _ _ _ h => he h)).symm
-    _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _) (fun b _ _ => hf b)
-
-lemma rooted_weight_sum_le (S : Finset (Card d)) (r : Fin A → ℕ) (ω : SwitchIndex d → Bool) :
-    (∑ c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r,
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2)) ≤
-    ∑ a : Fin A → S,
-      if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-        ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-      else 0 := by
-  classical
-  let p := butterflyPerm d (decodeButterfly d ω)
-  let g (a : Fin A → S) : ℝ :=
-    if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-    else 0
-  have he (c : RootedSlots p S r) :
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2) =
-      g (rootedStart c) := by
-    have hs := QueryTree.seekCycles_owned S r c.1.val c.2 ω c.1.property.2.1
-      (fun i => c.1.property.2.2 i (c.2 i).val (c.2 i).property) c.1.property.1
-    dsimp only at hs
-    unfold g
-    rw [ite_eq_left (show (QueryTree.seekCycles d S (List.ofFn (fun i => ((rootedStart c i).val,r i))) ∅).succeeds ω = true from hs.1)]
-    rw [← hs.2,neg_mul,Real.exp_neg,mul_comm (Real.log 2),Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
-    rfl
-  simp_rw [he]
-  exact sum_comp_injective_le rootedStart rootedStart_injective g (fun a => by dsimp [g]; split_ifs <;> positivity)
-
-noncomputable def rotationLower (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : ℝ :=
-  Real.exp (-Real.log 2 * ((A:ℝ)*d/2 + ∑ i : Fin A, Real.log (r i+1) / (2 * Real.log 2)))
-
-noncomputable def collectionWeight (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : ℝ :=
-  ((2:ℝ)^d)^A * (∏ i : Fin A, (r i+1 : ℝ)) * rotationLower d r
-
-lemma collectionWeight_pos (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : 0 < collectionWeight d r := by
-  unfold collectionWeight rotationLower
-  positivity
-
-lemma collection_rotation_lower (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r)
-    (ω : SwitchIndex d → Bool) (c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) :
-    collectionWeight d r ≤ ∑ z : Terminal c.val,
-      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z) := by
-  classical
-  let (i : Fin A) : Nonempty {x : Card d // c.val x = some i} := c.terminal_nonempty i
-  have hm : Monotone (fun i => (cycleFiber c.val i).card) := by
-    intro i j hij
-    dsimp only
-    rw [c.property.1,c.property.1]
-    exact Nat.add_le_add_right (hr hij) 1
-  have hb := rotation_exp_bound hm (decodeButterfly d ω)
-  have hcard : (Fintype.card (Terminal c.val) : ℝ) = ∏ i : Fin A, (r i+1 : ℝ) := by
-    rw [Fintype.card_pi,Nat.cast_prod]
-    apply Finset.prod_congr rfl
-    intro i _
-    rw [card_cycleFiber,c.property.1,Nat.cast_add,Nat.cast_one]
-  have hp : (0:ℝ) < Fintype.card (Terminal c.val) := Nat.cast_pos.mpr Fintype.card_pos
-  have hl : rotationLower d r ≤ finiteMean (fun z : Terminal c.val => Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z)) := by
-    simpa only [rotationLower,c.property.1,Nat.cast_add,Nat.cast_one] using hb
-  have hh := (le_div_iff₀ hp).mp hl
-  rw [hcard] at hh
-  unfold collectionWeight
-  rw [← Finset.mul_sum]
-  calc
-    _ = ((2:ℝ)^d)^A * (rotationLower d r * ∏ i : Fin A, (r i+1 : ℝ)) := by ring
-    _ ≤ _ := mul_le_mul_of_nonneg_left hh (by positivity)
-
-lemma cycleSlots_weighted_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) *
-      collectionWeight d r ≤ (S.card : ℝ)^A := by
-  classical
-  let g (a : Fin A → S) (ω : SwitchIndex d → Bool) : ℝ :=
-    if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
-      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
-    else 0
-  have hω (ω : SwitchIndex d → Bool) :
-      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ) * collectionWeight d r ≤
-        ∑ a : Fin A → S, g a ω := by
-    calc
-      _ = ∑ _c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r, collectionWeight d r := by simp
-      _ ≤ ∑ c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r,
-          ∑ z : Terminal c.val, ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z) :=
-        Finset.sum_le_sum (fun c _ => collection_rotation_lower S r hr ω c)
-      _ = ∑ c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r,
-          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2) :=
-        (Fintype.sum_sigma (fun c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r =>
-          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2))).symm
-      _ ≤ _ := rooted_weight_sum_le S r ω
-  have hb := finiteMean_mono hω
-  rw [finiteMean_mul_const,finiteMean_sum] at hb
-  calc
-    _ ≤ ∑ a : Fin A → S, finiteMean (g a) := hb
-    _ ≤ ∑ _a : Fin A → S, (1:ℝ) := by
-      apply Finset.sum_le_sum
-      intro a _
-      have hh := QueryTree.cycles_weighted_success d S (List.ofFn (fun i : Fin A => ((a i).val,r i)))
-      simpa only [List.length_ofFn] using hh
-    _ = _ := by simp
-
-lemma cycleSlots_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) ≤
-        (S.card : ℝ)^A / collectionWeight d r := by
-  exact (le_div_iff₀ (collectionWeight_pos d r)).mpr (cycleSlots_weighted_moment S r hr)
-
-lemma collectionWeight_eq_sqrt (d : ℕ) {A : ℕ} (r : Fin A → ℕ) :
-    collectionWeight d r = (Real.sqrt ((2:ℝ)^d))^A * ∏ i : Fin A, Real.sqrt (r i+1) := by
-  have hn : ((2:ℝ)^d)^A = Real.exp ((A:ℝ)*d*Real.log 2) := by
-    rw [mul_assoc,Real.exp_nat_mul,Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
-  have hp : (∏ i : Fin A, (r i+1 : ℝ)) = Real.exp (∑ i : Fin A, Real.log (r i+1)) := by
-    rw [Real.exp_sum]
-    apply Finset.prod_congr rfl
-    intro i _
-    exact (Real.exp_log (by positivity)).symm
-  have hs : Real.sqrt ((2:ℝ)^d) = Real.exp ((d:ℝ)*Real.log 2/2) := by
-    rw [Real.sqrt_eq_rpow,Real.rpow_def_of_pos (by positivity),Real.log_pow]
-    congr 1
-    ring
-  have hj (i : Fin A) : Real.sqrt (r i+1) = Real.exp (Real.log (r i+1)/2) := by
-    rw [Real.sqrt_eq_rpow,Real.rpow_def_of_pos (by positivity)]
-    congr 1
-    ring
-  simp_rw [collectionWeight,hn,hp,rotationLower,hs,hj]
-  rw [←Real.exp_add,←Real.exp_add,←Real.exp_nat_mul,←Real.exp_sum,←Real.exp_add]
-  congr 1
-  rw [←Finset.sum_div,←Finset.sum_div]
-  have hl : Real.log 2 ≠ 0 := ne_of_gt (Real.log_pos (by norm_num))
-  field_simp
-  ring
-
-lemma butterfly_cycleSlots_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
-    finiteMean (fun ω : SwitchIndex d → Bool =>
-      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) ≤
-        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
-          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
-  have h := cycleSlots_moment S r hr
-  rw [collectionWeight_eq_sqrt,div_mul_eq_div_mul_one_div] at h
-  rw [div_pow]
-  simpa only [one_div,Finset.prod_inv_distrib] using h
-
-end Collections
-
-variable {d A : ℕ}
-
-abbrev SelectedCycle (p : Equiv.Perm (Card d)) (S : Finset (Card d)) :=
-  {C : Finset (Card d) // (∃ x, C = orbitSet p x) ∧ C ⊆ S}
-
-noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) :
-    Fintype (SelectedCycle p S) := Fintype.ofFinite _
-
-lemma SelectedCycle.nonempty {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
-    (C : SelectedCycle p S) : C.val.Nonempty := by
-  obtain ⟨x,hx⟩ := C.property.1
-  rw [hx]
-  exact orbitSet_nonempty p x
-
-lemma SelectedCycle.orbit_eq {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
-    (C : SelectedCycle p S) {x : Card d} (hx : x ∈ C.val) : orbitSet p x = C.val := by
-  obtain ⟨y,hy⟩ := C.property.1
-  rw [hy] at hx ⊢
-  exact (orbitSet_eq p ((mem_orbitSet _ _ _).mp hx)).symm
-
-lemma SelectedCycle.eq_of_mem {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
-    (C D : SelectedCycle p S) {x : Card d} (hC : x ∈ C.val) (hD : x ∈ D.val) : C = D := by
-  apply Subtype.ext
-  exact (C.orbit_eq hC).symm.trans (D.orbit_eq hD)
-
-abbrev SlotAssignment (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
-  {f : Fin A → SelectedCycle p S // Function.Injective f ∧ ∀ i, (f i).val.card = r i+1}
-
-noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
-    Fintype (SlotAssignment p S r) := Fintype.ofFinite _
-
-noncomputable def slotsToAssignment {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (c : CycleSlots p S r) : SlotAssignment p S r := by
-  classical
-  let f (i : Fin A) : SelectedCycle p S := ⟨cycleFiber c.val i,by
-    obtain ⟨x,hx⟩ := c.terminal_nonempty i
-    exact ⟨⟨x,(c.property.2.2 i x hx).symm⟩,c.property.2.1 i⟩⟩
-  refine ⟨f,?_,fun i => c.property.1 i⟩
-  intro i j hij
-  have he : cycleFiber c.val i = cycleFiber c.val j := congrArg Subtype.val hij
-  obtain ⟨x,hx⟩ := c.terminal_nonempty i
-  have hxi : x ∈ cycleFiber c.val i := (mem_cycleFiber _ _ _).mpr hx
-  rw [he] at hxi
-  exact Option.some.inj (hx.symm.trans ((mem_cycleFiber _ _ _).mp hxi))
-
-noncomputable def assignmentOwner {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
-    (f : SlotAssignment p S r) (x : Card d) : Option (Fin A) := by
-  classical
-  exact if h : ∃ i, x ∈ (f.val i).val then some (Classical.choose h) else none
-
-end Thorp
-
-end ThorpNine.Tail
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Tail.ExposedCoins
+
+namespace OAI
+
+namespace ThorpNine.Tail
+
+namespace Thorp
+open scoped BigOperators
+open Filter
+section Rotation
+variable {d A : ℕ}
+
+noncomputable def rotationMeeting (o : Card d → Option (Fin A)) (z : Terminal o)
+    (u v : Card d) : ℝ :=
+  match o u, o v with
+  | some i, some j => if i < j then (if (z j).val = v then 1 else 0)
+      else if j < i then (if (z i).val = u then 1 else 0)
+      else (if (z i).val = u then 1 else 0) + (if (z i).val = v then 1 else 0)
+  | _, _ => 0
+
+lemma card_cycleFiber (o : Card d → Option (Fin A)) (i : Fin A) :
+    Fintype.card {x : Card d // o x = some i} = (cycleFiber o i).card := by
+  classical
+  simp [cycleFiber,Fintype.card_subtype]
+
+lemma terminal_mean {o : Card d → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
+    (i : Fin A) (x : Card d) (hx : o x = some i) :
+    finiteMean (fun z : Terminal o => if (z i).val = x then (1:ℝ) else 0) = slotWeight o i := by
+  classical
+  have hev := finiteMean_pi_eval (A := fun i : Fin A => {x : Card d // o x = some i})
+    i (fun z => if z.val = x then (1:ℝ) else 0)
+  rw [hev]
+  have he : (fun z : {x : Card d // o x = some i} => if z.val = x then (1:ℝ) else 0) =
+      (fun z => if z = ⟨x,hx⟩ then 1 else 0) := by
+    funext z
+    simp only [Subtype.ext_iff]
+  rw [he,finiteMean_eq_indicator,card_cycleFiber,slotWeight]
+
+lemma slotWeight_antitone (o : Card d → Option (Fin A))
+    (hn : ∀ i, (cycleFiber o i).Nonempty)
+    (hm : Monotone (fun i => (cycleFiber o i).card)) : Antitone (slotWeight o) := by
+  intro i j hij
+  unfold slotWeight
+  apply one_div_le_one_div_of_le
+  · exact_mod_cast Finset.card_pos.mpr (hn i)
+  · exact_mod_cast hm hij
+
+lemma rotationMeeting_mean {o : Card d → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
+    (hm : Monotone (fun i => (cycleFiber o i).card)) (u v : Card d) :
+    finiteMean (fun z : Terminal o => rotationMeeting o z u v) =
+      min (pathWeight o u) (pathWeight o v) + withinWeight o u v := by
+  classical
+  have hn : ∀ i, (cycleFiber o i).Nonempty := by
+    intro i
+    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : Card d // o x = some i})
+    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
+  have hw := slotWeight_antitone o hn hm
+  cases hu : o u with
+  | none =>
+    cases hv : o v with
+    | none => simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean]
+    | some j =>
+      have hj : 0 ≤ slotWeight o j := by unfold slotWeight; positivity
+      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_left hj]
+  | some i =>
+    cases hv : o v with
+    | none =>
+      have hi : 0 ≤ slotWeight o i := by unfold slotWeight; positivity
+      simp [rotationMeeting,pathWeight,withinWeight,hu,hv,finiteMean,min_eq_right hi]
+    | some j =>
+      simp only [rotationMeeting,pathWeight,withinWeight,hu,hv]
+      rcases lt_trichotomy i j with hij | rfl | hji
+      · simp only [hij,↓reduceIte,ne_of_lt hij]
+        rw [terminal_mean j v hv,min_eq_right (hw (le_of_lt hij)),add_zero]
+      · simp only [lt_self_iff_false,↓reduceIte,min_self]
+        rw [finiteMean_add,terminal_mean i u hu,terminal_mean i v hv]
+      · simp only [not_lt_of_ge (le_of_lt hji),hji,↓reduceIte,ne_of_gt hji]
+        rw [terminal_mean i u hu,min_eq_left (hw (le_of_lt hji)),add_zero]
+
+lemma pathWeight_sum (o : Card d → Option (Fin A)) (hn : ∀ i, (cycleFiber o i).Nonempty) :
+    ∑ x : Card d, pathWeight o x = (A : ℝ) := by
+  classical
+  have he (x : Card d) : pathWeight o x =
+      ∑ i : Fin A, if o x = some i then slotWeight o i else 0 := by
+    cases hx : o x with
+    | none => simp [pathWeight,hx]
+    | some j => simp [pathWeight,hx]
+  simp only [he]
+  rw [Finset.sum_comm]
+  have hf (i : Fin A) : (∑ x : Card d, if o x = some i then slotWeight o i else 0) = 1 := by
+    rw [← Finset.sum_filter]
+    change (∑ _x ∈ cycleFiber o i, slotWeight o i) = 1
+    simp only [Finset.sum_const,nsmul_eq_mul,slotWeight]
+    exact mul_one_div_cancel (by exact_mod_cast (Finset.card_pos.mpr (hn i)).ne')
+  simp only [hf,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,mul_one]
+
+lemma withinWeight_sum_bound (o : Card d → Option (Fin A))
+    (hn : ∀ i, (cycleFiber o i).Nonempty) (b : Butterfly d) :
+    (∑ s : SwitchIndex d, withinWeight o (switchUsers d b s).1 (switchUsers d b s).2) ≤
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
+  classical
+  have he (u v : Card d) : withinWeight o u v =
+      ∑ i : Fin A, (if u ∈ cycleFiber o i ∧ v ∈ cycleFiber o i then (1:ℝ) else 0) * slotWeight o i := by
+    simp only [mem_cycleFiber]
+    cases hu : o u <;> cases hv : o v <;> simp [withinWeight,hu,hv,ite_and,eq_comm]
+  simp only [he]
+  rw [Finset.sum_comm]
+  apply Finset.sum_le_sum
+  intro i _
+  rw [← Finset.sum_mul]
+  have hi := marked_switch_entropy_bound d b (cycleFiber o i)
+  have hpos : (0:ℝ) < (cycleFiber o i).card := by exact_mod_cast Finset.card_pos.mpr (hn i)
+  have hmul := mul_le_mul_of_nonneg_right hi (show 0 ≤ slotWeight o i by unfold slotWeight; positivity)
+  calc
+    _ ≤ ((cycleFiber o i).card : ℝ) * Real.log (cycleFiber o i).card /
+        (2 * Real.log 2) * slotWeight o i := hmul
+    _ = _ := by unfold slotWeight; field_simp
+
+noncomputable def rotationCharge (o : Card d → Option (Fin A)) (b : Butterfly d)
+    (z : Terminal o) : ℝ :=
+  ∑ s : SwitchIndex d, rotationMeeting o z (switchUsers d b s).1 (switchUsers d b s).2
+
+lemma rotationCharge_mean_bound {o : Card d → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
+    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : Butterfly d) :
+    finiteMean (rotationCharge o b) ≤ (A:ℝ) * d / 2 +
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2) := by
+  classical
+  have hn : ∀ i, (cycleFiber o i).Nonempty := by
+    intro i
+    obtain ⟨x,hx⟩ := (inferInstance : Nonempty {x : Card d // o x = some i})
+    exact ⟨x,mem_cycleFiber _ _ _ |>.mpr hx⟩
+  unfold rotationCharge
+  rw [finiteMean_sum]
+  simp only [rotationMeeting_mean hm,Finset.sum_add_distrib]
+  have h₁ := weighted_switch_min_bound d b (pathWeight o)
+  rw [pathWeight_sum o hn] at h₁
+  have h₂ := withinWeight_sum_bound o hn b
+  linarith
+
+lemma rotation_exp_bound {o : Card d → Option (Fin A)}
+    [∀ i : Fin A, Nonempty {x : Card d // o x = some i}]
+    (hm : Monotone (fun i => (cycleFiber o i).card)) (b : Butterfly d) :
+    Real.exp (-Real.log 2 * ((A:ℝ)*d/2 +
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2))) ≤
+      finiteMean (fun z : Terminal o => Real.exp (-Real.log 2 * rotationCharge o b z)) := by
+  have hb := rotationCharge_mean_bound hm b
+  have hlog : 0 ≤ Real.log 2 := Real.log_nonneg (by norm_num)
+  have hm' : -Real.log 2 * ((A:ℝ)*d/2 +
+      ∑ i : Fin A, Real.log (cycleFiber o i).card / (2 * Real.log 2)) ≤
+      finiteMean (fun z : Terminal o => -Real.log 2 * rotationCharge o b z) := by
+    have he := finiteMean_mul_const (rotationCharge o b) (-Real.log 2)
+    simp only [mul_comm] at he
+    rw [he]
+    exact mul_le_mul_of_nonpos_left hb (neg_nonpos.mpr hlog)
+  exact (Real.exp_le_exp.mpr hm').trans (exp_finiteMean_le _)
+
+end Rotation
+
+section OrbitCore
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+noncomputable def orbitSet (p : Equiv.Perm α) (x : α) : Finset α :=
+  Finset.univ.filter (p.SameCycle x)
+
+@[simp] lemma mem_orbitSet (p : Equiv.Perm α) (x y : α) :
+    y ∈ orbitSet p x ↔ p.SameCycle x y := by
+  classical
+  simp [orbitSet]
+
+lemma orbitSet_nonempty (p : Equiv.Perm α) (x : α) : (orbitSet p x).Nonempty :=
+  ⟨x,(mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.refl _ _)⟩
+
+lemma orbitSet_eq (p : Equiv.Perm α) {x y : α} (h : p.SameCycle x y) :
+    orbitSet p x = orbitSet p y := by
+  ext z
+  simp only [mem_orbitSet]
+  exact ⟨fun hz => h.symm.trans hz,fun hz => h.trans hz⟩
+
+@[simp] lemma orbitSet_apply (p : Equiv.Perm α) (x : α) : orbitSet p (p x) = orbitSet p x := by
+  ext y
+  simp only [mem_orbitSet,Equiv.Perm.sameCycle_apply_left]
+
+lemma orbitSet_eq_or_disjoint (p : Equiv.Perm α) (x y : α) :
+    orbitSet p x = orbitSet p y ∨ Disjoint (orbitSet p x) (orbitSet p y) := by
+  classical
+  by_cases h : p.SameCycle x y
+  · exact Or.inl (orbitSet_eq p h)
+  · right
+    apply Finset.disjoint_left.mpr
+    intro z hz hz'
+    exact h (((mem_orbitSet _ _ _).mp hz).trans ((mem_orbitSet _ _ _).mp hz').symm)
+
+omit [DecidableEq α] in
+lemma perm_minimalPeriod_pos (p : Equiv.Perm α) (x : α) :
+    0 < Function.minimalPeriod p x :=
+  Function.minimalPeriod_pos_of_mem_periodicPts (p.injective.mem_periodicPts x)
+
+lemma orbitSet_card (p : Equiv.Perm α) (x : α) :
+    (orbitSet p x).card = Function.minimalPeriod p x := by
+  classical
+  let f : Fin (Function.minimalPeriod p x) → {y // p.SameCycle x y} := fun n =>
+    ⟨p^[n.val] x,by rw [Equiv.Perm.iterate_eq_pow]; exact ⟨(n.val:ℤ),by simp⟩⟩
+  have hf : Function.Bijective f := by
+    constructor
+    · intro i j hij
+      apply Fin.ext
+      exact Function.iterate_injOn_Iio_minimalPeriod i.isLt j.isLt (congrArg Subtype.val hij)
+    · intro y
+      obtain ⟨n,hn⟩ := y.property.exists_nat_pow_eq
+      refine ⟨⟨n % Function.minimalPeriod p x, Nat.mod_lt _ (perm_minimalPeriod_pos p x)⟩,?_⟩
+      apply Subtype.ext
+      change p^[n % Function.minimalPeriod p x] x = y.val
+      rw [Function.iterate_mod_minimalPeriod_eq,Equiv.Perm.iterate_eq_pow,hn]
+  have hc := Fintype.card_congr (Equiv.ofBijective f hf)
+  simpa [orbitSet,Fintype.card_subtype] using hc.symm
+
+def walkSet (p : Equiv.Perm α) (r : ℕ) (x : α) : Finset α :=
+  (Finset.range r).image (fun n => p^[n] x)
+
+omit [Fintype α] in
+@[simp] lemma mem_walkSet (p : Equiv.Perm α) (r : ℕ) (x y : α) :
+    y ∈ walkSet p r x ↔ ∃ n < r, p^[n] x = y := by
+  simp [walkSet]
+
+omit [Fintype α] in
+@[simp] lemma walkSet_zero (p : Equiv.Perm α) (x : α) : walkSet p 0 x = ∅ := by
+  simp [walkSet]
+
+omit [Fintype α] in
+lemma walkSet_succ_last (p : Equiv.Perm α) (r : ℕ) (x : α) :
+    walkSet p (r+1) x = insert (p^[r] x) (walkSet p r x) := by
+  simp [walkSet,Finset.range_add_one]
+
+omit [Fintype α] in
+lemma walkSet_succ_first (p : Equiv.Perm α) (r : ℕ) (x : α) :
+    walkSet p (r+1) x = insert x (walkSet p r (p x)) := by
+  ext y
+  simp only [mem_walkSet,Finset.mem_insert]
+  constructor
+  · rintro ⟨n,hn,hny⟩
+    cases n with
+    | zero => exact Or.inl (by simpa using hny.symm)
+    | succ n => exact Or.inr ⟨n,by omega,by simpa only [Function.iterate_succ_apply] using hny⟩
+  · rintro (rfl | ⟨n,hn,hny⟩)
+    · exact ⟨0,by omega,rfl⟩
+    · exact ⟨n+1,by omega,by simpa only [Function.iterate_succ_apply] using hny⟩
+
+omit [Fintype α] in
+lemma not_mem_short_walk (p : Equiv.Perm α) (x : α) {r : ℕ}
+    (hr : r < Function.minimalPeriod p x) : x ∉ walkSet p r (p x) := by
+  rintro hx
+  obtain ⟨n,hn,hnx⟩ := (mem_walkSet _ _ _ _).mp hx
+  have hperiod : Function.IsPeriodicPt p (n+1) x := by
+    simpa only [Function.IsPeriodicPt,Function.IsFixedPt,Function.iterate_succ_apply] using hnx
+  have hp := hperiod.minimalPeriod_le (by omega : 0 < n+1)
+  omega
+
+lemma walkSet_period (p : Equiv.Perm α) (x : α) :
+    walkSet p (Function.minimalPeriod p x) x = orbitSet p x := by
+  ext y
+  simp only [mem_walkSet,mem_orbitSet]
+  constructor
+  · rintro ⟨n,_,rfl⟩
+    rw [Equiv.Perm.iterate_eq_pow]
+    exact ⟨(n:ℤ),by simp⟩
+  · intro h
+    obtain ⟨n,hn⟩ := h.exists_nat_pow_eq
+    refine ⟨n % Function.minimalPeriod p x,Nat.mod_lt _ (perm_minimalPeriod_pos p x),?_⟩
+    rw [Function.iterate_mod_minimalPeriod_eq,Equiv.Perm.iterate_eq_pow,hn]
+
+omit [Fintype α] in
+lemma walkSet_card (p : Equiv.Perm α) (x : α) {r : ℕ}
+    (hr : r ≤ Function.minimalPeriod p x) : (walkSet p r x).card = r := by
+  rw [walkSet,Finset.card_image_of_injOn]
+  · exact Finset.card_range _
+  · intro i hi j hj hij
+    exact Function.iterate_injOn_Iio_minimalPeriod
+      (lt_of_lt_of_le (Finset.mem_range.mp hi) hr)
+      (lt_of_lt_of_le (Finset.mem_range.mp hj) hr) hij
+
+lemma walkSet_subset_orbit (p : Equiv.Perm α) (r : ℕ) (x : α) :
+    walkSet p r x ⊆ orbitSet p x := by
+  intro y hy
+  obtain ⟨n,_,rfl⟩ := (mem_walkSet _ _ _ _).mp hy
+  rw [mem_orbitSet,Equiv.Perm.iterate_eq_pow]
+  exact ⟨(n:ℤ),by simp⟩
+
+def cycleTrace (p : Equiv.Perm α) (S : Finset α) (a : α) :
+    ℕ → α → Finset α → Option (Finset α × α)
+  | 0, x, V => if x ∈ S ∧ x ∉ V ∧ p x = a then some (insert x V,x) else none
+  | r+1, x, V => if x ∈ S ∧ x ∉ V then cycleTrace p S a r (p x) (insert x V) else none
+
+lemma cycleTrace_walk (p : Equiv.Perm α) (S : Finset α) (r : ℕ) (x : α) (V : Finset α)
+    (hr : r < Function.minimalPeriod p x) (hS : walkSet p (r+1) x ⊆ S)
+    (hV : Disjoint V (walkSet p (r+1) x)) :
+    cycleTrace p S (p^[r+1] x) r x V = some (V ∪ walkSet p (r+1) x,p^[r] x) := by
+  induction r generalizing x V with
+  | zero =>
+    have hxS : x ∈ S := hS (by simp [walkSet_succ_first])
+    have hxV : x ∉ V := fun h => Finset.disjoint_left.mp hV h (by simp [walkSet_succ_first])
+    simp [cycleTrace,hxS,hxV,walkSet_succ_first]
+  | succ r ih =>
+    have hxS : x ∈ S := hS (by rw [walkSet_succ_first]; exact Finset.mem_insert_self _ _)
+    have hxV : x ∉ V := fun h => Finset.disjoint_left.mp hV h
+      (by rw [walkSet_succ_first]; exact Finset.mem_insert_self _ _)
+    have hr' : r < Function.minimalPeriod p (p x) := by
+      rw [Function.minimalPeriod_apply (p.injective.mem_periodicPts x)]
+      omega
+    have hS' : walkSet p (r+1) (p x) ⊆ S := by
+      intro y hy
+      apply hS
+      rw [walkSet_succ_first]
+      exact Finset.mem_insert_of_mem hy
+    have hV' : Disjoint (insert x V) (walkSet p (r+1) (p x)) := by
+      apply Finset.disjoint_left.mpr
+      intro y hy hy'
+      rcases Finset.mem_insert.mp hy with hy | hy
+      · subst y
+        exact not_mem_short_walk p x hr hy'
+      · apply Finset.disjoint_left.mp hV hy
+        rw [walkSet_succ_first]
+        exact Finset.mem_insert_of_mem hy'
+    rw [cycleTrace,ite_eq_left ⟨hxS,hxV⟩,Function.iterate_succ_apply p (r+1) x,
+      ih _ _ hr' hS' hV',walkSet_succ_first p (r+1) x,Function.iterate_succ_apply]
+    simp only [Finset.insert_union,Finset.union_insert]
+
+lemma cycleTrace_orbit (p : Equiv.Perm α) (S V : Finset α) (z : α)
+    (hS : orbitSet p z ⊆ S) (hV : Disjoint V (orbitSet p z)) :
+    cycleTrace p S (p z) (Function.minimalPeriod p z - 1) (p z) V =
+      some (V ∪ orbitSet p z,z) := by
+  have hpos := perm_minimalPeriod_pos p z
+  have he : Function.minimalPeriod p (p z) = Function.minimalPeriod p z :=
+    Function.minimalPeriod_apply (p.injective.mem_periodicPts z)
+  have hp : Function.minimalPeriod p z - 1 + 1 = Function.minimalPeriod p z := by omega
+  have hw : walkSet p (Function.minimalPeriod p z - 1 + 1) (p z) = orbitSet p z := by
+    rw [hp,←he,walkSet_period,orbitSet_apply]
+  have hr : Function.minimalPeriod p z - 1 < Function.minimalPeriod p (p z) := by rw [he]; omega
+  have h := cycleTrace_walk p S (Function.minimalPeriod p z-1) (p z) V hr
+    (by simpa only [hw] using hS) (by simpa only [hw] using hV)
+  have ht : p^[Function.minimalPeriod p z-1] (p z) = z := by
+    rw [← Function.iterate_succ_apply,show (Function.minimalPeriod p z-1).succ =
+      Function.minimalPeriod p z by omega,Function.iterate_minimalPeriod]
+  have ha : p^[Function.minimalPeriod p z-1+1] (p z) = p z := by
+    rw [hp,←he,Function.iterate_minimalPeriod]
+  simpa only [ha,ht,hw] using h
+
+end OrbitCore
+
+noncomputable def exposedPaths (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) :
+    Finset (SwitchIndex d) :=
+  V.biUnion (fun x => routeDomain d x (butterflyPerm d b x))
+
+@[simp] lemma exposedPaths_empty (d : ℕ) (b : Butterfly d) : exposedPaths d b ∅ = ∅ := by
+  classical
+  simp [exposedPaths]
+
+lemma exposedPaths_insert (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) (x : Card d) :
+    exposedPaths d b (insert x V) = exposedPaths d b V ∪ routeDomain d x (butterflyPerm d b x) := by
+  classical
+  simp [exposedPaths,Finset.union_comm]
+
+noncomputable def terminalCharge (d : ℕ) (b : Butterfly d) (z : Card d) (V : Finset (Card d)) : ℕ :=
+  (routeDomain d z (butterflyPerm d b z) ∩ exposedPaths d b V).card
+
+namespace QueryTree
+
+lemma followPath_stale_charge {ι : Type*} [DecidableEq ι]
+    (d : ℕ) (e : SwitchIndex d → ι) (x : Card d)
+    (k : Card d → QueryTree ι) (R : Finset ι) (ω : ι → Bool) :
+    staleCharge (followPath d e x k) R ω =
+      staleCharge (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x))
+        (R ∪ (routeDomain d x (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).image e) ω := by
+  have h₀ := fresh_add_stale (followPath d e x k) R ω
+  have h₁ := fresh_add_stale (k (butterflyPerm d (decodeButterfly d (ω ∘ e)) x))
+    (R ∪ (routeDomain d x (butterflyPerm d (decodeButterfly d (ω ∘ e)) x)).image e) ω
+  rw [followPath_charge,followPath_raw_charge] at h₀
+  omega
+
+lemma seekCycle_semantics (d : ℕ) (S : Finset (Card d)) (a : Card d) (r : ℕ)
+    (x : Card d) (V W : Finset (Card d)) (z : Card d)
+    (k : Finset (Card d) → QueryTree (SwitchIndex d)) (ω : SwitchIndex d → Bool)
+    (h : cycleTrace (butterflyPerm d (decodeButterfly d ω)) S a r x V = some (W,z)) :
+    (seekCycle d S a r x V k).succeeds ω = (k W).succeeds ω ∧
+      (seekCycle d S a r x V k).staleCharge (exposedPaths d (decodeButterfly d ω) V) ω =
+        (k W).staleCharge (exposedPaths d (decodeButterfly d ω) W) ω +
+          terminalCharge d (decodeButterfly d ω) z (W.erase z) := by
+  classical
+  induction r generalizing x V with
+  | zero =>
+    simp only [cycleTrace] at h
+    split_ifs at h with hv
+    obtain ⟨hxS,hxV,hxa⟩ := hv
+    obtain ⟨rfl,rfl⟩ := Prod.mk.inj (Option.some.inj h)
+    simp only [seekCycle,ite_eq_left (show x ∈ S ∧ x ∉ V from ⟨hxS,hxV⟩),closePath_succeeds,ite_eq_left hxa]
+    constructor
+    · trivial
+    · rw [closePath_stale_charge _ _ _ _ _ _ hxa,exposedPaths_insert,
+        Finset.erase_insert hxV]
+      simp only [terminalCharge,hxa]
+  | succ r ih =>
+    simp only [cycleTrace] at h
+    split_ifs at h with hv
+    obtain ⟨hxS,hxV⟩ := hv
+    rw [seekCycle,ite_eq_left ⟨hxS,hxV⟩,followPath_succeeds,followPath_stale_charge]
+    simp only [Function.comp_id,Finset.image_id]
+    rw [← exposedPaths_insert]
+    exact ih _ _ h
+
+lemma seekCycle_orbit (d : ℕ) (S V : Finset (Card d)) (z : Card d)
+    (k : Finset (Card d) → QueryTree (SwitchIndex d)) (ω : SwitchIndex d → Bool)
+    (hS : orbitSet (butterflyPerm d (decodeButterfly d ω)) z ⊆ S)
+    (hV : Disjoint V (orbitSet (butterflyPerm d (decodeButterfly d ω)) z)) :
+    let p := butterflyPerm d (decodeButterfly d ω)
+    let W := V ∪ orbitSet p z
+    (seekCycle d S (p z) (Function.minimalPeriod p z-1) (p z) V k).succeeds ω = (k W).succeeds ω ∧
+      (seekCycle d S (p z) (Function.minimalPeriod p z-1) (p z) V k).staleCharge
+          (exposedPaths d (decodeButterfly d ω) V) ω =
+        (k W).staleCharge (exposedPaths d (decodeButterfly d ω) W) ω +
+          terminalCharge d (decodeButterfly d ω) z (W.erase z) := by
+  exact seekCycle_semantics _ _ _ _ _ _ _ _ _ _ (cycleTrace_orbit _ _ _ _ hS hV)
+
+end QueryTree
+
+lemma mem_exposedPaths (d : ℕ) (b : Butterfly d) (V : Finset (Card d)) (s : SwitchIndex d) :
+    s ∈ exposedPaths d b V ↔ (switchUsers d b s).1 ∈ V ∨ (switchUsers d b s).2 ∈ V := by
+  classical
+  simp only [exposedPaths,Finset.mem_biUnion,mem_routeDomain_iff_user]
+  constructor
+  · rintro ⟨x,hx,h | h⟩
+    · exact Or.inl (h ▸ hx)
+    · exact Or.inr (h ▸ hx)
+  · rintro (h | h)
+    · exact ⟨_,h,Or.inl rfl⟩
+    · exact ⟨_,h,Or.inr rfl⟩
+
+lemma terminalCharge_sum (d : ℕ) (b : Butterfly d) (z : Card d) (V : Finset (Card d))
+    (hz : z ∉ V) :
+    (terminalCharge d b z V : ℝ) = ∑ s : SwitchIndex d,
+      ((if (switchUsers d b s).1 = z ∧ (switchUsers d b s).2 ∈ V then (1:ℝ) else 0) +
+      (if (switchUsers d b s).2 = z ∧ (switchUsers d b s).1 ∈ V then (1:ℝ) else 0)) := by
+  classical
+  have he : routeDomain d z (butterflyPerm d b z) ∩ exposedPaths d b V =
+      Finset.univ.filter (fun s =>
+        ((switchUsers d b s).1 = z ∧ (switchUsers d b s).2 ∈ V) ∨
+        ((switchUsers d b s).2 = z ∧ (switchUsers d b s).1 ∈ V)) := by
+    ext s
+    simp only [Finset.mem_inter,mem_routeDomain_iff_user,mem_exposedPaths,
+      Finset.mem_filter,Finset.mem_univ,true_and]
+    by_cases h₁ : (switchUsers d b s).1 = z <;>
+      by_cases h₂ : (switchUsers d b s).2 = z <;> simp [h₁,h₂,hz]
+  rw [terminalCharge,he]
+  simp only [← Finset.sum_boole]
+  apply Finset.sum_congr rfl
+  intro s _
+  by_cases h₁ : (switchUsers d b s).1 = z <;>
+    by_cases h₂ : (switchUsers d b s).2 = z <;> simp [h₁,h₂,hz]
+
+noncomputable def priorPaths {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) : Finset (Card d) :=
+  Finset.univ.filter (fun x => ∃ i : Fin A, o x = some i ∧ i.val < t)
+
+@[simp] lemma mem_priorPaths {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) (x : Card d) :
+    x ∈ priorPaths o t ↔ ∃ i : Fin A, o x = some i ∧ i.val < t := by
+  classical
+  simp [priorPaths]
+
+lemma terminalCharge_prior_sum {d A : ℕ} (o : Card d → Option (Fin A))
+    (b : Butterfly d) (z : Terminal o) (i : Fin A) :
+    (terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) =
+      ∑ s : SwitchIndex d,
+        ((if (z i).val = (switchUsers d b s).1 ∧ (switchUsers d b s).2 ∈ priorPaths o (i.val+1)
+          then (1:ℝ) else 0) +
+        (if (z i).val = (switchUsers d b s).2 ∧ (switchUsers d b s).1 ∈ priorPaths o (i.val+1)
+          then (1:ℝ) else 0)) := by
+  classical
+  rw [terminalCharge_sum _ _ _ _ (Finset.notMem_erase _ _)]
+  apply Finset.sum_congr rfl
+  intro s _
+  have hn := switchUsers_ne d b s
+  simp only [Finset.mem_erase]
+  by_cases h₁ : (z i).val = (switchUsers d b s).1 <;>
+    by_cases h₂ : (z i).val = (switchUsers d b s).2 <;> simp_all [eq_comm]
+
+lemma rotationMeeting_eq_slot_sum {d A : ℕ} (o : Card d → Option (Fin A))
+    (z : Terminal o) (u v : Card d) :
+    rotationMeeting o z u v = ∑ i : Fin A,
+      ((if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) +
+      (if (z i).val = v ∧ u ∈ priorPaths o (i.val+1) then (1:ℝ) else 0)) := by
+  classical
+  have only_owner (u : Card d) (a : Fin A) (hou : o u = some a)
+      (i : Fin A) (hia : i ≠ a) : (z i).val ≠ u := by
+    intro h
+    have hi := (z i).property
+    rw [h,hou] at hi
+    exact hia (Option.some.inj hi).symm
+  have sum_one (u v : Card d) (a : Fin A) (hou : o u = some a) :
+      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) =
+      if (z a).val = u ∧ v ∈ priorPaths o (a.val+1) then 1 else 0 := by
+    apply Finset.sum_eq_single a
+    · intro i _ hia
+      simp [only_owner u a hou i hia]
+    · simp
+  have sum_none (u v : Card d) (hou : o u = none) :
+      (∑ i : Fin A, if (z i).val = u ∧ v ∈ priorPaths o (i.val+1) then (1:ℝ) else 0) = 0 := by
+    apply Finset.sum_eq_zero
+    intro i _
+    have hn : (z i).val ≠ u := by
+      intro h
+      have hi := (z i).property
+      rw [h,hou] at hi
+      contradiction
+    simp [hn]
+  rw [Finset.sum_add_distrib]
+  cases hou : o u with
+  | none =>
+    rw [sum_none u v hou]
+    simp [rotationMeeting,hou,mem_priorPaths]
+  | some a =>
+    rw [sum_one u v a hou]
+    cases hov : o v with
+    | none =>
+      rw [sum_none v u hov]
+      simp [rotationMeeting,hou,hov,mem_priorPaths]
+    | some b =>
+      rw [sum_one v u b hov]
+      simp only [rotationMeeting,hou,hov,mem_priorPaths,Option.some.injEq]
+      rcases lt_trichotomy a b with hab | rfl | hba
+      · have hn : ¬ b.val < a.val+1 := by omega
+        have hy : a.val < b.val+1 := by omega
+        simp [hab,hn,hy]
+      · simp
+      · have hn : ¬ a.val < b.val+1 := by omega
+        have hy : b.val < a.val+1 := by omega
+        simp [hba,not_lt_of_ge (le_of_lt hba),hn,hy]
+
+lemma rotationCharge_eq_terminal_sum {d A : ℕ} (o : Card d → Option (Fin A))
+    (b : Butterfly d) (z : Terminal o) :
+    rotationCharge o b z = ∑ i : Fin A,
+      (terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val) : ℝ) := by
+  simp only [rotationCharge,rotationMeeting_eq_slot_sum,terminalCharge_prior_sum]
+  exact Finset.sum_comm
+
+@[simp] lemma priorPaths_zero {d A : ℕ} (o : Card d → Option (Fin A)) : priorPaths o 0 = ∅ := by
+  classical
+  ext x
+  simp
+
+lemma priorPaths_succ {d A : ℕ} (o : Card d → Option (Fin A)) (t : ℕ) (ht : t < A) :
+    priorPaths o (t+1) = priorPaths o t ∪ cycleFiber o ⟨t,ht⟩ := by
+  classical
+  ext x
+  simp only [mem_priorPaths,Finset.mem_union,mem_cycleFiber]
+  constructor
+  · rintro ⟨i,hi,hit⟩
+    by_cases h : i.val < t
+    · exact Or.inl ⟨i,hi,h⟩
+    · have he : i = ⟨t,ht⟩ := Fin.ext (by change i.val = t; omega)
+      exact Or.inr (he ▸ hi)
+  · rintro (⟨i,hi,hit⟩ | hi)
+    · exact ⟨i,hi,by omega⟩
+    · exact ⟨⟨t,ht⟩,hi,by change t < t+1; omega⟩
+
+lemma priorPaths_disjoint {d A : ℕ} (o : Card d → Option (Fin A)) (i : Fin A) :
+    Disjoint (priorPaths o i.val) (cycleFiber o i) := by
+  classical
+  apply Finset.disjoint_left.mpr
+  intro x hx hx'
+  obtain ⟨j,hj,hji⟩ := (mem_priorPaths _ _ _).mp hx
+  have he : j = i := Option.some.inj (hj.symm.trans ((mem_cycleFiber _ _ _).mp hx'))
+  subst j
+  exact (lt_irrefl _ hji)
+
+namespace QueryTree
+
+lemma seekCycles_owned {d A : ℕ} (S : Finset (Card d)) (r : Fin A → ℕ)
+    (o : Card d → Option (Fin A)) (z : Terminal o) (ω : SwitchIndex d → Bool)
+    (hS : ∀ i, cycleFiber o i ⊆ S)
+    (hO : ∀ i, orbitSet (butterflyPerm d (decodeButterfly d ω)) (z i).val = cycleFiber o i)
+    (hC : ∀ i, (cycleFiber o i).card = r i+1) :
+    let p := butterflyPerm d (decodeButterfly d ω)
+    let L := List.ofFn (fun i : Fin A => (p (z i).val,r i))
+    (seekCycles d S L ∅).succeeds ω = true ∧
+      ((seekCycles d S L ∅).staleCharge ∅ ω : ℝ) = rotationCharge o (decodeButterfly d ω) z := by
+  classical
+  let b := decodeButterfly d ω
+  let p := butterflyPerm d b
+  let f (i : Fin A) := (p (z i).val,r i)
+  let c (i : Fin A) : ℝ := terminalCharge d b (z i).val ((priorPaths o (i.val+1)).erase (z i).val)
+  have hc (i : Fin A) : Function.minimalPeriod p (z i).val - 1 = r i := by
+    have he := orbitSet_card p (z i).val
+    rw [hO,hC] at he
+    omega
+  have hs (t : ℕ) (ht : t ≤ A) :
+      (seekCycles d S ((List.ofFn f).drop t) (priorPaths o t)).succeeds ω = true ∧
+        ((seekCycles d S ((List.ofFn f).drop t) (priorPaths o t)).staleCharge (exposedPaths d b (priorPaths o t)) ω : ℝ) =
+          ((List.ofFn c).drop t).sum := by
+    induction ht using Nat.decreasingInduction with
+    | self => simp only [List.length_ofFn,le_refl,List.drop_eq_nil_of_le,List.sum_nil,seekCycles,succeeds,staleCharge,Nat.cast_zero,and_self]
+    | of_succ t ht ih =>
+      have htf : t < (List.ofFn f).length := by simpa using ht
+      have htc : t < (List.ofFn c).length := by simpa using ht
+      rw [List.drop_eq_getElem_cons htf,List.drop_eq_getElem_cons htc]
+      simp only [List.getElem_ofFn,List.sum_cons,seekCycles]
+      have h := seekCycle_orbit d S (priorPaths o t) (z ⟨t,ht⟩).val
+        (fun V => seekCycles d S ((List.ofFn f).drop (t+1)) V) ω
+        (by rw [hO]; exact hS _) (by rw [hO]; exact priorPaths_disjoint o ⟨t,ht⟩)
+      dsimp only at h
+      rw [hc,hO,←priorPaths_succ o t ht] at h
+      dsimp only [f]
+      constructor
+      · exact h.1.trans ih.1
+      · rw [h.2,Nat.cast_add,ih.2]
+        exact add_comm _ _
+  have h := hs 0 (Nat.zero_le _)
+  simp only [List.drop_zero,priorPaths_zero,exposedPaths_empty,List.sum_ofFn] at h
+  dsimp only
+  exact ⟨h.1,h.2.trans (rotationCharge_eq_terminal_sum o b z).symm⟩
+
+end QueryTree
+
+section Collections
+variable {d A : ℕ}
+
+def ValidCycleSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ)
+    (o : Card d → Option (Fin A)) : Prop :=
+  (∀ i, (cycleFiber o i).card = r i + 1) ∧
+  (∀ i, cycleFiber o i ⊆ S) ∧
+  (∀ i x, o x = some i → orbitSet p x = cycleFiber o i)
+
+abbrev CycleSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
+  {o : Card d → Option (Fin A) // ValidCycleSlots p S r o}
+
+noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
+    Fintype (CycleSlots p S r) := Fintype.ofFinite _
+
+lemma CycleSlots.terminal_nonempty {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (c : CycleSlots p S r) (i : Fin A) : Nonempty {x : Card d // c.val x = some i} := by
+  have hn : (cycleFiber c.val i).Nonempty := Finset.card_pos.mp (by rw [c.property.1]; omega)
+  obtain ⟨x,hx⟩ := hn
+  exact ⟨x,(mem_cycleFiber _ _ _).mp hx⟩
+
+lemma owner_ext {o o' : Card d → Option (Fin A)}
+    (h : ∀ i, cycleFiber o i = cycleFiber o' i) : o = o' := by
+  funext x
+  have he (i : Fin A) : o x = some i ↔ o' x = some i := by
+    simpa only [mem_cycleFiber] using (Finset.ext_iff.mp (h i) x)
+  cases ho : o x with
+  | some i => exact ((he i).mp ho).symm
+  | none =>
+    cases ho' : o' x with
+    | none => rfl
+    | some i => have hn := (he i).mpr ho'; rw [ho] at hn; contradiction
+
+abbrev RootedSlots (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
+  Σ c : CycleSlots p S r, Terminal c.val
+
+noncomputable def rootedStart {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (c : RootedSlots p S r) : Fin A → S := fun i =>
+  ⟨p (c.2 i).val,c.1.property.2.1 i (by
+    rw [←c.1.property.2.2 i (c.2 i).val (c.2 i).property]
+    exact (mem_orbitSet _ _ _).mpr (Equiv.Perm.SameCycle.apply_right (Equiv.Perm.SameCycle.refl p _)))⟩
+
+lemma rootedStart_injective {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ} :
+    Function.Injective (rootedStart (p:=p) (S:=S) (r:=r)) := by
+  rintro ⟨c,z⟩ ⟨c',z'⟩ h
+  have hz (i : Fin A) : (z i).val = (z' i).val :=
+    p.injective (congrArg Subtype.val (congrFun h i))
+  have hc : c = c' := Subtype.ext (owner_ext (fun i => by
+    rw [←c.property.2.2 i (z i).val (z i).property,
+      ←c'.property.2.2 i (z' i).val (z' i).property,hz]))
+  subst c'
+  have hzz : z = z' := funext (fun i => Subtype.ext (hz i))
+  subst z'
+  rfl
+
+lemma sum_comp_injective_le {α β : Type*} [Fintype α] [Fintype β]
+    (e : α → β) (he : Function.Injective e) (f : β → ℝ) (hf : ∀ b, 0 ≤ f b) :
+    ∑ a, f (e a) ≤ ∑ b, f b := by
+  classical
+  calc
+    _ = ∑ b ∈ Finset.univ.image e, f b := (Finset.sum_image (fun _ _ _ _ h => he h)).symm
+    _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _) (fun b _ _ => hf b)
+
+lemma rooted_weight_sum_le (S : Finset (Card d)) (r : Fin A → ℕ) (ω : SwitchIndex d → Bool) :
+    (∑ c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r,
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2)) ≤
+    ∑ a : Fin A → S,
+      if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+        ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+      else 0 := by
+  classical
+  let p := butterflyPerm d (decodeButterfly d ω)
+  let g (a : Fin A → S) : ℝ :=
+    if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+    else 0
+  have he (c : RootedSlots p S r) :
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2) =
+      g (rootedStart c) := by
+    have hs := QueryTree.seekCycles_owned S r c.1.val c.2 ω c.1.property.2.1
+      (fun i => c.1.property.2.2 i (c.2 i).val (c.2 i).property) c.1.property.1
+    dsimp only at hs
+    unfold g
+    rw [ite_eq_left (show (QueryTree.seekCycles d S (List.ofFn (fun i => ((rootedStart c i).val,r i))) ∅).succeeds ω = true from hs.1)]
+    rw [← hs.2,neg_mul,Real.exp_neg,mul_comm (Real.log 2),Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
+    rfl
+  simp_rw [he]
+  exact sum_comp_injective_le rootedStart rootedStart_injective g (fun a => by dsimp [g]; split_ifs <;> positivity)
+
+noncomputable def rotationLower (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : ℝ :=
+  Real.exp (-Real.log 2 * ((A:ℝ)*d/2 + ∑ i : Fin A, Real.log (r i+1) / (2 * Real.log 2)))
+
+noncomputable def collectionWeight (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : ℝ :=
+  ((2:ℝ)^d)^A * (∏ i : Fin A, (r i+1 : ℝ)) * rotationLower d r
+
+lemma collectionWeight_pos (d : ℕ) {A : ℕ} (r : Fin A → ℕ) : 0 < collectionWeight d r := by
+  unfold collectionWeight rotationLower
+  positivity
+
+lemma collection_rotation_lower (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r)
+    (ω : SwitchIndex d → Bool) (c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) :
+    collectionWeight d r ≤ ∑ z : Terminal c.val,
+      ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z) := by
+  classical
+  let (i : Fin A) : Nonempty {x : Card d // c.val x = some i} := c.terminal_nonempty i
+  have hm : Monotone (fun i => (cycleFiber c.val i).card) := by
+    intro i j hij
+    dsimp only
+    rw [c.property.1,c.property.1]
+    exact Nat.add_le_add_right (hr hij) 1
+  have hb := rotation_exp_bound hm (decodeButterfly d ω)
+  have hcard : (Fintype.card (Terminal c.val) : ℝ) = ∏ i : Fin A, (r i+1 : ℝ) := by
+    rw [Fintype.card_pi,Nat.cast_prod]
+    apply Finset.prod_congr rfl
+    intro i _
+    rw [card_cycleFiber,c.property.1,Nat.cast_add,Nat.cast_one]
+  have hp : (0:ℝ) < Fintype.card (Terminal c.val) := Nat.cast_pos.mpr Fintype.card_pos
+  have hl : rotationLower d r ≤ finiteMean (fun z : Terminal c.val => Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z)) := by
+    simpa only [rotationLower,c.property.1,Nat.cast_add,Nat.cast_one] using hb
+  have hh := (le_div_iff₀ hp).mp hl
+  rw [hcard] at hh
+  unfold collectionWeight
+  rw [← Finset.mul_sum]
+  calc
+    _ = ((2:ℝ)^d)^A * (rotationLower d r * ∏ i : Fin A, (r i+1 : ℝ)) := by ring
+    _ ≤ _ := mul_le_mul_of_nonneg_left hh (by positivity)
+
+lemma cycleSlots_weighted_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) *
+      collectionWeight d r ≤ (S.card : ℝ)^A := by
+  classical
+  let g (a : Fin A → S) (ω : SwitchIndex d → Bool) : ℝ :=
+    if (QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).succeeds ω then
+      ((2:ℝ)^d)^A / (2:ℝ)^(QueryTree.seekCycles d S (List.ofFn (fun i => ((a i).val,r i))) ∅).staleCharge ∅ ω
+    else 0
+  have hω (ω : SwitchIndex d → Bool) :
+      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ) * collectionWeight d r ≤
+        ∑ a : Fin A → S, g a ω := by
+    calc
+      _ = ∑ _c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r, collectionWeight d r := by simp
+      _ ≤ ∑ c : CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r,
+          ∑ z : Terminal c.val, ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.val (decodeButterfly d ω) z) :=
+        Finset.sum_le_sum (fun c _ => collection_rotation_lower S r hr ω c)
+      _ = ∑ c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r,
+          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2) :=
+        (Fintype.sum_sigma (fun c : RootedSlots (butterflyPerm d (decodeButterfly d ω)) S r =>
+          ((2:ℝ)^d)^A * Real.exp (-Real.log 2 * rotationCharge c.1.val (decodeButterfly d ω) c.2))).symm
+      _ ≤ _ := rooted_weight_sum_le S r ω
+  have hb := finiteMean_mono hω
+  rw [finiteMean_mul_const,finiteMean_sum] at hb
+  calc
+    _ ≤ ∑ a : Fin A → S, finiteMean (g a) := hb
+    _ ≤ ∑ _a : Fin A → S, (1:ℝ) := by
+      apply Finset.sum_le_sum
+      intro a _
+      have hh := QueryTree.cycles_weighted_success d S (List.ofFn (fun i : Fin A => ((a i).val,r i)))
+      simpa only [List.length_ofFn] using hh
+    _ = _ := by simp
+
+lemma cycleSlots_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) ≤
+        (S.card : ℝ)^A / collectionWeight d r := by
+  exact (le_div_iff₀ (collectionWeight_pos d r)).mpr (cycleSlots_weighted_moment S r hr)
+
+lemma collectionWeight_eq_sqrt (d : ℕ) {A : ℕ} (r : Fin A → ℕ) :
+    collectionWeight d r = (Real.sqrt ((2:ℝ)^d))^A * ∏ i : Fin A, Real.sqrt (r i+1) := by
+  have hn : ((2:ℝ)^d)^A = Real.exp ((A:ℝ)*d*Real.log 2) := by
+    rw [mul_assoc,Real.exp_nat_mul,Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)]
+  have hp : (∏ i : Fin A, (r i+1 : ℝ)) = Real.exp (∑ i : Fin A, Real.log (r i+1)) := by
+    rw [Real.exp_sum]
+    apply Finset.prod_congr rfl
+    intro i _
+    exact (Real.exp_log (by positivity)).symm
+  have hs : Real.sqrt ((2:ℝ)^d) = Real.exp ((d:ℝ)*Real.log 2/2) := by
+    rw [Real.sqrt_eq_rpow,Real.rpow_def_of_pos (by positivity),Real.log_pow]
+    congr 1
+    ring
+  have hj (i : Fin A) : Real.sqrt (r i+1) = Real.exp (Real.log (r i+1)/2) := by
+    rw [Real.sqrt_eq_rpow,Real.rpow_def_of_pos (by positivity)]
+    congr 1
+    ring
+  simp_rw [collectionWeight,hn,hp,rotationLower,hs,hj]
+  rw [←Real.exp_add,←Real.exp_add,←Real.exp_nat_mul,←Real.exp_sum,←Real.exp_add]
+  congr 1
+  rw [←Finset.sum_div,←Finset.sum_div]
+  have hl : Real.log 2 ≠ 0 := ne_of_gt (Real.log_pos (by norm_num))
+  field_simp
+  ring
+
+lemma butterfly_cycleSlots_moment (S : Finset (Card d)) (r : Fin A → ℕ) (hr : Monotone r) :
+    finiteMean (fun ω : SwitchIndex d → Bool =>
+      (Fintype.card (CycleSlots (butterflyPerm d (decodeButterfly d ω)) S r) : ℝ)) ≤
+        ((S.card : ℝ) / Real.sqrt ((2:ℝ)^d))^A *
+          ∏ i : Fin A, (1 / Real.sqrt (r i+1)) := by
+  have h := cycleSlots_moment S r hr
+  rw [collectionWeight_eq_sqrt,div_mul_eq_div_mul_one_div] at h
+  rw [div_pow]
+  simpa only [one_div,Finset.prod_inv_distrib] using h
+
+end Collections
+
+variable {d A : ℕ}
+
+abbrev SelectedCycle (p : Equiv.Perm (Card d)) (S : Finset (Card d)) :=
+  {C : Finset (Card d) // (∃ x, C = orbitSet p x) ∧ C ⊆ S}
+
+noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) :
+    Fintype (SelectedCycle p S) := Fintype.ofFinite _
+
+lemma SelectedCycle.nonempty {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
+    (C : SelectedCycle p S) : C.val.Nonempty := by
+  obtain ⟨x,hx⟩ := C.property.1
+  rw [hx]
+  exact orbitSet_nonempty p x
+
+lemma SelectedCycle.orbit_eq {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
+    (C : SelectedCycle p S) {x : Card d} (hx : x ∈ C.val) : orbitSet p x = C.val := by
+  obtain ⟨y,hy⟩ := C.property.1
+  rw [hy] at hx ⊢
+  exact (orbitSet_eq p ((mem_orbitSet _ _ _).mp hx)).symm
+
+lemma SelectedCycle.eq_of_mem {p : Equiv.Perm (Card d)} {S : Finset (Card d)}
+    (C D : SelectedCycle p S) {x : Card d} (hC : x ∈ C.val) (hD : x ∈ D.val) : C = D := by
+  apply Subtype.ext
+  exact (C.orbit_eq hC).symm.trans (D.orbit_eq hD)
+
+abbrev SlotAssignment (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :=
+  {f : Fin A → SelectedCycle p S // Function.Injective f ∧ ∀ i, (f i).val.card = r i+1}
+
+noncomputable instance (p : Equiv.Perm (Card d)) (S : Finset (Card d)) (r : Fin A → ℕ) :
+    Fintype (SlotAssignment p S r) := Fintype.ofFinite _
+
+noncomputable def slotsToAssignment {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (c : CycleSlots p S r) : SlotAssignment p S r := by
+  classical
+  let f (i : Fin A) : SelectedCycle p S := ⟨cycleFiber c.val i,by
+    obtain ⟨x,hx⟩ := c.terminal_nonempty i
+    exact ⟨⟨x,(c.property.2.2 i x hx).symm⟩,c.property.2.1 i⟩⟩
+  refine ⟨f,?_,fun i => c.property.1 i⟩
+  intro i j hij
+  have he : cycleFiber c.val i = cycleFiber c.val j := congrArg Subtype.val hij
+  obtain ⟨x,hx⟩ := c.terminal_nonempty i
+  have hxi : x ∈ cycleFiber c.val i := (mem_cycleFiber _ _ _).mpr hx
+  rw [he] at hxi
+  exact Option.some.inj (hx.symm.trans ((mem_cycleFiber _ _ _).mp hxi))
+
+noncomputable def assignmentOwner {p : Equiv.Perm (Card d)} {S : Finset (Card d)} {r : Fin A → ℕ}
+    (f : SlotAssignment p S r) (x : Card d) : Option (Fin A) := by
+  classical
+  exact if h : ∃ i, x ∈ (f.val i).val then some (Classical.choose h) else none
+
+end Thorp
+
+end ThorpNine.Tail
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Tail/ShapePartitionEquiv.lean b/lean/OAI/Probability/ThorpRouting/Tail/ShapePartitionEquiv.lean
index 4090d6d2144c9e44d7ea86a8445a93c7a9905aaf..786acb094fab0bfdc8678847d8444fe265ab1549 100644
--- a/lean/OAI/Probability/ThorpRouting/Tail/ShapePartitionEquiv.lean
+++ b/lean/OAI/Probability/ThorpRouting/Tail/ShapePartitionEquiv.lean
@@ -1,875 +1,875 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Tail.ModifiedReverseEquiv
-
-namespace OAI
-
-namespace ThorpNine.Tail
-
-namespace Thorp.Specht
-open scoped BigOperators Classical
-
-noncomputable def shapePartitionEquiv (N : ℕ) : Shapes N ≃ Nat.Partition N where
-  toFun μ := { parts := μ.1.rowLens
-               parts_pos := fun {i} hi => μ.1.pos_of_mem_rowLens i (by simpa using hi)
-               parts_sum := by simpa using (sum_rowLens μ.1).trans μ.2 }
-  invFun p := ⟨partitionDiagram p,partitionDiagram_card p⟩
-  left_inv μ := by
-    apply Subtype.ext
-    obtain ⟨μ,rfl⟩ := μ
-    exact partitionDiagram_diagramPartition μ
-  right_inv p := by
-    apply Nat.Partition.ext
-    change ((partitionDiagram p).rowLens : Multiset ℕ) = p.parts
-    rw [partitionDiagram_rowLens,Multiset.sort_eq]
-
-noncomputable instance (N : ℕ) : Fintype (Shapes N) := Fintype.ofEquiv _ (shapePartitionEquiv N).symm
-
-noncomputable def conjugacyPartition {α : Type*} [Fintype α] [DecidableEq α] :
-    ConjClasses (Equiv.Perm α) → Nat.Partition (Fintype.card α) :=
-  Quotient.lift Equiv.Perm.partition (fun _ _ hpq => (Equiv.Perm.partition_eq_of_isConj).mp hpq)
-
-lemma conjugacyPartition_injective {α : Type*} [Fintype α] [DecidableEq α] :
-    Function.Injective (conjugacyPartition (α := α)) := by
-  intro p q
-  induction p using Quotient.inductionOn with | _ p =>
-    induction q using Quotient.inductionOn with | _ q =>
-      intro h
-      exact Quotient.sound ((Equiv.Perm.partition_eq_of_isConj).mpr h)
-
-lemma card_conjugacy_le_shapes (N : ℕ) :
-    Nat.card (ConjClasses (Equiv.Perm (Fin N))) ≤ Fintype.card (Shapes N) := by
-  have h := Nat.card_le_card_of_injective _ (conjugacyPartition_injective (α := Fin N))
-  simpa only [Fintype.card_fin, Nat.card_eq_fintype_card, Fintype.card_congr (shapePartitionEquiv N)] using h
-
-noncomputable def labels {N : ℕ} (μ : Shapes N) : Cell μ.1 ≃ Fin N :=
-  (Fintype.equivFin (Cell μ.1)).trans (finCongr ((card_cell μ.1).trans μ.2))
-
-noncomputable def labelledRepresentation {N : ℕ} (μ : Shapes N) :
-    Representation ℂ (Equiv.Perm (Fin N)) (space μ.1) :=
-  (representation μ.1).comp (labels μ).symm.permCongrHom.toMonoidHom
-
-lemma labelledRepresentation_irreducible {N : ℕ} (μ : Shapes N) :
-    Representation.IsIrreducible (labelledRepresentation μ) := by
-  let e := (labels μ).symm.permCongrHom
-  let ρ := representation μ.1
-  have hn : (⊥ : Subrepresentation (labelledRepresentation μ)) ≠ ⊤ := by
-    intro h
-    have he : (⟨polytabloid μ.1,polytabloid_mem_space μ.1⟩ : space μ.1) ∈
-        (⊥ : Subrepresentation (labelledRepresentation μ)) := by rw [h]; trivial
-    exact polytabloid_ne_zero μ.1 (congrArg Subtype.val he)
-  refine { exists_pair_ne := ⟨⊥,⊤,hn⟩,eq_bot_or_eq_top := ?_ }
-  intro U
-  let U' : Subrepresentation ρ :=
-    { toSubmodule := U.toSubmodule
-      apply_mem_toSubmodule := fun p v hv => by
-        have hh := U.apply_mem_toSubmodule (e.symm p) hv
-        change ρ (e (e.symm p)) v ∈ U.toSubmodule at hh
-        simpa using hh }
-  have := representation_irreducible μ.1
-  obtain h | h := IsSimpleOrder.eq_bot_or_eq_top U'
-  · left
-    apply Subrepresentation.toSubmodule_injective
-    exact congrArg (fun R : Subrepresentation ρ => R.toSubmodule) h
-  · right
-    apply Subrepresentation.toSubmodule_injective
-    exact congrArg (fun R : Subrepresentation ρ => R.toSubmodule) h
-
-lemma labelledRepresentation_equiv_shapes {N : ℕ} (μ ν : Shapes N)
-    (T : Representation.Equiv (labelledRepresentation μ) (labelledRepresentation ν)) : μ = ν := by
-  apply Subtype.ext
-  apply shapes_eq_of_equivariant_equiv μ.1 ν.1 ((labels μ).trans (labels ν).symm) T.toLinearEquiv
-  intro p v
-  have hh := Representation.IntertwiningMap.isIntertwining _ _ T.toIntertwiningMap ((labels μ).permCongr p) v
-  change T.toLinearEquiv
-    (representation μ.1 ((labels μ).symm.permCongr ((labels μ).permCongr p)) v) =
-      representation ν.1 ((labels ν).symm.permCongr ((labels μ).permCongr p)) (T.toLinearEquiv v) at hh
-  have hp : (labels μ).symm.permCongr ((labels μ).permCongr p) = p := by
-    apply Equiv.ext
-    intro x
-    simp [Equiv.permCongr_apply]
-  rw [hp] at hh
-  convert hh using 2
-  congr 1
-
-end Thorp.Specht
-
-namespace Thorp.FiniteCharacters
-open scoped BigOperators Classical
-
-section Group
-variable {G : Type*} [Group G] [Fintype G]
-
-noncomputable def classRep (c : ConjClasses G) : G := (ConjClasses.exists_rep c).choose
-omit [Fintype G] in
-lemma classRep_spec (c : ConjClasses G) : ConjClasses.mk (classRep c) = c :=
-  (ConjClasses.exists_rep c).choose_spec
-
-variable {V : Type*} [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
-
-noncomputable def characterClass (ρ : Representation ℂ G V) : ConjClasses G → ℂ :=
-  fun c => ρ.character (classRep c)
-
-omit [Fintype G] [FiniteDimensional ℂ V] in
-lemma characterClass_mk (ρ : Representation ℂ G V) (g : G) :
-    characterClass ρ (ConjClasses.mk g) = ρ.character g := by
-  have h := ConjClasses.mk_eq_mk_iff_isConj.mp (classRep_spec (ConjClasses.mk g))
-  obtain ⟨p,hp⟩ := isConj_iff.mp h
-  change ρ.character (classRep (ConjClasses.mk g)) = ρ.character g
-  exact (Representation.char_conj ρ _ p).symm.trans (congrArg ρ.character hp)
-
-noncomputable def characterDual (ρ : Representation ℂ G V) :
-    (ConjClasses G → ℂ) →ₗ[ℂ] ℂ where
-  toFun f := (Nat.card G : ℂ)⁻¹ * ∑ g, f (ConjClasses.mk g) * ρ.character g⁻¹
-  map_add' f h := by simp [Finset.sum_add_distrib,add_mul,mul_add]
-  map_smul' a f := by
-    simp only [Pi.smul_apply,smul_eq_mul,RingHom.id_apply]
-    simp_rw [mul_assoc]
-    rw [←Finset.mul_sum]
-    ring
-
-lemma characterDual_characterClass (ρ : Representation ℂ G V)
-    {W : Type*} [AddCommGroup W] [Module ℂ W] [FiniteDimensional ℂ W]
-    (σ : Representation ℂ G W) [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ] :
-    characterDual ρ (characterClass σ) = if Nonempty (Representation.Equiv ρ σ) then 1 else 0 := by
-  let : Invertible (Nat.card G : ℂ) := invertibleOfNonzero (by
-    exact_mod_cast (Nat.card_pos (α := G)).ne')
-  change (Nat.card G : ℂ)⁻¹ * ∑ g, characterClass σ (ConjClasses.mk g) * ρ.character g⁻¹ = _
-  simp only [characterClass_mk]
-  exact Representation.char_orthonormal σ ρ
-
-end Group
-
-theorem complete_of_card_conjugacy_le
-    {G ι : Type*} [Group G] [Fintype G] [Fintype ι]
-    (V : ι → Type*) [∀ i, AddCommGroup (V i)] [∀ i, Module ℂ (V i)]
-    [∀ i, FiniteDimensional ℂ (V i)]
-    (ρ : ∀ i, Representation ℂ G (V i)) [∀ i, Representation.IsIrreducible (ρ i)]
-    (hneq : ∀ i j, i ≠ j → ¬Nonempty (Representation.Equiv (ρ i) (ρ j)))
-    (hcard : Nat.card (ConjClasses G) ≤ Fintype.card ι)
-    {W : Type*} [AddCommGroup W] [Module ℂ W] [FiniteDimensional ℂ W]
-    (σ : Representation ℂ G W) [Representation.IsIrreducible σ] :
-    ∃ i, Nonempty (Representation.Equiv (ρ i) σ) := by
-  let : Fintype (ConjClasses G) := Fintype.ofFinite _
-  have horth (i j : ι) : characterDual (ρ i) (characterClass (ρ j)) = if i=j then 1 else 0 := by
-    rw [characterDual_characterClass]
-    by_cases hij : i=j
-    · subst j
-      simp only [ite_true]
-      exact ite_eq_left ⟨Representation.Equiv.refl _⟩
-    · rw [ite_eq_right hij,ite_eq_right (hneq i j hij)]
-  have hLI : LinearIndependent ℂ (fun i => characterClass (ρ i)) := by
-    apply LinearIndependent.of_pairwise_dual_eq_zero_one _ (fun i => characterDual (ρ i))
-    · intro i j hij
-      exact (horth i j).trans (ite_eq_right hij)
-    · intro i
-      simpa using horth i i
-  have heq : Fintype.card ι = Module.finrank ℂ (ConjClasses G → ℂ) := by
-    apply Nat.le_antisymm hLI.fintype_card_le_finrank
-    simpa only [Module.finrank_pi,Module.finrank_self,Finset.sum_const,Finset.card_univ,
-      smul_eq_mul,mul_one,Nat.card_eq_fintype_card] using hcard
-  have hspan := hLI.span_eq_top_of_card_eq_finrank' heq
-  by_contra h
-  push Not at h
-  have hker : Submodule.span ℂ (Set.range (fun i => characterClass (ρ i))) ≤
-      (characterDual σ).ker := by
-    apply Submodule.span_le.mpr
-    rintro f ⟨i,rfl⟩
-    change characterDual σ (characterClass (ρ i)) = 0
-    rw [characterDual_characterClass,ite_eq_right]
-    rintro ⟨T⟩
-    exact (h i).false T.symm
-  rw [hspan] at hker
-  have hz : characterDual σ (characterClass σ) = 0 := hker (show characterClass σ ∈ ⊤ from trivial)
-  rw [characterDual_characterClass,ite_eq_left ⟨Representation.Equiv.refl σ⟩] at hz
-  exact one_ne_zero hz
-
-end Thorp.FiniteCharacters
-
-namespace Thorp.Specht
-open scoped Classical
-
-theorem labelled_complete {N : ℕ}
-    {V : Type*} [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
-    (ρ : Representation ℂ (Equiv.Perm (Fin N)) V) [Representation.IsIrreducible ρ] :
-    ∃ μ : Shapes N, Nonempty (Representation.Equiv (labelledRepresentation μ) ρ) := by
-  let (μ : Shapes N) : Representation.IsIrreducible (labelledRepresentation μ) :=
-    labelledRepresentation_irreducible μ
-  apply FiniteCharacters.complete_of_card_conjugacy_le
-    (fun μ : Shapes N => space μ.1) labelledRepresentation _ (card_conjugacy_le_shapes N) ρ
-  intro μ ν h ⟨T⟩
-  exact h (labelledRepresentation_equiv_shapes μ ν T)
-
-end Thorp.Specht
-
-namespace Thorp.PermutationHilbert
-
-section
-open scoped BigOperators ComplexConjugate Classical
-open Complex
-
-variable {X : Type*} [Fintype X]
-
-noncomputable def act (p : Equiv.Perm X) : H X ≃ₗᵢ[ℂ] H X :=
-  LinearIsometryEquiv.piLpCongrLeft 2 ℂ ℂ p
-
-@[simp] lemma act_apply (p : Equiv.Perm X) (v : H X) (x : X) :
-    act p v x = v (p⁻¹ x) := rfl
-
-lemma act_single [DecidableEq X] (p : Equiv.Perm X) (x : X) :
-    act p (EuclideanSpace.single x (1:ℂ)) = EuclideanSpace.single (p x) 1 := by
-  exact EuclideanSpace.piLpCongrLeft_single p x 1
-
-lemma trace_projection (W : Submodule ℂ (H X)) :
-    ∑ x, (W.starProjection (EuclideanSpace.single x (1:ℂ))) x = Module.finrank ℂ W := by
-  classical
-  have hp : LinearMap.IsProj W W.starProjection.toLinearMap := by
-    constructor
-    · intro x; exact (W.orthogonalProjectionOnto x).2
-    · intro x hx; exact W.starProjection_eq_self_iff.mpr hx
-  have ht := hp.trace
-  rw [LinearMap.trace_eq_matrix_trace ℂ (EuclideanSpace.basisFun X ℂ).toBasis] at ht
-  simpa only [Matrix.trace,Matrix.diag_apply,LinearMap.toMatrix_apply,EuclideanSpace.basisFun_repr,
-    EuclideanSpace.basisFun_apply,OrthonormalBasis.coe_toBasis,
-    OrthonormalBasis.coe_toBasis_repr_apply,ContinuousLinearMap.coe_coe] using ht
-
-lemma projection_diagonal (W : Submodule ℂ (H X)) (x : X) :
-    (W.starProjection (EuclideanSpace.single x (1:ℂ))) x =
-      (‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 : ℝ) := by
-  classical
-  have hh := W.inner_starProjection_left_eq_right
-    (EuclideanSpace.single x (1:ℂ)) (W.starProjection (EuclideanSpace.single x (1:ℂ)))
-  have hf : W.starProjection (W.starProjection (EuclideanSpace.single x (1:ℂ))) =
-      W.starProjection (EuclideanSpace.single x (1:ℂ)) :=
-    W.starProjection_eq_self_iff.mpr (W.orthogonalProjectionOnto _).2
-  rw [hf] at hh
-  rw [inner_self_eq_norm_sq_to_K,EuclideanSpace.inner_single_left] at hh
-  simpa using hh.symm
-
-lemma projection_comm (W : Submodule ℂ (H X)) (p : Equiv.Perm X)
-    (hW : W.map (act p).toLinearEquiv.toLinearMap = W) (v : H X) :
-    act p (W.starProjection v) = W.starProjection (act p v) := by
-  have hh := (act p).toLinearIsometry.map_starProjection W v
-  change act p (W.starProjection v) =
-    (W.map (act p).toLinearEquiv.toLinearMap).starProjection (act p v) at hh
-  simpa only [hW] using hh
-
-lemma uniform_diagonal (W : Submodule ℂ (H X))
-    (htrans : ∀ x y : X, ∃ p : Equiv.Perm X, p x = y ∧
-      W.map (act p).toLinearEquiv.toLinearMap = W) (x : X) :
-    (Fintype.card X:ℝ) * ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 =
-      Module.finrank ℂ W := by
-  classical
-  have hh (y : X) : (W.starProjection (EuclideanSpace.single y (1:ℂ))) y =
-      (‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 : ℝ) := by
-    obtain ⟨p,hp,hW⟩ := htrans x y
-    subst y
-    rw [projection_diagonal,←act_single p x,←projection_comm W p hW,(act p).norm_map]
-  have ht := trace_projection W
-  simp_rw [hh] at ht
-  have hhreal : (∑ _y : X, ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2) =
-      (Module.finrank ℂ W:ℝ) := by exact_mod_cast ht
-  simpa using hhreal
-
-theorem uncertainty (W : Submodule ℂ (H X))
-    (htrans : ∀ x y : X, ∃ p : Equiv.Perm X, p x = y ∧
-      W.map (act p).toLinearEquiv.toLinearMap = W)
-    (v : H X) (hv : v ∈ W) (x : X) :
-    (Fintype.card X:ℝ) * ‖v x‖^2 ≤ (Module.finrank ℂ W:ℝ) * ‖v‖^2 := by
-  classical
-  have hh : inner ℂ (W.starProjection (EuclideanSpace.single x (1:ℂ))) v = v x := by
-    rw [W.inner_starProjection_left_eq_right,W.starProjection_eq_self_iff.mpr hv,
-      EuclideanSpace.inner_single_left]
-    simp
-  have hc := norm_inner_le_norm (𝕜 := ℂ) (W.starProjection (EuclideanSpace.single x (1:ℂ))) v
-  rw [hh] at hc
-  have hs : ‖v x‖^2 ≤ ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 * ‖v‖^2 := by
-    nlinarith [norm_nonneg (v x),norm_nonneg v,
-      norm_nonneg (W.starProjection (EuclideanSpace.single x (1:ℂ)))]
-  calc
-    (Fintype.card X:ℝ) * ‖v x‖^2 ≤
-      (Fintype.card X:ℝ) * (‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 * ‖v‖^2) :=
-      mul_le_mul_of_nonneg_left hs (Nat.cast_nonneg _)
-    _ = _ := by rw [←mul_assoc,uniform_diagonal W htrans x]
-
-end
-open scoped BigOperators Classical
-
-variable {X : Type*} [Fintype X]
-
-lemma trace_projection_real (W : Submodule ℂ (H X)) :
-    ∑ x, ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 = (Module.finrank ℂ W:ℝ) := by
-  have hh := trace_projection W
-  simp_rw [projection_diagonal] at hh
-  exact_mod_cast hh
-
-theorem footprint_dimension (W K : Submodule ℂ (H X)) (hKW : K ≤ W)
-    (htrans : ∀ x y : X, ∃ p : Equiv.Perm X, p x = y ∧
-      W.map (act p).toLinearEquiv.toLinearMap = W)
-    (S : Finset X) (hS : ∀ x ∈ S, W.starProjection (EuclideanSpace.single x (1:ℂ)) ∈ K) :
-    (S.card:ℝ) * (Module.finrank ℂ W:ℝ) ≤
-      (Fintype.card X:ℝ) * (Module.finrank ℂ K:ℝ) := by
-  have he (x : X) (hx : x ∈ S) :
-      K.starProjection (EuclideanSpace.single x (1:ℂ)) =
-        W.starProjection (EuclideanSpace.single x (1:ℂ)) := by
-    apply K.eq_starProjection_of_mem_of_inner_eq_zero (hS x hx)
-    intro v hv
-    exact W.starProjection_inner_eq_zero _ v (hKW hv)
-  have hs : ∑ x ∈ S, ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 ≤
-      (Module.finrank ℂ K:ℝ) := by
-    rw [←trace_projection_real K]
-    calc
-      _ = ∑ x ∈ S, ‖K.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 := by
-        apply Finset.sum_congr rfl
-        intro x hx
-        rw [he x hx]
-      _ ≤ _ := Finset.sum_le_univ_sum_of_nonneg (fun x => sq_nonneg _)
-  have hmul := mul_le_mul_of_nonneg_left hs (Nat.cast_nonneg (Fintype.card X): (0:ℝ) ≤ _)
-  rw [Finset.mul_sum] at hmul
-  simp_rw [uniform_diagonal W htrans] at hmul
-  simpa using hmul
-
-end Thorp.PermutationHilbert
-namespace Thorp.Specht
-open scoped BigOperators Classical
-
-lemma finrank_hilbertSpace (μ : YoungDiagram) :
-    Module.finrank ℂ (hilbertSpace μ) = Module.finrank ℂ (space μ) :=
-  (hilbertEquiv μ).finrank_map_eq (space μ)
-
-lemma hilbert_action (μ : YoungDiagram) (p : Equiv.Perm (Cell μ)) (v : Tabloid μ → ℂ) :
-    hilbertEquiv μ (tabloidRep μ p v) =
-      PermutationHilbert.act (tabloidAct p) (hilbertEquiv μ v) := by
-  ext f
-  simp [hilbertEquiv,PermutationHilbert.act_apply]
-
-lemma hilbertSpace_invariant (μ : YoungDiagram) (p : Equiv.Perm (Cell μ)) :
-    (hilbertSpace μ).map (PermutationHilbert.act (tabloidAct p)).toLinearEquiv.toLinearMap =
-      hilbertSpace μ := by
-  apply le_antisymm
-  · rintro _ ⟨v,⟨w,hw,rfl⟩,rfl⟩
-    exact ⟨tabloidRep μ p w,action_mem_space μ p w hw,hilbert_action μ p w⟩
-  · rintro _ ⟨w,hw,rfl⟩
-    refine ⟨hilbertEquiv μ (tabloidRep μ p⁻¹ w),
-      ⟨tabloidRep μ p⁻¹ w,action_mem_space μ p⁻¹ w hw,rfl⟩, ?_⟩
-    change PermutationHilbert.act (tabloidAct p) (hilbertEquiv μ (tabloidRep μ p⁻¹ w)) = hilbertEquiv μ w
-    rw [←hilbert_action]
-    change hilbertEquiv μ ((tabloidRep μ p * tabloidRep μ p⁻¹) w) = _
-    rw [←map_mul,mul_inv_cancel,map_one]
-    rfl
-
-lemma tabloidAct_transitive (μ : YoungDiagram) (f g : Tabloid μ) :
-    ∃ p : Equiv.Perm (Cell μ), tabloidAct p f = g := by
-  obtain ⟨q,hq⟩ := f.2
-  obtain ⟨r,hr⟩ := g.2
-  refine ⟨r⁻¹*q, ?_⟩
-  apply Subtype.ext
-  funext x
-  simp only [tabloidAct_apply,mul_inv_rev,Equiv.Perm.mul_apply,hq,hr,
-    Function.comp_apply,Equiv.Perm.inv_def,Equiv.apply_symm_apply,Equiv.symm_symm]
-
-lemma polytabloid_pairing (μ : YoungDiagram) :
-    pairing (polytabloid μ) (polytabloid μ) = Fintype.card (colGroup μ) := by
-  have h := alternator_pairing μ (polytabloid μ)
-  have he (c : colGroup μ) : tabloidRep μ c (polytabloid μ) =
-      permSign (c:Equiv.Perm (Cell μ)) • polytabloid μ :=
-    action_alternator_apply (colGroup μ) (tabloidRep μ) c _
-  have hb := congrArg (fun v : Tabloid μ → ℂ => v (baseTabloid μ)) h
-  simp only [alternator,LinearMap.sum_apply,LinearMap.smul_apply,he,
-    Pi.smul_apply,smul_eq_mul,Finset.sum_apply,polytabloid_base,mul_one] at hb
-  simp only [permSign_sq,Finset.sum_const,Finset.card_univ,nsmul_eq_mul,mul_one] at hb
-  exact hb.symm
-
-lemma polytabloid_norm_sq (μ : YoungDiagram) :
-    ‖hilbertEquiv μ (polytabloid μ)‖^2 = (Fintype.card (colGroup μ):ℝ) := by
-  have hp := polytabloid_pairing μ
-  change inner ℂ (hilbertEquiv μ (polytabloid μ)) (hilbertEquiv μ (polytabloid μ)) = _ at hp
-  rw [inner_self_eq_norm_sq_to_K] at hp
-  have hre := congrArg Complex.re hp
-  simpa [pow_two] using hre
-
-theorem dimension_times_column_ge_tabloids (μ : YoungDiagram) :
-    Fintype.card (Tabloid μ) ≤ Module.finrank ℂ (space μ) * Fintype.card (colGroup μ) := by
-  have ht (f g : Tabloid μ) : ∃ p : Equiv.Perm (Tabloid μ), p f = g ∧
-      (hilbertSpace μ).map (PermutationHilbert.act p).toLinearEquiv.toLinearMap = hilbertSpace μ := by
-    obtain ⟨p,hp⟩ := tabloidAct_transitive μ f g
-    exact ⟨tabloidAct p,hp,hilbertSpace_invariant μ p⟩
-  have hu := PermutationHilbert.uncertainty (hilbertSpace μ) ht
-    (hilbertEquiv μ (polytabloid μ)) ⟨polytabloid μ,polytabloid_mem_space μ,rfl⟩ (baseTabloid μ)
-  have he : hilbertEquiv μ (polytabloid μ) (baseTabloid μ) = 1 := polytabloid_base μ
-  rw [he,norm_one,one_pow,mul_one,finrank_hilbertSpace,polytabloid_norm_sq] at hu
-  exact_mod_cast hu
-
-
-section FiberGroup
-variable {α β γ : Type*} [Fintype α] [Fintype β] [Fintype γ]
-
-noncomputable def fiberGroupEquiv (f : α → β) :
-    fiberGroup f ≃ (∀ b, Equiv.Perm {x // f x = b}) where
-  toFun p b :=
-    { toFun x := ⟨p.1 x.1,(p.2 x.1).trans x.2⟩
-      invFun x := ⟨p.1⁻¹ x.1, by
-        have hh := p.2 (p.1⁻¹ x.1)
-        simp only [Equiv.Perm.inv_def,Equiv.apply_symm_apply] at hh
-        exact hh.symm.trans x.2⟩
-      left_inv x := by apply Subtype.ext; simp
-      right_inv x := by apply Subtype.ext; simp }
-  invFun q :=
-    ⟨(Equiv.sigmaFiberEquiv f).permCongr (Equiv.Perm.sigmaCongrRight q),
-      fun x => (q (f x) ⟨x,rfl⟩).2⟩
-  left_inv p := by
-    apply Subtype.ext
-    apply Equiv.ext
-    intro x
-    rfl
-  right_inv q := by
-    funext b
-    apply Equiv.ext
-    intro x
-    apply Subtype.ext
-    change (q (f x.1) ⟨x.1,rfl⟩).1 = (q b x).1
-    cases x with
-    | mk x hx => dsimp only at *; subst b; rfl
-
-lemma nat_card_fiberGroup (f : α → β) :
-    Nat.card (fiberGroup f) = ∏ b, (Nat.card {x // f x=b}).factorial := by
-  rw [Nat.card_congr (fiberGroupEquiv f),Nat.card_pi]
-  simp only [Nat.card_eq_fintype_card,Fintype.card_perm]
-
-omit [Fintype β] [Fintype γ] in
-
-lemma card_fiberGroups_mul_le (f : α → β) (g : α → γ)
-    (hinj : ∀ x y, f x = f y → g x = g y → x=y) :
-    Fintype.card (fiberGroup f) * Fintype.card (fiberGroup g) ≤ (Fintype.card α).factorial := by
-  have hi : Function.Injective (fun p : fiberGroup f × fiberGroup g => p.1.1*p.2.1) := by
-    intro p q hpq
-    dsimp only at hpq
-    have he : p.1.1⁻¹*q.1.1 = p.2.1*q.2.1⁻¹ := by
-      apply inv_mul_eq_iff_eq_mul.mpr
-      rw [←mul_assoc,hpq,mul_assoc,mul_inv_cancel,mul_one]
-    have hf : p.1.1⁻¹*q.1.1 ∈ fiberGroup f := (fiberGroup f).mul_mem ((fiberGroup f).inv_mem p.1.2) q.1.2
-    have hg : p.1.1⁻¹*q.1.1 ∈ fiberGroup g := he ▸ (fiberGroup g).mul_mem p.2.2 ((fiberGroup g).inv_mem q.2.2)
-    have h1 : p.1.1⁻¹*q.1.1 = 1 := by
-      apply Equiv.ext
-      intro x
-      exact hinj _ x (hf x) (hg x)
-    have hpf : p.1=q.1 := Subtype.ext (inv_mul_eq_one.mp h1)
-    have hpg : p.2=q.2 := by
-      apply Subtype.ext
-      have hh := hpq
-      rw [show p.1.1=q.1.1 from congrArg Subtype.val hpf] at hh
-      exact mul_left_cancel hh
-    exact Prod.ext hpf hpg
-  have hc := Fintype.card_le_of_injective _ hi
-  simpa only [Fintype.card_prod,Fintype.card_perm] using hc
-
-end FiberGroup
-
-noncomputable def rowGroup (μ : YoungDiagram) : Subgroup (Equiv.Perm (Cell μ)) :=
-  fiberGroup (rowIndex (μ := μ))
-
-noncomputable def colIndex {μ : YoungDiagram} (x : Cell μ) : Fin (μ.rowLen 0) :=
-  ⟨col x, (YoungDiagram.mem_iff_lt_rowLen.mp x.2).trans_le (μ.rowLen_anti 0 (row x) (Nat.zero_le _))⟩
-
-lemma colGroup_eq (μ : YoungDiagram) : colGroup μ = fiberGroup (colIndex (μ := μ)) := by
-  ext p
-  constructor
-  · intro hp x; exact Fin.ext (hp x)
-  · intro hp x; exact congrArg Fin.val (hp x)
-
-noncomputable instance tabloidMulAction (μ : YoungDiagram) : MulAction (Equiv.Perm (Cell μ)) (Tabloid μ) where
-  smul p f := tabloidAct p f
-  one_smul f := by change tabloidAct 1 f = f; rw [tabloidAct_one]; rfl
-  mul_smul p q f := by change tabloidAct (p*q) f = tabloidAct p (tabloidAct q f); rw [tabloidAct_mul]; rfl
-
-lemma tabloid_orbit_all (μ : YoungDiagram) :
-    MulAction.orbit (Equiv.Perm (Cell μ)) (baseTabloid μ) = Set.univ := by
-  ext f
-  simp only [Set.mem_univ,iff_true]
-  obtain ⟨p,hp⟩ := tabloidAct_transitive μ (baseTabloid μ) f
-  exact ⟨p,hp⟩
-
-lemma tabloid_stabilizer (μ : YoungDiagram) :
-    MulAction.stabilizer (Equiv.Perm (Cell μ)) (baseTabloid μ) = rowGroup μ := by
-  ext p
-  change tabloidAct p (baseTabloid μ) = baseTabloid μ ↔ ∀ x, rowIndex (p x) = rowIndex x
-  constructor
-  · intro h x
-    have hh := congrArg (fun f : Tabloid μ => f.1 (p x)) h
-    simpa only [tabloidAct_apply,baseTabloid,Equiv.Perm.inv_def,Equiv.symm_apply_apply] using hh.symm
-  · intro h
-    apply Subtype.ext
-    funext x
-    exact (h (p⁻¹ x)).symm.trans (by simp [baseTabloid])
-
-lemma card_tabloid_mul_row (μ : YoungDiagram) :
-    Fintype.card (Tabloid μ)*Fintype.card (rowGroup μ) = μ.card.factorial := by
-  have hh := MulAction.card_orbit_mul_card_stabilizer_eq_card_group
-    (Equiv.Perm (Cell μ)) (baseTabloid μ)
-  have hc : Fintype.card (MulAction.orbit (Equiv.Perm (Cell μ)) (baseTabloid μ)) =
-      Fintype.card (Tabloid μ) := by
-    rw [tabloid_orbit_all,Fintype.card_setUniv]
-  simpa only [hc,tabloid_stabilizer,Fintype.card_perm,card_cell] using hh
-
-theorem factorial_le_dimension_row_column (μ : YoungDiagram) :
-    μ.card.factorial ≤ Module.finrank ℂ (space μ)*Fintype.card (rowGroup μ)*Fintype.card (colGroup μ) := by
-  rw [←card_tabloid_mul_row]
-  nlinarith [dimension_times_column_ge_tabloids μ]
-
-
-lemma card_row_fiber (μ : YoungDiagram) (i : Fin (μ.colLen 0)) :
-    Fintype.card {x : Cell μ // rowIndex x = i} = μ.rowLen i := by
-  let e : {x : Cell μ // rowIndex x = i} ≃ {x : Cell μ // row x = i} :=
-    Equiv.subtypeEquivRight (fun x => Fin.ext_iff)
-  rw [Fintype.card_congr e,Fintype.card_congr (rowEquiv μ i),Fintype.card_fin]
-
-lemma card_col_fiber (μ : YoungDiagram) (j : Fin (μ.rowLen 0)) :
-    Fintype.card {x : Cell μ // colIndex x = j} = μ.colLen j := by
-  let e : {x : Cell μ // colIndex x = j} ≃ {x : Cell μ // col x = j} :=
-    Equiv.subtypeEquivRight (fun x => Fin.ext_iff)
-  rw [Fintype.card_congr e,Fintype.card_congr (columnEquiv μ j),Fintype.card_fin]
-
-lemma card_rowGroup (μ : YoungDiagram) :
-    Fintype.card (rowGroup μ) = ∏ i : Fin (μ.colLen 0), (μ.rowLen i).factorial := by
-  rw [←Nat.card_eq_fintype_card,rowGroup,nat_card_fiberGroup]
-  simp only [Nat.card_eq_fintype_card]
-  simp only [card_row_fiber]
-
-lemma card_colGroup (μ : YoungDiagram) :
-    Fintype.card (colGroup μ) = ∏ j : Fin (μ.rowLen 0), (μ.colLen j).factorial := by
-  rw [←Nat.card_eq_fintype_card,colGroup_eq,nat_card_fiberGroup]
-  simp only [Nat.card_eq_fintype_card,card_col_fiber]
-
-abbrev TailCell (μ : YoungDiagram) := {x : Cell μ // row x ≠ 0}
-
-lemma tail_card (μ : YoungDiagram) : Fintype.card (TailCell μ) = μ.card - μ.rowLen 0 := by
-  have hh := Fintype.card_subtype_compl (p := fun x : Cell μ => row x=0)
-  rw [Fintype.card_congr (rowEquiv μ 0),Fintype.card_fin,card_cell] at hh
-  exact hh
-
-lemma rowLen_zero_le_card (μ : YoungDiagram) : μ.rowLen 0 ≤ μ.card := by
-  have h := Fintype.card_subtype_le (fun x : Cell μ => row x=0)
-  rwa [Fintype.card_congr (rowEquiv μ 0),Fintype.card_fin,card_cell] at h
-
-noncomputable def tailColumnEquiv (μ : YoungDiagram) (j : Fin (μ.rowLen 0)) :
-    {x : TailCell μ // colIndex x.1 = j} ≃ Fin (μ.colLen j-1) where
-  toFun x := ⟨row x.1.1-1, by
-    have hx := row_lt_colLen x.1.1
-    have hc : col x.1.1 = j := congrArg Fin.val x.2
-    rw [hc] at hx
-    have hz := x.1.2
-    omega⟩
-  invFun i := ⟨⟨⟨(i+1,j),YoungDiagram.mem_iff_lt_colLen.mpr (by omega)⟩,by simp [row]⟩,rfl⟩
-  left_inv x := by
-    apply Subtype.ext
-    apply Subtype.ext
-    apply Subtype.ext
-    apply Prod.ext
-    · change row x.1.1 - 1 + 1 = row x.1.1
-      have := x.1.2
-      omega
-    · exact (congrArg Fin.val x.2).symm
-  right_inv i := by apply Fin.ext; simp [row]
-
-lemma card_tail_col (μ : YoungDiagram) (j : Fin (μ.rowLen 0)) :
-    Fintype.card {x : TailCell μ // colIndex x.1 = j} = μ.colLen j-1 := by
-  rw [Fintype.card_congr (tailColumnEquiv μ j),Fintype.card_fin]
-
-lemma card_tail_row (μ : YoungDiagram) (i : Fin (μ.colLen 0)) :
-    Fintype.card {x : TailCell μ // rowIndex x.1 = i} = if i.val=0 then 0 else μ.rowLen i := by
-  by_cases hi : i.val=0
-  · rw [ite_eq_left hi]
-    apply Fintype.card_eq_zero_iff.mpr
-    refine ⟨fun x => x.1.2 ?_⟩
-    exact (congrArg Fin.val x.2).trans hi
-  · rw [ite_eq_right hi]
-    let e : {x : TailCell μ // rowIndex x.1 = i} ≃ {x : Cell μ // rowIndex x=i} :=
-      { toFun := fun x => ⟨x.1.1,x.2⟩
-        invFun := fun x => ⟨⟨x.1,fun hz => hi ((congrArg Fin.val x.2).symm.trans hz)⟩,x.2⟩
-        left_inv := fun x => rfl
-        right_inv := fun x => rfl }
-    rw [Fintype.card_congr e,card_row_fiber]
-
-lemma tail_row_group (μ : YoungDiagram) :
-    (μ.rowLen 0).factorial * Fintype.card (fiberGroup (fun x : TailCell μ => rowIndex x.1)) =
-      Fintype.card (rowGroup μ) := by
-  rw [←Nat.card_eq_fintype_card (α := fiberGroup (fun x : TailCell μ => rowIndex x.1)),nat_card_fiberGroup,card_rowGroup]
-  simp only [Nat.card_eq_fintype_card]
-  simp only [card_tail_row]
-  by_cases hn : μ.colLen 0=0
-  · have hr : μ.rowLen 0=0 := by
-      by_contra h
-      have hmem : (0,0) ∈ μ := YoungDiagram.mem_iff_lt_rowLen.mpr (Nat.pos_of_ne_zero h)
-      have hh := YoungDiagram.mem_iff_lt_colLen.mp hmem
-      omega
-    let : IsEmpty (Fin (μ.colLen 0)) := ⟨fun i => by have := i.isLt; omega⟩
-    simp [hr]
-  · let z : Fin (μ.colLen 0) := ⟨0,Nat.pos_of_ne_zero hn⟩
-    have hz : ∀ i : Fin (μ.colLen 0), i.val=0 ↔ i=z := fun i => by simp [z,Fin.ext_iff]
-    simp only [hz]
-    have hprod : (∏ x : Fin (μ.colLen 0), (if x=z then 0 else μ.rowLen x).factorial) =
-        ∏ x ∈ Finset.univ.erase z, (μ.rowLen x).factorial := by
-      rw [←Finset.prod_erase_mul Finset.univ (fun x : Fin (μ.colLen 0) => (if x=z then 0 else μ.rowLen x).factorial) (Finset.mem_univ z)]
-      simp only [ite_true,Nat.factorial_zero,mul_one]
-      apply Finset.prod_congr rfl
-      intro x hx
-      rw [ite_eq_right (Finset.mem_erase.mp hx).1]
-    rw [hprod]
-    exact Finset.mul_prod_erase Finset.univ (fun i : Fin (μ.colLen 0) => (μ.rowLen i).factorial) (Finset.mem_univ z)
-
-lemma sum_tail_columns (μ : YoungDiagram) :
-    ∑ j : Fin (μ.rowLen 0), (μ.colLen j-1) = μ.card-μ.rowLen 0 := by
-  simp_rw [←card_tail_col]
-  rw [←Fintype.card_sigma,Fintype.card_congr (Equiv.sigmaFiberEquiv (fun x : TailCell μ => colIndex x.1)),tail_card]
-
-lemma column_pos (μ : YoungDiagram) (j : Fin (μ.rowLen 0)) : 0 < μ.colLen j :=
-  YoungDiagram.mem_iff_lt_colLen.mp (YoungDiagram.mem_iff_lt_rowLen.mpr j.isLt)
-
-lemma nat_le_two_pow_pred (n : ℕ) (hn : 0<n) : n ≤ 2^(n-1) := by
-  obtain ⟨n,rfl⟩ := Nat.exists_eq_succ_of_ne_zero hn.ne'
-  simp only [Nat.add_one_sub_one]
-  induction n with
-  | zero => norm_num
-  | succ n ih => rw [pow_succ]; omega
-
-lemma column_group_le_tail (μ : YoungDiagram) :
-    Fintype.card (colGroup μ) ≤ 2^(μ.card-μ.rowLen 0) *
-      Fintype.card (fiberGroup (fun x : TailCell μ => colIndex x.1)) := by
-  rw [card_colGroup,←Nat.card_eq_fintype_card (α := fiberGroup (fun x : TailCell μ => colIndex x.1)),nat_card_fiberGroup]
-  simp only [Nat.card_eq_fintype_card]
-  simp only [card_tail_col]
-  calc
-    _ = ∏ j : Fin (μ.rowLen 0), μ.colLen j * (μ.colLen j-1).factorial := by
-      apply Finset.prod_congr rfl
-      intro j hj
-      exact (Nat.mul_factorial_pred (column_pos μ j).ne').symm
-    _ ≤ ∏ j : Fin (μ.rowLen 0), 2^(μ.colLen j-1) * (μ.colLen j-1).factorial := by
-      apply Finset.prod_le_prod
-      intro j hj
-      exact Nat.mul_le_mul_right _ (nat_le_two_pow_pred _ (column_pos μ j))
-    _ = _ := by rw [Finset.prod_mul_distrib,Finset.prod_pow_eq_pow_sum,sum_tail_columns]
-
-theorem choose_tail_le_dimension (μ : YoungDiagram) :
-    μ.card.choose (μ.card-μ.rowLen 0) ≤ Module.finrank ℂ (space μ) * 2^(μ.card-μ.rowLen 0) := by
-  let j := μ.card-μ.rowLen 0
-  have htail := card_fiberGroups_mul_le
-    (fun x : TailCell μ => rowIndex x.1) (fun x : TailCell μ => colIndex x.1)
-    (fun x y hr hc => Subtype.ext (Subtype.ext (Prod.ext (congrArg Fin.val hr) (congrArg Fin.val hc))))
-  rw [tail_card] at htail
-  have hc := column_group_le_tail μ
-  have hr := tail_row_group μ
-  have hd := factorial_le_dimension_row_column μ
-  simp only [←Nat.card_eq_fintype_card] at htail hc hr hd
-  have hh : μ.card.factorial ≤ Module.finrank ℂ (space μ) * 2^j *
-      ((μ.rowLen 0).factorial * j.factorial) := by
-    calc
-      _ ≤ _ := hd
-      _ ≤ Module.finrank ℂ (space μ) * Nat.card (rowGroup μ) *
-          (2^j * Nat.card (fiberGroup (fun x : TailCell μ => colIndex x.1))) := Nat.mul_le_mul_left _ hc
-      _ = Module.finrank ℂ (space μ) * 2^j * (μ.rowLen 0).factorial *
-          (Nat.card (fiberGroup (fun x : TailCell μ => rowIndex x.1)) *
-           Nat.card (fiberGroup (fun x : TailCell μ => colIndex x.1))) := by rw [←hr]; ring
-      _ ≤ _ := by
-        calc
-          _ ≤ Module.finrank ℂ (space μ) * 2^j * (μ.rowLen 0).factorial * j.factorial :=
-            Nat.mul_le_mul_left _ htail
-          _ = _ := by ring
-  have hj : j ≤ μ.card := Nat.sub_le _ _
-  have hs : μ.card-j = μ.rowLen 0 := by dsimp [j]; have := rowLen_zero_le_card μ; omega
-  have hf := Nat.choose_mul_factorial_mul_factorial hj
-  rw [hs] at hf
-  rw [←hf] at hh
-  have hpos : 0 < (μ.rowLen 0).factorial * j.factorial := Nat.mul_pos (Nat.factorial_pos _) (Nat.factorial_pos _)
-  nlinarith
-
-
-lemma exists_perm_mem_iff {α : Type*} [Fintype α] (s t : Finset α) (hc : s.card=t.card) :
-    ∃ p : Equiv.Perm α, ∀ x, p x ∈ t ↔ x ∈ s := by
-  let e : s ≃ t := Finset.equivOfCardEq hc
-  let ec : {x : α // x ∉ s} ≃ {x : α // x ∉ t} := Fintype.equivOfCardEq (by
-    simp only [Fintype.card_subtype_compl,Fintype.card_coe,hc])
-  let p := (Equiv.sumCompl (· ∈ s)).symm.trans ((Equiv.sumCongr e ec).trans (Equiv.sumCompl (· ∈ t)))
-  refine ⟨p,fun x => ?_⟩
-  by_cases hx : x ∈ s
-  · have hp : p x ∈ t := by
-      change ((Equiv.sumCompl (· ∈ t)) ((Equiv.sumCongr e ec) ((Equiv.sumCompl (· ∈ s)).symm x))) ∈ t
-      rw [Equiv.sumCompl_symm_apply_of_pos hx]
-      exact (e ⟨x,hx⟩).2
-    exact iff_of_true hp hx
-  · have hp : p x ∉ t := by
-      change ((Equiv.sumCompl (· ∈ t)) ((Equiv.sumCongr e ec) ((Equiv.sumCompl (· ∈ s)).symm x))) ∉ t
-      rw [Equiv.sumCompl_symm_apply_of_neg hx]
-      exact (ec ⟨x,hx⟩).2
-    exact iff_of_false hp hx
-
-noncomputable def tailSet {μ : YoungDiagram} (f : Tabloid μ) : Finset (Cell μ) :=
-  Finset.univ.filter (fun x => (f.1 x).val ≠ 0)
-
-@[simp] lemma mem_tailSet {μ : YoungDiagram} (f : Tabloid μ) (x : Cell μ) :
-    x ∈ tailSet f ↔ (f.1 x).val ≠ 0 := by simp [tailSet]
-
-lemma tailSet_act {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) (f : Tabloid μ) :
-    tailSet (tabloidAct p f) = (tailSet f).image p := by
-  ext x
-  simp only [mem_tailSet,tabloidAct_apply,Finset.mem_image]
-  constructor
-  · intro h; exact ⟨p⁻¹ x,h,by simp⟩
-  · rintro ⟨y,hy,rfl⟩; simpa using hy
-
-lemma card_tailSet {μ : YoungDiagram} (f : Tabloid μ) :
-    (tailSet f).card = μ.card-μ.rowLen 0 := by
-  obtain ⟨p,rfl⟩ := tabloidAct_transitive μ (baseTabloid μ) f
-  rw [tailSet_act,Finset.card_image_of_injective _ p.injective]
-  change (Finset.univ.filter (fun x : Cell μ => row x ≠ 0)).card = _
-  rw [←Fintype.card_subtype]
-  exact tail_card μ
-
-noncomputable def footprint (μ : YoungDiagram) (A : Finset (Cell μ)) : Finset (Tabloid μ) :=
-  Finset.univ.filter (fun f => tailSet f ⊆ A)
-
-lemma tail_fiber_card_eq (μ : YoungDiagram) (s t : Finset (Cell μ)) (hst : s.card=t.card) :
-    (Finset.univ.filter (fun f : Tabloid μ => tailSet f=s)).card =
-      (Finset.univ.filter (fun f : Tabloid μ => tailSet f=t)).card := by
-  obtain ⟨p,hp⟩ := exists_perm_mem_iff s t hst
-  have he : s.image p=t := by
-    ext x
-    constructor
-    · intro hx
-      obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
-      exact (hp y).mpr hy
-    · intro hx
-      exact Finset.mem_image.mpr ⟨p⁻¹ x,(hp (p⁻¹ x)).mp (by simpa using hx),by simp⟩
-  apply Finset.card_equiv (tabloidAct p)
-  intro f
-  simp only [Finset.mem_filter,Finset.mem_univ,true_and,tailSet_act,←he]
-  exact (Finset.image_inj p.injective).symm
-
-lemma footprint_card_product (μ : YoungDiagram) (A : Finset (Cell μ)) :
-    (footprint μ A).card * μ.card.choose (μ.card-μ.rowLen 0) =
-      Fintype.card (Tabloid μ) * A.card.choose (μ.card-μ.rowLen 0) := by
-  let j := μ.card-μ.rowLen 0
-  let s := tailSet (baseTabloid μ)
-  let c := (Finset.univ.filter (fun f : Tabloid μ => tailSet f=s)).card
-  have hconstant (t : Finset (Cell μ)) (ht : t.card=j) :
-      (Finset.univ.filter (fun f : Tabloid μ => tailSet f=t)).card = c :=
-    tail_fiber_card_eq μ t s (ht.trans (card_tailSet _).symm)
-  have hcount (A : Finset (Cell μ)) : (footprint μ A).card = A.card.choose j*c := by
-    have h := Finset.sum_card_fiberwise_eq_card_filter
-      (Finset.univ : Finset (Tabloid μ)) (A.powersetCard j) tailSet
-    have he : (Finset.univ.filter (fun f : Tabloid μ => tailSet f ∈ A.powersetCard j)) = footprint μ A := by
-      ext f
-      simp only [Finset.mem_filter,Finset.mem_univ,true_and,Finset.mem_powersetCard,
-        card_tailSet,footprint]
-      simp [j]
-    rw [he] at h
-    rw [←h]
-    calc
-      _ = ∑ _t ∈ A.powersetCard j, c := by
-        apply Finset.sum_congr rfl
-        intro t ht
-        exact hconstant t (Finset.mem_powersetCard.mp ht).2
-      _ = _ := by simp [Finset.card_powersetCard]
-  have hfull : (footprint μ Finset.univ).card = Fintype.card (Tabloid μ) := by
-    simp only [footprint,Finset.subset_univ,Finset.filter_true,Finset.card_univ]
-  have hN := hcount Finset.univ
-  rw [hfull,Finset.card_univ,card_cell] at hN
-  rw [hcount,hN]
-  ring
-
-lemma footprint_fixed {μ : YoungDiagram} (A : Finset (Cell μ))
-    (f : Tabloid μ) (hf : f ∈ footprint μ A) (p : Equiv.Perm (Cell μ))
-    (hp : ∀ x ∈ A, p x=x) : tabloidAct p f=f := by
-  have htail : tailSet f ⊆ A := (Finset.mem_filter.mp hf).2
-  apply Subtype.ext
-  funext x
-  change f.1 (p⁻¹ x) = f.1 x
-  by_cases hx : x ∈ A
-  · rw [show p⁻¹ x=x from p.injective (by simpa using (hp x hx).symm)]
-  · have hy : p⁻¹ x ∉ A := by
-      intro h
-      have hh := hp (p⁻¹ x) h
-      have he : x=p⁻¹ x := by simpa using hh
-      exact hx (he ▸ h)
-    apply Fin.ext
-    have hx0 : (f.1 x).val=0 := by by_contra h; exact hx (htail (mem_tailSet f x |>.mpr h))
-    have hy0 : (f.1 (p⁻¹ x)).val=0 := by by_contra h; exact hy (htail (mem_tailSet f _ |>.mpr h))
-    exact hy0.trans hx0.symm
-
-noncomputable def fixedHilbert (μ : YoungDiagram) (A : Finset (Cell μ)) :
-    Submodule ℂ (PermutationHilbert.H (Tabloid μ)) where
-  carrier := {v | v ∈ hilbertSpace μ ∧ ∀ p : Equiv.Perm (Cell μ),
-    (∀ x ∈ A, p x=x) → PermutationHilbert.act (tabloidAct p) v=v}
-  zero_mem' := ⟨Submodule.zero_mem _,fun p hp => map_zero _⟩
-  add_mem' := fun hv hw => ⟨Submodule.add_mem _ hv.1 hw.1,fun p hp => by
-    rw [map_add,hv.2 p hp,hw.2 p hp]⟩
-  smul_mem' := fun c v hv => ⟨Submodule.smul_mem _ c hv.1,fun p hp => by
-    rw [map_smul,hv.2 p hp]⟩
-
-lemma fixedHilbert_le (μ : YoungDiagram) (A : Finset (Cell μ)) : fixedHilbert μ A ≤ hilbertSpace μ :=
-  fun _ hv => hv.1
-
-theorem fixedHilbert_dimension (μ : YoungDiagram) (A : Finset (Cell μ)) :
-    Module.finrank ℂ (space μ) * A.card.choose (μ.card-μ.rowLen 0) ≤
-      Module.finrank ℂ (fixedHilbert μ A) * μ.card.choose (μ.card-μ.rowLen 0) := by
-  let : DecidableEq (Tabloid μ) := Classical.decEq _
-  have ht (f g : Tabloid μ) : ∃ p : Equiv.Perm (Tabloid μ), p f=g ∧
-      (hilbertSpace μ).map (PermutationHilbert.act p).toLinearEquiv.toLinearMap = hilbertSpace μ := by
-    obtain ⟨p,hp⟩ := tabloidAct_transitive μ f g
-    exact ⟨tabloidAct p,hp,hilbertSpace_invariant μ p⟩
-  have hproj (f : Tabloid μ) (hf : f ∈ footprint μ A) :
-      (hilbertSpace μ).starProjection (EuclideanSpace.single f (1:ℂ)) ∈ fixedHilbert μ A := by
-    refine ⟨((hilbertSpace μ).orthogonalProjectionOnto _).2,fun p hp => ?_⟩
-    rw [PermutationHilbert.projection_comm _ _ (hilbertSpace_invariant μ p),
-      PermutationHilbert.act_single,footprint_fixed A f hf p hp]
-  have h := PermutationHilbert.footprint_dimension (hilbertSpace μ) (fixedHilbert μ A)
-    (fixedHilbert_le μ A) ht (footprint μ A) hproj
-  rw [finrank_hilbertSpace] at h
-  have hn : (footprint μ A).card * Module.finrank ℂ (space μ) ≤
-      Fintype.card (Tabloid μ)*Module.finrank ℂ (fixedHilbert μ A) := by exact_mod_cast h
-  have hc := footprint_card_product μ A
-  have hpos : 0 < Fintype.card (Tabloid μ) := Fintype.card_pos_iff.mpr ⟨baseTabloid μ⟩
-  have hh := Nat.mul_le_mul_right (μ.card.choose (μ.card-μ.rowLen 0)) hn
-  nlinarith
-
-end Thorp.Specht
-
-namespace Thorp.UnitaryFinite
-open scoped BigOperators ComplexConjugate Classical
-
-variable {G : Type*} [Group G] [Fintype G]
-variable {V W : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-  [NormedAddCommGroup W] [InnerProductSpace ℂ W] [FiniteDimensional ℂ W]
-
-def IsUnitary (ρ : Representation ℂ G V) : Prop :=
-  ∀ g v w, inner ℂ (ρ g v) (ρ g w) = inner ℂ v w
-
-omit [Fintype G] [FiniteDimensional ℂ V] in
-lemma unitary_inner_inv (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (g : G) (v w : V) :
-    inner ℂ v (ρ g⁻¹ w) = inner ℂ (ρ g v) w := by
-  have h := hρ g v (ρ g⁻¹ w)
-  have he : ρ g (ρ g⁻¹ w) = w := by
-    change (ρ g * ρ g⁻¹) w = w
-    rw [←map_mul,mul_inv_cancel,map_one]
-    rfl
-  rw [he] at h
-  exact h.symm
-
-noncomputable def averageMap (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
-    (A : V →ₗ[ℂ] W) : Representation.IntertwiningMap ρ σ where
-  toLinearMap := ∑ g, (σ g).comp (A.comp (ρ g⁻¹))
-  isIntertwining' h := by
-    ext v
-    simp only [LinearMap.comp_apply,LinearMap.sum_apply,map_sum]
-    apply Fintype.sum_equiv (Equiv.mulLeft h⁻¹)
-    intro g
-    simp only [Equiv.coe_mulLeft,mul_inv_rev,inv_inv,map_mul,Module.End.mul_apply]
-    have hh (w : W) : σ h (σ h⁻¹ w)=w := by
-      change (σ h*σ h⁻¹) w=w
-      rw [←map_mul,mul_inv_cancel,map_one]
-      rfl
-    rw [hh]
-
-end Thorp.UnitaryFinite
-
-end ThorpNine.Tail
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Tail.ModifiedReverseEquiv
+
+namespace OAI
+
+namespace ThorpNine.Tail
+
+namespace Thorp.Specht
+open scoped BigOperators Classical
+
+noncomputable def shapePartitionEquiv (N : ℕ) : Shapes N ≃ Nat.Partition N where
+  toFun μ := { parts := μ.1.rowLens
+               parts_pos := fun {i} hi => μ.1.pos_of_mem_rowLens i (by simpa using hi)
+               parts_sum := by simpa using (sum_rowLens μ.1).trans μ.2 }
+  invFun p := ⟨partitionDiagram p,partitionDiagram_card p⟩
+  left_inv μ := by
+    apply Subtype.ext
+    obtain ⟨μ,rfl⟩ := μ
+    exact partitionDiagram_diagramPartition μ
+  right_inv p := by
+    apply Nat.Partition.ext
+    change ((partitionDiagram p).rowLens : Multiset ℕ) = p.parts
+    rw [partitionDiagram_rowLens,Multiset.sort_eq]
+
+noncomputable instance (N : ℕ) : Fintype (Shapes N) := Fintype.ofEquiv _ (shapePartitionEquiv N).symm
+
+noncomputable def conjugacyPartition {α : Type*} [Fintype α] [DecidableEq α] :
+    ConjClasses (Equiv.Perm α) → Nat.Partition (Fintype.card α) :=
+  Quotient.lift Equiv.Perm.partition (fun _ _ hpq => (Equiv.Perm.partition_eq_of_isConj).mp hpq)
+
+lemma conjugacyPartition_injective {α : Type*} [Fintype α] [DecidableEq α] :
+    Function.Injective (conjugacyPartition (α := α)) := by
+  intro p q
+  induction p using Quotient.inductionOn with | _ p =>
+    induction q using Quotient.inductionOn with | _ q =>
+      intro h
+      exact Quotient.sound ((Equiv.Perm.partition_eq_of_isConj).mpr h)
+
+lemma card_conjugacy_le_shapes (N : ℕ) :
+    Nat.card (ConjClasses (Equiv.Perm (Fin N))) ≤ Fintype.card (Shapes N) := by
+  have h := Nat.card_le_card_of_injective _ (conjugacyPartition_injective (α := Fin N))
+  simpa only [Fintype.card_fin, Nat.card_eq_fintype_card, Fintype.card_congr (shapePartitionEquiv N)] using h
+
+noncomputable def labels {N : ℕ} (μ : Shapes N) : Cell μ.1 ≃ Fin N :=
+  (Fintype.equivFin (Cell μ.1)).trans (finCongr ((card_cell μ.1).trans μ.2))
+
+noncomputable def labelledRepresentation {N : ℕ} (μ : Shapes N) :
+    Representation ℂ (Equiv.Perm (Fin N)) (space μ.1) :=
+  (representation μ.1).comp (labels μ).symm.permCongrHom.toMonoidHom
+
+lemma labelledRepresentation_irreducible {N : ℕ} (μ : Shapes N) :
+    Representation.IsIrreducible (labelledRepresentation μ) := by
+  let e := (labels μ).symm.permCongrHom
+  let ρ := representation μ.1
+  have hn : (⊥ : Subrepresentation (labelledRepresentation μ)) ≠ ⊤ := by
+    intro h
+    have he : (⟨polytabloid μ.1,polytabloid_mem_space μ.1⟩ : space μ.1) ∈
+        (⊥ : Subrepresentation (labelledRepresentation μ)) := by rw [h]; trivial
+    exact polytabloid_ne_zero μ.1 (congrArg Subtype.val he)
+  refine { exists_pair_ne := ⟨⊥,⊤,hn⟩,eq_bot_or_eq_top := ?_ }
+  intro U
+  let U' : Subrepresentation ρ :=
+    { toSubmodule := U.toSubmodule
+      apply_mem_toSubmodule := fun p v hv => by
+        have hh := U.apply_mem_toSubmodule (e.symm p) hv
+        change ρ (e (e.symm p)) v ∈ U.toSubmodule at hh
+        simpa using hh }
+  have := representation_irreducible μ.1
+  obtain h | h := IsSimpleOrder.eq_bot_or_eq_top U'
+  · left
+    apply Subrepresentation.toSubmodule_injective
+    exact congrArg (fun R : Subrepresentation ρ => R.toSubmodule) h
+  · right
+    apply Subrepresentation.toSubmodule_injective
+    exact congrArg (fun R : Subrepresentation ρ => R.toSubmodule) h
+
+lemma labelledRepresentation_equiv_shapes {N : ℕ} (μ ν : Shapes N)
+    (T : Representation.Equiv (labelledRepresentation μ) (labelledRepresentation ν)) : μ = ν := by
+  apply Subtype.ext
+  apply shapes_eq_of_equivariant_equiv μ.1 ν.1 ((labels μ).trans (labels ν).symm) T.toLinearEquiv
+  intro p v
+  have hh := Representation.IntertwiningMap.isIntertwining _ _ T.toIntertwiningMap ((labels μ).permCongr p) v
+  change T.toLinearEquiv
+    (representation μ.1 ((labels μ).symm.permCongr ((labels μ).permCongr p)) v) =
+      representation ν.1 ((labels ν).symm.permCongr ((labels μ).permCongr p)) (T.toLinearEquiv v) at hh
+  have hp : (labels μ).symm.permCongr ((labels μ).permCongr p) = p := by
+    apply Equiv.ext
+    intro x
+    simp [Equiv.permCongr_apply]
+  rw [hp] at hh
+  convert hh using 2
+  congr 1
+
+end Thorp.Specht
+
+namespace Thorp.FiniteCharacters
+open scoped BigOperators Classical
+
+section Group
+variable {G : Type*} [Group G] [Fintype G]
+
+noncomputable def classRep (c : ConjClasses G) : G := (ConjClasses.exists_rep c).choose
+omit [Fintype G] in
+lemma classRep_spec (c : ConjClasses G) : ConjClasses.mk (classRep c) = c :=
+  (ConjClasses.exists_rep c).choose_spec
+
+variable {V : Type*} [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
+
+noncomputable def characterClass (ρ : Representation ℂ G V) : ConjClasses G → ℂ :=
+  fun c => ρ.character (classRep c)
+
+omit [Fintype G] [FiniteDimensional ℂ V] in
+lemma characterClass_mk (ρ : Representation ℂ G V) (g : G) :
+    characterClass ρ (ConjClasses.mk g) = ρ.character g := by
+  have h := ConjClasses.mk_eq_mk_iff_isConj.mp (classRep_spec (ConjClasses.mk g))
+  obtain ⟨p,hp⟩ := isConj_iff.mp h
+  change ρ.character (classRep (ConjClasses.mk g)) = ρ.character g
+  exact (Representation.char_conj ρ _ p).symm.trans (congrArg ρ.character hp)
+
+noncomputable def characterDual (ρ : Representation ℂ G V) :
+    (ConjClasses G → ℂ) →ₗ[ℂ] ℂ where
+  toFun f := (Nat.card G : ℂ)⁻¹ * ∑ g, f (ConjClasses.mk g) * ρ.character g⁻¹
+  map_add' f h := by simp [Finset.sum_add_distrib,add_mul,mul_add]
+  map_smul' a f := by
+    simp only [Pi.smul_apply,smul_eq_mul,RingHom.id_apply]
+    simp_rw [mul_assoc]
+    rw [←Finset.mul_sum]
+    ring
+
+lemma characterDual_characterClass (ρ : Representation ℂ G V)
+    {W : Type*} [AddCommGroup W] [Module ℂ W] [FiniteDimensional ℂ W]
+    (σ : Representation ℂ G W) [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ] :
+    characterDual ρ (characterClass σ) = if Nonempty (Representation.Equiv ρ σ) then 1 else 0 := by
+  let : Invertible (Nat.card G : ℂ) := invertibleOfNonzero (by
+    exact_mod_cast (Nat.card_pos (α := G)).ne')
+  change (Nat.card G : ℂ)⁻¹ * ∑ g, characterClass σ (ConjClasses.mk g) * ρ.character g⁻¹ = _
+  simp only [characterClass_mk]
+  exact Representation.char_orthonormal σ ρ
+
+end Group
+
+theorem complete_of_card_conjugacy_le
+    {G ι : Type*} [Group G] [Fintype G] [Fintype ι]
+    (V : ι → Type*) [∀ i, AddCommGroup (V i)] [∀ i, Module ℂ (V i)]
+    [∀ i, FiniteDimensional ℂ (V i)]
+    (ρ : ∀ i, Representation ℂ G (V i)) [∀ i, Representation.IsIrreducible (ρ i)]
+    (hneq : ∀ i j, i ≠ j → ¬Nonempty (Representation.Equiv (ρ i) (ρ j)))
+    (hcard : Nat.card (ConjClasses G) ≤ Fintype.card ι)
+    {W : Type*} [AddCommGroup W] [Module ℂ W] [FiniteDimensional ℂ W]
+    (σ : Representation ℂ G W) [Representation.IsIrreducible σ] :
+    ∃ i, Nonempty (Representation.Equiv (ρ i) σ) := by
+  let : Fintype (ConjClasses G) := Fintype.ofFinite _
+  have horth (i j : ι) : characterDual (ρ i) (characterClass (ρ j)) = if i=j then 1 else 0 := by
+    rw [characterDual_characterClass]
+    by_cases hij : i=j
+    · subst j
+      simp only [ite_true]
+      exact ite_eq_left ⟨Representation.Equiv.refl _⟩
+    · rw [ite_eq_right hij,ite_eq_right (hneq i j hij)]
+  have hLI : LinearIndependent ℂ (fun i => characterClass (ρ i)) := by
+    apply LinearIndependent.of_pairwise_dual_eq_zero_one _ (fun i => characterDual (ρ i))
+    · intro i j hij
+      exact (horth i j).trans (ite_eq_right hij)
+    · intro i
+      simpa using horth i i
+  have heq : Fintype.card ι = Module.finrank ℂ (ConjClasses G → ℂ) := by
+    apply Nat.le_antisymm hLI.fintype_card_le_finrank
+    simpa only [Module.finrank_pi,Module.finrank_self,Finset.sum_const,Finset.card_univ,
+      smul_eq_mul,mul_one,Nat.card_eq_fintype_card] using hcard
+  have hspan := hLI.span_eq_top_of_card_eq_finrank' heq
+  by_contra h
+  push Not at h
+  have hker : Submodule.span ℂ (Set.range (fun i => characterClass (ρ i))) ≤
+      (characterDual σ).ker := by
+    apply Submodule.span_le.mpr
+    rintro f ⟨i,rfl⟩
+    change characterDual σ (characterClass (ρ i)) = 0
+    rw [characterDual_characterClass,ite_eq_right]
+    rintro ⟨T⟩
+    exact (h i).false T.symm
+  rw [hspan] at hker
+  have hz : characterDual σ (characterClass σ) = 0 := hker (show characterClass σ ∈ ⊤ from trivial)
+  rw [characterDual_characterClass,ite_eq_left ⟨Representation.Equiv.refl σ⟩] at hz
+  exact one_ne_zero hz
+
+end Thorp.FiniteCharacters
+
+namespace Thorp.Specht
+open scoped Classical
+
+theorem labelled_complete {N : ℕ}
+    {V : Type*} [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
+    (ρ : Representation ℂ (Equiv.Perm (Fin N)) V) [Representation.IsIrreducible ρ] :
+    ∃ μ : Shapes N, Nonempty (Representation.Equiv (labelledRepresentation μ) ρ) := by
+  let (μ : Shapes N) : Representation.IsIrreducible (labelledRepresentation μ) :=
+    labelledRepresentation_irreducible μ
+  apply FiniteCharacters.complete_of_card_conjugacy_le
+    (fun μ : Shapes N => space μ.1) labelledRepresentation _ (card_conjugacy_le_shapes N) ρ
+  intro μ ν h ⟨T⟩
+  exact h (labelledRepresentation_equiv_shapes μ ν T)
+
+end Thorp.Specht
+
+namespace Thorp.PermutationHilbert
+
+section
+open scoped BigOperators ComplexConjugate Classical
+open Complex
+
+variable {X : Type*} [Fintype X]
+
+noncomputable def act (p : Equiv.Perm X) : H X ≃ₗᵢ[ℂ] H X :=
+  LinearIsometryEquiv.piLpCongrLeft 2 ℂ ℂ p
+
+@[simp] lemma act_apply (p : Equiv.Perm X) (v : H X) (x : X) :
+    act p v x = v (p⁻¹ x) := rfl
+
+lemma act_single [DecidableEq X] (p : Equiv.Perm X) (x : X) :
+    act p (EuclideanSpace.single x (1:ℂ)) = EuclideanSpace.single (p x) 1 := by
+  exact EuclideanSpace.piLpCongrLeft_single p x 1
+
+lemma trace_projection (W : Submodule ℂ (H X)) :
+    ∑ x, (W.starProjection (EuclideanSpace.single x (1:ℂ))) x = Module.finrank ℂ W := by
+  classical
+  have hp : LinearMap.IsProj W W.starProjection.toLinearMap := by
+    constructor
+    · intro x; exact (W.orthogonalProjectionOnto x).2
+    · intro x hx; exact W.starProjection_eq_self_iff.mpr hx
+  have ht := hp.trace
+  rw [LinearMap.trace_eq_matrix_trace ℂ (EuclideanSpace.basisFun X ℂ).toBasis] at ht
+  simpa only [Matrix.trace,Matrix.diag_apply,LinearMap.toMatrix_apply,EuclideanSpace.basisFun_repr,
+    EuclideanSpace.basisFun_apply,OrthonormalBasis.coe_toBasis,
+    OrthonormalBasis.coe_toBasis_repr_apply,ContinuousLinearMap.coe_coe] using ht
+
+lemma projection_diagonal (W : Submodule ℂ (H X)) (x : X) :
+    (W.starProjection (EuclideanSpace.single x (1:ℂ))) x =
+      (‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 : ℝ) := by
+  classical
+  have hh := W.inner_starProjection_left_eq_right
+    (EuclideanSpace.single x (1:ℂ)) (W.starProjection (EuclideanSpace.single x (1:ℂ)))
+  have hf : W.starProjection (W.starProjection (EuclideanSpace.single x (1:ℂ))) =
+      W.starProjection (EuclideanSpace.single x (1:ℂ)) :=
+    W.starProjection_eq_self_iff.mpr (W.orthogonalProjectionOnto _).2
+  rw [hf] at hh
+  rw [inner_self_eq_norm_sq_to_K,EuclideanSpace.inner_single_left] at hh
+  simpa using hh.symm
+
+lemma projection_comm (W : Submodule ℂ (H X)) (p : Equiv.Perm X)
+    (hW : W.map (act p).toLinearEquiv.toLinearMap = W) (v : H X) :
+    act p (W.starProjection v) = W.starProjection (act p v) := by
+  have hh := (act p).toLinearIsometry.map_starProjection W v
+  change act p (W.starProjection v) =
+    (W.map (act p).toLinearEquiv.toLinearMap).starProjection (act p v) at hh
+  simpa only [hW] using hh
+
+lemma uniform_diagonal (W : Submodule ℂ (H X))
+    (htrans : ∀ x y : X, ∃ p : Equiv.Perm X, p x = y ∧
+      W.map (act p).toLinearEquiv.toLinearMap = W) (x : X) :
+    (Fintype.card X:ℝ) * ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 =
+      Module.finrank ℂ W := by
+  classical
+  have hh (y : X) : (W.starProjection (EuclideanSpace.single y (1:ℂ))) y =
+      (‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 : ℝ) := by
+    obtain ⟨p,hp,hW⟩ := htrans x y
+    subst y
+    rw [projection_diagonal,←act_single p x,←projection_comm W p hW,(act p).norm_map]
+  have ht := trace_projection W
+  simp_rw [hh] at ht
+  have hhreal : (∑ _y : X, ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2) =
+      (Module.finrank ℂ W:ℝ) := by exact_mod_cast ht
+  simpa using hhreal
+
+theorem uncertainty (W : Submodule ℂ (H X))
+    (htrans : ∀ x y : X, ∃ p : Equiv.Perm X, p x = y ∧
+      W.map (act p).toLinearEquiv.toLinearMap = W)
+    (v : H X) (hv : v ∈ W) (x : X) :
+    (Fintype.card X:ℝ) * ‖v x‖^2 ≤ (Module.finrank ℂ W:ℝ) * ‖v‖^2 := by
+  classical
+  have hh : inner ℂ (W.starProjection (EuclideanSpace.single x (1:ℂ))) v = v x := by
+    rw [W.inner_starProjection_left_eq_right,W.starProjection_eq_self_iff.mpr hv,
+      EuclideanSpace.inner_single_left]
+    simp
+  have hc := norm_inner_le_norm (𝕜 := ℂ) (W.starProjection (EuclideanSpace.single x (1:ℂ))) v
+  rw [hh] at hc
+  have hs : ‖v x‖^2 ≤ ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 * ‖v‖^2 := by
+    nlinarith [norm_nonneg (v x),norm_nonneg v,
+      norm_nonneg (W.starProjection (EuclideanSpace.single x (1:ℂ)))]
+  calc
+    (Fintype.card X:ℝ) * ‖v x‖^2 ≤
+      (Fintype.card X:ℝ) * (‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 * ‖v‖^2) :=
+      mul_le_mul_of_nonneg_left hs (Nat.cast_nonneg _)
+    _ = _ := by rw [←mul_assoc,uniform_diagonal W htrans x]
+
+end
+open scoped BigOperators Classical
+
+variable {X : Type*} [Fintype X]
+
+lemma trace_projection_real (W : Submodule ℂ (H X)) :
+    ∑ x, ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 = (Module.finrank ℂ W:ℝ) := by
+  have hh := trace_projection W
+  simp_rw [projection_diagonal] at hh
+  exact_mod_cast hh
+
+theorem footprint_dimension (W K : Submodule ℂ (H X)) (hKW : K ≤ W)
+    (htrans : ∀ x y : X, ∃ p : Equiv.Perm X, p x = y ∧
+      W.map (act p).toLinearEquiv.toLinearMap = W)
+    (S : Finset X) (hS : ∀ x ∈ S, W.starProjection (EuclideanSpace.single x (1:ℂ)) ∈ K) :
+    (S.card:ℝ) * (Module.finrank ℂ W:ℝ) ≤
+      (Fintype.card X:ℝ) * (Module.finrank ℂ K:ℝ) := by
+  have he (x : X) (hx : x ∈ S) :
+      K.starProjection (EuclideanSpace.single x (1:ℂ)) =
+        W.starProjection (EuclideanSpace.single x (1:ℂ)) := by
+    apply K.eq_starProjection_of_mem_of_inner_eq_zero (hS x hx)
+    intro v hv
+    exact W.starProjection_inner_eq_zero _ v (hKW hv)
+  have hs : ∑ x ∈ S, ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 ≤
+      (Module.finrank ℂ K:ℝ) := by
+    rw [←trace_projection_real K]
+    calc
+      _ = ∑ x ∈ S, ‖K.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 := by
+        apply Finset.sum_congr rfl
+        intro x hx
+        rw [he x hx]
+      _ ≤ _ := Finset.sum_le_univ_sum_of_nonneg (fun x => sq_nonneg _)
+  have hmul := mul_le_mul_of_nonneg_left hs (Nat.cast_nonneg (Fintype.card X): (0:ℝ) ≤ _)
+  rw [Finset.mul_sum] at hmul
+  simp_rw [uniform_diagonal W htrans] at hmul
+  simpa using hmul
+
+end Thorp.PermutationHilbert
+namespace Thorp.Specht
+open scoped BigOperators Classical
+
+lemma finrank_hilbertSpace (μ : YoungDiagram) :
+    Module.finrank ℂ (hilbertSpace μ) = Module.finrank ℂ (space μ) :=
+  (hilbertEquiv μ).finrank_map_eq (space μ)
+
+lemma hilbert_action (μ : YoungDiagram) (p : Equiv.Perm (Cell μ)) (v : Tabloid μ → ℂ) :
+    hilbertEquiv μ (tabloidRep μ p v) =
+      PermutationHilbert.act (tabloidAct p) (hilbertEquiv μ v) := by
+  ext f
+  simp [hilbertEquiv,PermutationHilbert.act_apply]
+
+lemma hilbertSpace_invariant (μ : YoungDiagram) (p : Equiv.Perm (Cell μ)) :
+    (hilbertSpace μ).map (PermutationHilbert.act (tabloidAct p)).toLinearEquiv.toLinearMap =
+      hilbertSpace μ := by
+  apply le_antisymm
+  · rintro _ ⟨v,⟨w,hw,rfl⟩,rfl⟩
+    exact ⟨tabloidRep μ p w,action_mem_space μ p w hw,hilbert_action μ p w⟩
+  · rintro _ ⟨w,hw,rfl⟩
+    refine ⟨hilbertEquiv μ (tabloidRep μ p⁻¹ w),
+      ⟨tabloidRep μ p⁻¹ w,action_mem_space μ p⁻¹ w hw,rfl⟩, ?_⟩
+    change PermutationHilbert.act (tabloidAct p) (hilbertEquiv μ (tabloidRep μ p⁻¹ w)) = hilbertEquiv μ w
+    rw [←hilbert_action]
+    change hilbertEquiv μ ((tabloidRep μ p * tabloidRep μ p⁻¹) w) = _
+    rw [←map_mul,mul_inv_cancel,map_one]
+    rfl
+
+lemma tabloidAct_transitive (μ : YoungDiagram) (f g : Tabloid μ) :
+    ∃ p : Equiv.Perm (Cell μ), tabloidAct p f = g := by
+  obtain ⟨q,hq⟩ := f.2
+  obtain ⟨r,hr⟩ := g.2
+  refine ⟨r⁻¹*q, ?_⟩
+  apply Subtype.ext
+  funext x
+  simp only [tabloidAct_apply,mul_inv_rev,Equiv.Perm.mul_apply,hq,hr,
+    Function.comp_apply,Equiv.Perm.inv_def,Equiv.apply_symm_apply,Equiv.symm_symm]
+
+lemma polytabloid_pairing (μ : YoungDiagram) :
+    pairing (polytabloid μ) (polytabloid μ) = Fintype.card (colGroup μ) := by
+  have h := alternator_pairing μ (polytabloid μ)
+  have he (c : colGroup μ) : tabloidRep μ c (polytabloid μ) =
+      permSign (c:Equiv.Perm (Cell μ)) • polytabloid μ :=
+    action_alternator_apply (colGroup μ) (tabloidRep μ) c _
+  have hb := congrArg (fun v : Tabloid μ → ℂ => v (baseTabloid μ)) h
+  simp only [alternator,LinearMap.sum_apply,LinearMap.smul_apply,he,
+    Pi.smul_apply,smul_eq_mul,Finset.sum_apply,polytabloid_base,mul_one] at hb
+  simp only [permSign_sq,Finset.sum_const,Finset.card_univ,nsmul_eq_mul,mul_one] at hb
+  exact hb.symm
+
+lemma polytabloid_norm_sq (μ : YoungDiagram) :
+    ‖hilbertEquiv μ (polytabloid μ)‖^2 = (Fintype.card (colGroup μ):ℝ) := by
+  have hp := polytabloid_pairing μ
+  change inner ℂ (hilbertEquiv μ (polytabloid μ)) (hilbertEquiv μ (polytabloid μ)) = _ at hp
+  rw [inner_self_eq_norm_sq_to_K] at hp
+  have hre := congrArg Complex.re hp
+  simpa [pow_two] using hre
+
+theorem dimension_times_column_ge_tabloids (μ : YoungDiagram) :
+    Fintype.card (Tabloid μ) ≤ Module.finrank ℂ (space μ) * Fintype.card (colGroup μ) := by
+  have ht (f g : Tabloid μ) : ∃ p : Equiv.Perm (Tabloid μ), p f = g ∧
+      (hilbertSpace μ).map (PermutationHilbert.act p).toLinearEquiv.toLinearMap = hilbertSpace μ := by
+    obtain ⟨p,hp⟩ := tabloidAct_transitive μ f g
+    exact ⟨tabloidAct p,hp,hilbertSpace_invariant μ p⟩
+  have hu := PermutationHilbert.uncertainty (hilbertSpace μ) ht
+    (hilbertEquiv μ (polytabloid μ)) ⟨polytabloid μ,polytabloid_mem_space μ,rfl⟩ (baseTabloid μ)
+  have he : hilbertEquiv μ (polytabloid μ) (baseTabloid μ) = 1 := polytabloid_base μ
+  rw [he,norm_one,one_pow,mul_one,finrank_hilbertSpace,polytabloid_norm_sq] at hu
+  exact_mod_cast hu
+
+
+section FiberGroup
+variable {α β γ : Type*} [Fintype α] [Fintype β] [Fintype γ]
+
+noncomputable def fiberGroupEquiv (f : α → β) :
+    fiberGroup f ≃ (∀ b, Equiv.Perm {x // f x = b}) where
+  toFun p b :=
+    { toFun x := ⟨p.1 x.1,(p.2 x.1).trans x.2⟩
+      invFun x := ⟨p.1⁻¹ x.1, by
+        have hh := p.2 (p.1⁻¹ x.1)
+        simp only [Equiv.Perm.inv_def,Equiv.apply_symm_apply] at hh
+        exact hh.symm.trans x.2⟩
+      left_inv x := by apply Subtype.ext; simp
+      right_inv x := by apply Subtype.ext; simp }
+  invFun q :=
+    ⟨(Equiv.sigmaFiberEquiv f).permCongr (Equiv.Perm.sigmaCongrRight q),
+      fun x => (q (f x) ⟨x,rfl⟩).2⟩
+  left_inv p := by
+    apply Subtype.ext
+    apply Equiv.ext
+    intro x
+    rfl
+  right_inv q := by
+    funext b
+    apply Equiv.ext
+    intro x
+    apply Subtype.ext
+    change (q (f x.1) ⟨x.1,rfl⟩).1 = (q b x).1
+    cases x with
+    | mk x hx => dsimp only at *; subst b; rfl
+
+lemma nat_card_fiberGroup (f : α → β) :
+    Nat.card (fiberGroup f) = ∏ b, (Nat.card {x // f x=b}).factorial := by
+  rw [Nat.card_congr (fiberGroupEquiv f),Nat.card_pi]
+  simp only [Nat.card_eq_fintype_card,Fintype.card_perm]
+
+omit [Fintype β] [Fintype γ] in
+
+lemma card_fiberGroups_mul_le (f : α → β) (g : α → γ)
+    (hinj : ∀ x y, f x = f y → g x = g y → x=y) :
+    Fintype.card (fiberGroup f) * Fintype.card (fiberGroup g) ≤ (Fintype.card α).factorial := by
+  have hi : Function.Injective (fun p : fiberGroup f × fiberGroup g => p.1.1*p.2.1) := by
+    intro p q hpq
+    dsimp only at hpq
+    have he : p.1.1⁻¹*q.1.1 = p.2.1*q.2.1⁻¹ := by
+      apply inv_mul_eq_iff_eq_mul.mpr
+      rw [←mul_assoc,hpq,mul_assoc,mul_inv_cancel,mul_one]
+    have hf : p.1.1⁻¹*q.1.1 ∈ fiberGroup f := (fiberGroup f).mul_mem ((fiberGroup f).inv_mem p.1.2) q.1.2
+    have hg : p.1.1⁻¹*q.1.1 ∈ fiberGroup g := he ▸ (fiberGroup g).mul_mem p.2.2 ((fiberGroup g).inv_mem q.2.2)
+    have h1 : p.1.1⁻¹*q.1.1 = 1 := by
+      apply Equiv.ext
+      intro x
+      exact hinj _ x (hf x) (hg x)
+    have hpf : p.1=q.1 := Subtype.ext (inv_mul_eq_one.mp h1)
+    have hpg : p.2=q.2 := by
+      apply Subtype.ext
+      have hh := hpq
+      rw [show p.1.1=q.1.1 from congrArg Subtype.val hpf] at hh
+      exact mul_left_cancel hh
+    exact Prod.ext hpf hpg
+  have hc := Fintype.card_le_of_injective _ hi
+  simpa only [Fintype.card_prod,Fintype.card_perm] using hc
+
+end FiberGroup
+
+noncomputable def rowGroup (μ : YoungDiagram) : Subgroup (Equiv.Perm (Cell μ)) :=
+  fiberGroup (rowIndex (μ := μ))
+
+noncomputable def colIndex {μ : YoungDiagram} (x : Cell μ) : Fin (μ.rowLen 0) :=
+  ⟨col x, (YoungDiagram.mem_iff_lt_rowLen.mp x.2).trans_le (μ.rowLen_anti 0 (row x) (Nat.zero_le _))⟩
+
+lemma colGroup_eq (μ : YoungDiagram) : colGroup μ = fiberGroup (colIndex (μ := μ)) := by
+  ext p
+  constructor
+  · intro hp x; exact Fin.ext (hp x)
+  · intro hp x; exact congrArg Fin.val (hp x)
+
+noncomputable instance tabloidMulAction (μ : YoungDiagram) : MulAction (Equiv.Perm (Cell μ)) (Tabloid μ) where
+  smul p f := tabloidAct p f
+  one_smul f := by change tabloidAct 1 f = f; rw [tabloidAct_one]; rfl
+  mul_smul p q f := by change tabloidAct (p*q) f = tabloidAct p (tabloidAct q f); rw [tabloidAct_mul]; rfl
+
+lemma tabloid_orbit_all (μ : YoungDiagram) :
+    MulAction.orbit (Equiv.Perm (Cell μ)) (baseTabloid μ) = Set.univ := by
+  ext f
+  simp only [Set.mem_univ,iff_true]
+  obtain ⟨p,hp⟩ := tabloidAct_transitive μ (baseTabloid μ) f
+  exact ⟨p,hp⟩
+
+lemma tabloid_stabilizer (μ : YoungDiagram) :
+    MulAction.stabilizer (Equiv.Perm (Cell μ)) (baseTabloid μ) = rowGroup μ := by
+  ext p
+  change tabloidAct p (baseTabloid μ) = baseTabloid μ ↔ ∀ x, rowIndex (p x) = rowIndex x
+  constructor
+  · intro h x
+    have hh := congrArg (fun f : Tabloid μ => f.1 (p x)) h
+    simpa only [tabloidAct_apply,baseTabloid,Equiv.Perm.inv_def,Equiv.symm_apply_apply] using hh.symm
+  · intro h
+    apply Subtype.ext
+    funext x
+    exact (h (p⁻¹ x)).symm.trans (by simp [baseTabloid])
+
+lemma card_tabloid_mul_row (μ : YoungDiagram) :
+    Fintype.card (Tabloid μ)*Fintype.card (rowGroup μ) = μ.card.factorial := by
+  have hh := MulAction.card_orbit_mul_card_stabilizer_eq_card_group
+    (Equiv.Perm (Cell μ)) (baseTabloid μ)
+  have hc : Fintype.card (MulAction.orbit (Equiv.Perm (Cell μ)) (baseTabloid μ)) =
+      Fintype.card (Tabloid μ) := by
+    rw [tabloid_orbit_all,Fintype.card_setUniv]
+  simpa only [hc,tabloid_stabilizer,Fintype.card_perm,card_cell] using hh
+
+theorem factorial_le_dimension_row_column (μ : YoungDiagram) :
+    μ.card.factorial ≤ Module.finrank ℂ (space μ)*Fintype.card (rowGroup μ)*Fintype.card (colGroup μ) := by
+  rw [←card_tabloid_mul_row]
+  nlinarith [dimension_times_column_ge_tabloids μ]
+
+
+lemma card_row_fiber (μ : YoungDiagram) (i : Fin (μ.colLen 0)) :
+    Fintype.card {x : Cell μ // rowIndex x = i} = μ.rowLen i := by
+  let e : {x : Cell μ // rowIndex x = i} ≃ {x : Cell μ // row x = i} :=
+    Equiv.subtypeEquivRight (fun x => Fin.ext_iff)
+  rw [Fintype.card_congr e,Fintype.card_congr (rowEquiv μ i),Fintype.card_fin]
+
+lemma card_col_fiber (μ : YoungDiagram) (j : Fin (μ.rowLen 0)) :
+    Fintype.card {x : Cell μ // colIndex x = j} = μ.colLen j := by
+  let e : {x : Cell μ // colIndex x = j} ≃ {x : Cell μ // col x = j} :=
+    Equiv.subtypeEquivRight (fun x => Fin.ext_iff)
+  rw [Fintype.card_congr e,Fintype.card_congr (columnEquiv μ j),Fintype.card_fin]
+
+lemma card_rowGroup (μ : YoungDiagram) :
+    Fintype.card (rowGroup μ) = ∏ i : Fin (μ.colLen 0), (μ.rowLen i).factorial := by
+  rw [←Nat.card_eq_fintype_card,rowGroup,nat_card_fiberGroup]
+  simp only [Nat.card_eq_fintype_card]
+  simp only [card_row_fiber]
+
+lemma card_colGroup (μ : YoungDiagram) :
+    Fintype.card (colGroup μ) = ∏ j : Fin (μ.rowLen 0), (μ.colLen j).factorial := by
+  rw [←Nat.card_eq_fintype_card,colGroup_eq,nat_card_fiberGroup]
+  simp only [Nat.card_eq_fintype_card,card_col_fiber]
+
+abbrev TailCell (μ : YoungDiagram) := {x : Cell μ // row x ≠ 0}
+
+lemma tail_card (μ : YoungDiagram) : Fintype.card (TailCell μ) = μ.card - μ.rowLen 0 := by
+  have hh := Fintype.card_subtype_compl (p := fun x : Cell μ => row x=0)
+  rw [Fintype.card_congr (rowEquiv μ 0),Fintype.card_fin,card_cell] at hh
+  exact hh
+
+lemma rowLen_zero_le_card (μ : YoungDiagram) : μ.rowLen 0 ≤ μ.card := by
+  have h := Fintype.card_subtype_le (fun x : Cell μ => row x=0)
+  rwa [Fintype.card_congr (rowEquiv μ 0),Fintype.card_fin,card_cell] at h
+
+noncomputable def tailColumnEquiv (μ : YoungDiagram) (j : Fin (μ.rowLen 0)) :
+    {x : TailCell μ // colIndex x.1 = j} ≃ Fin (μ.colLen j-1) where
+  toFun x := ⟨row x.1.1-1, by
+    have hx := row_lt_colLen x.1.1
+    have hc : col x.1.1 = j := congrArg Fin.val x.2
+    rw [hc] at hx
+    have hz := x.1.2
+    omega⟩
+  invFun i := ⟨⟨⟨(i+1,j),YoungDiagram.mem_iff_lt_colLen.mpr (by omega)⟩,by simp [row]⟩,rfl⟩
+  left_inv x := by
+    apply Subtype.ext
+    apply Subtype.ext
+    apply Subtype.ext
+    apply Prod.ext
+    · change row x.1.1 - 1 + 1 = row x.1.1
+      have := x.1.2
+      omega
+    · exact (congrArg Fin.val x.2).symm
+  right_inv i := by apply Fin.ext; simp [row]
+
+lemma card_tail_col (μ : YoungDiagram) (j : Fin (μ.rowLen 0)) :
+    Fintype.card {x : TailCell μ // colIndex x.1 = j} = μ.colLen j-1 := by
+  rw [Fintype.card_congr (tailColumnEquiv μ j),Fintype.card_fin]
+
+lemma card_tail_row (μ : YoungDiagram) (i : Fin (μ.colLen 0)) :
+    Fintype.card {x : TailCell μ // rowIndex x.1 = i} = if i.val=0 then 0 else μ.rowLen i := by
+  by_cases hi : i.val=0
+  · rw [ite_eq_left hi]
+    apply Fintype.card_eq_zero_iff.mpr
+    refine ⟨fun x => x.1.2 ?_⟩
+    exact (congrArg Fin.val x.2).trans hi
+  · rw [ite_eq_right hi]
+    let e : {x : TailCell μ // rowIndex x.1 = i} ≃ {x : Cell μ // rowIndex x=i} :=
+      { toFun := fun x => ⟨x.1.1,x.2⟩
+        invFun := fun x => ⟨⟨x.1,fun hz => hi ((congrArg Fin.val x.2).symm.trans hz)⟩,x.2⟩
+        left_inv := fun x => rfl
+        right_inv := fun x => rfl }
+    rw [Fintype.card_congr e,card_row_fiber]
+
+lemma tail_row_group (μ : YoungDiagram) :
+    (μ.rowLen 0).factorial * Fintype.card (fiberGroup (fun x : TailCell μ => rowIndex x.1)) =
+      Fintype.card (rowGroup μ) := by
+  rw [←Nat.card_eq_fintype_card (α := fiberGroup (fun x : TailCell μ => rowIndex x.1)),nat_card_fiberGroup,card_rowGroup]
+  simp only [Nat.card_eq_fintype_card]
+  simp only [card_tail_row]
+  by_cases hn : μ.colLen 0=0
+  · have hr : μ.rowLen 0=0 := by
+      by_contra h
+      have hmem : (0,0) ∈ μ := YoungDiagram.mem_iff_lt_rowLen.mpr (Nat.pos_of_ne_zero h)
+      have hh := YoungDiagram.mem_iff_lt_colLen.mp hmem
+      omega
+    let : IsEmpty (Fin (μ.colLen 0)) := ⟨fun i => by have := i.isLt; omega⟩
+    simp [hr]
+  · let z : Fin (μ.colLen 0) := ⟨0,Nat.pos_of_ne_zero hn⟩
+    have hz : ∀ i : Fin (μ.colLen 0), i.val=0 ↔ i=z := fun i => by simp [z,Fin.ext_iff]
+    simp only [hz]
+    have hprod : (∏ x : Fin (μ.colLen 0), (if x=z then 0 else μ.rowLen x).factorial) =
+        ∏ x ∈ Finset.univ.erase z, (μ.rowLen x).factorial := by
+      rw [←Finset.prod_erase_mul Finset.univ (fun x : Fin (μ.colLen 0) => (if x=z then 0 else μ.rowLen x).factorial) (Finset.mem_univ z)]
+      simp only [ite_true,Nat.factorial_zero,mul_one]
+      apply Finset.prod_congr rfl
+      intro x hx
+      rw [ite_eq_right (Finset.mem_erase.mp hx).1]
+    rw [hprod]
+    exact Finset.mul_prod_erase Finset.univ (fun i : Fin (μ.colLen 0) => (μ.rowLen i).factorial) (Finset.mem_univ z)
+
+lemma sum_tail_columns (μ : YoungDiagram) :
+    ∑ j : Fin (μ.rowLen 0), (μ.colLen j-1) = μ.card-μ.rowLen 0 := by
+  simp_rw [←card_tail_col]
+  rw [←Fintype.card_sigma,Fintype.card_congr (Equiv.sigmaFiberEquiv (fun x : TailCell μ => colIndex x.1)),tail_card]
+
+lemma column_pos (μ : YoungDiagram) (j : Fin (μ.rowLen 0)) : 0 < μ.colLen j :=
+  YoungDiagram.mem_iff_lt_colLen.mp (YoungDiagram.mem_iff_lt_rowLen.mpr j.isLt)
+
+lemma nat_le_two_pow_pred (n : ℕ) (hn : 0<n) : n ≤ 2^(n-1) := by
+  obtain ⟨n,rfl⟩ := Nat.exists_eq_succ_of_ne_zero hn.ne'
+  simp only [Nat.add_one_sub_one]
+  induction n with
+  | zero => norm_num
+  | succ n ih => rw [pow_succ]; omega
+
+lemma column_group_le_tail (μ : YoungDiagram) :
+    Fintype.card (colGroup μ) ≤ 2^(μ.card-μ.rowLen 0) *
+      Fintype.card (fiberGroup (fun x : TailCell μ => colIndex x.1)) := by
+  rw [card_colGroup,←Nat.card_eq_fintype_card (α := fiberGroup (fun x : TailCell μ => colIndex x.1)),nat_card_fiberGroup]
+  simp only [Nat.card_eq_fintype_card]
+  simp only [card_tail_col]
+  calc
+    _ = ∏ j : Fin (μ.rowLen 0), μ.colLen j * (μ.colLen j-1).factorial := by
+      apply Finset.prod_congr rfl
+      intro j hj
+      exact (Nat.mul_factorial_pred (column_pos μ j).ne').symm
+    _ ≤ ∏ j : Fin (μ.rowLen 0), 2^(μ.colLen j-1) * (μ.colLen j-1).factorial := by
+      apply Finset.prod_le_prod
+      intro j hj
+      exact Nat.mul_le_mul_right _ (nat_le_two_pow_pred _ (column_pos μ j))
+    _ = _ := by rw [Finset.prod_mul_distrib,Finset.prod_pow_eq_pow_sum,sum_tail_columns]
+
+theorem choose_tail_le_dimension (μ : YoungDiagram) :
+    μ.card.choose (μ.card-μ.rowLen 0) ≤ Module.finrank ℂ (space μ) * 2^(μ.card-μ.rowLen 0) := by
+  let j := μ.card-μ.rowLen 0
+  have htail := card_fiberGroups_mul_le
+    (fun x : TailCell μ => rowIndex x.1) (fun x : TailCell μ => colIndex x.1)
+    (fun x y hr hc => Subtype.ext (Subtype.ext (Prod.ext (congrArg Fin.val hr) (congrArg Fin.val hc))))
+  rw [tail_card] at htail
+  have hc := column_group_le_tail μ
+  have hr := tail_row_group μ
+  have hd := factorial_le_dimension_row_column μ
+  simp only [←Nat.card_eq_fintype_card] at htail hc hr hd
+  have hh : μ.card.factorial ≤ Module.finrank ℂ (space μ) * 2^j *
+      ((μ.rowLen 0).factorial * j.factorial) := by
+    calc
+      _ ≤ _ := hd
+      _ ≤ Module.finrank ℂ (space μ) * Nat.card (rowGroup μ) *
+          (2^j * Nat.card (fiberGroup (fun x : TailCell μ => colIndex x.1))) := Nat.mul_le_mul_left _ hc
+      _ = Module.finrank ℂ (space μ) * 2^j * (μ.rowLen 0).factorial *
+          (Nat.card (fiberGroup (fun x : TailCell μ => rowIndex x.1)) *
+           Nat.card (fiberGroup (fun x : TailCell μ => colIndex x.1))) := by rw [←hr]; ring
+      _ ≤ _ := by
+        calc
+          _ ≤ Module.finrank ℂ (space μ) * 2^j * (μ.rowLen 0).factorial * j.factorial :=
+            Nat.mul_le_mul_left _ htail
+          _ = _ := by ring
+  have hj : j ≤ μ.card := Nat.sub_le _ _
+  have hs : μ.card-j = μ.rowLen 0 := by dsimp [j]; have := rowLen_zero_le_card μ; omega
+  have hf := Nat.choose_mul_factorial_mul_factorial hj
+  rw [hs] at hf
+  rw [←hf] at hh
+  have hpos : 0 < (μ.rowLen 0).factorial * j.factorial := Nat.mul_pos (Nat.factorial_pos _) (Nat.factorial_pos _)
+  nlinarith
+
+
+lemma exists_perm_mem_iff {α : Type*} [Fintype α] (s t : Finset α) (hc : s.card=t.card) :
+    ∃ p : Equiv.Perm α, ∀ x, p x ∈ t ↔ x ∈ s := by
+  let e : s ≃ t := Finset.equivOfCardEq hc
+  let ec : {x : α // x ∉ s} ≃ {x : α // x ∉ t} := Fintype.equivOfCardEq (by
+    simp only [Fintype.card_subtype_compl,Fintype.card_coe,hc])
+  let p := (Equiv.sumCompl (· ∈ s)).symm.trans ((Equiv.sumCongr e ec).trans (Equiv.sumCompl (· ∈ t)))
+  refine ⟨p,fun x => ?_⟩
+  by_cases hx : x ∈ s
+  · have hp : p x ∈ t := by
+      change ((Equiv.sumCompl (· ∈ t)) ((Equiv.sumCongr e ec) ((Equiv.sumCompl (· ∈ s)).symm x))) ∈ t
+      rw [Equiv.sumCompl_symm_apply_of_pos hx]
+      exact (e ⟨x,hx⟩).2
+    exact iff_of_true hp hx
+  · have hp : p x ∉ t := by
+      change ((Equiv.sumCompl (· ∈ t)) ((Equiv.sumCongr e ec) ((Equiv.sumCompl (· ∈ s)).symm x))) ∉ t
+      rw [Equiv.sumCompl_symm_apply_of_neg hx]
+      exact (ec ⟨x,hx⟩).2
+    exact iff_of_false hp hx
+
+noncomputable def tailSet {μ : YoungDiagram} (f : Tabloid μ) : Finset (Cell μ) :=
+  Finset.univ.filter (fun x => (f.1 x).val ≠ 0)
+
+@[simp] lemma mem_tailSet {μ : YoungDiagram} (f : Tabloid μ) (x : Cell μ) :
+    x ∈ tailSet f ↔ (f.1 x).val ≠ 0 := by simp [tailSet]
+
+lemma tailSet_act {μ : YoungDiagram} (p : Equiv.Perm (Cell μ)) (f : Tabloid μ) :
+    tailSet (tabloidAct p f) = (tailSet f).image p := by
+  ext x
+  simp only [mem_tailSet,tabloidAct_apply,Finset.mem_image]
+  constructor
+  · intro h; exact ⟨p⁻¹ x,h,by simp⟩
+  · rintro ⟨y,hy,rfl⟩; simpa using hy
+
+lemma card_tailSet {μ : YoungDiagram} (f : Tabloid μ) :
+    (tailSet f).card = μ.card-μ.rowLen 0 := by
+  obtain ⟨p,rfl⟩ := tabloidAct_transitive μ (baseTabloid μ) f
+  rw [tailSet_act,Finset.card_image_of_injective _ p.injective]
+  change (Finset.univ.filter (fun x : Cell μ => row x ≠ 0)).card = _
+  rw [←Fintype.card_subtype]
+  exact tail_card μ
+
+noncomputable def footprint (μ : YoungDiagram) (A : Finset (Cell μ)) : Finset (Tabloid μ) :=
+  Finset.univ.filter (fun f => tailSet f ⊆ A)
+
+lemma tail_fiber_card_eq (μ : YoungDiagram) (s t : Finset (Cell μ)) (hst : s.card=t.card) :
+    (Finset.univ.filter (fun f : Tabloid μ => tailSet f=s)).card =
+      (Finset.univ.filter (fun f : Tabloid μ => tailSet f=t)).card := by
+  obtain ⟨p,hp⟩ := exists_perm_mem_iff s t hst
+  have he : s.image p=t := by
+    ext x
+    constructor
+    · intro hx
+      obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx
+      exact (hp y).mpr hy
+    · intro hx
+      exact Finset.mem_image.mpr ⟨p⁻¹ x,(hp (p⁻¹ x)).mp (by simpa using hx),by simp⟩
+  apply Finset.card_equiv (tabloidAct p)
+  intro f
+  simp only [Finset.mem_filter,Finset.mem_univ,true_and,tailSet_act,←he]
+  exact (Finset.image_inj p.injective).symm
+
+lemma footprint_card_product (μ : YoungDiagram) (A : Finset (Cell μ)) :
+    (footprint μ A).card * μ.card.choose (μ.card-μ.rowLen 0) =
+      Fintype.card (Tabloid μ) * A.card.choose (μ.card-μ.rowLen 0) := by
+  let j := μ.card-μ.rowLen 0
+  let s := tailSet (baseTabloid μ)
+  let c := (Finset.univ.filter (fun f : Tabloid μ => tailSet f=s)).card
+  have hconstant (t : Finset (Cell μ)) (ht : t.card=j) :
+      (Finset.univ.filter (fun f : Tabloid μ => tailSet f=t)).card = c :=
+    tail_fiber_card_eq μ t s (ht.trans (card_tailSet _).symm)
+  have hcount (A : Finset (Cell μ)) : (footprint μ A).card = A.card.choose j*c := by
+    have h := Finset.sum_card_fiberwise_eq_card_filter
+      (Finset.univ : Finset (Tabloid μ)) (A.powersetCard j) tailSet
+    have he : (Finset.univ.filter (fun f : Tabloid μ => tailSet f ∈ A.powersetCard j)) = footprint μ A := by
+      ext f
+      simp only [Finset.mem_filter,Finset.mem_univ,true_and,Finset.mem_powersetCard,
+        card_tailSet,footprint]
+      simp [j]
+    rw [he] at h
+    rw [←h]
+    calc
+      _ = ∑ _t ∈ A.powersetCard j, c := by
+        apply Finset.sum_congr rfl
+        intro t ht
+        exact hconstant t (Finset.mem_powersetCard.mp ht).2
+      _ = _ := by simp [Finset.card_powersetCard]
+  have hfull : (footprint μ Finset.univ).card = Fintype.card (Tabloid μ) := by
+    simp only [footprint,Finset.subset_univ,Finset.filter_true,Finset.card_univ]
+  have hN := hcount Finset.univ
+  rw [hfull,Finset.card_univ,card_cell] at hN
+  rw [hcount,hN]
+  ring
+
+lemma footprint_fixed {μ : YoungDiagram} (A : Finset (Cell μ))
+    (f : Tabloid μ) (hf : f ∈ footprint μ A) (p : Equiv.Perm (Cell μ))
+    (hp : ∀ x ∈ A, p x=x) : tabloidAct p f=f := by
+  have htail : tailSet f ⊆ A := (Finset.mem_filter.mp hf).2
+  apply Subtype.ext
+  funext x
+  change f.1 (p⁻¹ x) = f.1 x
+  by_cases hx : x ∈ A
+  · rw [show p⁻¹ x=x from p.injective (by simpa using (hp x hx).symm)]
+  · have hy : p⁻¹ x ∉ A := by
+      intro h
+      have hh := hp (p⁻¹ x) h
+      have he : x=p⁻¹ x := by simpa using hh
+      exact hx (he ▸ h)
+    apply Fin.ext
+    have hx0 : (f.1 x).val=0 := by by_contra h; exact hx (htail (mem_tailSet f x |>.mpr h))
+    have hy0 : (f.1 (p⁻¹ x)).val=0 := by by_contra h; exact hy (htail (mem_tailSet f _ |>.mpr h))
+    exact hy0.trans hx0.symm
+
+noncomputable def fixedHilbert (μ : YoungDiagram) (A : Finset (Cell μ)) :
+    Submodule ℂ (PermutationHilbert.H (Tabloid μ)) where
+  carrier := {v | v ∈ hilbertSpace μ ∧ ∀ p : Equiv.Perm (Cell μ),
+    (∀ x ∈ A, p x=x) → PermutationHilbert.act (tabloidAct p) v=v}
+  zero_mem' := ⟨Submodule.zero_mem _,fun p hp => map_zero _⟩
+  add_mem' := fun hv hw => ⟨Submodule.add_mem _ hv.1 hw.1,fun p hp => by
+    rw [map_add,hv.2 p hp,hw.2 p hp]⟩
+  smul_mem' := fun c v hv => ⟨Submodule.smul_mem _ c hv.1,fun p hp => by
+    rw [map_smul,hv.2 p hp]⟩
+
+lemma fixedHilbert_le (μ : YoungDiagram) (A : Finset (Cell μ)) : fixedHilbert μ A ≤ hilbertSpace μ :=
+  fun _ hv => hv.1
+
+theorem fixedHilbert_dimension (μ : YoungDiagram) (A : Finset (Cell μ)) :
+    Module.finrank ℂ (space μ) * A.card.choose (μ.card-μ.rowLen 0) ≤
+      Module.finrank ℂ (fixedHilbert μ A) * μ.card.choose (μ.card-μ.rowLen 0) := by
+  let : DecidableEq (Tabloid μ) := Classical.decEq _
+  have ht (f g : Tabloid μ) : ∃ p : Equiv.Perm (Tabloid μ), p f=g ∧
+      (hilbertSpace μ).map (PermutationHilbert.act p).toLinearEquiv.toLinearMap = hilbertSpace μ := by
+    obtain ⟨p,hp⟩ := tabloidAct_transitive μ f g
+    exact ⟨tabloidAct p,hp,hilbertSpace_invariant μ p⟩
+  have hproj (f : Tabloid μ) (hf : f ∈ footprint μ A) :
+      (hilbertSpace μ).starProjection (EuclideanSpace.single f (1:ℂ)) ∈ fixedHilbert μ A := by
+    refine ⟨((hilbertSpace μ).orthogonalProjectionOnto _).2,fun p hp => ?_⟩
+    rw [PermutationHilbert.projection_comm _ _ (hilbertSpace_invariant μ p),
+      PermutationHilbert.act_single,footprint_fixed A f hf p hp]
+  have h := PermutationHilbert.footprint_dimension (hilbertSpace μ) (fixedHilbert μ A)
+    (fixedHilbert_le μ A) ht (footprint μ A) hproj
+  rw [finrank_hilbertSpace] at h
+  have hn : (footprint μ A).card * Module.finrank ℂ (space μ) ≤
+      Fintype.card (Tabloid μ)*Module.finrank ℂ (fixedHilbert μ A) := by exact_mod_cast h
+  have hc := footprint_card_product μ A
+  have hpos : 0 < Fintype.card (Tabloid μ) := Fintype.card_pos_iff.mpr ⟨baseTabloid μ⟩
+  have hh := Nat.mul_le_mul_right (μ.card.choose (μ.card-μ.rowLen 0)) hn
+  nlinarith
+
+end Thorp.Specht
+
+namespace Thorp.UnitaryFinite
+open scoped BigOperators ComplexConjugate Classical
+
+variable {G : Type*} [Group G] [Fintype G]
+variable {V W : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+  [NormedAddCommGroup W] [InnerProductSpace ℂ W] [FiniteDimensional ℂ W]
+
+def IsUnitary (ρ : Representation ℂ G V) : Prop :=
+  ∀ g v w, inner ℂ (ρ g v) (ρ g w) = inner ℂ v w
+
+omit [Fintype G] [FiniteDimensional ℂ V] in
+lemma unitary_inner_inv (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (g : G) (v w : V) :
+    inner ℂ v (ρ g⁻¹ w) = inner ℂ (ρ g v) w := by
+  have h := hρ g v (ρ g⁻¹ w)
+  have he : ρ g (ρ g⁻¹ w) = w := by
+    change (ρ g * ρ g⁻¹) w = w
+    rw [←map_mul,mul_inv_cancel,map_one]
+    rfl
+  rw [he] at h
+  exact h.symm
+
+noncomputable def averageMap (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
+    (A : V →ₗ[ℂ] W) : Representation.IntertwiningMap ρ σ where
+  toLinearMap := ∑ g, (σ g).comp (A.comp (ρ g⁻¹))
+  isIntertwining' h := by
+    ext v
+    simp only [LinearMap.comp_apply,LinearMap.sum_apply,map_sum]
+    apply Fintype.sum_equiv (Equiv.mulLeft h⁻¹)
+    intro g
+    simp only [Equiv.coe_mulLeft,mul_inv_rev,inv_inv,map_mul,Module.End.mul_apply]
+    have hh (w : W) : σ h (σ h⁻¹ w)=w := by
+      change (σ h*σ h⁻¹) w=w
+      rw [←map_mul,mul_inv_cancel,map_one]
+      rfl
+    rw [hh]
+
+end Thorp.UnitaryFinite
+
+end ThorpNine.Tail
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Tail/TailShapes.lean b/lean/OAI/Probability/ThorpRouting/Tail/TailShapes.lean
index a1f483db177428f6dd33537a5d930e5e417bec5e..561f96c297e39026ded1be4e7b6429ba14b77e5c 100644
--- a/lean/OAI/Probability/ThorpRouting/Tail/TailShapes.lean
+++ b/lean/OAI/Probability/ThorpRouting/Tail/TailShapes.lean
@@ -1,863 +1,863 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Tail.InsertionPerm
-
-namespace OAI
-
-namespace ThorpNine.Tail
-
-namespace Thorp.Specht
-open scoped BigOperators Classical
-
-abbrev TailShapes (N j : ℕ) := {μ : Shapes N // μ.1.card-μ.1.rowLen 0=j}
-
-noncomputable def tailShapePartition (N j : ℕ) (μ : TailShapes N j) : Nat.Partition j where
-  parts := μ.1.1.rowLens.tail
-  parts_pos := fun {x} hx => μ.1.1.pos_of_mem_rowLens x (List.mem_of_mem_tail (by simpa using hx))
-  parts_sum := by simpa using (rowLens_tail_sum μ.1.1).trans μ.2
-
-lemma tailShapePartition_injective (N j : ℕ) : Function.Injective (tailShapePartition N j) := by
-  intro μ ν he
-  have hh : (μ.1.1.rowLens.tail : Multiset ℕ)=(ν.1.1.rowLens.tail : Multiset ℕ) :=
-    congrArg Nat.Partition.parts he
-  have ht : μ.1.1.rowLens.tail=ν.1.1.rowLens.tail := by
-    have hm := congrArg (fun s : Multiset ℕ => s.sort (· ≥ ·)) hh
-    simpa only [Multiset.coe_sort,
-      List.mergeSort_eq_self _ μ.1.1.rowLens_sorted.pairwise.tail,
-      List.mergeSort_eq_self _ ν.1.1.rowLens_sorted.pairwise.tail] using hm
-  have hl := positive_list_eq_of_tail_sum (μ.1.1.pos_of_mem_rowLens) (ν.1.1.pos_of_mem_rowLens)
-    ht (by rw [sum_rowLens,sum_rowLens,μ.1.2,ν.1.2])
-  apply Subtype.ext
-  apply Subtype.ext
-  apply YoungDiagram.equivListRowLens.injective
-  exact Subtype.ext hl
-
-lemma tail_shapes_count (N j : ℕ) : Fintype.card (TailShapes N j)≤2^j := by
-  exact (Fintype.card_le_of_injective _ (tailShapePartition_injective N j)).trans (partition_count j)
-
-end Thorp.Specht
-
-namespace Thorp
-open scoped BigOperators Classical
-
-lemma tail_preserving_pairSwitch (d : ℕ) (p : Equiv.Perm (Card (d+1)))
-    (hp : ∀ x, Fin.tail (p x)=Fin.tail x) :
-    pairSwitch (fun u => p (Fin.cons false u) 0)=p := by
-  apply Equiv.ext
-  intro x
-  rw [←Fin.cons_self_tail x]
-  generalize x 0=b
-  generalize Fin.tail x=u
-  apply funext
-  intro i
-  refine Fin.cases ?_ (fun j => ?_) i
-  · change Bool.xor b (p (Fin.cons false u) 0)=p (Fin.cons b u) 0
-    cases b with
-    | false => simp
-    | true =>
-      have hn : p (Fin.cons false u) 0 ≠ p (Fin.cons true u) 0 := by
-        intro he
-        have hh : p (Fin.cons false u)=p (Fin.cons true u) := by
-          rw [←Fin.cons_self_tail (p (Fin.cons false u)),←Fin.cons_self_tail (p (Fin.cons true u)),
-            hp,hp,Fin.tail_cons,Fin.tail_cons,he]
-        have hh := congrFun (p.injective hh) 0
-        simp only [Fin.cons_zero] at hh
-        contradiction
-      cases h0 : p (Fin.cons false u) 0 <;> cases h1 : p (Fin.cons true u) 0 <;> simp_all
-  · change u j=p (Fin.cons b u) j.succ
-    have h := congrFun (hp (Fin.cons b u)) j
-    simpa only [Fin.tail_cons,Fin.tail,Fin.cons_succ] using h.symm
-
-lemma pairSwitch_injective (d : ℕ) : Function.Injective (pairSwitch (d := d)) := by
-  intro ξ η h
-  funext u
-  have hh := congrArg (fun p : Equiv.Perm (Card (d+1)) => p (Fin.cons false u) 0) h
-  change Bool.xor false (ξ u)=Bool.xor false (η u) at hh
-  simpa only [Bool.false_xor] using hh
-
-end Thorp
-namespace Thorp.Specht
-open scoped BigOperators Classical
-
-noncomputable def pairFiberEquiv (d : ℕ) (μ : YoungDiagram) (e : Card (d+1) ≃ Cell μ) :
-    (Card d → Bool) ≃ fiberGroup (fun x => Fin.tail (e.symm x)) :=
-  Equiv.ofBijective (fun ξ => ⟨e.permCongr (pairSwitch ξ),by
-    intro x
-    simp only [Equiv.permCongr_apply,e.symm_apply_apply]
-    exact Fin.tail_cons _ _⟩)
-    ⟨fun ξ η h => pairSwitch_injective d (e.permCongr.injective (congrArg Subtype.val h)),by
-      intro p
-      let q := e.symm.permCongr p.1
-      have hq : ∀ x, Fin.tail (q x)=Fin.tail x := by
-        intro x
-        have h := p.2 (e x)
-        simpa only [q,Equiv.permCongr_apply,Equiv.symm_symm,e.symm_apply_apply] using h
-      refine ⟨fun u => q (Fin.cons false u) 0,?_⟩
-      apply Subtype.ext
-      change e.permCongr (pairSwitch (fun u => q (Fin.cons false u) 0))=p.1
-      rw [tail_preserving_pairSwitch d q hq]
-      apply Equiv.ext
-      intro x
-      simp only [q,Equiv.permCongr_apply,Equiv.symm_symm,e.apply_symm_apply]⟩
-
-lemma pair_sum_taboid_zero (d : ℕ) (μ : YoungDiagram) (e : Card (d+1) ≃ Cell μ)
-    (hj : 2^d < μ.colLen 0) (v : space μ) :
-    (∑ ξ : Card d → Bool, tabloidRep μ (e.permCongr (pairSwitch ξ))) v.1=0 := by
-  have he : (∑ ξ : Card d → Bool, tabloidRep μ (e.permCongr (pairSwitch ξ)))=
-      symmetrizer (fiberGroup (fun x => Fin.tail (e.symm x))) (tabloidRep μ) :=
-    Fintype.sum_equiv (pairFiberEquiv d μ e) _ _ (fun _ => rfl)
-  rw [he]
-  exact tall_column_average_zero μ _ 0 (by simpa only [card_positions] using hj) v.1 v.2
-
-lemma pair_operator_tall_zero (d : ℕ) (μ : YoungDiagram) (e : Card (d+1) ≃ Cell μ)
-    (hj : 2^d < μ.colLen 0) :
-    UnitaryFinite.sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e) pairSwitch=0 := by
-  apply ContinuousLinearMap.ext
-  intro v
-  obtain ⟨w,rfl⟩ := (spaceHilbertEquiv μ).surjective v
-  have hh : (∑ ξ : Card d → Bool, representation μ (e.permCongr (pairSwitch ξ))) w=0 := by
-    apply Subtype.ext
-    simp only [LinearMap.sum_apply,Submodule.coe_sum,Submodule.coe_zero]
-    change (∑ ξ : Card d → Bool, tabloidRep μ (e.permCongr (pairSwitch ξ)) w.1)=0
-    simpa only [LinearMap.sum_apply] using pair_sum_taboid_zero d μ e hj w
-  have hh' := congrArg (spaceHilbertEquiv μ) hh
-  simp only [LinearMap.sum_apply,map_sum,map_zero] at hh'
-  change ((Fintype.card (Card d → Bool):ℂ)⁻¹ •
-    ∑ ξ : Card d → Bool, UnitaryFinite.continuousRepresentation (relabelledUnitary μ e) (pairSwitch ξ))
-      (spaceHilbertEquiv μ w)=0
-  rw [smul_apply,sum_apply]
-  change (Fintype.card (Card d → Bool):ℂ)⁻¹ •
-    (∑ ξ : Card d → Bool, spaceHilbertEquiv μ (representation μ (e.permCongr (pairSwitch ξ))
-      ((spaceHilbertEquiv μ).symm (spaceHilbertEquiv μ w))))=0
-  simp only [LinearEquiv.symm_apply_apply,hh',smul_zero]
-
-open UnitaryFinite
-
-lemma palindrome_tall_zero (d : ℕ) (μ : YoungDiagram) (e : Card (d+1) ≃ Cell μ)
-    (hj : 2^d<μ.colLen 0) :
-    sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e) (palindromePerm (d+1))=0 := by
-  let ρ := relabelledUnitary μ e
-  let : NormedAddCommGroup (hilbertSpace μ) := inferInstance
-  let : InnerProductSpace ℂ (hilbertSpace μ) := inferInstance
-  let : FiniteDimensional ℂ (hilbertSpace μ) := inferInstance
-  have hpair : sampleOperator (V := hilbertSpace μ) ρ (pairSwitch (d := d))=0 := pair_operator_tall_zero d μ e hj
-  have hi : sampleOperator ρ (fun ω => (butterflyPerm (d+1) (decodeButterfly (d+1) ω))⁻¹)=0 := by
-    rw [inverse_butterfly_average_step,hpair,mul_zero]
-  rw [sampleOperator_inv ρ (relabelledUnitary_unitary μ e)] at hi
-  rw [sampleOperator_palindrome _ _ (relabelledUnitary_unitary μ e),hi,mul_zero]
-
-lemma palindrome_tall_zero_general (d : ℕ) (hd : 0<d) (μ : YoungDiagram) (e : Card d ≃ Cell μ)
-    (hj : (μ.card:ℝ)/2<(μ.colLen 0:ℝ)) :
-    sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e) (palindromePerm d)=0 := by
-  obtain ⟨r,rfl⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : d≠0)
-  apply palindrome_tall_zero r μ e
-  have hc : μ.card=2^(r+1) := by rw [←card_cell,←Fintype.card_congr e,card_positions]
-  rw [hc,Nat.cast_pow,Nat.cast_ofNat,pow_succ] at hj
-  have hh : (2:ℝ)^r<(μ.colLen 0:ℝ) := by linarith
-  exact_mod_cast hh
-
-end Thorp.Specht
-
-namespace Thorp.DimensionEstimates
-open Filter
-
-lemma twopow_tendsto : Tendsto (fun d : ℕ => (2:ℝ)^d) atTop atTop :=
-  tendsto_pow_atTop_atTop_of_one_lt (by norm_num)
-
-lemma twopow_nat_tendsto : Tendsto (fun d : ℕ => (2:ℕ)^d) atTop atTop :=
-  tendsto_pow_atTop_atTop_of_one_lt (by norm_num)
-
-lemma twopow_mul_eventually (c : ℝ) (hc : 0<c) (B : ℝ) :
-    ∀ᶠ d : ℕ in atTop, B≤c*(2:ℝ)^d := by
-  filter_upwards [twopow_tendsto.eventually (eventually_ge_atTop (B/c))] with d hd
-  exact (div_le_iff₀ hc).mp hd |>.trans_eq (mul_comm _ _)
-
-lemma exp_two_absorb (β n : ℝ) (hx : Real.log 2≤β*n/2) :
-    Real.exp (-β*n)+Real.exp (-β*n)≤Real.exp (-β*n/2) := by
-  calc
-    _ = Real.exp (Real.log 2-β*n) := by rw [Real.exp_sub,Real.exp_log (by norm_num)]; rw [neg_mul,Real.exp_neg]; ring
-    _ ≤ _ := Real.exp_le_exp.mpr (by linarith)
-
-end Thorp.DimensionEstimates
-namespace Thorp.Specht
-open scoped BigOperators Classical
-open Filter UnitaryFinite DimensionEstimates
-
-theorem exponential_dimension_spectral (h : ℝ) (hh : 0<h) :
-    ∃ a : ℝ, 0<a ∧ ∀ᶠ d : ℕ in atTop, ∀ (μ : YoungDiagram) (e : Card d ≃ Cell μ),
-      h*(2:ℝ)^d≤Real.log (Module.finrank ℂ (space μ)) →
-      ‖sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e) (palindromePerm d)‖ ≤
-        Real.exp (-a*Real.log (Module.finrank ℂ (space μ))) := by
-  obtain ⟨L,hL,ε,t,hε,_,ht,hcount,hM⟩ := medium_parameters h hh
-  obtain ⟨R,hR⟩ := eventually_atTop.mp hM
-  let A := 4000*densityConstant+1
-  have hA : 0<A := by dsimp [A]; have := densityConstant_pos; positivity
-  let β := min (-Real.log (blockGap L)*ε/(2:ℝ)^L) (h/16000)
-  have hβ : 0<β := by
-    apply lt_min
-    · exact div_pos (mul_pos (neg_pos.mpr (Real.log_neg (blockGap_pos L) (blockGap_lt_one L))) hε) (by positivity)
-    · positivity
-  let a := min (1/8000:ℝ) (β/(2*A))
-  have ha : 0<a := lt_min (by norm_num) (div_pos hβ (by positivity))
-  refine ⟨a,ha,?_⟩
-  filter_upwards [eventually_ge_atTop (L+R),twopow_mul_eventually h hh (8000*Real.log 2),
-    twopow_mul_eventually (β/2) (by positivity) (Real.log 2)] with d hd hl hb
-  intro μ e hdim
-  let : NormedAddCommGroup (hilbertSpace μ) := inferInstance
-  let : InnerProductSpace ℂ (hilbertSpace μ) := inferInstance
-  let : FiniteDimensional ℂ (hilbertSpace μ) := inferInstance
-  let := relabelledUnitary_irreducible μ e
-  have hpos : 0<Module.finrank ℂ (hilbertSpace μ) := by rw [finrank_hilbertSpace]; exact dimension_pos μ
-  have hlarge : 8000*Real.log 2≤Real.log (Module.finrank ℂ (hilbertSpace μ)) := by rw [finrank_hilbertSpace]; exact hl.trans hdim
-  have hlog : 0≤Real.log (Module.finrank ℂ (space μ)) := (by positivity : (0:ℝ)≤h*(2:ℝ)^d).trans hdim
-  by_cases hhigh : A*(2:ℝ)^d≤Real.log (Module.finrank ℂ (space μ))
-  · have hden : densityConstant*(2:ℝ)^d≤Real.log (Module.finrank ℂ (hilbertSpace μ))/4000 := by
-      rw [finrank_hilbertSpace]
-      dsimp [A] at hhigh
-      nlinarith [pow_pos (by norm_num : (0:ℝ)<2) d]
-    have hspec := @full_palindrome_decay (hilbertSpace μ) _ _ _ d (relabelledUnitary μ e)
-      inferInstance (relabelledUnitary_unitary μ e) hpos hden hlarge
-    rw [finrank_hilbertSpace] at hspec
-    apply hspec.trans (Real.exp_le_exp.mpr ?_)
-    have := min_le_left (1/8000:ℝ) (β/(2*A))
-    nlinarith
-  · obtain ⟨r,rfl⟩ := Nat.exists_eq_add_of_le (show L≤d by omega)
-    have hr : R≤r := by omega
-    have hdim' : h*(2:ℝ)^(L+r)≤Real.log (Module.finrank ℂ (hilbertSpace μ)) := by rwa [finrank_hilbertSpace]
-    have hs := @medium_exception_decay (hilbertSpace μ) _ _ _ L r h ε t ht hcount
-      (relabelledUnitary μ e) inferInstance (relabelledUnitary_unitary μ e) hpos hdim' hlarge (hR r hr)
-    have hb1 : Real.log (blockGap L)*ε*(2:ℝ)^r≤-β*(2:ℝ)^(L+r) := by
-      have hb0 := min_le_left (-Real.log (blockGap L)*ε/(2:ℝ)^L) (h/16000)
-      have hp := (le_div_iff₀ (by positivity : (0:ℝ)<(2:ℝ)^L)).mp hb0
-      have hp := mul_le_mul_of_nonneg_right hp (by positivity : (0:ℝ)≤(2:ℝ)^r)
-      rw [pow_add]
-      dsimp [β]
-      nlinarith
-    have hb2 : -h*(2:ℝ)^(L+r)/16000≤-β*(2:ℝ)^(L+r) := by
-      have hp := mul_le_mul_of_nonneg_right (min_le_right (-Real.log (blockGap L)*ε/(2:ℝ)^L) (h/16000))
-        (by positivity : (0:ℝ)≤(2:ℝ)^(L+r))
-      dsimp [β]
-      linarith
-    apply hs.trans ((add_le_add (Real.exp_le_exp.mpr hb1) (Real.exp_le_exp.mpr hb2)).trans ?_)
-    apply (exp_two_absorb β ((2:ℝ)^(L+r)) (by nlinarith)).trans
-    apply Real.exp_le_exp.mpr
-    have hdmax := le_of_not_ge hhigh
-    have hp := mul_le_mul_of_nonneg_left hdmax (by positivity : 0≤β/(2*A))
-    have he : β/(2*A)*(A*(2:ℝ)^(L+r))=β*(2:ℝ)^(L+r)/2 := by field_simp
-    rw [he] at hp
-    have hba := mul_le_mul_of_nonneg_right (min_le_right (1/8000:ℝ) (β/(2*A))) hlog
-    dsimp [a] at *
-    linarith
-
-
-theorem spectral_regimes :
-    ∃ a h : ℝ, 0<a ∧ 0<h ∧ ∀ᶠ d : ℕ in atTop,
-      ∀ (μ : YoungDiagram) (e : Card d ≃ Cell μ), 0<μ.card-μ.rowLen 0 →
-        (μ.colLen 0:ℝ)≤(μ.card:ℝ)/2 →
-        ‖sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e) (palindromePerm d)‖≤
-          Real.exp (-a*Real.log (Module.finrank ℂ (space μ))) ∧
-        1≤Real.log (Module.finrank ℂ (space μ)) ∧
-        (((1/2:ℝ)*Real.log μ.card≤Real.log (Module.finrank ℂ (space μ)) ∧
-          (μ.card-μ.rowLen 0:ℕ)≤Real.log (Module.finrank ℂ (space μ))) ∨
-          h*(2:ℝ)^d≤Real.log (Module.finrank ℂ (space μ))) := by
-  obtain ⟨X,hX,hnear⟩ := near_row_spectral
-  have hXpos : 0<X := by linarith
-  have hδ : 0<(1:ℝ)/X := by positivity
-  have hδ2 : (1:ℝ)/X≤1/2 := (div_le_iff₀ hXpos).mpr (by linarith)
-  obtain ⟨h,hh,hmedium⟩ := medium_dimension (1/X) hδ hδ2
-  obtain ⟨a₀,ha₀,hlarge⟩ := exponential_dimension_spectral h hh
-  let a := min (nearFraction/8000) a₀
-  have ha : 0<a := lt_min (by norm_num [nearFraction]) ha₀
-  refine ⟨a,h,ha,hh,?_⟩
-  have hl := (Real.tendsto_log_atTop.comp twopow_tendsto).eventually
-    (eventually_ge_atTop (2*max (32768000*Real.log 2) 1))
-  filter_upwards [hlarge,twopow_nat_tendsto.eventually hmedium,hl,
-    twopow_mul_eventually h hh 1] with d hspec hdim hlog h1
-  intro μ e hj hc
-  simp only [Function.comp_apply] at hlog
-  have hcard : μ.card=2^d := by rw [←card_cell,←Fintype.card_congr e,card_positions]
-  have hcardR : (μ.card:ℝ)=(2:ℝ)^d := by exact_mod_cast hcard
-  have hμlog : max (32768000*Real.log 2) 1≤(1/2:ℝ)*Real.log μ.card := by rw [hcardR]; linarith
-  by_cases hx : X≤(μ.card:ℝ)/(μ.card-μ.rowLen 0:ℕ)
-  · obtain ⟨hs,hl,hjD⟩ := hnear d μ e hj hx ((le_max_left _ _).trans hμlog)
-    have h1D : 1≤Real.log (Module.finrank ℂ (space μ)) := ((le_max_right _ _).trans hμlog).trans hl
-    refine ⟨hs.trans (Real.exp_le_exp.mpr ?_),h1D,Or.inl ⟨hl,hjD⟩⟩
-    have hp := mul_le_mul_of_nonneg_right (min_le_left (nearFraction/8000) a₀) (by linarith : 0≤Real.log (Module.finrank ℂ (space μ)))
-    dsimp [a]
-    linarith
-  · have htail : (1/X)*(μ.card:ℝ)≤(μ.card:ℝ)-μ.rowLen 0 := by
-      have hjR : (0:ℝ)<(μ.card-μ.rowLen 0:ℕ) := by exact_mod_cast hj
-      have hh := (div_lt_iff₀ hjR).mp (lt_of_not_ge hx)
-      have hs : ((μ.card-μ.rowLen 0:ℕ):ℝ)=(μ.card:ℝ)-μ.rowLen 0 := by
-        rw [Nat.cast_sub (rowLen_zero_le_card μ)]
-      rw [hs] at hh
-      have hd : (μ.card:ℝ)/X≤(μ.card:ℝ)-μ.rowLen 0 :=
-        (div_le_iff₀ hXpos).mpr (by nlinarith)
-      convert hd using 1; ring
-    have hdim' : h*(2:ℝ)^d≤Real.log (Module.finrank ℂ (space μ)) := by
-      simpa only [Nat.cast_pow,Nat.cast_ofNat] using
-        hdim μ hcard (by simpa only [hcard] using htail) (by simpa only [hcard] using hc)
-    have h1D := h1.trans hdim'
-    refine ⟨(hspec μ e hdim').trans (Real.exp_le_exp.mpr ?_),h1D,Or.inr hdim'⟩
-    have hp := mul_le_mul_of_nonneg_right (min_le_right (nearFraction/8000) a₀) (by linarith : 0≤Real.log (Module.finrank ℂ (space μ)))
-    dsimp [a]
-    linarith
-
-end Thorp.Specht
-
-namespace Thorp.StrongSmoothing
-
-noncomputable def regularAbsoluteEvenTrace (d : ℕ) (order : Equiv.Perm (Fin d)) (u : ℕ) : ℝ :=
-  Matrix.trace ((reflectedKernel d (2^d) order false)^u)
-
-def MainStatement : Prop :=
-  ∃ u : ℕ, 1 ≤ u ∧ ∃ C : ℝ,
-    (∀ d l : ℕ, l ≤ 2^d → ∀ order : Equiv.Perm (Fin d), ∀ orientation : Bool,
-      ∀ x y : Tuples d l,
-        ((reflectedKernel d l order orientation)^u) x y ≤
-          Real.exp (C*l)/((2^d).descFactorial l:ℝ)) ∧
-    (∀ d : ℕ, ∀ order : Equiv.Perm (Fin d),
-      regularAbsoluteEvenTrace d order u ≤ Real.exp (C*(2:ℝ)^d))
-
-
-open scoped BigOperators
-
-variable {X : Type*} [Fintype X] [Nonempty X]
-
-omit [Nonempty X] in
-lemma mean_nonneg (f : X → ℝ) (hf : ∀ x, 0 ≤ f x) : 0 ≤ finiteMean f :=
-  finiteMean_nonneg hf
-
-lemma mean_rpow_concave (f : X → ℝ) (hf : ∀ x, 0 ≤ f x)
-    {a : ℝ} (ha : 0 ≤ a) (ha1 : a ≤ 1) :
-    finiteMean (fun x => (f x)^a) ≤ (finiteMean f)^a := by
-  have hN : (0:ℝ) < Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
-  have hh := (Real.concaveOn_rpow ha ha1).le_map_sum
-    (t := Finset.univ) (w := fun _ : X => 1/(Fintype.card X:ℝ))
-    (p := f) (fun _ _ => by positivity)
-    (by simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]; field_simp)
-    (fun x _ => hf x)
-  simpa only [smul_eq_mul,one_div,←Finset.mul_sum,mul_comm _ (∑ _ : X,_),
-    ←div_eq_mul_inv,finiteMean] using hh
-
-noncomputable def act (K : Matrix X X ℝ) (f : X → ℝ) (y : X) : ℝ := ∑ x, K x y*f x
-
-omit [Nonempty X] in
-lemma act_nonneg (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
-    (f : X → ℝ) (hf : ∀ x, 0 ≤ f x) (y : X) : 0 ≤ act K f y :=
-  Finset.sum_nonneg (fun x _ => mul_nonneg (hK x y) (hf x))
-
-omit [Nonempty X] in
-lemma act_rpow_le (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
-    (hcol : ∀ y, ∑ x, K x y = 1) (f : X → ℝ) (hf : ∀ x, 0 ≤ f x)
-    {b : ℝ} (hb : 1 ≤ b) (y : X) :
-    (act K f y)^b ≤ act K (fun x => (f x)^b) y := by
-  simpa only [smul_eq_mul,act] using
-    (convexOn_rpow hb).map_sum_le (t := Finset.univ) (w := fun x => K x y)
-      (p := f) (fun x _ => hK x y) (hcol y) (fun x _ => hf x)
-
-lemma act_one_to_p (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
-    {p A : ℝ} (hp : 1 ≤ p)
-    (hm : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) ≤ Real.exp A)
-    (f : X → ℝ) (hf : ∀ x, 0 ≤ f x) :
-    finiteMean (fun y => (act K f y)^p) ≤ Real.exp A*(finiteMean f)^p := by
-  classical
-  let s : ℝ := ∑ x, f x
-  have hs : 0 ≤ s := Finset.sum_nonneg (fun x _ => hf x)
-  have hN : (0:ℝ) < Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
-  by_cases hz : s=0
-  · have hfz : ∀ x, f x=0 := by
-      intro x
-      exact (Finset.sum_eq_zero_iff_of_nonneg (fun x _ => hf x)).mp hz x (Finset.mem_univ _)
-    simp only [act,hfz,mul_zero,Finset.sum_const_zero,finiteMean,zero_div,
-      Real.zero_rpow (show p ≠ 0 by linarith),mul_zero,le_refl]
-  · have hsp : 0 < s := lt_of_le_of_ne hs (Ne.symm hz)
-    let w : X → ℝ := fun x => f x/s
-    have hw : ∀ x, 0 ≤ w x := fun x => div_nonneg (hf x) hs
-    have hws : ∑ x, w x = 1 := by simp only [w,←Finset.sum_div]; exact div_self hz
-    have hact (y : X) : act K f y = finiteMean f * ∑ x, w x*((Fintype.card X:ℝ)*K x y) := by
-      simp only [act,w,finiteMean,Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro x _
-      have hz' : (∑ x, f x) ≠ 0 := hz
-      dsimp [s]
-      field_simp [hz',ne_of_gt hN]
-    have hj (y : X) : (act K f y)^p ≤ (finiteMean f)^p*
-        ∑ x, w x*((Fintype.card X:ℝ)*K x y)^p := by
-      rw [hact,Real.mul_rpow (mean_nonneg f hf)
-        (Finset.sum_nonneg (fun x _ => mul_nonneg (hw x) (mul_nonneg hN.le (hK x y))))]
-      apply mul_le_mul_of_nonneg_left _ (Real.rpow_nonneg (mean_nonneg f hf) _)
-      exact (convexOn_rpow hp).map_sum_le
-        (fun x _ => hw x) hws (fun x _ => mul_nonneg hN.le (hK x y))
-    have hh := finiteMean_mono hj
-    have he : finiteMean (fun y => (finiteMean f)^p*
-        ∑ x, w x*((Fintype.card X:ℝ)*K x y)^p) =
-        (finiteMean f)^p * ∑ x, w x*finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) := by
-      simp_rw [mul_comm ((finiteMean f)^p), finiteMean_mul_const,
-        finiteMean_sum, mul_comm (w _), finiteMean_mul_const]
-    rw [he] at hh
-    apply hh.trans
-    have hsum : (∑ x, w x*finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p)) ≤ Real.exp A := by
-      calc
-        _ ≤ ∑ x, w x*Real.exp A := Finset.sum_le_sum
-          (fun x _ => mul_le_mul_of_nonneg_left (hm x) (hw x))
-        _ = Real.exp A := by rw [←Finset.sum_mul,hws,one_mul]
-    simpa only [mul_comm (Real.exp A)] using
-      mul_le_mul_of_nonneg_left hsum (Real.rpow_nonneg (mean_nonneg f hf) _)
-
-lemma act_b_to_pb (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
-    (hcol : ∀ y, ∑ x, K x y = 1) {p A b : ℝ} (hp : 1 ≤ p) (hb : 1 ≤ b)
-    (hm : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) ≤ Real.exp A)
-    (f : X → ℝ) (hf : ∀ x, 0 ≤ f x) :
-    finiteMean (fun y => (act K f y)^(b*p)) ≤
-      Real.exp A*(finiteMean (fun x => (f x)^b))^p := by
-  calc
-    _ ≤ finiteMean (fun y => (act K (fun x => (f x)^b) y)^p) := by
-      apply finiteMean_mono
-      intro y
-      rw [Real.rpow_mul (act_nonneg K hK f hf y)]
-      exact Real.rpow_le_rpow (Real.rpow_nonneg (act_nonneg K hK f hf y) _) (act_rpow_le K hK hcol f hf hb y) (by linarith)
-    _ ≤ _ := act_one_to_p K hK hp hm _ (fun x => Real.rpow_nonneg (hf x) _)
-
-variable [DecidableEq X]
-
-omit [Nonempty X] in
-lemma kernel_pow_nonneg (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y) (r : ℕ) :
-    ∀ x y, 0 ≤ (K^r) x y := by
-  classical
-  induction r with
-  | zero => intro x y; simp only [pow_zero,Matrix.one_apply]; split_ifs <;> norm_num
-  | succ r ih =>
-    intro x y
-    rw [pow_succ,Matrix.mul_apply]
-    exact Finset.sum_nonneg (fun z _ => mul_nonneg (ih x z) (hK z y))
-
-omit [Nonempty X] in
-lemma kernel_pow_row (K : Matrix X X ℝ) (hrow : ∀ x, ∑ y, K x y = 1) (r : ℕ) :
-    ∀ x, ∑ y, (K^r) x y = 1 := by
-  classical
-  induction r with
-  | zero => intro x; simp [Matrix.one_apply]
-  | succ r ih =>
-    intro x
-    simp only [pow_succ,Matrix.mul_apply]
-    rw [Finset.sum_comm]
-    simp only [←Finset.mul_sum,hrow,mul_one,ih]
-
-noncomputable def rowDensity (K : Matrix X X ℝ) (r : ℕ) (x y : X) : ℝ :=
-  (Fintype.card X:ℝ)*(K^r) x y
-
-omit [Nonempty X] in
-lemma rowDensity_nonneg (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y) (r : ℕ) (x y : X) :
-    0 ≤ rowDensity K r x y := mul_nonneg (Nat.cast_nonneg _) (kernel_pow_nonneg K hK r x y)
-
-omit [Nonempty X] in
-lemma rowDensity_succ (K : Matrix X X ℝ) (r : ℕ) (x y : X) :
-    rowDensity K (r+1) x y = act K (rowDensity K r x) y := by
-  simp only [rowDensity,pow_succ,Matrix.mul_apply,act,Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro z _
-  ring
-
-lemma rowDensity_mean (K : Matrix X X ℝ) (hrow : ∀ x, ∑ y, K x y = 1) (r : ℕ) (x : X) :
-    finiteMean (rowDensity K r x) = 1 := by
-  have hN : (Fintype.card X:ℝ) ≠ 0 := ne_of_gt (Nat.cast_pos.mpr Fintype.card_pos)
-  simp only [finiteMean,rowDensity,←Finset.mul_sum,kernel_pow_row K hrow,mul_one,div_self hN]
-
-lemma rowDensity_moment_iter (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
-    (hrow : ∀ x, ∑ y, K x y = 1) (hcol : ∀ y, ∑ x, K x y = 1)
-    {p A : ℝ} (hp : 1 ≤ p) (hA : 0 ≤ A)
-    (hm : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) ≤ Real.exp A)
-    (r : ℕ) (x : X) :
-    finiteMean (fun y => (rowDensity K r x y)^(p^r)) ≤ Real.exp (A*r*p^r) := by
-  induction r with
-  | zero => simp [rowDensity_mean K hrow]
-  | succ r ih =>
-    have hpr : 1 ≤ p^r := one_le_pow₀ hp
-    have hps : 1 ≤ p^(r+1) := one_le_pow₀ hp
-    simp_rw [rowDensity_succ]
-    calc
-      _ ≤ Real.exp A*(finiteMean (fun y => (rowDensity K r x y)^(p^r)))^p := by
-        rw [pow_succ]
-        exact act_b_to_pb K hK hcol hp hpr hm _ (rowDensity_nonneg K hK r x)
-      _ ≤ Real.exp A*(Real.exp (A*r*p^r))^p := by
-        apply mul_le_mul_of_nonneg_left _ (Real.exp_pos _).le
-        exact Real.rpow_le_rpow (mean_nonneg _ (fun y => Real.rpow_nonneg
-          (rowDensity_nonneg K hK r x y) _)) ih (le_trans zero_le_one hp)
-      _ = Real.exp (A+A*r*p^(r+1)) := by rw [←Real.exp_mul,←Real.exp_add,pow_succ]; congr 1; ring
-      _ ≤ _ := by
-        apply Real.exp_le_exp.mpr
-        have hh := mul_le_mul_of_nonneg_left hps hA
-        push_cast
-        nlinarith
-
-lemma rowDensity_second_moment (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
-    (hrow : ∀ x, ∑ y, K x y = 1) (hcol : ∀ y, ∑ x, K x y = 1)
-    {p A : ℝ} (hp : 1 ≤ p) (hA : 0 ≤ A)
-    (hm : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) ≤ Real.exp A)
-    (r : ℕ) (hr : 2 ≤ p^r) (x : X) :
-    finiteMean (fun y => (rowDensity K r x y)^2) ≤ Real.exp (2*A*r) := by
-  have hpos : 0 < p^r := by linarith
-  have hj := mean_rpow_concave (fun y => (rowDensity K r x y)^(p^r))
-    (fun y => Real.rpow_nonneg (rowDensity_nonneg K hK r x y) _)
-    (a := 2/(p^r)) (by positivity) ((div_le_one hpos).mpr hr)
-  have he (y : X) : ((rowDensity K r x y)^(p^r))^(2/(p^r)) = (rowDensity K r x y)^2 := by
-    rw [←Real.rpow_mul (rowDensity_nonneg K hK r x y)]
-    rw [mul_div_cancel₀ _ (ne_of_gt hpos),Real.rpow_two]
-  simp_rw [he] at hj
-  apply hj.trans
-  calc
-    _ ≤ (Real.exp (A*r*p^r))^(2/(p^r)) :=
-      Real.rpow_le_rpow (mean_nonneg _ (fun y => Real.rpow_nonneg
-        (rowDensity_nonneg K hK r x y) _)) (rowDensity_moment_iter K hK hrow hcol hp hA hm r x)
-        (by positivity)
-    _ = _ := by rw [←Real.exp_mul]; congr 1; field_simp [ne_of_gt hpos]
-
-lemma symmetric_kernel_smoothing (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
-    (hrow : ∀ x, ∑ y, K x y = 1) (hcol : ∀ y, ∑ x, K x y = 1)
-    (hsym : K.IsSymm) {p A : ℝ} (hp : 1 ≤ p) (hA : 0 ≤ A)
-    (hm : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) ≤ Real.exp A)
-    (r : ℕ) (hr : 2 ≤ p^r) (x y : X) :
-    (K^(2*r)) x y ≤ Real.exp (2*A*r)/(Fintype.card X:ℝ) := by
-  have hN : (0:ℝ) < Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
-  have hs : ∀ z, (K^r) z y = (K^r) y z := fun z => (hsym.pow r).apply y z
-  have he : (Fintype.card X:ℝ)*(K^(2*r)) x y =
-      finiteMean (fun z => rowDensity K r x z*rowDensity K r y z) := by
-    simp only [finiteMean,rowDensity,two_mul,pow_add,Matrix.mul_apply,hs,Finset.mul_sum]
-    rw [Finset.sum_div]
-    apply Finset.sum_congr rfl
-    intro z _
-    field_simp
-  have hh : finiteMean (fun z => rowDensity K r x z*rowDensity K r y z) ≤
-      Real.exp (2*A*r) := by
-    calc
-      _ ≤ finiteMean (fun z => ((rowDensity K r x z)^2+(rowDensity K r y z)^2)/2) := by
-        apply finiteMean_mono
-        intro z
-        nlinarith [sq_nonneg (rowDensity K r x z-rowDensity K r y z)]
-      _ = (finiteMean (fun z => (rowDensity K r x z)^2)+
-          finiteMean (fun z => (rowDensity K r y z)^2))/2 := by
-        simp only [div_eq_mul_inv,finiteMean_mul_const,finiteMean_add]
-      _ ≤ _ := by
-        have hx := rowDensity_second_moment K hK hrow hcol hp hA hm r hr x
-        have hy := rowDensity_second_moment K hK hrow hcol hp hA hm r hr y
-        linarith
-  apply (le_div_iff₀ hN).mpr
-  simpa only [←he,mul_comm (Fintype.card X:ℝ)] using hh
-
-lemma density_exponent_numeric : 2 ≤ (1+(1/1000:ℝ))^(1000:ℕ) := by
-  calc
-    (2:ℝ) = 1+(1000:ℝ)*(1/1000) := by norm_num
-    _ ≤ _ := by simpa only [Nat.cast_ofNat] using
-      (one_add_mul_le_pow (a := (1/1000:ℝ)) (by norm_num) 1000)
-
-end Thorp.StrongSmoothing
-
-namespace Thorp.StrongSmoothing.PermLaw
-open scoped BigOperators
-variable {X Ω Ω' : Type*} [Fintype X] [DecidableEq X] [Fintype Ω] [Fintype Ω']
-
-noncomputable def kernel (P : Ω → Equiv.Perm X) : Matrix X X ℝ :=
-  fun x y => finiteMean (fun ω => if P ω x = y then (1:ℝ) else 0)
-
-omit [Fintype X] in
-lemma nonneg (P : Ω → Equiv.Perm X) (x y : X) : 0 ≤ kernel P x y := by
-  exact finiteMean_nonneg (fun ω => by split_ifs <;> norm_num)
-
-lemma row [Nonempty Ω] (P : Ω → Equiv.Perm X) (x : X) : ∑ y, kernel P x y = 1 := by
-  simp only [kernel,←finiteMean_sum]
-  simp only [Finset.sum_ite_eq,Finset.mem_univ,↓reduceIte,finiteMean_const]
-
-omit [Fintype X] in
-lemma inverse (P : Ω → Equiv.Perm X) : kernel (fun ω => (P ω)⁻¹) = (kernel P).transpose := by
-  ext x y
-  apply finiteMean_congr
-  intro ω
-  have he : (P ω)⁻¹ x=y ↔ P ω y=x := by
-    change (P ω).symm x=y ↔ P ω y=x
-    exact (Equiv.symm_apply_eq (P ω)).trans eq_comm
-  simp only [he]
-
-lemma col [Nonempty Ω] (P : Ω → Equiv.Perm X) (y : X) : ∑ x, kernel P x y = 1 := by
-  have hh := row (fun ω => (P ω)⁻¹) y
-  rw [inverse] at hh
-  exact hh
-
-lemma mul (P : Ω → Equiv.Perm X) (Q : Ω' → Equiv.Perm X) :
-    kernel (fun ω : Ω × Ω' => P ω.1*Q ω.2) = kernel Q*kernel P := by
-  ext x y
-  simp only [Matrix.mul_apply,kernel]
-  have he (ω : Ω × Ω') :
-      (if (P ω.1*Q ω.2) x=y then (1:ℝ) else 0) =
-        ∑ z : X, (if P ω.1 z=y then (1:ℝ) else 0)*(if Q ω.2 x=z then (1:ℝ) else 0) := by
-    simp only [mul_ite,mul_one,mul_zero,Finset.sum_ite_eq,Finset.mem_univ,↓reduceIte]
-    rfl
-  change finiteMean (fun ω : Ω × Ω' => if (P ω.1*Q ω.2) x=y then (1:ℝ) else 0) = _
-  simp_rw [he]
-  rw [finiteMean_sum]
-  apply Finset.sum_congr rfl
-  intro z _
-  exact (finiteMean_prod_mul (fun ω : Ω => if P ω z=y then (1:ℝ) else 0)
-    (fun ω : Ω' => if Q ω x=z then (1:ℝ) else 0)).trans (mul_comm _ _)
-
-omit [Fintype X] in
-lemma conjugate (P : Ω → Equiv.Perm X) (e : Equiv.Perm X) (x y : X) :
-    kernel (fun ω => e*P ω*e⁻¹) x y = kernel P (e.symm x) (e.symm y) := by
-  apply finiteMean_congr
-  intro ω
-  have he : (e*P ω*e⁻¹) x=y ↔ P ω (e.symm x)=e.symm y := by
-    change e (P ω (e.symm x))=y ↔ _
-    exact (Equiv.eq_symm_apply e).symm
-  simp only [he]
-
-end Thorp.StrongSmoothing.PermLaw
-
-namespace Thorp.StrongSmoothing
-open scoped BigOperators
-
-section MatrixHelpers
-variable {X : Type*} [Fintype X] [DecidableEq X]
-
-omit [DecidableEq X] in
-lemma mul_nonneg_entries (A B : Matrix X X ℝ) (hA : ∀ x y, 0 ≤ A x y)
-    (hB : ∀ x y, 0 ≤ B x y) (x y : X) : 0 ≤ (A*B) x y :=
-  Finset.sum_nonneg (fun z _ => mul_nonneg (hA x z) (hB z y))
-
-omit [DecidableEq X] in
-lemma mul_row (A B : Matrix X X ℝ) (hA : ∀ x, ∑ y, A x y = 1)
-    (hB : ∀ x, ∑ y, B x y = 1) (x : X) : ∑ y, (A*B) x y = 1 := by
-  simp only [Matrix.mul_apply]
-  rw [Finset.sum_comm]
-  simp only [←Finset.mul_sum,hB,mul_one,hA]
-
-omit [DecidableEq X] in
-lemma mul_col (A B : Matrix X X ℝ) (hA : ∀ y, ∑ x, A x y = 1)
-    (hB : ∀ y, ∑ x, B x y = 1) (y : X) : ∑ x, (A*B) x y = 1 := by
-  simp only [Matrix.mul_apply]
-  rw [Finset.sum_comm]
-  simp only [←Finset.sum_mul,hA,one_mul,hB]
-
-omit [DecidableEq X] in
-lemma bound_mul_left (A B : Matrix X X ℝ) (hA : ∀ x y, 0 ≤ A x y)
-    (hrow : ∀ x, ∑ y, A x y = 1) {D : ℝ} (hB : ∀ x y, B x y ≤ D) :
-    ∀ x y, (A*B) x y ≤ D := by
-  intro x y
-  calc
-    _ ≤ ∑ z, A x z*D := Finset.sum_le_sum (fun z _ => mul_le_mul_of_nonneg_left (hB z y) (hA x z))
-    _ = D := by rw [←Finset.sum_mul,hrow,one_mul]
-
-omit [DecidableEq X] in
-lemma bound_mul_right (A B : Matrix X X ℝ) (hB : ∀ x y, 0 ≤ B x y)
-    (hcol : ∀ y, ∑ x, B x y = 1) {D : ℝ} (hA : ∀ x y, A x y ≤ D) :
-    ∀ x y, (A*B) x y ≤ D := by
-  intro x y
-  calc
-    _ ≤ ∑ z, D*B z y := Finset.sum_le_sum (fun z _ => mul_le_mul_of_nonneg_right (hA x z) (hB z y))
-    _ = D := by rw [←Finset.mul_sum,hcol,mul_one]
-
-lemma sandwich_power (A B : Matrix X X ℝ) (r : ℕ) : A*(B*A)^r*B = (A*B)^(r+1) := by
-  induction r with
-  | zero => simp
-  | succ r ih =>
-    calc
-      _ = (A*(B*A)^r*B)*(A*B) := by rw [pow_succ]; noncomm_ring
-      _ = (A*B)^(r+1)*(A*B) := by rw [ih]
-      _ = _ := (pow_succ (A*B) (r+1)).symm
-
-end MatrixHelpers
-
-section TupleAction
-variable {ι β Ω : Type*} [Fintype ι] [Fintype β] [DecidableEq β] [Fintype Ω]
-
-def tuplePerm (P : Equiv.Perm β) : Equiv.Perm (ι ↪ β) where
-  toFun x := x.trans P.toEmbedding
-  invFun x := x.trans P.symm.toEmbedding
-  left_inv x := by ext i; simp
-  right_inv x := by ext i; simp
-
-omit [Fintype ι] [Fintype β] [DecidableEq β] in
-@[simp] lemma tuplePerm_apply (P : Equiv.Perm β) (x : ι ↪ β) (i : ι) :
-    tuplePerm P x i = P (x i) := rfl
-
-omit [Fintype ι] [Fintype β] [DecidableEq β] in
-lemma tuplePerm_mul (P Q : Equiv.Perm β) :
-    tuplePerm (ι := ι) (P*Q) = tuplePerm P*tuplePerm Q := by
-  ext x i
-  rfl
-
-omit [Fintype ι] [Fintype β] [DecidableEq β] in
-lemma tuplePerm_inverse (P : Equiv.Perm β) :
-    tuplePerm (ι := ι) P⁻¹ = (tuplePerm P)⁻¹ := by
-  ext x i
-  rfl
-
-omit [Fintype β] in
-lemma tupleProbability_kernel (P : Ω → Equiv.Perm β) (x y : ι ↪ β) :
-    PairRouting.tupleProbability P x y = PermLaw.kernel (fun ω => tuplePerm (P ω)) x y := by
-  classical
-  apply finiteMean_congr
-  intro ω
-  have he : (∀ i, P ω (x i) = y i) ↔ tuplePerm (P ω) x = y := by
-    constructor
-    · intro h; ext i; exact h i
-    · intro h i; exact congrArg (fun z : ι ↪ β => z i) h
-  simp only [he]
-end TupleAction
-
-lemma sweepKernel_law (d l : ℕ) (order : Equiv.Perm (Fin d)) :
-    sweepKernel d l order = PermLaw.kernel (fun ω => tuplePerm (ι := Fin l) (sweep d order ω)) := by
-  ext x y
-  exact tupleProbability_kernel _ x y
-
-lemma reflectedKernel_conjugate (d l : ℕ) (order : Equiv.Perm (Fin d)) (x y : Tuples d l) :
-    reflectedKernel d l order false x y =
-      PairRouting.tupleProbability (palindromePerm d)
-        ((tuplePerm (coordinateRelabel d order)).symm x)
-        ((tuplePerm (coordinateRelabel d order)).symm y) := by
-  classical
-  let e : Equiv.Perm (Tuples d l) := tuplePerm (coordinateRelabel d order)
-  let P := fun ω => tuplePerm (ι := Fin l) (butterflyPerm d (decodeButterfly d ω))
-  have hA : sweepKernel d l order = PermLaw.kernel (fun ω => e*P ω*e⁻¹) := by
-    rw [sweepKernel_law]
-    simp only [sweep,tuplePerm_mul,tuplePerm_inverse]
-    rfl
-  rw [reflectedKernel,ite_eq_right Bool.false_ne_true,hA]
-  rw [←PermLaw.inverse,←PermLaw.mul]
-  have hprod : (fun ω : BenesCoins d => (e*P ω.1*e⁻¹)*((e*P ω.2*e⁻¹)⁻¹)) =
-      fun ω : BenesCoins d => e*tuplePerm (ι := Fin l) (palindromePerm d ω)*e⁻¹ := by
-    funext ω
-    simp only [mul_inv_rev,inv_inv,palindromePerm,tuplePerm_mul]
-    change (e*P ω.1*e⁻¹)*(e*(P ω.2)⁻¹*e⁻¹) = e*(P ω.1*(P ω.2)⁻¹)*e⁻¹
-    group
-  rw [hprod,PermLaw.conjugate,tupleProbability_kernel]
-
-
-lemma sweepKernel_nonneg (d l : ℕ) (order : Equiv.Perm (Fin d)) (x y : Tuples d l) :
-    0 ≤ sweepKernel d l order x y := PairRouting.tupleProbability_nonneg _ _ _
-
-lemma sweepKernel_row (d l : ℕ) (order : Equiv.Perm (Fin d)) (x : Tuples d l) :
-    ∑ y, sweepKernel d l order x y = 1 := by
-  rw [sweepKernel_law]
-  exact PermLaw.row _ _
-
-lemma sweepKernel_col (d l : ℕ) (order : Equiv.Perm (Fin d)) (y : Tuples d l) :
-    ∑ x, sweepKernel d l order x y = 1 := by
-  rw [sweepKernel_law]
-  exact PermLaw.col _ _
-
-lemma reflected_nonneg (d l : ℕ) (order : Equiv.Perm (Fin d)) (x y : Tuples d l) :
-    0 ≤ reflectedKernel d l order false x y := by
-  exact mul_nonneg_entries _ _ (fun x y => sweepKernel_nonneg d l order y x)
-    (sweepKernel_nonneg d l order) x y
-
-lemma reflected_row (d l : ℕ) (order : Equiv.Perm (Fin d)) (x : Tuples d l) :
-    ∑ y, reflectedKernel d l order false x y = 1 :=
-  mul_row _ _ (sweepKernel_col d l order) (sweepKernel_row d l order) x
-
-lemma reflected_col (d l : ℕ) (order : Equiv.Perm (Fin d)) (y : Tuples d l) :
-    ∑ x, reflectedKernel d l order false x y = 1 :=
-  mul_col _ _ (sweepKernel_row d l order) (sweepKernel_col d l order) y
-
-lemma reflected_symmetric (d l : ℕ) (order : Equiv.Perm (Fin d)) :
-    (reflectedKernel d l order false).IsSymm := Matrix.isSymm_transpose_mul_self _
-
-lemma reflected_moment (d l : ℕ) (order : Equiv.Perm (Fin d)) (x : Tuples d l) :
-    finiteMean (fun y : Tuples d l =>
-      ((Fintype.card (Tuples d l):ℝ)*reflectedKernel d l order false x y)^(1+(1/1000:ℝ))) ≤
-      Real.exp (l*(1+heightDecay 1)) := by
-  classical
-  let e : Equiv.Perm (Tuples d l) := tuplePerm (coordinateRelabel d order)
-  simp_rw [reflectedKernel_conjugate]
-  have he := finiteMean_equiv e (fun y : Tuples d l =>
-    ((Fintype.card (Tuples d l):ℝ)*PairRouting.tupleProbability (palindromePerm d) (e.symm x) y)^
-      (1+(1/1000:ℝ)))
-  change finiteMean (fun y =>
-    ((Fintype.card (Tuples d l):ℝ)*PairRouting.tupleProbability (palindromePerm d) (e.symm x) (e.symm y))^
-      (1+(1/1000:ℝ))) ≤ _
-  rw [←he]
-  simpa only [palindromeRowMoment,Fintype.card_fin] using palindrome_row_moment_coarse d (e.symm x)
-
-lemma tuples_card (d l : ℕ) : Fintype.card (Tuples d l) = (2^d).descFactorial l := by
-  simp only [Tuples,Fintype.card_embedding_eq,Fintype.card_fin,card_positions]
-
-lemma canonical_pointwise (d l : ℕ) (order : Equiv.Perm (Fin d)) (x y : Tuples d l) :
-    ((reflectedKernel d l order false)^2000) x y ≤
-      Real.exp ((2000*(1+heightDecay 1))*l)/((2^d).descFactorial l:ℝ) := by
-  let : Nonempty (Tuples d l) := ⟨x⟩
-  have hh := symmetric_kernel_smoothing (reflectedKernel d l order false)
-    (reflected_nonneg d l order) (reflected_row d l order) (reflected_col d l order)
-    (reflected_symmetric d l order) (p := 1+(1/1000:ℝ)) (A := l*(1+heightDecay 1))
-    (by norm_num) (mul_nonneg (Nat.cast_nonneg _) (by linarith [heightDecay_nonneg 1]))
-    (reflected_moment d l order) 1000 density_exponent_numeric x y
-  have he : 2*((l:ℝ)*(1+heightDecay 1))*(1000:ℝ) = (2000*(1+heightDecay 1))*l := by ring
-  norm_num only [show 2*1000=2000 by norm_num,Nat.cast_ofNat] at hh
-  simpa only [he,tuples_card] using hh
-
-lemma both_pointwise (d l : ℕ) (order : Equiv.Perm (Fin d)) (orientation : Bool)
-    (x y : Tuples d l) :
-    ((reflectedKernel d l order orientation)^2001) x y ≤
-      Real.exp ((2000*(1+heightDecay 1))*l)/((2^d).descFactorial l:ℝ) := by
-  cases orientation
-  · change ((reflectedKernel d l order false)^(2000+1)) x y ≤ _
-    rw [pow_succ]
-    exact bound_mul_right _ _ (reflected_nonneg d l order) (reflected_col d l order)
-      (canonical_pointwise d l order) x y
-  · let A := sweepKernel d l order
-    change ((A*A.transpose)^(2000+1)) x y ≤ _
-    rw [←sandwich_power]
-    apply bound_mul_right _ _ (fun x y => sweepKernel_nonneg d l order y x)
-      (sweepKernel_row d l order)
-    exact bound_mul_left _ _ (sweepKernel_nonneg d l order) (sweepKernel_row d l order)
-      (canonical_pointwise d l order)
-
-lemma regular_trace_bound (d : ℕ) (order : Equiv.Perm (Fin d)) :
-    regularAbsoluteEvenTrace d order 2001 ≤ Real.exp ((2000*(1+heightDecay 1))*(2:ℝ)^d) := by
-  have hn : 0 < Fintype.card (Tuples d (2^d)) := by
-    rw [tuples_card,Nat.descFactorial_self]
-    exact Nat.factorial_pos _
-  have hn' : (Fintype.card (Tuples d (2^d)):ℝ) ≠ 0 := Nat.cast_ne_zero.mpr (ne_of_gt hn)
-  calc
-    _ ≤ ∑ x : Tuples d (2^d), Real.exp ((2000*(1+heightDecay 1))*(2^d:ℕ)) /
-        ((2^d).descFactorial (2^d):ℝ) := by
-      apply Finset.sum_le_sum
-      intro x _
-      exact both_pointwise d (2^d) order false x x
-    _ = _ := by
-      simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,←tuples_card]
-      push_cast
-      field_simp
-
-theorem strong_smoothing : MainStatement := by
-  refine ⟨2001,by norm_num,2000*(1+heightDecay 1),?_,regular_trace_bound⟩
-  intro d l _ order orientation x y
-  exact both_pointwise d l order orientation x y
-
-end Thorp.StrongSmoothing
-
-namespace Thorp.StrongTail
-open scoped BigOperators Classical
-open Specht
-
-lemma mem_tailDiagram (μ : YoungDiagram) (i j : ℕ) :
-    (i,j) ∈ tailDiagram μ ↔ (i+1,j) ∈ μ := by
-  rw [tailDiagram, YoungDiagram.mem_ofRowLens]
-  constructor
-  · rintro ⟨hi,hj⟩
-    rw [List.getElem_tail,YoungDiagram.get_rowLens] at hj
-    exact YoungDiagram.mem_iff_lt_rowLen.mpr hj
-  · intro h
-    have hi : i+1 < μ.colLen 0 :=
-      (YoungDiagram.mem_iff_lt_colLen.mp h).trans_le (μ.colLen_anti 0 j (Nat.zero_le _))
-    have hi' : i < μ.rowLens.tail.length := by simp only [List.length_tail,YoungDiagram.length_rowLens]; omega
-    refine ⟨hi', ?_⟩
-    simpa only [List.getElem_tail,YoungDiagram.get_rowLens] using YoungDiagram.mem_iff_lt_rowLen.mp h
-
-lemma tail_height (μ : YoungDiagram) : (tailDiagram μ).colLen 0 = μ.colLen 0 - 1 := by
-  rw [←YoungDiagram.length_rowLens]
-  exact (YoungDiagram.rowLens_length_ofRowLens (fun x hx => μ.pos_of_mem_rowLens x (List.mem_of_mem_tail hx))).trans
-    (by simp only [List.length_tail,YoungDiagram.length_rowLens])
-
-end Thorp.StrongTail
-
-end ThorpNine.Tail
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Tail.InsertionPerm
+
+namespace OAI
+
+namespace ThorpNine.Tail
+
+namespace Thorp.Specht
+open scoped BigOperators Classical
+
+abbrev TailShapes (N j : ℕ) := {μ : Shapes N // μ.1.card-μ.1.rowLen 0=j}
+
+noncomputable def tailShapePartition (N j : ℕ) (μ : TailShapes N j) : Nat.Partition j where
+  parts := μ.1.1.rowLens.tail
+  parts_pos := fun {x} hx => μ.1.1.pos_of_mem_rowLens x (List.mem_of_mem_tail (by simpa using hx))
+  parts_sum := by simpa using (rowLens_tail_sum μ.1.1).trans μ.2
+
+lemma tailShapePartition_injective (N j : ℕ) : Function.Injective (tailShapePartition N j) := by
+  intro μ ν he
+  have hh : (μ.1.1.rowLens.tail : Multiset ℕ)=(ν.1.1.rowLens.tail : Multiset ℕ) :=
+    congrArg Nat.Partition.parts he
+  have ht : μ.1.1.rowLens.tail=ν.1.1.rowLens.tail := by
+    have hm := congrArg (fun s : Multiset ℕ => s.sort (· ≥ ·)) hh
+    simpa only [Multiset.coe_sort,
+      List.mergeSort_eq_self _ μ.1.1.rowLens_sorted.pairwise.tail,
+      List.mergeSort_eq_self _ ν.1.1.rowLens_sorted.pairwise.tail] using hm
+  have hl := positive_list_eq_of_tail_sum (μ.1.1.pos_of_mem_rowLens) (ν.1.1.pos_of_mem_rowLens)
+    ht (by rw [sum_rowLens,sum_rowLens,μ.1.2,ν.1.2])
+  apply Subtype.ext
+  apply Subtype.ext
+  apply YoungDiagram.equivListRowLens.injective
+  exact Subtype.ext hl
+
+lemma tail_shapes_count (N j : ℕ) : Fintype.card (TailShapes N j)≤2^j := by
+  exact (Fintype.card_le_of_injective _ (tailShapePartition_injective N j)).trans (partition_count j)
+
+end Thorp.Specht
+
+namespace Thorp
+open scoped BigOperators Classical
+
+lemma tail_preserving_pairSwitch (d : ℕ) (p : Equiv.Perm (Card (d+1)))
+    (hp : ∀ x, Fin.tail (p x)=Fin.tail x) :
+    pairSwitch (fun u => p (Fin.cons false u) 0)=p := by
+  apply Equiv.ext
+  intro x
+  rw [←Fin.cons_self_tail x]
+  generalize x 0=b
+  generalize Fin.tail x=u
+  apply funext
+  intro i
+  refine Fin.cases ?_ (fun j => ?_) i
+  · change Bool.xor b (p (Fin.cons false u) 0)=p (Fin.cons b u) 0
+    cases b with
+    | false => simp
+    | true =>
+      have hn : p (Fin.cons false u) 0 ≠ p (Fin.cons true u) 0 := by
+        intro he
+        have hh : p (Fin.cons false u)=p (Fin.cons true u) := by
+          rw [←Fin.cons_self_tail (p (Fin.cons false u)),←Fin.cons_self_tail (p (Fin.cons true u)),
+            hp,hp,Fin.tail_cons,Fin.tail_cons,he]
+        have hh := congrFun (p.injective hh) 0
+        simp only [Fin.cons_zero] at hh
+        contradiction
+      cases h0 : p (Fin.cons false u) 0 <;> cases h1 : p (Fin.cons true u) 0 <;> simp_all
+  · change u j=p (Fin.cons b u) j.succ
+    have h := congrFun (hp (Fin.cons b u)) j
+    simpa only [Fin.tail_cons,Fin.tail,Fin.cons_succ] using h.symm
+
+lemma pairSwitch_injective (d : ℕ) : Function.Injective (pairSwitch (d := d)) := by
+  intro ξ η h
+  funext u
+  have hh := congrArg (fun p : Equiv.Perm (Card (d+1)) => p (Fin.cons false u) 0) h
+  change Bool.xor false (ξ u)=Bool.xor false (η u) at hh
+  simpa only [Bool.false_xor] using hh
+
+end Thorp
+namespace Thorp.Specht
+open scoped BigOperators Classical
+
+noncomputable def pairFiberEquiv (d : ℕ) (μ : YoungDiagram) (e : Card (d+1) ≃ Cell μ) :
+    (Card d → Bool) ≃ fiberGroup (fun x => Fin.tail (e.symm x)) :=
+  Equiv.ofBijective (fun ξ => ⟨e.permCongr (pairSwitch ξ),by
+    intro x
+    simp only [Equiv.permCongr_apply,e.symm_apply_apply]
+    exact Fin.tail_cons _ _⟩)
+    ⟨fun ξ η h => pairSwitch_injective d (e.permCongr.injective (congrArg Subtype.val h)),by
+      intro p
+      let q := e.symm.permCongr p.1
+      have hq : ∀ x, Fin.tail (q x)=Fin.tail x := by
+        intro x
+        have h := p.2 (e x)
+        simpa only [q,Equiv.permCongr_apply,Equiv.symm_symm,e.symm_apply_apply] using h
+      refine ⟨fun u => q (Fin.cons false u) 0,?_⟩
+      apply Subtype.ext
+      change e.permCongr (pairSwitch (fun u => q (Fin.cons false u) 0))=p.1
+      rw [tail_preserving_pairSwitch d q hq]
+      apply Equiv.ext
+      intro x
+      simp only [q,Equiv.permCongr_apply,Equiv.symm_symm,e.apply_symm_apply]⟩
+
+lemma pair_sum_taboid_zero (d : ℕ) (μ : YoungDiagram) (e : Card (d+1) ≃ Cell μ)
+    (hj : 2^d < μ.colLen 0) (v : space μ) :
+    (∑ ξ : Card d → Bool, tabloidRep μ (e.permCongr (pairSwitch ξ))) v.1=0 := by
+  have he : (∑ ξ : Card d → Bool, tabloidRep μ (e.permCongr (pairSwitch ξ)))=
+      symmetrizer (fiberGroup (fun x => Fin.tail (e.symm x))) (tabloidRep μ) :=
+    Fintype.sum_equiv (pairFiberEquiv d μ e) _ _ (fun _ => rfl)
+  rw [he]
+  exact tall_column_average_zero μ _ 0 (by simpa only [card_positions] using hj) v.1 v.2
+
+lemma pair_operator_tall_zero (d : ℕ) (μ : YoungDiagram) (e : Card (d+1) ≃ Cell μ)
+    (hj : 2^d < μ.colLen 0) :
+    UnitaryFinite.sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e) pairSwitch=0 := by
+  apply ContinuousLinearMap.ext
+  intro v
+  obtain ⟨w,rfl⟩ := (spaceHilbertEquiv μ).surjective v
+  have hh : (∑ ξ : Card d → Bool, representation μ (e.permCongr (pairSwitch ξ))) w=0 := by
+    apply Subtype.ext
+    simp only [LinearMap.sum_apply,Submodule.coe_sum,Submodule.coe_zero]
+    change (∑ ξ : Card d → Bool, tabloidRep μ (e.permCongr (pairSwitch ξ)) w.1)=0
+    simpa only [LinearMap.sum_apply] using pair_sum_taboid_zero d μ e hj w
+  have hh' := congrArg (spaceHilbertEquiv μ) hh
+  simp only [LinearMap.sum_apply,map_sum,map_zero] at hh'
+  change ((Fintype.card (Card d → Bool):ℂ)⁻¹ •
+    ∑ ξ : Card d → Bool, UnitaryFinite.continuousRepresentation (relabelledUnitary μ e) (pairSwitch ξ))
+      (spaceHilbertEquiv μ w)=0
+  rw [smul_apply,sum_apply]
+  change (Fintype.card (Card d → Bool):ℂ)⁻¹ •
+    (∑ ξ : Card d → Bool, spaceHilbertEquiv μ (representation μ (e.permCongr (pairSwitch ξ))
+      ((spaceHilbertEquiv μ).symm (spaceHilbertEquiv μ w))))=0
+  simp only [LinearEquiv.symm_apply_apply,hh',smul_zero]
+
+open UnitaryFinite
+
+lemma palindrome_tall_zero (d : ℕ) (μ : YoungDiagram) (e : Card (d+1) ≃ Cell μ)
+    (hj : 2^d<μ.colLen 0) :
+    sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e) (palindromePerm (d+1))=0 := by
+  let ρ := relabelledUnitary μ e
+  let : NormedAddCommGroup (hilbertSpace μ) := inferInstance
+  let : InnerProductSpace ℂ (hilbertSpace μ) := inferInstance
+  let : FiniteDimensional ℂ (hilbertSpace μ) := inferInstance
+  have hpair : sampleOperator (V := hilbertSpace μ) ρ (pairSwitch (d := d))=0 := pair_operator_tall_zero d μ e hj
+  have hi : sampleOperator ρ (fun ω => (butterflyPerm (d+1) (decodeButterfly (d+1) ω))⁻¹)=0 := by
+    rw [inverse_butterfly_average_step,hpair,mul_zero]
+  rw [sampleOperator_inv ρ (relabelledUnitary_unitary μ e)] at hi
+  rw [sampleOperator_palindrome _ _ (relabelledUnitary_unitary μ e),hi,mul_zero]
+
+lemma palindrome_tall_zero_general (d : ℕ) (hd : 0<d) (μ : YoungDiagram) (e : Card d ≃ Cell μ)
+    (hj : (μ.card:ℝ)/2<(μ.colLen 0:ℝ)) :
+    sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e) (palindromePerm d)=0 := by
+  obtain ⟨r,rfl⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : d≠0)
+  apply palindrome_tall_zero r μ e
+  have hc : μ.card=2^(r+1) := by rw [←card_cell,←Fintype.card_congr e,card_positions]
+  rw [hc,Nat.cast_pow,Nat.cast_ofNat,pow_succ] at hj
+  have hh : (2:ℝ)^r<(μ.colLen 0:ℝ) := by linarith
+  exact_mod_cast hh
+
+end Thorp.Specht
+
+namespace Thorp.DimensionEstimates
+open Filter
+
+lemma twopow_tendsto : Tendsto (fun d : ℕ => (2:ℝ)^d) atTop atTop :=
+  tendsto_pow_atTop_atTop_of_one_lt (by norm_num)
+
+lemma twopow_nat_tendsto : Tendsto (fun d : ℕ => (2:ℕ)^d) atTop atTop :=
+  tendsto_pow_atTop_atTop_of_one_lt (by norm_num)
+
+lemma twopow_mul_eventually (c : ℝ) (hc : 0<c) (B : ℝ) :
+    ∀ᶠ d : ℕ in atTop, B≤c*(2:ℝ)^d := by
+  filter_upwards [twopow_tendsto.eventually (eventually_ge_atTop (B/c))] with d hd
+  exact (div_le_iff₀ hc).mp hd |>.trans_eq (mul_comm _ _)
+
+lemma exp_two_absorb (β n : ℝ) (hx : Real.log 2≤β*n/2) :
+    Real.exp (-β*n)+Real.exp (-β*n)≤Real.exp (-β*n/2) := by
+  calc
+    _ = Real.exp (Real.log 2-β*n) := by rw [Real.exp_sub,Real.exp_log (by norm_num)]; rw [neg_mul,Real.exp_neg]; ring
+    _ ≤ _ := Real.exp_le_exp.mpr (by linarith)
+
+end Thorp.DimensionEstimates
+namespace Thorp.Specht
+open scoped BigOperators Classical
+open Filter UnitaryFinite DimensionEstimates
+
+theorem exponential_dimension_spectral (h : ℝ) (hh : 0<h) :
+    ∃ a : ℝ, 0<a ∧ ∀ᶠ d : ℕ in atTop, ∀ (μ : YoungDiagram) (e : Card d ≃ Cell μ),
+      h*(2:ℝ)^d≤Real.log (Module.finrank ℂ (space μ)) →
+      ‖sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e) (palindromePerm d)‖ ≤
+        Real.exp (-a*Real.log (Module.finrank ℂ (space μ))) := by
+  obtain ⟨L,hL,ε,t,hε,_,ht,hcount,hM⟩ := medium_parameters h hh
+  obtain ⟨R,hR⟩ := eventually_atTop.mp hM
+  let A := 4000*densityConstant+1
+  have hA : 0<A := by dsimp [A]; have := densityConstant_pos; positivity
+  let β := min (-Real.log (blockGap L)*ε/(2:ℝ)^L) (h/16000)
+  have hβ : 0<β := by
+    apply lt_min
+    · exact div_pos (mul_pos (neg_pos.mpr (Real.log_neg (blockGap_pos L) (blockGap_lt_one L))) hε) (by positivity)
+    · positivity
+  let a := min (1/8000:ℝ) (β/(2*A))
+  have ha : 0<a := lt_min (by norm_num) (div_pos hβ (by positivity))
+  refine ⟨a,ha,?_⟩
+  filter_upwards [eventually_ge_atTop (L+R),twopow_mul_eventually h hh (8000*Real.log 2),
+    twopow_mul_eventually (β/2) (by positivity) (Real.log 2)] with d hd hl hb
+  intro μ e hdim
+  let : NormedAddCommGroup (hilbertSpace μ) := inferInstance
+  let : InnerProductSpace ℂ (hilbertSpace μ) := inferInstance
+  let : FiniteDimensional ℂ (hilbertSpace μ) := inferInstance
+  let := relabelledUnitary_irreducible μ e
+  have hpos : 0<Module.finrank ℂ (hilbertSpace μ) := by rw [finrank_hilbertSpace]; exact dimension_pos μ
+  have hlarge : 8000*Real.log 2≤Real.log (Module.finrank ℂ (hilbertSpace μ)) := by rw [finrank_hilbertSpace]; exact hl.trans hdim
+  have hlog : 0≤Real.log (Module.finrank ℂ (space μ)) := (by positivity : (0:ℝ)≤h*(2:ℝ)^d).trans hdim
+  by_cases hhigh : A*(2:ℝ)^d≤Real.log (Module.finrank ℂ (space μ))
+  · have hden : densityConstant*(2:ℝ)^d≤Real.log (Module.finrank ℂ (hilbertSpace μ))/4000 := by
+      rw [finrank_hilbertSpace]
+      dsimp [A] at hhigh
+      nlinarith [pow_pos (by norm_num : (0:ℝ)<2) d]
+    have hspec := @full_palindrome_decay (hilbertSpace μ) _ _ _ d (relabelledUnitary μ e)
+      inferInstance (relabelledUnitary_unitary μ e) hpos hden hlarge
+    rw [finrank_hilbertSpace] at hspec
+    apply hspec.trans (Real.exp_le_exp.mpr ?_)
+    have := min_le_left (1/8000:ℝ) (β/(2*A))
+    nlinarith
+  · obtain ⟨r,rfl⟩ := Nat.exists_eq_add_of_le (show L≤d by omega)
+    have hr : R≤r := by omega
+    have hdim' : h*(2:ℝ)^(L+r)≤Real.log (Module.finrank ℂ (hilbertSpace μ)) := by rwa [finrank_hilbertSpace]
+    have hs := @medium_exception_decay (hilbertSpace μ) _ _ _ L r h ε t ht hcount
+      (relabelledUnitary μ e) inferInstance (relabelledUnitary_unitary μ e) hpos hdim' hlarge (hR r hr)
+    have hb1 : Real.log (blockGap L)*ε*(2:ℝ)^r≤-β*(2:ℝ)^(L+r) := by
+      have hb0 := min_le_left (-Real.log (blockGap L)*ε/(2:ℝ)^L) (h/16000)
+      have hp := (le_div_iff₀ (by positivity : (0:ℝ)<(2:ℝ)^L)).mp hb0
+      have hp := mul_le_mul_of_nonneg_right hp (by positivity : (0:ℝ)≤(2:ℝ)^r)
+      rw [pow_add]
+      dsimp [β]
+      nlinarith
+    have hb2 : -h*(2:ℝ)^(L+r)/16000≤-β*(2:ℝ)^(L+r) := by
+      have hp := mul_le_mul_of_nonneg_right (min_le_right (-Real.log (blockGap L)*ε/(2:ℝ)^L) (h/16000))
+        (by positivity : (0:ℝ)≤(2:ℝ)^(L+r))
+      dsimp [β]
+      linarith
+    apply hs.trans ((add_le_add (Real.exp_le_exp.mpr hb1) (Real.exp_le_exp.mpr hb2)).trans ?_)
+    apply (exp_two_absorb β ((2:ℝ)^(L+r)) (by nlinarith)).trans
+    apply Real.exp_le_exp.mpr
+    have hdmax := le_of_not_ge hhigh
+    have hp := mul_le_mul_of_nonneg_left hdmax (by positivity : 0≤β/(2*A))
+    have he : β/(2*A)*(A*(2:ℝ)^(L+r))=β*(2:ℝ)^(L+r)/2 := by field_simp
+    rw [he] at hp
+    have hba := mul_le_mul_of_nonneg_right (min_le_right (1/8000:ℝ) (β/(2*A))) hlog
+    dsimp [a] at *
+    linarith
+
+
+theorem spectral_regimes :
+    ∃ a h : ℝ, 0<a ∧ 0<h ∧ ∀ᶠ d : ℕ in atTop,
+      ∀ (μ : YoungDiagram) (e : Card d ≃ Cell μ), 0<μ.card-μ.rowLen 0 →
+        (μ.colLen 0:ℝ)≤(μ.card:ℝ)/2 →
+        ‖sampleOperator (V := hilbertSpace μ) (relabelledUnitary μ e) (palindromePerm d)‖≤
+          Real.exp (-a*Real.log (Module.finrank ℂ (space μ))) ∧
+        1≤Real.log (Module.finrank ℂ (space μ)) ∧
+        (((1/2:ℝ)*Real.log μ.card≤Real.log (Module.finrank ℂ (space μ)) ∧
+          (μ.card-μ.rowLen 0:ℕ)≤Real.log (Module.finrank ℂ (space μ))) ∨
+          h*(2:ℝ)^d≤Real.log (Module.finrank ℂ (space μ))) := by
+  obtain ⟨X,hX,hnear⟩ := near_row_spectral
+  have hXpos : 0<X := by linarith
+  have hδ : 0<(1:ℝ)/X := by positivity
+  have hδ2 : (1:ℝ)/X≤1/2 := (div_le_iff₀ hXpos).mpr (by linarith)
+  obtain ⟨h,hh,hmedium⟩ := medium_dimension (1/X) hδ hδ2
+  obtain ⟨a₀,ha₀,hlarge⟩ := exponential_dimension_spectral h hh
+  let a := min (nearFraction/8000) a₀
+  have ha : 0<a := lt_min (by norm_num [nearFraction]) ha₀
+  refine ⟨a,h,ha,hh,?_⟩
+  have hl := (Real.tendsto_log_atTop.comp twopow_tendsto).eventually
+    (eventually_ge_atTop (2*max (32768000*Real.log 2) 1))
+  filter_upwards [hlarge,twopow_nat_tendsto.eventually hmedium,hl,
+    twopow_mul_eventually h hh 1] with d hspec hdim hlog h1
+  intro μ e hj hc
+  simp only [Function.comp_apply] at hlog
+  have hcard : μ.card=2^d := by rw [←card_cell,←Fintype.card_congr e,card_positions]
+  have hcardR : (μ.card:ℝ)=(2:ℝ)^d := by exact_mod_cast hcard
+  have hμlog : max (32768000*Real.log 2) 1≤(1/2:ℝ)*Real.log μ.card := by rw [hcardR]; linarith
+  by_cases hx : X≤(μ.card:ℝ)/(μ.card-μ.rowLen 0:ℕ)
+  · obtain ⟨hs,hl,hjD⟩ := hnear d μ e hj hx ((le_max_left _ _).trans hμlog)
+    have h1D : 1≤Real.log (Module.finrank ℂ (space μ)) := ((le_max_right _ _).trans hμlog).trans hl
+    refine ⟨hs.trans (Real.exp_le_exp.mpr ?_),h1D,Or.inl ⟨hl,hjD⟩⟩
+    have hp := mul_le_mul_of_nonneg_right (min_le_left (nearFraction/8000) a₀) (by linarith : 0≤Real.log (Module.finrank ℂ (space μ)))
+    dsimp [a]
+    linarith
+  · have htail : (1/X)*(μ.card:ℝ)≤(μ.card:ℝ)-μ.rowLen 0 := by
+      have hjR : (0:ℝ)<(μ.card-μ.rowLen 0:ℕ) := by exact_mod_cast hj
+      have hh := (div_lt_iff₀ hjR).mp (lt_of_not_ge hx)
+      have hs : ((μ.card-μ.rowLen 0:ℕ):ℝ)=(μ.card:ℝ)-μ.rowLen 0 := by
+        rw [Nat.cast_sub (rowLen_zero_le_card μ)]
+      rw [hs] at hh
+      have hd : (μ.card:ℝ)/X≤(μ.card:ℝ)-μ.rowLen 0 :=
+        (div_le_iff₀ hXpos).mpr (by nlinarith)
+      convert hd using 1; ring
+    have hdim' : h*(2:ℝ)^d≤Real.log (Module.finrank ℂ (space μ)) := by
+      simpa only [Nat.cast_pow,Nat.cast_ofNat] using
+        hdim μ hcard (by simpa only [hcard] using htail) (by simpa only [hcard] using hc)
+    have h1D := h1.trans hdim'
+    refine ⟨(hspec μ e hdim').trans (Real.exp_le_exp.mpr ?_),h1D,Or.inr hdim'⟩
+    have hp := mul_le_mul_of_nonneg_right (min_le_right (nearFraction/8000) a₀) (by linarith : 0≤Real.log (Module.finrank ℂ (space μ)))
+    dsimp [a]
+    linarith
+
+end Thorp.Specht
+
+namespace Thorp.StrongSmoothing
+
+noncomputable def regularAbsoluteEvenTrace (d : ℕ) (order : Equiv.Perm (Fin d)) (u : ℕ) : ℝ :=
+  Matrix.trace ((reflectedKernel d (2^d) order false)^u)
+
+def MainStatement : Prop :=
+  ∃ u : ℕ, 1 ≤ u ∧ ∃ C : ℝ,
+    (∀ d l : ℕ, l ≤ 2^d → ∀ order : Equiv.Perm (Fin d), ∀ orientation : Bool,
+      ∀ x y : Tuples d l,
+        ((reflectedKernel d l order orientation)^u) x y ≤
+          Real.exp (C*l)/((2^d).descFactorial l:ℝ)) ∧
+    (∀ d : ℕ, ∀ order : Equiv.Perm (Fin d),
+      regularAbsoluteEvenTrace d order u ≤ Real.exp (C*(2:ℝ)^d))
+
+
+open scoped BigOperators
+
+variable {X : Type*} [Fintype X] [Nonempty X]
+
+omit [Nonempty X] in
+lemma mean_nonneg (f : X → ℝ) (hf : ∀ x, 0 ≤ f x) : 0 ≤ finiteMean f :=
+  finiteMean_nonneg hf
+
+lemma mean_rpow_concave (f : X → ℝ) (hf : ∀ x, 0 ≤ f x)
+    {a : ℝ} (ha : 0 ≤ a) (ha1 : a ≤ 1) :
+    finiteMean (fun x => (f x)^a) ≤ (finiteMean f)^a := by
+  have hN : (0:ℝ) < Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
+  have hh := (Real.concaveOn_rpow ha ha1).le_map_sum
+    (t := Finset.univ) (w := fun _ : X => 1/(Fintype.card X:ℝ))
+    (p := f) (fun _ _ => by positivity)
+    (by simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]; field_simp)
+    (fun x _ => hf x)
+  simpa only [smul_eq_mul,one_div,←Finset.mul_sum,mul_comm _ (∑ _ : X,_),
+    ←div_eq_mul_inv,finiteMean] using hh
+
+noncomputable def act (K : Matrix X X ℝ) (f : X → ℝ) (y : X) : ℝ := ∑ x, K x y*f x
+
+omit [Nonempty X] in
+lemma act_nonneg (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
+    (f : X → ℝ) (hf : ∀ x, 0 ≤ f x) (y : X) : 0 ≤ act K f y :=
+  Finset.sum_nonneg (fun x _ => mul_nonneg (hK x y) (hf x))
+
+omit [Nonempty X] in
+lemma act_rpow_le (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
+    (hcol : ∀ y, ∑ x, K x y = 1) (f : X → ℝ) (hf : ∀ x, 0 ≤ f x)
+    {b : ℝ} (hb : 1 ≤ b) (y : X) :
+    (act K f y)^b ≤ act K (fun x => (f x)^b) y := by
+  simpa only [smul_eq_mul,act] using
+    (convexOn_rpow hb).map_sum_le (t := Finset.univ) (w := fun x => K x y)
+      (p := f) (fun x _ => hK x y) (hcol y) (fun x _ => hf x)
+
+lemma act_one_to_p (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
+    {p A : ℝ} (hp : 1 ≤ p)
+    (hm : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) ≤ Real.exp A)
+    (f : X → ℝ) (hf : ∀ x, 0 ≤ f x) :
+    finiteMean (fun y => (act K f y)^p) ≤ Real.exp A*(finiteMean f)^p := by
+  classical
+  let s : ℝ := ∑ x, f x
+  have hs : 0 ≤ s := Finset.sum_nonneg (fun x _ => hf x)
+  have hN : (0:ℝ) < Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
+  by_cases hz : s=0
+  · have hfz : ∀ x, f x=0 := by
+      intro x
+      exact (Finset.sum_eq_zero_iff_of_nonneg (fun x _ => hf x)).mp hz x (Finset.mem_univ _)
+    simp only [act,hfz,mul_zero,Finset.sum_const_zero,finiteMean,zero_div,
+      Real.zero_rpow (show p ≠ 0 by linarith),mul_zero,le_refl]
+  · have hsp : 0 < s := lt_of_le_of_ne hs (Ne.symm hz)
+    let w : X → ℝ := fun x => f x/s
+    have hw : ∀ x, 0 ≤ w x := fun x => div_nonneg (hf x) hs
+    have hws : ∑ x, w x = 1 := by simp only [w,←Finset.sum_div]; exact div_self hz
+    have hact (y : X) : act K f y = finiteMean f * ∑ x, w x*((Fintype.card X:ℝ)*K x y) := by
+      simp only [act,w,finiteMean,Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro x _
+      have hz' : (∑ x, f x) ≠ 0 := hz
+      dsimp [s]
+      field_simp [hz',ne_of_gt hN]
+    have hj (y : X) : (act K f y)^p ≤ (finiteMean f)^p*
+        ∑ x, w x*((Fintype.card X:ℝ)*K x y)^p := by
+      rw [hact,Real.mul_rpow (mean_nonneg f hf)
+        (Finset.sum_nonneg (fun x _ => mul_nonneg (hw x) (mul_nonneg hN.le (hK x y))))]
+      apply mul_le_mul_of_nonneg_left _ (Real.rpow_nonneg (mean_nonneg f hf) _)
+      exact (convexOn_rpow hp).map_sum_le
+        (fun x _ => hw x) hws (fun x _ => mul_nonneg hN.le (hK x y))
+    have hh := finiteMean_mono hj
+    have he : finiteMean (fun y => (finiteMean f)^p*
+        ∑ x, w x*((Fintype.card X:ℝ)*K x y)^p) =
+        (finiteMean f)^p * ∑ x, w x*finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) := by
+      simp_rw [mul_comm ((finiteMean f)^p), finiteMean_mul_const,
+        finiteMean_sum, mul_comm (w _), finiteMean_mul_const]
+    rw [he] at hh
+    apply hh.trans
+    have hsum : (∑ x, w x*finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p)) ≤ Real.exp A := by
+      calc
+        _ ≤ ∑ x, w x*Real.exp A := Finset.sum_le_sum
+          (fun x _ => mul_le_mul_of_nonneg_left (hm x) (hw x))
+        _ = Real.exp A := by rw [←Finset.sum_mul,hws,one_mul]
+    simpa only [mul_comm (Real.exp A)] using
+      mul_le_mul_of_nonneg_left hsum (Real.rpow_nonneg (mean_nonneg f hf) _)
+
+lemma act_b_to_pb (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
+    (hcol : ∀ y, ∑ x, K x y = 1) {p A b : ℝ} (hp : 1 ≤ p) (hb : 1 ≤ b)
+    (hm : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) ≤ Real.exp A)
+    (f : X → ℝ) (hf : ∀ x, 0 ≤ f x) :
+    finiteMean (fun y => (act K f y)^(b*p)) ≤
+      Real.exp A*(finiteMean (fun x => (f x)^b))^p := by
+  calc
+    _ ≤ finiteMean (fun y => (act K (fun x => (f x)^b) y)^p) := by
+      apply finiteMean_mono
+      intro y
+      rw [Real.rpow_mul (act_nonneg K hK f hf y)]
+      exact Real.rpow_le_rpow (Real.rpow_nonneg (act_nonneg K hK f hf y) _) (act_rpow_le K hK hcol f hf hb y) (by linarith)
+    _ ≤ _ := act_one_to_p K hK hp hm _ (fun x => Real.rpow_nonneg (hf x) _)
+
+variable [DecidableEq X]
+
+omit [Nonempty X] in
+lemma kernel_pow_nonneg (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y) (r : ℕ) :
+    ∀ x y, 0 ≤ (K^r) x y := by
+  classical
+  induction r with
+  | zero => intro x y; simp only [pow_zero,Matrix.one_apply]; split_ifs <;> norm_num
+  | succ r ih =>
+    intro x y
+    rw [pow_succ,Matrix.mul_apply]
+    exact Finset.sum_nonneg (fun z _ => mul_nonneg (ih x z) (hK z y))
+
+omit [Nonempty X] in
+lemma kernel_pow_row (K : Matrix X X ℝ) (hrow : ∀ x, ∑ y, K x y = 1) (r : ℕ) :
+    ∀ x, ∑ y, (K^r) x y = 1 := by
+  classical
+  induction r with
+  | zero => intro x; simp [Matrix.one_apply]
+  | succ r ih =>
+    intro x
+    simp only [pow_succ,Matrix.mul_apply]
+    rw [Finset.sum_comm]
+    simp only [←Finset.mul_sum,hrow,mul_one,ih]
+
+noncomputable def rowDensity (K : Matrix X X ℝ) (r : ℕ) (x y : X) : ℝ :=
+  (Fintype.card X:ℝ)*(K^r) x y
+
+omit [Nonempty X] in
+lemma rowDensity_nonneg (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y) (r : ℕ) (x y : X) :
+    0 ≤ rowDensity K r x y := mul_nonneg (Nat.cast_nonneg _) (kernel_pow_nonneg K hK r x y)
+
+omit [Nonempty X] in
+lemma rowDensity_succ (K : Matrix X X ℝ) (r : ℕ) (x y : X) :
+    rowDensity K (r+1) x y = act K (rowDensity K r x) y := by
+  simp only [rowDensity,pow_succ,Matrix.mul_apply,act,Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro z _
+  ring
+
+lemma rowDensity_mean (K : Matrix X X ℝ) (hrow : ∀ x, ∑ y, K x y = 1) (r : ℕ) (x : X) :
+    finiteMean (rowDensity K r x) = 1 := by
+  have hN : (Fintype.card X:ℝ) ≠ 0 := ne_of_gt (Nat.cast_pos.mpr Fintype.card_pos)
+  simp only [finiteMean,rowDensity,←Finset.mul_sum,kernel_pow_row K hrow,mul_one,div_self hN]
+
+lemma rowDensity_moment_iter (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
+    (hrow : ∀ x, ∑ y, K x y = 1) (hcol : ∀ y, ∑ x, K x y = 1)
+    {p A : ℝ} (hp : 1 ≤ p) (hA : 0 ≤ A)
+    (hm : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) ≤ Real.exp A)
+    (r : ℕ) (x : X) :
+    finiteMean (fun y => (rowDensity K r x y)^(p^r)) ≤ Real.exp (A*r*p^r) := by
+  induction r with
+  | zero => simp [rowDensity_mean K hrow]
+  | succ r ih =>
+    have hpr : 1 ≤ p^r := one_le_pow₀ hp
+    have hps : 1 ≤ p^(r+1) := one_le_pow₀ hp
+    simp_rw [rowDensity_succ]
+    calc
+      _ ≤ Real.exp A*(finiteMean (fun y => (rowDensity K r x y)^(p^r)))^p := by
+        rw [pow_succ]
+        exact act_b_to_pb K hK hcol hp hpr hm _ (rowDensity_nonneg K hK r x)
+      _ ≤ Real.exp A*(Real.exp (A*r*p^r))^p := by
+        apply mul_le_mul_of_nonneg_left _ (Real.exp_pos _).le
+        exact Real.rpow_le_rpow (mean_nonneg _ (fun y => Real.rpow_nonneg
+          (rowDensity_nonneg K hK r x y) _)) ih (le_trans zero_le_one hp)
+      _ = Real.exp (A+A*r*p^(r+1)) := by rw [←Real.exp_mul,←Real.exp_add,pow_succ]; congr 1; ring
+      _ ≤ _ := by
+        apply Real.exp_le_exp.mpr
+        have hh := mul_le_mul_of_nonneg_left hps hA
+        push_cast
+        nlinarith
+
+lemma rowDensity_second_moment (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
+    (hrow : ∀ x, ∑ y, K x y = 1) (hcol : ∀ y, ∑ x, K x y = 1)
+    {p A : ℝ} (hp : 1 ≤ p) (hA : 0 ≤ A)
+    (hm : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) ≤ Real.exp A)
+    (r : ℕ) (hr : 2 ≤ p^r) (x : X) :
+    finiteMean (fun y => (rowDensity K r x y)^2) ≤ Real.exp (2*A*r) := by
+  have hpos : 0 < p^r := by linarith
+  have hj := mean_rpow_concave (fun y => (rowDensity K r x y)^(p^r))
+    (fun y => Real.rpow_nonneg (rowDensity_nonneg K hK r x y) _)
+    (a := 2/(p^r)) (by positivity) ((div_le_one hpos).mpr hr)
+  have he (y : X) : ((rowDensity K r x y)^(p^r))^(2/(p^r)) = (rowDensity K r x y)^2 := by
+    rw [←Real.rpow_mul (rowDensity_nonneg K hK r x y)]
+    rw [mul_div_cancel₀ _ (ne_of_gt hpos),Real.rpow_two]
+  simp_rw [he] at hj
+  apply hj.trans
+  calc
+    _ ≤ (Real.exp (A*r*p^r))^(2/(p^r)) :=
+      Real.rpow_le_rpow (mean_nonneg _ (fun y => Real.rpow_nonneg
+        (rowDensity_nonneg K hK r x y) _)) (rowDensity_moment_iter K hK hrow hcol hp hA hm r x)
+        (by positivity)
+    _ = _ := by rw [←Real.exp_mul]; congr 1; field_simp [ne_of_gt hpos]
+
+lemma symmetric_kernel_smoothing (K : Matrix X X ℝ) (hK : ∀ x y, 0 ≤ K x y)
+    (hrow : ∀ x, ∑ y, K x y = 1) (hcol : ∀ y, ∑ x, K x y = 1)
+    (hsym : K.IsSymm) {p A : ℝ} (hp : 1 ≤ p) (hA : 0 ≤ A)
+    (hm : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*K x y)^p) ≤ Real.exp A)
+    (r : ℕ) (hr : 2 ≤ p^r) (x y : X) :
+    (K^(2*r)) x y ≤ Real.exp (2*A*r)/(Fintype.card X:ℝ) := by
+  have hN : (0:ℝ) < Fintype.card X := Nat.cast_pos.mpr Fintype.card_pos
+  have hs : ∀ z, (K^r) z y = (K^r) y z := fun z => (hsym.pow r).apply y z
+  have he : (Fintype.card X:ℝ)*(K^(2*r)) x y =
+      finiteMean (fun z => rowDensity K r x z*rowDensity K r y z) := by
+    simp only [finiteMean,rowDensity,two_mul,pow_add,Matrix.mul_apply,hs,Finset.mul_sum]
+    rw [Finset.sum_div]
+    apply Finset.sum_congr rfl
+    intro z _
+    field_simp
+  have hh : finiteMean (fun z => rowDensity K r x z*rowDensity K r y z) ≤
+      Real.exp (2*A*r) := by
+    calc
+      _ ≤ finiteMean (fun z => ((rowDensity K r x z)^2+(rowDensity K r y z)^2)/2) := by
+        apply finiteMean_mono
+        intro z
+        nlinarith [sq_nonneg (rowDensity K r x z-rowDensity K r y z)]
+      _ = (finiteMean (fun z => (rowDensity K r x z)^2)+
+          finiteMean (fun z => (rowDensity K r y z)^2))/2 := by
+        simp only [div_eq_mul_inv,finiteMean_mul_const,finiteMean_add]
+      _ ≤ _ := by
+        have hx := rowDensity_second_moment K hK hrow hcol hp hA hm r hr x
+        have hy := rowDensity_second_moment K hK hrow hcol hp hA hm r hr y
+        linarith
+  apply (le_div_iff₀ hN).mpr
+  simpa only [←he,mul_comm (Fintype.card X:ℝ)] using hh
+
+lemma density_exponent_numeric : 2 ≤ (1+(1/1000:ℝ))^(1000:ℕ) := by
+  calc
+    (2:ℝ) = 1+(1000:ℝ)*(1/1000) := by norm_num
+    _ ≤ _ := by simpa only [Nat.cast_ofNat] using
+      (one_add_mul_le_pow (a := (1/1000:ℝ)) (by norm_num) 1000)
+
+end Thorp.StrongSmoothing
+
+namespace Thorp.StrongSmoothing.PermLaw
+open scoped BigOperators
+variable {X Ω Ω' : Type*} [Fintype X] [DecidableEq X] [Fintype Ω] [Fintype Ω']
+
+noncomputable def kernel (P : Ω → Equiv.Perm X) : Matrix X X ℝ :=
+  fun x y => finiteMean (fun ω => if P ω x = y then (1:ℝ) else 0)
+
+omit [Fintype X] in
+lemma nonneg (P : Ω → Equiv.Perm X) (x y : X) : 0 ≤ kernel P x y := by
+  exact finiteMean_nonneg (fun ω => by split_ifs <;> norm_num)
+
+lemma row [Nonempty Ω] (P : Ω → Equiv.Perm X) (x : X) : ∑ y, kernel P x y = 1 := by
+  simp only [kernel,←finiteMean_sum]
+  simp only [Finset.sum_ite_eq,Finset.mem_univ,↓reduceIte,finiteMean_const]
+
+omit [Fintype X] in
+lemma inverse (P : Ω → Equiv.Perm X) : kernel (fun ω => (P ω)⁻¹) = (kernel P).transpose := by
+  ext x y
+  apply finiteMean_congr
+  intro ω
+  have he : (P ω)⁻¹ x=y ↔ P ω y=x := by
+    change (P ω).symm x=y ↔ P ω y=x
+    exact (Equiv.symm_apply_eq (P ω)).trans eq_comm
+  simp only [he]
+
+lemma col [Nonempty Ω] (P : Ω → Equiv.Perm X) (y : X) : ∑ x, kernel P x y = 1 := by
+  have hh := row (fun ω => (P ω)⁻¹) y
+  rw [inverse] at hh
+  exact hh
+
+lemma mul (P : Ω → Equiv.Perm X) (Q : Ω' → Equiv.Perm X) :
+    kernel (fun ω : Ω × Ω' => P ω.1*Q ω.2) = kernel Q*kernel P := by
+  ext x y
+  simp only [Matrix.mul_apply,kernel]
+  have he (ω : Ω × Ω') :
+      (if (P ω.1*Q ω.2) x=y then (1:ℝ) else 0) =
+        ∑ z : X, (if P ω.1 z=y then (1:ℝ) else 0)*(if Q ω.2 x=z then (1:ℝ) else 0) := by
+    simp only [mul_ite,mul_one,mul_zero,Finset.sum_ite_eq,Finset.mem_univ,↓reduceIte]
+    rfl
+  change finiteMean (fun ω : Ω × Ω' => if (P ω.1*Q ω.2) x=y then (1:ℝ) else 0) = _
+  simp_rw [he]
+  rw [finiteMean_sum]
+  apply Finset.sum_congr rfl
+  intro z _
+  exact (finiteMean_prod_mul (fun ω : Ω => if P ω z=y then (1:ℝ) else 0)
+    (fun ω : Ω' => if Q ω x=z then (1:ℝ) else 0)).trans (mul_comm _ _)
+
+omit [Fintype X] in
+lemma conjugate (P : Ω → Equiv.Perm X) (e : Equiv.Perm X) (x y : X) :
+    kernel (fun ω => e*P ω*e⁻¹) x y = kernel P (e.symm x) (e.symm y) := by
+  apply finiteMean_congr
+  intro ω
+  have he : (e*P ω*e⁻¹) x=y ↔ P ω (e.symm x)=e.symm y := by
+    change e (P ω (e.symm x))=y ↔ _
+    exact (Equiv.eq_symm_apply e).symm
+  simp only [he]
+
+end Thorp.StrongSmoothing.PermLaw
+
+namespace Thorp.StrongSmoothing
+open scoped BigOperators
+
+section MatrixHelpers
+variable {X : Type*} [Fintype X] [DecidableEq X]
+
+omit [DecidableEq X] in
+lemma mul_nonneg_entries (A B : Matrix X X ℝ) (hA : ∀ x y, 0 ≤ A x y)
+    (hB : ∀ x y, 0 ≤ B x y) (x y : X) : 0 ≤ (A*B) x y :=
+  Finset.sum_nonneg (fun z _ => mul_nonneg (hA x z) (hB z y))
+
+omit [DecidableEq X] in
+lemma mul_row (A B : Matrix X X ℝ) (hA : ∀ x, ∑ y, A x y = 1)
+    (hB : ∀ x, ∑ y, B x y = 1) (x : X) : ∑ y, (A*B) x y = 1 := by
+  simp only [Matrix.mul_apply]
+  rw [Finset.sum_comm]
+  simp only [←Finset.mul_sum,hB,mul_one,hA]
+
+omit [DecidableEq X] in
+lemma mul_col (A B : Matrix X X ℝ) (hA : ∀ y, ∑ x, A x y = 1)
+    (hB : ∀ y, ∑ x, B x y = 1) (y : X) : ∑ x, (A*B) x y = 1 := by
+  simp only [Matrix.mul_apply]
+  rw [Finset.sum_comm]
+  simp only [←Finset.sum_mul,hA,one_mul,hB]
+
+omit [DecidableEq X] in
+lemma bound_mul_left (A B : Matrix X X ℝ) (hA : ∀ x y, 0 ≤ A x y)
+    (hrow : ∀ x, ∑ y, A x y = 1) {D : ℝ} (hB : ∀ x y, B x y ≤ D) :
+    ∀ x y, (A*B) x y ≤ D := by
+  intro x y
+  calc
+    _ ≤ ∑ z, A x z*D := Finset.sum_le_sum (fun z _ => mul_le_mul_of_nonneg_left (hB z y) (hA x z))
+    _ = D := by rw [←Finset.sum_mul,hrow,one_mul]
+
+omit [DecidableEq X] in
+lemma bound_mul_right (A B : Matrix X X ℝ) (hB : ∀ x y, 0 ≤ B x y)
+    (hcol : ∀ y, ∑ x, B x y = 1) {D : ℝ} (hA : ∀ x y, A x y ≤ D) :
+    ∀ x y, (A*B) x y ≤ D := by
+  intro x y
+  calc
+    _ ≤ ∑ z, D*B z y := Finset.sum_le_sum (fun z _ => mul_le_mul_of_nonneg_right (hA x z) (hB z y))
+    _ = D := by rw [←Finset.mul_sum,hcol,mul_one]
+
+lemma sandwich_power (A B : Matrix X X ℝ) (r : ℕ) : A*(B*A)^r*B = (A*B)^(r+1) := by
+  induction r with
+  | zero => simp
+  | succ r ih =>
+    calc
+      _ = (A*(B*A)^r*B)*(A*B) := by rw [pow_succ]; noncomm_ring
+      _ = (A*B)^(r+1)*(A*B) := by rw [ih]
+      _ = _ := (pow_succ (A*B) (r+1)).symm
+
+end MatrixHelpers
+
+section TupleAction
+variable {ι β Ω : Type*} [Fintype ι] [Fintype β] [DecidableEq β] [Fintype Ω]
+
+def tuplePerm (P : Equiv.Perm β) : Equiv.Perm (ι ↪ β) where
+  toFun x := x.trans P.toEmbedding
+  invFun x := x.trans P.symm.toEmbedding
+  left_inv x := by ext i; simp
+  right_inv x := by ext i; simp
+
+omit [Fintype ι] [Fintype β] [DecidableEq β] in
+@[simp] lemma tuplePerm_apply (P : Equiv.Perm β) (x : ι ↪ β) (i : ι) :
+    tuplePerm P x i = P (x i) := rfl
+
+omit [Fintype ι] [Fintype β] [DecidableEq β] in
+lemma tuplePerm_mul (P Q : Equiv.Perm β) :
+    tuplePerm (ι := ι) (P*Q) = tuplePerm P*tuplePerm Q := by
+  ext x i
+  rfl
+
+omit [Fintype ι] [Fintype β] [DecidableEq β] in
+lemma tuplePerm_inverse (P : Equiv.Perm β) :
+    tuplePerm (ι := ι) P⁻¹ = (tuplePerm P)⁻¹ := by
+  ext x i
+  rfl
+
+omit [Fintype β] in
+lemma tupleProbability_kernel (P : Ω → Equiv.Perm β) (x y : ι ↪ β) :
+    PairRouting.tupleProbability P x y = PermLaw.kernel (fun ω => tuplePerm (P ω)) x y := by
+  classical
+  apply finiteMean_congr
+  intro ω
+  have he : (∀ i, P ω (x i) = y i) ↔ tuplePerm (P ω) x = y := by
+    constructor
+    · intro h; ext i; exact h i
+    · intro h i; exact congrArg (fun z : ι ↪ β => z i) h
+  simp only [he]
+end TupleAction
+
+lemma sweepKernel_law (d l : ℕ) (order : Equiv.Perm (Fin d)) :
+    sweepKernel d l order = PermLaw.kernel (fun ω => tuplePerm (ι := Fin l) (sweep d order ω)) := by
+  ext x y
+  exact tupleProbability_kernel _ x y
+
+lemma reflectedKernel_conjugate (d l : ℕ) (order : Equiv.Perm (Fin d)) (x y : Tuples d l) :
+    reflectedKernel d l order false x y =
+      PairRouting.tupleProbability (palindromePerm d)
+        ((tuplePerm (coordinateRelabel d order)).symm x)
+        ((tuplePerm (coordinateRelabel d order)).symm y) := by
+  classical
+  let e : Equiv.Perm (Tuples d l) := tuplePerm (coordinateRelabel d order)
+  let P := fun ω => tuplePerm (ι := Fin l) (butterflyPerm d (decodeButterfly d ω))
+  have hA : sweepKernel d l order = PermLaw.kernel (fun ω => e*P ω*e⁻¹) := by
+    rw [sweepKernel_law]
+    simp only [sweep,tuplePerm_mul,tuplePerm_inverse]
+    rfl
+  rw [reflectedKernel,ite_eq_right Bool.false_ne_true,hA]
+  rw [←PermLaw.inverse,←PermLaw.mul]
+  have hprod : (fun ω : BenesCoins d => (e*P ω.1*e⁻¹)*((e*P ω.2*e⁻¹)⁻¹)) =
+      fun ω : BenesCoins d => e*tuplePerm (ι := Fin l) (palindromePerm d ω)*e⁻¹ := by
+    funext ω
+    simp only [mul_inv_rev,inv_inv,palindromePerm,tuplePerm_mul]
+    change (e*P ω.1*e⁻¹)*(e*(P ω.2)⁻¹*e⁻¹) = e*(P ω.1*(P ω.2)⁻¹)*e⁻¹
+    group
+  rw [hprod,PermLaw.conjugate,tupleProbability_kernel]
+
+
+lemma sweepKernel_nonneg (d l : ℕ) (order : Equiv.Perm (Fin d)) (x y : Tuples d l) :
+    0 ≤ sweepKernel d l order x y := PairRouting.tupleProbability_nonneg _ _ _
+
+lemma sweepKernel_row (d l : ℕ) (order : Equiv.Perm (Fin d)) (x : Tuples d l) :
+    ∑ y, sweepKernel d l order x y = 1 := by
+  rw [sweepKernel_law]
+  exact PermLaw.row _ _
+
+lemma sweepKernel_col (d l : ℕ) (order : Equiv.Perm (Fin d)) (y : Tuples d l) :
+    ∑ x, sweepKernel d l order x y = 1 := by
+  rw [sweepKernel_law]
+  exact PermLaw.col _ _
+
+lemma reflected_nonneg (d l : ℕ) (order : Equiv.Perm (Fin d)) (x y : Tuples d l) :
+    0 ≤ reflectedKernel d l order false x y := by
+  exact mul_nonneg_entries _ _ (fun x y => sweepKernel_nonneg d l order y x)
+    (sweepKernel_nonneg d l order) x y
+
+lemma reflected_row (d l : ℕ) (order : Equiv.Perm (Fin d)) (x : Tuples d l) :
+    ∑ y, reflectedKernel d l order false x y = 1 :=
+  mul_row _ _ (sweepKernel_col d l order) (sweepKernel_row d l order) x
+
+lemma reflected_col (d l : ℕ) (order : Equiv.Perm (Fin d)) (y : Tuples d l) :
+    ∑ x, reflectedKernel d l order false x y = 1 :=
+  mul_col _ _ (sweepKernel_row d l order) (sweepKernel_col d l order) y
+
+lemma reflected_symmetric (d l : ℕ) (order : Equiv.Perm (Fin d)) :
+    (reflectedKernel d l order false).IsSymm := Matrix.isSymm_transpose_mul_self _
+
+lemma reflected_moment (d l : ℕ) (order : Equiv.Perm (Fin d)) (x : Tuples d l) :
+    finiteMean (fun y : Tuples d l =>
+      ((Fintype.card (Tuples d l):ℝ)*reflectedKernel d l order false x y)^(1+(1/1000:ℝ))) ≤
+      Real.exp (l*(1+heightDecay 1)) := by
+  classical
+  let e : Equiv.Perm (Tuples d l) := tuplePerm (coordinateRelabel d order)
+  simp_rw [reflectedKernel_conjugate]
+  have he := finiteMean_equiv e (fun y : Tuples d l =>
+    ((Fintype.card (Tuples d l):ℝ)*PairRouting.tupleProbability (palindromePerm d) (e.symm x) y)^
+      (1+(1/1000:ℝ)))
+  change finiteMean (fun y =>
+    ((Fintype.card (Tuples d l):ℝ)*PairRouting.tupleProbability (palindromePerm d) (e.symm x) (e.symm y))^
+      (1+(1/1000:ℝ))) ≤ _
+  rw [←he]
+  simpa only [palindromeRowMoment,Fintype.card_fin] using palindrome_row_moment_coarse d (e.symm x)
+
+lemma tuples_card (d l : ℕ) : Fintype.card (Tuples d l) = (2^d).descFactorial l := by
+  simp only [Tuples,Fintype.card_embedding_eq,Fintype.card_fin,card_positions]
+
+lemma canonical_pointwise (d l : ℕ) (order : Equiv.Perm (Fin d)) (x y : Tuples d l) :
+    ((reflectedKernel d l order false)^2000) x y ≤
+      Real.exp ((2000*(1+heightDecay 1))*l)/((2^d).descFactorial l:ℝ) := by
+  let : Nonempty (Tuples d l) := ⟨x⟩
+  have hh := symmetric_kernel_smoothing (reflectedKernel d l order false)
+    (reflected_nonneg d l order) (reflected_row d l order) (reflected_col d l order)
+    (reflected_symmetric d l order) (p := 1+(1/1000:ℝ)) (A := l*(1+heightDecay 1))
+    (by norm_num) (mul_nonneg (Nat.cast_nonneg _) (by linarith [heightDecay_nonneg 1]))
+    (reflected_moment d l order) 1000 density_exponent_numeric x y
+  have he : 2*((l:ℝ)*(1+heightDecay 1))*(1000:ℝ) = (2000*(1+heightDecay 1))*l := by ring
+  norm_num only [show 2*1000=2000 by norm_num,Nat.cast_ofNat] at hh
+  simpa only [he,tuples_card] using hh
+
+lemma both_pointwise (d l : ℕ) (order : Equiv.Perm (Fin d)) (orientation : Bool)
+    (x y : Tuples d l) :
+    ((reflectedKernel d l order orientation)^2001) x y ≤
+      Real.exp ((2000*(1+heightDecay 1))*l)/((2^d).descFactorial l:ℝ) := by
+  cases orientation
+  · change ((reflectedKernel d l order false)^(2000+1)) x y ≤ _
+    rw [pow_succ]
+    exact bound_mul_right _ _ (reflected_nonneg d l order) (reflected_col d l order)
+      (canonical_pointwise d l order) x y
+  · let A := sweepKernel d l order
+    change ((A*A.transpose)^(2000+1)) x y ≤ _
+    rw [←sandwich_power]
+    apply bound_mul_right _ _ (fun x y => sweepKernel_nonneg d l order y x)
+      (sweepKernel_row d l order)
+    exact bound_mul_left _ _ (sweepKernel_nonneg d l order) (sweepKernel_row d l order)
+      (canonical_pointwise d l order)
+
+lemma regular_trace_bound (d : ℕ) (order : Equiv.Perm (Fin d)) :
+    regularAbsoluteEvenTrace d order 2001 ≤ Real.exp ((2000*(1+heightDecay 1))*(2:ℝ)^d) := by
+  have hn : 0 < Fintype.card (Tuples d (2^d)) := by
+    rw [tuples_card,Nat.descFactorial_self]
+    exact Nat.factorial_pos _
+  have hn' : (Fintype.card (Tuples d (2^d)):ℝ) ≠ 0 := Nat.cast_ne_zero.mpr (ne_of_gt hn)
+  calc
+    _ ≤ ∑ x : Tuples d (2^d), Real.exp ((2000*(1+heightDecay 1))*(2^d:ℕ)) /
+        ((2^d).descFactorial (2^d):ℝ) := by
+      apply Finset.sum_le_sum
+      intro x _
+      exact both_pointwise d (2^d) order false x x
+    _ = _ := by
+      simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,←tuples_card]
+      push_cast
+      field_simp
+
+theorem strong_smoothing : MainStatement := by
+  refine ⟨2001,by norm_num,2000*(1+heightDecay 1),?_,regular_trace_bound⟩
+  intro d l _ order orientation x y
+  exact both_pointwise d l order orientation x y
+
+end Thorp.StrongSmoothing
+
+namespace Thorp.StrongTail
+open scoped BigOperators Classical
+open Specht
+
+lemma mem_tailDiagram (μ : YoungDiagram) (i j : ℕ) :
+    (i,j) ∈ tailDiagram μ ↔ (i+1,j) ∈ μ := by
+  rw [tailDiagram, YoungDiagram.mem_ofRowLens]
+  constructor
+  · rintro ⟨hi,hj⟩
+    rw [List.getElem_tail,YoungDiagram.get_rowLens] at hj
+    exact YoungDiagram.mem_iff_lt_rowLen.mpr hj
+  · intro h
+    have hi : i+1 < μ.colLen 0 :=
+      (YoungDiagram.mem_iff_lt_colLen.mp h).trans_le (μ.colLen_anti 0 j (Nat.zero_le _))
+    have hi' : i < μ.rowLens.tail.length := by simp only [List.length_tail,YoungDiagram.length_rowLens]; omega
+    refine ⟨hi', ?_⟩
+    simpa only [List.getElem_tail,YoungDiagram.get_rowLens] using YoungDiagram.mem_iff_lt_rowLen.mp h
+
+lemma tail_height (μ : YoungDiagram) : (tailDiagram μ).colLen 0 = μ.colLen 0 - 1 := by
+  rw [←YoungDiagram.length_rowLens]
+  exact (YoungDiagram.rowLens_length_ofRowLens (fun x hx => μ.pos_of_mem_rowLens x (List.mem_of_mem_tail hx))).trans
+    (by simp only [List.length_tail,YoungDiagram.length_rowLens])
+
+end Thorp.StrongTail
+
+end ThorpNine.Tail
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/Tail/TypedMarks.lean b/lean/OAI/Probability/ThorpRouting/Tail/TypedMarks.lean
index dadf09c4a09fa6c471deb1240693cf5a607b990e..991ebfbebe6638ec6e2b805b88ccd79398ee2035 100644
--- a/lean/OAI/Probability/ThorpRouting/Tail/TypedMarks.lean
+++ b/lean/OAI/Probability/ThorpRouting/Tail/TypedMarks.lean
@@ -1,882 +1,882 @@
-import Mathlib
-import OAI.Probability.ThorpRouting.Tail.ModifiedStageEquiv
-
-namespace OAI
-
-namespace ThorpNine.Tail
-
-namespace Thorp.SparseContact
-open scoped BigOperators Classical
-
-noncomputable def typedMarks {N : Type*} [Fintype N] (d : ℕ)
-    (A : N → Finset (Card d)) (ω : N → (SwitchIndex d → Bool)) : Finset (N × Card d) :=
-  typedSet (fun t => butterflyMarks d (A t) (ω t))
-
-lemma typedMarks_card {N : Type*} [Fintype N] (d : ℕ)
-    (A : N → Finset (Card d)) (ω : N → (SwitchIndex d → Bool)) :
-    (typedMarks d A ω).card = ∑ t, (A t).card := by
-  classical
-  rw [typedMarks, typedSet_card]
-  apply Finset.sum_congr rfl
-  intro t _
-  exact Finset.card_image_of_injective _ (butterflyPerm d (decodeButterfly d (ω t))).injective
-
-lemma typedMarks_inclusion {N : Type*} [Fintype N] (d : ℕ)
-    (A : N → Finset (Card d)) (D : Finset (N × Card d)) :
-    inclusionMoment (typedMarks d A) D ≤
-      ∏ v ∈ D, ((A v.1).card:ℝ)/(2:ℝ)^d := by
-  classical
-  have he : inclusionMoment (typedMarks d A) D =
-      ∏ t, inclusionMoment (butterflyMarks d (A t)) (fiber D t) := by
-    rw [inclusionMoment_eq]
-    have hh (ω : N → (SwitchIndex d → Bool)) :
-        (if D ⊆ typedMarks d A ω then (1:ℝ) else 0) =
-        ∏ t, if fiber D t ⊆ butterflyMarks d (A t) (ω t) then (1:ℝ) else 0 := by
-      rw [Fintype.prod_boole]
-      simp only [typedMarks, subset_typedSet]
-    rw [finiteMean_congr hh]
-    simp only [inclusionMoment_eq]
-    exact finiteMean_pi_prod (ι := N) (Ω := fun _ => SwitchIndex d → Bool)
-      (fun t (ω : SwitchIndex d → Bool) =>
-        if fiber D t ⊆ butterflyMarks d (A t) ω then (1:ℝ) else 0)
-  have hprod : (∏ v ∈ D, ((A v.1).card:ℝ)/(2:ℝ)^d) =
-      ∏ t, ∏ x ∈ fiber D t, ((A t).card:ℝ)/(2:ℝ)^d := by
-    conv_lhs => rw [← typedSet_fiber D]
-    exact typedSet_prod _ _
-  rw [he, hprod]
-  apply Finset.prod_le_prod₀
-  · intro t _; exact inclusionMoment_nonneg _ _
-  · intro t _
-    simpa only [Finset.prod_const] using butterflyMarks_inclusion d (A t) (fiber D t)
-
-noncomputable def returnRoute {N : Type*} (d : ℕ) (b : N → Butterfly d)
-    (v : N × Card d) : Finset (SwitchIndex d) :=
-  routeDomain d ((butterflyPerm d (b v.1)).symm v.2) v.2
-
-noncomputable def returnGraph {N : Type*} (d : ℕ) (b : N → Butterfly d) :
-    SimpleGraph (N × Card d) where
-  Adj v w := v ≠ w ∧ ¬Disjoint (returnRoute d b v) (returnRoute d b w)
-  symm := ⟨fun _ _ h => ⟨h.1.symm, fun hh => h.2 hh.symm⟩⟩
-  loopless := ⟨fun _ h => h.1 rfl⟩
-
-lemma returnRoute_card {N : Type*} (d : ℕ) (b : N → Butterfly d) (v : N × Card d) :
-    (returnRoute d b v).card = d := card_routeDomain _ _ _
-
-lemma mem_returnRoute {N : Type*} (d : ℕ) (b : N → Butterfly d)
-    (t : N) (y : Card d) (i : SwitchIndex d) :
-    i ∈ returnRoute d b (t,y) ↔
-      y = butterflyPerm d (b t) (switchUsers d (b t) i).1 ∨
-      y = butterflyPerm d (b t) (switchUsers d (b t) i).2 := by
-  have h := mem_routeDomain_iff_user d (b t) ((butterflyPerm d (b t)).symm y) i
-  simpa only [Equiv.apply_symm_apply, returnRoute, Equiv.eq_symm_apply, eq_comm] using h
-
-lemma returnRoute_incidence {N : Type*} [Fintype N] (d : ℕ) (b : N → Butterfly d)
-    (p : N → ℝ) (i : SwitchIndex d) :
-    (∑ v : N × Card d, if i ∈ returnRoute d b v then p v.1 else 0) =
-      2 * ∑ t, p t := by
-  classical
-  rw [Fintype.sum_prod_type, Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro t _
-  have hne : butterflyPerm d (b t) (switchUsers d (b t) i).1 ≠
-      butterflyPerm d (b t) (switchUsers d (b t) i).2 :=
-    (butterflyPerm d (b t)).injective.ne (switchUsers_ne d (b t) i)
-  simp only [mem_returnRoute]
-  have he (y : Card d) :
-      (if y = butterflyPerm d (b t) (switchUsers d (b t) i).1 ∨
-        y = butterflyPerm d (b t) (switchUsers d (b t) i).2 then p t else 0) =
-      (if y = butterflyPerm d (b t) (switchUsers d (b t) i).1 then p t else 0) +
-      (if y = butterflyPerm d (b t) (switchUsers d (b t) i).2 then p t else 0) := by
-    split_ifs <;> simp_all
-  simp only [he, Finset.sum_add_distrib, Finset.sum_ite_eq', Finset.mem_univ, ite_true]
-  ring
-
-lemma returnGraph_degree {N : Type*} [Fintype N] (d : ℕ) (b : N → Butterfly d)
-    (p : N → ℝ) (hp : ∀ t, 0 ≤ p t) (v : N × Card d) :
-    (∑ w, if (returnGraph d b).Adj v w then p w.1 else 0) ≤
-      2*d*∑ t, p t := by
-  classical
-  calc
-    _ ≤ ∑ w, ∑ i ∈ returnRoute d b v, if i ∈ returnRoute d b w then p w.1 else 0 := by
-      apply Finset.sum_le_sum
-      intro w _
-      split_ifs with h
-      · obtain ⟨i, hi, hw⟩ := Finset.not_disjoint_iff.mp h.2
-        have hh := Finset.single_le_sum
-          (f := fun i => if i ∈ returnRoute d b w then p w.1 else 0)
-          (fun i _ => by split_ifs; exact hp _; exact le_rfl) hi
-        simpa only [ite_eq_left hw] using hh
-      · exact Finset.sum_nonneg (fun i _ => by split_ifs; exact hp _; exact le_rfl)
-    _ = ∑ i ∈ returnRoute d b v, ∑ w, if i ∈ returnRoute d b w then p w.1 else 0 :=
-      Finset.sum_comm
-    _ = _ := by simp only [returnRoute_incidence, Finset.sum_const, returnRoute_card,
-        nsmul_eq_mul]; ring
-
-lemma typed_weight_sum {N : Type*} [Fintype N] (d : ℕ)
-    (A : N → Finset (Card d)) :
-    (∑ v : N × Card d, ((A v.1).card:ℝ)/(2:ℝ)^d) = ∑ t, ((A t).card:ℝ) := by
-  rw [Fintype.sum_prod_type]
-  apply Finset.sum_congr rfl
-  intro t _
-  simp only [Finset.sum_const, Finset.card_univ, card_positions, Nat.cast_pow,
-    Nat.cast_ofNat, nsmul_eq_mul]
-  field_simp
-
-lemma mixed_return_noIsolated {N : Type*} [Fintype N] (d : ℕ)
-    (A : N → Finset (Card d)) (b : N → Butterfly d) (k : ℕ)
-    (hk : ∑ t, (A t).card = k)
-    (hδ : 0 < 2*(d:ℝ)*k/(2:ℝ)^d) (hδ1 : 2*(d:ℝ)*k/(2:ℝ)^d ≤ 1) :
-    finiteMean (fun ω : N → (SwitchIndex d → Bool) =>
-      if NoIsolated (returnGraph d b) (typedMarks d A ω) then (1:ℝ) else 0) ≤
-      (2*(d:ℝ)*k/(2:ℝ)^d)^((k:ℝ)/2) * Real.exp (Real.exp 1*k) := by
-  have hs : (∑ t, ((A t).card:ℝ)) = k := by exact_mod_cast hk
-  have hm := noIsolated_probability_le (returnGraph d b) (typedMarks d A) k
-    (fun ω => (typedMarks_card d A ω).trans hk)
-    (fun v => ((A v.1).card:ℝ)/(2:ℝ)^d) (fun v => by positivity)
-    (typedMarks_inclusion d A) _ hδ hδ1
-    (fun v => (returnGraph_degree d b (fun t => ((A t).card:ℝ)/(2:ℝ)^d)
-      (fun t => by positivity) v).trans_eq (by rw [← Finset.sum_div, hs]; ring))
-  simpa only [typed_weight_sum, hs] using hm
-
-
-lemma mean_fresh_cylinder {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (s : Finset ι) (v : ι → Bool) (f : (ι → Bool) → ℝ)
-    (hf : ∀ i ∈ s, ∀ ω, f (QueryTree.flipCoin i ω) = f ω) :
-    finiteMean (fun ω => if ∀ i ∈ s, ω i = v i then f ω else 0) =
-      finiteMean f / (2:ℝ)^s.card := by
-  induction s using Finset.induction_on with
-  | empty => simp [finiteMean]
-  | @insert i s hi ih =>
-    let g := fun ω => if ∀ j ∈ s, ω j = v j then f ω else 0
-    have hg : ∀ ω, g (QueryTree.flipCoin i ω) = g ω := by
-      intro ω
-      have he : (∀ j ∈ s, QueryTree.flipCoin i ω j = v j) ↔ ∀ j ∈ s, ω j = v j := by
-        apply forall₂_congr
-        intro j hj
-        rw [QueryTree.flipCoin_apply_ne i j (fun h => hi (h ▸ hj))]
-      simp only [g, he, hf i (Finset.mem_insert_self _ _) ω]
-    have he (ω : ι → Bool) :
-        (if ∀ j ∈ insert i s, ω j = v j then f ω else 0) =
-        (if ω i = v i then g ω else 0) := by
-      by_cases h : ω i = v i <;> simp [h,g]
-    rw [finiteMean_congr he, QueryTree.mean_indicator_bit g i (v i) hg]
-    have hs := ih (fun j hj => hf j (Finset.mem_insert_of_mem hj))
-    change finiteMean (fun ω => if ∀ j ∈ s, ω j = v j then f ω else 0) / 2 = _
-    rw [hs, Finset.card_insert_of_notMem hi, pow_succ]
-    ring
-
-noncomputable def endpointProb {L : Type*} (d : ℕ) (A : Finset L)
-    (x y : L → Card d) : ℝ := by
-  classical
-  exact finiteMean (fun ω : SwitchIndex d → Bool =>
-    if ∀ i ∈ A, butterflyPerm d (decodeButterfly d ω) (x i) = y i then (1:ℝ) else 0)
-
-lemma endpointProb_nonneg {L : Type*} (d : ℕ) (A : Finset L)
-    (x y : L → Card d) : 0 ≤ endpointProb d A x y := by
-  classical
-  exact finiteMean_nonneg (fun _ => by positivity)
-
-lemma endpointProb_insert_isolated {L : Type*} (d : ℕ) (A : Finset L)
-    (x y : L → Card d) (i : L)
-    (hiso : ∀ j ∈ A, Disjoint (routeDomain d (x i) (y i)) (routeDomain d (x j) (y j))) :
-    endpointProb d (insert i A) x y = endpointProb d A x y / (2:ℝ)^d := by
-  classical
-  let f := fun ω : SwitchIndex d → Bool =>
-    if ∀ j ∈ A, butterflyPerm d (decodeButterfly d ω) (x j) = y j then (1:ℝ) else 0
-  have hf : ∀ t ∈ routeDomain d (x i) (y i), ∀ ω, f (QueryTree.flipCoin t ω) = f ω := by
-    intro t ht ω
-    have he : (∀ j ∈ A, butterflyPerm d (decodeButterfly d (QueryTree.flipCoin t ω)) (x j) = y j) ↔
-        ∀ j ∈ A, butterflyPerm d (decodeButterfly d ω) (x j) = y j := by
-      apply forall₂_congr
-      intro j hj
-      rw [butterfly_endpoint_iff, butterfly_endpoint_iff]
-      apply forall₂_congr
-      intro s hs
-      rw [QueryTree.flipCoin_apply_ne t s (fun h =>
-        Finset.disjoint_left.mp (hiso j hj) ht (h ▸ hs))]
-    simp only [f, he]
-  have he (ω : SwitchIndex d → Bool) :
-      (if ∀ j ∈ insert i A, butterflyPerm d (decodeButterfly d ω) (x j) = y j then (1:ℝ) else 0) =
-      (if ∀ t ∈ routeDomain d (x i) (y i), ω t = routeValue d (x i) (y i) t then f ω else 0) := by
-    by_cases h : butterflyPerm d (decodeButterfly d ω) (x i) = y i
-    · have hh := (butterfly_endpoint_iff d ω (x i) (y i)).mp h
-      rw [ite_eq_left hh]
-      simp only [Finset.forall_mem_insert, h, true_and, f]
-    · have hh := mt (butterfly_endpoint_iff d ω (x i) (y i)).mpr h
-      simp only [Finset.forall_mem_insert, h, false_and, hh, ite_false]
-  change finiteMean _ = _
-  rw [finiteMean_congr he, mean_fresh_cylinder _ _ f hf, card_routeDomain]
-  rfl
-
-noncomputable def partialKernel {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x y : L → Card d) : ℝ :=
-  ((2:ℝ)^d)^A.card / ((2:ℝ)^d)^(Fintype.card L) * endpointProb d A x y
-
-lemma partialKernel_insert_isolated {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x y : L → Card d) (i : L) (hi : i ∉ A)
-    (hiso : ∀ j ∈ A, Disjoint (routeDomain d (x i) (y i)) (routeDomain d (x j) (y j))) :
-    partialKernel d (insert i A) x y = partialKernel d A x y := by
-  classical
-  rw [partialKernel, endpointProb_insert_isolated d A x y i hiso,
-    Finset.card_insert_of_notMem hi, pow_succ, partialKernel]
-  field_simp
-
-def ContactEvent {L : Type*} (d : ℕ) (x y : L → Card d) : Prop :=
-  ∀ i, ∃ j, i ≠ j ∧ ¬Disjoint (routeDomain d (x i) (y i)) (routeDomain d (x j) (y j))
-
-lemma alternating_cube_cancel {L : Type*} [Fintype L] (f : Finset L → ℝ)
-    (i : L) (hf : ∀ A, i ∉ A → f (insert i A) = f A) :
-    (∑ A : Finset L, (-1:ℝ)^A.card * f A) = 0 := by
-  classical
-  have hi : i ∈ (Finset.univ : Finset L) := Finset.mem_univ _
-  rw [← Finset.powerset_univ, ← Finset.insert_erase hi, Finset.sum_powerset_insert]
-  · rw [← Finset.sum_add_distrib]
-    apply Finset.sum_eq_zero
-    intro A hA
-    have hiA : i ∉ A := fun hh => Finset.notMem_erase i Finset.univ
-      ((Finset.mem_powerset.mp hA) hh)
-    rw [hf A hiA, Finset.card_insert_of_notMem hiA, pow_succ]
-    ring
-  · exact Finset.notMem_erase _ _
-
-lemma alternating_kernel_cancel {L : Type*} [Fintype L] (d : ℕ)
-    (x y : L → Card d) (h : ¬ContactEvent d x y) :
-    (∑ A : Finset L, (-1:ℝ)^A.card * partialKernel d A x y) = 0 := by
-  classical
-  unfold ContactEvent at h
-  push Not at h
-  obtain ⟨i, hi⟩ := h
-  apply alternating_cube_cancel _ i
-  intro A hiA
-  apply partialKernel_insert_isolated d A x y i hiA
-  intro j hj
-  exact hi j (fun hij => hiA (hij ▸ hj))
-
-
-noncomputable def networkOf {L : Type*} (A : Finset L) (i : L) : Option L := by
-  classical
-  exact if i ∈ A then none else some i
-
-noncomputable def mixedTuple {L : Type*} (d : ℕ) (A : Finset L)
-    (ω : Option L → (SwitchIndex d → Bool)) (x : L → Card d) : L → Card d :=
-  fun i => butterflyPerm d (decodeButterfly d (ω (networkOf A i))) (x i)
-
-noncomputable def mixedReturn {L : Type*} (d : ℕ) (A : Finset L)
-    (ω : Option L → (SwitchIndex d → Bool)) (y : L → Card d) : L → Card d :=
-  fun i => (butterflyPerm d (decodeButterfly d (ω (networkOf A i)))).symm (y i)
-
-noncomputable def labelsOf {L : Type*} (A : Finset L) : Option L → Finset L := by
-  classical
-  exact fun t => match t with
-    | none => A
-    | some i => if i ∈ A then ∅ else {i}
-
-lemma mem_labelsOf {L : Type*} (A : Finset L) (i : L) (t : Option L) :
-    i ∈ labelsOf A t ↔ networkOf A i = t := by
-  classical
-  cases t with
-  | none => simp [labelsOf, networkOf]
-  | some j => by_cases hj : j ∈ A <;> by_cases hi : i ∈ A <;> simp_all [labelsOf, networkOf] <;> intro h <;> subst j <;> contradiction
-
-lemma mixedTuple_eq_iff {L : Type*} (d : ℕ) (A : Finset L)
-    (ω : Option L → (SwitchIndex d → Bool)) (x y : L → Card d) :
-    mixedTuple d A ω x = y ↔ ∀ t, ∀ i ∈ labelsOf A t,
-      butterflyPerm d (decodeButterfly d (ω t)) (x i) = y i := by
-  constructor
-  · intro h t i hi
-    have he := congrFun h i
-    simpa only [mixedTuple, mem_labelsOf A i t |>.mp hi] using he
-  · intro h
-    funext i
-    exact h (networkOf A i) i (mem_labelsOf A i _ |>.mpr rfl)
-
-lemma mixedReturn_eq_iff {L : Type*} (d : ℕ) (A : Finset L)
-    (ω : Option L → (SwitchIndex d → Bool)) (y z : L → Card d) :
-    mixedReturn d A ω y = z ↔ mixedTuple d A ω z = y := by
-  constructor
-  · intro h
-    funext i
-    exact (Equiv.symm_apply_eq _).mp (congrFun h i) |>.symm
-  · intro h
-    funext i
-    exact (Equiv.symm_apply_eq _).mpr (congrFun h i).symm
-
-lemma endpointProb_singleton {L : Type*} (d : ℕ) (i : L) (x y : L → Card d) :
-    endpointProb d {i} x y = 1/(2:ℝ)^d := by
-  classical
-  simpa [endpointProb] using butterfly_single_card_uniform d (x i) (y i)
-
-lemma mixedTuple_mass {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x y : L → Card d) :
-    finiteMean (fun ω : Option L → (SwitchIndex d → Bool) =>
-      if mixedTuple d A ω x = y then (1:ℝ) else 0) = partialKernel d A x y := by
-  classical
-  have he (ω : Option L → (SwitchIndex d → Bool)) :
-      (if mixedTuple d A ω x = y then (1:ℝ) else 0) =
-      ∏ t, if ∀ i ∈ labelsOf A t, butterflyPerm d (decodeButterfly d (ω t)) (x i) = y i
-        then (1:ℝ) else 0 := by
-    simp only [Fintype.prod_boole, mixedTuple_eq_iff]
-  rw [finiteMean_congr he]
-  rw [finiteMean_pi_prod (fun t (w : SwitchIndex d → Bool) =>
-    if ∀ i ∈ labelsOf A t, butterflyPerm d (decodeButterfly d w) (x i) = y i then (1:ℝ) else 0)]
-  have hfac (t : Option L) : (finiteMean (fun w : SwitchIndex d → Bool =>
-      if ∀ i ∈ labelsOf A t, butterflyPerm d (decodeButterfly d w) (x i) = y i then (1:ℝ) else 0)) =
-      endpointProb d (labelsOf A t) x y := by
-        unfold endpointProb
-        apply finiteMean_congr
-        intro w
-        split_ifs <;> rfl
-  simp only [hfac]
-  rw [Fintype.prod_option]
-  have hi (i : L) : endpointProb d (labelsOf A (some i)) x y =
-      if i ∈ A then 1 else 1/(2:ℝ)^d := by
-    by_cases h : i ∈ A
-    · simp [labelsOf, h, endpointProb, finiteMean_const]
-    · simp only [labelsOf, ite_eq_right h, endpointProb_singleton]
-  rw [show labelsOf A none = A from rfl]
-  simp only [hi]
-  have hp : (∏ i : L, if i ∈ A then (1:ℝ) else 1/(2:ℝ)^d) =
-      (1/(2:ℝ)^d)^((Fintype.card L) - A.card) := by
-    rw [Finset.prod_ite]
-    simp only [Finset.prod_const_one, one_mul]
-    simp only [Finset.filter_not, Finset.filter_mem_eq_inter, Finset.univ_inter,
-      Finset.prod_const, Finset.card_sdiff, Finset.inter_univ, Finset.card_univ]
-  rw [hp, partialKernel, one_div, inv_pow]
-  have hc : A.card ≤ Fintype.card L := Finset.card_le_univ _
-  have hn : (((2:ℝ)^d)^A.card) * (((2:ℝ)^d)^(Fintype.card L - A.card)) =
-      ((2:ℝ)^d)^(Fintype.card L) := by rw [← pow_add, Nat.add_sub_of_le hc]
-  have hnz : ((2:ℝ)^d)^(Fintype.card L) ≠ 0 := by positivity
-  have hmz : ((2:ℝ)^d)^(Fintype.card L - A.card) ≠ 0 := by positivity
-  field_simp
-  rw [← hn]
-  ring
-
-lemma mixedReturn_mass {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (y z : L → Card d) :
-    finiteMean (fun ω : Option L → (SwitchIndex d → Bool) =>
-      if mixedReturn d A ω y = z then (1:ℝ) else 0) = partialKernel d A z y := by
-  simpa only [mixedReturn_eq_iff] using mixedTuple_mass d A z y
-
-noncomputable def mixedInitial {L : Type*} (d : ℕ) (A : Finset L)
-    (x : L → Card d) (t : Option L) : Finset (Card d) := by
-  classical
-  exact (labelsOf A t).image x
-
-lemma typedMarks_mixedInitial {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x : L → Card d) (ω : Option L → (SwitchIndex d → Bool)) :
-    typedMarks d (mixedInitial d A x) ω = Finset.univ.image
-      (fun i => (networkOf A i, mixedTuple d A ω x i)) := by
-  classical
-  ext v
-  simp only [typedMarks, mem_typedSet, butterflyMarks, mixedInitial,
-    Finset.mem_image, Finset.mem_univ, true_and]
-  constructor
-  · rintro ⟨z, ⟨i,hi,rfl⟩, hv⟩
-    have ht := mem_labelsOf A i v.1 |>.mp hi
-    exact ⟨i, Prod.ext ht (by simpa only [mixedTuple, ht] using hv)⟩
-  · rintro ⟨i, hi⟩
-    have ht := congrArg Prod.fst hi
-    have hv := congrArg Prod.snd hi
-    change networkOf A i = v.1 at ht
-    change mixedTuple d A ω x i = v.2 at hv
-    exact ⟨x i, ⟨i, mem_labelsOf A i v.1 |>.mpr ht, rfl⟩,
-      by simpa only [mixedTuple, ht] using hv⟩
-
-lemma typedTuple_injective {L : Type*} (d : ℕ) (A : Finset L)
-    (x : L → Card d) (hx : Function.Injective x)
-    (ω : Option L → (SwitchIndex d → Bool)) :
-    Function.Injective (fun i => (networkOf A i, mixedTuple d A ω x i)) := by
-  intro i j h
-  have ht := congrArg Prod.fst h
-  have hv := congrArg Prod.snd h
-  change networkOf A i = networkOf A j at ht
-  dsimp only [mixedTuple] at hv
-  rw [ht] at hv
-  exact hx ((butterflyPerm d (decodeButterfly d (ω (networkOf A j)))).injective hv)
-
-lemma mixedInitial_card {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x : L → Card d) (hx : Function.Injective x) :
-    (∑ t, (mixedInitial d A x t).card) = Fintype.card L := by
-  classical
-  let ω : Option L → (SwitchIndex d → Bool) := fun _ _ => false
-  rw [← typedMarks_card d (mixedInitial d A x) ω, typedMarks_mixedInitial,
-    Finset.card_image_of_injective _ (typedTuple_injective d A x hx ω), Finset.card_univ]
-
-lemma contactEvent_mixedReturn_iff {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x : L → Card d) (hx : Function.Injective x)
-    (ω b : Option L → (SwitchIndex d → Bool)) :
-    ContactEvent d (mixedReturn d A b (mixedTuple d A ω x)) (mixedTuple d A ω x) ↔
-      NoIsolated (returnGraph d (fun t => decodeButterfly d (b t)))
-        (typedMarks d (mixedInitial d A x) ω) := by
-  classical
-  rw [typedMarks_mixedInitial]
-  let f := fun i => (networkOf A i, mixedTuple d A ω x i)
-  have hf : Function.Injective f := typedTuple_injective d A x hx ω
-  constructor
-  · intro h v hv
-    obtain ⟨i,_,rfl⟩ := Finset.mem_image.mp hv
-    obtain ⟨j,hij,hc⟩ := h i
-    exact ⟨f j, Finset.mem_image.mpr ⟨j, Finset.mem_univ _,rfl⟩,
-      hf.ne hij, hc⟩
-  · intro h i
-    obtain ⟨v,hv,hij,hc⟩ := h (f i) (Finset.mem_image.mpr ⟨i,Finset.mem_univ _,rfl⟩)
-    obtain ⟨j,_,rfl⟩ := Finset.mem_image.mp hv
-    exact ⟨j, fun he => hij (congrArg f he), hc⟩
-
-lemma probability_mixed_contact {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x : L → Card d) (hx : Function.Injective x)
-    (b : Option L → (SwitchIndex d → Bool))
-    (hδ : 0 < 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)
-    (hδ1 : 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d ≤ 1) :
-    finiteMean (fun ω : Option L → (SwitchIndex d → Bool) =>
-      if ContactEvent d (mixedReturn d A b (mixedTuple d A ω x)) (mixedTuple d A ω x)
-        then (1:ℝ) else 0) ≤
-      (2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
-        Real.exp (Real.exp 1*(Fintype.card L)) := by
-  classical
-  simp only [contactEvent_mixedReturn_iff d A x hx]
-  convert mixed_return_noIsolated d (mixedInitial d A x) (fun t => decodeButterfly d (b t))
-      (Fintype.card L) (mixedInitial_card d A x hx) hδ hδ1 using 1
-  congr 1
-  exact Subsingleton.elim _ _
-
-
-lemma finiteMean_pushforward {Ω X : Type*} [Fintype Ω] [Fintype X]
-    (g : Ω → X) (f : X → ℝ) :
-    finiteMean (fun ω => f (g ω)) =
-      ∑ y, finiteMean (fun ω => if g ω = y then (1:ℝ) else 0) * f y := by
-  classical
-  simp_rw [← finiteMean_mul_const, ← finiteMean_sum]
-  apply finiteMean_congr
-  intro ω
-  simp
-
-lemma mixedTuple_expectation {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x : L → Card d) (f : (L → Card d) → ℝ) :
-    finiteMean (fun ω : Option L → (SwitchIndex d → Bool) => f (mixedTuple d A ω x)) =
-      ∑ y, partialKernel d A x y * f y := by
-  classical
-  rw [finiteMean_pushforward (fun ω : Option L → (SwitchIndex d → Bool) => mixedTuple d A ω x) f]
-  apply Finset.sum_congr rfl
-  intro y _
-  congr 1
-  convert mixedTuple_mass d A x y using 1
-  congr 1
-  funext ω
-  split_ifs <;> rfl
-
-lemma mixedReturn_expectation {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (y : L → Card d) (f : (L → Card d) → ℝ) :
-    finiteMean (fun ω : Option L → (SwitchIndex d → Bool) => f (mixedReturn d A ω y)) =
-      ∑ z, partialKernel d A z y * f z := by
-  classical
-  rw [finiteMean_pushforward (fun ω : Option L → (SwitchIndex d → Bool) => mixedReturn d A ω y) f]
-  apply Finset.sum_congr rfl
-  intro z _
-  congr 1
-  convert mixedReturn_mass d A y z using 1
-  congr 1
-  funext ω
-  split_ifs <;> rfl
-
-lemma partialKernel_nonneg {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x y : L → Card d) : 0 ≤ partialKernel d A x y :=
-  mul_nonneg (by positivity) (endpointProb_nonneg d A x y)
-
-noncomputable def retainedKernel {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x : L ↪ Card d) (y : L → Card d) : ℝ := by
-  classical
-  exact if ContactEvent d x y then partialKernel d A x y else 0
-
-lemma retainedKernel_nonneg {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x : L ↪ Card d) (y : L → Card d) : 0 ≤ retainedKernel d A x y := by
-  classical
-  unfold retainedKernel
-  split_ifs
-  · exact partialKernel_nonneg d A x y
-  · exact le_rfl
-
-lemma retainedKernel_le {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x : L ↪ Card d) (y : L → Card d) : retainedKernel d A x y ≤ partialKernel d A x y := by
-  classical
-  unfold retainedKernel
-  split_ifs
-  · exact le_rfl
-  · exact partialKernel_nonneg d A x y
-
-lemma retained_square_row {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (x : L ↪ Card d)
-    (hδ : 0 < 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)
-    (hδ1 : 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d ≤ 1) :
-    (∑ x' : L ↪ Card d, ∑ y : L → Card d, retainedKernel d A x y * retainedKernel d A x' y) ≤
-      (2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
-        Real.exp (Real.exp 1*(Fintype.card L)) := by
-  classical
-  let f := fun y z : L → Card d => if ContactEvent d z y then (1:ℝ) else 0
-  have hsub (y : L → Card d) : (∑ x' : L ↪ Card d, retainedKernel d A x' y) ≤
-      ∑ z : L → Card d, partialKernel d A z y * f y z := by
-    have he (x' : L ↪ Card d) : retainedKernel d A x' y = partialKernel d A x' y * f y x' := by
-      unfold retainedKernel f
-      split_ifs <;> ring
-    simp_rw [he]
-    rw [← Finset.sum_image (f := fun z => partialKernel d A z y * f y z)
-      (Function.Embedding.coe_injective.injOn)]
-    apply Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _)
-    intro z _ _
-    exact mul_nonneg (partialKernel_nonneg d A z y) (by dsimp [f]; positivity)
-  calc
-    _ = ∑ y, retainedKernel d A x y * ∑ x' : L ↪ Card d, retainedKernel d A x' y := by
-      rw [Finset.sum_comm]
-      simp only [Finset.mul_sum]
-    _ ≤ ∑ y, partialKernel d A x y * ∑ z : L → Card d, partialKernel d A z y * f y z := by
-      apply Finset.sum_le_sum
-      intro y _
-      exact mul_le_mul (retainedKernel_le d A x y) (hsub y)
-        (Finset.sum_nonneg (fun x' _ => retainedKernel_nonneg d A x' y)) (partialKernel_nonneg d A x y)
-    _ = finiteMean (fun ω : Option L → (SwitchIndex d → Bool) =>
-        finiteMean (fun b : Option L → (SwitchIndex d → Bool) =>
-          f (mixedTuple d A ω x) (mixedReturn d A b (mixedTuple d A ω x)))) := by
-      rw [mixedTuple_expectation d A x (fun y =>
-        finiteMean (fun b : Option L → (SwitchIndex d → Bool) => f y (mixedReturn d A b y)))]
-      simp only [mixedReturn_expectation]
-    _ = finiteMean (fun b : Option L → (SwitchIndex d → Bool) =>
-        finiteMean (fun ω : Option L → (SwitchIndex d → Bool) =>
-          f (mixedTuple d A ω x) (mixedReturn d A b (mixedTuple d A ω x)))) := finiteMean_comm _
-    _ ≤ _ := by
-      apply (finiteMean_mono (fun b => probability_mixed_contact d A x x.injective b hδ hδ1)).trans_eq
-      exact finiteMean_const _
-
-noncomputable def rectOperator {X Y : Type*} [Fintype X] [Fintype Y]
-    (P : X → Y → ℝ) : EuclideanSpace ℂ Y →L[ℂ] EuclideanSpace ℂ X :=
-  LinearMap.toContinuousLinearMap
-    { toFun := fun v => WithLp.toLp 2 (fun x => ∑ y, (P x y : ℂ)*v y)
-      map_add' := by
-        intro v w
-        ext x
-        simp only [PiLp.add_apply, mul_add, Finset.sum_add_distrib]
-      map_smul' := by
-        intro c v
-        ext x
-        simp only [PiLp.smul_apply, smul_eq_mul, RingHom.id_apply]
-        change (∑ y, (P x y : ℂ)*(c*v y)) = c*∑ y, (P x y : ℂ)*v y
-        rw [Finset.mul_sum]
-        apply Finset.sum_congr rfl
-        intro y _
-        ring }
-
-@[simp] lemma rectOperator_apply {X Y : Type*} [Fintype X] [Fintype Y]
-    (P : X → Y → ℝ) (v : EuclideanSpace ℂ Y) (x : X) :
-    rectOperator P v x = ∑ y, (P x y:ℂ)*v y := rfl
-
-lemma rectOperator_adjoint {X Y : Type*} [Fintype X] [Fintype Y]
-    (P : X → Y → ℝ) : ContinuousLinearMap.adjoint (rectOperator P) =
-      rectOperator (fun y x => P x y) := by
-  apply Eq.symm
-  rw [ContinuousLinearMap.eq_adjoint_iff]
-  intro v w
-  simp only [PiLp.inner_apply, RCLike.inner_apply, rectOperator_apply,
-    map_sum, map_mul, Complex.conj_ofReal, Finset.sum_mul, Finset.mul_sum]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro x _
-  apply Finset.sum_congr rfl
-  intro y _
-  ring
-
-lemma rectOperator_comp_transpose {X Y : Type*} [Fintype X] [Fintype Y]
-    (P : X → Y → ℝ) : (rectOperator P).comp (rectOperator (fun y x => P x y)) =
-      rectOperator (fun x x' => ∑ y, P x y * P x' y) := by
-  ext v x
-  simp only [ContinuousLinearMap.comp_apply, rectOperator_apply, Finset.mul_sum,
-    Complex.ofReal_sum, Complex.ofReal_mul, Finset.sum_mul]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro x' _
-  apply Finset.sum_congr rfl
-  intro y _
-  ring
-
-lemma rectOperator_norm_sq_le {X Y : Type*} [Fintype X] [Fintype Y]
-    (P : X → Y → ℝ) (hP : ∀ x y, 0 ≤ P x y) (B : ℝ) (hB : 0 ≤ B)
-    (hrow : ∀ x, (∑ x', ∑ y, P x y * P x' y) ≤ B) :
-    ‖rectOperator P‖^2 ≤ B := by
-  let S := fun x x' => ∑ y, P x y * P x' y
-  have hs : ∀ x x', S x x' = S x' x := by
-    intro x x'
-    unfold S
-    apply Finset.sum_congr rfl
-    intro y _
-    ring
-  have hbound : ‖rectOperator S‖ ≤ B := by
-    apply ContinuousLinearMap.opNorm_le_bound _ hB
-    intro v
-    exact DensityTransfer.schur_bound S
-      (fun x x' => Finset.sum_nonneg (fun y _ => mul_nonneg (hP x y) (hP x' y))) B hB
-      hrow (fun x' => by
-        calc (∑ x, S x x') = ∑ x, S x' x := Finset.sum_congr rfl (fun x _ => hs x x')
-             _ ≤ B := hrow x') v
-  have hn := ContinuousLinearMap.norm_adjoint_comp_self
-    (ContinuousLinearMap.adjoint (rectOperator P))
-  rw [ContinuousLinearMap.adjoint_adjoint, (ContinuousLinearMap.adjoint.norm_map _),
-    rectOperator_adjoint, rectOperator_comp_transpose] at hn
-  rw [pow_two, ← hn]
-  exact hbound
-
-lemma retained_norm_sq {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (hδ : 0 < 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)
-    (hδ1 : 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d ≤ 1) :
-    ‖rectOperator (retainedKernel d A)‖^2 ≤
-      (2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
-        Real.exp (Real.exp 1*(Fintype.card L)) :=
-  rectOperator_norm_sq_le _ (retainedKernel_nonneg d A) _ (by positivity)
-    (fun x => retained_square_row d A x hδ hδ1)
-
-
-noncomputable def pointStabilizer {α : Type*} (A : Finset α) : Subgroup (Equiv.Perm α) where
-  carrier := {p | ∀ x ∈ A, p x = x}
-  one_mem' := by intro x _; rfl
-  mul_mem' := by intro p q hp hq x hx; change p (q x) = x; rw [hq x hx, hp x hx]
-  inv_mem' := by intro p hp x hx; exact (Equiv.Perm.inv_eq_iff_eq).mpr (hp x hx).symm
-
-lemma swap_mem_pointStabilizer {α : Type*} [DecidableEq α] (A : Finset α)
-    (x y : α) (hx : x ∉ A) (hy : y ∉ A) : Equiv.swap x y ∈ pointStabilizer A := by
-  intro z hz
-  exact Equiv.swap_apply_of_ne_of_ne (fun h => hx (h ▸ hz)) (fun h => hy (h ▸ hz))
-
-lemma point_symmetrizer_zero (μ : YoungDiagram) (A : Finset (Specht.Cell μ))
-    (hA : A.card < μ.card - μ.rowLen 0) (v : Specht.Tabloid μ → ℂ) (hv : v ∈ Specht.space μ) :
-    Specht.symmetrizer (pointStabilizer A) (Specht.tabloidRep μ) v = 0 := by
-  classical
-  induction hv using Submodule.span_induction with
-  | mem v hv =>
-    obtain ⟨p,rfl⟩ := hv
-    let e : {x : Specht.Cell μ // p x ∉ A} ≃ {x : Specht.Cell μ // x ∉ A} :=
-      { toFun := fun x => ⟨p x.1,x.2⟩
-        invFun := fun x => ⟨p.symm x.1, by simpa only [p.apply_symm_apply] using x.2⟩
-        left_inv := fun x => by apply Subtype.ext; exact p.symm_apply_apply _
-        right_inv := fun x => by apply Subtype.ext; exact p.apply_symm_apply _ }
-    have hc : μ.rowLen 0 < Fintype.card {x : Specht.Cell μ // p x ∉ A} := by
-      rw [Fintype.card_congr e, Fintype.card_subtype_compl, Specht.card_cell, Fintype.card_coe]
-      have hr := Specht.rowLen_zero_le_card μ
-      omega
-    obtain ⟨x,y,hne,hcol⟩ := Fintype.exists_ne_map_eq_of_card_lt
-      (fun x : {x : Specht.Cell μ // p x ∉ A} => Specht.colIndex x.1)
-      (by simpa only [Fintype.card_fin] using hc)
-    have hxy : x.1 ≠ y.1 := fun h => hne (Subtype.ext h)
-    let h : pointStabilizer A := ⟨Equiv.swap (p x.1) (p y.1),
-      swap_mem_pointStabilizer A _ _ x.2 y.2⟩
-    apply Specht.symmetrizer_eq_zero_of_neg _ _ h
-    change Specht.tabloidRep μ (Equiv.swap (p x.1) (p y.1))
-      (Specht.tabloidRep μ p (Specht.polytabloid μ)) = -Specht.tabloidRep μ p (Specht.polytabloid μ)
-    rw [←Module.End.mul_apply, ←map_mul, ←Equiv.mul_swap_eq_swap_mul]
-    rw [map_mul, Module.End.mul_apply,
-      Specht.swap_polytabloid x.1 y.1 hxy (congrArg Fin.val hcol), map_neg]
-  | zero => exact map_zero _
-  | add v w hv hw ihv ihw => simp only [map_add,ihv,ihw,add_zero]
-  | smul a v hv ih => simp only [map_smul,ih,smul_zero]
-
-lemma point_fixed_zero (μ : YoungDiagram) (A : Finset (Specht.Cell μ))
-    (hA : A.card < μ.card - μ.rowLen 0) (v : Specht.Tabloid μ → ℂ) (hv : v ∈ Specht.space μ)
-    (hfix : ∀ p, p ∈ pointStabilizer A → Specht.tabloidRep μ p v = v) : v = 0 := by
-  classical
-  have hz := point_symmetrizer_zero μ A hA v hv
-  have he : Specht.symmetrizer (pointStabilizer A) (Specht.tabloidRep μ) v =
-      (Fintype.card (pointStabilizer A) : ℂ) • v := by
-    simp only [Specht.symmetrizer, LinearMap.sum_apply, hfix _ (Subtype.prop _),
-      Finset.sum_const, Finset.card_univ, nsmul_eq_mul, Nat.cast_smul_eq_nsmul]
-  rw [he] at hz
-  exact (smul_eq_zero.mp hz).resolve_left (by exact_mod_cast Fintype.card_ne_zero)
-
-lemma point_unitary_fixed_zero (μ : YoungDiagram) (A : Finset (Specht.Cell μ))
-    (hA : A.card < μ.card - μ.rowLen 0) (v : Specht.hilbertSpace μ)
-    (hfix : ∀ p, p ∈ pointStabilizer A → Specht.unitaryRepresentation μ p v = v) : v = 0 := by
-  let w := (Specht.spaceHilbertEquiv μ).symm v
-  have hw : ∀ p, p ∈ pointStabilizer A → Specht.tabloidRep μ p w.val = w.val := by
-    intro p hp
-    have he := congrArg ((Specht.spaceHilbertEquiv μ).symm) (hfix p hp)
-    change Specht.representation μ p w = w at he
-    exact congrArg Subtype.val he
-  have hz := point_fixed_zero μ A hA w.val w.prop hw
-  have hw0 : w = 0 := Subtype.ext hz
-  have he := congrArg (Specht.spaceHilbertEquiv μ) hw0
-  simpa only [w, LinearEquiv.apply_symm_apply, map_zero] using he
-
-lemma point_relabelled_fixed_zero {α : Type*} [Fintype α] (μ : YoungDiagram)
-    (e : α ≃ Specht.Cell μ) (A : Finset α) (hA : A.card < μ.card - μ.rowLen 0)
-    (v : Specht.hilbertSpace μ)
-    (hfix : ∀ p, p ∈ pointStabilizer A → Specht.relabelledUnitary μ e p v = v) : v = 0 := by
-  classical
-  apply point_unitary_fixed_zero μ (A.image e) (by simpa only [Finset.card_image_of_injective _ e.injective] using hA) v
-  intro p hp
-  have hf : e.symm.permCongr p ∈ pointStabilizer A := by
-    intro x hx
-    have hh := hp (e x) (Finset.mem_image.mpr ⟨x,hx,rfl⟩)
-    simp only [Equiv.permCongr_apply, Equiv.symm_symm, hh, e.symm_apply_apply]
-  have hh := hfix _ hf
-  change Specht.unitaryRepresentation μ (e.permCongr (e.symm.permCongr p)) v = v at hh
-  have he : e.permCongr (e.symm.permCongr p) = p := by
-    apply Equiv.ext
-    intro x
-    simp only [Equiv.permCongr_apply, Equiv.symm_symm, e.apply_symm_apply]
-  simpa only [he] using hh
-
-
-section Orbit
-variable {G X V : Type*} [Group G] [Fintype G] [Fintype X]
-    [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-    (a : G →* Equiv.Perm X) (x₀ : X) (ha : ∀ x, ∃ g, a g x₀ = x)
-    (ρ : Representation ℂ G V) (w : V) (hw : UnitaryFinite.IsFixed a x₀ ρ w)
-
-noncomputable def orbitVector (x : X) : V := ρ (UnitaryFinite.representative a x₀ ha x) w
-
-omit [Fintype G] [Fintype X] [FiniteDimensional ℂ V] in
-include hw in
-lemma orbitVector_at (g : G) : orbitVector a x₀ ha ρ w (a g x₀) = ρ g w := by
-  apply ext_inner_right ℂ
-  intro v
-  exact UnitaryFinite.coefficient_at a x₀ ha ρ w hw v g
-
-omit [Fintype G] [Fintype X] [FiniteDimensional ℂ V] in
-include hw in
-lemma orbitVector_action (g : G) (x : X) :
-    orbitVector a x₀ ha ρ w (a g x) = ρ g (orbitVector a x₀ ha ρ w x) := by
-  obtain ⟨h,rfl⟩ := ha x
-  rw [←Equiv.Perm.mul_apply, ←map_mul, orbitVector_at a x₀ ha ρ w hw,
-    orbitVector_at a x₀ ha ρ w hw, map_mul, Module.End.mul_apply]
-end Orbit
-
-def Harmonic {L α : Type*} [Fintype L] [Fintype α] (f : EuclideanSpace ℂ (L ↪ α)) : Prop :=
-  ∀ (A : Finset L), A ≠ Finset.univ → ∀ z : L → α,
-    (∑ y : L ↪ α, if ∀ i ∈ A, y i = z i then f y else 0) = 0
-
-lemma normalizedCoefficient_harmonic {L α : Type*} [Fintype L] [Fintype α]
-    (μ : YoungDiagram) (e : α ≃ Specht.Cell μ)
-    (hk : Fintype.card L = μ.card - μ.rowLen 0) (x₀ : L ↪ α)
-    (w : Specht.hilbertSpace μ)
-    (hw : UnitaryFinite.IsFixed UnitaryFinite.tupleAction x₀ (Specht.relabelledUnitary μ e) w)
-    (v : Specht.hilbertSpace μ) :
-    Harmonic (UnitaryFinite.normalizedCoefficient UnitaryFinite.tupleAction x₀
-      (UnitaryFinite.tupleAction_transitive x₀) (Specht.relabelledUnitary μ e) w v) := by
-  classical
-  intro A hA z
-  let ρ := Specht.relabelledUnitary μ e
-  let F := orbitVector UnitaryFinite.tupleAction x₀ (UnitaryFinite.tupleAction_transitive x₀) ρ w
-  let S : Specht.hilbertSpace μ := ∑ y : L ↪ α, if ∀ i ∈ A, y i = z i then F y else 0
-  have hcard : (A.image z).card < μ.card - μ.rowLen 0 := by
-    apply (Finset.card_image_le).trans_lt
-    rw [← hk]
-    exact Finset.card_lt_card (Finset.ssubset_univ_iff.mpr hA)
-  have hz : S = 0 := by
-    apply point_relabelled_fixed_zero μ e (A.image z) hcard
-    intro p hp
-    change ρ p S = S
-    simp only [S, map_sum, apply_ite, map_zero]
-    apply Fintype.sum_equiv (UnitaryFinite.tupleAction p)
-    intro y
-    have he : (∀ i ∈ A, UnitaryFinite.tupleAction p y i = z i) ↔ ∀ i ∈ A, y i = z i := by
-      apply forall₂_congr
-      intro i hi
-      have hpi : p (z i) = z i := hp _ (Finset.mem_image.mpr ⟨i, hi,rfl⟩)
-      change p (y i) = z i ↔ y i = z i
-      exact (show (p (y i) = z i) ↔ p (y i) = p (z i) by rw [hpi]).trans p.injective.eq_iff
-    simp only [he]
-    split_ifs
-    · exact (orbitVector_action UnitaryFinite.tupleAction x₀
-        (UnitaryFinite.tupleAction_transitive x₀) ρ w hw p y).symm
-    · rfl
-  let c : ℂ := Real.sqrt ((Module.finrank ℂ (Specht.hilbertSpace μ):ℝ)/(Fintype.card (L ↪ α):ℝ))
-  calc
-    _ = c * inner ℂ S v := by
-      dsimp only [S]
-      rw [sum_inner (𝕜 := ℂ) Finset.univ _ v, Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro y _
-      split_ifs <;> simp_all only [UnitaryFinite.normalizedCoefficient, PiLp.smul_apply,
-        smul_eq_mul, UnitaryFinite.coefficient, F, orbitVector, inner_zero_left (𝕜 := ℂ), mul_zero, c, ρ]
-    _ = 0 := by rw [hz, inner_zero_left (𝕜 := ℂ) v, mul_zero]
-
-
-lemma sum_injections {L α M : Type*} [Fintype L] [Fintype α] [AddCommMonoid M]
-    (F : (L → α) → M) (hF : ∀ y, ¬Function.Injective y → F y = 0) :
-    (∑ y : L → α, F y) = ∑ y : L ↪ α, F y := by
-  classical
-  rw [← Finset.sum_image Function.Embedding.coe_injective.injOn]
-  apply Eq.symm
-  apply Finset.sum_subset (Finset.subset_univ _)
-  intro y _ hy
-  apply hF y
-  intro h
-  exact hy (Finset.mem_image.mpr ⟨⟨y,h⟩, Finset.mem_univ _, rfl⟩)
-
-noncomputable def zeroExtend {L α : Type*} [Fintype L] [Fintype α]
-    (f : EuclideanSpace ℂ (L ↪ α)) : EuclideanSpace ℂ (L → α) := by
-  classical
-  exact WithLp.toLp 2 (fun y => if h : Function.Injective y then f ⟨y,h⟩ else 0)
-
-@[simp] lemma zeroExtend_inj {L α : Type*} [Fintype L] [Fintype α]
-    (f : EuclideanSpace ℂ (L ↪ α)) (y : L ↪ α) : zeroExtend f y = f y := by
-  classical
-  simp only [zeroExtend, WithLp.ofLp_toLp, dite_eq_left y.injective]
-  congr 1
-
-lemma zeroExtend_noninj {L α : Type*} [Fintype L] [Fintype α]
-    (f : EuclideanSpace ℂ (L ↪ α)) (y : L → α) (hy : ¬Function.Injective y) : zeroExtend f y = 0 := by
-  classical
-  simp only [zeroExtend, WithLp.ofLp_toLp, dite_eq_right hy]
-
-lemma sum_zeroExtend_mul {L α : Type*} [Fintype L] [Fintype α]
-    (f : EuclideanSpace ℂ (L ↪ α)) (g : (L → α) → ℂ) :
-    (∑ y : L → α, g y * zeroExtend f y) = ∑ y : L ↪ α, g y * f y := by
-  rw [sum_injections _ (fun y hy => by rw [zeroExtend_noninj f y hy, mul_zero])]
-  simp only [zeroExtend_inj]
-
-lemma norm_zeroExtend {L α : Type*} [Fintype L] [Fintype α]
-    (f : EuclideanSpace ℂ (L ↪ α)) : ‖zeroExtend f‖ = ‖f‖ := by
-  have hs : ‖zeroExtend f‖^2 = ‖f‖^2 := by
-    rw [EuclideanSpace.norm_sq_eq, EuclideanSpace.norm_sq_eq,
-      sum_injections _ (fun y hy => by rw [zeroExtend_noninj f y hy]; simp)]
-    simp only [zeroExtend_inj]
-  nlinarith [norm_nonneg (zeroExtend f), norm_nonneg f]
-
-lemma partial_proper_zero {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
-    (hA : A ≠ Finset.univ) (f : EuclideanSpace ℂ (L ↪ Card d)) (hf : Harmonic f) :
-    rectOperator (fun (x : L ↪ Card d) (y : L → Card d) => partialKernel d A x y) (zeroExtend f) = 0 := by
-  classical
-  ext x
-  rw [rectOperator_apply, sum_zeroExtend_mul]
-  change (∑ y : L ↪ Card d, (partialKernel d A x y : ℂ) * f y) = 0
-  simp only [partialKernel, endpointProb, finiteMean, Complex.ofReal_mul, Complex.ofReal_div,
-    Complex.ofReal_sum, Complex.ofReal_natCast]
-  simp_rw [div_eq_mul_inv, mul_assoc, ← Finset.mul_sum]
-  rw [mul_eq_zero]
-  right
-  simp only [Finset.sum_mul]
-  rw [Finset.sum_comm]
-  apply mul_eq_zero_of_right
-  apply Finset.sum_eq_zero
-  intro ω _
-  calc
-    _ = ∑ y : L ↪ Card d, (↑(Fintype.card (SwitchIndex d → Bool)) : ℂ)⁻¹ *
-        (if ∀ i ∈ A, y i = butterflyPerm d (decodeButterfly d ω) (x i) then f y else 0) :=
-      by
-        apply Finset.sum_congr rfl
-        intro y _
-        have hh : (∀ i ∈ A, butterflyPerm d (decodeButterfly d ω) (x i) = y i) ↔
-            (∀ i ∈ A, y i = butterflyPerm d (decodeButterfly d ω) (x i)) :=
-          forall₂_congr fun _ _ => eq_comm
-        by_cases h : ∀ i ∈ A, butterflyPerm d (decodeButterfly d ω) (x i) = y i
-        · simp only [ite_eq_left h, ite_eq_left (hh.mp h), Complex.ofReal_one, one_mul]
-        · simp only [ite_eq_right h, ite_eq_right (mt hh.mpr h), Complex.ofReal_zero, zero_mul, mul_zero]
-    _ = 0 := by
-      rw [← Finset.mul_sum]
-      apply mul_eq_zero_of_right
-      convert hf A hA (fun i => butterflyPerm d (decodeButterfly d ω) (x i)) using 1
-      apply Finset.sum_congr rfl
-      intro y _
-      split_ifs <;> rfl
-
-end Thorp.SparseContact
-
-end ThorpNine.Tail
-
-end OAI
+import Mathlib
+import OAI.Probability.ThorpRouting.Tail.ModifiedStageEquiv
+
+namespace OAI
+
+namespace ThorpNine.Tail
+
+namespace Thorp.SparseContact
+open scoped BigOperators Classical
+
+noncomputable def typedMarks {N : Type*} [Fintype N] (d : ℕ)
+    (A : N → Finset (Card d)) (ω : N → (SwitchIndex d → Bool)) : Finset (N × Card d) :=
+  typedSet (fun t => butterflyMarks d (A t) (ω t))
+
+lemma typedMarks_card {N : Type*} [Fintype N] (d : ℕ)
+    (A : N → Finset (Card d)) (ω : N → (SwitchIndex d → Bool)) :
+    (typedMarks d A ω).card = ∑ t, (A t).card := by
+  classical
+  rw [typedMarks, typedSet_card]
+  apply Finset.sum_congr rfl
+  intro t _
+  exact Finset.card_image_of_injective _ (butterflyPerm d (decodeButterfly d (ω t))).injective
+
+lemma typedMarks_inclusion {N : Type*} [Fintype N] (d : ℕ)
+    (A : N → Finset (Card d)) (D : Finset (N × Card d)) :
+    inclusionMoment (typedMarks d A) D ≤
+      ∏ v ∈ D, ((A v.1).card:ℝ)/(2:ℝ)^d := by
+  classical
+  have he : inclusionMoment (typedMarks d A) D =
+      ∏ t, inclusionMoment (butterflyMarks d (A t)) (fiber D t) := by
+    rw [inclusionMoment_eq]
+    have hh (ω : N → (SwitchIndex d → Bool)) :
+        (if D ⊆ typedMarks d A ω then (1:ℝ) else 0) =
+        ∏ t, if fiber D t ⊆ butterflyMarks d (A t) (ω t) then (1:ℝ) else 0 := by
+      rw [Fintype.prod_boole]
+      simp only [typedMarks, subset_typedSet]
+    rw [finiteMean_congr hh]
+    simp only [inclusionMoment_eq]
+    exact finiteMean_pi_prod (ι := N) (Ω := fun _ => SwitchIndex d → Bool)
+      (fun t (ω : SwitchIndex d → Bool) =>
+        if fiber D t ⊆ butterflyMarks d (A t) ω then (1:ℝ) else 0)
+  have hprod : (∏ v ∈ D, ((A v.1).card:ℝ)/(2:ℝ)^d) =
+      ∏ t, ∏ x ∈ fiber D t, ((A t).card:ℝ)/(2:ℝ)^d := by
+    conv_lhs => rw [← typedSet_fiber D]
+    exact typedSet_prod _ _
+  rw [he, hprod]
+  apply Finset.prod_le_prod₀
+  · intro t _; exact inclusionMoment_nonneg _ _
+  · intro t _
+    simpa only [Finset.prod_const] using butterflyMarks_inclusion d (A t) (fiber D t)
+
+noncomputable def returnRoute {N : Type*} (d : ℕ) (b : N → Butterfly d)
+    (v : N × Card d) : Finset (SwitchIndex d) :=
+  routeDomain d ((butterflyPerm d (b v.1)).symm v.2) v.2
+
+noncomputable def returnGraph {N : Type*} (d : ℕ) (b : N → Butterfly d) :
+    SimpleGraph (N × Card d) where
+  Adj v w := v ≠ w ∧ ¬Disjoint (returnRoute d b v) (returnRoute d b w)
+  symm := ⟨fun _ _ h => ⟨h.1.symm, fun hh => h.2 hh.symm⟩⟩
+  loopless := ⟨fun _ h => h.1 rfl⟩
+
+lemma returnRoute_card {N : Type*} (d : ℕ) (b : N → Butterfly d) (v : N × Card d) :
+    (returnRoute d b v).card = d := card_routeDomain _ _ _
+
+lemma mem_returnRoute {N : Type*} (d : ℕ) (b : N → Butterfly d)
+    (t : N) (y : Card d) (i : SwitchIndex d) :
+    i ∈ returnRoute d b (t,y) ↔
+      y = butterflyPerm d (b t) (switchUsers d (b t) i).1 ∨
+      y = butterflyPerm d (b t) (switchUsers d (b t) i).2 := by
+  have h := mem_routeDomain_iff_user d (b t) ((butterflyPerm d (b t)).symm y) i
+  simpa only [Equiv.apply_symm_apply, returnRoute, Equiv.eq_symm_apply, eq_comm] using h
+
+lemma returnRoute_incidence {N : Type*} [Fintype N] (d : ℕ) (b : N → Butterfly d)
+    (p : N → ℝ) (i : SwitchIndex d) :
+    (∑ v : N × Card d, if i ∈ returnRoute d b v then p v.1 else 0) =
+      2 * ∑ t, p t := by
+  classical
+  rw [Fintype.sum_prod_type, Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro t _
+  have hne : butterflyPerm d (b t) (switchUsers d (b t) i).1 ≠
+      butterflyPerm d (b t) (switchUsers d (b t) i).2 :=
+    (butterflyPerm d (b t)).injective.ne (switchUsers_ne d (b t) i)
+  simp only [mem_returnRoute]
+  have he (y : Card d) :
+      (if y = butterflyPerm d (b t) (switchUsers d (b t) i).1 ∨
+        y = butterflyPerm d (b t) (switchUsers d (b t) i).2 then p t else 0) =
+      (if y = butterflyPerm d (b t) (switchUsers d (b t) i).1 then p t else 0) +
+      (if y = butterflyPerm d (b t) (switchUsers d (b t) i).2 then p t else 0) := by
+    split_ifs <;> simp_all
+  simp only [he, Finset.sum_add_distrib, Finset.sum_ite_eq', Finset.mem_univ, ite_true]
+  ring
+
+lemma returnGraph_degree {N : Type*} [Fintype N] (d : ℕ) (b : N → Butterfly d)
+    (p : N → ℝ) (hp : ∀ t, 0 ≤ p t) (v : N × Card d) :
+    (∑ w, if (returnGraph d b).Adj v w then p w.1 else 0) ≤
+      2*d*∑ t, p t := by
+  classical
+  calc
+    _ ≤ ∑ w, ∑ i ∈ returnRoute d b v, if i ∈ returnRoute d b w then p w.1 else 0 := by
+      apply Finset.sum_le_sum
+      intro w _
+      split_ifs with h
+      · obtain ⟨i, hi, hw⟩ := Finset.not_disjoint_iff.mp h.2
+        have hh := Finset.single_le_sum
+          (f := fun i => if i ∈ returnRoute d b w then p w.1 else 0)
+          (fun i _ => by split_ifs; exact hp _; exact le_rfl) hi
+        simpa only [ite_eq_left hw] using hh
+      · exact Finset.sum_nonneg (fun i _ => by split_ifs; exact hp _; exact le_rfl)
+    _ = ∑ i ∈ returnRoute d b v, ∑ w, if i ∈ returnRoute d b w then p w.1 else 0 :=
+      Finset.sum_comm
+    _ = _ := by simp only [returnRoute_incidence, Finset.sum_const, returnRoute_card,
+        nsmul_eq_mul]; ring
+
+lemma typed_weight_sum {N : Type*} [Fintype N] (d : ℕ)
+    (A : N → Finset (Card d)) :
+    (∑ v : N × Card d, ((A v.1).card:ℝ)/(2:ℝ)^d) = ∑ t, ((A t).card:ℝ) := by
+  rw [Fintype.sum_prod_type]
+  apply Finset.sum_congr rfl
+  intro t _
+  simp only [Finset.sum_const, Finset.card_univ, card_positions, Nat.cast_pow,
+    Nat.cast_ofNat, nsmul_eq_mul]
+  field_simp
+
+lemma mixed_return_noIsolated {N : Type*} [Fintype N] (d : ℕ)
+    (A : N → Finset (Card d)) (b : N → Butterfly d) (k : ℕ)
+    (hk : ∑ t, (A t).card = k)
+    (hδ : 0 < 2*(d:ℝ)*k/(2:ℝ)^d) (hδ1 : 2*(d:ℝ)*k/(2:ℝ)^d ≤ 1) :
+    finiteMean (fun ω : N → (SwitchIndex d → Bool) =>
+      if NoIsolated (returnGraph d b) (typedMarks d A ω) then (1:ℝ) else 0) ≤
+      (2*(d:ℝ)*k/(2:ℝ)^d)^((k:ℝ)/2) * Real.exp (Real.exp 1*k) := by
+  have hs : (∑ t, ((A t).card:ℝ)) = k := by exact_mod_cast hk
+  have hm := noIsolated_probability_le (returnGraph d b) (typedMarks d A) k
+    (fun ω => (typedMarks_card d A ω).trans hk)
+    (fun v => ((A v.1).card:ℝ)/(2:ℝ)^d) (fun v => by positivity)
+    (typedMarks_inclusion d A) _ hδ hδ1
+    (fun v => (returnGraph_degree d b (fun t => ((A t).card:ℝ)/(2:ℝ)^d)
+      (fun t => by positivity) v).trans_eq (by rw [← Finset.sum_div, hs]; ring))
+  simpa only [typed_weight_sum, hs] using hm
+
+
+lemma mean_fresh_cylinder {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (s : Finset ι) (v : ι → Bool) (f : (ι → Bool) → ℝ)
+    (hf : ∀ i ∈ s, ∀ ω, f (QueryTree.flipCoin i ω) = f ω) :
+    finiteMean (fun ω => if ∀ i ∈ s, ω i = v i then f ω else 0) =
+      finiteMean f / (2:ℝ)^s.card := by
+  induction s using Finset.induction_on with
+  | empty => simp [finiteMean]
+  | @insert i s hi ih =>
+    let g := fun ω => if ∀ j ∈ s, ω j = v j then f ω else 0
+    have hg : ∀ ω, g (QueryTree.flipCoin i ω) = g ω := by
+      intro ω
+      have he : (∀ j ∈ s, QueryTree.flipCoin i ω j = v j) ↔ ∀ j ∈ s, ω j = v j := by
+        apply forall₂_congr
+        intro j hj
+        rw [QueryTree.flipCoin_apply_ne i j (fun h => hi (h ▸ hj))]
+      simp only [g, he, hf i (Finset.mem_insert_self _ _) ω]
+    have he (ω : ι → Bool) :
+        (if ∀ j ∈ insert i s, ω j = v j then f ω else 0) =
+        (if ω i = v i then g ω else 0) := by
+      by_cases h : ω i = v i <;> simp [h,g]
+    rw [finiteMean_congr he, QueryTree.mean_indicator_bit g i (v i) hg]
+    have hs := ih (fun j hj => hf j (Finset.mem_insert_of_mem hj))
+    change finiteMean (fun ω => if ∀ j ∈ s, ω j = v j then f ω else 0) / 2 = _
+    rw [hs, Finset.card_insert_of_notMem hi, pow_succ]
+    ring
+
+noncomputable def endpointProb {L : Type*} (d : ℕ) (A : Finset L)
+    (x y : L → Card d) : ℝ := by
+  classical
+  exact finiteMean (fun ω : SwitchIndex d → Bool =>
+    if ∀ i ∈ A, butterflyPerm d (decodeButterfly d ω) (x i) = y i then (1:ℝ) else 0)
+
+lemma endpointProb_nonneg {L : Type*} (d : ℕ) (A : Finset L)
+    (x y : L → Card d) : 0 ≤ endpointProb d A x y := by
+  classical
+  exact finiteMean_nonneg (fun _ => by positivity)
+
+lemma endpointProb_insert_isolated {L : Type*} (d : ℕ) (A : Finset L)
+    (x y : L → Card d) (i : L)
+    (hiso : ∀ j ∈ A, Disjoint (routeDomain d (x i) (y i)) (routeDomain d (x j) (y j))) :
+    endpointProb d (insert i A) x y = endpointProb d A x y / (2:ℝ)^d := by
+  classical
+  let f := fun ω : SwitchIndex d → Bool =>
+    if ∀ j ∈ A, butterflyPerm d (decodeButterfly d ω) (x j) = y j then (1:ℝ) else 0
+  have hf : ∀ t ∈ routeDomain d (x i) (y i), ∀ ω, f (QueryTree.flipCoin t ω) = f ω := by
+    intro t ht ω
+    have he : (∀ j ∈ A, butterflyPerm d (decodeButterfly d (QueryTree.flipCoin t ω)) (x j) = y j) ↔
+        ∀ j ∈ A, butterflyPerm d (decodeButterfly d ω) (x j) = y j := by
+      apply forall₂_congr
+      intro j hj
+      rw [butterfly_endpoint_iff, butterfly_endpoint_iff]
+      apply forall₂_congr
+      intro s hs
+      rw [QueryTree.flipCoin_apply_ne t s (fun h =>
+        Finset.disjoint_left.mp (hiso j hj) ht (h ▸ hs))]
+    simp only [f, he]
+  have he (ω : SwitchIndex d → Bool) :
+      (if ∀ j ∈ insert i A, butterflyPerm d (decodeButterfly d ω) (x j) = y j then (1:ℝ) else 0) =
+      (if ∀ t ∈ routeDomain d (x i) (y i), ω t = routeValue d (x i) (y i) t then f ω else 0) := by
+    by_cases h : butterflyPerm d (decodeButterfly d ω) (x i) = y i
+    · have hh := (butterfly_endpoint_iff d ω (x i) (y i)).mp h
+      rw [ite_eq_left hh]
+      simp only [Finset.forall_mem_insert, h, true_and, f]
+    · have hh := mt (butterfly_endpoint_iff d ω (x i) (y i)).mpr h
+      simp only [Finset.forall_mem_insert, h, false_and, hh, ite_false]
+  change finiteMean _ = _
+  rw [finiteMean_congr he, mean_fresh_cylinder _ _ f hf, card_routeDomain]
+  rfl
+
+noncomputable def partialKernel {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x y : L → Card d) : ℝ :=
+  ((2:ℝ)^d)^A.card / ((2:ℝ)^d)^(Fintype.card L) * endpointProb d A x y
+
+lemma partialKernel_insert_isolated {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x y : L → Card d) (i : L) (hi : i ∉ A)
+    (hiso : ∀ j ∈ A, Disjoint (routeDomain d (x i) (y i)) (routeDomain d (x j) (y j))) :
+    partialKernel d (insert i A) x y = partialKernel d A x y := by
+  classical
+  rw [partialKernel, endpointProb_insert_isolated d A x y i hiso,
+    Finset.card_insert_of_notMem hi, pow_succ, partialKernel]
+  field_simp
+
+def ContactEvent {L : Type*} (d : ℕ) (x y : L → Card d) : Prop :=
+  ∀ i, ∃ j, i ≠ j ∧ ¬Disjoint (routeDomain d (x i) (y i)) (routeDomain d (x j) (y j))
+
+lemma alternating_cube_cancel {L : Type*} [Fintype L] (f : Finset L → ℝ)
+    (i : L) (hf : ∀ A, i ∉ A → f (insert i A) = f A) :
+    (∑ A : Finset L, (-1:ℝ)^A.card * f A) = 0 := by
+  classical
+  have hi : i ∈ (Finset.univ : Finset L) := Finset.mem_univ _
+  rw [← Finset.powerset_univ, ← Finset.insert_erase hi, Finset.sum_powerset_insert]
+  · rw [← Finset.sum_add_distrib]
+    apply Finset.sum_eq_zero
+    intro A hA
+    have hiA : i ∉ A := fun hh => Finset.notMem_erase i Finset.univ
+      ((Finset.mem_powerset.mp hA) hh)
+    rw [hf A hiA, Finset.card_insert_of_notMem hiA, pow_succ]
+    ring
+  · exact Finset.notMem_erase _ _
+
+lemma alternating_kernel_cancel {L : Type*} [Fintype L] (d : ℕ)
+    (x y : L → Card d) (h : ¬ContactEvent d x y) :
+    (∑ A : Finset L, (-1:ℝ)^A.card * partialKernel d A x y) = 0 := by
+  classical
+  unfold ContactEvent at h
+  push Not at h
+  obtain ⟨i, hi⟩ := h
+  apply alternating_cube_cancel _ i
+  intro A hiA
+  apply partialKernel_insert_isolated d A x y i hiA
+  intro j hj
+  exact hi j (fun hij => hiA (hij ▸ hj))
+
+
+noncomputable def networkOf {L : Type*} (A : Finset L) (i : L) : Option L := by
+  classical
+  exact if i ∈ A then none else some i
+
+noncomputable def mixedTuple {L : Type*} (d : ℕ) (A : Finset L)
+    (ω : Option L → (SwitchIndex d → Bool)) (x : L → Card d) : L → Card d :=
+  fun i => butterflyPerm d (decodeButterfly d (ω (networkOf A i))) (x i)
+
+noncomputable def mixedReturn {L : Type*} (d : ℕ) (A : Finset L)
+    (ω : Option L → (SwitchIndex d → Bool)) (y : L → Card d) : L → Card d :=
+  fun i => (butterflyPerm d (decodeButterfly d (ω (networkOf A i)))).symm (y i)
+
+noncomputable def labelsOf {L : Type*} (A : Finset L) : Option L → Finset L := by
+  classical
+  exact fun t => match t with
+    | none => A
+    | some i => if i ∈ A then ∅ else {i}
+
+lemma mem_labelsOf {L : Type*} (A : Finset L) (i : L) (t : Option L) :
+    i ∈ labelsOf A t ↔ networkOf A i = t := by
+  classical
+  cases t with
+  | none => simp [labelsOf, networkOf]
+  | some j => by_cases hj : j ∈ A <;> by_cases hi : i ∈ A <;> simp_all [labelsOf, networkOf] <;> intro h <;> subst j <;> contradiction
+
+lemma mixedTuple_eq_iff {L : Type*} (d : ℕ) (A : Finset L)
+    (ω : Option L → (SwitchIndex d → Bool)) (x y : L → Card d) :
+    mixedTuple d A ω x = y ↔ ∀ t, ∀ i ∈ labelsOf A t,
+      butterflyPerm d (decodeButterfly d (ω t)) (x i) = y i := by
+  constructor
+  · intro h t i hi
+    have he := congrFun h i
+    simpa only [mixedTuple, mem_labelsOf A i t |>.mp hi] using he
+  · intro h
+    funext i
+    exact h (networkOf A i) i (mem_labelsOf A i _ |>.mpr rfl)
+
+lemma mixedReturn_eq_iff {L : Type*} (d : ℕ) (A : Finset L)
+    (ω : Option L → (SwitchIndex d → Bool)) (y z : L → Card d) :
+    mixedReturn d A ω y = z ↔ mixedTuple d A ω z = y := by
+  constructor
+  · intro h
+    funext i
+    exact (Equiv.symm_apply_eq _).mp (congrFun h i) |>.symm
+  · intro h
+    funext i
+    exact (Equiv.symm_apply_eq _).mpr (congrFun h i).symm
+
+lemma endpointProb_singleton {L : Type*} (d : ℕ) (i : L) (x y : L → Card d) :
+    endpointProb d {i} x y = 1/(2:ℝ)^d := by
+  classical
+  simpa [endpointProb] using butterfly_single_card_uniform d (x i) (y i)
+
+lemma mixedTuple_mass {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x y : L → Card d) :
+    finiteMean (fun ω : Option L → (SwitchIndex d → Bool) =>
+      if mixedTuple d A ω x = y then (1:ℝ) else 0) = partialKernel d A x y := by
+  classical
+  have he (ω : Option L → (SwitchIndex d → Bool)) :
+      (if mixedTuple d A ω x = y then (1:ℝ) else 0) =
+      ∏ t, if ∀ i ∈ labelsOf A t, butterflyPerm d (decodeButterfly d (ω t)) (x i) = y i
+        then (1:ℝ) else 0 := by
+    simp only [Fintype.prod_boole, mixedTuple_eq_iff]
+  rw [finiteMean_congr he]
+  rw [finiteMean_pi_prod (fun t (w : SwitchIndex d → Bool) =>
+    if ∀ i ∈ labelsOf A t, butterflyPerm d (decodeButterfly d w) (x i) = y i then (1:ℝ) else 0)]
+  have hfac (t : Option L) : (finiteMean (fun w : SwitchIndex d → Bool =>
+      if ∀ i ∈ labelsOf A t, butterflyPerm d (decodeButterfly d w) (x i) = y i then (1:ℝ) else 0)) =
+      endpointProb d (labelsOf A t) x y := by
+        unfold endpointProb
+        apply finiteMean_congr
+        intro w
+        split_ifs <;> rfl
+  simp only [hfac]
+  rw [Fintype.prod_option]
+  have hi (i : L) : endpointProb d (labelsOf A (some i)) x y =
+      if i ∈ A then 1 else 1/(2:ℝ)^d := by
+    by_cases h : i ∈ A
+    · simp [labelsOf, h, endpointProb, finiteMean_const]
+    · simp only [labelsOf, ite_eq_right h, endpointProb_singleton]
+  rw [show labelsOf A none = A from rfl]
+  simp only [hi]
+  have hp : (∏ i : L, if i ∈ A then (1:ℝ) else 1/(2:ℝ)^d) =
+      (1/(2:ℝ)^d)^((Fintype.card L) - A.card) := by
+    rw [Finset.prod_ite]
+    simp only [Finset.prod_const_one, one_mul]
+    simp only [Finset.filter_not, Finset.filter_mem_eq_inter, Finset.univ_inter,
+      Finset.prod_const, Finset.card_sdiff, Finset.inter_univ, Finset.card_univ]
+  rw [hp, partialKernel, one_div, inv_pow]
+  have hc : A.card ≤ Fintype.card L := Finset.card_le_univ _
+  have hn : (((2:ℝ)^d)^A.card) * (((2:ℝ)^d)^(Fintype.card L - A.card)) =
+      ((2:ℝ)^d)^(Fintype.card L) := by rw [← pow_add, Nat.add_sub_of_le hc]
+  have hnz : ((2:ℝ)^d)^(Fintype.card L) ≠ 0 := by positivity
+  have hmz : ((2:ℝ)^d)^(Fintype.card L - A.card) ≠ 0 := by positivity
+  field_simp
+  rw [← hn]
+  ring
+
+lemma mixedReturn_mass {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (y z : L → Card d) :
+    finiteMean (fun ω : Option L → (SwitchIndex d → Bool) =>
+      if mixedReturn d A ω y = z then (1:ℝ) else 0) = partialKernel d A z y := by
+  simpa only [mixedReturn_eq_iff] using mixedTuple_mass d A z y
+
+noncomputable def mixedInitial {L : Type*} (d : ℕ) (A : Finset L)
+    (x : L → Card d) (t : Option L) : Finset (Card d) := by
+  classical
+  exact (labelsOf A t).image x
+
+lemma typedMarks_mixedInitial {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x : L → Card d) (ω : Option L → (SwitchIndex d → Bool)) :
+    typedMarks d (mixedInitial d A x) ω = Finset.univ.image
+      (fun i => (networkOf A i, mixedTuple d A ω x i)) := by
+  classical
+  ext v
+  simp only [typedMarks, mem_typedSet, butterflyMarks, mixedInitial,
+    Finset.mem_image, Finset.mem_univ, true_and]
+  constructor
+  · rintro ⟨z, ⟨i,hi,rfl⟩, hv⟩
+    have ht := mem_labelsOf A i v.1 |>.mp hi
+    exact ⟨i, Prod.ext ht (by simpa only [mixedTuple, ht] using hv)⟩
+  · rintro ⟨i, hi⟩
+    have ht := congrArg Prod.fst hi
+    have hv := congrArg Prod.snd hi
+    change networkOf A i = v.1 at ht
+    change mixedTuple d A ω x i = v.2 at hv
+    exact ⟨x i, ⟨i, mem_labelsOf A i v.1 |>.mpr ht, rfl⟩,
+      by simpa only [mixedTuple, ht] using hv⟩
+
+lemma typedTuple_injective {L : Type*} (d : ℕ) (A : Finset L)
+    (x : L → Card d) (hx : Function.Injective x)
+    (ω : Option L → (SwitchIndex d → Bool)) :
+    Function.Injective (fun i => (networkOf A i, mixedTuple d A ω x i)) := by
+  intro i j h
+  have ht := congrArg Prod.fst h
+  have hv := congrArg Prod.snd h
+  change networkOf A i = networkOf A j at ht
+  dsimp only [mixedTuple] at hv
+  rw [ht] at hv
+  exact hx ((butterflyPerm d (decodeButterfly d (ω (networkOf A j)))).injective hv)
+
+lemma mixedInitial_card {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x : L → Card d) (hx : Function.Injective x) :
+    (∑ t, (mixedInitial d A x t).card) = Fintype.card L := by
+  classical
+  let ω : Option L → (SwitchIndex d → Bool) := fun _ _ => false
+  rw [← typedMarks_card d (mixedInitial d A x) ω, typedMarks_mixedInitial,
+    Finset.card_image_of_injective _ (typedTuple_injective d A x hx ω), Finset.card_univ]
+
+lemma contactEvent_mixedReturn_iff {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x : L → Card d) (hx : Function.Injective x)
+    (ω b : Option L → (SwitchIndex d → Bool)) :
+    ContactEvent d (mixedReturn d A b (mixedTuple d A ω x)) (mixedTuple d A ω x) ↔
+      NoIsolated (returnGraph d (fun t => decodeButterfly d (b t)))
+        (typedMarks d (mixedInitial d A x) ω) := by
+  classical
+  rw [typedMarks_mixedInitial]
+  let f := fun i => (networkOf A i, mixedTuple d A ω x i)
+  have hf : Function.Injective f := typedTuple_injective d A x hx ω
+  constructor
+  · intro h v hv
+    obtain ⟨i,_,rfl⟩ := Finset.mem_image.mp hv
+    obtain ⟨j,hij,hc⟩ := h i
+    exact ⟨f j, Finset.mem_image.mpr ⟨j, Finset.mem_univ _,rfl⟩,
+      hf.ne hij, hc⟩
+  · intro h i
+    obtain ⟨v,hv,hij,hc⟩ := h (f i) (Finset.mem_image.mpr ⟨i,Finset.mem_univ _,rfl⟩)
+    obtain ⟨j,_,rfl⟩ := Finset.mem_image.mp hv
+    exact ⟨j, fun he => hij (congrArg f he), hc⟩
+
+lemma probability_mixed_contact {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x : L → Card d) (hx : Function.Injective x)
+    (b : Option L → (SwitchIndex d → Bool))
+    (hδ : 0 < 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)
+    (hδ1 : 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d ≤ 1) :
+    finiteMean (fun ω : Option L → (SwitchIndex d → Bool) =>
+      if ContactEvent d (mixedReturn d A b (mixedTuple d A ω x)) (mixedTuple d A ω x)
+        then (1:ℝ) else 0) ≤
+      (2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
+        Real.exp (Real.exp 1*(Fintype.card L)) := by
+  classical
+  simp only [contactEvent_mixedReturn_iff d A x hx]
+  convert mixed_return_noIsolated d (mixedInitial d A x) (fun t => decodeButterfly d (b t))
+      (Fintype.card L) (mixedInitial_card d A x hx) hδ hδ1 using 1
+  congr 1
+  exact Subsingleton.elim _ _
+
+
+lemma finiteMean_pushforward {Ω X : Type*} [Fintype Ω] [Fintype X]
+    (g : Ω → X) (f : X → ℝ) :
+    finiteMean (fun ω => f (g ω)) =
+      ∑ y, finiteMean (fun ω => if g ω = y then (1:ℝ) else 0) * f y := by
+  classical
+  simp_rw [← finiteMean_mul_const, ← finiteMean_sum]
+  apply finiteMean_congr
+  intro ω
+  simp
+
+lemma mixedTuple_expectation {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x : L → Card d) (f : (L → Card d) → ℝ) :
+    finiteMean (fun ω : Option L → (SwitchIndex d → Bool) => f (mixedTuple d A ω x)) =
+      ∑ y, partialKernel d A x y * f y := by
+  classical
+  rw [finiteMean_pushforward (fun ω : Option L → (SwitchIndex d → Bool) => mixedTuple d A ω x) f]
+  apply Finset.sum_congr rfl
+  intro y _
+  congr 1
+  convert mixedTuple_mass d A x y using 1
+  congr 1
+  funext ω
+  split_ifs <;> rfl
+
+lemma mixedReturn_expectation {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (y : L → Card d) (f : (L → Card d) → ℝ) :
+    finiteMean (fun ω : Option L → (SwitchIndex d → Bool) => f (mixedReturn d A ω y)) =
+      ∑ z, partialKernel d A z y * f z := by
+  classical
+  rw [finiteMean_pushforward (fun ω : Option L → (SwitchIndex d → Bool) => mixedReturn d A ω y) f]
+  apply Finset.sum_congr rfl
+  intro z _
+  congr 1
+  convert mixedReturn_mass d A y z using 1
+  congr 1
+  funext ω
+  split_ifs <;> rfl
+
+lemma partialKernel_nonneg {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x y : L → Card d) : 0 ≤ partialKernel d A x y :=
+  mul_nonneg (by positivity) (endpointProb_nonneg d A x y)
+
+noncomputable def retainedKernel {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x : L ↪ Card d) (y : L → Card d) : ℝ := by
+  classical
+  exact if ContactEvent d x y then partialKernel d A x y else 0
+
+lemma retainedKernel_nonneg {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x : L ↪ Card d) (y : L → Card d) : 0 ≤ retainedKernel d A x y := by
+  classical
+  unfold retainedKernel
+  split_ifs
+  · exact partialKernel_nonneg d A x y
+  · exact le_rfl
+
+lemma retainedKernel_le {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x : L ↪ Card d) (y : L → Card d) : retainedKernel d A x y ≤ partialKernel d A x y := by
+  classical
+  unfold retainedKernel
+  split_ifs
+  · exact le_rfl
+  · exact partialKernel_nonneg d A x y
+
+lemma retained_square_row {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (x : L ↪ Card d)
+    (hδ : 0 < 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)
+    (hδ1 : 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d ≤ 1) :
+    (∑ x' : L ↪ Card d, ∑ y : L → Card d, retainedKernel d A x y * retainedKernel d A x' y) ≤
+      (2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
+        Real.exp (Real.exp 1*(Fintype.card L)) := by
+  classical
+  let f := fun y z : L → Card d => if ContactEvent d z y then (1:ℝ) else 0
+  have hsub (y : L → Card d) : (∑ x' : L ↪ Card d, retainedKernel d A x' y) ≤
+      ∑ z : L → Card d, partialKernel d A z y * f y z := by
+    have he (x' : L ↪ Card d) : retainedKernel d A x' y = partialKernel d A x' y * f y x' := by
+      unfold retainedKernel f
+      split_ifs <;> ring
+    simp_rw [he]
+    rw [← Finset.sum_image (f := fun z => partialKernel d A z y * f y z)
+      (Function.Embedding.coe_injective.injOn)]
+    apply Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _)
+    intro z _ _
+    exact mul_nonneg (partialKernel_nonneg d A z y) (by dsimp [f]; positivity)
+  calc
+    _ = ∑ y, retainedKernel d A x y * ∑ x' : L ↪ Card d, retainedKernel d A x' y := by
+      rw [Finset.sum_comm]
+      simp only [Finset.mul_sum]
+    _ ≤ ∑ y, partialKernel d A x y * ∑ z : L → Card d, partialKernel d A z y * f y z := by
+      apply Finset.sum_le_sum
+      intro y _
+      exact mul_le_mul (retainedKernel_le d A x y) (hsub y)
+        (Finset.sum_nonneg (fun x' _ => retainedKernel_nonneg d A x' y)) (partialKernel_nonneg d A x y)
+    _ = finiteMean (fun ω : Option L → (SwitchIndex d → Bool) =>
+        finiteMean (fun b : Option L → (SwitchIndex d → Bool) =>
+          f (mixedTuple d A ω x) (mixedReturn d A b (mixedTuple d A ω x)))) := by
+      rw [mixedTuple_expectation d A x (fun y =>
+        finiteMean (fun b : Option L → (SwitchIndex d → Bool) => f y (mixedReturn d A b y)))]
+      simp only [mixedReturn_expectation]
+    _ = finiteMean (fun b : Option L → (SwitchIndex d → Bool) =>
+        finiteMean (fun ω : Option L → (SwitchIndex d → Bool) =>
+          f (mixedTuple d A ω x) (mixedReturn d A b (mixedTuple d A ω x)))) := finiteMean_comm _
+    _ ≤ _ := by
+      apply (finiteMean_mono (fun b => probability_mixed_contact d A x x.injective b hδ hδ1)).trans_eq
+      exact finiteMean_const _
+
+noncomputable def rectOperator {X Y : Type*} [Fintype X] [Fintype Y]
+    (P : X → Y → ℝ) : EuclideanSpace ℂ Y →L[ℂ] EuclideanSpace ℂ X :=
+  LinearMap.toContinuousLinearMap
+    { toFun := fun v => WithLp.toLp 2 (fun x => ∑ y, (P x y : ℂ)*v y)
+      map_add' := by
+        intro v w
+        ext x
+        simp only [PiLp.add_apply, mul_add, Finset.sum_add_distrib]
+      map_smul' := by
+        intro c v
+        ext x
+        simp only [PiLp.smul_apply, smul_eq_mul, RingHom.id_apply]
+        change (∑ y, (P x y : ℂ)*(c*v y)) = c*∑ y, (P x y : ℂ)*v y
+        rw [Finset.mul_sum]
+        apply Finset.sum_congr rfl
+        intro y _
+        ring }
+
+@[simp] lemma rectOperator_apply {X Y : Type*} [Fintype X] [Fintype Y]
+    (P : X → Y → ℝ) (v : EuclideanSpace ℂ Y) (x : X) :
+    rectOperator P v x = ∑ y, (P x y:ℂ)*v y := rfl
+
+lemma rectOperator_adjoint {X Y : Type*} [Fintype X] [Fintype Y]
+    (P : X → Y → ℝ) : ContinuousLinearMap.adjoint (rectOperator P) =
+      rectOperator (fun y x => P x y) := by
+  apply Eq.symm
+  rw [ContinuousLinearMap.eq_adjoint_iff]
+  intro v w
+  simp only [PiLp.inner_apply, RCLike.inner_apply, rectOperator_apply,
+    map_sum, map_mul, Complex.conj_ofReal, Finset.sum_mul, Finset.mul_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro x _
+  apply Finset.sum_congr rfl
+  intro y _
+  ring
+
+lemma rectOperator_comp_transpose {X Y : Type*} [Fintype X] [Fintype Y]
+    (P : X → Y → ℝ) : (rectOperator P).comp (rectOperator (fun y x => P x y)) =
+      rectOperator (fun x x' => ∑ y, P x y * P x' y) := by
+  ext v x
+  simp only [ContinuousLinearMap.comp_apply, rectOperator_apply, Finset.mul_sum,
+    Complex.ofReal_sum, Complex.ofReal_mul, Finset.sum_mul]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro x' _
+  apply Finset.sum_congr rfl
+  intro y _
+  ring
+
+lemma rectOperator_norm_sq_le {X Y : Type*} [Fintype X] [Fintype Y]
+    (P : X → Y → ℝ) (hP : ∀ x y, 0 ≤ P x y) (B : ℝ) (hB : 0 ≤ B)
+    (hrow : ∀ x, (∑ x', ∑ y, P x y * P x' y) ≤ B) :
+    ‖rectOperator P‖^2 ≤ B := by
+  let S := fun x x' => ∑ y, P x y * P x' y
+  have hs : ∀ x x', S x x' = S x' x := by
+    intro x x'
+    unfold S
+    apply Finset.sum_congr rfl
+    intro y _
+    ring
+  have hbound : ‖rectOperator S‖ ≤ B := by
+    apply ContinuousLinearMap.opNorm_le_bound _ hB
+    intro v
+    exact DensityTransfer.schur_bound S
+      (fun x x' => Finset.sum_nonneg (fun y _ => mul_nonneg (hP x y) (hP x' y))) B hB
+      hrow (fun x' => by
+        calc (∑ x, S x x') = ∑ x, S x' x := Finset.sum_congr rfl (fun x _ => hs x x')
+             _ ≤ B := hrow x') v
+  have hn := ContinuousLinearMap.norm_adjoint_comp_self
+    (ContinuousLinearMap.adjoint (rectOperator P))
+  rw [ContinuousLinearMap.adjoint_adjoint, (ContinuousLinearMap.adjoint.norm_map _),
+    rectOperator_adjoint, rectOperator_comp_transpose] at hn
+  rw [pow_two, ← hn]
+  exact hbound
+
+lemma retained_norm_sq {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (hδ : 0 < 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)
+    (hδ1 : 2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d ≤ 1) :
+    ‖rectOperator (retainedKernel d A)‖^2 ≤
+      (2*(d:ℝ)*(Fintype.card L)/(2:ℝ)^d)^((Fintype.card L:ℝ)/2) *
+        Real.exp (Real.exp 1*(Fintype.card L)) :=
+  rectOperator_norm_sq_le _ (retainedKernel_nonneg d A) _ (by positivity)
+    (fun x => retained_square_row d A x hδ hδ1)
+
+
+noncomputable def pointStabilizer {α : Type*} (A : Finset α) : Subgroup (Equiv.Perm α) where
+  carrier := {p | ∀ x ∈ A, p x = x}
+  one_mem' := by intro x _; rfl
+  mul_mem' := by intro p q hp hq x hx; change p (q x) = x; rw [hq x hx, hp x hx]
+  inv_mem' := by intro p hp x hx; exact (Equiv.Perm.inv_eq_iff_eq).mpr (hp x hx).symm
+
+lemma swap_mem_pointStabilizer {α : Type*} [DecidableEq α] (A : Finset α)
+    (x y : α) (hx : x ∉ A) (hy : y ∉ A) : Equiv.swap x y ∈ pointStabilizer A := by
+  intro z hz
+  exact Equiv.swap_apply_of_ne_of_ne (fun h => hx (h ▸ hz)) (fun h => hy (h ▸ hz))
+
+lemma point_symmetrizer_zero (μ : YoungDiagram) (A : Finset (Specht.Cell μ))
+    (hA : A.card < μ.card - μ.rowLen 0) (v : Specht.Tabloid μ → ℂ) (hv : v ∈ Specht.space μ) :
+    Specht.symmetrizer (pointStabilizer A) (Specht.tabloidRep μ) v = 0 := by
+  classical
+  induction hv using Submodule.span_induction with
+  | mem v hv =>
+    obtain ⟨p,rfl⟩ := hv
+    let e : {x : Specht.Cell μ // p x ∉ A} ≃ {x : Specht.Cell μ // x ∉ A} :=
+      { toFun := fun x => ⟨p x.1,x.2⟩
+        invFun := fun x => ⟨p.symm x.1, by simpa only [p.apply_symm_apply] using x.2⟩
+        left_inv := fun x => by apply Subtype.ext; exact p.symm_apply_apply _
+        right_inv := fun x => by apply Subtype.ext; exact p.apply_symm_apply _ }
+    have hc : μ.rowLen 0 < Fintype.card {x : Specht.Cell μ // p x ∉ A} := by
+      rw [Fintype.card_congr e, Fintype.card_subtype_compl, Specht.card_cell, Fintype.card_coe]
+      have hr := Specht.rowLen_zero_le_card μ
+      omega
+    obtain ⟨x,y,hne,hcol⟩ := Fintype.exists_ne_map_eq_of_card_lt
+      (fun x : {x : Specht.Cell μ // p x ∉ A} => Specht.colIndex x.1)
+      (by simpa only [Fintype.card_fin] using hc)
+    have hxy : x.1 ≠ y.1 := fun h => hne (Subtype.ext h)
+    let h : pointStabilizer A := ⟨Equiv.swap (p x.1) (p y.1),
+      swap_mem_pointStabilizer A _ _ x.2 y.2⟩
+    apply Specht.symmetrizer_eq_zero_of_neg _ _ h
+    change Specht.tabloidRep μ (Equiv.swap (p x.1) (p y.1))
+      (Specht.tabloidRep μ p (Specht.polytabloid μ)) = -Specht.tabloidRep μ p (Specht.polytabloid μ)
+    rw [←Module.End.mul_apply, ←map_mul, ←Equiv.mul_swap_eq_swap_mul]
+    rw [map_mul, Module.End.mul_apply,
+      Specht.swap_polytabloid x.1 y.1 hxy (congrArg Fin.val hcol), map_neg]
+  | zero => exact map_zero _
+  | add v w hv hw ihv ihw => simp only [map_add,ihv,ihw,add_zero]
+  | smul a v hv ih => simp only [map_smul,ih,smul_zero]
+
+lemma point_fixed_zero (μ : YoungDiagram) (A : Finset (Specht.Cell μ))
+    (hA : A.card < μ.card - μ.rowLen 0) (v : Specht.Tabloid μ → ℂ) (hv : v ∈ Specht.space μ)
+    (hfix : ∀ p, p ∈ pointStabilizer A → Specht.tabloidRep μ p v = v) : v = 0 := by
+  classical
+  have hz := point_symmetrizer_zero μ A hA v hv
+  have he : Specht.symmetrizer (pointStabilizer A) (Specht.tabloidRep μ) v =
+      (Fintype.card (pointStabilizer A) : ℂ) • v := by
+    simp only [Specht.symmetrizer, LinearMap.sum_apply, hfix _ (Subtype.prop _),
+      Finset.sum_const, Finset.card_univ, nsmul_eq_mul, Nat.cast_smul_eq_nsmul]
+  rw [he] at hz
+  exact (smul_eq_zero.mp hz).resolve_left (by exact_mod_cast Fintype.card_ne_zero)
+
+lemma point_unitary_fixed_zero (μ : YoungDiagram) (A : Finset (Specht.Cell μ))
+    (hA : A.card < μ.card - μ.rowLen 0) (v : Specht.hilbertSpace μ)
+    (hfix : ∀ p, p ∈ pointStabilizer A → Specht.unitaryRepresentation μ p v = v) : v = 0 := by
+  let w := (Specht.spaceHilbertEquiv μ).symm v
+  have hw : ∀ p, p ∈ pointStabilizer A → Specht.tabloidRep μ p w.val = w.val := by
+    intro p hp
+    have he := congrArg ((Specht.spaceHilbertEquiv μ).symm) (hfix p hp)
+    change Specht.representation μ p w = w at he
+    exact congrArg Subtype.val he
+  have hz := point_fixed_zero μ A hA w.val w.prop hw
+  have hw0 : w = 0 := Subtype.ext hz
+  have he := congrArg (Specht.spaceHilbertEquiv μ) hw0
+  simpa only [w, LinearEquiv.apply_symm_apply, map_zero] using he
+
+lemma point_relabelled_fixed_zero {α : Type*} [Fintype α] (μ : YoungDiagram)
+    (e : α ≃ Specht.Cell μ) (A : Finset α) (hA : A.card < μ.card - μ.rowLen 0)
+    (v : Specht.hilbertSpace μ)
+    (hfix : ∀ p, p ∈ pointStabilizer A → Specht.relabelledUnitary μ e p v = v) : v = 0 := by
+  classical
+  apply point_unitary_fixed_zero μ (A.image e) (by simpa only [Finset.card_image_of_injective _ e.injective] using hA) v
+  intro p hp
+  have hf : e.symm.permCongr p ∈ pointStabilizer A := by
+    intro x hx
+    have hh := hp (e x) (Finset.mem_image.mpr ⟨x,hx,rfl⟩)
+    simp only [Equiv.permCongr_apply, Equiv.symm_symm, hh, e.symm_apply_apply]
+  have hh := hfix _ hf
+  change Specht.unitaryRepresentation μ (e.permCongr (e.symm.permCongr p)) v = v at hh
+  have he : e.permCongr (e.symm.permCongr p) = p := by
+    apply Equiv.ext
+    intro x
+    simp only [Equiv.permCongr_apply, Equiv.symm_symm, e.apply_symm_apply]
+  simpa only [he] using hh
+
+
+section Orbit
+variable {G X V : Type*} [Group G] [Fintype G] [Fintype X]
+    [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+    (a : G →* Equiv.Perm X) (x₀ : X) (ha : ∀ x, ∃ g, a g x₀ = x)
+    (ρ : Representation ℂ G V) (w : V) (hw : UnitaryFinite.IsFixed a x₀ ρ w)
+
+noncomputable def orbitVector (x : X) : V := ρ (UnitaryFinite.representative a x₀ ha x) w
+
+omit [Fintype G] [Fintype X] [FiniteDimensional ℂ V] in
+include hw in
+lemma orbitVector_at (g : G) : orbitVector a x₀ ha ρ w (a g x₀) = ρ g w := by
+  apply ext_inner_right ℂ
+  intro v
+  exact UnitaryFinite.coefficient_at a x₀ ha ρ w hw v g
+
+omit [Fintype G] [Fintype X] [FiniteDimensional ℂ V] in
+include hw in
+lemma orbitVector_action (g : G) (x : X) :
+    orbitVector a x₀ ha ρ w (a g x) = ρ g (orbitVector a x₀ ha ρ w x) := by
+  obtain ⟨h,rfl⟩ := ha x
+  rw [←Equiv.Perm.mul_apply, ←map_mul, orbitVector_at a x₀ ha ρ w hw,
+    orbitVector_at a x₀ ha ρ w hw, map_mul, Module.End.mul_apply]
+end Orbit
+
+def Harmonic {L α : Type*} [Fintype L] [Fintype α] (f : EuclideanSpace ℂ (L ↪ α)) : Prop :=
+  ∀ (A : Finset L), A ≠ Finset.univ → ∀ z : L → α,
+    (∑ y : L ↪ α, if ∀ i ∈ A, y i = z i then f y else 0) = 0
+
+lemma normalizedCoefficient_harmonic {L α : Type*} [Fintype L] [Fintype α]
+    (μ : YoungDiagram) (e : α ≃ Specht.Cell μ)
+    (hk : Fintype.card L = μ.card - μ.rowLen 0) (x₀ : L ↪ α)
+    (w : Specht.hilbertSpace μ)
+    (hw : UnitaryFinite.IsFixed UnitaryFinite.tupleAction x₀ (Specht.relabelledUnitary μ e) w)
+    (v : Specht.hilbertSpace μ) :
+    Harmonic (UnitaryFinite.normalizedCoefficient UnitaryFinite.tupleAction x₀
+      (UnitaryFinite.tupleAction_transitive x₀) (Specht.relabelledUnitary μ e) w v) := by
+  classical
+  intro A hA z
+  let ρ := Specht.relabelledUnitary μ e
+  let F := orbitVector UnitaryFinite.tupleAction x₀ (UnitaryFinite.tupleAction_transitive x₀) ρ w
+  let S : Specht.hilbertSpace μ := ∑ y : L ↪ α, if ∀ i ∈ A, y i = z i then F y else 0
+  have hcard : (A.image z).card < μ.card - μ.rowLen 0 := by
+    apply (Finset.card_image_le).trans_lt
+    rw [← hk]
+    exact Finset.card_lt_card (Finset.ssubset_univ_iff.mpr hA)
+  have hz : S = 0 := by
+    apply point_relabelled_fixed_zero μ e (A.image z) hcard
+    intro p hp
+    change ρ p S = S
+    simp only [S, map_sum, apply_ite, map_zero]
+    apply Fintype.sum_equiv (UnitaryFinite.tupleAction p)
+    intro y
+    have he : (∀ i ∈ A, UnitaryFinite.tupleAction p y i = z i) ↔ ∀ i ∈ A, y i = z i := by
+      apply forall₂_congr
+      intro i hi
+      have hpi : p (z i) = z i := hp _ (Finset.mem_image.mpr ⟨i, hi,rfl⟩)
+      change p (y i) = z i ↔ y i = z i
+      exact (show (p (y i) = z i) ↔ p (y i) = p (z i) by rw [hpi]).trans p.injective.eq_iff
+    simp only [he]
+    split_ifs
+    · exact (orbitVector_action UnitaryFinite.tupleAction x₀
+        (UnitaryFinite.tupleAction_transitive x₀) ρ w hw p y).symm
+    · rfl
+  let c : ℂ := Real.sqrt ((Module.finrank ℂ (Specht.hilbertSpace μ):ℝ)/(Fintype.card (L ↪ α):ℝ))
+  calc
+    _ = c * inner ℂ S v := by
+      dsimp only [S]
+      rw [sum_inner (𝕜 := ℂ) Finset.univ _ v, Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro y _
+      split_ifs <;> simp_all only [UnitaryFinite.normalizedCoefficient, PiLp.smul_apply,
+        smul_eq_mul, UnitaryFinite.coefficient, F, orbitVector, inner_zero_left (𝕜 := ℂ), mul_zero, c, ρ]
+    _ = 0 := by rw [hz, inner_zero_left (𝕜 := ℂ) v, mul_zero]
+
+
+lemma sum_injections {L α M : Type*} [Fintype L] [Fintype α] [AddCommMonoid M]
+    (F : (L → α) → M) (hF : ∀ y, ¬Function.Injective y → F y = 0) :
+    (∑ y : L → α, F y) = ∑ y : L ↪ α, F y := by
+  classical
+  rw [← Finset.sum_image Function.Embedding.coe_injective.injOn]
+  apply Eq.symm
+  apply Finset.sum_subset (Finset.subset_univ _)
+  intro y _ hy
+  apply hF y
+  intro h
+  exact hy (Finset.mem_image.mpr ⟨⟨y,h⟩, Finset.mem_univ _, rfl⟩)
+
+noncomputable def zeroExtend {L α : Type*} [Fintype L] [Fintype α]
+    (f : EuclideanSpace ℂ (L ↪ α)) : EuclideanSpace ℂ (L → α) := by
+  classical
+  exact WithLp.toLp 2 (fun y => if h : Function.Injective y then f ⟨y,h⟩ else 0)
+
+@[simp] lemma zeroExtend_inj {L α : Type*} [Fintype L] [Fintype α]
+    (f : EuclideanSpace ℂ (L ↪ α)) (y : L ↪ α) : zeroExtend f y = f y := by
+  classical
+  simp only [zeroExtend, WithLp.ofLp_toLp, dite_eq_left y.injective]
+  congr 1
+
+lemma zeroExtend_noninj {L α : Type*} [Fintype L] [Fintype α]
+    (f : EuclideanSpace ℂ (L ↪ α)) (y : L → α) (hy : ¬Function.Injective y) : zeroExtend f y = 0 := by
+  classical
+  simp only [zeroExtend, WithLp.ofLp_toLp, dite_eq_right hy]
+
+lemma sum_zeroExtend_mul {L α : Type*} [Fintype L] [Fintype α]
+    (f : EuclideanSpace ℂ (L ↪ α)) (g : (L → α) → ℂ) :
+    (∑ y : L → α, g y * zeroExtend f y) = ∑ y : L ↪ α, g y * f y := by
+  rw [sum_injections _ (fun y hy => by rw [zeroExtend_noninj f y hy, mul_zero])]
+  simp only [zeroExtend_inj]
+
+lemma norm_zeroExtend {L α : Type*} [Fintype L] [Fintype α]
+    (f : EuclideanSpace ℂ (L ↪ α)) : ‖zeroExtend f‖ = ‖f‖ := by
+  have hs : ‖zeroExtend f‖^2 = ‖f‖^2 := by
+    rw [EuclideanSpace.norm_sq_eq, EuclideanSpace.norm_sq_eq,
+      sum_injections _ (fun y hy => by rw [zeroExtend_noninj f y hy]; simp)]
+    simp only [zeroExtend_inj]
+  nlinarith [norm_nonneg (zeroExtend f), norm_nonneg f]
+
+lemma partial_proper_zero {L : Type*} [Fintype L] (d : ℕ) (A : Finset L)
+    (hA : A ≠ Finset.univ) (f : EuclideanSpace ℂ (L ↪ Card d)) (hf : Harmonic f) :
+    rectOperator (fun (x : L ↪ Card d) (y : L → Card d) => partialKernel d A x y) (zeroExtend f) = 0 := by
+  classical
+  ext x
+  rw [rectOperator_apply, sum_zeroExtend_mul]
+  change (∑ y : L ↪ Card d, (partialKernel d A x y : ℂ) * f y) = 0
+  simp only [partialKernel, endpointProb, finiteMean, Complex.ofReal_mul, Complex.ofReal_div,
+    Complex.ofReal_sum, Complex.ofReal_natCast]
+  simp_rw [div_eq_mul_inv, mul_assoc, ← Finset.mul_sum]
+  rw [mul_eq_zero]
+  right
+  simp only [Finset.sum_mul]
+  rw [Finset.sum_comm]
+  apply mul_eq_zero_of_right
+  apply Finset.sum_eq_zero
+  intro ω _
+  calc
+    _ = ∑ y : L ↪ Card d, (↑(Fintype.card (SwitchIndex d → Bool)) : ℂ)⁻¹ *
+        (if ∀ i ∈ A, y i = butterflyPerm d (decodeButterfly d ω) (x i) then f y else 0) :=
+      by
+        apply Finset.sum_congr rfl
+        intro y _
+        have hh : (∀ i ∈ A, butterflyPerm d (decodeButterfly d ω) (x i) = y i) ↔
+            (∀ i ∈ A, y i = butterflyPerm d (decodeButterfly d ω) (x i)) :=
+          forall₂_congr fun _ _ => eq_comm
+        by_cases h : ∀ i ∈ A, butterflyPerm d (decodeButterfly d ω) (x i) = y i
+        · simp only [ite_eq_left h, ite_eq_left (hh.mp h), Complex.ofReal_one, one_mul]
+        · simp only [ite_eq_right h, ite_eq_right (mt hh.mpr h), Complex.ofReal_zero, zero_mul, mul_zero]
+    _ = 0 := by
+      rw [← Finset.mul_sum]
+      apply mul_eq_zero_of_right
+      convert hf A hA (fun i => butterflyPerm d (decodeButterfly d ω) (x i)) using 1
+      apply Finset.sum_congr rfl
+      intro y _
+      split_ifs <;> rfl
+
+end Thorp.SparseContact
+
+end ThorpNine.Tail
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/TraceBound.lean b/lean/OAI/Probability/ThorpRouting/TraceBound.lean
index 0f5fc27fffb6402d6cfab7bdef0e0c0cdd6d883d..33065ea3a5cb37155f8547f2552e2f47dc51369c 100644
--- a/lean/OAI/Probability/ThorpRouting/TraceBound.lean
+++ b/lean/OAI/Probability/ThorpRouting/TraceBound.lean
@@ -1,105 +1,105 @@
-import OAI.Probability.ThorpRouting.TraceDecay
-
-namespace OAI
-
-namespace ThorpNine.Harmonic
-
-namespace Thorp.LowPlanes
-open scoped BigOperators Classical
-open Casimir Specht UnitaryFinite
-
-attribute [local instance] hsNorm hsInner hsFinite
-
-lemma trace_re_le_dimension_norm {V : Type*} [NormedAddCommGroup V]
-    [InnerProductSpace ℂ V] [FiniteDimensional ℂ V] (A : V →L[ℂ] V) :
-    (LinearMap.trace ℂ V A.toLinearMap).re ≤ (Module.finrank ℂ V : ℝ) * ‖A‖ := by
-  let B := stdOrthonormalBasis ℂ V
-  rw [LinearMap.trace_eq_matrix_trace ℂ B.toBasis]
-  simp only [Matrix.trace, Matrix.diag, Complex.re_sum, LinearMap.toMatrix_apply,
-    OrthonormalBasis.coe_toBasis_repr_apply, OrthonormalBasis.repr_apply_apply]
-  calc
-    _ ≤ ∑ j : Fin (Module.finrank ℂ V), ‖A‖ := by
-      apply Finset.sum_le_sum
-      intro j _
-      change (inner ℂ (B j) (A (B j))).re ≤ ‖A‖
-      apply (Complex.re_le_norm _).trans
-      apply (norm_inner_le_norm _ _).trans
-      simpa only [B.norm_eq_one, one_mul, mul_one] using A.le_opNorm (B j)
-    _ = _ := by simp
-
-lemma trace_power_le_dimension_norm {V : Type*} [NormedAddCommGroup V]
-    [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-    (A : V →L[ℂ] V) (u : ℕ) (hu : 0 < u) :
-    (LinearMap.trace ℂ V (A^u).toLinearMap).re ≤
-      (Module.finrank ℂ V : ℝ) * ‖A‖^u :=
-  (trace_re_le_dimension_norm (A^u)).trans
-    (mul_le_mul_of_nonneg_left (norm_pow_le' A hu) (Nat.cast_nonneg _))
-
-lemma full_row_dimension (μ : YoungDiagram) (hμ : μ.rowLen 0 = μ.card) :
-    Module.finrank ℂ (hilbertSpace μ) = 1 := by
-  rw [finrank_hilbertSpace]
-  have hle := Submodule.finrank_le (space μ)
-  rw [Module.finrank_fintype_fun_eq_card] at hle
-  have hc : Fintype.card (Tabloid μ) ≤ 1 := by
-    let := tabloid_subsingleton_of_full_row μ hμ
-    exact Fintype.card_le_one_iff.mpr (fun _ _ => Subsingleton.elim _ _)
-  exact Nat.le_antisymm (hle.trans hc) (dimension_pos μ)
-
-lemma full_row_unique (n : ℕ) (μ ν : Shapes n)
-    (hμ : μ.1.card - μ.1.rowLen 0 = 0) (hν : ν.1.card - ν.1.rowLen 0 = 0) : μ = ν := by
-  have hcells (γ : Shapes n) (hγ : γ.1.card - γ.1.rowLen 0 = 0) :
-      γ.1.cells = {0} ×ˢ Finset.range n := by
-    have hrow : γ.1.rowLen 0 = γ.1.card := by have := rowLen_zero_le_card γ.1; omega
-    have he : γ.1.row 0 = γ.1.cells := Finset.eq_of_subset_of_card_le
-      (Finset.filter_subset _ _) (by rw [←γ.1.rowLen_eq_card]; exact hrow.ge)
-    rw [←he, YoungDiagram.row_eq_prod, hrow, γ.2]
-  apply Subtype.ext
-  apply YoungDiagram.ext
-  exact (hcells μ hμ).trans (hcells ν hν).symm
-
-lemma full_row_count (n : ℕ) :
-    (∑ μ : Shapes n, if μ.1.card - μ.1.rowLen 0 = 0 then (1 : ℝ) else 0) ≤ 1 := by
-  have hc : (Finset.univ.filter (fun μ : Shapes n => μ.1.card - μ.1.rowLen 0 = 0)).card ≤ 1 := by
-    apply Finset.card_le_one.mpr
-    intro μ hμ ν hν
-    exact full_row_unique n μ ν (Finset.mem_filter.mp hμ).2 (Finset.mem_filter.mp hν).2
-  rw [Finset.sum_boole]
-  exact_mod_cast hc
-
-lemma full_row_trace (d u : ℕ) (μ : Shapes (2^d)) (hμ : μ.1.card - μ.1.rowLen 0 = 0) :
-    (Module.finrank ℂ (hilbertSpace μ.1) : ℝ) *
-      absoluteTrace (V := hilbertSpace μ.1) (Q d μ) (2 * (u : ℝ)) = 1 := by
-  have hrow : μ.1.rowLen 0 = μ.1.card := by have := rowLen_zero_le_card μ.1; omega
-  have hK : K d μ.1 (cardLabels d μ) = 1 :=
-    sampleOperator_trivial (V := hilbertSpace μ.1) _
-      (relabelled_full_row μ.1 hrow (cardLabels d μ)) _
-  rw [absoluteTrace_even, Q_square, hK, one_pow, full_row_dimension μ.1 hrow]
-  simp [full_row_dimension μ.1 hrow]
-
-lemma regularTrace_le (d u : ℕ) (hu : 0 < u) :
-    regularTrace d (2 * (u : ℝ)) ≤ 1 + tailBound d u := by
-  have hterm (μ : Shapes (2^d)) :
-      (Module.finrank ℂ (hilbertSpace μ.1) : ℝ) *
-        absoluteTrace (V := hilbertSpace μ.1) (Q d μ) (2 * (u : ℝ)) ≤
-      (if μ.1.card - μ.1.rowLen 0 = 0 then 1 else 0) +
-        (if μ.1.card - μ.1.rowLen 0 = 0 then 0 else
-          (Module.finrank ℂ (hilbertSpace μ.1) : ℝ)^2 * ‖K d μ.1 (cardLabels d μ)‖^u) := by
-    by_cases hz : μ.1.card - μ.1.rowLen 0 = 0
-    · simp only [hz, ite_true, add_zero, full_row_trace d u μ hz, le_refl]
-    · simp only [hz, ite_false, zero_add]
-      rw [absoluteTrace_even, Q_square, pow_two, mul_assoc]
-      exact mul_le_mul_of_nonneg_left
-        (trace_power_le_dimension_norm (V := hilbertSpace μ.1) (K d μ.1 (cardLabels d μ)) u hu) (Nat.cast_nonneg _)
-  calc
-    regularTrace d (2 * (u : ℝ)) ≤
-        (∑ μ : Shapes (2^d), if μ.1.card - μ.1.rowLen 0 = 0 then (1 : ℝ) else 0) +
-          tailBound d u := by
-      rw [tailBound, ←Finset.sum_add_distrib]
-      exact Finset.sum_le_sum (fun μ _ => hterm μ)
-    _ ≤ _ := add_le_add (full_row_count _) le_rfl
-
-end Thorp.LowPlanes
-
-end ThorpNine.Harmonic
-
-end OAI
+import OAI.Probability.ThorpRouting.TraceDecay
+
+namespace OAI
+
+namespace ThorpNine.Harmonic
+
+namespace Thorp.LowPlanes
+open scoped BigOperators Classical
+open Casimir Specht UnitaryFinite
+
+attribute [local instance] hsNorm hsInner hsFinite
+
+lemma trace_re_le_dimension_norm {V : Type*} [NormedAddCommGroup V]
+    [InnerProductSpace ℂ V] [FiniteDimensional ℂ V] (A : V →L[ℂ] V) :
+    (LinearMap.trace ℂ V A.toLinearMap).re ≤ (Module.finrank ℂ V : ℝ) * ‖A‖ := by
+  let B := stdOrthonormalBasis ℂ V
+  rw [LinearMap.trace_eq_matrix_trace ℂ B.toBasis]
+  simp only [Matrix.trace, Matrix.diag, Complex.re_sum, LinearMap.toMatrix_apply,
+    OrthonormalBasis.coe_toBasis_repr_apply, OrthonormalBasis.repr_apply_apply]
+  calc
+    _ ≤ ∑ j : Fin (Module.finrank ℂ V), ‖A‖ := by
+      apply Finset.sum_le_sum
+      intro j _
+      change (inner ℂ (B j) (A (B j))).re ≤ ‖A‖
+      apply (Complex.re_le_norm _).trans
+      apply (norm_inner_le_norm _ _).trans
+      simpa only [B.norm_eq_one, one_mul, mul_one] using A.le_opNorm (B j)
+    _ = _ := by simp
+
+lemma trace_power_le_dimension_norm {V : Type*} [NormedAddCommGroup V]
+    [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+    (A : V →L[ℂ] V) (u : ℕ) (hu : 0 < u) :
+    (LinearMap.trace ℂ V (A^u).toLinearMap).re ≤
+      (Module.finrank ℂ V : ℝ) * ‖A‖^u :=
+  (trace_re_le_dimension_norm (A^u)).trans
+    (mul_le_mul_of_nonneg_left (norm_pow_le' A hu) (Nat.cast_nonneg _))
+
+lemma full_row_dimension (μ : YoungDiagram) (hμ : μ.rowLen 0 = μ.card) :
+    Module.finrank ℂ (hilbertSpace μ) = 1 := by
+  rw [finrank_hilbertSpace]
+  have hle := Submodule.finrank_le (space μ)
+  rw [Module.finrank_fintype_fun_eq_card] at hle
+  have hc : Fintype.card (Tabloid μ) ≤ 1 := by
+    let := tabloid_subsingleton_of_full_row μ hμ
+    exact Fintype.card_le_one_iff.mpr (fun _ _ => Subsingleton.elim _ _)
+  exact Nat.le_antisymm (hle.trans hc) (dimension_pos μ)
+
+lemma full_row_unique (n : ℕ) (μ ν : Shapes n)
+    (hμ : μ.1.card - μ.1.rowLen 0 = 0) (hν : ν.1.card - ν.1.rowLen 0 = 0) : μ = ν := by
+  have hcells (γ : Shapes n) (hγ : γ.1.card - γ.1.rowLen 0 = 0) :
+      γ.1.cells = {0} ×ˢ Finset.range n := by
+    have hrow : γ.1.rowLen 0 = γ.1.card := by have := rowLen_zero_le_card γ.1; omega
+    have he : γ.1.row 0 = γ.1.cells := Finset.eq_of_subset_of_card_le
+      (Finset.filter_subset _ _) (by rw [←γ.1.rowLen_eq_card]; exact hrow.ge)
+    rw [←he, YoungDiagram.row_eq_prod, hrow, γ.2]
+  apply Subtype.ext
+  apply YoungDiagram.ext
+  exact (hcells μ hμ).trans (hcells ν hν).symm
+
+lemma full_row_count (n : ℕ) :
+    (∑ μ : Shapes n, if μ.1.card - μ.1.rowLen 0 = 0 then (1 : ℝ) else 0) ≤ 1 := by
+  have hc : (Finset.univ.filter (fun μ : Shapes n => μ.1.card - μ.1.rowLen 0 = 0)).card ≤ 1 := by
+    apply Finset.card_le_one.mpr
+    intro μ hμ ν hν
+    exact full_row_unique n μ ν (Finset.mem_filter.mp hμ).2 (Finset.mem_filter.mp hν).2
+  rw [Finset.sum_boole]
+  exact_mod_cast hc
+
+lemma full_row_trace (d u : ℕ) (μ : Shapes (2^d)) (hμ : μ.1.card - μ.1.rowLen 0 = 0) :
+    (Module.finrank ℂ (hilbertSpace μ.1) : ℝ) *
+      absoluteTrace (V := hilbertSpace μ.1) (Q d μ) (2 * (u : ℝ)) = 1 := by
+  have hrow : μ.1.rowLen 0 = μ.1.card := by have := rowLen_zero_le_card μ.1; omega
+  have hK : K d μ.1 (cardLabels d μ) = 1 :=
+    sampleOperator_trivial (V := hilbertSpace μ.1) _
+      (relabelled_full_row μ.1 hrow (cardLabels d μ)) _
+  rw [absoluteTrace_even, Q_square, hK, one_pow, full_row_dimension μ.1 hrow]
+  simp [full_row_dimension μ.1 hrow]
+
+lemma regularTrace_le (d u : ℕ) (hu : 0 < u) :
+    regularTrace d (2 * (u : ℝ)) ≤ 1 + tailBound d u := by
+  have hterm (μ : Shapes (2^d)) :
+      (Module.finrank ℂ (hilbertSpace μ.1) : ℝ) *
+        absoluteTrace (V := hilbertSpace μ.1) (Q d μ) (2 * (u : ℝ)) ≤
+      (if μ.1.card - μ.1.rowLen 0 = 0 then 1 else 0) +
+        (if μ.1.card - μ.1.rowLen 0 = 0 then 0 else
+          (Module.finrank ℂ (hilbertSpace μ.1) : ℝ)^2 * ‖K d μ.1 (cardLabels d μ)‖^u) := by
+    by_cases hz : μ.1.card - μ.1.rowLen 0 = 0
+    · simp only [hz, ite_true, add_zero, full_row_trace d u μ hz, le_refl]
+    · simp only [hz, ite_false, zero_add]
+      rw [absoluteTrace_even, Q_square, pow_two, mul_assoc]
+      exact mul_le_mul_of_nonneg_left
+        (trace_power_le_dimension_norm (V := hilbertSpace μ.1) (K d μ.1 (cardLabels d μ)) u hu) (Nat.cast_nonneg _)
+  calc
+    regularTrace d (2 * (u : ℝ)) ≤
+        (∑ μ : Shapes (2^d), if μ.1.card - μ.1.rowLen 0 = 0 then (1 : ℝ) else 0) +
+          tailBound d u := by
+      rw [tailBound, ←Finset.sum_add_distrib]
+      exact Finset.sum_le_sum (fun μ _ => hterm μ)
+    _ ≤ _ := add_le_add (full_row_count _) le_rfl
+
+end Thorp.LowPlanes
+
+end ThorpNine.Harmonic
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/TraceDecay.lean b/lean/OAI/Probability/ThorpRouting/TraceDecay.lean
index 183b3f6ae350bfa1f05f2f6806354e8757d375a3..c686074972f8f31b045929bcfd94724cd243511f 100644
--- a/lean/OAI/Probability/ThorpRouting/TraceDecay.lean
+++ b/lean/OAI/Probability/ThorpRouting/TraceDecay.lean
@@ -1,180 +1,180 @@
-import OAI.Probability.ThorpRouting.TraceModel
-
-namespace OAI
-
-namespace ThorpNine.Harmonic
-
-namespace Thorp.LowPlanes
-open scoped BigOperators Classical
-open Casimir Specht UnitaryFinite
-
-attribute [local instance] hsNorm hsInner hsFinite
-
-lemma absoluteTrace_even {V : Type*} [NormedAddCommGroup V]
-    [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-    (A : V →L[ℂ] V) (u : ℕ) :
-    absoluteTrace A (2 * (u : ℝ)) =
-      (LinearMap.trace ℂ V ((star A * A)^u).toLinearMap).re := by
-  have he : (2 : ℝ) * u = ((2*u : ℕ) : ℝ) := by simp
-  unfold absoluteTrace
-  rw [he, CFC.rpow_eq_pow, CFC.rpow_natCast _ _ (CFC.sqrt_nonneg _),
-    pow_mul, pow_two, CFC.sqrt_mul_sqrt_self _ (star_mul_self_nonneg A)]
-
-lemma Q_square (d : ℕ) (μ : Shapes (2^d)) :
-    star (Q d μ) * Q d μ = K d μ.1 (cardLabels d μ) := by
-  unfold Q
-  rw [physical_sweep_adjoint _ _ (cardRepresentation_unitary d μ)]
-  change _ = sampleOperator (cardRepresentation d μ) (palindromePerm d)
-  rw [sampleOperator_palindrome _ _ (cardRepresentation_unitary d μ)]
-  simp only [ContinuousLinearMap.star_eq_adjoint, ContinuousLinearMap.adjoint_adjoint]
-
-lemma levelScale_log (n k : ℕ) (hk : 1 ≤ k) (hkn : k ≤ n) :
-    Real.log n ≤ levelScale n k := by
-  have hkR : (1 : ℝ) ≤ k := by exact_mod_cast hk
-  have hkp : (0 : ℝ) < k := by exact_mod_cast (show 0 < k by omega)
-  have hn : (0 : ℝ) < n := by exact_mod_cast (show 0 < n by omega)
-  have hratio : 1 ≤ (n : ℝ) / k := (le_div_iff₀ hkp).mpr (by simpa using (show (k : ℝ) ≤ n by exact_mod_cast hkn))
-  have hp := mul_le_mul_of_nonneg_right hkR (Real.log_nonneg hratio)
-  have hl := Real.log_le_sub_one_of_pos hkp
-  have he : Real.log ((n : ℝ) / k) = Real.log n - Real.log k :=
-    Real.log_div hn.ne' hkp.ne'
-  dsimp [levelScale]
-  rw [he] at hp ⊢
-  nlinarith
-
-lemma level_shape_term (n k : ℕ) (hk : 1 ≤ k) (hkn : k ≤ n) :
-    Real.exp (-14 * levelScale n k) ≤ ((n : ℝ)^12)⁻¹ * (1/4 : ℝ)^k := by
-  have hn : (0 : ℝ) < n := by exact_mod_cast (show 0 < n by omega)
-  have hl := levelScale_log n k hk hkn
-  have hj := levelScale_ge n k hkn
-  have hlog4 : Real.log (4 : ℝ) ≤ 2 := by
-    have h2 : Real.log (2 : ℝ) ≤ 1 := by
-      have ht := Real.log_le_sub_one_of_pos (by norm_num : (0 : ℝ) < 2)
-      norm_num at ht
-      exact ht
-    rw [show (4 : ℝ) = 2^2 by norm_num, Real.log_pow]
-    norm_num
-    linarith
-  calc
-    _ ≤ Real.exp (-(12 * Real.log n + (k : ℝ) * Real.log 4)) := by
-      apply Real.exp_le_exp.mpr
-      have hm := mul_le_mul_of_nonneg_left hlog4 (Nat.cast_nonneg k : (0 : ℝ) ≤ k)
-      linarith
-    _ = _ := by
-      rw [neg_add, Real.exp_add]
-      congr 1
-      · have he : Real.log ((n : ℝ)^12) = 12 * Real.log n := by
-          norm_num [Real.log_pow]
-        rw [Real.exp_neg, ←he, Real.exp_log (pow_pos hn _)]
-      · rw [show -((k : ℝ) * Real.log 4) = (k : ℝ) * (-Real.log 4) by ring,
-          Real.exp_nat_mul, Real.exp_neg, Real.exp_log (by norm_num : (0 : ℝ) < 4)]
-        norm_num
-
-lemma uniform_nonconstant_decay :
-    ∃ u : ℕ, 1 ≤ u ∧ ∀ v : ℕ, u ≤ v →
-      ∀ (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Cell μ),
-        0 < μ.card - μ.rowLen 0 →
-        (Module.finrank ℂ (hilbertSpace μ) : ℝ)^2 * ‖K d μ e‖^v ≤
-          Real.exp (-14 * levelScale (2^d) (μ.card - μ.rowLen 0)) := by
-  obtain ⟨a, ha, hlevel⟩ := level_contraction
-  obtain ⟨b, hb, hdim⟩ := HarmonicBounds.dimension_contraction
-  obtain ⟨m, hm⟩ := exists_nat_ge (max (14/a) (2/b))
-  have hma : 14 ≤ a * m := by
-    have ht := (div_le_iff₀ ha).mp ((le_max_left _ _).trans hm)
-    linarith
-  have hmb : 2 ≤ b * m := by
-    have ht := (div_le_iff₀ hb).mp ((le_max_right _ _).trans hm)
-    linarith
-  have hmp : 0 < m := by
-    by_contra hn
-    have hz : m = 0 := by omega
-    simp only [hz, Nat.cast_zero, mul_zero] at hma
-    norm_num at hma
-  refine ⟨2*m, by omega, ?_⟩
-  intro v hv d μ e hk
-  let D : ℝ := Module.finrank ℂ (space μ)
-  let L : ℝ := levelScale (2^d) (μ.card - μ.rowLen 0)
-  have hD : 0 < D := by dsimp [D]; exact_mod_cast dimension_pos μ
-  have hlogD : 0 ≤ Real.log D := Real.log_nonneg (by
-    dsimp [D]; exact_mod_cast dimension_pos μ)
-  have hμ : μ.card = 2^d := by
-    rw [←card_cell, ←Fintype.card_congr e, card_positions]
-  have hkn : μ.card - μ.rowLen 0 ≤ 2^d := by omega
-  have hL : 0 ≤ L := (Nat.cast_nonneg _).trans (levelScale_ge _ _ hkn)
-  have h₁ := pow_le_pow_left₀ (norm_nonneg (K d μ e)) (hlevel d μ e hk) m
-  have h₂ := pow_le_pow_left₀ (norm_nonneg (K d μ e)) (hdim d μ e hk) m
-  change ‖K d μ e‖^m ≤ (Real.exp (-a*L))^m at h₁
-  change ‖K d μ e‖^m ≤ (Real.exp (-b*Real.log D))^m at h₂
-  have hp : ‖K d μ e‖^v ≤ (Real.exp (-a*L))^m * (Real.exp (-b*Real.log D))^m := by
-    apply (pow_le_pow_of_le_one (norm_nonneg _) (K_norm_le_one d μ e) hv).trans
-    rw [two_mul, pow_add]
-    exact mul_le_mul h₁ h₂ (pow_nonneg (norm_nonneg _) _) (by positivity)
-  rw [finrank_hilbertSpace]
-  change D^2 * ‖K d μ e‖^v ≤ Real.exp (-14*L)
-  calc
-    _ ≤ D^2 * ((Real.exp (-a*L))^m * (Real.exp (-b*Real.log D))^m) :=
-      mul_le_mul_of_nonneg_left hp (sq_nonneg _)
-    _ = Real.exp ((2-b*m)*Real.log D - (a*m)*L) := by
-      rw [← Real.exp_log hD, ←Real.exp_nat_mul, ←Real.exp_nat_mul,
-        ←Real.exp_nat_mul, ←Real.exp_add, ←Real.exp_add, Real.log_exp]
-      congr 1
-      ring
-    _ ≤ _ := by
-      apply Real.exp_le_exp.mpr
-      have hfirst : (2-b*m)*Real.log D ≤ 0 :=
-        mul_nonpos_of_nonpos_of_nonneg (by linarith) hlogD
-      have hlast := mul_le_mul_of_nonneg_right hma hL
-      linarith
-
-noncomputable def tailBound (d v : ℕ) : ℝ :=
-  ∑ μ : Shapes (2^d), if μ.1.card - μ.1.rowLen 0 = 0 then 0 else
-    (Module.finrank ℂ (hilbertSpace μ.1) : ℝ)^2 * ‖K d μ.1 (cardLabels d μ)‖^v
-
-lemma two_inverse_twelfth_le (x : ℝ) (hx : 2 ≤ x) :
-    (x^12)⁻¹ * 2 ≤ x^(-10 : ℤ) := by
-  have hxp : 0 < x := by linarith
-  have hsq : 2 ≤ x^2 := by nlinarith
-  rw [zpow_neg, zpow_ofNat]
-  change (x^12)⁻¹ * 2 ≤ (x^10)⁻¹
-  have hpow : x^12 = x^10 * x^2 := by ring
-  rw [inv_mul_eq_div, inv_eq_one_div]
-  apply (div_le_div_iff₀ (pow_pos hxp _) (pow_pos hxp _)).mpr
-  rw [hpow]
-  nlinarith [mul_le_mul_of_nonneg_left hsq (pow_nonneg hxp.le 10)]
-
-lemma uniform_tail_bound :
-    ∃ u : ℕ, 1 ≤ u ∧ ∀ d : ℕ, 1 ≤ d → ∀ v : ℕ, u ≤ v →
-      tailBound d v ≤ ((2 : ℝ)^d)^(-10 : ℤ) := by
-  obtain ⟨u, hu, hdec⟩ := uniform_nonconstant_decay
-  refine ⟨u, hu, ?_⟩
-  intro d hd v hv
-  have hn : (2 : ℝ) ≤ (2 : ℝ)^d := by
-    calc
-      (2 : ℝ) = 2^1 := by norm_num
-      _ ≤ 2^d := pow_le_pow_right₀ (by norm_num) hd
-  have hterm (μ : Shapes (2^d)) :
-      (if μ.1.card - μ.1.rowLen 0 = 0 then 0 else
-        (Module.finrank ℂ (hilbertSpace μ.1) : ℝ)^2 * ‖K d μ.1 (cardLabels d μ)‖^v) ≤
-      (((2 : ℝ)^d)^12)⁻¹ * (1/4 : ℝ)^(μ.1.card - μ.1.rowLen 0) := by
-    split_ifs with hz
-    · positivity
-    · have hk : 0 < μ.1.card - μ.1.rowLen 0 := by omega
-      have hkn : μ.1.card - μ.1.rowLen 0 ≤ 2^d := by have := μ.2; omega
-      apply (hdec v hv d μ.1 (cardLabels d μ) hk).trans
-      simpa only [Nat.cast_pow, Nat.cast_ofNat] using
-        level_shape_term (2^d) (μ.1.card - μ.1.rowLen 0) hk hkn
-  calc
-    tailBound d v ≤ ∑ μ : Shapes (2^d),
-        (((2 : ℝ)^d)^12)⁻¹ * (1/4 : ℝ)^(μ.1.card - μ.1.rowLen 0) :=
-      Finset.sum_le_sum (fun μ _ => hterm μ)
-    _ = (((2 : ℝ)^d)^12)⁻¹ * ∑ μ : Shapes (2^d), (1/4 : ℝ)^(μ.1.card - μ.1.rowLen 0) :=
-      (Finset.mul_sum ..).symm
-    _ ≤ (((2 : ℝ)^d)^12)⁻¹ * 2 :=
-      mul_le_mul_of_nonneg_left (shape_weight_sum _) (by positivity)
-    _ ≤ _ := two_inverse_twelfth_le _ hn
-
-end Thorp.LowPlanes
-
-end ThorpNine.Harmonic
-
-end OAI
+import OAI.Probability.ThorpRouting.TraceModel
+
+namespace OAI
+
+namespace ThorpNine.Harmonic
+
+namespace Thorp.LowPlanes
+open scoped BigOperators Classical
+open Casimir Specht UnitaryFinite
+
+attribute [local instance] hsNorm hsInner hsFinite
+
+lemma absoluteTrace_even {V : Type*} [NormedAddCommGroup V]
+    [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+    (A : V →L[ℂ] V) (u : ℕ) :
+    absoluteTrace A (2 * (u : ℝ)) =
+      (LinearMap.trace ℂ V ((star A * A)^u).toLinearMap).re := by
+  have he : (2 : ℝ) * u = ((2*u : ℕ) : ℝ) := by simp
+  unfold absoluteTrace
+  rw [he, CFC.rpow_eq_pow, CFC.rpow_natCast _ _ (CFC.sqrt_nonneg _),
+    pow_mul, pow_two, CFC.sqrt_mul_sqrt_self _ (star_mul_self_nonneg A)]
+
+lemma Q_square (d : ℕ) (μ : Shapes (2^d)) :
+    star (Q d μ) * Q d μ = K d μ.1 (cardLabels d μ) := by
+  unfold Q
+  rw [physical_sweep_adjoint _ _ (cardRepresentation_unitary d μ)]
+  change _ = sampleOperator (cardRepresentation d μ) (palindromePerm d)
+  rw [sampleOperator_palindrome _ _ (cardRepresentation_unitary d μ)]
+  simp only [ContinuousLinearMap.star_eq_adjoint, ContinuousLinearMap.adjoint_adjoint]
+
+lemma levelScale_log (n k : ℕ) (hk : 1 ≤ k) (hkn : k ≤ n) :
+    Real.log n ≤ levelScale n k := by
+  have hkR : (1 : ℝ) ≤ k := by exact_mod_cast hk
+  have hkp : (0 : ℝ) < k := by exact_mod_cast (show 0 < k by omega)
+  have hn : (0 : ℝ) < n := by exact_mod_cast (show 0 < n by omega)
+  have hratio : 1 ≤ (n : ℝ) / k := (le_div_iff₀ hkp).mpr (by simpa using (show (k : ℝ) ≤ n by exact_mod_cast hkn))
+  have hp := mul_le_mul_of_nonneg_right hkR (Real.log_nonneg hratio)
+  have hl := Real.log_le_sub_one_of_pos hkp
+  have he : Real.log ((n : ℝ) / k) = Real.log n - Real.log k :=
+    Real.log_div hn.ne' hkp.ne'
+  dsimp [levelScale]
+  rw [he] at hp ⊢
+  nlinarith
+
+lemma level_shape_term (n k : ℕ) (hk : 1 ≤ k) (hkn : k ≤ n) :
+    Real.exp (-14 * levelScale n k) ≤ ((n : ℝ)^12)⁻¹ * (1/4 : ℝ)^k := by
+  have hn : (0 : ℝ) < n := by exact_mod_cast (show 0 < n by omega)
+  have hl := levelScale_log n k hk hkn
+  have hj := levelScale_ge n k hkn
+  have hlog4 : Real.log (4 : ℝ) ≤ 2 := by
+    have h2 : Real.log (2 : ℝ) ≤ 1 := by
+      have ht := Real.log_le_sub_one_of_pos (by norm_num : (0 : ℝ) < 2)
+      norm_num at ht
+      exact ht
+    rw [show (4 : ℝ) = 2^2 by norm_num, Real.log_pow]
+    norm_num
+    linarith
+  calc
+    _ ≤ Real.exp (-(12 * Real.log n + (k : ℝ) * Real.log 4)) := by
+      apply Real.exp_le_exp.mpr
+      have hm := mul_le_mul_of_nonneg_left hlog4 (Nat.cast_nonneg k : (0 : ℝ) ≤ k)
+      linarith
+    _ = _ := by
+      rw [neg_add, Real.exp_add]
+      congr 1
+      · have he : Real.log ((n : ℝ)^12) = 12 * Real.log n := by
+          norm_num [Real.log_pow]
+        rw [Real.exp_neg, ←he, Real.exp_log (pow_pos hn _)]
+      · rw [show -((k : ℝ) * Real.log 4) = (k : ℝ) * (-Real.log 4) by ring,
+          Real.exp_nat_mul, Real.exp_neg, Real.exp_log (by norm_num : (0 : ℝ) < 4)]
+        norm_num
+
+lemma uniform_nonconstant_decay :
+    ∃ u : ℕ, 1 ≤ u ∧ ∀ v : ℕ, u ≤ v →
+      ∀ (d : ℕ) (μ : YoungDiagram) (e : Card d ≃ Cell μ),
+        0 < μ.card - μ.rowLen 0 →
+        (Module.finrank ℂ (hilbertSpace μ) : ℝ)^2 * ‖K d μ e‖^v ≤
+          Real.exp (-14 * levelScale (2^d) (μ.card - μ.rowLen 0)) := by
+  obtain ⟨a, ha, hlevel⟩ := level_contraction
+  obtain ⟨b, hb, hdim⟩ := HarmonicBounds.dimension_contraction
+  obtain ⟨m, hm⟩ := exists_nat_ge (max (14/a) (2/b))
+  have hma : 14 ≤ a * m := by
+    have ht := (div_le_iff₀ ha).mp ((le_max_left _ _).trans hm)
+    linarith
+  have hmb : 2 ≤ b * m := by
+    have ht := (div_le_iff₀ hb).mp ((le_max_right _ _).trans hm)
+    linarith
+  have hmp : 0 < m := by
+    by_contra hn
+    have hz : m = 0 := by omega
+    simp only [hz, Nat.cast_zero, mul_zero] at hma
+    norm_num at hma
+  refine ⟨2*m, by omega, ?_⟩
+  intro v hv d μ e hk
+  let D : ℝ := Module.finrank ℂ (space μ)
+  let L : ℝ := levelScale (2^d) (μ.card - μ.rowLen 0)
+  have hD : 0 < D := by dsimp [D]; exact_mod_cast dimension_pos μ
+  have hlogD : 0 ≤ Real.log D := Real.log_nonneg (by
+    dsimp [D]; exact_mod_cast dimension_pos μ)
+  have hμ : μ.card = 2^d := by
+    rw [←card_cell, ←Fintype.card_congr e, card_positions]
+  have hkn : μ.card - μ.rowLen 0 ≤ 2^d := by omega
+  have hL : 0 ≤ L := (Nat.cast_nonneg _).trans (levelScale_ge _ _ hkn)
+  have h₁ := pow_le_pow_left₀ (norm_nonneg (K d μ e)) (hlevel d μ e hk) m
+  have h₂ := pow_le_pow_left₀ (norm_nonneg (K d μ e)) (hdim d μ e hk) m
+  change ‖K d μ e‖^m ≤ (Real.exp (-a*L))^m at h₁
+  change ‖K d μ e‖^m ≤ (Real.exp (-b*Real.log D))^m at h₂
+  have hp : ‖K d μ e‖^v ≤ (Real.exp (-a*L))^m * (Real.exp (-b*Real.log D))^m := by
+    apply (pow_le_pow_of_le_one (norm_nonneg _) (K_norm_le_one d μ e) hv).trans
+    rw [two_mul, pow_add]
+    exact mul_le_mul h₁ h₂ (pow_nonneg (norm_nonneg _) _) (by positivity)
+  rw [finrank_hilbertSpace]
+  change D^2 * ‖K d μ e‖^v ≤ Real.exp (-14*L)
+  calc
+    _ ≤ D^2 * ((Real.exp (-a*L))^m * (Real.exp (-b*Real.log D))^m) :=
+      mul_le_mul_of_nonneg_left hp (sq_nonneg _)
+    _ = Real.exp ((2-b*m)*Real.log D - (a*m)*L) := by
+      rw [← Real.exp_log hD, ←Real.exp_nat_mul, ←Real.exp_nat_mul,
+        ←Real.exp_nat_mul, ←Real.exp_add, ←Real.exp_add, Real.log_exp]
+      congr 1
+      ring
+    _ ≤ _ := by
+      apply Real.exp_le_exp.mpr
+      have hfirst : (2-b*m)*Real.log D ≤ 0 :=
+        mul_nonpos_of_nonpos_of_nonneg (by linarith) hlogD
+      have hlast := mul_le_mul_of_nonneg_right hma hL
+      linarith
+
+noncomputable def tailBound (d v : ℕ) : ℝ :=
+  ∑ μ : Shapes (2^d), if μ.1.card - μ.1.rowLen 0 = 0 then 0 else
+    (Module.finrank ℂ (hilbertSpace μ.1) : ℝ)^2 * ‖K d μ.1 (cardLabels d μ)‖^v
+
+lemma two_inverse_twelfth_le (x : ℝ) (hx : 2 ≤ x) :
+    (x^12)⁻¹ * 2 ≤ x^(-10 : ℤ) := by
+  have hxp : 0 < x := by linarith
+  have hsq : 2 ≤ x^2 := by nlinarith
+  rw [zpow_neg, zpow_ofNat]
+  change (x^12)⁻¹ * 2 ≤ (x^10)⁻¹
+  have hpow : x^12 = x^10 * x^2 := by ring
+  rw [inv_mul_eq_div, inv_eq_one_div]
+  apply (div_le_div_iff₀ (pow_pos hxp _) (pow_pos hxp _)).mpr
+  rw [hpow]
+  nlinarith [mul_le_mul_of_nonneg_left hsq (pow_nonneg hxp.le 10)]
+
+lemma uniform_tail_bound :
+    ∃ u : ℕ, 1 ≤ u ∧ ∀ d : ℕ, 1 ≤ d → ∀ v : ℕ, u ≤ v →
+      tailBound d v ≤ ((2 : ℝ)^d)^(-10 : ℤ) := by
+  obtain ⟨u, hu, hdec⟩ := uniform_nonconstant_decay
+  refine ⟨u, hu, ?_⟩
+  intro d hd v hv
+  have hn : (2 : ℝ) ≤ (2 : ℝ)^d := by
+    calc
+      (2 : ℝ) = 2^1 := by norm_num
+      _ ≤ 2^d := pow_le_pow_right₀ (by norm_num) hd
+  have hterm (μ : Shapes (2^d)) :
+      (if μ.1.card - μ.1.rowLen 0 = 0 then 0 else
+        (Module.finrank ℂ (hilbertSpace μ.1) : ℝ)^2 * ‖K d μ.1 (cardLabels d μ)‖^v) ≤
+      (((2 : ℝ)^d)^12)⁻¹ * (1/4 : ℝ)^(μ.1.card - μ.1.rowLen 0) := by
+    split_ifs with hz
+    · positivity
+    · have hk : 0 < μ.1.card - μ.1.rowLen 0 := by omega
+      have hkn : μ.1.card - μ.1.rowLen 0 ≤ 2^d := by have := μ.2; omega
+      apply (hdec v hv d μ.1 (cardLabels d μ) hk).trans
+      simpa only [Nat.cast_pow, Nat.cast_ofNat] using
+        level_shape_term (2^d) (μ.1.card - μ.1.rowLen 0) hk hkn
+  calc
+    tailBound d v ≤ ∑ μ : Shapes (2^d),
+        (((2 : ℝ)^d)^12)⁻¹ * (1/4 : ℝ)^(μ.1.card - μ.1.rowLen 0) :=
+      Finset.sum_le_sum (fun μ _ => hterm μ)
+    _ = (((2 : ℝ)^d)^12)⁻¹ * ∑ μ : Shapes (2^d), (1/4 : ℝ)^(μ.1.card - μ.1.rowLen 0) :=
+      (Finset.mul_sum ..).symm
+    _ ≤ (((2 : ℝ)^d)^12)⁻¹ * 2 :=
+      mul_le_mul_of_nonneg_left (shape_weight_sum _) (by positivity)
+    _ ≤ _ := two_inverse_twelfth_le _ hn
+
+end Thorp.LowPlanes
+
+end ThorpNine.Harmonic
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpRouting/TraceModel.lean b/lean/OAI/Probability/ThorpRouting/TraceModel.lean
index f254d523b4d3de79827b0e71125f1637a994d80e..2626a7ec177edd1d73427bf8b62b0177b5f956f0 100644
--- a/lean/OAI/Probability/ThorpRouting/TraceModel.lean
+++ b/lean/OAI/Probability/ThorpRouting/TraceModel.lean
@@ -1,42 +1,42 @@
-import OAI.Probability.ThorpRouting.Harmonic.Main
-
-namespace OAI
-
-namespace ThorpNine.Harmonic
-
-namespace Thorp.LowPlanes
-open scoped BigOperators Classical
-open Casimir Specht UnitaryFinite
-
-noncomputable local instance hsNorm (μ : YoungDiagram) : NormedAddCommGroup (hilbertSpace μ) := inferInstance
-noncomputable local instance hsInner (μ : YoungDiagram) : InnerProductSpace ℂ (hilbertSpace μ) := inferInstance
-noncomputable local instance hsFinite (μ : YoungDiagram) : FiniteDimensional ℂ (hilbertSpace μ) := inferInstance
-
-noncomputable def Q (d : ℕ) (μ : Shapes (2^d)) :
-    hilbertSpace μ.1 →L[ℂ] hilbertSpace μ.1 :=
-  sampleOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) (run d d)
-
-noncomputable def absoluteTrace {V : Type*} [NormedAddCommGroup V]
-    [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-    (A : V →L[ℂ] V) (P : ℝ) : ℝ :=
-  (LinearMap.trace ℂ V (CFC.rpow (CFC.sqrt (star A * A)) P).toLinearMap).re
-
-noncomputable def regularTrace (d : ℕ) (P : ℝ) : ℝ :=
-  ∑ μ : Shapes (2^d), (Module.finrank ℂ (hilbertSpace μ.1) : ℝ) *
-    absoluteTrace (V := hilbertSpace μ.1) (Q d μ) P
-
-noncomputable def sweepDistance (d v : ℕ) (σ : Equiv.Perm (Card d)) : ℝ :=
-  (1/2:ℝ) * ∑ g : Equiv.Perm (Card d),
-    |law d (v*d) (g*σ⁻¹) - 1/(Nat.factorial (2^d):ℝ)|
-
-def MainStatement : Prop :=
-  ∃ P : ℝ, 2 ≤ P ∧ ∀ d : ℕ, 1 ≤ d →
-    regularTrace d P ≤ 1 + ((2 : ℝ)^d) ^ (-10 : ℤ) ∧
-    ∀ v : ℤ, P ≤ 2 * (v : ℝ) → ∀ σ : Equiv.Perm (Card d),
-      sweepDistance d v.toNat σ ≤ (1/2 : ℝ) * ((2 : ℝ)^d) ^ (-5 : ℤ)
-
-end Thorp.LowPlanes
-
-end ThorpNine.Harmonic
-
-end OAI
+import OAI.Probability.ThorpRouting.Harmonic.Main
+
+namespace OAI
+
+namespace ThorpNine.Harmonic
+
+namespace Thorp.LowPlanes
+open scoped BigOperators Classical
+open Casimir Specht UnitaryFinite
+
+noncomputable local instance hsNorm (μ : YoungDiagram) : NormedAddCommGroup (hilbertSpace μ) := inferInstance
+noncomputable local instance hsInner (μ : YoungDiagram) : InnerProductSpace ℂ (hilbertSpace μ) := inferInstance
+noncomputable local instance hsFinite (μ : YoungDiagram) : FiniteDimensional ℂ (hilbertSpace μ) := inferInstance
+
+noncomputable def Q (d : ℕ) (μ : Shapes (2^d)) :
+    hilbertSpace μ.1 →L[ℂ] hilbertSpace μ.1 :=
+  sampleOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) (run d d)
+
+noncomputable def absoluteTrace {V : Type*} [NormedAddCommGroup V]
+    [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+    (A : V →L[ℂ] V) (P : ℝ) : ℝ :=
+  (LinearMap.trace ℂ V (CFC.rpow (CFC.sqrt (star A * A)) P).toLinearMap).re
+
+noncomputable def regularTrace (d : ℕ) (P : ℝ) : ℝ :=
+  ∑ μ : Shapes (2^d), (Module.finrank ℂ (hilbertSpace μ.1) : ℝ) *
+    absoluteTrace (V := hilbertSpace μ.1) (Q d μ) P
+
+noncomputable def sweepDistance (d v : ℕ) (σ : Equiv.Perm (Card d)) : ℝ :=
+  (1/2:ℝ) * ∑ g : Equiv.Perm (Card d),
+    |law d (v*d) (g*σ⁻¹) - 1/(Nat.factorial (2^d):ℝ)|
+
+def MainStatement : Prop :=
+  ∃ P : ℝ, 2 ≤ P ∧ ∀ d : ℕ, 1 ≤ d →
+    regularTrace d P ≤ 1 + ((2 : ℝ)^d) ^ (-10 : ℤ) ∧
+    ∀ v : ℤ, P ≤ 2 * (v : ℝ) → ∀ σ : Equiv.Perm (Card d),
+      sweepDistance d v.toNat σ ≤ (1/2 : ℝ) * ((2 : ℝ)^d) ^ (-5 : ℤ)
+
+end Thorp.LowPlanes
+
+end ThorpNine.Harmonic
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/BlockEmbedding.lean b/lean/OAI/Probability/ThorpShuffle/BlockEmbedding.lean
index e53b46a92574929d0ff053dd743b724f817c5187..f6f61dc1240f942b88040e612774c855c5b0081d 100644
--- a/lean/OAI/Probability/ThorpShuffle/BlockEmbedding.lean
+++ b/lean/OAI/Probability/ThorpShuffle/BlockEmbedding.lean
@@ -1,135 +1,135 @@
-import OAI.Probability.ThorpShuffle.FourierPerturbation
-
-namespace OAI
-
-universe uI uβ uα
-
-noncomputable section
-
-open scoped BigOperators ComplexConjugate InnerProductSpace
-open Filter Topology
-
-namespace Thorp.Block
-open scoped Classical
-variable {I : Type uI} {β : Type uβ} {α : Type uα}
-
-attribute [-simp] Thorp.Specht.perm_apply_inv Thorp.Specht.perm_inv_apply in
-def perm (p : I → Equiv.Perm β) : Equiv.Perm (I × β) where
-  toFun x := (x.1, p x.1 x.2)
-  invFun x := (x.1, (p x.1)⁻¹ x.2)
-  left_inv x := by simp
-  right_inv x := by simp
-
-def hom : (I → Equiv.Perm β) →* Equiv.Perm (I × β) where
-  toFun := perm
-  map_one' := by ext ⟨i,b⟩ <;> rfl
-  map_mul' p q := by ext ⟨i,b⟩ <;> rfl
-
-def embedding (e : α ≃ I × β) : (I → Equiv.Perm β) →* Equiv.Perm α :=
-  (e.symm.permCongrHom).toMonoidHom.comp hom
-
-def complement (i : I) := {j : I // j ≠ i} × β
-
-instance [Fintype I] [Fintype β] (i : I) : Fintype (complement (β:=β) i) :=
-  inferInstanceAs (Fintype ({j : I // j ≠ i} × β))
-
-lemma card_complement [Fintype I] [Fintype β] (i : I) :
-    Fintype.card (complement (β:=β) i) = (Fintype.card I - 1) * Fintype.card β := by
-  change Fintype.card ({j : I // j ≠ i} × β) = _
-  rw [Fintype.card_prod, Fintype.card_subtype_compl, Fintype.card_subtype_eq]
-
-def complementIn (e : α ≃ I × β) (i : I) (x : complement (β:=β) i) : α :=
-  e.symm (x.1.val, x.2)
-
-lemma complementIn_injective (e : α ≃ I × β) (i : I) :
-    Function.Injective (complementIn (β:=β) e i) := by
-  intro x y h
-  have hh := e.symm.injective h
-  exact Prod.ext (Subtype.ext (congrArg Prod.fst hh)) (congrArg (fun p : I × β => p.2) hh)
-
-def restriction (e : α ≃ I × β) (i : I) (g : Equiv.Perm α) : complement (β:=β) i → α :=
-  g ∘ complementIn e i
-
-lemma perm_single_apply (i j : I) (h : Equiv.Perm β) (b : β) :
-    perm (Pi.mulSingle i h) (j,b) = if j = i then (j,h b) else (j,b) := by
-  by_cases hj : j = i
-  · subst j; simp [perm]
-  · simp [perm, hj]
-
-lemma embedding_apply (e : α ≃ I × β) (p : I → Equiv.Perm β) (x : α) :
-    embedding e p x = e.symm (perm p (e x)) := rfl
-
-lemma restriction_right (e : α ≃ I × β) (i : I) (g : Equiv.Perm α) (h : Equiv.Perm β) :
-    restriction e i (g * embedding e (Pi.mulSingle i h)) = restriction e i g := by
-  funext x
-  simp only [restriction, Function.comp_apply, Equiv.Perm.coe_mul, embedding_apply,
-    complementIn, e.apply_symm_apply, perm_single_apply, x.1.property, ↓reduceIte]
-
-lemma fixes_inside (i : I) (r : Equiv.Perm (I × β))
-    (hr : ∀ j b, j ≠ i → r (j,b) = (j,b)) (b : β) : (r (i,b)).1 = i := by
-  by_contra h
-  have hh := hr (r (i,b)).1 (r (i,b)).2 h
-  have he : (i,b) = r (i,b) := r.injective (by simpa only [Prod.mk.eta] using hh.symm)
-  exact h (congrArg Prod.fst he).symm
-
-lemma fixes_outside_inv (i : I) (r : Equiv.Perm (I × β))
-    (hr : ∀ j b, j ≠ i → r (j,b) = (j,b)) (j : I) (b : β) (hj : j ≠ i) :
-    r⁻¹ (j,b) = (j,b) := by
-  apply r.injective
-  simpa only [Equiv.Perm.inv_def, Equiv.apply_symm_apply] using (hr j b hj).symm
-
-def extract (i : I) (r : Equiv.Perm (I × β))
-    (hr : ∀ j b, j ≠ i → r (j,b) = (j,b)) : Equiv.Perm β where
-  toFun b := (r (i,b)).2
-  invFun b := (r⁻¹ (i,b)).2
-  left_inv b := by
-    have he : (i, (r (i,b)).2) = r (i,b) := Prod.ext (fixes_inside i r hr b).symm rfl
-    change (r⁻¹ (i, (r (i,b)).2)).2 = b
-    rw [he, Equiv.Perm.inv_def, Equiv.symm_apply_apply]
-  right_inv b := by
-    have he : (i, (r⁻¹ (i,b)).2) = r⁻¹ (i,b) :=
-      Prod.ext (fixes_inside i r⁻¹ (fixes_outside_inv i r hr) b).symm rfl
-    change (r (i, (r⁻¹ (i,b)).2)).2 = b
-    rw [he, Equiv.Perm.inv_def, Equiv.apply_symm_apply]
-
-lemma perm_extract (i : I) (r : Equiv.Perm (I × β))
-    (hr : ∀ j b, j ≠ i → r (j,b) = (j,b)) : perm (Pi.mulSingle i (extract i r hr)) = r := by
-  apply Equiv.ext
-  rintro ⟨j,b⟩
-  rw [perm_single_apply]
-  by_cases hj : j = i
-  · subst j
-    simp only [↓reduceIte]
-    exact Prod.ext (fixes_inside i r hr b).symm rfl
-  · rw [ite_eq_right hj, hr j b hj]
-
-lemma restriction_fiber (e : α ≃ I × β) (i : I) (g k : Equiv.Perm α)
-    (hk : restriction e i k = restriction e i g) :
-    ∃ h : Equiv.Perm β, g * embedding e (Pi.mulSingle i h) = k := by
-  let r : Equiv.Perm (I × β) := e.permCongr (g⁻¹ * k)
-  have hr (j : I) (b : β) (hj : j ≠ i) : r (j,b) = (j,b) := by
-    have hh := congrFun hk (⟨j,hj⟩,b)
-    change k (e.symm (j,b)) = g (e.symm (j,b)) at hh
-    change e (g⁻¹ (k (e.symm (j,b)))) = (j,b)
-    rw [hh, Equiv.Perm.inv_def, Equiv.symm_apply_apply, e.apply_symm_apply]
-  refine ⟨extract i r hr, ?_⟩
-  apply mul_left_cancel (a:=g⁻¹)
-  rw [← mul_assoc, inv_mul_cancel, one_mul]
-  have he := perm_extract i r hr
-  apply (Equiv.permCongrHom e).injective
-  change e.permCongr (e.symm.permCongr (perm (Pi.mulSingle i (extract i r hr)))) = r
-  simpa only [← Equiv.permCongr_symm, Equiv.apply_symm_apply] using he
-
-lemma single_embedding_injective (e : α ≃ I × β) (i : I) :
-    Function.Injective (fun h : Equiv.Perm β => embedding e (Pi.mulSingle i h)) := by
-  intro h k hh
-  ext b
-  have he := DFunLike.congr_fun hh (e.symm (i,b))
-  simp only [embedding_apply, e.apply_symm_apply, perm_single_apply, ↓reduceIte] at he
-  exact congrArg Prod.snd (e.symm.injective he)
-
-end Thorp.Block
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.FourierPerturbation
+
+namespace OAI
+
+universe uI uβ uα
+
+noncomputable section
+
+open scoped BigOperators ComplexConjugate InnerProductSpace
+open Filter Topology
+
+namespace Thorp.Block
+open scoped Classical
+variable {I : Type uI} {β : Type uβ} {α : Type uα}
+
+attribute [-simp] Thorp.Specht.perm_apply_inv Thorp.Specht.perm_inv_apply in
+def perm (p : I → Equiv.Perm β) : Equiv.Perm (I × β) where
+  toFun x := (x.1, p x.1 x.2)
+  invFun x := (x.1, (p x.1)⁻¹ x.2)
+  left_inv x := by simp
+  right_inv x := by simp
+
+def hom : (I → Equiv.Perm β) →* Equiv.Perm (I × β) where
+  toFun := perm
+  map_one' := by ext ⟨i,b⟩ <;> rfl
+  map_mul' p q := by ext ⟨i,b⟩ <;> rfl
+
+def embedding (e : α ≃ I × β) : (I → Equiv.Perm β) →* Equiv.Perm α :=
+  (e.symm.permCongrHom).toMonoidHom.comp hom
+
+def complement (i : I) := {j : I // j ≠ i} × β
+
+instance [Fintype I] [Fintype β] (i : I) : Fintype (complement (β:=β) i) :=
+  inferInstanceAs (Fintype ({j : I // j ≠ i} × β))
+
+lemma card_complement [Fintype I] [Fintype β] (i : I) :
+    Fintype.card (complement (β:=β) i) = (Fintype.card I - 1) * Fintype.card β := by
+  change Fintype.card ({j : I // j ≠ i} × β) = _
+  rw [Fintype.card_prod, Fintype.card_subtype_compl, Fintype.card_subtype_eq]
+
+def complementIn (e : α ≃ I × β) (i : I) (x : complement (β:=β) i) : α :=
+  e.symm (x.1.val, x.2)
+
+lemma complementIn_injective (e : α ≃ I × β) (i : I) :
+    Function.Injective (complementIn (β:=β) e i) := by
+  intro x y h
+  have hh := e.symm.injective h
+  exact Prod.ext (Subtype.ext (congrArg Prod.fst hh)) (congrArg (fun p : I × β => p.2) hh)
+
+def restriction (e : α ≃ I × β) (i : I) (g : Equiv.Perm α) : complement (β:=β) i → α :=
+  g ∘ complementIn e i
+
+lemma perm_single_apply (i j : I) (h : Equiv.Perm β) (b : β) :
+    perm (Pi.mulSingle i h) (j,b) = if j = i then (j,h b) else (j,b) := by
+  by_cases hj : j = i
+  · subst j; simp [perm]
+  · simp [perm, hj]
+
+lemma embedding_apply (e : α ≃ I × β) (p : I → Equiv.Perm β) (x : α) :
+    embedding e p x = e.symm (perm p (e x)) := rfl
+
+lemma restriction_right (e : α ≃ I × β) (i : I) (g : Equiv.Perm α) (h : Equiv.Perm β) :
+    restriction e i (g * embedding e (Pi.mulSingle i h)) = restriction e i g := by
+  funext x
+  simp only [restriction, Function.comp_apply, Equiv.Perm.coe_mul, embedding_apply,
+    complementIn, e.apply_symm_apply, perm_single_apply, x.1.property, ↓reduceIte]
+
+lemma fixes_inside (i : I) (r : Equiv.Perm (I × β))
+    (hr : ∀ j b, j ≠ i → r (j,b) = (j,b)) (b : β) : (r (i,b)).1 = i := by
+  by_contra h
+  have hh := hr (r (i,b)).1 (r (i,b)).2 h
+  have he : (i,b) = r (i,b) := r.injective (by simpa only [Prod.mk.eta] using hh.symm)
+  exact h (congrArg Prod.fst he).symm
+
+lemma fixes_outside_inv (i : I) (r : Equiv.Perm (I × β))
+    (hr : ∀ j b, j ≠ i → r (j,b) = (j,b)) (j : I) (b : β) (hj : j ≠ i) :
+    r⁻¹ (j,b) = (j,b) := by
+  apply r.injective
+  simpa only [Equiv.Perm.inv_def, Equiv.apply_symm_apply] using (hr j b hj).symm
+
+def extract (i : I) (r : Equiv.Perm (I × β))
+    (hr : ∀ j b, j ≠ i → r (j,b) = (j,b)) : Equiv.Perm β where
+  toFun b := (r (i,b)).2
+  invFun b := (r⁻¹ (i,b)).2
+  left_inv b := by
+    have he : (i, (r (i,b)).2) = r (i,b) := Prod.ext (fixes_inside i r hr b).symm rfl
+    change (r⁻¹ (i, (r (i,b)).2)).2 = b
+    rw [he, Equiv.Perm.inv_def, Equiv.symm_apply_apply]
+  right_inv b := by
+    have he : (i, (r⁻¹ (i,b)).2) = r⁻¹ (i,b) :=
+      Prod.ext (fixes_inside i r⁻¹ (fixes_outside_inv i r hr) b).symm rfl
+    change (r (i, (r⁻¹ (i,b)).2)).2 = b
+    rw [he, Equiv.Perm.inv_def, Equiv.apply_symm_apply]
+
+lemma perm_extract (i : I) (r : Equiv.Perm (I × β))
+    (hr : ∀ j b, j ≠ i → r (j,b) = (j,b)) : perm (Pi.mulSingle i (extract i r hr)) = r := by
+  apply Equiv.ext
+  rintro ⟨j,b⟩
+  rw [perm_single_apply]
+  by_cases hj : j = i
+  · subst j
+    simp only [↓reduceIte]
+    exact Prod.ext (fixes_inside i r hr b).symm rfl
+  · rw [ite_eq_right hj, hr j b hj]
+
+lemma restriction_fiber (e : α ≃ I × β) (i : I) (g k : Equiv.Perm α)
+    (hk : restriction e i k = restriction e i g) :
+    ∃ h : Equiv.Perm β, g * embedding e (Pi.mulSingle i h) = k := by
+  let r : Equiv.Perm (I × β) := e.permCongr (g⁻¹ * k)
+  have hr (j : I) (b : β) (hj : j ≠ i) : r (j,b) = (j,b) := by
+    have hh := congrFun hk (⟨j,hj⟩,b)
+    change k (e.symm (j,b)) = g (e.symm (j,b)) at hh
+    change e (g⁻¹ (k (e.symm (j,b)))) = (j,b)
+    rw [hh, Equiv.Perm.inv_def, Equiv.symm_apply_apply, e.apply_symm_apply]
+  refine ⟨extract i r hr, ?_⟩
+  apply mul_left_cancel (a:=g⁻¹)
+  rw [← mul_assoc, inv_mul_cancel, one_mul]
+  have he := perm_extract i r hr
+  apply (Equiv.permCongrHom e).injective
+  change e.permCongr (e.symm.permCongr (perm (Pi.mulSingle i (extract i r hr)))) = r
+  simpa only [← Equiv.permCongr_symm, Equiv.apply_symm_apply] using he
+
+lemma single_embedding_injective (e : α ≃ I × β) (i : I) :
+    Function.Injective (fun h : Equiv.Perm β => embedding e (Pi.mulSingle i h)) := by
+  intro h k hh
+  ext b
+  have he := DFunLike.congr_fun hh (e.symm (i,b))
+  simp only [embedding_apply, e.apply_symm_apply, perm_single_apply, ↓reduceIte] at he
+  exact congrArg Prod.snd (e.symm.injective he)
+
+end Thorp.Block
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/BlockFibers.lean b/lean/OAI/Probability/ThorpShuffle/BlockFibers.lean
index 30b5ecae230c3457ef1428df62cbdfdb22a5ab1f..5178ffd03870f8f679a7f81b1a57e335264dd02a 100644
--- a/lean/OAI/Probability/ThorpShuffle/BlockFibers.lean
+++ b/lean/OAI/Probability/ThorpShuffle/BlockFibers.lean
@@ -1,162 +1,162 @@
-import OAI.Probability.ThorpShuffle.BlockEmbedding
-
-namespace OAI
-
-universe uI uβ uα uK
-
-noncomputable section
-
-open scoped BigOperators ComplexConjugate InnerProductSpace
-open Filter Topology
-
-namespace Thorp
-
-namespace Block
-open scoped Classical
-variable {I : Type uI} {β : Type uβ} {α : Type uα} [Fintype I] [Fintype β] [Fintype α] [DecidableEq α] [DecidableEq β]
-
-def fiberEquiv (e : α ≃ I × β) (i : I) (g : Equiv.Perm α) :
-    Equiv.Perm β ≃ {k : Equiv.Perm α // restriction e i k = restriction e i g} :=
-  Equiv.ofBijective (fun h => ⟨g * embedding e (Pi.mulSingle i h), restriction_right e i g h⟩)
-    ⟨fun h k hh => single_embedding_injective e i (mul_left_cancel (congrArg Subtype.val hh)),
-      fun ⟨k,hk⟩ => by
-        obtain ⟨h, hh⟩ := restriction_fiber e i g k hk
-        exact ⟨h, Subtype.ext hh⟩⟩
-
-lemma push_restriction (e : α ≃ I × β) (i : I) (μ : Equiv.Perm α → ℝ) (g : Equiv.Perm α) :
-    Trim.push μ (restriction e i) (restriction e i g) = Trim.cosetMass (embedding e) μ i g := by
-  calc
-    _ = ∑ k : {k : Equiv.Perm α // restriction e i k = restriction e i g}, μ k.val := by
-      rw [← Finset.sum_subtype (Finset.univ.filter (fun k : Equiv.Perm α => restriction e i k = restriction e i g)) (by simp),
-        Finset.sum_filter]
-      unfold Trim.push
-      apply Finset.sum_congr rfl
-      intro k _
-      split_ifs <;> rfl
-    _ = _ := (Equiv.sum_comp (fiberEquiv e i g) (fun k => μ k.val)).symm
-
-lemma push_uniform_restriction (e : α ≃ I × β) (i : I) (g : Equiv.Perm α) :
-    Trim.push (fun _ : Equiv.Perm α => (Fintype.card (Equiv.Perm α) : ℝ)⁻¹)
-      (restriction e i) (restriction e i g) =
-      (Fintype.card (Equiv.Perm β) : ℝ) / Fintype.card (Equiv.Perm α) := by
-  rw [push_restriction]
-  simp only [Trim.cosetMass, Finset.sum_const, Finset.card_univ, nsmul_eq_mul, div_eq_mul_inv]
-
-variable {K : Type uK} [Fintype K]
-
-def orderedRestriction (e : α ≃ I × β) (i : I) (o : K ≃ complement (β:=β) i) (g : Equiv.Perm α) : K → α :=
-  restriction e i g ∘ o
-
-omit [Fintype I] [Fintype β] [Fintype α] [DecidableEq α] [DecidableEq β] [Fintype K] in
-lemma orderedRestriction_eq_iff (e : α ≃ I × β) (i : I) (o : K ≃ complement (β:=β) i)
-    (g k : Equiv.Perm α) : orderedRestriction e i o g = orderedRestriction e i o k ↔
-      restriction e i g = restriction e i k := by
-  constructor
-  · intro h
-    funext x
-    obtain ⟨k,rfl⟩ := o.surjective x
-    exact congrFun h k
-  · intro h
-    unfold orderedRestriction
-    rw [h]
-
-omit [Fintype K] in
-lemma push_orderedRestriction (e : α ≃ I × β) (i : I) (o : K ≃ complement (β:=β) i)
-    (μ : Equiv.Perm α → ℝ) (g : Equiv.Perm α) :
-    Trim.push μ (orderedRestriction e i o) (orderedRestriction e i o g) =
-      Trim.cosetMass (embedding e) μ i g := by
-  rw [← push_restriction e i μ g]
-  unfold Trim.push
-  apply Finset.sum_congr rfl
-  intro k _
-  by_cases hk : restriction e i k = restriction e i g
-  · have ho := (orderedRestriction_eq_iff e i o k g).mpr hk
-    simp only [ite_eq_left hk, ite_eq_left ho]
-  · have ho := mt (orderedRestriction_eq_iff e i o k g).mp hk
-    simp only [ite_eq_right hk, ite_eq_right ho]
-
-omit [Fintype K] in
-lemma uniform_orderedRestriction (e : α ≃ I × β) (i : I) (o : K ≃ complement (β:=β) i)
-    (g : Equiv.Perm α) :
-    Trim.push (fun _ : Equiv.Perm α => (Fintype.card (Equiv.Perm α) : ℝ)⁻¹)
-      (orderedRestriction e i o) (orderedRestriction e i o g) =
-      (Fintype.card (Equiv.Perm β) : ℝ) / Fintype.card (Equiv.Perm α) := by
-  rw [push_orderedRestriction]
-  simp only [Trim.cosetMass, Finset.sum_const, Finset.card_univ, nsmul_eq_mul, div_eq_mul_inv]
-
-lemma discarded_ordered_bound [DecidableEq K] (e : α ≃ I × β) (o : ∀ i, K ≃ complement (β:=β) i)
-    (μ : Equiv.Perm α → ℝ) (hμ : ∀ g, 0 ≤ μ g) (h1 : ∑ g, μ g = 1) :
-    Trim.discarded (embedding e) μ ≤
-      2 * ∑ i, tv (Trim.push μ (orderedRestriction e i (o i)))
-        (Trim.push (fun _ : Equiv.Perm α => (Fintype.card (Equiv.Perm α) : ℝ)⁻¹) (orderedRestriction e i (o i))) := by
-  exact Trim.discarded_bound (embedding e) μ hμ h1 (fun _ : I => K → α)
-    (fun i => orderedRestriction e i (o i))
-    (fun i g => push_orderedRestriction e i (o i) μ g)
-    (fun i g => uniform_orderedRestriction e i (o i) g)
-
-end Block
-
-lemma law_nonneg (d t : ℕ) (g : State d) : 0 ≤ law d t g := fairMass_nonneg (run d t) g
-
-lemma law_sum (d t : ℕ) : ∑ g, law d t g = 1 := fairMass_sum (run d t)
-
-namespace Block
-open scoped Classical
-
-def eightEquiv (d : ℕ) : Position (d+1+2) ≃ Fin 8 × Position d :=
-  Fintype.equivOfCardEq (by simp only [card_position, Fintype.card_prod, Fintype.card_fin, pow_add]; norm_num; omega)
-
-lemma complement_card (d : ℕ) (i : Fin 8) :
-    Fintype.card (complement (β:=Position d) i) = 7 * 2^d := by
-  rw [card_complement, Fintype.card_fin, card_position]
-
-def ordering (d : ℕ) (i : Fin 8) : Fin (7 * 2^d) ≃ complement (β:=Position d) i :=
-  Fintype.equivOfCardEq (by rw [Fintype.card_fin, complement_card])
-
-def domain (d : ℕ) (i : Fin 8) : Fin (7*2^d) → Position (d+1+2) :=
-  complementIn (eightEquiv d) i ∘ ordering d i
-
-lemma domain_injective (d : ℕ) (i : Fin 8) : Function.Injective (domain d i) :=
-  (complementIn_injective _ _).comp (ordering d i).injective
-
-lemma push_law_ordered (d : ℕ) (i : Fin 8) :
-    Trim.push (law (d+1+2) (200*(d+1+2))) (orderedRestriction (eightEquiv d) i (ordering d i)) =
-      fairMass (fun ω : History (d+1+2) (200*(d+1+2)) => run (d+1+2) (200*(d+1+2)) ω ∘ domain d i) := by
-  simpa only [law, Function.comp_def, orderedRestriction, restriction, domain] using
-    Trim.push_fairMass (run (d+1+2) (200*(d+1+2))) (orderedRestriction (eightEquiv d) i (ordering d i))
-
-lemma push_uniform_ordered (d : ℕ) (i : Fin 8) :
-    Trim.push (fun _ : State (d+1+2) => (Fintype.card (State (d+1+2)) : ℝ)⁻¹)
-      (orderedRestriction (eightEquiv d) i (ordering d i)) =
-      fairMass (fun g : State (d+1+2) => g ∘ domain d i) := by
-  rw [Trim.uniform_eq_fairMass_id]
-  simpa only [Function.comp_def, orderedRestriction, restriction, domain, id_eq] using
-    Trim.push_fairMass (id : State (d+1+2) → State (d+1+2)) (orderedRestriction (eightEquiv d) i (ordering d i))
-
-def discarded (d : ℕ) : ℝ :=
-  Trim.discarded (embedding (eightEquiv d)) (law (d+1+2) (200*(d+1+2)))
-
-lemma discarded_nonneg (d : ℕ) : 0 ≤ discarded d :=
-  Trim.discarded_nonneg _ _ (law_nonneg _ _)
-
-lemma discarded_le (d : ℕ) : discarded d ≤ 16 * Conditional.marginalRate d := by
-  have h := discarded_ordered_bound (eightEquiv d) (ordering d) (law (d+1+2) (200*(d+1+2)))
-    (law_nonneg (d+1+2) (200*(d+1+2))) (law_sum (d+1+2) (200*(d+1+2)))
-  simp only [push_law_ordered, push_uniform_ordered] at h
-  have hb := Finset.sum_le_sum (s:=Finset.univ) (fun i _ => Conditional.marginal_tv_rate d (domain d i) (domain_injective d i))
-  have hc : (∑ _ : Fin 8, Conditional.marginalRate d) = 8 * Conditional.marginalRate d := by simp
-  rw [hc] at hb
-  exact h.trans (by linarith)
-
-lemma discarded_tendsto : Tendsto discarded atTop (nhds 0) :=
-  tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds
-    (by simpa using Conditional.marginalRate_tendsto.const_mul 16)
-    discarded_nonneg discarded_le
-
-end Block
-
-end Thorp
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.BlockEmbedding
+
+namespace OAI
+
+universe uI uβ uα uK
+
+noncomputable section
+
+open scoped BigOperators ComplexConjugate InnerProductSpace
+open Filter Topology
+
+namespace Thorp
+
+namespace Block
+open scoped Classical
+variable {I : Type uI} {β : Type uβ} {α : Type uα} [Fintype I] [Fintype β] [Fintype α] [DecidableEq α] [DecidableEq β]
+
+def fiberEquiv (e : α ≃ I × β) (i : I) (g : Equiv.Perm α) :
+    Equiv.Perm β ≃ {k : Equiv.Perm α // restriction e i k = restriction e i g} :=
+  Equiv.ofBijective (fun h => ⟨g * embedding e (Pi.mulSingle i h), restriction_right e i g h⟩)
+    ⟨fun h k hh => single_embedding_injective e i (mul_left_cancel (congrArg Subtype.val hh)),
+      fun ⟨k,hk⟩ => by
+        obtain ⟨h, hh⟩ := restriction_fiber e i g k hk
+        exact ⟨h, Subtype.ext hh⟩⟩
+
+lemma push_restriction (e : α ≃ I × β) (i : I) (μ : Equiv.Perm α → ℝ) (g : Equiv.Perm α) :
+    Trim.push μ (restriction e i) (restriction e i g) = Trim.cosetMass (embedding e) μ i g := by
+  calc
+    _ = ∑ k : {k : Equiv.Perm α // restriction e i k = restriction e i g}, μ k.val := by
+      rw [← Finset.sum_subtype (Finset.univ.filter (fun k : Equiv.Perm α => restriction e i k = restriction e i g)) (by simp),
+        Finset.sum_filter]
+      unfold Trim.push
+      apply Finset.sum_congr rfl
+      intro k _
+      split_ifs <;> rfl
+    _ = _ := (Equiv.sum_comp (fiberEquiv e i g) (fun k => μ k.val)).symm
+
+lemma push_uniform_restriction (e : α ≃ I × β) (i : I) (g : Equiv.Perm α) :
+    Trim.push (fun _ : Equiv.Perm α => (Fintype.card (Equiv.Perm α) : ℝ)⁻¹)
+      (restriction e i) (restriction e i g) =
+      (Fintype.card (Equiv.Perm β) : ℝ) / Fintype.card (Equiv.Perm α) := by
+  rw [push_restriction]
+  simp only [Trim.cosetMass, Finset.sum_const, Finset.card_univ, nsmul_eq_mul, div_eq_mul_inv]
+
+variable {K : Type uK} [Fintype K]
+
+def orderedRestriction (e : α ≃ I × β) (i : I) (o : K ≃ complement (β:=β) i) (g : Equiv.Perm α) : K → α :=
+  restriction e i g ∘ o
+
+omit [Fintype I] [Fintype β] [Fintype α] [DecidableEq α] [DecidableEq β] [Fintype K] in
+lemma orderedRestriction_eq_iff (e : α ≃ I × β) (i : I) (o : K ≃ complement (β:=β) i)
+    (g k : Equiv.Perm α) : orderedRestriction e i o g = orderedRestriction e i o k ↔
+      restriction e i g = restriction e i k := by
+  constructor
+  · intro h
+    funext x
+    obtain ⟨k,rfl⟩ := o.surjective x
+    exact congrFun h k
+  · intro h
+    unfold orderedRestriction
+    rw [h]
+
+omit [Fintype K] in
+lemma push_orderedRestriction (e : α ≃ I × β) (i : I) (o : K ≃ complement (β:=β) i)
+    (μ : Equiv.Perm α → ℝ) (g : Equiv.Perm α) :
+    Trim.push μ (orderedRestriction e i o) (orderedRestriction e i o g) =
+      Trim.cosetMass (embedding e) μ i g := by
+  rw [← push_restriction e i μ g]
+  unfold Trim.push
+  apply Finset.sum_congr rfl
+  intro k _
+  by_cases hk : restriction e i k = restriction e i g
+  · have ho := (orderedRestriction_eq_iff e i o k g).mpr hk
+    simp only [ite_eq_left hk, ite_eq_left ho]
+  · have ho := mt (orderedRestriction_eq_iff e i o k g).mp hk
+    simp only [ite_eq_right hk, ite_eq_right ho]
+
+omit [Fintype K] in
+lemma uniform_orderedRestriction (e : α ≃ I × β) (i : I) (o : K ≃ complement (β:=β) i)
+    (g : Equiv.Perm α) :
+    Trim.push (fun _ : Equiv.Perm α => (Fintype.card (Equiv.Perm α) : ℝ)⁻¹)
+      (orderedRestriction e i o) (orderedRestriction e i o g) =
+      (Fintype.card (Equiv.Perm β) : ℝ) / Fintype.card (Equiv.Perm α) := by
+  rw [push_orderedRestriction]
+  simp only [Trim.cosetMass, Finset.sum_const, Finset.card_univ, nsmul_eq_mul, div_eq_mul_inv]
+
+lemma discarded_ordered_bound [DecidableEq K] (e : α ≃ I × β) (o : ∀ i, K ≃ complement (β:=β) i)
+    (μ : Equiv.Perm α → ℝ) (hμ : ∀ g, 0 ≤ μ g) (h1 : ∑ g, μ g = 1) :
+    Trim.discarded (embedding e) μ ≤
+      2 * ∑ i, tv (Trim.push μ (orderedRestriction e i (o i)))
+        (Trim.push (fun _ : Equiv.Perm α => (Fintype.card (Equiv.Perm α) : ℝ)⁻¹) (orderedRestriction e i (o i))) := by
+  exact Trim.discarded_bound (embedding e) μ hμ h1 (fun _ : I => K → α)
+    (fun i => orderedRestriction e i (o i))
+    (fun i g => push_orderedRestriction e i (o i) μ g)
+    (fun i g => uniform_orderedRestriction e i (o i) g)
+
+end Block
+
+lemma law_nonneg (d t : ℕ) (g : State d) : 0 ≤ law d t g := fairMass_nonneg (run d t) g
+
+lemma law_sum (d t : ℕ) : ∑ g, law d t g = 1 := fairMass_sum (run d t)
+
+namespace Block
+open scoped Classical
+
+def eightEquiv (d : ℕ) : Position (d+1+2) ≃ Fin 8 × Position d :=
+  Fintype.equivOfCardEq (by simp only [card_position, Fintype.card_prod, Fintype.card_fin, pow_add]; norm_num; omega)
+
+lemma complement_card (d : ℕ) (i : Fin 8) :
+    Fintype.card (complement (β:=Position d) i) = 7 * 2^d := by
+  rw [card_complement, Fintype.card_fin, card_position]
+
+def ordering (d : ℕ) (i : Fin 8) : Fin (7 * 2^d) ≃ complement (β:=Position d) i :=
+  Fintype.equivOfCardEq (by rw [Fintype.card_fin, complement_card])
+
+def domain (d : ℕ) (i : Fin 8) : Fin (7*2^d) → Position (d+1+2) :=
+  complementIn (eightEquiv d) i ∘ ordering d i
+
+lemma domain_injective (d : ℕ) (i : Fin 8) : Function.Injective (domain d i) :=
+  (complementIn_injective _ _).comp (ordering d i).injective
+
+lemma push_law_ordered (d : ℕ) (i : Fin 8) :
+    Trim.push (law (d+1+2) (200*(d+1+2))) (orderedRestriction (eightEquiv d) i (ordering d i)) =
+      fairMass (fun ω : History (d+1+2) (200*(d+1+2)) => run (d+1+2) (200*(d+1+2)) ω ∘ domain d i) := by
+  simpa only [law, Function.comp_def, orderedRestriction, restriction, domain] using
+    Trim.push_fairMass (run (d+1+2) (200*(d+1+2))) (orderedRestriction (eightEquiv d) i (ordering d i))
+
+lemma push_uniform_ordered (d : ℕ) (i : Fin 8) :
+    Trim.push (fun _ : State (d+1+2) => (Fintype.card (State (d+1+2)) : ℝ)⁻¹)
+      (orderedRestriction (eightEquiv d) i (ordering d i)) =
+      fairMass (fun g : State (d+1+2) => g ∘ domain d i) := by
+  rw [Trim.uniform_eq_fairMass_id]
+  simpa only [Function.comp_def, orderedRestriction, restriction, domain, id_eq] using
+    Trim.push_fairMass (id : State (d+1+2) → State (d+1+2)) (orderedRestriction (eightEquiv d) i (ordering d i))
+
+def discarded (d : ℕ) : ℝ :=
+  Trim.discarded (embedding (eightEquiv d)) (law (d+1+2) (200*(d+1+2)))
+
+lemma discarded_nonneg (d : ℕ) : 0 ≤ discarded d :=
+  Trim.discarded_nonneg _ _ (law_nonneg _ _)
+
+lemma discarded_le (d : ℕ) : discarded d ≤ 16 * Conditional.marginalRate d := by
+  have h := discarded_ordered_bound (eightEquiv d) (ordering d) (law (d+1+2) (200*(d+1+2)))
+    (law_nonneg (d+1+2) (200*(d+1+2))) (law_sum (d+1+2) (200*(d+1+2)))
+  simp only [push_law_ordered, push_uniform_ordered] at h
+  have hb := Finset.sum_le_sum (s:=Finset.univ) (fun i _ => Conditional.marginal_tv_rate d (domain d i) (domain_injective d i))
+  have hc : (∑ _ : Fin 8, Conditional.marginalRate d) = 8 * Conditional.marginalRate d := by simp
+  rw [hc] at hb
+  exact h.trans (by linarith)
+
+lemma discarded_tendsto : Tendsto discarded atTop (nhds 0) :=
+  tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds
+    (by simpa using Conditional.marginalRate_tendsto.const_mul 16)
+    discarded_nonneg discarded_le
+
+end Block
+
+end Thorp
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/BlockLaws.lean b/lean/OAI/Probability/ThorpShuffle/BlockLaws.lean
index 0e5a7e61f550368dd216df69cc5200a0616d3996..ef3f6637d2ef21b67a837d5b943e44b6a56bdad2 100644
--- a/lean/OAI/Probability/ThorpShuffle/BlockLaws.lean
+++ b/lean/OAI/Probability/ThorpShuffle/BlockLaws.lean
@@ -1,75 +1,75 @@
-import OAI.Probability.ThorpShuffle.BlockFibers
-
-namespace OAI
-
-noncomputable section
-
-open scoped BigOperators ComplexConjugate InnerProductSpace
-open Filter Topology
-
-namespace Thorp
-open scoped Classical
-
-theorem law_realPower (d t k : ℕ) :
-    law d (k*t) = Fourier.realPower (law d t) k := by
-  induction k with
-  | zero =>
-      funext g
-      simp only [Nat.zero_mul, law, run, List.ofFn_zero, List.reverse_nil, List.prod_nil,
-        fairMass, Fourier.realPower]
-      by_cases hg : g = 1
-      · subst g
-        simp
-      · simp [hg, Ne.symm hg]
-  | succ k ih =>
-      rw [Nat.succ_mul, law_add, ih]
-      rfl
-
-namespace Specht
-open Thorp.Fourier
-variable {α β : Type} [Fintype α] [Fintype β] [DecidableEq α] [DecidableEq β] [Nontrivial α]
-
-theorem trimmed_eight_tv (hn : 16 ≤ Fintype.card α)
-    (ψ : (Fin 8 → Equiv.Perm β) →* Equiv.Perm α) (μ : Equiv.Perm α → ℝ)
-    (hμ : ∀ g, 0 ≤ μ g) (h1 : ∑ g, μ g = 1)
-    (hb : ∑ g, (μ g : ℂ) * complexSign g = 0)
-    (hδ : Trim.discarded ψ μ ≤ 1/2) :
-    tv (realPower μ 8) (fun _ => (Fintype.card (Equiv.Perm α) : ℝ)⁻¹) ≤
-      8 * Trim.discarded ψ μ + (1/2 : ℝ) * Real.sqrt
-        (2 * (2 * Trim.discarded ψ μ)^16 +
-          (32 * (permutationFamily β).reciprocalSum)^8 * exceptionalReciprocalSum (Fintype.card α)) := by
-  let ν := Trim.normalized ψ μ
-  have hd : Trim.discarded ψ μ < 1 := by linarith
-  have hν : ∀ g, 0 ≤ ν g := Trim.normalized_nonneg ψ μ hμ hd
-  have hν1 : ∑ g, ν g = 1 := Trim.normalized_sum ψ μ h1 hd
-  have hcoset : ∀ i g, ∑ h, ν (g * ψ (Pi.mulSingle i h)) ≤
-      4 * (Fintype.card (Equiv.Perm β) : ℝ) / Fintype.card (Equiv.Perm α) := by
-    intro i g
-    convert Trim.normalized_coset ψ μ hμ hδ i g using 1
-    apply Finset.sum_congr rfl
-    intro h _
-    congr 4
-    exact Subsingleton.elim _ _
-  have hpar := Trim.normalized_bias ψ μ hμ h1 hd complexSign
-    (fun g => (complexSign_norm g).le) hb
-  have hfour := permutation_eight_bound hn ψ ν 4 hν hν1 (by norm_num) hcoset
-  have hpow := pow_le_pow_left₀ (norm_nonneg _) hpar 16
-  have hsq : (∑ g, |realPower ν 8 g - (Fintype.card (Equiv.Perm α) : ℝ)⁻¹|)^2 ≤
-      2 * (2 * Trim.discarded ψ μ)^16 +
-        (32 * (permutationFamily β).reciprocalSum)^8 * exceptionalReciprocalSum (Fintype.card α) := by
-    norm_num only [show (8:ℝ)*4 = 32 by norm_num] at hfour
-    exact hfour.trans (by gcongr)
-  have hnorm := Real.le_sqrt_of_sq_le hsq
-  calc
-    _ ≤ tv (realPower μ 8) (realPower ν 8) +
-        tv (realPower ν 8) (fun _ => (Fintype.card (Equiv.Perm α) : ℝ)⁻¹) := tv_triangle _ _ _
-    _ ≤ 8 * Trim.discarded ψ μ + _ := add_le_add
-      (by simpa only [ν, Nat.cast_ofNat, Trim.normalized_tv ψ μ hμ h1 hd] using tv_realPower μ ν hμ h1 hν hν1 8)
-      (mul_le_mul_of_nonneg_left hnorm (by norm_num : (0:ℝ) ≤ 1/2))
-
-end Specht
-end Thorp
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.BlockFibers
+
+namespace OAI
+
+noncomputable section
+
+open scoped BigOperators ComplexConjugate InnerProductSpace
+open Filter Topology
+
+namespace Thorp
+open scoped Classical
+
+theorem law_realPower (d t k : ℕ) :
+    law d (k*t) = Fourier.realPower (law d t) k := by
+  induction k with
+  | zero =>
+      funext g
+      simp only [Nat.zero_mul, law, run, List.ofFn_zero, List.reverse_nil, List.prod_nil,
+        fairMass, Fourier.realPower]
+      by_cases hg : g = 1
+      · subst g
+        simp
+      · simp [hg, Ne.symm hg]
+  | succ k ih =>
+      rw [Nat.succ_mul, law_add, ih]
+      rfl
+
+namespace Specht
+open Thorp.Fourier
+variable {α β : Type} [Fintype α] [Fintype β] [DecidableEq α] [DecidableEq β] [Nontrivial α]
+
+theorem trimmed_eight_tv (hn : 16 ≤ Fintype.card α)
+    (ψ : (Fin 8 → Equiv.Perm β) →* Equiv.Perm α) (μ : Equiv.Perm α → ℝ)
+    (hμ : ∀ g, 0 ≤ μ g) (h1 : ∑ g, μ g = 1)
+    (hb : ∑ g, (μ g : ℂ) * complexSign g = 0)
+    (hδ : Trim.discarded ψ μ ≤ 1/2) :
+    tv (realPower μ 8) (fun _ => (Fintype.card (Equiv.Perm α) : ℝ)⁻¹) ≤
+      8 * Trim.discarded ψ μ + (1/2 : ℝ) * Real.sqrt
+        (2 * (2 * Trim.discarded ψ μ)^16 +
+          (32 * (permutationFamily β).reciprocalSum)^8 * exceptionalReciprocalSum (Fintype.card α)) := by
+  let ν := Trim.normalized ψ μ
+  have hd : Trim.discarded ψ μ < 1 := by linarith
+  have hν : ∀ g, 0 ≤ ν g := Trim.normalized_nonneg ψ μ hμ hd
+  have hν1 : ∑ g, ν g = 1 := Trim.normalized_sum ψ μ h1 hd
+  have hcoset : ∀ i g, ∑ h, ν (g * ψ (Pi.mulSingle i h)) ≤
+      4 * (Fintype.card (Equiv.Perm β) : ℝ) / Fintype.card (Equiv.Perm α) := by
+    intro i g
+    convert Trim.normalized_coset ψ μ hμ hδ i g using 1
+    apply Finset.sum_congr rfl
+    intro h _
+    congr 4
+    exact Subsingleton.elim _ _
+  have hpar := Trim.normalized_bias ψ μ hμ h1 hd complexSign
+    (fun g => (complexSign_norm g).le) hb
+  have hfour := permutation_eight_bound hn ψ ν 4 hν hν1 (by norm_num) hcoset
+  have hpow := pow_le_pow_left₀ (norm_nonneg _) hpar 16
+  have hsq : (∑ g, |realPower ν 8 g - (Fintype.card (Equiv.Perm α) : ℝ)⁻¹|)^2 ≤
+      2 * (2 * Trim.discarded ψ μ)^16 +
+        (32 * (permutationFamily β).reciprocalSum)^8 * exceptionalReciprocalSum (Fintype.card α) := by
+    norm_num only [show (8:ℝ)*4 = 32 by norm_num] at hfour
+    exact hfour.trans (by gcongr)
+  have hnorm := Real.le_sqrt_of_sq_le hsq
+  calc
+    _ ≤ tv (realPower μ 8) (realPower ν 8) +
+        tv (realPower ν 8) (fun _ => (Fintype.card (Equiv.Perm α) : ℝ)⁻¹) := tv_triangle _ _ _
+    _ ≤ 8 * Trim.discarded ψ μ + _ := add_le_add
+      (by simpa only [ν, Nat.cast_ofNat, Trim.normalized_tv ψ μ hμ h1 hd] using tv_realPower μ ν hμ h1 hν hν1 8)
+      (mul_le_mul_of_nonneg_left hnorm (by norm_num : (0:ℝ) ≤ 1/2))
+
+end Specht
+end Thorp
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/Classification.lean b/lean/OAI/Probability/ThorpShuffle/Classification.lean
index b0d3725f35ee4704b9d82ac96ad417511be156c1..4e8e01e91dfeaba7c64e8d8007cc4b820e7b73dd 100644
--- a/lean/OAI/Probability/ThorpShuffle/Classification.lean
+++ b/lean/OAI/Probability/ThorpShuffle/Classification.lean
@@ -1,242 +1,242 @@
-import OAI.Probability.ThorpShuffle.Irreducibility
-
-namespace OAI
-
-universe uG uI uV uα
-
-noncomputable section
-
-open scoped BigOperators ComplexConjugate InnerProductSpace
-open Filter
-
-namespace Thorp.Young
-open scoped Classical
-
-def rowCellsEquiv (D : YoungDiagram) : Cells D ≃ (i : Fin (D.colLen 0)) × Fin (D.rowLen i) where
-  toFun x := ⟨⟨x.val.1, YoungDiagram.mem_iff_lt_colLen.mp (D.up_left_mem le_rfl (Nat.zero_le _) x.property)⟩,
-    ⟨x.val.2, YoungDiagram.mem_iff_lt_rowLen.mp x.property⟩⟩
-  invFun x := ⟨(x.1, x.2), YoungDiagram.mem_iff_lt_rowLen.mpr x.2.isLt⟩
-  left_inv x := rfl
-  right_inv x := by cases x; rfl
-
-lemma card_cells_eq_sum (D : YoungDiagram) : Fintype.card (Cells D) = D.rowLens.sum := by
-  rw [Fintype.card_congr (rowCellsEquiv D), Fintype.card_sigma]
-  simp only [Fintype.card_fin, YoungDiagram.rowLens]
-  rw [← List.sum_toFinset _ List.nodup_range, List.toFinset_range, Finset.sum_range]
-
-def ofPartition {n : ℕ} (p : Nat.Partition n) : YoungDiagram :=
-  YoungDiagram.ofRowLens (p.parts.sort (· ≥ ·)) (Multiset.pairwise_sort _ _).sortedGE
-
-lemma rowLens_ofPartition {n : ℕ} (p : Nat.Partition n) :
-    (ofPartition p).rowLens = p.parts.sort (· ≥ ·) := by
-  apply YoungDiagram.rowLens_ofRowLens_eq_self
-  intro x hx
-  exact p.parts_pos ((Multiset.mem_sort _).mp hx)
-
-lemma card_ofPartition {n : ℕ} (p : Nat.Partition n) : Fintype.card (Cells (ofPartition p)) = n := by
-  rw [card_cells_eq_sum, rowLens_ofPartition]
-  simpa only [← Multiset.sum_coe, Multiset.sort_eq] using p.parts_sum
-
-lemma ofPartition_injective {n : ℕ} : Function.Injective (ofPartition (n := n)) := by
-  intro p q hpq
-  apply Nat.Partition.ext
-  have hh := congrArg YoungDiagram.rowLens hpq
-  rw [rowLens_ofPartition, rowLens_ofPartition] at hh
-  simpa only [Multiset.sort_eq] using congrArg (fun x : List ℕ => (x : Multiset ℕ)) hh
-
-def toPartition (D : YoungDiagram) (n : ℕ) (hn : Fintype.card (Cells D) = n) : Nat.Partition n where
-  parts := D.rowLens
-  parts_pos := fun hx => D.pos_of_mem_rowLens _ hx
-  parts_sum := by simpa only [Multiset.sum_coe, ← card_cells_eq_sum] using hn
-
-lemma of_toPartition (D : YoungDiagram) (n : ℕ) (hn : Fintype.card (Cells D) = n) :
-    ofPartition (toPartition D n hn) = D := by
-  have hs : (D.rowLens : Multiset ℕ).sort (· ≥ ·) = D.rowLens := by
-    rw [Multiset.coe_sort]
-    exact List.mergeSort_eq_self _ D.rowLens_sorted.pairwise
-  unfold ofPartition toPartition
-  simp only [hs]
-  exact YoungDiagram.ofRowLens_to_rowLens_eq_self
-
-end Thorp.Young
-
-namespace Thorp.Fourier
-open scoped Classical
-
-variable {G : Type uG} [Group G] [Fintype G]
-
-def charPair (f h : G → ℂ) : ℂ := (Nat.card G : ℂ)⁻¹ * ∑ g, f g * h g⁻¹
-
-lemma charPair_sum {I : Type uI} (s : Finset I) (a : I → ℂ) (f : I → G → ℂ) (h : G → ℂ) :
-    charPair (∑ i ∈ s, a i • f i) h = ∑ i ∈ s, a i * charPair (f i) h := by
-  simp only [charPair, Finset.sum_apply, Pi.smul_apply, smul_eq_mul, Finset.sum_mul]
-  rw [Finset.sum_comm, Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro i _
-  simp_rw [mul_assoc (a i)]
-  rw [← Finset.mul_sum]
-  ring
-
-lemma charPair_zero (h : G → ℂ) : charPair (0 : G → ℂ) h = 0 := by simp [charPair]
-
-lemma irreducible_char_independent {I : Type uI} {V : I → Type uV}
-    [∀ i, AddCommGroup (V i)] [∀ i, Module ℂ (V i)] [∀ i, FiniteDimensional ℂ (V i)]
-    (ρ : ∀ i, Representation ℂ G (V i)) [∀ i, Representation.IsIrreducible (ρ i)]
-    (hne : ∀ i j, Nonempty (Representation.Equiv (ρ i) (ρ j)) → i = j) :
-    LinearIndependent ℂ (fun i => (ρ i).character) := by
-  let : Invertible (Nat.card G : ℂ) := invertibleOfNonzero (NeZero.ne _)
-  have ho (i j : I) : charPair (ρ i).character (ρ j).character = if i = j then 1 else 0 := by
-    rw [charPair, Representation.char_orthonormal]
-    congr 1
-    apply propext
-    constructor
-    · exact fun h => (hne j i h).symm
-    · intro h; subst j; exact ⟨Representation.Equiv.refl _⟩
-  rw [linearIndependent_iff']
-  intro s a ha i hi
-  have hh := congrArg (fun f => charPair f (ρ i).character) ha
-  rw [charPair_sum, charPair_zero] at hh
-  simp only [ho] at hh
-  simpa only [mul_ite, mul_one, mul_zero, Finset.sum_ite_eq', ite_eq_left hi] using hh
-
-end Thorp.Fourier
-
-namespace Thorp.Specht
-open scoped Classical
-variable {α : Type uα} [Fintype α]
-
-abbrev Shape (α : Type uα) [Fintype α] := Nat.Partition (Fintype.card α)
-
-def canonicalTableau (p : Shape α) : α ≃ Thorp.Young.Cells (Thorp.Young.ofPartition p) := by
-  have card_cells_eq_sum (D : YoungDiagram) : Fintype.card (Thorp.Young.Cells D) = D.rowLens.sum := by
-    rw [Fintype.card_congr (Thorp.Young.rowCellsEquiv D), Fintype.card_sigma]
-    simp only [Fintype.card_fin, YoungDiagram.rowLens]
-    rw [← List.sum_toFinset _ List.nodup_range, List.toFinset_range, Finset.sum_range]
-  have rowLens_ofPartition {n : ℕ} (p : Nat.Partition n) :
-      (Thorp.Young.ofPartition p).rowLens = p.parts.sort (· ≥ ·) := by
-    apply YoungDiagram.rowLens_ofRowLens_eq_self
-    intro x hx
-    exact p.parts_pos ((Multiset.mem_sort _).mp hx)
-  have card_ofPartition {n : ℕ} (p : Nat.Partition n) : Fintype.card (Thorp.Young.Cells (Thorp.Young.ofPartition p)) = n := by
-    rw [card_cells_eq_sum, rowLens_ofPartition]
-    simpa only [← Multiset.sum_coe, Multiset.sort_eq] using p.parts_sum
-  exact Fintype.equivOfCardEq (card_ofPartition p).symm
-
-abbrev Space (p : Shape α) := (module (Thorp.Young.ofPartition p) (canonicalTableau p)).toSubmodule
-
-def family (p : Shape α) : Representation ℂ (Equiv.Perm α) (Space p) :=
-  (module (Thorp.Young.ofPartition p) (canonicalTableau p)).toRepresentation
-
-instance family_irreducible (p : Shape α) : Representation.IsIrreducible (family p) :=
-  irreducible (Thorp.Young.ofPartition p) (canonicalTableau p)
-
-lemma family_inequiv (p q : Shape α) (he : Nonempty (Representation.Equiv (family p) (family q))) : p = q := by
-  obtain ⟨e⟩ := he
-  apply Thorp.Young.ofPartition_injective
-  exact shape_eq_of_equiv _ _ _ _ e
-
-lemma characters_independent : LinearIndependent ℂ (fun p : Shape α => (family p).character) :=
-  Thorp.Fourier.irreducible_char_independent family family_inequiv
-
-def classLift (f : Equiv.Perm α → ℂ) (p : Shape α) : ℂ :=
-  if h : ∃ g : Equiv.Perm α, g.partition = p then f h.choose else 0
-
-lemma classLift_apply (f : Equiv.Perm α → ℂ)
-    (hf : ∀ g h, IsConj g h → f g = f h) (g : Equiv.Perm α) :
-    classLift f g.partition = f g := by
-  unfold classLift
-  have hh : ∃ a : Equiv.Perm α, a.partition = g.partition := ⟨g, rfl⟩
-  rw [dite_eq_left hh]
-  exact hf _ g (Equiv.Perm.partition_eq_of_isConj.mpr hh.choose_spec)
-
-lemma character_isClass (p : Shape α) (g h : Equiv.Perm α) (hgh : IsConj g h) :
-    (family p).character g = (family p).character h := by
-  obtain ⟨x, hx⟩ := isConj_iff.mp hgh
-  rw [← hx]
-  exact (Representation.char_conj (family p) g x).symm
-
-lemma liftedCharacters_independent :
-    LinearIndependent ℂ (fun p : Shape α => classLift (family p).character) := by
-  let E : (Shape α → ℂ) →ₗ[ℂ] (Equiv.Perm α → ℂ) := {
-    toFun := fun f g => f g.partition
-    map_add' := by intro f h; rfl
-    map_smul' := by intro a f; rfl }
-  apply LinearIndependent.of_comp E
-  convert characters_independent (α := α) using 1
-  funext p g
-  exact classLift_apply _ (character_isClass p) g
-
-def characterBasis : Module.Basis (Shape α) ℂ (Shape α → ℂ) :=
-  basisOfPiSpaceOfLinearIndependent liftedCharacters_independent
-
-lemma characterBasis_apply (p : Shape α) : characterBasis p = classLift (family p).character := by
-  exact congrFun (coe_basisOfPiSpaceOfLinearIndependent liftedCharacters_independent) p
-
-end Thorp.Specht
-
-namespace Thorp.Specht
-open scoped Classical
-variable {α : Type uα} [Fintype α]
-
-lemma class_expansion (f : Equiv.Perm α → ℂ)
-    (hf : ∀ g h, IsConj g h → f g = f h) :
-    (∑ p : Shape α, (characterBasis.repr (classLift f)) p • (family p).character) = f := by
-  funext g
-  have he := congrFun (characterBasis.sum_repr (classLift f)) g.partition
-  simp only [Finset.sum_apply, Pi.smul_apply, smul_eq_mul, characterBasis_apply,
-    classLift_apply _ (character_isClass _), classLift_apply _ hf] at he
-  simpa only [Finset.sum_apply, Pi.smul_apply, smul_eq_mul] using he
-
-lemma char_orthogonal (p q : Shape α) :
-    Thorp.Fourier.charPair (family p).character (family q).character = if p = q then 1 else 0 := by
-  let : Invertible (Nat.card (Equiv.Perm α) : ℂ) := invertibleOfNonzero (NeZero.ne _)
-  rw [Thorp.Fourier.charPair, Representation.char_orthonormal]
-  congr 1
-  apply propext
-  exact ⟨fun h => (family_inequiv q p h).symm,
-    fun h => by subst q; exact ⟨Representation.Equiv.refl _⟩⟩
-
-lemma class_coeff (f : Equiv.Perm α → ℂ)
-    (hf : ∀ g h, IsConj g h → f g = f h) (q : Shape α) :
-    characterBasis.repr (classLift f) q = Thorp.Fourier.charPair f (family q).character := by
-  have he := congrArg (fun x => Thorp.Fourier.charPair x (family q).character) (class_expansion f hf)
-  rw [Thorp.Fourier.charPair_sum] at he
-  simp only [char_orthogonal, mul_ite, mul_one, mul_zero, Finset.sum_ite_eq', Finset.mem_univ,
-    ↓reduceIte] at he
-  exact he
-
-def degree (p : Shape α) : ℕ := Module.finrank ℂ (Space p)
-
-theorem regular_char (g : Equiv.Perm α) :
-    ∑ p : Shape α, (degree p : ℂ) * (family p).character g =
-      if g = 1 then (Nat.card (Equiv.Perm α) : ℂ) else 0 := by
-  let δ : Equiv.Perm α → ℂ := fun h => if h = 1 then Nat.card (Equiv.Perm α) else 0
-  have hδ : ∀ a b, IsConj a b → δ a = δ b := by
-    intro a b hab
-    have hh : a = 1 ↔ b = 1 := by
-      obtain ⟨c, hc⟩ := isConj_iff.mp hab
-      rw [← hc]
-      constructor
-      · intro ha; simp [ha]
-      · intro hb
-        have hh := congrArg (fun z => c⁻¹ * z * c) hb
-        simpa only [mul_assoc, inv_mul_cancel_left, inv_mul_cancel_right, inv_mul_cancel, one_mul, mul_one] using hh
-    simp only [δ, hh]
-  have hc (p : Shape α) : Thorp.Fourier.charPair δ (family p).character = degree p := by
-    rw [Thorp.Fourier.charPair]
-    simp only [δ, ite_mul, zero_mul, Finset.sum_ite_eq', Finset.mem_univ, ↓reduceIte, inv_one,
-      Representation.char_one]
-    rw [← mul_assoc, inv_mul_cancel₀ (NeZero.ne _), one_mul]
-    rfl
-  have he := congrFun (class_expansion δ hδ) g
-  simpa only [Finset.sum_apply, Pi.smul_apply, smul_eq_mul, class_coeff δ hδ, hc] using he
-
-theorem degree_sq_sum : ∑ p : Shape α, (degree p : ℂ) ^ 2 = Nat.card (Equiv.Perm α) := by
-  have hh := regular_char (α := α) 1
-  simpa only [Representation.char_one, degree, pow_two, ↓reduceIte] using hh
-
-end Thorp.Specht
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.Irreducibility
+
+namespace OAI
+
+universe uG uI uV uα
+
+noncomputable section
+
+open scoped BigOperators ComplexConjugate InnerProductSpace
+open Filter
+
+namespace Thorp.Young
+open scoped Classical
+
+def rowCellsEquiv (D : YoungDiagram) : Cells D ≃ (i : Fin (D.colLen 0)) × Fin (D.rowLen i) where
+  toFun x := ⟨⟨x.val.1, YoungDiagram.mem_iff_lt_colLen.mp (D.up_left_mem le_rfl (Nat.zero_le _) x.property)⟩,
+    ⟨x.val.2, YoungDiagram.mem_iff_lt_rowLen.mp x.property⟩⟩
+  invFun x := ⟨(x.1, x.2), YoungDiagram.mem_iff_lt_rowLen.mpr x.2.isLt⟩
+  left_inv x := rfl
+  right_inv x := by cases x; rfl
+
+lemma card_cells_eq_sum (D : YoungDiagram) : Fintype.card (Cells D) = D.rowLens.sum := by
+  rw [Fintype.card_congr (rowCellsEquiv D), Fintype.card_sigma]
+  simp only [Fintype.card_fin, YoungDiagram.rowLens]
+  rw [← List.sum_toFinset _ List.nodup_range, List.toFinset_range, Finset.sum_range]
+
+def ofPartition {n : ℕ} (p : Nat.Partition n) : YoungDiagram :=
+  YoungDiagram.ofRowLens (p.parts.sort (· ≥ ·)) (Multiset.pairwise_sort _ _).sortedGE
+
+lemma rowLens_ofPartition {n : ℕ} (p : Nat.Partition n) :
+    (ofPartition p).rowLens = p.parts.sort (· ≥ ·) := by
+  apply YoungDiagram.rowLens_ofRowLens_eq_self
+  intro x hx
+  exact p.parts_pos ((Multiset.mem_sort _).mp hx)
+
+lemma card_ofPartition {n : ℕ} (p : Nat.Partition n) : Fintype.card (Cells (ofPartition p)) = n := by
+  rw [card_cells_eq_sum, rowLens_ofPartition]
+  simpa only [← Multiset.sum_coe, Multiset.sort_eq] using p.parts_sum
+
+lemma ofPartition_injective {n : ℕ} : Function.Injective (ofPartition (n := n)) := by
+  intro p q hpq
+  apply Nat.Partition.ext
+  have hh := congrArg YoungDiagram.rowLens hpq
+  rw [rowLens_ofPartition, rowLens_ofPartition] at hh
+  simpa only [Multiset.sort_eq] using congrArg (fun x : List ℕ => (x : Multiset ℕ)) hh
+
+def toPartition (D : YoungDiagram) (n : ℕ) (hn : Fintype.card (Cells D) = n) : Nat.Partition n where
+  parts := D.rowLens
+  parts_pos := fun hx => D.pos_of_mem_rowLens _ hx
+  parts_sum := by simpa only [Multiset.sum_coe, ← card_cells_eq_sum] using hn
+
+lemma of_toPartition (D : YoungDiagram) (n : ℕ) (hn : Fintype.card (Cells D) = n) :
+    ofPartition (toPartition D n hn) = D := by
+  have hs : (D.rowLens : Multiset ℕ).sort (· ≥ ·) = D.rowLens := by
+    rw [Multiset.coe_sort]
+    exact List.mergeSort_eq_self _ D.rowLens_sorted.pairwise
+  unfold ofPartition toPartition
+  simp only [hs]
+  exact YoungDiagram.ofRowLens_to_rowLens_eq_self
+
+end Thorp.Young
+
+namespace Thorp.Fourier
+open scoped Classical
+
+variable {G : Type uG} [Group G] [Fintype G]
+
+def charPair (f h : G → ℂ) : ℂ := (Nat.card G : ℂ)⁻¹ * ∑ g, f g * h g⁻¹
+
+lemma charPair_sum {I : Type uI} (s : Finset I) (a : I → ℂ) (f : I → G → ℂ) (h : G → ℂ) :
+    charPair (∑ i ∈ s, a i • f i) h = ∑ i ∈ s, a i * charPair (f i) h := by
+  simp only [charPair, Finset.sum_apply, Pi.smul_apply, smul_eq_mul, Finset.sum_mul]
+  rw [Finset.sum_comm, Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro i _
+  simp_rw [mul_assoc (a i)]
+  rw [← Finset.mul_sum]
+  ring
+
+lemma charPair_zero (h : G → ℂ) : charPair (0 : G → ℂ) h = 0 := by simp [charPair]
+
+lemma irreducible_char_independent {I : Type uI} {V : I → Type uV}
+    [∀ i, AddCommGroup (V i)] [∀ i, Module ℂ (V i)] [∀ i, FiniteDimensional ℂ (V i)]
+    (ρ : ∀ i, Representation ℂ G (V i)) [∀ i, Representation.IsIrreducible (ρ i)]
+    (hne : ∀ i j, Nonempty (Representation.Equiv (ρ i) (ρ j)) → i = j) :
+    LinearIndependent ℂ (fun i => (ρ i).character) := by
+  let : Invertible (Nat.card G : ℂ) := invertibleOfNonzero (NeZero.ne _)
+  have ho (i j : I) : charPair (ρ i).character (ρ j).character = if i = j then 1 else 0 := by
+    rw [charPair, Representation.char_orthonormal]
+    congr 1
+    apply propext
+    constructor
+    · exact fun h => (hne j i h).symm
+    · intro h; subst j; exact ⟨Representation.Equiv.refl _⟩
+  rw [linearIndependent_iff']
+  intro s a ha i hi
+  have hh := congrArg (fun f => charPair f (ρ i).character) ha
+  rw [charPair_sum, charPair_zero] at hh
+  simp only [ho] at hh
+  simpa only [mul_ite, mul_one, mul_zero, Finset.sum_ite_eq', ite_eq_left hi] using hh
+
+end Thorp.Fourier
+
+namespace Thorp.Specht
+open scoped Classical
+variable {α : Type uα} [Fintype α]
+
+abbrev Shape (α : Type uα) [Fintype α] := Nat.Partition (Fintype.card α)
+
+def canonicalTableau (p : Shape α) : α ≃ Thorp.Young.Cells (Thorp.Young.ofPartition p) := by
+  have card_cells_eq_sum (D : YoungDiagram) : Fintype.card (Thorp.Young.Cells D) = D.rowLens.sum := by
+    rw [Fintype.card_congr (Thorp.Young.rowCellsEquiv D), Fintype.card_sigma]
+    simp only [Fintype.card_fin, YoungDiagram.rowLens]
+    rw [← List.sum_toFinset _ List.nodup_range, List.toFinset_range, Finset.sum_range]
+  have rowLens_ofPartition {n : ℕ} (p : Nat.Partition n) :
+      (Thorp.Young.ofPartition p).rowLens = p.parts.sort (· ≥ ·) := by
+    apply YoungDiagram.rowLens_ofRowLens_eq_self
+    intro x hx
+    exact p.parts_pos ((Multiset.mem_sort _).mp hx)
+  have card_ofPartition {n : ℕ} (p : Nat.Partition n) : Fintype.card (Thorp.Young.Cells (Thorp.Young.ofPartition p)) = n := by
+    rw [card_cells_eq_sum, rowLens_ofPartition]
+    simpa only [← Multiset.sum_coe, Multiset.sort_eq] using p.parts_sum
+  exact Fintype.equivOfCardEq (card_ofPartition p).symm
+
+abbrev Space (p : Shape α) := (module (Thorp.Young.ofPartition p) (canonicalTableau p)).toSubmodule
+
+def family (p : Shape α) : Representation ℂ (Equiv.Perm α) (Space p) :=
+  (module (Thorp.Young.ofPartition p) (canonicalTableau p)).toRepresentation
+
+instance family_irreducible (p : Shape α) : Representation.IsIrreducible (family p) :=
+  irreducible (Thorp.Young.ofPartition p) (canonicalTableau p)
+
+lemma family_inequiv (p q : Shape α) (he : Nonempty (Representation.Equiv (family p) (family q))) : p = q := by
+  obtain ⟨e⟩ := he
+  apply Thorp.Young.ofPartition_injective
+  exact shape_eq_of_equiv _ _ _ _ e
+
+lemma characters_independent : LinearIndependent ℂ (fun p : Shape α => (family p).character) :=
+  Thorp.Fourier.irreducible_char_independent family family_inequiv
+
+def classLift (f : Equiv.Perm α → ℂ) (p : Shape α) : ℂ :=
+  if h : ∃ g : Equiv.Perm α, g.partition = p then f h.choose else 0
+
+lemma classLift_apply (f : Equiv.Perm α → ℂ)
+    (hf : ∀ g h, IsConj g h → f g = f h) (g : Equiv.Perm α) :
+    classLift f g.partition = f g := by
+  unfold classLift
+  have hh : ∃ a : Equiv.Perm α, a.partition = g.partition := ⟨g, rfl⟩
+  rw [dite_eq_left hh]
+  exact hf _ g (Equiv.Perm.partition_eq_of_isConj.mpr hh.choose_spec)
+
+lemma character_isClass (p : Shape α) (g h : Equiv.Perm α) (hgh : IsConj g h) :
+    (family p).character g = (family p).character h := by
+  obtain ⟨x, hx⟩ := isConj_iff.mp hgh
+  rw [← hx]
+  exact (Representation.char_conj (family p) g x).symm
+
+lemma liftedCharacters_independent :
+    LinearIndependent ℂ (fun p : Shape α => classLift (family p).character) := by
+  let E : (Shape α → ℂ) →ₗ[ℂ] (Equiv.Perm α → ℂ) := {
+    toFun := fun f g => f g.partition
+    map_add' := by intro f h; rfl
+    map_smul' := by intro a f; rfl }
+  apply LinearIndependent.of_comp E
+  convert characters_independent (α := α) using 1
+  funext p g
+  exact classLift_apply _ (character_isClass p) g
+
+def characterBasis : Module.Basis (Shape α) ℂ (Shape α → ℂ) :=
+  basisOfPiSpaceOfLinearIndependent liftedCharacters_independent
+
+lemma characterBasis_apply (p : Shape α) : characterBasis p = classLift (family p).character := by
+  exact congrFun (coe_basisOfPiSpaceOfLinearIndependent liftedCharacters_independent) p
+
+end Thorp.Specht
+
+namespace Thorp.Specht
+open scoped Classical
+variable {α : Type uα} [Fintype α]
+
+lemma class_expansion (f : Equiv.Perm α → ℂ)
+    (hf : ∀ g h, IsConj g h → f g = f h) :
+    (∑ p : Shape α, (characterBasis.repr (classLift f)) p • (family p).character) = f := by
+  funext g
+  have he := congrFun (characterBasis.sum_repr (classLift f)) g.partition
+  simp only [Finset.sum_apply, Pi.smul_apply, smul_eq_mul, characterBasis_apply,
+    classLift_apply _ (character_isClass _), classLift_apply _ hf] at he
+  simpa only [Finset.sum_apply, Pi.smul_apply, smul_eq_mul] using he
+
+lemma char_orthogonal (p q : Shape α) :
+    Thorp.Fourier.charPair (family p).character (family q).character = if p = q then 1 else 0 := by
+  let : Invertible (Nat.card (Equiv.Perm α) : ℂ) := invertibleOfNonzero (NeZero.ne _)
+  rw [Thorp.Fourier.charPair, Representation.char_orthonormal]
+  congr 1
+  apply propext
+  exact ⟨fun h => (family_inequiv q p h).symm,
+    fun h => by subst q; exact ⟨Representation.Equiv.refl _⟩⟩
+
+lemma class_coeff (f : Equiv.Perm α → ℂ)
+    (hf : ∀ g h, IsConj g h → f g = f h) (q : Shape α) :
+    characterBasis.repr (classLift f) q = Thorp.Fourier.charPair f (family q).character := by
+  have he := congrArg (fun x => Thorp.Fourier.charPair x (family q).character) (class_expansion f hf)
+  rw [Thorp.Fourier.charPair_sum] at he
+  simp only [char_orthogonal, mul_ite, mul_one, mul_zero, Finset.sum_ite_eq', Finset.mem_univ,
+    ↓reduceIte] at he
+  exact he
+
+def degree (p : Shape α) : ℕ := Module.finrank ℂ (Space p)
+
+theorem regular_char (g : Equiv.Perm α) :
+    ∑ p : Shape α, (degree p : ℂ) * (family p).character g =
+      if g = 1 then (Nat.card (Equiv.Perm α) : ℂ) else 0 := by
+  let δ : Equiv.Perm α → ℂ := fun h => if h = 1 then Nat.card (Equiv.Perm α) else 0
+  have hδ : ∀ a b, IsConj a b → δ a = δ b := by
+    intro a b hab
+    have hh : a = 1 ↔ b = 1 := by
+      obtain ⟨c, hc⟩ := isConj_iff.mp hab
+      rw [← hc]
+      constructor
+      · intro ha; simp [ha]
+      · intro hb
+        have hh := congrArg (fun z => c⁻¹ * z * c) hb
+        simpa only [mul_assoc, inv_mul_cancel_left, inv_mul_cancel_right, inv_mul_cancel, one_mul, mul_one] using hh
+    simp only [δ, hh]
+  have hc (p : Shape α) : Thorp.Fourier.charPair δ (family p).character = degree p := by
+    rw [Thorp.Fourier.charPair]
+    simp only [δ, ite_mul, zero_mul, Finset.sum_ite_eq', Finset.mem_univ, ↓reduceIte, inv_one,
+      Representation.char_one]
+    rw [← mul_assoc, inv_mul_cancel₀ (NeZero.ne _), one_mul]
+    rfl
+  have he := congrFun (class_expansion δ hδ) g
+  simpa only [Finset.sum_apply, Pi.smul_apply, smul_eq_mul, class_coeff δ hδ, hc] using he
+
+theorem degree_sq_sum : ∑ p : Shape α, (degree p : ℂ) ^ 2 = Nat.card (Equiv.Perm α) := by
+  have hh := regular_char (α := α) 1
+  simpa only [Representation.char_one, degree, pow_two, ↓reduceIte] using hh
+
+end Thorp.Specht
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/ConditionalFlow.lean b/lean/OAI/Probability/ThorpShuffle/ConditionalFlow.lean
index 62dfc45e17bdcbc1d743d03f0a54af819ca26892..8016ad6cf1da7ca8854e6504c78c028d353efcbe 100644
--- a/lean/OAI/Probability/ThorpShuffle/ConditionalFlow.lean
+++ b/lean/OAI/Probability/ThorpShuffle/ConditionalFlow.lean
@@ -1,192 +1,192 @@
-import OAI.Probability.ThorpShuffle.SpectralDecay
-
-namespace OAI
-
-universe uα
-
-noncomputable section
-
-open scoped BigOperators
-open Filter
-
-namespace Thorp
-
-namespace Conditional
-
-def splitPosition (d : ℕ) : Position (d + 1) ≃ Bool × Position d :=
-  (Fin.consEquiv (fun _ : Fin (d + 1) => Bool)).symm
-
-def scatterPosition (d : ℕ) : Bool × Position d ≃ Position (d + 1) :=
-  (splitPosition d).symm.trans (rotate (d + 1))
-
-def physicalFlow (d : ℕ) (B : Position (d + 1) → Bool)
-    (w : Position (d + 1) → ℝ) (c : Coins (d + 1)) (x : Position (d + 1)) : ℝ :=
-  flow (B ∘ (splitPosition d).symm) (w ∘ (splitPosition d).symm) c
-    ((scatterPosition d).symm x)
-
-def physicalFree (d : ℕ) (B : Position (d + 1) → Bool)
-    (c : Coins (d + 1)) (x : Position (d + 1)) : Bool :=
-  B ((step (d + 1) c).symm x)
-
-theorem step_symm_scatter (d : ℕ) (c : Coins (d + 1)) (p : Bool × Position d) :
-    (step (d + 1) c).symm (scatterPosition d p) =
-      (splitPosition d).symm (pairMap c p) := by
-  simp [step, scatterPosition, splitPosition, pairSwitch, pairMap]
-
-@[simp] theorem physicalFlow_scatter (d : ℕ) (B : Position (d + 1) → Bool)
-    (w : Position (d + 1) → ℝ) (c : Coins (d + 1)) (p : Bool × Position d) :
-    physicalFlow d B w c (scatterPosition d p) =
-      flow (B ∘ (splitPosition d).symm) (w ∘ (splitPosition d).symm) c p := by
-  simp [physicalFlow]
-
-@[simp] theorem physicalFree_scatter (d : ℕ) (B : Position (d + 1) → Bool)
-    (c : Coins (d + 1)) (p : Bool × Position d) :
-    physicalFree d B c (scatterPosition d p) =
-      B ((splitPosition d).symm (pairMap c p)) := by
-  rw [physicalFree, step_symm_scatter]
-
-theorem physicalFlow_sum (d : ℕ) (B : Position (d + 1) → Bool)
-    (w : Position (d + 1) → ℝ) (c : Coins (d + 1)) :
-    (∑ x, physicalFlow d B w c x) = ∑ x, w x := by
-  calc
-    _ = ∑ p, physicalFlow d B w c (scatterPosition d p) :=
-      (Equiv.sum_comp (scatterPosition d) (physicalFlow d B w c)).symm
-    _ = ∑ p, w ((splitPosition d).symm p) := by
-      simp only [physicalFlow_scatter]
-      exact flow_sum (B ∘ (splitPosition d).symm)
-        (w ∘ (splitPosition d).symm) c
-    _ = _ := Equiv.sum_comp (splitPosition d).symm w
-
-theorem physicalFlow_energy_le (d : ℕ) (B : Position (d + 1) → Bool)
-    (w : Position (d + 1) → ℝ) (c : Coins (d + 1)) :
-    (∑ x, physicalFlow d B w c x ^ 2) ≤ ∑ x, w x ^ 2 := by
-  calc
-    _ = ∑ p, physicalFlow d B w c (scatterPosition d p) ^ 2 :=
-      (Equiv.sum_comp (scatterPosition d) (fun x => physicalFlow d B w c x ^ 2)).symm
-    _ ≤ ∑ p, w ((splitPosition d).symm p) ^ 2 := by
-      simp only [physicalFlow_scatter]
-      exact flow_energy_le (B ∘ (splitPosition d).symm)
-        (w ∘ (splitPosition d).symm) c
-    _ = _ := Equiv.sum_comp (splitPosition d).symm (fun x => w x ^ 2)
-
-theorem physicalFlow_support (d : ℕ) (B : Position (d + 1) → Bool)
-    (w : Position (d + 1) → ℝ) (hw : ∀ x, B x = false → w x = 0)
-    (c : Coins (d + 1)) (x : Position (d + 1))
-    (hx : physicalFree d B c x = false) : physicalFlow d B w c x = 0 := by
-  obtain ⟨p, rfl⟩ := (scatterPosition d).surjective x
-  rw [physicalFree_scatter] at hx
-  rw [physicalFlow_scatter]
-  exact flow_support _ _ (fun p => hw ((splitPosition d).symm p)) c p hx
-
-theorem physicalFlow_uniform (d : ℕ) (B : Position (d + 1) → Bool)
-    (r : ℝ) (c : Coins (d + 1)) (x : Position (d + 1)) :
-    physicalFlow d B (fun y => if B y then r else 0) c x =
-      if physicalFree d B c x then r else 0 := by
-  obtain ⟨p, rfl⟩ := (scatterPosition d).surjective x
-  rw [physicalFlow_scatter, physicalFree_scatter]
-  exact flow_uniform _ r c p
-
-def freeCount {α : Type uα} [Fintype α] (B : α → Bool) : ℕ :=
-  (Finset.univ.filter (fun x => B x = true)).card
-
-def centeredPoint {α : Type uα} [Fintype α] [DecidableEq α]
-    (B : α → Bool) (tag : α) (x : α) : ℝ :=
-  (if x = tag then 1 else 0) - (1 / (freeCount B : ℝ)) * (if B x then 1 else 0)
-
-theorem freeCount_pos {α : Type uα} [Fintype α] (B : α → Bool) (tag : α)
-    (ht : B tag = true) : 0 < freeCount B := by
-  apply Finset.card_pos.mpr
-  exact ⟨tag, Finset.mem_filter.mpr ⟨Finset.mem_univ _, ht⟩⟩
-
-theorem sum_free_indicator {α : Type uα} [Fintype α] (B : α → Bool) :
-    (∑ x : α, if B x then (1 : ℝ) else 0) = freeCount B := by
-  exact Finset.sum_boole _ _
-
-theorem centeredPoint_sum {α : Type uα} [Fintype α] [DecidableEq α]
-    (B : α → Bool) (tag : α) (ht : B tag = true) :
-    (∑ x, centeredPoint B tag x) = 0 := by
-  have hm : (freeCount B : ℝ) ≠ 0 := by exact_mod_cast (freeCount_pos B tag ht).ne'
-  unfold centeredPoint
-  rw [Finset.sum_sub_distrib, ← Finset.mul_sum, sum_free_indicator]
-  simp [hm]
-
-theorem centeredPoint_support {α : Type uα} [Fintype α] [DecidableEq α]
-    (B : α → Bool) (tag : α) (ht : B tag = true)
-    (x : α) (hx : B x = false) : centeredPoint B tag x = 0 := by
-  have hxt : x ≠ tag := by intro h; subst x; simp_all
-  simp [centeredPoint, hx, hxt]
-
-theorem centeredPoint_energy {α : Type uα} [Fintype α] [DecidableEq α]
-    (B : α → Bool) (tag : α) (ht : B tag = true) :
-    (∑ x, centeredPoint B tag x ^ 2) = 1 - 1 / (freeCount B : ℝ) := by
-  have hm : (freeCount B : ℝ) ≠ 0 := by exact_mod_cast (freeCount_pos B tag ht).ne'
-  have hpoint (x : α) : centeredPoint B tag x ^ 2 =
-      (if x = tag then (1 - 2 / (freeCount B : ℝ)) else 0) +
-        (1 / (freeCount B : ℝ)) ^ 2 * (if B x then 1 else 0) := by
-    by_cases hx : x = tag
-    · subst x
-      simp [centeredPoint, ht]
-      ring
-    · cases hB : B x <;> simp [centeredPoint, hx, hB]
-  simp_rw [hpoint]
-  rw [Finset.sum_add_distrib, ← Finset.mul_sum, sum_free_indicator]
-  simp only [Finset.sum_ite_eq', Finset.mem_univ, ite_true]
-  field_simp
-  ring
-
-structure CenteredState (d : ℕ) where
-  free : Position d → Bool
-  weight : Position d → ℝ
-  mass_zero : ∑ x, weight x = 0
-  supported : ∀ x, free x = false → weight x = 0
-  energy_le_one : ∑ x, weight x ^ 2 ≤ 1
-
-def initialState (d : ℕ) (B : Position d → Bool) (tag : Position d)
-    (ht : B tag = true) : CenteredState d where
-  free := B
-  weight := centeredPoint B tag
-  mass_zero := centeredPoint_sum B tag ht
-  supported := centeredPoint_support B tag ht
-  energy_le_one := by
-    rw [centeredPoint_energy B tag ht]
-    exact sub_le_self _ (by positivity)
-
-def nextState (d : ℕ) (v : CenteredState (d + 1)) (c : Coins (d + 1)) :
-    CenteredState (d + 1) where
-  free := physicalFree d v.free c
-  weight := physicalFlow d v.free v.weight c
-  mass_zero := by rw [physicalFlow_sum]; exact v.mass_zero
-  supported := physicalFlow_support d v.free v.weight v.supported c
-  energy_le_one := (physicalFlow_energy_le d v.free v.weight c).trans v.energy_le_one
-
-def iterateState (d : ℕ) (v : CenteredState (d + 1)) :
-    (t : ℕ) → History (d + 1) t → CenteredState (d + 1)
-  | 0, _ => v
-  | t + 1, ω => nextState d (iterateState d v t (fun i => ω i.castSucc)) (ω (Fin.last t))
-
-def expectedEnergy (d : ℕ) (v : CenteredState (d + 1)) (t : ℕ) : ℝ :=
-  mean (fun ω : History (d + 1) t => ∑ x, (iterateState d v t ω).weight x ^ 2)
-
-theorem expectedEnergy_nonneg (d : ℕ) (v : CenteredState (d + 1)) (t : ℕ) :
-    0 ≤ expectedEnergy d v t := by
-  unfold expectedEnergy mean
-  positivity
-
-theorem expectedEnergy_le_one (d : ℕ) (v : CenteredState (d + 1)) (t : ℕ) :
-    expectedEnergy d v t ≤ 1 := by
-  unfold expectedEnergy mean
-  have hc : (0 : ℝ) < Fintype.card (History (d + 1) t) := by
-    exact_mod_cast Fintype.card_pos
-  apply (div_le_iff₀ hc).mpr
-  calc
-    _ ≤ ∑ _ω : History (d + 1) t, (1 : ℝ) :=
-      Finset.sum_le_sum (fun ω _ => (iterateState d v t ω).energy_le_one)
-    _ = _ := by simp
-
-end Conditional
-
-end Thorp
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.SpectralDecay
+
+namespace OAI
+
+universe uα
+
+noncomputable section
+
+open scoped BigOperators
+open Filter
+
+namespace Thorp
+
+namespace Conditional
+
+def splitPosition (d : ℕ) : Position (d + 1) ≃ Bool × Position d :=
+  (Fin.consEquiv (fun _ : Fin (d + 1) => Bool)).symm
+
+def scatterPosition (d : ℕ) : Bool × Position d ≃ Position (d + 1) :=
+  (splitPosition d).symm.trans (rotate (d + 1))
+
+def physicalFlow (d : ℕ) (B : Position (d + 1) → Bool)
+    (w : Position (d + 1) → ℝ) (c : Coins (d + 1)) (x : Position (d + 1)) : ℝ :=
+  flow (B ∘ (splitPosition d).symm) (w ∘ (splitPosition d).symm) c
+    ((scatterPosition d).symm x)
+
+def physicalFree (d : ℕ) (B : Position (d + 1) → Bool)
+    (c : Coins (d + 1)) (x : Position (d + 1)) : Bool :=
+  B ((step (d + 1) c).symm x)
+
+theorem step_symm_scatter (d : ℕ) (c : Coins (d + 1)) (p : Bool × Position d) :
+    (step (d + 1) c).symm (scatterPosition d p) =
+      (splitPosition d).symm (pairMap c p) := by
+  simp [step, scatterPosition, splitPosition, pairSwitch, pairMap]
+
+@[simp] theorem physicalFlow_scatter (d : ℕ) (B : Position (d + 1) → Bool)
+    (w : Position (d + 1) → ℝ) (c : Coins (d + 1)) (p : Bool × Position d) :
+    physicalFlow d B w c (scatterPosition d p) =
+      flow (B ∘ (splitPosition d).symm) (w ∘ (splitPosition d).symm) c p := by
+  simp [physicalFlow]
+
+@[simp] theorem physicalFree_scatter (d : ℕ) (B : Position (d + 1) → Bool)
+    (c : Coins (d + 1)) (p : Bool × Position d) :
+    physicalFree d B c (scatterPosition d p) =
+      B ((splitPosition d).symm (pairMap c p)) := by
+  rw [physicalFree, step_symm_scatter]
+
+theorem physicalFlow_sum (d : ℕ) (B : Position (d + 1) → Bool)
+    (w : Position (d + 1) → ℝ) (c : Coins (d + 1)) :
+    (∑ x, physicalFlow d B w c x) = ∑ x, w x := by
+  calc
+    _ = ∑ p, physicalFlow d B w c (scatterPosition d p) :=
+      (Equiv.sum_comp (scatterPosition d) (physicalFlow d B w c)).symm
+    _ = ∑ p, w ((splitPosition d).symm p) := by
+      simp only [physicalFlow_scatter]
+      exact flow_sum (B ∘ (splitPosition d).symm)
+        (w ∘ (splitPosition d).symm) c
+    _ = _ := Equiv.sum_comp (splitPosition d).symm w
+
+theorem physicalFlow_energy_le (d : ℕ) (B : Position (d + 1) → Bool)
+    (w : Position (d + 1) → ℝ) (c : Coins (d + 1)) :
+    (∑ x, physicalFlow d B w c x ^ 2) ≤ ∑ x, w x ^ 2 := by
+  calc
+    _ = ∑ p, physicalFlow d B w c (scatterPosition d p) ^ 2 :=
+      (Equiv.sum_comp (scatterPosition d) (fun x => physicalFlow d B w c x ^ 2)).symm
+    _ ≤ ∑ p, w ((splitPosition d).symm p) ^ 2 := by
+      simp only [physicalFlow_scatter]
+      exact flow_energy_le (B ∘ (splitPosition d).symm)
+        (w ∘ (splitPosition d).symm) c
+    _ = _ := Equiv.sum_comp (splitPosition d).symm (fun x => w x ^ 2)
+
+theorem physicalFlow_support (d : ℕ) (B : Position (d + 1) → Bool)
+    (w : Position (d + 1) → ℝ) (hw : ∀ x, B x = false → w x = 0)
+    (c : Coins (d + 1)) (x : Position (d + 1))
+    (hx : physicalFree d B c x = false) : physicalFlow d B w c x = 0 := by
+  obtain ⟨p, rfl⟩ := (scatterPosition d).surjective x
+  rw [physicalFree_scatter] at hx
+  rw [physicalFlow_scatter]
+  exact flow_support _ _ (fun p => hw ((splitPosition d).symm p)) c p hx
+
+theorem physicalFlow_uniform (d : ℕ) (B : Position (d + 1) → Bool)
+    (r : ℝ) (c : Coins (d + 1)) (x : Position (d + 1)) :
+    physicalFlow d B (fun y => if B y then r else 0) c x =
+      if physicalFree d B c x then r else 0 := by
+  obtain ⟨p, rfl⟩ := (scatterPosition d).surjective x
+  rw [physicalFlow_scatter, physicalFree_scatter]
+  exact flow_uniform _ r c p
+
+def freeCount {α : Type uα} [Fintype α] (B : α → Bool) : ℕ :=
+  (Finset.univ.filter (fun x => B x = true)).card
+
+def centeredPoint {α : Type uα} [Fintype α] [DecidableEq α]
+    (B : α → Bool) (tag : α) (x : α) : ℝ :=
+  (if x = tag then 1 else 0) - (1 / (freeCount B : ℝ)) * (if B x then 1 else 0)
+
+theorem freeCount_pos {α : Type uα} [Fintype α] (B : α → Bool) (tag : α)
+    (ht : B tag = true) : 0 < freeCount B := by
+  apply Finset.card_pos.mpr
+  exact ⟨tag, Finset.mem_filter.mpr ⟨Finset.mem_univ _, ht⟩⟩
+
+theorem sum_free_indicator {α : Type uα} [Fintype α] (B : α → Bool) :
+    (∑ x : α, if B x then (1 : ℝ) else 0) = freeCount B := by
+  exact Finset.sum_boole _ _
+
+theorem centeredPoint_sum {α : Type uα} [Fintype α] [DecidableEq α]
+    (B : α → Bool) (tag : α) (ht : B tag = true) :
+    (∑ x, centeredPoint B tag x) = 0 := by
+  have hm : (freeCount B : ℝ) ≠ 0 := by exact_mod_cast (freeCount_pos B tag ht).ne'
+  unfold centeredPoint
+  rw [Finset.sum_sub_distrib, ← Finset.mul_sum, sum_free_indicator]
+  simp [hm]
+
+theorem centeredPoint_support {α : Type uα} [Fintype α] [DecidableEq α]
+    (B : α → Bool) (tag : α) (ht : B tag = true)
+    (x : α) (hx : B x = false) : centeredPoint B tag x = 0 := by
+  have hxt : x ≠ tag := by intro h; subst x; simp_all
+  simp [centeredPoint, hx, hxt]
+
+theorem centeredPoint_energy {α : Type uα} [Fintype α] [DecidableEq α]
+    (B : α → Bool) (tag : α) (ht : B tag = true) :
+    (∑ x, centeredPoint B tag x ^ 2) = 1 - 1 / (freeCount B : ℝ) := by
+  have hm : (freeCount B : ℝ) ≠ 0 := by exact_mod_cast (freeCount_pos B tag ht).ne'
+  have hpoint (x : α) : centeredPoint B tag x ^ 2 =
+      (if x = tag then (1 - 2 / (freeCount B : ℝ)) else 0) +
+        (1 / (freeCount B : ℝ)) ^ 2 * (if B x then 1 else 0) := by
+    by_cases hx : x = tag
+    · subst x
+      simp [centeredPoint, ht]
+      ring
+    · cases hB : B x <;> simp [centeredPoint, hx, hB]
+  simp_rw [hpoint]
+  rw [Finset.sum_add_distrib, ← Finset.mul_sum, sum_free_indicator]
+  simp only [Finset.sum_ite_eq', Finset.mem_univ, ite_true]
+  field_simp
+  ring
+
+structure CenteredState (d : ℕ) where
+  free : Position d → Bool
+  weight : Position d → ℝ
+  mass_zero : ∑ x, weight x = 0
+  supported : ∀ x, free x = false → weight x = 0
+  energy_le_one : ∑ x, weight x ^ 2 ≤ 1
+
+def initialState (d : ℕ) (B : Position d → Bool) (tag : Position d)
+    (ht : B tag = true) : CenteredState d where
+  free := B
+  weight := centeredPoint B tag
+  mass_zero := centeredPoint_sum B tag ht
+  supported := centeredPoint_support B tag ht
+  energy_le_one := by
+    rw [centeredPoint_energy B tag ht]
+    exact sub_le_self _ (by positivity)
+
+def nextState (d : ℕ) (v : CenteredState (d + 1)) (c : Coins (d + 1)) :
+    CenteredState (d + 1) where
+  free := physicalFree d v.free c
+  weight := physicalFlow d v.free v.weight c
+  mass_zero := by rw [physicalFlow_sum]; exact v.mass_zero
+  supported := physicalFlow_support d v.free v.weight v.supported c
+  energy_le_one := (physicalFlow_energy_le d v.free v.weight c).trans v.energy_le_one
+
+def iterateState (d : ℕ) (v : CenteredState (d + 1)) :
+    (t : ℕ) → History (d + 1) t → CenteredState (d + 1)
+  | 0, _ => v
+  | t + 1, ω => nextState d (iterateState d v t (fun i => ω i.castSucc)) (ω (Fin.last t))
+
+def expectedEnergy (d : ℕ) (v : CenteredState (d + 1)) (t : ℕ) : ℝ :=
+  mean (fun ω : History (d + 1) t => ∑ x, (iterateState d v t ω).weight x ^ 2)
+
+theorem expectedEnergy_nonneg (d : ℕ) (v : CenteredState (d + 1)) (t : ℕ) :
+    0 ≤ expectedEnergy d v t := by
+  unfold expectedEnergy mean
+  positivity
+
+theorem expectedEnergy_le_one (d : ℕ) (v : CenteredState (d + 1)) (t : ℕ) :
+    expectedEnergy d v t ≤ 1 := by
+  unfold expectedEnergy mean
+  have hc : (0 : ℝ) < Fintype.card (History (d + 1) t) := by
+    exact_mod_cast Fintype.card_pos
+  apply (div_le_iff₀ hc).mpr
+  calc
+    _ ≤ ∑ _ω : History (d + 1) t, (1 : ℝ) :=
+      Finset.sum_le_sum (fun ω _ => (iterateState d v t ω).energy_le_one)
+    _ = _ := by simp
+
+end Conditional
+
+end Thorp
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/ConditionalLaw.lean b/lean/OAI/Probability/ThorpShuffle/ConditionalLaw.lean
index 9e743044ee6cccf7ebf8b9c0958eab6df550993b..38d640c5c722d0ea8a20ad383707a542f50a37a2 100644
--- a/lean/OAI/Probability/ThorpShuffle/ConditionalLaw.lean
+++ b/lean/OAI/Probability/ThorpShuffle/ConditionalLaw.lean
@@ -1,113 +1,113 @@
-import OAI.Probability.ThorpShuffle.MaskedResampling
-
-namespace OAI
-
-universe uι uα
-
-noncomputable section
-
-open scoped BigOperators
-open Filter
-
-namespace Thorp
-
-namespace Conditional
-
-@[simp] theorem run_snoc (d t : ℕ) (ω : History d t) (c : Coins d) :
-    run d (t + 1) (Fin.snoc ω c) = step d c * run d t ω := by
-  rw [run_succ]
-  simp
-
-theorem flow_joint_identity {ι : Type uι} (d : ℕ) (v : RawState (d + 1))
-    (e : ι → Position (d + 1)) (he : ∀ i, v.free (e i) = false)
-    (t : ℕ) (f : (ι → Position (d + 1)) → Position (d + 1) → ℝ) :
-    mean (fun ω : History (d + 1) t =>
-      ∑ x, v.weight x * f ((run (d + 1) t ω) ∘ e) (run (d + 1) t ω x)) =
-    mean (fun ω : History (d + 1) t =>
-      ∑ y, (rawIterate d v t ω).weight y * f ((run (d + 1) t ω) ∘ e) y) := by
-  induction t generalizing f with
-  | zero => simp only [run_zero, rawIterate]; rfl
-  | succ t ih =>
-    let F (e' : ι → Position (d + 1)) (x : Position (d + 1)) : ℝ :=
-      mean (fun c : Coins (d + 1) => f ((step (d + 1) c) ∘ e') (step (d + 1) c x))
-    have hleft :
-        mean (fun ω : History (d + 1) (t + 1) =>
-          ∑ x, v.weight x * f ((run (d + 1) (t + 1) ω) ∘ e) (run (d + 1) (t + 1) ω x)) =
-        mean (fun ω : History (d + 1) t =>
-          ∑ x, v.weight x * F ((run (d + 1) t ω) ∘ e) (run (d + 1) t ω x)) := by
-      rw [mean_history_succ]
-      apply mean_congr
-      intro ω
-      simp only [run_snoc]
-      rw [mean_sum]
-      simp_rw [mean_const_mul]
-      rfl
-    have hright :
-        mean (fun ω : History (d + 1) (t + 1) =>
-          ∑ y, (rawIterate d v (t + 1) ω).weight y * f ((run (d + 1) (t + 1) ω) ∘ e) y) =
-        mean (fun ω : History (d + 1) t =>
-          ∑ y, (rawIterate d v t ω).weight y * F ((run (d + 1) t ω) ∘ e) y) := by
-      rw [mean_history_succ]
-      apply mean_congr
-      intro ω
-      simp only [run_snoc, rawIterate_snoc]
-      change mean (fun c : Position d → Bool =>
-        ∑ y, physicalFlow d (rawIterate d v t ω).free (rawIterate d v t ω).weight c y *
-          f ((step (d + 1) c) ∘ ((run (d + 1) t ω) ∘ e)) y) = _
-      rw [physicalFlow_joint d (rawIterate d v t ω).free (rawIterate d v t ω).weight
-        ((run (d + 1) t ω) ∘ e) (by
-          intro i
-          simp only [Function.comp_apply, rawIterate_free, Equiv.symm_apply_apply]
-          exact he i)]
-      rw [mean_sum]
-      simp_rw [mean_const_mul]
-      rfl
-    exact hleft.trans ((ih F).trans hright.symm)
-
-theorem centered_joint_identity {ι : Type uι} (d : ℕ) (v : CenteredState (d + 1))
-    (e : ι → Position (d + 1)) (he : ∀ i, v.free (e i) = false)
-    (t : ℕ) (f : (ι → Position (d + 1)) → Position (d + 1) → ℝ) :
-    mean (fun ω : History (d + 1) t =>
-      ∑ x, v.weight x * f ((run (d + 1) t ω) ∘ e) (run (d + 1) t ω x)) =
-    mean (fun ω : History (d + 1) t =>
-      ∑ y, (iterateState d v t ω).weight y * f ((run (d + 1) t ω) ∘ e) y) := by
-  have hh := flow_joint_identity d v.raw e he t f
-  simp_rw [← iterateState_raw] at hh
-  exact hh
-
-theorem centeredPoint_test {α : Type uα} [Fintype α] [DecidableEq α]
-    (B : α → Bool) (tag : α) (f : α → ℝ) :
-    (∑ x, centeredPoint B tag x * f x) =
-      f tag - (1 / (freeCount B : ℝ)) * (∑ x, if B x then f x else 0) := by
-  simp only [centeredPoint, sub_mul, mul_assoc, Finset.sum_sub_distrib,
-    ← Finset.mul_sum, ite_mul, one_mul, zero_mul]
-  simp
-
-theorem tagged_deviation_identity {ι : Type uι} (d : ℕ)
-    (B : Position (d + 1) → Bool) (tag : Position (d + 1)) (ht : B tag = true)
-    (e : ι → Position (d + 1)) (he : ∀ i, B (e i) = false)
-    (t : ℕ) (f : (ι → Position (d + 1)) → Position (d + 1) → ℝ) :
-    mean (fun ω : History (d + 1) t =>
-      f ((run (d + 1) t ω) ∘ e) (run (d + 1) t ω tag) -
-        (1 / (freeCount B : ℝ)) *
-          (∑ y, if B ((run (d + 1) t ω).symm y) then f ((run (d + 1) t ω) ∘ e) y else 0)) =
-    mean (fun ω : History (d + 1) t =>
-      ∑ y, (iterateState d (initialState (d + 1) B tag ht) t ω).weight y *
-        f ((run (d + 1) t ω) ∘ e) y) := by
-  rw [← centered_joint_identity d (initialState (d + 1) B tag ht) e he t f]
-  apply mean_congr
-  intro ω
-  change _ = ∑ x, centeredPoint B tag x * f ((run (d + 1) t ω) ∘ e) (run (d + 1) t ω x)
-  rw [centeredPoint_test]
-  congr 2
-  simpa only [Equiv.symm_apply_apply] using
-    (Equiv.sum_comp (run (d + 1) t ω)
-      (fun y => if B ((run (d + 1) t ω).symm y) then f ((run (d + 1) t ω) ∘ e) y else 0)).symm
-
-end Conditional
-
-end Thorp
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.MaskedResampling
+
+namespace OAI
+
+universe uι uα
+
+noncomputable section
+
+open scoped BigOperators
+open Filter
+
+namespace Thorp
+
+namespace Conditional
+
+@[simp] theorem run_snoc (d t : ℕ) (ω : History d t) (c : Coins d) :
+    run d (t + 1) (Fin.snoc ω c) = step d c * run d t ω := by
+  rw [run_succ]
+  simp
+
+theorem flow_joint_identity {ι : Type uι} (d : ℕ) (v : RawState (d + 1))
+    (e : ι → Position (d + 1)) (he : ∀ i, v.free (e i) = false)
+    (t : ℕ) (f : (ι → Position (d + 1)) → Position (d + 1) → ℝ) :
+    mean (fun ω : History (d + 1) t =>
+      ∑ x, v.weight x * f ((run (d + 1) t ω) ∘ e) (run (d + 1) t ω x)) =
+    mean (fun ω : History (d + 1) t =>
+      ∑ y, (rawIterate d v t ω).weight y * f ((run (d + 1) t ω) ∘ e) y) := by
+  induction t generalizing f with
+  | zero => simp only [run_zero, rawIterate]; rfl
+  | succ t ih =>
+    let F (e' : ι → Position (d + 1)) (x : Position (d + 1)) : ℝ :=
+      mean (fun c : Coins (d + 1) => f ((step (d + 1) c) ∘ e') (step (d + 1) c x))
+    have hleft :
+        mean (fun ω : History (d + 1) (t + 1) =>
+          ∑ x, v.weight x * f ((run (d + 1) (t + 1) ω) ∘ e) (run (d + 1) (t + 1) ω x)) =
+        mean (fun ω : History (d + 1) t =>
+          ∑ x, v.weight x * F ((run (d + 1) t ω) ∘ e) (run (d + 1) t ω x)) := by
+      rw [mean_history_succ]
+      apply mean_congr
+      intro ω
+      simp only [run_snoc]
+      rw [mean_sum]
+      simp_rw [mean_const_mul]
+      rfl
+    have hright :
+        mean (fun ω : History (d + 1) (t + 1) =>
+          ∑ y, (rawIterate d v (t + 1) ω).weight y * f ((run (d + 1) (t + 1) ω) ∘ e) y) =
+        mean (fun ω : History (d + 1) t =>
+          ∑ y, (rawIterate d v t ω).weight y * F ((run (d + 1) t ω) ∘ e) y) := by
+      rw [mean_history_succ]
+      apply mean_congr
+      intro ω
+      simp only [run_snoc, rawIterate_snoc]
+      change mean (fun c : Position d → Bool =>
+        ∑ y, physicalFlow d (rawIterate d v t ω).free (rawIterate d v t ω).weight c y *
+          f ((step (d + 1) c) ∘ ((run (d + 1) t ω) ∘ e)) y) = _
+      rw [physicalFlow_joint d (rawIterate d v t ω).free (rawIterate d v t ω).weight
+        ((run (d + 1) t ω) ∘ e) (by
+          intro i
+          simp only [Function.comp_apply, rawIterate_free, Equiv.symm_apply_apply]
+          exact he i)]
+      rw [mean_sum]
+      simp_rw [mean_const_mul]
+      rfl
+    exact hleft.trans ((ih F).trans hright.symm)
+
+theorem centered_joint_identity {ι : Type uι} (d : ℕ) (v : CenteredState (d + 1))
+    (e : ι → Position (d + 1)) (he : ∀ i, v.free (e i) = false)
+    (t : ℕ) (f : (ι → Position (d + 1)) → Position (d + 1) → ℝ) :
+    mean (fun ω : History (d + 1) t =>
+      ∑ x, v.weight x * f ((run (d + 1) t ω) ∘ e) (run (d + 1) t ω x)) =
+    mean (fun ω : History (d + 1) t =>
+      ∑ y, (iterateState d v t ω).weight y * f ((run (d + 1) t ω) ∘ e) y) := by
+  have hh := flow_joint_identity d v.raw e he t f
+  simp_rw [← iterateState_raw] at hh
+  exact hh
+
+theorem centeredPoint_test {α : Type uα} [Fintype α] [DecidableEq α]
+    (B : α → Bool) (tag : α) (f : α → ℝ) :
+    (∑ x, centeredPoint B tag x * f x) =
+      f tag - (1 / (freeCount B : ℝ)) * (∑ x, if B x then f x else 0) := by
+  simp only [centeredPoint, sub_mul, mul_assoc, Finset.sum_sub_distrib,
+    ← Finset.mul_sum, ite_mul, one_mul, zero_mul]
+  simp
+
+theorem tagged_deviation_identity {ι : Type uι} (d : ℕ)
+    (B : Position (d + 1) → Bool) (tag : Position (d + 1)) (ht : B tag = true)
+    (e : ι → Position (d + 1)) (he : ∀ i, B (e i) = false)
+    (t : ℕ) (f : (ι → Position (d + 1)) → Position (d + 1) → ℝ) :
+    mean (fun ω : History (d + 1) t =>
+      f ((run (d + 1) t ω) ∘ e) (run (d + 1) t ω tag) -
+        (1 / (freeCount B : ℝ)) *
+          (∑ y, if B ((run (d + 1) t ω).symm y) then f ((run (d + 1) t ω) ∘ e) y else 0)) =
+    mean (fun ω : History (d + 1) t =>
+      ∑ y, (iterateState d (initialState (d + 1) B tag ht) t ω).weight y *
+        f ((run (d + 1) t ω) ∘ e) y) := by
+  rw [← centered_joint_identity d (initialState (d + 1) B tag ht) e he t f]
+  apply mean_congr
+  intro ω
+  change _ = ∑ x, centeredPoint B tag x * f ((run (d + 1) t ω) ∘ e) (run (d + 1) t ω x)
+  rw [centeredPoint_test]
+  congr 2
+  simpa only [Equiv.symm_apply_apply] using
+    (Equiv.sum_comp (run (d + 1) t ω)
+      (fun y => if B ((run (d + 1) t ω).symm y) then f ((run (d + 1) t ω) ∘ e) y else 0)).symm
+
+end Conditional
+
+end Thorp
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/ConvolutionBounds.lean b/lean/OAI/Probability/ThorpShuffle/ConvolutionBounds.lean
index a29ad0108a6295301284aec6ce96347c3b27ed62..ba54a0ab38537338702a533a4eed33499438410d 100644
--- a/lean/OAI/Probability/ThorpShuffle/ConvolutionBounds.lean
+++ b/lean/OAI/Probability/ThorpShuffle/ConvolutionBounds.lean
@@ -1,154 +1,154 @@
-import OAI.Probability.ThorpShuffle.PermutationCharacters
-
-namespace OAI
-
-universe uE
-
-noncomputable section
-
-open scoped BigOperators ComplexConjugate InnerProductSpace
-open Filter Topology
-
-namespace Thorp.Fourier
-open scoped Classical
-variable {G H : Type} [Group G] [Fintype G] [Group H] [Fintype H]
-variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℂ E] [FiniteDimensional ℂ E]
-
-lemma hsSq_power_le (A : Module.End ℂ E) (R : ℝ) (hR : 0 ≤ R)
-    (hA : ∀ v, ‖A v‖ ^ 2 ≤ R * ‖v‖ ^ 2) (k : ℕ) :
-    hsSq (A ^ k) ≤ (Module.finrank ℂ E : ℝ) * R ^ k := by
-  unfold hsSq
-  calc
-    _ ≤ ∑ i, R ^ k * ‖stdOrthonormalBasis ℂ E i‖ ^ 2 :=
-      Finset.sum_le_sum (fun i _ => power_norm_sq A R hR hA k _)
-    _ = _ := by simp [(stdOrthonormalBasis ℂ E).norm_eq_one, mul_comm]
-
-lemma Family.eight_transform_bound (F : Family H) (ρ : Representation ℂ G E)
-    [Representation.IsIrreducible ρ]
-    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ)
-    (hd : 0 < Module.finrank ℂ E) (ψ : (Fin 8 → H) →* G) (ν : G → ℝ) (B : ℝ)
-    (hν : ∀ g, 0 ≤ ν g) (h1 : ∑ g, ν g = 1) (hB : 0 ≤ B)
-    (hcoset : ∀ i g, ∑ h, ν (g * ψ (Pi.mulSingle i h)) ≤
-      B * (Fintype.card H : ℝ) / Fintype.card G) :
-    (Module.finrank ℂ E : ℝ) * hsSq (integrated ρ (fun g => (realPower ν 8 g : ℂ))) ≤
-      (8 * B * F.reciprocalSum)^8 * ((Module.finrank ℂ E : ℝ)^2)⁻¹ := by
-  let S : ℝ := ∑ b ∈ Finset.univ.filter (fun b => F.degree b ^ 8 ≤ Module.finrank ℂ E), (F.degree b : ℝ)^2
-  let R : ℝ := 8 * B * S / Module.finrank ℂ E
-  have hD : (0 : ℝ) < Module.finrank ℂ E := by exact_mod_cast hd
-  have hR : 0 ≤ R := by dsimp [R, S]; positivity
-  have hA (v : E) : ‖integrated ρ (fun g => (ν g : ℂ)) v‖ ^ 2 ≤ R * ‖v‖ ^ 2 := by
-    simpa only [R, S, Fintype.card_fin, Nat.cast_ofNat] using F.integrated_norm_sq ρ hu (Nat.ne_of_gt hd) ψ ν B hν h1 hB (by
-      intro i g
-      convert hcoset i g using 1
-      apply Finset.sum_congr rfl
-      intro h _
-      congr 4
-      exact Subsingleton.elim _ _) v
-  rw [integrated_realPower]
-  calc
-    _ ≤ (Module.finrank ℂ E : ℝ) * ((Module.finrank ℂ E : ℝ) * R^8) :=
-      mul_le_mul_of_nonneg_left (hsSq_power_le _ R hR hA 8) hD.le
-    _ = (Module.finrank ℂ E : ℝ)^2 * R^8 := by ring
-    _ ≤ _ := eight_degree_bound _ hD B S F.reciprocalSum (F.low_degree_sq _)
-
-omit [FiniteDimensional ℂ E] in
-lemma integrated_uniform_real (ρ : Representation ℂ G E) [Representation.IsIrreducible ρ]
-    (hd : 1 < Module.finrank ℂ E) :
-    integrated ρ (fun _ => (((Fintype.card G : ℝ)⁻¹ : ℝ) : ℂ)) = 0 := by
-  simpa only [Complex.ofReal_inv, Complex.ofReal_natCast] using integrated_uniform_zero ρ hd
-
-omit [FiniteDimensional ℂ E] in
-lemma centered_trivial (ρ : Representation ℂ G E) (hρ : ∀ g v, ρ g v = v)
-    (ν : G → ℝ) (h1 : ∑ g, ν g = 1) :
-    integrated ρ (fun g => ((realPower ν 8 g - (Fintype.card G : ℝ)⁻¹ : ℝ) : ℂ)) = 0 := by
-  rw [integrated_real_sub]
-  have htr (g : G) (v : E) : ρ g v = (1 : ℂ) • v := by simpa using hρ g v
-  rw [integrated_scalar ρ (fun _ => 1) htr, integrated_scalar ρ (fun _ => 1) htr]
-  simp only [mul_one, ← Complex.ofReal_sum, realPower_sum ν h1, Complex.ofReal_one,
-    Finset.sum_const, Finset.card_univ, nsmul_eq_mul]
-  simp [Complex.ofReal_inv, Complex.ofReal_natCast, Fintype.card_ne_zero]
-
-omit [FiniteDimensional ℂ E] in
-lemma centered_character (ρ : Representation ℂ G E) (χ : G → ℂ)
-    (hρ : ∀ g v, ρ g v = χ g • v) (hχ : ∑ g, χ g = 0)
-    (ν : G → ℝ) :
-    integrated ρ (fun g => ((realPower ν 8 g - (Fintype.card G : ℝ)⁻¹ : ℝ) : ℂ)) =
-      (∑ g, (ν g : ℂ) * χ g)^8 • LinearMap.id := by
-  rw [integrated_real_sub, integrated_realPower, integrated_scalar ρ χ hρ,
-    integrated_scalar ρ χ hρ]
-  simp only [← Finset.mul_sum, hχ, mul_zero, zero_smul, sub_zero]
-  simp only [smul_pow, show (LinearMap.id : Module.End ℂ E) = 1 from rfl, one_pow]
-
-end Thorp.Fourier
-
-namespace Thorp.Specht
-open scoped Classical
-open Thorp.Fourier
-variable {α β : Type} [Fintype α] [Fintype β] [DecidableEq α] [DecidableEq β] [Nontrivial α]
-
-theorem permutation_eight_bound (hn : 16 ≤ Fintype.card α)
-    (ψ : (Fin 8 → Equiv.Perm β) →* Equiv.Perm α) (ν : Equiv.Perm α → ℝ) (B : ℝ)
-    (hν : ∀ g, 0 ≤ ν g) (h1 : ∑ g, ν g = 1) (hB : 0 ≤ B)
-    (hcoset : ∀ i g, ∑ h, ν (g * ψ (Pi.mulSingle i h)) ≤
-      B * (Fintype.card (Equiv.Perm β) : ℝ) / Fintype.card (Equiv.Perm α)) :
-    (∑ g, |realPower ν 8 g - (Fintype.card (Equiv.Perm α) : ℝ)⁻¹|)^2 ≤
-      2 * ‖∑ g, (ν g : ℂ) * complexSign g‖^16 +
-        (8 * B * (permutationFamily β).reciprocalSum)^8 * exceptionalReciprocalSum (Fintype.card α) := by
-  let F := permutationFamily α
-  let H := permutationFamily β
-  let bias : ℝ := ‖∑ g, (ν g : ℂ) * complexSign g‖^16
-  let K : ℝ := (8 * B * H.reciprocalSum)^8
-  let f : Equiv.Perm α → ℝ := fun g => realPower ν 8 g - (Fintype.card (Equiv.Perm α) : ℝ)⁻¹
-  have hp (p : F.Index) :
-      (F.degree p : ℝ) * hsSq (integrated (F.rep p) (fun g => (f g : ℂ))) ≤
-        if defect p = 0 then bias else K * inverseDegree p := by
-    by_cases hz : defect p = 0
-    · rw [ite_eq_left hz]
-      obtain ⟨hd, htr | hsign⟩ := permutation_exceptional_action (α:=α) p hz
-      · have hh := centered_trivial (F.rep p) htr ν h1
-        change integrated (F.rep p) (fun g => (f g : ℂ)) = 0 at hh
-        rw [hh]
-        simp only [hsSq, LinearMap.zero_apply, norm_zero, zero_pow (by decide : 2 ≠ 0),
-          Finset.sum_const_zero, mul_zero]
-        exact pow_nonneg (norm_nonneg _) _
-      · have hh := centered_character (F.rep p) complexSign hsign (complexSign_sum (α:=α)) ν
-        change integrated (F.rep p) (fun g => (f g : ℂ)) = _ at hh
-        rw [hh, hsSq_scalar]
-        change (F.degree p : ℝ) * ((F.degree p : ℝ) * ‖(∑ g, (ν g : ℂ) * complexSign g)^8‖^2) ≤ bias
-        rw [show F.degree p = 1 from hd]
-        simp only [Nat.cast_one, one_mul, norm_pow, ← pow_mul]
-        exact le_rfl
-    · rw [ite_eq_right hz]
-      have hgt : 1 < Module.finrank ℂ (F.Space p) := degree_gt_one p hn (Nat.pos_of_ne_zero hz)
-      have hu := integrated_uniform_real (F.rep p) hgt
-      have he : integrated (F.rep p) (fun g => (f g : ℂ)) =
-          integrated (F.rep p) (fun g => (realPower ν 8 g : ℂ)) := by
-        rw [show (fun g => (f g : ℂ)) = (fun g => ((realPower ν 8 g - (Fintype.card (Equiv.Perm α) : ℝ)⁻¹ : ℝ) : ℂ)) from rfl,
-          integrated_real_sub, hu, sub_zero]
-      rw [he]
-      exact H.eight_transform_bound (F.rep p) (F.unitary p) (by omega) ψ ν B hν h1 hB hcoset
-  have hs : (∑ p : NShape (Fintype.card α), if defect p = 0 then bias else K * inverseDegree p) =
-      (Fintype.card {p : NShape (Fintype.card α) // defect p = 0} : ℝ) * bias +
-        K * exceptionalReciprocalSum (Fintype.card α) := by
-    rw [show (Fintype.card {p : NShape (Fintype.card α) // defect p = 0} : ℝ) * bias =
-      ∑ p : NShape (Fintype.card α), if defect p = 0 then bias else 0 by rw [← Finset.sum_filter]; simp [Fintype.card_subtype],
-      exceptionalReciprocalSum, Finset.mul_sum, ← Finset.sum_add_distrib]
-    apply Finset.sum_congr rfl
-    intro p _
-    by_cases hz : defect p = 0
-    · simp only [hz, lt_self_iff_false, ↓reduceIte, mul_zero, add_zero]
-    · simp only [hz, Nat.pos_of_ne_zero hz, ↓reduceIte, zero_add]
-  have hc : (Fintype.card {p : NShape (Fintype.card α) // defect p = 0} : ℝ) ≤ 2 := by
-    exact_mod_cast exceptional_count (Fintype.card α)
-  calc
-    _ ≤ ∑ p, (F.degree p : ℝ) * hsSq (integrated (F.rep p) (fun g => (f g : ℂ))) := F.l1_sq_le_parseval f
-    _ ≤ ∑ p : NShape (Fintype.card α), if defect p = 0 then bias else K * inverseDegree p :=
-      Finset.sum_le_sum (fun p _ => hp p)
-    _ = _ := hs
-    _ ≤ _ := add_le_add (mul_le_mul_of_nonneg_right hc (pow_nonneg (norm_nonneg _) _)) le_rfl
-
-end Thorp.Specht
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.PermutationCharacters
+
+namespace OAI
+
+universe uE
+
+noncomputable section
+
+open scoped BigOperators ComplexConjugate InnerProductSpace
+open Filter Topology
+
+namespace Thorp.Fourier
+open scoped Classical
+variable {G H : Type} [Group G] [Fintype G] [Group H] [Fintype H]
+variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℂ E] [FiniteDimensional ℂ E]
+
+lemma hsSq_power_le (A : Module.End ℂ E) (R : ℝ) (hR : 0 ≤ R)
+    (hA : ∀ v, ‖A v‖ ^ 2 ≤ R * ‖v‖ ^ 2) (k : ℕ) :
+    hsSq (A ^ k) ≤ (Module.finrank ℂ E : ℝ) * R ^ k := by
+  unfold hsSq
+  calc
+    _ ≤ ∑ i, R ^ k * ‖stdOrthonormalBasis ℂ E i‖ ^ 2 :=
+      Finset.sum_le_sum (fun i _ => power_norm_sq A R hR hA k _)
+    _ = _ := by simp [(stdOrthonormalBasis ℂ E).norm_eq_one, mul_comm]
+
+lemma Family.eight_transform_bound (F : Family H) (ρ : Representation ℂ G E)
+    [Representation.IsIrreducible ρ]
+    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ)
+    (hd : 0 < Module.finrank ℂ E) (ψ : (Fin 8 → H) →* G) (ν : G → ℝ) (B : ℝ)
+    (hν : ∀ g, 0 ≤ ν g) (h1 : ∑ g, ν g = 1) (hB : 0 ≤ B)
+    (hcoset : ∀ i g, ∑ h, ν (g * ψ (Pi.mulSingle i h)) ≤
+      B * (Fintype.card H : ℝ) / Fintype.card G) :
+    (Module.finrank ℂ E : ℝ) * hsSq (integrated ρ (fun g => (realPower ν 8 g : ℂ))) ≤
+      (8 * B * F.reciprocalSum)^8 * ((Module.finrank ℂ E : ℝ)^2)⁻¹ := by
+  let S : ℝ := ∑ b ∈ Finset.univ.filter (fun b => F.degree b ^ 8 ≤ Module.finrank ℂ E), (F.degree b : ℝ)^2
+  let R : ℝ := 8 * B * S / Module.finrank ℂ E
+  have hD : (0 : ℝ) < Module.finrank ℂ E := by exact_mod_cast hd
+  have hR : 0 ≤ R := by dsimp [R, S]; positivity
+  have hA (v : E) : ‖integrated ρ (fun g => (ν g : ℂ)) v‖ ^ 2 ≤ R * ‖v‖ ^ 2 := by
+    simpa only [R, S, Fintype.card_fin, Nat.cast_ofNat] using F.integrated_norm_sq ρ hu (Nat.ne_of_gt hd) ψ ν B hν h1 hB (by
+      intro i g
+      convert hcoset i g using 1
+      apply Finset.sum_congr rfl
+      intro h _
+      congr 4
+      exact Subsingleton.elim _ _) v
+  rw [integrated_realPower]
+  calc
+    _ ≤ (Module.finrank ℂ E : ℝ) * ((Module.finrank ℂ E : ℝ) * R^8) :=
+      mul_le_mul_of_nonneg_left (hsSq_power_le _ R hR hA 8) hD.le
+    _ = (Module.finrank ℂ E : ℝ)^2 * R^8 := by ring
+    _ ≤ _ := eight_degree_bound _ hD B S F.reciprocalSum (F.low_degree_sq _)
+
+omit [FiniteDimensional ℂ E] in
+lemma integrated_uniform_real (ρ : Representation ℂ G E) [Representation.IsIrreducible ρ]
+    (hd : 1 < Module.finrank ℂ E) :
+    integrated ρ (fun _ => (((Fintype.card G : ℝ)⁻¹ : ℝ) : ℂ)) = 0 := by
+  simpa only [Complex.ofReal_inv, Complex.ofReal_natCast] using integrated_uniform_zero ρ hd
+
+omit [FiniteDimensional ℂ E] in
+lemma centered_trivial (ρ : Representation ℂ G E) (hρ : ∀ g v, ρ g v = v)
+    (ν : G → ℝ) (h1 : ∑ g, ν g = 1) :
+    integrated ρ (fun g => ((realPower ν 8 g - (Fintype.card G : ℝ)⁻¹ : ℝ) : ℂ)) = 0 := by
+  rw [integrated_real_sub]
+  have htr (g : G) (v : E) : ρ g v = (1 : ℂ) • v := by simpa using hρ g v
+  rw [integrated_scalar ρ (fun _ => 1) htr, integrated_scalar ρ (fun _ => 1) htr]
+  simp only [mul_one, ← Complex.ofReal_sum, realPower_sum ν h1, Complex.ofReal_one,
+    Finset.sum_const, Finset.card_univ, nsmul_eq_mul]
+  simp [Complex.ofReal_inv, Complex.ofReal_natCast, Fintype.card_ne_zero]
+
+omit [FiniteDimensional ℂ E] in
+lemma centered_character (ρ : Representation ℂ G E) (χ : G → ℂ)
+    (hρ : ∀ g v, ρ g v = χ g • v) (hχ : ∑ g, χ g = 0)
+    (ν : G → ℝ) :
+    integrated ρ (fun g => ((realPower ν 8 g - (Fintype.card G : ℝ)⁻¹ : ℝ) : ℂ)) =
+      (∑ g, (ν g : ℂ) * χ g)^8 • LinearMap.id := by
+  rw [integrated_real_sub, integrated_realPower, integrated_scalar ρ χ hρ,
+    integrated_scalar ρ χ hρ]
+  simp only [← Finset.mul_sum, hχ, mul_zero, zero_smul, sub_zero]
+  simp only [smul_pow, show (LinearMap.id : Module.End ℂ E) = 1 from rfl, one_pow]
+
+end Thorp.Fourier
+
+namespace Thorp.Specht
+open scoped Classical
+open Thorp.Fourier
+variable {α β : Type} [Fintype α] [Fintype β] [DecidableEq α] [DecidableEq β] [Nontrivial α]
+
+theorem permutation_eight_bound (hn : 16 ≤ Fintype.card α)
+    (ψ : (Fin 8 → Equiv.Perm β) →* Equiv.Perm α) (ν : Equiv.Perm α → ℝ) (B : ℝ)
+    (hν : ∀ g, 0 ≤ ν g) (h1 : ∑ g, ν g = 1) (hB : 0 ≤ B)
+    (hcoset : ∀ i g, ∑ h, ν (g * ψ (Pi.mulSingle i h)) ≤
+      B * (Fintype.card (Equiv.Perm β) : ℝ) / Fintype.card (Equiv.Perm α)) :
+    (∑ g, |realPower ν 8 g - (Fintype.card (Equiv.Perm α) : ℝ)⁻¹|)^2 ≤
+      2 * ‖∑ g, (ν g : ℂ) * complexSign g‖^16 +
+        (8 * B * (permutationFamily β).reciprocalSum)^8 * exceptionalReciprocalSum (Fintype.card α) := by
+  let F := permutationFamily α
+  let H := permutationFamily β
+  let bias : ℝ := ‖∑ g, (ν g : ℂ) * complexSign g‖^16
+  let K : ℝ := (8 * B * H.reciprocalSum)^8
+  let f : Equiv.Perm α → ℝ := fun g => realPower ν 8 g - (Fintype.card (Equiv.Perm α) : ℝ)⁻¹
+  have hp (p : F.Index) :
+      (F.degree p : ℝ) * hsSq (integrated (F.rep p) (fun g => (f g : ℂ))) ≤
+        if defect p = 0 then bias else K * inverseDegree p := by
+    by_cases hz : defect p = 0
+    · rw [ite_eq_left hz]
+      obtain ⟨hd, htr | hsign⟩ := permutation_exceptional_action (α:=α) p hz
+      · have hh := centered_trivial (F.rep p) htr ν h1
+        change integrated (F.rep p) (fun g => (f g : ℂ)) = 0 at hh
+        rw [hh]
+        simp only [hsSq, LinearMap.zero_apply, norm_zero, zero_pow (by decide : 2 ≠ 0),
+          Finset.sum_const_zero, mul_zero]
+        exact pow_nonneg (norm_nonneg _) _
+      · have hh := centered_character (F.rep p) complexSign hsign (complexSign_sum (α:=α)) ν
+        change integrated (F.rep p) (fun g => (f g : ℂ)) = _ at hh
+        rw [hh, hsSq_scalar]
+        change (F.degree p : ℝ) * ((F.degree p : ℝ) * ‖(∑ g, (ν g : ℂ) * complexSign g)^8‖^2) ≤ bias
+        rw [show F.degree p = 1 from hd]
+        simp only [Nat.cast_one, one_mul, norm_pow, ← pow_mul]
+        exact le_rfl
+    · rw [ite_eq_right hz]
+      have hgt : 1 < Module.finrank ℂ (F.Space p) := degree_gt_one p hn (Nat.pos_of_ne_zero hz)
+      have hu := integrated_uniform_real (F.rep p) hgt
+      have he : integrated (F.rep p) (fun g => (f g : ℂ)) =
+          integrated (F.rep p) (fun g => (realPower ν 8 g : ℂ)) := by
+        rw [show (fun g => (f g : ℂ)) = (fun g => ((realPower ν 8 g - (Fintype.card (Equiv.Perm α) : ℝ)⁻¹ : ℝ) : ℂ)) from rfl,
+          integrated_real_sub, hu, sub_zero]
+      rw [he]
+      exact H.eight_transform_bound (F.rep p) (F.unitary p) (by omega) ψ ν B hν h1 hB hcoset
+  have hs : (∑ p : NShape (Fintype.card α), if defect p = 0 then bias else K * inverseDegree p) =
+      (Fintype.card {p : NShape (Fintype.card α) // defect p = 0} : ℝ) * bias +
+        K * exceptionalReciprocalSum (Fintype.card α) := by
+    rw [show (Fintype.card {p : NShape (Fintype.card α) // defect p = 0} : ℝ) * bias =
+      ∑ p : NShape (Fintype.card α), if defect p = 0 then bias else 0 by rw [← Finset.sum_filter]; simp [Fintype.card_subtype],
+      exceptionalReciprocalSum, Finset.mul_sum, ← Finset.sum_add_distrib]
+    apply Finset.sum_congr rfl
+    intro p _
+    by_cases hz : defect p = 0
+    · simp only [hz, lt_self_iff_false, ↓reduceIte, mul_zero, add_zero]
+    · simp only [hz, Nat.pos_of_ne_zero hz, ↓reduceIte, zero_add]
+  have hc : (Fintype.card {p : NShape (Fintype.card α) // defect p = 0} : ℝ) ≤ 2 := by
+    exact_mod_cast exceptional_count (Fintype.card α)
+  calc
+    _ ≤ ∑ p, (F.degree p : ℝ) * hsSq (integrated (F.rep p) (fun g => (f g : ℂ))) := F.l1_sq_le_parseval f
+    _ ≤ ∑ p : NShape (Fintype.card α), if defect p = 0 then bias else K * inverseDegree p :=
+      Finset.sum_le_sum (fun p _ => hp p)
+    _ = _ := hs
+    _ ≤ _ := add_le_add (mul_le_mul_of_nonneg_right hc (pow_nonneg (norm_nonneg _) _)) le_rfl
+
+end Thorp.Specht
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/CosetBounds.lean b/lean/OAI/Probability/ThorpShuffle/CosetBounds.lean
index 71e0e606c8b4eb90c41b26d99a6c3c7c2443da3c..1a1e78385dae02e24ccb7d383c8f4a594b78f19e 100644
--- a/lean/OAI/Probability/ThorpShuffle/CosetBounds.lean
+++ b/lean/OAI/Probability/ThorpShuffle/CosetBounds.lean
@@ -1,192 +1,192 @@
-import OAI.Probability.ThorpShuffle.ProductProjectors
-
-namespace OAI
-
-universe uG uH uE uJ
-
-noncomputable section
-
-open scoped BigOperators ComplexConjugate InnerProductSpace
-open Filter Topology
-
-namespace Thorp.Fourier
-open scoped Classical
-variable {G : Type uG} {H : Type uH} [Group G] [Fintype G] [Group H] [Fintype H]
-
-lemma coset_sum_reindex (φ : H →* G) (ν f : G → ℝ) :
-    ∑ g, ν g * ∑ h, f (g * φ h) = ∑ g, (∑ h, ν (g * φ h)) * f g := by
-  simp only [Finset.mul_sum, Finset.sum_mul]
-  rw [Finset.sum_comm, Finset.sum_comm (f := fun g h => ν (g * φ h) * f g)]
-  have he := Equiv.sum_comp (Equiv.inv H) (fun h => ∑ g, ν (g * φ h) * f g)
-  rw [← he]
-  apply Finset.sum_congr rfl
-  intro h _
-  have hh := Equiv.sum_comp (Equiv.mulRight ((φ h)⁻¹)) (fun g => ν g * f (g * φ h))
-  rw [← hh]
-  simp only [Equiv.coe_mulRight, Equiv.inv_apply, map_inv, inv_mul_cancel_right]
-
-lemma weighted_coset_bound (φ : H →* G) (ν f : G → ℝ) (S C : ℝ)
-    (hν : ∀ g, 0 ≤ ν g) (hf : ∀ g, 0 ≤ f g) (hS : 0 ≤ S)
-    (heval : ∀ g, (Fintype.card H : ℝ) * f g ≤ S * ∑ h, f (g * φ h))
-    (hcoset : ∀ g, ∑ h, ν (g * φ h) ≤ C) :
-    (Fintype.card H : ℝ) * ∑ g, ν g * f g ≤ S * C * ∑ g, f g := by
-  calc
-    _ = ∑ g, ν g * ((Fintype.card H : ℝ) * f g) := by rw [Finset.mul_sum]; congr 1; funext g; ring
-    _ ≤ ∑ g, ν g * (S * ∑ h, f (g * φ h)) :=
-      Finset.sum_le_sum (fun g _ => mul_le_mul_of_nonneg_left (heval g) (hν g))
-    _ = S * ∑ g, (∑ h, ν (g * φ h)) * f g := by
-      rw [← coset_sum_reindex φ ν f, Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro g _
-      ring
-    _ ≤ S * ∑ g, C * f g := mul_le_mul_of_nonneg_left
-      (Finset.sum_le_sum (fun g _ => mul_le_mul_of_nonneg_right (hcoset g) (hf g))) hS
-    _ = _ := by rw [← Finset.mul_sum]; ring
-
-namespace Family
-variable {G H : Type} [Group G] [Fintype G] [Group H] [Fintype H] (F : Family H)
-variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℂ E] [FiniteDimensional ℂ E]
-
-omit [Fintype G] [FiniteDimensional ℂ E] in
-lemma coset_coefficient_evaluation (ρ : Representation ℂ G E)
-    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ)
-    (φ : H →* G) (s : Finset F.Index) (v w : E)
-    (hv : (∑ i ∈ s, F.projector (ρ.comp φ) i) v = v) (g : G) :
-    (Fintype.card H : ℝ) * ‖⟪w, ρ g v⟫_ℂ‖ ^ 2 ≤
-      (∑ i ∈ s, (F.degree i : ℝ)^2) * ∑ h, ‖⟪w, ρ (g * φ h) v⟫_ℂ‖ ^ 2 := by
-  have he := F.coefficient_evaluation (ρ.comp φ) s v (ρ g⁻¹ w) hv
-  have hh (x : E) : ⟪ρ g⁻¹ w, x⟫_ℂ = ⟪w, ρ g x⟫_ℂ := by
-    simpa only [inv_inv] using (inner_inv ρ hu g⁻¹ w x).symm
-  simpa only [hh, MonoidHom.comp_apply, map_mul, Module.End.mul_apply] using he
-
-theorem weighted_coefficient (ρ : Representation ℂ G E) [Representation.IsIrreducible ρ]
-    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ)
-    (hE : Module.finrank ℂ E ≠ 0) (φ : H →* G) (s : Finset F.Index) (ν : G → ℝ) (B : ℝ)
-    (hν : ∀ g, 0 ≤ ν g)
-    (hcoset : ∀ g, ∑ h, ν (g * φ h) ≤ B * (Fintype.card H : ℝ) / Fintype.card G)
-    (v w : E) (hv : (∑ i ∈ s, F.projector (ρ.comp φ) i) v = v) :
-    ∑ g, ν g * ‖⟪w, ρ g v⟫_ℂ‖ ^ 2 ≤
-      B * (∑ i ∈ s, (F.degree i : ℝ)^2) / Module.finrank ℂ E * (‖v‖ ^ 2 * ‖w‖ ^ 2) := by
-  have hh := weighted_coset_bound φ ν (fun g => ‖⟪w, ρ g v⟫_ℂ‖ ^ 2)
-    (∑ i ∈ s, (F.degree i : ℝ)^2) (B * (Fintype.card H : ℝ) / Fintype.card G)
-    hν (fun _ => sq_nonneg _) (Finset.sum_nonneg (fun _ _ => sq_nonneg _))
-    (F.coset_coefficient_evaluation ρ hu φ s v w hv) hcoset
-  have hcG : (Fintype.card G : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
-  have hcH : (0 : ℝ) < Fintype.card H := by exact_mod_cast Fintype.card_pos
-  have hd : (Module.finrank ℂ E : ℝ) ≠ 0 := by exact_mod_cast hE
-  have hs := coefficient_second_moment ρ hu hE v w
-  have hs' : ∑ g, ‖⟪w, ρ g v⟫_ℂ‖ ^ 2 =
-      (Fintype.card G : ℝ) * (‖v‖ ^ 2 * ‖w‖ ^ 2 / Module.finrank ℂ E) := by
-    calc
-      _ = (Fintype.card G : ℝ) * ((Fintype.card G : ℝ)⁻¹ * ∑ g, ‖⟪w, ρ g v⟫_ℂ‖ ^ 2) := by field_simp
-      _ = _ := by rw [hs]
-  rw [hs'] at hh
-  have he : (∑ i ∈ s, (F.degree i : ℝ)^2) * (B * (Fintype.card H : ℝ) / Fintype.card G) *
-      ((Fintype.card G : ℝ) * (‖v‖ ^ 2 * ‖w‖ ^ 2 / Module.finrank ℂ E)) =
-      (Fintype.card H : ℝ) * (B * (∑ i ∈ s, (F.degree i : ℝ)^2) /
-        Module.finrank ℂ E * (‖v‖ ^ 2 * ‖w‖ ^ 2)) := by field_simp
-  rw [he] at hh
-  exact (mul_le_mul_iff_right₀ hcH).mp hh
-
-end Family
-end Thorp.Fourier
-
-namespace Thorp.Fourier
-open scoped Classical
-
-lemma norm_weighted_sum_sq {J : Type uJ} [Fintype J] (ν : J → ℝ) (f : J → ℂ)
-    (hν : ∀ j, 0 ≤ ν j) (h1 : ∑ j, ν j = 1) :
-    ‖∑ j, (ν j : ℂ) * f j‖ ^ 2 ≤ ∑ j, ν j * ‖f j‖ ^ 2 := by
-  have ht : ‖∑ j, (ν j : ℂ) * f j‖ ≤ ∑ j, ν j * ‖f j‖ := by
-    simpa only [norm_mul, Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg (hν _)] using
-      norm_sum_le Finset.univ (fun j => (ν j : ℂ) * f j)
-  have hc := Finset.sum_sq_le_sum_mul_sum_of_sq_le_mul Finset.univ
-    (f := ν) (g := fun j => ν j * ‖f j‖ ^ 2) (r := fun j => ν j * ‖f j‖)
-    (fun j _ => hν j) (fun j _ => mul_nonneg (hν j) (sq_nonneg _))
-    (fun j _ => by nlinarith [sq_nonneg (ν j * ‖f j‖)])
-  rw [h1, one_mul] at hc
-  exact (pow_le_pow_left₀ (norm_nonneg _) ht 2).trans hc
-
-namespace Family
-variable {G H I : Type} [Group G] [Fintype G] [Group H] [Fintype H]
-  [Fintype I] [Nonempty I] (F : Family H)
-variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℂ E] [FiniteDimensional ℂ E]
-
-theorem split_weighted_coefficient (ρ : Representation ℂ G E) [Representation.IsIrreducible ρ]
-    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ)
-    (hE : Module.finrank ℂ E ≠ 0) (ψ : (I → H) →* G) (ν : G → ℝ) (B : ℝ)
-    (hν : ∀ g, 0 ≤ ν g)
-    (hcoset : ∀ i g, ∑ h, ν (g * ψ (Pi.mulSingle i h)) ≤
-      B * (Fintype.card H : ℝ) / Fintype.card G) (v w : E) :
-    ∑ g, ν g * ‖⟪w, ρ g v⟫_ℂ‖ ^ 2 ≤
-      (Fintype.card I : ℝ) * B *
-        (∑ b ∈ Finset.univ.filter (fun b => F.degree b ^ Fintype.card I ≤ Module.finrank ℂ E),
-          (F.degree b : ℝ)^2) / Module.finrank ℂ E * (‖v‖ ^ 2 * ‖w‖ ^ 2) := by
-  let z := F.splitVector (ρ.comp ψ) v
-  let s := Finset.univ.filter (fun b => F.degree b ^ Fintype.card I ≤ Module.finrank ℂ E)
-  let S : ℝ := ∑ b ∈ s, (F.degree b : ℝ)^2
-  have hz : ∑ i, z i = v := F.splitVector_sum (ρ.comp ψ) v
-  have hz2 : ∑ i, ‖z i‖ ^ 2 = ‖v‖ ^ 2 := F.splitVector_sq_sum (ρ.comp ψ) (fun g => hu (ψ g)) v
-  have hpoint (g : G) : ‖⟪w, ρ g v⟫_ℂ‖ ^ 2 ≤ (Fintype.card I : ℝ) * ∑ i, ‖⟪w, ρ g (z i)⟫_ℂ‖ ^ 2 := by
-    have hh := complex_cauchy Finset.univ (fun _ : I => 1) (fun i => ⟪w, ρ g (z i)⟫_ℂ)
-    simpa only [one_mul, norm_one, one_pow, Finset.sum_const, Finset.card_univ, nsmul_eq_mul,
-      mul_one, ← inner_sum, ← map_sum, hz] using hh
-  have hi (i : I) : ∑ g, ν g * ‖⟪w, ρ g (z i)⟫_ℂ‖ ^ 2 ≤
-      B * S / Module.finrank ℂ E * (‖z i‖ ^ 2 * ‖w‖ ^ 2) := by
-    exact F.weighted_coefficient ρ hu hE (ψ.comp (MonoidHom.mulSingle (fun _ : I => H) i))
-      s ν B hν (hcoset i) (z i) w (F.splitVector_low (ρ.comp ψ) v i)
-  calc
-    _ ≤ ∑ g, ν g * ((Fintype.card I : ℝ) * ∑ i, ‖⟪w, ρ g (z i)⟫_ℂ‖ ^ 2) :=
-      Finset.sum_le_sum (fun g _ => mul_le_mul_of_nonneg_left (hpoint g) (hν g))
-    _ = (Fintype.card I : ℝ) * ∑ i, ∑ g, ν g * ‖⟪w, ρ g (z i)⟫_ℂ‖ ^ 2 := by
-      rw [Finset.sum_comm, Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro g _
-      rw [← Finset.mul_sum]
-      ring
-    _ ≤ (Fintype.card I : ℝ) * ∑ i, B * S / Module.finrank ℂ E * (‖z i‖ ^ 2 * ‖w‖ ^ 2) :=
-      mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun i _ => hi i)) (Nat.cast_nonneg _)
-    _ = _ := by
-      simp only [← Finset.mul_sum, ← Finset.sum_mul, hz2]
-      dsimp [S, s]
-      ring
-
-theorem integrated_norm_sq (ρ : Representation ℂ G E) [Representation.IsIrreducible ρ]
-    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ)
-    (hE : Module.finrank ℂ E ≠ 0) (ψ : (I → H) →* G) (ν : G → ℝ) (B : ℝ)
-    (hν : ∀ g, 0 ≤ ν g) (h1 : ∑ g, ν g = 1) (hB : 0 ≤ B)
-    (hcoset : ∀ i g, ∑ h, ν (g * ψ (Pi.mulSingle i h)) ≤
-      B * (Fintype.card H : ℝ) / Fintype.card G) (v : E) :
-    ‖integrated ρ (fun g => (ν g : ℂ)) v‖ ^ 2 ≤
-      (Fintype.card I : ℝ) * B *
-        (∑ b ∈ Finset.univ.filter (fun b => F.degree b ^ Fintype.card I ≤ Module.finrank ℂ E),
-          (F.degree b : ℝ)^2) / Module.finrank ℂ E * ‖v‖ ^ 2 := by
-  let A := integrated ρ (fun g => (ν g : ℂ))
-  let R := (Fintype.card I : ℝ) * B *
-        (∑ b ∈ Finset.univ.filter (fun b => F.degree b ^ Fintype.card I ≤ Module.finrank ℂ E),
-          (F.degree b : ℝ)^2) / Module.finrank ℂ E
-  have hc : ‖⟪A v, A v⟫_ℂ‖ ^ 2 ≤ R * (‖v‖ ^ 2 * ‖A v‖ ^ 2) := by
-    calc
-      _ = ‖∑ g, (ν g : ℂ) * ⟪A v, ρ g v⟫_ℂ‖ ^ 2 := by
-        congr 2
-        simp only [A, integrated, LinearMap.sum_apply, LinearMap.smul_apply,
-          inner_sum, inner_smul_right]
-      _ ≤ ∑ g, ν g * ‖⟪A v, ρ g v⟫_ℂ‖ ^ 2 := norm_weighted_sum_sq ν _ hν h1
-      _ ≤ _ := F.split_weighted_coefficient ρ hu hE ψ ν B hν hcoset v (A v)
-  have hinner : ‖⟪A v, A v⟫_ℂ‖ = ‖A v‖ ^ 2 := by simp [inner_self_eq_norm_sq_to_K]
-  rw [hinner] at hc
-  have hR : 0 ≤ R := by positivity
-  by_cases hz : ‖A v‖ ^ 2 = 0
-  · change ‖A v‖ ^ 2 ≤ R * ‖v‖ ^ 2
-    rw [hz]
-    exact mul_nonneg hR (sq_nonneg _)
-  · have hp : 0 < ‖A v‖ ^ 2 := lt_of_le_of_ne (sq_nonneg _) (Ne.symm hz)
-    change ‖A v‖ ^ 2 ≤ R * ‖v‖ ^ 2
-    nlinarith
-
-end Family
-end Thorp.Fourier
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.ProductProjectors
+
+namespace OAI
+
+universe uG uH uE uJ
+
+noncomputable section
+
+open scoped BigOperators ComplexConjugate InnerProductSpace
+open Filter Topology
+
+namespace Thorp.Fourier
+open scoped Classical
+variable {G : Type uG} {H : Type uH} [Group G] [Fintype G] [Group H] [Fintype H]
+
+lemma coset_sum_reindex (φ : H →* G) (ν f : G → ℝ) :
+    ∑ g, ν g * ∑ h, f (g * φ h) = ∑ g, (∑ h, ν (g * φ h)) * f g := by
+  simp only [Finset.mul_sum, Finset.sum_mul]
+  rw [Finset.sum_comm, Finset.sum_comm (f := fun g h => ν (g * φ h) * f g)]
+  have he := Equiv.sum_comp (Equiv.inv H) (fun h => ∑ g, ν (g * φ h) * f g)
+  rw [← he]
+  apply Finset.sum_congr rfl
+  intro h _
+  have hh := Equiv.sum_comp (Equiv.mulRight ((φ h)⁻¹)) (fun g => ν g * f (g * φ h))
+  rw [← hh]
+  simp only [Equiv.coe_mulRight, Equiv.inv_apply, map_inv, inv_mul_cancel_right]
+
+lemma weighted_coset_bound (φ : H →* G) (ν f : G → ℝ) (S C : ℝ)
+    (hν : ∀ g, 0 ≤ ν g) (hf : ∀ g, 0 ≤ f g) (hS : 0 ≤ S)
+    (heval : ∀ g, (Fintype.card H : ℝ) * f g ≤ S * ∑ h, f (g * φ h))
+    (hcoset : ∀ g, ∑ h, ν (g * φ h) ≤ C) :
+    (Fintype.card H : ℝ) * ∑ g, ν g * f g ≤ S * C * ∑ g, f g := by
+  calc
+    _ = ∑ g, ν g * ((Fintype.card H : ℝ) * f g) := by rw [Finset.mul_sum]; congr 1; funext g; ring
+    _ ≤ ∑ g, ν g * (S * ∑ h, f (g * φ h)) :=
+      Finset.sum_le_sum (fun g _ => mul_le_mul_of_nonneg_left (heval g) (hν g))
+    _ = S * ∑ g, (∑ h, ν (g * φ h)) * f g := by
+      rw [← coset_sum_reindex φ ν f, Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro g _
+      ring
+    _ ≤ S * ∑ g, C * f g := mul_le_mul_of_nonneg_left
+      (Finset.sum_le_sum (fun g _ => mul_le_mul_of_nonneg_right (hcoset g) (hf g))) hS
+    _ = _ := by rw [← Finset.mul_sum]; ring
+
+namespace Family
+variable {G H : Type} [Group G] [Fintype G] [Group H] [Fintype H] (F : Family H)
+variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℂ E] [FiniteDimensional ℂ E]
+
+omit [Fintype G] [FiniteDimensional ℂ E] in
+lemma coset_coefficient_evaluation (ρ : Representation ℂ G E)
+    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ)
+    (φ : H →* G) (s : Finset F.Index) (v w : E)
+    (hv : (∑ i ∈ s, F.projector (ρ.comp φ) i) v = v) (g : G) :
+    (Fintype.card H : ℝ) * ‖⟪w, ρ g v⟫_ℂ‖ ^ 2 ≤
+      (∑ i ∈ s, (F.degree i : ℝ)^2) * ∑ h, ‖⟪w, ρ (g * φ h) v⟫_ℂ‖ ^ 2 := by
+  have he := F.coefficient_evaluation (ρ.comp φ) s v (ρ g⁻¹ w) hv
+  have hh (x : E) : ⟪ρ g⁻¹ w, x⟫_ℂ = ⟪w, ρ g x⟫_ℂ := by
+    simpa only [inv_inv] using (inner_inv ρ hu g⁻¹ w x).symm
+  simpa only [hh, MonoidHom.comp_apply, map_mul, Module.End.mul_apply] using he
+
+theorem weighted_coefficient (ρ : Representation ℂ G E) [Representation.IsIrreducible ρ]
+    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ)
+    (hE : Module.finrank ℂ E ≠ 0) (φ : H →* G) (s : Finset F.Index) (ν : G → ℝ) (B : ℝ)
+    (hν : ∀ g, 0 ≤ ν g)
+    (hcoset : ∀ g, ∑ h, ν (g * φ h) ≤ B * (Fintype.card H : ℝ) / Fintype.card G)
+    (v w : E) (hv : (∑ i ∈ s, F.projector (ρ.comp φ) i) v = v) :
+    ∑ g, ν g * ‖⟪w, ρ g v⟫_ℂ‖ ^ 2 ≤
+      B * (∑ i ∈ s, (F.degree i : ℝ)^2) / Module.finrank ℂ E * (‖v‖ ^ 2 * ‖w‖ ^ 2) := by
+  have hh := weighted_coset_bound φ ν (fun g => ‖⟪w, ρ g v⟫_ℂ‖ ^ 2)
+    (∑ i ∈ s, (F.degree i : ℝ)^2) (B * (Fintype.card H : ℝ) / Fintype.card G)
+    hν (fun _ => sq_nonneg _) (Finset.sum_nonneg (fun _ _ => sq_nonneg _))
+    (F.coset_coefficient_evaluation ρ hu φ s v w hv) hcoset
+  have hcG : (Fintype.card G : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
+  have hcH : (0 : ℝ) < Fintype.card H := by exact_mod_cast Fintype.card_pos
+  have hd : (Module.finrank ℂ E : ℝ) ≠ 0 := by exact_mod_cast hE
+  have hs := coefficient_second_moment ρ hu hE v w
+  have hs' : ∑ g, ‖⟪w, ρ g v⟫_ℂ‖ ^ 2 =
+      (Fintype.card G : ℝ) * (‖v‖ ^ 2 * ‖w‖ ^ 2 / Module.finrank ℂ E) := by
+    calc
+      _ = (Fintype.card G : ℝ) * ((Fintype.card G : ℝ)⁻¹ * ∑ g, ‖⟪w, ρ g v⟫_ℂ‖ ^ 2) := by field_simp
+      _ = _ := by rw [hs]
+  rw [hs'] at hh
+  have he : (∑ i ∈ s, (F.degree i : ℝ)^2) * (B * (Fintype.card H : ℝ) / Fintype.card G) *
+      ((Fintype.card G : ℝ) * (‖v‖ ^ 2 * ‖w‖ ^ 2 / Module.finrank ℂ E)) =
+      (Fintype.card H : ℝ) * (B * (∑ i ∈ s, (F.degree i : ℝ)^2) /
+        Module.finrank ℂ E * (‖v‖ ^ 2 * ‖w‖ ^ 2)) := by field_simp
+  rw [he] at hh
+  exact (mul_le_mul_iff_right₀ hcH).mp hh
+
+end Family
+end Thorp.Fourier
+
+namespace Thorp.Fourier
+open scoped Classical
+
+lemma norm_weighted_sum_sq {J : Type uJ} [Fintype J] (ν : J → ℝ) (f : J → ℂ)
+    (hν : ∀ j, 0 ≤ ν j) (h1 : ∑ j, ν j = 1) :
+    ‖∑ j, (ν j : ℂ) * f j‖ ^ 2 ≤ ∑ j, ν j * ‖f j‖ ^ 2 := by
+  have ht : ‖∑ j, (ν j : ℂ) * f j‖ ≤ ∑ j, ν j * ‖f j‖ := by
+    simpa only [norm_mul, Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg (hν _)] using
+      norm_sum_le Finset.univ (fun j => (ν j : ℂ) * f j)
+  have hc := Finset.sum_sq_le_sum_mul_sum_of_sq_le_mul Finset.univ
+    (f := ν) (g := fun j => ν j * ‖f j‖ ^ 2) (r := fun j => ν j * ‖f j‖)
+    (fun j _ => hν j) (fun j _ => mul_nonneg (hν j) (sq_nonneg _))
+    (fun j _ => by nlinarith [sq_nonneg (ν j * ‖f j‖)])
+  rw [h1, one_mul] at hc
+  exact (pow_le_pow_left₀ (norm_nonneg _) ht 2).trans hc
+
+namespace Family
+variable {G H I : Type} [Group G] [Fintype G] [Group H] [Fintype H]
+  [Fintype I] [Nonempty I] (F : Family H)
+variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℂ E] [FiniteDimensional ℂ E]
+
+theorem split_weighted_coefficient (ρ : Representation ℂ G E) [Representation.IsIrreducible ρ]
+    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ)
+    (hE : Module.finrank ℂ E ≠ 0) (ψ : (I → H) →* G) (ν : G → ℝ) (B : ℝ)
+    (hν : ∀ g, 0 ≤ ν g)
+    (hcoset : ∀ i g, ∑ h, ν (g * ψ (Pi.mulSingle i h)) ≤
+      B * (Fintype.card H : ℝ) / Fintype.card G) (v w : E) :
+    ∑ g, ν g * ‖⟪w, ρ g v⟫_ℂ‖ ^ 2 ≤
+      (Fintype.card I : ℝ) * B *
+        (∑ b ∈ Finset.univ.filter (fun b => F.degree b ^ Fintype.card I ≤ Module.finrank ℂ E),
+          (F.degree b : ℝ)^2) / Module.finrank ℂ E * (‖v‖ ^ 2 * ‖w‖ ^ 2) := by
+  let z := F.splitVector (ρ.comp ψ) v
+  let s := Finset.univ.filter (fun b => F.degree b ^ Fintype.card I ≤ Module.finrank ℂ E)
+  let S : ℝ := ∑ b ∈ s, (F.degree b : ℝ)^2
+  have hz : ∑ i, z i = v := F.splitVector_sum (ρ.comp ψ) v
+  have hz2 : ∑ i, ‖z i‖ ^ 2 = ‖v‖ ^ 2 := F.splitVector_sq_sum (ρ.comp ψ) (fun g => hu (ψ g)) v
+  have hpoint (g : G) : ‖⟪w, ρ g v⟫_ℂ‖ ^ 2 ≤ (Fintype.card I : ℝ) * ∑ i, ‖⟪w, ρ g (z i)⟫_ℂ‖ ^ 2 := by
+    have hh := complex_cauchy Finset.univ (fun _ : I => 1) (fun i => ⟪w, ρ g (z i)⟫_ℂ)
+    simpa only [one_mul, norm_one, one_pow, Finset.sum_const, Finset.card_univ, nsmul_eq_mul,
+      mul_one, ← inner_sum, ← map_sum, hz] using hh
+  have hi (i : I) : ∑ g, ν g * ‖⟪w, ρ g (z i)⟫_ℂ‖ ^ 2 ≤
+      B * S / Module.finrank ℂ E * (‖z i‖ ^ 2 * ‖w‖ ^ 2) := by
+    exact F.weighted_coefficient ρ hu hE (ψ.comp (MonoidHom.mulSingle (fun _ : I => H) i))
+      s ν B hν (hcoset i) (z i) w (F.splitVector_low (ρ.comp ψ) v i)
+  calc
+    _ ≤ ∑ g, ν g * ((Fintype.card I : ℝ) * ∑ i, ‖⟪w, ρ g (z i)⟫_ℂ‖ ^ 2) :=
+      Finset.sum_le_sum (fun g _ => mul_le_mul_of_nonneg_left (hpoint g) (hν g))
+    _ = (Fintype.card I : ℝ) * ∑ i, ∑ g, ν g * ‖⟪w, ρ g (z i)⟫_ℂ‖ ^ 2 := by
+      rw [Finset.sum_comm, Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro g _
+      rw [← Finset.mul_sum]
+      ring
+    _ ≤ (Fintype.card I : ℝ) * ∑ i, B * S / Module.finrank ℂ E * (‖z i‖ ^ 2 * ‖w‖ ^ 2) :=
+      mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun i _ => hi i)) (Nat.cast_nonneg _)
+    _ = _ := by
+      simp only [← Finset.mul_sum, ← Finset.sum_mul, hz2]
+      dsimp [S, s]
+      ring
+
+theorem integrated_norm_sq (ρ : Representation ℂ G E) [Representation.IsIrreducible ρ]
+    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ)
+    (hE : Module.finrank ℂ E ≠ 0) (ψ : (I → H) →* G) (ν : G → ℝ) (B : ℝ)
+    (hν : ∀ g, 0 ≤ ν g) (h1 : ∑ g, ν g = 1) (hB : 0 ≤ B)
+    (hcoset : ∀ i g, ∑ h, ν (g * ψ (Pi.mulSingle i h)) ≤
+      B * (Fintype.card H : ℝ) / Fintype.card G) (v : E) :
+    ‖integrated ρ (fun g => (ν g : ℂ)) v‖ ^ 2 ≤
+      (Fintype.card I : ℝ) * B *
+        (∑ b ∈ Finset.univ.filter (fun b => F.degree b ^ Fintype.card I ≤ Module.finrank ℂ E),
+          (F.degree b : ℝ)^2) / Module.finrank ℂ E * ‖v‖ ^ 2 := by
+  let A := integrated ρ (fun g => (ν g : ℂ))
+  let R := (Fintype.card I : ℝ) * B *
+        (∑ b ∈ Finset.univ.filter (fun b => F.degree b ^ Fintype.card I ≤ Module.finrank ℂ E),
+          (F.degree b : ℝ)^2) / Module.finrank ℂ E
+  have hc : ‖⟪A v, A v⟫_ℂ‖ ^ 2 ≤ R * (‖v‖ ^ 2 * ‖A v‖ ^ 2) := by
+    calc
+      _ = ‖∑ g, (ν g : ℂ) * ⟪A v, ρ g v⟫_ℂ‖ ^ 2 := by
+        congr 2
+        simp only [A, integrated, LinearMap.sum_apply, LinearMap.smul_apply,
+          inner_sum, inner_smul_right]
+      _ ≤ ∑ g, ν g * ‖⟪A v, ρ g v⟫_ℂ‖ ^ 2 := norm_weighted_sum_sq ν _ hν h1
+      _ ≤ _ := F.split_weighted_coefficient ρ hu hE ψ ν B hν hcoset v (A v)
+  have hinner : ‖⟪A v, A v⟫_ℂ‖ = ‖A v‖ ^ 2 := by simp [inner_self_eq_norm_sq_to_K]
+  rw [hinner] at hc
+  have hR : 0 ≤ R := by positivity
+  by_cases hz : ‖A v‖ ^ 2 = 0
+  · change ‖A v‖ ^ 2 ≤ R * ‖v‖ ^ 2
+    rw [hz]
+    exact mul_nonneg hR (sq_nonneg _)
+  · have hp : 0 < ‖A v‖ ^ 2 := lt_of_le_of_ne (sq_nonneg _) (Ne.symm hz)
+    change ‖A v‖ ^ 2 ≤ R * ‖v‖ ^ 2
+    nlinarith
+
+end Family
+end Thorp.Fourier
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/Cyclic.lean b/lean/OAI/Probability/ThorpShuffle/Cyclic.lean
index 4f00b5398dfca78f739d534016f88fdae81301ad..761c70faaf2c0f68298686bc8749cc39e00654bc 100644
--- a/lean/OAI/Probability/ThorpShuffle/Cyclic.lean
+++ b/lean/OAI/Probability/ThorpShuffle/Cyclic.lean
@@ -1,246 +1,246 @@
-import OAI.Probability.ThorpShuffle.Parity
-
-namespace OAI
-
-universe uG uE uX uH
-
-noncomputable section
-
-open scoped BigOperators ComplexConjugate InnerProductSpace
-open Filter
-
-namespace Thorp.Specht
-
-variable {G : Type uG} [Group G]
-variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
-
-def cyclic (ρ : Representation ℂ G E) (e : E) : Subrepresentation ρ where
-  toSubmodule := Submodule.span ℂ (Set.range fun g => ρ g e)
-  apply_mem_toSubmodule g := by
-    intro v hv
-    induction hv using Submodule.span_induction with
-    | mem v hv =>
-      obtain ⟨h, rfl⟩ := hv
-      exact Submodule.subset_span ⟨g * h, by simp⟩
-    | zero => simp
-    | add x y hx hy ihx ihy =>
-      simpa using (Submodule.span ℂ (Set.range fun h => ρ h e)).add_mem ihx ihy
-    | smul a x hx ih =>
-      simpa using (Submodule.span ℂ (Set.range fun h => ρ h e)).smul_mem a ih
-
-lemma self_mem_cyclic (ρ : Representation ℂ G E) (e : E) : e ∈ cyclic ρ e := by
-  exact Submodule.subset_span ⟨1, by simp⟩
-
-theorem submodule_alternative (ρ : Representation ℂ G E) (e : E)
-    (hunit : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ)
-    (K : Module.End ℂ E) (hK : ∀ v, K v = ⟪e, v⟫_ℂ • e)
-    (hstable : ∀ U : Subrepresentation ρ, ∀ v ∈ U, K v ∈ U)
-    (U : Subrepresentation ρ) :
-    cyclic ρ e ≤ U ∨ ∀ u ∈ U, ∀ v ∈ cyclic ρ e, ⟪v, u⟫_ℂ = 0 := by
-  classical
-  by_cases he : e ∈ U
-  · left
-    exact Submodule.span_le.mpr (by rintro _ ⟨g, rfl⟩; exact U.apply_mem_toSubmodule g he)
-  · right
-    intro u hu v hv
-    have horth (g : G) : ⟪ρ g e, u⟫_ℂ = 0 := by
-      have hgu : ρ g⁻¹ u ∈ U := U.apply_mem_toSubmodule g⁻¹ hu
-      have hk := hstable U _ hgu
-      rw [hK] at hk
-      have hz : ⟪e, ρ g⁻¹ u⟫_ℂ = 0 := by
-        by_contra hne
-        apply he
-        have hh := U.toSubmodule.smul_mem (⟪e, ρ g⁻¹ u⟫_ℂ)⁻¹ hk
-        change e ∈ U.toSubmodule
-        simpa [smul_smul, hne] using hh
-      have hi := hunit g e (ρ g⁻¹ u)
-      simpa only [← Module.End.mul_apply, ← map_mul, mul_inv_cancel, map_one,
-        Module.End.one_apply] using hi.trans hz
-    induction hv using Submodule.span_induction with
-    | mem v hv => obtain ⟨g, rfl⟩ := hv; exact horth g
-    | zero => simp
-    | add x y hx hy ihx ihy => simp [ihx, ihy]
-    | smul a x hx ih => simp [ih]
-
-theorem cyclic_irreducible (ρ : Representation ℂ G E) (e : E) (he : e ≠ 0)
-    (hunit : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ)
-    (K : Module.End ℂ E) (hK : ∀ v, K v = ⟪e, v⟫_ℂ • e)
-    (hstable : ∀ U : Subrepresentation ρ, ∀ v ∈ U, K v ∈ U) :
-    Representation.IsIrreducible (cyclic ρ e).toRepresentation := by
-  let S := cyclic ρ e
-  let eS : S.toSubmodule := ⟨e, self_mem_cyclic ρ e⟩
-  have heS : eS ≠ 0 := by intro hh; apply he; exact congrArg Subtype.val hh
-  have htb : (⊤ : Subrepresentation S.toRepresentation) ≠ ⊥ := by
-    intro hh
-    have hel : eS ∈ (⊤ : Subrepresentation S.toRepresentation) := by trivial
-    rw [hh] at hel
-    exact heS hel
-  let : Nontrivial (Subrepresentation S.toRepresentation) := ⟨⟨⊤, ⊥, htb⟩⟩
-  refine ⟨fun U => ?_⟩
-  let U' : Subrepresentation ρ := {
-    toSubmodule := U.toSubmodule.map S.toSubmodule.subtype
-    apply_mem_toSubmodule := by
-      intro g v hv
-      obtain ⟨u, hu, rfl⟩ := hv
-      exact ⟨S.toRepresentation g u, U.apply_mem_toSubmodule g hu, rfl⟩ }
-  rcases submodule_alternative ρ e hunit K hK hstable U' with h | h
-  · right
-    apply top_unique
-    intro u _
-    obtain ⟨v, hv, heq⟩ := h u.property
-    have hvu : v = u := Subtype.ext heq
-    exact hvu ▸ hv
-  · left
-    apply bot_unique
-    intro u hu
-    have hum : (u : E) ∈ U' := ⟨u, hu, rfl⟩
-    have hh := h _ hum _ u.property
-    have hz : (u : E) = 0 := inner_self_eq_zero.mp hh
-    exact Subtype.ext hz
-
-namespace PermutationSpace
-open scoped Classical
-
-variable {X : Type uX} [Fintype X] [MulAction G X]
-
-def rep : Representation ℂ G (EuclideanSpace ℂ X) where
-  toFun g := {
-    toFun v := WithLp.toLp 2 (fun x => v (g⁻¹ • x))
-    map_add' := by intro v w; rfl
-    map_smul' := by intro a v; rfl }
-  map_one' := by ext v x; simp
-  map_mul' := by
-    intro g h
-    ext v x
-    change v ((g * h)⁻¹ • x) = v (h⁻¹ • g⁻¹ • x)
-    simp [mul_smul]
-
-omit [Fintype X] in
-@[simp] lemma rep_apply (g : G) (v : EuclideanSpace ℂ X) (x : X) :
-    rep g v x = v (g⁻¹ • x) := rfl
-
-lemma rep_unitary (g : G) (v w : EuclideanSpace ℂ X) :
-    ⟪rep g v, rep g w⟫_ℂ = ⟪v, w⟫_ℂ := by
-  simp only [PiLp.inner_apply, rep_apply]
-  exact Equiv.sum_comp (MulAction.toPerm g⁻¹) (fun x => inner ℂ (v x) (w x))
-
-omit [Fintype X] in
-lemma rep_single (g : G) (x : X) :
-    rep g (EuclideanSpace.single x (1 : ℂ)) = EuclideanSpace.single (g • x) (1 : ℂ) := by
-  classical
-  ext y
-  simp only [rep_apply, PiLp.single_apply, inv_smul_eq_iff]
-
-variable {H : Type uH} [Group H] [Fintype H]
-
-def alternator (ι : H →* G) (χ : H →* ℂ) : Module.End ℂ (EuclideanSpace ℂ X) :=
-  ∑ h, χ h • rep (ι h)
-
-omit [Fintype X] in
-lemma alternator_apply (ι : H →* G) (χ : H →* ℂ) (v : EuclideanSpace ℂ X) :
-    alternator ι χ v = ∑ h, χ h • rep (ι h) v := by
-  simp [alternator, LinearMap.sum_apply]
-
-omit [Fintype X] in
-lemma alternator_stable (ι : H →* G) (χ : H →* ℂ)
-    (U : Subrepresentation (rep : Representation ℂ G (EuclideanSpace ℂ X)))
-    (v : EuclideanSpace ℂ X) (hv : v ∈ U) : alternator ι χ v ∈ U := by
-  rw [alternator_apply]
-  exact U.toSubmodule.sum_mem fun h _ => U.toSubmodule.smul_mem _ (U.apply_mem_toSubmodule _ hv)
-
-omit [Fintype X] in
-lemma alternator_translate (ι : H →* G) (χ : H →* ℂ) (a : H)
-    (v : EuclideanSpace ℂ X) :
-    alternator ι χ (rep (ι a) v) = χ a⁻¹ • alternator ι χ v := by
-  rw [alternator_apply, alternator_apply, Finset.smul_sum]
-  have he := Equiv.sum_comp (Equiv.mulRight a)
-    (fun h => χ a⁻¹ • (χ h • (rep (ι h) v : EuclideanSpace ℂ X)))
-  rw [← he]
-  apply Finset.sum_congr rfl
-  intro h _
-  simp only [Equiv.coe_mulRight, smul_smul, map_mul]
-  rw [show χ a⁻¹ * (χ h * χ a) = χ h by
-    calc
-      _ = χ h * (χ a⁻¹ * χ a) := by ring
-      _ = χ h := by rw [← map_mul, inv_mul_cancel, map_one, mul_one]]
-  simp only [Module.End.mul_apply]
-
-omit [Fintype X] in
-lemma alternator_single_zero (ι : H →* G) (χ : H →* ℂ)
-    (x : X) (a : H) (ha : ι a • x = x) (hs : χ a = -1) :
-    alternator ι χ (EuclideanSpace.single x (1 : ℂ)) = 0 := by
-  have hh := alternator_translate ι χ a (EuclideanSpace.single x (1 : ℂ))
-  rw [rep_single, ha, map_inv, hs, inv_neg, inv_one, neg_smul, one_smul] at hh
-  have hz := congrArg (fun z => z + alternator ι χ (EuclideanSpace.single x (1 : ℂ))) hh
-  have ht : (2 : ℂ) • alternator ι χ (EuclideanSpace.single x (1 : ℂ)) = 0 := by
-    simpa [two_smul] using hz
-  exact (smul_eq_zero.mp ht).resolve_left (by norm_num)
-
-omit [Fintype X] in
-lemma alternator_coeff (ι : H →* G) (χ : H →* ℂ) (v : EuclideanSpace ℂ X) (x : X) :
-    alternator ι χ v x = ∑ h, χ h * v ((ι h)⁻¹ • x) := by
-  simp [alternator_apply, WithLp.ofLp_sum, Finset.sum_apply]
-
-omit [Fintype X] in
-lemma polytabloid_base (ι : H →* G) (χ : H →* ℂ) (b : X)
-    (hfree : ∀ h : H, ι h • b = b → h = 1) :
-    alternator ι χ (EuclideanSpace.single b (1 : ℂ)) b = 1 := by
-  classical
-  rw [alternator_coeff, Finset.sum_eq_single 1]
-  · simp
-  · intro h _ hh
-    have hn : (ι h)⁻¹ • b ≠ b := by
-      intro he
-      have hi : h⁻¹ = 1 := hfree h⁻¹ (by simpa using he)
-      exact hh (inv_eq_one.mp hi)
-    simp [hn]
-  · simp
-
-lemma polytabloid_inner (ι : H →* G) (χ : H →* ℂ) (b : X)
-    (hreal : ∀ h, conj (χ h) = χ h) (hinv : ∀ h, χ h⁻¹ = χ h)
-    (v : EuclideanSpace ℂ X) :
-    ⟪alternator ι χ (EuclideanSpace.single b (1 : ℂ)), v⟫_ℂ = alternator ι χ v b := by
-  classical
-  rw [alternator_apply, sum_inner, alternator_coeff]
-  simp only [inner_smul_left, rep_single, EuclideanSpace.inner_single_left, map_one,
-    one_mul, hreal]
-  have hh := Equiv.sum_comp (Equiv.inv H) (fun h => χ h * v ((ι h)⁻¹ • b))
-  simpa only [Equiv.inv_apply, map_inv, inv_inv, hinv] using hh
-
-theorem alternator_rank_one (ι : H →* G) (χ : H →* ℂ) (b : X)
-    (hreal : ∀ h, conj (χ h) = χ h) (hinv : ∀ h, χ h⁻¹ = χ h)
-    (hfree : ∀ h : H, ι h • b = b → h = 1)
-    (halt : ∀ x : X, (∃ a : H, ι a • x = x ∧ χ a = -1) ∨
-      ∃ a : H, x = ι a • b) (v : EuclideanSpace ℂ X) :
-    alternator ι χ v =
-      ⟪alternator ι χ (EuclideanSpace.single b (1 : ℂ)), v⟫_ℂ •
-        alternator ι χ (EuclideanSpace.single b (1 : ℂ)) := by
-  classical
-  let e := alternator ι χ (EuclideanSpace.single b (1 : ℂ))
-  have hs (x : X) : alternator ι χ (EuclideanSpace.single x (1 : ℂ)) ∈ Submodule.span ℂ {e} := by
-    rcases halt x with ⟨a, ha, hχ⟩ | ⟨a, rfl⟩
-    · rw [alternator_single_zero ι χ x a ha hχ]
-      exact Submodule.zero_mem _
-    · rw [← rep_single, alternator_translate]
-      exact Submodule.smul_mem _ _ (Submodule.subset_span (Set.mem_singleton e))
-  have hv : alternator ι χ v ∈ Submodule.span ℂ {e} := by
-    have hexp := (EuclideanSpace.basisFun X ℂ).sum_repr v
-    simp only [EuclideanSpace.basisFun_repr, EuclideanSpace.basisFun_apply] at hexp
-    rw [← hexp, map_sum]
-    exact Submodule.sum_mem _ fun x _ => by
-      rw [map_smul]
-      exact Submodule.smul_mem _ _ (hs x)
-  obtain ⟨a, ha⟩ := Submodule.mem_span_singleton.mp hv
-  have heb : e b = 1 := polytabloid_base ι χ b hfree
-  have hab : a = alternator ι χ v b := by
-    have hh := congrArg (fun z : EuclideanSpace ℂ X => z b) ha
-    simpa only [PiLp.smul_apply, smul_eq_mul, heb, mul_one] using hh
-  rw [← ha, hab, polytabloid_inner ι χ b hreal hinv]
-
-end PermutationSpace
-
-end Thorp.Specht
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.Parity
+
+namespace OAI
+
+universe uG uE uX uH
+
+noncomputable section
+
+open scoped BigOperators ComplexConjugate InnerProductSpace
+open Filter
+
+namespace Thorp.Specht
+
+variable {G : Type uG} [Group G]
+variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
+
+def cyclic (ρ : Representation ℂ G E) (e : E) : Subrepresentation ρ where
+  toSubmodule := Submodule.span ℂ (Set.range fun g => ρ g e)
+  apply_mem_toSubmodule g := by
+    intro v hv
+    induction hv using Submodule.span_induction with
+    | mem v hv =>
+      obtain ⟨h, rfl⟩ := hv
+      exact Submodule.subset_span ⟨g * h, by simp⟩
+    | zero => simp
+    | add x y hx hy ihx ihy =>
+      simpa using (Submodule.span ℂ (Set.range fun h => ρ h e)).add_mem ihx ihy
+    | smul a x hx ih =>
+      simpa using (Submodule.span ℂ (Set.range fun h => ρ h e)).smul_mem a ih
+
+lemma self_mem_cyclic (ρ : Representation ℂ G E) (e : E) : e ∈ cyclic ρ e := by
+  exact Submodule.subset_span ⟨1, by simp⟩
+
+theorem submodule_alternative (ρ : Representation ℂ G E) (e : E)
+    (hunit : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ)
+    (K : Module.End ℂ E) (hK : ∀ v, K v = ⟪e, v⟫_ℂ • e)
+    (hstable : ∀ U : Subrepresentation ρ, ∀ v ∈ U, K v ∈ U)
+    (U : Subrepresentation ρ) :
+    cyclic ρ e ≤ U ∨ ∀ u ∈ U, ∀ v ∈ cyclic ρ e, ⟪v, u⟫_ℂ = 0 := by
+  classical
+  by_cases he : e ∈ U
+  · left
+    exact Submodule.span_le.mpr (by rintro _ ⟨g, rfl⟩; exact U.apply_mem_toSubmodule g he)
+  · right
+    intro u hu v hv
+    have horth (g : G) : ⟪ρ g e, u⟫_ℂ = 0 := by
+      have hgu : ρ g⁻¹ u ∈ U := U.apply_mem_toSubmodule g⁻¹ hu
+      have hk := hstable U _ hgu
+      rw [hK] at hk
+      have hz : ⟪e, ρ g⁻¹ u⟫_ℂ = 0 := by
+        by_contra hne
+        apply he
+        have hh := U.toSubmodule.smul_mem (⟪e, ρ g⁻¹ u⟫_ℂ)⁻¹ hk
+        change e ∈ U.toSubmodule
+        simpa [smul_smul, hne] using hh
+      have hi := hunit g e (ρ g⁻¹ u)
+      simpa only [← Module.End.mul_apply, ← map_mul, mul_inv_cancel, map_one,
+        Module.End.one_apply] using hi.trans hz
+    induction hv using Submodule.span_induction with
+    | mem v hv => obtain ⟨g, rfl⟩ := hv; exact horth g
+    | zero => simp
+    | add x y hx hy ihx ihy => simp [ihx, ihy]
+    | smul a x hx ih => simp [ih]
+
+theorem cyclic_irreducible (ρ : Representation ℂ G E) (e : E) (he : e ≠ 0)
+    (hunit : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ)
+    (K : Module.End ℂ E) (hK : ∀ v, K v = ⟪e, v⟫_ℂ • e)
+    (hstable : ∀ U : Subrepresentation ρ, ∀ v ∈ U, K v ∈ U) :
+    Representation.IsIrreducible (cyclic ρ e).toRepresentation := by
+  let S := cyclic ρ e
+  let eS : S.toSubmodule := ⟨e, self_mem_cyclic ρ e⟩
+  have heS : eS ≠ 0 := by intro hh; apply he; exact congrArg Subtype.val hh
+  have htb : (⊤ : Subrepresentation S.toRepresentation) ≠ ⊥ := by
+    intro hh
+    have hel : eS ∈ (⊤ : Subrepresentation S.toRepresentation) := by trivial
+    rw [hh] at hel
+    exact heS hel
+  let : Nontrivial (Subrepresentation S.toRepresentation) := ⟨⟨⊤, ⊥, htb⟩⟩
+  refine ⟨fun U => ?_⟩
+  let U' : Subrepresentation ρ := {
+    toSubmodule := U.toSubmodule.map S.toSubmodule.subtype
+    apply_mem_toSubmodule := by
+      intro g v hv
+      obtain ⟨u, hu, rfl⟩ := hv
+      exact ⟨S.toRepresentation g u, U.apply_mem_toSubmodule g hu, rfl⟩ }
+  rcases submodule_alternative ρ e hunit K hK hstable U' with h | h
+  · right
+    apply top_unique
+    intro u _
+    obtain ⟨v, hv, heq⟩ := h u.property
+    have hvu : v = u := Subtype.ext heq
+    exact hvu ▸ hv
+  · left
+    apply bot_unique
+    intro u hu
+    have hum : (u : E) ∈ U' := ⟨u, hu, rfl⟩
+    have hh := h _ hum _ u.property
+    have hz : (u : E) = 0 := inner_self_eq_zero.mp hh
+    exact Subtype.ext hz
+
+namespace PermutationSpace
+open scoped Classical
+
+variable {X : Type uX} [Fintype X] [MulAction G X]
+
+def rep : Representation ℂ G (EuclideanSpace ℂ X) where
+  toFun g := {
+    toFun v := WithLp.toLp 2 (fun x => v (g⁻¹ • x))
+    map_add' := by intro v w; rfl
+    map_smul' := by intro a v; rfl }
+  map_one' := by ext v x; simp
+  map_mul' := by
+    intro g h
+    ext v x
+    change v ((g * h)⁻¹ • x) = v (h⁻¹ • g⁻¹ • x)
+    simp [mul_smul]
+
+omit [Fintype X] in
+@[simp] lemma rep_apply (g : G) (v : EuclideanSpace ℂ X) (x : X) :
+    rep g v x = v (g⁻¹ • x) := rfl
+
+lemma rep_unitary (g : G) (v w : EuclideanSpace ℂ X) :
+    ⟪rep g v, rep g w⟫_ℂ = ⟪v, w⟫_ℂ := by
+  simp only [PiLp.inner_apply, rep_apply]
+  exact Equiv.sum_comp (MulAction.toPerm g⁻¹) (fun x => inner ℂ (v x) (w x))
+
+omit [Fintype X] in
+lemma rep_single (g : G) (x : X) :
+    rep g (EuclideanSpace.single x (1 : ℂ)) = EuclideanSpace.single (g • x) (1 : ℂ) := by
+  classical
+  ext y
+  simp only [rep_apply, PiLp.single_apply, inv_smul_eq_iff]
+
+variable {H : Type uH} [Group H] [Fintype H]
+
+def alternator (ι : H →* G) (χ : H →* ℂ) : Module.End ℂ (EuclideanSpace ℂ X) :=
+  ∑ h, χ h • rep (ι h)
+
+omit [Fintype X] in
+lemma alternator_apply (ι : H →* G) (χ : H →* ℂ) (v : EuclideanSpace ℂ X) :
+    alternator ι χ v = ∑ h, χ h • rep (ι h) v := by
+  simp [alternator, LinearMap.sum_apply]
+
+omit [Fintype X] in
+lemma alternator_stable (ι : H →* G) (χ : H →* ℂ)
+    (U : Subrepresentation (rep : Representation ℂ G (EuclideanSpace ℂ X)))
+    (v : EuclideanSpace ℂ X) (hv : v ∈ U) : alternator ι χ v ∈ U := by
+  rw [alternator_apply]
+  exact U.toSubmodule.sum_mem fun h _ => U.toSubmodule.smul_mem _ (U.apply_mem_toSubmodule _ hv)
+
+omit [Fintype X] in
+lemma alternator_translate (ι : H →* G) (χ : H →* ℂ) (a : H)
+    (v : EuclideanSpace ℂ X) :
+    alternator ι χ (rep (ι a) v) = χ a⁻¹ • alternator ι χ v := by
+  rw [alternator_apply, alternator_apply, Finset.smul_sum]
+  have he := Equiv.sum_comp (Equiv.mulRight a)
+    (fun h => χ a⁻¹ • (χ h • (rep (ι h) v : EuclideanSpace ℂ X)))
+  rw [← he]
+  apply Finset.sum_congr rfl
+  intro h _
+  simp only [Equiv.coe_mulRight, smul_smul, map_mul]
+  rw [show χ a⁻¹ * (χ h * χ a) = χ h by
+    calc
+      _ = χ h * (χ a⁻¹ * χ a) := by ring
+      _ = χ h := by rw [← map_mul, inv_mul_cancel, map_one, mul_one]]
+  simp only [Module.End.mul_apply]
+
+omit [Fintype X] in
+lemma alternator_single_zero (ι : H →* G) (χ : H →* ℂ)
+    (x : X) (a : H) (ha : ι a • x = x) (hs : χ a = -1) :
+    alternator ι χ (EuclideanSpace.single x (1 : ℂ)) = 0 := by
+  have hh := alternator_translate ι χ a (EuclideanSpace.single x (1 : ℂ))
+  rw [rep_single, ha, map_inv, hs, inv_neg, inv_one, neg_smul, one_smul] at hh
+  have hz := congrArg (fun z => z + alternator ι χ (EuclideanSpace.single x (1 : ℂ))) hh
+  have ht : (2 : ℂ) • alternator ι χ (EuclideanSpace.single x (1 : ℂ)) = 0 := by
+    simpa [two_smul] using hz
+  exact (smul_eq_zero.mp ht).resolve_left (by norm_num)
+
+omit [Fintype X] in
+lemma alternator_coeff (ι : H →* G) (χ : H →* ℂ) (v : EuclideanSpace ℂ X) (x : X) :
+    alternator ι χ v x = ∑ h, χ h * v ((ι h)⁻¹ • x) := by
+  simp [alternator_apply, WithLp.ofLp_sum, Finset.sum_apply]
+
+omit [Fintype X] in
+lemma polytabloid_base (ι : H →* G) (χ : H →* ℂ) (b : X)
+    (hfree : ∀ h : H, ι h • b = b → h = 1) :
+    alternator ι χ (EuclideanSpace.single b (1 : ℂ)) b = 1 := by
+  classical
+  rw [alternator_coeff, Finset.sum_eq_single 1]
+  · simp
+  · intro h _ hh
+    have hn : (ι h)⁻¹ • b ≠ b := by
+      intro he
+      have hi : h⁻¹ = 1 := hfree h⁻¹ (by simpa using he)
+      exact hh (inv_eq_one.mp hi)
+    simp [hn]
+  · simp
+
+lemma polytabloid_inner (ι : H →* G) (χ : H →* ℂ) (b : X)
+    (hreal : ∀ h, conj (χ h) = χ h) (hinv : ∀ h, χ h⁻¹ = χ h)
+    (v : EuclideanSpace ℂ X) :
+    ⟪alternator ι χ (EuclideanSpace.single b (1 : ℂ)), v⟫_ℂ = alternator ι χ v b := by
+  classical
+  rw [alternator_apply, sum_inner, alternator_coeff]
+  simp only [inner_smul_left, rep_single, EuclideanSpace.inner_single_left, map_one,
+    one_mul, hreal]
+  have hh := Equiv.sum_comp (Equiv.inv H) (fun h => χ h * v ((ι h)⁻¹ • b))
+  simpa only [Equiv.inv_apply, map_inv, inv_inv, hinv] using hh
+
+theorem alternator_rank_one (ι : H →* G) (χ : H →* ℂ) (b : X)
+    (hreal : ∀ h, conj (χ h) = χ h) (hinv : ∀ h, χ h⁻¹ = χ h)
+    (hfree : ∀ h : H, ι h • b = b → h = 1)
+    (halt : ∀ x : X, (∃ a : H, ι a • x = x ∧ χ a = -1) ∨
+      ∃ a : H, x = ι a • b) (v : EuclideanSpace ℂ X) :
+    alternator ι χ v =
+      ⟪alternator ι χ (EuclideanSpace.single b (1 : ℂ)), v⟫_ℂ •
+        alternator ι χ (EuclideanSpace.single b (1 : ℂ)) := by
+  classical
+  let e := alternator ι χ (EuclideanSpace.single b (1 : ℂ))
+  have hs (x : X) : alternator ι χ (EuclideanSpace.single x (1 : ℂ)) ∈ Submodule.span ℂ {e} := by
+    rcases halt x with ⟨a, ha, hχ⟩ | ⟨a, rfl⟩
+    · rw [alternator_single_zero ι χ x a ha hχ]
+      exact Submodule.zero_mem _
+    · rw [← rep_single, alternator_translate]
+      exact Submodule.smul_mem _ _ (Submodule.subset_span (Set.mem_singleton e))
+  have hv : alternator ι χ v ∈ Submodule.span ℂ {e} := by
+    have hexp := (EuclideanSpace.basisFun X ℂ).sum_repr v
+    simp only [EuclideanSpace.basisFun_repr, EuclideanSpace.basisFun_apply] at hexp
+    rw [← hexp, map_sum]
+    exact Submodule.sum_mem _ fun x _ => by
+      rw [map_smul]
+      exact Submodule.smul_mem _ _ (hs x)
+  obtain ⟨a, ha⟩ := Submodule.mem_span_singleton.mp hv
+  have heb : e b = 1 := polytabloid_base ι χ b hfree
+  have hab : a = alternator ι χ v b := by
+    have hh := congrArg (fun z : EuclideanSpace ℂ X => z b) ha
+    simpa only [PiLp.smul_apply, smul_eq_mul, heb, mul_one] using hh
+  rw [← ha, hab, polytabloid_inner ι χ b hreal hinv]
+
+end PermutationSpace
+
+end Thorp.Specht
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/DegreeBounds.lean b/lean/OAI/Probability/ThorpShuffle/DegreeBounds.lean
index 93051a355f6644d559e0d85613ee2f27acdad7c5..6b778df155333be7a9e63a0e45c7737e51be5a59 100644
--- a/lean/OAI/Probability/ThorpShuffle/DegreeBounds.lean
+++ b/lean/OAI/Probability/ThorpShuffle/DegreeBounds.lean
@@ -1,238 +1,238 @@
-import OAI.Probability.ThorpShuffle.PartitionBounds
-
-namespace OAI
-
-noncomputable section
-
-open scoped BigOperators ComplexConjugate InnerProductSpace
-open Filter
-
-namespace Thorp.Young
-open scoped Classical
-
-def rowTail (D : YoungDiagram) : YoungDiagram where
-  cells := (D.cells.filter (fun p => p.1 ≠ 0)).image (fun p => (p.1 - 1, p.2))
-  isLowerSet := by
-    intro y x hxy hy
-    obtain ⟨p, hp, rfl⟩ := Finset.mem_image.mp hy
-    obtain ⟨hpD, hp0⟩ := Finset.mem_filter.mp hp
-    apply Finset.mem_image.mpr
-    refine ⟨(x.1 + 1, x.2), Finset.mem_filter.mpr ⟨?_, by omega⟩, ?_⟩
-    · exact D.up_left_mem (by have := hxy.1; omega) hxy.2 hpD
-    · simp
-
-lemma mem_rowTail (D : YoungDiagram) (i j : ℕ) : (i, j) ∈ rowTail D ↔ (i + 1, j) ∈ D := by
-  change (i, j) ∈ (D.cells.filter (fun p => p.1 ≠ 0)).image (fun p => (p.1 - 1, p.2)) ↔ _
-  constructor
-  · intro h
-    obtain ⟨p, hp, he⟩ := Finset.mem_image.mp h
-    obtain ⟨hpD, hp0⟩ := Finset.mem_filter.mp hp
-    have h₁ := congrArg Prod.fst he
-    have h₂ := congrArg Prod.snd he
-    have hp : p = (i + 1, j) := Prod.ext (by dsimp at h₁; omega) h₂
-    simpa only [← hp, YoungDiagram.mem_cells] using hpD
-  · intro h
-    exact Finset.mem_image.mpr ⟨(i + 1, j), Finset.mem_filter.mpr ⟨h, by omega⟩, by simp⟩
-
-def rowTailCells (D : YoungDiagram) : Cells (rowTail D) ≃ Lower D where
-  toFun x := ⟨⟨(x.val.1 + 1, x.val.2), (mem_rowTail D _ _).mp x.property⟩, by simp⟩
-  invFun x := ⟨(x.val.val.1 - 1, x.val.val.2), (mem_rowTail D _ _).mpr (by
-    have h : x.val.val.1 - 1 + 1 = x.val.val.1 := by have := x.property; omega
-    simpa only [h, YoungDiagram.mem_cells, Prod.mk.eta] using x.val.property)⟩
-  left_inv x := by apply Subtype.ext; simp
-  right_inv x := by
-    apply Subtype.ext; apply Subtype.ext
-    exact Prod.ext (by have := x.property; dsimp; omega) rfl
-
-lemma rowTail_ext {D E : YoungDiagram} (hrow : D.rowLen 0 = E.rowLen 0) (ht : rowTail D = rowTail E) : D = E := by
-  apply YoungDiagram.ext
-  ext p
-  rcases p with ⟨i, j⟩
-  cases i with
-  | zero => simp only [YoungDiagram.mem_cells, YoungDiagram.mem_iff_lt_rowLen, hrow]
-  | succ i => simp only [YoungDiagram.mem_cells, ← mem_rowTail, ht]
-
-def orient (D : YoungDiagram) : YoungDiagram :=
-  if D.colLen 0 ≤ D.rowLen 0 then D else D.transpose
-
-def orientation (D : YoungDiagram) : Bool := decide (D.colLen 0 ≤ D.rowLen 0)
-
-lemma orient_rowLen (D : YoungDiagram) : (orient D).rowLen 0 = longest D := by
-  unfold orient longest
-  split_ifs with h
-  · exact (max_eq_left h).symm
-  · rw [YoungDiagram.rowLen_transpose, max_eq_right (by omega)]
-
-lemma orient_card (D : YoungDiagram) : Fintype.card (Cells (orient D)) = Fintype.card (Cells D) := by
-  unfold orient
-  split_ifs
-  · rfl
-  · exact (Fintype.card_congr (transposeCells D).toEquiv).symm
-
-lemma orient_extensions (D : YoungDiagram) :
-    Fintype.card (Extensions (Cells (orient D))) = Fintype.card (Extensions (Cells D)) := by
-  by_cases h : D.colLen 0 ≤ D.rowLen 0
-  · exact congrArg (fun E : YoungDiagram => Fintype.card (Extensions (Cells E))) (show orient D = D by simp [orient, h])
-  · exact (congrArg (fun E : YoungDiagram => Fintype.card (Extensions (Cells E)))
-      (show orient D = D.transpose by simp [orient, h])).trans
-      (extensions_card_orderIso (transposeCells D)).symm
-
-lemma recover_orientation (D : YoungDiagram) :
-    (if orientation D then orient D else (orient D).transpose) = D := by
-  unfold orientation orient
-  split_ifs <;> simp_all
-
-lemma longest_le_card (D : YoungDiagram) : longest D ≤ Fintype.card (Cells D) :=
-  max_le (rowLen_le_card D 0) (colLen_le_card D 0)
-
-theorem defect_count_le (n u : ℕ) :
-    Fintype.card {p : Nat.Partition n // n - longest (ofPartition p) = u} ≤
-      2 * Fintype.card (Nat.Partition u) := by
-  let T := {p : Nat.Partition n // n - longest (ofPartition p) = u}
-  have hcard (p : T) : Fintype.card (Cells (rowTail (orient (ofPartition p.val)))) = u := by
-    rw [Fintype.card_congr (rowTailCells _)]
-    have hh := card_lower_add (orient (ofPartition p.val))
-    rw [orient_rowLen, orient_card, card_ofPartition] at hh
-    have hp := p.property
-    omega
-  let f (p : T) : Bool × Nat.Partition u :=
-    (orientation (ofPartition p.val), toPartition (rowTail (orient (ofPartition p.val))) u (hcard p))
-  have hf : Function.Injective f := by
-    intro p q hpq
-    apply Subtype.ext
-    apply ofPartition_injective
-    have hflag : orientation (ofPartition p.val) = orientation (ofPartition q.val) := congrArg Prod.fst hpq
-    have ht : rowTail (orient (ofPartition p.val)) = rowTail (orient (ofPartition q.val)) := by
-      have he := congrArg (fun z : Bool × Nat.Partition u => ofPartition z.2) hpq
-      simpa only [f, of_toPartition] using he
-    have hr : (orient (ofPartition p.val)).rowLen 0 = (orient (ofPartition q.val)).rowLen 0 := by
-      simp only [orient_rowLen]
-      have hp := p.property
-      have hq := q.property
-      have hp' := longest_le_card (ofPartition p.val)
-      have hq' := longest_le_card (ofPartition q.val)
-      rw [card_ofPartition] at hp' hq'
-      omega
-    have ho := rowTail_ext hr ht
-    rw [← recover_orientation (ofPartition p.val), ← recover_orientation (ofPartition q.val), hflag, ho]
-  have hh := Fintype.card_le_of_injective f hf
-  simpa only [Fintype.card_prod, Fintype.card_bool] using hh
-
-end Thorp.Young
-
-namespace Thorp.Specht
-open scoped Classical
-open Thorp.Young
-
-abbrev NShape (n : ℕ) := Shape (Fin n)
-
-def defect {n : ℕ} (p : NShape n) : ℕ := n - longest (ofPartition p)
-
-lemma diagram_card {n : ℕ} (p : NShape n) : Fintype.card (Cells (ofPartition p)) = n := by
-  simpa only [Fintype.card_fin] using card_ofPartition p
-
-lemma extensions_le_degree {n : ℕ} (p : NShape n) :
-    Fintype.card (Extensions (Cells (ofPartition p))) ≤ degree p :=
-  extensions_card_le_degree (ofPartition p) (canonicalTableau p)
-
-lemma degree_pos {n : ℕ} (p : NShape n) : 0 < degree p := by
-  let := extensions_nonempty (Cells (ofPartition p))
-  exact (Fintype.card_pos).trans_le (extensions_le_degree p)
-
-lemma defect_add_longest {n : ℕ} (p : NShape n) : defect p + longest (ofPartition p) = n := by
-  have h := longest_le_card (ofPartition p)
-  rw [diagram_card] at h
-  exact Nat.sub_add_cancel h
-
-lemma choose_longest_le_degree {n : ℕ} (p : NShape n) (k : ℕ)
-    (hk : k ≤ defect p) (hL : k ≤ longest (ofPartition p)) :
-    (longest (ofPartition p)).choose k ≤ degree p := by
-  have hc : Fintype.card (Lower (orient (ofPartition p))) = defect p := by
-    have hh := card_lower_add (orient (ofPartition p))
-    rw [orient_rowLen, orient_card, diagram_card] at hh
-    have hd := defect_add_longest p
-    omega
-  have hh := choose_le_extensions (orient (ofPartition p)) k (by rwa [hc]) (by rwa [orient_rowLen])
-  rw [orient_rowLen, orient_extensions] at hh
-  exact hh.trans (extensions_le_degree p)
-
-lemma small_defect_degree {n : ℕ} (p : NShape n) (h : 4 * defect p ≤ n) :
-    3 ^ defect p ≤ degree p := by
-  have hd := defect_add_longest p
-  exact (pow_le_choose_of_mul_le 3 _ _ (by omega)).trans
-    (choose_longest_le_degree p _ le_rfl (by omega))
-
-lemma fixed_defect_degree {n : ℕ} (p : NShape n) (hp : 0 < defect p) (h : 2 * defect p ≤ n) :
-    n - 2 * defect p + 1 ≤ degree p := by
-  have hd := defect_add_longest p
-  have hh := sub_add_one_le_choose (longest (ofPartition p)) (defect p) hp (by omega)
-  have he : longest (ofPartition p) - defect p = n - 2 * defect p := by omega
-  rw [he] at hh
-  exact hh.trans (choose_longest_le_degree p _ le_rfl (by omega))
-
-lemma factorial_third_bound (n : ℕ) : ((n : ℝ) / 3) ^ n ≤ (n.factorial : ℝ) := by
-  rcases n with _ | n
-  · norm_num
-  have hn : (1 : ℝ) ≤ n + 1 := by have := Nat.cast_nonneg (α := ℝ) n; linarith
-  have hs : (1 : ℝ) ≤ Real.sqrt (2 * Real.pi * (n + 1)) := by
-    apply Real.le_sqrt_of_sq_le
-    nlinarith [Real.pi_gt_three]
-  have he : (n + 1 : ℝ) / 3 ≤ (n + 1) / Real.exp 1 := by
-    apply div_le_div_of_nonneg_left (by positivity) (Real.exp_pos 1) Real.exp_one_lt_three.le
-  have hh := Stirling.le_factorial_stirling (n + 1)
-  push_cast at hh
-  simpa only [Nat.cast_add, Nat.cast_one] using (pow_le_pow_left₀ (by positivity) he (n+1)).trans
-    ((le_mul_of_one_le_left (by positivity) hs).trans hh)
-
-lemma small_longest_degree {n : ℕ} (p : NShape n) (hn : 0 < n)
-    (hL : 8 * longest (ofPartition p) ≤ n) :
-    (4 / 3 : ℝ) ^ n ≤ degree p := by
-  have hf := factorial_le_rank_degree (ofPartition p) (canonicalTableau p)
-  have hf' : (n.factorial : ℝ) ≤ (2 * (longest (ofPartition p) : ℝ)) ^ n * (degree p : ℝ) := by
-    exact_mod_cast hf
-  have he : (2 * (longest (ofPartition p) : ℝ)) ≤ (n : ℝ) / 4 := by
-    have hh : (8 : ℝ) * longest (ofPartition p) ≤ n := by exact_mod_cast hL
-    linarith
-  have hd : (0 : ℝ) ≤ degree p := Nat.cast_nonneg _
-  have hh := (factorial_third_bound n).trans (hf'.trans (mul_le_mul_of_nonneg_right (pow_le_pow_left₀ (by positivity) he n) hd))
-  have hn' : (0 : ℝ) < n := by exact_mod_cast hn
-  have hid : ((n : ℝ) / 3) ^ n = ((n : ℝ) / 4) ^ n * (4 / 3 : ℝ)^n := by rw [← mul_pow]; congr 1; ring
-  rw [hid] at hh
-  exact le_of_mul_le_mul_left hh (pow_pos (by positivity : (0 : ℝ) < n / 4) n)
-
-lemma bulk_degree {n : ℕ} (p : NShape n) (hn : 0 < n) (hu : n < 4 * defect p) :
-    (2 : ℝ) ^ (n / 16) ≤ degree p := by
-  by_cases hL : 8 * longest (ofPartition p) ≤ n
-  · have hh : (2 : ℝ) ^ (n / 16) ≤ (4 / 3 : ℝ)^n := by
-      have hbase : (2 : ℝ) ≤ (4 / 3 : ℝ)^16 := by norm_num
-      calc
-        _ ≤ ((4 / 3 : ℝ)^16)^(n/16) := pow_le_pow_left₀ (by norm_num) hbase _
-        _ = (4 / 3 : ℝ)^(16 * (n/16)) := by rw [pow_mul]
-        _ ≤ _ := pow_le_pow_right₀ (by norm_num) (by omega)
-    exact hh.trans (small_longest_degree p hn hL)
-  · have hd := defect_add_longest p
-    have hk : n / 16 ≤ defect p := by omega
-    have hm : 2 * (n / 16) ≤ longest (ofPartition p) := by omega
-    exact_mod_cast (pow_le_choose_of_mul_le 2 _ _ hm).trans
-      (choose_longest_le_degree p _ hk (by omega))
-
-lemma defect_card_bound (n u : ℕ) :
-    Fintype.card {p : NShape n // defect p = u} ≤ 2 ^ (u + 1) := by
-  have hh : Nat.card {p : Nat.Partition n // n - longest (ofPartition p) = u} ≤ 2 ^ (u+1) := by
-    rw [← Fintype.card_eq_nat_card]
-    calc
-      _ ≤ 2 * Fintype.card (Nat.Partition u) := defect_count_le n u
-      _ ≤ 2 * 2^u := Nat.mul_le_mul_left 2 (partition_count_le_pow u)
-      _ = _ := by ring
-  rw [Fintype.card_eq_nat_card]
-  change Nat.card {p : Nat.Partition (Fintype.card (Fin n)) // n - longest (ofPartition p) = u} ≤ _
-  have he (m k : ℕ) (h : m = k) :
-      Nat.card {p : Nat.Partition m // n - longest (ofPartition p) = u} =
-      Nat.card {p : Nat.Partition k // n - longest (ofPartition p) = u} := by subst k; rfl
-  exact (he _ _ (Fintype.card_fin n)).trans_le hh
-
-end Thorp.Specht
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.PartitionBounds
+
+namespace OAI
+
+noncomputable section
+
+open scoped BigOperators ComplexConjugate InnerProductSpace
+open Filter
+
+namespace Thorp.Young
+open scoped Classical
+
+def rowTail (D : YoungDiagram) : YoungDiagram where
+  cells := (D.cells.filter (fun p => p.1 ≠ 0)).image (fun p => (p.1 - 1, p.2))
+  isLowerSet := by
+    intro y x hxy hy
+    obtain ⟨p, hp, rfl⟩ := Finset.mem_image.mp hy
+    obtain ⟨hpD, hp0⟩ := Finset.mem_filter.mp hp
+    apply Finset.mem_image.mpr
+    refine ⟨(x.1 + 1, x.2), Finset.mem_filter.mpr ⟨?_, by omega⟩, ?_⟩
+    · exact D.up_left_mem (by have := hxy.1; omega) hxy.2 hpD
+    · simp
+
+lemma mem_rowTail (D : YoungDiagram) (i j : ℕ) : (i, j) ∈ rowTail D ↔ (i + 1, j) ∈ D := by
+  change (i, j) ∈ (D.cells.filter (fun p => p.1 ≠ 0)).image (fun p => (p.1 - 1, p.2)) ↔ _
+  constructor
+  · intro h
+    obtain ⟨p, hp, he⟩ := Finset.mem_image.mp h
+    obtain ⟨hpD, hp0⟩ := Finset.mem_filter.mp hp
+    have h₁ := congrArg Prod.fst he
+    have h₂ := congrArg Prod.snd he
+    have hp : p = (i + 1, j) := Prod.ext (by dsimp at h₁; omega) h₂
+    simpa only [← hp, YoungDiagram.mem_cells] using hpD
+  · intro h
+    exact Finset.mem_image.mpr ⟨(i + 1, j), Finset.mem_filter.mpr ⟨h, by omega⟩, by simp⟩
+
+def rowTailCells (D : YoungDiagram) : Cells (rowTail D) ≃ Lower D where
+  toFun x := ⟨⟨(x.val.1 + 1, x.val.2), (mem_rowTail D _ _).mp x.property⟩, by simp⟩
+  invFun x := ⟨(x.val.val.1 - 1, x.val.val.2), (mem_rowTail D _ _).mpr (by
+    have h : x.val.val.1 - 1 + 1 = x.val.val.1 := by have := x.property; omega
+    simpa only [h, YoungDiagram.mem_cells, Prod.mk.eta] using x.val.property)⟩
+  left_inv x := by apply Subtype.ext; simp
+  right_inv x := by
+    apply Subtype.ext; apply Subtype.ext
+    exact Prod.ext (by have := x.property; dsimp; omega) rfl
+
+lemma rowTail_ext {D E : YoungDiagram} (hrow : D.rowLen 0 = E.rowLen 0) (ht : rowTail D = rowTail E) : D = E := by
+  apply YoungDiagram.ext
+  ext p
+  rcases p with ⟨i, j⟩
+  cases i with
+  | zero => simp only [YoungDiagram.mem_cells, YoungDiagram.mem_iff_lt_rowLen, hrow]
+  | succ i => simp only [YoungDiagram.mem_cells, ← mem_rowTail, ht]
+
+def orient (D : YoungDiagram) : YoungDiagram :=
+  if D.colLen 0 ≤ D.rowLen 0 then D else D.transpose
+
+def orientation (D : YoungDiagram) : Bool := decide (D.colLen 0 ≤ D.rowLen 0)
+
+lemma orient_rowLen (D : YoungDiagram) : (orient D).rowLen 0 = longest D := by
+  unfold orient longest
+  split_ifs with h
+  · exact (max_eq_left h).symm
+  · rw [YoungDiagram.rowLen_transpose, max_eq_right (by omega)]
+
+lemma orient_card (D : YoungDiagram) : Fintype.card (Cells (orient D)) = Fintype.card (Cells D) := by
+  unfold orient
+  split_ifs
+  · rfl
+  · exact (Fintype.card_congr (transposeCells D).toEquiv).symm
+
+lemma orient_extensions (D : YoungDiagram) :
+    Fintype.card (Extensions (Cells (orient D))) = Fintype.card (Extensions (Cells D)) := by
+  by_cases h : D.colLen 0 ≤ D.rowLen 0
+  · exact congrArg (fun E : YoungDiagram => Fintype.card (Extensions (Cells E))) (show orient D = D by simp [orient, h])
+  · exact (congrArg (fun E : YoungDiagram => Fintype.card (Extensions (Cells E)))
+      (show orient D = D.transpose by simp [orient, h])).trans
+      (extensions_card_orderIso (transposeCells D)).symm
+
+lemma recover_orientation (D : YoungDiagram) :
+    (if orientation D then orient D else (orient D).transpose) = D := by
+  unfold orientation orient
+  split_ifs <;> simp_all
+
+lemma longest_le_card (D : YoungDiagram) : longest D ≤ Fintype.card (Cells D) :=
+  max_le (rowLen_le_card D 0) (colLen_le_card D 0)
+
+theorem defect_count_le (n u : ℕ) :
+    Fintype.card {p : Nat.Partition n // n - longest (ofPartition p) = u} ≤
+      2 * Fintype.card (Nat.Partition u) := by
+  let T := {p : Nat.Partition n // n - longest (ofPartition p) = u}
+  have hcard (p : T) : Fintype.card (Cells (rowTail (orient (ofPartition p.val)))) = u := by
+    rw [Fintype.card_congr (rowTailCells _)]
+    have hh := card_lower_add (orient (ofPartition p.val))
+    rw [orient_rowLen, orient_card, card_ofPartition] at hh
+    have hp := p.property
+    omega
+  let f (p : T) : Bool × Nat.Partition u :=
+    (orientation (ofPartition p.val), toPartition (rowTail (orient (ofPartition p.val))) u (hcard p))
+  have hf : Function.Injective f := by
+    intro p q hpq
+    apply Subtype.ext
+    apply ofPartition_injective
+    have hflag : orientation (ofPartition p.val) = orientation (ofPartition q.val) := congrArg Prod.fst hpq
+    have ht : rowTail (orient (ofPartition p.val)) = rowTail (orient (ofPartition q.val)) := by
+      have he := congrArg (fun z : Bool × Nat.Partition u => ofPartition z.2) hpq
+      simpa only [f, of_toPartition] using he
+    have hr : (orient (ofPartition p.val)).rowLen 0 = (orient (ofPartition q.val)).rowLen 0 := by
+      simp only [orient_rowLen]
+      have hp := p.property
+      have hq := q.property
+      have hp' := longest_le_card (ofPartition p.val)
+      have hq' := longest_le_card (ofPartition q.val)
+      rw [card_ofPartition] at hp' hq'
+      omega
+    have ho := rowTail_ext hr ht
+    rw [← recover_orientation (ofPartition p.val), ← recover_orientation (ofPartition q.val), hflag, ho]
+  have hh := Fintype.card_le_of_injective f hf
+  simpa only [Fintype.card_prod, Fintype.card_bool] using hh
+
+end Thorp.Young
+
+namespace Thorp.Specht
+open scoped Classical
+open Thorp.Young
+
+abbrev NShape (n : ℕ) := Shape (Fin n)
+
+def defect {n : ℕ} (p : NShape n) : ℕ := n - longest (ofPartition p)
+
+lemma diagram_card {n : ℕ} (p : NShape n) : Fintype.card (Cells (ofPartition p)) = n := by
+  simpa only [Fintype.card_fin] using card_ofPartition p
+
+lemma extensions_le_degree {n : ℕ} (p : NShape n) :
+    Fintype.card (Extensions (Cells (ofPartition p))) ≤ degree p :=
+  extensions_card_le_degree (ofPartition p) (canonicalTableau p)
+
+lemma degree_pos {n : ℕ} (p : NShape n) : 0 < degree p := by
+  let := extensions_nonempty (Cells (ofPartition p))
+  exact (Fintype.card_pos).trans_le (extensions_le_degree p)
+
+lemma defect_add_longest {n : ℕ} (p : NShape n) : defect p + longest (ofPartition p) = n := by
+  have h := longest_le_card (ofPartition p)
+  rw [diagram_card] at h
+  exact Nat.sub_add_cancel h
+
+lemma choose_longest_le_degree {n : ℕ} (p : NShape n) (k : ℕ)
+    (hk : k ≤ defect p) (hL : k ≤ longest (ofPartition p)) :
+    (longest (ofPartition p)).choose k ≤ degree p := by
+  have hc : Fintype.card (Lower (orient (ofPartition p))) = defect p := by
+    have hh := card_lower_add (orient (ofPartition p))
+    rw [orient_rowLen, orient_card, diagram_card] at hh
+    have hd := defect_add_longest p
+    omega
+  have hh := choose_le_extensions (orient (ofPartition p)) k (by rwa [hc]) (by rwa [orient_rowLen])
+  rw [orient_rowLen, orient_extensions] at hh
+  exact hh.trans (extensions_le_degree p)
+
+lemma small_defect_degree {n : ℕ} (p : NShape n) (h : 4 * defect p ≤ n) :
+    3 ^ defect p ≤ degree p := by
+  have hd := defect_add_longest p
+  exact (pow_le_choose_of_mul_le 3 _ _ (by omega)).trans
+    (choose_longest_le_degree p _ le_rfl (by omega))
+
+lemma fixed_defect_degree {n : ℕ} (p : NShape n) (hp : 0 < defect p) (h : 2 * defect p ≤ n) :
+    n - 2 * defect p + 1 ≤ degree p := by
+  have hd := defect_add_longest p
+  have hh := sub_add_one_le_choose (longest (ofPartition p)) (defect p) hp (by omega)
+  have he : longest (ofPartition p) - defect p = n - 2 * defect p := by omega
+  rw [he] at hh
+  exact hh.trans (choose_longest_le_degree p _ le_rfl (by omega))
+
+lemma factorial_third_bound (n : ℕ) : ((n : ℝ) / 3) ^ n ≤ (n.factorial : ℝ) := by
+  rcases n with _ | n
+  · norm_num
+  have hn : (1 : ℝ) ≤ n + 1 := by have := Nat.cast_nonneg (α := ℝ) n; linarith
+  have hs : (1 : ℝ) ≤ Real.sqrt (2 * Real.pi * (n + 1)) := by
+    apply Real.le_sqrt_of_sq_le
+    nlinarith [Real.pi_gt_three]
+  have he : (n + 1 : ℝ) / 3 ≤ (n + 1) / Real.exp 1 := by
+    apply div_le_div_of_nonneg_left (by positivity) (Real.exp_pos 1) Real.exp_one_lt_three.le
+  have hh := Stirling.le_factorial_stirling (n + 1)
+  push_cast at hh
+  simpa only [Nat.cast_add, Nat.cast_one] using (pow_le_pow_left₀ (by positivity) he (n+1)).trans
+    ((le_mul_of_one_le_left (by positivity) hs).trans hh)
+
+lemma small_longest_degree {n : ℕ} (p : NShape n) (hn : 0 < n)
+    (hL : 8 * longest (ofPartition p) ≤ n) :
+    (4 / 3 : ℝ) ^ n ≤ degree p := by
+  have hf := factorial_le_rank_degree (ofPartition p) (canonicalTableau p)
+  have hf' : (n.factorial : ℝ) ≤ (2 * (longest (ofPartition p) : ℝ)) ^ n * (degree p : ℝ) := by
+    exact_mod_cast hf
+  have he : (2 * (longest (ofPartition p) : ℝ)) ≤ (n : ℝ) / 4 := by
+    have hh : (8 : ℝ) * longest (ofPartition p) ≤ n := by exact_mod_cast hL
+    linarith
+  have hd : (0 : ℝ) ≤ degree p := Nat.cast_nonneg _
+  have hh := (factorial_third_bound n).trans (hf'.trans (mul_le_mul_of_nonneg_right (pow_le_pow_left₀ (by positivity) he n) hd))
+  have hn' : (0 : ℝ) < n := by exact_mod_cast hn
+  have hid : ((n : ℝ) / 3) ^ n = ((n : ℝ) / 4) ^ n * (4 / 3 : ℝ)^n := by rw [← mul_pow]; congr 1; ring
+  rw [hid] at hh
+  exact le_of_mul_le_mul_left hh (pow_pos (by positivity : (0 : ℝ) < n / 4) n)
+
+lemma bulk_degree {n : ℕ} (p : NShape n) (hn : 0 < n) (hu : n < 4 * defect p) :
+    (2 : ℝ) ^ (n / 16) ≤ degree p := by
+  by_cases hL : 8 * longest (ofPartition p) ≤ n
+  · have hh : (2 : ℝ) ^ (n / 16) ≤ (4 / 3 : ℝ)^n := by
+      have hbase : (2 : ℝ) ≤ (4 / 3 : ℝ)^16 := by norm_num
+      calc
+        _ ≤ ((4 / 3 : ℝ)^16)^(n/16) := pow_le_pow_left₀ (by norm_num) hbase _
+        _ = (4 / 3 : ℝ)^(16 * (n/16)) := by rw [pow_mul]
+        _ ≤ _ := pow_le_pow_right₀ (by norm_num) (by omega)
+    exact hh.trans (small_longest_degree p hn hL)
+  · have hd := defect_add_longest p
+    have hk : n / 16 ≤ defect p := by omega
+    have hm : 2 * (n / 16) ≤ longest (ofPartition p) := by omega
+    exact_mod_cast (pow_le_choose_of_mul_le 2 _ _ hm).trans
+      (choose_longest_le_degree p _ hk (by omega))
+
+lemma defect_card_bound (n u : ℕ) :
+    Fintype.card {p : NShape n // defect p = u} ≤ 2 ^ (u + 1) := by
+  have hh : Nat.card {p : Nat.Partition n // n - longest (ofPartition p) = u} ≤ 2 ^ (u+1) := by
+    rw [← Fintype.card_eq_nat_card]
+    calc
+      _ ≤ 2 * Fintype.card (Nat.Partition u) := defect_count_le n u
+      _ ≤ 2 * 2^u := Nat.mul_le_mul_left 2 (partition_count_le_pow u)
+      _ = _ := by ring
+  rw [Fintype.card_eq_nat_card]
+  change Nat.card {p : Nat.Partition (Fintype.card (Fin n)) // n - longest (ofPartition p) = u} ≤ _
+  have he (m k : ℕ) (h : m = k) :
+      Nat.card {p : Nat.Partition m // n - longest (ofPartition p) = u} =
+      Nat.card {p : Nat.Partition k // n - longest (ofPartition p) = u} := by subst k; rfl
+  exact (he _ _ (Fintype.card_fin n)).trans_le hh
+
+end Thorp.Specht
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/EightBlocks.lean b/lean/OAI/Probability/ThorpShuffle/EightBlocks.lean
index 456e632eb209ca80ad1b896112dbb6a8e4bce7a0..2a42947dc82b39056f924c5cb0db4fc11720230b 100644
--- a/lean/OAI/Probability/ThorpShuffle/EightBlocks.lean
+++ b/lean/OAI/Probability/ThorpShuffle/EightBlocks.lean
@@ -1,98 +1,98 @@
-import OAI.Probability.ThorpShuffle.BlockLaws
-
-namespace OAI
-
-noncomputable section
-
-open scoped BigOperators ComplexConjugate InnerProductSpace
-open Filter Topology
-
-namespace Thorp
-open scoped Classical
-
-lemma law_complexSign_zero (d t : ℕ) (hd : 1 ≤ d) (ht : 1 ≤ t) :
-    (∑ g : State d, (law d t g : ℂ) * Specht.complexSign g) = 0 := by
-  obtain ⟨d,rfl⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : d ≠ 0)
-  obtain ⟨t,rfl⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : t ≠ 0)
-  have h := congrArg (fun x : ℝ => (x : ℂ)) (Parity.law_sign_zero d t)
-  simpa only [Complex.ofReal_sum, Complex.ofReal_mul, Complex.ofReal_zero, Parity.sign,
-    Specht.complexSign_eq, Complex.ofReal_intCast] using h
-
-namespace Block
-open Thorp.Fourier Thorp.Specht
-
-def fullRate (d : ℕ) : ℝ :=
-  8 * discarded d + (1/2 : ℝ) * Real.sqrt
-    (2 * (2 * discarded d)^16 + (96 : ℝ)^8 * exceptionalReciprocalSum (2^(d+1+2)))
-
-lemma exceptional_cube_tendsto :
-    Tendsto (fun d : ℕ => exceptionalReciprocalSum (2^d)) atTop (nhds 0) :=
-  exceptionalReciprocalSum_tendsto.comp
-    (tendsto_pow_atTop_atTop_of_one_lt (by norm_num : (1 : ℕ) < 2))
-
-lemma fullRate_tendsto : Tendsto fullRate atTop (nhds 0) := by
-  change Tendsto (fun d : ℕ => 8 * discarded d + (1/2 : ℝ) * Real.sqrt
-    (2 * (2 * discarded d)^16 + (96 : ℝ)^8 * exceptionalReciprocalSum (2^(d+1+2)))) atTop (nhds 0)
-  have he := exceptional_cube_tendsto.comp (tendsto_add_atTop_nat 3)
-  have hb := ((discarded_tendsto.const_mul 2).pow 16).const_mul 2
-  have hs := (hb.add (he.const_mul ((96:ℝ)^8))).sqrt
-  simpa only [Nat.add_assoc, Function.comp_def, mul_zero, zero_pow (by decide : 16 ≠ 0),
-    add_zero, Real.sqrt_zero] using (discarded_tendsto.const_mul 8).add (hs.const_mul (1/2 : ℝ))
-
-lemma reciprocal_atMost_three :
-    ∀ᶠ d : ℕ in atTop, (permutationFamily (Position d)).reciprocalSum ≤ 3 := by
-  have he : ∀ᶠ d : ℕ in atTop, exceptionalReciprocalSum (2^d) < 1 :=
-    (tendsto_order.1 exceptional_cube_tendsto).2 1 (by norm_num)
-  filter_upwards [he] with d hd
-  have h := full_reciprocal_bound (2^d)
-  have hc : (permutationFamily (Position d)).reciprocalSum = (symmetricFamily (2^d)).reciprocalSum := by
-    change (symmetricFamily (Fintype.card (Position d))).reciprocalSum = _
-    rw [card_position]
-  rw [hc]
-  linarith
-
-lemma distance_eight_eventually :
-    ∀ᶠ d : ℕ in atTop, distance (d+1+2) (1600*(d+1+2)) ≤ fullRate d := by
-  have hd : ∀ᶠ d : ℕ in atTop, discarded d ≤ 1/2 :=
-    ((tendsto_order.1 discarded_tendsto).2 (1/2) (by norm_num)).mono (fun _ h => h.le)
-  filter_upwards [hd, reciprocal_atMost_three, eventually_ge_atTop 1] with d hδ hR hd1
-  have hn : 16 ≤ Fintype.card (Position (d+1+2)) := by
-    rw [card_position]
-    calc
-      16 = 2^4 := by norm_num
-      _ ≤ 2^(d+1+2) := Nat.pow_le_pow_right (by norm_num) (by omega)
-  let : Nontrivial (Position (d+1+2)) := Fintype.one_lt_card_iff_nontrivial.mp (by omega)
-  have hbound := trimmed_eight_tv hn (embedding (eightEquiv d)) (law (d+1+2) (200*(d+1+2)))
-    (law_nonneg (d+1+2) (200*(d+1+2))) (law_sum (d+1+2) (200*(d+1+2)))
-    (law_complexSign_zero _ _ (by omega) (by omega)) hδ
-  have hpower : law (d+1+2) (1600*(d+1+2)) = realPower (law (d+1+2) (200*(d+1+2))) 8 := by
-    rw [show 1600*(d+1+2) = 8 * (200*(d+1+2)) by omega]
-    exact law_realPower (d+1+2) (200*(d+1+2)) 8
-  change tv (law (d+1+2) (1600*(d+1+2)))
-    (fun _ => 1 / (Fintype.card (State (d+1+2)) : ℝ)) ≤ fullRate d
-  simp only [one_div]
-  rw [hpower]
-  apply hbound.trans
-  unfold fullRate
-  refine add_le_add (le_refl _) ?_
-  apply mul_le_mul_of_nonneg_left _ (by norm_num)
-  apply Real.sqrt_le_sqrt
-  refine add_le_add (le_refl _) ?_
-  rw [card_position]
-  apply mul_le_mul_of_nonneg_right _ (Finset.sum_nonneg (fun p _ => by
-    change 0 ≤ (if 0 < defect p then inverseDegree p else 0)
-    split_ifs <;> first | exact inverseDegree_nonneg p | exact le_rfl))
-  apply pow_le_pow_left₀ (mul_nonneg (by norm_num) (permutationFamily (Position d)).reciprocalSum_nonneg)
-  linarith
-
-lemma full_mixing_shift :
-    Tendsto (fun d : ℕ => distance (d+1+2) (1600*(d+1+2))) atTop (nhds 0) := by
-  apply squeeze_zero' (Eventually.of_forall (fun d => by unfold distance tv; positivity))
-    distance_eight_eventually fullRate_tendsto
-
-end Block
-end Thorp
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.BlockLaws
+
+namespace OAI
+
+noncomputable section
+
+open scoped BigOperators ComplexConjugate InnerProductSpace
+open Filter Topology
+
+namespace Thorp
+open scoped Classical
+
+lemma law_complexSign_zero (d t : ℕ) (hd : 1 ≤ d) (ht : 1 ≤ t) :
+    (∑ g : State d, (law d t g : ℂ) * Specht.complexSign g) = 0 := by
+  obtain ⟨d,rfl⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : d ≠ 0)
+  obtain ⟨t,rfl⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : t ≠ 0)
+  have h := congrArg (fun x : ℝ => (x : ℂ)) (Parity.law_sign_zero d t)
+  simpa only [Complex.ofReal_sum, Complex.ofReal_mul, Complex.ofReal_zero, Parity.sign,
+    Specht.complexSign_eq, Complex.ofReal_intCast] using h
+
+namespace Block
+open Thorp.Fourier Thorp.Specht
+
+def fullRate (d : ℕ) : ℝ :=
+  8 * discarded d + (1/2 : ℝ) * Real.sqrt
+    (2 * (2 * discarded d)^16 + (96 : ℝ)^8 * exceptionalReciprocalSum (2^(d+1+2)))
+
+lemma exceptional_cube_tendsto :
+    Tendsto (fun d : ℕ => exceptionalReciprocalSum (2^d)) atTop (nhds 0) :=
+  exceptionalReciprocalSum_tendsto.comp
+    (tendsto_pow_atTop_atTop_of_one_lt (by norm_num : (1 : ℕ) < 2))
+
+lemma fullRate_tendsto : Tendsto fullRate atTop (nhds 0) := by
+  change Tendsto (fun d : ℕ => 8 * discarded d + (1/2 : ℝ) * Real.sqrt
+    (2 * (2 * discarded d)^16 + (96 : ℝ)^8 * exceptionalReciprocalSum (2^(d+1+2)))) atTop (nhds 0)
+  have he := exceptional_cube_tendsto.comp (tendsto_add_atTop_nat 3)
+  have hb := ((discarded_tendsto.const_mul 2).pow 16).const_mul 2
+  have hs := (hb.add (he.const_mul ((96:ℝ)^8))).sqrt
+  simpa only [Nat.add_assoc, Function.comp_def, mul_zero, zero_pow (by decide : 16 ≠ 0),
+    add_zero, Real.sqrt_zero] using (discarded_tendsto.const_mul 8).add (hs.const_mul (1/2 : ℝ))
+
+lemma reciprocal_atMost_three :
+    ∀ᶠ d : ℕ in atTop, (permutationFamily (Position d)).reciprocalSum ≤ 3 := by
+  have he : ∀ᶠ d : ℕ in atTop, exceptionalReciprocalSum (2^d) < 1 :=
+    (tendsto_order.1 exceptional_cube_tendsto).2 1 (by norm_num)
+  filter_upwards [he] with d hd
+  have h := full_reciprocal_bound (2^d)
+  have hc : (permutationFamily (Position d)).reciprocalSum = (symmetricFamily (2^d)).reciprocalSum := by
+    change (symmetricFamily (Fintype.card (Position d))).reciprocalSum = _
+    rw [card_position]
+  rw [hc]
+  linarith
+
+lemma distance_eight_eventually :
+    ∀ᶠ d : ℕ in atTop, distance (d+1+2) (1600*(d+1+2)) ≤ fullRate d := by
+  have hd : ∀ᶠ d : ℕ in atTop, discarded d ≤ 1/2 :=
+    ((tendsto_order.1 discarded_tendsto).2 (1/2) (by norm_num)).mono (fun _ h => h.le)
+  filter_upwards [hd, reciprocal_atMost_three, eventually_ge_atTop 1] with d hδ hR hd1
+  have hn : 16 ≤ Fintype.card (Position (d+1+2)) := by
+    rw [card_position]
+    calc
+      16 = 2^4 := by norm_num
+      _ ≤ 2^(d+1+2) := Nat.pow_le_pow_right (by norm_num) (by omega)
+  let : Nontrivial (Position (d+1+2)) := Fintype.one_lt_card_iff_nontrivial.mp (by omega)
+  have hbound := trimmed_eight_tv hn (embedding (eightEquiv d)) (law (d+1+2) (200*(d+1+2)))
+    (law_nonneg (d+1+2) (200*(d+1+2))) (law_sum (d+1+2) (200*(d+1+2)))
+    (law_complexSign_zero _ _ (by omega) (by omega)) hδ
+  have hpower : law (d+1+2) (1600*(d+1+2)) = realPower (law (d+1+2) (200*(d+1+2))) 8 := by
+    rw [show 1600*(d+1+2) = 8 * (200*(d+1+2)) by omega]
+    exact law_realPower (d+1+2) (200*(d+1+2)) 8
+  change tv (law (d+1+2) (1600*(d+1+2)))
+    (fun _ => 1 / (Fintype.card (State (d+1+2)) : ℝ)) ≤ fullRate d
+  simp only [one_div]
+  rw [hpower]
+  apply hbound.trans
+  unfold fullRate
+  refine add_le_add (le_refl _) ?_
+  apply mul_le_mul_of_nonneg_left _ (by norm_num)
+  apply Real.sqrt_le_sqrt
+  refine add_le_add (le_refl _) ?_
+  rw [card_position]
+  apply mul_le_mul_of_nonneg_right _ (Finset.sum_nonneg (fun p _ => by
+    change 0 ≤ (if 0 < defect p then inverseDegree p else 0)
+    split_ifs <;> first | exact inverseDegree_nonneg p | exact le_rfl))
+  apply pow_le_pow_left₀ (mul_nonneg (by norm_num) (permutationFamily (Position d)).reciprocalSum_nonneg)
+  linarith
+
+lemma full_mixing_shift :
+    Tendsto (fun d : ℕ => distance (d+1+2) (1600*(d+1+2))) atTop (nhds 0) := by
+  apply squeeze_zero' (Eventually.of_forall (fun d => by unfold distance tv; positivity))
+    distance_eight_eventually fullRate_tendsto
+
+end Block
+end Thorp
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/EnergyDecay.lean b/lean/OAI/Probability/ThorpShuffle/EnergyDecay.lean
index 44113119a6efedcb43f1c1be76bd4ddf5e2b8c3c..d2c749b7f2db418db9478bd43c60e49504f03fee 100644
--- a/lean/OAI/Probability/ThorpShuffle/EnergyDecay.lean
+++ b/lean/OAI/Probability/ThorpShuffle/EnergyDecay.lean
@@ -1,203 +1,203 @@
-import OAI.Probability.ThorpShuffle.NoiseEnergy
-
-namespace OAI
-
-noncomputable section
-
-open scoped BigOperators
-open Filter
-
-namespace Thorp
-
-namespace Conditional
-
-theorem mean_rawFree_sweep (d : ℕ) (v : RawState (d + 1)) (x : Position (d + 1)) :
-    mean (fun ω : History (d + 1) (d + 1) =>
-      if (rawIterate d v (d + 1) ω).free x then (1 : ℝ) else 0) =
-        (freeCount v.free : ℝ) / Fintype.card (Position (d + 1)) := by
-  simp only [rawIterate_free]
-  exact mean_free_after_sweep d v.free x
-
-theorem rawIterate_half_weight (d : ℕ) (v : RawState (d + 2)) (h : Bool)
-    (ω : History (d + 2) (d + 1)) (x : Position (d + 1)) :
-    (rawIterate (d + 1) v (d + 1) ω).weight (Fin.cons h x) =
-      (rawIterate d (halfRaw (d + 1) (Fin.last (d + 1)) h v) (d + 1)
-        (halfHistory d (d + 1) h ω)).weight x := by
-  have hz := congrArg (fun u : RawState (d + 1) => u.weight x)
-    (rawIterate_half d (d + 1) le_rfl h v ω)
-  simpa only [halfMarker_end, halfRaw, insertBit_zero] using hz
-
-theorem rawIterate_half_free (d : ℕ) (v : RawState (d + 2)) (h : Bool)
-    (ω : History (d + 2) (d + 1)) (x : Position (d + 1)) :
-    (rawIterate (d + 1) v (d + 1) ω).free (Fin.cons h x) =
-      (rawIterate d (halfRaw (d + 1) (Fin.last (d + 1)) h v) (d + 1)
-        (halfHistory d (d + 1) h ω)).free x := by
-  have hz := congrArg (fun u : RawState (d + 1) => u.free x)
-    (rawIterate_half d (d + 1) le_rfl h v ω)
-  simpa only [halfMarker_end, halfRaw, insertBit_zero] using hz
-
-theorem mean_half_single (d t : ℕ) (h : Bool) (f : History (d + 1) t → ℝ) :
-    mean (fun ω : History (d + 2) t => f (halfHistory d t h ω)) = mean f := by
-  simpa only [mean_const, mul_one] using
-    mean_opposite_half_product d t h f (fun _ => (1 : ℝ))
-
-theorem mean_half_factorization (d : ℕ) (v : RawState (d + 2)) (h : Bool)
-    (x : Position (d + 1)) :
-    mean (fun ω : History (d + 2) (d + 1) =>
-      (rawIterate (d + 1) v (d + 1) ω).weight (Fin.cons h x) ^ 2 *
-        (if (rawIterate (d + 1) v (d + 1) ω).free (Fin.cons (!h) x) then (1 : ℝ) else 0)) =
-      ((freeCount (halfRaw (d + 1) (Fin.last (d + 1)) (!h) v).free : ℝ) /
-        Fintype.card (Position (d + 1))) *
-      mean (fun ω : History (d + 2) (d + 1) =>
-        (rawIterate (d + 1) v (d + 1) ω).weight (Fin.cons h x) ^ 2) := by
-  simp_rw [rawIterate_half_weight, rawIterate_half_free]
-  rw [mean_opposite_half_product d (d + 1) h
-    (fun ω => (rawIterate d (halfRaw (d + 1) (Fin.last (d + 1)) h v) (d + 1) ω).weight x ^ 2)
-    (fun ω => if (rawIterate d (halfRaw (d + 1) (Fin.last (d + 1)) (!h) v)
-      (d + 1) ω).free x then (1 : ℝ) else 0)]
-  rw [mean_rawFree_sweep, mean_half_single d (d + 1) h
-    (fun ω => (rawIterate d (halfRaw (d + 1) (Fin.last (d + 1)) h v) (d + 1) ω).weight x ^ 2),
-    mul_comm]
-
- def rawEnergy {d : ℕ} (v : RawState d) : ℝ := ∑ x, v.weight x ^ 2
- def halfEnergy (d : ℕ) (v : RawState (d + 1)) (h : Bool) : ℝ :=
-   ∑ x : Position d, v.weight (Fin.cons h x) ^ 2
- def partnerEnergy (d : ℕ) (v : RawState (d + 1)) (h : Bool) : ℝ :=
-   ∑ x : Position d, v.weight (Fin.cons h x) ^ 2 *
-     (if v.free (Fin.cons (!h) x) then 1 else 0)
- def startHalfDensity (d : ℕ) (v : RawState (d + 1)) (h : Bool) : ℝ :=
-   (freeCount (halfRaw d (Fin.last d) h v).free : ℝ) / Fintype.card (Position d)
-
- theorem halfEnergy_nonneg (d : ℕ) (v : RawState (d + 1)) (h : Bool) :
-    0 ≤ halfEnergy d v h := Finset.sum_nonneg (fun _ _ => sq_nonneg _)
-
- theorem rawEnergy_halves (d : ℕ) (v : RawState (d + 1)) :
-    rawEnergy v = halfEnergy d v false + halfEnergy d v true := by
-  unfold rawEnergy halfEnergy
-  have he := Equiv.sum_comp (splitPosition d).symm (fun x => v.weight x ^ 2)
-  rw [← he, Fintype.sum_prod_type, Fintype.sum_bool, add_comm]
-  rfl
-
- theorem oneFreeEnergy_partner (d : ℕ) (v : RawState (d + 1)) :
-    oneFreeEnergy d v.free v.weight + partnerEnergy d v false + partnerEnergy d v true =
-      rawEnergy v := by
-  rw [rawEnergy_halves]
-  unfold oneFreeEnergy partnerEnergy halfEnergy
-  simp only [Bool.not_false, Bool.not_true, ← Finset.sum_add_distrib]
-  apply Finset.sum_congr rfl
-  intro x _
-  cases v.free (Fin.cons false x) <;> cases v.free (Fin.cons true x) <;> simp [add_comm]
-
- theorem mean_half_energy_factorization (d : ℕ) (v : RawState (d + 2)) (h : Bool) :
-    mean (fun ω : History (d + 2) (d + 1) =>
-      partnerEnergy (d + 1) (rawIterate (d + 1) v (d + 1) ω) h) =
-      startHalfDensity (d + 1) v (!h) *
-        mean (fun ω : History (d + 2) (d + 1) =>
-          halfEnergy (d + 1) (rawIterate (d + 1) v (d + 1) ω) h) := by
-  unfold partnerEnergy halfEnergy startHalfDensity
-  rw [mean_sum, mean_sum, Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro x _
-  exact mean_half_factorization d v h x
-
-theorem noise_bound_balanced_halves (d : ℕ) (v : RawState (d + 2))
-    (β : ℝ) (hb : ∀ h, β ≤ startHalfDensity (d + 1) v h) :
-    mean (fun ω : History (d + 2) (d + 1) =>
-      oneFreeEnergy (d + 1) (rawIterate (d + 1) v (d + 1) ω).free
-        (rawIterate (d + 1) v (d + 1) ω).weight) ≤
-      (1 - β) * mean (fun ω : History (d + 2) (d + 1) =>
-        rawEnergy (rawIterate (d + 1) v (d + 1) ω)) := by
-  have hid := mean_congr (fun ω : History (d + 2) (d + 1) =>
-    oneFreeEnergy_partner (d + 1) (rawIterate (d + 1) v (d + 1) ω))
-  rw [mean_add, mean_add, mean_half_energy_factorization,
-    mean_half_energy_factorization] at hid
-  have he := mean_congr (fun ω : History (d + 2) (d + 1) =>
-    rawEnergy_halves (d + 1) (rawIterate (d + 1) v (d + 1) ω))
-  rw [mean_add] at he
-  have hfalse := mul_le_mul_of_nonneg_right (hb true)
-    (mean_nonneg (fun ω : History (d + 2) (d + 1) =>
-      halfEnergy_nonneg (d + 1) (rawIterate (d + 1) v (d + 1) ω) false))
-  have htrue := mul_le_mul_of_nonneg_right (hb false)
-    (mean_nonneg (fun ω : History (d + 2) (d + 1) =>
-      halfEnergy_nonneg (d + 1) (rawIterate (d + 1) v (d + 1) ω) true))
-  simp only [Bool.not_false, Bool.not_true] at hid
-  rw [he] at hid ⊢
-  nlinarith
-
-end Conditional
-
-namespace Conditional
-
-def splitHistory (d s t : ℕ) : History d (s + t) ≃ History d s × History d t where
-  toFun ω := (fun i => ω (Fin.castAdd t i), fun i => ω (Fin.natAdd s i))
-  invFun p := Fin.append p.1 p.2
-  left_inv ω := by
-    funext i
-    refine Fin.addCases (fun j => ?_) (fun j => ?_) i
-    · exact Fin.append_left _ _ _
-    · exact Fin.append_right _ _ _
-  right_inv p := by
-    apply Prod.ext <;> funext i
-    · exact Fin.append_left _ _ _
-    · exact Fin.append_right _ _ _
-
-theorem mean_history_add (d s t : ℕ) (f : History d (s + t) → ℝ) :
-    mean f = mean (fun a : History d s =>
-      mean (fun b : History d t => f (Fin.append a b))) := by
-  have he := mean_equiv (splitHistory d s t).symm f
-  rw [mean_prod] at he
-  exact he.symm
-
-@[simp] theorem rawIterate_snoc (d t : ℕ) (v : RawState (d + 1))
-    (ω : History (d + 1) t) (c : Coins (d + 1)) :
-    rawIterate d v (t + 1) (Fin.snoc ω c) = rawNext d (rawIterate d v t ω) c := by
-  simp [rawIterate]
-
-@[simp] theorem iterateState_snoc (d t : ℕ) (v : CenteredState (d + 1))
-    (ω : History (d + 1) t) (c : Coins (d + 1)) :
-    iterateState d v (t + 1) (Fin.snoc ω c) = nextState d (iterateState d v t ω) c := by
-  simp [iterateState]
-
-theorem rawIterate_append (d s t : ℕ) (v : RawState (d + 1))
-    (a : History (d + 1) s) (b : History (d + 1) t) :
-    rawIterate d v (s + t) (Fin.append a b) = rawIterate d (rawIterate d v s a) t b := by
-  induction t with
-  | zero =>
-    have hz : Fin.append a b = a := by funext i; exact Fin.append_left a b i
-    rw [hz]; rfl
-  | succ t ih =>
-    obtain ⟨⟨c, b⟩, rfl⟩ := (Fin.snocEquiv (fun _ : Fin (t + 1) => Coins (d + 1))).surjective b
-    erw [Fin.append_snoc, rawIterate_snoc, rawIterate_snoc, ih]
-
-theorem iterateState_append (d s t : ℕ) (v : CenteredState (d + 1))
-    (a : History (d + 1) s) (b : History (d + 1) t) :
-    iterateState d v (s + t) (Fin.append a b) = iterateState d (iterateState d v s a) t b := by
-  induction t with
-  | zero =>
-    have hz : Fin.append a b = a := by funext i; exact Fin.append_left a b i
-    rw [hz]; rfl
-  | succ t ih =>
-    obtain ⟨⟨c, b⟩, rfl⟩ := (Fin.snocEquiv (fun _ : Fin (t + 1) => Coins (d + 1))).surjective b
-    erw [Fin.append_snoc, iterateState_snoc, iterateState_snoc, ih]
-
-theorem expectedEnergy_add (d s t : ℕ) (v : CenteredState (d + 1)) :
-    expectedEnergy d v (s + t) = mean (fun a : History (d + 1) s =>
-      expectedEnergy d (iterateState d v s a) t) := by
-  unfold expectedEnergy
-  rw [mean_history_add]
-  simp_rw [iterateState_append]
-
-theorem expectedNoise_add (d s t : ℕ) (v : CenteredState (d + 1)) :
-    expectedNoise d v (s + t) = mean (fun a : History (d + 1) s =>
-      expectedNoise d (iterateState d v s a) t) := by
-  unfold expectedNoise
-  rw [mean_history_add]
-  simp_rw [iterateState_append]
-
-end Conditional
-
-end Thorp
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.NoiseEnergy
+
+namespace OAI
+
+noncomputable section
+
+open scoped BigOperators
+open Filter
+
+namespace Thorp
+
+namespace Conditional
+
+theorem mean_rawFree_sweep (d : ℕ) (v : RawState (d + 1)) (x : Position (d + 1)) :
+    mean (fun ω : History (d + 1) (d + 1) =>
+      if (rawIterate d v (d + 1) ω).free x then (1 : ℝ) else 0) =
+        (freeCount v.free : ℝ) / Fintype.card (Position (d + 1)) := by
+  simp only [rawIterate_free]
+  exact mean_free_after_sweep d v.free x
+
+theorem rawIterate_half_weight (d : ℕ) (v : RawState (d + 2)) (h : Bool)
+    (ω : History (d + 2) (d + 1)) (x : Position (d + 1)) :
+    (rawIterate (d + 1) v (d + 1) ω).weight (Fin.cons h x) =
+      (rawIterate d (halfRaw (d + 1) (Fin.last (d + 1)) h v) (d + 1)
+        (halfHistory d (d + 1) h ω)).weight x := by
+  have hz := congrArg (fun u : RawState (d + 1) => u.weight x)
+    (rawIterate_half d (d + 1) le_rfl h v ω)
+  simpa only [halfMarker_end, halfRaw, insertBit_zero] using hz
+
+theorem rawIterate_half_free (d : ℕ) (v : RawState (d + 2)) (h : Bool)
+    (ω : History (d + 2) (d + 1)) (x : Position (d + 1)) :
+    (rawIterate (d + 1) v (d + 1) ω).free (Fin.cons h x) =
+      (rawIterate d (halfRaw (d + 1) (Fin.last (d + 1)) h v) (d + 1)
+        (halfHistory d (d + 1) h ω)).free x := by
+  have hz := congrArg (fun u : RawState (d + 1) => u.free x)
+    (rawIterate_half d (d + 1) le_rfl h v ω)
+  simpa only [halfMarker_end, halfRaw, insertBit_zero] using hz
+
+theorem mean_half_single (d t : ℕ) (h : Bool) (f : History (d + 1) t → ℝ) :
+    mean (fun ω : History (d + 2) t => f (halfHistory d t h ω)) = mean f := by
+  simpa only [mean_const, mul_one] using
+    mean_opposite_half_product d t h f (fun _ => (1 : ℝ))
+
+theorem mean_half_factorization (d : ℕ) (v : RawState (d + 2)) (h : Bool)
+    (x : Position (d + 1)) :
+    mean (fun ω : History (d + 2) (d + 1) =>
+      (rawIterate (d + 1) v (d + 1) ω).weight (Fin.cons h x) ^ 2 *
+        (if (rawIterate (d + 1) v (d + 1) ω).free (Fin.cons (!h) x) then (1 : ℝ) else 0)) =
+      ((freeCount (halfRaw (d + 1) (Fin.last (d + 1)) (!h) v).free : ℝ) /
+        Fintype.card (Position (d + 1))) *
+      mean (fun ω : History (d + 2) (d + 1) =>
+        (rawIterate (d + 1) v (d + 1) ω).weight (Fin.cons h x) ^ 2) := by
+  simp_rw [rawIterate_half_weight, rawIterate_half_free]
+  rw [mean_opposite_half_product d (d + 1) h
+    (fun ω => (rawIterate d (halfRaw (d + 1) (Fin.last (d + 1)) h v) (d + 1) ω).weight x ^ 2)
+    (fun ω => if (rawIterate d (halfRaw (d + 1) (Fin.last (d + 1)) (!h) v)
+      (d + 1) ω).free x then (1 : ℝ) else 0)]
+  rw [mean_rawFree_sweep, mean_half_single d (d + 1) h
+    (fun ω => (rawIterate d (halfRaw (d + 1) (Fin.last (d + 1)) h v) (d + 1) ω).weight x ^ 2),
+    mul_comm]
+
+ def rawEnergy {d : ℕ} (v : RawState d) : ℝ := ∑ x, v.weight x ^ 2
+ def halfEnergy (d : ℕ) (v : RawState (d + 1)) (h : Bool) : ℝ :=
+   ∑ x : Position d, v.weight (Fin.cons h x) ^ 2
+ def partnerEnergy (d : ℕ) (v : RawState (d + 1)) (h : Bool) : ℝ :=
+   ∑ x : Position d, v.weight (Fin.cons h x) ^ 2 *
+     (if v.free (Fin.cons (!h) x) then 1 else 0)
+ def startHalfDensity (d : ℕ) (v : RawState (d + 1)) (h : Bool) : ℝ :=
+   (freeCount (halfRaw d (Fin.last d) h v).free : ℝ) / Fintype.card (Position d)
+
+ theorem halfEnergy_nonneg (d : ℕ) (v : RawState (d + 1)) (h : Bool) :
+    0 ≤ halfEnergy d v h := Finset.sum_nonneg (fun _ _ => sq_nonneg _)
+
+ theorem rawEnergy_halves (d : ℕ) (v : RawState (d + 1)) :
+    rawEnergy v = halfEnergy d v false + halfEnergy d v true := by
+  unfold rawEnergy halfEnergy
+  have he := Equiv.sum_comp (splitPosition d).symm (fun x => v.weight x ^ 2)
+  rw [← he, Fintype.sum_prod_type, Fintype.sum_bool, add_comm]
+  rfl
+
+ theorem oneFreeEnergy_partner (d : ℕ) (v : RawState (d + 1)) :
+    oneFreeEnergy d v.free v.weight + partnerEnergy d v false + partnerEnergy d v true =
+      rawEnergy v := by
+  rw [rawEnergy_halves]
+  unfold oneFreeEnergy partnerEnergy halfEnergy
+  simp only [Bool.not_false, Bool.not_true, ← Finset.sum_add_distrib]
+  apply Finset.sum_congr rfl
+  intro x _
+  cases v.free (Fin.cons false x) <;> cases v.free (Fin.cons true x) <;> simp [add_comm]
+
+ theorem mean_half_energy_factorization (d : ℕ) (v : RawState (d + 2)) (h : Bool) :
+    mean (fun ω : History (d + 2) (d + 1) =>
+      partnerEnergy (d + 1) (rawIterate (d + 1) v (d + 1) ω) h) =
+      startHalfDensity (d + 1) v (!h) *
+        mean (fun ω : History (d + 2) (d + 1) =>
+          halfEnergy (d + 1) (rawIterate (d + 1) v (d + 1) ω) h) := by
+  unfold partnerEnergy halfEnergy startHalfDensity
+  rw [mean_sum, mean_sum, Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro x _
+  exact mean_half_factorization d v h x
+
+theorem noise_bound_balanced_halves (d : ℕ) (v : RawState (d + 2))
+    (β : ℝ) (hb : ∀ h, β ≤ startHalfDensity (d + 1) v h) :
+    mean (fun ω : History (d + 2) (d + 1) =>
+      oneFreeEnergy (d + 1) (rawIterate (d + 1) v (d + 1) ω).free
+        (rawIterate (d + 1) v (d + 1) ω).weight) ≤
+      (1 - β) * mean (fun ω : History (d + 2) (d + 1) =>
+        rawEnergy (rawIterate (d + 1) v (d + 1) ω)) := by
+  have hid := mean_congr (fun ω : History (d + 2) (d + 1) =>
+    oneFreeEnergy_partner (d + 1) (rawIterate (d + 1) v (d + 1) ω))
+  rw [mean_add, mean_add, mean_half_energy_factorization,
+    mean_half_energy_factorization] at hid
+  have he := mean_congr (fun ω : History (d + 2) (d + 1) =>
+    rawEnergy_halves (d + 1) (rawIterate (d + 1) v (d + 1) ω))
+  rw [mean_add] at he
+  have hfalse := mul_le_mul_of_nonneg_right (hb true)
+    (mean_nonneg (fun ω : History (d + 2) (d + 1) =>
+      halfEnergy_nonneg (d + 1) (rawIterate (d + 1) v (d + 1) ω) false))
+  have htrue := mul_le_mul_of_nonneg_right (hb false)
+    (mean_nonneg (fun ω : History (d + 2) (d + 1) =>
+      halfEnergy_nonneg (d + 1) (rawIterate (d + 1) v (d + 1) ω) true))
+  simp only [Bool.not_false, Bool.not_true] at hid
+  rw [he] at hid ⊢
+  nlinarith
+
+end Conditional
+
+namespace Conditional
+
+def splitHistory (d s t : ℕ) : History d (s + t) ≃ History d s × History d t where
+  toFun ω := (fun i => ω (Fin.castAdd t i), fun i => ω (Fin.natAdd s i))
+  invFun p := Fin.append p.1 p.2
+  left_inv ω := by
+    funext i
+    refine Fin.addCases (fun j => ?_) (fun j => ?_) i
+    · exact Fin.append_left _ _ _
+    · exact Fin.append_right _ _ _
+  right_inv p := by
+    apply Prod.ext <;> funext i
+    · exact Fin.append_left _ _ _
+    · exact Fin.append_right _ _ _
+
+theorem mean_history_add (d s t : ℕ) (f : History d (s + t) → ℝ) :
+    mean f = mean (fun a : History d s =>
+      mean (fun b : History d t => f (Fin.append a b))) := by
+  have he := mean_equiv (splitHistory d s t).symm f
+  rw [mean_prod] at he
+  exact he.symm
+
+@[simp] theorem rawIterate_snoc (d t : ℕ) (v : RawState (d + 1))
+    (ω : History (d + 1) t) (c : Coins (d + 1)) :
+    rawIterate d v (t + 1) (Fin.snoc ω c) = rawNext d (rawIterate d v t ω) c := by
+  simp [rawIterate]
+
+@[simp] theorem iterateState_snoc (d t : ℕ) (v : CenteredState (d + 1))
+    (ω : History (d + 1) t) (c : Coins (d + 1)) :
+    iterateState d v (t + 1) (Fin.snoc ω c) = nextState d (iterateState d v t ω) c := by
+  simp [iterateState]
+
+theorem rawIterate_append (d s t : ℕ) (v : RawState (d + 1))
+    (a : History (d + 1) s) (b : History (d + 1) t) :
+    rawIterate d v (s + t) (Fin.append a b) = rawIterate d (rawIterate d v s a) t b := by
+  induction t with
+  | zero =>
+    have hz : Fin.append a b = a := by funext i; exact Fin.append_left a b i
+    rw [hz]; rfl
+  | succ t ih =>
+    obtain ⟨⟨c, b⟩, rfl⟩ := (Fin.snocEquiv (fun _ : Fin (t + 1) => Coins (d + 1))).surjective b
+    erw [Fin.append_snoc, rawIterate_snoc, rawIterate_snoc, ih]
+
+theorem iterateState_append (d s t : ℕ) (v : CenteredState (d + 1))
+    (a : History (d + 1) s) (b : History (d + 1) t) :
+    iterateState d v (s + t) (Fin.append a b) = iterateState d (iterateState d v s a) t b := by
+  induction t with
+  | zero =>
+    have hz : Fin.append a b = a := by funext i; exact Fin.append_left a b i
+    rw [hz]; rfl
+  | succ t ih =>
+    obtain ⟨⟨c, b⟩, rfl⟩ := (Fin.snocEquiv (fun _ : Fin (t + 1) => Coins (d + 1))).surjective b
+    erw [Fin.append_snoc, iterateState_snoc, iterateState_snoc, ih]
+
+theorem expectedEnergy_add (d s t : ℕ) (v : CenteredState (d + 1)) :
+    expectedEnergy d v (s + t) = mean (fun a : History (d + 1) s =>
+      expectedEnergy d (iterateState d v s a) t) := by
+  unfold expectedEnergy
+  rw [mean_history_add]
+  simp_rw [iterateState_append]
+
+theorem expectedNoise_add (d s t : ℕ) (v : CenteredState (d + 1)) :
+    expectedNoise d v (s + t) = mean (fun a : History (d + 1) s =>
+      expectedNoise d (iterateState d v s a) t) := by
+  unfold expectedNoise
+  rw [mean_history_add]
+  simp_rw [iterateState_append]
+
+end Conditional
+
+end Thorp
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/ExceptionalShapes.lean b/lean/OAI/Probability/ThorpShuffle/ExceptionalShapes.lean
index 69cd79c87451d69692e3a152c3048477ade155ac..55d7a8c2ebb2a2b1c070d32962982c031e2783ac 100644
--- a/lean/OAI/Probability/ThorpShuffle/ExceptionalShapes.lean
+++ b/lean/OAI/Probability/ThorpShuffle/ExceptionalShapes.lean
@@ -1,70 +1,70 @@
-import OAI.Probability.ThorpShuffle.FirstMoment
-
-namespace OAI
-
-noncomputable section
-
-namespace Thorp.Current
-open scoped BigOperators Classical
-open Thorp.Specht Thorp.Young
-
-/-- Verification that the index predicate in `exceptionalFirst` excludes exactly
-the row and the column, rather than a merely degree-based surrogate. -/
-lemma full_row_iff (D : YoungDiagram) :
-    D.rowLen 0 = Fintype.card (Cells D) ↔ ∀ x : Cells D, x.val.1 = 0 := by
-  constructor
-  · intro h
-    have hc : Fintype.card (Lower D) = 0 := by have hh := card_lower_add D; omega
-    have he : IsEmpty (Lower D) := Fintype.card_eq_zero_iff.mp hc
-    intro x
-    by_contra hx
-    exact he.false ⟨x, hx⟩
-  · intro h
-    have he : IsEmpty (Lower D) := ⟨fun x => x.property (h x.val)⟩
-    have hc : Fintype.card (Lower D) = 0 := Fintype.card_eq_zero_iff.mpr he
-    have hh := card_lower_add D
-    omega
-
-lemma full_column_iff (D : YoungDiagram) :
-    D.colLen 0 = Fintype.card (Cells D) ↔ ∀ x : Cells D, x.val.2 = 0 := by
-  have hc := Fintype.card_congr (transposeCells D).toEquiv
-  constructor
-  · intro h x
-    have hT : D.transpose.rowLen 0 = Fintype.card (Cells D.transpose) := by
-      rw [YoungDiagram.rowLen_transpose, ← hc, h]
-    exact (full_row_iff D.transpose).mp hT (transposeCells D x)
-  · intro h
-    have hT : D.transpose.rowLen 0 = Fintype.card (Cells D.transpose) :=
-      (full_row_iff D.transpose).mpr (fun x => h ((transposeCells D).symm x))
-    simpa only [YoungDiagram.rowLen_transpose, ← hc] using hT
-
-lemma defect_zero_iff_row_or_column {n : ℕ} (p : NShape n) :
-    defect p = 0 ↔
-      (∀ x : Cells (ofPartition p), x.val.1 = 0) ∨
-      (∀ x : Cells (ofPartition p), x.val.2 = 0) := by
-  rw [← full_row_iff, ← full_column_iff, diagram_card]
-  have hd := defect_add_longest p
-  have hL := longest_le_card (ofPartition p)
-  rw [diagram_card] at hL
-  unfold longest at hd hL
-  omega
-
-lemma exceptionalFirst_exact_nonrow_noncolumn (n : ℕ) :
-    exceptionalFirst n = ∑ p : NShape n,
-      if (¬∀ x : Cells (ofPartition p), x.val.1 = 0) ∧
-         (¬∀ x : Cells (ofPartition p), x.val.2 = 0)
-      then (degree p : ℝ)⁻¹ else 0 := by
-  unfold exceptionalFirst
-  apply Finset.sum_congr rfl
-  intro p _
-  have h : 0 < defect p ↔
-      (¬∀ x : Cells (ofPartition p), x.val.1 = 0) ∧
-      (¬∀ x : Cells (ofPartition p), x.val.2 = 0) := by
-    rw [Nat.pos_iff_ne_zero, ne_eq, defect_zero_iff_row_or_column, not_or]
-  simp only [h]
-
-end Thorp.Current
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.FirstMoment
+
+namespace OAI
+
+noncomputable section
+
+namespace Thorp.Current
+open scoped BigOperators Classical
+open Thorp.Specht Thorp.Young
+
+/-- Verification that the index predicate in `exceptionalFirst` excludes exactly
+the row and the column, rather than a merely degree-based surrogate. -/
+lemma full_row_iff (D : YoungDiagram) :
+    D.rowLen 0 = Fintype.card (Cells D) ↔ ∀ x : Cells D, x.val.1 = 0 := by
+  constructor
+  · intro h
+    have hc : Fintype.card (Lower D) = 0 := by have hh := card_lower_add D; omega
+    have he : IsEmpty (Lower D) := Fintype.card_eq_zero_iff.mp hc
+    intro x
+    by_contra hx
+    exact he.false ⟨x, hx⟩
+  · intro h
+    have he : IsEmpty (Lower D) := ⟨fun x => x.property (h x.val)⟩
+    have hc : Fintype.card (Lower D) = 0 := Fintype.card_eq_zero_iff.mpr he
+    have hh := card_lower_add D
+    omega
+
+lemma full_column_iff (D : YoungDiagram) :
+    D.colLen 0 = Fintype.card (Cells D) ↔ ∀ x : Cells D, x.val.2 = 0 := by
+  have hc := Fintype.card_congr (transposeCells D).toEquiv
+  constructor
+  · intro h x
+    have hT : D.transpose.rowLen 0 = Fintype.card (Cells D.transpose) := by
+      rw [YoungDiagram.rowLen_transpose, ← hc, h]
+    exact (full_row_iff D.transpose).mp hT (transposeCells D x)
+  · intro h
+    have hT : D.transpose.rowLen 0 = Fintype.card (Cells D.transpose) :=
+      (full_row_iff D.transpose).mpr (fun x => h ((transposeCells D).symm x))
+    simpa only [YoungDiagram.rowLen_transpose, ← hc] using hT
+
+lemma defect_zero_iff_row_or_column {n : ℕ} (p : NShape n) :
+    defect p = 0 ↔
+      (∀ x : Cells (ofPartition p), x.val.1 = 0) ∨
+      (∀ x : Cells (ofPartition p), x.val.2 = 0) := by
+  rw [← full_row_iff, ← full_column_iff, diagram_card]
+  have hd := defect_add_longest p
+  have hL := longest_le_card (ofPartition p)
+  rw [diagram_card] at hL
+  unfold longest at hd hL
+  omega
+
+lemma exceptionalFirst_exact_nonrow_noncolumn (n : ℕ) :
+    exceptionalFirst n = ∑ p : NShape n,
+      if (¬∀ x : Cells (ofPartition p), x.val.1 = 0) ∧
+         (¬∀ x : Cells (ofPartition p), x.val.2 = 0)
+      then (degree p : ℝ)⁻¹ else 0 := by
+  unfold exceptionalFirst
+  apply Finset.sum_congr rfl
+  intro p _
+  have h : 0 < defect p ↔
+      (¬∀ x : Cells (ofPartition p), x.val.1 = 0) ∧
+      (¬∀ x : Cells (ofPartition p), x.val.2 = 0) := by
+    rw [Nat.pos_iff_ne_zero, ne_eq, defect_zero_iff_row_or_column, not_or]
+  simp only [h]
+
+end Thorp.Current
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/ExtensionBounds.lean b/lean/OAI/Probability/ThorpShuffle/ExtensionBounds.lean
index 31aaed4e55ad300a80dcc67dfaf65bbb7af58535..92113235ea824eb9927e95e724d2b6ac4d09a63d 100644
--- a/lean/OAI/Probability/ThorpShuffle/ExtensionBounds.lean
+++ b/lean/OAI/Probability/ThorpShuffle/ExtensionBounds.lean
@@ -1,237 +1,237 @@
-import OAI.Probability.ThorpShuffle.LinearExtensions
-
-namespace OAI
-
-universe uA
-
-noncomputable section
-
-open scoped BigOperators ComplexConjugate InnerProductSpace
-open Filter
-
-namespace Thorp.Young
-open scoped Classical
-variable {A : Type uA} [Fintype A] [PartialOrder A]
-
-lemma extensions_card_ideal (s : Set A) (hs : IsLowerSet s) :
-    Fintype.card (Extensions s) ≤ Fintype.card (Extensions A) := by
-  obtain ⟨c⟩ := extensions_nonempty (sᶜ : Set A)
-  have hc : Fintype.card s + Fintype.card (sᶜ : Set A) = Fintype.card A := by
-    simpa only [Fintype.card_sum] using Fintype.card_congr (Equiv.Set.sumCompl s)
-  let F (e : Extensions s) : A ≃ Fin (Fintype.card A) :=
-    ((Equiv.Set.sumCompl s).symm.trans (Equiv.sumCongr e.val c.val)).trans
-      (finSumFinEquiv.trans (finCongr hc))
-  have hFs (e : Extensions s) (x : s) : (F e x).val = (e.val x).val := by
-    simp [F, Equiv.Set.sumCompl_symm_apply_of_mem x.property]
-  have hFc (e : Extensions s) (x : (sᶜ : Set A)) :
-      (F e x).val = Fintype.card s + (c.val x).val := by
-    simp [F, Equiv.Set.sumCompl_symm_apply_of_notMem x.property]
-  have hF (e : Extensions s) : StrictMono (F e) := by
-    intro x y hxy
-    change (F e x).val < (F e y).val
-    by_cases hx : x ∈ s <;> by_cases hy : y ∈ s
-    · rw [hFs e ⟨x, hx⟩, hFs e ⟨y, hy⟩]
-      exact e.property hxy
-    · rw [hFs e ⟨x, hx⟩, hFc e ⟨y, hy⟩]
-      exact (e.val ⟨x, hx⟩).isLt.trans_le (Nat.le_add_right _ _)
-    · exact False.elim (hx (hs hxy.le hy))
-    · rw [hFc e ⟨x, hx⟩, hFc e ⟨y, hy⟩]
-      exact Nat.add_lt_add_left (c.property hxy) _
-  let f (e : Extensions s) : Extensions A := ⟨F e, hF e⟩
-  apply Fintype.card_le_of_injective f
-  intro e d hed
-  apply Subtype.ext
-  apply Equiv.ext
-  intro x
-  apply Fin.ext
-  have hh := congrArg (fun z : Extensions A => (z.val x).val) hed
-  change (F e x).val = (F d x).val at hh
-  simpa only [hFs] using hh
-
-lemma extensions_card_compl_ideal (s : Set A) (hs : IsLowerSet s) :
-    Fintype.card (Extensions (sᶜ : Set A)) ≤ Fintype.card (Extensions A) := by
-  obtain ⟨a⟩ := extensions_nonempty s
-  have hc : Fintype.card s + Fintype.card (sᶜ : Set A) = Fintype.card A := by
-    simpa only [Fintype.card_sum] using Fintype.card_congr (Equiv.Set.sumCompl s)
-  let F (e : Extensions (sᶜ : Set A)) : A ≃ Fin (Fintype.card A) :=
-    ((Equiv.Set.sumCompl s).symm.trans (Equiv.sumCongr a.val e.val)).trans
-      (finSumFinEquiv.trans (finCongr hc))
-  have hFs (e : Extensions (sᶜ : Set A)) (x : s) : (F e x).val = (a.val x).val := by
-    simp [F, Equiv.Set.sumCompl_symm_apply_of_mem x.property]
-  have hFc (e : Extensions (sᶜ : Set A)) (x : (sᶜ : Set A)) :
-      (F e x).val = Fintype.card s + (e.val x).val := by
-    simp [F, Equiv.Set.sumCompl_symm_apply_of_notMem x.property]
-  have hF (e : Extensions (sᶜ : Set A)) : StrictMono (F e) := by
-    intro x y hxy
-    change (F e x).val < (F e y).val
-    by_cases hx : x ∈ s <;> by_cases hy : y ∈ s
-    · rw [hFs e ⟨x, hx⟩, hFs e ⟨y, hy⟩]
-      exact a.property hxy
-    · rw [hFs e ⟨x, hx⟩, hFc e ⟨y, hy⟩]
-      exact (a.val ⟨x, hx⟩).isLt.trans_le (Nat.le_add_right _ _)
-    · exact False.elim (hx (hs hxy.le hy))
-    · rw [hFc e ⟨x, hx⟩, hFc e ⟨y, hy⟩]
-      exact Nat.add_lt_add_left (e.property hxy) _
-  let f (e : Extensions (sᶜ : Set A)) : Extensions A := ⟨F e, hF e⟩
-  apply Fintype.card_le_of_injective f
-  intro e d hed
-  apply Subtype.ext
-  apply Equiv.ext
-  intro x
-  apply Fin.ext
-  have hh := congrArg (fun z : Extensions A => (z.val x).val) hed
-  change (F e x).val = (F d x).val at hh
-  have h := congrArg (fun z : ℕ => z - Fintype.card s) hh
-  simpa only [hFc, Nat.add_sub_cancel_left] using h
-
-end Thorp.Young
-
-namespace Thorp.Young
-open scoped Classical
-variable {A : Type uA} [Fintype A] [PartialOrder A]
-
-lemma choose_le_extensions_split (s : Set A)
-    (hs : ∀ x y : A, x < y → (x ∈ s ↔ y ∈ s)) :
-    (Fintype.card A).choose (Fintype.card s) ≤ Fintype.card (Extensions A) := by
-  obtain ⟨a⟩ := extensions_nonempty s
-  obtain ⟨b⟩ := extensions_nonempty (sᶜ : Set A)
-  let I := {S : Finset (Fin (Fintype.card A)) // S.card = Fintype.card s}
-  have hi : Fintype.card I = (Fintype.card A).choose (Fintype.card s) := by
-    calc
-      Fintype.card I = ((Finset.univ : Finset (Fin (Fintype.card A))).powersetCard (Fintype.card s)).card := by
-        rw [Fintype.card_subtype]
-        congr 1
-        ext S
-        simp
-      _ = _ := by rw [Finset.card_powersetCard, Finset.card_univ, Fintype.card_fin]
-  let P (S : I) : Set (Fin (Fintype.card A)) := S.val
-  have hP (S : I) : Fintype.card (P S) = Fintype.card s := by
-    simpa [P] using S.property
-  have hC (S : I) : Fintype.card ((P S)ᶜ : Set (Fin (Fintype.card A))) =
-      Fintype.card (sᶜ : Set A) := by
-    have h₁ := Fintype.card_congr (Equiv.Set.sumCompl (P S))
-    have h₂ := Fintype.card_congr (Equiv.Set.sumCompl s)
-    simp only [Fintype.card_sum, Fintype.card_fin, hP] at h₁ h₂
-    omega
-  let ea (S : I) := (Fintype.orderIsoFinOfCardEq (P S) (hP S))
-  let eb (S : I) := (Fintype.orderIsoFinOfCardEq ((P S)ᶜ : Set _) (hC S))
-  let F (S : I) : A ≃ Fin (Fintype.card A) :=
-    ((Equiv.Set.sumCompl s).symm.trans
-      (Equiv.sumCongr (a.val.trans (ea S).toEquiv) (b.val.trans (eb S).toEquiv))).trans
-        (Equiv.Set.sumCompl (P S))
-  have hFa (S : I) (x : s) : F S x = (ea S (a.val x)).val := by
-    simp [F, Equiv.Set.sumCompl_symm_apply_of_mem x.property]
-  have hFb (S : I) (x : (sᶜ : Set A)) : F S x = (eb S (b.val x)).val := by
-    simp [F, Equiv.Set.sumCompl_symm_apply_of_notMem x.property]
-  have hFm (S : I) (x : A) : F S x ∈ P S ↔ x ∈ s := by
-    by_cases hx : x ∈ s
-    · rw [hFa S ⟨x, hx⟩]
-      exact iff_of_true (ea S (a.val ⟨x, hx⟩)).property hx
-    · rw [hFb S ⟨x, hx⟩]
-      exact iff_of_false (eb S (b.val ⟨x, hx⟩)).property hx
-  have hF (S : I) : StrictMono (F S) := by
-    intro x y hxy
-    by_cases hx : x ∈ s
-    · have hy := (hs x y hxy).mp hx
-      rw [hFa S ⟨x, hx⟩, hFa S ⟨y, hy⟩]
-      exact (ea S).strictMono (a.property hxy)
-    · have hy : y ∉ s := fun h => hx ((hs x y hxy).mpr h)
-      rw [hFb S ⟨x, hx⟩, hFb S ⟨y, hy⟩]
-      exact (eb S).strictMono (b.property hxy)
-  let f (S : I) : Extensions A := ⟨F S, hF S⟩
-  rw [← hi]
-  apply Fintype.card_le_of_injective f
-  intro S T hST
-  apply Subtype.ext
-  ext z
-  obtain ⟨x, rfl⟩ := (F S).surjective z
-  have he : F S x = F T x := congrArg (fun e : Extensions A => e.val x) hST
-  change F S x ∈ P S ↔ F S x ∈ P T
-  rw [hFm, he, hFm]
-
-end Thorp.Young
-
-namespace Thorp.Young
-open scoped Classical
-
-abbrev Lower (D : YoungDiagram) := {x : Cells D // x.val.1 ≠ 0}
-
-def topPrefix (D : YoungDiagram) (u : ℕ) : Set (Cells D) :=
-  {x | x.val.1 = 0 ∧ x.val.2 < u}
-
-lemma topPrefix_lower (D : YoungDiagram) (u : ℕ) : IsLowerSet (topPrefix D u) := by
-  intro y x hxy hy
-  change y.val.1 = 0 ∧ y.val.2 < u at hy
-  change x.val.1 = 0 ∧ x.val.2 < u
-  exact ⟨by have := hxy.1; omega, lt_of_le_of_lt hxy.2 hy.2⟩
-
-lemma card_topPrefix (D : YoungDiagram) (u : ℕ) (hu : u ≤ D.rowLen 0) :
-    Fintype.card (topPrefix D u) = u := by
-  let e : topPrefix D u ≃ Fin u := {
-    toFun := fun x => ⟨x.val.val.2, x.property.2⟩
-    invFun := fun j => ⟨⟨(0, j), YoungDiagram.mem_iff_lt_rowLen.mpr (j.isLt.trans_le hu)⟩, rfl, j.isLt⟩
-    left_inv := by intro x; apply Subtype.ext; apply Subtype.ext; exact Prod.ext x.property.1.symm rfl
-    right_inv := fun _ => rfl }
-  exact (Fintype.card_congr e).trans (Fintype.card_fin u)
-
-lemma card_lower_add (D : YoungDiagram) :
-    D.rowLen 0 + Fintype.card (Lower D) = Fintype.card (Cells D) := by
-  have h := Fintype.card_congr (Equiv.sumCompl (fun x : Cells D => x.val.1 = 0))
-  simpa only [Fintype.card_sum, card_row_subtype] using h
-
-lemma lower_col_lt (D : YoungDiagram) (x : Lower D) : x.val.val.2 < Fintype.card (Lower D) := by
-  let f : Fin (x.val.val.2 + 1) → Lower D := fun j =>
-    ⟨⟨(1, j), D.up_left_mem (i2 := x.val.val.1) (j2 := x.val.val.2) (by have := x.property; omega) (by have := j.isLt; omega) x.val.property⟩, by simp⟩
-  have hf : Function.Injective f := by
-    intro j k hjk
-    exact Fin.ext (congrArg (fun z : Lower D => z.val.val.2) hjk)
-  have hh := Fintype.card_le_of_injective f hf
-  rw [Fintype.card_fin] at hh
-  omega
-
-theorem choose_row_le_extensions (D : YoungDiagram)
-    (hu : Fintype.card (Lower D) ≤ D.rowLen 0) :
-    (D.rowLen 0).choose (Fintype.card (Lower D)) ≤ Fintype.card (Extensions (Cells D)) := by
-  let u := Fintype.card (Lower D)
-  let s := topPrefix D u
-  let A := (sᶜ : Set (Cells D))
-  let r : Set A := {x | x.val.val.1 ≠ 0}
-  have hc : Fintype.card A = D.rowLen 0 := by
-    have hh := Fintype.card_congr (Equiv.Set.sumCompl s)
-    simp only [Fintype.card_sum] at hh
-    have hs : Fintype.card s = u := card_topPrefix D u hu
-    have hl := card_lower_add D
-    change D.rowLen 0 + u = Fintype.card (Cells D) at hl
-    change Fintype.card s + Fintype.card A = Fintype.card (Cells D) at hh
-    omega
-  let e : r ≃ Lower D := {
-    toFun := fun x => ⟨x.val.val, x.property⟩
-    invFun := fun x => ⟨⟨x.val, fun h => x.property h.1⟩, x.property⟩
-    left_inv := fun _ => rfl
-    right_inv := fun _ => rfl }
-  have hr : Fintype.card r = u := Fintype.card_congr e
-  have hsplit : ∀ x y : A, x < y → (x ∈ r ↔ y ∈ r) := by
-    intro x y hxy
-    have h₁ := hxy.le.1
-    have h₂ := hxy.le.2
-    change x.val.val.1 ≠ 0 ↔ y.val.val.1 ≠ 0
-    constructor
-    · intro hx hy; exact hx (by omega)
-    · intro hy hx
-      have hcol := lower_col_lt D ⟨y.val, hy⟩
-      have hp := x.property
-      change ¬(x.val.val.1 = 0 ∧ x.val.val.2 < u) at hp
-      change y.val.val.2 < u at hcol
-      exact hp ⟨hx, lt_of_le_of_lt h₂ hcol⟩
-  have hh := choose_le_extensions_split r hsplit
-  have hrN : Nat.card r = u := by simpa only [Fintype.card_eq_nat_card] using hr
-  have hcN : Nat.card A = D.rowLen 0 := by simpa only [Fintype.card_eq_nat_card] using hc
-  have hh' : (D.rowLen 0).choose u ≤ Fintype.card (Extensions A) := by
-    simpa only [Fintype.card_eq_nat_card, hrN, hcN] using hh
-  exact hh'.trans (extensions_card_compl_ideal s (topPrefix_lower D u))
-
-end Thorp.Young
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.LinearExtensions
+
+namespace OAI
+
+universe uA
+
+noncomputable section
+
+open scoped BigOperators ComplexConjugate InnerProductSpace
+open Filter
+
+namespace Thorp.Young
+open scoped Classical
+variable {A : Type uA} [Fintype A] [PartialOrder A]
+
+lemma extensions_card_ideal (s : Set A) (hs : IsLowerSet s) :
+    Fintype.card (Extensions s) ≤ Fintype.card (Extensions A) := by
+  obtain ⟨c⟩ := extensions_nonempty (sᶜ : Set A)
+  have hc : Fintype.card s + Fintype.card (sᶜ : Set A) = Fintype.card A := by
+    simpa only [Fintype.card_sum] using Fintype.card_congr (Equiv.Set.sumCompl s)
+  let F (e : Extensions s) : A ≃ Fin (Fintype.card A) :=
+    ((Equiv.Set.sumCompl s).symm.trans (Equiv.sumCongr e.val c.val)).trans
+      (finSumFinEquiv.trans (finCongr hc))
+  have hFs (e : Extensions s) (x : s) : (F e x).val = (e.val x).val := by
+    simp [F, Equiv.Set.sumCompl_symm_apply_of_mem x.property]
+  have hFc (e : Extensions s) (x : (sᶜ : Set A)) :
+      (F e x).val = Fintype.card s + (c.val x).val := by
+    simp [F, Equiv.Set.sumCompl_symm_apply_of_notMem x.property]
+  have hF (e : Extensions s) : StrictMono (F e) := by
+    intro x y hxy
+    change (F e x).val < (F e y).val
+    by_cases hx : x ∈ s <;> by_cases hy : y ∈ s
+    · rw [hFs e ⟨x, hx⟩, hFs e ⟨y, hy⟩]
+      exact e.property hxy
+    · rw [hFs e ⟨x, hx⟩, hFc e ⟨y, hy⟩]
+      exact (e.val ⟨x, hx⟩).isLt.trans_le (Nat.le_add_right _ _)
+    · exact False.elim (hx (hs hxy.le hy))
+    · rw [hFc e ⟨x, hx⟩, hFc e ⟨y, hy⟩]
+      exact Nat.add_lt_add_left (c.property hxy) _
+  let f (e : Extensions s) : Extensions A := ⟨F e, hF e⟩
+  apply Fintype.card_le_of_injective f
+  intro e d hed
+  apply Subtype.ext
+  apply Equiv.ext
+  intro x
+  apply Fin.ext
+  have hh := congrArg (fun z : Extensions A => (z.val x).val) hed
+  change (F e x).val = (F d x).val at hh
+  simpa only [hFs] using hh
+
+lemma extensions_card_compl_ideal (s : Set A) (hs : IsLowerSet s) :
+    Fintype.card (Extensions (sᶜ : Set A)) ≤ Fintype.card (Extensions A) := by
+  obtain ⟨a⟩ := extensions_nonempty s
+  have hc : Fintype.card s + Fintype.card (sᶜ : Set A) = Fintype.card A := by
+    simpa only [Fintype.card_sum] using Fintype.card_congr (Equiv.Set.sumCompl s)
+  let F (e : Extensions (sᶜ : Set A)) : A ≃ Fin (Fintype.card A) :=
+    ((Equiv.Set.sumCompl s).symm.trans (Equiv.sumCongr a.val e.val)).trans
+      (finSumFinEquiv.trans (finCongr hc))
+  have hFs (e : Extensions (sᶜ : Set A)) (x : s) : (F e x).val = (a.val x).val := by
+    simp [F, Equiv.Set.sumCompl_symm_apply_of_mem x.property]
+  have hFc (e : Extensions (sᶜ : Set A)) (x : (sᶜ : Set A)) :
+      (F e x).val = Fintype.card s + (e.val x).val := by
+    simp [F, Equiv.Set.sumCompl_symm_apply_of_notMem x.property]
+  have hF (e : Extensions (sᶜ : Set A)) : StrictMono (F e) := by
+    intro x y hxy
+    change (F e x).val < (F e y).val
+    by_cases hx : x ∈ s <;> by_cases hy : y ∈ s
+    · rw [hFs e ⟨x, hx⟩, hFs e ⟨y, hy⟩]
+      exact a.property hxy
+    · rw [hFs e ⟨x, hx⟩, hFc e ⟨y, hy⟩]
+      exact (a.val ⟨x, hx⟩).isLt.trans_le (Nat.le_add_right _ _)
+    · exact False.elim (hx (hs hxy.le hy))
+    · rw [hFc e ⟨x, hx⟩, hFc e ⟨y, hy⟩]
+      exact Nat.add_lt_add_left (e.property hxy) _
+  let f (e : Extensions (sᶜ : Set A)) : Extensions A := ⟨F e, hF e⟩
+  apply Fintype.card_le_of_injective f
+  intro e d hed
+  apply Subtype.ext
+  apply Equiv.ext
+  intro x
+  apply Fin.ext
+  have hh := congrArg (fun z : Extensions A => (z.val x).val) hed
+  change (F e x).val = (F d x).val at hh
+  have h := congrArg (fun z : ℕ => z - Fintype.card s) hh
+  simpa only [hFc, Nat.add_sub_cancel_left] using h
+
+end Thorp.Young
+
+namespace Thorp.Young
+open scoped Classical
+variable {A : Type uA} [Fintype A] [PartialOrder A]
+
+lemma choose_le_extensions_split (s : Set A)
+    (hs : ∀ x y : A, x < y → (x ∈ s ↔ y ∈ s)) :
+    (Fintype.card A).choose (Fintype.card s) ≤ Fintype.card (Extensions A) := by
+  obtain ⟨a⟩ := extensions_nonempty s
+  obtain ⟨b⟩ := extensions_nonempty (sᶜ : Set A)
+  let I := {S : Finset (Fin (Fintype.card A)) // S.card = Fintype.card s}
+  have hi : Fintype.card I = (Fintype.card A).choose (Fintype.card s) := by
+    calc
+      Fintype.card I = ((Finset.univ : Finset (Fin (Fintype.card A))).powersetCard (Fintype.card s)).card := by
+        rw [Fintype.card_subtype]
+        congr 1
+        ext S
+        simp
+      _ = _ := by rw [Finset.card_powersetCard, Finset.card_univ, Fintype.card_fin]
+  let P (S : I) : Set (Fin (Fintype.card A)) := S.val
+  have hP (S : I) : Fintype.card (P S) = Fintype.card s := by
+    simpa [P] using S.property
+  have hC (S : I) : Fintype.card ((P S)ᶜ : Set (Fin (Fintype.card A))) =
+      Fintype.card (sᶜ : Set A) := by
+    have h₁ := Fintype.card_congr (Equiv.Set.sumCompl (P S))
+    have h₂ := Fintype.card_congr (Equiv.Set.sumCompl s)
+    simp only [Fintype.card_sum, Fintype.card_fin, hP] at h₁ h₂
+    omega
+  let ea (S : I) := (Fintype.orderIsoFinOfCardEq (P S) (hP S))
+  let eb (S : I) := (Fintype.orderIsoFinOfCardEq ((P S)ᶜ : Set _) (hC S))
+  let F (S : I) : A ≃ Fin (Fintype.card A) :=
+    ((Equiv.Set.sumCompl s).symm.trans
+      (Equiv.sumCongr (a.val.trans (ea S).toEquiv) (b.val.trans (eb S).toEquiv))).trans
+        (Equiv.Set.sumCompl (P S))
+  have hFa (S : I) (x : s) : F S x = (ea S (a.val x)).val := by
+    simp [F, Equiv.Set.sumCompl_symm_apply_of_mem x.property]
+  have hFb (S : I) (x : (sᶜ : Set A)) : F S x = (eb S (b.val x)).val := by
+    simp [F, Equiv.Set.sumCompl_symm_apply_of_notMem x.property]
+  have hFm (S : I) (x : A) : F S x ∈ P S ↔ x ∈ s := by
+    by_cases hx : x ∈ s
+    · rw [hFa S ⟨x, hx⟩]
+      exact iff_of_true (ea S (a.val ⟨x, hx⟩)).property hx
+    · rw [hFb S ⟨x, hx⟩]
+      exact iff_of_false (eb S (b.val ⟨x, hx⟩)).property hx
+  have hF (S : I) : StrictMono (F S) := by
+    intro x y hxy
+    by_cases hx : x ∈ s
+    · have hy := (hs x y hxy).mp hx
+      rw [hFa S ⟨x, hx⟩, hFa S ⟨y, hy⟩]
+      exact (ea S).strictMono (a.property hxy)
+    · have hy : y ∉ s := fun h => hx ((hs x y hxy).mpr h)
+      rw [hFb S ⟨x, hx⟩, hFb S ⟨y, hy⟩]
+      exact (eb S).strictMono (b.property hxy)
+  let f (S : I) : Extensions A := ⟨F S, hF S⟩
+  rw [← hi]
+  apply Fintype.card_le_of_injective f
+  intro S T hST
+  apply Subtype.ext
+  ext z
+  obtain ⟨x, rfl⟩ := (F S).surjective z
+  have he : F S x = F T x := congrArg (fun e : Extensions A => e.val x) hST
+  change F S x ∈ P S ↔ F S x ∈ P T
+  rw [hFm, he, hFm]
+
+end Thorp.Young
+
+namespace Thorp.Young
+open scoped Classical
+
+abbrev Lower (D : YoungDiagram) := {x : Cells D // x.val.1 ≠ 0}
+
+def topPrefix (D : YoungDiagram) (u : ℕ) : Set (Cells D) :=
+  {x | x.val.1 = 0 ∧ x.val.2 < u}
+
+lemma topPrefix_lower (D : YoungDiagram) (u : ℕ) : IsLowerSet (topPrefix D u) := by
+  intro y x hxy hy
+  change y.val.1 = 0 ∧ y.val.2 < u at hy
+  change x.val.1 = 0 ∧ x.val.2 < u
+  exact ⟨by have := hxy.1; omega, lt_of_le_of_lt hxy.2 hy.2⟩
+
+lemma card_topPrefix (D : YoungDiagram) (u : ℕ) (hu : u ≤ D.rowLen 0) :
+    Fintype.card (topPrefix D u) = u := by
+  let e : topPrefix D u ≃ Fin u := {
+    toFun := fun x => ⟨x.val.val.2, x.property.2⟩
+    invFun := fun j => ⟨⟨(0, j), YoungDiagram.mem_iff_lt_rowLen.mpr (j.isLt.trans_le hu)⟩, rfl, j.isLt⟩
+    left_inv := by intro x; apply Subtype.ext; apply Subtype.ext; exact Prod.ext x.property.1.symm rfl
+    right_inv := fun _ => rfl }
+  exact (Fintype.card_congr e).trans (Fintype.card_fin u)
+
+lemma card_lower_add (D : YoungDiagram) :
+    D.rowLen 0 + Fintype.card (Lower D) = Fintype.card (Cells D) := by
+  have h := Fintype.card_congr (Equiv.sumCompl (fun x : Cells D => x.val.1 = 0))
+  simpa only [Fintype.card_sum, card_row_subtype] using h
+
+lemma lower_col_lt (D : YoungDiagram) (x : Lower D) : x.val.val.2 < Fintype.card (Lower D) := by
+  let f : Fin (x.val.val.2 + 1) → Lower D := fun j =>
+    ⟨⟨(1, j), D.up_left_mem (i2 := x.val.val.1) (j2 := x.val.val.2) (by have := x.property; omega) (by have := j.isLt; omega) x.val.property⟩, by simp⟩
+  have hf : Function.Injective f := by
+    intro j k hjk
+    exact Fin.ext (congrArg (fun z : Lower D => z.val.val.2) hjk)
+  have hh := Fintype.card_le_of_injective f hf
+  rw [Fintype.card_fin] at hh
+  omega
+
+theorem choose_row_le_extensions (D : YoungDiagram)
+    (hu : Fintype.card (Lower D) ≤ D.rowLen 0) :
+    (D.rowLen 0).choose (Fintype.card (Lower D)) ≤ Fintype.card (Extensions (Cells D)) := by
+  let u := Fintype.card (Lower D)
+  let s := topPrefix D u
+  let A := (sᶜ : Set (Cells D))
+  let r : Set A := {x | x.val.val.1 ≠ 0}
+  have hc : Fintype.card A = D.rowLen 0 := by
+    have hh := Fintype.card_congr (Equiv.Set.sumCompl s)
+    simp only [Fintype.card_sum] at hh
+    have hs : Fintype.card s = u := card_topPrefix D u hu
+    have hl := card_lower_add D
+    change D.rowLen 0 + u = Fintype.card (Cells D) at hl
+    change Fintype.card s + Fintype.card A = Fintype.card (Cells D) at hh
+    omega
+  let e : r ≃ Lower D := {
+    toFun := fun x => ⟨x.val.val, x.property⟩
+    invFun := fun x => ⟨⟨x.val, fun h => x.property h.1⟩, x.property⟩
+    left_inv := fun _ => rfl
+    right_inv := fun _ => rfl }
+  have hr : Fintype.card r = u := Fintype.card_congr e
+  have hsplit : ∀ x y : A, x < y → (x ∈ r ↔ y ∈ r) := by
+    intro x y hxy
+    have h₁ := hxy.le.1
+    have h₂ := hxy.le.2
+    change x.val.val.1 ≠ 0 ↔ y.val.val.1 ≠ 0
+    constructor
+    · intro hx hy; exact hx (by omega)
+    · intro hy hx
+      have hcol := lower_col_lt D ⟨y.val, hy⟩
+      have hp := x.property
+      change ¬(x.val.val.1 = 0 ∧ x.val.val.2 < u) at hp
+      change y.val.val.2 < u at hcol
+      exact hp ⟨hx, lt_of_le_of_lt h₂ hcol⟩
+  have hh := choose_le_extensions_split r hsplit
+  have hrN : Nat.card r = u := by simpa only [Fintype.card_eq_nat_card] using hr
+  have hcN : Nat.card A = D.rowLen 0 := by simpa only [Fintype.card_eq_nat_card] using hc
+  have hh' : (D.rowLen 0).choose u ≤ Fintype.card (Extensions A) := by
+    simpa only [Fintype.card_eq_nat_card, hrN, hcN] using hh
+  exact hh'.trans (extensions_card_compl_ideal s (topPrefix_lower D u))
+
+end Thorp.Young
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/FiniteExpectation.lean b/lean/OAI/Probability/ThorpShuffle/FiniteExpectation.lean
index f6a1fa60b314495edeed1be614d0d8e5ec3caecf..ff62f4f4a65ddc8725be16d61d9385cac4e662b8 100644
--- a/lean/OAI/Probability/ThorpShuffle/FiniteExpectation.lean
+++ b/lean/OAI/Probability/ThorpShuffle/FiniteExpectation.lean
@@ -1,134 +1,134 @@
-import OAI.Probability.ThorpShuffle.ConditionalFlow
-
-namespace OAI
-
-universe uΩ uι uΩ'
-
-noncomputable section
-
-open scoped BigOperators
-open Filter
-
-namespace Thorp
-
-namespace Conditional
-
-theorem mean_congr {Ω : Type uΩ} [Fintype Ω] {f g : Ω → ℝ} (h : ∀ ω, f ω = g ω) :
-    mean f = mean g := by
-  unfold mean
-  congr 1
-  exact Finset.sum_congr rfl (fun ω _ => h ω)
-
-theorem mean_zero {Ω : Type uΩ} [Fintype Ω] : mean (fun _ : Ω => (0 : ℝ)) = 0 := by
-  simp [mean]
-
-theorem mean_const {Ω : Type uΩ} [Fintype Ω] [Nonempty Ω] (a : ℝ) :
-    mean (fun _ : Ω => a) = a := by
-  have hc : (Fintype.card Ω : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
-  simp [mean, hc]
-
-theorem mean_add {Ω : Type uΩ} [Fintype Ω] (f g : Ω → ℝ) :
-    mean (fun ω => f ω + g ω) = mean f + mean g := by
-  simp only [mean, Finset.sum_add_distrib, add_div]
-
-theorem mean_mul_const {Ω : Type uΩ} [Fintype Ω] (f : Ω → ℝ) (a : ℝ) :
-    mean (fun ω => f ω * a) = mean f * a := by
-  simp only [mean, ← Finset.sum_mul]
-  ring
-
-theorem mean_const_mul {Ω : Type uΩ} [Fintype Ω] (f : Ω → ℝ) (a : ℝ) :
-    mean (fun ω => a * f ω) = a * mean f := by
-  simpa only [mul_comm] using mean_mul_const f a
-
-theorem mean_sum {Ω : Type uΩ} {ι : Type uι} [Fintype Ω] (s : Finset ι) (f : Ω → ι → ℝ) :
-    mean (fun ω => ∑ i ∈ s, f ω i) = ∑ i ∈ s, mean (fun ω => f ω i) := by
-  unfold mean
-  rw [Finset.sum_comm, Finset.sum_div]
-
-theorem mean_nonneg {Ω : Type uΩ} [Fintype Ω] {f : Ω → ℝ} (hf : ∀ ω, 0 ≤ f ω) :
-    0 ≤ mean f := by
-  exact div_nonneg (Finset.sum_nonneg (fun ω _ => hf ω)) (Nat.cast_nonneg _)
-
-theorem mean_le_mean {Ω : Type uΩ} [Fintype Ω] {f g : Ω → ℝ} (h : ∀ ω, f ω ≤ g ω) :
-    mean f ≤ mean g := by
-  exact div_le_div_of_nonneg_right (Finset.sum_le_sum (fun ω _ => h ω)) (Nat.cast_nonneg _)
-
-theorem mean_le_const {Ω : Type uΩ} [Fintype Ω] [Nonempty Ω] {f : Ω → ℝ} {a : ℝ}
-    (h : ∀ ω, f ω ≤ a) : mean f ≤ a := by
-  simpa only [mean_const] using mean_le_mean h
-
-theorem mean_equiv {Ω : Type uΩ} {Ω' : Type uΩ'} [Fintype Ω] [Fintype Ω']
-    (e : Ω ≃ Ω') (f : Ω' → ℝ) : mean (f ∘ e) = mean f := by
-  unfold mean
-  rw [Fintype.card_congr e]
-  congr 1
-  exact Equiv.sum_comp e f
-
-theorem mean_prod {Ω : Type uΩ} {Ω' : Type uΩ'} [Fintype Ω] [Fintype Ω']
-    (f : Ω × Ω' → ℝ) : mean f = mean (fun ω => mean (fun ω' => f (ω, ω'))) := by
-  simp only [mean, Fintype.sum_prod_type, Fintype.card_prod, Nat.cast_mul,
-    ← Finset.sum_div]
-  ring
-
-theorem mean_comm {Ω : Type uΩ} {Ω' : Type uΩ'} [Fintype Ω] [Fintype Ω'] (f : Ω → Ω' → ℝ) :
-    mean (fun ω => mean (f ω)) = mean (fun ω' => mean (fun ω => f ω ω')) := by
-  simp only [mean, ← Finset.sum_div]
-  rw [Finset.sum_comm]
-  ring
-
-theorem mean_history_succ (d t : ℕ) (f : History d (t + 1) → ℝ) :
-    mean f = mean (fun ω : History d t => mean (fun c : Coins d => f (Fin.snoc ω c))) := by
-  have h := mean_equiv (Fin.snocEquiv (fun _ : Fin (t + 1) => Coins d)) f
-  rw [mean_prod] at h
-  rw [← h, mean_comm]
-  rfl
-
-end Conditional
-
-@[simp] theorem run_zero (d : ℕ) (ω : History d 0) : run d 0 ω = 1 := by
-  simp [run]
-
-theorem run_succ (d t : ℕ) (ω : History d (t + 1)) :
-    run d (t + 1) ω = step d (ω (Fin.last t)) *
-      run d t (fun i => ω i.castSucc) := by
-  simp only [run, List.ofFn_succ']
-  rw [List.concat_eq_append, List.reverse_concat', List.prod_cons]
-
-namespace Conditional
-
-theorem iterateState_free (d : ℕ) (v : CenteredState (d + 1)) (t : ℕ)
-    (ω : History (d + 1) t) (x : Position (d + 1)) :
-    (iterateState d v t ω).free x = v.free ((run (d + 1) t ω).symm x) := by
-  induction t generalizing x with
-  | zero => simp only [iterateState, run_zero]; rfl
-  | succ t ih =>
-    change (iterateState d v t (fun i => ω i.castSucc)).free
-      ((step (d + 1) (ω (Fin.last t))).symm x) = _
-    rw [ih, run_succ]
-    rfl
-
-theorem iterateState_freeCount (d : ℕ) (v : CenteredState (d + 1)) (t : ℕ)
-    (ω : History (d + 1) t) : freeCount (iterateState d v t ω).free = freeCount v.free := by
-  apply Nat.cast_injective (R := ℝ)
-  rw [← sum_free_indicator, ← sum_free_indicator]
-  simp only [iterateState_free]
-  exact Equiv.sum_comp (run (d + 1) t ω).symm (fun x => if v.free x then (1 : ℝ) else 0)
-
-theorem expectedEnergy_succ (d : ℕ) (v : CenteredState (d + 1)) (t : ℕ) :
-    expectedEnergy d v (t + 1) = mean (fun ω : History (d + 1) t =>
-      mean (fun c : Coins (d + 1) => ∑ x, (nextState d (iterateState d v t ω) c).weight x ^ 2)) := by
-  unfold expectedEnergy
-  rw [mean_history_succ]
-  apply mean_congr
-  intro ω
-  apply mean_congr
-  intro c
-  simp [iterateState]
-
-end Conditional
-
-end Thorp
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.ConditionalFlow
+
+namespace OAI
+
+universe uΩ uι uΩ'
+
+noncomputable section
+
+open scoped BigOperators
+open Filter
+
+namespace Thorp
+
+namespace Conditional
+
+theorem mean_congr {Ω : Type uΩ} [Fintype Ω] {f g : Ω → ℝ} (h : ∀ ω, f ω = g ω) :
+    mean f = mean g := by
+  unfold mean
+  congr 1
+  exact Finset.sum_congr rfl (fun ω _ => h ω)
+
+theorem mean_zero {Ω : Type uΩ} [Fintype Ω] : mean (fun _ : Ω => (0 : ℝ)) = 0 := by
+  simp [mean]
+
+theorem mean_const {Ω : Type uΩ} [Fintype Ω] [Nonempty Ω] (a : ℝ) :
+    mean (fun _ : Ω => a) = a := by
+  have hc : (Fintype.card Ω : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
+  simp [mean, hc]
+
+theorem mean_add {Ω : Type uΩ} [Fintype Ω] (f g : Ω → ℝ) :
+    mean (fun ω => f ω + g ω) = mean f + mean g := by
+  simp only [mean, Finset.sum_add_distrib, add_div]
+
+theorem mean_mul_const {Ω : Type uΩ} [Fintype Ω] (f : Ω → ℝ) (a : ℝ) :
+    mean (fun ω => f ω * a) = mean f * a := by
+  simp only [mean, ← Finset.sum_mul]
+  ring
+
+theorem mean_const_mul {Ω : Type uΩ} [Fintype Ω] (f : Ω → ℝ) (a : ℝ) :
+    mean (fun ω => a * f ω) = a * mean f := by
+  simpa only [mul_comm] using mean_mul_const f a
+
+theorem mean_sum {Ω : Type uΩ} {ι : Type uι} [Fintype Ω] (s : Finset ι) (f : Ω → ι → ℝ) :
+    mean (fun ω => ∑ i ∈ s, f ω i) = ∑ i ∈ s, mean (fun ω => f ω i) := by
+  unfold mean
+  rw [Finset.sum_comm, Finset.sum_div]
+
+theorem mean_nonneg {Ω : Type uΩ} [Fintype Ω] {f : Ω → ℝ} (hf : ∀ ω, 0 ≤ f ω) :
+    0 ≤ mean f := by
+  exact div_nonneg (Finset.sum_nonneg (fun ω _ => hf ω)) (Nat.cast_nonneg _)
+
+theorem mean_le_mean {Ω : Type uΩ} [Fintype Ω] {f g : Ω → ℝ} (h : ∀ ω, f ω ≤ g ω) :
+    mean f ≤ mean g := by
+  exact div_le_div_of_nonneg_right (Finset.sum_le_sum (fun ω _ => h ω)) (Nat.cast_nonneg _)
+
+theorem mean_le_const {Ω : Type uΩ} [Fintype Ω] [Nonempty Ω] {f : Ω → ℝ} {a : ℝ}
+    (h : ∀ ω, f ω ≤ a) : mean f ≤ a := by
+  simpa only [mean_const] using mean_le_mean h
+
+theorem mean_equiv {Ω : Type uΩ} {Ω' : Type uΩ'} [Fintype Ω] [Fintype Ω']
+    (e : Ω ≃ Ω') (f : Ω' → ℝ) : mean (f ∘ e) = mean f := by
+  unfold mean
+  rw [Fintype.card_congr e]
+  congr 1
+  exact Equiv.sum_comp e f
+
+theorem mean_prod {Ω : Type uΩ} {Ω' : Type uΩ'} [Fintype Ω] [Fintype Ω']
+    (f : Ω × Ω' → ℝ) : mean f = mean (fun ω => mean (fun ω' => f (ω, ω'))) := by
+  simp only [mean, Fintype.sum_prod_type, Fintype.card_prod, Nat.cast_mul,
+    ← Finset.sum_div]
+  ring
+
+theorem mean_comm {Ω : Type uΩ} {Ω' : Type uΩ'} [Fintype Ω] [Fintype Ω'] (f : Ω → Ω' → ℝ) :
+    mean (fun ω => mean (f ω)) = mean (fun ω' => mean (fun ω => f ω ω')) := by
+  simp only [mean, ← Finset.sum_div]
+  rw [Finset.sum_comm]
+  ring
+
+theorem mean_history_succ (d t : ℕ) (f : History d (t + 1) → ℝ) :
+    mean f = mean (fun ω : History d t => mean (fun c : Coins d => f (Fin.snoc ω c))) := by
+  have h := mean_equiv (Fin.snocEquiv (fun _ : Fin (t + 1) => Coins d)) f
+  rw [mean_prod] at h
+  rw [← h, mean_comm]
+  rfl
+
+end Conditional
+
+@[simp] theorem run_zero (d : ℕ) (ω : History d 0) : run d 0 ω = 1 := by
+  simp [run]
+
+theorem run_succ (d t : ℕ) (ω : History d (t + 1)) :
+    run d (t + 1) ω = step d (ω (Fin.last t)) *
+      run d t (fun i => ω i.castSucc) := by
+  simp only [run, List.ofFn_succ']
+  rw [List.concat_eq_append, List.reverse_concat', List.prod_cons]
+
+namespace Conditional
+
+theorem iterateState_free (d : ℕ) (v : CenteredState (d + 1)) (t : ℕ)
+    (ω : History (d + 1) t) (x : Position (d + 1)) :
+    (iterateState d v t ω).free x = v.free ((run (d + 1) t ω).symm x) := by
+  induction t generalizing x with
+  | zero => simp only [iterateState, run_zero]; rfl
+  | succ t ih =>
+    change (iterateState d v t (fun i => ω i.castSucc)).free
+      ((step (d + 1) (ω (Fin.last t))).symm x) = _
+    rw [ih, run_succ]
+    rfl
+
+theorem iterateState_freeCount (d : ℕ) (v : CenteredState (d + 1)) (t : ℕ)
+    (ω : History (d + 1) t) : freeCount (iterateState d v t ω).free = freeCount v.free := by
+  apply Nat.cast_injective (R := ℝ)
+  rw [← sum_free_indicator, ← sum_free_indicator]
+  simp only [iterateState_free]
+  exact Equiv.sum_comp (run (d + 1) t ω).symm (fun x => if v.free x then (1 : ℝ) else 0)
+
+theorem expectedEnergy_succ (d : ℕ) (v : CenteredState (d + 1)) (t : ℕ) :
+    expectedEnergy d v (t + 1) = mean (fun ω : History (d + 1) t =>
+      mean (fun c : Coins (d + 1) => ∑ x, (nextState d (iterateState d v t ω) c).weight x ^ 2)) := by
+  unfold expectedEnergy
+  rw [mean_history_succ]
+  apply mean_congr
+  intro ω
+  apply mean_congr
+  intro c
+  simp [iterateState]
+
+end Conditional
+
+end Thorp
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/FiniteLaw.lean b/lean/OAI/Probability/ThorpShuffle/FiniteLaw.lean
index 45ec9541772878ed2fc5e6da07e6d2ee5f0aec72..5a1a160d7591b3f32fdde38f2da0b159cfa0fdbb 100644
--- a/lean/OAI/Probability/ThorpShuffle/FiniteLaw.lean
+++ b/lean/OAI/Probability/ThorpShuffle/FiniteLaw.lean
@@ -1,141 +1,141 @@
-import OAI.Probability.ThorpShuffle.Reachability
-
-namespace OAI
-
-universe uG uΩ uΩ' uα
-
-noncomputable section
-
-open scoped BigOperators
-open Filter
-
-namespace Thorp
-
-namespace FiniteLaw
-open Conditional
-
-variable {G : Type uG} [Fintype G] [Group G]
-
-def conv (μ ν : G → ℝ) (z : G) : ℝ := ∑ g, μ g * ν (g⁻¹ * z)
-
-theorem conv_fairMass {Ω : Type uΩ} {Ω' : Type uΩ'} [Fintype Ω] [Fintype Ω']
-    (a : Ω → G) (b : Ω' → G) :
-    conv (fairMass a) (fairMass b) = fairMass (fun p : Ω × Ω' => a p.1 * b p.2) := by
-  classical
-  funext z
-  unfold conv
-  rw [fairMass_test]
-  change mean (fun ω => mean (fun ω' => if b ω' = (a ω)⁻¹ * z then 1 else 0)) =
-    mean (fun p : Ω × Ω' => if a p.1 * b p.2 = z then 1 else 0)
-  rw [mean_prod]
-  apply mean_congr
-  intro ω
-  apply mean_congr
-  intro ω'
-  rw [eq_inv_mul_iff_mul_eq]
-
-theorem sum_inv_mul (f : G → ℝ) (z : G) : ∑ g, f (g⁻¹ * z) = ∑ g, f g := by
-  exact Equiv.sum_comp ((Equiv.inv G).trans (Equiv.mulRight z)) f
-
-theorem sum_mul_left (f : G → ℝ) (g : G) : ∑ z, f (g * z) = ∑ z, f z :=
-  Equiv.sum_comp (Equiv.mulLeft g) f
-
-theorem conv_uniform (μ : G → ℝ) (hμ : ∑ g, μ g = 1) :
-    conv μ (fun _ => (1 : ℝ) / Fintype.card G) = fun _ => (1 : ℝ) / Fintype.card G := by
-  funext z
-  simp only [conv, ← Finset.sum_mul, hμ, one_mul]
-
-theorem tv_conv_contract (μ ν : G → ℝ) (hμ : ∑ g, μ g = 1)
-    (hν : ∑ g, ν g = 1) (c : ℝ) (hc : ∀ g, c ≤ μ g) :
-    tv (conv μ ν) (fun _ => (1 : ℝ) / Fintype.card G) ≤
-      (1 - (Fintype.card G : ℝ) * c) * tv ν (fun _ => (1 : ℝ) / Fintype.card G) := by
-  let u : ℝ := 1 / Fintype.card G
-  have hu : ∑ _ : G, u = 1 := by simp [u, Fintype.card_ne_zero]
-  have hcenter (z : G) : ∑ g, (ν (g⁻¹ * z) - u) = 0 := by
-    rw [Finset.sum_sub_distrib, sum_inv_mul, hν, hu, sub_self]
-  have hid (z : G) : conv μ ν z - u = ∑ g, (μ g - c) * (ν (g⁻¹ * z) - u) := by
-    simp only [sub_mul, Finset.sum_sub_distrib]
-    rw [← Finset.mul_sum, hcenter, mul_zero, sub_zero]
-    simp only [mul_sub, Finset.sum_sub_distrib, ← Finset.sum_mul, hμ, one_mul, conv]
-  have hb : ∑ z, |conv μ ν z - u| ≤
-      (1 - (Fintype.card G : ℝ) * c) * ∑ z, |ν z - u| := by
-    calc
-      _ = ∑ z, |∑ g, (μ g - c) * (ν (g⁻¹ * z) - u)| := by simp_rw [hid]
-      _ ≤ ∑ z, ∑ g, (μ g - c) * |ν (g⁻¹ * z) - u| := by
-        apply Finset.sum_le_sum
-        intro z _
-        have h := Finset.abs_sum_le_sum_abs (fun g => (μ g - c) * (ν (g⁻¹ * z) - u)) Finset.univ
-        simpa only [abs_mul, abs_of_nonneg (sub_nonneg.mpr (hc _))] using h
-      _ = (∑ g, (μ g - c)) * ∑ z, |ν z - u| := by
-        rw [Finset.sum_comm, Finset.sum_mul]
-        apply Finset.sum_congr rfl
-        intro g _
-        rw [← Finset.mul_sum, sum_mul_left (fun z => |ν z - u|)]
-      _ = _ := by simp [Finset.sum_sub_distrib, hμ]
-  unfold tv
-  change (1 / 2 : ℝ) * (∑ z, |conv μ ν z - u|) ≤ _
-  nlinarith
-
-omit [Fintype G] [Group G] in
-
-theorem fairMass_lower_of_surjective {Ω : Type uΩ} [Fintype Ω] [Nonempty Ω]
-    (a : Ω → G) (ha : Function.Surjective a) (g : G) :
-    1 / (Fintype.card Ω : ℝ) ≤ fairMass a g := by
-  classical
-  obtain ⟨ω, hω⟩ := ha g
-  unfold fairMass
-  apply div_le_div_of_nonneg_right _ (Nat.cast_nonneg _)
-  have h := Finset.single_le_sum (f := fun ω => if a ω = g then (1 : ℝ) else 0)
-    (fun _ _ => by split_ifs <;> norm_num) (Finset.mem_univ ω)
-  simpa only [hω, ↓reduceIte] using h
-
-end FiniteLaw
-
-theorem fairMass_eq_mean {Ω : Type uΩ} {α : Type uα} [Fintype Ω] [DecidableEq α] (f : Ω → α) (a : α) :
-    fairMass f a = Conditional.mean (fun ω => if f ω = a then (1 : ℝ) else 0) := by
-  classical
-  unfold fairMass Conditional.mean
-  congr 1
-  apply Finset.sum_congr rfl
-  intro ω _
-  dsimp only
-  split_ifs <;> rfl
-
-theorem law_add (d s t : ℕ) :
-    law d (s + t) = FiniteLaw.conv (law d t) (law d s) := by
-  classical
-  rw [law, law, law, FiniteLaw.conv_fairMass]
-  funext z
-  rw [fairMass_eq_mean, fairMass_eq_mean]
-  rw [Conditional.mean_history_add, Conditional.mean_prod, Conditional.mean_comm]
-  simp only [Reachability.run_append]
-
-namespace Reachability
-
-theorem exists_surjective_run (d : ℕ) :
-    ∃ T : ℕ, Function.Surjective (run (d + 1) T) := by
-  classical
-  choose t ht ω hω using attainable_return d
-  let T : ℕ := ∑ g : State (d + 1), t g
-  have hT : orderOf (rotate (d + 1)) ∣ T := Finset.dvd_sum (fun g _ => ht g)
-  refine ⟨T, fun g => ?_⟩
-  have hle : t g ≤ T := Finset.single_le_sum (fun _ _ => Nat.zero_le _) (Finset.mem_univ g)
-  have hdvd : orderOf (rotate (d + 1)) ∣ T - t g := Nat.dvd_sub hT (ht g)
-  have hid : run (d + 1) (T - t g) (fun _ => zeroCoins d) = 1 := by
-    rw [run_false]
-    exact orderOf_dvd_iff_pow_eq_one.mp hdvd
-  have hh : ∃ a : History (d + 1) (t g + (T - t g)),
-      run (d + 1) (t g + (T - t g)) a = g := by
-    refine ⟨Fin.append (ω g) (fun _ => zeroCoins d), ?_⟩
-    rw [run_append, hid, one_mul, hω]
-  have he : t g + (T - t g) = T := by omega
-  rw [he] at hh
-  exact hh
-
-end Reachability
-
-end Thorp
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.Reachability
+
+namespace OAI
+
+universe uG uΩ uΩ' uα
+
+noncomputable section
+
+open scoped BigOperators
+open Filter
+
+namespace Thorp
+
+namespace FiniteLaw
+open Conditional
+
+variable {G : Type uG} [Fintype G] [Group G]
+
+def conv (μ ν : G → ℝ) (z : G) : ℝ := ∑ g, μ g * ν (g⁻¹ * z)
+
+theorem conv_fairMass {Ω : Type uΩ} {Ω' : Type uΩ'} [Fintype Ω] [Fintype Ω']
+    (a : Ω → G) (b : Ω' → G) :
+    conv (fairMass a) (fairMass b) = fairMass (fun p : Ω × Ω' => a p.1 * b p.2) := by
+  classical
+  funext z
+  unfold conv
+  rw [fairMass_test]
+  change mean (fun ω => mean (fun ω' => if b ω' = (a ω)⁻¹ * z then 1 else 0)) =
+    mean (fun p : Ω × Ω' => if a p.1 * b p.2 = z then 1 else 0)
+  rw [mean_prod]
+  apply mean_congr
+  intro ω
+  apply mean_congr
+  intro ω'
+  rw [eq_inv_mul_iff_mul_eq]
+
+theorem sum_inv_mul (f : G → ℝ) (z : G) : ∑ g, f (g⁻¹ * z) = ∑ g, f g := by
+  exact Equiv.sum_comp ((Equiv.inv G).trans (Equiv.mulRight z)) f
+
+theorem sum_mul_left (f : G → ℝ) (g : G) : ∑ z, f (g * z) = ∑ z, f z :=
+  Equiv.sum_comp (Equiv.mulLeft g) f
+
+theorem conv_uniform (μ : G → ℝ) (hμ : ∑ g, μ g = 1) :
+    conv μ (fun _ => (1 : ℝ) / Fintype.card G) = fun _ => (1 : ℝ) / Fintype.card G := by
+  funext z
+  simp only [conv, ← Finset.sum_mul, hμ, one_mul]
+
+theorem tv_conv_contract (μ ν : G → ℝ) (hμ : ∑ g, μ g = 1)
+    (hν : ∑ g, ν g = 1) (c : ℝ) (hc : ∀ g, c ≤ μ g) :
+    tv (conv μ ν) (fun _ => (1 : ℝ) / Fintype.card G) ≤
+      (1 - (Fintype.card G : ℝ) * c) * tv ν (fun _ => (1 : ℝ) / Fintype.card G) := by
+  let u : ℝ := 1 / Fintype.card G
+  have hu : ∑ _ : G, u = 1 := by simp [u, Fintype.card_ne_zero]
+  have hcenter (z : G) : ∑ g, (ν (g⁻¹ * z) - u) = 0 := by
+    rw [Finset.sum_sub_distrib, sum_inv_mul, hν, hu, sub_self]
+  have hid (z : G) : conv μ ν z - u = ∑ g, (μ g - c) * (ν (g⁻¹ * z) - u) := by
+    simp only [sub_mul, Finset.sum_sub_distrib]
+    rw [← Finset.mul_sum, hcenter, mul_zero, sub_zero]
+    simp only [mul_sub, Finset.sum_sub_distrib, ← Finset.sum_mul, hμ, one_mul, conv]
+  have hb : ∑ z, |conv μ ν z - u| ≤
+      (1 - (Fintype.card G : ℝ) * c) * ∑ z, |ν z - u| := by
+    calc
+      _ = ∑ z, |∑ g, (μ g - c) * (ν (g⁻¹ * z) - u)| := by simp_rw [hid]
+      _ ≤ ∑ z, ∑ g, (μ g - c) * |ν (g⁻¹ * z) - u| := by
+        apply Finset.sum_le_sum
+        intro z _
+        have h := Finset.abs_sum_le_sum_abs (fun g => (μ g - c) * (ν (g⁻¹ * z) - u)) Finset.univ
+        simpa only [abs_mul, abs_of_nonneg (sub_nonneg.mpr (hc _))] using h
+      _ = (∑ g, (μ g - c)) * ∑ z, |ν z - u| := by
+        rw [Finset.sum_comm, Finset.sum_mul]
+        apply Finset.sum_congr rfl
+        intro g _
+        rw [← Finset.mul_sum, sum_mul_left (fun z => |ν z - u|)]
+      _ = _ := by simp [Finset.sum_sub_distrib, hμ]
+  unfold tv
+  change (1 / 2 : ℝ) * (∑ z, |conv μ ν z - u|) ≤ _
+  nlinarith
+
+omit [Fintype G] [Group G] in
+
+theorem fairMass_lower_of_surjective {Ω : Type uΩ} [Fintype Ω] [Nonempty Ω]
+    (a : Ω → G) (ha : Function.Surjective a) (g : G) :
+    1 / (Fintype.card Ω : ℝ) ≤ fairMass a g := by
+  classical
+  obtain ⟨ω, hω⟩ := ha g
+  unfold fairMass
+  apply div_le_div_of_nonneg_right _ (Nat.cast_nonneg _)
+  have h := Finset.single_le_sum (f := fun ω => if a ω = g then (1 : ℝ) else 0)
+    (fun _ _ => by split_ifs <;> norm_num) (Finset.mem_univ ω)
+  simpa only [hω, ↓reduceIte] using h
+
+end FiniteLaw
+
+theorem fairMass_eq_mean {Ω : Type uΩ} {α : Type uα} [Fintype Ω] [DecidableEq α] (f : Ω → α) (a : α) :
+    fairMass f a = Conditional.mean (fun ω => if f ω = a then (1 : ℝ) else 0) := by
+  classical
+  unfold fairMass Conditional.mean
+  congr 1
+  apply Finset.sum_congr rfl
+  intro ω _
+  dsimp only
+  split_ifs <;> rfl
+
+theorem law_add (d s t : ℕ) :
+    law d (s + t) = FiniteLaw.conv (law d t) (law d s) := by
+  classical
+  rw [law, law, law, FiniteLaw.conv_fairMass]
+  funext z
+  rw [fairMass_eq_mean, fairMass_eq_mean]
+  rw [Conditional.mean_history_add, Conditional.mean_prod, Conditional.mean_comm]
+  simp only [Reachability.run_append]
+
+namespace Reachability
+
+theorem exists_surjective_run (d : ℕ) :
+    ∃ T : ℕ, Function.Surjective (run (d + 1) T) := by
+  classical
+  choose t ht ω hω using attainable_return d
+  let T : ℕ := ∑ g : State (d + 1), t g
+  have hT : orderOf (rotate (d + 1)) ∣ T := Finset.dvd_sum (fun g _ => ht g)
+  refine ⟨T, fun g => ?_⟩
+  have hle : t g ≤ T := Finset.single_le_sum (fun _ _ => Nat.zero_le _) (Finset.mem_univ g)
+  have hdvd : orderOf (rotate (d + 1)) ∣ T - t g := Nat.dvd_sub hT (ht g)
+  have hid : run (d + 1) (T - t g) (fun _ => zeroCoins d) = 1 := by
+    rw [run_false]
+    exact orderOf_dvd_iff_pow_eq_one.mp hdvd
+  have hh : ∃ a : History (d + 1) (t g + (T - t g)),
+      run (d + 1) (t g + (T - t g)) a = g := by
+    refine ⟨Fin.append (ω g) (fun _ => zeroCoins d), ?_⟩
+    rw [run_append, hid, one_mul, hω]
+  have he : t g + (T - t g) = T := by omega
+  rw [he] at hh
+  exact hh
+
+end Reachability
+
+end Thorp
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/FirstFourier.lean b/lean/OAI/Probability/ThorpShuffle/FirstFourier.lean
index 44f529eebb91047c2532689ed0f3d7e566ff245f..22a10bed1a388aaaa921cf78953ed9594b38f098 100644
--- a/lean/OAI/Probability/ThorpShuffle/FirstFourier.lean
+++ b/lean/OAI/Probability/ThorpShuffle/FirstFourier.lean
@@ -1,121 +1,121 @@
-import OAI.Probability.ThorpShuffle.ExceptionalShapes
-import OAI.Probability.ThorpShuffle.BlockEmbedding
-import OAI.Probability.ThorpShuffle.Model
-
-namespace OAI
-
-universe uE
-
-noncomputable section
-
-namespace Thorp.Current
-open scoped BigOperators Classical ComplexConjugate InnerProductSpace
-open Thorp.Fourier
-
-/-- The first inverse-degree moment yields a `D^(3/8)` orbit bound.
-The filter is precisely `degree ≤ D^(1/8)`, written without a root. -/
-lemma low_degree_first {H : Type} [Group H] [Fintype H]
-    (F : Fourier.Family H) (D : ℕ) :
-    (∑ b ∈ Finset.univ.filter (fun b => F.degree b ^ 8 ≤ D), (F.degree b : ℝ)^2) ≤
-      (D : ℝ) ^ (3 / 8 : ℝ) * ∑ b, (F.degree b : ℝ)⁻¹ := by
-  let s := Finset.univ.filter (fun b => F.degree b ^ 8 ≤ D)
-  have hp (b : F.Index) : (0 : ℝ) < F.degree b := by exact_mod_cast F.positive b
-  have hsmall : ∑ b ∈ s, (F.degree b : ℝ)⁻¹ ≤ ∑ b, (F.degree b : ℝ)⁻¹ :=
-    Finset.sum_le_sum_of_subset_of_nonneg (Finset.filter_subset _ _)
-      (fun b _ _ => inv_nonneg.mpr (hp b).le)
-  calc
-    _ ≤ ∑ b ∈ s, (D : ℝ) ^ (3 / 8 : ℝ) * (F.degree b : ℝ)⁻¹ := by
-      apply Finset.sum_le_sum
-      intro b hb
-      have hbD : (F.degree b : ℝ)^8 ≤ D := by
-        exact_mod_cast (Finset.mem_filter.mp hb).2
-      apply (le_mul_inv_iff₀ (hp b)).mpr
-      calc
-        (F.degree b : ℝ)^2 * F.degree b = (F.degree b : ℝ)^3 := by ring
-        _ = ((F.degree b : ℝ)^8) ^ (3 / 8 : ℝ) := by
-          rw [← Real.rpow_natCast_mul (hp b).le]
-          norm_num
-        _ ≤ _ := Real.rpow_le_rpow (by positivity) hbD (by norm_num)
-    _ = (D : ℝ) ^ (3 / 8 : ℝ) * ∑ b ∈ s, (F.degree b : ℝ)⁻¹ := by
-      rw [Finset.mul_sum]
-    _ ≤ _ := mul_le_mul_of_nonneg_left hsmall (Real.rpow_nonneg (Nat.cast_nonneg D) _)
-
-/-- The integrated operator norm bound for an eight-fold block action
-with a common probability mass function satisfying all eight coset caps. -/
-lemma eight_block_of_cap {G : Type} [Group G] [Fintype G]
-    (s : ℕ) (hs : 1 ≤ s)
-    {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
-    [FiniteDimensional ℂ E]
-    (ρ : Representation ℂ G E) [Representation.IsIrreducible ρ]
-    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ)
-    (ψ : (Fin 8 → Equiv.Perm (Fin s)) →* G)
-    (ν : G → ℝ) (hν : ∀ g, 0 ≤ ν g) (h1 : ∑ g, ν g = 1)
-    (hcoset : ∀ (i : Fin 8) g,
-      ∑ h : Equiv.Perm (Fin s), ν (g * ψ (Pi.mulSingle i h)) ≤
-        4 * (Fintype.card (Equiv.Perm (Fin s)) : ℝ) / Fintype.card G) :
-    ‖LinearMap.toContinuousLinearMap (integrated ρ (fun g => (ν g : ℂ)))‖ ≤
-      Real.sqrt (32 * C1) * (Module.finrank ℂ E : ℝ) ^ (-5 / 16 : ℝ) := by
-  let : Nontrivial E := IsSimpleModule.nontrivial (MonoidAlgebra ℂ G) ρ.asModule
-  have hD : (0 : ℝ) < Module.finrank ℂ E := by exact_mod_cast (Module.finrank_pos (R:=ℂ) (M:=E))
-  let D := Module.finrank ℂ E
-  let F := Specht.symmetricFamily s
-  have hsum : (∑ b ∈ Finset.univ.filter (fun b => F.degree b ^ 8 ≤ D), (F.degree b : ℝ)^2) ≤
-      (D : ℝ) ^ (3 / 8 : ℝ) * C1 := by
-    refine (low_degree_first F D).trans (mul_le_mul_of_nonneg_left ?_ (by positivity))
-    exact reciprocalFirst_le_C1 s hs
-  have hconst : 0 ≤ Real.sqrt (32 * C1) * (D : ℝ) ^ (-5 / 16 : ℝ) := by positivity
-  have hconst_sq : (Real.sqrt (32 * C1) * (D : ℝ) ^ (-5 / 16 : ℝ))^2 =
-      8 * 4 * ((D : ℝ) ^ (3 / 8 : ℝ) * C1) / D := by
-    rw [mul_pow, Real.sq_sqrt (mul_nonneg (by norm_num) C1_nonneg),
-      ← Real.rpow_mul_natCast (Nat.cast_nonneg D)]
-    rw [show (-5 / 16 : ℝ) * (2 : ℕ) = (3 / 8 : ℝ) - 1 by norm_num,
-      Real.rpow_sub hD, Real.rpow_one]
-    ring
-  apply ContinuousLinearMap.opNorm_le_bound _ hconst
-  intro v
-  have hb := F.integrated_norm_sq ρ hu (Nat.ne_of_gt Module.finrank_pos) ψ ν 4 hν h1
-    (by norm_num) (fun i g => by
-      convert hcoset i g using 1
-      apply Finset.sum_congr rfl
-      intro h _
-      congr 3
-      funext j
-      by_cases hij : j = i <;> simp [hij]) v
-  simp only [Fintype.card_fin, Nat.cast_ofNat] at hb
-  have hv : ‖integrated ρ (fun g => (ν g : ℂ)) v‖ ^ 2 ≤
-      (Real.sqrt (32 * C1) * (D : ℝ) ^ (-5 / 16 : ℝ) * ‖v‖)^2 := by
-    rw [mul_pow, hconst_sq]
-    exact hb.trans (mul_le_mul_of_nonneg_right
-      (div_le_div_of_nonneg_right
-        (mul_le_mul_of_nonneg_left hsum (by norm_num)) hD.le) (sq_nonneg _))
-  change ‖integrated ρ (fun g => (ν g : ℂ)) v‖ ≤ _
-  nlinarith [norm_nonneg (integrated ρ (fun g => (ν g : ℂ)) v),
-    mul_nonneg hconst (norm_nonneg v)]
-
-/-- The integrated operator norm bound for permutations of binary positions
-under any equivalence with eight equal blocks. -/
-theorem eight_block_fourier_bound
-    (d : ℕ) (_hd : 3 ≤ d)
-    (e : Position d ≃ Fin 8 × Fin (2 ^ (d - 3)))
-    {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
-    [FiniteDimensional ℂ E]
-    (ρ : Representation ℂ (Equiv.Perm (Position d)) E)
-    [Representation.IsIrreducible ρ]
-    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ)
-    (ν : Equiv.Perm (Position d) → ℝ)
-    (hν : ∀ g, 0 ≤ ν g) (h1 : ∑ g, ν g = 1)
-    (hcoset : ∀ (i : Fin 8) (g : Equiv.Perm (Position d)),
-      ∑ h : Equiv.Perm (Fin (2 ^ (d - 3))),
-          ν (g * Block.embedding e (Pi.mulSingle i h)) ≤
-        4 * (Fintype.card (Equiv.Perm (Fin (2 ^ (d - 3)))) : ℝ) /
-          Fintype.card (Equiv.Perm (Position d))) :
-    ‖LinearMap.toContinuousLinearMap (Fourier.integrated ρ (fun g => (ν g : ℂ)))‖ ≤
-      Real.sqrt (32 * C1) * (Module.finrank ℂ E : ℝ) ^ (-5 / 16 : ℝ) := by
-  exact eight_block_of_cap (2 ^ (d - 3)) (Nat.one_le_pow _ _ (by norm_num)) ρ hu
-    (Block.embedding e) ν hν h1 hcoset
-
-end Thorp.Current
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.ExceptionalShapes
+import OAI.Probability.ThorpShuffle.BlockEmbedding
+import OAI.Probability.ThorpShuffle.Model
+
+namespace OAI
+
+universe uE
+
+noncomputable section
+
+namespace Thorp.Current
+open scoped BigOperators Classical ComplexConjugate InnerProductSpace
+open Thorp.Fourier
+
+/-- The first inverse-degree moment yields a `D^(3/8)` orbit bound.
+The filter is precisely `degree ≤ D^(1/8)`, written without a root. -/
+lemma low_degree_first {H : Type} [Group H] [Fintype H]
+    (F : Fourier.Family H) (D : ℕ) :
+    (∑ b ∈ Finset.univ.filter (fun b => F.degree b ^ 8 ≤ D), (F.degree b : ℝ)^2) ≤
+      (D : ℝ) ^ (3 / 8 : ℝ) * ∑ b, (F.degree b : ℝ)⁻¹ := by
+  let s := Finset.univ.filter (fun b => F.degree b ^ 8 ≤ D)
+  have hp (b : F.Index) : (0 : ℝ) < F.degree b := by exact_mod_cast F.positive b
+  have hsmall : ∑ b ∈ s, (F.degree b : ℝ)⁻¹ ≤ ∑ b, (F.degree b : ℝ)⁻¹ :=
+    Finset.sum_le_sum_of_subset_of_nonneg (Finset.filter_subset _ _)
+      (fun b _ _ => inv_nonneg.mpr (hp b).le)
+  calc
+    _ ≤ ∑ b ∈ s, (D : ℝ) ^ (3 / 8 : ℝ) * (F.degree b : ℝ)⁻¹ := by
+      apply Finset.sum_le_sum
+      intro b hb
+      have hbD : (F.degree b : ℝ)^8 ≤ D := by
+        exact_mod_cast (Finset.mem_filter.mp hb).2
+      apply (le_mul_inv_iff₀ (hp b)).mpr
+      calc
+        (F.degree b : ℝ)^2 * F.degree b = (F.degree b : ℝ)^3 := by ring
+        _ = ((F.degree b : ℝ)^8) ^ (3 / 8 : ℝ) := by
+          rw [← Real.rpow_natCast_mul (hp b).le]
+          norm_num
+        _ ≤ _ := Real.rpow_le_rpow (by positivity) hbD (by norm_num)
+    _ = (D : ℝ) ^ (3 / 8 : ℝ) * ∑ b ∈ s, (F.degree b : ℝ)⁻¹ := by
+      rw [Finset.mul_sum]
+    _ ≤ _ := mul_le_mul_of_nonneg_left hsmall (Real.rpow_nonneg (Nat.cast_nonneg D) _)
+
+/-- The integrated operator norm bound for an eight-fold block action
+with a common probability mass function satisfying all eight coset caps. -/
+lemma eight_block_of_cap {G : Type} [Group G] [Fintype G]
+    (s : ℕ) (hs : 1 ≤ s)
+    {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
+    [FiniteDimensional ℂ E]
+    (ρ : Representation ℂ G E) [Representation.IsIrreducible ρ]
+    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ)
+    (ψ : (Fin 8 → Equiv.Perm (Fin s)) →* G)
+    (ν : G → ℝ) (hν : ∀ g, 0 ≤ ν g) (h1 : ∑ g, ν g = 1)
+    (hcoset : ∀ (i : Fin 8) g,
+      ∑ h : Equiv.Perm (Fin s), ν (g * ψ (Pi.mulSingle i h)) ≤
+        4 * (Fintype.card (Equiv.Perm (Fin s)) : ℝ) / Fintype.card G) :
+    ‖LinearMap.toContinuousLinearMap (integrated ρ (fun g => (ν g : ℂ)))‖ ≤
+      Real.sqrt (32 * C1) * (Module.finrank ℂ E : ℝ) ^ (-5 / 16 : ℝ) := by
+  let : Nontrivial E := IsSimpleModule.nontrivial (MonoidAlgebra ℂ G) ρ.asModule
+  have hD : (0 : ℝ) < Module.finrank ℂ E := by exact_mod_cast (Module.finrank_pos (R:=ℂ) (M:=E))
+  let D := Module.finrank ℂ E
+  let F := Specht.symmetricFamily s
+  have hsum : (∑ b ∈ Finset.univ.filter (fun b => F.degree b ^ 8 ≤ D), (F.degree b : ℝ)^2) ≤
+      (D : ℝ) ^ (3 / 8 : ℝ) * C1 := by
+    refine (low_degree_first F D).trans (mul_le_mul_of_nonneg_left ?_ (by positivity))
+    exact reciprocalFirst_le_C1 s hs
+  have hconst : 0 ≤ Real.sqrt (32 * C1) * (D : ℝ) ^ (-5 / 16 : ℝ) := by positivity
+  have hconst_sq : (Real.sqrt (32 * C1) * (D : ℝ) ^ (-5 / 16 : ℝ))^2 =
+      8 * 4 * ((D : ℝ) ^ (3 / 8 : ℝ) * C1) / D := by
+    rw [mul_pow, Real.sq_sqrt (mul_nonneg (by norm_num) C1_nonneg),
+      ← Real.rpow_mul_natCast (Nat.cast_nonneg D)]
+    rw [show (-5 / 16 : ℝ) * (2 : ℕ) = (3 / 8 : ℝ) - 1 by norm_num,
+      Real.rpow_sub hD, Real.rpow_one]
+    ring
+  apply ContinuousLinearMap.opNorm_le_bound _ hconst
+  intro v
+  have hb := F.integrated_norm_sq ρ hu (Nat.ne_of_gt Module.finrank_pos) ψ ν 4 hν h1
+    (by norm_num) (fun i g => by
+      convert hcoset i g using 1
+      apply Finset.sum_congr rfl
+      intro h _
+      congr 3
+      funext j
+      by_cases hij : j = i <;> simp [hij]) v
+  simp only [Fintype.card_fin, Nat.cast_ofNat] at hb
+  have hv : ‖integrated ρ (fun g => (ν g : ℂ)) v‖ ^ 2 ≤
+      (Real.sqrt (32 * C1) * (D : ℝ) ^ (-5 / 16 : ℝ) * ‖v‖)^2 := by
+    rw [mul_pow, hconst_sq]
+    exact hb.trans (mul_le_mul_of_nonneg_right
+      (div_le_div_of_nonneg_right
+        (mul_le_mul_of_nonneg_left hsum (by norm_num)) hD.le) (sq_nonneg _))
+  change ‖integrated ρ (fun g => (ν g : ℂ)) v‖ ≤ _
+  nlinarith [norm_nonneg (integrated ρ (fun g => (ν g : ℂ)) v),
+    mul_nonneg hconst (norm_nonneg v)]
+
+/-- The integrated operator norm bound for permutations of binary positions
+under any equivalence with eight equal blocks. -/
+theorem eight_block_fourier_bound
+    (d : ℕ) (_hd : 3 ≤ d)
+    (e : Position d ≃ Fin 8 × Fin (2 ^ (d - 3)))
+    {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
+    [FiniteDimensional ℂ E]
+    (ρ : Representation ℂ (Equiv.Perm (Position d)) E)
+    [Representation.IsIrreducible ρ]
+    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ)
+    (ν : Equiv.Perm (Position d) → ℝ)
+    (hν : ∀ g, 0 ≤ ν g) (h1 : ∑ g, ν g = 1)
+    (hcoset : ∀ (i : Fin 8) (g : Equiv.Perm (Position d)),
+      ∑ h : Equiv.Perm (Fin (2 ^ (d - 3))),
+          ν (g * Block.embedding e (Pi.mulSingle i h)) ≤
+        4 * (Fintype.card (Equiv.Perm (Fin (2 ^ (d - 3)))) : ℝ) /
+          Fintype.card (Equiv.Perm (Position d))) :
+    ‖LinearMap.toContinuousLinearMap (Fourier.integrated ρ (fun g => (ν g : ℂ)))‖ ≤
+      Real.sqrt (32 * C1) * (Module.finrank ℂ E : ℝ) ^ (-5 / 16 : ℝ) := by
+  exact eight_block_of_cap (2 ^ (d - 3)) (Nat.one_le_pow _ _ (by norm_num)) ρ hu
+    (Block.embedding e) ν hν h1 hcoset
+
+end Thorp.Current
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/FirstMain.lean b/lean/OAI/Probability/ThorpShuffle/FirstMain.lean
index bea01810ccaf0a02f22e03b81479b0650fc88d85..16d534400804ba0f7ee07e4eae7c11f51fdb6149 100644
--- a/lean/OAI/Probability/ThorpShuffle/FirstMain.lean
+++ b/lean/OAI/Probability/ThorpShuffle/FirstMain.lean
@@ -1,39 +1,39 @@
-import OAI.Probability.ThorpShuffle.FirstFourier
-
-namespace OAI
-
-universe uE
-
-noncomputable section
-
-namespace Thorp.Current
-open scoped BigOperators Classical ComplexConjugate InnerProductSpace
-open Filter Topology
-
-theorem first_reciprocal_main :
-    (BddAbove (Set.range (fun n : {n : ℕ // 1 ≤ n} => reciprocalFirst n.val)) ∧
-    (∀ n : ℕ, 1 ≤ n → reciprocalFirst n ≤ C1) ∧
-    Tendsto exceptionalFirst atTop (nhds 0)) ∧
-    (∀ (d : ℕ) (_hd : 3 ≤ d)
-    (e : Position d ≃ Fin 8 × Fin (2 ^ (d - 3)))
-    {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
-    [FiniteDimensional ℂ E]
-    (ρ : Representation ℂ (Equiv.Perm (Position d)) E)
-    [Representation.IsIrreducible ρ]
-    (_ : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ)
-    (ν : Equiv.Perm (Position d) → ℝ)
-    (_ : ∀ g, 0 ≤ ν g) (_ : ∑ g, ν g = 1)
-    (_ : ∀ (i : Fin 8) (g : Equiv.Perm (Position d)),
-      ∑ h : Equiv.Perm (Fin (2 ^ (d - 3))),
-          ν (g * Block.embedding e (Pi.mulSingle i h)) ≤
-        4 * (Fintype.card (Equiv.Perm (Fin (2 ^ (d - 3)))) : ℝ) /
-          Fintype.card (Equiv.Perm (Position d))),
-    ‖LinearMap.toContinuousLinearMap (Fourier.integrated ρ (fun g => (ν g : ℂ)))‖ ≤
-      Real.sqrt (32 * C1) * (Module.finrank ℂ E : ℝ) ^ (-5 / 16 : ℝ)) := by
-  exact ⟨reciprocal_degrees, eight_block_fourier_bound⟩
-
-end Thorp.Current
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.FirstFourier
+
+namespace OAI
+
+universe uE
+
+noncomputable section
+
+namespace Thorp.Current
+open scoped BigOperators Classical ComplexConjugate InnerProductSpace
+open Filter Topology
+
+theorem first_reciprocal_main :
+    (BddAbove (Set.range (fun n : {n : ℕ // 1 ≤ n} => reciprocalFirst n.val)) ∧
+    (∀ n : ℕ, 1 ≤ n → reciprocalFirst n ≤ C1) ∧
+    Tendsto exceptionalFirst atTop (nhds 0)) ∧
+    (∀ (d : ℕ) (_hd : 3 ≤ d)
+    (e : Position d ≃ Fin 8 × Fin (2 ^ (d - 3)))
+    {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
+    [FiniteDimensional ℂ E]
+    (ρ : Representation ℂ (Equiv.Perm (Position d)) E)
+    [Representation.IsIrreducible ρ]
+    (_ : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ)
+    (ν : Equiv.Perm (Position d) → ℝ)
+    (_ : ∀ g, 0 ≤ ν g) (_ : ∑ g, ν g = 1)
+    (_ : ∀ (i : Fin 8) (g : Equiv.Perm (Position d)),
+      ∑ h : Equiv.Perm (Fin (2 ^ (d - 3))),
+          ν (g * Block.embedding e (Pi.mulSingle i h)) ≤
+        4 * (Fintype.card (Equiv.Perm (Fin (2 ^ (d - 3)))) : ℝ) /
+          Fintype.card (Equiv.Perm (Position d))),
+    ‖LinearMap.toContinuousLinearMap (Fourier.integrated ρ (fun g => (ν g : ℂ)))‖ ≤
+      Real.sqrt (32 * C1) * (Module.finrank ℂ E : ℝ) ^ (-5 / 16 : ℝ)) := by
+  exact ⟨reciprocal_degrees, eight_block_fourier_bound⟩
+
+end Thorp.Current
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/FirstMoment.lean b/lean/OAI/Probability/ThorpShuffle/FirstMoment.lean
index f77e58520dbc372bcfc790dc923278b87966c34a..2cafec6e32bf34496bc17d191b682715c126397b 100644
--- a/lean/OAI/Probability/ThorpShuffle/FirstMoment.lean
+++ b/lean/OAI/Probability/ThorpShuffle/FirstMoment.lean
@@ -1,272 +1,272 @@
-import OAI.Probability.ThorpShuffle.ReciprocalDegrees
-
-namespace OAI
-
-noncomputable section
-
-namespace Thorp.Current
-open scoped BigOperators Classical ComplexConjugate InnerProductSpace
-open Filter Topology
-
-/-- The FIRST reciprocal degree moment of the concrete symmetric-group
-irreducible family. `Specht.degree` is the actual complex dimension. -/
-def reciprocalFirst (n : ℕ) : ℝ :=
-  ∑ p : Specht.NShape n, (Specht.degree p : ℝ)⁻¹
-
-/-- Exactly the non-row, non-column shapes: defect is size minus the maximum
-of first row length and first column length. For size≥1 defect=0 precisely
-for the row or column; at size=1 they coincide and are counted once. -/
-def exceptionalFirst (n : ℕ) : ℝ :=
-  ∑ p : Specht.NShape n, if 0 < Specht.defect p then (Specht.degree p : ℝ)⁻¹ else 0
-
-/-- The supremum of the first reciprocal degree moments over positive sizes. Boundedness is proved in `reciprocal_degrees`. -/
-def C1 : ℝ := sSup (Set.range (fun n : {n : ℕ // 1 ≤ n} => reciprocalFirst n.val))
-
-open Thorp.Specht Thorp.Young
-variable {n : ℕ}
-
-lemma partition_first_exponential_decay :
-    Tendsto (fun n : ℕ => ((n : ℝ) + 1) ^ (2 * Nat.sqrt n) / (2 : ℝ) ^ (n / 16)) atTop (𝓝 0) := by
-  have hc : 0 < Real.log 2 / 256 := by positivity
-  have he := Thorp.Analysis.sqrt_log_small (Real.log 2 / 256) hc
-  have ht : Tendsto (fun n : ℕ => Real.exp (-(Real.log 2 / 64) * n)) atTop (𝓝 0) := by
-    apply Real.tendsto_exp_atBot.comp
-    exact tendsto_natCast_atTop_atTop.const_mul_atTop_of_neg (neg_neg_of_pos (by positivity))
-  apply squeeze_zero' (Eventually.of_forall (fun n => by positivity)) _ ht
-  filter_upwards [he, eventually_ge_atTop 32] with n hn hn32
-  have hn1 : (1 : ℝ) ≤ n := by exact_mod_cast (show 1 ≤ n by omega)
-  have hs : (Nat.sqrt n : ℝ) ≤ Real.sqrt (n + 1) := by
-    apply Real.le_sqrt_of_sq_le
-    have hh : ((Nat.sqrt n : ℝ)^2) ≤ n := by exact_mod_cast (show (Nat.sqrt n)^2 ≤ n by simpa only [pow_two] using Nat.sqrt_le n)
-    linarith
-  have hdiv : (n : ℝ) / 32 ≤ (n / 16 : ℕ) := by
-    have hh : n ≤ 32 * (n / 16) := by omega
-    exact (div_le_iff₀ (by norm_num : (0 : ℝ) < 32)).mpr (by exact_mod_cast (by omega : n ≤ n / 16 * 32))
-  have hp : 2 * (Nat.sqrt n : ℝ) * Real.log (n + 1) ≤ (Real.log 2 / 64) * n := by
-    have hl : 0 ≤ Real.log (n + 1) := Real.log_nonneg (by linarith)
-    have h₁ := mul_le_mul_of_nonneg_right hs hl
-    have h₂ := mul_le_mul_of_nonneg_left hn (Real.sqrt_nonneg ((n : ℝ) + 1))
-    have hh := Real.sq_sqrt (by positivity : (0 : ℝ) ≤ n + 1)
-    have hc0 : 0 ≤ Real.log 2 / 256 := hc.le
-    have hsq : Real.sqrt (n + 1) * ((Real.log 2 / 256) * Real.sqrt (n + 1)) = (Real.log 2 / 256) * (n + 1) := by calc
-      _ = (Real.log 2 / 256) * (Real.sqrt (n + 1))^2 := by ring
-      _ = _ := by rw [hh]
-    rw [hsq] at h₂
-    have h₃ := mul_le_mul_of_nonneg_left (show (n : ℝ) + 1 ≤ 2 * n by linarith) hc0
-    linarith only [h₁, h₂, h₃]
-  have hq : -(Real.log 2 / 64) * n ≥
-      (2 * Nat.sqrt n : ℕ) * Real.log (n + 1) - (n / 16 : ℕ) * Real.log 2 := by
-    push_cast
-    have hh := mul_le_mul_of_nonneg_right hdiv (Real.log_nonneg (by norm_num : (1 : ℝ) ≤ 2))
-    linarith
-  calc
-    _ = Real.exp ((2 * Nat.sqrt n : ℕ) * Real.log (n + 1) - (n / 16 : ℕ) * Real.log 2) := by
-      rw [Real.exp_sub, Real.exp_nat_mul, Real.exp_nat_mul, Real.exp_log (by positivity), Real.exp_log (by norm_num : (0 : ℝ) < 2)]
-    _ ≤ _ := Real.exp_le_exp.mpr hq
-
-
-def inverseFirst (p : NShape n) : ℝ := (degree p : ℝ)⁻¹
-
-def slice (n u : ℕ) : ℝ := ∑ p : {p : NShape n // defect p = u}, inverseFirst p.val
-
-def smallSlice (n u : ℕ) : ℝ := if 0 < u ∧ 4 * u ≤ n then slice n u else 0
-
-
-lemma inverseFirst_nonneg (p : NShape n) : 0 ≤ inverseFirst p := by unfold inverseFirst; positivity
-lemma slice_nonneg (n u : ℕ) : 0 ≤ slice n u := Finset.sum_nonneg (fun _ _ => inverseFirst_nonneg _)
-lemma smallSlice_nonneg (n u : ℕ) : 0 ≤ smallSlice n u := by unfold smallSlice; split_ifs; exact slice_nonneg _ _; rfl
-
-lemma slice_bound (n u : ℕ) (K : ℝ) (hK : 0 < K)
-    (hk : ∀ p : NShape n, defect p = u → K ≤ degree p) :
-    slice n u ≤ (2 : ℝ)^(u+1) / K := by
-  have hI (p : {p : NShape n // defect p = u}) : inverseFirst p.val ≤ K⁻¹ := by
-    exact inv_anti₀ hK (hk p.val p.property)
-  calc
-    _ ≤ ∑ _ : {p : NShape n // defect p = u}, K⁻¹ := Finset.sum_le_sum (fun p _ => hI p)
-    _ = (Fintype.card {p : NShape n // defect p = u} : ℝ) * K⁻¹ := by simp
-    _ ≤ (2 : ℝ)^(u+1) * K⁻¹ := mul_le_mul_of_nonneg_right (by exact_mod_cast defect_card_bound n u) (by positivity)
-    _ = _ := rfl
-
-lemma smallSlice_bound (n u : ℕ) : smallSlice n u ≤ 2 * (2 / 3 : ℝ)^u := by
-  unfold smallSlice
-  split_ifs with h
-  · have hk : ∀ p : NShape n, defect p = u → (3 : ℝ)^u ≤ degree p := by
-      intro p hp
-      have hh := small_defect_degree p (by rw [hp]; exact h.2)
-      rw [hp] at hh
-      exact_mod_cast hh
-    have hh := slice_bound n u (3^u) (by positivity) hk
-    have he : (2 : ℝ)^(u+1) / 3^u = 2 * (2/3 :ℝ)^u := by
-      rw [pow_succ, div_pow]
-      ring
-    exact hh.trans_eq he
-  · positivity
-
-lemma smallSlice_tendsto (u : ℕ) : Tendsto (fun n => smallSlice n u) atTop (𝓝 0) := by
-  by_cases hu : 0 < u
-  · have ht : Tendsto (fun n : ℕ => ((n - 2*u + 1 : ℕ) : ℝ)) atTop atTop := by
-      exact (tendsto_natCast_atTop_atTop : Tendsto (fun n : ℕ => (n : ℝ)) atTop atTop).comp
-        ((tendsto_add_atTop_nat 1).comp (tendsto_sub_atTop_nat (2*u)))
-    have hi : Tendsto (fun n : ℕ => (((n - 2*u + 1 : ℕ) : ℝ))⁻¹) atTop (𝓝 0) :=
-      tendsto_inv_atTop_zero.comp ht
-    have he := hi.const_mul ((2 : ℝ)^(u+1))
-    rw [mul_zero] at he
-    apply squeeze_zero' (Eventually.of_forall (fun n => smallSlice_nonneg n u)) _ he
-    filter_upwards [eventually_ge_atTop (4*u)] with n hn
-    rw [smallSlice, ite_eq_left ⟨hu, hn⟩]
-    exact slice_bound n u _ (by positivity) (fun p hp => by
-      have hh := fixed_defect_degree p (by rwa [hp]) (by rw [hp]; omega)
-      rw [hp] at hh
-      exact_mod_cast hh)
-  · simp only [smallSlice, hu, false_and, ↓reduceIte]
-    exact tendsto_const_nhds
-
-lemma smallSum_tendsto : Tendsto (fun n => ∑' u, smallSlice n u) atTop (𝓝 0) := by
-  have hs : Summable (fun u : ℕ => 2 * (2/3 : ℝ)^u) :=
-    (summable_geometric_of_norm_lt_one (by norm_num : ‖(2/3 : ℝ)‖ < 1)).mul_left 2
-  have h := tendsto_tsum_of_dominated_convergence hs smallSlice_tendsto
-    (Eventually.of_forall (fun n u => by rw [Real.norm_of_nonneg (smallSlice_nonneg n u)]; exact smallSlice_bound n u))
-  simpa only [tsum_zero] using h
-
-end Thorp.Current
-
-namespace Thorp.Current
-open scoped Classical
-open Filter Topology Thorp.Young Thorp.Specht
-
-lemma smallSlice_as_sum (n u : ℕ) :
-    smallSlice n u = ∑ p : NShape n,
-      if defect p = u ∧ 0 < defect p ∧ 4 * defect p ≤ n then inverseFirst p else 0 := by
-  unfold smallSlice slice
-  rw [← Finset.sum_subtype (Finset.univ.filter (fun p : NShape n => defect p = u)) (by simp), Finset.sum_filter]
-  by_cases h : 0 < u ∧ 4 * u ≤ n
-  · rw [ite_eq_left h]
-    apply Finset.sum_congr rfl
-    intro p _
-    by_cases hp : defect p = u
-    · simp only [hp, h.1, h.2, true_and, ↓reduceIte]
-    · simp only [hp, false_and, ↓reduceIte]
-  · rw [ite_eq_right h]
-    apply Eq.symm
-    apply Finset.sum_eq_zero
-    intro p _
-    split_ifs with hp
-    · exact (h (by simpa only [hp.1] using hp.2)).elim
-    · rfl
-
-lemma smallSlice_support (n u : ℕ) (hu : u ∉ Finset.range (n+1)) : smallSlice n u = 0 := by
-  unfold smallSlice
-  have hh : ¬(0 < u ∧ 4*u ≤ n) := by have := (Finset.mem_range.not.mp hu); omega
-  exact ite_eq_right hh
-
-lemma smallSum_eq (n : ℕ) :
-    (∑' u, smallSlice n u) = ∑ p : NShape n,
-      if 0 < defect p ∧ 4 * defect p ≤ n then inverseFirst p else 0 := by
-  rw [tsum_eq_sum (smallSlice_support n)]
-  simp_rw [smallSlice_as_sum]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro p _
-  have hd : defect p ∈ Finset.range (n+1) := Finset.mem_range.mpr (by have := defect_add_longest p; omega)
-  have he (u : ℕ) : (if defect p = u ∧ 0 < defect p ∧ 4 * defect p ≤ n then inverseFirst p else 0) =
-      if defect p = u then (if 0 < defect p ∧ 4 * defect p ≤ n then inverseFirst p else 0) else 0 := by
-    split_ifs <;> tauto
-  simp only [he, Finset.sum_ite_eq, hd, ↓reduceIte]
-
-lemma bulkSum_bound (n : ℕ) (hn : 0 < n) :
-    (∑ p : NShape n, if n < 4 * defect p then inverseFirst p else 0) ≤
-      ((n : ℝ)+1)^(2*Nat.sqrt n) / (2:ℝ)^(n/16) := by
-  have hb (p : NShape n) : (if n < 4*defect p then inverseFirst p else 0) ≤ ((2:ℝ)^(n/16))⁻¹ := by
-    by_cases hp : n < 4*defect p
-    · rw [ite_eq_left hp]
-      have hh := bulk_degree p hn hp
-      exact inv_anti₀ (by positivity) hh
-    · rw [ite_eq_right hp]; positivity
-  have hc : Fintype.card (NShape n) ≤ (n+1)^(2*Nat.sqrt n) := by
-    rw [Fintype.card_eq_nat_card]
-    change Nat.card (Nat.Partition (Fintype.card (Fin n))) ≤ _
-    rw [Fintype.card_fin, ← Fintype.card_eq_nat_card]
-    exact partition_count_le_sqrt_pow n
-  calc
-    _ ≤ ∑ _ : NShape n, ((2:ℝ)^(n/16))⁻¹ := Finset.sum_le_sum (fun p _ => hb p)
-    _ = (Fintype.card (NShape n) : ℝ) * ((2:ℝ)^(n/16))⁻¹ := by simp
-    _ ≤ ((n:ℝ)+1)^(2*Nat.sqrt n) * ((2:ℝ)^(n/16))⁻¹ :=
-      mul_le_mul_of_nonneg_right (by exact_mod_cast hc) (by positivity)
-    _ = _ := rfl
-
-lemma exceptionalFirst_split (n : ℕ) : exceptionalFirst n =
-    (∑' u, smallSlice n u) + (∑ p : NShape n, if n < 4 * defect p then inverseFirst p else 0) := by
-  unfold exceptionalFirst
-  change (∑ p : NShape n, if 0 < defect p then inverseFirst p else 0) = _
-  rw [smallSum_eq, ← Finset.sum_add_distrib]
-  apply Finset.sum_congr rfl
-  intro p _
-  split_ifs <;> simp_all
-  all_goals omega
-
-/-- The contribution of positive-defect Specht modules tends to zero. -/
-theorem exceptionalFirst_tendsto : Tendsto exceptionalFirst atTop (𝓝 0) := by
-  have hb : Tendsto (fun n => ∑ p : NShape n, if n < 4 * defect p then inverseFirst p else 0) atTop (𝓝 0) := by
-    apply squeeze_zero' (Eventually.of_forall (fun n => Finset.sum_nonneg (fun p _ => by split_ifs; exact inverseFirst_nonneg p; rfl)))
-      ((eventually_gt_atTop 0).mono (fun n hn => bulkSum_bound n hn)) Thorp.Current.partition_first_exponential_decay
-  have he : exceptionalFirst = (fun n => (∑' u, smallSlice n u) + ∑ p : NShape n, if n < 4 * defect p then inverseFirst p else 0) := funext exceptionalFirst_split
-  rw [he]
-  simpa only [zero_add] using smallSum_tendsto.add hb
-
-end Thorp.Current
-
-namespace Thorp.Current
-open scoped Classical
-open Thorp.Specht Filter Topology
-
-lemma reciprocalFirst_nonneg (n : ℕ) : 0 ≤ reciprocalFirst n :=
-  Finset.sum_nonneg (fun _p _ => inv_nonneg.mpr (Nat.cast_nonneg _))
-
-lemma full_reciprocalFirst_bound (n : ℕ) :
-    reciprocalFirst n ≤ 2 + exceptionalFirst n := by
-  have hzero : slice n 0 ≤ 2 := by
-    have h := slice_bound n 0 1 (by norm_num)
-      (fun p _ => by exact_mod_cast degree_pos p)
-    norm_num at h
-    exact h
-  have he : reciprocalFirst n = slice n 0 + exceptionalFirst n := by
-    unfold slice
-    rw [← Finset.sum_subtype (Finset.univ.filter (fun p : NShape n => defect p = 0))
-      (by simp), Finset.sum_filter]
-    unfold exceptionalFirst reciprocalFirst
-    rw [← Finset.sum_add_distrib]
-    apply Finset.sum_congr rfl
-    intro p _
-    change inverseFirst p = _
-    by_cases hp : defect p = 0
-    · simp only [hp, lt_self_iff_false, ↓reduceIte, add_zero]
-    · simp only [hp, Nat.pos_of_ne_zero hp, ↓reduceIte, zero_add]
-      rfl
-  rw [he]
-  linarith
-
-/-- Uniform first-moment bound, including the finitely many initial sizes. -/
-lemma reciprocalFirst_bddAbove :
-    BddAbove (Set.range (fun n : {n : ℕ // 1 ≤ n} => reciprocalFirst n.val)) := by
-  obtain ⟨B, hB⟩ := exceptionalFirst_tendsto.bddAbove_range
-  refine ⟨2 + B, ?_⟩
-  rintro _ ⟨n, rfl⟩
-  exact (full_reciprocalFirst_bound n).trans (add_le_add le_rfl (hB ⟨n.val, rfl⟩))
-
-lemma reciprocalFirst_le_C1 (n : ℕ) (hn : 1 ≤ n) : reciprocalFirst n ≤ C1 :=
-  le_csSup reciprocalFirst_bddAbove ⟨⟨n, hn⟩, rfl⟩
-
-lemma C1_nonneg : 0 ≤ C1 :=
-  (reciprocalFirst_nonneg 1).trans (reciprocalFirst_le_C1 1 le_rfl)
-
-theorem reciprocal_degrees :
-    BddAbove (Set.range (fun n : {n : ℕ // 1 ≤ n} => reciprocalFirst n.val)) ∧
-    (∀ n : ℕ, 1 ≤ n → reciprocalFirst n ≤ C1) ∧
-    Tendsto exceptionalFirst atTop (nhds 0) :=
-  ⟨reciprocalFirst_bddAbove, reciprocalFirst_le_C1, exceptionalFirst_tendsto⟩
-
-end Thorp.Current
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.ReciprocalDegrees
+
+namespace OAI
+
+noncomputable section
+
+namespace Thorp.Current
+open scoped BigOperators Classical ComplexConjugate InnerProductSpace
+open Filter Topology
+
+/-- The FIRST reciprocal degree moment of the concrete symmetric-group
+irreducible family. `Specht.degree` is the actual complex dimension. -/
+def reciprocalFirst (n : ℕ) : ℝ :=
+  ∑ p : Specht.NShape n, (Specht.degree p : ℝ)⁻¹
+
+/-- Exactly the non-row, non-column shapes: defect is size minus the maximum
+of first row length and first column length. For size≥1 defect=0 precisely
+for the row or column; at size=1 they coincide and are counted once. -/
+def exceptionalFirst (n : ℕ) : ℝ :=
+  ∑ p : Specht.NShape n, if 0 < Specht.defect p then (Specht.degree p : ℝ)⁻¹ else 0
+
+/-- The supremum of the first reciprocal degree moments over positive sizes. Boundedness is proved in `reciprocal_degrees`. -/
+def C1 : ℝ := sSup (Set.range (fun n : {n : ℕ // 1 ≤ n} => reciprocalFirst n.val))
+
+open Thorp.Specht Thorp.Young
+variable {n : ℕ}
+
+lemma partition_first_exponential_decay :
+    Tendsto (fun n : ℕ => ((n : ℝ) + 1) ^ (2 * Nat.sqrt n) / (2 : ℝ) ^ (n / 16)) atTop (𝓝 0) := by
+  have hc : 0 < Real.log 2 / 256 := by positivity
+  have he := Thorp.Analysis.sqrt_log_small (Real.log 2 / 256) hc
+  have ht : Tendsto (fun n : ℕ => Real.exp (-(Real.log 2 / 64) * n)) atTop (𝓝 0) := by
+    apply Real.tendsto_exp_atBot.comp
+    exact tendsto_natCast_atTop_atTop.const_mul_atTop_of_neg (neg_neg_of_pos (by positivity))
+  apply squeeze_zero' (Eventually.of_forall (fun n => by positivity)) _ ht
+  filter_upwards [he, eventually_ge_atTop 32] with n hn hn32
+  have hn1 : (1 : ℝ) ≤ n := by exact_mod_cast (show 1 ≤ n by omega)
+  have hs : (Nat.sqrt n : ℝ) ≤ Real.sqrt (n + 1) := by
+    apply Real.le_sqrt_of_sq_le
+    have hh : ((Nat.sqrt n : ℝ)^2) ≤ n := by exact_mod_cast (show (Nat.sqrt n)^2 ≤ n by simpa only [pow_two] using Nat.sqrt_le n)
+    linarith
+  have hdiv : (n : ℝ) / 32 ≤ (n / 16 : ℕ) := by
+    have hh : n ≤ 32 * (n / 16) := by omega
+    exact (div_le_iff₀ (by norm_num : (0 : ℝ) < 32)).mpr (by exact_mod_cast (by omega : n ≤ n / 16 * 32))
+  have hp : 2 * (Nat.sqrt n : ℝ) * Real.log (n + 1) ≤ (Real.log 2 / 64) * n := by
+    have hl : 0 ≤ Real.log (n + 1) := Real.log_nonneg (by linarith)
+    have h₁ := mul_le_mul_of_nonneg_right hs hl
+    have h₂ := mul_le_mul_of_nonneg_left hn (Real.sqrt_nonneg ((n : ℝ) + 1))
+    have hh := Real.sq_sqrt (by positivity : (0 : ℝ) ≤ n + 1)
+    have hc0 : 0 ≤ Real.log 2 / 256 := hc.le
+    have hsq : Real.sqrt (n + 1) * ((Real.log 2 / 256) * Real.sqrt (n + 1)) = (Real.log 2 / 256) * (n + 1) := by calc
+      _ = (Real.log 2 / 256) * (Real.sqrt (n + 1))^2 := by ring
+      _ = _ := by rw [hh]
+    rw [hsq] at h₂
+    have h₃ := mul_le_mul_of_nonneg_left (show (n : ℝ) + 1 ≤ 2 * n by linarith) hc0
+    linarith only [h₁, h₂, h₃]
+  have hq : -(Real.log 2 / 64) * n ≥
+      (2 * Nat.sqrt n : ℕ) * Real.log (n + 1) - (n / 16 : ℕ) * Real.log 2 := by
+    push_cast
+    have hh := mul_le_mul_of_nonneg_right hdiv (Real.log_nonneg (by norm_num : (1 : ℝ) ≤ 2))
+    linarith
+  calc
+    _ = Real.exp ((2 * Nat.sqrt n : ℕ) * Real.log (n + 1) - (n / 16 : ℕ) * Real.log 2) := by
+      rw [Real.exp_sub, Real.exp_nat_mul, Real.exp_nat_mul, Real.exp_log (by positivity), Real.exp_log (by norm_num : (0 : ℝ) < 2)]
+    _ ≤ _ := Real.exp_le_exp.mpr hq
+
+
+def inverseFirst (p : NShape n) : ℝ := (degree p : ℝ)⁻¹
+
+def slice (n u : ℕ) : ℝ := ∑ p : {p : NShape n // defect p = u}, inverseFirst p.val
+
+def smallSlice (n u : ℕ) : ℝ := if 0 < u ∧ 4 * u ≤ n then slice n u else 0
+
+
+lemma inverseFirst_nonneg (p : NShape n) : 0 ≤ inverseFirst p := by unfold inverseFirst; positivity
+lemma slice_nonneg (n u : ℕ) : 0 ≤ slice n u := Finset.sum_nonneg (fun _ _ => inverseFirst_nonneg _)
+lemma smallSlice_nonneg (n u : ℕ) : 0 ≤ smallSlice n u := by unfold smallSlice; split_ifs; exact slice_nonneg _ _; rfl
+
+lemma slice_bound (n u : ℕ) (K : ℝ) (hK : 0 < K)
+    (hk : ∀ p : NShape n, defect p = u → K ≤ degree p) :
+    slice n u ≤ (2 : ℝ)^(u+1) / K := by
+  have hI (p : {p : NShape n // defect p = u}) : inverseFirst p.val ≤ K⁻¹ := by
+    exact inv_anti₀ hK (hk p.val p.property)
+  calc
+    _ ≤ ∑ _ : {p : NShape n // defect p = u}, K⁻¹ := Finset.sum_le_sum (fun p _ => hI p)
+    _ = (Fintype.card {p : NShape n // defect p = u} : ℝ) * K⁻¹ := by simp
+    _ ≤ (2 : ℝ)^(u+1) * K⁻¹ := mul_le_mul_of_nonneg_right (by exact_mod_cast defect_card_bound n u) (by positivity)
+    _ = _ := rfl
+
+lemma smallSlice_bound (n u : ℕ) : smallSlice n u ≤ 2 * (2 / 3 : ℝ)^u := by
+  unfold smallSlice
+  split_ifs with h
+  · have hk : ∀ p : NShape n, defect p = u → (3 : ℝ)^u ≤ degree p := by
+      intro p hp
+      have hh := small_defect_degree p (by rw [hp]; exact h.2)
+      rw [hp] at hh
+      exact_mod_cast hh
+    have hh := slice_bound n u (3^u) (by positivity) hk
+    have he : (2 : ℝ)^(u+1) / 3^u = 2 * (2/3 :ℝ)^u := by
+      rw [pow_succ, div_pow]
+      ring
+    exact hh.trans_eq he
+  · positivity
+
+lemma smallSlice_tendsto (u : ℕ) : Tendsto (fun n => smallSlice n u) atTop (𝓝 0) := by
+  by_cases hu : 0 < u
+  · have ht : Tendsto (fun n : ℕ => ((n - 2*u + 1 : ℕ) : ℝ)) atTop atTop := by
+      exact (tendsto_natCast_atTop_atTop : Tendsto (fun n : ℕ => (n : ℝ)) atTop atTop).comp
+        ((tendsto_add_atTop_nat 1).comp (tendsto_sub_atTop_nat (2*u)))
+    have hi : Tendsto (fun n : ℕ => (((n - 2*u + 1 : ℕ) : ℝ))⁻¹) atTop (𝓝 0) :=
+      tendsto_inv_atTop_zero.comp ht
+    have he := hi.const_mul ((2 : ℝ)^(u+1))
+    rw [mul_zero] at he
+    apply squeeze_zero' (Eventually.of_forall (fun n => smallSlice_nonneg n u)) _ he
+    filter_upwards [eventually_ge_atTop (4*u)] with n hn
+    rw [smallSlice, ite_eq_left ⟨hu, hn⟩]
+    exact slice_bound n u _ (by positivity) (fun p hp => by
+      have hh := fixed_defect_degree p (by rwa [hp]) (by rw [hp]; omega)
+      rw [hp] at hh
+      exact_mod_cast hh)
+  · simp only [smallSlice, hu, false_and, ↓reduceIte]
+    exact tendsto_const_nhds
+
+lemma smallSum_tendsto : Tendsto (fun n => ∑' u, smallSlice n u) atTop (𝓝 0) := by
+  have hs : Summable (fun u : ℕ => 2 * (2/3 : ℝ)^u) :=
+    (summable_geometric_of_norm_lt_one (by norm_num : ‖(2/3 : ℝ)‖ < 1)).mul_left 2
+  have h := tendsto_tsum_of_dominated_convergence hs smallSlice_tendsto
+    (Eventually.of_forall (fun n u => by rw [Real.norm_of_nonneg (smallSlice_nonneg n u)]; exact smallSlice_bound n u))
+  simpa only [tsum_zero] using h
+
+end Thorp.Current
+
+namespace Thorp.Current
+open scoped Classical
+open Filter Topology Thorp.Young Thorp.Specht
+
+lemma smallSlice_as_sum (n u : ℕ) :
+    smallSlice n u = ∑ p : NShape n,
+      if defect p = u ∧ 0 < defect p ∧ 4 * defect p ≤ n then inverseFirst p else 0 := by
+  unfold smallSlice slice
+  rw [← Finset.sum_subtype (Finset.univ.filter (fun p : NShape n => defect p = u)) (by simp), Finset.sum_filter]
+  by_cases h : 0 < u ∧ 4 * u ≤ n
+  · rw [ite_eq_left h]
+    apply Finset.sum_congr rfl
+    intro p _
+    by_cases hp : defect p = u
+    · simp only [hp, h.1, h.2, true_and, ↓reduceIte]
+    · simp only [hp, false_and, ↓reduceIte]
+  · rw [ite_eq_right h]
+    apply Eq.symm
+    apply Finset.sum_eq_zero
+    intro p _
+    split_ifs with hp
+    · exact (h (by simpa only [hp.1] using hp.2)).elim
+    · rfl
+
+lemma smallSlice_support (n u : ℕ) (hu : u ∉ Finset.range (n+1)) : smallSlice n u = 0 := by
+  unfold smallSlice
+  have hh : ¬(0 < u ∧ 4*u ≤ n) := by have := (Finset.mem_range.not.mp hu); omega
+  exact ite_eq_right hh
+
+lemma smallSum_eq (n : ℕ) :
+    (∑' u, smallSlice n u) = ∑ p : NShape n,
+      if 0 < defect p ∧ 4 * defect p ≤ n then inverseFirst p else 0 := by
+  rw [tsum_eq_sum (smallSlice_support n)]
+  simp_rw [smallSlice_as_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro p _
+  have hd : defect p ∈ Finset.range (n+1) := Finset.mem_range.mpr (by have := defect_add_longest p; omega)
+  have he (u : ℕ) : (if defect p = u ∧ 0 < defect p ∧ 4 * defect p ≤ n then inverseFirst p else 0) =
+      if defect p = u then (if 0 < defect p ∧ 4 * defect p ≤ n then inverseFirst p else 0) else 0 := by
+    split_ifs <;> tauto
+  simp only [he, Finset.sum_ite_eq, hd, ↓reduceIte]
+
+lemma bulkSum_bound (n : ℕ) (hn : 0 < n) :
+    (∑ p : NShape n, if n < 4 * defect p then inverseFirst p else 0) ≤
+      ((n : ℝ)+1)^(2*Nat.sqrt n) / (2:ℝ)^(n/16) := by
+  have hb (p : NShape n) : (if n < 4*defect p then inverseFirst p else 0) ≤ ((2:ℝ)^(n/16))⁻¹ := by
+    by_cases hp : n < 4*defect p
+    · rw [ite_eq_left hp]
+      have hh := bulk_degree p hn hp
+      exact inv_anti₀ (by positivity) hh
+    · rw [ite_eq_right hp]; positivity
+  have hc : Fintype.card (NShape n) ≤ (n+1)^(2*Nat.sqrt n) := by
+    rw [Fintype.card_eq_nat_card]
+    change Nat.card (Nat.Partition (Fintype.card (Fin n))) ≤ _
+    rw [Fintype.card_fin, ← Fintype.card_eq_nat_card]
+    exact partition_count_le_sqrt_pow n
+  calc
+    _ ≤ ∑ _ : NShape n, ((2:ℝ)^(n/16))⁻¹ := Finset.sum_le_sum (fun p _ => hb p)
+    _ = (Fintype.card (NShape n) : ℝ) * ((2:ℝ)^(n/16))⁻¹ := by simp
+    _ ≤ ((n:ℝ)+1)^(2*Nat.sqrt n) * ((2:ℝ)^(n/16))⁻¹ :=
+      mul_le_mul_of_nonneg_right (by exact_mod_cast hc) (by positivity)
+    _ = _ := rfl
+
+lemma exceptionalFirst_split (n : ℕ) : exceptionalFirst n =
+    (∑' u, smallSlice n u) + (∑ p : NShape n, if n < 4 * defect p then inverseFirst p else 0) := by
+  unfold exceptionalFirst
+  change (∑ p : NShape n, if 0 < defect p then inverseFirst p else 0) = _
+  rw [smallSum_eq, ← Finset.sum_add_distrib]
+  apply Finset.sum_congr rfl
+  intro p _
+  split_ifs <;> simp_all
+  all_goals omega
+
+/-- The contribution of positive-defect Specht modules tends to zero. -/
+theorem exceptionalFirst_tendsto : Tendsto exceptionalFirst atTop (𝓝 0) := by
+  have hb : Tendsto (fun n => ∑ p : NShape n, if n < 4 * defect p then inverseFirst p else 0) atTop (𝓝 0) := by
+    apply squeeze_zero' (Eventually.of_forall (fun n => Finset.sum_nonneg (fun p _ => by split_ifs; exact inverseFirst_nonneg p; rfl)))
+      ((eventually_gt_atTop 0).mono (fun n hn => bulkSum_bound n hn)) Thorp.Current.partition_first_exponential_decay
+  have he : exceptionalFirst = (fun n => (∑' u, smallSlice n u) + ∑ p : NShape n, if n < 4 * defect p then inverseFirst p else 0) := funext exceptionalFirst_split
+  rw [he]
+  simpa only [zero_add] using smallSum_tendsto.add hb
+
+end Thorp.Current
+
+namespace Thorp.Current
+open scoped Classical
+open Thorp.Specht Filter Topology
+
+lemma reciprocalFirst_nonneg (n : ℕ) : 0 ≤ reciprocalFirst n :=
+  Finset.sum_nonneg (fun _p _ => inv_nonneg.mpr (Nat.cast_nonneg _))
+
+lemma full_reciprocalFirst_bound (n : ℕ) :
+    reciprocalFirst n ≤ 2 + exceptionalFirst n := by
+  have hzero : slice n 0 ≤ 2 := by
+    have h := slice_bound n 0 1 (by norm_num)
+      (fun p _ => by exact_mod_cast degree_pos p)
+    norm_num at h
+    exact h
+  have he : reciprocalFirst n = slice n 0 + exceptionalFirst n := by
+    unfold slice
+    rw [← Finset.sum_subtype (Finset.univ.filter (fun p : NShape n => defect p = 0))
+      (by simp), Finset.sum_filter]
+    unfold exceptionalFirst reciprocalFirst
+    rw [← Finset.sum_add_distrib]
+    apply Finset.sum_congr rfl
+    intro p _
+    change inverseFirst p = _
+    by_cases hp : defect p = 0
+    · simp only [hp, lt_self_iff_false, ↓reduceIte, add_zero]
+    · simp only [hp, Nat.pos_of_ne_zero hp, ↓reduceIte, zero_add]
+      rfl
+  rw [he]
+  linarith
+
+/-- Uniform first-moment bound, including the finitely many initial sizes. -/
+lemma reciprocalFirst_bddAbove :
+    BddAbove (Set.range (fun n : {n : ℕ // 1 ≤ n} => reciprocalFirst n.val)) := by
+  obtain ⟨B, hB⟩ := exceptionalFirst_tendsto.bddAbove_range
+  refine ⟨2 + B, ?_⟩
+  rintro _ ⟨n, rfl⟩
+  exact (full_reciprocalFirst_bound n).trans (add_le_add le_rfl (hB ⟨n.val, rfl⟩))
+
+lemma reciprocalFirst_le_C1 (n : ℕ) (hn : 1 ≤ n) : reciprocalFirst n ≤ C1 :=
+  le_csSup reciprocalFirst_bddAbove ⟨⟨n, hn⟩, rfl⟩
+
+lemma C1_nonneg : 0 ≤ C1 :=
+  (reciprocalFirst_nonneg 1).trans (reciprocalFirst_le_C1 1 le_rfl)
+
+theorem reciprocal_degrees :
+    BddAbove (Set.range (fun n : {n : ℕ // 1 ≤ n} => reciprocalFirst n.val)) ∧
+    (∀ n : ℕ, 1 ≤ n → reciprocalFirst n ≤ C1) ∧
+    Tendsto exceptionalFirst atTop (nhds 0) :=
+  ⟨reciprocalFirst_bddAbove, reciprocalFirst_le_C1, exceptionalFirst_tendsto⟩
+
+end Thorp.Current
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/FixedLists/Asymptotics.lean b/lean/OAI/Probability/ThorpShuffle/FixedLists/Asymptotics.lean
index 88e8269fea2f0f481a556cfa9ab582820ff2a434..d8cf37c008b91533b8389bdda614573238bb232d 100644
--- a/lean/OAI/Probability/ThorpShuffle/FixedLists/Asymptotics.lean
+++ b/lean/OAI/Probability/ThorpShuffle/FixedLists/Asymptotics.lean
@@ -1,97 +1,97 @@
-import OAI.Probability.ThorpShuffle.FixedLists.Energy
-
-namespace OAI
-
-
-noncomputable section
-open scoped BigOperators
-open Filter
-namespace Thorp.FixedLists
-
-def energyEnvelope (C d : ℕ) : ℝ :=
-  (63 / 64 : ℝ) ^ (C * (d + 4)) + 64 * (9 / 10 : ℝ) ^ (2 ^ d)
-
-lemma energyEnvelope_nonneg (C d : ℕ) : 0 ≤ energyEnvelope C d := by
-  unfold energyEnvelope
-  positivity
-
-lemma geometric_base_256 : (2 : ℝ) ^ 3 * (63 / 64 : ℝ) ^ 254 < 1 := by
-  have h : (63 / 64 : ℝ) ^ 64 ≤ 1 / 2 := by norm_num
-  have h' : (63 / 64 : ℝ) ^ 192 ≤ (1 / 2) ^ 3 := by
-    rw [show 192 = 64 * 3 by norm_num, pow_mul]
-    exact pow_le_pow_left₀ (by positivity) h 3
-  have h'' : (63 / 64 : ℝ) ^ 254 ≤ (63 / 64 : ℝ) ^ 193 :=
-    pow_le_pow_of_le_one (by norm_num) (by norm_num) (by norm_num)
-  have h''' : (63 / 64 : ℝ) ^ 193 ≤ (1 / 2) ^ 3 * (63 / 64) := by
-    rw [show 193 = 192 + 1 by norm_num, pow_succ]
-    exact mul_le_mul_of_nonneg_right h' (by norm_num)
-  have he := h''.trans h'''
-  calc
-    _ ≤ (2 : ℝ) ^ 3 * ((1 / 2) ^ 3 * (63 / 64)) :=
-      mul_le_mul_of_nonneg_left he (by norm_num)
-    _ < 1 := by norm_num
-
-lemma geometric_base_1024 : (2 : ℝ) ^ 6 * (63 / 64 : ℝ) ^ 1022 < 1 := by
-  have h : (63 / 64 : ℝ) ^ 64 ≤ 1 / 2 := by norm_num
-  have h' : (63 / 64 : ℝ) ^ 448 ≤ (1 / 2) ^ 7 := by
-    rw [show 448 = 64 * 7 by norm_num, pow_mul]
-    exact pow_le_pow_left₀ (by positivity) h 7
-  have h'' : (63 / 64 : ℝ) ^ 1022 ≤ (63 / 64 : ℝ) ^ 448 :=
-    pow_le_pow_of_le_one (by norm_num) (by norm_num) (by norm_num)
-  have hb : (1 / 2 : ℝ) ^ 7 = 1 / 128 := by norm_num
-  rw [hb] at h'
-  calc
-    _ ≤ (2 : ℝ) ^ 6 * (1 / 128 : ℝ) := mul_le_mul_of_nonneg_left (h''.trans h') (by norm_num)
-    _ < 1 := by norm_num
-
-lemma scaledEnergy_tendsto (C p : ℕ)
-    (hbase : (2 : ℝ) ^ p * (63 / 64 : ℝ) ^ C < 1) :
-    Tendsto (fun d : ℕ => ((2 : ℝ) ^ (d + 4)) ^ p * energyEnvelope C d)
-      atTop (nhds 0) := by
-  have h₁ := (tendsto_pow_atTop_nhds_zero_of_lt_one
-    (by positivity : (0 : ℝ) ≤ 2 ^ p * (63 / 64 : ℝ) ^ C) hbase).comp
-    (tendsto_add_atTop_nat 4)
-  have h₂ := (tendsto_pow_const_mul_const_pow_of_lt_one p
-    (by norm_num : (0 : ℝ) ≤ 9 / 10) (by norm_num : (9 / 10 : ℝ) < 1)).comp
-    (tendsto_pow_atTop_atTop_of_one_lt (by norm_num : (1 : ℕ) < 2))
-  have hh := h₁.add (h₂.const_mul (64 * (16 : ℝ) ^ p))
-  convert hh using 1
-  · funext d
-    simp only [energyEnvelope, Function.comp_def, Nat.cast_pow, Nat.cast_ofNat]
-    rw [mul_add]
-    congr 1
-    · rw [← pow_mul, Nat.mul_comm (d + 4) p, pow_mul, pow_mul, ← mul_pow]
-    · rw [pow_add, mul_pow]
-      norm_num
-      ring
-  · simp
-
-def listRate (d : ℕ) : ℝ :=
-  Real.sqrt (((2 : ℝ) ^ (d + 4)) ^ 3 * energyEnvelope 254 d)
-
-lemma listRate_tendsto : Tendsto listRate atTop (nhds 0) := by
-  unfold listRate
-  simpa only [Real.sqrt_zero] using (scaledEnergy_tendsto 254 3 geometric_base_256).sqrt
-
-lemma scalar_tv_finish (n a : ℝ) (hn : 0 < n) (ha : 0 ≤ a)
-    (h : n ^ 6 * a ≤ 2) : n / 2 * Real.sqrt (n * a) ≤ Real.rpow n (-(3 : ℝ) / 2) := by
-  simp only [Real.rpow_eq_pow]
-  have hr : (n ^ (-(3 : ℝ) / 2)) ^ (2 : ℕ) = 1 / n ^ 3 := by
-    rw [← Real.rpow_mul_natCast hn.le]
-    norm_num only [Nat.cast_ofNat, neg_div, div_mul_cancel₀ _ (by norm_num : (2 : ℝ) ≠ 0)]
-    rw [Real.rpow_neg hn.le, Real.rpow_ofNat]
-    simp only [one_div]
-  apply (sq_le_sq₀ (by positivity) (Real.rpow_nonneg hn.le _)).mp
-  rw [hr]
-  apply (le_div_iff₀ (pow_pos hn 3)).mpr
-  calc
-    (n / 2 * Real.sqrt (n * a)) ^ 2 * n ^ 3 = n ^ 6 * a / 4 := by
-      rw [mul_pow, div_pow, Real.sq_sqrt (mul_nonneg hn.le ha)]
-      ring
-    _ ≤ 1 := by linarith
-
-end Thorp.FixedLists
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.FixedLists.Energy
+
+namespace OAI
+
+
+noncomputable section
+open scoped BigOperators
+open Filter
+namespace Thorp.FixedLists
+
+def energyEnvelope (C d : ℕ) : ℝ :=
+  (63 / 64 : ℝ) ^ (C * (d + 4)) + 64 * (9 / 10 : ℝ) ^ (2 ^ d)
+
+lemma energyEnvelope_nonneg (C d : ℕ) : 0 ≤ energyEnvelope C d := by
+  unfold energyEnvelope
+  positivity
+
+lemma geometric_base_256 : (2 : ℝ) ^ 3 * (63 / 64 : ℝ) ^ 254 < 1 := by
+  have h : (63 / 64 : ℝ) ^ 64 ≤ 1 / 2 := by norm_num
+  have h' : (63 / 64 : ℝ) ^ 192 ≤ (1 / 2) ^ 3 := by
+    rw [show 192 = 64 * 3 by norm_num, pow_mul]
+    exact pow_le_pow_left₀ (by positivity) h 3
+  have h'' : (63 / 64 : ℝ) ^ 254 ≤ (63 / 64 : ℝ) ^ 193 :=
+    pow_le_pow_of_le_one (by norm_num) (by norm_num) (by norm_num)
+  have h''' : (63 / 64 : ℝ) ^ 193 ≤ (1 / 2) ^ 3 * (63 / 64) := by
+    rw [show 193 = 192 + 1 by norm_num, pow_succ]
+    exact mul_le_mul_of_nonneg_right h' (by norm_num)
+  have he := h''.trans h'''
+  calc
+    _ ≤ (2 : ℝ) ^ 3 * ((1 / 2) ^ 3 * (63 / 64)) :=
+      mul_le_mul_of_nonneg_left he (by norm_num)
+    _ < 1 := by norm_num
+
+lemma geometric_base_1024 : (2 : ℝ) ^ 6 * (63 / 64 : ℝ) ^ 1022 < 1 := by
+  have h : (63 / 64 : ℝ) ^ 64 ≤ 1 / 2 := by norm_num
+  have h' : (63 / 64 : ℝ) ^ 448 ≤ (1 / 2) ^ 7 := by
+    rw [show 448 = 64 * 7 by norm_num, pow_mul]
+    exact pow_le_pow_left₀ (by positivity) h 7
+  have h'' : (63 / 64 : ℝ) ^ 1022 ≤ (63 / 64 : ℝ) ^ 448 :=
+    pow_le_pow_of_le_one (by norm_num) (by norm_num) (by norm_num)
+  have hb : (1 / 2 : ℝ) ^ 7 = 1 / 128 := by norm_num
+  rw [hb] at h'
+  calc
+    _ ≤ (2 : ℝ) ^ 6 * (1 / 128 : ℝ) := mul_le_mul_of_nonneg_left (h''.trans h') (by norm_num)
+    _ < 1 := by norm_num
+
+lemma scaledEnergy_tendsto (C p : ℕ)
+    (hbase : (2 : ℝ) ^ p * (63 / 64 : ℝ) ^ C < 1) :
+    Tendsto (fun d : ℕ => ((2 : ℝ) ^ (d + 4)) ^ p * energyEnvelope C d)
+      atTop (nhds 0) := by
+  have h₁ := (tendsto_pow_atTop_nhds_zero_of_lt_one
+    (by positivity : (0 : ℝ) ≤ 2 ^ p * (63 / 64 : ℝ) ^ C) hbase).comp
+    (tendsto_add_atTop_nat 4)
+  have h₂ := (tendsto_pow_const_mul_const_pow_of_lt_one p
+    (by norm_num : (0 : ℝ) ≤ 9 / 10) (by norm_num : (9 / 10 : ℝ) < 1)).comp
+    (tendsto_pow_atTop_atTop_of_one_lt (by norm_num : (1 : ℕ) < 2))
+  have hh := h₁.add (h₂.const_mul (64 * (16 : ℝ) ^ p))
+  convert hh using 1
+  · funext d
+    simp only [energyEnvelope, Function.comp_def, Nat.cast_pow, Nat.cast_ofNat]
+    rw [mul_add]
+    congr 1
+    · rw [← pow_mul, Nat.mul_comm (d + 4) p, pow_mul, pow_mul, ← mul_pow]
+    · rw [pow_add, mul_pow]
+      norm_num
+      ring
+  · simp
+
+def listRate (d : ℕ) : ℝ :=
+  Real.sqrt (((2 : ℝ) ^ (d + 4)) ^ 3 * energyEnvelope 254 d)
+
+lemma listRate_tendsto : Tendsto listRate atTop (nhds 0) := by
+  unfold listRate
+  simpa only [Real.sqrt_zero] using (scaledEnergy_tendsto 254 3 geometric_base_256).sqrt
+
+lemma scalar_tv_finish (n a : ℝ) (hn : 0 < n) (ha : 0 ≤ a)
+    (h : n ^ 6 * a ≤ 2) : n / 2 * Real.sqrt (n * a) ≤ Real.rpow n (-(3 : ℝ) / 2) := by
+  simp only [Real.rpow_eq_pow]
+  have hr : (n ^ (-(3 : ℝ) / 2)) ^ (2 : ℕ) = 1 / n ^ 3 := by
+    rw [← Real.rpow_mul_natCast hn.le]
+    norm_num only [Nat.cast_ofNat, neg_div, div_mul_cancel₀ _ (by norm_num : (2 : ℝ) ≠ 0)]
+    rw [Real.rpow_neg hn.le, Real.rpow_ofNat]
+    simp only [one_div]
+  apply (sq_le_sq₀ (by positivity) (Real.rpow_nonneg hn.le _)).mp
+  rw [hr]
+  apply (le_div_iff₀ (pow_pos hn 3)).mpr
+  calc
+    (n / 2 * Real.sqrt (n * a)) ^ 2 * n ^ 3 = n ^ 6 * a / 4 := by
+      rw [mul_pow, div_pow, Real.sq_sqrt (mul_nonneg hn.le ha)]
+      ring
+    _ ≤ 1 := by linarith
+
+end Thorp.FixedLists
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/FixedLists/Energy.lean b/lean/OAI/Probability/ThorpShuffle/FixedLists/Energy.lean
index 59ffb067058c7e017cf08c023610cad7a445ac47..b4cf2b0782774e696bfe9e2aa075e04c3dc69f45 100644
--- a/lean/OAI/Probability/ThorpShuffle/FixedLists/Energy.lean
+++ b/lean/OAI/Probability/ThorpShuffle/FixedLists/Energy.lean
@@ -1,160 +1,160 @@
-import OAI.Probability.ThorpShuffle.Main
-
-namespace OAI
-
-
-noncomputable section
-open scoped BigOperators
-open Filter
-namespace Thorp
-namespace Conditional
-
-theorem halfDensity_after_step_sixteen (d : ℕ) (v : RawState (d + 1))
-    (c : Coins (d + 1)) (h : Bool)
-    (hm : Fintype.card (Position (d + 1)) ≤ 16 * freeCount v.free)
-    (hc : freeCount v.free ≤ 4 * pairHalfCount (v.free ∘ (splitPosition d).symm) c h) :
-    (1 / 32 : ℝ) ≤ startHalfDensity d (rawNext d v c) h := by
-  unfold startHalfDensity
-  rw [halfCount_after_step]
-  have hn : Fintype.card (Position (d + 1)) = 2 * Fintype.card (Position d) := by
-    simp [Fintype.card_pi, Fintype.card_bool, Fintype.card_fin, pow_succ, Nat.mul_comm]
-  rw [hn] at hm
-  have hm' : (2 : ℝ) * Fintype.card (Position d) ≤ 16 * (freeCount v.free : ℝ) := by
-    exact_mod_cast hm
-  have hc' : (freeCount v.free : ℝ) ≤
-      4 * (pairHalfCount (v.free ∘ (splitPosition d).symm) c h : ℝ) := by exact_mod_cast hc
-  apply (le_div_iff₀ (by exact_mod_cast Fintype.card_pos (α := Position d))).mpr
-  nlinarith
-
-theorem expectedNoise_dimension_sixteen (d : ℕ) (v : CenteredState (d + 2))
-    (hm : Fintype.card (Position (d + 2)) ≤ 16 * freeCount v.free) :
-    expectedNoise (d + 1) v (d + 2) ≤ (31 / 32 : ℝ) * expectedEnergy (d + 1) v (d + 2) +
-      2 * (9 / 10 : ℝ) ^ freeCount v.free := by
-  let B := v.free ∘ (splitPosition (d + 1)).symm
-  let bad (c : Coins (d + 2)) : Prop :=
-    4 * pairHalfCount B c false < pairFreeCount B ∨
-    4 * pairHalfCount B c true < pairFreeCount B
-  have hlocal (c : Coins (d + 2)) :
-      expectedNoise (d + 1) (nextState (d + 1) v c) (d + 1) ≤
-      (31 / 32 : ℝ) * expectedEnergy (d + 1) (nextState (d + 1) v c) (d + 1) +
-        (if bad c then 1 else 0) := by
-    by_cases hc : bad c
-    · rw [ite_eq_left hc]
-      have hle := expectedNoise_le_energy (d + 1) (nextState (d + 1) v c) (d + 1)
-      have hone := expectedEnergy_le_one (d + 1) (nextState (d + 1) v c) (d + 1)
-      linarith
-    · rw [ite_eq_right hc, add_zero]
-      have hb (h : Bool) : (1 / 32 : ℝ) ≤
-          startHalfDensity (d + 1) (nextState (d + 1) v c).raw h := by
-        apply halfDensity_after_step_sixteen (d + 1) v.raw c h hm
-        have hf : ¬ 4 * pairHalfCount B c false < freeCount v.free := by
-          simpa only [B, pairFreeCount_physical] using not_or.mp hc |>.1
-        have ht : ¬ 4 * pairHalfCount B c true < freeCount v.free := by
-          simpa only [B, pairFreeCount_physical] using not_or.mp hc |>.2
-        change freeCount v.free ≤ 4 * pairHalfCount B c h
-        cases h
-        · exact Nat.le_of_not_gt hf
-        · exact Nat.le_of_not_gt ht
-      convert noise_bound_balanced_state d (nextState (d + 1) v c) (1 / 32) hb using 1; norm_num
-  have hmean := mean_le_mean hlocal
-  rw [mean_add, mean_const_mul, ← expectedNoise_first, ← expectedEnergy_first] at hmean
-  have hbad := pair_both_half_balance B
-  have hbad' : mean (fun c : Coins (d + 2) => if bad c then (1 : ℝ) else 0) ≤
-      2 * (9 / 10 : ℝ) ^ freeCount v.free := by
-    change mean (fun c : Position (d + 1) → Bool => if bad c then (1 : ℝ) else 0) ≤ _
-    simpa only [bad, B, pairFreeCount_physical] using hbad
-  exact hmean.trans (add_le_add (le_refl _) hbad')
-
-theorem expectedNoise_bound_sixteen (d : ℕ) (v : CenteredState (d + 2))
-    (hm : Fintype.card (Position (d + 2)) ≤ 16 * freeCount v.free)
-    (s : ℕ) (hs : d + 2 ≤ s) :
-    expectedNoise (d + 1) v s ≤ (31 / 32 : ℝ) * expectedEnergy (d + 1) v s +
-      2 * (9 / 10 : ℝ) ^ freeCount v.free := by
-  obtain ⟨a, rfl⟩ := Nat.exists_eq_add_of_le hs
-  rw [Nat.add_comm (d + 2) a, expectedNoise_add, expectedEnergy_add]
-  have he := mean_le_mean (fun ω : History (d + 2) a =>
-    expectedNoise_dimension_sixteen d (iterateState (d + 1) v a ω)
-      (by simpa only [iterateState_freeCount] using hm))
-  simpa only [iterateState_freeCount, mean_add, mean_const_mul, mean_const] using he
-
-theorem expectedEnergy_decay_sixteen (d : ℕ) (v : CenteredState (d + 2))
-    (hm : Fintype.card (Position (d + 2)) ≤ 16 * freeCount v.free) (t : ℕ) :
-    expectedEnergy (d + 1) v t ≤
-      (63 / 64 : ℝ) ^ t / (63 / 64 : ℝ) ^ (2 * (d + 2)) +
-        64 * (9 / 10 : ℝ) ^ freeCount v.free := by
-  have he := scalar_decay (d + 2) (expectedEnergy (d + 1) v)
-    (expectedNoise (d + 1) v) (1 / 32) (63 / 64)
-    (2 * (9 / 10 : ℝ) ^ freeCount v.free)
-    (by norm_num) (by norm_num) (by norm_num) (by norm_num)
-    (by norm_num) (by positivity) (expectedEnergy_le_one (d + 1) v)
-    (expectedEnergy_recurrence (d + 1) v)
-    (by intro s hs; convert expectedNoise_bound_sixteen d v hm s hs using 1; norm_num) t
-  convert he using 1; ring
-
-theorem expectedEnergy_long (d C : ℕ) (v : CenteredState (d + 2))
-    (hm : Fintype.card (Position (d + 2)) ≤ 16 * freeCount v.free) :
-    expectedEnergy (d + 1) v ((C + 2) * (d + 2)) ≤
-      (63 / 64 : ℝ) ^ (C * (d + 2)) + 64 * (9 / 10 : ℝ) ^ freeCount v.free := by
-  have he := expectedEnergy_decay_sixteen d v hm ((C + 2) * (d + 2))
-  rw [Nat.add_mul, pow_add, mul_div_cancel_right₀ _ (by positivity)] at he
-  simpa only [Nat.add_mul] using he
-
-theorem prefix_test_bound_sixteen (d C m : ℕ)
-    (hm : Fintype.card (Position (d + 2)) ≤ 16 * m)
-    (k : ℕ) (hk : k + m ≤ Fintype.card (Position (d + 2)))
-    (e : Fin k → Position (d + 2)) (he : Function.Injective e)
-    (f : (Fin k → Position (d + 2)) → ℝ) (hf : ∀ z, |f z| ≤ 1) :
-    |mean (fun ω : History (d + 2) ((C + 2) * (d + 2)) =>
-        f ((run (d + 2) ((C + 2) * (d + 2)) ω) ∘ e)) -
-      mean (fun g : State (d + 2) => f (g ∘ e))| ≤
-    (k : ℝ) * Real.sqrt ((Fintype.card (Position (d + 2)) : ℝ) *
-      ((63 / 64 : ℝ) ^ (C * (d + 2)) + 64 * (9 / 10 : ℝ) ^ m)) := by
-  induction k with
-  | zero =>
-    have hz (g : State (d + 2)) : g ∘ e = e := by funext i; exact Fin.elim0 i
-    simp only [hz, mean_const, sub_self, abs_zero, Nat.cast_zero, zero_mul, le_refl]
-  | succ k ih =>
-    have he0 : Function.Injective (Fin.init e) := by
-      intro i j hij
-      exact Fin.castSucc_inj.mp (he hij)
-    have ht : freeOf (Fin.init e) (e (Fin.last k)) = true := by
-      apply (freeOf_true _ _).mpr
-      intro i hi
-      exact Fin.castSucc_ne_last i (he hi)
-    have hcount : m ≤ freeCount (freeOf (Fin.init e)) := by
-      rw [freeCount_freeOf _ he0]
-      omega
-    have henergy := expectedEnergy_long d C
-      (initialState (d + 2) (freeOf (Fin.init e)) (e (Fin.last k)) ht) (by
-        change Fintype.card (Position (d + 2)) ≤ 16 * freeCount (freeOf (Fin.init e))
-        omega)
-    have hpower : (9 / 10 : ℝ) ^ freeCount (freeOf (Fin.init e)) ≤ (9 / 10 : ℝ) ^ m :=
-      pow_le_pow_of_le_one (by norm_num) (by norm_num) hcount
-    have hz := prefix_one_step_bound (d + 1) ((C + 2) * (d + 2)) k e he f hf
-    have hestimate :
-        Real.sqrt ((Fintype.card (Position (d + 2)) : ℝ) *
-          expectedEnergy (d + 1) (initialState (d + 2) (freeOf (Fin.init e)) (e (Fin.last k)) ht)
-            ((C + 2) * (d + 2))) ≤
-        Real.sqrt ((Fintype.card (Position (d + 2)) : ℝ) *
-          ((63 / 64 : ℝ) ^ (C * (d + 2)) + 64 * (9 / 10 : ℝ) ^ m)) := by
-      apply Real.sqrt_le_sqrt
-      apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
-      exact henergy.trans (add_le_add (le_refl _) (mul_le_mul_of_nonneg_left hpower (by norm_num : (0 : ℝ) ≤ 64)))
-    have hnext := ih (by omega) (Fin.init e) he0 (completeTest f) (completeTest_bound f hf)
-    rw [uniform_prefix_step k e he f]
-    have htri := abs_sub_le
-      (mean (fun ω : History (d + 2) ((C + 2) * (d + 2)) =>
-        f ((run (d + 2) ((C + 2) * (d + 2)) ω) ∘ e)))
-      (mean (fun ω : History (d + 2) ((C + 2) * (d + 2)) =>
-        completeTest f ((run (d + 2) ((C + 2) * (d + 2)) ω) ∘ Fin.init e)))
-      (mean (fun g : State (d + 2) => completeTest f (g ∘ Fin.init e)))
-    have hz' := hz.trans hestimate
-    push_cast
-    nlinarith
-
-end Conditional
-end Thorp
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.Main
+
+namespace OAI
+
+
+noncomputable section
+open scoped BigOperators
+open Filter
+namespace Thorp
+namespace Conditional
+
+theorem halfDensity_after_step_sixteen (d : ℕ) (v : RawState (d + 1))
+    (c : Coins (d + 1)) (h : Bool)
+    (hm : Fintype.card (Position (d + 1)) ≤ 16 * freeCount v.free)
+    (hc : freeCount v.free ≤ 4 * pairHalfCount (v.free ∘ (splitPosition d).symm) c h) :
+    (1 / 32 : ℝ) ≤ startHalfDensity d (rawNext d v c) h := by
+  unfold startHalfDensity
+  rw [halfCount_after_step]
+  have hn : Fintype.card (Position (d + 1)) = 2 * Fintype.card (Position d) := by
+    simp [Fintype.card_pi, Fintype.card_bool, Fintype.card_fin, pow_succ, Nat.mul_comm]
+  rw [hn] at hm
+  have hm' : (2 : ℝ) * Fintype.card (Position d) ≤ 16 * (freeCount v.free : ℝ) := by
+    exact_mod_cast hm
+  have hc' : (freeCount v.free : ℝ) ≤
+      4 * (pairHalfCount (v.free ∘ (splitPosition d).symm) c h : ℝ) := by exact_mod_cast hc
+  apply (le_div_iff₀ (by exact_mod_cast Fintype.card_pos (α := Position d))).mpr
+  nlinarith
+
+theorem expectedNoise_dimension_sixteen (d : ℕ) (v : CenteredState (d + 2))
+    (hm : Fintype.card (Position (d + 2)) ≤ 16 * freeCount v.free) :
+    expectedNoise (d + 1) v (d + 2) ≤ (31 / 32 : ℝ) * expectedEnergy (d + 1) v (d + 2) +
+      2 * (9 / 10 : ℝ) ^ freeCount v.free := by
+  let B := v.free ∘ (splitPosition (d + 1)).symm
+  let bad (c : Coins (d + 2)) : Prop :=
+    4 * pairHalfCount B c false < pairFreeCount B ∨
+    4 * pairHalfCount B c true < pairFreeCount B
+  have hlocal (c : Coins (d + 2)) :
+      expectedNoise (d + 1) (nextState (d + 1) v c) (d + 1) ≤
+      (31 / 32 : ℝ) * expectedEnergy (d + 1) (nextState (d + 1) v c) (d + 1) +
+        (if bad c then 1 else 0) := by
+    by_cases hc : bad c
+    · rw [ite_eq_left hc]
+      have hle := expectedNoise_le_energy (d + 1) (nextState (d + 1) v c) (d + 1)
+      have hone := expectedEnergy_le_one (d + 1) (nextState (d + 1) v c) (d + 1)
+      linarith
+    · rw [ite_eq_right hc, add_zero]
+      have hb (h : Bool) : (1 / 32 : ℝ) ≤
+          startHalfDensity (d + 1) (nextState (d + 1) v c).raw h := by
+        apply halfDensity_after_step_sixteen (d + 1) v.raw c h hm
+        have hf : ¬ 4 * pairHalfCount B c false < freeCount v.free := by
+          simpa only [B, pairFreeCount_physical] using not_or.mp hc |>.1
+        have ht : ¬ 4 * pairHalfCount B c true < freeCount v.free := by
+          simpa only [B, pairFreeCount_physical] using not_or.mp hc |>.2
+        change freeCount v.free ≤ 4 * pairHalfCount B c h
+        cases h
+        · exact Nat.le_of_not_gt hf
+        · exact Nat.le_of_not_gt ht
+      convert noise_bound_balanced_state d (nextState (d + 1) v c) (1 / 32) hb using 1; norm_num
+  have hmean := mean_le_mean hlocal
+  rw [mean_add, mean_const_mul, ← expectedNoise_first, ← expectedEnergy_first] at hmean
+  have hbad := pair_both_half_balance B
+  have hbad' : mean (fun c : Coins (d + 2) => if bad c then (1 : ℝ) else 0) ≤
+      2 * (9 / 10 : ℝ) ^ freeCount v.free := by
+    change mean (fun c : Position (d + 1) → Bool => if bad c then (1 : ℝ) else 0) ≤ _
+    simpa only [bad, B, pairFreeCount_physical] using hbad
+  exact hmean.trans (add_le_add (le_refl _) hbad')
+
+theorem expectedNoise_bound_sixteen (d : ℕ) (v : CenteredState (d + 2))
+    (hm : Fintype.card (Position (d + 2)) ≤ 16 * freeCount v.free)
+    (s : ℕ) (hs : d + 2 ≤ s) :
+    expectedNoise (d + 1) v s ≤ (31 / 32 : ℝ) * expectedEnergy (d + 1) v s +
+      2 * (9 / 10 : ℝ) ^ freeCount v.free := by
+  obtain ⟨a, rfl⟩ := Nat.exists_eq_add_of_le hs
+  rw [Nat.add_comm (d + 2) a, expectedNoise_add, expectedEnergy_add]
+  have he := mean_le_mean (fun ω : History (d + 2) a =>
+    expectedNoise_dimension_sixteen d (iterateState (d + 1) v a ω)
+      (by simpa only [iterateState_freeCount] using hm))
+  simpa only [iterateState_freeCount, mean_add, mean_const_mul, mean_const] using he
+
+theorem expectedEnergy_decay_sixteen (d : ℕ) (v : CenteredState (d + 2))
+    (hm : Fintype.card (Position (d + 2)) ≤ 16 * freeCount v.free) (t : ℕ) :
+    expectedEnergy (d + 1) v t ≤
+      (63 / 64 : ℝ) ^ t / (63 / 64 : ℝ) ^ (2 * (d + 2)) +
+        64 * (9 / 10 : ℝ) ^ freeCount v.free := by
+  have he := scalar_decay (d + 2) (expectedEnergy (d + 1) v)
+    (expectedNoise (d + 1) v) (1 / 32) (63 / 64)
+    (2 * (9 / 10 : ℝ) ^ freeCount v.free)
+    (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+    (by norm_num) (by positivity) (expectedEnergy_le_one (d + 1) v)
+    (expectedEnergy_recurrence (d + 1) v)
+    (by intro s hs; convert expectedNoise_bound_sixteen d v hm s hs using 1; norm_num) t
+  convert he using 1; ring
+
+theorem expectedEnergy_long (d C : ℕ) (v : CenteredState (d + 2))
+    (hm : Fintype.card (Position (d + 2)) ≤ 16 * freeCount v.free) :
+    expectedEnergy (d + 1) v ((C + 2) * (d + 2)) ≤
+      (63 / 64 : ℝ) ^ (C * (d + 2)) + 64 * (9 / 10 : ℝ) ^ freeCount v.free := by
+  have he := expectedEnergy_decay_sixteen d v hm ((C + 2) * (d + 2))
+  rw [Nat.add_mul, pow_add, mul_div_cancel_right₀ _ (by positivity)] at he
+  simpa only [Nat.add_mul] using he
+
+theorem prefix_test_bound_sixteen (d C m : ℕ)
+    (hm : Fintype.card (Position (d + 2)) ≤ 16 * m)
+    (k : ℕ) (hk : k + m ≤ Fintype.card (Position (d + 2)))
+    (e : Fin k → Position (d + 2)) (he : Function.Injective e)
+    (f : (Fin k → Position (d + 2)) → ℝ) (hf : ∀ z, |f z| ≤ 1) :
+    |mean (fun ω : History (d + 2) ((C + 2) * (d + 2)) =>
+        f ((run (d + 2) ((C + 2) * (d + 2)) ω) ∘ e)) -
+      mean (fun g : State (d + 2) => f (g ∘ e))| ≤
+    (k : ℝ) * Real.sqrt ((Fintype.card (Position (d + 2)) : ℝ) *
+      ((63 / 64 : ℝ) ^ (C * (d + 2)) + 64 * (9 / 10 : ℝ) ^ m)) := by
+  induction k with
+  | zero =>
+    have hz (g : State (d + 2)) : g ∘ e = e := by funext i; exact Fin.elim0 i
+    simp only [hz, mean_const, sub_self, abs_zero, Nat.cast_zero, zero_mul, le_refl]
+  | succ k ih =>
+    have he0 : Function.Injective (Fin.init e) := by
+      intro i j hij
+      exact Fin.castSucc_inj.mp (he hij)
+    have ht : freeOf (Fin.init e) (e (Fin.last k)) = true := by
+      apply (freeOf_true _ _).mpr
+      intro i hi
+      exact Fin.castSucc_ne_last i (he hi)
+    have hcount : m ≤ freeCount (freeOf (Fin.init e)) := by
+      rw [freeCount_freeOf _ he0]
+      omega
+    have henergy := expectedEnergy_long d C
+      (initialState (d + 2) (freeOf (Fin.init e)) (e (Fin.last k)) ht) (by
+        change Fintype.card (Position (d + 2)) ≤ 16 * freeCount (freeOf (Fin.init e))
+        omega)
+    have hpower : (9 / 10 : ℝ) ^ freeCount (freeOf (Fin.init e)) ≤ (9 / 10 : ℝ) ^ m :=
+      pow_le_pow_of_le_one (by norm_num) (by norm_num) hcount
+    have hz := prefix_one_step_bound (d + 1) ((C + 2) * (d + 2)) k e he f hf
+    have hestimate :
+        Real.sqrt ((Fintype.card (Position (d + 2)) : ℝ) *
+          expectedEnergy (d + 1) (initialState (d + 2) (freeOf (Fin.init e)) (e (Fin.last k)) ht)
+            ((C + 2) * (d + 2))) ≤
+        Real.sqrt ((Fintype.card (Position (d + 2)) : ℝ) *
+          ((63 / 64 : ℝ) ^ (C * (d + 2)) + 64 * (9 / 10 : ℝ) ^ m)) := by
+      apply Real.sqrt_le_sqrt
+      apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
+      exact henergy.trans (add_le_add (le_refl _) (mul_le_mul_of_nonneg_left hpower (by norm_num : (0 : ℝ) ≤ 64)))
+    have hnext := ih (by omega) (Fin.init e) he0 (completeTest f) (completeTest_bound f hf)
+    rw [uniform_prefix_step k e he f]
+    have htri := abs_sub_le
+      (mean (fun ω : History (d + 2) ((C + 2) * (d + 2)) =>
+        f ((run (d + 2) ((C + 2) * (d + 2)) ω) ∘ e)))
+      (mean (fun ω : History (d + 2) ((C + 2) * (d + 2)) =>
+        completeTest f ((run (d + 2) ((C + 2) * (d + 2)) ω) ∘ Fin.init e)))
+      (mean (fun g : State (d + 2) => completeTest f (g ∘ Fin.init e)))
+    have hz' := hz.trans hestimate
+    push_cast
+    nlinarith
+
+end Conditional
+end Thorp
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/FixedLists/FullDeck.lean b/lean/OAI/Probability/ThorpShuffle/FixedLists/FullDeck.lean
index 07feb6a7c4ca7158eeb150288ef44e8102c39ee7..f7dceafa63853180768f4e5a5461be88dda577e5 100644
--- a/lean/OAI/Probability/ThorpShuffle/FixedLists/FullDeck.lean
+++ b/lean/OAI/Probability/ThorpShuffle/FixedLists/FullDeck.lean
@@ -1,32 +1,32 @@
-import OAI.Probability.ThorpShuffle.Main
-
-namespace OAI
-
-
-noncomputable section
-open scoped BigOperators
-open Filter
-namespace Thorp
-
-lemma distance_add_le (d s t : ℕ) : distance d (s + t) ≤ distance d s := by
-  unfold distance
-  rw [law_add]
-  have h := FiniteLaw.tv_conv_contract (law d t) (law d s)
-    (law_sum d t) (law_sum d s) 0 (law_nonneg d t)
-  change tv (FiniteLaw.conv (law d t) (law d s))
-    (fun _ => (1 : ℝ) / Fintype.card (State d)) ≤
-    tv (law d s) (fun _ => (1 : ℝ) / Fintype.card (State d))
-  simpa only [mul_zero, sub_zero, one_mul] using h
-
-theorem full_mixing_2048 :
-    Tendsto (fun d : ℕ => distance d (2048 * d)) atTop (nhds 0) := by
-  apply squeeze_zero (fun d => by unfold distance tv; positivity) _ full_mixing
-  intro d
-  rw [show 2048 * d = 1600 * d + 448 * d by omega]
-  exact distance_add_le d (1600 * d) (448 * d)
-
-end Thorp
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.Main
+
+namespace OAI
+
+
+noncomputable section
+open scoped BigOperators
+open Filter
+namespace Thorp
+
+lemma distance_add_le (d s t : ℕ) : distance d (s + t) ≤ distance d s := by
+  unfold distance
+  rw [law_add]
+  have h := FiniteLaw.tv_conv_contract (law d t) (law d s)
+    (law_sum d t) (law_sum d s) 0 (law_nonneg d t)
+  change tv (FiniteLaw.conv (law d t) (law d s))
+    (fun _ => (1 : ℝ) / Fintype.card (State d)) ≤
+    tv (law d s) (fun _ => (1 : ℝ) / Fintype.card (State d))
+  simpa only [mul_zero, sub_zero, one_mul] using h
+
+theorem full_mixing_2048 :
+    Tendsto (fun d : ℕ => distance d (2048 * d)) atTop (nhds 0) := by
+  apply squeeze_zero (fun d => by unfold distance tv; positivity) _ full_mixing
+  intro d
+  rw [show 2048 * d = 1600 * d + 448 * d by omega]
+  exact distance_add_le d (1600 * d) (448 * d)
+
+end Thorp
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/FixedLists/Lists.lean b/lean/OAI/Probability/ThorpShuffle/FixedLists/Lists.lean
index 8faa4d280463676fa4c9446c43ea9ab543f987a2..0ccf2197ed7451383062f202309934152010a06e 100644
--- a/lean/OAI/Probability/ThorpShuffle/FixedLists/Lists.lean
+++ b/lean/OAI/Probability/ThorpShuffle/FixedLists/Lists.lean
@@ -1,142 +1,142 @@
-import OAI.Probability.ThorpShuffle.FixedLists.Objects
-import OAI.Probability.ThorpShuffle.FixedLists.Asymptotics
-
-namespace OAI
-
-
-noncomputable section
-open scoped BigOperators
-open Filter
-namespace Thorp
-namespace FixedLists
-open Conditional
-
-theorem list_test_bound_raw (d C m k : ℕ)
-    (hm : Fintype.card (Position (d + 2)) ≤ 16 * m)
-    (hk : k + m ≤ Fintype.card (Position (d + 2)))
-    (e : LabelList (d + 2) k) (r : State (d + 2))
-    (f : LabelList (d + 2) k → ℝ) (hf : ∀ z, |f z| ≤ 1) :
-    |mean (fun ω : History (d + 2) ((C + 2) * (d + 2)) =>
-        f (e.trans ((r * run (d + 2) ((C + 2) * (d + 2)) ω).toEmbedding))) -
-      mean (fun g : State (d + 2) => f (e.trans ((r * g).toEmbedding)))| ≤
-      (k : ℝ) * Real.sqrt ((Fintype.card (Position (d + 2)) : ℝ) *
-        ((63 / 64 : ℝ) ^ (C * (d + 2)) + 64 * (9 / 10 : ℝ) ^ m)) := by
-  let F : (Fin k → Position (d + 2)) → ℝ := fun z =>
-    if hz : Function.Injective z then f ((⟨z, hz⟩ : LabelList (d + 2) k).trans r.toEmbedding)
-    else 0
-  have hF : ∀ z, |F z| ≤ 1 := by
-    intro z
-    dsimp only [F]
-    split_ifs
-    · exact hf _
-    · norm_num
-  have hmap (g : State (d + 2)) :
-      F (g ∘ e) = f (e.trans ((r * g).toEmbedding)) := by
-    dsimp only [F]
-    rw [dite_eq_left (g.injective.comp e.injective)]
-    rfl
-  have hb := prefix_test_bound_sixteen d C m hm k hk e e.injective F hF
-  simpa only [hmap] using hb
-
-theorem list_tv_bound_raw (d C m k : ℕ)
-    (hm : Fintype.card (Position (d + 2)) ≤ 16 * m)
-    (hk : k + m ≤ Fintype.card (Position (d + 2)))
-    (e : LabelList (d + 2) k) (r : State (d + 2)) :
-    tv (fairMass (fun ω : History (d + 2) ((C + 2) * (d + 2)) =>
-        e.trans ((r * run (d + 2) ((C + 2) * (d + 2)) ω).toEmbedding)))
-      (fun _ => (1 : ℝ) / Fintype.card (LabelList (d + 2) k)) ≤
-      (k : ℝ) / 2 * Real.sqrt ((Fintype.card (Position (d + 2)) : ℝ) *
-        ((63 / 64 : ℝ) ^ (C * (d + 2)) + 64 * (9 / 10 : ℝ) ^ m)) := by
-  rw [← uniform_list_mass_frame e r]
-  have hh := fairMass_tv_le_of_test
-    (fun ω : History (d + 2) ((C + 2) * (d + 2)) =>
-      e.trans ((r * run (d + 2) ((C + 2) * (d + 2)) ω).toEmbedding))
-    (fun g : State (d + 2) => e.trans ((r * g).toEmbedding))
-    ((k : ℝ) * Real.sqrt ((Fintype.card (Position (d + 2)) : ℝ) *
-      ((63 / 64 : ℝ) ^ (C * (d + 2)) + 64 * (9 / 10 : ℝ) ^ m)))
-    (list_test_bound_raw d C m k hm hk e r)
-  calc
-    _ ≤ (↑k * Real.sqrt ((Fintype.card (Position (d + 2)) : ℝ) *
-      ((63 / 64 : ℝ) ^ (C * (d + 2)) + 64 * (9 / 10 : ℝ) ^ m))) / 2 := hh
-    _ = _ := by ring
-
-lemma images_factor {d k : ℕ} (t : ℕ) (start : State d) (labels : LabelList d k)
-    (ω : History d t) :
-    images t start labels ω =
-      (labels.trans start.toEmbedding).trans
-        (((rotate d)⁻¹ ^ t * run d t ω).toEmbedding) := by
-  ext i
-  rfl
-
-theorem listTV_bound_raw (d C m k : ℕ)
-    (hm : Fintype.card (Position (d + 2)) ≤ 16 * m)
-    (hk : k + m ≤ Fintype.card (Position (d + 2)))
-    (start : State (d + 2)) (labels : LabelList (d + 2) k) :
-    listTV ((C + 2) * (d + 2)) start labels ≤
-      (k : ℝ) / 2 * Real.sqrt ((Fintype.card (Position (d + 2)) : ℝ) *
-        ((63 / 64 : ℝ) ^ (C * (d + 2)) + 64 * (9 / 10 : ℝ) ^ m)) := by
-  simp only [listTV]
-  exact list_tv_bound_raw d C m k hm hk (labels.trans start.toEmbedding)
-    ((rotate (d + 2))⁻¹ ^ ((C + 2) * (d + 2)))
-
-theorem listTV_bound (d C k : ℕ) (hk : 16 * k ≤ 15 * 2 ^ (d + 4))
-    (start : State (d + 4)) (labels : LabelList (d + 4) k) :
-    listTV ((C + 2) * (d + 4)) start labels ≤
-      (k : ℝ) / 2 * Real.sqrt ((2 : ℝ) ^ (d + 4) * energyEnvelope C d) := by
-  have hn : Fintype.card (Position (d + 2 + 2)) = 16 * 2 ^ d := by
-    rw [card_position]
-    simp only [pow_add]
-    norm_num
-    omega
-  have hk' : k + 2 ^ d ≤ Fintype.card (Position (d + 2 + 2)) := by
-    have hn' : 2 ^ (d + 4) = 16 * 2 ^ d := by simp [pow_add, Nat.mul_comm]
-    rw [hn' ] at hk
-    rw [hn]
-    omega
-  have hh := listTV_bound_raw (d + 2) C (2 ^ d) k (by rw [hn]) hk' start labels
-  simpa only [Nat.add_assoc, card_position, Nat.cast_pow, Nat.cast_ofNat, energyEnvelope] using hh
-
-lemma k_le_size (d k : ℕ) (hk : 16 * k ≤ 15 * 2 ^ d) : (k : ℝ) ≤ (2 : ℝ) ^ d := by
-  have h : k ≤ 2 ^ d := by omega
-  exact_mod_cast h
-
-theorem listTV_256 (d k : ℕ) (hk : 16 * k ≤ 15 * 2 ^ (d + 4))
-    (start : State (d + 4)) (labels : LabelList (d + 4) k) :
-    listTV (256 * (d + 4)) start labels ≤ listRate d := by
-  have h := listTV_bound d 254 k hk start labels
-  have hk' := k_le_size (d + 4) k hk
-  have ha := energyEnvelope_nonneg 254 d
-  have hb : (k : ℝ) / 2 * Real.sqrt ((2 : ℝ) ^ (d + 4) * energyEnvelope 254 d) ≤ listRate d := by
-    calc
-      _ ≤ ((2 : ℝ) ^ (d + 4)) / 2 * Real.sqrt ((2 : ℝ) ^ (d + 4) * energyEnvelope 254 d) :=
-        mul_le_mul_of_nonneg_right (div_le_div_of_nonneg_right hk' (by norm_num)) (Real.sqrt_nonneg _)
-      _ ≤ listRate d := by
-        unfold listRate
-        apply Real.le_sqrt_of_sq_le
-        rw [mul_pow, div_pow, Real.sq_sqrt (mul_nonneg (by positivity) ha)]
-        have hp : 0 ≤ ((2 : ℝ) ^ (d + 4)) ^ 3 * energyEnvelope 254 d := mul_nonneg (by positivity) ha
-        nlinarith
-  exact h.trans hb
-
-theorem listTV_1024_eventually :
-    ∀ᶠ d : ℕ in atTop, ∀ k, 16 * k ≤ 15 * 2 ^ (d + 4) →
-      ∀ (start : State (d + 4)) (labels : LabelList (d + 4) k),
-      listTV (1024 * (d + 4)) start labels ≤ Real.rpow ((2 : ℝ) ^ (d + 4)) (-(3 : ℝ) / 2) := by
-  have he : ∀ᶠ d : ℕ in atTop,
-      ((2 : ℝ) ^ (d + 4)) ^ 6 * energyEnvelope 1022 d ≤ 2 :=
-    (scaledEnergy_tendsto 1022 6 geometric_base_1024).eventually
-      (eventually_le_nhds (by norm_num : (0 : ℝ) < 2))
-  filter_upwards [he] with d hd
-  intro k hk start labels
-  have h := listTV_bound d 1022 k hk start labels
-  have hk' := k_le_size (d + 4) k hk
-  exact h.trans ((mul_le_mul_of_nonneg_right
-    (div_le_div_of_nonneg_right hk' (by norm_num)) (Real.sqrt_nonneg _)).trans
-      (scalar_tv_finish _ _ (by positivity) (energyEnvelope_nonneg _ _) hd))
-
-end FixedLists
-end Thorp
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.FixedLists.Objects
+import OAI.Probability.ThorpShuffle.FixedLists.Asymptotics
+
+namespace OAI
+
+
+noncomputable section
+open scoped BigOperators
+open Filter
+namespace Thorp
+namespace FixedLists
+open Conditional
+
+theorem list_test_bound_raw (d C m k : ℕ)
+    (hm : Fintype.card (Position (d + 2)) ≤ 16 * m)
+    (hk : k + m ≤ Fintype.card (Position (d + 2)))
+    (e : LabelList (d + 2) k) (r : State (d + 2))
+    (f : LabelList (d + 2) k → ℝ) (hf : ∀ z, |f z| ≤ 1) :
+    |mean (fun ω : History (d + 2) ((C + 2) * (d + 2)) =>
+        f (e.trans ((r * run (d + 2) ((C + 2) * (d + 2)) ω).toEmbedding))) -
+      mean (fun g : State (d + 2) => f (e.trans ((r * g).toEmbedding)))| ≤
+      (k : ℝ) * Real.sqrt ((Fintype.card (Position (d + 2)) : ℝ) *
+        ((63 / 64 : ℝ) ^ (C * (d + 2)) + 64 * (9 / 10 : ℝ) ^ m)) := by
+  let F : (Fin k → Position (d + 2)) → ℝ := fun z =>
+    if hz : Function.Injective z then f ((⟨z, hz⟩ : LabelList (d + 2) k).trans r.toEmbedding)
+    else 0
+  have hF : ∀ z, |F z| ≤ 1 := by
+    intro z
+    dsimp only [F]
+    split_ifs
+    · exact hf _
+    · norm_num
+  have hmap (g : State (d + 2)) :
+      F (g ∘ e) = f (e.trans ((r * g).toEmbedding)) := by
+    dsimp only [F]
+    rw [dite_eq_left (g.injective.comp e.injective)]
+    rfl
+  have hb := prefix_test_bound_sixteen d C m hm k hk e e.injective F hF
+  simpa only [hmap] using hb
+
+theorem list_tv_bound_raw (d C m k : ℕ)
+    (hm : Fintype.card (Position (d + 2)) ≤ 16 * m)
+    (hk : k + m ≤ Fintype.card (Position (d + 2)))
+    (e : LabelList (d + 2) k) (r : State (d + 2)) :
+    tv (fairMass (fun ω : History (d + 2) ((C + 2) * (d + 2)) =>
+        e.trans ((r * run (d + 2) ((C + 2) * (d + 2)) ω).toEmbedding)))
+      (fun _ => (1 : ℝ) / Fintype.card (LabelList (d + 2) k)) ≤
+      (k : ℝ) / 2 * Real.sqrt ((Fintype.card (Position (d + 2)) : ℝ) *
+        ((63 / 64 : ℝ) ^ (C * (d + 2)) + 64 * (9 / 10 : ℝ) ^ m)) := by
+  rw [← uniform_list_mass_frame e r]
+  have hh := fairMass_tv_le_of_test
+    (fun ω : History (d + 2) ((C + 2) * (d + 2)) =>
+      e.trans ((r * run (d + 2) ((C + 2) * (d + 2)) ω).toEmbedding))
+    (fun g : State (d + 2) => e.trans ((r * g).toEmbedding))
+    ((k : ℝ) * Real.sqrt ((Fintype.card (Position (d + 2)) : ℝ) *
+      ((63 / 64 : ℝ) ^ (C * (d + 2)) + 64 * (9 / 10 : ℝ) ^ m)))
+    (list_test_bound_raw d C m k hm hk e r)
+  calc
+    _ ≤ (↑k * Real.sqrt ((Fintype.card (Position (d + 2)) : ℝ) *
+      ((63 / 64 : ℝ) ^ (C * (d + 2)) + 64 * (9 / 10 : ℝ) ^ m))) / 2 := hh
+    _ = _ := by ring
+
+lemma images_factor {d k : ℕ} (t : ℕ) (start : State d) (labels : LabelList d k)
+    (ω : History d t) :
+    images t start labels ω =
+      (labels.trans start.toEmbedding).trans
+        (((rotate d)⁻¹ ^ t * run d t ω).toEmbedding) := by
+  ext i
+  rfl
+
+theorem listTV_bound_raw (d C m k : ℕ)
+    (hm : Fintype.card (Position (d + 2)) ≤ 16 * m)
+    (hk : k + m ≤ Fintype.card (Position (d + 2)))
+    (start : State (d + 2)) (labels : LabelList (d + 2) k) :
+    listTV ((C + 2) * (d + 2)) start labels ≤
+      (k : ℝ) / 2 * Real.sqrt ((Fintype.card (Position (d + 2)) : ℝ) *
+        ((63 / 64 : ℝ) ^ (C * (d + 2)) + 64 * (9 / 10 : ℝ) ^ m)) := by
+  simp only [listTV]
+  exact list_tv_bound_raw d C m k hm hk (labels.trans start.toEmbedding)
+    ((rotate (d + 2))⁻¹ ^ ((C + 2) * (d + 2)))
+
+theorem listTV_bound (d C k : ℕ) (hk : 16 * k ≤ 15 * 2 ^ (d + 4))
+    (start : State (d + 4)) (labels : LabelList (d + 4) k) :
+    listTV ((C + 2) * (d + 4)) start labels ≤
+      (k : ℝ) / 2 * Real.sqrt ((2 : ℝ) ^ (d + 4) * energyEnvelope C d) := by
+  have hn : Fintype.card (Position (d + 2 + 2)) = 16 * 2 ^ d := by
+    rw [card_position]
+    simp only [pow_add]
+    norm_num
+    omega
+  have hk' : k + 2 ^ d ≤ Fintype.card (Position (d + 2 + 2)) := by
+    have hn' : 2 ^ (d + 4) = 16 * 2 ^ d := by simp [pow_add, Nat.mul_comm]
+    rw [hn' ] at hk
+    rw [hn]
+    omega
+  have hh := listTV_bound_raw (d + 2) C (2 ^ d) k (by rw [hn]) hk' start labels
+  simpa only [Nat.add_assoc, card_position, Nat.cast_pow, Nat.cast_ofNat, energyEnvelope] using hh
+
+lemma k_le_size (d k : ℕ) (hk : 16 * k ≤ 15 * 2 ^ d) : (k : ℝ) ≤ (2 : ℝ) ^ d := by
+  have h : k ≤ 2 ^ d := by omega
+  exact_mod_cast h
+
+theorem listTV_256 (d k : ℕ) (hk : 16 * k ≤ 15 * 2 ^ (d + 4))
+    (start : State (d + 4)) (labels : LabelList (d + 4) k) :
+    listTV (256 * (d + 4)) start labels ≤ listRate d := by
+  have h := listTV_bound d 254 k hk start labels
+  have hk' := k_le_size (d + 4) k hk
+  have ha := energyEnvelope_nonneg 254 d
+  have hb : (k : ℝ) / 2 * Real.sqrt ((2 : ℝ) ^ (d + 4) * energyEnvelope 254 d) ≤ listRate d := by
+    calc
+      _ ≤ ((2 : ℝ) ^ (d + 4)) / 2 * Real.sqrt ((2 : ℝ) ^ (d + 4) * energyEnvelope 254 d) :=
+        mul_le_mul_of_nonneg_right (div_le_div_of_nonneg_right hk' (by norm_num)) (Real.sqrt_nonneg _)
+      _ ≤ listRate d := by
+        unfold listRate
+        apply Real.le_sqrt_of_sq_le
+        rw [mul_pow, div_pow, Real.sq_sqrt (mul_nonneg (by positivity) ha)]
+        have hp : 0 ≤ ((2 : ℝ) ^ (d + 4)) ^ 3 * energyEnvelope 254 d := mul_nonneg (by positivity) ha
+        nlinarith
+  exact h.trans hb
+
+theorem listTV_1024_eventually :
+    ∀ᶠ d : ℕ in atTop, ∀ k, 16 * k ≤ 15 * 2 ^ (d + 4) →
+      ∀ (start : State (d + 4)) (labels : LabelList (d + 4) k),
+      listTV (1024 * (d + 4)) start labels ≤ Real.rpow ((2 : ℝ) ^ (d + 4)) (-(3 : ℝ) / 2) := by
+  have he : ∀ᶠ d : ℕ in atTop,
+      ((2 : ℝ) ^ (d + 4)) ^ 6 * energyEnvelope 1022 d ≤ 2 :=
+    (scaledEnergy_tendsto 1022 6 geometric_base_1024).eventually
+      (eventually_le_nhds (by norm_num : (0 : ℝ) < 2))
+  filter_upwards [he] with d hd
+  intro k hk start labels
+  have h := listTV_bound d 1022 k hk start labels
+  have hk' := k_le_size (d + 4) k hk
+  exact h.trans ((mul_le_mul_of_nonneg_right
+    (div_le_div_of_nonneg_right hk' (by norm_num)) (Real.sqrt_nonneg _)).trans
+      (scalar_tv_finish _ _ (by positivity) (energyEnvelope_nonneg _ _) hd))
+
+end FixedLists
+end Thorp
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/FixedLists/Main.lean b/lean/OAI/Probability/ThorpShuffle/FixedLists/Main.lean
index 0e97f25f67ca55850f2d6531f6971462e7226626..51d7f8a7f66fb3d38129827e488923531c93edec 100644
--- a/lean/OAI/Probability/ThorpShuffle/FixedLists/Main.lean
+++ b/lean/OAI/Probability/ThorpShuffle/FixedLists/Main.lean
@@ -1,61 +1,61 @@
-import OAI.Probability.ThorpShuffle.FixedLists.Lists
-import OAI.Probability.ThorpShuffle.FixedLists.FullDeck
-
-namespace OAI
-
-
-noncomputable section
-open scoped BigOperators
-open Filter
-
-namespace Thorp
-namespace FixedLists
-
-lemma eventually_unshift {P : ℕ → Prop}
-    (h : ∀ᶠ d : ℕ in atTop, P (d + 4)) : ∀ᶠ d : ℕ in atTop, P d := by
-  rw [← map_add_atTop_eq_nat 4]
-  exact h
-
-lemma lists_256_uniform :
-    ∀ ε : ℝ, 0 < ε → ∀ᶠ d : ℕ in atTop,
-      ∀ k : ℕ, 8 * k ≤ 7 * 2 ^ d →
-      ∀ (start : State d) (labels : LabelList d k),
-        listTV (256 * d) start labels < ε := by
-  intro ε hε
-  apply eventually_unshift
-  have he : ∀ᶠ d : ℕ in atTop, listRate d < ε :=
-    listRate_tendsto.eventually (eventually_lt_nhds hε)
-  filter_upwards [he] with d hd
-  intro k hk start labels
-  have hk' : 16 * k ≤ 15 * 2 ^ (d + 4) := by omega
-  exact (listTV_256 d k hk' start labels).trans_lt hd
-
-lemma lists_1024_uniform :
-    ∀ᶠ d : ℕ in atTop,
-      ∀ k : ℕ, 16 * k ≤ 15 * 2 ^ d →
-      ∀ (start : State d) (labels : LabelList d k),
-        listTV (1024 * d) start labels ≤ Real.rpow ((2 : ℝ) ^ d) (-(3 : ℝ) / 2) :=
-  eventually_unshift listTV_1024_eventually
-
-end FixedLists
-
-theorem first_main :
-    (∀ ε : ℝ, 0 < ε → ∀ᶠ d : ℕ in atTop,
-      ∀ k : ℕ, 8 * k ≤ 7 * 2 ^ d →
-      ∀ (start : State d) (labels : LabelList d k),
-        listTV (256 * d) start labels < ε) ∧
-    (∀ᶠ d : ℕ in atTop,
-      ∀ k : ℕ, 16 * k ≤ 15 * 2 ^ d →
-      ∀ (start : State d) (labels : LabelList d k),
-        listTV (1024 * d) start labels ≤ Real.rpow ((2 : ℝ) ^ d) (-(3 : ℝ) / 2)) ∧
-    (Tendsto (fun d : ℕ => distance d (2048 * d)) atTop (nhds 0) ∧
-      ∀ (d : ℕ) (start : State d),
-        tv (lawFrom d (2048 * d) start) (uniform d) = distance d (2048 * d)) := by
-  exact ⟨FixedLists.lists_256_uniform, FixedLists.lists_1024_uniform,
-    full_mixing_2048, fun d start => worst_start_eq d (2048 * d) start⟩
-
-end Thorp
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.FixedLists.Lists
+import OAI.Probability.ThorpShuffle.FixedLists.FullDeck
+
+namespace OAI
+
+
+noncomputable section
+open scoped BigOperators
+open Filter
+
+namespace Thorp
+namespace FixedLists
+
+lemma eventually_unshift {P : ℕ → Prop}
+    (h : ∀ᶠ d : ℕ in atTop, P (d + 4)) : ∀ᶠ d : ℕ in atTop, P d := by
+  rw [← map_add_atTop_eq_nat 4]
+  exact h
+
+lemma lists_256_uniform :
+    ∀ ε : ℝ, 0 < ε → ∀ᶠ d : ℕ in atTop,
+      ∀ k : ℕ, 8 * k ≤ 7 * 2 ^ d →
+      ∀ (start : State d) (labels : LabelList d k),
+        listTV (256 * d) start labels < ε := by
+  intro ε hε
+  apply eventually_unshift
+  have he : ∀ᶠ d : ℕ in atTop, listRate d < ε :=
+    listRate_tendsto.eventually (eventually_lt_nhds hε)
+  filter_upwards [he] with d hd
+  intro k hk start labels
+  have hk' : 16 * k ≤ 15 * 2 ^ (d + 4) := by omega
+  exact (listTV_256 d k hk' start labels).trans_lt hd
+
+lemma lists_1024_uniform :
+    ∀ᶠ d : ℕ in atTop,
+      ∀ k : ℕ, 16 * k ≤ 15 * 2 ^ d →
+      ∀ (start : State d) (labels : LabelList d k),
+        listTV (1024 * d) start labels ≤ Real.rpow ((2 : ℝ) ^ d) (-(3 : ℝ) / 2) :=
+  eventually_unshift listTV_1024_eventually
+
+end FixedLists
+
+theorem first_main :
+    (∀ ε : ℝ, 0 < ε → ∀ᶠ d : ℕ in atTop,
+      ∀ k : ℕ, 8 * k ≤ 7 * 2 ^ d →
+      ∀ (start : State d) (labels : LabelList d k),
+        listTV (256 * d) start labels < ε) ∧
+    (∀ᶠ d : ℕ in atTop,
+      ∀ k : ℕ, 16 * k ≤ 15 * 2 ^ d →
+      ∀ (start : State d) (labels : LabelList d k),
+        listTV (1024 * d) start labels ≤ Real.rpow ((2 : ℝ) ^ d) (-(3 : ℝ) / 2)) ∧
+    (Tendsto (fun d : ℕ => distance d (2048 * d)) atTop (nhds 0) ∧
+      ∀ (d : ℕ) (start : State d),
+        tv (lawFrom d (2048 * d) start) (uniform d) = distance d (2048 * d)) := by
+  exact ⟨FixedLists.lists_256_uniform, FixedLists.lists_1024_uniform,
+    full_mixing_2048, fun d start => worst_start_eq d (2048 * d) start⟩
+
+end Thorp
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/FixedLists/Objects.lean b/lean/OAI/Probability/ThorpShuffle/FixedLists/Objects.lean
index 578347babc008ae2d8341b22ab901d38153c6051..1d20a4dd6ec7cea786f92d3a3864f233402050be 100644
--- a/lean/OAI/Probability/ThorpShuffle/FixedLists/Objects.lean
+++ b/lean/OAI/Probability/ThorpShuffle/FixedLists/Objects.lean
@@ -1,78 +1,78 @@
-import OAI.Probability.ThorpShuffle.Main
-
-namespace OAI
-
-
-noncomputable section
-open scoped BigOperators
-open Filter
-
-namespace Thorp
-
-abbrev LabelList (d k : ℕ) := Fin k ↪ Position d
-
-def coordinateRun (d t : ℕ) (ω : History d t) : State d :=
-  (rotate d)⁻¹ ^ t * run d t ω
-
-def images {d k : ℕ} (t : ℕ) (start : State d) (labels : LabelList d k)
-    (ω : History d t) : LabelList d k :=
-  labels.trans ((coordinateRun d t ω * start).toEmbedding)
-
-def listTV {d k : ℕ} (t : ℕ) (start : State d) (labels : LabelList d k) : ℝ :=
-  tv (fairMass (images t start labels)) (fun _ => 1 / Fintype.card (LabelList d k))
-
-namespace FixedLists
-open Conditional
-
-theorem uniform_list_mass {α : Type*} [Fintype α] [DecidableEq α] {k : ℕ} (e : Fin k ↪ α) :
-    fairMass (fun g : Equiv.Perm α => e.trans g.toEmbedding) =
-      fun _ => (1 : ℝ) / Fintype.card (Fin k ↪ α) := by
-  classical
-  let : Nonempty (Fin k ↪ α) := ⟨e⟩
-  let μ := fairMass (fun g : Equiv.Perm α => e.trans g.toEmbedding)
-  have hconst (z w : Fin k ↪ α) : μ z = μ w := by
-    obtain ⟨σ, hσ⟩ := Equiv.Perm.exists_extending_pair z w z.injective w.injective
-    have heq (g : Equiv.Perm α) : e.trans ((σ * g).toEmbedding) = w ↔ e.trans g.toEmbedding = z := by
-      constructor
-      · intro h
-        ext i
-        apply σ.injective
-        exact (congrArg (fun a : Fin k ↪ α => a i) h).trans (hσ i).symm
-      · intro h
-        ext i
-        change σ (g (e i)) = w i
-        rw [show g (e i) = z i from congrArg (fun a : Fin k ↪ α => a i) h, hσ]
-    have hh := mean_equiv (Equiv.mulLeft σ)
-      (fun g : Equiv.Perm α => if e.trans g.toEmbedding = w then (1 : ℝ) else 0)
-    change mean (fun g : Equiv.Perm α => if e.trans ((σ * g).toEmbedding) = w then (1 : ℝ) else 0) = _ at hh
-    dsimp only [μ]
-    simp only [fairMass_eq_mean]
-    simpa only [heq] using hh
-  have hs : ∑ z, μ z = 1 := fairMass_sum _
-  funext z
-  have hz : (Fintype.card (Fin k ↪ α) : ℝ) * μ z = 1 := by
-    have hh : (∑ w, μ w) = (Fintype.card (Fin k ↪ α) : ℝ) * μ z := by
-      simp only [hconst _ z, Finset.sum_const, Finset.card_univ, nsmul_eq_mul]
-    linarith
-  change μ z = _
-  exact (eq_div_iff (by exact_mod_cast Fintype.card_ne_zero (α := Fin k ↪ α))).mpr (by linarith)
-
-theorem uniform_list_mass_frame {α : Type*} [Fintype α] [DecidableEq α] {k : ℕ}
-    (e : Fin k ↪ α) (r : Equiv.Perm α) :
-    fairMass (fun g : Equiv.Perm α => e.trans ((r * g).toEmbedding)) =
-      fun _ => (1 : ℝ) / Fintype.card (Fin k ↪ α) := by
-  classical
-  calc
-    _ = fairMass (fun g : Equiv.Perm α => e.trans g.toEmbedding) := by
-      funext z
-      simp only [fairMass_eq_mean]
-      exact mean_equiv (Equiv.mulLeft r)
-        (fun g : Equiv.Perm α => if e.trans g.toEmbedding = z then (1 : ℝ) else 0)
-    _ = _ := uniform_list_mass e
-
-end FixedLists
-end Thorp
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.Main
+
+namespace OAI
+
+
+noncomputable section
+open scoped BigOperators
+open Filter
+
+namespace Thorp
+
+abbrev LabelList (d k : ℕ) := Fin k ↪ Position d
+
+def coordinateRun (d t : ℕ) (ω : History d t) : State d :=
+  (rotate d)⁻¹ ^ t * run d t ω
+
+def images {d k : ℕ} (t : ℕ) (start : State d) (labels : LabelList d k)
+    (ω : History d t) : LabelList d k :=
+  labels.trans ((coordinateRun d t ω * start).toEmbedding)
+
+def listTV {d k : ℕ} (t : ℕ) (start : State d) (labels : LabelList d k) : ℝ :=
+  tv (fairMass (images t start labels)) (fun _ => 1 / Fintype.card (LabelList d k))
+
+namespace FixedLists
+open Conditional
+
+theorem uniform_list_mass {α : Type*} [Fintype α] [DecidableEq α] {k : ℕ} (e : Fin k ↪ α) :
+    fairMass (fun g : Equiv.Perm α => e.trans g.toEmbedding) =
+      fun _ => (1 : ℝ) / Fintype.card (Fin k ↪ α) := by
+  classical
+  let : Nonempty (Fin k ↪ α) := ⟨e⟩
+  let μ := fairMass (fun g : Equiv.Perm α => e.trans g.toEmbedding)
+  have hconst (z w : Fin k ↪ α) : μ z = μ w := by
+    obtain ⟨σ, hσ⟩ := Equiv.Perm.exists_extending_pair z w z.injective w.injective
+    have heq (g : Equiv.Perm α) : e.trans ((σ * g).toEmbedding) = w ↔ e.trans g.toEmbedding = z := by
+      constructor
+      · intro h
+        ext i
+        apply σ.injective
+        exact (congrArg (fun a : Fin k ↪ α => a i) h).trans (hσ i).symm
+      · intro h
+        ext i
+        change σ (g (e i)) = w i
+        rw [show g (e i) = z i from congrArg (fun a : Fin k ↪ α => a i) h, hσ]
+    have hh := mean_equiv (Equiv.mulLeft σ)
+      (fun g : Equiv.Perm α => if e.trans g.toEmbedding = w then (1 : ℝ) else 0)
+    change mean (fun g : Equiv.Perm α => if e.trans ((σ * g).toEmbedding) = w then (1 : ℝ) else 0) = _ at hh
+    dsimp only [μ]
+    simp only [fairMass_eq_mean]
+    simpa only [heq] using hh
+  have hs : ∑ z, μ z = 1 := fairMass_sum _
+  funext z
+  have hz : (Fintype.card (Fin k ↪ α) : ℝ) * μ z = 1 := by
+    have hh : (∑ w, μ w) = (Fintype.card (Fin k ↪ α) : ℝ) * μ z := by
+      simp only [hconst _ z, Finset.sum_const, Finset.card_univ, nsmul_eq_mul]
+    linarith
+  change μ z = _
+  exact (eq_div_iff (by exact_mod_cast Fintype.card_ne_zero (α := Fin k ↪ α))).mpr (by linarith)
+
+theorem uniform_list_mass_frame {α : Type*} [Fintype α] [DecidableEq α] {k : ℕ}
+    (e : Fin k ↪ α) (r : Equiv.Perm α) :
+    fairMass (fun g : Equiv.Perm α => e.trans ((r * g).toEmbedding)) =
+      fun _ => (1 : ℝ) / Fintype.card (Fin k ↪ α) := by
+  classical
+  calc
+    _ = fairMass (fun g : Equiv.Perm α => e.trans g.toEmbedding) := by
+      funext z
+      simp only [fairMass_eq_mean]
+      exact mean_equiv (Equiv.mulLeft r)
+        (fun g : Equiv.Perm α => if e.trans g.toEmbedding = z then (1 : ℝ) else 0)
+    _ = _ := uniform_list_mass e
+
+end FixedLists
+end Thorp
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/FourierBounds.lean b/lean/OAI/Probability/ThorpShuffle/FourierBounds.lean
index a084d7b05605457589f98b9aa485690af92a26cd..f162c672d5b4b388f0ec45268378a02b7fc3a553 100644
--- a/lean/OAI/Probability/ThorpShuffle/FourierBounds.lean
+++ b/lean/OAI/Probability/ThorpShuffle/FourierBounds.lean
@@ -1,172 +1,172 @@
-import OAI.Probability.ThorpShuffle.CosetBounds
-
-namespace OAI
-
-universe uG uE
-
-noncomputable section
-
-open scoped BigOperators ComplexConjugate InnerProductSpace
-open Filter Topology
-
-namespace Thorp.Fourier.Family
-open scoped Classical
-variable {H : Type} [Group H] [Fintype H] (F : Family H)
-
-def reciprocalSum : ℝ := ∑ b, ((F.degree b : ℝ) ^ 2)⁻¹
-
-lemma reciprocalSum_nonneg : 0 ≤ F.reciprocalSum :=
-  Finset.sum_nonneg (fun _ _ => inv_nonneg.mpr (sq_nonneg _))
-
-lemma low_degree_sq (D : ℕ) :
-    (∑ b ∈ Finset.univ.filter (fun b => F.degree b ^ 8 ≤ D), (F.degree b : ℝ)^2)^2 ≤
-      (D : ℝ) * F.reciprocalSum ^ 2 := by
-  let s := Finset.univ.filter (fun b => F.degree b ^ 8 ≤ D)
-  have hpos (b : F.Index) : (0 : ℝ) < F.degree b := by exact_mod_cast F.positive b
-  have hcs := Finset.sum_sq_le_sum_mul_sum_of_sq_le_mul s
-    (r := fun b => (F.degree b : ℝ)^2) (f := fun b => ((F.degree b : ℝ)^2)⁻¹)
-    (g := fun b => (F.degree b : ℝ)^6)
-    (fun b _ => inv_nonneg.mpr (sq_nonneg _)) (fun b _ => pow_nonneg (hpos b).le _)
-    (fun b _ => by apply le_of_eq; field_simp)
-  have hsmall : ∑ b ∈ s, ((F.degree b : ℝ)^2)⁻¹ ≤ F.reciprocalSum :=
-    Finset.sum_le_sum_of_subset_of_nonneg (Finset.filter_subset _ _) (fun b _ _ => inv_nonneg.mpr (sq_nonneg _))
-  have hlarge : ∑ b ∈ s, (F.degree b : ℝ)^6 ≤ (D : ℝ) * F.reciprocalSum := by
-    calc
-      _ ≤ ∑ b ∈ s, (D : ℝ) * ((F.degree b : ℝ)^2)⁻¹ := by
-        apply Finset.sum_le_sum
-        intro b hb
-        have hh : (F.degree b : ℝ)^8 ≤ D := by exact_mod_cast (Finset.mem_filter.mp hb).2
-        apply (le_mul_inv_iff₀ (pow_pos (hpos b) 2)).mpr
-        calc
-          _ = (F.degree b : ℝ)^8 := by ring
-          _ ≤ _ := hh
-      _ = (D : ℝ) * ∑ b ∈ s, ((F.degree b : ℝ)^2)⁻¹ := by rw [Finset.mul_sum]
-      _ ≤ _ := mul_le_mul_of_nonneg_left hsmall (Nat.cast_nonneg _)
-  calc
-    _ ≤ _ := hcs
-    _ ≤ F.reciprocalSum * ((D : ℝ) * F.reciprocalSum) :=
-      mul_le_mul hsmall hlarge (Finset.sum_nonneg (fun b _ => pow_nonneg (hpos b).le _)) F.reciprocalSum_nonneg
-    _ = _ := by ring
-
-end Thorp.Fourier.Family
-
-namespace Thorp.Fourier
-open scoped Classical
-variable {G : Type uG} [Group G] [Fintype G]
-variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℂ E] [FiniteDimensional ℂ E]
-
-noncomputable def hsSq (A : Module.End ℂ E) : ℝ :=
-  ∑ k, ‖A (stdOrthonormalBasis ℂ E k)‖ ^ 2
-
-lemma hsSq_nonneg (A : Module.End ℂ E) : 0 ≤ hsSq A :=
-  Finset.sum_nonneg (fun _ _ => sq_nonneg _)
-
-lemma hsSq_le (A : Module.End ℂ E) (R : ℝ) (hA : ∀ v, ‖A v‖ ^ 2 ≤ R * ‖v‖ ^ 2) :
-    hsSq A ≤ (Module.finrank ℂ E : ℝ) * R := by
-  calc
-    _ ≤ ∑ k, R * ‖stdOrthonormalBasis ℂ E k‖ ^ 2 := Finset.sum_le_sum (fun k _ => hA _)
-    _ = _ := by simp
-
-omit [FiniteDimensional ℂ E] in
-lemma norm_integrated_sq (ρ : Representation ℂ G E)
-    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ) (f : G → ℂ) (v : E) :
-    ((‖integrated ρ f v‖ ^ 2 : ℝ) : ℂ) =
-      ∑ g, ∑ h, conj (f g) * f h * ⟪v, ρ (g⁻¹ * h) v⟫_ℂ := by
-  rw [Complex.ofReal_pow, ← RCLike.ofReal_eq_complex_ofReal, ← inner_self_eq_norm_sq_to_K]
-  simp only [integrated, LinearMap.sum_apply, LinearMap.smul_apply]
-  rw [sum_inner]
-  simp only [inner_sum, inner_smul_left, inner_smul_right]
-  apply Finset.sum_congr rfl
-  intro g _
-  apply Finset.sum_congr rfl
-  intro h _
-  rw [map_mul, Module.End.mul_apply, inner_inv ρ hu]
-  ring
-
-lemma hsSq_integrated (ρ : Representation ℂ G E)
-    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ) (f : G → ℂ) :
-    (hsSq (integrated ρ f) : ℂ) =
-      ∑ g, ∑ h, conj (f g) * f h * ρ.character (g⁻¹ * h) := by
-  simp only [hsSq, Complex.ofReal_sum, norm_integrated_sq ρ hu]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro g _
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro h _
-  rw [← Finset.mul_sum]
-  congr 1
-  exact (LinearMap.trace_eq_sum_inner _ (stdOrthonormalBasis ℂ E)).symm
-
-namespace Family
-variable {G : Type} [Group G] [Fintype G] (F : Family G)
-
-theorem parseval (f : G → ℂ) :
-    (Fintype.card G : ℝ) * ∑ g, ‖f g‖ ^ 2 =
-      ∑ i, (F.degree i : ℝ) * hsSq (integrated (F.rep i) f) := by
-  apply Complex.ofReal_injective
-  push_cast
-  symm
-  simp only [hsSq_integrated _ (F.unitary _), Finset.mul_sum]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro g _
-  rw [Finset.sum_comm]
-  have he (h : G) : ∑ i, (F.degree i : ℂ) *
-      (conj (f g) * f h * (F.rep i).character (g⁻¹ * h)) =
-      conj (f g) * f h * (if g⁻¹ * h = 1 then (Fintype.card G : ℂ) else 0) := by
-    rw [← F.regular, Finset.mul_sum]
-    apply Finset.sum_congr rfl
-    intro i _
-    dsimp [degree]
-    ring
-  simp only [he, inv_mul_eq_one, mul_ite, mul_zero, Finset.sum_ite_eq, Finset.mem_univ,
-    ↓reduceIte, Complex.conj_mul']
-  ring
-
-end Family
-end Thorp.Fourier
-
-namespace Thorp.Fourier
-open scoped Classical
-variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℂ E] [FiniteDimensional ℂ E]
-
-omit [FiniteDimensional ℂ E] in
-lemma power_norm_sq (A : Module.End ℂ E) (R : ℝ) (hR : 0 ≤ R)
-    (hA : ∀ v, ‖A v‖ ^ 2 ≤ R * ‖v‖ ^ 2) (k : ℕ) (v : E) :
-    ‖(A ^ k) v‖ ^ 2 ≤ R ^ k * ‖v‖ ^ 2 := by
-  induction k with
-  | zero => simp
-  | succ k ih =>
-      rw [pow_succ' A k, Module.End.mul_apply, pow_succ' R k]
-      exact (hA _).trans (by nlinarith [mul_le_mul_of_nonneg_left ih hR])
-
-lemma eight_degree_bound (D : ℝ) (hD : 0 < D) (B S C : ℝ)
-    (hs : S ^ 2 ≤ D * C ^ 2) :
-    D ^ 2 * (8 * B * S / D) ^ 8 ≤ (8 * B * C) ^ 8 * (D ^ 2)⁻¹ := by
-  have hh := pow_le_pow_left₀ (sq_nonneg S) hs 4
-  have hscalar : 0 ≤ (8 * B)^8 := by positivity
-  have ht := mul_le_mul_of_nonneg_left hh hscalar
-  apply (le_mul_inv_iff₀ (pow_pos hD 2)).mpr
-  field_simp
-  field_simp at ht
-  nlinarith
-
-namespace Family
-variable {G : Type} [Group G] [Fintype G] (F : Family G)
-
-lemma l1_sq_le_parseval (f : G → ℝ) :
-    (∑ g, |f g|)^2 ≤ ∑ i, (F.degree i : ℝ) * hsSq (integrated (F.rep i) (fun g => (f g : ℂ))) := by
-  have hc := Finset.sum_mul_sq_le_sq_mul_sq Finset.univ (fun _ : G => (1 : ℝ)) (fun g => |f g|)
-  simp only [one_mul, one_pow, Finset.sum_const, Finset.card_univ, nsmul_eq_mul, mul_one,
-    sq_abs] at hc
-  have hp := F.parseval (fun g => (f g : ℂ))
-  simp only [Complex.norm_real, Real.norm_eq_abs, sq_abs] at hp
-  exact hc.trans_eq hp
-
-end Family
-end Thorp.Fourier
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.CosetBounds
+
+namespace OAI
+
+universe uG uE
+
+noncomputable section
+
+open scoped BigOperators ComplexConjugate InnerProductSpace
+open Filter Topology
+
+namespace Thorp.Fourier.Family
+open scoped Classical
+variable {H : Type} [Group H] [Fintype H] (F : Family H)
+
+def reciprocalSum : ℝ := ∑ b, ((F.degree b : ℝ) ^ 2)⁻¹
+
+lemma reciprocalSum_nonneg : 0 ≤ F.reciprocalSum :=
+  Finset.sum_nonneg (fun _ _ => inv_nonneg.mpr (sq_nonneg _))
+
+lemma low_degree_sq (D : ℕ) :
+    (∑ b ∈ Finset.univ.filter (fun b => F.degree b ^ 8 ≤ D), (F.degree b : ℝ)^2)^2 ≤
+      (D : ℝ) * F.reciprocalSum ^ 2 := by
+  let s := Finset.univ.filter (fun b => F.degree b ^ 8 ≤ D)
+  have hpos (b : F.Index) : (0 : ℝ) < F.degree b := by exact_mod_cast F.positive b
+  have hcs := Finset.sum_sq_le_sum_mul_sum_of_sq_le_mul s
+    (r := fun b => (F.degree b : ℝ)^2) (f := fun b => ((F.degree b : ℝ)^2)⁻¹)
+    (g := fun b => (F.degree b : ℝ)^6)
+    (fun b _ => inv_nonneg.mpr (sq_nonneg _)) (fun b _ => pow_nonneg (hpos b).le _)
+    (fun b _ => by apply le_of_eq; field_simp)
+  have hsmall : ∑ b ∈ s, ((F.degree b : ℝ)^2)⁻¹ ≤ F.reciprocalSum :=
+    Finset.sum_le_sum_of_subset_of_nonneg (Finset.filter_subset _ _) (fun b _ _ => inv_nonneg.mpr (sq_nonneg _))
+  have hlarge : ∑ b ∈ s, (F.degree b : ℝ)^6 ≤ (D : ℝ) * F.reciprocalSum := by
+    calc
+      _ ≤ ∑ b ∈ s, (D : ℝ) * ((F.degree b : ℝ)^2)⁻¹ := by
+        apply Finset.sum_le_sum
+        intro b hb
+        have hh : (F.degree b : ℝ)^8 ≤ D := by exact_mod_cast (Finset.mem_filter.mp hb).2
+        apply (le_mul_inv_iff₀ (pow_pos (hpos b) 2)).mpr
+        calc
+          _ = (F.degree b : ℝ)^8 := by ring
+          _ ≤ _ := hh
+      _ = (D : ℝ) * ∑ b ∈ s, ((F.degree b : ℝ)^2)⁻¹ := by rw [Finset.mul_sum]
+      _ ≤ _ := mul_le_mul_of_nonneg_left hsmall (Nat.cast_nonneg _)
+  calc
+    _ ≤ _ := hcs
+    _ ≤ F.reciprocalSum * ((D : ℝ) * F.reciprocalSum) :=
+      mul_le_mul hsmall hlarge (Finset.sum_nonneg (fun b _ => pow_nonneg (hpos b).le _)) F.reciprocalSum_nonneg
+    _ = _ := by ring
+
+end Thorp.Fourier.Family
+
+namespace Thorp.Fourier
+open scoped Classical
+variable {G : Type uG} [Group G] [Fintype G]
+variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℂ E] [FiniteDimensional ℂ E]
+
+noncomputable def hsSq (A : Module.End ℂ E) : ℝ :=
+  ∑ k, ‖A (stdOrthonormalBasis ℂ E k)‖ ^ 2
+
+lemma hsSq_nonneg (A : Module.End ℂ E) : 0 ≤ hsSq A :=
+  Finset.sum_nonneg (fun _ _ => sq_nonneg _)
+
+lemma hsSq_le (A : Module.End ℂ E) (R : ℝ) (hA : ∀ v, ‖A v‖ ^ 2 ≤ R * ‖v‖ ^ 2) :
+    hsSq A ≤ (Module.finrank ℂ E : ℝ) * R := by
+  calc
+    _ ≤ ∑ k, R * ‖stdOrthonormalBasis ℂ E k‖ ^ 2 := Finset.sum_le_sum (fun k _ => hA _)
+    _ = _ := by simp
+
+omit [FiniteDimensional ℂ E] in
+lemma norm_integrated_sq (ρ : Representation ℂ G E)
+    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ) (f : G → ℂ) (v : E) :
+    ((‖integrated ρ f v‖ ^ 2 : ℝ) : ℂ) =
+      ∑ g, ∑ h, conj (f g) * f h * ⟪v, ρ (g⁻¹ * h) v⟫_ℂ := by
+  rw [Complex.ofReal_pow, ← RCLike.ofReal_eq_complex_ofReal, ← inner_self_eq_norm_sq_to_K]
+  simp only [integrated, LinearMap.sum_apply, LinearMap.smul_apply]
+  rw [sum_inner]
+  simp only [inner_sum, inner_smul_left, inner_smul_right]
+  apply Finset.sum_congr rfl
+  intro g _
+  apply Finset.sum_congr rfl
+  intro h _
+  rw [map_mul, Module.End.mul_apply, inner_inv ρ hu]
+  ring
+
+lemma hsSq_integrated (ρ : Representation ℂ G E)
+    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ) (f : G → ℂ) :
+    (hsSq (integrated ρ f) : ℂ) =
+      ∑ g, ∑ h, conj (f g) * f h * ρ.character (g⁻¹ * h) := by
+  simp only [hsSq, Complex.ofReal_sum, norm_integrated_sq ρ hu]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro g _
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro h _
+  rw [← Finset.mul_sum]
+  congr 1
+  exact (LinearMap.trace_eq_sum_inner _ (stdOrthonormalBasis ℂ E)).symm
+
+namespace Family
+variable {G : Type} [Group G] [Fintype G] (F : Family G)
+
+theorem parseval (f : G → ℂ) :
+    (Fintype.card G : ℝ) * ∑ g, ‖f g‖ ^ 2 =
+      ∑ i, (F.degree i : ℝ) * hsSq (integrated (F.rep i) f) := by
+  apply Complex.ofReal_injective
+  push_cast
+  symm
+  simp only [hsSq_integrated _ (F.unitary _), Finset.mul_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro g _
+  rw [Finset.sum_comm]
+  have he (h : G) : ∑ i, (F.degree i : ℂ) *
+      (conj (f g) * f h * (F.rep i).character (g⁻¹ * h)) =
+      conj (f g) * f h * (if g⁻¹ * h = 1 then (Fintype.card G : ℂ) else 0) := by
+    rw [← F.regular, Finset.mul_sum]
+    apply Finset.sum_congr rfl
+    intro i _
+    dsimp [degree]
+    ring
+  simp only [he, inv_mul_eq_one, mul_ite, mul_zero, Finset.sum_ite_eq, Finset.mem_univ,
+    ↓reduceIte, Complex.conj_mul']
+  ring
+
+end Family
+end Thorp.Fourier
+
+namespace Thorp.Fourier
+open scoped Classical
+variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℂ E] [FiniteDimensional ℂ E]
+
+omit [FiniteDimensional ℂ E] in
+lemma power_norm_sq (A : Module.End ℂ E) (R : ℝ) (hR : 0 ≤ R)
+    (hA : ∀ v, ‖A v‖ ^ 2 ≤ R * ‖v‖ ^ 2) (k : ℕ) (v : E) :
+    ‖(A ^ k) v‖ ^ 2 ≤ R ^ k * ‖v‖ ^ 2 := by
+  induction k with
+  | zero => simp
+  | succ k ih =>
+      rw [pow_succ' A k, Module.End.mul_apply, pow_succ' R k]
+      exact (hA _).trans (by nlinarith [mul_le_mul_of_nonneg_left ih hR])
+
+lemma eight_degree_bound (D : ℝ) (hD : 0 < D) (B S C : ℝ)
+    (hs : S ^ 2 ≤ D * C ^ 2) :
+    D ^ 2 * (8 * B * S / D) ^ 8 ≤ (8 * B * C) ^ 8 * (D ^ 2)⁻¹ := by
+  have hh := pow_le_pow_left₀ (sq_nonneg S) hs 4
+  have hscalar : 0 ≤ (8 * B)^8 := by positivity
+  have ht := mul_le_mul_of_nonneg_left hh hscalar
+  apply (le_mul_inv_iff₀ (pow_pos hD 2)).mpr
+  field_simp
+  field_simp at ht
+  nlinarith
+
+namespace Family
+variable {G : Type} [Group G] [Fintype G] (F : Family G)
+
+lemma l1_sq_le_parseval (f : G → ℝ) :
+    (∑ g, |f g|)^2 ≤ ∑ i, (F.degree i : ℝ) * hsSq (integrated (F.rep i) (fun g => (f g : ℂ))) := by
+  have hc := Finset.sum_mul_sq_le_sq_mul_sq Finset.univ (fun _ : G => (1 : ℝ)) (fun g => |f g|)
+  simp only [one_mul, one_pow, Finset.sum_const, Finset.card_univ, nsmul_eq_mul, mul_one,
+    sq_abs] at hc
+  have hp := F.parseval (fun g => (f g : ℂ))
+  simp only [Complex.norm_real, Real.norm_eq_abs, sq_abs] at hp
+  exact hc.trans_eq hp
+
+end Family
+end Thorp.Fourier
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/FourierFamilies.lean b/lean/OAI/Probability/ThorpShuffle/FourierFamilies.lean
index 51f4f5730ffd724cf411b0297dbbeff2b56e3e26..39189a1b7d31bf13e9a53d962b6c6050fc86e226 100644
--- a/lean/OAI/Probability/ThorpShuffle/FourierFamilies.lean
+++ b/lean/OAI/Probability/ThorpShuffle/FourierFamilies.lean
@@ -1,227 +1,227 @@
-import OAI.Probability.ThorpShuffle.Reynolds
-
-namespace OAI
-
-universe uG uE uV
-
-noncomputable section
-
-open scoped BigOperators ComplexConjugate InnerProductSpace
-open Filter Topology
-
-namespace Thorp.Fourier
-open scoped Classical
-
-structure Family (G : Type) [Group G] [Fintype G] where
-  Index : Type
-  [finiteIndex : Fintype Index]
-  Space : Index → Type
-  [normed : ∀ i, NormedAddCommGroup (Space i)]
-  [innerSpace : ∀ i, InnerProductSpace ℂ (Space i)]
-  [finiteDim : ∀ i, FiniteDimensional ℂ (Space i)]
-  rep : ∀ i, Representation ℂ G (Space i)
-  [irreducible : ∀ i, Representation.IsIrreducible (rep i)]
-  positive : ∀ i, 0 < Module.finrank ℂ (Space i)
-  unitary : ∀ i g x y, ⟪rep i g x, rep i g y⟫_ℂ = ⟪x, y⟫_ℂ
-  distinct : ∀ i j, Nonempty (Representation.Equiv (rep i) (rep j)) → i = j
-  regular : ∀ g, ∑ i, (Module.finrank ℂ (Space i) : ℂ) * (rep i).character g =
-    if g = 1 then (Fintype.card G : ℂ) else 0
-
-attribute [instance] Family.finiteIndex Family.normed Family.innerSpace Family.finiteDim Family.irreducible
-
-namespace Family
-variable {G : Type} [Group G] [Fintype G] (F : Family G)
-
-def degree (i : F.Index) : ℕ := Module.finrank ℂ (F.Space i)
-
-lemma orthogonal (i j : F.Index) :
-    (Fintype.card G : ℂ)⁻¹ * ∑ g, (F.rep i).character g * (F.rep j).character g⁻¹ =
-      if i = j then 1 else 0 := by
-  let : Invertible (Nat.card G : ℂ) := invertibleOfNonzero (NeZero.ne _)
-  rw [Fintype.card_eq_nat_card, Representation.char_orthonormal]
-  congr 1
-  apply propext
-  exact ⟨fun h => (F.distinct j i h).symm, fun h => by subst j; exact ⟨Representation.Equiv.refl _⟩⟩
-
-lemma integrated_faithful (f : G → ℂ) (hf : ∀ i, integrated (F.rep i) f = 0) : f = 0 := by
-  funext x
-  have hh : ∑ i, (F.degree i : ℂ) * LinearMap.trace ℂ (F.Space i)
-      (integrated (F.rep i) f * F.rep i x⁻¹) = 0 := by simp [hf]
-  simp only [integrated, Finset.sum_mul, smul_mul_assoc, ← map_mul, map_sum, map_smul, smul_eq_mul] at hh
-  change ∑ i, (F.degree i : ℂ) * ∑ g, f g * (F.rep i).character (g * x⁻¹) = 0 at hh
-  simp_rw [Finset.mul_sum] at hh
-  rw [Finset.sum_comm] at hh
-  have he (g : G) : ∑ i, (F.degree i : ℂ) * (f g * (F.rep i).character (g * x⁻¹)) =
-      f g * (if g * x⁻¹ = 1 then (Fintype.card G : ℂ) else 0) := by
-    rw [← F.regular]
-    simp only [Finset.mul_sum, degree]
-    apply Finset.sum_congr rfl
-    intro i _
-    ring
-  simp only [he, mul_inv_eq_one, mul_ite, mul_zero, Finset.sum_ite_eq', Finset.mem_univ,
-    ↓reduceIte] at hh
-  exact (mul_eq_zero.mp hh).resolve_right (by exact_mod_cast Fintype.card_ne_zero)
-
-lemma integrated_injective (f k : G → ℂ) (h : ∀ i, integrated (F.rep i) f = integrated (F.rep i) k) : f = k := by
-  apply sub_eq_zero.mp
-  apply F.integrated_faithful
-  intro i
-  rw [integrated_sub, h i, sub_self]
-
-def kernel (i : F.Index) (g : G) : ℂ :=
-  (F.degree i : ℂ) * (Fintype.card G : ℂ)⁻¹ * (F.rep i).character g⁻¹
-
-lemma kernel_central (i : F.Index) (g h : G) : F.kernel i (h * g * h⁻¹) = F.kernel i g := by
-  simp only [kernel, mul_inv_rev, inv_inv]
-  rw [← mul_assoc, Representation.char_conj]
-
-lemma integrated_kernel (i j : F.Index) : integrated (F.rep j) (F.kernel i) = if i = j then 1 else 0 := by
-  obtain ⟨c, hc⟩ := central_scalar (F.rep j) (F.kernel i) (F.kernel_central i)
-  have ht := congrArg (LinearMap.trace ℂ (F.Space j)) hc
-  rw [trace_integrated, map_smul, LinearMap.trace_id, smul_eq_mul] at ht
-  have hh : ∑ g, F.kernel i g * (F.rep j).character g =
-      (F.degree i : ℂ) * (if j = i then 1 else 0) := by
-    rw [← F.orthogonal j i]
-    simp only [kernel, Finset.mul_sum]
-    apply Finset.sum_congr rfl
-    intro g _
-    ring
-  rw [hh] at ht
-  have hd : (F.degree j : ℂ) ≠ 0 := by exact_mod_cast (F.positive j).ne'
-  by_cases hij : i = j
-  · subst j
-    simp only [↓reduceIte, mul_one] at ht ⊢
-    have he : c = 1 := (mul_right_cancel₀ hd (ht.symm.trans (one_mul _).symm))
-    simpa only [he, one_smul, Module.End.one_eq_id] using hc
-  · simp only [ite_eq_right (Ne.symm hij), mul_zero] at ht
-    have he : c = 0 := (mul_eq_zero.mp ht.symm).resolve_right hd
-    simp only [ite_eq_right hij, hc, he, zero_smul]
-
-lemma kernel_convolution (i j : F.Index) : convolution (F.kernel i) (F.kernel j) =
-    if i = j then F.kernel i else 0 := by
-  apply F.integrated_injective
-  intro k
-  rw [integrated_convolution, F.integrated_kernel, F.integrated_kernel]
-  by_cases hij : i = j
-  · subst j
-    simp only [↓reduceIte, integrated_kernel]
-    split_ifs <;> simp
-  · rw [ite_eq_right hij]
-    simp only [integrated, Pi.zero_apply, zero_smul, Finset.sum_const_zero]
-    by_cases hik : i = k
-    · subst k; simp [Ne.symm hij]
-    · simp [hik]
-
-lemma kernel_sum : (∑ i, F.kernel i) = delta := by
-  apply F.integrated_injective
-  intro j
-  rw [integrated_sum, integrated_delta]
-  simp only [integrated_kernel, Finset.sum_ite_eq', Finset.mem_univ, ↓reduceIte]
-
-end Family
-end Thorp.Fourier
-
-namespace Thorp.Fourier
-open scoped Classical
-variable {G : Type uG} [Group G] [Fintype G]
-variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℂ E] [FiniteDimensional ℂ E]
-
-omit [Fintype G] in
-lemma character_inv (ρ : Representation ℂ G E)
-    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ) (g : G) :
-    ρ.character g⁻¹ = conj (ρ.character g) := by
-  let b := stdOrthonormalBasis ℂ E
-  simp only [Representation.character, LinearMap.trace_eq_sum_inner _ b, map_sum,
-    inner_inv ρ hu, inner_conj_symm]
-
-omit [FiniteDimensional ℂ E] in
-lemma integrated_symmetric (ρ : Representation ℂ G E)
-    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ)
-    (f : G → ℂ) (hf : ∀ g, conj (f g⁻¹) = f g) : (integrated ρ f).IsSymmetric := by
-  intro x y
-  simp only [integrated, LinearMap.sum_apply, LinearMap.smul_apply, sum_inner,
-    inner_sum, inner_smul_left, inner_smul_right]
-  have he := Equiv.sum_comp (Equiv.inv G) (fun g => conj (f g) * ⟪ρ g x, y⟫_ℂ)
-  rw [← he]
-  apply Finset.sum_congr rfl
-  intro g _
-  simp only [Equiv.inv_apply, hf]
-  rw [← inner_inv ρ hu, inv_inv]
-
-namespace Family
-variable {G : Type} [Group G] [Fintype G] (F : Family G)
-
-lemma kernel_star (i : F.Index) (g : G) : conj (F.kernel i g⁻¹) = F.kernel i g := by
-  simp only [kernel, inv_inv, map_mul, map_inv₀, map_natCast, character_inv _ (F.unitary i)]
-
-variable {V : Type uV} [AddCommGroup V] [Module ℂ V]
-
-def projector (ρ : Representation ℂ G V) (i : F.Index) : Module.End ℂ V := integrated ρ (F.kernel i)
-
-lemma projector_mul (ρ : Representation ℂ G V) (i j : F.Index) :
-    F.projector ρ i * F.projector ρ j = if i = j then F.projector ρ i else 0 := by
-  rw [projector, projector, ← integrated_convolution, F.kernel_convolution]
-  split_ifs <;> simp only [integrated, Pi.zero_apply, zero_smul, Finset.sum_const_zero]
-
-lemma projector_idempotent (ρ : Representation ℂ G V) (i : F.Index) :
-    IsIdempotentElem (F.projector ρ i) := by
-  simpa only [IsIdempotentElem, ↓reduceIte] using F.projector_mul ρ i i
-
-lemma projector_sum (ρ : Representation ℂ G V) : ∑ i, F.projector ρ i = 1 := by
-  simp only [projector]
-  rw [← integrated_sum, F.kernel_sum, integrated_delta]
-
-lemma projector_commute (ρ : Representation ℂ G V) (i : F.Index) (g : G) :
-    Commute (F.projector ρ i) (ρ g) := integrated_central ρ _ (F.kernel_central i) g
-
-omit [FiniteDimensional ℂ E] in
-lemma projector_symmetric (ρ : Representation ℂ G E)
-    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ) (i : F.Index) :
-    (F.projector ρ i).IsSymmetric := integrated_symmetric ρ hu _ (F.kernel_star i)
-
-lemma kernel_sum_sq (s : Finset F.Index) :
-    (Fintype.card G : ℝ) * ∑ g, ‖∑ i ∈ s, F.kernel i g‖ ^ 2 = ∑ i ∈ s, (F.degree i : ℝ) ^ 2 := by
-  have hh (i j : F.Index) : (Fintype.card G : ℂ) * ∑ g, conj (F.kernel i g) * F.kernel j g =
-      if i = j then (F.degree i : ℂ) ^ 2 else 0 := by
-    have hc : ∀ g, conj (F.kernel i g) =
-        (F.degree i : ℂ) * (Fintype.card G : ℂ)⁻¹ * (F.rep i).character g := by
-      intro g
-      calc
-        _ = F.kernel i g⁻¹ := by simpa only [inv_inv] using F.kernel_star i g⁻¹
-        _ = _ := by rw [kernel, inv_inv]
-    simp_rw [hc]
-    simp only [kernel]
-    have he : (Fintype.card G : ℂ) * ∑ g,
-        ((F.degree i : ℂ) * (Fintype.card G : ℂ)⁻¹ * (F.rep i).character g) *
-          ((F.degree j : ℂ) * (Fintype.card G : ℂ)⁻¹ * (F.rep j).character g⁻¹) =
-        (F.degree i : ℂ) * (F.degree j : ℂ) *
-          ((Fintype.card G : ℂ)⁻¹ * ∑ g, (F.rep i).character g * (F.rep j).character g⁻¹) := by
-      rw [Finset.mul_sum, Finset.mul_sum, Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro g _
-      field_simp
-    rw [he, F.orthogonal]
-    split_ifs with h
-    · subst j; ring
-    · ring
-  have he : (Fintype.card G : ℂ) * ∑ g, conj (∑ i ∈ s, F.kernel i g) * (∑ i ∈ s, F.kernel i g) =
-      ∑ i ∈ s, (F.degree i : ℂ) ^ 2 := by
-    simp only [map_sum, Finset.sum_mul, Finset.mul_sum]
-    rw [Finset.sum_comm]
-    apply Finset.sum_congr rfl
-    intro j hj
-    rw [Finset.sum_comm]
-    have hz (i : F.Index) : ∑ g, (Fintype.card G : ℂ) * (conj (F.kernel i g) * F.kernel j g) =
-        if i = j then (F.degree i : ℂ)^2 else 0 := by rw [← Finset.mul_sum, hh]
-    simp only [hz, Finset.sum_ite_eq', hj, ↓reduceIte]
-  simp only [Complex.conj_mul'] at he
-  apply Complex.ofReal_injective
-  push_cast
-  exact he
-
-end Family
-end Thorp.Fourier
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.Reynolds
+
+namespace OAI
+
+universe uG uE uV
+
+noncomputable section
+
+open scoped BigOperators ComplexConjugate InnerProductSpace
+open Filter Topology
+
+namespace Thorp.Fourier
+open scoped Classical
+
+structure Family (G : Type) [Group G] [Fintype G] where
+  Index : Type
+  [finiteIndex : Fintype Index]
+  Space : Index → Type
+  [normed : ∀ i, NormedAddCommGroup (Space i)]
+  [innerSpace : ∀ i, InnerProductSpace ℂ (Space i)]
+  [finiteDim : ∀ i, FiniteDimensional ℂ (Space i)]
+  rep : ∀ i, Representation ℂ G (Space i)
+  [irreducible : ∀ i, Representation.IsIrreducible (rep i)]
+  positive : ∀ i, 0 < Module.finrank ℂ (Space i)
+  unitary : ∀ i g x y, ⟪rep i g x, rep i g y⟫_ℂ = ⟪x, y⟫_ℂ
+  distinct : ∀ i j, Nonempty (Representation.Equiv (rep i) (rep j)) → i = j
+  regular : ∀ g, ∑ i, (Module.finrank ℂ (Space i) : ℂ) * (rep i).character g =
+    if g = 1 then (Fintype.card G : ℂ) else 0
+
+attribute [instance] Family.finiteIndex Family.normed Family.innerSpace Family.finiteDim Family.irreducible
+
+namespace Family
+variable {G : Type} [Group G] [Fintype G] (F : Family G)
+
+def degree (i : F.Index) : ℕ := Module.finrank ℂ (F.Space i)
+
+lemma orthogonal (i j : F.Index) :
+    (Fintype.card G : ℂ)⁻¹ * ∑ g, (F.rep i).character g * (F.rep j).character g⁻¹ =
+      if i = j then 1 else 0 := by
+  let : Invertible (Nat.card G : ℂ) := invertibleOfNonzero (NeZero.ne _)
+  rw [Fintype.card_eq_nat_card, Representation.char_orthonormal]
+  congr 1
+  apply propext
+  exact ⟨fun h => (F.distinct j i h).symm, fun h => by subst j; exact ⟨Representation.Equiv.refl _⟩⟩
+
+lemma integrated_faithful (f : G → ℂ) (hf : ∀ i, integrated (F.rep i) f = 0) : f = 0 := by
+  funext x
+  have hh : ∑ i, (F.degree i : ℂ) * LinearMap.trace ℂ (F.Space i)
+      (integrated (F.rep i) f * F.rep i x⁻¹) = 0 := by simp [hf]
+  simp only [integrated, Finset.sum_mul, smul_mul_assoc, ← map_mul, map_sum, map_smul, smul_eq_mul] at hh
+  change ∑ i, (F.degree i : ℂ) * ∑ g, f g * (F.rep i).character (g * x⁻¹) = 0 at hh
+  simp_rw [Finset.mul_sum] at hh
+  rw [Finset.sum_comm] at hh
+  have he (g : G) : ∑ i, (F.degree i : ℂ) * (f g * (F.rep i).character (g * x⁻¹)) =
+      f g * (if g * x⁻¹ = 1 then (Fintype.card G : ℂ) else 0) := by
+    rw [← F.regular]
+    simp only [Finset.mul_sum, degree]
+    apply Finset.sum_congr rfl
+    intro i _
+    ring
+  simp only [he, mul_inv_eq_one, mul_ite, mul_zero, Finset.sum_ite_eq', Finset.mem_univ,
+    ↓reduceIte] at hh
+  exact (mul_eq_zero.mp hh).resolve_right (by exact_mod_cast Fintype.card_ne_zero)
+
+lemma integrated_injective (f k : G → ℂ) (h : ∀ i, integrated (F.rep i) f = integrated (F.rep i) k) : f = k := by
+  apply sub_eq_zero.mp
+  apply F.integrated_faithful
+  intro i
+  rw [integrated_sub, h i, sub_self]
+
+def kernel (i : F.Index) (g : G) : ℂ :=
+  (F.degree i : ℂ) * (Fintype.card G : ℂ)⁻¹ * (F.rep i).character g⁻¹
+
+lemma kernel_central (i : F.Index) (g h : G) : F.kernel i (h * g * h⁻¹) = F.kernel i g := by
+  simp only [kernel, mul_inv_rev, inv_inv]
+  rw [← mul_assoc, Representation.char_conj]
+
+lemma integrated_kernel (i j : F.Index) : integrated (F.rep j) (F.kernel i) = if i = j then 1 else 0 := by
+  obtain ⟨c, hc⟩ := central_scalar (F.rep j) (F.kernel i) (F.kernel_central i)
+  have ht := congrArg (LinearMap.trace ℂ (F.Space j)) hc
+  rw [trace_integrated, map_smul, LinearMap.trace_id, smul_eq_mul] at ht
+  have hh : ∑ g, F.kernel i g * (F.rep j).character g =
+      (F.degree i : ℂ) * (if j = i then 1 else 0) := by
+    rw [← F.orthogonal j i]
+    simp only [kernel, Finset.mul_sum]
+    apply Finset.sum_congr rfl
+    intro g _
+    ring
+  rw [hh] at ht
+  have hd : (F.degree j : ℂ) ≠ 0 := by exact_mod_cast (F.positive j).ne'
+  by_cases hij : i = j
+  · subst j
+    simp only [↓reduceIte, mul_one] at ht ⊢
+    have he : c = 1 := (mul_right_cancel₀ hd (ht.symm.trans (one_mul _).symm))
+    simpa only [he, one_smul, Module.End.one_eq_id] using hc
+  · simp only [ite_eq_right (Ne.symm hij), mul_zero] at ht
+    have he : c = 0 := (mul_eq_zero.mp ht.symm).resolve_right hd
+    simp only [ite_eq_right hij, hc, he, zero_smul]
+
+lemma kernel_convolution (i j : F.Index) : convolution (F.kernel i) (F.kernel j) =
+    if i = j then F.kernel i else 0 := by
+  apply F.integrated_injective
+  intro k
+  rw [integrated_convolution, F.integrated_kernel, F.integrated_kernel]
+  by_cases hij : i = j
+  · subst j
+    simp only [↓reduceIte, integrated_kernel]
+    split_ifs <;> simp
+  · rw [ite_eq_right hij]
+    simp only [integrated, Pi.zero_apply, zero_smul, Finset.sum_const_zero]
+    by_cases hik : i = k
+    · subst k; simp [Ne.symm hij]
+    · simp [hik]
+
+lemma kernel_sum : (∑ i, F.kernel i) = delta := by
+  apply F.integrated_injective
+  intro j
+  rw [integrated_sum, integrated_delta]
+  simp only [integrated_kernel, Finset.sum_ite_eq', Finset.mem_univ, ↓reduceIte]
+
+end Family
+end Thorp.Fourier
+
+namespace Thorp.Fourier
+open scoped Classical
+variable {G : Type uG} [Group G] [Fintype G]
+variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℂ E] [FiniteDimensional ℂ E]
+
+omit [Fintype G] in
+lemma character_inv (ρ : Representation ℂ G E)
+    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ) (g : G) :
+    ρ.character g⁻¹ = conj (ρ.character g) := by
+  let b := stdOrthonormalBasis ℂ E
+  simp only [Representation.character, LinearMap.trace_eq_sum_inner _ b, map_sum,
+    inner_inv ρ hu, inner_conj_symm]
+
+omit [FiniteDimensional ℂ E] in
+lemma integrated_symmetric (ρ : Representation ℂ G E)
+    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ)
+    (f : G → ℂ) (hf : ∀ g, conj (f g⁻¹) = f g) : (integrated ρ f).IsSymmetric := by
+  intro x y
+  simp only [integrated, LinearMap.sum_apply, LinearMap.smul_apply, sum_inner,
+    inner_sum, inner_smul_left, inner_smul_right]
+  have he := Equiv.sum_comp (Equiv.inv G) (fun g => conj (f g) * ⟪ρ g x, y⟫_ℂ)
+  rw [← he]
+  apply Finset.sum_congr rfl
+  intro g _
+  simp only [Equiv.inv_apply, hf]
+  rw [← inner_inv ρ hu, inv_inv]
+
+namespace Family
+variable {G : Type} [Group G] [Fintype G] (F : Family G)
+
+lemma kernel_star (i : F.Index) (g : G) : conj (F.kernel i g⁻¹) = F.kernel i g := by
+  simp only [kernel, inv_inv, map_mul, map_inv₀, map_natCast, character_inv _ (F.unitary i)]
+
+variable {V : Type uV} [AddCommGroup V] [Module ℂ V]
+
+def projector (ρ : Representation ℂ G V) (i : F.Index) : Module.End ℂ V := integrated ρ (F.kernel i)
+
+lemma projector_mul (ρ : Representation ℂ G V) (i j : F.Index) :
+    F.projector ρ i * F.projector ρ j = if i = j then F.projector ρ i else 0 := by
+  rw [projector, projector, ← integrated_convolution, F.kernel_convolution]
+  split_ifs <;> simp only [integrated, Pi.zero_apply, zero_smul, Finset.sum_const_zero]
+
+lemma projector_idempotent (ρ : Representation ℂ G V) (i : F.Index) :
+    IsIdempotentElem (F.projector ρ i) := by
+  simpa only [IsIdempotentElem, ↓reduceIte] using F.projector_mul ρ i i
+
+lemma projector_sum (ρ : Representation ℂ G V) : ∑ i, F.projector ρ i = 1 := by
+  simp only [projector]
+  rw [← integrated_sum, F.kernel_sum, integrated_delta]
+
+lemma projector_commute (ρ : Representation ℂ G V) (i : F.Index) (g : G) :
+    Commute (F.projector ρ i) (ρ g) := integrated_central ρ _ (F.kernel_central i) g
+
+omit [FiniteDimensional ℂ E] in
+lemma projector_symmetric (ρ : Representation ℂ G E)
+    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ) (i : F.Index) :
+    (F.projector ρ i).IsSymmetric := integrated_symmetric ρ hu _ (F.kernel_star i)
+
+lemma kernel_sum_sq (s : Finset F.Index) :
+    (Fintype.card G : ℝ) * ∑ g, ‖∑ i ∈ s, F.kernel i g‖ ^ 2 = ∑ i ∈ s, (F.degree i : ℝ) ^ 2 := by
+  have hh (i j : F.Index) : (Fintype.card G : ℂ) * ∑ g, conj (F.kernel i g) * F.kernel j g =
+      if i = j then (F.degree i : ℂ) ^ 2 else 0 := by
+    have hc : ∀ g, conj (F.kernel i g) =
+        (F.degree i : ℂ) * (Fintype.card G : ℂ)⁻¹ * (F.rep i).character g := by
+      intro g
+      calc
+        _ = F.kernel i g⁻¹ := by simpa only [inv_inv] using F.kernel_star i g⁻¹
+        _ = _ := by rw [kernel, inv_inv]
+    simp_rw [hc]
+    simp only [kernel]
+    have he : (Fintype.card G : ℂ) * ∑ g,
+        ((F.degree i : ℂ) * (Fintype.card G : ℂ)⁻¹ * (F.rep i).character g) *
+          ((F.degree j : ℂ) * (Fintype.card G : ℂ)⁻¹ * (F.rep j).character g⁻¹) =
+        (F.degree i : ℂ) * (F.degree j : ℂ) *
+          ((Fintype.card G : ℂ)⁻¹ * ∑ g, (F.rep i).character g * (F.rep j).character g⁻¹) := by
+      rw [Finset.mul_sum, Finset.mul_sum, Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro g _
+      field_simp
+    rw [he, F.orthogonal]
+    split_ifs with h
+    · subst j; ring
+    · ring
+  have he : (Fintype.card G : ℂ) * ∑ g, conj (∑ i ∈ s, F.kernel i g) * (∑ i ∈ s, F.kernel i g) =
+      ∑ i ∈ s, (F.degree i : ℂ) ^ 2 := by
+    simp only [map_sum, Finset.sum_mul, Finset.mul_sum]
+    rw [Finset.sum_comm]
+    apply Finset.sum_congr rfl
+    intro j hj
+    rw [Finset.sum_comm]
+    have hz (i : F.Index) : ∑ g, (Fintype.card G : ℂ) * (conj (F.kernel i g) * F.kernel j g) =
+        if i = j then (F.degree i : ℂ)^2 else 0 := by rw [← Finset.mul_sum, hh]
+    simp only [hz, Finset.sum_ite_eq', hj, ↓reduceIte]
+  simp only [Complex.conj_mul'] at he
+  apply Complex.ofReal_injective
+  push_cast
+  exact he
+
+end Family
+end Thorp.Fourier
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/FourierPerturbation.lean b/lean/OAI/Probability/ThorpShuffle/FourierPerturbation.lean
index fa606f589b97e9409c39e82f0bbe2634e3c1cfb6..5be4a08f8dfdbe9bfaa8ed81bc3467af195e676c 100644
--- a/lean/OAI/Probability/ThorpShuffle/FourierPerturbation.lean
+++ b/lean/OAI/Probability/ThorpShuffle/FourierPerturbation.lean
@@ -1,92 +1,92 @@
-import OAI.Probability.ThorpShuffle.Trimming
-
-namespace OAI
-
-noncomputable section
-
-open scoped BigOperators ComplexConjugate InnerProductSpace
-open Filter Topology
-
-namespace Thorp
-
-namespace Fourier
-open scoped Classical
-variable {G : Type} [Fintype G] [Group G]
-
-omit [Group G] in
-lemma tv_nonneg (μ ν : G → ℝ) : 0 ≤ tv μ ν := by unfold tv; positivity
-
-omit [Group G] in
-lemma tv_triangle (μ ν η : G → ℝ) : tv μ η ≤ tv μ ν + tv ν η := by
-  unfold tv
-  rw [← mul_add, ← Finset.sum_add_distrib]
-  apply mul_le_mul_of_nonneg_left _ (by norm_num)
-  exact Finset.sum_le_sum (fun g _ => by simpa only [sub_add_sub_cancel] using abs_add_le (μ g-ν g) (ν g-η g))
-
-lemma tv_realConv_left (μ ν η : G → ℝ) (hμ : ∀ g, 0 ≤ μ g) (h1 : ∑ g, μ g = 1) :
-    tv (realConv μ ν) (realConv μ η) ≤ tv ν η := by
-  have hb : (∑ z, |realConv μ ν z - realConv μ η z|) ≤ ∑ z, |ν z-η z| := by
-    calc
-      _ = ∑ z, |∑ g, μ g * (ν (g⁻¹*z)-η (g⁻¹*z))| := by
-        simp only [realConv, mul_sub, Finset.sum_sub_distrib]
-      _ ≤ ∑ z, ∑ g, μ g * |ν (g⁻¹*z)-η (g⁻¹*z)| := by
-        apply Finset.sum_le_sum
-        intro z _
-        simpa only [abs_mul, abs_of_nonneg (hμ _)] using
-          Finset.abs_sum_le_sum_abs (fun g => μ g * (ν (g⁻¹*z)-η (g⁻¹*z))) Finset.univ
-      _ = (∑ g, μ g) * ∑ z, |ν z-η z| := by
-        rw [Finset.sum_comm, Finset.sum_mul]
-        apply Finset.sum_congr rfl
-        intro g _
-        rw [← Finset.mul_sum]
-        congr 1
-        exact Equiv.sum_comp (Equiv.mulLeft g⁻¹) (fun z => |ν z-η z|)
-      _ = _ := by rw [h1, one_mul]
-  exact mul_le_mul_of_nonneg_left hb (by norm_num : (0:ℝ) ≤ 1/2)
-
-lemma tv_realConv_right (μ ν η : G → ℝ) (hη : ∀ g, 0 ≤ η g) (h1 : ∑ g, η g = 1) :
-    tv (realConv μ η) (realConv ν η) ≤ tv μ ν := by
-  have hb : (∑ z, |realConv μ η z - realConv ν η z|) ≤ ∑ z, |μ z-ν z| := by
-    calc
-      _ = ∑ z, |∑ g, (μ g-ν g) * η (g⁻¹*z)| := by
-        simp only [realConv, sub_mul, Finset.sum_sub_distrib]
-      _ ≤ ∑ z, ∑ g, |μ g-ν g| * η (g⁻¹*z) := by
-        apply Finset.sum_le_sum
-        intro z _
-        simpa only [abs_mul, abs_of_nonneg (hη _)] using
-          Finset.abs_sum_le_sum_abs (fun g => (μ g-ν g) * η (g⁻¹*z)) Finset.univ
-      _ = (∑ g, |μ g-ν g|) * ∑ z, η z := by
-        rw [Finset.sum_comm, Finset.sum_mul]
-        apply Finset.sum_congr rfl
-        intro g _
-        rw [← Finset.mul_sum]
-        congr 1
-        exact Equiv.sum_comp (Equiv.mulLeft g⁻¹) η
-      _ = _ := by rw [h1, mul_one]
-  exact mul_le_mul_of_nonneg_left hb (by norm_num : (0:ℝ) ≤ 1/2)
-
-lemma tv_realConv (μ ν μ' ν' : G → ℝ) (hμ' : ∀ g, 0 ≤ μ' g) (hμ1 : ∑ g, μ' g = 1)
-    (hν : ∀ g, 0 ≤ ν g) (hν1 : ∑ g, ν g = 1) :
-    tv (realConv μ ν) (realConv μ' ν') ≤ tv μ μ' + tv ν ν' := by
-  exact (tv_triangle _ (realConv μ' ν) _).trans
-    (add_le_add (tv_realConv_right μ μ' ν hν hν1) (tv_realConv_left μ' ν ν' hμ' hμ1))
-
-lemma tv_realPower (μ ν : G → ℝ) (hμ : ∀ g, 0 ≤ μ g) (hμ1 : ∑ g, μ g = 1)
-    (hν : ∀ g, 0 ≤ ν g) (hν1 : ∑ g, ν g = 1) (k : ℕ) :
-    tv (realPower μ k) (realPower ν k) ≤ (k : ℝ) * tv μ ν := by
-  induction k with
-  | zero => simp [realPower, tv]
-  | succ k ih =>
-      calc
-        _ ≤ tv μ ν + tv (realPower μ k) (realPower ν k) :=
-          tv_realConv μ (realPower μ k) ν (realPower ν k) hν hν1 (realPower_nonneg μ hμ k) (realPower_sum μ hμ1 k)
-        _ ≤ tv μ ν + (k : ℝ) * tv μ ν := add_le_add_right ih _
-        _ = _ := by push_cast; ring
-
-end Fourier
-
-end Thorp
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.Trimming
+
+namespace OAI
+
+noncomputable section
+
+open scoped BigOperators ComplexConjugate InnerProductSpace
+open Filter Topology
+
+namespace Thorp
+
+namespace Fourier
+open scoped Classical
+variable {G : Type} [Fintype G] [Group G]
+
+omit [Group G] in
+lemma tv_nonneg (μ ν : G → ℝ) : 0 ≤ tv μ ν := by unfold tv; positivity
+
+omit [Group G] in
+lemma tv_triangle (μ ν η : G → ℝ) : tv μ η ≤ tv μ ν + tv ν η := by
+  unfold tv
+  rw [← mul_add, ← Finset.sum_add_distrib]
+  apply mul_le_mul_of_nonneg_left _ (by norm_num)
+  exact Finset.sum_le_sum (fun g _ => by simpa only [sub_add_sub_cancel] using abs_add_le (μ g-ν g) (ν g-η g))
+
+lemma tv_realConv_left (μ ν η : G → ℝ) (hμ : ∀ g, 0 ≤ μ g) (h1 : ∑ g, μ g = 1) :
+    tv (realConv μ ν) (realConv μ η) ≤ tv ν η := by
+  have hb : (∑ z, |realConv μ ν z - realConv μ η z|) ≤ ∑ z, |ν z-η z| := by
+    calc
+      _ = ∑ z, |∑ g, μ g * (ν (g⁻¹*z)-η (g⁻¹*z))| := by
+        simp only [realConv, mul_sub, Finset.sum_sub_distrib]
+      _ ≤ ∑ z, ∑ g, μ g * |ν (g⁻¹*z)-η (g⁻¹*z)| := by
+        apply Finset.sum_le_sum
+        intro z _
+        simpa only [abs_mul, abs_of_nonneg (hμ _)] using
+          Finset.abs_sum_le_sum_abs (fun g => μ g * (ν (g⁻¹*z)-η (g⁻¹*z))) Finset.univ
+      _ = (∑ g, μ g) * ∑ z, |ν z-η z| := by
+        rw [Finset.sum_comm, Finset.sum_mul]
+        apply Finset.sum_congr rfl
+        intro g _
+        rw [← Finset.mul_sum]
+        congr 1
+        exact Equiv.sum_comp (Equiv.mulLeft g⁻¹) (fun z => |ν z-η z|)
+      _ = _ := by rw [h1, one_mul]
+  exact mul_le_mul_of_nonneg_left hb (by norm_num : (0:ℝ) ≤ 1/2)
+
+lemma tv_realConv_right (μ ν η : G → ℝ) (hη : ∀ g, 0 ≤ η g) (h1 : ∑ g, η g = 1) :
+    tv (realConv μ η) (realConv ν η) ≤ tv μ ν := by
+  have hb : (∑ z, |realConv μ η z - realConv ν η z|) ≤ ∑ z, |μ z-ν z| := by
+    calc
+      _ = ∑ z, |∑ g, (μ g-ν g) * η (g⁻¹*z)| := by
+        simp only [realConv, sub_mul, Finset.sum_sub_distrib]
+      _ ≤ ∑ z, ∑ g, |μ g-ν g| * η (g⁻¹*z) := by
+        apply Finset.sum_le_sum
+        intro z _
+        simpa only [abs_mul, abs_of_nonneg (hη _)] using
+          Finset.abs_sum_le_sum_abs (fun g => (μ g-ν g) * η (g⁻¹*z)) Finset.univ
+      _ = (∑ g, |μ g-ν g|) * ∑ z, η z := by
+        rw [Finset.sum_comm, Finset.sum_mul]
+        apply Finset.sum_congr rfl
+        intro g _
+        rw [← Finset.mul_sum]
+        congr 1
+        exact Equiv.sum_comp (Equiv.mulLeft g⁻¹) η
+      _ = _ := by rw [h1, mul_one]
+  exact mul_le_mul_of_nonneg_left hb (by norm_num : (0:ℝ) ≤ 1/2)
+
+lemma tv_realConv (μ ν μ' ν' : G → ℝ) (hμ' : ∀ g, 0 ≤ μ' g) (hμ1 : ∑ g, μ' g = 1)
+    (hν : ∀ g, 0 ≤ ν g) (hν1 : ∑ g, ν g = 1) :
+    tv (realConv μ ν) (realConv μ' ν') ≤ tv μ μ' + tv ν ν' := by
+  exact (tv_triangle _ (realConv μ' ν) _).trans
+    (add_le_add (tv_realConv_right μ μ' ν hν hν1) (tv_realConv_left μ' ν ν' hμ' hμ1))
+
+lemma tv_realPower (μ ν : G → ℝ) (hμ : ∀ g, 0 ≤ μ g) (hμ1 : ∑ g, μ g = 1)
+    (hν : ∀ g, 0 ≤ ν g) (hν1 : ∑ g, ν g = 1) (k : ℕ) :
+    tv (realPower μ k) (realPower ν k) ≤ (k : ℝ) * tv μ ν := by
+  induction k with
+  | zero => simp [realPower, tv]
+  | succ k ih =>
+      calc
+        _ ≤ tv μ ν + tv (realPower μ k) (realPower ν k) :=
+          tv_realConv μ (realPower μ k) ν (realPower ν k) hν hν1 (realPower_nonneg μ hμ k) (realPower_sum μ hμ1 k)
+        _ ≤ tv μ ν + (k : ℝ) * tv μ ν := add_le_add_right ih _
+        _ = _ := by push_cast; ring
+
+end Fourier
+
+end Thorp
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/FreshCoins.lean b/lean/OAI/Probability/ThorpShuffle/FreshCoins.lean
index 3e2f5d769d40e5cdcdcefd4b844220321127bfbd..fb58ab88f6546ced8810e4c84421fde25feac7ea 100644
--- a/lean/OAI/Probability/ThorpShuffle/FreshCoins.lean
+++ b/lean/OAI/Probability/ThorpShuffle/FreshCoins.lean
@@ -1,175 +1,175 @@
-import OAI.Probability.ThorpShuffle.WalshFlow
-
-namespace OAI
-
-universe uΩ uι uα
-
-noncomputable section
-
-open scoped BigOperators
-open Filter
-
-namespace Thorp
-
-namespace Conditional
-
-def expectedBand (d : ℕ) (v : CenteredState (d + 1)) (t j : ℕ) : ℝ :=
-  mean (fun ω : History (d + 1) t =>
-    bandEnergy (d + 1) (iterateState d v t ω).weight j / 2 ^ (d + 1))
-
-def expectedNoise (d : ℕ) (v : CenteredState (d + 1)) (t : ℕ) : ℝ :=
-  mean (fun ω : History (d + 1) t =>
-    oneFreeEnergy d (iterateState d v t ω).free (iterateState d v t ω).weight)
-
-theorem mean_div_const {Ω : Type uΩ} [Fintype Ω] (f : Ω → ℝ) (a : ℝ) :
-    mean (fun ω => f ω / a) = mean f / a := by
-  simpa only [div_eq_mul_inv] using mean_mul_const f a⁻¹
-
-@[simp] theorem expectedBand_zero (d : ℕ) (v : CenteredState (d + 1)) (t : ℕ) :
-    expectedBand d v t 0 = expectedEnergy d v t := by
-  unfold expectedBand expectedEnergy
-  apply mean_congr
-  intro ω
-  rw [bandEnergy_zero]
-  simp [Fintype.card_pi, Fintype.card_bool, Fintype.card_fin]
-
-@[simp] theorem expectedBand_full (d : ℕ) (v : CenteredState (d + 1)) (t : ℕ) :
-    expectedBand d v t (d + 1) = 0 := by
-  unfold expectedBand
-  have hz (ω : History (d + 1) t) :
-      bandEnergy (d + 1) (iterateState d v t ω).weight (d + 1) / 2 ^ (d + 1) = 0 := by
-    rw [bandEnergy_full, (iterateState d v t ω).mass_zero]
-    simp
-  simp_rw [hz]
-  exact mean_zero
-
-theorem two_pow_ratio (d j : ℕ) (hj : j ≤ d) :
-    (2 : ℝ) ^ (d - j) / 2 ^ (d + 1) = (1 / 2 : ℝ) ^ (j + 1) := by
-  have hd : d + 1 = (d - j) + (j + 1) := by omega
-  rw [hd, pow_add, div_mul_eq_div_div, div_self (by positivity), div_pow, one_pow]
-
-theorem expectedBand_succ (d : ℕ) (v : CenteredState (d + 1)) (t j : ℕ)
-    (hj : j < d + 1) :
-    expectedBand d v (t + 1) j = expectedBand d v t (j + 1) +
-      (1 / 2 : ℝ) ^ (j + 1) * expectedNoise d v t := by
-  unfold expectedBand expectedNoise
-  rw [mean_history_succ]
-  have hstep (ω : History (d + 1) t) :
-      mean (fun c : Coins (d + 1) =>
-        bandEnergy (d + 1) (iterateState d v (t + 1) (Fin.snoc ω c)).weight j /
-          2 ^ (d + 1)) =
-      bandEnergy (d + 1) (iterateState d v t ω).weight (j + 1) / 2 ^ (d + 1) +
-        (1 / 2 : ℝ) ^ (j + 1) *
-          oneFreeEnergy d (iterateState d v t ω).free (iterateState d v t ω).weight := by
-    simp only [iterateState, Fin.snoc_last, Fin.snoc_castSucc, nextState]
-    rw [mean_div_const, mean_bandEnergy_physicalFlow d j (by omega)
-      _ _ (iterateState d v t ω).supported, add_div]
-    rw [← div_mul_eq_mul_div, two_pow_ratio d j (by omega)]
-  simp_rw [hstep]
-  rw [mean_add, mean_const_mul]
-
-theorem expectedEnergy_recurrence (d : ℕ) (v : CenteredState (d + 1))
-    (t : ℕ) (ht : d + 1 ≤ t) :
-    expectedEnergy d v t = ∑ j ∈ Finset.range (d + 1),
-      (1 / 2 : ℝ) ^ (j + 1) * expectedNoise d v (t - 1 - j) := by
-  simpa only [expectedBand_zero] using
-    energy_of_spectral_tracks (d + 1) (expectedBand d v) (expectedNoise d v)
-      (expectedBand_succ d v) (expectedBand_full d v) t ht
-
-end Conditional
-
-namespace Conditional
-
-theorem mean_product_bits {ι : Type uι} [Fintype ι] [DecidableEq ι] (f : ι → Bool → ℝ) :
-    mean (fun c : ι → Bool => ∏ i, f i (c i)) =
-      ∏ i, (f i false + f i true) / 2 := by
-  classical
-  unfold mean
-  rw [← Fintype.prod_sum]
-  simp only [Fintype.sum_bool, add_comm (f _ true)]
-  rw [Finset.prod_div_distrib]
-  simp [Fintype.card_pi, Fintype.card_bool]
-
-def pairFreeCount {α : Type uα} [Fintype α] (B : Bool × α → Bool) : ℕ :=
-  ∑ i, ((B (false, i)).toNat + (B (true, i)).toNat)
-
-def pairHalfCount {α : Type uα} [Fintype α] (B : Bool × α → Bool)
-    (c : α → Bool) (h : Bool) : ℕ :=
-  ∑ i, (B (h ^^ c i, i)).toNat
-
-theorem local_balance_factor (u v h : Bool) :
-    mean (fun c : Bool => (5 / 4 : ℝ) ^ (u.toNat + v.toNat) *
-      (4 / 5 : ℝ) ^ (4 * (if h ^^ c then v else u).toNat)) ≤
-        (9 / 10 : ℝ) ^ (u.toNat + v.toNat) := by
-  cases u <;> cases v <;> cases h <;> norm_num [mean, Fintype.sum_bool]
-
-theorem balance_exponential_mean {α : Type uα} [Fintype α] [DecidableEq α]
-    (B : Bool × α → Bool) (h : Bool) :
-    mean (fun c : α → Bool => (5 / 4 : ℝ) ^ pairFreeCount B *
-      (4 / 5 : ℝ) ^ (4 * pairHalfCount B c h)) ≤
-        (9 / 10 : ℝ) ^ pairFreeCount B := by
-  classical
-  have hprod (c : α → Bool) :
-      (5 / 4 : ℝ) ^ pairFreeCount B * (4 / 5 : ℝ) ^ (4 * pairHalfCount B c h) =
-      ∏ i, ((5 / 4 : ℝ) ^ ((B (false, i)).toNat + (B (true, i)).toNat) *
-        (4 / 5 : ℝ) ^ (4 * (B (h ^^ c i, i)).toNat)) := by
-    simp only [pairFreeCount, pairHalfCount, Finset.mul_sum, ← Finset.prod_pow_eq_pow_sum,
-      Finset.prod_mul_distrib]
-  simp_rw [hprod]
-  rw [mean_product_bits (fun i (c : Bool) =>
-    (5 / 4 : ℝ) ^ ((B (false, i)).toNat + (B (true, i)).toNat) *
-        (4 / 5 : ℝ) ^ (4 * (B (h ^^ c, i)).toNat))]
-  rw [pairFreeCount, ← Finset.prod_pow_eq_pow_sum]
-  apply Finset.prod_le_prod₀
-  · intro i _
-    positivity
-  · intro i _
-    have hlocal := local_balance_factor (B (false, i)) (B (true, i)) h
-    have hbit (c : Bool) : B (h ^^ c, i) = if h ^^ c then B (true, i) else B (false, i) := by
-      cases (h ^^ c) <;> rfl
-    simp only [hbit]
-    cases h <;> simpa [mean, Fintype.sum_bool, add_comm] using hlocal
-
-theorem pair_half_balance {α : Type uα} [Fintype α] [DecidableEq α]
-    (B : Bool × α → Bool) (h : Bool) :
-    mean (fun c : α → Bool =>
-      if 4 * pairHalfCount B c h < pairFreeCount B then (1 : ℝ) else 0) ≤
-        (9 / 10 : ℝ) ^ pairFreeCount B := by
-  apply le_trans (mean_le_mean (fun c => ?_)) (balance_exponential_mean B h)
-  by_cases hc : 4 * pairHalfCount B c h < pairFreeCount B
-  · rw [ite_eq_left hc]
-    have hp : (5 / 4 : ℝ) ^ (4 * pairHalfCount B c h) ≤
-        (5 / 4 : ℝ) ^ pairFreeCount B :=
-      pow_le_pow_right₀ (by norm_num) hc.le
-    have hr : (4 / 5 : ℝ) = (5 / 4 : ℝ)⁻¹ := by norm_num
-    rw [hr, inv_pow, ← div_eq_mul_inv]
-    exact (le_div_iff₀ (by positivity)).mpr (by simpa using hp)
-  · rw [ite_eq_right hc]
-    positivity
-
-theorem pair_both_half_balance {α : Type uα} [Fintype α] [DecidableEq α]
-    (B : Bool × α → Bool) :
-    mean (fun c : α → Bool =>
-      if (4 * pairHalfCount B c false < pairFreeCount B) ∨
-        (4 * pairHalfCount B c true < pairFreeCount B) then (1 : ℝ) else 0) ≤
-          2 * (9 / 10 : ℝ) ^ pairFreeCount B := by
-  have hm : mean (fun c : α → Bool =>
-      if (4 * pairHalfCount B c false < pairFreeCount B) ∨
-        (4 * pairHalfCount B c true < pairFreeCount B) then (1 : ℝ) else 0) ≤
-      mean (fun c : α → Bool =>
-        (if 4 * pairHalfCount B c false < pairFreeCount B then (1 : ℝ) else 0) +
-        (if 4 * pairHalfCount B c true < pairFreeCount B then (1 : ℝ) else 0)) := by
-    apply mean_le_mean
-    intro c
-    split_ifs <;> norm_num at *
-    all_goals omega
-  rw [mean_add] at hm
-  linarith [pair_half_balance B false, pair_half_balance B true]
-
-end Conditional
-
-end Thorp
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.WalshFlow
+
+namespace OAI
+
+universe uΩ uι uα
+
+noncomputable section
+
+open scoped BigOperators
+open Filter
+
+namespace Thorp
+
+namespace Conditional
+
+def expectedBand (d : ℕ) (v : CenteredState (d + 1)) (t j : ℕ) : ℝ :=
+  mean (fun ω : History (d + 1) t =>
+    bandEnergy (d + 1) (iterateState d v t ω).weight j / 2 ^ (d + 1))
+
+def expectedNoise (d : ℕ) (v : CenteredState (d + 1)) (t : ℕ) : ℝ :=
+  mean (fun ω : History (d + 1) t =>
+    oneFreeEnergy d (iterateState d v t ω).free (iterateState d v t ω).weight)
+
+theorem mean_div_const {Ω : Type uΩ} [Fintype Ω] (f : Ω → ℝ) (a : ℝ) :
+    mean (fun ω => f ω / a) = mean f / a := by
+  simpa only [div_eq_mul_inv] using mean_mul_const f a⁻¹
+
+@[simp] theorem expectedBand_zero (d : ℕ) (v : CenteredState (d + 1)) (t : ℕ) :
+    expectedBand d v t 0 = expectedEnergy d v t := by
+  unfold expectedBand expectedEnergy
+  apply mean_congr
+  intro ω
+  rw [bandEnergy_zero]
+  simp [Fintype.card_pi, Fintype.card_bool, Fintype.card_fin]
+
+@[simp] theorem expectedBand_full (d : ℕ) (v : CenteredState (d + 1)) (t : ℕ) :
+    expectedBand d v t (d + 1) = 0 := by
+  unfold expectedBand
+  have hz (ω : History (d + 1) t) :
+      bandEnergy (d + 1) (iterateState d v t ω).weight (d + 1) / 2 ^ (d + 1) = 0 := by
+    rw [bandEnergy_full, (iterateState d v t ω).mass_zero]
+    simp
+  simp_rw [hz]
+  exact mean_zero
+
+theorem two_pow_ratio (d j : ℕ) (hj : j ≤ d) :
+    (2 : ℝ) ^ (d - j) / 2 ^ (d + 1) = (1 / 2 : ℝ) ^ (j + 1) := by
+  have hd : d + 1 = (d - j) + (j + 1) := by omega
+  rw [hd, pow_add, div_mul_eq_div_div, div_self (by positivity), div_pow, one_pow]
+
+theorem expectedBand_succ (d : ℕ) (v : CenteredState (d + 1)) (t j : ℕ)
+    (hj : j < d + 1) :
+    expectedBand d v (t + 1) j = expectedBand d v t (j + 1) +
+      (1 / 2 : ℝ) ^ (j + 1) * expectedNoise d v t := by
+  unfold expectedBand expectedNoise
+  rw [mean_history_succ]
+  have hstep (ω : History (d + 1) t) :
+      mean (fun c : Coins (d + 1) =>
+        bandEnergy (d + 1) (iterateState d v (t + 1) (Fin.snoc ω c)).weight j /
+          2 ^ (d + 1)) =
+      bandEnergy (d + 1) (iterateState d v t ω).weight (j + 1) / 2 ^ (d + 1) +
+        (1 / 2 : ℝ) ^ (j + 1) *
+          oneFreeEnergy d (iterateState d v t ω).free (iterateState d v t ω).weight := by
+    simp only [iterateState, Fin.snoc_last, Fin.snoc_castSucc, nextState]
+    rw [mean_div_const, mean_bandEnergy_physicalFlow d j (by omega)
+      _ _ (iterateState d v t ω).supported, add_div]
+    rw [← div_mul_eq_mul_div, two_pow_ratio d j (by omega)]
+  simp_rw [hstep]
+  rw [mean_add, mean_const_mul]
+
+theorem expectedEnergy_recurrence (d : ℕ) (v : CenteredState (d + 1))
+    (t : ℕ) (ht : d + 1 ≤ t) :
+    expectedEnergy d v t = ∑ j ∈ Finset.range (d + 1),
+      (1 / 2 : ℝ) ^ (j + 1) * expectedNoise d v (t - 1 - j) := by
+  simpa only [expectedBand_zero] using
+    energy_of_spectral_tracks (d + 1) (expectedBand d v) (expectedNoise d v)
+      (expectedBand_succ d v) (expectedBand_full d v) t ht
+
+end Conditional
+
+namespace Conditional
+
+theorem mean_product_bits {ι : Type uι} [Fintype ι] [DecidableEq ι] (f : ι → Bool → ℝ) :
+    mean (fun c : ι → Bool => ∏ i, f i (c i)) =
+      ∏ i, (f i false + f i true) / 2 := by
+  classical
+  unfold mean
+  rw [← Fintype.prod_sum]
+  simp only [Fintype.sum_bool, add_comm (f _ true)]
+  rw [Finset.prod_div_distrib]
+  simp [Fintype.card_pi, Fintype.card_bool]
+
+def pairFreeCount {α : Type uα} [Fintype α] (B : Bool × α → Bool) : ℕ :=
+  ∑ i, ((B (false, i)).toNat + (B (true, i)).toNat)
+
+def pairHalfCount {α : Type uα} [Fintype α] (B : Bool × α → Bool)
+    (c : α → Bool) (h : Bool) : ℕ :=
+  ∑ i, (B (h ^^ c i, i)).toNat
+
+theorem local_balance_factor (u v h : Bool) :
+    mean (fun c : Bool => (5 / 4 : ℝ) ^ (u.toNat + v.toNat) *
+      (4 / 5 : ℝ) ^ (4 * (if h ^^ c then v else u).toNat)) ≤
+        (9 / 10 : ℝ) ^ (u.toNat + v.toNat) := by
+  cases u <;> cases v <;> cases h <;> norm_num [mean, Fintype.sum_bool]
+
+theorem balance_exponential_mean {α : Type uα} [Fintype α] [DecidableEq α]
+    (B : Bool × α → Bool) (h : Bool) :
+    mean (fun c : α → Bool => (5 / 4 : ℝ) ^ pairFreeCount B *
+      (4 / 5 : ℝ) ^ (4 * pairHalfCount B c h)) ≤
+        (9 / 10 : ℝ) ^ pairFreeCount B := by
+  classical
+  have hprod (c : α → Bool) :
+      (5 / 4 : ℝ) ^ pairFreeCount B * (4 / 5 : ℝ) ^ (4 * pairHalfCount B c h) =
+      ∏ i, ((5 / 4 : ℝ) ^ ((B (false, i)).toNat + (B (true, i)).toNat) *
+        (4 / 5 : ℝ) ^ (4 * (B (h ^^ c i, i)).toNat)) := by
+    simp only [pairFreeCount, pairHalfCount, Finset.mul_sum, ← Finset.prod_pow_eq_pow_sum,
+      Finset.prod_mul_distrib]
+  simp_rw [hprod]
+  rw [mean_product_bits (fun i (c : Bool) =>
+    (5 / 4 : ℝ) ^ ((B (false, i)).toNat + (B (true, i)).toNat) *
+        (4 / 5 : ℝ) ^ (4 * (B (h ^^ c, i)).toNat))]
+  rw [pairFreeCount, ← Finset.prod_pow_eq_pow_sum]
+  apply Finset.prod_le_prod₀
+  · intro i _
+    positivity
+  · intro i _
+    have hlocal := local_balance_factor (B (false, i)) (B (true, i)) h
+    have hbit (c : Bool) : B (h ^^ c, i) = if h ^^ c then B (true, i) else B (false, i) := by
+      cases (h ^^ c) <;> rfl
+    simp only [hbit]
+    cases h <;> simpa [mean, Fintype.sum_bool, add_comm] using hlocal
+
+theorem pair_half_balance {α : Type uα} [Fintype α] [DecidableEq α]
+    (B : Bool × α → Bool) (h : Bool) :
+    mean (fun c : α → Bool =>
+      if 4 * pairHalfCount B c h < pairFreeCount B then (1 : ℝ) else 0) ≤
+        (9 / 10 : ℝ) ^ pairFreeCount B := by
+  apply le_trans (mean_le_mean (fun c => ?_)) (balance_exponential_mean B h)
+  by_cases hc : 4 * pairHalfCount B c h < pairFreeCount B
+  · rw [ite_eq_left hc]
+    have hp : (5 / 4 : ℝ) ^ (4 * pairHalfCount B c h) ≤
+        (5 / 4 : ℝ) ^ pairFreeCount B :=
+      pow_le_pow_right₀ (by norm_num) hc.le
+    have hr : (4 / 5 : ℝ) = (5 / 4 : ℝ)⁻¹ := by norm_num
+    rw [hr, inv_pow, ← div_eq_mul_inv]
+    exact (le_div_iff₀ (by positivity)).mpr (by simpa using hp)
+  · rw [ite_eq_right hc]
+    positivity
+
+theorem pair_both_half_balance {α : Type uα} [Fintype α] [DecidableEq α]
+    (B : Bool × α → Bool) :
+    mean (fun c : α → Bool =>
+      if (4 * pairHalfCount B c false < pairFreeCount B) ∨
+        (4 * pairHalfCount B c true < pairFreeCount B) then (1 : ℝ) else 0) ≤
+          2 * (9 / 10 : ℝ) ^ pairFreeCount B := by
+  have hm : mean (fun c : α → Bool =>
+      if (4 * pairHalfCount B c false < pairFreeCount B) ∨
+        (4 * pairHalfCount B c true < pairFreeCount B) then (1 : ℝ) else 0) ≤
+      mean (fun c : α → Bool =>
+        (if 4 * pairHalfCount B c false < pairFreeCount B then (1 : ℝ) else 0) +
+        (if 4 * pairHalfCount B c true < pairFreeCount B then (1 : ℝ) else 0)) := by
+    apply mean_le_mean
+    intro c
+    split_ifs <;> norm_num at *
+    all_goals omega
+  rw [mean_add] at hm
+  linarith [pair_half_balance B false, pair_half_balance B true]
+
+end Conditional
+
+end Thorp
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/HalfHistories.lean b/lean/OAI/Probability/ThorpShuffle/HalfHistories.lean
index f442d88234344f690c54a65b4d500dd55090d191..0f0ee21f065a9b549a77da73112cd17ced43e902 100644
--- a/lean/OAI/Probability/ThorpShuffle/HalfHistories.lean
+++ b/lean/OAI/Probability/ThorpShuffle/HalfHistories.lean
@@ -1,225 +1,225 @@
-import OAI.Probability.ThorpShuffle.FreshCoins
-
-namespace OAI
-
-universe uα
-
-noncomputable section
-
-open scoped BigOperators
-open Filter
-
-namespace Thorp
-
-namespace Conditional
-
-structure RawState (d : ℕ) where
-  free : Position d → Bool
-  weight : Position d → ℝ
-
-def rawNext (d : ℕ) (v : RawState (d + 1)) (c : Coins (d + 1)) : RawState (d + 1) where
-  free := physicalFree d v.free c
-  weight := physicalFlow d v.free v.weight c
-
-def rawIterate (d : ℕ) (v : RawState (d + 1)) :
-    (t : ℕ) → History (d + 1) t → RawState (d + 1)
-  | 0, _ => v
-  | t + 1, ω => rawNext d (rawIterate d v t (fun i => ω i.castSucc)) (ω (Fin.last t))
-
-def CenteredState.raw {d : ℕ} (v : CenteredState d) : RawState d := ⟨v.free, v.weight⟩
-
-theorem iterateState_raw (d : ℕ) (v : CenteredState (d + 1)) (t : ℕ)
-    (ω : History (d + 1) t) :
-    (iterateState d v t ω).raw = rawIterate d v.raw t ω := by
-  induction t with
-  | zero => rfl
-  | succ t ih =>
-    change rawNext d (iterateState d v t (fun i => ω i.castSucc)).raw (ω (Fin.last t)) = _
-    rw [ih]
-    rfl
-
-theorem rawIterate_free (d : ℕ) (v : RawState (d + 1)) (t : ℕ)
-    (ω : History (d + 1) t) (x : Position (d + 1)) :
-    (rawIterate d v t ω).free x = v.free ((run (d + 1) t ω).symm x) := by
-  induction t generalizing x with
-  | zero => simp only [rawIterate, run_zero]; rfl
-  | succ t ih =>
-    change (rawIterate d v t (fun i => ω i.castSucc)).free
-      ((step (d + 1) (ω (Fin.last t))).symm x) = _
-    rw [ih, run_succ]
-    rfl
-
-theorem mean_bool (f : Bool → ℝ) : mean f = (f false + f true) / 2 := by
-  simp [mean, add_comm]
-
-theorem pairUpdate_mean_fst (u v : Bool) (a b : ℝ) :
-    mean (fun c : Bool => (pairUpdate u v a b c).1) = (a + b) / 2 := by
-  cases u <;> cases v <;> simp [mean_bool, pairUpdate]
-
-theorem pairUpdate_mean_snd (u v : Bool) (a b : ℝ) :
-    mean (fun c : Bool => (pairUpdate u v a b c).2) = (a + b) / 2 := by
-  cases u <;> cases v <;> simp [mean_bool, pairUpdate]
-  all_goals ring
-
-theorem mean_sign {α : Type uα} [Fintype α] [DecidableEq α] (i : α) :
-    mean (fun c : α → Bool => sign (c i)) = 0 := by
-  simp [mean, sum_sign]
-
-theorem mean_coin_eval {α : Type uα} [Fintype α] [DecidableEq α]
-    (i : α) (f : Bool → ℝ) : mean (fun c : α → Bool => f (c i)) = mean f := by
-  have hfun (b : Bool) : f b = (f false + f true) / 2 +
-      sign b * ((f false - f true) / 2) := by
-    cases b <;> simp [sign] <;> ring
-  calc
-    _ = mean (fun c : α → Bool => (f false + f true) / 2 +
-        sign (c i) * ((f false - f true) / 2)) := mean_congr (fun c => hfun (c i))
-    _ = mean f := by
-      rw [mean_add, mean_const, mean_mul_const]
-      simp only [mean_sign, zero_mul, add_zero, mean_bool]
-
-theorem mean_physicalFlow_snoc (d : ℕ) (B : Position (d + 1) → Bool)
-    (w : Position (d + 1) → ℝ) (x : Position d) (b : Bool) :
-    mean (fun c : Coins (d + 1) => physicalFlow d B w c (Fin.snoc x b)) =
-      (w (Fin.cons false x) + w (Fin.cons true x)) / 2 := by
-  rw [← scatterPosition_eq_snoc]
-  simp only [physicalFlow_scatter]
-  cases b
-  · change mean (fun c : Position d → Bool =>
-      (pairUpdate (B (Fin.cons false x)) (B (Fin.cons true x))
-        (w (Fin.cons false x)) (w (Fin.cons true x)) (c x)).1) = _
-    rw [mean_coin_eval x (fun c => (pairUpdate (B (Fin.cons false x))
-      (B (Fin.cons true x)) (w (Fin.cons false x)) (w (Fin.cons true x)) c).1),
-      pairUpdate_mean_fst]
-  · change mean (fun c : Position d → Bool =>
-      (pairUpdate (B (Fin.cons false x)) (B (Fin.cons true x))
-        (w (Fin.cons false x)) (w (Fin.cons true x)) (c x)).2) = _
-    rw [mean_coin_eval x (fun c => (pairUpdate (B (Fin.cons false x))
-      (B (Fin.cons true x)) (w (Fin.cons false x)) (w (Fin.cons true x)) c).2),
-      pairUpdate_mean_snd]
-
-end Conditional
-
-namespace Conditional
-
-def firstMean (d : ℕ) (w : Position (d + 1) → ℝ) (x : Position d) : ℝ :=
-  mean (fun b : Bool => w (Fin.cons b x))
-
-def meanLayer : (d : ℕ) → (Position d → ℝ) → Position d → ℝ
-  | 0, w => w
-  | d + 1, w => fun x => firstMean d w (Fin.init x)
-
-theorem mean_physicalFlow (d : ℕ) (B : Position (d + 1) → Bool)
-    (w : Position (d + 1) → ℝ) (x : Position (d + 1)) :
-    mean (fun c : Coins (d + 1) => physicalFlow d B w c x) =
-      meanLayer (d + 1) w x := by
-  conv_lhs => rw [← Fin.snoc_init_self x]
-  rw [mean_physicalFlow_snoc]
-  change _ = mean (fun b : Bool => w (Fin.cons b (Fin.init x)))
-  exact (mean_bool (fun b : Bool => w (Fin.cons b (Fin.init x)))).symm
-
-theorem init_cons (d : ℕ) (b : Bool) (x : Position (d + 1)) :
-    (Fin.init (Fin.cons b x : Position (d + 2)) : Position (d + 1)) =
-      (Fin.cons b (Fin.init x) : Position (d + 1)) := by
-  funext i
-  refine Fin.cases ?_ (fun j => ?_) i <;> rfl
-
-theorem meanLayer_lift (d : ℕ) (w : Position d → ℝ) :
-    meanLayer (d + 1) (fun x => w (Fin.init x)) =
-      fun x => meanLayer d w (Fin.init x) := by
-  funext x
-  cases d with
-  | zero =>
-    have hall (b : Bool) : Fin.init (Fin.cons b (Fin.init x) : Position 1) = Fin.init x :=
-      Subsingleton.elim _ _
-    simp only [meanLayer, firstMean, hall, mean_const]
-  | succ d =>
-    simp only [meanLayer, firstMean, init_cons]
-
-theorem mean_iterations_dimension (d t : ℕ) (w : Position (d + 1) → ℝ)
-    (x : Position (d + 1)) :
-    (meanLayer (d + 1))^[t + 1] w x =
-      (meanLayer d)^[t] (firstMean d w) (Fin.init x) := by
-  have hs : Function.Semiconj (fun w : Position d → ℝ => fun x : Position (d + 1) => w (Fin.init x))
-      (meanLayer d) (meanLayer (d + 1)) := fun w => (meanLayer_lift d w).symm
-  rw [Function.iterate_succ_apply]
-  exact (congrFun (hs.iterate_right t (firstMean d w)) x).symm
-
-theorem mean_firstMean (d : ℕ) (w : Position (d + 1) → ℝ) :
-    mean (firstMean d w) = mean w := by
-  have h := mean_equiv (splitPosition d).symm w
-  rw [mean_prod] at h
-  rw [← h, mean_comm]
-  rfl
-
-theorem mean_iterations_sweep (d : ℕ) (w : Position d → ℝ) (x : Position d) :
-    (meanLayer d)^[d] w x = mean w := by
-  induction d with
-  | zero =>
-    change w x = mean w
-    have hw : w = fun _ => w x := funext (fun y => congrArg w (Subsingleton.elim y x))
-    rw [hw, mean_const]
-  | succ d ih =>
-    rw [mean_iterations_dimension, ih, mean_firstMean]
-
-theorem mean_rawIterate_weight (d : ℕ) (v : RawState (d + 1)) (t : ℕ)
-    (x : Position (d + 1)) :
-    mean (fun ω : History (d + 1) t => (rawIterate d v t ω).weight x) =
-      (meanLayer (d + 1))^[t] v.weight x := by
-  induction t generalizing x with
-  | zero => simp [rawIterate, mean_const]
-  | succ t ih =>
-    rw [mean_history_succ]
-    have hi (ω : History (d + 1) t) :
-        mean (fun c : Coins (d + 1) =>
-          (rawIterate d v (t + 1) (Fin.snoc ω c)).weight x) =
-            meanLayer (d + 1) (rawIterate d v t ω).weight x := by
-      simp only [rawIterate, Fin.snoc_castSucc, Fin.snoc_last, rawNext]
-      exact mean_physicalFlow _ _ _ _
-    simp_rw [hi]
-    change mean (fun ω : History (d + 1) t =>
-      mean (fun b : Bool => (rawIterate d v t ω).weight (Fin.cons b (Fin.init x)))) = _
-    rw [mean_comm]
-    simp_rw [ih]
-    rw [Function.iterate_succ_apply']
-    rfl
-
-theorem mean_rawIterate_sweep (d : ℕ) (v : RawState (d + 1))
-    (x : Position (d + 1)) :
-    mean (fun ω : History (d + 1) (d + 1) => (rawIterate d v (d + 1) ω).weight x) =
-      mean v.weight := by
-  rw [mean_rawIterate_weight, mean_iterations_sweep]
-
-theorem rawIterate_indicator (d : ℕ) (B : Position (d + 1) → Bool) (t : ℕ)
-    (ω : History (d + 1) t) (x : Position (d + 1)) :
-    (rawIterate d ⟨B, fun y => if B y then 1 else 0⟩ t ω).weight x =
-      if (rawIterate d ⟨B, fun y => if B y then 1 else 0⟩ t ω).free x then 1 else 0 := by
-  induction t generalizing x with
-  | zero => rfl
-  | succ t ih =>
-    change physicalFlow d _ (rawIterate d _ t (fun i => ω i.castSucc)).weight
-      (ω (Fin.last t)) x = _
-    rw [funext (ih (fun i => ω i.castSucc))]
-    exact physicalFlow_uniform _ _ _ _ _
-
-theorem mean_free_after_sweep (d : ℕ) (B : Position (d + 1) → Bool)
-    (x : Position (d + 1)) :
-    mean (fun ω : History (d + 1) (d + 1) =>
-      if B ((run (d + 1) (d + 1) ω).symm x) then (1 : ℝ) else 0) =
-        (freeCount B : ℝ) / Fintype.card (Position (d + 1)) := by
-  let v : RawState (d + 1) := ⟨B, fun y => if B y then 1 else 0⟩
-  have h (ω : History (d + 1) (d + 1)) :
-      (if B ((run (d + 1) (d + 1) ω).symm x) then (1 : ℝ) else 0) =
-        (rawIterate d v (d + 1) ω).weight x := by
-    rw [rawIterate_indicator, rawIterate_free]
-  simp_rw [h]
-  rw [mean_rawIterate_sweep]
-  change (∑ y, if B y then (1 : ℝ) else 0) / _ = _
-  rw [sum_free_indicator]
-
-end Conditional
-
-end Thorp
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.FreshCoins
+
+namespace OAI
+
+universe uα
+
+noncomputable section
+
+open scoped BigOperators
+open Filter
+
+namespace Thorp
+
+namespace Conditional
+
+structure RawState (d : ℕ) where
+  free : Position d → Bool
+  weight : Position d → ℝ
+
+def rawNext (d : ℕ) (v : RawState (d + 1)) (c : Coins (d + 1)) : RawState (d + 1) where
+  free := physicalFree d v.free c
+  weight := physicalFlow d v.free v.weight c
+
+def rawIterate (d : ℕ) (v : RawState (d + 1)) :
+    (t : ℕ) → History (d + 1) t → RawState (d + 1)
+  | 0, _ => v
+  | t + 1, ω => rawNext d (rawIterate d v t (fun i => ω i.castSucc)) (ω (Fin.last t))
+
+def CenteredState.raw {d : ℕ} (v : CenteredState d) : RawState d := ⟨v.free, v.weight⟩
+
+theorem iterateState_raw (d : ℕ) (v : CenteredState (d + 1)) (t : ℕ)
+    (ω : History (d + 1) t) :
+    (iterateState d v t ω).raw = rawIterate d v.raw t ω := by
+  induction t with
+  | zero => rfl
+  | succ t ih =>
+    change rawNext d (iterateState d v t (fun i => ω i.castSucc)).raw (ω (Fin.last t)) = _
+    rw [ih]
+    rfl
+
+theorem rawIterate_free (d : ℕ) (v : RawState (d + 1)) (t : ℕ)
+    (ω : History (d + 1) t) (x : Position (d + 1)) :
+    (rawIterate d v t ω).free x = v.free ((run (d + 1) t ω).symm x) := by
+  induction t generalizing x with
+  | zero => simp only [rawIterate, run_zero]; rfl
+  | succ t ih =>
+    change (rawIterate d v t (fun i => ω i.castSucc)).free
+      ((step (d + 1) (ω (Fin.last t))).symm x) = _
+    rw [ih, run_succ]
+    rfl
+
+theorem mean_bool (f : Bool → ℝ) : mean f = (f false + f true) / 2 := by
+  simp [mean, add_comm]
+
+theorem pairUpdate_mean_fst (u v : Bool) (a b : ℝ) :
+    mean (fun c : Bool => (pairUpdate u v a b c).1) = (a + b) / 2 := by
+  cases u <;> cases v <;> simp [mean_bool, pairUpdate]
+
+theorem pairUpdate_mean_snd (u v : Bool) (a b : ℝ) :
+    mean (fun c : Bool => (pairUpdate u v a b c).2) = (a + b) / 2 := by
+  cases u <;> cases v <;> simp [mean_bool, pairUpdate]
+  all_goals ring
+
+theorem mean_sign {α : Type uα} [Fintype α] [DecidableEq α] (i : α) :
+    mean (fun c : α → Bool => sign (c i)) = 0 := by
+  simp [mean, sum_sign]
+
+theorem mean_coin_eval {α : Type uα} [Fintype α] [DecidableEq α]
+    (i : α) (f : Bool → ℝ) : mean (fun c : α → Bool => f (c i)) = mean f := by
+  have hfun (b : Bool) : f b = (f false + f true) / 2 +
+      sign b * ((f false - f true) / 2) := by
+    cases b <;> simp [sign] <;> ring
+  calc
+    _ = mean (fun c : α → Bool => (f false + f true) / 2 +
+        sign (c i) * ((f false - f true) / 2)) := mean_congr (fun c => hfun (c i))
+    _ = mean f := by
+      rw [mean_add, mean_const, mean_mul_const]
+      simp only [mean_sign, zero_mul, add_zero, mean_bool]
+
+theorem mean_physicalFlow_snoc (d : ℕ) (B : Position (d + 1) → Bool)
+    (w : Position (d + 1) → ℝ) (x : Position d) (b : Bool) :
+    mean (fun c : Coins (d + 1) => physicalFlow d B w c (Fin.snoc x b)) =
+      (w (Fin.cons false x) + w (Fin.cons true x)) / 2 := by
+  rw [← scatterPosition_eq_snoc]
+  simp only [physicalFlow_scatter]
+  cases b
+  · change mean (fun c : Position d → Bool =>
+      (pairUpdate (B (Fin.cons false x)) (B (Fin.cons true x))
+        (w (Fin.cons false x)) (w (Fin.cons true x)) (c x)).1) = _
+    rw [mean_coin_eval x (fun c => (pairUpdate (B (Fin.cons false x))
+      (B (Fin.cons true x)) (w (Fin.cons false x)) (w (Fin.cons true x)) c).1),
+      pairUpdate_mean_fst]
+  · change mean (fun c : Position d → Bool =>
+      (pairUpdate (B (Fin.cons false x)) (B (Fin.cons true x))
+        (w (Fin.cons false x)) (w (Fin.cons true x)) (c x)).2) = _
+    rw [mean_coin_eval x (fun c => (pairUpdate (B (Fin.cons false x))
+      (B (Fin.cons true x)) (w (Fin.cons false x)) (w (Fin.cons true x)) c).2),
+      pairUpdate_mean_snd]
+
+end Conditional
+
+namespace Conditional
+
+def firstMean (d : ℕ) (w : Position (d + 1) → ℝ) (x : Position d) : ℝ :=
+  mean (fun b : Bool => w (Fin.cons b x))
+
+def meanLayer : (d : ℕ) → (Position d → ℝ) → Position d → ℝ
+  | 0, w => w
+  | d + 1, w => fun x => firstMean d w (Fin.init x)
+
+theorem mean_physicalFlow (d : ℕ) (B : Position (d + 1) → Bool)
+    (w : Position (d + 1) → ℝ) (x : Position (d + 1)) :
+    mean (fun c : Coins (d + 1) => physicalFlow d B w c x) =
+      meanLayer (d + 1) w x := by
+  conv_lhs => rw [← Fin.snoc_init_self x]
+  rw [mean_physicalFlow_snoc]
+  change _ = mean (fun b : Bool => w (Fin.cons b (Fin.init x)))
+  exact (mean_bool (fun b : Bool => w (Fin.cons b (Fin.init x)))).symm
+
+theorem init_cons (d : ℕ) (b : Bool) (x : Position (d + 1)) :
+    (Fin.init (Fin.cons b x : Position (d + 2)) : Position (d + 1)) =
+      (Fin.cons b (Fin.init x) : Position (d + 1)) := by
+  funext i
+  refine Fin.cases ?_ (fun j => ?_) i <;> rfl
+
+theorem meanLayer_lift (d : ℕ) (w : Position d → ℝ) :
+    meanLayer (d + 1) (fun x => w (Fin.init x)) =
+      fun x => meanLayer d w (Fin.init x) := by
+  funext x
+  cases d with
+  | zero =>
+    have hall (b : Bool) : Fin.init (Fin.cons b (Fin.init x) : Position 1) = Fin.init x :=
+      Subsingleton.elim _ _
+    simp only [meanLayer, firstMean, hall, mean_const]
+  | succ d =>
+    simp only [meanLayer, firstMean, init_cons]
+
+theorem mean_iterations_dimension (d t : ℕ) (w : Position (d + 1) → ℝ)
+    (x : Position (d + 1)) :
+    (meanLayer (d + 1))^[t + 1] w x =
+      (meanLayer d)^[t] (firstMean d w) (Fin.init x) := by
+  have hs : Function.Semiconj (fun w : Position d → ℝ => fun x : Position (d + 1) => w (Fin.init x))
+      (meanLayer d) (meanLayer (d + 1)) := fun w => (meanLayer_lift d w).symm
+  rw [Function.iterate_succ_apply]
+  exact (congrFun (hs.iterate_right t (firstMean d w)) x).symm
+
+theorem mean_firstMean (d : ℕ) (w : Position (d + 1) → ℝ) :
+    mean (firstMean d w) = mean w := by
+  have h := mean_equiv (splitPosition d).symm w
+  rw [mean_prod] at h
+  rw [← h, mean_comm]
+  rfl
+
+theorem mean_iterations_sweep (d : ℕ) (w : Position d → ℝ) (x : Position d) :
+    (meanLayer d)^[d] w x = mean w := by
+  induction d with
+  | zero =>
+    change w x = mean w
+    have hw : w = fun _ => w x := funext (fun y => congrArg w (Subsingleton.elim y x))
+    rw [hw, mean_const]
+  | succ d ih =>
+    rw [mean_iterations_dimension, ih, mean_firstMean]
+
+theorem mean_rawIterate_weight (d : ℕ) (v : RawState (d + 1)) (t : ℕ)
+    (x : Position (d + 1)) :
+    mean (fun ω : History (d + 1) t => (rawIterate d v t ω).weight x) =
+      (meanLayer (d + 1))^[t] v.weight x := by
+  induction t generalizing x with
+  | zero => simp [rawIterate, mean_const]
+  | succ t ih =>
+    rw [mean_history_succ]
+    have hi (ω : History (d + 1) t) :
+        mean (fun c : Coins (d + 1) =>
+          (rawIterate d v (t + 1) (Fin.snoc ω c)).weight x) =
+            meanLayer (d + 1) (rawIterate d v t ω).weight x := by
+      simp only [rawIterate, Fin.snoc_castSucc, Fin.snoc_last, rawNext]
+      exact mean_physicalFlow _ _ _ _
+    simp_rw [hi]
+    change mean (fun ω : History (d + 1) t =>
+      mean (fun b : Bool => (rawIterate d v t ω).weight (Fin.cons b (Fin.init x)))) = _
+    rw [mean_comm]
+    simp_rw [ih]
+    rw [Function.iterate_succ_apply']
+    rfl
+
+theorem mean_rawIterate_sweep (d : ℕ) (v : RawState (d + 1))
+    (x : Position (d + 1)) :
+    mean (fun ω : History (d + 1) (d + 1) => (rawIterate d v (d + 1) ω).weight x) =
+      mean v.weight := by
+  rw [mean_rawIterate_weight, mean_iterations_sweep]
+
+theorem rawIterate_indicator (d : ℕ) (B : Position (d + 1) → Bool) (t : ℕ)
+    (ω : History (d + 1) t) (x : Position (d + 1)) :
+    (rawIterate d ⟨B, fun y => if B y then 1 else 0⟩ t ω).weight x =
+      if (rawIterate d ⟨B, fun y => if B y then 1 else 0⟩ t ω).free x then 1 else 0 := by
+  induction t generalizing x with
+  | zero => rfl
+  | succ t ih =>
+    change physicalFlow d _ (rawIterate d _ t (fun i => ω i.castSucc)).weight
+      (ω (Fin.last t)) x = _
+    rw [funext (ih (fun i => ω i.castSucc))]
+    exact physicalFlow_uniform _ _ _ _ _
+
+theorem mean_free_after_sweep (d : ℕ) (B : Position (d + 1) → Bool)
+    (x : Position (d + 1)) :
+    mean (fun ω : History (d + 1) (d + 1) =>
+      if B ((run (d + 1) (d + 1) ω).symm x) then (1 : ℝ) else 0) =
+        (freeCount B : ℝ) / Fintype.card (Position (d + 1)) := by
+  let v : RawState (d + 1) := ⟨B, fun y => if B y then 1 else 0⟩
+  have h (ω : History (d + 1) (d + 1)) :
+      (if B ((run (d + 1) (d + 1) ω).symm x) then (1 : ℝ) else 0) =
+        (rawIterate d v (d + 1) ω).weight x := by
+    rw [rawIterate_indicator, rawIterate_free]
+  simp_rw [h]
+  rw [mean_rawIterate_sweep]
+  change (∑ y, if B y then (1 : ℝ) else 0) / _ = _
+  rw [sum_free_indicator]
+
+end Conditional
+
+end Thorp
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/Irreducibility.lean b/lean/OAI/Probability/ThorpShuffle/Irreducibility.lean
index 3885d1c845dbed16569bad7da4dff15ac5afc138..5333473ef82eacc00d65b2b2fb6d28baf6cb9503 100644
--- a/lean/OAI/Probability/ThorpShuffle/Irreducibility.lean
+++ b/lean/OAI/Probability/ThorpShuffle/Irreducibility.lean
@@ -1,230 +1,230 @@
-import OAI.Probability.ThorpShuffle.Polytabloids
-
-namespace OAI
-
-universe uI uX uK
-
-noncomputable section
-
-open scoped BigOperators ComplexConjugate InnerProductSpace
-open Filter
-
-namespace Thorp.Young
-open scoped Classical
-
-variable {n : ℕ} {D : YoungDiagram}
-
-def Standard (t : Fin n ≃ Cells D) : Prop := StrictMono t.symm
-
-lemma standard_row_lt_iff (t : Fin n ≃ Cells D) (ht : Standard t) (x y : Fin n)
-    (hr : (t x).val.1 = (t y).val.1) : x < y ↔ (t x).val.2 < (t y).val.2 := by
-  constructor
-  · intro hxy
-    by_contra hn
-    have hc : (t y).val.2 ≤ (t x).val.2 := Nat.le_of_not_gt hn
-    have hyx : t y ≤ t x := ⟨by omega, hc⟩
-    have hh : y ≤ x := by simpa only [Equiv.symm_apply_apply] using ht.monotone hyx
-    exact not_le_of_gt hxy hh
-  · intro hc
-    have hxy : t x < t y := by
-      refine lt_iff_le_not_ge.mpr ⟨⟨by omega, hc.le⟩, ?_⟩
-      intro hh
-      exact not_le_of_gt hc hh.2
-    simpa only [Equiv.symm_apply_apply] using ht hxy
-
-lemma standard_col_lt_iff (t : Fin n ≃ Cells D) (ht : Standard t) (x y : Fin n)
-    (hc : (t x).val.2 = (t y).val.2) : x < y ↔ (t x).val.1 < (t y).val.1 := by
-  constructor
-  · intro hxy
-    by_contra hn
-    have hr : (t y).val.1 ≤ (t x).val.1 := Nat.le_of_not_gt hn
-    have hyx : t y ≤ t x := ⟨hr, by omega⟩
-    have hh : y ≤ x := by simpa only [Equiv.symm_apply_apply] using ht.monotone hyx
-    exact not_le_of_gt hxy hh
-  · intro hr
-    have hxy : t x < t y := by
-      refine lt_iff_le_not_ge.mpr ⟨⟨hr.le, by omega⟩, ?_⟩
-      intro hh
-      exact not_le_of_gt hr hh.1
-    simpa only [Equiv.symm_apply_apply] using ht hxy
-
-lemma column_eq_rowRank (t : Fin n ≃ Cells D) (ht : Standard t) (x : Fin n) :
-    Fintype.card {y : Fin n // y < x ∧ (t y).val.1 = (t x).val.1} = (t x).val.2 := by
-  let e : {y : Fin n // y < x ∧ (t y).val.1 = (t x).val.1} ≃ Fin ((t x).val.2) := {
-    toFun := fun y => ⟨(t y.val).val.2, (standard_row_lt_iff t ht y.val x y.property.2).mp y.property.1⟩
-    invFun := fun j => ⟨t.symm ⟨((t x).val.1, j), D.up_left_mem le_rfl j.isLt.le (t x).property⟩, by
-      constructor
-      · apply (standard_row_lt_iff t ht _ x (by simp)).mpr
-        simpa only [Equiv.apply_symm_apply] using j.isLt
-      · simp⟩
-    left_inv := by
-      intro y
-      apply Subtype.ext
-      apply t.injective
-      simp only [Equiv.apply_symm_apply]
-      apply Subtype.ext
-      exact Prod.ext y.property.2.symm rfl
-    right_inv := by intro j; apply Fin.ext; simp }
-  exact (Fintype.card_congr e).trans (Fintype.card_fin _)
-
-lemma standard_eq_of_rows (t u : Fin n ≃ Cells D) (ht : Standard t) (hu : Standard u)
-    (hr : ∀ x, (t x).val.1 = (u x).val.1) : t = u := by
-  apply Equiv.ext
-  intro x
-  apply Subtype.ext
-  apply Prod.ext (hr x)
-  rw [← column_eq_rowRank t ht x, ← column_eq_rowRank u hu x]
-  exact Fintype.card_congr ((Equiv.refl (Fin n)).subtypeEquiv (fun y => by simp only [Equiv.refl_apply, hr]))
-
-end Thorp.Young
-
-namespace Thorp.Specht
-open scoped Classical
-
-theorem independent_of_leading {I : Type uI} {X : Type uX} {K : Type uK} [Fintype I] [Fintype X] [LinearOrder K]
-    (e : I → EuclideanSpace ℂ X) (b : I → X) (key : X → K)
-    (hinj : Function.Injective (key ∘ b)) (hdiag : ∀ i, e i (b i) ≠ 0)
-    (htri : ∀ i x, e i x ≠ 0 → key (b i) ≤ key x) : LinearIndependent ℂ e := by
-  let : LinearOrder I := LinearOrder.lift' (key ∘ b) hinj
-  rw [linearIndependent_iff']
-  intro s a ha i hi
-  by_contra hai
-  let S := s.filter fun j => a j ≠ 0
-  have hs : S.Nonempty := ⟨i, Finset.mem_filter.mpr ⟨hi, hai⟩⟩
-  let j := S.min' hs
-  have hj : j ∈ S := S.min'_mem hs
-  have hj' : j ∈ s := (Finset.mem_filter.mp hj).1
-  have haz : a j ≠ 0 := (Finset.mem_filter.mp hj).2
-  have hval := congrArg (fun v : EuclideanSpace ℂ X => v (b j)) ha
-  simp only [WithLp.ofLp_sum, Finset.sum_apply, PiLp.smul_apply, smul_eq_mul, PiLp.zero_apply] at hval
-  rw [Finset.sum_eq_single_of_mem j hj'] at hval
-  · exact (mul_ne_zero haz (hdiag j)) hval
-  · intro k hk hkj
-    by_cases hak : a k = 0
-    · simp [hak]
-    have hks : k ∈ S := Finset.mem_filter.mpr ⟨hk, hak⟩
-    have hjk : j ≤ k := S.min'_le k hks
-    have hjk' : j < k := lt_of_le_of_ne hjk (Ne.symm hkj)
-    have hzero : e k (b j) = 0 := by
-      by_contra hn
-      have hh := htri k (b j) hn
-      exact not_le_of_gt hjk' hh
-    simp [hzero]
-
-end Thorp.Specht
-
-namespace Thorp.Specht
-open scoped Classical
-
-variable {n : ℕ} (D : YoungDiagram) (t : Fin n ≃ Thorp.Young.Cells D)
-
-lemma column_word_le (ht : Thorp.Young.Standard t) (c : fiberGroup (tableauCol D t)) :
-    toLex (tableauRow D t) ≤ toLex (fun x => tableauRow D t (c.val⁻¹ x)) := by
-  by_cases hc : c.val = 1
-  · simp [hc]
-  let S := Finset.univ.filter fun x : Fin n => c.val⁻¹ x ≠ x
-  have hs : S.Nonempty := by
-    by_contra hn
-    apply hc
-    have he : c.val⁻¹ = 1 := by
-      apply Equiv.ext
-      intro x
-      by_contra hx
-      exact hn ⟨x, Finset.mem_filter.mpr ⟨Finset.mem_univ x, hx⟩⟩
-    exact inv_eq_one.mp he
-  let x := S.min' hs
-  have hx : c.val⁻¹ x ≠ x := (Finset.mem_filter.mp (S.min'_mem hs)).2
-  have hbefore (y : Fin n) (hy : y < x) : c.val⁻¹ y = y := by
-    by_contra hn
-    have hm : y ∈ S := Finset.mem_filter.mpr ⟨Finset.mem_univ y, hn⟩
-    exact not_le_of_gt hy (S.min'_le y hm)
-  have hlt : x < c.val⁻¹ x := by
-    apply lt_of_le_of_ne
-    · by_contra hn
-      have hh := hbefore (c.val⁻¹ x) (lt_of_not_ge hn)
-      exact hx (c.val.symm.injective hh)
-    · exact Ne.symm hx
-  have hcol : (t x).val.2 = (t (c.val⁻¹ x)).val.2 := by
-    have hh := c.property (c.val⁻¹ x)
-    simpa only [tableauCol, perm_apply_inv] using hh
-  have hrow := (Thorp.Young.standard_col_lt_iff t ht x (c.val⁻¹ x) hcol).mp hlt
-  apply le_of_lt
-  change Pi.Lex (· < ·) (· < ·) (tableauRow D t) (fun y => tableauRow D t (c.val⁻¹ y))
-  exact ⟨x, fun y hy => congrArg (tableauRow D t) (hbefore y hy).symm, hrow⟩
-
-def movedTableau (g : Equiv.Perm (Fin n)) : Fin n ≃ Thorp.Young.Cells D := (g⁻¹).trans t
-
-def StandardPerm := {g : Equiv.Perm (Fin n) // Thorp.Young.Standard (movedTableau D t g)}
-
-instance : Fintype (StandardPerm D t) := inferInstanceAs (Fintype {g : Equiv.Perm (Fin n) // Thorp.Young.Standard (movedTableau D t g)})
-
-def movedWord (g : Equiv.Perm (Fin n)) : Tabloid (tableauRow D t) := g • baseTabloid (tableauRow D t)
-
-lemma movedWord_apply (g : Equiv.Perm (Fin n)) (x : Fin n) :
-    (movedWord D t g).val x = tableauRow D (movedTableau D t g) x := rfl
-
-lemma moved_diag (g : Equiv.Perm (Fin n)) :
-    tabloidRep D t g (polytabloid D t) (movedWord D t g) = 1 := by
-  change polytabloid D t (g⁻¹ • (g • baseTabloid (tableauRow D t))) = 1
-  rw [inv_smul_smul]
-  exact polytabloid_diag D t
-
-lemma moved_leading (g : Equiv.Perm (Fin n)) (hg : Thorp.Young.Standard (movedTableau D t g))
-    (f : Tabloid (tableauRow D t)) (hf : tabloidRep D t g (polytabloid D t) f ≠ 0) :
-    toLex (movedWord D t g).val ≤ toLex f.val := by
-  obtain ⟨h, hh⟩ := polytabloid_support D t (g⁻¹ • f) hf
-  have he : f = (g * h.val) • baseTabloid (tableauRow D t) := by
-    rw [mul_smul]
-    exact inv_smul_eq_iff.mp hh
-  let c : Equiv.Perm (Fin n) := g * h.val * g⁻¹
-  have hc : c ∈ fiberGroup (tableauCol D (movedTableau D t g)) := by
-    intro x
-    change (t (g⁻¹ (g (h.val (g⁻¹ x))))).val.2 = (t (g⁻¹ x)).val.2
-    rw [perm_inv_apply]
-    exact h.property (g⁻¹ x)
-  have hcword : c • movedWord D t g = f := by
-    rw [he]
-    change (g * h.val * g⁻¹) • (g • baseTabloid _) = (g * h.val) • baseTabloid _
-    simp only [← mul_smul, inv_mul_cancel_right]
-  have hw := column_word_le D (movedTableau D t g) hg ⟨c, hc⟩
-  change toLex (movedWord D t g).val ≤ toLex (c • movedWord D t g).val at hw
-  rw [hcword] at hw
-  exact hw
-
-lemma standardWord_injective : Function.Injective (fun g : StandardPerm D t => toLex (movedWord D t g.val).val) := by
-  intro g h he
-  have hr : ∀ x, (movedTableau D t g.val x).val.1 = (movedTableau D t h.val x).val.1 :=
-    fun x => congrFun he x
-  have ht := Thorp.Young.standard_eq_of_rows _ _ g.property h.property hr
-  apply Subtype.ext
-  apply inv_injective
-  apply Equiv.ext
-  intro x
-  apply t.injective
-  exact congrArg (fun z : Fin n ≃ Thorp.Young.Cells D => z x) ht
-
-theorem standard_polytabloids_independent :
-    LinearIndependent ℂ (fun g : StandardPerm D t => tabloidRep D t g.val (polytabloid D t)) := by
-  apply independent_of_leading _ (fun g => movedWord D t g.val) (fun f => toLex f.val)
-    (standardWord_injective D t)
-  · intro i
-    rw [moved_diag]
-    exact one_ne_zero
-  · intro i f hf
-    exact moved_leading D t i.val i.property f hf
-
-def orbitVector (g : Equiv.Perm (Fin n)) : (module D t).toSubmodule :=
-  ⟨tabloidRep D t g (polytabloid D t),
-    (module D t).apply_mem_toSubmodule g (self_mem_cyclic _ _)⟩
-
-theorem standardPerm_card_le_degree :
-    Fintype.card (StandardPerm D t) ≤ Module.finrank ℂ (module D t).toSubmodule := by
-  have hi : LinearIndependent ℂ (fun g : StandardPerm D t => orbitVector D t g.val) :=
-    LinearIndependent.of_comp (module D t).toSubmodule.subtype (standard_polytabloids_independent D t)
-  exact hi.fintype_card_le_finrank
-
-end Thorp.Specht
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.Polytabloids
+
+namespace OAI
+
+universe uI uX uK
+
+noncomputable section
+
+open scoped BigOperators ComplexConjugate InnerProductSpace
+open Filter
+
+namespace Thorp.Young
+open scoped Classical
+
+variable {n : ℕ} {D : YoungDiagram}
+
+def Standard (t : Fin n ≃ Cells D) : Prop := StrictMono t.symm
+
+lemma standard_row_lt_iff (t : Fin n ≃ Cells D) (ht : Standard t) (x y : Fin n)
+    (hr : (t x).val.1 = (t y).val.1) : x < y ↔ (t x).val.2 < (t y).val.2 := by
+  constructor
+  · intro hxy
+    by_contra hn
+    have hc : (t y).val.2 ≤ (t x).val.2 := Nat.le_of_not_gt hn
+    have hyx : t y ≤ t x := ⟨by omega, hc⟩
+    have hh : y ≤ x := by simpa only [Equiv.symm_apply_apply] using ht.monotone hyx
+    exact not_le_of_gt hxy hh
+  · intro hc
+    have hxy : t x < t y := by
+      refine lt_iff_le_not_ge.mpr ⟨⟨by omega, hc.le⟩, ?_⟩
+      intro hh
+      exact not_le_of_gt hc hh.2
+    simpa only [Equiv.symm_apply_apply] using ht hxy
+
+lemma standard_col_lt_iff (t : Fin n ≃ Cells D) (ht : Standard t) (x y : Fin n)
+    (hc : (t x).val.2 = (t y).val.2) : x < y ↔ (t x).val.1 < (t y).val.1 := by
+  constructor
+  · intro hxy
+    by_contra hn
+    have hr : (t y).val.1 ≤ (t x).val.1 := Nat.le_of_not_gt hn
+    have hyx : t y ≤ t x := ⟨hr, by omega⟩
+    have hh : y ≤ x := by simpa only [Equiv.symm_apply_apply] using ht.monotone hyx
+    exact not_le_of_gt hxy hh
+  · intro hr
+    have hxy : t x < t y := by
+      refine lt_iff_le_not_ge.mpr ⟨⟨hr.le, by omega⟩, ?_⟩
+      intro hh
+      exact not_le_of_gt hr hh.1
+    simpa only [Equiv.symm_apply_apply] using ht hxy
+
+lemma column_eq_rowRank (t : Fin n ≃ Cells D) (ht : Standard t) (x : Fin n) :
+    Fintype.card {y : Fin n // y < x ∧ (t y).val.1 = (t x).val.1} = (t x).val.2 := by
+  let e : {y : Fin n // y < x ∧ (t y).val.1 = (t x).val.1} ≃ Fin ((t x).val.2) := {
+    toFun := fun y => ⟨(t y.val).val.2, (standard_row_lt_iff t ht y.val x y.property.2).mp y.property.1⟩
+    invFun := fun j => ⟨t.symm ⟨((t x).val.1, j), D.up_left_mem le_rfl j.isLt.le (t x).property⟩, by
+      constructor
+      · apply (standard_row_lt_iff t ht _ x (by simp)).mpr
+        simpa only [Equiv.apply_symm_apply] using j.isLt
+      · simp⟩
+    left_inv := by
+      intro y
+      apply Subtype.ext
+      apply t.injective
+      simp only [Equiv.apply_symm_apply]
+      apply Subtype.ext
+      exact Prod.ext y.property.2.symm rfl
+    right_inv := by intro j; apply Fin.ext; simp }
+  exact (Fintype.card_congr e).trans (Fintype.card_fin _)
+
+lemma standard_eq_of_rows (t u : Fin n ≃ Cells D) (ht : Standard t) (hu : Standard u)
+    (hr : ∀ x, (t x).val.1 = (u x).val.1) : t = u := by
+  apply Equiv.ext
+  intro x
+  apply Subtype.ext
+  apply Prod.ext (hr x)
+  rw [← column_eq_rowRank t ht x, ← column_eq_rowRank u hu x]
+  exact Fintype.card_congr ((Equiv.refl (Fin n)).subtypeEquiv (fun y => by simp only [Equiv.refl_apply, hr]))
+
+end Thorp.Young
+
+namespace Thorp.Specht
+open scoped Classical
+
+theorem independent_of_leading {I : Type uI} {X : Type uX} {K : Type uK} [Fintype I] [Fintype X] [LinearOrder K]
+    (e : I → EuclideanSpace ℂ X) (b : I → X) (key : X → K)
+    (hinj : Function.Injective (key ∘ b)) (hdiag : ∀ i, e i (b i) ≠ 0)
+    (htri : ∀ i x, e i x ≠ 0 → key (b i) ≤ key x) : LinearIndependent ℂ e := by
+  let : LinearOrder I := LinearOrder.lift' (key ∘ b) hinj
+  rw [linearIndependent_iff']
+  intro s a ha i hi
+  by_contra hai
+  let S := s.filter fun j => a j ≠ 0
+  have hs : S.Nonempty := ⟨i, Finset.mem_filter.mpr ⟨hi, hai⟩⟩
+  let j := S.min' hs
+  have hj : j ∈ S := S.min'_mem hs
+  have hj' : j ∈ s := (Finset.mem_filter.mp hj).1
+  have haz : a j ≠ 0 := (Finset.mem_filter.mp hj).2
+  have hval := congrArg (fun v : EuclideanSpace ℂ X => v (b j)) ha
+  simp only [WithLp.ofLp_sum, Finset.sum_apply, PiLp.smul_apply, smul_eq_mul, PiLp.zero_apply] at hval
+  rw [Finset.sum_eq_single_of_mem j hj'] at hval
+  · exact (mul_ne_zero haz (hdiag j)) hval
+  · intro k hk hkj
+    by_cases hak : a k = 0
+    · simp [hak]
+    have hks : k ∈ S := Finset.mem_filter.mpr ⟨hk, hak⟩
+    have hjk : j ≤ k := S.min'_le k hks
+    have hjk' : j < k := lt_of_le_of_ne hjk (Ne.symm hkj)
+    have hzero : e k (b j) = 0 := by
+      by_contra hn
+      have hh := htri k (b j) hn
+      exact not_le_of_gt hjk' hh
+    simp [hzero]
+
+end Thorp.Specht
+
+namespace Thorp.Specht
+open scoped Classical
+
+variable {n : ℕ} (D : YoungDiagram) (t : Fin n ≃ Thorp.Young.Cells D)
+
+lemma column_word_le (ht : Thorp.Young.Standard t) (c : fiberGroup (tableauCol D t)) :
+    toLex (tableauRow D t) ≤ toLex (fun x => tableauRow D t (c.val⁻¹ x)) := by
+  by_cases hc : c.val = 1
+  · simp [hc]
+  let S := Finset.univ.filter fun x : Fin n => c.val⁻¹ x ≠ x
+  have hs : S.Nonempty := by
+    by_contra hn
+    apply hc
+    have he : c.val⁻¹ = 1 := by
+      apply Equiv.ext
+      intro x
+      by_contra hx
+      exact hn ⟨x, Finset.mem_filter.mpr ⟨Finset.mem_univ x, hx⟩⟩
+    exact inv_eq_one.mp he
+  let x := S.min' hs
+  have hx : c.val⁻¹ x ≠ x := (Finset.mem_filter.mp (S.min'_mem hs)).2
+  have hbefore (y : Fin n) (hy : y < x) : c.val⁻¹ y = y := by
+    by_contra hn
+    have hm : y ∈ S := Finset.mem_filter.mpr ⟨Finset.mem_univ y, hn⟩
+    exact not_le_of_gt hy (S.min'_le y hm)
+  have hlt : x < c.val⁻¹ x := by
+    apply lt_of_le_of_ne
+    · by_contra hn
+      have hh := hbefore (c.val⁻¹ x) (lt_of_not_ge hn)
+      exact hx (c.val.symm.injective hh)
+    · exact Ne.symm hx
+  have hcol : (t x).val.2 = (t (c.val⁻¹ x)).val.2 := by
+    have hh := c.property (c.val⁻¹ x)
+    simpa only [tableauCol, perm_apply_inv] using hh
+  have hrow := (Thorp.Young.standard_col_lt_iff t ht x (c.val⁻¹ x) hcol).mp hlt
+  apply le_of_lt
+  change Pi.Lex (· < ·) (· < ·) (tableauRow D t) (fun y => tableauRow D t (c.val⁻¹ y))
+  exact ⟨x, fun y hy => congrArg (tableauRow D t) (hbefore y hy).symm, hrow⟩
+
+def movedTableau (g : Equiv.Perm (Fin n)) : Fin n ≃ Thorp.Young.Cells D := (g⁻¹).trans t
+
+def StandardPerm := {g : Equiv.Perm (Fin n) // Thorp.Young.Standard (movedTableau D t g)}
+
+instance : Fintype (StandardPerm D t) := inferInstanceAs (Fintype {g : Equiv.Perm (Fin n) // Thorp.Young.Standard (movedTableau D t g)})
+
+def movedWord (g : Equiv.Perm (Fin n)) : Tabloid (tableauRow D t) := g • baseTabloid (tableauRow D t)
+
+lemma movedWord_apply (g : Equiv.Perm (Fin n)) (x : Fin n) :
+    (movedWord D t g).val x = tableauRow D (movedTableau D t g) x := rfl
+
+lemma moved_diag (g : Equiv.Perm (Fin n)) :
+    tabloidRep D t g (polytabloid D t) (movedWord D t g) = 1 := by
+  change polytabloid D t (g⁻¹ • (g • baseTabloid (tableauRow D t))) = 1
+  rw [inv_smul_smul]
+  exact polytabloid_diag D t
+
+lemma moved_leading (g : Equiv.Perm (Fin n)) (hg : Thorp.Young.Standard (movedTableau D t g))
+    (f : Tabloid (tableauRow D t)) (hf : tabloidRep D t g (polytabloid D t) f ≠ 0) :
+    toLex (movedWord D t g).val ≤ toLex f.val := by
+  obtain ⟨h, hh⟩ := polytabloid_support D t (g⁻¹ • f) hf
+  have he : f = (g * h.val) • baseTabloid (tableauRow D t) := by
+    rw [mul_smul]
+    exact inv_smul_eq_iff.mp hh
+  let c : Equiv.Perm (Fin n) := g * h.val * g⁻¹
+  have hc : c ∈ fiberGroup (tableauCol D (movedTableau D t g)) := by
+    intro x
+    change (t (g⁻¹ (g (h.val (g⁻¹ x))))).val.2 = (t (g⁻¹ x)).val.2
+    rw [perm_inv_apply]
+    exact h.property (g⁻¹ x)
+  have hcword : c • movedWord D t g = f := by
+    rw [he]
+    change (g * h.val * g⁻¹) • (g • baseTabloid _) = (g * h.val) • baseTabloid _
+    simp only [← mul_smul, inv_mul_cancel_right]
+  have hw := column_word_le D (movedTableau D t g) hg ⟨c, hc⟩
+  change toLex (movedWord D t g).val ≤ toLex (c • movedWord D t g).val at hw
+  rw [hcword] at hw
+  exact hw
+
+lemma standardWord_injective : Function.Injective (fun g : StandardPerm D t => toLex (movedWord D t g.val).val) := by
+  intro g h he
+  have hr : ∀ x, (movedTableau D t g.val x).val.1 = (movedTableau D t h.val x).val.1 :=
+    fun x => congrFun he x
+  have ht := Thorp.Young.standard_eq_of_rows _ _ g.property h.property hr
+  apply Subtype.ext
+  apply inv_injective
+  apply Equiv.ext
+  intro x
+  apply t.injective
+  exact congrArg (fun z : Fin n ≃ Thorp.Young.Cells D => z x) ht
+
+theorem standard_polytabloids_independent :
+    LinearIndependent ℂ (fun g : StandardPerm D t => tabloidRep D t g.val (polytabloid D t)) := by
+  apply independent_of_leading _ (fun g => movedWord D t g.val) (fun f => toLex f.val)
+    (standardWord_injective D t)
+  · intro i
+    rw [moved_diag]
+    exact one_ne_zero
+  · intro i f hf
+    exact moved_leading D t i.val i.property f hf
+
+def orbitVector (g : Equiv.Perm (Fin n)) : (module D t).toSubmodule :=
+  ⟨tabloidRep D t g (polytabloid D t),
+    (module D t).apply_mem_toSubmodule g (self_mem_cyclic _ _)⟩
+
+theorem standardPerm_card_le_degree :
+    Fintype.card (StandardPerm D t) ≤ Module.finrank ℂ (module D t).toSubmodule := by
+  have hi : LinearIndependent ℂ (fun g : StandardPerm D t => orbitVector D t g.val) :=
+    LinearIndependent.of_comp (module D t).toSubmodule.subtype (standard_polytabloids_independent D t)
+  exact hi.fintype_card_le_finrank
+
+end Thorp.Specht
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/JointProjectors.lean b/lean/OAI/Probability/ThorpShuffle/JointProjectors.lean
index 4a16375087a203d39144b68d550cf890f98528b3..50fe0aac680dc9427577a9a4030d0f01d8c3223c 100644
--- a/lean/OAI/Probability/ThorpShuffle/JointProjectors.lean
+++ b/lean/OAI/Probability/ThorpShuffle/JointProjectors.lean
@@ -1,177 +1,177 @@
-import OAI.Probability.ThorpShuffle.FourierFamilies
-
-namespace OAI
-
-universe uI uE
-
-noncomputable section
-
-open scoped BigOperators ComplexConjugate InnerProductSpace
-open Filter Topology
-
-namespace Thorp.Fourier
-open scoped Classical
-
-lemma complex_cauchy {I : Type uI} (s : Finset I) (a b : I → ℂ) :
-    ‖∑ i ∈ s, a i * b i‖ ^ 2 ≤ (∑ i ∈ s, ‖a i‖ ^ 2) * ∑ i ∈ s, ‖b i‖ ^ 2 := by
-  have ht : ‖∑ i ∈ s, a i * b i‖ ≤ ∑ i ∈ s, ‖a i‖ * ‖b i‖ := by
-    simpa only [norm_mul] using norm_sum_le s (fun i => a i * b i)
-  exact (pow_le_pow_left₀ (norm_nonneg _) ht 2).trans
-    (Finset.sum_mul_sq_le_sq_mul_sq s (fun i => ‖a i‖) (fun i => ‖b i‖))
-
-namespace Family
-variable {G : Type} [Group G] [Fintype G] (F : Family G)
-variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
-
-theorem coefficient_evaluation (ρ : Representation ℂ G E) (s : Finset F.Index)
-    (v w : E) (hv : (∑ i ∈ s, F.projector ρ i) v = v) :
-    (Fintype.card G : ℝ) * ‖⟪w, v⟫_ℂ‖ ^ 2 ≤
-      (∑ i ∈ s, (F.degree i : ℝ)^2) * ∑ g, ‖⟪w, ρ g v⟫_ℂ‖ ^ 2 := by
-  have he : ⟪w, v⟫_ℂ = ∑ g, (∑ i ∈ s, F.kernel i g) * ⟪w, ρ g v⟫_ℂ := by
-    conv_lhs => rw [← hv]
-    conv_lhs => rw [show ∑ i ∈ s, F.projector ρ i = integrated ρ (∑ i ∈ s, F.kernel i) by
-      simp only [projector, integrated_sum]]
-    simp only [integrated, LinearMap.sum_apply, LinearMap.smul_apply, inner_sum, inner_smul_right,
-      Finset.sum_apply]
-  have hh := mul_le_mul_of_nonneg_left
-    (complex_cauchy Finset.univ (fun g => ∑ i ∈ s, F.kernel i g) (fun g => ⟪w, ρ g v⟫_ℂ))
-    (Nat.cast_nonneg (α := ℝ) (Fintype.card G))
-  rw [← he, ← mul_assoc, F.kernel_sum_sq] at hh
-  exact hh
-
-end Family
-end Thorp.Fourier
-
-namespace Thorp.Fourier.Family
-open scoped Classical
-variable {H : Type} [Group H] [Fintype H] (F : Family H)
-variable {I : Type} [Fintype I]
-
-def jointKernel (a : I → F.Index) (g : I → H) : ℂ := ∏ i, F.kernel (a i) (g i)
-
-lemma jointKernel_sum : (∑ a : I → F.Index, F.jointKernel a) = delta := by
-  funext g
-  simp only [Finset.sum_apply, jointKernel]
-  rw [← Fintype.prod_sum (fun i (j : F.Index) => F.kernel j (g i))]
-  have hh (i : I) : ∑ j : F.Index, F.kernel j (g i) = if g i = 1 then 1 else 0 :=
-    by simpa only [Finset.sum_apply, delta] using congrFun F.kernel_sum (g i)
-  simp only [hh]
-  by_cases hg : g = 1
-  · subst g; simp [delta]
-  · have hn : ∃ i, g i ≠ 1 := by
-      by_contra hn
-      push Not at hn
-      exact hg (funext hn)
-    obtain ⟨i, hi⟩ := hn
-    rw [Finset.prod_eq_zero (Finset.mem_univ i) (ite_eq_right hi)]
-    simp [delta, hg]
-
-lemma jointKernel_convolution (a b : I → F.Index) :
-    convolution (F.jointKernel a) (F.jointKernel b) = if a = b then F.jointKernel a else 0 := by
-  funext g
-  simp only [convolution, jointKernel, Pi.mul_apply, Pi.inv_apply, ← Finset.prod_mul_distrib]
-  rw [← Fintype.prod_sum (fun i (h : H) => F.kernel (a i) h * F.kernel (b i) (h⁻¹ * g i))]
-  have hh (i : I) : ∑ h, F.kernel (a i) h * F.kernel (b i) (h⁻¹ * g i) =
-      if a i = b i then F.kernel (a i) (g i) else 0 := by
-    have he := congrFun (F.kernel_convolution (a i) (b i)) (g i)
-    simpa only [convolution, ite_apply, Pi.zero_apply] using he
-  simp only [hh]
-  by_cases hab : a = b
-  · subst b; simp only [↓reduceIte, jointKernel]
-  · rw [ite_eq_right hab]
-    obtain ⟨i, hi⟩ : ∃ i, a i ≠ b i := by
-      by_contra hn; push Not at hn; exact hab (funext hn)
-    exact Finset.prod_eq_zero (Finset.mem_univ i) (ite_eq_right hi)
-
-lemma jointKernel_star (a : I → F.Index) (g : I → H) :
-    conj (F.jointKernel a g⁻¹) = F.jointKernel a g := by
-  simp only [jointKernel, map_prod, Pi.inv_apply, F.kernel_star]
-
-variable {E : Type uE} [AddCommGroup E] [Module ℂ E]
-
-def jointProjector (ρ : Representation ℂ (I → H) E) (a : I → F.Index) : Module.End ℂ E :=
-  integrated ρ (F.jointKernel a)
-
-lemma jointProjector_mul (ρ : Representation ℂ (I → H) E) (a b : I → F.Index) :
-    F.jointProjector ρ a * F.jointProjector ρ b = if a = b then F.jointProjector ρ a else 0 := by
-  rw [jointProjector, jointProjector, ← integrated_convolution, jointKernel_convolution]
-  split_ifs <;> simp only [integrated, Pi.zero_apply, zero_smul, Finset.sum_const_zero]
-
-lemma jointProjector_idempotent (ρ : Representation ℂ (I → H) E) (a : I → F.Index) :
-    IsIdempotentElem (F.jointProjector ρ a) := by
-  simpa only [IsIdempotentElem, ↓reduceIte] using F.jointProjector_mul ρ a a
-
-lemma jointProjector_sum (ρ : Representation ℂ (I → H) E) : ∑ a, F.jointProjector ρ a = 1 := by
-  simp only [jointProjector]
-  rw [← integrated_sum, F.jointKernel_sum, integrated_delta]
-
-def tensorList (a : I → F.Index) : List I → FDRep ℂ (I → H)
-  | [] => FDRep.of (Representation.trivial ℂ (I → H) ℂ)
-  | i :: l => FDRep.of (Representation.tprod ((F.rep (a i)).comp (Pi.evalMonoidHom (fun _ : I => H) i)) (tensorList a l).ρ)
-
-omit [Fintype I] in
-lemma tensorList_character (a : I → F.Index) (l : List I) (g : I → H) :
-    Representation.character (F.tensorList a l).ρ g = (l.map (fun i => (F.rep (a i)).character (g i))).prod := by
-  induction l with
-  | nil =>
-    change LinearMap.trace ℂ ℂ (1 : Module.End ℂ ℂ) = 1
-    simp
-  | cons i l ih =>
-    change (Representation.tprod ((F.rep (a i)).comp (Pi.evalMonoidHom (fun _ : I => H) i))
-      (F.tensorList a l).ρ).character g = _
-    rw [Representation.char_tensor ((F.rep (a i)).comp (Pi.evalMonoidHom (fun _ : I => H) i))
-      (F.tensorList a l).ρ]
-    change (F.rep (a i)).character (g i) * Representation.character (F.tensorList a l).ρ g = _
-    rw [ih]
-    rfl
-
-def tensorFamily (a : I → F.Index) : FDRep ℂ (I → H) := F.tensorList a Finset.univ.toList
-
-lemma tensorFamily_character (a : I → F.Index) (g : I → H) :
-    Representation.character (F.tensorFamily a).ρ g = ∏ i, (F.rep (a i)).character (g i) := by
-  rw [tensorFamily, tensorList_character]
-  exact Finset.prod_map_toList _ _
-
-lemma jointKernel_formula (a : I → F.Index) (g : I → H) :
-    F.jointKernel a g = (∏ i, (F.degree (a i) : ℂ)) * (Fintype.card (I → H) : ℂ)⁻¹ *
-      Representation.character (F.tensorFamily a).ρ g⁻¹ := by
-  simp only [jointKernel, kernel, Finset.prod_mul_distrib, Finset.prod_const,
-    Finset.card_univ, tensorFamily_character, Pi.inv_apply, Fintype.card_fun, Nat.cast_pow, inv_pow]
-
-lemma trace_jointProjector [FiniteDimensional ℂ E] (ρ : Representation ℂ (I → H) E)
-    (a : I → F.Index) : LinearMap.trace ℂ E (F.jointProjector ρ a) =
-      (∏ i, (F.degree (a i) : ℂ)) * Module.finrank ℂ (Representation.IntertwiningMap (F.tensorFamily a).ρ ρ) := by
-  let : Invertible (Nat.card (I → H) : ℂ) := invertibleOfNonzero (NeZero.ne _)
-  rw [jointProjector, trace_integrated]
-  simp only [jointKernel_formula]
-  rw [← Representation.card_inv_mul_sum_char_mul_char_eq_finrank]
-  simp only [Finset.mul_sum, Fintype.card_eq_nat_card]
-  apply Finset.sum_congr rfl
-  intro g _
-  ring
-
-theorem joint_degree_le [FiniteDimensional ℂ E] (ρ : Representation ℂ (I → H) E)
-    (a : I → F.Index) (ha : F.jointProjector ρ a ≠ 0) :
-    (∏ i, F.degree (a i)) ≤ Module.finrank ℂ E := by
-  have hp := F.jointProjector_idempotent ρ a
-  have ht := F.trace_jointProjector ρ a
-  rw [(LinearMap.IsIdempotentElem.isProj_range _ hp).trace] at ht
-  have hm : Module.finrank ℂ (Representation.IntertwiningMap (F.tensorFamily a).ρ ρ) ≠ 0 := by
-    intro hz
-    have hh := F.trace_jointProjector ρ a
-    rw [hz, Nat.cast_zero, mul_zero] at hh
-    exact ha (LinearMap.IsIdempotentElem.eq_zero_of_trace_eq_zero hp hh)
-  have hn : Module.finrank ℂ (LinearMap.range (F.jointProjector ρ a)) =
-      (∏ i, F.degree (a i)) * Module.finrank ℂ (Representation.IntertwiningMap (F.tensorFamily a).ρ ρ) := by
-    exact_mod_cast ht
-  calc
-    _ ≤ (∏ i, F.degree (a i)) * Module.finrank ℂ (Representation.IntertwiningMap (F.tensorFamily a).ρ ρ) :=
-      Nat.le_mul_of_pos_right _ (Nat.pos_of_ne_zero hm)
-    _ = _ := hn.symm
-    _ ≤ Module.finrank ℂ E := Submodule.finrank_le _
-
-end Thorp.Fourier.Family
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.FourierFamilies
+
+namespace OAI
+
+universe uI uE
+
+noncomputable section
+
+open scoped BigOperators ComplexConjugate InnerProductSpace
+open Filter Topology
+
+namespace Thorp.Fourier
+open scoped Classical
+
+lemma complex_cauchy {I : Type uI} (s : Finset I) (a b : I → ℂ) :
+    ‖∑ i ∈ s, a i * b i‖ ^ 2 ≤ (∑ i ∈ s, ‖a i‖ ^ 2) * ∑ i ∈ s, ‖b i‖ ^ 2 := by
+  have ht : ‖∑ i ∈ s, a i * b i‖ ≤ ∑ i ∈ s, ‖a i‖ * ‖b i‖ := by
+    simpa only [norm_mul] using norm_sum_le s (fun i => a i * b i)
+  exact (pow_le_pow_left₀ (norm_nonneg _) ht 2).trans
+    (Finset.sum_mul_sq_le_sq_mul_sq s (fun i => ‖a i‖) (fun i => ‖b i‖))
+
+namespace Family
+variable {G : Type} [Group G] [Fintype G] (F : Family G)
+variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
+
+theorem coefficient_evaluation (ρ : Representation ℂ G E) (s : Finset F.Index)
+    (v w : E) (hv : (∑ i ∈ s, F.projector ρ i) v = v) :
+    (Fintype.card G : ℝ) * ‖⟪w, v⟫_ℂ‖ ^ 2 ≤
+      (∑ i ∈ s, (F.degree i : ℝ)^2) * ∑ g, ‖⟪w, ρ g v⟫_ℂ‖ ^ 2 := by
+  have he : ⟪w, v⟫_ℂ = ∑ g, (∑ i ∈ s, F.kernel i g) * ⟪w, ρ g v⟫_ℂ := by
+    conv_lhs => rw [← hv]
+    conv_lhs => rw [show ∑ i ∈ s, F.projector ρ i = integrated ρ (∑ i ∈ s, F.kernel i) by
+      simp only [projector, integrated_sum]]
+    simp only [integrated, LinearMap.sum_apply, LinearMap.smul_apply, inner_sum, inner_smul_right,
+      Finset.sum_apply]
+  have hh := mul_le_mul_of_nonneg_left
+    (complex_cauchy Finset.univ (fun g => ∑ i ∈ s, F.kernel i g) (fun g => ⟪w, ρ g v⟫_ℂ))
+    (Nat.cast_nonneg (α := ℝ) (Fintype.card G))
+  rw [← he, ← mul_assoc, F.kernel_sum_sq] at hh
+  exact hh
+
+end Family
+end Thorp.Fourier
+
+namespace Thorp.Fourier.Family
+open scoped Classical
+variable {H : Type} [Group H] [Fintype H] (F : Family H)
+variable {I : Type} [Fintype I]
+
+def jointKernel (a : I → F.Index) (g : I → H) : ℂ := ∏ i, F.kernel (a i) (g i)
+
+lemma jointKernel_sum : (∑ a : I → F.Index, F.jointKernel a) = delta := by
+  funext g
+  simp only [Finset.sum_apply, jointKernel]
+  rw [← Fintype.prod_sum (fun i (j : F.Index) => F.kernel j (g i))]
+  have hh (i : I) : ∑ j : F.Index, F.kernel j (g i) = if g i = 1 then 1 else 0 :=
+    by simpa only [Finset.sum_apply, delta] using congrFun F.kernel_sum (g i)
+  simp only [hh]
+  by_cases hg : g = 1
+  · subst g; simp [delta]
+  · have hn : ∃ i, g i ≠ 1 := by
+      by_contra hn
+      push Not at hn
+      exact hg (funext hn)
+    obtain ⟨i, hi⟩ := hn
+    rw [Finset.prod_eq_zero (Finset.mem_univ i) (ite_eq_right hi)]
+    simp [delta, hg]
+
+lemma jointKernel_convolution (a b : I → F.Index) :
+    convolution (F.jointKernel a) (F.jointKernel b) = if a = b then F.jointKernel a else 0 := by
+  funext g
+  simp only [convolution, jointKernel, Pi.mul_apply, Pi.inv_apply, ← Finset.prod_mul_distrib]
+  rw [← Fintype.prod_sum (fun i (h : H) => F.kernel (a i) h * F.kernel (b i) (h⁻¹ * g i))]
+  have hh (i : I) : ∑ h, F.kernel (a i) h * F.kernel (b i) (h⁻¹ * g i) =
+      if a i = b i then F.kernel (a i) (g i) else 0 := by
+    have he := congrFun (F.kernel_convolution (a i) (b i)) (g i)
+    simpa only [convolution, ite_apply, Pi.zero_apply] using he
+  simp only [hh]
+  by_cases hab : a = b
+  · subst b; simp only [↓reduceIte, jointKernel]
+  · rw [ite_eq_right hab]
+    obtain ⟨i, hi⟩ : ∃ i, a i ≠ b i := by
+      by_contra hn; push Not at hn; exact hab (funext hn)
+    exact Finset.prod_eq_zero (Finset.mem_univ i) (ite_eq_right hi)
+
+lemma jointKernel_star (a : I → F.Index) (g : I → H) :
+    conj (F.jointKernel a g⁻¹) = F.jointKernel a g := by
+  simp only [jointKernel, map_prod, Pi.inv_apply, F.kernel_star]
+
+variable {E : Type uE} [AddCommGroup E] [Module ℂ E]
+
+def jointProjector (ρ : Representation ℂ (I → H) E) (a : I → F.Index) : Module.End ℂ E :=
+  integrated ρ (F.jointKernel a)
+
+lemma jointProjector_mul (ρ : Representation ℂ (I → H) E) (a b : I → F.Index) :
+    F.jointProjector ρ a * F.jointProjector ρ b = if a = b then F.jointProjector ρ a else 0 := by
+  rw [jointProjector, jointProjector, ← integrated_convolution, jointKernel_convolution]
+  split_ifs <;> simp only [integrated, Pi.zero_apply, zero_smul, Finset.sum_const_zero]
+
+lemma jointProjector_idempotent (ρ : Representation ℂ (I → H) E) (a : I → F.Index) :
+    IsIdempotentElem (F.jointProjector ρ a) := by
+  simpa only [IsIdempotentElem, ↓reduceIte] using F.jointProjector_mul ρ a a
+
+lemma jointProjector_sum (ρ : Representation ℂ (I → H) E) : ∑ a, F.jointProjector ρ a = 1 := by
+  simp only [jointProjector]
+  rw [← integrated_sum, F.jointKernel_sum, integrated_delta]
+
+def tensorList (a : I → F.Index) : List I → FDRep ℂ (I → H)
+  | [] => FDRep.of (Representation.trivial ℂ (I → H) ℂ)
+  | i :: l => FDRep.of (Representation.tprod ((F.rep (a i)).comp (Pi.evalMonoidHom (fun _ : I => H) i)) (tensorList a l).ρ)
+
+omit [Fintype I] in
+lemma tensorList_character (a : I → F.Index) (l : List I) (g : I → H) :
+    Representation.character (F.tensorList a l).ρ g = (l.map (fun i => (F.rep (a i)).character (g i))).prod := by
+  induction l with
+  | nil =>
+    change LinearMap.trace ℂ ℂ (1 : Module.End ℂ ℂ) = 1
+    simp
+  | cons i l ih =>
+    change (Representation.tprod ((F.rep (a i)).comp (Pi.evalMonoidHom (fun _ : I => H) i))
+      (F.tensorList a l).ρ).character g = _
+    rw [Representation.char_tensor ((F.rep (a i)).comp (Pi.evalMonoidHom (fun _ : I => H) i))
+      (F.tensorList a l).ρ]
+    change (F.rep (a i)).character (g i) * Representation.character (F.tensorList a l).ρ g = _
+    rw [ih]
+    rfl
+
+def tensorFamily (a : I → F.Index) : FDRep ℂ (I → H) := F.tensorList a Finset.univ.toList
+
+lemma tensorFamily_character (a : I → F.Index) (g : I → H) :
+    Representation.character (F.tensorFamily a).ρ g = ∏ i, (F.rep (a i)).character (g i) := by
+  rw [tensorFamily, tensorList_character]
+  exact Finset.prod_map_toList _ _
+
+lemma jointKernel_formula (a : I → F.Index) (g : I → H) :
+    F.jointKernel a g = (∏ i, (F.degree (a i) : ℂ)) * (Fintype.card (I → H) : ℂ)⁻¹ *
+      Representation.character (F.tensorFamily a).ρ g⁻¹ := by
+  simp only [jointKernel, kernel, Finset.prod_mul_distrib, Finset.prod_const,
+    Finset.card_univ, tensorFamily_character, Pi.inv_apply, Fintype.card_fun, Nat.cast_pow, inv_pow]
+
+lemma trace_jointProjector [FiniteDimensional ℂ E] (ρ : Representation ℂ (I → H) E)
+    (a : I → F.Index) : LinearMap.trace ℂ E (F.jointProjector ρ a) =
+      (∏ i, (F.degree (a i) : ℂ)) * Module.finrank ℂ (Representation.IntertwiningMap (F.tensorFamily a).ρ ρ) := by
+  let : Invertible (Nat.card (I → H) : ℂ) := invertibleOfNonzero (NeZero.ne _)
+  rw [jointProjector, trace_integrated]
+  simp only [jointKernel_formula]
+  rw [← Representation.card_inv_mul_sum_char_mul_char_eq_finrank]
+  simp only [Finset.mul_sum, Fintype.card_eq_nat_card]
+  apply Finset.sum_congr rfl
+  intro g _
+  ring
+
+theorem joint_degree_le [FiniteDimensional ℂ E] (ρ : Representation ℂ (I → H) E)
+    (a : I → F.Index) (ha : F.jointProjector ρ a ≠ 0) :
+    (∏ i, F.degree (a i)) ≤ Module.finrank ℂ E := by
+  have hp := F.jointProjector_idempotent ρ a
+  have ht := F.trace_jointProjector ρ a
+  rw [(LinearMap.IsIdempotentElem.isProj_range _ hp).trace] at ht
+  have hm : Module.finrank ℂ (Representation.IntertwiningMap (F.tensorFamily a).ρ ρ) ≠ 0 := by
+    intro hz
+    have hh := F.trace_jointProjector ρ a
+    rw [hz, Nat.cast_zero, mul_zero] at hh
+    exact ha (LinearMap.IsIdempotentElem.eq_zero_of_trace_eq_zero hp hh)
+  have hn : Module.finrank ℂ (LinearMap.range (F.jointProjector ρ a)) =
+      (∏ i, F.degree (a i)) * Module.finrank ℂ (Representation.IntertwiningMap (F.tensorFamily a).ρ ρ) := by
+    exact_mod_cast ht
+  calc
+    _ ≤ (∏ i, F.degree (a i)) * Module.finrank ℂ (Representation.IntertwiningMap (F.tensorFamily a).ρ ρ) :=
+      Nat.le_mul_of_pos_right _ (Nat.pos_of_ne_zero hm)
+    _ = _ := hn.symm
+    _ ≤ Module.finrank ℂ E := Submodule.finrank_le _
+
+end Thorp.Fourier.Family
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/LinearExtensions.lean b/lean/OAI/Probability/ThorpShuffle/LinearExtensions.lean
index cb1e94f24fc8451f8f3f3c9bb77efd7408bdd963..5cc265127ad1fa4c64abc93807c8f0c7959d913d 100644
--- a/lean/OAI/Probability/ThorpShuffle/LinearExtensions.lean
+++ b/lean/OAI/Probability/ThorpShuffle/LinearExtensions.lean
@@ -1,227 +1,227 @@
-import OAI.Probability.ThorpShuffle.Classification
-
-namespace OAI
-
-universe uA uα uI uB
-
-noncomputable section
-
-open scoped BigOperators ComplexConjugate InnerProductSpace
-open Filter
-
-namespace Thorp.Young
-open scoped Classical
-
-variable (A : Type uA) [Fintype A] [PartialOrder A]
-
-def Extensions := {e : A ≃ Fin (Fintype.card A) // StrictMono e}
-
-instance : Fintype (Extensions A) := inferInstanceAs (Fintype {e : A ≃ Fin (Fintype.card A) // StrictMono e})
-
-omit [PartialOrder A] in
-
-lemma labelByKey (k : A → ℕ) (hk : Function.Injective k) :
-    ∃ e : A ≃ Fin (Fintype.card A), ∀ x y, e x < e y ↔ k x < k y := by
-  let e : A ≃ Set.range k := Equiv.ofInjective k hk
-  let : Fintype (Set.range k) := Fintype.ofEquiv A e
-  let o := (Fintype.orderIsoFinOfCardEq (Set.range k) (Fintype.card_congr e.symm)).symm
-  exact ⟨e.trans o.toEquiv, fun x y => o.lt_iff_lt⟩
-
-omit [PartialOrder A] in
-
-lemma labelByRank (r : A → ℕ) :
-    ∃ e : A ≃ Fin (Fintype.card A), ∀ x y, r x < r y → e x < e y := by
-  let b := Fintype.equivFin A
-  let k (x : A) := r x * (Fintype.card A + 1) + (b x).val
-  have hk : Function.Injective k := by
-    intro x y h
-    have hx := (b x).isLt
-    have hy := (b y).isLt
-    have hr : r x = r y := by
-      change r x * (Fintype.card A + 1) + (b x).val = r y * (Fintype.card A + 1) + (b y).val at h
-      nlinarith
-    apply b.injective
-    apply Fin.ext
-    change r x * (Fintype.card A + 1) + (b x).val = r y * (Fintype.card A + 1) + (b y).val at h
-    rw [hr] at h
-    omega
-  obtain ⟨e, he⟩ := labelByKey A k hk
-  refine ⟨e, fun x y hxy => (he x y).mpr ?_⟩
-  dsimp [k]
-  have hx := (b x).isLt
-  nlinarith
-
-lemma extensions_nonempty : Nonempty (Extensions A) := by
-  let h : A ≃ LinearExtension A := Equiv.refl A
-  let : Fintype (LinearExtension A) := Fintype.ofEquiv A h
-  let e := (Fintype.orderIsoFinOfCardEq (LinearExtension A) (Fintype.card_congr h.symm)).symm
-  refine ⟨⟨h.trans e.toEquiv, ?_⟩⟩
-  exact e.strictMono.comp
-    (toLinearExtension.monotone.strictMono_of_injective fun _ _ hh => hh)
-
-lemma fiberPerm_le_extensions (r : A → ℕ) (hr : StrictMono r) :
-    Fintype.card (Thorp.Specht.fiberGroup r) ≤ Fintype.card (Extensions A) := by
-  obtain ⟨e, he⟩ := labelByRank A r
-  let F (g : Thorp.Specht.fiberGroup r) : Extensions A := ⟨g.val.trans e, by
-    intro x y hxy
-    apply he
-    rw [g.property x, g.property y]
-    exact hr hxy⟩
-  apply Fintype.card_le_of_injective F
-  intro g h hgh
-  apply Subtype.ext
-  apply Equiv.ext
-  intro x
-  apply e.injective
-  exact congrArg (fun z : Extensions A => z.val x) hgh
-
-end Thorp.Young
-
-namespace Thorp.Specht
-open scoped Classical
-variable {α : Type uα} [Fintype α]
-
-lemma tabloid_card_mul (r : α → ℕ) :
-    Fintype.card (Tabloid r) * Fintype.card (fiberGroup r) = (Fintype.card α).factorial := by
-  let G := Equiv.Perm α
-  let b := baseTabloid r
-  let e : MulAction.orbit G b ≃ Tabloid r := {
-    toFun := Subtype.val
-    invFun := fun f => ⟨f, by
-      obtain ⟨g, hg⟩ := f.property
-      refine ⟨g⁻¹, ?_⟩
-      apply Subtype.ext
-      funext x
-      change r ((g⁻¹ : Equiv.Perm α)⁻¹ x) = f.val x
-      rw [inv_inv]
-      exact (congrFun hg x).symm⟩
-    left_inv := fun _ => rfl
-    right_inv := fun _ => rfl }
-  let u : MulAction.stabilizer G b ≃ fiberGroup r := {
-    toFun := fun g => ⟨g.val, (stabilizes_tabloid_iff r g.val b).mp g.property⟩
-    invFun := fun g => ⟨g.val, (stabilizes_tabloid_iff r g.val b).mpr g.property⟩
-    left_inv := fun _ => rfl
-    right_inv := fun _ => rfl }
-  rw [← Fintype.card_congr e, ← Fintype.card_congr u,
-    MulAction.card_orbit_mul_card_stabilizer_eq_card_group, Fintype.card_perm]
-
-lemma tabloid_card_le (r : α → ℕ) (m : ℕ) (hr : ∀ x, r x < m) :
-    Fintype.card (Tabloid r) ≤ m ^ Fintype.card α := by
-  have hb (f : Tabloid r) (x : α) : f.val x < m := by
-    obtain ⟨g, hg⟩ := f.property
-    rw [hg]
-    exact hr (g x)
-  let F (f : Tabloid r) : α → Fin m := fun x => ⟨f.val x, hb f x⟩
-  have hi : Function.Injective F := by
-    intro f h hfh
-    apply Subtype.ext
-    funext x
-    exact congrArg Fin.val (congrFun hfh x)
-  have hh := Fintype.card_le_of_injective F hi
-  simpa only [Fintype.card_fun, Fintype.card_fin] using hh
-
-end Thorp.Specht
-
-namespace Thorp.Young
-open scoped Classical
-variable (A : Type uA) [Fintype A] [PartialOrder A]
-
-theorem rank_extension_bound (r : A → ℕ) (hr : StrictMono r) (m : ℕ) (hm : ∀ x, r x < m) :
-    (Fintype.card A).factorial ≤ m ^ Fintype.card A * Fintype.card (Extensions A) := by
-  rw [← Thorp.Specht.tabloid_card_mul r]
-  exact Nat.mul_le_mul (Thorp.Specht.tabloid_card_le r m hm) (fiberPerm_le_extensions A r hr)
-
-end Thorp.Young
-
-namespace Thorp.Specht
-open scoped Classical
-variable {n : ℕ} (D : YoungDiagram) (t : Fin n ≃ Thorp.Young.Cells D)
-
-lemma labels_card_le_degree {I : Type uI} [Fintype I]
-    (l : I → Thorp.Young.Cells D ≃ Fin n) (hl : ∀ i, StrictMono (l i))
-    (hi : Function.Injective l) : Fintype.card I ≤ Module.finrank ℂ (module D t).toSubmodule := by
-  let F (i : I) : StandardPerm D t := ⟨t.trans (l i), by
-    intro x y hxy
-    change l i (t (t.symm x)) < l i (t (t.symm y))
-    simpa only [Equiv.apply_symm_apply] using hl i hxy⟩
-  have hF : Function.Injective F := by
-    intro i j hij
-    apply hi
-    apply Equiv.ext
-    intro x
-    have hh := congrArg (fun g : StandardPerm D t => g.val (t.symm x)) hij
-    simpa only [F, Equiv.trans_apply, Equiv.apply_symm_apply] using hh
-  exact (Fintype.card_le_of_injective F hF).trans (standardPerm_card_le_degree D t)
-
-lemma extensions_card_le_degree :
-    Fintype.card (Thorp.Young.Extensions (Thorp.Young.Cells D)) ≤
-      Module.finrank ℂ (module D t).toSubmodule := by
-  have hc : Fintype.card (Thorp.Young.Cells D) = n := (Fintype.card_congr t.symm).trans (Fintype.card_fin n)
-  let c := finCongr hc
-  apply labels_card_le_degree D t (fun e => e.val.trans c)
-  · intro e x y hxy
-    exact e.property hxy
-  · intro e f hef
-    apply Subtype.ext
-    apply Equiv.ext
-    intro x
-    apply c.injective
-    exact congrArg (fun z : Thorp.Young.Cells D ≃ Fin n => z x) hef
-
-end Thorp.Specht
-
-namespace Thorp.Young
-open scoped Classical
-
-def transposeCells (D : YoungDiagram) : Cells D ≃o Cells D.transpose where
-  toFun x := ⟨x.val.swap, by
-    change x.val.swap ∈ D.transpose
-    rw [YoungDiagram.mem_transpose, Prod.swap_swap]
-    exact x.property⟩
-  invFun x := ⟨x.val.swap, (YoungDiagram.mem_transpose.mp x.property)⟩
-  left_inv x := by apply Subtype.ext; exact Prod.swap_swap x.val
-  right_inv x := by apply Subtype.ext; exact Prod.swap_swap x.val
-  map_rel_iff' := by intro x y; exact and_comm
-
-lemma extensions_card_orderIso {A : Type uA} {B : Type uB} [Fintype A] [Fintype B] [PartialOrder A] [PartialOrder B]
-    (e : A ≃o B) : Fintype.card (Extensions A) = Fintype.card (Extensions B) := by
-  let c := finCongr (Fintype.card_congr e.toEquiv)
-  let F : Extensions A ≃ Extensions B := {
-    toFun := fun l => ⟨(e.symm.toEquiv.trans l.val).trans c, fun _ _ h => l.property (e.symm.strictMono h)⟩
-    invFun := fun l => ⟨(e.toEquiv.trans l.val).trans c.symm, fun _ _ h => l.property (e.strictMono h)⟩
-    left_inv := by intro l; apply Subtype.ext; ext x; simp [c]
-    right_inv := by intro l; apply Subtype.ext; ext x; simp [c] }
-  exact Fintype.card_congr F
-
-lemma rank_strictMono (D : YoungDiagram) : StrictMono (fun x : Cells D => x.val.1 + x.val.2) := by
-  intro x y hxy
-  have hle := hxy.le
-  have hn : x.val ≠ y.val := fun h => hxy.ne (Subtype.ext h)
-  have he : x.val.1 ≠ y.val.1 ∨ x.val.2 ≠ y.val.2 := by
-    by_contra hh
-    push Not at hh
-    exact hn (Prod.ext hh.1 hh.2)
-  change x.val.1 + x.val.2 < y.val.1 + y.val.2
-  rcases he with he | he <;> have := hle.1 <;> have := hle.2 <;> omega
-
-def longest (D : YoungDiagram) : ℕ := max (D.rowLen 0) (D.colLen 0)
-
-lemma coord_lt_longest (D : YoungDiagram) (x : Cells D) :
-    x.val.1 < longest D ∧ x.val.2 < longest D := by
-  constructor
-  · exact (YoungDiagram.mem_iff_lt_colLen.mp (D.up_left_mem le_rfl (Nat.zero_le _) x.property)).trans_le (le_max_right _ _)
-  · exact (YoungDiagram.mem_iff_lt_rowLen.mp (D.up_left_mem (Nat.zero_le _) le_rfl x.property)).trans_le (le_max_left _ _)
-
-theorem factorial_le_rank_degree {n : ℕ} (D : YoungDiagram) (t : Fin n ≃ Cells D) :
-    n.factorial ≤ (2 * longest D) ^ n * Module.finrank ℂ (Thorp.Specht.module D t).toSubmodule := by
-  have hh := rank_extension_bound (Cells D) (fun x => x.val.1 + x.val.2) (rank_strictMono D)
-    (2 * longest D) (fun x => by have := coord_lt_longest D x; omega)
-  have hc : Fintype.card (Cells D) = n := (Fintype.card_congr t.symm).trans (Fintype.card_fin n)
-  rw [hc] at hh
-  exact hh.trans (Nat.mul_le_mul_left _ (Thorp.Specht.extensions_card_le_degree D t))
-
-end Thorp.Young
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.Classification
+
+namespace OAI
+
+universe uA uα uI uB
+
+noncomputable section
+
+open scoped BigOperators ComplexConjugate InnerProductSpace
+open Filter
+
+namespace Thorp.Young
+open scoped Classical
+
+variable (A : Type uA) [Fintype A] [PartialOrder A]
+
+def Extensions := {e : A ≃ Fin (Fintype.card A) // StrictMono e}
+
+instance : Fintype (Extensions A) := inferInstanceAs (Fintype {e : A ≃ Fin (Fintype.card A) // StrictMono e})
+
+omit [PartialOrder A] in
+
+lemma labelByKey (k : A → ℕ) (hk : Function.Injective k) :
+    ∃ e : A ≃ Fin (Fintype.card A), ∀ x y, e x < e y ↔ k x < k y := by
+  let e : A ≃ Set.range k := Equiv.ofInjective k hk
+  let : Fintype (Set.range k) := Fintype.ofEquiv A e
+  let o := (Fintype.orderIsoFinOfCardEq (Set.range k) (Fintype.card_congr e.symm)).symm
+  exact ⟨e.trans o.toEquiv, fun x y => o.lt_iff_lt⟩
+
+omit [PartialOrder A] in
+
+lemma labelByRank (r : A → ℕ) :
+    ∃ e : A ≃ Fin (Fintype.card A), ∀ x y, r x < r y → e x < e y := by
+  let b := Fintype.equivFin A
+  let k (x : A) := r x * (Fintype.card A + 1) + (b x).val
+  have hk : Function.Injective k := by
+    intro x y h
+    have hx := (b x).isLt
+    have hy := (b y).isLt
+    have hr : r x = r y := by
+      change r x * (Fintype.card A + 1) + (b x).val = r y * (Fintype.card A + 1) + (b y).val at h
+      nlinarith
+    apply b.injective
+    apply Fin.ext
+    change r x * (Fintype.card A + 1) + (b x).val = r y * (Fintype.card A + 1) + (b y).val at h
+    rw [hr] at h
+    omega
+  obtain ⟨e, he⟩ := labelByKey A k hk
+  refine ⟨e, fun x y hxy => (he x y).mpr ?_⟩
+  dsimp [k]
+  have hx := (b x).isLt
+  nlinarith
+
+lemma extensions_nonempty : Nonempty (Extensions A) := by
+  let h : A ≃ LinearExtension A := Equiv.refl A
+  let : Fintype (LinearExtension A) := Fintype.ofEquiv A h
+  let e := (Fintype.orderIsoFinOfCardEq (LinearExtension A) (Fintype.card_congr h.symm)).symm
+  refine ⟨⟨h.trans e.toEquiv, ?_⟩⟩
+  exact e.strictMono.comp
+    (toLinearExtension.monotone.strictMono_of_injective fun _ _ hh => hh)
+
+lemma fiberPerm_le_extensions (r : A → ℕ) (hr : StrictMono r) :
+    Fintype.card (Thorp.Specht.fiberGroup r) ≤ Fintype.card (Extensions A) := by
+  obtain ⟨e, he⟩ := labelByRank A r
+  let F (g : Thorp.Specht.fiberGroup r) : Extensions A := ⟨g.val.trans e, by
+    intro x y hxy
+    apply he
+    rw [g.property x, g.property y]
+    exact hr hxy⟩
+  apply Fintype.card_le_of_injective F
+  intro g h hgh
+  apply Subtype.ext
+  apply Equiv.ext
+  intro x
+  apply e.injective
+  exact congrArg (fun z : Extensions A => z.val x) hgh
+
+end Thorp.Young
+
+namespace Thorp.Specht
+open scoped Classical
+variable {α : Type uα} [Fintype α]
+
+lemma tabloid_card_mul (r : α → ℕ) :
+    Fintype.card (Tabloid r) * Fintype.card (fiberGroup r) = (Fintype.card α).factorial := by
+  let G := Equiv.Perm α
+  let b := baseTabloid r
+  let e : MulAction.orbit G b ≃ Tabloid r := {
+    toFun := Subtype.val
+    invFun := fun f => ⟨f, by
+      obtain ⟨g, hg⟩ := f.property
+      refine ⟨g⁻¹, ?_⟩
+      apply Subtype.ext
+      funext x
+      change r ((g⁻¹ : Equiv.Perm α)⁻¹ x) = f.val x
+      rw [inv_inv]
+      exact (congrFun hg x).symm⟩
+    left_inv := fun _ => rfl
+    right_inv := fun _ => rfl }
+  let u : MulAction.stabilizer G b ≃ fiberGroup r := {
+    toFun := fun g => ⟨g.val, (stabilizes_tabloid_iff r g.val b).mp g.property⟩
+    invFun := fun g => ⟨g.val, (stabilizes_tabloid_iff r g.val b).mpr g.property⟩
+    left_inv := fun _ => rfl
+    right_inv := fun _ => rfl }
+  rw [← Fintype.card_congr e, ← Fintype.card_congr u,
+    MulAction.card_orbit_mul_card_stabilizer_eq_card_group, Fintype.card_perm]
+
+lemma tabloid_card_le (r : α → ℕ) (m : ℕ) (hr : ∀ x, r x < m) :
+    Fintype.card (Tabloid r) ≤ m ^ Fintype.card α := by
+  have hb (f : Tabloid r) (x : α) : f.val x < m := by
+    obtain ⟨g, hg⟩ := f.property
+    rw [hg]
+    exact hr (g x)
+  let F (f : Tabloid r) : α → Fin m := fun x => ⟨f.val x, hb f x⟩
+  have hi : Function.Injective F := by
+    intro f h hfh
+    apply Subtype.ext
+    funext x
+    exact congrArg Fin.val (congrFun hfh x)
+  have hh := Fintype.card_le_of_injective F hi
+  simpa only [Fintype.card_fun, Fintype.card_fin] using hh
+
+end Thorp.Specht
+
+namespace Thorp.Young
+open scoped Classical
+variable (A : Type uA) [Fintype A] [PartialOrder A]
+
+theorem rank_extension_bound (r : A → ℕ) (hr : StrictMono r) (m : ℕ) (hm : ∀ x, r x < m) :
+    (Fintype.card A).factorial ≤ m ^ Fintype.card A * Fintype.card (Extensions A) := by
+  rw [← Thorp.Specht.tabloid_card_mul r]
+  exact Nat.mul_le_mul (Thorp.Specht.tabloid_card_le r m hm) (fiberPerm_le_extensions A r hr)
+
+end Thorp.Young
+
+namespace Thorp.Specht
+open scoped Classical
+variable {n : ℕ} (D : YoungDiagram) (t : Fin n ≃ Thorp.Young.Cells D)
+
+lemma labels_card_le_degree {I : Type uI} [Fintype I]
+    (l : I → Thorp.Young.Cells D ≃ Fin n) (hl : ∀ i, StrictMono (l i))
+    (hi : Function.Injective l) : Fintype.card I ≤ Module.finrank ℂ (module D t).toSubmodule := by
+  let F (i : I) : StandardPerm D t := ⟨t.trans (l i), by
+    intro x y hxy
+    change l i (t (t.symm x)) < l i (t (t.symm y))
+    simpa only [Equiv.apply_symm_apply] using hl i hxy⟩
+  have hF : Function.Injective F := by
+    intro i j hij
+    apply hi
+    apply Equiv.ext
+    intro x
+    have hh := congrArg (fun g : StandardPerm D t => g.val (t.symm x)) hij
+    simpa only [F, Equiv.trans_apply, Equiv.apply_symm_apply] using hh
+  exact (Fintype.card_le_of_injective F hF).trans (standardPerm_card_le_degree D t)
+
+lemma extensions_card_le_degree :
+    Fintype.card (Thorp.Young.Extensions (Thorp.Young.Cells D)) ≤
+      Module.finrank ℂ (module D t).toSubmodule := by
+  have hc : Fintype.card (Thorp.Young.Cells D) = n := (Fintype.card_congr t.symm).trans (Fintype.card_fin n)
+  let c := finCongr hc
+  apply labels_card_le_degree D t (fun e => e.val.trans c)
+  · intro e x y hxy
+    exact e.property hxy
+  · intro e f hef
+    apply Subtype.ext
+    apply Equiv.ext
+    intro x
+    apply c.injective
+    exact congrArg (fun z : Thorp.Young.Cells D ≃ Fin n => z x) hef
+
+end Thorp.Specht
+
+namespace Thorp.Young
+open scoped Classical
+
+def transposeCells (D : YoungDiagram) : Cells D ≃o Cells D.transpose where
+  toFun x := ⟨x.val.swap, by
+    change x.val.swap ∈ D.transpose
+    rw [YoungDiagram.mem_transpose, Prod.swap_swap]
+    exact x.property⟩
+  invFun x := ⟨x.val.swap, (YoungDiagram.mem_transpose.mp x.property)⟩
+  left_inv x := by apply Subtype.ext; exact Prod.swap_swap x.val
+  right_inv x := by apply Subtype.ext; exact Prod.swap_swap x.val
+  map_rel_iff' := by intro x y; exact and_comm
+
+lemma extensions_card_orderIso {A : Type uA} {B : Type uB} [Fintype A] [Fintype B] [PartialOrder A] [PartialOrder B]
+    (e : A ≃o B) : Fintype.card (Extensions A) = Fintype.card (Extensions B) := by
+  let c := finCongr (Fintype.card_congr e.toEquiv)
+  let F : Extensions A ≃ Extensions B := {
+    toFun := fun l => ⟨(e.symm.toEquiv.trans l.val).trans c, fun _ _ h => l.property (e.symm.strictMono h)⟩
+    invFun := fun l => ⟨(e.toEquiv.trans l.val).trans c.symm, fun _ _ h => l.property (e.strictMono h)⟩
+    left_inv := by intro l; apply Subtype.ext; ext x; simp [c]
+    right_inv := by intro l; apply Subtype.ext; ext x; simp [c] }
+  exact Fintype.card_congr F
+
+lemma rank_strictMono (D : YoungDiagram) : StrictMono (fun x : Cells D => x.val.1 + x.val.2) := by
+  intro x y hxy
+  have hle := hxy.le
+  have hn : x.val ≠ y.val := fun h => hxy.ne (Subtype.ext h)
+  have he : x.val.1 ≠ y.val.1 ∨ x.val.2 ≠ y.val.2 := by
+    by_contra hh
+    push Not at hh
+    exact hn (Prod.ext hh.1 hh.2)
+  change x.val.1 + x.val.2 < y.val.1 + y.val.2
+  rcases he with he | he <;> have := hle.1 <;> have := hle.2 <;> omega
+
+def longest (D : YoungDiagram) : ℕ := max (D.rowLen 0) (D.colLen 0)
+
+lemma coord_lt_longest (D : YoungDiagram) (x : Cells D) :
+    x.val.1 < longest D ∧ x.val.2 < longest D := by
+  constructor
+  · exact (YoungDiagram.mem_iff_lt_colLen.mp (D.up_left_mem le_rfl (Nat.zero_le _) x.property)).trans_le (le_max_right _ _)
+  · exact (YoungDiagram.mem_iff_lt_rowLen.mp (D.up_left_mem (Nat.zero_le _) le_rfl x.property)).trans_le (le_max_left _ _)
+
+theorem factorial_le_rank_degree {n : ℕ} (D : YoungDiagram) (t : Fin n ≃ Cells D) :
+    n.factorial ≤ (2 * longest D) ^ n * Module.finrank ℂ (Thorp.Specht.module D t).toSubmodule := by
+  have hh := rank_extension_bound (Cells D) (fun x => x.val.1 + x.val.2) (rank_strictMono D)
+    (2 * longest D) (fun x => by have := coord_lt_longest D x; omega)
+  have hc : Fintype.card (Cells D) = n := (Fintype.card_congr t.symm).trans (Fintype.card_fin n)
+  rw [hc] at hh
+  exact hh.trans (Nat.mul_le_mul_left _ (Thorp.Specht.extensions_card_le_degree D t))
+
+end Thorp.Young
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/Main.lean b/lean/OAI/Probability/ThorpShuffle/Main.lean
index 1591c20cc523c7fab723bf2797c8a29110ed70b4..53a54949d4e9f65f28d13952c30675e1d0a68781 100644
--- a/lean/OAI/Probability/ThorpShuffle/Main.lean
+++ b/lean/OAI/Probability/ThorpShuffle/Main.lean
@@ -1,59 +1,59 @@
-import OAI.Probability.ThorpShuffle.EightBlocks
-
-namespace OAI
-
-noncomputable section
-
-open scoped BigOperators ComplexConjugate InnerProductSpace
-open Filter Topology
-
-namespace Thorp
-
-theorem full_mixing :
-    Tendsto (fun d : ℕ => distance d (1600 * d)) atTop (nhds 0) := by
-  apply (tendsto_add_atTop_iff_nat 3).mp
-  simpa only [Nat.add_assoc] using Block.full_mixing_shift
-
-theorem mixingTime_isBigO_orderFunction :
-    Asymptotics.IsBigO atTop (fun d : ℕ => (mixingTime d : ℝ)) orderFunction := by
-  rw [Asymptotics.isBigO_iff]
-  refine ⟨1600, ?_⟩
-  have he : ∀ᶠ d : ℕ in atTop, distance d (1600*d) < 1/4 :=
-    full_mixing.eventually (eventually_lt_nhds (by norm_num : (0:ℝ) < 1/4))
-  filter_upwards [he] with d hd
-  have ht : mixingTime d ≤ 1600*d := Nat.sInf_le hd.le
-  simp only [orderFunction, Real.norm_eq_abs, abs_of_nonneg (Nat.cast_nonneg (α:=ℝ) (mixingTime d)),
-    abs_of_nonneg (Nat.cast_nonneg (α:=ℝ) d)]
-  exact_mod_cast ht
-
-theorem mixingTime_isTheta_orderFunction :
-    Asymptotics.IsTheta atTop (fun d : ℕ => (mixingTime d : ℝ)) orderFunction :=
-  ⟨mixingTime_isBigO_orderFunction, orderFunction_isBigO_mixingTime⟩
-
-theorem mixingTime_isTheta_log :
-    Asymptotics.IsTheta atTop (fun d : ℕ => (mixingTime d : ℝ))
-      (fun d : ℕ => Real.log (2^d : ℝ)) := by
-  have h := Asymptotics.IsTheta.const_mul_right
-    (show Real.log 2 ≠ 0 from (Real.log_pos (by norm_num : (1:ℝ) < 2)).ne')
-    mixingTime_isTheta_orderFunction
-  simpa only [orderFunction, Real.log_pow, mul_comm] using h
-
-theorem main :
-    Tendsto (fun d : ℕ => distance d (1600 * d)) atTop (nhds 0) ∧
-    (∀ d : ℕ, 1 ≤ d → ∃ t : ℕ, distance d t ≤ 1 / 4) ∧
-    (∀ (d t : ℕ) (g₀ : State d), tv (lawFrom d t g₀) (uniform d) = distance d t) ∧
-    (∀ d : ℕ, 1 ≤ d → supportThreshold d ≤ (mixingTime d : ℤ)) ∧
-    Asymptotics.IsBigO atTop
-      (fun d : ℕ => (supportThreshold d : ℝ) - 2 * (d : ℝ)) (fun _ => (1 : ℝ)) ∧
-    Asymptotics.IsTheta atTop (fun d : ℕ => (mixingTime d : ℝ)) orderFunction ∧
-    Asymptotics.IsTheta atTop (fun d : ℕ => (mixingTime d : ℝ))
-      (fun d : ℕ => Real.log (2 ^ d : ℝ)) := by
-  exact ⟨full_mixing, fun _ hd => exists_mixing_time hd, worst_start_eq,
-    fun _ hd => exact_support_lower hd, supportThreshold_isBigO,
-    mixingTime_isTheta_orderFunction, mixingTime_isTheta_log⟩
-
-end Thorp
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.EightBlocks
+
+namespace OAI
+
+noncomputable section
+
+open scoped BigOperators ComplexConjugate InnerProductSpace
+open Filter Topology
+
+namespace Thorp
+
+theorem full_mixing :
+    Tendsto (fun d : ℕ => distance d (1600 * d)) atTop (nhds 0) := by
+  apply (tendsto_add_atTop_iff_nat 3).mp
+  simpa only [Nat.add_assoc] using Block.full_mixing_shift
+
+theorem mixingTime_isBigO_orderFunction :
+    Asymptotics.IsBigO atTop (fun d : ℕ => (mixingTime d : ℝ)) orderFunction := by
+  rw [Asymptotics.isBigO_iff]
+  refine ⟨1600, ?_⟩
+  have he : ∀ᶠ d : ℕ in atTop, distance d (1600*d) < 1/4 :=
+    full_mixing.eventually (eventually_lt_nhds (by norm_num : (0:ℝ) < 1/4))
+  filter_upwards [he] with d hd
+  have ht : mixingTime d ≤ 1600*d := Nat.sInf_le hd.le
+  simp only [orderFunction, Real.norm_eq_abs, abs_of_nonneg (Nat.cast_nonneg (α:=ℝ) (mixingTime d)),
+    abs_of_nonneg (Nat.cast_nonneg (α:=ℝ) d)]
+  exact_mod_cast ht
+
+theorem mixingTime_isTheta_orderFunction :
+    Asymptotics.IsTheta atTop (fun d : ℕ => (mixingTime d : ℝ)) orderFunction :=
+  ⟨mixingTime_isBigO_orderFunction, orderFunction_isBigO_mixingTime⟩
+
+theorem mixingTime_isTheta_log :
+    Asymptotics.IsTheta atTop (fun d : ℕ => (mixingTime d : ℝ))
+      (fun d : ℕ => Real.log (2^d : ℝ)) := by
+  have h := Asymptotics.IsTheta.const_mul_right
+    (show Real.log 2 ≠ 0 from (Real.log_pos (by norm_num : (1:ℝ) < 2)).ne')
+    mixingTime_isTheta_orderFunction
+  simpa only [orderFunction, Real.log_pow, mul_comm] using h
+
+theorem main :
+    Tendsto (fun d : ℕ => distance d (1600 * d)) atTop (nhds 0) ∧
+    (∀ d : ℕ, 1 ≤ d → ∃ t : ℕ, distance d t ≤ 1 / 4) ∧
+    (∀ (d t : ℕ) (g₀ : State d), tv (lawFrom d t g₀) (uniform d) = distance d t) ∧
+    (∀ d : ℕ, 1 ≤ d → supportThreshold d ≤ (mixingTime d : ℤ)) ∧
+    Asymptotics.IsBigO atTop
+      (fun d : ℕ => (supportThreshold d : ℝ) - 2 * (d : ℝ)) (fun _ => (1 : ℝ)) ∧
+    Asymptotics.IsTheta atTop (fun d : ℕ => (mixingTime d : ℝ)) orderFunction ∧
+    Asymptotics.IsTheta atTop (fun d : ℕ => (mixingTime d : ℝ))
+      (fun d : ℕ => Real.log (2 ^ d : ℝ)) := by
+  exact ⟨full_mixing, fun _ hd => exists_mixing_time hd, worst_start_eq,
+    fun _ hd => exact_support_lower hd, supportThreshold_isBigO,
+    mixingTime_isTheta_orderFunction, mixingTime_isTheta_log⟩
+
+end Thorp
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/MarginalBounds.lean b/lean/OAI/Probability/ThorpShuffle/MarginalBounds.lean
index 3f2dc96aca856999805624515a96f1d84ba329a7..486f1aa8aac72a17f183cf8462e22b5cc45ded82 100644
--- a/lean/OAI/Probability/ThorpShuffle/MarginalBounds.lean
+++ b/lean/OAI/Probability/ThorpShuffle/MarginalBounds.lean
@@ -1,141 +1,141 @@
-import OAI.Probability.ThorpShuffle.PrefixTests
-
-namespace OAI
-
-universe uΩ uα uΩ'
-
-noncomputable section
-
-open scoped BigOperators
-open Filter
-
-namespace Thorp
-
-namespace Conditional
-
-theorem fairMass_test {Ω : Type uΩ} {α : Type uα} [Fintype Ω] [Fintype α]
-    (a : Ω → α) (f : α → ℝ) :
-    (∑ x, fairMass a x * f x) = mean (fun ω => f (a ω)) := by
-  classical
-  unfold fairMass mean
-  simp only [div_mul_eq_mul_div, ← Finset.sum_div, Finset.sum_mul]
-  congr 1
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro ω _
-  simp
-
-theorem fairMass_tv_le_of_test {Ω : Type uΩ} {Ω' : Type uΩ'} {α : Type uα} [Fintype Ω] [Fintype Ω'] [Fintype α]
-    (a : Ω → α) (b : Ω' → α) (δ : ℝ)
-    (h : ∀ f : α → ℝ, (∀ x, |f x| ≤ 1) →
-      |mean (fun ω => f (a ω)) - mean (fun ω => f (b ω))| ≤ δ) :
-    tv (fairMass a) (fairMass b) ≤ δ / 2 := by
-  classical
-  let f : α → ℝ := fun x => if 0 ≤ fairMass a x - fairMass b x then 1 else -1
-  have hf : ∀ x, |f x| ≤ 1 := by intro x; dsimp [f]; split_ifs <;> norm_num
-  have hh := h f hf
-  rw [← fairMass_test, ← fairMass_test, ← Finset.sum_sub_distrib] at hh
-  have hid (x : α) : fairMass a x * f x - fairMass b x * f x = |fairMass a x - fairMass b x| := by
-    dsimp [f]
-    split_ifs with hx
-    · rw [abs_of_nonneg hx]; ring
-    · rw [abs_of_neg (lt_of_not_ge hx)]; ring
-  simp_rw [hid] at hh
-  rw [abs_of_nonneg (Finset.sum_nonneg (fun _ _ => abs_nonneg _))] at hh
-  unfold tv
-  linarith
-
-theorem marginal_tv_200 (d m k : ℕ)
-    (hm : Fintype.card (Position (d + 2)) ≤ 8 * m)
-    (hk : k + m ≤ Fintype.card (Position (d + 2)))
-    (e : Fin k → Position (d + 2)) (he : Function.Injective e) :
-    tv (fairMass (fun ω : History (d + 2) (200 * (d + 2)) =>
-          (run (d + 2) (200 * (d + 2)) ω) ∘ e))
-      (fairMass (fun g : State (d + 2) => g ∘ e)) ≤
-    ((k : ℝ) / 2) * Real.sqrt ((Fintype.card (Position (d + 2)) : ℝ) *
-      ((1 / 2 : ℝ) ^ (8 * (d + 2)) + 32 * (9 / 10 : ℝ) ^ m)) := by
-  have h := fairMass_tv_le_of_test
-    (fun ω : History (d + 2) (200 * (d + 2)) => (run (d + 2) (200 * (d + 2)) ω) ∘ e)
-    (fun g : State (d + 2) => g ∘ e)
-    ((k : ℝ) * Real.sqrt ((Fintype.card (Position (d + 2)) : ℝ) *
-      ((1 / 2 : ℝ) ^ (8 * (d + 2)) + 32 * (9 / 10 : ℝ) ^ m)))
-    (prefix_test_bound d m hm k hk e he)
-  convert h using 1; ring
-
-end Conditional
-
-namespace Conditional
-
-def marginalRate (d : ℕ) : ℝ :=
-  Real.sqrt (3136 * ((1 / 32 : ℝ) ^ d + (2 ^ d : ℝ) ^ 3 * (9 / 10 : ℝ) ^ (2 ^ d)))
-
-theorem marginalRate_tendsto : Tendsto marginalRate atTop (nhds 0) := by
-  have h₁ := tendsto_pow_atTop_nhds_zero_of_lt_one
-    (by norm_num : (0 : ℝ) ≤ 1 / 32) (by norm_num : (1 / 32 : ℝ) < 1)
-  have h₂ := (tendsto_pow_const_mul_const_pow_of_lt_one 3
-    (by norm_num : (0 : ℝ) ≤ 9 / 10) (by norm_num : (9 / 10 : ℝ) < 1)).comp
-    (tendsto_pow_atTop_atTop_of_one_lt (by norm_num : (1 : ℕ) < 2))
-  have hh := ((h₁.add h₂).const_mul 3136).sqrt
-  unfold marginalRate
-  simpa only [Nat.cast_pow, Nat.cast_ofNat, Function.comp_def,
-    add_zero, mul_zero, Real.sqrt_zero] using hh
-
-theorem marginal_tv_rate (d : ℕ) (e : Fin (7 * 2 ^ d) → Position (d + 1 + 2))
-    (he : Function.Injective e) :
-    tv (fairMass (fun ω : History (d + 1 + 2) (200 * (d + 1 + 2)) =>
-          (run (d + 1 + 2) (200 * (d + 1 + 2)) ω) ∘ e))
-      (fairMass (fun g : State (d + 1 + 2) => g ∘ e)) ≤ marginalRate d := by
-  have hc : Fintype.card (Position (d + 1 + 2)) = 8 * 2 ^ d := by
-    simp only [card_position, pow_add]
-    norm_num
-    omega
-  have hh := marginal_tv_200 (d + 1) (2 ^ d) (7 * 2 ^ d)
-    (by rw [hc]) (by rw [hc]; omega) e he
-  simp only [Nat.add_assoc] at hh
-  have hbound :
-      (((7 * 2 ^ d : ℕ) : ℝ) / 2) *
-        Real.sqrt ((Fintype.card (Position (d + 1 + 2)) : ℝ) *
-          ((1 / 2 : ℝ) ^ (8 * (d + 3)) + 32 * (9 / 10 : ℝ) ^ (2 ^ d))) ≤ marginalRate d := by
-    unfold marginalRate
-    apply Real.le_sqrt_of_sq_le
-    have hp : (1 / 2 : ℝ) ^ (8 * (d + 3)) ≤ (1 / 2 : ℝ) ^ (8 * d) :=
-      pow_le_pow_of_le_one (by norm_num) (by norm_num) (by omega)
-    have hid : (2 ^ d : ℝ) ^ 3 * (1 / 2 : ℝ) ^ (8 * d) = (1 / 32 : ℝ) ^ d := by
-      rw [← pow_mul, Nat.mul_comm d 3, pow_mul, pow_mul, ← mul_pow]
-      norm_num
-    have hp' := mul_le_mul_of_nonneg_left hp (by positivity : (0 : ℝ) ≤ (2 ^ d : ℝ) ^ 3)
-    rw [hid] at hp'
-    have hsq :
-        (((7 * 2 ^ d : ℕ) : ℝ) / 2 *
-          Real.sqrt ((Fintype.card (Position (d + 1 + 2)) : ℝ) *
-            ((1 / 2 : ℝ) ^ (8 * (d + 3)) + 32 * (9 / 10 : ℝ) ^ (2 ^ d)))) ^ 2 =
-        98 * (2 ^ d : ℝ) ^ 3 *
-          ((1 / 2 : ℝ) ^ (8 * (d + 3)) + 32 * (9 / 10 : ℝ) ^ (2 ^ d)) := by
-      rw [mul_pow, Real.sq_sqrt (by positivity), hc]
-      push_cast
-      ring
-    rw [hsq]
-    nlinarith [pow_nonneg (by norm_num : (0 : ℝ) ≤ 1 / 32) d]
-  have hone := le_trans hh hbound
-  exact hone
-
-theorem marginal_tendsto (e : ∀ d : ℕ, Fin (7 * 2 ^ d) → Position (d + 1 + 2))
-    (he : ∀ d, Function.Injective (e d)) :
-    Tendsto (fun d : ℕ =>
-      tv (fairMass (fun ω : History (d + 1 + 2) (200 * (d + 1 + 2)) =>
-          (run (d + 1 + 2) (200 * (d + 1 + 2)) ω) ∘ e d))
-        (fairMass (fun g : State (d + 1 + 2) => g ∘ e d))) atTop (nhds 0) := by
-  apply tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds marginalRate_tendsto
-  · intro d
-    unfold tv
-    positivity
-  · intro d
-    exact marginal_tv_rate d (e d) (he d)
-
-end Conditional
-
-end Thorp
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.PrefixTests
+
+namespace OAI
+
+universe uΩ uα uΩ'
+
+noncomputable section
+
+open scoped BigOperators
+open Filter
+
+namespace Thorp
+
+namespace Conditional
+
+theorem fairMass_test {Ω : Type uΩ} {α : Type uα} [Fintype Ω] [Fintype α]
+    (a : Ω → α) (f : α → ℝ) :
+    (∑ x, fairMass a x * f x) = mean (fun ω => f (a ω)) := by
+  classical
+  unfold fairMass mean
+  simp only [div_mul_eq_mul_div, ← Finset.sum_div, Finset.sum_mul]
+  congr 1
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro ω _
+  simp
+
+theorem fairMass_tv_le_of_test {Ω : Type uΩ} {Ω' : Type uΩ'} {α : Type uα} [Fintype Ω] [Fintype Ω'] [Fintype α]
+    (a : Ω → α) (b : Ω' → α) (δ : ℝ)
+    (h : ∀ f : α → ℝ, (∀ x, |f x| ≤ 1) →
+      |mean (fun ω => f (a ω)) - mean (fun ω => f (b ω))| ≤ δ) :
+    tv (fairMass a) (fairMass b) ≤ δ / 2 := by
+  classical
+  let f : α → ℝ := fun x => if 0 ≤ fairMass a x - fairMass b x then 1 else -1
+  have hf : ∀ x, |f x| ≤ 1 := by intro x; dsimp [f]; split_ifs <;> norm_num
+  have hh := h f hf
+  rw [← fairMass_test, ← fairMass_test, ← Finset.sum_sub_distrib] at hh
+  have hid (x : α) : fairMass a x * f x - fairMass b x * f x = |fairMass a x - fairMass b x| := by
+    dsimp [f]
+    split_ifs with hx
+    · rw [abs_of_nonneg hx]; ring
+    · rw [abs_of_neg (lt_of_not_ge hx)]; ring
+  simp_rw [hid] at hh
+  rw [abs_of_nonneg (Finset.sum_nonneg (fun _ _ => abs_nonneg _))] at hh
+  unfold tv
+  linarith
+
+theorem marginal_tv_200 (d m k : ℕ)
+    (hm : Fintype.card (Position (d + 2)) ≤ 8 * m)
+    (hk : k + m ≤ Fintype.card (Position (d + 2)))
+    (e : Fin k → Position (d + 2)) (he : Function.Injective e) :
+    tv (fairMass (fun ω : History (d + 2) (200 * (d + 2)) =>
+          (run (d + 2) (200 * (d + 2)) ω) ∘ e))
+      (fairMass (fun g : State (d + 2) => g ∘ e)) ≤
+    ((k : ℝ) / 2) * Real.sqrt ((Fintype.card (Position (d + 2)) : ℝ) *
+      ((1 / 2 : ℝ) ^ (8 * (d + 2)) + 32 * (9 / 10 : ℝ) ^ m)) := by
+  have h := fairMass_tv_le_of_test
+    (fun ω : History (d + 2) (200 * (d + 2)) => (run (d + 2) (200 * (d + 2)) ω) ∘ e)
+    (fun g : State (d + 2) => g ∘ e)
+    ((k : ℝ) * Real.sqrt ((Fintype.card (Position (d + 2)) : ℝ) *
+      ((1 / 2 : ℝ) ^ (8 * (d + 2)) + 32 * (9 / 10 : ℝ) ^ m)))
+    (prefix_test_bound d m hm k hk e he)
+  convert h using 1; ring
+
+end Conditional
+
+namespace Conditional
+
+def marginalRate (d : ℕ) : ℝ :=
+  Real.sqrt (3136 * ((1 / 32 : ℝ) ^ d + (2 ^ d : ℝ) ^ 3 * (9 / 10 : ℝ) ^ (2 ^ d)))
+
+theorem marginalRate_tendsto : Tendsto marginalRate atTop (nhds 0) := by
+  have h₁ := tendsto_pow_atTop_nhds_zero_of_lt_one
+    (by norm_num : (0 : ℝ) ≤ 1 / 32) (by norm_num : (1 / 32 : ℝ) < 1)
+  have h₂ := (tendsto_pow_const_mul_const_pow_of_lt_one 3
+    (by norm_num : (0 : ℝ) ≤ 9 / 10) (by norm_num : (9 / 10 : ℝ) < 1)).comp
+    (tendsto_pow_atTop_atTop_of_one_lt (by norm_num : (1 : ℕ) < 2))
+  have hh := ((h₁.add h₂).const_mul 3136).sqrt
+  unfold marginalRate
+  simpa only [Nat.cast_pow, Nat.cast_ofNat, Function.comp_def,
+    add_zero, mul_zero, Real.sqrt_zero] using hh
+
+theorem marginal_tv_rate (d : ℕ) (e : Fin (7 * 2 ^ d) → Position (d + 1 + 2))
+    (he : Function.Injective e) :
+    tv (fairMass (fun ω : History (d + 1 + 2) (200 * (d + 1 + 2)) =>
+          (run (d + 1 + 2) (200 * (d + 1 + 2)) ω) ∘ e))
+      (fairMass (fun g : State (d + 1 + 2) => g ∘ e)) ≤ marginalRate d := by
+  have hc : Fintype.card (Position (d + 1 + 2)) = 8 * 2 ^ d := by
+    simp only [card_position, pow_add]
+    norm_num
+    omega
+  have hh := marginal_tv_200 (d + 1) (2 ^ d) (7 * 2 ^ d)
+    (by rw [hc]) (by rw [hc]; omega) e he
+  simp only [Nat.add_assoc] at hh
+  have hbound :
+      (((7 * 2 ^ d : ℕ) : ℝ) / 2) *
+        Real.sqrt ((Fintype.card (Position (d + 1 + 2)) : ℝ) *
+          ((1 / 2 : ℝ) ^ (8 * (d + 3)) + 32 * (9 / 10 : ℝ) ^ (2 ^ d))) ≤ marginalRate d := by
+    unfold marginalRate
+    apply Real.le_sqrt_of_sq_le
+    have hp : (1 / 2 : ℝ) ^ (8 * (d + 3)) ≤ (1 / 2 : ℝ) ^ (8 * d) :=
+      pow_le_pow_of_le_one (by norm_num) (by norm_num) (by omega)
+    have hid : (2 ^ d : ℝ) ^ 3 * (1 / 2 : ℝ) ^ (8 * d) = (1 / 32 : ℝ) ^ d := by
+      rw [← pow_mul, Nat.mul_comm d 3, pow_mul, pow_mul, ← mul_pow]
+      norm_num
+    have hp' := mul_le_mul_of_nonneg_left hp (by positivity : (0 : ℝ) ≤ (2 ^ d : ℝ) ^ 3)
+    rw [hid] at hp'
+    have hsq :
+        (((7 * 2 ^ d : ℕ) : ℝ) / 2 *
+          Real.sqrt ((Fintype.card (Position (d + 1 + 2)) : ℝ) *
+            ((1 / 2 : ℝ) ^ (8 * (d + 3)) + 32 * (9 / 10 : ℝ) ^ (2 ^ d)))) ^ 2 =
+        98 * (2 ^ d : ℝ) ^ 3 *
+          ((1 / 2 : ℝ) ^ (8 * (d + 3)) + 32 * (9 / 10 : ℝ) ^ (2 ^ d)) := by
+      rw [mul_pow, Real.sq_sqrt (by positivity), hc]
+      push_cast
+      ring
+    rw [hsq]
+    nlinarith [pow_nonneg (by norm_num : (0 : ℝ) ≤ 1 / 32) d]
+  have hone := le_trans hh hbound
+  exact hone
+
+theorem marginal_tendsto (e : ∀ d : ℕ, Fin (7 * 2 ^ d) → Position (d + 1 + 2))
+    (he : ∀ d, Function.Injective (e d)) :
+    Tendsto (fun d : ℕ =>
+      tv (fairMass (fun ω : History (d + 1 + 2) (200 * (d + 1 + 2)) =>
+          (run (d + 1 + 2) (200 * (d + 1 + 2)) ω) ∘ e d))
+        (fairMass (fun g : State (d + 1 + 2) => g ∘ e d))) atTop (nhds 0) := by
+  apply tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds marginalRate_tendsto
+  · intro d
+    unfold tv
+    positivity
+  · intro d
+    exact marginal_tv_rate d (e d) (he d)
+
+end Conditional
+
+end Thorp
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/MarginalEnergy.lean b/lean/OAI/Probability/ThorpShuffle/MarginalEnergy.lean
index 07b63596e928beba9df65748b1b5e6770835d6b6..467212277683d3ca8af89ab9bcaa02a82c213697 100644
--- a/lean/OAI/Probability/ThorpShuffle/MarginalEnergy.lean
+++ b/lean/OAI/Probability/ThorpShuffle/MarginalEnergy.lean
@@ -1,208 +1,208 @@
-import OAI.Probability.ThorpShuffle.EnergyDecay
-
-namespace OAI
-
-universe uα
-
-noncomputable section
-
-open scoped BigOperators
-open Filter
-
-namespace Thorp
-
-namespace Conditional
-
-theorem freeCount_eq_sum {α : Type uα} [Fintype α] (B : α → Bool) :
-    freeCount B = ∑ x, (B x).toNat := by
-  classical
-  calc
-    freeCount B = ∑ x, if B x = true then (1 : ℕ) else 0 := (Finset.sum_boole _ _).symm
-    _ = _ := Finset.sum_congr rfl (fun x _ => by cases B x <;> rfl)
-
-theorem pairFreeCount_physical (d : ℕ) (B : Position (d + 1) → Bool) :
-    pairFreeCount (B ∘ (splitPosition d).symm) = freeCount B := by
-  rw [freeCount_eq_sum]
-  have he := Equiv.sum_comp (splitPosition d).symm (fun x => (B x).toNat)
-  rw [← he, Fintype.sum_prod_type, Fintype.sum_bool]
-  simp only [pairFreeCount, Function.comp_apply, Finset.sum_add_distrib]
-  exact Nat.add_comm _ _
-
-theorem halfCount_after_step (d : ℕ) (v : RawState (d + 1))
-    (c : Coins (d + 1)) (h : Bool) :
-    freeCount (halfRaw d (Fin.last d) h (rawNext d v c)).free =
-      pairHalfCount (v.free ∘ (splitPosition d).symm) c h := by
-  rw [freeCount_eq_sum]
-  unfold halfRaw rawNext pairHalfCount
-  simp only [insertBit_last]
-  apply Finset.sum_congr rfl
-  intro x _
-  erw [physicalFree_snoc]
-  rfl
-
-theorem halfDensity_after_step (d : ℕ) (v : RawState (d + 1))
-    (c : Coins (d + 1)) (h : Bool)
-    (hm : Fintype.card (Position (d + 1)) ≤ 8 * freeCount v.free)
-    (hc : freeCount v.free ≤ 4 * pairHalfCount (v.free ∘ (splitPosition d).symm) c h) :
-    (1 / 16 : ℝ) ≤ startHalfDensity d (rawNext d v c) h := by
-  unfold startHalfDensity
-  rw [halfCount_after_step]
-  have hn : Fintype.card (Position (d + 1)) = 2 * Fintype.card (Position d) := by
-    simp [Fintype.card_pi, Fintype.card_bool, Fintype.card_fin, pow_succ, Nat.mul_comm]
-  rw [hn] at hm
-  have hm' : (2 : ℝ) * Fintype.card (Position d) ≤ 8 * (freeCount v.free : ℝ) := by
-    exact_mod_cast hm
-  have hc' : (freeCount v.free : ℝ) ≤
-      4 * (pairHalfCount (v.free ∘ (splitPosition d).symm) c h : ℝ) := by exact_mod_cast hc
-  apply (le_div_iff₀ (by exact_mod_cast Fintype.card_pos (α := Position d))).mpr
-  nlinarith
-
-theorem partnerEnergy_nonneg (d : ℕ) (v : RawState (d + 1)) (h : Bool) :
-    0 ≤ partnerEnergy d v h := by
-  unfold partnerEnergy
-  exact Finset.sum_nonneg (fun _ _ => mul_nonneg (sq_nonneg _) (by split_ifs <;> norm_num))
-
-theorem oneFreeEnergy_le (d : ℕ) (v : RawState (d + 1)) :
-    oneFreeEnergy d v.free v.weight ≤ rawEnergy v := by
-  have hid := oneFreeEnergy_partner d v
-  linarith [partnerEnergy_nonneg d v false, partnerEnergy_nonneg d v true]
-
-theorem expectedNoise_le_energy (d : ℕ) (v : CenteredState (d + 1)) (t : ℕ) :
-    expectedNoise d v t ≤ expectedEnergy d v t := by
-  apply mean_le_mean
-  intro ω
-  exact oneFreeEnergy_le d (iterateState d v t ω).raw
-
-theorem noise_bound_balanced_state (d : ℕ) (v : CenteredState (d + 2))
-    (β : ℝ) (hb : ∀ h, β ≤ startHalfDensity (d + 1) v.raw h) :
-    expectedNoise (d + 1) v (d + 1) ≤ (1 - β) * expectedEnergy (d + 1) v (d + 1) := by
-  have he := noise_bound_balanced_halves d v.raw β hb
-  simp_rw [← iterateState_raw] at he
-  exact he
-
-def singleHistory (d : ℕ) : Coins d ≃ History d 1 where
-  toFun c := fun _ => c
-  invFun ω := ω 0
-  left_inv _ := rfl
-  right_inv ω := by
-    funext i
-    change ω 0 = ω i
-    rw [Subsingleton.elim i 0]
-
-theorem mean_history_one (d : ℕ) (f : History d 1 → ℝ) :
-    mean f = mean (fun c : Coins d => f (fun _ => c)) :=
-  (mean_equiv (singleHistory d) f).symm
-
-@[simp] theorem iterateState_one (d : ℕ) (v : CenteredState (d + 1)) (c : Coins (d + 1)) :
-    iterateState d v 1 (fun _ => c) = nextState d v c := rfl
-
-theorem expectedEnergy_first (d t : ℕ) (v : CenteredState (d + 1)) :
-    expectedEnergy d v (t + 1) =
-      mean (fun c : Coins (d + 1) => expectedEnergy d (nextState d v c) t) := by
-  rw [Nat.add_comm t 1, expectedEnergy_add, mean_history_one]
-  rfl
-
-theorem expectedNoise_first (d t : ℕ) (v : CenteredState (d + 1)) :
-    expectedNoise d v (t + 1) =
-      mean (fun c : Coins (d + 1) => expectedNoise d (nextState d v c) t) := by
-  rw [Nat.add_comm t 1, expectedNoise_add, mean_history_one]
-  rfl
-
-theorem expectedNoise_dimension (d : ℕ) (v : CenteredState (d + 2))
-    (hm : Fintype.card (Position (d + 2)) ≤ 8 * freeCount v.free) :
-    expectedNoise (d + 1) v (d + 2) ≤ (15 / 16 : ℝ) * expectedEnergy (d + 1) v (d + 2) +
-      2 * (9 / 10 : ℝ) ^ freeCount v.free := by
-  let B := v.free ∘ (splitPosition (d + 1)).symm
-  let bad (c : Coins (d + 2)) : Prop :=
-    4 * pairHalfCount B c false < pairFreeCount B ∨
-    4 * pairHalfCount B c true < pairFreeCount B
-  have hlocal (c : Coins (d + 2)) :
-      expectedNoise (d + 1) (nextState (d + 1) v c) (d + 1) ≤
-      (15 / 16 : ℝ) * expectedEnergy (d + 1) (nextState (d + 1) v c) (d + 1) +
-        (if bad c then 1 else 0) := by
-    by_cases hc : bad c
-    · rw [ite_eq_left hc]
-      have hle := expectedNoise_le_energy (d + 1) (nextState (d + 1) v c) (d + 1)
-      have hone := expectedEnergy_le_one (d + 1) (nextState (d + 1) v c) (d + 1)
-      linarith
-    · rw [ite_eq_right hc, add_zero]
-      have hb (h : Bool) : (1 / 16 : ℝ) ≤
-          startHalfDensity (d + 1) (nextState (d + 1) v c).raw h := by
-        apply halfDensity_after_step (d + 1) v.raw c h hm
-        have hf : ¬ 4 * pairHalfCount B c false < freeCount v.free := by
-          simpa only [B, pairFreeCount_physical] using not_or.mp hc |>.1
-        have ht : ¬ 4 * pairHalfCount B c true < freeCount v.free := by
-          simpa only [B, pairFreeCount_physical] using not_or.mp hc |>.2
-        change freeCount v.free ≤ 4 * pairHalfCount B c h
-        cases h
-        · exact Nat.le_of_not_gt hf
-        · exact Nat.le_of_not_gt ht
-      convert noise_bound_balanced_state d (nextState (d + 1) v c) (1 / 16) hb using 1; norm_num
-  have hmean := mean_le_mean hlocal
-  rw [mean_add, mean_const_mul, ← expectedNoise_first, ← expectedEnergy_first] at hmean
-  have hbad := pair_both_half_balance B
-  have hbad' : mean (fun c : Coins (d + 2) => if bad c then (1 : ℝ) else 0) ≤
-      2 * (9 / 10 : ℝ) ^ freeCount v.free := by
-    change mean (fun c : Position (d + 1) → Bool => if bad c then (1 : ℝ) else 0) ≤ _
-    simpa only [bad, B, pairFreeCount_physical] using hbad
-  exact hmean.trans (add_le_add (le_refl _) hbad')
-
-theorem expectedNoise_bound (d : ℕ) (v : CenteredState (d + 2))
-    (hm : Fintype.card (Position (d + 2)) ≤ 8 * freeCount v.free)
-    (s : ℕ) (hs : d + 2 ≤ s) :
-    expectedNoise (d + 1) v s ≤ (15 / 16 : ℝ) * expectedEnergy (d + 1) v s +
-      2 * (9 / 10 : ℝ) ^ freeCount v.free := by
-  obtain ⟨a, rfl⟩ := Nat.exists_eq_add_of_le hs
-  rw [Nat.add_comm (d + 2) a, expectedNoise_add, expectedEnergy_add]
-  have he := mean_le_mean (fun ω : History (d + 2) a =>
-    expectedNoise_dimension d (iterateState (d + 1) v a ω)
-      (by simpa only [iterateState_freeCount] using hm))
-  simpa only [iterateState_freeCount, mean_add, mean_const_mul, mean_const] using he
-
-end Conditional
-
-namespace Conditional
-
-theorem expectedEnergy_decay (d : ℕ) (v : CenteredState (d + 2))
-    (hm : Fintype.card (Position (d + 2)) ≤ 8 * freeCount v.free) (t : ℕ) :
-    expectedEnergy (d + 1) v t ≤
-      (31 / 32 : ℝ) ^ t / (31 / 32 : ℝ) ^ (2 * (d + 2)) +
-        32 * (9 / 10 : ℝ) ^ freeCount v.free := by
-  have he := scalar_decay (d + 2) (expectedEnergy (d + 1) v)
-    (expectedNoise (d + 1) v) (1 / 16) (31 / 32)
-    (2 * (9 / 10 : ℝ) ^ freeCount v.free)
-    (by norm_num) (by norm_num) (by norm_num) (by norm_num)
-    (by norm_num) (by positivity) (expectedEnergy_le_one (d + 1) v)
-    (expectedEnergy_recurrence (d + 1) v)
-    (by intro s hs; convert expectedNoise_bound d v hm s hs using 1; norm_num) t
-  convert he using 1; ring
-
-theorem expectedEnergy_200 (d : ℕ) (v : CenteredState (d + 2))
-    (hm : Fintype.card (Position (d + 2)) ≤ 8 * freeCount v.free) :
-    expectedEnergy (d + 1) v (200 * (d + 2)) ≤
-      (1 / 2 : ℝ) ^ (8 * (d + 2)) + 32 * (9 / 10 : ℝ) ^ freeCount v.free := by
-  have he := expectedEnergy_decay d v hm (200 * (d + 2))
-  have hp : (31 / 32 : ℝ) ^ (200 * (d + 2)) / (31 / 32 : ℝ) ^ (2 * (d + 2)) =
-      (31 / 32 : ℝ) ^ (198 * (d + 2)) := by
-    have hn : 200 * (d + 2) = 198 * (d + 2) + 2 * (d + 2) := by omega
-    rw [hn, pow_add, mul_div_cancel_right₀ _ (by positivity)]
-  rw [hp] at he
-  have hpow := pow_le_pow_left₀ (by positivity : (0 : ℝ) ≤ (31 / 32) ^ 198)
-    eight_block_geometric_choice (d + 2)
-  rw [← pow_mul, ← pow_mul] at hpow
-  exact he.trans (add_le_add hpow (le_refl _))
-
-theorem centeredPoint_energy_200 (d : ℕ) (B : Position (d + 2) → Bool)
-    (tag : Position (d + 2)) (ht : B tag = true)
-    (hm : Fintype.card (Position (d + 2)) ≤ 8 * freeCount B) :
-    expectedEnergy (d + 1) (initialState (d + 2) B tag ht) (200 * (d + 2)) ≤
-      (1 / 2 : ℝ) ^ (8 * (d + 2)) + 32 * (9 / 10 : ℝ) ^ freeCount B :=
-  expectedEnergy_200 d _ hm
-
-end Conditional
-
-end Thorp
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.EnergyDecay
+
+namespace OAI
+
+universe uα
+
+noncomputable section
+
+open scoped BigOperators
+open Filter
+
+namespace Thorp
+
+namespace Conditional
+
+theorem freeCount_eq_sum {α : Type uα} [Fintype α] (B : α → Bool) :
+    freeCount B = ∑ x, (B x).toNat := by
+  classical
+  calc
+    freeCount B = ∑ x, if B x = true then (1 : ℕ) else 0 := (Finset.sum_boole _ _).symm
+    _ = _ := Finset.sum_congr rfl (fun x _ => by cases B x <;> rfl)
+
+theorem pairFreeCount_physical (d : ℕ) (B : Position (d + 1) → Bool) :
+    pairFreeCount (B ∘ (splitPosition d).symm) = freeCount B := by
+  rw [freeCount_eq_sum]
+  have he := Equiv.sum_comp (splitPosition d).symm (fun x => (B x).toNat)
+  rw [← he, Fintype.sum_prod_type, Fintype.sum_bool]
+  simp only [pairFreeCount, Function.comp_apply, Finset.sum_add_distrib]
+  exact Nat.add_comm _ _
+
+theorem halfCount_after_step (d : ℕ) (v : RawState (d + 1))
+    (c : Coins (d + 1)) (h : Bool) :
+    freeCount (halfRaw d (Fin.last d) h (rawNext d v c)).free =
+      pairHalfCount (v.free ∘ (splitPosition d).symm) c h := by
+  rw [freeCount_eq_sum]
+  unfold halfRaw rawNext pairHalfCount
+  simp only [insertBit_last]
+  apply Finset.sum_congr rfl
+  intro x _
+  erw [physicalFree_snoc]
+  rfl
+
+theorem halfDensity_after_step (d : ℕ) (v : RawState (d + 1))
+    (c : Coins (d + 1)) (h : Bool)
+    (hm : Fintype.card (Position (d + 1)) ≤ 8 * freeCount v.free)
+    (hc : freeCount v.free ≤ 4 * pairHalfCount (v.free ∘ (splitPosition d).symm) c h) :
+    (1 / 16 : ℝ) ≤ startHalfDensity d (rawNext d v c) h := by
+  unfold startHalfDensity
+  rw [halfCount_after_step]
+  have hn : Fintype.card (Position (d + 1)) = 2 * Fintype.card (Position d) := by
+    simp [Fintype.card_pi, Fintype.card_bool, Fintype.card_fin, pow_succ, Nat.mul_comm]
+  rw [hn] at hm
+  have hm' : (2 : ℝ) * Fintype.card (Position d) ≤ 8 * (freeCount v.free : ℝ) := by
+    exact_mod_cast hm
+  have hc' : (freeCount v.free : ℝ) ≤
+      4 * (pairHalfCount (v.free ∘ (splitPosition d).symm) c h : ℝ) := by exact_mod_cast hc
+  apply (le_div_iff₀ (by exact_mod_cast Fintype.card_pos (α := Position d))).mpr
+  nlinarith
+
+theorem partnerEnergy_nonneg (d : ℕ) (v : RawState (d + 1)) (h : Bool) :
+    0 ≤ partnerEnergy d v h := by
+  unfold partnerEnergy
+  exact Finset.sum_nonneg (fun _ _ => mul_nonneg (sq_nonneg _) (by split_ifs <;> norm_num))
+
+theorem oneFreeEnergy_le (d : ℕ) (v : RawState (d + 1)) :
+    oneFreeEnergy d v.free v.weight ≤ rawEnergy v := by
+  have hid := oneFreeEnergy_partner d v
+  linarith [partnerEnergy_nonneg d v false, partnerEnergy_nonneg d v true]
+
+theorem expectedNoise_le_energy (d : ℕ) (v : CenteredState (d + 1)) (t : ℕ) :
+    expectedNoise d v t ≤ expectedEnergy d v t := by
+  apply mean_le_mean
+  intro ω
+  exact oneFreeEnergy_le d (iterateState d v t ω).raw
+
+theorem noise_bound_balanced_state (d : ℕ) (v : CenteredState (d + 2))
+    (β : ℝ) (hb : ∀ h, β ≤ startHalfDensity (d + 1) v.raw h) :
+    expectedNoise (d + 1) v (d + 1) ≤ (1 - β) * expectedEnergy (d + 1) v (d + 1) := by
+  have he := noise_bound_balanced_halves d v.raw β hb
+  simp_rw [← iterateState_raw] at he
+  exact he
+
+def singleHistory (d : ℕ) : Coins d ≃ History d 1 where
+  toFun c := fun _ => c
+  invFun ω := ω 0
+  left_inv _ := rfl
+  right_inv ω := by
+    funext i
+    change ω 0 = ω i
+    rw [Subsingleton.elim i 0]
+
+theorem mean_history_one (d : ℕ) (f : History d 1 → ℝ) :
+    mean f = mean (fun c : Coins d => f (fun _ => c)) :=
+  (mean_equiv (singleHistory d) f).symm
+
+@[simp] theorem iterateState_one (d : ℕ) (v : CenteredState (d + 1)) (c : Coins (d + 1)) :
+    iterateState d v 1 (fun _ => c) = nextState d v c := rfl
+
+theorem expectedEnergy_first (d t : ℕ) (v : CenteredState (d + 1)) :
+    expectedEnergy d v (t + 1) =
+      mean (fun c : Coins (d + 1) => expectedEnergy d (nextState d v c) t) := by
+  rw [Nat.add_comm t 1, expectedEnergy_add, mean_history_one]
+  rfl
+
+theorem expectedNoise_first (d t : ℕ) (v : CenteredState (d + 1)) :
+    expectedNoise d v (t + 1) =
+      mean (fun c : Coins (d + 1) => expectedNoise d (nextState d v c) t) := by
+  rw [Nat.add_comm t 1, expectedNoise_add, mean_history_one]
+  rfl
+
+theorem expectedNoise_dimension (d : ℕ) (v : CenteredState (d + 2))
+    (hm : Fintype.card (Position (d + 2)) ≤ 8 * freeCount v.free) :
+    expectedNoise (d + 1) v (d + 2) ≤ (15 / 16 : ℝ) * expectedEnergy (d + 1) v (d + 2) +
+      2 * (9 / 10 : ℝ) ^ freeCount v.free := by
+  let B := v.free ∘ (splitPosition (d + 1)).symm
+  let bad (c : Coins (d + 2)) : Prop :=
+    4 * pairHalfCount B c false < pairFreeCount B ∨
+    4 * pairHalfCount B c true < pairFreeCount B
+  have hlocal (c : Coins (d + 2)) :
+      expectedNoise (d + 1) (nextState (d + 1) v c) (d + 1) ≤
+      (15 / 16 : ℝ) * expectedEnergy (d + 1) (nextState (d + 1) v c) (d + 1) +
+        (if bad c then 1 else 0) := by
+    by_cases hc : bad c
+    · rw [ite_eq_left hc]
+      have hle := expectedNoise_le_energy (d + 1) (nextState (d + 1) v c) (d + 1)
+      have hone := expectedEnergy_le_one (d + 1) (nextState (d + 1) v c) (d + 1)
+      linarith
+    · rw [ite_eq_right hc, add_zero]
+      have hb (h : Bool) : (1 / 16 : ℝ) ≤
+          startHalfDensity (d + 1) (nextState (d + 1) v c).raw h := by
+        apply halfDensity_after_step (d + 1) v.raw c h hm
+        have hf : ¬ 4 * pairHalfCount B c false < freeCount v.free := by
+          simpa only [B, pairFreeCount_physical] using not_or.mp hc |>.1
+        have ht : ¬ 4 * pairHalfCount B c true < freeCount v.free := by
+          simpa only [B, pairFreeCount_physical] using not_or.mp hc |>.2
+        change freeCount v.free ≤ 4 * pairHalfCount B c h
+        cases h
+        · exact Nat.le_of_not_gt hf
+        · exact Nat.le_of_not_gt ht
+      convert noise_bound_balanced_state d (nextState (d + 1) v c) (1 / 16) hb using 1; norm_num
+  have hmean := mean_le_mean hlocal
+  rw [mean_add, mean_const_mul, ← expectedNoise_first, ← expectedEnergy_first] at hmean
+  have hbad := pair_both_half_balance B
+  have hbad' : mean (fun c : Coins (d + 2) => if bad c then (1 : ℝ) else 0) ≤
+      2 * (9 / 10 : ℝ) ^ freeCount v.free := by
+    change mean (fun c : Position (d + 1) → Bool => if bad c then (1 : ℝ) else 0) ≤ _
+    simpa only [bad, B, pairFreeCount_physical] using hbad
+  exact hmean.trans (add_le_add (le_refl _) hbad')
+
+theorem expectedNoise_bound (d : ℕ) (v : CenteredState (d + 2))
+    (hm : Fintype.card (Position (d + 2)) ≤ 8 * freeCount v.free)
+    (s : ℕ) (hs : d + 2 ≤ s) :
+    expectedNoise (d + 1) v s ≤ (15 / 16 : ℝ) * expectedEnergy (d + 1) v s +
+      2 * (9 / 10 : ℝ) ^ freeCount v.free := by
+  obtain ⟨a, rfl⟩ := Nat.exists_eq_add_of_le hs
+  rw [Nat.add_comm (d + 2) a, expectedNoise_add, expectedEnergy_add]
+  have he := mean_le_mean (fun ω : History (d + 2) a =>
+    expectedNoise_dimension d (iterateState (d + 1) v a ω)
+      (by simpa only [iterateState_freeCount] using hm))
+  simpa only [iterateState_freeCount, mean_add, mean_const_mul, mean_const] using he
+
+end Conditional
+
+namespace Conditional
+
+theorem expectedEnergy_decay (d : ℕ) (v : CenteredState (d + 2))
+    (hm : Fintype.card (Position (d + 2)) ≤ 8 * freeCount v.free) (t : ℕ) :
+    expectedEnergy (d + 1) v t ≤
+      (31 / 32 : ℝ) ^ t / (31 / 32 : ℝ) ^ (2 * (d + 2)) +
+        32 * (9 / 10 : ℝ) ^ freeCount v.free := by
+  have he := scalar_decay (d + 2) (expectedEnergy (d + 1) v)
+    (expectedNoise (d + 1) v) (1 / 16) (31 / 32)
+    (2 * (9 / 10 : ℝ) ^ freeCount v.free)
+    (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+    (by norm_num) (by positivity) (expectedEnergy_le_one (d + 1) v)
+    (expectedEnergy_recurrence (d + 1) v)
+    (by intro s hs; convert expectedNoise_bound d v hm s hs using 1; norm_num) t
+  convert he using 1; ring
+
+theorem expectedEnergy_200 (d : ℕ) (v : CenteredState (d + 2))
+    (hm : Fintype.card (Position (d + 2)) ≤ 8 * freeCount v.free) :
+    expectedEnergy (d + 1) v (200 * (d + 2)) ≤
+      (1 / 2 : ℝ) ^ (8 * (d + 2)) + 32 * (9 / 10 : ℝ) ^ freeCount v.free := by
+  have he := expectedEnergy_decay d v hm (200 * (d + 2))
+  have hp : (31 / 32 : ℝ) ^ (200 * (d + 2)) / (31 / 32 : ℝ) ^ (2 * (d + 2)) =
+      (31 / 32 : ℝ) ^ (198 * (d + 2)) := by
+    have hn : 200 * (d + 2) = 198 * (d + 2) + 2 * (d + 2) := by omega
+    rw [hn, pow_add, mul_div_cancel_right₀ _ (by positivity)]
+  rw [hp] at he
+  have hpow := pow_le_pow_left₀ (by positivity : (0 : ℝ) ≤ (31 / 32) ^ 198)
+    eight_block_geometric_choice (d + 2)
+  rw [← pow_mul, ← pow_mul] at hpow
+  exact he.trans (add_le_add hpow (le_refl _))
+
+theorem centeredPoint_energy_200 (d : ℕ) (B : Position (d + 2) → Bool)
+    (tag : Position (d + 2)) (ht : B tag = true)
+    (hm : Fintype.card (Position (d + 2)) ≤ 8 * freeCount B) :
+    expectedEnergy (d + 1) (initialState (d + 2) B tag ht) (200 * (d + 2)) ≤
+      (1 / 2 : ℝ) ^ (8 * (d + 2)) + 32 * (9 / 10 : ℝ) ^ freeCount B :=
+  expectedEnergy_200 d _ hm
+
+end Conditional
+
+end Thorp
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/MaskedResampling.lean b/lean/OAI/Probability/ThorpShuffle/MaskedResampling.lean
index b26da45440ccc6b448bd04b2c34ecf0a6e94f4cc..7c43a5e603ae568e09f34f9a053909d9fcfaa240 100644
--- a/lean/OAI/Probability/ThorpShuffle/MaskedResampling.lean
+++ b/lean/OAI/Probability/ThorpShuffle/MaskedResampling.lean
@@ -1,129 +1,129 @@
-import OAI.Probability.ThorpShuffle.MarginalEnergy
-
-namespace OAI
-
-universe uα uι
-
-noncomputable section
-
-open scoped BigOperators
-open Filter
-
-namespace Thorp
-
-namespace Conditional
-
-def maskedCoin {α : Type uα} (mask : α → Bool) (c r : α → Bool) : α → Bool :=
-  fun x => if mask x then r x else c x
-
-def maskSwap {α : Type uα} (mask : α → Bool) :
-    Equiv.Perm ((α → Bool) × (α → Bool)) :=
-  Function.Involutive.toPerm
-    (fun p => (maskedCoin mask p.1 p.2, maskedCoin mask p.2 p.1)) (by
-      intro p
-      apply Prod.ext <;> funext x <;>
-        cases hm : mask x <;> simp [maskedCoin, hm])
-
-theorem mean_maskedCoin {α : Type uα} [Fintype α] [DecidableEq α]
-    (mask : α → Bool) (f : (α → Bool) → ℝ) :
-    mean (fun c : α → Bool => mean (fun r : α → Bool => f (maskedCoin mask c r))) =
-      mean f := by
-  have he := mean_equiv (maskSwap mask) (fun p => f p.1)
-  change mean (fun p : (α → Bool) × (α → Bool) => f (maskedCoin mask p.1 p.2)) = _ at he
-  rw [mean_prod, mean_prod] at he
-  simpa only [mean_const] using he
-
-def freePairMask (d : ℕ) (B : Position (d + 1) → Bool) : Position d → Bool :=
-  fun x => B (Fin.cons false x) && B (Fin.cons true x)
-
-theorem step_cons (d : ℕ) (c : Position d → Bool) (b : Bool) (x : Position d) :
-    step (d + 1) c (Fin.cons b x) = scatterPosition d (b ^^ c x, x) := by
-  simp [step, scatterPosition, splitPosition, pairSwitch]
-
-theorem masked_step_occupied (d : ℕ) (B : Position (d + 1) → Bool)
-    (c r : Position d → Bool) (x : Position (d + 1)) (hx : B x = false) :
-    step (d + 1) (maskedCoin (freePairMask d B) c r) x = step (d + 1) c x := by
-  obtain ⟨⟨b, y⟩, rfl⟩ := (splitPosition d).symm.surjective x
-  change B (Fin.cons b y) = false at hx
-  change step (d + 1) (maskedCoin (freePairMask d B) c r) (Fin.cons b y) =
-    step (d + 1) c (Fin.cons b y)
-  rw [step_cons, step_cons]
-  cases b <;> simp [maskedCoin, freePairMask, hx]
-
-theorem physicalFlow_resampling (d : ℕ) (B : Position (d + 1) → Bool)
-    (w : Position (d + 1) → ℝ) (c : Position d → Bool) (y : Position (d + 1)) :
-    physicalFlow d B w c y = mean (fun r : Position d → Bool =>
-      w ((step (d + 1) (maskedCoin (freePairMask d B) c r)).symm y)) := by
-  obtain ⟨⟨b, x⟩, rfl⟩ := (scatterPosition d).surjective y
-  have hs : mean (fun r : Position d → Bool =>
-      w ((step (d + 1) (maskedCoin (freePairMask d B) c r)).symm
-        (scatterPosition d (b, x)))) =
-      mean (fun r : Position d → Bool =>
-        w (Fin.cons (b ^^ maskedCoin (freePairMask d B) c r x) x)) := by
-    apply mean_congr
-    intro r
-    exact congrArg w (step_symm_scatter d (maskedCoin (freePairMask d B) c r) (b, x))
-  rw [hs]
-  erw [physicalFlow_scatter]
-  change (if b then (pairUpdate (B (Fin.cons false x)) (B (Fin.cons true x))
-      (w (Fin.cons false x)) (w (Fin.cons true x)) (c x)).2
-    else (pairUpdate (B (Fin.cons false x)) (B (Fin.cons true x))
-      (w (Fin.cons false x)) (w (Fin.cons true x)) (c x)).1) =
-    mean (fun r : Position d → Bool =>
-      w (Fin.cons (b ^^ (if freePairMask d B x then r x else c x)) x))
-  rw [mean_coin_eval x (fun z => w (Fin.cons (b ^^ (if freePairMask d B x then z else c x)) x))]
-  cases h0 : B (Fin.cons false x) <;> cases h1 : B (Fin.cons true x) <;>
-    cases b <;> cases c x <;>
-    simp [pairUpdate, freePairMask, h0, h1, mean_bool] <;> ring
-
-theorem masked_step_exposed {ι : Type uι} (d : ℕ) (B : Position (d + 1) → Bool)
-    (e : ι → Position (d + 1)) (he : ∀ i, B (e i) = false) (c r : Position d → Bool) :
-    (step (d + 1) (maskedCoin (freePairMask d B) c r)) ∘ e = (step (d + 1) c) ∘ e := by
-  funext i
-  exact masked_step_occupied d B c r (e i) (he i)
-
-theorem physicalFlow_joint {ι : Type uι} (d : ℕ) (B : Position (d + 1) → Bool)
-    (w : Position (d + 1) → ℝ) (e : ι → Position (d + 1))
-    (he : ∀ i, B (e i) = false) (f : (ι → Position (d + 1)) → Position (d + 1) → ℝ) :
-    mean (fun c : Position d → Bool => ∑ y, physicalFlow d B w c y * f ((step (d + 1) c) ∘ e) y) =
-      mean (fun c : Position d → Bool => ∑ x, w x * f ((step (d + 1) c) ∘ e) (step (d + 1) c x)) := by
-  calc
-    _ = mean (fun c : Position d → Bool => mean (fun r : Position d → Bool =>
-        ∑ y, w ((step (d + 1) (maskedCoin (freePairMask d B) c r)).symm y) *
-          f ((step (d + 1) c) ∘ e) y)) := by
-      apply mean_congr
-      intro c
-      rw [mean_sum]
-      apply Finset.sum_congr rfl
-      intro y _
-      rw [mean_mul_const, physicalFlow_resampling]
-    _ = mean (fun c : Position d → Bool => mean (fun r : Position d → Bool =>
-        ∑ x, w x * f ((step (d + 1) c) ∘ e)
-          (step (d + 1) (maskedCoin (freePairMask d B) c r) x))) := by
-      apply mean_congr
-      intro c
-      apply mean_congr
-      intro r
-      simpa only [Equiv.symm_apply_apply] using
-        (Equiv.sum_comp (step (d + 1) (maskedCoin (freePairMask d B) c r))
-          (fun y => w ((step (d + 1) (maskedCoin (freePairMask d B) c r)).symm y) *
-            f ((step (d + 1) c) ∘ e) y)).symm
-    _ = mean (fun c : Position d → Bool => mean (fun r : Position d → Bool =>
-        ∑ x, w x * f ((step (d + 1) (maskedCoin (freePairMask d B) c r)) ∘ e)
-          (step (d + 1) (maskedCoin (freePairMask d B) c r) x))) := by
-      apply mean_congr
-      intro c
-      apply mean_congr
-      intro r
-      rw [masked_step_exposed d B e he c r]
-    _ = _ := by
-      exact mean_maskedCoin (freePairMask d B) (fun c => ∑ x, w x *
-        f ((step (d + 1) c) ∘ e) (step (d + 1) c x))
-
-end Conditional
-
-end Thorp
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.MarginalEnergy
+
+namespace OAI
+
+universe uα uι
+
+noncomputable section
+
+open scoped BigOperators
+open Filter
+
+namespace Thorp
+
+namespace Conditional
+
+def maskedCoin {α : Type uα} (mask : α → Bool) (c r : α → Bool) : α → Bool :=
+  fun x => if mask x then r x else c x
+
+def maskSwap {α : Type uα} (mask : α → Bool) :
+    Equiv.Perm ((α → Bool) × (α → Bool)) :=
+  Function.Involutive.toPerm
+    (fun p => (maskedCoin mask p.1 p.2, maskedCoin mask p.2 p.1)) (by
+      intro p
+      apply Prod.ext <;> funext x <;>
+        cases hm : mask x <;> simp [maskedCoin, hm])
+
+theorem mean_maskedCoin {α : Type uα} [Fintype α] [DecidableEq α]
+    (mask : α → Bool) (f : (α → Bool) → ℝ) :
+    mean (fun c : α → Bool => mean (fun r : α → Bool => f (maskedCoin mask c r))) =
+      mean f := by
+  have he := mean_equiv (maskSwap mask) (fun p => f p.1)
+  change mean (fun p : (α → Bool) × (α → Bool) => f (maskedCoin mask p.1 p.2)) = _ at he
+  rw [mean_prod, mean_prod] at he
+  simpa only [mean_const] using he
+
+def freePairMask (d : ℕ) (B : Position (d + 1) → Bool) : Position d → Bool :=
+  fun x => B (Fin.cons false x) && B (Fin.cons true x)
+
+theorem step_cons (d : ℕ) (c : Position d → Bool) (b : Bool) (x : Position d) :
+    step (d + 1) c (Fin.cons b x) = scatterPosition d (b ^^ c x, x) := by
+  simp [step, scatterPosition, splitPosition, pairSwitch]
+
+theorem masked_step_occupied (d : ℕ) (B : Position (d + 1) → Bool)
+    (c r : Position d → Bool) (x : Position (d + 1)) (hx : B x = false) :
+    step (d + 1) (maskedCoin (freePairMask d B) c r) x = step (d + 1) c x := by
+  obtain ⟨⟨b, y⟩, rfl⟩ := (splitPosition d).symm.surjective x
+  change B (Fin.cons b y) = false at hx
+  change step (d + 1) (maskedCoin (freePairMask d B) c r) (Fin.cons b y) =
+    step (d + 1) c (Fin.cons b y)
+  rw [step_cons, step_cons]
+  cases b <;> simp [maskedCoin, freePairMask, hx]
+
+theorem physicalFlow_resampling (d : ℕ) (B : Position (d + 1) → Bool)
+    (w : Position (d + 1) → ℝ) (c : Position d → Bool) (y : Position (d + 1)) :
+    physicalFlow d B w c y = mean (fun r : Position d → Bool =>
+      w ((step (d + 1) (maskedCoin (freePairMask d B) c r)).symm y)) := by
+  obtain ⟨⟨b, x⟩, rfl⟩ := (scatterPosition d).surjective y
+  have hs : mean (fun r : Position d → Bool =>
+      w ((step (d + 1) (maskedCoin (freePairMask d B) c r)).symm
+        (scatterPosition d (b, x)))) =
+      mean (fun r : Position d → Bool =>
+        w (Fin.cons (b ^^ maskedCoin (freePairMask d B) c r x) x)) := by
+    apply mean_congr
+    intro r
+    exact congrArg w (step_symm_scatter d (maskedCoin (freePairMask d B) c r) (b, x))
+  rw [hs]
+  erw [physicalFlow_scatter]
+  change (if b then (pairUpdate (B (Fin.cons false x)) (B (Fin.cons true x))
+      (w (Fin.cons false x)) (w (Fin.cons true x)) (c x)).2
+    else (pairUpdate (B (Fin.cons false x)) (B (Fin.cons true x))
+      (w (Fin.cons false x)) (w (Fin.cons true x)) (c x)).1) =
+    mean (fun r : Position d → Bool =>
+      w (Fin.cons (b ^^ (if freePairMask d B x then r x else c x)) x))
+  rw [mean_coin_eval x (fun z => w (Fin.cons (b ^^ (if freePairMask d B x then z else c x)) x))]
+  cases h0 : B (Fin.cons false x) <;> cases h1 : B (Fin.cons true x) <;>
+    cases b <;> cases c x <;>
+    simp [pairUpdate, freePairMask, h0, h1, mean_bool] <;> ring
+
+theorem masked_step_exposed {ι : Type uι} (d : ℕ) (B : Position (d + 1) → Bool)
+    (e : ι → Position (d + 1)) (he : ∀ i, B (e i) = false) (c r : Position d → Bool) :
+    (step (d + 1) (maskedCoin (freePairMask d B) c r)) ∘ e = (step (d + 1) c) ∘ e := by
+  funext i
+  exact masked_step_occupied d B c r (e i) (he i)
+
+theorem physicalFlow_joint {ι : Type uι} (d : ℕ) (B : Position (d + 1) → Bool)
+    (w : Position (d + 1) → ℝ) (e : ι → Position (d + 1))
+    (he : ∀ i, B (e i) = false) (f : (ι → Position (d + 1)) → Position (d + 1) → ℝ) :
+    mean (fun c : Position d → Bool => ∑ y, physicalFlow d B w c y * f ((step (d + 1) c) ∘ e) y) =
+      mean (fun c : Position d → Bool => ∑ x, w x * f ((step (d + 1) c) ∘ e) (step (d + 1) c x)) := by
+  calc
+    _ = mean (fun c : Position d → Bool => mean (fun r : Position d → Bool =>
+        ∑ y, w ((step (d + 1) (maskedCoin (freePairMask d B) c r)).symm y) *
+          f ((step (d + 1) c) ∘ e) y)) := by
+      apply mean_congr
+      intro c
+      rw [mean_sum]
+      apply Finset.sum_congr rfl
+      intro y _
+      rw [mean_mul_const, physicalFlow_resampling]
+    _ = mean (fun c : Position d → Bool => mean (fun r : Position d → Bool =>
+        ∑ x, w x * f ((step (d + 1) c) ∘ e)
+          (step (d + 1) (maskedCoin (freePairMask d B) c r) x))) := by
+      apply mean_congr
+      intro c
+      apply mean_congr
+      intro r
+      simpa only [Equiv.symm_apply_apply] using
+        (Equiv.sum_comp (step (d + 1) (maskedCoin (freePairMask d B) c r))
+          (fun y => w ((step (d + 1) (maskedCoin (freePairMask d B) c r)).symm y) *
+            f ((step (d + 1) c) ∘ e) y)).symm
+    _ = mean (fun c : Position d → Bool => mean (fun r : Position d → Bool =>
+        ∑ x, w x * f ((step (d + 1) (maskedCoin (freePairMask d B) c r)) ∘ e)
+          (step (d + 1) (maskedCoin (freePairMask d B) c r) x))) := by
+      apply mean_congr
+      intro c
+      apply mean_congr
+      intro r
+      rw [masked_step_exposed d B e he c r]
+    _ = _ := by
+      exact mean_maskedCoin (freePairMask d B) (fun c => ∑ x, w x *
+        f ((step (d + 1) c) ∘ e) (step (d + 1) c x))
+
+end Conditional
+
+end Thorp
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/MixingBounds.lean b/lean/OAI/Probability/ThorpShuffle/MixingBounds.lean
index 21bd6fb71f9384e1631c2e971b02143b81d4f7df..cdd34f24afa6d0e61f2b10412ec7ebd7a468769e 100644
--- a/lean/OAI/Probability/ThorpShuffle/MixingBounds.lean
+++ b/lean/OAI/Probability/ThorpShuffle/MixingBounds.lean
@@ -1,155 +1,155 @@
-import OAI.Probability.ThorpShuffle.FiniteLaw
-
-namespace OAI
-
-noncomputable section
-
-open scoped BigOperators
-open Filter
-
-namespace Thorp
-
-theorem exists_mixing_time_succ (d : ℕ) : ∃ t : ℕ, distance (d + 1) t ≤ 1 / 4 := by
-  classical
-  obtain ⟨T, hT⟩ := Reachability.exists_surjective_run d
-  let c : ℝ := 1 / Fintype.card (History (d + 1) T)
-  let r : ℝ := 1 - (Fintype.card (State (d + 1)) : ℝ) * c
-  have hcpos : 0 < c := by dsimp [c]; positivity
-  have hminor (g : State (d + 1)) : c ≤ law (d + 1) T g :=
-    FiniteLaw.fairMass_lower_of_surjective _ hT g
-  have hr : 0 ≤ r := by
-    have hh := Finset.sum_le_sum (s := Finset.univ) (fun g _ => hminor g)
-    simp only [law, fairMass_sum] at hh
-    simpa only [Finset.sum_const, Finset.card_univ, nsmul_eq_mul] using sub_nonneg.mpr hh
-  have hrlt : r < 1 := by
-    dsimp [r]
-    have hn : (0 : ℝ) < Fintype.card (State (d + 1)) := by positivity
-    nlinarith
-  have hcontract (s : ℕ) : distance (d + 1) (s + T) ≤ r * distance (d + 1) s := by
-    unfold distance uniform
-    rw [law_add]
-    exact FiniteLaw.tv_conv_contract _ _ (fairMass_sum _) (fairMass_sum _) c hminor
-  have hgeom (n : ℕ) : distance (d + 1) (n * T) ≤ r ^ n * distance (d + 1) 0 := by
-    induction n with
-    | zero => simp
-    | succ n ih =>
-      rw [Nat.succ_mul]
-      calc
-        _ ≤ r * distance (d + 1) (n * T) := hcontract _
-        _ ≤ r * (r ^ n * distance (d + 1) 0) := mul_le_mul_of_nonneg_left ih hr
-        _ = _ := by rw [pow_succ']; ring
-  have hlim : Tendsto (fun n : ℕ => r ^ n * distance (d + 1) 0) atTop (nhds 0) := by
-    simpa using (tendsto_pow_atTop_nhds_zero_of_lt_one hr hrlt).mul_const (distance (d + 1) 0)
-  have hev : ∀ᶠ n : ℕ in atTop, r ^ n * distance (d + 1) 0 < 1 / 4 :=
-    hlim.eventually (eventually_lt_nhds (by norm_num : (0 : ℝ) < 1 / 4))
-  obtain ⟨n, hn⟩ := hev.exists
-  exact ⟨n * T, (hgeom n).trans hn.le⟩
-
-theorem exists_mixing_time {d : ℕ} (hd : 1 ≤ d) : ∃ t : ℕ, distance d t ≤ 1 / 4 := by
-  obtain ⟨k, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : d ≠ 0)
-  exact exists_mixing_time_succ k
-
-theorem exact_support_lower {d : ℕ} (hd : 1 ≤ d) :
-    supportThreshold d ≤ (mixingTime d : ℤ) :=
-  supportThreshold_le_mixingTime hd (exists_mixing_time hd)
-
-theorem log_factorial_lower (n : ℕ) (hn : 0 < n) :
-    (n : ℝ) * Real.log n - n ≤ Real.log (n.factorial : ℝ) := by
-  have hpos : (0 : ℝ) < n := by exact_mod_cast hn
-  have hf : (0 : ℝ) < n.factorial := by exact_mod_cast Nat.factorial_pos n
-  have hh := Real.pow_div_factorial_le_exp (n : ℝ) hpos.le n
-  have hlog := Real.log_le_log (by positivity : (0 : ℝ) < (n : ℝ) ^ n / (n.factorial : ℝ)) hh
-  rw [Real.log_exp, Real.log_div (by positivity) hf.ne', Real.log_pow] at hlog
-  linarith
-
-theorem supportThreshold_bounds (d : ℕ) :
-    2 * (d : ℝ) - 6 ≤ (supportThreshold d : ℝ) ∧
-      (supportThreshold d : ℝ) ≤ 2 * (d : ℝ) := by
-  let n : ℕ := 2 ^ d
-  have hn : 0 < n := by dsimp [n]; positivity
-  have hnR : (0 : ℝ) < n := by exact_mod_cast hn
-  have hn1 : (1 : ℝ) ≤ n := by exact_mod_cast hn
-  have hfact : (0 : ℝ) < n.factorial := by exact_mod_cast Nat.factorial_pos n
-  have hl2 : (1 / 2 : ℝ) ≤ Real.log 2 := by
-    have hh := Real.one_sub_inv_le_log_of_pos (by norm_num : (0 : ℝ) < 2)
-    norm_num at hh ⊢
-    exact hh
-  have hl2pos : 0 < Real.log 2 := by linarith
-  have hlfrac : -(1 / 3 : ℝ) ≤ Real.log (3 / 4 : ℝ) := by
-    have hh := Real.one_sub_inv_le_log_of_pos (by norm_num : (0 : ℝ) < 3 / 4)
-    norm_num at hh ⊢
-    exact hh
-  have hlogn : Real.log (n : ℝ) = (d : ℝ) * Real.log 2 := by
-    simp only [n, Nat.cast_pow, Nat.cast_ofNat, Real.log_pow]
-  have hlow := log_factorial_lower n hn
-  have hupp : Real.log (n.factorial : ℝ) ≤ (n : ℝ) * Real.log n := by
-    have hh : (n.factorial : ℝ) ≤ (n : ℝ) ^ n := by exact_mod_cast Nat.factorial_le_pow n
-    simpa only [Real.log_pow] using Real.log_le_log hfact hh
-  have hprod : Real.log (3 * (n.factorial : ℝ) / 4) =
-      Real.log (3 / 4 : ℝ) + Real.log (n.factorial : ℝ) := by
-    rw [show 3 * (n.factorial : ℝ) / 4 = (3 / 4) * n.factorial by ring]
-    exact Real.log_mul (by norm_num) hfact.ne'
-  have hlowlog : (n : ℝ) * ((d : ℝ) * Real.log 2) - n - 1 / 3 ≤
-      Real.log (3 * (n.factorial : ℝ) / 4) := by
-    rw [hprod, ← hlogn]
-    linarith
-  have hupplog : Real.log (3 * (n.factorial : ℝ) / 4) ≤
-      (n : ℝ) * ((d : ℝ) * Real.log 2) := by
-    rw [hprod, ← hlogn]
-    have hh : Real.log (3 / 4 : ℝ) ≤ 0 := Real.log_nonpos (by norm_num) (by norm_num)
-    linarith
-  let a : ℝ := (2 / (n : ℝ)) * Real.logb 2 (3 * (n.factorial : ℝ) / 4)
-  have ha_low : 2 * (d : ℝ) - 6 ≤ a := by
-    dsimp [a]
-    rw [Real.logb]
-    have he : (2 / (n : ℝ)) *
-        (Real.log (3 * (n.factorial : ℝ) / 4) / Real.log 2) =
-        (2 * Real.log (3 * (n.factorial : ℝ) / 4)) / ((n : ℝ) * Real.log 2) := by ring
-    rw [he]
-    apply (le_div_iff₀ (mul_pos hnR hl2pos)).mpr
-    nlinarith [mul_le_mul_of_nonneg_left hl2 (by positivity : (0 : ℝ) ≤ n)]
-  have ha_upp : a ≤ 2 * (d : ℝ) := by
-    dsimp [a]
-    rw [Real.logb]
-    have hscale := mul_le_mul_of_nonneg_left hupplog (by positivity : (0 : ℝ) ≤ 2 / n / Real.log 2)
-    have hid : (2 / (n : ℝ) / Real.log 2) * ((n : ℝ) * ((d : ℝ) * Real.log 2)) = 2 * d := by
-      field_simp
-    rw [hid] at hscale
-    calc
-      _ = (2 / (n : ℝ) / Real.log 2) * Real.log (3 * (n.factorial : ℝ) / 4) := by ring
-      _ ≤ 2 * (d : ℝ) := hscale
-  have hceil : supportThreshold d = ⌈a⌉ := by
-    simp only [supportThreshold, a, n, Nat.cast_pow, Nat.cast_ofNat]
-  rw [hceil]
-  constructor
-  · exact ha_low.trans (Int.le_ceil a)
-  · have hh : ⌈a⌉ ≤ (2 * d : ℤ) := Int.ceil_le.mpr (by exact_mod_cast ha_upp)
-    exact_mod_cast hh
-
-theorem supportThreshold_isBigO :
-    Asymptotics.IsBigO atTop
-      (fun d : ℕ => (supportThreshold d : ℝ) - 2 * (d : ℝ)) (fun _ => (1 : ℝ)) := by
-  rw [Asymptotics.isBigO_iff]
-  refine ⟨6, Filter.Eventually.of_forall (fun d => ?_)⟩
-  have h := supportThreshold_bounds d
-  simp only [Real.norm_eq_abs, norm_one, mul_one]
-  apply abs_le.mpr
-  constructor <;> linarith
-
-theorem orderFunction_isBigO_mixingTime :
-    Asymptotics.IsBigO atTop orderFunction (fun d : ℕ => (mixingTime d : ℝ)) := by
-  rw [Asymptotics.isBigO_iff]
-  refine ⟨1, Filter.eventually_atTop.2 ⟨6, fun d hd => ?_⟩⟩
-  have hs := (supportThreshold_bounds d).1
-  have hl : (supportThreshold d : ℝ) ≤ mixingTime d := by
-    exact_mod_cast exact_support_lower (by omega : 1 ≤ d)
-  have hdR : (6 : ℝ) ≤ d := by exact_mod_cast hd
-  simp only [orderFunction, Real.norm_eq_abs, one_mul]
-  rw [abs_of_nonneg (Nat.cast_nonneg d), abs_of_nonneg (Nat.cast_nonneg (mixingTime d))]
-  linarith
-
-end Thorp
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.FiniteLaw
+
+namespace OAI
+
+noncomputable section
+
+open scoped BigOperators
+open Filter
+
+namespace Thorp
+
+theorem exists_mixing_time_succ (d : ℕ) : ∃ t : ℕ, distance (d + 1) t ≤ 1 / 4 := by
+  classical
+  obtain ⟨T, hT⟩ := Reachability.exists_surjective_run d
+  let c : ℝ := 1 / Fintype.card (History (d + 1) T)
+  let r : ℝ := 1 - (Fintype.card (State (d + 1)) : ℝ) * c
+  have hcpos : 0 < c := by dsimp [c]; positivity
+  have hminor (g : State (d + 1)) : c ≤ law (d + 1) T g :=
+    FiniteLaw.fairMass_lower_of_surjective _ hT g
+  have hr : 0 ≤ r := by
+    have hh := Finset.sum_le_sum (s := Finset.univ) (fun g _ => hminor g)
+    simp only [law, fairMass_sum] at hh
+    simpa only [Finset.sum_const, Finset.card_univ, nsmul_eq_mul] using sub_nonneg.mpr hh
+  have hrlt : r < 1 := by
+    dsimp [r]
+    have hn : (0 : ℝ) < Fintype.card (State (d + 1)) := by positivity
+    nlinarith
+  have hcontract (s : ℕ) : distance (d + 1) (s + T) ≤ r * distance (d + 1) s := by
+    unfold distance uniform
+    rw [law_add]
+    exact FiniteLaw.tv_conv_contract _ _ (fairMass_sum _) (fairMass_sum _) c hminor
+  have hgeom (n : ℕ) : distance (d + 1) (n * T) ≤ r ^ n * distance (d + 1) 0 := by
+    induction n with
+    | zero => simp
+    | succ n ih =>
+      rw [Nat.succ_mul]
+      calc
+        _ ≤ r * distance (d + 1) (n * T) := hcontract _
+        _ ≤ r * (r ^ n * distance (d + 1) 0) := mul_le_mul_of_nonneg_left ih hr
+        _ = _ := by rw [pow_succ']; ring
+  have hlim : Tendsto (fun n : ℕ => r ^ n * distance (d + 1) 0) atTop (nhds 0) := by
+    simpa using (tendsto_pow_atTop_nhds_zero_of_lt_one hr hrlt).mul_const (distance (d + 1) 0)
+  have hev : ∀ᶠ n : ℕ in atTop, r ^ n * distance (d + 1) 0 < 1 / 4 :=
+    hlim.eventually (eventually_lt_nhds (by norm_num : (0 : ℝ) < 1 / 4))
+  obtain ⟨n, hn⟩ := hev.exists
+  exact ⟨n * T, (hgeom n).trans hn.le⟩
+
+theorem exists_mixing_time {d : ℕ} (hd : 1 ≤ d) : ∃ t : ℕ, distance d t ≤ 1 / 4 := by
+  obtain ⟨k, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : d ≠ 0)
+  exact exists_mixing_time_succ k
+
+theorem exact_support_lower {d : ℕ} (hd : 1 ≤ d) :
+    supportThreshold d ≤ (mixingTime d : ℤ) :=
+  supportThreshold_le_mixingTime hd (exists_mixing_time hd)
+
+theorem log_factorial_lower (n : ℕ) (hn : 0 < n) :
+    (n : ℝ) * Real.log n - n ≤ Real.log (n.factorial : ℝ) := by
+  have hpos : (0 : ℝ) < n := by exact_mod_cast hn
+  have hf : (0 : ℝ) < n.factorial := by exact_mod_cast Nat.factorial_pos n
+  have hh := Real.pow_div_factorial_le_exp (n : ℝ) hpos.le n
+  have hlog := Real.log_le_log (by positivity : (0 : ℝ) < (n : ℝ) ^ n / (n.factorial : ℝ)) hh
+  rw [Real.log_exp, Real.log_div (by positivity) hf.ne', Real.log_pow] at hlog
+  linarith
+
+theorem supportThreshold_bounds (d : ℕ) :
+    2 * (d : ℝ) - 6 ≤ (supportThreshold d : ℝ) ∧
+      (supportThreshold d : ℝ) ≤ 2 * (d : ℝ) := by
+  let n : ℕ := 2 ^ d
+  have hn : 0 < n := by dsimp [n]; positivity
+  have hnR : (0 : ℝ) < n := by exact_mod_cast hn
+  have hn1 : (1 : ℝ) ≤ n := by exact_mod_cast hn
+  have hfact : (0 : ℝ) < n.factorial := by exact_mod_cast Nat.factorial_pos n
+  have hl2 : (1 / 2 : ℝ) ≤ Real.log 2 := by
+    have hh := Real.one_sub_inv_le_log_of_pos (by norm_num : (0 : ℝ) < 2)
+    norm_num at hh ⊢
+    exact hh
+  have hl2pos : 0 < Real.log 2 := by linarith
+  have hlfrac : -(1 / 3 : ℝ) ≤ Real.log (3 / 4 : ℝ) := by
+    have hh := Real.one_sub_inv_le_log_of_pos (by norm_num : (0 : ℝ) < 3 / 4)
+    norm_num at hh ⊢
+    exact hh
+  have hlogn : Real.log (n : ℝ) = (d : ℝ) * Real.log 2 := by
+    simp only [n, Nat.cast_pow, Nat.cast_ofNat, Real.log_pow]
+  have hlow := log_factorial_lower n hn
+  have hupp : Real.log (n.factorial : ℝ) ≤ (n : ℝ) * Real.log n := by
+    have hh : (n.factorial : ℝ) ≤ (n : ℝ) ^ n := by exact_mod_cast Nat.factorial_le_pow n
+    simpa only [Real.log_pow] using Real.log_le_log hfact hh
+  have hprod : Real.log (3 * (n.factorial : ℝ) / 4) =
+      Real.log (3 / 4 : ℝ) + Real.log (n.factorial : ℝ) := by
+    rw [show 3 * (n.factorial : ℝ) / 4 = (3 / 4) * n.factorial by ring]
+    exact Real.log_mul (by norm_num) hfact.ne'
+  have hlowlog : (n : ℝ) * ((d : ℝ) * Real.log 2) - n - 1 / 3 ≤
+      Real.log (3 * (n.factorial : ℝ) / 4) := by
+    rw [hprod, ← hlogn]
+    linarith
+  have hupplog : Real.log (3 * (n.factorial : ℝ) / 4) ≤
+      (n : ℝ) * ((d : ℝ) * Real.log 2) := by
+    rw [hprod, ← hlogn]
+    have hh : Real.log (3 / 4 : ℝ) ≤ 0 := Real.log_nonpos (by norm_num) (by norm_num)
+    linarith
+  let a : ℝ := (2 / (n : ℝ)) * Real.logb 2 (3 * (n.factorial : ℝ) / 4)
+  have ha_low : 2 * (d : ℝ) - 6 ≤ a := by
+    dsimp [a]
+    rw [Real.logb]
+    have he : (2 / (n : ℝ)) *
+        (Real.log (3 * (n.factorial : ℝ) / 4) / Real.log 2) =
+        (2 * Real.log (3 * (n.factorial : ℝ) / 4)) / ((n : ℝ) * Real.log 2) := by ring
+    rw [he]
+    apply (le_div_iff₀ (mul_pos hnR hl2pos)).mpr
+    nlinarith [mul_le_mul_of_nonneg_left hl2 (by positivity : (0 : ℝ) ≤ n)]
+  have ha_upp : a ≤ 2 * (d : ℝ) := by
+    dsimp [a]
+    rw [Real.logb]
+    have hscale := mul_le_mul_of_nonneg_left hupplog (by positivity : (0 : ℝ) ≤ 2 / n / Real.log 2)
+    have hid : (2 / (n : ℝ) / Real.log 2) * ((n : ℝ) * ((d : ℝ) * Real.log 2)) = 2 * d := by
+      field_simp
+    rw [hid] at hscale
+    calc
+      _ = (2 / (n : ℝ) / Real.log 2) * Real.log (3 * (n.factorial : ℝ) / 4) := by ring
+      _ ≤ 2 * (d : ℝ) := hscale
+  have hceil : supportThreshold d = ⌈a⌉ := by
+    simp only [supportThreshold, a, n, Nat.cast_pow, Nat.cast_ofNat]
+  rw [hceil]
+  constructor
+  · exact ha_low.trans (Int.le_ceil a)
+  · have hh : ⌈a⌉ ≤ (2 * d : ℤ) := Int.ceil_le.mpr (by exact_mod_cast ha_upp)
+    exact_mod_cast hh
+
+theorem supportThreshold_isBigO :
+    Asymptotics.IsBigO atTop
+      (fun d : ℕ => (supportThreshold d : ℝ) - 2 * (d : ℝ)) (fun _ => (1 : ℝ)) := by
+  rw [Asymptotics.isBigO_iff]
+  refine ⟨6, Filter.Eventually.of_forall (fun d => ?_)⟩
+  have h := supportThreshold_bounds d
+  simp only [Real.norm_eq_abs, norm_one, mul_one]
+  apply abs_le.mpr
+  constructor <;> linarith
+
+theorem orderFunction_isBigO_mixingTime :
+    Asymptotics.IsBigO atTop orderFunction (fun d : ℕ => (mixingTime d : ℝ)) := by
+  rw [Asymptotics.isBigO_iff]
+  refine ⟨1, Filter.eventually_atTop.2 ⟨6, fun d hd => ?_⟩⟩
+  have hs := (supportThreshold_bounds d).1
+  have hl : (supportThreshold d : ℝ) ≤ mixingTime d := by
+    exact_mod_cast exact_support_lower (by omega : 1 ≤ d)
+  have hdR : (6 : ℝ) ≤ d := by exact_mod_cast hd
+  simp only [orderFunction, Real.norm_eq_abs, one_mul]
+  rw [abs_of_nonneg (Nat.cast_nonneg d), abs_of_nonneg (Nat.cast_nonneg (mixingTime d))]
+  linarith
+
+end Thorp
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/Model.lean b/lean/OAI/Probability/ThorpShuffle/Model.lean
index ffb3d0030db96ad9164777198407176cb34964e3..b1d74acafdaaa5fdda4bdf49c67f3e22ab88dffb 100644
--- a/lean/OAI/Probability/ThorpShuffle/Model.lean
+++ b/lean/OAI/Probability/ThorpShuffle/Model.lean
@@ -1,78 +1,78 @@
-import Mathlib
-
-namespace OAI
-
-universe u v
-
-noncomputable section
-open scoped BigOperators
-open Filter
-
-namespace Thorp
-
-abbrev Position (d : ℕ) := Fin d → Bool
-abbrev State (d : ℕ) := Equiv.Perm (Position d)
-
-def Coins : ℕ → Type
-  | 0 => Unit
-  | d + 1 => Position d → Bool
-
-instance coinsFintype (d : ℕ) : Fintype (Coins d) := by
-  cases d <;> simp only [Coins] <;> infer_instance
-
-instance coinsInhabited (d : ℕ) : Inhabited (Coins d) := by
-  cases d <;> simp only [Coins] <;> infer_instance
-
-def pairSwitch (d : ℕ) (c : Position d → Bool) : Equiv.Perm (Bool × Position d) where
-  toFun p := (p.1 ^^ c p.2, p.2)
-  invFun p := (p.1 ^^ c p.2, p.2)
-  left_inv p := by
-    rcases p with ⟨b, s⟩
-    cases b <;> cases h : c s <;> simp [h]
-  right_inv p := by
-    rcases p with ⟨b, s⟩
-    cases b <;> cases h : c s <;> simp [h]
-
-def rotate (d : ℕ) : State d :=
-  Equiv.piCongrLeft (fun _ : Fin d => Bool) (finRotate d).symm
-
-def step : (d : ℕ) → Coins d → State d
-  | 0, _ => Equiv.refl _
-  | d + 1, c =>
-    (((Fin.consEquiv (fun _ : Fin (d + 1) => Bool)).symm.trans
-      (pairSwitch d c)).trans
-      (Fin.consEquiv (fun _ : Fin (d + 1) => Bool))).trans (rotate (d + 1))
-
-abbrev History (d t : ℕ) := Fin t → Coins d
-
-def run (d t : ℕ) (ω : History d t) : State d :=
-  ((List.ofFn fun i : Fin t => step d (ω i)).reverse).prod
-
-def fairMass {Ω : Type u} {α : Type v} [Fintype Ω] (f : Ω → α) (a : α) : ℝ := by
-  classical
-  exact (∑ ω : Ω, if f ω = a then (1 : ℝ) else 0) / Fintype.card Ω
-
-def law (d t : ℕ) : State d → ℝ := fairMass (run d t)
-
-def uniform (d : ℕ) (_ : State d) : ℝ := 1 / Fintype.card (State d)
-
-def tv {α : Type u} [Fintype α] (μ ν : α → ℝ) : ℝ :=
-  (1 / 2 : ℝ) * ∑ a : α, |μ a - ν a|
-
-def distance (d t : ℕ) : ℝ := tv (law d t) (uniform d)
-
-def lawFrom (d t : ℕ) (g₀ : State d) : State d → ℝ :=
-  fairMass (fun ω : History d t => run d t ω * g₀)
-
-def mixingTime (d : ℕ) : ℕ := sInf {t : ℕ | distance d t ≤ 1 / 4}
-
-def supportThreshold (d : ℕ) : ℤ :=
-  ⌈(2 / (2 ^ d : ℝ)) * Real.logb 2 (3 * ((2 ^ d).factorial : ℝ) / 4)⌉
-
-def orderFunction (d : ℕ) : ℝ := d
-
-end Thorp
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+universe u v
+
+noncomputable section
+open scoped BigOperators
+open Filter
+
+namespace Thorp
+
+abbrev Position (d : ℕ) := Fin d → Bool
+abbrev State (d : ℕ) := Equiv.Perm (Position d)
+
+def Coins : ℕ → Type
+  | 0 => Unit
+  | d + 1 => Position d → Bool
+
+instance coinsFintype (d : ℕ) : Fintype (Coins d) := by
+  cases d <;> simp only [Coins] <;> infer_instance
+
+instance coinsInhabited (d : ℕ) : Inhabited (Coins d) := by
+  cases d <;> simp only [Coins] <;> infer_instance
+
+def pairSwitch (d : ℕ) (c : Position d → Bool) : Equiv.Perm (Bool × Position d) where
+  toFun p := (p.1 ^^ c p.2, p.2)
+  invFun p := (p.1 ^^ c p.2, p.2)
+  left_inv p := by
+    rcases p with ⟨b, s⟩
+    cases b <;> cases h : c s <;> simp [h]
+  right_inv p := by
+    rcases p with ⟨b, s⟩
+    cases b <;> cases h : c s <;> simp [h]
+
+def rotate (d : ℕ) : State d :=
+  Equiv.piCongrLeft (fun _ : Fin d => Bool) (finRotate d).symm
+
+def step : (d : ℕ) → Coins d → State d
+  | 0, _ => Equiv.refl _
+  | d + 1, c =>
+    (((Fin.consEquiv (fun _ : Fin (d + 1) => Bool)).symm.trans
+      (pairSwitch d c)).trans
+      (Fin.consEquiv (fun _ : Fin (d + 1) => Bool))).trans (rotate (d + 1))
+
+abbrev History (d t : ℕ) := Fin t → Coins d
+
+def run (d t : ℕ) (ω : History d t) : State d :=
+  ((List.ofFn fun i : Fin t => step d (ω i)).reverse).prod
+
+def fairMass {Ω : Type u} {α : Type v} [Fintype Ω] (f : Ω → α) (a : α) : ℝ := by
+  classical
+  exact (∑ ω : Ω, if f ω = a then (1 : ℝ) else 0) / Fintype.card Ω
+
+def law (d t : ℕ) : State d → ℝ := fairMass (run d t)
+
+def uniform (d : ℕ) (_ : State d) : ℝ := 1 / Fintype.card (State d)
+
+def tv {α : Type u} [Fintype α] (μ ν : α → ℝ) : ℝ :=
+  (1 / 2 : ℝ) * ∑ a : α, |μ a - ν a|
+
+def distance (d t : ℕ) : ℝ := tv (law d t) (uniform d)
+
+def lawFrom (d t : ℕ) (g₀ : State d) : State d → ℝ :=
+  fairMass (fun ω : History d t => run d t ω * g₀)
+
+def mixingTime (d : ℕ) : ℕ := sInf {t : ℕ | distance d t ≤ 1 / 4}
+
+def supportThreshold (d : ℕ) : ℤ :=
+  ⌈(2 / (2 ^ d : ℝ)) * Real.logb 2 (3 * ((2 ^ d).factorial : ℝ) / 4)⌉
+
+def orderFunction (d : ℕ) : ℝ := d
+
+end Thorp
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/NoiseEnergy.lean b/lean/OAI/Probability/ThorpShuffle/NoiseEnergy.lean
index 1cae4d611c1311e2c97b5d2849ca9f92705b9c67..10e479bcb229dd7188ec4bc215022f7d9ed7b068 100644
--- a/lean/OAI/Probability/ThorpShuffle/NoiseEnergy.lean
+++ b/lean/OAI/Probability/ThorpShuffle/NoiseEnergy.lean
@@ -1,188 +1,188 @@
-import OAI.Probability.ThorpShuffle.HalfHistories
-
-namespace OAI
-
-noncomputable section
-
-open scoped BigOperators
-open Filter
-
-namespace Thorp
-
-namespace Conditional
-
-def insertBit (d : ℕ) (i : Fin (d + 1)) (b : Bool) (x : Position d) : Position (d + 1) :=
-  Fin.insertNth i b x
-
-def halfPositionEquiv (d : ℕ) (i : Fin (d + 1)) : Bool × Position d ≃ Position (d + 1) :=
-  Fin.insertNthEquiv (fun _ => Bool) i
-
-@[simp] theorem halfPositionEquiv_apply (d : ℕ) (i : Fin (d + 1)) (p : Bool × Position d) :
-    halfPositionEquiv d i p = insertBit d i p.1 p.2 := rfl
-
-@[simp] theorem insertBit_zero (d : ℕ) (b : Bool) (x : Position d) :
-    insertBit d 0 b x = Fin.cons b x := Fin.insertNth_zero' _ _
-
-@[simp] theorem insertBit_last (d : ℕ) (b : Bool) (x : Position d) :
-    insertBit d (Fin.last d) b x = Fin.snoc x b := Fin.insertNth_last' _ _
-
- theorem insertBit_cons (d : ℕ) (k : Fin (d + 1)) (h b : Bool) (x : Position d) :
-    insertBit (d + 1) k.succ h (Fin.cons b x) = Fin.cons b (insertBit d k h x) := by
-  change Fin.insertNth (α := fun _ => Bool) k.succ h (Fin.cons b x) = _
-  apply Fin.insertNth_eq_iff.mpr
-  constructor
-  · simp [insertBit]
-  · change Fin.cons b x = Fin.cons b (insertBit d k h x) ∘ k.succ.succAbove
-    erw [Fin.cons_comp_succ_succAbove]
-    simp [insertBit]
-
- theorem insertBit_snoc (d : ℕ) (k : Fin (d + 1)) (h b : Bool) (x : Position d) :
-    insertBit (d + 1) k.castSucc h (Fin.snoc x b) = Fin.snoc (insertBit d k h x) b := by
-  change Fin.insertNth (α := fun _ => Bool) k.castSucc h (Fin.snoc x b) = _
-  apply Fin.insertNth_eq_iff.mpr
-  constructor
-  · simp [insertBit]
-  · funext j
-    refine Fin.lastCases ?_ (fun i => ?_) j
-    · simp [Fin.removeNth, Fin.succAbove_ne_last_last, insertBit]
-    · simp [Fin.removeNth, Fin.castSucc_succAbove_castSucc, insertBit]
-
- theorem physicalFree_snoc (d : ℕ) (B : Position (d + 1) → Bool)
-    (c : Position d → Bool) (x : Position d) (b : Bool) :
-    physicalFree d B c (Fin.snoc x b) = B (Fin.cons (b ^^ c x) x) := by
-  erw [← scatterPosition_eq_snoc, physicalFree_scatter]
-  rfl
-
- theorem physicalFlow_snoc (d : ℕ) (B : Position (d + 1) → Bool)
-    (w : Position (d + 1) → ℝ) (c : Position d → Bool) (x : Position d) (b : Bool) :
-    physicalFlow d B w c (Fin.snoc x b) =
-      if b then (pairUpdate (B (Fin.cons false x)) (B (Fin.cons true x))
-        (w (Fin.cons false x)) (w (Fin.cons true x)) (c x)).2
-      else (pairUpdate (B (Fin.cons false x)) (B (Fin.cons true x))
-        (w (Fin.cons false x)) (w (Fin.cons true x)) (c x)).1 := by
-  erw [← scatterPosition_eq_snoc, physicalFlow_scatter]
-  rfl
-
-@[ext] theorem RawState.ext {d : ℕ} {v u : RawState d}
-    (hf : v.free = u.free) (hw : v.weight = u.weight) : v = u := by
-  cases v; cases u; simp_all
-
-def halfRaw (d : ℕ) (i : Fin (d + 1)) (h : Bool) (v : RawState (d + 1)) : RawState d where
-  free x := v.free (insertBit d i h x)
-  weight x := v.weight (insertBit d i h x)
-
-theorem rawNext_half (d : ℕ) (k : Fin (d + 1)) (h : Bool)
-    (v : RawState (d + 2)) (c : Position (d + 1) → Bool) :
-    halfRaw (d + 1) k.castSucc h (rawNext (d + 1) v c) =
-      rawNext d (halfRaw (d + 1) k.succ h v) (fun x => c (insertBit d k h x)) := by
-  apply RawState.ext
-  · funext x
-    obtain ⟨⟨b, y⟩, rfl⟩ := (Fin.snocEquiv (fun _ : Fin (d + 1) => Bool)).surjective x
-    change physicalFree (d + 1) v.free c (insertBit (d + 1) k.castSucc h (Fin.snoc y b)) =
-      physicalFree d (fun z => v.free (insertBit (d + 1) k.succ h z))
-        (fun z => c (insertBit d k h z)) (Fin.snoc y b)
-    erw [insertBit_snoc, physicalFree_snoc, physicalFree_snoc, insertBit_cons]
-  · funext x
-    obtain ⟨⟨b, y⟩, rfl⟩ := (Fin.snocEquiv (fun _ : Fin (d + 1) => Bool)).surjective x
-    change physicalFlow (d + 1) v.free v.weight c
-      (insertBit (d + 1) k.castSucc h (Fin.snoc y b)) =
-      physicalFlow d (fun z => v.free (insertBit (d + 1) k.succ h z))
-        (fun z => v.weight (insertBit (d + 1) k.succ h z))
-        (fun z => c (insertBit d k h z)) (Fin.snoc y b)
-    erw [insertBit_snoc, physicalFlow_snoc, physicalFlow_snoc]
-    simp only [insertBit_cons]
-
-end Conditional
-
-namespace Conditional
-
-def halfMarker (d t : ℕ) : Fin (d + 2) := ⟨d + 1 - t, by omega⟩
-def coinHalfMarker (d t : ℕ) : Fin (d + 1) := ⟨d - t, by omega⟩
-
-@[simp] theorem halfMarker_zero (d : ℕ) : halfMarker d 0 = Fin.last (d + 1) := rfl
-@[simp] theorem halfMarker_end (d : ℕ) : halfMarker d (d + 1) = 0 := by
-  apply Fin.ext; simp [halfMarker]
-
-theorem halfMarker_next (d t : ℕ) (ht : t ≤ d) :
-    halfMarker d (t + 1) = (coinHalfMarker d t).castSucc := by
-  apply Fin.ext; simp only [halfMarker, coinHalfMarker, Fin.val_castSucc]; omega
-
-theorem halfMarker_previous (d t : ℕ) (ht : t ≤ d) :
-    halfMarker d t = (coinHalfMarker d t).succ := by
-  apply Fin.ext; simp only [halfMarker, coinHalfMarker, Fin.val_succ]; omega
-
-def halfHistory (d t : ℕ) (h : Bool) (ω : History (d + 2) t) : History (d + 1) t :=
-  fun i x => ω i (insertBit d (coinHalfMarker d i.val) h x)
-
-@[simp] theorem halfHistory_snoc (d t : ℕ) (h : Bool)
-    (ω : History (d + 2) t) (c : Coins (d + 2)) :
-    halfHistory d (t + 1) h (Fin.snoc ω c) =
-      Fin.snoc (halfHistory d t h ω) (fun x => c (insertBit d (coinHalfMarker d t) h x)) := by
-  funext i x
-  refine Fin.lastCases ?_ (fun j => ?_) i
-  · change (Fin.snoc (α := fun _ => Coins (d + 2)) ω c (Fin.last t)) (insertBit d (coinHalfMarker d t) h x) =
-      (Fin.snoc (α := fun _ => Coins (d + 1)) (halfHistory d t h ω)
-        (fun z => c (insertBit d (coinHalfMarker d t) h z)) (Fin.last t)) x
-    erw [Fin.snoc_last, Fin.snoc_last]
-  · change (Fin.snoc (α := fun _ => Coins (d + 2)) ω c j.castSucc) (insertBit d (coinHalfMarker d j.val) h x) =
-      (Fin.snoc (α := fun _ => Coins (d + 1)) (halfHistory d t h ω)
-        (fun z => c (insertBit d (coinHalfMarker d t) h z)) j.castSucc) x
-    erw [Fin.snoc_castSucc, Fin.snoc_castSucc]
-    rfl
-
-theorem rawIterate_half (d t : ℕ) (ht : t ≤ d + 1) (h : Bool)
-    (v : RawState (d + 2)) (ω : History (d + 2) t) :
-    halfRaw (d + 1) (halfMarker d t) h (rawIterate (d + 1) v t ω) =
-      rawIterate d (halfRaw (d + 1) (Fin.last (d + 1)) h v) t (halfHistory d t h ω) := by
-  induction t with
-  | zero => rfl
-  | succ t ih =>
-    change halfRaw (d + 1) (halfMarker d (t + 1)) h
-      (rawNext (d + 1) (rawIterate (d + 1) v t (fun i => ω i.castSucc)) (ω (Fin.last t))) = _
-    erw [halfMarker_next d t (by omega), rawNext_half,
-      ← halfMarker_previous d t (by omega), ih (by omega)]
-    rfl
-
-def splitHalfHistory (d t : ℕ) :
-    History (d + 2) t ≃ History (d + 1) t × History (d + 1) t where
-  toFun ω := (halfHistory d t false ω, halfHistory d t true ω)
-  invFun p := fun i y =>
-    if y (coinHalfMarker d i.val) then
-      p.2 i (Fin.removeNth (coinHalfMarker d i.val) y)
-    else p.1 i (Fin.removeNth (coinHalfMarker d i.val) y)
-  left_inv ω := by
-    funext i y
-    obtain ⟨⟨h, x⟩, rfl⟩ := (halfPositionEquiv d (coinHalfMarker d i.val)).surjective y
-    simp only [halfPositionEquiv_apply, insertBit, Fin.insertNth_apply_same,
-      halfHistory, Fin.removeNth_insertNth]
-    cases h <;> rfl
-  right_inv p := by
-    apply Prod.ext <;> funext i x <;>
-      simp [halfHistory, insertBit, Fin.insertNth_apply_same, Fin.removeNth_insertNth]
-
-theorem mean_half_product (d t : ℕ) (f g : History (d + 1) t → ℝ) :
-    mean (fun ω : History (d + 2) t => f (halfHistory d t false ω) *
-      g (halfHistory d t true ω)) = mean f * mean g := by
-  have he := mean_equiv (splitHalfHistory d t) (fun p => f p.1 * g p.2)
-  change mean (fun ω : History (d + 2) t => f (halfHistory d t false ω) *
-      g (halfHistory d t true ω)) = _ at he
-  erw [he, mean_prod]
-  simp_rw [mean_const_mul]
-  exact mean_mul_const _ _
-
-theorem mean_opposite_half_product (d t : ℕ) (h : Bool)
-    (f g : History (d + 1) t → ℝ) :
-    mean (fun ω : History (d + 2) t => f (halfHistory d t h ω) *
-      g (halfHistory d t (!h) ω)) = mean f * mean g := by
-  cases h
-  · exact mean_half_product d t f g
-  · simp only [Bool.not_true]
-    simpa only [mul_comm] using mean_half_product d t g f
-
-end Conditional
-
-end Thorp
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.HalfHistories
+
+namespace OAI
+
+noncomputable section
+
+open scoped BigOperators
+open Filter
+
+namespace Thorp
+
+namespace Conditional
+
+def insertBit (d : ℕ) (i : Fin (d + 1)) (b : Bool) (x : Position d) : Position (d + 1) :=
+  Fin.insertNth i b x
+
+def halfPositionEquiv (d : ℕ) (i : Fin (d + 1)) : Bool × Position d ≃ Position (d + 1) :=
+  Fin.insertNthEquiv (fun _ => Bool) i
+
+@[simp] theorem halfPositionEquiv_apply (d : ℕ) (i : Fin (d + 1)) (p : Bool × Position d) :
+    halfPositionEquiv d i p = insertBit d i p.1 p.2 := rfl
+
+@[simp] theorem insertBit_zero (d : ℕ) (b : Bool) (x : Position d) :
+    insertBit d 0 b x = Fin.cons b x := Fin.insertNth_zero' _ _
+
+@[simp] theorem insertBit_last (d : ℕ) (b : Bool) (x : Position d) :
+    insertBit d (Fin.last d) b x = Fin.snoc x b := Fin.insertNth_last' _ _
+
+ theorem insertBit_cons (d : ℕ) (k : Fin (d + 1)) (h b : Bool) (x : Position d) :
+    insertBit (d + 1) k.succ h (Fin.cons b x) = Fin.cons b (insertBit d k h x) := by
+  change Fin.insertNth (α := fun _ => Bool) k.succ h (Fin.cons b x) = _
+  apply Fin.insertNth_eq_iff.mpr
+  constructor
+  · simp [insertBit]
+  · change Fin.cons b x = Fin.cons b (insertBit d k h x) ∘ k.succ.succAbove
+    erw [Fin.cons_comp_succ_succAbove]
+    simp [insertBit]
+
+ theorem insertBit_snoc (d : ℕ) (k : Fin (d + 1)) (h b : Bool) (x : Position d) :
+    insertBit (d + 1) k.castSucc h (Fin.snoc x b) = Fin.snoc (insertBit d k h x) b := by
+  change Fin.insertNth (α := fun _ => Bool) k.castSucc h (Fin.snoc x b) = _
+  apply Fin.insertNth_eq_iff.mpr
+  constructor
+  · simp [insertBit]
+  · funext j
+    refine Fin.lastCases ?_ (fun i => ?_) j
+    · simp [Fin.removeNth, Fin.succAbove_ne_last_last, insertBit]
+    · simp [Fin.removeNth, Fin.castSucc_succAbove_castSucc, insertBit]
+
+ theorem physicalFree_snoc (d : ℕ) (B : Position (d + 1) → Bool)
+    (c : Position d → Bool) (x : Position d) (b : Bool) :
+    physicalFree d B c (Fin.snoc x b) = B (Fin.cons (b ^^ c x) x) := by
+  erw [← scatterPosition_eq_snoc, physicalFree_scatter]
+  rfl
+
+ theorem physicalFlow_snoc (d : ℕ) (B : Position (d + 1) → Bool)
+    (w : Position (d + 1) → ℝ) (c : Position d → Bool) (x : Position d) (b : Bool) :
+    physicalFlow d B w c (Fin.snoc x b) =
+      if b then (pairUpdate (B (Fin.cons false x)) (B (Fin.cons true x))
+        (w (Fin.cons false x)) (w (Fin.cons true x)) (c x)).2
+      else (pairUpdate (B (Fin.cons false x)) (B (Fin.cons true x))
+        (w (Fin.cons false x)) (w (Fin.cons true x)) (c x)).1 := by
+  erw [← scatterPosition_eq_snoc, physicalFlow_scatter]
+  rfl
+
+@[ext] theorem RawState.ext {d : ℕ} {v u : RawState d}
+    (hf : v.free = u.free) (hw : v.weight = u.weight) : v = u := by
+  cases v; cases u; simp_all
+
+def halfRaw (d : ℕ) (i : Fin (d + 1)) (h : Bool) (v : RawState (d + 1)) : RawState d where
+  free x := v.free (insertBit d i h x)
+  weight x := v.weight (insertBit d i h x)
+
+theorem rawNext_half (d : ℕ) (k : Fin (d + 1)) (h : Bool)
+    (v : RawState (d + 2)) (c : Position (d + 1) → Bool) :
+    halfRaw (d + 1) k.castSucc h (rawNext (d + 1) v c) =
+      rawNext d (halfRaw (d + 1) k.succ h v) (fun x => c (insertBit d k h x)) := by
+  apply RawState.ext
+  · funext x
+    obtain ⟨⟨b, y⟩, rfl⟩ := (Fin.snocEquiv (fun _ : Fin (d + 1) => Bool)).surjective x
+    change physicalFree (d + 1) v.free c (insertBit (d + 1) k.castSucc h (Fin.snoc y b)) =
+      physicalFree d (fun z => v.free (insertBit (d + 1) k.succ h z))
+        (fun z => c (insertBit d k h z)) (Fin.snoc y b)
+    erw [insertBit_snoc, physicalFree_snoc, physicalFree_snoc, insertBit_cons]
+  · funext x
+    obtain ⟨⟨b, y⟩, rfl⟩ := (Fin.snocEquiv (fun _ : Fin (d + 1) => Bool)).surjective x
+    change physicalFlow (d + 1) v.free v.weight c
+      (insertBit (d + 1) k.castSucc h (Fin.snoc y b)) =
+      physicalFlow d (fun z => v.free (insertBit (d + 1) k.succ h z))
+        (fun z => v.weight (insertBit (d + 1) k.succ h z))
+        (fun z => c (insertBit d k h z)) (Fin.snoc y b)
+    erw [insertBit_snoc, physicalFlow_snoc, physicalFlow_snoc]
+    simp only [insertBit_cons]
+
+end Conditional
+
+namespace Conditional
+
+def halfMarker (d t : ℕ) : Fin (d + 2) := ⟨d + 1 - t, by omega⟩
+def coinHalfMarker (d t : ℕ) : Fin (d + 1) := ⟨d - t, by omega⟩
+
+@[simp] theorem halfMarker_zero (d : ℕ) : halfMarker d 0 = Fin.last (d + 1) := rfl
+@[simp] theorem halfMarker_end (d : ℕ) : halfMarker d (d + 1) = 0 := by
+  apply Fin.ext; simp [halfMarker]
+
+theorem halfMarker_next (d t : ℕ) (ht : t ≤ d) :
+    halfMarker d (t + 1) = (coinHalfMarker d t).castSucc := by
+  apply Fin.ext; simp only [halfMarker, coinHalfMarker, Fin.val_castSucc]; omega
+
+theorem halfMarker_previous (d t : ℕ) (ht : t ≤ d) :
+    halfMarker d t = (coinHalfMarker d t).succ := by
+  apply Fin.ext; simp only [halfMarker, coinHalfMarker, Fin.val_succ]; omega
+
+def halfHistory (d t : ℕ) (h : Bool) (ω : History (d + 2) t) : History (d + 1) t :=
+  fun i x => ω i (insertBit d (coinHalfMarker d i.val) h x)
+
+@[simp] theorem halfHistory_snoc (d t : ℕ) (h : Bool)
+    (ω : History (d + 2) t) (c : Coins (d + 2)) :
+    halfHistory d (t + 1) h (Fin.snoc ω c) =
+      Fin.snoc (halfHistory d t h ω) (fun x => c (insertBit d (coinHalfMarker d t) h x)) := by
+  funext i x
+  refine Fin.lastCases ?_ (fun j => ?_) i
+  · change (Fin.snoc (α := fun _ => Coins (d + 2)) ω c (Fin.last t)) (insertBit d (coinHalfMarker d t) h x) =
+      (Fin.snoc (α := fun _ => Coins (d + 1)) (halfHistory d t h ω)
+        (fun z => c (insertBit d (coinHalfMarker d t) h z)) (Fin.last t)) x
+    erw [Fin.snoc_last, Fin.snoc_last]
+  · change (Fin.snoc (α := fun _ => Coins (d + 2)) ω c j.castSucc) (insertBit d (coinHalfMarker d j.val) h x) =
+      (Fin.snoc (α := fun _ => Coins (d + 1)) (halfHistory d t h ω)
+        (fun z => c (insertBit d (coinHalfMarker d t) h z)) j.castSucc) x
+    erw [Fin.snoc_castSucc, Fin.snoc_castSucc]
+    rfl
+
+theorem rawIterate_half (d t : ℕ) (ht : t ≤ d + 1) (h : Bool)
+    (v : RawState (d + 2)) (ω : History (d + 2) t) :
+    halfRaw (d + 1) (halfMarker d t) h (rawIterate (d + 1) v t ω) =
+      rawIterate d (halfRaw (d + 1) (Fin.last (d + 1)) h v) t (halfHistory d t h ω) := by
+  induction t with
+  | zero => rfl
+  | succ t ih =>
+    change halfRaw (d + 1) (halfMarker d (t + 1)) h
+      (rawNext (d + 1) (rawIterate (d + 1) v t (fun i => ω i.castSucc)) (ω (Fin.last t))) = _
+    erw [halfMarker_next d t (by omega), rawNext_half,
+      ← halfMarker_previous d t (by omega), ih (by omega)]
+    rfl
+
+def splitHalfHistory (d t : ℕ) :
+    History (d + 2) t ≃ History (d + 1) t × History (d + 1) t where
+  toFun ω := (halfHistory d t false ω, halfHistory d t true ω)
+  invFun p := fun i y =>
+    if y (coinHalfMarker d i.val) then
+      p.2 i (Fin.removeNth (coinHalfMarker d i.val) y)
+    else p.1 i (Fin.removeNth (coinHalfMarker d i.val) y)
+  left_inv ω := by
+    funext i y
+    obtain ⟨⟨h, x⟩, rfl⟩ := (halfPositionEquiv d (coinHalfMarker d i.val)).surjective y
+    simp only [halfPositionEquiv_apply, insertBit, Fin.insertNth_apply_same,
+      halfHistory, Fin.removeNth_insertNth]
+    cases h <;> rfl
+  right_inv p := by
+    apply Prod.ext <;> funext i x <;>
+      simp [halfHistory, insertBit, Fin.insertNth_apply_same, Fin.removeNth_insertNth]
+
+theorem mean_half_product (d t : ℕ) (f g : History (d + 1) t → ℝ) :
+    mean (fun ω : History (d + 2) t => f (halfHistory d t false ω) *
+      g (halfHistory d t true ω)) = mean f * mean g := by
+  have he := mean_equiv (splitHalfHistory d t) (fun p => f p.1 * g p.2)
+  change mean (fun ω : History (d + 2) t => f (halfHistory d t false ω) *
+      g (halfHistory d t true ω)) = _ at he
+  erw [he, mean_prod]
+  simp_rw [mean_const_mul]
+  exact mean_mul_const _ _
+
+theorem mean_opposite_half_product (d t : ℕ) (h : Bool)
+    (f g : History (d + 1) t → ℝ) :
+    mean (fun ω : History (d + 2) t => f (halfHistory d t h ω) *
+      g (halfHistory d t (!h) ω)) = mean f * mean g := by
+  cases h
+  · exact mean_half_product d t f g
+  · simp only [Bool.not_true]
+    simpa only [mul_comm] using mean_half_product d t g f
+
+end Conditional
+
+end Thorp
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/PairAveraging.lean b/lean/OAI/Probability/ThorpShuffle/PairAveraging.lean
index c68172483d97682b936390b0a6341225b357c4b5..de51433df0a4c8a8dd058d1629c63a181e12f9c1 100644
--- a/lean/OAI/Probability/ThorpShuffle/PairAveraging.lean
+++ b/lean/OAI/Probability/ThorpShuffle/PairAveraging.lean
@@ -1,232 +1,232 @@
-import OAI.Probability.ThorpShuffle.Model
-
-namespace OAI
-
-universe uΩ uα
-
-noncomputable section
-
-open scoped BigOperators
-open Filter
-
-namespace Thorp
-
-namespace Conditional
-
-def mean {Ω : Type uΩ} [Fintype Ω] (f : Ω → ℝ) : ℝ :=
-  (∑ ω, f ω) / Fintype.card Ω
-
-def sign (b : Bool) : ℝ := if b then -1 else 1
-
-@[simp] theorem sign_not (b : Bool) : sign (!b) = -sign b := by
-  cases b <;> norm_num [sign]
-
-@[simp] theorem sign_sq (b : Bool) : sign b ^ 2 = 1 := by
-  cases b <;> norm_num [sign]
-
-@[simp] theorem sign_mul_self (b : Bool) : sign b * sign b = 1 := by
-  simpa only [pow_two] using sign_sq b
-
-def toggleCoin {α : Type uα} [DecidableEq α] (i : α) : Equiv.Perm (α → Bool) :=
-  Function.Involutive.toPerm (fun c => Function.update c i (!(c i))) (by
-    intro c
-    funext j
-    by_cases h : j = i
-    · subst j
-      simp
-    · simp [h])
-
-@[simp] theorem toggleCoin_apply {α : Type uα} [DecidableEq α]
-    (i : α) (c : α → Bool) (j : α) :
-    toggleCoin i c j = Function.update c i (!(c i)) j := rfl
-
-theorem sum_sign {α : Type uα} [Fintype α] [DecidableEq α] (i : α) :
-    ∑ c : α → Bool, sign (c i) = 0 := by
-  have h := Equiv.sum_comp (toggleCoin i) (fun c : α → Bool => sign (c i))
-  simp only [toggleCoin_apply, Function.update_self, sign_not, Finset.sum_neg_distrib] at h
-  linarith
-
-theorem sum_sign_pair {α : Type uα} [Fintype α] [DecidableEq α] (i j : α) :
-    (∑ c : α → Bool, sign (c i) * sign (c j)) =
-      if i = j then (Fintype.card (α → Bool) : ℝ) else 0 := by
-  by_cases h : i = j
-  · subst j
-    simp
-  · have he := Equiv.sum_comp (toggleCoin i) (fun c : α → Bool => sign (c i) * sign (c j))
-    simp only [toggleCoin_apply, Function.update_self,
-      Function.update_of_ne (Ne.symm h), sign_not, neg_mul,
-      Finset.sum_neg_distrib] at he
-    rw [ite_eq_right h]
-    linarith
-
-theorem mean_signed_sum_sq {α : Type uα} [Fintype α] [DecidableEq α] (a : α → ℝ) :
-    mean (fun c : α → Bool => (∑ i, sign (c i) * a i) ^ 2) = ∑ i, a i ^ 2 := by
-  have hs :
-      (∑ c : α → Bool, (∑ i, sign (c i) * a i) ^ 2) =
-        (Fintype.card (α → Bool) : ℝ) * ∑ i, a i ^ 2 := by
-    calc
-      _ = ∑ c : α → Bool, ∑ i, ∑ j,
-          (sign (c i) * sign (c j)) * (a i * a j) := by
-        apply Finset.sum_congr rfl
-        intro c _
-        simp only [pow_two, Finset.sum_mul, Finset.mul_sum]
-        apply Finset.sum_congr rfl
-        intro i _
-        apply Finset.sum_congr rfl
-        intro j _
-        ring
-      _ = ∑ i, ∑ j, (∑ c : α → Bool, sign (c i) * sign (c j)) * (a i * a j) := by
-        rw [Finset.sum_comm]
-        apply Finset.sum_congr rfl
-        intro i _
-        rw [Finset.sum_comm]
-        simp only [Finset.sum_mul]
-      _ = _ := by
-        simp only [sum_sign_pair, ite_mul, zero_mul]
-        simp [Finset.mul_sum, pow_two]
-  unfold mean
-  rw [hs]
-  have hcard : (Fintype.card (α → Bool) : ℝ) ≠ 0 := by
-    exact_mod_cast Fintype.card_ne_zero
-  exact mul_div_cancel_left₀ _ hcard
-
-def pairUpdate (free₀ free₁ : Bool) (a b : ℝ) (coin : Bool) : ℝ × ℝ :=
-  if free₀ && free₁ then ((a + b) / 2, (a + b) / 2)
-  else if coin then (b, a) else (a, b)
-
-@[simp] theorem pairUpdate_sum (free₀ free₁ : Bool) (a b : ℝ) (coin : Bool) :
-    (pairUpdate free₀ free₁ a b coin).1 + (pairUpdate free₀ free₁ a b coin).2 = a + b := by
-  cases free₀ <;> cases free₁ <;> cases coin <;> simp [pairUpdate] <;> ring
-
-theorem pairUpdate_difference (free₀ free₁ : Bool) (a b : ℝ) (coin : Bool) :
-    (pairUpdate free₀ free₁ a b coin).1 - (pairUpdate free₀ free₁ a b coin).2 =
-      sign coin * (if free₀ && free₁ then 0 else a - b) := by
-  cases free₀ <;> cases free₁ <;> cases coin <;> simp [pairUpdate, sign]
-
-theorem pairUpdate_mean_first (free₀ free₁ : Bool) (a b : ℝ) :
-    mean (fun coin : Bool => (pairUpdate free₀ free₁ a b coin).1) = (a + b) / 2 := by
-  cases free₀ <;> cases free₁ <;> simp [mean, pairUpdate] <;> ring
-
-theorem pairUpdate_difference_sq (free₀ free₁ : Bool) (a b : ℝ)
-    (ha : free₀ = false → a = 0) (hb : free₁ = false → b = 0) :
-    (if free₀ && free₁ then (0 : ℝ) else a - b) ^ 2 =
-      (if free₁ then 0 else a ^ 2) + (if free₀ then 0 else b ^ 2) := by
-  cases free₀ <;> cases free₁ <;> simp_all
-
-def flow {α : Type uα} (B : Bool × α → Bool) (w : Bool × α → ℝ)
-    (c : α → Bool) (x : Bool × α) : ℝ :=
-  let p := pairUpdate (B (false, x.2)) (B (true, x.2))
-    (w (false, x.2)) (w (true, x.2)) (c x.2)
-  if x.1 then p.2 else p.1
-
-def oddCoefficient {α : Type uα} [Fintype α] (ε : α → ℝ) (w : Bool × α → ℝ) : ℝ :=
-  ∑ i, ε i * (w (false, i) - w (true, i))
-
-theorem mean_oddCoefficient_sq {α : Type uα} [Fintype α] [DecidableEq α]
-    (B : Bool × α → Bool) (w : Bool × α → ℝ) (ε : α → ℝ)
-    (hw : ∀ x, B x = false → w x = 0) (hε : ∀ i, ε i ^ 2 = 1) :
-    mean (fun c : α → Bool => oddCoefficient ε (flow B w c) ^ 2) =
-      ∑ i, ((if B (true, i) then 0 else w (false, i) ^ 2) +
-        (if B (false, i) then 0 else w (true, i) ^ 2)) := by
-  have he (c : α → Bool) :
-      oddCoefficient ε (flow B w c) =
-        ∑ i, sign (c i) * (ε i *
-          (if B (false, i) && B (true, i) then 0 else w (false, i) - w (true, i))) := by
-    unfold oddCoefficient
-    apply Finset.sum_congr rfl
-    intro i _
-    simp only [flow, Bool.false_eq_true, ↓reduceIte]
-    rw [pairUpdate_difference]
-    ring
-  simp_rw [he]
-  rw [mean_signed_sum_sq]
-  apply Finset.sum_congr rfl
-  intro i _
-  rw [mul_pow, hε, one_mul]
-  exact pairUpdate_difference_sq _ _ _ _ (hw _) (hw _)
-
-end Conditional
-
-namespace Conditional
-
-def pairMap {α : Type uα} (c : α → Bool) (x : Bool × α) : Bool × α :=
-  (x.1 ^^ c x.2, x.2)
-
-theorem pairMap_involutive {α : Type uα} (c : α → Bool) :
-    Function.Involutive (pairMap c) := by
-  rintro ⟨b, i⟩
-  cases b <;> cases h : c i <;> simp [pairMap, h]
-
-theorem flow_sub {α : Type uα} (B : Bool × α → Bool) (u v : Bool × α → ℝ)
-    (c : α → Bool) (x : Bool × α) :
-    flow B (fun y => u y - v y) c x = flow B u c x - flow B v c x := by
-  rcases x with ⟨b, i⟩
-  cases b <;> cases h₀ : B (false, i) <;> cases h₁ : B (true, i) <;>
-    cases hc : c i <;> simp [flow, pairUpdate, h₀, h₁, hc] <;> ring
-
-theorem flow_uniform {α : Type uα} (B : Bool × α → Bool) (r : ℝ)
-    (c : α → Bool) (x : Bool × α) :
-    flow B (fun y => if B y then r else 0) c x =
-      if B (pairMap c x) then r else 0 := by
-  rcases x with ⟨b, i⟩
-  cases b <;> cases h₀ : B (false, i) <;> cases h₁ : B (true, i) <;>
-    cases hc : c i <;> simp [flow, pairUpdate, pairMap, h₀, h₁, hc]
-
-theorem flow_support {α : Type uα} (B : Bool × α → Bool) (w : Bool × α → ℝ)
-    (hw : ∀ x, B x = false → w x = 0) (c : α → Bool) (x : Bool × α)
-    (hx : B (pairMap c x) = false) : flow B w c x = 0 := by
-  rcases x with ⟨b, i⟩
-  have h₀ := hw (false, i)
-  have h₁ := hw (true, i)
-  cases b <;> cases hb₀ : B (false, i) <;> cases hb₁ : B (true, i) <;>
-    cases hc : c i <;> simp_all [flow, pairUpdate, pairMap]
-
-theorem flow_evenCoefficient {α : Type uα} [Fintype α]
-    (B : Bool × α → Bool) (w : Bool × α → ℝ) (ε : α → ℝ) (c : α → Bool) :
-    (∑ i, ε i * (flow B w c (false, i) + flow B w c (true, i))) =
-      ∑ i, ε i * (w (false, i) + w (true, i)) := by
-  apply Finset.sum_congr rfl
-  intro i _
-  simp only [flow, Bool.false_eq_true, ↓reduceIte, pairUpdate_sum]
-
-theorem flow_sum {α : Type uα} [Fintype α] (B : Bool × α → Bool)
-    (w : Bool × α → ℝ) (c : α → Bool) :
-    (∑ x, flow B w c x) = ∑ x, w x := by
-  simp only [Fintype.sum_prod_type, Fintype.sum_bool, ← Finset.sum_add_distrib]
-  apply Finset.sum_congr rfl
-  intro i _
-  simpa only [flow, Bool.false_eq_true, ↓reduceIte, add_comm] using
-    (pairUpdate_sum (B (false, i)) (B (true, i)) (w (false, i)) (w (true, i)) (c i))
-
-theorem pairUpdate_energy_le (free₀ free₁ : Bool) (a b : ℝ) (coin : Bool) :
-    (pairUpdate free₀ free₁ a b coin).1 ^ 2 +
-        (pairUpdate free₀ free₁ a b coin).2 ^ 2 ≤ a ^ 2 + b ^ 2 := by
-  have h := sq_nonneg (a - b)
-  cases free₀ <;> cases free₁ <;> cases coin <;> simp [pairUpdate] <;> nlinarith
-
-theorem flow_energy_le {α : Type uα} [Fintype α] (B : Bool × α → Bool)
-    (w : Bool × α → ℝ) (c : α → Bool) :
-    (∑ x, flow B w c x ^ 2) ≤ ∑ x, w x ^ 2 := by
-  simp only [Fintype.sum_prod_type, Fintype.sum_bool, ← Finset.sum_add_distrib]
-  apply Finset.sum_le_sum
-  intro i _
-  simpa only [flow, Bool.false_eq_true, ↓reduceIte, add_comm] using
-    pairUpdate_energy_le (B (false, i)) (B (true, i)) (w (false, i)) (w (true, i)) (c i)
-
-theorem pairUpdate_first_noise (free₀ free₁ : Bool) (a b : ℝ) (coin : Bool) :
-    (pairUpdate free₀ free₁ a b coin).1 = (a + b) / 2 +
-      sign coin * (if free₀ && free₁ then 0 else (a - b) / 2) := by
-  cases free₀ <;> cases free₁ <;> cases coin <;> simp [pairUpdate, sign] <;> ring
-
-theorem pairUpdate_second_noise (free₀ free₁ : Bool) (a b : ℝ) (coin : Bool) :
-    (pairUpdate free₀ free₁ a b coin).2 = (a + b) / 2 -
-      sign coin * (if free₀ && free₁ then 0 else (a - b) / 2) := by
-  cases free₀ <;> cases free₁ <;> cases coin <;> simp [pairUpdate, sign] <;> ring
-
-end Conditional
-
-end Thorp
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.Model
+
+namespace OAI
+
+universe uΩ uα
+
+noncomputable section
+
+open scoped BigOperators
+open Filter
+
+namespace Thorp
+
+namespace Conditional
+
+def mean {Ω : Type uΩ} [Fintype Ω] (f : Ω → ℝ) : ℝ :=
+  (∑ ω, f ω) / Fintype.card Ω
+
+def sign (b : Bool) : ℝ := if b then -1 else 1
+
+@[simp] theorem sign_not (b : Bool) : sign (!b) = -sign b := by
+  cases b <;> norm_num [sign]
+
+@[simp] theorem sign_sq (b : Bool) : sign b ^ 2 = 1 := by
+  cases b <;> norm_num [sign]
+
+@[simp] theorem sign_mul_self (b : Bool) : sign b * sign b = 1 := by
+  simpa only [pow_two] using sign_sq b
+
+def toggleCoin {α : Type uα} [DecidableEq α] (i : α) : Equiv.Perm (α → Bool) :=
+  Function.Involutive.toPerm (fun c => Function.update c i (!(c i))) (by
+    intro c
+    funext j
+    by_cases h : j = i
+    · subst j
+      simp
+    · simp [h])
+
+@[simp] theorem toggleCoin_apply {α : Type uα} [DecidableEq α]
+    (i : α) (c : α → Bool) (j : α) :
+    toggleCoin i c j = Function.update c i (!(c i)) j := rfl
+
+theorem sum_sign {α : Type uα} [Fintype α] [DecidableEq α] (i : α) :
+    ∑ c : α → Bool, sign (c i) = 0 := by
+  have h := Equiv.sum_comp (toggleCoin i) (fun c : α → Bool => sign (c i))
+  simp only [toggleCoin_apply, Function.update_self, sign_not, Finset.sum_neg_distrib] at h
+  linarith
+
+theorem sum_sign_pair {α : Type uα} [Fintype α] [DecidableEq α] (i j : α) :
+    (∑ c : α → Bool, sign (c i) * sign (c j)) =
+      if i = j then (Fintype.card (α → Bool) : ℝ) else 0 := by
+  by_cases h : i = j
+  · subst j
+    simp
+  · have he := Equiv.sum_comp (toggleCoin i) (fun c : α → Bool => sign (c i) * sign (c j))
+    simp only [toggleCoin_apply, Function.update_self,
+      Function.update_of_ne (Ne.symm h), sign_not, neg_mul,
+      Finset.sum_neg_distrib] at he
+    rw [ite_eq_right h]
+    linarith
+
+theorem mean_signed_sum_sq {α : Type uα} [Fintype α] [DecidableEq α] (a : α → ℝ) :
+    mean (fun c : α → Bool => (∑ i, sign (c i) * a i) ^ 2) = ∑ i, a i ^ 2 := by
+  have hs :
+      (∑ c : α → Bool, (∑ i, sign (c i) * a i) ^ 2) =
+        (Fintype.card (α → Bool) : ℝ) * ∑ i, a i ^ 2 := by
+    calc
+      _ = ∑ c : α → Bool, ∑ i, ∑ j,
+          (sign (c i) * sign (c j)) * (a i * a j) := by
+        apply Finset.sum_congr rfl
+        intro c _
+        simp only [pow_two, Finset.sum_mul, Finset.mul_sum]
+        apply Finset.sum_congr rfl
+        intro i _
+        apply Finset.sum_congr rfl
+        intro j _
+        ring
+      _ = ∑ i, ∑ j, (∑ c : α → Bool, sign (c i) * sign (c j)) * (a i * a j) := by
+        rw [Finset.sum_comm]
+        apply Finset.sum_congr rfl
+        intro i _
+        rw [Finset.sum_comm]
+        simp only [Finset.sum_mul]
+      _ = _ := by
+        simp only [sum_sign_pair, ite_mul, zero_mul]
+        simp [Finset.mul_sum, pow_two]
+  unfold mean
+  rw [hs]
+  have hcard : (Fintype.card (α → Bool) : ℝ) ≠ 0 := by
+    exact_mod_cast Fintype.card_ne_zero
+  exact mul_div_cancel_left₀ _ hcard
+
+def pairUpdate (free₀ free₁ : Bool) (a b : ℝ) (coin : Bool) : ℝ × ℝ :=
+  if free₀ && free₁ then ((a + b) / 2, (a + b) / 2)
+  else if coin then (b, a) else (a, b)
+
+@[simp] theorem pairUpdate_sum (free₀ free₁ : Bool) (a b : ℝ) (coin : Bool) :
+    (pairUpdate free₀ free₁ a b coin).1 + (pairUpdate free₀ free₁ a b coin).2 = a + b := by
+  cases free₀ <;> cases free₁ <;> cases coin <;> simp [pairUpdate] <;> ring
+
+theorem pairUpdate_difference (free₀ free₁ : Bool) (a b : ℝ) (coin : Bool) :
+    (pairUpdate free₀ free₁ a b coin).1 - (pairUpdate free₀ free₁ a b coin).2 =
+      sign coin * (if free₀ && free₁ then 0 else a - b) := by
+  cases free₀ <;> cases free₁ <;> cases coin <;> simp [pairUpdate, sign]
+
+theorem pairUpdate_mean_first (free₀ free₁ : Bool) (a b : ℝ) :
+    mean (fun coin : Bool => (pairUpdate free₀ free₁ a b coin).1) = (a + b) / 2 := by
+  cases free₀ <;> cases free₁ <;> simp [mean, pairUpdate] <;> ring
+
+theorem pairUpdate_difference_sq (free₀ free₁ : Bool) (a b : ℝ)
+    (ha : free₀ = false → a = 0) (hb : free₁ = false → b = 0) :
+    (if free₀ && free₁ then (0 : ℝ) else a - b) ^ 2 =
+      (if free₁ then 0 else a ^ 2) + (if free₀ then 0 else b ^ 2) := by
+  cases free₀ <;> cases free₁ <;> simp_all
+
+def flow {α : Type uα} (B : Bool × α → Bool) (w : Bool × α → ℝ)
+    (c : α → Bool) (x : Bool × α) : ℝ :=
+  let p := pairUpdate (B (false, x.2)) (B (true, x.2))
+    (w (false, x.2)) (w (true, x.2)) (c x.2)
+  if x.1 then p.2 else p.1
+
+def oddCoefficient {α : Type uα} [Fintype α] (ε : α → ℝ) (w : Bool × α → ℝ) : ℝ :=
+  ∑ i, ε i * (w (false, i) - w (true, i))
+
+theorem mean_oddCoefficient_sq {α : Type uα} [Fintype α] [DecidableEq α]
+    (B : Bool × α → Bool) (w : Bool × α → ℝ) (ε : α → ℝ)
+    (hw : ∀ x, B x = false → w x = 0) (hε : ∀ i, ε i ^ 2 = 1) :
+    mean (fun c : α → Bool => oddCoefficient ε (flow B w c) ^ 2) =
+      ∑ i, ((if B (true, i) then 0 else w (false, i) ^ 2) +
+        (if B (false, i) then 0 else w (true, i) ^ 2)) := by
+  have he (c : α → Bool) :
+      oddCoefficient ε (flow B w c) =
+        ∑ i, sign (c i) * (ε i *
+          (if B (false, i) && B (true, i) then 0 else w (false, i) - w (true, i))) := by
+    unfold oddCoefficient
+    apply Finset.sum_congr rfl
+    intro i _
+    simp only [flow, Bool.false_eq_true, ↓reduceIte]
+    rw [pairUpdate_difference]
+    ring
+  simp_rw [he]
+  rw [mean_signed_sum_sq]
+  apply Finset.sum_congr rfl
+  intro i _
+  rw [mul_pow, hε, one_mul]
+  exact pairUpdate_difference_sq _ _ _ _ (hw _) (hw _)
+
+end Conditional
+
+namespace Conditional
+
+def pairMap {α : Type uα} (c : α → Bool) (x : Bool × α) : Bool × α :=
+  (x.1 ^^ c x.2, x.2)
+
+theorem pairMap_involutive {α : Type uα} (c : α → Bool) :
+    Function.Involutive (pairMap c) := by
+  rintro ⟨b, i⟩
+  cases b <;> cases h : c i <;> simp [pairMap, h]
+
+theorem flow_sub {α : Type uα} (B : Bool × α → Bool) (u v : Bool × α → ℝ)
+    (c : α → Bool) (x : Bool × α) :
+    flow B (fun y => u y - v y) c x = flow B u c x - flow B v c x := by
+  rcases x with ⟨b, i⟩
+  cases b <;> cases h₀ : B (false, i) <;> cases h₁ : B (true, i) <;>
+    cases hc : c i <;> simp [flow, pairUpdate, h₀, h₁, hc] <;> ring
+
+theorem flow_uniform {α : Type uα} (B : Bool × α → Bool) (r : ℝ)
+    (c : α → Bool) (x : Bool × α) :
+    flow B (fun y => if B y then r else 0) c x =
+      if B (pairMap c x) then r else 0 := by
+  rcases x with ⟨b, i⟩
+  cases b <;> cases h₀ : B (false, i) <;> cases h₁ : B (true, i) <;>
+    cases hc : c i <;> simp [flow, pairUpdate, pairMap, h₀, h₁, hc]
+
+theorem flow_support {α : Type uα} (B : Bool × α → Bool) (w : Bool × α → ℝ)
+    (hw : ∀ x, B x = false → w x = 0) (c : α → Bool) (x : Bool × α)
+    (hx : B (pairMap c x) = false) : flow B w c x = 0 := by
+  rcases x with ⟨b, i⟩
+  have h₀ := hw (false, i)
+  have h₁ := hw (true, i)
+  cases b <;> cases hb₀ : B (false, i) <;> cases hb₁ : B (true, i) <;>
+    cases hc : c i <;> simp_all [flow, pairUpdate, pairMap]
+
+theorem flow_evenCoefficient {α : Type uα} [Fintype α]
+    (B : Bool × α → Bool) (w : Bool × α → ℝ) (ε : α → ℝ) (c : α → Bool) :
+    (∑ i, ε i * (flow B w c (false, i) + flow B w c (true, i))) =
+      ∑ i, ε i * (w (false, i) + w (true, i)) := by
+  apply Finset.sum_congr rfl
+  intro i _
+  simp only [flow, Bool.false_eq_true, ↓reduceIte, pairUpdate_sum]
+
+theorem flow_sum {α : Type uα} [Fintype α] (B : Bool × α → Bool)
+    (w : Bool × α → ℝ) (c : α → Bool) :
+    (∑ x, flow B w c x) = ∑ x, w x := by
+  simp only [Fintype.sum_prod_type, Fintype.sum_bool, ← Finset.sum_add_distrib]
+  apply Finset.sum_congr rfl
+  intro i _
+  simpa only [flow, Bool.false_eq_true, ↓reduceIte, add_comm] using
+    (pairUpdate_sum (B (false, i)) (B (true, i)) (w (false, i)) (w (true, i)) (c i))
+
+theorem pairUpdate_energy_le (free₀ free₁ : Bool) (a b : ℝ) (coin : Bool) :
+    (pairUpdate free₀ free₁ a b coin).1 ^ 2 +
+        (pairUpdate free₀ free₁ a b coin).2 ^ 2 ≤ a ^ 2 + b ^ 2 := by
+  have h := sq_nonneg (a - b)
+  cases free₀ <;> cases free₁ <;> cases coin <;> simp [pairUpdate] <;> nlinarith
+
+theorem flow_energy_le {α : Type uα} [Fintype α] (B : Bool × α → Bool)
+    (w : Bool × α → ℝ) (c : α → Bool) :
+    (∑ x, flow B w c x ^ 2) ≤ ∑ x, w x ^ 2 := by
+  simp only [Fintype.sum_prod_type, Fintype.sum_bool, ← Finset.sum_add_distrib]
+  apply Finset.sum_le_sum
+  intro i _
+  simpa only [flow, Bool.false_eq_true, ↓reduceIte, add_comm] using
+    pairUpdate_energy_le (B (false, i)) (B (true, i)) (w (false, i)) (w (true, i)) (c i)
+
+theorem pairUpdate_first_noise (free₀ free₁ : Bool) (a b : ℝ) (coin : Bool) :
+    (pairUpdate free₀ free₁ a b coin).1 = (a + b) / 2 +
+      sign coin * (if free₀ && free₁ then 0 else (a - b) / 2) := by
+  cases free₀ <;> cases free₁ <;> cases coin <;> simp [pairUpdate, sign] <;> ring
+
+theorem pairUpdate_second_noise (free₀ free₁ : Bool) (a b : ℝ) (coin : Bool) :
+    (pairUpdate free₀ free₁ a b coin).2 = (a + b) / 2 -
+      sign coin * (if free₀ && free₁ then 0 else (a - b) / 2) := by
+  cases free₀ <;> cases free₁ <;> cases coin <;> simp [pairUpdate, sign] <;> ring
+
+end Conditional
+
+end Thorp
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/Parity.lean b/lean/OAI/Probability/ThorpShuffle/Parity.lean
index 9e7f486ef25b6de7cd920bbfcd664d6f45d82093..6bccef7bd6483b2c152867ec3e1c6551051e8f0e 100644
--- a/lean/OAI/Probability/ThorpShuffle/Parity.lean
+++ b/lean/OAI/Probability/ThorpShuffle/Parity.lean
@@ -1,101 +1,101 @@
-import OAI.Probability.ThorpShuffle.MixingBounds
-
-namespace OAI
-
-noncomputable section
-
-open scoped BigOperators
-open Filter
-
-namespace Thorp
-
-namespace Parity
-open Conditional
-
-def sign (d : ℕ) (g : State d) : ℝ := ((Equiv.Perm.sign g : ℤˣ) : ℤ)
-
-theorem sign_mul (d : ℕ) (g h : State d) : sign d (g * h) = sign d g * sign d h := by
-  simp [sign]
-
-theorem step_toggle (d : ℕ) (s : Position d) (c : Coins (d + 1)) :
-    step (d + 1) (toggleCoin s c) =
-      step (d + 1) c * Equiv.swap (Fin.cons false s) (Fin.cons true s) := by
-  classical
-  change Position d → Bool at c
-  apply Equiv.Perm.ext
-  intro x
-  obtain ⟨⟨b, t⟩, rfl⟩ := (splitPosition d).symm.surjective x
-  change (rotate (d + 1)) (Fin.cons (b ^^ toggleCoin s c t) t) =
-    step (d + 1) c (Equiv.swap (Fin.cons false s) (Fin.cons true s) (Fin.cons b t))
-  by_cases ht : t = s
-  · subst t
-    cases b <;> simp only [Equiv.swap_apply_left, Equiv.swap_apply_right]
-    · change rotate (d + 1) (Fin.cons (false ^^ toggleCoin s c s) s) =
-        rotate (d + 1) (Fin.cons (true ^^ c s) s)
-      have hh : toggleCoin s c s = !(c s) := by
-        change Function.update c s (!(c s)) s = !(c s)
-        simp
-      rw [hh]
-      simp
-    · change rotate (d + 1) (Fin.cons (true ^^ toggleCoin s c s) s) =
-        rotate (d + 1) (Fin.cons (false ^^ c s) s)
-      have hh : toggleCoin s c s = !(c s) := by
-        change Function.update c s (!(c s)) s = !(c s)
-        simp
-      rw [hh]
-      simp
-  · have hf : (Fin.cons b t : Position (d + 1)) ≠ Fin.cons false s := by
-      intro he; apply ht; simpa using congrArg Fin.tail he
-    have htr : (Fin.cons b t : Position (d + 1)) ≠ Fin.cons true s := by
-      intro he; apply ht; simpa using congrArg Fin.tail he
-    rw [Equiv.swap_apply_of_ne_of_ne hf htr]
-    change rotate (d + 1) (Fin.cons (b ^^ toggleCoin s c t) t) =
-      rotate (d + 1) (Fin.cons (b ^^ c t) t)
-    have hh : toggleCoin s c t = c t := by
-      change Function.update c s (!(c s)) t = c t
-      exact Function.update_of_ne ht _ _
-    rw [hh]
-
-theorem sign_step_toggle (d : ℕ) (s : Position d) (c : Coins (d + 1)) :
-    sign (d + 1) (step (d + 1) (toggleCoin s c)) =
-      -sign (d + 1) (step (d + 1) c) := by
-  classical
-  rw [step_toggle, sign_mul]
-  have hne : (Fin.cons false s : Position (d + 1)) ≠ Fin.cons true s := by
-    intro he
-    have hh := congrFun he 0
-    simp at hh
-  simp [sign, Equiv.Perm.sign_swap hne]
-
-theorem mean_sign_step (d : ℕ) :
-    mean (fun c : Coins (d + 1) => sign (d + 1) (step (d + 1) c)) = 0 := by
-  classical
-  let s : Position d := fun _ => false
-  have hh := mean_equiv (toggleCoin s)
-    (fun c : Coins (d + 1) => sign (d + 1) (step (d + 1) c))
-  change mean (fun c : Coins (d + 1) => sign (d + 1) (step (d + 1) (toggleCoin s c))) =
-    mean (fun c : Coins (d + 1) => sign (d + 1) (step (d + 1) c)) at hh
-  simp_rw [sign_step_toggle] at hh
-  have hn : mean (fun c : Coins (d + 1) => -sign (d + 1) (step (d + 1) c)) =
-      -mean (fun c : Coins (d + 1) => sign (d + 1) (step (d + 1) c)) := by
-    simp only [mean, Finset.sum_neg_distrib, neg_div]
-  rw [hn] at hh
-  linarith
-
-theorem mean_sign_run (d t : ℕ) :
-    mean (fun ω : History (d + 1) (t + 1) => sign (d + 1) (run (d + 1) (t + 1) ω)) = 0 := by
-  rw [mean_history_succ]
-  simp only [run_snoc, sign_mul, mean_mul_const, mean_sign_step, zero_mul, mean_zero]
-
-theorem law_sign_zero (d t : ℕ) :
-    (∑ g : State (d + 1), law (d + 1) (t + 1) g * sign (d + 1) g) = 0 := by
-  rw [law, fairMass_test]
-  exact mean_sign_run d t
-
-end Parity
-
-end Thorp
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.MixingBounds
+
+namespace OAI
+
+noncomputable section
+
+open scoped BigOperators
+open Filter
+
+namespace Thorp
+
+namespace Parity
+open Conditional
+
+def sign (d : ℕ) (g : State d) : ℝ := ((Equiv.Perm.sign g : ℤˣ) : ℤ)
+
+theorem sign_mul (d : ℕ) (g h : State d) : sign d (g * h) = sign d g * sign d h := by
+  simp [sign]
+
+theorem step_toggle (d : ℕ) (s : Position d) (c : Coins (d + 1)) :
+    step (d + 1) (toggleCoin s c) =
+      step (d + 1) c * Equiv.swap (Fin.cons false s) (Fin.cons true s) := by
+  classical
+  change Position d → Bool at c
+  apply Equiv.Perm.ext
+  intro x
+  obtain ⟨⟨b, t⟩, rfl⟩ := (splitPosition d).symm.surjective x
+  change (rotate (d + 1)) (Fin.cons (b ^^ toggleCoin s c t) t) =
+    step (d + 1) c (Equiv.swap (Fin.cons false s) (Fin.cons true s) (Fin.cons b t))
+  by_cases ht : t = s
+  · subst t
+    cases b <;> simp only [Equiv.swap_apply_left, Equiv.swap_apply_right]
+    · change rotate (d + 1) (Fin.cons (false ^^ toggleCoin s c s) s) =
+        rotate (d + 1) (Fin.cons (true ^^ c s) s)
+      have hh : toggleCoin s c s = !(c s) := by
+        change Function.update c s (!(c s)) s = !(c s)
+        simp
+      rw [hh]
+      simp
+    · change rotate (d + 1) (Fin.cons (true ^^ toggleCoin s c s) s) =
+        rotate (d + 1) (Fin.cons (false ^^ c s) s)
+      have hh : toggleCoin s c s = !(c s) := by
+        change Function.update c s (!(c s)) s = !(c s)
+        simp
+      rw [hh]
+      simp
+  · have hf : (Fin.cons b t : Position (d + 1)) ≠ Fin.cons false s := by
+      intro he; apply ht; simpa using congrArg Fin.tail he
+    have htr : (Fin.cons b t : Position (d + 1)) ≠ Fin.cons true s := by
+      intro he; apply ht; simpa using congrArg Fin.tail he
+    rw [Equiv.swap_apply_of_ne_of_ne hf htr]
+    change rotate (d + 1) (Fin.cons (b ^^ toggleCoin s c t) t) =
+      rotate (d + 1) (Fin.cons (b ^^ c t) t)
+    have hh : toggleCoin s c t = c t := by
+      change Function.update c s (!(c s)) t = c t
+      exact Function.update_of_ne ht _ _
+    rw [hh]
+
+theorem sign_step_toggle (d : ℕ) (s : Position d) (c : Coins (d + 1)) :
+    sign (d + 1) (step (d + 1) (toggleCoin s c)) =
+      -sign (d + 1) (step (d + 1) c) := by
+  classical
+  rw [step_toggle, sign_mul]
+  have hne : (Fin.cons false s : Position (d + 1)) ≠ Fin.cons true s := by
+    intro he
+    have hh := congrFun he 0
+    simp at hh
+  simp [sign, Equiv.Perm.sign_swap hne]
+
+theorem mean_sign_step (d : ℕ) :
+    mean (fun c : Coins (d + 1) => sign (d + 1) (step (d + 1) c)) = 0 := by
+  classical
+  let s : Position d := fun _ => false
+  have hh := mean_equiv (toggleCoin s)
+    (fun c : Coins (d + 1) => sign (d + 1) (step (d + 1) c))
+  change mean (fun c : Coins (d + 1) => sign (d + 1) (step (d + 1) (toggleCoin s c))) =
+    mean (fun c : Coins (d + 1) => sign (d + 1) (step (d + 1) c)) at hh
+  simp_rw [sign_step_toggle] at hh
+  have hn : mean (fun c : Coins (d + 1) => -sign (d + 1) (step (d + 1) c)) =
+      -mean (fun c : Coins (d + 1) => sign (d + 1) (step (d + 1) c)) := by
+    simp only [mean, Finset.sum_neg_distrib, neg_div]
+  rw [hn] at hh
+  linarith
+
+theorem mean_sign_run (d t : ℕ) :
+    mean (fun ω : History (d + 1) (t + 1) => sign (d + 1) (run (d + 1) (t + 1) ω)) = 0 := by
+  rw [mean_history_succ]
+  simp only [run_snoc, sign_mul, mean_mul_const, mean_sign_step, zero_mul, mean_zero]
+
+theorem law_sign_zero (d t : ℕ) :
+    (∑ g : State (d + 1), law (d + 1) (t + 1) g * sign (d + 1) g) = 0 := by
+  rw [law, fairMass_test]
+  exact mean_sign_run d t
+
+end Parity
+
+end Thorp
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/PartitionBounds.lean b/lean/OAI/Probability/ThorpShuffle/PartitionBounds.lean
index 55de68fede73d0c73b049eae6951fa8300bf0a6a..5c0a3d73bf52a2a38512a61df45ca2eeee1d8354 100644
--- a/lean/OAI/Probability/ThorpShuffle/PartitionBounds.lean
+++ b/lean/OAI/Probability/ThorpShuffle/PartitionBounds.lean
@@ -1,221 +1,221 @@
-import OAI.Probability.ThorpShuffle.ExtensionBounds
-
-namespace OAI
-
-noncomputable section
-
-open scoped BigOperators ComplexConjugate InnerProductSpace
-open Filter
-
-namespace Thorp.Young
-open scoped Classical
-
-def idealDiagram (D : YoungDiagram) (s : Set (Cells D)) (hs : IsLowerSet s) : YoungDiagram where
-  cells := (Finset.univ.filter (· ∈ s)).image Subtype.val
-  isLowerSet := by
-    intro y x hxy hy
-    obtain ⟨z, hz, hzy⟩ := Finset.mem_image.mp hy
-    have hyD : y ∈ D := by rw [← hzy]; exact z.property
-    have hxD : x ∈ D := D.isLowerSet hxy hyD
-    refine Finset.mem_image.mpr ⟨⟨x, hxD⟩, Finset.mem_filter.mpr ⟨Finset.mem_univ _, ?_⟩, rfl⟩
-    apply hs (show (⟨x, hxD⟩ : Cells D) ≤ z from by change x ≤ z.val; rwa [hzy])
-    exact (Finset.mem_filter.mp hz).2
-
-lemma mem_idealDiagram (D : YoungDiagram) (s : Set (Cells D)) (hs : IsLowerSet s) (p : ℕ × ℕ) :
-    p ∈ idealDiagram D s hs ↔ ∃ h : p ∈ D, (⟨p, h⟩ : Cells D) ∈ s := by
-  constructor
-  · intro hp
-    obtain ⟨x, hx, hxp⟩ := Finset.mem_image.mp hp
-    subst p
-    exact ⟨x.property, (Finset.mem_filter.mp hx).2⟩
-  · rintro ⟨h, hp⟩
-    exact Finset.mem_image.mpr ⟨⟨p, h⟩, Finset.mem_filter.mpr ⟨Finset.mem_univ _, hp⟩, rfl⟩
-
-def idealCells (D : YoungDiagram) (s : Set (Cells D)) (hs : IsLowerSet s) :
-    Cells (idealDiagram D s hs) ≃o s where
-  toFun x := ⟨⟨x.val, (mem_idealDiagram D s hs x.val).mp x.property |>.choose⟩,
-    (mem_idealDiagram D s hs x.val).mp x.property |>.choose_spec⟩
-  invFun x := ⟨x.val.val, (mem_idealDiagram D s hs x.val.val).mpr ⟨x.val.property, x.property⟩⟩
-  left_inv _ := rfl
-  right_inv _ := rfl
-  map_rel_iff' := Iff.rfl
-
-lemma idealDiagram_extensions_le (D : YoungDiagram) (s : Set (Cells D)) (hs : IsLowerSet s) :
-    Fintype.card (Extensions (Cells (idealDiagram D s hs))) ≤ Fintype.card (Extensions (Cells D)) := by
-  rw [extensions_card_orderIso (idealCells D s hs)]
-  exact extensions_card_ideal s hs
-
-theorem exists_trimLower (D : YoungDiagram) (k : ℕ) (hk : k ≤ Fintype.card (Lower D)) :
-    ∃ E : YoungDiagram, E.rowLen 0 = D.rowLen 0 ∧ Fintype.card (Lower E) = k ∧
-      Fintype.card (Extensions (Cells E)) ≤ Fintype.card (Extensions (Cells D)) := by
-  obtain ⟨e⟩ := extensions_nonempty (Lower D)
-  let s : Set (Cells D) := {x | ∀ h : x.val.1 ≠ 0, (e.val ⟨x, h⟩).val < k}
-  have hs : IsLowerSet s := by
-    intro y x hxy hy hx
-    have hy' : y.val.1 ≠ 0 := by have := hxy.1; omega
-    exact lt_of_le_of_lt (e.property.monotone (show (⟨x, hx⟩ : Lower D) ≤ ⟨y, hy'⟩ from hxy)) (hy hy')
-  let E := idealDiagram D s hs
-  have htop : E.rowLen 0 = D.rowLen 0 := by
-    apply eq_of_forall_lt_iff
-    intro j
-    rw [← YoungDiagram.mem_iff_lt_rowLen, ← YoungDiagram.mem_iff_lt_rowLen]
-    change (0, j) ∈ idealDiagram D s hs ↔ (0, j) ∈ D
-    rw [mem_idealDiagram]
-    exact ⟨fun h => h.choose, fun h => ⟨h, fun hn => (hn rfl).elim⟩⟩
-  let f : Lower E ≃ Fin k := {
-    toFun := fun x =>
-      let z := idealCells D s hs x.val
-      ⟨(e.val ⟨z.val, x.property⟩).val, z.property x.property⟩
-    invFun := fun j =>
-      let z := e.val.symm ⟨j.val, j.isLt.trans_le hk⟩
-      ⟨(idealCells D s hs).symm ⟨z.val, by
-        intro h
-        have hh : (⟨z.val, h⟩ : Lower D) = z := rfl
-        rw [hh, Equiv.apply_symm_apply]
-        exact j.isLt⟩, z.property⟩
-    left_inv := by
-      intro x
-      apply Subtype.ext
-      apply (idealCells D s hs).injective
-      apply Subtype.ext
-      have hh := e.val.symm_apply_apply ⟨(idealCells D s hs x.val).val, x.property⟩
-      simpa only [OrderIso.apply_symm_apply] using congrArg Subtype.val hh
-    right_inv := by
-      intro j
-      apply Fin.ext
-      simpa only [OrderIso.apply_symm_apply] using congrArg Fin.val (e.val.apply_symm_apply ⟨j.val, j.isLt.trans_le hk⟩) }
-  exact ⟨E, htop, (Fintype.card_congr f).trans (Fintype.card_fin k), idealDiagram_extensions_le D s hs⟩
-
-theorem choose_le_extensions (D : YoungDiagram) (k : ℕ)
-    (hk : k ≤ Fintype.card (Lower D)) (hL : k ≤ D.rowLen 0) :
-    (D.rowLen 0).choose k ≤ Fintype.card (Extensions (Cells D)) := by
-  obtain ⟨E, htop, hsize, hle⟩ := exists_trimLower D k hk
-  have hh := choose_row_le_extensions E (by rw [hsize, htop]; exact hL)
-  rw [htop, hsize] at hh
-  exact hh.trans hle
-
-end Thorp.Young
-
-namespace Thorp.Young
-open scoped Classical
-
-lemma partition_count_le_pow (n : ℕ) : Fintype.card (Nat.Partition n) ≤ 2 ^ n := by
-  have h := Fintype.card_le_of_surjective (Nat.Partition.ofComposition n) Nat.Partition.ofComposition_surj
-  rw [composition_card] at h
-  exact h.trans (Nat.pow_le_pow_right (by omega) (Nat.sub_le _ _))
-
-lemma rectangle_card_le (D : YoungDiagram) (x : Cells D) :
-    (x.val.1 + 1) * (x.val.2 + 1) ≤ Fintype.card (Cells D) := by
-  let f : Fin (x.val.1 + 1) × Fin (x.val.2 + 1) → Cells D := fun p =>
-    ⟨(p.1, p.2), D.up_left_mem (i2 := x.val.1) (j2 := x.val.2)
-      (by have := p.1.isLt; omega) (by have := p.2.isLt; omega) x.property⟩
-  have hf : Function.Injective f := by
-    intro p q hpq
-    apply Prod.ext <;> apply Fin.ext
-    · exact congrArg (fun z : Cells D => z.val.1) hpq
-    · exact congrArg (fun z : Cells D => z.val.2) hpq
-  simpa only [Fintype.card_prod, Fintype.card_fin] using Fintype.card_le_of_injective f hf
-
-lemma cell_small_coord (D : YoungDiagram) (x : Cells D) :
-    x.val.1 < Nat.sqrt (Fintype.card (Cells D)) ∨ x.val.2 < Nat.sqrt (Fintype.card (Cells D)) := by
-  by_contra h
-  push Not at h
-  have hh := rectangle_card_le D x
-  have hs := Nat.lt_succ_sqrt' (Fintype.card (Cells D))
-  have hm := Nat.mul_le_mul (Nat.succ_le_succ h.1) (Nat.succ_le_succ h.2)
-  nlinarith
-
-lemma rowLen_le_card (D : YoungDiagram) (i : ℕ) : D.rowLen i ≤ Fintype.card (Cells D) := by
-  rw [← card_row_subtype D i]
-  exact Fintype.card_le_of_injective Subtype.val Subtype.val_injective
-
-lemma colLen_le_card (D : YoungDiagram) (j : ℕ) : D.colLen j ≤ Fintype.card (Cells D) := by
-  rw [← card_column D j]
-  exact (Finset.card_le_card (Finset.filter_subset _ _)).trans_eq (Finset.card_univ)
-
-theorem partition_count_le_sqrt_pow (n : ℕ) :
-    Fintype.card (Nat.Partition n) ≤ (n + 1) ^ (2 * Nat.sqrt n) := by
-  let F (p : Nat.Partition n) : (Fin (Nat.sqrt n) → Fin (n + 1)) × (Fin (Nat.sqrt n) → Fin (n + 1)) :=
-    (fun i => ⟨(ofPartition p).rowLen i, by
-      have := rowLen_le_card (ofPartition p) i
-      rw [card_ofPartition] at this
-      omega⟩,
-     fun j => ⟨(ofPartition p).colLen j, by
-      have := colLen_le_card (ofPartition p) j
-      rw [card_ofPartition] at this
-      omega⟩)
-  have hf : Function.Injective F := by
-    intro p q hpq
-    apply ofPartition_injective
-    apply YoungDiagram.ext
-    ext z
-    have aux (p q : Nat.Partition n) (hpq : F p = F q) (hz : z ∈ ofPartition p) : z ∈ ofPartition q := by
-      have hs := cell_small_coord (ofPartition p) ⟨z, hz⟩
-      rw [card_ofPartition] at hs
-      rcases hs with hs | hs
-      · have he := congrArg (fun v : (Fin (Nat.sqrt n) → Fin (n + 1)) × (Fin (Nat.sqrt n) → Fin (n + 1)) =>
-            (v.1 ⟨z.1, hs⟩).val) hpq
-        change (ofPartition p).rowLen z.1 = (ofPartition q).rowLen z.1 at he
-        apply YoungDiagram.mem_iff_lt_rowLen.mpr
-        rw [← he]
-        exact YoungDiagram.mem_iff_lt_rowLen.mp hz
-      · have he := congrArg (fun v : (Fin (Nat.sqrt n) → Fin (n + 1)) × (Fin (Nat.sqrt n) → Fin (n + 1)) =>
-            (v.2 ⟨z.2, hs⟩).val) hpq
-        change (ofPartition p).colLen z.2 = (ofPartition q).colLen z.2 at he
-        apply YoungDiagram.mem_iff_lt_colLen.mpr
-        rw [← he]
-        exact YoungDiagram.mem_iff_lt_colLen.mp hz
-    exact ⟨aux p q hpq, aux q p hpq.symm⟩
-  have hh := Fintype.card_le_of_injective F hf
-  simpa only [Fintype.card_prod, Fintype.card_fun, Fintype.card_fin, two_mul, pow_add] using hh
-
-end Thorp.Young
-
-namespace Thorp.Young
-open scoped Classical
-
-lemma pow_le_choose_mul (a k : ℕ) : a ^ k ≤ (k * a).choose k := by
-  let X := Fin k × Fin a
-  let S := {s : Finset X // s.card = k}
-  let f (v : Fin k → Fin a) : S := ⟨Finset.univ.image (fun i => (i, v i)), by
-    rw [Finset.card_image_of_injective _ (by intro i j h; exact congrArg Prod.fst h), Finset.card_univ, Fintype.card_fin]⟩
-  have hf : Function.Injective f := by
-    intro v w hvw
-    funext i
-    have he := congrArg Subtype.val hvw
-    have hm : (i, v i) ∈ (f w).val := by rw [← he]; exact Finset.mem_image.mpr ⟨i, Finset.mem_univ _, rfl⟩
-    obtain ⟨j, _, hj⟩ := Finset.mem_image.mp hm
-    have hji : j = i := congrArg Prod.fst hj
-    subst j
-    exact (congrArg Prod.snd hj).symm
-  have hS : Fintype.card S = (k * a).choose k := by
-    calc
-      _ = ((Finset.univ : Finset X).powersetCard k).card := by
-        rw [Fintype.card_subtype]
-        congr 1
-        ext s
-        simp
-      _ = _ := by simp [X]
-  have hh := Fintype.card_le_of_injective f hf
-  simpa only [Fintype.card_fun, Fintype.card_fin, hS] using hh
-
-lemma pow_le_choose_of_mul_le (a k L : ℕ) (h : a * k ≤ L) : a ^ k ≤ L.choose k := by
-  exact (pow_le_choose_mul a k).trans (Nat.choose_le_choose k (by simpa only [Nat.mul_comm] using h))
-
-lemma sub_add_one_le_choose (n k : ℕ) (hk : 1 ≤ k) (hkn : k ≤ n) : n - k + 1 ≤ n.choose k := by
-  obtain ⟨r, rfl⟩ := Nat.exists_eq_add_of_le hkn
-  simp only [Nat.add_sub_cancel_left]
-  induction r with
-  | zero => simp
-  | succ r ih =>
-      obtain ⟨j, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : k ≠ 0)
-      rw [show j + 1 + (r + 1) = (j + 1 + r) + 1 by omega, Nat.choose_succ_succ]
-      have hp : 1 ≤ (j + 1 + r).choose j := Nat.one_le_iff_ne_zero.mpr (Nat.choose_ne_zero (by omega))
-      simp only [Nat.succ_eq_add_one] at ih ⊢
-      omega
-
-end Thorp.Young
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.ExtensionBounds
+
+namespace OAI
+
+noncomputable section
+
+open scoped BigOperators ComplexConjugate InnerProductSpace
+open Filter
+
+namespace Thorp.Young
+open scoped Classical
+
+def idealDiagram (D : YoungDiagram) (s : Set (Cells D)) (hs : IsLowerSet s) : YoungDiagram where
+  cells := (Finset.univ.filter (· ∈ s)).image Subtype.val
+  isLowerSet := by
+    intro y x hxy hy
+    obtain ⟨z, hz, hzy⟩ := Finset.mem_image.mp hy
+    have hyD : y ∈ D := by rw [← hzy]; exact z.property
+    have hxD : x ∈ D := D.isLowerSet hxy hyD
+    refine Finset.mem_image.mpr ⟨⟨x, hxD⟩, Finset.mem_filter.mpr ⟨Finset.mem_univ _, ?_⟩, rfl⟩
+    apply hs (show (⟨x, hxD⟩ : Cells D) ≤ z from by change x ≤ z.val; rwa [hzy])
+    exact (Finset.mem_filter.mp hz).2
+
+lemma mem_idealDiagram (D : YoungDiagram) (s : Set (Cells D)) (hs : IsLowerSet s) (p : ℕ × ℕ) :
+    p ∈ idealDiagram D s hs ↔ ∃ h : p ∈ D, (⟨p, h⟩ : Cells D) ∈ s := by
+  constructor
+  · intro hp
+    obtain ⟨x, hx, hxp⟩ := Finset.mem_image.mp hp
+    subst p
+    exact ⟨x.property, (Finset.mem_filter.mp hx).2⟩
+  · rintro ⟨h, hp⟩
+    exact Finset.mem_image.mpr ⟨⟨p, h⟩, Finset.mem_filter.mpr ⟨Finset.mem_univ _, hp⟩, rfl⟩
+
+def idealCells (D : YoungDiagram) (s : Set (Cells D)) (hs : IsLowerSet s) :
+    Cells (idealDiagram D s hs) ≃o s where
+  toFun x := ⟨⟨x.val, (mem_idealDiagram D s hs x.val).mp x.property |>.choose⟩,
+    (mem_idealDiagram D s hs x.val).mp x.property |>.choose_spec⟩
+  invFun x := ⟨x.val.val, (mem_idealDiagram D s hs x.val.val).mpr ⟨x.val.property, x.property⟩⟩
+  left_inv _ := rfl
+  right_inv _ := rfl
+  map_rel_iff' := Iff.rfl
+
+lemma idealDiagram_extensions_le (D : YoungDiagram) (s : Set (Cells D)) (hs : IsLowerSet s) :
+    Fintype.card (Extensions (Cells (idealDiagram D s hs))) ≤ Fintype.card (Extensions (Cells D)) := by
+  rw [extensions_card_orderIso (idealCells D s hs)]
+  exact extensions_card_ideal s hs
+
+theorem exists_trimLower (D : YoungDiagram) (k : ℕ) (hk : k ≤ Fintype.card (Lower D)) :
+    ∃ E : YoungDiagram, E.rowLen 0 = D.rowLen 0 ∧ Fintype.card (Lower E) = k ∧
+      Fintype.card (Extensions (Cells E)) ≤ Fintype.card (Extensions (Cells D)) := by
+  obtain ⟨e⟩ := extensions_nonempty (Lower D)
+  let s : Set (Cells D) := {x | ∀ h : x.val.1 ≠ 0, (e.val ⟨x, h⟩).val < k}
+  have hs : IsLowerSet s := by
+    intro y x hxy hy hx
+    have hy' : y.val.1 ≠ 0 := by have := hxy.1; omega
+    exact lt_of_le_of_lt (e.property.monotone (show (⟨x, hx⟩ : Lower D) ≤ ⟨y, hy'⟩ from hxy)) (hy hy')
+  let E := idealDiagram D s hs
+  have htop : E.rowLen 0 = D.rowLen 0 := by
+    apply eq_of_forall_lt_iff
+    intro j
+    rw [← YoungDiagram.mem_iff_lt_rowLen, ← YoungDiagram.mem_iff_lt_rowLen]
+    change (0, j) ∈ idealDiagram D s hs ↔ (0, j) ∈ D
+    rw [mem_idealDiagram]
+    exact ⟨fun h => h.choose, fun h => ⟨h, fun hn => (hn rfl).elim⟩⟩
+  let f : Lower E ≃ Fin k := {
+    toFun := fun x =>
+      let z := idealCells D s hs x.val
+      ⟨(e.val ⟨z.val, x.property⟩).val, z.property x.property⟩
+    invFun := fun j =>
+      let z := e.val.symm ⟨j.val, j.isLt.trans_le hk⟩
+      ⟨(idealCells D s hs).symm ⟨z.val, by
+        intro h
+        have hh : (⟨z.val, h⟩ : Lower D) = z := rfl
+        rw [hh, Equiv.apply_symm_apply]
+        exact j.isLt⟩, z.property⟩
+    left_inv := by
+      intro x
+      apply Subtype.ext
+      apply (idealCells D s hs).injective
+      apply Subtype.ext
+      have hh := e.val.symm_apply_apply ⟨(idealCells D s hs x.val).val, x.property⟩
+      simpa only [OrderIso.apply_symm_apply] using congrArg Subtype.val hh
+    right_inv := by
+      intro j
+      apply Fin.ext
+      simpa only [OrderIso.apply_symm_apply] using congrArg Fin.val (e.val.apply_symm_apply ⟨j.val, j.isLt.trans_le hk⟩) }
+  exact ⟨E, htop, (Fintype.card_congr f).trans (Fintype.card_fin k), idealDiagram_extensions_le D s hs⟩
+
+theorem choose_le_extensions (D : YoungDiagram) (k : ℕ)
+    (hk : k ≤ Fintype.card (Lower D)) (hL : k ≤ D.rowLen 0) :
+    (D.rowLen 0).choose k ≤ Fintype.card (Extensions (Cells D)) := by
+  obtain ⟨E, htop, hsize, hle⟩ := exists_trimLower D k hk
+  have hh := choose_row_le_extensions E (by rw [hsize, htop]; exact hL)
+  rw [htop, hsize] at hh
+  exact hh.trans hle
+
+end Thorp.Young
+
+namespace Thorp.Young
+open scoped Classical
+
+lemma partition_count_le_pow (n : ℕ) : Fintype.card (Nat.Partition n) ≤ 2 ^ n := by
+  have h := Fintype.card_le_of_surjective (Nat.Partition.ofComposition n) Nat.Partition.ofComposition_surj
+  rw [composition_card] at h
+  exact h.trans (Nat.pow_le_pow_right (by omega) (Nat.sub_le _ _))
+
+lemma rectangle_card_le (D : YoungDiagram) (x : Cells D) :
+    (x.val.1 + 1) * (x.val.2 + 1) ≤ Fintype.card (Cells D) := by
+  let f : Fin (x.val.1 + 1) × Fin (x.val.2 + 1) → Cells D := fun p =>
+    ⟨(p.1, p.2), D.up_left_mem (i2 := x.val.1) (j2 := x.val.2)
+      (by have := p.1.isLt; omega) (by have := p.2.isLt; omega) x.property⟩
+  have hf : Function.Injective f := by
+    intro p q hpq
+    apply Prod.ext <;> apply Fin.ext
+    · exact congrArg (fun z : Cells D => z.val.1) hpq
+    · exact congrArg (fun z : Cells D => z.val.2) hpq
+  simpa only [Fintype.card_prod, Fintype.card_fin] using Fintype.card_le_of_injective f hf
+
+lemma cell_small_coord (D : YoungDiagram) (x : Cells D) :
+    x.val.1 < Nat.sqrt (Fintype.card (Cells D)) ∨ x.val.2 < Nat.sqrt (Fintype.card (Cells D)) := by
+  by_contra h
+  push Not at h
+  have hh := rectangle_card_le D x
+  have hs := Nat.lt_succ_sqrt' (Fintype.card (Cells D))
+  have hm := Nat.mul_le_mul (Nat.succ_le_succ h.1) (Nat.succ_le_succ h.2)
+  nlinarith
+
+lemma rowLen_le_card (D : YoungDiagram) (i : ℕ) : D.rowLen i ≤ Fintype.card (Cells D) := by
+  rw [← card_row_subtype D i]
+  exact Fintype.card_le_of_injective Subtype.val Subtype.val_injective
+
+lemma colLen_le_card (D : YoungDiagram) (j : ℕ) : D.colLen j ≤ Fintype.card (Cells D) := by
+  rw [← card_column D j]
+  exact (Finset.card_le_card (Finset.filter_subset _ _)).trans_eq (Finset.card_univ)
+
+theorem partition_count_le_sqrt_pow (n : ℕ) :
+    Fintype.card (Nat.Partition n) ≤ (n + 1) ^ (2 * Nat.sqrt n) := by
+  let F (p : Nat.Partition n) : (Fin (Nat.sqrt n) → Fin (n + 1)) × (Fin (Nat.sqrt n) → Fin (n + 1)) :=
+    (fun i => ⟨(ofPartition p).rowLen i, by
+      have := rowLen_le_card (ofPartition p) i
+      rw [card_ofPartition] at this
+      omega⟩,
+     fun j => ⟨(ofPartition p).colLen j, by
+      have := colLen_le_card (ofPartition p) j
+      rw [card_ofPartition] at this
+      omega⟩)
+  have hf : Function.Injective F := by
+    intro p q hpq
+    apply ofPartition_injective
+    apply YoungDiagram.ext
+    ext z
+    have aux (p q : Nat.Partition n) (hpq : F p = F q) (hz : z ∈ ofPartition p) : z ∈ ofPartition q := by
+      have hs := cell_small_coord (ofPartition p) ⟨z, hz⟩
+      rw [card_ofPartition] at hs
+      rcases hs with hs | hs
+      · have he := congrArg (fun v : (Fin (Nat.sqrt n) → Fin (n + 1)) × (Fin (Nat.sqrt n) → Fin (n + 1)) =>
+            (v.1 ⟨z.1, hs⟩).val) hpq
+        change (ofPartition p).rowLen z.1 = (ofPartition q).rowLen z.1 at he
+        apply YoungDiagram.mem_iff_lt_rowLen.mpr
+        rw [← he]
+        exact YoungDiagram.mem_iff_lt_rowLen.mp hz
+      · have he := congrArg (fun v : (Fin (Nat.sqrt n) → Fin (n + 1)) × (Fin (Nat.sqrt n) → Fin (n + 1)) =>
+            (v.2 ⟨z.2, hs⟩).val) hpq
+        change (ofPartition p).colLen z.2 = (ofPartition q).colLen z.2 at he
+        apply YoungDiagram.mem_iff_lt_colLen.mpr
+        rw [← he]
+        exact YoungDiagram.mem_iff_lt_colLen.mp hz
+    exact ⟨aux p q hpq, aux q p hpq.symm⟩
+  have hh := Fintype.card_le_of_injective F hf
+  simpa only [Fintype.card_prod, Fintype.card_fun, Fintype.card_fin, two_mul, pow_add] using hh
+
+end Thorp.Young
+
+namespace Thorp.Young
+open scoped Classical
+
+lemma pow_le_choose_mul (a k : ℕ) : a ^ k ≤ (k * a).choose k := by
+  let X := Fin k × Fin a
+  let S := {s : Finset X // s.card = k}
+  let f (v : Fin k → Fin a) : S := ⟨Finset.univ.image (fun i => (i, v i)), by
+    rw [Finset.card_image_of_injective _ (by intro i j h; exact congrArg Prod.fst h), Finset.card_univ, Fintype.card_fin]⟩
+  have hf : Function.Injective f := by
+    intro v w hvw
+    funext i
+    have he := congrArg Subtype.val hvw
+    have hm : (i, v i) ∈ (f w).val := by rw [← he]; exact Finset.mem_image.mpr ⟨i, Finset.mem_univ _, rfl⟩
+    obtain ⟨j, _, hj⟩ := Finset.mem_image.mp hm
+    have hji : j = i := congrArg Prod.fst hj
+    subst j
+    exact (congrArg Prod.snd hj).symm
+  have hS : Fintype.card S = (k * a).choose k := by
+    calc
+      _ = ((Finset.univ : Finset X).powersetCard k).card := by
+        rw [Fintype.card_subtype]
+        congr 1
+        ext s
+        simp
+      _ = _ := by simp [X]
+  have hh := Fintype.card_le_of_injective f hf
+  simpa only [Fintype.card_fun, Fintype.card_fin, hS] using hh
+
+lemma pow_le_choose_of_mul_le (a k L : ℕ) (h : a * k ≤ L) : a ^ k ≤ L.choose k := by
+  exact (pow_le_choose_mul a k).trans (Nat.choose_le_choose k (by simpa only [Nat.mul_comm] using h))
+
+lemma sub_add_one_le_choose (n k : ℕ) (hk : 1 ≤ k) (hkn : k ≤ n) : n - k + 1 ≤ n.choose k := by
+  obtain ⟨r, rfl⟩ := Nat.exists_eq_add_of_le hkn
+  simp only [Nat.add_sub_cancel_left]
+  induction r with
+  | zero => simp
+  | succ r ih =>
+      obtain ⟨j, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : k ≠ 0)
+      rw [show j + 1 + (r + 1) = (j + 1 + r) + 1 by omega, Nat.choose_succ_succ]
+      have hp : 1 ≤ (j + 1 + r).choose j := Nat.one_le_iff_ne_zero.mpr (Nat.choose_ne_zero (by omega))
+      simp only [Nat.succ_eq_add_one] at ih ⊢
+      omega
+
+end Thorp.Young
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/PermutationCharacters.lean b/lean/OAI/Probability/ThorpShuffle/PermutationCharacters.lean
index db26504bdded22929ffe612eab6ae6f11a9059f7..e78e0abbb11bb3fa0c342ee7e019c386f5633200 100644
--- a/lean/OAI/Probability/ThorpShuffle/PermutationCharacters.lean
+++ b/lean/OAI/Probability/ThorpShuffle/PermutationCharacters.lean
@@ -1,153 +1,153 @@
-import OAI.Probability.ThorpShuffle.VectorOrbits
-
-namespace OAI
-
-universe uE
-
-noncomputable section
-
-open scoped BigOperators ComplexConjugate InnerProductSpace
-open Filter Topology
-
-namespace Thorp.Specht
-open scoped Classical
-open Thorp.Young
-
-lemma degree_gt_one {n : ℕ} (p : NShape n) (hn : 16 ≤ n) (hp : 0 < defect p) :
-    1 < degree p := by
-  by_cases hs : 4 * defect p ≤ n
-  · have hh := small_defect_degree p hs
-    have h3 : 3 ≤ 3 ^ defect p := by simpa using Nat.pow_le_pow_right (by omega : 0 < 3) (show 1 ≤ defect p by omega)
-    omega
-  · have hb := bulk_degree p (by omega) (by omega)
-    have h2 : (2 : ℝ) ≤ (2 : ℝ)^(n/16) := by
-      simpa using pow_le_pow_right₀ (by norm_num : (1:ℝ) ≤ 2) (show 1 ≤ n/16 by omega)
-    have : (1 : ℝ) < degree p := lt_of_lt_of_le (by norm_num) (h2.trans hb)
-    exact_mod_cast this
-
-lemma full_reciprocal_bound (n : ℕ) :
-    (symmetricFamily n).reciprocalSum ≤ 2 + exceptionalReciprocalSum n := by
-  have hzero : slice n 0 ≤ 2 := by
-    have h := slice_bound n 0 1 (by norm_num) (fun p _ => by exact_mod_cast degree_pos p)
-    norm_num at h
-    exact h
-  have he : (symmetricFamily n).reciprocalSum = slice n 0 + exceptionalReciprocalSum n := by
-    unfold slice
-    rw [← Finset.sum_subtype (Finset.univ.filter (fun p : NShape n => defect p = 0)) (by simp), Finset.sum_filter]
-    unfold exceptionalReciprocalSum
-    rw [← Finset.sum_add_distrib]
-    apply Finset.sum_congr rfl
-    intro p _
-    change inverseDegree p = _
-    by_cases hp : defect p = 0
-    · simp only [hp, lt_self_iff_false, ↓reduceIte, add_zero]
-    · simp only [hp, Nat.pos_of_ne_zero hp, ↓reduceIte, zero_add]
-  rw [he]
-  linarith
-
-lemma exceptional_count (n : ℕ) : Fintype.card {p : NShape n // defect p = 0} ≤ 2 := by
-  simpa using defect_card_bound n 0
-
-end Thorp.Specht
-
-namespace Thorp.Specht
-open scoped Classical
-
-variable {α : Type} [Fintype α]
-
-def permutationFamily (α : Type) [Fintype α] [DecidableEq α] : Thorp.Fourier.Family (Equiv.Perm α) :=
-  (symmetricFamily (Fintype.card α)).pull (Equiv.permCongrHom (Fintype.equivFin α))
-
-lemma complexSign_permCongr {β : Type} [Fintype β] (e : α ≃ β) (g : Equiv.Perm α) :
-    complexSign (e.permCongr g) = complexSign g := by
-  simp only [complexSign, MonoidHom.coe_mk, OneHom.coe_mk, Equiv.Perm.sign_permCongr]
-
-lemma complexSign_norm (g : Equiv.Perm α) : ‖complexSign g‖ = 1 := by
-  rcases Int.units_eq_one_or (Equiv.Perm.sign g) with h | h <;> simp [complexSign, h]
-
-lemma complexSign_eq [DecidableEq α] (g : Equiv.Perm α) :
-    complexSign g = (((Equiv.Perm.sign g : ℤˣ) : ℤ) : ℂ) := by
-  change (((@Equiv.Perm.sign α (Classical.typeDecidableEq α) _ g : ℤˣ) : ℤ) : ℂ) = _
-  rw [Subsingleton.elim (Classical.typeDecidableEq α) (inferInstance : DecidableEq α)]
-
-lemma complexSign_sum [DecidableEq α] [Nontrivial α] : (∑ g : Equiv.Perm α, complexSign g) = 0 := by
-  obtain ⟨a, ha⟩ := Equiv.Perm.sign_surjective α (-1)
-  have he : complexSign a = -1 := by rw [complexSign_eq, ha]; simp
-  have hh := Equiv.sum_comp (Equiv.mulLeft a) (fun g : Equiv.Perm α => complexSign g)
-  simp only [Equiv.coe_mulLeft, map_mul, he, neg_one_mul, Finset.sum_neg_distrib] at hh
-  linear_combination - (1/2 : ℂ) * hh
-
-lemma permutation_exceptional_action [DecidableEq α] (p : NShape (Fintype.card α)) (hp : defect p = 0) :
-    (permutationFamily α).degree p = 1 ∧
-      ((∀ (g : Equiv.Perm α) (v : Space (α:=Fin (Fintype.card α)) p), (permutationFamily α).rep p g v = v) ∨
-       (∀ (g : Equiv.Perm α) (v : Space (α:=Fin (Fintype.card α)) p), (permutationFamily α).rep p g v = complexSign g • v)) := by
-  obtain ⟨hd, htr | hsign⟩ := exceptional_action p hp
-  · exact ⟨hd, Or.inl fun g v => htr _ v⟩
-  · refine ⟨hd, Or.inr ?_⟩
-    intro g v
-    change family p ((Fintype.equivFin α).permCongr g) v = _
-    rw [hsign, complexSign_permCongr]
-
-end Thorp.Specht
-
-namespace Thorp.Fourier
-open scoped Classical
-variable {G : Type} [Group G] [Fintype G]
-
-def realConv (μ ν : G → ℝ) (g : G) : ℝ := ∑ h, μ h * ν (h⁻¹ * g)
-
-def realPower (ν : G → ℝ) : ℕ → G → ℝ
-  | 0 => fun g => if g = 1 then 1 else 0
-  | k+1 => realConv ν (realPower ν k)
-
-lemma realConv_nonneg (μ ν : G → ℝ) (hμ : ∀ g, 0 ≤ μ g) (hν : ∀ g, 0 ≤ ν g) :
-    ∀ g, 0 ≤ realConv μ ν g := fun _ => Finset.sum_nonneg (fun h _ => mul_nonneg (hμ h) (hν _))
-
-lemma realConv_sum (μ ν : G → ℝ) : ∑ g, realConv μ ν g = (∑ g, μ g) * (∑ g, ν g) := by
-  unfold realConv
-  rw [Finset.sum_comm, Finset.sum_mul]
-  apply Finset.sum_congr rfl
-  intro g _
-  rw [← Finset.mul_sum]
-  congr 1
-  exact Equiv.sum_comp (Equiv.mulLeft g⁻¹) ν
-
-lemma realPower_nonneg (ν : G → ℝ) (hν : ∀ g, 0 ≤ ν g) (k : ℕ) : ∀ g, 0 ≤ realPower ν k g := by
-  induction k with
-  | zero => intro g; simp only [realPower]; split_ifs <;> norm_num
-  | succ k ih => exact realConv_nonneg ν (realPower ν k) hν ih
-
-lemma realPower_sum (ν : G → ℝ) (hν : ∑ g, ν g = 1) (k : ℕ) : ∑ g, realPower ν k g = 1 := by
-  induction k with
-  | zero => simp [realPower]
-  | succ k ih => rw [realPower, realConv_sum, hν, ih, mul_one]
-
-variable {E : Type uE} [AddCommGroup E] [Module ℂ E]
-
-lemma integrated_realConv (ρ : Representation ℂ G E) (μ ν : G → ℝ) :
-    integrated ρ (fun g => (realConv μ ν g : ℂ)) = integrated ρ (fun g => (μ g : ℂ)) *
-      integrated ρ (fun g => (ν g : ℂ)) := by
-  rw [← integrated_convolution]
-  congr 1
-  funext g
-  simp only [realConv, convolution, Complex.ofReal_sum, Complex.ofReal_mul]
-
-lemma integrated_realPower (ρ : Representation ℂ G E) (ν : G → ℝ) (k : ℕ) :
-    integrated ρ (fun g => (realPower ν k g : ℂ)) = (integrated ρ (fun g => (ν g : ℂ)))^k := by
-  induction k with
-  | zero =>
-    rw [pow_zero]
-    have he : (fun g => (realPower ν 0 g : ℂ)) = delta := by funext g; simp only [realPower, delta]; split_ifs <;> norm_num
-    rw [he, integrated_delta]
-  | succ k ih => rw [realPower, integrated_realConv, ih, pow_succ']
-
-lemma integrated_real_sub (ρ : Representation ℂ G E) (μ ν : G → ℝ) :
-    integrated ρ (fun g => ((μ g - ν g : ℝ) : ℂ)) =
-      integrated ρ (fun g => (μ g : ℂ)) - integrated ρ (fun g => (ν g : ℂ)) := by
-  simp only [integrated, Complex.ofReal_sub, sub_smul, Finset.sum_sub_distrib]
-
-end Thorp.Fourier
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.VectorOrbits
+
+namespace OAI
+
+universe uE
+
+noncomputable section
+
+open scoped BigOperators ComplexConjugate InnerProductSpace
+open Filter Topology
+
+namespace Thorp.Specht
+open scoped Classical
+open Thorp.Young
+
+lemma degree_gt_one {n : ℕ} (p : NShape n) (hn : 16 ≤ n) (hp : 0 < defect p) :
+    1 < degree p := by
+  by_cases hs : 4 * defect p ≤ n
+  · have hh := small_defect_degree p hs
+    have h3 : 3 ≤ 3 ^ defect p := by simpa using Nat.pow_le_pow_right (by omega : 0 < 3) (show 1 ≤ defect p by omega)
+    omega
+  · have hb := bulk_degree p (by omega) (by omega)
+    have h2 : (2 : ℝ) ≤ (2 : ℝ)^(n/16) := by
+      simpa using pow_le_pow_right₀ (by norm_num : (1:ℝ) ≤ 2) (show 1 ≤ n/16 by omega)
+    have : (1 : ℝ) < degree p := lt_of_lt_of_le (by norm_num) (h2.trans hb)
+    exact_mod_cast this
+
+lemma full_reciprocal_bound (n : ℕ) :
+    (symmetricFamily n).reciprocalSum ≤ 2 + exceptionalReciprocalSum n := by
+  have hzero : slice n 0 ≤ 2 := by
+    have h := slice_bound n 0 1 (by norm_num) (fun p _ => by exact_mod_cast degree_pos p)
+    norm_num at h
+    exact h
+  have he : (symmetricFamily n).reciprocalSum = slice n 0 + exceptionalReciprocalSum n := by
+    unfold slice
+    rw [← Finset.sum_subtype (Finset.univ.filter (fun p : NShape n => defect p = 0)) (by simp), Finset.sum_filter]
+    unfold exceptionalReciprocalSum
+    rw [← Finset.sum_add_distrib]
+    apply Finset.sum_congr rfl
+    intro p _
+    change inverseDegree p = _
+    by_cases hp : defect p = 0
+    · simp only [hp, lt_self_iff_false, ↓reduceIte, add_zero]
+    · simp only [hp, Nat.pos_of_ne_zero hp, ↓reduceIte, zero_add]
+  rw [he]
+  linarith
+
+lemma exceptional_count (n : ℕ) : Fintype.card {p : NShape n // defect p = 0} ≤ 2 := by
+  simpa using defect_card_bound n 0
+
+end Thorp.Specht
+
+namespace Thorp.Specht
+open scoped Classical
+
+variable {α : Type} [Fintype α]
+
+def permutationFamily (α : Type) [Fintype α] [DecidableEq α] : Thorp.Fourier.Family (Equiv.Perm α) :=
+  (symmetricFamily (Fintype.card α)).pull (Equiv.permCongrHom (Fintype.equivFin α))
+
+lemma complexSign_permCongr {β : Type} [Fintype β] (e : α ≃ β) (g : Equiv.Perm α) :
+    complexSign (e.permCongr g) = complexSign g := by
+  simp only [complexSign, MonoidHom.coe_mk, OneHom.coe_mk, Equiv.Perm.sign_permCongr]
+
+lemma complexSign_norm (g : Equiv.Perm α) : ‖complexSign g‖ = 1 := by
+  rcases Int.units_eq_one_or (Equiv.Perm.sign g) with h | h <;> simp [complexSign, h]
+
+lemma complexSign_eq [DecidableEq α] (g : Equiv.Perm α) :
+    complexSign g = (((Equiv.Perm.sign g : ℤˣ) : ℤ) : ℂ) := by
+  change (((@Equiv.Perm.sign α (Classical.typeDecidableEq α) _ g : ℤˣ) : ℤ) : ℂ) = _
+  rw [Subsingleton.elim (Classical.typeDecidableEq α) (inferInstance : DecidableEq α)]
+
+lemma complexSign_sum [DecidableEq α] [Nontrivial α] : (∑ g : Equiv.Perm α, complexSign g) = 0 := by
+  obtain ⟨a, ha⟩ := Equiv.Perm.sign_surjective α (-1)
+  have he : complexSign a = -1 := by rw [complexSign_eq, ha]; simp
+  have hh := Equiv.sum_comp (Equiv.mulLeft a) (fun g : Equiv.Perm α => complexSign g)
+  simp only [Equiv.coe_mulLeft, map_mul, he, neg_one_mul, Finset.sum_neg_distrib] at hh
+  linear_combination - (1/2 : ℂ) * hh
+
+lemma permutation_exceptional_action [DecidableEq α] (p : NShape (Fintype.card α)) (hp : defect p = 0) :
+    (permutationFamily α).degree p = 1 ∧
+      ((∀ (g : Equiv.Perm α) (v : Space (α:=Fin (Fintype.card α)) p), (permutationFamily α).rep p g v = v) ∨
+       (∀ (g : Equiv.Perm α) (v : Space (α:=Fin (Fintype.card α)) p), (permutationFamily α).rep p g v = complexSign g • v)) := by
+  obtain ⟨hd, htr | hsign⟩ := exceptional_action p hp
+  · exact ⟨hd, Or.inl fun g v => htr _ v⟩
+  · refine ⟨hd, Or.inr ?_⟩
+    intro g v
+    change family p ((Fintype.equivFin α).permCongr g) v = _
+    rw [hsign, complexSign_permCongr]
+
+end Thorp.Specht
+
+namespace Thorp.Fourier
+open scoped Classical
+variable {G : Type} [Group G] [Fintype G]
+
+def realConv (μ ν : G → ℝ) (g : G) : ℝ := ∑ h, μ h * ν (h⁻¹ * g)
+
+def realPower (ν : G → ℝ) : ℕ → G → ℝ
+  | 0 => fun g => if g = 1 then 1 else 0
+  | k+1 => realConv ν (realPower ν k)
+
+lemma realConv_nonneg (μ ν : G → ℝ) (hμ : ∀ g, 0 ≤ μ g) (hν : ∀ g, 0 ≤ ν g) :
+    ∀ g, 0 ≤ realConv μ ν g := fun _ => Finset.sum_nonneg (fun h _ => mul_nonneg (hμ h) (hν _))
+
+lemma realConv_sum (μ ν : G → ℝ) : ∑ g, realConv μ ν g = (∑ g, μ g) * (∑ g, ν g) := by
+  unfold realConv
+  rw [Finset.sum_comm, Finset.sum_mul]
+  apply Finset.sum_congr rfl
+  intro g _
+  rw [← Finset.mul_sum]
+  congr 1
+  exact Equiv.sum_comp (Equiv.mulLeft g⁻¹) ν
+
+lemma realPower_nonneg (ν : G → ℝ) (hν : ∀ g, 0 ≤ ν g) (k : ℕ) : ∀ g, 0 ≤ realPower ν k g := by
+  induction k with
+  | zero => intro g; simp only [realPower]; split_ifs <;> norm_num
+  | succ k ih => exact realConv_nonneg ν (realPower ν k) hν ih
+
+lemma realPower_sum (ν : G → ℝ) (hν : ∑ g, ν g = 1) (k : ℕ) : ∑ g, realPower ν k g = 1 := by
+  induction k with
+  | zero => simp [realPower]
+  | succ k ih => rw [realPower, realConv_sum, hν, ih, mul_one]
+
+variable {E : Type uE} [AddCommGroup E] [Module ℂ E]
+
+lemma integrated_realConv (ρ : Representation ℂ G E) (μ ν : G → ℝ) :
+    integrated ρ (fun g => (realConv μ ν g : ℂ)) = integrated ρ (fun g => (μ g : ℂ)) *
+      integrated ρ (fun g => (ν g : ℂ)) := by
+  rw [← integrated_convolution]
+  congr 1
+  funext g
+  simp only [realConv, convolution, Complex.ofReal_sum, Complex.ofReal_mul]
+
+lemma integrated_realPower (ρ : Representation ℂ G E) (ν : G → ℝ) (k : ℕ) :
+    integrated ρ (fun g => (realPower ν k g : ℂ)) = (integrated ρ (fun g => (ν g : ℂ)))^k := by
+  induction k with
+  | zero =>
+    rw [pow_zero]
+    have he : (fun g => (realPower ν 0 g : ℂ)) = delta := by funext g; simp only [realPower, delta]; split_ifs <;> norm_num
+    rw [he, integrated_delta]
+  | succ k ih => rw [realPower, integrated_realConv, ih, pow_succ']
+
+lemma integrated_real_sub (ρ : Representation ℂ G E) (μ ν : G → ℝ) :
+    integrated ρ (fun g => ((μ g - ν g : ℝ) : ℂ)) =
+      integrated ρ (fun g => (μ g : ℂ)) - integrated ρ (fun g => (ν g : ℂ)) := by
+  simp only [integrated, Complex.ofReal_sub, sub_smul, Finset.sum_sub_distrib]
+
+end Thorp.Fourier
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/Polytabloids.lean b/lean/OAI/Probability/ThorpShuffle/Polytabloids.lean
index eb78e715c56233fd363de06e7d56ce72021bbb94..33a6807af77b1ae51455fd24e4b7a69c2431cc2a 100644
--- a/lean/OAI/Probability/ThorpShuffle/Polytabloids.lean
+++ b/lean/OAI/Probability/ThorpShuffle/Polytabloids.lean
@@ -1,142 +1,142 @@
-import OAI.Probability.ThorpShuffle.Specht
-
-namespace OAI
-
-universe uα
-
-noncomputable section
-
-open scoped BigOperators ComplexConjugate InnerProductSpace
-open Filter
-
-namespace Thorp.Specht
-open scoped Classical
-
-variable {α : Type uα} [Fintype α]
-
-lemma tabloid_sum (r : α → ℕ) (f : Tabloid r) : ∑ x, f.val x = ∑ x, r x := by
-  obtain ⟨g, hg⟩ := f.property
-  rw [hg]
-  exact Equiv.sum_comp g r
-
-lemma collision_zero (r c : α → ℕ) (f : Tabloid r)
-    (x y : α) (hxy : x ≠ y) (hc : c x = c y) (hf : f.val x = f.val y) :
-    PermutationSpace.alternator (fiberGroup c).subtype
-      (complexSign.comp (fiberGroup c).subtype) (EuclideanSpace.single f 1) = 0 := by
-  apply PermutationSpace.alternator_single_zero _ _ f (⟨Equiv.swap x y, swap_preserves hc⟩ : fiberGroup c)
-  · rw [stabilizes_tabloid_iff]
-    exact swap_preserves hf
-  · exact complexSign_swap x y hxy
-
-lemma alternator_nonzero_nocollision (r c : α → ℕ)
-    (v : EuclideanSpace ℂ (Tabloid r))
-    (hv : PermutationSpace.alternator (fiberGroup c).subtype
-      (complexSign.comp (fiberGroup c).subtype) v ≠ 0) :
-    ∃ f : Tabloid r, ∀ x y, c x = c y → f.val x = f.val y → x = y := by
-  by_contra hn
-  push Not at hn
-  apply hv
-  have hexp := (EuclideanSpace.basisFun (Tabloid r) ℂ).sum_repr v
-  simp only [EuclideanSpace.basisFun_repr, EuclideanSpace.basisFun_apply] at hexp
-  rw [← hexp, map_sum]
-  apply Finset.sum_eq_zero
-  intro f _
-  obtain ⟨x, y, hcol, hrow, hne⟩ := hn f
-  rw [map_smul, collision_zero r c f x y hne hcol hrow, smul_zero]
-
-lemma rowRankSum_tableau (D : YoungDiagram) (t : α ≃ Thorp.Young.Cells D) :
-    ∑ x : α, tableauRow D t x = ∑ z : Thorp.Young.Cells D, z.val.1 :=
-  Equiv.sum_comp t (fun z => z.val.1)
-
-lemma rankSum_le_of_nocollision (D E : YoungDiagram)
-    (t : α ≃ Thorp.Young.Cells D) (u : α ≃ Thorp.Young.Cells E)
-    (f : Tabloid (tableauRow E u))
-    (hf : ∀ x y, tableauCol D t x = tableauCol D t y → f.val x = f.val y → x = y) :
-    ∑ z : Thorp.Young.Cells D, z.val.1 ≤ ∑ z : Thorp.Young.Cells E, z.val.1 := by
-  have hi : ∀ x y : Thorp.Young.Cells D, x.val.2 = y.val.2 →
-      f.val (t.symm x) = f.val (t.symm y) → x = y := by
-    intro x y hc hr
-    apply t.symm.injective
-    apply hf _ _ _ hr
-    simpa only [tableauCol, Equiv.apply_symm_apply] using hc
-  have hh := Thorp.Young.rowRankSum_le D (fun z => f.val (t.symm z)) hi
-  rw [Equiv.sum_comp t.symm, tabloid_sum, rowRankSum_tableau] at hh
-  exact hh
-
-lemma shape_eq_of_nocollision_equal_sum (D E : YoungDiagram)
-    (t : α ≃ Thorp.Young.Cells D) (u : α ≃ Thorp.Young.Cells E)
-    (f : Tabloid (tableauRow E u))
-    (hf : ∀ x y, tableauCol D t x = tableauCol D t y → f.val x = f.val y → x = y)
-    (hs : ∑ z : Thorp.Young.Cells D, z.val.1 = ∑ z : Thorp.Young.Cells E, z.val.1) : D = E := by
-  have hi : ∀ x y : Thorp.Young.Cells D, x.val.2 = y.val.2 →
-      f.val (t.symm x) = f.val (t.symm y) → x = y := by
-    intro x y hc hr
-    apply t.symm.injective
-    apply hf _ _ _ hr
-    simpa only [tableauCol, Equiv.apply_symm_apply] using hc
-  have hsum : ∑ z : Thorp.Young.Cells D, f.val (t.symm z) = ∑ z : Thorp.Young.Cells D, z.val.1 := by
-    rw [Equiv.sum_comp t.symm, tabloid_sum, rowRankSum_tableau, hs]
-  obtain ⟨σ, -, hσ⟩ := Thorp.Young.column_sort D (fun z => f.val (t.symm z)) hi hsum
-  obtain ⟨g, hg⟩ := f.property
-  apply Thorp.Young.eq_of_rowPreserving D E (((σ.trans t.symm).trans g).trans u)
-  intro x
-  have hh := hσ x
-  rw [hg] at hh
-  exact hh
-
-lemma mapped_column_nonzero (D E : YoungDiagram)
-    (t : α ≃ Thorp.Young.Cells D) (u : α ≃ Thorp.Young.Cells E)
-    (F : Representation.IntertwiningMap (module D t).toRepresentation (module E u).toRepresentation)
-    (hF : Function.Injective F) :
-    ∃ v : EuclideanSpace ℂ (Tabloid (tableauRow E u)),
-      PermutationSpace.alternator (fiberGroup (tableauCol D t)).subtype
-        (complexSign.comp (fiberGroup (tableauCol D t)).subtype) v ≠ 0 := by
-  let e : (module D t).toSubmodule := ⟨polytabloid D t, self_mem_cyclic _ _⟩
-  have he : e ≠ 0 := by
-    intro h
-    exact polytabloid_ne_zero D t (congrArg Subtype.val h)
-  have heig : ∑ h : fiberGroup (tableauCol D t), complexSign h.val •
-      (module D t).toRepresentation h.val e = ⟪polytabloid D t, polytabloid D t⟫_ℂ • e := by
-    apply Subtype.ext
-    simp only [Submodule.coe_sum, Submodule.coe_smul]
-    change (∑ h : fiberGroup (tableauCol D t), complexSign h.val •
-      tabloidRep D t h.val (polytabloid D t)) = _
-    exact (PermutationSpace.alternator_apply _ _ _).symm.trans (columnAlternator_rank_one D t _)
-  have hh := congrArg (fun z => (F z : EuclideanSpace ℂ (Tabloid (tableauRow E u)))) heig
-  simp only [map_sum, map_smul, F.isIntertwining, Submodule.coe_sum, Submodule.coe_smul] at hh
-  refine ⟨F e, ?_⟩
-  intro hz
-  have heq : PermutationSpace.alternator (fiberGroup (tableauCol D t)).subtype
-      (complexSign.comp (fiberGroup (tableauCol D t)).subtype) (F e : EuclideanSpace ℂ (Tabloid (tableauRow E u))) =
-      ⟪polytabloid D t, polytabloid D t⟫_ℂ • (F e : EuclideanSpace ℂ (Tabloid (tableauRow E u))) := by
-    rw [PermutationSpace.alternator_apply]
-    exact hh
-  rw [hz] at heq
-  rcases smul_eq_zero.mp heq.symm with hs | hs
-  · exact polytabloid_ne_zero D t (inner_self_eq_zero.mp hs)
-  · have hh' : F e = 0 := Subtype.ext hs
-    exact he (hF (by simpa using hh'))
-
-lemma rankSum_le_of_injective_intertwiner (D E : YoungDiagram)
-    (t : α ≃ Thorp.Young.Cells D) (u : α ≃ Thorp.Young.Cells E)
-    (F : Representation.IntertwiningMap (module D t).toRepresentation (module E u).toRepresentation)
-    (hF : Function.Injective F) :
-    ∑ z : Thorp.Young.Cells D, z.val.1 ≤ ∑ z : Thorp.Young.Cells E, z.val.1 := by
-  obtain ⟨v, hv⟩ := mapped_column_nonzero D E t u F hF
-  obtain ⟨f, hf⟩ := alternator_nonzero_nocollision _ _ v hv
-  exact rankSum_le_of_nocollision D E t u f hf
-
-theorem shape_eq_of_equiv (D E : YoungDiagram)
-    (t : α ≃ Thorp.Young.Cells D) (u : α ≃ Thorp.Young.Cells E)
-    (F : Representation.Equiv (module D t).toRepresentation (module E u).toRepresentation) : D = E := by
-  have hle := rankSum_le_of_injective_intertwiner D E t u F.toIntertwiningMap F.toLinearEquiv.injective
-  have hge := rankSum_le_of_injective_intertwiner E D u t F.symm.toIntertwiningMap F.symm.toLinearEquiv.injective
-  obtain ⟨v, hv⟩ := mapped_column_nonzero D E t u F.toIntertwiningMap F.toLinearEquiv.injective
-  obtain ⟨f, hf⟩ := alternator_nonzero_nocollision _ _ v hv
-  exact shape_eq_of_nocollision_equal_sum D E t u f hf (le_antisymm hle hge)
-
-end Thorp.Specht
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.Specht
+
+namespace OAI
+
+universe uα
+
+noncomputable section
+
+open scoped BigOperators ComplexConjugate InnerProductSpace
+open Filter
+
+namespace Thorp.Specht
+open scoped Classical
+
+variable {α : Type uα} [Fintype α]
+
+lemma tabloid_sum (r : α → ℕ) (f : Tabloid r) : ∑ x, f.val x = ∑ x, r x := by
+  obtain ⟨g, hg⟩ := f.property
+  rw [hg]
+  exact Equiv.sum_comp g r
+
+lemma collision_zero (r c : α → ℕ) (f : Tabloid r)
+    (x y : α) (hxy : x ≠ y) (hc : c x = c y) (hf : f.val x = f.val y) :
+    PermutationSpace.alternator (fiberGroup c).subtype
+      (complexSign.comp (fiberGroup c).subtype) (EuclideanSpace.single f 1) = 0 := by
+  apply PermutationSpace.alternator_single_zero _ _ f (⟨Equiv.swap x y, swap_preserves hc⟩ : fiberGroup c)
+  · rw [stabilizes_tabloid_iff]
+    exact swap_preserves hf
+  · exact complexSign_swap x y hxy
+
+lemma alternator_nonzero_nocollision (r c : α → ℕ)
+    (v : EuclideanSpace ℂ (Tabloid r))
+    (hv : PermutationSpace.alternator (fiberGroup c).subtype
+      (complexSign.comp (fiberGroup c).subtype) v ≠ 0) :
+    ∃ f : Tabloid r, ∀ x y, c x = c y → f.val x = f.val y → x = y := by
+  by_contra hn
+  push Not at hn
+  apply hv
+  have hexp := (EuclideanSpace.basisFun (Tabloid r) ℂ).sum_repr v
+  simp only [EuclideanSpace.basisFun_repr, EuclideanSpace.basisFun_apply] at hexp
+  rw [← hexp, map_sum]
+  apply Finset.sum_eq_zero
+  intro f _
+  obtain ⟨x, y, hcol, hrow, hne⟩ := hn f
+  rw [map_smul, collision_zero r c f x y hne hcol hrow, smul_zero]
+
+lemma rowRankSum_tableau (D : YoungDiagram) (t : α ≃ Thorp.Young.Cells D) :
+    ∑ x : α, tableauRow D t x = ∑ z : Thorp.Young.Cells D, z.val.1 :=
+  Equiv.sum_comp t (fun z => z.val.1)
+
+lemma rankSum_le_of_nocollision (D E : YoungDiagram)
+    (t : α ≃ Thorp.Young.Cells D) (u : α ≃ Thorp.Young.Cells E)
+    (f : Tabloid (tableauRow E u))
+    (hf : ∀ x y, tableauCol D t x = tableauCol D t y → f.val x = f.val y → x = y) :
+    ∑ z : Thorp.Young.Cells D, z.val.1 ≤ ∑ z : Thorp.Young.Cells E, z.val.1 := by
+  have hi : ∀ x y : Thorp.Young.Cells D, x.val.2 = y.val.2 →
+      f.val (t.symm x) = f.val (t.symm y) → x = y := by
+    intro x y hc hr
+    apply t.symm.injective
+    apply hf _ _ _ hr
+    simpa only [tableauCol, Equiv.apply_symm_apply] using hc
+  have hh := Thorp.Young.rowRankSum_le D (fun z => f.val (t.symm z)) hi
+  rw [Equiv.sum_comp t.symm, tabloid_sum, rowRankSum_tableau] at hh
+  exact hh
+
+lemma shape_eq_of_nocollision_equal_sum (D E : YoungDiagram)
+    (t : α ≃ Thorp.Young.Cells D) (u : α ≃ Thorp.Young.Cells E)
+    (f : Tabloid (tableauRow E u))
+    (hf : ∀ x y, tableauCol D t x = tableauCol D t y → f.val x = f.val y → x = y)
+    (hs : ∑ z : Thorp.Young.Cells D, z.val.1 = ∑ z : Thorp.Young.Cells E, z.val.1) : D = E := by
+  have hi : ∀ x y : Thorp.Young.Cells D, x.val.2 = y.val.2 →
+      f.val (t.symm x) = f.val (t.symm y) → x = y := by
+    intro x y hc hr
+    apply t.symm.injective
+    apply hf _ _ _ hr
+    simpa only [tableauCol, Equiv.apply_symm_apply] using hc
+  have hsum : ∑ z : Thorp.Young.Cells D, f.val (t.symm z) = ∑ z : Thorp.Young.Cells D, z.val.1 := by
+    rw [Equiv.sum_comp t.symm, tabloid_sum, rowRankSum_tableau, hs]
+  obtain ⟨σ, -, hσ⟩ := Thorp.Young.column_sort D (fun z => f.val (t.symm z)) hi hsum
+  obtain ⟨g, hg⟩ := f.property
+  apply Thorp.Young.eq_of_rowPreserving D E (((σ.trans t.symm).trans g).trans u)
+  intro x
+  have hh := hσ x
+  rw [hg] at hh
+  exact hh
+
+lemma mapped_column_nonzero (D E : YoungDiagram)
+    (t : α ≃ Thorp.Young.Cells D) (u : α ≃ Thorp.Young.Cells E)
+    (F : Representation.IntertwiningMap (module D t).toRepresentation (module E u).toRepresentation)
+    (hF : Function.Injective F) :
+    ∃ v : EuclideanSpace ℂ (Tabloid (tableauRow E u)),
+      PermutationSpace.alternator (fiberGroup (tableauCol D t)).subtype
+        (complexSign.comp (fiberGroup (tableauCol D t)).subtype) v ≠ 0 := by
+  let e : (module D t).toSubmodule := ⟨polytabloid D t, self_mem_cyclic _ _⟩
+  have he : e ≠ 0 := by
+    intro h
+    exact polytabloid_ne_zero D t (congrArg Subtype.val h)
+  have heig : ∑ h : fiberGroup (tableauCol D t), complexSign h.val •
+      (module D t).toRepresentation h.val e = ⟪polytabloid D t, polytabloid D t⟫_ℂ • e := by
+    apply Subtype.ext
+    simp only [Submodule.coe_sum, Submodule.coe_smul]
+    change (∑ h : fiberGroup (tableauCol D t), complexSign h.val •
+      tabloidRep D t h.val (polytabloid D t)) = _
+    exact (PermutationSpace.alternator_apply _ _ _).symm.trans (columnAlternator_rank_one D t _)
+  have hh := congrArg (fun z => (F z : EuclideanSpace ℂ (Tabloid (tableauRow E u)))) heig
+  simp only [map_sum, map_smul, F.isIntertwining, Submodule.coe_sum, Submodule.coe_smul] at hh
+  refine ⟨F e, ?_⟩
+  intro hz
+  have heq : PermutationSpace.alternator (fiberGroup (tableauCol D t)).subtype
+      (complexSign.comp (fiberGroup (tableauCol D t)).subtype) (F e : EuclideanSpace ℂ (Tabloid (tableauRow E u))) =
+      ⟪polytabloid D t, polytabloid D t⟫_ℂ • (F e : EuclideanSpace ℂ (Tabloid (tableauRow E u))) := by
+    rw [PermutationSpace.alternator_apply]
+    exact hh
+  rw [hz] at heq
+  rcases smul_eq_zero.mp heq.symm with hs | hs
+  · exact polytabloid_ne_zero D t (inner_self_eq_zero.mp hs)
+  · have hh' : F e = 0 := Subtype.ext hs
+    exact he (hF (by simpa using hh'))
+
+lemma rankSum_le_of_injective_intertwiner (D E : YoungDiagram)
+    (t : α ≃ Thorp.Young.Cells D) (u : α ≃ Thorp.Young.Cells E)
+    (F : Representation.IntertwiningMap (module D t).toRepresentation (module E u).toRepresentation)
+    (hF : Function.Injective F) :
+    ∑ z : Thorp.Young.Cells D, z.val.1 ≤ ∑ z : Thorp.Young.Cells E, z.val.1 := by
+  obtain ⟨v, hv⟩ := mapped_column_nonzero D E t u F hF
+  obtain ⟨f, hf⟩ := alternator_nonzero_nocollision _ _ v hv
+  exact rankSum_le_of_nocollision D E t u f hf
+
+theorem shape_eq_of_equiv (D E : YoungDiagram)
+    (t : α ≃ Thorp.Young.Cells D) (u : α ≃ Thorp.Young.Cells E)
+    (F : Representation.Equiv (module D t).toRepresentation (module E u).toRepresentation) : D = E := by
+  have hle := rankSum_le_of_injective_intertwiner D E t u F.toIntertwiningMap F.toLinearEquiv.injective
+  have hge := rankSum_le_of_injective_intertwiner E D u t F.symm.toIntertwiningMap F.symm.toLinearEquiv.injective
+  obtain ⟨v, hv⟩ := mapped_column_nonzero D E t u F.toIntertwiningMap F.toLinearEquiv.injective
+  obtain ⟨f, hf⟩ := alternator_nonzero_nocollision _ _ v hv
+  exact shape_eq_of_nocollision_equal_sum D E t u f hf (le_antisymm hle hge)
+
+end Thorp.Specht
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/PrefixTests.lean b/lean/OAI/Probability/ThorpShuffle/PrefixTests.lean
index 504d173ecd93c36c9dff73c018027a1259c33e5d..40fb20c696b202945b8b95f9fb01470fc910a1af 100644
--- a/lean/OAI/Probability/ThorpShuffle/PrefixTests.lean
+++ b/lean/OAI/Probability/ThorpShuffle/PrefixTests.lean
@@ -1,109 +1,109 @@
-import OAI.Probability.ThorpShuffle.TaggedCompletion
-
-namespace OAI
-
-universe uα
-
-noncomputable section
-
-open scoped BigOperators
-open Filter
-
-namespace Thorp
-
-namespace Conditional
-
-theorem prefix_one_step_bound (d t k : ℕ) (e : Fin (k + 1) → Position (d + 1))
-    (he : Function.Injective e) (f : (Fin (k + 1) → Position (d + 1)) → ℝ)
-    (hf : ∀ z, |f z| ≤ 1) :
-    |mean (fun ω : History (d + 1) t => f ((run (d + 1) t ω) ∘ e)) -
-      mean (fun ω : History (d + 1) t => completeTest f ((run (d + 1) t ω) ∘ Fin.init e))| ≤
-    Real.sqrt ((Fintype.card (Position (d + 1)) : ℝ) *
-      expectedEnergy d (initialState (d + 1) (freeOf (Fin.init e)) (e (Fin.last k))
-        (by
-          apply (freeOf_true _ _).mpr
-          intro i hi
-          exact Fin.castSucc_ne_last i (he hi))) t) := by
-  have ht : freeOf (Fin.init e) (e (Fin.last k)) = true := by
-    apply (freeOf_true _ _).mpr
-    intro i hi
-    exact Fin.castSucc_ne_last i (he hi)
-  have hh := tagged_deviation_bound d (freeOf (Fin.init e)) (e (Fin.last k)) ht
-    (Fin.init e) (freeOf_exposed _) t (fun z y => f (Fin.snoc z y)) (fun z y => hf _)
-  convert hh using 1
-  congr 1
-  rw [← mean_sub]
-  apply mean_congr
-  intro ω
-  rw [← Fin.comp_snoc, Fin.snoc_init_self]
-  unfold completeTest
-  rw [freeCount_perm]
-  simp only [freeOf_perm]
-
-theorem uniform_prefix_step {α : Type uα} [Fintype α] [DecidableEq α]
-    (k : ℕ) (e : Fin (k + 1) → α) (he : Function.Injective e)
-    (f : (Fin (k + 1) → α) → ℝ) :
-    mean (fun g : Equiv.Perm α => f (g ∘ e)) =
-      mean (fun g : Equiv.Perm α => completeTest f (g ∘ Fin.init e)) := by
-  have ht : freeOf (Fin.init e) (e (Fin.last k)) = true := by
-    apply (freeOf_true _ _).mpr
-    intro i hi
-    exact Fin.castSucc_ne_last i (he hi)
-  have h := uniform_tagged_completion (Fin.init e) (e (Fin.last k)) ht
-    (fun z y => f (Fin.snoc z y))
-  simpa only [← Fin.comp_snoc, Fin.snoc_init_self, completeTest, freeCount_perm] using h
-
-theorem prefix_test_bound (d m : ℕ)
-    (hm : Fintype.card (Position (d + 2)) ≤ 8 * m)
-    (k : ℕ) (hk : k + m ≤ Fintype.card (Position (d + 2)))
-    (e : Fin k → Position (d + 2)) (he : Function.Injective e)
-    (f : (Fin k → Position (d + 2)) → ℝ) (hf : ∀ z, |f z| ≤ 1) :
-    |mean (fun ω : History (d + 2) (200 * (d + 2)) => f ((run (d + 2) (200 * (d + 2)) ω) ∘ e)) -
-      mean (fun g : State (d + 2) => f (g ∘ e))| ≤
-    (k : ℝ) * Real.sqrt ((Fintype.card (Position (d + 2)) : ℝ) *
-      ((1 / 2 : ℝ) ^ (8 * (d + 2)) + 32 * (9 / 10 : ℝ) ^ m)) := by
-  induction k with
-  | zero =>
-    have hz (g : State (d + 2)) : g ∘ e = e := by funext i; exact Fin.elim0 i
-    simp only [hz, mean_const, sub_self, abs_zero, Nat.cast_zero, zero_mul, le_refl]
-  | succ k ih =>
-    have he0 : Function.Injective (Fin.init e) := by
-      intro i j hij
-      exact Fin.castSucc_inj.mp (he hij)
-    have ht : freeOf (Fin.init e) (e (Fin.last k)) = true := by
-      apply (freeOf_true _ _).mpr
-      intro i hi
-      exact Fin.castSucc_ne_last i (he hi)
-    have hcount : m ≤ freeCount (freeOf (Fin.init e)) := by
-      rw [freeCount_freeOf _ he0]
-      omega
-    have henergy := centeredPoint_energy_200 d (freeOf (Fin.init e)) (e (Fin.last k)) ht (by omega)
-    have hpower : (9 / 10 : ℝ) ^ freeCount (freeOf (Fin.init e)) ≤ (9 / 10 : ℝ) ^ m :=
-      pow_le_pow_of_le_one (by norm_num) (by norm_num) hcount
-    have hz := prefix_one_step_bound (d + 1) (200 * (d + 2)) k e he f hf
-    have hestimate :
-        Real.sqrt ((Fintype.card (Position (d + 2)) : ℝ) *
-          expectedEnergy (d + 1) (initialState (d + 2) (freeOf (Fin.init e)) (e (Fin.last k)) ht)
-            (200 * (d + 2))) ≤
-        Real.sqrt ((Fintype.card (Position (d + 2)) : ℝ) *
-          ((1 / 2 : ℝ) ^ (8 * (d + 2)) + 32 * (9 / 10 : ℝ) ^ m)) := by
-      apply Real.sqrt_le_sqrt
-      apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
-      exact henergy.trans (by linarith)
-    have hnext := ih (by omega) (Fin.init e) he0 (completeTest f) (completeTest_bound f hf)
-    rw [uniform_prefix_step k e he f]
-    have htri := abs_sub_le
-      (mean (fun ω : History (d + 2) (200 * (d + 2)) => f ((run (d + 2) (200 * (d + 2)) ω) ∘ e)))
-      (mean (fun ω : History (d + 2) (200 * (d + 2)) => completeTest f ((run (d + 2) (200 * (d + 2)) ω) ∘ Fin.init e)))
-      (mean (fun g : State (d + 2) => completeTest f (g ∘ Fin.init e)))
-    have hz' := hz.trans hestimate
-    push_cast
-    nlinarith
-
-end Conditional
-
-end Thorp
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.TaggedCompletion
+
+namespace OAI
+
+universe uα
+
+noncomputable section
+
+open scoped BigOperators
+open Filter
+
+namespace Thorp
+
+namespace Conditional
+
+theorem prefix_one_step_bound (d t k : ℕ) (e : Fin (k + 1) → Position (d + 1))
+    (he : Function.Injective e) (f : (Fin (k + 1) → Position (d + 1)) → ℝ)
+    (hf : ∀ z, |f z| ≤ 1) :
+    |mean (fun ω : History (d + 1) t => f ((run (d + 1) t ω) ∘ e)) -
+      mean (fun ω : History (d + 1) t => completeTest f ((run (d + 1) t ω) ∘ Fin.init e))| ≤
+    Real.sqrt ((Fintype.card (Position (d + 1)) : ℝ) *
+      expectedEnergy d (initialState (d + 1) (freeOf (Fin.init e)) (e (Fin.last k))
+        (by
+          apply (freeOf_true _ _).mpr
+          intro i hi
+          exact Fin.castSucc_ne_last i (he hi))) t) := by
+  have ht : freeOf (Fin.init e) (e (Fin.last k)) = true := by
+    apply (freeOf_true _ _).mpr
+    intro i hi
+    exact Fin.castSucc_ne_last i (he hi)
+  have hh := tagged_deviation_bound d (freeOf (Fin.init e)) (e (Fin.last k)) ht
+    (Fin.init e) (freeOf_exposed _) t (fun z y => f (Fin.snoc z y)) (fun z y => hf _)
+  convert hh using 1
+  congr 1
+  rw [← mean_sub]
+  apply mean_congr
+  intro ω
+  rw [← Fin.comp_snoc, Fin.snoc_init_self]
+  unfold completeTest
+  rw [freeCount_perm]
+  simp only [freeOf_perm]
+
+theorem uniform_prefix_step {α : Type uα} [Fintype α] [DecidableEq α]
+    (k : ℕ) (e : Fin (k + 1) → α) (he : Function.Injective e)
+    (f : (Fin (k + 1) → α) → ℝ) :
+    mean (fun g : Equiv.Perm α => f (g ∘ e)) =
+      mean (fun g : Equiv.Perm α => completeTest f (g ∘ Fin.init e)) := by
+  have ht : freeOf (Fin.init e) (e (Fin.last k)) = true := by
+    apply (freeOf_true _ _).mpr
+    intro i hi
+    exact Fin.castSucc_ne_last i (he hi)
+  have h := uniform_tagged_completion (Fin.init e) (e (Fin.last k)) ht
+    (fun z y => f (Fin.snoc z y))
+  simpa only [← Fin.comp_snoc, Fin.snoc_init_self, completeTest, freeCount_perm] using h
+
+theorem prefix_test_bound (d m : ℕ)
+    (hm : Fintype.card (Position (d + 2)) ≤ 8 * m)
+    (k : ℕ) (hk : k + m ≤ Fintype.card (Position (d + 2)))
+    (e : Fin k → Position (d + 2)) (he : Function.Injective e)
+    (f : (Fin k → Position (d + 2)) → ℝ) (hf : ∀ z, |f z| ≤ 1) :
+    |mean (fun ω : History (d + 2) (200 * (d + 2)) => f ((run (d + 2) (200 * (d + 2)) ω) ∘ e)) -
+      mean (fun g : State (d + 2) => f (g ∘ e))| ≤
+    (k : ℝ) * Real.sqrt ((Fintype.card (Position (d + 2)) : ℝ) *
+      ((1 / 2 : ℝ) ^ (8 * (d + 2)) + 32 * (9 / 10 : ℝ) ^ m)) := by
+  induction k with
+  | zero =>
+    have hz (g : State (d + 2)) : g ∘ e = e := by funext i; exact Fin.elim0 i
+    simp only [hz, mean_const, sub_self, abs_zero, Nat.cast_zero, zero_mul, le_refl]
+  | succ k ih =>
+    have he0 : Function.Injective (Fin.init e) := by
+      intro i j hij
+      exact Fin.castSucc_inj.mp (he hij)
+    have ht : freeOf (Fin.init e) (e (Fin.last k)) = true := by
+      apply (freeOf_true _ _).mpr
+      intro i hi
+      exact Fin.castSucc_ne_last i (he hi)
+    have hcount : m ≤ freeCount (freeOf (Fin.init e)) := by
+      rw [freeCount_freeOf _ he0]
+      omega
+    have henergy := centeredPoint_energy_200 d (freeOf (Fin.init e)) (e (Fin.last k)) ht (by omega)
+    have hpower : (9 / 10 : ℝ) ^ freeCount (freeOf (Fin.init e)) ≤ (9 / 10 : ℝ) ^ m :=
+      pow_le_pow_of_le_one (by norm_num) (by norm_num) hcount
+    have hz := prefix_one_step_bound (d + 1) (200 * (d + 2)) k e he f hf
+    have hestimate :
+        Real.sqrt ((Fintype.card (Position (d + 2)) : ℝ) *
+          expectedEnergy (d + 1) (initialState (d + 2) (freeOf (Fin.init e)) (e (Fin.last k)) ht)
+            (200 * (d + 2))) ≤
+        Real.sqrt ((Fintype.card (Position (d + 2)) : ℝ) *
+          ((1 / 2 : ℝ) ^ (8 * (d + 2)) + 32 * (9 / 10 : ℝ) ^ m)) := by
+      apply Real.sqrt_le_sqrt
+      apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
+      exact henergy.trans (by linarith)
+    have hnext := ih (by omega) (Fin.init e) he0 (completeTest f) (completeTest_bound f hf)
+    rw [uniform_prefix_step k e he f]
+    have htri := abs_sub_le
+      (mean (fun ω : History (d + 2) (200 * (d + 2)) => f ((run (d + 2) (200 * (d + 2)) ω) ∘ e)))
+      (mean (fun ω : History (d + 2) (200 * (d + 2)) => completeTest f ((run (d + 2) (200 * (d + 2)) ω) ∘ Fin.init e)))
+      (mean (fun g : State (d + 2) => completeTest f (g ∘ Fin.init e)))
+    have hz' := hz.trans hestimate
+    push_cast
+    nlinarith
+
+end Conditional
+
+end Thorp
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/ProductProjectors.lean b/lean/OAI/Probability/ThorpShuffle/ProductProjectors.lean
index e077164d23a875a473dab585e0d7c9ccf27b9480..e39d80f63414cf42e84672e2cae5391d886bf53c 100644
--- a/lean/OAI/Probability/ThorpShuffle/ProductProjectors.lean
+++ b/lean/OAI/Probability/ThorpShuffle/ProductProjectors.lean
@@ -1,193 +1,193 @@
-import OAI.Probability.ThorpShuffle.JointProjectors
-
-namespace OAI
-
-universe uH uI uE uJ
-
-noncomputable section
-
-open scoped BigOperators ComplexConjugate InnerProductSpace
-open Filter Topology
-
-namespace Thorp.Fourier
-open scoped Classical
-variable {H : Type uH} {I : Type uI} {E : Type uE} [Group H] [Fintype H] [Fintype I] [AddCommGroup E] [Module ℂ E]
-
-lemma integrated_single_mul (ρ : Representation ℂ (I → H) E) (i : I) (k : H → ℂ) (f : (I → H) → ℂ) :
-    integrated (ρ.comp (MonoidHom.mulSingle (fun _ : I => H) i)) k * integrated ρ f =
-      integrated ρ (fun g => ∑ h, k h * f (Pi.mulSingle i h⁻¹ * g)) := by
-  simp only [integrated, Finset.sum_mul, Finset.mul_sum, Finset.sum_smul]
-  conv_rhs => rw [Finset.sum_comm]
-  conv_lhs => rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro h _
-  have he := Equiv.sum_comp (Equiv.mulLeft (Pi.mulSingle i h))
-    (fun g : I → H => (k h * f (Pi.mulSingle i h⁻¹ * g)) • ρ g)
-  rw [← he]
-  apply Finset.sum_congr rfl
-  intro g _
-  have hi : Pi.mulSingle i h⁻¹ = (Pi.mulSingle i h : I → H)⁻¹ :=
-    (map_inv (MonoidHom.mulSingle (fun _ : I => H) i) h).symm.symm
-  simp only [Equiv.coe_mulLeft, hi, inv_mul_cancel_left, smul_mul_smul,
-    MonoidHom.comp_apply, MonoidHom.mulSingle_apply, map_mul]
-
-end Thorp.Fourier
-
-namespace Thorp.Fourier.Family
-open scoped Classical
-variable {H : Type} [Group H] [Fintype H] (F : Family H)
-variable {I : Type} [Fintype I]
-
-lemma jointKernel_single (a : I → F.Index) (i : I) (h : H) (g : I → H) :
-    F.jointKernel a (Pi.mulSingle i h * g) =
-      F.kernel (a i) (h * g i) * ∏ j ∈ Finset.univ.erase i, F.kernel (a j) (g j) := by
-  rw [jointKernel, ← Finset.mul_prod_erase _ _ (Finset.mem_univ i)]
-  simp only [Pi.mul_apply, Pi.mulSingle_eq_same]
-  congr 1
-  apply Finset.prod_congr rfl
-  intro j hj
-  have hji := (Finset.mem_erase.mp hj).1
-  simp only [Pi.mulSingle_eq_of_ne hji, one_mul]
-
-lemma jointKernel_split (a : I → F.Index) (i : I) (g : I → H) :
-    F.jointKernel a g = F.kernel (a i) (g i) * ∏ j ∈ Finset.univ.erase i, F.kernel (a j) (g j) :=
-  (Finset.mul_prod_erase Finset.univ (fun j => F.kernel (a j) (g j)) (Finset.mem_univ i)).symm
-
-lemma jointKernel_single_convolution (a : I → F.Index) (i : I) (b : F.Index) (g : I → H) :
-    ∑ h, F.kernel b h * F.jointKernel a (Pi.mulSingle i h⁻¹ * g) =
-      if b = a i then F.jointKernel a g else 0 := by
-  simp only [F.jointKernel_single, ← mul_assoc, ← Finset.sum_mul]
-  change convolution (F.kernel b) (F.kernel (a i)) (g i) * _ = _
-  rw [F.kernel_convolution]
-  split_ifs with hb
-  · simp only [hb]
-    exact (F.jointKernel_split a i g).symm
-  · simp
-
-variable {E : Type uE} [AddCommGroup E] [Module ℂ E]
-
-lemma projector_jointProjector (ρ : Representation ℂ (I → H) E) (a : I → F.Index)
-    (i : I) (b : F.Index) :
-    F.projector (ρ.comp (MonoidHom.mulSingle (fun _ : I => H) i)) b * F.jointProjector ρ a =
-      if b = a i then F.jointProjector ρ a else 0 := by
-  rw [projector, jointProjector, integrated_single_mul]
-  simp only [F.jointKernel_single_convolution]
-  split_ifs <;> simp only [integrated, zero_smul, Finset.sum_const_zero]
-
-lemma low_fixes_joint (ρ : Representation ℂ (I → H) E) (a : I → F.Index)
-    (i : I) (s : Finset F.Index) (ha : a i ∈ s) :
-    (∑ b ∈ s, F.projector (ρ.comp (MonoidHom.mulSingle (fun _ : I => H) i)) b) * F.jointProjector ρ a =
-      F.jointProjector ρ a := by
-  rw [Finset.sum_mul]
-  simp only [F.projector_jointProjector, Finset.sum_ite_eq', ha, ↓reduceIte]
-
-end Thorp.Fourier.Family
-
-namespace Thorp.Fourier
-open scoped Classical
-
-lemma norm_sum_sq_of_orthogonal {J : Type uJ} {E : Type uE} [Fintype J]
-    [NormedAddCommGroup E] [InnerProductSpace ℂ E] (v : J → E)
-    (hv : ∀ i j, i ≠ j → ⟪v i, v j⟫_ℂ = 0) :
-    ‖∑ i, v i‖ ^ 2 = ∑ i, ‖v i‖ ^ 2 := by
-  have he : ⟪∑ i, v i, ∑ i, v i⟫_ℂ = ∑ i, ⟪v i, v i⟫_ℂ := by
-    simp only [sum_inner, inner_sum]
-    apply Finset.sum_congr rfl
-    intro i _
-    exact Finset.sum_eq_single i (fun j _ hji => hv j i hji) (by simp)
-  have hr := congrArg Complex.re he
-  simpa only [inner_self_eq_norm_sq_to_K, RCLike.ofReal_eq_complex_ofReal, ← Complex.ofReal_pow,
-    ← Complex.ofReal_sum, Complex.ofReal_re] using hr
-
-namespace Family
-variable {H I : Type} [Group H] [Fintype H] [Fintype I] [Nonempty I] (F : Family H)
-
-noncomputable def leastFactor (a : I → F.Index) : I :=
-  Classical.choose (Finset.exists_min_image Finset.univ (fun i => F.degree (a i)) Finset.univ_nonempty)
-
-lemma leastFactor_le (a : I → F.Index) (i : I) :
-    F.degree (a (F.leastFactor a)) ≤ F.degree (a i) :=
-  (Classical.choose_spec (Finset.exists_min_image Finset.univ
-    (fun i => F.degree (a i)) Finset.univ_nonempty)).2 i (Finset.mem_univ i)
-
-variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℂ E] [FiniteDimensional ℂ E]
-
-omit [Nonempty I] [FiniteDimensional ℂ E] in
-lemma jointProjector_symmetric (ρ : Representation ℂ (I → H) E)
-    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ) (a : I → F.Index) :
-    (F.jointProjector ρ a).IsSymmetric := integrated_symmetric ρ hu _ (F.jointKernel_star a)
-
-omit [Nonempty I] [FiniteDimensional ℂ E] in
-lemma jointProjector_orthogonal (ρ : Representation ℂ (I → H) E)
-    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ)
-    (a b : I → F.Index) (hab : a ≠ b) (x y : E) :
-    ⟪F.jointProjector ρ a x, F.jointProjector ρ b y⟫_ℂ = 0 := by
-  rw [F.jointProjector_symmetric ρ hu a]
-  change ⟪x, (F.jointProjector ρ a * F.jointProjector ρ b) y⟫_ℂ = 0
-  rw [F.jointProjector_mul, ite_eq_right hab]
-  simp
-
-noncomputable def splitVector (ρ : Representation ℂ (I → H) E) (v : E) (i : I) : E :=
-  ∑ a : I → F.Index, if F.leastFactor a = i then F.jointProjector ρ a v else 0
-
-omit [FiniteDimensional ℂ E] in
-lemma splitVector_sum (ρ : Representation ℂ (I → H) E) (v : E) :
-    ∑ i, F.splitVector ρ v i = v := by
-  simp only [splitVector]
-  rw [Finset.sum_comm]
-  simp only [Finset.sum_ite_eq, Finset.mem_univ, ↓reduceIte]
-  rw [← LinearMap.sum_apply, F.jointProjector_sum]
-  rfl
-
-omit [FiniteDimensional ℂ E] in
-lemma splitVector_orthogonal (ρ : Representation ℂ (I → H) E)
-    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ)
-    (v : E) (i j : I) (hij : i ≠ j) :
-    ⟪F.splitVector ρ v i, F.splitVector ρ v j⟫_ℂ = 0 := by
-  simp only [splitVector, sum_inner, inner_sum]
-  apply Finset.sum_eq_zero
-  intro a _
-  apply Finset.sum_eq_zero
-  intro b _
-  split_ifs with hai hbj
-  · exact F.jointProjector_orthogonal ρ hu b a (by intro h; subst b; exact hij (hai.symm.trans hbj)) v v
-  all_goals simp
-
-omit [FiniteDimensional ℂ E] in
-lemma splitVector_sq_sum (ρ : Representation ℂ (I → H) E)
-    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ) (v : E) :
-    ∑ i, ‖F.splitVector ρ v i‖ ^ 2 = ‖v‖ ^ 2 := by
-  rw [← norm_sum_sq_of_orthogonal _ (F.splitVector_orthogonal ρ hu v), F.splitVector_sum]
-
-lemma leastFactor_pow_le (ρ : Representation ℂ (I → H) E)
-    (a : I → F.Index) (ha : F.jointProjector ρ a ≠ 0) :
-    F.degree (a (F.leastFactor a)) ^ Fintype.card I ≤ Module.finrank ℂ E := by
-  calc
-    _ = ∏ _ : I, F.degree (a (F.leastFactor a)) := by simp
-    _ ≤ ∏ i, F.degree (a i) := Finset.prod_le_prod (fun i _ => F.leastFactor_le a i)
-    _ ≤ _ := F.joint_degree_le ρ a ha
-
-lemma splitVector_low (ρ : Representation ℂ (I → H) E) (v : E) (i : I) :
-    (∑ b ∈ Finset.univ.filter (fun b => F.degree b ^ Fintype.card I ≤ Module.finrank ℂ E),
-      F.projector (ρ.comp (MonoidHom.mulSingle (fun _ : I => H) i)) b) (F.splitVector ρ v i) =
-        F.splitVector ρ v i := by
-  unfold splitVector
-  rw [map_sum]
-  apply Finset.sum_congr rfl
-  intro a _
-  by_cases hai : F.leastFactor a = i
-  · simp only [ite_eq_left hai]
-    by_cases ha : F.jointProjector ρ a = 0
-    · simp [ha]
-    · have hb : a i ∈ Finset.univ.filter (fun b => F.degree b ^ Fintype.card I ≤ Module.finrank ℂ E) := by
-        simp only [Finset.mem_filter, Finset.mem_univ, true_and]
-        simpa only [hai] using F.leastFactor_pow_le ρ a ha
-      exact congrArg (fun f : Module.End ℂ E => f v) (F.low_fixes_joint ρ a i _ hb)
-  · simp [hai]
-
-end Family
-end Thorp.Fourier
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.JointProjectors
+
+namespace OAI
+
+universe uH uI uE uJ
+
+noncomputable section
+
+open scoped BigOperators ComplexConjugate InnerProductSpace
+open Filter Topology
+
+namespace Thorp.Fourier
+open scoped Classical
+variable {H : Type uH} {I : Type uI} {E : Type uE} [Group H] [Fintype H] [Fintype I] [AddCommGroup E] [Module ℂ E]
+
+lemma integrated_single_mul (ρ : Representation ℂ (I → H) E) (i : I) (k : H → ℂ) (f : (I → H) → ℂ) :
+    integrated (ρ.comp (MonoidHom.mulSingle (fun _ : I => H) i)) k * integrated ρ f =
+      integrated ρ (fun g => ∑ h, k h * f (Pi.mulSingle i h⁻¹ * g)) := by
+  simp only [integrated, Finset.sum_mul, Finset.mul_sum, Finset.sum_smul]
+  conv_rhs => rw [Finset.sum_comm]
+  conv_lhs => rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro h _
+  have he := Equiv.sum_comp (Equiv.mulLeft (Pi.mulSingle i h))
+    (fun g : I → H => (k h * f (Pi.mulSingle i h⁻¹ * g)) • ρ g)
+  rw [← he]
+  apply Finset.sum_congr rfl
+  intro g _
+  have hi : Pi.mulSingle i h⁻¹ = (Pi.mulSingle i h : I → H)⁻¹ :=
+    (map_inv (MonoidHom.mulSingle (fun _ : I => H) i) h).symm.symm
+  simp only [Equiv.coe_mulLeft, hi, inv_mul_cancel_left, smul_mul_smul,
+    MonoidHom.comp_apply, MonoidHom.mulSingle_apply, map_mul]
+
+end Thorp.Fourier
+
+namespace Thorp.Fourier.Family
+open scoped Classical
+variable {H : Type} [Group H] [Fintype H] (F : Family H)
+variable {I : Type} [Fintype I]
+
+lemma jointKernel_single (a : I → F.Index) (i : I) (h : H) (g : I → H) :
+    F.jointKernel a (Pi.mulSingle i h * g) =
+      F.kernel (a i) (h * g i) * ∏ j ∈ Finset.univ.erase i, F.kernel (a j) (g j) := by
+  rw [jointKernel, ← Finset.mul_prod_erase _ _ (Finset.mem_univ i)]
+  simp only [Pi.mul_apply, Pi.mulSingle_eq_same]
+  congr 1
+  apply Finset.prod_congr rfl
+  intro j hj
+  have hji := (Finset.mem_erase.mp hj).1
+  simp only [Pi.mulSingle_eq_of_ne hji, one_mul]
+
+lemma jointKernel_split (a : I → F.Index) (i : I) (g : I → H) :
+    F.jointKernel a g = F.kernel (a i) (g i) * ∏ j ∈ Finset.univ.erase i, F.kernel (a j) (g j) :=
+  (Finset.mul_prod_erase Finset.univ (fun j => F.kernel (a j) (g j)) (Finset.mem_univ i)).symm
+
+lemma jointKernel_single_convolution (a : I → F.Index) (i : I) (b : F.Index) (g : I → H) :
+    ∑ h, F.kernel b h * F.jointKernel a (Pi.mulSingle i h⁻¹ * g) =
+      if b = a i then F.jointKernel a g else 0 := by
+  simp only [F.jointKernel_single, ← mul_assoc, ← Finset.sum_mul]
+  change convolution (F.kernel b) (F.kernel (a i)) (g i) * _ = _
+  rw [F.kernel_convolution]
+  split_ifs with hb
+  · simp only [hb]
+    exact (F.jointKernel_split a i g).symm
+  · simp
+
+variable {E : Type uE} [AddCommGroup E] [Module ℂ E]
+
+lemma projector_jointProjector (ρ : Representation ℂ (I → H) E) (a : I → F.Index)
+    (i : I) (b : F.Index) :
+    F.projector (ρ.comp (MonoidHom.mulSingle (fun _ : I => H) i)) b * F.jointProjector ρ a =
+      if b = a i then F.jointProjector ρ a else 0 := by
+  rw [projector, jointProjector, integrated_single_mul]
+  simp only [F.jointKernel_single_convolution]
+  split_ifs <;> simp only [integrated, zero_smul, Finset.sum_const_zero]
+
+lemma low_fixes_joint (ρ : Representation ℂ (I → H) E) (a : I → F.Index)
+    (i : I) (s : Finset F.Index) (ha : a i ∈ s) :
+    (∑ b ∈ s, F.projector (ρ.comp (MonoidHom.mulSingle (fun _ : I => H) i)) b) * F.jointProjector ρ a =
+      F.jointProjector ρ a := by
+  rw [Finset.sum_mul]
+  simp only [F.projector_jointProjector, Finset.sum_ite_eq', ha, ↓reduceIte]
+
+end Thorp.Fourier.Family
+
+namespace Thorp.Fourier
+open scoped Classical
+
+lemma norm_sum_sq_of_orthogonal {J : Type uJ} {E : Type uE} [Fintype J]
+    [NormedAddCommGroup E] [InnerProductSpace ℂ E] (v : J → E)
+    (hv : ∀ i j, i ≠ j → ⟪v i, v j⟫_ℂ = 0) :
+    ‖∑ i, v i‖ ^ 2 = ∑ i, ‖v i‖ ^ 2 := by
+  have he : ⟪∑ i, v i, ∑ i, v i⟫_ℂ = ∑ i, ⟪v i, v i⟫_ℂ := by
+    simp only [sum_inner, inner_sum]
+    apply Finset.sum_congr rfl
+    intro i _
+    exact Finset.sum_eq_single i (fun j _ hji => hv j i hji) (by simp)
+  have hr := congrArg Complex.re he
+  simpa only [inner_self_eq_norm_sq_to_K, RCLike.ofReal_eq_complex_ofReal, ← Complex.ofReal_pow,
+    ← Complex.ofReal_sum, Complex.ofReal_re] using hr
+
+namespace Family
+variable {H I : Type} [Group H] [Fintype H] [Fintype I] [Nonempty I] (F : Family H)
+
+noncomputable def leastFactor (a : I → F.Index) : I :=
+  Classical.choose (Finset.exists_min_image Finset.univ (fun i => F.degree (a i)) Finset.univ_nonempty)
+
+lemma leastFactor_le (a : I → F.Index) (i : I) :
+    F.degree (a (F.leastFactor a)) ≤ F.degree (a i) :=
+  (Classical.choose_spec (Finset.exists_min_image Finset.univ
+    (fun i => F.degree (a i)) Finset.univ_nonempty)).2 i (Finset.mem_univ i)
+
+variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℂ E] [FiniteDimensional ℂ E]
+
+omit [Nonempty I] [FiniteDimensional ℂ E] in
+lemma jointProjector_symmetric (ρ : Representation ℂ (I → H) E)
+    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ) (a : I → F.Index) :
+    (F.jointProjector ρ a).IsSymmetric := integrated_symmetric ρ hu _ (F.jointKernel_star a)
+
+omit [Nonempty I] [FiniteDimensional ℂ E] in
+lemma jointProjector_orthogonal (ρ : Representation ℂ (I → H) E)
+    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ)
+    (a b : I → F.Index) (hab : a ≠ b) (x y : E) :
+    ⟪F.jointProjector ρ a x, F.jointProjector ρ b y⟫_ℂ = 0 := by
+  rw [F.jointProjector_symmetric ρ hu a]
+  change ⟪x, (F.jointProjector ρ a * F.jointProjector ρ b) y⟫_ℂ = 0
+  rw [F.jointProjector_mul, ite_eq_right hab]
+  simp
+
+noncomputable def splitVector (ρ : Representation ℂ (I → H) E) (v : E) (i : I) : E :=
+  ∑ a : I → F.Index, if F.leastFactor a = i then F.jointProjector ρ a v else 0
+
+omit [FiniteDimensional ℂ E] in
+lemma splitVector_sum (ρ : Representation ℂ (I → H) E) (v : E) :
+    ∑ i, F.splitVector ρ v i = v := by
+  simp only [splitVector]
+  rw [Finset.sum_comm]
+  simp only [Finset.sum_ite_eq, Finset.mem_univ, ↓reduceIte]
+  rw [← LinearMap.sum_apply, F.jointProjector_sum]
+  rfl
+
+omit [FiniteDimensional ℂ E] in
+lemma splitVector_orthogonal (ρ : Representation ℂ (I → H) E)
+    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ)
+    (v : E) (i j : I) (hij : i ≠ j) :
+    ⟪F.splitVector ρ v i, F.splitVector ρ v j⟫_ℂ = 0 := by
+  simp only [splitVector, sum_inner, inner_sum]
+  apply Finset.sum_eq_zero
+  intro a _
+  apply Finset.sum_eq_zero
+  intro b _
+  split_ifs with hai hbj
+  · exact F.jointProjector_orthogonal ρ hu b a (by intro h; subst b; exact hij (hai.symm.trans hbj)) v v
+  all_goals simp
+
+omit [FiniteDimensional ℂ E] in
+lemma splitVector_sq_sum (ρ : Representation ℂ (I → H) E)
+    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ) (v : E) :
+    ∑ i, ‖F.splitVector ρ v i‖ ^ 2 = ‖v‖ ^ 2 := by
+  rw [← norm_sum_sq_of_orthogonal _ (F.splitVector_orthogonal ρ hu v), F.splitVector_sum]
+
+lemma leastFactor_pow_le (ρ : Representation ℂ (I → H) E)
+    (a : I → F.Index) (ha : F.jointProjector ρ a ≠ 0) :
+    F.degree (a (F.leastFactor a)) ^ Fintype.card I ≤ Module.finrank ℂ E := by
+  calc
+    _ = ∏ _ : I, F.degree (a (F.leastFactor a)) := by simp
+    _ ≤ ∏ i, F.degree (a i) := Finset.prod_le_prod (fun i _ => F.leastFactor_le a i)
+    _ ≤ _ := F.joint_degree_le ρ a ha
+
+lemma splitVector_low (ρ : Representation ℂ (I → H) E) (v : E) (i : I) :
+    (∑ b ∈ Finset.univ.filter (fun b => F.degree b ^ Fintype.card I ≤ Module.finrank ℂ E),
+      F.projector (ρ.comp (MonoidHom.mulSingle (fun _ : I => H) i)) b) (F.splitVector ρ v i) =
+        F.splitVector ρ v i := by
+  unfold splitVector
+  rw [map_sum]
+  apply Finset.sum_congr rfl
+  intro a _
+  by_cases hai : F.leastFactor a = i
+  · simp only [ite_eq_left hai]
+    by_cases ha : F.jointProjector ρ a = 0
+    · simp [ha]
+    · have hb : a i ∈ Finset.univ.filter (fun b => F.degree b ^ Fintype.card I ≤ Module.finrank ℂ E) := by
+        simp only [Finset.mem_filter, Finset.mem_univ, true_and]
+        simpa only [hai] using F.leastFactor_pow_le ρ a ha
+      exact congrArg (fun f : Module.End ℂ E => f v) (F.low_fixes_joint ρ a i _ hb)
+  · simp [hai]
+
+end Family
+end Thorp.Fourier
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/Reachability.lean b/lean/OAI/Probability/ThorpShuffle/Reachability.lean
index 47591790909e0eb93ea8fef8768907cdd72ae3b1..e0cbd3f0931d90520d794a70b271b28b8ca4913f 100644
--- a/lean/OAI/Probability/ThorpShuffle/Reachability.lean
+++ b/lean/OAI/Probability/ThorpShuffle/Reachability.lean
@@ -1,224 +1,224 @@
-import OAI.Probability.ThorpShuffle.MarginalBounds
-
-namespace OAI
-
-noncomputable section
-
-open scoped BigOperators
-open Filter
-
-namespace Thorp
-
-namespace Reachability
-open Conditional
-
-theorem run_append (d s t : ℕ) (a : History d s) (b : History d t) :
-    run d (s + t) (Fin.append a b) = run d t b * run d s a := by
-  induction t with
-  | zero =>
-    have hz : Fin.append a b = a := by funext i; exact Fin.append_left a b i
-    rw [hz, run_zero, one_mul]
-    rfl
-  | succ t ih =>
-    obtain ⟨⟨c, b⟩, rfl⟩ := (Fin.snocEquiv (fun _ : Fin (t + 1) => Coins d)).surjective b
-    erw [Fin.append_snoc, run_snoc, run_snoc, ih, mul_assoc]
-
-def zeroCoins (d : ℕ) : Coins (d + 1) := fun _ => false
-
-theorem step_false (d : ℕ) : step (d + 1) (zeroCoins d) = rotate (d + 1) := by
-  ext x i
-  simp only [step, Equiv.trans_apply, Fin.consEquiv_symm_apply, pairSwitch, zeroCoins, Bool.xor_false]
-  change (rotate (d + 1)) (Fin.cons (x 0) (Fin.tail x)) i = _
-  rw [Fin.cons_self_tail]
-
-theorem run_false (d t : ℕ) :
-    run (d + 1) t (fun _ => zeroCoins d) = rotate (d + 1) ^ t := by
-  induction t with
-  | zero => simp
-  | succ t ih =>
-    rw [run_succ]
-    change step (d + 1) (zeroCoins d) * run (d + 1) t (fun _ => zeroCoins d) = _
-    rw [step_false, ih, pow_succ']
-
-def returnMonoid (d : ℕ) : Submonoid (State (d + 1)) where
-  carrier := {g | ∃ t, orderOf (rotate (d + 1)) ∣ t ∧ ∃ ω : History (d + 1) t, run (d + 1) t ω = g}
-  one_mem' := ⟨0, dvd_zero _, fun i => i.elim0, run_zero _ _⟩
-  mul_mem' := by
-    rintro g h ⟨s, hs, a, rfl⟩ ⟨t, ht, b, rfl⟩
-    exact ⟨t + s, dvd_add ht hs, Fin.append b a, run_append _ _ _ _ _⟩
-
-def returnGroup (d : ℕ) : Subgroup (State (d + 1)) :=
-  Subgroup.closure (returnMonoid d : Set (State (d + 1)))
-
-@[simp] theorem mem_returnGroup (d : ℕ) (g : State (d + 1)) :
-    g ∈ returnGroup d ↔ g ∈ returnMonoid d := by
-  change g ∈ (Subgroup.closure (returnMonoid d : Set (State (d + 1)))).toSubmonoid ↔ _
-  rw [Subgroup.closure_toSubmonoid_of_finite, Submonoid.closure_eq]
-
-theorem rotate_inv_pow (d : ℕ) :
-    rotate (d + 1) ^ (orderOf (rotate (d + 1)) - 1) = (rotate (d + 1))⁻¹ := by
-  apply eq_inv_iff_mul_eq_one.mpr
-  rw [← pow_succ, Nat.sub_add_cancel (orderOf_pos _), pow_orderOf_eq_one]
-
-theorem inverse_conj_mem (d : ℕ) {g : State (d + 1)} (hg : g ∈ returnGroup d) :
-    (rotate (d + 1))⁻¹ * g * rotate (d + 1) ∈ returnGroup d := by
-  rw [mem_returnGroup] at hg ⊢
-  rcases hg with ⟨t, ht, ω, rfl⟩
-  refine ⟨(1 + t) + (orderOf (rotate (d + 1)) - 1), ?_,
-    Fin.append (Fin.append (fun _ => zeroCoins d) ω) (fun _ => zeroCoins d), ?_⟩
-  · have hp := orderOf_pos (rotate (d + 1))
-    have he : (1 + t) + (orderOf (rotate (d + 1)) - 1) = t + orderOf (rotate (d + 1)) := by omega
-    rw [he]; exact dvd_add ht (dvd_refl _)
-  · rw [run_append, run_append, run_false, run_false, pow_one, rotate_inv_pow, mul_assoc]
-
-theorem inverse_conj_pow_mem (d t : ℕ) {g : State (d + 1)} (hg : g ∈ returnGroup d) :
-    (rotate (d + 1) ^ t)⁻¹ * g * rotate (d + 1) ^ t ∈ returnGroup d := by
-  induction t with
-  | zero => simpa using hg
-  | succ t ih =>
-    have h := inverse_conj_mem d ih
-    simpa only [pow_succ, mul_inv_rev, mul_assoc] using h
-
-theorem canceled_step_mem (d : ℕ) (c : Coins (d + 1)) :
-    (rotate (d + 1))⁻¹ * step (d + 1) c ∈ returnGroup d := by
-  rw [mem_returnGroup]
-  refine ⟨1 + (orderOf (rotate (d + 1)) - 1), ?_,
-    Fin.append (fun _ => c) (fun _ => zeroCoins d), ?_⟩
-  · rw [Nat.add_sub_of_le (orderOf_pos _)]
-  · rw [run_append, run_false, rotate_inv_pow, run_succ, run_zero, mul_one]
-
-end Reachability
-
-namespace Reachability
-open Conditional
-
-theorem first_pair_swap (d : ℕ) (s : Position d) :
-    (rotate (d + 1))⁻¹ * step (d + 1) (fun x => decide (x = s)) =
-      Equiv.swap (Fin.cons false s) (Fin.cons true s) := by
-  classical
-  apply Equiv.Perm.ext
-  intro x
-  obtain ⟨⟨b, t⟩, rfl⟩ := (splitPosition d).symm.surjective x
-  have hc : ((rotate (d + 1))⁻¹ * step (d + 1) (fun x => decide (x = s)))
-      ((splitPosition d).symm (b, t)) =
-      (splitPosition d).symm (b ^^ decide (t = s), t) := by
-    simp [step, splitPosition, pairSwitch]
-  rw [hc]
-  change (Fin.cons (b ^^ decide (t = s)) t : Position (d + 1)) =
-    Equiv.swap (Fin.cons false s) (Fin.cons true s) (Fin.cons b t)
-  by_cases ht : t = s
-  · subst t; cases b <;> simp
-  · have hf : (Fin.cons b t : Position (d + 1)) ≠ Fin.cons false s := by
-      intro h; apply ht; simpa using congrArg Fin.tail h
-    have htr : (Fin.cons b t : Position (d + 1)) ≠ Fin.cons true s := by
-      intro h; apply ht; simpa using congrArg Fin.tail h
-    simp [ht, Equiv.swap_apply_of_ne_of_ne hf htr]
-
-theorem first_pair_mem (d : ℕ) (s : Position d) :
-    Equiv.swap (Fin.cons false s) (Fin.cons true s) ∈ returnGroup d := by
-  classical
-  rw [← first_pair_swap]
-  exact canceled_step_mem d _
-
-theorem first_edge_mem (d : ℕ) (x y : Position (d + 1))
-    (h : Fin.tail x = Fin.tail y) : Equiv.swap x y ∈ returnGroup d := by
-  classical
-  rw [← Fin.cons_self_tail x, ← Fin.cons_self_tail y]
-  rw [h]
-  cases hx : x 0 <;> cases hy : y 0
-  · simpa only [Equiv.swap_self, Equiv.Perm.one_def] using (returnGroup d).one_mem
-  · exact first_pair_mem d _
-  · rw [Equiv.swap_comm]; exact first_pair_mem d _
-  · simpa only [Equiv.swap_self, Equiv.Perm.one_def] using (returnGroup d).one_mem
-
-theorem rotate_pow_apply (d t : ℕ) (x : Position d) (j : Fin d) :
-    (rotate d ^ t) x j = x ((finRotate d ^ t) j) := by
-  induction t generalizing j with
-  | zero => rfl
-  | succ t ih =>
-    rw [pow_succ', Equiv.Perm.mul_apply]
-    simp only [rotate, Equiv.piCongrLeft_apply, Equiv.symm_symm] at *
-    simp only [ih, pow_succ, Equiv.Perm.mul_apply]
-    simp
-
-theorem finRotate_pow_zero (d : ℕ) (i : Fin (d + 1)) :
-    (finRotate (d + 1) ^ i.val) 0 = i := by
-  rw [Equiv.Perm.coe_pow, ← finCycle_eq_finRotate_iterate]
-  simp
-
-theorem edge_swap_mem (d : ℕ) (i : Fin (d + 1)) (x y : Position (d + 1))
-    (h : ∀ j, j ≠ i → x j = y j) : Equiv.swap x y ∈ returnGroup d := by
-  classical
-  have ht : Fin.tail ((rotate (d + 1) ^ i.val) x) =
-      Fin.tail ((rotate (d + 1) ^ i.val) y) := by
-    funext j
-    simp only [Fin.tail, rotate_pow_apply]
-    apply h
-    intro he
-    have hz : j.succ = 0 := (finRotate (d + 1) ^ i.val).injective
-      (he.trans (finRotate_pow_zero d i).symm)
-    exact Fin.succ_ne_zero j hz
-  have hm := inverse_conj_pow_mem d i.val (first_edge_mem d _ _ ht)
-  rw [Equiv.mul_swap_eq_swap_mul, mul_assoc, inv_mul_cancel, mul_one] at hm
-  simpa using hm
-
-theorem cube_connected (d : ℕ) (R : Position d → Position d → Prop)
-    (hrefl : ∀ x, R x x) (htrans : ∀ x y z, R x y → R y z → R x z)
-    (hedge : ∀ (i : Fin d) x y, (∀ j, j ≠ i → x j = y j) → R x y)
-    (x y : Position d) : R x y := by
-  induction d with
-  | zero => have : x = y := Subsingleton.elim _ _; subst y; exact hrefl x
-  | succ d ih =>
-    let z : Position (d + 1) := Fin.cons (y 0) (Fin.tail x)
-    apply htrans x z y
-    · apply hedge 0
-      intro j hj
-      obtain ⟨j, rfl⟩ := Fin.eq_succ_of_ne_zero hj
-      rfl
-    · have hh := ih (fun a b => R (Fin.cons (y 0) a) (Fin.cons (y 0) b))
-        (fun a => hrefl _) (fun a b c hab hbc => htrans _ _ _ hab hbc)
-        (fun i a b hab => hedge i.succ _ _ (by
-          intro j
-          refine Fin.cases (fun _ => rfl) (fun j hj => ?_) j
-          exact hab j (by intro he; subst j; exact hj rfl)))
-        (Fin.tail x) (Fin.tail y)
-      simpa only [Fin.cons_self_tail] using hh
-
-theorem all_swap_mem (d : ℕ) (x y : Position (d + 1)) :
-    Equiv.swap x y ∈ returnGroup d := by
-  classical
-  apply cube_connected (d + 1) (fun x y => Equiv.swap x y ∈ returnGroup d)
-    (fun x => by simpa only [Equiv.swap_self, Equiv.Perm.one_def] using (returnGroup d).one_mem) ?_ (edge_swap_mem d) x y
-  intro a b c hab hbc
-  by_cases hac : a = c
-  · subst c; simpa only [Equiv.swap_self, Equiv.Perm.one_def] using (returnGroup d).one_mem
-  by_cases hab' : a = b
-  · subst b; exact hbc
-  by_cases hbc' : b = c
-  · subst c; exact hab
-  have hh := (returnGroup d).mul_mem ((returnGroup d).mul_mem hab hbc) hab
-  rw [Equiv.mul_swap_eq_swap_mul (Equiv.swap a b) b c, Equiv.swap_apply_right,
-    Equiv.swap_apply_of_ne_of_ne (Ne.symm hac) (Ne.symm hbc'),
-    mul_assoc, Equiv.swap_mul_self, mul_one] at hh
-  exact hh
-
-theorem returnGroup_eq_top (d : ℕ) : returnGroup d = ⊤ := by
-  apply top_unique
-  intro g hg
-  clear hg
-  induction g using Equiv.Perm.swap_induction_on with
-  | one => exact (returnGroup d).one_mem
-  | swap_mul g x y _ hg => exact (returnGroup d).mul_mem (all_swap_mem d x y) hg
-
-theorem attainable_return (d : ℕ) (g : State (d + 1)) :
-    ∃ t, orderOf (rotate (d + 1)) ∣ t ∧ ∃ ω : History (d + 1) t, run (d + 1) t ω = g := by
-  have hg : g ∈ returnGroup d := by rw [returnGroup_eq_top]; trivial
-  rwa [mem_returnGroup] at hg
-
-end Reachability
-
-end Thorp
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.MarginalBounds
+
+namespace OAI
+
+noncomputable section
+
+open scoped BigOperators
+open Filter
+
+namespace Thorp
+
+namespace Reachability
+open Conditional
+
+theorem run_append (d s t : ℕ) (a : History d s) (b : History d t) :
+    run d (s + t) (Fin.append a b) = run d t b * run d s a := by
+  induction t with
+  | zero =>
+    have hz : Fin.append a b = a := by funext i; exact Fin.append_left a b i
+    rw [hz, run_zero, one_mul]
+    rfl
+  | succ t ih =>
+    obtain ⟨⟨c, b⟩, rfl⟩ := (Fin.snocEquiv (fun _ : Fin (t + 1) => Coins d)).surjective b
+    erw [Fin.append_snoc, run_snoc, run_snoc, ih, mul_assoc]
+
+def zeroCoins (d : ℕ) : Coins (d + 1) := fun _ => false
+
+theorem step_false (d : ℕ) : step (d + 1) (zeroCoins d) = rotate (d + 1) := by
+  ext x i
+  simp only [step, Equiv.trans_apply, Fin.consEquiv_symm_apply, pairSwitch, zeroCoins, Bool.xor_false]
+  change (rotate (d + 1)) (Fin.cons (x 0) (Fin.tail x)) i = _
+  rw [Fin.cons_self_tail]
+
+theorem run_false (d t : ℕ) :
+    run (d + 1) t (fun _ => zeroCoins d) = rotate (d + 1) ^ t := by
+  induction t with
+  | zero => simp
+  | succ t ih =>
+    rw [run_succ]
+    change step (d + 1) (zeroCoins d) * run (d + 1) t (fun _ => zeroCoins d) = _
+    rw [step_false, ih, pow_succ']
+
+def returnMonoid (d : ℕ) : Submonoid (State (d + 1)) where
+  carrier := {g | ∃ t, orderOf (rotate (d + 1)) ∣ t ∧ ∃ ω : History (d + 1) t, run (d + 1) t ω = g}
+  one_mem' := ⟨0, dvd_zero _, fun i => i.elim0, run_zero _ _⟩
+  mul_mem' := by
+    rintro g h ⟨s, hs, a, rfl⟩ ⟨t, ht, b, rfl⟩
+    exact ⟨t + s, dvd_add ht hs, Fin.append b a, run_append _ _ _ _ _⟩
+
+def returnGroup (d : ℕ) : Subgroup (State (d + 1)) :=
+  Subgroup.closure (returnMonoid d : Set (State (d + 1)))
+
+@[simp] theorem mem_returnGroup (d : ℕ) (g : State (d + 1)) :
+    g ∈ returnGroup d ↔ g ∈ returnMonoid d := by
+  change g ∈ (Subgroup.closure (returnMonoid d : Set (State (d + 1)))).toSubmonoid ↔ _
+  rw [Subgroup.closure_toSubmonoid_of_finite, Submonoid.closure_eq]
+
+theorem rotate_inv_pow (d : ℕ) :
+    rotate (d + 1) ^ (orderOf (rotate (d + 1)) - 1) = (rotate (d + 1))⁻¹ := by
+  apply eq_inv_iff_mul_eq_one.mpr
+  rw [← pow_succ, Nat.sub_add_cancel (orderOf_pos _), pow_orderOf_eq_one]
+
+theorem inverse_conj_mem (d : ℕ) {g : State (d + 1)} (hg : g ∈ returnGroup d) :
+    (rotate (d + 1))⁻¹ * g * rotate (d + 1) ∈ returnGroup d := by
+  rw [mem_returnGroup] at hg ⊢
+  rcases hg with ⟨t, ht, ω, rfl⟩
+  refine ⟨(1 + t) + (orderOf (rotate (d + 1)) - 1), ?_,
+    Fin.append (Fin.append (fun _ => zeroCoins d) ω) (fun _ => zeroCoins d), ?_⟩
+  · have hp := orderOf_pos (rotate (d + 1))
+    have he : (1 + t) + (orderOf (rotate (d + 1)) - 1) = t + orderOf (rotate (d + 1)) := by omega
+    rw [he]; exact dvd_add ht (dvd_refl _)
+  · rw [run_append, run_append, run_false, run_false, pow_one, rotate_inv_pow, mul_assoc]
+
+theorem inverse_conj_pow_mem (d t : ℕ) {g : State (d + 1)} (hg : g ∈ returnGroup d) :
+    (rotate (d + 1) ^ t)⁻¹ * g * rotate (d + 1) ^ t ∈ returnGroup d := by
+  induction t with
+  | zero => simpa using hg
+  | succ t ih =>
+    have h := inverse_conj_mem d ih
+    simpa only [pow_succ, mul_inv_rev, mul_assoc] using h
+
+theorem canceled_step_mem (d : ℕ) (c : Coins (d + 1)) :
+    (rotate (d + 1))⁻¹ * step (d + 1) c ∈ returnGroup d := by
+  rw [mem_returnGroup]
+  refine ⟨1 + (orderOf (rotate (d + 1)) - 1), ?_,
+    Fin.append (fun _ => c) (fun _ => zeroCoins d), ?_⟩
+  · rw [Nat.add_sub_of_le (orderOf_pos _)]
+  · rw [run_append, run_false, rotate_inv_pow, run_succ, run_zero, mul_one]
+
+end Reachability
+
+namespace Reachability
+open Conditional
+
+theorem first_pair_swap (d : ℕ) (s : Position d) :
+    (rotate (d + 1))⁻¹ * step (d + 1) (fun x => decide (x = s)) =
+      Equiv.swap (Fin.cons false s) (Fin.cons true s) := by
+  classical
+  apply Equiv.Perm.ext
+  intro x
+  obtain ⟨⟨b, t⟩, rfl⟩ := (splitPosition d).symm.surjective x
+  have hc : ((rotate (d + 1))⁻¹ * step (d + 1) (fun x => decide (x = s)))
+      ((splitPosition d).symm (b, t)) =
+      (splitPosition d).symm (b ^^ decide (t = s), t) := by
+    simp [step, splitPosition, pairSwitch]
+  rw [hc]
+  change (Fin.cons (b ^^ decide (t = s)) t : Position (d + 1)) =
+    Equiv.swap (Fin.cons false s) (Fin.cons true s) (Fin.cons b t)
+  by_cases ht : t = s
+  · subst t; cases b <;> simp
+  · have hf : (Fin.cons b t : Position (d + 1)) ≠ Fin.cons false s := by
+      intro h; apply ht; simpa using congrArg Fin.tail h
+    have htr : (Fin.cons b t : Position (d + 1)) ≠ Fin.cons true s := by
+      intro h; apply ht; simpa using congrArg Fin.tail h
+    simp [ht, Equiv.swap_apply_of_ne_of_ne hf htr]
+
+theorem first_pair_mem (d : ℕ) (s : Position d) :
+    Equiv.swap (Fin.cons false s) (Fin.cons true s) ∈ returnGroup d := by
+  classical
+  rw [← first_pair_swap]
+  exact canceled_step_mem d _
+
+theorem first_edge_mem (d : ℕ) (x y : Position (d + 1))
+    (h : Fin.tail x = Fin.tail y) : Equiv.swap x y ∈ returnGroup d := by
+  classical
+  rw [← Fin.cons_self_tail x, ← Fin.cons_self_tail y]
+  rw [h]
+  cases hx : x 0 <;> cases hy : y 0
+  · simpa only [Equiv.swap_self, Equiv.Perm.one_def] using (returnGroup d).one_mem
+  · exact first_pair_mem d _
+  · rw [Equiv.swap_comm]; exact first_pair_mem d _
+  · simpa only [Equiv.swap_self, Equiv.Perm.one_def] using (returnGroup d).one_mem
+
+theorem rotate_pow_apply (d t : ℕ) (x : Position d) (j : Fin d) :
+    (rotate d ^ t) x j = x ((finRotate d ^ t) j) := by
+  induction t generalizing j with
+  | zero => rfl
+  | succ t ih =>
+    rw [pow_succ', Equiv.Perm.mul_apply]
+    simp only [rotate, Equiv.piCongrLeft_apply, Equiv.symm_symm] at *
+    simp only [ih, pow_succ, Equiv.Perm.mul_apply]
+    simp
+
+theorem finRotate_pow_zero (d : ℕ) (i : Fin (d + 1)) :
+    (finRotate (d + 1) ^ i.val) 0 = i := by
+  rw [Equiv.Perm.coe_pow, ← finCycle_eq_finRotate_iterate]
+  simp
+
+theorem edge_swap_mem (d : ℕ) (i : Fin (d + 1)) (x y : Position (d + 1))
+    (h : ∀ j, j ≠ i → x j = y j) : Equiv.swap x y ∈ returnGroup d := by
+  classical
+  have ht : Fin.tail ((rotate (d + 1) ^ i.val) x) =
+      Fin.tail ((rotate (d + 1) ^ i.val) y) := by
+    funext j
+    simp only [Fin.tail, rotate_pow_apply]
+    apply h
+    intro he
+    have hz : j.succ = 0 := (finRotate (d + 1) ^ i.val).injective
+      (he.trans (finRotate_pow_zero d i).symm)
+    exact Fin.succ_ne_zero j hz
+  have hm := inverse_conj_pow_mem d i.val (first_edge_mem d _ _ ht)
+  rw [Equiv.mul_swap_eq_swap_mul, mul_assoc, inv_mul_cancel, mul_one] at hm
+  simpa using hm
+
+theorem cube_connected (d : ℕ) (R : Position d → Position d → Prop)
+    (hrefl : ∀ x, R x x) (htrans : ∀ x y z, R x y → R y z → R x z)
+    (hedge : ∀ (i : Fin d) x y, (∀ j, j ≠ i → x j = y j) → R x y)
+    (x y : Position d) : R x y := by
+  induction d with
+  | zero => have : x = y := Subsingleton.elim _ _; subst y; exact hrefl x
+  | succ d ih =>
+    let z : Position (d + 1) := Fin.cons (y 0) (Fin.tail x)
+    apply htrans x z y
+    · apply hedge 0
+      intro j hj
+      obtain ⟨j, rfl⟩ := Fin.eq_succ_of_ne_zero hj
+      rfl
+    · have hh := ih (fun a b => R (Fin.cons (y 0) a) (Fin.cons (y 0) b))
+        (fun a => hrefl _) (fun a b c hab hbc => htrans _ _ _ hab hbc)
+        (fun i a b hab => hedge i.succ _ _ (by
+          intro j
+          refine Fin.cases (fun _ => rfl) (fun j hj => ?_) j
+          exact hab j (by intro he; subst j; exact hj rfl)))
+        (Fin.tail x) (Fin.tail y)
+      simpa only [Fin.cons_self_tail] using hh
+
+theorem all_swap_mem (d : ℕ) (x y : Position (d + 1)) :
+    Equiv.swap x y ∈ returnGroup d := by
+  classical
+  apply cube_connected (d + 1) (fun x y => Equiv.swap x y ∈ returnGroup d)
+    (fun x => by simpa only [Equiv.swap_self, Equiv.Perm.one_def] using (returnGroup d).one_mem) ?_ (edge_swap_mem d) x y
+  intro a b c hab hbc
+  by_cases hac : a = c
+  · subst c; simpa only [Equiv.swap_self, Equiv.Perm.one_def] using (returnGroup d).one_mem
+  by_cases hab' : a = b
+  · subst b; exact hbc
+  by_cases hbc' : b = c
+  · subst c; exact hab
+  have hh := (returnGroup d).mul_mem ((returnGroup d).mul_mem hab hbc) hab
+  rw [Equiv.mul_swap_eq_swap_mul (Equiv.swap a b) b c, Equiv.swap_apply_right,
+    Equiv.swap_apply_of_ne_of_ne (Ne.symm hac) (Ne.symm hbc'),
+    mul_assoc, Equiv.swap_mul_self, mul_one] at hh
+  exact hh
+
+theorem returnGroup_eq_top (d : ℕ) : returnGroup d = ⊤ := by
+  apply top_unique
+  intro g hg
+  clear hg
+  induction g using Equiv.Perm.swap_induction_on with
+  | one => exact (returnGroup d).one_mem
+  | swap_mul g x y _ hg => exact (returnGroup d).mul_mem (all_swap_mem d x y) hg
+
+theorem attainable_return (d : ℕ) (g : State (d + 1)) :
+    ∃ t, orderOf (rotate (d + 1)) ∣ t ∧ ∃ ω : History (d + 1) t, run (d + 1) t ω = g := by
+  have hg : g ∈ returnGroup d := by rw [returnGroup_eq_top]; trivial
+  rwa [mem_returnGroup] at hg
+
+end Reachability
+
+end Thorp
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/ReciprocalDegrees.lean b/lean/OAI/Probability/ThorpShuffle/ReciprocalDegrees.lean
index 3ec3353387c3404e380cc2f35baa73e3db8f9772..ea9f7d66380c85dcb923af4da1de85a8d90b61e6 100644
--- a/lean/OAI/Probability/ThorpShuffle/ReciprocalDegrees.lean
+++ b/lean/OAI/Probability/ThorpShuffle/ReciprocalDegrees.lean
@@ -1,230 +1,230 @@
-import OAI.Probability.ThorpShuffle.DegreeBounds
-
-namespace OAI
-
-noncomputable section
-
-open scoped BigOperators ComplexConjugate InnerProductSpace
-open Filter Topology
-
-open Filter Topology
-
-namespace Thorp.Analysis
-
-lemma sqrt_log_small (c : ℝ) (hc : 0 < c) :
-    ∀ᶠ n : ℕ in atTop, Real.log (n + 1) ≤ c * Real.sqrt (n + 1) := by
-  have ht : Tendsto (fun n : ℕ => (n : ℝ) + 1) atTop atTop := tendsto_atTop_add_const_right atTop 1 tendsto_natCast_atTop_atTop
-  have h := (isLittleO_log_rpow_atTop (by norm_num : (0 : ℝ) < 1/2)).tendsto_div_nhds_zero.comp ht
-  have he := h.eventually (gt_mem_nhds hc)
-  filter_upwards [he] with n hn
-  dsimp only [Function.comp_apply] at hn
-  rw [← Real.sqrt_eq_rpow] at hn
-  exact ((div_lt_iff₀ (Real.sqrt_pos.mpr (by positivity))).mp hn).le
-
-lemma partition_exponential_decay :
-    Tendsto (fun n : ℕ => ((n : ℝ) + 1) ^ (2 * Nat.sqrt n) / (4 : ℝ) ^ (n / 16)) atTop (𝓝 0) := by
-  have hc : 0 < Real.log 4 / 256 := by positivity
-  have he := sqrt_log_small (Real.log 4 / 256) hc
-  have ht : Tendsto (fun n : ℕ => Real.exp (-(Real.log 4 / 64) * n)) atTop (𝓝 0) := by
-    apply Real.tendsto_exp_atBot.comp
-    exact tendsto_natCast_atTop_atTop.const_mul_atTop_of_neg (neg_neg_of_pos (by positivity))
-  apply squeeze_zero' (Eventually.of_forall (fun n => by positivity)) _ ht
-  filter_upwards [he, eventually_ge_atTop 32] with n hn hn32
-  have hn1 : (1 : ℝ) ≤ n := by exact_mod_cast (show 1 ≤ n by omega)
-  have hs : (Nat.sqrt n : ℝ) ≤ Real.sqrt (n + 1) := by
-    apply Real.le_sqrt_of_sq_le
-    have hh : ((Nat.sqrt n : ℝ)^2) ≤ n := by exact_mod_cast (show (Nat.sqrt n)^2 ≤ n by simpa only [pow_two] using Nat.sqrt_le n)
-    linarith
-  have hdiv : (n : ℝ) / 32 ≤ (n / 16 : ℕ) := by
-    have hh : n ≤ 32 * (n / 16) := by omega
-    exact (div_le_iff₀ (by norm_num : (0 : ℝ) < 32)).mpr (by exact_mod_cast (by omega : n ≤ n / 16 * 32))
-  have hp : 2 * (Nat.sqrt n : ℝ) * Real.log (n + 1) ≤ (Real.log 4 / 64) * n := by
-    have hl : 0 ≤ Real.log (n + 1) := Real.log_nonneg (by linarith)
-    have h₁ := mul_le_mul_of_nonneg_right hs hl
-    have h₂ := mul_le_mul_of_nonneg_left hn (Real.sqrt_nonneg ((n : ℝ) + 1))
-    have hh := Real.sq_sqrt (by positivity : (0 : ℝ) ≤ n + 1)
-    have hc0 : 0 ≤ Real.log 4 / 256 := hc.le
-    have hsq : Real.sqrt (n + 1) * ((Real.log 4 / 256) * Real.sqrt (n + 1)) = (Real.log 4 / 256) * (n + 1) := by calc
-      _ = (Real.log 4 / 256) * (Real.sqrt (n + 1))^2 := by ring
-      _ = _ := by rw [hh]
-    rw [hsq] at h₂
-    have h₃ := mul_le_mul_of_nonneg_left (show (n : ℝ) + 1 ≤ 2 * n by linarith) hc0
-    linarith only [h₁, h₂, h₃]
-  have hq : -(Real.log 4 / 64) * n ≥
-      (2 * Nat.sqrt n : ℕ) * Real.log (n + 1) - (n / 16 : ℕ) * Real.log 4 := by
-    push_cast
-    have hh := mul_le_mul_of_nonneg_right hdiv (Real.log_nonneg (by norm_num : (1 : ℝ) ≤ 4))
-    linarith
-  calc
-    _ = Real.exp ((2 * Nat.sqrt n : ℕ) * Real.log (n + 1) - (n / 16 : ℕ) * Real.log 4) := by
-      rw [Real.exp_sub, Real.exp_nat_mul, Real.exp_nat_mul, Real.exp_log (by positivity), Real.exp_log (by norm_num : (0 : ℝ) < 4)]
-    _ ≤ _ := Real.exp_le_exp.mpr hq
-
-end Thorp.Analysis
-
-namespace Thorp.Specht
-open scoped Classical
-open Filter Topology Thorp.Young
-
-variable {n : ℕ}
-
-def inverseDegree (p : NShape n) : ℝ := ((degree p : ℝ)^2)⁻¹
-
-def slice (n u : ℕ) : ℝ := ∑ p : {p : NShape n // defect p = u}, inverseDegree p.val
-
-def smallSlice (n u : ℕ) : ℝ := if 0 < u ∧ 4 * u ≤ n then slice n u else 0
-
-def exceptionalReciprocalSum (n : ℕ) : ℝ := ∑ p : NShape n, if 0 < defect p then inverseDegree p else 0
-
-lemma inverseDegree_nonneg (p : NShape n) : 0 ≤ inverseDegree p := by unfold inverseDegree; positivity
-lemma slice_nonneg (n u : ℕ) : 0 ≤ slice n u := Finset.sum_nonneg (fun _ _ => inverseDegree_nonneg _)
-lemma smallSlice_nonneg (n u : ℕ) : 0 ≤ smallSlice n u := by unfold smallSlice; split_ifs; exact slice_nonneg _ _; rfl
-
-lemma slice_bound (n u : ℕ) (K : ℝ) (hK : 0 < K)
-    (hk : ∀ p : NShape n, defect p = u → K ≤ degree p) :
-    slice n u ≤ (2 : ℝ)^(u+1) / K^2 := by
-  have hI (p : {p : NShape n // defect p = u}) : inverseDegree p.val ≤ (K^2)⁻¹ := by
-    apply inv_anti₀ (pow_pos hK 2)
-    exact pow_le_pow_left₀ hK.le (hk p.val p.property) 2
-  calc
-    _ ≤ ∑ _ : {p : NShape n // defect p = u}, (K^2)⁻¹ := Finset.sum_le_sum (fun p _ => hI p)
-    _ = (Fintype.card {p : NShape n // defect p = u} : ℝ) * (K^2)⁻¹ := by simp
-    _ ≤ (2 : ℝ)^(u+1) * (K^2)⁻¹ := mul_le_mul_of_nonneg_right (by exact_mod_cast defect_card_bound n u) (by positivity)
-    _ = _ := rfl
-
-lemma smallSlice_bound (n u : ℕ) : smallSlice n u ≤ 2 * (2 / 9 : ℝ)^u := by
-  unfold smallSlice
-  split_ifs with h
-  · have hk : ∀ p : NShape n, defect p = u → (3 : ℝ)^u ≤ degree p := by
-      intro p hp
-      have hh := small_defect_degree p (by rw [hp]; exact h.2)
-      rw [hp] at hh
-      exact_mod_cast hh
-    have hh := slice_bound n u (3^u) (by positivity) hk
-    have he : (2 : ℝ)^(u+1) / (3^u)^2 = 2 * (2/9 :ℝ)^u := by
-      rw [pow_succ, div_pow]
-      have hp : ((3 : ℝ)^u) ^ 2 = 9^u := by rw [← pow_mul, Nat.mul_comm, pow_mul]; norm_num
-      rw [hp]
-      ring
-    exact hh.trans_eq he
-  · positivity
-
-lemma smallSlice_tendsto (u : ℕ) : Tendsto (fun n => smallSlice n u) atTop (𝓝 0) := by
-  by_cases hu : 0 < u
-  · have ht : Tendsto (fun n : ℕ => ((n - 2*u + 1 : ℕ) : ℝ)) atTop atTop := by
-      exact (tendsto_natCast_atTop_atTop : Tendsto (fun n : ℕ => (n : ℝ)) atTop atTop).comp
-        ((tendsto_add_atTop_nat 1).comp (tendsto_sub_atTop_nat (2*u)))
-    have hi : Tendsto (fun n : ℕ => (((n - 2*u + 1 : ℕ) : ℝ)^2)⁻¹) atTop (𝓝 0) := by
-      apply tendsto_inv_atTop_zero.comp
-      simpa only [pow_two] using ht.atTop_mul_atTop₀ ht
-    have he := hi.const_mul ((2 : ℝ)^(u+1))
-    rw [mul_zero] at he
-    apply squeeze_zero' (Eventually.of_forall (fun n => smallSlice_nonneg n u)) _ he
-    filter_upwards [eventually_ge_atTop (4*u)] with n hn
-    rw [smallSlice, ite_eq_left ⟨hu, hn⟩]
-    exact slice_bound n u _ (by positivity) (fun p hp => by
-      have hh := fixed_defect_degree p (by rwa [hp]) (by rw [hp]; omega)
-      rw [hp] at hh
-      exact_mod_cast hh)
-  · simp only [smallSlice, hu, false_and, ↓reduceIte]
-    exact tendsto_const_nhds
-
-lemma smallSum_tendsto : Tendsto (fun n => ∑' u, smallSlice n u) atTop (𝓝 0) := by
-  have hs : Summable (fun u : ℕ => 2 * (2/9 : ℝ)^u) :=
-    (summable_geometric_of_norm_lt_one (by norm_num : ‖(2/9 : ℝ)‖ < 1)).mul_left 2
-  have h := tendsto_tsum_of_dominated_convergence hs smallSlice_tendsto
-    (Eventually.of_forall (fun n u => by rw [Real.norm_of_nonneg (smallSlice_nonneg n u)]; exact smallSlice_bound n u))
-  simpa only [tsum_zero] using h
-
-end Thorp.Specht
-
-namespace Thorp.Specht
-open scoped Classical
-open Filter Topology Thorp.Young
-
-lemma smallSlice_as_sum (n u : ℕ) :
-    smallSlice n u = ∑ p : NShape n,
-      if defect p = u ∧ 0 < defect p ∧ 4 * defect p ≤ n then inverseDegree p else 0 := by
-  unfold smallSlice slice
-  rw [← Finset.sum_subtype (Finset.univ.filter (fun p : NShape n => defect p = u)) (by simp), Finset.sum_filter]
-  by_cases h : 0 < u ∧ 4 * u ≤ n
-  · rw [ite_eq_left h]
-    apply Finset.sum_congr rfl
-    intro p _
-    by_cases hp : defect p = u
-    · simp only [hp, h.1, h.2, true_and, ↓reduceIte]
-    · simp only [hp, false_and, ↓reduceIte]
-  · rw [ite_eq_right h]
-    apply Eq.symm
-    apply Finset.sum_eq_zero
-    intro p _
-    split_ifs with hp
-    · exact (h (by simpa only [hp.1] using hp.2)).elim
-    · rfl
-
-lemma smallSlice_support (n u : ℕ) (hu : u ∉ Finset.range (n+1)) : smallSlice n u = 0 := by
-  unfold smallSlice
-  have hh : ¬(0 < u ∧ 4*u ≤ n) := by have := (Finset.mem_range.not.mp hu); omega
-  exact ite_eq_right hh
-
-lemma smallSum_eq (n : ℕ) :
-    (∑' u, smallSlice n u) = ∑ p : NShape n,
-      if 0 < defect p ∧ 4 * defect p ≤ n then inverseDegree p else 0 := by
-  rw [tsum_eq_sum (smallSlice_support n)]
-  simp_rw [smallSlice_as_sum]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro p _
-  have hd : defect p ∈ Finset.range (n+1) := Finset.mem_range.mpr (by have := defect_add_longest p; omega)
-  have he (u : ℕ) : (if defect p = u ∧ 0 < defect p ∧ 4 * defect p ≤ n then inverseDegree p else 0) =
-      if defect p = u then (if 0 < defect p ∧ 4 * defect p ≤ n then inverseDegree p else 0) else 0 := by
-    split_ifs <;> tauto
-  simp only [he, Finset.sum_ite_eq, hd, ↓reduceIte]
-
-lemma bulkSum_bound (n : ℕ) (hn : 0 < n) :
-    (∑ p : NShape n, if n < 4 * defect p then inverseDegree p else 0) ≤
-      ((n : ℝ)+1)^(2*Nat.sqrt n) / (4:ℝ)^(n/16) := by
-  have hb (p : NShape n) : (if n < 4*defect p then inverseDegree p else 0) ≤ ((4:ℝ)^(n/16))⁻¹ := by
-    by_cases hp : n < 4*defect p
-    · rw [ite_eq_left hp]
-      have hh := bulk_degree p hn hp
-      have he : ((2 : ℝ)^(n/16))^2 = (4:ℝ)^(n/16) := by
-        rw [← pow_mul, Nat.mul_comm, pow_mul]; norm_num
-      apply inv_anti₀ (by positivity)
-      change (4:ℝ)^(n/16) ≤ (degree p : ℝ)^2
-      rw [← he]
-      exact pow_le_pow_left₀ (by positivity) hh 2
-    · rw [ite_eq_right hp]; positivity
-  have hc : Fintype.card (NShape n) ≤ (n+1)^(2*Nat.sqrt n) := by
-    rw [Fintype.card_eq_nat_card]
-    change Nat.card (Nat.Partition (Fintype.card (Fin n))) ≤ _
-    rw [Fintype.card_fin, ← Fintype.card_eq_nat_card]
-    exact partition_count_le_sqrt_pow n
-  calc
-    _ ≤ ∑ _ : NShape n, ((4:ℝ)^(n/16))⁻¹ := Finset.sum_le_sum (fun p _ => hb p)
-    _ = (Fintype.card (NShape n) : ℝ) * ((4:ℝ)^(n/16))⁻¹ := by simp
-    _ ≤ ((n:ℝ)+1)^(2*Nat.sqrt n) * ((4:ℝ)^(n/16))⁻¹ :=
-      mul_le_mul_of_nonneg_right (by exact_mod_cast hc) (by positivity)
-    _ = _ := rfl
-
-lemma exceptionalReciprocalSum_split (n : ℕ) : exceptionalReciprocalSum n =
-    (∑' u, smallSlice n u) + (∑ p : NShape n, if n < 4 * defect p then inverseDegree p else 0) := by
-  rw [smallSum_eq, ← Finset.sum_add_distrib]
-  apply Finset.sum_congr rfl
-  intro p _
-  split_ifs <;> simp_all
-  all_goals omega
-
-theorem exceptionalReciprocalSum_tendsto : Tendsto exceptionalReciprocalSum atTop (𝓝 0) := by
-  have hb : Tendsto (fun n => ∑ p : NShape n, if n < 4 * defect p then inverseDegree p else 0) atTop (𝓝 0) := by
-    apply squeeze_zero' (Eventually.of_forall (fun n => Finset.sum_nonneg (fun p _ => by split_ifs; exact inverseDegree_nonneg p; rfl)))
-      ((eventually_gt_atTop 0).mono (fun n hn => bulkSum_bound n hn)) Thorp.Analysis.partition_exponential_decay
-  have he : exceptionalReciprocalSum = (fun n => (∑' u, smallSlice n u) + ∑ p : NShape n, if n < 4 * defect p then inverseDegree p else 0) := funext exceptionalReciprocalSum_split
-  rw [he]
-  simpa only [zero_add] using smallSum_tendsto.add hb
-
-end Thorp.Specht
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.DegreeBounds
+
+namespace OAI
+
+noncomputable section
+
+open scoped BigOperators ComplexConjugate InnerProductSpace
+open Filter Topology
+
+open Filter Topology
+
+namespace Thorp.Analysis
+
+lemma sqrt_log_small (c : ℝ) (hc : 0 < c) :
+    ∀ᶠ n : ℕ in atTop, Real.log (n + 1) ≤ c * Real.sqrt (n + 1) := by
+  have ht : Tendsto (fun n : ℕ => (n : ℝ) + 1) atTop atTop := tendsto_atTop_add_const_right atTop 1 tendsto_natCast_atTop_atTop
+  have h := (isLittleO_log_rpow_atTop (by norm_num : (0 : ℝ) < 1/2)).tendsto_div_nhds_zero.comp ht
+  have he := h.eventually (gt_mem_nhds hc)
+  filter_upwards [he] with n hn
+  dsimp only [Function.comp_apply] at hn
+  rw [← Real.sqrt_eq_rpow] at hn
+  exact ((div_lt_iff₀ (Real.sqrt_pos.mpr (by positivity))).mp hn).le
+
+lemma partition_exponential_decay :
+    Tendsto (fun n : ℕ => ((n : ℝ) + 1) ^ (2 * Nat.sqrt n) / (4 : ℝ) ^ (n / 16)) atTop (𝓝 0) := by
+  have hc : 0 < Real.log 4 / 256 := by positivity
+  have he := sqrt_log_small (Real.log 4 / 256) hc
+  have ht : Tendsto (fun n : ℕ => Real.exp (-(Real.log 4 / 64) * n)) atTop (𝓝 0) := by
+    apply Real.tendsto_exp_atBot.comp
+    exact tendsto_natCast_atTop_atTop.const_mul_atTop_of_neg (neg_neg_of_pos (by positivity))
+  apply squeeze_zero' (Eventually.of_forall (fun n => by positivity)) _ ht
+  filter_upwards [he, eventually_ge_atTop 32] with n hn hn32
+  have hn1 : (1 : ℝ) ≤ n := by exact_mod_cast (show 1 ≤ n by omega)
+  have hs : (Nat.sqrt n : ℝ) ≤ Real.sqrt (n + 1) := by
+    apply Real.le_sqrt_of_sq_le
+    have hh : ((Nat.sqrt n : ℝ)^2) ≤ n := by exact_mod_cast (show (Nat.sqrt n)^2 ≤ n by simpa only [pow_two] using Nat.sqrt_le n)
+    linarith
+  have hdiv : (n : ℝ) / 32 ≤ (n / 16 : ℕ) := by
+    have hh : n ≤ 32 * (n / 16) := by omega
+    exact (div_le_iff₀ (by norm_num : (0 : ℝ) < 32)).mpr (by exact_mod_cast (by omega : n ≤ n / 16 * 32))
+  have hp : 2 * (Nat.sqrt n : ℝ) * Real.log (n + 1) ≤ (Real.log 4 / 64) * n := by
+    have hl : 0 ≤ Real.log (n + 1) := Real.log_nonneg (by linarith)
+    have h₁ := mul_le_mul_of_nonneg_right hs hl
+    have h₂ := mul_le_mul_of_nonneg_left hn (Real.sqrt_nonneg ((n : ℝ) + 1))
+    have hh := Real.sq_sqrt (by positivity : (0 : ℝ) ≤ n + 1)
+    have hc0 : 0 ≤ Real.log 4 / 256 := hc.le
+    have hsq : Real.sqrt (n + 1) * ((Real.log 4 / 256) * Real.sqrt (n + 1)) = (Real.log 4 / 256) * (n + 1) := by calc
+      _ = (Real.log 4 / 256) * (Real.sqrt (n + 1))^2 := by ring
+      _ = _ := by rw [hh]
+    rw [hsq] at h₂
+    have h₃ := mul_le_mul_of_nonneg_left (show (n : ℝ) + 1 ≤ 2 * n by linarith) hc0
+    linarith only [h₁, h₂, h₃]
+  have hq : -(Real.log 4 / 64) * n ≥
+      (2 * Nat.sqrt n : ℕ) * Real.log (n + 1) - (n / 16 : ℕ) * Real.log 4 := by
+    push_cast
+    have hh := mul_le_mul_of_nonneg_right hdiv (Real.log_nonneg (by norm_num : (1 : ℝ) ≤ 4))
+    linarith
+  calc
+    _ = Real.exp ((2 * Nat.sqrt n : ℕ) * Real.log (n + 1) - (n / 16 : ℕ) * Real.log 4) := by
+      rw [Real.exp_sub, Real.exp_nat_mul, Real.exp_nat_mul, Real.exp_log (by positivity), Real.exp_log (by norm_num : (0 : ℝ) < 4)]
+    _ ≤ _ := Real.exp_le_exp.mpr hq
+
+end Thorp.Analysis
+
+namespace Thorp.Specht
+open scoped Classical
+open Filter Topology Thorp.Young
+
+variable {n : ℕ}
+
+def inverseDegree (p : NShape n) : ℝ := ((degree p : ℝ)^2)⁻¹
+
+def slice (n u : ℕ) : ℝ := ∑ p : {p : NShape n // defect p = u}, inverseDegree p.val
+
+def smallSlice (n u : ℕ) : ℝ := if 0 < u ∧ 4 * u ≤ n then slice n u else 0
+
+def exceptionalReciprocalSum (n : ℕ) : ℝ := ∑ p : NShape n, if 0 < defect p then inverseDegree p else 0
+
+lemma inverseDegree_nonneg (p : NShape n) : 0 ≤ inverseDegree p := by unfold inverseDegree; positivity
+lemma slice_nonneg (n u : ℕ) : 0 ≤ slice n u := Finset.sum_nonneg (fun _ _ => inverseDegree_nonneg _)
+lemma smallSlice_nonneg (n u : ℕ) : 0 ≤ smallSlice n u := by unfold smallSlice; split_ifs; exact slice_nonneg _ _; rfl
+
+lemma slice_bound (n u : ℕ) (K : ℝ) (hK : 0 < K)
+    (hk : ∀ p : NShape n, defect p = u → K ≤ degree p) :
+    slice n u ≤ (2 : ℝ)^(u+1) / K^2 := by
+  have hI (p : {p : NShape n // defect p = u}) : inverseDegree p.val ≤ (K^2)⁻¹ := by
+    apply inv_anti₀ (pow_pos hK 2)
+    exact pow_le_pow_left₀ hK.le (hk p.val p.property) 2
+  calc
+    _ ≤ ∑ _ : {p : NShape n // defect p = u}, (K^2)⁻¹ := Finset.sum_le_sum (fun p _ => hI p)
+    _ = (Fintype.card {p : NShape n // defect p = u} : ℝ) * (K^2)⁻¹ := by simp
+    _ ≤ (2 : ℝ)^(u+1) * (K^2)⁻¹ := mul_le_mul_of_nonneg_right (by exact_mod_cast defect_card_bound n u) (by positivity)
+    _ = _ := rfl
+
+lemma smallSlice_bound (n u : ℕ) : smallSlice n u ≤ 2 * (2 / 9 : ℝ)^u := by
+  unfold smallSlice
+  split_ifs with h
+  · have hk : ∀ p : NShape n, defect p = u → (3 : ℝ)^u ≤ degree p := by
+      intro p hp
+      have hh := small_defect_degree p (by rw [hp]; exact h.2)
+      rw [hp] at hh
+      exact_mod_cast hh
+    have hh := slice_bound n u (3^u) (by positivity) hk
+    have he : (2 : ℝ)^(u+1) / (3^u)^2 = 2 * (2/9 :ℝ)^u := by
+      rw [pow_succ, div_pow]
+      have hp : ((3 : ℝ)^u) ^ 2 = 9^u := by rw [← pow_mul, Nat.mul_comm, pow_mul]; norm_num
+      rw [hp]
+      ring
+    exact hh.trans_eq he
+  · positivity
+
+lemma smallSlice_tendsto (u : ℕ) : Tendsto (fun n => smallSlice n u) atTop (𝓝 0) := by
+  by_cases hu : 0 < u
+  · have ht : Tendsto (fun n : ℕ => ((n - 2*u + 1 : ℕ) : ℝ)) atTop atTop := by
+      exact (tendsto_natCast_atTop_atTop : Tendsto (fun n : ℕ => (n : ℝ)) atTop atTop).comp
+        ((tendsto_add_atTop_nat 1).comp (tendsto_sub_atTop_nat (2*u)))
+    have hi : Tendsto (fun n : ℕ => (((n - 2*u + 1 : ℕ) : ℝ)^2)⁻¹) atTop (𝓝 0) := by
+      apply tendsto_inv_atTop_zero.comp
+      simpa only [pow_two] using ht.atTop_mul_atTop₀ ht
+    have he := hi.const_mul ((2 : ℝ)^(u+1))
+    rw [mul_zero] at he
+    apply squeeze_zero' (Eventually.of_forall (fun n => smallSlice_nonneg n u)) _ he
+    filter_upwards [eventually_ge_atTop (4*u)] with n hn
+    rw [smallSlice, ite_eq_left ⟨hu, hn⟩]
+    exact slice_bound n u _ (by positivity) (fun p hp => by
+      have hh := fixed_defect_degree p (by rwa [hp]) (by rw [hp]; omega)
+      rw [hp] at hh
+      exact_mod_cast hh)
+  · simp only [smallSlice, hu, false_and, ↓reduceIte]
+    exact tendsto_const_nhds
+
+lemma smallSum_tendsto : Tendsto (fun n => ∑' u, smallSlice n u) atTop (𝓝 0) := by
+  have hs : Summable (fun u : ℕ => 2 * (2/9 : ℝ)^u) :=
+    (summable_geometric_of_norm_lt_one (by norm_num : ‖(2/9 : ℝ)‖ < 1)).mul_left 2
+  have h := tendsto_tsum_of_dominated_convergence hs smallSlice_tendsto
+    (Eventually.of_forall (fun n u => by rw [Real.norm_of_nonneg (smallSlice_nonneg n u)]; exact smallSlice_bound n u))
+  simpa only [tsum_zero] using h
+
+end Thorp.Specht
+
+namespace Thorp.Specht
+open scoped Classical
+open Filter Topology Thorp.Young
+
+lemma smallSlice_as_sum (n u : ℕ) :
+    smallSlice n u = ∑ p : NShape n,
+      if defect p = u ∧ 0 < defect p ∧ 4 * defect p ≤ n then inverseDegree p else 0 := by
+  unfold smallSlice slice
+  rw [← Finset.sum_subtype (Finset.univ.filter (fun p : NShape n => defect p = u)) (by simp), Finset.sum_filter]
+  by_cases h : 0 < u ∧ 4 * u ≤ n
+  · rw [ite_eq_left h]
+    apply Finset.sum_congr rfl
+    intro p _
+    by_cases hp : defect p = u
+    · simp only [hp, h.1, h.2, true_and, ↓reduceIte]
+    · simp only [hp, false_and, ↓reduceIte]
+  · rw [ite_eq_right h]
+    apply Eq.symm
+    apply Finset.sum_eq_zero
+    intro p _
+    split_ifs with hp
+    · exact (h (by simpa only [hp.1] using hp.2)).elim
+    · rfl
+
+lemma smallSlice_support (n u : ℕ) (hu : u ∉ Finset.range (n+1)) : smallSlice n u = 0 := by
+  unfold smallSlice
+  have hh : ¬(0 < u ∧ 4*u ≤ n) := by have := (Finset.mem_range.not.mp hu); omega
+  exact ite_eq_right hh
+
+lemma smallSum_eq (n : ℕ) :
+    (∑' u, smallSlice n u) = ∑ p : NShape n,
+      if 0 < defect p ∧ 4 * defect p ≤ n then inverseDegree p else 0 := by
+  rw [tsum_eq_sum (smallSlice_support n)]
+  simp_rw [smallSlice_as_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro p _
+  have hd : defect p ∈ Finset.range (n+1) := Finset.mem_range.mpr (by have := defect_add_longest p; omega)
+  have he (u : ℕ) : (if defect p = u ∧ 0 < defect p ∧ 4 * defect p ≤ n then inverseDegree p else 0) =
+      if defect p = u then (if 0 < defect p ∧ 4 * defect p ≤ n then inverseDegree p else 0) else 0 := by
+    split_ifs <;> tauto
+  simp only [he, Finset.sum_ite_eq, hd, ↓reduceIte]
+
+lemma bulkSum_bound (n : ℕ) (hn : 0 < n) :
+    (∑ p : NShape n, if n < 4 * defect p then inverseDegree p else 0) ≤
+      ((n : ℝ)+1)^(2*Nat.sqrt n) / (4:ℝ)^(n/16) := by
+  have hb (p : NShape n) : (if n < 4*defect p then inverseDegree p else 0) ≤ ((4:ℝ)^(n/16))⁻¹ := by
+    by_cases hp : n < 4*defect p
+    · rw [ite_eq_left hp]
+      have hh := bulk_degree p hn hp
+      have he : ((2 : ℝ)^(n/16))^2 = (4:ℝ)^(n/16) := by
+        rw [← pow_mul, Nat.mul_comm, pow_mul]; norm_num
+      apply inv_anti₀ (by positivity)
+      change (4:ℝ)^(n/16) ≤ (degree p : ℝ)^2
+      rw [← he]
+      exact pow_le_pow_left₀ (by positivity) hh 2
+    · rw [ite_eq_right hp]; positivity
+  have hc : Fintype.card (NShape n) ≤ (n+1)^(2*Nat.sqrt n) := by
+    rw [Fintype.card_eq_nat_card]
+    change Nat.card (Nat.Partition (Fintype.card (Fin n))) ≤ _
+    rw [Fintype.card_fin, ← Fintype.card_eq_nat_card]
+    exact partition_count_le_sqrt_pow n
+  calc
+    _ ≤ ∑ _ : NShape n, ((4:ℝ)^(n/16))⁻¹ := Finset.sum_le_sum (fun p _ => hb p)
+    _ = (Fintype.card (NShape n) : ℝ) * ((4:ℝ)^(n/16))⁻¹ := by simp
+    _ ≤ ((n:ℝ)+1)^(2*Nat.sqrt n) * ((4:ℝ)^(n/16))⁻¹ :=
+      mul_le_mul_of_nonneg_right (by exact_mod_cast hc) (by positivity)
+    _ = _ := rfl
+
+lemma exceptionalReciprocalSum_split (n : ℕ) : exceptionalReciprocalSum n =
+    (∑' u, smallSlice n u) + (∑ p : NShape n, if n < 4 * defect p then inverseDegree p else 0) := by
+  rw [smallSum_eq, ← Finset.sum_add_distrib]
+  apply Finset.sum_congr rfl
+  intro p _
+  split_ifs <;> simp_all
+  all_goals omega
+
+theorem exceptionalReciprocalSum_tendsto : Tendsto exceptionalReciprocalSum atTop (𝓝 0) := by
+  have hb : Tendsto (fun n => ∑ p : NShape n, if n < 4 * defect p then inverseDegree p else 0) atTop (𝓝 0) := by
+    apply squeeze_zero' (Eventually.of_forall (fun n => Finset.sum_nonneg (fun p _ => by split_ifs; exact inverseDegree_nonneg p; rfl)))
+      ((eventually_gt_atTop 0).mono (fun n hn => bulkSum_bound n hn)) Thorp.Analysis.partition_exponential_decay
+  have he : exceptionalReciprocalSum = (fun n => (∑' u, smallSlice n u) + ∑ p : NShape n, if n < 4 * defect p then inverseDegree p else 0) := funext exceptionalReciprocalSum_split
+  rw [he]
+  simpa only [zero_add] using smallSum_tendsto.add hb
+
+end Thorp.Specht
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/Reynolds.lean b/lean/OAI/Probability/ThorpShuffle/Reynolds.lean
index 20ad85a87029aff63c48c59c8115b8a4520c0e87..ab2cd5858413e26aecd730f77d030dabbefc9679 100644
--- a/lean/OAI/Probability/ThorpShuffle/Reynolds.lean
+++ b/lean/OAI/Probability/ThorpShuffle/Reynolds.lean
@@ -1,183 +1,183 @@
-import OAI.Probability.ThorpShuffle.ReciprocalDegrees
-
-namespace OAI
-
-universe uG uV uW uE uI
-
-noncomputable section
-
-open scoped BigOperators ComplexConjugate InnerProductSpace
-open Filter Topology
-
-open scoped BigOperators ComplexConjugate InnerProductSpace
-
-namespace Thorp.Fourier
-open scoped Classical
-variable {G : Type uG} [Group G] [Fintype G]
-variable {V : Type uV} {W : Type uW} [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
-
-def integrated (ρ : Representation ℂ G V) (f : G → ℂ) : Module.End ℂ V :=
-  ∑ g, f g • ρ g
-
-def reynolds (ρ : Representation ℂ G V) (σ : Representation ℂ G W) (A : V →ₗ[ℂ] W) : V →ₗ[ℂ] W :=
-  (Fintype.card G : ℂ)⁻¹ • ∑ g : G, σ g ∘ₗ A ∘ₗ ρ g⁻¹
-
-lemma reynolds_intertwines (ρ : Representation ℂ G V) (σ : Representation ℂ G W) (A : V →ₗ[ℂ] W)
-    (h : G) (v : V) : reynolds ρ σ A (ρ h v) = σ h (reynolds ρ σ A v) := by
-  simp only [reynolds, LinearMap.smul_apply, LinearMap.sum_apply, LinearMap.comp_apply,
-    map_smul, map_sum]
-  congr 1
-  have he := Equiv.sum_comp (Equiv.mulLeft h) (fun g : G => σ g (A (ρ g⁻¹ (ρ h v))))
-  rw [← he]
-  apply Finset.sum_congr rfl
-  intro g _
-  change σ (h * g) (A (ρ (h * g)⁻¹ (ρ h v))) = σ h (σ g (A (ρ g⁻¹ v)))
-  simp only [mul_inv_rev, map_mul, Module.End.mul_apply]
-  congr 2
-  simp only [← Module.End.mul_apply, ← map_mul, inv_mul_cancel, map_one, Module.End.one_apply]
-
-def reynoldsMap (ρ : Representation ℂ G V) (σ : Representation ℂ G W) (A : V →ₗ[ℂ] W) :
-    Representation.IntertwiningMap ρ σ :=
-  (reynolds ρ σ A).intertwiningMap_of_isIntertwiningMap ρ σ (reynolds_intertwines ρ σ A)
-
-lemma reynolds_scalar (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
-    [FiniteDimensional ℂ V] (A : Module.End ℂ V) :
-    ∃ c : ℂ, reynolds ρ ρ A = c • LinearMap.id := by
-  obtain ⟨c, hc⟩ := (Representation.IsIrreducible.algebraMap_intertwiningMap_bijective_of_isAlgClosed
-    (ρ := ρ)).surjective (reynoldsMap ρ ρ A)
-  refine ⟨c, ?_⟩
-  have hh := congrArg Representation.IntertwiningMap.toLinearMap hc
-  exact hh.symm
-
-lemma reynolds_zero (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
-    [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ]
-    (hne : ¬ Nonempty (Representation.Equiv ρ σ)) (A : V →ₗ[ℂ] W) : reynolds ρ σ A = 0 := by
-  rcases Representation.IsIrreducible.bijective_or_eq_zero (reynoldsMap ρ σ A) with h | h
-  · exact (hne ⟨(reynoldsMap ρ σ A).ofBijective h⟩).elim
-  · exact congrArg Representation.IntertwiningMap.toLinearMap h
-
-lemma trace_reynolds (ρ : Representation ℂ G V) [FiniteDimensional ℂ V] (A : Module.End ℂ V) :
-    LinearMap.trace ℂ V (reynolds ρ ρ A) = LinearMap.trace ℂ V A := by
-  simp only [reynolds, map_smul, map_sum]
-  have hh (g : G) : LinearMap.trace ℂ V (ρ g ∘ₗ A ∘ₗ ρ g⁻¹) = LinearMap.trace ℂ V A := by
-    rw [LinearMap.trace_comp_comm']
-    rw [LinearMap.comp_assoc]
-    have he : ρ g⁻¹ ∘ₗ ρ g = LinearMap.id := by
-      change ρ g⁻¹ * ρ g = 1
-      rw [← map_mul, inv_mul_cancel, map_one]
-    rw [he, LinearMap.comp_id]
-  simp only [hh, Finset.sum_const, Finset.card_univ, nsmul_eq_mul, smul_eq_mul]
-  rw [← mul_assoc, inv_mul_cancel₀ (by exact_mod_cast Fintype.card_ne_zero), one_mul]
-
-lemma reynolds_eq_trace (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
-    [FiniteDimensional ℂ V] (hV : Module.finrank ℂ V ≠ 0) (A : Module.End ℂ V) :
-    reynolds ρ ρ A = (LinearMap.trace ℂ V A / Module.finrank ℂ V) • LinearMap.id := by
-  obtain ⟨c, hc⟩ := reynolds_scalar ρ A
-  have ht := trace_reynolds ρ A
-  rw [hc, map_smul, LinearMap.trace_id, smul_eq_mul] at ht
-  have hD : (Module.finrank ℂ V : ℂ) ≠ 0 := by exact_mod_cast hV
-  rw [hc, ← ht, mul_div_cancel_right₀ _ hD]
-
-section Unitary
-variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
-  [FiniteDimensional ℂ E]
-
-omit [Fintype G] [FiniteDimensional ℂ E] in
-lemma inner_inv (ρ : Representation ℂ G E)
-    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ) (g : G) (x y : E) :
-    ⟪x, ρ g⁻¹ y⟫_ℂ = ⟪ρ g x, y⟫_ℂ := by
-  have hh := hu g x (ρ g⁻¹ y)
-  simpa only [← Module.End.mul_apply, ← map_mul, mul_inv_cancel, map_one,
-    Module.End.one_apply] using hh.symm
-
-theorem coefficient_second_moment (ρ : Representation ℂ G E)
-    [Representation.IsIrreducible ρ]
-    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ)
-    (hE : Module.finrank ℂ E ≠ 0) (v w : E) :
-    (Fintype.card G : ℝ)⁻¹ * ∑ g : G, ‖⟪w, ρ g v⟫_ℂ‖ ^ 2 =
-      ‖v‖ ^ 2 * ‖w‖ ^ 2 / Module.finrank ℂ E := by
-  have hh := congrArg (fun A : Module.End ℂ E => ⟪w, A w⟫_ℂ)
-    (reynolds_eq_trace ρ hE (InnerProductSpace.rankOne ℂ v v).toLinearMap)
-  simp only [reynolds, LinearMap.smul_apply, LinearMap.sum_apply, LinearMap.comp_apply,
-    inner_smul_right, inner_sum, ContinuousLinearMap.coe_coe,
-    InnerProductSpace.rankOne_apply, map_smul, inner_inv ρ hu,
-    InnerProductSpace.trace_rankOne, LinearMap.id_apply] at hh
-  have hc (g : G) : ⟪ρ g v, w⟫_ℂ * ⟪w, ρ g v⟫_ℂ = (‖⟪w, ρ g v⟫_ℂ‖ ^ 2 : ℝ) := by
-    rw [← inner_conj_symm (ρ g v) w, Complex.conj_mul', Complex.ofReal_pow]
-  simp only [hc, inner_self_eq_norm_sq_to_K] at hh
-  have hr : (Fintype.card G : ℝ)⁻¹ * ∑ g : G, ‖⟪w, ρ g v⟫_ℂ‖ ^ 2 =
-      ‖v‖ ^ 2 / Module.finrank ℂ E * ‖w‖ ^ 2 := by
-    apply Complex.ofReal_injective
-    push_cast
-    simpa only [Complex.ofReal_pow, RCLike.ofReal_eq_complex_ofReal] using hh
-  simpa only [div_mul_eq_mul_div] using hr
-
-end Unitary
-end Thorp.Fourier
-
-namespace Thorp.Fourier
-open scoped Classical
-variable {G : Type uG} [Group G] [Fintype G]
-variable {V : Type uV} [AddCommGroup V] [Module ℂ V]
-
-def convolution (f k : G → ℂ) (x : G) : ℂ := ∑ g, f g * k (g⁻¹ * x)
-
-def delta : G → ℂ := fun g => if g = 1 then 1 else 0
-
-lemma integrated_delta (ρ : Representation ℂ G V) : integrated ρ delta = 1 := by
-  simp [integrated, delta]
-
-lemma integrated_sum {I : Type uI} (ρ : Representation ℂ G V) (s : Finset I) (f : I → G → ℂ) :
-    integrated ρ (∑ i ∈ s, f i) = ∑ i ∈ s, integrated ρ (f i) := by
-  simp only [integrated, Finset.sum_apply, Finset.sum_smul]
-  rw [Finset.sum_comm]
-
-lemma integrated_sub (ρ : Representation ℂ G V) (f k : G → ℂ) :
-    integrated ρ (f - k) = integrated ρ f - integrated ρ k := by
-  simp only [integrated, Pi.sub_apply, sub_smul, Finset.sum_sub_distrib]
-
-lemma integrated_convolution (ρ : Representation ℂ G V) (f k : G → ℂ) :
-    integrated ρ (convolution f k) = integrated ρ f * integrated ρ k := by
-  simp only [integrated, convolution, Finset.sum_smul, Finset.sum_mul, Finset.mul_sum]
-  rw [Finset.sum_comm]
-  conv_rhs => rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro g _
-  have he := Equiv.sum_comp (Equiv.mulLeft g) (fun x => (f g * k (g⁻¹ * x)) • ρ x)
-  rw [← he]
-  apply Finset.sum_congr rfl
-  intro h _
-  simp only [Equiv.coe_mulLeft, inv_mul_cancel_left, map_mul, smul_mul_smul]
-
-lemma integrated_central (ρ : Representation ℂ G V) (f : G → ℂ)
-    (hf : ∀ g h, f (h * g * h⁻¹) = f g) (h : G) :
-    integrated ρ f * ρ h = ρ h * integrated ρ f := by
-  rw [integrated, Finset.sum_mul, Finset.mul_sum]
-  have he := Equiv.sum_comp (Equiv.mulLeft h |>.trans (Equiv.mulRight h⁻¹))
-    (fun g => (f g • ρ g) * ρ h)
-  rw [← he]
-  apply Finset.sum_congr rfl
-  intro g _
-  simp only [Equiv.trans_apply, Equiv.coe_mulLeft, Equiv.coe_mulRight, hf,
-    smul_mul_assoc, mul_smul_comm, ← map_mul, inv_mul_cancel_right]
-
-lemma central_scalar (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
-    [FiniteDimensional ℂ V] (f : G → ℂ) (hf : ∀ g h, f (h * g * h⁻¹) = f g) :
-    ∃ c : ℂ, integrated ρ f = c • LinearMap.id := by
-  let A : Representation.IntertwiningMap ρ ρ :=
-    (integrated ρ f).intertwiningMap_of_isIntertwiningMap ρ ρ (by
-      intro g v
-      exact congrArg (fun A : Module.End ℂ V => A v) (integrated_central ρ f hf g))
-  obtain ⟨c, hc⟩ := (Representation.IsIrreducible.algebraMap_intertwiningMap_bijective_of_isAlgClosed
-    (ρ := ρ)).surjective A
-  exact ⟨c, (congrArg Representation.IntertwiningMap.toLinearMap hc).symm⟩
-
-lemma trace_integrated (ρ : Representation ℂ G V) (f : G → ℂ) :
-    LinearMap.trace ℂ V (integrated ρ f) = ∑ g, f g * ρ.character g := by
-  simp only [integrated, map_sum, map_smul, smul_eq_mul, Representation.character]
-
-end Thorp.Fourier
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.ReciprocalDegrees
+
+namespace OAI
+
+universe uG uV uW uE uI
+
+noncomputable section
+
+open scoped BigOperators ComplexConjugate InnerProductSpace
+open Filter Topology
+
+open scoped BigOperators ComplexConjugate InnerProductSpace
+
+namespace Thorp.Fourier
+open scoped Classical
+variable {G : Type uG} [Group G] [Fintype G]
+variable {V : Type uV} {W : Type uW} [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
+
+def integrated (ρ : Representation ℂ G V) (f : G → ℂ) : Module.End ℂ V :=
+  ∑ g, f g • ρ g
+
+def reynolds (ρ : Representation ℂ G V) (σ : Representation ℂ G W) (A : V →ₗ[ℂ] W) : V →ₗ[ℂ] W :=
+  (Fintype.card G : ℂ)⁻¹ • ∑ g : G, σ g ∘ₗ A ∘ₗ ρ g⁻¹
+
+lemma reynolds_intertwines (ρ : Representation ℂ G V) (σ : Representation ℂ G W) (A : V →ₗ[ℂ] W)
+    (h : G) (v : V) : reynolds ρ σ A (ρ h v) = σ h (reynolds ρ σ A v) := by
+  simp only [reynolds, LinearMap.smul_apply, LinearMap.sum_apply, LinearMap.comp_apply,
+    map_smul, map_sum]
+  congr 1
+  have he := Equiv.sum_comp (Equiv.mulLeft h) (fun g : G => σ g (A (ρ g⁻¹ (ρ h v))))
+  rw [← he]
+  apply Finset.sum_congr rfl
+  intro g _
+  change σ (h * g) (A (ρ (h * g)⁻¹ (ρ h v))) = σ h (σ g (A (ρ g⁻¹ v)))
+  simp only [mul_inv_rev, map_mul, Module.End.mul_apply]
+  congr 2
+  simp only [← Module.End.mul_apply, ← map_mul, inv_mul_cancel, map_one, Module.End.one_apply]
+
+def reynoldsMap (ρ : Representation ℂ G V) (σ : Representation ℂ G W) (A : V →ₗ[ℂ] W) :
+    Representation.IntertwiningMap ρ σ :=
+  (reynolds ρ σ A).intertwiningMap_of_isIntertwiningMap ρ σ (reynolds_intertwines ρ σ A)
+
+lemma reynolds_scalar (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
+    [FiniteDimensional ℂ V] (A : Module.End ℂ V) :
+    ∃ c : ℂ, reynolds ρ ρ A = c • LinearMap.id := by
+  obtain ⟨c, hc⟩ := (Representation.IsIrreducible.algebraMap_intertwiningMap_bijective_of_isAlgClosed
+    (ρ := ρ)).surjective (reynoldsMap ρ ρ A)
+  refine ⟨c, ?_⟩
+  have hh := congrArg Representation.IntertwiningMap.toLinearMap hc
+  exact hh.symm
+
+lemma reynolds_zero (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
+    [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ]
+    (hne : ¬ Nonempty (Representation.Equiv ρ σ)) (A : V →ₗ[ℂ] W) : reynolds ρ σ A = 0 := by
+  rcases Representation.IsIrreducible.bijective_or_eq_zero (reynoldsMap ρ σ A) with h | h
+  · exact (hne ⟨(reynoldsMap ρ σ A).ofBijective h⟩).elim
+  · exact congrArg Representation.IntertwiningMap.toLinearMap h
+
+lemma trace_reynolds (ρ : Representation ℂ G V) [FiniteDimensional ℂ V] (A : Module.End ℂ V) :
+    LinearMap.trace ℂ V (reynolds ρ ρ A) = LinearMap.trace ℂ V A := by
+  simp only [reynolds, map_smul, map_sum]
+  have hh (g : G) : LinearMap.trace ℂ V (ρ g ∘ₗ A ∘ₗ ρ g⁻¹) = LinearMap.trace ℂ V A := by
+    rw [LinearMap.trace_comp_comm']
+    rw [LinearMap.comp_assoc]
+    have he : ρ g⁻¹ ∘ₗ ρ g = LinearMap.id := by
+      change ρ g⁻¹ * ρ g = 1
+      rw [← map_mul, inv_mul_cancel, map_one]
+    rw [he, LinearMap.comp_id]
+  simp only [hh, Finset.sum_const, Finset.card_univ, nsmul_eq_mul, smul_eq_mul]
+  rw [← mul_assoc, inv_mul_cancel₀ (by exact_mod_cast Fintype.card_ne_zero), one_mul]
+
+lemma reynolds_eq_trace (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
+    [FiniteDimensional ℂ V] (hV : Module.finrank ℂ V ≠ 0) (A : Module.End ℂ V) :
+    reynolds ρ ρ A = (LinearMap.trace ℂ V A / Module.finrank ℂ V) • LinearMap.id := by
+  obtain ⟨c, hc⟩ := reynolds_scalar ρ A
+  have ht := trace_reynolds ρ A
+  rw [hc, map_smul, LinearMap.trace_id, smul_eq_mul] at ht
+  have hD : (Module.finrank ℂ V : ℂ) ≠ 0 := by exact_mod_cast hV
+  rw [hc, ← ht, mul_div_cancel_right₀ _ hD]
+
+section Unitary
+variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
+  [FiniteDimensional ℂ E]
+
+omit [Fintype G] [FiniteDimensional ℂ E] in
+lemma inner_inv (ρ : Representation ℂ G E)
+    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ) (g : G) (x y : E) :
+    ⟪x, ρ g⁻¹ y⟫_ℂ = ⟪ρ g x, y⟫_ℂ := by
+  have hh := hu g x (ρ g⁻¹ y)
+  simpa only [← Module.End.mul_apply, ← map_mul, mul_inv_cancel, map_one,
+    Module.End.one_apply] using hh.symm
+
+theorem coefficient_second_moment (ρ : Representation ℂ G E)
+    [Representation.IsIrreducible ρ]
+    (hu : ∀ g x y, ⟪ρ g x, ρ g y⟫_ℂ = ⟪x, y⟫_ℂ)
+    (hE : Module.finrank ℂ E ≠ 0) (v w : E) :
+    (Fintype.card G : ℝ)⁻¹ * ∑ g : G, ‖⟪w, ρ g v⟫_ℂ‖ ^ 2 =
+      ‖v‖ ^ 2 * ‖w‖ ^ 2 / Module.finrank ℂ E := by
+  have hh := congrArg (fun A : Module.End ℂ E => ⟪w, A w⟫_ℂ)
+    (reynolds_eq_trace ρ hE (InnerProductSpace.rankOne ℂ v v).toLinearMap)
+  simp only [reynolds, LinearMap.smul_apply, LinearMap.sum_apply, LinearMap.comp_apply,
+    inner_smul_right, inner_sum, ContinuousLinearMap.coe_coe,
+    InnerProductSpace.rankOne_apply, map_smul, inner_inv ρ hu,
+    InnerProductSpace.trace_rankOne, LinearMap.id_apply] at hh
+  have hc (g : G) : ⟪ρ g v, w⟫_ℂ * ⟪w, ρ g v⟫_ℂ = (‖⟪w, ρ g v⟫_ℂ‖ ^ 2 : ℝ) := by
+    rw [← inner_conj_symm (ρ g v) w, Complex.conj_mul', Complex.ofReal_pow]
+  simp only [hc, inner_self_eq_norm_sq_to_K] at hh
+  have hr : (Fintype.card G : ℝ)⁻¹ * ∑ g : G, ‖⟪w, ρ g v⟫_ℂ‖ ^ 2 =
+      ‖v‖ ^ 2 / Module.finrank ℂ E * ‖w‖ ^ 2 := by
+    apply Complex.ofReal_injective
+    push_cast
+    simpa only [Complex.ofReal_pow, RCLike.ofReal_eq_complex_ofReal] using hh
+  simpa only [div_mul_eq_mul_div] using hr
+
+end Unitary
+end Thorp.Fourier
+
+namespace Thorp.Fourier
+open scoped Classical
+variable {G : Type uG} [Group G] [Fintype G]
+variable {V : Type uV} [AddCommGroup V] [Module ℂ V]
+
+def convolution (f k : G → ℂ) (x : G) : ℂ := ∑ g, f g * k (g⁻¹ * x)
+
+def delta : G → ℂ := fun g => if g = 1 then 1 else 0
+
+lemma integrated_delta (ρ : Representation ℂ G V) : integrated ρ delta = 1 := by
+  simp [integrated, delta]
+
+lemma integrated_sum {I : Type uI} (ρ : Representation ℂ G V) (s : Finset I) (f : I → G → ℂ) :
+    integrated ρ (∑ i ∈ s, f i) = ∑ i ∈ s, integrated ρ (f i) := by
+  simp only [integrated, Finset.sum_apply, Finset.sum_smul]
+  rw [Finset.sum_comm]
+
+lemma integrated_sub (ρ : Representation ℂ G V) (f k : G → ℂ) :
+    integrated ρ (f - k) = integrated ρ f - integrated ρ k := by
+  simp only [integrated, Pi.sub_apply, sub_smul, Finset.sum_sub_distrib]
+
+lemma integrated_convolution (ρ : Representation ℂ G V) (f k : G → ℂ) :
+    integrated ρ (convolution f k) = integrated ρ f * integrated ρ k := by
+  simp only [integrated, convolution, Finset.sum_smul, Finset.sum_mul, Finset.mul_sum]
+  rw [Finset.sum_comm]
+  conv_rhs => rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro g _
+  have he := Equiv.sum_comp (Equiv.mulLeft g) (fun x => (f g * k (g⁻¹ * x)) • ρ x)
+  rw [← he]
+  apply Finset.sum_congr rfl
+  intro h _
+  simp only [Equiv.coe_mulLeft, inv_mul_cancel_left, map_mul, smul_mul_smul]
+
+lemma integrated_central (ρ : Representation ℂ G V) (f : G → ℂ)
+    (hf : ∀ g h, f (h * g * h⁻¹) = f g) (h : G) :
+    integrated ρ f * ρ h = ρ h * integrated ρ f := by
+  rw [integrated, Finset.sum_mul, Finset.mul_sum]
+  have he := Equiv.sum_comp (Equiv.mulLeft h |>.trans (Equiv.mulRight h⁻¹))
+    (fun g => (f g • ρ g) * ρ h)
+  rw [← he]
+  apply Finset.sum_congr rfl
+  intro g _
+  simp only [Equiv.trans_apply, Equiv.coe_mulLeft, Equiv.coe_mulRight, hf,
+    smul_mul_assoc, mul_smul_comm, ← map_mul, inv_mul_cancel_right]
+
+lemma central_scalar (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
+    [FiniteDimensional ℂ V] (f : G → ℂ) (hf : ∀ g h, f (h * g * h⁻¹) = f g) :
+    ∃ c : ℂ, integrated ρ f = c • LinearMap.id := by
+  let A : Representation.IntertwiningMap ρ ρ :=
+    (integrated ρ f).intertwiningMap_of_isIntertwiningMap ρ ρ (by
+      intro g v
+      exact congrArg (fun A : Module.End ℂ V => A v) (integrated_central ρ f hf g))
+  obtain ⟨c, hc⟩ := (Representation.IsIrreducible.algebraMap_intertwiningMap_bijective_of_isAlgClosed
+    (ρ := ρ)).surjective A
+  exact ⟨c, (congrArg Representation.IntertwiningMap.toLinearMap hc).symm⟩
+
+lemma trace_integrated (ρ : Representation ℂ G V) (f : G → ℂ) :
+    LinearMap.trace ℂ V (integrated ρ f) = ∑ g, f g * ρ.character g := by
+  simp only [integrated, map_sum, map_smul, smul_eq_mul, Representation.character]
+
+end Thorp.Fourier
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/Specht.lean b/lean/OAI/Probability/ThorpShuffle/Specht.lean
index 35b686a6066633a655c1324003760e0c36f65f69..f30bfbee8ee8e294573f32f0ddb1ac96ea3fe191 100644
--- a/lean/OAI/Probability/ThorpShuffle/Specht.lean
+++ b/lean/OAI/Probability/ThorpShuffle/Specht.lean
@@ -1,227 +1,227 @@
-import OAI.Probability.ThorpShuffle.Tableaux
-
-namespace OAI
-
-universe uα
-
-noncomputable section
-
-open scoped BigOperators ComplexConjugate InnerProductSpace
-open Filter
-
-namespace Thorp.Specht
-open scoped Classical
-
-variable {α : Type uα} [Fintype α]
-
-omit [Fintype α] in
-@[simp] lemma perm_apply_inv (g : Equiv.Perm α) (x : α) : g (g⁻¹ x) = x := g.apply_symm_apply x
-
-omit [Fintype α] in
-@[simp] lemma perm_inv_apply (g : Equiv.Perm α) (x : α) : g⁻¹ (g x) = x := g.symm_apply_apply x
-
-def Tabloid (r : α → ℕ) := {f : α → ℕ // ∃ g : Equiv.Perm α, f = r ∘ g}
-
-instance tabloidFinite (r : α → ℕ) : Finite (Tabloid r) := by
-  apply Finite.of_surjective (fun g : Equiv.Perm α => (⟨r ∘ g, g, rfl⟩ : Tabloid r))
-  rintro ⟨f, g, rfl⟩
-  exact ⟨g, rfl⟩
-
-instance tabloidFintype (r : α → ℕ) : Fintype (Tabloid r) := Fintype.ofFinite _
-
-instance tabloidAction (r : α → ℕ) : MulAction (Equiv.Perm α) (Tabloid r) where
-  smul g f := ⟨f.val ∘ ⇑(g⁻¹ : Equiv.Perm α), by
-    obtain ⟨h, hh⟩ := f.property
-    refine ⟨h * g⁻¹, ?_⟩
-    funext x
-    simp only [Function.comp_apply, hh, Equiv.Perm.mul_apply]⟩
-  one_smul f := by apply Subtype.ext; funext x; rfl
-  mul_smul g h f := by
-    apply Subtype.ext
-    funext x
-    change f.val ((g * h)⁻¹ x) = f.val (h⁻¹ (g⁻¹ x))
-    rw [mul_inv_rev, Equiv.Perm.mul_apply]
-
-omit [Fintype α] in
-@[simp] lemma tabloid_smul_apply (r : α → ℕ) (g : Equiv.Perm α) (f : Tabloid r) (x : α) :
-    (g • f).val x = f.val (g⁻¹ x) := rfl
-
-def baseTabloid (r : α → ℕ) : Tabloid r := ⟨r, 1, by funext x; rfl⟩
-
-omit [Fintype α] in
-@[simp] lemma baseTabloid_apply (r : α → ℕ) (x : α) : (baseTabloid r).val x = r x := rfl
-
-def fiberGroup (c : α → ℕ) : Subgroup (Equiv.Perm α) where
-  carrier := {g | ∀ x, c (g x) = c x}
-  one_mem' := by intro x; rfl
-  mul_mem' := by intro g h hg hh x; exact (hg (h x)).trans (hh x)
-  inv_mem' := by
-    intro g hg x
-    have hh := hg (g⁻¹ x)
-    change c (g (g.symm x)) = c (g.symm x) at hh
-    rw [g.apply_symm_apply] at hh
-    exact hh.symm
-
-omit [Fintype α] in
-lemma stabilizes_tabloid_iff (r : α → ℕ) (g : Equiv.Perm α) (f : Tabloid r) :
-    g • f = f ↔ ∀ x, f.val (g x) = f.val x := by
-  constructor
-  · intro h x
-    have hh := congrArg (fun t : Tabloid r => t.val (g x)) h
-    simpa only [tabloid_smul_apply, perm_inv_apply] using hh.symm
-  · intro h
-    apply Subtype.ext
-    funext x
-    have hh := h (g⁻¹ x)
-    simpa only [tabloid_smul_apply, perm_apply_inv] using hh.symm
-
-omit [Fintype α] in
-lemma swap_preserves {c : α → ℕ} {x y : α} (hc : c x = c y) :
-    Equiv.swap x y ∈ fiberGroup c := by
-  intro z
-  by_cases hzx : z = x
-  · subst z; simp [hc]
-  by_cases hzy : z = y
-  · subst z; simp [hc]
-  rw [Equiv.swap_apply_of_ne_of_ne hzx hzy]
-
-def complexSign : Equiv.Perm α →* ℂ where
-  toFun g := ((Equiv.Perm.sign g : ℤˣ) : ℤ)
-  map_one' := by simp
-  map_mul' := by intro g h; simp
-
-@[simp] lemma complexSign_conj (g : Equiv.Perm α) : conj (complexSign g) = complexSign g := by
-  simp [complexSign]
-
-@[simp] lemma complexSign_inv (g : Equiv.Perm α) : complexSign g⁻¹ = complexSign g := by
-  simp [complexSign]
-
-@[simp] lemma complexSign_swap (x y : α) (h : x ≠ y) : complexSign (Equiv.swap x y) = -1 := by
-  simp [complexSign, Equiv.Perm.sign_swap h]
-
-variable (D : YoungDiagram) (t : α ≃ Thorp.Young.Cells D)
-
-def tableauRow (x : α) : ℕ := (t x).val.1
-
-def tableauCol (x : α) : ℕ := (t x).val.2
-
-omit [Fintype α] in
-
-lemma row_col_free (g : fiberGroup (tableauCol D t))
-    (hg : (g : Equiv.Perm α) • baseTabloid (tableauRow D t) = baseTabloid (tableauRow D t)) : g = 1 := by
-  apply Subtype.ext
-  apply Equiv.Perm.ext
-  intro x
-  apply t.injective
-  apply Subtype.ext
-  apply Prod.ext
-  · exact (stabilizes_tabloid_iff _ _ _).mp hg x
-  · exact g.property x
-
-lemma collision_odd (f : Tabloid (tableauRow D t)) (x y : α) (hxy : x ≠ y)
-    (hc : tableauCol D t x = tableauCol D t y) (hr : f.val x = f.val y) :
-    ∃ g : fiberGroup (tableauCol D t), (g : Equiv.Perm α) • f = f ∧ complexSign (g : Equiv.Perm α) = -1 := by
-  refine ⟨⟨Equiv.swap x y, swap_preserves hc⟩, ?_, complexSign_swap x y hxy⟩
-  rw [stabilizes_tabloid_iff]
-  exact swap_preserves hr
-
-lemma collision_or_column (f : Tabloid (tableauRow D t)) :
-    (∃ a : fiberGroup (tableauCol D t), (a : Equiv.Perm α) • f = f ∧
-      complexSign (a : Equiv.Perm α) = -1) ∨
-    ∃ a : fiberGroup (tableauCol D t), f = (a : Equiv.Perm α) • baseTabloid (tableauRow D t) := by
-  classical
-  by_cases hc : ∃ x y : α, x ≠ y ∧ tableauCol D t x = tableauCol D t y ∧ f.val x = f.val y
-  · obtain ⟨x, y, hxy, hx, hy⟩ := hc
-    exact Or.inl (collision_odd D t f x y hxy hx hy)
-  right
-  let r : Thorp.Young.Cells D → ℕ := fun z => f.val (t.symm z)
-  have hi : ∀ x y : Thorp.Young.Cells D, x.val.2 = y.val.2 → r x = r y → x = y := by
-    intro x y hcol hr
-    by_contra hxy
-    apply hc
-    refine ⟨t.symm x, t.symm y, ?_, ?_, hr⟩
-    · exact fun h => hxy (t.symm.injective h)
-    · simpa only [tableauCol, Equiv.apply_symm_apply] using hcol
-  have hs : ∑ z : Thorp.Young.Cells D, r z = ∑ z : Thorp.Young.Cells D, z.val.1 := by
-    change ∑ z, f.val (t.symm z) = _
-    rw [Equiv.sum_comp t.symm]
-    obtain ⟨g, hg⟩ := f.property
-    rw [hg]
-    change ∑ x, tableauRow D t (g x) = _
-    rw [Equiv.sum_comp g]
-    exact Equiv.sum_comp t (fun z : Thorp.Young.Cells D => z.val.1)
-  obtain ⟨σ, hσc, hσr⟩ := Thorp.Young.column_sort D r hi hs
-  let a : Equiv.Perm α := (t.trans σ).trans t.symm
-  have hac : a ∈ fiberGroup (tableauCol D t) := by
-    intro x
-    change (t (t.symm (σ (t x)))).val.2 = (t x).val.2
-    simpa only [Equiv.apply_symm_apply] using hσc (t x)
-  have har (x : α) : f.val (a x) = tableauRow D t x := hσr (t x)
-  refine ⟨⟨a, hac⟩, ?_⟩
-  apply Subtype.ext
-  funext x
-  have hh := har (a⁻¹ x)
-  simpa only [tabloid_smul_apply, baseTabloid_apply, perm_apply_inv] using hh
-
-def tabloidRep : Representation ℂ (Equiv.Perm α) (EuclideanSpace ℂ (Tabloid (tableauRow D t))) :=
-  PermutationSpace.rep
-
-def columnAlternator : Module.End ℂ (EuclideanSpace ℂ (Tabloid (tableauRow D t))) :=
-  PermutationSpace.alternator (fiberGroup (tableauCol D t)).subtype
-    (complexSign.comp (fiberGroup (tableauCol D t)).subtype)
-
-def polytabloid : EuclideanSpace ℂ (Tabloid (tableauRow D t)) :=
-  columnAlternator D t (EuclideanSpace.single (baseTabloid (tableauRow D t)) (1 : ℂ))
-
-lemma polytabloid_support (f : Tabloid (tableauRow D t)) (hf : polytabloid D t f ≠ 0) :
-    ∃ h : fiberGroup (tableauCol D t), f = h.val • baseTabloid (tableauRow D t) := by
-  by_contra hn
-  push Not at hn
-  apply hf
-  change PermutationSpace.alternator (fiberGroup (tableauCol D t)).subtype
-    (complexSign.comp (fiberGroup (tableauCol D t)).subtype)
-    (EuclideanSpace.single (baseTabloid (tableauRow D t)) (1 : ℂ)) f = 0
-  rw [PermutationSpace.alternator_coeff]
-  apply Finset.sum_eq_zero
-  intro h _
-  have hne : h.val⁻¹ • f ≠ baseTabloid (tableauRow D t) := by
-    intro he
-    exact hn h (inv_smul_eq_iff.mp he)
-  rw [PiLp.single_apply]
-  change complexSign h.val * (if h.val⁻¹ • f = baseTabloid (tableauRow D t) then 1 else 0) = 0
-  rw [ite_eq_right hne, mul_zero]
-
-lemma polytabloid_diag : polytabloid D t (baseTabloid (tableauRow D t)) = 1 := by
-  exact PermutationSpace.polytabloid_base (fiberGroup (tableauCol D t)).subtype
-    (complexSign.comp (fiberGroup (tableauCol D t)).subtype)
-    (baseTabloid (tableauRow D t)) (row_col_free D t)
-
-lemma polytabloid_ne_zero : polytabloid D t ≠ 0 := by
-  have hh := PermutationSpace.polytabloid_base (fiberGroup (tableauCol D t)).subtype
-    (complexSign.comp (fiberGroup (tableauCol D t)).subtype)
-    (baseTabloid (tableauRow D t)) (row_col_free D t)
-  intro hz
-  change polytabloid D t (baseTabloid (tableauRow D t)) = 1 at hh
-  rw [hz] at hh
-  norm_num at hh
-
-lemma columnAlternator_rank_one (v : EuclideanSpace ℂ (Tabloid (tableauRow D t))) :
-    columnAlternator D t v = ⟪polytabloid D t, v⟫_ℂ • polytabloid D t := by
-  exact PermutationSpace.alternator_rank_one (fiberGroup (tableauCol D t)).subtype
-    (complexSign.comp (fiberGroup (tableauCol D t)).subtype)
-    (baseTabloid (tableauRow D t)) (fun h => complexSign_conj h.val)
-    (fun h => complexSign_inv h.val) (row_col_free D t) (collision_or_column D t) v
-
-def module : Subrepresentation (tabloidRep D t) := cyclic (tabloidRep D t) (polytabloid D t)
-
-theorem irreducible : Representation.IsIrreducible (module D t).toRepresentation := by
-  exact cyclic_irreducible (tabloidRep D t) (polytabloid D t) (polytabloid_ne_zero D t)
-    PermutationSpace.rep_unitary (columnAlternator D t) (columnAlternator_rank_one D t)
-    (PermutationSpace.alternator_stable (fiberGroup (tableauCol D t)).subtype
-      (complexSign.comp (fiberGroup (tableauCol D t)).subtype))
-
-end Thorp.Specht
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.Tableaux
+
+namespace OAI
+
+universe uα
+
+noncomputable section
+
+open scoped BigOperators ComplexConjugate InnerProductSpace
+open Filter
+
+namespace Thorp.Specht
+open scoped Classical
+
+variable {α : Type uα} [Fintype α]
+
+omit [Fintype α] in
+@[simp] lemma perm_apply_inv (g : Equiv.Perm α) (x : α) : g (g⁻¹ x) = x := g.apply_symm_apply x
+
+omit [Fintype α] in
+@[simp] lemma perm_inv_apply (g : Equiv.Perm α) (x : α) : g⁻¹ (g x) = x := g.symm_apply_apply x
+
+def Tabloid (r : α → ℕ) := {f : α → ℕ // ∃ g : Equiv.Perm α, f = r ∘ g}
+
+instance tabloidFinite (r : α → ℕ) : Finite (Tabloid r) := by
+  apply Finite.of_surjective (fun g : Equiv.Perm α => (⟨r ∘ g, g, rfl⟩ : Tabloid r))
+  rintro ⟨f, g, rfl⟩
+  exact ⟨g, rfl⟩
+
+instance tabloidFintype (r : α → ℕ) : Fintype (Tabloid r) := Fintype.ofFinite _
+
+instance tabloidAction (r : α → ℕ) : MulAction (Equiv.Perm α) (Tabloid r) where
+  smul g f := ⟨f.val ∘ ⇑(g⁻¹ : Equiv.Perm α), by
+    obtain ⟨h, hh⟩ := f.property
+    refine ⟨h * g⁻¹, ?_⟩
+    funext x
+    simp only [Function.comp_apply, hh, Equiv.Perm.mul_apply]⟩
+  one_smul f := by apply Subtype.ext; funext x; rfl
+  mul_smul g h f := by
+    apply Subtype.ext
+    funext x
+    change f.val ((g * h)⁻¹ x) = f.val (h⁻¹ (g⁻¹ x))
+    rw [mul_inv_rev, Equiv.Perm.mul_apply]
+
+omit [Fintype α] in
+@[simp] lemma tabloid_smul_apply (r : α → ℕ) (g : Equiv.Perm α) (f : Tabloid r) (x : α) :
+    (g • f).val x = f.val (g⁻¹ x) := rfl
+
+def baseTabloid (r : α → ℕ) : Tabloid r := ⟨r, 1, by funext x; rfl⟩
+
+omit [Fintype α] in
+@[simp] lemma baseTabloid_apply (r : α → ℕ) (x : α) : (baseTabloid r).val x = r x := rfl
+
+def fiberGroup (c : α → ℕ) : Subgroup (Equiv.Perm α) where
+  carrier := {g | ∀ x, c (g x) = c x}
+  one_mem' := by intro x; rfl
+  mul_mem' := by intro g h hg hh x; exact (hg (h x)).trans (hh x)
+  inv_mem' := by
+    intro g hg x
+    have hh := hg (g⁻¹ x)
+    change c (g (g.symm x)) = c (g.symm x) at hh
+    rw [g.apply_symm_apply] at hh
+    exact hh.symm
+
+omit [Fintype α] in
+lemma stabilizes_tabloid_iff (r : α → ℕ) (g : Equiv.Perm α) (f : Tabloid r) :
+    g • f = f ↔ ∀ x, f.val (g x) = f.val x := by
+  constructor
+  · intro h x
+    have hh := congrArg (fun t : Tabloid r => t.val (g x)) h
+    simpa only [tabloid_smul_apply, perm_inv_apply] using hh.symm
+  · intro h
+    apply Subtype.ext
+    funext x
+    have hh := h (g⁻¹ x)
+    simpa only [tabloid_smul_apply, perm_apply_inv] using hh.symm
+
+omit [Fintype α] in
+lemma swap_preserves {c : α → ℕ} {x y : α} (hc : c x = c y) :
+    Equiv.swap x y ∈ fiberGroup c := by
+  intro z
+  by_cases hzx : z = x
+  · subst z; simp [hc]
+  by_cases hzy : z = y
+  · subst z; simp [hc]
+  rw [Equiv.swap_apply_of_ne_of_ne hzx hzy]
+
+def complexSign : Equiv.Perm α →* ℂ where
+  toFun g := ((Equiv.Perm.sign g : ℤˣ) : ℤ)
+  map_one' := by simp
+  map_mul' := by intro g h; simp
+
+@[simp] lemma complexSign_conj (g : Equiv.Perm α) : conj (complexSign g) = complexSign g := by
+  simp [complexSign]
+
+@[simp] lemma complexSign_inv (g : Equiv.Perm α) : complexSign g⁻¹ = complexSign g := by
+  simp [complexSign]
+
+@[simp] lemma complexSign_swap (x y : α) (h : x ≠ y) : complexSign (Equiv.swap x y) = -1 := by
+  simp [complexSign, Equiv.Perm.sign_swap h]
+
+variable (D : YoungDiagram) (t : α ≃ Thorp.Young.Cells D)
+
+def tableauRow (x : α) : ℕ := (t x).val.1
+
+def tableauCol (x : α) : ℕ := (t x).val.2
+
+omit [Fintype α] in
+
+lemma row_col_free (g : fiberGroup (tableauCol D t))
+    (hg : (g : Equiv.Perm α) • baseTabloid (tableauRow D t) = baseTabloid (tableauRow D t)) : g = 1 := by
+  apply Subtype.ext
+  apply Equiv.Perm.ext
+  intro x
+  apply t.injective
+  apply Subtype.ext
+  apply Prod.ext
+  · exact (stabilizes_tabloid_iff _ _ _).mp hg x
+  · exact g.property x
+
+lemma collision_odd (f : Tabloid (tableauRow D t)) (x y : α) (hxy : x ≠ y)
+    (hc : tableauCol D t x = tableauCol D t y) (hr : f.val x = f.val y) :
+    ∃ g : fiberGroup (tableauCol D t), (g : Equiv.Perm α) • f = f ∧ complexSign (g : Equiv.Perm α) = -1 := by
+  refine ⟨⟨Equiv.swap x y, swap_preserves hc⟩, ?_, complexSign_swap x y hxy⟩
+  rw [stabilizes_tabloid_iff]
+  exact swap_preserves hr
+
+lemma collision_or_column (f : Tabloid (tableauRow D t)) :
+    (∃ a : fiberGroup (tableauCol D t), (a : Equiv.Perm α) • f = f ∧
+      complexSign (a : Equiv.Perm α) = -1) ∨
+    ∃ a : fiberGroup (tableauCol D t), f = (a : Equiv.Perm α) • baseTabloid (tableauRow D t) := by
+  classical
+  by_cases hc : ∃ x y : α, x ≠ y ∧ tableauCol D t x = tableauCol D t y ∧ f.val x = f.val y
+  · obtain ⟨x, y, hxy, hx, hy⟩ := hc
+    exact Or.inl (collision_odd D t f x y hxy hx hy)
+  right
+  let r : Thorp.Young.Cells D → ℕ := fun z => f.val (t.symm z)
+  have hi : ∀ x y : Thorp.Young.Cells D, x.val.2 = y.val.2 → r x = r y → x = y := by
+    intro x y hcol hr
+    by_contra hxy
+    apply hc
+    refine ⟨t.symm x, t.symm y, ?_, ?_, hr⟩
+    · exact fun h => hxy (t.symm.injective h)
+    · simpa only [tableauCol, Equiv.apply_symm_apply] using hcol
+  have hs : ∑ z : Thorp.Young.Cells D, r z = ∑ z : Thorp.Young.Cells D, z.val.1 := by
+    change ∑ z, f.val (t.symm z) = _
+    rw [Equiv.sum_comp t.symm]
+    obtain ⟨g, hg⟩ := f.property
+    rw [hg]
+    change ∑ x, tableauRow D t (g x) = _
+    rw [Equiv.sum_comp g]
+    exact Equiv.sum_comp t (fun z : Thorp.Young.Cells D => z.val.1)
+  obtain ⟨σ, hσc, hσr⟩ := Thorp.Young.column_sort D r hi hs
+  let a : Equiv.Perm α := (t.trans σ).trans t.symm
+  have hac : a ∈ fiberGroup (tableauCol D t) := by
+    intro x
+    change (t (t.symm (σ (t x)))).val.2 = (t x).val.2
+    simpa only [Equiv.apply_symm_apply] using hσc (t x)
+  have har (x : α) : f.val (a x) = tableauRow D t x := hσr (t x)
+  refine ⟨⟨a, hac⟩, ?_⟩
+  apply Subtype.ext
+  funext x
+  have hh := har (a⁻¹ x)
+  simpa only [tabloid_smul_apply, baseTabloid_apply, perm_apply_inv] using hh
+
+def tabloidRep : Representation ℂ (Equiv.Perm α) (EuclideanSpace ℂ (Tabloid (tableauRow D t))) :=
+  PermutationSpace.rep
+
+def columnAlternator : Module.End ℂ (EuclideanSpace ℂ (Tabloid (tableauRow D t))) :=
+  PermutationSpace.alternator (fiberGroup (tableauCol D t)).subtype
+    (complexSign.comp (fiberGroup (tableauCol D t)).subtype)
+
+def polytabloid : EuclideanSpace ℂ (Tabloid (tableauRow D t)) :=
+  columnAlternator D t (EuclideanSpace.single (baseTabloid (tableauRow D t)) (1 : ℂ))
+
+lemma polytabloid_support (f : Tabloid (tableauRow D t)) (hf : polytabloid D t f ≠ 0) :
+    ∃ h : fiberGroup (tableauCol D t), f = h.val • baseTabloid (tableauRow D t) := by
+  by_contra hn
+  push Not at hn
+  apply hf
+  change PermutationSpace.alternator (fiberGroup (tableauCol D t)).subtype
+    (complexSign.comp (fiberGroup (tableauCol D t)).subtype)
+    (EuclideanSpace.single (baseTabloid (tableauRow D t)) (1 : ℂ)) f = 0
+  rw [PermutationSpace.alternator_coeff]
+  apply Finset.sum_eq_zero
+  intro h _
+  have hne : h.val⁻¹ • f ≠ baseTabloid (tableauRow D t) := by
+    intro he
+    exact hn h (inv_smul_eq_iff.mp he)
+  rw [PiLp.single_apply]
+  change complexSign h.val * (if h.val⁻¹ • f = baseTabloid (tableauRow D t) then 1 else 0) = 0
+  rw [ite_eq_right hne, mul_zero]
+
+lemma polytabloid_diag : polytabloid D t (baseTabloid (tableauRow D t)) = 1 := by
+  exact PermutationSpace.polytabloid_base (fiberGroup (tableauCol D t)).subtype
+    (complexSign.comp (fiberGroup (tableauCol D t)).subtype)
+    (baseTabloid (tableauRow D t)) (row_col_free D t)
+
+lemma polytabloid_ne_zero : polytabloid D t ≠ 0 := by
+  have hh := PermutationSpace.polytabloid_base (fiberGroup (tableauCol D t)).subtype
+    (complexSign.comp (fiberGroup (tableauCol D t)).subtype)
+    (baseTabloid (tableauRow D t)) (row_col_free D t)
+  intro hz
+  change polytabloid D t (baseTabloid (tableauRow D t)) = 1 at hh
+  rw [hz] at hh
+  norm_num at hh
+
+lemma columnAlternator_rank_one (v : EuclideanSpace ℂ (Tabloid (tableauRow D t))) :
+    columnAlternator D t v = ⟪polytabloid D t, v⟫_ℂ • polytabloid D t := by
+  exact PermutationSpace.alternator_rank_one (fiberGroup (tableauCol D t)).subtype
+    (complexSign.comp (fiberGroup (tableauCol D t)).subtype)
+    (baseTabloid (tableauRow D t)) (fun h => complexSign_conj h.val)
+    (fun h => complexSign_inv h.val) (row_col_free D t) (collision_or_column D t) v
+
+def module : Subrepresentation (tabloidRep D t) := cyclic (tabloidRep D t) (polytabloid D t)
+
+theorem irreducible : Representation.IsIrreducible (module D t).toRepresentation := by
+  exact cyclic_irreducible (tabloidRep D t) (polytabloid D t) (polytabloid_ne_zero D t)
+    PermutationSpace.rep_unitary (columnAlternator D t) (columnAlternator_rank_one D t)
+    (PermutationSpace.alternator_stable (fiberGroup (tableauCol D t)).subtype
+      (complexSign.comp (fiberGroup (tableauCol D t)).subtype))
+
+end Thorp.Specht
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/SpectralDecay.lean b/lean/OAI/Probability/ThorpShuffle/SpectralDecay.lean
index 27b05982dc48b190212599703caf5eb43562ccf2..1f4bfc6b7cd7cf6a414816038698caa05054b7a1 100644
--- a/lean/OAI/Probability/ThorpShuffle/SpectralDecay.lean
+++ b/lean/OAI/Probability/ThorpShuffle/SpectralDecay.lean
@@ -1,147 +1,147 @@
-import OAI.Probability.ThorpShuffle.Support
-
-namespace OAI
-
-noncomputable section
-
-open scoped BigOperators
-open Filter
-
-namespace Thorp
-
-namespace Conditional
-
-theorem half_weight_sum (d : ℕ) :
-    (∑ j ∈ Finset.range d, (1 / 2 : ℝ) ^ (j + 1)) = 1 - (1 / 2 : ℝ) ^ d := by
-  induction d with
-  | zero => simp
-  | succ d ih =>
-    rw [Finset.sum_range_succ, ih, pow_succ]
-    ring
-
-theorem half_weight_geometric_identity (γ : ℝ) (d t : ℕ) (hdt : d ≤ t) :
-    (2 * γ - 1) *
-        (∑ j ∈ Finset.range d, (1 / 2 : ℝ) ^ (j + 1) * γ ^ (t - 1 - j)) =
-      γ ^ t - (1 / 2 : ℝ) ^ d * γ ^ (t - d) := by
-  induction d with
-  | zero => simp
-  | succ d ih =>
-    have hd : d ≤ t := by omega
-    have ht : t - d = (t - 1 - d) + 1 := by omega
-    have ht' : t - (d + 1) = t - 1 - d := by omega
-    rw [Finset.sum_range_succ, mul_add, ih hd, ht, ht', pow_succ, pow_succ]
-    ring
-
-theorem half_weight_geometric_bound (β γ : ℝ) (d t : ℕ)
-    (hγ : 0 ≤ γ) (hrel : 1 - β ≤ 2 * γ - 1) (hdt : d ≤ t) :
-    (1 - β) *
-        (∑ j ∈ Finset.range d, (1 / 2 : ℝ) ^ (j + 1) * γ ^ (t - 1 - j)) ≤ γ ^ t := by
-  have hs : 0 ≤ ∑ j ∈ Finset.range d,
-      (1 / 2 : ℝ) ^ (j + 1) * γ ^ (t - 1 - j) := by positivity
-  calc
-    _ ≤ (2 * γ - 1) *
-        (∑ j ∈ Finset.range d, (1 / 2 : ℝ) ^ (j + 1) * γ ^ (t - 1 - j)) :=
-      mul_le_mul_of_nonneg_right hrel hs
-    _ = γ ^ t - (1 / 2 : ℝ) ^ d * γ ^ (t - d) :=
-      half_weight_geometric_identity γ d t hdt
-    _ ≤ γ ^ t := sub_le_self _ (by positivity)
-
-theorem spectral_track_unroll (d : ℕ) (H : ℕ → ℕ → ℝ) (b : ℕ → ℝ)
-    (hstep : ∀ t j, j < d → H (t + 1) j =
-      H t (j + 1) + (1 / 2 : ℝ) ^ (j + 1) * b t)
-    (t j k : ℕ) (hjk : j + k ≤ d) (hkt : k ≤ t) :
-    H t j = H (t - k) (j + k) +
-      ∑ r ∈ Finset.range k, (1 / 2 : ℝ) ^ (j + r + 1) * b (t - 1 - r) := by
-  induction k with
-  | zero => simp
-  | succ k ih =>
-    have hk : k ≤ t := by omega
-    have hjk' : j + k ≤ d := by omega
-    rw [ih hjk' hk, Finset.sum_range_succ]
-    have ht : t - k = (t - 1 - k) + 1 := by omega
-    have hj : j + k < d := by omega
-    rw [ht, hstep _ _ hj]
-    have ht' : t - (k + 1) = t - 1 - k := by omega
-    rw [ht']
-    simp only [Nat.add_assoc]
-    ring
-
-theorem energy_of_spectral_tracks (d : ℕ) (H : ℕ → ℕ → ℝ) (b : ℕ → ℝ)
-    (hstep : ∀ t j, j < d → H (t + 1) j =
-      H t (j + 1) + (1 / 2 : ℝ) ^ (j + 1) * b t)
-    (hzero : ∀ t, H t d = 0) (t : ℕ) (ht : d ≤ t) :
-    H t 0 = ∑ j ∈ Finset.range d, (1 / 2 : ℝ) ^ (j + 1) * b (t - 1 - j) := by
-  simpa only [zero_add, hzero] using
-    spectral_track_unroll d H b hstep t 0 d (by omega) ht
-
-theorem scalar_decay (d : ℕ) (E b : ℕ → ℝ) (β γ η : ℝ)
-    (hβ : 0 < β) (hβ₁ : β ≤ 1) (hγ : 0 < γ) (hγ₁ : γ ≤ 1)
-    (hrel : 1 - β ≤ 2 * γ - 1) (hη : 0 ≤ η)
-    (hE : ∀ t, E t ≤ 1)
-    (henergy : ∀ t, d ≤ t → E t =
-      ∑ j ∈ Finset.range d, (1 / 2 : ℝ) ^ (j + 1) * b (t - 1 - j))
-    (hnoise : ∀ s, d ≤ s → b s ≤ (1 - β) * E s + η) :
-    ∀ t, E t ≤ γ ^ t / γ ^ (2 * d) + η / β := by
-  intro t
-  induction t using Nat.strong_induction_on with
-  | h t ih =>
-    by_cases ht : t ≤ 2 * d
-    · have hp : γ ^ (2 * d) ≤ γ ^ t := pow_le_pow_of_le_one hγ.le hγ₁ ht
-      have hquot : (1 : ℝ) ≤ γ ^ t / γ ^ (2 * d) :=
-        (le_div_iff₀ (pow_pos hγ _)).mpr (by simpa using hp)
-      have hc : 0 ≤ η / β := div_nonneg hη hβ.le
-      linarith [hE t]
-    · have hdt : d ≤ t := by omega
-      rw [henergy t hdt]
-      have hbound :
-          (∑ j ∈ Finset.range d, (1 / 2 : ℝ) ^ (j + 1) * b (t - 1 - j)) ≤
-          ∑ j ∈ Finset.range d, (1 / 2 : ℝ) ^ (j + 1) *
-            ((1 - β) * (γ ^ (t - 1 - j) / γ ^ (2 * d) + η / β) + η) := by
-        apply Finset.sum_le_sum
-        intro j hj
-        have hjd := Finset.mem_range.mp hj
-        have hsd : d ≤ t - 1 - j := by omega
-        have hst : t - 1 - j < t := by omega
-        apply mul_le_mul_of_nonneg_left _ (by positivity)
-        exact (hnoise _ hsd).trans
-          (add_le_add (mul_le_mul_of_nonneg_left (ih _ hst)
-            (show 0 ≤ 1 - β by linarith)) (le_refl η))
-      have hconstant : (1 - β) * (η / β) + η = η / β := by
-        field_simp
-        ring
-      have hsplit :
-          (∑ j ∈ Finset.range d, (1 / 2 : ℝ) ^ (j + 1) *
-            ((1 - β) * (γ ^ (t - 1 - j) / γ ^ (2 * d) + η / β) + η)) =
-          ((1 - β) * (∑ j ∈ Finset.range d,
-            (1 / 2 : ℝ) ^ (j + 1) * γ ^ (t - 1 - j))) / γ ^ (2 * d) +
-          (∑ j ∈ Finset.range d, (1 / 2 : ℝ) ^ (j + 1)) * (η / β) := by
-        calc
-          _ = ∑ j ∈ Finset.range d,
-              (((1 - β) * ((1 / 2 : ℝ) ^ (j + 1) * γ ^ (t - 1 - j))) /
-                γ ^ (2 * d) + (1 / 2 : ℝ) ^ (j + 1) * (η / β)) := by
-            apply Finset.sum_congr rfl
-            intro j _
-            linear_combination (1 / 2 : ℝ) ^ (j + 1) * hconstant
-          _ = _ := by
-            rw [Finset.sum_add_distrib, ← Finset.sum_div, ← Finset.mul_sum,
-              ← Finset.sum_mul]
-      rw [hsplit] at hbound
-      have hgeom := half_weight_geometric_bound β γ d t hγ.le hrel hdt
-      have hgeom' := div_le_div_of_nonneg_right hgeom (pow_nonneg hγ.le (2 * d))
-      have hc : (∑ j ∈ Finset.range d, (1 / 2 : ℝ) ^ (j + 1)) * (η / β) ≤ η / β := by
-        rw [half_weight_sum]
-        have hz : 0 ≤ (1 / 2 : ℝ) ^ d := by positivity
-        have hηβ : 0 ≤ η / β := div_nonneg hη hβ.le
-        nlinarith
-      linarith
-
-theorem eight_block_geometric_choice : (31 / 32 : ℝ) ^ 198 ≤ (1 / 2 : ℝ) ^ 8 := by
-  norm_num
-
-end Conditional
-
-end Thorp
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.Support
+
+namespace OAI
+
+noncomputable section
+
+open scoped BigOperators
+open Filter
+
+namespace Thorp
+
+namespace Conditional
+
+theorem half_weight_sum (d : ℕ) :
+    (∑ j ∈ Finset.range d, (1 / 2 : ℝ) ^ (j + 1)) = 1 - (1 / 2 : ℝ) ^ d := by
+  induction d with
+  | zero => simp
+  | succ d ih =>
+    rw [Finset.sum_range_succ, ih, pow_succ]
+    ring
+
+theorem half_weight_geometric_identity (γ : ℝ) (d t : ℕ) (hdt : d ≤ t) :
+    (2 * γ - 1) *
+        (∑ j ∈ Finset.range d, (1 / 2 : ℝ) ^ (j + 1) * γ ^ (t - 1 - j)) =
+      γ ^ t - (1 / 2 : ℝ) ^ d * γ ^ (t - d) := by
+  induction d with
+  | zero => simp
+  | succ d ih =>
+    have hd : d ≤ t := by omega
+    have ht : t - d = (t - 1 - d) + 1 := by omega
+    have ht' : t - (d + 1) = t - 1 - d := by omega
+    rw [Finset.sum_range_succ, mul_add, ih hd, ht, ht', pow_succ, pow_succ]
+    ring
+
+theorem half_weight_geometric_bound (β γ : ℝ) (d t : ℕ)
+    (hγ : 0 ≤ γ) (hrel : 1 - β ≤ 2 * γ - 1) (hdt : d ≤ t) :
+    (1 - β) *
+        (∑ j ∈ Finset.range d, (1 / 2 : ℝ) ^ (j + 1) * γ ^ (t - 1 - j)) ≤ γ ^ t := by
+  have hs : 0 ≤ ∑ j ∈ Finset.range d,
+      (1 / 2 : ℝ) ^ (j + 1) * γ ^ (t - 1 - j) := by positivity
+  calc
+    _ ≤ (2 * γ - 1) *
+        (∑ j ∈ Finset.range d, (1 / 2 : ℝ) ^ (j + 1) * γ ^ (t - 1 - j)) :=
+      mul_le_mul_of_nonneg_right hrel hs
+    _ = γ ^ t - (1 / 2 : ℝ) ^ d * γ ^ (t - d) :=
+      half_weight_geometric_identity γ d t hdt
+    _ ≤ γ ^ t := sub_le_self _ (by positivity)
+
+theorem spectral_track_unroll (d : ℕ) (H : ℕ → ℕ → ℝ) (b : ℕ → ℝ)
+    (hstep : ∀ t j, j < d → H (t + 1) j =
+      H t (j + 1) + (1 / 2 : ℝ) ^ (j + 1) * b t)
+    (t j k : ℕ) (hjk : j + k ≤ d) (hkt : k ≤ t) :
+    H t j = H (t - k) (j + k) +
+      ∑ r ∈ Finset.range k, (1 / 2 : ℝ) ^ (j + r + 1) * b (t - 1 - r) := by
+  induction k with
+  | zero => simp
+  | succ k ih =>
+    have hk : k ≤ t := by omega
+    have hjk' : j + k ≤ d := by omega
+    rw [ih hjk' hk, Finset.sum_range_succ]
+    have ht : t - k = (t - 1 - k) + 1 := by omega
+    have hj : j + k < d := by omega
+    rw [ht, hstep _ _ hj]
+    have ht' : t - (k + 1) = t - 1 - k := by omega
+    rw [ht']
+    simp only [Nat.add_assoc]
+    ring
+
+theorem energy_of_spectral_tracks (d : ℕ) (H : ℕ → ℕ → ℝ) (b : ℕ → ℝ)
+    (hstep : ∀ t j, j < d → H (t + 1) j =
+      H t (j + 1) + (1 / 2 : ℝ) ^ (j + 1) * b t)
+    (hzero : ∀ t, H t d = 0) (t : ℕ) (ht : d ≤ t) :
+    H t 0 = ∑ j ∈ Finset.range d, (1 / 2 : ℝ) ^ (j + 1) * b (t - 1 - j) := by
+  simpa only [zero_add, hzero] using
+    spectral_track_unroll d H b hstep t 0 d (by omega) ht
+
+theorem scalar_decay (d : ℕ) (E b : ℕ → ℝ) (β γ η : ℝ)
+    (hβ : 0 < β) (hβ₁ : β ≤ 1) (hγ : 0 < γ) (hγ₁ : γ ≤ 1)
+    (hrel : 1 - β ≤ 2 * γ - 1) (hη : 0 ≤ η)
+    (hE : ∀ t, E t ≤ 1)
+    (henergy : ∀ t, d ≤ t → E t =
+      ∑ j ∈ Finset.range d, (1 / 2 : ℝ) ^ (j + 1) * b (t - 1 - j))
+    (hnoise : ∀ s, d ≤ s → b s ≤ (1 - β) * E s + η) :
+    ∀ t, E t ≤ γ ^ t / γ ^ (2 * d) + η / β := by
+  intro t
+  induction t using Nat.strong_induction_on with
+  | h t ih =>
+    by_cases ht : t ≤ 2 * d
+    · have hp : γ ^ (2 * d) ≤ γ ^ t := pow_le_pow_of_le_one hγ.le hγ₁ ht
+      have hquot : (1 : ℝ) ≤ γ ^ t / γ ^ (2 * d) :=
+        (le_div_iff₀ (pow_pos hγ _)).mpr (by simpa using hp)
+      have hc : 0 ≤ η / β := div_nonneg hη hβ.le
+      linarith [hE t]
+    · have hdt : d ≤ t := by omega
+      rw [henergy t hdt]
+      have hbound :
+          (∑ j ∈ Finset.range d, (1 / 2 : ℝ) ^ (j + 1) * b (t - 1 - j)) ≤
+          ∑ j ∈ Finset.range d, (1 / 2 : ℝ) ^ (j + 1) *
+            ((1 - β) * (γ ^ (t - 1 - j) / γ ^ (2 * d) + η / β) + η) := by
+        apply Finset.sum_le_sum
+        intro j hj
+        have hjd := Finset.mem_range.mp hj
+        have hsd : d ≤ t - 1 - j := by omega
+        have hst : t - 1 - j < t := by omega
+        apply mul_le_mul_of_nonneg_left _ (by positivity)
+        exact (hnoise _ hsd).trans
+          (add_le_add (mul_le_mul_of_nonneg_left (ih _ hst)
+            (show 0 ≤ 1 - β by linarith)) (le_refl η))
+      have hconstant : (1 - β) * (η / β) + η = η / β := by
+        field_simp
+        ring
+      have hsplit :
+          (∑ j ∈ Finset.range d, (1 / 2 : ℝ) ^ (j + 1) *
+            ((1 - β) * (γ ^ (t - 1 - j) / γ ^ (2 * d) + η / β) + η)) =
+          ((1 - β) * (∑ j ∈ Finset.range d,
+            (1 / 2 : ℝ) ^ (j + 1) * γ ^ (t - 1 - j))) / γ ^ (2 * d) +
+          (∑ j ∈ Finset.range d, (1 / 2 : ℝ) ^ (j + 1)) * (η / β) := by
+        calc
+          _ = ∑ j ∈ Finset.range d,
+              (((1 - β) * ((1 / 2 : ℝ) ^ (j + 1) * γ ^ (t - 1 - j))) /
+                γ ^ (2 * d) + (1 / 2 : ℝ) ^ (j + 1) * (η / β)) := by
+            apply Finset.sum_congr rfl
+            intro j _
+            linear_combination (1 / 2 : ℝ) ^ (j + 1) * hconstant
+          _ = _ := by
+            rw [Finset.sum_add_distrib, ← Finset.sum_div, ← Finset.mul_sum,
+              ← Finset.sum_mul]
+      rw [hsplit] at hbound
+      have hgeom := half_weight_geometric_bound β γ d t hγ.le hrel hdt
+      have hgeom' := div_le_div_of_nonneg_right hgeom (pow_nonneg hγ.le (2 * d))
+      have hc : (∑ j ∈ Finset.range d, (1 / 2 : ℝ) ^ (j + 1)) * (η / β) ≤ η / β := by
+        rw [half_weight_sum]
+        have hz : 0 ≤ (1 / 2 : ℝ) ^ d := by positivity
+        have hηβ : 0 ≤ η / β := div_nonneg hη hβ.le
+        nlinarith
+      linarith
+
+theorem eight_block_geometric_choice : (31 / 32 : ℝ) ^ 198 ≤ (1 / 2 : ℝ) ^ 8 := by
+  norm_num
+
+end Conditional
+
+end Thorp
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/Support.lean b/lean/OAI/Probability/ThorpShuffle/Support.lean
index cbd8dacdd4c5b13223e62e791eae3c4786267c18..d422e36cfdeb1047fa854c65370f367a734d41ee 100644
--- a/lean/OAI/Probability/ThorpShuffle/Support.lean
+++ b/lean/OAI/Probability/ThorpShuffle/Support.lean
@@ -1,145 +1,145 @@
-import OAI.Probability.ThorpShuffle.Walsh
-
-namespace OAI
-
-universe uΩ uα
-
-noncomputable section
-
-open scoped BigOperators
-open Filter
-
-namespace Thorp
-
-theorem fairMass_nonneg {Ω : Type uΩ} {α : Type uα} [Fintype Ω] (f : Ω → α) (a : α) :
-    0 ≤ fairMass f a := by
-  classical
-  exact div_nonneg (Finset.sum_nonneg (fun _ _ => by split_ifs <;> norm_num))
-    (Nat.cast_nonneg _)
-
-theorem fairMass_sum {Ω : Type uΩ} {α : Type uα} [Fintype Ω] [Nonempty Ω] [Fintype α]
-    (f : Ω → α) : ∑ a, fairMass f a = 1 := by
-  classical
-  unfold fairMass
-  rw [← Finset.sum_div, Finset.sum_comm]
-  simp [Fintype.card_ne_zero]
-
-theorem sum_sub_le_tv {α : Type uα} [Fintype α] [DecidableEq α]
-    (μ ν : α → ℝ) (hm : ∑ a, μ a = ∑ a, ν a) (s : Finset α) :
-    (∑ a ∈ s, (μ a - ν a)) ≤ tv μ ν := by
-  have h0 : ∑ a, (μ a - ν a) = 0 := by rw [Finset.sum_sub_distrib, hm, sub_self]
-  have hsplit := s.sum_add_sum_compl (fun a => μ a - ν a)
-  have hsplitabs := s.sum_add_sum_compl (fun a => |μ a - ν a|)
-  have h₁ : (∑ a ∈ s, (μ a - ν a)) ≤ ∑ a ∈ s, |μ a - ν a| :=
-    Finset.sum_le_sum (fun a _ => le_abs_self (μ a - ν a))
-  have h₂ : -(∑ a ∈ sᶜ, (μ a - ν a)) ≤ ∑ a ∈ sᶜ, |μ a - ν a| := by
-    rw [← Finset.sum_neg_distrib]
-    exact Finset.sum_le_sum (fun a _ => neg_le_abs (μ a - ν a))
-  unfold tv
-  linarith
-
-theorem fairMass_tv_lower {Ω : Type uΩ} {α : Type uα} [Fintype Ω] [Nonempty Ω]
-    [Fintype α] [Nonempty α] (f : Ω → α) :
-    1 - (Fintype.card Ω : ℝ) / Fintype.card α ≤ tv (fairMass f)
-      (fun _ => 1 / Fintype.card α) := by
-  classical
-  let s : Finset α := Finset.univ.image f
-  have hm : ∑ a, fairMass f a = ∑ a : α, (1 / Fintype.card α : ℝ) := by
-    rw [fairMass_sum]
-    simp [Fintype.card_ne_zero]
-  have hz (a : α) (ha : a ∉ s) : fairMass f a = 0 := by
-    unfold fairMass
-    have hne (ω : Ω) : f ω ≠ a := by
-      intro h
-      apply ha
-      exact Finset.mem_image.mpr ⟨ω, Finset.mem_univ _, h⟩
-    simp [hne]
-  have hs : ∑ a ∈ s, fairMass f a = 1 := by
-    rw [← fairMass_sum f]
-    exact Finset.sum_subset (Finset.subset_univ _) (fun a _ ha => hz a ha)
-  have hbound := sum_sub_le_tv (fairMass f) (fun _ => 1 / Fintype.card α) hm s
-  rw [Finset.sum_sub_distrib, hs] at hbound
-  simp only [Finset.sum_const, nsmul_eq_mul, mul_one_div] at hbound
-  have hc : (s.card : ℝ) ≤ Fintype.card Ω := by
-    exact_mod_cast (Finset.card_image_le (s := Finset.univ) (f := f))
-  have hd : 0 ≤ (Fintype.card α : ℝ) := Nat.cast_nonneg _
-  have hdiv := div_le_div_of_nonneg_right hc hd
-  linarith
-
-@[simp] theorem card_position (d : ℕ) : Fintype.card (Position d) = 2 ^ d := by
-  simp [Position]
-
-@[simp] theorem card_state (d : ℕ) : Fintype.card (State d) = (2 ^ d).factorial := by
-  classical
-  rw [Fintype.card_perm, card_position]
-
-@[simp] theorem card_coins_succ (d : ℕ) : Fintype.card (Coins (d + 1)) = 2 ^ (2 ^ d) := by
-  simp [Coins]
-
-@[simp] theorem card_history_succ (d t : ℕ) :
-    Fintype.card (History (d + 1) t) = 2 ^ (2 ^ d * t) := by
-  simp [History, ← pow_mul]
-
-theorem distance_support_lower (d t : ℕ) :
-    1 - (2 : ℝ) ^ (2 ^ d * t) / ((2 ^ (d + 1)).factorial : ℝ) ≤
-      distance (d + 1) t := by
-  let : Nonempty (State (d + 1)) := ⟨Equiv.refl _⟩
-  have h := fairMass_tv_lower (run (d + 1) t)
-  change 1 - _ / _ ≤ tv (fairMass (run (d + 1) t))
-    (fun _ => 1 / Fintype.card (State (d + 1)))
-  simpa only [card_history_succ, card_state, Nat.cast_pow, Nat.cast_ofNat] using h
-
-theorem worst_start_eq (d t : ℕ) (g₀ : State d) :
-    tv (lawFrom d t g₀) (uniform d) = distance d t := by
-  classical
-  have hmass (g : State d) : lawFrom d t g₀ (g * g₀) = law d t g := by
-    unfold lawFrom law fairMass
-    congr 1
-    apply Finset.sum_congr rfl
-    intro ω _
-    by_cases h : run d t ω = g <;> simp [h]
-  unfold tv distance
-  congr 1
-  have h := Equiv.sum_comp (Equiv.mulRight g₀)
-    (fun g : State d => |lawFrom d t g₀ g - uniform d g|)
-  change (∑ g, |lawFrom d t g₀ (g * g₀) - (1 / Fintype.card (State d) : ℝ)|) = _ at h
-  simp only [hmass] at h
-  exact h.symm
-
-theorem supportThreshold_le_of_distance {d t : ℕ} (hd : 1 ≤ d)
-    (ht : distance d t ≤ 1 / 4) : supportThreshold d ≤ (t : ℤ) := by
-  obtain ⟨k, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : d ≠ 0)
-  have hfact : (0 : ℝ) < ((2 ^ (k + 1)).factorial : ℝ) := by
-    exact_mod_cast Nat.factorial_pos _
-  have hlower := distance_support_lower k t
-  have hdiv : (3 : ℝ) / 4 ≤ 2 ^ (2 ^ k * t) / ((2 ^ (k + 1)).factorial : ℝ) := by
-    linarith
-  have hpow : 3 * ((2 ^ (k + 1)).factorial : ℝ) / 4 ≤ (2 : ℝ) ^ (2 ^ k * t) := by
-    have h := (le_div_iff₀ hfact).mp hdiv
-    nlinarith
-  have hlog := (Real.logb_le_logb (by norm_num : (1 : ℝ) < 2)
-    (by positivity : (0 : ℝ) < 3 * ((2 ^ (k + 1)).factorial : ℝ) / 4)
-    (by positivity : (0 : ℝ) < 2 ^ (2 ^ k * t))).mpr hpow
-  rw [Real.logb_pow, Real.logb_self_eq_one (by norm_num)] at hlog
-  simp only [mul_one] at hlog
-  unfold supportThreshold
-  apply Int.ceil_le.mpr
-  push_cast
-  have hden : (0 : ℝ) < 2 ^ (k + 1) := by positivity
-  have hscale := mul_le_mul_of_nonneg_left hlog (by positivity : (0 : ℝ) ≤ 2 / 2 ^ (k + 1))
-  have hid : (2 / (2 : ℝ) ^ (k + 1)) * ((2 ^ k * t : ℕ) : ℝ) = t := by
-    push_cast
-    rw [pow_succ]
-    field_simp
-  rwa [hid] at hscale
-
-theorem supportThreshold_le_mixingTime {d : ℕ} (hd : 1 ≤ d)
-    (hex : ∃ t, distance d t ≤ 1 / 4) : supportThreshold d ≤ (mixingTime d : ℤ) := by
-  apply supportThreshold_le_of_distance hd
-  exact Nat.sInf_mem hex
-
-end Thorp
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.Walsh
+
+namespace OAI
+
+universe uΩ uα
+
+noncomputable section
+
+open scoped BigOperators
+open Filter
+
+namespace Thorp
+
+theorem fairMass_nonneg {Ω : Type uΩ} {α : Type uα} [Fintype Ω] (f : Ω → α) (a : α) :
+    0 ≤ fairMass f a := by
+  classical
+  exact div_nonneg (Finset.sum_nonneg (fun _ _ => by split_ifs <;> norm_num))
+    (Nat.cast_nonneg _)
+
+theorem fairMass_sum {Ω : Type uΩ} {α : Type uα} [Fintype Ω] [Nonempty Ω] [Fintype α]
+    (f : Ω → α) : ∑ a, fairMass f a = 1 := by
+  classical
+  unfold fairMass
+  rw [← Finset.sum_div, Finset.sum_comm]
+  simp [Fintype.card_ne_zero]
+
+theorem sum_sub_le_tv {α : Type uα} [Fintype α] [DecidableEq α]
+    (μ ν : α → ℝ) (hm : ∑ a, μ a = ∑ a, ν a) (s : Finset α) :
+    (∑ a ∈ s, (μ a - ν a)) ≤ tv μ ν := by
+  have h0 : ∑ a, (μ a - ν a) = 0 := by rw [Finset.sum_sub_distrib, hm, sub_self]
+  have hsplit := s.sum_add_sum_compl (fun a => μ a - ν a)
+  have hsplitabs := s.sum_add_sum_compl (fun a => |μ a - ν a|)
+  have h₁ : (∑ a ∈ s, (μ a - ν a)) ≤ ∑ a ∈ s, |μ a - ν a| :=
+    Finset.sum_le_sum (fun a _ => le_abs_self (μ a - ν a))
+  have h₂ : -(∑ a ∈ sᶜ, (μ a - ν a)) ≤ ∑ a ∈ sᶜ, |μ a - ν a| := by
+    rw [← Finset.sum_neg_distrib]
+    exact Finset.sum_le_sum (fun a _ => neg_le_abs (μ a - ν a))
+  unfold tv
+  linarith
+
+theorem fairMass_tv_lower {Ω : Type uΩ} {α : Type uα} [Fintype Ω] [Nonempty Ω]
+    [Fintype α] [Nonempty α] (f : Ω → α) :
+    1 - (Fintype.card Ω : ℝ) / Fintype.card α ≤ tv (fairMass f)
+      (fun _ => 1 / Fintype.card α) := by
+  classical
+  let s : Finset α := Finset.univ.image f
+  have hm : ∑ a, fairMass f a = ∑ a : α, (1 / Fintype.card α : ℝ) := by
+    rw [fairMass_sum]
+    simp [Fintype.card_ne_zero]
+  have hz (a : α) (ha : a ∉ s) : fairMass f a = 0 := by
+    unfold fairMass
+    have hne (ω : Ω) : f ω ≠ a := by
+      intro h
+      apply ha
+      exact Finset.mem_image.mpr ⟨ω, Finset.mem_univ _, h⟩
+    simp [hne]
+  have hs : ∑ a ∈ s, fairMass f a = 1 := by
+    rw [← fairMass_sum f]
+    exact Finset.sum_subset (Finset.subset_univ _) (fun a _ ha => hz a ha)
+  have hbound := sum_sub_le_tv (fairMass f) (fun _ => 1 / Fintype.card α) hm s
+  rw [Finset.sum_sub_distrib, hs] at hbound
+  simp only [Finset.sum_const, nsmul_eq_mul, mul_one_div] at hbound
+  have hc : (s.card : ℝ) ≤ Fintype.card Ω := by
+    exact_mod_cast (Finset.card_image_le (s := Finset.univ) (f := f))
+  have hd : 0 ≤ (Fintype.card α : ℝ) := Nat.cast_nonneg _
+  have hdiv := div_le_div_of_nonneg_right hc hd
+  linarith
+
+@[simp] theorem card_position (d : ℕ) : Fintype.card (Position d) = 2 ^ d := by
+  simp [Position]
+
+@[simp] theorem card_state (d : ℕ) : Fintype.card (State d) = (2 ^ d).factorial := by
+  classical
+  rw [Fintype.card_perm, card_position]
+
+@[simp] theorem card_coins_succ (d : ℕ) : Fintype.card (Coins (d + 1)) = 2 ^ (2 ^ d) := by
+  simp [Coins]
+
+@[simp] theorem card_history_succ (d t : ℕ) :
+    Fintype.card (History (d + 1) t) = 2 ^ (2 ^ d * t) := by
+  simp [History, ← pow_mul]
+
+theorem distance_support_lower (d t : ℕ) :
+    1 - (2 : ℝ) ^ (2 ^ d * t) / ((2 ^ (d + 1)).factorial : ℝ) ≤
+      distance (d + 1) t := by
+  let : Nonempty (State (d + 1)) := ⟨Equiv.refl _⟩
+  have h := fairMass_tv_lower (run (d + 1) t)
+  change 1 - _ / _ ≤ tv (fairMass (run (d + 1) t))
+    (fun _ => 1 / Fintype.card (State (d + 1)))
+  simpa only [card_history_succ, card_state, Nat.cast_pow, Nat.cast_ofNat] using h
+
+theorem worst_start_eq (d t : ℕ) (g₀ : State d) :
+    tv (lawFrom d t g₀) (uniform d) = distance d t := by
+  classical
+  have hmass (g : State d) : lawFrom d t g₀ (g * g₀) = law d t g := by
+    unfold lawFrom law fairMass
+    congr 1
+    apply Finset.sum_congr rfl
+    intro ω _
+    by_cases h : run d t ω = g <;> simp [h]
+  unfold tv distance
+  congr 1
+  have h := Equiv.sum_comp (Equiv.mulRight g₀)
+    (fun g : State d => |lawFrom d t g₀ g - uniform d g|)
+  change (∑ g, |lawFrom d t g₀ (g * g₀) - (1 / Fintype.card (State d) : ℝ)|) = _ at h
+  simp only [hmass] at h
+  exact h.symm
+
+theorem supportThreshold_le_of_distance {d t : ℕ} (hd : 1 ≤ d)
+    (ht : distance d t ≤ 1 / 4) : supportThreshold d ≤ (t : ℤ) := by
+  obtain ⟨k, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : d ≠ 0)
+  have hfact : (0 : ℝ) < ((2 ^ (k + 1)).factorial : ℝ) := by
+    exact_mod_cast Nat.factorial_pos _
+  have hlower := distance_support_lower k t
+  have hdiv : (3 : ℝ) / 4 ≤ 2 ^ (2 ^ k * t) / ((2 ^ (k + 1)).factorial : ℝ) := by
+    linarith
+  have hpow : 3 * ((2 ^ (k + 1)).factorial : ℝ) / 4 ≤ (2 : ℝ) ^ (2 ^ k * t) := by
+    have h := (le_div_iff₀ hfact).mp hdiv
+    nlinarith
+  have hlog := (Real.logb_le_logb (by norm_num : (1 : ℝ) < 2)
+    (by positivity : (0 : ℝ) < 3 * ((2 ^ (k + 1)).factorial : ℝ) / 4)
+    (by positivity : (0 : ℝ) < 2 ^ (2 ^ k * t))).mpr hpow
+  rw [Real.logb_pow, Real.logb_self_eq_one (by norm_num)] at hlog
+  simp only [mul_one] at hlog
+  unfold supportThreshold
+  apply Int.ceil_le.mpr
+  push_cast
+  have hden : (0 : ℝ) < 2 ^ (k + 1) := by positivity
+  have hscale := mul_le_mul_of_nonneg_left hlog (by positivity : (0 : ℝ) ≤ 2 / 2 ^ (k + 1))
+  have hid : (2 / (2 : ℝ) ^ (k + 1)) * ((2 ^ k * t : ℕ) : ℝ) = t := by
+    push_cast
+    rw [pow_succ]
+    field_simp
+  rwa [hid] at hscale
+
+theorem supportThreshold_le_mixingTime {d : ℕ} (hd : 1 ≤ d)
+    (hex : ∃ t, distance d t ≤ 1 / 4) : supportThreshold d ≤ (mixingTime d : ℤ) := by
+  apply supportThreshold_le_of_distance hd
+  exact Nat.sInf_mem hex
+
+end Thorp
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/SymmetricFamilies.lean b/lean/OAI/Probability/ThorpShuffle/SymmetricFamilies.lean
index 4c59c6908fe4fd8d229e19a0e49dca15381a3d8f..4011a283f816167abb6f575761b791a9d5a05438 100644
--- a/lean/OAI/Probability/ThorpShuffle/SymmetricFamilies.lean
+++ b/lean/OAI/Probability/ThorpShuffle/SymmetricFamilies.lean
@@ -1,92 +1,92 @@
-import OAI.Probability.ThorpShuffle.FourierBounds
-
-namespace OAI
-
-noncomputable section
-
-open scoped BigOperators ComplexConjugate InnerProductSpace
-open Filter Topology
-
-namespace Thorp.Specht
-open scoped Classical
-open Thorp.Young
-
-lemma family_unitary {α : Type} [Fintype α] (p : Shape α) (g : Equiv.Perm α) (v w : Space (α:=α) p) :
-    ⟪family p g v, family p g w⟫_ℂ = ⟪v, w⟫_ℂ := by
-  change ⟪tabloidRep _ _ g v.val, tabloidRep _ _ g w.val⟫_ℂ = ⟪v.val, w.val⟫_ℂ
-  exact PermutationSpace.rep_unitary g v.val w.val
-
-lemma regular_char_fin (n : ℕ) (g : Equiv.Perm (Fin n)) :
-    ∑ p : NShape n, (Module.finrank ℂ (Space (α:=Fin n) p) : ℂ) * (family (α:=Fin n) p).character g =
-      if g = 1 then (Fintype.card (Equiv.Perm (Fin n)) : ℂ) else 0 := by
-  have hc : Fintype.card (Equiv.Perm (Fin n)) = Nat.card (Equiv.Perm (Fin n)) := Fintype.card_eq_nat_card
-  rw [hc]
-  by_cases h : g = 1
-  · simpa only [degree, ite_eq_left h] using regular_char g
-  · simpa only [degree, ite_eq_right h] using regular_char g
-
-def symmetricFamily (n : ℕ) : Thorp.Fourier.Family (Equiv.Perm (Fin n)) where
-  Index := NShape n
-  Space := fun p => Space (α:=Fin n) p
-  rep := fun p => family (α:=Fin n) p
-  positive := by
-    intro p
-    simpa only [degree] using degree_pos p
-  unitary := fun p g v w => family_unitary (α:=Fin n) p g v w
-  distinct := fun p q h => family_inequiv (α:=Fin n) p q h
-  regular := by
-    intro g
-    by_cases h : g = 1
-    · simpa only [ite_eq_left h] using regular_char_fin n g
-    · simpa only [ite_eq_right h] using regular_char_fin n g
-
-end Thorp.Specht
-
-namespace Thorp.Fourier
-open scoped Classical
-
-variable {G H : Type} [Group G] [Fintype G] [Group H] [Fintype H]
-
-variable {V : Type} [AddCommGroup V] [Module ℂ V]
-
-def subrepPull (ρ : Representation ℂ G V) (e : H ≃* G) :
-    Subrepresentation ρ ≃o Subrepresentation (ρ.comp e.toMonoidHom) where
-  toFun U := { toSubmodule := U.toSubmodule
-               apply_mem_toSubmodule := fun h => U.apply_mem_toSubmodule (e h) }
-  invFun U := { toSubmodule := U.toSubmodule
-                apply_mem_toSubmodule := by
-                  intro g v hv
-                  have hh := U.apply_mem_toSubmodule (e.symm g) hv
-                  simpa only [MonoidHom.comp_apply, MulEquiv.coe_toMonoidHom, e.apply_symm_apply] using hh }
-  left_inv U := by rfl
-  right_inv U := by rfl
-  map_rel_iff' := by rfl
-
-def Family.pull (F : Family G) (e : H ≃* G) : Family H where
-  Index := F.Index
-  Space := F.Space
-  rep i := (F.rep i).comp e.toMonoidHom
-  irreducible i := (subrepPull (F.rep i) e).symm.isSimpleOrder
-  positive := F.positive
-  unitary := fun i h => F.unitary i (e h)
-  distinct := by
-    intro i j hh
-    obtain ⟨A⟩ := hh
-    apply F.distinct i j
-    refine ⟨{ A.toLinearEquiv with isIntertwining' := ?_ }⟩
-    intro g
-    ext v
-    have ha := DFunLike.congr_fun (A.isIntertwining' (e.symm g)) v
-    simpa only [MonoidHom.comp_apply, MulEquiv.coe_toMonoidHom, e.apply_symm_apply] using ha
-  regular := by
-    intro h
-    have hh := F.regular (e h)
-    have hc : Fintype.card G = Fintype.card H := (Fintype.card_congr e.toEquiv).symm
-    change ∑ i, (Module.finrank ℂ (F.Space i) : ℂ) * (F.rep i).character (e h) = _
-    simpa only [e.map_eq_one_iff, hc] using hh
-
-end Thorp.Fourier
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.FourierBounds
+
+namespace OAI
+
+noncomputable section
+
+open scoped BigOperators ComplexConjugate InnerProductSpace
+open Filter Topology
+
+namespace Thorp.Specht
+open scoped Classical
+open Thorp.Young
+
+lemma family_unitary {α : Type} [Fintype α] (p : Shape α) (g : Equiv.Perm α) (v w : Space (α:=α) p) :
+    ⟪family p g v, family p g w⟫_ℂ = ⟪v, w⟫_ℂ := by
+  change ⟪tabloidRep _ _ g v.val, tabloidRep _ _ g w.val⟫_ℂ = ⟪v.val, w.val⟫_ℂ
+  exact PermutationSpace.rep_unitary g v.val w.val
+
+lemma regular_char_fin (n : ℕ) (g : Equiv.Perm (Fin n)) :
+    ∑ p : NShape n, (Module.finrank ℂ (Space (α:=Fin n) p) : ℂ) * (family (α:=Fin n) p).character g =
+      if g = 1 then (Fintype.card (Equiv.Perm (Fin n)) : ℂ) else 0 := by
+  have hc : Fintype.card (Equiv.Perm (Fin n)) = Nat.card (Equiv.Perm (Fin n)) := Fintype.card_eq_nat_card
+  rw [hc]
+  by_cases h : g = 1
+  · simpa only [degree, ite_eq_left h] using regular_char g
+  · simpa only [degree, ite_eq_right h] using regular_char g
+
+def symmetricFamily (n : ℕ) : Thorp.Fourier.Family (Equiv.Perm (Fin n)) where
+  Index := NShape n
+  Space := fun p => Space (α:=Fin n) p
+  rep := fun p => family (α:=Fin n) p
+  positive := by
+    intro p
+    simpa only [degree] using degree_pos p
+  unitary := fun p g v w => family_unitary (α:=Fin n) p g v w
+  distinct := fun p q h => family_inequiv (α:=Fin n) p q h
+  regular := by
+    intro g
+    by_cases h : g = 1
+    · simpa only [ite_eq_left h] using regular_char_fin n g
+    · simpa only [ite_eq_right h] using regular_char_fin n g
+
+end Thorp.Specht
+
+namespace Thorp.Fourier
+open scoped Classical
+
+variable {G H : Type} [Group G] [Fintype G] [Group H] [Fintype H]
+
+variable {V : Type} [AddCommGroup V] [Module ℂ V]
+
+def subrepPull (ρ : Representation ℂ G V) (e : H ≃* G) :
+    Subrepresentation ρ ≃o Subrepresentation (ρ.comp e.toMonoidHom) where
+  toFun U := { toSubmodule := U.toSubmodule
+               apply_mem_toSubmodule := fun h => U.apply_mem_toSubmodule (e h) }
+  invFun U := { toSubmodule := U.toSubmodule
+                apply_mem_toSubmodule := by
+                  intro g v hv
+                  have hh := U.apply_mem_toSubmodule (e.symm g) hv
+                  simpa only [MonoidHom.comp_apply, MulEquiv.coe_toMonoidHom, e.apply_symm_apply] using hh }
+  left_inv U := by rfl
+  right_inv U := by rfl
+  map_rel_iff' := by rfl
+
+def Family.pull (F : Family G) (e : H ≃* G) : Family H where
+  Index := F.Index
+  Space := F.Space
+  rep i := (F.rep i).comp e.toMonoidHom
+  irreducible i := (subrepPull (F.rep i) e).symm.isSimpleOrder
+  positive := F.positive
+  unitary := fun i h => F.unitary i (e h)
+  distinct := by
+    intro i j hh
+    obtain ⟨A⟩ := hh
+    apply F.distinct i j
+    refine ⟨{ A.toLinearEquiv with isIntertwining' := ?_ }⟩
+    intro g
+    ext v
+    have ha := DFunLike.congr_fun (A.isIntertwining' (e.symm g)) v
+    simpa only [MonoidHom.comp_apply, MulEquiv.coe_toMonoidHom, e.apply_symm_apply] using ha
+  regular := by
+    intro h
+    have hh := F.regular (e h)
+    have hc : Fintype.card G = Fintype.card H := (Fintype.card_congr e.toEquiv).symm
+    change ∑ i, (Module.finrank ℂ (F.Space i) : ℂ) * (F.rep i).character (e h) = _
+    simpa only [e.map_eq_one_iff, hc] using hh
+
+end Thorp.Fourier
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/Tableaux.lean b/lean/OAI/Probability/ThorpShuffle/Tableaux.lean
index 4fd819b2095e39b201430f9192a305e634861d2f..d5e9ea416b3f56f21393481d5c00ba09b6569936 100644
--- a/lean/OAI/Probability/ThorpShuffle/Tableaux.lean
+++ b/lean/OAI/Probability/ThorpShuffle/Tableaux.lean
@@ -1,130 +1,130 @@
-import OAI.Probability.ThorpShuffle.Cyclic
-
-namespace OAI
-
-noncomputable section
-
-open scoped BigOperators ComplexConjugate InnerProductSpace
-open Filter
-
-namespace Thorp.Young
-
-abbrev Cells (D : YoungDiagram) := {p : ℕ × ℕ // p ∈ D.cells}
-
-lemma fin_le_strictMono {n : ℕ} (f : Fin n → ℕ) (hf : StrictMono f) (i : Fin n) :
-    i.val ≤ f i := by
-  have aux : ∀ k (hk : k < n), k ≤ f ⟨k, hk⟩ := by
-    intro k
-    induction k with
-    | zero => intro hk; exact Nat.zero_le _
-    | succ k ih =>
-      intro hk
-      have hk' : k < n := by omega
-      have hl := hf (show (⟨k, hk'⟩ : Fin n) < ⟨k + 1, hk⟩ from by simp)
-      have hi := ih hk'
-      omega
-  exact aux i.val i.isLt
-
-lemma card_column (D : YoungDiagram) (j : ℕ) :
-    (Finset.univ.filter fun x : Cells D => x.val.2 = j).card = D.colLen j := by
-  classical
-  let e : {x : Cells D // x.val.2 = j} ≃ Fin (D.colLen j) := {
-    toFun := fun x => ⟨x.val.val.1, YoungDiagram.mem_iff_lt_colLen.mp (by
-      have hx : (x.val.val.1, x.val.val.2) ∈ D := x.val.property
-      simpa only [x.property] using hx)⟩
-    invFun := fun i => ⟨⟨(i, j), YoungDiagram.mem_iff_lt_colLen.mpr i.isLt⟩, rfl⟩
-    left_inv := by intro x; apply Subtype.ext; apply Subtype.ext; exact Prod.ext rfl x.property.symm
-    right_inv := by intro i; rfl }
-  rw [← Fintype.card_subtype]
-  exact (Fintype.card_congr e).trans (Fintype.card_fin _)
-
-lemma card_row_subtype (D : YoungDiagram) (i : ℕ) :
-    Fintype.card {x : Cells D // x.val.1 = i} = D.rowLen i := by
-  let e : {x : Cells D // x.val.1 = i} ≃ Fin (D.rowLen i) := {
-    toFun := fun x => ⟨x.val.val.2, YoungDiagram.mem_iff_lt_rowLen.mp (by
-      have hx : (x.val.val.1, x.val.val.2) ∈ D := x.val.property
-      simpa only [x.property] using hx)⟩
-    invFun := fun j => ⟨⟨(i, j), YoungDiagram.mem_iff_lt_rowLen.mpr j.isLt⟩, rfl⟩
-    left_inv := by intro x; apply Subtype.ext; apply Subtype.ext; exact Prod.ext x.property.symm rfl
-    right_inv := by intro j; rfl }
-  exact (Fintype.card_congr e).trans (Fintype.card_fin _)
-
-lemma eq_of_rowPreserving (D E : YoungDiagram) (e : Cells D ≃ Cells E)
-    (he : ∀ x, (e x).val.1 = x.val.1) : D = E := by
-  have hr (i : ℕ) : D.rowLen i = E.rowLen i := by
-    rw [← card_row_subtype D i, ← card_row_subtype E i]
-    exact Fintype.card_congr (e.subtypeEquiv (fun x => by rw [he]))
-  apply YoungDiagram.ext
-  ext p
-  obtain ⟨i, j⟩ := p
-  change (i, j) ∈ D ↔ (i, j) ∈ E
-  simp only [YoungDiagram.mem_iff_lt_rowLen, hr]
-
-theorem column_sort_le (D : YoungDiagram) (r : Cells D → ℕ)
-    (hinj : ∀ x y : Cells D, x.val.2 = y.val.2 → r x = r y → x = y)
-    :
-    ∃ σ : Equiv.Perm (Cells D), (∀ x, (σ x).val.2 = x.val.2) ∧
-      ∀ x, x.val.1 ≤ r (σ x) := by
-  classical
-  let S (j : ℕ) : Finset ℕ := (Finset.univ.filter fun x : Cells D => x.val.2 = j).image r
-  have hc (j : ℕ) : (S j).card = D.colLen j := by
-    change ((Finset.univ.filter fun x : Cells D => x.val.2 = j).image r).card = _
-    rw [Finset.card_image_iff.mpr (show Set.InjOn r ↑(Finset.univ.filter fun x : Cells D => x.val.2 = j) from by
-      intro x hx y hy hxy
-      exact hinj x y ((Finset.mem_filter.mp hx).2.trans (Finset.mem_filter.mp hy).2.symm) hxy)]
-    exact card_column D j
-  let F (j : ℕ) := (S j).orderEmbOfFin (hc j)
-  have hpre (x : Cells D) : ∃ y : Cells D, y.val.2 = x.val.2 ∧
-      r y = F x.val.2 ⟨x.val.1, YoungDiagram.mem_iff_lt_colLen.mp x.property⟩ := by
-    have hm := (S x.val.2).orderEmbOfFin_mem (hc _) ⟨x.val.1, YoungDiagram.mem_iff_lt_colLen.mp x.property⟩
-    obtain ⟨y, hy, he⟩ := Finset.mem_image.mp hm
-    exact ⟨y, (Finset.mem_filter.mp hy).2, he⟩
-  choose f hcol hr using hpre
-  have hf : Function.Injective f := by
-    intro x y he
-    have hj : x.val.2 = y.val.2 := (hcol x).symm.trans ((congrArg (fun z : Cells D => z.val.2) he).trans (hcol y))
-    have hi : x.val.1 = y.val.1 := by
-      have hval : F x.val.2 ⟨x.val.1, YoungDiagram.mem_iff_lt_colLen.mp x.property⟩ =
-          F y.val.2 ⟨y.val.1, YoungDiagram.mem_iff_lt_colLen.mp y.property⟩ := by
-        rw [← hr x, ← hr y, he]
-      have hy : y.val.1 < D.colLen x.val.2 := by rw [hj]; exact YoungDiagram.mem_iff_lt_colLen.mp y.property
-      have hid : F y.val.2 ⟨y.val.1, YoungDiagram.mem_iff_lt_colLen.mp y.property⟩ = F x.val.2 ⟨y.val.1, hy⟩ := by
-        have aux (j k i : ℕ) (hk : j = k) (hi : i < D.colLen j) (hi' : i < D.colLen k) :
-            F j ⟨i, hi⟩ = F k ⟨i, hi'⟩ := by subst k; rfl
-        exact aux _ _ _ hj.symm _ _
-      rw [hid] at hval
-      exact congrArg Fin.val ((F x.val.2).injective hval)
-    exact Subtype.ext (Prod.ext hi hj)
-  let σ : Equiv.Perm (Cells D) := Equiv.ofBijective f ⟨hf, Finite.surjective_of_injective hf⟩
-  refine ⟨σ, hcol, ?_⟩
-  have hle (x : Cells D) : x.val.1 ≤ r (σ x) := by
-    change x.val.1 ≤ r (f x)
-    rw [hr]
-    exact fin_le_strictMono _ (F x.val.2).strictMono ⟨x.val.1, YoungDiagram.mem_iff_lt_colLen.mp x.property⟩
-  exact hle
-
-lemma rowRankSum_le (D : YoungDiagram) (r : Cells D → ℕ)
-    (hinj : ∀ x y : Cells D, x.val.2 = y.val.2 → r x = r y → x = y) :
-    ∑ x : Cells D, x.val.1 ≤ ∑ x : Cells D, r x := by
-  obtain ⟨σ, -, hσ⟩ := column_sort_le D r hinj
-  rw [← Equiv.sum_comp σ r]
-  exact Finset.sum_le_sum fun x _ => hσ x
-
-theorem column_sort (D : YoungDiagram) (r : Cells D → ℕ)
-    (hinj : ∀ x y : Cells D, x.val.2 = y.val.2 → r x = r y → x = y)
-    (hsum : ∑ x : Cells D, r x = ∑ x : Cells D, x.val.1) :
-    ∃ σ : Equiv.Perm (Cells D), (∀ x, (σ x).val.2 = x.val.2) ∧
-      ∀ x, r (σ x) = x.val.1 := by
-  obtain ⟨σ, hc, hle⟩ := column_sort_le D r hinj
-  refine ⟨σ, hc, ?_⟩
-  have heq : ∑ x : Cells D, x.val.1 = ∑ x : Cells D, r (σ x) := by
-    rw [Equiv.sum_comp σ r, hsum]
-  have hh := (Finset.sum_eq_sum_iff_of_le (s := Finset.univ) (fun x _ => hle x)).mp heq
-  intro x
-  exact (hh x (Finset.mem_univ x)).symm
-
-end Thorp.Young
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.Cyclic
+
+namespace OAI
+
+noncomputable section
+
+open scoped BigOperators ComplexConjugate InnerProductSpace
+open Filter
+
+namespace Thorp.Young
+
+abbrev Cells (D : YoungDiagram) := {p : ℕ × ℕ // p ∈ D.cells}
+
+lemma fin_le_strictMono {n : ℕ} (f : Fin n → ℕ) (hf : StrictMono f) (i : Fin n) :
+    i.val ≤ f i := by
+  have aux : ∀ k (hk : k < n), k ≤ f ⟨k, hk⟩ := by
+    intro k
+    induction k with
+    | zero => intro hk; exact Nat.zero_le _
+    | succ k ih =>
+      intro hk
+      have hk' : k < n := by omega
+      have hl := hf (show (⟨k, hk'⟩ : Fin n) < ⟨k + 1, hk⟩ from by simp)
+      have hi := ih hk'
+      omega
+  exact aux i.val i.isLt
+
+lemma card_column (D : YoungDiagram) (j : ℕ) :
+    (Finset.univ.filter fun x : Cells D => x.val.2 = j).card = D.colLen j := by
+  classical
+  let e : {x : Cells D // x.val.2 = j} ≃ Fin (D.colLen j) := {
+    toFun := fun x => ⟨x.val.val.1, YoungDiagram.mem_iff_lt_colLen.mp (by
+      have hx : (x.val.val.1, x.val.val.2) ∈ D := x.val.property
+      simpa only [x.property] using hx)⟩
+    invFun := fun i => ⟨⟨(i, j), YoungDiagram.mem_iff_lt_colLen.mpr i.isLt⟩, rfl⟩
+    left_inv := by intro x; apply Subtype.ext; apply Subtype.ext; exact Prod.ext rfl x.property.symm
+    right_inv := by intro i; rfl }
+  rw [← Fintype.card_subtype]
+  exact (Fintype.card_congr e).trans (Fintype.card_fin _)
+
+lemma card_row_subtype (D : YoungDiagram) (i : ℕ) :
+    Fintype.card {x : Cells D // x.val.1 = i} = D.rowLen i := by
+  let e : {x : Cells D // x.val.1 = i} ≃ Fin (D.rowLen i) := {
+    toFun := fun x => ⟨x.val.val.2, YoungDiagram.mem_iff_lt_rowLen.mp (by
+      have hx : (x.val.val.1, x.val.val.2) ∈ D := x.val.property
+      simpa only [x.property] using hx)⟩
+    invFun := fun j => ⟨⟨(i, j), YoungDiagram.mem_iff_lt_rowLen.mpr j.isLt⟩, rfl⟩
+    left_inv := by intro x; apply Subtype.ext; apply Subtype.ext; exact Prod.ext x.property.symm rfl
+    right_inv := by intro j; rfl }
+  exact (Fintype.card_congr e).trans (Fintype.card_fin _)
+
+lemma eq_of_rowPreserving (D E : YoungDiagram) (e : Cells D ≃ Cells E)
+    (he : ∀ x, (e x).val.1 = x.val.1) : D = E := by
+  have hr (i : ℕ) : D.rowLen i = E.rowLen i := by
+    rw [← card_row_subtype D i, ← card_row_subtype E i]
+    exact Fintype.card_congr (e.subtypeEquiv (fun x => by rw [he]))
+  apply YoungDiagram.ext
+  ext p
+  obtain ⟨i, j⟩ := p
+  change (i, j) ∈ D ↔ (i, j) ∈ E
+  simp only [YoungDiagram.mem_iff_lt_rowLen, hr]
+
+theorem column_sort_le (D : YoungDiagram) (r : Cells D → ℕ)
+    (hinj : ∀ x y : Cells D, x.val.2 = y.val.2 → r x = r y → x = y)
+    :
+    ∃ σ : Equiv.Perm (Cells D), (∀ x, (σ x).val.2 = x.val.2) ∧
+      ∀ x, x.val.1 ≤ r (σ x) := by
+  classical
+  let S (j : ℕ) : Finset ℕ := (Finset.univ.filter fun x : Cells D => x.val.2 = j).image r
+  have hc (j : ℕ) : (S j).card = D.colLen j := by
+    change ((Finset.univ.filter fun x : Cells D => x.val.2 = j).image r).card = _
+    rw [Finset.card_image_iff.mpr (show Set.InjOn r ↑(Finset.univ.filter fun x : Cells D => x.val.2 = j) from by
+      intro x hx y hy hxy
+      exact hinj x y ((Finset.mem_filter.mp hx).2.trans (Finset.mem_filter.mp hy).2.symm) hxy)]
+    exact card_column D j
+  let F (j : ℕ) := (S j).orderEmbOfFin (hc j)
+  have hpre (x : Cells D) : ∃ y : Cells D, y.val.2 = x.val.2 ∧
+      r y = F x.val.2 ⟨x.val.1, YoungDiagram.mem_iff_lt_colLen.mp x.property⟩ := by
+    have hm := (S x.val.2).orderEmbOfFin_mem (hc _) ⟨x.val.1, YoungDiagram.mem_iff_lt_colLen.mp x.property⟩
+    obtain ⟨y, hy, he⟩ := Finset.mem_image.mp hm
+    exact ⟨y, (Finset.mem_filter.mp hy).2, he⟩
+  choose f hcol hr using hpre
+  have hf : Function.Injective f := by
+    intro x y he
+    have hj : x.val.2 = y.val.2 := (hcol x).symm.trans ((congrArg (fun z : Cells D => z.val.2) he).trans (hcol y))
+    have hi : x.val.1 = y.val.1 := by
+      have hval : F x.val.2 ⟨x.val.1, YoungDiagram.mem_iff_lt_colLen.mp x.property⟩ =
+          F y.val.2 ⟨y.val.1, YoungDiagram.mem_iff_lt_colLen.mp y.property⟩ := by
+        rw [← hr x, ← hr y, he]
+      have hy : y.val.1 < D.colLen x.val.2 := by rw [hj]; exact YoungDiagram.mem_iff_lt_colLen.mp y.property
+      have hid : F y.val.2 ⟨y.val.1, YoungDiagram.mem_iff_lt_colLen.mp y.property⟩ = F x.val.2 ⟨y.val.1, hy⟩ := by
+        have aux (j k i : ℕ) (hk : j = k) (hi : i < D.colLen j) (hi' : i < D.colLen k) :
+            F j ⟨i, hi⟩ = F k ⟨i, hi'⟩ := by subst k; rfl
+        exact aux _ _ _ hj.symm _ _
+      rw [hid] at hval
+      exact congrArg Fin.val ((F x.val.2).injective hval)
+    exact Subtype.ext (Prod.ext hi hj)
+  let σ : Equiv.Perm (Cells D) := Equiv.ofBijective f ⟨hf, Finite.surjective_of_injective hf⟩
+  refine ⟨σ, hcol, ?_⟩
+  have hle (x : Cells D) : x.val.1 ≤ r (σ x) := by
+    change x.val.1 ≤ r (f x)
+    rw [hr]
+    exact fin_le_strictMono _ (F x.val.2).strictMono ⟨x.val.1, YoungDiagram.mem_iff_lt_colLen.mp x.property⟩
+  exact hle
+
+lemma rowRankSum_le (D : YoungDiagram) (r : Cells D → ℕ)
+    (hinj : ∀ x y : Cells D, x.val.2 = y.val.2 → r x = r y → x = y) :
+    ∑ x : Cells D, x.val.1 ≤ ∑ x : Cells D, r x := by
+  obtain ⟨σ, -, hσ⟩ := column_sort_le D r hinj
+  rw [← Equiv.sum_comp σ r]
+  exact Finset.sum_le_sum fun x _ => hσ x
+
+theorem column_sort (D : YoungDiagram) (r : Cells D → ℕ)
+    (hinj : ∀ x y : Cells D, x.val.2 = y.val.2 → r x = r y → x = y)
+    (hsum : ∑ x : Cells D, r x = ∑ x : Cells D, x.val.1) :
+    ∃ σ : Equiv.Perm (Cells D), (∀ x, (σ x).val.2 = x.val.2) ∧
+      ∀ x, r (σ x) = x.val.1 := by
+  obtain ⟨σ, hc, hle⟩ := column_sort_le D r hinj
+  refine ⟨σ, hc, ?_⟩
+  have heq : ∑ x : Cells D, x.val.1 = ∑ x : Cells D, r (σ x) := by
+    rw [Equiv.sum_comp σ r, hsum]
+  have hh := (Finset.sum_eq_sum_iff_of_le (s := Finset.univ) (fun x _ => hle x)).mp heq
+  intro x
+  exact (hh x (Finset.mem_univ x)).symm
+
+end Thorp.Young
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/TaggedCompletion.lean b/lean/OAI/Probability/ThorpShuffle/TaggedCompletion.lean
index 46bce02174ce9a214bcf406fa671d51be140251b..42f3b0e29f2c5c5da38402a868f943c7da8627b0 100644
--- a/lean/OAI/Probability/ThorpShuffle/TaggedCompletion.lean
+++ b/lean/OAI/Probability/ThorpShuffle/TaggedCompletion.lean
@@ -1,129 +1,129 @@
-import OAI.Probability.ThorpShuffle.TaggedDeviation
-
-namespace OAI
-
-universe uα
-
-noncomputable section
-
-open scoped BigOperators
-open Filter
-
-namespace Thorp
-
-namespace Conditional
-
-def freeOf {α : Type uα} [DecidableEq α] {k : ℕ} (e : Fin k → α) : α → Bool :=
-  fun y => decide (∀ i, e i ≠ y)
-
-@[simp] theorem freeOf_exposed {α : Type uα} [DecidableEq α] {k : ℕ}
-    (e : Fin k → α) (i : Fin k) : freeOf e (e i) = false := by
-  simp [freeOf]
-
-@[simp] theorem freeOf_true {α : Type uα} [DecidableEq α] {k : ℕ}
-    (e : Fin k → α) (y : α) : freeOf e y = true ↔ ∀ i, e i ≠ y := by
-  simp [freeOf]
-
-theorem freeOf_perm {α : Type uα} [DecidableEq α] {k : ℕ}
-    (e : Fin k → α) (g : Equiv.Perm α) (y : α) :
-    freeOf (g ∘ e) y = freeOf e (g.symm y) := by
-  simp [freeOf, ← g.eq_symm_apply]
-
-theorem freeCount_perm {α : Type uα} [Fintype α] [DecidableEq α] {k : ℕ}
-    (e : Fin k → α) (g : Equiv.Perm α) : freeCount (freeOf (g ∘ e)) = freeCount (freeOf e) := by
-  apply Nat.cast_injective (R := ℝ)
-  rw [← sum_free_indicator, ← sum_free_indicator]
-  simp only [freeOf_perm]
-  exact Equiv.sum_comp g.symm (fun x => if freeOf e x then (1 : ℝ) else 0)
-
-theorem freeCount_freeOf {α : Type uα} [Fintype α] [DecidableEq α] {k : ℕ}
-    (e : Fin k → α) (he : Function.Injective e) : freeCount (freeOf e) = Fintype.card α - k := by
-  have hs : Finset.univ.filter (fun x => freeOf e x = true) = (Finset.univ.image e)ᶜ := by
-    ext x
-    simp [freeOf]
-  unfold freeCount
-  rw [hs, Finset.card_compl, Finset.card_image_of_injective _ he]
-  simp
-
-def completeTest {α : Type uα} [Fintype α] [DecidableEq α] {k : ℕ}
-    (f : (Fin (k + 1) → α) → ℝ) (e : Fin k → α) : ℝ :=
-  (1 / (freeCount (freeOf e) : ℝ)) * ∑ y, if freeOf e y then f (Fin.snoc e y) else 0
-
-theorem completeTest_bound {α : Type uα} [Fintype α] [DecidableEq α] {k : ℕ}
-    (f : (Fin (k + 1) → α) → ℝ) (hf : ∀ e, |f e| ≤ 1) (e : Fin k → α) :
-    |completeTest f e| ≤ 1 := by
-  unfold completeTest
-  rw [abs_mul, abs_of_nonneg (by positivity : (0 : ℝ) ≤ 1 / (freeCount (freeOf e) : ℝ))]
-  have hs : |∑ y, if freeOf e y then f (Fin.snoc e y) else 0| ≤ freeCount (freeOf e) := by
-    calc
-      _ ≤ ∑ y, |if freeOf e y then f (Fin.snoc e y) else 0| := Finset.abs_sum_le_sum_abs _ _
-      _ ≤ ∑ y, if freeOf e y then (1 : ℝ) else 0 := Finset.sum_le_sum (fun y _ => by
-        by_cases h : freeOf e y = true
-        · simpa only [h, ↓reduceIte] using hf (Fin.snoc e y)
-        · simp [h])
-      _ = _ := sum_free_indicator _
-  have hh := mul_le_mul_of_nonneg_left hs
-    (by positivity : (0 : ℝ) ≤ 1 / (freeCount (freeOf e) : ℝ))
-  by_cases hz : freeCount (freeOf e) = 0
-  · simp [hz]
-  · have hn : (freeCount (freeOf e) : ℝ) ≠ 0 := by exact_mod_cast hz
-    simpa [hn] using hh
-
-theorem uniform_tagged_completion {α : Type uα} [Fintype α] [DecidableEq α]
-    {k : ℕ} (e : Fin k → α) (tag : α) (ht : freeOf e tag = true)
-    (f : (Fin k → α) → α → ℝ) :
-    mean (fun g : Equiv.Perm α => f (g ∘ e) (g tag)) =
-    mean (fun g : Equiv.Perm α => (1 / (freeCount (freeOf e) : ℝ)) *
-      ∑ y, if freeOf (g ∘ e) y then f (g ∘ e) y else 0) := by
-  have hm : (freeCount (freeOf e) : ℝ) ≠ 0 := by
-    exact_mod_cast (freeCount_pos _ tag ht).ne'
-  have hs (x : α) (hx : freeOf e x = true) :
-      mean (fun g : Equiv.Perm α => f (g ∘ e) (g x)) =
-        mean (fun g : Equiv.Perm α => f (g ∘ e) (g tag)) := by
-    have heq : (Equiv.swap tag x) ∘ e = e := by
-      funext i
-      exact Equiv.swap_apply_of_ne_of_ne ((freeOf_true e tag).mp ht i)
-        ((freeOf_true e x).mp hx i)
-    have h := mean_equiv (Equiv.mulRight (Equiv.swap tag x))
-      (fun g : Equiv.Perm α => f (g ∘ e) (g tag))
-    convert h using 1
-    apply mean_congr
-    intro g
-    change f (g ∘ e) (g x) = f (g ∘ ((Equiv.swap tag x) ∘ e)) (g (Equiv.swap tag x tag))
-    rw [heq, Equiv.swap_apply_left]
-  symm
-  calc
-    _ = mean (fun g : Equiv.Perm α => (1 / (freeCount (freeOf e) : ℝ)) *
-        ∑ x, if freeOf e x then f (g ∘ e) (g x) else 0) := by
-      apply mean_congr
-      intro g
-      congr 1
-      simp only [freeOf_perm]
-      exact (Equiv.sum_comp g (fun y => if freeOf e (g.symm y) then f (g ∘ e) y else 0)).symm.trans
-        (by simp only [Equiv.symm_apply_apply])
-    _ = (1 / (freeCount (freeOf e) : ℝ)) *
-        ∑ x, mean (fun g : Equiv.Perm α => if freeOf e x then f (g ∘ e) (g x) else 0) := by
-      rw [mean_const_mul, mean_sum]
-    _ = (1 / (freeCount (freeOf e) : ℝ)) *
-        ∑ x, if freeOf e x then mean (fun g : Equiv.Perm α => f (g ∘ e) (g tag)) else 0 := by
-      congr 1
-      apply Finset.sum_congr rfl
-      intro x _
-      cases hx : freeOf e x
-      · simp [mean_zero]
-      · simpa only [hx, ↓reduceIte] using hs x hx
-    _ = _ := by
-      have hid (x : α) : (if freeOf e x then mean (fun g : Equiv.Perm α => f (g ∘ e) (g tag)) else 0) =
-          (if freeOf e x then (1 : ℝ) else 0) * mean (fun g : Equiv.Perm α => f (g ∘ e) (g tag)) := by
-        cases freeOf e x <;> simp
-      simp_rw [hid]
-      rw [← Finset.sum_mul, sum_free_indicator]
-      field_simp
-
-end Conditional
-
-end Thorp
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.TaggedDeviation
+
+namespace OAI
+
+universe uα
+
+noncomputable section
+
+open scoped BigOperators
+open Filter
+
+namespace Thorp
+
+namespace Conditional
+
+def freeOf {α : Type uα} [DecidableEq α] {k : ℕ} (e : Fin k → α) : α → Bool :=
+  fun y => decide (∀ i, e i ≠ y)
+
+@[simp] theorem freeOf_exposed {α : Type uα} [DecidableEq α] {k : ℕ}
+    (e : Fin k → α) (i : Fin k) : freeOf e (e i) = false := by
+  simp [freeOf]
+
+@[simp] theorem freeOf_true {α : Type uα} [DecidableEq α] {k : ℕ}
+    (e : Fin k → α) (y : α) : freeOf e y = true ↔ ∀ i, e i ≠ y := by
+  simp [freeOf]
+
+theorem freeOf_perm {α : Type uα} [DecidableEq α] {k : ℕ}
+    (e : Fin k → α) (g : Equiv.Perm α) (y : α) :
+    freeOf (g ∘ e) y = freeOf e (g.symm y) := by
+  simp [freeOf, ← g.eq_symm_apply]
+
+theorem freeCount_perm {α : Type uα} [Fintype α] [DecidableEq α] {k : ℕ}
+    (e : Fin k → α) (g : Equiv.Perm α) : freeCount (freeOf (g ∘ e)) = freeCount (freeOf e) := by
+  apply Nat.cast_injective (R := ℝ)
+  rw [← sum_free_indicator, ← sum_free_indicator]
+  simp only [freeOf_perm]
+  exact Equiv.sum_comp g.symm (fun x => if freeOf e x then (1 : ℝ) else 0)
+
+theorem freeCount_freeOf {α : Type uα} [Fintype α] [DecidableEq α] {k : ℕ}
+    (e : Fin k → α) (he : Function.Injective e) : freeCount (freeOf e) = Fintype.card α - k := by
+  have hs : Finset.univ.filter (fun x => freeOf e x = true) = (Finset.univ.image e)ᶜ := by
+    ext x
+    simp [freeOf]
+  unfold freeCount
+  rw [hs, Finset.card_compl, Finset.card_image_of_injective _ he]
+  simp
+
+def completeTest {α : Type uα} [Fintype α] [DecidableEq α] {k : ℕ}
+    (f : (Fin (k + 1) → α) → ℝ) (e : Fin k → α) : ℝ :=
+  (1 / (freeCount (freeOf e) : ℝ)) * ∑ y, if freeOf e y then f (Fin.snoc e y) else 0
+
+theorem completeTest_bound {α : Type uα} [Fintype α] [DecidableEq α] {k : ℕ}
+    (f : (Fin (k + 1) → α) → ℝ) (hf : ∀ e, |f e| ≤ 1) (e : Fin k → α) :
+    |completeTest f e| ≤ 1 := by
+  unfold completeTest
+  rw [abs_mul, abs_of_nonneg (by positivity : (0 : ℝ) ≤ 1 / (freeCount (freeOf e) : ℝ))]
+  have hs : |∑ y, if freeOf e y then f (Fin.snoc e y) else 0| ≤ freeCount (freeOf e) := by
+    calc
+      _ ≤ ∑ y, |if freeOf e y then f (Fin.snoc e y) else 0| := Finset.abs_sum_le_sum_abs _ _
+      _ ≤ ∑ y, if freeOf e y then (1 : ℝ) else 0 := Finset.sum_le_sum (fun y _ => by
+        by_cases h : freeOf e y = true
+        · simpa only [h, ↓reduceIte] using hf (Fin.snoc e y)
+        · simp [h])
+      _ = _ := sum_free_indicator _
+  have hh := mul_le_mul_of_nonneg_left hs
+    (by positivity : (0 : ℝ) ≤ 1 / (freeCount (freeOf e) : ℝ))
+  by_cases hz : freeCount (freeOf e) = 0
+  · simp [hz]
+  · have hn : (freeCount (freeOf e) : ℝ) ≠ 0 := by exact_mod_cast hz
+    simpa [hn] using hh
+
+theorem uniform_tagged_completion {α : Type uα} [Fintype α] [DecidableEq α]
+    {k : ℕ} (e : Fin k → α) (tag : α) (ht : freeOf e tag = true)
+    (f : (Fin k → α) → α → ℝ) :
+    mean (fun g : Equiv.Perm α => f (g ∘ e) (g tag)) =
+    mean (fun g : Equiv.Perm α => (1 / (freeCount (freeOf e) : ℝ)) *
+      ∑ y, if freeOf (g ∘ e) y then f (g ∘ e) y else 0) := by
+  have hm : (freeCount (freeOf e) : ℝ) ≠ 0 := by
+    exact_mod_cast (freeCount_pos _ tag ht).ne'
+  have hs (x : α) (hx : freeOf e x = true) :
+      mean (fun g : Equiv.Perm α => f (g ∘ e) (g x)) =
+        mean (fun g : Equiv.Perm α => f (g ∘ e) (g tag)) := by
+    have heq : (Equiv.swap tag x) ∘ e = e := by
+      funext i
+      exact Equiv.swap_apply_of_ne_of_ne ((freeOf_true e tag).mp ht i)
+        ((freeOf_true e x).mp hx i)
+    have h := mean_equiv (Equiv.mulRight (Equiv.swap tag x))
+      (fun g : Equiv.Perm α => f (g ∘ e) (g tag))
+    convert h using 1
+    apply mean_congr
+    intro g
+    change f (g ∘ e) (g x) = f (g ∘ ((Equiv.swap tag x) ∘ e)) (g (Equiv.swap tag x tag))
+    rw [heq, Equiv.swap_apply_left]
+  symm
+  calc
+    _ = mean (fun g : Equiv.Perm α => (1 / (freeCount (freeOf e) : ℝ)) *
+        ∑ x, if freeOf e x then f (g ∘ e) (g x) else 0) := by
+      apply mean_congr
+      intro g
+      congr 1
+      simp only [freeOf_perm]
+      exact (Equiv.sum_comp g (fun y => if freeOf e (g.symm y) then f (g ∘ e) y else 0)).symm.trans
+        (by simp only [Equiv.symm_apply_apply])
+    _ = (1 / (freeCount (freeOf e) : ℝ)) *
+        ∑ x, mean (fun g : Equiv.Perm α => if freeOf e x then f (g ∘ e) (g x) else 0) := by
+      rw [mean_const_mul, mean_sum]
+    _ = (1 / (freeCount (freeOf e) : ℝ)) *
+        ∑ x, if freeOf e x then mean (fun g : Equiv.Perm α => f (g ∘ e) (g tag)) else 0 := by
+      congr 1
+      apply Finset.sum_congr rfl
+      intro x _
+      cases hx : freeOf e x
+      · simp [mean_zero]
+      · simpa only [hx, ↓reduceIte] using hs x hx
+    _ = _ := by
+      have hid (x : α) : (if freeOf e x then mean (fun g : Equiv.Perm α => f (g ∘ e) (g tag)) else 0) =
+          (if freeOf e x then (1 : ℝ) else 0) * mean (fun g : Equiv.Perm α => f (g ∘ e) (g tag)) := by
+        cases freeOf e x <;> simp
+      simp_rw [hid]
+      rw [← Finset.sum_mul, sum_free_indicator]
+      field_simp
+
+end Conditional
+
+end Thorp
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/TaggedDeviation.lean b/lean/OAI/Probability/ThorpShuffle/TaggedDeviation.lean
index 0f8d4575551f1d792434b9bc67c72aa8683433bb..ffa6a7d66641fcd5eb53b7b0559e3590f2918388 100644
--- a/lean/OAI/Probability/ThorpShuffle/TaggedDeviation.lean
+++ b/lean/OAI/Probability/ThorpShuffle/TaggedDeviation.lean
@@ -1,84 +1,84 @@
-import OAI.Probability.ThorpShuffle.ConditionalLaw
-
-namespace OAI
-
-universe uΩ uα uι
-
-noncomputable section
-
-open scoped BigOperators
-open Filter
-
-namespace Thorp
-
-namespace Conditional
-
-theorem mean_sub {Ω : Type uΩ} [Fintype Ω] (f g : Ω → ℝ) :
-    mean (fun ω => f ω - g ω) = mean f - mean g := by
-  simp only [mean, Finset.sum_sub_distrib, sub_div]
-
-theorem abs_mean_le {Ω : Type uΩ} [Fintype Ω] (f : Ω → ℝ) :
-    |mean f| ≤ mean (fun ω => |f ω|) := by
-  unfold mean
-  rw [abs_div, abs_of_nonneg (Nat.cast_nonneg (Fintype.card Ω) : (0 : ℝ) ≤ Fintype.card Ω)]
-  exact div_le_div_of_nonneg_right (Finset.abs_sum_le_sum_abs _ _) (Nat.cast_nonneg _)
-
-theorem mean_sq_le {Ω : Type uΩ} [Fintype Ω] [Nonempty Ω] (f : Ω → ℝ) :
-    mean f ^ 2 ≤ mean (fun ω => f ω ^ 2) := by
-  have h := Finset.expect_mul_sq_le_sq_mul_sq Finset.univ f (fun _ => (1 : ℝ))
-  simpa only [mul_one, one_pow, Finset.expect_const (Finset.univ_nonempty),
-    Finset.expect_eq_sum_div_card, Finset.card_univ, mean] using h
-
-theorem mean_l1_le_sqrt {Ω : Type uΩ} {α : Type uα} [Fintype Ω] [Nonempty Ω] [Fintype α]
-    (w : Ω → α → ℝ) :
-    mean (fun ω => ∑ x, |w ω x|) ≤
-      Real.sqrt ((Fintype.card α : ℝ) * mean (fun ω => ∑ x, w ω x ^ 2)) := by
-  apply Real.le_sqrt_of_sq_le
-  calc
-    _ ≤ mean (fun ω => (∑ x, |w ω x|) ^ 2) := mean_sq_le _
-    _ ≤ mean (fun ω => (Fintype.card α : ℝ) * ∑ x, w ω x ^ 2) := by
-      apply mean_le_mean
-      intro ω
-      have h := Finset.sum_mul_sq_le_sq_mul_sq Finset.univ
-        (fun _ : α => (1 : ℝ)) (fun x => |w ω x|)
-      simpa only [one_mul, one_pow, Finset.sum_const, Finset.card_univ,
-        nsmul_eq_mul, mul_one, sq_abs] using h
-    _ = _ := mean_const_mul _ _
-
-theorem mean_weighted_test_le {Ω : Type uΩ} {α : Type uα} [Fintype Ω] [Nonempty Ω] [Fintype α]
-    (w f : Ω → α → ℝ) (hf : ∀ ω x, |f ω x| ≤ 1) :
-    |mean (fun ω => ∑ x, w ω x * f ω x)| ≤
-      Real.sqrt ((Fintype.card α : ℝ) * mean (fun ω => ∑ x, w ω x ^ 2)) := by
-  calc
-    _ ≤ mean (fun ω => |∑ x, w ω x * f ω x|) := abs_mean_le _
-    _ ≤ mean (fun ω => ∑ x, |w ω x|) := by
-      apply mean_le_mean
-      intro ω
-      calc
-        _ ≤ ∑ x, |w ω x * f ω x| := Finset.abs_sum_le_sum_abs _ _
-        _ ≤ _ := Finset.sum_le_sum (fun x _ => by
-          rw [abs_mul]
-          simpa only [mul_one] using mul_le_mul_of_nonneg_left (hf ω x) (abs_nonneg _))
-    _ ≤ _ := mean_l1_le_sqrt w
-
-theorem tagged_deviation_bound {ι : Type uι} (d : ℕ)
-    (B : Position (d + 1) → Bool) (tag : Position (d + 1)) (ht : B tag = true)
-    (e : ι → Position (d + 1)) (he : ∀ i, B (e i) = false)
-    (t : ℕ) (f : (ι → Position (d + 1)) → Position (d + 1) → ℝ)
-    (hf : ∀ e' y, |f e' y| ≤ 1) :
-    |mean (fun ω : History (d + 1) t =>
-      f ((run (d + 1) t ω) ∘ e) (run (d + 1) t ω tag) -
-        (1 / (freeCount B : ℝ)) *
-          (∑ y, if B ((run (d + 1) t ω).symm y) then f ((run (d + 1) t ω) ∘ e) y else 0))| ≤
-    Real.sqrt ((Fintype.card (Position (d + 1)) : ℝ) *
-      expectedEnergy d (initialState (d + 1) B tag ht) t) := by
-  rw [tagged_deviation_identity d B tag ht e he]
-  exact mean_weighted_test_le _ _ (fun ω y => hf _ y)
-
-end Conditional
-
-end Thorp
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.ConditionalLaw
+
+namespace OAI
+
+universe uΩ uα uι
+
+noncomputable section
+
+open scoped BigOperators
+open Filter
+
+namespace Thorp
+
+namespace Conditional
+
+theorem mean_sub {Ω : Type uΩ} [Fintype Ω] (f g : Ω → ℝ) :
+    mean (fun ω => f ω - g ω) = mean f - mean g := by
+  simp only [mean, Finset.sum_sub_distrib, sub_div]
+
+theorem abs_mean_le {Ω : Type uΩ} [Fintype Ω] (f : Ω → ℝ) :
+    |mean f| ≤ mean (fun ω => |f ω|) := by
+  unfold mean
+  rw [abs_div, abs_of_nonneg (Nat.cast_nonneg (Fintype.card Ω) : (0 : ℝ) ≤ Fintype.card Ω)]
+  exact div_le_div_of_nonneg_right (Finset.abs_sum_le_sum_abs _ _) (Nat.cast_nonneg _)
+
+theorem mean_sq_le {Ω : Type uΩ} [Fintype Ω] [Nonempty Ω] (f : Ω → ℝ) :
+    mean f ^ 2 ≤ mean (fun ω => f ω ^ 2) := by
+  have h := Finset.expect_mul_sq_le_sq_mul_sq Finset.univ f (fun _ => (1 : ℝ))
+  simpa only [mul_one, one_pow, Finset.expect_const (Finset.univ_nonempty),
+    Finset.expect_eq_sum_div_card, Finset.card_univ, mean] using h
+
+theorem mean_l1_le_sqrt {Ω : Type uΩ} {α : Type uα} [Fintype Ω] [Nonempty Ω] [Fintype α]
+    (w : Ω → α → ℝ) :
+    mean (fun ω => ∑ x, |w ω x|) ≤
+      Real.sqrt ((Fintype.card α : ℝ) * mean (fun ω => ∑ x, w ω x ^ 2)) := by
+  apply Real.le_sqrt_of_sq_le
+  calc
+    _ ≤ mean (fun ω => (∑ x, |w ω x|) ^ 2) := mean_sq_le _
+    _ ≤ mean (fun ω => (Fintype.card α : ℝ) * ∑ x, w ω x ^ 2) := by
+      apply mean_le_mean
+      intro ω
+      have h := Finset.sum_mul_sq_le_sq_mul_sq Finset.univ
+        (fun _ : α => (1 : ℝ)) (fun x => |w ω x|)
+      simpa only [one_mul, one_pow, Finset.sum_const, Finset.card_univ,
+        nsmul_eq_mul, mul_one, sq_abs] using h
+    _ = _ := mean_const_mul _ _
+
+theorem mean_weighted_test_le {Ω : Type uΩ} {α : Type uα} [Fintype Ω] [Nonempty Ω] [Fintype α]
+    (w f : Ω → α → ℝ) (hf : ∀ ω x, |f ω x| ≤ 1) :
+    |mean (fun ω => ∑ x, w ω x * f ω x)| ≤
+      Real.sqrt ((Fintype.card α : ℝ) * mean (fun ω => ∑ x, w ω x ^ 2)) := by
+  calc
+    _ ≤ mean (fun ω => |∑ x, w ω x * f ω x|) := abs_mean_le _
+    _ ≤ mean (fun ω => ∑ x, |w ω x|) := by
+      apply mean_le_mean
+      intro ω
+      calc
+        _ ≤ ∑ x, |w ω x * f ω x| := Finset.abs_sum_le_sum_abs _ _
+        _ ≤ _ := Finset.sum_le_sum (fun x _ => by
+          rw [abs_mul]
+          simpa only [mul_one] using mul_le_mul_of_nonneg_left (hf ω x) (abs_nonneg _))
+    _ ≤ _ := mean_l1_le_sqrt w
+
+theorem tagged_deviation_bound {ι : Type uι} (d : ℕ)
+    (B : Position (d + 1) → Bool) (tag : Position (d + 1)) (ht : B tag = true)
+    (e : ι → Position (d + 1)) (he : ∀ i, B (e i) = false)
+    (t : ℕ) (f : (ι → Position (d + 1)) → Position (d + 1) → ℝ)
+    (hf : ∀ e' y, |f e' y| ≤ 1) :
+    |mean (fun ω : History (d + 1) t =>
+      f ((run (d + 1) t ω) ∘ e) (run (d + 1) t ω tag) -
+        (1 / (freeCount B : ℝ)) *
+          (∑ y, if B ((run (d + 1) t ω).symm y) then f ((run (d + 1) t ω) ∘ e) y else 0))| ≤
+    Real.sqrt ((Fintype.card (Position (d + 1)) : ℝ) *
+      expectedEnergy d (initialState (d + 1) B tag ht) t) := by
+  rw [tagged_deviation_identity d B tag ht e he]
+  exact mean_weighted_test_le _ _ (fun ω y => hf _ y)
+
+end Conditional
+
+end Thorp
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/Trimming.lean b/lean/OAI/Probability/ThorpShuffle/Trimming.lean
index 11a8986ea7b6138036cf6d9e78170aae332d5fd1..3bbaf45f5e97a7835c12bec44b584c0eee24ddfc 100644
--- a/lean/OAI/Probability/ThorpShuffle/Trimming.lean
+++ b/lean/OAI/Probability/ThorpShuffle/Trimming.lean
@@ -1,235 +1,235 @@
-import OAI.Probability.ThorpShuffle.ConvolutionBounds
-
-namespace OAI
-
-universe uX uY uΩ uI uG uH uA
-
-noncomputable section
-
-open scoped BigOperators ComplexConjugate InnerProductSpace
-open Filter Topology
-
-namespace Thorp
-
-namespace Trim
-open scoped Classical
-
-variable {X : Type uX} {Y : Type uY} [Fintype X] [Fintype Y]
-
-def push (μ : X → ℝ) (f : X → Y) (y : Y) : ℝ := ∑ x, if f x = y then μ x else 0
-
-omit [Fintype Y] in
-lemma push_nonneg (μ : X → ℝ) (hμ : ∀ x, 0 ≤ μ x) (f : X → Y) (y : Y) :
-    0 ≤ push μ f y := Finset.sum_nonneg (fun x _ => by split_ifs <;> first | exact hμ _ | exact le_rfl)
-
-lemma push_sum (μ : X → ℝ) (f : X → Y) : ∑ y, push μ f y = ∑ x, μ x := by
-  unfold push
-  rw [Finset.sum_comm]
-  simp
-
-lemma push_test (μ : X → ℝ) (f : X → Y) (s : Y → ℝ) :
-    ∑ y, push μ f y * s y = ∑ x, μ x * s (f x) := by
-  unfold push
-  simp only [Finset.sum_mul, ite_mul, zero_mul]
-  rw [Finset.sum_comm]
-  simp
-
-omit [Fintype Y] in
-lemma push_fairMass {Ω : Type uΩ} [Fintype Ω] (a : Ω → X) (f : X → Y) :
-    push (fairMass a) f = fairMass (f ∘ a) := by
-  funext y
-  have he (x : X) : (if f x = y then fairMass a x else 0) =
-      (∑ ω, if a ω = x then (if f x = y then (1:ℝ) else 0) else 0) / Fintype.card Ω := by
-    by_cases h : f x = y <;> simp [h, fairMass]
-  unfold push
-  simp_rw [he]
-  rw [← Finset.sum_div]
-  unfold fairMass
-  congr 1
-  rw [Finset.sum_comm]
-  simp
-
-lemma push_bad_mass (μ υ : X → ℝ) (f : X → Y)
-    (hm : ∑ x, μ x = ∑ x, υ x) :
-    (∑ x, if 2 * push υ f (f x) < push μ f (f x) then μ x else 0) ≤
-      2 * tv (push μ f) (push υ f) := by
-  let s := Finset.univ.filter (fun y => 2 * push υ f y < push μ f y)
-  have he : (∑ x, if 2 * push υ f (f x) < push μ f (f x) then μ x else 0) =
-      ∑ y ∈ s, push μ f y := by
-    rw [Finset.sum_filter]
-    have ht := push_test μ f (fun y => if 2 * push υ f y < push μ f y then 1 else 0)
-    simpa only [mul_ite, mul_one, mul_zero] using ht.symm
-  have ht := sum_sub_le_tv (push μ f) (push υ f) (by simp only [push_sum, hm]) s
-  have hs : ∑ y ∈ s, push μ f y ≤ 2 * ∑ y ∈ s, (push μ f y - push υ f y) := by
-    rw [Finset.mul_sum]
-    exact Finset.sum_le_sum (fun y hy => by have hh := (Finset.mem_filter.mp hy).2; linarith)
-  rw [he]
-  linarith
-
-lemma uniform_eq_fairMass_id : (fun _ : X => (Fintype.card X : ℝ)⁻¹) = fairMass id := by
-  funext x
-  simp [fairMass]
-
-variable {I : Type uI} [Fintype I]
-variable {G : Type uG} {H : Type uH} [Group G] [Fintype G] [Group H] [Fintype H]
-
-def cosetMass (ψ : (I → H) →* G) (μ : G → ℝ) (i : I) (g : G) : ℝ :=
-  ∑ h, μ (g * ψ (Pi.mulSingle i h))
-
-omit [Fintype I] [Fintype G] in
-lemma cosetMass_right (ψ : (I → H) →* G) (μ : G → ℝ) (i : I) (g : G) (a : H) :
-    cosetMass ψ μ i (g * ψ (Pi.mulSingle i a)) = cosetMass ψ μ i g := by
-  unfold cosetMass
-  have he (h : H) : g * ψ (Pi.mulSingle i a) * ψ (Pi.mulSingle i h) =
-      g * ψ (Pi.mulSingle i (a * h)) := by rw [Pi.mulSingle_mul, map_mul, mul_assoc]
-  simp_rw [he]
-  exact Equiv.sum_comp (Equiv.mulLeft a) (fun h => μ (g * ψ (Pi.mulSingle i h)))
-
-def good (ψ : (I → H) →* G) (μ : G → ℝ) (g : G) : Prop :=
-  ∀ i, cosetMass ψ μ i g ≤ 2 * ((Fintype.card H : ℝ) / Fintype.card G)
-
-def discarded (ψ : (I → H) →* G) (μ : G → ℝ) : ℝ :=
-  ∑ g, if good ψ μ g then 0 else μ g
-
-def kept (ψ : (I → H) →* G) (μ : G → ℝ) (g : G) : ℝ :=
-  if good ψ μ g then μ g else 0
-
-def normalized (ψ : (I → H) →* G) (μ : G → ℝ) (g : G) : ℝ :=
-  kept ψ μ g / (1 - discarded ψ μ)
-
-omit [Fintype I] in
-lemma discarded_nonneg (ψ : (I → H) →* G) (μ : G → ℝ) (hμ : ∀ g, 0 ≤ μ g) :
-    0 ≤ discarded ψ μ := Finset.sum_nonneg (fun g _ => by split_ifs <;> first | exact hμ _ | exact le_rfl)
-
-omit [Fintype I] in
-lemma kept_add_discarded (ψ : (I → H) →* G) (μ : G → ℝ) :
-    (∑ g, kept ψ μ g) + discarded ψ μ = ∑ g, μ g := by
-  rw [discarded, ← Finset.sum_add_distrib]
-  apply Finset.sum_congr rfl
-  intro g _
-  unfold kept
-  split_ifs <;> ring
-
-lemma discarded_bound (ψ : (I → H) →* G) (μ : G → ℝ) (hμ : ∀ g, 0 ≤ μ g)
-    (h1 : ∑ g, μ g = 1) (A : I → Type uA) [∀ i, Fintype (A i)] (f : ∀ i, G → A i)
-    (hM : ∀ i g, push μ (f i) (f i g) = cosetMass ψ μ i g)
-    (hU : ∀ i g, push (fun _ : G => (Fintype.card G : ℝ)⁻¹) (f i) (f i g) =
-      (Fintype.card H : ℝ) / Fintype.card G) :
-    discarded ψ μ ≤ 2 * ∑ i, tv (push μ (f i)) (push (fun _ : G => (Fintype.card G : ℝ)⁻¹) (f i)) := by
-  have hu : (∑ _ : G, (Fintype.card G : ℝ)⁻¹) = 1 := by simp [Fintype.card_ne_zero]
-  have hpoint (g : G) : (if good ψ μ g then 0 else μ g) ≤
-      ∑ i, if 2 * push (fun _ : G => (Fintype.card G : ℝ)⁻¹) (f i) (f i g) < push μ (f i) (f i g)
-        then μ g else 0 := by
-    simp only [hM, hU]
-    by_cases hg : good ψ μ g
-    · rw [ite_eq_left hg]
-      exact Finset.sum_nonneg (fun i _ => by split_ifs <;> first | exact hμ _ | exact le_rfl)
-    · rw [ite_eq_right hg]
-      have ⟨i, hi⟩ : ∃ i, 2 * ((Fintype.card H : ℝ) / Fintype.card G) < cosetMass ψ μ i g := by
-        simpa only [good, not_forall, not_le] using hg
-      have hh := Finset.single_le_sum (s := Finset.univ)
-        (f := fun i => if 2 * ((Fintype.card H : ℝ) / Fintype.card G) < cosetMass ψ μ i g then μ g else 0)
-        (fun j _ => by split_ifs <;> first | exact hμ _ | exact le_rfl) (Finset.mem_univ i)
-      simpa only [ite_eq_left hi] using hh
-  calc
-    _ ≤ ∑ g, ∑ i, if 2 * push (fun _ : G => (Fintype.card G : ℝ)⁻¹) (f i) (f i g) < push μ (f i) (f i g)
-          then μ g else 0 := Finset.sum_le_sum (fun g _ => hpoint g)
-    _ = ∑ i, ∑ g, if 2 * push (fun _ : G => (Fintype.card G : ℝ)⁻¹) (f i) (f i g) < push μ (f i) (f i g)
-          then μ g else 0 := Finset.sum_comm
-    _ ≤ ∑ i, 2 * tv (push μ (f i)) (push (fun _ : G => (Fintype.card G : ℝ)⁻¹) (f i)) :=
-      Finset.sum_le_sum (fun i _ => push_bad_mass _ _ (f i) (h1.trans hu.symm))
-    _ = _ := by rw [Finset.mul_sum]
-
-end Trim
-
-namespace Trim
-open scoped Classical
-variable {I : Type uI} {G : Type uG} {H : Type uH} [Group G] [Fintype G] [Group H] [Fintype H]
-
-lemma normalized_nonneg (ψ : (I → H) →* G) (μ : G → ℝ) (hμ : ∀ g, 0 ≤ μ g)
-    (hδ : discarded ψ μ < 1) (g : G) : 0 ≤ normalized ψ μ g := by
-  unfold normalized kept
-  exact div_nonneg (by split_ifs <;> first | exact hμ _ | exact le_rfl) (by linarith)
-
-lemma normalized_sum (ψ : (I → H) →* G) (μ : G → ℝ) (h1 : ∑ g, μ g = 1)
-    (hδ : discarded ψ μ < 1) : ∑ g, normalized ψ μ g = 1 := by
-  have hk : ∑ g, kept ψ μ g = 1 - discarded ψ μ := by
-    have hh := kept_add_discarded ψ μ
-    rw [h1] at hh
-    linarith
-  simp only [normalized, ← Finset.sum_div, hk]
-  exact div_self (by linarith)
-
-lemma normalized_tv (ψ : (I → H) →* G) (μ : G → ℝ) (hμ : ∀ g, 0 ≤ μ g)
-    (h1 : ∑ g, μ g = 1) (hδ : discarded ψ μ < 1) :
-    tv μ (normalized ψ μ) = discarded ψ μ := by
-  have hden : 0 < 1 - discarded ψ μ := by linarith
-  have hd0 := discarded_nonneg ψ μ hμ
-  have hpt (g : G) : |μ g - normalized ψ μ g| =
-      normalized ψ μ g - μ g + 2 * (if good ψ μ g then 0 else μ g) := by
-    by_cases hg : good ψ μ g
-    · have he : normalized ψ μ g = μ g / (1 - discarded ψ μ) := by simp only [normalized, kept, ite_eq_left hg]
-      have hle : μ g ≤ normalized ψ μ g := by rw [he]; apply (le_div_iff₀ hden).mpr; nlinarith [hμ g]
-      rw [abs_of_nonpos (sub_nonpos.mpr hle), ite_eq_left hg]
-      ring
-    · simp only [normalized, kept, ite_eq_right hg, zero_div, sub_zero, abs_of_nonneg (hμ g)]
-      ring
-  unfold tv
-  simp only [hpt, Finset.sum_add_distrib, Finset.sum_sub_distrib, ← Finset.mul_sum,
-    normalized_sum ψ μ h1 hδ, h1]
-  change (1/2 : ℝ) * (1-1+2*discarded ψ μ) = discarded ψ μ
-  ring
-
-lemma normalized_coset (ψ : (I → H) →* G) (μ : G → ℝ) (hμ : ∀ g, 0 ≤ μ g)
-    (hδ : discarded ψ μ ≤ 1/2) (i : I) (g : G) :
-    cosetMass ψ (normalized ψ μ) i g ≤ 4 * (Fintype.card H : ℝ) / Fintype.card G := by
-  have hden : 0 < 1 - discarded ψ μ := by linarith
-  have hu : 0 ≤ (Fintype.card H : ℝ) / Fintype.card G := by positivity
-  by_cases hg : cosetMass ψ μ i g ≤ 2 * ((Fintype.card H : ℝ) / Fintype.card G)
-  · have hle : cosetMass ψ (normalized ψ μ) i g ≤ cosetMass ψ μ i g / (1-discarded ψ μ) := by
-      simp only [cosetMass, normalized, ← Finset.sum_div]
-      apply div_le_div_of_nonneg_right _ hden.le
-      apply Finset.sum_le_sum
-      intro h _
-      unfold kept
-      split_ifs <;> simp [hμ]
-    apply hle.trans
-    apply (div_le_iff₀ hden).mpr
-    have hh := mul_le_mul_of_nonneg_left (show (1/2 : ℝ) ≤ 1-discarded ψ μ by linarith) (show 0 ≤ 4*((Fintype.card H : ℝ)/Fintype.card G) by positivity)
-    rw [mul_div_assoc]
-    nlinarith
-  · have hz (h : H) : normalized ψ μ (g * ψ (Pi.mulSingle i h)) = 0 := by
-      have hn : ¬ good ψ μ (g * ψ (Pi.mulSingle i h)) := by
-        intro hh
-        have hi := hh i
-        rw [cosetMass_right] at hi
-        exact hg hi
-      simp only [normalized, kept, ite_eq_right hn, zero_div]
-    simp only [cosetMass, hz, Finset.sum_const_zero]
-    positivity
-
-lemma normalized_bias (ψ : (I → H) →* G) (μ : G → ℝ) (hμ : ∀ g, 0 ≤ μ g)
-    (h1 : ∑ g, μ g = 1) (hδ : discarded ψ μ < 1) (χ : G → ℂ)
-    (hχ : ∀ g, ‖χ g‖ ≤ 1) (hb : ∑ g, (μ g : ℂ) * χ g = 0) :
-    ‖∑ g, (normalized ψ μ g : ℂ) * χ g‖ ≤ 2 * discarded ψ μ := by
-  have he : (∑ g, (normalized ψ μ g : ℂ) * χ g) =
-      ∑ g, ((normalized ψ μ g - μ g : ℝ) : ℂ) * χ g := by
-    simp only [Complex.ofReal_sub, sub_mul, Finset.sum_sub_distrib, hb, sub_zero]
-  rw [he]
-  calc
-    _ ≤ ∑ g, ‖((normalized ψ μ g - μ g : ℝ) : ℂ) * χ g‖ := norm_sum_le _ _
-    _ ≤ ∑ g, |normalized ψ μ g - μ g| := by
-      apply Finset.sum_le_sum
-      intro g _
-      simp only [norm_mul, Complex.norm_real, Real.norm_eq_abs]
-      simpa only [mul_one] using mul_le_mul_of_nonneg_left (hχ g) (abs_nonneg (normalized ψ μ g - μ g))
-    _ = 2 * tv μ (normalized ψ μ) := by simp only [tv, abs_sub_comm]; ring
-    _ = _ := by rw [normalized_tv ψ μ hμ h1 hδ]
-
-end Trim
-
-end Thorp
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.ConvolutionBounds
+
+namespace OAI
+
+universe uX uY uΩ uI uG uH uA
+
+noncomputable section
+
+open scoped BigOperators ComplexConjugate InnerProductSpace
+open Filter Topology
+
+namespace Thorp
+
+namespace Trim
+open scoped Classical
+
+variable {X : Type uX} {Y : Type uY} [Fintype X] [Fintype Y]
+
+def push (μ : X → ℝ) (f : X → Y) (y : Y) : ℝ := ∑ x, if f x = y then μ x else 0
+
+omit [Fintype Y] in
+lemma push_nonneg (μ : X → ℝ) (hμ : ∀ x, 0 ≤ μ x) (f : X → Y) (y : Y) :
+    0 ≤ push μ f y := Finset.sum_nonneg (fun x _ => by split_ifs <;> first | exact hμ _ | exact le_rfl)
+
+lemma push_sum (μ : X → ℝ) (f : X → Y) : ∑ y, push μ f y = ∑ x, μ x := by
+  unfold push
+  rw [Finset.sum_comm]
+  simp
+
+lemma push_test (μ : X → ℝ) (f : X → Y) (s : Y → ℝ) :
+    ∑ y, push μ f y * s y = ∑ x, μ x * s (f x) := by
+  unfold push
+  simp only [Finset.sum_mul, ite_mul, zero_mul]
+  rw [Finset.sum_comm]
+  simp
+
+omit [Fintype Y] in
+lemma push_fairMass {Ω : Type uΩ} [Fintype Ω] (a : Ω → X) (f : X → Y) :
+    push (fairMass a) f = fairMass (f ∘ a) := by
+  funext y
+  have he (x : X) : (if f x = y then fairMass a x else 0) =
+      (∑ ω, if a ω = x then (if f x = y then (1:ℝ) else 0) else 0) / Fintype.card Ω := by
+    by_cases h : f x = y <;> simp [h, fairMass]
+  unfold push
+  simp_rw [he]
+  rw [← Finset.sum_div]
+  unfold fairMass
+  congr 1
+  rw [Finset.sum_comm]
+  simp
+
+lemma push_bad_mass (μ υ : X → ℝ) (f : X → Y)
+    (hm : ∑ x, μ x = ∑ x, υ x) :
+    (∑ x, if 2 * push υ f (f x) < push μ f (f x) then μ x else 0) ≤
+      2 * tv (push μ f) (push υ f) := by
+  let s := Finset.univ.filter (fun y => 2 * push υ f y < push μ f y)
+  have he : (∑ x, if 2 * push υ f (f x) < push μ f (f x) then μ x else 0) =
+      ∑ y ∈ s, push μ f y := by
+    rw [Finset.sum_filter]
+    have ht := push_test μ f (fun y => if 2 * push υ f y < push μ f y then 1 else 0)
+    simpa only [mul_ite, mul_one, mul_zero] using ht.symm
+  have ht := sum_sub_le_tv (push μ f) (push υ f) (by simp only [push_sum, hm]) s
+  have hs : ∑ y ∈ s, push μ f y ≤ 2 * ∑ y ∈ s, (push μ f y - push υ f y) := by
+    rw [Finset.mul_sum]
+    exact Finset.sum_le_sum (fun y hy => by have hh := (Finset.mem_filter.mp hy).2; linarith)
+  rw [he]
+  linarith
+
+lemma uniform_eq_fairMass_id : (fun _ : X => (Fintype.card X : ℝ)⁻¹) = fairMass id := by
+  funext x
+  simp [fairMass]
+
+variable {I : Type uI} [Fintype I]
+variable {G : Type uG} {H : Type uH} [Group G] [Fintype G] [Group H] [Fintype H]
+
+def cosetMass (ψ : (I → H) →* G) (μ : G → ℝ) (i : I) (g : G) : ℝ :=
+  ∑ h, μ (g * ψ (Pi.mulSingle i h))
+
+omit [Fintype I] [Fintype G] in
+lemma cosetMass_right (ψ : (I → H) →* G) (μ : G → ℝ) (i : I) (g : G) (a : H) :
+    cosetMass ψ μ i (g * ψ (Pi.mulSingle i a)) = cosetMass ψ μ i g := by
+  unfold cosetMass
+  have he (h : H) : g * ψ (Pi.mulSingle i a) * ψ (Pi.mulSingle i h) =
+      g * ψ (Pi.mulSingle i (a * h)) := by rw [Pi.mulSingle_mul, map_mul, mul_assoc]
+  simp_rw [he]
+  exact Equiv.sum_comp (Equiv.mulLeft a) (fun h => μ (g * ψ (Pi.mulSingle i h)))
+
+def good (ψ : (I → H) →* G) (μ : G → ℝ) (g : G) : Prop :=
+  ∀ i, cosetMass ψ μ i g ≤ 2 * ((Fintype.card H : ℝ) / Fintype.card G)
+
+def discarded (ψ : (I → H) →* G) (μ : G → ℝ) : ℝ :=
+  ∑ g, if good ψ μ g then 0 else μ g
+
+def kept (ψ : (I → H) →* G) (μ : G → ℝ) (g : G) : ℝ :=
+  if good ψ μ g then μ g else 0
+
+def normalized (ψ : (I → H) →* G) (μ : G → ℝ) (g : G) : ℝ :=
+  kept ψ μ g / (1 - discarded ψ μ)
+
+omit [Fintype I] in
+lemma discarded_nonneg (ψ : (I → H) →* G) (μ : G → ℝ) (hμ : ∀ g, 0 ≤ μ g) :
+    0 ≤ discarded ψ μ := Finset.sum_nonneg (fun g _ => by split_ifs <;> first | exact hμ _ | exact le_rfl)
+
+omit [Fintype I] in
+lemma kept_add_discarded (ψ : (I → H) →* G) (μ : G → ℝ) :
+    (∑ g, kept ψ μ g) + discarded ψ μ = ∑ g, μ g := by
+  rw [discarded, ← Finset.sum_add_distrib]
+  apply Finset.sum_congr rfl
+  intro g _
+  unfold kept
+  split_ifs <;> ring
+
+lemma discarded_bound (ψ : (I → H) →* G) (μ : G → ℝ) (hμ : ∀ g, 0 ≤ μ g)
+    (h1 : ∑ g, μ g = 1) (A : I → Type uA) [∀ i, Fintype (A i)] (f : ∀ i, G → A i)
+    (hM : ∀ i g, push μ (f i) (f i g) = cosetMass ψ μ i g)
+    (hU : ∀ i g, push (fun _ : G => (Fintype.card G : ℝ)⁻¹) (f i) (f i g) =
+      (Fintype.card H : ℝ) / Fintype.card G) :
+    discarded ψ μ ≤ 2 * ∑ i, tv (push μ (f i)) (push (fun _ : G => (Fintype.card G : ℝ)⁻¹) (f i)) := by
+  have hu : (∑ _ : G, (Fintype.card G : ℝ)⁻¹) = 1 := by simp [Fintype.card_ne_zero]
+  have hpoint (g : G) : (if good ψ μ g then 0 else μ g) ≤
+      ∑ i, if 2 * push (fun _ : G => (Fintype.card G : ℝ)⁻¹) (f i) (f i g) < push μ (f i) (f i g)
+        then μ g else 0 := by
+    simp only [hM, hU]
+    by_cases hg : good ψ μ g
+    · rw [ite_eq_left hg]
+      exact Finset.sum_nonneg (fun i _ => by split_ifs <;> first | exact hμ _ | exact le_rfl)
+    · rw [ite_eq_right hg]
+      have ⟨i, hi⟩ : ∃ i, 2 * ((Fintype.card H : ℝ) / Fintype.card G) < cosetMass ψ μ i g := by
+        simpa only [good, not_forall, not_le] using hg
+      have hh := Finset.single_le_sum (s := Finset.univ)
+        (f := fun i => if 2 * ((Fintype.card H : ℝ) / Fintype.card G) < cosetMass ψ μ i g then μ g else 0)
+        (fun j _ => by split_ifs <;> first | exact hμ _ | exact le_rfl) (Finset.mem_univ i)
+      simpa only [ite_eq_left hi] using hh
+  calc
+    _ ≤ ∑ g, ∑ i, if 2 * push (fun _ : G => (Fintype.card G : ℝ)⁻¹) (f i) (f i g) < push μ (f i) (f i g)
+          then μ g else 0 := Finset.sum_le_sum (fun g _ => hpoint g)
+    _ = ∑ i, ∑ g, if 2 * push (fun _ : G => (Fintype.card G : ℝ)⁻¹) (f i) (f i g) < push μ (f i) (f i g)
+          then μ g else 0 := Finset.sum_comm
+    _ ≤ ∑ i, 2 * tv (push μ (f i)) (push (fun _ : G => (Fintype.card G : ℝ)⁻¹) (f i)) :=
+      Finset.sum_le_sum (fun i _ => push_bad_mass _ _ (f i) (h1.trans hu.symm))
+    _ = _ := by rw [Finset.mul_sum]
+
+end Trim
+
+namespace Trim
+open scoped Classical
+variable {I : Type uI} {G : Type uG} {H : Type uH} [Group G] [Fintype G] [Group H] [Fintype H]
+
+lemma normalized_nonneg (ψ : (I → H) →* G) (μ : G → ℝ) (hμ : ∀ g, 0 ≤ μ g)
+    (hδ : discarded ψ μ < 1) (g : G) : 0 ≤ normalized ψ μ g := by
+  unfold normalized kept
+  exact div_nonneg (by split_ifs <;> first | exact hμ _ | exact le_rfl) (by linarith)
+
+lemma normalized_sum (ψ : (I → H) →* G) (μ : G → ℝ) (h1 : ∑ g, μ g = 1)
+    (hδ : discarded ψ μ < 1) : ∑ g, normalized ψ μ g = 1 := by
+  have hk : ∑ g, kept ψ μ g = 1 - discarded ψ μ := by
+    have hh := kept_add_discarded ψ μ
+    rw [h1] at hh
+    linarith
+  simp only [normalized, ← Finset.sum_div, hk]
+  exact div_self (by linarith)
+
+lemma normalized_tv (ψ : (I → H) →* G) (μ : G → ℝ) (hμ : ∀ g, 0 ≤ μ g)
+    (h1 : ∑ g, μ g = 1) (hδ : discarded ψ μ < 1) :
+    tv μ (normalized ψ μ) = discarded ψ μ := by
+  have hden : 0 < 1 - discarded ψ μ := by linarith
+  have hd0 := discarded_nonneg ψ μ hμ
+  have hpt (g : G) : |μ g - normalized ψ μ g| =
+      normalized ψ μ g - μ g + 2 * (if good ψ μ g then 0 else μ g) := by
+    by_cases hg : good ψ μ g
+    · have he : normalized ψ μ g = μ g / (1 - discarded ψ μ) := by simp only [normalized, kept, ite_eq_left hg]
+      have hle : μ g ≤ normalized ψ μ g := by rw [he]; apply (le_div_iff₀ hden).mpr; nlinarith [hμ g]
+      rw [abs_of_nonpos (sub_nonpos.mpr hle), ite_eq_left hg]
+      ring
+    · simp only [normalized, kept, ite_eq_right hg, zero_div, sub_zero, abs_of_nonneg (hμ g)]
+      ring
+  unfold tv
+  simp only [hpt, Finset.sum_add_distrib, Finset.sum_sub_distrib, ← Finset.mul_sum,
+    normalized_sum ψ μ h1 hδ, h1]
+  change (1/2 : ℝ) * (1-1+2*discarded ψ μ) = discarded ψ μ
+  ring
+
+lemma normalized_coset (ψ : (I → H) →* G) (μ : G → ℝ) (hμ : ∀ g, 0 ≤ μ g)
+    (hδ : discarded ψ μ ≤ 1/2) (i : I) (g : G) :
+    cosetMass ψ (normalized ψ μ) i g ≤ 4 * (Fintype.card H : ℝ) / Fintype.card G := by
+  have hden : 0 < 1 - discarded ψ μ := by linarith
+  have hu : 0 ≤ (Fintype.card H : ℝ) / Fintype.card G := by positivity
+  by_cases hg : cosetMass ψ μ i g ≤ 2 * ((Fintype.card H : ℝ) / Fintype.card G)
+  · have hle : cosetMass ψ (normalized ψ μ) i g ≤ cosetMass ψ μ i g / (1-discarded ψ μ) := by
+      simp only [cosetMass, normalized, ← Finset.sum_div]
+      apply div_le_div_of_nonneg_right _ hden.le
+      apply Finset.sum_le_sum
+      intro h _
+      unfold kept
+      split_ifs <;> simp [hμ]
+    apply hle.trans
+    apply (div_le_iff₀ hden).mpr
+    have hh := mul_le_mul_of_nonneg_left (show (1/2 : ℝ) ≤ 1-discarded ψ μ by linarith) (show 0 ≤ 4*((Fintype.card H : ℝ)/Fintype.card G) by positivity)
+    rw [mul_div_assoc]
+    nlinarith
+  · have hz (h : H) : normalized ψ μ (g * ψ (Pi.mulSingle i h)) = 0 := by
+      have hn : ¬ good ψ μ (g * ψ (Pi.mulSingle i h)) := by
+        intro hh
+        have hi := hh i
+        rw [cosetMass_right] at hi
+        exact hg hi
+      simp only [normalized, kept, ite_eq_right hn, zero_div]
+    simp only [cosetMass, hz, Finset.sum_const_zero]
+    positivity
+
+lemma normalized_bias (ψ : (I → H) →* G) (μ : G → ℝ) (hμ : ∀ g, 0 ≤ μ g)
+    (h1 : ∑ g, μ g = 1) (hδ : discarded ψ μ < 1) (χ : G → ℂ)
+    (hχ : ∀ g, ‖χ g‖ ≤ 1) (hb : ∑ g, (μ g : ℂ) * χ g = 0) :
+    ‖∑ g, (normalized ψ μ g : ℂ) * χ g‖ ≤ 2 * discarded ψ μ := by
+  have he : (∑ g, (normalized ψ μ g : ℂ) * χ g) =
+      ∑ g, ((normalized ψ μ g - μ g : ℝ) : ℂ) * χ g := by
+    simp only [Complex.ofReal_sub, sub_mul, Finset.sum_sub_distrib, hb, sub_zero]
+  rw [he]
+  calc
+    _ ≤ ∑ g, ‖((normalized ψ μ g - μ g : ℝ) : ℂ) * χ g‖ := norm_sum_le _ _
+    _ ≤ ∑ g, |normalized ψ μ g - μ g| := by
+      apply Finset.sum_le_sum
+      intro g _
+      simp only [norm_mul, Complex.norm_real, Real.norm_eq_abs]
+      simpa only [mul_one] using mul_le_mul_of_nonneg_left (hχ g) (abs_nonneg (normalized ψ μ g - μ g))
+    _ = 2 * tv μ (normalized ψ μ) := by simp only [tv, abs_sub_comm]; ring
+    _ = _ := by rw [normalized_tv ψ μ hμ h1 hδ]
+
+end Trim
+
+end Thorp
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/VectorOrbits.lean b/lean/OAI/Probability/ThorpShuffle/VectorOrbits.lean
index 8d4077cfe38a1fa46af0ce8cfcd66c9d373fa11e..b86552a0e78698ace96a977b39920c79f4d0dc1c 100644
--- a/lean/OAI/Probability/ThorpShuffle/VectorOrbits.lean
+++ b/lean/OAI/Probability/ThorpShuffle/VectorOrbits.lean
@@ -1,187 +1,187 @@
-import OAI.Probability.ThorpShuffle.SymmetricFamilies
-
-namespace OAI
-
-universe uG uE uX uH uα
-
-noncomputable section
-
-open scoped BigOperators ComplexConjugate InnerProductSpace
-open Filter Topology
-
-namespace Thorp.Specht
-open scoped Classical
-open Thorp.Young
-
-variable {G : Type uG} [Group G]
-variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
-
-lemma cyclic_scalar_mem (ρ : Representation ℂ G E) (e : E) (χ : G → ℂ)
-    (he : ∀ g, ρ g e = χ g • e) (v : E) (hv : v ∈ cyclic ρ e) :
-    ∃ c : ℂ, c • e = v := by
-  induction hv using Submodule.span_induction with
-  | mem v hv => obtain ⟨g, rfl⟩ := hv; exact ⟨χ g, (he g).symm⟩
-  | zero => exact ⟨0, zero_smul _ _⟩
-  | add x y hx hy ihx ihy =>
-    obtain ⟨a, rfl⟩ := ihx
-    obtain ⟨b, rfl⟩ := ihy
-    exact ⟨a+b, add_smul _ _ _⟩
-  | smul a x hx ih =>
-    obtain ⟨b, rfl⟩ := ih
-    exact ⟨a*b, mul_smul _ _ _⟩
-
-lemma cyclic_scalar_action (ρ : Representation ℂ G E) (e : E) (χ : G → ℂ)
-    (he : ∀ g, ρ g e = χ g • e) (g : G) (v : (cyclic ρ e).toSubmodule) :
-    (cyclic ρ e).toRepresentation g v = χ g • v := by
-  obtain ⟨a, ha⟩ := cyclic_scalar_mem ρ e χ he v.val v.property
-  apply Subtype.ext
-  change ρ g v.val = χ g • v.val
-  rw [← ha, map_smul, he, smul_comm]
-
-lemma cyclic_scalar_finrank (ρ : Representation ℂ G E) (e : E) (χ : G → ℂ)
-    (he : ∀ g, ρ g e = χ g • e) : Module.finrank ℂ (cyclic ρ e).toSubmodule ≤ 1 := by
-  apply finrank_le_one (⟨e, self_mem_cyclic ρ e⟩ : (cyclic ρ e).toSubmodule)
-  intro v
-  obtain ⟨a, ha⟩ := cyclic_scalar_mem ρ e χ he v.val v.property
-  exact ⟨a, Subtype.ext ha⟩
-
-namespace PermutationSpace
-variable {X : Type uX} {H : Type uH} [Fintype X] [MulAction G X] [Group H] [Fintype H]
-omit [Fintype X] in
-lemma translate_alternator (ι : H →* G) (χ : H →* ℂ) (a : H)
-    (v : EuclideanSpace ℂ X) :
-    rep (ι a) (alternator ι χ v) = χ a⁻¹ • alternator ι χ v := by
-  simp only [alternator_apply, map_sum, Finset.smul_sum]
-  simp only [map_smul]
-  have hh := Equiv.sum_comp (Equiv.mulLeft a)
-    (fun h => χ a⁻¹ • (χ h • (rep (ι h) v : EuclideanSpace ℂ X)))
-  rw [← hh]
-  apply Finset.sum_congr rfl
-  intro h _
-  simp only [Equiv.coe_mulLeft, smul_smul, map_mul]
-  rw [show χ a⁻¹ * (χ a * χ h) = χ h by rw [← mul_assoc, ← map_mul, inv_mul_cancel, map_one, one_mul]]
-  rfl
-end PermutationSpace
-
-lemma row_zero_of_full (D : YoungDiagram) (h : D.rowLen 0 = Fintype.card (Cells D)) :
-    ∀ x : Cells D, x.val.1 = 0 := by
-  have hc : Fintype.card (Lower D) = 0 := by have hh := card_lower_add D; omega
-  have he : IsEmpty (Lower D) := Fintype.card_eq_zero_iff.mp hc
-  intro x
-  by_contra hx
-  exact he.false ⟨x, hx⟩
-
-lemma col_zero_of_full (D : YoungDiagram) (h : D.colLen 0 = Fintype.card (Cells D)) :
-    ∀ x : Cells D, x.val.2 = 0 := by
-  have hc : D.transpose.rowLen 0 = Fintype.card (Cells D.transpose) := by
-    rw [YoungDiagram.rowLen_transpose, ← Fintype.card_congr (transposeCells D).toEquiv, h]
-  intro x
-  exact row_zero_of_full D.transpose hc (transposeCells D x)
-
-variable {α : Type uα} [Fintype α]
-
-omit [Fintype α] in
-lemma row_rep_action (D : YoungDiagram) (t : α ≃ Cells D)
-    (h : ∀ x : Cells D, x.val.1 = 0) (g : Equiv.Perm α)
-    (v : EuclideanSpace ℂ (Tabloid (tableauRow D t))) : tabloidRep D t g v = v := by
-  ext f
-  change v (g⁻¹ • f) = v f
-  apply congrArg (fun a => v a)
-  apply Subtype.ext
-  funext x
-  obtain ⟨a, ha⟩ := f.property
-  simp only [tabloid_smul_apply, ha, Function.comp_apply, tableauRow, h]
-
-lemma column_poly_action (D : YoungDiagram) (t : α ≃ Cells D)
-    (h : ∀ x : Cells D, x.val.2 = 0) (g : Equiv.Perm α) :
-    tabloidRep D t g (polytabloid D t) = complexSign g • polytabloid D t := by
-  have hg : g ∈ fiberGroup (tableauCol D t) := by intro x; simp only [tableauCol, h]
-  have hh := PermutationSpace.translate_alternator (fiberGroup (tableauCol D t)).subtype
-    (complexSign.comp (fiberGroup (tableauCol D t)).subtype) ⟨g, hg⟩
-    (EuclideanSpace.single (baseTabloid (tableauRow D t)) (1 : ℂ))
-  change tabloidRep D t g (polytabloid D t) = complexSign g⁻¹ • polytabloid D t at hh
-  rwa [complexSign_inv] at hh
-
-lemma exceptional_action {n : ℕ} (p : NShape n) (hp : defect p = 0) :
-    degree p = 1 ∧ ((∀ g v, family p g v = v) ∨
-      (∀ g v, family p g v = complexSign g • v)) := by
-  let D := ofPartition p
-  let t := canonicalTableau p
-  have hL : longest D = Fintype.card (Cells D) := by
-    have hh := defect_add_longest p
-    rw [hp, zero_add] at hh
-    exact hh.trans (diagram_card p).symm
-  have hor : D.rowLen 0 = Fintype.card (Cells D) ∨ D.colLen 0 = Fintype.card (Cells D) := by
-    unfold longest at hL
-    rcases le_total (D.colLen 0) (D.rowLen 0) with h | h
-    · exact Or.inl (by rwa [max_eq_left h] at hL)
-    · exact Or.inr (by rwa [max_eq_right h] at hL)
-  rcases hor with hr | hc
-  · have he : ∀ g, tabloidRep D t g (polytabloid D t) = (1 : ℂ) • polytabloid D t := by
-      intro g; rw [one_smul]; exact row_rep_action D t (row_zero_of_full D hr) g _
-    have hd := cyclic_scalar_finrank (tabloidRep D t) (polytabloid D t) (fun _ => 1) he
-    refine ⟨Nat.le_antisymm hd (degree_pos p), Or.inl ?_⟩
-    intro g v
-    apply Subtype.ext
-    exact row_rep_action D t (row_zero_of_full D hr) g v.val
-  · have he := column_poly_action D t (col_zero_of_full D hc)
-    have hd := cyclic_scalar_finrank (tabloidRep D t) (polytabloid D t) complexSign he
-    refine ⟨Nat.le_antisymm hd (degree_pos p), Or.inr ?_⟩
-    exact cyclic_scalar_action (tabloidRep D t) (polytabloid D t) complexSign he
-
-end Thorp.Specht
-
-namespace Thorp.Fourier
-open scoped Classical
-variable {G : Type} [Group G] [Fintype G]
-variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℂ E] [FiniteDimensional ℂ E]
-
-omit [FiniteDimensional ℂ E] in
-lemma sum_rep_invariant (ρ : Representation ℂ G E) (g : G) (v : E) :
-    ρ g (∑ h, ρ h v) = ∑ h, ρ h v := by
-  rw [map_sum]
-  simp only [← Module.End.mul_apply, ← map_mul]
-  exact Equiv.sum_comp (Equiv.mulLeft g) (fun h => ρ h v)
-
-omit [Fintype G] [FiniteDimensional ℂ E] in
-lemma invariant_finrank_le_one (ρ : Representation ℂ G E) [Representation.IsIrreducible ρ]
-    (v : E) (hv : v ≠ 0) (hinv : ∀ g, ρ g v = v) : Module.finrank ℂ E ≤ 1 := by
-  let A : Representation.IntertwiningMap (Representation.trivial ℂ G ℂ) ρ := {
-    toLinearMap := LinearMap.toSpanSingleton ℂ E v
-    isIntertwining' := by intro g; ext; simp [Representation.trivial, hinv] }
-  have hA : A ≠ 0 := by
-    intro ha
-    have hh := DFunLike.congr_fun ha 1
-    exact hv (by simpa [A] using hh)
-  have hs := (Representation.IsIrreducible.surjective_or_eq_zero A).resolve_right hA
-  exact finrank_le_one v hs
-
-omit [FiniteDimensional ℂ E] in
-lemma integrated_uniform_zero (ρ : Representation ℂ G E) [Representation.IsIrreducible ρ]
-    (hd : 1 < Module.finrank ℂ E) :
-    integrated ρ (fun _ => (Fintype.card G : ℂ)⁻¹) = 0 := by
-  have hz (v : E) : (∑ h, ρ h v) = 0 := by
-    by_contra hn
-    have hh := invariant_finrank_le_one ρ _ hn (fun g => sum_rep_invariant ρ g v)
-    omega
-  ext v
-  simp only [integrated, LinearMap.sum_apply, LinearMap.smul_apply, ← Finset.smul_sum, hz, smul_zero,
-    LinearMap.zero_apply]
-
-omit [FiniteDimensional ℂ E] in
-lemma integrated_scalar (ρ : Representation ℂ G E) (χ : G → ℂ)
-    (hχ : ∀ g v, ρ g v = χ g • v) (f : G → ℂ) :
-    integrated ρ f = (∑ g, f g * χ g) • LinearMap.id := by
-  ext v
-  simp only [integrated, LinearMap.sum_apply, LinearMap.smul_apply, hχ, smul_smul,
-    Finset.sum_smul, LinearMap.id_apply]
-
-lemma hsSq_scalar (c : ℂ) : hsSq (c • (LinearMap.id : Module.End ℂ E)) =
-    (Module.finrank ℂ E : ℝ) * ‖c‖^2 := by
-  simp [hsSq, norm_smul, (stdOrthonormalBasis ℂ E).norm_eq_one]
-
-end Thorp.Fourier
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.SymmetricFamilies
+
+namespace OAI
+
+universe uG uE uX uH uα
+
+noncomputable section
+
+open scoped BigOperators ComplexConjugate InnerProductSpace
+open Filter Topology
+
+namespace Thorp.Specht
+open scoped Classical
+open Thorp.Young
+
+variable {G : Type uG} [Group G]
+variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
+
+lemma cyclic_scalar_mem (ρ : Representation ℂ G E) (e : E) (χ : G → ℂ)
+    (he : ∀ g, ρ g e = χ g • e) (v : E) (hv : v ∈ cyclic ρ e) :
+    ∃ c : ℂ, c • e = v := by
+  induction hv using Submodule.span_induction with
+  | mem v hv => obtain ⟨g, rfl⟩ := hv; exact ⟨χ g, (he g).symm⟩
+  | zero => exact ⟨0, zero_smul _ _⟩
+  | add x y hx hy ihx ihy =>
+    obtain ⟨a, rfl⟩ := ihx
+    obtain ⟨b, rfl⟩ := ihy
+    exact ⟨a+b, add_smul _ _ _⟩
+  | smul a x hx ih =>
+    obtain ⟨b, rfl⟩ := ih
+    exact ⟨a*b, mul_smul _ _ _⟩
+
+lemma cyclic_scalar_action (ρ : Representation ℂ G E) (e : E) (χ : G → ℂ)
+    (he : ∀ g, ρ g e = χ g • e) (g : G) (v : (cyclic ρ e).toSubmodule) :
+    (cyclic ρ e).toRepresentation g v = χ g • v := by
+  obtain ⟨a, ha⟩ := cyclic_scalar_mem ρ e χ he v.val v.property
+  apply Subtype.ext
+  change ρ g v.val = χ g • v.val
+  rw [← ha, map_smul, he, smul_comm]
+
+lemma cyclic_scalar_finrank (ρ : Representation ℂ G E) (e : E) (χ : G → ℂ)
+    (he : ∀ g, ρ g e = χ g • e) : Module.finrank ℂ (cyclic ρ e).toSubmodule ≤ 1 := by
+  apply finrank_le_one (⟨e, self_mem_cyclic ρ e⟩ : (cyclic ρ e).toSubmodule)
+  intro v
+  obtain ⟨a, ha⟩ := cyclic_scalar_mem ρ e χ he v.val v.property
+  exact ⟨a, Subtype.ext ha⟩
+
+namespace PermutationSpace
+variable {X : Type uX} {H : Type uH} [Fintype X] [MulAction G X] [Group H] [Fintype H]
+omit [Fintype X] in
+lemma translate_alternator (ι : H →* G) (χ : H →* ℂ) (a : H)
+    (v : EuclideanSpace ℂ X) :
+    rep (ι a) (alternator ι χ v) = χ a⁻¹ • alternator ι χ v := by
+  simp only [alternator_apply, map_sum, Finset.smul_sum]
+  simp only [map_smul]
+  have hh := Equiv.sum_comp (Equiv.mulLeft a)
+    (fun h => χ a⁻¹ • (χ h • (rep (ι h) v : EuclideanSpace ℂ X)))
+  rw [← hh]
+  apply Finset.sum_congr rfl
+  intro h _
+  simp only [Equiv.coe_mulLeft, smul_smul, map_mul]
+  rw [show χ a⁻¹ * (χ a * χ h) = χ h by rw [← mul_assoc, ← map_mul, inv_mul_cancel, map_one, one_mul]]
+  rfl
+end PermutationSpace
+
+lemma row_zero_of_full (D : YoungDiagram) (h : D.rowLen 0 = Fintype.card (Cells D)) :
+    ∀ x : Cells D, x.val.1 = 0 := by
+  have hc : Fintype.card (Lower D) = 0 := by have hh := card_lower_add D; omega
+  have he : IsEmpty (Lower D) := Fintype.card_eq_zero_iff.mp hc
+  intro x
+  by_contra hx
+  exact he.false ⟨x, hx⟩
+
+lemma col_zero_of_full (D : YoungDiagram) (h : D.colLen 0 = Fintype.card (Cells D)) :
+    ∀ x : Cells D, x.val.2 = 0 := by
+  have hc : D.transpose.rowLen 0 = Fintype.card (Cells D.transpose) := by
+    rw [YoungDiagram.rowLen_transpose, ← Fintype.card_congr (transposeCells D).toEquiv, h]
+  intro x
+  exact row_zero_of_full D.transpose hc (transposeCells D x)
+
+variable {α : Type uα} [Fintype α]
+
+omit [Fintype α] in
+lemma row_rep_action (D : YoungDiagram) (t : α ≃ Cells D)
+    (h : ∀ x : Cells D, x.val.1 = 0) (g : Equiv.Perm α)
+    (v : EuclideanSpace ℂ (Tabloid (tableauRow D t))) : tabloidRep D t g v = v := by
+  ext f
+  change v (g⁻¹ • f) = v f
+  apply congrArg (fun a => v a)
+  apply Subtype.ext
+  funext x
+  obtain ⟨a, ha⟩ := f.property
+  simp only [tabloid_smul_apply, ha, Function.comp_apply, tableauRow, h]
+
+lemma column_poly_action (D : YoungDiagram) (t : α ≃ Cells D)
+    (h : ∀ x : Cells D, x.val.2 = 0) (g : Equiv.Perm α) :
+    tabloidRep D t g (polytabloid D t) = complexSign g • polytabloid D t := by
+  have hg : g ∈ fiberGroup (tableauCol D t) := by intro x; simp only [tableauCol, h]
+  have hh := PermutationSpace.translate_alternator (fiberGroup (tableauCol D t)).subtype
+    (complexSign.comp (fiberGroup (tableauCol D t)).subtype) ⟨g, hg⟩
+    (EuclideanSpace.single (baseTabloid (tableauRow D t)) (1 : ℂ))
+  change tabloidRep D t g (polytabloid D t) = complexSign g⁻¹ • polytabloid D t at hh
+  rwa [complexSign_inv] at hh
+
+lemma exceptional_action {n : ℕ} (p : NShape n) (hp : defect p = 0) :
+    degree p = 1 ∧ ((∀ g v, family p g v = v) ∨
+      (∀ g v, family p g v = complexSign g • v)) := by
+  let D := ofPartition p
+  let t := canonicalTableau p
+  have hL : longest D = Fintype.card (Cells D) := by
+    have hh := defect_add_longest p
+    rw [hp, zero_add] at hh
+    exact hh.trans (diagram_card p).symm
+  have hor : D.rowLen 0 = Fintype.card (Cells D) ∨ D.colLen 0 = Fintype.card (Cells D) := by
+    unfold longest at hL
+    rcases le_total (D.colLen 0) (D.rowLen 0) with h | h
+    · exact Or.inl (by rwa [max_eq_left h] at hL)
+    · exact Or.inr (by rwa [max_eq_right h] at hL)
+  rcases hor with hr | hc
+  · have he : ∀ g, tabloidRep D t g (polytabloid D t) = (1 : ℂ) • polytabloid D t := by
+      intro g; rw [one_smul]; exact row_rep_action D t (row_zero_of_full D hr) g _
+    have hd := cyclic_scalar_finrank (tabloidRep D t) (polytabloid D t) (fun _ => 1) he
+    refine ⟨Nat.le_antisymm hd (degree_pos p), Or.inl ?_⟩
+    intro g v
+    apply Subtype.ext
+    exact row_rep_action D t (row_zero_of_full D hr) g v.val
+  · have he := column_poly_action D t (col_zero_of_full D hc)
+    have hd := cyclic_scalar_finrank (tabloidRep D t) (polytabloid D t) complexSign he
+    refine ⟨Nat.le_antisymm hd (degree_pos p), Or.inr ?_⟩
+    exact cyclic_scalar_action (tabloidRep D t) (polytabloid D t) complexSign he
+
+end Thorp.Specht
+
+namespace Thorp.Fourier
+open scoped Classical
+variable {G : Type} [Group G] [Fintype G]
+variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℂ E] [FiniteDimensional ℂ E]
+
+omit [FiniteDimensional ℂ E] in
+lemma sum_rep_invariant (ρ : Representation ℂ G E) (g : G) (v : E) :
+    ρ g (∑ h, ρ h v) = ∑ h, ρ h v := by
+  rw [map_sum]
+  simp only [← Module.End.mul_apply, ← map_mul]
+  exact Equiv.sum_comp (Equiv.mulLeft g) (fun h => ρ h v)
+
+omit [Fintype G] [FiniteDimensional ℂ E] in
+lemma invariant_finrank_le_one (ρ : Representation ℂ G E) [Representation.IsIrreducible ρ]
+    (v : E) (hv : v ≠ 0) (hinv : ∀ g, ρ g v = v) : Module.finrank ℂ E ≤ 1 := by
+  let A : Representation.IntertwiningMap (Representation.trivial ℂ G ℂ) ρ := {
+    toLinearMap := LinearMap.toSpanSingleton ℂ E v
+    isIntertwining' := by intro g; ext; simp [Representation.trivial, hinv] }
+  have hA : A ≠ 0 := by
+    intro ha
+    have hh := DFunLike.congr_fun ha 1
+    exact hv (by simpa [A] using hh)
+  have hs := (Representation.IsIrreducible.surjective_or_eq_zero A).resolve_right hA
+  exact finrank_le_one v hs
+
+omit [FiniteDimensional ℂ E] in
+lemma integrated_uniform_zero (ρ : Representation ℂ G E) [Representation.IsIrreducible ρ]
+    (hd : 1 < Module.finrank ℂ E) :
+    integrated ρ (fun _ => (Fintype.card G : ℂ)⁻¹) = 0 := by
+  have hz (v : E) : (∑ h, ρ h v) = 0 := by
+    by_contra hn
+    have hh := invariant_finrank_le_one ρ _ hn (fun g => sum_rep_invariant ρ g v)
+    omega
+  ext v
+  simp only [integrated, LinearMap.sum_apply, LinearMap.smul_apply, ← Finset.smul_sum, hz, smul_zero,
+    LinearMap.zero_apply]
+
+omit [FiniteDimensional ℂ E] in
+lemma integrated_scalar (ρ : Representation ℂ G E) (χ : G → ℂ)
+    (hχ : ∀ g v, ρ g v = χ g • v) (f : G → ℂ) :
+    integrated ρ f = (∑ g, f g * χ g) • LinearMap.id := by
+  ext v
+  simp only [integrated, LinearMap.sum_apply, LinearMap.smul_apply, hχ, smul_smul,
+    Finset.sum_smul, LinearMap.id_apply]
+
+lemma hsSq_scalar (c : ℂ) : hsSq (c • (LinearMap.id : Module.End ℂ E)) =
+    (Module.finrank ℂ E : ℝ) * ‖c‖^2 := by
+  simp [hsSq, norm_smul, (stdOrthonormalBasis ℂ E).norm_eq_one]
+
+end Thorp.Fourier
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/Walsh.lean b/lean/OAI/Probability/ThorpShuffle/Walsh.lean
index ec528adc915029628e1ab5503207d0232b0b1a38..3245991198375b7bf33d017c2d120e4a3889b50d 100644
--- a/lean/OAI/Probability/ThorpShuffle/Walsh.lean
+++ b/lean/OAI/Probability/ThorpShuffle/Walsh.lean
@@ -1,132 +1,132 @@
-import OAI.Probability.ThorpShuffle.PairAveraging
-
-namespace OAI
-
-universe uι uκ
-
-noncomputable section
-
-open scoped BigOperators
-open Filter
-
-namespace Thorp
-
-namespace Conditional
-
-def character {ι : Type uι} [Fintype ι] (a x : ι → Bool) : ℝ :=
-  ∏ i, if a i then sign (x i) else 1
-
-theorem character_symm {ι : Type uι} [Fintype ι] (a x : ι → Bool) :
-    character a x = character x a := by
-  apply Finset.prod_congr rfl
-  intro i _
-  cases a i <;> cases x i <;> rfl
-
-@[simp] theorem character_false {ι : Type uι} [Fintype ι] (x : ι → Bool) :
-    character (fun _ => false) x = 1 := by
-  simp [character]
-
-@[simp] theorem character_sq {ι : Type uι} [Fintype ι] (a x : ι → Bool) :
-    character a x ^ 2 = 1 := by
-  unfold character
-  rw [← Finset.prod_pow]
-  have h (i : ι) : (if a i then sign (x i) else (1 : ℝ)) ^ 2 = 1 := by
-    cases a i <;> simp
-  simp only [h, Finset.prod_const_one]
-
-theorem sum_character_pair {ι : Type uι} [Fintype ι] [DecidableEq ι]
-    (a b : ι → Bool) :
-    (∑ x : ι → Bool, character a x * character b x) =
-      if a = b then (Fintype.card (ι → Bool) : ℝ) else 0 := by
-  have hlocal (u v : Bool) :
-      (∑ x : Bool, (if u then sign x else (1 : ℝ)) * (if v then sign x else 1)) =
-        if u = v then 2 else 0 := by
-    cases u <;> cases v <;> norm_num [sign]
-  have hf : (∑ x : ι → Bool, character a x * character b x) =
-      ∏ i, (if a i = b i then (2 : ℝ) else 0) := by
-    calc
-      _ = ∑ x : ι → Bool, ∏ i,
-          (if a i then sign (x i) else (1 : ℝ)) * (if b i then sign (x i) else 1) := by
-        apply Finset.sum_congr rfl
-        intro x _
-        exact (Finset.prod_mul_distrib).symm
-      _ = ∏ i, ∑ x : Bool,
-          (if a i then sign x else (1 : ℝ)) * (if b i then sign x else 1) :=
-        (Fintype.prod_sum (fun (i : ι) (x : Bool) =>
-          (if a i then sign x else (1 : ℝ)) * (if b i then sign x else 1))).symm
-      _ = _ := by simp only [hlocal]
-  rw [hf]
-  by_cases h : a = b
-  · subst b
-    simp
-  · rw [ite_eq_right h]
-    have hn : ¬ ∀ i, a i = b i := fun he => h (funext he)
-    obtain ⟨i, hi⟩ := not_forall.mp hn
-    exact Finset.prod_eq_zero (Finset.mem_univ i) (by simp [hi])
-
-theorem sum_character_pair_dual {ι : Type uι} [Fintype ι] [DecidableEq ι]
-    (x y : ι → Bool) :
-    (∑ a : ι → Bool, character a x * character a y) =
-      if x = y then (Fintype.card (ι → Bool) : ℝ) else 0 := by
-  calc
-    _ = ∑ a : ι → Bool, character x a * character y a := by
-      apply Finset.sum_congr rfl
-      intro a _
-      rw [character_symm a x, character_symm a y]
-    _ = _ := sum_character_pair x y
-
-def coefficient {ι : Type uι} [Fintype ι] [DecidableEq ι] (w : (ι → Bool) → ℝ) (a : ι → Bool) : ℝ :=
-  ∑ x, w x * character a x
-
-theorem parseval {ι : Type uι} [Fintype ι] [DecidableEq ι] (w : (ι → Bool) → ℝ) :
-    mean (fun a : ι → Bool => coefficient w a ^ 2) = ∑ x, w x ^ 2 := by
-  have hs :
-      (∑ a : ι → Bool, coefficient w a ^ 2) =
-        (Fintype.card (ι → Bool) : ℝ) * ∑ x, w x ^ 2 := by
-    calc
-      _ = ∑ a : ι → Bool, ∑ x, ∑ y,
-          (character a x * character a y) * (w x * w y) := by
-        apply Finset.sum_congr rfl
-        intro a _
-        simp only [coefficient, pow_two, Finset.sum_mul, Finset.mul_sum]
-        apply Finset.sum_congr rfl
-        intro x _
-        apply Finset.sum_congr rfl
-        intro y _
-        ring
-      _ = ∑ x, ∑ y, (∑ a : ι → Bool, character a x * character a y) * (w x * w y) := by
-        rw [Finset.sum_comm]
-        apply Finset.sum_congr rfl
-        intro x _
-        rw [Finset.sum_comm]
-        simp only [Finset.sum_mul]
-      _ = _ := by
-        simp only [sum_character_pair_dual, ite_mul, zero_mul]
-        simp [Finset.mul_sum, pow_two]
-  unfold mean
-  rw [hs]
-  have hcard : (Fintype.card (ι → Bool) : ℝ) ≠ 0 := by
-    exact_mod_cast Fintype.card_ne_zero
-  exact mul_div_cancel_left₀ _ hcard
-
-@[simp] theorem coefficient_false {ι : Type uι} [Fintype ι] [DecidableEq ι] (w : (ι → Bool) → ℝ) :
-    coefficient w (fun _ => false) = ∑ x, w x := by
-  simp [coefficient]
-
-theorem character_reindex {ι : Type uι} {κ : Type uκ} [Fintype ι] [Fintype κ]
-    (e : ι ≃ κ) (a x : κ → Bool) :
-    character (a ∘ e) (x ∘ e) = character a x := by
-  exact Equiv.prod_comp e (fun i => if a i then sign (x i) else 1)
-
-theorem character_cons (d : ℕ) (h b : Bool) (a x : Position d) :
-    character (Fin.cons h a) (Fin.cons b x) =
-      (if h then sign b else 1) * character a x := by
-  simp [character, Fin.prod_univ_succ]
-
-end Conditional
-
-end Thorp
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.PairAveraging
+
+namespace OAI
+
+universe uι uκ
+
+noncomputable section
+
+open scoped BigOperators
+open Filter
+
+namespace Thorp
+
+namespace Conditional
+
+def character {ι : Type uι} [Fintype ι] (a x : ι → Bool) : ℝ :=
+  ∏ i, if a i then sign (x i) else 1
+
+theorem character_symm {ι : Type uι} [Fintype ι] (a x : ι → Bool) :
+    character a x = character x a := by
+  apply Finset.prod_congr rfl
+  intro i _
+  cases a i <;> cases x i <;> rfl
+
+@[simp] theorem character_false {ι : Type uι} [Fintype ι] (x : ι → Bool) :
+    character (fun _ => false) x = 1 := by
+  simp [character]
+
+@[simp] theorem character_sq {ι : Type uι} [Fintype ι] (a x : ι → Bool) :
+    character a x ^ 2 = 1 := by
+  unfold character
+  rw [← Finset.prod_pow]
+  have h (i : ι) : (if a i then sign (x i) else (1 : ℝ)) ^ 2 = 1 := by
+    cases a i <;> simp
+  simp only [h, Finset.prod_const_one]
+
+theorem sum_character_pair {ι : Type uι} [Fintype ι] [DecidableEq ι]
+    (a b : ι → Bool) :
+    (∑ x : ι → Bool, character a x * character b x) =
+      if a = b then (Fintype.card (ι → Bool) : ℝ) else 0 := by
+  have hlocal (u v : Bool) :
+      (∑ x : Bool, (if u then sign x else (1 : ℝ)) * (if v then sign x else 1)) =
+        if u = v then 2 else 0 := by
+    cases u <;> cases v <;> norm_num [sign]
+  have hf : (∑ x : ι → Bool, character a x * character b x) =
+      ∏ i, (if a i = b i then (2 : ℝ) else 0) := by
+    calc
+      _ = ∑ x : ι → Bool, ∏ i,
+          (if a i then sign (x i) else (1 : ℝ)) * (if b i then sign (x i) else 1) := by
+        apply Finset.sum_congr rfl
+        intro x _
+        exact (Finset.prod_mul_distrib).symm
+      _ = ∏ i, ∑ x : Bool,
+          (if a i then sign x else (1 : ℝ)) * (if b i then sign x else 1) :=
+        (Fintype.prod_sum (fun (i : ι) (x : Bool) =>
+          (if a i then sign x else (1 : ℝ)) * (if b i then sign x else 1))).symm
+      _ = _ := by simp only [hlocal]
+  rw [hf]
+  by_cases h : a = b
+  · subst b
+    simp
+  · rw [ite_eq_right h]
+    have hn : ¬ ∀ i, a i = b i := fun he => h (funext he)
+    obtain ⟨i, hi⟩ := not_forall.mp hn
+    exact Finset.prod_eq_zero (Finset.mem_univ i) (by simp [hi])
+
+theorem sum_character_pair_dual {ι : Type uι} [Fintype ι] [DecidableEq ι]
+    (x y : ι → Bool) :
+    (∑ a : ι → Bool, character a x * character a y) =
+      if x = y then (Fintype.card (ι → Bool) : ℝ) else 0 := by
+  calc
+    _ = ∑ a : ι → Bool, character x a * character y a := by
+      apply Finset.sum_congr rfl
+      intro a _
+      rw [character_symm a x, character_symm a y]
+    _ = _ := sum_character_pair x y
+
+def coefficient {ι : Type uι} [Fintype ι] [DecidableEq ι] (w : (ι → Bool) → ℝ) (a : ι → Bool) : ℝ :=
+  ∑ x, w x * character a x
+
+theorem parseval {ι : Type uι} [Fintype ι] [DecidableEq ι] (w : (ι → Bool) → ℝ) :
+    mean (fun a : ι → Bool => coefficient w a ^ 2) = ∑ x, w x ^ 2 := by
+  have hs :
+      (∑ a : ι → Bool, coefficient w a ^ 2) =
+        (Fintype.card (ι → Bool) : ℝ) * ∑ x, w x ^ 2 := by
+    calc
+      _ = ∑ a : ι → Bool, ∑ x, ∑ y,
+          (character a x * character a y) * (w x * w y) := by
+        apply Finset.sum_congr rfl
+        intro a _
+        simp only [coefficient, pow_two, Finset.sum_mul, Finset.mul_sum]
+        apply Finset.sum_congr rfl
+        intro x _
+        apply Finset.sum_congr rfl
+        intro y _
+        ring
+      _ = ∑ x, ∑ y, (∑ a : ι → Bool, character a x * character a y) * (w x * w y) := by
+        rw [Finset.sum_comm]
+        apply Finset.sum_congr rfl
+        intro x _
+        rw [Finset.sum_comm]
+        simp only [Finset.sum_mul]
+      _ = _ := by
+        simp only [sum_character_pair_dual, ite_mul, zero_mul]
+        simp [Finset.mul_sum, pow_two]
+  unfold mean
+  rw [hs]
+  have hcard : (Fintype.card (ι → Bool) : ℝ) ≠ 0 := by
+    exact_mod_cast Fintype.card_ne_zero
+  exact mul_div_cancel_left₀ _ hcard
+
+@[simp] theorem coefficient_false {ι : Type uι} [Fintype ι] [DecidableEq ι] (w : (ι → Bool) → ℝ) :
+    coefficient w (fun _ => false) = ∑ x, w x := by
+  simp [coefficient]
+
+theorem character_reindex {ι : Type uι} {κ : Type uκ} [Fintype ι] [Fintype κ]
+    (e : ι ≃ κ) (a x : κ → Bool) :
+    character (a ∘ e) (x ∘ e) = character a x := by
+  exact Equiv.prod_comp e (fun i => if a i then sign (x i) else 1)
+
+theorem character_cons (d : ℕ) (h b : Bool) (a x : Position d) :
+    character (Fin.cons h a) (Fin.cons b x) =
+      (if h then sign b else 1) * character a x := by
+  simp [character, Fin.prod_univ_succ]
+
+end Conditional
+
+end Thorp
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThorpShuffle/WalshFlow.lean b/lean/OAI/Probability/ThorpShuffle/WalshFlow.lean
index dc5bc6423d3a7d6b6fe6d205d0a870e41141b6a1..258c232a796d4187ec206cc49185226793483be2 100644
--- a/lean/OAI/Probability/ThorpShuffle/WalshFlow.lean
+++ b/lean/OAI/Probability/ThorpShuffle/WalshFlow.lean
@@ -1,231 +1,231 @@
-import OAI.Probability.ThorpShuffle.FiniteExpectation
-
-namespace OAI
-
-universe uΩ
-
-noncomputable section
-
-open scoped BigOperators
-open Filter
-
-namespace Thorp
-
-namespace Conditional
-
-theorem scatterPosition_eq_snoc (d : ℕ) (b : Bool) (x : Position d) :
-    scatterPosition d (b, x) = Fin.snoc x b := by
-  rw [Fin.snoc_eq_cons_rotate]
-  ext i
-  simp only [scatterPosition, splitPosition, Equiv.trans_apply, Equiv.symm_symm,
-    Fin.consEquiv_apply, rotate, Equiv.piCongrLeft_apply]
-  simp
-
-theorem character_snoc (d : ℕ) (h b : Bool) (a x : Position d) :
-    character (Fin.snoc a h) (Fin.snoc x b) =
-      character a x * (if h then sign b else 1) := by
-  simp [character, Fin.prod_univ_castSucc]
-
-theorem coefficient_physicalFlow_even (d : ℕ) (B : Position (d + 1) → Bool)
-    (w : Position (d + 1) → ℝ) (c : Coins (d + 1)) (a : Position d) :
-    coefficient (physicalFlow d B w c) (Fin.snoc a false) =
-      coefficient w (Fin.cons false a) := by
-  unfold coefficient
-  rw [← Equiv.sum_comp (scatterPosition d)
-    (fun x => physicalFlow d B w c x * character (Fin.snoc a false) x)]
-  rw [← Equiv.sum_comp (splitPosition d).symm
-    (fun x => w x * character (Fin.cons false a) x)]
-  simp only [Fintype.sum_prod_type]
-  simp only [Fintype.sum_bool]
-  rw [← Finset.sum_add_distrib, ← Finset.sum_add_distrib]
-  apply Finset.sum_congr rfl
-  intro x _
-  simp only [physicalFlow_scatter]
-  simp only [scatterPosition_eq_snoc, character_snoc,
-    Bool.false_eq_true, ↓reduceIte, mul_one]
-  change flow _ _ c (true, x) * character a x +
-    flow _ _ c (false, x) * character a x =
-    w (Fin.cons true x) * character (Fin.cons false a) (Fin.cons true x) +
-    w (Fin.cons false x) * character (Fin.cons false a) (Fin.cons false x)
-  rw [character_cons, character_cons]
-  simp only [Bool.false_eq_true, ↓reduceIte, one_mul]
-  rw [← add_mul, ← add_mul]
-  congr 1
-  simpa only [flow, Bool.false_eq_true, ↓reduceIte, add_comm, Function.comp_apply,
-    splitPosition, Equiv.symm_symm, Fin.consEquiv, Equiv.coe_fn_mk] using
-    pairUpdate_sum (B (Fin.cons false x)) (B (Fin.cons true x))
-      (w (Fin.cons false x)) (w (Fin.cons true x)) (c x)
-
-theorem coefficient_physicalFlow_odd (d : ℕ) (B : Position (d + 1) → Bool)
-    (w : Position (d + 1) → ℝ) (c : Coins (d + 1)) (a : Position d) :
-    coefficient (physicalFlow d B w c) (Fin.snoc a true) =
-      oddCoefficient (character a) (flow (B ∘ (splitPosition d).symm)
-        (w ∘ (splitPosition d).symm) c) := by
-  unfold coefficient
-  rw [← Equiv.sum_comp (scatterPosition d)
-    (fun x => physicalFlow d B w c x * character (Fin.snoc a true) x)]
-  simp only [Fintype.sum_prod_type, Fintype.sum_bool]
-  rw [← Finset.sum_add_distrib]
-  unfold oddCoefficient
-  apply Finset.sum_congr rfl
-  intro x _
-  simp only [physicalFlow_scatter]
-  simp only [scatterPosition_eq_snoc, character_snoc,
-    ↓reduceIte, sign, Bool.false_eq_true]
-  ring
-
-def oneFreeEnergy (d : ℕ) (B : Position (d + 1) → Bool)
-    (w : Position (d + 1) → ℝ) : ℝ :=
-  ∑ x : Position d,
-    ((if B (Fin.cons true x) then 0 else w (Fin.cons false x) ^ 2) +
-      (if B (Fin.cons false x) then 0 else w (Fin.cons true x) ^ 2))
-
-theorem mean_coefficient_physicalFlow_odd_sq (d : ℕ)
-    (B : Position (d + 1) → Bool) (w : Position (d + 1) → ℝ)
-    (hw : ∀ x, B x = false → w x = 0) (a : Position d) :
-    mean (fun c : Coins (d + 1) =>
-      coefficient (physicalFlow d B w c) (Fin.snoc a true) ^ 2) =
-        oneFreeEnergy d B w := by
-  simp_rw [coefficient_physicalFlow_odd]
-  exact mean_oddCoefficient_sq _ _ _ (fun p => hw ((splitPosition d).symm p))
-    (character_sq a)
-
-end Conditional
-
-namespace Conditional
-
-def prefixZero {d : ℕ} (j : ℕ) (a : Position d) : Prop :=
-  ∀ i : Fin d, i.val < j → a i = false
-
-instance prefixZeroDecidable {d : ℕ} (j : ℕ) (a : Position d) :
-    Decidable (prefixZero j a) := by
-  unfold prefixZero
-  infer_instance
-
-@[simp] theorem prefixZero_zero {d : ℕ} (a : Position d) : prefixZero 0 a := by
-  intro i hi
-  omega
-
-theorem prefixZero_full {d : ℕ} (a : Position d) :
-    prefixZero d a ↔ a = fun _ => false := by
-  constructor
-  · intro h
-    funext i
-    exact h i i.isLt
-  · intro h
-    subst a
-    intro i _
-    rfl
-
-theorem prefixZero_cons (j d : ℕ) (h : Bool) (a : Position d) :
-    prefixZero (j + 1) (Fin.cons h a) ↔ h = false ∧ prefixZero j a := by
-  constructor
-  · intro hp
-    refine ⟨?_, ?_⟩
-    · simpa using hp 0 (by simp)
-    · intro i hi
-      simpa using hp i.succ (by simpa using hi)
-  · rintro ⟨rfl, hp⟩ i
-    refine Fin.cases ?_ (fun k => ?_) i
-    · intro _; rfl
-    · intro hi
-      exact hp k (by simpa using hi)
-
-theorem prefixZero_snoc (j d : ℕ) (hj : j ≤ d) (h : Bool) (a : Position d) :
-    prefixZero j (Fin.snoc a h) ↔ prefixZero j a := by
-  constructor
-  · intro hp i hi
-    simpa using hp i.castSucc hi
-  · intro hp i hi
-    have hid : i.val < d := lt_of_lt_of_le hi hj
-    rw [Fin.snoc, dite_eq_left hid]
-    exact hp _ hi
-
-theorem prefixZero_count (j d : ℕ) (hj : j ≤ d) :
-    (∑ a : Position d, if prefixZero j a then (1 : ℝ) else 0) = 2 ^ (d - j) := by
-  classical
-  induction d generalizing j with
-  | zero =>
-    have : j = 0 := by omega
-    subst j
-    simp
-  | succ d ih =>
-    cases j with
-    | zero => simp [Fintype.card_pi, Fintype.card_bool]
-    | succ j =>
-      rw [← Equiv.sum_comp (Fin.consEquiv (fun _ : Fin (d + 1) => Bool))
-        (fun a => if prefixZero (j + 1) a then (1 : ℝ) else 0)]
-      simp only [Fin.consEquiv, Equiv.coe_fn_mk, Fintype.sum_prod_type, Fintype.sum_bool,
-        prefixZero_cons]
-      simpa using ih j (by omega)
-
-def bandEnergy (d : ℕ) (w : Position d → ℝ) (j : ℕ) : ℝ := by
-  classical
-  exact ∑ a : Position d, if prefixZero j a then coefficient w a ^ 2 else 0
-
-@[simp] theorem bandEnergy_zero (d : ℕ) (w : Position d → ℝ) :
-    bandEnergy d w 0 = (Fintype.card (Position d) : ℝ) * ∑ x, w x ^ 2 := by
-  have hp := parseval w
-  unfold mean at hp
-  have hc : (Fintype.card (Position d) : ℝ) ≠ 0 := by
-    exact_mod_cast Fintype.card_ne_zero
-  have hsum : (∑ a, coefficient w a ^ 2) =
-      (Fintype.card (Position d) : ℝ) * ∑ x, w x ^ 2 := by
-    rw [div_eq_iff hc] at hp
-    linarith
-  simpa [bandEnergy] using hsum
-
-@[simp] theorem bandEnergy_full (d : ℕ) (w : Position d → ℝ) :
-    bandEnergy d w d = (∑ x, w x) ^ 2 := by
-  classical
-  simp [bandEnergy, prefixZero_full]
-
-theorem bandEnergy_cons (d j : ℕ) (w : Position (d + 1) → ℝ) :
-    bandEnergy (d + 1) w (j + 1) =
-      ∑ a : Position d, if prefixZero j a then coefficient w (Fin.cons false a) ^ 2 else 0 := by
-  classical
-  unfold bandEnergy
-  rw [← Equiv.sum_comp (Fin.consEquiv (fun _ : Fin (d + 1) => Bool))
-    (fun a => if prefixZero (j + 1) a then coefficient w a ^ 2 else 0)]
-  simp [Fintype.sum_prod_type, prefixZero_cons, Fin.consEquiv]
-
-theorem mean_indicator {Ω : Type uΩ} [Fintype Ω] (p : Prop) [Decidable p] (f : Ω → ℝ) :
-    mean (fun ω => if p then f ω else 0) = if p then mean f else 0 := by
-  by_cases hp : p <;> simp [hp, mean_zero]
-
-theorem mean_bandEnergy_physicalFlow (d j : ℕ) (hj : j ≤ d)
-    (B : Position (d + 1) → Bool) (w : Position (d + 1) → ℝ)
-    (hw : ∀ x, B x = false → w x = 0) :
-    mean (fun c : Coins (d + 1) => bandEnergy (d + 1) (physicalFlow d B w c) j) =
-      bandEnergy (d + 1) w (j + 1) + 2 ^ (d - j) * oneFreeEnergy d B w := by
-  classical
-  have hs (c : Coins (d + 1)) :
-      bandEnergy (d + 1) (physicalFlow d B w c) j =
-        (∑ a : Position d, if prefixZero j a then
-          coefficient (physicalFlow d B w c) (Fin.snoc a true) ^ 2 else 0) +
-        (∑ a : Position d, if prefixZero j a then
-          coefficient w (Fin.cons false a) ^ 2 else 0) := by
-    unfold bandEnergy
-    rw [← Equiv.sum_comp (Fin.snocEquiv (fun _ : Fin (d + 1) => Bool))
-      (fun a => if prefixZero j a then coefficient (physicalFlow d B w c) a ^ 2 else 0)]
-    simp only [Fintype.sum_prod_type, Fintype.sum_bool, Fin.snocEquiv, Equiv.coe_fn_mk,
-      prefixZero_snoc j d hj, coefficient_physicalFlow_even]
-  simp_rw [hs]
-  rw [mean_add, mean_sum, mean_const, bandEnergy_cons]
-  simp_rw [mean_indicator, mean_coefficient_physicalFlow_odd_sq d B w hw]
-  have hcount :
-      (∑ a : Position d, if prefixZero j a then oneFreeEnergy d B w else 0) =
-        2 ^ (d - j) * oneFreeEnergy d B w := by
-    rw [← prefixZero_count j d hj, Finset.sum_mul]
-    apply Finset.sum_congr rfl
-    intro a _
-    split_ifs <;> simp
-  rw [hcount, add_comm]
-
-end Conditional
-
-end Thorp
-
-end
-
-end OAI
+import OAI.Probability.ThorpShuffle.FiniteExpectation
+
+namespace OAI
+
+universe uΩ
+
+noncomputable section
+
+open scoped BigOperators
+open Filter
+
+namespace Thorp
+
+namespace Conditional
+
+theorem scatterPosition_eq_snoc (d : ℕ) (b : Bool) (x : Position d) :
+    scatterPosition d (b, x) = Fin.snoc x b := by
+  rw [Fin.snoc_eq_cons_rotate]
+  ext i
+  simp only [scatterPosition, splitPosition, Equiv.trans_apply, Equiv.symm_symm,
+    Fin.consEquiv_apply, rotate, Equiv.piCongrLeft_apply]
+  simp
+
+theorem character_snoc (d : ℕ) (h b : Bool) (a x : Position d) :
+    character (Fin.snoc a h) (Fin.snoc x b) =
+      character a x * (if h then sign b else 1) := by
+  simp [character, Fin.prod_univ_castSucc]
+
+theorem coefficient_physicalFlow_even (d : ℕ) (B : Position (d + 1) → Bool)
+    (w : Position (d + 1) → ℝ) (c : Coins (d + 1)) (a : Position d) :
+    coefficient (physicalFlow d B w c) (Fin.snoc a false) =
+      coefficient w (Fin.cons false a) := by
+  unfold coefficient
+  rw [← Equiv.sum_comp (scatterPosition d)
+    (fun x => physicalFlow d B w c x * character (Fin.snoc a false) x)]
+  rw [← Equiv.sum_comp (splitPosition d).symm
+    (fun x => w x * character (Fin.cons false a) x)]
+  simp only [Fintype.sum_prod_type]
+  simp only [Fintype.sum_bool]
+  rw [← Finset.sum_add_distrib, ← Finset.sum_add_distrib]
+  apply Finset.sum_congr rfl
+  intro x _
+  simp only [physicalFlow_scatter]
+  simp only [scatterPosition_eq_snoc, character_snoc,
+    Bool.false_eq_true, ↓reduceIte, mul_one]
+  change flow _ _ c (true, x) * character a x +
+    flow _ _ c (false, x) * character a x =
+    w (Fin.cons true x) * character (Fin.cons false a) (Fin.cons true x) +
+    w (Fin.cons false x) * character (Fin.cons false a) (Fin.cons false x)
+  rw [character_cons, character_cons]
+  simp only [Bool.false_eq_true, ↓reduceIte, one_mul]
+  rw [← add_mul, ← add_mul]
+  congr 1
+  simpa only [flow, Bool.false_eq_true, ↓reduceIte, add_comm, Function.comp_apply,
+    splitPosition, Equiv.symm_symm, Fin.consEquiv, Equiv.coe_fn_mk] using
+    pairUpdate_sum (B (Fin.cons false x)) (B (Fin.cons true x))
+      (w (Fin.cons false x)) (w (Fin.cons true x)) (c x)
+
+theorem coefficient_physicalFlow_odd (d : ℕ) (B : Position (d + 1) → Bool)
+    (w : Position (d + 1) → ℝ) (c : Coins (d + 1)) (a : Position d) :
+    coefficient (physicalFlow d B w c) (Fin.snoc a true) =
+      oddCoefficient (character a) (flow (B ∘ (splitPosition d).symm)
+        (w ∘ (splitPosition d).symm) c) := by
+  unfold coefficient
+  rw [← Equiv.sum_comp (scatterPosition d)
+    (fun x => physicalFlow d B w c x * character (Fin.snoc a true) x)]
+  simp only [Fintype.sum_prod_type, Fintype.sum_bool]
+  rw [← Finset.sum_add_distrib]
+  unfold oddCoefficient
+  apply Finset.sum_congr rfl
+  intro x _
+  simp only [physicalFlow_scatter]
+  simp only [scatterPosition_eq_snoc, character_snoc,
+    ↓reduceIte, sign, Bool.false_eq_true]
+  ring
+
+def oneFreeEnergy (d : ℕ) (B : Position (d + 1) → Bool)
+    (w : Position (d + 1) → ℝ) : ℝ :=
+  ∑ x : Position d,
+    ((if B (Fin.cons true x) then 0 else w (Fin.cons false x) ^ 2) +
+      (if B (Fin.cons false x) then 0 else w (Fin.cons true x) ^ 2))
+
+theorem mean_coefficient_physicalFlow_odd_sq (d : ℕ)
+    (B : Position (d + 1) → Bool) (w : Position (d + 1) → ℝ)
+    (hw : ∀ x, B x = false → w x = 0) (a : Position d) :
+    mean (fun c : Coins (d + 1) =>
+      coefficient (physicalFlow d B w c) (Fin.snoc a true) ^ 2) =
+        oneFreeEnergy d B w := by
+  simp_rw [coefficient_physicalFlow_odd]
+  exact mean_oddCoefficient_sq _ _ _ (fun p => hw ((splitPosition d).symm p))
+    (character_sq a)
+
+end Conditional
+
+namespace Conditional
+
+def prefixZero {d : ℕ} (j : ℕ) (a : Position d) : Prop :=
+  ∀ i : Fin d, i.val < j → a i = false
+
+instance prefixZeroDecidable {d : ℕ} (j : ℕ) (a : Position d) :
+    Decidable (prefixZero j a) := by
+  unfold prefixZero
+  infer_instance
+
+@[simp] theorem prefixZero_zero {d : ℕ} (a : Position d) : prefixZero 0 a := by
+  intro i hi
+  omega
+
+theorem prefixZero_full {d : ℕ} (a : Position d) :
+    prefixZero d a ↔ a = fun _ => false := by
+  constructor
+  · intro h
+    funext i
+    exact h i i.isLt
+  · intro h
+    subst a
+    intro i _
+    rfl
+
+theorem prefixZero_cons (j d : ℕ) (h : Bool) (a : Position d) :
+    prefixZero (j + 1) (Fin.cons h a) ↔ h = false ∧ prefixZero j a := by
+  constructor
+  · intro hp
+    refine ⟨?_, ?_⟩
+    · simpa using hp 0 (by simp)
+    · intro i hi
+      simpa using hp i.succ (by simpa using hi)
+  · rintro ⟨rfl, hp⟩ i
+    refine Fin.cases ?_ (fun k => ?_) i
+    · intro _; rfl
+    · intro hi
+      exact hp k (by simpa using hi)
+
+theorem prefixZero_snoc (j d : ℕ) (hj : j ≤ d) (h : Bool) (a : Position d) :
+    prefixZero j (Fin.snoc a h) ↔ prefixZero j a := by
+  constructor
+  · intro hp i hi
+    simpa using hp i.castSucc hi
+  · intro hp i hi
+    have hid : i.val < d := lt_of_lt_of_le hi hj
+    rw [Fin.snoc, dite_eq_left hid]
+    exact hp _ hi
+
+theorem prefixZero_count (j d : ℕ) (hj : j ≤ d) :
+    (∑ a : Position d, if prefixZero j a then (1 : ℝ) else 0) = 2 ^ (d - j) := by
+  classical
+  induction d generalizing j with
+  | zero =>
+    have : j = 0 := by omega
+    subst j
+    simp
+  | succ d ih =>
+    cases j with
+    | zero => simp [Fintype.card_pi, Fintype.card_bool]
+    | succ j =>
+      rw [← Equiv.sum_comp (Fin.consEquiv (fun _ : Fin (d + 1) => Bool))
+        (fun a => if prefixZero (j + 1) a then (1 : ℝ) else 0)]
+      simp only [Fin.consEquiv, Equiv.coe_fn_mk, Fintype.sum_prod_type, Fintype.sum_bool,
+        prefixZero_cons]
+      simpa using ih j (by omega)
+
+def bandEnergy (d : ℕ) (w : Position d → ℝ) (j : ℕ) : ℝ := by
+  classical
+  exact ∑ a : Position d, if prefixZero j a then coefficient w a ^ 2 else 0
+
+@[simp] theorem bandEnergy_zero (d : ℕ) (w : Position d → ℝ) :
+    bandEnergy d w 0 = (Fintype.card (Position d) : ℝ) * ∑ x, w x ^ 2 := by
+  have hp := parseval w
+  unfold mean at hp
+  have hc : (Fintype.card (Position d) : ℝ) ≠ 0 := by
+    exact_mod_cast Fintype.card_ne_zero
+  have hsum : (∑ a, coefficient w a ^ 2) =
+      (Fintype.card (Position d) : ℝ) * ∑ x, w x ^ 2 := by
+    rw [div_eq_iff hc] at hp
+    linarith
+  simpa [bandEnergy] using hsum
+
+@[simp] theorem bandEnergy_full (d : ℕ) (w : Position d → ℝ) :
+    bandEnergy d w d = (∑ x, w x) ^ 2 := by
+  classical
+  simp [bandEnergy, prefixZero_full]
+
+theorem bandEnergy_cons (d j : ℕ) (w : Position (d + 1) → ℝ) :
+    bandEnergy (d + 1) w (j + 1) =
+      ∑ a : Position d, if prefixZero j a then coefficient w (Fin.cons false a) ^ 2 else 0 := by
+  classical
+  unfold bandEnergy
+  rw [← Equiv.sum_comp (Fin.consEquiv (fun _ : Fin (d + 1) => Bool))
+    (fun a => if prefixZero (j + 1) a then coefficient w a ^ 2 else 0)]
+  simp [Fintype.sum_prod_type, prefixZero_cons, Fin.consEquiv]
+
+theorem mean_indicator {Ω : Type uΩ} [Fintype Ω] (p : Prop) [Decidable p] (f : Ω → ℝ) :
+    mean (fun ω => if p then f ω else 0) = if p then mean f else 0 := by
+  by_cases hp : p <;> simp [hp, mean_zero]
+
+theorem mean_bandEnergy_physicalFlow (d j : ℕ) (hj : j ≤ d)
+    (B : Position (d + 1) → Bool) (w : Position (d + 1) → ℝ)
+    (hw : ∀ x, B x = false → w x = 0) :
+    mean (fun c : Coins (d + 1) => bandEnergy (d + 1) (physicalFlow d B w c) j) =
+      bandEnergy (d + 1) w (j + 1) + 2 ^ (d - j) * oneFreeEnergy d B w := by
+  classical
+  have hs (c : Coins (d + 1)) :
+      bandEnergy (d + 1) (physicalFlow d B w c) j =
+        (∑ a : Position d, if prefixZero j a then
+          coefficient (physicalFlow d B w c) (Fin.snoc a true) ^ 2 else 0) +
+        (∑ a : Position d, if prefixZero j a then
+          coefficient w (Fin.cons false a) ^ 2 else 0) := by
+    unfold bandEnergy
+    rw [← Equiv.sum_comp (Fin.snocEquiv (fun _ : Fin (d + 1) => Bool))
+      (fun a => if prefixZero j a then coefficient (physicalFlow d B w c) a ^ 2 else 0)]
+    simp only [Fintype.sum_prod_type, Fintype.sum_bool, Fin.snocEquiv, Equiv.coe_fn_mk,
+      prefixZero_snoc j d hj, coefficient_physicalFlow_even]
+  simp_rw [hs]
+  rw [mean_add, mean_sum, mean_const, bandEnergy_cons]
+  simp_rw [mean_indicator, mean_coefficient_physicalFlow_odd_sq d B w hw]
+  have hcount :
+      (∑ a : Position d, if prefixZero j a then oneFreeEnergy d B w else 0) =
+        2 ^ (d - j) * oneFreeEnergy d B w := by
+    rw [← prefixZero_count j d hj, Finset.sum_mul]
+    apply Finset.sum_congr rfl
+    intro a _
+    split_ifs <;> simp
+  rw [hcount, add_comm]
+
+end Conditional
+
+end Thorp
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeState/Basic.lean b/lean/OAI/Probability/ThreeState/Basic.lean
index 9d016426a824db3aae7a54ff89cf30301aebeec9..f7d00d16fbca29621cc2e6fd8c848035c8e742d2 100644
--- a/lean/OAI/Probability/ThreeState/Basic.lean
+++ b/lean/OAI/Probability/ThreeState/Basic.lean
@@ -1,285 +1,285 @@
-import Mathlib
-
-namespace OAI
-
-/-! Three-state symmetric broadcast experiments, posterior advantage and total variation. -/
-
-namespace ThreeState
-
-abbrev Spin := Fin 3
-
- 
-def Admissible (lam : ℝ) : Prop := -(1 / 2 : ℝ) ≤ lam ∧ lam ≤ 1
-
- 
-noncomputable def channel (lam : ℝ) (i j : Spin) : ℝ :=
-  if i = j then (1 + 2 * lam) / 3 else (1 - lam) / 3
-
- 
-
-def Observation : ℕ → Type
-  | 0 => Spin
-  | n + 1 => Multiset (Observation n)
-
-theorem channel_nonneg {lam : ℝ} (h : Admissible lam) (i j : Spin) :
-    0 ≤ channel lam i j := by
-  unfold channel
-  split_ifs
-  · dsimp [Admissible] at h
-    linarith [h.1]
-  · dsimp [Admissible] at h
-    linarith [h.2]
-
-theorem channel_sum (lam : ℝ) (i : Spin) : ∑ j, channel lam i j = 1 := by
-  fin_cases i <;> simp [channel, Fin.sum_univ_succ] <;> ring
-
-noncomputable def channelPMF (lam : ℝ) (h : Admissible lam) (i : Spin) : PMF Spin :=
-  PMF.ofFintype (fun j => ENNReal.ofReal (channel lam i j)) (by
-    rw [← ENNReal.ofReal_sum_of_nonneg (fun j _ => channel_nonneg h i j)]
-    simp [channel_sum])
-
- 
-noncomputable def iidList {α : Type*} (p : PMF α) : ℕ → PMF (List α)
-  | 0 => PMF.pure []
-  | n + 1 => p.bind fun a => (iidList p n).map (List.cons a)
-
- 
-
-noncomputable def observedLaw (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) :
-    (n : ℕ) → Spin → PMF (Observation n)
-  | 0, i => PMF.pure i
-  | n + 1, i => offspring.bind fun k =>
-      (iidList ((channelPMF lam h i).bind (observedLaw offspring lam h n)) k).map
-        (fun xs : List (Observation n) => (xs : Multiset (Observation n)))
-
-noncomputable def uniformSpin : PMF Spin := PMF.ofFintype (fun _ => 1 / 3) (by
-  simp
-  exact ENNReal.mul_inv_cancel (by norm_num) (by finiteness))
-
- 
-noncomputable def marginal {α : Type*} (law : Spin → PMF α) : PMF α :=
-  uniformSpin.bind law
-
- 
-noncomputable def posterior {α : Type*} (law : Spin → PMF α) (y : α) (i : Spin) : ℝ :=
-  if marginal law y = 0 then 1 / 3
-  else (law i y).toReal / (3 * (marginal law y).toReal)
-
- 
-
-noncomputable def advantage {α : Type*} (law : Spin → PMF α) : ℝ :=
-  ∑' y, (marginal law y).toReal *
-    ((∑ i : Spin, |posterior law y i - 1 / 3|) / 2)
-
-noncomputable def regularLaw (b : ℕ) (lam : ℝ) (h : Admissible lam) (n : ℕ) :=
-  observedLaw (PMF.pure b) lam h n
-
- 
-noncomputable def poissonOffspring (d : ℝ) (hd : 0 ≤ d) : PMF ℕ := by
-  refine ⟨fun k => ENNReal.ofReal (Real.exp (-d) * d ^ k / (k.factorial : ℝ)), ?_⟩
-  apply ENNReal.hasSum_coe.2
-  have hsum : HasSum (fun k : ℕ => Real.exp (-d) * d ^ k / (k.factorial : ℝ)) 1 :=
-    ProbabilityTheory.hasSum_one_poissonMeasure ⟨d, hd⟩
-  simpa using hsum.toNNReal (fun k => by positivity)
-
-noncomputable def poissonLaw (d : ℝ) (hd : 0 ≤ d) (lam : ℝ)
-    (h : Admissible lam) (n : ℕ) :=
-  observedLaw (poissonOffspring d hd) lam h n
-
-noncomputable def regularAdvantage (b : ℕ) (lam : ℝ) (h : Admissible lam) (n : ℕ) : ℝ :=
-  advantage (regularLaw b lam h n)
-
-noncomputable def poissonAdvantage (d : ℝ) (hd : 1 < d) (lam : ℝ)
-    (h : Admissible lam) (n : ℕ) : ℝ :=
-  advantage (poissonLaw d (le_trans (by norm_num) hd.le) lam h n)
-
- 
-def Reconstructs (a : ℕ → ℝ) : Prop :=
-  ∃ L : ℝ, 0 < L ∧ Filter.Tendsto a Filter.atTop (nhds L)
-
- 
-def Nonreconstructs (a : ℕ → ℝ) : Prop :=
-  Filter.Tendsto a Filter.atTop (nhds 0)
-
-end ThreeState
-namespace ThreeState
-
-noncomputable def mass {α : Type*} (p : PMF α) (a : α) : ℝ := (p a).toReal
-
-lemma mass_nonneg {α : Type*} (p : PMF α) (a : α) : 0 ≤ mass p a := ENNReal.toReal_nonneg
-lemma mass_sum {α : Type*} (p : PMF α) : ∑' a, mass p a = 1 := by
-  simp only [mass, ← ENNReal.tsum_toReal_eq (fun a => p.apply_ne_top a), p.tsum_coe,
-    ENNReal.toReal_one]
-lemma mass_summable {α : Type*} (p : PMF α) : Summable (mass p) :=
-  ENNReal.summable_toReal p.tsum_coe_ne_top
-
-lemma mass_bind {α β : Type*} (p : PMF α) (k : α → PMF β) (b : β) :
-    mass (p.bind k) b = ∑' a, mass p a * mass (k a) b := by
-  simp only [mass, PMF.bind_apply, ENNReal.tsum_toReal_eq
-    (fun a => ENNReal.mul_ne_top (p.apply_ne_top a) ((k a).apply_ne_top b)), ENNReal.toReal_mul]
-
-lemma mass_uniform (i : Spin) : mass uniformSpin i = 1/3 := by
-  simp [mass, uniformSpin, PMF.ofFintype_apply]
-
-lemma mass_marginal {α : Type*} (law : Spin → PMF α) (y : α) :
-    mass (marginal law) y = (∑ i : Spin, mass (law i) y)/3 := by
-  simp only [marginal, mass_bind, tsum_fintype, mass_uniform]
-  rw [← Finset.mul_sum]
-  ring
-
-lemma mass_eq_zero {α : Type*} (p : PMF α) (a : α) : mass p a = 0 ↔ p a = 0 := by
-  simp [mass, ENNReal.toReal_eq_zero_iff, p.apply_ne_top a]
-
-lemma marginal_zero {α : Type*} (law : Spin → PMF α) (y : α)
-    (h : marginal law y = 0) (i : Spin) : law i y = 0 := by
-  apply (mass_eq_zero (law i) y).mp
-  have hh : mass (marginal law) y = 0 := (mass_eq_zero _ _).mpr h
-  rw [mass_marginal] at hh
-  have hs : ∑ j : Spin, mass (law j) y = 0 := by linarith
-  exact (Finset.sum_eq_zero_iff_of_nonneg (fun j _ => mass_nonneg (law j) y)).mp hs i (by simp)
-
-lemma posterior_nonneg {α : Type*} (law : Spin → PMF α) (y : α) (i : Spin) :
-    0 ≤ posterior law y i := by
-  unfold posterior
-  split_ifs <;> positivity
-
-lemma posterior_sum {α : Type*} (law : Spin → PMF α) (y : α) :
-    ∑ i : Spin, posterior law y i = 1 := by
-  classical
-  have hn := (marginal law).apply_ne_top y
-  by_cases h : marginal law y = 0
-  · simp [posterior, h]
-  · have hm : mass (marginal law) y ≠ 0 := mt (mass_eq_zero _ _).mp h
-    simp only [posterior, ite_eq_right h, ← Finset.sum_div]
-    change (∑ i : Spin, mass (law i) y)/(3*mass (marginal law) y) = 1
-    have he := mass_marginal law y
-    field_simp
-    linarith
-
- 
-lemma weighted_posterior {α : Type*} (law : Spin → PMF α) (y : α) (i : Spin) :
-    mass (marginal law) y * (posterior law y i-1/3) =
-      (mass (law i) y-mass (marginal law) y)/3 := by
-  classical
-  by_cases h : marginal law y = 0
-  · have hi := marginal_zero law y h i
-    simp [mass, h, hi, posterior]
-  · have hm : mass (marginal law) y ≠ 0 := mt (mass_eq_zero _ _).mp h
-    simp only [posterior, ite_eq_right h]
-    change mass (marginal law) y * (mass (law i) y / (3*mass (marginal law) y)-1/3) = _
-    field_simp
-
-lemma weighted_abs_posterior {α : Type*} (law : Spin → PMF α) (y : α) (i : Spin) :
-    mass (marginal law) y * |posterior law y i-1/3| =
-      |mass (law i) y-mass (marginal law) y|/3 := by
-  calc
-    mass (marginal law) y * |posterior law y i-1/3| =
-        |mass (marginal law) y * (posterior law y i-1/3)| := by
-      rw [abs_mul, abs_of_nonneg (mass_nonneg (marginal law) y)]
-    _ = _ := by rw [weighted_posterior, abs_div]; norm_num
-
-lemma mass_le_one {α : Type*} (p : PMF α) (a : α) : mass p a ≤ 1 := by
-  simpa [mass] using ENNReal.toReal_mono (by finiteness) (p.coe_le_one a)
-
-noncomputable def tvMass {α : Type*} (p q : PMF α) : ℝ :=
-  (∑' a, |mass p a-mass q a|)/2
-
-lemma mass_diff_summable {α : Type*} (p q : PMF α) :
-    Summable (fun a => |mass p a-mass q a|) :=
-  ((mass_summable p).sub (mass_summable q)).abs
-
-lemma tvMass_nonneg {α : Type*} (p q : PMF α) : 0 ≤ tvMass p q := by
-  exact div_nonneg (tsum_nonneg (fun a => abs_nonneg _)) (by norm_num)
-
-lemma tvMass_le_one {α : Type*} (p q : PMF α) : tvMass p q ≤ 1 := by
-  have h := (mass_diff_summable p q).tsum_le_tsum
-    (fun a => (abs_sub (mass p a) (mass q a)).trans_eq (by
-      rw [abs_of_nonneg (mass_nonneg p a), abs_of_nonneg (mass_nonneg q a)]))
-    ((mass_summable p).add (mass_summable q))
-  rw [(mass_summable p).tsum_add (mass_summable q), mass_sum, mass_sum] at h
-  dsimp [tvMass]
-  linarith
-
-lemma advantage_eq_tvMass {α : Type*} (law : Spin → PMF α) :
-    advantage law = (∑ i : Spin, tvMass (law i) (marginal law))/3 := by
-  have ht (y : α) : mass (marginal law) y *
-      ((∑ i : Spin, |posterior law y i-1/3|)/2) =
-      (∑ i : Spin, |mass (law i) y-mass (marginal law) y|)/6 := by
-    rw [← mul_div_assoc, Finset.mul_sum]
-    simp_rw [weighted_abs_posterior]
-    rw [← Finset.sum_div]
-    ring
-  change (∑' y, mass (marginal law) y *
-    ((∑ i : Spin, |posterior law y i-1/3|)/2)) = _
-  simp_rw [ht]
-  rw [tsum_div_const, Summable.tsum_finsetSum (fun i _ => mass_diff_summable (law i) (marginal law))]
-  simp only [tvMass, ← Finset.sum_div]
-  ring
-
-lemma advantage_nonneg {α : Type*} (law : Spin → PMF α) : 0 ≤ advantage law := by
-  rw [advantage_eq_tvMass]
-  exact div_nonneg (Finset.sum_nonneg (fun i _ => tvMass_nonneg _ _)) (by norm_num)
-
-lemma advantage_le_one {α : Type*} (law : Spin → PMF α) : advantage law ≤ 1 := by
-  rw [advantage_eq_tvMass]
-  have h := Finset.sum_le_sum (s := Finset.univ) (fun (i : Spin) _ => tvMass_le_one (law i) (marginal law))
-  norm_num at h
-  linarith
-
-lemma kernel_mass_summable {α β : Type*} (p : PMF α) (k : α → PMF β) (b : β) :
-    Summable (fun a => mass p a*mass (k a) b) := by
-  apply Summable.of_nonneg_of_le (fun a => mul_nonneg (mass_nonneg p a) (mass_nonneg (k a) b))
-    (fun a => mul_le_of_le_one_right (mass_nonneg p a) (mass_le_one (k a) b))
-    (mass_summable p)
-
-lemma weighted_kernel_summable {α β : Type*} (p q : PMF α) (k : α → PMF β) (b : β) :
-    Summable (fun a => |mass p a-mass q a| *mass (k a) b) := by
-  apply Summable.of_nonneg_of_le (fun a => mul_nonneg (abs_nonneg _) (mass_nonneg (k a) b))
-    (fun a => mul_le_of_le_one_right (abs_nonneg _) (mass_le_one (k a) b))
-    (mass_diff_summable p q)
-
-lemma mass_diff_bind_le {α β : Type*} (p q : PMF α) (k : α → PMF β) (b : β) :
-    |mass (p.bind k) b-mass (q.bind k) b| ≤
-      ∑' a, |mass p a-mass q a| *mass (k a) b := by
-  rw [mass_bind, mass_bind, ← (kernel_mass_summable p k b).tsum_sub (kernel_mass_summable q k b)]
-  simp_rw [← sub_mul]
-  have hnorm : Summable (fun a => ‖(mass p a-mass q a)*mass (k a) b‖) :=
-    (weighted_kernel_summable p q k b).congr (fun a => by
-      rw [Real.norm_eq_abs, abs_mul, abs_of_nonneg (mass_nonneg (k a) b)])
-  simpa only [Real.norm_eq_abs, abs_mul, abs_of_nonneg (mass_nonneg (k _) b)] using
-    norm_tsum_le_tsum_norm hnorm
-
-lemma tvMass_bind_le {α β : Type*} (p q : PMF α) (k : α → PMF β) :
-    tvMass (p.bind k) (q.bind k) ≤ tvMass p q := by
-  have hrow (a : α) : (∑' b, |mass p a-mass q a| *mass (k a) b) =
-      |mass p a-mass q a| := by
-    rw [(mass_summable (k a)).tsum_mul_left, mass_sum, mul_one]
-  have hs : Summable (Function.uncurry (fun a b => |mass p a-mass q a| *mass (k a) b)) := by
-    apply (summable_prod_of_nonneg (fun ab => mul_nonneg (abs_nonneg _) (mass_nonneg _ _))).mpr
-    change (∀ a, Summable (fun b => |mass p a-mass q a| * mass (k a) b)) ∧
-      Summable (fun a => ∑' b, |mass p a-mass q a| * mass (k a) b)
-    constructor
-    · intro a
-      exact (mass_summable (k a)).mul_left |mass p a-mass q a|
-    ·
-      simpa only [hrow] using mass_diff_summable p q
-  have h := Summable.tsum_le_tsum (mass_diff_bind_le p q k)
-    (mass_diff_summable (p.bind k) (q.bind k)) hs.prod_symm.prod
-  rw [hs.tsum_comm] at h
-  simp_rw [hrow] at h
-  exact div_le_div_of_nonneg_right h (by norm_num)
-
-lemma marginal_bind {α β : Type*} (law : Spin → PMF α) (k : α → PMF β) :
-    marginal (fun i => (law i).bind k) = (marginal law).bind k := by
-  simp only [marginal, PMF.bind_bind]
-
- 
-theorem advantage_bind_le {α β : Type*} (law : Spin → PMF α) (k : α → PMF β) :
-    advantage (fun i => (law i).bind k) ≤ advantage law := by
-  rw [advantage_eq_tvMass, advantage_eq_tvMass, marginal_bind]
-  exact div_le_div_of_nonneg_right (Finset.sum_le_sum (fun i _ => tvMass_bind_le (law i) (marginal law) k))
-    (by norm_num)
-
-end ThreeState
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-! Three-state symmetric broadcast experiments, posterior advantage and total variation. -/
+
+namespace ThreeState
+
+abbrev Spin := Fin 3
+
+ 
+def Admissible (lam : ℝ) : Prop := -(1 / 2 : ℝ) ≤ lam ∧ lam ≤ 1
+
+ 
+noncomputable def channel (lam : ℝ) (i j : Spin) : ℝ :=
+  if i = j then (1 + 2 * lam) / 3 else (1 - lam) / 3
+
+ 
+
+def Observation : ℕ → Type
+  | 0 => Spin
+  | n + 1 => Multiset (Observation n)
+
+theorem channel_nonneg {lam : ℝ} (h : Admissible lam) (i j : Spin) :
+    0 ≤ channel lam i j := by
+  unfold channel
+  split_ifs
+  · dsimp [Admissible] at h
+    linarith [h.1]
+  · dsimp [Admissible] at h
+    linarith [h.2]
+
+theorem channel_sum (lam : ℝ) (i : Spin) : ∑ j, channel lam i j = 1 := by
+  fin_cases i <;> simp [channel, Fin.sum_univ_succ] <;> ring
+
+noncomputable def channelPMF (lam : ℝ) (h : Admissible lam) (i : Spin) : PMF Spin :=
+  PMF.ofFintype (fun j => ENNReal.ofReal (channel lam i j)) (by
+    rw [← ENNReal.ofReal_sum_of_nonneg (fun j _ => channel_nonneg h i j)]
+    simp [channel_sum])
+
+ 
+noncomputable def iidList {α : Type*} (p : PMF α) : ℕ → PMF (List α)
+  | 0 => PMF.pure []
+  | n + 1 => p.bind fun a => (iidList p n).map (List.cons a)
+
+ 
+
+noncomputable def observedLaw (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) :
+    (n : ℕ) → Spin → PMF (Observation n)
+  | 0, i => PMF.pure i
+  | n + 1, i => offspring.bind fun k =>
+      (iidList ((channelPMF lam h i).bind (observedLaw offspring lam h n)) k).map
+        (fun xs : List (Observation n) => (xs : Multiset (Observation n)))
+
+noncomputable def uniformSpin : PMF Spin := PMF.ofFintype (fun _ => 1 / 3) (by
+  simp
+  exact ENNReal.mul_inv_cancel (by norm_num) (by finiteness))
+
+ 
+noncomputable def marginal {α : Type*} (law : Spin → PMF α) : PMF α :=
+  uniformSpin.bind law
+
+ 
+noncomputable def posterior {α : Type*} (law : Spin → PMF α) (y : α) (i : Spin) : ℝ :=
+  if marginal law y = 0 then 1 / 3
+  else (law i y).toReal / (3 * (marginal law y).toReal)
+
+ 
+
+noncomputable def advantage {α : Type*} (law : Spin → PMF α) : ℝ :=
+  ∑' y, (marginal law y).toReal *
+    ((∑ i : Spin, |posterior law y i - 1 / 3|) / 2)
+
+noncomputable def regularLaw (b : ℕ) (lam : ℝ) (h : Admissible lam) (n : ℕ) :=
+  observedLaw (PMF.pure b) lam h n
+
+ 
+noncomputable def poissonOffspring (d : ℝ) (hd : 0 ≤ d) : PMF ℕ := by
+  refine ⟨fun k => ENNReal.ofReal (Real.exp (-d) * d ^ k / (k.factorial : ℝ)), ?_⟩
+  apply ENNReal.hasSum_coe.2
+  have hsum : HasSum (fun k : ℕ => Real.exp (-d) * d ^ k / (k.factorial : ℝ)) 1 :=
+    ProbabilityTheory.hasSum_one_poissonMeasure ⟨d, hd⟩
+  simpa using hsum.toNNReal (fun k => by positivity)
+
+noncomputable def poissonLaw (d : ℝ) (hd : 0 ≤ d) (lam : ℝ)
+    (h : Admissible lam) (n : ℕ) :=
+  observedLaw (poissonOffspring d hd) lam h n
+
+noncomputable def regularAdvantage (b : ℕ) (lam : ℝ) (h : Admissible lam) (n : ℕ) : ℝ :=
+  advantage (regularLaw b lam h n)
+
+noncomputable def poissonAdvantage (d : ℝ) (hd : 1 < d) (lam : ℝ)
+    (h : Admissible lam) (n : ℕ) : ℝ :=
+  advantage (poissonLaw d (le_trans (by norm_num) hd.le) lam h n)
+
+ 
+def Reconstructs (a : ℕ → ℝ) : Prop :=
+  ∃ L : ℝ, 0 < L ∧ Filter.Tendsto a Filter.atTop (nhds L)
+
+ 
+def Nonreconstructs (a : ℕ → ℝ) : Prop :=
+  Filter.Tendsto a Filter.atTop (nhds 0)
+
+end ThreeState
+namespace ThreeState
+
+noncomputable def mass {α : Type*} (p : PMF α) (a : α) : ℝ := (p a).toReal
+
+lemma mass_nonneg {α : Type*} (p : PMF α) (a : α) : 0 ≤ mass p a := ENNReal.toReal_nonneg
+lemma mass_sum {α : Type*} (p : PMF α) : ∑' a, mass p a = 1 := by
+  simp only [mass, ← ENNReal.tsum_toReal_eq (fun a => p.apply_ne_top a), p.tsum_coe,
+    ENNReal.toReal_one]
+lemma mass_summable {α : Type*} (p : PMF α) : Summable (mass p) :=
+  ENNReal.summable_toReal p.tsum_coe_ne_top
+
+lemma mass_bind {α β : Type*} (p : PMF α) (k : α → PMF β) (b : β) :
+    mass (p.bind k) b = ∑' a, mass p a * mass (k a) b := by
+  simp only [mass, PMF.bind_apply, ENNReal.tsum_toReal_eq
+    (fun a => ENNReal.mul_ne_top (p.apply_ne_top a) ((k a).apply_ne_top b)), ENNReal.toReal_mul]
+
+lemma mass_uniform (i : Spin) : mass uniformSpin i = 1/3 := by
+  simp [mass, uniformSpin, PMF.ofFintype_apply]
+
+lemma mass_marginal {α : Type*} (law : Spin → PMF α) (y : α) :
+    mass (marginal law) y = (∑ i : Spin, mass (law i) y)/3 := by
+  simp only [marginal, mass_bind, tsum_fintype, mass_uniform]
+  rw [← Finset.mul_sum]
+  ring
+
+lemma mass_eq_zero {α : Type*} (p : PMF α) (a : α) : mass p a = 0 ↔ p a = 0 := by
+  simp [mass, ENNReal.toReal_eq_zero_iff, p.apply_ne_top a]
+
+lemma marginal_zero {α : Type*} (law : Spin → PMF α) (y : α)
+    (h : marginal law y = 0) (i : Spin) : law i y = 0 := by
+  apply (mass_eq_zero (law i) y).mp
+  have hh : mass (marginal law) y = 0 := (mass_eq_zero _ _).mpr h
+  rw [mass_marginal] at hh
+  have hs : ∑ j : Spin, mass (law j) y = 0 := by linarith
+  exact (Finset.sum_eq_zero_iff_of_nonneg (fun j _ => mass_nonneg (law j) y)).mp hs i (by simp)
+
+lemma posterior_nonneg {α : Type*} (law : Spin → PMF α) (y : α) (i : Spin) :
+    0 ≤ posterior law y i := by
+  unfold posterior
+  split_ifs <;> positivity
+
+lemma posterior_sum {α : Type*} (law : Spin → PMF α) (y : α) :
+    ∑ i : Spin, posterior law y i = 1 := by
+  classical
+  have hn := (marginal law).apply_ne_top y
+  by_cases h : marginal law y = 0
+  · simp [posterior, h]
+  · have hm : mass (marginal law) y ≠ 0 := mt (mass_eq_zero _ _).mp h
+    simp only [posterior, ite_eq_right h, ← Finset.sum_div]
+    change (∑ i : Spin, mass (law i) y)/(3*mass (marginal law) y) = 1
+    have he := mass_marginal law y
+    field_simp
+    linarith
+
+ 
+lemma weighted_posterior {α : Type*} (law : Spin → PMF α) (y : α) (i : Spin) :
+    mass (marginal law) y * (posterior law y i-1/3) =
+      (mass (law i) y-mass (marginal law) y)/3 := by
+  classical
+  by_cases h : marginal law y = 0
+  · have hi := marginal_zero law y h i
+    simp [mass, h, hi, posterior]
+  · have hm : mass (marginal law) y ≠ 0 := mt (mass_eq_zero _ _).mp h
+    simp only [posterior, ite_eq_right h]
+    change mass (marginal law) y * (mass (law i) y / (3*mass (marginal law) y)-1/3) = _
+    field_simp
+
+lemma weighted_abs_posterior {α : Type*} (law : Spin → PMF α) (y : α) (i : Spin) :
+    mass (marginal law) y * |posterior law y i-1/3| =
+      |mass (law i) y-mass (marginal law) y|/3 := by
+  calc
+    mass (marginal law) y * |posterior law y i-1/3| =
+        |mass (marginal law) y * (posterior law y i-1/3)| := by
+      rw [abs_mul, abs_of_nonneg (mass_nonneg (marginal law) y)]
+    _ = _ := by rw [weighted_posterior, abs_div]; norm_num
+
+lemma mass_le_one {α : Type*} (p : PMF α) (a : α) : mass p a ≤ 1 := by
+  simpa [mass] using ENNReal.toReal_mono (by finiteness) (p.coe_le_one a)
+
+noncomputable def tvMass {α : Type*} (p q : PMF α) : ℝ :=
+  (∑' a, |mass p a-mass q a|)/2
+
+lemma mass_diff_summable {α : Type*} (p q : PMF α) :
+    Summable (fun a => |mass p a-mass q a|) :=
+  ((mass_summable p).sub (mass_summable q)).abs
+
+lemma tvMass_nonneg {α : Type*} (p q : PMF α) : 0 ≤ tvMass p q := by
+  exact div_nonneg (tsum_nonneg (fun a => abs_nonneg _)) (by norm_num)
+
+lemma tvMass_le_one {α : Type*} (p q : PMF α) : tvMass p q ≤ 1 := by
+  have h := (mass_diff_summable p q).tsum_le_tsum
+    (fun a => (abs_sub (mass p a) (mass q a)).trans_eq (by
+      rw [abs_of_nonneg (mass_nonneg p a), abs_of_nonneg (mass_nonneg q a)]))
+    ((mass_summable p).add (mass_summable q))
+  rw [(mass_summable p).tsum_add (mass_summable q), mass_sum, mass_sum] at h
+  dsimp [tvMass]
+  linarith
+
+lemma advantage_eq_tvMass {α : Type*} (law : Spin → PMF α) :
+    advantage law = (∑ i : Spin, tvMass (law i) (marginal law))/3 := by
+  have ht (y : α) : mass (marginal law) y *
+      ((∑ i : Spin, |posterior law y i-1/3|)/2) =
+      (∑ i : Spin, |mass (law i) y-mass (marginal law) y|)/6 := by
+    rw [← mul_div_assoc, Finset.mul_sum]
+    simp_rw [weighted_abs_posterior]
+    rw [← Finset.sum_div]
+    ring
+  change (∑' y, mass (marginal law) y *
+    ((∑ i : Spin, |posterior law y i-1/3|)/2)) = _
+  simp_rw [ht]
+  rw [tsum_div_const, Summable.tsum_finsetSum (fun i _ => mass_diff_summable (law i) (marginal law))]
+  simp only [tvMass, ← Finset.sum_div]
+  ring
+
+lemma advantage_nonneg {α : Type*} (law : Spin → PMF α) : 0 ≤ advantage law := by
+  rw [advantage_eq_tvMass]
+  exact div_nonneg (Finset.sum_nonneg (fun i _ => tvMass_nonneg _ _)) (by norm_num)
+
+lemma advantage_le_one {α : Type*} (law : Spin → PMF α) : advantage law ≤ 1 := by
+  rw [advantage_eq_tvMass]
+  have h := Finset.sum_le_sum (s := Finset.univ) (fun (i : Spin) _ => tvMass_le_one (law i) (marginal law))
+  norm_num at h
+  linarith
+
+lemma kernel_mass_summable {α β : Type*} (p : PMF α) (k : α → PMF β) (b : β) :
+    Summable (fun a => mass p a*mass (k a) b) := by
+  apply Summable.of_nonneg_of_le (fun a => mul_nonneg (mass_nonneg p a) (mass_nonneg (k a) b))
+    (fun a => mul_le_of_le_one_right (mass_nonneg p a) (mass_le_one (k a) b))
+    (mass_summable p)
+
+lemma weighted_kernel_summable {α β : Type*} (p q : PMF α) (k : α → PMF β) (b : β) :
+    Summable (fun a => |mass p a-mass q a| *mass (k a) b) := by
+  apply Summable.of_nonneg_of_le (fun a => mul_nonneg (abs_nonneg _) (mass_nonneg (k a) b))
+    (fun a => mul_le_of_le_one_right (abs_nonneg _) (mass_le_one (k a) b))
+    (mass_diff_summable p q)
+
+lemma mass_diff_bind_le {α β : Type*} (p q : PMF α) (k : α → PMF β) (b : β) :
+    |mass (p.bind k) b-mass (q.bind k) b| ≤
+      ∑' a, |mass p a-mass q a| *mass (k a) b := by
+  rw [mass_bind, mass_bind, ← (kernel_mass_summable p k b).tsum_sub (kernel_mass_summable q k b)]
+  simp_rw [← sub_mul]
+  have hnorm : Summable (fun a => ‖(mass p a-mass q a)*mass (k a) b‖) :=
+    (weighted_kernel_summable p q k b).congr (fun a => by
+      rw [Real.norm_eq_abs, abs_mul, abs_of_nonneg (mass_nonneg (k a) b)])
+  simpa only [Real.norm_eq_abs, abs_mul, abs_of_nonneg (mass_nonneg (k _) b)] using
+    norm_tsum_le_tsum_norm hnorm
+
+lemma tvMass_bind_le {α β : Type*} (p q : PMF α) (k : α → PMF β) :
+    tvMass (p.bind k) (q.bind k) ≤ tvMass p q := by
+  have hrow (a : α) : (∑' b, |mass p a-mass q a| *mass (k a) b) =
+      |mass p a-mass q a| := by
+    rw [(mass_summable (k a)).tsum_mul_left, mass_sum, mul_one]
+  have hs : Summable (Function.uncurry (fun a b => |mass p a-mass q a| *mass (k a) b)) := by
+    apply (summable_prod_of_nonneg (fun ab => mul_nonneg (abs_nonneg _) (mass_nonneg _ _))).mpr
+    change (∀ a, Summable (fun b => |mass p a-mass q a| * mass (k a) b)) ∧
+      Summable (fun a => ∑' b, |mass p a-mass q a| * mass (k a) b)
+    constructor
+    · intro a
+      exact (mass_summable (k a)).mul_left |mass p a-mass q a|
+    ·
+      simpa only [hrow] using mass_diff_summable p q
+  have h := Summable.tsum_le_tsum (mass_diff_bind_le p q k)
+    (mass_diff_summable (p.bind k) (q.bind k)) hs.prod_symm.prod
+  rw [hs.tsum_comm] at h
+  simp_rw [hrow] at h
+  exact div_le_div_of_nonneg_right h (by norm_num)
+
+lemma marginal_bind {α β : Type*} (law : Spin → PMF α) (k : α → PMF β) :
+    marginal (fun i => (law i).bind k) = (marginal law).bind k := by
+  simp only [marginal, PMF.bind_bind]
+
+ 
+theorem advantage_bind_le {α β : Type*} (law : Spin → PMF α) (k : α → PMF β) :
+    advantage (fun i => (law i).bind k) ≤ advantage law := by
+  rw [advantage_eq_tvMass, advantage_eq_tvMass, marginal_bind]
+  exact div_le_div_of_nonneg_right (Finset.sum_le_sum (fun i _ => tvMass_bind_le (law i) (marginal law) k))
+    (by norm_num)
+
+end ThreeState
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeState/Couplings.lean b/lean/OAI/Probability/ThreeState/Couplings.lean
index 964b4dceea1a34e0f4d0c6c5961b1f306d8c92f8..b3992be6fd4f4d8fa284629ec4ac41c39ba424a3 100644
--- a/lean/OAI/Probability/ThreeState/Couplings.lean
+++ b/lean/OAI/Probability/ThreeState/Couplings.lean
@@ -1,164 +1,164 @@
-import Mathlib
-import OAI.Probability.ThreeState.Update
-
-namespace OAI
-
-/-! Common-mass couplings and total variation bounds for products and mixtures. -/
-
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-noncomputable section
-
- 
-
-def commonMass {α : Type*} (p q:PMF α) : ℝ := ∑'a,min (mass p a) (mass q a)
-lemma commonMass_summable {α : Type*} (p q:PMF α) :
-    Summable (fun a=>min (mass p a) (mass q a)) :=
-  Summable.of_nonneg_of_le (fun a=>le_min (mass_nonneg p a) (mass_nonneg q a))
-    (fun value => min_le_left (mass p value) (mass q value)) (mass_summable p)
-lemma commonMass_nonneg {α : Type*} (p q:PMF α) : 0≤commonMass p q :=
-  tsum_nonneg (fun a=>le_min (mass_nonneg p a) (mass_nonneg q a))
-lemma commonMass_eq {α : Type*} (p q:PMF α) : commonMass p q=1-tvMass p q := by
-  have he (a:α) : min (mass p a) (mass q a)=(mass p a+mass q a-|mass p a-mass q a|)/2 := by
-    rw [min_def];split_ifs with h
-    · rw [abs_of_nonpos (sub_nonpos.mpr h)];ring
-    · rw [abs_of_nonneg (sub_nonneg.mpr (le_of_not_ge h))];ring
-  simp_rw [commonMass,he]
-  rw [tsum_div_const,((mass_summable p).add (mass_summable q)).tsum_sub (mass_diff_summable p q),
-    (mass_summable p).tsum_add (mass_summable q),mass_sum,mass_sum]
-  unfold tvMass
-  ring
-lemma commonMass_le_one {α : Type*} (p q:PMF α) : commonMass p q≤1 := by
-  rw [commonMass_eq];linarith [tvMass_nonneg p q]
-
- 
-
-def sharedPMF {α : Type*} (p q:PMF α) (hc:0<commonMass p q) : PMF α := by
-  refine ⟨fun a=>ENNReal.ofReal (min (mass p a) (mass q a)/commonMass p q),?_⟩
-  have hs : HasSum (fun a=>min (mass p a) (mass q a)/commonMass p q) 1 := by
-    convert! (commonMass_summable p q).hasSum.div_const (commonMass p q) using 1
-    exact (div_self hc.ne').symm
-  apply ENNReal.hasSum_coe.2
-  simpa using (hs.toNNReal (fun a=>div_nonneg
-    (le_min (mass_nonneg p a) (mass_nonneg q a)) hc.le))
-lemma mass_sharedPMF {α : Type*} (p q:PMF α) (hc:0<commonMass p q) (a:α) :
-    mass (sharedPMF p q hc) a=min (mass p a) (mass q a)/commonMass p q := by
-  change (ENNReal.ofReal _).toReal=_
-  exact ENNReal.toReal_ofReal (div_nonneg (le_min (mass_nonneg p a) (mass_nonneg q a)) hc.le)
-lemma sharedPMF_scaled {α : Type*} (p q:PMF α) (hc:0<commonMass p q) (a:α) :
-    commonMass p q*mass (sharedPMF p q hc) a=min (mass p a) (mass q a) := by
-  rw [mass_sharedPMF];field_simp
-
-lemma commonMass_iidVector {α : Type*} (p q:PMF α) (n:ℕ) :
-    (commonMass p q)^n≤commonMass (iidVector p n) (iidVector q n) := by
-  rcases n with _|n
-  · simp only [iidVector,commonMass,min_self,mass_sum,pow_zero,le_refl]
-  by_cases hc:commonMass p q=0
-  · rw [hc,zero_pow (Nat.succ_ne_zero _)]
-    exact commonMass_nonneg _ _
-  have hcpos:0<commonMass p q:=lt_of_le_of_ne (commonMass_nonneg p q) (Ne.symm hc)
-  let r:=sharedPMF p q hcpos
-  have hpoint (v:Fin (n+1)→α) : (commonMass p q)^(n+1)*mass (iidVector r (n+1)) v ≤
-      min (mass (iidVector p (n+1)) v) (mass (iidVector q (n+1)) v) := by
-    simp only [mass_iidVector]
-    have he : (commonMass p q)^(n+1)*∏j:Fin (n+1),mass r (v j)=
-        ∏j:Fin (n+1),min (mass p (v j)) (mass q (v j)) := by
-      rw [←show (∏j:Fin (n+1),commonMass p q)=(commonMass p q)^(n+1) by simp,
-        ←Finset.prod_mul_distrib]
-      apply Finset.prod_congr rfl
-      intro j _
-      exact sharedPMF_scaled p q hcpos (v j)
-    rw [he]
-    exact le_min
-      (Finset.prod_le_prod₀
-        (fun index _ => le_min (mass_nonneg p (v index)) (mass_nonneg q (v index)))
-        (fun index _ => min_le_left (mass p (v index)) (mass q (v index))))
-      (Finset.prod_le_prod₀
-        (fun index _ => le_min (mass_nonneg p (v index)) (mass_nonneg q (v index)))
-        (fun index _ => min_le_right (mass p (v index)) (mass q (v index))))
-  have hh:=(mass_summable (iidVector r (n+1))).mul_left ((commonMass p q)^(n+1))
-  have ht:=Summable.tsum_le_tsum hpoint hh (commonMass_summable _ _)
-  rw [tsum_mul_left] at ht
-  simpa only [mass_sum,mul_one,commonMass] using ht
-
-lemma tvMass_iidVector_le {α : Type*} (p q:PMF α) (n:ℕ) :
-    tvMass (iidVector p n) (iidVector q n)≤1-(1-tvMass p q)^n := by
-  have hh:=commonMass_iidVector p q n
-  rw [commonMass_eq,commonMass_eq] at hh
-  linarith
-end
-end ThreeState
-
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-noncomputable section
-
-lemma tvMass_map_le {α β:Type*} (p q:PMF α) (f:α→β) : tvMass (p.map f) (q.map f)≤tvMass p q :=
-  tvMass_bind_le p q (fun a=>PMF.pure (f a))
-
-lemma mass_edge_difference {α:Type*} (law:Spin→PMF α) (lam:ℝ) (h:Admissible lam)
-    (i j:Spin) (a:α) :
-    mass (edgeLaw law lam h i) a-mass (edgeLaw law lam h j) a=
-      lam*(mass (law i) a-mass (law j) a) := by
-  rw [mass_edgeLaw,mass_edgeLaw,←weighted_likelihood law a i,←weighted_likelihood law a j]
-  change _*(1+lam*(_-1))-_*(1+lam*(_-1))=_
-  ring
-lemma tvMass_edge {α:Type*} (law:Spin→PMF α) (lam:ℝ) (h:Admissible lam) (i j:Spin) :
-    tvMass (edgeLaw law lam h i) (edgeLaw law lam h j)= |lam| *tvMass (law i) (law j) := by
-  unfold tvMass
-  simp_rw [mass_edge_difference,abs_mul]
-  rw [tsum_mul_left]
-  ring
-
-lemma tvMass_marginal_le {α:Type*} (law:Spin→PMF α) (i:Spin) :
-    tvMass (law i) (marginal law)≤(∑j:Spin,tvMass (law i) (law j))/3 := by
-  have he (a:α) : mass (law i) a-mass (marginal law) a=
-      (∑j:Spin,(mass (law i) a-mass (law j) a))/3 := by
-    rw [mass_marginal,Finset.sum_sub_distrib]
-    simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]
-    ring
-  have hp (a:α) : |mass (law i) a-mass (marginal law) a|≤
-      (∑j:Spin,|mass (law i) a-mass (law j) a|)/3 := by
-    rw [he,abs_div]
-    rw [abs_of_nonneg (by norm_num : (0:ℝ)≤3)]
-    exact div_le_div_of_nonneg_right (Finset.abs_sum_le_sum_abs _ _) (by norm_num)
-  have hs : Summable (fun a=>∑j:Spin,|mass (law i) a-mass (law j) a|) :=
-    summable_sum (fun j _=>mass_diff_summable _ _)
-  have hh:=Summable.tsum_le_tsum hp (mass_diff_summable _ _) (hs.div_const 3)
-  rw [tsum_div_const,Summable.tsum_finsetSum (fun j _=>mass_diff_summable (law i) (law j))] at hh
-  unfold tvMass
-  rw [←Finset.sum_div]
-  exact (div_le_div_of_nonneg_right hh (by norm_num : (0:ℝ)≤2)).trans_eq (by ring)
-
-lemma advantage_le_pair_bound {α:Type*} (law:Spin→PMF α) (x:ℝ)
-    (hx:∀i j,tvMass (law i) (law j)≤x) : advantage law≤x := by
-  rw [advantage_eq_tvMass]
-  have hb (i:Spin):tvMass (law i) (marginal law)≤x := by
-    calc
-      _ ≤ (∑j:Spin,tvMass (law i) (law j))/3 := tvMass_marginal_le law i
-      _ ≤ (∑j:Spin,x)/3 := div_le_div_of_nonneg_right (Finset.sum_le_sum (fun j _=>hx i j)) (by norm_num)
-      _ = x := by simp
-  have hh:=Finset.sum_le_sum (s:=Finset.univ) (fun i _=>hb i)
-  have hd:=div_le_div_of_nonneg_right hh (by norm_num : (0:ℝ)≤3)
-  simpa using hd
-
-lemma tvMass_sigmaPMF {ι:Type*} {α:ι→Type*} (p:PMF ι) (q r:∀i,PMF (α i)) :
-    tvMass (sigmaPMF p q) (sigmaPMF p r)=∑'i,mass p i*tvMass (q i) (r i) := by
-  unfold tvMass
-  rw [(mass_diff_summable (sigmaPMF p q) (sigmaPMF p r)).tsum_sigma]
-  simp_rw [mass_sigmaPMF,←mul_sub,abs_mul,abs_of_nonneg (mass_nonneg p _),tsum_mul_left]
-  rw [←tsum_div_const]
-  congr 1
-  funext i
-  ring
-
-lemma summable_weighted_tv {ι:Type*} {α:ι→Type*} (p:PMF ι) (q r:∀i,PMF (α i)) :
-    Summable (fun i=>mass p i*tvMass (q i) (r i)) :=
-  Summable.of_nonneg_of_le (fun i=>mul_nonneg (mass_nonneg p i) (tvMass_nonneg _ _))
-    (fun i=>mul_le_of_le_one_right (mass_nonneg p i) (tvMass_le_one _ _)) (mass_summable p)
-end
-end ThreeState
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeState.Update
+
+namespace OAI
+
+/-! Common-mass couplings and total variation bounds for products and mixtures. -/
+
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+noncomputable section
+
+ 
+
+def commonMass {α : Type*} (p q:PMF α) : ℝ := ∑'a,min (mass p a) (mass q a)
+lemma commonMass_summable {α : Type*} (p q:PMF α) :
+    Summable (fun a=>min (mass p a) (mass q a)) :=
+  Summable.of_nonneg_of_le (fun a=>le_min (mass_nonneg p a) (mass_nonneg q a))
+    (fun value => min_le_left (mass p value) (mass q value)) (mass_summable p)
+lemma commonMass_nonneg {α : Type*} (p q:PMF α) : 0≤commonMass p q :=
+  tsum_nonneg (fun a=>le_min (mass_nonneg p a) (mass_nonneg q a))
+lemma commonMass_eq {α : Type*} (p q:PMF α) : commonMass p q=1-tvMass p q := by
+  have he (a:α) : min (mass p a) (mass q a)=(mass p a+mass q a-|mass p a-mass q a|)/2 := by
+    rw [min_def];split_ifs with h
+    · rw [abs_of_nonpos (sub_nonpos.mpr h)];ring
+    · rw [abs_of_nonneg (sub_nonneg.mpr (le_of_not_ge h))];ring
+  simp_rw [commonMass,he]
+  rw [tsum_div_const,((mass_summable p).add (mass_summable q)).tsum_sub (mass_diff_summable p q),
+    (mass_summable p).tsum_add (mass_summable q),mass_sum,mass_sum]
+  unfold tvMass
+  ring
+lemma commonMass_le_one {α : Type*} (p q:PMF α) : commonMass p q≤1 := by
+  rw [commonMass_eq];linarith [tvMass_nonneg p q]
+
+ 
+
+def sharedPMF {α : Type*} (p q:PMF α) (hc:0<commonMass p q) : PMF α := by
+  refine ⟨fun a=>ENNReal.ofReal (min (mass p a) (mass q a)/commonMass p q),?_⟩
+  have hs : HasSum (fun a=>min (mass p a) (mass q a)/commonMass p q) 1 := by
+    convert! (commonMass_summable p q).hasSum.div_const (commonMass p q) using 1
+    exact (div_self hc.ne').symm
+  apply ENNReal.hasSum_coe.2
+  simpa using (hs.toNNReal (fun a=>div_nonneg
+    (le_min (mass_nonneg p a) (mass_nonneg q a)) hc.le))
+lemma mass_sharedPMF {α : Type*} (p q:PMF α) (hc:0<commonMass p q) (a:α) :
+    mass (sharedPMF p q hc) a=min (mass p a) (mass q a)/commonMass p q := by
+  change (ENNReal.ofReal _).toReal=_
+  exact ENNReal.toReal_ofReal (div_nonneg (le_min (mass_nonneg p a) (mass_nonneg q a)) hc.le)
+lemma sharedPMF_scaled {α : Type*} (p q:PMF α) (hc:0<commonMass p q) (a:α) :
+    commonMass p q*mass (sharedPMF p q hc) a=min (mass p a) (mass q a) := by
+  rw [mass_sharedPMF];field_simp
+
+lemma commonMass_iidVector {α : Type*} (p q:PMF α) (n:ℕ) :
+    (commonMass p q)^n≤commonMass (iidVector p n) (iidVector q n) := by
+  rcases n with _|n
+  · simp only [iidVector,commonMass,min_self,mass_sum,pow_zero,le_refl]
+  by_cases hc:commonMass p q=0
+  · rw [hc,zero_pow (Nat.succ_ne_zero _)]
+    exact commonMass_nonneg _ _
+  have hcpos:0<commonMass p q:=lt_of_le_of_ne (commonMass_nonneg p q) (Ne.symm hc)
+  let r:=sharedPMF p q hcpos
+  have hpoint (v:Fin (n+1)→α) : (commonMass p q)^(n+1)*mass (iidVector r (n+1)) v ≤
+      min (mass (iidVector p (n+1)) v) (mass (iidVector q (n+1)) v) := by
+    simp only [mass_iidVector]
+    have he : (commonMass p q)^(n+1)*∏j:Fin (n+1),mass r (v j)=
+        ∏j:Fin (n+1),min (mass p (v j)) (mass q (v j)) := by
+      rw [←show (∏j:Fin (n+1),commonMass p q)=(commonMass p q)^(n+1) by simp,
+        ←Finset.prod_mul_distrib]
+      apply Finset.prod_congr rfl
+      intro j _
+      exact sharedPMF_scaled p q hcpos (v j)
+    rw [he]
+    exact le_min
+      (Finset.prod_le_prod₀
+        (fun index _ => le_min (mass_nonneg p (v index)) (mass_nonneg q (v index)))
+        (fun index _ => min_le_left (mass p (v index)) (mass q (v index))))
+      (Finset.prod_le_prod₀
+        (fun index _ => le_min (mass_nonneg p (v index)) (mass_nonneg q (v index)))
+        (fun index _ => min_le_right (mass p (v index)) (mass q (v index))))
+  have hh:=(mass_summable (iidVector r (n+1))).mul_left ((commonMass p q)^(n+1))
+  have ht:=Summable.tsum_le_tsum hpoint hh (commonMass_summable _ _)
+  rw [tsum_mul_left] at ht
+  simpa only [mass_sum,mul_one,commonMass] using ht
+
+lemma tvMass_iidVector_le {α : Type*} (p q:PMF α) (n:ℕ) :
+    tvMass (iidVector p n) (iidVector q n)≤1-(1-tvMass p q)^n := by
+  have hh:=commonMass_iidVector p q n
+  rw [commonMass_eq,commonMass_eq] at hh
+  linarith
+end
+end ThreeState
+
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+noncomputable section
+
+lemma tvMass_map_le {α β:Type*} (p q:PMF α) (f:α→β) : tvMass (p.map f) (q.map f)≤tvMass p q :=
+  tvMass_bind_le p q (fun a=>PMF.pure (f a))
+
+lemma mass_edge_difference {α:Type*} (law:Spin→PMF α) (lam:ℝ) (h:Admissible lam)
+    (i j:Spin) (a:α) :
+    mass (edgeLaw law lam h i) a-mass (edgeLaw law lam h j) a=
+      lam*(mass (law i) a-mass (law j) a) := by
+  rw [mass_edgeLaw,mass_edgeLaw,←weighted_likelihood law a i,←weighted_likelihood law a j]
+  change _*(1+lam*(_-1))-_*(1+lam*(_-1))=_
+  ring
+lemma tvMass_edge {α:Type*} (law:Spin→PMF α) (lam:ℝ) (h:Admissible lam) (i j:Spin) :
+    tvMass (edgeLaw law lam h i) (edgeLaw law lam h j)= |lam| *tvMass (law i) (law j) := by
+  unfold tvMass
+  simp_rw [mass_edge_difference,abs_mul]
+  rw [tsum_mul_left]
+  ring
+
+lemma tvMass_marginal_le {α:Type*} (law:Spin→PMF α) (i:Spin) :
+    tvMass (law i) (marginal law)≤(∑j:Spin,tvMass (law i) (law j))/3 := by
+  have he (a:α) : mass (law i) a-mass (marginal law) a=
+      (∑j:Spin,(mass (law i) a-mass (law j) a))/3 := by
+    rw [mass_marginal,Finset.sum_sub_distrib]
+    simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]
+    ring
+  have hp (a:α) : |mass (law i) a-mass (marginal law) a|≤
+      (∑j:Spin,|mass (law i) a-mass (law j) a|)/3 := by
+    rw [he,abs_div]
+    rw [abs_of_nonneg (by norm_num : (0:ℝ)≤3)]
+    exact div_le_div_of_nonneg_right (Finset.abs_sum_le_sum_abs _ _) (by norm_num)
+  have hs : Summable (fun a=>∑j:Spin,|mass (law i) a-mass (law j) a|) :=
+    summable_sum (fun j _=>mass_diff_summable _ _)
+  have hh:=Summable.tsum_le_tsum hp (mass_diff_summable _ _) (hs.div_const 3)
+  rw [tsum_div_const,Summable.tsum_finsetSum (fun j _=>mass_diff_summable (law i) (law j))] at hh
+  unfold tvMass
+  rw [←Finset.sum_div]
+  exact (div_le_div_of_nonneg_right hh (by norm_num : (0:ℝ)≤2)).trans_eq (by ring)
+
+lemma advantage_le_pair_bound {α:Type*} (law:Spin→PMF α) (x:ℝ)
+    (hx:∀i j,tvMass (law i) (law j)≤x) : advantage law≤x := by
+  rw [advantage_eq_tvMass]
+  have hb (i:Spin):tvMass (law i) (marginal law)≤x := by
+    calc
+      _ ≤ (∑j:Spin,tvMass (law i) (law j))/3 := tvMass_marginal_le law i
+      _ ≤ (∑j:Spin,x)/3 := div_le_div_of_nonneg_right (Finset.sum_le_sum (fun j _=>hx i j)) (by norm_num)
+      _ = x := by simp
+  have hh:=Finset.sum_le_sum (s:=Finset.univ) (fun i _=>hb i)
+  have hd:=div_le_div_of_nonneg_right hh (by norm_num : (0:ℝ)≤3)
+  simpa using hd
+
+lemma tvMass_sigmaPMF {ι:Type*} {α:ι→Type*} (p:PMF ι) (q r:∀i,PMF (α i)) :
+    tvMass (sigmaPMF p q) (sigmaPMF p r)=∑'i,mass p i*tvMass (q i) (r i) := by
+  unfold tvMass
+  rw [(mass_diff_summable (sigmaPMF p q) (sigmaPMF p r)).tsum_sigma]
+  simp_rw [mass_sigmaPMF,←mul_sub,abs_mul,abs_of_nonneg (mass_nonneg p _),tsum_mul_left]
+  rw [←tsum_div_const]
+  congr 1
+  funext i
+  ring
+
+lemma summable_weighted_tv {ι:Type*} {α:ι→Type*} (p:PMF ι) (q r:∀i,PMF (α i)) :
+    Summable (fun i=>mass p i*tvMass (q i) (r i)) :=
+  Summable.of_nonneg_of_le (fun i=>mul_nonneg (mass_nonneg p i) (tvMass_nonneg _ _))
+    (fun i=>mul_le_of_le_one_right (mass_nonneg p i) (tvMass_le_one _ _)) (mass_summable p)
+end
+end ThreeState
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeState/DegreePosterior.lean b/lean/OAI/Probability/ThreeState/DegreePosterior.lean
index 1883acae8ce5450763fb68d505e3500d4a3397f7..7b04d1082e95f6ef13d946a6becaf94b5ddd085e 100644
--- a/lean/OAI/Probability/ThreeState/DegreePosterior.lean
+++ b/lean/OAI/Probability/ThreeState/DegreePosterior.lean
@@ -1,205 +1,205 @@
-import Mathlib
-import OAI.Probability.ThreeState.PairCorrelation
-
-namespace OAI
-
-/-! Posterior laws by offspring degree and monotonicity of their moments. -/
-
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-open Radial (avg)
-
-lemma continuous_integral_message {f : Message → Message → ℝ} (hf : Continuous f.uncurry)
-    (Q : ProbabilityMeasure Message) : Continuous (fun a => ∫ b, f a b ∂(Q : Measure Message)) := by
-  simpa only [Measure.restrict_univ] using continuous_parametric_integral_of_continuous hf isCompact_univ
-
-lemma continuous_pairIncrement (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) : Continuous (pairIncrement lam h Q) :=
-  continuous_integral_message (f := fun a b => pairNormalizer lam h a b*Real.log (pairNormalizer lam h a b)) (continuous_pairIntegrand lam h hl0 hl1) Q
-
-noncomputable def degreePosterior (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message)
-    (hQ : Balanced Q) (n : ℕ) : ProbabilityMeasure Message :=
-  ⟨degreeOutput lam h Q n, degreeOutput_probability lam h Q hQ n⟩
-
-lemma integral_iid_cons (Q : ProbabilityMeasure Message) (n : ℕ)
-    (f : (Fin (n+1) → Message) → ℝ) (hf : Continuous f) :
-    (∫ m, f m ∂(iidMeasure Q (n+1) : Measure (Fin (n+1) → Message))) =
-      ∫ m : Fin n → Message, ∫ b, f (Fin.cons b m) ∂(Q : Measure Message)
-        ∂(iidMeasure Q n : Measure (Fin n → Message)) := by
-  change (∫ m, f m ∂Measure.pi (fun _ : Fin (n+1) => (Q : Measure Message))) = _
-  rw [← ((measurePreserving_piFinSuccAbove (fun _ : Fin (n+1) => (Q : Measure Message)) 0).symm).integral_comp']
-  simp only [MeasurableEquiv.piFinSuccAbove_symm_apply, Fin.insertNthEquiv,
-    Fin.insertNth_zero, Equiv.coe_fn_mk, Fin.zero_succAbove, cast_eq]
-  apply integral_prod_symm
-  exact (show Continuous (fun p : Message × (Fin n → Message) => f (Fin.cons p.1 p.2)) by
-    apply hf.comp
-    apply continuous_pi
-    intro j
-    refine Fin.cases ?_ (fun i => ?_) j
-    · simpa using (continuous_fst : Continuous (Prod.fst : Message × (Fin n → Message) → Message))
-    · simpa only [Fin.cons_succ, Function.comp_def] using ((continuous_apply i).comp continuous_snd : Continuous (fun p : Message × (Fin n → Message) => p.2 i))).integrable_of_hasCompactSupport
-        (HasCompactSupport.of_compactSpace _)
-
-lemma normalizer_cons (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    {n : ℕ} (m : Fin n → Message) (b : Message) :
-    normalizer lam h (Fin.cons b m) = normalizer lam h m * pairNormalizer lam h (combineMessage lam h m) b := by
-  have he (i : Spin) : productWeight lam h (Fin.cons b m) i = edgeMessage lam h b i*productWeight lam h m i := by
-    simp [productWeight, Fin.prod_univ_succ]
-  have hn := ne_of_gt (normalizer_pos lam h hl0 hl1 m)
-  change avg (productWeight lam h (Fin.cons b m)) = normalizer lam h m*avg (fun i => combineMessage lam h m i*edgeMessage lam h b i)
-  simp only [Radial.avg_expand, he, combineMessage, dite_eq_right hn]
-  field_simp
-
-lemma combineMessage_cons (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    {n : ℕ} (m : Fin n → Message) (b : Message) :
-    combineMessage lam h (Fin.cons b m) = pairMessage lam h hl0 hl1 (combineMessage lam h m) b := by
-  apply Subtype.ext
-  funext i
-  have hp := ne_of_gt (normalizer_pos lam h hl0 hl1 m)
-  have hp' := ne_of_gt (normalizer_pos lam h hl0 hl1 (Fin.cons b m))
-  change (combineMessage lam h (Fin.cons b m)) i = (combineMessage lam h m) i*edgeMessage lam h b i / pairNormalizer lam h (combineMessage lam h m) b
-  conv_lhs => rw [combineMessage, dite_eq_right hp']
-  change productWeight lam h (Fin.cons b m) i/normalizer lam h (Fin.cons b m) = _
-  rw [normalizer_cons lam h hl0 hl1]
-  have hw := weighted_combineMessage lam h m i
-  have he : productWeight lam h (Fin.cons b m) i = edgeMessage lam h b i*productWeight lam h m i := by
-    simp [productWeight, Fin.prod_univ_succ]
-  rw [he, ← hw]
-  field_simp [hp, ne_of_gt (pairNormalizer_pos lam h hl0 hl1 _ _)]
-
-noncomputable def pairOperator (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) (f : C(Message,ℝ)) : C(Message,ℝ) :=
-  ⟨fun a => ∫ b, pairNormalizer lam h a b*f (pairMessage lam h hl0 hl1 a b) ∂(Q : Measure Message),
-    continuous_integral_message (f := fun a b => pairNormalizer lam h a b*f (pairMessage lam h hl0 hl1 a b)) ((continuous_pairNormalizer lam h).mul (f.continuous.comp (continuous_pairMessage lam h hl0 hl1))) Q⟩
-
-lemma integral_degree_succ (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) (n : ℕ) (f : C(Message,ℝ)) :
-    (∫ m, f m ∂degreeOutput lam h Q (n+1)) =
-      ∫ a, pairOperator lam h hl0 hl1 Q f a ∂degreeOutput lam h Q n := by
-  rw [integral_degreeOutput lam h Q (n+1) f f.continuous.measurable,
-    integral_iid_cons Q n (fun m => normalizer lam h m*f (combineMessage lam h m)) ((continuous_normalizer lam h (n+1)).mul
-      (f.continuous.comp (continuous_combineMessage_positive lam h hl0 hl1 (n+1)))),
-    integral_degreeOutput lam h Q n _ (pairOperator lam h hl0 hl1 Q f).continuous.measurable]
-  apply integral_congr_ae
-  filter_upwards with m
-  simp only [normalizer_cons lam h hl0 hl1, combineMessage_cons lam h hl0 hl1, mul_assoc, pairOperator]
-  exact integral_const_mul _ _
-
-lemma correlationEntropy_zero (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message) :
-    correlationEntropy lam h Q 0 = 0 := by
-  simp [correlationEntropy, normalizer, productWeight]
-
-lemma correlationEntropy_succ (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (n : ℕ) :
-    correlationEntropy lam h Q (n+1) = correlationEntropy lam h Q n +
-      ∫ a, pairIncrement lam h Q a ∂degreeOutput lam h Q n := by
-  unfold correlationEntropy
-  rw [integral_iid_cons Q n (fun m => normalizer lam h m*Real.log (normalizer lam h m)) ((continuous_normalizer lam h (n+1)).mul (continuous_log_normalizer lam h hl0 hl1 (n+1)))]
-  have he (m : Fin n → Message) :
-      (∫ b, normalizer lam h (Fin.cons b m)*Real.log (normalizer lam h (Fin.cons b m)) ∂(Q : Measure Message)) =
-        normalizer lam h m*Real.log (normalizer lam h m)+normalizer lam h m*pairIncrement lam h Q (combineMessage lam h m) := by
-    let a := combineMessage lam h m
-    have hpn : Continuous (fun b => pairNormalizer lam h a b) :=
-      (continuous_pairNormalizer lam h).comp (continuous_const.prodMk continuous_id)
-    have hpi : Continuous (fun b => pairNormalizer lam h a b*Real.log (pairNormalizer lam h a b)) :=
-      (continuous_pairIntegrand lam h hl0 hl1).comp (continuous_const.prodMk continuous_id)
-    have hfun (b : Message) : normalizer lam h (Fin.cons b m)*Real.log (normalizer lam h (Fin.cons b m)) =
-        (normalizer lam h m*Real.log (normalizer lam h m))*pairNormalizer lam h a b+
-          normalizer lam h m*(pairNormalizer lam h a b*Real.log (pairNormalizer lam h a b)) := by
-      rw [normalizer_cons lam h hl0 hl1, Real.log_mul (ne_of_gt (normalizer_pos lam h hl0 hl1 m))
-        (ne_of_gt (pairNormalizer_pos lam h hl0 hl1 _ _))]
-      dsimp only [a]
-      ring
-    simp_rw [hfun]
-    rw [integral_add ((integrable_continuous_message Q hpn).const_mul _) ((integrable_continuous_message Q hpi).const_mul _),
-      integral_const_mul, integral_const_mul, integral_pairNormalizer lam h Q hQ, mul_one]
-    rfl
-  simp_rw [he]
-  rw [integral_add]
-  · rw [integral_degreeOutput lam h Q n _ (continuous_pairIncrement lam h hl0 hl1 Q).measurable]
-  · exact ((continuous_normalizer lam h n).mul (continuous_log_normalizer lam h hl0 hl1 n)).integrable_of_hasCompactSupport
-      (HasCompactSupport.of_compactSpace _)
-  · exact ((continuous_normalizer lam h n).mul ((continuous_pairIncrement lam h hl0 hl1 Q).comp
-      (continuous_combineMessage_positive lam h hl0 hl1 n))).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
-
-end ThreeState
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-open Radial (avg)
-
-lemma messageX_sq_tangent (a c : Message) :
-    (messageX a)^2+2*messageX a*avg (fun i => (a i-1)*(c i-a i)) ≤ (messageX c)^2 := by
-  let D := avg (fun i => (c i-a i)^2)/2
-  have hD : 0 ≤ D := div_nonneg (avg_nonneg (fun i => sq_nonneg (c i-a i))) (by norm_num)
-  have he : messageX c-messageX a = avg (fun i => (a i-1)*(c i-a i))+D := by
-    dsimp only [messageX, Radial.momentX, messageDeviation, D]
-    simp only [Radial.avg_expand]
-    ring
-  have hm := mul_nonneg (messageX_nonneg a) hD
-  have hx := congrArg (fun t : ℝ => 2*messageX a*t) he
-  nlinarith only [hx, hm, sq_nonneg (messageX c-messageX a)]
-
-lemma integral_weighted_pair_deviation (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (a : Message) (i : Spin) :
-    (∫ b, pairNormalizer lam h a b*(pairMessage lam h hl0 hl1 a b i-a i) ∂(Q : Measure Message)) = 0 := by
-  have he (b : Message) : pairNormalizer lam h a b*(pairMessage lam h hl0 hl1 a b i-a i) =
-      a i*(edgeMessage lam h b i-pairNormalizer lam h a b) := by
-    change pairNormalizer lam h a b*(a i*edgeMessage lam h b i/pairNormalizer lam h a b-a i) = _
-    field_simp [ne_of_gt (pairNormalizer_pos lam h hl0 hl1 a b)]
-  simp_rw [he]
-  have hi : Integrable (fun b => edgeMessage lam h b i) (Q : Measure Message) :=
-    integrable_continuous_message Q (((continuous_apply i).comp continuous_subtype_val).comp (continuous_edgeMessage lam h))
-  have hp : Integrable (fun b => pairNormalizer lam h a b) (Q : Measure Message) :=
-    integrable_continuous_message Q ((continuous_pairNormalizer lam h).comp (continuous_const.prodMk continuous_id))
-  rw [integral_const_mul, integral_sub hi hp, integral_edgeMessage lam h Q hQ,
-    integral_pairNormalizer lam h Q hQ]
-  ring
-
-lemma pairOperator_square_lower (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (a : Message) :
-    (messageX a)^2 ≤ pairOperator lam h hl0 hl1 Q ⟨fun m => (messageX m)^2, continuous_messageX.pow 2⟩ a := by
-  let N : Message → ℝ := fun b => pairNormalizer lam h a b
-  let P : Message → Message := fun b => pairMessage lam h hl0 hl1 a b
-  let G : Message → Spin → ℝ := fun b i => N b*(P b i-a i)
-  have hn : Continuous N := (continuous_pairNormalizer lam h).comp (continuous_const.prodMk continuous_id)
-  have hp : Continuous P := (continuous_pairMessage lam h hl0 hl1).comp (continuous_const.prodMk continuous_id)
-  have hg (i : Spin) : Continuous (fun b => G b i) := hn.mul ((((continuous_apply i).comp continuous_subtype_val).comp hp).sub continuous_const)
-  have hl : Continuous (fun b => N b*(messageX a)^2+2*messageX a*avg (fun i => (a i-1)*G b i)) :=
-    (hn.mul continuous_const).add (continuous_const.mul (continuous_avg _ (fun i => continuous_const.mul (hg i))))
-  have hr : Continuous (fun b => N b*(messageX (P b))^2) := hn.mul ((continuous_messageX.comp hp).pow 2)
-  have hin (b : Message) : N b*(messageX a)^2+2*messageX a*avg (fun i => (a i-1)*G b i) ≤ N b*(messageX (P b))^2 := by
-    have hh := mul_le_mul_of_nonneg_left (messageX_sq_tangent a (P b)) (pairNormalizer_pos lam h hl0 hl1 a b).le
-    convert hh using 1
-    first | rfl | (simp only [G, Radial.avg_expand]; ring)
-  have hh := integral_mono (integrable_continuous_message Q hl) (integrable_continuous_message Q hr) hin
-  have hil : Integrable (fun b => N b*(messageX a)^2) (Q : Measure Message) := integrable_continuous_message Q (hn.mul continuous_const)
-  have hig : Integrable (fun b => 2*messageX a*avg (fun i => (a i-1)*G b i)) (Q : Measure Message) :=
-    integrable_continuous_message Q (continuous_const.mul (continuous_avg _ (fun i => continuous_const.mul (hg i))))
-  rw [integral_add hil hig, integral_mul_const, integral_const_mul,
-    integral_avg _ _ (fun i => (integrable_continuous_message Q (hg i)).const_mul _)] at hh
-  simp only [G,N,P, integral_const_mul, integral_weighted_pair_deviation lam h hl0 hl1 Q hQ,
-    integral_pairNormalizer lam h Q hQ, mul_zero, avg, Finset.sum_const_zero, zero_div, one_mul, add_zero] at hh
-  exact hh
-
-lemma degree_nu_monotone (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) :
-    Monotone (fun n => posteriorNu (degreePosterior lam h Q hQ n)) := by
-  apply monotone_nat_of_le_succ
-  intro n
-  change (∫ m, (messageX m)^2 ∂degreeOutput lam h Q n) ≤ ∫ m, (messageX m)^2 ∂degreeOutput lam h Q (n+1)
-  have he : (∫ m, (messageX m)^2 ∂degreeOutput lam h Q (n+1)) =
-      ∫ a, pairOperator lam h hl0 hl1 Q ⟨fun m => (messageX m)^2, continuous_messageX.pow 2⟩ a ∂degreeOutput lam h Q n := by
-    exact integral_degree_succ lam h hl0 hl1 Q n ⟨fun m => (messageX m)^2, continuous_messageX.pow 2⟩
-  rw [he]
-  let := degreeOutput_probability lam h Q hQ n
-  apply integral_mono
-  · exact (continuous_messageX.pow 2).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
-  · exact (pairOperator lam h hl0 hl1 Q ⟨fun m => (messageX m)^2, continuous_messageX.pow 2⟩).continuous.integrable_of_hasCompactSupport
-      (HasCompactSupport.of_compactSpace _)
-  · exact pairOperator_square_lower lam h hl0 hl1 Q hQ
-
-end ThreeState
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeState.PairCorrelation
+
+namespace OAI
+
+/-! Posterior laws by offspring degree and monotonicity of their moments. -/
+
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+open Radial (avg)
+
+lemma continuous_integral_message {f : Message → Message → ℝ} (hf : Continuous f.uncurry)
+    (Q : ProbabilityMeasure Message) : Continuous (fun a => ∫ b, f a b ∂(Q : Measure Message)) := by
+  simpa only [Measure.restrict_univ] using continuous_parametric_integral_of_continuous hf isCompact_univ
+
+lemma continuous_pairIncrement (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) : Continuous (pairIncrement lam h Q) :=
+  continuous_integral_message (f := fun a b => pairNormalizer lam h a b*Real.log (pairNormalizer lam h a b)) (continuous_pairIntegrand lam h hl0 hl1) Q
+
+noncomputable def degreePosterior (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message)
+    (hQ : Balanced Q) (n : ℕ) : ProbabilityMeasure Message :=
+  ⟨degreeOutput lam h Q n, degreeOutput_probability lam h Q hQ n⟩
+
+lemma integral_iid_cons (Q : ProbabilityMeasure Message) (n : ℕ)
+    (f : (Fin (n+1) → Message) → ℝ) (hf : Continuous f) :
+    (∫ m, f m ∂(iidMeasure Q (n+1) : Measure (Fin (n+1) → Message))) =
+      ∫ m : Fin n → Message, ∫ b, f (Fin.cons b m) ∂(Q : Measure Message)
+        ∂(iidMeasure Q n : Measure (Fin n → Message)) := by
+  change (∫ m, f m ∂Measure.pi (fun _ : Fin (n+1) => (Q : Measure Message))) = _
+  rw [← ((measurePreserving_piFinSuccAbove (fun _ : Fin (n+1) => (Q : Measure Message)) 0).symm).integral_comp']
+  simp only [MeasurableEquiv.piFinSuccAbove_symm_apply, Fin.insertNthEquiv,
+    Fin.insertNth_zero, Equiv.coe_fn_mk, Fin.zero_succAbove, cast_eq]
+  apply integral_prod_symm
+  exact (show Continuous (fun p : Message × (Fin n → Message) => f (Fin.cons p.1 p.2)) by
+    apply hf.comp
+    apply continuous_pi
+    intro j
+    refine Fin.cases ?_ (fun i => ?_) j
+    · simpa using (continuous_fst : Continuous (Prod.fst : Message × (Fin n → Message) → Message))
+    · simpa only [Fin.cons_succ, Function.comp_def] using ((continuous_apply i).comp continuous_snd : Continuous (fun p : Message × (Fin n → Message) => p.2 i))).integrable_of_hasCompactSupport
+        (HasCompactSupport.of_compactSpace _)
+
+lemma normalizer_cons (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    {n : ℕ} (m : Fin n → Message) (b : Message) :
+    normalizer lam h (Fin.cons b m) = normalizer lam h m * pairNormalizer lam h (combineMessage lam h m) b := by
+  have he (i : Spin) : productWeight lam h (Fin.cons b m) i = edgeMessage lam h b i*productWeight lam h m i := by
+    simp [productWeight, Fin.prod_univ_succ]
+  have hn := ne_of_gt (normalizer_pos lam h hl0 hl1 m)
+  change avg (productWeight lam h (Fin.cons b m)) = normalizer lam h m*avg (fun i => combineMessage lam h m i*edgeMessage lam h b i)
+  simp only [Radial.avg_expand, he, combineMessage, dite_eq_right hn]
+  field_simp
+
+lemma combineMessage_cons (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    {n : ℕ} (m : Fin n → Message) (b : Message) :
+    combineMessage lam h (Fin.cons b m) = pairMessage lam h hl0 hl1 (combineMessage lam h m) b := by
+  apply Subtype.ext
+  funext i
+  have hp := ne_of_gt (normalizer_pos lam h hl0 hl1 m)
+  have hp' := ne_of_gt (normalizer_pos lam h hl0 hl1 (Fin.cons b m))
+  change (combineMessage lam h (Fin.cons b m)) i = (combineMessage lam h m) i*edgeMessage lam h b i / pairNormalizer lam h (combineMessage lam h m) b
+  conv_lhs => rw [combineMessage, dite_eq_right hp']
+  change productWeight lam h (Fin.cons b m) i/normalizer lam h (Fin.cons b m) = _
+  rw [normalizer_cons lam h hl0 hl1]
+  have hw := weighted_combineMessage lam h m i
+  have he : productWeight lam h (Fin.cons b m) i = edgeMessage lam h b i*productWeight lam h m i := by
+    simp [productWeight, Fin.prod_univ_succ]
+  rw [he, ← hw]
+  field_simp [hp, ne_of_gt (pairNormalizer_pos lam h hl0 hl1 _ _)]
+
+noncomputable def pairOperator (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) (f : C(Message,ℝ)) : C(Message,ℝ) :=
+  ⟨fun a => ∫ b, pairNormalizer lam h a b*f (pairMessage lam h hl0 hl1 a b) ∂(Q : Measure Message),
+    continuous_integral_message (f := fun a b => pairNormalizer lam h a b*f (pairMessage lam h hl0 hl1 a b)) ((continuous_pairNormalizer lam h).mul (f.continuous.comp (continuous_pairMessage lam h hl0 hl1))) Q⟩
+
+lemma integral_degree_succ (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) (n : ℕ) (f : C(Message,ℝ)) :
+    (∫ m, f m ∂degreeOutput lam h Q (n+1)) =
+      ∫ a, pairOperator lam h hl0 hl1 Q f a ∂degreeOutput lam h Q n := by
+  rw [integral_degreeOutput lam h Q (n+1) f f.continuous.measurable,
+    integral_iid_cons Q n (fun m => normalizer lam h m*f (combineMessage lam h m)) ((continuous_normalizer lam h (n+1)).mul
+      (f.continuous.comp (continuous_combineMessage_positive lam h hl0 hl1 (n+1)))),
+    integral_degreeOutput lam h Q n _ (pairOperator lam h hl0 hl1 Q f).continuous.measurable]
+  apply integral_congr_ae
+  filter_upwards with m
+  simp only [normalizer_cons lam h hl0 hl1, combineMessage_cons lam h hl0 hl1, mul_assoc, pairOperator]
+  exact integral_const_mul _ _
+
+lemma correlationEntropy_zero (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message) :
+    correlationEntropy lam h Q 0 = 0 := by
+  simp [correlationEntropy, normalizer, productWeight]
+
+lemma correlationEntropy_succ (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (n : ℕ) :
+    correlationEntropy lam h Q (n+1) = correlationEntropy lam h Q n +
+      ∫ a, pairIncrement lam h Q a ∂degreeOutput lam h Q n := by
+  unfold correlationEntropy
+  rw [integral_iid_cons Q n (fun m => normalizer lam h m*Real.log (normalizer lam h m)) ((continuous_normalizer lam h (n+1)).mul (continuous_log_normalizer lam h hl0 hl1 (n+1)))]
+  have he (m : Fin n → Message) :
+      (∫ b, normalizer lam h (Fin.cons b m)*Real.log (normalizer lam h (Fin.cons b m)) ∂(Q : Measure Message)) =
+        normalizer lam h m*Real.log (normalizer lam h m)+normalizer lam h m*pairIncrement lam h Q (combineMessage lam h m) := by
+    let a := combineMessage lam h m
+    have hpn : Continuous (fun b => pairNormalizer lam h a b) :=
+      (continuous_pairNormalizer lam h).comp (continuous_const.prodMk continuous_id)
+    have hpi : Continuous (fun b => pairNormalizer lam h a b*Real.log (pairNormalizer lam h a b)) :=
+      (continuous_pairIntegrand lam h hl0 hl1).comp (continuous_const.prodMk continuous_id)
+    have hfun (b : Message) : normalizer lam h (Fin.cons b m)*Real.log (normalizer lam h (Fin.cons b m)) =
+        (normalizer lam h m*Real.log (normalizer lam h m))*pairNormalizer lam h a b+
+          normalizer lam h m*(pairNormalizer lam h a b*Real.log (pairNormalizer lam h a b)) := by
+      rw [normalizer_cons lam h hl0 hl1, Real.log_mul (ne_of_gt (normalizer_pos lam h hl0 hl1 m))
+        (ne_of_gt (pairNormalizer_pos lam h hl0 hl1 _ _))]
+      dsimp only [a]
+      ring
+    simp_rw [hfun]
+    rw [integral_add ((integrable_continuous_message Q hpn).const_mul _) ((integrable_continuous_message Q hpi).const_mul _),
+      integral_const_mul, integral_const_mul, integral_pairNormalizer lam h Q hQ, mul_one]
+    rfl
+  simp_rw [he]
+  rw [integral_add]
+  · rw [integral_degreeOutput lam h Q n _ (continuous_pairIncrement lam h hl0 hl1 Q).measurable]
+  · exact ((continuous_normalizer lam h n).mul (continuous_log_normalizer lam h hl0 hl1 n)).integrable_of_hasCompactSupport
+      (HasCompactSupport.of_compactSpace _)
+  · exact ((continuous_normalizer lam h n).mul ((continuous_pairIncrement lam h hl0 hl1 Q).comp
+      (continuous_combineMessage_positive lam h hl0 hl1 n))).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
+
+end ThreeState
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+open Radial (avg)
+
+lemma messageX_sq_tangent (a c : Message) :
+    (messageX a)^2+2*messageX a*avg (fun i => (a i-1)*(c i-a i)) ≤ (messageX c)^2 := by
+  let D := avg (fun i => (c i-a i)^2)/2
+  have hD : 0 ≤ D := div_nonneg (avg_nonneg (fun i => sq_nonneg (c i-a i))) (by norm_num)
+  have he : messageX c-messageX a = avg (fun i => (a i-1)*(c i-a i))+D := by
+    dsimp only [messageX, Radial.momentX, messageDeviation, D]
+    simp only [Radial.avg_expand]
+    ring
+  have hm := mul_nonneg (messageX_nonneg a) hD
+  have hx := congrArg (fun t : ℝ => 2*messageX a*t) he
+  nlinarith only [hx, hm, sq_nonneg (messageX c-messageX a)]
+
+lemma integral_weighted_pair_deviation (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (a : Message) (i : Spin) :
+    (∫ b, pairNormalizer lam h a b*(pairMessage lam h hl0 hl1 a b i-a i) ∂(Q : Measure Message)) = 0 := by
+  have he (b : Message) : pairNormalizer lam h a b*(pairMessage lam h hl0 hl1 a b i-a i) =
+      a i*(edgeMessage lam h b i-pairNormalizer lam h a b) := by
+    change pairNormalizer lam h a b*(a i*edgeMessage lam h b i/pairNormalizer lam h a b-a i) = _
+    field_simp [ne_of_gt (pairNormalizer_pos lam h hl0 hl1 a b)]
+  simp_rw [he]
+  have hi : Integrable (fun b => edgeMessage lam h b i) (Q : Measure Message) :=
+    integrable_continuous_message Q (((continuous_apply i).comp continuous_subtype_val).comp (continuous_edgeMessage lam h))
+  have hp : Integrable (fun b => pairNormalizer lam h a b) (Q : Measure Message) :=
+    integrable_continuous_message Q ((continuous_pairNormalizer lam h).comp (continuous_const.prodMk continuous_id))
+  rw [integral_const_mul, integral_sub hi hp, integral_edgeMessage lam h Q hQ,
+    integral_pairNormalizer lam h Q hQ]
+  ring
+
+lemma pairOperator_square_lower (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (a : Message) :
+    (messageX a)^2 ≤ pairOperator lam h hl0 hl1 Q ⟨fun m => (messageX m)^2, continuous_messageX.pow 2⟩ a := by
+  let N : Message → ℝ := fun b => pairNormalizer lam h a b
+  let P : Message → Message := fun b => pairMessage lam h hl0 hl1 a b
+  let G : Message → Spin → ℝ := fun b i => N b*(P b i-a i)
+  have hn : Continuous N := (continuous_pairNormalizer lam h).comp (continuous_const.prodMk continuous_id)
+  have hp : Continuous P := (continuous_pairMessage lam h hl0 hl1).comp (continuous_const.prodMk continuous_id)
+  have hg (i : Spin) : Continuous (fun b => G b i) := hn.mul ((((continuous_apply i).comp continuous_subtype_val).comp hp).sub continuous_const)
+  have hl : Continuous (fun b => N b*(messageX a)^2+2*messageX a*avg (fun i => (a i-1)*G b i)) :=
+    (hn.mul continuous_const).add (continuous_const.mul (continuous_avg _ (fun i => continuous_const.mul (hg i))))
+  have hr : Continuous (fun b => N b*(messageX (P b))^2) := hn.mul ((continuous_messageX.comp hp).pow 2)
+  have hin (b : Message) : N b*(messageX a)^2+2*messageX a*avg (fun i => (a i-1)*G b i) ≤ N b*(messageX (P b))^2 := by
+    have hh := mul_le_mul_of_nonneg_left (messageX_sq_tangent a (P b)) (pairNormalizer_pos lam h hl0 hl1 a b).le
+    convert hh using 1
+    first | rfl | (simp only [G, Radial.avg_expand]; ring)
+  have hh := integral_mono (integrable_continuous_message Q hl) (integrable_continuous_message Q hr) hin
+  have hil : Integrable (fun b => N b*(messageX a)^2) (Q : Measure Message) := integrable_continuous_message Q (hn.mul continuous_const)
+  have hig : Integrable (fun b => 2*messageX a*avg (fun i => (a i-1)*G b i)) (Q : Measure Message) :=
+    integrable_continuous_message Q (continuous_const.mul (continuous_avg _ (fun i => continuous_const.mul (hg i))))
+  rw [integral_add hil hig, integral_mul_const, integral_const_mul,
+    integral_avg _ _ (fun i => (integrable_continuous_message Q (hg i)).const_mul _)] at hh
+  simp only [G,N,P, integral_const_mul, integral_weighted_pair_deviation lam h hl0 hl1 Q hQ,
+    integral_pairNormalizer lam h Q hQ, mul_zero, avg, Finset.sum_const_zero, zero_div, one_mul, add_zero] at hh
+  exact hh
+
+lemma degree_nu_monotone (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) :
+    Monotone (fun n => posteriorNu (degreePosterior lam h Q hQ n)) := by
+  apply monotone_nat_of_le_succ
+  intro n
+  change (∫ m, (messageX m)^2 ∂degreeOutput lam h Q n) ≤ ∫ m, (messageX m)^2 ∂degreeOutput lam h Q (n+1)
+  have he : (∫ m, (messageX m)^2 ∂degreeOutput lam h Q (n+1)) =
+      ∫ a, pairOperator lam h hl0 hl1 Q ⟨fun m => (messageX m)^2, continuous_messageX.pow 2⟩ a ∂degreeOutput lam h Q n := by
+    exact integral_degree_succ lam h hl0 hl1 Q n ⟨fun m => (messageX m)^2, continuous_messageX.pow 2⟩
+  rw [he]
+  let := degreeOutput_probability lam h Q hQ n
+  apply integral_mono
+  · exact (continuous_messageX.pow 2).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
+  · exact (pairOperator lam h hl0 hl1 Q ⟨fun m => (messageX m)^2, continuous_messageX.pow 2⟩).continuous.integrable_of_hasCompactSupport
+      (HasCompactSupport.of_compactSpace _)
+  · exact pairOperator_square_lower lam h hl0 hl1 Q hQ
+
+end ThreeState
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeState/Entropy.lean b/lean/OAI/Probability/ThreeState/Entropy.lean
index cae795d41acbe5d69d85ec76e2cacd04b44566b1..98ab8c0e78cb9c1e802c4a265efe0d3aceed5d11 100644
--- a/lean/OAI/Probability/ThreeState/Entropy.lean
+++ b/lean/OAI/Probability/ThreeState/Entropy.lean
@@ -1,326 +1,326 @@
-import Mathlib
-import OAI.Probability.ThreeState.RadialDerivatives
-import OAI.Probability.ThreeState.Symmetry
-
-namespace OAI
-
-/-! Message entropy and its integrability. -/
-
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-open Radial (avg)
-
-lemma average_message (m : Message) : avg (fun i => m i) = 1 := by
-  simp [avg, Message.sum]
-
-lemma message_last (m : Message) : m 2 = 3-m 0-m 1 := by
-  have hh := average_message m
-  rw [Radial.avg_expand] at hh
-  linarith
-
-lemma avg_nonneg {v : Spin → ℝ} (h : ∀ i, 0 ≤ v i) : 0 ≤ avg v :=
-  div_nonneg (Finset.sum_nonneg (fun i _ => h i)) (by norm_num)
-
-lemma avg_le_const {v : Spin → ℝ} {c : ℝ} (h : ∀ i, v i ≤ c) : avg v ≤ c := by
-  rw [Radial.avg_expand]
-  linarith [h 0,h 1,h 2]
-
-lemma abs_avg_le {v : Spin → ℝ} {c : ℝ} (h : ∀ i, |v i| ≤ c) : |avg v| ≤ c := by
-  apply abs_le.mpr
-  constructor
-  · have h0 := (abs_le.mp (h 0)).1
-    have h1 := (abs_le.mp (h 1)).1
-    have h2 := (abs_le.mp (h 2)).1
-    rw [Radial.avg_expand]
-    linarith
-  · exact avg_le_const (fun i => (abs_le.mp (h i)).2)
-
-lemma edgeMessage_bounds (lam : ℝ) (h : Admissible lam) (hl : 0 ≤ lam) (m : Message) (i : Spin) :
-    1-lam ≤ edgeMessage lam h m i ∧ edgeMessage lam h m i ≤ 1+2*lam := by
-  change 1-lam ≤ 1+lam*(m i-1) ∧ 1+lam*(m i-1) ≤ 1+2*lam
-  constructor <;> nlinarith [Message.nonneg m i, Message.le_three m i]
-
-lemma edgeMessage_pos (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (m : Message) (i : Spin) : 0 < edgeMessage lam h m i := by
-  have hh := (edgeMessage_bounds lam h hl0 m i).1
-  linarith
-
-lemma productWeight_bounds (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    {n : ℕ} (m : Fin n → Message) (i : Spin) :
-    (1-lam)^n ≤ productWeight lam h m i ∧ productWeight lam h m i ≤ (1+2*lam)^n := by
-  constructor
-  · calc
-      (1-lam)^n = ∏ _j : Fin n, (1-lam) := by simp
-      _ ≤ productWeight lam h m i := Finset.prod_le_prod₀ (fun _ _ => by linarith)
-        (fun j _ => (edgeMessage_bounds lam h hl0 (m j) i).1)
-  · calc
-      productWeight lam h m i ≤ ∏ _j : Fin n, (1+2*lam) :=
-        Finset.prod_le_prod₀ (fun j _ => Message.nonneg (edgeMessage lam h (m j)) i)
-          (fun j _ => (edgeMessage_bounds lam h hl0 (m j) i).2)
-      _ = (1+2*lam)^n := by simp
-
-lemma normalizer_bounds (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    {n : ℕ} (m : Fin n → Message) :
-    (1-lam)^n ≤ normalizer lam h m ∧ normalizer lam h m ≤ (1+2*lam)^n := by
-  have hh := productWeight_bounds lam h hl0 hl1 m
-  change (1-lam)^n ≤ avg (productWeight lam h m) ∧ avg (productWeight lam h m) ≤ (1+2*lam)^n
-  constructor
-  · rw [Radial.avg_expand]
-    linarith [(hh 0).1,(hh 1).1,(hh 2).1]
-  · exact avg_le_const (fun i => (hh i).2)
-
-lemma normalizer_pos (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    {n : ℕ} (m : Fin n → Message) : 0 < normalizer lam h m :=
-  lt_of_lt_of_le (pow_pos (by linarith) _) (normalizer_bounds lam h hl0 hl1 m).1
-
-lemma combineMessage_pos (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    {n : ℕ} (m : Fin n → Message) (i : Spin) : 0 < combineMessage lam h m i := by
-  have hz := normalizer_pos lam h hl0 hl1 m
-  rw [combineMessage, dite_eq_right (ne_of_gt hz)]
-  exact div_pos (lt_of_lt_of_le (pow_pos (by linarith) n) (productWeight_bounds lam h hl0 hl1 m i).1) hz
-
-lemma continuous_combineMessage_positive (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam)
-    (hl1 : lam < 1) (n : ℕ) : Continuous (combineMessage lam h : (Fin n → Message) → Message) :=
-  continuous_iff_continuousAt.mpr (fun m => continuousAt_combineMessage lam h m
-    (ne_of_gt (normalizer_pos lam h hl0 hl1 m)))
-
-lemma positive_fixedpoint (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
-    (hl0 : 0 ≤ lam) (hl1 : lam < 1) (Q : ProbabilityMeasure Message)
-    (hQ : IsPosteriorFixedPoint offspring lam h Q) :
-    ∀ᵐ m ∂(Q : Measure Message), ∀ i : Spin, 0 < m i := by
-  apply ae_fixedpoint_of_output offspring lam h Q hQ
-  · simpa only [Set.ofPred_forall, Function.comp_def] using MeasurableSet.iInter (fun i : Spin =>
-      measurableSet_lt (f := fun _ : Message => (0:ℝ)) measurable_const
-        ((continuous_apply i).comp continuous_subtype_val).measurable)
-  · intro n m i
-    exact combineMessage_pos lam h hl0 hl1 m i
-
-noncomputable def edgeLogBound (lam : ℝ) : ℝ := |Real.log (1-lam)|+|Real.log (1+2*lam)|
-
-lemma edgeLogBound_nonneg (lam : ℝ) : 0 ≤ edgeLogBound lam := add_nonneg (abs_nonneg _) (abs_nonneg _)
-
-lemma abs_log_of_bounds (lam : ℝ) (hl0 : 0 ≤ lam) (hl1 : lam < 1) (n : ℕ) (a : ℝ)
-    (ha : (1-lam)^n ≤ a) (hb : a ≤ (1+2*lam)^n) : |Real.log a| ≤ n*edgeLogBound lam := by
-  have ha0 : 0 < (1-lam)^n := pow_pos (by linarith) _
-  have hap : 0 < a := lt_of_lt_of_le ha0 ha
-  have hlo := Real.log_le_log ha0 ha
-  have hhi : Real.log a ≤ Real.log ((1+2*lam)^n) :=
-    (Real.log_le_log_iff hap (pow_pos (by linarith [hl0]) n)).mpr hb
-  rw [Real.log_pow] at hlo hhi
-  have hn : (0:ℝ) ≤ n := Nat.cast_nonneg n
-  have ha1 := neg_abs_le (Real.log (1-lam))
-  have ha2 := le_abs_self (Real.log (1+2*lam))
-  dsimp [edgeLogBound]
-  apply abs_le.mpr
-  constructor
-  · nlinarith [mul_nonneg hn (abs_nonneg (Real.log (1+2*lam)))]
-  · nlinarith [mul_nonneg hn (abs_nonneg (Real.log (1-lam)))]
-
-lemma log_combineMessage (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    {n : ℕ} (m : Fin n → Message) (i : Spin) :
-    Real.log (combineMessage lam h m i) =
-      (∑ j, Real.log (edgeMessage lam h (m j) i))-Real.log (normalizer lam h m) := by
-  have hz := normalizer_pos lam h hl0 hl1 m
-  have hw : productWeight lam h m i ≠ 0 := ne_of_gt
-    (lt_of_lt_of_le (pow_pos (by linarith) n) (productWeight_bounds lam h hl0 hl1 m i).1)
-  rw [combineMessage, dite_eq_right (ne_of_gt hz), Real.log_div hw (ne_of_gt hz)]
-  rw [productWeight, Real.log_prod (fun j _ => ne_of_gt (edgeMessage_pos lam h hl0 hl1 (m j) i))]
-
-lemma log_combineMessage_bound (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    {n : ℕ} (m : Fin n → Message) (i : Spin) :
-    |Real.log (combineMessage lam h m i)| ≤ 2*n*edgeLogBound lam := by
-  have hz := normalizer_pos lam h hl0 hl1 m
-  have hw := productWeight_bounds lam h hl0 hl1 m i
-  have hn := normalizer_bounds lam h hl0 hl1 m
-  have hw0 : productWeight lam h m i ≠ 0 := ne_of_gt (lt_of_lt_of_le (pow_pos (by linarith) n) hw.1)
-  rw [combineMessage, dite_eq_right (ne_of_gt hz), Real.log_div hw0 (ne_of_gt hz)]
-  calc
-    |Real.log (productWeight lam h m i)-Real.log (normalizer lam h m)| ≤
-      |Real.log (productWeight lam h m i)|+|Real.log (normalizer lam h m)| := abs_sub _ _
-    _ ≤ n*edgeLogBound lam+n*edgeLogBound lam := add_le_add
-      (abs_log_of_bounds lam hl0 hl1 n _ hw.1 hw.2) (abs_log_of_bounds lam hl0 hl1 n _ hn.1 hn.2)
-    _ = _ := by ring
-
-noncomputable def mLog (m : Message) (i : Spin) : ℝ := Real.log (m i)
-noncomputable def mCenteredLog (m : Message) (i : Spin) : ℝ := mLog m i-avg (mLog m)
-noncomputable def entropyI (m : Message) : ℝ := avg (fun i => m i*mLog m i)
-noncomputable def entropyJ (m : Message) : ℝ := avg (fun i => (m i-1)*mLog m i)/2
-noncomputable def entropyB (m : Message) : ℝ := avg (fun i => m i*(mCenteredLog m i)^2)/2
-noncomputable def entropyF (m : Message) : ℝ := 3*entropyI m-2*entropyJ m+(4/5:ℝ)*entropyB m
-
-lemma measurable_mLog (i : Spin) : Measurable (fun m : Message => mLog m i) :=
-  Real.measurable_log.comp ((continuous_apply i).comp continuous_subtype_val).measurable
-lemma measurable_avg (f : Message → Spin → ℝ) (hf : ∀ i, Measurable (fun m => f m i)) :
-    Measurable (fun m => avg (f m)) := by
-  simp only [Radial.avg_expand]
-  exact (((hf 0).add (hf 1)).add (hf 2)).div_const 3
-lemma measurable_mCenteredLog (i : Spin) : Measurable (fun m : Message => mCenteredLog m i) :=
-  (measurable_mLog i).sub (measurable_avg mLog measurable_mLog)
-lemma measurable_entropyI : Measurable entropyI :=
-  measurable_avg _ (fun i => ((continuous_apply i).comp continuous_subtype_val).measurable.mul (measurable_mLog i))
-lemma measurable_entropyJ : Measurable entropyJ :=
-  (measurable_avg _ (fun i => (((continuous_apply i).comp continuous_subtype_val).measurable.sub_const 1).mul (measurable_mLog i))).div_const 2
-lemma measurable_entropyB : Measurable entropyB :=
-  (measurable_avg _ (fun i => ((continuous_apply i).comp continuous_subtype_val).measurable.mul ((measurable_mCenteredLog i).pow_const 2))).div_const 2
-lemma measurable_entropyF : Measurable entropyF := by
-  exact ((measurable_entropyI.const_mul 3).sub (measurable_entropyJ.const_mul 2)).add
-    (measurable_entropyB.const_mul (4/5:ℝ))
-
-lemma centeredLog_combine (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    {n : ℕ} (m : Fin n → Message) (i : Spin) :
-    mCenteredLog (combineMessage lam h m) i = ∑ j, mCenteredLog (edgeMessage lam h (m j)) i := by
-  simp only [mCenteredLog, mLog, Radial.avg_expand, log_combineMessage lam h hl0 hl1,
-    Finset.sum_sub_distrib, add_div, ← Finset.sum_div, Finset.sum_add_distrib]
-  ring
-
-lemma centeredLog_J (m : Message) : avg (fun i => m i*mCenteredLog m i) = 2*entropyJ m := by
-  simp only [mCenteredLog, entropyJ, mLog, Radial.avg_expand, message_last m]
-  ring
-
-lemma entropyF_radial (m : Message) : entropyF m = Radial.sourceF (fun i => m i) := by
-  simp only [entropyF, entropyI, entropyJ, entropyB, mCenteredLog, mLog, Radial.sourceF,
-    Radial.avg_expand]
-  ring
-
-lemma centeredLog_bound (m : Message) (C : ℝ) (hc : ∀ i, |mLog m i| ≤ C) (i : Spin) :
-    |mCenteredLog m i| ≤ 2*C := by
-  calc
-    |mCenteredLog m i| ≤ |mLog m i|+|avg (mLog m)| := abs_sub _ _
-    _ ≤ C+C := add_le_add (hc i) (abs_avg_le hc)
-    _ = _ := by ring
-
-lemma entropyI_bound (m : Message) (C : ℝ) (hc : ∀ i, |mLog m i| ≤ C) : |entropyI m| ≤ 3*C := by
-  apply abs_avg_le
-  intro i
-  rw [abs_mul, abs_of_nonneg (Message.nonneg m i)]
-  exact mul_le_mul (Message.le_three m i) (hc i) (abs_nonneg _) (by norm_num)
-
-lemma entropyJ_bound (m : Message) (C : ℝ) (hc : ∀ i, |mLog m i| ≤ C) : |entropyJ m| ≤ C := by
-  have hv (i : Spin) : |m i-1| ≤ 2 := abs_le.mpr ⟨by linarith [Message.nonneg m i], by linarith [Message.le_three m i]⟩
-  have ha : |avg (fun i => (m i-1)*mLog m i)| ≤ 2*C := by
-    apply abs_avg_le
-    intro i
-    rw [abs_mul]
-    exact mul_le_mul (hv i) (hc i) (abs_nonneg _) (by norm_num)
-  dsimp only [entropyJ]
-  rw [abs_div, abs_of_pos (by norm_num : (0:ℝ)<2)]
-  linarith
-
-lemma entropyB_nonneg (m : Message) : 0 ≤ entropyB m :=
-  div_nonneg (avg_nonneg (fun i => mul_nonneg (Message.nonneg m i) (sq_nonneg _))) (by norm_num)
-
-lemma entropyB_bound (m : Message) (C : ℝ) (hc : ∀ i, |mLog m i| ≤ C) : |entropyB m| ≤ 6*C^2 := by
-  have hC : 0 ≤ C := le_trans (abs_nonneg _) (hc 0)
-  have hu (i : Spin) : (mCenteredLog m i)^2 ≤ (2*C)^2 := by
-    have hh := (sq_le_sq₀ (abs_nonneg (mCenteredLog m i)) (by positivity : 0 ≤ 2*C)).mpr (centeredLog_bound m C hc i)
-    simpa only [sq_abs] using hh
-  have ha : avg (fun i => m i*(mCenteredLog m i)^2) ≤ 12*C^2 := by
-    apply avg_le_const
-    intro i
-    have hh := mul_le_mul (Message.le_three m i) (hu i) (sq_nonneg _) (by norm_num : (0:ℝ)≤3)
-    nlinarith
-  rw [abs_of_nonneg (entropyB_nonneg m)]
-  dsimp only [entropyB]
-  linarith
-
-end ThreeState
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-open Radial (avg)
-
-lemma continuous_avg {α : Type*} [TopologicalSpace α] (f : α → Spin → ℝ)
-    (hf : ∀ i, Continuous (fun m => f m i)) : Continuous (fun m => avg (f m)) := by
-  simp only [Radial.avg_expand]
-  exact (((hf 0).add (hf 1)).add (hf 2)).div_const 3
-
-lemma continuous_mLog_edge (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1) (i : Spin) :
-    Continuous (fun m => mLog (edgeMessage lam h m) i) :=
-  (((continuous_apply i).comp continuous_subtype_val).comp (continuous_edgeMessage lam h)).log
-    (fun m => ne_of_gt (edgeMessage_pos lam h hl0 hl1 m i))
-
-lemma continuous_mCenteredLog_edge (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1) (i : Spin) :
-    Continuous (fun m => mCenteredLog (edgeMessage lam h m) i) :=
-  (continuous_mLog_edge lam h hl0 hl1 i).sub
-    (continuous_avg _ (continuous_mLog_edge lam h hl0 hl1))
-
-lemma continuous_entropyI_edge (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1) :
-    Continuous (fun m => entropyI (edgeMessage lam h m)) :=
-  continuous_avg _ (fun i =>
-    (((continuous_apply i).comp continuous_subtype_val).comp (continuous_edgeMessage lam h)).mul
-      (continuous_mLog_edge lam h hl0 hl1 i))
-
-lemma continuous_entropyJ_edge (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1) :
-    Continuous (fun m => entropyJ (edgeMessage lam h m)) :=
-  (continuous_avg _ (fun i =>
-    ((((continuous_apply i).comp continuous_subtype_val).comp (continuous_edgeMessage lam h)).sub continuous_const).mul
-      (continuous_mLog_edge lam h hl0 hl1 i))).div_const 2
-
-lemma continuous_entropyB_edge (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1) :
-    Continuous (fun m => entropyB (edgeMessage lam h m)) :=
-  (continuous_avg _ (fun i =>
-    (((continuous_apply i).comp continuous_subtype_val).comp (continuous_edgeMessage lam h)).mul
-      ((continuous_mCenteredLog_edge lam h hl0 hl1 i).pow 2))).div_const 2
-
-lemma integrable_mLog_fixedpoint (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
-    (hl0 : 0 ≤ lam) (hl1 : lam < 1) (Q : ProbabilityMeasure Message)
-    (hQ : IsPosteriorFixedPoint offspring lam h Q)
-    (hD : HasMean offspring (fun n : ℕ => (n:ℝ))) (i : Spin) :
-    Integrable (fun m => mLog m i) (Q : Measure Message) := by
-  apply integrable_fixedpoint_of_growth offspring lam h Q hQ _ (measurable_mLog i)
-    (fun n => (n:ℝ)*(2*edgeLogBound lam)) (hD.mul_const _)
-  intro n m
-  have hh := log_combineMessage_bound lam h hl0 hl1 m i
-  dsimp only [mLog]
-  nlinarith
-
-lemma integrable_entropyI_fixedpoint (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
-    (hl0 : 0 ≤ lam) (hl1 : lam < 1) (Q : ProbabilityMeasure Message)
-    (hQ : IsPosteriorFixedPoint offspring lam h Q)
-    (hD : HasMean offspring (fun n : ℕ => (n:ℝ))) :
-    Integrable entropyI (Q : Measure Message) := by
-  apply integrable_fixedpoint_of_growth offspring lam h Q hQ _ measurable_entropyI
-    (fun n => (n:ℝ)*(6*edgeLogBound lam)) (hD.mul_const _)
-  intro n m
-  have hh := entropyI_bound (combineMessage lam h m) (2*n*edgeLogBound lam)
-    (log_combineMessage_bound lam h hl0 hl1 m)
-  nlinarith
-
-lemma integrable_entropyJ_fixedpoint (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
-    (hl0 : 0 ≤ lam) (hl1 : lam < 1) (Q : ProbabilityMeasure Message)
-    (hQ : IsPosteriorFixedPoint offspring lam h Q)
-    (hD : HasMean offspring (fun n : ℕ => (n:ℝ))) :
-    Integrable entropyJ (Q : Measure Message) := by
-  apply integrable_fixedpoint_of_growth offspring lam h Q hQ _ measurable_entropyJ
-    (fun n => (n:ℝ)*(2*edgeLogBound lam)) (hD.mul_const _)
-  intro n m
-  have hh := entropyJ_bound (combineMessage lam h m) (2*n*edgeLogBound lam)
-    (log_combineMessage_bound lam h hl0 hl1 m)
-  nlinarith
-
-lemma integrable_entropyB_fixedpoint (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
-    (hl0 : 0 ≤ lam) (hl1 : lam < 1) (Q : ProbabilityMeasure Message)
-    (hQ : IsPosteriorFixedPoint offspring lam h Q)
-    (hD2 : HasMean offspring (fun n : ℕ => (n:ℝ)^2)) :
-    Integrable entropyB (Q : Measure Message) := by
-  apply integrable_fixedpoint_of_growth offspring lam h Q hQ _ measurable_entropyB
-    (fun n => (n:ℝ)^2*(24*(edgeLogBound lam)^2)) (hD2.mul_const _)
-  intro n m
-  have hh := entropyB_bound (combineMessage lam h m) (2*n*edgeLogBound lam)
-    (log_combineMessage_bound lam h hl0 hl1 m)
-  nlinarith
-
-lemma integrable_entropyF_fixedpoint (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
-    (hl0 : 0 ≤ lam) (hl1 : lam < 1) (Q : ProbabilityMeasure Message)
-    (hQ : IsPosteriorFixedPoint offspring lam h Q)
-    (hD2 : HasMean offspring (fun n : ℕ => (n:ℝ)^2)) :
-    Integrable entropyF (Q : Measure Message) := by
-  have hD := hasMean_of_square hD2
-  exact (((integrable_entropyI_fixedpoint offspring lam h hl0 hl1 Q hQ hD).const_mul 3).sub
-    ((integrable_entropyJ_fixedpoint offspring lam h hl0 hl1 Q hQ hD).const_mul 2)).add
-      ((integrable_entropyB_fixedpoint offspring lam h hl0 hl1 Q hQ hD2).const_mul (4/5:ℝ))
-
-end ThreeState
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeState.RadialDerivatives
+import OAI.Probability.ThreeState.Symmetry
+
+namespace OAI
+
+/-! Message entropy and its integrability. -/
+
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+open Radial (avg)
+
+lemma average_message (m : Message) : avg (fun i => m i) = 1 := by
+  simp [avg, Message.sum]
+
+lemma message_last (m : Message) : m 2 = 3-m 0-m 1 := by
+  have hh := average_message m
+  rw [Radial.avg_expand] at hh
+  linarith
+
+lemma avg_nonneg {v : Spin → ℝ} (h : ∀ i, 0 ≤ v i) : 0 ≤ avg v :=
+  div_nonneg (Finset.sum_nonneg (fun i _ => h i)) (by norm_num)
+
+lemma avg_le_const {v : Spin → ℝ} {c : ℝ} (h : ∀ i, v i ≤ c) : avg v ≤ c := by
+  rw [Radial.avg_expand]
+  linarith [h 0,h 1,h 2]
+
+lemma abs_avg_le {v : Spin → ℝ} {c : ℝ} (h : ∀ i, |v i| ≤ c) : |avg v| ≤ c := by
+  apply abs_le.mpr
+  constructor
+  · have h0 := (abs_le.mp (h 0)).1
+    have h1 := (abs_le.mp (h 1)).1
+    have h2 := (abs_le.mp (h 2)).1
+    rw [Radial.avg_expand]
+    linarith
+  · exact avg_le_const (fun i => (abs_le.mp (h i)).2)
+
+lemma edgeMessage_bounds (lam : ℝ) (h : Admissible lam) (hl : 0 ≤ lam) (m : Message) (i : Spin) :
+    1-lam ≤ edgeMessage lam h m i ∧ edgeMessage lam h m i ≤ 1+2*lam := by
+  change 1-lam ≤ 1+lam*(m i-1) ∧ 1+lam*(m i-1) ≤ 1+2*lam
+  constructor <;> nlinarith [Message.nonneg m i, Message.le_three m i]
+
+lemma edgeMessage_pos (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (m : Message) (i : Spin) : 0 < edgeMessage lam h m i := by
+  have hh := (edgeMessage_bounds lam h hl0 m i).1
+  linarith
+
+lemma productWeight_bounds (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    {n : ℕ} (m : Fin n → Message) (i : Spin) :
+    (1-lam)^n ≤ productWeight lam h m i ∧ productWeight lam h m i ≤ (1+2*lam)^n := by
+  constructor
+  · calc
+      (1-lam)^n = ∏ _j : Fin n, (1-lam) := by simp
+      _ ≤ productWeight lam h m i := Finset.prod_le_prod₀ (fun _ _ => by linarith)
+        (fun j _ => (edgeMessage_bounds lam h hl0 (m j) i).1)
+  · calc
+      productWeight lam h m i ≤ ∏ _j : Fin n, (1+2*lam) :=
+        Finset.prod_le_prod₀ (fun j _ => Message.nonneg (edgeMessage lam h (m j)) i)
+          (fun j _ => (edgeMessage_bounds lam h hl0 (m j) i).2)
+      _ = (1+2*lam)^n := by simp
+
+lemma normalizer_bounds (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    {n : ℕ} (m : Fin n → Message) :
+    (1-lam)^n ≤ normalizer lam h m ∧ normalizer lam h m ≤ (1+2*lam)^n := by
+  have hh := productWeight_bounds lam h hl0 hl1 m
+  change (1-lam)^n ≤ avg (productWeight lam h m) ∧ avg (productWeight lam h m) ≤ (1+2*lam)^n
+  constructor
+  · rw [Radial.avg_expand]
+    linarith [(hh 0).1,(hh 1).1,(hh 2).1]
+  · exact avg_le_const (fun i => (hh i).2)
+
+lemma normalizer_pos (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    {n : ℕ} (m : Fin n → Message) : 0 < normalizer lam h m :=
+  lt_of_lt_of_le (pow_pos (by linarith) _) (normalizer_bounds lam h hl0 hl1 m).1
+
+lemma combineMessage_pos (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    {n : ℕ} (m : Fin n → Message) (i : Spin) : 0 < combineMessage lam h m i := by
+  have hz := normalizer_pos lam h hl0 hl1 m
+  rw [combineMessage, dite_eq_right (ne_of_gt hz)]
+  exact div_pos (lt_of_lt_of_le (pow_pos (by linarith) n) (productWeight_bounds lam h hl0 hl1 m i).1) hz
+
+lemma continuous_combineMessage_positive (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam)
+    (hl1 : lam < 1) (n : ℕ) : Continuous (combineMessage lam h : (Fin n → Message) → Message) :=
+  continuous_iff_continuousAt.mpr (fun m => continuousAt_combineMessage lam h m
+    (ne_of_gt (normalizer_pos lam h hl0 hl1 m)))
+
+lemma positive_fixedpoint (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
+    (hl0 : 0 ≤ lam) (hl1 : lam < 1) (Q : ProbabilityMeasure Message)
+    (hQ : IsPosteriorFixedPoint offspring lam h Q) :
+    ∀ᵐ m ∂(Q : Measure Message), ∀ i : Spin, 0 < m i := by
+  apply ae_fixedpoint_of_output offspring lam h Q hQ
+  · simpa only [Set.ofPred_forall, Function.comp_def] using MeasurableSet.iInter (fun i : Spin =>
+      measurableSet_lt (f := fun _ : Message => (0:ℝ)) measurable_const
+        ((continuous_apply i).comp continuous_subtype_val).measurable)
+  · intro n m i
+    exact combineMessage_pos lam h hl0 hl1 m i
+
+noncomputable def edgeLogBound (lam : ℝ) : ℝ := |Real.log (1-lam)|+|Real.log (1+2*lam)|
+
+lemma edgeLogBound_nonneg (lam : ℝ) : 0 ≤ edgeLogBound lam := add_nonneg (abs_nonneg _) (abs_nonneg _)
+
+lemma abs_log_of_bounds (lam : ℝ) (hl0 : 0 ≤ lam) (hl1 : lam < 1) (n : ℕ) (a : ℝ)
+    (ha : (1-lam)^n ≤ a) (hb : a ≤ (1+2*lam)^n) : |Real.log a| ≤ n*edgeLogBound lam := by
+  have ha0 : 0 < (1-lam)^n := pow_pos (by linarith) _
+  have hap : 0 < a := lt_of_lt_of_le ha0 ha
+  have hlo := Real.log_le_log ha0 ha
+  have hhi : Real.log a ≤ Real.log ((1+2*lam)^n) :=
+    (Real.log_le_log_iff hap (pow_pos (by linarith [hl0]) n)).mpr hb
+  rw [Real.log_pow] at hlo hhi
+  have hn : (0:ℝ) ≤ n := Nat.cast_nonneg n
+  have ha1 := neg_abs_le (Real.log (1-lam))
+  have ha2 := le_abs_self (Real.log (1+2*lam))
+  dsimp [edgeLogBound]
+  apply abs_le.mpr
+  constructor
+  · nlinarith [mul_nonneg hn (abs_nonneg (Real.log (1+2*lam)))]
+  · nlinarith [mul_nonneg hn (abs_nonneg (Real.log (1-lam)))]
+
+lemma log_combineMessage (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    {n : ℕ} (m : Fin n → Message) (i : Spin) :
+    Real.log (combineMessage lam h m i) =
+      (∑ j, Real.log (edgeMessage lam h (m j) i))-Real.log (normalizer lam h m) := by
+  have hz := normalizer_pos lam h hl0 hl1 m
+  have hw : productWeight lam h m i ≠ 0 := ne_of_gt
+    (lt_of_lt_of_le (pow_pos (by linarith) n) (productWeight_bounds lam h hl0 hl1 m i).1)
+  rw [combineMessage, dite_eq_right (ne_of_gt hz), Real.log_div hw (ne_of_gt hz)]
+  rw [productWeight, Real.log_prod (fun j _ => ne_of_gt (edgeMessage_pos lam h hl0 hl1 (m j) i))]
+
+lemma log_combineMessage_bound (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    {n : ℕ} (m : Fin n → Message) (i : Spin) :
+    |Real.log (combineMessage lam h m i)| ≤ 2*n*edgeLogBound lam := by
+  have hz := normalizer_pos lam h hl0 hl1 m
+  have hw := productWeight_bounds lam h hl0 hl1 m i
+  have hn := normalizer_bounds lam h hl0 hl1 m
+  have hw0 : productWeight lam h m i ≠ 0 := ne_of_gt (lt_of_lt_of_le (pow_pos (by linarith) n) hw.1)
+  rw [combineMessage, dite_eq_right (ne_of_gt hz), Real.log_div hw0 (ne_of_gt hz)]
+  calc
+    |Real.log (productWeight lam h m i)-Real.log (normalizer lam h m)| ≤
+      |Real.log (productWeight lam h m i)|+|Real.log (normalizer lam h m)| := abs_sub _ _
+    _ ≤ n*edgeLogBound lam+n*edgeLogBound lam := add_le_add
+      (abs_log_of_bounds lam hl0 hl1 n _ hw.1 hw.2) (abs_log_of_bounds lam hl0 hl1 n _ hn.1 hn.2)
+    _ = _ := by ring
+
+noncomputable def mLog (m : Message) (i : Spin) : ℝ := Real.log (m i)
+noncomputable def mCenteredLog (m : Message) (i : Spin) : ℝ := mLog m i-avg (mLog m)
+noncomputable def entropyI (m : Message) : ℝ := avg (fun i => m i*mLog m i)
+noncomputable def entropyJ (m : Message) : ℝ := avg (fun i => (m i-1)*mLog m i)/2
+noncomputable def entropyB (m : Message) : ℝ := avg (fun i => m i*(mCenteredLog m i)^2)/2
+noncomputable def entropyF (m : Message) : ℝ := 3*entropyI m-2*entropyJ m+(4/5:ℝ)*entropyB m
+
+lemma measurable_mLog (i : Spin) : Measurable (fun m : Message => mLog m i) :=
+  Real.measurable_log.comp ((continuous_apply i).comp continuous_subtype_val).measurable
+lemma measurable_avg (f : Message → Spin → ℝ) (hf : ∀ i, Measurable (fun m => f m i)) :
+    Measurable (fun m => avg (f m)) := by
+  simp only [Radial.avg_expand]
+  exact (((hf 0).add (hf 1)).add (hf 2)).div_const 3
+lemma measurable_mCenteredLog (i : Spin) : Measurable (fun m : Message => mCenteredLog m i) :=
+  (measurable_mLog i).sub (measurable_avg mLog measurable_mLog)
+lemma measurable_entropyI : Measurable entropyI :=
+  measurable_avg _ (fun i => ((continuous_apply i).comp continuous_subtype_val).measurable.mul (measurable_mLog i))
+lemma measurable_entropyJ : Measurable entropyJ :=
+  (measurable_avg _ (fun i => (((continuous_apply i).comp continuous_subtype_val).measurable.sub_const 1).mul (measurable_mLog i))).div_const 2
+lemma measurable_entropyB : Measurable entropyB :=
+  (measurable_avg _ (fun i => ((continuous_apply i).comp continuous_subtype_val).measurable.mul ((measurable_mCenteredLog i).pow_const 2))).div_const 2
+lemma measurable_entropyF : Measurable entropyF := by
+  exact ((measurable_entropyI.const_mul 3).sub (measurable_entropyJ.const_mul 2)).add
+    (measurable_entropyB.const_mul (4/5:ℝ))
+
+lemma centeredLog_combine (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    {n : ℕ} (m : Fin n → Message) (i : Spin) :
+    mCenteredLog (combineMessage lam h m) i = ∑ j, mCenteredLog (edgeMessage lam h (m j)) i := by
+  simp only [mCenteredLog, mLog, Radial.avg_expand, log_combineMessage lam h hl0 hl1,
+    Finset.sum_sub_distrib, add_div, ← Finset.sum_div, Finset.sum_add_distrib]
+  ring
+
+lemma centeredLog_J (m : Message) : avg (fun i => m i*mCenteredLog m i) = 2*entropyJ m := by
+  simp only [mCenteredLog, entropyJ, mLog, Radial.avg_expand, message_last m]
+  ring
+
+lemma entropyF_radial (m : Message) : entropyF m = Radial.sourceF (fun i => m i) := by
+  simp only [entropyF, entropyI, entropyJ, entropyB, mCenteredLog, mLog, Radial.sourceF,
+    Radial.avg_expand]
+  ring
+
+lemma centeredLog_bound (m : Message) (C : ℝ) (hc : ∀ i, |mLog m i| ≤ C) (i : Spin) :
+    |mCenteredLog m i| ≤ 2*C := by
+  calc
+    |mCenteredLog m i| ≤ |mLog m i|+|avg (mLog m)| := abs_sub _ _
+    _ ≤ C+C := add_le_add (hc i) (abs_avg_le hc)
+    _ = _ := by ring
+
+lemma entropyI_bound (m : Message) (C : ℝ) (hc : ∀ i, |mLog m i| ≤ C) : |entropyI m| ≤ 3*C := by
+  apply abs_avg_le
+  intro i
+  rw [abs_mul, abs_of_nonneg (Message.nonneg m i)]
+  exact mul_le_mul (Message.le_three m i) (hc i) (abs_nonneg _) (by norm_num)
+
+lemma entropyJ_bound (m : Message) (C : ℝ) (hc : ∀ i, |mLog m i| ≤ C) : |entropyJ m| ≤ C := by
+  have hv (i : Spin) : |m i-1| ≤ 2 := abs_le.mpr ⟨by linarith [Message.nonneg m i], by linarith [Message.le_three m i]⟩
+  have ha : |avg (fun i => (m i-1)*mLog m i)| ≤ 2*C := by
+    apply abs_avg_le
+    intro i
+    rw [abs_mul]
+    exact mul_le_mul (hv i) (hc i) (abs_nonneg _) (by norm_num)
+  dsimp only [entropyJ]
+  rw [abs_div, abs_of_pos (by norm_num : (0:ℝ)<2)]
+  linarith
+
+lemma entropyB_nonneg (m : Message) : 0 ≤ entropyB m :=
+  div_nonneg (avg_nonneg (fun i => mul_nonneg (Message.nonneg m i) (sq_nonneg _))) (by norm_num)
+
+lemma entropyB_bound (m : Message) (C : ℝ) (hc : ∀ i, |mLog m i| ≤ C) : |entropyB m| ≤ 6*C^2 := by
+  have hC : 0 ≤ C := le_trans (abs_nonneg _) (hc 0)
+  have hu (i : Spin) : (mCenteredLog m i)^2 ≤ (2*C)^2 := by
+    have hh := (sq_le_sq₀ (abs_nonneg (mCenteredLog m i)) (by positivity : 0 ≤ 2*C)).mpr (centeredLog_bound m C hc i)
+    simpa only [sq_abs] using hh
+  have ha : avg (fun i => m i*(mCenteredLog m i)^2) ≤ 12*C^2 := by
+    apply avg_le_const
+    intro i
+    have hh := mul_le_mul (Message.le_three m i) (hu i) (sq_nonneg _) (by norm_num : (0:ℝ)≤3)
+    nlinarith
+  rw [abs_of_nonneg (entropyB_nonneg m)]
+  dsimp only [entropyB]
+  linarith
+
+end ThreeState
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+open Radial (avg)
+
+lemma continuous_avg {α : Type*} [TopologicalSpace α] (f : α → Spin → ℝ)
+    (hf : ∀ i, Continuous (fun m => f m i)) : Continuous (fun m => avg (f m)) := by
+  simp only [Radial.avg_expand]
+  exact (((hf 0).add (hf 1)).add (hf 2)).div_const 3
+
+lemma continuous_mLog_edge (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1) (i : Spin) :
+    Continuous (fun m => mLog (edgeMessage lam h m) i) :=
+  (((continuous_apply i).comp continuous_subtype_val).comp (continuous_edgeMessage lam h)).log
+    (fun m => ne_of_gt (edgeMessage_pos lam h hl0 hl1 m i))
+
+lemma continuous_mCenteredLog_edge (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1) (i : Spin) :
+    Continuous (fun m => mCenteredLog (edgeMessage lam h m) i) :=
+  (continuous_mLog_edge lam h hl0 hl1 i).sub
+    (continuous_avg _ (continuous_mLog_edge lam h hl0 hl1))
+
+lemma continuous_entropyI_edge (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1) :
+    Continuous (fun m => entropyI (edgeMessage lam h m)) :=
+  continuous_avg _ (fun i =>
+    (((continuous_apply i).comp continuous_subtype_val).comp (continuous_edgeMessage lam h)).mul
+      (continuous_mLog_edge lam h hl0 hl1 i))
+
+lemma continuous_entropyJ_edge (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1) :
+    Continuous (fun m => entropyJ (edgeMessage lam h m)) :=
+  (continuous_avg _ (fun i =>
+    ((((continuous_apply i).comp continuous_subtype_val).comp (continuous_edgeMessage lam h)).sub continuous_const).mul
+      (continuous_mLog_edge lam h hl0 hl1 i))).div_const 2
+
+lemma continuous_entropyB_edge (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1) :
+    Continuous (fun m => entropyB (edgeMessage lam h m)) :=
+  (continuous_avg _ (fun i =>
+    (((continuous_apply i).comp continuous_subtype_val).comp (continuous_edgeMessage lam h)).mul
+      ((continuous_mCenteredLog_edge lam h hl0 hl1 i).pow 2))).div_const 2
+
+lemma integrable_mLog_fixedpoint (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
+    (hl0 : 0 ≤ lam) (hl1 : lam < 1) (Q : ProbabilityMeasure Message)
+    (hQ : IsPosteriorFixedPoint offspring lam h Q)
+    (hD : HasMean offspring (fun n : ℕ => (n:ℝ))) (i : Spin) :
+    Integrable (fun m => mLog m i) (Q : Measure Message) := by
+  apply integrable_fixedpoint_of_growth offspring lam h Q hQ _ (measurable_mLog i)
+    (fun n => (n:ℝ)*(2*edgeLogBound lam)) (hD.mul_const _)
+  intro n m
+  have hh := log_combineMessage_bound lam h hl0 hl1 m i
+  dsimp only [mLog]
+  nlinarith
+
+lemma integrable_entropyI_fixedpoint (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
+    (hl0 : 0 ≤ lam) (hl1 : lam < 1) (Q : ProbabilityMeasure Message)
+    (hQ : IsPosteriorFixedPoint offspring lam h Q)
+    (hD : HasMean offspring (fun n : ℕ => (n:ℝ))) :
+    Integrable entropyI (Q : Measure Message) := by
+  apply integrable_fixedpoint_of_growth offspring lam h Q hQ _ measurable_entropyI
+    (fun n => (n:ℝ)*(6*edgeLogBound lam)) (hD.mul_const _)
+  intro n m
+  have hh := entropyI_bound (combineMessage lam h m) (2*n*edgeLogBound lam)
+    (log_combineMessage_bound lam h hl0 hl1 m)
+  nlinarith
+
+lemma integrable_entropyJ_fixedpoint (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
+    (hl0 : 0 ≤ lam) (hl1 : lam < 1) (Q : ProbabilityMeasure Message)
+    (hQ : IsPosteriorFixedPoint offspring lam h Q)
+    (hD : HasMean offspring (fun n : ℕ => (n:ℝ))) :
+    Integrable entropyJ (Q : Measure Message) := by
+  apply integrable_fixedpoint_of_growth offspring lam h Q hQ _ measurable_entropyJ
+    (fun n => (n:ℝ)*(2*edgeLogBound lam)) (hD.mul_const _)
+  intro n m
+  have hh := entropyJ_bound (combineMessage lam h m) (2*n*edgeLogBound lam)
+    (log_combineMessage_bound lam h hl0 hl1 m)
+  nlinarith
+
+lemma integrable_entropyB_fixedpoint (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
+    (hl0 : 0 ≤ lam) (hl1 : lam < 1) (Q : ProbabilityMeasure Message)
+    (hQ : IsPosteriorFixedPoint offspring lam h Q)
+    (hD2 : HasMean offspring (fun n : ℕ => (n:ℝ)^2)) :
+    Integrable entropyB (Q : Measure Message) := by
+  apply integrable_fixedpoint_of_growth offspring lam h Q hQ _ measurable_entropyB
+    (fun n => (n:ℝ)^2*(24*(edgeLogBound lam)^2)) (hD2.mul_const _)
+  intro n m
+  have hh := entropyB_bound (combineMessage lam h m) (2*n*edgeLogBound lam)
+    (log_combineMessage_bound lam h hl0 hl1 m)
+  nlinarith
+
+lemma integrable_entropyF_fixedpoint (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
+    (hl0 : 0 ≤ lam) (hl1 : lam < 1) (Q : ProbabilityMeasure Message)
+    (hQ : IsPosteriorFixedPoint offspring lam h Q)
+    (hD2 : HasMean offspring (fun n : ℕ => (n:ℝ)^2)) :
+    Integrable entropyF (Q : Measure Message) := by
+  have hD := hasMean_of_square hD2
+  exact (((integrable_entropyI_fixedpoint offspring lam h hl0 hl1 Q hQ hD).const_mul 3).sub
+    ((integrable_entropyJ_fixedpoint offspring lam h hl0 hl1 Q hQ hD).const_mul 2)).add
+      ((integrable_entropyB_fixedpoint offspring lam h hl0 hl1 Q hQ hD2).const_mul (4/5:ℝ))
+
+end ThreeState
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeState/EntropyRecursion.lean b/lean/OAI/Probability/ThreeState/EntropyRecursion.lean
index 7ad525ada548b5b47e9db62f5d502441ba02dab3..0c44c5d6dd6902ee1cd5ed7c033b213890bab6ae 100644
--- a/lean/OAI/Probability/ThreeState/EntropyRecursion.lean
+++ b/lean/OAI/Probability/ThreeState/EntropyRecursion.lean
@@ -1,464 +1,464 @@
-import Mathlib
-import OAI.Probability.ThreeState.Entropy
-
-namespace OAI
-
-/-! Independent-branch entropy identities and correlation corrections. -/
-
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-
-lemma iid_integral_weighted_single (Q : ProbabilityMeasure Message) (n : ℕ)
-    (w g : C(Message,ℝ)) (hw : ∫ m, w m ∂(Q : Measure Message) = 1) (a : Fin n) :
-    ∫ m : Fin n → Message, (∏ j, w (m j))*g (m a) ∂(iidMeasure Q n : Measure (Fin n → Message)) =
-      ∫ m, w m*g m ∂(Q : Measure Message) := by
-  have he (m : Fin n → Message) : (∏ j, w (m j))*g (m a) =
-      ∏ j, w (m j)*(if j=a then g (m j) else 1) := by
-    rw [Finset.prod_mul_distrib]
-    simp
-  simp_rw [he]
-  change (∫ m : Fin n → Message, ∏ j, w (m j)*(if j=a then g (m j) else 1)
-    ∂Measure.pi (fun _ : Fin n => (Q : Measure Message))) = _
-  rw [integral_fintype_prod_eq_prod (fun j (m : Message) => w m*(if j=a then g m else 1))]
-  have hi (j : Fin n) : (∫ m, w m*(if j=a then g m else 1) ∂(Q : Measure Message)) =
-      if j=a then ∫ m, w m*g m ∂(Q : Measure Message) else 1 := by
-    by_cases hj : j=a <;> simp [hj, hw]
-  simp_rw [hi]
-  simp
-
-lemma iid_integral_weighted_pair (Q : ProbabilityMeasure Message) (n : ℕ)
-    (w g : C(Message,ℝ)) (hw : ∫ m, w m ∂(Q : Measure Message) = 1) (a b : Fin n) (hab : a ≠ b) :
-    ∫ m : Fin n → Message, (∏ j, w (m j))*g (m a)*g (m b) ∂(iidMeasure Q n : Measure (Fin n → Message)) =
-      (∫ m, w m*g m ∂(Q : Measure Message))^2 := by
-  have he (m : Fin n → Message) : (∏ j, w (m j))*g (m a)*g (m b) =
-      ∏ j, w (m j)*(if j=a then g (m j) else 1)*(if j=b then g (m j) else 1) := by
-    simp only [Finset.prod_mul_distrib, Finset.prod_ite_eq', Finset.mem_univ, ite_true]
-  simp_rw [he]
-  change (∫ m : Fin n → Message, ∏ j, w (m j)*(if j=a then g (m j) else 1)*(if j=b then g (m j) else 1)
-    ∂Measure.pi (fun _ : Fin n => (Q : Measure Message))) = _
-  rw [integral_fintype_prod_eq_prod (fun j (m : Message) => w m*(if j=a then g m else 1)*(if j=b then g m else 1))]
-  have hi (j : Fin n) :
-      (∫ m, w m*(if j=a then g m else 1)*(if j=b then g m else 1) ∂(Q : Measure Message)) =
-      (if j=a then ∫ m, w m*g m ∂(Q : Measure Message) else 1)*
-      (if j=b then ∫ m, w m*g m ∂(Q : Measure Message) else 1) := by
-    by_cases hja : j=a
-    · subst j
-      simp [hab]
-    · by_cases hjb : j=b
-      · subst j
-        simp [hja]
-      · simp [hja, hjb, hw]
-  simp_rw [hi]
-  simp only [Finset.prod_mul_distrib, Finset.prod_ite_eq', Finset.mem_univ, ite_true, pow_two]
-
-lemma iid_integral_weighted_sum (Q : ProbabilityMeasure Message) (n : ℕ)
-    (w g : C(Message,ℝ)) (hw : ∫ m, w m ∂(Q : Measure Message) = 1) :
-    ∫ m : Fin n → Message, (∏ j, w (m j))*(∑ j, g (m j)) ∂(iidMeasure Q n : Measure (Fin n → Message)) =
-      n*(∫ m, w m*g m ∂(Q : Measure Message)) := by
-  simp_rw [Finset.mul_sum]
-  rw [integral_finsetSum]
-  · simp_rw [iid_integral_weighted_single Q n w g hw]
-    simp
-  · intro j _
-    exact (show Continuous (fun m : Fin n → Message => (∏ k, w (m k))*g (m j)) by fun_prop).integrable_of_hasCompactSupport
-      (HasCompactSupport.of_compactSpace _)
-
-lemma iid_integral_weighted_sum_sq (Q : ProbabilityMeasure Message) (n : ℕ)
-    (w g : C(Message,ℝ)) (hw : ∫ m, w m ∂(Q : Measure Message) = 1) :
-    ∫ m : Fin n → Message, (∏ j, w (m j))*(∑ j, g (m j))^2 ∂(iidMeasure Q n : Measure (Fin n → Message)) =
-      n*(∫ m, w m*(g m)^2 ∂(Q : Measure Message))+
-      (n:ℝ)*(n-1)*(∫ m, w m*g m ∂(Q : Measure Message))^2 := by
-  have he (m : Fin n → Message) : (∏ j, w (m j))*(∑ j, g (m j))^2 =
-      ∑ a, ∑ b, (∏ j, w (m j))*g (m a)*g (m b) := by
-    simp only [pow_two, Finset.mul_sum, Finset.sum_mul, mul_assoc]
-    apply Finset.sum_congr rfl
-    intro a _
-    apply Finset.sum_congr rfl
-    intro b _
-    ring
-  simp_rw [he]
-  have hi (a b : Fin n) : Integrable
-      (fun m : Fin n → Message => (∏ j, w (m j))*g (m a)*g (m b))
-      (iidMeasure Q n : Measure (Fin n → Message)) :=
-    (show Continuous _ by fun_prop).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
-  rw [integral_finsetSum Finset.univ (fun a _ => integrable_finsetSum _ (fun b _ => hi a b))]
-  simp_rw [integral_finsetSum Finset.univ (fun b _ => hi _ b)]
-  have hj (a b : Fin n) :
-      (∫ m : Fin n → Message, (∏ j, w (m j))*g (m a)*g (m b) ∂(iidMeasure Q n : Measure (Fin n → Message))) =
-      (∫ m, w m*g m ∂(Q : Measure Message))^2 +
-      if a=b then (∫ m, w m*(g m)^2 ∂(Q : Measure Message))-(∫ m, w m*g m ∂(Q : Measure Message))^2 else 0 := by
-    by_cases hab : a=b
-    · subst b
-      have hp : (fun m : Fin n → Message => (∏ j, w (m j))*g (m a)*g (m a)) =
-        (fun m => (∏ j, w (m j))*(g (m a))^2) := by funext m; ring
-      rw [hp]
-      have hh := iid_integral_weighted_single Q n w (g*g) hw a
-      change (∫ m : Fin n → Message, (∏ j, w (m j))*(g (m a)*g (m a)) ∂(iidMeasure Q n : Measure (Fin n → Message))) =
-        (∫ m, w m*(g m*g m) ∂(Q : Measure Message)) at hh
-      simpa only [pow_two, ite_true, add_sub_cancel] using hh
-    · rw [iid_integral_weighted_pair Q n w g hw a b hab]
-      simp [hab]
-  simp_rw [hj]
-  simp only [Finset.sum_add_distrib, Finset.sum_const, Finset.card_univ,
-    Fintype.card_fin, nsmul_eq_mul, Finset.sum_ite_eq, Finset.mem_univ, ite_true]
-  ring
-
-end ThreeState
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-open Radial (avg)
-
-lemma integral_avg {α : Type*} [MeasurableSpace α] (μ : Measure α) (f : α → Spin → ℝ)
-    (hf : ∀ i, Integrable (fun m => f m i) μ) :
-    ∫ m, avg (f m) ∂μ = avg (fun i => ∫ m, f m i ∂μ) := by
-  unfold avg
-  rw [integral_div, integral_finsetSum Finset.univ (fun i _ => hf i)]
-
-lemma integral_permMessage (Q : ProbabilityMeasure Message) (hQ : SpinSymmetric Q)
-    (π : Equiv.Perm Spin) (f : Message → ℝ) (hf : Measurable f) :
-    ∫ m, f (permMessage π m) ∂(Q : Measure Message) = ∫ m, f m ∂(Q : Measure Message) := by
-  have he : Measure.map (permMessage π) (Q : Measure Message) = (Q : Measure Message) :=
-    congrArg ProbabilityMeasure.toMeasure (hQ π)
-  rw [← integral_map (continuous_permMessage π).measurable.aemeasurable hf.aestronglyMeasurable, he]
-
-lemma symmetric_integral_coordinate (Q : ProbabilityMeasure Message) (hQ : SpinSymmetric Q)
-    (f : Message → Spin → ℝ) (hf : ∀ i, Measurable (fun m => f m i))
-    (he : ∀ (π : Equiv.Perm Spin) m i, f (permMessage π m) i = f m (π.symm i)) (i j : Spin) :
-    ∫ m, f m i ∂(Q : Measure Message) = ∫ m, f m j ∂(Q : Measure Message) := by
-  rw [← integral_permMessage Q hQ (Equiv.swap i j) (fun m => f m i) (hf i)]
-  apply integral_congr_ae
-  filter_upwards with m
-  rw [he]
-  simp
-
-lemma mLog_perm (π : Equiv.Perm Spin) (m : Message) (i : Spin) :
-    mLog (permMessage π m) i = mLog m (π.symm i) := rfl
-
-lemma mCenteredLog_perm (π : Equiv.Perm Spin) (m : Message) (i : Spin) :
-    mCenteredLog (permMessage π m) i = mCenteredLog m (π.symm i) := by
-  simp only [mCenteredLog, mLog_perm, avg, Equiv.sum_comp π.symm]
-
-lemma edgeMessage_perm (lam : ℝ) (h : Admissible lam) (π : Equiv.Perm Spin) (m : Message) :
-    edgeMessage lam h (permMessage π m) = permMessage π (edgeMessage lam h m) := rfl
-
-lemma integral_avg_eq_coordinate (Q : ProbabilityMeasure Message) (hQ : SpinSymmetric Q)
-    (f : Message → Spin → ℝ) (hf : ∀ i, Continuous (fun m => f m i))
-    (he : ∀ (π : Equiv.Perm Spin) m i, f (permMessage π m) i = f m (π.symm i)) (i : Spin) :
-    ∫ m, avg (f m) ∂(Q : Measure Message) = ∫ m, f m i ∂(Q : Measure Message) := by
-  rw [integral_avg _ _ (fun j => (hf j).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _))]
-  have hh (j : Spin) := symmetric_integral_coordinate Q hQ f (fun j => (hf j).measurable) he j i
-  simp_rw [hh]
-  simp [avg]
-
-lemma integral_edge_centered (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) (hQ : SpinSymmetric Q) (i : Spin) :
-    ∫ m, edgeMessage lam h m i*mCenteredLog (edgeMessage lam h m) i ∂(Q : Measure Message) =
-      2*(∫ m, entropyJ (edgeMessage lam h m) ∂(Q : Measure Message)) := by
-  have hc (j : Spin) : Continuous (fun m => edgeMessage lam h m j*mCenteredLog (edgeMessage lam h m) j) :=
-    (((continuous_apply j).comp continuous_subtype_val).comp (continuous_edgeMessage lam h)).mul
-      (continuous_mCenteredLog_edge lam h hl0 hl1 j)
-  rw [← integral_avg_eq_coordinate Q hQ _ hc]
-  · simp_rw [centeredLog_J]
-    exact integral_const_mul _ _
-  · intro π m j
-    rw [edgeMessage_perm, mCenteredLog_perm]
-    rfl
-
-lemma integral_edge_centered_sq (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) (hQ : SpinSymmetric Q) (i : Spin) :
-    ∫ m, edgeMessage lam h m i*(mCenteredLog (edgeMessage lam h m) i)^2 ∂(Q : Measure Message) =
-      2*(∫ m, entropyB (edgeMessage lam h m) ∂(Q : Measure Message)) := by
-  have hc (j : Spin) : Continuous (fun m => edgeMessage lam h m j*(mCenteredLog (edgeMessage lam h m) j)^2) :=
-    (((continuous_apply j).comp continuous_subtype_val).comp (continuous_edgeMessage lam h)).mul
-      ((continuous_mCenteredLog_edge lam h hl0 hl1 j).pow 2)
-  rw [← integral_avg_eq_coordinate Q hQ _ hc]
-  · have he (m : Message) : avg (fun j => edgeMessage lam h m j*(mCenteredLog (edgeMessage lam h m) j)^2) =
-        2*entropyB (edgeMessage lam h m) := by unfold entropyB; ring
-    simp_rw [he]
-    exact integral_const_mul _ _
-  · intro π m j
-    rw [edgeMessage_perm, mCenteredLog_perm]
-    rfl
-
-end ThreeState
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-open Radial (avg)
-
-lemma weighted_avg_combine (lam : ℝ) (h : Admissible lam) {n : ℕ} (m : Fin n → Message) (v : Spin → ℝ) :
-    normalizer lam h m * avg (fun i => combineMessage lam h m i*v i) =
-      avg (fun i => productWeight lam h m i*v i) := by
-  unfold avg
-  rw [← mul_div_assoc, Finset.mul_sum]
-  congr 1
-  apply Finset.sum_congr rfl
-  intro i _
-  rw [← mul_assoc, weighted_combineMessage]
-
-lemma weighted_entropyJ (lam : ℝ) (h : Admissible lam) {n : ℕ} (m : Fin n → Message) :
-    normalizer lam h m * entropyJ (combineMessage lam h m) =
-      avg (fun i => productWeight lam h m i*mCenteredLog (combineMessage lam h m) i)/2 := by
-  have hh := weighted_avg_combine lam h m (mCenteredLog (combineMessage lam h m))
-  rw [centeredLog_J] at hh
-  linarith
-
-lemma weighted_entropyB (lam : ℝ) (h : Admissible lam) {n : ℕ} (m : Fin n → Message) :
-    normalizer lam h m * entropyB (combineMessage lam h m) =
-      avg (fun i => productWeight lam h m i*(mCenteredLog (combineMessage lam h m) i)^2)/2 := by
-  rw [entropyB, ← mul_div_assoc, weighted_avg_combine]
-
-noncomputable def edgeCoordinateMap (lam : ℝ) (h : Admissible lam) (i : Spin) : C(Message,ℝ) :=
-  ⟨fun m => edgeMessage lam h m i, ((continuous_apply i).comp continuous_subtype_val).comp (continuous_edgeMessage lam h)⟩
-
-noncomputable def edgeCenteredMap (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (i : Spin) : C(Message,ℝ) :=
-  ⟨fun m => mCenteredLog (edgeMessage lam h m) i, continuous_mCenteredLog_edge lam h hl0 hl1 i⟩
-
-lemma integral_degree_entropyJ (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (hs : SpinSymmetric Q) (n : ℕ) :
-    ∫ m : Fin n → Message, normalizer lam h m*entropyJ (combineMessage lam h m)
-      ∂(iidMeasure Q n : Measure (Fin n → Message)) =
-      n*(∫ m, entropyJ (edgeMessage lam h m) ∂(Q : Measure Message)) := by
-  simp_rw [weighted_entropyJ, centeredLog_combine lam h hl0 hl1]
-  rw [integral_div, integral_avg]
-  · have hi (i : Spin) :
-        (∫ m : Fin n → Message, productWeight lam h m i*(∑ j, mCenteredLog (edgeMessage lam h (m j)) i)
-          ∂(iidMeasure Q n : Measure (Fin n → Message))) =
-        n*(2*(∫ m, entropyJ (edgeMessage lam h m) ∂(Q : Measure Message))) := by
-      have hh := iid_integral_weighted_sum Q n (edgeCoordinateMap lam h i) (edgeCenteredMap lam h hl0 hl1 i)
-        (integral_edgeMessage lam h Q hQ i)
-      change _ = n*(∫ m, edgeMessage lam h m i*mCenteredLog (edgeMessage lam h m) i ∂(Q : Measure Message)) at hh
-      rw [integral_edge_centered lam h hl0 hl1 Q hs i] at hh
-      exact hh
-    simp_rw [hi]
-    simp [avg]
-    ring
-  · intro i
-    apply Continuous.integrable_of_hasCompactSupport _ (HasCompactSupport.of_compactSpace _)
-    exact (continuous_productWeight lam h n i).mul
-      (continuous_finsetSum _ (fun j _ => (continuous_mCenteredLog_edge lam h hl0 hl1 i).comp (continuous_apply j)))
-
-lemma integral_degree_entropyB (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (hs : SpinSymmetric Q) (n : ℕ) :
-    ∫ m : Fin n → Message, normalizer lam h m*entropyB (combineMessage lam h m)
-      ∂(iidMeasure Q n : Measure (Fin n → Message)) =
-      n*(∫ m, entropyB (edgeMessage lam h m) ∂(Q : Measure Message))+
-      2*(n:ℝ)*(n-1)*(∫ m, entropyJ (edgeMessage lam h m) ∂(Q : Measure Message))^2 := by
-  simp_rw [weighted_entropyB, centeredLog_combine lam h hl0 hl1]
-  rw [integral_div, integral_avg]
-  · have hi (i : Spin) :
-        (∫ m : Fin n → Message, productWeight lam h m i*(∑ j, mCenteredLog (edgeMessage lam h (m j)) i)^2
-          ∂(iidMeasure Q n : Measure (Fin n → Message))) =
-        n*(2*(∫ m, entropyB (edgeMessage lam h m) ∂(Q : Measure Message)))+
-        (n:ℝ)*(n-1)*(2*(∫ m, entropyJ (edgeMessage lam h m) ∂(Q : Measure Message)))^2 := by
-      have hh := iid_integral_weighted_sum_sq Q n (edgeCoordinateMap lam h i) (edgeCenteredMap lam h hl0 hl1 i)
-        (integral_edgeMessage lam h Q hQ i)
-      change _ = n*(∫ m, edgeMessage lam h m i*(mCenteredLog (edgeMessage lam h m) i)^2 ∂(Q : Measure Message))+
-        (n:ℝ)*(n-1)*(∫ m, edgeMessage lam h m i*mCenteredLog (edgeMessage lam h m) i ∂(Q : Measure Message))^2 at hh
-      rw [integral_edge_centered lam h hl0 hl1 Q hs i, integral_edge_centered_sq lam h hl0 hl1 Q hs i] at hh
-      exact hh
-    simp_rw [hi]
-    simp [avg]
-    ring
-  · intro i
-    apply Continuous.integrable_of_hasCompactSupport _ (HasCompactSupport.of_compactSpace _)
-    exact (continuous_productWeight lam h n i).mul
-      ((continuous_finsetSum _ (fun j _ => (continuous_mCenteredLog_edge lam h hl0 hl1 i).comp (continuous_apply j))).pow 2)
-
-lemma additive_entropyJ (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint offspring lam h Q) (hs : SpinSymmetric Q)
-    (hD : HasMean offspring (fun n : ℕ => (n:ℝ))) :
-    ∫ m, entropyJ m ∂(Q : Measure Message) = mean offspring (fun n : ℕ => (n:ℝ))*
-      (∫ m, entropyJ (edgeMessage lam h m) ∂(Q : Measure Message)) := by
-  rw [integral_fixedpoint offspring lam h Q hQ _ measurable_entropyJ
-    (integrable_entropyJ_fixedpoint offspring lam h hl0 hl1 Q hQ hD)]
-  simp_rw [integral_degree_entropyJ lam h hl0 hl1 Q hQ.1 hs]
-  exact mean_mul_const _ _ _
-
-lemma hasMean_factorialSecond {offspring : PMF ℕ} (hD2 : HasMean offspring (fun n : ℕ => (n:ℝ)^2)) :
-    HasMean offspring (fun n : ℕ => (n:ℝ)*(n-1)) := by
-  have hD := hasMean_of_square hD2
-  exact (hD2.add (hD.mul_const (-1))).congr (fun n => by ring)
-
-lemma additive_entropyB (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint offspring lam h Q) (hs : SpinSymmetric Q)
-    (hD2 : HasMean offspring (fun n : ℕ => (n:ℝ)^2)) :
-    ∫ m, entropyB m ∂(Q : Measure Message) = mean offspring (fun n : ℕ => (n:ℝ))*
-      (∫ m, entropyB (edgeMessage lam h m) ∂(Q : Measure Message))+
-      2*mean offspring (fun n : ℕ => (n:ℝ)*(n-1))*(∫ m, entropyJ (edgeMessage lam h m) ∂(Q : Measure Message))^2 := by
-  rw [integral_fixedpoint offspring lam h Q hQ _ measurable_entropyB
-    (integrable_entropyB_fixedpoint offspring lam h hl0 hl1 Q hQ hD2)]
-  simp_rw [integral_degree_entropyB lam h hl0 hl1 Q hQ.1 hs]
-  have he (n : ℕ) :
-      (n:ℝ)*(∫ m, entropyB (edgeMessage lam h m) ∂(Q : Measure Message))+
-      2*(n:ℝ)*(n-1)*(∫ m, entropyJ (edgeMessage lam h m) ∂(Q : Measure Message))^2 =
-      (n:ℝ)*(∫ m, entropyB (edgeMessage lam h m) ∂(Q : Measure Message))+
-      ((n:ℝ)*(n-1))*(2*(∫ m, entropyJ (edgeMessage lam h m) ∂(Q : Measure Message))^2) := by ring
-  simp_rw [he]
-  rw [mean_add ((hasMean_of_square hD2).mul_const _) ((hasMean_factorialSecond hD2).mul_const _),
-    mean_mul_const, mean_mul_const]
-  ring
-
-end ThreeState
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-open Radial (avg)
-
-noncomputable def correlationEntropy (lam : ℝ) (h : Admissible lam)
-    (Q : ProbabilityMeasure Message) (n : ℕ) : ℝ :=
-  ∫ m : Fin n → Message, normalizer lam h m * Real.log (normalizer lam h m)
-    ∂(iidMeasure Q n : Measure (Fin n → Message))
-
-lemma continuous_log_normalizer (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1) (n : ℕ) :
-    Continuous (fun m : Fin n → Message => Real.log (normalizer lam h m)) :=
-  (continuous_normalizer lam h n).log (fun m => ne_of_gt (normalizer_pos lam h hl0 hl1 m))
-
-lemma continuous_entropyI_combine (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1) (n : ℕ) :
-    Continuous (fun m : Fin n → Message => entropyI (combineMessage lam h m)) := by
-  apply continuous_avg
-  intro i
-  have hc := ((continuous_apply i).comp continuous_subtype_val).comp
-    (continuous_combineMessage_positive lam h hl0 hl1 n)
-  exact hc.mul (hc.log (fun m => ne_of_gt (combineMessage_pos lam h hl0 hl1 m i)))
-
-lemma weighted_entropyI (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    {n : ℕ} (m : Fin n → Message) :
-    normalizer lam h m * entropyI (combineMessage lam h m) =
-      avg (fun i => productWeight lam h m i * (∑ j, mLog (edgeMessage lam h (m j)) i)) -
-      normalizer lam h m * Real.log (normalizer lam h m) := by
-  rw [entropyI, weighted_avg_combine]
-  simp only [mLog, log_combineMessage lam h hl0 hl1, Radial.avg_expand]
-  have hn : normalizer lam h m = (productWeight lam h m 0+productWeight lam h m 1+productWeight lam h m 2)/3 := by
-    simp [normalizer, Fin.sum_univ_succ]; ring
-  rw [hn]
-  ring
-
-noncomputable def edgeLogMap (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (i : Spin) : C(Message,ℝ) :=
-  ⟨fun m => mLog (edgeMessage lam h m) i, continuous_mLog_edge lam h hl0 hl1 i⟩
-
-lemma integral_degree_entropyI (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (n : ℕ) :
-    ∫ m : Fin n → Message, normalizer lam h m*entropyI (combineMessage lam h m)
-      ∂(iidMeasure Q n : Measure (Fin n → Message)) =
-      n*(∫ m, entropyI (edgeMessage lam h m) ∂(Q : Measure Message))-correlationEntropy lam h Q n := by
-  have hc (i : Spin) : Continuous (fun m : Fin n → Message => productWeight lam h m i *
-      (∑ j, mLog (edgeMessage lam h (m j)) i)) :=
-    (continuous_productWeight lam h n i).mul
-      (continuous_finsetSum _ (fun j _ => (continuous_mLog_edge lam h hl0 hl1 i).comp (continuous_apply j)))
-  have hz : Continuous (fun m : Fin n → Message => normalizer lam h m*Real.log (normalizer lam h m)) :=
-    (continuous_normalizer lam h n).mul (continuous_log_normalizer lam h hl0 hl1 n)
-  simp_rw [weighted_entropyI lam h hl0 hl1]
-  rw [integral_sub ((continuous_avg _ hc).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _))
-    (hz.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _))]
-  congr 1
-  rw [integral_avg _ _ (fun i => (hc i).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _))]
-  have hi (i : Spin) := iid_integral_weighted_sum Q n (edgeCoordinateMap lam h i) (edgeLogMap lam h hl0 hl1 i)
-    (integral_edgeMessage lam h Q hQ i)
-  change ∀ i : Spin, (∫ m : Fin n → Message, productWeight lam h m i*(∑ j, mLog (edgeMessage lam h (m j)) i)
-    ∂(iidMeasure Q n : Measure (Fin n → Message))) = n*(∫ m, edgeMessage lam h m i*mLog (edgeMessage lam h m) i
-    ∂(Q : Measure Message)) at hi
-  simp_rw [hi]
-  simp only [entropyI]
-  rw [integral_avg]
-  · simp only [Radial.avg_expand]; ring
-  · intro i
-    exact (((edgeCoordinateMap lam h i).continuous).mul (continuous_mLog_edge lam h hl0 hl1 i)).integrable_of_hasCompactSupport
-      (HasCompactSupport.of_compactSpace _)
-
-lemma correlationEntropy_nonneg (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (n : ℕ) : 0 ≤ correlationEntropy lam h Q n := by
-  have hc : Continuous (fun m : Fin n → Message => normalizer lam h m*Real.log (normalizer lam h m)) :=
-    (continuous_normalizer lam h n).mul (continuous_log_normalizer lam h hl0 hl1 n)
-  have hi : Integrable (normalizer lam h) (iidMeasure Q n : Measure (Fin n → Message)) := (continuous_normalizer lam h n).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
-  have hs (m : Fin n → Message) : normalizer lam h m-1 ≤ normalizer lam h m*Real.log (normalizer lam h m) := by
-    have hz := normalizer_pos lam h hl0 hl1 m
-    have hh := Real.one_sub_inv_le_log_of_pos hz
-    have hm := mul_le_mul_of_nonneg_left hh hz.le
-    rw [mul_sub, mul_one, mul_inv_cancel₀ (ne_of_gt hz)] at hm
-    exact hm
-  have hh := integral_mono (hi.sub (integrable_const 1)) (hc.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)) hs
-  simp only [Pi.sub_apply] at hh
-  rw [integral_sub hi (integrable_const 1), integral_normalizer lam h Q hQ n] at hh
-  simpa [correlationEntropy] using hh
-
-lemma correlationEntropy_bound (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (n : ℕ) : |correlationEntropy lam h Q n| ≤ n*edgeLogBound lam := by
-  have hb (m : Fin n → Message) : |Real.log (normalizer lam h m)| ≤ n*edgeLogBound lam :=
-    abs_log_of_bounds lam hl0 hl1 n _ (normalizer_bounds lam h hl0 hl1 m).1 (normalizer_bounds lam h hl0 hl1 m).2
-  have hc : Continuous (fun m : Fin n → Message => normalizer lam h m*Real.log (normalizer lam h m)) :=
-    (continuous_normalizer lam h n).mul (continuous_log_normalizer lam h hl0 hl1 n)
-  have hi : Integrable (fun m : Fin n → Message => |normalizer lam h m*Real.log (normalizer lam h m)|)
-      (iidMeasure Q n : Measure (Fin n → Message)) := hc.abs.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
-  have hj : Integrable (fun m : Fin n → Message => normalizer lam h m*(n*edgeLogBound lam))
-      (iidMeasure Q n : Measure (Fin n → Message)) := ((continuous_normalizer lam h n).mul_const (n*edgeLogBound lam)).integrable_of_hasCompactSupport
-    (HasCompactSupport.of_compactSpace _)
-  calc
-    |correlationEntropy lam h Q n| ≤ ∫ m, |normalizer lam h m*Real.log (normalizer lam h m)|
-      ∂(iidMeasure Q n : Measure (Fin n → Message)) := abs_integral_le_integral_abs
-    _ ≤ ∫ m, normalizer lam h m*(n*edgeLogBound lam) ∂(iidMeasure Q n : Measure (Fin n → Message)) := by
-      apply integral_mono hi hj
-      intro m
-      dsimp only
-      rw [abs_mul, abs_of_nonneg (normalizer_nonneg lam h m)]
-      exact mul_le_mul_of_nonneg_left (hb m) (normalizer_nonneg lam h m)
-    _ = _ := by rw [integral_mul_const, integral_normalizer lam h Q hQ n, one_mul]
-
-lemma hasMean_correlationEntropy (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
-    (hl0 : 0 ≤ lam) (hl1 : lam < 1) (Q : ProbabilityMeasure Message) (hQ : Balanced Q)
-    (hD : HasMean offspring (fun n : ℕ => (n:ℝ))) : HasMean offspring (correlationEntropy lam h Q) := by
-  apply HasMean.of_abs
-  exact (hD.mul_const (edgeLogBound lam)).mono (fun n => abs_nonneg _)
-    (correlationEntropy_bound lam h hl0 hl1 Q hQ)
-
-lemma additive_entropyI (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint offspring lam h Q)
-    (hD : HasMean offspring (fun n : ℕ => (n:ℝ))) :
-    ∫ m, entropyI m ∂(Q : Measure Message) = mean offspring (fun n : ℕ => (n:ℝ))*
-      (∫ m, entropyI (edgeMessage lam h m) ∂(Q : Measure Message))-mean offspring (correlationEntropy lam h Q) := by
-  rw [integral_fixedpoint offspring lam h Q hQ _ measurable_entropyI
-    (integrable_entropyI_fixedpoint offspring lam h hl0 hl1 Q hQ hD)]
-  simp_rw [integral_degree_entropyI lam h hl0 hl1 Q hQ.1]
-  have he (n : ℕ) : (n:ℝ)*(∫ m, entropyI (edgeMessage lam h m) ∂(Q : Measure Message))-correlationEntropy lam h Q n =
-    (n:ℝ)*(∫ m, entropyI (edgeMessage lam h m) ∂(Q : Measure Message))+correlationEntropy lam h Q n*(-1) := by ring
-  simp_rw [he]
-  rw [mean_add (hD.mul_const _) ((hasMean_correlationEntropy offspring lam h hl0 hl1 Q hQ.1 hD).mul_const _),
-    mean_mul_const, mean_mul_const]
-  ring
-
-lemma entropy_identity (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint offspring lam h Q) (hs : SpinSymmetric Q)
-    (hD2 : HasMean offspring (fun n : ℕ => (n:ℝ)^2)) :
-    (∫ m, entropyF m ∂(Q : Measure Message)) - mean offspring (fun n : ℕ => (n:ℝ))*
-      (∫ m, entropyF (edgeMessage lam h m) ∂(Q : Measure Message)) =
-      -3*mean offspring (correlationEntropy lam h Q) +
-      (8/5:ℝ)*mean offspring (fun n : ℕ => (n:ℝ)*(n-1))*(∫ m, entropyJ (edgeMessage lam h m) ∂(Q : Measure Message))^2 := by
-  have hD := hasMean_of_square hD2
-  have hi := integrable_entropyI_fixedpoint offspring lam h hl0 hl1 Q hQ hD
-  have hj := integrable_entropyJ_fixedpoint offspring lam h hl0 hl1 Q hQ hD
-  have hb := integrable_entropyB_fixedpoint offspring lam h hl0 hl1 Q hQ hD2
-  have hie : Integrable (fun m => entropyI (edgeMessage lam h m)) (Q : Measure Message) := (continuous_entropyI_edge lam h hl0 hl1).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
-  have hje : Integrable (fun m => entropyJ (edgeMessage lam h m)) (Q : Measure Message) := (continuous_entropyJ_edge lam h hl0 hl1).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
-  have hbe : Integrable (fun m => entropyB (edgeMessage lam h m)) (Q : Measure Message) := (continuous_entropyB_edge lam h hl0 hl1).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
-  have hlin (f g k : Message → ℝ) (hf : Integrable f (Q : Measure Message))
-      (hg : Integrable g (Q : Measure Message)) (hk : Integrable k (Q : Measure Message)) :
-      (∫ m, 3*f m-2*g m+(4/5:ℝ)*k m ∂(Q : Measure Message)) =
-        3*(∫ m, f m ∂(Q : Measure Message))-2*(∫ m, g m ∂(Q : Measure Message))+
-        (4/5:ℝ)*(∫ m, k m ∂(Q : Measure Message)) := by
-    have hsub : Integrable (fun m => 3*f m-2*g m) (Q : Measure Message) := (hf.const_mul 3).sub (hg.const_mul 2)
-    rw [integral_add hsub (hk.const_mul (4/5:ℝ)), integral_sub (hf.const_mul 3) (hg.const_mul 2)]
-    simp only [integral_const_mul]
-  simp only [entropyF]
-  rw [hlin _ _ _ hi hj hb, hlin _ _ _ hie hje hbe]
-  rw [additive_entropyI offspring lam h hl0 hl1 Q hQ hD,
-    additive_entropyJ offspring lam h hl0 hl1 Q hQ hs hD,
-    additive_entropyB offspring lam h hl0 hl1 Q hQ hs hD2]
-  ring
-
-end ThreeState
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeState.Entropy
+
+namespace OAI
+
+/-! Independent-branch entropy identities and correlation corrections. -/
+
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+
+lemma iid_integral_weighted_single (Q : ProbabilityMeasure Message) (n : ℕ)
+    (w g : C(Message,ℝ)) (hw : ∫ m, w m ∂(Q : Measure Message) = 1) (a : Fin n) :
+    ∫ m : Fin n → Message, (∏ j, w (m j))*g (m a) ∂(iidMeasure Q n : Measure (Fin n → Message)) =
+      ∫ m, w m*g m ∂(Q : Measure Message) := by
+  have he (m : Fin n → Message) : (∏ j, w (m j))*g (m a) =
+      ∏ j, w (m j)*(if j=a then g (m j) else 1) := by
+    rw [Finset.prod_mul_distrib]
+    simp
+  simp_rw [he]
+  change (∫ m : Fin n → Message, ∏ j, w (m j)*(if j=a then g (m j) else 1)
+    ∂Measure.pi (fun _ : Fin n => (Q : Measure Message))) = _
+  rw [integral_fintype_prod_eq_prod (fun j (m : Message) => w m*(if j=a then g m else 1))]
+  have hi (j : Fin n) : (∫ m, w m*(if j=a then g m else 1) ∂(Q : Measure Message)) =
+      if j=a then ∫ m, w m*g m ∂(Q : Measure Message) else 1 := by
+    by_cases hj : j=a <;> simp [hj, hw]
+  simp_rw [hi]
+  simp
+
+lemma iid_integral_weighted_pair (Q : ProbabilityMeasure Message) (n : ℕ)
+    (w g : C(Message,ℝ)) (hw : ∫ m, w m ∂(Q : Measure Message) = 1) (a b : Fin n) (hab : a ≠ b) :
+    ∫ m : Fin n → Message, (∏ j, w (m j))*g (m a)*g (m b) ∂(iidMeasure Q n : Measure (Fin n → Message)) =
+      (∫ m, w m*g m ∂(Q : Measure Message))^2 := by
+  have he (m : Fin n → Message) : (∏ j, w (m j))*g (m a)*g (m b) =
+      ∏ j, w (m j)*(if j=a then g (m j) else 1)*(if j=b then g (m j) else 1) := by
+    simp only [Finset.prod_mul_distrib, Finset.prod_ite_eq', Finset.mem_univ, ite_true]
+  simp_rw [he]
+  change (∫ m : Fin n → Message, ∏ j, w (m j)*(if j=a then g (m j) else 1)*(if j=b then g (m j) else 1)
+    ∂Measure.pi (fun _ : Fin n => (Q : Measure Message))) = _
+  rw [integral_fintype_prod_eq_prod (fun j (m : Message) => w m*(if j=a then g m else 1)*(if j=b then g m else 1))]
+  have hi (j : Fin n) :
+      (∫ m, w m*(if j=a then g m else 1)*(if j=b then g m else 1) ∂(Q : Measure Message)) =
+      (if j=a then ∫ m, w m*g m ∂(Q : Measure Message) else 1)*
+      (if j=b then ∫ m, w m*g m ∂(Q : Measure Message) else 1) := by
+    by_cases hja : j=a
+    · subst j
+      simp [hab]
+    · by_cases hjb : j=b
+      · subst j
+        simp [hja]
+      · simp [hja, hjb, hw]
+  simp_rw [hi]
+  simp only [Finset.prod_mul_distrib, Finset.prod_ite_eq', Finset.mem_univ, ite_true, pow_two]
+
+lemma iid_integral_weighted_sum (Q : ProbabilityMeasure Message) (n : ℕ)
+    (w g : C(Message,ℝ)) (hw : ∫ m, w m ∂(Q : Measure Message) = 1) :
+    ∫ m : Fin n → Message, (∏ j, w (m j))*(∑ j, g (m j)) ∂(iidMeasure Q n : Measure (Fin n → Message)) =
+      n*(∫ m, w m*g m ∂(Q : Measure Message)) := by
+  simp_rw [Finset.mul_sum]
+  rw [integral_finsetSum]
+  · simp_rw [iid_integral_weighted_single Q n w g hw]
+    simp
+  · intro j _
+    exact (show Continuous (fun m : Fin n → Message => (∏ k, w (m k))*g (m j)) by fun_prop).integrable_of_hasCompactSupport
+      (HasCompactSupport.of_compactSpace _)
+
+lemma iid_integral_weighted_sum_sq (Q : ProbabilityMeasure Message) (n : ℕ)
+    (w g : C(Message,ℝ)) (hw : ∫ m, w m ∂(Q : Measure Message) = 1) :
+    ∫ m : Fin n → Message, (∏ j, w (m j))*(∑ j, g (m j))^2 ∂(iidMeasure Q n : Measure (Fin n → Message)) =
+      n*(∫ m, w m*(g m)^2 ∂(Q : Measure Message))+
+      (n:ℝ)*(n-1)*(∫ m, w m*g m ∂(Q : Measure Message))^2 := by
+  have he (m : Fin n → Message) : (∏ j, w (m j))*(∑ j, g (m j))^2 =
+      ∑ a, ∑ b, (∏ j, w (m j))*g (m a)*g (m b) := by
+    simp only [pow_two, Finset.mul_sum, Finset.sum_mul, mul_assoc]
+    apply Finset.sum_congr rfl
+    intro a _
+    apply Finset.sum_congr rfl
+    intro b _
+    ring
+  simp_rw [he]
+  have hi (a b : Fin n) : Integrable
+      (fun m : Fin n → Message => (∏ j, w (m j))*g (m a)*g (m b))
+      (iidMeasure Q n : Measure (Fin n → Message)) :=
+    (show Continuous _ by fun_prop).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
+  rw [integral_finsetSum Finset.univ (fun a _ => integrable_finsetSum _ (fun b _ => hi a b))]
+  simp_rw [integral_finsetSum Finset.univ (fun b _ => hi _ b)]
+  have hj (a b : Fin n) :
+      (∫ m : Fin n → Message, (∏ j, w (m j))*g (m a)*g (m b) ∂(iidMeasure Q n : Measure (Fin n → Message))) =
+      (∫ m, w m*g m ∂(Q : Measure Message))^2 +
+      if a=b then (∫ m, w m*(g m)^2 ∂(Q : Measure Message))-(∫ m, w m*g m ∂(Q : Measure Message))^2 else 0 := by
+    by_cases hab : a=b
+    · subst b
+      have hp : (fun m : Fin n → Message => (∏ j, w (m j))*g (m a)*g (m a)) =
+        (fun m => (∏ j, w (m j))*(g (m a))^2) := by funext m; ring
+      rw [hp]
+      have hh := iid_integral_weighted_single Q n w (g*g) hw a
+      change (∫ m : Fin n → Message, (∏ j, w (m j))*(g (m a)*g (m a)) ∂(iidMeasure Q n : Measure (Fin n → Message))) =
+        (∫ m, w m*(g m*g m) ∂(Q : Measure Message)) at hh
+      simpa only [pow_two, ite_true, add_sub_cancel] using hh
+    · rw [iid_integral_weighted_pair Q n w g hw a b hab]
+      simp [hab]
+  simp_rw [hj]
+  simp only [Finset.sum_add_distrib, Finset.sum_const, Finset.card_univ,
+    Fintype.card_fin, nsmul_eq_mul, Finset.sum_ite_eq, Finset.mem_univ, ite_true]
+  ring
+
+end ThreeState
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+open Radial (avg)
+
+lemma integral_avg {α : Type*} [MeasurableSpace α] (μ : Measure α) (f : α → Spin → ℝ)
+    (hf : ∀ i, Integrable (fun m => f m i) μ) :
+    ∫ m, avg (f m) ∂μ = avg (fun i => ∫ m, f m i ∂μ) := by
+  unfold avg
+  rw [integral_div, integral_finsetSum Finset.univ (fun i _ => hf i)]
+
+lemma integral_permMessage (Q : ProbabilityMeasure Message) (hQ : SpinSymmetric Q)
+    (π : Equiv.Perm Spin) (f : Message → ℝ) (hf : Measurable f) :
+    ∫ m, f (permMessage π m) ∂(Q : Measure Message) = ∫ m, f m ∂(Q : Measure Message) := by
+  have he : Measure.map (permMessage π) (Q : Measure Message) = (Q : Measure Message) :=
+    congrArg ProbabilityMeasure.toMeasure (hQ π)
+  rw [← integral_map (continuous_permMessage π).measurable.aemeasurable hf.aestronglyMeasurable, he]
+
+lemma symmetric_integral_coordinate (Q : ProbabilityMeasure Message) (hQ : SpinSymmetric Q)
+    (f : Message → Spin → ℝ) (hf : ∀ i, Measurable (fun m => f m i))
+    (he : ∀ (π : Equiv.Perm Spin) m i, f (permMessage π m) i = f m (π.symm i)) (i j : Spin) :
+    ∫ m, f m i ∂(Q : Measure Message) = ∫ m, f m j ∂(Q : Measure Message) := by
+  rw [← integral_permMessage Q hQ (Equiv.swap i j) (fun m => f m i) (hf i)]
+  apply integral_congr_ae
+  filter_upwards with m
+  rw [he]
+  simp
+
+lemma mLog_perm (π : Equiv.Perm Spin) (m : Message) (i : Spin) :
+    mLog (permMessage π m) i = mLog m (π.symm i) := rfl
+
+lemma mCenteredLog_perm (π : Equiv.Perm Spin) (m : Message) (i : Spin) :
+    mCenteredLog (permMessage π m) i = mCenteredLog m (π.symm i) := by
+  simp only [mCenteredLog, mLog_perm, avg, Equiv.sum_comp π.symm]
+
+lemma edgeMessage_perm (lam : ℝ) (h : Admissible lam) (π : Equiv.Perm Spin) (m : Message) :
+    edgeMessage lam h (permMessage π m) = permMessage π (edgeMessage lam h m) := rfl
+
+lemma integral_avg_eq_coordinate (Q : ProbabilityMeasure Message) (hQ : SpinSymmetric Q)
+    (f : Message → Spin → ℝ) (hf : ∀ i, Continuous (fun m => f m i))
+    (he : ∀ (π : Equiv.Perm Spin) m i, f (permMessage π m) i = f m (π.symm i)) (i : Spin) :
+    ∫ m, avg (f m) ∂(Q : Measure Message) = ∫ m, f m i ∂(Q : Measure Message) := by
+  rw [integral_avg _ _ (fun j => (hf j).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _))]
+  have hh (j : Spin) := symmetric_integral_coordinate Q hQ f (fun j => (hf j).measurable) he j i
+  simp_rw [hh]
+  simp [avg]
+
+lemma integral_edge_centered (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) (hQ : SpinSymmetric Q) (i : Spin) :
+    ∫ m, edgeMessage lam h m i*mCenteredLog (edgeMessage lam h m) i ∂(Q : Measure Message) =
+      2*(∫ m, entropyJ (edgeMessage lam h m) ∂(Q : Measure Message)) := by
+  have hc (j : Spin) : Continuous (fun m => edgeMessage lam h m j*mCenteredLog (edgeMessage lam h m) j) :=
+    (((continuous_apply j).comp continuous_subtype_val).comp (continuous_edgeMessage lam h)).mul
+      (continuous_mCenteredLog_edge lam h hl0 hl1 j)
+  rw [← integral_avg_eq_coordinate Q hQ _ hc]
+  · simp_rw [centeredLog_J]
+    exact integral_const_mul _ _
+  · intro π m j
+    rw [edgeMessage_perm, mCenteredLog_perm]
+    rfl
+
+lemma integral_edge_centered_sq (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) (hQ : SpinSymmetric Q) (i : Spin) :
+    ∫ m, edgeMessage lam h m i*(mCenteredLog (edgeMessage lam h m) i)^2 ∂(Q : Measure Message) =
+      2*(∫ m, entropyB (edgeMessage lam h m) ∂(Q : Measure Message)) := by
+  have hc (j : Spin) : Continuous (fun m => edgeMessage lam h m j*(mCenteredLog (edgeMessage lam h m) j)^2) :=
+    (((continuous_apply j).comp continuous_subtype_val).comp (continuous_edgeMessage lam h)).mul
+      ((continuous_mCenteredLog_edge lam h hl0 hl1 j).pow 2)
+  rw [← integral_avg_eq_coordinate Q hQ _ hc]
+  · have he (m : Message) : avg (fun j => edgeMessage lam h m j*(mCenteredLog (edgeMessage lam h m) j)^2) =
+        2*entropyB (edgeMessage lam h m) := by unfold entropyB; ring
+    simp_rw [he]
+    exact integral_const_mul _ _
+  · intro π m j
+    rw [edgeMessage_perm, mCenteredLog_perm]
+    rfl
+
+end ThreeState
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+open Radial (avg)
+
+lemma weighted_avg_combine (lam : ℝ) (h : Admissible lam) {n : ℕ} (m : Fin n → Message) (v : Spin → ℝ) :
+    normalizer lam h m * avg (fun i => combineMessage lam h m i*v i) =
+      avg (fun i => productWeight lam h m i*v i) := by
+  unfold avg
+  rw [← mul_div_assoc, Finset.mul_sum]
+  congr 1
+  apply Finset.sum_congr rfl
+  intro i _
+  rw [← mul_assoc, weighted_combineMessage]
+
+lemma weighted_entropyJ (lam : ℝ) (h : Admissible lam) {n : ℕ} (m : Fin n → Message) :
+    normalizer lam h m * entropyJ (combineMessage lam h m) =
+      avg (fun i => productWeight lam h m i*mCenteredLog (combineMessage lam h m) i)/2 := by
+  have hh := weighted_avg_combine lam h m (mCenteredLog (combineMessage lam h m))
+  rw [centeredLog_J] at hh
+  linarith
+
+lemma weighted_entropyB (lam : ℝ) (h : Admissible lam) {n : ℕ} (m : Fin n → Message) :
+    normalizer lam h m * entropyB (combineMessage lam h m) =
+      avg (fun i => productWeight lam h m i*(mCenteredLog (combineMessage lam h m) i)^2)/2 := by
+  rw [entropyB, ← mul_div_assoc, weighted_avg_combine]
+
+noncomputable def edgeCoordinateMap (lam : ℝ) (h : Admissible lam) (i : Spin) : C(Message,ℝ) :=
+  ⟨fun m => edgeMessage lam h m i, ((continuous_apply i).comp continuous_subtype_val).comp (continuous_edgeMessage lam h)⟩
+
+noncomputable def edgeCenteredMap (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (i : Spin) : C(Message,ℝ) :=
+  ⟨fun m => mCenteredLog (edgeMessage lam h m) i, continuous_mCenteredLog_edge lam h hl0 hl1 i⟩
+
+lemma integral_degree_entropyJ (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (hs : SpinSymmetric Q) (n : ℕ) :
+    ∫ m : Fin n → Message, normalizer lam h m*entropyJ (combineMessage lam h m)
+      ∂(iidMeasure Q n : Measure (Fin n → Message)) =
+      n*(∫ m, entropyJ (edgeMessage lam h m) ∂(Q : Measure Message)) := by
+  simp_rw [weighted_entropyJ, centeredLog_combine lam h hl0 hl1]
+  rw [integral_div, integral_avg]
+  · have hi (i : Spin) :
+        (∫ m : Fin n → Message, productWeight lam h m i*(∑ j, mCenteredLog (edgeMessage lam h (m j)) i)
+          ∂(iidMeasure Q n : Measure (Fin n → Message))) =
+        n*(2*(∫ m, entropyJ (edgeMessage lam h m) ∂(Q : Measure Message))) := by
+      have hh := iid_integral_weighted_sum Q n (edgeCoordinateMap lam h i) (edgeCenteredMap lam h hl0 hl1 i)
+        (integral_edgeMessage lam h Q hQ i)
+      change _ = n*(∫ m, edgeMessage lam h m i*mCenteredLog (edgeMessage lam h m) i ∂(Q : Measure Message)) at hh
+      rw [integral_edge_centered lam h hl0 hl1 Q hs i] at hh
+      exact hh
+    simp_rw [hi]
+    simp [avg]
+    ring
+  · intro i
+    apply Continuous.integrable_of_hasCompactSupport _ (HasCompactSupport.of_compactSpace _)
+    exact (continuous_productWeight lam h n i).mul
+      (continuous_finsetSum _ (fun j _ => (continuous_mCenteredLog_edge lam h hl0 hl1 i).comp (continuous_apply j)))
+
+lemma integral_degree_entropyB (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (hs : SpinSymmetric Q) (n : ℕ) :
+    ∫ m : Fin n → Message, normalizer lam h m*entropyB (combineMessage lam h m)
+      ∂(iidMeasure Q n : Measure (Fin n → Message)) =
+      n*(∫ m, entropyB (edgeMessage lam h m) ∂(Q : Measure Message))+
+      2*(n:ℝ)*(n-1)*(∫ m, entropyJ (edgeMessage lam h m) ∂(Q : Measure Message))^2 := by
+  simp_rw [weighted_entropyB, centeredLog_combine lam h hl0 hl1]
+  rw [integral_div, integral_avg]
+  · have hi (i : Spin) :
+        (∫ m : Fin n → Message, productWeight lam h m i*(∑ j, mCenteredLog (edgeMessage lam h (m j)) i)^2
+          ∂(iidMeasure Q n : Measure (Fin n → Message))) =
+        n*(2*(∫ m, entropyB (edgeMessage lam h m) ∂(Q : Measure Message)))+
+        (n:ℝ)*(n-1)*(2*(∫ m, entropyJ (edgeMessage lam h m) ∂(Q : Measure Message)))^2 := by
+      have hh := iid_integral_weighted_sum_sq Q n (edgeCoordinateMap lam h i) (edgeCenteredMap lam h hl0 hl1 i)
+        (integral_edgeMessage lam h Q hQ i)
+      change _ = n*(∫ m, edgeMessage lam h m i*(mCenteredLog (edgeMessage lam h m) i)^2 ∂(Q : Measure Message))+
+        (n:ℝ)*(n-1)*(∫ m, edgeMessage lam h m i*mCenteredLog (edgeMessage lam h m) i ∂(Q : Measure Message))^2 at hh
+      rw [integral_edge_centered lam h hl0 hl1 Q hs i, integral_edge_centered_sq lam h hl0 hl1 Q hs i] at hh
+      exact hh
+    simp_rw [hi]
+    simp [avg]
+    ring
+  · intro i
+    apply Continuous.integrable_of_hasCompactSupport _ (HasCompactSupport.of_compactSpace _)
+    exact (continuous_productWeight lam h n i).mul
+      ((continuous_finsetSum _ (fun j _ => (continuous_mCenteredLog_edge lam h hl0 hl1 i).comp (continuous_apply j))).pow 2)
+
+lemma additive_entropyJ (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint offspring lam h Q) (hs : SpinSymmetric Q)
+    (hD : HasMean offspring (fun n : ℕ => (n:ℝ))) :
+    ∫ m, entropyJ m ∂(Q : Measure Message) = mean offspring (fun n : ℕ => (n:ℝ))*
+      (∫ m, entropyJ (edgeMessage lam h m) ∂(Q : Measure Message)) := by
+  rw [integral_fixedpoint offspring lam h Q hQ _ measurable_entropyJ
+    (integrable_entropyJ_fixedpoint offspring lam h hl0 hl1 Q hQ hD)]
+  simp_rw [integral_degree_entropyJ lam h hl0 hl1 Q hQ.1 hs]
+  exact mean_mul_const _ _ _
+
+lemma hasMean_factorialSecond {offspring : PMF ℕ} (hD2 : HasMean offspring (fun n : ℕ => (n:ℝ)^2)) :
+    HasMean offspring (fun n : ℕ => (n:ℝ)*(n-1)) := by
+  have hD := hasMean_of_square hD2
+  exact (hD2.add (hD.mul_const (-1))).congr (fun n => by ring)
+
+lemma additive_entropyB (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint offspring lam h Q) (hs : SpinSymmetric Q)
+    (hD2 : HasMean offspring (fun n : ℕ => (n:ℝ)^2)) :
+    ∫ m, entropyB m ∂(Q : Measure Message) = mean offspring (fun n : ℕ => (n:ℝ))*
+      (∫ m, entropyB (edgeMessage lam h m) ∂(Q : Measure Message))+
+      2*mean offspring (fun n : ℕ => (n:ℝ)*(n-1))*(∫ m, entropyJ (edgeMessage lam h m) ∂(Q : Measure Message))^2 := by
+  rw [integral_fixedpoint offspring lam h Q hQ _ measurable_entropyB
+    (integrable_entropyB_fixedpoint offspring lam h hl0 hl1 Q hQ hD2)]
+  simp_rw [integral_degree_entropyB lam h hl0 hl1 Q hQ.1 hs]
+  have he (n : ℕ) :
+      (n:ℝ)*(∫ m, entropyB (edgeMessage lam h m) ∂(Q : Measure Message))+
+      2*(n:ℝ)*(n-1)*(∫ m, entropyJ (edgeMessage lam h m) ∂(Q : Measure Message))^2 =
+      (n:ℝ)*(∫ m, entropyB (edgeMessage lam h m) ∂(Q : Measure Message))+
+      ((n:ℝ)*(n-1))*(2*(∫ m, entropyJ (edgeMessage lam h m) ∂(Q : Measure Message))^2) := by ring
+  simp_rw [he]
+  rw [mean_add ((hasMean_of_square hD2).mul_const _) ((hasMean_factorialSecond hD2).mul_const _),
+    mean_mul_const, mean_mul_const]
+  ring
+
+end ThreeState
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+open Radial (avg)
+
+noncomputable def correlationEntropy (lam : ℝ) (h : Admissible lam)
+    (Q : ProbabilityMeasure Message) (n : ℕ) : ℝ :=
+  ∫ m : Fin n → Message, normalizer lam h m * Real.log (normalizer lam h m)
+    ∂(iidMeasure Q n : Measure (Fin n → Message))
+
+lemma continuous_log_normalizer (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1) (n : ℕ) :
+    Continuous (fun m : Fin n → Message => Real.log (normalizer lam h m)) :=
+  (continuous_normalizer lam h n).log (fun m => ne_of_gt (normalizer_pos lam h hl0 hl1 m))
+
+lemma continuous_entropyI_combine (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1) (n : ℕ) :
+    Continuous (fun m : Fin n → Message => entropyI (combineMessage lam h m)) := by
+  apply continuous_avg
+  intro i
+  have hc := ((continuous_apply i).comp continuous_subtype_val).comp
+    (continuous_combineMessage_positive lam h hl0 hl1 n)
+  exact hc.mul (hc.log (fun m => ne_of_gt (combineMessage_pos lam h hl0 hl1 m i)))
+
+lemma weighted_entropyI (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    {n : ℕ} (m : Fin n → Message) :
+    normalizer lam h m * entropyI (combineMessage lam h m) =
+      avg (fun i => productWeight lam h m i * (∑ j, mLog (edgeMessage lam h (m j)) i)) -
+      normalizer lam h m * Real.log (normalizer lam h m) := by
+  rw [entropyI, weighted_avg_combine]
+  simp only [mLog, log_combineMessage lam h hl0 hl1, Radial.avg_expand]
+  have hn : normalizer lam h m = (productWeight lam h m 0+productWeight lam h m 1+productWeight lam h m 2)/3 := by
+    simp [normalizer, Fin.sum_univ_succ]; ring
+  rw [hn]
+  ring
+
+noncomputable def edgeLogMap (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (i : Spin) : C(Message,ℝ) :=
+  ⟨fun m => mLog (edgeMessage lam h m) i, continuous_mLog_edge lam h hl0 hl1 i⟩
+
+lemma integral_degree_entropyI (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (n : ℕ) :
+    ∫ m : Fin n → Message, normalizer lam h m*entropyI (combineMessage lam h m)
+      ∂(iidMeasure Q n : Measure (Fin n → Message)) =
+      n*(∫ m, entropyI (edgeMessage lam h m) ∂(Q : Measure Message))-correlationEntropy lam h Q n := by
+  have hc (i : Spin) : Continuous (fun m : Fin n → Message => productWeight lam h m i *
+      (∑ j, mLog (edgeMessage lam h (m j)) i)) :=
+    (continuous_productWeight lam h n i).mul
+      (continuous_finsetSum _ (fun j _ => (continuous_mLog_edge lam h hl0 hl1 i).comp (continuous_apply j)))
+  have hz : Continuous (fun m : Fin n → Message => normalizer lam h m*Real.log (normalizer lam h m)) :=
+    (continuous_normalizer lam h n).mul (continuous_log_normalizer lam h hl0 hl1 n)
+  simp_rw [weighted_entropyI lam h hl0 hl1]
+  rw [integral_sub ((continuous_avg _ hc).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _))
+    (hz.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _))]
+  congr 1
+  rw [integral_avg _ _ (fun i => (hc i).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _))]
+  have hi (i : Spin) := iid_integral_weighted_sum Q n (edgeCoordinateMap lam h i) (edgeLogMap lam h hl0 hl1 i)
+    (integral_edgeMessage lam h Q hQ i)
+  change ∀ i : Spin, (∫ m : Fin n → Message, productWeight lam h m i*(∑ j, mLog (edgeMessage lam h (m j)) i)
+    ∂(iidMeasure Q n : Measure (Fin n → Message))) = n*(∫ m, edgeMessage lam h m i*mLog (edgeMessage lam h m) i
+    ∂(Q : Measure Message)) at hi
+  simp_rw [hi]
+  simp only [entropyI]
+  rw [integral_avg]
+  · simp only [Radial.avg_expand]; ring
+  · intro i
+    exact (((edgeCoordinateMap lam h i).continuous).mul (continuous_mLog_edge lam h hl0 hl1 i)).integrable_of_hasCompactSupport
+      (HasCompactSupport.of_compactSpace _)
+
+lemma correlationEntropy_nonneg (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (n : ℕ) : 0 ≤ correlationEntropy lam h Q n := by
+  have hc : Continuous (fun m : Fin n → Message => normalizer lam h m*Real.log (normalizer lam h m)) :=
+    (continuous_normalizer lam h n).mul (continuous_log_normalizer lam h hl0 hl1 n)
+  have hi : Integrable (normalizer lam h) (iidMeasure Q n : Measure (Fin n → Message)) := (continuous_normalizer lam h n).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
+  have hs (m : Fin n → Message) : normalizer lam h m-1 ≤ normalizer lam h m*Real.log (normalizer lam h m) := by
+    have hz := normalizer_pos lam h hl0 hl1 m
+    have hh := Real.one_sub_inv_le_log_of_pos hz
+    have hm := mul_le_mul_of_nonneg_left hh hz.le
+    rw [mul_sub, mul_one, mul_inv_cancel₀ (ne_of_gt hz)] at hm
+    exact hm
+  have hh := integral_mono (hi.sub (integrable_const 1)) (hc.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)) hs
+  simp only [Pi.sub_apply] at hh
+  rw [integral_sub hi (integrable_const 1), integral_normalizer lam h Q hQ n] at hh
+  simpa [correlationEntropy] using hh
+
+lemma correlationEntropy_bound (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (n : ℕ) : |correlationEntropy lam h Q n| ≤ n*edgeLogBound lam := by
+  have hb (m : Fin n → Message) : |Real.log (normalizer lam h m)| ≤ n*edgeLogBound lam :=
+    abs_log_of_bounds lam hl0 hl1 n _ (normalizer_bounds lam h hl0 hl1 m).1 (normalizer_bounds lam h hl0 hl1 m).2
+  have hc : Continuous (fun m : Fin n → Message => normalizer lam h m*Real.log (normalizer lam h m)) :=
+    (continuous_normalizer lam h n).mul (continuous_log_normalizer lam h hl0 hl1 n)
+  have hi : Integrable (fun m : Fin n → Message => |normalizer lam h m*Real.log (normalizer lam h m)|)
+      (iidMeasure Q n : Measure (Fin n → Message)) := hc.abs.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
+  have hj : Integrable (fun m : Fin n → Message => normalizer lam h m*(n*edgeLogBound lam))
+      (iidMeasure Q n : Measure (Fin n → Message)) := ((continuous_normalizer lam h n).mul_const (n*edgeLogBound lam)).integrable_of_hasCompactSupport
+    (HasCompactSupport.of_compactSpace _)
+  calc
+    |correlationEntropy lam h Q n| ≤ ∫ m, |normalizer lam h m*Real.log (normalizer lam h m)|
+      ∂(iidMeasure Q n : Measure (Fin n → Message)) := abs_integral_le_integral_abs
+    _ ≤ ∫ m, normalizer lam h m*(n*edgeLogBound lam) ∂(iidMeasure Q n : Measure (Fin n → Message)) := by
+      apply integral_mono hi hj
+      intro m
+      dsimp only
+      rw [abs_mul, abs_of_nonneg (normalizer_nonneg lam h m)]
+      exact mul_le_mul_of_nonneg_left (hb m) (normalizer_nonneg lam h m)
+    _ = _ := by rw [integral_mul_const, integral_normalizer lam h Q hQ n, one_mul]
+
+lemma hasMean_correlationEntropy (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
+    (hl0 : 0 ≤ lam) (hl1 : lam < 1) (Q : ProbabilityMeasure Message) (hQ : Balanced Q)
+    (hD : HasMean offspring (fun n : ℕ => (n:ℝ))) : HasMean offspring (correlationEntropy lam h Q) := by
+  apply HasMean.of_abs
+  exact (hD.mul_const (edgeLogBound lam)).mono (fun n => abs_nonneg _)
+    (correlationEntropy_bound lam h hl0 hl1 Q hQ)
+
+lemma additive_entropyI (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint offspring lam h Q)
+    (hD : HasMean offspring (fun n : ℕ => (n:ℝ))) :
+    ∫ m, entropyI m ∂(Q : Measure Message) = mean offspring (fun n : ℕ => (n:ℝ))*
+      (∫ m, entropyI (edgeMessage lam h m) ∂(Q : Measure Message))-mean offspring (correlationEntropy lam h Q) := by
+  rw [integral_fixedpoint offspring lam h Q hQ _ measurable_entropyI
+    (integrable_entropyI_fixedpoint offspring lam h hl0 hl1 Q hQ hD)]
+  simp_rw [integral_degree_entropyI lam h hl0 hl1 Q hQ.1]
+  have he (n : ℕ) : (n:ℝ)*(∫ m, entropyI (edgeMessage lam h m) ∂(Q : Measure Message))-correlationEntropy lam h Q n =
+    (n:ℝ)*(∫ m, entropyI (edgeMessage lam h m) ∂(Q : Measure Message))+correlationEntropy lam h Q n*(-1) := by ring
+  simp_rw [he]
+  rw [mean_add (hD.mul_const _) ((hasMean_correlationEntropy offspring lam h hl0 hl1 Q hQ.1 hD).mul_const _),
+    mean_mul_const, mean_mul_const]
+  ring
+
+lemma entropy_identity (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint offspring lam h Q) (hs : SpinSymmetric Q)
+    (hD2 : HasMean offspring (fun n : ℕ => (n:ℝ)^2)) :
+    (∫ m, entropyF m ∂(Q : Measure Message)) - mean offspring (fun n : ℕ => (n:ℝ))*
+      (∫ m, entropyF (edgeMessage lam h m) ∂(Q : Measure Message)) =
+      -3*mean offspring (correlationEntropy lam h Q) +
+      (8/5:ℝ)*mean offspring (fun n : ℕ => (n:ℝ)*(n-1))*(∫ m, entropyJ (edgeMessage lam h m) ∂(Q : Measure Message))^2 := by
+  have hD := hasMean_of_square hD2
+  have hi := integrable_entropyI_fixedpoint offspring lam h hl0 hl1 Q hQ hD
+  have hj := integrable_entropyJ_fixedpoint offspring lam h hl0 hl1 Q hQ hD
+  have hb := integrable_entropyB_fixedpoint offspring lam h hl0 hl1 Q hQ hD2
+  have hie : Integrable (fun m => entropyI (edgeMessage lam h m)) (Q : Measure Message) := (continuous_entropyI_edge lam h hl0 hl1).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
+  have hje : Integrable (fun m => entropyJ (edgeMessage lam h m)) (Q : Measure Message) := (continuous_entropyJ_edge lam h hl0 hl1).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
+  have hbe : Integrable (fun m => entropyB (edgeMessage lam h m)) (Q : Measure Message) := (continuous_entropyB_edge lam h hl0 hl1).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
+  have hlin (f g k : Message → ℝ) (hf : Integrable f (Q : Measure Message))
+      (hg : Integrable g (Q : Measure Message)) (hk : Integrable k (Q : Measure Message)) :
+      (∫ m, 3*f m-2*g m+(4/5:ℝ)*k m ∂(Q : Measure Message)) =
+        3*(∫ m, f m ∂(Q : Measure Message))-2*(∫ m, g m ∂(Q : Measure Message))+
+        (4/5:ℝ)*(∫ m, k m ∂(Q : Measure Message)) := by
+    have hsub : Integrable (fun m => 3*f m-2*g m) (Q : Measure Message) := (hf.const_mul 3).sub (hg.const_mul 2)
+    rw [integral_add hsub (hk.const_mul (4/5:ℝ)), integral_sub (hf.const_mul 3) (hg.const_mul 2)]
+    simp only [integral_const_mul]
+  simp only [entropyF]
+  rw [hlin _ _ _ hi hj hb, hlin _ _ _ hie hje hbe]
+  rw [additive_entropyI offspring lam h hl0 hl1 Q hQ hD,
+    additive_entropyJ offspring lam h hl0 hl1 Q hQ hs hD,
+    additive_entropyB offspring lam h hl0 hl1 Q hQ hs hD2]
+  ring
+
+end ThreeState
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeState/Expectations.lean b/lean/OAI/Probability/ThreeState/Expectations.lean
index e8b5370678a2baf21631a53c64dd216bc6289e87..b0c93d36b48ea7ac882047e4a3ae62ee9aab0a05 100644
--- a/lean/OAI/Probability/ThreeState/Expectations.lean
+++ b/lean/OAI/Probability/ThreeState/Expectations.lean
@@ -1,384 +1,384 @@
-import Mathlib
-import OAI.Probability.ThreeState.Basic
-
-namespace OAI
-
-/-! Expectations, second moments and the regular and Poisson offspring laws. -/
-
-namespace ThreeState
-open scoped Classical
-
- 
-noncomputable def mean {α : Type*} (p : PMF α) (f : α → ℝ) : ℝ :=
-  ∑' a, mass p a * f a
-
-def HasMean {α : Type*} (p : PMF α) (f : α → ℝ) : Prop :=
-  Summable (fun a => mass p a * f a)
-
-lemma HasMean.abs {α : Type*} {p : PMF α} {f : α → ℝ} (h : HasMean p f) :
-    HasMean p (fun a => |f a|) := by
-  simpa only [HasMean, abs_mul, abs_of_nonneg (mass_nonneg p _)] using Summable.abs h
-
-lemma HasMean.of_abs {α : Type*} {p : PMF α} {f : α → ℝ}
-    (h : HasMean p (fun a => |f a|)) : HasMean p f := by
-  apply summable_norm_iff.mp
-  simpa only [HasMean, Real.norm_eq_abs, abs_mul, abs_of_nonneg (mass_nonneg p _)] using h
-
-lemma hasMean_const {α : Type*} (p : PMF α) (c : ℝ) : HasMean p (fun _ => c) :=
-  (mass_summable p).mul_right c
-
-lemma mean_const {α : Type*} (p : PMF α) (c : ℝ) : mean p (fun _ => c) = c := by
-  rw [mean, (mass_summable p).tsum_mul_right, mass_sum, one_mul]
-
-lemma HasMean.add {α : Type*} {p : PMF α} {f g : α → ℝ} (hf : HasMean p f) (hg : HasMean p g) :
-    HasMean p (fun a => f a+g a) := by
-  simpa only [HasMean, mul_add] using Summable.add hf hg
-
-lemma mean_add {α : Type*} {p : PMF α} {f g : α → ℝ} (hf : HasMean p f) (hg : HasMean p g) :
-    mean p (fun a => f a+g a) = mean p f+mean p g := by
-  simp only [mean, mul_add, hf.tsum_add hg]
-
-lemma HasMean.mul_const {α : Type*} {p : PMF α} {f : α → ℝ} (hf : HasMean p f) (c : ℝ) :
-    HasMean p (fun a => f a*c) := by
-  simpa only [HasMean, mul_assoc] using hf.mul_right c
-
-lemma mean_mul_const {α : Type*} (p : PMF α) (f : α → ℝ) (c : ℝ) :
-    mean p (fun a => f a*c) = mean p f*c := by
-  simp only [mean, ← mul_assoc, tsum_mul_right]
-
-lemma HasMean.const_mul {α : Type*} {p : PMF α} {f : α → ℝ} (hf : HasMean p f) (c : ℝ) :
-    HasMean p (fun a => c*f a) := by
-  simpa only [mul_comm c] using hf.mul_const c
-
-lemma mean_const_mul {α : Type*} (p : PMF α) (f : α → ℝ) (c : ℝ) :
-    mean p (fun a => c*f a) = c*mean p f := by
-  simp only [mul_comm c, mean_mul_const]
-
-lemma hasMean_of_square {α : Type*} {p : PMF α} {f : α → ℝ}
-    (h : HasMean p (fun a => (f a)^2)) : HasMean p f := by
-  apply HasMean.of_abs
-  apply Summable.of_nonneg_of_le (fun a => mul_nonneg (mass_nonneg p a) (abs_nonneg _))
-    (fun a => mul_le_mul_of_nonneg_left (by nlinarith [sq_nonneg (|f a|-1), sq_abs (f a)])
-      (mass_nonneg p a)) ((hasMean_const p 1).add h)
-
-lemma mass_pure {α : Type*} (a b : α) : mass (PMF.pure a) b = if b=a then 1 else 0 := by
-  classical
-  by_cases h : b=a <;> simp [mass, PMF.pure_apply, h]
-
-lemma mean_pure {α : Type*} (a : α) (f : α → ℝ) : mean (PMF.pure a) f = f a := by
-  classical
-  simp [mean, mass_pure]
-
-lemma hasMean_pure {α : Type*} (a : α) (f : α → ℝ) : HasMean (PMF.pure a) f := by
-  classical
-  simp only [HasMean, mass_pure, ite_mul, one_mul, zero_mul]
-  exact summable_of_ne_finset_zero (s := {a}) (fun b hb => by simp at hb; simp [hb])
-
-lemma mean_nonneg {α : Type*} (p : PMF α) {f : α → ℝ} (hf : ∀ a, 0 ≤ f a) :
-    0 ≤ mean p f := tsum_nonneg (fun a => mul_nonneg (mass_nonneg p a) (hf a))
-
-lemma HasMean.bind {α β : Type*} {p : PMF α} {k : α → PMF β} {f : β → ℝ}
-    (hk : ∀ a, HasMean (k a) f)
-    (hp : HasMean p (fun a => mean (k a) (fun b => |f b|))) :
-    HasMean (p.bind k) f := by
-  have hs : Summable (Function.uncurry (fun a b => mass p a * (mass (k a) b * |f b|))) := by
-    apply (summable_prod_of_nonneg (fun ab =>
-      mul_nonneg (mass_nonneg p ab.1) (mul_nonneg (mass_nonneg _ _) (abs_nonneg _)))).mpr
-    change (∀ a, Summable (fun b => mass p a * (mass (k a) b * |f b|))) ∧
-      Summable (fun a => ∑' b, mass p a * (mass (k a) b * |f b|))
-    constructor
-    · intro a
-      exact (hk a).abs.mul_left (mass p a)
-    · simpa only [tsum_mul_left, mean, HasMean] using hp
-  apply HasMean.of_abs
-  apply hs.prod_symm.prod.congr
-  intro b
-  change (∑' a, mass p a * (mass (k a) b * |f b|)) = mass (p.bind k) b * |f b|
-  rw [mass_bind, ← tsum_mul_right]
-  congr 1
-  funext a
-  ring
-
-lemma mean_bind {α β : Type*} {p : PMF α} {k : α → PMF β} {f : β → ℝ}
-    (hk : ∀ a, HasMean (k a) f)
-    (hp : HasMean p (fun a => mean (k a) (fun b => |f b|))) :
-    mean (p.bind k) f = mean p (fun a => mean (k a) f) := by
-  have hsabs : Summable (Function.uncurry (fun a b => mass p a * (mass (k a) b * |f b|))) := by
-    apply (summable_prod_of_nonneg (fun ab =>
-      mul_nonneg (mass_nonneg p ab.1) (mul_nonneg (mass_nonneg _ _) (abs_nonneg _)))).mpr
-    change (∀ a, Summable (fun b => mass p a * (mass (k a) b * |f b|))) ∧
-      Summable (fun a => ∑' b, mass p a * (mass (k a) b * |f b|))
-    constructor
-    · intro a
-      exact (hk a).abs.mul_left (mass p a)
-    · simpa only [tsum_mul_left, mean, HasMean] using hp
-  have hs : Summable (Function.uncurry (fun a b => mass p a * (mass (k a) b * f b))) := by
-    apply summable_norm_iff.mp
-    apply hsabs.congr
-    rintro ⟨a,b⟩
-    change mass p a * (mass (k a) b * |f b|) = ‖mass p a * (mass (k a) b * f b)‖
-    rw [Real.norm_eq_abs, abs_mul, abs_mul, abs_of_nonneg (mass_nonneg p a),
-      abs_of_nonneg (mass_nonneg (k a) b)]
-  unfold mean
-  simp_rw [mass_bind, ← tsum_mul_right, mul_assoc]
-  rw [hs.tsum_comm]
-  simp only [tsum_mul_left]
-
-lemma HasMean.map {α β : Type*} {p : PMF α} {g : α → β} {f : β → ℝ}
-    (h : HasMean p (fun a => f (g a))) : HasMean (p.map g) f := by
-  change HasMean (p.bind (fun a => PMF.pure (g a))) f
-  apply HasMean.bind (fun a => hasMean_pure _ _)
-  simpa only [mean_pure] using h.abs
-
-lemma mean_map {α β : Type*} {p : PMF α} {g : α → β} {f : β → ℝ}
-    (h : HasMean p (fun a => f (g a))) : mean (p.map g) f = mean p (fun a => f (g a)) := by
-  change mean (p.bind (fun a => PMF.pure (g a))) f = _
-  rw [mean_bind (fun a => hasMean_pure _ _) (by simpa only [mean_pure] using h.abs)]
-  simp only [mean_pure]
-
-end ThreeState
-namespace ThreeState
-open scoped Classical
-
-lemma HasMean.mono {α : Type*} {p : PMF α} {f g : α → ℝ} (hg : HasMean p g)
-    (hf : ∀ a, 0 ≤ f a) (hle : ∀ a, f a ≤ g a) : HasMean p f := by
-  exact Summable.of_nonneg_of_le (fun a => mul_nonneg (mass_nonneg p a) (hf a))
-    (fun a => mul_le_mul_of_nonneg_left (hle a) (mass_nonneg p a)) hg
-
-lemma mean_le {α : Type*} {p : PMF α} {f g : α → ℝ}
-    (hf : HasMean p f) (hg : HasMean p g) (hle : ∀ a, f a ≤ g a) : mean p f ≤ mean p g := by
-  exact Summable.tsum_le_tsum (fun a => mul_le_mul_of_nonneg_left (hle a) (mass_nonneg p a)) hf hg
-
-lemma mean_abs_le_one_add_square {α : Type*} {p : PMF α} {f : α → ℝ}
-    (hf : HasMean p (fun a => (f a)^2)) :
-    mean p (fun a => |f a|) ≤ 1+mean p (fun a => (f a)^2) := by
-  have h := mean_le (hasMean_of_square hf).abs ((hasMean_const p 1).add hf)
-    (fun a => (show |f a| ≤ 1+(f a)^2 by nlinarith [sq_nonneg (|f a|-1), sq_abs (f a)]))
-  simpa only [mean_add (hasMean_const p 1) hf, mean_const] using h
-
-lemma hasMean_bind_square {α β : Type*} {p : PMF α} {k : α → PMF β} {f : β → ℝ}
-    (hk : ∀ a, HasMean (k a) (fun b => (f b)^2))
-    (hp : HasMean p (fun a => mean (k a) (fun b => (f b)^2))) :
-    HasMean (p.bind k) (fun b => (f b)^2) := by
-  apply HasMean.bind hk
-  simpa only [abs_sq] using hp
-
-lemma hasMean_bind_abs_of_square {α β : Type*} {p : PMF α} {k : α → PMF β} {f : β → ℝ}
-    (hk : ∀ a, HasMean (k a) (fun b => (f b)^2))
-    (hp : HasMean p (fun a => mean (k a) (fun b => (f b)^2))) :
-    HasMean p (fun a => mean (k a) (fun b => |f b|)) := by
-  exact ((hasMean_const p 1).add hp).mono
-    (fun a => mean_nonneg _ (fun b => abs_nonneg _))
-    (fun a => mean_abs_le_one_add_square (hk a))
-
-lemma mean_bind_of_square {α β : Type*} {p : PMF α} {k : α → PMF β} {f : β → ℝ}
-    (hk : ∀ a, HasMean (k a) (fun b => (f b)^2))
-    (hp : HasMean p (fun a => mean (k a) (fun b => (f b)^2))) :
-    mean (p.bind k) f = mean p (fun a => mean (k a) f) :=
-  mean_bind (fun a => hasMean_of_square (hk a)) (hasMean_bind_abs_of_square hk hp)
-
-lemma mean_bind_square {α β : Type*} {p : PMF α} {k : α → PMF β} {f : β → ℝ}
-    (hk : ∀ a, HasMean (k a) (fun b => (f b)^2))
-    (hp : HasMean p (fun a => mean (k a) (fun b => (f b)^2))) :
-    mean (p.bind k) (fun b => (f b)^2) = mean p (fun a => mean (k a) (fun b => (f b)^2)) := by
-  apply mean_bind hk
-  simpa only [abs_sq] using hp
-
-lemma HasMean.congr {α : Type*} {p : PMF α} {f g : α → ℝ} (hf : HasMean p f)
-    (hfg : ∀ a, f a = g a) : HasMean p g := by
-  exact Summable.congr hf (fun a => congrArg (fun x => mass p a*x) (hfg a))
-
-lemma mean_congr {α : Type*} (p : PMF α) {f g : α → ℝ} (hfg : ∀ a, f a = g a) :
-    mean p f = mean p g := tsum_congr (fun a => congrArg (fun x => mass p a*x) (hfg a))
-
-lemma hasMean_shift_square {α : Type*} {p : PMF α} {f : α → ℝ}
-    (hf : HasMean p (fun a => (f a)^2)) (c : ℝ) : HasMean p (fun a => (c+f a)^2) := by
-  exact (((hasMean_const p (c^2)).add ((hasMean_of_square hf).const_mul (2*c))).add hf).congr
-    (fun a => by ring)
-
-lemma mean_shift_square {α : Type*} {p : PMF α} {f : α → ℝ}
-    (hf : HasMean p (fun a => (f a)^2)) (c : ℝ) :
-    mean p (fun a => (c+f a)^2) = c^2+2*c*mean p f+mean p (fun a => (f a)^2) := by
-  rw [mean_congr p (g := fun a => c^2+(2*c)*f a+(f a)^2) (fun a => by ring),
-    mean_add ((hasMean_const p (c^2)).add ((hasMean_of_square hf).const_mul (2*c))) hf,
-    mean_add (hasMean_const p (c^2)) ((hasMean_of_square hf).const_mul (2*c)),
-    mean_const, mean_const_mul]
-
-lemma hasMean_fintype {α : Type*} [Fintype α] (p : PMF α) (f : α → ℝ) : HasMean p f :=
-  summable_of_hasFiniteSupport (Set.toFinite _)
-
-lemma mean_fintype {α : Type*} [Fintype α] (p : PMF α) (f : α → ℝ) :
-    mean p f = ∑ a, mass p a * f a := tsum_fintype _
-
- 
-
-structure Moments {α : Type*} (p : PMF α) (f : α → ℝ) (m s : ℝ) : Prop where
-  square : HasMean p (fun a => (f a)^2)
-  first : mean p f = m
-  second : mean p (fun a => (f a)^2) = s
-
-lemma Moments.hasMean {α : Type*} {p : PMF α} {f : α → ℝ} {m s : ℝ}
-    (h : Moments p f m s) : HasMean p f := hasMean_of_square h.square
-
-lemma Moments.map {α β : Type*} {p : PMF α} {f : β → ℝ} {g : α → β} {m s : ℝ}
-    (h : Moments p (fun a => f (g a)) m s) : Moments (p.map g) f m s := by
-  refine ⟨HasMean.map (g := g) (f := fun b => (f b)^2) h.square, ?_, ?_⟩
-  · exact (mean_map (g := g) (f := f) h.hasMean).trans h.first
-  · exact (mean_map (g := g) (f := fun b => (f b)^2) h.square).trans h.second
-
-lemma Moments.pure {α : Type*} (a : α) (f : α → ℝ) :
-    Moments (PMF.pure a) f (f a) ((f a)^2) :=
-  ⟨hasMean_pure _ _, mean_pure _ _, mean_pure _ _⟩
-
-lemma Moments.shift {α : Type*} {p : PMF α} {f : α → ℝ} {m s : ℝ}
-    (h : Moments p f m s) (c : ℝ) :
-    Moments p (fun a => c+f a) (c+m) (c^2+2*c*m+s) := by
-  refine ⟨hasMean_shift_square h.square c, ?_, ?_⟩
-  · rw [mean_add (hasMean_const p c) h.hasMean, mean_const, h.first]
-  · rw [mean_shift_square h.square, h.first, h.second]
-
-lemma Moments.bind {α β : Type*} {p : PMF α} {k : α → PMF β} {f : β → ℝ}
-    {m s : α → ℝ} {M S : ℝ} (hk : ∀ a, Moments (k a) f (m a) (s a))
-    (hs : HasMean p s) (hm : mean p m = M) (hS : mean p s = S) :
-    Moments (p.bind k) f M S := by
-  have hp : HasMean p (fun a => mean (k a) (fun b => (f b)^2)) :=
-    hs.congr (fun a => (hk a).second.symm)
-  refine ⟨hasMean_bind_square (fun a => (hk a).square) hp, ?_, ?_⟩
-  · rw [mean_bind_of_square (fun a => (hk a).square) hp]
-    simpa only [(hk _).first] using hm
-  · rw [mean_bind_square (fun a => (hk a).square) hp]
-    simpa only [(hk _).second] using hS
-
-lemma Moments.independent_add {α β : Type*} {p : PMF α} {q : PMF β}
-    {f : α → ℝ} {g : β → ℝ} {m s n t : ℝ}
-    (hf : Moments p f m s) (hg : Moments q g n t) :
-    Moments (p.bind (fun a => q.map (fun b => (a,b))))
-      (fun ab => f ab.1+g ab.2) (m+n) (s+2*m*n+t) := by
-  apply Moments.bind (m := fun a => f a+n) (s := fun a => (f a)^2+2*f a*n+t)
-    (fun a => Moments.map (g := fun b => (a,b)) (f := fun ab => f ab.1+g ab.2) (hg.shift (f a)))
-  · exact (hf.square.add (hf.hasMean.mul_const (2*n))).add (hasMean_const p t) |>.congr
-      (fun a => by ring)
-  · rw [mean_add hf.hasMean (hasMean_const p n), hf.first, mean_const]
-  · have he := mean_add (hf.square.add (hf.hasMean.mul_const (2*n))) (hasMean_const p t)
-    rw [mean_congr p (g := fun a => ((f a)^2+f a*(2*n))+t) (fun a => by ring), he,
-      mean_add hf.square (hf.hasMean.mul_const (2*n)), mean_mul_const, hf.first, hf.second, mean_const]
-    ring
-
-end ThreeState
-namespace ThreeState
-
-lemma hasMean_bounded {α : Type*} (p : PMF α) {f : α → ℝ} (C : ℝ)
-    (hf : ∀ a, |f a| ≤ C) : HasMean p f :=
-  HasMean.of_abs ((hasMean_const p C).mono (fun value => abs_nonneg (f value)) hf)
-
-lemma abs_mean_le {α : Type*} (p : PMF α) {f : α → ℝ} (C : ℝ)
-    (hf : ∀ a, |f a| ≤ C) : |mean p f| ≤ C := by
-  have hi := hasMean_bounded p C hf
-  rw [abs_le]
-  constructor
-  · simpa only [mean_const] using mean_le (hasMean_const p (-C)) hi (fun a => (abs_le.mp (hf a)).1)
-  · simpa only [mean_const] using mean_le hi (hasMean_const p C) (fun a => (abs_le.mp (hf a)).2)
-
-lemma mean_bind_bounded {α β : Type*} (p : PMF α) (k : α → PMF β) (f : β → ℝ)
-    (C : ℝ) (hf : ∀ b, |f b| ≤ C) :
-    mean (p.bind k) f = mean p (fun a => mean (k a) f) := by
-  apply mean_bind (fun a => hasMean_bounded (k a) C hf)
-  exact hasMean_bounded p C (fun a => abs_mean_le (k a) C (fun b => by simpa using hf b))
-
-lemma mean_map_bounded {α β : Type*} (p : PMF α) (g : α → β) (f : β → ℝ)
-    (C : ℝ) (hf : ∀ a, |f (g a)| ≤ C) :
-    mean (p.map g) f = mean p (fun a => f (g a)) :=
-  mean_map (hasMean_bounded p C hf)
-
- 
-noncomputable def joint {α β : Type*} (p : PMF α) (k : α → PMF β) : PMF (α × β) :=
-  p.bind (fun a => (k a).map (fun b => (a,b)))
-
-lemma mean_joint_bounded {α β : Type*} (p : PMF α) (k : α → PMF β) (f : α × β → ℝ)
-    (C : ℝ) (hf : ∀ ab, |f ab| ≤ C) :
-    mean (joint p k) f = mean p (fun a => mean (k a) (fun b => f (a,b))) := by
-  rw [joint, mean_bind_bounded p _ f C hf]
-  apply mean_congr
-  intro a
-  exact mean_map_bounded (k a) _ f C (fun b => hf (a,b))
-
-lemma mean_joint_fst {α β : Type*} (p : PMF α) (k : α → PMF β) (f : α → ℝ)
-    (C : ℝ) (hf : ∀ a, |f a| ≤ C) :
-    mean (joint p k) (fun ab => f ab.1) = mean p f := by
-  rw [mean_joint_bounded p k _ C (fun ab => hf ab.1)]
-  simp only [mean_const]
-
-lemma mean_joint_snd {α β : Type*} (p : PMF α) (k : α → PMF β) (f : β → ℝ)
-    (C : ℝ) (hf : ∀ b, |f b| ≤ C) :
-    mean (joint p k) (fun ab => f ab.2) = mean (p.bind k) f := by
-  rw [mean_joint_bounded p k _ C (fun ab => hf ab.2), mean_bind_bounded p k f C hf]
-
-end ThreeState
-
-namespace ThreeState
-
-lemma HasMean.sub {α : Type*} {p : PMF α} {f g : α → ℝ}
-    (hf : HasMean p f) (hg : HasMean p g) : HasMean p (fun a => f a-g a) := by
-  simpa only [HasMean, mul_sub] using Summable.sub hf hg
-
-lemma mean_sub {α : Type*} {p : PMF α} {f g : α → ℝ}
-    (hf : HasMean p f) (hg : HasMean p g) :
-    mean p (fun a => f a-g a) = mean p f-mean p g := by
-  simp only [mean, mul_sub, hf.tsum_sub hg]
-
-lemma poisson_mass (d : ℝ) (hd : 0 ≤ d) (k : ℕ) :
-    mass (poissonOffspring d hd) k = Real.exp (-d)*d^k/(k.factorial : ℝ) := by
-  change (ENNReal.ofReal _).toReal = _
-  apply ENNReal.toReal_ofReal
-  positivity
-
-lemma poisson_mass_succ (d : ℝ) (hd : 0 ≤ d) (k : ℕ) :
-    mass (poissonOffspring d hd) (k+1) * ((k+1 : ℕ) : ℝ) =
-      d*mass (poissonOffspring d hd) k := by
-  rw [poisson_mass, poisson_mass, Nat.factorial_succ, Nat.cast_mul, pow_succ]
-  have hk : ((k+1 : ℕ) : ℝ) ≠ 0 := by positivity
-  have hf : (k.factorial : ℝ) ≠ 0 := by positivity
-  field_simp
-
-lemma poisson_first_hasSum (d : ℝ) (hd : 0 ≤ d) :
-    HasSum (fun k : ℕ => mass (poissonOffspring d hd) k * (k : ℝ)) d := by
-  have h : HasSum (fun k : ℕ => mass (poissonOffspring d hd) (k+1) * ((k+1 : ℕ) : ℝ)) d := by
-    simpa only [poisson_mass_succ, mass_sum, mul_one] using
-      (mass_summable (poissonOffspring d hd)).hasSum.mul_left d
-  have hz : mass (poissonOffspring d hd) 0*(0:ℝ) = 0 := mul_zero _
-  exact (hasSum_nat_add_iff' 1).mp (by simpa [hz, mass_sum] using h)
-
-lemma poisson_second_factorial_hasSum (d : ℝ) (hd : 0 ≤ d) :
-    HasSum (fun k : ℕ => mass (poissonOffspring d hd) k * ((k : ℝ)*((k : ℝ)-1))) (d^2) := by
-  have hh (k : ℕ) :
-      mass (poissonOffspring d hd) (k+2) * (((k+2 : ℕ) : ℝ)*(((k+2 : ℕ) : ℝ)-1)) =
-        d*(mass (poissonOffspring d hd) (k+1) * ((k+1 : ℕ) : ℝ)) := by
-    have he := poisson_mass_succ d hd (k+1)
-    push_cast at he ⊢
-    nlinarith [congrArg (fun x => x*((k:ℝ)+1)) he]
-  have h : HasSum (fun k : ℕ =>
-      mass (poissonOffspring d hd) (k+2) * (((k+2 : ℕ) : ℝ)*(((k+2 : ℕ) : ℝ)-1))) (d^2) := by
-    have h1 : HasSum (fun k : ℕ => mass (poissonOffspring d hd) (k+1) * ((k+1 : ℕ) : ℝ)) d := by
-      simpa only [poisson_mass_succ, mass_sum, mul_one] using
-        (mass_summable (poissonOffspring d hd)).hasSum.mul_left d
-    simpa only [hh, pow_two] using h1.mul_left d
-  exact (hasSum_nat_add_iff' 2).mp (by simpa [Finset.sum_range_succ] using h)
-
- 
-theorem poisson_moments (d : ℝ) (hd : 0 ≤ d) :
-    Moments (poissonOffspring d hd) (fun k : ℕ => (k : ℝ)) d (d^2+d) := by
-  have h1 := poisson_first_hasSum d hd
-  have h2 := poisson_second_factorial_hasSum d hd
-  have hs : HasSum (fun k : ℕ => mass (poissonOffspring d hd) k*(k:ℝ)^2) (d^2+d) := by
-    convert h2.add h1 using 1
-    funext k
-    ring
-  exact ⟨hs.summable, h1.tsum_eq, hs.tsum_eq⟩
-
-lemma regular_moments (b : ℕ) :
-    Moments (PMF.pure b) (fun k : ℕ => (k : ℝ)) (b:ℝ) ((b:ℝ)^2) :=
-  Moments.pure b _
-
-end ThreeState
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeState.Basic
+
+namespace OAI
+
+/-! Expectations, second moments and the regular and Poisson offspring laws. -/
+
+namespace ThreeState
+open scoped Classical
+
+ 
+noncomputable def mean {α : Type*} (p : PMF α) (f : α → ℝ) : ℝ :=
+  ∑' a, mass p a * f a
+
+def HasMean {α : Type*} (p : PMF α) (f : α → ℝ) : Prop :=
+  Summable (fun a => mass p a * f a)
+
+lemma HasMean.abs {α : Type*} {p : PMF α} {f : α → ℝ} (h : HasMean p f) :
+    HasMean p (fun a => |f a|) := by
+  simpa only [HasMean, abs_mul, abs_of_nonneg (mass_nonneg p _)] using Summable.abs h
+
+lemma HasMean.of_abs {α : Type*} {p : PMF α} {f : α → ℝ}
+    (h : HasMean p (fun a => |f a|)) : HasMean p f := by
+  apply summable_norm_iff.mp
+  simpa only [HasMean, Real.norm_eq_abs, abs_mul, abs_of_nonneg (mass_nonneg p _)] using h
+
+lemma hasMean_const {α : Type*} (p : PMF α) (c : ℝ) : HasMean p (fun _ => c) :=
+  (mass_summable p).mul_right c
+
+lemma mean_const {α : Type*} (p : PMF α) (c : ℝ) : mean p (fun _ => c) = c := by
+  rw [mean, (mass_summable p).tsum_mul_right, mass_sum, one_mul]
+
+lemma HasMean.add {α : Type*} {p : PMF α} {f g : α → ℝ} (hf : HasMean p f) (hg : HasMean p g) :
+    HasMean p (fun a => f a+g a) := by
+  simpa only [HasMean, mul_add] using Summable.add hf hg
+
+lemma mean_add {α : Type*} {p : PMF α} {f g : α → ℝ} (hf : HasMean p f) (hg : HasMean p g) :
+    mean p (fun a => f a+g a) = mean p f+mean p g := by
+  simp only [mean, mul_add, hf.tsum_add hg]
+
+lemma HasMean.mul_const {α : Type*} {p : PMF α} {f : α → ℝ} (hf : HasMean p f) (c : ℝ) :
+    HasMean p (fun a => f a*c) := by
+  simpa only [HasMean, mul_assoc] using hf.mul_right c
+
+lemma mean_mul_const {α : Type*} (p : PMF α) (f : α → ℝ) (c : ℝ) :
+    mean p (fun a => f a*c) = mean p f*c := by
+  simp only [mean, ← mul_assoc, tsum_mul_right]
+
+lemma HasMean.const_mul {α : Type*} {p : PMF α} {f : α → ℝ} (hf : HasMean p f) (c : ℝ) :
+    HasMean p (fun a => c*f a) := by
+  simpa only [mul_comm c] using hf.mul_const c
+
+lemma mean_const_mul {α : Type*} (p : PMF α) (f : α → ℝ) (c : ℝ) :
+    mean p (fun a => c*f a) = c*mean p f := by
+  simp only [mul_comm c, mean_mul_const]
+
+lemma hasMean_of_square {α : Type*} {p : PMF α} {f : α → ℝ}
+    (h : HasMean p (fun a => (f a)^2)) : HasMean p f := by
+  apply HasMean.of_abs
+  apply Summable.of_nonneg_of_le (fun a => mul_nonneg (mass_nonneg p a) (abs_nonneg _))
+    (fun a => mul_le_mul_of_nonneg_left (by nlinarith [sq_nonneg (|f a|-1), sq_abs (f a)])
+      (mass_nonneg p a)) ((hasMean_const p 1).add h)
+
+lemma mass_pure {α : Type*} (a b : α) : mass (PMF.pure a) b = if b=a then 1 else 0 := by
+  classical
+  by_cases h : b=a <;> simp [mass, PMF.pure_apply, h]
+
+lemma mean_pure {α : Type*} (a : α) (f : α → ℝ) : mean (PMF.pure a) f = f a := by
+  classical
+  simp [mean, mass_pure]
+
+lemma hasMean_pure {α : Type*} (a : α) (f : α → ℝ) : HasMean (PMF.pure a) f := by
+  classical
+  simp only [HasMean, mass_pure, ite_mul, one_mul, zero_mul]
+  exact summable_of_ne_finset_zero (s := {a}) (fun b hb => by simp at hb; simp [hb])
+
+lemma mean_nonneg {α : Type*} (p : PMF α) {f : α → ℝ} (hf : ∀ a, 0 ≤ f a) :
+    0 ≤ mean p f := tsum_nonneg (fun a => mul_nonneg (mass_nonneg p a) (hf a))
+
+lemma HasMean.bind {α β : Type*} {p : PMF α} {k : α → PMF β} {f : β → ℝ}
+    (hk : ∀ a, HasMean (k a) f)
+    (hp : HasMean p (fun a => mean (k a) (fun b => |f b|))) :
+    HasMean (p.bind k) f := by
+  have hs : Summable (Function.uncurry (fun a b => mass p a * (mass (k a) b * |f b|))) := by
+    apply (summable_prod_of_nonneg (fun ab =>
+      mul_nonneg (mass_nonneg p ab.1) (mul_nonneg (mass_nonneg _ _) (abs_nonneg _)))).mpr
+    change (∀ a, Summable (fun b => mass p a * (mass (k a) b * |f b|))) ∧
+      Summable (fun a => ∑' b, mass p a * (mass (k a) b * |f b|))
+    constructor
+    · intro a
+      exact (hk a).abs.mul_left (mass p a)
+    · simpa only [tsum_mul_left, mean, HasMean] using hp
+  apply HasMean.of_abs
+  apply hs.prod_symm.prod.congr
+  intro b
+  change (∑' a, mass p a * (mass (k a) b * |f b|)) = mass (p.bind k) b * |f b|
+  rw [mass_bind, ← tsum_mul_right]
+  congr 1
+  funext a
+  ring
+
+lemma mean_bind {α β : Type*} {p : PMF α} {k : α → PMF β} {f : β → ℝ}
+    (hk : ∀ a, HasMean (k a) f)
+    (hp : HasMean p (fun a => mean (k a) (fun b => |f b|))) :
+    mean (p.bind k) f = mean p (fun a => mean (k a) f) := by
+  have hsabs : Summable (Function.uncurry (fun a b => mass p a * (mass (k a) b * |f b|))) := by
+    apply (summable_prod_of_nonneg (fun ab =>
+      mul_nonneg (mass_nonneg p ab.1) (mul_nonneg (mass_nonneg _ _) (abs_nonneg _)))).mpr
+    change (∀ a, Summable (fun b => mass p a * (mass (k a) b * |f b|))) ∧
+      Summable (fun a => ∑' b, mass p a * (mass (k a) b * |f b|))
+    constructor
+    · intro a
+      exact (hk a).abs.mul_left (mass p a)
+    · simpa only [tsum_mul_left, mean, HasMean] using hp
+  have hs : Summable (Function.uncurry (fun a b => mass p a * (mass (k a) b * f b))) := by
+    apply summable_norm_iff.mp
+    apply hsabs.congr
+    rintro ⟨a,b⟩
+    change mass p a * (mass (k a) b * |f b|) = ‖mass p a * (mass (k a) b * f b)‖
+    rw [Real.norm_eq_abs, abs_mul, abs_mul, abs_of_nonneg (mass_nonneg p a),
+      abs_of_nonneg (mass_nonneg (k a) b)]
+  unfold mean
+  simp_rw [mass_bind, ← tsum_mul_right, mul_assoc]
+  rw [hs.tsum_comm]
+  simp only [tsum_mul_left]
+
+lemma HasMean.map {α β : Type*} {p : PMF α} {g : α → β} {f : β → ℝ}
+    (h : HasMean p (fun a => f (g a))) : HasMean (p.map g) f := by
+  change HasMean (p.bind (fun a => PMF.pure (g a))) f
+  apply HasMean.bind (fun a => hasMean_pure _ _)
+  simpa only [mean_pure] using h.abs
+
+lemma mean_map {α β : Type*} {p : PMF α} {g : α → β} {f : β → ℝ}
+    (h : HasMean p (fun a => f (g a))) : mean (p.map g) f = mean p (fun a => f (g a)) := by
+  change mean (p.bind (fun a => PMF.pure (g a))) f = _
+  rw [mean_bind (fun a => hasMean_pure _ _) (by simpa only [mean_pure] using h.abs)]
+  simp only [mean_pure]
+
+end ThreeState
+namespace ThreeState
+open scoped Classical
+
+lemma HasMean.mono {α : Type*} {p : PMF α} {f g : α → ℝ} (hg : HasMean p g)
+    (hf : ∀ a, 0 ≤ f a) (hle : ∀ a, f a ≤ g a) : HasMean p f := by
+  exact Summable.of_nonneg_of_le (fun a => mul_nonneg (mass_nonneg p a) (hf a))
+    (fun a => mul_le_mul_of_nonneg_left (hle a) (mass_nonneg p a)) hg
+
+lemma mean_le {α : Type*} {p : PMF α} {f g : α → ℝ}
+    (hf : HasMean p f) (hg : HasMean p g) (hle : ∀ a, f a ≤ g a) : mean p f ≤ mean p g := by
+  exact Summable.tsum_le_tsum (fun a => mul_le_mul_of_nonneg_left (hle a) (mass_nonneg p a)) hf hg
+
+lemma mean_abs_le_one_add_square {α : Type*} {p : PMF α} {f : α → ℝ}
+    (hf : HasMean p (fun a => (f a)^2)) :
+    mean p (fun a => |f a|) ≤ 1+mean p (fun a => (f a)^2) := by
+  have h := mean_le (hasMean_of_square hf).abs ((hasMean_const p 1).add hf)
+    (fun a => (show |f a| ≤ 1+(f a)^2 by nlinarith [sq_nonneg (|f a|-1), sq_abs (f a)]))
+  simpa only [mean_add (hasMean_const p 1) hf, mean_const] using h
+
+lemma hasMean_bind_square {α β : Type*} {p : PMF α} {k : α → PMF β} {f : β → ℝ}
+    (hk : ∀ a, HasMean (k a) (fun b => (f b)^2))
+    (hp : HasMean p (fun a => mean (k a) (fun b => (f b)^2))) :
+    HasMean (p.bind k) (fun b => (f b)^2) := by
+  apply HasMean.bind hk
+  simpa only [abs_sq] using hp
+
+lemma hasMean_bind_abs_of_square {α β : Type*} {p : PMF α} {k : α → PMF β} {f : β → ℝ}
+    (hk : ∀ a, HasMean (k a) (fun b => (f b)^2))
+    (hp : HasMean p (fun a => mean (k a) (fun b => (f b)^2))) :
+    HasMean p (fun a => mean (k a) (fun b => |f b|)) := by
+  exact ((hasMean_const p 1).add hp).mono
+    (fun a => mean_nonneg _ (fun b => abs_nonneg _))
+    (fun a => mean_abs_le_one_add_square (hk a))
+
+lemma mean_bind_of_square {α β : Type*} {p : PMF α} {k : α → PMF β} {f : β → ℝ}
+    (hk : ∀ a, HasMean (k a) (fun b => (f b)^2))
+    (hp : HasMean p (fun a => mean (k a) (fun b => (f b)^2))) :
+    mean (p.bind k) f = mean p (fun a => mean (k a) f) :=
+  mean_bind (fun a => hasMean_of_square (hk a)) (hasMean_bind_abs_of_square hk hp)
+
+lemma mean_bind_square {α β : Type*} {p : PMF α} {k : α → PMF β} {f : β → ℝ}
+    (hk : ∀ a, HasMean (k a) (fun b => (f b)^2))
+    (hp : HasMean p (fun a => mean (k a) (fun b => (f b)^2))) :
+    mean (p.bind k) (fun b => (f b)^2) = mean p (fun a => mean (k a) (fun b => (f b)^2)) := by
+  apply mean_bind hk
+  simpa only [abs_sq] using hp
+
+lemma HasMean.congr {α : Type*} {p : PMF α} {f g : α → ℝ} (hf : HasMean p f)
+    (hfg : ∀ a, f a = g a) : HasMean p g := by
+  exact Summable.congr hf (fun a => congrArg (fun x => mass p a*x) (hfg a))
+
+lemma mean_congr {α : Type*} (p : PMF α) {f g : α → ℝ} (hfg : ∀ a, f a = g a) :
+    mean p f = mean p g := tsum_congr (fun a => congrArg (fun x => mass p a*x) (hfg a))
+
+lemma hasMean_shift_square {α : Type*} {p : PMF α} {f : α → ℝ}
+    (hf : HasMean p (fun a => (f a)^2)) (c : ℝ) : HasMean p (fun a => (c+f a)^2) := by
+  exact (((hasMean_const p (c^2)).add ((hasMean_of_square hf).const_mul (2*c))).add hf).congr
+    (fun a => by ring)
+
+lemma mean_shift_square {α : Type*} {p : PMF α} {f : α → ℝ}
+    (hf : HasMean p (fun a => (f a)^2)) (c : ℝ) :
+    mean p (fun a => (c+f a)^2) = c^2+2*c*mean p f+mean p (fun a => (f a)^2) := by
+  rw [mean_congr p (g := fun a => c^2+(2*c)*f a+(f a)^2) (fun a => by ring),
+    mean_add ((hasMean_const p (c^2)).add ((hasMean_of_square hf).const_mul (2*c))) hf,
+    mean_add (hasMean_const p (c^2)) ((hasMean_of_square hf).const_mul (2*c)),
+    mean_const, mean_const_mul]
+
+lemma hasMean_fintype {α : Type*} [Fintype α] (p : PMF α) (f : α → ℝ) : HasMean p f :=
+  summable_of_hasFiniteSupport (Set.toFinite _)
+
+lemma mean_fintype {α : Type*} [Fintype α] (p : PMF α) (f : α → ℝ) :
+    mean p f = ∑ a, mass p a * f a := tsum_fintype _
+
+ 
+
+structure Moments {α : Type*} (p : PMF α) (f : α → ℝ) (m s : ℝ) : Prop where
+  square : HasMean p (fun a => (f a)^2)
+  first : mean p f = m
+  second : mean p (fun a => (f a)^2) = s
+
+lemma Moments.hasMean {α : Type*} {p : PMF α} {f : α → ℝ} {m s : ℝ}
+    (h : Moments p f m s) : HasMean p f := hasMean_of_square h.square
+
+lemma Moments.map {α β : Type*} {p : PMF α} {f : β → ℝ} {g : α → β} {m s : ℝ}
+    (h : Moments p (fun a => f (g a)) m s) : Moments (p.map g) f m s := by
+  refine ⟨HasMean.map (g := g) (f := fun b => (f b)^2) h.square, ?_, ?_⟩
+  · exact (mean_map (g := g) (f := f) h.hasMean).trans h.first
+  · exact (mean_map (g := g) (f := fun b => (f b)^2) h.square).trans h.second
+
+lemma Moments.pure {α : Type*} (a : α) (f : α → ℝ) :
+    Moments (PMF.pure a) f (f a) ((f a)^2) :=
+  ⟨hasMean_pure _ _, mean_pure _ _, mean_pure _ _⟩
+
+lemma Moments.shift {α : Type*} {p : PMF α} {f : α → ℝ} {m s : ℝ}
+    (h : Moments p f m s) (c : ℝ) :
+    Moments p (fun a => c+f a) (c+m) (c^2+2*c*m+s) := by
+  refine ⟨hasMean_shift_square h.square c, ?_, ?_⟩
+  · rw [mean_add (hasMean_const p c) h.hasMean, mean_const, h.first]
+  · rw [mean_shift_square h.square, h.first, h.second]
+
+lemma Moments.bind {α β : Type*} {p : PMF α} {k : α → PMF β} {f : β → ℝ}
+    {m s : α → ℝ} {M S : ℝ} (hk : ∀ a, Moments (k a) f (m a) (s a))
+    (hs : HasMean p s) (hm : mean p m = M) (hS : mean p s = S) :
+    Moments (p.bind k) f M S := by
+  have hp : HasMean p (fun a => mean (k a) (fun b => (f b)^2)) :=
+    hs.congr (fun a => (hk a).second.symm)
+  refine ⟨hasMean_bind_square (fun a => (hk a).square) hp, ?_, ?_⟩
+  · rw [mean_bind_of_square (fun a => (hk a).square) hp]
+    simpa only [(hk _).first] using hm
+  · rw [mean_bind_square (fun a => (hk a).square) hp]
+    simpa only [(hk _).second] using hS
+
+lemma Moments.independent_add {α β : Type*} {p : PMF α} {q : PMF β}
+    {f : α → ℝ} {g : β → ℝ} {m s n t : ℝ}
+    (hf : Moments p f m s) (hg : Moments q g n t) :
+    Moments (p.bind (fun a => q.map (fun b => (a,b))))
+      (fun ab => f ab.1+g ab.2) (m+n) (s+2*m*n+t) := by
+  apply Moments.bind (m := fun a => f a+n) (s := fun a => (f a)^2+2*f a*n+t)
+    (fun a => Moments.map (g := fun b => (a,b)) (f := fun ab => f ab.1+g ab.2) (hg.shift (f a)))
+  · exact (hf.square.add (hf.hasMean.mul_const (2*n))).add (hasMean_const p t) |>.congr
+      (fun a => by ring)
+  · rw [mean_add hf.hasMean (hasMean_const p n), hf.first, mean_const]
+  · have he := mean_add (hf.square.add (hf.hasMean.mul_const (2*n))) (hasMean_const p t)
+    rw [mean_congr p (g := fun a => ((f a)^2+f a*(2*n))+t) (fun a => by ring), he,
+      mean_add hf.square (hf.hasMean.mul_const (2*n)), mean_mul_const, hf.first, hf.second, mean_const]
+    ring
+
+end ThreeState
+namespace ThreeState
+
+lemma hasMean_bounded {α : Type*} (p : PMF α) {f : α → ℝ} (C : ℝ)
+    (hf : ∀ a, |f a| ≤ C) : HasMean p f :=
+  HasMean.of_abs ((hasMean_const p C).mono (fun value => abs_nonneg (f value)) hf)
+
+lemma abs_mean_le {α : Type*} (p : PMF α) {f : α → ℝ} (C : ℝ)
+    (hf : ∀ a, |f a| ≤ C) : |mean p f| ≤ C := by
+  have hi := hasMean_bounded p C hf
+  rw [abs_le]
+  constructor
+  · simpa only [mean_const] using mean_le (hasMean_const p (-C)) hi (fun a => (abs_le.mp (hf a)).1)
+  · simpa only [mean_const] using mean_le hi (hasMean_const p C) (fun a => (abs_le.mp (hf a)).2)
+
+lemma mean_bind_bounded {α β : Type*} (p : PMF α) (k : α → PMF β) (f : β → ℝ)
+    (C : ℝ) (hf : ∀ b, |f b| ≤ C) :
+    mean (p.bind k) f = mean p (fun a => mean (k a) f) := by
+  apply mean_bind (fun a => hasMean_bounded (k a) C hf)
+  exact hasMean_bounded p C (fun a => abs_mean_le (k a) C (fun b => by simpa using hf b))
+
+lemma mean_map_bounded {α β : Type*} (p : PMF α) (g : α → β) (f : β → ℝ)
+    (C : ℝ) (hf : ∀ a, |f (g a)| ≤ C) :
+    mean (p.map g) f = mean p (fun a => f (g a)) :=
+  mean_map (hasMean_bounded p C hf)
+
+ 
+noncomputable def joint {α β : Type*} (p : PMF α) (k : α → PMF β) : PMF (α × β) :=
+  p.bind (fun a => (k a).map (fun b => (a,b)))
+
+lemma mean_joint_bounded {α β : Type*} (p : PMF α) (k : α → PMF β) (f : α × β → ℝ)
+    (C : ℝ) (hf : ∀ ab, |f ab| ≤ C) :
+    mean (joint p k) f = mean p (fun a => mean (k a) (fun b => f (a,b))) := by
+  rw [joint, mean_bind_bounded p _ f C hf]
+  apply mean_congr
+  intro a
+  exact mean_map_bounded (k a) _ f C (fun b => hf (a,b))
+
+lemma mean_joint_fst {α β : Type*} (p : PMF α) (k : α → PMF β) (f : α → ℝ)
+    (C : ℝ) (hf : ∀ a, |f a| ≤ C) :
+    mean (joint p k) (fun ab => f ab.1) = mean p f := by
+  rw [mean_joint_bounded p k _ C (fun ab => hf ab.1)]
+  simp only [mean_const]
+
+lemma mean_joint_snd {α β : Type*} (p : PMF α) (k : α → PMF β) (f : β → ℝ)
+    (C : ℝ) (hf : ∀ b, |f b| ≤ C) :
+    mean (joint p k) (fun ab => f ab.2) = mean (p.bind k) f := by
+  rw [mean_joint_bounded p k _ C (fun ab => hf ab.2), mean_bind_bounded p k f C hf]
+
+end ThreeState
+
+namespace ThreeState
+
+lemma HasMean.sub {α : Type*} {p : PMF α} {f g : α → ℝ}
+    (hf : HasMean p f) (hg : HasMean p g) : HasMean p (fun a => f a-g a) := by
+  simpa only [HasMean, mul_sub] using Summable.sub hf hg
+
+lemma mean_sub {α : Type*} {p : PMF α} {f g : α → ℝ}
+    (hf : HasMean p f) (hg : HasMean p g) :
+    mean p (fun a => f a-g a) = mean p f-mean p g := by
+  simp only [mean, mul_sub, hf.tsum_sub hg]
+
+lemma poisson_mass (d : ℝ) (hd : 0 ≤ d) (k : ℕ) :
+    mass (poissonOffspring d hd) k = Real.exp (-d)*d^k/(k.factorial : ℝ) := by
+  change (ENNReal.ofReal _).toReal = _
+  apply ENNReal.toReal_ofReal
+  positivity
+
+lemma poisson_mass_succ (d : ℝ) (hd : 0 ≤ d) (k : ℕ) :
+    mass (poissonOffspring d hd) (k+1) * ((k+1 : ℕ) : ℝ) =
+      d*mass (poissonOffspring d hd) k := by
+  rw [poisson_mass, poisson_mass, Nat.factorial_succ, Nat.cast_mul, pow_succ]
+  have hk : ((k+1 : ℕ) : ℝ) ≠ 0 := by positivity
+  have hf : (k.factorial : ℝ) ≠ 0 := by positivity
+  field_simp
+
+lemma poisson_first_hasSum (d : ℝ) (hd : 0 ≤ d) :
+    HasSum (fun k : ℕ => mass (poissonOffspring d hd) k * (k : ℝ)) d := by
+  have h : HasSum (fun k : ℕ => mass (poissonOffspring d hd) (k+1) * ((k+1 : ℕ) : ℝ)) d := by
+    simpa only [poisson_mass_succ, mass_sum, mul_one] using
+      (mass_summable (poissonOffspring d hd)).hasSum.mul_left d
+  have hz : mass (poissonOffspring d hd) 0*(0:ℝ) = 0 := mul_zero _
+  exact (hasSum_nat_add_iff' 1).mp (by simpa [hz, mass_sum] using h)
+
+lemma poisson_second_factorial_hasSum (d : ℝ) (hd : 0 ≤ d) :
+    HasSum (fun k : ℕ => mass (poissonOffspring d hd) k * ((k : ℝ)*((k : ℝ)-1))) (d^2) := by
+  have hh (k : ℕ) :
+      mass (poissonOffspring d hd) (k+2) * (((k+2 : ℕ) : ℝ)*(((k+2 : ℕ) : ℝ)-1)) =
+        d*(mass (poissonOffspring d hd) (k+1) * ((k+1 : ℕ) : ℝ)) := by
+    have he := poisson_mass_succ d hd (k+1)
+    push_cast at he ⊢
+    nlinarith [congrArg (fun x => x*((k:ℝ)+1)) he]
+  have h : HasSum (fun k : ℕ =>
+      mass (poissonOffspring d hd) (k+2) * (((k+2 : ℕ) : ℝ)*(((k+2 : ℕ) : ℝ)-1))) (d^2) := by
+    have h1 : HasSum (fun k : ℕ => mass (poissonOffspring d hd) (k+1) * ((k+1 : ℕ) : ℝ)) d := by
+      simpa only [poisson_mass_succ, mass_sum, mul_one] using
+        (mass_summable (poissonOffspring d hd)).hasSum.mul_left d
+    simpa only [hh, pow_two] using h1.mul_left d
+  exact (hasSum_nat_add_iff' 2).mp (by simpa [Finset.sum_range_succ] using h)
+
+ 
+theorem poisson_moments (d : ℝ) (hd : 0 ≤ d) :
+    Moments (poissonOffspring d hd) (fun k : ℕ => (k : ℝ)) d (d^2+d) := by
+  have h1 := poisson_first_hasSum d hd
+  have h2 := poisson_second_factorial_hasSum d hd
+  have hs : HasSum (fun k : ℕ => mass (poissonOffspring d hd) k*(k:ℝ)^2) (d^2+d) := by
+    convert h2.add h1 using 1
+    funext k
+    ring
+  exact ⟨hs.summable, h1.tsum_eq, hs.tsum_eq⟩
+
+lemma regular_moments (b : ℕ) :
+    Moments (PMF.pure b) (fun k : ℕ => (k : ℝ)) (b:ℝ) ((b:ℝ)^2) :=
+  Moments.pure b _
+
+end ThreeState
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeState/FixedPoints.lean b/lean/OAI/Probability/ThreeState/FixedPoints.lean
index c59636a8655ef94dea7134f88f54e2b8179a4e66..c2e1671ba202351d807fefd702acff05535127b9 100644
--- a/lean/OAI/Probability/ThreeState/FixedPoints.lean
+++ b/lean/OAI/Probability/ThreeState/FixedPoints.lean
@@ -1,290 +1,290 @@
-import Mathlib
-import OAI.Probability.ThreeState.Update
-
-namespace OAI
-
-/-! Continuity of posterior updates and fixed points of the likelihood law. -/
-
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-
-lemma continuous_productWeight (lam : ℝ) (h : Admissible lam) (n : ℕ) (i : Spin) :
-    Continuous (fun m : Fin n → Message => productWeight lam h m i) := by
-  apply continuous_finsetProd
-  intro j _
-  exact (continuous_apply i).comp
-    (continuous_subtype_val.comp ((continuous_edgeMessage lam h).comp (continuous_apply j)))
-
-lemma continuous_normalizer (lam : ℝ) (h : Admissible lam) (n : ℕ) :
-    Continuous (fun m : Fin n → Message => normalizer lam h m) :=
-  (continuous_finsetSum _ (fun i _ => continuous_productWeight lam h n i)).div_const 3
-
-lemma continuousAt_combineMessage (lam : ℝ) (h : Admissible lam) {n : ℕ}
-    (m : Fin n → Message) (hm : normalizer lam h m ≠ 0) :
-    ContinuousAt (combineMessage lam h) m := by
-  apply tendsto_subtype_rng.mpr
-  apply tendsto_pi_nhds.mpr
-  intro i
-  have hn : ContinuousAt (fun v : Fin n → Message => normalizer lam h v) m :=
-    (continuous_normalizer lam h n).continuousAt
-  have hw : ContinuousAt (fun v : Fin n → Message => productWeight lam h v i) m :=
-    (continuous_productWeight lam h n i).continuousAt
-  have ht := hw.div hn hm
-  have he : ∀ᶠ v in 𝓝 m, normalizer lam h v ≠ 0 := hn.eventually (eventually_ne_nhds hm)
-  have heq : (fun v => productWeight lam h v i / normalizer lam h v) =ᶠ[𝓝 m]
-      (fun v => combineMessage lam h v i) := by
-    filter_upwards [he] with v hv
-    simp only [combineMessage, dite_eq_right hv]
-  have hp : productWeight lam h m i / normalizer lam h m = combineMessage lam h m i := by
-    simp only [combineMessage, dite_eq_right hm]
-  change Tendsto _ (𝓝 m) (𝓝 (productWeight lam h m i / normalizer lam h m)) at ht
-  rw [hp] at ht
-  exact ht.congr' heq
-
- 
-noncomputable def weightedTest (lam : ℝ) (h : Admissible lam) (n : ℕ)
-    (f : C(Message,ℝ)) (m : Fin n → Message) : ℝ :=
-  normalizer lam h m * f (combineMessage lam h m)
-
-lemma weightedTest_bound (lam : ℝ) (h : Admissible lam) (n : ℕ)
-    (f : C(Message,ℝ)) (C : ℝ) (hC : ∀ m, |f m| ≤ C) (m : Fin n → Message) :
-    |weightedTest lam h n f m| ≤ C * normalizer lam h m := by
-  dsimp only [weightedTest]
-  rw [abs_mul, abs_of_nonneg (normalizer_nonneg _ _ _), mul_comm C]
-  exact mul_le_mul_of_nonneg_left (hC _) (normalizer_nonneg _ _ _)
-
-lemma continuous_weightedTest (lam : ℝ) (h : Admissible lam) (n : ℕ)
-    (f : C(Message,ℝ)) : Continuous (weightedTest lam h n f) := by
-  apply continuous_iff_continuousAt.mpr
-  intro m
-  by_cases hm : normalizer lam h m = 0
-  · obtain ⟨C,hC⟩ := isCompact_univ.exists_bound_of_continuousOn f.continuous.continuousOn
-    have hh : ∀ v, |weightedTest lam h n f v| ≤ C * normalizer lam h v :=
-      weightedTest_bound lam h n f C (fun v => by simpa only [Real.norm_eq_abs] using hC v (Set.mem_univ _))
-    have ht : Tendsto (fun v : Fin n → Message => C * normalizer lam h v) (𝓝 m) (𝓝 0) := by
-      simpa only [hm, mul_zero] using
-        (tendsto_const_nhds.mul (continuous_normalizer lam h n).continuousAt :
-          Tendsto (fun v => C * normalizer lam h v) (𝓝 m) (𝓝 (C*normalizer lam h m)))
-    have hz : weightedTest lam h n f m = 0 := by simp [weightedTest, hm]
-    rw [ContinuousAt, hz]
-    exact squeeze_zero_norm (fun v => by simpa only [Real.norm_eq_abs] using hh v) ht
-  · exact (continuous_normalizer lam h n).continuousAt.mul
-      (f.continuous.continuousAt.comp (continuousAt_combineMessage lam h m hm))
-
-noncomputable def weightedTestMap (lam : ℝ) (h : Admissible lam) (n : ℕ)
-    (f : C(Message,ℝ)) : C((Fin n → Message),ℝ) :=
-  ⟨weightedTest lam h n f, continuous_weightedTest lam h n f⟩
-
-noncomputable def degreeTest (lam : ℝ) (h : Admissible lam) (n : ℕ)
-    (f : C(Message,ℝ)) (Q : ProbabilityMeasure Message) : ℝ :=
-  ∫ m, weightedTest lam h n f m ∂(iidMeasure Q n : Measure (Fin n → Message))
-
-lemma continuous_degreeTest (lam : ℝ) (h : Admissible lam) (n : ℕ)
-    (f : C(Message,ℝ)) : Continuous (degreeTest lam h n f) :=
-  (ProbabilityMeasure.continuous_integral_continuousMap (weightedTestMap lam h n f)).comp
-    (continuous_iidMeasure n)
-
-end ThreeState
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-
-lemma mean_sigmaPMF {ι : Type*} {α : ι → Type*} (p : PMF ι) (q : ∀ i, PMF (α i))
-    (f : Sigma α → ℝ) (hf : HasMean (sigmaPMF p q) f) :
-    mean (sigmaPMF p q) f = mean p (fun i => mean (q i) (fun a => f ⟨i,a⟩)) := by
-  change (∑' x, mass (sigmaPMF p q) x * f x) = _
-  rw [hf.tsum_sigma]
-  unfold mean
-  apply tsum_congr
-  intro i
-  simp only [mass_sigmaPMF, mul_assoc, tsum_mul_left]
-
-lemma degreeTest_likelihood {α : Type*} (law : Spin → PMF α) (lam : ℝ)
-    (h : Admissible lam) (n : ℕ) (f : C(Message,ℝ)) :
-    degreeTest lam h n f (likelihoodLaw law) =
-      mean (iidVector (marginal law) n) (fun v =>
-        normalizer lam h (fun j => likelihood law (v j)) *
-          f (combineMessage lam h (fun j => likelihood law (v j)))) :=
-  integral_iid_likelihood law n (weightedTestMap lam h n f)
-
- 
-
-lemma test_compose_identity {α : Type*} (offspring : PMF ℕ) (law : Spin → PMF α)
-    (lam : ℝ) (h : Admissible lam) (f : C(Message,ℝ)) :
-    ∫ m, f m ∂(likelihoodLaw (composeLaw offspring law lam h) : Measure Message) =
-      mean offspring (fun n => degreeTest lam h n f (likelihoodLaw law)) := by
-  rw [integral_likelihoodLaw, mean_test_compose]
-  obtain ⟨C,hC⟩ := isCompact_univ.exists_bound_of_continuousOn f.continuous.continuousOn
-  have hi := hasMean_weighted_compose offspring law lam h f C
-    (fun m => by simpa only [Real.norm_eq_abs] using hC m (Set.mem_univ _))
-  rw [productReference, mean_sigmaPMF _ _ _ hi]
-  apply mean_congr
-  intro n
-  exact (degreeTest_likelihood law lam h n f).symm
-
-noncomputable def coordinateMap (i : Spin) : C(Message,ℝ) :=
-  ⟨fun m => m i, (continuous_apply i).comp continuous_subtype_val⟩
-
- 
-def Balanced (Q : ProbabilityMeasure Message) : Prop :=
-  ∀ i : Spin, ∫ m, m i ∂(Q : Measure Message) = 1
-
-lemma balanced_likelihoodLaw {α : Type*} (law : Spin → PMF α) : Balanced (likelihoodLaw law) := by
-  intro i
-  change (∫ m, coordinateMap i m ∂(likelihoodLaw law : Measure Message)) = 1
-  rw [integral_likelihoodLaw]
-  exact mean_likelihood law i
-
-lemma balanced_limit {Qn : ℕ → ProbabilityMeasure Message} {Q : ProbabilityMeasure Message}
-    (hn : ∀ n, Balanced (Qn n)) (hlim : Tendsto Qn atTop (𝓝 Q)) : Balanced Q := by
-  intro i
-  have hh := ((ProbabilityMeasure.continuous_integral_continuousMap (coordinateMap i)).tendsto Q).comp hlim
-  have he : (fun n => ∫ m, coordinateMap i m ∂(Qn n : Measure Message)) = (fun _ : ℕ => (1:ℝ)) :=
-    funext (fun n => hn n i)
-  simp only [Function.comp_def, he] at hh
-  exact tendsto_nhds_unique hh tendsto_const_nhds
-
-lemma integral_edgeMessage (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message)
-    (hQ : Balanced Q) (i : Spin) : ∫ m, edgeMessage lam h m i ∂(Q : Measure Message) = 1 := by
-  have hi : Integrable (fun m : Message => m i) (Q : Measure Message) :=
-    (coordinateMap i).continuous.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
-  change (∫ m, 1+lam*(m i-1) ∂(Q : Measure Message)) = 1
-  have hs : Integrable (fun m : Message => m i-1) (Q : Measure Message) :=
-    hi.sub (integrable_const (1:ℝ))
-  have ht : Integrable (fun m : Message => lam*(m i-1)) (Q : Measure Message) := hs.const_mul lam
-  rw [integral_add (integrable_const (1:ℝ)) ht, integral_const_mul,
-    integral_sub hi (integrable_const (1:ℝ)), integral_const, hQ i]
-  simp
-
-lemma integral_productWeight (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message)
-    (hQ : Balanced Q) (n : ℕ) (i : Spin) :
-    ∫ m, productWeight lam h m i ∂(iidMeasure Q n : Measure (Fin n → Message)) = 1 := by
-  change (∫ m : Fin n → Message, ∏ j, edgeMessage lam h (m j) i
-    ∂Measure.pi (fun _ : Fin n => (Q : Measure Message))) = 1
-  rw [integral_fintype_prod_eq_prod (fun (_ : Fin n) (m : Message) => edgeMessage lam h m i)]
-  simp [integral_edgeMessage lam h Q hQ i]
-
-lemma integral_normalizer (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message)
-    (hQ : Balanced Q) (n : ℕ) :
-    ∫ m, normalizer lam h m ∂(iidMeasure Q n : Measure (Fin n → Message)) = 1 := by
-  change (∫ m, (∑ i, productWeight lam h m i)/3 ∂(iidMeasure Q n : Measure (Fin n → Message))) = 1
-  rw [integral_div, integral_finsetSum]
-  · simp [integral_productWeight lam h Q hQ n]
-  · intro i _
-    exact (continuous_productWeight lam h n i).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
-
-lemma degreeTest_bound (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message)
-    (hQ : Balanced Q) (n : ℕ) (f : C(Message,ℝ)) (C : ℝ) (hC : ∀ m, |f m| ≤ C) :
-    |degreeTest lam h n f Q| ≤ C := by
-  have hi : Integrable (fun m : Fin n → Message => |weightedTest lam h n f m|)
-      (iidMeasure Q n : Measure (Fin n → Message)) :=
-    (continuous_weightedTest lam h n f).abs.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
-  have hj : Integrable (fun m : Fin n → Message => C*normalizer lam h m)
-      (iidMeasure Q n : Measure (Fin n → Message)) :=
-    (continuous_const.mul (continuous_normalizer lam h n)).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
-  calc
-    |degreeTest lam h n f Q| ≤ ∫ m, |weightedTest lam h n f m| ∂(iidMeasure Q n : Measure (Fin n → Message)) :=
-      abs_integral_le_integral_abs
-    _ ≤ ∫ m, C*normalizer lam h m ∂(iidMeasure Q n : Measure (Fin n → Message)) :=
-      integral_mono hi hj (weightedTest_bound lam h n f C hC)
-    _ = C := by rw [integral_const_mul, integral_normalizer lam h Q hQ n, mul_one]
-
-lemma mean_degreeTest_tendsto (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
-    (f : C(Message,ℝ)) {Qn : ℕ → ProbabilityMeasure Message} {Q : ProbabilityMeasure Message}
-    (hn : ∀ n, Balanced (Qn n)) (hlim : Tendsto Qn atTop (𝓝 Q)) :
-    Tendsto (fun k => mean offspring (fun n => degreeTest lam h n f (Qn k))) atTop
-      (𝓝 (mean offspring (fun n => degreeTest lam h n f Q))) := by
-  obtain ⟨C,hC⟩ := isCompact_univ.exists_bound_of_continuousOn f.continuous.continuousOn
-  apply tendsto_tsum_of_dominated_convergence ((mass_summable offspring).mul_right C)
-  · intro n
-    exact tendsto_const_nhds.mul (((continuous_degreeTest lam h n f).tendsto Q).comp hlim)
-  · apply Eventually.of_forall
-    intro k n
-    rw [Real.norm_eq_abs, abs_mul, abs_of_nonneg (mass_nonneg offspring n)]
-    exact mul_le_mul_of_nonneg_left (degreeTest_bound lam h (Qn k) (hn k) n f C
-      (fun m => by simpa only [Real.norm_eq_abs] using hC m (Set.mem_univ _))) (mass_nonneg offspring n)
-
-end ThreeState
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-
- 
-def IsPosteriorFixedPoint (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
-    (Q : ProbabilityMeasure Message) : Prop :=
-  Balanced Q ∧ ∀ f : C(Message,ℝ), ∫ m, f m ∂(Q : Measure Message) =
-    mean offspring (fun n => degreeTest lam h n f Q)
-
-lemma orderedLaw_succ (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) (n : ℕ) :
-    orderedLaw offspring lam h (n+1) = composeLaw offspring (orderedLaw offspring lam h n) lam h := rfl
-
-lemma ordered_exists_fixedpoint (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) :
-    ∃ Q : ProbabilityMeasure Message, IsPosteriorFixedPoint offspring lam h Q ∧
-      ∃ φ : ℕ → ℕ, StrictMono φ ∧
-        Tendsto (fun n => likelihoodLaw (orderedLaw offspring lam h (φ n))) atTop (𝓝 Q) := by
-  obtain ⟨Q,φ,hφ,hleft,hright⟩ := ordered_likelihood_common_subsequence offspring lam h
-  refine ⟨Q, ⟨balanced_limit (fun n => balanced_likelihoodLaw _) hleft, ?_⟩,φ,hφ,hleft⟩
-  intro f
-  have hl := ((ProbabilityMeasure.continuous_integral_continuousMap f).tendsto Q).comp hright
-  have hr := mean_degreeTest_tendsto offspring lam h f
-    (fun n => balanced_likelihoodLaw (orderedLaw offspring lam h (φ n))) hleft
-  have he : (fun n => ∫ m, f m ∂(likelihoodLaw (orderedLaw offspring lam h (φ n+1)) : Measure Message)) =
-      (fun n => mean offspring (fun k => degreeTest lam h k f (likelihoodLaw (orderedLaw offspring lam h (φ n))))) := by
-    funext n
-    rw [orderedLaw_succ, test_compose_identity]
-  simp only [Function.comp_def, he] at hl
-  exact tendsto_nhds_unique hl hr
-
-noncomputable def messageTV (m : Message) : ℝ := (∑ i : Spin, |m i-1|)/6
-
-lemma continuous_messageTV : Continuous messageTV := by
-  apply Continuous.div_const
-  apply continuous_finsetSum
-  intro i _
-  exact (((continuous_apply i).comp continuous_subtype_val).sub continuous_const).abs
-
-noncomputable def messageTVMap : C(Message,ℝ) := ⟨messageTV, continuous_messageTV⟩
-
-lemma posterior_tv_likelihood {α : Type*} (law : Spin → PMF α) (y : α) :
-    (∑ i : Spin, |posterior law y i - 1/3|)/2 = messageTV (likelihood law y) := by
-  have he (i : Spin) : |likelihood law y i - 1| = 3 * |posterior law y i - 1/3| := by
-    rw [likelihood_apply]
-    have hr : 3*posterior law y i-1 = 3*(posterior law y i-1/3) := by ring
-    rw [hr, abs_mul, abs_of_nonneg (by norm_num : (0:ℝ) ≤ 3)]
-  simp only [messageTV, he, ← Finset.mul_sum]
-  ring
-
-lemma integral_messageTV_likelihood {α : Type*} (law : Spin → PMF α) :
-    ∫ m, messageTV m ∂(likelihoodLaw law : Measure Message) = advantage law := by
-  change (∫ m, messageTVMap m ∂(likelihoodLaw law : Measure Message)) = _
-  rw [integral_likelihoodLaw]
-  unfold mean advantage
-  apply tsum_congr
-  intro y
-  rw [posterior_tv_likelihood]
-  rfl
-
-lemma advantage_tendsto_of_likelihood {α : ℕ → Type*} (law : ∀ n, Spin → PMF (α n))
-    {Q : ProbabilityMeasure Message} (hlim : Tendsto (fun n => likelihoodLaw (law n)) atTop (𝓝 Q)) :
-    Tendsto (fun n => advantage (law n)) atTop (𝓝 (∫ m, messageTV m ∂(Q : Measure Message))) := by
-  have hh := ((ProbabilityMeasure.continuous_integral_continuousMap messageTVMap).tendsto Q).comp hlim
-  simpa only [Function.comp_def, messageTVMap, ContinuousMap.coe_mk, integral_messageTV_likelihood] using hh
-
- 
-
-lemma exists_fixedpoint_advantage (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) :
-    ∃ L : ℝ, ∃ Q : ProbabilityMeasure Message,
-      0 ≤ L ∧ Tendsto (fun n => advantage (observedLaw offspring lam h n)) atTop (𝓝 L) ∧
-      IsPosteriorFixedPoint offspring lam h Q ∧
-      L ≤ ∫ m, messageTV m ∂(Q : Measure Message) := by
-  obtain ⟨Q,hQ,φ,hφ,hlim⟩ := ordered_exists_fixedpoint offspring lam h
-  have htop := advantage_tendsto_of_likelihood (fun n => orderedLaw offspring lam h (φ n)) hlim
-  obtain ⟨L,hL,_hL1,hactual⟩ := observedAdvantage_tendsto offspring lam h
-  refine ⟨L,Q,hL,hactual,hQ,?_⟩
-  exact le_of_tendsto_of_tendsto (hactual.comp hφ.tendsto_atTop) htop
-    (Eventually.of_forall (fun n => orderedAdvantage_dominates offspring lam h (φ n)))
-
-end ThreeState
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeState.Update
+
+namespace OAI
+
+/-! Continuity of posterior updates and fixed points of the likelihood law. -/
+
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+
+lemma continuous_productWeight (lam : ℝ) (h : Admissible lam) (n : ℕ) (i : Spin) :
+    Continuous (fun m : Fin n → Message => productWeight lam h m i) := by
+  apply continuous_finsetProd
+  intro j _
+  exact (continuous_apply i).comp
+    (continuous_subtype_val.comp ((continuous_edgeMessage lam h).comp (continuous_apply j)))
+
+lemma continuous_normalizer (lam : ℝ) (h : Admissible lam) (n : ℕ) :
+    Continuous (fun m : Fin n → Message => normalizer lam h m) :=
+  (continuous_finsetSum _ (fun i _ => continuous_productWeight lam h n i)).div_const 3
+
+lemma continuousAt_combineMessage (lam : ℝ) (h : Admissible lam) {n : ℕ}
+    (m : Fin n → Message) (hm : normalizer lam h m ≠ 0) :
+    ContinuousAt (combineMessage lam h) m := by
+  apply tendsto_subtype_rng.mpr
+  apply tendsto_pi_nhds.mpr
+  intro i
+  have hn : ContinuousAt (fun v : Fin n → Message => normalizer lam h v) m :=
+    (continuous_normalizer lam h n).continuousAt
+  have hw : ContinuousAt (fun v : Fin n → Message => productWeight lam h v i) m :=
+    (continuous_productWeight lam h n i).continuousAt
+  have ht := hw.div hn hm
+  have he : ∀ᶠ v in 𝓝 m, normalizer lam h v ≠ 0 := hn.eventually (eventually_ne_nhds hm)
+  have heq : (fun v => productWeight lam h v i / normalizer lam h v) =ᶠ[𝓝 m]
+      (fun v => combineMessage lam h v i) := by
+    filter_upwards [he] with v hv
+    simp only [combineMessage, dite_eq_right hv]
+  have hp : productWeight lam h m i / normalizer lam h m = combineMessage lam h m i := by
+    simp only [combineMessage, dite_eq_right hm]
+  change Tendsto _ (𝓝 m) (𝓝 (productWeight lam h m i / normalizer lam h m)) at ht
+  rw [hp] at ht
+  exact ht.congr' heq
+
+ 
+noncomputable def weightedTest (lam : ℝ) (h : Admissible lam) (n : ℕ)
+    (f : C(Message,ℝ)) (m : Fin n → Message) : ℝ :=
+  normalizer lam h m * f (combineMessage lam h m)
+
+lemma weightedTest_bound (lam : ℝ) (h : Admissible lam) (n : ℕ)
+    (f : C(Message,ℝ)) (C : ℝ) (hC : ∀ m, |f m| ≤ C) (m : Fin n → Message) :
+    |weightedTest lam h n f m| ≤ C * normalizer lam h m := by
+  dsimp only [weightedTest]
+  rw [abs_mul, abs_of_nonneg (normalizer_nonneg _ _ _), mul_comm C]
+  exact mul_le_mul_of_nonneg_left (hC _) (normalizer_nonneg _ _ _)
+
+lemma continuous_weightedTest (lam : ℝ) (h : Admissible lam) (n : ℕ)
+    (f : C(Message,ℝ)) : Continuous (weightedTest lam h n f) := by
+  apply continuous_iff_continuousAt.mpr
+  intro m
+  by_cases hm : normalizer lam h m = 0
+  · obtain ⟨C,hC⟩ := isCompact_univ.exists_bound_of_continuousOn f.continuous.continuousOn
+    have hh : ∀ v, |weightedTest lam h n f v| ≤ C * normalizer lam h v :=
+      weightedTest_bound lam h n f C (fun v => by simpa only [Real.norm_eq_abs] using hC v (Set.mem_univ _))
+    have ht : Tendsto (fun v : Fin n → Message => C * normalizer lam h v) (𝓝 m) (𝓝 0) := by
+      simpa only [hm, mul_zero] using
+        (tendsto_const_nhds.mul (continuous_normalizer lam h n).continuousAt :
+          Tendsto (fun v => C * normalizer lam h v) (𝓝 m) (𝓝 (C*normalizer lam h m)))
+    have hz : weightedTest lam h n f m = 0 := by simp [weightedTest, hm]
+    rw [ContinuousAt, hz]
+    exact squeeze_zero_norm (fun v => by simpa only [Real.norm_eq_abs] using hh v) ht
+  · exact (continuous_normalizer lam h n).continuousAt.mul
+      (f.continuous.continuousAt.comp (continuousAt_combineMessage lam h m hm))
+
+noncomputable def weightedTestMap (lam : ℝ) (h : Admissible lam) (n : ℕ)
+    (f : C(Message,ℝ)) : C((Fin n → Message),ℝ) :=
+  ⟨weightedTest lam h n f, continuous_weightedTest lam h n f⟩
+
+noncomputable def degreeTest (lam : ℝ) (h : Admissible lam) (n : ℕ)
+    (f : C(Message,ℝ)) (Q : ProbabilityMeasure Message) : ℝ :=
+  ∫ m, weightedTest lam h n f m ∂(iidMeasure Q n : Measure (Fin n → Message))
+
+lemma continuous_degreeTest (lam : ℝ) (h : Admissible lam) (n : ℕ)
+    (f : C(Message,ℝ)) : Continuous (degreeTest lam h n f) :=
+  (ProbabilityMeasure.continuous_integral_continuousMap (weightedTestMap lam h n f)).comp
+    (continuous_iidMeasure n)
+
+end ThreeState
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+
+lemma mean_sigmaPMF {ι : Type*} {α : ι → Type*} (p : PMF ι) (q : ∀ i, PMF (α i))
+    (f : Sigma α → ℝ) (hf : HasMean (sigmaPMF p q) f) :
+    mean (sigmaPMF p q) f = mean p (fun i => mean (q i) (fun a => f ⟨i,a⟩)) := by
+  change (∑' x, mass (sigmaPMF p q) x * f x) = _
+  rw [hf.tsum_sigma]
+  unfold mean
+  apply tsum_congr
+  intro i
+  simp only [mass_sigmaPMF, mul_assoc, tsum_mul_left]
+
+lemma degreeTest_likelihood {α : Type*} (law : Spin → PMF α) (lam : ℝ)
+    (h : Admissible lam) (n : ℕ) (f : C(Message,ℝ)) :
+    degreeTest lam h n f (likelihoodLaw law) =
+      mean (iidVector (marginal law) n) (fun v =>
+        normalizer lam h (fun j => likelihood law (v j)) *
+          f (combineMessage lam h (fun j => likelihood law (v j)))) :=
+  integral_iid_likelihood law n (weightedTestMap lam h n f)
+
+ 
+
+lemma test_compose_identity {α : Type*} (offspring : PMF ℕ) (law : Spin → PMF α)
+    (lam : ℝ) (h : Admissible lam) (f : C(Message,ℝ)) :
+    ∫ m, f m ∂(likelihoodLaw (composeLaw offspring law lam h) : Measure Message) =
+      mean offspring (fun n => degreeTest lam h n f (likelihoodLaw law)) := by
+  rw [integral_likelihoodLaw, mean_test_compose]
+  obtain ⟨C,hC⟩ := isCompact_univ.exists_bound_of_continuousOn f.continuous.continuousOn
+  have hi := hasMean_weighted_compose offspring law lam h f C
+    (fun m => by simpa only [Real.norm_eq_abs] using hC m (Set.mem_univ _))
+  rw [productReference, mean_sigmaPMF _ _ _ hi]
+  apply mean_congr
+  intro n
+  exact (degreeTest_likelihood law lam h n f).symm
+
+noncomputable def coordinateMap (i : Spin) : C(Message,ℝ) :=
+  ⟨fun m => m i, (continuous_apply i).comp continuous_subtype_val⟩
+
+ 
+def Balanced (Q : ProbabilityMeasure Message) : Prop :=
+  ∀ i : Spin, ∫ m, m i ∂(Q : Measure Message) = 1
+
+lemma balanced_likelihoodLaw {α : Type*} (law : Spin → PMF α) : Balanced (likelihoodLaw law) := by
+  intro i
+  change (∫ m, coordinateMap i m ∂(likelihoodLaw law : Measure Message)) = 1
+  rw [integral_likelihoodLaw]
+  exact mean_likelihood law i
+
+lemma balanced_limit {Qn : ℕ → ProbabilityMeasure Message} {Q : ProbabilityMeasure Message}
+    (hn : ∀ n, Balanced (Qn n)) (hlim : Tendsto Qn atTop (𝓝 Q)) : Balanced Q := by
+  intro i
+  have hh := ((ProbabilityMeasure.continuous_integral_continuousMap (coordinateMap i)).tendsto Q).comp hlim
+  have he : (fun n => ∫ m, coordinateMap i m ∂(Qn n : Measure Message)) = (fun _ : ℕ => (1:ℝ)) :=
+    funext (fun n => hn n i)
+  simp only [Function.comp_def, he] at hh
+  exact tendsto_nhds_unique hh tendsto_const_nhds
+
+lemma integral_edgeMessage (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message)
+    (hQ : Balanced Q) (i : Spin) : ∫ m, edgeMessage lam h m i ∂(Q : Measure Message) = 1 := by
+  have hi : Integrable (fun m : Message => m i) (Q : Measure Message) :=
+    (coordinateMap i).continuous.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
+  change (∫ m, 1+lam*(m i-1) ∂(Q : Measure Message)) = 1
+  have hs : Integrable (fun m : Message => m i-1) (Q : Measure Message) :=
+    hi.sub (integrable_const (1:ℝ))
+  have ht : Integrable (fun m : Message => lam*(m i-1)) (Q : Measure Message) := hs.const_mul lam
+  rw [integral_add (integrable_const (1:ℝ)) ht, integral_const_mul,
+    integral_sub hi (integrable_const (1:ℝ)), integral_const, hQ i]
+  simp
+
+lemma integral_productWeight (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message)
+    (hQ : Balanced Q) (n : ℕ) (i : Spin) :
+    ∫ m, productWeight lam h m i ∂(iidMeasure Q n : Measure (Fin n → Message)) = 1 := by
+  change (∫ m : Fin n → Message, ∏ j, edgeMessage lam h (m j) i
+    ∂Measure.pi (fun _ : Fin n => (Q : Measure Message))) = 1
+  rw [integral_fintype_prod_eq_prod (fun (_ : Fin n) (m : Message) => edgeMessage lam h m i)]
+  simp [integral_edgeMessage lam h Q hQ i]
+
+lemma integral_normalizer (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message)
+    (hQ : Balanced Q) (n : ℕ) :
+    ∫ m, normalizer lam h m ∂(iidMeasure Q n : Measure (Fin n → Message)) = 1 := by
+  change (∫ m, (∑ i, productWeight lam h m i)/3 ∂(iidMeasure Q n : Measure (Fin n → Message))) = 1
+  rw [integral_div, integral_finsetSum]
+  · simp [integral_productWeight lam h Q hQ n]
+  · intro i _
+    exact (continuous_productWeight lam h n i).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
+
+lemma degreeTest_bound (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message)
+    (hQ : Balanced Q) (n : ℕ) (f : C(Message,ℝ)) (C : ℝ) (hC : ∀ m, |f m| ≤ C) :
+    |degreeTest lam h n f Q| ≤ C := by
+  have hi : Integrable (fun m : Fin n → Message => |weightedTest lam h n f m|)
+      (iidMeasure Q n : Measure (Fin n → Message)) :=
+    (continuous_weightedTest lam h n f).abs.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
+  have hj : Integrable (fun m : Fin n → Message => C*normalizer lam h m)
+      (iidMeasure Q n : Measure (Fin n → Message)) :=
+    (continuous_const.mul (continuous_normalizer lam h n)).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
+  calc
+    |degreeTest lam h n f Q| ≤ ∫ m, |weightedTest lam h n f m| ∂(iidMeasure Q n : Measure (Fin n → Message)) :=
+      abs_integral_le_integral_abs
+    _ ≤ ∫ m, C*normalizer lam h m ∂(iidMeasure Q n : Measure (Fin n → Message)) :=
+      integral_mono hi hj (weightedTest_bound lam h n f C hC)
+    _ = C := by rw [integral_const_mul, integral_normalizer lam h Q hQ n, mul_one]
+
+lemma mean_degreeTest_tendsto (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
+    (f : C(Message,ℝ)) {Qn : ℕ → ProbabilityMeasure Message} {Q : ProbabilityMeasure Message}
+    (hn : ∀ n, Balanced (Qn n)) (hlim : Tendsto Qn atTop (𝓝 Q)) :
+    Tendsto (fun k => mean offspring (fun n => degreeTest lam h n f (Qn k))) atTop
+      (𝓝 (mean offspring (fun n => degreeTest lam h n f Q))) := by
+  obtain ⟨C,hC⟩ := isCompact_univ.exists_bound_of_continuousOn f.continuous.continuousOn
+  apply tendsto_tsum_of_dominated_convergence ((mass_summable offspring).mul_right C)
+  · intro n
+    exact tendsto_const_nhds.mul (((continuous_degreeTest lam h n f).tendsto Q).comp hlim)
+  · apply Eventually.of_forall
+    intro k n
+    rw [Real.norm_eq_abs, abs_mul, abs_of_nonneg (mass_nonneg offspring n)]
+    exact mul_le_mul_of_nonneg_left (degreeTest_bound lam h (Qn k) (hn k) n f C
+      (fun m => by simpa only [Real.norm_eq_abs] using hC m (Set.mem_univ _))) (mass_nonneg offspring n)
+
+end ThreeState
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+
+ 
+def IsPosteriorFixedPoint (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
+    (Q : ProbabilityMeasure Message) : Prop :=
+  Balanced Q ∧ ∀ f : C(Message,ℝ), ∫ m, f m ∂(Q : Measure Message) =
+    mean offspring (fun n => degreeTest lam h n f Q)
+
+lemma orderedLaw_succ (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) (n : ℕ) :
+    orderedLaw offspring lam h (n+1) = composeLaw offspring (orderedLaw offspring lam h n) lam h := rfl
+
+lemma ordered_exists_fixedpoint (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) :
+    ∃ Q : ProbabilityMeasure Message, IsPosteriorFixedPoint offspring lam h Q ∧
+      ∃ φ : ℕ → ℕ, StrictMono φ ∧
+        Tendsto (fun n => likelihoodLaw (orderedLaw offspring lam h (φ n))) atTop (𝓝 Q) := by
+  obtain ⟨Q,φ,hφ,hleft,hright⟩ := ordered_likelihood_common_subsequence offspring lam h
+  refine ⟨Q, ⟨balanced_limit (fun n => balanced_likelihoodLaw _) hleft, ?_⟩,φ,hφ,hleft⟩
+  intro f
+  have hl := ((ProbabilityMeasure.continuous_integral_continuousMap f).tendsto Q).comp hright
+  have hr := mean_degreeTest_tendsto offspring lam h f
+    (fun n => balanced_likelihoodLaw (orderedLaw offspring lam h (φ n))) hleft
+  have he : (fun n => ∫ m, f m ∂(likelihoodLaw (orderedLaw offspring lam h (φ n+1)) : Measure Message)) =
+      (fun n => mean offspring (fun k => degreeTest lam h k f (likelihoodLaw (orderedLaw offspring lam h (φ n))))) := by
+    funext n
+    rw [orderedLaw_succ, test_compose_identity]
+  simp only [Function.comp_def, he] at hl
+  exact tendsto_nhds_unique hl hr
+
+noncomputable def messageTV (m : Message) : ℝ := (∑ i : Spin, |m i-1|)/6
+
+lemma continuous_messageTV : Continuous messageTV := by
+  apply Continuous.div_const
+  apply continuous_finsetSum
+  intro i _
+  exact (((continuous_apply i).comp continuous_subtype_val).sub continuous_const).abs
+
+noncomputable def messageTVMap : C(Message,ℝ) := ⟨messageTV, continuous_messageTV⟩
+
+lemma posterior_tv_likelihood {α : Type*} (law : Spin → PMF α) (y : α) :
+    (∑ i : Spin, |posterior law y i - 1/3|)/2 = messageTV (likelihood law y) := by
+  have he (i : Spin) : |likelihood law y i - 1| = 3 * |posterior law y i - 1/3| := by
+    rw [likelihood_apply]
+    have hr : 3*posterior law y i-1 = 3*(posterior law y i-1/3) := by ring
+    rw [hr, abs_mul, abs_of_nonneg (by norm_num : (0:ℝ) ≤ 3)]
+  simp only [messageTV, he, ← Finset.mul_sum]
+  ring
+
+lemma integral_messageTV_likelihood {α : Type*} (law : Spin → PMF α) :
+    ∫ m, messageTV m ∂(likelihoodLaw law : Measure Message) = advantage law := by
+  change (∫ m, messageTVMap m ∂(likelihoodLaw law : Measure Message)) = _
+  rw [integral_likelihoodLaw]
+  unfold mean advantage
+  apply tsum_congr
+  intro y
+  rw [posterior_tv_likelihood]
+  rfl
+
+lemma advantage_tendsto_of_likelihood {α : ℕ → Type*} (law : ∀ n, Spin → PMF (α n))
+    {Q : ProbabilityMeasure Message} (hlim : Tendsto (fun n => likelihoodLaw (law n)) atTop (𝓝 Q)) :
+    Tendsto (fun n => advantage (law n)) atTop (𝓝 (∫ m, messageTV m ∂(Q : Measure Message))) := by
+  have hh := ((ProbabilityMeasure.continuous_integral_continuousMap messageTVMap).tendsto Q).comp hlim
+  simpa only [Function.comp_def, messageTVMap, ContinuousMap.coe_mk, integral_messageTV_likelihood] using hh
+
+ 
+
+lemma exists_fixedpoint_advantage (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) :
+    ∃ L : ℝ, ∃ Q : ProbabilityMeasure Message,
+      0 ≤ L ∧ Tendsto (fun n => advantage (observedLaw offspring lam h n)) atTop (𝓝 L) ∧
+      IsPosteriorFixedPoint offspring lam h Q ∧
+      L ≤ ∫ m, messageTV m ∂(Q : Measure Message) := by
+  obtain ⟨Q,hQ,φ,hφ,hlim⟩ := ordered_exists_fixedpoint offspring lam h
+  have htop := advantage_tendsto_of_likelihood (fun n => orderedLaw offspring lam h (φ n)) hlim
+  obtain ⟨L,hL,_hL1,hactual⟩ := observedAdvantage_tendsto offspring lam h
+  refine ⟨L,Q,hL,hactual,hQ,?_⟩
+  exact le_of_tendsto_of_tendsto (hactual.comp hφ.tendsto_atTop) htop
+    (Eventually.of_forall (fun n => orderedAdvantage_dominates offspring lam h (φ n)))
+
+end ThreeState
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeState/Main.lean b/lean/OAI/Probability/ThreeState/Main.lean
index cbf89478d95464b86817e4fc4edf2f53b04a9a73..fcdfb5f0cacb148882249426c169b38b92729ac7 100644
--- a/lean/OAI/Probability/ThreeState/Main.lean
+++ b/lean/OAI/Probability/ThreeState/Main.lean
@@ -1 +1 @@
-import OAI.Probability.ThreeState.Region
+import OAI.Probability.ThreeState.Region
diff --git a/lean/OAI/Probability/ThreeState/Messages.lean b/lean/OAI/Probability/ThreeState/Messages.lean
index b2ca98973567af21f954e780c513eabf75c32644..c0af0fc3841f1a7b562187e850713bec8aee3957 100644
--- a/lean/OAI/Probability/ThreeState/Messages.lean
+++ b/lean/OAI/Probability/ThreeState/Messages.lean
@@ -1,307 +1,307 @@
-import Mathlib
-import OAI.Probability.ThreeState.Expectations
-
-namespace OAI
-
-/-! Likelihood messages, compact couplings and subsequential posterior limits. -/
-
-namespace ThreeState
-open MeasureTheory
-
- 
-abbrev Message := {m : Spin → ℝ // (∀ i, 0 ≤ m i) ∧ ∑ i, m i = 3}
-
-instance : CoeFun Message (fun _ => Spin → ℝ) := ⟨Subtype.val⟩
-
-lemma Message.nonneg (m : Message) (i : Spin) : 0 ≤ m i := m.2.1 i
-lemma Message.sum (m : Message) : ∑ i, m i = 3 := m.2.2
-lemma Message.le_three (m : Message) (i : Spin) : m i ≤ 3 := by
-  calc
-    m i ≤ ∑ j, m j := Finset.single_le_sum (fun j _ => Message.nonneg m j) (Finset.mem_univ i)
-    _ = 3 := Message.sum m
-
-lemma message_isCompact : IsCompact {m : Spin → ℝ | (∀ i, 0 ≤ m i) ∧ ∑ i, m i = 3} := by
-  have hc : IsClosed {m : Spin → ℝ | (∀ i, 0 ≤ m i) ∧ ∑ i, m i = 3} := by
-    apply IsClosed.inter
-    · have he : {m : Spin → ℝ | ∀ i, 0 ≤ m i} =
-          ⋂ i : Spin, {m : Spin → ℝ | 0 ≤ m i} := by ext m; simp
-      change IsClosed {m : Spin → ℝ | ∀ i, 0 ≤ m i}
-      rw [he]
-      exact isClosed_iInter (fun i : Spin => isClosed_le (continuous_const (y := (0:ℝ)))
-          (continuous_apply i : Continuous (fun m : Spin → ℝ => m i)))
-    · exact isClosed_eq (continuous_finsetSum _ (fun i _ => continuous_apply i)) continuous_const
-  apply (isCompact_Icc (a := (fun _ : Spin => (0:ℝ))) (b := fun _ => (3:ℝ))).of_isClosed_subset hc
-  intro m hm
-  exact ⟨hm.1, fun i => Message.le_three ⟨m,hm⟩ i⟩
-
-instance : CompactSpace Message := isCompact_iff_compactSpace.mp message_isCompact
-noncomputable def uninformative : Message := ⟨fun _ => 1, by simp⟩
-
- 
-noncomputable def likelihood {α : Type*} (law : Spin → PMF α) (y : α) : Message :=
-  ⟨fun i => 3*posterior law y i, by
-    constructor
-    · intro i; exact mul_nonneg (by norm_num) (posterior_nonneg law y i)
-    · rw [← Finset.mul_sum, posterior_sum]; norm_num⟩
-
-lemma likelihood_apply {α : Type*} (law : Spin → PMF α) (y : α) (i : Spin) :
-    likelihood law y i = 3*posterior law y i := rfl
-
-lemma weighted_likelihood {α : Type*} (law : Spin → PMF α) (y : α) (i : Spin) :
-    mass (marginal law) y * likelihood law y i = mass (law i) y := by
-  rw [likelihood_apply]
-  have hh := weighted_posterior law y i
-  nlinarith
-
-lemma hasMean_likelihood {α : Type*} (law : Spin → PMF α) (i : Spin) :
-    HasMean (marginal law) (fun y => likelihood law y i) := by
-  exact (mass_summable (law i)).congr (fun y => (weighted_likelihood law y i).symm)
-
-lemma mean_likelihood {α : Type*} (law : Spin → PMF α) (i : Spin) :
-    mean (marginal law) (fun y => likelihood law y i) = 1 := by
-  unfold mean
-  simp_rw [weighted_likelihood]
-  exact mass_sum _
-
-noncomputable def likelihoodPMF {α : Type*} (law : Spin → PMF α) : PMF Message :=
-  (marginal law).map (likelihood law)
-
-noncomputable def likelihoodLaw {α : Type*} (law : Spin → PMF α) : ProbabilityMeasure Message :=
-  ⟨(likelihoodPMF law).toMeasure, inferInstance⟩
-
-lemma mean_likelihoodPMF {α : Type*} (law : Spin → PMF α) (f : Message → ℝ)
-    (hf : HasMean (marginal law) (fun y => f (likelihood law y))) :
-    mean (likelihoodPMF law) f = mean (marginal law) (fun y => f (likelihood law y)) :=
-  mean_map hf
-
-lemma integral_likelihoodLaw {α : Type*} (law : Spin → PMF α) (f : C(Message,ℝ)) :
-    ∫ m, f m ∂(likelihoodLaw law : Measure Message) =
-      mean (marginal law) (fun y => f (likelihood law y)) := by
-  have hf : Integrable f (likelihoodPMF law).toMeasure := f.continuous.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace f)
-  change (∫ m, f m ∂(likelihoodPMF law).toMeasure) = _
-  rw [PMF.integral_eq_tsum _ _ hf]
-  change mean (likelihoodPMF law) f = _
-  apply mean_likelihoodPMF
-  obtain ⟨C,hC⟩ := isCompact_univ.exists_bound_of_continuousOn f.continuous.continuousOn
-  apply HasMean.of_abs
-  apply (hasMean_const (marginal law) C).mono (fun y => abs_nonneg _)
-  intro y
-  simpa only [Real.norm_eq_abs] using hC (likelihood law y) (Set.mem_univ _)
-
-end ThreeState
-
-namespace ThreeState
-open MeasureTheory
-
-lemma abs_likelihood {α : Type*} (law : Spin → PMF α) (y : α) (i : Spin) :
-    |likelihood law y i| ≤ 3 := by
-  rw [abs_of_nonneg (Message.nonneg _ _)]
-  exact Message.le_three _ _
-
-lemma abs_likelihood_square {α : Type*} (law : Spin → PMF α) (y : α) (i : Spin) :
-    |(likelihood law y i)^2| ≤ 9 := by
-  rw [abs_sq]
-  have h := abs_likelihood law y i
-  have hl := (abs_le.mp h).1
-  have hu := (abs_le.mp h).2
-  nlinarith
-
-lemma mean_likelihood_mul {α : Type*} (law : Spin → PMF α) (i : Spin) (f : α → ℝ) :
-    mean (marginal law) (fun y => likelihood law y i * f y) = mean (law i) f := by
-  unfold mean
-  apply tsum_congr
-  intro y
-  rw [← mul_assoc, weighted_likelihood]
-
- 
-lemma mean_joint_likelihood_mul {α β : Type*} (law : Spin → PMF α) (k : α → PMF β)
-    (i : Spin) (f : β → ℝ) (C : ℝ) (hC : 0 ≤ C) (hf : ∀ b, |f b| ≤ C) :
-    mean (joint (marginal law) k) (fun ab => likelihood law ab.1 i * f ab.2) =
-      mean (marginal (fun j => (law j).bind k))
-        (fun b => likelihood (fun j => (law j).bind k) b i * f b) := by
-  rw [mean_joint_bounded _ _ _ (3*C) (fun ab => by
-    rw [abs_mul]
-    exact (mul_le_mul_of_nonneg_left (hf _) (abs_nonneg _)).trans
-      (mul_le_mul_of_nonneg_right (abs_likelihood _ _ _) hC))]
-  simp_rw [mean_const_mul]
-  rw [mean_likelihood_mul, mean_likelihood_mul, mean_bind_bounded _ _ _ C hf]
-
-lemma mean_joint_likelihood_square {α β : Type*} (law : Spin → PMF α) (k : α → PMF β)
-    (i : Spin) :
-    mean (joint (marginal law) k) (fun ab =>
-      (likelihood law ab.1 i - likelihood (fun j => (law j).bind k) ab.2 i)^2) =
-      mean (marginal law) (fun a => (likelihood law a i)^2) -
-      mean (marginal (fun j => (law j).bind k))
-        (fun b => (likelihood (fun j => (law j).bind k) b i)^2) := by
-  let p := joint (marginal law) k
-  let u := fun ab : α × β => likelihood law ab.1 i
-  let v := fun ab : α × β => likelihood (fun j => (law j).bind k) ab.2 i
-  have hu : ∀ ab, |u ab| ≤ 3 := fun ab => abs_likelihood _ _ _
-  have hv : ∀ ab, |v ab| ≤ 3 := fun ab => abs_likelihood _ _ _
-  have husq : ∀ ab, |(u ab)^2| ≤ 9 := fun ab => by
-    rw [abs_sq]; nlinarith [sq_le_sq₀ (abs_nonneg (u ab)) (by norm_num : (0:ℝ) ≤ 3) |>.2 (hu ab), sq_abs (u ab)]
-  have hvsq : ∀ ab, |(v ab)^2| ≤ 9 := fun ab => by
-    rw [abs_sq]; nlinarith [sq_le_sq₀ (abs_nonneg (v ab)) (by norm_num : (0:ℝ) ≤ 3) |>.2 (hv ab), sq_abs (v ab)]
-  have huv : ∀ ab, |u ab * v ab| ≤ 9 := fun ab => by
-    rw [abs_mul]; exact (mul_le_mul (hu ab) (hv ab) (abs_nonneg _) (by norm_num)).trans_eq (by norm_num)
-  change mean p (fun ab => (u ab-v ab)^2) = _
-  rw [mean_congr p (g := fun ab => ((u ab)^2+(v ab)^2)+ (-2)*(u ab*v ab)) (fun ab => by ring),
-    mean_add ((hasMean_bounded p 9 husq).add (hasMean_bounded p 9 hvsq))
-      ((hasMean_bounded p 9 huv).const_mul (-2)),
-    mean_add (hasMean_bounded p 9 husq) (hasMean_bounded p 9 hvsq), mean_const_mul]
-  change mean (joint (marginal law) k) (fun ab => (likelihood law ab.1 i)^2) +
-    mean (joint (marginal law) k) (fun ab => (likelihood (fun j => (law j).bind k) ab.2 i)^2) +
-    -2 * mean (joint (marginal law) k) (fun ab => likelihood law ab.1 i * likelihood (fun j => (law j).bind k) ab.2 i) = _
-  rw [mean_joint_fst (marginal law) k (fun a => (likelihood law a i)^2) 9
-      (fun a => abs_likelihood_square law a i),
-    mean_joint_snd (marginal law) k (fun b => (likelihood (fun j => (law j).bind k) b i)^2) 9
-      (fun b => abs_likelihood_square _ b i),
-    mean_joint_likelihood_mul law k i (fun b => likelihood (fun j => (law j).bind k) b i) 3
-      (by norm_num) (fun b => abs_likelihood (fun j => (law j).bind k) b i),
-    ← marginal_bind]
-  simp only [← sq]
-  ring
-
-end ThreeState
-
-namespace ThreeState
-open MeasureTheory Filter Topology
-
-lemma joint_map_fst {α β : Type*} (p : PMF α) (k : α → PMF β) :
-    (joint p k).map Prod.fst = p := by
-  simp only [joint, PMF.map_bind, PMF.map_comp, Function.comp_def]
-  change (p.bind (fun a => (k a).map (Function.const β a))) = _
-  simp only [PMF.map_const, PMF.bind_pure]
-
-lemma joint_map_snd {α β : Type*} (p : PMF α) (k : α → PMF β) :
-    (joint p k).map Prod.snd = p.bind k := by
-  simp only [joint, PMF.map_bind, PMF.map_comp, Function.comp_def]
-  change (p.bind (fun a => (k a).map id)) = _
-  simp only [PMF.map_id]
-
-noncomputable def messageCouplingPMF {α β : Type*} (law : Spin → PMF α) (k : α → PMF β) :
-    PMF (Message × Message) :=
-  (joint (marginal law) k).map (fun ab => (likelihood law ab.1, likelihood (fun j => (law j).bind k) ab.2))
-
-noncomputable def messageCoupling {α β : Type*} (law : Spin → PMF α) (k : α → PMF β) :
-    ProbabilityMeasure (Message × Message) := ⟨(messageCouplingPMF law k).toMeasure, inferInstance⟩
-
-lemma messageCouplingPMF_fst {α β : Type*} (law : Spin → PMF α) (k : α → PMF β) :
-    (messageCouplingPMF law k).map Prod.fst = likelihoodPMF law := by
-  rw [messageCouplingPMF, PMF.map_comp]
-  change (joint (marginal law) k).map (likelihood law ∘ Prod.fst) = _
-  rw [← PMF.map_comp, joint_map_fst]
-  rfl
-
-lemma messageCouplingPMF_snd {α β : Type*} (law : Spin → PMF α) (k : α → PMF β) :
-    (messageCouplingPMF law k).map Prod.snd = likelihoodPMF (fun j => (law j).bind k) := by
-  rw [messageCouplingPMF, PMF.map_comp]
-  change (joint (marginal law) k).map (likelihood (fun j => (law j).bind k) ∘ Prod.snd) = _
-  rw [← PMF.map_comp, joint_map_snd, ← marginal_bind]
-  rfl
-
-lemma messageCoupling_fst {α β : Type*} (law : Spin → PMF α) (k : α → PMF β) :
-    (messageCoupling law k).map Prod.fst = likelihoodLaw law := by
-  apply Subtype.ext
-  change (messageCouplingPMF law k).toMeasure.map Prod.fst = (likelihoodPMF law).toMeasure
-  rw [PMF.toMeasure_map _ _ measurable_fst, messageCouplingPMF_fst]
-
-lemma messageCoupling_snd {α β : Type*} (law : Spin → PMF α) (k : α → PMF β) :
-    (messageCoupling law k).map Prod.snd = likelihoodLaw (fun j => (law j).bind k) := by
-  apply Subtype.ext
-  change (messageCouplingPMF law k).toMeasure.map Prod.snd = (likelihoodPMF _).toMeasure
-  rw [PMF.toMeasure_map _ _ measurable_snd, messageCouplingPMF_snd]
-
-lemma integral_messageCoupling {α β : Type*} (law : Spin → PMF α) (k : α → PMF β)
-    (f : C(Message × Message,ℝ)) :
-    ∫ mm, f mm ∂(messageCoupling law k : Measure (Message × Message)) =
-      mean (joint (marginal law) k) (fun ab => f (likelihood law ab.1, likelihood (fun j => (law j).bind k) ab.2)) := by
-  have hi : Integrable f (messageCouplingPMF law k).toMeasure :=
-    f.continuous.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace f)
-  change (∫ mm, f mm ∂(messageCouplingPMF law k).toMeasure) = _
-  rw [PMF.integral_eq_tsum _ _ hi]
-  change mean ((joint (marginal law) k).map _) f = _
-  obtain ⟨C,hC⟩ := isCompact_univ.exists_bound_of_continuousOn f.continuous.continuousOn
-  exact mean_map_bounded _ _ _ C (fun ab => by simpa only [Real.norm_eq_abs] using hC _ (Set.mem_univ _))
-
-noncomputable def coordinateSecond {α : Type*} (law : Spin → PMF α) (i : Spin) : ℝ :=
-  mean (marginal law) (fun a => (likelihood law a i)^2)
-
-lemma coordinateSecond_nonneg {α : Type*} (law : Spin → PMF α) (i : Spin) :
-    0 ≤ coordinateSecond law i := mean_nonneg _ (fun value => sq_nonneg (likelihood law value i))
-
-lemma coordinateSecond_bind_le {α β : Type*} (law : Spin → PMF α) (k : α → PMF β) (i : Spin) :
-    coordinateSecond (fun j => (law j).bind k) i ≤ coordinateSecond law i := by
-  have hn := mean_nonneg (joint (marginal law) k) (fun ab =>
-    sq_nonneg (likelihood law ab.1 i - likelihood (fun j => (law j).bind k) ab.2 i))
-  rw [mean_joint_likelihood_square] at hn
-  exact sub_nonneg.mp hn
-
-noncomputable def squareJump (i : Spin) : C(Message × Message,ℝ) :=
-  ⟨fun mm => (mm.1 i-mm.2 i)^2,
-    (((continuous_apply i).comp (continuous_subtype_val.comp continuous_fst)).sub
-      ((continuous_apply i).comp (continuous_subtype_val.comp continuous_snd))).pow 2⟩
-
-lemma squareJump_nonneg (i : Spin) (mm : Message × Message) : 0 ≤ squareJump i mm := sq_nonneg _
-
-lemma integral_squareJump {α β : Type*} (law : Spin → PMF α) (k : α → PMF β) (i : Spin) :
-    ∫ mm, squareJump i mm ∂(messageCoupling law k : Measure (Message × Message)) =
-      coordinateSecond law i - coordinateSecond (fun j => (law j).bind k) i := by
-  rw [integral_messageCoupling]
-  exact mean_joint_likelihood_square law k i
-
- 
-
-theorem likelihood_common_subsequence {α : ℕ → Type*} (law : ∀ n, Spin → PMF (α n))
-    (k : ∀ n, α n → PMF (α (n+1)))
-    (hstep : ∀ n i, law (n+1) i = (law n i).bind (k n)) :
-    ∃ Q : ProbabilityMeasure Message, ∃ φ : ℕ → ℕ, StrictMono φ ∧
-      Tendsto (fun n => likelihoodLaw (law (φ n))) atTop (𝓝 Q) ∧
-      Tendsto (fun n => likelihoodLaw (law (φ n+1))) atTop (𝓝 Q) := by
-  have he (n : ℕ) : law (n+1) = (fun i => (law n i).bind (k n)) := funext (hstep n)
-  have ha (i : Spin) : Antitone (fun n => coordinateSecond (law n) i) := by
-    apply antitone_nat_of_succ_le
-    intro n
-    rw [he n]
-    exact coordinateSecond_bind_le _ _ _
-  have hb (i : Spin) : BddBelow (Set.range (fun n => coordinateSecond (law n) i)) :=
-    ⟨0, by rintro _ ⟨n,rfl⟩; exact coordinateSecond_nonneg _ _⟩
-  have ht (i : Spin) := tendsto_atTop_ciInf (ha i) (hb i)
-  let Pn := fun n => messageCoupling (law n) (k n)
-  obtain ⟨P,_,φ,hφ,hlim⟩ := isCompact_univ.tendsto_subseq (fun n => Set.mem_univ (Pn n))
-  have hzero (i : Spin) : ∫ mm, squareJump i mm ∂(P : Measure (Message × Message)) = 0 := by
-    have hdiff : Tendsto (fun n => coordinateSecond (law n) i - coordinateSecond (law (n+1)) i)
-        atTop (𝓝 0) := by
-      convert (ht i).sub ((ht i).comp (tendsto_add_atTop_nat 1)) using 1 <;> simp
-    have hint := (ProbabilityMeasure.continuous_integral_continuousMap (squareJump i)).continuousAt.tendsto.comp hlim
-    have hn : (fun n => ∫ mm, squareJump i mm ∂(Pn (φ n) : Measure (Message × Message))) =
-        (fun n => coordinateSecond (law (φ n)) i - coordinateSecond (law (φ n+1)) i) := by
-      funext n
-      dsimp [Pn]
-      rw [integral_squareJump, ← he]
-    simp only [Function.comp_def] at hint
-    rw [hn] at hint
-    exact tendsto_nhds_unique hint (hdiff.comp hφ.tendsto_atTop)
-  have hdiag : ∀ᵐ mm ∂(P : Measure (Message × Message)), mm.1 = mm.2 := by
-    have hi (i : Spin) : ∀ᵐ mm ∂(P : Measure (Message × Message)), squareJump i mm = 0 :=
-      (integral_eq_zero_iff_of_nonneg (squareJump_nonneg i)
-        ((squareJump i).continuous.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _))).mp (hzero i)
-    filter_upwards [ae_all_iff.mpr hi] with mm hmm
-    apply Subtype.ext
-    funext i
-    have h := hmm i
-    change (mm.1 i-mm.2 i)^2 = 0 at h
-    exact sub_eq_zero.mp (sq_eq_zero_iff.mp h)
-  have hmaps : P.map Prod.snd = P.map Prod.fst := by
-    apply Subtype.ext
-    exact Measure.map_congr (hdiag.mono (fun mm hm => hm.symm))
-  refine ⟨P.map Prod.fst, φ, hφ, ?_, ?_⟩
-  · have hh := (ProbabilityMeasure.continuous_map (continuous_fst : Continuous (@Prod.fst Message Message))).tendsto P |>.comp hlim
-    simpa only [Function.comp_def, Pn, messageCoupling_fst] using hh
-  · have hh := (ProbabilityMeasure.continuous_map (continuous_snd : Continuous (@Prod.snd Message Message))).tendsto P |>.comp hlim
-    simp only [Function.comp_def, Pn, messageCoupling_snd, ← he, hmaps] at hh
-    exact hh
-
-end ThreeState
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeState.Expectations
+
+namespace OAI
+
+/-! Likelihood messages, compact couplings and subsequential posterior limits. -/
+
+namespace ThreeState
+open MeasureTheory
+
+ 
+abbrev Message := {m : Spin → ℝ // (∀ i, 0 ≤ m i) ∧ ∑ i, m i = 3}
+
+instance : CoeFun Message (fun _ => Spin → ℝ) := ⟨Subtype.val⟩
+
+lemma Message.nonneg (m : Message) (i : Spin) : 0 ≤ m i := m.2.1 i
+lemma Message.sum (m : Message) : ∑ i, m i = 3 := m.2.2
+lemma Message.le_three (m : Message) (i : Spin) : m i ≤ 3 := by
+  calc
+    m i ≤ ∑ j, m j := Finset.single_le_sum (fun j _ => Message.nonneg m j) (Finset.mem_univ i)
+    _ = 3 := Message.sum m
+
+lemma message_isCompact : IsCompact {m : Spin → ℝ | (∀ i, 0 ≤ m i) ∧ ∑ i, m i = 3} := by
+  have hc : IsClosed {m : Spin → ℝ | (∀ i, 0 ≤ m i) ∧ ∑ i, m i = 3} := by
+    apply IsClosed.inter
+    · have he : {m : Spin → ℝ | ∀ i, 0 ≤ m i} =
+          ⋂ i : Spin, {m : Spin → ℝ | 0 ≤ m i} := by ext m; simp
+      change IsClosed {m : Spin → ℝ | ∀ i, 0 ≤ m i}
+      rw [he]
+      exact isClosed_iInter (fun i : Spin => isClosed_le (continuous_const (y := (0:ℝ)))
+          (continuous_apply i : Continuous (fun m : Spin → ℝ => m i)))
+    · exact isClosed_eq (continuous_finsetSum _ (fun i _ => continuous_apply i)) continuous_const
+  apply (isCompact_Icc (a := (fun _ : Spin => (0:ℝ))) (b := fun _ => (3:ℝ))).of_isClosed_subset hc
+  intro m hm
+  exact ⟨hm.1, fun i => Message.le_three ⟨m,hm⟩ i⟩
+
+instance : CompactSpace Message := isCompact_iff_compactSpace.mp message_isCompact
+noncomputable def uninformative : Message := ⟨fun _ => 1, by simp⟩
+
+ 
+noncomputable def likelihood {α : Type*} (law : Spin → PMF α) (y : α) : Message :=
+  ⟨fun i => 3*posterior law y i, by
+    constructor
+    · intro i; exact mul_nonneg (by norm_num) (posterior_nonneg law y i)
+    · rw [← Finset.mul_sum, posterior_sum]; norm_num⟩
+
+lemma likelihood_apply {α : Type*} (law : Spin → PMF α) (y : α) (i : Spin) :
+    likelihood law y i = 3*posterior law y i := rfl
+
+lemma weighted_likelihood {α : Type*} (law : Spin → PMF α) (y : α) (i : Spin) :
+    mass (marginal law) y * likelihood law y i = mass (law i) y := by
+  rw [likelihood_apply]
+  have hh := weighted_posterior law y i
+  nlinarith
+
+lemma hasMean_likelihood {α : Type*} (law : Spin → PMF α) (i : Spin) :
+    HasMean (marginal law) (fun y => likelihood law y i) := by
+  exact (mass_summable (law i)).congr (fun y => (weighted_likelihood law y i).symm)
+
+lemma mean_likelihood {α : Type*} (law : Spin → PMF α) (i : Spin) :
+    mean (marginal law) (fun y => likelihood law y i) = 1 := by
+  unfold mean
+  simp_rw [weighted_likelihood]
+  exact mass_sum _
+
+noncomputable def likelihoodPMF {α : Type*} (law : Spin → PMF α) : PMF Message :=
+  (marginal law).map (likelihood law)
+
+noncomputable def likelihoodLaw {α : Type*} (law : Spin → PMF α) : ProbabilityMeasure Message :=
+  ⟨(likelihoodPMF law).toMeasure, inferInstance⟩
+
+lemma mean_likelihoodPMF {α : Type*} (law : Spin → PMF α) (f : Message → ℝ)
+    (hf : HasMean (marginal law) (fun y => f (likelihood law y))) :
+    mean (likelihoodPMF law) f = mean (marginal law) (fun y => f (likelihood law y)) :=
+  mean_map hf
+
+lemma integral_likelihoodLaw {α : Type*} (law : Spin → PMF α) (f : C(Message,ℝ)) :
+    ∫ m, f m ∂(likelihoodLaw law : Measure Message) =
+      mean (marginal law) (fun y => f (likelihood law y)) := by
+  have hf : Integrable f (likelihoodPMF law).toMeasure := f.continuous.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace f)
+  change (∫ m, f m ∂(likelihoodPMF law).toMeasure) = _
+  rw [PMF.integral_eq_tsum _ _ hf]
+  change mean (likelihoodPMF law) f = _
+  apply mean_likelihoodPMF
+  obtain ⟨C,hC⟩ := isCompact_univ.exists_bound_of_continuousOn f.continuous.continuousOn
+  apply HasMean.of_abs
+  apply (hasMean_const (marginal law) C).mono (fun y => abs_nonneg _)
+  intro y
+  simpa only [Real.norm_eq_abs] using hC (likelihood law y) (Set.mem_univ _)
+
+end ThreeState
+
+namespace ThreeState
+open MeasureTheory
+
+lemma abs_likelihood {α : Type*} (law : Spin → PMF α) (y : α) (i : Spin) :
+    |likelihood law y i| ≤ 3 := by
+  rw [abs_of_nonneg (Message.nonneg _ _)]
+  exact Message.le_three _ _
+
+lemma abs_likelihood_square {α : Type*} (law : Spin → PMF α) (y : α) (i : Spin) :
+    |(likelihood law y i)^2| ≤ 9 := by
+  rw [abs_sq]
+  have h := abs_likelihood law y i
+  have hl := (abs_le.mp h).1
+  have hu := (abs_le.mp h).2
+  nlinarith
+
+lemma mean_likelihood_mul {α : Type*} (law : Spin → PMF α) (i : Spin) (f : α → ℝ) :
+    mean (marginal law) (fun y => likelihood law y i * f y) = mean (law i) f := by
+  unfold mean
+  apply tsum_congr
+  intro y
+  rw [← mul_assoc, weighted_likelihood]
+
+ 
+lemma mean_joint_likelihood_mul {α β : Type*} (law : Spin → PMF α) (k : α → PMF β)
+    (i : Spin) (f : β → ℝ) (C : ℝ) (hC : 0 ≤ C) (hf : ∀ b, |f b| ≤ C) :
+    mean (joint (marginal law) k) (fun ab => likelihood law ab.1 i * f ab.2) =
+      mean (marginal (fun j => (law j).bind k))
+        (fun b => likelihood (fun j => (law j).bind k) b i * f b) := by
+  rw [mean_joint_bounded _ _ _ (3*C) (fun ab => by
+    rw [abs_mul]
+    exact (mul_le_mul_of_nonneg_left (hf _) (abs_nonneg _)).trans
+      (mul_le_mul_of_nonneg_right (abs_likelihood _ _ _) hC))]
+  simp_rw [mean_const_mul]
+  rw [mean_likelihood_mul, mean_likelihood_mul, mean_bind_bounded _ _ _ C hf]
+
+lemma mean_joint_likelihood_square {α β : Type*} (law : Spin → PMF α) (k : α → PMF β)
+    (i : Spin) :
+    mean (joint (marginal law) k) (fun ab =>
+      (likelihood law ab.1 i - likelihood (fun j => (law j).bind k) ab.2 i)^2) =
+      mean (marginal law) (fun a => (likelihood law a i)^2) -
+      mean (marginal (fun j => (law j).bind k))
+        (fun b => (likelihood (fun j => (law j).bind k) b i)^2) := by
+  let p := joint (marginal law) k
+  let u := fun ab : α × β => likelihood law ab.1 i
+  let v := fun ab : α × β => likelihood (fun j => (law j).bind k) ab.2 i
+  have hu : ∀ ab, |u ab| ≤ 3 := fun ab => abs_likelihood _ _ _
+  have hv : ∀ ab, |v ab| ≤ 3 := fun ab => abs_likelihood _ _ _
+  have husq : ∀ ab, |(u ab)^2| ≤ 9 := fun ab => by
+    rw [abs_sq]; nlinarith [sq_le_sq₀ (abs_nonneg (u ab)) (by norm_num : (0:ℝ) ≤ 3) |>.2 (hu ab), sq_abs (u ab)]
+  have hvsq : ∀ ab, |(v ab)^2| ≤ 9 := fun ab => by
+    rw [abs_sq]; nlinarith [sq_le_sq₀ (abs_nonneg (v ab)) (by norm_num : (0:ℝ) ≤ 3) |>.2 (hv ab), sq_abs (v ab)]
+  have huv : ∀ ab, |u ab * v ab| ≤ 9 := fun ab => by
+    rw [abs_mul]; exact (mul_le_mul (hu ab) (hv ab) (abs_nonneg _) (by norm_num)).trans_eq (by norm_num)
+  change mean p (fun ab => (u ab-v ab)^2) = _
+  rw [mean_congr p (g := fun ab => ((u ab)^2+(v ab)^2)+ (-2)*(u ab*v ab)) (fun ab => by ring),
+    mean_add ((hasMean_bounded p 9 husq).add (hasMean_bounded p 9 hvsq))
+      ((hasMean_bounded p 9 huv).const_mul (-2)),
+    mean_add (hasMean_bounded p 9 husq) (hasMean_bounded p 9 hvsq), mean_const_mul]
+  change mean (joint (marginal law) k) (fun ab => (likelihood law ab.1 i)^2) +
+    mean (joint (marginal law) k) (fun ab => (likelihood (fun j => (law j).bind k) ab.2 i)^2) +
+    -2 * mean (joint (marginal law) k) (fun ab => likelihood law ab.1 i * likelihood (fun j => (law j).bind k) ab.2 i) = _
+  rw [mean_joint_fst (marginal law) k (fun a => (likelihood law a i)^2) 9
+      (fun a => abs_likelihood_square law a i),
+    mean_joint_snd (marginal law) k (fun b => (likelihood (fun j => (law j).bind k) b i)^2) 9
+      (fun b => abs_likelihood_square _ b i),
+    mean_joint_likelihood_mul law k i (fun b => likelihood (fun j => (law j).bind k) b i) 3
+      (by norm_num) (fun b => abs_likelihood (fun j => (law j).bind k) b i),
+    ← marginal_bind]
+  simp only [← sq]
+  ring
+
+end ThreeState
+
+namespace ThreeState
+open MeasureTheory Filter Topology
+
+lemma joint_map_fst {α β : Type*} (p : PMF α) (k : α → PMF β) :
+    (joint p k).map Prod.fst = p := by
+  simp only [joint, PMF.map_bind, PMF.map_comp, Function.comp_def]
+  change (p.bind (fun a => (k a).map (Function.const β a))) = _
+  simp only [PMF.map_const, PMF.bind_pure]
+
+lemma joint_map_snd {α β : Type*} (p : PMF α) (k : α → PMF β) :
+    (joint p k).map Prod.snd = p.bind k := by
+  simp only [joint, PMF.map_bind, PMF.map_comp, Function.comp_def]
+  change (p.bind (fun a => (k a).map id)) = _
+  simp only [PMF.map_id]
+
+noncomputable def messageCouplingPMF {α β : Type*} (law : Spin → PMF α) (k : α → PMF β) :
+    PMF (Message × Message) :=
+  (joint (marginal law) k).map (fun ab => (likelihood law ab.1, likelihood (fun j => (law j).bind k) ab.2))
+
+noncomputable def messageCoupling {α β : Type*} (law : Spin → PMF α) (k : α → PMF β) :
+    ProbabilityMeasure (Message × Message) := ⟨(messageCouplingPMF law k).toMeasure, inferInstance⟩
+
+lemma messageCouplingPMF_fst {α β : Type*} (law : Spin → PMF α) (k : α → PMF β) :
+    (messageCouplingPMF law k).map Prod.fst = likelihoodPMF law := by
+  rw [messageCouplingPMF, PMF.map_comp]
+  change (joint (marginal law) k).map (likelihood law ∘ Prod.fst) = _
+  rw [← PMF.map_comp, joint_map_fst]
+  rfl
+
+lemma messageCouplingPMF_snd {α β : Type*} (law : Spin → PMF α) (k : α → PMF β) :
+    (messageCouplingPMF law k).map Prod.snd = likelihoodPMF (fun j => (law j).bind k) := by
+  rw [messageCouplingPMF, PMF.map_comp]
+  change (joint (marginal law) k).map (likelihood (fun j => (law j).bind k) ∘ Prod.snd) = _
+  rw [← PMF.map_comp, joint_map_snd, ← marginal_bind]
+  rfl
+
+lemma messageCoupling_fst {α β : Type*} (law : Spin → PMF α) (k : α → PMF β) :
+    (messageCoupling law k).map Prod.fst = likelihoodLaw law := by
+  apply Subtype.ext
+  change (messageCouplingPMF law k).toMeasure.map Prod.fst = (likelihoodPMF law).toMeasure
+  rw [PMF.toMeasure_map _ _ measurable_fst, messageCouplingPMF_fst]
+
+lemma messageCoupling_snd {α β : Type*} (law : Spin → PMF α) (k : α → PMF β) :
+    (messageCoupling law k).map Prod.snd = likelihoodLaw (fun j => (law j).bind k) := by
+  apply Subtype.ext
+  change (messageCouplingPMF law k).toMeasure.map Prod.snd = (likelihoodPMF _).toMeasure
+  rw [PMF.toMeasure_map _ _ measurable_snd, messageCouplingPMF_snd]
+
+lemma integral_messageCoupling {α β : Type*} (law : Spin → PMF α) (k : α → PMF β)
+    (f : C(Message × Message,ℝ)) :
+    ∫ mm, f mm ∂(messageCoupling law k : Measure (Message × Message)) =
+      mean (joint (marginal law) k) (fun ab => f (likelihood law ab.1, likelihood (fun j => (law j).bind k) ab.2)) := by
+  have hi : Integrable f (messageCouplingPMF law k).toMeasure :=
+    f.continuous.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace f)
+  change (∫ mm, f mm ∂(messageCouplingPMF law k).toMeasure) = _
+  rw [PMF.integral_eq_tsum _ _ hi]
+  change mean ((joint (marginal law) k).map _) f = _
+  obtain ⟨C,hC⟩ := isCompact_univ.exists_bound_of_continuousOn f.continuous.continuousOn
+  exact mean_map_bounded _ _ _ C (fun ab => by simpa only [Real.norm_eq_abs] using hC _ (Set.mem_univ _))
+
+noncomputable def coordinateSecond {α : Type*} (law : Spin → PMF α) (i : Spin) : ℝ :=
+  mean (marginal law) (fun a => (likelihood law a i)^2)
+
+lemma coordinateSecond_nonneg {α : Type*} (law : Spin → PMF α) (i : Spin) :
+    0 ≤ coordinateSecond law i := mean_nonneg _ (fun value => sq_nonneg (likelihood law value i))
+
+lemma coordinateSecond_bind_le {α β : Type*} (law : Spin → PMF α) (k : α → PMF β) (i : Spin) :
+    coordinateSecond (fun j => (law j).bind k) i ≤ coordinateSecond law i := by
+  have hn := mean_nonneg (joint (marginal law) k) (fun ab =>
+    sq_nonneg (likelihood law ab.1 i - likelihood (fun j => (law j).bind k) ab.2 i))
+  rw [mean_joint_likelihood_square] at hn
+  exact sub_nonneg.mp hn
+
+noncomputable def squareJump (i : Spin) : C(Message × Message,ℝ) :=
+  ⟨fun mm => (mm.1 i-mm.2 i)^2,
+    (((continuous_apply i).comp (continuous_subtype_val.comp continuous_fst)).sub
+      ((continuous_apply i).comp (continuous_subtype_val.comp continuous_snd))).pow 2⟩
+
+lemma squareJump_nonneg (i : Spin) (mm : Message × Message) : 0 ≤ squareJump i mm := sq_nonneg _
+
+lemma integral_squareJump {α β : Type*} (law : Spin → PMF α) (k : α → PMF β) (i : Spin) :
+    ∫ mm, squareJump i mm ∂(messageCoupling law k : Measure (Message × Message)) =
+      coordinateSecond law i - coordinateSecond (fun j => (law j).bind k) i := by
+  rw [integral_messageCoupling]
+  exact mean_joint_likelihood_square law k i
+
+ 
+
+theorem likelihood_common_subsequence {α : ℕ → Type*} (law : ∀ n, Spin → PMF (α n))
+    (k : ∀ n, α n → PMF (α (n+1)))
+    (hstep : ∀ n i, law (n+1) i = (law n i).bind (k n)) :
+    ∃ Q : ProbabilityMeasure Message, ∃ φ : ℕ → ℕ, StrictMono φ ∧
+      Tendsto (fun n => likelihoodLaw (law (φ n))) atTop (𝓝 Q) ∧
+      Tendsto (fun n => likelihoodLaw (law (φ n+1))) atTop (𝓝 Q) := by
+  have he (n : ℕ) : law (n+1) = (fun i => (law n i).bind (k n)) := funext (hstep n)
+  have ha (i : Spin) : Antitone (fun n => coordinateSecond (law n) i) := by
+    apply antitone_nat_of_succ_le
+    intro n
+    rw [he n]
+    exact coordinateSecond_bind_le _ _ _
+  have hb (i : Spin) : BddBelow (Set.range (fun n => coordinateSecond (law n) i)) :=
+    ⟨0, by rintro _ ⟨n,rfl⟩; exact coordinateSecond_nonneg _ _⟩
+  have ht (i : Spin) := tendsto_atTop_ciInf (ha i) (hb i)
+  let Pn := fun n => messageCoupling (law n) (k n)
+  obtain ⟨P,_,φ,hφ,hlim⟩ := isCompact_univ.tendsto_subseq (fun n => Set.mem_univ (Pn n))
+  have hzero (i : Spin) : ∫ mm, squareJump i mm ∂(P : Measure (Message × Message)) = 0 := by
+    have hdiff : Tendsto (fun n => coordinateSecond (law n) i - coordinateSecond (law (n+1)) i)
+        atTop (𝓝 0) := by
+      convert (ht i).sub ((ht i).comp (tendsto_add_atTop_nat 1)) using 1 <;> simp
+    have hint := (ProbabilityMeasure.continuous_integral_continuousMap (squareJump i)).continuousAt.tendsto.comp hlim
+    have hn : (fun n => ∫ mm, squareJump i mm ∂(Pn (φ n) : Measure (Message × Message))) =
+        (fun n => coordinateSecond (law (φ n)) i - coordinateSecond (law (φ n+1)) i) := by
+      funext n
+      dsimp [Pn]
+      rw [integral_squareJump, ← he]
+    simp only [Function.comp_def] at hint
+    rw [hn] at hint
+    exact tendsto_nhds_unique hint (hdiff.comp hφ.tendsto_atTop)
+  have hdiag : ∀ᵐ mm ∂(P : Measure (Message × Message)), mm.1 = mm.2 := by
+    have hi (i : Spin) : ∀ᵐ mm ∂(P : Measure (Message × Message)), squareJump i mm = 0 :=
+      (integral_eq_zero_iff_of_nonneg (squareJump_nonneg i)
+        ((squareJump i).continuous.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _))).mp (hzero i)
+    filter_upwards [ae_all_iff.mpr hi] with mm hmm
+    apply Subtype.ext
+    funext i
+    have h := hmm i
+    change (mm.1 i-mm.2 i)^2 = 0 at h
+    exact sub_eq_zero.mp (sq_eq_zero_iff.mp h)
+  have hmaps : P.map Prod.snd = P.map Prod.fst := by
+    apply Subtype.ext
+    exact Measure.map_congr (hdiag.mono (fun mm hm => hm.symm))
+  refine ⟨P.map Prod.fst, φ, hφ, ?_, ?_⟩
+  · have hh := (ProbabilityMeasure.continuous_map (continuous_fst : Continuous (@Prod.fst Message Message))).tendsto P |>.comp hlim
+    simpa only [Function.comp_def, Pn, messageCoupling_fst] using hh
+  · have hh := (ProbabilityMeasure.continuous_map (continuous_snd : Continuous (@Prod.snd Message Message))).tendsto P |>.comp hlim
+    simp only [Function.comp_def, Pn, messageCoupling_snd, ← he, hmaps] at hh
+    exact hh
+
+end ThreeState
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeState/Negative.lean b/lean/OAI/Probability/ThreeState/Negative.lean
index 47bb29db6841a5ee6a4c65185c31b6ad3ec3447a..d092c1e3c3bbd10324e08892406c43d86a1b3a77 100644
--- a/lean/OAI/Probability/ThreeState/Negative.lean
+++ b/lean/OAI/Probability/ThreeState/Negative.lean
@@ -1,222 +1,222 @@
-import Mathlib
-import OAI.Probability.ThreeState.Couplings
-import OAI.Probability.ThreeState.Noise
-
-namespace OAI
-
-/-! Signed small-channel and Dobrushin non-reconstruction regions. -/
-
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-
- 
-
-theorem regular_small_negative_nonreconstruction (b : ℕ) (hb : 2 ≤ b) (lam : ℝ)
-    (h : Admissible lam) (hl : lam ≤ 0) (hc : 4*(b:ℝ)*lam^2 ≤ 1) :
-    Tendsto (regularAdvantage b lam h) atTop (𝓝 0) := by
-  have hb0 : (0:ℝ) < b := by exact_mod_cast (show 0<b by omega)
-  let L := Real.sqrt ((b:ℝ)⁻¹)
-  have hL0 : 0 < L := Real.sqrt_pos.mpr (inv_pos.mpr hb0)
-  have hLsq : L^2=(b:ℝ)⁻¹ := Real.sq_sqrt (inv_nonneg.mpr hb0.le)
-  have hK : (b:ℝ)*L^2=1 := by rw [hLsq, mul_inv_cancel₀ hb0.ne']
-  have hbr : (2:ℝ)≤b := by exact_mod_cast hb
-  have hL1 : L≤1 := by nlinarith [sq_nonneg L]
-  have hL : Admissible L := ⟨by linarith, hL1⟩
-  have hlow : -L/2≤lam := by
-    have hh : 4*lam^2≤L^2 := (mul_le_mul_iff_right₀ hb0).mp (by nlinarith only [hc,hK])
-    nlinarith
-  let r := lam/L
-  have hr0 : -(1/2:ℝ)≤r := (le_div_iff₀ hL0).mpr (by linarith)
-  have hr1 : r≤1 := (div_le_one hL0).mpr (by linarith)
-  have hr : Admissible r := ⟨hr0,hr1⟩
-  have heq : L*r=lam := by dsimp [r]; field_simp
-  have hp : Admissible (L*r) := by rwa [heq]
-  have ht := ordered_tendsto_zero_of_uniform_fixedpoints (PMF.pure b) L hL
-    (regular_positive_critical_uniform b hb L hL hL0 hK)
-  apply squeeze_zero (fun n => advantage_nonneg (regularLaw b lam h n))
-    (g := fun n => advantage (orderedLaw (PMF.pure b) L hL n)) _ ht
-  intro n
-  have hg := orderedAdvantage_degrade (PMF.pure b) L r hL hr hp n
-  have heLaw : orderedLaw (PMF.pure b) (L*r) hp n = orderedLaw (PMF.pure b) lam h n := by congr 1
-  rw [heLaw] at hg
-  exact (orderedAdvantage_dominates (PMF.pure b) lam h n).trans hg
-
- 
-
-theorem poisson_small_negative_nonreconstruction (d : ℝ) (hd : 1 < d) (lam : ℝ)
-    (h : Admissible lam) (hl : lam ≤ 0) (hc : 4*d*lam^2 ≤ 1) :
-    Tendsto (poissonAdvantage d hd lam h) atTop (𝓝 0) := by
-  have hd0 : 0 < d := by linarith
-  let L := Real.sqrt d⁻¹
-  have hL0 : 0 < L := Real.sqrt_pos.mpr (inv_pos.mpr hd0)
-  have hLsq : L^2=d⁻¹ := Real.sq_sqrt (inv_nonneg.mpr hd0.le)
-  have hK : d*L^2=1 := by rw [hLsq, mul_inv_cancel₀ hd0.ne']
-  have hL1 : L≤1 := by
-    have hi : d⁻¹≤1 := (inv_le_one₀ hd0).mpr hd.le
-    nlinarith [sq_nonneg L]
-  have hL : Admissible L := ⟨by linarith, hL1⟩
-  have hlow : -L/2≤lam := by
-    have hh : 4*lam^2≤L^2 := (mul_le_mul_iff_right₀ hd0).mp (by nlinarith only [hc,hK])
-    nlinarith
-  let r := lam/L
-  have hr0 : -(1/2:ℝ)≤r := (le_div_iff₀ hL0).mpr (by linarith)
-  have hr1 : r≤1 := (div_le_one hL0).mpr (by linarith)
-  have hr : Admissible r := ⟨hr0,hr1⟩
-  have heq : L*r=lam := by dsimp [r]; field_simp
-  have hp : Admissible (L*r) := by rwa [heq]
-  have ht := ordered_tendsto_zero_of_uniform_fixedpoints (poissonOffspring d hd0.le) L hL
-    (poisson_positive_critical_uniform d hd L hL hL0 hK)
-  apply squeeze_zero (fun n => advantage_nonneg (poissonLaw d hd0.le lam h n))
-    (g := fun n => advantage (orderedLaw (poissonOffspring d hd0.le) L hL n)) _ ht
-  intro n
-  have hg := orderedAdvantage_degrade (poissonOffspring d hd0.le) L r hL hr hp n
-  have heLaw : orderedLaw (poissonOffspring d hd0.le) (L*r) hp n = orderedLaw (poissonOffspring d hd0.le) lam h n := by congr 1
-  rw [heLaw] at hg
-  exact (orderedAdvantage_dominates (poissonOffspring d hd0.le) lam h n).trans hg
-
-end ThreeState
-
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-noncomputable section
-
- 
-
-lemma compose_pairTV_bound {α:Type*} (p:PMF ℕ) (law:Spin→PMF α) (d lam x:ℝ)
-    (h:Admissible lam) (hD:HasMean p (fun k:ℕ=>(k:ℝ)))
-    (hmean:mean p (fun k:ℕ=>(k:ℝ))=d) (K:ℕ) (hK:2≤K)
-    (hx0:0≤x) (hx1:x≤1) (hx:∀i j,tvMass (law i) (law j)≤x) (i j:Spin) :
-    tvMass (composeLaw p law lam h i) (composeLaw p law lam h j) ≤
-      d*|lam| *x-mass p K*(|lam| *x)^2 := by
-  let t:ℝ:=|lam| *x
-  have hl:|lam|≤1:=abs_le.mpr ⟨by linarith [h.1],h.2⟩
-  have ht0:0≤t:=mul_nonneg (abs_nonneg _) hx0
-  have ht1:t≤1 := calc
-    t ≤ 1*x := mul_le_mul_of_nonneg_right hl hx0
-    _ = x := one_mul x
-    _ ≤ 1 := hx1
-  have he:tvMass (edgeLaw law lam h i) (edgeLaw law lam h j)≤t := by
-    rw [tvMass_edge]
-    exact mul_le_mul_of_nonneg_left (hx i j) (abs_nonneg _)
-  have hpoint (k:ℕ) : tvMass (iidVector (edgeLaw law lam h i) k) (iidVector (edgeLaw law lam h j) k)≤
-      1-(1-t)^k := by
-    apply (tvMass_iidVector_le _ _ k).trans
-    have hp:=pow_le_pow_left₀ (by linarith : 0≤1-t) (by linarith : 1-t≤1-tvMass (edgeLaw law lam h i) (edgeLaw law lam h j)) k
-    linarith
-  have hs1:Summable (fun k:ℕ=>mass p k*((k:ℝ)*t)):=hD.mul_const t
-  have hs2:Summable (fun k:ℕ=>if k=K then mass p K*t^2 else 0):=(hasSum_ite_eq K (mass p K*t^2)).summable
-  have hu (k:ℕ) : mass p k*tvMass (iidVector (edgeLaw law lam h i) k) (iidVector (edgeLaw law lam h j) k)≤
-      mass p k*((k:ℝ)*t)-(if k=K then mass p K*t^2 else 0) := by
-    by_cases hk:k=K
-    · subst k
-      simp only [ite_true]
-      have hb:=bernoulli_quadratic_lower t ht0 ht1 K hK
-      have hv:=hpoint K
-      have hh:=mul_le_mul_of_nonneg_left (show tvMass (iidVector (edgeLaw law lam h i) K) (iidVector (edgeLaw law lam h j) K)≤(K:ℝ)*t-t^2 by linarith) (mass_nonneg p K)
-      nlinarith only [hh]
-    · rw [ite_eq_right hk,sub_zero]
-      have hb:=bernoulli_lower t ht0 ht1 k
-      exact mul_le_mul_of_nonneg_left (by linarith [hpoint k]) (mass_nonneg p k)
-  have hh:=Summable.tsum_le_tsum hu (summable_weighted_tv p
-    (fun k=>iidVector (edgeLaw law lam h i) k) (fun k=>iidVector (edgeLaw law lam h j) k)) (hs1.sub hs2)
-  have hsum: (∑'k:ℕ,mass p k*((k:ℝ)*t))=d*t := by
-    change mean p (fun k:ℕ=>(k:ℝ)*t)=_
-    rw [mean_mul_const,hmean]
-  rw [hs1.tsum_sub hs2,hsum,tsum_ite_eq] at hh
-  simpa only [composeLaw,tvMass_sigmaPMF,t,mul_assoc] using hh
-
-lemma ordered_pairTV_decay (p:PMF ℕ) (d lam:ℝ) (h:Admissible lam)
-    (hD:HasMean p (fun k:ℕ=>(k:ℝ))) (hmean:mean p (fun k:ℕ=>(k:ℝ))=d)
-    (K:ℕ) (hK:2≤K) (hd:d*|lam|≤1) (n:ℕ) (i j:Spin) :
-    tvMass (orderedLaw p lam h n i) (orderedLaw p lam h n j)≤decaySeq (mass p K*lam^2) n := by
-  have hl:|lam|≤1:=abs_le.mpr ⟨by linarith [h.1],h.2⟩
-  have hl2:lam^2≤1:=by nlinarith [sq_abs lam,abs_nonneg lam]
-  have hc0:0 ≤ mass p K*lam^2:=mul_nonneg (mass_nonneg p K) (sq_nonneg lam)
-  have hc1:mass p K*lam^2≤1 := calc
-    mass p K*lam^2 ≤ 1*lam^2 := mul_le_mul_of_nonneg_right (mass_le_one p K) (sq_nonneg lam)
-    _ = lam^2 := one_mul (lam^2)
-    _ ≤ 1 := hl2
-  induction n generalizing i j with
-  | zero => exact tvMass_le_one _ _
-  | succ n ih =>
-    have hb:=decaySeq_bounds (mass p K*lam^2) hc0 hc1 n
-    have hh:=compose_pairTV_bound p (orderedLaw p lam h n) d lam (decaySeq (mass p K*lam^2) n)
-      h hD hmean K hK hb.1 hb.2 (fun i j=>ih i j) i j
-    have hg:=mul_le_mul_of_nonneg_right hd hb.1
-    change tvMass (composeLaw p (orderedLaw p lam h n) lam h i)
-      (composeLaw p (orderedLaw p lam h n) lam h j)≤_
-    change _≤decaySeq (mass p K*lam^2) n-(mass p K*lam^2)*(decaySeq (mass p K*lam^2) n)^2
-    simp only [mul_pow,sq_abs] at hh
-    nlinarith only [hh,hg]
-
- 
-
-theorem nonreconstruction_dobrushin (p:PMF ℕ) (d lam:ℝ) (h:Admissible lam)
-    (hD:HasMean p (fun k:ℕ=>(k:ℝ))) (hmean:mean p (fun k:ℕ=>(k:ℝ))=d)
-    (K:ℕ) (hK:2≤K) (hp:0 < mass p K) (hl:lam≠0) (hd:d*|lam|≤1) :
-    Tendsto (fun n=>advantage (observedLaw p lam h n)) atTop (𝓝 0) := by
-  have hal:|lam|≤1:=abs_le.mpr ⟨by linarith [h.1],h.2⟩
-  have hl2:lam^2≤1:=by nlinarith [sq_abs lam,abs_nonneg lam]
-  have hc:0 < mass p K*lam^2:=mul_pos hp (sq_pos_of_ne_zero hl)
-  have hc1:mass p K*lam^2≤1 := calc
-    mass p K*lam^2 ≤ 1*lam^2 := mul_le_mul_of_nonneg_right (mass_le_one p K) (sq_nonneg lam)
-    _ = lam^2 := one_mul (lam^2)
-    _ ≤ 1 := hl2
-  apply squeeze_zero (fun n=>advantage_nonneg (observedLaw p lam h n)) _ (decaySeq_tendsto _ hc hc1)
-  intro n
-  exact (orderedAdvantage_dominates p lam h n).trans (advantage_le_pair_bound _ _
-    (ordered_pairTV_decay p d lam h hD hmean K hK hd n))
-end
-end ThreeState
-
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-noncomputable section
-
- 
-theorem regular_dobrushin_nonreconstruction (b:ℕ) (hb:2≤b) (lam:ℝ)
-    (h:Admissible lam) (hd:(b:ℝ)*|lam|≤1) :
-    Tendsto (regularAdvantage b lam h) atTop (𝓝 0) := by
-  by_cases hl:lam=0
-  · apply regular_small_negative_nonreconstruction b hb lam h (by simp [hl])
-    simp [hl]
-  · exact nonreconstruction_dobrushin (PMF.pure b) (b:ℝ) lam h
-      (regular_moments b).hasMean (regular_moments b).first b hb
-      (by simp [mass_pure]) hl hd
-
- 
-
-theorem poisson_dobrushin_nonreconstruction (d:ℝ) (hd:1<d) (lam:ℝ)
-    (h:Admissible lam) (hD:d*|lam|≤1) :
-    Tendsto (poissonAdvantage d hd lam h) atTop (𝓝 0) := by
-  by_cases hl:lam=0
-  · apply poisson_nonnegative_nonreconstruction d hd lam h (by simp [hl])
-    simp [hl]
-  · exact nonreconstruction_dobrushin (poissonOffspring d (by linarith)) d lam h
-      (poisson_moments d (by linarith)).hasMean (poisson_moments d (by linarith)).first
-      2 (by omega) (by rw [poisson_mass]; positivity) hl hD
-
- 
-
-theorem binary_negative_nonreconstruction (lam:ℝ) (h:Admissible lam) (hl:lam≤0) :
-    Tendsto (regularAdvantage 2 lam h) atTop (𝓝 0) := by
-  apply regular_dobrushin_nonreconstruction 2 (by omega) lam h
-  rw [abs_of_nonpos hl]
-  norm_num
-  linarith [h.1]
-
- 
-
-theorem poisson_le_two_negative_nonreconstruction (d:ℝ) (hd:1<d) (hd2:d≤2)
-    (lam:ℝ) (h:Admissible lam) (hl:lam≤0) :
-    Tendsto (poissonAdvantage d hd lam h) atTop (𝓝 0) := by
-  apply poisson_dobrushin_nonreconstruction d hd lam h
-  have ha:|lam|≤1/2:=by rw [abs_of_nonpos hl];linarith [h.1]
-  nlinarith [mul_nonneg (show 0≤d by linarith) (show 0≤1/2-|lam| by linarith)]
-end
-end ThreeState
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeState.Couplings
+import OAI.Probability.ThreeState.Noise
+
+namespace OAI
+
+/-! Signed small-channel and Dobrushin non-reconstruction regions. -/
+
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+
+ 
+
+theorem regular_small_negative_nonreconstruction (b : ℕ) (hb : 2 ≤ b) (lam : ℝ)
+    (h : Admissible lam) (hl : lam ≤ 0) (hc : 4*(b:ℝ)*lam^2 ≤ 1) :
+    Tendsto (regularAdvantage b lam h) atTop (𝓝 0) := by
+  have hb0 : (0:ℝ) < b := by exact_mod_cast (show 0<b by omega)
+  let L := Real.sqrt ((b:ℝ)⁻¹)
+  have hL0 : 0 < L := Real.sqrt_pos.mpr (inv_pos.mpr hb0)
+  have hLsq : L^2=(b:ℝ)⁻¹ := Real.sq_sqrt (inv_nonneg.mpr hb0.le)
+  have hK : (b:ℝ)*L^2=1 := by rw [hLsq, mul_inv_cancel₀ hb0.ne']
+  have hbr : (2:ℝ)≤b := by exact_mod_cast hb
+  have hL1 : L≤1 := by nlinarith [sq_nonneg L]
+  have hL : Admissible L := ⟨by linarith, hL1⟩
+  have hlow : -L/2≤lam := by
+    have hh : 4*lam^2≤L^2 := (mul_le_mul_iff_right₀ hb0).mp (by nlinarith only [hc,hK])
+    nlinarith
+  let r := lam/L
+  have hr0 : -(1/2:ℝ)≤r := (le_div_iff₀ hL0).mpr (by linarith)
+  have hr1 : r≤1 := (div_le_one hL0).mpr (by linarith)
+  have hr : Admissible r := ⟨hr0,hr1⟩
+  have heq : L*r=lam := by dsimp [r]; field_simp
+  have hp : Admissible (L*r) := by rwa [heq]
+  have ht := ordered_tendsto_zero_of_uniform_fixedpoints (PMF.pure b) L hL
+    (regular_positive_critical_uniform b hb L hL hL0 hK)
+  apply squeeze_zero (fun n => advantage_nonneg (regularLaw b lam h n))
+    (g := fun n => advantage (orderedLaw (PMF.pure b) L hL n)) _ ht
+  intro n
+  have hg := orderedAdvantage_degrade (PMF.pure b) L r hL hr hp n
+  have heLaw : orderedLaw (PMF.pure b) (L*r) hp n = orderedLaw (PMF.pure b) lam h n := by congr 1
+  rw [heLaw] at hg
+  exact (orderedAdvantage_dominates (PMF.pure b) lam h n).trans hg
+
+ 
+
+theorem poisson_small_negative_nonreconstruction (d : ℝ) (hd : 1 < d) (lam : ℝ)
+    (h : Admissible lam) (hl : lam ≤ 0) (hc : 4*d*lam^2 ≤ 1) :
+    Tendsto (poissonAdvantage d hd lam h) atTop (𝓝 0) := by
+  have hd0 : 0 < d := by linarith
+  let L := Real.sqrt d⁻¹
+  have hL0 : 0 < L := Real.sqrt_pos.mpr (inv_pos.mpr hd0)
+  have hLsq : L^2=d⁻¹ := Real.sq_sqrt (inv_nonneg.mpr hd0.le)
+  have hK : d*L^2=1 := by rw [hLsq, mul_inv_cancel₀ hd0.ne']
+  have hL1 : L≤1 := by
+    have hi : d⁻¹≤1 := (inv_le_one₀ hd0).mpr hd.le
+    nlinarith [sq_nonneg L]
+  have hL : Admissible L := ⟨by linarith, hL1⟩
+  have hlow : -L/2≤lam := by
+    have hh : 4*lam^2≤L^2 := (mul_le_mul_iff_right₀ hd0).mp (by nlinarith only [hc,hK])
+    nlinarith
+  let r := lam/L
+  have hr0 : -(1/2:ℝ)≤r := (le_div_iff₀ hL0).mpr (by linarith)
+  have hr1 : r≤1 := (div_le_one hL0).mpr (by linarith)
+  have hr : Admissible r := ⟨hr0,hr1⟩
+  have heq : L*r=lam := by dsimp [r]; field_simp
+  have hp : Admissible (L*r) := by rwa [heq]
+  have ht := ordered_tendsto_zero_of_uniform_fixedpoints (poissonOffspring d hd0.le) L hL
+    (poisson_positive_critical_uniform d hd L hL hL0 hK)
+  apply squeeze_zero (fun n => advantage_nonneg (poissonLaw d hd0.le lam h n))
+    (g := fun n => advantage (orderedLaw (poissonOffspring d hd0.le) L hL n)) _ ht
+  intro n
+  have hg := orderedAdvantage_degrade (poissonOffspring d hd0.le) L r hL hr hp n
+  have heLaw : orderedLaw (poissonOffspring d hd0.le) (L*r) hp n = orderedLaw (poissonOffspring d hd0.le) lam h n := by congr 1
+  rw [heLaw] at hg
+  exact (orderedAdvantage_dominates (poissonOffspring d hd0.le) lam h n).trans hg
+
+end ThreeState
+
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+noncomputable section
+
+ 
+
+lemma compose_pairTV_bound {α:Type*} (p:PMF ℕ) (law:Spin→PMF α) (d lam x:ℝ)
+    (h:Admissible lam) (hD:HasMean p (fun k:ℕ=>(k:ℝ)))
+    (hmean:mean p (fun k:ℕ=>(k:ℝ))=d) (K:ℕ) (hK:2≤K)
+    (hx0:0≤x) (hx1:x≤1) (hx:∀i j,tvMass (law i) (law j)≤x) (i j:Spin) :
+    tvMass (composeLaw p law lam h i) (composeLaw p law lam h j) ≤
+      d*|lam| *x-mass p K*(|lam| *x)^2 := by
+  let t:ℝ:=|lam| *x
+  have hl:|lam|≤1:=abs_le.mpr ⟨by linarith [h.1],h.2⟩
+  have ht0:0≤t:=mul_nonneg (abs_nonneg _) hx0
+  have ht1:t≤1 := calc
+    t ≤ 1*x := mul_le_mul_of_nonneg_right hl hx0
+    _ = x := one_mul x
+    _ ≤ 1 := hx1
+  have he:tvMass (edgeLaw law lam h i) (edgeLaw law lam h j)≤t := by
+    rw [tvMass_edge]
+    exact mul_le_mul_of_nonneg_left (hx i j) (abs_nonneg _)
+  have hpoint (k:ℕ) : tvMass (iidVector (edgeLaw law lam h i) k) (iidVector (edgeLaw law lam h j) k)≤
+      1-(1-t)^k := by
+    apply (tvMass_iidVector_le _ _ k).trans
+    have hp:=pow_le_pow_left₀ (by linarith : 0≤1-t) (by linarith : 1-t≤1-tvMass (edgeLaw law lam h i) (edgeLaw law lam h j)) k
+    linarith
+  have hs1:Summable (fun k:ℕ=>mass p k*((k:ℝ)*t)):=hD.mul_const t
+  have hs2:Summable (fun k:ℕ=>if k=K then mass p K*t^2 else 0):=(hasSum_ite_eq K (mass p K*t^2)).summable
+  have hu (k:ℕ) : mass p k*tvMass (iidVector (edgeLaw law lam h i) k) (iidVector (edgeLaw law lam h j) k)≤
+      mass p k*((k:ℝ)*t)-(if k=K then mass p K*t^2 else 0) := by
+    by_cases hk:k=K
+    · subst k
+      simp only [ite_true]
+      have hb:=bernoulli_quadratic_lower t ht0 ht1 K hK
+      have hv:=hpoint K
+      have hh:=mul_le_mul_of_nonneg_left (show tvMass (iidVector (edgeLaw law lam h i) K) (iidVector (edgeLaw law lam h j) K)≤(K:ℝ)*t-t^2 by linarith) (mass_nonneg p K)
+      nlinarith only [hh]
+    · rw [ite_eq_right hk,sub_zero]
+      have hb:=bernoulli_lower t ht0 ht1 k
+      exact mul_le_mul_of_nonneg_left (by linarith [hpoint k]) (mass_nonneg p k)
+  have hh:=Summable.tsum_le_tsum hu (summable_weighted_tv p
+    (fun k=>iidVector (edgeLaw law lam h i) k) (fun k=>iidVector (edgeLaw law lam h j) k)) (hs1.sub hs2)
+  have hsum: (∑'k:ℕ,mass p k*((k:ℝ)*t))=d*t := by
+    change mean p (fun k:ℕ=>(k:ℝ)*t)=_
+    rw [mean_mul_const,hmean]
+  rw [hs1.tsum_sub hs2,hsum,tsum_ite_eq] at hh
+  simpa only [composeLaw,tvMass_sigmaPMF,t,mul_assoc] using hh
+
+lemma ordered_pairTV_decay (p:PMF ℕ) (d lam:ℝ) (h:Admissible lam)
+    (hD:HasMean p (fun k:ℕ=>(k:ℝ))) (hmean:mean p (fun k:ℕ=>(k:ℝ))=d)
+    (K:ℕ) (hK:2≤K) (hd:d*|lam|≤1) (n:ℕ) (i j:Spin) :
+    tvMass (orderedLaw p lam h n i) (orderedLaw p lam h n j)≤decaySeq (mass p K*lam^2) n := by
+  have hl:|lam|≤1:=abs_le.mpr ⟨by linarith [h.1],h.2⟩
+  have hl2:lam^2≤1:=by nlinarith [sq_abs lam,abs_nonneg lam]
+  have hc0:0 ≤ mass p K*lam^2:=mul_nonneg (mass_nonneg p K) (sq_nonneg lam)
+  have hc1:mass p K*lam^2≤1 := calc
+    mass p K*lam^2 ≤ 1*lam^2 := mul_le_mul_of_nonneg_right (mass_le_one p K) (sq_nonneg lam)
+    _ = lam^2 := one_mul (lam^2)
+    _ ≤ 1 := hl2
+  induction n generalizing i j with
+  | zero => exact tvMass_le_one _ _
+  | succ n ih =>
+    have hb:=decaySeq_bounds (mass p K*lam^2) hc0 hc1 n
+    have hh:=compose_pairTV_bound p (orderedLaw p lam h n) d lam (decaySeq (mass p K*lam^2) n)
+      h hD hmean K hK hb.1 hb.2 (fun i j=>ih i j) i j
+    have hg:=mul_le_mul_of_nonneg_right hd hb.1
+    change tvMass (composeLaw p (orderedLaw p lam h n) lam h i)
+      (composeLaw p (orderedLaw p lam h n) lam h j)≤_
+    change _≤decaySeq (mass p K*lam^2) n-(mass p K*lam^2)*(decaySeq (mass p K*lam^2) n)^2
+    simp only [mul_pow,sq_abs] at hh
+    nlinarith only [hh,hg]
+
+ 
+
+theorem nonreconstruction_dobrushin (p:PMF ℕ) (d lam:ℝ) (h:Admissible lam)
+    (hD:HasMean p (fun k:ℕ=>(k:ℝ))) (hmean:mean p (fun k:ℕ=>(k:ℝ))=d)
+    (K:ℕ) (hK:2≤K) (hp:0 < mass p K) (hl:lam≠0) (hd:d*|lam|≤1) :
+    Tendsto (fun n=>advantage (observedLaw p lam h n)) atTop (𝓝 0) := by
+  have hal:|lam|≤1:=abs_le.mpr ⟨by linarith [h.1],h.2⟩
+  have hl2:lam^2≤1:=by nlinarith [sq_abs lam,abs_nonneg lam]
+  have hc:0 < mass p K*lam^2:=mul_pos hp (sq_pos_of_ne_zero hl)
+  have hc1:mass p K*lam^2≤1 := calc
+    mass p K*lam^2 ≤ 1*lam^2 := mul_le_mul_of_nonneg_right (mass_le_one p K) (sq_nonneg lam)
+    _ = lam^2 := one_mul (lam^2)
+    _ ≤ 1 := hl2
+  apply squeeze_zero (fun n=>advantage_nonneg (observedLaw p lam h n)) _ (decaySeq_tendsto _ hc hc1)
+  intro n
+  exact (orderedAdvantage_dominates p lam h n).trans (advantage_le_pair_bound _ _
+    (ordered_pairTV_decay p d lam h hD hmean K hK hd n))
+end
+end ThreeState
+
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+noncomputable section
+
+ 
+theorem regular_dobrushin_nonreconstruction (b:ℕ) (hb:2≤b) (lam:ℝ)
+    (h:Admissible lam) (hd:(b:ℝ)*|lam|≤1) :
+    Tendsto (regularAdvantage b lam h) atTop (𝓝 0) := by
+  by_cases hl:lam=0
+  · apply regular_small_negative_nonreconstruction b hb lam h (by simp [hl])
+    simp [hl]
+  · exact nonreconstruction_dobrushin (PMF.pure b) (b:ℝ) lam h
+      (regular_moments b).hasMean (regular_moments b).first b hb
+      (by simp [mass_pure]) hl hd
+
+ 
+
+theorem poisson_dobrushin_nonreconstruction (d:ℝ) (hd:1<d) (lam:ℝ)
+    (h:Admissible lam) (hD:d*|lam|≤1) :
+    Tendsto (poissonAdvantage d hd lam h) atTop (𝓝 0) := by
+  by_cases hl:lam=0
+  · apply poisson_nonnegative_nonreconstruction d hd lam h (by simp [hl])
+    simp [hl]
+  · exact nonreconstruction_dobrushin (poissonOffspring d (by linarith)) d lam h
+      (poisson_moments d (by linarith)).hasMean (poisson_moments d (by linarith)).first
+      2 (by omega) (by rw [poisson_mass]; positivity) hl hD
+
+ 
+
+theorem binary_negative_nonreconstruction (lam:ℝ) (h:Admissible lam) (hl:lam≤0) :
+    Tendsto (regularAdvantage 2 lam h) atTop (𝓝 0) := by
+  apply regular_dobrushin_nonreconstruction 2 (by omega) lam h
+  rw [abs_of_nonpos hl]
+  norm_num
+  linarith [h.1]
+
+ 
+
+theorem poisson_le_two_negative_nonreconstruction (d:ℝ) (hd:1<d) (hd2:d≤2)
+    (lam:ℝ) (h:Admissible lam) (hl:lam≤0) :
+    Tendsto (poissonAdvantage d hd lam h) atTop (𝓝 0) := by
+  apply poisson_dobrushin_nonreconstruction d hd lam h
+  have ha:|lam|≤1/2:=by rw [abs_of_nonpos hl];linarith [h.1]
+  nlinarith [mul_nonneg (show 0≤d by linarith) (show 0≤1/2-|lam| by linarith)]
+end
+end ThreeState
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeState/Noise.lean b/lean/OAI/Probability/ThreeState/Noise.lean
index 956e43ba99f43963ef01ed32a51fe1c5ee61e400..1463fb4df97350af4684fe27e2152bc2174f129d 100644
--- a/lean/OAI/Probability/ThreeState/Noise.lean
+++ b/lean/OAI/Probability/ThreeState/Noise.lean
@@ -1,182 +1,182 @@
-import Mathlib
-import OAI.Probability.ThreeState.PoissonCritical
-
-namespace OAI
-
-/-! Additional channel noise and nonnegative-channel non-reconstruction. -/
-
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-
-lemma pmf_ext_mass {α : Type*} {p q : PMF α} (he : ∀ a, mass p a=mass q a) : p=q := by
-  apply PMF.ext
-  intro a
-  rcases (ENNReal.toReal_eq_toReal_iff (p a) (q a)).mp (he a) with he | he | he
-  · exact he
-  · exact False.elim ((q.apply_ne_top a) he.2)
-  · exact False.elim ((p.apply_ne_top a) he.1)
-
-lemma channelPMF_comp (a b : ℝ) (ha : Admissible a) (hb : Admissible b) (hab : Admissible (a*b)) (i : Spin) :
-    (channelPMF a ha i).bind (channelPMF b hb) = channelPMF (a*b) hab i := by
-  apply pmf_ext_mass
-  intro j
-  rw [mass_bind]
-  simp only [tsum_fintype, mass_channelPMF]
-  fin_cases i <;> fin_cases j <;> simp [channel, Fin.sum_univ_succ] <;> ring
-
-noncomputable def orderedExtraNoise (r : ℝ) (hr : Admissible r) (n : ℕ)
-    (y : OrderedObservation n) : PMF (OrderedObservation n) :=
-  (channelPMF r hr 0).map (fun j : Spin => renameOrdered n (Equiv.addRight j) y)
-
-lemma orderedLaw_extraNoise (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
-    (r : ℝ) (hr : Admissible r) (n : ℕ) (i : Spin) :
-    (orderedLaw offspring lam h n i).bind (orderedExtraNoise r hr n) =
-      (channelPMF r hr i).bind (orderedLaw offspring lam h n) := by
-  unfold orderedExtraNoise
-  simp only [PMF.map, Function.comp_def]
-  rw [PMF.bind_comm]
-  have he (j : Spin) : (orderedLaw offspring lam h n i).bind
-      (fun y => PMF.pure (renameOrdered n (Equiv.addRight j) y)) = orderedLaw offspring lam h n (i+j) := by
-    exact orderedLaw_rename offspring lam h (Equiv.addRight j) n i
-  simp_rw [he]
-  have hm : (channelPMF r hr 0).map (Equiv.addLeft i) = channelPMF r hr i := by
-    simpa using channelPMF_map r hr (Equiv.addLeft i) 0
-  rw [← hm, PMF.bind_map]
-  rfl
-
- 
-
-noncomputable def orderedDegrade (r : ℝ) (hr : Admissible r) :
-    (n : ℕ) → OrderedObservation n → PMF (OrderedObservation n)
-  | 0 => PMF.pure
-  | n+1 => fun ⟨k,v⟩ => (vectorTraverse (fun y => (orderedDegrade r hr n y).bind (orderedExtraNoise r hr n)) k v).map (Sigma.mk k)
-
-lemma orderedLaw_degrade (offspring : PMF ℕ) (lam r : ℝ) (h : Admissible lam)
-    (hr : Admissible r) (hp : Admissible (lam*r)) (n : ℕ) (i : Spin) :
-    (orderedLaw offspring lam h n i).bind (orderedDegrade r hr n) =
-      orderedLaw offspring (lam*r) hp n i := by
-  induction n generalizing i with
-  | zero => simp only [orderedLaw, orderedDegrade, PMF.bind_pure]
-  | succ n ih =>
-    simp only [orderedLaw, orderedDegrade, PMF.bind_bind, PMF.bind_map, Function.comp_def]
-    apply congrArg (PMF.bind offspring)
-    funext k
-    rw [← PMF.map_bind, iidVector_bind_traverse, PMF.bind_bind]
-    have he (j : Spin) : (orderedLaw offspring lam h n j).bind
-        (fun y => (orderedDegrade r hr n y).bind (orderedExtraNoise r hr n)) =
-          (channelPMF r hr j).bind (orderedLaw offspring (lam*r) hp n) := by
-      rw [← PMF.bind_bind, ih, orderedLaw_extraNoise]
-    simp_rw [he]
-    rw [← PMF.bind_bind, channelPMF_comp lam r h hr hp]
-
-lemma orderedAdvantage_degrade (offspring : PMF ℕ) (lam r : ℝ) (h : Admissible lam)
-    (hr : Admissible r) (hp : Admissible (lam*r)) (n : ℕ) :
-    advantage (orderedLaw offspring (lam*r) hp n) ≤ advantage (orderedLaw offspring lam h n) := by
-  have he : orderedLaw offspring (lam*r) hp n = fun i =>
-      (orderedLaw offspring lam h n i).bind (orderedDegrade r hr n) := funext (fun i => (orderedLaw_degrade offspring lam r h hr hp n i).symm)
-  rw [he]
-  exact advantage_bind_le _ _
-
-lemma orderedAdvantage_antitone (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) :
-    Antitone (fun n => advantage (orderedLaw offspring lam h n)) := by
-  apply antitone_nat_of_succ_le
-  intro n
-  have he : orderedLaw offspring lam h (n+1) = fun i =>
-      (orderedLaw offspring lam h n i).bind (orderedGrow offspring lam h n) := funext (fun i => (orderedLaw_grow offspring lam h n i).symm)
-  rw [he]
-  exact advantage_bind_le _ _
-
-lemma ordered_tendsto_zero_of_uniform_fixedpoints (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
-    (hunif : ∀ Q : ProbabilityMeasure Message, IsPosteriorFixedPoint offspring lam h Q → SpinSymmetric Q → posteriorMu Q=0) :
-    Tendsto (fun n => advantage (orderedLaw offspring lam h n)) atTop (𝓝 0) := by
-  obtain ⟨Q,hQ,hs,φ,hφ,hlim⟩ := ordered_exists_symmetric_fixedpoint offspring lam h
-  have htv := integral_tv_zero_of_mu_zero Q (hunif Q hQ hs)
-  have hordered := advantage_tendsto_of_likelihood (fun n => orderedLaw offspring lam h (φ n)) hlim
-  rw [htv] at hordered
-  let a := fun n => advantage (orderedLaw offspring lam h n)
-  have hb : BddBelow (Set.range a) := ⟨0, by rintro x ⟨n,rfl⟩; exact advantage_nonneg _⟩
-  have hall := tendsto_atTop_ciInf (orderedAdvantage_antitone offspring lam h) hb
-  have hz : (⨅ n, a n)=0 := tendsto_nhds_unique (hall.comp hφ.tendsto_atTop) hordered
-  rwa [hz] at hall
-
-end ThreeState
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-
- 
-
-theorem poisson_nonnegative_nonreconstruction (d : ℝ) (hd : 1 < d) (lam : ℝ)
-    (h : Admissible lam) (hl0 : 0 ≤ lam) (hc : d*lam^2 ≤ 1) :
-    Tendsto (poissonAdvantage d (by linarith) lam h) atTop (𝓝 0) := by
-  have hd0 : 0 < d := by linarith
-  let L := Real.sqrt d⁻¹
-  have hL0 : 0 < L := Real.sqrt_pos.mpr (inv_pos.mpr hd0)
-  have hLsq : L^2=d⁻¹ := Real.sq_sqrt (inv_nonneg.mpr hd0.le)
-  have hK : d*L^2=1 := by rw [hLsq, mul_inv_cancel₀ hd0.ne']
-  have hL1 : L≤1 := by
-    have hi : d⁻¹≤1 := (inv_le_one₀ hd0).mpr hd.le
-    nlinarith [sq_nonneg L]
-  have hL : Admissible L := ⟨by linarith, hL1⟩
-  have hlL : lam≤L := by
-    have hh : lam^2≤L^2 := (mul_le_mul_iff_right₀ hd0).mp (by nlinarith only [hc,hK])
-    nlinarith
-  let r := lam/L
-  have hr0 : 0≤r := div_nonneg hl0 hL0.le
-  have hr1 : r≤1 := (div_le_one hL0).mpr hlL
-  have hr : Admissible r := ⟨by linarith, hr1⟩
-  have heq : L*r=lam := by dsimp [r]; field_simp
-  have hp : Admissible (L*r) := by rwa [heq]
-  have ht := ordered_tendsto_zero_of_uniform_fixedpoints (poissonOffspring d hd0.le) L hL
-    (poisson_positive_critical_uniform d hd L hL hL0 hK)
-  apply squeeze_zero (fun n => advantage_nonneg _)
-    (g := fun n => advantage (orderedLaw (poissonOffspring d hd0.le) L hL n)) _ ht
-  intro n
-  have hg := orderedAdvantage_degrade (poissonOffspring d hd0.le) L r hL hr hp n
-  have heLaw : orderedLaw (poissonOffspring d hd0.le) (L*r) hp n = orderedLaw (poissonOffspring d hd0.le) lam h n := by
-    congr 1
-  rw [heLaw] at hg
-  exact (orderedAdvantage_dominates (poissonOffspring d hd0.le) lam h n).trans hg
-
-end ThreeState
-
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-
- 
-theorem regular_nonnegative_nonreconstruction (b : ℕ) (hb : 2 ≤ b) (lam : ℝ)
-    (h : Admissible lam) (hl0 : 0 ≤ lam) (hc : (b:ℝ)*lam^2 ≤ 1) :
-    Tendsto (regularAdvantage b lam h) atTop (𝓝 0) := by
-  have hb0 : (0:ℝ) < b := by exact_mod_cast (show 0<b by omega)
-  let L := Real.sqrt ((b:ℝ)⁻¹)
-  have hL0 : 0 < L := Real.sqrt_pos.mpr (inv_pos.mpr hb0)
-  have hLsq : L^2=(b:ℝ)⁻¹ := Real.sq_sqrt (inv_nonneg.mpr hb0.le)
-  have hK : (b:ℝ)*L^2=1 := by rw [hLsq, mul_inv_cancel₀ hb0.ne']
-  have hbr : (2:ℝ)≤b := by exact_mod_cast hb
-  have hL1 : L≤1 := by nlinarith [sq_nonneg L]
-  have hL : Admissible L := ⟨by linarith, hL1⟩
-  have hlL : lam≤L := by
-    have hh : lam^2≤L^2 := (mul_le_mul_iff_right₀ hb0).mp (by nlinarith only [hc,hK])
-    nlinarith
-  let r := lam/L
-  have hr0 : 0≤r := div_nonneg hl0 hL0.le
-  have hr1 : r≤1 := (div_le_one hL0).mpr hlL
-  have hr : Admissible r := ⟨by linarith, hr1⟩
-  have heq : L*r=lam := by dsimp [r]; field_simp
-  have hp : Admissible (L*r) := by rwa [heq]
-  have ht := ordered_tendsto_zero_of_uniform_fixedpoints (PMF.pure b) L hL
-    (regular_positive_critical_uniform b hb L hL hL0 hK)
-  apply squeeze_zero (fun n => advantage_nonneg _)
-    (g := fun n => advantage (orderedLaw (PMF.pure b) L hL n)) _ ht
-  intro n
-  have hg := orderedAdvantage_degrade (PMF.pure b) L r hL hr hp n
-  have heLaw : orderedLaw (PMF.pure b) (L*r) hp n = orderedLaw (PMF.pure b) lam h n := by
-    congr 1
-  rw [heLaw] at hg
-  exact (orderedAdvantage_dominates (PMF.pure b) lam h n).trans hg
-
-end ThreeState
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeState.PoissonCritical
+
+namespace OAI
+
+/-! Additional channel noise and nonnegative-channel non-reconstruction. -/
+
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+
+lemma pmf_ext_mass {α : Type*} {p q : PMF α} (he : ∀ a, mass p a=mass q a) : p=q := by
+  apply PMF.ext
+  intro a
+  rcases (ENNReal.toReal_eq_toReal_iff (p a) (q a)).mp (he a) with he | he | he
+  · exact he
+  · exact False.elim ((q.apply_ne_top a) he.2)
+  · exact False.elim ((p.apply_ne_top a) he.1)
+
+lemma channelPMF_comp (a b : ℝ) (ha : Admissible a) (hb : Admissible b) (hab : Admissible (a*b)) (i : Spin) :
+    (channelPMF a ha i).bind (channelPMF b hb) = channelPMF (a*b) hab i := by
+  apply pmf_ext_mass
+  intro j
+  rw [mass_bind]
+  simp only [tsum_fintype, mass_channelPMF]
+  fin_cases i <;> fin_cases j <;> simp [channel, Fin.sum_univ_succ] <;> ring
+
+noncomputable def orderedExtraNoise (r : ℝ) (hr : Admissible r) (n : ℕ)
+    (y : OrderedObservation n) : PMF (OrderedObservation n) :=
+  (channelPMF r hr 0).map (fun j : Spin => renameOrdered n (Equiv.addRight j) y)
+
+lemma orderedLaw_extraNoise (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
+    (r : ℝ) (hr : Admissible r) (n : ℕ) (i : Spin) :
+    (orderedLaw offspring lam h n i).bind (orderedExtraNoise r hr n) =
+      (channelPMF r hr i).bind (orderedLaw offspring lam h n) := by
+  unfold orderedExtraNoise
+  simp only [PMF.map, Function.comp_def]
+  rw [PMF.bind_comm]
+  have he (j : Spin) : (orderedLaw offspring lam h n i).bind
+      (fun y => PMF.pure (renameOrdered n (Equiv.addRight j) y)) = orderedLaw offspring lam h n (i+j) := by
+    exact orderedLaw_rename offspring lam h (Equiv.addRight j) n i
+  simp_rw [he]
+  have hm : (channelPMF r hr 0).map (Equiv.addLeft i) = channelPMF r hr i := by
+    simpa using channelPMF_map r hr (Equiv.addLeft i) 0
+  rw [← hm, PMF.bind_map]
+  rfl
+
+ 
+
+noncomputable def orderedDegrade (r : ℝ) (hr : Admissible r) :
+    (n : ℕ) → OrderedObservation n → PMF (OrderedObservation n)
+  | 0 => PMF.pure
+  | n+1 => fun ⟨k,v⟩ => (vectorTraverse (fun y => (orderedDegrade r hr n y).bind (orderedExtraNoise r hr n)) k v).map (Sigma.mk k)
+
+lemma orderedLaw_degrade (offspring : PMF ℕ) (lam r : ℝ) (h : Admissible lam)
+    (hr : Admissible r) (hp : Admissible (lam*r)) (n : ℕ) (i : Spin) :
+    (orderedLaw offspring lam h n i).bind (orderedDegrade r hr n) =
+      orderedLaw offspring (lam*r) hp n i := by
+  induction n generalizing i with
+  | zero => simp only [orderedLaw, orderedDegrade, PMF.bind_pure]
+  | succ n ih =>
+    simp only [orderedLaw, orderedDegrade, PMF.bind_bind, PMF.bind_map, Function.comp_def]
+    apply congrArg (PMF.bind offspring)
+    funext k
+    rw [← PMF.map_bind, iidVector_bind_traverse, PMF.bind_bind]
+    have he (j : Spin) : (orderedLaw offspring lam h n j).bind
+        (fun y => (orderedDegrade r hr n y).bind (orderedExtraNoise r hr n)) =
+          (channelPMF r hr j).bind (orderedLaw offspring (lam*r) hp n) := by
+      rw [← PMF.bind_bind, ih, orderedLaw_extraNoise]
+    simp_rw [he]
+    rw [← PMF.bind_bind, channelPMF_comp lam r h hr hp]
+
+lemma orderedAdvantage_degrade (offspring : PMF ℕ) (lam r : ℝ) (h : Admissible lam)
+    (hr : Admissible r) (hp : Admissible (lam*r)) (n : ℕ) :
+    advantage (orderedLaw offspring (lam*r) hp n) ≤ advantage (orderedLaw offspring lam h n) := by
+  have he : orderedLaw offspring (lam*r) hp n = fun i =>
+      (orderedLaw offspring lam h n i).bind (orderedDegrade r hr n) := funext (fun i => (orderedLaw_degrade offspring lam r h hr hp n i).symm)
+  rw [he]
+  exact advantage_bind_le _ _
+
+lemma orderedAdvantage_antitone (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) :
+    Antitone (fun n => advantage (orderedLaw offspring lam h n)) := by
+  apply antitone_nat_of_succ_le
+  intro n
+  have he : orderedLaw offspring lam h (n+1) = fun i =>
+      (orderedLaw offspring lam h n i).bind (orderedGrow offspring lam h n) := funext (fun i => (orderedLaw_grow offspring lam h n i).symm)
+  rw [he]
+  exact advantage_bind_le _ _
+
+lemma ordered_tendsto_zero_of_uniform_fixedpoints (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
+    (hunif : ∀ Q : ProbabilityMeasure Message, IsPosteriorFixedPoint offspring lam h Q → SpinSymmetric Q → posteriorMu Q=0) :
+    Tendsto (fun n => advantage (orderedLaw offspring lam h n)) atTop (𝓝 0) := by
+  obtain ⟨Q,hQ,hs,φ,hφ,hlim⟩ := ordered_exists_symmetric_fixedpoint offspring lam h
+  have htv := integral_tv_zero_of_mu_zero Q (hunif Q hQ hs)
+  have hordered := advantage_tendsto_of_likelihood (fun n => orderedLaw offspring lam h (φ n)) hlim
+  rw [htv] at hordered
+  let a := fun n => advantage (orderedLaw offspring lam h n)
+  have hb : BddBelow (Set.range a) := ⟨0, by rintro x ⟨n,rfl⟩; exact advantage_nonneg _⟩
+  have hall := tendsto_atTop_ciInf (orderedAdvantage_antitone offspring lam h) hb
+  have hz : (⨅ n, a n)=0 := tendsto_nhds_unique (hall.comp hφ.tendsto_atTop) hordered
+  rwa [hz] at hall
+
+end ThreeState
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+
+ 
+
+theorem poisson_nonnegative_nonreconstruction (d : ℝ) (hd : 1 < d) (lam : ℝ)
+    (h : Admissible lam) (hl0 : 0 ≤ lam) (hc : d*lam^2 ≤ 1) :
+    Tendsto (poissonAdvantage d (by linarith) lam h) atTop (𝓝 0) := by
+  have hd0 : 0 < d := by linarith
+  let L := Real.sqrt d⁻¹
+  have hL0 : 0 < L := Real.sqrt_pos.mpr (inv_pos.mpr hd0)
+  have hLsq : L^2=d⁻¹ := Real.sq_sqrt (inv_nonneg.mpr hd0.le)
+  have hK : d*L^2=1 := by rw [hLsq, mul_inv_cancel₀ hd0.ne']
+  have hL1 : L≤1 := by
+    have hi : d⁻¹≤1 := (inv_le_one₀ hd0).mpr hd.le
+    nlinarith [sq_nonneg L]
+  have hL : Admissible L := ⟨by linarith, hL1⟩
+  have hlL : lam≤L := by
+    have hh : lam^2≤L^2 := (mul_le_mul_iff_right₀ hd0).mp (by nlinarith only [hc,hK])
+    nlinarith
+  let r := lam/L
+  have hr0 : 0≤r := div_nonneg hl0 hL0.le
+  have hr1 : r≤1 := (div_le_one hL0).mpr hlL
+  have hr : Admissible r := ⟨by linarith, hr1⟩
+  have heq : L*r=lam := by dsimp [r]; field_simp
+  have hp : Admissible (L*r) := by rwa [heq]
+  have ht := ordered_tendsto_zero_of_uniform_fixedpoints (poissonOffspring d hd0.le) L hL
+    (poisson_positive_critical_uniform d hd L hL hL0 hK)
+  apply squeeze_zero (fun n => advantage_nonneg _)
+    (g := fun n => advantage (orderedLaw (poissonOffspring d hd0.le) L hL n)) _ ht
+  intro n
+  have hg := orderedAdvantage_degrade (poissonOffspring d hd0.le) L r hL hr hp n
+  have heLaw : orderedLaw (poissonOffspring d hd0.le) (L*r) hp n = orderedLaw (poissonOffspring d hd0.le) lam h n := by
+    congr 1
+  rw [heLaw] at hg
+  exact (orderedAdvantage_dominates (poissonOffspring d hd0.le) lam h n).trans hg
+
+end ThreeState
+
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+
+ 
+theorem regular_nonnegative_nonreconstruction (b : ℕ) (hb : 2 ≤ b) (lam : ℝ)
+    (h : Admissible lam) (hl0 : 0 ≤ lam) (hc : (b:ℝ)*lam^2 ≤ 1) :
+    Tendsto (regularAdvantage b lam h) atTop (𝓝 0) := by
+  have hb0 : (0:ℝ) < b := by exact_mod_cast (show 0<b by omega)
+  let L := Real.sqrt ((b:ℝ)⁻¹)
+  have hL0 : 0 < L := Real.sqrt_pos.mpr (inv_pos.mpr hb0)
+  have hLsq : L^2=(b:ℝ)⁻¹ := Real.sq_sqrt (inv_nonneg.mpr hb0.le)
+  have hK : (b:ℝ)*L^2=1 := by rw [hLsq, mul_inv_cancel₀ hb0.ne']
+  have hbr : (2:ℝ)≤b := by exact_mod_cast hb
+  have hL1 : L≤1 := by nlinarith [sq_nonneg L]
+  have hL : Admissible L := ⟨by linarith, hL1⟩
+  have hlL : lam≤L := by
+    have hh : lam^2≤L^2 := (mul_le_mul_iff_right₀ hb0).mp (by nlinarith only [hc,hK])
+    nlinarith
+  let r := lam/L
+  have hr0 : 0≤r := div_nonneg hl0 hL0.le
+  have hr1 : r≤1 := (div_le_one hL0).mpr hlL
+  have hr : Admissible r := ⟨by linarith, hr1⟩
+  have heq : L*r=lam := by dsimp [r]; field_simp
+  have hp : Admissible (L*r) := by rwa [heq]
+  have ht := ordered_tendsto_zero_of_uniform_fixedpoints (PMF.pure b) L hL
+    (regular_positive_critical_uniform b hb L hL hL0 hK)
+  apply squeeze_zero (fun n => advantage_nonneg _)
+    (g := fun n => advantage (orderedLaw (PMF.pure b) L hL n)) _ ht
+  intro n
+  have hg := orderedAdvantage_degrade (PMF.pure b) L r hL hr hp n
+  have heLaw : orderedLaw (PMF.pure b) (L*r) hp n = orderedLaw (PMF.pure b) lam h n := by
+    congr 1
+  rw [heLaw] at hg
+  exact (orderedAdvantage_dominates (PMF.pure b) lam h n).trans hg
+
+end ThreeState
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeState/Observation.lean b/lean/OAI/Probability/ThreeState/Observation.lean
index 7b60dc631ee4d9eb367fed570b348f51022e70bf..1542b1566cd2f8b58da84cf1d82605a153bb7ecb 100644
--- a/lean/OAI/Probability/ThreeState/Observation.lean
+++ b/lean/OAI/Probability/ThreeState/Observation.lean
@@ -1,271 +1,271 @@
-import Mathlib
-import OAI.Probability.ThreeState.Messages
-
-namespace OAI
-
-/-! Ordered and unlabelled boundary observations and information degradation. -/
-
-namespace ThreeState
-
- 
-
-noncomputable def iidVector {α : Type*} (p : PMF α) : (n : ℕ) → PMF (Fin n → α)
-  | 0 => PMF.pure Fin.elim0
-  | n+1 => p.bind (fun a => (iidVector p n).map (Fin.cons a))
-
-lemma iidVector_map {α β : Type*} (p : PMF α) (f : α → β) (n : ℕ) :
-    (iidVector p n).map (fun as => f ∘ as) = iidVector (p.map f) n := by
-  induction n with
-  | zero =>
-    simp only [iidVector, PMF.pure_map]
-    congr 1
-    funext i
-    exact Fin.elim0 i
-  | succ n ih =>
-    simp only [iidVector, PMF.map_bind, PMF.map_comp, PMF.bind_map]
-    apply congrArg (PMF.bind p)
-    funext a
-    rw [← ih]
-    dsimp only [Function.comp_def]
-    rw [PMF.map_comp]
-    congr 1
-    funext v j
-    cases j using Fin.cases <;> simp
-
-lemma iidVector_ofFn {α : Type*} (p : PMF α) (n : ℕ) :
-    (iidVector p n).map List.ofFn = iidList p n := by
-  induction n with
-  | zero => simp [iidVector, iidList, PMF.pure_map]
-  | succ n ih =>
-    simp only [iidVector, iidList, PMF.map_bind, PMF.map_comp]
-    apply congrArg (PMF.bind p)
-    funext a
-    rw [← ih, PMF.map_comp]
-    congr 1
-    funext v
-    simp [List.ofFn_succ]
-
-lemma iidVector_apply {α : Type*} (p : PMF α) (n : ℕ) (v : Fin n → α) :
-    iidVector p n v = ∏ j, p (v j) := by
-  classical
-  induction n with
-  | zero =>
-    have he : v = Fin.elim0 := by funext j; exact Fin.elim0 j
-    simp [iidVector, he]
-  | succ n ih =>
-    simp only [iidVector, PMF.bind_apply, PMF.map_apply]
-    have he (a : α) (w : Fin n → α) : (v = Fin.cons a w) ↔ (a=v 0 ∧ w=Fin.tail v) := by
-      constructor
-      · intro h
-        subst v
-        simp
-      · rintro ⟨rfl,rfl⟩
-        exact (Fin.cons_self_tail v).symm
-    simp_rw [he]
-    simp only [ite_and]
-    rw [tsum_eq_single (v 0)]
-    · simp [ih, Fin.prod_univ_succ, Fin.tail]
-    · intro a ha
-      simp [ha]
-
- 
-abbrev OrderedObservation : ℕ → Type
-  | 0 => Spin
-  | n+1 => Σ k : ℕ, Fin k → OrderedObservation n
-
-noncomputable def orderedLaw (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) :
-    (n : ℕ) → Spin → PMF (OrderedObservation n)
-  | 0, i => PMF.pure i
-  | n+1, i => offspring.bind (fun k =>
-      (iidVector ((channelPMF lam h i).bind (orderedLaw offspring lam h n)) k).map (Sigma.mk k))
-
-def forgetOrder : (n : ℕ) → OrderedObservation n → Observation n
-  | 0, i => i
-  | n+1, ⟨k,v⟩ => (List.ofFn (fun j : Fin k => forgetOrder n (v j)) : Multiset (Observation n))
-
-lemma orderedLaw_forget (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) (n : ℕ) (i : Spin) :
-    (orderedLaw offspring lam h n i).map (forgetOrder n) = observedLaw offspring lam h n i := by
-  induction n generalizing i with
-  | zero =>
-    change (PMF.pure i).map id = PMF.pure i
-    exact PMF.map_id _
-  | succ n ih =>
-    simp only [orderedLaw, observedLaw, PMF.map_bind, PMF.map_comp]
-    apply congrArg (PMF.bind offspring)
-    funext k
-    let p := (channelPMF lam h i).bind (orderedLaw offspring lam h n)
-    change (iidVector p k).map
-      ((fun xs : List (Observation n) => (xs : Multiset (Observation n))) ∘ List.ofFn ∘
-        (fun v => forgetOrder n ∘ v)) = _
-    rw [← PMF.map_comp, ← PMF.map_comp, iidVector_map, iidVector_ofFn]
-    congr 2
-    rw [PMF.map_bind]
-    congr 1
-    funext j
-    exact ih j
-
-lemma orderedAdvantage_dominates (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) (n : ℕ) :
-    advantage (observedLaw offspring lam h n) ≤ advantage (orderedLaw offspring lam h n) := by
-  have hh := advantage_bind_le (orderedLaw offspring lam h n) (fun y => PMF.pure (forgetOrder n y))
-  have he : (fun i => (orderedLaw offspring lam h n i).bind (fun y => PMF.pure (forgetOrder n y))) =
-      observedLaw offspring lam h n := by
-    funext i
-    exact orderedLaw_forget offspring lam h n i
-  rw [he] at hh
-  exact hh
-
-end ThreeState
-
-namespace ThreeState
-
-noncomputable def vectorTraverse {α β : Type*} (k : α → PMF β) :
-    (n : ℕ) → (Fin n → α) → PMF (Fin n → β)
-  | 0, _ => PMF.pure Fin.elim0
-  | n+1, v => (k (v 0)).bind (fun b => (vectorTraverse k n (Fin.tail v)).map (Fin.cons b))
-
-lemma vectorTraverse_cons {α β : Type*} (k : α → PMF β) (n : ℕ) (a : α) (v : Fin n → α) :
-    vectorTraverse k (n+1) (Fin.cons a v) =
-      (k a).bind (fun b => (vectorTraverse k n v).map (Fin.cons b)) := by
-  simp only [vectorTraverse, Fin.cons_zero, Fin.tail_cons]
-
-lemma iidVector_bind_traverse {α β : Type*} (p : PMF α) (k : α → PMF β) (n : ℕ) :
-    (iidVector p n).bind (vectorTraverse k n) = iidVector (p.bind k) n := by
-  induction n with
-  | zero => simp only [iidVector, PMF.pure_bind, vectorTraverse]
-  | succ n ih =>
-    simp only [iidVector, PMF.bind_bind, PMF.bind_map, Function.comp_def, vectorTraverse_cons]
-    apply congrArg (PMF.bind p)
-    funext a
-    rw [PMF.bind_comm]
-    apply congrArg (PMF.bind (k a))
-    funext b
-    rw [← PMF.map_bind, ih]
-
-noncomputable def orderedGrow (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) :
-    (n : ℕ) → OrderedObservation n → PMF (OrderedObservation (n+1))
-  | 0 => orderedLaw offspring lam h 1
-  | n+1 => fun ⟨k,v⟩ => (vectorTraverse (orderedGrow offspring lam h n) k v).map (Sigma.mk k)
-
-lemma orderedLaw_grow (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) (n : ℕ) (i : Spin) :
-    (orderedLaw offspring lam h n i).bind (orderedGrow offspring lam h n) =
-      orderedLaw offspring lam h (n+1) i := by
-  induction n generalizing i with
-  | zero => exact PMF.pure_bind i _
-  | succ n ih =>
-    simp only [orderedLaw, orderedGrow, PMF.bind_bind, PMF.bind_map, Function.comp_def]
-    apply congrArg (PMF.bind offspring)
-    funext k
-    rw [← PMF.map_bind, iidVector_bind_traverse, PMF.bind_bind]
-    have he : (fun j => (orderedLaw offspring lam h n j).bind (orderedGrow offspring lam h n)) =
-        orderedLaw offspring lam h (n+1) := funext ih
-    rw [he]
-    rfl
-
-lemma ordered_likelihood_common_subsequence (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) :
-    ∃ Q : MeasureTheory.ProbabilityMeasure Message, ∃ φ : ℕ → ℕ, StrictMono φ ∧
-      Filter.Tendsto (fun n => likelihoodLaw (orderedLaw offspring lam h (φ n))) Filter.atTop (nhds Q) ∧
-      Filter.Tendsto (fun n => likelihoodLaw (orderedLaw offspring lam h (φ n+1))) Filter.atTop (nhds Q) := by
-  exact likelihood_common_subsequence (orderedLaw offspring lam h) (orderedGrow offspring lam h)
-    (fun n i => (orderedLaw_grow offspring lam h n i).symm)
-
-end ThreeState
-universe u
-namespace ThreeState
-
-instance pmfCommApplicative : CommApplicative PMF where
-  commutative_prod a b := by
-    change ((a.map Prod.mk).bind (fun f => b.map f)) =
-      ((b.map (fun y x => (x,y))).bind (fun f => a.map f))
-    simp only [PMF.map, PMF.bind_bind, PMF.pure_bind, Function.comp_def]
-    exact PMF.bind_comm a b (fun x y => PMF.pure (x,y))
-
-lemma list_traverse_nil {α β : Type u} (k : α → PMF β) :
-    Traversable.traverse k [] = PMF.pure [] := rfl
-
-lemma list_traverse_cons {α β : Type u} (k : α → PMF β) (a : α) (as : List α) :
-    Traversable.traverse k (a::as) = (k a).bind (fun b => (Traversable.traverse k as).map (List.cons b)) := by
-  rw [List.traverse_cons, seq_eq_bind_map]
-  change ((k a).map List.cons).bind _ = _
-  rw [PMF.bind_map]
-  rfl
-
-lemma multiset_traverse_list {α β : Type u} (k : α → PMF β) (as : List α) :
-    Multiset.traverse k (as : Multiset α) =
-      (Traversable.traverse k as).map (fun bs : List β => (bs : Multiset β)) := rfl
-
-lemma iidList_bind_traverse {α β : Type u} (p : PMF α) (k : α → PMF β) (n : ℕ) :
-    (iidList p n).bind (Traversable.traverse k) = iidList (p.bind k) n := by
-  induction n with
-  | zero => simp only [iidList, PMF.pure_bind, list_traverse_nil]
-  | succ n ih =>
-    simp only [iidList, PMF.bind_bind, PMF.bind_map, Function.comp_def, list_traverse_cons]
-    apply congrArg (PMF.bind p)
-    funext a
-    rw [PMF.bind_comm]
-    apply congrArg (PMF.bind (k a))
-    funext b
-    rw [← PMF.map_bind, ih]
-
-lemma iidList_multiset_bind {α β : Type u} (p : PMF α) (k : α → PMF β) (n : ℕ) :
-    ((iidList p n).map (fun as : List α => (as : Multiset α))).bind (Multiset.traverse k) =
-      (iidList (p.bind k) n).map (fun bs : List β => (bs : Multiset β)) := by
-  rw [PMF.bind_map]
-  change (iidList p n).bind (fun as => ((Traversable.traverse k as).map (fun bs : List β => (bs : Multiset β)))) = _
-  rw [← PMF.map_bind, iidList_bind_traverse]
-
- 
-
-noncomputable def grow (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) :
-    (n : ℕ) → Observation n → PMF (Observation (n+1))
-  | 0 => observedLaw offspring lam h 1
-  | n+1 => Multiset.traverse (grow offspring lam h n)
-
- 
-theorem observedLaw_grow (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
-    (n : ℕ) (i : Spin) :
-    (observedLaw offspring lam h n i).bind (grow offspring lam h n) =
-      observedLaw offspring lam h (n+1) i := by
-  induction n generalizing i with
-  | zero => exact PMF.pure_bind i _
-  | succ n ih =>
-    change (offspring.bind (fun k =>
-      (iidList ((channelPMF lam h i).bind (observedLaw offspring lam h n)) k).map
-        (fun xs : List (Observation n) => (xs : Multiset (Observation n))))).bind
-        (Multiset.traverse (grow offspring lam h n)) =
-      offspring.bind (fun k =>
-        (iidList ((channelPMF lam h i).bind (observedLaw offspring lam h (n+1))) k).map
-          (fun xs : List (Observation (n+1)) => (xs : Multiset (Observation (n+1)))))
-    rw [PMF.bind_bind]
-    apply congrArg (PMF.bind offspring)
-    funext k
-    rw [iidList_multiset_bind, PMF.bind_bind]
-    have he : (fun j => (observedLaw offspring lam h n j).bind (grow offspring lam h n)) =
-        observedLaw offspring lam h (n+1) := funext ih
-    rw [he]
-
- 
-theorem observedAdvantage_antitone (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) :
-    Antitone (fun n => advantage (observedLaw offspring lam h n)) := by
-  apply antitone_nat_of_succ_le
-  intro n
-  have he : observedLaw offspring lam h (n+1) =
-      fun i => (observedLaw offspring lam h n i).bind (grow offspring lam h n) := by
-    funext i
-    exact (observedLaw_grow offspring lam h n i).symm
-  rw [he]
-  exact advantage_bind_le _ _
-
- 
-theorem observedAdvantage_tendsto (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) :
-    ∃ L : ℝ, 0 ≤ L ∧ L ≤ 1 ∧
-      Filter.Tendsto (fun n => advantage (observedLaw offspring lam h n))
-        Filter.atTop (nhds L) := by
-  let a := fun n => advantage (observedLaw offspring lam h n)
-  have hb : BddBelow (Set.range a) := ⟨0, by rintro x ⟨n,rfl⟩; exact advantage_nonneg _⟩
-  refine ⟨⨅ n, a n, ?_, ?_, tendsto_atTop_ciInf (observedAdvantage_antitone offspring lam h) hb⟩
-  · exact le_ciInf (fun n => advantage_nonneg _)
-  · exact (ciInf_le hb 0).trans (advantage_le_one _)
-
-end ThreeState
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeState.Messages
+
+namespace OAI
+
+/-! Ordered and unlabelled boundary observations and information degradation. -/
+
+namespace ThreeState
+
+ 
+
+noncomputable def iidVector {α : Type*} (p : PMF α) : (n : ℕ) → PMF (Fin n → α)
+  | 0 => PMF.pure Fin.elim0
+  | n+1 => p.bind (fun a => (iidVector p n).map (Fin.cons a))
+
+lemma iidVector_map {α β : Type*} (p : PMF α) (f : α → β) (n : ℕ) :
+    (iidVector p n).map (fun as => f ∘ as) = iidVector (p.map f) n := by
+  induction n with
+  | zero =>
+    simp only [iidVector, PMF.pure_map]
+    congr 1
+    funext i
+    exact Fin.elim0 i
+  | succ n ih =>
+    simp only [iidVector, PMF.map_bind, PMF.map_comp, PMF.bind_map]
+    apply congrArg (PMF.bind p)
+    funext a
+    rw [← ih]
+    dsimp only [Function.comp_def]
+    rw [PMF.map_comp]
+    congr 1
+    funext v j
+    cases j using Fin.cases <;> simp
+
+lemma iidVector_ofFn {α : Type*} (p : PMF α) (n : ℕ) :
+    (iidVector p n).map List.ofFn = iidList p n := by
+  induction n with
+  | zero => simp [iidVector, iidList, PMF.pure_map]
+  | succ n ih =>
+    simp only [iidVector, iidList, PMF.map_bind, PMF.map_comp]
+    apply congrArg (PMF.bind p)
+    funext a
+    rw [← ih, PMF.map_comp]
+    congr 1
+    funext v
+    simp [List.ofFn_succ]
+
+lemma iidVector_apply {α : Type*} (p : PMF α) (n : ℕ) (v : Fin n → α) :
+    iidVector p n v = ∏ j, p (v j) := by
+  classical
+  induction n with
+  | zero =>
+    have he : v = Fin.elim0 := by funext j; exact Fin.elim0 j
+    simp [iidVector, he]
+  | succ n ih =>
+    simp only [iidVector, PMF.bind_apply, PMF.map_apply]
+    have he (a : α) (w : Fin n → α) : (v = Fin.cons a w) ↔ (a=v 0 ∧ w=Fin.tail v) := by
+      constructor
+      · intro h
+        subst v
+        simp
+      · rintro ⟨rfl,rfl⟩
+        exact (Fin.cons_self_tail v).symm
+    simp_rw [he]
+    simp only [ite_and]
+    rw [tsum_eq_single (v 0)]
+    · simp [ih, Fin.prod_univ_succ, Fin.tail]
+    · intro a ha
+      simp [ha]
+
+ 
+abbrev OrderedObservation : ℕ → Type
+  | 0 => Spin
+  | n+1 => Σ k : ℕ, Fin k → OrderedObservation n
+
+noncomputable def orderedLaw (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) :
+    (n : ℕ) → Spin → PMF (OrderedObservation n)
+  | 0, i => PMF.pure i
+  | n+1, i => offspring.bind (fun k =>
+      (iidVector ((channelPMF lam h i).bind (orderedLaw offspring lam h n)) k).map (Sigma.mk k))
+
+def forgetOrder : (n : ℕ) → OrderedObservation n → Observation n
+  | 0, i => i
+  | n+1, ⟨k,v⟩ => (List.ofFn (fun j : Fin k => forgetOrder n (v j)) : Multiset (Observation n))
+
+lemma orderedLaw_forget (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) (n : ℕ) (i : Spin) :
+    (orderedLaw offspring lam h n i).map (forgetOrder n) = observedLaw offspring lam h n i := by
+  induction n generalizing i with
+  | zero =>
+    change (PMF.pure i).map id = PMF.pure i
+    exact PMF.map_id _
+  | succ n ih =>
+    simp only [orderedLaw, observedLaw, PMF.map_bind, PMF.map_comp]
+    apply congrArg (PMF.bind offspring)
+    funext k
+    let p := (channelPMF lam h i).bind (orderedLaw offspring lam h n)
+    change (iidVector p k).map
+      ((fun xs : List (Observation n) => (xs : Multiset (Observation n))) ∘ List.ofFn ∘
+        (fun v => forgetOrder n ∘ v)) = _
+    rw [← PMF.map_comp, ← PMF.map_comp, iidVector_map, iidVector_ofFn]
+    congr 2
+    rw [PMF.map_bind]
+    congr 1
+    funext j
+    exact ih j
+
+lemma orderedAdvantage_dominates (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) (n : ℕ) :
+    advantage (observedLaw offspring lam h n) ≤ advantage (orderedLaw offspring lam h n) := by
+  have hh := advantage_bind_le (orderedLaw offspring lam h n) (fun y => PMF.pure (forgetOrder n y))
+  have he : (fun i => (orderedLaw offspring lam h n i).bind (fun y => PMF.pure (forgetOrder n y))) =
+      observedLaw offspring lam h n := by
+    funext i
+    exact orderedLaw_forget offspring lam h n i
+  rw [he] at hh
+  exact hh
+
+end ThreeState
+
+namespace ThreeState
+
+noncomputable def vectorTraverse {α β : Type*} (k : α → PMF β) :
+    (n : ℕ) → (Fin n → α) → PMF (Fin n → β)
+  | 0, _ => PMF.pure Fin.elim0
+  | n+1, v => (k (v 0)).bind (fun b => (vectorTraverse k n (Fin.tail v)).map (Fin.cons b))
+
+lemma vectorTraverse_cons {α β : Type*} (k : α → PMF β) (n : ℕ) (a : α) (v : Fin n → α) :
+    vectorTraverse k (n+1) (Fin.cons a v) =
+      (k a).bind (fun b => (vectorTraverse k n v).map (Fin.cons b)) := by
+  simp only [vectorTraverse, Fin.cons_zero, Fin.tail_cons]
+
+lemma iidVector_bind_traverse {α β : Type*} (p : PMF α) (k : α → PMF β) (n : ℕ) :
+    (iidVector p n).bind (vectorTraverse k n) = iidVector (p.bind k) n := by
+  induction n with
+  | zero => simp only [iidVector, PMF.pure_bind, vectorTraverse]
+  | succ n ih =>
+    simp only [iidVector, PMF.bind_bind, PMF.bind_map, Function.comp_def, vectorTraverse_cons]
+    apply congrArg (PMF.bind p)
+    funext a
+    rw [PMF.bind_comm]
+    apply congrArg (PMF.bind (k a))
+    funext b
+    rw [← PMF.map_bind, ih]
+
+noncomputable def orderedGrow (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) :
+    (n : ℕ) → OrderedObservation n → PMF (OrderedObservation (n+1))
+  | 0 => orderedLaw offspring lam h 1
+  | n+1 => fun ⟨k,v⟩ => (vectorTraverse (orderedGrow offspring lam h n) k v).map (Sigma.mk k)
+
+lemma orderedLaw_grow (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) (n : ℕ) (i : Spin) :
+    (orderedLaw offspring lam h n i).bind (orderedGrow offspring lam h n) =
+      orderedLaw offspring lam h (n+1) i := by
+  induction n generalizing i with
+  | zero => exact PMF.pure_bind i _
+  | succ n ih =>
+    simp only [orderedLaw, orderedGrow, PMF.bind_bind, PMF.bind_map, Function.comp_def]
+    apply congrArg (PMF.bind offspring)
+    funext k
+    rw [← PMF.map_bind, iidVector_bind_traverse, PMF.bind_bind]
+    have he : (fun j => (orderedLaw offspring lam h n j).bind (orderedGrow offspring lam h n)) =
+        orderedLaw offspring lam h (n+1) := funext ih
+    rw [he]
+    rfl
+
+lemma ordered_likelihood_common_subsequence (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) :
+    ∃ Q : MeasureTheory.ProbabilityMeasure Message, ∃ φ : ℕ → ℕ, StrictMono φ ∧
+      Filter.Tendsto (fun n => likelihoodLaw (orderedLaw offspring lam h (φ n))) Filter.atTop (nhds Q) ∧
+      Filter.Tendsto (fun n => likelihoodLaw (orderedLaw offspring lam h (φ n+1))) Filter.atTop (nhds Q) := by
+  exact likelihood_common_subsequence (orderedLaw offspring lam h) (orderedGrow offspring lam h)
+    (fun n i => (orderedLaw_grow offspring lam h n i).symm)
+
+end ThreeState
+universe u
+namespace ThreeState
+
+instance pmfCommApplicative : CommApplicative PMF where
+  commutative_prod a b := by
+    change ((a.map Prod.mk).bind (fun f => b.map f)) =
+      ((b.map (fun y x => (x,y))).bind (fun f => a.map f))
+    simp only [PMF.map, PMF.bind_bind, PMF.pure_bind, Function.comp_def]
+    exact PMF.bind_comm a b (fun x y => PMF.pure (x,y))
+
+lemma list_traverse_nil {α β : Type u} (k : α → PMF β) :
+    Traversable.traverse k [] = PMF.pure [] := rfl
+
+lemma list_traverse_cons {α β : Type u} (k : α → PMF β) (a : α) (as : List α) :
+    Traversable.traverse k (a::as) = (k a).bind (fun b => (Traversable.traverse k as).map (List.cons b)) := by
+  rw [List.traverse_cons, seq_eq_bind_map]
+  change ((k a).map List.cons).bind _ = _
+  rw [PMF.bind_map]
+  rfl
+
+lemma multiset_traverse_list {α β : Type u} (k : α → PMF β) (as : List α) :
+    Multiset.traverse k (as : Multiset α) =
+      (Traversable.traverse k as).map (fun bs : List β => (bs : Multiset β)) := rfl
+
+lemma iidList_bind_traverse {α β : Type u} (p : PMF α) (k : α → PMF β) (n : ℕ) :
+    (iidList p n).bind (Traversable.traverse k) = iidList (p.bind k) n := by
+  induction n with
+  | zero => simp only [iidList, PMF.pure_bind, list_traverse_nil]
+  | succ n ih =>
+    simp only [iidList, PMF.bind_bind, PMF.bind_map, Function.comp_def, list_traverse_cons]
+    apply congrArg (PMF.bind p)
+    funext a
+    rw [PMF.bind_comm]
+    apply congrArg (PMF.bind (k a))
+    funext b
+    rw [← PMF.map_bind, ih]
+
+lemma iidList_multiset_bind {α β : Type u} (p : PMF α) (k : α → PMF β) (n : ℕ) :
+    ((iidList p n).map (fun as : List α => (as : Multiset α))).bind (Multiset.traverse k) =
+      (iidList (p.bind k) n).map (fun bs : List β => (bs : Multiset β)) := by
+  rw [PMF.bind_map]
+  change (iidList p n).bind (fun as => ((Traversable.traverse k as).map (fun bs : List β => (bs : Multiset β)))) = _
+  rw [← PMF.map_bind, iidList_bind_traverse]
+
+ 
+
+noncomputable def grow (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) :
+    (n : ℕ) → Observation n → PMF (Observation (n+1))
+  | 0 => observedLaw offspring lam h 1
+  | n+1 => Multiset.traverse (grow offspring lam h n)
+
+ 
+theorem observedLaw_grow (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
+    (n : ℕ) (i : Spin) :
+    (observedLaw offspring lam h n i).bind (grow offspring lam h n) =
+      observedLaw offspring lam h (n+1) i := by
+  induction n generalizing i with
+  | zero => exact PMF.pure_bind i _
+  | succ n ih =>
+    change (offspring.bind (fun k =>
+      (iidList ((channelPMF lam h i).bind (observedLaw offspring lam h n)) k).map
+        (fun xs : List (Observation n) => (xs : Multiset (Observation n))))).bind
+        (Multiset.traverse (grow offspring lam h n)) =
+      offspring.bind (fun k =>
+        (iidList ((channelPMF lam h i).bind (observedLaw offspring lam h (n+1))) k).map
+          (fun xs : List (Observation (n+1)) => (xs : Multiset (Observation (n+1)))))
+    rw [PMF.bind_bind]
+    apply congrArg (PMF.bind offspring)
+    funext k
+    rw [iidList_multiset_bind, PMF.bind_bind]
+    have he : (fun j => (observedLaw offspring lam h n j).bind (grow offspring lam h n)) =
+        observedLaw offspring lam h (n+1) := funext ih
+    rw [he]
+
+ 
+theorem observedAdvantage_antitone (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) :
+    Antitone (fun n => advantage (observedLaw offspring lam h n)) := by
+  apply antitone_nat_of_succ_le
+  intro n
+  have he : observedLaw offspring lam h (n+1) =
+      fun i => (observedLaw offspring lam h n i).bind (grow offspring lam h n) := by
+    funext i
+    exact (observedLaw_grow offspring lam h n i).symm
+  rw [he]
+  exact advantage_bind_le _ _
+
+ 
+theorem observedAdvantage_tendsto (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) :
+    ∃ L : ℝ, 0 ≤ L ∧ L ≤ 1 ∧
+      Filter.Tendsto (fun n => advantage (observedLaw offspring lam h n))
+        Filter.atTop (nhds L) := by
+  let a := fun n => advantage (observedLaw offspring lam h n)
+  have hb : BddBelow (Set.range a) := ⟨0, by rintro x ⟨n,rfl⟩; exact advantage_nonneg _⟩
+  refine ⟨⨅ n, a n, ?_, ?_, tendsto_atTop_ciInf (observedAdvantage_antitone offspring lam h) hb⟩
+  · exact le_ciInf (fun n => advantage_nonneg _)
+  · exact (ciInf_le hb 0).trans (advantage_le_one _)
+
+end ThreeState
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeState/PairCorrelation.lean b/lean/OAI/Probability/ThreeState/PairCorrelation.lean
index c4af19d3aec030746acdf0ee65bfaac3042233d3..9a6a3ab5040761023b748a02cc2cd884d23729e9 100644
--- a/lean/OAI/Probability/ThreeState/PairCorrelation.lean
+++ b/lean/OAI/Probability/ThreeState/PairCorrelation.lean
@@ -1,258 +1,258 @@
-import Mathlib
-import OAI.Probability.ThreeState.PosteriorMoments
-
-namespace OAI
-
-/-! Two-message correlation and quantitative entropy increments. -/
-
-namespace ThreeState
-open Set
-
-lemma log_correlation_cubic (a : ℝ) (ha : 0 < a) :
-    (a-1)+(a-1)^2/2-(a-1)^3/6 ≤ a*Real.log a := by
-  let f : ℝ → ℝ := fun x => x*Real.log x-((x-1)+(x-1)^2/2-(x-1)^3/6)
-  let g : ℝ → ℝ := fun x => Real.log x-(x-1)+(x-1)^2/2
-  have hd (x : ℝ) (hx : 0 < x) : HasDerivAt f (g x) x := by
-    have hh := ((hasDerivAt_id x).mul (Real.hasDerivAt_log hx.ne')).sub
-      ((((hasDerivAt_id x).sub_const 1).add (((hasDerivAt_id x).sub_const 1).pow 2 |>.div_const 2)).sub
-        (((hasDerivAt_id x).sub_const 1).pow 3 |>.div_const 6))
-    convert hh using 1 <;> first | rfl | (dsimp only [g, id]; field_simp; ring)
-  have hdg (x : ℝ) (hx : 0 < x) : HasDerivAt g ((x-1)^2/x) x := by
-    have hh := ((Real.hasDerivAt_log hx.ne').sub ((hasDerivAt_id x).sub_const 1)).add
-      (((hasDerivAt_id x).sub_const 1).pow 2 |>.div_const 2)
-    convert hh using 1 <;> first | rfl | (simp only [id_eq]; field_simp; ring)
-  have hconv : ConvexOn ℝ (Ioi 0) f := by
-    apply convexOn_of_hasDerivWithinAt2_nonneg (f' := g) (f'' := fun x => (x-1)^2/x) (convex_Ioi 0)
-    · exact fun x hx => (hd x hx).continuousAt.continuousWithinAt
-    · intro x hx
-      rw [interior_Ioi] at hx
-      exact (hd x hx).hasDerivWithinAt
-    · intro x hx
-      rw [interior_Ioi] at hx
-      exact (hdg x hx).hasDerivWithinAt
-    · intro x hx
-      rw [interior_Ioi] at hx
-      exact div_nonneg (sq_nonneg _) hx.le
-  have hf1 : f 1 = 0 := by simp [f]
-  have hg1 : g 1 = 0 := by simp [g]
-  have hf0 : 0 ≤ f a := by
-    rcases lt_trichotomy a 1 with hl | he | hg
-    · have hh := hconv.slope_le_of_hasDerivAt ha (by norm_num : (1:ℝ) ∈ Ioi 0) hl (hd 1 (by norm_num))
-      rw [hg1, slope_def_field, hf1] at hh
-      have hp := (div_le_iff₀ (sub_pos.mpr hl)).mp hh
-      linarith
-    · simp [he, hf1]
-    · have hh := hconv.le_slope_of_hasDerivAt (by norm_num : (1:ℝ) ∈ Ioi 0) ha hg (hd 1 (by norm_num))
-      rw [hg1, slope_def_field, hf1] at hh
-      have hp := (le_div_iff₀ (sub_pos.mpr hg)).mp hh
-      linarith
-  dsimp only [f] at hf0
-  linarith
-
-end ThreeState
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-open Radial (avg)
-
-noncomputable def correlationZ (a b : Message) : ℝ := avg (fun i => messageDeviation a i * messageDeviation b i)
-lemma continuous_correlationZ : Continuous (fun p : Message × Message => correlationZ p.1 p.2) := by
-  apply continuous_avg
-  intro i
-  exact ((((continuous_apply i).comp continuous_subtype_val).comp continuous_fst).sub continuous_const).mul
-    ((((continuous_apply i).comp continuous_subtype_val).comp continuous_snd).sub continuous_const)
-
-noncomputable def symPerm : Fin 6 → Equiv.Perm Spin :=
-  ![Equiv.refl _, Equiv.swap 0 1, Equiv.swap 0 2, Equiv.swap 1 2,
-    (Equiv.swap 0 1).trans (Equiv.swap 1 2), (Equiv.swap 1 2).trans (Equiv.swap 0 1)]
-noncomputable def symAvg (f : Message → ℝ) (m : Message) : ℝ :=
-  (∑ i : Fin 6, f (permMessage (symPerm i) m))/6
-
-lemma integral_symAvg (Q : ProbabilityMeasure Message) (hs : SpinSymmetric Q)
-    (f : Message → ℝ) (hf : Continuous f) :
-    (∫ m, symAvg f m ∂(Q : Measure Message)) = ∫ m, f m ∂(Q : Measure Message) := by
-  simp only [symAvg, integral_div]
-  have hi (i : Fin 6) : Integrable (fun m => f (permMessage (symPerm i) m)) (Q : Measure Message) :=
-    integrable_continuous_message Q (hf.comp (continuous_permMessage _))
-  rw [integral_finsetSum _ (fun i _ => hi i)]
-  simp only [integral_permMessage Q hs _ _ hf.measurable]
-  simp
-
-lemma symAvg_correlationZ (a b : Message) : symAvg (fun m => correlationZ m b) a = 0 := by
-  have ha := average_message a
-  have hb := average_message b
-  simp only [Radial.avg_expand] at ha hb
-  have ha2 : a 2 = 3-a 0-a 1 := by linarith
-  have hb2 : b 2 = 3-b 0-b 1 := by linarith
-  norm_num [symAvg, symPerm, Fin.sum_univ_succ, correlationZ, messageDeviation,
-    Radial.avg_expand, permMessage, Equiv.swap_apply_def, Equiv.trans_apply, Equiv.symm_trans_apply, Fin.ext_iff]
-  rw [ha2, hb2]
-  ring
-
-lemma symAvg_correlationZ_sq (a b : Message) :
-    symAvg (fun m => (correlationZ m b)^2) a = 2*messageX a*messageX b := by
-  have ha := average_message a
-  have hb := average_message b
-  simp only [Radial.avg_expand] at ha hb
-  have ha2 : a 2 = 3-a 0-a 1 := by linarith
-  have hb2 : b 2 = 3-b 0-b 1 := by linarith
-  norm_num [symAvg, symPerm, Fin.sum_univ_succ, correlationZ, messageDeviation,
-    Radial.avg_expand, permMessage, Equiv.swap_apply_def, Equiv.trans_apply, Equiv.symm_trans_apply, Fin.ext_iff,
-    messageX, Radial.momentX]
-  rw [ha2, hb2]
-  ring
-
-lemma symAvg_correlationZ_cube (a b : Message) :
-    symAvg (fun m => (correlationZ m b)^3) a = 2*messageY a*messageY b := by
-  have ha := average_message a
-  have hb := average_message b
-  simp only [Radial.avg_expand] at ha hb
-  have ha2 : a 2 = 3-a 0-a 1 := by linarith
-  have hb2 : b 2 = 3-b 0-b 1 := by linarith
-  norm_num [symAvg, symPerm, Fin.sum_univ_succ, correlationZ, messageDeviation,
-    Radial.avg_expand, permMessage, Equiv.swap_apply_def, Equiv.trans_apply, Equiv.symm_trans_apply, Fin.ext_iff,
-    messageY, Radial.momentY]
-  rw [ha2, hb2]
-  ring
-
-lemma integral_correlationZ (Q : ProbabilityMeasure Message) (hs : SpinSymmetric Q) (b : Message) :
-    (∫ m, correlationZ m b ∂(Q : Measure Message)) = 0 := by
-  rw [← integral_symAvg Q hs (fun m => correlationZ m b) (continuous_correlationZ.comp (continuous_id.prodMk continuous_const))]
-  simp only [symAvg_correlationZ, integral_zero]
-
-lemma integral_correlationZ_sq (Q : ProbabilityMeasure Message) (hs : SpinSymmetric Q) (b : Message) :
-    (∫ m, (correlationZ m b)^2 ∂(Q : Measure Message)) = 2*posteriorMu Q*messageX b := by
-  rw [← integral_symAvg Q hs (fun m => (correlationZ m b)^2) ((continuous_correlationZ.comp (continuous_id.prodMk continuous_const)).pow 2)]
-  simp only [symAvg_correlationZ_sq, integral_mul_const, integral_const_mul, posteriorMu]
-
-lemma integral_correlationZ_cube (Q : ProbabilityMeasure Message) (hs : SpinSymmetric Q) (b : Message) :
-    (∫ m, (correlationZ m b)^3 ∂(Q : Measure Message)) = 2*posteriorEta Q*messageY b := by
-  rw [← integral_symAvg Q hs (fun m => (correlationZ m b)^3) ((continuous_correlationZ.comp (continuous_id.prodMk continuous_const)).pow 3)]
-  simp only [symAvg_correlationZ_cube, integral_mul_const, integral_const_mul, posteriorEta]
-
-end ThreeState
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-open Radial (avg)
-
-lemma correlationZ_comm (a b : Message) : correlationZ a b = correlationZ b a := by
-  simp [correlationZ, mul_comm]
-
-noncomputable def pairNormalizer (lam : ℝ) (h : Admissible lam) (a b : Message) : ℝ :=
-  avg (fun i => a i*edgeMessage lam h b i)
-
-lemma pairNormalizer_eq (lam : ℝ) (h : Admissible lam) (a b : Message) :
-    pairNormalizer lam h a b = 1+lam*correlationZ a b := by
-  have ha := average_message a
-  have hb := average_message b
-  simp only [Radial.avg_expand] at ha hb
-  simp only [pairNormalizer, correlationZ, messageDeviation, edgeMessage, Radial.avg_expand]
-  rw [message_last a, message_last b]
-  ring
-
-lemma continuous_pairNormalizer (lam : ℝ) (h : Admissible lam) :
-    Continuous (fun p : Message × Message => pairNormalizer lam h p.1 p.2) := by
-  simp only [pairNormalizer_eq]
-  exact continuous_const.add (continuous_const.mul continuous_correlationZ)
-
-lemma pairNormalizer_pos (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (a b : Message) : 0 < pairNormalizer lam h a b := by
-  have he (i : Spin) : (1-lam)*a i ≤ a i*edgeMessage lam h b i := by
-    nlinarith [mul_le_mul_of_nonneg_left (edgeMessage_bounds lam h hl0 b i).1 (Message.nonneg a i)]
-  have ha := average_message a
-  simp only [pairNormalizer, Radial.avg_expand] at *
-  nlinarith [he 0, he 1, he 2]
-
-noncomputable def pairMessage (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (a b : Message) : Message :=
-  ⟨fun i => a i*edgeMessage lam h b i/pairNormalizer lam h a b, by
-    constructor
-    · intro i
-      exact div_nonneg (mul_nonneg (Message.nonneg a i) (Message.nonneg _ _)) (pairNormalizer_pos lam h hl0 hl1 a b).le
-    · rw [← Finset.sum_div]
-      change (∑ i, a i*edgeMessage lam h b i)/((∑ i, a i*edgeMessage lam h b i)/3) = 3
-      have hh := ne_of_gt (pairNormalizer_pos lam h hl0 hl1 a b)
-      dsimp only [pairNormalizer, avg] at hh
-      have hs : (∑ i, a i*edgeMessage lam h b i) ≠ 0 := by
-        intro he
-        apply hh
-        rw [he]
-        norm_num
-      field_simp [hs]⟩
-
-lemma continuous_pairMessage (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1) :
-    Continuous (fun p : Message × Message => pairMessage lam h hl0 hl1 p.1 p.2) := by
-  apply Continuous.subtype_mk
-  apply continuous_pi
-  intro i
-  exact ((((continuous_apply i).comp continuous_subtype_val).comp continuous_fst).mul
-    (((continuous_apply i).comp continuous_subtype_val).comp ((continuous_edgeMessage lam h).comp continuous_snd))).div
-      (continuous_pairNormalizer lam h) (fun p => ne_of_gt (pairNormalizer_pos lam h hl0 hl1 p.1 p.2))
-
-lemma integral_pairNormalizer (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message)
-    (hQ : Balanced Q) (a : Message) :
-    (∫ b, pairNormalizer lam h a b ∂(Q : Measure Message)) = 1 := by
-  change (∫ b, avg (fun i => a i*edgeMessage lam h b i) ∂(Q : Measure Message)) = 1
-  rw [integral_avg]
-  · simp only [integral_const_mul, integral_edgeMessage lam h Q hQ, mul_one]
-    exact average_message a
-  · intro i
-    exact integrable_continuous_message Q (continuous_const.mul (((continuous_apply i).comp continuous_subtype_val).comp (continuous_edgeMessage lam h)))
-
-noncomputable def pairIncrement (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message) (a : Message) : ℝ :=
-  ∫ b, pairNormalizer lam h a b*Real.log (pairNormalizer lam h a b) ∂(Q : Measure Message)
-
-lemma continuous_pairIntegrand (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1) :
-    Continuous (fun p : Message × Message => pairNormalizer lam h p.1 p.2*Real.log (pairNormalizer lam h p.1 p.2)) :=
-  (continuous_pairNormalizer lam h).mul ((continuous_pairNormalizer lam h).log
-    (fun p => ne_of_gt (pairNormalizer_pos lam h hl0 hl1 p.1 p.2)))
-
-lemma pairIncrement_nonneg (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (a : Message) : 0 ≤ pairIncrement lam h Q a := by
-  have hf : Continuous (fun b => pairNormalizer lam h a b) :=
-    (continuous_pairNormalizer lam h).comp (continuous_const.prodMk continuous_id)
-  have hg : Continuous (fun b => pairNormalizer lam h a b*Real.log (pairNormalizer lam h a b)) :=
-    (continuous_pairIntegrand lam h hl0 hl1).comp (continuous_const.prodMk continuous_id)
-  have hin (b : Message) : pairNormalizer lam h a b-1 ≤ pairNormalizer lam h a b*Real.log (pairNormalizer lam h a b) := by
-    have hp := pairNormalizer_pos lam h hl0 hl1 a b
-    have hh := mul_le_mul_of_nonneg_left (Real.one_sub_inv_le_log_of_pos hp) hp.le
-    rw [mul_sub, mul_one, mul_inv_cancel₀ hp.ne'] at hh
-    exact hh
-  have hif : Integrable (fun b => pairNormalizer lam h a b-1) (Q : Measure Message) :=
-    integrable_continuous_message Q (hf.sub continuous_const)
-  have hh := integral_mono hif (integrable_continuous_message Q hg) hin
-  rw [integral_sub (integrable_continuous_message Q hf) (integrable_const 1), integral_pairNormalizer lam h Q hQ] at hh
-  simpa [pairIncrement] using hh
-
-lemma pairIncrement_lower (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) (hs : SpinSymmetric Q) (a : Message) :
-    lam^2*posteriorMu Q*messageX a-lam^3*posteriorEta Q*messageY a/3 ≤ pairIncrement lam h Q a := by
-  let z : Message → ℝ := fun b => correlationZ b a
-  have hz : Continuous z := continuous_correlationZ.comp (continuous_id.prodMk continuous_const)
-  have h1 : (∫ b, z b ∂(Q : Measure Message)) = 0 := integral_correlationZ Q hs a
-  have h2 : (∫ b, (z b)^2 ∂(Q : Measure Message)) = 2*posteriorMu Q*messageX a := integral_correlationZ_sq Q hs a
-  have h3 : (∫ b, (z b)^3 ∂(Q : Measure Message)) = 2*posteriorEta Q*messageY a := integral_correlationZ_cube Q hs a
-  have hin (b : Message) : lam*z b+lam^2*(z b)^2/2-lam^3*(z b)^3/6 ≤ pairNormalizer lam h a b*Real.log (pairNormalizer lam h a b) := by
-    have hh := log_correlation_cubic _ (pairNormalizer_pos lam h hl0 hl1 a b)
-    rw [pairNormalizer_eq, correlationZ_comm a b] at hh ⊢
-    dsimp only [z]
-    convert hh using 1
-    ring
-  have hf : Continuous (fun b => lam*z b+lam^2*(z b)^2/2-lam^3*(z b)^3/6) := by fun_prop
-  have hg : Continuous (fun b => pairNormalizer lam h a b*Real.log (pairNormalizer lam h a b)) :=
-    (continuous_pairIntegrand lam h hl0 hl1).comp (continuous_const.prodMk continuous_id)
-  have hh := integral_mono (integrable_continuous_message Q hf) (integrable_continuous_message Q hg) hin
-  have hi1 := integrable_continuous_message Q hz
-  have hi2 : Integrable (fun b => (z b)^2) (Q : Measure Message) := integrable_continuous_message Q (hz.pow 2)
-  have hi3 : Integrable (fun b => (z b)^3) (Q : Measure Message) := integrable_continuous_message Q (hz.pow 3)
-  have hiadd : Integrable (fun b => lam*z b+lam^2*(z b)^2/2) (Q : Measure Message) :=
-    (hi1.const_mul _).add ((hi2.const_mul _).div_const _)
-  rw [integral_sub hiadd ((hi3.const_mul _).div_const _), integral_add (hi1.const_mul _) ((hi2.const_mul _).div_const _),
-    integral_div, integral_div, integral_const_mul, integral_const_mul, integral_const_mul, h1,h2,h3] at hh
-  dsimp only [pairIncrement]
-  linarith
-
-end ThreeState
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeState.PosteriorMoments
+
+namespace OAI
+
+/-! Two-message correlation and quantitative entropy increments. -/
+
+namespace ThreeState
+open Set
+
+lemma log_correlation_cubic (a : ℝ) (ha : 0 < a) :
+    (a-1)+(a-1)^2/2-(a-1)^3/6 ≤ a*Real.log a := by
+  let f : ℝ → ℝ := fun x => x*Real.log x-((x-1)+(x-1)^2/2-(x-1)^3/6)
+  let g : ℝ → ℝ := fun x => Real.log x-(x-1)+(x-1)^2/2
+  have hd (x : ℝ) (hx : 0 < x) : HasDerivAt f (g x) x := by
+    have hh := ((hasDerivAt_id x).mul (Real.hasDerivAt_log hx.ne')).sub
+      ((((hasDerivAt_id x).sub_const 1).add (((hasDerivAt_id x).sub_const 1).pow 2 |>.div_const 2)).sub
+        (((hasDerivAt_id x).sub_const 1).pow 3 |>.div_const 6))
+    convert hh using 1 <;> first | rfl | (dsimp only [g, id]; field_simp; ring)
+  have hdg (x : ℝ) (hx : 0 < x) : HasDerivAt g ((x-1)^2/x) x := by
+    have hh := ((Real.hasDerivAt_log hx.ne').sub ((hasDerivAt_id x).sub_const 1)).add
+      (((hasDerivAt_id x).sub_const 1).pow 2 |>.div_const 2)
+    convert hh using 1 <;> first | rfl | (simp only [id_eq]; field_simp; ring)
+  have hconv : ConvexOn ℝ (Ioi 0) f := by
+    apply convexOn_of_hasDerivWithinAt2_nonneg (f' := g) (f'' := fun x => (x-1)^2/x) (convex_Ioi 0)
+    · exact fun x hx => (hd x hx).continuousAt.continuousWithinAt
+    · intro x hx
+      rw [interior_Ioi] at hx
+      exact (hd x hx).hasDerivWithinAt
+    · intro x hx
+      rw [interior_Ioi] at hx
+      exact (hdg x hx).hasDerivWithinAt
+    · intro x hx
+      rw [interior_Ioi] at hx
+      exact div_nonneg (sq_nonneg _) hx.le
+  have hf1 : f 1 = 0 := by simp [f]
+  have hg1 : g 1 = 0 := by simp [g]
+  have hf0 : 0 ≤ f a := by
+    rcases lt_trichotomy a 1 with hl | he | hg
+    · have hh := hconv.slope_le_of_hasDerivAt ha (by norm_num : (1:ℝ) ∈ Ioi 0) hl (hd 1 (by norm_num))
+      rw [hg1, slope_def_field, hf1] at hh
+      have hp := (div_le_iff₀ (sub_pos.mpr hl)).mp hh
+      linarith
+    · simp [he, hf1]
+    · have hh := hconv.le_slope_of_hasDerivAt (by norm_num : (1:ℝ) ∈ Ioi 0) ha hg (hd 1 (by norm_num))
+      rw [hg1, slope_def_field, hf1] at hh
+      have hp := (le_div_iff₀ (sub_pos.mpr hg)).mp hh
+      linarith
+  dsimp only [f] at hf0
+  linarith
+
+end ThreeState
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+open Radial (avg)
+
+noncomputable def correlationZ (a b : Message) : ℝ := avg (fun i => messageDeviation a i * messageDeviation b i)
+lemma continuous_correlationZ : Continuous (fun p : Message × Message => correlationZ p.1 p.2) := by
+  apply continuous_avg
+  intro i
+  exact ((((continuous_apply i).comp continuous_subtype_val).comp continuous_fst).sub continuous_const).mul
+    ((((continuous_apply i).comp continuous_subtype_val).comp continuous_snd).sub continuous_const)
+
+noncomputable def symPerm : Fin 6 → Equiv.Perm Spin :=
+  ![Equiv.refl _, Equiv.swap 0 1, Equiv.swap 0 2, Equiv.swap 1 2,
+    (Equiv.swap 0 1).trans (Equiv.swap 1 2), (Equiv.swap 1 2).trans (Equiv.swap 0 1)]
+noncomputable def symAvg (f : Message → ℝ) (m : Message) : ℝ :=
+  (∑ i : Fin 6, f (permMessage (symPerm i) m))/6
+
+lemma integral_symAvg (Q : ProbabilityMeasure Message) (hs : SpinSymmetric Q)
+    (f : Message → ℝ) (hf : Continuous f) :
+    (∫ m, symAvg f m ∂(Q : Measure Message)) = ∫ m, f m ∂(Q : Measure Message) := by
+  simp only [symAvg, integral_div]
+  have hi (i : Fin 6) : Integrable (fun m => f (permMessage (symPerm i) m)) (Q : Measure Message) :=
+    integrable_continuous_message Q (hf.comp (continuous_permMessage _))
+  rw [integral_finsetSum _ (fun i _ => hi i)]
+  simp only [integral_permMessage Q hs _ _ hf.measurable]
+  simp
+
+lemma symAvg_correlationZ (a b : Message) : symAvg (fun m => correlationZ m b) a = 0 := by
+  have ha := average_message a
+  have hb := average_message b
+  simp only [Radial.avg_expand] at ha hb
+  have ha2 : a 2 = 3-a 0-a 1 := by linarith
+  have hb2 : b 2 = 3-b 0-b 1 := by linarith
+  norm_num [symAvg, symPerm, Fin.sum_univ_succ, correlationZ, messageDeviation,
+    Radial.avg_expand, permMessage, Equiv.swap_apply_def, Equiv.trans_apply, Equiv.symm_trans_apply, Fin.ext_iff]
+  rw [ha2, hb2]
+  ring
+
+lemma symAvg_correlationZ_sq (a b : Message) :
+    symAvg (fun m => (correlationZ m b)^2) a = 2*messageX a*messageX b := by
+  have ha := average_message a
+  have hb := average_message b
+  simp only [Radial.avg_expand] at ha hb
+  have ha2 : a 2 = 3-a 0-a 1 := by linarith
+  have hb2 : b 2 = 3-b 0-b 1 := by linarith
+  norm_num [symAvg, symPerm, Fin.sum_univ_succ, correlationZ, messageDeviation,
+    Radial.avg_expand, permMessage, Equiv.swap_apply_def, Equiv.trans_apply, Equiv.symm_trans_apply, Fin.ext_iff,
+    messageX, Radial.momentX]
+  rw [ha2, hb2]
+  ring
+
+lemma symAvg_correlationZ_cube (a b : Message) :
+    symAvg (fun m => (correlationZ m b)^3) a = 2*messageY a*messageY b := by
+  have ha := average_message a
+  have hb := average_message b
+  simp only [Radial.avg_expand] at ha hb
+  have ha2 : a 2 = 3-a 0-a 1 := by linarith
+  have hb2 : b 2 = 3-b 0-b 1 := by linarith
+  norm_num [symAvg, symPerm, Fin.sum_univ_succ, correlationZ, messageDeviation,
+    Radial.avg_expand, permMessage, Equiv.swap_apply_def, Equiv.trans_apply, Equiv.symm_trans_apply, Fin.ext_iff,
+    messageY, Radial.momentY]
+  rw [ha2, hb2]
+  ring
+
+lemma integral_correlationZ (Q : ProbabilityMeasure Message) (hs : SpinSymmetric Q) (b : Message) :
+    (∫ m, correlationZ m b ∂(Q : Measure Message)) = 0 := by
+  rw [← integral_symAvg Q hs (fun m => correlationZ m b) (continuous_correlationZ.comp (continuous_id.prodMk continuous_const))]
+  simp only [symAvg_correlationZ, integral_zero]
+
+lemma integral_correlationZ_sq (Q : ProbabilityMeasure Message) (hs : SpinSymmetric Q) (b : Message) :
+    (∫ m, (correlationZ m b)^2 ∂(Q : Measure Message)) = 2*posteriorMu Q*messageX b := by
+  rw [← integral_symAvg Q hs (fun m => (correlationZ m b)^2) ((continuous_correlationZ.comp (continuous_id.prodMk continuous_const)).pow 2)]
+  simp only [symAvg_correlationZ_sq, integral_mul_const, integral_const_mul, posteriorMu]
+
+lemma integral_correlationZ_cube (Q : ProbabilityMeasure Message) (hs : SpinSymmetric Q) (b : Message) :
+    (∫ m, (correlationZ m b)^3 ∂(Q : Measure Message)) = 2*posteriorEta Q*messageY b := by
+  rw [← integral_symAvg Q hs (fun m => (correlationZ m b)^3) ((continuous_correlationZ.comp (continuous_id.prodMk continuous_const)).pow 3)]
+  simp only [symAvg_correlationZ_cube, integral_mul_const, integral_const_mul, posteriorEta]
+
+end ThreeState
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+open Radial (avg)
+
+lemma correlationZ_comm (a b : Message) : correlationZ a b = correlationZ b a := by
+  simp [correlationZ, mul_comm]
+
+noncomputable def pairNormalizer (lam : ℝ) (h : Admissible lam) (a b : Message) : ℝ :=
+  avg (fun i => a i*edgeMessage lam h b i)
+
+lemma pairNormalizer_eq (lam : ℝ) (h : Admissible lam) (a b : Message) :
+    pairNormalizer lam h a b = 1+lam*correlationZ a b := by
+  have ha := average_message a
+  have hb := average_message b
+  simp only [Radial.avg_expand] at ha hb
+  simp only [pairNormalizer, correlationZ, messageDeviation, edgeMessage, Radial.avg_expand]
+  rw [message_last a, message_last b]
+  ring
+
+lemma continuous_pairNormalizer (lam : ℝ) (h : Admissible lam) :
+    Continuous (fun p : Message × Message => pairNormalizer lam h p.1 p.2) := by
+  simp only [pairNormalizer_eq]
+  exact continuous_const.add (continuous_const.mul continuous_correlationZ)
+
+lemma pairNormalizer_pos (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (a b : Message) : 0 < pairNormalizer lam h a b := by
+  have he (i : Spin) : (1-lam)*a i ≤ a i*edgeMessage lam h b i := by
+    nlinarith [mul_le_mul_of_nonneg_left (edgeMessage_bounds lam h hl0 b i).1 (Message.nonneg a i)]
+  have ha := average_message a
+  simp only [pairNormalizer, Radial.avg_expand] at *
+  nlinarith [he 0, he 1, he 2]
+
+noncomputable def pairMessage (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (a b : Message) : Message :=
+  ⟨fun i => a i*edgeMessage lam h b i/pairNormalizer lam h a b, by
+    constructor
+    · intro i
+      exact div_nonneg (mul_nonneg (Message.nonneg a i) (Message.nonneg _ _)) (pairNormalizer_pos lam h hl0 hl1 a b).le
+    · rw [← Finset.sum_div]
+      change (∑ i, a i*edgeMessage lam h b i)/((∑ i, a i*edgeMessage lam h b i)/3) = 3
+      have hh := ne_of_gt (pairNormalizer_pos lam h hl0 hl1 a b)
+      dsimp only [pairNormalizer, avg] at hh
+      have hs : (∑ i, a i*edgeMessage lam h b i) ≠ 0 := by
+        intro he
+        apply hh
+        rw [he]
+        norm_num
+      field_simp [hs]⟩
+
+lemma continuous_pairMessage (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1) :
+    Continuous (fun p : Message × Message => pairMessage lam h hl0 hl1 p.1 p.2) := by
+  apply Continuous.subtype_mk
+  apply continuous_pi
+  intro i
+  exact ((((continuous_apply i).comp continuous_subtype_val).comp continuous_fst).mul
+    (((continuous_apply i).comp continuous_subtype_val).comp ((continuous_edgeMessage lam h).comp continuous_snd))).div
+      (continuous_pairNormalizer lam h) (fun p => ne_of_gt (pairNormalizer_pos lam h hl0 hl1 p.1 p.2))
+
+lemma integral_pairNormalizer (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message)
+    (hQ : Balanced Q) (a : Message) :
+    (∫ b, pairNormalizer lam h a b ∂(Q : Measure Message)) = 1 := by
+  change (∫ b, avg (fun i => a i*edgeMessage lam h b i) ∂(Q : Measure Message)) = 1
+  rw [integral_avg]
+  · simp only [integral_const_mul, integral_edgeMessage lam h Q hQ, mul_one]
+    exact average_message a
+  · intro i
+    exact integrable_continuous_message Q (continuous_const.mul (((continuous_apply i).comp continuous_subtype_val).comp (continuous_edgeMessage lam h)))
+
+noncomputable def pairIncrement (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message) (a : Message) : ℝ :=
+  ∫ b, pairNormalizer lam h a b*Real.log (pairNormalizer lam h a b) ∂(Q : Measure Message)
+
+lemma continuous_pairIntegrand (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1) :
+    Continuous (fun p : Message × Message => pairNormalizer lam h p.1 p.2*Real.log (pairNormalizer lam h p.1 p.2)) :=
+  (continuous_pairNormalizer lam h).mul ((continuous_pairNormalizer lam h).log
+    (fun p => ne_of_gt (pairNormalizer_pos lam h hl0 hl1 p.1 p.2)))
+
+lemma pairIncrement_nonneg (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (a : Message) : 0 ≤ pairIncrement lam h Q a := by
+  have hf : Continuous (fun b => pairNormalizer lam h a b) :=
+    (continuous_pairNormalizer lam h).comp (continuous_const.prodMk continuous_id)
+  have hg : Continuous (fun b => pairNormalizer lam h a b*Real.log (pairNormalizer lam h a b)) :=
+    (continuous_pairIntegrand lam h hl0 hl1).comp (continuous_const.prodMk continuous_id)
+  have hin (b : Message) : pairNormalizer lam h a b-1 ≤ pairNormalizer lam h a b*Real.log (pairNormalizer lam h a b) := by
+    have hp := pairNormalizer_pos lam h hl0 hl1 a b
+    have hh := mul_le_mul_of_nonneg_left (Real.one_sub_inv_le_log_of_pos hp) hp.le
+    rw [mul_sub, mul_one, mul_inv_cancel₀ hp.ne'] at hh
+    exact hh
+  have hif : Integrable (fun b => pairNormalizer lam h a b-1) (Q : Measure Message) :=
+    integrable_continuous_message Q (hf.sub continuous_const)
+  have hh := integral_mono hif (integrable_continuous_message Q hg) hin
+  rw [integral_sub (integrable_continuous_message Q hf) (integrable_const 1), integral_pairNormalizer lam h Q hQ] at hh
+  simpa [pairIncrement] using hh
+
+lemma pairIncrement_lower (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) (hs : SpinSymmetric Q) (a : Message) :
+    lam^2*posteriorMu Q*messageX a-lam^3*posteriorEta Q*messageY a/3 ≤ pairIncrement lam h Q a := by
+  let z : Message → ℝ := fun b => correlationZ b a
+  have hz : Continuous z := continuous_correlationZ.comp (continuous_id.prodMk continuous_const)
+  have h1 : (∫ b, z b ∂(Q : Measure Message)) = 0 := integral_correlationZ Q hs a
+  have h2 : (∫ b, (z b)^2 ∂(Q : Measure Message)) = 2*posteriorMu Q*messageX a := integral_correlationZ_sq Q hs a
+  have h3 : (∫ b, (z b)^3 ∂(Q : Measure Message)) = 2*posteriorEta Q*messageY a := integral_correlationZ_cube Q hs a
+  have hin (b : Message) : lam*z b+lam^2*(z b)^2/2-lam^3*(z b)^3/6 ≤ pairNormalizer lam h a b*Real.log (pairNormalizer lam h a b) := by
+    have hh := log_correlation_cubic _ (pairNormalizer_pos lam h hl0 hl1 a b)
+    rw [pairNormalizer_eq, correlationZ_comm a b] at hh ⊢
+    dsimp only [z]
+    convert hh using 1
+    ring
+  have hf : Continuous (fun b => lam*z b+lam^2*(z b)^2/2-lam^3*(z b)^3/6) := by fun_prop
+  have hg : Continuous (fun b => pairNormalizer lam h a b*Real.log (pairNormalizer lam h a b)) :=
+    (continuous_pairIntegrand lam h hl0 hl1).comp (continuous_const.prodMk continuous_id)
+  have hh := integral_mono (integrable_continuous_message Q hf) (integrable_continuous_message Q hg) hin
+  have hi1 := integrable_continuous_message Q hz
+  have hi2 : Integrable (fun b => (z b)^2) (Q : Measure Message) := integrable_continuous_message Q (hz.pow 2)
+  have hi3 : Integrable (fun b => (z b)^3) (Q : Measure Message) := integrable_continuous_message Q (hz.pow 3)
+  have hiadd : Integrable (fun b => lam*z b+lam^2*(z b)^2/2) (Q : Measure Message) :=
+    (hi1.const_mul _).add ((hi2.const_mul _).div_const _)
+  rw [integral_sub hiadd ((hi3.const_mul _).div_const _), integral_add (hi1.const_mul _) ((hi2.const_mul _).div_const _),
+    integral_div, integral_div, integral_const_mul, integral_const_mul, integral_const_mul, h1,h2,h3] at hh
+  dsimp only [pairIncrement]
+  linarith
+
+end ThreeState
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeState/PoissonBounds.lean b/lean/OAI/Probability/ThreeState/PoissonBounds.lean
index e8673d458f02f4b4322f190bff4253a9ede827ce..8cdfed7605831e642e54e2f2dcdc12d29bc3ee85 100644
--- a/lean/OAI/Probability/ThreeState/PoissonBounds.lean
+++ b/lean/OAI/Probability/ThreeState/PoissonBounds.lean
@@ -1,263 +1,263 @@
-import Mathlib
-import OAI.Probability.ThreeState.PoissonCalculus
-import OAI.Probability.ThreeState.RegularCritical
-
-namespace OAI
-
-/-! Poisson posterior projection and entropy bounds. -/
-
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-
-noncomputable def offspringPosterior (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
-    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) : ProbabilityMeasure Message :=
-  ⟨densityEvolution offspring lam h Q, densityEvolution_probability offspring lam h Q hQ⟩
-
-lemma integral_offspringPosterior (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
-    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (f : C(Message,ℝ)) :
-    (∫ m, f m ∂(offspringPosterior offspring lam h Q hQ : Measure Message)) =
-      mean offspring (fun n => ∫ m, f m ∂(degreePosterior lam h Q hQ n : Measure Message)) := by
-  rw [show (offspringPosterior offspring lam h Q hQ : Measure Message) = densityEvolution offspring lam h Q by rfl,
-    integral_densityEvolution_continuous offspring lam h Q hQ f]
-  apply mean_congr
-  intro n
-  exact (integral_degreeOutput lam h Q n f f.continuous.measurable).symm
-
-lemma posteriorMu_le_one (Q : ProbabilityMeasure Message) : posteriorMu Q ≤ 1 := by
-  simpa [posteriorMu, posteriorNu] using
-    integral_mono (integrable_continuous_message Q continuous_messageX) (integrable_const (1:ℝ)) messageX_le_one
-
-lemma posteriorNu_le_one (Q : ProbabilityMeasure Message) : posteriorNu Q ≤ 1 := by
-  have hp (m : Message) : (messageX m)^2≤1 := by nlinarith [messageX_nonneg m, messageX_le_one m]
-  simpa [posteriorMu, posteriorNu] using
-    integral_mono (integrable_continuous_message Q (continuous_messageX.pow 2)) (integrable_const (1:ℝ)) hp
-
-lemma posteriorMu_offspring (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
-    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) :
-    posteriorMu (offspringPosterior offspring lam h Q hQ) = mean offspring (fun n => posteriorMu (degreePosterior lam h Q hQ n)) :=
-  integral_offspringPosterior offspring lam h Q hQ ⟨messageX,continuous_messageX⟩
-
-lemma posteriorNu_offspring (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
-    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) :
-    posteriorNu (offspringPosterior offspring lam h Q hQ) = mean offspring (fun n => posteriorNu (degreePosterior lam h Q hQ n)) := by
-  unfold posteriorNu
-  exact integral_offspringPosterior offspring lam h Q hQ ⟨fun m => (messageX m)^2,continuous_messageX.pow 2⟩
-
-lemma pairCorrelation_offspring (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) :
-    pairCorrelation lam h Q (offspringPosterior offspring lam h Q hQ) = mean offspring (fun n => pairCorrelation lam h Q (degreePosterior lam h Q hQ n)) :=
-  integral_offspringPosterior offspring lam h Q hQ ⟨pairIncrement lam h Q,continuous_pairIncrement lam h hl0 hl1 Q⟩
-
-lemma poisson_projection_quadratic (t : ℝ) (ht : 0 ≤ t) (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (hs : SpinSymmetric Q) (s : ℝ) :
-    0 ≤ posteriorMu (offspringPosterior (poissonOffspring t ht) lam h Q hQ)*s^2-
-      2*(t*(lam^2*posteriorMu Q))*s+(t*(lam^2*posteriorMu Q)+t^2*(lam^2*posteriorMu Q)^2) := by
-  let p := poissonOffspring t ht
-  let beta := lam^2*posteriorMu Q
-  let muj := fun n => posteriorMu (degreePosterior lam h Q hQ n)
-  have hi : HasMean p muj := hasMean_bounded p 1 (fun n => by rw [abs_of_nonneg (posteriorMu_nonneg _)]; exact posteriorMu_le_one _)
-  have h1 : HasMean p (fun n : ℕ => (n:ℝ)) := (poisson_moments t ht).hasMean
-  have h2 : HasMean p (fun n : ℕ => (n:ℝ)*((n:ℝ)-1)) := (poisson_second_factorial_hasSum t ht).summable
-  have hh := mean_nonneg p (fun n => degree_projection_quadratic lam h hl0 hl1 Q hQ hs n s)
-  change 0 ≤ mean p (fun n => muj n*s^2-2*((n:ℝ)*beta)*s+((n:ℝ)*beta+(n:ℝ)*(n-1)*beta^2)) at hh
-  have ha : HasMean p (fun n : ℕ => 2*((n:ℝ)*beta)*s) := ((h1.mul_const beta).const_mul 2).mul_const s
-  rw [mean_add ((hi.mul_const _).sub ha) ((h1.mul_const _).add (h2.mul_const _)),
-    mean_sub (hi.mul_const _) ha, mean_mul_const, mean_mul_const, mean_const_mul, mean_mul_const,
-    mean_add (h1.mul_const _) (h2.mul_const _), mean_mul_const, mean_mul_const] at hh
-  have hm1 : mean p (fun n : ℕ => (n:ℝ))=t := (poisson_moments t ht).first
-  have hm2 : mean p (fun n : ℕ => (n:ℝ)*((n:ℝ)-1))=t^2 := (poisson_second_factorial_hasSum t ht).tsum_eq
-  rw [hm1,hm2, ← posteriorMu_offspring (poissonOffspring t ht) lam h Q hQ] at hh
-  exact hh
-
-lemma poisson_projection (t : ℝ) (ht : 0 ≤ t) (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (hs : SpinSymmetric Q) :
-    t*(lam^2*posteriorMu Q)/(1+t*(lam^2*posteriorMu Q)) ≤ posteriorMu (offspringPosterior (poissonOffspring t ht) lam h Q hQ) := by
-  let x := t*(lam^2*posteriorMu Q)
-  have hx : 0 ≤ x := mul_nonneg ht (mul_nonneg (sq_nonneg _) (posteriorMu_nonneg Q))
-  have hd : 0 < 1+x := by positivity
-  by_cases hx0 : x=0
-  · change x/(1+x) ≤ _
-    rw [hx0,zero_div]
-    exact posteriorMu_nonneg _
-  · have hxpos : 0<x := lt_of_le_of_ne hx (Ne.symm hx0)
-    have hp := poisson_projection_quadratic t ht lam h hl0 hl1 Q hQ hs
-    have hdisc := discrim_le_zero (show ∀ s, 0 ≤ posteriorMu (offspringPosterior (poissonOffspring t ht) lam h Q hQ)*(s*s)+(-2*x)*s+(x+x^2) by
-      intro s
-      convert hp s using 1
-      dsimp [x]
-      ring)
-    simp only [discrim] at hdisc
-    change x/(1+x) ≤ _
-    apply (div_le_iff₀ hd).mpr
-    apply (mul_le_mul_iff_right₀ hxpos).mp
-    nlinarith only [hdisc]
-
-end ThreeState
-
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-
-lemma poissonAverage_eq_mean (f : ℕ → ℝ) (t : ℝ) (ht : 0 ≤ t) :
-    poissonAverage f t = mean (poissonOffspring t ht) f := by
-  unfold poissonAverage expGenerating mean
-  rw [← tsum_mul_left]
-  apply tsum_congr
-  intro n
-  rw [poisson_mass]
-  ring
-
-lemma poissonAverage_zero (f : ℕ → ℝ) : poissonAverage f 0 = f 0 := by
-  unfold poissonAverage expGenerating
-  rw [neg_zero,Real.exp_zero,one_mul, tsum_eq_single 0]
-  · norm_num
-  · intro n hn
-    simp [hn]
-
-lemma poissonAverage_monotoneOn (f : ℕ → ℝ) (C : ℝ) (hC : 0 ≤ C)
-    (hf : ∀ n, |f n| ≤ C*((n:ℝ)+1)) (hmono : Monotone f) :
-    MonotoneOn (poissonAverage f) (Set.Ici 0) := by
-  apply monotoneOn_of_hasDerivWithinAt_nonneg (convex_Ici 0)
-    (f' := poissonAverage (fun n => f (n+1)-f n))
-  · intro t _
-    exact (poissonAverage_hasDerivAt f C hC hf t).continuousAt.continuousWithinAt
-  · intro t _
-    exact (poissonAverage_hasDerivAt f C hC hf t).hasDerivWithinAt
-  · intro t ht
-    have ht0 : 0 ≤ t := interior_subset ht
-    rw [poissonAverage_eq_mean _ t ht0]
-    apply mean_nonneg
-    intro n
-    exact sub_nonneg.mpr (hmono (Nat.le_succ n))
-
-lemma poisson_nu_le_fixedpoint (d : ℝ) (hd : 0 ≤ d) (t : ℝ) (ht : 0 ≤ t) (htd : t ≤ d)
-    (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint (poissonOffspring d hd) lam h Q) :
-    posteriorNu (offspringPosterior (poissonOffspring t ht) lam h Q hQ.1) ≤ posteriorNu Q := by
-  let f := fun n => posteriorNu (degreePosterior lam h Q hQ.1 n)
-  have hf (n : ℕ) : |f n| ≤ 1*((n:ℝ)+1) := by
-    rw [abs_of_nonneg (posteriorNu_nonneg _)]
-    have hh := posteriorNu_le_one (degreePosterior lam h Q hQ.1 n)
-    change posteriorNu (degreePosterior lam h Q hQ.1 n) ≤ _
-    nlinarith [show (0:ℝ)≤n by positivity]
-  have he : offspringPosterior (poissonOffspring d hd) lam h Q hQ.1 = Q := by
-    apply Subtype.ext
-    exact (fixedpoint_measure (poissonOffspring d hd) lam h Q hQ).symm
-  have hm := poissonAverage_monotoneOn f 1 (by norm_num) hf (degree_nu_monotone lam h hl0 hl1 Q hQ.1) ht hd htd
-  rw [poissonAverage_eq_mean _ t ht, poissonAverage_eq_mean _ d hd] at hm
-  change mean (poissonOffspring t ht) (fun n => posteriorNu (degreePosterior lam h Q hQ.1 n)) ≤
-    mean (poissonOffspring d hd) (fun n => posteriorNu (degreePosterior lam h Q hQ.1 n)) at hm
-  rwa [← posteriorNu_offspring, ← posteriorNu_offspring, he] at hm
-
-lemma poisson_correlation_hasDerivAt (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (t : ℝ) (ht : 0 ≤ t) :
-    HasDerivAt (poissonAverage (correlationEntropy lam h Q))
-      (pairCorrelation lam h Q (offspringPosterior (poissonOffspring t ht) lam h Q hQ)) t := by
-  have hf (n : ℕ) : |correlationEntropy lam h Q n| ≤ edgeLogBound lam*((n:ℝ)+1) := by
-    have hh := correlationEntropy_bound lam h hl0 hl1 Q hQ n
-    nlinarith only [hh,edgeLogBound_nonneg lam]
-  have hh := poissonAverage_hasDerivAt (correlationEntropy lam h Q) (edgeLogBound lam) (edgeLogBound_nonneg lam) hf t
-  rw [poissonAverage_eq_mean _ t ht] at hh
-  convert hh using 1
-  rw [pairCorrelation_offspring (poissonOffspring t ht) lam h hl0 hl1 Q hQ]
-  apply mean_congr
-  intro n
-  have he := correlationEntropy_succ lam h hl0 hl1 Q hQ n
-  change _ = _ at he
-  change (∫ a, pairIncrement lam h Q a ∂degreeOutput lam h Q n) = _
-  linarith only [he]
-
-end ThreeState
-
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-
-lemma poisson_increment_bound (d : ℝ) (hd : 0 ≤ d) (t : ℝ) (ht : 0 ≤ t) (htd : t ≤ d)
-    (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint (poissonOffspring d hd) lam h Q)
-    (hs : SpinSymmetric Q) (he : 0 ≤ posteriorEta Q) :
-    3*lam^4*(posteriorMu Q)^2*t/(1+t*lam^2*posteriorMu Q)-
-      lam^4*posteriorMu Q*posteriorNu Q*Real.sqrt t ≤
-      3*pairCorrelation lam h Q (offspringPosterior (poissonOffspring t ht) lam h Q hQ.1) := by
-  have hm := posteriorMu_nonneg Q
-  have hn := posteriorNu_nonneg Q
-  let L := t*(lam^2*posteriorMu Q)/(1+t*(lam^2*posteriorMu Q))
-  have hb : 0 ≤ lam^2*posteriorMu Q := mul_nonneg (sq_nonneg _) hm
-  have hL : 0 ≤ L := div_nonneg (mul_nonneg ht hb) (by positivity)
-  have hh := pairCorrelation_truncated lam h hl0 hl1 Q _ hQ.1 hs he
-    (poisson_nu_le_fixedpoint d hd t ht htd lam h hl0 hl1 Q hQ) hL
-    (poisson_projection t ht lam h hl0 hl1 Q hQ.1 hs)
-  have hsqrt : Real.sqrt L ≤ lam*Real.sqrt (posteriorMu Q)*Real.sqrt t := by
-    have hh := Real.sqrt_le_sqrt (div_le_self (mul_nonneg ht hb) (show 1 ≤ 1+t*(lam^2*posteriorMu Q) by nlinarith only [mul_nonneg ht hb]))
-    rw [Real.sqrt_mul ht, Real.sqrt_mul (sq_nonneg lam), Real.sqrt_sq_eq_abs, abs_of_nonneg hl0] at hh
-    dsimp only [L]
-    nlinarith only [hh]
-  have hmul := mul_le_mul_of_nonneg_left hsqrt (show 0 ≤ lam^3*Real.sqrt (posteriorMu Q)*posteriorNu Q by positivity)
-  have hsq := Real.sq_sqrt hm
-  have hlead : 3*lam^2*posteriorMu Q*L =
-      3*lam^4*(posteriorMu Q)^2*t/(1+t*lam^2*posteriorMu Q) := by dsimp only [L]; ring
-  rw [hlead] at hh
-  have hloss : lam^3*Real.sqrt (posteriorMu Q)*posteriorNu Q*(lam*Real.sqrt (posteriorMu Q)*Real.sqrt t) =
-      lam^4*posteriorMu Q*posteriorNu Q*Real.sqrt t := by
-    calc
-      _ = lam^4*(Real.sqrt (posteriorMu Q))^2*posteriorNu Q*Real.sqrt t := by ring
-      _ = _ := by rw [hsq]
-  rw [hloss] at hmul
-  linarith only [hh, hmul]
-
-lemma poisson_correlation_bound (d : ℝ) (hd : 0 < d) (lam : ℝ) (h : Admissible lam)
-    (hl0 : 0 < lam) (hl1 : lam < 1) (hc : d*lam^2=1)
-    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint (poissonOffspring d hd.le) lam h Q)
-    (hs : SpinSymmetric Q) (he : 0 ≤ posteriorEta Q) :
-    3*(posteriorMu Q)^2/(2*(1+2*posteriorMu Q/3))-
-      lam*posteriorMu Q*posteriorNu Q/Real.sqrt 2 ≤ 3*mean (poissonOffspring d hd.le) (correlationEntropy lam h Q) := by
-  let H := poissonAverage (correlationEntropy lam h Q)
-  let C := poissonAverage (fun n => correlationEntropy lam h Q (n+1)-correlationEntropy lam h Q n)
-  have hderiv (t : ℝ) : HasDerivAt H (C t) t :=
-    poissonAverage_hasDerivAt (correlationEntropy lam h Q) (edgeLogBound lam) (edgeLogBound_nonneg lam)
-      (fun n => by
-        have hh := correlationEntropy_bound lam h hl0.le hl1 Q hQ.1 n
-        nlinarith only [hh,edgeLogBound_nonneg lam]) t
-  have heq (t : ℝ) (ht : 0 ≤ t) : C t=pairCorrelation lam h Q (offspringPosterior (poissonOffspring t ht) lam h Q hQ.1) :=
-    (hderiv t).unique (poisson_correlation_hasDerivAt lam h hl0.le hl1 Q hQ.1 t ht)
-  have hh := poisson_correlation_scalar d lam (posteriorMu Q) (posteriorNu Q) hd hl0
-    (posteriorMu_nonneg Q) (posteriorNu_nonneg Q) hc H C
-    (by dsimp only [H]; rw [poissonAverage_zero,correlationEntropy_zero])
-    (fun t _ => hderiv t) (fun t ht => by
-      rw [heq t ht.1]
-      exact poisson_increment_bound d hd.le t ht.1 ht.2 lam h hl0.le hl1 Q hQ hs he)
-  simpa only [H,poissonAverage_eq_mean _ d hd.le] using hh
-
-lemma poisson_entropy_bound (d : ℝ) (hd : 0 < d) (lam : ℝ) (h : Admissible lam)
-    (hl0 : 0 < lam) (hl1 : lam < 1) (hc : d*lam^2=1)
-    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint (poissonOffspring d hd.le) lam h Q)
-    (hs : SpinSymmetric Q) (he : 0 ≤ posteriorEta Q) :
-    (1-lam^2)/2*posteriorNu Q+3*(posteriorMu Q)^2/(2*(1+2*posteriorMu Q/3))-
-      lam*posteriorMu Q*posteriorNu Q/Real.sqrt 2 ≤
-      (8/5:ℝ)*(∫ m, entropyJ m ∂(Q : Measure Message))^2 := by
-  have hD := (poisson_moments d hd.le).hasMean
-  have hD2 := (poisson_moments d hd.le).square
-  have hid := entropy_identity (poissonOffspring d hd.le) lam h hl0.le hl1 Q hQ hs hD2
-  have hrad := integral_entropyF_radial (poissonOffspring d hd.le) lam h hl0 hl1 Q hQ hD2
-  have hcorr := poisson_correlation_bound d hd lam h hl0 hl1 hc Q hQ hs he
-  have hJ := additive_entropyJ (poissonOffspring d hd.le) lam h hl0.le hl1 Q hQ hs hD
-  have hfacmean : mean (poissonOffspring d hd.le) (fun n : ℕ => (n:ℝ)*((n:ℝ)-1)) = d^2 := (poisson_second_factorial_hasSum d hd.le).tsum_eq
-  rw [(poisson_moments d hd.le).first, hfacmean] at hid
-  rw [(poisson_moments d hd.le).first] at hJ
-  have hl : lam⁻¹^2 = d := by field_simp; nlinarith only [hc]
-  rw [hl] at hrad
-  have hJe : (∫ m, entropyJ (edgeMessage lam h m) ∂(Q : Measure Message)) =
-      lam^2*(∫ m, entropyJ m ∂(Q : Measure Message)) := by
-    nlinarith only [hJ, congrArg (fun x : ℝ => x*(∫ m, entropyJ (edgeMessage lam h m) ∂(Q : Measure Message))) hc]
-  rw [hJe] at hid
-  have hfac : d^2*lam^4=1 := by
-    calc _ = (d*lam^2)^2 := by ring
-         _ = 1 := by rw [hc]; norm_num
-  nlinarith only [hid,hrad,hcorr,
-    congrArg (fun x : ℝ => (8/5:ℝ)*x*(∫ m, entropyJ m ∂(Q : Measure Message))^2) hfac]
-
-end ThreeState
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeState.PoissonCalculus
+import OAI.Probability.ThreeState.RegularCritical
+
+namespace OAI
+
+/-! Poisson posterior projection and entropy bounds. -/
+
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+
+noncomputable def offspringPosterior (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
+    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) : ProbabilityMeasure Message :=
+  ⟨densityEvolution offspring lam h Q, densityEvolution_probability offspring lam h Q hQ⟩
+
+lemma integral_offspringPosterior (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
+    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (f : C(Message,ℝ)) :
+    (∫ m, f m ∂(offspringPosterior offspring lam h Q hQ : Measure Message)) =
+      mean offspring (fun n => ∫ m, f m ∂(degreePosterior lam h Q hQ n : Measure Message)) := by
+  rw [show (offspringPosterior offspring lam h Q hQ : Measure Message) = densityEvolution offspring lam h Q by rfl,
+    integral_densityEvolution_continuous offspring lam h Q hQ f]
+  apply mean_congr
+  intro n
+  exact (integral_degreeOutput lam h Q n f f.continuous.measurable).symm
+
+lemma posteriorMu_le_one (Q : ProbabilityMeasure Message) : posteriorMu Q ≤ 1 := by
+  simpa [posteriorMu, posteriorNu] using
+    integral_mono (integrable_continuous_message Q continuous_messageX) (integrable_const (1:ℝ)) messageX_le_one
+
+lemma posteriorNu_le_one (Q : ProbabilityMeasure Message) : posteriorNu Q ≤ 1 := by
+  have hp (m : Message) : (messageX m)^2≤1 := by nlinarith [messageX_nonneg m, messageX_le_one m]
+  simpa [posteriorMu, posteriorNu] using
+    integral_mono (integrable_continuous_message Q (continuous_messageX.pow 2)) (integrable_const (1:ℝ)) hp
+
+lemma posteriorMu_offspring (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
+    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) :
+    posteriorMu (offspringPosterior offspring lam h Q hQ) = mean offspring (fun n => posteriorMu (degreePosterior lam h Q hQ n)) :=
+  integral_offspringPosterior offspring lam h Q hQ ⟨messageX,continuous_messageX⟩
+
+lemma posteriorNu_offspring (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
+    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) :
+    posteriorNu (offspringPosterior offspring lam h Q hQ) = mean offspring (fun n => posteriorNu (degreePosterior lam h Q hQ n)) := by
+  unfold posteriorNu
+  exact integral_offspringPosterior offspring lam h Q hQ ⟨fun m => (messageX m)^2,continuous_messageX.pow 2⟩
+
+lemma pairCorrelation_offspring (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) :
+    pairCorrelation lam h Q (offspringPosterior offspring lam h Q hQ) = mean offspring (fun n => pairCorrelation lam h Q (degreePosterior lam h Q hQ n)) :=
+  integral_offspringPosterior offspring lam h Q hQ ⟨pairIncrement lam h Q,continuous_pairIncrement lam h hl0 hl1 Q⟩
+
+lemma poisson_projection_quadratic (t : ℝ) (ht : 0 ≤ t) (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (hs : SpinSymmetric Q) (s : ℝ) :
+    0 ≤ posteriorMu (offspringPosterior (poissonOffspring t ht) lam h Q hQ)*s^2-
+      2*(t*(lam^2*posteriorMu Q))*s+(t*(lam^2*posteriorMu Q)+t^2*(lam^2*posteriorMu Q)^2) := by
+  let p := poissonOffspring t ht
+  let beta := lam^2*posteriorMu Q
+  let muj := fun n => posteriorMu (degreePosterior lam h Q hQ n)
+  have hi : HasMean p muj := hasMean_bounded p 1 (fun n => by rw [abs_of_nonneg (posteriorMu_nonneg _)]; exact posteriorMu_le_one _)
+  have h1 : HasMean p (fun n : ℕ => (n:ℝ)) := (poisson_moments t ht).hasMean
+  have h2 : HasMean p (fun n : ℕ => (n:ℝ)*((n:ℝ)-1)) := (poisson_second_factorial_hasSum t ht).summable
+  have hh := mean_nonneg p (fun n => degree_projection_quadratic lam h hl0 hl1 Q hQ hs n s)
+  change 0 ≤ mean p (fun n => muj n*s^2-2*((n:ℝ)*beta)*s+((n:ℝ)*beta+(n:ℝ)*(n-1)*beta^2)) at hh
+  have ha : HasMean p (fun n : ℕ => 2*((n:ℝ)*beta)*s) := ((h1.mul_const beta).const_mul 2).mul_const s
+  rw [mean_add ((hi.mul_const _).sub ha) ((h1.mul_const _).add (h2.mul_const _)),
+    mean_sub (hi.mul_const _) ha, mean_mul_const, mean_mul_const, mean_const_mul, mean_mul_const,
+    mean_add (h1.mul_const _) (h2.mul_const _), mean_mul_const, mean_mul_const] at hh
+  have hm1 : mean p (fun n : ℕ => (n:ℝ))=t := (poisson_moments t ht).first
+  have hm2 : mean p (fun n : ℕ => (n:ℝ)*((n:ℝ)-1))=t^2 := (poisson_second_factorial_hasSum t ht).tsum_eq
+  rw [hm1,hm2, ← posteriorMu_offspring (poissonOffspring t ht) lam h Q hQ] at hh
+  exact hh
+
+lemma poisson_projection (t : ℝ) (ht : 0 ≤ t) (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (hs : SpinSymmetric Q) :
+    t*(lam^2*posteriorMu Q)/(1+t*(lam^2*posteriorMu Q)) ≤ posteriorMu (offspringPosterior (poissonOffspring t ht) lam h Q hQ) := by
+  let x := t*(lam^2*posteriorMu Q)
+  have hx : 0 ≤ x := mul_nonneg ht (mul_nonneg (sq_nonneg _) (posteriorMu_nonneg Q))
+  have hd : 0 < 1+x := by positivity
+  by_cases hx0 : x=0
+  · change x/(1+x) ≤ _
+    rw [hx0,zero_div]
+    exact posteriorMu_nonneg _
+  · have hxpos : 0<x := lt_of_le_of_ne hx (Ne.symm hx0)
+    have hp := poisson_projection_quadratic t ht lam h hl0 hl1 Q hQ hs
+    have hdisc := discrim_le_zero (show ∀ s, 0 ≤ posteriorMu (offspringPosterior (poissonOffspring t ht) lam h Q hQ)*(s*s)+(-2*x)*s+(x+x^2) by
+      intro s
+      convert hp s using 1
+      dsimp [x]
+      ring)
+    simp only [discrim] at hdisc
+    change x/(1+x) ≤ _
+    apply (div_le_iff₀ hd).mpr
+    apply (mul_le_mul_iff_right₀ hxpos).mp
+    nlinarith only [hdisc]
+
+end ThreeState
+
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+
+lemma poissonAverage_eq_mean (f : ℕ → ℝ) (t : ℝ) (ht : 0 ≤ t) :
+    poissonAverage f t = mean (poissonOffspring t ht) f := by
+  unfold poissonAverage expGenerating mean
+  rw [← tsum_mul_left]
+  apply tsum_congr
+  intro n
+  rw [poisson_mass]
+  ring
+
+lemma poissonAverage_zero (f : ℕ → ℝ) : poissonAverage f 0 = f 0 := by
+  unfold poissonAverage expGenerating
+  rw [neg_zero,Real.exp_zero,one_mul, tsum_eq_single 0]
+  · norm_num
+  · intro n hn
+    simp [hn]
+
+lemma poissonAverage_monotoneOn (f : ℕ → ℝ) (C : ℝ) (hC : 0 ≤ C)
+    (hf : ∀ n, |f n| ≤ C*((n:ℝ)+1)) (hmono : Monotone f) :
+    MonotoneOn (poissonAverage f) (Set.Ici 0) := by
+  apply monotoneOn_of_hasDerivWithinAt_nonneg (convex_Ici 0)
+    (f' := poissonAverage (fun n => f (n+1)-f n))
+  · intro t _
+    exact (poissonAverage_hasDerivAt f C hC hf t).continuousAt.continuousWithinAt
+  · intro t _
+    exact (poissonAverage_hasDerivAt f C hC hf t).hasDerivWithinAt
+  · intro t ht
+    have ht0 : 0 ≤ t := interior_subset ht
+    rw [poissonAverage_eq_mean _ t ht0]
+    apply mean_nonneg
+    intro n
+    exact sub_nonneg.mpr (hmono (Nat.le_succ n))
+
+lemma poisson_nu_le_fixedpoint (d : ℝ) (hd : 0 ≤ d) (t : ℝ) (ht : 0 ≤ t) (htd : t ≤ d)
+    (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint (poissonOffspring d hd) lam h Q) :
+    posteriorNu (offspringPosterior (poissonOffspring t ht) lam h Q hQ.1) ≤ posteriorNu Q := by
+  let f := fun n => posteriorNu (degreePosterior lam h Q hQ.1 n)
+  have hf (n : ℕ) : |f n| ≤ 1*((n:ℝ)+1) := by
+    rw [abs_of_nonneg (posteriorNu_nonneg _)]
+    have hh := posteriorNu_le_one (degreePosterior lam h Q hQ.1 n)
+    change posteriorNu (degreePosterior lam h Q hQ.1 n) ≤ _
+    nlinarith [show (0:ℝ)≤n by positivity]
+  have he : offspringPosterior (poissonOffspring d hd) lam h Q hQ.1 = Q := by
+    apply Subtype.ext
+    exact (fixedpoint_measure (poissonOffspring d hd) lam h Q hQ).symm
+  have hm := poissonAverage_monotoneOn f 1 (by norm_num) hf (degree_nu_monotone lam h hl0 hl1 Q hQ.1) ht hd htd
+  rw [poissonAverage_eq_mean _ t ht, poissonAverage_eq_mean _ d hd] at hm
+  change mean (poissonOffspring t ht) (fun n => posteriorNu (degreePosterior lam h Q hQ.1 n)) ≤
+    mean (poissonOffspring d hd) (fun n => posteriorNu (degreePosterior lam h Q hQ.1 n)) at hm
+  rwa [← posteriorNu_offspring, ← posteriorNu_offspring, he] at hm
+
+lemma poisson_correlation_hasDerivAt (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (t : ℝ) (ht : 0 ≤ t) :
+    HasDerivAt (poissonAverage (correlationEntropy lam h Q))
+      (pairCorrelation lam h Q (offspringPosterior (poissonOffspring t ht) lam h Q hQ)) t := by
+  have hf (n : ℕ) : |correlationEntropy lam h Q n| ≤ edgeLogBound lam*((n:ℝ)+1) := by
+    have hh := correlationEntropy_bound lam h hl0 hl1 Q hQ n
+    nlinarith only [hh,edgeLogBound_nonneg lam]
+  have hh := poissonAverage_hasDerivAt (correlationEntropy lam h Q) (edgeLogBound lam) (edgeLogBound_nonneg lam) hf t
+  rw [poissonAverage_eq_mean _ t ht] at hh
+  convert hh using 1
+  rw [pairCorrelation_offspring (poissonOffspring t ht) lam h hl0 hl1 Q hQ]
+  apply mean_congr
+  intro n
+  have he := correlationEntropy_succ lam h hl0 hl1 Q hQ n
+  change _ = _ at he
+  change (∫ a, pairIncrement lam h Q a ∂degreeOutput lam h Q n) = _
+  linarith only [he]
+
+end ThreeState
+
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+
+lemma poisson_increment_bound (d : ℝ) (hd : 0 ≤ d) (t : ℝ) (ht : 0 ≤ t) (htd : t ≤ d)
+    (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint (poissonOffspring d hd) lam h Q)
+    (hs : SpinSymmetric Q) (he : 0 ≤ posteriorEta Q) :
+    3*lam^4*(posteriorMu Q)^2*t/(1+t*lam^2*posteriorMu Q)-
+      lam^4*posteriorMu Q*posteriorNu Q*Real.sqrt t ≤
+      3*pairCorrelation lam h Q (offspringPosterior (poissonOffspring t ht) lam h Q hQ.1) := by
+  have hm := posteriorMu_nonneg Q
+  have hn := posteriorNu_nonneg Q
+  let L := t*(lam^2*posteriorMu Q)/(1+t*(lam^2*posteriorMu Q))
+  have hb : 0 ≤ lam^2*posteriorMu Q := mul_nonneg (sq_nonneg _) hm
+  have hL : 0 ≤ L := div_nonneg (mul_nonneg ht hb) (by positivity)
+  have hh := pairCorrelation_truncated lam h hl0 hl1 Q _ hQ.1 hs he
+    (poisson_nu_le_fixedpoint d hd t ht htd lam h hl0 hl1 Q hQ) hL
+    (poisson_projection t ht lam h hl0 hl1 Q hQ.1 hs)
+  have hsqrt : Real.sqrt L ≤ lam*Real.sqrt (posteriorMu Q)*Real.sqrt t := by
+    have hh := Real.sqrt_le_sqrt (div_le_self (mul_nonneg ht hb) (show 1 ≤ 1+t*(lam^2*posteriorMu Q) by nlinarith only [mul_nonneg ht hb]))
+    rw [Real.sqrt_mul ht, Real.sqrt_mul (sq_nonneg lam), Real.sqrt_sq_eq_abs, abs_of_nonneg hl0] at hh
+    dsimp only [L]
+    nlinarith only [hh]
+  have hmul := mul_le_mul_of_nonneg_left hsqrt (show 0 ≤ lam^3*Real.sqrt (posteriorMu Q)*posteriorNu Q by positivity)
+  have hsq := Real.sq_sqrt hm
+  have hlead : 3*lam^2*posteriorMu Q*L =
+      3*lam^4*(posteriorMu Q)^2*t/(1+t*lam^2*posteriorMu Q) := by dsimp only [L]; ring
+  rw [hlead] at hh
+  have hloss : lam^3*Real.sqrt (posteriorMu Q)*posteriorNu Q*(lam*Real.sqrt (posteriorMu Q)*Real.sqrt t) =
+      lam^4*posteriorMu Q*posteriorNu Q*Real.sqrt t := by
+    calc
+      _ = lam^4*(Real.sqrt (posteriorMu Q))^2*posteriorNu Q*Real.sqrt t := by ring
+      _ = _ := by rw [hsq]
+  rw [hloss] at hmul
+  linarith only [hh, hmul]
+
+lemma poisson_correlation_bound (d : ℝ) (hd : 0 < d) (lam : ℝ) (h : Admissible lam)
+    (hl0 : 0 < lam) (hl1 : lam < 1) (hc : d*lam^2=1)
+    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint (poissonOffspring d hd.le) lam h Q)
+    (hs : SpinSymmetric Q) (he : 0 ≤ posteriorEta Q) :
+    3*(posteriorMu Q)^2/(2*(1+2*posteriorMu Q/3))-
+      lam*posteriorMu Q*posteriorNu Q/Real.sqrt 2 ≤ 3*mean (poissonOffspring d hd.le) (correlationEntropy lam h Q) := by
+  let H := poissonAverage (correlationEntropy lam h Q)
+  let C := poissonAverage (fun n => correlationEntropy lam h Q (n+1)-correlationEntropy lam h Q n)
+  have hderiv (t : ℝ) : HasDerivAt H (C t) t :=
+    poissonAverage_hasDerivAt (correlationEntropy lam h Q) (edgeLogBound lam) (edgeLogBound_nonneg lam)
+      (fun n => by
+        have hh := correlationEntropy_bound lam h hl0.le hl1 Q hQ.1 n
+        nlinarith only [hh,edgeLogBound_nonneg lam]) t
+  have heq (t : ℝ) (ht : 0 ≤ t) : C t=pairCorrelation lam h Q (offspringPosterior (poissonOffspring t ht) lam h Q hQ.1) :=
+    (hderiv t).unique (poisson_correlation_hasDerivAt lam h hl0.le hl1 Q hQ.1 t ht)
+  have hh := poisson_correlation_scalar d lam (posteriorMu Q) (posteriorNu Q) hd hl0
+    (posteriorMu_nonneg Q) (posteriorNu_nonneg Q) hc H C
+    (by dsimp only [H]; rw [poissonAverage_zero,correlationEntropy_zero])
+    (fun t _ => hderiv t) (fun t ht => by
+      rw [heq t ht.1]
+      exact poisson_increment_bound d hd.le t ht.1 ht.2 lam h hl0.le hl1 Q hQ hs he)
+  simpa only [H,poissonAverage_eq_mean _ d hd.le] using hh
+
+lemma poisson_entropy_bound (d : ℝ) (hd : 0 < d) (lam : ℝ) (h : Admissible lam)
+    (hl0 : 0 < lam) (hl1 : lam < 1) (hc : d*lam^2=1)
+    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint (poissonOffspring d hd.le) lam h Q)
+    (hs : SpinSymmetric Q) (he : 0 ≤ posteriorEta Q) :
+    (1-lam^2)/2*posteriorNu Q+3*(posteriorMu Q)^2/(2*(1+2*posteriorMu Q/3))-
+      lam*posteriorMu Q*posteriorNu Q/Real.sqrt 2 ≤
+      (8/5:ℝ)*(∫ m, entropyJ m ∂(Q : Measure Message))^2 := by
+  have hD := (poisson_moments d hd.le).hasMean
+  have hD2 := (poisson_moments d hd.le).square
+  have hid := entropy_identity (poissonOffspring d hd.le) lam h hl0.le hl1 Q hQ hs hD2
+  have hrad := integral_entropyF_radial (poissonOffspring d hd.le) lam h hl0 hl1 Q hQ hD2
+  have hcorr := poisson_correlation_bound d hd lam h hl0 hl1 hc Q hQ hs he
+  have hJ := additive_entropyJ (poissonOffspring d hd.le) lam h hl0.le hl1 Q hQ hs hD
+  have hfacmean : mean (poissonOffspring d hd.le) (fun n : ℕ => (n:ℝ)*((n:ℝ)-1)) = d^2 := (poisson_second_factorial_hasSum d hd.le).tsum_eq
+  rw [(poisson_moments d hd.le).first, hfacmean] at hid
+  rw [(poisson_moments d hd.le).first] at hJ
+  have hl : lam⁻¹^2 = d := by field_simp; nlinarith only [hc]
+  rw [hl] at hrad
+  have hJe : (∫ m, entropyJ (edgeMessage lam h m) ∂(Q : Measure Message)) =
+      lam^2*(∫ m, entropyJ m ∂(Q : Measure Message)) := by
+    nlinarith only [hJ, congrArg (fun x : ℝ => x*(∫ m, entropyJ (edgeMessage lam h m) ∂(Q : Measure Message))) hc]
+  rw [hJe] at hid
+  have hfac : d^2*lam^4=1 := by
+    calc _ = (d*lam^2)^2 := by ring
+         _ = 1 := by rw [hc]; norm_num
+  nlinarith only [hid,hrad,hcorr,
+    congrArg (fun x : ℝ => (8/5:ℝ)*x*(∫ m, entropyJ m ∂(Q : Measure Message))^2) hfac]
+
+end ThreeState
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeState/PoissonCalculus.lean b/lean/OAI/Probability/ThreeState/PoissonCalculus.lean
index c00ac103c93208a9253f7e1ef69af2f9481fbc49..52dd5adbe28d00d286db3a5cfc5549d91c617c1f 100644
--- a/lean/OAI/Probability/ThreeState/PoissonCalculus.lean
+++ b/lean/OAI/Probability/ThreeState/PoissonCalculus.lean
@@ -1,118 +1,118 @@
-import Mathlib
-
-namespace OAI
-
-/-! Exponential generating functions and differentiation of Poisson averages. -/
-
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-
-noncomputable def expGenerating (f : ℕ → ℝ) (t : ℝ) : ℝ :=
-  ∑' n : ℕ, t^n/(n.factorial:ℝ)*f n
-noncomputable def poissonAverage (f : ℕ → ℝ) (t : ℝ) : ℝ :=
-  Real.exp (-t)*expGenerating f t
-
-lemma summable_expGenerating_linear (R : ℝ) :
-    Summable (fun n : ℕ => R^n/(n.factorial:ℝ)*((n:ℝ)+2)) := by
-  have h0 := (NormedSpace.expSeries_div_hasSum_exp R).summable
-  have h1 : Summable (fun n : ℕ => R^n/(n.factorial:ℝ)*(n:ℝ)) := by
-    apply (summable_nat_add_iff 1).mp
-    convert! h0.mul_left R using 1
-    funext n
-    rw [pow_succ, Nat.factorial_succ, Nat.cast_mul]
-    have hn : ((n+1:ℕ):ℝ) ≠ 0 := by positivity
-    have hf : (n.factorial:ℝ) ≠ 0 := by positivity
-    field_simp
-  convert h1.add (h0.mul_right 2) using 1
-  funext n
-  ring
-
-lemma summable_expGenerating_deriv_bound (R C : ℝ) :
-    Summable (fun n : ℕ => (n:ℝ)*R^(n-1)/(n.factorial:ℝ)*(C*((n:ℝ)+1))) := by
-  apply (summable_nat_add_iff 1).mp
-  convert! (summable_expGenerating_linear R).mul_left C using 1
-  funext n
-  rw [Nat.add_sub_cancel, Nat.factorial_succ, Nat.cast_mul]
-  push_cast
-  have hn : (n:ℝ)+1 ≠ 0 := by positivity
-  field_simp
-  ring
-
-lemma expGenerating_hasDerivAt (f : ℕ → ℝ) (C : ℝ) (hC : 0 ≤ C)
-    (hf : ∀ n, |f n| ≤ C*((n:ℝ)+1)) (t : ℝ) :
-    HasDerivAt (expGenerating f) (expGenerating (fun n => f (n+1)) t) t := by
-  let R := |t|+1
-  have hR : 0 < R := by dsimp [R]; positivity
-  have ht : t ∈ Set.Ioo (-R) R := by dsimp [R]; constructor <;> linarith [neg_abs_le t, le_abs_self t]
-  have hzero : (0:ℝ) ∈ Set.Ioo (-R) R := by constructor <;> linarith
-  have hsum0 : Summable (fun n : ℕ => (0:ℝ)^n/(n.factorial:ℝ)*f n) := by
-    apply summable_of_ne_finset_zero (s := {0})
-    intro n hn
-    have hn0 : n ≠ 0 := by simpa using hn
-    simp [hn0]
-  have hb (n : ℕ) (y : ℝ) (hy : y ∈ Set.Ioo (-R) R) :
-      ‖(n:ℝ)*y^(n-1)/(n.factorial:ℝ)*f n‖ ≤
-        (n:ℝ)*R^(n-1)/(n.factorial:ℝ)*(C*((n:ℝ)+1)) := by
-    rw [Real.norm_eq_abs, abs_mul, abs_div, abs_mul, abs_of_nonneg (by positivity : (0:ℝ)≤n),
-      abs_pow, abs_of_nonneg (by positivity : (0:ℝ)≤n.factorial)]
-    calc
-      _ ≤ (n:ℝ)*|y|^(n-1)/(n.factorial:ℝ)*(C*((n:ℝ)+1)) :=
-        mul_le_mul_of_nonneg_left (hf n) (by positivity)
-      _ ≤ _ := by
-        apply mul_le_mul_of_nonneg_right _ (mul_nonneg hC (by positivity))
-        gcongr
-        exact le_of_lt (abs_lt.mpr hy)
-  have hh := hasDerivAt_tsum_of_isPreconnected (summable_expGenerating_deriv_bound R C)
-    isOpen_Ioo (convex_Ioo (-R) R).isPreconnected
-    (g := fun n y => y^n/(n.factorial:ℝ)*f n)
-    (g' := fun n y => (n:ℝ)*y^(n-1)/(n.factorial:ℝ)*f n)
-    (fun n y _ => ((hasDerivAt_pow n y).div_const (n.factorial:ℝ)).mul_const (f n))
-    hb hzero hsum0 ht
-  have hs : Summable (fun n : ℕ => (n:ℝ)*t^(n-1)/(n.factorial:ℝ)*f n) :=
-    Summable.of_norm_bounded (summable_expGenerating_deriv_bound R C) (fun n => hb n t ht)
-  have hid : (∑' n : ℕ, (n:ℝ)*t^(n-1)/(n.factorial:ℝ)*f n) = expGenerating (fun n => f (n+1)) t := by
-    rw [← hs.sum_add_tsum_nat_add 1]
-    simp only [Finset.sum_range_one, Nat.cast_zero, zero_mul, zero_div, zero_add]
-    apply tsum_congr
-    intro n
-    rw [Nat.add_sub_cancel, Nat.factorial_succ, Nat.cast_mul]
-    have hn : ((n+1:ℕ):ℝ) ≠ 0 := by positivity
-    have hc : (n.factorial:ℝ) ≠ 0 := by positivity
-    field_simp
-  rwa [hid] at hh
-
-lemma expGenerating_summable (f : ℕ → ℝ) (C : ℝ) (hC : 0 ≤ C)
-    (hf : ∀ n, |f n| ≤ C*((n:ℝ)+1)) (t : ℝ) :
-    Summable (fun n : ℕ => t^n/(n.factorial:ℝ)*f n) := by
-  apply Summable.of_norm_bounded ((summable_expGenerating_linear |t|).mul_left C)
-  intro n
-  rw [Real.norm_eq_abs, abs_mul, abs_div, abs_pow, abs_of_nonneg (by positivity : (0:ℝ)≤n.factorial)]
-  calc
-    _ ≤ |t|^n/(n.factorial:ℝ)*(C*((n:ℝ)+1)) := mul_le_mul_of_nonneg_left (hf n) (by positivity)
-    _ ≤ _ := by nlinarith [mul_nonneg (show 0 ≤ |t|^n/(n.factorial:ℝ) by positivity) hC]
-
-lemma poissonAverage_hasDerivAt (f : ℕ → ℝ) (C : ℝ) (hC : 0 ≤ C)
-    (hf : ∀ n, |f n| ≤ C*((n:ℝ)+1)) (t : ℝ) :
-    HasDerivAt (poissonAverage f) (poissonAverage (fun n => f (n+1)-f n) t) t := by
-  have hf' (n : ℕ) : |f (n+1)| ≤ (2*C)*((n:ℝ)+1) := by
-    have hh := hf (n+1)
-    push_cast at hh
-    nlinarith [mul_nonneg hC (show (0:ℝ)≤n by positivity)]
-  have hs := expGenerating_summable f C hC hf t
-  have hs' := expGenerating_summable (fun n => f (n+1)) (2*C) (by positivity) hf' t
-  have hdiff : expGenerating (fun n => f (n+1)-f n) t =
-      expGenerating (fun n => f (n+1)) t-expGenerating f t := by
-    unfold expGenerating
-    simp only [mul_sub, hs'.tsum_sub hs]
-  have hh := ((Real.hasDerivAt_exp (-t)).comp t (hasDerivAt_neg t)).mul (expGenerating_hasDerivAt f C hC hf t)
-  change HasDerivAt (poissonAverage f) _ t at hh
-  convert hh using 1
-  dsimp only [poissonAverage]
-  rw [hdiff]
-  simp only [Function.comp_apply]
-  ring
-
-end ThreeState
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-! Exponential generating functions and differentiation of Poisson averages. -/
+
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+
+noncomputable def expGenerating (f : ℕ → ℝ) (t : ℝ) : ℝ :=
+  ∑' n : ℕ, t^n/(n.factorial:ℝ)*f n
+noncomputable def poissonAverage (f : ℕ → ℝ) (t : ℝ) : ℝ :=
+  Real.exp (-t)*expGenerating f t
+
+lemma summable_expGenerating_linear (R : ℝ) :
+    Summable (fun n : ℕ => R^n/(n.factorial:ℝ)*((n:ℝ)+2)) := by
+  have h0 := (NormedSpace.expSeries_div_hasSum_exp R).summable
+  have h1 : Summable (fun n : ℕ => R^n/(n.factorial:ℝ)*(n:ℝ)) := by
+    apply (summable_nat_add_iff 1).mp
+    convert! h0.mul_left R using 1
+    funext n
+    rw [pow_succ, Nat.factorial_succ, Nat.cast_mul]
+    have hn : ((n+1:ℕ):ℝ) ≠ 0 := by positivity
+    have hf : (n.factorial:ℝ) ≠ 0 := by positivity
+    field_simp
+  convert h1.add (h0.mul_right 2) using 1
+  funext n
+  ring
+
+lemma summable_expGenerating_deriv_bound (R C : ℝ) :
+    Summable (fun n : ℕ => (n:ℝ)*R^(n-1)/(n.factorial:ℝ)*(C*((n:ℝ)+1))) := by
+  apply (summable_nat_add_iff 1).mp
+  convert! (summable_expGenerating_linear R).mul_left C using 1
+  funext n
+  rw [Nat.add_sub_cancel, Nat.factorial_succ, Nat.cast_mul]
+  push_cast
+  have hn : (n:ℝ)+1 ≠ 0 := by positivity
+  field_simp
+  ring
+
+lemma expGenerating_hasDerivAt (f : ℕ → ℝ) (C : ℝ) (hC : 0 ≤ C)
+    (hf : ∀ n, |f n| ≤ C*((n:ℝ)+1)) (t : ℝ) :
+    HasDerivAt (expGenerating f) (expGenerating (fun n => f (n+1)) t) t := by
+  let R := |t|+1
+  have hR : 0 < R := by dsimp [R]; positivity
+  have ht : t ∈ Set.Ioo (-R) R := by dsimp [R]; constructor <;> linarith [neg_abs_le t, le_abs_self t]
+  have hzero : (0:ℝ) ∈ Set.Ioo (-R) R := by constructor <;> linarith
+  have hsum0 : Summable (fun n : ℕ => (0:ℝ)^n/(n.factorial:ℝ)*f n) := by
+    apply summable_of_ne_finset_zero (s := {0})
+    intro n hn
+    have hn0 : n ≠ 0 := by simpa using hn
+    simp [hn0]
+  have hb (n : ℕ) (y : ℝ) (hy : y ∈ Set.Ioo (-R) R) :
+      ‖(n:ℝ)*y^(n-1)/(n.factorial:ℝ)*f n‖ ≤
+        (n:ℝ)*R^(n-1)/(n.factorial:ℝ)*(C*((n:ℝ)+1)) := by
+    rw [Real.norm_eq_abs, abs_mul, abs_div, abs_mul, abs_of_nonneg (by positivity : (0:ℝ)≤n),
+      abs_pow, abs_of_nonneg (by positivity : (0:ℝ)≤n.factorial)]
+    calc
+      _ ≤ (n:ℝ)*|y|^(n-1)/(n.factorial:ℝ)*(C*((n:ℝ)+1)) :=
+        mul_le_mul_of_nonneg_left (hf n) (by positivity)
+      _ ≤ _ := by
+        apply mul_le_mul_of_nonneg_right _ (mul_nonneg hC (by positivity))
+        gcongr
+        exact le_of_lt (abs_lt.mpr hy)
+  have hh := hasDerivAt_tsum_of_isPreconnected (summable_expGenerating_deriv_bound R C)
+    isOpen_Ioo (convex_Ioo (-R) R).isPreconnected
+    (g := fun n y => y^n/(n.factorial:ℝ)*f n)
+    (g' := fun n y => (n:ℝ)*y^(n-1)/(n.factorial:ℝ)*f n)
+    (fun n y _ => ((hasDerivAt_pow n y).div_const (n.factorial:ℝ)).mul_const (f n))
+    hb hzero hsum0 ht
+  have hs : Summable (fun n : ℕ => (n:ℝ)*t^(n-1)/(n.factorial:ℝ)*f n) :=
+    Summable.of_norm_bounded (summable_expGenerating_deriv_bound R C) (fun n => hb n t ht)
+  have hid : (∑' n : ℕ, (n:ℝ)*t^(n-1)/(n.factorial:ℝ)*f n) = expGenerating (fun n => f (n+1)) t := by
+    rw [← hs.sum_add_tsum_nat_add 1]
+    simp only [Finset.sum_range_one, Nat.cast_zero, zero_mul, zero_div, zero_add]
+    apply tsum_congr
+    intro n
+    rw [Nat.add_sub_cancel, Nat.factorial_succ, Nat.cast_mul]
+    have hn : ((n+1:ℕ):ℝ) ≠ 0 := by positivity
+    have hc : (n.factorial:ℝ) ≠ 0 := by positivity
+    field_simp
+  rwa [hid] at hh
+
+lemma expGenerating_summable (f : ℕ → ℝ) (C : ℝ) (hC : 0 ≤ C)
+    (hf : ∀ n, |f n| ≤ C*((n:ℝ)+1)) (t : ℝ) :
+    Summable (fun n : ℕ => t^n/(n.factorial:ℝ)*f n) := by
+  apply Summable.of_norm_bounded ((summable_expGenerating_linear |t|).mul_left C)
+  intro n
+  rw [Real.norm_eq_abs, abs_mul, abs_div, abs_pow, abs_of_nonneg (by positivity : (0:ℝ)≤n.factorial)]
+  calc
+    _ ≤ |t|^n/(n.factorial:ℝ)*(C*((n:ℝ)+1)) := mul_le_mul_of_nonneg_left (hf n) (by positivity)
+    _ ≤ _ := by nlinarith [mul_nonneg (show 0 ≤ |t|^n/(n.factorial:ℝ) by positivity) hC]
+
+lemma poissonAverage_hasDerivAt (f : ℕ → ℝ) (C : ℝ) (hC : 0 ≤ C)
+    (hf : ∀ n, |f n| ≤ C*((n:ℝ)+1)) (t : ℝ) :
+    HasDerivAt (poissonAverage f) (poissonAverage (fun n => f (n+1)-f n) t) t := by
+  have hf' (n : ℕ) : |f (n+1)| ≤ (2*C)*((n:ℝ)+1) := by
+    have hh := hf (n+1)
+    push_cast at hh
+    nlinarith [mul_nonneg hC (show (0:ℝ)≤n by positivity)]
+  have hs := expGenerating_summable f C hC hf t
+  have hs' := expGenerating_summable (fun n => f (n+1)) (2*C) (by positivity) hf' t
+  have hdiff : expGenerating (fun n => f (n+1)-f n) t =
+      expGenerating (fun n => f (n+1)) t-expGenerating f t := by
+    unfold expGenerating
+    simp only [mul_sub, hs'.tsum_sub hs]
+  have hh := ((Real.hasDerivAt_exp (-t)).comp t (hasDerivAt_neg t)).mul (expGenerating_hasDerivAt f C hC hf t)
+  change HasDerivAt (poissonAverage f) _ t at hh
+  convert hh using 1
+  dsimp only [poissonAverage]
+  rw [hdiff]
+  simp only [Function.comp_apply]
+  ring
+
+end ThreeState
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeState/PoissonCritical.lean b/lean/OAI/Probability/ThreeState/PoissonCritical.lean
index f22b9cfb3446c592bf1eca55724ebf635c152b3a..ec0a3a9793ebf62d693c077bbae219bbe5cb5578 100644
--- a/lean/OAI/Probability/ThreeState/PoissonCritical.lean
+++ b/lean/OAI/Probability/ThreeState/PoissonCritical.lean
@@ -1,207 +1,207 @@
-import Mathlib
-import OAI.Probability.ThreeState.PoissonBounds
-
-namespace OAI
-
-/-! Informative posterior mass and positive critical Poisson non-reconstruction. -/
-
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-
-@[simp] lemma normalizer_uninformative (lam : ℝ) (h : Admissible lam) (n : ℕ) :
-    normalizer lam h (fun _ : Fin n => uninformative) = 1 := by
-  simp [normalizer,productWeight,edgeMessage,uninformative]
-
-@[simp] lemma combineMessage_uninformative (lam : ℝ) (h : Admissible lam) (n : ℕ) :
-    combineMessage lam h (fun _ : Fin n => uninformative) = uninformative := by
-  simp only [combineMessage,normalizer_uninformative,one_ne_zero,↓reduceDIte,div_one]
-  apply Subtype.ext
-  funext i
-  simp [productWeight,edgeMessage,uninformative]
-
-lemma degreeOutput_atom (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message) (n : ℕ) :
-    (Q : Measure Message) {uninformative}^n ≤ degreeOutput lam h Q n {uninformative} := by
-  have hsub : {fun _ : Fin n => uninformative} ⊆ (combineMessage lam h) ⁻¹' {uninformative} := by
-    intro m hm
-    rcases Set.mem_singleton_iff.mp hm with rfl
-    exact combineMessage_uninformative lam h n
-  rw [degreeOutput, Measure.map_apply (measurable_combineMessage lam h n) (measurableSet_singleton _)]
-  have hh := measure_mono (μ := degreeInput lam h Q n) hsub
-  rw [degreeInput,withDensity_apply _ (measurableSet_singleton _),lintegral_singleton,
-    normalizer_uninformative,ENNReal.ofReal_one,one_mul] at hh
-  change (Measure.pi (fun _ : Fin n => (Q : Measure Message))) {fun _ => uninformative} ≤ _ at hh
-  simpa [degreeInput] using hh
-
-lemma densityEvolution_real (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
-    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (s : Set Message) (hs : MeasurableSet s) :
-    (densityEvolution offspring lam h Q).real s = mean offspring (fun n => (degreeOutput lam h Q n).real s) := by
-  rw [Measure.real,densityEvolution,Measure.sum_apply _ hs,ENNReal.tsum_toReal_eq]
-  · simp only [Measure.smul_apply,smul_eq_mul,ENNReal.toReal_mul,mean,mass,Measure.real]
-  · intro n
-    let := degreeOutput_probability lam h Q hQ n
-    simp only [Measure.smul_apply,smul_eq_mul]
-    exact ENNReal.mul_ne_top (offspring.apply_ne_top n) (measure_ne_top _ _)
-
-lemma poisson_power_mean (d a : ℝ) (hd : 0 ≤ d) :
-    mean (poissonOffspring d hd) (fun n => a^n) = Real.exp (d*(a-1)) := by
-  have he (n : ℕ) : mass (poissonOffspring d hd) n*a^n = Real.exp (-d)*((d*a)^n/(n.factorial : ℝ)) := by
-    rw [poisson_mass,mul_pow]
-    ring
-  unfold mean
-  simp_rw [he]
-  rw [tsum_mul_left]
-  have hh := NormedSpace.expSeries_div_hasSum_exp (d*a)
-  have he : (∑' n : ℕ, (d*a)^n/(n.factorial : ℝ)) = Real.exp (d*a) := by
-    simpa only [← Real.exp_eq_exp_ℝ] using hh.tsum_eq
-  rw [he,← Real.exp_add]
-  congr 1
-  ring
-
-noncomputable def informativeMass (Q : ProbabilityMeasure Message) : ℝ :=
-  1-(Q : Measure Message).real {uninformative}
-
-lemma informativeMass_nonneg (Q : ProbabilityMeasure Message) : 0 ≤ informativeMass Q := by
-  simp [informativeMass]
-
-lemma informativeMass_le_one (Q : ProbabilityMeasure Message) : informativeMass Q ≤ 1 := by
-  have hh := measureReal_nonneg (μ := (Q : Measure Message)) (s := {uninformative})
-  dsimp only [informativeMass]
-  linarith
-
-lemma poisson_informative_recursion (d : ℝ) (hd : 0 ≤ d) (lam : ℝ) (h : Admissible lam)
-    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint (poissonOffspring d hd) lam h Q) :
-    Real.exp (-d*informativeMass Q) ≤ 1-informativeMass Q := by
-  have hdegree (n : ℕ) : ((Q : Measure Message).real {uninformative})^n ≤ (degreeOutput lam h Q n).real {uninformative} := by
-    let := degreeOutput_probability lam h Q hQ.1 n
-    have ht := ENNReal.toReal_mono (measure_ne_top (degreeOutput lam h Q n) {uninformative}) (degreeOutput_atom lam h Q n)
-    simpa only [ENNReal.toReal_pow,Measure.real] using ht
-  have hf : HasMean (poissonOffspring d hd) (fun n => ((Q : Measure Message).real {uninformative})^n) := by
-    apply hasMean_bounded _ 1
-    intro n
-    rw [abs_of_nonneg (by positivity)]
-    apply pow_le_one₀ (measureReal_nonneg) _
-    simp
-  have hg : HasMean (poissonOffspring d hd) (fun n => (degreeOutput lam h Q n).real {uninformative}) := by
-    apply hasMean_bounded _ 1
-    intro n
-    let := degreeOutput_probability lam h Q hQ.1 n
-    rw [abs_of_nonneg measureReal_nonneg]
-    simp
-  have hh := mean_nonneg (poissonOffspring d hd) (fun n => sub_nonneg.mpr (hdegree n))
-  rw [mean_sub hg hf] at hh
-  replace hh := sub_nonneg.mp hh
-  rw [poisson_power_mean] at hh
-  have he := congrArg (fun M : Measure Message => M.real {uninformative}) (fixedpoint_measure (poissonOffspring d hd) lam h Q hQ)
-  rw [densityEvolution_real _ _ _ _ hQ.1 _ (measurableSet_singleton _)] at he
-  rw [← he] at hh
-  convert hh using 1 <;> dsimp only [informativeMass] <;> congr 1 <;> ring
-
-end ThreeState
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-
-lemma integral_real_cauchy {α : Type*} [MeasurableSpace α] (M : Measure α) (f g : α → ℝ)
-    (hf2 : Integrable (fun a => (f a)^2) M) (hg2 : Integrable (fun a => (g a)^2) M)
-    (hfg : Integrable (fun a => f a*g a) M) :
-    (∫ a, f a*g a ∂M)^2 ≤ (∫ a, (f a)^2 ∂M)*(∫ a, (g a)^2 ∂M) := by
-  have hp (t : ℝ) : 0 ≤
-      (∫ a, (f a)^2 ∂M)*(t*t)+(-2*(∫ a, f a*g a ∂M))*t+(∫ a, (g a)^2 ∂M) := by
-    have hnon : 0 ≤ ∫ a, (t*f a-g a)^2 ∂M := integral_nonneg (fun value => sq_nonneg (t*f value-g value))
-    have he (a : α) : (t*f a-g a)^2 = t^2*(f a)^2+(-2*t)*(f a*g a)+(g a)^2 := by ring
-    simp_rw [he] at hnon
-    have his : Integrable (fun a => t^2*(f a)^2+(-2*t)*(f a*g a)) M := (hf2.const_mul _).add (hfg.const_mul _)
-    rw [integral_add his hg2, integral_add (hf2.const_mul _) (hfg.const_mul _), integral_const_mul, integral_const_mul] at hnon
-    nlinarith only [hnon]
-  have hh := discrim_le_zero hp
-  simp only [discrim] at hh
-  nlinarith only [hh]
-
-@[simp] lemma messageX_uninformative : messageX uninformative = 0 := by
-  simp [messageX,Radial.momentX,Radial.avg_expand,messageDeviation,uninformative]
-
-lemma posteriorMu_sq_le_informative (Q : ProbabilityMeasure Message) :
-    (posteriorMu Q)^2 ≤ informativeMass Q*posteriorNu Q := by
-  let s : Set Message := {uninformative}ᶜ
-  have hs : MeasurableSet s := (measurableSet_singleton _).compl
-  let g : Message → ℝ := s.indicator (fun _ => 1)
-  have hg : Integrable g (Q : Measure Message) := (integrable_const (1:ℝ)).indicator hs
-  have hg2 : (fun m => (g m)^2) = g := by
-    funext m
-    by_cases hm : m∈s <;> simp [g,Set.indicator_of_mem,Set.indicator_of_notMem,hm]
-  have hfg : (fun m => messageX m*g m) = messageX := by
-    funext m
-    by_cases hm : m∈s
-    · simp [g,hm]
-    · have he : m=uninformative := by simpa only [s,Set.mem_compl_iff,Set.mem_singleton_iff,not_not] using hm
-      subst m
-      simp
-  have hh := integral_real_cauchy (Q : Measure Message) messageX g
-    (integrable_continuous_message Q (continuous_messageX.pow 2))
-    (by rw [hg2]; exact hg) (by rw [hfg]; exact integrable_continuous_message Q continuous_messageX)
-  rw [hg2,hfg] at hh
-  have hgint : (∫ m, g m ∂(Q : Measure Message))=informativeMass Q := by
-    rw [show g=s.indicator (fun _ => (1:ℝ)) by rfl,integral_indicator hs]
-    simp only [integral_const,smul_eq_mul,mul_one,measureReal_restrict_apply_univ]
-    change (Q : Measure Message).real {uninformative}ᶜ = 1-(Q : Measure Message).real {uninformative}
-    rw [measureReal_compl (measurableSet_singleton _)]
-    simp
-  rw [hgint] at hh
-  simpa only [posteriorMu,posteriorNu,mul_comm] using hh
-
-lemma informativeMass_pos_of_mu_pos (Q : ProbabilityMeasure Message) (hm : 0 < posteriorMu Q) :
-    0 < informativeMass Q := by
-  have hh := posteriorMu_sq_le_informative Q
-  apply lt_of_le_of_ne (informativeMass_nonneg Q)
-  intro hzero
-  rw [← hzero,zero_mul] at hh
-  nlinarith only [hh,sq_pos_of_pos hm]
-
-end ThreeState
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-
-lemma poisson_positive_critical_uniform (d : ℝ) (hd : 1 < d) (lam : ℝ) (h : Admissible lam)
-    (hl0 : 0 < lam) (hc : d*lam^2=1)
-    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint (poissonOffspring d (by linarith)) lam h Q)
-    (hs : SpinSymmetric Q) : posteriorMu Q = 0 := by
-  have hd0 : 0 < d := by linarith
-  have hl1 : lam < 1 := by
-    by_contra hn
-    have hh : 1 ≤ lam^2 := by nlinarith [sq_nonneg (lam-1)]
-    have hm := mul_le_mul_of_nonneg_left hh hd0.le
-    nlinarith only [hm,hc,hd]
-  have hD := (poisson_moments d hd0.le).hasMean
-  have hc' : mean (poissonOffspring d hd0.le) (fun n : ℕ => (n:ℝ))*lam^2=1 := by
-    simpa only [(poisson_moments d hd0.le).first] using hc
-  have hh := positive_moment_estimates (poissonOffspring d hd0.le) lam h hl0 hl1 Q hQ hs hD hc'
-  dsimp only at hh
-  have hm := posteriorMu_nonneg Q
-  have hn := posteriorNu_nonneg Q
-  have he : 0 ≤ posteriorEta Q := (show 0 ≤ ((37/27:ℝ)+2*lam*((37/27:ℝ)/(2+lam)))*posteriorNu Q by positivity).trans hh.2.2.1
-  have hent := poisson_entropy_bound d hd0 lam h hl0 hl1 hc Q hQ hs he
-  have hcnu := critical_moment_nu_bound (poissonOffspring d hd0.le) lam h hl0 hl1 Q hQ hs hD hc'
-  by_contra hzero
-  have hmpos : 0 < posteriorMu Q := lt_of_le_of_ne hm (Ne.symm hzero)
-  have hsp := informativeMass_pos_of_mu_pos Q hmpos
-  apply PositiveScalar.moment_scalar_contradiction lam (posteriorMu Q) (posteriorNu Q)
-    (∫ m, entropyJ m ∂(Q : Measure Message)) (informativeMass Q) hl0 hl1 hmpos hn hsp
-    (informativeMass_le_one Q) (posteriorMu_sq_le_informative Q) hcnu hh.1 hh.2.1 hent
-  intro _
-  have hsb := poisson_support_bound d (informativeMass Q) hd0 hsp
-    (poisson_informative_recursion d hd0.le lam h Q hQ)
-  have heq : d⁻¹=lam^2 := by field_simp; nlinarith only [hc]
-  rwa [heq] at hsb
-
- 
-theorem poisson_positive_critical_nonreconstruction (d : ℝ) (hd : 1 < d) (lam : ℝ)
-    (h : Admissible lam) (hl0 : 0 < lam) (hc : d*lam^2=1) :
-    Tendsto (poissonAdvantage d (by linarith) lam h) atTop (𝓝 0) := by
-  exact observed_tendsto_zero_of_uniform_fixedpoints (poissonOffspring d (by linarith)) lam h
-    (poisson_positive_critical_uniform d hd lam h hl0 hc)
-
-end ThreeState
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeState.PoissonBounds
+
+namespace OAI
+
+/-! Informative posterior mass and positive critical Poisson non-reconstruction. -/
+
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+
+@[simp] lemma normalizer_uninformative (lam : ℝ) (h : Admissible lam) (n : ℕ) :
+    normalizer lam h (fun _ : Fin n => uninformative) = 1 := by
+  simp [normalizer,productWeight,edgeMessage,uninformative]
+
+@[simp] lemma combineMessage_uninformative (lam : ℝ) (h : Admissible lam) (n : ℕ) :
+    combineMessage lam h (fun _ : Fin n => uninformative) = uninformative := by
+  simp only [combineMessage,normalizer_uninformative,one_ne_zero,↓reduceDIte,div_one]
+  apply Subtype.ext
+  funext i
+  simp [productWeight,edgeMessage,uninformative]
+
+lemma degreeOutput_atom (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message) (n : ℕ) :
+    (Q : Measure Message) {uninformative}^n ≤ degreeOutput lam h Q n {uninformative} := by
+  have hsub : {fun _ : Fin n => uninformative} ⊆ (combineMessage lam h) ⁻¹' {uninformative} := by
+    intro m hm
+    rcases Set.mem_singleton_iff.mp hm with rfl
+    exact combineMessage_uninformative lam h n
+  rw [degreeOutput, Measure.map_apply (measurable_combineMessage lam h n) (measurableSet_singleton _)]
+  have hh := measure_mono (μ := degreeInput lam h Q n) hsub
+  rw [degreeInput,withDensity_apply _ (measurableSet_singleton _),lintegral_singleton,
+    normalizer_uninformative,ENNReal.ofReal_one,one_mul] at hh
+  change (Measure.pi (fun _ : Fin n => (Q : Measure Message))) {fun _ => uninformative} ≤ _ at hh
+  simpa [degreeInput] using hh
+
+lemma densityEvolution_real (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
+    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (s : Set Message) (hs : MeasurableSet s) :
+    (densityEvolution offspring lam h Q).real s = mean offspring (fun n => (degreeOutput lam h Q n).real s) := by
+  rw [Measure.real,densityEvolution,Measure.sum_apply _ hs,ENNReal.tsum_toReal_eq]
+  · simp only [Measure.smul_apply,smul_eq_mul,ENNReal.toReal_mul,mean,mass,Measure.real]
+  · intro n
+    let := degreeOutput_probability lam h Q hQ n
+    simp only [Measure.smul_apply,smul_eq_mul]
+    exact ENNReal.mul_ne_top (offspring.apply_ne_top n) (measure_ne_top _ _)
+
+lemma poisson_power_mean (d a : ℝ) (hd : 0 ≤ d) :
+    mean (poissonOffspring d hd) (fun n => a^n) = Real.exp (d*(a-1)) := by
+  have he (n : ℕ) : mass (poissonOffspring d hd) n*a^n = Real.exp (-d)*((d*a)^n/(n.factorial : ℝ)) := by
+    rw [poisson_mass,mul_pow]
+    ring
+  unfold mean
+  simp_rw [he]
+  rw [tsum_mul_left]
+  have hh := NormedSpace.expSeries_div_hasSum_exp (d*a)
+  have he : (∑' n : ℕ, (d*a)^n/(n.factorial : ℝ)) = Real.exp (d*a) := by
+    simpa only [← Real.exp_eq_exp_ℝ] using hh.tsum_eq
+  rw [he,← Real.exp_add]
+  congr 1
+  ring
+
+noncomputable def informativeMass (Q : ProbabilityMeasure Message) : ℝ :=
+  1-(Q : Measure Message).real {uninformative}
+
+lemma informativeMass_nonneg (Q : ProbabilityMeasure Message) : 0 ≤ informativeMass Q := by
+  simp [informativeMass]
+
+lemma informativeMass_le_one (Q : ProbabilityMeasure Message) : informativeMass Q ≤ 1 := by
+  have hh := measureReal_nonneg (μ := (Q : Measure Message)) (s := {uninformative})
+  dsimp only [informativeMass]
+  linarith
+
+lemma poisson_informative_recursion (d : ℝ) (hd : 0 ≤ d) (lam : ℝ) (h : Admissible lam)
+    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint (poissonOffspring d hd) lam h Q) :
+    Real.exp (-d*informativeMass Q) ≤ 1-informativeMass Q := by
+  have hdegree (n : ℕ) : ((Q : Measure Message).real {uninformative})^n ≤ (degreeOutput lam h Q n).real {uninformative} := by
+    let := degreeOutput_probability lam h Q hQ.1 n
+    have ht := ENNReal.toReal_mono (measure_ne_top (degreeOutput lam h Q n) {uninformative}) (degreeOutput_atom lam h Q n)
+    simpa only [ENNReal.toReal_pow,Measure.real] using ht
+  have hf : HasMean (poissonOffspring d hd) (fun n => ((Q : Measure Message).real {uninformative})^n) := by
+    apply hasMean_bounded _ 1
+    intro n
+    rw [abs_of_nonneg (by positivity)]
+    apply pow_le_one₀ (measureReal_nonneg) _
+    simp
+  have hg : HasMean (poissonOffspring d hd) (fun n => (degreeOutput lam h Q n).real {uninformative}) := by
+    apply hasMean_bounded _ 1
+    intro n
+    let := degreeOutput_probability lam h Q hQ.1 n
+    rw [abs_of_nonneg measureReal_nonneg]
+    simp
+  have hh := mean_nonneg (poissonOffspring d hd) (fun n => sub_nonneg.mpr (hdegree n))
+  rw [mean_sub hg hf] at hh
+  replace hh := sub_nonneg.mp hh
+  rw [poisson_power_mean] at hh
+  have he := congrArg (fun M : Measure Message => M.real {uninformative}) (fixedpoint_measure (poissonOffspring d hd) lam h Q hQ)
+  rw [densityEvolution_real _ _ _ _ hQ.1 _ (measurableSet_singleton _)] at he
+  rw [← he] at hh
+  convert hh using 1 <;> dsimp only [informativeMass] <;> congr 1 <;> ring
+
+end ThreeState
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+
+lemma integral_real_cauchy {α : Type*} [MeasurableSpace α] (M : Measure α) (f g : α → ℝ)
+    (hf2 : Integrable (fun a => (f a)^2) M) (hg2 : Integrable (fun a => (g a)^2) M)
+    (hfg : Integrable (fun a => f a*g a) M) :
+    (∫ a, f a*g a ∂M)^2 ≤ (∫ a, (f a)^2 ∂M)*(∫ a, (g a)^2 ∂M) := by
+  have hp (t : ℝ) : 0 ≤
+      (∫ a, (f a)^2 ∂M)*(t*t)+(-2*(∫ a, f a*g a ∂M))*t+(∫ a, (g a)^2 ∂M) := by
+    have hnon : 0 ≤ ∫ a, (t*f a-g a)^2 ∂M := integral_nonneg (fun value => sq_nonneg (t*f value-g value))
+    have he (a : α) : (t*f a-g a)^2 = t^2*(f a)^2+(-2*t)*(f a*g a)+(g a)^2 := by ring
+    simp_rw [he] at hnon
+    have his : Integrable (fun a => t^2*(f a)^2+(-2*t)*(f a*g a)) M := (hf2.const_mul _).add (hfg.const_mul _)
+    rw [integral_add his hg2, integral_add (hf2.const_mul _) (hfg.const_mul _), integral_const_mul, integral_const_mul] at hnon
+    nlinarith only [hnon]
+  have hh := discrim_le_zero hp
+  simp only [discrim] at hh
+  nlinarith only [hh]
+
+@[simp] lemma messageX_uninformative : messageX uninformative = 0 := by
+  simp [messageX,Radial.momentX,Radial.avg_expand,messageDeviation,uninformative]
+
+lemma posteriorMu_sq_le_informative (Q : ProbabilityMeasure Message) :
+    (posteriorMu Q)^2 ≤ informativeMass Q*posteriorNu Q := by
+  let s : Set Message := {uninformative}ᶜ
+  have hs : MeasurableSet s := (measurableSet_singleton _).compl
+  let g : Message → ℝ := s.indicator (fun _ => 1)
+  have hg : Integrable g (Q : Measure Message) := (integrable_const (1:ℝ)).indicator hs
+  have hg2 : (fun m => (g m)^2) = g := by
+    funext m
+    by_cases hm : m∈s <;> simp [g,Set.indicator_of_mem,Set.indicator_of_notMem,hm]
+  have hfg : (fun m => messageX m*g m) = messageX := by
+    funext m
+    by_cases hm : m∈s
+    · simp [g,hm]
+    · have he : m=uninformative := by simpa only [s,Set.mem_compl_iff,Set.mem_singleton_iff,not_not] using hm
+      subst m
+      simp
+  have hh := integral_real_cauchy (Q : Measure Message) messageX g
+    (integrable_continuous_message Q (continuous_messageX.pow 2))
+    (by rw [hg2]; exact hg) (by rw [hfg]; exact integrable_continuous_message Q continuous_messageX)
+  rw [hg2,hfg] at hh
+  have hgint : (∫ m, g m ∂(Q : Measure Message))=informativeMass Q := by
+    rw [show g=s.indicator (fun _ => (1:ℝ)) by rfl,integral_indicator hs]
+    simp only [integral_const,smul_eq_mul,mul_one,measureReal_restrict_apply_univ]
+    change (Q : Measure Message).real {uninformative}ᶜ = 1-(Q : Measure Message).real {uninformative}
+    rw [measureReal_compl (measurableSet_singleton _)]
+    simp
+  rw [hgint] at hh
+  simpa only [posteriorMu,posteriorNu,mul_comm] using hh
+
+lemma informativeMass_pos_of_mu_pos (Q : ProbabilityMeasure Message) (hm : 0 < posteriorMu Q) :
+    0 < informativeMass Q := by
+  have hh := posteriorMu_sq_le_informative Q
+  apply lt_of_le_of_ne (informativeMass_nonneg Q)
+  intro hzero
+  rw [← hzero,zero_mul] at hh
+  nlinarith only [hh,sq_pos_of_pos hm]
+
+end ThreeState
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+
+lemma poisson_positive_critical_uniform (d : ℝ) (hd : 1 < d) (lam : ℝ) (h : Admissible lam)
+    (hl0 : 0 < lam) (hc : d*lam^2=1)
+    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint (poissonOffspring d (by linarith)) lam h Q)
+    (hs : SpinSymmetric Q) : posteriorMu Q = 0 := by
+  have hd0 : 0 < d := by linarith
+  have hl1 : lam < 1 := by
+    by_contra hn
+    have hh : 1 ≤ lam^2 := by nlinarith [sq_nonneg (lam-1)]
+    have hm := mul_le_mul_of_nonneg_left hh hd0.le
+    nlinarith only [hm,hc,hd]
+  have hD := (poisson_moments d hd0.le).hasMean
+  have hc' : mean (poissonOffspring d hd0.le) (fun n : ℕ => (n:ℝ))*lam^2=1 := by
+    simpa only [(poisson_moments d hd0.le).first] using hc
+  have hh := positive_moment_estimates (poissonOffspring d hd0.le) lam h hl0 hl1 Q hQ hs hD hc'
+  dsimp only at hh
+  have hm := posteriorMu_nonneg Q
+  have hn := posteriorNu_nonneg Q
+  have he : 0 ≤ posteriorEta Q := (show 0 ≤ ((37/27:ℝ)+2*lam*((37/27:ℝ)/(2+lam)))*posteriorNu Q by positivity).trans hh.2.2.1
+  have hent := poisson_entropy_bound d hd0 lam h hl0 hl1 hc Q hQ hs he
+  have hcnu := critical_moment_nu_bound (poissonOffspring d hd0.le) lam h hl0 hl1 Q hQ hs hD hc'
+  by_contra hzero
+  have hmpos : 0 < posteriorMu Q := lt_of_le_of_ne hm (Ne.symm hzero)
+  have hsp := informativeMass_pos_of_mu_pos Q hmpos
+  apply PositiveScalar.moment_scalar_contradiction lam (posteriorMu Q) (posteriorNu Q)
+    (∫ m, entropyJ m ∂(Q : Measure Message)) (informativeMass Q) hl0 hl1 hmpos hn hsp
+    (informativeMass_le_one Q) (posteriorMu_sq_le_informative Q) hcnu hh.1 hh.2.1 hent
+  intro _
+  have hsb := poisson_support_bound d (informativeMass Q) hd0 hsp
+    (poisson_informative_recursion d hd0.le lam h Q hQ)
+  have heq : d⁻¹=lam^2 := by field_simp; nlinarith only [hc]
+  rwa [heq] at hsb
+
+ 
+theorem poisson_positive_critical_nonreconstruction (d : ℝ) (hd : 1 < d) (lam : ℝ)
+    (h : Admissible lam) (hl0 : 0 < lam) (hc : d*lam^2=1) :
+    Tendsto (poissonAdvantage d (by linarith) lam h) atTop (𝓝 0) := by
+  exact observed_tendsto_zero_of_uniform_fixedpoints (poissonOffspring d (by linarith)) lam h
+    (poisson_positive_critical_uniform d hd lam h hl0 hc)
+
+end ThreeState
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeState/PosteriorMoments.lean b/lean/OAI/Probability/ThreeState/PosteriorMoments.lean
index 20ec6970838ddda406f04f87f38db588b60187d1..d28cdb9c4f41155a038744d37470b9bb47063bc8 100644
--- a/lean/OAI/Probability/ThreeState/PosteriorMoments.lean
+++ b/lean/OAI/Probability/ThreeState/PosteriorMoments.lean
@@ -1,216 +1,216 @@
-import Mathlib
-import OAI.Probability.ThreeState.EntropyRecursion
-import OAI.Probability.ThreeState.Resolvents
-
-namespace OAI
-
-/-! Posterior deviation moments and positive-channel estimates. -/
-
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-open Radial (avg)
-
-noncomputable def messageDeviation (m : Message) : Spin → ℝ := fun i => m i-1
-noncomputable def messageX (m : Message) : ℝ := Radial.momentX (messageDeviation m)
-noncomputable def messageY (m : Message) : ℝ := Radial.momentY (messageDeviation m)
-
-lemma avg_messageDeviation (m : Message) : avg (messageDeviation m) = 0 := by
-  have hh := average_message m
-  simp only [messageDeviation, Radial.avg_expand] at *
-  linarith
-
-lemma messageX_nonneg (m : Message) : 0 ≤ messageX m := Radial.momentX_nonneg _
-
-lemma messageX_le_one (m : Message) : messageX m ≤ 1 := by
-  have hi (i : Spin) : (m i)^2 ≤ 3*m i := by nlinarith [Message.nonneg m i, Message.le_three m i]
-  have hh := average_message m
-  simp only [messageX, messageDeviation, Radial.momentX, Radial.avg_expand] at *
-  nlinarith [hi 0,hi 1,hi 2]
-
-lemma continuous_messageX : Continuous messageX := by
-  change Continuous (fun m : Message => avg (fun i => (m i-1)^2)/2)
-  exact (continuous_avg _ (fun i => (((continuous_apply i).comp continuous_subtype_val).sub continuous_const).pow 2)).div_const 2
-
-lemma continuous_messageY : Continuous messageY := by
-  change Continuous (fun m : Message => avg (fun i => (m i-1)^3)/2)
-  exact (continuous_avg _ (fun i => (((continuous_apply i).comp continuous_subtype_val).sub continuous_const).pow 3)).div_const 2
-
-lemma integrable_continuous_message (Q : ProbabilityMeasure Message) {f : Message → ℝ} (hf : Continuous f) :
-    Integrable f (Q : Measure Message) := hf.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
-
-lemma entropyJ_functionJ (m : Message) : entropyJ m = Radial.functionJ (messageDeviation m) := by
-  simp [entropyJ, Radial.functionJ, messageDeviation, mLog]
-
-lemma messageDeviation_edge (lam : ℝ) (h : Admissible lam) (m : Message) :
-    messageDeviation (edgeMessage lam h m) = fun i => lam*messageDeviation m i := by
-  funext i
-  simp [messageDeviation, edgeMessage]
-
-lemma entropyJ_nonneg (m : Message) (hp : ∀ i, 0 < m i) : 0 ≤ entropyJ m := by
-  apply div_nonneg (avg_nonneg _) (by norm_num)
-  intro i
-  dsimp only [mLog]
-  rcases le_total (m i) 1 with hi | hi
-  · exact mul_nonneg_of_nonpos_of_nonpos (by linarith) (Real.log_nonpos (le_of_lt (hp i)) hi)
-  · exact mul_nonneg (by linarith) (Real.log_nonneg hi)
-
- 
-
-noncomputable def messageA (m : Message) : ℝ := 2*entropyJ m-2*messageX m+messageY m
-noncomputable def messageQ (lam : ℝ) (h : Admissible lam) (m : Message) : ℝ :=
-  (lam^3*entropyJ m-entropyJ (edgeMessage lam h m)+lam^2*(1-lam)*messageX m)/(lam^3*(1-lam))
-
-lemma messageA_eq_resolvent (m : Message) (hp : ∀ i, 0 < m i) :
-    messageA m = Radial.resolventA (messageDeviation m) := by
-  rw [Radial.resolventA_identity _ (by simpa [messageDeviation] using hp)]
-  simp [messageA, entropyJ_functionJ, messageX, messageY]
-
-lemma messageQ_eq_resolvent (lam : ℝ) (h : Admissible lam) (hl0 : 0 < lam) (hl1 : lam < 1)
-    (m : Message) (hp : ∀ i, 0 < m i) :
-    messageQ lam h m = Radial.resolventQ (messageDeviation m) lam := by
-  apply (div_eq_iff (ne_of_gt (mul_pos (pow_pos hl0 _) (sub_pos.mpr hl1)))).mpr
-  have hh := Radial.resolventQ_identity (messageDeviation m) (by simpa [messageDeviation] using hp) lam hl0.le hl1.le
-  simp only [entropyJ_functionJ, messageDeviation_edge, messageX]
-  linarith only [hh]
-
-lemma messageA_lower (m : Message) (hp : ∀ i, 0 < m i) :
-    (37/27:ℝ)*(messageX m)^2 ≤ messageA m := by
-  rw [messageA_eq_resolvent m hp]
-  exact Radial.resolventA_lower _ (avg_messageDeviation m) (by simpa [messageDeviation] using hp)
-
-lemma messageQ_lower (lam : ℝ) (h : Admissible lam) (hl0 : 0 < lam) (hl1 : lam < 1)
-    (m : Message) (hp : ∀ i, 0 < m i) : messageA m/(2+lam) ≤ messageQ lam h m := by
-  rw [messageA_eq_resolvent m hp, messageQ_eq_resolvent lam h hl0 hl1 m hp]
-  exact Radial.resolventQ_lower _ (avg_messageDeviation m) (by simpa [messageDeviation] using hp) lam hl0.le hl1.le
-
-lemma messageY_upper (m : Message) : messageY m ≤ messageX m*Real.sqrt (messageX m) := by
-  have hh := (Radial.momentY_sqrt_bound _ (avg_messageDeviation m)).2
-  change messageY m ≤ (Real.sqrt (messageX m))^3 at hh
-  have he : (Real.sqrt (messageX m))^3 = messageX m*Real.sqrt (messageX m) := by
-    calc
-      (Real.sqrt (messageX m))^3 = (Real.sqrt (messageX m))^2*Real.sqrt (messageX m) := by ring
-      _ = _ := by rw [Real.sq_sqrt (messageX_nonneg m)]
-  rwa [he] at hh
-
-lemma integral_continuous_cauchy (Q : ProbabilityMeasure Message) (f g : Message → ℝ)
-    (hf : Continuous f) (hg : Continuous g) :
-    (∫ m, f m*g m ∂(Q : Measure Message))^2 ≤
-      (∫ m, (f m)^2 ∂(Q : Measure Message))*(∫ m, (g m)^2 ∂(Q : Measure Message)) := by
-  have hf2 : Integrable (fun m => (f m)^2) (Q : Measure Message) := integrable_continuous_message Q (hf.pow 2)
-  have hg2 : Integrable (fun m => (g m)^2) (Q : Measure Message) := integrable_continuous_message Q (hg.pow 2)
-  have hfg : Integrable (fun m => f m*g m) (Q : Measure Message) := integrable_continuous_message Q (hf.mul hg)
-  have hp (t : ℝ) : 0 ≤
-      (∫ m, (f m)^2 ∂(Q : Measure Message))*(t*t)+
-      (-2*(∫ m, f m*g m ∂(Q : Measure Message)))*t+(∫ m, (g m)^2 ∂(Q : Measure Message)) := by
-    have hnon : 0 ≤ ∫ m, (t*f m-g m)^2 ∂(Q : Measure Message) := integral_nonneg (fun message => sq_nonneg (t*f message-g message))
-    have he (m : Message) : (t*f m-g m)^2 = t^2*(f m)^2+(-2*t)*(f m*g m)+(g m)^2 := by ring
-    simp_rw [he] at hnon
-    have his : Integrable (fun m => t^2*(f m)^2+(-2*t)*(f m*g m)) (Q : Measure Message) := (hf2.const_mul _).add (hfg.const_mul _)
-    rw [integral_add his hg2,
-      integral_add (hf2.const_mul _) (hfg.const_mul _), integral_const_mul, integral_const_mul] at hnon
-    nlinarith only [hnon]
-  have hh := discrim_le_zero hp
-  simp only [discrim] at hh
-  nlinarith only [hh]
-
-noncomputable def posteriorMu (Q : ProbabilityMeasure Message) : ℝ := ∫ m, messageX m ∂(Q : Measure Message)
-noncomputable def posteriorNu (Q : ProbabilityMeasure Message) : ℝ := ∫ m, (messageX m)^2 ∂(Q : Measure Message)
-noncomputable def posteriorEta (Q : ProbabilityMeasure Message) : ℝ := ∫ m, messageY m ∂(Q : Measure Message)
-
-lemma posteriorMu_nonneg (Q : ProbabilityMeasure Message) : 0 ≤ posteriorMu Q := integral_nonneg messageX_nonneg
-lemma posteriorNu_nonneg (Q : ProbabilityMeasure Message) : 0 ≤ posteriorNu Q := integral_nonneg (fun message => sq_nonneg (messageX message))
-
-lemma posteriorMu_sq_le (Q : ProbabilityMeasure Message) : (posteriorMu Q)^2 ≤ posteriorNu Q := by
-  have hh := integral_continuous_cauchy Q messageX (fun _ => 1) continuous_messageX continuous_const
-  simpa [posteriorMu, posteriorNu] using hh
-
-lemma posteriorEta_upper (Q : ProbabilityMeasure Message) : posteriorEta Q ≤ Real.sqrt (posteriorMu Q*posteriorNu Q) := by
-  let r : Message → ℝ := fun m => Real.sqrt (messageX m)
-  have hc : Continuous r := continuous_messageX.sqrt
-  have hh := integral_continuous_cauchy Q r messageX hc continuous_messageX
-  have hs : ∀ m, (r m)^2 = messageX m := fun m => Real.sq_sqrt (messageX_nonneg m)
-  simp_rw [hs] at hh
-  have hi : Integrable (fun m => messageX m*r m) (Q : Measure Message) := integrable_continuous_message Q (continuous_messageX.mul hc)
-  have h0 : 0 ≤ ∫ m, messageX m*r m ∂(Q : Measure Message) := integral_nonneg (fun m => mul_nonneg (messageX_nonneg m) (Real.sqrt_nonneg _))
-  have he : (∫ m, messageX m*r m ∂(Q : Measure Message)) ≤ Real.sqrt (posteriorMu Q*posteriorNu Q) := by
-    apply (Real.le_sqrt h0 (mul_nonneg (posteriorMu_nonneg Q) (posteriorNu_nonneg Q))).mpr
-    simpa only [posteriorMu, posteriorNu, mul_comm] using hh
-  exact (integral_mono (integrable_continuous_message Q continuous_messageY) hi messageY_upper).trans he
-
-lemma integrable_messageA (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint offspring lam h Q)
-    (hD : HasMean offspring (fun n : ℕ => (n:ℝ))) : Integrable messageA (Q : Measure Message) := by
-  exact (((integrable_entropyJ_fixedpoint offspring lam h hl0 hl1 Q hQ hD).const_mul 2).sub
-    ((integrable_continuous_message Q continuous_messageX).const_mul 2)).add
-      (integrable_continuous_message Q continuous_messageY)
-
-lemma integrable_messageQ (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint offspring lam h Q)
-    (hD : HasMean offspring (fun n : ℕ => (n:ℝ))) : Integrable (messageQ lam h) (Q : Measure Message) := by
-  exact ((((integrable_entropyJ_fixedpoint offspring lam h hl0 hl1 Q hQ hD).const_mul _).sub
-    (integrable_continuous_message Q (continuous_entropyJ_edge lam h hl0 hl1))).add
-      ((integrable_continuous_message Q continuous_messageX).const_mul _)).div_const _
-
-lemma integral_messageA (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint offspring lam h Q)
-    (hD : HasMean offspring (fun n : ℕ => (n:ℝ))) :
-    (∫ m, messageA m ∂(Q : Measure Message)) = 2*(∫ m, entropyJ m ∂(Q : Measure Message))-2*posteriorMu Q+posteriorEta Q := by
-  have hj := integrable_entropyJ_fixedpoint offspring lam h hl0 hl1 Q hQ hD
-  have hx := integrable_continuous_message Q continuous_messageX
-  have hy := integrable_continuous_message Q continuous_messageY
-  change (∫ m, 2*entropyJ m-2*messageX m+messageY m ∂(Q : Measure Message)) = _
-  have his : Integrable (fun m => 2*entropyJ m-2*messageX m) (Q : Measure Message) := (hj.const_mul 2).sub (hx.const_mul 2)
-  rw [integral_add his hy, integral_sub (hj.const_mul 2) (hx.const_mul 2), integral_const_mul, integral_const_mul]
-  rfl
-
-lemma integral_messageQ (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) (hl0 : 0 < lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint offspring lam h Q) (hs : SpinSymmetric Q)
-    (hD : HasMean offspring (fun n : ℕ => (n:ℝ))) (hc : mean offspring (fun n : ℕ => (n:ℝ))*lam^2 = 1) :
-    (∫ m, entropyJ m ∂(Q : Measure Message)) = posteriorMu Q-lam*(∫ m, messageQ lam h m ∂(Q : Measure Message)) := by
-  have hj := integrable_entropyJ_fixedpoint offspring lam h hl0.le hl1 Q hQ hD
-  have hje := integrable_continuous_message Q (continuous_entropyJ_edge lam h hl0.le hl1)
-  have hx := integrable_continuous_message Q continuous_messageX
-  have hadd := additive_entropyJ offspring lam h hl0.le hl1 Q hQ hs hD
-  have he : (∫ m, entropyJ (edgeMessage lam h m) ∂(Q : Measure Message)) =
-      lam^2*(∫ m, entropyJ m ∂(Q : Measure Message)) := by nlinarith [congrArg (fun x : ℝ => x*(∫ m, entropyJ (edgeMessage lam h m) ∂(Q : Measure Message))) hc]
-  simp only [messageQ]
-  have his : Integrable (fun m => lam^3*entropyJ m-entropyJ (edgeMessage lam h m)) (Q : Measure Message) := (hj.const_mul _).sub hje
-  rw [integral_div, integral_add his (hx.const_mul _),
-    integral_sub (hj.const_mul _) hje, integral_const_mul, integral_const_mul, he]
-  dsimp only [posteriorMu]
-  field_simp [ne_of_gt hl0, ne_of_gt (sub_pos.mpr hl1)]
-  ring
-
-lemma positive_moment_estimates (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) (hl0 : 0 < lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint offspring lam h Q) (hs : SpinSymmetric Q)
-    (hD : HasMean offspring (fun n : ℕ => (n:ℝ))) (hc : mean offspring (fun n : ℕ => (n:ℝ))*lam^2 = 1) :
-    let q : ℝ := (37/27:ℝ)/(2+lam)
-    let c : ℝ := (37/27:ℝ)+2*lam*q
-    0 ≤ (∫ m, entropyJ m ∂(Q : Measure Message)) ∧
-    (∫ m, entropyJ m ∂(Q : Measure Message)) ≤ posteriorMu Q-lam*q*posteriorNu Q ∧
-    c*posteriorNu Q ≤ posteriorEta Q ∧ posteriorEta Q ≤ Real.sqrt (posteriorMu Q*posteriorNu Q) := by
-  dsimp only
-  have hp := positive_fixedpoint offspring lam h hl0.le hl1 Q hQ
-  have hiA := integrable_messageA offspring lam h hl0.le hl1 Q hQ hD
-  have hiQ := integrable_messageQ offspring lam h hl0.le hl1 Q hQ hD
-  have hA : (37/27:ℝ)*posteriorNu Q ≤ ∫ m, messageA m ∂(Q : Measure Message) := by
-    have hh := integral_mono_ae ((integrable_continuous_message Q (continuous_messageX.pow 2)).const_mul (37/27:ℝ)) hiA
-      (hp.mono (fun m hm => messageA_lower m hm))
-    simpa only [integral_const_mul, posteriorNu, Pi.pow_apply] using hh
-  have hQ' : (∫ m, messageA m ∂(Q : Measure Message))/(2+lam) ≤ ∫ m, messageQ lam h m ∂(Q : Measure Message) := by
-    have hh := integral_mono_ae (hiA.div_const (2+lam)) hiQ
-      (hp.mono (fun m hm => messageQ_lower lam h hl0 hl1 m hm))
-    simpa only [integral_div] using hh
-  have hQl : (37/27:ℝ)/(2+lam)*posteriorNu Q ≤ ∫ m, messageQ lam h m ∂(Q : Measure Message) := by
-    have hh := (div_le_div_of_nonneg_right hA (by linarith : 0 ≤ 2+lam)).trans hQ'
-    simpa only [div_mul_eq_mul_div] using hh
-  have hJ := integral_messageQ offspring lam h hl0 hl1 Q hQ hs hD hc
-  have hAi := integral_messageA offspring lam h hl0.le hl1 Q hQ hD
-  refine ⟨integral_nonneg_of_ae (hp.mono (fun m hm => entropyJ_nonneg m hm)), ?_, ?_, posteriorEta_upper Q⟩
-  · nlinarith only [hQl, hJ, hl0]
-  · nlinarith only [hQl, hJ, hAi, hA, hl0]
-
-end ThreeState
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeState.EntropyRecursion
+import OAI.Probability.ThreeState.Resolvents
+
+namespace OAI
+
+/-! Posterior deviation moments and positive-channel estimates. -/
+
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+open Radial (avg)
+
+noncomputable def messageDeviation (m : Message) : Spin → ℝ := fun i => m i-1
+noncomputable def messageX (m : Message) : ℝ := Radial.momentX (messageDeviation m)
+noncomputable def messageY (m : Message) : ℝ := Radial.momentY (messageDeviation m)
+
+lemma avg_messageDeviation (m : Message) : avg (messageDeviation m) = 0 := by
+  have hh := average_message m
+  simp only [messageDeviation, Radial.avg_expand] at *
+  linarith
+
+lemma messageX_nonneg (m : Message) : 0 ≤ messageX m := Radial.momentX_nonneg _
+
+lemma messageX_le_one (m : Message) : messageX m ≤ 1 := by
+  have hi (i : Spin) : (m i)^2 ≤ 3*m i := by nlinarith [Message.nonneg m i, Message.le_three m i]
+  have hh := average_message m
+  simp only [messageX, messageDeviation, Radial.momentX, Radial.avg_expand] at *
+  nlinarith [hi 0,hi 1,hi 2]
+
+lemma continuous_messageX : Continuous messageX := by
+  change Continuous (fun m : Message => avg (fun i => (m i-1)^2)/2)
+  exact (continuous_avg _ (fun i => (((continuous_apply i).comp continuous_subtype_val).sub continuous_const).pow 2)).div_const 2
+
+lemma continuous_messageY : Continuous messageY := by
+  change Continuous (fun m : Message => avg (fun i => (m i-1)^3)/2)
+  exact (continuous_avg _ (fun i => (((continuous_apply i).comp continuous_subtype_val).sub continuous_const).pow 3)).div_const 2
+
+lemma integrable_continuous_message (Q : ProbabilityMeasure Message) {f : Message → ℝ} (hf : Continuous f) :
+    Integrable f (Q : Measure Message) := hf.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
+
+lemma entropyJ_functionJ (m : Message) : entropyJ m = Radial.functionJ (messageDeviation m) := by
+  simp [entropyJ, Radial.functionJ, messageDeviation, mLog]
+
+lemma messageDeviation_edge (lam : ℝ) (h : Admissible lam) (m : Message) :
+    messageDeviation (edgeMessage lam h m) = fun i => lam*messageDeviation m i := by
+  funext i
+  simp [messageDeviation, edgeMessage]
+
+lemma entropyJ_nonneg (m : Message) (hp : ∀ i, 0 < m i) : 0 ≤ entropyJ m := by
+  apply div_nonneg (avg_nonneg _) (by norm_num)
+  intro i
+  dsimp only [mLog]
+  rcases le_total (m i) 1 with hi | hi
+  · exact mul_nonneg_of_nonpos_of_nonpos (by linarith) (Real.log_nonpos (le_of_lt (hp i)) hi)
+  · exact mul_nonneg (by linarith) (Real.log_nonneg hi)
+
+ 
+
+noncomputable def messageA (m : Message) : ℝ := 2*entropyJ m-2*messageX m+messageY m
+noncomputable def messageQ (lam : ℝ) (h : Admissible lam) (m : Message) : ℝ :=
+  (lam^3*entropyJ m-entropyJ (edgeMessage lam h m)+lam^2*(1-lam)*messageX m)/(lam^3*(1-lam))
+
+lemma messageA_eq_resolvent (m : Message) (hp : ∀ i, 0 < m i) :
+    messageA m = Radial.resolventA (messageDeviation m) := by
+  rw [Radial.resolventA_identity _ (by simpa [messageDeviation] using hp)]
+  simp [messageA, entropyJ_functionJ, messageX, messageY]
+
+lemma messageQ_eq_resolvent (lam : ℝ) (h : Admissible lam) (hl0 : 0 < lam) (hl1 : lam < 1)
+    (m : Message) (hp : ∀ i, 0 < m i) :
+    messageQ lam h m = Radial.resolventQ (messageDeviation m) lam := by
+  apply (div_eq_iff (ne_of_gt (mul_pos (pow_pos hl0 _) (sub_pos.mpr hl1)))).mpr
+  have hh := Radial.resolventQ_identity (messageDeviation m) (by simpa [messageDeviation] using hp) lam hl0.le hl1.le
+  simp only [entropyJ_functionJ, messageDeviation_edge, messageX]
+  linarith only [hh]
+
+lemma messageA_lower (m : Message) (hp : ∀ i, 0 < m i) :
+    (37/27:ℝ)*(messageX m)^2 ≤ messageA m := by
+  rw [messageA_eq_resolvent m hp]
+  exact Radial.resolventA_lower _ (avg_messageDeviation m) (by simpa [messageDeviation] using hp)
+
+lemma messageQ_lower (lam : ℝ) (h : Admissible lam) (hl0 : 0 < lam) (hl1 : lam < 1)
+    (m : Message) (hp : ∀ i, 0 < m i) : messageA m/(2+lam) ≤ messageQ lam h m := by
+  rw [messageA_eq_resolvent m hp, messageQ_eq_resolvent lam h hl0 hl1 m hp]
+  exact Radial.resolventQ_lower _ (avg_messageDeviation m) (by simpa [messageDeviation] using hp) lam hl0.le hl1.le
+
+lemma messageY_upper (m : Message) : messageY m ≤ messageX m*Real.sqrt (messageX m) := by
+  have hh := (Radial.momentY_sqrt_bound _ (avg_messageDeviation m)).2
+  change messageY m ≤ (Real.sqrt (messageX m))^3 at hh
+  have he : (Real.sqrt (messageX m))^3 = messageX m*Real.sqrt (messageX m) := by
+    calc
+      (Real.sqrt (messageX m))^3 = (Real.sqrt (messageX m))^2*Real.sqrt (messageX m) := by ring
+      _ = _ := by rw [Real.sq_sqrt (messageX_nonneg m)]
+  rwa [he] at hh
+
+lemma integral_continuous_cauchy (Q : ProbabilityMeasure Message) (f g : Message → ℝ)
+    (hf : Continuous f) (hg : Continuous g) :
+    (∫ m, f m*g m ∂(Q : Measure Message))^2 ≤
+      (∫ m, (f m)^2 ∂(Q : Measure Message))*(∫ m, (g m)^2 ∂(Q : Measure Message)) := by
+  have hf2 : Integrable (fun m => (f m)^2) (Q : Measure Message) := integrable_continuous_message Q (hf.pow 2)
+  have hg2 : Integrable (fun m => (g m)^2) (Q : Measure Message) := integrable_continuous_message Q (hg.pow 2)
+  have hfg : Integrable (fun m => f m*g m) (Q : Measure Message) := integrable_continuous_message Q (hf.mul hg)
+  have hp (t : ℝ) : 0 ≤
+      (∫ m, (f m)^2 ∂(Q : Measure Message))*(t*t)+
+      (-2*(∫ m, f m*g m ∂(Q : Measure Message)))*t+(∫ m, (g m)^2 ∂(Q : Measure Message)) := by
+    have hnon : 0 ≤ ∫ m, (t*f m-g m)^2 ∂(Q : Measure Message) := integral_nonneg (fun message => sq_nonneg (t*f message-g message))
+    have he (m : Message) : (t*f m-g m)^2 = t^2*(f m)^2+(-2*t)*(f m*g m)+(g m)^2 := by ring
+    simp_rw [he] at hnon
+    have his : Integrable (fun m => t^2*(f m)^2+(-2*t)*(f m*g m)) (Q : Measure Message) := (hf2.const_mul _).add (hfg.const_mul _)
+    rw [integral_add his hg2,
+      integral_add (hf2.const_mul _) (hfg.const_mul _), integral_const_mul, integral_const_mul] at hnon
+    nlinarith only [hnon]
+  have hh := discrim_le_zero hp
+  simp only [discrim] at hh
+  nlinarith only [hh]
+
+noncomputable def posteriorMu (Q : ProbabilityMeasure Message) : ℝ := ∫ m, messageX m ∂(Q : Measure Message)
+noncomputable def posteriorNu (Q : ProbabilityMeasure Message) : ℝ := ∫ m, (messageX m)^2 ∂(Q : Measure Message)
+noncomputable def posteriorEta (Q : ProbabilityMeasure Message) : ℝ := ∫ m, messageY m ∂(Q : Measure Message)
+
+lemma posteriorMu_nonneg (Q : ProbabilityMeasure Message) : 0 ≤ posteriorMu Q := integral_nonneg messageX_nonneg
+lemma posteriorNu_nonneg (Q : ProbabilityMeasure Message) : 0 ≤ posteriorNu Q := integral_nonneg (fun message => sq_nonneg (messageX message))
+
+lemma posteriorMu_sq_le (Q : ProbabilityMeasure Message) : (posteriorMu Q)^2 ≤ posteriorNu Q := by
+  have hh := integral_continuous_cauchy Q messageX (fun _ => 1) continuous_messageX continuous_const
+  simpa [posteriorMu, posteriorNu] using hh
+
+lemma posteriorEta_upper (Q : ProbabilityMeasure Message) : posteriorEta Q ≤ Real.sqrt (posteriorMu Q*posteriorNu Q) := by
+  let r : Message → ℝ := fun m => Real.sqrt (messageX m)
+  have hc : Continuous r := continuous_messageX.sqrt
+  have hh := integral_continuous_cauchy Q r messageX hc continuous_messageX
+  have hs : ∀ m, (r m)^2 = messageX m := fun m => Real.sq_sqrt (messageX_nonneg m)
+  simp_rw [hs] at hh
+  have hi : Integrable (fun m => messageX m*r m) (Q : Measure Message) := integrable_continuous_message Q (continuous_messageX.mul hc)
+  have h0 : 0 ≤ ∫ m, messageX m*r m ∂(Q : Measure Message) := integral_nonneg (fun m => mul_nonneg (messageX_nonneg m) (Real.sqrt_nonneg _))
+  have he : (∫ m, messageX m*r m ∂(Q : Measure Message)) ≤ Real.sqrt (posteriorMu Q*posteriorNu Q) := by
+    apply (Real.le_sqrt h0 (mul_nonneg (posteriorMu_nonneg Q) (posteriorNu_nonneg Q))).mpr
+    simpa only [posteriorMu, posteriorNu, mul_comm] using hh
+  exact (integral_mono (integrable_continuous_message Q continuous_messageY) hi messageY_upper).trans he
+
+lemma integrable_messageA (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint offspring lam h Q)
+    (hD : HasMean offspring (fun n : ℕ => (n:ℝ))) : Integrable messageA (Q : Measure Message) := by
+  exact (((integrable_entropyJ_fixedpoint offspring lam h hl0 hl1 Q hQ hD).const_mul 2).sub
+    ((integrable_continuous_message Q continuous_messageX).const_mul 2)).add
+      (integrable_continuous_message Q continuous_messageY)
+
+lemma integrable_messageQ (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint offspring lam h Q)
+    (hD : HasMean offspring (fun n : ℕ => (n:ℝ))) : Integrable (messageQ lam h) (Q : Measure Message) := by
+  exact ((((integrable_entropyJ_fixedpoint offspring lam h hl0 hl1 Q hQ hD).const_mul _).sub
+    (integrable_continuous_message Q (continuous_entropyJ_edge lam h hl0 hl1))).add
+      ((integrable_continuous_message Q continuous_messageX).const_mul _)).div_const _
+
+lemma integral_messageA (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint offspring lam h Q)
+    (hD : HasMean offspring (fun n : ℕ => (n:ℝ))) :
+    (∫ m, messageA m ∂(Q : Measure Message)) = 2*(∫ m, entropyJ m ∂(Q : Measure Message))-2*posteriorMu Q+posteriorEta Q := by
+  have hj := integrable_entropyJ_fixedpoint offspring lam h hl0 hl1 Q hQ hD
+  have hx := integrable_continuous_message Q continuous_messageX
+  have hy := integrable_continuous_message Q continuous_messageY
+  change (∫ m, 2*entropyJ m-2*messageX m+messageY m ∂(Q : Measure Message)) = _
+  have his : Integrable (fun m => 2*entropyJ m-2*messageX m) (Q : Measure Message) := (hj.const_mul 2).sub (hx.const_mul 2)
+  rw [integral_add his hy, integral_sub (hj.const_mul 2) (hx.const_mul 2), integral_const_mul, integral_const_mul]
+  rfl
+
+lemma integral_messageQ (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) (hl0 : 0 < lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint offspring lam h Q) (hs : SpinSymmetric Q)
+    (hD : HasMean offspring (fun n : ℕ => (n:ℝ))) (hc : mean offspring (fun n : ℕ => (n:ℝ))*lam^2 = 1) :
+    (∫ m, entropyJ m ∂(Q : Measure Message)) = posteriorMu Q-lam*(∫ m, messageQ lam h m ∂(Q : Measure Message)) := by
+  have hj := integrable_entropyJ_fixedpoint offspring lam h hl0.le hl1 Q hQ hD
+  have hje := integrable_continuous_message Q (continuous_entropyJ_edge lam h hl0.le hl1)
+  have hx := integrable_continuous_message Q continuous_messageX
+  have hadd := additive_entropyJ offspring lam h hl0.le hl1 Q hQ hs hD
+  have he : (∫ m, entropyJ (edgeMessage lam h m) ∂(Q : Measure Message)) =
+      lam^2*(∫ m, entropyJ m ∂(Q : Measure Message)) := by nlinarith [congrArg (fun x : ℝ => x*(∫ m, entropyJ (edgeMessage lam h m) ∂(Q : Measure Message))) hc]
+  simp only [messageQ]
+  have his : Integrable (fun m => lam^3*entropyJ m-entropyJ (edgeMessage lam h m)) (Q : Measure Message) := (hj.const_mul _).sub hje
+  rw [integral_div, integral_add his (hx.const_mul _),
+    integral_sub (hj.const_mul _) hje, integral_const_mul, integral_const_mul, he]
+  dsimp only [posteriorMu]
+  field_simp [ne_of_gt hl0, ne_of_gt (sub_pos.mpr hl1)]
+  ring
+
+lemma positive_moment_estimates (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) (hl0 : 0 < lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint offspring lam h Q) (hs : SpinSymmetric Q)
+    (hD : HasMean offspring (fun n : ℕ => (n:ℝ))) (hc : mean offspring (fun n : ℕ => (n:ℝ))*lam^2 = 1) :
+    let q : ℝ := (37/27:ℝ)/(2+lam)
+    let c : ℝ := (37/27:ℝ)+2*lam*q
+    0 ≤ (∫ m, entropyJ m ∂(Q : Measure Message)) ∧
+    (∫ m, entropyJ m ∂(Q : Measure Message)) ≤ posteriorMu Q-lam*q*posteriorNu Q ∧
+    c*posteriorNu Q ≤ posteriorEta Q ∧ posteriorEta Q ≤ Real.sqrt (posteriorMu Q*posteriorNu Q) := by
+  dsimp only
+  have hp := positive_fixedpoint offspring lam h hl0.le hl1 Q hQ
+  have hiA := integrable_messageA offspring lam h hl0.le hl1 Q hQ hD
+  have hiQ := integrable_messageQ offspring lam h hl0.le hl1 Q hQ hD
+  have hA : (37/27:ℝ)*posteriorNu Q ≤ ∫ m, messageA m ∂(Q : Measure Message) := by
+    have hh := integral_mono_ae ((integrable_continuous_message Q (continuous_messageX.pow 2)).const_mul (37/27:ℝ)) hiA
+      (hp.mono (fun m hm => messageA_lower m hm))
+    simpa only [integral_const_mul, posteriorNu, Pi.pow_apply] using hh
+  have hQ' : (∫ m, messageA m ∂(Q : Measure Message))/(2+lam) ≤ ∫ m, messageQ lam h m ∂(Q : Measure Message) := by
+    have hh := integral_mono_ae (hiA.div_const (2+lam)) hiQ
+      (hp.mono (fun m hm => messageQ_lower lam h hl0 hl1 m hm))
+    simpa only [integral_div] using hh
+  have hQl : (37/27:ℝ)/(2+lam)*posteriorNu Q ≤ ∫ m, messageQ lam h m ∂(Q : Measure Message) := by
+    have hh := (div_le_div_of_nonneg_right hA (by linarith : 0 ≤ 2+lam)).trans hQ'
+    simpa only [div_mul_eq_mul_div] using hh
+  have hJ := integral_messageQ offspring lam h hl0 hl1 Q hQ hs hD hc
+  have hAi := integral_messageA offspring lam h hl0.le hl1 Q hQ hD
+  refine ⟨integral_nonneg_of_ae (hp.mono (fun m hm => entropyJ_nonneg m hm)), ?_, ?_, posteriorEta_upper Q⟩
+  · nlinarith only [hQl, hJ, hl0]
+  · nlinarith only [hQl, hJ, hAi, hA, hl0]
+
+end ThreeState
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeState/Projection.lean b/lean/OAI/Probability/ThreeState/Projection.lean
index 77e17207688eca63f16a93d3b438472d58299e9c..af9089850c06cf0b18027d29e100d11ac8438992 100644
--- a/lean/OAI/Probability/ThreeState/Projection.lean
+++ b/lean/OAI/Probability/ThreeState/Projection.lean
@@ -1,370 +1,370 @@
-import Mathlib
-import OAI.Probability.ThreeState.DegreePosterior
-
-namespace OAI
-
-/-! Quadratic projections, covariance and truncated correlation estimates. -/
-
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-open Radial (avg)
-
-lemma symAvg_deviation_mul (a : Message) (i j : Spin) :
-    symAvg (fun m => messageDeviation m i*messageDeviation m j) a =
-      (if i=j then 2 else -1)*messageX a := by
-  fin_cases i <;> fin_cases j <;>
-    norm_num [symAvg, symPerm, Fin.sum_univ_succ, messageDeviation,
-      Radial.avg_expand, permMessage, Equiv.swap_apply_def, Equiv.trans_apply,
-      Equiv.symm_trans_apply, Fin.ext_iff, messageX, Radial.momentX] <;>
-    (try rw [message_last a]) <;> ring
-
-lemma continuous_messageDeviation (i : Spin) : Continuous (fun m => messageDeviation m i) :=
-  ((continuous_apply i).comp continuous_subtype_val).sub continuous_const
-
-lemma integral_deviation (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (i : Spin) :
-    (∫ m, messageDeviation m i ∂(Q : Measure Message)) = 0 := by
-  change (∫ m : Message, m i-1 ∂(Q : Measure Message)) = 0
-  have hi : Integrable (fun m : Message => m i) (Q : Measure Message) :=
-    integrable_continuous_message Q (coordinateMap i).continuous
-  rw [integral_sub hi (integrable_const (1:ℝ)), hQ i]
-  simp
-
-lemma integral_deviation_mul (Q : ProbabilityMeasure Message) (hs : SpinSymmetric Q) (i j : Spin) :
-    (∫ m, messageDeviation m i*messageDeviation m j ∂(Q : Measure Message)) =
-      (if i=j then 2 else -1)*posteriorMu Q := by
-  rw [← integral_symAvg Q hs (fun m => messageDeviation m i*messageDeviation m j) ((continuous_messageDeviation i).mul (continuous_messageDeviation j))]
-  simp only [symAvg_deviation_mul, integral_const_mul, posteriorMu]
-
-noncomputable def edgeDeviationMap (lam : ℝ) (h : Admissible lam) (i : Spin) : C(Message,ℝ) :=
-  ⟨fun m => messageDeviation (edgeMessage lam h m) i,
-    (continuous_messageDeviation i).comp (continuous_edgeMessage lam h)⟩
-
-lemma edge_deviation (lam : ℝ) (h : Admissible lam) (m : Message) (i : Spin) :
-    messageDeviation (edgeMessage lam h m) i = lam*messageDeviation m i := by
-  change 1+lam*(m i-1)-1 = lam*(m i-1)
-  ring
-
-lemma integral_edge_covariance (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message)
-    (hQ : Balanced Q) (hs : SpinSymmetric Q) (i j : Spin) :
-    (∫ m, edgeMessage lam h m i*messageDeviation (edgeMessage lam h m) j ∂(Q : Measure Message)) =
-      lam^2*posteriorMu Q*(if i=j then 2 else -1) := by
-  have he (m : Message) : edgeMessage lam h m i*messageDeviation (edgeMessage lam h m) j =
-      lam*messageDeviation m j + lam^2*(messageDeviation m i*messageDeviation m j) := by
-    rw [edge_deviation]
-    change (1+lam*(m i-1))*(lam*(m j-1)) = _
-    simp only [messageDeviation]
-    ring
-  simp_rw [he]
-  have hi : Integrable (fun m => lam*messageDeviation m j) (Q : Measure Message) :=
-    (integrable_continuous_message Q (continuous_messageDeviation j)).const_mul _
-  have hj : Integrable (fun m => lam^2*(messageDeviation m i*messageDeviation m j)) (Q : Measure Message) :=
-    (integrable_continuous_message Q ((continuous_messageDeviation i).mul (continuous_messageDeviation j))).const_mul _
-  rw [integral_add hi hj,
-    integral_const_mul, integral_const_mul, integral_deviation Q hQ,
-    integral_deviation_mul Q hs]
-  ring
-
-lemma integral_edge_deviation_sq (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message)
-    (hs : SpinSymmetric Q) (i : Spin) :
-    (∫ m, (messageDeviation (edgeMessage lam h m) i)^2 ∂(Q : Measure Message)) = 2*lam^2*posteriorMu Q := by
-  have he (m : Message) : (messageDeviation (edgeMessage lam h m) i)^2 =
-      lam^2*(messageDeviation m i*messageDeviation m i) := by rw [edge_deviation]; ring
-  simp_rw [he]
-  rw [integral_const_mul, integral_deviation_mul Q hs]
-  simp only [ite_true]
-  ring
-
-lemma avg_swap (f : Spin → Spin → ℝ) : avg (fun i => avg (f i)) = avg (fun j => avg (fun i => f i j)) := by
-  simp only [Radial.avg_expand]
-  ring
-lemma avg_add (a b : Spin → ℝ) : avg (fun i => a i+b i) = avg a+avg b := by
-  simp only [Radial.avg_expand]
-  ring
-lemma avg_mul_const (a : Spin → ℝ) (c : ℝ) : avg (fun i => a i*c) = avg a*c := by
-  simp only [Radial.avg_expand]
-  ring
-lemma avg_const_mul (c : ℝ) (a : Spin → ℝ) : avg (fun i => c*a i) = c*avg a := by
-  simp only [Radial.avg_expand]
-  ring
-lemma avg_covariance_coeff (j : Spin) : avg (fun i : Spin => (if i=j then (2:ℝ) else -1)^2) = 2 := by
-  have he (i : Spin) : (if i=j then (2:ℝ) else -1)^2 = 1+(if i=j then (3:ℝ) else 0) := by
-    split_ifs <;> norm_num
-  simp only [he, avg, Finset.sum_add_distrib, Finset.sum_const, Finset.card_univ,
-    Fintype.card_fin, nsmul_eq_mul, Finset.sum_ite_eq', Finset.mem_univ, ite_true]
-  norm_num
-
-end ThreeState
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-open Radial (avg)
-
-noncomputable def branchSum (lam : ℝ) (h : Admissible lam) {n : ℕ} (m : Fin n → Message) (i : Spin) : ℝ :=
-  ∑ j, messageDeviation (edgeMessage lam h (m j)) i
-lemma continuous_branchSum (lam : ℝ) (h : Admissible lam) (n : ℕ) (i : Spin) :
-    Continuous (fun m : Fin n → Message => branchSum lam h m i) := by
-  apply continuous_finsetSum
-  intro j _
-  exact ((continuous_messageDeviation i).comp (continuous_edgeMessage lam h)).comp (continuous_apply j)
-lemma avg_branchSum (lam : ℝ) (h : Admissible lam) {n : ℕ} (m : Fin n → Message) :
-    avg (branchSum lam h m) = 0 := by
-  have he : avg (branchSum lam h m) = ∑ j, avg (messageDeviation (edgeMessage lam h (m j))) := by
-    simp only [branchSum, Radial.avg_expand]
-    rw [← Finset.sum_div, Finset.sum_add_distrib, Finset.sum_add_distrib]
-  rw [he]
-  simp only [avg_messageDeviation, Finset.sum_const_zero]
-
-lemma weighted_deviation_branchSum (lam : ℝ) (h : Admissible lam) {n : ℕ} (m : Fin n → Message) :
-    normalizer lam h m * avg (fun i => messageDeviation (combineMessage lam h m) i*branchSum lam h m i) =
-      avg (fun i => productWeight lam h m i*branchSum lam h m i) := by
-  have hh := weighted_avg_combine lam h m (branchSum lam h m)
-  have hz := avg_branchSum lam h m
-  simp only [messageDeviation, Radial.avg_expand] at *
-  nlinarith only [hh, mul_eq_zero_of_right (normalizer lam h m) hz]
-
-lemma integral_projection_cross (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message)
-    (hQ : Balanced Q) (hs : SpinSymmetric Q) (n : ℕ) :
-    (∫ m : Fin n → Message, normalizer lam h m *
-      (avg (fun i => messageDeviation (combineMessage lam h m) i*branchSum lam h m i)/2)
-        ∂(iidMeasure Q n : Measure (Fin n → Message))) = n*(lam^2*posteriorMu Q) := by
-  have he (m : Fin n → Message) : normalizer lam h m *
-      (avg (fun i => messageDeviation (combineMessage lam h m) i*branchSum lam h m i)/2) =
-      avg (fun i => productWeight lam h m i*branchSum lam h m i)/2 := by
-    rw [← mul_div_assoc, weighted_deviation_branchSum]
-  simp_rw [he]
-  rw [integral_div, integral_avg]
-  · have hi (i : Spin) :
-        (∫ m : Fin n → Message, productWeight lam h m i*branchSum lam h m i
-          ∂(iidMeasure Q n : Measure (Fin n → Message))) = (n:ℝ)*(lam^2*posteriorMu Q*2) := by
-      have hh := iid_integral_weighted_sum Q n (edgeCoordinateMap lam h i) (edgeDeviationMap lam h i)
-        (integral_edgeMessage lam h Q hQ i)
-      change _ = n*(∫ m, edgeMessage lam h m i*messageDeviation (edgeMessage lam h m) i ∂(Q : Measure Message)) at hh
-      rw [integral_edge_covariance lam h Q hQ hs] at hh
-      simpa only [edgeCoordinateMap, edgeDeviationMap, ContinuousMap.coe_mk, productWeight, branchSum, ite_true] using hh
-    simp_rw [hi]
-    simp [Radial.avg_expand]
-    ring
-  · intro i
-    exact ((continuous_productWeight lam h n i).mul (continuous_branchSum lam h n i)).integrable_of_hasCompactSupport
-      (HasCompactSupport.of_compactSpace _)
-
-lemma avg_integral_edge_sq (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message)
-    (hs : SpinSymmetric Q) (i : Spin) :
-    avg (fun k => ∫ m, edgeMessage lam h m k*(messageDeviation (edgeMessage lam h m) i)^2 ∂(Q : Measure Message)) =
-      2*lam^2*posteriorMu Q := by
-  rw [← integral_avg]
-  · have he (m : Message) : avg (fun k => edgeMessage lam h m k*(messageDeviation (edgeMessage lam h m) i)^2) =
-        (messageDeviation (edgeMessage lam h m) i)^2 := by
-      rw [avg_mul_const, average_message, one_mul]
-    simp_rw [he]
-    exact integral_edge_deviation_sq lam h Q hs i
-  · intro k
-    exact integrable_continuous_message Q ((edgeCoordinateMap lam h k).continuous.mul ((edgeDeviationMap lam h i).continuous.pow 2))
-
-lemma integral_projection_variance_coordinate (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message)
-    (hQ : Balanced Q) (hs : SpinSymmetric Q) (n : ℕ) (i : Spin) :
-    (∫ m : Fin n → Message, normalizer lam h m*(branchSum lam h m i)^2
-      ∂(iidMeasure Q n : Measure (Fin n → Message))) =
-      2*((n:ℝ)*(lam^2*posteriorMu Q)+(n:ℝ)*(n-1)*(lam^2*posteriorMu Q)^2) := by
-  have he (m : Fin n → Message) : normalizer lam h m*(branchSum lam h m i)^2 =
-      avg (fun k => productWeight lam h m k*(branchSum lam h m i)^2) := (avg_mul_const _ _).symm
-  simp_rw [he]
-  rw [integral_avg]
-  · have hi (k : Spin) :
-        (∫ m : Fin n → Message, productWeight lam h m k*(branchSum lam h m i)^2
-          ∂(iidMeasure Q n : Measure (Fin n → Message))) =
-        (n:ℝ)*(∫ m, edgeMessage lam h m k*(messageDeviation (edgeMessage lam h m) i)^2 ∂(Q : Measure Message))+
-        ((n:ℝ)*(n-1)*(lam^2*posteriorMu Q)^2)*(if k=i then (2:ℝ) else -1)^2 := by
-      have hh := iid_integral_weighted_sum_sq Q n (edgeCoordinateMap lam h k) (edgeDeviationMap lam h i)
-        (integral_edgeMessage lam h Q hQ k)
-      change _ = n*(∫ m, edgeMessage lam h m k*(messageDeviation (edgeMessage lam h m) i)^2 ∂(Q : Measure Message))+
-        (n:ℝ)*(n-1)*(∫ m, edgeMessage lam h m k*messageDeviation (edgeMessage lam h m) i ∂(Q : Measure Message))^2 at hh
-      rw [integral_edge_covariance lam h Q hQ hs] at hh
-      change (∫ m : Fin n → Message, productWeight lam h m k*(branchSum lam h m i)^2
-        ∂(iidMeasure Q n : Measure (Fin n → Message))) = _ at hh
-      rw [hh]
-      ring
-    simp_rw [hi]
-    rw [avg_add, avg_const_mul, avg_const_mul, avg_integral_edge_sq lam h Q hs, avg_covariance_coeff]
-    ring
-  · intro k
-    exact ((continuous_productWeight lam h n k).mul ((continuous_branchSum lam h n i).pow 2)).integrable_of_hasCompactSupport
-      (HasCompactSupport.of_compactSpace _)
-
-lemma integral_projection_variance (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message)
-    (hQ : Balanced Q) (hs : SpinSymmetric Q) (n : ℕ) :
-    (∫ m : Fin n → Message, normalizer lam h m*(avg (fun i => (branchSum lam h m i)^2)/2)
-      ∂(iidMeasure Q n : Measure (Fin n → Message))) =
-      (n:ℝ)*(lam^2*posteriorMu Q)+(n:ℝ)*(n-1)*(lam^2*posteriorMu Q)^2 := by
-  have he (m : Fin n → Message) : normalizer lam h m*(avg (fun i => (branchSum lam h m i)^2)/2) =
-      avg (fun i => normalizer lam h m*(branchSum lam h m i)^2)/2 := by rw [avg_const_mul]; ring
-  simp_rw [he]
-  rw [integral_div, integral_avg]
-  · simp_rw [integral_projection_variance_coordinate lam h Q hQ hs]
-    simp [Radial.avg_expand]
-    ring
-  · intro i
-    exact ((continuous_normalizer lam h n).mul ((continuous_branchSum lam h n i).pow 2)).integrable_of_hasCompactSupport
-      (HasCompactSupport.of_compactSpace _)
-
-end ThreeState
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-open Radial (avg)
-
-lemma weighted_vector_quadratic {α : Type*} [TopologicalSpace α] [CompactSpace α]
-    [MeasurableSpace α] [BorelSpace α] (M : Measure α) [IsFiniteMeasure M]
-    (w : α → ℝ) (hw : Continuous w) (hw0 : ∀ a, 0 ≤ w a)
-    (f g : α → Spin → ℝ) (hf : ∀ i, Continuous (fun a => f a i)) (hg : ∀ i, Continuous (fun a => g a i)) (t : ℝ) :
-    0 ≤ (∫ a, w a*(avg (fun i => (f a i)^2)/2) ∂M)*t^2-
-      2*(∫ a, w a*(avg (fun i => f a i*g a i)/2) ∂M)*t+
-      (∫ a, w a*(avg (fun i => (g a i)^2)/2) ∂M) := by
-  have hA : Integrable (fun a => w a*(avg (fun i => (f a i)^2)/2)) M :=
-    (hw.mul ((continuous_avg _ (fun i => (hf i).pow 2)).div_const 2)).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
-  have hB : Integrable (fun a => w a*(avg (fun i => (g a i)^2)/2)) M :=
-    (hw.mul ((continuous_avg _ (fun i => (hg i).pow 2)).div_const 2)).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
-  have hC : Integrable (fun a => w a*(avg (fun i => f a i*g a i)/2)) M :=
-    (hw.mul ((continuous_avg _ (fun i => (hf i).mul (hg i))).div_const 2)).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
-  have hp : 0 ≤ ∫ a, w a*(avg (fun i => (t*f a i-g a i)^2)/2) ∂M := by
-    apply integral_nonneg
-    intro a
-    apply mul_nonneg (hw0 a)
-    rw [Radial.avg_expand]
-    positivity
-  have he (a : α) : w a*(avg (fun i => (t*f a i-g a i)^2)/2) =
-      t^2*(w a*(avg (fun i => (f a i)^2)/2))+(-2*t)*(w a*(avg (fun i => f a i*g a i)/2))+
-        w a*(avg (fun i => (g a i)^2)/2) := by simp only [Radial.avg_expand]; ring
-  simp_rw [he] at hp
-  have hAC : Integrable (fun a => t^2*(w a*(avg (fun i => (f a i)^2)/2))+
-      (-2*t)*(w a*(avg (fun i => f a i*g a i)/2))) M := (hA.const_mul _).add (hC.const_mul _)
-  rw [integral_add hAC hB,
-    integral_add (hA.const_mul _) (hC.const_mul _), integral_const_mul, integral_const_mul] at hp
-  nlinarith only [hp]
-
-lemma degree_projection_quadratic (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (hs : SpinSymmetric Q) (n : ℕ) (t : ℝ) :
-    0 ≤ posteriorMu (degreePosterior lam h Q hQ n)*t^2-2*((n:ℝ)*(lam^2*posteriorMu Q))*t+
-      ((n:ℝ)*(lam^2*posteriorMu Q)+(n:ℝ)*(n-1)*(lam^2*posteriorMu Q)^2) := by
-  have hh := weighted_vector_quadratic (iidMeasure Q n : Measure (Fin n → Message)) (normalizer lam h)
-    (continuous_normalizer lam h n) (normalizer_nonneg lam h)
-    (fun m i => messageDeviation (combineMessage lam h m) i) (branchSum lam h)
-    (fun i => (continuous_messageDeviation i).comp (continuous_combineMessage_positive lam h hl0 hl1 n))
-    (continuous_branchSum lam h n) t
-  rw [integral_projection_cross lam h Q hQ hs, integral_projection_variance lam h Q hQ hs] at hh
-  have he : (∫ m : Fin n → Message, normalizer lam h m *
-      (avg (fun i => (messageDeviation (combineMessage lam h m) i)^2)/2)
-      ∂(iidMeasure Q n : Measure (Fin n → Message))) = posteriorMu (degreePosterior lam h Q hQ n) := by
-    change _ = ∫ m, messageX m ∂degreeOutput lam h Q n
-    rw [integral_degreeOutput lam h Q n messageX continuous_messageX.measurable]
-    rfl
-  rwa [he] at hh
-
-lemma degree_projection (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (hs : SpinSymmetric Q) (n : ℕ) (hn : 1 ≤ n) :
-    (n:ℝ)*(lam^2*posteriorMu Q)/(1+((n:ℝ)-1)*(lam^2*posteriorMu Q)) ≤
-      posteriorMu (degreePosterior lam h Q hQ n) := by
-  let beta := lam^2*posteriorMu Q
-  have hb : 0 ≤ beta := mul_nonneg (sq_nonneg _) (posteriorMu_nonneg Q)
-  have hn0 : (0:ℝ)<n := by exact_mod_cast (show 0<n by omega)
-  have hn1 : (1:ℝ)≤n := by exact_mod_cast hn
-  have hd : 0<1+((n:ℝ)-1)*beta := by positivity
-  by_cases hb0 : beta=0
-  · change (n:ℝ)*beta/(1+((n:ℝ)-1)*beta) ≤ _
-    simp only [hb0, mul_zero, zero_div]
-    exact posteriorMu_nonneg _
-  · have hbpos : 0<beta := lt_of_le_of_ne hb (Ne.symm hb0)
-    have hnb : 0<(n:ℝ)*beta := mul_pos hn0 hbpos
-    have hp := degree_projection_quadratic lam h hl0 hl1 Q hQ hs n
-    have hdisc := discrim_le_zero (show ∀ t, 0 ≤ posteriorMu (degreePosterior lam h Q hQ n)*(t*t)+
-        (-2*((n:ℝ)*beta))*t+((n:ℝ)*beta+(n:ℝ)*(n-1)*beta^2) by intro t; simpa [beta, pow_two, sub_eq_add_neg] using hp t)
-    simp only [discrim] at hdisc
-    change (n:ℝ)*beta/(1+((n:ℝ)-1)*beta) ≤ _
-    apply (div_le_iff₀ hd).mpr
-    apply (mul_le_mul_iff_right₀ hnb).mp
-    nlinarith only [hdisc]
-
-end ThreeState
-
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-
-lemma truncated_quadratic_lower {A B L u C : ℝ} (hA : 0 ≤ A) (hL : 0 ≤ L)
-    (hLu : L ≤ u) (hC : 0 ≤ C) (hCu : A*u-B*Real.sqrt u ≤ C) :
-    A*L-B*Real.sqrt L ≤ C := by
-  by_cases ht : A*L-B*Real.sqrt L ≤ 0
-  · exact ht.trans hC
-  · have hl := Real.sq_sqrt hL
-    have hu := Real.sq_sqrt (hL.trans hLu)
-    have hroot := Real.sqrt_le_sqrt hLu
-    have hf : 0 ≤ A*Real.sqrt L-B := by
-      by_contra hh
-      have hp : Real.sqrt L*(A*Real.sqrt L-B) ≤ 0 :=
-        mul_nonpos_of_nonneg_of_nonpos (Real.sqrt_nonneg _) (le_of_lt (lt_of_not_ge hh))
-      apply ht
-      nlinarith only [hp, congrArg (fun x => A*x) hl]
-    have hg : 0 ≤ A*(Real.sqrt u+Real.sqrt L)-B := by
-      nlinarith only [hf, mul_nonneg hA (Real.sqrt_nonneg u)]
-    have hp := mul_nonneg (sub_nonneg.mpr hroot) hg
-    nlinarith only [hp, congrArg (fun x => A*x) hl, congrArg (fun x => A*x) hu, hCu]
-
-noncomputable def pairCorrelation (lam : ℝ) (h : Admissible lam) (Q P : ProbabilityMeasure Message) : ℝ :=
-  ∫ a, pairIncrement lam h Q a ∂(P : Measure Message)
-
-lemma pairCorrelation_nonneg (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q P : ProbabilityMeasure Message) (hQ : Balanced Q) : 0 ≤ pairCorrelation lam h Q P :=
-  integral_nonneg (pairIncrement_nonneg lam h hl0 hl1 Q hQ)
-
-lemma pairCorrelation_lower (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q P : ProbabilityMeasure Message) (hs : SpinSymmetric Q) :
-    3*lam^2*posteriorMu Q*posteriorMu P-lam^3*posteriorEta Q*posteriorEta P ≤ 3*pairCorrelation lam h Q P := by
-  have hi : Integrable (fun a => lam^2*posteriorMu Q*messageX a-lam^3*posteriorEta Q*messageY a/3) (P : Measure Message) :=
-    ((integrable_continuous_message P continuous_messageX).const_mul _).sub
-      (((integrable_continuous_message P continuous_messageY).const_mul _).div_const _)
-  have hh := integral_mono hi (integrable_continuous_message P (continuous_pairIncrement lam h hl0 hl1 Q))
-    (pairIncrement_lower lam h hl0 hl1 Q hs)
-  have hix : Integrable (fun a => lam^2*posteriorMu Q*messageX a) (P : Measure Message) :=
-    (integrable_continuous_message P continuous_messageX).const_mul _
-  have hiy : Integrable (fun a => lam^3*posteriorEta Q*messageY a/3) (P : Measure Message) :=
-    ((integrable_continuous_message P continuous_messageY).const_mul _).div_const _
-  rw [integral_sub hix hiy, integral_div, integral_const_mul, integral_const_mul] at hh
-  change lam^2*posteriorMu Q*posteriorMu P-lam^3*posteriorEta Q*posteriorEta P/3 ≤ pairCorrelation lam h Q P at hh
-  linarith
-
-lemma eta_product_lower_control (Q P : ProbabilityMeasure Message) (he : 0 ≤ posteriorEta Q)
-    (hn : posteriorNu P ≤ posteriorNu Q) :
-    posteriorEta Q*posteriorEta P ≤ Real.sqrt (posteriorMu Q)*posteriorNu Q*Real.sqrt (posteriorMu P) := by
-  have hQ := posteriorEta_upper Q
-  have hP := posteriorEta_upper P
-  rw [Real.sqrt_mul (posteriorMu_nonneg Q)] at hQ
-  rw [Real.sqrt_mul (posteriorMu_nonneg P)] at hP
-  have hP' : posteriorEta P ≤ Real.sqrt (posteriorMu P)*Real.sqrt (posteriorNu Q) :=
-    hP.trans (mul_le_mul_of_nonneg_left (Real.sqrt_le_sqrt hn) (Real.sqrt_nonneg _))
-  have hh : posteriorEta Q*posteriorEta P ≤
-      (Real.sqrt (posteriorMu Q)*Real.sqrt (posteriorNu Q))*(Real.sqrt (posteriorMu P)*Real.sqrt (posteriorNu Q)) :=
-    (mul_le_mul_of_nonneg_left hP' he).trans
-      (mul_le_mul_of_nonneg_right hQ (mul_nonneg (Real.sqrt_nonneg _) (Real.sqrt_nonneg _)))
-  have heq : (Real.sqrt (posteriorMu Q)*Real.sqrt (posteriorNu Q))*(Real.sqrt (posteriorMu P)*Real.sqrt (posteriorNu Q)) =
-      Real.sqrt (posteriorMu Q)*posteriorNu Q*Real.sqrt (posteriorMu P) := by
-    calc
-      _ = Real.sqrt (posteriorMu Q)*(Real.sqrt (posteriorNu Q))^2*Real.sqrt (posteriorMu P) := by ring
-      _ = _ := by rw [Real.sq_sqrt (posteriorNu_nonneg Q)]
-  rwa [heq] at hh
-
-lemma pairCorrelation_truncated (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q P : ProbabilityMeasure Message) (hQ : Balanced Q) (hs : SpinSymmetric Q)
-    (he : 0 ≤ posteriorEta Q) (hn : posteriorNu P ≤ posteriorNu Q) {L : ℝ} (hL : 0 ≤ L) (hLP : L ≤ posteriorMu P) :
-    3*lam^2*posteriorMu Q*L-lam^3*Real.sqrt (posteriorMu Q)*posteriorNu Q*Real.sqrt L ≤
-      3*pairCorrelation lam h Q P := by
-  have hm := posteriorMu_nonneg Q
-  have hh := pairCorrelation_lower lam h hl0 hl1 Q P hs
-  have hg := mul_le_mul_of_nonneg_left (eta_product_lower_control Q P he hn) (pow_nonneg hl0 3)
-  apply truncated_quadratic_lower (show 0 ≤ 3*lam^2*posteriorMu Q by positivity) hL hLP
-    (mul_nonneg (by norm_num) (pairCorrelation_nonneg lam h hl0 hl1 Q P hQ))
-  nlinarith only [hh,hg]
-
-end ThreeState
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeState.DegreePosterior
+
+namespace OAI
+
+/-! Quadratic projections, covariance and truncated correlation estimates. -/
+
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+open Radial (avg)
+
+lemma symAvg_deviation_mul (a : Message) (i j : Spin) :
+    symAvg (fun m => messageDeviation m i*messageDeviation m j) a =
+      (if i=j then 2 else -1)*messageX a := by
+  fin_cases i <;> fin_cases j <;>
+    norm_num [symAvg, symPerm, Fin.sum_univ_succ, messageDeviation,
+      Radial.avg_expand, permMessage, Equiv.swap_apply_def, Equiv.trans_apply,
+      Equiv.symm_trans_apply, Fin.ext_iff, messageX, Radial.momentX] <;>
+    (try rw [message_last a]) <;> ring
+
+lemma continuous_messageDeviation (i : Spin) : Continuous (fun m => messageDeviation m i) :=
+  ((continuous_apply i).comp continuous_subtype_val).sub continuous_const
+
+lemma integral_deviation (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (i : Spin) :
+    (∫ m, messageDeviation m i ∂(Q : Measure Message)) = 0 := by
+  change (∫ m : Message, m i-1 ∂(Q : Measure Message)) = 0
+  have hi : Integrable (fun m : Message => m i) (Q : Measure Message) :=
+    integrable_continuous_message Q (coordinateMap i).continuous
+  rw [integral_sub hi (integrable_const (1:ℝ)), hQ i]
+  simp
+
+lemma integral_deviation_mul (Q : ProbabilityMeasure Message) (hs : SpinSymmetric Q) (i j : Spin) :
+    (∫ m, messageDeviation m i*messageDeviation m j ∂(Q : Measure Message)) =
+      (if i=j then 2 else -1)*posteriorMu Q := by
+  rw [← integral_symAvg Q hs (fun m => messageDeviation m i*messageDeviation m j) ((continuous_messageDeviation i).mul (continuous_messageDeviation j))]
+  simp only [symAvg_deviation_mul, integral_const_mul, posteriorMu]
+
+noncomputable def edgeDeviationMap (lam : ℝ) (h : Admissible lam) (i : Spin) : C(Message,ℝ) :=
+  ⟨fun m => messageDeviation (edgeMessage lam h m) i,
+    (continuous_messageDeviation i).comp (continuous_edgeMessage lam h)⟩
+
+lemma edge_deviation (lam : ℝ) (h : Admissible lam) (m : Message) (i : Spin) :
+    messageDeviation (edgeMessage lam h m) i = lam*messageDeviation m i := by
+  change 1+lam*(m i-1)-1 = lam*(m i-1)
+  ring
+
+lemma integral_edge_covariance (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message)
+    (hQ : Balanced Q) (hs : SpinSymmetric Q) (i j : Spin) :
+    (∫ m, edgeMessage lam h m i*messageDeviation (edgeMessage lam h m) j ∂(Q : Measure Message)) =
+      lam^2*posteriorMu Q*(if i=j then 2 else -1) := by
+  have he (m : Message) : edgeMessage lam h m i*messageDeviation (edgeMessage lam h m) j =
+      lam*messageDeviation m j + lam^2*(messageDeviation m i*messageDeviation m j) := by
+    rw [edge_deviation]
+    change (1+lam*(m i-1))*(lam*(m j-1)) = _
+    simp only [messageDeviation]
+    ring
+  simp_rw [he]
+  have hi : Integrable (fun m => lam*messageDeviation m j) (Q : Measure Message) :=
+    (integrable_continuous_message Q (continuous_messageDeviation j)).const_mul _
+  have hj : Integrable (fun m => lam^2*(messageDeviation m i*messageDeviation m j)) (Q : Measure Message) :=
+    (integrable_continuous_message Q ((continuous_messageDeviation i).mul (continuous_messageDeviation j))).const_mul _
+  rw [integral_add hi hj,
+    integral_const_mul, integral_const_mul, integral_deviation Q hQ,
+    integral_deviation_mul Q hs]
+  ring
+
+lemma integral_edge_deviation_sq (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message)
+    (hs : SpinSymmetric Q) (i : Spin) :
+    (∫ m, (messageDeviation (edgeMessage lam h m) i)^2 ∂(Q : Measure Message)) = 2*lam^2*posteriorMu Q := by
+  have he (m : Message) : (messageDeviation (edgeMessage lam h m) i)^2 =
+      lam^2*(messageDeviation m i*messageDeviation m i) := by rw [edge_deviation]; ring
+  simp_rw [he]
+  rw [integral_const_mul, integral_deviation_mul Q hs]
+  simp only [ite_true]
+  ring
+
+lemma avg_swap (f : Spin → Spin → ℝ) : avg (fun i => avg (f i)) = avg (fun j => avg (fun i => f i j)) := by
+  simp only [Radial.avg_expand]
+  ring
+lemma avg_add (a b : Spin → ℝ) : avg (fun i => a i+b i) = avg a+avg b := by
+  simp only [Radial.avg_expand]
+  ring
+lemma avg_mul_const (a : Spin → ℝ) (c : ℝ) : avg (fun i => a i*c) = avg a*c := by
+  simp only [Radial.avg_expand]
+  ring
+lemma avg_const_mul (c : ℝ) (a : Spin → ℝ) : avg (fun i => c*a i) = c*avg a := by
+  simp only [Radial.avg_expand]
+  ring
+lemma avg_covariance_coeff (j : Spin) : avg (fun i : Spin => (if i=j then (2:ℝ) else -1)^2) = 2 := by
+  have he (i : Spin) : (if i=j then (2:ℝ) else -1)^2 = 1+(if i=j then (3:ℝ) else 0) := by
+    split_ifs <;> norm_num
+  simp only [he, avg, Finset.sum_add_distrib, Finset.sum_const, Finset.card_univ,
+    Fintype.card_fin, nsmul_eq_mul, Finset.sum_ite_eq', Finset.mem_univ, ite_true]
+  norm_num
+
+end ThreeState
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+open Radial (avg)
+
+noncomputable def branchSum (lam : ℝ) (h : Admissible lam) {n : ℕ} (m : Fin n → Message) (i : Spin) : ℝ :=
+  ∑ j, messageDeviation (edgeMessage lam h (m j)) i
+lemma continuous_branchSum (lam : ℝ) (h : Admissible lam) (n : ℕ) (i : Spin) :
+    Continuous (fun m : Fin n → Message => branchSum lam h m i) := by
+  apply continuous_finsetSum
+  intro j _
+  exact ((continuous_messageDeviation i).comp (continuous_edgeMessage lam h)).comp (continuous_apply j)
+lemma avg_branchSum (lam : ℝ) (h : Admissible lam) {n : ℕ} (m : Fin n → Message) :
+    avg (branchSum lam h m) = 0 := by
+  have he : avg (branchSum lam h m) = ∑ j, avg (messageDeviation (edgeMessage lam h (m j))) := by
+    simp only [branchSum, Radial.avg_expand]
+    rw [← Finset.sum_div, Finset.sum_add_distrib, Finset.sum_add_distrib]
+  rw [he]
+  simp only [avg_messageDeviation, Finset.sum_const_zero]
+
+lemma weighted_deviation_branchSum (lam : ℝ) (h : Admissible lam) {n : ℕ} (m : Fin n → Message) :
+    normalizer lam h m * avg (fun i => messageDeviation (combineMessage lam h m) i*branchSum lam h m i) =
+      avg (fun i => productWeight lam h m i*branchSum lam h m i) := by
+  have hh := weighted_avg_combine lam h m (branchSum lam h m)
+  have hz := avg_branchSum lam h m
+  simp only [messageDeviation, Radial.avg_expand] at *
+  nlinarith only [hh, mul_eq_zero_of_right (normalizer lam h m) hz]
+
+lemma integral_projection_cross (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message)
+    (hQ : Balanced Q) (hs : SpinSymmetric Q) (n : ℕ) :
+    (∫ m : Fin n → Message, normalizer lam h m *
+      (avg (fun i => messageDeviation (combineMessage lam h m) i*branchSum lam h m i)/2)
+        ∂(iidMeasure Q n : Measure (Fin n → Message))) = n*(lam^2*posteriorMu Q) := by
+  have he (m : Fin n → Message) : normalizer lam h m *
+      (avg (fun i => messageDeviation (combineMessage lam h m) i*branchSum lam h m i)/2) =
+      avg (fun i => productWeight lam h m i*branchSum lam h m i)/2 := by
+    rw [← mul_div_assoc, weighted_deviation_branchSum]
+  simp_rw [he]
+  rw [integral_div, integral_avg]
+  · have hi (i : Spin) :
+        (∫ m : Fin n → Message, productWeight lam h m i*branchSum lam h m i
+          ∂(iidMeasure Q n : Measure (Fin n → Message))) = (n:ℝ)*(lam^2*posteriorMu Q*2) := by
+      have hh := iid_integral_weighted_sum Q n (edgeCoordinateMap lam h i) (edgeDeviationMap lam h i)
+        (integral_edgeMessage lam h Q hQ i)
+      change _ = n*(∫ m, edgeMessage lam h m i*messageDeviation (edgeMessage lam h m) i ∂(Q : Measure Message)) at hh
+      rw [integral_edge_covariance lam h Q hQ hs] at hh
+      simpa only [edgeCoordinateMap, edgeDeviationMap, ContinuousMap.coe_mk, productWeight, branchSum, ite_true] using hh
+    simp_rw [hi]
+    simp [Radial.avg_expand]
+    ring
+  · intro i
+    exact ((continuous_productWeight lam h n i).mul (continuous_branchSum lam h n i)).integrable_of_hasCompactSupport
+      (HasCompactSupport.of_compactSpace _)
+
+lemma avg_integral_edge_sq (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message)
+    (hs : SpinSymmetric Q) (i : Spin) :
+    avg (fun k => ∫ m, edgeMessage lam h m k*(messageDeviation (edgeMessage lam h m) i)^2 ∂(Q : Measure Message)) =
+      2*lam^2*posteriorMu Q := by
+  rw [← integral_avg]
+  · have he (m : Message) : avg (fun k => edgeMessage lam h m k*(messageDeviation (edgeMessage lam h m) i)^2) =
+        (messageDeviation (edgeMessage lam h m) i)^2 := by
+      rw [avg_mul_const, average_message, one_mul]
+    simp_rw [he]
+    exact integral_edge_deviation_sq lam h Q hs i
+  · intro k
+    exact integrable_continuous_message Q ((edgeCoordinateMap lam h k).continuous.mul ((edgeDeviationMap lam h i).continuous.pow 2))
+
+lemma integral_projection_variance_coordinate (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message)
+    (hQ : Balanced Q) (hs : SpinSymmetric Q) (n : ℕ) (i : Spin) :
+    (∫ m : Fin n → Message, normalizer lam h m*(branchSum lam h m i)^2
+      ∂(iidMeasure Q n : Measure (Fin n → Message))) =
+      2*((n:ℝ)*(lam^2*posteriorMu Q)+(n:ℝ)*(n-1)*(lam^2*posteriorMu Q)^2) := by
+  have he (m : Fin n → Message) : normalizer lam h m*(branchSum lam h m i)^2 =
+      avg (fun k => productWeight lam h m k*(branchSum lam h m i)^2) := (avg_mul_const _ _).symm
+  simp_rw [he]
+  rw [integral_avg]
+  · have hi (k : Spin) :
+        (∫ m : Fin n → Message, productWeight lam h m k*(branchSum lam h m i)^2
+          ∂(iidMeasure Q n : Measure (Fin n → Message))) =
+        (n:ℝ)*(∫ m, edgeMessage lam h m k*(messageDeviation (edgeMessage lam h m) i)^2 ∂(Q : Measure Message))+
+        ((n:ℝ)*(n-1)*(lam^2*posteriorMu Q)^2)*(if k=i then (2:ℝ) else -1)^2 := by
+      have hh := iid_integral_weighted_sum_sq Q n (edgeCoordinateMap lam h k) (edgeDeviationMap lam h i)
+        (integral_edgeMessage lam h Q hQ k)
+      change _ = n*(∫ m, edgeMessage lam h m k*(messageDeviation (edgeMessage lam h m) i)^2 ∂(Q : Measure Message))+
+        (n:ℝ)*(n-1)*(∫ m, edgeMessage lam h m k*messageDeviation (edgeMessage lam h m) i ∂(Q : Measure Message))^2 at hh
+      rw [integral_edge_covariance lam h Q hQ hs] at hh
+      change (∫ m : Fin n → Message, productWeight lam h m k*(branchSum lam h m i)^2
+        ∂(iidMeasure Q n : Measure (Fin n → Message))) = _ at hh
+      rw [hh]
+      ring
+    simp_rw [hi]
+    rw [avg_add, avg_const_mul, avg_const_mul, avg_integral_edge_sq lam h Q hs, avg_covariance_coeff]
+    ring
+  · intro k
+    exact ((continuous_productWeight lam h n k).mul ((continuous_branchSum lam h n i).pow 2)).integrable_of_hasCompactSupport
+      (HasCompactSupport.of_compactSpace _)
+
+lemma integral_projection_variance (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message)
+    (hQ : Balanced Q) (hs : SpinSymmetric Q) (n : ℕ) :
+    (∫ m : Fin n → Message, normalizer lam h m*(avg (fun i => (branchSum lam h m i)^2)/2)
+      ∂(iidMeasure Q n : Measure (Fin n → Message))) =
+      (n:ℝ)*(lam^2*posteriorMu Q)+(n:ℝ)*(n-1)*(lam^2*posteriorMu Q)^2 := by
+  have he (m : Fin n → Message) : normalizer lam h m*(avg (fun i => (branchSum lam h m i)^2)/2) =
+      avg (fun i => normalizer lam h m*(branchSum lam h m i)^2)/2 := by rw [avg_const_mul]; ring
+  simp_rw [he]
+  rw [integral_div, integral_avg]
+  · simp_rw [integral_projection_variance_coordinate lam h Q hQ hs]
+    simp [Radial.avg_expand]
+    ring
+  · intro i
+    exact ((continuous_normalizer lam h n).mul ((continuous_branchSum lam h n i).pow 2)).integrable_of_hasCompactSupport
+      (HasCompactSupport.of_compactSpace _)
+
+end ThreeState
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+open Radial (avg)
+
+lemma weighted_vector_quadratic {α : Type*} [TopologicalSpace α] [CompactSpace α]
+    [MeasurableSpace α] [BorelSpace α] (M : Measure α) [IsFiniteMeasure M]
+    (w : α → ℝ) (hw : Continuous w) (hw0 : ∀ a, 0 ≤ w a)
+    (f g : α → Spin → ℝ) (hf : ∀ i, Continuous (fun a => f a i)) (hg : ∀ i, Continuous (fun a => g a i)) (t : ℝ) :
+    0 ≤ (∫ a, w a*(avg (fun i => (f a i)^2)/2) ∂M)*t^2-
+      2*(∫ a, w a*(avg (fun i => f a i*g a i)/2) ∂M)*t+
+      (∫ a, w a*(avg (fun i => (g a i)^2)/2) ∂M) := by
+  have hA : Integrable (fun a => w a*(avg (fun i => (f a i)^2)/2)) M :=
+    (hw.mul ((continuous_avg _ (fun i => (hf i).pow 2)).div_const 2)).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
+  have hB : Integrable (fun a => w a*(avg (fun i => (g a i)^2)/2)) M :=
+    (hw.mul ((continuous_avg _ (fun i => (hg i).pow 2)).div_const 2)).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
+  have hC : Integrable (fun a => w a*(avg (fun i => f a i*g a i)/2)) M :=
+    (hw.mul ((continuous_avg _ (fun i => (hf i).mul (hg i))).div_const 2)).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
+  have hp : 0 ≤ ∫ a, w a*(avg (fun i => (t*f a i-g a i)^2)/2) ∂M := by
+    apply integral_nonneg
+    intro a
+    apply mul_nonneg (hw0 a)
+    rw [Radial.avg_expand]
+    positivity
+  have he (a : α) : w a*(avg (fun i => (t*f a i-g a i)^2)/2) =
+      t^2*(w a*(avg (fun i => (f a i)^2)/2))+(-2*t)*(w a*(avg (fun i => f a i*g a i)/2))+
+        w a*(avg (fun i => (g a i)^2)/2) := by simp only [Radial.avg_expand]; ring
+  simp_rw [he] at hp
+  have hAC : Integrable (fun a => t^2*(w a*(avg (fun i => (f a i)^2)/2))+
+      (-2*t)*(w a*(avg (fun i => f a i*g a i)/2))) M := (hA.const_mul _).add (hC.const_mul _)
+  rw [integral_add hAC hB,
+    integral_add (hA.const_mul _) (hC.const_mul _), integral_const_mul, integral_const_mul] at hp
+  nlinarith only [hp]
+
+lemma degree_projection_quadratic (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (hs : SpinSymmetric Q) (n : ℕ) (t : ℝ) :
+    0 ≤ posteriorMu (degreePosterior lam h Q hQ n)*t^2-2*((n:ℝ)*(lam^2*posteriorMu Q))*t+
+      ((n:ℝ)*(lam^2*posteriorMu Q)+(n:ℝ)*(n-1)*(lam^2*posteriorMu Q)^2) := by
+  have hh := weighted_vector_quadratic (iidMeasure Q n : Measure (Fin n → Message)) (normalizer lam h)
+    (continuous_normalizer lam h n) (normalizer_nonneg lam h)
+    (fun m i => messageDeviation (combineMessage lam h m) i) (branchSum lam h)
+    (fun i => (continuous_messageDeviation i).comp (continuous_combineMessage_positive lam h hl0 hl1 n))
+    (continuous_branchSum lam h n) t
+  rw [integral_projection_cross lam h Q hQ hs, integral_projection_variance lam h Q hQ hs] at hh
+  have he : (∫ m : Fin n → Message, normalizer lam h m *
+      (avg (fun i => (messageDeviation (combineMessage lam h m) i)^2)/2)
+      ∂(iidMeasure Q n : Measure (Fin n → Message))) = posteriorMu (degreePosterior lam h Q hQ n) := by
+    change _ = ∫ m, messageX m ∂degreeOutput lam h Q n
+    rw [integral_degreeOutput lam h Q n messageX continuous_messageX.measurable]
+    rfl
+  rwa [he] at hh
+
+lemma degree_projection (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (hs : SpinSymmetric Q) (n : ℕ) (hn : 1 ≤ n) :
+    (n:ℝ)*(lam^2*posteriorMu Q)/(1+((n:ℝ)-1)*(lam^2*posteriorMu Q)) ≤
+      posteriorMu (degreePosterior lam h Q hQ n) := by
+  let beta := lam^2*posteriorMu Q
+  have hb : 0 ≤ beta := mul_nonneg (sq_nonneg _) (posteriorMu_nonneg Q)
+  have hn0 : (0:ℝ)<n := by exact_mod_cast (show 0<n by omega)
+  have hn1 : (1:ℝ)≤n := by exact_mod_cast hn
+  have hd : 0<1+((n:ℝ)-1)*beta := by positivity
+  by_cases hb0 : beta=0
+  · change (n:ℝ)*beta/(1+((n:ℝ)-1)*beta) ≤ _
+    simp only [hb0, mul_zero, zero_div]
+    exact posteriorMu_nonneg _
+  · have hbpos : 0<beta := lt_of_le_of_ne hb (Ne.symm hb0)
+    have hnb : 0<(n:ℝ)*beta := mul_pos hn0 hbpos
+    have hp := degree_projection_quadratic lam h hl0 hl1 Q hQ hs n
+    have hdisc := discrim_le_zero (show ∀ t, 0 ≤ posteriorMu (degreePosterior lam h Q hQ n)*(t*t)+
+        (-2*((n:ℝ)*beta))*t+((n:ℝ)*beta+(n:ℝ)*(n-1)*beta^2) by intro t; simpa [beta, pow_two, sub_eq_add_neg] using hp t)
+    simp only [discrim] at hdisc
+    change (n:ℝ)*beta/(1+((n:ℝ)-1)*beta) ≤ _
+    apply (div_le_iff₀ hd).mpr
+    apply (mul_le_mul_iff_right₀ hnb).mp
+    nlinarith only [hdisc]
+
+end ThreeState
+
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+
+lemma truncated_quadratic_lower {A B L u C : ℝ} (hA : 0 ≤ A) (hL : 0 ≤ L)
+    (hLu : L ≤ u) (hC : 0 ≤ C) (hCu : A*u-B*Real.sqrt u ≤ C) :
+    A*L-B*Real.sqrt L ≤ C := by
+  by_cases ht : A*L-B*Real.sqrt L ≤ 0
+  · exact ht.trans hC
+  · have hl := Real.sq_sqrt hL
+    have hu := Real.sq_sqrt (hL.trans hLu)
+    have hroot := Real.sqrt_le_sqrt hLu
+    have hf : 0 ≤ A*Real.sqrt L-B := by
+      by_contra hh
+      have hp : Real.sqrt L*(A*Real.sqrt L-B) ≤ 0 :=
+        mul_nonpos_of_nonneg_of_nonpos (Real.sqrt_nonneg _) (le_of_lt (lt_of_not_ge hh))
+      apply ht
+      nlinarith only [hp, congrArg (fun x => A*x) hl]
+    have hg : 0 ≤ A*(Real.sqrt u+Real.sqrt L)-B := by
+      nlinarith only [hf, mul_nonneg hA (Real.sqrt_nonneg u)]
+    have hp := mul_nonneg (sub_nonneg.mpr hroot) hg
+    nlinarith only [hp, congrArg (fun x => A*x) hl, congrArg (fun x => A*x) hu, hCu]
+
+noncomputable def pairCorrelation (lam : ℝ) (h : Admissible lam) (Q P : ProbabilityMeasure Message) : ℝ :=
+  ∫ a, pairIncrement lam h Q a ∂(P : Measure Message)
+
+lemma pairCorrelation_nonneg (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q P : ProbabilityMeasure Message) (hQ : Balanced Q) : 0 ≤ pairCorrelation lam h Q P :=
+  integral_nonneg (pairIncrement_nonneg lam h hl0 hl1 Q hQ)
+
+lemma pairCorrelation_lower (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q P : ProbabilityMeasure Message) (hs : SpinSymmetric Q) :
+    3*lam^2*posteriorMu Q*posteriorMu P-lam^3*posteriorEta Q*posteriorEta P ≤ 3*pairCorrelation lam h Q P := by
+  have hi : Integrable (fun a => lam^2*posteriorMu Q*messageX a-lam^3*posteriorEta Q*messageY a/3) (P : Measure Message) :=
+    ((integrable_continuous_message P continuous_messageX).const_mul _).sub
+      (((integrable_continuous_message P continuous_messageY).const_mul _).div_const _)
+  have hh := integral_mono hi (integrable_continuous_message P (continuous_pairIncrement lam h hl0 hl1 Q))
+    (pairIncrement_lower lam h hl0 hl1 Q hs)
+  have hix : Integrable (fun a => lam^2*posteriorMu Q*messageX a) (P : Measure Message) :=
+    (integrable_continuous_message P continuous_messageX).const_mul _
+  have hiy : Integrable (fun a => lam^3*posteriorEta Q*messageY a/3) (P : Measure Message) :=
+    ((integrable_continuous_message P continuous_messageY).const_mul _).div_const _
+  rw [integral_sub hix hiy, integral_div, integral_const_mul, integral_const_mul] at hh
+  change lam^2*posteriorMu Q*posteriorMu P-lam^3*posteriorEta Q*posteriorEta P/3 ≤ pairCorrelation lam h Q P at hh
+  linarith
+
+lemma eta_product_lower_control (Q P : ProbabilityMeasure Message) (he : 0 ≤ posteriorEta Q)
+    (hn : posteriorNu P ≤ posteriorNu Q) :
+    posteriorEta Q*posteriorEta P ≤ Real.sqrt (posteriorMu Q)*posteriorNu Q*Real.sqrt (posteriorMu P) := by
+  have hQ := posteriorEta_upper Q
+  have hP := posteriorEta_upper P
+  rw [Real.sqrt_mul (posteriorMu_nonneg Q)] at hQ
+  rw [Real.sqrt_mul (posteriorMu_nonneg P)] at hP
+  have hP' : posteriorEta P ≤ Real.sqrt (posteriorMu P)*Real.sqrt (posteriorNu Q) :=
+    hP.trans (mul_le_mul_of_nonneg_left (Real.sqrt_le_sqrt hn) (Real.sqrt_nonneg _))
+  have hh : posteriorEta Q*posteriorEta P ≤
+      (Real.sqrt (posteriorMu Q)*Real.sqrt (posteriorNu Q))*(Real.sqrt (posteriorMu P)*Real.sqrt (posteriorNu Q)) :=
+    (mul_le_mul_of_nonneg_left hP' he).trans
+      (mul_le_mul_of_nonneg_right hQ (mul_nonneg (Real.sqrt_nonneg _) (Real.sqrt_nonneg _)))
+  have heq : (Real.sqrt (posteriorMu Q)*Real.sqrt (posteriorNu Q))*(Real.sqrt (posteriorMu P)*Real.sqrt (posteriorNu Q)) =
+      Real.sqrt (posteriorMu Q)*posteriorNu Q*Real.sqrt (posteriorMu P) := by
+    calc
+      _ = Real.sqrt (posteriorMu Q)*(Real.sqrt (posteriorNu Q))^2*Real.sqrt (posteriorMu P) := by ring
+      _ = _ := by rw [Real.sq_sqrt (posteriorNu_nonneg Q)]
+  rwa [heq] at hh
+
+lemma pairCorrelation_truncated (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q P : ProbabilityMeasure Message) (hQ : Balanced Q) (hs : SpinSymmetric Q)
+    (he : 0 ≤ posteriorEta Q) (hn : posteriorNu P ≤ posteriorNu Q) {L : ℝ} (hL : 0 ≤ L) (hLP : L ≤ posteriorMu P) :
+    3*lam^2*posteriorMu Q*L-lam^3*Real.sqrt (posteriorMu Q)*posteriorNu Q*Real.sqrt L ≤
+      3*pairCorrelation lam h Q P := by
+  have hm := posteriorMu_nonneg Q
+  have hh := pairCorrelation_lower lam h hl0 hl1 Q P hs
+  have hg := mul_le_mul_of_nonneg_left (eta_product_lower_control Q P he hn) (pow_nonneg hl0 3)
+  apply truncated_quadratic_lower (show 0 ≤ 3*lam^2*posteriorMu Q by positivity) hL hLP
+    (mul_nonneg (by norm_num) (pairCorrelation_nonneg lam h hl0 hl1 Q P hQ))
+  nlinarith only [hh,hg]
+
+end ThreeState
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeState/RadialDerivatives.lean b/lean/OAI/Probability/ThreeState/RadialDerivatives.lean
index 2acee507e3ebb14d26395e8ae8e0c123b363fdb7..c10844e8dc8f8515b7d0d10a0b68a730b23b905e 100644
--- a/lean/OAI/Probability/ThreeState/RadialDerivatives.lean
+++ b/lean/OAI/Probability/ThreeState/RadialDerivatives.lean
@@ -1,452 +1,452 @@
-import Mathlib
-import OAI.Probability.ThreeState.RadialMoments
-
-namespace OAI
-
-/-! Radial logarithmic derivatives and the entropy inequality. -/
-
-namespace ThreeState.Radial
-
-noncomputable def ray (v : Fin 3 → ℝ) (t : ℝ) : Fin 3 → ℝ := fun i => t*v i
-noncomputable def logAvg (v : Fin 3 → ℝ) := avg (fun i => Real.log (1+v i))
-noncomputable def logVariance (v : Fin 3 → ℝ) := avg (fun i => (centeredLog v i)^2)
-noncomputable def weightedLogVariance (v : Fin 3 → ℝ) :=
-  avg (fun i => (1+v i)*(centeredLog v i)^2)
- 
-noncomputable def energy (v : Fin 3 → ℝ) :=
-  3*logAvg v+2*logMoment v+(2/5 : ℝ)*weightedLogVariance v
- 
-noncomputable def gapY (v : Fin 3 → ℝ) :=
-  1-invMoment v 1-6*logAvg v-2*logMoment v-(2/5 : ℝ)*weightedLogVariance v-
-    (2/5 : ℝ)*logVariance v+(4/5 : ℝ)*invMoment v 1*logMoment v-(momentX v)^2
- 
-noncomputable def gapZ (v : Fin 3 → ℝ) :=
-  6*logAvg v-5+(26/5 : ℝ)*invMoment v 1-invMoment v 2+
-    (4/5 : ℝ)*logVariance v+(4/5 : ℝ)*(invMoment v 2-2*invMoment v 1)*logMoment v+
-    (4/5 : ℝ)*logInvMoment v 1+(4/5 : ℝ)*(invMoment v 1)^2-3*(momentX v)^2
-
-lemma avg_hasDerivAt (f : Fin 3 → ℝ → ℝ) (f' : Fin 3 → ℝ) (t : ℝ)
-    (hf : ∀ i, HasDerivAt (f i) (f' i) t) :
-    HasDerivAt (fun t => avg (fun i => f i t)) (avg f') t := by
-  simp only [avg_expand]
-  exact (((hf 0).add (hf 1)).add (hf 2)).div_const 3
-
-lemma avg_ray (v : Fin 3 → ℝ) (t : ℝ) : avg (ray v t) = t*avg v := by
-  simp only [avg_expand, ray]; ring
-lemma X_ray (v : Fin 3 → ℝ) (t : ℝ) : momentX (ray v t) = t^2*momentX v := by
-  simp only [momentX, avg_expand, ray]; ring
-lemma centeredLog_avg_zero (v : Fin 3 → ℝ) : avg (centeredLog v) = 0 := by
-  simp only [centeredLog, avg_expand]; ring
-
-lemma ray_log_hasDerivAt (v : Fin 3 → ℝ) (t : ℝ) (hp : ∀ i, 1+t*v i ≠ 0)
-    (i : Fin 3) : HasDerivAt (fun t => Real.log (1+t*v i)) (v i/(1+t*v i)) t := by
-  simpa using (((hasDerivAt_id t).mul_const (v i)).const_add 1).log (hp i)
-lemma ray_inv_hasDerivAt (v : Fin 3 → ℝ) (t : ℝ) (hp : ∀ i, 1+t*v i ≠ 0)
-    (i : Fin 3) : HasDerivAt (fun t => (1+t*v i)⁻¹) (-v i/(1+t*v i)^2) t := by
-  convert (((hasDerivAt_id t).mul_const (v i)).const_add 1).inv (hp i) using 1 <;>
-    first | rfl | norm_num
-
-lemma logAvg_hasDerivAt (v : Fin 3 → ℝ) (t : ℝ) (ht : t ≠ 0)
-    (hp : ∀ i, 1+t*v i ≠ 0) :
-    HasDerivAt (fun t => logAvg (ray v t)) ((1-invMoment (ray v t) 1)/t) t := by
-  have hh := avg_hasDerivAt (fun i t => Real.log (1+t*v i)) (fun i => v i/(1+t*v i)) t
-    (ray_log_hasDerivAt v t hp)
-  convert hh using 1
-  · rfl
-  · simp only [invMoment, avg_expand, ray, pow_one]
-    field_simp [hp 0, hp 1, hp 2]
-    ring
-
-lemma centeredLog_hasDerivAt (v : Fin 3 → ℝ) (t : ℝ) (ht : t ≠ 0)
-    (hp : ∀ i, 1+t*v i ≠ 0) (i : Fin 3) :
-    HasDerivAt (fun t => centeredLog (ray v t) i)
-      ((invMoment (ray v t) 1-(1+t*v i)⁻¹)/t) t := by
-  have hh := (ray_log_hasDerivAt v t hp i).sub (logAvg_hasDerivAt v t ht hp)
-  convert hh using 1 <;> first | rfl | (field_simp [hp i]; ring)
-
-lemma invMoment_one_hasDerivAt (v : Fin 3 → ℝ) (t : ℝ) (ht : t ≠ 0)
-    (hp : ∀ i, 1+t*v i ≠ 0) :
-    HasDerivAt (fun t => invMoment (ray v t) 1)
-      ((invMoment (ray v t) 2-invMoment (ray v t) 1)/t) t := by
-  have hh := avg_hasDerivAt (fun i t => (1+t*v i)⁻¹)
-    (fun i => -v i/(1+t*v i)^2) t (ray_inv_hasDerivAt v t hp)
-  convert hh using 1
-  · simp only [invMoment, ray, pow_one]
-  · simp only [invMoment, avg_expand, ray, pow_one]
-    field_simp [hp 0, hp 1, hp 2]
-    ring
-
-lemma invMoment_two_hasDerivAt (v : Fin 3 → ℝ) (t : ℝ) (ht : t ≠ 0)
-    (hp : ∀ i, 1+t*v i ≠ 0) :
-    HasDerivAt (fun t => invMoment (ray v t) 2)
-      ((2*(invMoment (ray v t) 3-invMoment (ray v t) 2))/t) t := by
-  have hh := avg_hasDerivAt (fun i t => ((1+t*v i)⁻¹)^2)
-    (fun i => 2*(1+t*v i)⁻¹*(-v i/(1+t*v i)^2)) t
-    (fun i => by
-      convert (ray_inv_hasDerivAt v t hp i).pow 2 using 1; first | rfl | norm_num)
-  convert hh using 1
-  · rfl
-  · simp only [invMoment, avg_expand, ray]
-    field_simp [hp 0, hp 1, hp 2]
-    ring
-
-lemma logMoment_hasDerivAt (v : Fin 3 → ℝ) (hv : avg v = 0) (t : ℝ) (ht : t ≠ 0)
-    (hp : ∀ i, 1+t*v i ≠ 0) :
-    HasDerivAt (fun t => logMoment (ray v t))
-      ((logMoment (ray v t)+invMoment (ray v t) 1-1)/t) t := by
-  have hh := avg_hasDerivAt
-    (fun i t => t*v i*centeredLog (ray v t) i)
-    (fun i => v i*centeredLog (ray v t) i+t*v i*((invMoment (ray v t) 1-(1+t*v i)⁻¹)/t)) t
-    (fun i => by
-      convert ((hasDerivAt_id t).mul_const (v i)).mul (centeredLog_hasDerivAt v t ht hp i) using 1 <;>
-        first | rfl | simp)
-  have hv2 := zero_sum_elim v hv
-  convert hh using 1
-  · rfl
-  · simp only [logMoment, invMoment, avg_expand, ray, pow_one]
-    field_simp [hp 0, hp 1, hp 2]
-    rw [hv2]
-    ring
-
-lemma logVariance_hasDerivAt (v : Fin 3 → ℝ) (t : ℝ) (ht : t ≠ 0)
-    (hp : ∀ i, 1+t*v i ≠ 0) :
-    HasDerivAt (fun t => logVariance (ray v t))
-      ((-2*logInvMoment (ray v t) 1)/t) t := by
-  have hh := avg_hasDerivAt
-    (fun i t => (centeredLog (ray v t) i)^2)
-    (fun i => 2*centeredLog (ray v t) i*((invMoment (ray v t) 1-(1+t*v i)⁻¹)/t)) t
-    (fun i => by
-      convert (centeredLog_hasDerivAt v t ht hp i).pow 2 using 1; first | rfl | norm_num)
-  have hu2 := zero_sum_elim (centeredLog (ray v t)) (centeredLog_avg_zero (ray v t))
-  convert hh using 1
-  · rfl
-  · simp only [logInvMoment, invMoment, avg_expand, pow_one]
-    simp only [hu2, ray]
-    ring
-
-lemma weightedLogVariance_hasDerivAt (v : Fin 3 → ℝ) (t : ℝ) (ht : t ≠ 0)
-    (hp : ∀ i, 1+t*v i ≠ 0) :
-    HasDerivAt (fun t => weightedLogVariance (ray v t))
-      ((weightedLogVariance (ray v t)-logVariance (ray v t)+
-        2*invMoment (ray v t) 1*logMoment (ray v t))/t) t := by
-  have hh := avg_hasDerivAt
-    (fun i t => (1+t*v i)*(centeredLog (ray v t) i)^2)
-    (fun i => v i*(centeredLog (ray v t) i)^2+(1+t*v i)*
-      (2*centeredLog (ray v t) i*((invMoment (ray v t) 1-(1+t*v i)⁻¹)/t))) t
-    (fun i => by
-      convert (((hasDerivAt_id t).mul_const (v i)).const_add 1).mul
-        ((centeredLog_hasDerivAt v t ht hp i).pow 2) using 1 <;> first | rfl | norm_num)
-  have hu2 := zero_sum_elim (centeredLog (ray v t)) (centeredLog_avg_zero (ray v t))
-  convert hh using 1
-  · rfl
-  · simp only [weightedLogVariance, logVariance, logMoment, invMoment, avg_expand, pow_one]
-    simp only [hu2, ray]
-    field_simp [hp 0, hp 1, hp 2]
-    ring
-
-lemma logInvMoment_one_hasDerivAt (v : Fin 3 → ℝ) (t : ℝ) (ht : t ≠ 0)
-    (hp : ∀ i, 1+t*v i ≠ 0) :
-    HasDerivAt (fun t => logInvMoment (ray v t) 1)
-      (((invMoment (ray v t) 1)^2-invMoment (ray v t) 2-
-        logInvMoment (ray v t) 1+logInvMoment (ray v t) 2)/t) t := by
-  have hh := avg_hasDerivAt
-    (fun i t => centeredLog (ray v t) i*(1+t*v i)⁻¹)
-    (fun i => ((invMoment (ray v t) 1-(1+t*v i)⁻¹)/t)*(1+t*v i)⁻¹+
-      centeredLog (ray v t) i*(-v i/(1+t*v i)^2)) t
-    (fun i => (centeredLog_hasDerivAt v t ht hp i).mul (ray_inv_hasDerivAt v t hp i))
-  convert hh using 1
-  · simp only [logInvMoment, ray, pow_one]
-  · simp only [logInvMoment, invMoment, avg_expand, pow_one, ray]
-    field_simp [hp 0, hp 1, hp 2]
-    ring
-
-lemma X_hasDerivAt (v : Fin 3 → ℝ) (t : ℝ) :
-    HasDerivAt (fun t => momentX (ray v t)) (2*t*momentX v) t := by
-  simp_rw [X_ray]
-  convert ((hasDerivAt_id t).pow 2).mul_const (momentX v) using 1 <;> first | rfl | norm_num
-
-lemma energy_hasDerivAt (v : Fin 3 → ℝ) (hv : avg v = 0) (t : ℝ) (ht : t ≠ 0)
-    (hp : ∀ i, 1+t*v i ≠ 0) :
-    HasDerivAt (fun t => energy (ray v t))
-      ((gapY (ray v t)+2*energy (ray v t)+(momentX (ray v t))^2)/t) t := by
-  have hh := (((logAvg_hasDerivAt v t ht hp).const_mul 3).add
-    ((logMoment_hasDerivAt v hv t ht hp).const_mul 2)).add
-    ((weightedLogVariance_hasDerivAt v t ht hp).const_mul (2/5 : ℝ))
-  convert hh using 1 <;> first | rfl | (simp only [gapY, energy]; ring)
-
-lemma gapY_hasDerivAt (v : Fin 3 → ℝ) (hv : avg v = 0) (t : ℝ) (ht : t ≠ 0)
-    (hp : ∀ i, 1+t*v i ≠ 0) :
-    HasDerivAt (fun t => gapY (ray v t))
-      ((gapZ (ray v t)+gapY (ray v t))/t) t := by
-  have hg := logAvg_hasDerivAt v t ht hp
-  have hm := logMoment_hasDerivAt v hv t ht hp
-  have hu := invMoment_one_hasDerivAt v t ht hp
-  have hvv := weightedLogVariance_hasDerivAt v t ht hp
-  have hw := logVariance_hasDerivAt v t ht hp
-  have hx := X_hasDerivAt v t
-  have hh := (((((((hasDerivAt_const t (1 : ℝ)).sub hu).sub (hg.const_mul 6)).sub
-    (hm.const_mul 2)).sub (hvv.const_mul (2/5 : ℝ))).sub (hw.const_mul (2/5 : ℝ))).add
-    ((hu.const_mul (4/5 : ℝ)).mul hm)).sub (hx.pow 2)
-  convert hh using 1 <;> first | rfl | (simp only [gapY, gapZ, X_ray]; field_simp; ring)
-
-lemma gapZ_hasDerivAt (v : Fin 3 → ℝ) (hv : avg v = 0) (t : ℝ) (ht : t ≠ 0)
-    (hp : ∀ i, 1+t*v i ≠ 0) :
-    HasDerivAt (fun t => gapZ (ray v t))
-      ((gExpr (ray v t)-12*(momentX (ray v t))^2)/t) t := by
-  have hg := logAvg_hasDerivAt v t ht hp
-  have hm := logMoment_hasDerivAt v hv t ht hp
-  have hu := invMoment_one_hasDerivAt v t ht hp
-  have hu2 := invMoment_two_hasDerivAt v t ht hp
-  have hw := logVariance_hasDerivAt v t ht hp
-  have hs := logInvMoment_one_hasDerivAt v t ht hp
-  have hx := X_hasDerivAt v t
-  have hh := ((((((((hg.const_mul 6).sub_const 5).add (hu.const_mul (26/5 : ℝ))).sub hu2).add
-    (hw.const_mul (4/5 : ℝ))).add
-    (((hu2.sub (hu.const_mul 2)).const_mul (4/5 : ℝ)).mul hm)).add
-    (hs.const_mul (4/5 : ℝ))).add ((hu.pow 2).const_mul (4/5 : ℝ))).sub
-    ((hx.pow 2).const_mul 3)
-  convert hh using 1 <;> first | rfl | (simp only [gExpr, X_ray, Pi.sub_apply]; field_simp; ring)
-
-end ThreeState.Radial
-namespace ThreeState.Radial
-
-@[simp] lemma ray_zero (v : Fin 3 → ℝ) : ray v 0 = 0 := by ext i; simp [ray]
-@[simp] lemma ray_one (v : Fin 3 → ℝ) : ray v 1 = v := by ext i; simp [ray]
-@[simp] lemma logAvg_zero : logAvg 0 = 0 := by norm_num [logAvg, avg_expand]
-@[simp] lemma centeredLog_zero (i : Fin 3) : centeredLog 0 i = 0 := by
-  norm_num [centeredLog, avg_expand]
-@[simp] lemma logMoment_zero : logMoment 0 = 0 := by norm_num [logMoment, avg_expand]
-@[simp] lemma logVariance_zero : logVariance 0 = 0 := by norm_num [logVariance, avg_expand]
-@[simp] lemma weightedLogVariance_zero : weightedLogVariance 0 = 0 := by
-  norm_num [weightedLogVariance, avg_expand]
-@[simp] lemma invMoment_zero (n : ℕ) : invMoment 0 n = 1 := by
-  norm_num [invMoment, avg_expand]
-@[simp] lemma logInvMoment_zero (n : ℕ) : logInvMoment 0 n = 0 := by
-  norm_num [logInvMoment, avg_expand]
-@[simp] lemma momentX_zero : momentX 0 = 0 := by norm_num [momentX, avg_expand]
-@[simp] lemma energy_zero : energy 0 = 0 := by norm_num [energy]
-@[simp] lemma gapY_zero : gapY 0 = 0 := by norm_num [gapY]
-@[simp] lemma gapZ_zero : gapZ 0 = 0 := by norm_num [gapZ]
-
-lemma logAvg_hasDerivAt_zero (v : Fin 3 → ℝ) (hv : avg v = 0) :
-    HasDerivAt (fun t => logAvg (ray v t)) 0 0 := by
-  have hh := avg_hasDerivAt (fun i t => Real.log (1+t*v i)) v 0 (fun i => by
-    simpa using ray_log_hasDerivAt v 0 (by simp) i)
-  simpa [logAvg, ray, hv] using hh
-
-lemma centeredLog_hasDerivAt_zero (v : Fin 3 → ℝ) (hv : avg v = 0) (i : Fin 3) :
-    HasDerivAt (fun t => centeredLog (ray v t) i) (v i) 0 := by
-  have hh := (ray_log_hasDerivAt v 0 (by simp) i).sub (logAvg_hasDerivAt_zero v hv)
-  convert hh using 1 <;> first | rfl | simp
-
-lemma invMoment_one_hasDerivAt_zero (v : Fin 3 → ℝ) (hv : avg v = 0) :
-    HasDerivAt (fun t => invMoment (ray v t) 1) 0 0 := by
-  have hh := avg_hasDerivAt (fun i t => (1+t*v i)⁻¹) (fun i => -v i) 0
-    (fun i => by simpa using ray_inv_hasDerivAt v 0 (by simp) i)
-  have hz : avg (fun i => -v i) = 0 := by simp only [avg_expand] at *; linarith
-  simpa [invMoment, ray, hz] using hh
-
-lemma logMoment_hasDerivAt_zero (v : Fin 3 → ℝ) (hv : avg v = 0) :
-    HasDerivAt (fun t => logMoment (ray v t)) 0 0 := by
-  have hh := avg_hasDerivAt (fun i t => t*v i*centeredLog (ray v t) i) (fun _ => 0) 0
-    (fun i => by
-      convert ((hasDerivAt_id (0 : ℝ)).mul_const (v i)).mul
-        (centeredLog_hasDerivAt_zero v hv i) using 1 <;> first | rfl | simp)
-  simpa [logMoment, ray, avg_expand, centeredLog] using hh
-
-lemma logVariance_hasDerivAt_zero (v : Fin 3 → ℝ) (hv : avg v = 0) :
-    HasDerivAt (fun t => logVariance (ray v t)) 0 0 := by
-  have hh := avg_hasDerivAt (fun i t => (centeredLog (ray v t) i)^2) (fun _ => 0) 0
-    (fun i => by
-      convert (centeredLog_hasDerivAt_zero v hv i).pow 2 using 1; first | rfl | simp)
-  simpa [logVariance, avg_expand] using hh
-
-lemma weightedLogVariance_hasDerivAt_zero (v : Fin 3 → ℝ) (hv : avg v = 0) :
-    HasDerivAt (fun t => weightedLogVariance (ray v t)) 0 0 := by
-  have hh := avg_hasDerivAt (fun i t => (1+t*v i)*(centeredLog (ray v t) i)^2) (fun _ => 0) 0
-    (fun i => by
-      convert (((hasDerivAt_id (0 : ℝ)).mul_const (v i)).const_add 1).mul
-        ((centeredLog_hasDerivAt_zero v hv i).pow 2) using 1 <;> first | rfl | simp)
-  simpa [weightedLogVariance, ray, avg_expand, centeredLog] using hh
-
-lemma gapY_hasDerivAt_zero (v : Fin 3 → ℝ) (hv : avg v = 0) :
-    HasDerivAt (fun t => gapY (ray v t)) 0 0 := by
-  have hu := invMoment_one_hasDerivAt_zero v hv
-  have hm := logMoment_hasDerivAt_zero v hv
-  have hh := (((((((hasDerivAt_const (0 : ℝ) (1 : ℝ)).sub hu).sub
-    ((logAvg_hasDerivAt_zero v hv).const_mul 6)).sub (hm.const_mul 2)).sub
-    ((weightedLogVariance_hasDerivAt_zero v hv).const_mul (2/5 : ℝ))).sub
-    ((logVariance_hasDerivAt_zero v hv).const_mul (2/5 : ℝ))).add
-    ((hu.const_mul (4/5 : ℝ)).mul hm)).sub ((X_hasDerivAt v 0).pow 2)
-  convert hh using 1 <;> first | rfl | simp
-
-lemma logAvg_continuousAt (v : Fin 3 → ℝ) (t : ℝ) (hp : ∀ i, 1+t*v i ≠ 0) :
-    ContinuousAt (fun t => logAvg (ray v t)) t := by
-  have hh (i : Fin 3) := (ray_log_hasDerivAt v t hp i).continuousAt
-  simp only [logAvg, avg_expand, ray]
-  exact (((hh 0).add (hh 1)).add (hh 2)).div_const 3
-
-lemma centeredLog_continuousAt (v : Fin 3 → ℝ) (t : ℝ)
-    (hp : ∀ i, 1+t*v i ≠ 0) (i : Fin 3) :
-    ContinuousAt (fun t => centeredLog (ray v t) i) t :=
-  ((ray_log_hasDerivAt v t hp i).continuousAt).sub (logAvg_continuousAt v t hp)
-
-lemma gapZ_continuousAt (v : Fin 3 → ℝ) (t : ℝ) (hp : ∀ i, 1+t*v i ≠ 0) :
-    ContinuousAt (fun t => gapZ (ray v t)) t := by
-  have hg := logAvg_continuousAt v t hp
-  have hu (i : Fin 3) := centeredLog_continuousAt v t hp i
-  have hinv (i : Fin 3) := (ray_inv_hasDerivAt v t hp i).continuousAt
-  unfold gapZ invMoment logMoment logVariance logInvMoment momentX
-  simp only [avg_expand, ray]
-  fun_prop
-
-lemma normalized_gapY_continuousAt_zero (v : Fin 3 → ℝ) (hv : avg v = 0) :
-    ContinuousAt (fun t => gapY (ray v t)/t) 0 := by
-  have hh := (gapY_hasDerivAt_zero v hv).continuousAt_div
-  simp only [ray_zero, gapY_zero, sub_zero] at hh
-  have heq : Function.update (fun t => gapY (ray v t)/t) 0 0 =
-      (fun t => gapY (ray v t)/t) := by
-    ext t
-    by_cases ht : t = 0
-    · simp [ht]
-    · simp [Function.update, ht]
-  rwa [heq] at hh
-
-lemma gapZ_nonneg (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i)
-    (t : ℝ) (ht0 : 0 ≤ t) (ht1 : t ≤ 1) : 0 ≤ gapZ (ray v t) := by
-  have hmono : MonotoneOn (fun t => gapZ (ray v t)) (Set.Icc (0 : ℝ) 1) := by
-    apply monotoneOn_of_hasDerivWithinAt_nonneg (f' := fun t =>
-      (gExpr (ray v t)-12*(momentX (ray v t))^2)/t) (convex_Icc _ _)
-    · intro s hs
-      exact (gapZ_continuousAt v s (fun i =>
-        ne_of_gt (message_segment_pos v hp s hs.1 hs.2 i))).continuousWithinAt
-    · intro s hs
-      rw [interior_Icc] at hs
-      exact (gapZ_hasDerivAt v hv s hs.1.ne' (fun i =>
-        ne_of_gt (message_segment_pos v hp s hs.1.le hs.2.le i))).hasDerivWithinAt
-    · intro s hs
-      rw [interior_Icc] at hs
-      apply div_nonneg _ hs.1.le
-      apply sub_nonneg.mpr
-      apply gExpr_lower (ray v s)
-      · rw [avg_ray, hv, mul_zero]
-      · exact fun i => message_segment_pos v hp s hs.1.le hs.2.le i
-  simpa using hmono (by simp) ⟨ht0, ht1⟩ ht0
-
-lemma normalized_gapY_hasDerivAt (v : Fin 3 → ℝ) (hv : avg v = 0)
-    (t : ℝ) (ht : t ≠ 0) (hp : ∀ i, 1+t*v i ≠ 0) :
-    HasDerivAt (fun t => gapY (ray v t)/t) (gapZ (ray v t)/t^2) t := by
-  have hh := (gapY_hasDerivAt v hv t ht hp).div (hasDerivAt_id t) ht
-  convert hh using 1 <;> first | rfl | (simp only [id_eq]; field_simp; ring)
-
-lemma gapY_nonneg (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i)
-    (t : ℝ) (ht0 : 0 ≤ t) (ht1 : t ≤ 1) : 0 ≤ gapY (ray v t) := by
-  have hmono : MonotoneOn (fun t => gapY (ray v t)/t) (Set.Icc (0 : ℝ) 1) := by
-    apply monotoneOn_of_hasDerivWithinAt_nonneg (f' := fun t =>
-      gapZ (ray v t)/t^2) (convex_Icc _ _)
-    · intro s hs
-      rcases eq_or_lt_of_le hs.1 with h | h
-      · rw [← h]
-        exact (normalized_gapY_continuousAt_zero v hv).continuousWithinAt
-      · exact (normalized_gapY_hasDerivAt v hv s h.ne' (fun i =>
-          ne_of_gt (message_segment_pos v hp s hs.1 hs.2 i))).continuousAt.continuousWithinAt
-    · intro s hs
-      rw [interior_Icc] at hs
-      exact (normalized_gapY_hasDerivAt v hv s hs.1.ne' (fun i =>
-        ne_of_gt (message_segment_pos v hp s hs.1.le hs.2.le i))).hasDerivWithinAt
-    · intro s hs
-      rw [interior_Icc] at hs
-      exact div_nonneg (gapZ_nonneg v hv hp s hs.1.le hs.2.le) (sq_nonneg s)
-  have h := hmono (by simp) ⟨ht0, ht1⟩ ht0
-  simp only [div_zero] at h
-  rcases eq_or_lt_of_le ht0 with h0 | h0
-  · simp [← h0]
-  · simpa using (le_div_iff₀ h0).mp h
-
-lemma normalized_energy_hasDerivAt (v : Fin 3 → ℝ) (hv : avg v = 0)
-    (t : ℝ) (ht : t ≠ 0) (hp : ∀ i, 1+t*v i ≠ 0) :
-    HasDerivAt (fun t => (energy (ray v t)-(1/2 : ℝ)*(momentX (ray v t))^2)/t^2)
-      (gapY (ray v t)/t^3) t := by
-  have hh := ((energy_hasDerivAt v hv t ht hp).sub
-    (((X_hasDerivAt v t).pow 2).const_mul (1/2 : ℝ))).div ((hasDerivAt_id t).pow 2)
-    (pow_ne_zero 2 ht)
-  convert hh using 1 <;> first | rfl | (simp only [X_ray, id_eq, Pi.pow_apply, Pi.sub_apply]; field_simp; ring)
-
- 
-theorem radial_inequality (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i)
-    (l : ℝ) (hl0 : 0 < l) (hl1 : l < 1) :
-    (1-l^2)/2*(momentX v)^2 ≤ energy v-l⁻¹^2*energy (ray v l) := by
-  have hmono : MonotoneOn
-      (fun t => (energy (ray v t)-(1/2 : ℝ)*(momentX (ray v t))^2)/t^2)
-      (Set.Icc l 1) := by
-    apply monotoneOn_of_hasDerivWithinAt_nonneg (f' := fun t =>
-      gapY (ray v t)/t^3) (convex_Icc _ _)
-    · intro s hs
-      exact (normalized_energy_hasDerivAt v hv s (lt_of_lt_of_le hl0 hs.1).ne' (fun i =>
-        ne_of_gt (message_segment_pos v hp s (le_trans hl0.le hs.1) hs.2 i))).continuousAt.continuousWithinAt
-    · intro s hs
-      rw [interior_Icc] at hs
-      exact (normalized_energy_hasDerivAt v hv s (lt_trans hl0 hs.1).ne' (fun i =>
-        ne_of_gt (message_segment_pos v hp s (lt_trans hl0 hs.1).le hs.2.le i))).hasDerivWithinAt
-    · intro s hs
-      rw [interior_Icc] at hs
-      exact div_nonneg (gapY_nonneg v hv hp s (lt_trans hl0 hs.1).le hs.2.le)
-        (pow_nonneg (lt_trans hl0 hs.1).le 3)
-  have hh := hmono ⟨le_rfl, hl1.le⟩ ⟨hl1.le, le_rfl⟩ hl1.le
-  simp only [ray_one, X_ray, one_pow, div_one] at hh
-  have hl : l ≠ 0 := hl0.ne'
-  have hsq : 0 < l^2 := sq_pos_of_pos hl0
-  rw [div_le_iff₀ hsq] at hh
-  apply (mul_le_mul_iff_left₀ hsq).mp
-  field_simp [hl]
-  nlinarith [hh]
-
-end ThreeState.Radial
-namespace ThreeState.Radial
-
- 
-noncomputable def sourceF (m : Fin 3 → ℝ) : ℝ :=
-  3*avg (fun i => Real.log (m i)) +
-  4*((1/2 : ℝ)*avg (fun i => (m i-1)*Real.log (m i))) +
-  (4/5 : ℝ)*((1/2 : ℝ)*avg (fun i => m i*
-    (Real.log (m i)-avg (fun j => Real.log (m j)))^2))
-
-lemma avg_message_deviation (m : Fin 3 → ℝ) (hm : avg m = 1) :
-    avg (fun i => m i-1) = 0 := by
-  simp only [avg_expand] at *
-  linarith
-
-lemma logMoment_uncentered (v : Fin 3 → ℝ) (hv : avg v = 0) :
-    logMoment v = avg (fun i => v i*Real.log (1+v i)) := by
-  have hv2 := zero_sum_elim v hv
-  simp only [logMoment, centeredLog, avg_expand, hv2]
-  ring
-
-lemma energy_sourceF (m : Fin 3 → ℝ) (hm : avg m = 1) :
-    energy (fun i => m i-1) = sourceF m := by
-  rw [energy, logMoment_uncentered _ (avg_message_deviation m hm)]
-  simp only [logAvg, weightedLogVariance, centeredLog, sourceF]
-  have he (i : Fin 3) : 1+(m i-1) = m i := by ring
-  simp only [he]
-  ring
-
- 
-theorem radial_source (m : Fin 3 → ℝ) (hm : avg m = 1) (hp : ∀ i, 0 < m i)
-    (l : ℝ) (hl0 : 0 < l) (hl1 : l < 1) :
-    sourceF m-l⁻¹^2*sourceF (fun i => 1+l*(m i-1)) ≥
-      (1-l^2)/2*((1/2 : ℝ)*avg (fun i => (m i-1)^2))^2 := by
-  have hv := avg_message_deviation m hm
-  have h := radial_inequality (fun i => m i-1) hv (by simpa using hp) l hl0 hl1
-  have he : avg (fun i => 1+l*(m i-1)) = 1 := by
-    simp only [avg_expand] at *
-    nlinarith [hm]
-  have hr : ray (fun i => m i-1) l = (fun i => (1+l*(m i-1))-1) := by
-    ext i; simp [ray]
-  rw [energy_sourceF m hm, hr, energy_sourceF _ he] at h
-  convert h using 1
-  simp only [momentX]
-  ring
-
-end ThreeState.Radial
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeState.RadialMoments
+
+namespace OAI
+
+/-! Radial logarithmic derivatives and the entropy inequality. -/
+
+namespace ThreeState.Radial
+
+noncomputable def ray (v : Fin 3 → ℝ) (t : ℝ) : Fin 3 → ℝ := fun i => t*v i
+noncomputable def logAvg (v : Fin 3 → ℝ) := avg (fun i => Real.log (1+v i))
+noncomputable def logVariance (v : Fin 3 → ℝ) := avg (fun i => (centeredLog v i)^2)
+noncomputable def weightedLogVariance (v : Fin 3 → ℝ) :=
+  avg (fun i => (1+v i)*(centeredLog v i)^2)
+ 
+noncomputable def energy (v : Fin 3 → ℝ) :=
+  3*logAvg v+2*logMoment v+(2/5 : ℝ)*weightedLogVariance v
+ 
+noncomputable def gapY (v : Fin 3 → ℝ) :=
+  1-invMoment v 1-6*logAvg v-2*logMoment v-(2/5 : ℝ)*weightedLogVariance v-
+    (2/5 : ℝ)*logVariance v+(4/5 : ℝ)*invMoment v 1*logMoment v-(momentX v)^2
+ 
+noncomputable def gapZ (v : Fin 3 → ℝ) :=
+  6*logAvg v-5+(26/5 : ℝ)*invMoment v 1-invMoment v 2+
+    (4/5 : ℝ)*logVariance v+(4/5 : ℝ)*(invMoment v 2-2*invMoment v 1)*logMoment v+
+    (4/5 : ℝ)*logInvMoment v 1+(4/5 : ℝ)*(invMoment v 1)^2-3*(momentX v)^2
+
+lemma avg_hasDerivAt (f : Fin 3 → ℝ → ℝ) (f' : Fin 3 → ℝ) (t : ℝ)
+    (hf : ∀ i, HasDerivAt (f i) (f' i) t) :
+    HasDerivAt (fun t => avg (fun i => f i t)) (avg f') t := by
+  simp only [avg_expand]
+  exact (((hf 0).add (hf 1)).add (hf 2)).div_const 3
+
+lemma avg_ray (v : Fin 3 → ℝ) (t : ℝ) : avg (ray v t) = t*avg v := by
+  simp only [avg_expand, ray]; ring
+lemma X_ray (v : Fin 3 → ℝ) (t : ℝ) : momentX (ray v t) = t^2*momentX v := by
+  simp only [momentX, avg_expand, ray]; ring
+lemma centeredLog_avg_zero (v : Fin 3 → ℝ) : avg (centeredLog v) = 0 := by
+  simp only [centeredLog, avg_expand]; ring
+
+lemma ray_log_hasDerivAt (v : Fin 3 → ℝ) (t : ℝ) (hp : ∀ i, 1+t*v i ≠ 0)
+    (i : Fin 3) : HasDerivAt (fun t => Real.log (1+t*v i)) (v i/(1+t*v i)) t := by
+  simpa using (((hasDerivAt_id t).mul_const (v i)).const_add 1).log (hp i)
+lemma ray_inv_hasDerivAt (v : Fin 3 → ℝ) (t : ℝ) (hp : ∀ i, 1+t*v i ≠ 0)
+    (i : Fin 3) : HasDerivAt (fun t => (1+t*v i)⁻¹) (-v i/(1+t*v i)^2) t := by
+  convert (((hasDerivAt_id t).mul_const (v i)).const_add 1).inv (hp i) using 1 <;>
+    first | rfl | norm_num
+
+lemma logAvg_hasDerivAt (v : Fin 3 → ℝ) (t : ℝ) (ht : t ≠ 0)
+    (hp : ∀ i, 1+t*v i ≠ 0) :
+    HasDerivAt (fun t => logAvg (ray v t)) ((1-invMoment (ray v t) 1)/t) t := by
+  have hh := avg_hasDerivAt (fun i t => Real.log (1+t*v i)) (fun i => v i/(1+t*v i)) t
+    (ray_log_hasDerivAt v t hp)
+  convert hh using 1
+  · rfl
+  · simp only [invMoment, avg_expand, ray, pow_one]
+    field_simp [hp 0, hp 1, hp 2]
+    ring
+
+lemma centeredLog_hasDerivAt (v : Fin 3 → ℝ) (t : ℝ) (ht : t ≠ 0)
+    (hp : ∀ i, 1+t*v i ≠ 0) (i : Fin 3) :
+    HasDerivAt (fun t => centeredLog (ray v t) i)
+      ((invMoment (ray v t) 1-(1+t*v i)⁻¹)/t) t := by
+  have hh := (ray_log_hasDerivAt v t hp i).sub (logAvg_hasDerivAt v t ht hp)
+  convert hh using 1 <;> first | rfl | (field_simp [hp i]; ring)
+
+lemma invMoment_one_hasDerivAt (v : Fin 3 → ℝ) (t : ℝ) (ht : t ≠ 0)
+    (hp : ∀ i, 1+t*v i ≠ 0) :
+    HasDerivAt (fun t => invMoment (ray v t) 1)
+      ((invMoment (ray v t) 2-invMoment (ray v t) 1)/t) t := by
+  have hh := avg_hasDerivAt (fun i t => (1+t*v i)⁻¹)
+    (fun i => -v i/(1+t*v i)^2) t (ray_inv_hasDerivAt v t hp)
+  convert hh using 1
+  · simp only [invMoment, ray, pow_one]
+  · simp only [invMoment, avg_expand, ray, pow_one]
+    field_simp [hp 0, hp 1, hp 2]
+    ring
+
+lemma invMoment_two_hasDerivAt (v : Fin 3 → ℝ) (t : ℝ) (ht : t ≠ 0)
+    (hp : ∀ i, 1+t*v i ≠ 0) :
+    HasDerivAt (fun t => invMoment (ray v t) 2)
+      ((2*(invMoment (ray v t) 3-invMoment (ray v t) 2))/t) t := by
+  have hh := avg_hasDerivAt (fun i t => ((1+t*v i)⁻¹)^2)
+    (fun i => 2*(1+t*v i)⁻¹*(-v i/(1+t*v i)^2)) t
+    (fun i => by
+      convert (ray_inv_hasDerivAt v t hp i).pow 2 using 1; first | rfl | norm_num)
+  convert hh using 1
+  · rfl
+  · simp only [invMoment, avg_expand, ray]
+    field_simp [hp 0, hp 1, hp 2]
+    ring
+
+lemma logMoment_hasDerivAt (v : Fin 3 → ℝ) (hv : avg v = 0) (t : ℝ) (ht : t ≠ 0)
+    (hp : ∀ i, 1+t*v i ≠ 0) :
+    HasDerivAt (fun t => logMoment (ray v t))
+      ((logMoment (ray v t)+invMoment (ray v t) 1-1)/t) t := by
+  have hh := avg_hasDerivAt
+    (fun i t => t*v i*centeredLog (ray v t) i)
+    (fun i => v i*centeredLog (ray v t) i+t*v i*((invMoment (ray v t) 1-(1+t*v i)⁻¹)/t)) t
+    (fun i => by
+      convert ((hasDerivAt_id t).mul_const (v i)).mul (centeredLog_hasDerivAt v t ht hp i) using 1 <;>
+        first | rfl | simp)
+  have hv2 := zero_sum_elim v hv
+  convert hh using 1
+  · rfl
+  · simp only [logMoment, invMoment, avg_expand, ray, pow_one]
+    field_simp [hp 0, hp 1, hp 2]
+    rw [hv2]
+    ring
+
+lemma logVariance_hasDerivAt (v : Fin 3 → ℝ) (t : ℝ) (ht : t ≠ 0)
+    (hp : ∀ i, 1+t*v i ≠ 0) :
+    HasDerivAt (fun t => logVariance (ray v t))
+      ((-2*logInvMoment (ray v t) 1)/t) t := by
+  have hh := avg_hasDerivAt
+    (fun i t => (centeredLog (ray v t) i)^2)
+    (fun i => 2*centeredLog (ray v t) i*((invMoment (ray v t) 1-(1+t*v i)⁻¹)/t)) t
+    (fun i => by
+      convert (centeredLog_hasDerivAt v t ht hp i).pow 2 using 1; first | rfl | norm_num)
+  have hu2 := zero_sum_elim (centeredLog (ray v t)) (centeredLog_avg_zero (ray v t))
+  convert hh using 1
+  · rfl
+  · simp only [logInvMoment, invMoment, avg_expand, pow_one]
+    simp only [hu2, ray]
+    ring
+
+lemma weightedLogVariance_hasDerivAt (v : Fin 3 → ℝ) (t : ℝ) (ht : t ≠ 0)
+    (hp : ∀ i, 1+t*v i ≠ 0) :
+    HasDerivAt (fun t => weightedLogVariance (ray v t))
+      ((weightedLogVariance (ray v t)-logVariance (ray v t)+
+        2*invMoment (ray v t) 1*logMoment (ray v t))/t) t := by
+  have hh := avg_hasDerivAt
+    (fun i t => (1+t*v i)*(centeredLog (ray v t) i)^2)
+    (fun i => v i*(centeredLog (ray v t) i)^2+(1+t*v i)*
+      (2*centeredLog (ray v t) i*((invMoment (ray v t) 1-(1+t*v i)⁻¹)/t))) t
+    (fun i => by
+      convert (((hasDerivAt_id t).mul_const (v i)).const_add 1).mul
+        ((centeredLog_hasDerivAt v t ht hp i).pow 2) using 1 <;> first | rfl | norm_num)
+  have hu2 := zero_sum_elim (centeredLog (ray v t)) (centeredLog_avg_zero (ray v t))
+  convert hh using 1
+  · rfl
+  · simp only [weightedLogVariance, logVariance, logMoment, invMoment, avg_expand, pow_one]
+    simp only [hu2, ray]
+    field_simp [hp 0, hp 1, hp 2]
+    ring
+
+lemma logInvMoment_one_hasDerivAt (v : Fin 3 → ℝ) (t : ℝ) (ht : t ≠ 0)
+    (hp : ∀ i, 1+t*v i ≠ 0) :
+    HasDerivAt (fun t => logInvMoment (ray v t) 1)
+      (((invMoment (ray v t) 1)^2-invMoment (ray v t) 2-
+        logInvMoment (ray v t) 1+logInvMoment (ray v t) 2)/t) t := by
+  have hh := avg_hasDerivAt
+    (fun i t => centeredLog (ray v t) i*(1+t*v i)⁻¹)
+    (fun i => ((invMoment (ray v t) 1-(1+t*v i)⁻¹)/t)*(1+t*v i)⁻¹+
+      centeredLog (ray v t) i*(-v i/(1+t*v i)^2)) t
+    (fun i => (centeredLog_hasDerivAt v t ht hp i).mul (ray_inv_hasDerivAt v t hp i))
+  convert hh using 1
+  · simp only [logInvMoment, ray, pow_one]
+  · simp only [logInvMoment, invMoment, avg_expand, pow_one, ray]
+    field_simp [hp 0, hp 1, hp 2]
+    ring
+
+lemma X_hasDerivAt (v : Fin 3 → ℝ) (t : ℝ) :
+    HasDerivAt (fun t => momentX (ray v t)) (2*t*momentX v) t := by
+  simp_rw [X_ray]
+  convert ((hasDerivAt_id t).pow 2).mul_const (momentX v) using 1 <;> first | rfl | norm_num
+
+lemma energy_hasDerivAt (v : Fin 3 → ℝ) (hv : avg v = 0) (t : ℝ) (ht : t ≠ 0)
+    (hp : ∀ i, 1+t*v i ≠ 0) :
+    HasDerivAt (fun t => energy (ray v t))
+      ((gapY (ray v t)+2*energy (ray v t)+(momentX (ray v t))^2)/t) t := by
+  have hh := (((logAvg_hasDerivAt v t ht hp).const_mul 3).add
+    ((logMoment_hasDerivAt v hv t ht hp).const_mul 2)).add
+    ((weightedLogVariance_hasDerivAt v t ht hp).const_mul (2/5 : ℝ))
+  convert hh using 1 <;> first | rfl | (simp only [gapY, energy]; ring)
+
+lemma gapY_hasDerivAt (v : Fin 3 → ℝ) (hv : avg v = 0) (t : ℝ) (ht : t ≠ 0)
+    (hp : ∀ i, 1+t*v i ≠ 0) :
+    HasDerivAt (fun t => gapY (ray v t))
+      ((gapZ (ray v t)+gapY (ray v t))/t) t := by
+  have hg := logAvg_hasDerivAt v t ht hp
+  have hm := logMoment_hasDerivAt v hv t ht hp
+  have hu := invMoment_one_hasDerivAt v t ht hp
+  have hvv := weightedLogVariance_hasDerivAt v t ht hp
+  have hw := logVariance_hasDerivAt v t ht hp
+  have hx := X_hasDerivAt v t
+  have hh := (((((((hasDerivAt_const t (1 : ℝ)).sub hu).sub (hg.const_mul 6)).sub
+    (hm.const_mul 2)).sub (hvv.const_mul (2/5 : ℝ))).sub (hw.const_mul (2/5 : ℝ))).add
+    ((hu.const_mul (4/5 : ℝ)).mul hm)).sub (hx.pow 2)
+  convert hh using 1 <;> first | rfl | (simp only [gapY, gapZ, X_ray]; field_simp; ring)
+
+lemma gapZ_hasDerivAt (v : Fin 3 → ℝ) (hv : avg v = 0) (t : ℝ) (ht : t ≠ 0)
+    (hp : ∀ i, 1+t*v i ≠ 0) :
+    HasDerivAt (fun t => gapZ (ray v t))
+      ((gExpr (ray v t)-12*(momentX (ray v t))^2)/t) t := by
+  have hg := logAvg_hasDerivAt v t ht hp
+  have hm := logMoment_hasDerivAt v hv t ht hp
+  have hu := invMoment_one_hasDerivAt v t ht hp
+  have hu2 := invMoment_two_hasDerivAt v t ht hp
+  have hw := logVariance_hasDerivAt v t ht hp
+  have hs := logInvMoment_one_hasDerivAt v t ht hp
+  have hx := X_hasDerivAt v t
+  have hh := ((((((((hg.const_mul 6).sub_const 5).add (hu.const_mul (26/5 : ℝ))).sub hu2).add
+    (hw.const_mul (4/5 : ℝ))).add
+    (((hu2.sub (hu.const_mul 2)).const_mul (4/5 : ℝ)).mul hm)).add
+    (hs.const_mul (4/5 : ℝ))).add ((hu.pow 2).const_mul (4/5 : ℝ))).sub
+    ((hx.pow 2).const_mul 3)
+  convert hh using 1 <;> first | rfl | (simp only [gExpr, X_ray, Pi.sub_apply]; field_simp; ring)
+
+end ThreeState.Radial
+namespace ThreeState.Radial
+
+@[simp] lemma ray_zero (v : Fin 3 → ℝ) : ray v 0 = 0 := by ext i; simp [ray]
+@[simp] lemma ray_one (v : Fin 3 → ℝ) : ray v 1 = v := by ext i; simp [ray]
+@[simp] lemma logAvg_zero : logAvg 0 = 0 := by norm_num [logAvg, avg_expand]
+@[simp] lemma centeredLog_zero (i : Fin 3) : centeredLog 0 i = 0 := by
+  norm_num [centeredLog, avg_expand]
+@[simp] lemma logMoment_zero : logMoment 0 = 0 := by norm_num [logMoment, avg_expand]
+@[simp] lemma logVariance_zero : logVariance 0 = 0 := by norm_num [logVariance, avg_expand]
+@[simp] lemma weightedLogVariance_zero : weightedLogVariance 0 = 0 := by
+  norm_num [weightedLogVariance, avg_expand]
+@[simp] lemma invMoment_zero (n : ℕ) : invMoment 0 n = 1 := by
+  norm_num [invMoment, avg_expand]
+@[simp] lemma logInvMoment_zero (n : ℕ) : logInvMoment 0 n = 0 := by
+  norm_num [logInvMoment, avg_expand]
+@[simp] lemma momentX_zero : momentX 0 = 0 := by norm_num [momentX, avg_expand]
+@[simp] lemma energy_zero : energy 0 = 0 := by norm_num [energy]
+@[simp] lemma gapY_zero : gapY 0 = 0 := by norm_num [gapY]
+@[simp] lemma gapZ_zero : gapZ 0 = 0 := by norm_num [gapZ]
+
+lemma logAvg_hasDerivAt_zero (v : Fin 3 → ℝ) (hv : avg v = 0) :
+    HasDerivAt (fun t => logAvg (ray v t)) 0 0 := by
+  have hh := avg_hasDerivAt (fun i t => Real.log (1+t*v i)) v 0 (fun i => by
+    simpa using ray_log_hasDerivAt v 0 (by simp) i)
+  simpa [logAvg, ray, hv] using hh
+
+lemma centeredLog_hasDerivAt_zero (v : Fin 3 → ℝ) (hv : avg v = 0) (i : Fin 3) :
+    HasDerivAt (fun t => centeredLog (ray v t) i) (v i) 0 := by
+  have hh := (ray_log_hasDerivAt v 0 (by simp) i).sub (logAvg_hasDerivAt_zero v hv)
+  convert hh using 1 <;> first | rfl | simp
+
+lemma invMoment_one_hasDerivAt_zero (v : Fin 3 → ℝ) (hv : avg v = 0) :
+    HasDerivAt (fun t => invMoment (ray v t) 1) 0 0 := by
+  have hh := avg_hasDerivAt (fun i t => (1+t*v i)⁻¹) (fun i => -v i) 0
+    (fun i => by simpa using ray_inv_hasDerivAt v 0 (by simp) i)
+  have hz : avg (fun i => -v i) = 0 := by simp only [avg_expand] at *; linarith
+  simpa [invMoment, ray, hz] using hh
+
+lemma logMoment_hasDerivAt_zero (v : Fin 3 → ℝ) (hv : avg v = 0) :
+    HasDerivAt (fun t => logMoment (ray v t)) 0 0 := by
+  have hh := avg_hasDerivAt (fun i t => t*v i*centeredLog (ray v t) i) (fun _ => 0) 0
+    (fun i => by
+      convert ((hasDerivAt_id (0 : ℝ)).mul_const (v i)).mul
+        (centeredLog_hasDerivAt_zero v hv i) using 1 <;> first | rfl | simp)
+  simpa [logMoment, ray, avg_expand, centeredLog] using hh
+
+lemma logVariance_hasDerivAt_zero (v : Fin 3 → ℝ) (hv : avg v = 0) :
+    HasDerivAt (fun t => logVariance (ray v t)) 0 0 := by
+  have hh := avg_hasDerivAt (fun i t => (centeredLog (ray v t) i)^2) (fun _ => 0) 0
+    (fun i => by
+      convert (centeredLog_hasDerivAt_zero v hv i).pow 2 using 1; first | rfl | simp)
+  simpa [logVariance, avg_expand] using hh
+
+lemma weightedLogVariance_hasDerivAt_zero (v : Fin 3 → ℝ) (hv : avg v = 0) :
+    HasDerivAt (fun t => weightedLogVariance (ray v t)) 0 0 := by
+  have hh := avg_hasDerivAt (fun i t => (1+t*v i)*(centeredLog (ray v t) i)^2) (fun _ => 0) 0
+    (fun i => by
+      convert (((hasDerivAt_id (0 : ℝ)).mul_const (v i)).const_add 1).mul
+        ((centeredLog_hasDerivAt_zero v hv i).pow 2) using 1 <;> first | rfl | simp)
+  simpa [weightedLogVariance, ray, avg_expand, centeredLog] using hh
+
+lemma gapY_hasDerivAt_zero (v : Fin 3 → ℝ) (hv : avg v = 0) :
+    HasDerivAt (fun t => gapY (ray v t)) 0 0 := by
+  have hu := invMoment_one_hasDerivAt_zero v hv
+  have hm := logMoment_hasDerivAt_zero v hv
+  have hh := (((((((hasDerivAt_const (0 : ℝ) (1 : ℝ)).sub hu).sub
+    ((logAvg_hasDerivAt_zero v hv).const_mul 6)).sub (hm.const_mul 2)).sub
+    ((weightedLogVariance_hasDerivAt_zero v hv).const_mul (2/5 : ℝ))).sub
+    ((logVariance_hasDerivAt_zero v hv).const_mul (2/5 : ℝ))).add
+    ((hu.const_mul (4/5 : ℝ)).mul hm)).sub ((X_hasDerivAt v 0).pow 2)
+  convert hh using 1 <;> first | rfl | simp
+
+lemma logAvg_continuousAt (v : Fin 3 → ℝ) (t : ℝ) (hp : ∀ i, 1+t*v i ≠ 0) :
+    ContinuousAt (fun t => logAvg (ray v t)) t := by
+  have hh (i : Fin 3) := (ray_log_hasDerivAt v t hp i).continuousAt
+  simp only [logAvg, avg_expand, ray]
+  exact (((hh 0).add (hh 1)).add (hh 2)).div_const 3
+
+lemma centeredLog_continuousAt (v : Fin 3 → ℝ) (t : ℝ)
+    (hp : ∀ i, 1+t*v i ≠ 0) (i : Fin 3) :
+    ContinuousAt (fun t => centeredLog (ray v t) i) t :=
+  ((ray_log_hasDerivAt v t hp i).continuousAt).sub (logAvg_continuousAt v t hp)
+
+lemma gapZ_continuousAt (v : Fin 3 → ℝ) (t : ℝ) (hp : ∀ i, 1+t*v i ≠ 0) :
+    ContinuousAt (fun t => gapZ (ray v t)) t := by
+  have hg := logAvg_continuousAt v t hp
+  have hu (i : Fin 3) := centeredLog_continuousAt v t hp i
+  have hinv (i : Fin 3) := (ray_inv_hasDerivAt v t hp i).continuousAt
+  unfold gapZ invMoment logMoment logVariance logInvMoment momentX
+  simp only [avg_expand, ray]
+  fun_prop
+
+lemma normalized_gapY_continuousAt_zero (v : Fin 3 → ℝ) (hv : avg v = 0) :
+    ContinuousAt (fun t => gapY (ray v t)/t) 0 := by
+  have hh := (gapY_hasDerivAt_zero v hv).continuousAt_div
+  simp only [ray_zero, gapY_zero, sub_zero] at hh
+  have heq : Function.update (fun t => gapY (ray v t)/t) 0 0 =
+      (fun t => gapY (ray v t)/t) := by
+    ext t
+    by_cases ht : t = 0
+    · simp [ht]
+    · simp [Function.update, ht]
+  rwa [heq] at hh
+
+lemma gapZ_nonneg (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i)
+    (t : ℝ) (ht0 : 0 ≤ t) (ht1 : t ≤ 1) : 0 ≤ gapZ (ray v t) := by
+  have hmono : MonotoneOn (fun t => gapZ (ray v t)) (Set.Icc (0 : ℝ) 1) := by
+    apply monotoneOn_of_hasDerivWithinAt_nonneg (f' := fun t =>
+      (gExpr (ray v t)-12*(momentX (ray v t))^2)/t) (convex_Icc _ _)
+    · intro s hs
+      exact (gapZ_continuousAt v s (fun i =>
+        ne_of_gt (message_segment_pos v hp s hs.1 hs.2 i))).continuousWithinAt
+    · intro s hs
+      rw [interior_Icc] at hs
+      exact (gapZ_hasDerivAt v hv s hs.1.ne' (fun i =>
+        ne_of_gt (message_segment_pos v hp s hs.1.le hs.2.le i))).hasDerivWithinAt
+    · intro s hs
+      rw [interior_Icc] at hs
+      apply div_nonneg _ hs.1.le
+      apply sub_nonneg.mpr
+      apply gExpr_lower (ray v s)
+      · rw [avg_ray, hv, mul_zero]
+      · exact fun i => message_segment_pos v hp s hs.1.le hs.2.le i
+  simpa using hmono (by simp) ⟨ht0, ht1⟩ ht0
+
+lemma normalized_gapY_hasDerivAt (v : Fin 3 → ℝ) (hv : avg v = 0)
+    (t : ℝ) (ht : t ≠ 0) (hp : ∀ i, 1+t*v i ≠ 0) :
+    HasDerivAt (fun t => gapY (ray v t)/t) (gapZ (ray v t)/t^2) t := by
+  have hh := (gapY_hasDerivAt v hv t ht hp).div (hasDerivAt_id t) ht
+  convert hh using 1 <;> first | rfl | (simp only [id_eq]; field_simp; ring)
+
+lemma gapY_nonneg (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i)
+    (t : ℝ) (ht0 : 0 ≤ t) (ht1 : t ≤ 1) : 0 ≤ gapY (ray v t) := by
+  have hmono : MonotoneOn (fun t => gapY (ray v t)/t) (Set.Icc (0 : ℝ) 1) := by
+    apply monotoneOn_of_hasDerivWithinAt_nonneg (f' := fun t =>
+      gapZ (ray v t)/t^2) (convex_Icc _ _)
+    · intro s hs
+      rcases eq_or_lt_of_le hs.1 with h | h
+      · rw [← h]
+        exact (normalized_gapY_continuousAt_zero v hv).continuousWithinAt
+      · exact (normalized_gapY_hasDerivAt v hv s h.ne' (fun i =>
+          ne_of_gt (message_segment_pos v hp s hs.1 hs.2 i))).continuousAt.continuousWithinAt
+    · intro s hs
+      rw [interior_Icc] at hs
+      exact (normalized_gapY_hasDerivAt v hv s hs.1.ne' (fun i =>
+        ne_of_gt (message_segment_pos v hp s hs.1.le hs.2.le i))).hasDerivWithinAt
+    · intro s hs
+      rw [interior_Icc] at hs
+      exact div_nonneg (gapZ_nonneg v hv hp s hs.1.le hs.2.le) (sq_nonneg s)
+  have h := hmono (by simp) ⟨ht0, ht1⟩ ht0
+  simp only [div_zero] at h
+  rcases eq_or_lt_of_le ht0 with h0 | h0
+  · simp [← h0]
+  · simpa using (le_div_iff₀ h0).mp h
+
+lemma normalized_energy_hasDerivAt (v : Fin 3 → ℝ) (hv : avg v = 0)
+    (t : ℝ) (ht : t ≠ 0) (hp : ∀ i, 1+t*v i ≠ 0) :
+    HasDerivAt (fun t => (energy (ray v t)-(1/2 : ℝ)*(momentX (ray v t))^2)/t^2)
+      (gapY (ray v t)/t^3) t := by
+  have hh := ((energy_hasDerivAt v hv t ht hp).sub
+    (((X_hasDerivAt v t).pow 2).const_mul (1/2 : ℝ))).div ((hasDerivAt_id t).pow 2)
+    (pow_ne_zero 2 ht)
+  convert hh using 1 <;> first | rfl | (simp only [X_ray, id_eq, Pi.pow_apply, Pi.sub_apply]; field_simp; ring)
+
+ 
+theorem radial_inequality (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i)
+    (l : ℝ) (hl0 : 0 < l) (hl1 : l < 1) :
+    (1-l^2)/2*(momentX v)^2 ≤ energy v-l⁻¹^2*energy (ray v l) := by
+  have hmono : MonotoneOn
+      (fun t => (energy (ray v t)-(1/2 : ℝ)*(momentX (ray v t))^2)/t^2)
+      (Set.Icc l 1) := by
+    apply monotoneOn_of_hasDerivWithinAt_nonneg (f' := fun t =>
+      gapY (ray v t)/t^3) (convex_Icc _ _)
+    · intro s hs
+      exact (normalized_energy_hasDerivAt v hv s (lt_of_lt_of_le hl0 hs.1).ne' (fun i =>
+        ne_of_gt (message_segment_pos v hp s (le_trans hl0.le hs.1) hs.2 i))).continuousAt.continuousWithinAt
+    · intro s hs
+      rw [interior_Icc] at hs
+      exact (normalized_energy_hasDerivAt v hv s (lt_trans hl0 hs.1).ne' (fun i =>
+        ne_of_gt (message_segment_pos v hp s (lt_trans hl0 hs.1).le hs.2.le i))).hasDerivWithinAt
+    · intro s hs
+      rw [interior_Icc] at hs
+      exact div_nonneg (gapY_nonneg v hv hp s (lt_trans hl0 hs.1).le hs.2.le)
+        (pow_nonneg (lt_trans hl0 hs.1).le 3)
+  have hh := hmono ⟨le_rfl, hl1.le⟩ ⟨hl1.le, le_rfl⟩ hl1.le
+  simp only [ray_one, X_ray, one_pow, div_one] at hh
+  have hl : l ≠ 0 := hl0.ne'
+  have hsq : 0 < l^2 := sq_pos_of_pos hl0
+  rw [div_le_iff₀ hsq] at hh
+  apply (mul_le_mul_iff_left₀ hsq).mp
+  field_simp [hl]
+  nlinarith [hh]
+
+end ThreeState.Radial
+namespace ThreeState.Radial
+
+ 
+noncomputable def sourceF (m : Fin 3 → ℝ) : ℝ :=
+  3*avg (fun i => Real.log (m i)) +
+  4*((1/2 : ℝ)*avg (fun i => (m i-1)*Real.log (m i))) +
+  (4/5 : ℝ)*((1/2 : ℝ)*avg (fun i => m i*
+    (Real.log (m i)-avg (fun j => Real.log (m j)))^2))
+
+lemma avg_message_deviation (m : Fin 3 → ℝ) (hm : avg m = 1) :
+    avg (fun i => m i-1) = 0 := by
+  simp only [avg_expand] at *
+  linarith
+
+lemma logMoment_uncentered (v : Fin 3 → ℝ) (hv : avg v = 0) :
+    logMoment v = avg (fun i => v i*Real.log (1+v i)) := by
+  have hv2 := zero_sum_elim v hv
+  simp only [logMoment, centeredLog, avg_expand, hv2]
+  ring
+
+lemma energy_sourceF (m : Fin 3 → ℝ) (hm : avg m = 1) :
+    energy (fun i => m i-1) = sourceF m := by
+  rw [energy, logMoment_uncentered _ (avg_message_deviation m hm)]
+  simp only [logAvg, weightedLogVariance, centeredLog, sourceF]
+  have he (i : Fin 3) : 1+(m i-1) = m i := by ring
+  simp only [he]
+  ring
+
+ 
+theorem radial_source (m : Fin 3 → ℝ) (hm : avg m = 1) (hp : ∀ i, 0 < m i)
+    (l : ℝ) (hl0 : 0 < l) (hl1 : l < 1) :
+    sourceF m-l⁻¹^2*sourceF (fun i => 1+l*(m i-1)) ≥
+      (1-l^2)/2*((1/2 : ℝ)*avg (fun i => (m i-1)^2))^2 := by
+  have hv := avg_message_deviation m hm
+  have h := radial_inequality (fun i => m i-1) hv (by simpa using hp) l hl0 hl1
+  have he : avg (fun i => 1+l*(m i-1)) = 1 := by
+    simp only [avg_expand] at *
+    nlinarith [hm]
+  have hr : ray (fun i => m i-1) l = (fun i => (1+l*(m i-1))-1) := by
+    ext i; simp [ray]
+  rw [energy_sourceF m hm, hr, energy_sourceF _ he] at h
+  convert h using 1
+  simp only [momentX]
+  ring
+
+end ThreeState.Radial
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeState/RadialMoments.lean b/lean/OAI/Probability/ThreeState/RadialMoments.lean
index 6a87ca4cdd441ef1948317cd4daa3321b3e50b81..1c0e76d64f530dc865960be0ec39f5217cb53412 100644
--- a/lean/OAI/Probability/ThreeState/RadialMoments.lean
+++ b/lean/OAI/Probability/ThreeState/RadialMoments.lean
@@ -1,585 +1,585 @@
-import Mathlib
-
-namespace OAI
-
-/-! Three-coordinate moment identities and positive radial polynomials. -/
-
-namespace ThreeState.Radial
-
-noncomputable def f0 (r : ℝ) : ℝ := 480*r^4-842*r^3-117*r^2+445*r+160
-noncomputable def f1 (r : ℝ) : ℝ :=
-  336*r^6-1008*r^5+1096*r^4-896*r^3+63*r^2+685*r+240
-noncomputable def f2 (r : ℝ) : ℝ := 440*r^4-1104*r^3+368*r^2+413*r+210
-
-lemma tail0_pos (p : ℝ) : 0 < 12267*p^2-1161*p+174 := by
-  nlinarith [sq_nonneg (24534*p-1161)]
-lemma tail1_pos (p : ℝ) : 0 < 20184*p^2-2724*p+522 := by
-  nlinarith [sq_nonneg (40368*p-2724)]
-lemma tail2_pos (p : ℝ) : 0 < 12366*p^2-1842*p+1044 := by
-  nlinarith [sq_nonneg (24732*p-1842)]
-
-lemma transform_f0 (p : ℝ) (hp : 1+p ≠ 0) :
-    4*(1+p)^4*f0 ((p-1/2)/(1+p)) =
-      504*p^4+4386*p^3+12267*p^2-1161*p+174 := by
-  unfold f0
-  field_simp
-  ring
-
-lemma transform_f1 (p : ℝ) (hp : 1+p ≠ 0) :
-    4*(1+p)^6*f1 ((p-1/2)/(1+p)) =
-      2064*p^6+15486*p^5+36669*p^4+31368*p^3+20184*p^2-2724*p+522 := by
-  unfold f1
-  field_simp
-  ring
-
-lemma transform_f2 (p : ℝ) (hp : 1+p ≠ 0) :
-    4*(1+p)^4*f2 ((p-1/2)/(1+p)) =
-      1308*p^4+7650*p^3+12366*p^2-1842*p+1044 := by
-  unfold f2
-  field_simp
-  ring
-
-lemma f0_pos (r : ℝ) (hl : -(1/2 : ℝ) ≤ r) (hu : r < 1) : 0 < f0 r := by
-  let p := (r+1/2)/(1-r)
-  have hp : 0 ≤ p := div_nonneg (by linarith) (by linarith)
-  have hden : 1+p ≠ 0 := ne_of_gt (by linarith)
-  have hr : (p-1/2)/(1+p) = r := by
-    apply (div_eq_iff hden).2
-    dsimp [p]
-    have h1 : 1-r ≠ 0 := by linarith
-    field_simp
-    ring
-  have ht := transform_f0 p hden
-  rw [hr] at ht
-  have hrest : 0 ≤ 504*p^4+4386*p^3 := by positivity
-  have hmul : 0 < 4*(1+p)^4 * f0 r := by linarith [tail0_pos p]
-  exact pos_of_mul_pos_right hmul (by positivity)
-
-lemma f1_pos (r : ℝ) (hl : -(1/2 : ℝ) ≤ r) (hu : r < 1) : 0 < f1 r := by
-  let p := (r+1/2)/(1-r)
-  have hp : 0 ≤ p := div_nonneg (by linarith) (by linarith)
-  have hden : 1+p ≠ 0 := ne_of_gt (by linarith)
-  have hr : (p-1/2)/(1+p) = r := by
-    apply (div_eq_iff hden).2
-    dsimp [p]
-    have h1 : 1-r ≠ 0 := by linarith
-    field_simp
-    ring
-  have ht := transform_f1 p hden
-  rw [hr] at ht
-  have hrest : 0 ≤ 2064*p^6+15486*p^5+36669*p^4+31368*p^3 := by positivity
-  have hmul : 0 < 4*(1+p)^6 * f1 r := by linarith [tail1_pos p]
-  exact pos_of_mul_pos_right hmul (by positivity)
-
-lemma f2_pos (r : ℝ) (hl : -(1/2 : ℝ) ≤ r) (hu : r < 1) : 0 < f2 r := by
-  let p := (r+1/2)/(1-r)
-  have hp : 0 ≤ p := div_nonneg (by linarith) (by linarith)
-  have hden : 1+p ≠ 0 := ne_of_gt (by linarith)
-  have hr : (p-1/2)/(1+p) = r := by
-    apply (div_eq_iff hden).2
-    dsimp [p]
-    have h1 : 1-r ≠ 0 := by linarith
-    field_simp
-    ring
-  have ht := transform_f2 p hden
-  rw [hr] at ht
-  have hrest : 0 ≤ 1308*p^4+7650*p^3 := by positivity
-  have hmul : 0 < 4*(1+p)^4 * f2 r := by linarith [tail2_pos p]
-  exact pos_of_mul_pos_right hmul (by positivity)
-
-noncomputable def A (x y : ℝ) : ℝ :=
-  3*x^4+12*x^3+x^2+(-6*x^3-28*x^2-2*x)*y+(26*x-2)*y^2-4*y^3
-noncomputable def C (x y : ℝ) : ℝ :=
-  6*x^4-2*x^3+(-13*x^3-10*x^2-x)*y+(10*x^2+22*x+4)*y^2-16*y^3
-noncomputable def Lscaled (x y : ℝ) : ℝ :=
-  1500*x^5-1020*x^4+240*x^3+168*y^4+(-480*x^2-756*x-52)*y^3+
-  (2114*x^3+710*x^2+48*x-80)*y^2+(-3093*x^4+736*x^3-35*x^2)*y
-
-lemma N1_factor (x y : ℝ) :
-    A x y + C x y = (x-y)*(9*x-10*y+1)*(x^2+x-2*y) := by
-  unfold A C
-  ring
-
-noncomputable def xa (r z : ℝ) : ℝ := (r^2+z/(3-2*r))/(1+z)
-noncomputable def a0 (r : ℝ) : ℝ := (1-r)^3*(3-2*r)^2*(10*r^2+r+1)
-noncomputable def a1 (r : ℝ) : ℝ := 2*(1-r)^3*(3-2*r)*(9+5*r+20*r^2-4*r^3)
-noncomputable def a2 (r : ℝ) : ℝ := 48*(1-r)^3
-noncomputable def b0 (r : ℝ) : ℝ := r^2*(1-r)^3*(3-2*r)^3*f0 r
-noncomputable def b1 (r : ℝ) : ℝ := (1-r)^3*(3-2*r)^2*f1 r
-noncomputable def b2 (r : ℝ) : ℝ := 2*(1-r)^3*(3-2*r)*f2 r
-noncomputable def b3 (r : ℝ) : ℝ := 24*(1-r)^3*(25-4*r^2-10*r)
-
-lemma a_nonneg (r : ℝ) (hl : -(1/2 : ℝ) ≤ r) (hu : r < 1) :
-    0 ≤ a0 r ∧ 0 ≤ a1 r ∧ 0 ≤ a2 r := by
-  have h1 : 0 ≤ 1-r := by linarith
-  have h2 : 0 ≤ 3-2*r := by linarith
-  have h3 : 0 ≤ 10*r^2+r+1 := by nlinarith [sq_nonneg (20*r+1)]
-  have h4a : 0 ≤ r^2*(20-4*r) := mul_nonneg (sq_nonneg _) (by linarith)
-  have h4 : 0 ≤ 9+5*r+20*r^2-4*r^3 := by nlinarith
-  dsimp [a0, a1, a2]
-  exact ⟨by positivity, by positivity, by positivity⟩
-
-lemma b_nonneg (r : ℝ) (hl : -(1/2 : ℝ) ≤ r) (hu : r < 1) :
-    0 ≤ b0 r ∧ 0 ≤ b1 r ∧ 0 ≤ b2 r ∧ 0 ≤ b3 r := by
-  have h1 : 0 ≤ 1-r := by linarith
-  have h2 : 0 ≤ 3-2*r := by linarith
-  have h3a : 0 ≤ (1-r)*(4*r+14) := mul_nonneg h1 (by linarith)
-  have h3 : 0 ≤ 25-4*r^2-10*r := by nlinarith
-  have hf0 := (f0_pos r hl hu).le
-  have hf1 := (f1_pos r hl hu).le
-  have hf2 := (f2_pos r hl hu).le
-  dsimp [b0, b1, b2, b3]
-  exact ⟨by positivity, by positivity, by positivity, by positivity⟩
-
-lemma A_certificate (r z : ℝ) (hr : 3-2*r ≠ 0) (hz : 1+z ≠ 0) :
-    (1+z)^2*(3-2*r)^2 * A (xa r z) (r*xa r z) =
-      (xa r z)^2*(a0 r+a1 r*z+a2 r*z^2) := by
-  dsimp [A, xa, a0, a1, a2]
-  field_simp
-  ring
-
-lemma L_certificate (r z : ℝ) (hr : 3-2*r ≠ 0) (hz : 1+z ≠ 0) :
-    (1+z)^3*(3-2*r)^3 * Lscaled (xa r z) (r*xa r z) =
-      (xa r z)^2*(b0 r+b1 r*z+b2 r*z^2+b3 r*z^3) := by
-  dsimp [Lscaled, xa, b0, b1, b2, b3, f0, f1, f2]
-  field_simp
-  ring
-
-lemma A_nonneg_param (r z : ℝ) (hl : -(1/2 : ℝ) ≤ r) (hu : r < 1)
-    (hz : 0 ≤ z) : 0 ≤ A (xa r z) (r*xa r z) := by
-  have h1 : 0 < 1+z := by linarith
-  have h2 : 0 < 3-2*r := by linarith
-  obtain ⟨h0, ha1, ha2⟩ := a_nonneg r hl hu
-  have hrhs : 0 ≤ (xa r z)^2*(a0 r+a1 r*z+a2 r*z^2) := by positivity
-  rw [← A_certificate r z (ne_of_gt h2) (ne_of_gt h1)] at hrhs
-  exact nonneg_of_mul_nonneg_right hrhs (by positivity)
-
-lemma L_nonneg_param (r z : ℝ) (hl : -(1/2 : ℝ) ≤ r) (hu : r < 1)
-    (hz : 0 ≤ z) : 0 ≤ Lscaled (xa r z) (r*xa r z) := by
-  have h1 : 0 < 1+z := by linarith
-  have h2 : 0 < 3-2*r := by linarith
-  obtain ⟨h0, hb1, hb2, hb3⟩ := b_nonneg r hl hu
-  have hrhs : 0 ≤ (xa r z)^2*(b0 r+b1 r*z+b2 r*z^2+b3 r*z^3) := by positivity
-  rw [← L_certificate r z (ne_of_gt h2) (ne_of_gt h1)] at hrhs
-  exact nonneg_of_mul_nonneg_right hrhs (by positivity)
-
-lemma physical_parameterization (x y : ℝ) (hx : 0 < x) (hx1 : x < 1)
-    (hdisc : y^2 ≤ x^3) (hD : 0 < 1-3*x+2*y) :
-    ∃ r z : ℝ, -(1/2 : ℝ) ≤ r ∧ r < 1 ∧ 0 ≤ z ∧
-      x = xa r z ∧ y = r * x := by
-  let r := y/x
-  have hylt : y < x := by
-    nlinarith [mul_pos (sq_pos_of_pos hx) (sub_pos.mpr hx1)]
-  have hr1 : r < 1 := (div_lt_one hx).2 hylt
-  have hyr : y = r*x := by dsimp [r]; field_simp
-  have hrsq : r^2 ≤ x := by
-    dsimp [r]
-    rw [div_pow]
-    apply (div_le_iff₀ (sq_pos_of_pos hx)).2
-    nlinarith
-  have hden : 0 < 3-2*r := by linarith
-  have hupper : x < 1/(3-2*r) := by
-    apply (lt_div_iff₀ hden).2
-    nlinarith [hD]
-  have hdsq : r^2*(3-2*r) < 1 := lt_of_le_of_lt
-    (mul_le_mul_of_nonneg_right hrsq hden.le) ((lt_div_iff₀ hden).1 hupper)
-  have hpos : 0 < (1-r)^2*(1+2*r) := by nlinarith
-  have hrlo : -(1/2 : ℝ) ≤ r := by
-    have := pos_of_mul_pos_right hpos (sq_nonneg (1-r))
-    linarith
-  let z := (x-r^2)/(1/(3-2*r)-x)
-  have hz : 0 ≤ z := div_nonneg (by linarith) (by linarith)
-  have hfrep : x = xa r z := by
-    dsimp [xa]
-    apply (eq_div_iff (ne_of_gt (show 0 < 1+z by linarith))).2
-    have hn0 : 1/(3-2*r)-x ≠ 0 := ne_of_gt (sub_pos.mpr hupper)
-    have hzEq : z * (1/(3-2*r)-x) = x-r^2 := div_mul_cancel₀ _ hn0
-    simp only [div_eq_mul_inv, one_mul] at hzEq ⊢
-    nlinarith only [hzEq]
-  exact ⟨r, z, hrlo, hr1, hz, hfrep, hyr⟩
-
-lemma physical_signs (x y : ℝ) (hx : 0 ≤ x) (hx1 : x < 1)
-    (hdisc : y^2 ≤ x^3) (hD : 0 < 1-3*x+2*y) :
-    0 ≤ A x y ∧ 0 ≤ A x y + C x y ∧ 0 ≤ Lscaled x y := by
-  have hyx : y ≤ x := by
-    have hpow : x^3 ≤ x^2 := by nlinarith [mul_nonneg (sq_nonneg x) (by linarith : 0 ≤ 1-x)]
-    nlinarith [sq_nonneg (y-x)]
-  have hlast : 0 ≤ x^2+x-2*y := by
-    by_contra hn
-    have hn' : x^2+x < 2*y := by linarith
-    have hy : 0 < y := by nlinarith [sq_nonneg x]
-    have hp : (x^2+x)^2 < (2*y)^2 := by
-      nlinarith [mul_pos (by linarith : 0 < 2*y-(x^2+x))
-        (by nlinarith [sq_nonneg x] : 0 < 2*y+(x^2+x))]
-    nlinarith [sq_nonneg (x^2-x)]
-  have hN1 : 0 ≤ A x y + C x y := by
-    rw [N1_factor]
-    exact mul_nonneg (mul_nonneg (by linarith) (by linarith)) hlast
-  rcases eq_or_lt_of_le hx with hzero | hpos
-  · have hxzero : x = 0 := hzero.symm
-    have hyzero : y = 0 := by rw [hxzero] at hdisc; nlinarith [sq_nonneg y]
-    subst x; subst y
-    norm_num [A, C, Lscaled]
-  · obtain ⟨r, z, hl, hu, hz, hxr, hyr⟩ := physical_parameterization x y hpos hx1 hdisc hD
-    have hA : 0 ≤ A x y := by rw [hyr, hxr]; exact A_nonneg_param r z hl hu hz
-    have hL : 0 ≤ Lscaled x y := by rw [hyr, hxr]; exact L_nonneg_param r z hl hu hz
-    exact ⟨hA, hN1, hL⟩
-
-end ThreeState.Radial
-
- 
-
-namespace ThreeState.Radial
-
-noncomputable def avg (v : Fin 3 → ℝ) : ℝ := (∑ i, v i) / 3
-noncomputable def momentX (v : Fin 3 → ℝ) : ℝ := avg (fun i => (v i)^2) / 2
-noncomputable def momentY (v : Fin 3 → ℝ) : ℝ := avg (fun i => (v i)^3) / 2
-noncomputable def denom (v : Fin 3 → ℝ) (t : ℝ) : ℝ :=
-  1 - 3*t^2*momentX v + 2*t^3*momentY v
-noncomputable def cofactor (v : Fin 3 → ℝ) (i : Fin 3) : ℝ :=
-  1-v i+(v i)^2-3*momentX v
-noncomputable def kernel (v : Fin 3 → ℝ) (t : ℝ) (i : Fin 3) : ℝ :=
-  v i-t*((v i)^2-2*momentX v)
-noncomputable def centeredLog (v : Fin 3 → ℝ) (i : Fin 3) : ℝ :=
-  Real.log (1+v i) - avg (fun j => Real.log (1+v j))
-
-lemma avg_expand (v : Fin 3 → ℝ) : avg v = (v 0+v 1+v 2)/3 := by
-  simp [avg, Fin.sum_univ_succ]; ring
-
-lemma zero_sum_elim (v : Fin 3 → ℝ) (hv : avg v = 0) : v 2 = -v 0-v 1 := by
-  rw [avg_expand] at hv; linarith
-
-lemma cubic_identity (v : Fin 3 → ℝ) (hv : avg v = 0) (i : Fin 3) :
-    (v i)^3 = 3*momentX v*v i+2*momentY v := by
-  have h2 := zero_sum_elim v hv
-  fin_cases i
-  · change (v 0)^3 = 3*momentX v*v 0+2*momentY v
-    simp only [momentX, momentY, avg_expand, h2]; ring
-  · change (v 1)^3 = 3*momentX v*v 1+2*momentY v
-    simp only [momentX, momentY, avg_expand, h2]; ring
-  · change (v 2)^3 = 3*momentX v*v 2+2*momentY v
-    simp only [momentX, momentY, avg_expand, h2]; ring
-
-lemma denom_product (v : Fin 3 → ℝ) (hv : avg v = 0) (t : ℝ) :
-    denom v t = (1+t*v 0)*(1+t*v 1)*(1+t*v 2) := by
-  have h2 := zero_sum_elim v hv
-  simp only [denom, momentX, momentY, avg_expand, h2]
-  ring
-
-lemma cofactor_identity (v : Fin 3 → ℝ) (hv : avg v = 0) (i : Fin 3) :
-    (1+v i) * cofactor v i = denom v 1 := by
-  dsimp [cofactor, denom]
-  nlinarith only [cubic_identity v hv i]
-
-lemma message_segment_pos (v : Fin 3 → ℝ) (hp : ∀ i, 0 < 1+v i)
-    (t : ℝ) (ht0 : 0 ≤ t) (ht1 : t ≤ 1) (i : Fin 3) : 0 < 1+t*v i := by
-  have hv := hp i
-  rcases eq_or_lt_of_le ht0 with h | h
-  · rw [← h]; norm_num
-  · have : 0 < t*(1+v i) := mul_pos h hv
-    nlinarith
-
-lemma denom_pos (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i)
-    (t : ℝ) (ht0 : 0 ≤ t) (ht1 : t ≤ 1) : 0 < denom v t := by
-  rw [denom_product v hv t]
-  exact mul_pos (mul_pos (message_segment_pos v hp t ht0 ht1 0)
-    (message_segment_pos v hp t ht0 ht1 1)) (message_segment_pos v hp t ht0 ht1 2)
-
-lemma momentX_nonneg (v : Fin 3 → ℝ) : 0 ≤ momentX v := by
-  simp only [momentX, avg_expand]; positivity
-
-lemma discriminant (v : Fin 3 → ℝ) (hv : avg v = 0) :
-    (v 0-v 1)^2*(v 0-v 2)^2*(v 1-v 2)^2 =
-      108*((momentX v)^3-(momentY v)^2) := by
-  have h2 := zero_sum_elim v hv
-  simp only [momentX, momentY, avg_expand, h2]
-  ring
-
-lemma discriminant_nonneg (v : Fin 3 → ℝ) (hv : avg v = 0) :
-    (momentY v)^2 ≤ (momentX v)^3 := by
-  have h := discriminant v hv
-  have : 0 ≤ (v 0-v 1)^2*(v 0-v 2)^2*(v 1-v 2)^2 := by positivity
-  linarith
-
-lemma momentX_lt_one (v : Fin 3 → ℝ) (hv : avg v = 0)
-    (hp : ∀ i, 0 < 1+v i) : momentX v < 1 := by
-  have hu := zero_sum_elim v hv
-  have h01 := mul_pos (hp 0) (hp 1)
-  have h02 := mul_pos (hp 0) (hp 2)
-  have h12 := mul_pos (hp 1) (hp 2)
-  simp only [momentX, avg_expand]
-  rw [hu] at *
-  nlinarith
-
-lemma momentY_le_X (v : Fin 3 → ℝ) (hv : avg v = 0)
-    (hp : ∀ i, 0 < 1+v i) : momentY v ≤ momentX v := by
-  have hX := momentX_nonneg v
-  have hX1 := momentX_lt_one v hv hp
-  have hdisc := discriminant_nonneg v hv
-  have hpow : (momentX v)^3 ≤ (momentX v)^2 := by
-    nlinarith [mul_nonneg (sq_nonneg (momentX v)) (by linarith : 0 ≤ 1-momentX v)]
-  nlinarith [sq_nonneg (momentY v-momentX v)]
-
-lemma denom_le_one (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i)
-    (t : ℝ) (ht0 : 0 ≤ t) (ht1 : t ≤ 1) : denom v t ≤ 1 := by
-  have h1 := mul_le_mul_of_nonneg_left (momentY_le_X v hv hp) (pow_nonneg ht0 3)
-  have h2 := mul_nonneg (mul_nonneg (sq_nonneg t) (momentX_nonneg v))
-    (show 0 ≤ 1-t by linarith)
-  have h3 : 0 ≤ t^2*momentX v := mul_nonneg (sq_nonneg t) (momentX_nonneg v)
-  dsimp [denom]
-  nlinarith
-
-noncomputable def cofMoment (v : Fin 3 → ℝ) (j : ℕ) :=
-  avg (fun i => (cofactor v i)^j)
-
-lemma cofMoment_one (v : Fin 3 → ℝ) (hv : avg v = 0) :
-    cofMoment v 1 = 1-momentX v := by
-  have h2 := zero_sum_elim v hv
-  simp only [cofMoment, cofactor, momentX, avg_expand, h2]
-  ring
-
-lemma cofMoment_two (v : Fin 3 → ℝ) (hv : avg v = 0) :
-    cofMoment v 2 = 3*(momentX v)^2-4*momentY v+1 := by
-  have h2 := zero_sum_elim v hv
-  simp only [cofMoment, cofactor, momentX, momentY, avg_expand, h2]
-  ring
-
-lemma cofMoment_three (v : Fin 3 → ℝ) (hv : avg v = 0) :
-    cofMoment v 3 = -9*(momentX v)^3+9*(momentX v)^2+
-      6*momentX v*momentY v+3*momentX v+4*(momentY v)^2-14*momentY v+1 := by
-  have h2 := zero_sum_elim v hv
-  simp only [cofMoment, cofactor, momentX, momentY, avg_expand, h2]
-  ring
-
-noncomputable def numerator (v : Fin 3 → ℝ) (t : ℝ) :=
-  (4/5 : ℝ) * avg (fun i => kernel v t i *
-    ((cofactor v i)^2*denom v 1-3*cofactor v i*(denom v 1)^2+
-      (2*cofMoment v 3-3*cofMoment v 2*denom v 1)*v i))
-
-lemma numerator_expansion (v : Fin 3 → ℝ) (hv : avg v = 0) (t : ℝ) :
-    numerator v t = (24/5 : ℝ)*(A (momentX v) (momentY v)+t*C (momentX v) (momentY v)) := by
-  unfold numerator
-  rw [cofMoment_two v hv, cofMoment_three v hv]
-  have h2 := zero_sum_elim v hv
-  simp only [cofactor, kernel, denom, A, C, momentX, momentY, avg_expand, h2]
-  ring
-
-lemma integral_avg (f : Fin 3 → ℝ → ℝ)
-    (hf : ∀ i, IntervalIntegrable (f i) MeasureTheory.volume 0 1) :
-    (∫ t in (0 : ℝ)..1, avg (fun i => f i t)) = avg (fun i => ∫ t in (0 : ℝ)..1, f i t) := by
-  unfold avg
-  rw [intervalIntegral.integral_div, intervalIntegral.integral_finsetSum (fun i _ => hf i)]
-
-lemma log_integral (a : ℝ) (ha : 0 < 1+a) :
-    Real.log (1+a) = ∫ t in (0 : ℝ)..1, a/(1+t*a) := by
-  have hp : ∀ t ∈ Set.Icc (0 : ℝ) 1, 0 < 1+t*a := by
-    intro t ht
-    rcases eq_or_lt_of_le ht.1 with hz | hz
-    · rw [← hz]; norm_num
-    · nlinarith [mul_pos hz ha, ht.2]
-  have hc : ContinuousOn (fun t : ℝ => a/(1+t*a)) (Set.Icc 0 1) :=
-    continuousOn_const.div (by fun_prop) (fun t ht => ne_of_gt (hp t ht))
-  have hi : IntervalIntegrable (fun t : ℝ => a/(1+t*a)) MeasureTheory.volume 0 1 :=
-    hc.intervalIntegrable_of_Icc (by norm_num)
-  have hD : ∀ t ∈ Set.uIcc (0 : ℝ) 1,
-      HasDerivAt (fun t : ℝ => Real.log (1+t*a)) (a/(1+t*a)) t := by
-    intro t ht
-    have ht' : t ∈ Set.Icc (0 : ℝ) 1 := by simpa using ht
-    simpa using (((hasDerivAt_id t).mul_const a).const_add 1).log (ne_of_gt (hp t ht'))
-  simpa using (intervalIntegral.integral_eq_sub_of_hasDerivAt hD hi).symm
-
-lemma kernel_div_identity (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i)
-    (t : ℝ) (ht0 : 0 ≤ t) (ht1 : t ≤ 1) (i : Fin 3) :
-    v i/(1+t*v i)-avg (fun j => v j/(1+t*v j)) = kernel v t i/denom v t := by
-  have h0 := ne_of_gt (message_segment_pos v hp t ht0 ht1 0)
-  have h1 := ne_of_gt (message_segment_pos v hp t ht0 ht1 1)
-  have h2 := ne_of_gt (message_segment_pos v hp t ht0 ht1 2)
-  have h0c : 1+v 0*t ≠ 0 := by simpa [mul_comm] using h0
-  have h1c : 1+v 1*t ≠ 0 := by simpa [mul_comm] using h1
-  have h2c : 1+v 2*t ≠ 0 := by simpa [mul_comm] using h2
-  have hv2 := zero_sum_elim v hv
-  rw [denom_product v hv t]
-  fin_cases i
-  · change v 0/(1+t*v 0)-avg (fun j => v j/(1+t*v j)) =
-      kernel v t 0/((1+t*v 0)*(1+t*v 1)*(1+t*v 2))
-    simp only [avg_expand, kernel, momentX]
-    field_simp [h0, h1, h2, h0c, h1c, h2c]
-    rw [hv2]
-    ring
-  · change v 1/(1+t*v 1)-avg (fun j => v j/(1+t*v j)) =
-      kernel v t 1/((1+t*v 0)*(1+t*v 1)*(1+t*v 2))
-    simp only [avg_expand, kernel, momentX]
-    field_simp [h0, h1, h2, h0c, h1c, h2c]
-    rw [hv2]
-    ring
-  · change v 2/(1+t*v 2)-avg (fun j => v j/(1+t*v j)) =
-      kernel v t 2/((1+t*v 0)*(1+t*v 1)*(1+t*v 2))
-    simp only [avg_expand, kernel, momentX]
-    field_simp [h0, h1, h2, h0c, h1c, h2c]
-    rw [hv2]
-    ring
-
-lemma centeredLog_integral (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i)
-    (i : Fin 3) : centeredLog v i = ∫ t in (0 : ℝ)..1, kernel v t i / denom v t := by
-  have hi : ∀ j, IntervalIntegrable (fun t : ℝ => v j/(1+t*v j)) MeasureTheory.volume 0 1 := by
-    intro j
-    apply ContinuousOn.intervalIntegrable_of_Icc (by norm_num)
-    refine continuousOn_const.div (by fun_prop) ?_
-    intro t ht
-    exact ne_of_gt (message_segment_pos v hp t ht.1 ht.2 j)
-  have ha : IntervalIntegrable (fun t : ℝ => avg (fun j => v j/(1+t*v j)))
-      MeasureTheory.volume 0 1 := by
-    unfold avg
-    exact (IntervalIntegrable.sum Finset.univ (fun j _ => hi j)).div_const 3
-  unfold centeredLog
-  rw [log_integral (v i) (hp i)]
-  have hlogs : (fun j => Real.log (1+v j)) = (fun j => ∫ t in (0 : ℝ)..1, v j/(1+t*v j)) := by
-    funext j; exact log_integral (v j) (hp j)
-  rw [hlogs, ← integral_avg _ hi, ← intervalIntegral.integral_sub (hi i) ha]
-  apply intervalIntegral.integral_congr
-  intro t ht
-  have ht' : t ∈ Set.Icc (0 : ℝ) 1 := by simpa using ht
-  exact kernel_div_identity v hv hp t ht'.1 ht'.2 i
-
-noncomputable def invMoment (v : Fin 3 → ℝ) (j : ℕ) :=
-  avg (fun i => ((1+v i)⁻¹)^j)
-noncomputable def logInvMoment (v : Fin 3 → ℝ) (j : ℕ) :=
-  avg (fun i => centeredLog v i*((1+v i)⁻¹)^j)
-noncomputable def logMoment (v : Fin 3 → ℝ) := avg (fun i => v i*centeredLog v i)
-
- 
-noncomputable def gExpr (v : Fin 3 → ℝ) :=
-  6-12*invMoment v 1+8*invMoment v 2-2*invMoment v 3+
-  (4/5 : ℝ)*(logInvMoment v 2-3*logInvMoment v 1+
-    (2*invMoment v 3-3*invMoment v 2)*logMoment v+
-    3*(invMoment v 1-1)*(invMoment v 2-invMoment v 1))
-noncomputable def rCore (v : Fin 3 → ℝ) :=
-  (6-12*(momentX v)^2)*(denom v 1)^3-12*cofMoment v 1*(denom v 1)^2+
-  8*cofMoment v 2*denom v 1-2*cofMoment v 3+
-  (12/5 : ℝ)*(cofMoment v 1-denom v 1)*(cofMoment v 2-cofMoment v 1*denom v 1)
-noncomputable def logCoeff (v : Fin 3 → ℝ) (i : Fin 3) :=
-  (cofactor v i)^2*denom v 1-3*cofactor v i*(denom v 1)^2+
-    (2*cofMoment v 3-3*cofMoment v 2*denom v 1)*v i
-
-lemma inv_cofactor (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i)
-    (i : Fin 3) : (1+v i)⁻¹ = cofactor v i / denom v 1 := by
-  apply (eq_div_iff (ne_of_gt (denom_pos v hv hp 1 (by norm_num) le_rfl))).2
-  rw [← cofactor_identity v hv i]
-  field_simp [ne_of_gt (hp i)]
-
-lemma RN_algebra (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i) :
-    (denom v 1)^3*(gExpr v-12*(momentX v)^2) =
-      rCore v + (4/5 : ℝ)*avg (fun i => centeredLog v i*logCoeff v i) := by
-  have hD := ne_of_gt (denom_pos v hv hp 1 (by norm_num) le_rfl)
-  unfold gExpr invMoment logInvMoment logMoment rCore logCoeff
-  simp_rw [inv_cofactor v hv hp]
-  simp only [cofMoment, avg_expand]
-  field_simp
-  ring
-
-lemma kernel_div_integrable (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i)
-    (i : Fin 3) : IntervalIntegrable (fun t : ℝ => kernel v t i/denom v t)
-      MeasureTheory.volume 0 1 := by
-  apply ContinuousOn.intervalIntegrable_of_Icc (by norm_num)
-  unfold kernel denom
-  refine ContinuousOn.div (by fun_prop) (by fun_prop) ?_
-  intro t ht
-  exact ne_of_gt (denom_pos v hv hp t ht.1 ht.2)
-
-lemma RN_representation (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i) :
-    (denom v 1)^3*(gExpr v-12*(momentX v)^2) =
-      rCore v + ∫ t in (0 : ℝ)..1, numerator v t / denom v t := by
-  rw [RN_algebra v hv hp]
-  congr 1
-  have hg : ∀ t : ℝ, numerator v t/denom v t = (4/5 : ℝ)*
-      avg (fun i => (kernel v t i/denom v t)*logCoeff v i) := by
-    intro t
-    simp only [numerator, logCoeff, avg_expand]
-    ring
-  simp_rw [hg]
-  rw [intervalIntegral.integral_const_mul, integral_avg _
-    (fun i => (kernel_div_integrable v hv hp i).mul_const (logCoeff v i))]
-  congr 1
-  congr 1
-  funext i
-  rw [intervalIntegral.integral_mul_const, centeredLog_integral v hv hp i]
-
-lemma integrate_N_lower (a c x y : ℝ) :
-    (∫ t in (0 : ℝ)..1, (24/5 : ℝ)*(a+t*c)*(1+3*t^2*x-2*t^3*y)) =
-      (24/5 : ℝ)*(a+c/2+a*x+(3*c*x-2*a*y)/4-2*c*y/5) := by
-  let P := fun t : ℝ => (24/5 : ℝ)*(a*t+c*t^2/2+a*x*t^3+
-      (3*c*x-2*a*y)*t^4/4-2*c*y*t^5/5)
-  have hd : ∀ t : ℝ, HasDerivAt P ((24/5 : ℝ)*(a+t*c)*(1+3*t^2*x-2*t^3*y)) t := by
-    intro t
-    dsimp [P]
-    convert (((((hasDerivAt_id t).const_mul a).add
-      (((hasDerivAt_id t).pow 2).const_mul c |>.div_const 2)).add
-      (((hasDerivAt_id t).pow 3).const_mul (a*x))).add
-      (((hasDerivAt_id t).pow 4).const_mul (3*c*x-2*a*y) |>.div_const 4) |>.sub
-      (((hasDerivAt_id t).pow 5).const_mul (2*c*y) |>.div_const 5)).const_mul (24/5 : ℝ) using 1 <;>
-      first | rfl | (norm_num; ring)
-  have hi : IntervalIntegrable (fun t : ℝ => (24/5 : ℝ)*(a+t*c)*(1+3*t^2*x-2*t^3*y))
-      MeasureTheory.volume 0 1 := Continuous.intervalIntegrable (by fun_prop) 0 1
-  rw [intervalIntegral.integral_eq_sub_of_hasDerivAt (fun t _ => hd t) hi]
-  dsimp [P]; ring
-
-lemma L_expansion (v : Fin 3 → ℝ) (hv : avg v = 0) :
-    rCore v + (∫ t in (0 : ℝ)..1, numerator v t*(2-denom v t)) =
-      (6/25 : ℝ)*Lscaled (momentX v) (momentY v) := by
-  have hg : (fun t : ℝ => numerator v t*(2-denom v t)) =
-    (fun t : ℝ => (24/5 : ℝ)*(A (momentX v) (momentY v)+t*C (momentX v) (momentY v))*
-      (1+3*t^2*momentX v-2*t^3*momentY v)) := by
-    funext t
-    rw [numerator_expansion v hv t]
-    dsimp [denom]; ring
-  rw [hg, integrate_N_lower]
-  unfold rCore
-  rw [cofMoment_one v hv, cofMoment_two v hv, cofMoment_three v hv]
-  dsimp [denom, A, C, Lscaled]
-  ring
-
-lemma numerator_nonneg (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i)
-    (t : ℝ) (ht0 : 0 ≤ t) (ht1 : t ≤ 1) : 0 ≤ numerator v t := by
-  have hD : 0 < 1-3*momentX v+2*momentY v := by
-    simpa [denom] using denom_pos v hv hp 1 (by norm_num) le_rfl
-  obtain ⟨ha, hac, _⟩ := physical_signs (momentX v) (momentY v)
-    (momentX_nonneg v) (momentX_lt_one v hv hp) (discriminant_nonneg v hv) hD
-  rw [numerator_expansion v hv t]
-  have h := add_nonneg (mul_nonneg (show 0 ≤ 1-t by linarith) ha) (mul_nonneg ht0 hac)
-  nlinarith only [h]
-
- 
-
-lemma gExpr_lower (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i) :
-    12*(momentX v)^2 ≤ gExpr v := by
-  have hi1 : IntervalIntegrable (fun t => numerator v t*(2-denom v t))
-      MeasureTheory.volume 0 1 := by
-    apply Continuous.intervalIntegrable
-    unfold numerator kernel denom avg
-    fun_prop
-  have hi2 : IntervalIntegrable (fun t => numerator v t/denom v t)
-      MeasureTheory.volume 0 1 := by
-    apply ContinuousOn.intervalIntegrable_of_Icc (by norm_num)
-    apply ContinuousOn.div
-    · unfold numerator kernel avg; fun_prop
-    · unfold denom; fun_prop
-    · intro t ht; exact ne_of_gt (denom_pos v hv hp t ht.1 ht.2)
-  have hbound : ∀ t ∈ Set.Icc (0 : ℝ) 1, numerator v t*(2-denom v t) ≤ numerator v t/denom v t := by
-    intro t ht
-    have hD := denom_pos v hv hp t ht.1 ht.2
-    apply (le_div_iff₀ hD).2
-    nlinarith [mul_nonneg (numerator_nonneg v hv hp t ht.1 ht.2) (sq_nonneg (1-denom v t))]
-  have hint := intervalIntegral.integral_mono_on (by norm_num) hi1 hi2 hbound
-  have hR := RN_representation v hv hp
-  have hL := L_expansion v hv
-  have hd := denom_pos v hv hp 1 (by norm_num) le_rfl
-  have hdom : 0 < 1-3*momentX v+2*momentY v := by simpa [denom] using hd
-  have hLpos := (physical_signs (momentX v) (momentY v) (momentX_nonneg v)
-    (momentX_lt_one v hv hp) (discriminant_nonneg v hv) hdom).2.2
-  have hprod : 0 ≤ (denom v 1)^3*(gExpr v-12*(momentX v)^2) := by linarith
-  have := nonneg_of_mul_nonneg_right hprod (pow_pos hd 3)
-  linarith
-
-end ThreeState.Radial
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-! Three-coordinate moment identities and positive radial polynomials. -/
+
+namespace ThreeState.Radial
+
+noncomputable def f0 (r : ℝ) : ℝ := 480*r^4-842*r^3-117*r^2+445*r+160
+noncomputable def f1 (r : ℝ) : ℝ :=
+  336*r^6-1008*r^5+1096*r^4-896*r^3+63*r^2+685*r+240
+noncomputable def f2 (r : ℝ) : ℝ := 440*r^4-1104*r^3+368*r^2+413*r+210
+
+lemma tail0_pos (p : ℝ) : 0 < 12267*p^2-1161*p+174 := by
+  nlinarith [sq_nonneg (24534*p-1161)]
+lemma tail1_pos (p : ℝ) : 0 < 20184*p^2-2724*p+522 := by
+  nlinarith [sq_nonneg (40368*p-2724)]
+lemma tail2_pos (p : ℝ) : 0 < 12366*p^2-1842*p+1044 := by
+  nlinarith [sq_nonneg (24732*p-1842)]
+
+lemma transform_f0 (p : ℝ) (hp : 1+p ≠ 0) :
+    4*(1+p)^4*f0 ((p-1/2)/(1+p)) =
+      504*p^4+4386*p^3+12267*p^2-1161*p+174 := by
+  unfold f0
+  field_simp
+  ring
+
+lemma transform_f1 (p : ℝ) (hp : 1+p ≠ 0) :
+    4*(1+p)^6*f1 ((p-1/2)/(1+p)) =
+      2064*p^6+15486*p^5+36669*p^4+31368*p^3+20184*p^2-2724*p+522 := by
+  unfold f1
+  field_simp
+  ring
+
+lemma transform_f2 (p : ℝ) (hp : 1+p ≠ 0) :
+    4*(1+p)^4*f2 ((p-1/2)/(1+p)) =
+      1308*p^4+7650*p^3+12366*p^2-1842*p+1044 := by
+  unfold f2
+  field_simp
+  ring
+
+lemma f0_pos (r : ℝ) (hl : -(1/2 : ℝ) ≤ r) (hu : r < 1) : 0 < f0 r := by
+  let p := (r+1/2)/(1-r)
+  have hp : 0 ≤ p := div_nonneg (by linarith) (by linarith)
+  have hden : 1+p ≠ 0 := ne_of_gt (by linarith)
+  have hr : (p-1/2)/(1+p) = r := by
+    apply (div_eq_iff hden).2
+    dsimp [p]
+    have h1 : 1-r ≠ 0 := by linarith
+    field_simp
+    ring
+  have ht := transform_f0 p hden
+  rw [hr] at ht
+  have hrest : 0 ≤ 504*p^4+4386*p^3 := by positivity
+  have hmul : 0 < 4*(1+p)^4 * f0 r := by linarith [tail0_pos p]
+  exact pos_of_mul_pos_right hmul (by positivity)
+
+lemma f1_pos (r : ℝ) (hl : -(1/2 : ℝ) ≤ r) (hu : r < 1) : 0 < f1 r := by
+  let p := (r+1/2)/(1-r)
+  have hp : 0 ≤ p := div_nonneg (by linarith) (by linarith)
+  have hden : 1+p ≠ 0 := ne_of_gt (by linarith)
+  have hr : (p-1/2)/(1+p) = r := by
+    apply (div_eq_iff hden).2
+    dsimp [p]
+    have h1 : 1-r ≠ 0 := by linarith
+    field_simp
+    ring
+  have ht := transform_f1 p hden
+  rw [hr] at ht
+  have hrest : 0 ≤ 2064*p^6+15486*p^5+36669*p^4+31368*p^3 := by positivity
+  have hmul : 0 < 4*(1+p)^6 * f1 r := by linarith [tail1_pos p]
+  exact pos_of_mul_pos_right hmul (by positivity)
+
+lemma f2_pos (r : ℝ) (hl : -(1/2 : ℝ) ≤ r) (hu : r < 1) : 0 < f2 r := by
+  let p := (r+1/2)/(1-r)
+  have hp : 0 ≤ p := div_nonneg (by linarith) (by linarith)
+  have hden : 1+p ≠ 0 := ne_of_gt (by linarith)
+  have hr : (p-1/2)/(1+p) = r := by
+    apply (div_eq_iff hden).2
+    dsimp [p]
+    have h1 : 1-r ≠ 0 := by linarith
+    field_simp
+    ring
+  have ht := transform_f2 p hden
+  rw [hr] at ht
+  have hrest : 0 ≤ 1308*p^4+7650*p^3 := by positivity
+  have hmul : 0 < 4*(1+p)^4 * f2 r := by linarith [tail2_pos p]
+  exact pos_of_mul_pos_right hmul (by positivity)
+
+noncomputable def A (x y : ℝ) : ℝ :=
+  3*x^4+12*x^3+x^2+(-6*x^3-28*x^2-2*x)*y+(26*x-2)*y^2-4*y^3
+noncomputable def C (x y : ℝ) : ℝ :=
+  6*x^4-2*x^3+(-13*x^3-10*x^2-x)*y+(10*x^2+22*x+4)*y^2-16*y^3
+noncomputable def Lscaled (x y : ℝ) : ℝ :=
+  1500*x^5-1020*x^4+240*x^3+168*y^4+(-480*x^2-756*x-52)*y^3+
+  (2114*x^3+710*x^2+48*x-80)*y^2+(-3093*x^4+736*x^3-35*x^2)*y
+
+lemma N1_factor (x y : ℝ) :
+    A x y + C x y = (x-y)*(9*x-10*y+1)*(x^2+x-2*y) := by
+  unfold A C
+  ring
+
+noncomputable def xa (r z : ℝ) : ℝ := (r^2+z/(3-2*r))/(1+z)
+noncomputable def a0 (r : ℝ) : ℝ := (1-r)^3*(3-2*r)^2*(10*r^2+r+1)
+noncomputable def a1 (r : ℝ) : ℝ := 2*(1-r)^3*(3-2*r)*(9+5*r+20*r^2-4*r^3)
+noncomputable def a2 (r : ℝ) : ℝ := 48*(1-r)^3
+noncomputable def b0 (r : ℝ) : ℝ := r^2*(1-r)^3*(3-2*r)^3*f0 r
+noncomputable def b1 (r : ℝ) : ℝ := (1-r)^3*(3-2*r)^2*f1 r
+noncomputable def b2 (r : ℝ) : ℝ := 2*(1-r)^3*(3-2*r)*f2 r
+noncomputable def b3 (r : ℝ) : ℝ := 24*(1-r)^3*(25-4*r^2-10*r)
+
+lemma a_nonneg (r : ℝ) (hl : -(1/2 : ℝ) ≤ r) (hu : r < 1) :
+    0 ≤ a0 r ∧ 0 ≤ a1 r ∧ 0 ≤ a2 r := by
+  have h1 : 0 ≤ 1-r := by linarith
+  have h2 : 0 ≤ 3-2*r := by linarith
+  have h3 : 0 ≤ 10*r^2+r+1 := by nlinarith [sq_nonneg (20*r+1)]
+  have h4a : 0 ≤ r^2*(20-4*r) := mul_nonneg (sq_nonneg _) (by linarith)
+  have h4 : 0 ≤ 9+5*r+20*r^2-4*r^3 := by nlinarith
+  dsimp [a0, a1, a2]
+  exact ⟨by positivity, by positivity, by positivity⟩
+
+lemma b_nonneg (r : ℝ) (hl : -(1/2 : ℝ) ≤ r) (hu : r < 1) :
+    0 ≤ b0 r ∧ 0 ≤ b1 r ∧ 0 ≤ b2 r ∧ 0 ≤ b3 r := by
+  have h1 : 0 ≤ 1-r := by linarith
+  have h2 : 0 ≤ 3-2*r := by linarith
+  have h3a : 0 ≤ (1-r)*(4*r+14) := mul_nonneg h1 (by linarith)
+  have h3 : 0 ≤ 25-4*r^2-10*r := by nlinarith
+  have hf0 := (f0_pos r hl hu).le
+  have hf1 := (f1_pos r hl hu).le
+  have hf2 := (f2_pos r hl hu).le
+  dsimp [b0, b1, b2, b3]
+  exact ⟨by positivity, by positivity, by positivity, by positivity⟩
+
+lemma A_certificate (r z : ℝ) (hr : 3-2*r ≠ 0) (hz : 1+z ≠ 0) :
+    (1+z)^2*(3-2*r)^2 * A (xa r z) (r*xa r z) =
+      (xa r z)^2*(a0 r+a1 r*z+a2 r*z^2) := by
+  dsimp [A, xa, a0, a1, a2]
+  field_simp
+  ring
+
+lemma L_certificate (r z : ℝ) (hr : 3-2*r ≠ 0) (hz : 1+z ≠ 0) :
+    (1+z)^3*(3-2*r)^3 * Lscaled (xa r z) (r*xa r z) =
+      (xa r z)^2*(b0 r+b1 r*z+b2 r*z^2+b3 r*z^3) := by
+  dsimp [Lscaled, xa, b0, b1, b2, b3, f0, f1, f2]
+  field_simp
+  ring
+
+lemma A_nonneg_param (r z : ℝ) (hl : -(1/2 : ℝ) ≤ r) (hu : r < 1)
+    (hz : 0 ≤ z) : 0 ≤ A (xa r z) (r*xa r z) := by
+  have h1 : 0 < 1+z := by linarith
+  have h2 : 0 < 3-2*r := by linarith
+  obtain ⟨h0, ha1, ha2⟩ := a_nonneg r hl hu
+  have hrhs : 0 ≤ (xa r z)^2*(a0 r+a1 r*z+a2 r*z^2) := by positivity
+  rw [← A_certificate r z (ne_of_gt h2) (ne_of_gt h1)] at hrhs
+  exact nonneg_of_mul_nonneg_right hrhs (by positivity)
+
+lemma L_nonneg_param (r z : ℝ) (hl : -(1/2 : ℝ) ≤ r) (hu : r < 1)
+    (hz : 0 ≤ z) : 0 ≤ Lscaled (xa r z) (r*xa r z) := by
+  have h1 : 0 < 1+z := by linarith
+  have h2 : 0 < 3-2*r := by linarith
+  obtain ⟨h0, hb1, hb2, hb3⟩ := b_nonneg r hl hu
+  have hrhs : 0 ≤ (xa r z)^2*(b0 r+b1 r*z+b2 r*z^2+b3 r*z^3) := by positivity
+  rw [← L_certificate r z (ne_of_gt h2) (ne_of_gt h1)] at hrhs
+  exact nonneg_of_mul_nonneg_right hrhs (by positivity)
+
+lemma physical_parameterization (x y : ℝ) (hx : 0 < x) (hx1 : x < 1)
+    (hdisc : y^2 ≤ x^3) (hD : 0 < 1-3*x+2*y) :
+    ∃ r z : ℝ, -(1/2 : ℝ) ≤ r ∧ r < 1 ∧ 0 ≤ z ∧
+      x = xa r z ∧ y = r * x := by
+  let r := y/x
+  have hylt : y < x := by
+    nlinarith [mul_pos (sq_pos_of_pos hx) (sub_pos.mpr hx1)]
+  have hr1 : r < 1 := (div_lt_one hx).2 hylt
+  have hyr : y = r*x := by dsimp [r]; field_simp
+  have hrsq : r^2 ≤ x := by
+    dsimp [r]
+    rw [div_pow]
+    apply (div_le_iff₀ (sq_pos_of_pos hx)).2
+    nlinarith
+  have hden : 0 < 3-2*r := by linarith
+  have hupper : x < 1/(3-2*r) := by
+    apply (lt_div_iff₀ hden).2
+    nlinarith [hD]
+  have hdsq : r^2*(3-2*r) < 1 := lt_of_le_of_lt
+    (mul_le_mul_of_nonneg_right hrsq hden.le) ((lt_div_iff₀ hden).1 hupper)
+  have hpos : 0 < (1-r)^2*(1+2*r) := by nlinarith
+  have hrlo : -(1/2 : ℝ) ≤ r := by
+    have := pos_of_mul_pos_right hpos (sq_nonneg (1-r))
+    linarith
+  let z := (x-r^2)/(1/(3-2*r)-x)
+  have hz : 0 ≤ z := div_nonneg (by linarith) (by linarith)
+  have hfrep : x = xa r z := by
+    dsimp [xa]
+    apply (eq_div_iff (ne_of_gt (show 0 < 1+z by linarith))).2
+    have hn0 : 1/(3-2*r)-x ≠ 0 := ne_of_gt (sub_pos.mpr hupper)
+    have hzEq : z * (1/(3-2*r)-x) = x-r^2 := div_mul_cancel₀ _ hn0
+    simp only [div_eq_mul_inv, one_mul] at hzEq ⊢
+    nlinarith only [hzEq]
+  exact ⟨r, z, hrlo, hr1, hz, hfrep, hyr⟩
+
+lemma physical_signs (x y : ℝ) (hx : 0 ≤ x) (hx1 : x < 1)
+    (hdisc : y^2 ≤ x^3) (hD : 0 < 1-3*x+2*y) :
+    0 ≤ A x y ∧ 0 ≤ A x y + C x y ∧ 0 ≤ Lscaled x y := by
+  have hyx : y ≤ x := by
+    have hpow : x^3 ≤ x^2 := by nlinarith [mul_nonneg (sq_nonneg x) (by linarith : 0 ≤ 1-x)]
+    nlinarith [sq_nonneg (y-x)]
+  have hlast : 0 ≤ x^2+x-2*y := by
+    by_contra hn
+    have hn' : x^2+x < 2*y := by linarith
+    have hy : 0 < y := by nlinarith [sq_nonneg x]
+    have hp : (x^2+x)^2 < (2*y)^2 := by
+      nlinarith [mul_pos (by linarith : 0 < 2*y-(x^2+x))
+        (by nlinarith [sq_nonneg x] : 0 < 2*y+(x^2+x))]
+    nlinarith [sq_nonneg (x^2-x)]
+  have hN1 : 0 ≤ A x y + C x y := by
+    rw [N1_factor]
+    exact mul_nonneg (mul_nonneg (by linarith) (by linarith)) hlast
+  rcases eq_or_lt_of_le hx with hzero | hpos
+  · have hxzero : x = 0 := hzero.symm
+    have hyzero : y = 0 := by rw [hxzero] at hdisc; nlinarith [sq_nonneg y]
+    subst x; subst y
+    norm_num [A, C, Lscaled]
+  · obtain ⟨r, z, hl, hu, hz, hxr, hyr⟩ := physical_parameterization x y hpos hx1 hdisc hD
+    have hA : 0 ≤ A x y := by rw [hyr, hxr]; exact A_nonneg_param r z hl hu hz
+    have hL : 0 ≤ Lscaled x y := by rw [hyr, hxr]; exact L_nonneg_param r z hl hu hz
+    exact ⟨hA, hN1, hL⟩
+
+end ThreeState.Radial
+
+ 
+
+namespace ThreeState.Radial
+
+noncomputable def avg (v : Fin 3 → ℝ) : ℝ := (∑ i, v i) / 3
+noncomputable def momentX (v : Fin 3 → ℝ) : ℝ := avg (fun i => (v i)^2) / 2
+noncomputable def momentY (v : Fin 3 → ℝ) : ℝ := avg (fun i => (v i)^3) / 2
+noncomputable def denom (v : Fin 3 → ℝ) (t : ℝ) : ℝ :=
+  1 - 3*t^2*momentX v + 2*t^3*momentY v
+noncomputable def cofactor (v : Fin 3 → ℝ) (i : Fin 3) : ℝ :=
+  1-v i+(v i)^2-3*momentX v
+noncomputable def kernel (v : Fin 3 → ℝ) (t : ℝ) (i : Fin 3) : ℝ :=
+  v i-t*((v i)^2-2*momentX v)
+noncomputable def centeredLog (v : Fin 3 → ℝ) (i : Fin 3) : ℝ :=
+  Real.log (1+v i) - avg (fun j => Real.log (1+v j))
+
+lemma avg_expand (v : Fin 3 → ℝ) : avg v = (v 0+v 1+v 2)/3 := by
+  simp [avg, Fin.sum_univ_succ]; ring
+
+lemma zero_sum_elim (v : Fin 3 → ℝ) (hv : avg v = 0) : v 2 = -v 0-v 1 := by
+  rw [avg_expand] at hv; linarith
+
+lemma cubic_identity (v : Fin 3 → ℝ) (hv : avg v = 0) (i : Fin 3) :
+    (v i)^3 = 3*momentX v*v i+2*momentY v := by
+  have h2 := zero_sum_elim v hv
+  fin_cases i
+  · change (v 0)^3 = 3*momentX v*v 0+2*momentY v
+    simp only [momentX, momentY, avg_expand, h2]; ring
+  · change (v 1)^3 = 3*momentX v*v 1+2*momentY v
+    simp only [momentX, momentY, avg_expand, h2]; ring
+  · change (v 2)^3 = 3*momentX v*v 2+2*momentY v
+    simp only [momentX, momentY, avg_expand, h2]; ring
+
+lemma denom_product (v : Fin 3 → ℝ) (hv : avg v = 0) (t : ℝ) :
+    denom v t = (1+t*v 0)*(1+t*v 1)*(1+t*v 2) := by
+  have h2 := zero_sum_elim v hv
+  simp only [denom, momentX, momentY, avg_expand, h2]
+  ring
+
+lemma cofactor_identity (v : Fin 3 → ℝ) (hv : avg v = 0) (i : Fin 3) :
+    (1+v i) * cofactor v i = denom v 1 := by
+  dsimp [cofactor, denom]
+  nlinarith only [cubic_identity v hv i]
+
+lemma message_segment_pos (v : Fin 3 → ℝ) (hp : ∀ i, 0 < 1+v i)
+    (t : ℝ) (ht0 : 0 ≤ t) (ht1 : t ≤ 1) (i : Fin 3) : 0 < 1+t*v i := by
+  have hv := hp i
+  rcases eq_or_lt_of_le ht0 with h | h
+  · rw [← h]; norm_num
+  · have : 0 < t*(1+v i) := mul_pos h hv
+    nlinarith
+
+lemma denom_pos (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i)
+    (t : ℝ) (ht0 : 0 ≤ t) (ht1 : t ≤ 1) : 0 < denom v t := by
+  rw [denom_product v hv t]
+  exact mul_pos (mul_pos (message_segment_pos v hp t ht0 ht1 0)
+    (message_segment_pos v hp t ht0 ht1 1)) (message_segment_pos v hp t ht0 ht1 2)
+
+lemma momentX_nonneg (v : Fin 3 → ℝ) : 0 ≤ momentX v := by
+  simp only [momentX, avg_expand]; positivity
+
+lemma discriminant (v : Fin 3 → ℝ) (hv : avg v = 0) :
+    (v 0-v 1)^2*(v 0-v 2)^2*(v 1-v 2)^2 =
+      108*((momentX v)^3-(momentY v)^2) := by
+  have h2 := zero_sum_elim v hv
+  simp only [momentX, momentY, avg_expand, h2]
+  ring
+
+lemma discriminant_nonneg (v : Fin 3 → ℝ) (hv : avg v = 0) :
+    (momentY v)^2 ≤ (momentX v)^3 := by
+  have h := discriminant v hv
+  have : 0 ≤ (v 0-v 1)^2*(v 0-v 2)^2*(v 1-v 2)^2 := by positivity
+  linarith
+
+lemma momentX_lt_one (v : Fin 3 → ℝ) (hv : avg v = 0)
+    (hp : ∀ i, 0 < 1+v i) : momentX v < 1 := by
+  have hu := zero_sum_elim v hv
+  have h01 := mul_pos (hp 0) (hp 1)
+  have h02 := mul_pos (hp 0) (hp 2)
+  have h12 := mul_pos (hp 1) (hp 2)
+  simp only [momentX, avg_expand]
+  rw [hu] at *
+  nlinarith
+
+lemma momentY_le_X (v : Fin 3 → ℝ) (hv : avg v = 0)
+    (hp : ∀ i, 0 < 1+v i) : momentY v ≤ momentX v := by
+  have hX := momentX_nonneg v
+  have hX1 := momentX_lt_one v hv hp
+  have hdisc := discriminant_nonneg v hv
+  have hpow : (momentX v)^3 ≤ (momentX v)^2 := by
+    nlinarith [mul_nonneg (sq_nonneg (momentX v)) (by linarith : 0 ≤ 1-momentX v)]
+  nlinarith [sq_nonneg (momentY v-momentX v)]
+
+lemma denom_le_one (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i)
+    (t : ℝ) (ht0 : 0 ≤ t) (ht1 : t ≤ 1) : denom v t ≤ 1 := by
+  have h1 := mul_le_mul_of_nonneg_left (momentY_le_X v hv hp) (pow_nonneg ht0 3)
+  have h2 := mul_nonneg (mul_nonneg (sq_nonneg t) (momentX_nonneg v))
+    (show 0 ≤ 1-t by linarith)
+  have h3 : 0 ≤ t^2*momentX v := mul_nonneg (sq_nonneg t) (momentX_nonneg v)
+  dsimp [denom]
+  nlinarith
+
+noncomputable def cofMoment (v : Fin 3 → ℝ) (j : ℕ) :=
+  avg (fun i => (cofactor v i)^j)
+
+lemma cofMoment_one (v : Fin 3 → ℝ) (hv : avg v = 0) :
+    cofMoment v 1 = 1-momentX v := by
+  have h2 := zero_sum_elim v hv
+  simp only [cofMoment, cofactor, momentX, avg_expand, h2]
+  ring
+
+lemma cofMoment_two (v : Fin 3 → ℝ) (hv : avg v = 0) :
+    cofMoment v 2 = 3*(momentX v)^2-4*momentY v+1 := by
+  have h2 := zero_sum_elim v hv
+  simp only [cofMoment, cofactor, momentX, momentY, avg_expand, h2]
+  ring
+
+lemma cofMoment_three (v : Fin 3 → ℝ) (hv : avg v = 0) :
+    cofMoment v 3 = -9*(momentX v)^3+9*(momentX v)^2+
+      6*momentX v*momentY v+3*momentX v+4*(momentY v)^2-14*momentY v+1 := by
+  have h2 := zero_sum_elim v hv
+  simp only [cofMoment, cofactor, momentX, momentY, avg_expand, h2]
+  ring
+
+noncomputable def numerator (v : Fin 3 → ℝ) (t : ℝ) :=
+  (4/5 : ℝ) * avg (fun i => kernel v t i *
+    ((cofactor v i)^2*denom v 1-3*cofactor v i*(denom v 1)^2+
+      (2*cofMoment v 3-3*cofMoment v 2*denom v 1)*v i))
+
+lemma numerator_expansion (v : Fin 3 → ℝ) (hv : avg v = 0) (t : ℝ) :
+    numerator v t = (24/5 : ℝ)*(A (momentX v) (momentY v)+t*C (momentX v) (momentY v)) := by
+  unfold numerator
+  rw [cofMoment_two v hv, cofMoment_three v hv]
+  have h2 := zero_sum_elim v hv
+  simp only [cofactor, kernel, denom, A, C, momentX, momentY, avg_expand, h2]
+  ring
+
+lemma integral_avg (f : Fin 3 → ℝ → ℝ)
+    (hf : ∀ i, IntervalIntegrable (f i) MeasureTheory.volume 0 1) :
+    (∫ t in (0 : ℝ)..1, avg (fun i => f i t)) = avg (fun i => ∫ t in (0 : ℝ)..1, f i t) := by
+  unfold avg
+  rw [intervalIntegral.integral_div, intervalIntegral.integral_finsetSum (fun i _ => hf i)]
+
+lemma log_integral (a : ℝ) (ha : 0 < 1+a) :
+    Real.log (1+a) = ∫ t in (0 : ℝ)..1, a/(1+t*a) := by
+  have hp : ∀ t ∈ Set.Icc (0 : ℝ) 1, 0 < 1+t*a := by
+    intro t ht
+    rcases eq_or_lt_of_le ht.1 with hz | hz
+    · rw [← hz]; norm_num
+    · nlinarith [mul_pos hz ha, ht.2]
+  have hc : ContinuousOn (fun t : ℝ => a/(1+t*a)) (Set.Icc 0 1) :=
+    continuousOn_const.div (by fun_prop) (fun t ht => ne_of_gt (hp t ht))
+  have hi : IntervalIntegrable (fun t : ℝ => a/(1+t*a)) MeasureTheory.volume 0 1 :=
+    hc.intervalIntegrable_of_Icc (by norm_num)
+  have hD : ∀ t ∈ Set.uIcc (0 : ℝ) 1,
+      HasDerivAt (fun t : ℝ => Real.log (1+t*a)) (a/(1+t*a)) t := by
+    intro t ht
+    have ht' : t ∈ Set.Icc (0 : ℝ) 1 := by simpa using ht
+    simpa using (((hasDerivAt_id t).mul_const a).const_add 1).log (ne_of_gt (hp t ht'))
+  simpa using (intervalIntegral.integral_eq_sub_of_hasDerivAt hD hi).symm
+
+lemma kernel_div_identity (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i)
+    (t : ℝ) (ht0 : 0 ≤ t) (ht1 : t ≤ 1) (i : Fin 3) :
+    v i/(1+t*v i)-avg (fun j => v j/(1+t*v j)) = kernel v t i/denom v t := by
+  have h0 := ne_of_gt (message_segment_pos v hp t ht0 ht1 0)
+  have h1 := ne_of_gt (message_segment_pos v hp t ht0 ht1 1)
+  have h2 := ne_of_gt (message_segment_pos v hp t ht0 ht1 2)
+  have h0c : 1+v 0*t ≠ 0 := by simpa [mul_comm] using h0
+  have h1c : 1+v 1*t ≠ 0 := by simpa [mul_comm] using h1
+  have h2c : 1+v 2*t ≠ 0 := by simpa [mul_comm] using h2
+  have hv2 := zero_sum_elim v hv
+  rw [denom_product v hv t]
+  fin_cases i
+  · change v 0/(1+t*v 0)-avg (fun j => v j/(1+t*v j)) =
+      kernel v t 0/((1+t*v 0)*(1+t*v 1)*(1+t*v 2))
+    simp only [avg_expand, kernel, momentX]
+    field_simp [h0, h1, h2, h0c, h1c, h2c]
+    rw [hv2]
+    ring
+  · change v 1/(1+t*v 1)-avg (fun j => v j/(1+t*v j)) =
+      kernel v t 1/((1+t*v 0)*(1+t*v 1)*(1+t*v 2))
+    simp only [avg_expand, kernel, momentX]
+    field_simp [h0, h1, h2, h0c, h1c, h2c]
+    rw [hv2]
+    ring
+  · change v 2/(1+t*v 2)-avg (fun j => v j/(1+t*v j)) =
+      kernel v t 2/((1+t*v 0)*(1+t*v 1)*(1+t*v 2))
+    simp only [avg_expand, kernel, momentX]
+    field_simp [h0, h1, h2, h0c, h1c, h2c]
+    rw [hv2]
+    ring
+
+lemma centeredLog_integral (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i)
+    (i : Fin 3) : centeredLog v i = ∫ t in (0 : ℝ)..1, kernel v t i / denom v t := by
+  have hi : ∀ j, IntervalIntegrable (fun t : ℝ => v j/(1+t*v j)) MeasureTheory.volume 0 1 := by
+    intro j
+    apply ContinuousOn.intervalIntegrable_of_Icc (by norm_num)
+    refine continuousOn_const.div (by fun_prop) ?_
+    intro t ht
+    exact ne_of_gt (message_segment_pos v hp t ht.1 ht.2 j)
+  have ha : IntervalIntegrable (fun t : ℝ => avg (fun j => v j/(1+t*v j)))
+      MeasureTheory.volume 0 1 := by
+    unfold avg
+    exact (IntervalIntegrable.sum Finset.univ (fun j _ => hi j)).div_const 3
+  unfold centeredLog
+  rw [log_integral (v i) (hp i)]
+  have hlogs : (fun j => Real.log (1+v j)) = (fun j => ∫ t in (0 : ℝ)..1, v j/(1+t*v j)) := by
+    funext j; exact log_integral (v j) (hp j)
+  rw [hlogs, ← integral_avg _ hi, ← intervalIntegral.integral_sub (hi i) ha]
+  apply intervalIntegral.integral_congr
+  intro t ht
+  have ht' : t ∈ Set.Icc (0 : ℝ) 1 := by simpa using ht
+  exact kernel_div_identity v hv hp t ht'.1 ht'.2 i
+
+noncomputable def invMoment (v : Fin 3 → ℝ) (j : ℕ) :=
+  avg (fun i => ((1+v i)⁻¹)^j)
+noncomputable def logInvMoment (v : Fin 3 → ℝ) (j : ℕ) :=
+  avg (fun i => centeredLog v i*((1+v i)⁻¹)^j)
+noncomputable def logMoment (v : Fin 3 → ℝ) := avg (fun i => v i*centeredLog v i)
+
+ 
+noncomputable def gExpr (v : Fin 3 → ℝ) :=
+  6-12*invMoment v 1+8*invMoment v 2-2*invMoment v 3+
+  (4/5 : ℝ)*(logInvMoment v 2-3*logInvMoment v 1+
+    (2*invMoment v 3-3*invMoment v 2)*logMoment v+
+    3*(invMoment v 1-1)*(invMoment v 2-invMoment v 1))
+noncomputable def rCore (v : Fin 3 → ℝ) :=
+  (6-12*(momentX v)^2)*(denom v 1)^3-12*cofMoment v 1*(denom v 1)^2+
+  8*cofMoment v 2*denom v 1-2*cofMoment v 3+
+  (12/5 : ℝ)*(cofMoment v 1-denom v 1)*(cofMoment v 2-cofMoment v 1*denom v 1)
+noncomputable def logCoeff (v : Fin 3 → ℝ) (i : Fin 3) :=
+  (cofactor v i)^2*denom v 1-3*cofactor v i*(denom v 1)^2+
+    (2*cofMoment v 3-3*cofMoment v 2*denom v 1)*v i
+
+lemma inv_cofactor (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i)
+    (i : Fin 3) : (1+v i)⁻¹ = cofactor v i / denom v 1 := by
+  apply (eq_div_iff (ne_of_gt (denom_pos v hv hp 1 (by norm_num) le_rfl))).2
+  rw [← cofactor_identity v hv i]
+  field_simp [ne_of_gt (hp i)]
+
+lemma RN_algebra (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i) :
+    (denom v 1)^3*(gExpr v-12*(momentX v)^2) =
+      rCore v + (4/5 : ℝ)*avg (fun i => centeredLog v i*logCoeff v i) := by
+  have hD := ne_of_gt (denom_pos v hv hp 1 (by norm_num) le_rfl)
+  unfold gExpr invMoment logInvMoment logMoment rCore logCoeff
+  simp_rw [inv_cofactor v hv hp]
+  simp only [cofMoment, avg_expand]
+  field_simp
+  ring
+
+lemma kernel_div_integrable (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i)
+    (i : Fin 3) : IntervalIntegrable (fun t : ℝ => kernel v t i/denom v t)
+      MeasureTheory.volume 0 1 := by
+  apply ContinuousOn.intervalIntegrable_of_Icc (by norm_num)
+  unfold kernel denom
+  refine ContinuousOn.div (by fun_prop) (by fun_prop) ?_
+  intro t ht
+  exact ne_of_gt (denom_pos v hv hp t ht.1 ht.2)
+
+lemma RN_representation (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i) :
+    (denom v 1)^3*(gExpr v-12*(momentX v)^2) =
+      rCore v + ∫ t in (0 : ℝ)..1, numerator v t / denom v t := by
+  rw [RN_algebra v hv hp]
+  congr 1
+  have hg : ∀ t : ℝ, numerator v t/denom v t = (4/5 : ℝ)*
+      avg (fun i => (kernel v t i/denom v t)*logCoeff v i) := by
+    intro t
+    simp only [numerator, logCoeff, avg_expand]
+    ring
+  simp_rw [hg]
+  rw [intervalIntegral.integral_const_mul, integral_avg _
+    (fun i => (kernel_div_integrable v hv hp i).mul_const (logCoeff v i))]
+  congr 1
+  congr 1
+  funext i
+  rw [intervalIntegral.integral_mul_const, centeredLog_integral v hv hp i]
+
+lemma integrate_N_lower (a c x y : ℝ) :
+    (∫ t in (0 : ℝ)..1, (24/5 : ℝ)*(a+t*c)*(1+3*t^2*x-2*t^3*y)) =
+      (24/5 : ℝ)*(a+c/2+a*x+(3*c*x-2*a*y)/4-2*c*y/5) := by
+  let P := fun t : ℝ => (24/5 : ℝ)*(a*t+c*t^2/2+a*x*t^3+
+      (3*c*x-2*a*y)*t^4/4-2*c*y*t^5/5)
+  have hd : ∀ t : ℝ, HasDerivAt P ((24/5 : ℝ)*(a+t*c)*(1+3*t^2*x-2*t^3*y)) t := by
+    intro t
+    dsimp [P]
+    convert (((((hasDerivAt_id t).const_mul a).add
+      (((hasDerivAt_id t).pow 2).const_mul c |>.div_const 2)).add
+      (((hasDerivAt_id t).pow 3).const_mul (a*x))).add
+      (((hasDerivAt_id t).pow 4).const_mul (3*c*x-2*a*y) |>.div_const 4) |>.sub
+      (((hasDerivAt_id t).pow 5).const_mul (2*c*y) |>.div_const 5)).const_mul (24/5 : ℝ) using 1 <;>
+      first | rfl | (norm_num; ring)
+  have hi : IntervalIntegrable (fun t : ℝ => (24/5 : ℝ)*(a+t*c)*(1+3*t^2*x-2*t^3*y))
+      MeasureTheory.volume 0 1 := Continuous.intervalIntegrable (by fun_prop) 0 1
+  rw [intervalIntegral.integral_eq_sub_of_hasDerivAt (fun t _ => hd t) hi]
+  dsimp [P]; ring
+
+lemma L_expansion (v : Fin 3 → ℝ) (hv : avg v = 0) :
+    rCore v + (∫ t in (0 : ℝ)..1, numerator v t*(2-denom v t)) =
+      (6/25 : ℝ)*Lscaled (momentX v) (momentY v) := by
+  have hg : (fun t : ℝ => numerator v t*(2-denom v t)) =
+    (fun t : ℝ => (24/5 : ℝ)*(A (momentX v) (momentY v)+t*C (momentX v) (momentY v))*
+      (1+3*t^2*momentX v-2*t^3*momentY v)) := by
+    funext t
+    rw [numerator_expansion v hv t]
+    dsimp [denom]; ring
+  rw [hg, integrate_N_lower]
+  unfold rCore
+  rw [cofMoment_one v hv, cofMoment_two v hv, cofMoment_three v hv]
+  dsimp [denom, A, C, Lscaled]
+  ring
+
+lemma numerator_nonneg (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i)
+    (t : ℝ) (ht0 : 0 ≤ t) (ht1 : t ≤ 1) : 0 ≤ numerator v t := by
+  have hD : 0 < 1-3*momentX v+2*momentY v := by
+    simpa [denom] using denom_pos v hv hp 1 (by norm_num) le_rfl
+  obtain ⟨ha, hac, _⟩ := physical_signs (momentX v) (momentY v)
+    (momentX_nonneg v) (momentX_lt_one v hv hp) (discriminant_nonneg v hv) hD
+  rw [numerator_expansion v hv t]
+  have h := add_nonneg (mul_nonneg (show 0 ≤ 1-t by linarith) ha) (mul_nonneg ht0 hac)
+  nlinarith only [h]
+
+ 
+
+lemma gExpr_lower (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i) :
+    12*(momentX v)^2 ≤ gExpr v := by
+  have hi1 : IntervalIntegrable (fun t => numerator v t*(2-denom v t))
+      MeasureTheory.volume 0 1 := by
+    apply Continuous.intervalIntegrable
+    unfold numerator kernel denom avg
+    fun_prop
+  have hi2 : IntervalIntegrable (fun t => numerator v t/denom v t)
+      MeasureTheory.volume 0 1 := by
+    apply ContinuousOn.intervalIntegrable_of_Icc (by norm_num)
+    apply ContinuousOn.div
+    · unfold numerator kernel avg; fun_prop
+    · unfold denom; fun_prop
+    · intro t ht; exact ne_of_gt (denom_pos v hv hp t ht.1 ht.2)
+  have hbound : ∀ t ∈ Set.Icc (0 : ℝ) 1, numerator v t*(2-denom v t) ≤ numerator v t/denom v t := by
+    intro t ht
+    have hD := denom_pos v hv hp t ht.1 ht.2
+    apply (le_div_iff₀ hD).2
+    nlinarith [mul_nonneg (numerator_nonneg v hv hp t ht.1 ht.2) (sq_nonneg (1-denom v t))]
+  have hint := intervalIntegral.integral_mono_on (by norm_num) hi1 hi2 hbound
+  have hR := RN_representation v hv hp
+  have hL := L_expansion v hv
+  have hd := denom_pos v hv hp 1 (by norm_num) le_rfl
+  have hdom : 0 < 1-3*momentX v+2*momentY v := by simpa [denom] using hd
+  have hLpos := (physical_signs (momentX v) (momentY v) (momentX_nonneg v)
+    (momentX_lt_one v hv hp) (discriminant_nonneg v hv) hdom).2.2
+  have hprod : 0 ≤ (denom v 1)^3*(gExpr v-12*(momentX v)^2) := by linarith
+  have := nonneg_of_mul_nonneg_right hprod (pow_pos hd 3)
+  linarith
+
+end ThreeState.Radial
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeState/Region.lean b/lean/OAI/Probability/ThreeState/Region.lean
index 25e7426a23c14f7e6ab8dc0abf4ff2ba39233496..491426d25eee97eebdfa41739d2089d74c9794e5 100644
--- a/lean/OAI/Probability/ThreeState/Region.lean
+++ b/lean/OAI/Probability/ThreeState/Region.lean
@@ -1,139 +1,139 @@
-import Mathlib
-import OAI.Probability.ThreeState.Negative
-import OAI.Probability.ThreeState.Supercritical
-
-namespace OAI
-
-/-! Threshold consequences on the proved parameter regions. -/
-
-namespace ThreeState
-open Filter Topology
-
-lemma Nonreconstructs.not_reconstructs {a : ℕ → ℝ} (hz : Nonreconstructs a) :
-    ¬ Reconstructs a := by
-  rintro ⟨L, hL, hlim⟩
-  have he : L = 0 := tendsto_nhds_unique hlim hz
-  exact (ne_of_gt hL) he
-
- 
-
-theorem observed_nonreconstructs_iff_not_reconstructs (offspring : PMF ℕ)
-    (lam : ℝ) (h : Admissible lam) :
-    Nonreconstructs (fun n => advantage (observedLaw offspring lam h n)) ↔
-      ¬ Reconstructs (fun n => advantage (observedLaw offspring lam h n)) := by
-  refine ⟨Nonreconstructs.not_reconstructs, ?_⟩
-  intro hn
-  obtain ⟨L, hL, _, hlim⟩ := observedAdvantage_tendsto offspring lam h
-  have he : L = 0 := by
-    by_contra hne
-    exact hn ⟨L, lt_of_le_of_ne hL (Ne.symm hne), hlim⟩
-  simpa only [Nonreconstructs, he] using hlim
-
- 
-
-theorem regular_nonreconstruction_in_proved_region (b : ℕ) (hb : 2 ≤ b)
-    (lam : ℝ) (h : Admissible lam) (hc : (b : ℝ) * lam ^ 2 ≤ 1)
-    (hregion : 0 ≤ lam ∨ (b : ℝ) * |lam| ≤ 1 ∨ 4 * (b : ℝ) * lam ^ 2 ≤ 1) :
-    Nonreconstructs (regularAdvantage b lam h) := by
-  rcases hregion with hn | hD | hs
-  · exact regular_nonnegative_nonreconstruction b hb lam h hn hc
-  · exact regular_dobrushin_nonreconstruction b hb lam h hD
-  · by_cases hn : 0 ≤ lam
-    · exact regular_nonnegative_nonreconstruction b hb lam h hn hc
-    · exact regular_small_negative_nonreconstruction b hb lam h (lt_of_not_ge hn).le hs
-
- 
-
-theorem poisson_nonreconstruction_in_proved_region (d : ℝ) (hd : 1 < d)
-    (lam : ℝ) (h : Admissible lam) (hc : d * lam ^ 2 ≤ 1)
-    (hregion : 0 ≤ lam ∨ d * |lam| ≤ 1 ∨ 4 * d * lam ^ 2 ≤ 1) :
-    Nonreconstructs (poissonAdvantage d hd lam h) := by
-  rcases hregion with hn | hD | hs
-  · exact poisson_nonnegative_nonreconstruction d hd lam h hn hc
-  · exact poisson_dobrushin_nonreconstruction d hd lam h hD
-  · by_cases hn : 0 ≤ lam
-    · exact poisson_nonnegative_nonreconstruction d hd lam h hn hc
-    · exact poisson_small_negative_nonreconstruction d hd lam h (lt_of_not_ge hn).le hs
-
- 
-
-theorem regular_threshold_in_proved_region (b : ℕ) (hb : 2 ≤ b) (lam : ℝ)
-    (h : Admissible lam)
-    (hregion : 1 < (b : ℝ) * lam ^ 2 ∨ 0 ≤ lam ∨
-      (b : ℝ) * |lam| ≤ 1 ∨ 4 * (b : ℝ) * lam ^ 2 ≤ 1) :
-    (Reconstructs (regularAdvantage b lam h) ↔ 1 < (b : ℝ) * lam ^ 2) ∧
-    ((b : ℝ) * lam ^ 2 ≤ 1 → Nonreconstructs (regularAdvantage b lam h)) := by
-  have hz (hc : (b : ℝ) * lam ^ 2 ≤ 1) :
-      Nonreconstructs (regularAdvantage b lam h) := by
-    rcases hregion with hsuper | hknown
-    · exact False.elim ((not_lt_of_ge hc) hsuper)
-    · exact regular_nonreconstruction_in_proved_region b hb lam h hc hknown
-  refine ⟨⟨?_, regular_supercritical b hb lam h⟩, hz⟩
-  intro hr
-  by_contra hcrit
-  exact (hz (le_of_not_gt hcrit)).not_reconstructs hr
-
- 
-
-theorem poisson_threshold_in_proved_region (d : ℝ) (hd : 1 < d) (lam : ℝ)
-    (h : Admissible lam)
-    (hregion : 1 < d * lam ^ 2 ∨ 0 ≤ lam ∨ d * |lam| ≤ 1 ∨ 4 * d * lam ^ 2 ≤ 1) :
-    (Reconstructs (poissonAdvantage d hd lam h) ↔ 1 < d * lam ^ 2) ∧
-    (d * lam ^ 2 ≤ 1 → Nonreconstructs (poissonAdvantage d hd lam h)) := by
-  have hz (hc : d * lam ^ 2 ≤ 1) : Nonreconstructs (poissonAdvantage d hd lam h) := by
-    rcases hregion with hsuper | hknown
-    · exact False.elim ((not_lt_of_ge hc) hsuper)
-    · exact poisson_nonreconstruction_in_proved_region d hd lam h hc hknown
-  refine ⟨⟨?_, poisson_supercritical d hd lam h⟩, hz⟩
-  intro hr
-  by_contra hcrit
-  exact (hz (le_of_not_gt hcrit)).not_reconstructs hr
-
- 
-
-theorem regular_nonnegative_threshold (b : ℕ) (hb : 2 ≤ b) (lam : ℝ)
-    (h : Admissible lam) (hn : 0 ≤ lam) :
-    (Reconstructs (regularAdvantage b lam h) ↔ 1 < (b : ℝ) * lam ^ 2) ∧
-    ((b : ℝ) * lam ^ 2 ≤ 1 → Nonreconstructs (regularAdvantage b lam h)) :=
-  regular_threshold_in_proved_region b hb lam h (Or.inr (Or.inl hn))
-
- 
-
-theorem poisson_nonnegative_threshold (d : ℝ) (hd : 1 < d) (lam : ℝ)
-    (h : Admissible lam) (hn : 0 ≤ lam) :
-    (Reconstructs (poissonAdvantage d hd lam h) ↔ 1 < d * lam ^ 2) ∧
-    (d * lam ^ 2 ≤ 1 → Nonreconstructs (poissonAdvantage d hd lam h)) :=
-  poisson_threshold_in_proved_region d hd lam h (Or.inr (Or.inl hn))
-
- 
-
-theorem binary_exact_threshold (lam : ℝ) (h : Admissible lam) :
-    (Reconstructs (regularAdvantage 2 lam h) ↔ 1 < (2 : ℝ) * lam ^ 2) ∧
-    ((2 : ℝ) * lam ^ 2 ≤ 1 → Nonreconstructs (regularAdvantage 2 lam h)) := by
-  apply regular_threshold_in_proved_region 2 (by omega) lam h
-  by_cases hn : 0 ≤ lam
-  · exact Or.inr (Or.inl hn)
-  · refine Or.inr (Or.inr (Or.inl ?_))
-    rw [abs_of_nonpos (lt_of_not_ge hn).le]
-    norm_num
-    linarith [h.1]
-
- 
-
-theorem poisson_le_two_exact_threshold (d : ℝ) (hd : 1 < d) (hupper : d ≤ 2)
-    (lam : ℝ) (h : Admissible lam) :
-    (Reconstructs (poissonAdvantage d hd lam h) ↔ 1 < d * lam ^ 2) ∧
-    (d * lam ^ 2 ≤ 1 → Nonreconstructs (poissonAdvantage d hd lam h)) := by
-  apply poisson_threshold_in_proved_region d hd lam h
-  by_cases hn : 0 ≤ lam
-  · exact Or.inr (Or.inl hn)
-  · refine Or.inr (Or.inr (Or.inl ?_))
-    have hs : |lam| ≤ 1 / 2 := by
-      rw [abs_of_nonpos (lt_of_not_ge hn).le]
-      linarith [h.1]
-    nlinarith [mul_nonneg (show 0 ≤ d by linarith)
-      (show 0 ≤ 1 / 2 - |lam| by linarith)]
-
-end ThreeState
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeState.Negative
+import OAI.Probability.ThreeState.Supercritical
+
+namespace OAI
+
+/-! Threshold consequences on the proved parameter regions. -/
+
+namespace ThreeState
+open Filter Topology
+
+lemma Nonreconstructs.not_reconstructs {a : ℕ → ℝ} (hz : Nonreconstructs a) :
+    ¬ Reconstructs a := by
+  rintro ⟨L, hL, hlim⟩
+  have he : L = 0 := tendsto_nhds_unique hlim hz
+  exact (ne_of_gt hL) he
+
+ 
+
+theorem observed_nonreconstructs_iff_not_reconstructs (offspring : PMF ℕ)
+    (lam : ℝ) (h : Admissible lam) :
+    Nonreconstructs (fun n => advantage (observedLaw offspring lam h n)) ↔
+      ¬ Reconstructs (fun n => advantage (observedLaw offspring lam h n)) := by
+  refine ⟨Nonreconstructs.not_reconstructs, ?_⟩
+  intro hn
+  obtain ⟨L, hL, _, hlim⟩ := observedAdvantage_tendsto offspring lam h
+  have he : L = 0 := by
+    by_contra hne
+    exact hn ⟨L, lt_of_le_of_ne hL (Ne.symm hne), hlim⟩
+  simpa only [Nonreconstructs, he] using hlim
+
+ 
+
+theorem regular_nonreconstruction_in_proved_region (b : ℕ) (hb : 2 ≤ b)
+    (lam : ℝ) (h : Admissible lam) (hc : (b : ℝ) * lam ^ 2 ≤ 1)
+    (hregion : 0 ≤ lam ∨ (b : ℝ) * |lam| ≤ 1 ∨ 4 * (b : ℝ) * lam ^ 2 ≤ 1) :
+    Nonreconstructs (regularAdvantage b lam h) := by
+  rcases hregion with hn | hD | hs
+  · exact regular_nonnegative_nonreconstruction b hb lam h hn hc
+  · exact regular_dobrushin_nonreconstruction b hb lam h hD
+  · by_cases hn : 0 ≤ lam
+    · exact regular_nonnegative_nonreconstruction b hb lam h hn hc
+    · exact regular_small_negative_nonreconstruction b hb lam h (lt_of_not_ge hn).le hs
+
+ 
+
+theorem poisson_nonreconstruction_in_proved_region (d : ℝ) (hd : 1 < d)
+    (lam : ℝ) (h : Admissible lam) (hc : d * lam ^ 2 ≤ 1)
+    (hregion : 0 ≤ lam ∨ d * |lam| ≤ 1 ∨ 4 * d * lam ^ 2 ≤ 1) :
+    Nonreconstructs (poissonAdvantage d hd lam h) := by
+  rcases hregion with hn | hD | hs
+  · exact poisson_nonnegative_nonreconstruction d hd lam h hn hc
+  · exact poisson_dobrushin_nonreconstruction d hd lam h hD
+  · by_cases hn : 0 ≤ lam
+    · exact poisson_nonnegative_nonreconstruction d hd lam h hn hc
+    · exact poisson_small_negative_nonreconstruction d hd lam h (lt_of_not_ge hn).le hs
+
+ 
+
+theorem regular_threshold_in_proved_region (b : ℕ) (hb : 2 ≤ b) (lam : ℝ)
+    (h : Admissible lam)
+    (hregion : 1 < (b : ℝ) * lam ^ 2 ∨ 0 ≤ lam ∨
+      (b : ℝ) * |lam| ≤ 1 ∨ 4 * (b : ℝ) * lam ^ 2 ≤ 1) :
+    (Reconstructs (regularAdvantage b lam h) ↔ 1 < (b : ℝ) * lam ^ 2) ∧
+    ((b : ℝ) * lam ^ 2 ≤ 1 → Nonreconstructs (regularAdvantage b lam h)) := by
+  have hz (hc : (b : ℝ) * lam ^ 2 ≤ 1) :
+      Nonreconstructs (regularAdvantage b lam h) := by
+    rcases hregion with hsuper | hknown
+    · exact False.elim ((not_lt_of_ge hc) hsuper)
+    · exact regular_nonreconstruction_in_proved_region b hb lam h hc hknown
+  refine ⟨⟨?_, regular_supercritical b hb lam h⟩, hz⟩
+  intro hr
+  by_contra hcrit
+  exact (hz (le_of_not_gt hcrit)).not_reconstructs hr
+
+ 
+
+theorem poisson_threshold_in_proved_region (d : ℝ) (hd : 1 < d) (lam : ℝ)
+    (h : Admissible lam)
+    (hregion : 1 < d * lam ^ 2 ∨ 0 ≤ lam ∨ d * |lam| ≤ 1 ∨ 4 * d * lam ^ 2 ≤ 1) :
+    (Reconstructs (poissonAdvantage d hd lam h) ↔ 1 < d * lam ^ 2) ∧
+    (d * lam ^ 2 ≤ 1 → Nonreconstructs (poissonAdvantage d hd lam h)) := by
+  have hz (hc : d * lam ^ 2 ≤ 1) : Nonreconstructs (poissonAdvantage d hd lam h) := by
+    rcases hregion with hsuper | hknown
+    · exact False.elim ((not_lt_of_ge hc) hsuper)
+    · exact poisson_nonreconstruction_in_proved_region d hd lam h hc hknown
+  refine ⟨⟨?_, poisson_supercritical d hd lam h⟩, hz⟩
+  intro hr
+  by_contra hcrit
+  exact (hz (le_of_not_gt hcrit)).not_reconstructs hr
+
+ 
+
+theorem regular_nonnegative_threshold (b : ℕ) (hb : 2 ≤ b) (lam : ℝ)
+    (h : Admissible lam) (hn : 0 ≤ lam) :
+    (Reconstructs (regularAdvantage b lam h) ↔ 1 < (b : ℝ) * lam ^ 2) ∧
+    ((b : ℝ) * lam ^ 2 ≤ 1 → Nonreconstructs (regularAdvantage b lam h)) :=
+  regular_threshold_in_proved_region b hb lam h (Or.inr (Or.inl hn))
+
+ 
+
+theorem poisson_nonnegative_threshold (d : ℝ) (hd : 1 < d) (lam : ℝ)
+    (h : Admissible lam) (hn : 0 ≤ lam) :
+    (Reconstructs (poissonAdvantage d hd lam h) ↔ 1 < d * lam ^ 2) ∧
+    (d * lam ^ 2 ≤ 1 → Nonreconstructs (poissonAdvantage d hd lam h)) :=
+  poisson_threshold_in_proved_region d hd lam h (Or.inr (Or.inl hn))
+
+ 
+
+theorem binary_exact_threshold (lam : ℝ) (h : Admissible lam) :
+    (Reconstructs (regularAdvantage 2 lam h) ↔ 1 < (2 : ℝ) * lam ^ 2) ∧
+    ((2 : ℝ) * lam ^ 2 ≤ 1 → Nonreconstructs (regularAdvantage 2 lam h)) := by
+  apply regular_threshold_in_proved_region 2 (by omega) lam h
+  by_cases hn : 0 ≤ lam
+  · exact Or.inr (Or.inl hn)
+  · refine Or.inr (Or.inr (Or.inl ?_))
+    rw [abs_of_nonpos (lt_of_not_ge hn).le]
+    norm_num
+    linarith [h.1]
+
+ 
+
+theorem poisson_le_two_exact_threshold (d : ℝ) (hd : 1 < d) (hupper : d ≤ 2)
+    (lam : ℝ) (h : Admissible lam) :
+    (Reconstructs (poissonAdvantage d hd lam h) ↔ 1 < d * lam ^ 2) ∧
+    (d * lam ^ 2 ≤ 1 → Nonreconstructs (poissonAdvantage d hd lam h)) := by
+  apply poisson_threshold_in_proved_region d hd lam h
+  by_cases hn : 0 ≤ lam
+  · exact Or.inr (Or.inl hn)
+  · refine Or.inr (Or.inr (Or.inl ?_))
+    have hs : |lam| ≤ 1 / 2 := by
+      rw [abs_of_nonpos (lt_of_not_ge hn).le]
+      linarith [h.1]
+    nlinarith [mul_nonneg (show 0 ≤ d by linarith)
+      (show 0 ≤ 1 / 2 - |lam| by linarith)]
+
+end ThreeState
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeState/RegularCritical.lean b/lean/OAI/Probability/ThreeState/RegularCritical.lean
index 68a375045ca1af1222ea7f4ba107995bcbf89990..ef9e48eec73c93fd0ac6301a12d0ca434a73e2e4 100644
--- a/lean/OAI/Probability/ThreeState/RegularCritical.lean
+++ b/lean/OAI/Probability/ThreeState/RegularCritical.lean
@@ -1,373 +1,373 @@
-import Mathlib
-import OAI.Probability.ThreeState.Projection
-import OAI.Probability.ThreeState.ScalarBounds
-
-namespace OAI
-
-/-! Entropy bounds and non-reconstruction at the positive regular-tree threshold. -/
-
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-
-lemma regular_fixedpoint_degree_nu (b : ℕ) (lam : ℝ) (h : Admissible lam)
-    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint (PMF.pure b) lam h Q) :
-    posteriorNu (degreePosterior lam h Q hQ.1 b) = posteriorNu Q := by
-  have hh := hQ.2 ⟨fun m => (messageX m)^2, continuous_messageX.pow 2⟩
-  rw [mean_pure] at hh
-  change (∫ m, (messageX m)^2 ∂(Q : Measure Message)) =
-    ∫ m : Fin b → Message, normalizer lam h m*(messageX (combineMessage lam h m))^2 ∂(iidMeasure Q b : Measure (Fin b → Message)) at hh
-  change (∫ m, (messageX m)^2 ∂degreeOutput lam h Q b) = _
-  rw [integral_degreeOutput lam h Q b (fun m => (messageX m)^2) (continuous_messageX.pow 2).measurable]
-  exact hh.symm
-
-lemma partial_nu_le_regular (b : ℕ) (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint (PMF.pure b) lam h Q) {j : ℕ} (hj : j ≤ b) :
-    posteriorNu (degreePosterior lam h Q hQ.1 j) ≤ posteriorNu Q := by
-  rw [← regular_fixedpoint_degree_nu b lam h Q hQ]
-  exact degree_nu_monotone lam h hl0 hl1 Q hQ.1 hj
-
-lemma correlationEntropy_sum (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (b : ℕ) :
-    correlationEntropy lam h Q b =
-      ∑ j ∈ Finset.range b, pairCorrelation lam h Q (degreePosterior lam h Q hQ j) := by
-  induction b with
-  | zero => simp [correlationEntropy_zero]
-  | succ b ih =>
-    rw [Finset.sum_range_succ, ← ih]
-    have hh := correlationEntropy_succ lam h hl0 hl1 Q hQ b
-    change correlationEntropy lam h Q (b+1) = correlationEntropy lam h Q b +
-      pairCorrelation lam h Q (degreePosterior lam h Q hQ b) at hh
-    exact hh
-
-lemma scalar_sqrt_projection_bound {j beta : ℝ} (hj : 1 ≤ j) (hb : 0 ≤ beta) :
-    Real.sqrt (j*beta/(1+(j-1)*beta)) ≤ Real.sqrt (j*beta) := by
-  apply Real.sqrt_le_sqrt
-  have hd : 1 ≤ 1+(j-1)*beta := by nlinarith
-  exact div_le_self (mul_nonneg (by linarith) hb) hd
-
-lemma regular_increment_bound (b : ℕ) (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
-    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint (PMF.pure b) lam h Q) (hs : SpinSymmetric Q)
-    (he : 0 ≤ posteriorEta Q) (j : ℕ) (hj : j < b) :
-    3*lam^4*(posteriorMu Q)^2*((j:ℝ)/(1+((j:ℝ)-1)*(lam^2*posteriorMu Q)))-
-      lam^4*posteriorMu Q*posteriorNu Q*Real.sqrt j ≤
-      3*pairCorrelation lam h Q (degreePosterior lam h Q hQ.1 j) := by
-  by_cases hj0 : j=0
-  · subst j
-    simp only [Nat.cast_zero, zero_div, mul_zero, Real.sqrt_zero, sub_zero]
-    exact mul_nonneg (by norm_num) (pairCorrelation_nonneg lam h hl0 hl1 Q _ hQ.1)
-  · have hj1 : 1 ≤ j := by omega
-    have hjr : (1:ℝ) ≤ j := by exact_mod_cast hj1
-    have hm := posteriorMu_nonneg Q
-    have hn := posteriorNu_nonneg Q
-    let L := (j:ℝ)*(lam^2*posteriorMu Q)/(1+((j:ℝ)-1)*(lam^2*posteriorMu Q))
-    have hb : 0 ≤ lam^2*posteriorMu Q := mul_nonneg (sq_nonneg _) hm
-    have hL : 0 ≤ L := div_nonneg (mul_nonneg (by positivity) hb) (by positivity)
-    have hh := pairCorrelation_truncated lam h hl0 hl1 Q (degreePosterior lam h Q hQ.1 j) hQ.1 hs he
-      (partial_nu_le_regular b lam h hl0 hl1 Q hQ (Nat.le_of_lt hj)) hL
-      (degree_projection lam h hl0 hl1 Q hQ.1 hs j hj1)
-    have hsqrt : Real.sqrt L ≤ lam*Real.sqrt (posteriorMu Q)*Real.sqrt j := by
-      have hh := scalar_sqrt_projection_bound hjr hb
-      rw [Real.sqrt_mul (show (0:ℝ)≤j by positivity), Real.sqrt_mul (sq_nonneg lam), Real.sqrt_sq_eq_abs, abs_of_nonneg hl0] at hh
-      dsimp only [L]
-      nlinarith only [hh]
-    have hmul := mul_le_mul_of_nonneg_left hsqrt (show 0 ≤ lam^3*Real.sqrt (posteriorMu Q)*posteriorNu Q by positivity)
-    have hsq := Real.sq_sqrt hm
-    have hlead : 3*lam^2*posteriorMu Q*L =
-        3*lam^4*(posteriorMu Q)^2*((j:ℝ)/(1+((j:ℝ)-1)*(lam^2*posteriorMu Q))) := by dsimp only [L]; ring
-    rw [hlead] at hh
-    have hloss : lam^3*Real.sqrt (posteriorMu Q)*posteriorNu Q*(lam*Real.sqrt (posteriorMu Q)*Real.sqrt j) =
-        lam^4*posteriorMu Q*posteriorNu Q*Real.sqrt j := by
-      calc
-        _ = lam^4*(Real.sqrt (posteriorMu Q))^2*posteriorNu Q*Real.sqrt j := by ring
-        _ = _ := by rw [hsq]
-    rw [hloss] at hmul
-    linarith only [hh, hmul]
-
-end ThreeState
-
-namespace ThreeState
-open scoped BigOperators
-
-lemma sum_range_real (b : ℕ) : (∑ j ∈ Finset.range b, (j:ℝ)) = (b:ℝ)*(b-1)/2 := by
-  induction b with
-  | zero => norm_num
-  | succ b ih => rw [Finset.sum_range_succ, ih]; push_cast; ring
-
-lemma sum_range_factorial_two (b : ℕ) :
-    (∑ j ∈ Finset.range b, (j:ℝ)*((j:ℝ)-1)) = (b:ℝ)*(b-1)*(b-2)/3 := by
-  induction b with
-  | zero => norm_num
-  | succ b ih => rw [Finset.sum_range_succ, ih]; push_cast; ring
-
-lemma reciprocal_tangent {x y D : ℝ} (hx : 0 ≤ x) (hy : 0 < y) (hD : 0 < D) :
-    x*(2*D-y)/D^2 ≤ x/y := by
-  apply (div_le_div_iff₀ (sq_pos_of_pos hD) hy).mpr
-  nlinarith only [mul_nonneg hx (sq_nonneg (D-y))]
-
-lemma regular_rational_sum (b : ℕ) (hb : 2 ≤ b) (lam mu : ℝ) (hc : (b:ℝ)*lam^2=1) (hm : 0 ≤ mu) :
-    (1-lam^2)/(2*(1+2*mu/3)) ≤
-      lam^4*(∑ j ∈ Finset.range b, (j:ℝ)/(1+((j:ℝ)-1)*(lam^2*mu))) := by
-  let D := 1+2*mu/3
-  have hD : 0 < D := by dsimp [D]; positivity
-  have hbr : (2:ℝ) ≤ b := by exact_mod_cast hb
-  have hb0 : (0:ℝ) ≤ b := by positivity
-  have hbm : 0 ≤ (b:ℝ)-1 := by linarith
-  have hterm (j : ℕ) :
-      ((2*D-1)/D^2)*(j:ℝ)-((lam^2*mu)/D^2)*((j:ℝ)*((j:ℝ)-1)) ≤
-        (j:ℝ)/(1+((j:ℝ)-1)*(lam^2*mu)) := by
-    by_cases hj : j=0
-    · subst j; norm_num
-    · have hjr : (1:ℝ) ≤ j := by exact_mod_cast (show 1≤j by omega)
-      have hd : 0<1+((j:ℝ)-1)*(lam^2*mu) := by positivity
-      have hh := reciprocal_tangent (show (0:ℝ)≤j by positivity) hd hD
-      convert hh using 1
-      ring
-  have hs := Finset.sum_le_sum (s := Finset.range b) (fun j _ => hterm j)
-  rw [Finset.sum_sub_distrib, ← Finset.mul_sum, ← Finset.mul_sum,
-    sum_range_real, sum_range_factorial_two] at hs
-  have haux : (b:ℝ)*(b-1)/(2*D) ≤
-      ((2*D-1)/D^2)*((b:ℝ)*(b-1)/2)-((lam^2*mu)/D^2)*((b:ℝ)*(b-1)*(b-2)/3) := by
-    apply (le_sub_iff_add_le).mpr
-    field_simp [ne_of_gt hD]
-    dsimp only [D]
-    have hp : 0 ≤ mu*((b:ℝ)-1)*lam^2 := by positivity
-    nlinarith only [hp, congrArg (fun x => mu*((b:ℝ)-1)*x) hc]
-  have hh := mul_le_mul_of_nonneg_left (haux.trans hs) (show (0:ℝ) ≤ lam^4 by positivity)
-  have he : lam^4*((b:ℝ)*(b-1)/(2*D)) = (1-lam^2)/(2*D) := by
-    have ht : lam^4*((b:ℝ)*(b-1)) = 1-lam^2 := by
-      calc
-        _ = ((b:ℝ)*lam^2)*(((b:ℝ)*lam^2)-lam^2) := by ring
-        _ = _ := by rw [hc]; ring
-    calc
-      _ = (lam^4*((b:ℝ)*(b-1)))/(2*D) := by ring
-      _ = _ := by rw [ht]
-  change lam^4*((b:ℝ)*(b-1)/(2*D)) ≤ lam^4*(∑ j ∈ Finset.range b, (j:ℝ)/(1+((j:ℝ)-1)*(lam^2*mu))) at hh
-  rwa [he] at hh
-
-lemma regular_sqrt_sum_sq (b : ℕ) (hb : 1 ≤ b) :
-    (∑ j ∈ Finset.range b, Real.sqrt (j:ℝ))^2 ≤ (b:ℝ)*(b-1)^2/2 := by
-  let f : ℕ → ℝ := fun j => if j=0 then 0 else 1
-  let g : ℕ → ℝ := fun j => Real.sqrt (j:ℝ)
-  have hf : (∑ j ∈ Finset.range b, (f j)^2) = (b:ℝ)-1 := by
-    have hzero : 0 ∈ Finset.range b := Finset.mem_range.mpr (by omega)
-    have he (j : ℕ) : (f j)^2 = 1-(if j=0 then (1:ℝ) else 0) := by dsimp only [f]; split_ifs <;> norm_num
-    simp only [he, Finset.sum_sub_distrib, Finset.sum_const, Finset.card_range, nsmul_eq_mul,
-      Finset.sum_ite_eq', hzero, ite_true, mul_one]
-  have hg : (∑ j ∈ Finset.range b, (g j)^2) = (b:ℝ)*(b-1)/2 := by
-    simp only [g, Real.sq_sqrt (Nat.cast_nonneg _), sum_range_real]
-  have hfg : (∑ j ∈ Finset.range b, f j*g j) = ∑ j ∈ Finset.range b, Real.sqrt (j:ℝ) := by
-    apply Finset.sum_congr rfl
-    intro j _
-    by_cases hj : j=0 <;> simp [f,g,hj]
-  have hh := Finset.sum_mul_sq_le_sq_mul_sq (Finset.range b) f g
-  rw [hf,hg,hfg] at hh
-  nlinarith only [hh]
-
-lemma regular_sqrt_sum (b : ℕ) (hb : 2 ≤ b) (lam : ℝ) (hl : 0 ≤ lam) (hc : (b:ℝ)*lam^2=1) :
-    lam^4*(∑ j ∈ Finset.range b, Real.sqrt (j:ℝ)) ≤ (1-lam^2)*lam/Real.sqrt 2 := by
-  have hbr : (2:ℝ) ≤ b := by exact_mod_cast hb
-  have ha : 0 ≤ 1-lam^2 := by nlinarith [sq_nonneg lam]
-  have hs := regular_sqrt_sum_sq b (by omega)
-  have hp := mul_le_mul_of_nonneg_left hs (show 0 ≤ 2*lam^8 by positivity)
-  have he : 2*lam^8*((b:ℝ)*(b-1)^2/2) = ((1-lam^2)*lam)^2 := by
-    calc
-      _ = lam^2*((b:ℝ)*lam^2)*(((b:ℝ)*lam^2)-lam^2)^2 := by ring
-      _ = _ := by rw [hc]; ring
-  rw [he] at hp
-  have hs0 : 0 ≤ ∑ j ∈ Finset.range b, Real.sqrt (j:ℝ) := Finset.sum_nonneg (fun j _ => Real.sqrt_nonneg _)
-  have hr : 0 < Real.sqrt 2 := by positivity
-  apply (le_div_iff₀ hr).mpr
-  have hroot : (Real.sqrt 2)^2=2 := Real.sq_sqrt (by norm_num)
-  have hx : 0 ≤ lam^4*(∑ j ∈ Finset.range b, Real.sqrt (j:ℝ))*Real.sqrt 2 := by positivity
-  have hy : 0 ≤ (1-lam^2)*lam := mul_nonneg ha hl
-  apply (sq_le_sq₀ hx hy).mp
-  have hsq : (lam^4*(∑ j ∈ Finset.range b, Real.sqrt (j:ℝ))*Real.sqrt 2)^2 =
-      2*lam^8*(∑ j ∈ Finset.range b, Real.sqrt (j:ℝ))^2 := by
-    calc
-      _ = lam^8*(∑ j ∈ Finset.range b, Real.sqrt (j:ℝ))^2*(Real.sqrt 2)^2 := by ring
-      _ = _ := by rw [hroot]; ring
-  rwa [hsq]
-
-end ThreeState
-
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-
-lemma regular_correlation_bound (b : ℕ) (hb : 2 ≤ b) (lam : ℝ) (h : Admissible lam)
-    (hl0 : 0 ≤ lam) (hl1 : lam < 1) (hc : (b:ℝ)*lam^2=1)
-    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint (PMF.pure b) lam h Q)
-    (hs : SpinSymmetric Q) (he : 0 ≤ posteriorEta Q) :
-    (1-lam^2)*(3*(posteriorMu Q)^2/(2*(1+2*posteriorMu Q/3))-
-      lam*posteriorMu Q*posteriorNu Q/Real.sqrt 2) ≤ 3*correlationEntropy lam h Q b := by
-  have hm := posteriorMu_nonneg Q
-  have hn := posteriorNu_nonneg Q
-  have hsum := Finset.sum_le_sum (s := Finset.range b) (fun j hj =>
-    regular_increment_bound b lam h hl0 hl1 Q hQ hs he j (Finset.mem_range.mp hj))
-  rw [Finset.sum_sub_distrib, ← Finset.mul_sum, ← Finset.mul_sum, ← Finset.mul_sum,
-    ← correlationEntropy_sum lam h hl0 hl1 Q hQ.1 b] at hsum
-  have hpos := mul_le_mul_of_nonneg_left (regular_rational_sum b hb lam (posteriorMu Q) hc hm)
-    (show 0 ≤ 3*(posteriorMu Q)^2 by positivity)
-  have hneg := mul_le_mul_of_nonneg_left (regular_sqrt_sum b hb lam hl0 hc)
-    (mul_nonneg hm hn)
-  calc
-    _ = 3*(posteriorMu Q)^2*((1-lam^2)/(2*(1+2*posteriorMu Q/3)))-
-        (posteriorMu Q*posteriorNu Q)*((1-lam^2)*lam/Real.sqrt 2) := by ring
-    _ ≤ 3*(posteriorMu Q)^2*(lam^4*(∑ j ∈ Finset.range b, (j:ℝ)/(1+((j:ℝ)-1)*(lam^2*posteriorMu Q))))-
-        (posteriorMu Q*posteriorNu Q)*(lam^4*(∑ j ∈ Finset.range b, Real.sqrt (j:ℝ))) := sub_le_sub hpos hneg
-    _ = _ := by ring
-    _ ≤ _ := hsum
-
-lemma entropyF_sourceF (m : Message) : entropyF m = Radial.sourceF (fun i => m i) := by
-  simp only [entropyF, entropyI, entropyJ, entropyB, Radial.sourceF, mLog, mCenteredLog,
-    Radial.avg_expand]
-  ring
-
-lemma continuous_entropyF_edge (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1) :
-    Continuous (fun m => entropyF (edgeMessage lam h m)) :=
-  (((continuous_entropyI_edge lam h hl0 hl1).const_mul 3).sub
-    ((continuous_entropyJ_edge lam h hl0 hl1).const_mul 2)).add
-      ((continuous_entropyB_edge lam h hl0 hl1).const_mul (4/5:ℝ))
-
-lemma integral_entropyF_radial (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
-    (hl0 : 0 < lam) (hl1 : lam < 1) (Q : ProbabilityMeasure Message)
-    (hQ : IsPosteriorFixedPoint offspring lam h Q)
-    (hD2 : HasMean offspring (fun n : ℕ => (n:ℝ)^2)) :
-    (1-lam^2)/2*posteriorNu Q ≤ (∫ m, entropyF m ∂(Q : Measure Message))-
-      lam⁻¹^2*(∫ m, entropyF (edgeMessage lam h m) ∂(Q : Measure Message)) := by
-  have hi := integrable_entropyF_fixedpoint offspring lam h hl0.le hl1 Q hQ hD2
-  have hie := integrable_continuous_message Q (continuous_entropyF_edge lam h hl0.le hl1)
-  have hp := positive_fixedpoint offspring lam h hl0.le hl1 Q hQ
-  have hh := integral_mono_ae ((integrable_continuous_message Q (continuous_messageX.pow 2)).const_mul ((1-lam^2)/2))
-    (show Integrable (fun m => entropyF m-lam⁻¹^2*entropyF (edgeMessage lam h m)) (Q : Measure Message) from hi.sub (hie.const_mul _))
-    (hp.mono (fun m hm => by
-      have hh := Radial.radial_source (fun i => m i) (average_message m) hm lam hl0 hl1
-      change (1-lam^2)/2*(messageX m)^2 ≤ entropyF m-lam⁻¹^2*entropyF (edgeMessage lam h m)
-      rw [entropyF_sourceF, entropyF_sourceF]
-      convert hh using 1 <;> simp only [edgeMessage, messageX, messageDeviation, Radial.momentX]
-      ring))
-  rw [integral_const_mul, integral_sub hi (hie.const_mul _), integral_const_mul] at hh
-  exact hh
-
-lemma regular_entropy_bound (b : ℕ) (hb : 2 ≤ b) (lam : ℝ) (h : Admissible lam)
-    (hl0 : 0 < lam) (hl1 : lam < 1) (hc : (b:ℝ)*lam^2=1)
-    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint (PMF.pure b) lam h Q)
-    (hs : SpinSymmetric Q) (he : 0 ≤ posteriorEta Q) :
-    posteriorNu Q/2+3*(posteriorMu Q)^2/(2*(1+2*posteriorMu Q/3))-
-      lam*posteriorMu Q*posteriorNu Q/Real.sqrt 2 ≤
-        (8/5:ℝ)*(∫ m, entropyJ m ∂(Q : Measure Message))^2 := by
-  have hD2 : HasMean (PMF.pure b) (fun n : ℕ => (n:ℝ)^2) := hasMean_pure _ _
-  have hD := hasMean_of_square hD2
-  have hid := entropy_identity (PMF.pure b) lam h hl0.le hl1 Q hQ hs hD2
-  have hrad := integral_entropyF_radial (PMF.pure b) lam h hl0 hl1 Q hQ hD2
-  have hcorr := regular_correlation_bound b hb lam h hl0.le hl1 hc Q hQ hs he
-  have hJ := additive_entropyJ (PMF.pure b) lam h hl0.le hl1 Q hQ hs hD
-  simp only [mean_pure] at hid hJ
-  have hl : lam⁻¹^2 = (b:ℝ) := by field_simp; nlinarith only [hc]
-  rw [hl] at hrad
-  have hJe : (∫ m, entropyJ (edgeMessage lam h m) ∂(Q : Measure Message)) =
-      lam^2*(∫ m, entropyJ m ∂(Q : Measure Message)) := by
-    nlinarith only [hJ, congrArg (fun x : ℝ => x*(∫ m, entropyJ (edgeMessage lam h m) ∂(Q : Measure Message))) hc]
-  rw [hJe] at hid
-  have hfac : (b:ℝ)*(b-1)*lam^4=1-lam^2 := by
-    calc
-      _ = ((b:ℝ)*lam^2)*(((b:ℝ)*lam^2)-lam^2) := by ring
-      _ = _ := by rw [hc]; ring
-  have hid' : (1-lam^2)/2*posteriorNu Q+3*correlationEntropy lam h Q b ≤
-      (8/5:ℝ)*(1-lam^2)*(∫ m, entropyJ m ∂(Q : Measure Message))^2 := by
-    nlinarith only [hid, hrad, congrArg (fun x : ℝ => (8/5:ℝ)*x*(∫ m, entropyJ m ∂(Q : Measure Message))^2) hfac]
-  have ha : 0 < 1-lam^2 := by nlinarith
-  apply (mul_le_mul_iff_left₀ ha).mp
-  nlinarith only [hid', hcorr]
-
-end ThreeState
-
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-
-lemma critical_moment_nu_bound (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
-    (hl0 : 0 < lam) (hl1 : lam < 1) (Q : ProbabilityMeasure Message)
-    (hQ : IsPosteriorFixedPoint offspring lam h Q) (hs : SpinSymmetric Q)
-    (hD : HasMean offspring (fun n : ℕ => (n:ℝ)))
-    (hc : mean offspring (fun n : ℕ => (n:ℝ))*lam^2=1) :
-    ((37/27:ℝ)+2*lam*((37/27:ℝ)/(2+lam)))^2*posteriorNu Q ≤ posteriorMu Q := by
-  have hh := positive_moment_estimates offspring lam h hl0 hl1 Q hQ hs hD hc
-  dsimp only at hh
-  have hnu := posteriorNu_nonneg Q
-  have hmu := posteriorMu_nonneg Q
-  by_cases hzero : posteriorNu Q=0
-  · simp only [hzero,mul_zero]
-    exact hmu
-  · have hnupos : 0 < posteriorNu Q := lt_of_le_of_ne hnu (Ne.symm hzero)
-    have he : 0 ≤ ((37/27:ℝ)+2*lam*((37/27:ℝ)/(2+lam)))*posteriorNu Q := by positivity
-    have hsq := (Real.le_sqrt he (mul_nonneg hmu hnu)).mp (hh.2.2.1.trans hh.2.2.2)
-    apply (mul_le_mul_iff_left₀ hnupos).mp
-    nlinarith only [hsq]
-
-lemma regular_positive_critical_uniform (b : ℕ) (hb : 2 ≤ b) (lam : ℝ) (h : Admissible lam)
-    (hl0 : 0 < lam) (hc : (b:ℝ)*lam^2=1)
-    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint (PMF.pure b) lam h Q)
-    (hs : SpinSymmetric Q) : posteriorMu Q = 0 := by
-  have hbr : (2:ℝ) ≤ b := by exact_mod_cast hb
-  have hlhalf : lam^2 ≤ 1/2 := by nlinarith [sq_nonneg lam]
-  have hl1 : lam < 1 := by nlinarith
-  have hD : HasMean (PMF.pure b) (fun n : ℕ => (n:ℝ)) := hasMean_pure _ _
-  have hc' : mean (PMF.pure b) (fun n : ℕ => (n:ℝ))*lam^2=1 := by simpa only [mean_pure] using hc
-  have hh := positive_moment_estimates (PMF.pure b) lam h hl0 hl1 Q hQ hs hD hc'
-  dsimp only at hh
-  have hm := posteriorMu_nonneg Q
-  have hn := posteriorNu_nonneg Q
-  have he : 0 ≤ posteriorEta Q := (show 0 ≤ ((37/27:ℝ)+2*lam*((37/27:ℝ)/(2+lam)))*posteriorNu Q by positivity).trans hh.2.2.1
-  have hent := regular_entropy_bound b hb lam h hl0 hl1 hc Q hQ hs he
-  have hcnu := critical_moment_nu_bound (PMF.pure b) lam h hl0 hl1 Q hQ hs hD hc'
-  by_contra hzero
-  have hmpos : 0 < posteriorMu Q := lt_of_le_of_ne hm (Ne.symm hzero)
-  apply PositiveScalar.moment_scalar_contradiction lam (posteriorMu Q) (posteriorNu Q)
-    (∫ m, entropyJ m ∂(Q : Measure Message)) 1 hl0 hl1 hmpos hn (by norm_num) (by norm_num)
-    (by simpa only [one_mul] using posteriorMu_sq_le Q) hcnu hh.1 hh.2.1
-  · nlinarith only [hent, mul_nonneg (sq_nonneg lam) hn]
-  · intro hhigher
-    linarith only [hhigher,hlhalf]
-
-lemma messageX_zero_tv {m : Message} (hm : messageX m=0) : messageTV m=0 := by
-  have hc : ∀ i : Spin, m i=1 := by
-    have h0 := sq_nonneg (m 0-1)
-    have h1 := sq_nonneg (m 1-1)
-    have h2 := sq_nonneg (m 2-1)
-    simp only [messageX, Radial.momentX, Radial.avg_expand, messageDeviation] at hm
-    intro i
-    fin_cases i <;> (try change m 0=1) <;> (try change m 1=1) <;> (try change m 2=1) <;> nlinarith
-  simp [messageTV,hc]
-
-lemma integral_tv_zero_of_mu_zero (Q : ProbabilityMeasure Message) (hmu : posteriorMu Q=0) :
-    ∫ m, messageTV m ∂(Q : Measure Message) = 0 := by
-  have ha := (integral_eq_zero_iff_of_nonneg messageX_nonneg (integrable_continuous_message Q continuous_messageX)).mp hmu
-  calc
-    _ = ∫ _m : Message, (0:ℝ) ∂(Q : Measure Message) := integral_congr_ae (ha.mono (fun m hm => messageX_zero_tv hm))
-    _ = 0 := integral_zero _ _
-
-lemma observed_tendsto_zero_of_uniform_fixedpoints (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
-    (hunif : ∀ Q : ProbabilityMeasure Message, IsPosteriorFixedPoint offspring lam h Q → SpinSymmetric Q → posteriorMu Q=0) :
-    Tendsto (fun n => advantage (observedLaw offspring lam h n)) atTop (𝓝 0) := by
-  obtain ⟨Q,hQ,hs,φ,hφ,hlim⟩ := ordered_exists_symmetric_fixedpoint offspring lam h
-  have htv := integral_tv_zero_of_mu_zero Q (hunif Q hQ hs)
-  have hordered := advantage_tendsto_of_likelihood (fun n => orderedLaw offspring lam h (φ n)) hlim
-  rw [htv] at hordered
-  obtain ⟨L,hL,_hL1,hactual⟩ := observedAdvantage_tendsto offspring lam h
-  have hle := le_of_tendsto_of_tendsto (hactual.comp hφ.tendsto_atTop) hordered
-    (Eventually.of_forall (fun n => orderedAdvantage_dominates offspring lam h (φ n)))
-  have heq : L=0 := le_antisymm hle hL
-  simpa only [heq] using hactual
-
- 
-theorem regular_positive_critical_nonreconstruction (b : ℕ) (hb : 2 ≤ b) (lam : ℝ)
-    (h : Admissible lam) (hl0 : 0 < lam) (hc : (b:ℝ)*lam^2=1) :
-    Tendsto (regularAdvantage b lam h) atTop (𝓝 0) := by
-  exact observed_tendsto_zero_of_uniform_fixedpoints (PMF.pure b) lam h
-    (regular_positive_critical_uniform b hb lam h hl0 hc)
-
-end ThreeState
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeState.Projection
+import OAI.Probability.ThreeState.ScalarBounds
+
+namespace OAI
+
+/-! Entropy bounds and non-reconstruction at the positive regular-tree threshold. -/
+
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+
+lemma regular_fixedpoint_degree_nu (b : ℕ) (lam : ℝ) (h : Admissible lam)
+    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint (PMF.pure b) lam h Q) :
+    posteriorNu (degreePosterior lam h Q hQ.1 b) = posteriorNu Q := by
+  have hh := hQ.2 ⟨fun m => (messageX m)^2, continuous_messageX.pow 2⟩
+  rw [mean_pure] at hh
+  change (∫ m, (messageX m)^2 ∂(Q : Measure Message)) =
+    ∫ m : Fin b → Message, normalizer lam h m*(messageX (combineMessage lam h m))^2 ∂(iidMeasure Q b : Measure (Fin b → Message)) at hh
+  change (∫ m, (messageX m)^2 ∂degreeOutput lam h Q b) = _
+  rw [integral_degreeOutput lam h Q b (fun m => (messageX m)^2) (continuous_messageX.pow 2).measurable]
+  exact hh.symm
+
+lemma partial_nu_le_regular (b : ℕ) (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint (PMF.pure b) lam h Q) {j : ℕ} (hj : j ≤ b) :
+    posteriorNu (degreePosterior lam h Q hQ.1 j) ≤ posteriorNu Q := by
+  rw [← regular_fixedpoint_degree_nu b lam h Q hQ]
+  exact degree_nu_monotone lam h hl0 hl1 Q hQ.1 hj
+
+lemma correlationEntropy_sum (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (b : ℕ) :
+    correlationEntropy lam h Q b =
+      ∑ j ∈ Finset.range b, pairCorrelation lam h Q (degreePosterior lam h Q hQ j) := by
+  induction b with
+  | zero => simp [correlationEntropy_zero]
+  | succ b ih =>
+    rw [Finset.sum_range_succ, ← ih]
+    have hh := correlationEntropy_succ lam h hl0 hl1 Q hQ b
+    change correlationEntropy lam h Q (b+1) = correlationEntropy lam h Q b +
+      pairCorrelation lam h Q (degreePosterior lam h Q hQ b) at hh
+    exact hh
+
+lemma scalar_sqrt_projection_bound {j beta : ℝ} (hj : 1 ≤ j) (hb : 0 ≤ beta) :
+    Real.sqrt (j*beta/(1+(j-1)*beta)) ≤ Real.sqrt (j*beta) := by
+  apply Real.sqrt_le_sqrt
+  have hd : 1 ≤ 1+(j-1)*beta := by nlinarith
+  exact div_le_self (mul_nonneg (by linarith) hb) hd
+
+lemma regular_increment_bound (b : ℕ) (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1)
+    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint (PMF.pure b) lam h Q) (hs : SpinSymmetric Q)
+    (he : 0 ≤ posteriorEta Q) (j : ℕ) (hj : j < b) :
+    3*lam^4*(posteriorMu Q)^2*((j:ℝ)/(1+((j:ℝ)-1)*(lam^2*posteriorMu Q)))-
+      lam^4*posteriorMu Q*posteriorNu Q*Real.sqrt j ≤
+      3*pairCorrelation lam h Q (degreePosterior lam h Q hQ.1 j) := by
+  by_cases hj0 : j=0
+  · subst j
+    simp only [Nat.cast_zero, zero_div, mul_zero, Real.sqrt_zero, sub_zero]
+    exact mul_nonneg (by norm_num) (pairCorrelation_nonneg lam h hl0 hl1 Q _ hQ.1)
+  · have hj1 : 1 ≤ j := by omega
+    have hjr : (1:ℝ) ≤ j := by exact_mod_cast hj1
+    have hm := posteriorMu_nonneg Q
+    have hn := posteriorNu_nonneg Q
+    let L := (j:ℝ)*(lam^2*posteriorMu Q)/(1+((j:ℝ)-1)*(lam^2*posteriorMu Q))
+    have hb : 0 ≤ lam^2*posteriorMu Q := mul_nonneg (sq_nonneg _) hm
+    have hL : 0 ≤ L := div_nonneg (mul_nonneg (by positivity) hb) (by positivity)
+    have hh := pairCorrelation_truncated lam h hl0 hl1 Q (degreePosterior lam h Q hQ.1 j) hQ.1 hs he
+      (partial_nu_le_regular b lam h hl0 hl1 Q hQ (Nat.le_of_lt hj)) hL
+      (degree_projection lam h hl0 hl1 Q hQ.1 hs j hj1)
+    have hsqrt : Real.sqrt L ≤ lam*Real.sqrt (posteriorMu Q)*Real.sqrt j := by
+      have hh := scalar_sqrt_projection_bound hjr hb
+      rw [Real.sqrt_mul (show (0:ℝ)≤j by positivity), Real.sqrt_mul (sq_nonneg lam), Real.sqrt_sq_eq_abs, abs_of_nonneg hl0] at hh
+      dsimp only [L]
+      nlinarith only [hh]
+    have hmul := mul_le_mul_of_nonneg_left hsqrt (show 0 ≤ lam^3*Real.sqrt (posteriorMu Q)*posteriorNu Q by positivity)
+    have hsq := Real.sq_sqrt hm
+    have hlead : 3*lam^2*posteriorMu Q*L =
+        3*lam^4*(posteriorMu Q)^2*((j:ℝ)/(1+((j:ℝ)-1)*(lam^2*posteriorMu Q))) := by dsimp only [L]; ring
+    rw [hlead] at hh
+    have hloss : lam^3*Real.sqrt (posteriorMu Q)*posteriorNu Q*(lam*Real.sqrt (posteriorMu Q)*Real.sqrt j) =
+        lam^4*posteriorMu Q*posteriorNu Q*Real.sqrt j := by
+      calc
+        _ = lam^4*(Real.sqrt (posteriorMu Q))^2*posteriorNu Q*Real.sqrt j := by ring
+        _ = _ := by rw [hsq]
+    rw [hloss] at hmul
+    linarith only [hh, hmul]
+
+end ThreeState
+
+namespace ThreeState
+open scoped BigOperators
+
+lemma sum_range_real (b : ℕ) : (∑ j ∈ Finset.range b, (j:ℝ)) = (b:ℝ)*(b-1)/2 := by
+  induction b with
+  | zero => norm_num
+  | succ b ih => rw [Finset.sum_range_succ, ih]; push_cast; ring
+
+lemma sum_range_factorial_two (b : ℕ) :
+    (∑ j ∈ Finset.range b, (j:ℝ)*((j:ℝ)-1)) = (b:ℝ)*(b-1)*(b-2)/3 := by
+  induction b with
+  | zero => norm_num
+  | succ b ih => rw [Finset.sum_range_succ, ih]; push_cast; ring
+
+lemma reciprocal_tangent {x y D : ℝ} (hx : 0 ≤ x) (hy : 0 < y) (hD : 0 < D) :
+    x*(2*D-y)/D^2 ≤ x/y := by
+  apply (div_le_div_iff₀ (sq_pos_of_pos hD) hy).mpr
+  nlinarith only [mul_nonneg hx (sq_nonneg (D-y))]
+
+lemma regular_rational_sum (b : ℕ) (hb : 2 ≤ b) (lam mu : ℝ) (hc : (b:ℝ)*lam^2=1) (hm : 0 ≤ mu) :
+    (1-lam^2)/(2*(1+2*mu/3)) ≤
+      lam^4*(∑ j ∈ Finset.range b, (j:ℝ)/(1+((j:ℝ)-1)*(lam^2*mu))) := by
+  let D := 1+2*mu/3
+  have hD : 0 < D := by dsimp [D]; positivity
+  have hbr : (2:ℝ) ≤ b := by exact_mod_cast hb
+  have hb0 : (0:ℝ) ≤ b := by positivity
+  have hbm : 0 ≤ (b:ℝ)-1 := by linarith
+  have hterm (j : ℕ) :
+      ((2*D-1)/D^2)*(j:ℝ)-((lam^2*mu)/D^2)*((j:ℝ)*((j:ℝ)-1)) ≤
+        (j:ℝ)/(1+((j:ℝ)-1)*(lam^2*mu)) := by
+    by_cases hj : j=0
+    · subst j; norm_num
+    · have hjr : (1:ℝ) ≤ j := by exact_mod_cast (show 1≤j by omega)
+      have hd : 0<1+((j:ℝ)-1)*(lam^2*mu) := by positivity
+      have hh := reciprocal_tangent (show (0:ℝ)≤j by positivity) hd hD
+      convert hh using 1
+      ring
+  have hs := Finset.sum_le_sum (s := Finset.range b) (fun j _ => hterm j)
+  rw [Finset.sum_sub_distrib, ← Finset.mul_sum, ← Finset.mul_sum,
+    sum_range_real, sum_range_factorial_two] at hs
+  have haux : (b:ℝ)*(b-1)/(2*D) ≤
+      ((2*D-1)/D^2)*((b:ℝ)*(b-1)/2)-((lam^2*mu)/D^2)*((b:ℝ)*(b-1)*(b-2)/3) := by
+    apply (le_sub_iff_add_le).mpr
+    field_simp [ne_of_gt hD]
+    dsimp only [D]
+    have hp : 0 ≤ mu*((b:ℝ)-1)*lam^2 := by positivity
+    nlinarith only [hp, congrArg (fun x => mu*((b:ℝ)-1)*x) hc]
+  have hh := mul_le_mul_of_nonneg_left (haux.trans hs) (show (0:ℝ) ≤ lam^4 by positivity)
+  have he : lam^4*((b:ℝ)*(b-1)/(2*D)) = (1-lam^2)/(2*D) := by
+    have ht : lam^4*((b:ℝ)*(b-1)) = 1-lam^2 := by
+      calc
+        _ = ((b:ℝ)*lam^2)*(((b:ℝ)*lam^2)-lam^2) := by ring
+        _ = _ := by rw [hc]; ring
+    calc
+      _ = (lam^4*((b:ℝ)*(b-1)))/(2*D) := by ring
+      _ = _ := by rw [ht]
+  change lam^4*((b:ℝ)*(b-1)/(2*D)) ≤ lam^4*(∑ j ∈ Finset.range b, (j:ℝ)/(1+((j:ℝ)-1)*(lam^2*mu))) at hh
+  rwa [he] at hh
+
+lemma regular_sqrt_sum_sq (b : ℕ) (hb : 1 ≤ b) :
+    (∑ j ∈ Finset.range b, Real.sqrt (j:ℝ))^2 ≤ (b:ℝ)*(b-1)^2/2 := by
+  let f : ℕ → ℝ := fun j => if j=0 then 0 else 1
+  let g : ℕ → ℝ := fun j => Real.sqrt (j:ℝ)
+  have hf : (∑ j ∈ Finset.range b, (f j)^2) = (b:ℝ)-1 := by
+    have hzero : 0 ∈ Finset.range b := Finset.mem_range.mpr (by omega)
+    have he (j : ℕ) : (f j)^2 = 1-(if j=0 then (1:ℝ) else 0) := by dsimp only [f]; split_ifs <;> norm_num
+    simp only [he, Finset.sum_sub_distrib, Finset.sum_const, Finset.card_range, nsmul_eq_mul,
+      Finset.sum_ite_eq', hzero, ite_true, mul_one]
+  have hg : (∑ j ∈ Finset.range b, (g j)^2) = (b:ℝ)*(b-1)/2 := by
+    simp only [g, Real.sq_sqrt (Nat.cast_nonneg _), sum_range_real]
+  have hfg : (∑ j ∈ Finset.range b, f j*g j) = ∑ j ∈ Finset.range b, Real.sqrt (j:ℝ) := by
+    apply Finset.sum_congr rfl
+    intro j _
+    by_cases hj : j=0 <;> simp [f,g,hj]
+  have hh := Finset.sum_mul_sq_le_sq_mul_sq (Finset.range b) f g
+  rw [hf,hg,hfg] at hh
+  nlinarith only [hh]
+
+lemma regular_sqrt_sum (b : ℕ) (hb : 2 ≤ b) (lam : ℝ) (hl : 0 ≤ lam) (hc : (b:ℝ)*lam^2=1) :
+    lam^4*(∑ j ∈ Finset.range b, Real.sqrt (j:ℝ)) ≤ (1-lam^2)*lam/Real.sqrt 2 := by
+  have hbr : (2:ℝ) ≤ b := by exact_mod_cast hb
+  have ha : 0 ≤ 1-lam^2 := by nlinarith [sq_nonneg lam]
+  have hs := regular_sqrt_sum_sq b (by omega)
+  have hp := mul_le_mul_of_nonneg_left hs (show 0 ≤ 2*lam^8 by positivity)
+  have he : 2*lam^8*((b:ℝ)*(b-1)^2/2) = ((1-lam^2)*lam)^2 := by
+    calc
+      _ = lam^2*((b:ℝ)*lam^2)*(((b:ℝ)*lam^2)-lam^2)^2 := by ring
+      _ = _ := by rw [hc]; ring
+  rw [he] at hp
+  have hs0 : 0 ≤ ∑ j ∈ Finset.range b, Real.sqrt (j:ℝ) := Finset.sum_nonneg (fun j _ => Real.sqrt_nonneg _)
+  have hr : 0 < Real.sqrt 2 := by positivity
+  apply (le_div_iff₀ hr).mpr
+  have hroot : (Real.sqrt 2)^2=2 := Real.sq_sqrt (by norm_num)
+  have hx : 0 ≤ lam^4*(∑ j ∈ Finset.range b, Real.sqrt (j:ℝ))*Real.sqrt 2 := by positivity
+  have hy : 0 ≤ (1-lam^2)*lam := mul_nonneg ha hl
+  apply (sq_le_sq₀ hx hy).mp
+  have hsq : (lam^4*(∑ j ∈ Finset.range b, Real.sqrt (j:ℝ))*Real.sqrt 2)^2 =
+      2*lam^8*(∑ j ∈ Finset.range b, Real.sqrt (j:ℝ))^2 := by
+    calc
+      _ = lam^8*(∑ j ∈ Finset.range b, Real.sqrt (j:ℝ))^2*(Real.sqrt 2)^2 := by ring
+      _ = _ := by rw [hroot]; ring
+  rwa [hsq]
+
+end ThreeState
+
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+
+lemma regular_correlation_bound (b : ℕ) (hb : 2 ≤ b) (lam : ℝ) (h : Admissible lam)
+    (hl0 : 0 ≤ lam) (hl1 : lam < 1) (hc : (b:ℝ)*lam^2=1)
+    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint (PMF.pure b) lam h Q)
+    (hs : SpinSymmetric Q) (he : 0 ≤ posteriorEta Q) :
+    (1-lam^2)*(3*(posteriorMu Q)^2/(2*(1+2*posteriorMu Q/3))-
+      lam*posteriorMu Q*posteriorNu Q/Real.sqrt 2) ≤ 3*correlationEntropy lam h Q b := by
+  have hm := posteriorMu_nonneg Q
+  have hn := posteriorNu_nonneg Q
+  have hsum := Finset.sum_le_sum (s := Finset.range b) (fun j hj =>
+    regular_increment_bound b lam h hl0 hl1 Q hQ hs he j (Finset.mem_range.mp hj))
+  rw [Finset.sum_sub_distrib, ← Finset.mul_sum, ← Finset.mul_sum, ← Finset.mul_sum,
+    ← correlationEntropy_sum lam h hl0 hl1 Q hQ.1 b] at hsum
+  have hpos := mul_le_mul_of_nonneg_left (regular_rational_sum b hb lam (posteriorMu Q) hc hm)
+    (show 0 ≤ 3*(posteriorMu Q)^2 by positivity)
+  have hneg := mul_le_mul_of_nonneg_left (regular_sqrt_sum b hb lam hl0 hc)
+    (mul_nonneg hm hn)
+  calc
+    _ = 3*(posteriorMu Q)^2*((1-lam^2)/(2*(1+2*posteriorMu Q/3)))-
+        (posteriorMu Q*posteriorNu Q)*((1-lam^2)*lam/Real.sqrt 2) := by ring
+    _ ≤ 3*(posteriorMu Q)^2*(lam^4*(∑ j ∈ Finset.range b, (j:ℝ)/(1+((j:ℝ)-1)*(lam^2*posteriorMu Q))))-
+        (posteriorMu Q*posteriorNu Q)*(lam^4*(∑ j ∈ Finset.range b, Real.sqrt (j:ℝ))) := sub_le_sub hpos hneg
+    _ = _ := by ring
+    _ ≤ _ := hsum
+
+lemma entropyF_sourceF (m : Message) : entropyF m = Radial.sourceF (fun i => m i) := by
+  simp only [entropyF, entropyI, entropyJ, entropyB, Radial.sourceF, mLog, mCenteredLog,
+    Radial.avg_expand]
+  ring
+
+lemma continuous_entropyF_edge (lam : ℝ) (h : Admissible lam) (hl0 : 0 ≤ lam) (hl1 : lam < 1) :
+    Continuous (fun m => entropyF (edgeMessage lam h m)) :=
+  (((continuous_entropyI_edge lam h hl0 hl1).const_mul 3).sub
+    ((continuous_entropyJ_edge lam h hl0 hl1).const_mul 2)).add
+      ((continuous_entropyB_edge lam h hl0 hl1).const_mul (4/5:ℝ))
+
+lemma integral_entropyF_radial (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
+    (hl0 : 0 < lam) (hl1 : lam < 1) (Q : ProbabilityMeasure Message)
+    (hQ : IsPosteriorFixedPoint offspring lam h Q)
+    (hD2 : HasMean offspring (fun n : ℕ => (n:ℝ)^2)) :
+    (1-lam^2)/2*posteriorNu Q ≤ (∫ m, entropyF m ∂(Q : Measure Message))-
+      lam⁻¹^2*(∫ m, entropyF (edgeMessage lam h m) ∂(Q : Measure Message)) := by
+  have hi := integrable_entropyF_fixedpoint offspring lam h hl0.le hl1 Q hQ hD2
+  have hie := integrable_continuous_message Q (continuous_entropyF_edge lam h hl0.le hl1)
+  have hp := positive_fixedpoint offspring lam h hl0.le hl1 Q hQ
+  have hh := integral_mono_ae ((integrable_continuous_message Q (continuous_messageX.pow 2)).const_mul ((1-lam^2)/2))
+    (show Integrable (fun m => entropyF m-lam⁻¹^2*entropyF (edgeMessage lam h m)) (Q : Measure Message) from hi.sub (hie.const_mul _))
+    (hp.mono (fun m hm => by
+      have hh := Radial.radial_source (fun i => m i) (average_message m) hm lam hl0 hl1
+      change (1-lam^2)/2*(messageX m)^2 ≤ entropyF m-lam⁻¹^2*entropyF (edgeMessage lam h m)
+      rw [entropyF_sourceF, entropyF_sourceF]
+      convert hh using 1 <;> simp only [edgeMessage, messageX, messageDeviation, Radial.momentX]
+      ring))
+  rw [integral_const_mul, integral_sub hi (hie.const_mul _), integral_const_mul] at hh
+  exact hh
+
+lemma regular_entropy_bound (b : ℕ) (hb : 2 ≤ b) (lam : ℝ) (h : Admissible lam)
+    (hl0 : 0 < lam) (hl1 : lam < 1) (hc : (b:ℝ)*lam^2=1)
+    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint (PMF.pure b) lam h Q)
+    (hs : SpinSymmetric Q) (he : 0 ≤ posteriorEta Q) :
+    posteriorNu Q/2+3*(posteriorMu Q)^2/(2*(1+2*posteriorMu Q/3))-
+      lam*posteriorMu Q*posteriorNu Q/Real.sqrt 2 ≤
+        (8/5:ℝ)*(∫ m, entropyJ m ∂(Q : Measure Message))^2 := by
+  have hD2 : HasMean (PMF.pure b) (fun n : ℕ => (n:ℝ)^2) := hasMean_pure _ _
+  have hD := hasMean_of_square hD2
+  have hid := entropy_identity (PMF.pure b) lam h hl0.le hl1 Q hQ hs hD2
+  have hrad := integral_entropyF_radial (PMF.pure b) lam h hl0 hl1 Q hQ hD2
+  have hcorr := regular_correlation_bound b hb lam h hl0.le hl1 hc Q hQ hs he
+  have hJ := additive_entropyJ (PMF.pure b) lam h hl0.le hl1 Q hQ hs hD
+  simp only [mean_pure] at hid hJ
+  have hl : lam⁻¹^2 = (b:ℝ) := by field_simp; nlinarith only [hc]
+  rw [hl] at hrad
+  have hJe : (∫ m, entropyJ (edgeMessage lam h m) ∂(Q : Measure Message)) =
+      lam^2*(∫ m, entropyJ m ∂(Q : Measure Message)) := by
+    nlinarith only [hJ, congrArg (fun x : ℝ => x*(∫ m, entropyJ (edgeMessage lam h m) ∂(Q : Measure Message))) hc]
+  rw [hJe] at hid
+  have hfac : (b:ℝ)*(b-1)*lam^4=1-lam^2 := by
+    calc
+      _ = ((b:ℝ)*lam^2)*(((b:ℝ)*lam^2)-lam^2) := by ring
+      _ = _ := by rw [hc]; ring
+  have hid' : (1-lam^2)/2*posteriorNu Q+3*correlationEntropy lam h Q b ≤
+      (8/5:ℝ)*(1-lam^2)*(∫ m, entropyJ m ∂(Q : Measure Message))^2 := by
+    nlinarith only [hid, hrad, congrArg (fun x : ℝ => (8/5:ℝ)*x*(∫ m, entropyJ m ∂(Q : Measure Message))^2) hfac]
+  have ha : 0 < 1-lam^2 := by nlinarith
+  apply (mul_le_mul_iff_left₀ ha).mp
+  nlinarith only [hid', hcorr]
+
+end ThreeState
+
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+
+lemma critical_moment_nu_bound (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
+    (hl0 : 0 < lam) (hl1 : lam < 1) (Q : ProbabilityMeasure Message)
+    (hQ : IsPosteriorFixedPoint offspring lam h Q) (hs : SpinSymmetric Q)
+    (hD : HasMean offspring (fun n : ℕ => (n:ℝ)))
+    (hc : mean offspring (fun n : ℕ => (n:ℝ))*lam^2=1) :
+    ((37/27:ℝ)+2*lam*((37/27:ℝ)/(2+lam)))^2*posteriorNu Q ≤ posteriorMu Q := by
+  have hh := positive_moment_estimates offspring lam h hl0 hl1 Q hQ hs hD hc
+  dsimp only at hh
+  have hnu := posteriorNu_nonneg Q
+  have hmu := posteriorMu_nonneg Q
+  by_cases hzero : posteriorNu Q=0
+  · simp only [hzero,mul_zero]
+    exact hmu
+  · have hnupos : 0 < posteriorNu Q := lt_of_le_of_ne hnu (Ne.symm hzero)
+    have he : 0 ≤ ((37/27:ℝ)+2*lam*((37/27:ℝ)/(2+lam)))*posteriorNu Q := by positivity
+    have hsq := (Real.le_sqrt he (mul_nonneg hmu hnu)).mp (hh.2.2.1.trans hh.2.2.2)
+    apply (mul_le_mul_iff_left₀ hnupos).mp
+    nlinarith only [hsq]
+
+lemma regular_positive_critical_uniform (b : ℕ) (hb : 2 ≤ b) (lam : ℝ) (h : Admissible lam)
+    (hl0 : 0 < lam) (hc : (b:ℝ)*lam^2=1)
+    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint (PMF.pure b) lam h Q)
+    (hs : SpinSymmetric Q) : posteriorMu Q = 0 := by
+  have hbr : (2:ℝ) ≤ b := by exact_mod_cast hb
+  have hlhalf : lam^2 ≤ 1/2 := by nlinarith [sq_nonneg lam]
+  have hl1 : lam < 1 := by nlinarith
+  have hD : HasMean (PMF.pure b) (fun n : ℕ => (n:ℝ)) := hasMean_pure _ _
+  have hc' : mean (PMF.pure b) (fun n : ℕ => (n:ℝ))*lam^2=1 := by simpa only [mean_pure] using hc
+  have hh := positive_moment_estimates (PMF.pure b) lam h hl0 hl1 Q hQ hs hD hc'
+  dsimp only at hh
+  have hm := posteriorMu_nonneg Q
+  have hn := posteriorNu_nonneg Q
+  have he : 0 ≤ posteriorEta Q := (show 0 ≤ ((37/27:ℝ)+2*lam*((37/27:ℝ)/(2+lam)))*posteriorNu Q by positivity).trans hh.2.2.1
+  have hent := regular_entropy_bound b hb lam h hl0 hl1 hc Q hQ hs he
+  have hcnu := critical_moment_nu_bound (PMF.pure b) lam h hl0 hl1 Q hQ hs hD hc'
+  by_contra hzero
+  have hmpos : 0 < posteriorMu Q := lt_of_le_of_ne hm (Ne.symm hzero)
+  apply PositiveScalar.moment_scalar_contradiction lam (posteriorMu Q) (posteriorNu Q)
+    (∫ m, entropyJ m ∂(Q : Measure Message)) 1 hl0 hl1 hmpos hn (by norm_num) (by norm_num)
+    (by simpa only [one_mul] using posteriorMu_sq_le Q) hcnu hh.1 hh.2.1
+  · nlinarith only [hent, mul_nonneg (sq_nonneg lam) hn]
+  · intro hhigher
+    linarith only [hhigher,hlhalf]
+
+lemma messageX_zero_tv {m : Message} (hm : messageX m=0) : messageTV m=0 := by
+  have hc : ∀ i : Spin, m i=1 := by
+    have h0 := sq_nonneg (m 0-1)
+    have h1 := sq_nonneg (m 1-1)
+    have h2 := sq_nonneg (m 2-1)
+    simp only [messageX, Radial.momentX, Radial.avg_expand, messageDeviation] at hm
+    intro i
+    fin_cases i <;> (try change m 0=1) <;> (try change m 1=1) <;> (try change m 2=1) <;> nlinarith
+  simp [messageTV,hc]
+
+lemma integral_tv_zero_of_mu_zero (Q : ProbabilityMeasure Message) (hmu : posteriorMu Q=0) :
+    ∫ m, messageTV m ∂(Q : Measure Message) = 0 := by
+  have ha := (integral_eq_zero_iff_of_nonneg messageX_nonneg (integrable_continuous_message Q continuous_messageX)).mp hmu
+  calc
+    _ = ∫ _m : Message, (0:ℝ) ∂(Q : Measure Message) := integral_congr_ae (ha.mono (fun m hm => messageX_zero_tv hm))
+    _ = 0 := integral_zero _ _
+
+lemma observed_tendsto_zero_of_uniform_fixedpoints (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
+    (hunif : ∀ Q : ProbabilityMeasure Message, IsPosteriorFixedPoint offspring lam h Q → SpinSymmetric Q → posteriorMu Q=0) :
+    Tendsto (fun n => advantage (observedLaw offspring lam h n)) atTop (𝓝 0) := by
+  obtain ⟨Q,hQ,hs,φ,hφ,hlim⟩ := ordered_exists_symmetric_fixedpoint offspring lam h
+  have htv := integral_tv_zero_of_mu_zero Q (hunif Q hQ hs)
+  have hordered := advantage_tendsto_of_likelihood (fun n => orderedLaw offspring lam h (φ n)) hlim
+  rw [htv] at hordered
+  obtain ⟨L,hL,_hL1,hactual⟩ := observedAdvantage_tendsto offspring lam h
+  have hle := le_of_tendsto_of_tendsto (hactual.comp hφ.tendsto_atTop) hordered
+    (Eventually.of_forall (fun n => orderedAdvantage_dominates offspring lam h (φ n)))
+  have heq : L=0 := le_antisymm hle hL
+  simpa only [heq] using hactual
+
+ 
+theorem regular_positive_critical_nonreconstruction (b : ℕ) (hb : 2 ≤ b) (lam : ℝ)
+    (h : Admissible lam) (hl0 : 0 < lam) (hc : (b:ℝ)*lam^2=1) :
+    Tendsto (regularAdvantage b lam h) atTop (𝓝 0) := by
+  exact observed_tendsto_zero_of_uniform_fixedpoints (PMF.pure b) lam h
+    (regular_positive_critical_uniform b hb lam h hl0 hc)
+
+end ThreeState
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeState/Resolvents.lean b/lean/OAI/Probability/ThreeState/Resolvents.lean
index 0ce424a9e5711f65388269d6f87ff8419576fd12..1f6e30eb047857c0bc987aeed6612cc8be5486d5 100644
--- a/lean/OAI/Probability/ThreeState/Resolvents.lean
+++ b/lean/OAI/Probability/ThreeState/Resolvents.lean
@@ -1,286 +1,286 @@
-import Mathlib
-import OAI.Probability.ThreeState.RadialMoments
-
-namespace OAI
-
-/-! Fourth-moment resolvents and their lower bounds. -/
-
-namespace ThreeState.Radial
-open MeasureTheory
-
-noncomputable def fourthResolvent (v : Fin 3 → ℝ) (t : ℝ) : ℝ := avg (fun i => (v i)^4/(1+t*v i))
-noncomputable def doubleFourthResolvent (v : Fin 3 → ℝ) (lam t : ℝ) : ℝ :=
-  avg (fun i => (v i)^4/((1+t*v i)*(1+lam*t*v i)))
-
-lemma fourth_moment (v : Fin 3 → ℝ) (hv : avg v = 0) : avg (fun i => (v i)^4) = 6*(momentX v)^2 := by
-  simp only [momentX, avg_expand, zero_sum_elim v hv]
-  ring
-
-lemma fourthResolvent_eq (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i)
-    (t : ℝ) (ht0 : 0 ≤ t) (ht1 : t ≤ 1) :
-    fourthResolvent v t = (6*(momentX v)^2-10*t*momentX v*momentY v+4*t^2*(momentY v)^2)/denom v t := by
-  have h0 := ne_of_gt (message_segment_pos v hp t ht0 ht1 0)
-  have h1 := ne_of_gt (message_segment_pos v hp t ht0 ht1 1)
-  have h2 := ne_of_gt (message_segment_pos v hp t ht0 ht1 2)
-  have h0c : 1+v 0*t ≠ 0 := by simpa [mul_comm] using h0
-  have h1c : 1+v 1*t ≠ 0 := by simpa [mul_comm] using h1
-  have h2c : 1+v 2*t ≠ 0 := by simpa [mul_comm] using h2
-  rw [denom_product v hv t]
-  simp only [fourthResolvent, momentX, momentY, avg_expand]
-  field_simp [h0, h1, h2, h0c, h1c, h2c]
-  rw [zero_sum_elim v hv]
-  ring
-
-lemma h_polynomial_bound (z : ℝ) (hz0 : 0 ≤ z) (hz1 : z < 1) :
-    (1-3*z^2+2*z^3)*(4+(64/9:ℝ)*(z-1/2)^2) ≤ 6-10*z+4*z^2 := by
-  rcases hz0.eq_or_lt with hzero | _
-  · rw [← hzero]
-    norm_num
-  have hE : 1+z-2*z^2 ≤ (9/8:ℝ) := by nlinarith [sq_nonneg (z-1/4)]
-  have hh : (1+z-2*z^2)*(4+(64/9:ℝ)*(z-1/2)^2) ≤ 6-4*z := by
-    nlinarith [mul_nonneg (show 0 ≤ (9/8:ℝ)-(1+z-2*z^2) by linarith) (sq_nonneg (z-1/2))]
-  have he := mul_le_mul_of_nonneg_left hh (show 0 ≤ 1-z by linarith)
-  nlinarith only [he]
-
-lemma momentY_sqrt_bound (v : Fin 3 → ℝ) (hv : avg v = 0) :
-    -(Real.sqrt (momentX v))^3 ≤ momentY v ∧ momentY v ≤ (Real.sqrt (momentX v))^3 := by
-  have h := discriminant_nonneg v hv
-  have hs := Real.sq_sqrt (momentX_nonneg v)
-  have he : (momentX v)^3 = ((Real.sqrt (momentX v))^3)^2 := by
-    calc
-      (momentX v)^3 = ((Real.sqrt (momentX v))^2)^3 := congrArg (fun x : ℝ => x^3) hs.symm
-      _ = _ := by ring
-  rw [he] at h
-  have hr : 0 ≤ (Real.sqrt (momentX v))^3 := by positivity
-  exact abs_le.mp ((sq_le_sq₀ (abs_nonneg (momentY v)) hr).mp (by simpa only [sq_abs] using h))
-
-lemma fourthResolvent_lower (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i)
-    (t : ℝ) (ht0 : 0 ≤ t) (ht1 : t ≤ 1) :
-    (momentX v)^2*(4+(64/9:ℝ)*(t*Real.sqrt (momentX v)-1/2)^2) ≤ fourthResolvent v t := by
-  let r := Real.sqrt (momentX v)
-  have hr0 : 0 ≤ r := Real.sqrt_nonneg _
-  have hr2 : r^2 = momentX v := Real.sq_sqrt (momentX_nonneg v)
-  have hr1 : r < 1 := by nlinarith [momentX_lt_one v hv hp]
-  have hy := momentY_sqrt_bound v hv
-  change -r^3 ≤ momentY v ∧ momentY v ≤ r^3 at hy
-  have hz0 : 0 ≤ t*r := mul_nonneg ht0 hr0
-  have hz1 : t*r < 1 := lt_of_le_of_lt (mul_le_of_le_one_left hr0 ht1) hr1
-  have hn : (momentX v)^2*(6-10*(t*r)+4*(t*r)^2) ≤
-      6*(momentX v)^2-10*t*momentX v*momentY v+4*t^2*(momentY v)^2 := by
-    have htr : t*r ≤ 1 := hz1.le
-    have hfac : 0 ≤ 10*momentX v-4*t*(momentY v+r^3) := by
-      have h1 := mul_le_mul_of_nonneg_left hy.2 (by positivity : 0 ≤ 4*t)
-      have h2 := mul_le_mul_of_nonneg_left htr (by positivity : 0 ≤ 8*r^2)
-      nlinarith only [h1,h2, sq_nonneg r,hr2]
-    have hh := mul_nonneg (mul_nonneg ht0 (sub_nonneg.mpr hy.2)) hfac
-    rw [← hr2] at hh ⊢
-    nlinarith only [hh]
-  have hd : denom v t ≤ 1-3*(t*r)^2+2*(t*r)^3 := by
-    have hh := mul_le_mul_of_nonneg_left hy.2 (by positivity : 0 ≤ 2*t^3)
-    dsimp only [denom]
-    rw [← hr2]
-    nlinarith only [hh]
-  have hg : 0 ≤ (momentX v)^2*(4+(64/9:ℝ)*(t*r-1/2)^2) := by positivity
-  have hh := mul_le_mul_of_nonneg_left hd hg
-  have hpoly := mul_le_mul_of_nonneg_left (h_polynomial_bound (t*r) hz0 hz1) (sq_nonneg (momentX v))
-  rw [fourthResolvent_eq v hv hp t ht0 ht1]
-  apply (le_div_iff₀ (denom_pos v hv hp t ht0 ht1)).mpr
-  dsimp only [r] at hh hpoly
-  nlinarith only [hh,hpoly,hn]
-
-lemma fourthResolvent_four (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i)
-    (t : ℝ) (ht0 : 0 ≤ t) (ht1 : t ≤ 1) : 4*(momentX v)^2 ≤ fourthResolvent v t := by
-  have hh := fourthResolvent_lower v hv hp t ht0 ht1
-  nlinarith [mul_nonneg (sq_nonneg (momentX v)) (sq_nonneg (t*Real.sqrt (momentX v)-1/2))]
-
-end ThreeState.Radial
-namespace ThreeState.Radial
-open MeasureTheory
-
-lemma doubleFourthResolvent_lower (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i)
-    (lam : ℝ) (hl0 : 0 ≤ lam) (hl1 : lam ≤ 1) (t : ℝ) (ht0 : 0 ≤ t) (ht1 : t ≤ 1) :
-    2*fourthResolvent v t ≤ (2+lam)*doubleFourthResolvent v lam t := by
-  let c := 1+lam/2
-  have hc : 0 < c := by dsimp [c]; linarith
-  have hlt0 : 0 ≤ lam*t := mul_nonneg hl0 ht0
-  have hlt1 : lam*t ≤ 1 := (mul_le_of_le_one_left ht0 hl1).trans ht1
-  have hi (i : Fin 3) : (2*c-1+lam)*(v i)^4/(1+t*v i)-lam*(v i)^4 ≤
-      c^2*(v i)^4/((1+t*v i)*(1+lam*t*v i)) := by
-    have hd := message_segment_pos v hp t ht0 ht1 i
-    have he := message_segment_pos v hp (lam*t) hlt0 hlt1 i
-    have hdc : 1+v i*t ≠ 0 := by simpa [mul_comm] using (ne_of_gt hd)
-    have hec : 1+v i*t*lam ≠ 0 := by convert ne_of_gt he using 1; ring
-    have hh : 0 ≤ (v i)^4*(c-(1+lam*t*v i))^2/((1+t*v i)*(1+lam*t*v i)) := by positivity
-    have hid : c^2*(v i)^4/((1+t*v i)*(1+lam*t*v i))-
-      ((2*c-1+lam)*(v i)^4/(1+t*v i)-lam*(v i)^4) =
-      (v i)^4*(c-(1+lam*t*v i))^2/((1+t*v i)*(1+lam*t*v i)) := by
-      field_simp [ne_of_gt hd, ne_of_gt he, hdc, hec]
-      ring
-    linarith
-  have hh : (2*c-1+lam)*fourthResolvent v t-lam*(6*(momentX v)^2) ≤
-      c^2*doubleFourthResolvent v lam t := by
-    rw [← fourth_moment v hv]
-    simp only [fourthResolvent, doubleFourthResolvent, avg_expand]
-    have hi0 := hi 0
-    have hi1 := hi 1
-    have hi2 := hi 2
-    simp only [mul_div_assoc] at hi0 hi1 hi2
-    nlinarith only [hi0,hi1,hi2]
-  have hw := fourthResolvent_four v hv hp t ht0 ht1
-  have hnon := mul_nonneg hl0 (show 0 ≤ fourthResolvent v t-4*(momentX v)^2 by linarith)
-  have hcW : c*fourthResolvent v t ≤ c^2*doubleFourthResolvent v lam t := by
-    dsimp only [c] at *
-    nlinarith only [hh,hnon]
-  have hres : fourthResolvent v t ≤ c*doubleFourthResolvent v lam t := by
-    apply (mul_le_mul_iff_right₀ hc).mp
-    nlinarith only [hcW]
-  dsimp only [c] at hres
-  linarith
-
-lemma continuousOn_fourthResolvent (v : Fin 3 → ℝ) (hp : ∀ i, 0 < 1+v i) :
-    ContinuousOn (fourthResolvent v) (Set.Icc 0 1) := by
-  have hi (i : Fin 3) : ContinuousOn (fun t : ℝ => (v i)^4/(1+t*v i)) (Set.Icc 0 1) :=
-    continuousOn_const.div (by fun_prop) (fun t ht => ne_of_gt (message_segment_pos v hp t ht.1 ht.2 i))
-  convert (((hi 0).add (hi 1)).add (hi 2)).div_const 3 using 1
-  funext t
-  exact avg_expand _
-
-lemma continuousOn_doubleFourthResolvent (v : Fin 3 → ℝ) (hp : ∀ i, 0 < 1+v i)
-    (lam : ℝ) (hl0 : 0 ≤ lam) (hl1 : lam ≤ 1) :
-    ContinuousOn (doubleFourthResolvent v lam) (Set.Icc 0 1) := by
-  have hi (i : Fin 3) : ContinuousOn (fun t : ℝ => (v i)^4/((1+t*v i)*(1+lam*t*v i))) (Set.Icc 0 1) :=
-    continuousOn_const.div (by fun_prop) (fun t ht => ne_of_gt (mul_pos
-      (message_segment_pos v hp t ht.1 ht.2 i)
-      (message_segment_pos v hp (lam*t) (mul_nonneg hl0 ht.1) ((mul_le_of_le_one_left ht.1 hl1).trans ht.2) i)))
-  convert (((hi 0).add (hi 1)).add (hi 2)).div_const 3 using 1
-  funext t
-  exact avg_expand _
-
-noncomputable def resolventA (v : Fin 3 → ℝ) : ℝ := ∫ t in (0:ℝ)..1, t^2*fourthResolvent v t
-noncomputable def resolventQ (v : Fin 3 → ℝ) (lam : ℝ) : ℝ := (∫ t in (0:ℝ)..1, t^2*doubleFourthResolvent v lam t)/2
-noncomputable def functionJ (v : Fin 3 → ℝ) : ℝ := avg (fun i => v i*Real.log (1+v i))/2
-
-lemma integral_h_lower (r : ℝ) :
-    (∫ t in (0:ℝ)..1, t^2*(4+(64/9:ℝ)*(t*r-1/2)^2)) = (37/27:ℝ)+(64/45:ℝ)*(r-5/8)^2 := by
-  let P := fun t : ℝ => (52/27:ℝ)*t^3-(16/9:ℝ)*r*t^4+(64/45:ℝ)*r^2*t^5
-  have hd (t : ℝ) : HasDerivAt P (t^2*(4+(64/9:ℝ)*(t*r-1/2)^2)) t := by
-    dsimp only [P]
-    convert ((((hasDerivAt_id t).pow 3).const_mul (52/27:ℝ)).sub
-      (((hasDerivAt_id t).pow 4).const_mul ((16/9:ℝ)*r))).add
-      (((hasDerivAt_id t).pow 5).const_mul ((64/45:ℝ)*r^2)) using 1 <;> first | rfl | (norm_num; ring)
-  rw [intervalIntegral.integral_eq_sub_of_hasDerivAt (fun t _ => hd t)
-    (Continuous.intervalIntegrable (by fun_prop) 0 1)]
-  dsimp [P]
-  ring
-
-lemma resolventA_lower (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i) :
-    (37/27:ℝ)*(momentX v)^2 ≤ resolventA v := by
-  have hi : IntervalIntegrable (fun t : ℝ => (momentX v)^2*(t^2*(4+(64/9:ℝ)*(t*Real.sqrt (momentX v)-1/2)^2)))
-      volume 0 1 := Continuous.intervalIntegrable (by fun_prop) 0 1
-  have hj : IntervalIntegrable (fun t : ℝ => t^2*fourthResolvent v t) volume 0 1 :=
-    ((by fun_prop : ContinuousOn (fun t : ℝ => t^2) (Set.Icc 0 1)).mul (continuousOn_fourthResolvent v hp)).intervalIntegrable_of_Icc (by norm_num)
-  have hh := intervalIntegral.integral_mono_on (by norm_num : (0:ℝ) ≤ 1) hi hj (fun t ht => by
-    have he := mul_le_mul_of_nonneg_left (fourthResolvent_lower v hv hp t ht.1 ht.2) (sq_nonneg t)
-    nlinarith only [he])
-  rw [intervalIntegral.integral_const_mul, integral_h_lower] at hh
-  dsimp only [resolventA]
-  nlinarith [mul_nonneg (sq_nonneg (momentX v)) (sq_nonneg (Real.sqrt (momentX v)-5/8))]
-
-lemma resolventQ_lower (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i)
-    (lam : ℝ) (hl0 : 0 ≤ lam) (hl1 : lam ≤ 1) : resolventA v/(2+lam) ≤ resolventQ v lam := by
-  have hi : IntervalIntegrable (fun t : ℝ => 2*(t^2*fourthResolvent v t)) volume 0 1 :=
-    (continuousOn_const.mul ((continuousOn_id.pow 2).mul (continuousOn_fourthResolvent v hp))).intervalIntegrable_of_Icc (by norm_num)
-  have hj : IntervalIntegrable (fun t : ℝ => (2+lam)*(t^2*doubleFourthResolvent v lam t)) volume 0 1 :=
-    (continuousOn_const.mul ((continuousOn_id.pow 2).mul (continuousOn_doubleFourthResolvent v hp lam hl0 hl1))).intervalIntegrable_of_Icc (by norm_num)
-  have hh := intervalIntegral.integral_mono_on (by norm_num : (0:ℝ) ≤ 1) hi hj (fun t ht => by
-    have he := mul_le_mul_of_nonneg_left (doubleFourthResolvent_lower v hv hp lam hl0 hl1 t ht.1 ht.2) (sq_nonneg t)
-    nlinarith only [he])
-  rw [intervalIntegral.integral_const_mul, intervalIntegral.integral_const_mul] at hh
-  apply (div_le_iff₀ (by linarith : 0 < 2+lam)).mpr
-  dsimp only [resolventQ, resolventA]
-  linarith
-
-lemma scalar_resolventA (a : ℝ) (ha : 0 < 1+a) :
-    (∫ t in (0:ℝ)..1, t^2*a^4/(1+t*a)) = a^3/2-a^2+a*Real.log (1+a) := by
-  have hp (t : ℝ) (ht : t ∈ Set.Icc (0:ℝ) 1) : 0 < 1+t*a := by
-    rcases eq_or_lt_of_le ht.1 with hz | hz
-    · rw [← hz]; norm_num
-    · nlinarith [mul_pos hz ha, ht.2]
-  have hc : ContinuousOn (fun t : ℝ => a/(1+t*a)) (Set.Icc 0 1) :=
-    continuousOn_const.div (by fun_prop) (fun t ht => ne_of_gt (hp t ht))
-  have hi : IntervalIntegrable (fun t : ℝ => a/(1+t*a)) volume 0 1 := hc.intervalIntegrable_of_Icc (by norm_num : (0:ℝ) ≤ 1)
-  have he : (∫ t in (0:ℝ)..1, t^2*a^4/(1+t*a)) =
-      ∫ t in (0:ℝ)..1, (a^3*t-a^2)+a*(a/(1+t*a)) := by
-    apply intervalIntegral.integral_congr
-    intro t ht
-    have ht' : t ∈ Set.Icc (0:ℝ) 1 := by simpa using ht
-    field_simp [ne_of_gt (hp t ht')]
-    ring
-  rw [he, intervalIntegral.integral_add (Continuous.intervalIntegrable (by fun_prop) 0 1) (hi.const_mul a),
-    intervalIntegral.integral_sub (Continuous.intervalIntegrable (by fun_prop) 0 1) (intervalIntegrable_const),
-    intervalIntegral.integral_const_mul, intervalIntegral.integral_const_mul, integral_id,
-    intervalIntegral.integral_const, ← log_integral a ha]
-  norm_num
-  ring
-
-lemma resolventA_identity (v : Fin 3 → ℝ) (hp : ∀ i, 0 < 1+v i) :
-    resolventA v = 2*functionJ v-2*momentX v+momentY v := by
-  have he (t : ℝ) : t^2*fourthResolvent v t = avg (fun i => t^2*(v i)^4/(1+t*v i)) := by
-    simp only [fourthResolvent, avg_expand]; ring
-  dsimp only [resolventA]
-  simp_rw [he]
-  rw [integral_avg]
-  · simp_rw [scalar_resolventA _ (hp _)]
-    simp only [functionJ, momentX, momentY, avg_expand]
-    ring
-  · intro i
-    exact ((by fun_prop : ContinuousOn (fun t : ℝ => t^2*(v i)^4) (Set.Icc 0 1)).div
-      (by fun_prop) (fun t ht => ne_of_gt (message_segment_pos v hp t ht.1 ht.2 i))).intervalIntegrable_of_Icc (by norm_num)
-
-lemma resolvent_scaling (v : Fin 3 → ℝ) (hp : ∀ i, 0 < 1+v i)
-    (lam : ℝ) (hl0 : 0 ≤ lam) (hl1 : lam ≤ 1) :
-    lam^3*resolventA v-resolventA (fun i => lam*v i) = 2*lam^3*(1-lam)*resolventQ v lam := by
-  have hpe (i : Fin 3) : 0 < 1+lam*v i := message_segment_pos v hp lam hl0 hl1 i
-  have hvI : IntervalIntegrable (fun t : ℝ => t^2*fourthResolvent v t) volume 0 1 :=
-    ((continuousOn_id.pow 2).mul (continuousOn_fourthResolvent v hp)).intervalIntegrable_of_Icc (by norm_num)
-  have heI : IntervalIntegrable (fun t : ℝ => t^2*fourthResolvent (fun i => lam*v i) t) volume 0 1 :=
-    ((continuousOn_id.pow 2).mul (continuousOn_fourthResolvent _ hpe)).intervalIntegrable_of_Icc (by norm_num)
-  have he : (∫ t in (0:ℝ)..1, lam^3*(t^2*fourthResolvent v t)-t^2*fourthResolvent (fun i => lam*v i) t) =
-      ∫ t in (0:ℝ)..1, (lam^3*(1-lam))*(t^2*doubleFourthResolvent v lam t) := by
-    apply intervalIntegral.integral_congr
-    intro t ht
-    have ht' : t ∈ Set.Icc (0:ℝ) 1 := by simpa using ht
-    have hi (i : Fin 3) : lam^3*((v i)^4/(1+t*v i))-(lam*v i)^4/(1+t*(lam*v i)) =
-        lam^3*(1-lam)*((v i)^4/((1+t*v i)*(1+lam*t*v i))) := by
-      have ha := ne_of_gt (message_segment_pos v hp t ht'.1 ht'.2 i)
-      have hb := ne_of_gt (message_segment_pos _ hpe t ht'.1 ht'.2 i)
-      have ha' : 1+v i*t ≠ 0 := by simpa [mul_comm] using ha
-      have hb' : 1+v i*t*lam ≠ 0 := by convert hb using 1; ring
-      have hb'' : 1+lam*v i*t ≠ 0 := by convert hb using 1; ring
-      have hb''' : 1+lam*t*v i ≠ 0 := by convert hb using 1; ring
-      field_simp [ha, hb, ha', hb', hb'', hb''']
-      ring
-    simp only [fourthResolvent, doubleFourthResolvent, avg_expand]
-    linear_combination t^2/3*(hi 0)+t^2/3*(hi 1)+t^2/3*(hi 2)
-  rw [intervalIntegral.integral_sub (hvI.const_mul _) heI,
-    intervalIntegral.integral_const_mul, intervalIntegral.integral_const_mul] at he
-  dsimp only [resolventA, resolventQ]
-  linarith only [he]
-
-lemma momentX_scale (v : Fin 3 → ℝ) (lam : ℝ) : momentX (fun i => lam*v i) = lam^2*momentX v := by
-  simp only [momentX, avg_expand]; ring
-
-lemma momentY_scale (v : Fin 3 → ℝ) (lam : ℝ) : momentY (fun i => lam*v i) = lam^3*momentY v := by
-  simp only [momentY, avg_expand]; ring
-
-lemma resolventQ_identity (v : Fin 3 → ℝ) (hp : ∀ i, 0 < 1+v i)
-    (lam : ℝ) (hl0 : 0 ≤ lam) (hl1 : lam ≤ 1) :
-    lam^3*(1-lam)*resolventQ v lam = lam^3*functionJ v-functionJ (fun i => lam*v i)+lam^2*(1-lam)*momentX v := by
-  have hh := resolvent_scaling v hp lam hl0 hl1
-  rw [resolventA_identity v hp, resolventA_identity _ (fun i => message_segment_pos v hp lam hl0 hl1 i),
-    momentX_scale, momentY_scale] at hh
-  nlinarith only [hh]
-
-end ThreeState.Radial
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeState.RadialMoments
+
+namespace OAI
+
+/-! Fourth-moment resolvents and their lower bounds. -/
+
+namespace ThreeState.Radial
+open MeasureTheory
+
+noncomputable def fourthResolvent (v : Fin 3 → ℝ) (t : ℝ) : ℝ := avg (fun i => (v i)^4/(1+t*v i))
+noncomputable def doubleFourthResolvent (v : Fin 3 → ℝ) (lam t : ℝ) : ℝ :=
+  avg (fun i => (v i)^4/((1+t*v i)*(1+lam*t*v i)))
+
+lemma fourth_moment (v : Fin 3 → ℝ) (hv : avg v = 0) : avg (fun i => (v i)^4) = 6*(momentX v)^2 := by
+  simp only [momentX, avg_expand, zero_sum_elim v hv]
+  ring
+
+lemma fourthResolvent_eq (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i)
+    (t : ℝ) (ht0 : 0 ≤ t) (ht1 : t ≤ 1) :
+    fourthResolvent v t = (6*(momentX v)^2-10*t*momentX v*momentY v+4*t^2*(momentY v)^2)/denom v t := by
+  have h0 := ne_of_gt (message_segment_pos v hp t ht0 ht1 0)
+  have h1 := ne_of_gt (message_segment_pos v hp t ht0 ht1 1)
+  have h2 := ne_of_gt (message_segment_pos v hp t ht0 ht1 2)
+  have h0c : 1+v 0*t ≠ 0 := by simpa [mul_comm] using h0
+  have h1c : 1+v 1*t ≠ 0 := by simpa [mul_comm] using h1
+  have h2c : 1+v 2*t ≠ 0 := by simpa [mul_comm] using h2
+  rw [denom_product v hv t]
+  simp only [fourthResolvent, momentX, momentY, avg_expand]
+  field_simp [h0, h1, h2, h0c, h1c, h2c]
+  rw [zero_sum_elim v hv]
+  ring
+
+lemma h_polynomial_bound (z : ℝ) (hz0 : 0 ≤ z) (hz1 : z < 1) :
+    (1-3*z^2+2*z^3)*(4+(64/9:ℝ)*(z-1/2)^2) ≤ 6-10*z+4*z^2 := by
+  rcases hz0.eq_or_lt with hzero | _
+  · rw [← hzero]
+    norm_num
+  have hE : 1+z-2*z^2 ≤ (9/8:ℝ) := by nlinarith [sq_nonneg (z-1/4)]
+  have hh : (1+z-2*z^2)*(4+(64/9:ℝ)*(z-1/2)^2) ≤ 6-4*z := by
+    nlinarith [mul_nonneg (show 0 ≤ (9/8:ℝ)-(1+z-2*z^2) by linarith) (sq_nonneg (z-1/2))]
+  have he := mul_le_mul_of_nonneg_left hh (show 0 ≤ 1-z by linarith)
+  nlinarith only [he]
+
+lemma momentY_sqrt_bound (v : Fin 3 → ℝ) (hv : avg v = 0) :
+    -(Real.sqrt (momentX v))^3 ≤ momentY v ∧ momentY v ≤ (Real.sqrt (momentX v))^3 := by
+  have h := discriminant_nonneg v hv
+  have hs := Real.sq_sqrt (momentX_nonneg v)
+  have he : (momentX v)^3 = ((Real.sqrt (momentX v))^3)^2 := by
+    calc
+      (momentX v)^3 = ((Real.sqrt (momentX v))^2)^3 := congrArg (fun x : ℝ => x^3) hs.symm
+      _ = _ := by ring
+  rw [he] at h
+  have hr : 0 ≤ (Real.sqrt (momentX v))^3 := by positivity
+  exact abs_le.mp ((sq_le_sq₀ (abs_nonneg (momentY v)) hr).mp (by simpa only [sq_abs] using h))
+
+lemma fourthResolvent_lower (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i)
+    (t : ℝ) (ht0 : 0 ≤ t) (ht1 : t ≤ 1) :
+    (momentX v)^2*(4+(64/9:ℝ)*(t*Real.sqrt (momentX v)-1/2)^2) ≤ fourthResolvent v t := by
+  let r := Real.sqrt (momentX v)
+  have hr0 : 0 ≤ r := Real.sqrt_nonneg _
+  have hr2 : r^2 = momentX v := Real.sq_sqrt (momentX_nonneg v)
+  have hr1 : r < 1 := by nlinarith [momentX_lt_one v hv hp]
+  have hy := momentY_sqrt_bound v hv
+  change -r^3 ≤ momentY v ∧ momentY v ≤ r^3 at hy
+  have hz0 : 0 ≤ t*r := mul_nonneg ht0 hr0
+  have hz1 : t*r < 1 := lt_of_le_of_lt (mul_le_of_le_one_left hr0 ht1) hr1
+  have hn : (momentX v)^2*(6-10*(t*r)+4*(t*r)^2) ≤
+      6*(momentX v)^2-10*t*momentX v*momentY v+4*t^2*(momentY v)^2 := by
+    have htr : t*r ≤ 1 := hz1.le
+    have hfac : 0 ≤ 10*momentX v-4*t*(momentY v+r^3) := by
+      have h1 := mul_le_mul_of_nonneg_left hy.2 (by positivity : 0 ≤ 4*t)
+      have h2 := mul_le_mul_of_nonneg_left htr (by positivity : 0 ≤ 8*r^2)
+      nlinarith only [h1,h2, sq_nonneg r,hr2]
+    have hh := mul_nonneg (mul_nonneg ht0 (sub_nonneg.mpr hy.2)) hfac
+    rw [← hr2] at hh ⊢
+    nlinarith only [hh]
+  have hd : denom v t ≤ 1-3*(t*r)^2+2*(t*r)^3 := by
+    have hh := mul_le_mul_of_nonneg_left hy.2 (by positivity : 0 ≤ 2*t^3)
+    dsimp only [denom]
+    rw [← hr2]
+    nlinarith only [hh]
+  have hg : 0 ≤ (momentX v)^2*(4+(64/9:ℝ)*(t*r-1/2)^2) := by positivity
+  have hh := mul_le_mul_of_nonneg_left hd hg
+  have hpoly := mul_le_mul_of_nonneg_left (h_polynomial_bound (t*r) hz0 hz1) (sq_nonneg (momentX v))
+  rw [fourthResolvent_eq v hv hp t ht0 ht1]
+  apply (le_div_iff₀ (denom_pos v hv hp t ht0 ht1)).mpr
+  dsimp only [r] at hh hpoly
+  nlinarith only [hh,hpoly,hn]
+
+lemma fourthResolvent_four (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i)
+    (t : ℝ) (ht0 : 0 ≤ t) (ht1 : t ≤ 1) : 4*(momentX v)^2 ≤ fourthResolvent v t := by
+  have hh := fourthResolvent_lower v hv hp t ht0 ht1
+  nlinarith [mul_nonneg (sq_nonneg (momentX v)) (sq_nonneg (t*Real.sqrt (momentX v)-1/2))]
+
+end ThreeState.Radial
+namespace ThreeState.Radial
+open MeasureTheory
+
+lemma doubleFourthResolvent_lower (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i)
+    (lam : ℝ) (hl0 : 0 ≤ lam) (hl1 : lam ≤ 1) (t : ℝ) (ht0 : 0 ≤ t) (ht1 : t ≤ 1) :
+    2*fourthResolvent v t ≤ (2+lam)*doubleFourthResolvent v lam t := by
+  let c := 1+lam/2
+  have hc : 0 < c := by dsimp [c]; linarith
+  have hlt0 : 0 ≤ lam*t := mul_nonneg hl0 ht0
+  have hlt1 : lam*t ≤ 1 := (mul_le_of_le_one_left ht0 hl1).trans ht1
+  have hi (i : Fin 3) : (2*c-1+lam)*(v i)^4/(1+t*v i)-lam*(v i)^4 ≤
+      c^2*(v i)^4/((1+t*v i)*(1+lam*t*v i)) := by
+    have hd := message_segment_pos v hp t ht0 ht1 i
+    have he := message_segment_pos v hp (lam*t) hlt0 hlt1 i
+    have hdc : 1+v i*t ≠ 0 := by simpa [mul_comm] using (ne_of_gt hd)
+    have hec : 1+v i*t*lam ≠ 0 := by convert ne_of_gt he using 1; ring
+    have hh : 0 ≤ (v i)^4*(c-(1+lam*t*v i))^2/((1+t*v i)*(1+lam*t*v i)) := by positivity
+    have hid : c^2*(v i)^4/((1+t*v i)*(1+lam*t*v i))-
+      ((2*c-1+lam)*(v i)^4/(1+t*v i)-lam*(v i)^4) =
+      (v i)^4*(c-(1+lam*t*v i))^2/((1+t*v i)*(1+lam*t*v i)) := by
+      field_simp [ne_of_gt hd, ne_of_gt he, hdc, hec]
+      ring
+    linarith
+  have hh : (2*c-1+lam)*fourthResolvent v t-lam*(6*(momentX v)^2) ≤
+      c^2*doubleFourthResolvent v lam t := by
+    rw [← fourth_moment v hv]
+    simp only [fourthResolvent, doubleFourthResolvent, avg_expand]
+    have hi0 := hi 0
+    have hi1 := hi 1
+    have hi2 := hi 2
+    simp only [mul_div_assoc] at hi0 hi1 hi2
+    nlinarith only [hi0,hi1,hi2]
+  have hw := fourthResolvent_four v hv hp t ht0 ht1
+  have hnon := mul_nonneg hl0 (show 0 ≤ fourthResolvent v t-4*(momentX v)^2 by linarith)
+  have hcW : c*fourthResolvent v t ≤ c^2*doubleFourthResolvent v lam t := by
+    dsimp only [c] at *
+    nlinarith only [hh,hnon]
+  have hres : fourthResolvent v t ≤ c*doubleFourthResolvent v lam t := by
+    apply (mul_le_mul_iff_right₀ hc).mp
+    nlinarith only [hcW]
+  dsimp only [c] at hres
+  linarith
+
+lemma continuousOn_fourthResolvent (v : Fin 3 → ℝ) (hp : ∀ i, 0 < 1+v i) :
+    ContinuousOn (fourthResolvent v) (Set.Icc 0 1) := by
+  have hi (i : Fin 3) : ContinuousOn (fun t : ℝ => (v i)^4/(1+t*v i)) (Set.Icc 0 1) :=
+    continuousOn_const.div (by fun_prop) (fun t ht => ne_of_gt (message_segment_pos v hp t ht.1 ht.2 i))
+  convert (((hi 0).add (hi 1)).add (hi 2)).div_const 3 using 1
+  funext t
+  exact avg_expand _
+
+lemma continuousOn_doubleFourthResolvent (v : Fin 3 → ℝ) (hp : ∀ i, 0 < 1+v i)
+    (lam : ℝ) (hl0 : 0 ≤ lam) (hl1 : lam ≤ 1) :
+    ContinuousOn (doubleFourthResolvent v lam) (Set.Icc 0 1) := by
+  have hi (i : Fin 3) : ContinuousOn (fun t : ℝ => (v i)^4/((1+t*v i)*(1+lam*t*v i))) (Set.Icc 0 1) :=
+    continuousOn_const.div (by fun_prop) (fun t ht => ne_of_gt (mul_pos
+      (message_segment_pos v hp t ht.1 ht.2 i)
+      (message_segment_pos v hp (lam*t) (mul_nonneg hl0 ht.1) ((mul_le_of_le_one_left ht.1 hl1).trans ht.2) i)))
+  convert (((hi 0).add (hi 1)).add (hi 2)).div_const 3 using 1
+  funext t
+  exact avg_expand _
+
+noncomputable def resolventA (v : Fin 3 → ℝ) : ℝ := ∫ t in (0:ℝ)..1, t^2*fourthResolvent v t
+noncomputable def resolventQ (v : Fin 3 → ℝ) (lam : ℝ) : ℝ := (∫ t in (0:ℝ)..1, t^2*doubleFourthResolvent v lam t)/2
+noncomputable def functionJ (v : Fin 3 → ℝ) : ℝ := avg (fun i => v i*Real.log (1+v i))/2
+
+lemma integral_h_lower (r : ℝ) :
+    (∫ t in (0:ℝ)..1, t^2*(4+(64/9:ℝ)*(t*r-1/2)^2)) = (37/27:ℝ)+(64/45:ℝ)*(r-5/8)^2 := by
+  let P := fun t : ℝ => (52/27:ℝ)*t^3-(16/9:ℝ)*r*t^4+(64/45:ℝ)*r^2*t^5
+  have hd (t : ℝ) : HasDerivAt P (t^2*(4+(64/9:ℝ)*(t*r-1/2)^2)) t := by
+    dsimp only [P]
+    convert ((((hasDerivAt_id t).pow 3).const_mul (52/27:ℝ)).sub
+      (((hasDerivAt_id t).pow 4).const_mul ((16/9:ℝ)*r))).add
+      (((hasDerivAt_id t).pow 5).const_mul ((64/45:ℝ)*r^2)) using 1 <;> first | rfl | (norm_num; ring)
+  rw [intervalIntegral.integral_eq_sub_of_hasDerivAt (fun t _ => hd t)
+    (Continuous.intervalIntegrable (by fun_prop) 0 1)]
+  dsimp [P]
+  ring
+
+lemma resolventA_lower (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i) :
+    (37/27:ℝ)*(momentX v)^2 ≤ resolventA v := by
+  have hi : IntervalIntegrable (fun t : ℝ => (momentX v)^2*(t^2*(4+(64/9:ℝ)*(t*Real.sqrt (momentX v)-1/2)^2)))
+      volume 0 1 := Continuous.intervalIntegrable (by fun_prop) 0 1
+  have hj : IntervalIntegrable (fun t : ℝ => t^2*fourthResolvent v t) volume 0 1 :=
+    ((by fun_prop : ContinuousOn (fun t : ℝ => t^2) (Set.Icc 0 1)).mul (continuousOn_fourthResolvent v hp)).intervalIntegrable_of_Icc (by norm_num)
+  have hh := intervalIntegral.integral_mono_on (by norm_num : (0:ℝ) ≤ 1) hi hj (fun t ht => by
+    have he := mul_le_mul_of_nonneg_left (fourthResolvent_lower v hv hp t ht.1 ht.2) (sq_nonneg t)
+    nlinarith only [he])
+  rw [intervalIntegral.integral_const_mul, integral_h_lower] at hh
+  dsimp only [resolventA]
+  nlinarith [mul_nonneg (sq_nonneg (momentX v)) (sq_nonneg (Real.sqrt (momentX v)-5/8))]
+
+lemma resolventQ_lower (v : Fin 3 → ℝ) (hv : avg v = 0) (hp : ∀ i, 0 < 1+v i)
+    (lam : ℝ) (hl0 : 0 ≤ lam) (hl1 : lam ≤ 1) : resolventA v/(2+lam) ≤ resolventQ v lam := by
+  have hi : IntervalIntegrable (fun t : ℝ => 2*(t^2*fourthResolvent v t)) volume 0 1 :=
+    (continuousOn_const.mul ((continuousOn_id.pow 2).mul (continuousOn_fourthResolvent v hp))).intervalIntegrable_of_Icc (by norm_num)
+  have hj : IntervalIntegrable (fun t : ℝ => (2+lam)*(t^2*doubleFourthResolvent v lam t)) volume 0 1 :=
+    (continuousOn_const.mul ((continuousOn_id.pow 2).mul (continuousOn_doubleFourthResolvent v hp lam hl0 hl1))).intervalIntegrable_of_Icc (by norm_num)
+  have hh := intervalIntegral.integral_mono_on (by norm_num : (0:ℝ) ≤ 1) hi hj (fun t ht => by
+    have he := mul_le_mul_of_nonneg_left (doubleFourthResolvent_lower v hv hp lam hl0 hl1 t ht.1 ht.2) (sq_nonneg t)
+    nlinarith only [he])
+  rw [intervalIntegral.integral_const_mul, intervalIntegral.integral_const_mul] at hh
+  apply (div_le_iff₀ (by linarith : 0 < 2+lam)).mpr
+  dsimp only [resolventQ, resolventA]
+  linarith
+
+lemma scalar_resolventA (a : ℝ) (ha : 0 < 1+a) :
+    (∫ t in (0:ℝ)..1, t^2*a^4/(1+t*a)) = a^3/2-a^2+a*Real.log (1+a) := by
+  have hp (t : ℝ) (ht : t ∈ Set.Icc (0:ℝ) 1) : 0 < 1+t*a := by
+    rcases eq_or_lt_of_le ht.1 with hz | hz
+    · rw [← hz]; norm_num
+    · nlinarith [mul_pos hz ha, ht.2]
+  have hc : ContinuousOn (fun t : ℝ => a/(1+t*a)) (Set.Icc 0 1) :=
+    continuousOn_const.div (by fun_prop) (fun t ht => ne_of_gt (hp t ht))
+  have hi : IntervalIntegrable (fun t : ℝ => a/(1+t*a)) volume 0 1 := hc.intervalIntegrable_of_Icc (by norm_num : (0:ℝ) ≤ 1)
+  have he : (∫ t in (0:ℝ)..1, t^2*a^4/(1+t*a)) =
+      ∫ t in (0:ℝ)..1, (a^3*t-a^2)+a*(a/(1+t*a)) := by
+    apply intervalIntegral.integral_congr
+    intro t ht
+    have ht' : t ∈ Set.Icc (0:ℝ) 1 := by simpa using ht
+    field_simp [ne_of_gt (hp t ht')]
+    ring
+  rw [he, intervalIntegral.integral_add (Continuous.intervalIntegrable (by fun_prop) 0 1) (hi.const_mul a),
+    intervalIntegral.integral_sub (Continuous.intervalIntegrable (by fun_prop) 0 1) (intervalIntegrable_const),
+    intervalIntegral.integral_const_mul, intervalIntegral.integral_const_mul, integral_id,
+    intervalIntegral.integral_const, ← log_integral a ha]
+  norm_num
+  ring
+
+lemma resolventA_identity (v : Fin 3 → ℝ) (hp : ∀ i, 0 < 1+v i) :
+    resolventA v = 2*functionJ v-2*momentX v+momentY v := by
+  have he (t : ℝ) : t^2*fourthResolvent v t = avg (fun i => t^2*(v i)^4/(1+t*v i)) := by
+    simp only [fourthResolvent, avg_expand]; ring
+  dsimp only [resolventA]
+  simp_rw [he]
+  rw [integral_avg]
+  · simp_rw [scalar_resolventA _ (hp _)]
+    simp only [functionJ, momentX, momentY, avg_expand]
+    ring
+  · intro i
+    exact ((by fun_prop : ContinuousOn (fun t : ℝ => t^2*(v i)^4) (Set.Icc 0 1)).div
+      (by fun_prop) (fun t ht => ne_of_gt (message_segment_pos v hp t ht.1 ht.2 i))).intervalIntegrable_of_Icc (by norm_num)
+
+lemma resolvent_scaling (v : Fin 3 → ℝ) (hp : ∀ i, 0 < 1+v i)
+    (lam : ℝ) (hl0 : 0 ≤ lam) (hl1 : lam ≤ 1) :
+    lam^3*resolventA v-resolventA (fun i => lam*v i) = 2*lam^3*(1-lam)*resolventQ v lam := by
+  have hpe (i : Fin 3) : 0 < 1+lam*v i := message_segment_pos v hp lam hl0 hl1 i
+  have hvI : IntervalIntegrable (fun t : ℝ => t^2*fourthResolvent v t) volume 0 1 :=
+    ((continuousOn_id.pow 2).mul (continuousOn_fourthResolvent v hp)).intervalIntegrable_of_Icc (by norm_num)
+  have heI : IntervalIntegrable (fun t : ℝ => t^2*fourthResolvent (fun i => lam*v i) t) volume 0 1 :=
+    ((continuousOn_id.pow 2).mul (continuousOn_fourthResolvent _ hpe)).intervalIntegrable_of_Icc (by norm_num)
+  have he : (∫ t in (0:ℝ)..1, lam^3*(t^2*fourthResolvent v t)-t^2*fourthResolvent (fun i => lam*v i) t) =
+      ∫ t in (0:ℝ)..1, (lam^3*(1-lam))*(t^2*doubleFourthResolvent v lam t) := by
+    apply intervalIntegral.integral_congr
+    intro t ht
+    have ht' : t ∈ Set.Icc (0:ℝ) 1 := by simpa using ht
+    have hi (i : Fin 3) : lam^3*((v i)^4/(1+t*v i))-(lam*v i)^4/(1+t*(lam*v i)) =
+        lam^3*(1-lam)*((v i)^4/((1+t*v i)*(1+lam*t*v i))) := by
+      have ha := ne_of_gt (message_segment_pos v hp t ht'.1 ht'.2 i)
+      have hb := ne_of_gt (message_segment_pos _ hpe t ht'.1 ht'.2 i)
+      have ha' : 1+v i*t ≠ 0 := by simpa [mul_comm] using ha
+      have hb' : 1+v i*t*lam ≠ 0 := by convert hb using 1; ring
+      have hb'' : 1+lam*v i*t ≠ 0 := by convert hb using 1; ring
+      have hb''' : 1+lam*t*v i ≠ 0 := by convert hb using 1; ring
+      field_simp [ha, hb, ha', hb', hb'', hb''']
+      ring
+    simp only [fourthResolvent, doubleFourthResolvent, avg_expand]
+    linear_combination t^2/3*(hi 0)+t^2/3*(hi 1)+t^2/3*(hi 2)
+  rw [intervalIntegral.integral_sub (hvI.const_mul _) heI,
+    intervalIntegral.integral_const_mul, intervalIntegral.integral_const_mul] at he
+  dsimp only [resolventA, resolventQ]
+  linarith only [he]
+
+lemma momentX_scale (v : Fin 3 → ℝ) (lam : ℝ) : momentX (fun i => lam*v i) = lam^2*momentX v := by
+  simp only [momentX, avg_expand]; ring
+
+lemma momentY_scale (v : Fin 3 → ℝ) (lam : ℝ) : momentY (fun i => lam*v i) = lam^3*momentY v := by
+  simp only [momentY, avg_expand]; ring
+
+lemma resolventQ_identity (v : Fin 3 → ℝ) (hp : ∀ i, 0 < 1+v i)
+    (lam : ℝ) (hl0 : 0 ≤ lam) (hl1 : lam ≤ 1) :
+    lam^3*(1-lam)*resolventQ v lam = lam^3*functionJ v-functionJ (fun i => lam*v i)+lam^2*(1-lam)*momentX v := by
+  have hh := resolvent_scaling v hp lam hl0 hl1
+  rw [resolventA_identity v hp, resolventA_identity _ (fun i => message_segment_pos v hp lam hl0 hl1 i),
+    momentX_scale, momentY_scale] at hh
+  nlinarith only [hh]
+
+end ThreeState.Radial
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeState/ScalarBounds.lean b/lean/OAI/Probability/ThreeState/ScalarBounds.lean
index 28c8a987850efbdb09c63b1cd8c2ff1f801a9eaa..0f3d00d9472b08dbe4f0242b46add0877d5f4adc 100644
--- a/lean/OAI/Probability/ThreeState/ScalarBounds.lean
+++ b/lean/OAI/Probability/ThreeState/ScalarBounds.lean
@@ -1,387 +1,387 @@
-import Mathlib
-
-namespace OAI
-
-/-! Scalar inequalities for criticality, Poisson survival and nonlinear decay. -/
-
-namespace ThreeState.PositiveScalar
-
-noncomputable def lowerC (lam : ℝ) : ℝ := (37/27:ℝ)*(1+2*lam/3)
-
-lemma small_polynomial (lam : ℝ) (h0 : 0 < lam) (h1 : lam ≤ 3/10) :
-    (2/5-lam^2/2)*(lowerC lam)^2 > (2/3:ℝ)*(11/10+lam^2/2) := by
-  have hnon : 0 ≤ (3/10:ℝ)-lam := by linarith
-  have he : (2/5-lam^2/2)*(lowerC lam)^2-(2/3:ℝ)*(11/10+lam^2/2) =
-    (130000/59049:ℝ)*(3/10-lam)^4+(2710400/59049:ℝ)*lam*(3/10-lam)^3+
-    (6735490/59049:ℝ)*lam^2*(3/10-lam)^2+(1871120/19683:ℝ)*lam^3*(3/10-lam)+
-    (159292/6561:ℝ)*lam^4 := by dsimp only [lowerC]; ring
-  have hp : 0 < (130000/59049:ℝ)*(3/10-lam)^4+(2710400/59049:ℝ)*lam*(3/10-lam)^3+
-    (6735490/59049:ℝ)*lam^2*(3/10-lam)^2+(1871120/19683:ℝ)*lam^3*(3/10-lam)+
-    (159292/6561:ℝ)*lam^4 := by positivity
-  linarith
-
-lemma middle_polynomial (lam : ℝ) (h0 : 3/10 ≤ lam) (h1 : lam^2 ≤ 1/2) :
-    1-17*lam/25 < (2/5-lam^2/2)*(lowerC lam)^2 := by
-  have ha : 0 ≤ lam-(3/10:ℝ) := by linarith
-  have hb : 0 < (71/100:ℝ)-lam := by nlinarith
-  have he : (2/5-lam^2/2)*(lowerC lam)^2-(1-17*lam/25) =
-    (1328320000/228886641:ℝ)*(71/100-lam)^4+
-    (75777584000/2059979769:ℝ)*(lam-3/10)*(71/100-lam)^3+
-    (1184570339800/18539817921:ℝ)*(lam-3/10)^2*(71/100-lam)^2+
-    (674265198440/18539817921:ℝ)*(lam-3/10)^3*(71/100-lam)+
-    (56362261022/18539817921:ℝ)*(lam-3/10)^4 := by dsimp only [lowerC]; ring
-  have hp : 0 < (1328320000/228886641:ℝ)*(71/100-lam)^4+
-    (75777584000/2059979769:ℝ)*(lam-3/10)*(71/100-lam)^3+
-    (1184570339800/18539817921:ℝ)*(lam-3/10)^2*(71/100-lam)^2+
-    (674265198440/18539817921:ℝ)*(lam-3/10)^3*(71/100-lam)+
-    (56362261022/18539817921:ℝ)*(lam-3/10)^4 := by positivity
-  linarith
-
-lemma rational_lower (mu : ℝ) (hm : 0 ≤ mu) : 3/2-mu ≤ 3/(2*(1+2*mu/3)) := by
-  apply (le_div_iff₀ (by positivity : 0 < 2*(1+2*mu/3))).mpr
-  nlinarith [sq_nonneg mu]
-
-lemma regular_scalar (lam mu c : ℝ) (h0 : 0 < lam) (hl : lam^2 ≤ 1/2)
-    (hm : 0 ≤ mu) (hc : lowerC lam ≤ c) (hb : mu*c^2 ≤ 1) :
-    8/5 < (1-lam^2)/2+3/(2*(1+2*mu/3))+17*lam*mu/25 := by
-  have hl1 : lam < 1 := by nlinarith
-  have hc0 : 0 < lowerC lam := by dsimp only [lowerC]; positivity
-  have hc2 : (lowerC lam)^2 ≤ c^2 := sq_le_sq₀ hc0.le (hc0.trans_le hc).le |>.mpr hc
-  have hcpos : 0 < c := hc0.trans_le hc
-  have hden : 0 < 2*(1+2*mu/3) := by positivity
-  rcases le_total lam (3/10) with hs | hs
-  · have hp := small_polynomial lam h0 hs
-    have hfac : 0 < 2/5-lam^2/2 := by nlinarith
-    have hpc : (2/3:ℝ)*(11/10+lam^2/2) < (2/5-lam^2/2)*c^2 :=
-      hp.trans_le (mul_le_mul_of_nonneg_left hc2 hfac.le)
-    have ht : (11/10+lam^2/2)*mu < (3/2:ℝ)*(2/5-lam^2/2) := by
-      have hh := mul_le_mul_of_nonneg_left hb (show 0 ≤ 11/10+lam^2/2 by positivity)
-      have hh' := mul_lt_mul_of_pos_left hpc (show 0 < (3/2:ℝ) by norm_num)
-      have he : ((11/10+lam^2/2)*mu)*c^2 < ((3/2:ℝ)*(2/5-lam^2/2))*c^2 := by nlinarith only [hh,hh']
-      exact (mul_lt_mul_iff_left₀ (sq_pos_of_pos hcpos)).mp he
-    have he : 11/10+lam^2/2 < 3/(2*(1+2*mu/3)) := by
-      apply (lt_div_iff₀ hden).mpr
-      nlinarith only [ht]
-    have hlast : 0 ≤ 17*lam*mu/25 := by positivity
-    linarith
-  · have hpoly := middle_polynomial lam hs hl
-    have hfac : 0 < 2/5-lam^2/2 := by nlinarith
-    have hpc : 1-17*lam/25 < (2/5-lam^2/2)*c^2 :=
-      hpoly.trans_le (mul_le_mul_of_nonneg_left hc2 hfac.le)
-    have hcoef : 0 < 1-17*lam/25 := by linarith
-    have hh := mul_le_mul_of_nonneg_left hb hcoef.le
-    have ht : (1-17*lam/25)*mu < 2/5-lam^2/2 := by
-      apply (mul_lt_mul_iff_left₀ (sq_pos_of_pos hcpos)).mp
-      nlinarith only [hpc,hh]
-    have hr := rational_lower mu hm
-    nlinarith only [ht,hr]
-
-lemma high_scalar (lam mu c : ℝ) (h0 : 0 < lam) (hl1 : lam < 1) (hl : 1/2 < lam^2)
-    (hm : 0 ≤ mu) (hc : lowerC lam ≤ c) (hb : mu*c^2 ≤ 1) :
-    8/5 < (1/4:ℝ)+3/(2*(1+2*mu/3))+17*lam*mu/25 := by
-  have hx : (707/1000:ℝ) < lam := by nlinarith
-  have hc' : (37/27:ℝ)*(1+2*(707/1000:ℝ)/3) < c := by dsimp only [lowerC] at hc; linarith
-  have hcpos : 0 < c := by linarith
-  have hp : 1-17*lam/25 < (3/20:ℝ)*c^2 := by nlinarith [sq_nonneg (c-(37/27:ℝ)*(1+2*(707/1000:ℝ)/3))]
-  have hcoef : 0 < 1-17*lam/25 := by linarith
-  have hh := mul_le_mul_of_nonneg_left hb hcoef.le
-  have ht : (1-17*lam/25)*mu < (3/20:ℝ) := by
-    apply (mul_lt_mul_iff_left₀ (sq_pos_of_pos hcpos)).mp
-    nlinarith only [hp,hh]
-  have hr := rational_lower mu hm
-  nlinarith only [ht,hr]
-
- 
-
-theorem scalar_contradiction (lam mu c s : ℝ) (h0 : 0 < lam) (h1 : lam < 1)
-    (hm : 0 ≤ mu) (hs0 : 0 < s) (hs1 : s ≤ 1) (hc : lowerC lam ≤ c)
-    (hb : mu*c^2 ≤ s) (hsurv : 1/2 < lam^2 → s ≤ 2*(1-lam^2)) :
-    8/5 < (1-lam^2)/(2*s)+3/(2*(1+2*mu/3))+17*lam*mu/(25*s) := by
-  have hlast : 17*lam*mu/25 ≤ 17*lam*mu/(25*s) := by
-    apply (le_div_iff₀ (by positivity : 0 < 25*s)).mpr
-    nlinarith [mul_nonneg (show 0 ≤ 17*lam*mu by positivity) (show 0 ≤ 1-s by linarith)]
-  rcases le_or_gt (lam^2) (1/2) with hl | hl
-  · have hh := regular_scalar lam mu c h0 hl hm hc (hb.trans hs1)
-    have hf : (1-lam^2)/2 ≤ (1-lam^2)/(2*s) := by
-      apply (le_div_iff₀ (by positivity : 0 < 2*s)).mpr
-      nlinarith [mul_nonneg (show 0 ≤ 1-lam^2 by nlinarith) (show 0 ≤ 1-s by linarith)]
-    linarith
-  · have hh := high_scalar lam mu c h0 h1 hl hm hc (hb.trans hs1)
-    have hf : (1/4:ℝ) ≤ (1-lam^2)/(2*s) := by
-      apply (le_div_iff₀ (by positivity : 0 < 2*s)).mpr
-      linarith [hsurv hl]
-    linarith
-
-end ThreeState.PositiveScalar
-
-namespace ThreeState.PositiveScalar
-
-lemma q_lower (lam : ℝ) (h0 : 0 ≤ lam) (h1 : lam ≤ 1) :
-    (37/81:ℝ) ≤ (37/27:ℝ)/(2+lam) := by
-  apply (le_div_iff₀ (by linarith : 0 < 2+lam)).mpr
-  linarith
-
-lemma root_loss : (1:ℝ)/Real.sqrt 2 < 113/81-17/25 := by
-  have hr : 0 < Real.sqrt 2 := by positivity
-  have hs : (Real.sqrt 2)^2=2 := Real.sq_sqrt (by norm_num)
-  apply (div_lt_iff₀ hr).mpr
-  nlinarith only [hs, Real.sqrt_nonneg 2]
-
-lemma J_square_bound (lam mu nu J : ℝ) (h0 : 0 < lam) (h1 : lam < 1)
-    (hm : 0 ≤ mu) (hn : 0 ≤ nu) (hJ0 : 0 ≤ J)
-    (hJ : J ≤ mu-lam*((37/27:ℝ)/(2+lam))*nu)
-    (hc : ((37/27:ℝ)+2*lam*((37/27:ℝ)/(2+lam)))^2*nu ≤ mu) :
-    (8/5:ℝ)*J^2 ≤ (8/5:ℝ)*mu^2-(113/81:ℝ)*lam*mu*nu := by
-  let q : ℝ := (37/27:ℝ)/(2+lam)
-  let A := lam*q*nu
-  have hq := q_lower lam h0.le h1.le
-  change (37/81:ℝ) ≤ q at hq
-  have hA0 : 0 ≤ A := by dsimp only [A]; positivity
-  have hAc : 8*(37/27:ℝ)*A ≤ mu := by
-    have hh := mul_le_mul_of_nonneg_right (sq_nonneg ((37/27:ℝ)-2*lam*q)) hn
-    dsimp only [A,q] at *
-    nlinarith only [hh,hc]
-  have hA : A ≤ (27/296:ℝ)*mu := by linarith only [hAc]
-  have hs := mul_le_mul_of_nonneg_right hA hA0
-  change J ≤ mu-A at hJ
-  have hj2 : J^2 ≤ (mu-A)^2 := (sq_le_sq₀ hJ0 (hJ0.trans hJ)).mpr hJ
-  have hqmu := mul_le_mul_of_nonneg_right hq (show 0 ≤ lam*mu*nu by positivity)
-  have hlead : mu*A = q*(lam*mu*nu) := by dsimp only [A]; ring
-  rw [← hlead] at hqmu
-  nlinarith only [hj2,hs,hqmu]
-
-lemma scalar_from_moments (lam mu nu J s : ℝ) (h0 : 0 < lam) (h1 : lam < 1)
-    (hm : 0 < mu) (hn : 0 ≤ nu) (hs0 : 0 < s) (_ : s ≤ 1)
-    (hsecond : mu^2 ≤ s*nu)
-    (hcnu : ((37/27:ℝ)+2*lam*((37/27:ℝ)/(2+lam)))^2*nu ≤ mu)
-    (hJ0 : 0 ≤ J) (hJ : J ≤ mu-lam*((37/27:ℝ)/(2+lam))*nu)
-    (hentropy : (1-lam^2)/2*nu+3*mu^2/(2*(1+2*mu/3))-lam*mu*nu/Real.sqrt 2 ≤ (8/5:ℝ)*J^2) :
-    8/5 ≥ (1-lam^2)/(2*s)+3/(2*(1+2*mu/3))+17*lam*mu/(25*s) := by
-  have hj2 := J_square_bound lam mu nu J h0 h1 hm.le hn hJ0 hJ hcnu
-  have hroot := mul_le_mul_of_nonneg_right root_loss.le (show 0 ≤ lam*mu*nu by positivity)
-  have he : ((1-lam^2)/2+17*lam*mu/25)*nu+3*mu^2/(2*(1+2*mu/3)) ≤ (8/5:ℝ)*mu^2 := by
-    simp only [div_eq_mul_inv] at hentropy hj2 hroot ⊢
-    nlinarith only [hentropy,hj2,hroot]
-  have ha : 0 ≤ (1-lam^2)/2+17*lam*mu/25 := by
-    have hh : 0 ≤ 1-lam^2 := by nlinarith
-    positivity
-  have hh := mul_le_mul_of_nonneg_left ((div_le_iff₀ hs0).mpr (show mu^2 ≤ nu*s by nlinarith only [hsecond])) ha
-  have hneed : ((1-lam^2)/(2*s)+3/(2*(1+2*mu/3))+17*lam*mu/(25*s))*mu^2 ≤ (8/5:ℝ)*mu^2 := by
-    simp only [div_eq_mul_inv, mul_inv_rev] at he hh ⊢
-    nlinarith only [he,hh]
-  exact (mul_le_mul_iff_left₀ (sq_pos_of_pos hm)).mp hneed
-
-lemma moment_scalar_contradiction (lam mu nu J s : ℝ) (h0 : 0 < lam) (h1 : lam < 1)
-    (hm : 0 < mu) (hn : 0 ≤ nu) (hs0 : 0 < s) (hs1 : s ≤ 1)
-    (hsecond : mu^2 ≤ s*nu)
-    (hcnu : ((37/27:ℝ)+2*lam*((37/27:ℝ)/(2+lam)))^2*nu ≤ mu)
-    (hJ0 : 0 ≤ J) (hJ : J ≤ mu-lam*((37/27:ℝ)/(2+lam))*nu)
-    (hentropy : (1-lam^2)/2*nu+3*mu^2/(2*(1+2*mu/3))-lam*mu*nu/Real.sqrt 2 ≤ (8/5:ℝ)*J^2)
-    (hsurv : 1/2 < lam^2 → s ≤ 2*(1-lam^2)) : False := by
-  let c : ℝ := (37/27:ℝ)+2*lam*((37/27:ℝ)/(2+lam))
-  have hc : 0 < c := by dsimp only [c]; positivity
-  have hcL : lowerC lam ≤ c := by
-    have hh := q_lower lam h0.le h1.le
-    dsimp only [lowerC,c]
-    nlinarith only [hh,h0]
-  have hb : mu*c^2 ≤ s := by
-    have hh := mul_le_mul_of_nonneg_right hsecond (sq_nonneg c)
-    have hh' := mul_le_mul_of_nonneg_left hcnu hs0.le
-    apply (mul_le_mul_iff_left₀ hm).mp
-    dsimp only [c] at *
-    nlinarith only [hh,hh']
-  have hfalse := scalar_contradiction lam mu c s h0 h1 hm.le hs0 hs1 hcL hb hsurv
-  have hneed := scalar_from_moments lam mu nu J s h0 h1 hm hn hs0 hs1 hsecond hcnu hJ0 hJ hentropy
-  linarith only [hfalse,hneed]
-
-end ThreeState.PositiveScalar
-
-namespace ThreeState
-open Filter Topology
-
-lemma poisson_sqrt_tangent (d lam t : ℝ) (_ : 0<d) (hl : 0<lam) (ht : 0≤t)
-    (hc : d*lam^2=1) : Real.sqrt t ≤ lam*(t+d/2)/Real.sqrt 2 := by
-  have hs2 : (Real.sqrt 2)^2=(2:ℝ) := Real.sq_sqrt (by norm_num)
-  have hst : (Real.sqrt t)^2=t := Real.sq_sqrt ht
-  have hsp : 0 < Real.sqrt (2:ℝ) := by positivity
-  apply (le_div_iff₀ hsp).mpr
-  apply (mul_le_mul_iff_right₀ hl).mp
-  have hp := sq_nonneg (Real.sqrt 2*lam*Real.sqrt t-1)
-  have he : (Real.sqrt 2*lam*Real.sqrt t)^2=2*lam^2*t := by
-    calc _ = (Real.sqrt 2)^2*lam^2*(Real.sqrt t)^2 := by ring
-         _ = _ := by rw [hs2,hst]
-  nlinarith only [hp,he,hc]
-
-lemma poisson_reciprocal_tangent (t beta D : ℝ) (ht : 0≤t) (hb : 0≤beta) (hD : 0<D) :
-    t*(2*D-1-beta*t)/D^2 ≤ t/(1+beta*t) := by
-  have hy : 0<1+beta*t := by positivity
-  apply (div_le_div_iff₀ (sq_pos_of_pos hD) hy).mpr
-  nlinarith only [mul_nonneg ht (sq_nonneg (D-(1+beta*t)))]
-
- 
-
-lemma poisson_correlation_scalar (d lam mu nu : ℝ) (hd : 0<d) (hl : 0<lam)
-    (hm : 0 ≤ mu) (hn : 0≤nu) (hc : d*lam^2=1)
-    (H C : ℝ → ℝ) (hH0 : H 0=0)
-    (hderiv : ∀ t ∈ Set.Icc 0 d, HasDerivAt H (C t) t)
-    (hlower : ∀ t ∈ Set.Icc 0 d,
-      3*lam^4*mu^2*t/(1+t*lam^2*mu)-lam^4*mu*nu*Real.sqrt t ≤ 3*C t) :
-    3*mu^2/(2*(1+2*mu/3))-lam*mu*nu/Real.sqrt 2 ≤ 3*H d := by
-  let D := 1+2*mu/3
-  have hD : 0<D := by dsimp [D]; positivity
-  let P := fun t : ℝ => (3*lam^4*mu^2/D^2)*((2*D-1)*t^2/2-(lam^2*mu)*t^3/3)-
-    (lam^5*mu*nu/Real.sqrt 2)*(t^2/2+d*t/2)
-  let P' := fun t : ℝ => (3*lam^4*mu^2/D^2)*((2*D-1)*t-(lam^2*mu)*t^2)-
-    (lam^5*mu*nu/Real.sqrt 2)*(t+d/2)
-  have hp (t : ℝ) : HasDerivAt P (P' t) t := by
-    have hh := (((((hasDerivAt_pow 2 t).const_mul (2*D-1)).div_const 2).sub
-      (((hasDerivAt_pow 3 t).const_mul (lam^2*mu)).div_const 3)).const_mul (3*lam^4*mu^2/D^2)).sub
-      ((((hasDerivAt_pow 2 t).div_const 2).add (((hasDerivAt_id t).const_mul d).div_const 2)).const_mul (lam^5*mu*nu/Real.sqrt 2))
-    convert hh using 1 <;> try rfl
-    dsimp only [P']
-    ring
-  have hpd (t : ℝ) (ht : t ∈ Set.Icc 0 d) : P' t ≤ 3*C t := by
-    have hr := mul_le_mul_of_nonneg_left (poisson_reciprocal_tangent t (lam^2*mu) D ht.1 (by positivity) hD)
-      (show 0≤3*lam^4*mu^2 by positivity)
-    have hs := mul_le_mul_of_nonneg_left (poisson_sqrt_tangent d lam t hd hl ht.1 hc)
-      (show 0≤lam^4*mu*nu by positivity)
-    calc
-      P' t = 3*lam^4*mu^2*(t*(2*D-1-(lam^2*mu)*t)/D^2)-lam^4*mu*nu*(lam*(t+d/2)/Real.sqrt 2) := by dsimp only [P']; ring
-      _ ≤ 3*lam^4*mu^2*(t/(1+(lam^2*mu)*t))-lam^4*mu*nu*Real.sqrt t := sub_le_sub hr hs
-      _ = 3*lam^4*mu^2*t/(1+t*lam^2*mu)-lam^4*mu*nu*Real.sqrt t := by ring
-      _ ≤ _ := hlower t ht
-  have hmono : MonotoneOn (fun t => 3*H t-P t) (Set.Icc 0 d) := by
-    apply monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc 0 d) (f' := fun t => 3*C t-P' t)
-    · intro t ht
-      exact (((hderiv t ht).const_mul 3).sub (hp t)).continuousAt.continuousWithinAt
-    · intro t ht
-      exact (((hderiv t (interior_subset ht)).const_mul 3).sub (hp t)).hasDerivWithinAt
-    · intro t ht
-      exact sub_nonneg.mpr (hpd t (interior_subset ht))
-  have hh := hmono (show (0:ℝ)∈Set.Icc 0 d from ⟨le_rfl,hd.le⟩) (show d∈Set.Icc 0 d from ⟨hd.le,le_rfl⟩) hd.le
-  have hp0 : P 0=0 := by simp [P]
-  have hpend : P d=3*mu^2/(2*D)-lam*mu*nu/Real.sqrt 2 := by
-    have hd2 : d^2*lam^4=1 := by nlinarith [sq_nonneg (d*lam^2-1)]
-    have hd3 : d^3*lam^6=1 := by
-      calc
-        _ = (d*lam^2)^3 := by ring
-        _ = 1 := by rw [hc]; norm_num
-    dsimp only [P]
-    field_simp [ne_of_gt hD]
-    dsimp only [D]
-    nlinarith only [congrArg (fun x : ℝ => x*(Real.sqrt 2)*mu^2) hd2,
-      congrArg (fun x : ℝ => x*(Real.sqrt 2)*mu^3) hd2,
-      congrArg (fun x : ℝ => x*(Real.sqrt 2)*mu^3) hd3,
-      congrArg (fun x : ℝ => x*lam*mu*nu) hd2,
-      congrArg (fun x : ℝ => x*lam*mu^2*nu) hd2,
-      congrArg (fun x : ℝ => x*lam*mu^3*nu) hd2]
-  change 3*H 0-P 0 ≤ 3*H d-P d at hh
-  rw [hH0,hp0,hpend] at hh
-  dsimp only [D] at hh
-  linarith
-
-end ThreeState
-
-namespace ThreeState
-open Filter Topology
-
-lemma log_survival_bound (s : ℝ) (hs0 : 0 ≤ s) (hs1 : s < 1) :
-    2*s/(2-s) ≤ -Real.log (1-s) := by
-  let F := fun x : ℝ => -Real.log (1-x)-2*x/(2-x)
-  have hd (x : ℝ) (hx : x ∈ Set.Icc 0 s) : HasDerivAt F
-      (1/(1-x)-4/(2-x)^2) x := by
-    have h1 : 1-x ≠ 0 := by linarith [hx.2]
-    have h2 : 2-x ≠ 0 := by linarith [hx.2]
-    convert (((hasDerivAt_const x (1:ℝ)).sub (hasDerivAt_id x)).log h1).neg.sub
-      (((hasDerivAt_id x).const_mul 2).div ((hasDerivAt_const x (2:ℝ)).sub (hasDerivAt_id x)) h2) using 1 <;> try rfl
-    dsimp
-    field_simp
-    ring
-  have hder (x : ℝ) (hx : x ∈ Set.Icc 0 s) : 0 ≤ 1/(1-x)-4/(2-x)^2 := by
-    have h1 : 0 < 1-x := by linarith [hx.2]
-    have h2 : 0 < 2-x := by linarith [hx.2]
-    apply sub_nonneg.mpr
-    apply (div_le_div_iff₀ (sq_pos_of_pos h2) h1).mpr
-    nlinarith [sq_nonneg x]
-  have hmono : MonotoneOn F (Set.Icc 0 s) := by
-    apply monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc 0 s)
-      (f' := fun x => 1/(1-x)-4/(2-x)^2)
-    · intro x hx; exact (hd x hx).continuousAt.continuousWithinAt
-    · intro x hx; exact (hd x (interior_subset hx)).hasDerivWithinAt
-    · intro x hx; exact hder x (interior_subset hx)
-  have hh := hmono (show (0:ℝ)∈Set.Icc 0 s from ⟨le_rfl,hs0⟩) (show s∈Set.Icc 0 s from ⟨hs0,le_rfl⟩) hs0
-  dsimp only [F] at hh
-  norm_num at hh
-  linarith
-
-lemma poisson_support_bound (d s : ℝ) (hd : 0 < d) (hs : 0 < s)
-    (hfp : Real.exp (-d*s) ≤ 1-s) : s ≤ 2*(1-d⁻¹) := by
-  have hs1 : s < 1 := by linarith [Real.exp_pos (-d*s)]
-  have hlog := Real.log_le_log (Real.exp_pos (-d*s)) hfp
-  rw [Real.log_exp] at hlog
-  have hh := log_survival_bound s hs.le hs1
-  have hden : 0 < 2-s := by linarith
-  have hrat : 2*s/(2-s) ≤ d*s := by linarith
-  have hr := (div_le_iff₀ hden).mp hrat
-  have hsd : 2 ≤ d*(2-s) := (mul_le_mul_iff_right₀ hs).mp (by nlinarith only [hr])
-  apply (mul_le_mul_iff_left₀ hd).mp
-  have he : (2*(1-d⁻¹))*d=2*d-2 := by field_simp
-  rw [he]
-  nlinarith only [hsd]
-end ThreeState
-
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-noncomputable section
-
-lemma bernoulli_lower (t:ℝ) (ht0:0≤t) (ht1:t≤1) (k:ℕ) : 1-k*t≤(1-t)^k := by
-  induction k with
-  | zero => simp
-  | succ k ih =>
-    have hh:=mul_le_mul_of_nonneg_right ih (by linarith : 0≤1-t)
-    rw [pow_succ (1-t) k,Nat.cast_succ]
-    nlinarith [mul_nonneg (Nat.cast_nonneg (α:=ℝ) k) (mul_nonneg ht0 ht0)]
-
-lemma bernoulli_quadratic_lower (t:ℝ) (ht0:0≤t) (ht1:t≤1) (k:ℕ) (hk:2≤k) :
-    1-k*t+t^2≤(1-t)^k := by
-  induction k,hk using Nat.le_induction with
-  | base => norm_num;ring_nf;exact le_refl _
-  | succ k hk ih =>
-    have hh:=mul_le_mul_of_nonneg_right ih (by linarith : 0≤1-t)
-    have hkr:(2:ℝ)≤k:=by exact_mod_cast hk
-    have hp:=mul_nonneg (show 0≤(k:ℝ)-t by linarith) (show 0≤t^2 by exact pow_nonneg ht0 2)
-    rw [pow_succ (1-t) k,Nat.cast_succ]
-    nlinarith
-
- 
-def decaySeq (c:ℝ) : ℕ→ℝ
-  | 0=>1
-  | n+1=>decaySeq c n-c*(decaySeq c n)^2
-lemma decaySeq_bounds (c:ℝ) (hc0:0≤c) (hc1:c≤1) (n:ℕ) :
-    0≤decaySeq c n ∧ decaySeq c n≤1 := by
-  induction n with
-  | zero => simp [decaySeq]
-  | succ n ih =>
-    have hp:0≤c*(decaySeq c n)^2:=mul_nonneg hc0 (sq_nonneg _)
-    have ht:c*decaySeq c n≤1:=(mul_le_mul_of_nonneg_right hc1 ih.1).trans (by simpa using ih.2)
-    have hh:=mul_nonneg ih.1 (show 0≤1-c*decaySeq c n by linarith)
-    simp only [decaySeq]
-    constructor <;> nlinarith
-lemma decaySeq_antitone (c:ℝ) (hc:0≤c) : Antitone (decaySeq c) :=
-  antitone_nat_of_succ_le (fun n=>by dsimp [decaySeq];nlinarith [mul_nonneg hc (sq_nonneg (decaySeq c n))])
-lemma decaySeq_tendsto (c:ℝ) (hc:0<c) (hc1:c≤1) : Tendsto (decaySeq c) atTop (𝓝 0) := by
-  have hb : BddBelow (Set.range (decaySeq c)) := ⟨0,by rintro x ⟨n,rfl⟩;exact (decaySeq_bounds c hc.le hc1 n).1⟩
-  have hl:=tendsto_atTop_ciInf (decaySeq_antitone c hc.le) hb
-  let L:ℝ:=⨅n,decaySeq c n
-  have hl':Tendsto (fun n=>decaySeq c (n+1)) atTop (𝓝 L):=hl.comp (Filter.tendsto_add_atTop_nat 1)
-  have hrec:Tendsto (fun n=>decaySeq c (n+1)) atTop (𝓝 (L-c*L^2)):=hl.sub (tendsto_const_nhds.mul (hl.pow 2))
-  have he:L=L-c*L^2:=tendsto_nhds_unique hl' hrec
-  have hzero:c*L^2=0:=by linarith
-  have hz:L=0:= (eq_zero_of_pow_eq_zero ((mul_eq_zero.mp hzero).resolve_left hc.ne'))
-  have hlast:Tendsto (decaySeq c) atTop (𝓝 L):=hl
-  rwa [hz] at hlast
-end
-end ThreeState
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-! Scalar inequalities for criticality, Poisson survival and nonlinear decay. -/
+
+namespace ThreeState.PositiveScalar
+
+noncomputable def lowerC (lam : ℝ) : ℝ := (37/27:ℝ)*(1+2*lam/3)
+
+lemma small_polynomial (lam : ℝ) (h0 : 0 < lam) (h1 : lam ≤ 3/10) :
+    (2/5-lam^2/2)*(lowerC lam)^2 > (2/3:ℝ)*(11/10+lam^2/2) := by
+  have hnon : 0 ≤ (3/10:ℝ)-lam := by linarith
+  have he : (2/5-lam^2/2)*(lowerC lam)^2-(2/3:ℝ)*(11/10+lam^2/2) =
+    (130000/59049:ℝ)*(3/10-lam)^4+(2710400/59049:ℝ)*lam*(3/10-lam)^3+
+    (6735490/59049:ℝ)*lam^2*(3/10-lam)^2+(1871120/19683:ℝ)*lam^3*(3/10-lam)+
+    (159292/6561:ℝ)*lam^4 := by dsimp only [lowerC]; ring
+  have hp : 0 < (130000/59049:ℝ)*(3/10-lam)^4+(2710400/59049:ℝ)*lam*(3/10-lam)^3+
+    (6735490/59049:ℝ)*lam^2*(3/10-lam)^2+(1871120/19683:ℝ)*lam^3*(3/10-lam)+
+    (159292/6561:ℝ)*lam^4 := by positivity
+  linarith
+
+lemma middle_polynomial (lam : ℝ) (h0 : 3/10 ≤ lam) (h1 : lam^2 ≤ 1/2) :
+    1-17*lam/25 < (2/5-lam^2/2)*(lowerC lam)^2 := by
+  have ha : 0 ≤ lam-(3/10:ℝ) := by linarith
+  have hb : 0 < (71/100:ℝ)-lam := by nlinarith
+  have he : (2/5-lam^2/2)*(lowerC lam)^2-(1-17*lam/25) =
+    (1328320000/228886641:ℝ)*(71/100-lam)^4+
+    (75777584000/2059979769:ℝ)*(lam-3/10)*(71/100-lam)^3+
+    (1184570339800/18539817921:ℝ)*(lam-3/10)^2*(71/100-lam)^2+
+    (674265198440/18539817921:ℝ)*(lam-3/10)^3*(71/100-lam)+
+    (56362261022/18539817921:ℝ)*(lam-3/10)^4 := by dsimp only [lowerC]; ring
+  have hp : 0 < (1328320000/228886641:ℝ)*(71/100-lam)^4+
+    (75777584000/2059979769:ℝ)*(lam-3/10)*(71/100-lam)^3+
+    (1184570339800/18539817921:ℝ)*(lam-3/10)^2*(71/100-lam)^2+
+    (674265198440/18539817921:ℝ)*(lam-3/10)^3*(71/100-lam)+
+    (56362261022/18539817921:ℝ)*(lam-3/10)^4 := by positivity
+  linarith
+
+lemma rational_lower (mu : ℝ) (hm : 0 ≤ mu) : 3/2-mu ≤ 3/(2*(1+2*mu/3)) := by
+  apply (le_div_iff₀ (by positivity : 0 < 2*(1+2*mu/3))).mpr
+  nlinarith [sq_nonneg mu]
+
+lemma regular_scalar (lam mu c : ℝ) (h0 : 0 < lam) (hl : lam^2 ≤ 1/2)
+    (hm : 0 ≤ mu) (hc : lowerC lam ≤ c) (hb : mu*c^2 ≤ 1) :
+    8/5 < (1-lam^2)/2+3/(2*(1+2*mu/3))+17*lam*mu/25 := by
+  have hl1 : lam < 1 := by nlinarith
+  have hc0 : 0 < lowerC lam := by dsimp only [lowerC]; positivity
+  have hc2 : (lowerC lam)^2 ≤ c^2 := sq_le_sq₀ hc0.le (hc0.trans_le hc).le |>.mpr hc
+  have hcpos : 0 < c := hc0.trans_le hc
+  have hden : 0 < 2*(1+2*mu/3) := by positivity
+  rcases le_total lam (3/10) with hs | hs
+  · have hp := small_polynomial lam h0 hs
+    have hfac : 0 < 2/5-lam^2/2 := by nlinarith
+    have hpc : (2/3:ℝ)*(11/10+lam^2/2) < (2/5-lam^2/2)*c^2 :=
+      hp.trans_le (mul_le_mul_of_nonneg_left hc2 hfac.le)
+    have ht : (11/10+lam^2/2)*mu < (3/2:ℝ)*(2/5-lam^2/2) := by
+      have hh := mul_le_mul_of_nonneg_left hb (show 0 ≤ 11/10+lam^2/2 by positivity)
+      have hh' := mul_lt_mul_of_pos_left hpc (show 0 < (3/2:ℝ) by norm_num)
+      have he : ((11/10+lam^2/2)*mu)*c^2 < ((3/2:ℝ)*(2/5-lam^2/2))*c^2 := by nlinarith only [hh,hh']
+      exact (mul_lt_mul_iff_left₀ (sq_pos_of_pos hcpos)).mp he
+    have he : 11/10+lam^2/2 < 3/(2*(1+2*mu/3)) := by
+      apply (lt_div_iff₀ hden).mpr
+      nlinarith only [ht]
+    have hlast : 0 ≤ 17*lam*mu/25 := by positivity
+    linarith
+  · have hpoly := middle_polynomial lam hs hl
+    have hfac : 0 < 2/5-lam^2/2 := by nlinarith
+    have hpc : 1-17*lam/25 < (2/5-lam^2/2)*c^2 :=
+      hpoly.trans_le (mul_le_mul_of_nonneg_left hc2 hfac.le)
+    have hcoef : 0 < 1-17*lam/25 := by linarith
+    have hh := mul_le_mul_of_nonneg_left hb hcoef.le
+    have ht : (1-17*lam/25)*mu < 2/5-lam^2/2 := by
+      apply (mul_lt_mul_iff_left₀ (sq_pos_of_pos hcpos)).mp
+      nlinarith only [hpc,hh]
+    have hr := rational_lower mu hm
+    nlinarith only [ht,hr]
+
+lemma high_scalar (lam mu c : ℝ) (h0 : 0 < lam) (hl1 : lam < 1) (hl : 1/2 < lam^2)
+    (hm : 0 ≤ mu) (hc : lowerC lam ≤ c) (hb : mu*c^2 ≤ 1) :
+    8/5 < (1/4:ℝ)+3/(2*(1+2*mu/3))+17*lam*mu/25 := by
+  have hx : (707/1000:ℝ) < lam := by nlinarith
+  have hc' : (37/27:ℝ)*(1+2*(707/1000:ℝ)/3) < c := by dsimp only [lowerC] at hc; linarith
+  have hcpos : 0 < c := by linarith
+  have hp : 1-17*lam/25 < (3/20:ℝ)*c^2 := by nlinarith [sq_nonneg (c-(37/27:ℝ)*(1+2*(707/1000:ℝ)/3))]
+  have hcoef : 0 < 1-17*lam/25 := by linarith
+  have hh := mul_le_mul_of_nonneg_left hb hcoef.le
+  have ht : (1-17*lam/25)*mu < (3/20:ℝ) := by
+    apply (mul_lt_mul_iff_left₀ (sq_pos_of_pos hcpos)).mp
+    nlinarith only [hp,hh]
+  have hr := rational_lower mu hm
+  nlinarith only [ht,hr]
+
+ 
+
+theorem scalar_contradiction (lam mu c s : ℝ) (h0 : 0 < lam) (h1 : lam < 1)
+    (hm : 0 ≤ mu) (hs0 : 0 < s) (hs1 : s ≤ 1) (hc : lowerC lam ≤ c)
+    (hb : mu*c^2 ≤ s) (hsurv : 1/2 < lam^2 → s ≤ 2*(1-lam^2)) :
+    8/5 < (1-lam^2)/(2*s)+3/(2*(1+2*mu/3))+17*lam*mu/(25*s) := by
+  have hlast : 17*lam*mu/25 ≤ 17*lam*mu/(25*s) := by
+    apply (le_div_iff₀ (by positivity : 0 < 25*s)).mpr
+    nlinarith [mul_nonneg (show 0 ≤ 17*lam*mu by positivity) (show 0 ≤ 1-s by linarith)]
+  rcases le_or_gt (lam^2) (1/2) with hl | hl
+  · have hh := regular_scalar lam mu c h0 hl hm hc (hb.trans hs1)
+    have hf : (1-lam^2)/2 ≤ (1-lam^2)/(2*s) := by
+      apply (le_div_iff₀ (by positivity : 0 < 2*s)).mpr
+      nlinarith [mul_nonneg (show 0 ≤ 1-lam^2 by nlinarith) (show 0 ≤ 1-s by linarith)]
+    linarith
+  · have hh := high_scalar lam mu c h0 h1 hl hm hc (hb.trans hs1)
+    have hf : (1/4:ℝ) ≤ (1-lam^2)/(2*s) := by
+      apply (le_div_iff₀ (by positivity : 0 < 2*s)).mpr
+      linarith [hsurv hl]
+    linarith
+
+end ThreeState.PositiveScalar
+
+namespace ThreeState.PositiveScalar
+
+lemma q_lower (lam : ℝ) (h0 : 0 ≤ lam) (h1 : lam ≤ 1) :
+    (37/81:ℝ) ≤ (37/27:ℝ)/(2+lam) := by
+  apply (le_div_iff₀ (by linarith : 0 < 2+lam)).mpr
+  linarith
+
+lemma root_loss : (1:ℝ)/Real.sqrt 2 < 113/81-17/25 := by
+  have hr : 0 < Real.sqrt 2 := by positivity
+  have hs : (Real.sqrt 2)^2=2 := Real.sq_sqrt (by norm_num)
+  apply (div_lt_iff₀ hr).mpr
+  nlinarith only [hs, Real.sqrt_nonneg 2]
+
+lemma J_square_bound (lam mu nu J : ℝ) (h0 : 0 < lam) (h1 : lam < 1)
+    (hm : 0 ≤ mu) (hn : 0 ≤ nu) (hJ0 : 0 ≤ J)
+    (hJ : J ≤ mu-lam*((37/27:ℝ)/(2+lam))*nu)
+    (hc : ((37/27:ℝ)+2*lam*((37/27:ℝ)/(2+lam)))^2*nu ≤ mu) :
+    (8/5:ℝ)*J^2 ≤ (8/5:ℝ)*mu^2-(113/81:ℝ)*lam*mu*nu := by
+  let q : ℝ := (37/27:ℝ)/(2+lam)
+  let A := lam*q*nu
+  have hq := q_lower lam h0.le h1.le
+  change (37/81:ℝ) ≤ q at hq
+  have hA0 : 0 ≤ A := by dsimp only [A]; positivity
+  have hAc : 8*(37/27:ℝ)*A ≤ mu := by
+    have hh := mul_le_mul_of_nonneg_right (sq_nonneg ((37/27:ℝ)-2*lam*q)) hn
+    dsimp only [A,q] at *
+    nlinarith only [hh,hc]
+  have hA : A ≤ (27/296:ℝ)*mu := by linarith only [hAc]
+  have hs := mul_le_mul_of_nonneg_right hA hA0
+  change J ≤ mu-A at hJ
+  have hj2 : J^2 ≤ (mu-A)^2 := (sq_le_sq₀ hJ0 (hJ0.trans hJ)).mpr hJ
+  have hqmu := mul_le_mul_of_nonneg_right hq (show 0 ≤ lam*mu*nu by positivity)
+  have hlead : mu*A = q*(lam*mu*nu) := by dsimp only [A]; ring
+  rw [← hlead] at hqmu
+  nlinarith only [hj2,hs,hqmu]
+
+lemma scalar_from_moments (lam mu nu J s : ℝ) (h0 : 0 < lam) (h1 : lam < 1)
+    (hm : 0 < mu) (hn : 0 ≤ nu) (hs0 : 0 < s) (_ : s ≤ 1)
+    (hsecond : mu^2 ≤ s*nu)
+    (hcnu : ((37/27:ℝ)+2*lam*((37/27:ℝ)/(2+lam)))^2*nu ≤ mu)
+    (hJ0 : 0 ≤ J) (hJ : J ≤ mu-lam*((37/27:ℝ)/(2+lam))*nu)
+    (hentropy : (1-lam^2)/2*nu+3*mu^2/(2*(1+2*mu/3))-lam*mu*nu/Real.sqrt 2 ≤ (8/5:ℝ)*J^2) :
+    8/5 ≥ (1-lam^2)/(2*s)+3/(2*(1+2*mu/3))+17*lam*mu/(25*s) := by
+  have hj2 := J_square_bound lam mu nu J h0 h1 hm.le hn hJ0 hJ hcnu
+  have hroot := mul_le_mul_of_nonneg_right root_loss.le (show 0 ≤ lam*mu*nu by positivity)
+  have he : ((1-lam^2)/2+17*lam*mu/25)*nu+3*mu^2/(2*(1+2*mu/3)) ≤ (8/5:ℝ)*mu^2 := by
+    simp only [div_eq_mul_inv] at hentropy hj2 hroot ⊢
+    nlinarith only [hentropy,hj2,hroot]
+  have ha : 0 ≤ (1-lam^2)/2+17*lam*mu/25 := by
+    have hh : 0 ≤ 1-lam^2 := by nlinarith
+    positivity
+  have hh := mul_le_mul_of_nonneg_left ((div_le_iff₀ hs0).mpr (show mu^2 ≤ nu*s by nlinarith only [hsecond])) ha
+  have hneed : ((1-lam^2)/(2*s)+3/(2*(1+2*mu/3))+17*lam*mu/(25*s))*mu^2 ≤ (8/5:ℝ)*mu^2 := by
+    simp only [div_eq_mul_inv, mul_inv_rev] at he hh ⊢
+    nlinarith only [he,hh]
+  exact (mul_le_mul_iff_left₀ (sq_pos_of_pos hm)).mp hneed
+
+lemma moment_scalar_contradiction (lam mu nu J s : ℝ) (h0 : 0 < lam) (h1 : lam < 1)
+    (hm : 0 < mu) (hn : 0 ≤ nu) (hs0 : 0 < s) (hs1 : s ≤ 1)
+    (hsecond : mu^2 ≤ s*nu)
+    (hcnu : ((37/27:ℝ)+2*lam*((37/27:ℝ)/(2+lam)))^2*nu ≤ mu)
+    (hJ0 : 0 ≤ J) (hJ : J ≤ mu-lam*((37/27:ℝ)/(2+lam))*nu)
+    (hentropy : (1-lam^2)/2*nu+3*mu^2/(2*(1+2*mu/3))-lam*mu*nu/Real.sqrt 2 ≤ (8/5:ℝ)*J^2)
+    (hsurv : 1/2 < lam^2 → s ≤ 2*(1-lam^2)) : False := by
+  let c : ℝ := (37/27:ℝ)+2*lam*((37/27:ℝ)/(2+lam))
+  have hc : 0 < c := by dsimp only [c]; positivity
+  have hcL : lowerC lam ≤ c := by
+    have hh := q_lower lam h0.le h1.le
+    dsimp only [lowerC,c]
+    nlinarith only [hh,h0]
+  have hb : mu*c^2 ≤ s := by
+    have hh := mul_le_mul_of_nonneg_right hsecond (sq_nonneg c)
+    have hh' := mul_le_mul_of_nonneg_left hcnu hs0.le
+    apply (mul_le_mul_iff_left₀ hm).mp
+    dsimp only [c] at *
+    nlinarith only [hh,hh']
+  have hfalse := scalar_contradiction lam mu c s h0 h1 hm.le hs0 hs1 hcL hb hsurv
+  have hneed := scalar_from_moments lam mu nu J s h0 h1 hm hn hs0 hs1 hsecond hcnu hJ0 hJ hentropy
+  linarith only [hfalse,hneed]
+
+end ThreeState.PositiveScalar
+
+namespace ThreeState
+open Filter Topology
+
+lemma poisson_sqrt_tangent (d lam t : ℝ) (_ : 0<d) (hl : 0<lam) (ht : 0≤t)
+    (hc : d*lam^2=1) : Real.sqrt t ≤ lam*(t+d/2)/Real.sqrt 2 := by
+  have hs2 : (Real.sqrt 2)^2=(2:ℝ) := Real.sq_sqrt (by norm_num)
+  have hst : (Real.sqrt t)^2=t := Real.sq_sqrt ht
+  have hsp : 0 < Real.sqrt (2:ℝ) := by positivity
+  apply (le_div_iff₀ hsp).mpr
+  apply (mul_le_mul_iff_right₀ hl).mp
+  have hp := sq_nonneg (Real.sqrt 2*lam*Real.sqrt t-1)
+  have he : (Real.sqrt 2*lam*Real.sqrt t)^2=2*lam^2*t := by
+    calc _ = (Real.sqrt 2)^2*lam^2*(Real.sqrt t)^2 := by ring
+         _ = _ := by rw [hs2,hst]
+  nlinarith only [hp,he,hc]
+
+lemma poisson_reciprocal_tangent (t beta D : ℝ) (ht : 0≤t) (hb : 0≤beta) (hD : 0<D) :
+    t*(2*D-1-beta*t)/D^2 ≤ t/(1+beta*t) := by
+  have hy : 0<1+beta*t := by positivity
+  apply (div_le_div_iff₀ (sq_pos_of_pos hD) hy).mpr
+  nlinarith only [mul_nonneg ht (sq_nonneg (D-(1+beta*t)))]
+
+ 
+
+lemma poisson_correlation_scalar (d lam mu nu : ℝ) (hd : 0<d) (hl : 0<lam)
+    (hm : 0 ≤ mu) (hn : 0≤nu) (hc : d*lam^2=1)
+    (H C : ℝ → ℝ) (hH0 : H 0=0)
+    (hderiv : ∀ t ∈ Set.Icc 0 d, HasDerivAt H (C t) t)
+    (hlower : ∀ t ∈ Set.Icc 0 d,
+      3*lam^4*mu^2*t/(1+t*lam^2*mu)-lam^4*mu*nu*Real.sqrt t ≤ 3*C t) :
+    3*mu^2/(2*(1+2*mu/3))-lam*mu*nu/Real.sqrt 2 ≤ 3*H d := by
+  let D := 1+2*mu/3
+  have hD : 0<D := by dsimp [D]; positivity
+  let P := fun t : ℝ => (3*lam^4*mu^2/D^2)*((2*D-1)*t^2/2-(lam^2*mu)*t^3/3)-
+    (lam^5*mu*nu/Real.sqrt 2)*(t^2/2+d*t/2)
+  let P' := fun t : ℝ => (3*lam^4*mu^2/D^2)*((2*D-1)*t-(lam^2*mu)*t^2)-
+    (lam^5*mu*nu/Real.sqrt 2)*(t+d/2)
+  have hp (t : ℝ) : HasDerivAt P (P' t) t := by
+    have hh := (((((hasDerivAt_pow 2 t).const_mul (2*D-1)).div_const 2).sub
+      (((hasDerivAt_pow 3 t).const_mul (lam^2*mu)).div_const 3)).const_mul (3*lam^4*mu^2/D^2)).sub
+      ((((hasDerivAt_pow 2 t).div_const 2).add (((hasDerivAt_id t).const_mul d).div_const 2)).const_mul (lam^5*mu*nu/Real.sqrt 2))
+    convert hh using 1 <;> try rfl
+    dsimp only [P']
+    ring
+  have hpd (t : ℝ) (ht : t ∈ Set.Icc 0 d) : P' t ≤ 3*C t := by
+    have hr := mul_le_mul_of_nonneg_left (poisson_reciprocal_tangent t (lam^2*mu) D ht.1 (by positivity) hD)
+      (show 0≤3*lam^4*mu^2 by positivity)
+    have hs := mul_le_mul_of_nonneg_left (poisson_sqrt_tangent d lam t hd hl ht.1 hc)
+      (show 0≤lam^4*mu*nu by positivity)
+    calc
+      P' t = 3*lam^4*mu^2*(t*(2*D-1-(lam^2*mu)*t)/D^2)-lam^4*mu*nu*(lam*(t+d/2)/Real.sqrt 2) := by dsimp only [P']; ring
+      _ ≤ 3*lam^4*mu^2*(t/(1+(lam^2*mu)*t))-lam^4*mu*nu*Real.sqrt t := sub_le_sub hr hs
+      _ = 3*lam^4*mu^2*t/(1+t*lam^2*mu)-lam^4*mu*nu*Real.sqrt t := by ring
+      _ ≤ _ := hlower t ht
+  have hmono : MonotoneOn (fun t => 3*H t-P t) (Set.Icc 0 d) := by
+    apply monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc 0 d) (f' := fun t => 3*C t-P' t)
+    · intro t ht
+      exact (((hderiv t ht).const_mul 3).sub (hp t)).continuousAt.continuousWithinAt
+    · intro t ht
+      exact (((hderiv t (interior_subset ht)).const_mul 3).sub (hp t)).hasDerivWithinAt
+    · intro t ht
+      exact sub_nonneg.mpr (hpd t (interior_subset ht))
+  have hh := hmono (show (0:ℝ)∈Set.Icc 0 d from ⟨le_rfl,hd.le⟩) (show d∈Set.Icc 0 d from ⟨hd.le,le_rfl⟩) hd.le
+  have hp0 : P 0=0 := by simp [P]
+  have hpend : P d=3*mu^2/(2*D)-lam*mu*nu/Real.sqrt 2 := by
+    have hd2 : d^2*lam^4=1 := by nlinarith [sq_nonneg (d*lam^2-1)]
+    have hd3 : d^3*lam^6=1 := by
+      calc
+        _ = (d*lam^2)^3 := by ring
+        _ = 1 := by rw [hc]; norm_num
+    dsimp only [P]
+    field_simp [ne_of_gt hD]
+    dsimp only [D]
+    nlinarith only [congrArg (fun x : ℝ => x*(Real.sqrt 2)*mu^2) hd2,
+      congrArg (fun x : ℝ => x*(Real.sqrt 2)*mu^3) hd2,
+      congrArg (fun x : ℝ => x*(Real.sqrt 2)*mu^3) hd3,
+      congrArg (fun x : ℝ => x*lam*mu*nu) hd2,
+      congrArg (fun x : ℝ => x*lam*mu^2*nu) hd2,
+      congrArg (fun x : ℝ => x*lam*mu^3*nu) hd2]
+  change 3*H 0-P 0 ≤ 3*H d-P d at hh
+  rw [hH0,hp0,hpend] at hh
+  dsimp only [D] at hh
+  linarith
+
+end ThreeState
+
+namespace ThreeState
+open Filter Topology
+
+lemma log_survival_bound (s : ℝ) (hs0 : 0 ≤ s) (hs1 : s < 1) :
+    2*s/(2-s) ≤ -Real.log (1-s) := by
+  let F := fun x : ℝ => -Real.log (1-x)-2*x/(2-x)
+  have hd (x : ℝ) (hx : x ∈ Set.Icc 0 s) : HasDerivAt F
+      (1/(1-x)-4/(2-x)^2) x := by
+    have h1 : 1-x ≠ 0 := by linarith [hx.2]
+    have h2 : 2-x ≠ 0 := by linarith [hx.2]
+    convert (((hasDerivAt_const x (1:ℝ)).sub (hasDerivAt_id x)).log h1).neg.sub
+      (((hasDerivAt_id x).const_mul 2).div ((hasDerivAt_const x (2:ℝ)).sub (hasDerivAt_id x)) h2) using 1 <;> try rfl
+    dsimp
+    field_simp
+    ring
+  have hder (x : ℝ) (hx : x ∈ Set.Icc 0 s) : 0 ≤ 1/(1-x)-4/(2-x)^2 := by
+    have h1 : 0 < 1-x := by linarith [hx.2]
+    have h2 : 0 < 2-x := by linarith [hx.2]
+    apply sub_nonneg.mpr
+    apply (div_le_div_iff₀ (sq_pos_of_pos h2) h1).mpr
+    nlinarith [sq_nonneg x]
+  have hmono : MonotoneOn F (Set.Icc 0 s) := by
+    apply monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc 0 s)
+      (f' := fun x => 1/(1-x)-4/(2-x)^2)
+    · intro x hx; exact (hd x hx).continuousAt.continuousWithinAt
+    · intro x hx; exact (hd x (interior_subset hx)).hasDerivWithinAt
+    · intro x hx; exact hder x (interior_subset hx)
+  have hh := hmono (show (0:ℝ)∈Set.Icc 0 s from ⟨le_rfl,hs0⟩) (show s∈Set.Icc 0 s from ⟨hs0,le_rfl⟩) hs0
+  dsimp only [F] at hh
+  norm_num at hh
+  linarith
+
+lemma poisson_support_bound (d s : ℝ) (hd : 0 < d) (hs : 0 < s)
+    (hfp : Real.exp (-d*s) ≤ 1-s) : s ≤ 2*(1-d⁻¹) := by
+  have hs1 : s < 1 := by linarith [Real.exp_pos (-d*s)]
+  have hlog := Real.log_le_log (Real.exp_pos (-d*s)) hfp
+  rw [Real.log_exp] at hlog
+  have hh := log_survival_bound s hs.le hs1
+  have hden : 0 < 2-s := by linarith
+  have hrat : 2*s/(2-s) ≤ d*s := by linarith
+  have hr := (div_le_iff₀ hden).mp hrat
+  have hsd : 2 ≤ d*(2-s) := (mul_le_mul_iff_right₀ hs).mp (by nlinarith only [hr])
+  apply (mul_le_mul_iff_left₀ hd).mp
+  have he : (2*(1-d⁻¹))*d=2*d-2 := by field_simp
+  rw [he]
+  nlinarith only [hsd]
+end ThreeState
+
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+noncomputable section
+
+lemma bernoulli_lower (t:ℝ) (ht0:0≤t) (ht1:t≤1) (k:ℕ) : 1-k*t≤(1-t)^k := by
+  induction k with
+  | zero => simp
+  | succ k ih =>
+    have hh:=mul_le_mul_of_nonneg_right ih (by linarith : 0≤1-t)
+    rw [pow_succ (1-t) k,Nat.cast_succ]
+    nlinarith [mul_nonneg (Nat.cast_nonneg (α:=ℝ) k) (mul_nonneg ht0 ht0)]
+
+lemma bernoulli_quadratic_lower (t:ℝ) (ht0:0≤t) (ht1:t≤1) (k:ℕ) (hk:2≤k) :
+    1-k*t+t^2≤(1-t)^k := by
+  induction k,hk using Nat.le_induction with
+  | base => norm_num;ring_nf;exact le_refl _
+  | succ k hk ih =>
+    have hh:=mul_le_mul_of_nonneg_right ih (by linarith : 0≤1-t)
+    have hkr:(2:ℝ)≤k:=by exact_mod_cast hk
+    have hp:=mul_nonneg (show 0≤(k:ℝ)-t by linarith) (show 0≤t^2 by exact pow_nonneg ht0 2)
+    rw [pow_succ (1-t) k,Nat.cast_succ]
+    nlinarith
+
+ 
+def decaySeq (c:ℝ) : ℕ→ℝ
+  | 0=>1
+  | n+1=>decaySeq c n-c*(decaySeq c n)^2
+lemma decaySeq_bounds (c:ℝ) (hc0:0≤c) (hc1:c≤1) (n:ℕ) :
+    0≤decaySeq c n ∧ decaySeq c n≤1 := by
+  induction n with
+  | zero => simp [decaySeq]
+  | succ n ih =>
+    have hp:0≤c*(decaySeq c n)^2:=mul_nonneg hc0 (sq_nonneg _)
+    have ht:c*decaySeq c n≤1:=(mul_le_mul_of_nonneg_right hc1 ih.1).trans (by simpa using ih.2)
+    have hh:=mul_nonneg ih.1 (show 0≤1-c*decaySeq c n by linarith)
+    simp only [decaySeq]
+    constructor <;> nlinarith
+lemma decaySeq_antitone (c:ℝ) (hc:0≤c) : Antitone (decaySeq c) :=
+  antitone_nat_of_succ_le (fun n=>by dsimp [decaySeq];nlinarith [mul_nonneg hc (sq_nonneg (decaySeq c n))])
+lemma decaySeq_tendsto (c:ℝ) (hc:0<c) (hc1:c≤1) : Tendsto (decaySeq c) atTop (𝓝 0) := by
+  have hb : BddBelow (Set.range (decaySeq c)) := ⟨0,by rintro x ⟨n,rfl⟩;exact (decaySeq_bounds c hc.le hc1 n).1⟩
+  have hl:=tendsto_atTop_ciInf (decaySeq_antitone c hc.le) hb
+  let L:ℝ:=⨅n,decaySeq c n
+  have hl':Tendsto (fun n=>decaySeq c (n+1)) atTop (𝓝 L):=hl.comp (Filter.tendsto_add_atTop_nat 1)
+  have hrec:Tendsto (fun n=>decaySeq c (n+1)) atTop (𝓝 (L-c*L^2)):=hl.sub (tendsto_const_nhds.mul (hl.pow 2))
+  have he:L=L-c*L^2:=tendsto_nhds_unique hl' hrec
+  have hzero:c*L^2=0:=by linarith
+  have hz:L=0:= (eq_zero_of_pow_eq_zero ((mul_eq_zero.mp hzero).resolve_left hc.ne'))
+  have hlast:Tendsto (decaySeq c) atTop (𝓝 L):=hl
+  rwa [hz] at hlast
+end
+end ThreeState
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeState/Supercritical.lean b/lean/OAI/Probability/ThreeState/Supercritical.lean
index 95ddeb6ab3daf44ed0e86593cfb2a8e32aaca5f1..db425d942267c4424d32576e24cea60e7f5e9c63 100644
--- a/lean/OAI/Probability/ThreeState/Supercritical.lean
+++ b/lean/OAI/Probability/ThreeState/Supercritical.lean
@@ -1,373 +1,373 @@
-import Mathlib
-import OAI.Probability.ThreeState.Observation
-
-namespace OAI
-
-/-! Boundary score moments and reconstruction above the Kesten–Stigum threshold. -/
-
-namespace ThreeState
-
-noncomputable def listScore {α : Type*} (f : α → ℝ) (as : List α) : ℝ := (as.map f).sum
-noncomputable def multisetScore {α : Type*} (f : α → ℝ) (as : Multiset α) : ℝ := (as.map f).sum
-
-lemma listScore_cons {α : Type*} (f : α → ℝ) (a : α) (as : List α) :
-    listScore f (a::as) = f a+listScore f as := by simp [listScore]
-
-lemma multisetScore_coe {α : Type*} (f : α → ℝ) (as : List α) :
-    multisetScore f (as : Multiset α) = listScore f as := by
-  simp [multisetScore, listScore]
-
-lemma iidList_moments {α : Type*} {p : PMF α} {f : α → ℝ} {m s : ℝ}
-    (h : Moments p f m s) (n : ℕ) :
-    Moments (iidList p n) (listScore f)
-      ((n:ℝ)*m) ((n:ℝ)*s+(n:ℝ)*((n:ℝ)-1)*m^2) := by
-  induction n with
-  | zero => simpa [iidList, listScore] using Moments.pure ([] : List α) (listScore f)
-  | succ n ih =>
-    have hi := Moments.independent_add h ih
-    have hm := Moments.map (g := fun ab : α × List α => ab.1::ab.2) (f := listScore f) hi
-    simp only [PMF.map_bind, PMF.map_comp, Function.comp_def] at hm
-    change Moments (iidList p (n+1)) (listScore f)
-      (m+(n:ℝ)*m) (s+2*m*((n:ℝ)*m)+((n:ℝ)*s+(n:ℝ)*((n:ℝ)-1)*m^2)) at hm
-    convert hm using 1 <;> push_cast <;> ring
-
-lemma iidMultiset_moments {α : Type*} {p : PMF α} {f : α → ℝ} {m s : ℝ}
-    (h : Moments p f m s) (n : ℕ) :
-    Moments ((iidList p n).map (fun as : List α => (as : Multiset α))) (multisetScore f)
-      ((n:ℝ)*m) ((n:ℝ)*s+(n:ℝ)*((n:ℝ)-1)*m^2) := by
-  apply Moments.map (g := fun as : List α => (as : Multiset α))
-  simpa only [multisetScore_coe] using iidList_moments h n
-
- 
-
-lemma compound_moments {α : Type*} {p : PMF α} {f : α → ℝ} {m s d e : ℝ}
-    {offspring : PMF ℕ} (hp : Moments p f m s)
-    (hD : Moments offspring (fun k : ℕ => (k:ℝ)) d (e+d)) :
-    Moments (offspring.bind (fun k =>
-      (iidList p k).map (fun as : List α => (as : Multiset α)))) (multisetScore f)
-      (d*m) (d*s+e*m^2) := by
-  apply Moments.bind (fun k => iidMultiset_moments hp k)
-  · exact ((hD.hasMean.mul_const s).add ((hD.square.sub hD.hasMean).mul_const (m^2))).congr
-      (fun k => by ring)
-  · rw [mean_mul_const, hD.first]
-  · rw [mean_congr offspring (g := fun k => (k:ℝ)*s+((k:ℝ)^2-(k:ℝ))*m^2) (fun k => by ring),
-      mean_add (hD.hasMean.mul_const s) ((hD.square.sub hD.hasMean).mul_const (m^2)),
-      mean_mul_const, mean_mul_const, mean_sub hD.square hD.hasMean, hD.first, hD.second]
-    ring
-
- 
-
-def spinScore (i : Spin) : ℝ := if i=0 then 2 else -1
-
-noncomputable def boundaryScore : (n : ℕ) → Observation n → ℝ
-  | 0 => spinScore
-  | n+1 => multisetScore (boundaryScore n)
-
-lemma mass_channel (lam : ℝ) (h : Admissible lam) (i j : Spin) :
-    mass (channelPMF lam h i) j = channel lam i j := by
-  exact ENNReal.toReal_ofReal (channel_nonneg h i j)
-
-lemma channel_score (lam : ℝ) (h : Admissible lam) (i : Spin) :
-    mean (channelPMF lam h i) spinScore = lam*spinScore i := by
-  rw [mean_fintype]
-  simp only [mass_channel]
-  fin_cases i <;> simp [Fin.sum_univ_succ, channel, spinScore] <;> ring
-
-noncomputable def scoreSecond (d e lam : ℝ) (h : Admissible lam) : ℕ → Spin → ℝ
-  | 0, i => (spinScore i)^2
-  | n+1, i => d*mean (channelPMF lam h i) (scoreSecond d e lam h n) +
-      e*((d*lam)^n*lam*spinScore i)^2
-
- 
-theorem boundaryScore_moments (offspring : PMF ℕ) (d e lam : ℝ) (h : Admissible lam)
-    (hD : Moments offspring (fun k : ℕ => (k:ℝ)) d (e+d)) (n : ℕ) (i : Spin) :
-    Moments (observedLaw offspring lam h n i) (boundaryScore n)
-      ((d*lam)^n*spinScore i) (scoreSecond d e lam h n i) := by
-  induction n generalizing i with
-  | zero =>
-    change Moments (PMF.pure i) spinScore ((d*lam)^0*spinScore i) ((spinScore i)^2)
-    simpa only [pow_zero, one_mul] using Moments.pure i spinScore
-  | succ n ih =>
-    have hedge : Moments ((channelPMF lam h i).bind (observedLaw offspring lam h n))
-        (boundaryScore n) ((d*lam)^n*lam*spinScore i)
-        (mean (channelPMF lam h i) (scoreSecond d e lam h n)) := by
-      apply Moments.bind ih (hasMean_fintype _ _) ?_ rfl
-      rw [mean_const_mul, channel_score]
-      ring
-    have hh := compound_moments hedge hD
-    change Moments (observedLaw offspring lam h (n+1) i) (boundaryScore (n+1))
-      (d*((d*lam)^n*lam*spinScore i)) (scoreSecond d e lam h (n+1) i) at hh
-    convert hh using 1
-    rw [pow_succ]
-    ring
-
-lemma hasMean_marginal {α : Type*} {law : Spin → PMF α} {f : α → ℝ}
-    (h : ∀ i, HasMean (law i) f) : HasMean (marginal law) f :=
-  HasMean.bind h (hasMean_fintype _ _)
-
-lemma mean_uniform (f : Spin → ℝ) : mean uniformSpin f = (∑ i : Spin, f i)/3 := by
-  rw [mean_fintype]
-  simp only [mass_uniform, ← Finset.mul_sum]
-  ring
-
-lemma mean_marginal {α : Type*} {law : Spin → PMF α} {f : α → ℝ}
-    (h : ∀ i, HasMean (law i) f) :
-    mean (marginal law) f = mean uniformSpin (fun i => mean (law i) f) :=
-  mean_bind h (hasMean_fintype _ _)
-
-lemma mean_uniform_spinScore : mean uniformSpin spinScore = 0 := by
-  simp [mean_uniform, Fin.sum_univ_succ, spinScore]
-
-lemma mean_uniform_spinScore_square : mean uniformSpin (fun i => (spinScore i)^2) = 2 := by
-  norm_num [mean_uniform, Fin.sum_univ_succ, spinScore]
-
-lemma mean_uniform_channel (lam : ℝ) (h : Admissible lam) (f : Spin → ℝ) :
-    mean uniformSpin (fun i => mean (channelPMF lam h i) f) = mean uniformSpin f := by
-  simp [mean_fintype, mass_uniform, mass_channel, Fin.sum_univ_succ, channel]
-  ring
-
-lemma pow_square_comm (a : ℝ) (n : ℕ) : (a^n)^2 = (a^2)^n := by
-  rw [← pow_mul, ← pow_mul, Nat.mul_comm n 2]
-
-lemma scoreSecond_average_zero (d e lam : ℝ) (h : Admissible lam) :
-    mean uniformSpin (scoreSecond d e lam h 0) = 2 := mean_uniform_spinScore_square
-
-lemma scoreSecond_average_succ (d e lam : ℝ) (h : Admissible lam) (n : ℕ) :
-    mean uniformSpin (scoreSecond d e lam h (n+1)) =
-      d*mean uniformSpin (scoreSecond d e lam h n)+2*e*lam^2*((d*lam)^2)^n := by
-  change mean uniformSpin (fun i =>
-    d*mean (channelPMF lam h i) (scoreSecond d e lam h n) +
-      e*((d*lam)^n*lam*spinScore i)^2) = _
-  rw [mean_add (hasMean_fintype _ _) (hasMean_fintype _ _), mean_const_mul,
-    mean_uniform_channel]
-  congr 1
-  rw [mean_congr uniformSpin (g := fun i => (e*lam^2*((d*lam)^2)^n)*(spinScore i)^2)
-    (fun i => by rw [mul_pow, mul_pow, pow_square_comm]; ring), mean_const_mul,
-    mean_uniform_spinScore_square]
-  ring
-
- 
-
-lemma scoreSecond_bound (d e lam C : ℝ) (h : Admissible lam)
-    (hd : 0 ≤ d) (hC : 2 ≤ C) (hstep : d*C+2*e*lam^2 ≤ C*(d*lam)^2) (n : ℕ) :
-    mean uniformSpin (scoreSecond d e lam h n) ≤ C*((d*lam)^2)^n := by
-  induction n with
-  | zero => simpa only [scoreSecond_average_zero, pow_zero, mul_one] using hC
-  | succ n ih =>
-    rw [scoreSecond_average_succ, pow_succ ((d*lam)^2) n]
-    have h1 := mul_le_mul_of_nonneg_left ih hd
-    have h2 := mul_le_mul_of_nonneg_right hstep (pow_nonneg (sq_nonneg (d*lam)) n)
-    nlinarith
-
-lemma boundaryScore_marginal_moments (offspring : PMF ℕ) (d e lam : ℝ) (h : Admissible lam)
-    (hD : Moments offspring (fun k : ℕ => (k:ℝ)) d (e+d)) (n : ℕ) :
-    Moments (marginal (observedLaw offspring lam h n)) (boundaryScore n) 0
-      (mean uniformSpin (scoreSecond d e lam h n)) := by
-  apply Moments.bind (fun i => boundaryScore_moments offspring d e lam h hD n i)
-    (hasMean_fintype _ _) ?_ rfl
-  rw [mean_const_mul, mean_uniform_spinScore, mul_zero]
-
-end ThreeState
-
-namespace ThreeState
-
-lemma HasMean.mul_of_square {α : Type*} {p : PMF α} {f g : α → ℝ}
-    (hf : HasMean p (fun a => (f a)^2)) (hg : HasMean p (fun a => (g a)^2)) :
-    HasMean p (fun a => f a*g a) := by
-  apply HasMean.of_abs
-  apply ((hf.add hg).mul_const (1/2)).mono (fun a => abs_nonneg _)
-  intro a
-  rw [abs_mul]
-  nlinarith [sq_nonneg (|f a|-|g a|), sq_abs (f a), sq_abs (g a)]
-
- 
-lemma mean_cauchy_square {α : Type*} {p : PMF α} {f g : α → ℝ}
-    (hf : HasMean p (fun a => (f a)^2)) (hg : HasMean p (fun a => (g a)^2)) :
-    (mean p (fun a => f a*g a))^2 ≤
-      mean p (fun a => (f a)^2)*mean p (fun a => (g a)^2) := by
-  have hfg := hf.mul_of_square hg
-  have hp (t : ℝ) : 0 ≤ mean p (fun a => (g a)^2)*(t*t) +
-      (-2*mean p (fun a => f a*g a))*t + mean p (fun a => (f a)^2) := by
-    have hn := mean_nonneg p (fun a => sq_nonneg (f a-t*g a))
-    have he : mean p (fun a => (f a-t*g a)^2) =
-        mean p (fun a => (g a)^2)*(t*t) +
-        (-2*mean p (fun a => f a*g a))*t + mean p (fun a => (f a)^2) := by
-      rw [mean_congr p (g := fun a => (g a)^2*(t*t)+(f a*g a)*(-2*t)+(f a)^2)
-        (fun a => by ring),
-        mean_add ((hg.mul_const (t*t)).add (hfg.mul_const (-2*t))) hf,
-        mean_add (hg.mul_const (t*t)) (hfg.mul_const (-2*t)), mean_mul_const, mean_mul_const]
-      ring
-    rwa [he] at hn
-  have hd := discrim_le_zero hp
-  dsimp [discrim] at hd
-  nlinarith
-
-lemma posterior_le_one {α : Type*} (law : Spin → PMF α) (y : α) (i : Spin) :
-    posterior law y i ≤ 1 := by
-  have h := Finset.single_le_sum (fun j (_ : j ∈ Finset.univ) => posterior_nonneg law y j)
-    (Finset.mem_univ i)
-  simpa only [posterior_sum] using h
-
-noncomputable def posteriorTV {α : Type*} (law : Spin → PMF α) (y : α) : ℝ :=
-  (∑ i : Spin, |posterior law y i-1/3|)/2
-
-lemma posteriorTV_nonneg {α : Type*} (law : Spin → PMF α) (y : α) :
-    0 ≤ posteriorTV law y := by
-  exact div_nonneg (Finset.sum_nonneg (fun i _ => abs_nonneg _)) (by norm_num)
-
-lemma posteriorTV_le_two {α : Type*} (law : Spin → PMF α) (y : α) :
-    posteriorTV law y ≤ 2 := by
-  have hh (i : Spin) : |posterior law y i-1/3| ≤ 1 := by
-    rw [abs_le]
-    constructor <;> linarith [posterior_nonneg law y i, posterior_le_one law y i]
-  have h := Finset.sum_le_sum (s := Finset.univ) (fun i _ => hh i)
-  norm_num at h
-  dsimp [posteriorTV]
-  linarith
-
-lemma hasMean_posteriorTV {α : Type*} (law : Spin → PMF α) :
-    HasMean (marginal law) (posteriorTV law) :=
-  (hasMean_const (marginal law) 2).mono (posteriorTV_nonneg law) (posteriorTV_le_two law)
-
-lemma mean_posteriorTV {α : Type*} (law : Spin → PMF α) :
-    mean (marginal law) (posteriorTV law) = advantage law := rfl
-
-noncomputable def posteriorScore {α : Type*} (law : Spin → PMF α) (y : α) : ℝ :=
-  3*posterior law y 0-1
-
-lemma posteriorScore_square_le {α : Type*} (law : Spin → PMF α) (y : α) :
-    (posteriorScore law y)^2 ≤ 12*posteriorTV law y := by
-  have hn := posterior_nonneg law y 0
-  have hu := posterior_le_one law y 0
-  have hr : |posteriorScore law y| ≤ 2 := by
-    rw [abs_le]
-    dsimp [posteriorScore]
-    constructor <;> linarith
-  have he : |posteriorScore law y| = 3*|posterior law y 0-1/3| := by
-    rw [show posteriorScore law y = 3*(posterior law y 0-1/3) by dsimp [posteriorScore]; ring, abs_mul]
-    norm_num
-  have h0 := Finset.single_le_sum (fun i (_ : i ∈ Finset.univ) => abs_nonneg (posterior law y i-1/3))
-    (Finset.mem_univ (0:Spin))
-  dsimp [posteriorTV]
-  nlinarith [sq_abs (posteriorScore law y),
-    mul_nonneg (abs_nonneg (posteriorScore law y)) (sub_nonneg.mpr hr)]
-
-lemma hasMean_posteriorScore_square {α : Type*} (law : Spin → PMF α) :
-    HasMean (marginal law) (fun y => (posteriorScore law y)^2) :=
-  ((hasMean_posteriorTV law).const_mul 12).mono
-    (fun y => sq_nonneg (posteriorScore law y)) (posteriorScore_square_le law)
-
-lemma mean_posteriorScore_square_le {α : Type*} (law : Spin → PMF α) :
-    mean (marginal law) (fun y => (posteriorScore law y)^2) ≤ 12*advantage law := by
-  have h := mean_le (hasMean_posteriorScore_square law) ((hasMean_posteriorTV law).const_mul 12)
-    (posteriorScore_square_le law)
-  simpa only [mean_const_mul, mean_posteriorTV] using h
-
-lemma weighted_posteriorScore {α : Type*} (law : Spin → PMF α) (y : α) :
-    mass (marginal law) y*posteriorScore law y = mass (law 0) y-mass (marginal law) y := by
-  have h := weighted_posterior law y 0
-  dsimp [posteriorScore]
-  linarith
-
-lemma mean_score_correlation {α : Type*} (law : Spin → PMF α) (f : α → ℝ)
-    (h0 : HasMean (law 0) f) (hm : HasMean (marginal law) f) :
-    mean (marginal law) (fun y => posteriorScore law y*f y) = mean (law 0) f-mean (marginal law) f := by
-  unfold mean
-  rw [← h0.tsum_sub hm]
-  apply tsum_congr
-  intro y
-  rw [← mul_assoc, weighted_posteriorScore]
-  ring
-
- 
-
-theorem advantage_lower_from_score {α : Type*} (law : Spin → PMF α) (f : α → ℝ)
-    (h0 : HasMean (law 0) f) (hf : HasMean (marginal law) (fun y => (f y)^2)) :
-    (mean (law 0) f-mean (marginal law) f)^2 ≤
-      12*advantage law*mean (marginal law) (fun y => (f y)^2) := by
-  have h := mean_cauchy_square (hasMean_posteriorScore_square law) hf
-  rw [mean_score_correlation law f h0 (hasMean_of_square hf)] at h
-  exact h.trans (mul_le_mul_of_nonneg_right (mean_posteriorScore_square_le law)
-    (mean_nonneg _ (fun y => sq_nonneg _)))
-
-end ThreeState
-
-namespace ThreeState
-
- 
-
-lemma supercritical_advantage_bound (offspring : PMF ℕ) (d e lam C : ℝ)
-    (h : Admissible lam) (hD : Moments offspring (fun k : ℕ => (k:ℝ)) d (e+d))
-    (hd : 0 ≤ d) (hq : 0 < (d*lam)^2) (hC : 2 ≤ C)
-    (hstep : d*C+2*e*lam^2 ≤ C*(d*lam)^2) (n : ℕ) :
-    1/(3*C) ≤ advantage (observedLaw offspring lam h n) := by
-  have hm := boundaryScore_marginal_moments offspring d e lam h hD n
-  have h0 := boundaryScore_moments offspring d e lam h hD n (0:Spin)
-  have hc := advantage_lower_from_score (observedLaw offspring lam h n) (boundaryScore n)
-    h0.hasMean hm.square
-  rw [h0.first, hm.first, hm.second] at hc
-  have hs := scoreSecond_bound d e lam C h hd hC hstep n
-  have ha := advantage_nonneg (observedLaw offspring lam h n)
-  have hp := pow_pos hq n
-  have hu := mul_le_mul_of_nonneg_left hs (mul_nonneg (by norm_num : (0:ℝ) ≤ 12) ha)
-  have he : (((d*lam)^n*spinScore (0:Spin))-0)^2 = 4*((d*lam)^2)^n := by
-    simp only [spinScore, ↓reduceIte, sub_zero, mul_pow, pow_square_comm]
-    ring
-  rw [he] at hc
-  have hh : (4:ℝ)*((d*lam)^2)^n ≤
-      (12*C*advantage (observedLaw offspring lam h n))*((d*lam)^2)^n := by
-    calc
-      _ ≤ 12*advantage (observedLaw offspring lam h n) *
-          (C*((d*lam)^2)^n) := hc.trans hu
-      _ = _ := by ring
-  have hh' := (mul_le_mul_iff_left₀ hp).mp hh
-  apply (div_le_iff₀ (show 0 < 3*C by linarith)).mpr
-  nlinarith
-
- 
-
-theorem reconstructs_supercritical_of_moments (offspring : PMF ℕ) (d e lam : ℝ)
-    (h : Admissible lam) (hD : Moments offspring (fun k : ℕ => (k:ℝ)) d (e+d))
-    (hd : 0 < d) (he : 0 ≤ e) (hcrit : 1 < d*lam^2) :
-    Reconstructs (fun n => advantage (observedLaw offspring lam h n)) := by
-  have hgap : 0 < (d*lam)^2-d := by
-    have hp := mul_pos hd (sub_pos.mpr hcrit)
-    nlinarith
-  have hq : 0 < (d*lam)^2 := by linarith
-  let C : ℝ := 2+2*e*lam^2/((d*lam)^2-d)
-  have hC : 2 ≤ C := by
-    dsimp [C]
-    have hh : 0 ≤ 2*e*lam^2/((d*lam)^2-d) := by positivity
-    linarith
-  have hCpos : 0 < C := by linarith
-  have hCeq : C*((d*lam)^2-d) = 2*((d*lam)^2-d)+2*e*lam^2 := by
-    dsimp [C]
-    rw [add_mul, div_mul_cancel₀ _ (ne_of_gt hgap)]
-  have hstep : d*C+2*e*lam^2 ≤ C*(d*lam)^2 := by nlinarith
-  obtain ⟨L,hL0,hL1,hL⟩ := observedAdvantage_tendsto offspring lam h
-  have hb : 1/(3*C) ≤ L := ge_of_tendsto' hL
-    (fun n => supercritical_advantage_bound offspring d e lam C h hD hd.le hq hC hstep n)
-  exact ⟨L, (by positivity : 0 < 1/(3*C)).trans_le hb, hL⟩
-
- 
-
-theorem regular_supercritical (b : ℕ) (hb : 2 ≤ b) (lam : ℝ) (h : Admissible lam)
-    (hcrit : 1 < (b:ℝ)*lam^2) : Reconstructs (regularAdvantage b lam h) := by
-  have hd : 0 < (b:ℝ) := by exact_mod_cast (show 0 < b by omega)
-  have he : 0 ≤ (b:ℝ)^2-(b:ℝ) := by
-    have hb' : (2:ℝ) ≤ (b:ℝ) := by exact_mod_cast hb
-    nlinarith
-  apply reconstructs_supercritical_of_moments (PMF.pure b) (b:ℝ) ((b:ℝ)^2-(b:ℝ)) lam h
-    ?_ hd he hcrit
-  simpa only [sub_add_cancel] using regular_moments b
-
- 
-
-theorem poisson_supercritical (d : ℝ) (hd : 1 < d) (lam : ℝ) (h : Admissible lam)
-    (hcrit : 1 < d*lam^2) : Reconstructs (poissonAdvantage d hd lam h) := by
-  apply reconstructs_supercritical_of_moments (poissonOffspring d (le_trans (by norm_num) hd.le))
-    d (d^2) lam h (poisson_moments d _) (by linarith) (sq_nonneg d) hcrit
-
-end ThreeState
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeState.Observation
+
+namespace OAI
+
+/-! Boundary score moments and reconstruction above the Kesten–Stigum threshold. -/
+
+namespace ThreeState
+
+noncomputable def listScore {α : Type*} (f : α → ℝ) (as : List α) : ℝ := (as.map f).sum
+noncomputable def multisetScore {α : Type*} (f : α → ℝ) (as : Multiset α) : ℝ := (as.map f).sum
+
+lemma listScore_cons {α : Type*} (f : α → ℝ) (a : α) (as : List α) :
+    listScore f (a::as) = f a+listScore f as := by simp [listScore]
+
+lemma multisetScore_coe {α : Type*} (f : α → ℝ) (as : List α) :
+    multisetScore f (as : Multiset α) = listScore f as := by
+  simp [multisetScore, listScore]
+
+lemma iidList_moments {α : Type*} {p : PMF α} {f : α → ℝ} {m s : ℝ}
+    (h : Moments p f m s) (n : ℕ) :
+    Moments (iidList p n) (listScore f)
+      ((n:ℝ)*m) ((n:ℝ)*s+(n:ℝ)*((n:ℝ)-1)*m^2) := by
+  induction n with
+  | zero => simpa [iidList, listScore] using Moments.pure ([] : List α) (listScore f)
+  | succ n ih =>
+    have hi := Moments.independent_add h ih
+    have hm := Moments.map (g := fun ab : α × List α => ab.1::ab.2) (f := listScore f) hi
+    simp only [PMF.map_bind, PMF.map_comp, Function.comp_def] at hm
+    change Moments (iidList p (n+1)) (listScore f)
+      (m+(n:ℝ)*m) (s+2*m*((n:ℝ)*m)+((n:ℝ)*s+(n:ℝ)*((n:ℝ)-1)*m^2)) at hm
+    convert hm using 1 <;> push_cast <;> ring
+
+lemma iidMultiset_moments {α : Type*} {p : PMF α} {f : α → ℝ} {m s : ℝ}
+    (h : Moments p f m s) (n : ℕ) :
+    Moments ((iidList p n).map (fun as : List α => (as : Multiset α))) (multisetScore f)
+      ((n:ℝ)*m) ((n:ℝ)*s+(n:ℝ)*((n:ℝ)-1)*m^2) := by
+  apply Moments.map (g := fun as : List α => (as : Multiset α))
+  simpa only [multisetScore_coe] using iidList_moments h n
+
+ 
+
+lemma compound_moments {α : Type*} {p : PMF α} {f : α → ℝ} {m s d e : ℝ}
+    {offspring : PMF ℕ} (hp : Moments p f m s)
+    (hD : Moments offspring (fun k : ℕ => (k:ℝ)) d (e+d)) :
+    Moments (offspring.bind (fun k =>
+      (iidList p k).map (fun as : List α => (as : Multiset α)))) (multisetScore f)
+      (d*m) (d*s+e*m^2) := by
+  apply Moments.bind (fun k => iidMultiset_moments hp k)
+  · exact ((hD.hasMean.mul_const s).add ((hD.square.sub hD.hasMean).mul_const (m^2))).congr
+      (fun k => by ring)
+  · rw [mean_mul_const, hD.first]
+  · rw [mean_congr offspring (g := fun k => (k:ℝ)*s+((k:ℝ)^2-(k:ℝ))*m^2) (fun k => by ring),
+      mean_add (hD.hasMean.mul_const s) ((hD.square.sub hD.hasMean).mul_const (m^2)),
+      mean_mul_const, mean_mul_const, mean_sub hD.square hD.hasMean, hD.first, hD.second]
+    ring
+
+ 
+
+def spinScore (i : Spin) : ℝ := if i=0 then 2 else -1
+
+noncomputable def boundaryScore : (n : ℕ) → Observation n → ℝ
+  | 0 => spinScore
+  | n+1 => multisetScore (boundaryScore n)
+
+lemma mass_channel (lam : ℝ) (h : Admissible lam) (i j : Spin) :
+    mass (channelPMF lam h i) j = channel lam i j := by
+  exact ENNReal.toReal_ofReal (channel_nonneg h i j)
+
+lemma channel_score (lam : ℝ) (h : Admissible lam) (i : Spin) :
+    mean (channelPMF lam h i) spinScore = lam*spinScore i := by
+  rw [mean_fintype]
+  simp only [mass_channel]
+  fin_cases i <;> simp [Fin.sum_univ_succ, channel, spinScore] <;> ring
+
+noncomputable def scoreSecond (d e lam : ℝ) (h : Admissible lam) : ℕ → Spin → ℝ
+  | 0, i => (spinScore i)^2
+  | n+1, i => d*mean (channelPMF lam h i) (scoreSecond d e lam h n) +
+      e*((d*lam)^n*lam*spinScore i)^2
+
+ 
+theorem boundaryScore_moments (offspring : PMF ℕ) (d e lam : ℝ) (h : Admissible lam)
+    (hD : Moments offspring (fun k : ℕ => (k:ℝ)) d (e+d)) (n : ℕ) (i : Spin) :
+    Moments (observedLaw offspring lam h n i) (boundaryScore n)
+      ((d*lam)^n*spinScore i) (scoreSecond d e lam h n i) := by
+  induction n generalizing i with
+  | zero =>
+    change Moments (PMF.pure i) spinScore ((d*lam)^0*spinScore i) ((spinScore i)^2)
+    simpa only [pow_zero, one_mul] using Moments.pure i spinScore
+  | succ n ih =>
+    have hedge : Moments ((channelPMF lam h i).bind (observedLaw offspring lam h n))
+        (boundaryScore n) ((d*lam)^n*lam*spinScore i)
+        (mean (channelPMF lam h i) (scoreSecond d e lam h n)) := by
+      apply Moments.bind ih (hasMean_fintype _ _) ?_ rfl
+      rw [mean_const_mul, channel_score]
+      ring
+    have hh := compound_moments hedge hD
+    change Moments (observedLaw offspring lam h (n+1) i) (boundaryScore (n+1))
+      (d*((d*lam)^n*lam*spinScore i)) (scoreSecond d e lam h (n+1) i) at hh
+    convert hh using 1
+    rw [pow_succ]
+    ring
+
+lemma hasMean_marginal {α : Type*} {law : Spin → PMF α} {f : α → ℝ}
+    (h : ∀ i, HasMean (law i) f) : HasMean (marginal law) f :=
+  HasMean.bind h (hasMean_fintype _ _)
+
+lemma mean_uniform (f : Spin → ℝ) : mean uniformSpin f = (∑ i : Spin, f i)/3 := by
+  rw [mean_fintype]
+  simp only [mass_uniform, ← Finset.mul_sum]
+  ring
+
+lemma mean_marginal {α : Type*} {law : Spin → PMF α} {f : α → ℝ}
+    (h : ∀ i, HasMean (law i) f) :
+    mean (marginal law) f = mean uniformSpin (fun i => mean (law i) f) :=
+  mean_bind h (hasMean_fintype _ _)
+
+lemma mean_uniform_spinScore : mean uniformSpin spinScore = 0 := by
+  simp [mean_uniform, Fin.sum_univ_succ, spinScore]
+
+lemma mean_uniform_spinScore_square : mean uniformSpin (fun i => (spinScore i)^2) = 2 := by
+  norm_num [mean_uniform, Fin.sum_univ_succ, spinScore]
+
+lemma mean_uniform_channel (lam : ℝ) (h : Admissible lam) (f : Spin → ℝ) :
+    mean uniformSpin (fun i => mean (channelPMF lam h i) f) = mean uniformSpin f := by
+  simp [mean_fintype, mass_uniform, mass_channel, Fin.sum_univ_succ, channel]
+  ring
+
+lemma pow_square_comm (a : ℝ) (n : ℕ) : (a^n)^2 = (a^2)^n := by
+  rw [← pow_mul, ← pow_mul, Nat.mul_comm n 2]
+
+lemma scoreSecond_average_zero (d e lam : ℝ) (h : Admissible lam) :
+    mean uniformSpin (scoreSecond d e lam h 0) = 2 := mean_uniform_spinScore_square
+
+lemma scoreSecond_average_succ (d e lam : ℝ) (h : Admissible lam) (n : ℕ) :
+    mean uniformSpin (scoreSecond d e lam h (n+1)) =
+      d*mean uniformSpin (scoreSecond d e lam h n)+2*e*lam^2*((d*lam)^2)^n := by
+  change mean uniformSpin (fun i =>
+    d*mean (channelPMF lam h i) (scoreSecond d e lam h n) +
+      e*((d*lam)^n*lam*spinScore i)^2) = _
+  rw [mean_add (hasMean_fintype _ _) (hasMean_fintype _ _), mean_const_mul,
+    mean_uniform_channel]
+  congr 1
+  rw [mean_congr uniformSpin (g := fun i => (e*lam^2*((d*lam)^2)^n)*(spinScore i)^2)
+    (fun i => by rw [mul_pow, mul_pow, pow_square_comm]; ring), mean_const_mul,
+    mean_uniform_spinScore_square]
+  ring
+
+ 
+
+lemma scoreSecond_bound (d e lam C : ℝ) (h : Admissible lam)
+    (hd : 0 ≤ d) (hC : 2 ≤ C) (hstep : d*C+2*e*lam^2 ≤ C*(d*lam)^2) (n : ℕ) :
+    mean uniformSpin (scoreSecond d e lam h n) ≤ C*((d*lam)^2)^n := by
+  induction n with
+  | zero => simpa only [scoreSecond_average_zero, pow_zero, mul_one] using hC
+  | succ n ih =>
+    rw [scoreSecond_average_succ, pow_succ ((d*lam)^2) n]
+    have h1 := mul_le_mul_of_nonneg_left ih hd
+    have h2 := mul_le_mul_of_nonneg_right hstep (pow_nonneg (sq_nonneg (d*lam)) n)
+    nlinarith
+
+lemma boundaryScore_marginal_moments (offspring : PMF ℕ) (d e lam : ℝ) (h : Admissible lam)
+    (hD : Moments offspring (fun k : ℕ => (k:ℝ)) d (e+d)) (n : ℕ) :
+    Moments (marginal (observedLaw offspring lam h n)) (boundaryScore n) 0
+      (mean uniformSpin (scoreSecond d e lam h n)) := by
+  apply Moments.bind (fun i => boundaryScore_moments offspring d e lam h hD n i)
+    (hasMean_fintype _ _) ?_ rfl
+  rw [mean_const_mul, mean_uniform_spinScore, mul_zero]
+
+end ThreeState
+
+namespace ThreeState
+
+lemma HasMean.mul_of_square {α : Type*} {p : PMF α} {f g : α → ℝ}
+    (hf : HasMean p (fun a => (f a)^2)) (hg : HasMean p (fun a => (g a)^2)) :
+    HasMean p (fun a => f a*g a) := by
+  apply HasMean.of_abs
+  apply ((hf.add hg).mul_const (1/2)).mono (fun a => abs_nonneg _)
+  intro a
+  rw [abs_mul]
+  nlinarith [sq_nonneg (|f a|-|g a|), sq_abs (f a), sq_abs (g a)]
+
+ 
+lemma mean_cauchy_square {α : Type*} {p : PMF α} {f g : α → ℝ}
+    (hf : HasMean p (fun a => (f a)^2)) (hg : HasMean p (fun a => (g a)^2)) :
+    (mean p (fun a => f a*g a))^2 ≤
+      mean p (fun a => (f a)^2)*mean p (fun a => (g a)^2) := by
+  have hfg := hf.mul_of_square hg
+  have hp (t : ℝ) : 0 ≤ mean p (fun a => (g a)^2)*(t*t) +
+      (-2*mean p (fun a => f a*g a))*t + mean p (fun a => (f a)^2) := by
+    have hn := mean_nonneg p (fun a => sq_nonneg (f a-t*g a))
+    have he : mean p (fun a => (f a-t*g a)^2) =
+        mean p (fun a => (g a)^2)*(t*t) +
+        (-2*mean p (fun a => f a*g a))*t + mean p (fun a => (f a)^2) := by
+      rw [mean_congr p (g := fun a => (g a)^2*(t*t)+(f a*g a)*(-2*t)+(f a)^2)
+        (fun a => by ring),
+        mean_add ((hg.mul_const (t*t)).add (hfg.mul_const (-2*t))) hf,
+        mean_add (hg.mul_const (t*t)) (hfg.mul_const (-2*t)), mean_mul_const, mean_mul_const]
+      ring
+    rwa [he] at hn
+  have hd := discrim_le_zero hp
+  dsimp [discrim] at hd
+  nlinarith
+
+lemma posterior_le_one {α : Type*} (law : Spin → PMF α) (y : α) (i : Spin) :
+    posterior law y i ≤ 1 := by
+  have h := Finset.single_le_sum (fun j (_ : j ∈ Finset.univ) => posterior_nonneg law y j)
+    (Finset.mem_univ i)
+  simpa only [posterior_sum] using h
+
+noncomputable def posteriorTV {α : Type*} (law : Spin → PMF α) (y : α) : ℝ :=
+  (∑ i : Spin, |posterior law y i-1/3|)/2
+
+lemma posteriorTV_nonneg {α : Type*} (law : Spin → PMF α) (y : α) :
+    0 ≤ posteriorTV law y := by
+  exact div_nonneg (Finset.sum_nonneg (fun i _ => abs_nonneg _)) (by norm_num)
+
+lemma posteriorTV_le_two {α : Type*} (law : Spin → PMF α) (y : α) :
+    posteriorTV law y ≤ 2 := by
+  have hh (i : Spin) : |posterior law y i-1/3| ≤ 1 := by
+    rw [abs_le]
+    constructor <;> linarith [posterior_nonneg law y i, posterior_le_one law y i]
+  have h := Finset.sum_le_sum (s := Finset.univ) (fun i _ => hh i)
+  norm_num at h
+  dsimp [posteriorTV]
+  linarith
+
+lemma hasMean_posteriorTV {α : Type*} (law : Spin → PMF α) :
+    HasMean (marginal law) (posteriorTV law) :=
+  (hasMean_const (marginal law) 2).mono (posteriorTV_nonneg law) (posteriorTV_le_two law)
+
+lemma mean_posteriorTV {α : Type*} (law : Spin → PMF α) :
+    mean (marginal law) (posteriorTV law) = advantage law := rfl
+
+noncomputable def posteriorScore {α : Type*} (law : Spin → PMF α) (y : α) : ℝ :=
+  3*posterior law y 0-1
+
+lemma posteriorScore_square_le {α : Type*} (law : Spin → PMF α) (y : α) :
+    (posteriorScore law y)^2 ≤ 12*posteriorTV law y := by
+  have hn := posterior_nonneg law y 0
+  have hu := posterior_le_one law y 0
+  have hr : |posteriorScore law y| ≤ 2 := by
+    rw [abs_le]
+    dsimp [posteriorScore]
+    constructor <;> linarith
+  have he : |posteriorScore law y| = 3*|posterior law y 0-1/3| := by
+    rw [show posteriorScore law y = 3*(posterior law y 0-1/3) by dsimp [posteriorScore]; ring, abs_mul]
+    norm_num
+  have h0 := Finset.single_le_sum (fun i (_ : i ∈ Finset.univ) => abs_nonneg (posterior law y i-1/3))
+    (Finset.mem_univ (0:Spin))
+  dsimp [posteriorTV]
+  nlinarith [sq_abs (posteriorScore law y),
+    mul_nonneg (abs_nonneg (posteriorScore law y)) (sub_nonneg.mpr hr)]
+
+lemma hasMean_posteriorScore_square {α : Type*} (law : Spin → PMF α) :
+    HasMean (marginal law) (fun y => (posteriorScore law y)^2) :=
+  ((hasMean_posteriorTV law).const_mul 12).mono
+    (fun y => sq_nonneg (posteriorScore law y)) (posteriorScore_square_le law)
+
+lemma mean_posteriorScore_square_le {α : Type*} (law : Spin → PMF α) :
+    mean (marginal law) (fun y => (posteriorScore law y)^2) ≤ 12*advantage law := by
+  have h := mean_le (hasMean_posteriorScore_square law) ((hasMean_posteriorTV law).const_mul 12)
+    (posteriorScore_square_le law)
+  simpa only [mean_const_mul, mean_posteriorTV] using h
+
+lemma weighted_posteriorScore {α : Type*} (law : Spin → PMF α) (y : α) :
+    mass (marginal law) y*posteriorScore law y = mass (law 0) y-mass (marginal law) y := by
+  have h := weighted_posterior law y 0
+  dsimp [posteriorScore]
+  linarith
+
+lemma mean_score_correlation {α : Type*} (law : Spin → PMF α) (f : α → ℝ)
+    (h0 : HasMean (law 0) f) (hm : HasMean (marginal law) f) :
+    mean (marginal law) (fun y => posteriorScore law y*f y) = mean (law 0) f-mean (marginal law) f := by
+  unfold mean
+  rw [← h0.tsum_sub hm]
+  apply tsum_congr
+  intro y
+  rw [← mul_assoc, weighted_posteriorScore]
+  ring
+
+ 
+
+theorem advantage_lower_from_score {α : Type*} (law : Spin → PMF α) (f : α → ℝ)
+    (h0 : HasMean (law 0) f) (hf : HasMean (marginal law) (fun y => (f y)^2)) :
+    (mean (law 0) f-mean (marginal law) f)^2 ≤
+      12*advantage law*mean (marginal law) (fun y => (f y)^2) := by
+  have h := mean_cauchy_square (hasMean_posteriorScore_square law) hf
+  rw [mean_score_correlation law f h0 (hasMean_of_square hf)] at h
+  exact h.trans (mul_le_mul_of_nonneg_right (mean_posteriorScore_square_le law)
+    (mean_nonneg _ (fun y => sq_nonneg _)))
+
+end ThreeState
+
+namespace ThreeState
+
+ 
+
+lemma supercritical_advantage_bound (offspring : PMF ℕ) (d e lam C : ℝ)
+    (h : Admissible lam) (hD : Moments offspring (fun k : ℕ => (k:ℝ)) d (e+d))
+    (hd : 0 ≤ d) (hq : 0 < (d*lam)^2) (hC : 2 ≤ C)
+    (hstep : d*C+2*e*lam^2 ≤ C*(d*lam)^2) (n : ℕ) :
+    1/(3*C) ≤ advantage (observedLaw offspring lam h n) := by
+  have hm := boundaryScore_marginal_moments offspring d e lam h hD n
+  have h0 := boundaryScore_moments offspring d e lam h hD n (0:Spin)
+  have hc := advantage_lower_from_score (observedLaw offspring lam h n) (boundaryScore n)
+    h0.hasMean hm.square
+  rw [h0.first, hm.first, hm.second] at hc
+  have hs := scoreSecond_bound d e lam C h hd hC hstep n
+  have ha := advantage_nonneg (observedLaw offspring lam h n)
+  have hp := pow_pos hq n
+  have hu := mul_le_mul_of_nonneg_left hs (mul_nonneg (by norm_num : (0:ℝ) ≤ 12) ha)
+  have he : (((d*lam)^n*spinScore (0:Spin))-0)^2 = 4*((d*lam)^2)^n := by
+    simp only [spinScore, ↓reduceIte, sub_zero, mul_pow, pow_square_comm]
+    ring
+  rw [he] at hc
+  have hh : (4:ℝ)*((d*lam)^2)^n ≤
+      (12*C*advantage (observedLaw offspring lam h n))*((d*lam)^2)^n := by
+    calc
+      _ ≤ 12*advantage (observedLaw offspring lam h n) *
+          (C*((d*lam)^2)^n) := hc.trans hu
+      _ = _ := by ring
+  have hh' := (mul_le_mul_iff_left₀ hp).mp hh
+  apply (div_le_iff₀ (show 0 < 3*C by linarith)).mpr
+  nlinarith
+
+ 
+
+theorem reconstructs_supercritical_of_moments (offspring : PMF ℕ) (d e lam : ℝ)
+    (h : Admissible lam) (hD : Moments offspring (fun k : ℕ => (k:ℝ)) d (e+d))
+    (hd : 0 < d) (he : 0 ≤ e) (hcrit : 1 < d*lam^2) :
+    Reconstructs (fun n => advantage (observedLaw offspring lam h n)) := by
+  have hgap : 0 < (d*lam)^2-d := by
+    have hp := mul_pos hd (sub_pos.mpr hcrit)
+    nlinarith
+  have hq : 0 < (d*lam)^2 := by linarith
+  let C : ℝ := 2+2*e*lam^2/((d*lam)^2-d)
+  have hC : 2 ≤ C := by
+    dsimp [C]
+    have hh : 0 ≤ 2*e*lam^2/((d*lam)^2-d) := by positivity
+    linarith
+  have hCpos : 0 < C := by linarith
+  have hCeq : C*((d*lam)^2-d) = 2*((d*lam)^2-d)+2*e*lam^2 := by
+    dsimp [C]
+    rw [add_mul, div_mul_cancel₀ _ (ne_of_gt hgap)]
+  have hstep : d*C+2*e*lam^2 ≤ C*(d*lam)^2 := by nlinarith
+  obtain ⟨L,hL0,hL1,hL⟩ := observedAdvantage_tendsto offspring lam h
+  have hb : 1/(3*C) ≤ L := ge_of_tendsto' hL
+    (fun n => supercritical_advantage_bound offspring d e lam C h hD hd.le hq hC hstep n)
+  exact ⟨L, (by positivity : 0 < 1/(3*C)).trans_le hb, hL⟩
+
+ 
+
+theorem regular_supercritical (b : ℕ) (hb : 2 ≤ b) (lam : ℝ) (h : Admissible lam)
+    (hcrit : 1 < (b:ℝ)*lam^2) : Reconstructs (regularAdvantage b lam h) := by
+  have hd : 0 < (b:ℝ) := by exact_mod_cast (show 0 < b by omega)
+  have he : 0 ≤ (b:ℝ)^2-(b:ℝ) := by
+    have hb' : (2:ℝ) ≤ (b:ℝ) := by exact_mod_cast hb
+    nlinarith
+  apply reconstructs_supercritical_of_moments (PMF.pure b) (b:ℝ) ((b:ℝ)^2-(b:ℝ)) lam h
+    ?_ hd he hcrit
+  simpa only [sub_add_cancel] using regular_moments b
+
+ 
+
+theorem poisson_supercritical (d : ℝ) (hd : 1 < d) (lam : ℝ) (h : Admissible lam)
+    (hcrit : 1 < d*lam^2) : Reconstructs (poissonAdvantage d hd lam h) := by
+  apply reconstructs_supercritical_of_moments (poissonOffspring d (le_trans (by norm_num) hd.le))
+    d (d^2) lam h (poisson_moments d _) (by linarith) (sq_nonneg d) hcrit
+
+end ThreeState
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeState/Symmetry.lean b/lean/OAI/Probability/ThreeState/Symmetry.lean
index bd9ec88f30f081c2e7f47322994352fca2f20de6..fb22f2d374bafa472dcad8e55f30bdd1b6714166 100644
--- a/lean/OAI/Probability/ThreeState/Symmetry.lean
+++ b/lean/OAI/Probability/ThreeState/Symmetry.lean
@@ -1,305 +1,305 @@
-import Mathlib
-import OAI.Probability.ThreeState.FixedPoints
-
-namespace OAI
-
-/-! Spin symmetry and measure identities for posterior fixed points. -/
-
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-
-lemma pmf_map_equiv_apply {α β : Type*} (p : PMF α) (e : α ≃ β) (a : α) :
-    p.map e (e a) = p a := by simp [PMF.map_apply]
-
-lemma uniformSpin_map (π : Equiv.Perm Spin) : uniformSpin.map π = uniformSpin := by
-  apply PMF.ext
-  intro j
-  obtain ⟨i,rfl⟩ := π.surjective j
-  rw [pmf_map_equiv_apply]
-  simp [uniformSpin, PMF.ofFintype_apply]
-
-lemma channelPMF_map (lam : ℝ) (h : Admissible lam) (π : Equiv.Perm Spin) (i : Spin) :
-    (channelPMF lam h i).map π = channelPMF lam h (π i) := by
-  apply PMF.ext
-  intro j
-  obtain ⟨k,rfl⟩ := π.surjective j
-  rw [pmf_map_equiv_apply]
-  simp [channelPMF, PMF.ofFintype_apply, channel]
-
-noncomputable def permMessage (π : Equiv.Perm Spin) (m : Message) : Message :=
-  ⟨fun i => m (π.symm i), by
-    constructor
-    · intro i; exact Message.nonneg m _
-    · rw [Equiv.sum_comp π.symm]
-      exact Message.sum m⟩
-
-lemma continuous_permMessage (π : Equiv.Perm Spin) : Continuous (permMessage π) := by
-  apply Continuous.subtype_mk
-  apply continuous_pi
-  intro i
-  exact (continuous_apply (π.symm i)).comp continuous_subtype_val
-
-noncomputable def renameOrdered : (n : ℕ) → Equiv.Perm Spin → Equiv.Perm (OrderedObservation n)
-  | 0, π => π
-  | n+1, π => Equiv.sigmaCongrRight (fun k => Equiv.piCongrRight (fun _ : Fin k => renameOrdered n π))
-
-lemma orderedLaw_rename (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
-    (π : Equiv.Perm Spin) (n : ℕ) (i : Spin) :
-    (orderedLaw offspring lam h n i).map (renameOrdered n π) = orderedLaw offspring lam h n (π i) := by
-  induction n generalizing i with
-  | zero => exact PMF.pure_map _ _
-  | succ n ih =>
-    simp only [orderedLaw, PMF.map_bind, PMF.map_comp]
-    apply congrArg (PMF.bind offspring)
-    funext k
-    change (iidVector ((channelPMF lam h i).bind (orderedLaw offspring lam h n)) k).map
-      ((Sigma.mk (β := fun k => Fin k → OrderedObservation n) k) ∘
-        (fun v : Fin k → OrderedObservation n => renameOrdered n π ∘ v)) = _
-    rw [← PMF.map_comp, iidVector_map]
-    congr 2
-    rw [PMF.map_bind]
-    simp_rw [ih]
-    change (channelPMF lam h i).bind ((orderedLaw offspring lam h n) ∘ π) = _
-    rw [← PMF.bind_map, channelPMF_map]
-
-lemma marginal_equivariant {α : Type*} (law : Spin → PMF α) (π : Equiv.Perm Spin)
-    (e : Equiv.Perm α) (he : ∀ i, (law i).map e = law (π i)) :
-    (marginal law).map e = marginal law := by
-  rw [marginal, PMF.map_bind]
-  simp_rw [he]
-  change uniformSpin.bind (law ∘ π) = _
-  rw [← PMF.bind_map, uniformSpin_map]
-
-lemma posterior_equivariant {α : Type*} (law : Spin → PMF α) (π : Equiv.Perm Spin)
-    (e : Equiv.Perm α) (he : ∀ i, (law i).map e = law (π i)) (y : α) (i : Spin) :
-    posterior law (e y) (π i) = posterior law y i := by
-  have hm : marginal law (e y) = marginal law y := by
-    calc
-      marginal law (e y) = (marginal law).map e (e y) := by rw [marginal_equivariant law π e he]
-      _ = _ := pmf_map_equiv_apply _ _ _
-  have hi : law (π i) (e y) = law i y := by
-    rw [← he i]
-    exact pmf_map_equiv_apply _ _ _
-  simp only [posterior, hm, hi]
-
-lemma likelihood_equivariant {α : Type*} (law : Spin → PMF α) (π : Equiv.Perm Spin)
-    (e : Equiv.Perm α) (he : ∀ i, (law i).map e = law (π i)) (y : α) :
-    likelihood law (e y) = permMessage π (likelihood law y) := by
-  apply Subtype.ext
-  funext i
-  have hh := posterior_equivariant law π e he y (π.symm i)
-  simp only [Equiv.apply_symm_apply] at hh
-  change 3*posterior law (e y) i = 3*posterior law y (π.symm i)
-  rw [hh]
-
-lemma likelihoodPMF_equivariant {α : Type*} (law : Spin → PMF α) (π : Equiv.Perm Spin)
-    (e : Equiv.Perm α) (he : ∀ i, (law i).map e = law (π i)) :
-    (likelihoodPMF law).map (permMessage π) = likelihoodPMF law := by
-  rw [likelihoodPMF, PMF.map_comp]
-  have hh : permMessage π ∘ likelihood law = likelihood law ∘ e := by
-    funext y
-    exact (likelihood_equivariant law π e he y).symm
-  rw [hh, ← PMF.map_comp, marginal_equivariant law π e he]
-
- 
-def SpinSymmetric (Q : ProbabilityMeasure Message) : Prop :=
-  ∀ π : Equiv.Perm Spin, Q.map (permMessage π) = Q
-
-lemma ordered_likelihood_symmetric (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
-    (n : ℕ) : SpinSymmetric (likelihoodLaw (orderedLaw offspring lam h n)) := by
-  intro π
-  apply Subtype.ext
-  change (likelihoodPMF (orderedLaw offspring lam h n)).toMeasure.map (permMessage π) = _
-  rw [PMF.toMeasure_map _ _ (continuous_permMessage π).measurable]
-  rw [likelihoodPMF_equivariant _ π (renameOrdered n π)
-    (fun i => orderedLaw_rename offspring lam h π n i)]
-  rfl
-
-lemma symmetric_limit {Qn : ℕ → ProbabilityMeasure Message} {Q : ProbabilityMeasure Message}
-    (hn : ∀ n, SpinSymmetric (Qn n)) (hlim : Tendsto Qn atTop (𝓝 Q)) : SpinSymmetric Q := by
-  intro π
-  have hh := ((ProbabilityMeasure.continuous_map (continuous_permMessage π)).tendsto Q).comp hlim
-  have he : (fun n => (Qn n).map (permMessage π)) = Qn :=
-    funext (fun n => hn n π)
-  simp only [Function.comp_def, he] at hh
-  exact tendsto_nhds_unique hh hlim
-
-lemma ordered_exists_symmetric_fixedpoint (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) :
-    ∃ Q : ProbabilityMeasure Message, IsPosteriorFixedPoint offspring lam h Q ∧ SpinSymmetric Q ∧
-      ∃ φ : ℕ → ℕ, StrictMono φ ∧
-        Tendsto (fun n => likelihoodLaw (orderedLaw offspring lam h (φ n))) atTop (𝓝 Q) := by
-  obtain ⟨Q,hQ,φ,hφ,hlim⟩ := ordered_exists_fixedpoint offspring lam h
-  exact ⟨Q,hQ,symmetric_limit (fun n => ordered_likelihood_symmetric offspring lam h (φ n)) hlim,φ,hφ,hlim⟩
-
-end ThreeState
-
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-
-lemma measurable_combineMessage (lam : ℝ) (h : Admissible lam) (n : ℕ) :
-    Measurable (combineMessage lam h : (Fin n → Message) → Message) := by
-  apply Measurable.subtype_mk
-  apply Measurable.of_eval
-  intro i
-  have he : (fun m : Fin n → Message => combineMessage lam h m i) =
-      (fun m => if normalizer lam h m = 0 then uninformative i else
-        productWeight lam h m i / normalizer lam h m) := by
-    funext m
-    by_cases hm : normalizer lam h m = 0 <;> simp [combineMessage, hm]
-  change Measurable (fun m : Fin n → Message => combineMessage lam h m i)
-  rw [he]
-  exact Measurable.ite ((continuous_normalizer lam h n).measurable (measurableSet_singleton 0))
-    measurable_const ((continuous_productWeight lam h n i).measurable.div
-      (continuous_normalizer lam h n).measurable)
-
- 
-noncomputable def degreeInput (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message) (n : ℕ) :
-    Measure (Fin n → Message) :=
-  (iidMeasure Q n : Measure (Fin n → Message)).withDensity (fun m => ENNReal.ofReal (normalizer lam h m))
-
-lemma degreeInput_probability (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message)
-    (hQ : Balanced Q) (n : ℕ) : IsProbabilityMeasure (degreeInput lam h Q n) := by
-  constructor
-  rw [degreeInput, withDensity_apply _ MeasurableSet.univ, Measure.restrict_univ,
-    ← ofReal_integral_eq_lintegral_ofReal
-      ((continuous_normalizer lam h n).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _))
-      (Eventually.of_forall (normalizer_nonneg lam h)), integral_normalizer lam h Q hQ n]
-  simp
-
-noncomputable def degreeOutput (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message) (n : ℕ) :
-    Measure Message := Measure.map (combineMessage lam h) (degreeInput lam h Q n)
-
-lemma degreeOutput_probability (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message)
-    (hQ : Balanced Q) (n : ℕ) : IsProbabilityMeasure (degreeOutput lam h Q n) := by
-  let := degreeInput_probability lam h Q hQ n
-  constructor
-  rw [degreeOutput, Measure.map_apply (measurable_combineMessage lam h n) MeasurableSet.univ]
-  simp
-
- 
-noncomputable def densityEvolution (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
-    (Q : ProbabilityMeasure Message) : Measure Message :=
-  Measure.sum (fun n => offspring n • degreeOutput lam h Q n)
-
-lemma densityEvolution_probability (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
-    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) : IsProbabilityMeasure (densityEvolution offspring lam h Q) := by
-  constructor
-  rw [densityEvolution, Measure.sum_apply _ MeasurableSet.univ]
-  convert offspring.tsum_coe using 1
-  apply tsum_congr
-  intro n
-  let := degreeOutput_probability lam h Q hQ n
-  simp
-
-lemma integral_degreeOutput (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message) (n : ℕ)
-    (f : Message → ℝ) (hf : Measurable f) :
-    ∫ m, f m ∂degreeOutput lam h Q n =
-      ∫ m, normalizer lam h m * f (combineMessage lam h m) ∂(iidMeasure Q n : Measure (Fin n → Message)) := by
-  rw [degreeOutput, integral_map (measurable_combineMessage lam h n).aemeasurable hf.aestronglyMeasurable,
-    degreeInput, integral_withDensity_eq_integral_toReal_smul
-    ((continuous_normalizer lam h n).measurable.ennreal_ofReal)
-    (Eventually.of_forall (fun _ => ENNReal.ofReal_lt_top))]
-  apply integral_congr_ae
-  filter_upwards with m
-  simp [ENNReal.toReal_ofReal (normalizer_nonneg lam h m)]
-
-lemma integral_densityEvolution_continuous (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
-    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (f : C(Message,ℝ)) :
-    ∫ m, f m ∂densityEvolution offspring lam h Q =
-      mean offspring (fun n => degreeTest lam h n f Q) := by
-  let := densityEvolution_probability offspring lam h Q hQ
-  have hi : Integrable f (densityEvolution offspring lam h Q) :=
-    f.continuous.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
-  rw [densityEvolution, integral_sum_measure hi]
-  unfold mean
-  apply tsum_congr
-  intro n
-  rw [integral_smul_measure, integral_degreeOutput lam h Q n f f.continuous.measurable]
-  rfl
-
-lemma fixedpoint_measure (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
-    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint offspring lam h Q) :
-    (Q : Measure Message) = densityEvolution offspring lam h Q := by
-  let := densityEvolution_probability offspring lam h Q hQ.1
-  apply ext_of_forall_integral_eq_of_IsFiniteMeasure
-  intro f
-  change (∫ m, f.toContinuousMap m ∂(Q : Measure Message)) =
-    ∫ m, f.toContinuousMap m ∂densityEvolution offspring lam h Q
-  rw [integral_densityEvolution_continuous offspring lam h Q hQ.1 f.toContinuousMap]
-  exact hQ.2 f.toContinuousMap
-
-end ThreeState
-
-namespace ThreeState
-open MeasureTheory Filter Topology
-open scoped Classical
-
-lemma ae_degreeOutput_of_forall (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message) (n : ℕ)
-    (P : Message → Prop) (hP : MeasurableSet {m | P m})
-    (hp : ∀ m : Fin n → Message, P (combineMessage lam h m)) :
-    ∀ᵐ m ∂degreeOutput lam h Q n, P m := by
-  exact (ae_map_iff (measurable_combineMessage lam h n).aemeasurable hP).mpr
-    (Eventually.of_forall hp)
-
-lemma ae_fixedpoint_of_output (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
-    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint offspring lam h Q)
-    (P : Message → Prop) (hP : MeasurableSet {m | P m})
-    (hp : ∀ n (m : Fin n → Message), P (combineMessage lam h m)) :
-    ∀ᵐ m ∂(Q : Measure Message), P m := by
-  rw [fixedpoint_measure offspring lam h Q hQ, densityEvolution]
-  apply Measure.ae_sum_iff.mpr
-  intro n
-  exact Measure.ae_smul_measure (ae_degreeOutput_of_forall lam h Q n P hP (hp n)) (offspring n)
-
-lemma integrable_degreeOutput_bound (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message)
-    (hQ : Balanced Q) (n : ℕ) (f : Message → ℝ) (hf : Measurable f) (C : ℝ)
-    (hC : ∀ m : Fin n → Message, |f (combineMessage lam h m)| ≤ C) :
-    Integrable f (degreeOutput lam h Q n) ∧
-      (∫ m, ‖f m‖ ∂degreeOutput lam h Q n) ≤ C := by
-  let := degreeOutput_probability lam h Q hQ n
-  have hb : ∀ᵐ m ∂degreeOutput lam h Q n, ‖f m‖ ≤ C :=
-    ae_degreeOutput_of_forall lam h Q n _ (measurableSet_le hf.norm measurable_const)
-      (fun m => by simpa only [Real.norm_eq_abs] using hC m)
-  have hi : Integrable f (degreeOutput lam h Q n) :=
-    (integrable_const C).mono' hf.aestronglyMeasurable hb
-  refine ⟨hi,?_⟩
-  calc
-    (∫ m, ‖f m‖ ∂degreeOutput lam h Q n) ≤ ∫ _, C ∂degreeOutput lam h Q n :=
-      integral_mono_ae hi.norm (integrable_const C) hb
-    _ = C := by simp
-
-lemma integrable_fixedpoint_of_growth (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
-    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint offspring lam h Q)
-    (f : Message → ℝ) (hf : Measurable f) (C : ℕ → ℝ) (hC : HasMean offspring C)
-    (hb : ∀ n (m : Fin n → Message), |f (combineMessage lam h m)| ≤ C n) :
-    Integrable f (Q : Measure Message) := by
-  rw [fixedpoint_measure offspring lam h Q hQ, densityEvolution]
-  apply integrable_sum_measure
-  · intro n
-    exact (integrable_degreeOutput_bound lam h Q hQ.1 n f hf (C n) (hb n)).1.smul_measure (offspring.apply_ne_top n)
-  · apply hC.of_norm_bounded
-    intro n
-    rw [integral_smul_measure, norm_smul, Real.norm_eq_abs,
-      abs_of_nonneg (ENNReal.toReal_nonneg)]
-    rw [Real.norm_eq_abs, abs_of_nonneg (integral_nonneg (fun _ => norm_nonneg _))]
-    exact mul_le_mul_of_nonneg_left
-      (integrable_degreeOutput_bound lam h Q hQ.1 n f hf (C n) (hb n)).2 (mass_nonneg offspring n)
-
-lemma integral_fixedpoint (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
-    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint offspring lam h Q)
-    (f : Message → ℝ) (hf : Measurable f) (hi : Integrable f (Q : Measure Message)) :
-    ∫ m, f m ∂(Q : Measure Message) = mean offspring (fun n =>
-      ∫ m, normalizer lam h m * f (combineMessage lam h m) ∂(iidMeasure Q n : Measure (Fin n → Message))) := by
-  rw [fixedpoint_measure offspring lam h Q hQ] at hi ⊢
-  rw [densityEvolution, integral_sum_measure hi]
-  unfold mean
-  apply tsum_congr
-  intro n
-  rw [integral_smul_measure, integral_degreeOutput lam h Q n f hf]
-  rfl
-
-end ThreeState
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeState.FixedPoints
+
+namespace OAI
+
+/-! Spin symmetry and measure identities for posterior fixed points. -/
+
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+
+lemma pmf_map_equiv_apply {α β : Type*} (p : PMF α) (e : α ≃ β) (a : α) :
+    p.map e (e a) = p a := by simp [PMF.map_apply]
+
+lemma uniformSpin_map (π : Equiv.Perm Spin) : uniformSpin.map π = uniformSpin := by
+  apply PMF.ext
+  intro j
+  obtain ⟨i,rfl⟩ := π.surjective j
+  rw [pmf_map_equiv_apply]
+  simp [uniformSpin, PMF.ofFintype_apply]
+
+lemma channelPMF_map (lam : ℝ) (h : Admissible lam) (π : Equiv.Perm Spin) (i : Spin) :
+    (channelPMF lam h i).map π = channelPMF lam h (π i) := by
+  apply PMF.ext
+  intro j
+  obtain ⟨k,rfl⟩ := π.surjective j
+  rw [pmf_map_equiv_apply]
+  simp [channelPMF, PMF.ofFintype_apply, channel]
+
+noncomputable def permMessage (π : Equiv.Perm Spin) (m : Message) : Message :=
+  ⟨fun i => m (π.symm i), by
+    constructor
+    · intro i; exact Message.nonneg m _
+    · rw [Equiv.sum_comp π.symm]
+      exact Message.sum m⟩
+
+lemma continuous_permMessage (π : Equiv.Perm Spin) : Continuous (permMessage π) := by
+  apply Continuous.subtype_mk
+  apply continuous_pi
+  intro i
+  exact (continuous_apply (π.symm i)).comp continuous_subtype_val
+
+noncomputable def renameOrdered : (n : ℕ) → Equiv.Perm Spin → Equiv.Perm (OrderedObservation n)
+  | 0, π => π
+  | n+1, π => Equiv.sigmaCongrRight (fun k => Equiv.piCongrRight (fun _ : Fin k => renameOrdered n π))
+
+lemma orderedLaw_rename (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
+    (π : Equiv.Perm Spin) (n : ℕ) (i : Spin) :
+    (orderedLaw offspring lam h n i).map (renameOrdered n π) = orderedLaw offspring lam h n (π i) := by
+  induction n generalizing i with
+  | zero => exact PMF.pure_map _ _
+  | succ n ih =>
+    simp only [orderedLaw, PMF.map_bind, PMF.map_comp]
+    apply congrArg (PMF.bind offspring)
+    funext k
+    change (iidVector ((channelPMF lam h i).bind (orderedLaw offspring lam h n)) k).map
+      ((Sigma.mk (β := fun k => Fin k → OrderedObservation n) k) ∘
+        (fun v : Fin k → OrderedObservation n => renameOrdered n π ∘ v)) = _
+    rw [← PMF.map_comp, iidVector_map]
+    congr 2
+    rw [PMF.map_bind]
+    simp_rw [ih]
+    change (channelPMF lam h i).bind ((orderedLaw offspring lam h n) ∘ π) = _
+    rw [← PMF.bind_map, channelPMF_map]
+
+lemma marginal_equivariant {α : Type*} (law : Spin → PMF α) (π : Equiv.Perm Spin)
+    (e : Equiv.Perm α) (he : ∀ i, (law i).map e = law (π i)) :
+    (marginal law).map e = marginal law := by
+  rw [marginal, PMF.map_bind]
+  simp_rw [he]
+  change uniformSpin.bind (law ∘ π) = _
+  rw [← PMF.bind_map, uniformSpin_map]
+
+lemma posterior_equivariant {α : Type*} (law : Spin → PMF α) (π : Equiv.Perm Spin)
+    (e : Equiv.Perm α) (he : ∀ i, (law i).map e = law (π i)) (y : α) (i : Spin) :
+    posterior law (e y) (π i) = posterior law y i := by
+  have hm : marginal law (e y) = marginal law y := by
+    calc
+      marginal law (e y) = (marginal law).map e (e y) := by rw [marginal_equivariant law π e he]
+      _ = _ := pmf_map_equiv_apply _ _ _
+  have hi : law (π i) (e y) = law i y := by
+    rw [← he i]
+    exact pmf_map_equiv_apply _ _ _
+  simp only [posterior, hm, hi]
+
+lemma likelihood_equivariant {α : Type*} (law : Spin → PMF α) (π : Equiv.Perm Spin)
+    (e : Equiv.Perm α) (he : ∀ i, (law i).map e = law (π i)) (y : α) :
+    likelihood law (e y) = permMessage π (likelihood law y) := by
+  apply Subtype.ext
+  funext i
+  have hh := posterior_equivariant law π e he y (π.symm i)
+  simp only [Equiv.apply_symm_apply] at hh
+  change 3*posterior law (e y) i = 3*posterior law y (π.symm i)
+  rw [hh]
+
+lemma likelihoodPMF_equivariant {α : Type*} (law : Spin → PMF α) (π : Equiv.Perm Spin)
+    (e : Equiv.Perm α) (he : ∀ i, (law i).map e = law (π i)) :
+    (likelihoodPMF law).map (permMessage π) = likelihoodPMF law := by
+  rw [likelihoodPMF, PMF.map_comp]
+  have hh : permMessage π ∘ likelihood law = likelihood law ∘ e := by
+    funext y
+    exact (likelihood_equivariant law π e he y).symm
+  rw [hh, ← PMF.map_comp, marginal_equivariant law π e he]
+
+ 
+def SpinSymmetric (Q : ProbabilityMeasure Message) : Prop :=
+  ∀ π : Equiv.Perm Spin, Q.map (permMessage π) = Q
+
+lemma ordered_likelihood_symmetric (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
+    (n : ℕ) : SpinSymmetric (likelihoodLaw (orderedLaw offspring lam h n)) := by
+  intro π
+  apply Subtype.ext
+  change (likelihoodPMF (orderedLaw offspring lam h n)).toMeasure.map (permMessage π) = _
+  rw [PMF.toMeasure_map _ _ (continuous_permMessage π).measurable]
+  rw [likelihoodPMF_equivariant _ π (renameOrdered n π)
+    (fun i => orderedLaw_rename offspring lam h π n i)]
+  rfl
+
+lemma symmetric_limit {Qn : ℕ → ProbabilityMeasure Message} {Q : ProbabilityMeasure Message}
+    (hn : ∀ n, SpinSymmetric (Qn n)) (hlim : Tendsto Qn atTop (𝓝 Q)) : SpinSymmetric Q := by
+  intro π
+  have hh := ((ProbabilityMeasure.continuous_map (continuous_permMessage π)).tendsto Q).comp hlim
+  have he : (fun n => (Qn n).map (permMessage π)) = Qn :=
+    funext (fun n => hn n π)
+  simp only [Function.comp_def, he] at hh
+  exact tendsto_nhds_unique hh hlim
+
+lemma ordered_exists_symmetric_fixedpoint (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam) :
+    ∃ Q : ProbabilityMeasure Message, IsPosteriorFixedPoint offspring lam h Q ∧ SpinSymmetric Q ∧
+      ∃ φ : ℕ → ℕ, StrictMono φ ∧
+        Tendsto (fun n => likelihoodLaw (orderedLaw offspring lam h (φ n))) atTop (𝓝 Q) := by
+  obtain ⟨Q,hQ,φ,hφ,hlim⟩ := ordered_exists_fixedpoint offspring lam h
+  exact ⟨Q,hQ,symmetric_limit (fun n => ordered_likelihood_symmetric offspring lam h (φ n)) hlim,φ,hφ,hlim⟩
+
+end ThreeState
+
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+
+lemma measurable_combineMessage (lam : ℝ) (h : Admissible lam) (n : ℕ) :
+    Measurable (combineMessage lam h : (Fin n → Message) → Message) := by
+  apply Measurable.subtype_mk
+  apply Measurable.of_eval
+  intro i
+  have he : (fun m : Fin n → Message => combineMessage lam h m i) =
+      (fun m => if normalizer lam h m = 0 then uninformative i else
+        productWeight lam h m i / normalizer lam h m) := by
+    funext m
+    by_cases hm : normalizer lam h m = 0 <;> simp [combineMessage, hm]
+  change Measurable (fun m : Fin n → Message => combineMessage lam h m i)
+  rw [he]
+  exact Measurable.ite ((continuous_normalizer lam h n).measurable (measurableSet_singleton 0))
+    measurable_const ((continuous_productWeight lam h n i).measurable.div
+      (continuous_normalizer lam h n).measurable)
+
+ 
+noncomputable def degreeInput (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message) (n : ℕ) :
+    Measure (Fin n → Message) :=
+  (iidMeasure Q n : Measure (Fin n → Message)).withDensity (fun m => ENNReal.ofReal (normalizer lam h m))
+
+lemma degreeInput_probability (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message)
+    (hQ : Balanced Q) (n : ℕ) : IsProbabilityMeasure (degreeInput lam h Q n) := by
+  constructor
+  rw [degreeInput, withDensity_apply _ MeasurableSet.univ, Measure.restrict_univ,
+    ← ofReal_integral_eq_lintegral_ofReal
+      ((continuous_normalizer lam h n).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _))
+      (Eventually.of_forall (normalizer_nonneg lam h)), integral_normalizer lam h Q hQ n]
+  simp
+
+noncomputable def degreeOutput (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message) (n : ℕ) :
+    Measure Message := Measure.map (combineMessage lam h) (degreeInput lam h Q n)
+
+lemma degreeOutput_probability (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message)
+    (hQ : Balanced Q) (n : ℕ) : IsProbabilityMeasure (degreeOutput lam h Q n) := by
+  let := degreeInput_probability lam h Q hQ n
+  constructor
+  rw [degreeOutput, Measure.map_apply (measurable_combineMessage lam h n) MeasurableSet.univ]
+  simp
+
+ 
+noncomputable def densityEvolution (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
+    (Q : ProbabilityMeasure Message) : Measure Message :=
+  Measure.sum (fun n => offspring n • degreeOutput lam h Q n)
+
+lemma densityEvolution_probability (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
+    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) : IsProbabilityMeasure (densityEvolution offspring lam h Q) := by
+  constructor
+  rw [densityEvolution, Measure.sum_apply _ MeasurableSet.univ]
+  convert offspring.tsum_coe using 1
+  apply tsum_congr
+  intro n
+  let := degreeOutput_probability lam h Q hQ n
+  simp
+
+lemma integral_degreeOutput (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message) (n : ℕ)
+    (f : Message → ℝ) (hf : Measurable f) :
+    ∫ m, f m ∂degreeOutput lam h Q n =
+      ∫ m, normalizer lam h m * f (combineMessage lam h m) ∂(iidMeasure Q n : Measure (Fin n → Message)) := by
+  rw [degreeOutput, integral_map (measurable_combineMessage lam h n).aemeasurable hf.aestronglyMeasurable,
+    degreeInput, integral_withDensity_eq_integral_toReal_smul
+    ((continuous_normalizer lam h n).measurable.ennreal_ofReal)
+    (Eventually.of_forall (fun _ => ENNReal.ofReal_lt_top))]
+  apply integral_congr_ae
+  filter_upwards with m
+  simp [ENNReal.toReal_ofReal (normalizer_nonneg lam h m)]
+
+lemma integral_densityEvolution_continuous (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
+    (Q : ProbabilityMeasure Message) (hQ : Balanced Q) (f : C(Message,ℝ)) :
+    ∫ m, f m ∂densityEvolution offspring lam h Q =
+      mean offspring (fun n => degreeTest lam h n f Q) := by
+  let := densityEvolution_probability offspring lam h Q hQ
+  have hi : Integrable f (densityEvolution offspring lam h Q) :=
+    f.continuous.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
+  rw [densityEvolution, integral_sum_measure hi]
+  unfold mean
+  apply tsum_congr
+  intro n
+  rw [integral_smul_measure, integral_degreeOutput lam h Q n f f.continuous.measurable]
+  rfl
+
+lemma fixedpoint_measure (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
+    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint offspring lam h Q) :
+    (Q : Measure Message) = densityEvolution offspring lam h Q := by
+  let := densityEvolution_probability offspring lam h Q hQ.1
+  apply ext_of_forall_integral_eq_of_IsFiniteMeasure
+  intro f
+  change (∫ m, f.toContinuousMap m ∂(Q : Measure Message)) =
+    ∫ m, f.toContinuousMap m ∂densityEvolution offspring lam h Q
+  rw [integral_densityEvolution_continuous offspring lam h Q hQ.1 f.toContinuousMap]
+  exact hQ.2 f.toContinuousMap
+
+end ThreeState
+
+namespace ThreeState
+open MeasureTheory Filter Topology
+open scoped Classical
+
+lemma ae_degreeOutput_of_forall (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message) (n : ℕ)
+    (P : Message → Prop) (hP : MeasurableSet {m | P m})
+    (hp : ∀ m : Fin n → Message, P (combineMessage lam h m)) :
+    ∀ᵐ m ∂degreeOutput lam h Q n, P m := by
+  exact (ae_map_iff (measurable_combineMessage lam h n).aemeasurable hP).mpr
+    (Eventually.of_forall hp)
+
+lemma ae_fixedpoint_of_output (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
+    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint offspring lam h Q)
+    (P : Message → Prop) (hP : MeasurableSet {m | P m})
+    (hp : ∀ n (m : Fin n → Message), P (combineMessage lam h m)) :
+    ∀ᵐ m ∂(Q : Measure Message), P m := by
+  rw [fixedpoint_measure offspring lam h Q hQ, densityEvolution]
+  apply Measure.ae_sum_iff.mpr
+  intro n
+  exact Measure.ae_smul_measure (ae_degreeOutput_of_forall lam h Q n P hP (hp n)) (offspring n)
+
+lemma integrable_degreeOutput_bound (lam : ℝ) (h : Admissible lam) (Q : ProbabilityMeasure Message)
+    (hQ : Balanced Q) (n : ℕ) (f : Message → ℝ) (hf : Measurable f) (C : ℝ)
+    (hC : ∀ m : Fin n → Message, |f (combineMessage lam h m)| ≤ C) :
+    Integrable f (degreeOutput lam h Q n) ∧
+      (∫ m, ‖f m‖ ∂degreeOutput lam h Q n) ≤ C := by
+  let := degreeOutput_probability lam h Q hQ n
+  have hb : ∀ᵐ m ∂degreeOutput lam h Q n, ‖f m‖ ≤ C :=
+    ae_degreeOutput_of_forall lam h Q n _ (measurableSet_le hf.norm measurable_const)
+      (fun m => by simpa only [Real.norm_eq_abs] using hC m)
+  have hi : Integrable f (degreeOutput lam h Q n) :=
+    (integrable_const C).mono' hf.aestronglyMeasurable hb
+  refine ⟨hi,?_⟩
+  calc
+    (∫ m, ‖f m‖ ∂degreeOutput lam h Q n) ≤ ∫ _, C ∂degreeOutput lam h Q n :=
+      integral_mono_ae hi.norm (integrable_const C) hb
+    _ = C := by simp
+
+lemma integrable_fixedpoint_of_growth (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
+    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint offspring lam h Q)
+    (f : Message → ℝ) (hf : Measurable f) (C : ℕ → ℝ) (hC : HasMean offspring C)
+    (hb : ∀ n (m : Fin n → Message), |f (combineMessage lam h m)| ≤ C n) :
+    Integrable f (Q : Measure Message) := by
+  rw [fixedpoint_measure offspring lam h Q hQ, densityEvolution]
+  apply integrable_sum_measure
+  · intro n
+    exact (integrable_degreeOutput_bound lam h Q hQ.1 n f hf (C n) (hb n)).1.smul_measure (offspring.apply_ne_top n)
+  · apply hC.of_norm_bounded
+    intro n
+    rw [integral_smul_measure, norm_smul, Real.norm_eq_abs,
+      abs_of_nonneg (ENNReal.toReal_nonneg)]
+    rw [Real.norm_eq_abs, abs_of_nonneg (integral_nonneg (fun _ => norm_nonneg _))]
+    exact mul_le_mul_of_nonneg_left
+      (integrable_degreeOutput_bound lam h Q hQ.1 n f hf (C n) (hb n)).2 (mass_nonneg offspring n)
+
+lemma integral_fixedpoint (offspring : PMF ℕ) (lam : ℝ) (h : Admissible lam)
+    (Q : ProbabilityMeasure Message) (hQ : IsPosteriorFixedPoint offspring lam h Q)
+    (f : Message → ℝ) (hf : Measurable f) (hi : Integrable f (Q : Measure Message)) :
+    ∫ m, f m ∂(Q : Measure Message) = mean offspring (fun n =>
+      ∫ m, normalizer lam h m * f (combineMessage lam h m) ∂(iidMeasure Q n : Measure (Fin n → Message))) := by
+  rw [fixedpoint_measure offspring lam h Q hQ] at hi ⊢
+  rw [densityEvolution, integral_sum_measure hi]
+  unfold mean
+  apply tsum_congr
+  intro n
+  rw [integral_smul_measure, integral_degreeOutput lam h Q n f hf]
+  rfl
+
+end ThreeState
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeState/Update.lean b/lean/OAI/Probability/ThreeState/Update.lean
index 51a691fa098c01f965aada130b1900a185c4e32e..54bba587883b0d38d90076d4d17ac5e88bd6fd70 100644
--- a/lean/OAI/Probability/ThreeState/Update.lean
+++ b/lean/OAI/Probability/ThreeState/Update.lean
@@ -1,258 +1,258 @@
-import Mathlib
-import OAI.Probability.ThreeState.Observation
-
-namespace OAI
-
-/-! Branchwise likelihood updates and their independent product measures. -/
-
-namespace ThreeState
-open MeasureTheory
-open scoped Classical
-
-noncomputable def edgeMessage (lam : ℝ) (h : Admissible lam) (m : Message) : Message :=
-  ⟨fun i => 1+lam*(m i-1), by
-    constructor
-    · intro i
-      have hl := Message.nonneg m i
-      have hu := Message.le_three m i
-      by_cases hp : 0 ≤ lam
-      · nlinarith [h.2]
-      · have hn : lam ≤ 0 := le_of_not_ge hp
-        nlinarith [h.1]
-    · simp only [Finset.sum_add_distrib, ← Finset.mul_sum, Finset.sum_sub_distrib,
-        Message.sum, Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul]
-      norm_num⟩
-
-lemma continuous_edgeMessage (lam : ℝ) (h : Admissible lam) : Continuous (edgeMessage lam h) := by
-  apply Continuous.subtype_mk
-  apply continuous_pi
-  intro i
-  exact continuous_const.add (continuous_const.mul (((continuous_apply i).comp continuous_subtype_val).sub continuous_const))
-
-lemma channel_decompose (lam : ℝ) (i j : Spin) :
-    channel lam i j = (1-lam)/3 + if i=j then lam else 0 := by
-  unfold channel
-  split_ifs <;> ring
-
-lemma channel_message (lam : ℝ) (m : Message) (i : Spin) :
-    ∑ j, channel lam i j * m j = 1+lam*(m i-1) := by
-  simp_rw [channel_decompose, add_mul, ite_mul, zero_mul]
-  rw [Finset.sum_add_distrib, ← Finset.mul_sum, Message.sum]
-  simp only [Finset.sum_ite_eq, Finset.mem_univ, ite_true]
-  ring
-
-lemma mass_channelPMF (lam : ℝ) (h : Admissible lam) (i j : Spin) :
-    mass (channelPMF lam h i) j = channel lam i j := by
-  simp [mass, channelPMF, PMF.ofFintype_apply, ENNReal.toReal_ofReal (channel_nonneg h i j)]
-
-noncomputable def edgeLaw {α : Type*} (law : Spin → PMF α) (lam : ℝ) (h : Admissible lam) : Spin → PMF α :=
-  fun i => (channelPMF lam h i).bind law
-
-lemma mass_edgeLaw {α : Type*} (law : Spin → PMF α) (lam : ℝ) (h : Admissible lam) (i : Spin) (y : α) :
-    mass (edgeLaw law lam h i) y = mass (marginal law) y * edgeMessage lam h (likelihood law y) i := by
-  rw [edgeLaw, mass_bind, tsum_fintype]
-  simp_rw [mass_channelPMF, ← weighted_likelihood law y]
-  calc
-    (∑ j, channel lam i j * (mass (marginal law) y * likelihood law y j)) =
-      mass (marginal law) y * (∑ j, channel lam i j * likelihood law y j) := by
-        rw [Finset.mul_sum]
-        congr 1
-        funext j
-        ring
-    _ = _ := by rw [channel_message]; rfl
-
-noncomputable def sigmaPMF {ι : Type*} {α : ι → Type*} (p : PMF ι) (q : ∀ i, PMF (α i)) : PMF (Sigma α) :=
-  p.bind (fun i => (q i).map (Sigma.mk i))
-
-lemma sigmaPMF_apply {ι : Type*} {α : ι → Type*} (p : PMF ι) (q : ∀ i, PMF (α i))
-    (i : ι) (a : α i) : sigmaPMF p q ⟨i,a⟩ = p i * q i a := by
-  classical
-  rw [sigmaPMF, PMF.bind_apply, tsum_eq_single i]
-  · simp [PMF.map_apply]
-  · intro j hj
-    have he (b : α j) : (⟨i,a⟩ : Sigma α) ≠ ⟨j,b⟩ := by
-      intro hh
-      exact hj (congrArg Sigma.fst hh).symm
-    simp [PMF.map_apply, he]
-
-lemma mass_sigmaPMF {ι : Type*} {α : ι → Type*} (p : PMF ι) (q : ∀ i, PMF (α i))
-    (i : ι) (a : α i) : mass (sigmaPMF p q) ⟨i,a⟩ = mass p i * mass (q i) a := by
-  simp only [mass, sigmaPMF_apply, ENNReal.toReal_mul]
-
-lemma mass_iidVector {α : Type*} (p : PMF α) (n : ℕ) (v : Fin n → α) :
-    mass (iidVector p n) v = ∏ j, mass p (v j) := by
-  simp only [mass, iidVector_apply, ENNReal.toReal_prod]
-
- 
-noncomputable def productReference {α : Type*} (offspring : PMF ℕ) (law : Spin → PMF α) :
-    PMF (Σ n : ℕ, Fin n → α) := sigmaPMF offspring (iidVector (marginal law))
-
-noncomputable def composeLaw {α : Type*} (offspring : PMF ℕ) (law : Spin → PMF α)
-    (lam : ℝ) (h : Admissible lam) : Spin → PMF (Σ n : ℕ, Fin n → α) :=
-  fun i => sigmaPMF offspring (iidVector (edgeLaw law lam h i))
-
-noncomputable def productWeight (lam : ℝ) (h : Admissible lam) {n : ℕ} (m : Fin n → Message) (i : Spin) : ℝ :=
-  ∏ j, edgeMessage lam h (m j) i
-
-noncomputable def normalizer (lam : ℝ) (h : Admissible lam) {n : ℕ} (m : Fin n → Message) : ℝ :=
-  (∑ i : Spin, productWeight lam h m i)/3
-
-lemma productWeight_nonneg (lam : ℝ) (h : Admissible lam) {n : ℕ} (m : Fin n → Message) (i : Spin) :
-    0 ≤ productWeight lam h m i :=
-  Finset.prod_nonneg (fun index _ => Message.nonneg (edgeMessage lam h (m index)) i)
-
-lemma normalizer_nonneg (lam : ℝ) (h : Admissible lam) {n : ℕ} (m : Fin n → Message) :
-    0 ≤ normalizer lam h m := div_nonneg (Finset.sum_nonneg (fun i _ => productWeight_nonneg _ _ _ _)) (by norm_num)
-
-lemma productWeight_sum (lam : ℝ) (h : Admissible lam) {n : ℕ} (m : Fin n → Message) :
-    ∑ i, productWeight lam h m i = 3*normalizer lam h m := by dsimp [normalizer]; ring
-
-noncomputable def combineMessage (lam : ℝ) (h : Admissible lam) {n : ℕ} (m : Fin n → Message) : Message :=
-  if hz : normalizer lam h m = 0 then uninformative else
-    ⟨fun i => productWeight lam h m i / normalizer lam h m, by
-      constructor
-      · intro i
-        exact div_nonneg (productWeight_nonneg _ _ _ _) (normalizer_nonneg _ _ _)
-      · rw [← Finset.sum_div, productWeight_sum]
-        field_simp⟩
-
-lemma mass_composeLaw {α : Type*} (offspring : PMF ℕ) (law : Spin → PMF α)
-    (lam : ℝ) (h : Admissible lam) (n : ℕ) (v : Fin n → α) (i : Spin) :
-    mass (composeLaw offspring law lam h i) ⟨n,v⟩ =
-      mass (productReference offspring law) ⟨n,v⟩ * productWeight lam h (fun j => likelihood law (v j)) i := by
-  simp only [composeLaw, productReference, mass_sigmaPMF, mass_iidVector, mass_edgeLaw,
-    Finset.prod_mul_distrib, productWeight, mul_assoc]
-
-lemma mass_marginal_composeLaw {α : Type*} (offspring : PMF ℕ) (law : Spin → PMF α)
-    (lam : ℝ) (h : Admissible lam) (n : ℕ) (v : Fin n → α) :
-    mass (marginal (composeLaw offspring law lam h)) ⟨n,v⟩ =
-      mass (productReference offspring law) ⟨n,v⟩ * normalizer lam h (fun j => likelihood law (v j)) := by
-  rw [mass_marginal]
-  simp_rw [mass_composeLaw]
-  rw [← Finset.mul_sum]
-  simp [normalizer]
-  ring
-
-lemma weighted_combineMessage (lam : ℝ) (h : Admissible lam) {n : ℕ}
-    (m : Fin n → Message) (i : Spin) :
-    normalizer lam h m * combineMessage lam h m i = productWeight lam h m i := by
-  by_cases hz : normalizer lam h m = 0
-  · have hs := productWeight_sum lam h m
-    have hi := Finset.single_le_sum (fun j _ => productWeight_nonneg lam h m j) (Finset.mem_univ i)
-    have hn := productWeight_nonneg lam h m i
-    simp only [hz, mul_zero] at hs
-    have hw : productWeight lam h m i = 0 := le_antisymm (by simpa [hs] using hi) hn
-    simp [hz, hw]
-  · simp only [combineMessage, dite_eq_right hz]
-    field_simp
-
-lemma likelihood_composeLaw {α : Type*} (offspring : PMF ℕ) (law : Spin → PMF α)
-    (lam : ℝ) (h : Admissible lam) (n : ℕ) (v : Fin n → α)
-    (hm : mass (marginal (composeLaw offspring law lam h)) ⟨n,v⟩ ≠ 0) :
-    likelihood (composeLaw offspring law lam h) ⟨n,v⟩ =
-      combineMessage lam h (fun j => likelihood law (v j)) := by
-  apply Subtype.ext
-  funext i
-  apply mul_left_cancel₀ hm
-  rw [weighted_likelihood, mass_composeLaw, mass_marginal_composeLaw, mul_assoc,
-    weighted_combineMessage]
-
-lemma weighted_test_composeLaw {α : Type*} (offspring : PMF ℕ) (law : Spin → PMF α)
-    (lam : ℝ) (h : Admissible lam) (f : Message → ℝ) (n : ℕ) (v : Fin n → α) :
-    mass (marginal (composeLaw offspring law lam h)) ⟨n,v⟩ *
-      f (likelihood (composeLaw offspring law lam h) ⟨n,v⟩) =
-    mass (productReference offspring law) ⟨n,v⟩ *
-      (normalizer lam h (fun j => likelihood law (v j)) *
-        f (combineMessage lam h (fun j => likelihood law (v j)))) := by
-  by_cases hm : mass (marginal (composeLaw offspring law lam h)) ⟨n,v⟩ = 0
-  · rw [mass_marginal_composeLaw] at hm
-    rw [← mul_assoc, ← mass_marginal_composeLaw, mass_marginal_composeLaw, hm]
-    simp
-  · rw [likelihood_composeLaw offspring law lam h n v hm, mass_marginal_composeLaw, mul_assoc]
-
-lemma hasMean_weighted_compose {α : Type*} (offspring : PMF ℕ) (law : Spin → PMF α)
-    (lam : ℝ) (h : Admissible lam) (f : Message → ℝ) (C : ℝ) (hf : ∀ m, |f m| ≤ C) :
-    HasMean (productReference offspring law) (fun nv =>
-      normalizer lam h (fun j => likelihood law (nv.2 j)) *
-        f (combineMessage lam h (fun j => likelihood law (nv.2 j)))) := by
-  have hs : Summable (fun nv => mass (marginal (composeLaw offspring law lam h)) nv *
-      f (likelihood (composeLaw offspring law lam h) nv)) :=
-    hasMean_bounded (marginal (composeLaw offspring law lam h)) C
-      (fun nv => hf (likelihood (composeLaw offspring law lam h) nv))
-  exact hs.congr (fun ⟨n,v⟩ => weighted_test_composeLaw offspring law lam h f n v)
-
-lemma mean_test_compose {α : Type*} (offspring : PMF ℕ) (law : Spin → PMF α)
-    (lam : ℝ) (h : Admissible lam) (f : Message → ℝ) :
-    mean (marginal (composeLaw offspring law lam h)) (fun nv => f (likelihood (composeLaw offspring law lam h) nv)) =
-    mean (productReference offspring law) (fun nv =>
-      normalizer lam h (fun j => likelihood law (nv.2 j)) *
-        f (combineMessage lam h (fun j => likelihood law (nv.2 j)))) := by
-  apply tsum_congr
-  rintro ⟨n,v⟩
-  exact weighted_test_composeLaw offspring law lam h f n v
-
-end ThreeState
-namespace ThreeState
-open MeasureTheory
-open scoped Classical
-
-lemma iidVector_outer_pi {α : Type*} (p : PMF α) (n : ℕ) (s : Fin n → Set α) :
-    (iidVector p n).toOuterMeasure (Set.univ.pi s) = ∏ j, p.toOuterMeasure (s j) := by
-  induction n with
-  | zero =>
-    simp [iidVector]
-    funext i
-    exact Fin.elim0 i
-  | succ n ih =>
-    rw [iidVector, PMF.toOuterMeasure_bind_apply]
-    simp only [PMF.toOuterMeasure_map_apply]
-    have he (a : α) : Fin.cons a ⁻¹' (Set.univ.pi s) =
-        if a ∈ s 0 then Set.univ.pi (fun j : Fin n => s j.succ) else ∅ := by
-      ext v
-      simp only [Set.mem_preimage, Set.mem_pi, Set.mem_univ, forall_true_left]
-      rw [Fin.forall_fin_succ]
-      by_cases ha : a ∈ s 0 <;> simp [ha]
-    simp_rw [he]
-    have hh (a : α) : p a * (iidVector p n).toOuterMeasure
-        (if a ∈ s 0 then Set.univ.pi (fun j : Fin n => s j.succ) else ∅) =
-        (s 0).indicator p a * ∏ j : Fin n, p.toOuterMeasure (s j.succ) := by
-      by_cases ha : a ∈ s 0 <;> simp [ha, ih]
-    simp_rw [hh]
-    rw [ENNReal.tsum_mul_right, ← PMF.toOuterMeasure_apply, Fin.prod_univ_succ]
-
-lemma iidVector_toMeasure {α : Type*} [MeasurableSpace α] (p : PMF α) (n : ℕ) :
-    (iidVector p n).toMeasure = Measure.pi (fun _ : Fin n => p.toMeasure) := by
-  symm
-  apply Measure.pi_eq
-  intro s hs
-  rw [PMF.toMeasure_apply_eq_toOuterMeasure_apply _ (MeasurableSet.univ_pi hs), iidVector_outer_pi]
-  congr 1
-  funext j
-  exact (PMF.toMeasure_apply_eq_toOuterMeasure_apply p (hs j)).symm
-
-noncomputable def iidMeasure (Q : ProbabilityMeasure Message) (n : ℕ) :
-    ProbabilityMeasure (Fin n → Message) := ProbabilityMeasure.pi (fun _ => Q)
-
-lemma continuous_iidMeasure (n : ℕ) : Continuous (fun Q => iidMeasure Q n) :=
-  ProbabilityMeasure.continuous_pi.comp (continuous_pi (fun _ => continuous_id))
-
-lemma likelihoodPMF_iid_toMeasure {α : Type*} (law : Spin → PMF α) (n : ℕ) :
-    (iidVector (likelihoodPMF law) n).toMeasure =
-      (iidMeasure (likelihoodLaw law) n : Measure (Fin n → Message)) := by
-  exact iidVector_toMeasure _ n
-
-lemma integral_iid_likelihood {α : Type*} (law : Spin → PMF α) (n : ℕ)
-    (f : C((Fin n → Message),ℝ)) :
-    ∫ m, f m ∂(iidMeasure (likelihoodLaw law) n : Measure (Fin n → Message)) =
-      mean (iidVector (marginal law) n) (fun v => f (fun j => likelihood law (v j))) := by
-  rw [← likelihoodPMF_iid_toMeasure]
-  rw [PMF.integral_eq_tsum _ _
-    (f.continuous.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _))]
-  change mean (iidVector ((marginal law).map (likelihood law)) n) f = _
-  rw [← iidVector_map]
-  obtain ⟨C,hC⟩ := isCompact_univ.exists_bound_of_continuousOn f.continuous.continuousOn
-  exact mean_map_bounded _ _ _ C (fun v => by simpa only [Real.norm_eq_abs] using hC _ (Set.mem_univ _))
-
-end ThreeState
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeState.Observation
+
+namespace OAI
+
+/-! Branchwise likelihood updates and their independent product measures. -/
+
+namespace ThreeState
+open MeasureTheory
+open scoped Classical
+
+noncomputable def edgeMessage (lam : ℝ) (h : Admissible lam) (m : Message) : Message :=
+  ⟨fun i => 1+lam*(m i-1), by
+    constructor
+    · intro i
+      have hl := Message.nonneg m i
+      have hu := Message.le_three m i
+      by_cases hp : 0 ≤ lam
+      · nlinarith [h.2]
+      · have hn : lam ≤ 0 := le_of_not_ge hp
+        nlinarith [h.1]
+    · simp only [Finset.sum_add_distrib, ← Finset.mul_sum, Finset.sum_sub_distrib,
+        Message.sum, Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul]
+      norm_num⟩
+
+lemma continuous_edgeMessage (lam : ℝ) (h : Admissible lam) : Continuous (edgeMessage lam h) := by
+  apply Continuous.subtype_mk
+  apply continuous_pi
+  intro i
+  exact continuous_const.add (continuous_const.mul (((continuous_apply i).comp continuous_subtype_val).sub continuous_const))
+
+lemma channel_decompose (lam : ℝ) (i j : Spin) :
+    channel lam i j = (1-lam)/3 + if i=j then lam else 0 := by
+  unfold channel
+  split_ifs <;> ring
+
+lemma channel_message (lam : ℝ) (m : Message) (i : Spin) :
+    ∑ j, channel lam i j * m j = 1+lam*(m i-1) := by
+  simp_rw [channel_decompose, add_mul, ite_mul, zero_mul]
+  rw [Finset.sum_add_distrib, ← Finset.mul_sum, Message.sum]
+  simp only [Finset.sum_ite_eq, Finset.mem_univ, ite_true]
+  ring
+
+lemma mass_channelPMF (lam : ℝ) (h : Admissible lam) (i j : Spin) :
+    mass (channelPMF lam h i) j = channel lam i j := by
+  simp [mass, channelPMF, PMF.ofFintype_apply, ENNReal.toReal_ofReal (channel_nonneg h i j)]
+
+noncomputable def edgeLaw {α : Type*} (law : Spin → PMF α) (lam : ℝ) (h : Admissible lam) : Spin → PMF α :=
+  fun i => (channelPMF lam h i).bind law
+
+lemma mass_edgeLaw {α : Type*} (law : Spin → PMF α) (lam : ℝ) (h : Admissible lam) (i : Spin) (y : α) :
+    mass (edgeLaw law lam h i) y = mass (marginal law) y * edgeMessage lam h (likelihood law y) i := by
+  rw [edgeLaw, mass_bind, tsum_fintype]
+  simp_rw [mass_channelPMF, ← weighted_likelihood law y]
+  calc
+    (∑ j, channel lam i j * (mass (marginal law) y * likelihood law y j)) =
+      mass (marginal law) y * (∑ j, channel lam i j * likelihood law y j) := by
+        rw [Finset.mul_sum]
+        congr 1
+        funext j
+        ring
+    _ = _ := by rw [channel_message]; rfl
+
+noncomputable def sigmaPMF {ι : Type*} {α : ι → Type*} (p : PMF ι) (q : ∀ i, PMF (α i)) : PMF (Sigma α) :=
+  p.bind (fun i => (q i).map (Sigma.mk i))
+
+lemma sigmaPMF_apply {ι : Type*} {α : ι → Type*} (p : PMF ι) (q : ∀ i, PMF (α i))
+    (i : ι) (a : α i) : sigmaPMF p q ⟨i,a⟩ = p i * q i a := by
+  classical
+  rw [sigmaPMF, PMF.bind_apply, tsum_eq_single i]
+  · simp [PMF.map_apply]
+  · intro j hj
+    have he (b : α j) : (⟨i,a⟩ : Sigma α) ≠ ⟨j,b⟩ := by
+      intro hh
+      exact hj (congrArg Sigma.fst hh).symm
+    simp [PMF.map_apply, he]
+
+lemma mass_sigmaPMF {ι : Type*} {α : ι → Type*} (p : PMF ι) (q : ∀ i, PMF (α i))
+    (i : ι) (a : α i) : mass (sigmaPMF p q) ⟨i,a⟩ = mass p i * mass (q i) a := by
+  simp only [mass, sigmaPMF_apply, ENNReal.toReal_mul]
+
+lemma mass_iidVector {α : Type*} (p : PMF α) (n : ℕ) (v : Fin n → α) :
+    mass (iidVector p n) v = ∏ j, mass p (v j) := by
+  simp only [mass, iidVector_apply, ENNReal.toReal_prod]
+
+ 
+noncomputable def productReference {α : Type*} (offspring : PMF ℕ) (law : Spin → PMF α) :
+    PMF (Σ n : ℕ, Fin n → α) := sigmaPMF offspring (iidVector (marginal law))
+
+noncomputable def composeLaw {α : Type*} (offspring : PMF ℕ) (law : Spin → PMF α)
+    (lam : ℝ) (h : Admissible lam) : Spin → PMF (Σ n : ℕ, Fin n → α) :=
+  fun i => sigmaPMF offspring (iidVector (edgeLaw law lam h i))
+
+noncomputable def productWeight (lam : ℝ) (h : Admissible lam) {n : ℕ} (m : Fin n → Message) (i : Spin) : ℝ :=
+  ∏ j, edgeMessage lam h (m j) i
+
+noncomputable def normalizer (lam : ℝ) (h : Admissible lam) {n : ℕ} (m : Fin n → Message) : ℝ :=
+  (∑ i : Spin, productWeight lam h m i)/3
+
+lemma productWeight_nonneg (lam : ℝ) (h : Admissible lam) {n : ℕ} (m : Fin n → Message) (i : Spin) :
+    0 ≤ productWeight lam h m i :=
+  Finset.prod_nonneg (fun index _ => Message.nonneg (edgeMessage lam h (m index)) i)
+
+lemma normalizer_nonneg (lam : ℝ) (h : Admissible lam) {n : ℕ} (m : Fin n → Message) :
+    0 ≤ normalizer lam h m := div_nonneg (Finset.sum_nonneg (fun i _ => productWeight_nonneg _ _ _ _)) (by norm_num)
+
+lemma productWeight_sum (lam : ℝ) (h : Admissible lam) {n : ℕ} (m : Fin n → Message) :
+    ∑ i, productWeight lam h m i = 3*normalizer lam h m := by dsimp [normalizer]; ring
+
+noncomputable def combineMessage (lam : ℝ) (h : Admissible lam) {n : ℕ} (m : Fin n → Message) : Message :=
+  if hz : normalizer lam h m = 0 then uninformative else
+    ⟨fun i => productWeight lam h m i / normalizer lam h m, by
+      constructor
+      · intro i
+        exact div_nonneg (productWeight_nonneg _ _ _ _) (normalizer_nonneg _ _ _)
+      · rw [← Finset.sum_div, productWeight_sum]
+        field_simp⟩
+
+lemma mass_composeLaw {α : Type*} (offspring : PMF ℕ) (law : Spin → PMF α)
+    (lam : ℝ) (h : Admissible lam) (n : ℕ) (v : Fin n → α) (i : Spin) :
+    mass (composeLaw offspring law lam h i) ⟨n,v⟩ =
+      mass (productReference offspring law) ⟨n,v⟩ * productWeight lam h (fun j => likelihood law (v j)) i := by
+  simp only [composeLaw, productReference, mass_sigmaPMF, mass_iidVector, mass_edgeLaw,
+    Finset.prod_mul_distrib, productWeight, mul_assoc]
+
+lemma mass_marginal_composeLaw {α : Type*} (offspring : PMF ℕ) (law : Spin → PMF α)
+    (lam : ℝ) (h : Admissible lam) (n : ℕ) (v : Fin n → α) :
+    mass (marginal (composeLaw offspring law lam h)) ⟨n,v⟩ =
+      mass (productReference offspring law) ⟨n,v⟩ * normalizer lam h (fun j => likelihood law (v j)) := by
+  rw [mass_marginal]
+  simp_rw [mass_composeLaw]
+  rw [← Finset.mul_sum]
+  simp [normalizer]
+  ring
+
+lemma weighted_combineMessage (lam : ℝ) (h : Admissible lam) {n : ℕ}
+    (m : Fin n → Message) (i : Spin) :
+    normalizer lam h m * combineMessage lam h m i = productWeight lam h m i := by
+  by_cases hz : normalizer lam h m = 0
+  · have hs := productWeight_sum lam h m
+    have hi := Finset.single_le_sum (fun j _ => productWeight_nonneg lam h m j) (Finset.mem_univ i)
+    have hn := productWeight_nonneg lam h m i
+    simp only [hz, mul_zero] at hs
+    have hw : productWeight lam h m i = 0 := le_antisymm (by simpa [hs] using hi) hn
+    simp [hz, hw]
+  · simp only [combineMessage, dite_eq_right hz]
+    field_simp
+
+lemma likelihood_composeLaw {α : Type*} (offspring : PMF ℕ) (law : Spin → PMF α)
+    (lam : ℝ) (h : Admissible lam) (n : ℕ) (v : Fin n → α)
+    (hm : mass (marginal (composeLaw offspring law lam h)) ⟨n,v⟩ ≠ 0) :
+    likelihood (composeLaw offspring law lam h) ⟨n,v⟩ =
+      combineMessage lam h (fun j => likelihood law (v j)) := by
+  apply Subtype.ext
+  funext i
+  apply mul_left_cancel₀ hm
+  rw [weighted_likelihood, mass_composeLaw, mass_marginal_composeLaw, mul_assoc,
+    weighted_combineMessage]
+
+lemma weighted_test_composeLaw {α : Type*} (offspring : PMF ℕ) (law : Spin → PMF α)
+    (lam : ℝ) (h : Admissible lam) (f : Message → ℝ) (n : ℕ) (v : Fin n → α) :
+    mass (marginal (composeLaw offspring law lam h)) ⟨n,v⟩ *
+      f (likelihood (composeLaw offspring law lam h) ⟨n,v⟩) =
+    mass (productReference offspring law) ⟨n,v⟩ *
+      (normalizer lam h (fun j => likelihood law (v j)) *
+        f (combineMessage lam h (fun j => likelihood law (v j)))) := by
+  by_cases hm : mass (marginal (composeLaw offspring law lam h)) ⟨n,v⟩ = 0
+  · rw [mass_marginal_composeLaw] at hm
+    rw [← mul_assoc, ← mass_marginal_composeLaw, mass_marginal_composeLaw, hm]
+    simp
+  · rw [likelihood_composeLaw offspring law lam h n v hm, mass_marginal_composeLaw, mul_assoc]
+
+lemma hasMean_weighted_compose {α : Type*} (offspring : PMF ℕ) (law : Spin → PMF α)
+    (lam : ℝ) (h : Admissible lam) (f : Message → ℝ) (C : ℝ) (hf : ∀ m, |f m| ≤ C) :
+    HasMean (productReference offspring law) (fun nv =>
+      normalizer lam h (fun j => likelihood law (nv.2 j)) *
+        f (combineMessage lam h (fun j => likelihood law (nv.2 j)))) := by
+  have hs : Summable (fun nv => mass (marginal (composeLaw offspring law lam h)) nv *
+      f (likelihood (composeLaw offspring law lam h) nv)) :=
+    hasMean_bounded (marginal (composeLaw offspring law lam h)) C
+      (fun nv => hf (likelihood (composeLaw offspring law lam h) nv))
+  exact hs.congr (fun ⟨n,v⟩ => weighted_test_composeLaw offspring law lam h f n v)
+
+lemma mean_test_compose {α : Type*} (offspring : PMF ℕ) (law : Spin → PMF α)
+    (lam : ℝ) (h : Admissible lam) (f : Message → ℝ) :
+    mean (marginal (composeLaw offspring law lam h)) (fun nv => f (likelihood (composeLaw offspring law lam h) nv)) =
+    mean (productReference offspring law) (fun nv =>
+      normalizer lam h (fun j => likelihood law (nv.2 j)) *
+        f (combineMessage lam h (fun j => likelihood law (nv.2 j)))) := by
+  apply tsum_congr
+  rintro ⟨n,v⟩
+  exact weighted_test_composeLaw offspring law lam h f n v
+
+end ThreeState
+namespace ThreeState
+open MeasureTheory
+open scoped Classical
+
+lemma iidVector_outer_pi {α : Type*} (p : PMF α) (n : ℕ) (s : Fin n → Set α) :
+    (iidVector p n).toOuterMeasure (Set.univ.pi s) = ∏ j, p.toOuterMeasure (s j) := by
+  induction n with
+  | zero =>
+    simp [iidVector]
+    funext i
+    exact Fin.elim0 i
+  | succ n ih =>
+    rw [iidVector, PMF.toOuterMeasure_bind_apply]
+    simp only [PMF.toOuterMeasure_map_apply]
+    have he (a : α) : Fin.cons a ⁻¹' (Set.univ.pi s) =
+        if a ∈ s 0 then Set.univ.pi (fun j : Fin n => s j.succ) else ∅ := by
+      ext v
+      simp only [Set.mem_preimage, Set.mem_pi, Set.mem_univ, forall_true_left]
+      rw [Fin.forall_fin_succ]
+      by_cases ha : a ∈ s 0 <;> simp [ha]
+    simp_rw [he]
+    have hh (a : α) : p a * (iidVector p n).toOuterMeasure
+        (if a ∈ s 0 then Set.univ.pi (fun j : Fin n => s j.succ) else ∅) =
+        (s 0).indicator p a * ∏ j : Fin n, p.toOuterMeasure (s j.succ) := by
+      by_cases ha : a ∈ s 0 <;> simp [ha, ih]
+    simp_rw [hh]
+    rw [ENNReal.tsum_mul_right, ← PMF.toOuterMeasure_apply, Fin.prod_univ_succ]
+
+lemma iidVector_toMeasure {α : Type*} [MeasurableSpace α] (p : PMF α) (n : ℕ) :
+    (iidVector p n).toMeasure = Measure.pi (fun _ : Fin n => p.toMeasure) := by
+  symm
+  apply Measure.pi_eq
+  intro s hs
+  rw [PMF.toMeasure_apply_eq_toOuterMeasure_apply _ (MeasurableSet.univ_pi hs), iidVector_outer_pi]
+  congr 1
+  funext j
+  exact (PMF.toMeasure_apply_eq_toOuterMeasure_apply p (hs j)).symm
+
+noncomputable def iidMeasure (Q : ProbabilityMeasure Message) (n : ℕ) :
+    ProbabilityMeasure (Fin n → Message) := ProbabilityMeasure.pi (fun _ => Q)
+
+lemma continuous_iidMeasure (n : ℕ) : Continuous (fun Q => iidMeasure Q n) :=
+  ProbabilityMeasure.continuous_pi.comp (continuous_pi (fun _ => continuous_id))
+
+lemma likelihoodPMF_iid_toMeasure {α : Type*} (law : Spin → PMF α) (n : ℕ) :
+    (iidVector (likelihoodPMF law) n).toMeasure =
+      (iidMeasure (likelihoodLaw law) n : Measure (Fin n → Message)) := by
+  exact iidVector_toMeasure _ n
+
+lemma integral_iid_likelihood {α : Type*} (law : Spin → PMF α) (n : ℕ)
+    (f : C((Fin n → Message),ℝ)) :
+    ∫ m, f m ∂(iidMeasure (likelihoodLaw law) n : Measure (Fin n → Message)) =
+      mean (iidVector (marginal law) n) (fun v => f (fun j => likelihood law (v j))) := by
+  rw [← likelihoodPMF_iid_toMeasure]
+  rw [PMF.integral_eq_tsum _ _
+    (f.continuous.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _))]
+  change mean (iidVector ((marginal law).map (likelihood law)) n) f = _
+  rw [← iidVector_map]
+  obtain ⟨C,hC⟩ := isCompact_univ.exists_bound_of_continuousOn f.continuous.continuousOn
+  exact mean_map_bounded _ _ _ C (fun v => by simpa only [Real.norm_eq_abs] using hC _ (Set.mem_univ _))
+
+end ThreeState
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeStateClauses/Basic.lean b/lean/OAI/Probability/ThreeStateClauses/Basic.lean
index 5d2615d8c018606c98b38a64f8ed67413488f5f4..d863af141acb1a9183db6999caf069db71c69725 100644
--- a/lean/OAI/Probability/ThreeStateClauses/Basic.lean
+++ b/lean/OAI/Probability/ThreeStateClauses/Basic.lean
@@ -1,89 +1,89 @@
-import Mathlib
-
-namespace OAI
-
-/-! Basic. -/
-
-open scoped BigOperators ENNReal NNReal Topology
-open Filter
-noncomputable section
-open scoped BigOperators ENNReal NNReal
-open Filter
-
-namespace ThreeState.TreeClauses
-
-abbrev Spin := Fin 3
-
-def Admissible (lam : ℝ) : Prop := -(1 / 2 : ℝ) ≤ lam ∧ lam ≤ 1
-
-def channelWeight (lam : ℝ) (i j : Spin) : ℝ :=
-  if i = j then (1 + 2 * lam) / 3 else (1 - lam) / 3
-
-lemma channelWeight_nonneg {lam : ℝ} (hlam : Admissible lam) (i j : Spin) :
-    0 ≤ channelWeight lam i j := by
-  unfold channelWeight
-  split_ifs <;> dsimp [Admissible] at hlam <;> linarith [hlam.1, hlam.2]
-
-lemma channelWeight_sum (lam : ℝ) (i : Spin) : ∑ j, channelWeight lam i j = 1 := by
-  fin_cases i <;> simp [channelWeight, Fin.sum_univ_three] <;> ring
-
-def channel (lam : ℝ) (hlam : Admissible lam) (i : Spin) : PMF Spin :=
-  PMF.ofFintype (fun j ↦ ENNReal.ofReal (channelWeight lam i j)) (by
-    rw [← ENNReal.ofReal_sum_of_nonneg (fun j _ ↦ channelWeight_nonneg hlam i j),
-      channelWeight_sum]
-    norm_num)
-
-def Observation : ℕ → Type
-  | 0 => Spin
-  | n + 1 => Multiset (Observation n)
-
-def iidList {α : Type} (p : PMF α) : ℕ → PMF (List α)
-  | 0 => PMF.pure []
-  | n + 1 => p.bind fun x ↦ (iidList p n).map (List.cons x)
-
-def observationLaw (lam : ℝ) (hlam : Admissible lam) (offspring : PMF ℕ) :
-    (ℓ : ℕ) → Spin → PMF (Observation ℓ)
-  | 0, i => PMF.pure i
-  | ℓ + 1, i => offspring.bind fun n ↦
-      (iidList ((channel lam hlam i).bind (observationLaw lam hlam offspring ℓ)) n).map
-        (fun xs ↦ (xs : Multiset (Observation ℓ)))
-
-def jointWeight (lam : ℝ) (hlam : Admissible lam) (offspring : PMF ℕ)
-    (ℓ : ℕ) (i : Spin) (o : Observation ℓ) : ℝ≥0∞ :=
-  (1 / 3 : ℝ≥0∞) * observationLaw lam hlam offspring ℓ i o
-
-def marginalWeight (lam : ℝ) (hlam : Admissible lam) (offspring : PMF ℕ)
-    (ℓ : ℕ) (o : Observation ℓ) : ℝ≥0∞ :=
-  ∑ i, jointWeight lam hlam offspring ℓ i o
-
-def posterior (lam : ℝ) (hlam : Admissible lam) (offspring : PMF ℕ)
-    (ℓ : ℕ) (o : Observation ℓ) (i : Spin) : ℝ :=
-  (jointWeight lam hlam offspring ℓ i o / marginalWeight lam hlam offspring ℓ o).toReal
-
-def advantage (lam : ℝ) (hlam : Admissible lam) (offspring : PMF ℕ) (ℓ : ℕ) : ℝ :=
-  ∑' o : Observation ℓ, (marginalWeight lam hlam offspring ℓ o).toReal *
-    ((∑ i : Spin, |posterior lam hlam offspring ℓ o i - (1 / 3 : ℝ)|) / 2)
-
-def Reconstructs (lam : ℝ) (hlam : Admissible lam) (offspring : PMF ℕ) : Prop :=
-  ∃ a : ℝ, 0 < a ∧ Tendsto (advantage lam hlam offspring) atTop (nhds a)
-
-def ThresholdAt (d lam : ℝ) (hlam : Admissible lam) (offspring : PMF ℕ) : Prop :=
-  (Reconstructs lam hlam offspring ↔ 1 < d * lam ^ 2) ∧
-  (d * lam ^ 2 ≤ 1 → Tendsto (advantage lam hlam offspring) atTop (nhds 0))
-
-def ExactThreshold : Prop :=
-  (∀ (b : ℕ), 2 ≤ b → ∀ (lam : ℝ) (hlam : Admissible lam),
-    ThresholdAt (b : ℝ) lam hlam (PMF.pure b)) ∧
-  (∀ (d : ℝ) (hd : 1 < d) (lam : ℝ) (hlam : Admissible lam),
-    ThresholdAt d lam hlam (by
-      let rate : ℝ≥0 := ⟨d, le_trans (by norm_num) hd.le⟩
-      haveI : MeasureTheory.IsProbabilityMeasure (ProbabilityTheory.poissonMeasure rate) :=
-        (ProbabilityTheory.hasSum_one_poissonMeasure rate).isProbabilityMeasure_sum_dirac
-          (fun _ ↦ by positivity)
-      exact (ProbabilityTheory.poissonMeasure rate).toPMF))
-
-end ThreeState.TreeClauses
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-! Basic. -/
+
+open scoped BigOperators ENNReal NNReal Topology
+open Filter
+noncomputable section
+open scoped BigOperators ENNReal NNReal
+open Filter
+
+namespace ThreeState.TreeClauses
+
+abbrev Spin := Fin 3
+
+def Admissible (lam : ℝ) : Prop := -(1 / 2 : ℝ) ≤ lam ∧ lam ≤ 1
+
+def channelWeight (lam : ℝ) (i j : Spin) : ℝ :=
+  if i = j then (1 + 2 * lam) / 3 else (1 - lam) / 3
+
+lemma channelWeight_nonneg {lam : ℝ} (hlam : Admissible lam) (i j : Spin) :
+    0 ≤ channelWeight lam i j := by
+  unfold channelWeight
+  split_ifs <;> dsimp [Admissible] at hlam <;> linarith [hlam.1, hlam.2]
+
+lemma channelWeight_sum (lam : ℝ) (i : Spin) : ∑ j, channelWeight lam i j = 1 := by
+  fin_cases i <;> simp [channelWeight, Fin.sum_univ_three] <;> ring
+
+def channel (lam : ℝ) (hlam : Admissible lam) (i : Spin) : PMF Spin :=
+  PMF.ofFintype (fun j ↦ ENNReal.ofReal (channelWeight lam i j)) (by
+    rw [← ENNReal.ofReal_sum_of_nonneg (fun j _ ↦ channelWeight_nonneg hlam i j),
+      channelWeight_sum]
+    norm_num)
+
+def Observation : ℕ → Type
+  | 0 => Spin
+  | n + 1 => Multiset (Observation n)
+
+def iidList {α : Type} (p : PMF α) : ℕ → PMF (List α)
+  | 0 => PMF.pure []
+  | n + 1 => p.bind fun x ↦ (iidList p n).map (List.cons x)
+
+def observationLaw (lam : ℝ) (hlam : Admissible lam) (offspring : PMF ℕ) :
+    (ℓ : ℕ) → Spin → PMF (Observation ℓ)
+  | 0, i => PMF.pure i
+  | ℓ + 1, i => offspring.bind fun n ↦
+      (iidList ((channel lam hlam i).bind (observationLaw lam hlam offspring ℓ)) n).map
+        (fun xs ↦ (xs : Multiset (Observation ℓ)))
+
+def jointWeight (lam : ℝ) (hlam : Admissible lam) (offspring : PMF ℕ)
+    (ℓ : ℕ) (i : Spin) (o : Observation ℓ) : ℝ≥0∞ :=
+  (1 / 3 : ℝ≥0∞) * observationLaw lam hlam offspring ℓ i o
+
+def marginalWeight (lam : ℝ) (hlam : Admissible lam) (offspring : PMF ℕ)
+    (ℓ : ℕ) (o : Observation ℓ) : ℝ≥0∞ :=
+  ∑ i, jointWeight lam hlam offspring ℓ i o
+
+def posterior (lam : ℝ) (hlam : Admissible lam) (offspring : PMF ℕ)
+    (ℓ : ℕ) (o : Observation ℓ) (i : Spin) : ℝ :=
+  (jointWeight lam hlam offspring ℓ i o / marginalWeight lam hlam offspring ℓ o).toReal
+
+def advantage (lam : ℝ) (hlam : Admissible lam) (offspring : PMF ℕ) (ℓ : ℕ) : ℝ :=
+  ∑' o : Observation ℓ, (marginalWeight lam hlam offspring ℓ o).toReal *
+    ((∑ i : Spin, |posterior lam hlam offspring ℓ o i - (1 / 3 : ℝ)|) / 2)
+
+def Reconstructs (lam : ℝ) (hlam : Admissible lam) (offspring : PMF ℕ) : Prop :=
+  ∃ a : ℝ, 0 < a ∧ Tendsto (advantage lam hlam offspring) atTop (nhds a)
+
+def ThresholdAt (d lam : ℝ) (hlam : Admissible lam) (offspring : PMF ℕ) : Prop :=
+  (Reconstructs lam hlam offspring ↔ 1 < d * lam ^ 2) ∧
+  (d * lam ^ 2 ≤ 1 → Tendsto (advantage lam hlam offspring) atTop (nhds 0))
+
+def ExactThreshold : Prop :=
+  (∀ (b : ℕ), 2 ≤ b → ∀ (lam : ℝ) (hlam : Admissible lam),
+    ThresholdAt (b : ℝ) lam hlam (PMF.pure b)) ∧
+  (∀ (d : ℝ) (hd : 1 < d) (lam : ℝ) (hlam : Admissible lam),
+    ThresholdAt d lam hlam (by
+      let rate : ℝ≥0 := ⟨d, le_trans (by norm_num) hd.le⟩
+      haveI : MeasureTheory.IsProbabilityMeasure (ProbabilityTheory.poissonMeasure rate) :=
+        (ProbabilityTheory.hasSum_one_poissonMeasure rate).isProbabilityMeasure_sum_dirac
+          (fun _ ↦ by positivity)
+      exact (ProbabilityTheory.poissonMeasure rate).toPMF))
+
+end ThreeState.TreeClauses
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeStateClauses/BranchGrowth.lean b/lean/OAI/Probability/ThreeStateClauses/BranchGrowth.lean
index de5615470138ca380329c19bb8330c21db25f17d..7ba0f858fac5c335edbd598e44cacb7d9fcf2511 100644
--- a/lean/OAI/Probability/ThreeStateClauses/BranchGrowth.lean
+++ b/lean/OAI/Probability/ThreeStateClauses/BranchGrowth.lean
@@ -1,211 +1,211 @@
-import Mathlib
-import OAI.Probability.ThreeStateClauses.Projection
-
-namespace OAI
-
-/-! Branch Growth. -/
-
-open scoped BigOperators ENNReal NNReal Topology
-open Filter
-noncomputable section
-open Set MeasureTheory
-open scoped BigOperators
-namespace ThreeState.TreeClauses.Experiment
-open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
-
-lemma continuous_compact_integral {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
-    [MeasurableSpace Y] [BorelSpace Y] [CompactSpace Y] {μ : Measure Y} [IsFiniteMeasure μ]
-    {f : X × Y → ℝ} (hf : Continuous f) : Continuous (fun x ↦ ∫ y, f (x,y) ∂μ) := by
-  apply continuousOn_univ.mp
-  apply continuousOn_integral_of_compact_support (k := (univ : Set Y)) isCompact_univ
-  · exact hf.continuousOn
-  · simp
-
-lemma continuous_cons (n : ℕ) : Continuous (fun p : Message × (Fin n → Message) ↦ (Fin.cons p.1 p.2 : Fin (n+1) → Message)) := by
-  apply continuous_pi
-  intro j
-  refine Fin.cases ?_ (fun k ↦ ?_) j
-  · simpa using (continuous_fst : Continuous (fun p : Message × (Fin n → Message) ↦ p.1))
-  · change Continuous (fun p : Message × (Fin n → Message) ↦ p.2 k)
-    fun_prop
-
-lemma integral_pi_cons (Q : Law) (n : ℕ) {f : (Fin (n+1) → Message) → ℝ} (hf : Continuous f) :
-    (∫ m, f m ∂(Measure.pi fun _ : Fin (n+1) ↦ Q.probability.toMeasure)) =
-      ∫ m : Fin n → Message, (∫ e, f (Fin.cons e m) ∂Q.probability.toMeasure)
-        ∂(Measure.pi fun _ ↦ Q.probability.toMeasure) := by
-  have he : (∫ m, f m ∂(Measure.pi fun _ : Fin (n+1) ↦ Q.probability.toMeasure)) =
-      ∫ p : Message × (Fin n → Message), f (Fin.cons p.1 p.2)
-        ∂Q.probability.toMeasure.prod (Measure.pi fun _ ↦ Q.probability.toMeasure) := by
-    rw [← (measurePreserving_piFinSuccAbove (fun _ : Fin (n+1) ↦ Q.probability.toMeasure) 0).symm.integral_comp']
-    simp only [MeasurableEquiv.piFinSuccAbove_symm_apply, Fin.insertNthEquiv,
-      Fin.insertNth_zero, Equiv.coe_fn_mk, Fin.zero_succAbove, cast_eq]
-  rw [he]
-  exact integral_prod_symm _ (compact_integrable (hf.comp (continuous_cons n)))
-
-lemma branchProduct_cons (lam : ℝ) {n : ℕ} (e : Message) (m : Fin n → Message) (i : Fin 3) :
-    branchProduct (branchEdges lam (Fin.cons e m)) i = edgeMessage lam e i*branchProduct (branchEdges lam m) i := by
-  simp [branchProduct, branchEdges, Fin.prod_univ_succ]
-
-lemma normalizer_cons {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) {n : ℕ} (e : Message) (m : Fin n → Message) :
-    normalizer (branchEdges lam (Fin.cons e m)) =
-      normalizer (branchEdges lam m)*pairZ lam (branchOutput hl₀ hl₁ m) e := by
-  unfold normalizer pairZ
-  change avg (fun i ↦ branchProduct (branchEdges lam (Fin.cons e m)) i) = _
-  simp_rw [branchProduct_cons]
-  change avg (fun i ↦ edgeMessage lam e i*branchProduct (branchEdges lam m) i) =
-    avg (branchProduct (branchEdges lam m))*avg (fun i ↦
-      (branchProduct (branchEdges lam m) i/normalizer (branchEdges lam m))*edgeMessage lam e i)
-  have hz := ne_of_gt (normalizer_positive (branchEdges_positive hl₀ hl₁ m))
-  change normalizer (branchEdges lam m) ≠ 0 at hz
-  have hid : avg (branchProduct (branchEdges lam m)) = normalizer (branchEdges lam m) := rfl
-  rw [hid]
-  dsimp only [avg]
-  field_simp [hz]
-
-lemma branchOutput_cons {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) {n : ℕ} (e : Message) (m : Fin n → Message) :
-    branchOutput hl₀ hl₁ (Fin.cons e m) = pairPosterior hl₀ hl₁ (branchOutput hl₀ hl₁ m) e := by
-  apply Subtype.ext
-  funext i
-  change branchProduct (branchEdges lam (Fin.cons e m)) i/normalizer (branchEdges lam (Fin.cons e m)) =
-    (branchProduct (branchEdges lam m) i/normalizer (branchEdges lam m))*edgeMessage lam e i/
-      pairZ lam (branchOutput hl₀ hl₁ m) e
-  rw [branchProduct_cons, normalizer_cons hl₀ hl₁]
-  ring
-
-lemma finite_succ_integral (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ)
-    {f : Message → ℝ} (hf : Continuous f) :
-    (∫ m, f m ∂(finiteProbability Q hl₀ hl₁ (n+1)).toMeasure) =
-      ∫ r, (∫ m, pairZ lam r m*f (pairPosterior hl₀ hl₁ r m) ∂Q.probability.toMeasure)
-        ∂(finiteProbability Q hl₀ hl₁ n).toMeasure := by
-  have hk : Continuous (fun p : Message × Message ↦ pairZ lam p.1 p.2*f (pairPosterior hl₀ hl₁ p.1 p.2)) :=
-    (continuous_pairZ lam).mul (hf.comp (continuous_pairPosterior hl₀ hl₁))
-  rw [integral_finiteProbability Q hl₀ hl₁ (n+1) hf.measurable,
-    integral_pi_cons Q n (f := fun m ↦ normalizer (branchEdges lam m)*f (branchOutput hl₀ hl₁ m)) ((continuous_normalizer (n+1) lam).mul (hf.comp (continuous_branchOutput hl₀ hl₁ (n+1)))),
-    integral_finiteProbability Q hl₀ hl₁ n (continuous_compact_integral hk).measurable]
-  congr 1
-  funext m
-  simp_rw [normalizer_cons hl₀ hl₁, branchOutput_cons hl₀ hl₁, mul_assoc, integral_const_mul]
-
-theorem correlationEntropy_succ (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
-    correlationEntropy Q lam (n+1) = correlationEntropy Q lam n + pairCorrelation (finiteLaw Q hl₀ hl₁ n) Q lam := by
-  let c : Message → ℝ := fun r ↦ ∫ e, pairZ lam r e*Real.log (pairZ lam r e) ∂Q.probability.toMeasure
-  have hc : Continuous c := continuous_compact_integral (continuous_pairEntropy hl₀ hl₁)
-  have he (m : Fin n → Message) :
-      (∫ e, normalizer (branchEdges lam (Fin.cons e m))*Real.log (normalizer (branchEdges lam (Fin.cons e m)))
-        ∂Q.probability.toMeasure) =
-      normalizer (branchEdges lam m)*Real.log (normalizer (branchEdges lam m))+
-      normalizer (branchEdges lam m)*c (branchOutput hl₀ hl₁ m) := by
-    have hp := normalizer_positive (branchEdges_positive hl₀ hl₁ m)
-    have hpp (e : Message) := pairZ_positive hl₀ hl₁ (branchOutput hl₀ hl₁ m) e
-    have hid (e : Message) :
-        normalizer (branchEdges lam (Fin.cons e m))*Real.log (normalizer (branchEdges lam (Fin.cons e m))) =
-        (normalizer (branchEdges lam m)*Real.log (normalizer (branchEdges lam m)))*pairZ lam (branchOutput hl₀ hl₁ m) e+
-        normalizer (branchEdges lam m)*(pairZ lam (branchOutput hl₀ hl₁ m) e*Real.log (pairZ lam (branchOutput hl₀ hl₁ m) e)) := by
-      rw [normalizer_cons hl₀ hl₁, Real.log_mul (ne_of_gt hp) (ne_of_gt (hpp e))]
-      ring
-    simp_rw [hid]
-    have hz : Continuous (fun e ↦ pairZ lam (branchOutput hl₀ hl₁ m) e) :=
-      (continuous_pairZ lam).comp (continuous_const.prodMk continuous_id)
-    have hj : Continuous (fun e ↦ pairZ lam (branchOutput hl₀ hl₁ m) e*Real.log (pairZ lam (branchOutput hl₀ hl₁ m) e)) :=
-      (continuous_pairEntropy hl₀ hl₁).comp (continuous_const.prodMk continuous_id)
-    rw [integral_add (f := fun e ↦
-      (normalizer (branchEdges lam m)*Real.log (normalizer (branchEdges lam m)))*pairZ lam (branchOutput hl₀ hl₁ m) e)
-      (g := fun e ↦ normalizer (branchEdges lam m)*(pairZ lam (branchOutput hl₀ hl₁ m) e*Real.log (pairZ lam (branchOutput hl₀ hl₁ m) e)))
-      ((compact_integrable hz).const_mul _) ((compact_integrable hj).const_mul _),
-      integral_const_mul, integral_const_mul, integral_pairZ, mul_one]
-  unfold correlationEntropy
-  rw [integral_pi_cons Q n (continuous_correlationIntegrand hl₀ hl₁ (n+1))]
-  simp_rw [he]
-  rw [integral_add (f := fun m : Fin n → Message ↦ normalizer (branchEdges lam m)*Real.log (normalizer (branchEdges lam m)))
-    (g := fun m ↦ normalizer (branchEdges lam m)*c (branchOutput hl₀ hl₁ m))
-    (compact_integrable (continuous_correlationIntegrand hl₀ hl₁ n))
-    (compact_integrable ((continuous_normalizer n lam).mul (hc.comp (continuous_branchOutput hl₀ hl₁ n))))]
-  congr 1
-  exact (integral_finiteProbability Q hl₀ hl₁ n hc.measurable).symm
-
-lemma xMoment_tangent (r s : Message) :
-    xMoment r+avg (fun i ↦ (r.1 i-1)*(s.1 i-r.1 i)) ≤ xMoment s := by
-  have hp : 0 ≤ avg (fun i ↦ (s.1 i-r.1 i)^2)/2 :=
-    by
-      dsimp [avg]
-      positivity
-  have he : xMoment s-xMoment r-avg (fun i ↦ (r.1 i-1)*(s.1 i-r.1 i)) =
-      avg (fun i ↦ (s.1 i-r.1 i)^2)/2 := by
-    dsimp [xMoment, momentX, centered, avg]; ring
-  linarith
-
-lemma xMoment_sq_tangent (r s : Message) :
-    xMoment r^2+avg (fun i ↦ 2*xMoment r*(r.1 i-1)*(s.1 i-r.1 i)) ≤ xMoment s^2 := by
-  have ht := xMoment_tangent r s
-  have he : avg (fun i ↦ 2*xMoment r*(r.1 i-1)*(s.1 i-r.1 i)) =
-      2*xMoment r*avg (fun i ↦ (r.1 i-1)*(s.1 i-r.1 i)) := by
-    dsimp [avg]; ring
-  rw [he]
-  nlinarith [mul_nonneg (xMoment_nonneg r) (sub_nonneg.mpr ht), sq_nonneg (xMoment s-xMoment r)]
-
-lemma integral_pair_linear (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
-    (r : Message) (a : Vec) :
-    (∫ m, pairZ lam r m*avg (fun i ↦ a i*((pairPosterior hl₀ hl₁ r m).1 i-r.1 i))
-      ∂Q.probability.toMeasure) = 0 := by
-  have hc (i : Fin 3) : Continuous (fun m : Message ↦ (pairPosterior hl₀ hl₁ r m).1 i) :=
-    (continuous_coordinate i).comp ((continuous_pairPosterior hl₀ hl₁).comp (continuous_const.prodMk continuous_id))
-  have hz : Continuous (fun m : Message ↦ pairZ lam r m) :=
-    (continuous_pairZ lam).comp (continuous_const.prodMk continuous_id)
-  have he (m : Message) : pairZ lam r m*avg (fun i ↦ a i*((pairPosterior hl₀ hl₁ r m).1 i-r.1 i)) =
-      avg (fun i ↦ a i*(pairZ lam r m*(pairPosterior hl₀ hl₁ r m).1 i-r.1 i*pairZ lam r m)) := by
-    dsimp [avg]; ring
-  simp_rw [he]
-  rw [integral_avg_measure (f := fun m : Message ↦ fun i ↦ a i*(pairZ lam r m*(pairPosterior hl₀ hl₁ r m).1 i-r.1 i*pairZ lam r m))
-    (fun i ↦ compact_integrable (continuous_const.mul ((hz.mul (hc i)).sub (continuous_const.mul hz))))]
-  have hzero (i : Fin 3) : (∫ m, a i*(pairZ lam r m*(pairPosterior hl₀ hl₁ r m).1 i-r.1 i*pairZ lam r m)
-      ∂Q.probability.toMeasure) = 0 := by
-    rw [integral_const_mul, integral_sub (f := fun m ↦ pairZ lam r m*(pairPosterior hl₀ hl₁ r m).1 i)
-      (g := fun m ↦ r.1 i*pairZ lam r m)
-      (compact_integrable (hz.mul (hc i))) ((compact_integrable hz).const_mul _),
-      integral_pair_coordinate, integral_const_mul, integral_pairZ]
-    ring
-  simp_rw [hzero]
-  exact avg_const 0
-
-lemma integral_pair_tangent_lower (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
-    {f : Message → ℝ} (hf : Continuous f) (r : Message) (a : Vec)
-    (ht : ∀ s, f r+avg (fun i ↦ a i*(s.1 i-r.1 i)) ≤ f s) :
-    f r ≤ ∫ m, pairZ lam r m*f (pairPosterior hl₀ hl₁ r m) ∂Q.probability.toMeasure := by
-  have hz : Continuous (fun m : Message ↦ pairZ lam r m) :=
-    (continuous_pairZ lam).comp (continuous_const.prodMk continuous_id)
-  have hp : Continuous (fun m : Message ↦ pairPosterior hl₀ hl₁ r m) :=
-    (continuous_pairPosterior hl₀ hl₁).comp (continuous_const.prodMk continuous_id)
-  have hc : Continuous (fun m : Message ↦ avg (fun i ↦ a i*((pairPosterior hl₀ hl₁ r m).1 i-r.1 i))) :=
-    continuous_avg.comp (continuous_pi (fun i ↦ continuous_const.mul (((continuous_coordinate i).comp hp).sub continuous_const)))
-  have hh := integral_mono (μ := Q.probability.toMeasure)
-    (f := fun m ↦ pairZ lam r m*(f r+avg (fun i ↦ a i*((pairPosterior hl₀ hl₁ r m).1 i-r.1 i))))
-    (g := fun m ↦ pairZ lam r m*f (pairPosterior hl₀ hl₁ r m))
-    (compact_integrable (hz.mul (continuous_const.add hc)))
-    (compact_integrable (hz.mul (hf.comp hp)))
-    (fun m ↦ mul_le_mul_of_nonneg_left (ht _) (pairZ_positive hl₀ hl₁ r m).le)
-  simp_rw [mul_add] at hh
-  rw [integral_add (f := fun m ↦ pairZ lam r m*f r)
-    (g := fun m ↦ pairZ lam r m*avg (fun i ↦ a i*((pairPosterior hl₀ hl₁ r m).1 i-r.1 i)))
-    ((compact_integrable hz).mul_const _) (compact_integrable (hz.mul hc)),
-    integral_mul_const, integral_pairZ, integral_pair_linear, one_mul, add_zero] at hh
-  exact hh
-
-theorem finite_nu_mono (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) :
-    Monotone (fun n ↦ (finiteLaw Q hl₀ hl₁ n).nu) := by
-  apply monotone_nat_of_le_succ
-  intro n
-  change (∫ m, xMoment m^2 ∂(finiteProbability Q hl₀ hl₁ n).toMeasure) ≤
-    ∫ m, xMoment m^2 ∂(finiteProbability Q hl₀ hl₁ (n+1)).toMeasure
-  rw [finite_succ_integral Q hl₀ hl₁ n (f := fun m ↦ xMoment m^2) (continuous_xMoment.pow 2)]
-  apply integral_mono (compact_integrable (continuous_xMoment.pow 2))
-    (compact_integrable (continuous_compact_integral
-      ((continuous_pairZ lam).mul ((continuous_xMoment.pow 2).comp (continuous_pairPosterior hl₀ hl₁)))))
-  intro r
-  exact integral_pair_tangent_lower Q hl₀ hl₁ (continuous_xMoment.pow 2) r
-    (fun i ↦ 2*xMoment r*(r.1 i-1)) (xMoment_sq_tangent r)
-
-end ThreeState.TreeClauses.Experiment
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeStateClauses.Projection
+
+namespace OAI
+
+/-! Branch Growth. -/
+
+open scoped BigOperators ENNReal NNReal Topology
+open Filter
+noncomputable section
+open Set MeasureTheory
+open scoped BigOperators
+namespace ThreeState.TreeClauses.Experiment
+open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
+
+lemma continuous_compact_integral {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
+    [MeasurableSpace Y] [BorelSpace Y] [CompactSpace Y] {μ : Measure Y} [IsFiniteMeasure μ]
+    {f : X × Y → ℝ} (hf : Continuous f) : Continuous (fun x ↦ ∫ y, f (x,y) ∂μ) := by
+  apply continuousOn_univ.mp
+  apply continuousOn_integral_of_compact_support (k := (univ : Set Y)) isCompact_univ
+  · exact hf.continuousOn
+  · simp
+
+lemma continuous_cons (n : ℕ) : Continuous (fun p : Message × (Fin n → Message) ↦ (Fin.cons p.1 p.2 : Fin (n+1) → Message)) := by
+  apply continuous_pi
+  intro j
+  refine Fin.cases ?_ (fun k ↦ ?_) j
+  · simpa using (continuous_fst : Continuous (fun p : Message × (Fin n → Message) ↦ p.1))
+  · change Continuous (fun p : Message × (Fin n → Message) ↦ p.2 k)
+    fun_prop
+
+lemma integral_pi_cons (Q : Law) (n : ℕ) {f : (Fin (n+1) → Message) → ℝ} (hf : Continuous f) :
+    (∫ m, f m ∂(Measure.pi fun _ : Fin (n+1) ↦ Q.probability.toMeasure)) =
+      ∫ m : Fin n → Message, (∫ e, f (Fin.cons e m) ∂Q.probability.toMeasure)
+        ∂(Measure.pi fun _ ↦ Q.probability.toMeasure) := by
+  have he : (∫ m, f m ∂(Measure.pi fun _ : Fin (n+1) ↦ Q.probability.toMeasure)) =
+      ∫ p : Message × (Fin n → Message), f (Fin.cons p.1 p.2)
+        ∂Q.probability.toMeasure.prod (Measure.pi fun _ ↦ Q.probability.toMeasure) := by
+    rw [← (measurePreserving_piFinSuccAbove (fun _ : Fin (n+1) ↦ Q.probability.toMeasure) 0).symm.integral_comp']
+    simp only [MeasurableEquiv.piFinSuccAbove_symm_apply, Fin.insertNthEquiv,
+      Fin.insertNth_zero, Equiv.coe_fn_mk, Fin.zero_succAbove, cast_eq]
+  rw [he]
+  exact integral_prod_symm _ (compact_integrable (hf.comp (continuous_cons n)))
+
+lemma branchProduct_cons (lam : ℝ) {n : ℕ} (e : Message) (m : Fin n → Message) (i : Fin 3) :
+    branchProduct (branchEdges lam (Fin.cons e m)) i = edgeMessage lam e i*branchProduct (branchEdges lam m) i := by
+  simp [branchProduct, branchEdges, Fin.prod_univ_succ]
+
+lemma normalizer_cons {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) {n : ℕ} (e : Message) (m : Fin n → Message) :
+    normalizer (branchEdges lam (Fin.cons e m)) =
+      normalizer (branchEdges lam m)*pairZ lam (branchOutput hl₀ hl₁ m) e := by
+  unfold normalizer pairZ
+  change avg (fun i ↦ branchProduct (branchEdges lam (Fin.cons e m)) i) = _
+  simp_rw [branchProduct_cons]
+  change avg (fun i ↦ edgeMessage lam e i*branchProduct (branchEdges lam m) i) =
+    avg (branchProduct (branchEdges lam m))*avg (fun i ↦
+      (branchProduct (branchEdges lam m) i/normalizer (branchEdges lam m))*edgeMessage lam e i)
+  have hz := ne_of_gt (normalizer_positive (branchEdges_positive hl₀ hl₁ m))
+  change normalizer (branchEdges lam m) ≠ 0 at hz
+  have hid : avg (branchProduct (branchEdges lam m)) = normalizer (branchEdges lam m) := rfl
+  rw [hid]
+  dsimp only [avg]
+  field_simp [hz]
+
+lemma branchOutput_cons {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) {n : ℕ} (e : Message) (m : Fin n → Message) :
+    branchOutput hl₀ hl₁ (Fin.cons e m) = pairPosterior hl₀ hl₁ (branchOutput hl₀ hl₁ m) e := by
+  apply Subtype.ext
+  funext i
+  change branchProduct (branchEdges lam (Fin.cons e m)) i/normalizer (branchEdges lam (Fin.cons e m)) =
+    (branchProduct (branchEdges lam m) i/normalizer (branchEdges lam m))*edgeMessage lam e i/
+      pairZ lam (branchOutput hl₀ hl₁ m) e
+  rw [branchProduct_cons, normalizer_cons hl₀ hl₁]
+  ring
+
+lemma finite_succ_integral (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ)
+    {f : Message → ℝ} (hf : Continuous f) :
+    (∫ m, f m ∂(finiteProbability Q hl₀ hl₁ (n+1)).toMeasure) =
+      ∫ r, (∫ m, pairZ lam r m*f (pairPosterior hl₀ hl₁ r m) ∂Q.probability.toMeasure)
+        ∂(finiteProbability Q hl₀ hl₁ n).toMeasure := by
+  have hk : Continuous (fun p : Message × Message ↦ pairZ lam p.1 p.2*f (pairPosterior hl₀ hl₁ p.1 p.2)) :=
+    (continuous_pairZ lam).mul (hf.comp (continuous_pairPosterior hl₀ hl₁))
+  rw [integral_finiteProbability Q hl₀ hl₁ (n+1) hf.measurable,
+    integral_pi_cons Q n (f := fun m ↦ normalizer (branchEdges lam m)*f (branchOutput hl₀ hl₁ m)) ((continuous_normalizer (n+1) lam).mul (hf.comp (continuous_branchOutput hl₀ hl₁ (n+1)))),
+    integral_finiteProbability Q hl₀ hl₁ n (continuous_compact_integral hk).measurable]
+  congr 1
+  funext m
+  simp_rw [normalizer_cons hl₀ hl₁, branchOutput_cons hl₀ hl₁, mul_assoc, integral_const_mul]
+
+theorem correlationEntropy_succ (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
+    correlationEntropy Q lam (n+1) = correlationEntropy Q lam n + pairCorrelation (finiteLaw Q hl₀ hl₁ n) Q lam := by
+  let c : Message → ℝ := fun r ↦ ∫ e, pairZ lam r e*Real.log (pairZ lam r e) ∂Q.probability.toMeasure
+  have hc : Continuous c := continuous_compact_integral (continuous_pairEntropy hl₀ hl₁)
+  have he (m : Fin n → Message) :
+      (∫ e, normalizer (branchEdges lam (Fin.cons e m))*Real.log (normalizer (branchEdges lam (Fin.cons e m)))
+        ∂Q.probability.toMeasure) =
+      normalizer (branchEdges lam m)*Real.log (normalizer (branchEdges lam m))+
+      normalizer (branchEdges lam m)*c (branchOutput hl₀ hl₁ m) := by
+    have hp := normalizer_positive (branchEdges_positive hl₀ hl₁ m)
+    have hpp (e : Message) := pairZ_positive hl₀ hl₁ (branchOutput hl₀ hl₁ m) e
+    have hid (e : Message) :
+        normalizer (branchEdges lam (Fin.cons e m))*Real.log (normalizer (branchEdges lam (Fin.cons e m))) =
+        (normalizer (branchEdges lam m)*Real.log (normalizer (branchEdges lam m)))*pairZ lam (branchOutput hl₀ hl₁ m) e+
+        normalizer (branchEdges lam m)*(pairZ lam (branchOutput hl₀ hl₁ m) e*Real.log (pairZ lam (branchOutput hl₀ hl₁ m) e)) := by
+      rw [normalizer_cons hl₀ hl₁, Real.log_mul (ne_of_gt hp) (ne_of_gt (hpp e))]
+      ring
+    simp_rw [hid]
+    have hz : Continuous (fun e ↦ pairZ lam (branchOutput hl₀ hl₁ m) e) :=
+      (continuous_pairZ lam).comp (continuous_const.prodMk continuous_id)
+    have hj : Continuous (fun e ↦ pairZ lam (branchOutput hl₀ hl₁ m) e*Real.log (pairZ lam (branchOutput hl₀ hl₁ m) e)) :=
+      (continuous_pairEntropy hl₀ hl₁).comp (continuous_const.prodMk continuous_id)
+    rw [integral_add (f := fun e ↦
+      (normalizer (branchEdges lam m)*Real.log (normalizer (branchEdges lam m)))*pairZ lam (branchOutput hl₀ hl₁ m) e)
+      (g := fun e ↦ normalizer (branchEdges lam m)*(pairZ lam (branchOutput hl₀ hl₁ m) e*Real.log (pairZ lam (branchOutput hl₀ hl₁ m) e)))
+      ((compact_integrable hz).const_mul _) ((compact_integrable hj).const_mul _),
+      integral_const_mul, integral_const_mul, integral_pairZ, mul_one]
+  unfold correlationEntropy
+  rw [integral_pi_cons Q n (continuous_correlationIntegrand hl₀ hl₁ (n+1))]
+  simp_rw [he]
+  rw [integral_add (f := fun m : Fin n → Message ↦ normalizer (branchEdges lam m)*Real.log (normalizer (branchEdges lam m)))
+    (g := fun m ↦ normalizer (branchEdges lam m)*c (branchOutput hl₀ hl₁ m))
+    (compact_integrable (continuous_correlationIntegrand hl₀ hl₁ n))
+    (compact_integrable ((continuous_normalizer n lam).mul (hc.comp (continuous_branchOutput hl₀ hl₁ n))))]
+  congr 1
+  exact (integral_finiteProbability Q hl₀ hl₁ n hc.measurable).symm
+
+lemma xMoment_tangent (r s : Message) :
+    xMoment r+avg (fun i ↦ (r.1 i-1)*(s.1 i-r.1 i)) ≤ xMoment s := by
+  have hp : 0 ≤ avg (fun i ↦ (s.1 i-r.1 i)^2)/2 :=
+    by
+      dsimp [avg]
+      positivity
+  have he : xMoment s-xMoment r-avg (fun i ↦ (r.1 i-1)*(s.1 i-r.1 i)) =
+      avg (fun i ↦ (s.1 i-r.1 i)^2)/2 := by
+    dsimp [xMoment, momentX, centered, avg]; ring
+  linarith
+
+lemma xMoment_sq_tangent (r s : Message) :
+    xMoment r^2+avg (fun i ↦ 2*xMoment r*(r.1 i-1)*(s.1 i-r.1 i)) ≤ xMoment s^2 := by
+  have ht := xMoment_tangent r s
+  have he : avg (fun i ↦ 2*xMoment r*(r.1 i-1)*(s.1 i-r.1 i)) =
+      2*xMoment r*avg (fun i ↦ (r.1 i-1)*(s.1 i-r.1 i)) := by
+    dsimp [avg]; ring
+  rw [he]
+  nlinarith [mul_nonneg (xMoment_nonneg r) (sub_nonneg.mpr ht), sq_nonneg (xMoment s-xMoment r)]
+
+lemma integral_pair_linear (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
+    (r : Message) (a : Vec) :
+    (∫ m, pairZ lam r m*avg (fun i ↦ a i*((pairPosterior hl₀ hl₁ r m).1 i-r.1 i))
+      ∂Q.probability.toMeasure) = 0 := by
+  have hc (i : Fin 3) : Continuous (fun m : Message ↦ (pairPosterior hl₀ hl₁ r m).1 i) :=
+    (continuous_coordinate i).comp ((continuous_pairPosterior hl₀ hl₁).comp (continuous_const.prodMk continuous_id))
+  have hz : Continuous (fun m : Message ↦ pairZ lam r m) :=
+    (continuous_pairZ lam).comp (continuous_const.prodMk continuous_id)
+  have he (m : Message) : pairZ lam r m*avg (fun i ↦ a i*((pairPosterior hl₀ hl₁ r m).1 i-r.1 i)) =
+      avg (fun i ↦ a i*(pairZ lam r m*(pairPosterior hl₀ hl₁ r m).1 i-r.1 i*pairZ lam r m)) := by
+    dsimp [avg]; ring
+  simp_rw [he]
+  rw [integral_avg_measure (f := fun m : Message ↦ fun i ↦ a i*(pairZ lam r m*(pairPosterior hl₀ hl₁ r m).1 i-r.1 i*pairZ lam r m))
+    (fun i ↦ compact_integrable (continuous_const.mul ((hz.mul (hc i)).sub (continuous_const.mul hz))))]
+  have hzero (i : Fin 3) : (∫ m, a i*(pairZ lam r m*(pairPosterior hl₀ hl₁ r m).1 i-r.1 i*pairZ lam r m)
+      ∂Q.probability.toMeasure) = 0 := by
+    rw [integral_const_mul, integral_sub (f := fun m ↦ pairZ lam r m*(pairPosterior hl₀ hl₁ r m).1 i)
+      (g := fun m ↦ r.1 i*pairZ lam r m)
+      (compact_integrable (hz.mul (hc i))) ((compact_integrable hz).const_mul _),
+      integral_pair_coordinate, integral_const_mul, integral_pairZ]
+    ring
+  simp_rw [hzero]
+  exact avg_const 0
+
+lemma integral_pair_tangent_lower (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
+    {f : Message → ℝ} (hf : Continuous f) (r : Message) (a : Vec)
+    (ht : ∀ s, f r+avg (fun i ↦ a i*(s.1 i-r.1 i)) ≤ f s) :
+    f r ≤ ∫ m, pairZ lam r m*f (pairPosterior hl₀ hl₁ r m) ∂Q.probability.toMeasure := by
+  have hz : Continuous (fun m : Message ↦ pairZ lam r m) :=
+    (continuous_pairZ lam).comp (continuous_const.prodMk continuous_id)
+  have hp : Continuous (fun m : Message ↦ pairPosterior hl₀ hl₁ r m) :=
+    (continuous_pairPosterior hl₀ hl₁).comp (continuous_const.prodMk continuous_id)
+  have hc : Continuous (fun m : Message ↦ avg (fun i ↦ a i*((pairPosterior hl₀ hl₁ r m).1 i-r.1 i))) :=
+    continuous_avg.comp (continuous_pi (fun i ↦ continuous_const.mul (((continuous_coordinate i).comp hp).sub continuous_const)))
+  have hh := integral_mono (μ := Q.probability.toMeasure)
+    (f := fun m ↦ pairZ lam r m*(f r+avg (fun i ↦ a i*((pairPosterior hl₀ hl₁ r m).1 i-r.1 i))))
+    (g := fun m ↦ pairZ lam r m*f (pairPosterior hl₀ hl₁ r m))
+    (compact_integrable (hz.mul (continuous_const.add hc)))
+    (compact_integrable (hz.mul (hf.comp hp)))
+    (fun m ↦ mul_le_mul_of_nonneg_left (ht _) (pairZ_positive hl₀ hl₁ r m).le)
+  simp_rw [mul_add] at hh
+  rw [integral_add (f := fun m ↦ pairZ lam r m*f r)
+    (g := fun m ↦ pairZ lam r m*avg (fun i ↦ a i*((pairPosterior hl₀ hl₁ r m).1 i-r.1 i)))
+    ((compact_integrable hz).mul_const _) (compact_integrable (hz.mul hc)),
+    integral_mul_const, integral_pairZ, integral_pair_linear, one_mul, add_zero] at hh
+  exact hh
+
+theorem finite_nu_mono (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) :
+    Monotone (fun n ↦ (finiteLaw Q hl₀ hl₁ n).nu) := by
+  apply monotone_nat_of_le_succ
+  intro n
+  change (∫ m, xMoment m^2 ∂(finiteProbability Q hl₀ hl₁ n).toMeasure) ≤
+    ∫ m, xMoment m^2 ∂(finiteProbability Q hl₀ hl₁ (n+1)).toMeasure
+  rw [finite_succ_integral Q hl₀ hl₁ n (f := fun m ↦ xMoment m^2) (continuous_xMoment.pow 2)]
+  apply integral_mono (compact_integrable (continuous_xMoment.pow 2))
+    (compact_integrable (continuous_compact_integral
+      ((continuous_pairZ lam).mul ((continuous_xMoment.pow 2).comp (continuous_pairPosterior hl₀ hl₁)))))
+  intro r
+  exact integral_pair_tangent_lower Q hl₀ hl₁ (continuous_xMoment.pow 2) r
+    (fun i ↦ 2*xMoment r*(r.1 i-1)) (xMoment_sq_tangent r)
+
+end ThreeState.TreeClauses.Experiment
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeStateClauses/BranchMoments.lean b/lean/OAI/Probability/ThreeStateClauses/BranchMoments.lean
index baf994be3b922ba38641c2ff5d90c28c12c04240..8e46de22c79fba2bf4e64280ed6be36503f0bbe1 100644
--- a/lean/OAI/Probability/ThreeStateClauses/BranchMoments.lean
+++ b/lean/OAI/Probability/ThreeStateClauses/BranchMoments.lean
@@ -1,362 +1,362 @@
-import Mathlib
-import OAI.Probability.ThreeStateClauses.PosteriorLaw
-
-namespace OAI
-
-/-! Branch Moments. -/
-
-open scoped BigOperators ENNReal NNReal Topology
-open Filter
-noncomputable section
-open Set MeasureTheory
-open scoped BigOperators
-namespace ThreeState.TreeClauses.Experiment
-open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
-
-variable {ι : Type*} [Fintype ι]
-
-def branchProduct (a : ι → Vec) : Vec := fun i ↦ ∏ j, a j i
-
-def normalizer (a : ι → Vec) : ℝ := avg (branchProduct a)
-
-def normalizedProduct (a : ι → Vec) : Vec := fun i ↦ branchProduct a i / normalizer a
-
-lemma avg_sum (f : ι → Vec) : avg (fun i ↦ ∑ j, f j i) = ∑ j, avg (f j) := by
-  dsimp only [avg]
-  rw [← Finset.sum_div]
-  congr 1
-  simp only [Finset.sum_add_distrib]
-
-lemma avg_div (f : Vec) (c : ℝ) : avg (fun i ↦ f i/c) = avg f/c := by
-  simp only [avg]; ring
-
-lemma normalizer_positive {a : ι → Vec} (ha : ∀ j i, 0 < a j i) : 0 < normalizer a := by
-  have h (i) : 0 < branchProduct a i := Finset.prod_pos (fun j _ ↦ ha j i)
-  dsimp only [normalizer, avg]
-  exact div_pos (add_pos (add_pos (h 0) (h 1)) (h 2)) (by norm_num)
-
-lemma normalizedProduct_positive {a : ι → Vec} (ha : ∀ j i, 0 < a j i) :
-    PositiveMessage (normalizedProduct a) := by
-  refine ⟨fun i ↦ div_pos (Finset.prod_pos (fun j _ ↦ ha j i)) (normalizer_positive ha), ?_⟩
-  change avg (fun i ↦ branchProduct a i/normalizer a) = 1
-  rw [avg_div]
-  exact div_self (ne_of_gt (normalizer_positive ha))
-
-lemma log_normalizedProduct {a : ι → Vec} (ha : ∀ j i, 0 < a j i) (i : Fin 3) :
-    Real.log (normalizedProduct a i) = (∑ j, Real.log (a j i))-Real.log (normalizer a) := by
-  rw [normalizedProduct, branchProduct, Real.log_div (ne_of_gt (Finset.prod_pos (fun j _ ↦ ha j i)))
-    (ne_of_gt (normalizer_positive ha))]
-  congr 1
-  exact Real.log_prod (fun j _ ↦ ne_of_gt (ha j i))
-
-lemma logCentered_normalizedProduct {a : ι → Vec} (ha : ∀ j i, 0 < a j i) (i : Fin 3) :
-    logCentered (normalizedProduct a) i = ∑ j, logCentered (a j) i := by
-  simp only [logCentered, logAverage, log_normalizedProduct ha]
-  rw [avg_sub, avg_sum, avg_const, Finset.sum_sub_distrib]
-  ring
-
-lemma symEntropy_centeredLog {m : Vec} (hm : avg m = 1) :
-    symEntropy m = avg (fun i ↦ m i*logCentered m i)/2 := by
-  have h : avg (fun i ↦ m i*logAverage m) = logAverage m := by
-    calc
-      _ = logAverage m*avg m := by simp only [avg]; ring
-      _ = _ := by rw [hm]; ring
-  unfold symEntropy logCentered
-  simp_rw [mul_sub, sub_mul, one_mul]
-  rw [avg_sub, avg_sub, h]
-  rfl
-
-lemma weighted_symEntropy {a : ι → Vec} (ha : ∀ j i, 0 < a j i) :
-    normalizer a*symEntropy (normalizedProduct a) =
-      avg (fun i ↦ branchProduct a i*(∑ j, logCentered (a j) i))/2 := by
-  rw [symEntropy_centeredLog (normalizedProduct_positive ha).2]
-  simp_rw [logCentered_normalizedProduct ha]
-  change normalizer a*(avg (fun i ↦ (branchProduct a i/normalizer a)*(∑ j, logCentered (a j) i))/2) = _
-  dsimp only [avg]
-  field_simp [ne_of_gt (normalizer_positive ha)]
-
-lemma weighted_correction {a : ι → Vec} (ha : ∀ j i, 0 < a j i) :
-    normalizer a*correction (normalizedProduct a) =
-      avg (fun i ↦ branchProduct a i*(∑ j, logCentered (a j) i)^2)/2 := by
-  simp only [correction, logCentered_normalizedProduct ha, normalizedProduct, avg]
-  field_simp [ne_of_gt (normalizer_positive ha)]
-
-lemma weighted_info {a : ι → Vec} (ha : ∀ j i, 0 < a j i) :
-    normalizer a*info (normalizedProduct a) =
-      avg (fun i ↦ branchProduct a i*(∑ j, Real.log (a j i))) -
-      normalizer a*Real.log (normalizer a) := by
-  unfold info
-  simp_rw [log_normalizedProduct ha]
-  simp only [normalizedProduct, avg]
-  field_simp [ne_of_gt (normalizer_positive ha)]
-  dsimp only [normalizer, avg]
-  ring
-
-end ThreeState.TreeClauses.Experiment
-
-end 
-
-noncomputable section
-open MeasureTheory
-open scoped BigOperators
-namespace ThreeState.TreeClauses.Experiment
-
-variable {ι X : Type*} [Fintype ι] [DecidableEq ι] [MeasurableSpace X]
-  {μ : Measure X} [SigmaFinite μ] {a u : X → ℝ}
-
-def weightProduct (a : X → ℝ) (x : ι → X) : ℝ := ∏ i, a (x i)
-
-omit [MeasurableSpace X] in
-lemma select_product (a u : X → ℝ) (x : ι → X) (j : ι) :
-    weightProduct a x*u (x j) = ∏ i, a (x i)*(if i=j then u (x i) else 1) := by
-  rw [Finset.prod_mul_distrib]
-  simp only [weightProduct, Fintype.prod_ite_eq']
-
-lemma integrable_weighted_single (ha : Integrable a μ) (hau : Integrable (fun x ↦ a x*u x) μ)
-    (j : ι) : Integrable (fun x : ι → X ↦ weightProduct a x*u (x j)) (Measure.pi fun _ ↦ μ) := by
-  have hi : ∀ i : ι, Integrable (fun x ↦ a x*(if i=j then u x else 1)) μ := by
-    intro i
-    by_cases h : i=j
-    · simpa [h] using hau
-    · simpa [h] using ha
-  simpa only [← select_product] using Integrable.fintype_prod hi
-
-lemma integral_weighted_single (ha : ∫ x, a x ∂μ = 1) (j : ι) :
-    (∫ x : ι → X, weightProduct a x*u (x j) ∂(Measure.pi fun _ ↦ μ)) = ∫ x, a x*u x ∂μ := by
-  simp_rw [select_product]
-  rw [integral_fintype_prod_eq_prod (fun i x ↦ a x*(if i=j then u x else 1))]
-  have he (i : ι) : (∫ x, a x*(if i=j then u x else 1) ∂μ) =
-      if i=j then (∫ x, a x*u x ∂μ) else 1 := by
-    by_cases h : i=j <;> simp [h, ha]
-  simp_rw [he]
-  simp
-
-omit [MeasurableSpace X] in
-lemma select_pair_product (a u : X → ℝ) (x : ι → X) (j k : ι) :
-    weightProduct a x*(u (x j)*u (x k)) =
-      ∏ i, a (x i)*(if i=j then u (x i) else 1)*(if i=k then u (x i) else 1) := by
-  rw [Finset.prod_mul_distrib, Finset.prod_mul_distrib]
-  simp only [weightProduct, Fintype.prod_ite_eq']
-  ring
-
-lemma integrable_weighted_pair (ha : Integrable a μ) (hau : Integrable (fun x ↦ a x*u x) μ)
-    (hau₂ : Integrable (fun x ↦ a x*u x^2) μ) (j k : ι) :
-    Integrable (fun x : ι → X ↦ weightProduct a x*(u (x j)*u (x k))) (Measure.pi fun _ ↦ μ) := by
-  have hi : ∀ i : ι, Integrable (fun x ↦ a x*(if i=j then u x else 1)*(if i=k then u x else 1)) μ := by
-    intro i
-    by_cases h : i=j <;> by_cases h' : i=k
-    · simpa only [ite_eq_left h, ite_eq_left h', pow_two, mul_assoc] using hau₂
-    · simpa only [ite_eq_left h, ite_eq_right h', mul_one] using hau
-    · simpa only [ite_eq_right h, ite_eq_left h', mul_one] using hau
-    · simpa [h, h'] using ha
-  simpa only [← select_pair_product] using Integrable.fintype_prod hi
-
-lemma integral_weighted_pair (ha : ∫ x, a x ∂μ = 1) (j k : ι) :
-    (∫ x : ι → X, weightProduct a x*(u (x j)*u (x k)) ∂(Measure.pi fun _ ↦ μ)) =
-      if j=k then (∫ x, a x*u x^2 ∂μ) else (∫ x, a x*u x ∂μ)^2 := by
-  by_cases hjk : j=k
-  · subst k
-    simp only [ite_true, pow_two]
-    exact integral_weighted_single (u := fun x ↦ u x*u x) ha j
-  · simp_rw [select_pair_product]
-    rw [integral_fintype_prod_eq_prod (fun i x ↦ a x*(if i=j then u x else 1)*(if i=k then u x else 1)), ite_eq_right hjk]
-    have he (i : ι) : (∫ x, a x*(if i=j then u x else 1)*(if i=k then u x else 1) ∂μ) =
-        (if i=j then (∫ x, a x*u x ∂μ) else 1)*(if i=k then (∫ x, a x*u x ∂μ) else 1) := by
-      by_cases h : i=j <;> by_cases h' : i=k
-      · exact (hjk (h.symm.trans h')).elim
-      · simp only [ite_eq_left h, ite_eq_right h', mul_one]
-      · simp only [ite_eq_right h, ite_eq_left h', mul_one, one_mul]
-      · simp only [ite_eq_right h, ite_eq_right h', mul_one, ha]
-    simp_rw [he]
-    rw [Finset.prod_mul_distrib]
-    simp [sq]
-
-lemma integrable_weighted_sum (ha : Integrable a μ) (hau : Integrable (fun x ↦ a x*u x) μ) :
-    Integrable (fun x : ι → X ↦ weightProduct a x*(∑ j, u (x j))) (Measure.pi fun _ ↦ μ) := by
-  simp_rw [Finset.mul_sum]
-  exact integrable_finsetSum _ (fun j _ ↦ integrable_weighted_single ha hau j)
-
-lemma integral_weighted_sum (ha : Integrable a μ) (hau : Integrable (fun x ↦ a x*u x) μ)
-    (hbal : ∫ x, a x ∂μ = 1) :
-    (∫ x : ι → X, weightProduct a x*(∑ j, u (x j)) ∂(Measure.pi fun _ ↦ μ)) =
-      (Fintype.card ι : ℝ)*(∫ x, a x*u x ∂μ) := by
-  simp_rw [Finset.mul_sum]
-  rw [integral_finsetSum _ (fun j _ ↦ integrable_weighted_single ha hau j)]
-  simp only [integral_weighted_single hbal, Finset.sum_const, Finset.card_univ, nsmul_eq_mul]
-
-lemma integrable_weighted_sq_sum (ha : Integrable a μ) (hau : Integrable (fun x ↦ a x*u x) μ)
-    (hau₂ : Integrable (fun x ↦ a x*u x^2) μ) :
-    Integrable (fun x : ι → X ↦ weightProduct a x*(∑ j, u (x j))^2) (Measure.pi fun _ ↦ μ) := by
-  have he (x : ι → X) : weightProduct a x*(∑ j, u (x j))^2 =
-      ∑ j, ∑ k, weightProduct a x*(u (x j)*u (x k)) := by
-    rw [sq, Finset.sum_mul_sum]
-    simp_rw [Finset.mul_sum]
-  simp_rw [he]
-  exact integrable_finsetSum _ (fun j _ ↦ integrable_finsetSum _ (fun k _ ↦ integrable_weighted_pair ha hau hau₂ j k))
-
-theorem integral_weighted_sq_sum (ha : Integrable a μ) (hau : Integrable (fun x ↦ a x*u x) μ)
-    (hau₂ : Integrable (fun x ↦ a x*u x^2) μ) (hbal : ∫ x, a x ∂μ = 1) :
-    (∫ x : ι → X, weightProduct a x*(∑ j, u (x j))^2 ∂(Measure.pi fun _ ↦ μ)) =
-      (Fintype.card ι : ℝ)*(∫ x, a x*u x^2 ∂μ)+
-      ((Fintype.card ι : ℝ)^2-(Fintype.card ι : ℝ))*(∫ x, a x*u x ∂μ)^2 := by
-  have he (x : ι → X) : weightProduct a x*(∑ j, u (x j))^2 =
-      ∑ j, ∑ k, weightProduct a x*(u (x j)*u (x k)) := by
-    rw [sq, Finset.sum_mul_sum]
-    simp_rw [Finset.mul_sum]
-  simp_rw [he]
-  rw [integral_finsetSum _ (fun j _ ↦ integrable_finsetSum _ (fun k _ ↦ integrable_weighted_pair ha hau hau₂ j k))]
-  simp_rw [integral_finsetSum _ (fun k _ ↦ integrable_weighted_pair ha hau hau₂ _ k), integral_weighted_pair hbal]
-  have he' (j k : ι) : (if j=k then (∫ x, a x*u x^2 ∂μ) else (∫ x, a x*u x ∂μ)^2) =
-      (if j=k then (∫ x, a x*u x^2 ∂μ)-(∫ x, a x*u x ∂μ)^2 else 0)+(∫ x, a x*u x ∂μ)^2 := by
-    split_ifs <;> ring
-  simp_rw [he', Finset.sum_add_distrib]
-  simp only [Finset.sum_ite_eq, Finset.mem_univ, ite_true, Finset.sum_const, Finset.card_univ, nsmul_eq_mul]
-  ring
-
-end ThreeState.TreeClauses.Experiment
-
-end 
-
-noncomputable section
-open Set MeasureTheory
-open scoped BigOperators
-namespace ThreeState.TreeClauses.Experiment
-open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
-
-abbrev branchEdges {n : ℕ} (lam : ℝ) (m : Fin n → Message) : Fin n → Vec := fun j ↦ edgeMessage lam (m j)
-
-def finiteBranchExpectation (Q : Law) (lam : ℝ) (n : ℕ) (f : Vec → ℝ) : ℝ :=
-  ∫ m : Fin n → Message, normalizer (branchEdges lam m)*f (normalizedProduct (branchEdges lam m))
-    ∂(Measure.pi fun _ ↦ Q.probability.toMeasure)
-
-def correlationEntropy (Q : Law) (lam : ℝ) (n : ℕ) : ℝ :=
-  ∫ m : Fin n → Message, normalizer (branchEdges lam m)*Real.log (normalizer (branchEdges lam m))
-    ∂(Measure.pi fun _ ↦ Q.probability.toMeasure)
-
-lemma branchEdges_positive {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) {n : ℕ} (m : Fin n → Message) :
-    ∀ j i, 0 < branchEdges lam m j i := fun j i ↦ (edge_positive hl₀ hl₁ (m j)).1 i
-
-lemma continuous_branchProduct {n : ℕ} (lam : ℝ) (i : Fin 3) :
-    Continuous (fun m : Fin n → Message ↦ branchProduct (branchEdges lam m) i) := by
-  unfold branchProduct branchEdges
-  exact continuous_finsetProd _ (fun j _ ↦ ((continuous_apply i).comp (continuous_edgeMessage lam)).comp (continuous_apply j))
-
-lemma continuous_normalizer (n : ℕ) (lam : ℝ) :
-    Continuous (fun m : Fin n → Message ↦ normalizer (branchEdges lam m)) :=
-  continuous_avg.comp (continuous_pi (continuous_branchProduct lam))
-
-lemma continuous_correlationIntegrand {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
-    Continuous (fun m : Fin n → Message ↦ normalizer (branchEdges lam m)*Real.log (normalizer (branchEdges lam m))) :=
-  (continuous_normalizer n lam).mul ((continuous_normalizer n lam).log
-    (fun m ↦ ne_of_gt (normalizer_positive (branchEdges_positive hl₀ hl₁ m))))
-
-lemma integral_normalizer (Q : Law) (lam : ℝ) (n : ℕ) :
-    (∫ m : Fin n → Message, normalizer (branchEdges lam m) ∂(Measure.pi fun _ ↦ Q.probability.toMeasure)) = 1 := by
-  unfold normalizer
-  rw [integral_avg_measure (fun i ↦ compact_integrable (continuous_branchProduct lam i))]
-  have he (i : Fin 3) : (∫ m : Fin n → Message, branchProduct (branchEdges lam m) i
-      ∂(Measure.pi fun _ ↦ Q.probability.toMeasure)) = 1 := by
-    change (∫ m : Fin n → Message, ∏ j, edgeMessage lam (m j) i ∂(Measure.pi fun _ ↦ Q.probability.toMeasure)) = 1
-    rw [integral_fintype_prod_eq_pow (fun m ↦ edgeMessage lam m i), Q.edge_balanced]
-    simp
-  simp_rw [he]
-  exact avg_const 1
-
-lemma correlationEntropy_nonneg (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
-    0 ≤ correlationEntropy Q lam n := by
-  have he (m : Fin n → Message) : normalizer (branchEdges lam m)-1 ≤
-      normalizer (branchEdges lam m)*Real.log (normalizer (branchEdges lam m)) := by
-    have hp := normalizer_positive (branchEdges_positive hl₀ hl₁ m)
-    have h := mul_le_mul_of_nonneg_left (Real.one_sub_inv_le_log_of_pos hp) hp.le
-    simpa [mul_sub, ne_of_gt hp] using h
-  have h := integral_mono (μ := Measure.pi fun _ : Fin n ↦ Q.probability.toMeasure)
-    (f := fun m ↦ normalizer (branchEdges lam m)-1)
-    (g := fun m ↦ normalizer (branchEdges lam m)*Real.log (normalizer (branchEdges lam m)))
-    ((compact_integrable (continuous_normalizer n lam)).sub (integrable_const 1))
-    (compact_integrable (continuous_correlationIntegrand hl₀ hl₁ n)) he
-  rw [integral_sub (f := fun m : Fin n → Message ↦ normalizer (branchEdges lam m))
-    (g := fun _ ↦ (1:ℝ)) (compact_integrable (continuous_normalizer n lam)) (integrable_const _),
-    integral_normalizer] at h
-  simpa [correlationEntropy] using h
-
-theorem finite_symEntropy (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
-    finiteBranchExpectation Q lam n symEntropy =
-      n*(∫ m, symEntropy (edgeMessage lam m) ∂Q.probability.toMeasure) := by
-  have ha (i : Fin 3) : Integrable (fun m ↦ edgeMessage lam m i) Q.probability.toMeasure :=
-    compact_integrable ((continuous_apply i).comp (continuous_edgeMessage lam))
-  have hau (i : Fin 3) : Integrable (fun m ↦ edgeMessage lam m i*logCentered (edgeMessage lam m) i) Q.probability.toMeasure :=
-    compact_integrable (((continuous_apply i).comp (continuous_edgeMessage lam)).mul
-      (continuous_edge_centeredLog hl₀ hl₁ i))
-  unfold finiteBranchExpectation
-  simp_rw [weighted_symEntropy (branchEdges_positive hl₀ hl₁ _)]
-  rw [integral_div]
-  rw [integral_avg_measure (f := fun m : Fin n → Message ↦ fun i ↦ branchProduct (branchEdges lam m) i*(∑ j, logCentered (branchEdges lam m j) i))
-    (fun i ↦ integrable_weighted_sum (ι := Fin n) (ha i) (hau i))]
-  have he (i : Fin 3) : (∫ m : Fin n → Message, branchProduct (branchEdges lam m) i*(∑ j, logCentered (branchEdges lam m j) i)
-      ∂(Measure.pi fun _ ↦ Q.probability.toMeasure)) =
-      n*(2*∫ m, symEntropy (edgeMessage lam m) ∂Q.probability.toMeasure) := by
-    have h := integral_weighted_sum (ι := Fin n) (ha i) (hau i) (Q.edge_balanced lam i)
-    rw [Q.edge_centeredLog_mean hl₀ hl₁] at h
-    simpa only [Fintype.card_fin, weightProduct, branchProduct, branchEdges] using h
-  simp_rw [he]
-  rw [avg_const]
-  ring
-
-theorem finite_correction (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
-    finiteBranchExpectation Q lam n correction =
-      n*(∫ m, correction (edgeMessage lam m) ∂Q.probability.toMeasure)+
-      2*((n:ℝ)^2-n)*(∫ m, symEntropy (edgeMessage lam m) ∂Q.probability.toMeasure)^2 := by
-  have ha (i : Fin 3) : Integrable (fun m ↦ edgeMessage lam m i) Q.probability.toMeasure :=
-    compact_integrable ((continuous_apply i).comp (continuous_edgeMessage lam))
-  have hau (i : Fin 3) : Integrable (fun m ↦ edgeMessage lam m i*logCentered (edgeMessage lam m) i) Q.probability.toMeasure :=
-    compact_integrable (((continuous_apply i).comp (continuous_edgeMessage lam)).mul
-      (continuous_edge_centeredLog hl₀ hl₁ i))
-  have hau₂ (i : Fin 3) : Integrable (fun m ↦ edgeMessage lam m i*logCentered (edgeMessage lam m) i^2) Q.probability.toMeasure :=
-    compact_integrable (((continuous_apply i).comp (continuous_edgeMessage lam)).mul
-      ((continuous_edge_centeredLog hl₀ hl₁ i).pow 2))
-  unfold finiteBranchExpectation
-  simp_rw [weighted_correction (branchEdges_positive hl₀ hl₁ _)]
-  rw [integral_div]
-  rw [integral_avg_measure (f := fun m : Fin n → Message ↦ fun i ↦ branchProduct (branchEdges lam m) i*(∑ j, logCentered (branchEdges lam m j) i)^2)
-    (fun i ↦ integrable_weighted_sq_sum (ι := Fin n) (ha i) (hau i) (hau₂ i))]
-  have he (i : Fin 3) : (∫ m : Fin n → Message, branchProduct (branchEdges lam m) i*(∑ j, logCentered (branchEdges lam m j) i)^2
-      ∂(Measure.pi fun _ ↦ Q.probability.toMeasure)) =
-      n*(2*∫ m, correction (edgeMessage lam m) ∂Q.probability.toMeasure)+
-      ((n:ℝ)^2-n)*(2*∫ m, symEntropy (edgeMessage lam m) ∂Q.probability.toMeasure)^2 := by
-    have h := integral_weighted_sq_sum (ι := Fin n) (ha i) (hau i) (hau₂ i) (Q.edge_balanced lam i)
-    rw [Q.edge_centeredLog_mean hl₀ hl₁, Q.edge_centeredLog_sq hl₀ hl₁] at h
-    simpa only [Fintype.card_fin, weightProduct, branchProduct, branchEdges] using h
-  simp_rw [he]
-  rw [avg_const]
-  ring
-
-theorem finite_info (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
-    finiteBranchExpectation Q lam n info =
-      n*(∫ m, info (edgeMessage lam m) ∂Q.probability.toMeasure)-correlationEntropy Q lam n := by
-  have ha (i : Fin 3) : Integrable (fun m ↦ edgeMessage lam m i) Q.probability.toMeasure :=
-    compact_integrable ((continuous_apply i).comp (continuous_edgeMessage lam))
-  have hau (i : Fin 3) : Integrable (fun m ↦ edgeMessage lam m i*Real.log (edgeMessage lam m i)) Q.probability.toMeasure :=
-    compact_integrable (((continuous_apply i).comp (continuous_edgeMessage lam)).mul
-      (continuous_edge_log hl₀ hl₁ i))
-  unfold finiteBranchExpectation
-  simp_rw [weighted_info (branchEdges_positive hl₀ hl₁ _)]
-  rw [integral_sub (f := fun m : Fin n → Message ↦ avg (fun i ↦ branchProduct (branchEdges lam m) i*(∑ j, Real.log (branchEdges lam m j i))))
-    (g := fun m ↦ normalizer (branchEdges lam m)*Real.log (normalizer (branchEdges lam m)))
-    (integrable_avg (fun i ↦ integrable_weighted_sum (ι := Fin n) (ha i) (hau i)))
-    (compact_integrable (continuous_correlationIntegrand hl₀ hl₁ n))]
-  congr 1
-  rw [integral_avg_measure (f := fun m : Fin n → Message ↦ fun i ↦ branchProduct (branchEdges lam m) i*(∑ j, Real.log (branchEdges lam m j i)))
-    (fun i ↦ integrable_weighted_sum (ι := Fin n) (ha i) (hau i))]
-  have he (i : Fin 3) : (∫ m : Fin n → Message, branchProduct (branchEdges lam m) i*(∑ j, Real.log (branchEdges lam m j i))
-      ∂(Measure.pi fun _ ↦ Q.probability.toMeasure)) =
-      n*(∫ m, edgeMessage lam m i*Real.log (edgeMessage lam m i) ∂Q.probability.toMeasure) := by
-    simpa only [Fintype.card_fin, weightProduct, branchProduct, branchEdges] using integral_weighted_sum (ι := Fin n) (ha i) (hau i) (Q.edge_balanced lam i)
-  simp_rw [he]
-  rw [avg_mul, ← integral_avg_measure hau]
-  rfl
-
-end ThreeState.TreeClauses.Experiment
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeStateClauses.PosteriorLaw
+
+namespace OAI
+
+/-! Branch Moments. -/
+
+open scoped BigOperators ENNReal NNReal Topology
+open Filter
+noncomputable section
+open Set MeasureTheory
+open scoped BigOperators
+namespace ThreeState.TreeClauses.Experiment
+open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
+
+variable {ι : Type*} [Fintype ι]
+
+def branchProduct (a : ι → Vec) : Vec := fun i ↦ ∏ j, a j i
+
+def normalizer (a : ι → Vec) : ℝ := avg (branchProduct a)
+
+def normalizedProduct (a : ι → Vec) : Vec := fun i ↦ branchProduct a i / normalizer a
+
+lemma avg_sum (f : ι → Vec) : avg (fun i ↦ ∑ j, f j i) = ∑ j, avg (f j) := by
+  dsimp only [avg]
+  rw [← Finset.sum_div]
+  congr 1
+  simp only [Finset.sum_add_distrib]
+
+lemma avg_div (f : Vec) (c : ℝ) : avg (fun i ↦ f i/c) = avg f/c := by
+  simp only [avg]; ring
+
+lemma normalizer_positive {a : ι → Vec} (ha : ∀ j i, 0 < a j i) : 0 < normalizer a := by
+  have h (i) : 0 < branchProduct a i := Finset.prod_pos (fun j _ ↦ ha j i)
+  dsimp only [normalizer, avg]
+  exact div_pos (add_pos (add_pos (h 0) (h 1)) (h 2)) (by norm_num)
+
+lemma normalizedProduct_positive {a : ι → Vec} (ha : ∀ j i, 0 < a j i) :
+    PositiveMessage (normalizedProduct a) := by
+  refine ⟨fun i ↦ div_pos (Finset.prod_pos (fun j _ ↦ ha j i)) (normalizer_positive ha), ?_⟩
+  change avg (fun i ↦ branchProduct a i/normalizer a) = 1
+  rw [avg_div]
+  exact div_self (ne_of_gt (normalizer_positive ha))
+
+lemma log_normalizedProduct {a : ι → Vec} (ha : ∀ j i, 0 < a j i) (i : Fin 3) :
+    Real.log (normalizedProduct a i) = (∑ j, Real.log (a j i))-Real.log (normalizer a) := by
+  rw [normalizedProduct, branchProduct, Real.log_div (ne_of_gt (Finset.prod_pos (fun j _ ↦ ha j i)))
+    (ne_of_gt (normalizer_positive ha))]
+  congr 1
+  exact Real.log_prod (fun j _ ↦ ne_of_gt (ha j i))
+
+lemma logCentered_normalizedProduct {a : ι → Vec} (ha : ∀ j i, 0 < a j i) (i : Fin 3) :
+    logCentered (normalizedProduct a) i = ∑ j, logCentered (a j) i := by
+  simp only [logCentered, logAverage, log_normalizedProduct ha]
+  rw [avg_sub, avg_sum, avg_const, Finset.sum_sub_distrib]
+  ring
+
+lemma symEntropy_centeredLog {m : Vec} (hm : avg m = 1) :
+    symEntropy m = avg (fun i ↦ m i*logCentered m i)/2 := by
+  have h : avg (fun i ↦ m i*logAverage m) = logAverage m := by
+    calc
+      _ = logAverage m*avg m := by simp only [avg]; ring
+      _ = _ := by rw [hm]; ring
+  unfold symEntropy logCentered
+  simp_rw [mul_sub, sub_mul, one_mul]
+  rw [avg_sub, avg_sub, h]
+  rfl
+
+lemma weighted_symEntropy {a : ι → Vec} (ha : ∀ j i, 0 < a j i) :
+    normalizer a*symEntropy (normalizedProduct a) =
+      avg (fun i ↦ branchProduct a i*(∑ j, logCentered (a j) i))/2 := by
+  rw [symEntropy_centeredLog (normalizedProduct_positive ha).2]
+  simp_rw [logCentered_normalizedProduct ha]
+  change normalizer a*(avg (fun i ↦ (branchProduct a i/normalizer a)*(∑ j, logCentered (a j) i))/2) = _
+  dsimp only [avg]
+  field_simp [ne_of_gt (normalizer_positive ha)]
+
+lemma weighted_correction {a : ι → Vec} (ha : ∀ j i, 0 < a j i) :
+    normalizer a*correction (normalizedProduct a) =
+      avg (fun i ↦ branchProduct a i*(∑ j, logCentered (a j) i)^2)/2 := by
+  simp only [correction, logCentered_normalizedProduct ha, normalizedProduct, avg]
+  field_simp [ne_of_gt (normalizer_positive ha)]
+
+lemma weighted_info {a : ι → Vec} (ha : ∀ j i, 0 < a j i) :
+    normalizer a*info (normalizedProduct a) =
+      avg (fun i ↦ branchProduct a i*(∑ j, Real.log (a j i))) -
+      normalizer a*Real.log (normalizer a) := by
+  unfold info
+  simp_rw [log_normalizedProduct ha]
+  simp only [normalizedProduct, avg]
+  field_simp [ne_of_gt (normalizer_positive ha)]
+  dsimp only [normalizer, avg]
+  ring
+
+end ThreeState.TreeClauses.Experiment
+
+end 
+
+noncomputable section
+open MeasureTheory
+open scoped BigOperators
+namespace ThreeState.TreeClauses.Experiment
+
+variable {ι X : Type*} [Fintype ι] [DecidableEq ι] [MeasurableSpace X]
+  {μ : Measure X} [SigmaFinite μ] {a u : X → ℝ}
+
+def weightProduct (a : X → ℝ) (x : ι → X) : ℝ := ∏ i, a (x i)
+
+omit [MeasurableSpace X] in
+lemma select_product (a u : X → ℝ) (x : ι → X) (j : ι) :
+    weightProduct a x*u (x j) = ∏ i, a (x i)*(if i=j then u (x i) else 1) := by
+  rw [Finset.prod_mul_distrib]
+  simp only [weightProduct, Fintype.prod_ite_eq']
+
+lemma integrable_weighted_single (ha : Integrable a μ) (hau : Integrable (fun x ↦ a x*u x) μ)
+    (j : ι) : Integrable (fun x : ι → X ↦ weightProduct a x*u (x j)) (Measure.pi fun _ ↦ μ) := by
+  have hi : ∀ i : ι, Integrable (fun x ↦ a x*(if i=j then u x else 1)) μ := by
+    intro i
+    by_cases h : i=j
+    · simpa [h] using hau
+    · simpa [h] using ha
+  simpa only [← select_product] using Integrable.fintype_prod hi
+
+lemma integral_weighted_single (ha : ∫ x, a x ∂μ = 1) (j : ι) :
+    (∫ x : ι → X, weightProduct a x*u (x j) ∂(Measure.pi fun _ ↦ μ)) = ∫ x, a x*u x ∂μ := by
+  simp_rw [select_product]
+  rw [integral_fintype_prod_eq_prod (fun i x ↦ a x*(if i=j then u x else 1))]
+  have he (i : ι) : (∫ x, a x*(if i=j then u x else 1) ∂μ) =
+      if i=j then (∫ x, a x*u x ∂μ) else 1 := by
+    by_cases h : i=j <;> simp [h, ha]
+  simp_rw [he]
+  simp
+
+omit [MeasurableSpace X] in
+lemma select_pair_product (a u : X → ℝ) (x : ι → X) (j k : ι) :
+    weightProduct a x*(u (x j)*u (x k)) =
+      ∏ i, a (x i)*(if i=j then u (x i) else 1)*(if i=k then u (x i) else 1) := by
+  rw [Finset.prod_mul_distrib, Finset.prod_mul_distrib]
+  simp only [weightProduct, Fintype.prod_ite_eq']
+  ring
+
+lemma integrable_weighted_pair (ha : Integrable a μ) (hau : Integrable (fun x ↦ a x*u x) μ)
+    (hau₂ : Integrable (fun x ↦ a x*u x^2) μ) (j k : ι) :
+    Integrable (fun x : ι → X ↦ weightProduct a x*(u (x j)*u (x k))) (Measure.pi fun _ ↦ μ) := by
+  have hi : ∀ i : ι, Integrable (fun x ↦ a x*(if i=j then u x else 1)*(if i=k then u x else 1)) μ := by
+    intro i
+    by_cases h : i=j <;> by_cases h' : i=k
+    · simpa only [ite_eq_left h, ite_eq_left h', pow_two, mul_assoc] using hau₂
+    · simpa only [ite_eq_left h, ite_eq_right h', mul_one] using hau
+    · simpa only [ite_eq_right h, ite_eq_left h', mul_one] using hau
+    · simpa [h, h'] using ha
+  simpa only [← select_pair_product] using Integrable.fintype_prod hi
+
+lemma integral_weighted_pair (ha : ∫ x, a x ∂μ = 1) (j k : ι) :
+    (∫ x : ι → X, weightProduct a x*(u (x j)*u (x k)) ∂(Measure.pi fun _ ↦ μ)) =
+      if j=k then (∫ x, a x*u x^2 ∂μ) else (∫ x, a x*u x ∂μ)^2 := by
+  by_cases hjk : j=k
+  · subst k
+    simp only [ite_true, pow_two]
+    exact integral_weighted_single (u := fun x ↦ u x*u x) ha j
+  · simp_rw [select_pair_product]
+    rw [integral_fintype_prod_eq_prod (fun i x ↦ a x*(if i=j then u x else 1)*(if i=k then u x else 1)), ite_eq_right hjk]
+    have he (i : ι) : (∫ x, a x*(if i=j then u x else 1)*(if i=k then u x else 1) ∂μ) =
+        (if i=j then (∫ x, a x*u x ∂μ) else 1)*(if i=k then (∫ x, a x*u x ∂μ) else 1) := by
+      by_cases h : i=j <;> by_cases h' : i=k
+      · exact (hjk (h.symm.trans h')).elim
+      · simp only [ite_eq_left h, ite_eq_right h', mul_one]
+      · simp only [ite_eq_right h, ite_eq_left h', mul_one, one_mul]
+      · simp only [ite_eq_right h, ite_eq_right h', mul_one, ha]
+    simp_rw [he]
+    rw [Finset.prod_mul_distrib]
+    simp [sq]
+
+lemma integrable_weighted_sum (ha : Integrable a μ) (hau : Integrable (fun x ↦ a x*u x) μ) :
+    Integrable (fun x : ι → X ↦ weightProduct a x*(∑ j, u (x j))) (Measure.pi fun _ ↦ μ) := by
+  simp_rw [Finset.mul_sum]
+  exact integrable_finsetSum _ (fun j _ ↦ integrable_weighted_single ha hau j)
+
+lemma integral_weighted_sum (ha : Integrable a μ) (hau : Integrable (fun x ↦ a x*u x) μ)
+    (hbal : ∫ x, a x ∂μ = 1) :
+    (∫ x : ι → X, weightProduct a x*(∑ j, u (x j)) ∂(Measure.pi fun _ ↦ μ)) =
+      (Fintype.card ι : ℝ)*(∫ x, a x*u x ∂μ) := by
+  simp_rw [Finset.mul_sum]
+  rw [integral_finsetSum _ (fun j _ ↦ integrable_weighted_single ha hau j)]
+  simp only [integral_weighted_single hbal, Finset.sum_const, Finset.card_univ, nsmul_eq_mul]
+
+lemma integrable_weighted_sq_sum (ha : Integrable a μ) (hau : Integrable (fun x ↦ a x*u x) μ)
+    (hau₂ : Integrable (fun x ↦ a x*u x^2) μ) :
+    Integrable (fun x : ι → X ↦ weightProduct a x*(∑ j, u (x j))^2) (Measure.pi fun _ ↦ μ) := by
+  have he (x : ι → X) : weightProduct a x*(∑ j, u (x j))^2 =
+      ∑ j, ∑ k, weightProduct a x*(u (x j)*u (x k)) := by
+    rw [sq, Finset.sum_mul_sum]
+    simp_rw [Finset.mul_sum]
+  simp_rw [he]
+  exact integrable_finsetSum _ (fun j _ ↦ integrable_finsetSum _ (fun k _ ↦ integrable_weighted_pair ha hau hau₂ j k))
+
+theorem integral_weighted_sq_sum (ha : Integrable a μ) (hau : Integrable (fun x ↦ a x*u x) μ)
+    (hau₂ : Integrable (fun x ↦ a x*u x^2) μ) (hbal : ∫ x, a x ∂μ = 1) :
+    (∫ x : ι → X, weightProduct a x*(∑ j, u (x j))^2 ∂(Measure.pi fun _ ↦ μ)) =
+      (Fintype.card ι : ℝ)*(∫ x, a x*u x^2 ∂μ)+
+      ((Fintype.card ι : ℝ)^2-(Fintype.card ι : ℝ))*(∫ x, a x*u x ∂μ)^2 := by
+  have he (x : ι → X) : weightProduct a x*(∑ j, u (x j))^2 =
+      ∑ j, ∑ k, weightProduct a x*(u (x j)*u (x k)) := by
+    rw [sq, Finset.sum_mul_sum]
+    simp_rw [Finset.mul_sum]
+  simp_rw [he]
+  rw [integral_finsetSum _ (fun j _ ↦ integrable_finsetSum _ (fun k _ ↦ integrable_weighted_pair ha hau hau₂ j k))]
+  simp_rw [integral_finsetSum _ (fun k _ ↦ integrable_weighted_pair ha hau hau₂ _ k), integral_weighted_pair hbal]
+  have he' (j k : ι) : (if j=k then (∫ x, a x*u x^2 ∂μ) else (∫ x, a x*u x ∂μ)^2) =
+      (if j=k then (∫ x, a x*u x^2 ∂μ)-(∫ x, a x*u x ∂μ)^2 else 0)+(∫ x, a x*u x ∂μ)^2 := by
+    split_ifs <;> ring
+  simp_rw [he', Finset.sum_add_distrib]
+  simp only [Finset.sum_ite_eq, Finset.mem_univ, ite_true, Finset.sum_const, Finset.card_univ, nsmul_eq_mul]
+  ring
+
+end ThreeState.TreeClauses.Experiment
+
+end 
+
+noncomputable section
+open Set MeasureTheory
+open scoped BigOperators
+namespace ThreeState.TreeClauses.Experiment
+open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
+
+abbrev branchEdges {n : ℕ} (lam : ℝ) (m : Fin n → Message) : Fin n → Vec := fun j ↦ edgeMessage lam (m j)
+
+def finiteBranchExpectation (Q : Law) (lam : ℝ) (n : ℕ) (f : Vec → ℝ) : ℝ :=
+  ∫ m : Fin n → Message, normalizer (branchEdges lam m)*f (normalizedProduct (branchEdges lam m))
+    ∂(Measure.pi fun _ ↦ Q.probability.toMeasure)
+
+def correlationEntropy (Q : Law) (lam : ℝ) (n : ℕ) : ℝ :=
+  ∫ m : Fin n → Message, normalizer (branchEdges lam m)*Real.log (normalizer (branchEdges lam m))
+    ∂(Measure.pi fun _ ↦ Q.probability.toMeasure)
+
+lemma branchEdges_positive {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) {n : ℕ} (m : Fin n → Message) :
+    ∀ j i, 0 < branchEdges lam m j i := fun j i ↦ (edge_positive hl₀ hl₁ (m j)).1 i
+
+lemma continuous_branchProduct {n : ℕ} (lam : ℝ) (i : Fin 3) :
+    Continuous (fun m : Fin n → Message ↦ branchProduct (branchEdges lam m) i) := by
+  unfold branchProduct branchEdges
+  exact continuous_finsetProd _ (fun j _ ↦ ((continuous_apply i).comp (continuous_edgeMessage lam)).comp (continuous_apply j))
+
+lemma continuous_normalizer (n : ℕ) (lam : ℝ) :
+    Continuous (fun m : Fin n → Message ↦ normalizer (branchEdges lam m)) :=
+  continuous_avg.comp (continuous_pi (continuous_branchProduct lam))
+
+lemma continuous_correlationIntegrand {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
+    Continuous (fun m : Fin n → Message ↦ normalizer (branchEdges lam m)*Real.log (normalizer (branchEdges lam m))) :=
+  (continuous_normalizer n lam).mul ((continuous_normalizer n lam).log
+    (fun m ↦ ne_of_gt (normalizer_positive (branchEdges_positive hl₀ hl₁ m))))
+
+lemma integral_normalizer (Q : Law) (lam : ℝ) (n : ℕ) :
+    (∫ m : Fin n → Message, normalizer (branchEdges lam m) ∂(Measure.pi fun _ ↦ Q.probability.toMeasure)) = 1 := by
+  unfold normalizer
+  rw [integral_avg_measure (fun i ↦ compact_integrable (continuous_branchProduct lam i))]
+  have he (i : Fin 3) : (∫ m : Fin n → Message, branchProduct (branchEdges lam m) i
+      ∂(Measure.pi fun _ ↦ Q.probability.toMeasure)) = 1 := by
+    change (∫ m : Fin n → Message, ∏ j, edgeMessage lam (m j) i ∂(Measure.pi fun _ ↦ Q.probability.toMeasure)) = 1
+    rw [integral_fintype_prod_eq_pow (fun m ↦ edgeMessage lam m i), Q.edge_balanced]
+    simp
+  simp_rw [he]
+  exact avg_const 1
+
+lemma correlationEntropy_nonneg (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
+    0 ≤ correlationEntropy Q lam n := by
+  have he (m : Fin n → Message) : normalizer (branchEdges lam m)-1 ≤
+      normalizer (branchEdges lam m)*Real.log (normalizer (branchEdges lam m)) := by
+    have hp := normalizer_positive (branchEdges_positive hl₀ hl₁ m)
+    have h := mul_le_mul_of_nonneg_left (Real.one_sub_inv_le_log_of_pos hp) hp.le
+    simpa [mul_sub, ne_of_gt hp] using h
+  have h := integral_mono (μ := Measure.pi fun _ : Fin n ↦ Q.probability.toMeasure)
+    (f := fun m ↦ normalizer (branchEdges lam m)-1)
+    (g := fun m ↦ normalizer (branchEdges lam m)*Real.log (normalizer (branchEdges lam m)))
+    ((compact_integrable (continuous_normalizer n lam)).sub (integrable_const 1))
+    (compact_integrable (continuous_correlationIntegrand hl₀ hl₁ n)) he
+  rw [integral_sub (f := fun m : Fin n → Message ↦ normalizer (branchEdges lam m))
+    (g := fun _ ↦ (1:ℝ)) (compact_integrable (continuous_normalizer n lam)) (integrable_const _),
+    integral_normalizer] at h
+  simpa [correlationEntropy] using h
+
+theorem finite_symEntropy (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
+    finiteBranchExpectation Q lam n symEntropy =
+      n*(∫ m, symEntropy (edgeMessage lam m) ∂Q.probability.toMeasure) := by
+  have ha (i : Fin 3) : Integrable (fun m ↦ edgeMessage lam m i) Q.probability.toMeasure :=
+    compact_integrable ((continuous_apply i).comp (continuous_edgeMessage lam))
+  have hau (i : Fin 3) : Integrable (fun m ↦ edgeMessage lam m i*logCentered (edgeMessage lam m) i) Q.probability.toMeasure :=
+    compact_integrable (((continuous_apply i).comp (continuous_edgeMessage lam)).mul
+      (continuous_edge_centeredLog hl₀ hl₁ i))
+  unfold finiteBranchExpectation
+  simp_rw [weighted_symEntropy (branchEdges_positive hl₀ hl₁ _)]
+  rw [integral_div]
+  rw [integral_avg_measure (f := fun m : Fin n → Message ↦ fun i ↦ branchProduct (branchEdges lam m) i*(∑ j, logCentered (branchEdges lam m j) i))
+    (fun i ↦ integrable_weighted_sum (ι := Fin n) (ha i) (hau i))]
+  have he (i : Fin 3) : (∫ m : Fin n → Message, branchProduct (branchEdges lam m) i*(∑ j, logCentered (branchEdges lam m j) i)
+      ∂(Measure.pi fun _ ↦ Q.probability.toMeasure)) =
+      n*(2*∫ m, symEntropy (edgeMessage lam m) ∂Q.probability.toMeasure) := by
+    have h := integral_weighted_sum (ι := Fin n) (ha i) (hau i) (Q.edge_balanced lam i)
+    rw [Q.edge_centeredLog_mean hl₀ hl₁] at h
+    simpa only [Fintype.card_fin, weightProduct, branchProduct, branchEdges] using h
+  simp_rw [he]
+  rw [avg_const]
+  ring
+
+theorem finite_correction (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
+    finiteBranchExpectation Q lam n correction =
+      n*(∫ m, correction (edgeMessage lam m) ∂Q.probability.toMeasure)+
+      2*((n:ℝ)^2-n)*(∫ m, symEntropy (edgeMessage lam m) ∂Q.probability.toMeasure)^2 := by
+  have ha (i : Fin 3) : Integrable (fun m ↦ edgeMessage lam m i) Q.probability.toMeasure :=
+    compact_integrable ((continuous_apply i).comp (continuous_edgeMessage lam))
+  have hau (i : Fin 3) : Integrable (fun m ↦ edgeMessage lam m i*logCentered (edgeMessage lam m) i) Q.probability.toMeasure :=
+    compact_integrable (((continuous_apply i).comp (continuous_edgeMessage lam)).mul
+      (continuous_edge_centeredLog hl₀ hl₁ i))
+  have hau₂ (i : Fin 3) : Integrable (fun m ↦ edgeMessage lam m i*logCentered (edgeMessage lam m) i^2) Q.probability.toMeasure :=
+    compact_integrable (((continuous_apply i).comp (continuous_edgeMessage lam)).mul
+      ((continuous_edge_centeredLog hl₀ hl₁ i).pow 2))
+  unfold finiteBranchExpectation
+  simp_rw [weighted_correction (branchEdges_positive hl₀ hl₁ _)]
+  rw [integral_div]
+  rw [integral_avg_measure (f := fun m : Fin n → Message ↦ fun i ↦ branchProduct (branchEdges lam m) i*(∑ j, logCentered (branchEdges lam m j) i)^2)
+    (fun i ↦ integrable_weighted_sq_sum (ι := Fin n) (ha i) (hau i) (hau₂ i))]
+  have he (i : Fin 3) : (∫ m : Fin n → Message, branchProduct (branchEdges lam m) i*(∑ j, logCentered (branchEdges lam m j) i)^2
+      ∂(Measure.pi fun _ ↦ Q.probability.toMeasure)) =
+      n*(2*∫ m, correction (edgeMessage lam m) ∂Q.probability.toMeasure)+
+      ((n:ℝ)^2-n)*(2*∫ m, symEntropy (edgeMessage lam m) ∂Q.probability.toMeasure)^2 := by
+    have h := integral_weighted_sq_sum (ι := Fin n) (ha i) (hau i) (hau₂ i) (Q.edge_balanced lam i)
+    rw [Q.edge_centeredLog_mean hl₀ hl₁, Q.edge_centeredLog_sq hl₀ hl₁] at h
+    simpa only [Fintype.card_fin, weightProduct, branchProduct, branchEdges] using h
+  simp_rw [he]
+  rw [avg_const]
+  ring
+
+theorem finite_info (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
+    finiteBranchExpectation Q lam n info =
+      n*(∫ m, info (edgeMessage lam m) ∂Q.probability.toMeasure)-correlationEntropy Q lam n := by
+  have ha (i : Fin 3) : Integrable (fun m ↦ edgeMessage lam m i) Q.probability.toMeasure :=
+    compact_integrable ((continuous_apply i).comp (continuous_edgeMessage lam))
+  have hau (i : Fin 3) : Integrable (fun m ↦ edgeMessage lam m i*Real.log (edgeMessage lam m i)) Q.probability.toMeasure :=
+    compact_integrable (((continuous_apply i).comp (continuous_edgeMessage lam)).mul
+      (continuous_edge_log hl₀ hl₁ i))
+  unfold finiteBranchExpectation
+  simp_rw [weighted_info (branchEdges_positive hl₀ hl₁ _)]
+  rw [integral_sub (f := fun m : Fin n → Message ↦ avg (fun i ↦ branchProduct (branchEdges lam m) i*(∑ j, Real.log (branchEdges lam m j i))))
+    (g := fun m ↦ normalizer (branchEdges lam m)*Real.log (normalizer (branchEdges lam m)))
+    (integrable_avg (fun i ↦ integrable_weighted_sum (ι := Fin n) (ha i) (hau i)))
+    (compact_integrable (continuous_correlationIntegrand hl₀ hl₁ n))]
+  congr 1
+  rw [integral_avg_measure (f := fun m : Fin n → Message ↦ fun i ↦ branchProduct (branchEdges lam m) i*(∑ j, Real.log (branchEdges lam m j i)))
+    (fun i ↦ integrable_weighted_sum (ι := Fin n) (ha i) (hau i))]
+  have he (i : Fin 3) : (∫ m : Fin n → Message, branchProduct (branchEdges lam m) i*(∑ j, Real.log (branchEdges lam m j i))
+      ∂(Measure.pi fun _ ↦ Q.probability.toMeasure)) =
+      n*(∫ m, edgeMessage lam m i*Real.log (edgeMessage lam m i) ∂Q.probability.toMeasure) := by
+    simpa only [Fintype.card_fin, weightProduct, branchProduct, branchEdges] using integral_weighted_sum (ι := Fin n) (ha i) (hau i) (Q.edge_balanced lam i)
+  simp_rw [he]
+  rw [avg_mul, ← integral_avg_measure hau]
+  rfl
+
+end ThreeState.TreeClauses.Experiment
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeStateClauses/CompactLimits.lean b/lean/OAI/Probability/ThreeStateClauses/CompactLimits.lean
index 6b510c94f4933169e4f332b298be1ba776ab83df..6eee225578f4bbbe7352774632ced76c7873ff77 100644
--- a/lean/OAI/Probability/ThreeStateClauses/CompactLimits.lean
+++ b/lean/OAI/Probability/ThreeStateClauses/CompactLimits.lean
@@ -1,438 +1,438 @@
-import Mathlib
-import OAI.Probability.ThreeStateClauses.Degradation
-import OAI.Probability.ThreeStateClauses.RegularLaw
-import OAI.Probability.ThreeStateClauses.Projection
-
-namespace OAI
-
-/-! Compact Limits. -/
-
-open scoped BigOperators ENNReal NNReal Topology
-open Filter
-noncomputable section
-open Set MeasureTheory
-open scoped BigOperators ENNReal
-namespace ThreeState.TreeClauses.Experiment
-open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
-
-def quadraticGap (z : Message × Message) : ℝ := avg (fun i ↦ (z.1.1 i-z.2.1 i)^2)/2
-
-lemma continuous_quadraticGap : Continuous quadraticGap := by
-  unfold quadraticGap
-  exact (continuous_avg.comp (continuous_pi (fun i ↦
-    ((continuous_coordinate i).comp continuous_fst |>.sub ((continuous_coordinate i).comp continuous_snd)).pow 2))).div_const 2
-
-lemma quadraticGap_nonneg (z : Message × Message) : 0 ≤ quadraticGap z := by
-  dsimp [quadraticGap, avg]; positivity
-
-lemma quadraticGap_zero_iff (z : Message × Message) : quadraticGap z = 0 ↔ z.1 = z.2 := by
-  constructor
-  · intro h
-    apply Subtype.ext
-    funext i
-    have h₀ := sq_nonneg (z.1.1 0-z.2.1 0)
-    have h₁ := sq_nonneg (z.1.1 1-z.2.1 1)
-    have h₂ := sq_nonneg (z.1.1 2-z.2.1 2)
-    dsimp [quadraticGap, avg] at h
-    fin_cases i <;> dsimp <;> nlinarith
-  · intro h; simp [quadraticGap, h, avg]
-
-lemma quadraticGap_expand (z : Message × Message) :
-    quadraticGap z = xMoment z.1-xMoment z.2-
-      avg (fun i ↦ (z.1.1 i-z.2.1 i)*(z.2.1 i-1)) := by
-  dsimp [quadraticGap, xMoment, momentX, avg, centered]; ring
-
-lemma continuous_centeredProduct (i : Fin 3) :
-    Continuous (fun z : Message × Message ↦ (z.1.1 i-z.2.1 i)*(z.2.1 i-1)) :=
-  (((continuous_coordinate i).comp continuous_fst).sub ((continuous_coordinate i).comp continuous_snd)).mul
-    (((continuous_coordinate i).comp continuous_snd).sub continuous_const)
-
-section Discrete
-variable {α β : Type*} [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α]
-  [Countable β] [MeasurableSpace β] [MeasurableSingletonClass β]
-
-def degradationCoupling (p : Fin 3 → PMF α) (K : α → PMF β) : ProbabilityMeasure (Message × Message) :=
-  ⟨((kernelJoint (discreteMarginal p) K).map (fun z ↦
-    (discreteMessage p z.1, discreteMessage (fun i ↦ (p i).bind K) z.2))).toMeasure, inferInstance⟩
-
-lemma integral_degradationCoupling (p : Fin 3 → PMF α) (K : α → PMF β)
-    {f : Message × Message → ℝ} (hf : Measurable f) :
-    (∫ z, f z ∂(degradationCoupling p K).toMeasure) =
-      ∫ z, f (discreteMessage p z.1, discreteMessage (fun i ↦ (p i).bind K) z.2)
-        ∂(kernelJoint (discreteMarginal p) K).toMeasure := by
-  change (∫ z, f z ∂(PMF.map _ _).toMeasure) = _
-  rw [← PMF.toMeasure_map _ _ (measurable_of_countable _)]
-  exact integral_map (measurable_of_countable _).aemeasurable hf.aestronglyMeasurable
-
-omit [Countable β] [MeasurableSpace β] [MeasurableSingletonClass β] in
-lemma degradationCoupling_fst (p : Fin 3 → PMF α) (K : α → PMF β) :
-    (degradationCoupling p K).map Prod.fst = discreteProbability p := by
-  apply Subtype.ext
-  change Measure.map Prod.fst (PMF.map _ _).toMeasure = Measure.map _ (discreteMarginal p).toMeasure
-  rw [PMF.toMeasure_map _ _ measurable_fst, PMF.map_comp]
-  change (PMF.map ((discreteMessage p) ∘ Prod.fst) _).toMeasure = _
-  rw [← PMF.map_comp, kernelJoint_fst, PMF.toMeasure_map _ _ (measurable_of_countable _)]
-
-omit [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α] in
-lemma degradationCoupling_snd (p : Fin 3 → PMF α) (K : α → PMF β) :
-    (degradationCoupling p K).map Prod.snd =
-      discreteProbability (fun i ↦ (p i).bind K) := by
-  apply Subtype.ext
-  change Measure.map Prod.snd (PMF.map _ _).toMeasure = Measure.map _ (discreteMarginal _).toMeasure
-  rw [PMF.toMeasure_map _ _ measurable_snd, PMF.map_comp]
-  change (PMF.map ((discreteMessage (fun i ↦ (p i).bind K)) ∘ Prod.snd) _).toMeasure = _
-  rw [← PMF.map_comp, kernelJoint_snd, ← discreteMarginal_bind,
-    PMF.toMeasure_map _ _ (measurable_of_countable _)]
-
-lemma degradationCoupling_centered (p : Fin 3 → PMF α) (K : α → PMF β) (i : Fin 3) :
-    (∫ z, (z.1.1 i-z.2.1 i)*(z.2.1 i-1) ∂(degradationCoupling p K).toMeasure) = 0 := by
-  rw [integral_degradationCoupling _ _ (continuous_centeredProduct i).measurable]
-  exact discrete_degradation_centered p K i
-    (g := fun b ↦ (discreteMessage (fun j ↦ (p j).bind K) b).1 i-1) (C := 2) (fun b ↦ by
-    apply abs_le.mpr
-    constructor <;> linarith [coordinate_nonneg (discreteMessage (fun j ↦ (p j).bind K) b) i,
-      coordinate_le_three (discreteMessage (fun j ↦ (p j).bind K) b) i])
-
-lemma degradationCoupling_loss (p : Fin 3 → PMF α) (K : α → PMF β) :
-    (∫ z, quadraticGap z ∂(degradationCoupling p K).toMeasure) =
-      (∫ m, xMoment m ∂(discreteProbability p).toMeasure)-
-      ∫ m, xMoment m ∂(discreteProbability (fun i ↦ (p i).bind K)).toMeasure := by
-  let C := degradationCoupling p K
-  have hx : Integrable (fun z : Message × Message ↦ xMoment z.1) C.toMeasure :=
-    compact_integrable (continuous_xMoment.comp continuous_fst)
-  have hy : Integrable (fun z : Message × Message ↦ xMoment z.2) C.toMeasure :=
-    compact_integrable (continuous_xMoment.comp continuous_snd)
-  have hc (i : Fin 3) : Integrable (fun z : Message × Message ↦ (z.1.1 i-z.2.1 i)*(z.2.1 i-1)) C.toMeasure :=
-    compact_integrable (continuous_centeredProduct i)
-  have hz : (∫ z, avg (fun i ↦ (z.1.1 i-z.2.1 i)*(z.2.1 i-1)) ∂C.toMeasure) = 0 := by
-    rw [integral_avg_measure hc]
-    simp only [C, degradationCoupling_centered, avg]
-    norm_num
-  simp_rw [quadraticGap_expand]
-  rw [integral_sub (f := fun z : Message × Message ↦ xMoment z.1-xMoment z.2)
-    (g := fun z : Message × Message ↦ avg (fun i ↦ (z.1.1 i-z.2.1 i)*(z.2.1 i-1)))
-    (hx.sub hy) (integrable_avg hc),
-    integral_sub (f := fun z : Message × Message ↦ xMoment z.1)
-    (g := fun z : Message × Message ↦ xMoment z.2) hx hy, hz, sub_zero]
-  congr 1
-  · rw [← degradationCoupling_fst p K]
-    exact (integral_map measurable_fst.aemeasurable continuous_xMoment.measurable.aestronglyMeasurable).symm
-  · rw [← degradationCoupling_snd p K]
-    exact (integral_map measurable_snd.aemeasurable continuous_xMoment.measurable.aestronglyMeasurable).symm
-
-lemma discrete_degradation_mu (p : Fin 3 → PMF α) (K : α → PMF β) :
-    (∫ m, xMoment m ∂(discreteProbability (fun i ↦ (p i).bind K)).toMeasure) ≤
-      ∫ m, xMoment m ∂(discreteProbability p).toMeasure := by
-  have h := MeasureTheory.integral_nonneg (μ := (degradationCoupling p K).toMeasure)
-    (f := quadraticGap) quadraticGap_nonneg
-  rw [degradationCoupling_loss] at h
-  linarith
-
-end Discrete
-end ThreeState.TreeClauses.Experiment
-
-end 
-
-noncomputable section
-open Set MeasureTheory
-open scoped BigOperators ENNReal
-namespace ThreeState.TreeClauses.Tree
-open ThreeState.TreeClauses.Experiment ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
-
-def vertex (i : Spin) : Message :=
-  ⟨fun j ↦ if i = j then 3 else 0, fun j ↦ by dsimp; split_ifs <;> norm_num, by
-    fin_cases i <;> norm_num [avg, Fin.ext_iff]⟩
-
-lemma discreteMessage_pure (i : Spin) : discreteMessage (fun j : Spin ↦ PMF.pure j) i = vertex i := by
-  apply Subtype.ext
-  funext j
-  fin_cases i <;> fin_cases j <;> norm_num [discreteMessage, discreteWeight, avg, PMF.pure_apply, vertex, Fin.ext_iff]
-
-lemma vertex_permute (σ : Equiv.Perm Spin) (i : Spin) : permute σ (vertex i) = vertex (σ.symm i) := by
-  apply Subtype.ext
-  funext j
-  change (if i = σ j then (3:ℝ) else 0) = if σ.symm i = j then 3 else 0
-  simp only [σ.symm_apply_eq]
-
-lemma pureMarginal : discreteMarginal (fun i : Spin ↦ PMF.pure i) = uniformSpin := PMF.bind_pure _
-
-lemma integral_rootProbability {f : Message → ℝ} (hf : Measurable f) :
-    (∫ m, f m ∂(discreteProbability (fun i : Spin ↦ PMF.pure i)).toMeasure) =
-      (∑ i : Spin, f (vertex i))/3 := by
-  rw [integral_discreteProbability _ hf, pureMarginal, PMF.integral_eq_sum]
-  simp only [discreteMessage_pure, smul_eq_mul, uniformSpin_apply, ENNReal.toReal_div,
-    ENNReal.toReal_one, ENNReal.toReal_ofNat]
-  rw [← Finset.mul_sum]
-  ring
-
-lemma rootProbability_symmetric (σ : Equiv.Perm Spin) :
-    MeasurePreserving (permute σ) (discreteProbability (fun i : Spin ↦ PMF.pure i)).toMeasure
-      (discreteProbability (fun i : Spin ↦ PMF.pure i)).toMeasure := by
-  refine ⟨(continuous_permute σ).measurable, ?_⟩
-  have h : (discreteProbability (fun i : Spin ↦ PMF.pure i)).map
-      (permute σ) = discreteProbability (fun i : Spin ↦ PMF.pure i) := by
-    apply probability_ext_continuous
-    intro f hf
-    change (∫ m, f m ∂Measure.map (permute σ) _) = _
-    rw [integral_map (continuous_permute σ).measurable.aemeasurable hf.measurable.aestronglyMeasurable,
-      integral_rootProbability (f := fun m ↦ f (permute σ m)) (hf.comp (continuous_permute σ)).measurable,
-      integral_rootProbability hf.measurable]
-    simp only [vertex_permute]
-    exact congrArg (fun x : ℝ ↦ x/3) (Equiv.sum_comp σ.symm (fun i ↦ f (vertex i)))
-  exact congrArg ProbabilityMeasure.toMeasure h
-
-def rootLaw : Law where
-  probability := discreteProbability (fun i : Spin ↦ PMF.pure i)
-  balanced := discreteProbability_balanced _
-  symmetric := rootProbability_symmetric
-
-def regularPosteriorLaw (b : ℕ) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) : ℕ → Law
-  | 0 => rootLaw
-  | ℓ+1 => finiteLaw (regularPosteriorLaw b hl₀ hl₁ ℓ) hl₀ hl₁ b
-
-lemma regularPosteriorLaw_probability (b : ℕ) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
-    (hlam : Admissible lam) (ℓ : ℕ) :
-    (regularPosteriorLaw b hl₀ hl₁ ℓ).probability = discreteProbability (regularLaw b lam hlam ℓ) := by
-  induction ℓ with
-  | zero => rfl
-  | succ ℓ ih =>
-    exact (discreteBranchLaw_probability (regularLaw b lam hlam ℓ)
-      (regularPosteriorLaw b hl₀ hl₁ ℓ) ih hl₀ hl₁ hlam b).symm
-
-lemma regularPosteriorLaw_antitone (b : ℕ) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
-    (hlam : Admissible lam) : Antitone (fun ℓ ↦ (regularPosteriorLaw b hl₀ hl₁ ℓ).mu) := by
-  apply antitone_nat_of_succ_le
-  intro ℓ
-  unfold Law.mu
-  rw [regularPosteriorLaw_probability b hl₀ hl₁ hlam ℓ,
-    regularPosteriorLaw_probability b hl₀ hl₁ hlam (ℓ+1)]
-  have h := discrete_degradation_mu (regularLaw b lam hlam ℓ) (regularExpansion b lam hlam ℓ)
-  simpa only [regularLaw_degrades] using h
-
-end ThreeState.TreeClauses.Tree
-
-end 
-
-noncomputable section
-open Set MeasureTheory Filter
-open scoped Topology
-namespace ThreeState.TreeClauses.Experiment
-open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
-
-lemma probability_continuous_integral {X : Type*} [TopologicalSpace X] [CompactSpace X]
-    [MeasurableSpace X] [OpensMeasurableSpace X] {f : X → ℝ} (hf : Continuous f) :
-    Continuous (fun P : ProbabilityMeasure X ↦ ∫ x, f x ∂P.toMeasure) :=
-  ProbabilityMeasure.continuous_integral_continuousMap (⟨f, hf⟩ : C(X, ℝ))
-
-section Limits
-variable {ι : Type*} {F : Filter ι} [F.NeBot] {Q : ι → Law} {P : ProbabilityMeasure Message}
-
-lemma limit_balanced (hP : Tendsto (fun i ↦ (Q i).probability) F (𝓝 P)) (j : Fin 3) :
-    (∫ m, m.1 j ∂P.toMeasure) = 1 := by
-  have h := (probability_continuous_integral (continuous_coordinate j)).tendsto P |>.comp hP
-  simp only [Function.comp_def, Law.balanced] at h
-  exact tendsto_nhds_unique h tendsto_const_nhds
-
-lemma limit_symmetric (hP : Tendsto (fun i ↦ (Q i).probability) F (𝓝 P))
-    (σ : Equiv.Perm (Fin 3)) : MeasurePreserving (permute σ) P.toMeasure P.toMeasure := by
-  refine ⟨(continuous_permute σ).measurable, ?_⟩
-  have hm := (ProbabilityMeasure.continuous_map (continuous_permute σ)).tendsto P |>.comp hP
-  have he (i : ι) : (Q i).probability.map (permute σ) =
-      (Q i).probability := Subtype.ext ((Q i).symmetric σ).map_eq
-  simp only [Function.comp_def, he] at hm
-  exact congrArg ProbabilityMeasure.toMeasure (tendsto_nhds_unique hm hP)
-
-def limitLaw (hP : Tendsto (fun i ↦ (Q i).probability) F (𝓝 P)) : Law where
-  probability := P
-  balanced := limit_balanced hP
-  symmetric := limit_symmetric hP
-
-end Limits
-
-lemma finiteProbability_tendsto {ι : Type*} {F : Filter ι} {Q : ι → Law} {R : Law}
-    (hQ : Tendsto (fun i ↦ (Q i).probability) F (𝓝 R.probability))
-    {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
-    Tendsto (fun i ↦ finiteProbability (Q i) hl₀ hl₁ n) F (𝓝 (finiteProbability R hl₀ hl₁ n)) := by
-  apply ProbabilityMeasure.tendsto_iff_forall_integral_tendsto.mpr
-  intro f
-  have hp : Tendsto (fun i ↦ ProbabilityMeasure.pi (fun _ : Fin n ↦ (Q i).probability)) F
-      (𝓝 (ProbabilityMeasure.pi (fun _ : Fin n ↦ R.probability))) :=
-    ProbabilityMeasure.continuous_pi.tendsto _ |>.comp (tendsto_pi_nhds.mpr (fun _ ↦ hQ))
-  have hfun : Continuous (fun m : Fin n → Message ↦ normalizer (branchEdges lam m)*f (branchOutput hl₀ hl₁ m)) :=
-    (continuous_normalizer n lam).mul (f.continuous.comp (continuous_branchOutput hl₀ hl₁ n))
-  have h := (probability_continuous_integral hfun).tendsto _ |>.comp hp
-  simpa only [Function.comp_def, ProbabilityMeasure.toMeasure_pi,
-    integral_finiteProbability _ hl₀ hl₁ n f.continuous.measurable] using h
-
-lemma coupling_diagonal_of_loss_zero (C : ProbabilityMeasure (Message × Message))
-    (h : (∫ z, quadraticGap z ∂C.toMeasure) = 0) :
-    C.map Prod.fst = C.map Prod.snd := by
-  have he := (integral_eq_zero_iff_of_nonneg quadraticGap_nonneg
-    (compact_integrable continuous_quadraticGap)).mp h
-  apply Subtype.ext
-  exact Measure.map_congr (he.mono (fun z hz ↦ (quadraticGap_zero_iff z).mp hz))
-
-end ThreeState.TreeClauses.Experiment
-
-end 
-
-noncomputable section
-open Set MeasureTheory Filter
-open scoped Topology
-namespace ThreeState.TreeClauses.Experiment
-open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
-
-lemma degrading_finite_fixed_point (Q : ℕ → Law) (C : ℕ → ProbabilityMeasure (Message × Message))
-    (hfst : ∀ ℓ, (C ℓ).map Prod.fst = (Q ℓ).probability)
-    (hsnd : ∀ ℓ, (C ℓ).map Prod.snd = (Q (ℓ+1)).probability)
-    (hloss : ∀ ℓ, (∫ z, quadraticGap z ∂(C ℓ).toMeasure) = (Q ℓ).mu-(Q (ℓ+1)).mu)
-    {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ)
-    (hrec : ∀ ℓ, (Q (ℓ+1)).probability = finiteProbability (Q ℓ) hl₀ hl₁ n) :
-    ∃ R : Law, R.probability = finiteProbability R hl₀ hl₁ n ∧
-      Tendsto (fun ℓ ↦ (Q ℓ).mu) atTop (𝓝 R.mu) := by
-  have hanti : Antitone (fun ℓ ↦ (Q ℓ).mu) := by
-    apply antitone_nat_of_succ_le
-    intro ℓ
-    have h := MeasureTheory.integral_nonneg (μ := (C ℓ).toMeasure) (f := quadraticGap) quadraticGap_nonneg
-    rw [hloss] at h
-    linarith
-  let a := ⨅ ℓ, (Q ℓ).mu
-  have hlim : Tendsto (fun ℓ ↦ (Q ℓ).mu) atTop (𝓝 a) :=
-    tendsto_atTop_ciInf hanti ⟨0, by rintro _ ⟨ℓ,rfl⟩; exact (Q ℓ).mu_nonneg⟩
-  have hdiff : Tendsto (fun ℓ ↦ (Q ℓ).mu-(Q (ℓ+1)).mu) atTop (𝓝 0) := by
-    simpa using hlim.sub (hlim.comp (tendsto_add_atTop_nat 1))
-  obtain ⟨D, hD⟩ := exists_clusterPt_of_compactSpace (Filter.map C atTop)
-  let F : Filter ℕ := Filter.comap C (𝓝 D) ⊓ atTop
-  have : F.NeBot := neBot_inf_comap_iff_map'.mpr hD
-  have hF : F ≤ atTop := inf_le_right
-  have hC : Tendsto C F (𝓝 D) := tendsto_comap.mono_left inf_le_left
-  have hgap : (∫ z, quadraticGap z ∂D.toMeasure) = 0 := by
-    have hg := (probability_continuous_integral continuous_quadraticGap).tendsto D |>.comp hC
-    simp only [Function.comp_def, hloss] at hg
-    exact tendsto_nhds_unique hg (hdiff.mono_left hF)
-  have hdiag := coupling_diagonal_of_loss_zero D hgap
-  have hP := (ProbabilityMeasure.continuous_map (continuous_fst : Continuous (Prod.fst : Message × Message → Message))).tendsto D |>.comp hC
-  simp only [Function.comp_def, hfst] at hP
-  let R : Law := limitLaw hP
-  have hS := (ProbabilityMeasure.continuous_map (continuous_snd : Continuous (Prod.snd : Message × Message → Message))).tendsto D |>.comp hC
-  simp only [Function.comp_def, hsnd] at hS
-  rw [← hdiag] at hS
-  have hR : R.probability = finiteProbability R hl₀ hl₁ n := by
-    have hr := finiteProbability_tendsto (R := R) hP hl₀ hl₁ n
-    simp only [← hrec] at hr
-    exact tendsto_nhds_unique hS hr
-  have hm : a = R.mu := by
-    have hp := (probability_continuous_integral continuous_xMoment).tendsto R.probability |>.comp hP
-    exact tendsto_nhds_unique (hlim.mono_left hF) hp
-  exact ⟨R, hR, hm ▸ hlim⟩
-
-end ThreeState.TreeClauses.Experiment
-
-end 
-
-noncomputable section
-open Set MeasureTheory
-open scoped BigOperators ENNReal
-namespace ThreeState.TreeClauses.Experiment
-open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
-
-def tvMessage (m : Message) : ℝ := avg (fun i ↦ |m.1 i-1|)/2
-
-lemma continuous_tvMessage : Continuous tvMessage :=
-  (continuous_avg.comp (continuous_pi (fun i ↦ ((continuous_coordinate i).sub continuous_const).abs))).div_const 2
-
-lemma tvMessage_nonneg (m : Message) : 0 ≤ tvMessage m := by
-  dsimp [tvMessage, avg]; positivity
-
-lemma tvMessage_formula (m : Message) :
-    tvMessage m = (∑ i : Spin, |m.1 i/3-(1/3:ℝ)|)/2 := by
-  simp only [tvMessage, avg, ← sub_div, abs_div, abs_of_pos (by norm_num : (0:ℝ) < 3), Fin.sum_univ_three]
-  ring
-
-lemma tvMessage_sq_bound (m : Message) : tvMessage m^2 ≤ xMoment m/2 := by
-  have h₀ := sq_nonneg (|m.1 0-1|-|m.1 1-1|)
-  have h₁ := sq_nonneg (|m.1 1-1|-|m.1 2-1|)
-  have h₂ := sq_nonneg (|m.1 2-1|-|m.1 0-1|)
-  dsimp [tvMessage, xMoment, momentX, centered, avg]
-  nlinarith only [h₀,h₁,h₂, sq_abs (m.1 0-1), sq_abs (m.1 1-1), sq_abs (m.1 2-1)]
-
-lemma integral_tvMessage_sq_bound (P : ProbabilityMeasure Message) :
-    (∫ m, tvMessage m ∂P.toMeasure)^2 ≤ (∫ m, xMoment m ∂P.toMeasure)/2 := by
-  let t := ∫ m, tvMessage m ∂P.toMeasure
-  have hi := compact_integrable continuous_tvMessage (μ := P.toMeasure)
-  have hi₂ := compact_integrable (continuous_tvMessage.pow 2) (μ := P.toMeasure)
-  have h := MeasureTheory.integral_nonneg (μ := P.toMeasure) (f := fun m ↦ (tvMessage m-t)^2)
-    (fun m ↦ sq_nonneg (tvMessage m-t))
-  have he : (fun m : Message ↦ (tvMessage m-t)^2) = fun m ↦ tvMessage m^2-2*t*tvMessage m+t^2 := by
-    funext m; ring
-  rw [he, integral_add (f := fun m ↦ tvMessage m^2-2*t*tvMessage m) (g := fun _ ↦ t^2)
-    (hi₂.sub (hi.const_mul _)) (integrable_const _),
-    integral_sub (f := fun m ↦ tvMessage m^2) (g := fun m ↦ 2*t*tvMessage m) hi₂ (hi.const_mul _),
-    integral_const_mul] at h
-  simp only [integral_const, measure_univ, Measure.real, ENNReal.toReal_one, smul_eq_mul, one_mul] at h
-  have hb := integral_mono (μ := P.toMeasure) hi₂ (compact_integrable (continuous_xMoment.div_const 2)) tvMessage_sq_bound
-  rw [integral_div] at hb
-  change (∫ m, tvMessage m^2 ∂P.toMeasure) ≤ (∫ m, xMoment m ∂P.toMeasure)/2 at hb
-  change 0 ≤ (∫ m, tvMessage m^2 ∂P.toMeasure)-2*t*t+t^2 at h
-  change t^2 ≤ _
-  nlinarith
-
-lemma main_marginal_eq (lam : ℝ) (hlam : Admissible lam) (offspring : PMF ℕ)
-    (ℓ : ℕ) (o : Observation ℓ) :
-    marginalWeight lam hlam offspring ℓ o = discreteMarginal (observationLaw lam hlam offspring ℓ) o := by
-  simp only [marginalWeight, jointWeight, discreteMarginal, PMF.bind_apply, tsum_fintype, uniformSpin_apply]
-
-lemma main_posterior_eq (lam : ℝ) (hlam : Admissible lam) (offspring : PMF ℕ)
-    (ℓ : ℕ) (o : Observation ℓ) (ho : discreteWeight (observationLaw lam hlam offspring ℓ) o ≠ 0) (i : Spin) :
-    posterior lam hlam offspring ℓ o i = (discreteMessage (observationLaw lam hlam offspring ℓ) o).1 i/3 := by
-  rw [posterior, ENNReal.toReal_div, main_marginal_eq, discreteMarginal_real, jointWeight,
-    ENNReal.toReal_mul, discrete_coord ho]
-  norm_num
-  ring
-
-lemma tvMessage_le_one (m : Message) : tvMessage m ≤ 1 := by
-  have hb (i : Spin) : |m.1 i-1| ≤ 2 := abs_le.mpr ⟨by linarith [coordinate_nonneg m i],
-    by linarith [coordinate_le_three m i]⟩
-  dsimp [tvMessage, avg]
-  linarith [hb 0, hb 1, hb 2]
-
-lemma integrable_tv_discrete {α : Type*} [Countable α] [MeasurableSpace α]
-    [MeasurableSingletonClass α] (p : Spin → PMF α) :
-    Integrable (fun o ↦ tvMessage (discreteMessage p o)) (discreteMarginal p).toMeasure := by
-  apply Integrable.of_bound (measurable_of_countable _).aestronglyMeasurable 1
-  exact Filter.Eventually.of_forall (fun a ↦ by
-    rw [Real.norm_eq_abs, abs_of_nonneg (tvMessage_nonneg _)]
-    exact tvMessage_le_one _)
-
-lemma advantage_eq_integral (lam : ℝ) (hlam : Admissible lam) (offspring : PMF ℕ) (ℓ : ℕ) :
-    advantage lam hlam offspring ℓ =
-      ∫ m, tvMessage m ∂(discreteProbability (observationLaw lam hlam offspring ℓ)).toMeasure := by
-  have hi : Integrable (fun o ↦ tvMessage (discreteMessage (observationLaw lam hlam offspring ℓ) o))
-      (discreteMarginal (observationLaw lam hlam offspring ℓ)).toMeasure :=
-    integrable_tv_discrete (observationLaw lam hlam offspring ℓ)
-  rw [integral_discreteProbability _ continuous_tvMessage.measurable, PMF.integral_eq_tsum _ _ hi]
-  unfold advantage
-  apply tsum_congr
-  intro o
-  rw [main_marginal_eq, discreteMarginal_real]
-  simp only [smul_eq_mul]
-  by_cases ho : discreteWeight (observationLaw lam hlam offspring ℓ) o = 0
-  · simp only [ho, zero_mul]
-  · simp only [main_posterior_eq lam hlam offspring ℓ o ho, tvMessage_formula]
-
-lemma advantage_nonneg (lam : ℝ) (hlam : Admissible lam) (offspring : PMF ℕ) (ℓ : ℕ) :
-    0 ≤ advantage lam hlam offspring ℓ := by
-  rw [advantage_eq_integral]
-  exact MeasureTheory.integral_nonneg tvMessage_nonneg
-
-lemma advantage_sq_bound (lam : ℝ) (hlam : Admissible lam) (offspring : PMF ℕ) (ℓ : ℕ) :
-    (advantage lam hlam offspring ℓ)^2 ≤
-      (∫ m, xMoment m ∂(discreteProbability (observationLaw lam hlam offspring ℓ)).toMeasure)/2 := by
-  rw [advantage_eq_integral]
-  exact integral_tvMessage_sq_bound _
-
-end ThreeState.TreeClauses.Experiment
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeStateClauses.Degradation
+import OAI.Probability.ThreeStateClauses.RegularLaw
+import OAI.Probability.ThreeStateClauses.Projection
+
+namespace OAI
+
+/-! Compact Limits. -/
+
+open scoped BigOperators ENNReal NNReal Topology
+open Filter
+noncomputable section
+open Set MeasureTheory
+open scoped BigOperators ENNReal
+namespace ThreeState.TreeClauses.Experiment
+open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
+
+def quadraticGap (z : Message × Message) : ℝ := avg (fun i ↦ (z.1.1 i-z.2.1 i)^2)/2
+
+lemma continuous_quadraticGap : Continuous quadraticGap := by
+  unfold quadraticGap
+  exact (continuous_avg.comp (continuous_pi (fun i ↦
+    ((continuous_coordinate i).comp continuous_fst |>.sub ((continuous_coordinate i).comp continuous_snd)).pow 2))).div_const 2
+
+lemma quadraticGap_nonneg (z : Message × Message) : 0 ≤ quadraticGap z := by
+  dsimp [quadraticGap, avg]; positivity
+
+lemma quadraticGap_zero_iff (z : Message × Message) : quadraticGap z = 0 ↔ z.1 = z.2 := by
+  constructor
+  · intro h
+    apply Subtype.ext
+    funext i
+    have h₀ := sq_nonneg (z.1.1 0-z.2.1 0)
+    have h₁ := sq_nonneg (z.1.1 1-z.2.1 1)
+    have h₂ := sq_nonneg (z.1.1 2-z.2.1 2)
+    dsimp [quadraticGap, avg] at h
+    fin_cases i <;> dsimp <;> nlinarith
+  · intro h; simp [quadraticGap, h, avg]
+
+lemma quadraticGap_expand (z : Message × Message) :
+    quadraticGap z = xMoment z.1-xMoment z.2-
+      avg (fun i ↦ (z.1.1 i-z.2.1 i)*(z.2.1 i-1)) := by
+  dsimp [quadraticGap, xMoment, momentX, avg, centered]; ring
+
+lemma continuous_centeredProduct (i : Fin 3) :
+    Continuous (fun z : Message × Message ↦ (z.1.1 i-z.2.1 i)*(z.2.1 i-1)) :=
+  (((continuous_coordinate i).comp continuous_fst).sub ((continuous_coordinate i).comp continuous_snd)).mul
+    (((continuous_coordinate i).comp continuous_snd).sub continuous_const)
+
+section Discrete
+variable {α β : Type*} [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α]
+  [Countable β] [MeasurableSpace β] [MeasurableSingletonClass β]
+
+def degradationCoupling (p : Fin 3 → PMF α) (K : α → PMF β) : ProbabilityMeasure (Message × Message) :=
+  ⟨((kernelJoint (discreteMarginal p) K).map (fun z ↦
+    (discreteMessage p z.1, discreteMessage (fun i ↦ (p i).bind K) z.2))).toMeasure, inferInstance⟩
+
+lemma integral_degradationCoupling (p : Fin 3 → PMF α) (K : α → PMF β)
+    {f : Message × Message → ℝ} (hf : Measurable f) :
+    (∫ z, f z ∂(degradationCoupling p K).toMeasure) =
+      ∫ z, f (discreteMessage p z.1, discreteMessage (fun i ↦ (p i).bind K) z.2)
+        ∂(kernelJoint (discreteMarginal p) K).toMeasure := by
+  change (∫ z, f z ∂(PMF.map _ _).toMeasure) = _
+  rw [← PMF.toMeasure_map _ _ (measurable_of_countable _)]
+  exact integral_map (measurable_of_countable _).aemeasurable hf.aestronglyMeasurable
+
+omit [Countable β] [MeasurableSpace β] [MeasurableSingletonClass β] in
+lemma degradationCoupling_fst (p : Fin 3 → PMF α) (K : α → PMF β) :
+    (degradationCoupling p K).map Prod.fst = discreteProbability p := by
+  apply Subtype.ext
+  change Measure.map Prod.fst (PMF.map _ _).toMeasure = Measure.map _ (discreteMarginal p).toMeasure
+  rw [PMF.toMeasure_map _ _ measurable_fst, PMF.map_comp]
+  change (PMF.map ((discreteMessage p) ∘ Prod.fst) _).toMeasure = _
+  rw [← PMF.map_comp, kernelJoint_fst, PMF.toMeasure_map _ _ (measurable_of_countable _)]
+
+omit [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α] in
+lemma degradationCoupling_snd (p : Fin 3 → PMF α) (K : α → PMF β) :
+    (degradationCoupling p K).map Prod.snd =
+      discreteProbability (fun i ↦ (p i).bind K) := by
+  apply Subtype.ext
+  change Measure.map Prod.snd (PMF.map _ _).toMeasure = Measure.map _ (discreteMarginal _).toMeasure
+  rw [PMF.toMeasure_map _ _ measurable_snd, PMF.map_comp]
+  change (PMF.map ((discreteMessage (fun i ↦ (p i).bind K)) ∘ Prod.snd) _).toMeasure = _
+  rw [← PMF.map_comp, kernelJoint_snd, ← discreteMarginal_bind,
+    PMF.toMeasure_map _ _ (measurable_of_countable _)]
+
+lemma degradationCoupling_centered (p : Fin 3 → PMF α) (K : α → PMF β) (i : Fin 3) :
+    (∫ z, (z.1.1 i-z.2.1 i)*(z.2.1 i-1) ∂(degradationCoupling p K).toMeasure) = 0 := by
+  rw [integral_degradationCoupling _ _ (continuous_centeredProduct i).measurable]
+  exact discrete_degradation_centered p K i
+    (g := fun b ↦ (discreteMessage (fun j ↦ (p j).bind K) b).1 i-1) (C := 2) (fun b ↦ by
+    apply abs_le.mpr
+    constructor <;> linarith [coordinate_nonneg (discreteMessage (fun j ↦ (p j).bind K) b) i,
+      coordinate_le_three (discreteMessage (fun j ↦ (p j).bind K) b) i])
+
+lemma degradationCoupling_loss (p : Fin 3 → PMF α) (K : α → PMF β) :
+    (∫ z, quadraticGap z ∂(degradationCoupling p K).toMeasure) =
+      (∫ m, xMoment m ∂(discreteProbability p).toMeasure)-
+      ∫ m, xMoment m ∂(discreteProbability (fun i ↦ (p i).bind K)).toMeasure := by
+  let C := degradationCoupling p K
+  have hx : Integrable (fun z : Message × Message ↦ xMoment z.1) C.toMeasure :=
+    compact_integrable (continuous_xMoment.comp continuous_fst)
+  have hy : Integrable (fun z : Message × Message ↦ xMoment z.2) C.toMeasure :=
+    compact_integrable (continuous_xMoment.comp continuous_snd)
+  have hc (i : Fin 3) : Integrable (fun z : Message × Message ↦ (z.1.1 i-z.2.1 i)*(z.2.1 i-1)) C.toMeasure :=
+    compact_integrable (continuous_centeredProduct i)
+  have hz : (∫ z, avg (fun i ↦ (z.1.1 i-z.2.1 i)*(z.2.1 i-1)) ∂C.toMeasure) = 0 := by
+    rw [integral_avg_measure hc]
+    simp only [C, degradationCoupling_centered, avg]
+    norm_num
+  simp_rw [quadraticGap_expand]
+  rw [integral_sub (f := fun z : Message × Message ↦ xMoment z.1-xMoment z.2)
+    (g := fun z : Message × Message ↦ avg (fun i ↦ (z.1.1 i-z.2.1 i)*(z.2.1 i-1)))
+    (hx.sub hy) (integrable_avg hc),
+    integral_sub (f := fun z : Message × Message ↦ xMoment z.1)
+    (g := fun z : Message × Message ↦ xMoment z.2) hx hy, hz, sub_zero]
+  congr 1
+  · rw [← degradationCoupling_fst p K]
+    exact (integral_map measurable_fst.aemeasurable continuous_xMoment.measurable.aestronglyMeasurable).symm
+  · rw [← degradationCoupling_snd p K]
+    exact (integral_map measurable_snd.aemeasurable continuous_xMoment.measurable.aestronglyMeasurable).symm
+
+lemma discrete_degradation_mu (p : Fin 3 → PMF α) (K : α → PMF β) :
+    (∫ m, xMoment m ∂(discreteProbability (fun i ↦ (p i).bind K)).toMeasure) ≤
+      ∫ m, xMoment m ∂(discreteProbability p).toMeasure := by
+  have h := MeasureTheory.integral_nonneg (μ := (degradationCoupling p K).toMeasure)
+    (f := quadraticGap) quadraticGap_nonneg
+  rw [degradationCoupling_loss] at h
+  linarith
+
+end Discrete
+end ThreeState.TreeClauses.Experiment
+
+end 
+
+noncomputable section
+open Set MeasureTheory
+open scoped BigOperators ENNReal
+namespace ThreeState.TreeClauses.Tree
+open ThreeState.TreeClauses.Experiment ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
+
+def vertex (i : Spin) : Message :=
+  ⟨fun j ↦ if i = j then 3 else 0, fun j ↦ by dsimp; split_ifs <;> norm_num, by
+    fin_cases i <;> norm_num [avg, Fin.ext_iff]⟩
+
+lemma discreteMessage_pure (i : Spin) : discreteMessage (fun j : Spin ↦ PMF.pure j) i = vertex i := by
+  apply Subtype.ext
+  funext j
+  fin_cases i <;> fin_cases j <;> norm_num [discreteMessage, discreteWeight, avg, PMF.pure_apply, vertex, Fin.ext_iff]
+
+lemma vertex_permute (σ : Equiv.Perm Spin) (i : Spin) : permute σ (vertex i) = vertex (σ.symm i) := by
+  apply Subtype.ext
+  funext j
+  change (if i = σ j then (3:ℝ) else 0) = if σ.symm i = j then 3 else 0
+  simp only [σ.symm_apply_eq]
+
+lemma pureMarginal : discreteMarginal (fun i : Spin ↦ PMF.pure i) = uniformSpin := PMF.bind_pure _
+
+lemma integral_rootProbability {f : Message → ℝ} (hf : Measurable f) :
+    (∫ m, f m ∂(discreteProbability (fun i : Spin ↦ PMF.pure i)).toMeasure) =
+      (∑ i : Spin, f (vertex i))/3 := by
+  rw [integral_discreteProbability _ hf, pureMarginal, PMF.integral_eq_sum]
+  simp only [discreteMessage_pure, smul_eq_mul, uniformSpin_apply, ENNReal.toReal_div,
+    ENNReal.toReal_one, ENNReal.toReal_ofNat]
+  rw [← Finset.mul_sum]
+  ring
+
+lemma rootProbability_symmetric (σ : Equiv.Perm Spin) :
+    MeasurePreserving (permute σ) (discreteProbability (fun i : Spin ↦ PMF.pure i)).toMeasure
+      (discreteProbability (fun i : Spin ↦ PMF.pure i)).toMeasure := by
+  refine ⟨(continuous_permute σ).measurable, ?_⟩
+  have h : (discreteProbability (fun i : Spin ↦ PMF.pure i)).map
+      (permute σ) = discreteProbability (fun i : Spin ↦ PMF.pure i) := by
+    apply probability_ext_continuous
+    intro f hf
+    change (∫ m, f m ∂Measure.map (permute σ) _) = _
+    rw [integral_map (continuous_permute σ).measurable.aemeasurable hf.measurable.aestronglyMeasurable,
+      integral_rootProbability (f := fun m ↦ f (permute σ m)) (hf.comp (continuous_permute σ)).measurable,
+      integral_rootProbability hf.measurable]
+    simp only [vertex_permute]
+    exact congrArg (fun x : ℝ ↦ x/3) (Equiv.sum_comp σ.symm (fun i ↦ f (vertex i)))
+  exact congrArg ProbabilityMeasure.toMeasure h
+
+def rootLaw : Law where
+  probability := discreteProbability (fun i : Spin ↦ PMF.pure i)
+  balanced := discreteProbability_balanced _
+  symmetric := rootProbability_symmetric
+
+def regularPosteriorLaw (b : ℕ) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) : ℕ → Law
+  | 0 => rootLaw
+  | ℓ+1 => finiteLaw (regularPosteriorLaw b hl₀ hl₁ ℓ) hl₀ hl₁ b
+
+lemma regularPosteriorLaw_probability (b : ℕ) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
+    (hlam : Admissible lam) (ℓ : ℕ) :
+    (regularPosteriorLaw b hl₀ hl₁ ℓ).probability = discreteProbability (regularLaw b lam hlam ℓ) := by
+  induction ℓ with
+  | zero => rfl
+  | succ ℓ ih =>
+    exact (discreteBranchLaw_probability (regularLaw b lam hlam ℓ)
+      (regularPosteriorLaw b hl₀ hl₁ ℓ) ih hl₀ hl₁ hlam b).symm
+
+lemma regularPosteriorLaw_antitone (b : ℕ) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
+    (hlam : Admissible lam) : Antitone (fun ℓ ↦ (regularPosteriorLaw b hl₀ hl₁ ℓ).mu) := by
+  apply antitone_nat_of_succ_le
+  intro ℓ
+  unfold Law.mu
+  rw [regularPosteriorLaw_probability b hl₀ hl₁ hlam ℓ,
+    regularPosteriorLaw_probability b hl₀ hl₁ hlam (ℓ+1)]
+  have h := discrete_degradation_mu (regularLaw b lam hlam ℓ) (regularExpansion b lam hlam ℓ)
+  simpa only [regularLaw_degrades] using h
+
+end ThreeState.TreeClauses.Tree
+
+end 
+
+noncomputable section
+open Set MeasureTheory Filter
+open scoped Topology
+namespace ThreeState.TreeClauses.Experiment
+open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
+
+lemma probability_continuous_integral {X : Type*} [TopologicalSpace X] [CompactSpace X]
+    [MeasurableSpace X] [OpensMeasurableSpace X] {f : X → ℝ} (hf : Continuous f) :
+    Continuous (fun P : ProbabilityMeasure X ↦ ∫ x, f x ∂P.toMeasure) :=
+  ProbabilityMeasure.continuous_integral_continuousMap (⟨f, hf⟩ : C(X, ℝ))
+
+section Limits
+variable {ι : Type*} {F : Filter ι} [F.NeBot] {Q : ι → Law} {P : ProbabilityMeasure Message}
+
+lemma limit_balanced (hP : Tendsto (fun i ↦ (Q i).probability) F (𝓝 P)) (j : Fin 3) :
+    (∫ m, m.1 j ∂P.toMeasure) = 1 := by
+  have h := (probability_continuous_integral (continuous_coordinate j)).tendsto P |>.comp hP
+  simp only [Function.comp_def, Law.balanced] at h
+  exact tendsto_nhds_unique h tendsto_const_nhds
+
+lemma limit_symmetric (hP : Tendsto (fun i ↦ (Q i).probability) F (𝓝 P))
+    (σ : Equiv.Perm (Fin 3)) : MeasurePreserving (permute σ) P.toMeasure P.toMeasure := by
+  refine ⟨(continuous_permute σ).measurable, ?_⟩
+  have hm := (ProbabilityMeasure.continuous_map (continuous_permute σ)).tendsto P |>.comp hP
+  have he (i : ι) : (Q i).probability.map (permute σ) =
+      (Q i).probability := Subtype.ext ((Q i).symmetric σ).map_eq
+  simp only [Function.comp_def, he] at hm
+  exact congrArg ProbabilityMeasure.toMeasure (tendsto_nhds_unique hm hP)
+
+def limitLaw (hP : Tendsto (fun i ↦ (Q i).probability) F (𝓝 P)) : Law where
+  probability := P
+  balanced := limit_balanced hP
+  symmetric := limit_symmetric hP
+
+end Limits
+
+lemma finiteProbability_tendsto {ι : Type*} {F : Filter ι} {Q : ι → Law} {R : Law}
+    (hQ : Tendsto (fun i ↦ (Q i).probability) F (𝓝 R.probability))
+    {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
+    Tendsto (fun i ↦ finiteProbability (Q i) hl₀ hl₁ n) F (𝓝 (finiteProbability R hl₀ hl₁ n)) := by
+  apply ProbabilityMeasure.tendsto_iff_forall_integral_tendsto.mpr
+  intro f
+  have hp : Tendsto (fun i ↦ ProbabilityMeasure.pi (fun _ : Fin n ↦ (Q i).probability)) F
+      (𝓝 (ProbabilityMeasure.pi (fun _ : Fin n ↦ R.probability))) :=
+    ProbabilityMeasure.continuous_pi.tendsto _ |>.comp (tendsto_pi_nhds.mpr (fun _ ↦ hQ))
+  have hfun : Continuous (fun m : Fin n → Message ↦ normalizer (branchEdges lam m)*f (branchOutput hl₀ hl₁ m)) :=
+    (continuous_normalizer n lam).mul (f.continuous.comp (continuous_branchOutput hl₀ hl₁ n))
+  have h := (probability_continuous_integral hfun).tendsto _ |>.comp hp
+  simpa only [Function.comp_def, ProbabilityMeasure.toMeasure_pi,
+    integral_finiteProbability _ hl₀ hl₁ n f.continuous.measurable] using h
+
+lemma coupling_diagonal_of_loss_zero (C : ProbabilityMeasure (Message × Message))
+    (h : (∫ z, quadraticGap z ∂C.toMeasure) = 0) :
+    C.map Prod.fst = C.map Prod.snd := by
+  have he := (integral_eq_zero_iff_of_nonneg quadraticGap_nonneg
+    (compact_integrable continuous_quadraticGap)).mp h
+  apply Subtype.ext
+  exact Measure.map_congr (he.mono (fun z hz ↦ (quadraticGap_zero_iff z).mp hz))
+
+end ThreeState.TreeClauses.Experiment
+
+end 
+
+noncomputable section
+open Set MeasureTheory Filter
+open scoped Topology
+namespace ThreeState.TreeClauses.Experiment
+open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
+
+lemma degrading_finite_fixed_point (Q : ℕ → Law) (C : ℕ → ProbabilityMeasure (Message × Message))
+    (hfst : ∀ ℓ, (C ℓ).map Prod.fst = (Q ℓ).probability)
+    (hsnd : ∀ ℓ, (C ℓ).map Prod.snd = (Q (ℓ+1)).probability)
+    (hloss : ∀ ℓ, (∫ z, quadraticGap z ∂(C ℓ).toMeasure) = (Q ℓ).mu-(Q (ℓ+1)).mu)
+    {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ)
+    (hrec : ∀ ℓ, (Q (ℓ+1)).probability = finiteProbability (Q ℓ) hl₀ hl₁ n) :
+    ∃ R : Law, R.probability = finiteProbability R hl₀ hl₁ n ∧
+      Tendsto (fun ℓ ↦ (Q ℓ).mu) atTop (𝓝 R.mu) := by
+  have hanti : Antitone (fun ℓ ↦ (Q ℓ).mu) := by
+    apply antitone_nat_of_succ_le
+    intro ℓ
+    have h := MeasureTheory.integral_nonneg (μ := (C ℓ).toMeasure) (f := quadraticGap) quadraticGap_nonneg
+    rw [hloss] at h
+    linarith
+  let a := ⨅ ℓ, (Q ℓ).mu
+  have hlim : Tendsto (fun ℓ ↦ (Q ℓ).mu) atTop (𝓝 a) :=
+    tendsto_atTop_ciInf hanti ⟨0, by rintro _ ⟨ℓ,rfl⟩; exact (Q ℓ).mu_nonneg⟩
+  have hdiff : Tendsto (fun ℓ ↦ (Q ℓ).mu-(Q (ℓ+1)).mu) atTop (𝓝 0) := by
+    simpa using hlim.sub (hlim.comp (tendsto_add_atTop_nat 1))
+  obtain ⟨D, hD⟩ := exists_clusterPt_of_compactSpace (Filter.map C atTop)
+  let F : Filter ℕ := Filter.comap C (𝓝 D) ⊓ atTop
+  have : F.NeBot := neBot_inf_comap_iff_map'.mpr hD
+  have hF : F ≤ atTop := inf_le_right
+  have hC : Tendsto C F (𝓝 D) := tendsto_comap.mono_left inf_le_left
+  have hgap : (∫ z, quadraticGap z ∂D.toMeasure) = 0 := by
+    have hg := (probability_continuous_integral continuous_quadraticGap).tendsto D |>.comp hC
+    simp only [Function.comp_def, hloss] at hg
+    exact tendsto_nhds_unique hg (hdiff.mono_left hF)
+  have hdiag := coupling_diagonal_of_loss_zero D hgap
+  have hP := (ProbabilityMeasure.continuous_map (continuous_fst : Continuous (Prod.fst : Message × Message → Message))).tendsto D |>.comp hC
+  simp only [Function.comp_def, hfst] at hP
+  let R : Law := limitLaw hP
+  have hS := (ProbabilityMeasure.continuous_map (continuous_snd : Continuous (Prod.snd : Message × Message → Message))).tendsto D |>.comp hC
+  simp only [Function.comp_def, hsnd] at hS
+  rw [← hdiag] at hS
+  have hR : R.probability = finiteProbability R hl₀ hl₁ n := by
+    have hr := finiteProbability_tendsto (R := R) hP hl₀ hl₁ n
+    simp only [← hrec] at hr
+    exact tendsto_nhds_unique hS hr
+  have hm : a = R.mu := by
+    have hp := (probability_continuous_integral continuous_xMoment).tendsto R.probability |>.comp hP
+    exact tendsto_nhds_unique (hlim.mono_left hF) hp
+  exact ⟨R, hR, hm ▸ hlim⟩
+
+end ThreeState.TreeClauses.Experiment
+
+end 
+
+noncomputable section
+open Set MeasureTheory
+open scoped BigOperators ENNReal
+namespace ThreeState.TreeClauses.Experiment
+open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
+
+def tvMessage (m : Message) : ℝ := avg (fun i ↦ |m.1 i-1|)/2
+
+lemma continuous_tvMessage : Continuous tvMessage :=
+  (continuous_avg.comp (continuous_pi (fun i ↦ ((continuous_coordinate i).sub continuous_const).abs))).div_const 2
+
+lemma tvMessage_nonneg (m : Message) : 0 ≤ tvMessage m := by
+  dsimp [tvMessage, avg]; positivity
+
+lemma tvMessage_formula (m : Message) :
+    tvMessage m = (∑ i : Spin, |m.1 i/3-(1/3:ℝ)|)/2 := by
+  simp only [tvMessage, avg, ← sub_div, abs_div, abs_of_pos (by norm_num : (0:ℝ) < 3), Fin.sum_univ_three]
+  ring
+
+lemma tvMessage_sq_bound (m : Message) : tvMessage m^2 ≤ xMoment m/2 := by
+  have h₀ := sq_nonneg (|m.1 0-1|-|m.1 1-1|)
+  have h₁ := sq_nonneg (|m.1 1-1|-|m.1 2-1|)
+  have h₂ := sq_nonneg (|m.1 2-1|-|m.1 0-1|)
+  dsimp [tvMessage, xMoment, momentX, centered, avg]
+  nlinarith only [h₀,h₁,h₂, sq_abs (m.1 0-1), sq_abs (m.1 1-1), sq_abs (m.1 2-1)]
+
+lemma integral_tvMessage_sq_bound (P : ProbabilityMeasure Message) :
+    (∫ m, tvMessage m ∂P.toMeasure)^2 ≤ (∫ m, xMoment m ∂P.toMeasure)/2 := by
+  let t := ∫ m, tvMessage m ∂P.toMeasure
+  have hi := compact_integrable continuous_tvMessage (μ := P.toMeasure)
+  have hi₂ := compact_integrable (continuous_tvMessage.pow 2) (μ := P.toMeasure)
+  have h := MeasureTheory.integral_nonneg (μ := P.toMeasure) (f := fun m ↦ (tvMessage m-t)^2)
+    (fun m ↦ sq_nonneg (tvMessage m-t))
+  have he : (fun m : Message ↦ (tvMessage m-t)^2) = fun m ↦ tvMessage m^2-2*t*tvMessage m+t^2 := by
+    funext m; ring
+  rw [he, integral_add (f := fun m ↦ tvMessage m^2-2*t*tvMessage m) (g := fun _ ↦ t^2)
+    (hi₂.sub (hi.const_mul _)) (integrable_const _),
+    integral_sub (f := fun m ↦ tvMessage m^2) (g := fun m ↦ 2*t*tvMessage m) hi₂ (hi.const_mul _),
+    integral_const_mul] at h
+  simp only [integral_const, measure_univ, Measure.real, ENNReal.toReal_one, smul_eq_mul, one_mul] at h
+  have hb := integral_mono (μ := P.toMeasure) hi₂ (compact_integrable (continuous_xMoment.div_const 2)) tvMessage_sq_bound
+  rw [integral_div] at hb
+  change (∫ m, tvMessage m^2 ∂P.toMeasure) ≤ (∫ m, xMoment m ∂P.toMeasure)/2 at hb
+  change 0 ≤ (∫ m, tvMessage m^2 ∂P.toMeasure)-2*t*t+t^2 at h
+  change t^2 ≤ _
+  nlinarith
+
+lemma main_marginal_eq (lam : ℝ) (hlam : Admissible lam) (offspring : PMF ℕ)
+    (ℓ : ℕ) (o : Observation ℓ) :
+    marginalWeight lam hlam offspring ℓ o = discreteMarginal (observationLaw lam hlam offspring ℓ) o := by
+  simp only [marginalWeight, jointWeight, discreteMarginal, PMF.bind_apply, tsum_fintype, uniformSpin_apply]
+
+lemma main_posterior_eq (lam : ℝ) (hlam : Admissible lam) (offspring : PMF ℕ)
+    (ℓ : ℕ) (o : Observation ℓ) (ho : discreteWeight (observationLaw lam hlam offspring ℓ) o ≠ 0) (i : Spin) :
+    posterior lam hlam offspring ℓ o i = (discreteMessage (observationLaw lam hlam offspring ℓ) o).1 i/3 := by
+  rw [posterior, ENNReal.toReal_div, main_marginal_eq, discreteMarginal_real, jointWeight,
+    ENNReal.toReal_mul, discrete_coord ho]
+  norm_num
+  ring
+
+lemma tvMessage_le_one (m : Message) : tvMessage m ≤ 1 := by
+  have hb (i : Spin) : |m.1 i-1| ≤ 2 := abs_le.mpr ⟨by linarith [coordinate_nonneg m i],
+    by linarith [coordinate_le_three m i]⟩
+  dsimp [tvMessage, avg]
+  linarith [hb 0, hb 1, hb 2]
+
+lemma integrable_tv_discrete {α : Type*} [Countable α] [MeasurableSpace α]
+    [MeasurableSingletonClass α] (p : Spin → PMF α) :
+    Integrable (fun o ↦ tvMessage (discreteMessage p o)) (discreteMarginal p).toMeasure := by
+  apply Integrable.of_bound (measurable_of_countable _).aestronglyMeasurable 1
+  exact Filter.Eventually.of_forall (fun a ↦ by
+    rw [Real.norm_eq_abs, abs_of_nonneg (tvMessage_nonneg _)]
+    exact tvMessage_le_one _)
+
+lemma advantage_eq_integral (lam : ℝ) (hlam : Admissible lam) (offspring : PMF ℕ) (ℓ : ℕ) :
+    advantage lam hlam offspring ℓ =
+      ∫ m, tvMessage m ∂(discreteProbability (observationLaw lam hlam offspring ℓ)).toMeasure := by
+  have hi : Integrable (fun o ↦ tvMessage (discreteMessage (observationLaw lam hlam offspring ℓ) o))
+      (discreteMarginal (observationLaw lam hlam offspring ℓ)).toMeasure :=
+    integrable_tv_discrete (observationLaw lam hlam offspring ℓ)
+  rw [integral_discreteProbability _ continuous_tvMessage.measurable, PMF.integral_eq_tsum _ _ hi]
+  unfold advantage
+  apply tsum_congr
+  intro o
+  rw [main_marginal_eq, discreteMarginal_real]
+  simp only [smul_eq_mul]
+  by_cases ho : discreteWeight (observationLaw lam hlam offspring ℓ) o = 0
+  · simp only [ho, zero_mul]
+  · simp only [main_posterior_eq lam hlam offspring ℓ o ho, tvMessage_formula]
+
+lemma advantage_nonneg (lam : ℝ) (hlam : Admissible lam) (offspring : PMF ℕ) (ℓ : ℕ) :
+    0 ≤ advantage lam hlam offspring ℓ := by
+  rw [advantage_eq_integral]
+  exact MeasureTheory.integral_nonneg tvMessage_nonneg
+
+lemma advantage_sq_bound (lam : ℝ) (hlam : Admissible lam) (offspring : PMF ℕ) (ℓ : ℕ) :
+    (advantage lam hlam offspring ℓ)^2 ≤
+      (∫ m, xMoment m ∂(discreteProbability (observationLaw lam hlam offspring ℓ)).toMeasure)/2 := by
+  rw [advantage_eq_integral]
+  exact integral_tvMessage_sq_bound _
+
+end ThreeState.TreeClauses.Experiment
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeStateClauses/Correlation.lean b/lean/OAI/Probability/ThreeStateClauses/Correlation.lean
index 1156b1116601f5cfae9acab11ab3a0d211d52c7d..f0c96e3d5bf81e56c3c802088e3faa410a899baa 100644
--- a/lean/OAI/Probability/ThreeStateClauses/Correlation.lean
+++ b/lean/OAI/Probability/ThreeStateClauses/Correlation.lean
@@ -1,316 +1,316 @@
-import Mathlib
-import OAI.Probability.ThreeStateClauses.RadialCalculus
-import OAI.Probability.ThreeStateClauses.BranchMoments
-
-namespace OAI
-
-/-! Correlation. -/
-
-open scoped BigOperators ENNReal NNReal Topology
-open Filter
-noncomputable section
-open Set
-namespace ThreeState.TreeClauses.Positive
-open Radial
-
-lemma log_cubic_lower {z : ℝ} (hz : -1 < z) :
-    z+z^2/2-z^3/6 ≤ (1+z)*Real.log (1+z) := by
-  let f : ℝ → ℝ := fun r ↦ r*Real.log r-(r-1)-(r-1)^2/2+(r-1)^3/6
-  let g : ℝ → ℝ := fun r ↦ Real.log r-(r-1)+(r-1)^2/2
-  have df (r : ℝ) (hr : 0 < r) : HasDerivAt f (g r) r := by
-    have hd := ((((hasDerivAt_id r).mul (Real.hasDerivAt_log (ne_of_gt hr))).sub
-      ((hasDerivAt_id r).sub_const 1)).sub
-      ((((hasDerivAt_id r).sub_const 1).pow 2).div_const 2)).add
-      ((((hasDerivAt_id r).sub_const 1).pow 3).div_const 6)
-    apply hd.congr_deriv
-    dsimp [g]
-    field_simp [ne_of_gt hr]
-    ring
-  have dg (r : ℝ) (hr : 0 < r) : HasDerivAt g ((r-1)^2/r) r := by
-    have hd := ((Real.hasDerivAt_log (ne_of_gt hr)).sub ((hasDerivAt_id r).sub_const 1)).add
-      ((((hasDerivAt_id r).sub_const 1).pow 2).div_const 2)
-    apply hd.congr_deriv
-    dsimp only [id]
-    field_simp [ne_of_gt hr]
-    ring
-  have gm {a b : ℝ} (ha : 0 < a) : MonotoneOn g (Icc a b) := by
-    apply monotoneOn_Icc_of_hasDerivAt_nonneg (fun r hr ↦ dg r (lt_of_lt_of_le ha hr.1))
-    intro r hr
-    exact div_nonneg (sq_nonneg _) (lt_trans ha hr.1).le
-  have hg₁ : g 1 = 0 := by simp [g]
-  have hf₁ : f 1 = 0 := by simp [f]
-  have hmin (r : ℝ) (hr : 0 < r) : 0 ≤ f r := by
-    rcases le_total 1 r with h | h
-    · have hg (t : ℝ) (ht : t ∈ Icc 1 r) : 0 ≤ g t := by
-        have hh := gm (b := r) (by norm_num : (0:ℝ) < 1) ⟨le_rfl, h⟩ ht ht.1
-        rwa [hg₁] at hh
-      have hmono : MonotoneOn f (Icc 1 r) :=
-        monotoneOn_Icc_of_hasDerivAt_nonneg (fun t ht ↦ df t (by linarith [ht.1]))
-          (fun t ht ↦ hg t ⟨ht.1.le, ht.2.le⟩)
-      have hh := hmono ⟨le_rfl, h⟩ ⟨h, le_rfl⟩ h
-      rwa [hf₁] at hh
-    · have hg (t : ℝ) (ht : t ∈ Icc r 1) : g t ≤ 0 := by
-        have hh := gm (b := 1) hr ht ⟨h, le_rfl⟩ ht.2
-        rwa [hg₁] at hh
-      have hmono : MonotoneOn (fun t ↦ -f t) (Icc r 1) :=
-        monotoneOn_Icc_of_hasDerivAt_nonneg (fun t ht ↦ (df t (lt_of_lt_of_le hr ht.1)).neg)
-          (fun t ht ↦ neg_nonneg.mpr (hg t ⟨ht.1.le, ht.2.le⟩))
-      have hh := hmono ⟨le_rfl, h⟩ ⟨h, le_rfl⟩ h
-      change -f r ≤ -f 1 at hh
-      rw [hf₁] at hh
-      linarith
-  have hh := hmin (1+z) (by linarith)
-  dsimp [f] at hh
-  nlinarith
-
-def orbit (v : Vec) : Fin 6 → Vec :=
-  ![ ![v 0,v 1,v 2], ![v 0,v 2,v 1], ![v 1,v 0,v 2],
-     ![v 1,v 2,v 0], ![v 2,v 0,v 1], ![v 2,v 1,v 0] ]
-
-def avgSix (f : Fin 6 → ℝ) : ℝ :=
-  (f 0+f 1+f 2+f 3+f 4+f 5)/6
-
-def dotAvg (v w : Vec) : ℝ := avg (fun i ↦ v i*w i)
-
-lemma avgSix_dot {v w : Vec} (hv : avg v = 0) (hw : avg w = 0) :
-    avgSix (fun k ↦ dotAvg v (orbit w k)) = 0 := by
-  simp [avgSix, dotAvg, orbit, avg, third_eq hv, third_eq hw]
-  ring
-
-lemma avgSix_dot_sq {v w : Vec} (hv : avg v = 0) (hw : avg w = 0) :
-    avgSix (fun k ↦ dotAvg v (orbit w k)^2) = 2*momentX v*momentX w := by
-  simp [avgSix, dotAvg, orbit, momentX, avg, third_eq hv, third_eq hw]
-  ring
-
-lemma avgSix_dot_cube {v w : Vec} (hv : avg v = 0) (hw : avg w = 0) :
-    avgSix (fun k ↦ dotAvg v (orbit w k)^3) = 2*momentY v*momentY w := by
-  simp [avgSix, dotAvg, orbit, momentY, avg, third_eq hv, third_eq hw]
-  ring
-
-lemma avgSix_mono {f g : Fin 6 → ℝ} (h : ∀ k, f k ≤ g k) : avgSix f ≤ avgSix g := by
-  dsimp [avgSix]
-  linarith [h 0, h 1, h 2, h 3, h 4, h 5]
-
-theorem correlation_orbit_lower {v w : Vec} (hv : avg v = 0) (hw : avg w = 0)
-    (hZ : ∀ k, -1 < dotAvg v (orbit w k)) :
-    3*momentX v*momentX w-momentY v*momentY w ≤
-      3*avgSix (fun k ↦ (1+dotAvg v (orbit w k))*Real.log (1+dotAvg v (orbit w k))) := by
-  have h := avgSix_mono (fun k ↦ log_cubic_lower (hZ k))
-  have he : avgSix (fun k ↦ dotAvg v (orbit w k)+dotAvg v (orbit w k)^2/2-dotAvg v (orbit w k)^3/6) =
-      avgSix (fun k ↦ dotAvg v (orbit w k))+
-      avgSix (fun k ↦ dotAvg v (orbit w k)^2)/2-
-      avgSix (fun k ↦ dotAvg v (orbit w k)^3)/6 := by
-    dsimp [avgSix]; ring
-  rw [he, avgSix_dot hv hw, avgSix_dot_sq hv hw, avgSix_dot_cube hv hw] at h
-  linarith
-
-end ThreeState.TreeClauses.Positive
-
-end 
-
-noncomputable section
-open Set MeasureTheory
-namespace ThreeState.TreeClauses.Experiment
-open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
-
-def pairZ (lam : ℝ) (r m : Message) : ℝ := avg (fun i ↦ r.1 i*edgeMessage lam m i)
-
-def pairPosterior {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (r m : Message) : Message := by
-  have hz : 0 < pairZ lam r m := by
-    have h := avg_mono (fun i ↦ mul_le_mul_of_nonneg_left (edge_lower hl₀ m i) (coordinate_nonneg r i))
-    have he : avg (fun i ↦ r.1 i*(1-lam)) = 1-lam := by
-      calc
-        _ = (1-lam)*avg r.1 := by simp only [avg]; ring
-        _ = _ := by rw [r.2.2]; ring
-    rw [he] at h
-    exact lt_of_lt_of_le (sub_pos.mpr hl₁) h
-  refine ⟨fun i ↦ r.1 i*edgeMessage lam m i/pairZ lam r m, ?_, ?_⟩
-  · intro i
-    exact div_nonneg (mul_nonneg (coordinate_nonneg r i) ((edge_positive hl₀ hl₁ m).1 i).le) hz.le
-  · rw [avg_div]
-    exact div_self (ne_of_gt hz)
-
-lemma pairZ_lower {lam : ℝ} (hl₀ : 0 ≤ lam) (r m : Message) : 1-lam ≤ pairZ lam r m := by
-  have h := avg_mono (fun i ↦ mul_le_mul_of_nonneg_left (edge_lower hl₀ m i) (coordinate_nonneg r i))
-  have he : avg (fun i ↦ r.1 i*(1-lam)) = 1-lam := by
-    calc
-      _ = (1-lam)*avg r.1 := by simp only [avg]; ring
-      _ = _ := by rw [r.2.2]; ring
-  rwa [he] at h
-
-lemma pairZ_positive {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (r m : Message) :
-    0 < pairZ lam r m := lt_of_lt_of_le (sub_pos.mpr hl₁) (pairZ_lower hl₀ r m)
-
-lemma continuous_pairZ (lam : ℝ) : Continuous (fun p : Message × Message ↦ pairZ lam p.1 p.2) :=
-  continuous_avg.comp (continuous_pi (fun i ↦ ((continuous_coordinate i).comp continuous_fst).mul
-    (((continuous_apply i).comp (continuous_edgeMessage lam)).comp continuous_snd)))
-
-lemma continuous_pairPosterior {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) :
-    Continuous (fun p : Message × Message ↦ pairPosterior hl₀ hl₁ p.1 p.2) := by
-  apply Continuous.subtype_mk
-  exact continuous_pi (fun i ↦ (((continuous_coordinate i).comp continuous_fst).mul
-    (((continuous_apply i).comp (continuous_edgeMessage lam)).comp continuous_snd)).div
-      (continuous_pairZ lam) (fun p ↦ ne_of_gt (pairZ_positive hl₀ hl₁ p.1 p.2)))
-
-lemma pairZ_dot (lam : ℝ) (r m : Message) : pairZ lam r m =
-    1+dotAvg (centered r.1) (centered (edgeMessage lam m)) := by
-  have hr := r.2.2
-  have hm := m.2.2
-  dsimp only [pairZ, dotAvg, centered, edgeMessage, edge, avg] at *
-  linear_combination hr + lam*hm
-
-lemma integral_pairZ (Q : Law) (lam : ℝ) (r : Message) :
-    (∫ m, pairZ lam r m ∂Q.probability.toMeasure) = 1 := by
-  have hi (i : Fin 3) : Integrable (fun m ↦ r.1 i*edgeMessage lam m i) Q.probability.toMeasure :=
-    (compact_integrable ((continuous_apply i).comp (continuous_edgeMessage lam))).const_mul _
-  unfold pairZ
-  rw [integral_avg_measure hi]
-  simp_rw [integral_const_mul, Q.edge_balanced, mul_one]
-  exact r.2.2
-
-lemma integral_pair_coordinate (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
-    (r : Message) (i : Fin 3) :
-    (∫ m, pairZ lam r m*(pairPosterior hl₀ hl₁ r m).1 i ∂Q.probability.toMeasure) = r.1 i := by
-  have he (m : Message) : pairZ lam r m*(pairPosterior hl₀ hl₁ r m).1 i = r.1 i*edgeMessage lam m i := by
-    change pairZ lam r m*(r.1 i*edgeMessage lam m i/pairZ lam r m) = _
-    field_simp [ne_of_gt (pairZ_positive hl₀ hl₁ r m)]
-  simp_rw [he]
-  rw [integral_const_mul, Q.edge_balanced, mul_one]
-
-lemma continuous_pairEntropy {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) :
-    Continuous (fun p : Message × Message ↦ pairZ lam p.1 p.2*Real.log (pairZ lam p.1 p.2)) :=
-  (continuous_pairZ lam).mul ((continuous_pairZ lam).log (fun p ↦ ne_of_gt (pairZ_positive hl₀ hl₁ p.1 p.2)))
-
-def pairCorrelation (R Q : Law) (lam : ℝ) : ℝ :=
-  ∫ r, (∫ m, pairZ lam r m*Real.log (pairZ lam r m) ∂Q.probability.toMeasure) ∂ R.probability.toMeasure
-
-lemma pairCorrelation_nonneg (R Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) :
-    0 ≤ pairCorrelation R Q lam := by
-  apply integral_nonneg
-  intro r
-  have hc : Continuous (fun m ↦ pairZ lam r m) := (continuous_pairZ lam).comp (continuous_const.prodMk continuous_id)
-  have hi := compact_integrable hc (μ := Q.probability.toMeasure)
-  have he : Integrable (fun m : Message ↦ pairZ lam r m*Real.log (pairZ lam r m)) Q.probability.toMeasure :=
-    compact_integrable ((continuous_pairEntropy hl₀ hl₁).comp (continuous_const.prodMk continuous_id))
-  have h := integral_mono (f := fun m : Message ↦ pairZ lam r m-1)
-    (g := fun m ↦ pairZ lam r m*Real.log (pairZ lam r m)) (hi.sub (integrable_const _)) he
-    (fun m ↦ by
-      have hp := pairZ_positive hl₀ hl₁ r m
-      have hh := mul_le_mul_of_nonneg_left (Real.one_sub_inv_le_log_of_pos hp) hp.le
-      simpa [mul_sub, ne_of_gt hp] using hh)
-  rw [integral_sub (f := fun m ↦ pairZ lam r m) (g := fun _ ↦ (1:ℝ)) hi (integrable_const _), integral_pairZ] at h
-  simpa using h
-
-def orbitIndex : Fin 6 → (Fin 3 → Fin 3) :=
-  ![ ![0,1,2], ![0,2,1], ![1,0,2], ![1,2,0], ![2,0,1], ![2,1,0] ]
-
-def orbitPerm (k : Fin 6) : Equiv.Perm (Fin 3) :=
-  Equiv.ofBijective (orbitIndex k) (by fin_cases k <;> decide)
-
-lemma orbitPerm_apply (v : Vec) (k : Fin 6) (i : Fin 3) :
-    v (orbitPerm k i) = orbit v k i := by
-  fin_cases k <;> fin_cases i <;> rfl
-
-lemma pairZ_orbit (lam : ℝ) (r m : Message) (k : Fin 6) :
-    pairZ lam r (permute (orbitPerm k) m) =
-      1+dotAvg (centered r.1) (orbit (centered (edgeMessage lam m)) k) := by
-  rw [pairZ_dot]
-  congr 2
-  funext i
-  change edgeMessage lam m (orbitPerm k i)-1 = _
-  exact orbitPerm_apply (centered (edgeMessage lam m)) k i
-
-lemma integral_avgSix_measure {X : Type*} [MeasurableSpace X] {ρ : Measure X}
-    {f : X → Fin 6 → ℝ} (hf : ∀ k, Integrable (fun x ↦ f x k) ρ) :
-    (∫ x, avgSix (f x) ∂ρ) = avgSix (fun k ↦ ∫ x, f x k ∂ρ) := by
-  unfold avgSix
-  rw [integral_div,
-    integral_add (f := fun x ↦ f x 0+f x 1+f x 2+f x 3+f x 4) (g := fun x ↦ f x 5)
-      (((((hf 0).add (hf 1)).add (hf 2)).add (hf 3)).add (hf 4)) (hf 5),
-    integral_add (f := fun x ↦ f x 0+f x 1+f x 2+f x 3) (g := fun x ↦ f x 4)
-      ((((hf 0).add (hf 1)).add (hf 2)).add (hf 3)) (hf 4),
-    integral_add (f := fun x ↦ f x 0+f x 1+f x 2) (g := fun x ↦ f x 3)
-      (((hf 0).add (hf 1)).add (hf 2)) (hf 3),
-    integral_add (f := fun x ↦ f x 0+f x 1) (g := fun x ↦ f x 2)
-      ((hf 0).add (hf 1)) (hf 2),
-    integral_add (f := fun x ↦ f x 0) (g := fun x ↦ f x 1) (hf 0) (hf 1)]
-
-lemma xMoment_edge (lam : ℝ) (m : Message) :
-    momentX (centered (edgeMessage lam m)) = lam^2*xMoment m := by
-  dsimp [momentX, centered, edgeMessage, edge, xMoment, avg]
-  ring
-
-lemma yMoment_edge (lam : ℝ) (m : Message) :
-    momentY (centered (edgeMessage lam m)) = lam^3*yMoment m := by
-  dsimp [momentY, centered, edgeMessage, edge, yMoment, avg]
-  ring
-
-lemma pairCorrelation_increment_at (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
-    (r : Message) :
-    3*lam^2*Q.mu*xMoment r-lam^3*Q.eta*yMoment r ≤
-      3*(∫ m, pairZ lam r m*Real.log (pairZ lam r m) ∂Q.probability.toMeasure) := by
-  let f : Message → Fin 6 → ℝ := fun m k ↦
-    pairZ lam r (permute (orbitPerm k) m)*Real.log (pairZ lam r (permute (orbitPerm k) m))
-  have hi (k : Fin 6) : Integrable (fun m ↦ f m k) Q.probability.toMeasure :=
-    compact_integrable ((continuous_pairEntropy hl₀ hl₁).comp
-      (continuous_const.prodMk (continuous_permute (orbitPerm k))))
-  have havg : Integrable (fun m ↦ avgSix (f m)) Q.probability.toMeasure :=
-    (((((hi 0).add (hi 1)).add (hi 2)).add (hi 3)).add (hi 4) |>.add (hi 5)).div_const 6
-  have hpoint (m : Message) : 3*lam^2*xMoment r*xMoment m-lam^3*yMoment r*yMoment m ≤
-      3*avgSix (f m) := by
-    have h := correlation_orbit_lower (message_centered_avg r)
-      (show avg (centered (edgeMessage lam m)) = 0 from by
-        change avg (fun i ↦ edgeMessage lam m i-1) = 0
-        rw [avg_sub, avg_const, (edge_positive hl₀ hl₁ m).2]; ring)
-      (fun k ↦ by
-        have hh := pairZ_positive hl₀ hl₁ r (permute (orbitPerm k) m)
-        rw [pairZ_orbit] at hh; linarith)
-    simp only [xMoment_edge, yMoment_edge] at h
-    have he : (fun k ↦ (1+dotAvg (centered r.1) (orbit (centered (edgeMessage lam m)) k))*
-        Real.log (1+dotAvg (centered r.1) (orbit (centered (edgeMessage lam m)) k))) = f m := by
-      funext k; dsimp only [f]; rw [pairZ_orbit]
-    rw [he] at h
-    change 3*xMoment r*(lam^2*xMoment m)-yMoment r*(lam^3*yMoment m) ≤ _ at h
-    nlinarith [h]
-  have h := integral_mono
-    (((compact_integrable continuous_xMoment).const_mul (3*lam^2*xMoment r)).sub
-      ((compact_integrable continuous_yMoment).const_mul (lam^3*yMoment r)))
-    (havg.const_mul 3) hpoint
-  simp only [Pi.sub_apply] at h
-  rw [integral_sub (f := fun m ↦ 3*lam^2*xMoment r*xMoment m)
-    (g := fun m ↦ lam^3*yMoment r*yMoment m)
-      ((compact_integrable continuous_xMoment).const_mul _)
-      ((compact_integrable continuous_yMoment).const_mul _),
-    integral_const_mul, integral_const_mul, integral_const_mul, integral_avgSix_measure hi] at h
-  have hev (k : Fin 6) : (∫ m, f m k ∂Q.probability.toMeasure) =
-      ∫ m, pairZ lam r m*Real.log (pairZ lam r m) ∂Q.probability.toMeasure := by
-    exact (show MeasurePreserving (permuteMeasurableEquiv (orbitPerm k))
-      Q.probability.toMeasure Q.probability.toMeasure from Q.symmetric _).integral_comp' (fun m ↦ pairZ lam r m*Real.log (pairZ lam r m))
-  simp_rw [hev] at h
-  dsimp only [avgSix] at h
-  change 3*lam^2*xMoment r*Q.mu-lam^3*yMoment r*Q.eta ≤ _ at h
-  linarith
-
-theorem pairCorrelation_increment (R Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) :
-    3*lam^2*Q.mu*R.mu-lam^3*Q.eta*R.eta ≤ 3*pairCorrelation R Q lam := by
-  have hc : Integrable (fun r ↦ ∫ m, pairZ lam r m*Real.log (pairZ lam r m) ∂Q.probability.toMeasure)
-      R.probability.toMeasure :=
-    (compact_integrable (continuous_pairEntropy hl₀ hl₁)
-      (μ := R.probability.toMeasure.prod Q.probability.toMeasure)).integral_prod_left
-  have h := integral_mono
-    (((compact_integrable continuous_xMoment).const_mul (3*lam^2*Q.mu)).sub
-      ((compact_integrable continuous_yMoment).const_mul (lam^3*Q.eta))) (hc.const_mul 3)
-      (pairCorrelation_increment_at Q hl₀ hl₁)
-  simp only [Pi.sub_apply] at h
-  rw [integral_sub (f := fun r ↦ 3*lam^2*Q.mu*xMoment r)
-      (g := fun r ↦ lam^3*Q.eta*yMoment r)
-      ((compact_integrable continuous_xMoment).const_mul _)
-      ((compact_integrable continuous_yMoment).const_mul _),
-    integral_const_mul, integral_const_mul, integral_const_mul] at h
-  exact h
-
-end ThreeState.TreeClauses.Experiment
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeStateClauses.RadialCalculus
+import OAI.Probability.ThreeStateClauses.BranchMoments
+
+namespace OAI
+
+/-! Correlation. -/
+
+open scoped BigOperators ENNReal NNReal Topology
+open Filter
+noncomputable section
+open Set
+namespace ThreeState.TreeClauses.Positive
+open Radial
+
+lemma log_cubic_lower {z : ℝ} (hz : -1 < z) :
+    z+z^2/2-z^3/6 ≤ (1+z)*Real.log (1+z) := by
+  let f : ℝ → ℝ := fun r ↦ r*Real.log r-(r-1)-(r-1)^2/2+(r-1)^3/6
+  let g : ℝ → ℝ := fun r ↦ Real.log r-(r-1)+(r-1)^2/2
+  have df (r : ℝ) (hr : 0 < r) : HasDerivAt f (g r) r := by
+    have hd := ((((hasDerivAt_id r).mul (Real.hasDerivAt_log (ne_of_gt hr))).sub
+      ((hasDerivAt_id r).sub_const 1)).sub
+      ((((hasDerivAt_id r).sub_const 1).pow 2).div_const 2)).add
+      ((((hasDerivAt_id r).sub_const 1).pow 3).div_const 6)
+    apply hd.congr_deriv
+    dsimp [g]
+    field_simp [ne_of_gt hr]
+    ring
+  have dg (r : ℝ) (hr : 0 < r) : HasDerivAt g ((r-1)^2/r) r := by
+    have hd := ((Real.hasDerivAt_log (ne_of_gt hr)).sub ((hasDerivAt_id r).sub_const 1)).add
+      ((((hasDerivAt_id r).sub_const 1).pow 2).div_const 2)
+    apply hd.congr_deriv
+    dsimp only [id]
+    field_simp [ne_of_gt hr]
+    ring
+  have gm {a b : ℝ} (ha : 0 < a) : MonotoneOn g (Icc a b) := by
+    apply monotoneOn_Icc_of_hasDerivAt_nonneg (fun r hr ↦ dg r (lt_of_lt_of_le ha hr.1))
+    intro r hr
+    exact div_nonneg (sq_nonneg _) (lt_trans ha hr.1).le
+  have hg₁ : g 1 = 0 := by simp [g]
+  have hf₁ : f 1 = 0 := by simp [f]
+  have hmin (r : ℝ) (hr : 0 < r) : 0 ≤ f r := by
+    rcases le_total 1 r with h | h
+    · have hg (t : ℝ) (ht : t ∈ Icc 1 r) : 0 ≤ g t := by
+        have hh := gm (b := r) (by norm_num : (0:ℝ) < 1) ⟨le_rfl, h⟩ ht ht.1
+        rwa [hg₁] at hh
+      have hmono : MonotoneOn f (Icc 1 r) :=
+        monotoneOn_Icc_of_hasDerivAt_nonneg (fun t ht ↦ df t (by linarith [ht.1]))
+          (fun t ht ↦ hg t ⟨ht.1.le, ht.2.le⟩)
+      have hh := hmono ⟨le_rfl, h⟩ ⟨h, le_rfl⟩ h
+      rwa [hf₁] at hh
+    · have hg (t : ℝ) (ht : t ∈ Icc r 1) : g t ≤ 0 := by
+        have hh := gm (b := 1) hr ht ⟨h, le_rfl⟩ ht.2
+        rwa [hg₁] at hh
+      have hmono : MonotoneOn (fun t ↦ -f t) (Icc r 1) :=
+        monotoneOn_Icc_of_hasDerivAt_nonneg (fun t ht ↦ (df t (lt_of_lt_of_le hr ht.1)).neg)
+          (fun t ht ↦ neg_nonneg.mpr (hg t ⟨ht.1.le, ht.2.le⟩))
+      have hh := hmono ⟨le_rfl, h⟩ ⟨h, le_rfl⟩ h
+      change -f r ≤ -f 1 at hh
+      rw [hf₁] at hh
+      linarith
+  have hh := hmin (1+z) (by linarith)
+  dsimp [f] at hh
+  nlinarith
+
+def orbit (v : Vec) : Fin 6 → Vec :=
+  ![ ![v 0,v 1,v 2], ![v 0,v 2,v 1], ![v 1,v 0,v 2],
+     ![v 1,v 2,v 0], ![v 2,v 0,v 1], ![v 2,v 1,v 0] ]
+
+def avgSix (f : Fin 6 → ℝ) : ℝ :=
+  (f 0+f 1+f 2+f 3+f 4+f 5)/6
+
+def dotAvg (v w : Vec) : ℝ := avg (fun i ↦ v i*w i)
+
+lemma avgSix_dot {v w : Vec} (hv : avg v = 0) (hw : avg w = 0) :
+    avgSix (fun k ↦ dotAvg v (orbit w k)) = 0 := by
+  simp [avgSix, dotAvg, orbit, avg, third_eq hv, third_eq hw]
+  ring
+
+lemma avgSix_dot_sq {v w : Vec} (hv : avg v = 0) (hw : avg w = 0) :
+    avgSix (fun k ↦ dotAvg v (orbit w k)^2) = 2*momentX v*momentX w := by
+  simp [avgSix, dotAvg, orbit, momentX, avg, third_eq hv, third_eq hw]
+  ring
+
+lemma avgSix_dot_cube {v w : Vec} (hv : avg v = 0) (hw : avg w = 0) :
+    avgSix (fun k ↦ dotAvg v (orbit w k)^3) = 2*momentY v*momentY w := by
+  simp [avgSix, dotAvg, orbit, momentY, avg, third_eq hv, third_eq hw]
+  ring
+
+lemma avgSix_mono {f g : Fin 6 → ℝ} (h : ∀ k, f k ≤ g k) : avgSix f ≤ avgSix g := by
+  dsimp [avgSix]
+  linarith [h 0, h 1, h 2, h 3, h 4, h 5]
+
+theorem correlation_orbit_lower {v w : Vec} (hv : avg v = 0) (hw : avg w = 0)
+    (hZ : ∀ k, -1 < dotAvg v (orbit w k)) :
+    3*momentX v*momentX w-momentY v*momentY w ≤
+      3*avgSix (fun k ↦ (1+dotAvg v (orbit w k))*Real.log (1+dotAvg v (orbit w k))) := by
+  have h := avgSix_mono (fun k ↦ log_cubic_lower (hZ k))
+  have he : avgSix (fun k ↦ dotAvg v (orbit w k)+dotAvg v (orbit w k)^2/2-dotAvg v (orbit w k)^3/6) =
+      avgSix (fun k ↦ dotAvg v (orbit w k))+
+      avgSix (fun k ↦ dotAvg v (orbit w k)^2)/2-
+      avgSix (fun k ↦ dotAvg v (orbit w k)^3)/6 := by
+    dsimp [avgSix]; ring
+  rw [he, avgSix_dot hv hw, avgSix_dot_sq hv hw, avgSix_dot_cube hv hw] at h
+  linarith
+
+end ThreeState.TreeClauses.Positive
+
+end 
+
+noncomputable section
+open Set MeasureTheory
+namespace ThreeState.TreeClauses.Experiment
+open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
+
+def pairZ (lam : ℝ) (r m : Message) : ℝ := avg (fun i ↦ r.1 i*edgeMessage lam m i)
+
+def pairPosterior {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (r m : Message) : Message := by
+  have hz : 0 < pairZ lam r m := by
+    have h := avg_mono (fun i ↦ mul_le_mul_of_nonneg_left (edge_lower hl₀ m i) (coordinate_nonneg r i))
+    have he : avg (fun i ↦ r.1 i*(1-lam)) = 1-lam := by
+      calc
+        _ = (1-lam)*avg r.1 := by simp only [avg]; ring
+        _ = _ := by rw [r.2.2]; ring
+    rw [he] at h
+    exact lt_of_lt_of_le (sub_pos.mpr hl₁) h
+  refine ⟨fun i ↦ r.1 i*edgeMessage lam m i/pairZ lam r m, ?_, ?_⟩
+  · intro i
+    exact div_nonneg (mul_nonneg (coordinate_nonneg r i) ((edge_positive hl₀ hl₁ m).1 i).le) hz.le
+  · rw [avg_div]
+    exact div_self (ne_of_gt hz)
+
+lemma pairZ_lower {lam : ℝ} (hl₀ : 0 ≤ lam) (r m : Message) : 1-lam ≤ pairZ lam r m := by
+  have h := avg_mono (fun i ↦ mul_le_mul_of_nonneg_left (edge_lower hl₀ m i) (coordinate_nonneg r i))
+  have he : avg (fun i ↦ r.1 i*(1-lam)) = 1-lam := by
+    calc
+      _ = (1-lam)*avg r.1 := by simp only [avg]; ring
+      _ = _ := by rw [r.2.2]; ring
+  rwa [he] at h
+
+lemma pairZ_positive {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (r m : Message) :
+    0 < pairZ lam r m := lt_of_lt_of_le (sub_pos.mpr hl₁) (pairZ_lower hl₀ r m)
+
+lemma continuous_pairZ (lam : ℝ) : Continuous (fun p : Message × Message ↦ pairZ lam p.1 p.2) :=
+  continuous_avg.comp (continuous_pi (fun i ↦ ((continuous_coordinate i).comp continuous_fst).mul
+    (((continuous_apply i).comp (continuous_edgeMessage lam)).comp continuous_snd)))
+
+lemma continuous_pairPosterior {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) :
+    Continuous (fun p : Message × Message ↦ pairPosterior hl₀ hl₁ p.1 p.2) := by
+  apply Continuous.subtype_mk
+  exact continuous_pi (fun i ↦ (((continuous_coordinate i).comp continuous_fst).mul
+    (((continuous_apply i).comp (continuous_edgeMessage lam)).comp continuous_snd)).div
+      (continuous_pairZ lam) (fun p ↦ ne_of_gt (pairZ_positive hl₀ hl₁ p.1 p.2)))
+
+lemma pairZ_dot (lam : ℝ) (r m : Message) : pairZ lam r m =
+    1+dotAvg (centered r.1) (centered (edgeMessage lam m)) := by
+  have hr := r.2.2
+  have hm := m.2.2
+  dsimp only [pairZ, dotAvg, centered, edgeMessage, edge, avg] at *
+  linear_combination hr + lam*hm
+
+lemma integral_pairZ (Q : Law) (lam : ℝ) (r : Message) :
+    (∫ m, pairZ lam r m ∂Q.probability.toMeasure) = 1 := by
+  have hi (i : Fin 3) : Integrable (fun m ↦ r.1 i*edgeMessage lam m i) Q.probability.toMeasure :=
+    (compact_integrable ((continuous_apply i).comp (continuous_edgeMessage lam))).const_mul _
+  unfold pairZ
+  rw [integral_avg_measure hi]
+  simp_rw [integral_const_mul, Q.edge_balanced, mul_one]
+  exact r.2.2
+
+lemma integral_pair_coordinate (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
+    (r : Message) (i : Fin 3) :
+    (∫ m, pairZ lam r m*(pairPosterior hl₀ hl₁ r m).1 i ∂Q.probability.toMeasure) = r.1 i := by
+  have he (m : Message) : pairZ lam r m*(pairPosterior hl₀ hl₁ r m).1 i = r.1 i*edgeMessage lam m i := by
+    change pairZ lam r m*(r.1 i*edgeMessage lam m i/pairZ lam r m) = _
+    field_simp [ne_of_gt (pairZ_positive hl₀ hl₁ r m)]
+  simp_rw [he]
+  rw [integral_const_mul, Q.edge_balanced, mul_one]
+
+lemma continuous_pairEntropy {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) :
+    Continuous (fun p : Message × Message ↦ pairZ lam p.1 p.2*Real.log (pairZ lam p.1 p.2)) :=
+  (continuous_pairZ lam).mul ((continuous_pairZ lam).log (fun p ↦ ne_of_gt (pairZ_positive hl₀ hl₁ p.1 p.2)))
+
+def pairCorrelation (R Q : Law) (lam : ℝ) : ℝ :=
+  ∫ r, (∫ m, pairZ lam r m*Real.log (pairZ lam r m) ∂Q.probability.toMeasure) ∂ R.probability.toMeasure
+
+lemma pairCorrelation_nonneg (R Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) :
+    0 ≤ pairCorrelation R Q lam := by
+  apply integral_nonneg
+  intro r
+  have hc : Continuous (fun m ↦ pairZ lam r m) := (continuous_pairZ lam).comp (continuous_const.prodMk continuous_id)
+  have hi := compact_integrable hc (μ := Q.probability.toMeasure)
+  have he : Integrable (fun m : Message ↦ pairZ lam r m*Real.log (pairZ lam r m)) Q.probability.toMeasure :=
+    compact_integrable ((continuous_pairEntropy hl₀ hl₁).comp (continuous_const.prodMk continuous_id))
+  have h := integral_mono (f := fun m : Message ↦ pairZ lam r m-1)
+    (g := fun m ↦ pairZ lam r m*Real.log (pairZ lam r m)) (hi.sub (integrable_const _)) he
+    (fun m ↦ by
+      have hp := pairZ_positive hl₀ hl₁ r m
+      have hh := mul_le_mul_of_nonneg_left (Real.one_sub_inv_le_log_of_pos hp) hp.le
+      simpa [mul_sub, ne_of_gt hp] using hh)
+  rw [integral_sub (f := fun m ↦ pairZ lam r m) (g := fun _ ↦ (1:ℝ)) hi (integrable_const _), integral_pairZ] at h
+  simpa using h
+
+def orbitIndex : Fin 6 → (Fin 3 → Fin 3) :=
+  ![ ![0,1,2], ![0,2,1], ![1,0,2], ![1,2,0], ![2,0,1], ![2,1,0] ]
+
+def orbitPerm (k : Fin 6) : Equiv.Perm (Fin 3) :=
+  Equiv.ofBijective (orbitIndex k) (by fin_cases k <;> decide)
+
+lemma orbitPerm_apply (v : Vec) (k : Fin 6) (i : Fin 3) :
+    v (orbitPerm k i) = orbit v k i := by
+  fin_cases k <;> fin_cases i <;> rfl
+
+lemma pairZ_orbit (lam : ℝ) (r m : Message) (k : Fin 6) :
+    pairZ lam r (permute (orbitPerm k) m) =
+      1+dotAvg (centered r.1) (orbit (centered (edgeMessage lam m)) k) := by
+  rw [pairZ_dot]
+  congr 2
+  funext i
+  change edgeMessage lam m (orbitPerm k i)-1 = _
+  exact orbitPerm_apply (centered (edgeMessage lam m)) k i
+
+lemma integral_avgSix_measure {X : Type*} [MeasurableSpace X] {ρ : Measure X}
+    {f : X → Fin 6 → ℝ} (hf : ∀ k, Integrable (fun x ↦ f x k) ρ) :
+    (∫ x, avgSix (f x) ∂ρ) = avgSix (fun k ↦ ∫ x, f x k ∂ρ) := by
+  unfold avgSix
+  rw [integral_div,
+    integral_add (f := fun x ↦ f x 0+f x 1+f x 2+f x 3+f x 4) (g := fun x ↦ f x 5)
+      (((((hf 0).add (hf 1)).add (hf 2)).add (hf 3)).add (hf 4)) (hf 5),
+    integral_add (f := fun x ↦ f x 0+f x 1+f x 2+f x 3) (g := fun x ↦ f x 4)
+      ((((hf 0).add (hf 1)).add (hf 2)).add (hf 3)) (hf 4),
+    integral_add (f := fun x ↦ f x 0+f x 1+f x 2) (g := fun x ↦ f x 3)
+      (((hf 0).add (hf 1)).add (hf 2)) (hf 3),
+    integral_add (f := fun x ↦ f x 0+f x 1) (g := fun x ↦ f x 2)
+      ((hf 0).add (hf 1)) (hf 2),
+    integral_add (f := fun x ↦ f x 0) (g := fun x ↦ f x 1) (hf 0) (hf 1)]
+
+lemma xMoment_edge (lam : ℝ) (m : Message) :
+    momentX (centered (edgeMessage lam m)) = lam^2*xMoment m := by
+  dsimp [momentX, centered, edgeMessage, edge, xMoment, avg]
+  ring
+
+lemma yMoment_edge (lam : ℝ) (m : Message) :
+    momentY (centered (edgeMessage lam m)) = lam^3*yMoment m := by
+  dsimp [momentY, centered, edgeMessage, edge, yMoment, avg]
+  ring
+
+lemma pairCorrelation_increment_at (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
+    (r : Message) :
+    3*lam^2*Q.mu*xMoment r-lam^3*Q.eta*yMoment r ≤
+      3*(∫ m, pairZ lam r m*Real.log (pairZ lam r m) ∂Q.probability.toMeasure) := by
+  let f : Message → Fin 6 → ℝ := fun m k ↦
+    pairZ lam r (permute (orbitPerm k) m)*Real.log (pairZ lam r (permute (orbitPerm k) m))
+  have hi (k : Fin 6) : Integrable (fun m ↦ f m k) Q.probability.toMeasure :=
+    compact_integrable ((continuous_pairEntropy hl₀ hl₁).comp
+      (continuous_const.prodMk (continuous_permute (orbitPerm k))))
+  have havg : Integrable (fun m ↦ avgSix (f m)) Q.probability.toMeasure :=
+    (((((hi 0).add (hi 1)).add (hi 2)).add (hi 3)).add (hi 4) |>.add (hi 5)).div_const 6
+  have hpoint (m : Message) : 3*lam^2*xMoment r*xMoment m-lam^3*yMoment r*yMoment m ≤
+      3*avgSix (f m) := by
+    have h := correlation_orbit_lower (message_centered_avg r)
+      (show avg (centered (edgeMessage lam m)) = 0 from by
+        change avg (fun i ↦ edgeMessage lam m i-1) = 0
+        rw [avg_sub, avg_const, (edge_positive hl₀ hl₁ m).2]; ring)
+      (fun k ↦ by
+        have hh := pairZ_positive hl₀ hl₁ r (permute (orbitPerm k) m)
+        rw [pairZ_orbit] at hh; linarith)
+    simp only [xMoment_edge, yMoment_edge] at h
+    have he : (fun k ↦ (1+dotAvg (centered r.1) (orbit (centered (edgeMessage lam m)) k))*
+        Real.log (1+dotAvg (centered r.1) (orbit (centered (edgeMessage lam m)) k))) = f m := by
+      funext k; dsimp only [f]; rw [pairZ_orbit]
+    rw [he] at h
+    change 3*xMoment r*(lam^2*xMoment m)-yMoment r*(lam^3*yMoment m) ≤ _ at h
+    nlinarith [h]
+  have h := integral_mono
+    (((compact_integrable continuous_xMoment).const_mul (3*lam^2*xMoment r)).sub
+      ((compact_integrable continuous_yMoment).const_mul (lam^3*yMoment r)))
+    (havg.const_mul 3) hpoint
+  simp only [Pi.sub_apply] at h
+  rw [integral_sub (f := fun m ↦ 3*lam^2*xMoment r*xMoment m)
+    (g := fun m ↦ lam^3*yMoment r*yMoment m)
+      ((compact_integrable continuous_xMoment).const_mul _)
+      ((compact_integrable continuous_yMoment).const_mul _),
+    integral_const_mul, integral_const_mul, integral_const_mul, integral_avgSix_measure hi] at h
+  have hev (k : Fin 6) : (∫ m, f m k ∂Q.probability.toMeasure) =
+      ∫ m, pairZ lam r m*Real.log (pairZ lam r m) ∂Q.probability.toMeasure := by
+    exact (show MeasurePreserving (permuteMeasurableEquiv (orbitPerm k))
+      Q.probability.toMeasure Q.probability.toMeasure from Q.symmetric _).integral_comp' (fun m ↦ pairZ lam r m*Real.log (pairZ lam r m))
+  simp_rw [hev] at h
+  dsimp only [avgSix] at h
+  change 3*lam^2*xMoment r*Q.mu-lam^3*yMoment r*Q.eta ≤ _ at h
+  linarith
+
+theorem pairCorrelation_increment (R Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) :
+    3*lam^2*Q.mu*R.mu-lam^3*Q.eta*R.eta ≤ 3*pairCorrelation R Q lam := by
+  have hc : Integrable (fun r ↦ ∫ m, pairZ lam r m*Real.log (pairZ lam r m) ∂Q.probability.toMeasure)
+      R.probability.toMeasure :=
+    (compact_integrable (continuous_pairEntropy hl₀ hl₁)
+      (μ := R.probability.toMeasure.prod Q.probability.toMeasure)).integral_prod_left
+  have h := integral_mono
+    (((compact_integrable continuous_xMoment).const_mul (3*lam^2*Q.mu)).sub
+      ((compact_integrable continuous_yMoment).const_mul (lam^3*Q.eta))) (hc.const_mul 3)
+      (pairCorrelation_increment_at Q hl₀ hl₁)
+  simp only [Pi.sub_apply] at h
+  rw [integral_sub (f := fun r ↦ 3*lam^2*Q.mu*xMoment r)
+      (g := fun r ↦ lam^3*Q.eta*yMoment r)
+      ((compact_integrable continuous_xMoment).const_mul _)
+      ((compact_integrable continuous_yMoment).const_mul _),
+    integral_const_mul, integral_const_mul, integral_const_mul] at h
+  exact h
+
+end ThreeState.TreeClauses.Experiment
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeStateClauses/CountableStatistics.lean b/lean/OAI/Probability/ThreeStateClauses/CountableStatistics.lean
index 9ecea8337d74eaf2635a0607a3962aa87f954058..1a769f7177da05c7d0f4be977276ad6248036a00 100644
--- a/lean/OAI/Probability/ThreeStateClauses/CountableStatistics.lean
+++ b/lean/OAI/Probability/ThreeStateClauses/CountableStatistics.lean
@@ -1,208 +1,208 @@
-import Mathlib
-import OAI.Probability.ThreeStateClauses.InformationLoss
-import OAI.Probability.ThreeStateClauses.ObservedMixtures
-
-namespace OAI
-
-/-! Countable Statistics. -/
-
-open scoped BigOperators ENNReal NNReal Topology
-open Filter
-noncomputable section
-open Set MeasureTheory
-open scoped BigOperators
-namespace ThreeState.TreeClauses.Experiment
-open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
-
-variable {α β : Type*} [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α]
-  [Countable β] [MeasurableSpace β] [MeasurableSingletonClass β]
-
-lemma pmf_measure_sum_dirac (p : PMF α) :
-    p.toMeasure = Measure.sum (fun a ↦ p a • Measure.dirac a) := by
-  simpa only [PMF.toMeasure_apply_singleton _ _ (measurableSet_singleton _)] using
-    (Measure.sum_smul_dirac p.toMeasure).symm
-
-lemma pmf_integrable_iff (p : PMF α) (f : α → ℝ) :
-    Integrable f p.toMeasure ↔ Summable (fun a ↦ (p a).toReal*|f a|) := by
-  rw [pmf_measure_sum_dirac, integrable_sum_dirac_iff (fun a ↦ p.apply_ne_top a)]
-  rfl
-
-lemma integral_pmf_tsum (p : PMF α) (f : α → ℝ) :
-    (∫ a, f a ∂p.toMeasure) = ∑' a, (p a).toReal*f a := by
-  rw [pmf_measure_sum_dirac, integral_sum_dirac (fun a ↦ p.apply_ne_top a)]
-  rfl
-
-omit [Countable β] [MeasurableSingletonClass β] in
-lemma pmf_integrable_bind (p : PMF α) (K : α → PMF β) {f : β → ℝ}
-    (hf : ∀ a, Integrable f (K a).toMeasure)
-    (hn : Integrable (fun a ↦ ∫ b, |f b| ∂(K a).toMeasure) p.toMeasure) :
-    Integrable f (p.bind K).toMeasure := by
-  rw [pmf_measure_bind]
-  apply integrable_sum_measure (fun a ↦ (hf a).smul_measure (p.apply_ne_top a))
-  have h := (pmf_integrable_iff p _).mp hn
-  simpa only [integral_smul_measure, smul_eq_mul, Real.norm_eq_abs,
-    abs_of_nonneg (integral_nonneg (fun _ ↦ abs_nonneg _))] using h
-
-omit [Countable β] [MeasurableSingletonClass β] in
-lemma integral_pmf_bind_integrable (p : PMF α) (K : α → PMF β) {f : β → ℝ}
-    (hf : Integrable f (p.bind K).toMeasure) :
-    (∫ b, f b ∂(p.bind K).toMeasure) = ∫ a, (∫ b, f b ∂(K a).toMeasure) ∂p.toMeasure := by
-  rw [pmf_measure_bind] at hf ⊢
-  rw [integral_sum_measure hf, integral_pmf_tsum]
-  simp only [integral_smul_measure, smul_eq_mul]
-
-omit [Countable α] [MeasurableSingletonClass α] in
-lemma pmf_integrable_bind_finite {ι : Type*} [Fintype ι] [MeasurableSpace ι]
-    [MeasurableSingletonClass ι] (p : PMF ι) (K : ι → PMF α) {f : α → ℝ}
-    (hf : ∀ i, Integrable f (K i).toMeasure) : Integrable f (p.bind K).toMeasure :=
-  pmf_integrable_bind p K hf Integrable.of_finite
-
-lemma pmf_integrable_map_iff (p : PMF α) (g : α → β) (f : β → ℝ) :
-    Integrable f (p.map g).toMeasure ↔ Integrable (fun a ↦ f (g a)) p.toMeasure := by
-  rw [← PMF.toMeasure_map g p (measurable_of_countable g)]
-  exact integrable_map_measure (measurable_of_countable _).aestronglyMeasurable
-    (measurable_of_countable _).aemeasurable
-
-lemma pmf_integrable_of_square (p : PMF α) {f : α → ℝ}
-    (hf : Integrable (fun a ↦ f a^2) p.toMeasure) : Integrable f p.toMeasure := by
-  apply ((integrable_const (1:ℝ)).add hf).mono' (measurable_of_countable _).aestronglyMeasurable
-  exact Filter.Eventually.of_forall (fun a ↦ by
-    simp only [Real.norm_eq_abs, Pi.add_apply]
-    nlinarith [sq_nonneg (|f a|-1), sq_abs (f a)])
-
-lemma discrete_bayes_integral_integrable (p : Spin → PMF α) (i : Spin) {f : α → ℝ}
-    (hf : Integrable f (discreteMarginal p).toMeasure) (hi : Integrable f (p i).toMeasure) :
-    (∫ a, (discreteMessage p a).1 i*f a ∂(discreteMarginal p).toMeasure) =
-      ∫ a, f a ∂(p i).toMeasure := by
-  have hl := hf.bdd_mul (measurable_of_countable (fun a ↦ (discreteMessage p a).1 i)).aestronglyMeasurable
-    (Filter.Eventually.of_forall (fun a ↦ show ‖(discreteMessage p a).1 i‖ ≤ (3:ℝ) by
-      rw [Real.norm_eq_abs, abs_of_nonneg (coordinate_nonneg _ _)]; exact coordinate_le_three _ _))
-  rw [PMF.integral_eq_tsum _ _ hl, PMF.integral_eq_tsum _ _ hi]
-  apply tsum_congr
-  intro a
-  simp only [smul_eq_mul, discreteMarginal_real, ← mul_assoc, discrete_bayes]
-
-omit [Countable α] [MeasurableSingletonClass α] in
-lemma integral_discreteMarginal_integrable (p : Spin → PMF α) {f : α → ℝ}
-    (hf : ∀ i, Integrable f (p i).toMeasure) :
-    (∫ a, f a ∂(discreteMarginal p).toMeasure) =
-      ((∫ a, f a ∂(p 0).toMeasure)+(∫ a, f a ∂(p 1).toMeasure)+(∫ a, f a ∂(p 2).toMeasure))/3 := by
-  rw [discreteMarginal, integral_pmf_bind_integrable _ _ (pmf_integrable_bind_finite _ _ hf), PMF.integral_eq_sum]
-  simp only [Fin.sum_univ_three, uniformSpin_apply, ENNReal.toReal_div,
-    ENNReal.toReal_one, ENNReal.toReal_ofNat, smul_eq_mul]
-  ring
-
-lemma estimator_tv_lower_integrable (p : Spin → PMF α) {f : α → ℝ} {C : ℝ}
-    (hf : ∀ i, Integrable (fun a ↦ f a^2) (p i).toMeasure) (hC : 0 < C)
-    (hzero : (∫ a, f a ∂(discreteMarginal p).toMeasure) = 0)
-    (hroot : (∫ a, f a ∂(p 0).toMeasure) = 2)
-    (hsecond : (∫ a, f a^2 ∂(discreteMarginal p).toMeasure) ≤ C) :
-    1/(3*C) ≤ ∫ m, tvMessage m ∂(discreteProbability p).toMeasure := by
-  let v : α → ℝ := fun a ↦ (discreteMessage p a).1 0-1
-  have hv (a : α) : |v a| ≤ 2 := coord_center_abs_le_two _ _
-  have hf₂ : Integrable (fun a ↦ f a^2) (discreteMarginal p).toMeasure :=
-    pmf_integrable_bind_finite _ _ hf
-  have hf₁ := pmf_integrable_of_square _ hf₂
-  have hv₂ := pmf_bounded_integrable (discreteMarginal p) (fun a ↦ show |v a^2| ≤ (2:ℝ)^2 by
-    rw [abs_of_nonneg (sq_nonneg _)]; nlinarith [hv a, abs_nonneg (v a), sq_abs (v a)])
-  have hvf := hf₁.bdd_mul (measurable_of_countable v).aestronglyMeasurable
-    (Filter.Eventually.of_forall (fun a ↦ show ‖v a‖ ≤ (2:ℝ) from hv a))
-  have hcross : (∫ a, v a*f a ∂(discreteMarginal p).toMeasure) = 2 := by
-    have he (a : α) : v a*f a = (discreteMessage p a).1 0*f a-f a := by dsimp [v]; ring
-    simp_rw [he]
-    rw [integral_sub (hf₁.bdd_mul (measurable_of_countable _).aestronglyMeasurable
-      (Filter.Eventually.of_forall (fun a ↦ show ‖(discreteMessage p a).1 0‖ ≤ (3:ℝ) by
-        rw [Real.norm_eq_abs, abs_of_nonneg (coordinate_nonneg _ _)]; exact coordinate_le_three _ _))) hf₁,
-      discrete_bayes_integral_integrable p 0 hf₁ (pmf_integrable_of_square _ (hf 0)), hroot, hzero, sub_zero]
-  let r := 2/C
-  have hr : r*C = 2 := div_mul_cancel₀ 2 hC.ne'
-  have h := MeasureTheory.integral_nonneg (μ := (discreteMarginal p).toMeasure)
-    (f := fun a ↦ (v a-r*f a)^2) (fun _ ↦ sq_nonneg _)
-  rw [integral_scaled_square _ r hv₂ hf₂ hvf, hcross] at h
-  have hb := mul_le_mul_of_nonneg_left hsecond (sq_nonneg r)
-  have hr2 : r^2*C = 2*r := by
-    have hh := congrArg (fun x : ℝ ↦ r*x) hr
-    nlinarith only [hh]
-  have hD : 4 ≤ C*(∫ a, v a^2 ∂(discreteMarginal p).toMeasure) := by
-    have hs : 2*r ≤ ∫ a, v a^2 ∂(discreteMarginal p).toMeasure := by
-      nlinarith only [h, hb, hr2]
-    have hh := mul_le_mul_of_nonneg_left hs hC.le
-    nlinarith only [hh, hr]
-  have hT := integral_mono hv₂ (pmf_bounded_integrable (discreteMarginal p)
-    (fun a ↦ show |12*tvMessage (discreteMessage p a)| ≤ 12 by
-      rw [abs_mul, abs_of_nonneg (by norm_num : (0:ℝ) ≤ 12), abs_of_nonneg (tvMessage_nonneg _)];
-      nlinarith [tvMessage_le_one (discreteMessage p a)]))
-    (fun a ↦ coord_sq_le_tv (discreteMessage p a) 0)
-  rw [integral_const_mul, ← integral_discreteProbability p continuous_tvMessage.measurable] at hT
-  apply (div_le_iff₀ (by positivity : 0 < 3*C)).2
-  have hh := mul_le_mul_of_nonneg_left hT hC.le
-  nlinarith only [hD, hh]
-
-end ThreeState.TreeClauses.Experiment
-
-end 
-
-noncomputable section
-open Set MeasureTheory
-open scoped BigOperators
-namespace ThreeState.TreeClauses.Experiment
-
-variable {α : Type*} [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α]
-
-lemma product_sum_integrable (p : PMF α) {f : α → ℝ} (hf : Integrable f p.toMeasure) (n : ℕ) :
-    Integrable (fun v : Fin n → α ↦ ∑ j, f (v j)) (productPMF (fun _ ↦ p)).toMeasure := by
-  rw [productPMF_measure]
-  simpa [weightProduct] using integrable_weighted_sum (ι := Fin n) (a := fun _ : α ↦ (1:ℝ))
-    (u := f) (integrable_const 1) (by simpa using hf)
-
-lemma product_square_sum_integrable (p : PMF α) {f : α → ℝ}
-    (hf : Integrable (fun a ↦ f a^2) p.toMeasure) (n : ℕ) :
-    Integrable (fun v : Fin n → α ↦ (∑ j, f (v j))^2) (productPMF (fun _ ↦ p)).toMeasure := by
-  rw [productPMF_measure]
-  simpa [weightProduct] using integrable_weighted_sq_sum (ι := Fin n) (a := fun _ : α ↦ (1:ℝ))
-    (u := f) (integrable_const 1) (by simpa using pmf_integrable_of_square p hf) (by simpa using hf)
-
-lemma integral_product_sum_integrable (p : PMF α) {f : α → ℝ}
-    (hf : Integrable f p.toMeasure) (n : ℕ) :
-    (∫ v : Fin n → α, (∑ j, f (v j)) ∂(productPMF (fun _ ↦ p)).toMeasure) =
-      (n:ℝ)*(∫ a, f a ∂p.toMeasure) := by
-  rw [productPMF_measure]
-  simpa [weightProduct] using integral_weighted_sum (ι := Fin n) (a := fun _ : α ↦ (1:ℝ))
-    (u := f) (integrable_const 1) (by simpa using hf) (by simp)
-
-lemma integral_product_sq_sum_integrable (p : PMF α) {f : α → ℝ}
-    (hf : Integrable (fun a ↦ f a^2) p.toMeasure) (n : ℕ) :
-    (∫ v : Fin n → α, (∑ j, f (v j))^2 ∂(productPMF (fun _ ↦ p)).toMeasure) =
-      (n:ℝ)*(∫ a, f a^2 ∂p.toMeasure)+((n:ℝ)^2-n)*(∫ a, f a ∂p.toMeasure)^2 := by
-  rw [productPMF_measure]
-  simpa [weightProduct] using integral_weighted_sq_sum (ι := Fin n) (a := fun _ : α ↦ (1:ℝ))
-    (u := f) (integrable_const 1) (by simpa using pmf_integrable_of_square p hf)
-    (by simpa using hf) (by simp)
-
-section Sigma
-variable {β : ℕ → Type*} [∀ n, Countable (β n)] [∀ n, MeasurableSpace (β n)]
-  [∀ n, MeasurableSingletonClass (β n)]
-
-lemma sigmaPMF_integrable (ρ : PMF ℕ) (p : ∀ n, PMF (β n)) {f : Sigma β → ℝ}
-    (hf : ∀ n, Integrable (fun a ↦ f ⟨n,a⟩) (p n).toMeasure)
-    (hn : Integrable (fun n ↦ ∫ a, |f ⟨n,a⟩| ∂(p n).toMeasure) ρ.toMeasure) :
-    Integrable f (sigmaPMF ρ p).toMeasure := by
-  rw [sigmaPMF_eq_bind]
-  apply pmf_integrable_bind
-  · intro n
-    exact (pmf_integrable_map_iff (p n) (Sigma.mk n) f).2 (hf n)
-  · simpa only [integral_pmf_map] using hn
-
-lemma integral_sigmaPMF_integrable (ρ : PMF ℕ) (p : ∀ n, PMF (β n)) {f : Sigma β → ℝ}
-    (hf : Integrable f (sigmaPMF ρ p).toMeasure) :
-    (∫ z, f z ∂(sigmaPMF ρ p).toMeasure) = ∫ n, (∫ a, f ⟨n,a⟩ ∂(p n).toMeasure) ∂ρ.toMeasure := by
-  rw [sigmaPMF_eq_bind] at hf ⊢
-  rw [integral_pmf_bind_integrable _ _ hf]
-  simp only [integral_pmf_map]
-
-end Sigma
-end ThreeState.TreeClauses.Experiment
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeStateClauses.InformationLoss
+import OAI.Probability.ThreeStateClauses.ObservedMixtures
+
+namespace OAI
+
+/-! Countable Statistics. -/
+
+open scoped BigOperators ENNReal NNReal Topology
+open Filter
+noncomputable section
+open Set MeasureTheory
+open scoped BigOperators
+namespace ThreeState.TreeClauses.Experiment
+open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
+
+variable {α β : Type*} [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α]
+  [Countable β] [MeasurableSpace β] [MeasurableSingletonClass β]
+
+lemma pmf_measure_sum_dirac (p : PMF α) :
+    p.toMeasure = Measure.sum (fun a ↦ p a • Measure.dirac a) := by
+  simpa only [PMF.toMeasure_apply_singleton _ _ (measurableSet_singleton _)] using
+    (Measure.sum_smul_dirac p.toMeasure).symm
+
+lemma pmf_integrable_iff (p : PMF α) (f : α → ℝ) :
+    Integrable f p.toMeasure ↔ Summable (fun a ↦ (p a).toReal*|f a|) := by
+  rw [pmf_measure_sum_dirac, integrable_sum_dirac_iff (fun a ↦ p.apply_ne_top a)]
+  rfl
+
+lemma integral_pmf_tsum (p : PMF α) (f : α → ℝ) :
+    (∫ a, f a ∂p.toMeasure) = ∑' a, (p a).toReal*f a := by
+  rw [pmf_measure_sum_dirac, integral_sum_dirac (fun a ↦ p.apply_ne_top a)]
+  rfl
+
+omit [Countable β] [MeasurableSingletonClass β] in
+lemma pmf_integrable_bind (p : PMF α) (K : α → PMF β) {f : β → ℝ}
+    (hf : ∀ a, Integrable f (K a).toMeasure)
+    (hn : Integrable (fun a ↦ ∫ b, |f b| ∂(K a).toMeasure) p.toMeasure) :
+    Integrable f (p.bind K).toMeasure := by
+  rw [pmf_measure_bind]
+  apply integrable_sum_measure (fun a ↦ (hf a).smul_measure (p.apply_ne_top a))
+  have h := (pmf_integrable_iff p _).mp hn
+  simpa only [integral_smul_measure, smul_eq_mul, Real.norm_eq_abs,
+    abs_of_nonneg (integral_nonneg (fun _ ↦ abs_nonneg _))] using h
+
+omit [Countable β] [MeasurableSingletonClass β] in
+lemma integral_pmf_bind_integrable (p : PMF α) (K : α → PMF β) {f : β → ℝ}
+    (hf : Integrable f (p.bind K).toMeasure) :
+    (∫ b, f b ∂(p.bind K).toMeasure) = ∫ a, (∫ b, f b ∂(K a).toMeasure) ∂p.toMeasure := by
+  rw [pmf_measure_bind] at hf ⊢
+  rw [integral_sum_measure hf, integral_pmf_tsum]
+  simp only [integral_smul_measure, smul_eq_mul]
+
+omit [Countable α] [MeasurableSingletonClass α] in
+lemma pmf_integrable_bind_finite {ι : Type*} [Fintype ι] [MeasurableSpace ι]
+    [MeasurableSingletonClass ι] (p : PMF ι) (K : ι → PMF α) {f : α → ℝ}
+    (hf : ∀ i, Integrable f (K i).toMeasure) : Integrable f (p.bind K).toMeasure :=
+  pmf_integrable_bind p K hf Integrable.of_finite
+
+lemma pmf_integrable_map_iff (p : PMF α) (g : α → β) (f : β → ℝ) :
+    Integrable f (p.map g).toMeasure ↔ Integrable (fun a ↦ f (g a)) p.toMeasure := by
+  rw [← PMF.toMeasure_map g p (measurable_of_countable g)]
+  exact integrable_map_measure (measurable_of_countable _).aestronglyMeasurable
+    (measurable_of_countable _).aemeasurable
+
+lemma pmf_integrable_of_square (p : PMF α) {f : α → ℝ}
+    (hf : Integrable (fun a ↦ f a^2) p.toMeasure) : Integrable f p.toMeasure := by
+  apply ((integrable_const (1:ℝ)).add hf).mono' (measurable_of_countable _).aestronglyMeasurable
+  exact Filter.Eventually.of_forall (fun a ↦ by
+    simp only [Real.norm_eq_abs, Pi.add_apply]
+    nlinarith [sq_nonneg (|f a|-1), sq_abs (f a)])
+
+lemma discrete_bayes_integral_integrable (p : Spin → PMF α) (i : Spin) {f : α → ℝ}
+    (hf : Integrable f (discreteMarginal p).toMeasure) (hi : Integrable f (p i).toMeasure) :
+    (∫ a, (discreteMessage p a).1 i*f a ∂(discreteMarginal p).toMeasure) =
+      ∫ a, f a ∂(p i).toMeasure := by
+  have hl := hf.bdd_mul (measurable_of_countable (fun a ↦ (discreteMessage p a).1 i)).aestronglyMeasurable
+    (Filter.Eventually.of_forall (fun a ↦ show ‖(discreteMessage p a).1 i‖ ≤ (3:ℝ) by
+      rw [Real.norm_eq_abs, abs_of_nonneg (coordinate_nonneg _ _)]; exact coordinate_le_three _ _))
+  rw [PMF.integral_eq_tsum _ _ hl, PMF.integral_eq_tsum _ _ hi]
+  apply tsum_congr
+  intro a
+  simp only [smul_eq_mul, discreteMarginal_real, ← mul_assoc, discrete_bayes]
+
+omit [Countable α] [MeasurableSingletonClass α] in
+lemma integral_discreteMarginal_integrable (p : Spin → PMF α) {f : α → ℝ}
+    (hf : ∀ i, Integrable f (p i).toMeasure) :
+    (∫ a, f a ∂(discreteMarginal p).toMeasure) =
+      ((∫ a, f a ∂(p 0).toMeasure)+(∫ a, f a ∂(p 1).toMeasure)+(∫ a, f a ∂(p 2).toMeasure))/3 := by
+  rw [discreteMarginal, integral_pmf_bind_integrable _ _ (pmf_integrable_bind_finite _ _ hf), PMF.integral_eq_sum]
+  simp only [Fin.sum_univ_three, uniformSpin_apply, ENNReal.toReal_div,
+    ENNReal.toReal_one, ENNReal.toReal_ofNat, smul_eq_mul]
+  ring
+
+lemma estimator_tv_lower_integrable (p : Spin → PMF α) {f : α → ℝ} {C : ℝ}
+    (hf : ∀ i, Integrable (fun a ↦ f a^2) (p i).toMeasure) (hC : 0 < C)
+    (hzero : (∫ a, f a ∂(discreteMarginal p).toMeasure) = 0)
+    (hroot : (∫ a, f a ∂(p 0).toMeasure) = 2)
+    (hsecond : (∫ a, f a^2 ∂(discreteMarginal p).toMeasure) ≤ C) :
+    1/(3*C) ≤ ∫ m, tvMessage m ∂(discreteProbability p).toMeasure := by
+  let v : α → ℝ := fun a ↦ (discreteMessage p a).1 0-1
+  have hv (a : α) : |v a| ≤ 2 := coord_center_abs_le_two _ _
+  have hf₂ : Integrable (fun a ↦ f a^2) (discreteMarginal p).toMeasure :=
+    pmf_integrable_bind_finite _ _ hf
+  have hf₁ := pmf_integrable_of_square _ hf₂
+  have hv₂ := pmf_bounded_integrable (discreteMarginal p) (fun a ↦ show |v a^2| ≤ (2:ℝ)^2 by
+    rw [abs_of_nonneg (sq_nonneg _)]; nlinarith [hv a, abs_nonneg (v a), sq_abs (v a)])
+  have hvf := hf₁.bdd_mul (measurable_of_countable v).aestronglyMeasurable
+    (Filter.Eventually.of_forall (fun a ↦ show ‖v a‖ ≤ (2:ℝ) from hv a))
+  have hcross : (∫ a, v a*f a ∂(discreteMarginal p).toMeasure) = 2 := by
+    have he (a : α) : v a*f a = (discreteMessage p a).1 0*f a-f a := by dsimp [v]; ring
+    simp_rw [he]
+    rw [integral_sub (hf₁.bdd_mul (measurable_of_countable _).aestronglyMeasurable
+      (Filter.Eventually.of_forall (fun a ↦ show ‖(discreteMessage p a).1 0‖ ≤ (3:ℝ) by
+        rw [Real.norm_eq_abs, abs_of_nonneg (coordinate_nonneg _ _)]; exact coordinate_le_three _ _))) hf₁,
+      discrete_bayes_integral_integrable p 0 hf₁ (pmf_integrable_of_square _ (hf 0)), hroot, hzero, sub_zero]
+  let r := 2/C
+  have hr : r*C = 2 := div_mul_cancel₀ 2 hC.ne'
+  have h := MeasureTheory.integral_nonneg (μ := (discreteMarginal p).toMeasure)
+    (f := fun a ↦ (v a-r*f a)^2) (fun _ ↦ sq_nonneg _)
+  rw [integral_scaled_square _ r hv₂ hf₂ hvf, hcross] at h
+  have hb := mul_le_mul_of_nonneg_left hsecond (sq_nonneg r)
+  have hr2 : r^2*C = 2*r := by
+    have hh := congrArg (fun x : ℝ ↦ r*x) hr
+    nlinarith only [hh]
+  have hD : 4 ≤ C*(∫ a, v a^2 ∂(discreteMarginal p).toMeasure) := by
+    have hs : 2*r ≤ ∫ a, v a^2 ∂(discreteMarginal p).toMeasure := by
+      nlinarith only [h, hb, hr2]
+    have hh := mul_le_mul_of_nonneg_left hs hC.le
+    nlinarith only [hh, hr]
+  have hT := integral_mono hv₂ (pmf_bounded_integrable (discreteMarginal p)
+    (fun a ↦ show |12*tvMessage (discreteMessage p a)| ≤ 12 by
+      rw [abs_mul, abs_of_nonneg (by norm_num : (0:ℝ) ≤ 12), abs_of_nonneg (tvMessage_nonneg _)];
+      nlinarith [tvMessage_le_one (discreteMessage p a)]))
+    (fun a ↦ coord_sq_le_tv (discreteMessage p a) 0)
+  rw [integral_const_mul, ← integral_discreteProbability p continuous_tvMessage.measurable] at hT
+  apply (div_le_iff₀ (by positivity : 0 < 3*C)).2
+  have hh := mul_le_mul_of_nonneg_left hT hC.le
+  nlinarith only [hD, hh]
+
+end ThreeState.TreeClauses.Experiment
+
+end 
+
+noncomputable section
+open Set MeasureTheory
+open scoped BigOperators
+namespace ThreeState.TreeClauses.Experiment
+
+variable {α : Type*} [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α]
+
+lemma product_sum_integrable (p : PMF α) {f : α → ℝ} (hf : Integrable f p.toMeasure) (n : ℕ) :
+    Integrable (fun v : Fin n → α ↦ ∑ j, f (v j)) (productPMF (fun _ ↦ p)).toMeasure := by
+  rw [productPMF_measure]
+  simpa [weightProduct] using integrable_weighted_sum (ι := Fin n) (a := fun _ : α ↦ (1:ℝ))
+    (u := f) (integrable_const 1) (by simpa using hf)
+
+lemma product_square_sum_integrable (p : PMF α) {f : α → ℝ}
+    (hf : Integrable (fun a ↦ f a^2) p.toMeasure) (n : ℕ) :
+    Integrable (fun v : Fin n → α ↦ (∑ j, f (v j))^2) (productPMF (fun _ ↦ p)).toMeasure := by
+  rw [productPMF_measure]
+  simpa [weightProduct] using integrable_weighted_sq_sum (ι := Fin n) (a := fun _ : α ↦ (1:ℝ))
+    (u := f) (integrable_const 1) (by simpa using pmf_integrable_of_square p hf) (by simpa using hf)
+
+lemma integral_product_sum_integrable (p : PMF α) {f : α → ℝ}
+    (hf : Integrable f p.toMeasure) (n : ℕ) :
+    (∫ v : Fin n → α, (∑ j, f (v j)) ∂(productPMF (fun _ ↦ p)).toMeasure) =
+      (n:ℝ)*(∫ a, f a ∂p.toMeasure) := by
+  rw [productPMF_measure]
+  simpa [weightProduct] using integral_weighted_sum (ι := Fin n) (a := fun _ : α ↦ (1:ℝ))
+    (u := f) (integrable_const 1) (by simpa using hf) (by simp)
+
+lemma integral_product_sq_sum_integrable (p : PMF α) {f : α → ℝ}
+    (hf : Integrable (fun a ↦ f a^2) p.toMeasure) (n : ℕ) :
+    (∫ v : Fin n → α, (∑ j, f (v j))^2 ∂(productPMF (fun _ ↦ p)).toMeasure) =
+      (n:ℝ)*(∫ a, f a^2 ∂p.toMeasure)+((n:ℝ)^2-n)*(∫ a, f a ∂p.toMeasure)^2 := by
+  rw [productPMF_measure]
+  simpa [weightProduct] using integral_weighted_sq_sum (ι := Fin n) (a := fun _ : α ↦ (1:ℝ))
+    (u := f) (integrable_const 1) (by simpa using pmf_integrable_of_square p hf)
+    (by simpa using hf) (by simp)
+
+section Sigma
+variable {β : ℕ → Type*} [∀ n, Countable (β n)] [∀ n, MeasurableSpace (β n)]
+  [∀ n, MeasurableSingletonClass (β n)]
+
+lemma sigmaPMF_integrable (ρ : PMF ℕ) (p : ∀ n, PMF (β n)) {f : Sigma β → ℝ}
+    (hf : ∀ n, Integrable (fun a ↦ f ⟨n,a⟩) (p n).toMeasure)
+    (hn : Integrable (fun n ↦ ∫ a, |f ⟨n,a⟩| ∂(p n).toMeasure) ρ.toMeasure) :
+    Integrable f (sigmaPMF ρ p).toMeasure := by
+  rw [sigmaPMF_eq_bind]
+  apply pmf_integrable_bind
+  · intro n
+    exact (pmf_integrable_map_iff (p n) (Sigma.mk n) f).2 (hf n)
+  · simpa only [integral_pmf_map] using hn
+
+lemma integral_sigmaPMF_integrable (ρ : PMF ℕ) (p : ∀ n, PMF (β n)) {f : Sigma β → ℝ}
+    (hf : Integrable f (sigmaPMF ρ p).toMeasure) :
+    (∫ z, f z ∂(sigmaPMF ρ p).toMeasure) = ∫ n, (∫ a, f ⟨n,a⟩ ∂(p n).toMeasure) ∂ρ.toMeasure := by
+  rw [sigmaPMF_eq_bind] at hf ⊢
+  rw [integral_pmf_bind_integrable _ _ hf]
+  simp only [integral_pmf_map]
+
+end Sigma
+end ThreeState.TreeClauses.Experiment
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeStateClauses/Degradation.lean b/lean/OAI/Probability/ThreeStateClauses/Degradation.lean
index 066c3217ca481b641ce1b7da883828fff3521a57..c0b0181ec752850509a93d9df1f5790f9b9a1711 100644
--- a/lean/OAI/Probability/ThreeStateClauses/Degradation.lean
+++ b/lean/OAI/Probability/ThreeStateClauses/Degradation.lean
@@ -1,97 +1,97 @@
-import Mathlib
-import OAI.Probability.ThreeStateClauses.DiscreteLaw
-
-namespace OAI
-
-/-! Degradation. -/
-
-open scoped BigOperators ENNReal NNReal Topology
-open Filter
-noncomputable section
-open Set MeasureTheory
-open scoped BigOperators ENNReal
-namespace ThreeState.TreeClauses.Experiment
-open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
-
-variable {α β : Type*}
-
-def kernelJoint (p : PMF α) (K : α → PMF β) : PMF (α × β) :=
-  p.bind (fun a ↦ (K a).map (fun b ↦ (a,b)))
-
-lemma kernelJoint_fst (p : PMF α) (K : α → PMF β) :
-    (kernelJoint p K).map Prod.fst = p := by
-  simp only [kernelJoint, PMF.map_bind, PMF.map_comp, Function.comp_def]
-  have h (a : α) : ((K a).map (fun _ ↦ a)) = PMF.pure a := PMF.map_const _ _
-  simp only [h, PMF.bind_pure]
-
-lemma kernelJoint_snd (p : PMF α) (K : α → PMF β) :
-    (kernelJoint p K).map Prod.snd = p.bind K := by
-  simp only [kernelJoint, PMF.map_bind, PMF.map_comp, Function.comp_def]
-  have h (a : α) : ((K a).map (fun b ↦ b)) = K a := PMF.map_id _
-  simp only [h]
-
-lemma discreteMarginal_bind (p : Fin 3 → PMF α) (K : α → PMF β) :
-    discreteMarginal (fun i ↦ (p i).bind K) = (discreteMarginal p).bind K := by
-  simp only [discreteMarginal, PMF.bind_bind]
-
-section Measures
-variable [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α]
-  [Countable β] [MeasurableSpace β] [MeasurableSingletonClass β]
-
-lemma integral_kernelJoint (p : PMF α) (K : α → PMF β) {f : α × β → ℝ} {C : ℝ}
-    (hf : ∀ z, |f z| ≤ C) :
-    (∫ z, f z ∂(kernelJoint p K).toMeasure) =
-      ∫ a, (∫ b, f (a,b) ∂(K a).toMeasure) ∂p.toMeasure := by
-  rw [kernelJoint, integral_pmf_bind p _ hf]
-  apply integral_congr_ae
-  exact Filter.Eventually.of_forall (fun a ↦ integral_pmf_map (K a) _ f)
-
-lemma integral_kernelJoint_fst (p : PMF α) (K : α → PMF β) (f : α → ℝ) :
-    (∫ z, f z.1 ∂(kernelJoint p K).toMeasure) = ∫ a, f a ∂p.toMeasure := by
-  rw [← integral_pmf_map, kernelJoint_fst]
-
-lemma integral_kernelJoint_snd (p : PMF α) (K : α → PMF β) (f : β → ℝ) :
-    (∫ z, f z.2 ∂(kernelJoint p K).toMeasure) = ∫ b, f b ∂(p.bind K).toMeasure := by
-  rw [← integral_pmf_map, kernelJoint_snd]
-
-lemma discrete_degradation_identity (p : Fin 3 → PMF α) (K : α → PMF β) (i : Fin 3)
-    {g : β → ℝ} {C : ℝ} (hg : ∀ b, |g b| ≤ C) :
-    (∫ z, (discreteMessage p z.1).1 i*g z.2 ∂(kernelJoint (discreteMarginal p) K).toMeasure) =
-      ∫ b, (discreteMessage (fun j ↦ (p j).bind K) b).1 i*g b
-        ∂(discreteMarginal (fun j ↦ (p j).bind K)).toMeasure := by
-  have hb (z : α × β) : |(discreteMessage p z.1).1 i*g z.2| ≤ 3*C := by
-    rw [abs_mul, abs_of_nonneg (coordinate_nonneg _ _)]
-    exact mul_le_mul (coordinate_le_three _ _) (hg z.2) (abs_nonneg _) (by norm_num)
-  rw [integral_kernelJoint _ _ hb]
-  simp only [integral_const_mul]
-  have hbound (a : α) : |∫ b, g b ∂(K a).toMeasure| ≤ C := by
-    calc
-      |∫ b, g b ∂(K a).toMeasure| ≤ ∫ b, |g b| ∂(K a).toMeasure := abs_integral_le_integral_abs
-      _ ≤ ∫ _, C ∂(K a).toMeasure := integral_mono (pmf_bounded_integrable (K a) hg).abs
-        (integrable_const C) hg
-      _ = C := by simp
-  rw [discrete_bayes_integral p i hbound, discrete_bayes_integral _ i hg,
-    integral_pmf_bind (p i) K hg]
-
-lemma discrete_degradation_centered (p : Fin 3 → PMF α) (K : α → PMF β) (i : Fin 3)
-    {g : β → ℝ} {C : ℝ} (hg : ∀ b, |g b| ≤ C) :
-    (∫ z, ((discreteMessage p z.1).1 i-(discreteMessage (fun j ↦ (p j).bind K) z.2).1 i)*g z.2
-      ∂(kernelJoint (discreteMarginal p) K).toMeasure) = 0 := by
-  have hb (m : Message) (b : β) : |m.1 i*g b| ≤ 3*C := by
-    rw [abs_mul, abs_of_nonneg (coordinate_nonneg _ _)]
-    exact mul_le_mul (coordinate_le_three _ _) (hg b) (abs_nonneg _) (by norm_num)
-  simp_rw [sub_mul]
-  rw [integral_sub (f := fun z : α × β ↦ (discreteMessage p z.1).1 i*g z.2)
-    (g := fun z : α × β ↦ (discreteMessage (fun j ↦ (p j).bind K) z.2).1 i*g z.2)
-    (pmf_bounded_integrable _ (fun z ↦ hb _ z.2))
-    (pmf_bounded_integrable _ (fun z ↦ hb _ z.2)),
-    discrete_degradation_identity p K i hg,
-    integral_kernelJoint_snd (f := fun b ↦ (discreteMessage (fun j ↦ (p j).bind K) b).1 i*g b),
-    ← discreteMarginal_bind, sub_self]
-
-end Measures
-end ThreeState.TreeClauses.Experiment
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeStateClauses.DiscreteLaw
+
+namespace OAI
+
+/-! Degradation. -/
+
+open scoped BigOperators ENNReal NNReal Topology
+open Filter
+noncomputable section
+open Set MeasureTheory
+open scoped BigOperators ENNReal
+namespace ThreeState.TreeClauses.Experiment
+open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
+
+variable {α β : Type*}
+
+def kernelJoint (p : PMF α) (K : α → PMF β) : PMF (α × β) :=
+  p.bind (fun a ↦ (K a).map (fun b ↦ (a,b)))
+
+lemma kernelJoint_fst (p : PMF α) (K : α → PMF β) :
+    (kernelJoint p K).map Prod.fst = p := by
+  simp only [kernelJoint, PMF.map_bind, PMF.map_comp, Function.comp_def]
+  have h (a : α) : ((K a).map (fun _ ↦ a)) = PMF.pure a := PMF.map_const _ _
+  simp only [h, PMF.bind_pure]
+
+lemma kernelJoint_snd (p : PMF α) (K : α → PMF β) :
+    (kernelJoint p K).map Prod.snd = p.bind K := by
+  simp only [kernelJoint, PMF.map_bind, PMF.map_comp, Function.comp_def]
+  have h (a : α) : ((K a).map (fun b ↦ b)) = K a := PMF.map_id _
+  simp only [h]
+
+lemma discreteMarginal_bind (p : Fin 3 → PMF α) (K : α → PMF β) :
+    discreteMarginal (fun i ↦ (p i).bind K) = (discreteMarginal p).bind K := by
+  simp only [discreteMarginal, PMF.bind_bind]
+
+section Measures
+variable [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α]
+  [Countable β] [MeasurableSpace β] [MeasurableSingletonClass β]
+
+lemma integral_kernelJoint (p : PMF α) (K : α → PMF β) {f : α × β → ℝ} {C : ℝ}
+    (hf : ∀ z, |f z| ≤ C) :
+    (∫ z, f z ∂(kernelJoint p K).toMeasure) =
+      ∫ a, (∫ b, f (a,b) ∂(K a).toMeasure) ∂p.toMeasure := by
+  rw [kernelJoint, integral_pmf_bind p _ hf]
+  apply integral_congr_ae
+  exact Filter.Eventually.of_forall (fun a ↦ integral_pmf_map (K a) _ f)
+
+lemma integral_kernelJoint_fst (p : PMF α) (K : α → PMF β) (f : α → ℝ) :
+    (∫ z, f z.1 ∂(kernelJoint p K).toMeasure) = ∫ a, f a ∂p.toMeasure := by
+  rw [← integral_pmf_map, kernelJoint_fst]
+
+lemma integral_kernelJoint_snd (p : PMF α) (K : α → PMF β) (f : β → ℝ) :
+    (∫ z, f z.2 ∂(kernelJoint p K).toMeasure) = ∫ b, f b ∂(p.bind K).toMeasure := by
+  rw [← integral_pmf_map, kernelJoint_snd]
+
+lemma discrete_degradation_identity (p : Fin 3 → PMF α) (K : α → PMF β) (i : Fin 3)
+    {g : β → ℝ} {C : ℝ} (hg : ∀ b, |g b| ≤ C) :
+    (∫ z, (discreteMessage p z.1).1 i*g z.2 ∂(kernelJoint (discreteMarginal p) K).toMeasure) =
+      ∫ b, (discreteMessage (fun j ↦ (p j).bind K) b).1 i*g b
+        ∂(discreteMarginal (fun j ↦ (p j).bind K)).toMeasure := by
+  have hb (z : α × β) : |(discreteMessage p z.1).1 i*g z.2| ≤ 3*C := by
+    rw [abs_mul, abs_of_nonneg (coordinate_nonneg _ _)]
+    exact mul_le_mul (coordinate_le_three _ _) (hg z.2) (abs_nonneg _) (by norm_num)
+  rw [integral_kernelJoint _ _ hb]
+  simp only [integral_const_mul]
+  have hbound (a : α) : |∫ b, g b ∂(K a).toMeasure| ≤ C := by
+    calc
+      |∫ b, g b ∂(K a).toMeasure| ≤ ∫ b, |g b| ∂(K a).toMeasure := abs_integral_le_integral_abs
+      _ ≤ ∫ _, C ∂(K a).toMeasure := integral_mono (pmf_bounded_integrable (K a) hg).abs
+        (integrable_const C) hg
+      _ = C := by simp
+  rw [discrete_bayes_integral p i hbound, discrete_bayes_integral _ i hg,
+    integral_pmf_bind (p i) K hg]
+
+lemma discrete_degradation_centered (p : Fin 3 → PMF α) (K : α → PMF β) (i : Fin 3)
+    {g : β → ℝ} {C : ℝ} (hg : ∀ b, |g b| ≤ C) :
+    (∫ z, ((discreteMessage p z.1).1 i-(discreteMessage (fun j ↦ (p j).bind K) z.2).1 i)*g z.2
+      ∂(kernelJoint (discreteMarginal p) K).toMeasure) = 0 := by
+  have hb (m : Message) (b : β) : |m.1 i*g b| ≤ 3*C := by
+    rw [abs_mul, abs_of_nonneg (coordinate_nonneg _ _)]
+    exact mul_le_mul (coordinate_le_three _ _) (hg b) (abs_nonneg _) (by norm_num)
+  simp_rw [sub_mul]
+  rw [integral_sub (f := fun z : α × β ↦ (discreteMessage p z.1).1 i*g z.2)
+    (g := fun z : α × β ↦ (discreteMessage (fun j ↦ (p j).bind K) z.2).1 i*g z.2)
+    (pmf_bounded_integrable _ (fun z ↦ hb _ z.2))
+    (pmf_bounded_integrable _ (fun z ↦ hb _ z.2)),
+    discrete_degradation_identity p K i hg,
+    integral_kernelJoint_snd (f := fun b ↦ (discreteMessage (fun j ↦ (p j).bind K) b).1 i*g b),
+    ← discreteMarginal_bind, sub_self]
+
+end Measures
+end ThreeState.TreeClauses.Experiment
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeStateClauses/DiscreteLaw.lean b/lean/OAI/Probability/ThreeStateClauses/DiscreteLaw.lean
index 677e1bcdd2a0a6838385ffdf9d89951f8adeda39..f6322ad25f5b598ede4e0367170ffde119f4326a 100644
--- a/lean/OAI/Probability/ThreeStateClauses/DiscreteLaw.lean
+++ b/lean/OAI/Probability/ThreeStateClauses/DiscreteLaw.lean
@@ -1,172 +1,172 @@
-import Mathlib
-import OAI.Probability.ThreeStateClauses.PosteriorLaw
-
-namespace OAI
-
-/-! Discrete Law. -/
-
-open scoped BigOperators ENNReal NNReal Topology
-open Filter
-noncomputable section
-open Set MeasureTheory
-open scoped BigOperators ENNReal
-namespace ThreeState.TreeClauses.Experiment
-open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
-
-variable {α : Type*}
-
-def uniformSpin : PMF (Fin 3) := PMF.ofFintype (fun _ ↦ (1/3 : ℝ≥0∞)) (by
-  norm_num
-  exact ENNReal.mul_inv_cancel (by norm_num) (by norm_num))
-
-@[simp] lemma uniformSpin_apply (i : Fin 3) : uniformSpin i = (1/3 : ℝ≥0∞) := rfl
-
-def discreteMarginal (p : Fin 3 → PMF α) : PMF α := uniformSpin.bind p
-
-def discreteWeight (p : Fin 3 → PMF α) (a : α) : ℝ := avg (fun i ↦ (p i a).toReal)
-
-lemma discreteWeight_nonneg (p : Fin 3 → PMF α) (a : α) : 0 ≤ discreteWeight p a := by
-  unfold discreteWeight avg; positivity
-
-lemma discreteMarginal_apply (p : Fin 3 → PMF α) (a : α) :
-    discreteMarginal p a = (p 0 a+p 1 a+p 2 a)/3 := by
-  simp [discreteMarginal, PMF.bind_apply, tsum_fintype, Fin.sum_univ_three, div_eq_mul_inv]
-  ring
-
-lemma discreteMarginal_real (p : Fin 3 → PMF α) (a : α) :
-    (discreteMarginal p a).toReal = discreteWeight p a := by
-  rw [discreteMarginal_apply, ENNReal.toReal_div,
-    ENNReal.toReal_add (ENNReal.add_ne_top.mpr ⟨(p 0).apply_ne_top a, (p 1).apply_ne_top a⟩)
-      ((p 2).apply_ne_top a),
-    ENNReal.toReal_add ((p 0).apply_ne_top a) ((p 1).apply_ne_top a)]
-  norm_num [discreteWeight, avg]
-
-lemma discreteWeight_zero (p : Fin 3 → PMF α) (a : α) (h : discreteWeight p a = 0) (i : Fin 3) :
-    (p i a).toReal = 0 := by
-  dsimp [discreteWeight, avg] at h
-  fin_cases i <;> dsimp <;> linarith [ENNReal.toReal_nonneg (a := p 0 a),
-    ENNReal.toReal_nonneg (a := p 1 a), ENNReal.toReal_nonneg (a := p 2 a)]
-
-def discreteMessage (p : Fin 3 → PMF α) (a : α) : Message := by
-  by_cases h : discreteWeight p a = 0
-  · exact ⟨fun _ ↦ 1, fun _ ↦ by norm_num, by norm_num [avg]⟩
-  · refine ⟨fun i ↦ (p i a).toReal/discreteWeight p a, fun _ ↦ div_nonneg (by positivity)
-      (discreteWeight_nonneg p a), ?_⟩
-    change ((p 0 a).toReal/discreteWeight p a+(p 1 a).toReal/discreteWeight p a+
-      (p 2 a).toReal/discreteWeight p a)/3 = 1
-    rw [← add_div, ← add_div, div_div, mul_comm]
-    have hw : (p 0 a).toReal+(p 1 a).toReal+(p 2 a).toReal = 3*discreteWeight p a := by
-      dsimp [discreteWeight, avg]; ring
-    rw [hw, div_self (mul_ne_zero (by norm_num) h)]
-
-lemma discrete_bayes (p : Fin 3 → PMF α) (a : α) (i : Fin 3) :
-    discreteWeight p a*(discreteMessage p a).1 i = (p i a).toReal := by
-  unfold discreteMessage
-  split_ifs with h
-  · simpa [h] using (discreteWeight_zero p a h i).symm
-  · exact mul_div_cancel₀ _ h
-
-lemma discrete_coord {p : Fin 3 → PMF α} {a : α} (h : discreteWeight p a ≠ 0) (i : Fin 3) :
-    (discreteMessage p a).1 i = (p i a).toReal/discreteWeight p a := by
-  simp only [discreteMessage, dite_eq_right h]
-
-lemma pmf_real_sum (p : PMF α) : ∑' a, (p a).toReal = 1 := by
-  rw [← ENNReal.tsum_toReal_eq p.apply_ne_top, p.tsum_coe]
-  norm_num
-
-section Measures
-variable [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α]
-
-lemma discreteCoordinate_integrable (p : Fin 3 → PMF α) (i : Fin 3) :
-    Integrable (fun a ↦ (discreteMessage p a).1 i) (discreteMarginal p).toMeasure :=
-  Integrable.of_bound (measurable_of_countable _).aestronglyMeasurable 3
-    (Filter.Eventually.of_forall (fun a ↦ by
-      rw [Real.norm_eq_abs, abs_of_nonneg (coordinate_nonneg _ _)]
-      exact coordinate_le_three _ _))
-
-def discreteProbability (p : Fin 3 → PMF α) : ProbabilityMeasure Message :=
-  ProbabilityMeasure.map (⟨(discreteMarginal p).toMeasure, inferInstance⟩ : ProbabilityMeasure α)
-    (discreteMessage p)
-
-lemma integral_discreteProbability (p : Fin 3 → PMF α) {f : Message → ℝ} (hf : Measurable f) :
-    (∫ m, f m ∂(discreteProbability p).toMeasure) =
-      ∫ a, f (discreteMessage p a) ∂(discreteMarginal p).toMeasure := by
-  change (∫ m, f m ∂Measure.map (discreteMessage p) (discreteMarginal p).toMeasure) = _
-  exact integral_map (measurable_of_countable _).aemeasurable hf.aestronglyMeasurable
-
-lemma discreteProbability_balanced (p : Fin 3 → PMF α) (i : Fin 3) :
-    (∫ m, m.1 i ∂(discreteProbability p).toMeasure) = 1 := by
-  rw [integral_discreteProbability p (continuous_coordinate i).measurable,
-    PMF.integral_eq_tsum _ _ (discreteCoordinate_integrable p i)]
-  simpa only [smul_eq_mul, discreteMarginal_real, discrete_bayes] using pmf_real_sum (p i)
-
-end Measures
-
-end ThreeState.TreeClauses.Experiment
-
-end 
-
-noncomputable section
-open Set MeasureTheory
-open scoped BigOperators ENNReal
-namespace ThreeState.TreeClauses.Experiment
-open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
-
-variable {α β : Type*} [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α]
-  [Countable β] [MeasurableSpace β] [MeasurableSingletonClass β]
-
-lemma pmf_bounded_integrable (p : PMF α) {f : α → ℝ} {C : ℝ} (hf : ∀ a, |f a| ≤ C) :
-    Integrable f p.toMeasure :=
-  Integrable.of_bound (measurable_of_countable _).aestronglyMeasurable C
-    (Filter.Eventually.of_forall (fun a ↦ by simpa only [Real.norm_eq_abs] using hf a))
-
-omit [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α]
-  [Countable β] [MeasurableSingletonClass β] in
-lemma pmf_measure_bind (p : PMF α) (K : α → PMF β) :
-    (p.bind K).toMeasure = Measure.sum (fun a ↦ p a • (K a).toMeasure) := by
-  apply Measure.ext
-  intro s hs
-  rw [PMF.toMeasure_bind_apply _ _ _ hs, Measure.sum_apply _ hs]
-  simp only [Measure.smul_apply, smul_eq_mul]
-
-lemma integral_pmf_bind (p : PMF α) (K : α → PMF β) {f : β → ℝ} {C : ℝ}
-    (hf : ∀ b, |f b| ≤ C) :
-    (∫ b, f b ∂(p.bind K).toMeasure) = ∫ a, (∫ b, f b ∂(K a).toMeasure) ∂p.toMeasure := by
-  have hi := pmf_bounded_integrable (p.bind K) hf
-  rw [pmf_measure_bind] at hi ⊢
-  rw [integral_sum_measure hi]
-  simp only [integral_smul_measure, smul_eq_mul]
-  rw [PMF.integral_eq_tsum]
-  · rfl
-  · apply pmf_bounded_integrable
-    intro a
-    calc
-      |∫ b, f b ∂(K a).toMeasure| ≤ ∫ b, |f b| ∂(K a).toMeasure := abs_integral_le_integral_abs
-      _ ≤ ∫ _, C ∂(K a).toMeasure := integral_mono (pmf_bounded_integrable (K a) hf).abs
-        (integrable_const C) hf
-      _ = C := by simp
-
-lemma integral_pmf_map (p : PMF α) (T : α → β) (f : β → ℝ) :
-    (∫ b, f b ∂(p.map T).toMeasure) = ∫ a, f (T a) ∂p.toMeasure := by
-  rw [← PMF.toMeasure_map T p (measurable_of_countable T)]
-  exact integral_map (measurable_of_countable _).aemeasurable
-    (measurable_of_countable _).aestronglyMeasurable
-
-lemma discrete_bayes_integral (p : Fin 3 → PMF α) (i : Fin 3) {g : α → ℝ} {C : ℝ}
-    (hg : ∀ a, |g a| ≤ C) :
-    (∫ a, (discreteMessage p a).1 i*g a ∂(discreteMarginal p).toMeasure) =
-      ∫ a, g a ∂(p i).toMeasure := by
-  rw [PMF.integral_eq_tsum, PMF.integral_eq_tsum _ _ (pmf_bounded_integrable (p i) hg)]
-  · apply tsum_congr
-    intro a
-    simp only [smul_eq_mul, discreteMarginal_real, ← mul_assoc, discrete_bayes]
-  · apply pmf_bounded_integrable (C := 3*C)
-    intro a
-    rw [abs_mul, abs_of_nonneg (coordinate_nonneg _ _)]
-    exact mul_le_mul (coordinate_le_three _ _) (hg a) (abs_nonneg _) (by norm_num)
-
-end ThreeState.TreeClauses.Experiment
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeStateClauses.PosteriorLaw
+
+namespace OAI
+
+/-! Discrete Law. -/
+
+open scoped BigOperators ENNReal NNReal Topology
+open Filter
+noncomputable section
+open Set MeasureTheory
+open scoped BigOperators ENNReal
+namespace ThreeState.TreeClauses.Experiment
+open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
+
+variable {α : Type*}
+
+def uniformSpin : PMF (Fin 3) := PMF.ofFintype (fun _ ↦ (1/3 : ℝ≥0∞)) (by
+  norm_num
+  exact ENNReal.mul_inv_cancel (by norm_num) (by norm_num))
+
+@[simp] lemma uniformSpin_apply (i : Fin 3) : uniformSpin i = (1/3 : ℝ≥0∞) := rfl
+
+def discreteMarginal (p : Fin 3 → PMF α) : PMF α := uniformSpin.bind p
+
+def discreteWeight (p : Fin 3 → PMF α) (a : α) : ℝ := avg (fun i ↦ (p i a).toReal)
+
+lemma discreteWeight_nonneg (p : Fin 3 → PMF α) (a : α) : 0 ≤ discreteWeight p a := by
+  unfold discreteWeight avg; positivity
+
+lemma discreteMarginal_apply (p : Fin 3 → PMF α) (a : α) :
+    discreteMarginal p a = (p 0 a+p 1 a+p 2 a)/3 := by
+  simp [discreteMarginal, PMF.bind_apply, tsum_fintype, Fin.sum_univ_three, div_eq_mul_inv]
+  ring
+
+lemma discreteMarginal_real (p : Fin 3 → PMF α) (a : α) :
+    (discreteMarginal p a).toReal = discreteWeight p a := by
+  rw [discreteMarginal_apply, ENNReal.toReal_div,
+    ENNReal.toReal_add (ENNReal.add_ne_top.mpr ⟨(p 0).apply_ne_top a, (p 1).apply_ne_top a⟩)
+      ((p 2).apply_ne_top a),
+    ENNReal.toReal_add ((p 0).apply_ne_top a) ((p 1).apply_ne_top a)]
+  norm_num [discreteWeight, avg]
+
+lemma discreteWeight_zero (p : Fin 3 → PMF α) (a : α) (h : discreteWeight p a = 0) (i : Fin 3) :
+    (p i a).toReal = 0 := by
+  dsimp [discreteWeight, avg] at h
+  fin_cases i <;> dsimp <;> linarith [ENNReal.toReal_nonneg (a := p 0 a),
+    ENNReal.toReal_nonneg (a := p 1 a), ENNReal.toReal_nonneg (a := p 2 a)]
+
+def discreteMessage (p : Fin 3 → PMF α) (a : α) : Message := by
+  by_cases h : discreteWeight p a = 0
+  · exact ⟨fun _ ↦ 1, fun _ ↦ by norm_num, by norm_num [avg]⟩
+  · refine ⟨fun i ↦ (p i a).toReal/discreteWeight p a, fun _ ↦ div_nonneg (by positivity)
+      (discreteWeight_nonneg p a), ?_⟩
+    change ((p 0 a).toReal/discreteWeight p a+(p 1 a).toReal/discreteWeight p a+
+      (p 2 a).toReal/discreteWeight p a)/3 = 1
+    rw [← add_div, ← add_div, div_div, mul_comm]
+    have hw : (p 0 a).toReal+(p 1 a).toReal+(p 2 a).toReal = 3*discreteWeight p a := by
+      dsimp [discreteWeight, avg]; ring
+    rw [hw, div_self (mul_ne_zero (by norm_num) h)]
+
+lemma discrete_bayes (p : Fin 3 → PMF α) (a : α) (i : Fin 3) :
+    discreteWeight p a*(discreteMessage p a).1 i = (p i a).toReal := by
+  unfold discreteMessage
+  split_ifs with h
+  · simpa [h] using (discreteWeight_zero p a h i).symm
+  · exact mul_div_cancel₀ _ h
+
+lemma discrete_coord {p : Fin 3 → PMF α} {a : α} (h : discreteWeight p a ≠ 0) (i : Fin 3) :
+    (discreteMessage p a).1 i = (p i a).toReal/discreteWeight p a := by
+  simp only [discreteMessage, dite_eq_right h]
+
+lemma pmf_real_sum (p : PMF α) : ∑' a, (p a).toReal = 1 := by
+  rw [← ENNReal.tsum_toReal_eq p.apply_ne_top, p.tsum_coe]
+  norm_num
+
+section Measures
+variable [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α]
+
+lemma discreteCoordinate_integrable (p : Fin 3 → PMF α) (i : Fin 3) :
+    Integrable (fun a ↦ (discreteMessage p a).1 i) (discreteMarginal p).toMeasure :=
+  Integrable.of_bound (measurable_of_countable _).aestronglyMeasurable 3
+    (Filter.Eventually.of_forall (fun a ↦ by
+      rw [Real.norm_eq_abs, abs_of_nonneg (coordinate_nonneg _ _)]
+      exact coordinate_le_three _ _))
+
+def discreteProbability (p : Fin 3 → PMF α) : ProbabilityMeasure Message :=
+  ProbabilityMeasure.map (⟨(discreteMarginal p).toMeasure, inferInstance⟩ : ProbabilityMeasure α)
+    (discreteMessage p)
+
+lemma integral_discreteProbability (p : Fin 3 → PMF α) {f : Message → ℝ} (hf : Measurable f) :
+    (∫ m, f m ∂(discreteProbability p).toMeasure) =
+      ∫ a, f (discreteMessage p a) ∂(discreteMarginal p).toMeasure := by
+  change (∫ m, f m ∂Measure.map (discreteMessage p) (discreteMarginal p).toMeasure) = _
+  exact integral_map (measurable_of_countable _).aemeasurable hf.aestronglyMeasurable
+
+lemma discreteProbability_balanced (p : Fin 3 → PMF α) (i : Fin 3) :
+    (∫ m, m.1 i ∂(discreteProbability p).toMeasure) = 1 := by
+  rw [integral_discreteProbability p (continuous_coordinate i).measurable,
+    PMF.integral_eq_tsum _ _ (discreteCoordinate_integrable p i)]
+  simpa only [smul_eq_mul, discreteMarginal_real, discrete_bayes] using pmf_real_sum (p i)
+
+end Measures
+
+end ThreeState.TreeClauses.Experiment
+
+end 
+
+noncomputable section
+open Set MeasureTheory
+open scoped BigOperators ENNReal
+namespace ThreeState.TreeClauses.Experiment
+open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
+
+variable {α β : Type*} [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α]
+  [Countable β] [MeasurableSpace β] [MeasurableSingletonClass β]
+
+lemma pmf_bounded_integrable (p : PMF α) {f : α → ℝ} {C : ℝ} (hf : ∀ a, |f a| ≤ C) :
+    Integrable f p.toMeasure :=
+  Integrable.of_bound (measurable_of_countable _).aestronglyMeasurable C
+    (Filter.Eventually.of_forall (fun a ↦ by simpa only [Real.norm_eq_abs] using hf a))
+
+omit [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α]
+  [Countable β] [MeasurableSingletonClass β] in
+lemma pmf_measure_bind (p : PMF α) (K : α → PMF β) :
+    (p.bind K).toMeasure = Measure.sum (fun a ↦ p a • (K a).toMeasure) := by
+  apply Measure.ext
+  intro s hs
+  rw [PMF.toMeasure_bind_apply _ _ _ hs, Measure.sum_apply _ hs]
+  simp only [Measure.smul_apply, smul_eq_mul]
+
+lemma integral_pmf_bind (p : PMF α) (K : α → PMF β) {f : β → ℝ} {C : ℝ}
+    (hf : ∀ b, |f b| ≤ C) :
+    (∫ b, f b ∂(p.bind K).toMeasure) = ∫ a, (∫ b, f b ∂(K a).toMeasure) ∂p.toMeasure := by
+  have hi := pmf_bounded_integrable (p.bind K) hf
+  rw [pmf_measure_bind] at hi ⊢
+  rw [integral_sum_measure hi]
+  simp only [integral_smul_measure, smul_eq_mul]
+  rw [PMF.integral_eq_tsum]
+  · rfl
+  · apply pmf_bounded_integrable
+    intro a
+    calc
+      |∫ b, f b ∂(K a).toMeasure| ≤ ∫ b, |f b| ∂(K a).toMeasure := abs_integral_le_integral_abs
+      _ ≤ ∫ _, C ∂(K a).toMeasure := integral_mono (pmf_bounded_integrable (K a) hf).abs
+        (integrable_const C) hf
+      _ = C := by simp
+
+lemma integral_pmf_map (p : PMF α) (T : α → β) (f : β → ℝ) :
+    (∫ b, f b ∂(p.map T).toMeasure) = ∫ a, f (T a) ∂p.toMeasure := by
+  rw [← PMF.toMeasure_map T p (measurable_of_countable T)]
+  exact integral_map (measurable_of_countable _).aemeasurable
+    (measurable_of_countable _).aestronglyMeasurable
+
+lemma discrete_bayes_integral (p : Fin 3 → PMF α) (i : Fin 3) {g : α → ℝ} {C : ℝ}
+    (hg : ∀ a, |g a| ≤ C) :
+    (∫ a, (discreteMessage p a).1 i*g a ∂(discreteMarginal p).toMeasure) =
+      ∫ a, g a ∂(p i).toMeasure := by
+  rw [PMF.integral_eq_tsum, PMF.integral_eq_tsum _ _ (pmf_bounded_integrable (p i) hg)]
+  · apply tsum_congr
+    intro a
+    simp only [smul_eq_mul, discreteMarginal_real, ← mul_assoc, discrete_bayes]
+  · apply pmf_bounded_integrable (C := 3*C)
+    intro a
+    rw [abs_mul, abs_of_nonneg (coordinate_nonneg _ _)]
+    exact mul_le_mul (coordinate_le_three _ _) (hg a) (abs_nonneg _) (by norm_num)
+
+end ThreeState.TreeClauses.Experiment
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeStateClauses/FiniteUpdate.lean b/lean/OAI/Probability/ThreeStateClauses/FiniteUpdate.lean
index 56e9e3e985d1fd708d5f65e21d175401e4ceec34..b49620c4f725d84ff2e3fed883ca3155f8994227 100644
--- a/lean/OAI/Probability/ThreeStateClauses/FiniteUpdate.lean
+++ b/lean/OAI/Probability/ThreeStateClauses/FiniteUpdate.lean
@@ -1,372 +1,372 @@
-import Mathlib
-import OAI.Probability.ThreeStateClauses.Basic
-import OAI.Probability.ThreeStateClauses.DiscreteLaw
-import OAI.Probability.ThreeStateClauses.Products
-import OAI.Probability.ThreeStateClauses.BranchMoments
-
-namespace OAI
-
-/-! Finite Update. -/
-
-open scoped BigOperators ENNReal NNReal Topology
-open Filter
-noncomputable section
-open Set MeasureTheory
-open scoped BigOperators
-namespace ThreeState.TreeClauses.Experiment
-open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
-
-def branchOutput {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) {n : ℕ}
-    (m : Fin n → Message) : Message :=
-  ⟨normalizedProduct (branchEdges lam m),
-    (fun i ↦ (normalizedProduct_positive (branchEdges_positive hl₀ hl₁ m)).1 i |>.le),
-    (normalizedProduct_positive (branchEdges_positive hl₀ hl₁ m)).2⟩
-
-lemma continuous_branchOutput {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
-    Continuous (branchOutput hl₀ hl₁ (n := n)) := by
-  apply Continuous.subtype_mk
-  exact continuous_pi (fun i ↦ (continuous_branchProduct lam i).div (continuous_normalizer n lam)
-    (fun m ↦ ne_of_gt (normalizer_positive (branchEdges_positive hl₀ hl₁ m))))
-
-def tiltedProduct (Q : Law) (lam : ℝ) (n : ℕ) : Measure (Fin n → Message) :=
-  (Measure.pi fun _ ↦ Q.probability.toMeasure).withDensity
-    (fun m ↦ ENNReal.ofReal (normalizer (branchEdges lam m)))
-
-lemma tiltedProduct_isProbability (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
-    IsProbabilityMeasure (tiltedProduct Q lam n) := by
-  constructor
-  rw [tiltedProduct, withDensity_apply _ MeasurableSet.univ, setLIntegral_univ,
-    ← ofReal_integral_eq_lintegral_ofReal (compact_integrable (continuous_normalizer n lam))
-      (Filter.Eventually.of_forall (fun m ↦ (normalizer_positive (branchEdges_positive hl₀ hl₁ m)).le)),
-    integral_normalizer]
-  norm_num
-
-lemma integral_tiltedProduct (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ)
-    (f : (Fin n → Message) → ℝ) :
-    (∫ m, f m ∂tiltedProduct Q lam n) =
-      ∫ m, normalizer (branchEdges lam m)*f m ∂(Measure.pi fun _ ↦ Q.probability.toMeasure) := by
-  rw [tiltedProduct, integral_withDensity_eq_integral_toReal_smul
-    (show Measurable (fun m : Fin n → Message ↦ ENNReal.ofReal (normalizer (branchEdges lam m))) from
-      ENNReal.measurable_ofReal.comp (continuous_normalizer n lam).measurable)
-    (Filter.Eventually.of_forall (fun _ ↦ ENNReal.ofReal_lt_top))]
-  congr 1; funext m
-  rw [ENNReal.toReal_ofReal (normalizer_positive (branchEdges_positive hl₀ hl₁ m)).le]
-  rfl
-
-lemma measurePreserving_density {X : Type*} [MeasurableSpace X] {μ : Measure X}
-    {T : X → X} (hT : MeasurePreserving T μ μ) {w : X → ENNReal} (hw : Measurable w)
-    (hinv : ∀ x, w (T x) = w x) :
-    MeasurePreserving T (μ.withDensity w) (μ.withDensity w) := by
-  refine ⟨hT.measurable, Measure.ext (fun s hs ↦ ?_)⟩
-  rw [Measure.map_apply hT.measurable hs, withDensity_apply _ (hT.measurable hs), withDensity_apply _ hs]
-  have h := hT.setLIntegral_comp_preimage hs hw
-  simpa only [hinv] using h
-
-lemma branchProduct_permute {n : ℕ} (lam : ℝ) (σ : Equiv.Perm (Fin 3))
-    (m : Fin n → Message) (i : Fin 3) :
-    branchProduct (branchEdges lam (fun j ↦ permute σ (m j))) i =
-      branchProduct (branchEdges lam m) (σ i) := rfl
-
-lemma normalizer_permute {n : ℕ} (lam : ℝ) (σ : Equiv.Perm (Fin 3)) (m : Fin n → Message) :
-    normalizer (branchEdges lam (fun j ↦ permute σ (m j))) = normalizer (branchEdges lam m) := by
-  unfold normalizer
-  change avg (fun i ↦ branchProduct (branchEdges lam m) (σ i)) = _
-  exact avg_permute_vec σ (branchProduct (branchEdges lam m))
-
-lemma branchOutput_permute {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) {n : ℕ}
-    (σ : Equiv.Perm (Fin 3)) (m : Fin n → Message) :
-    branchOutput hl₀ hl₁ (fun j ↦ permute σ (m j)) = permute σ (branchOutput hl₀ hl₁ m) := by
-  apply Subtype.ext; funext i
-  change branchProduct (branchEdges lam (fun j ↦ permute σ (m j))) i /
-    normalizer (branchEdges lam (fun j ↦ permute σ (m j))) =
-    branchProduct (branchEdges lam m) (σ i) / normalizer (branchEdges lam m)
-  rw [branchProduct_permute, normalizer_permute]
-
-lemma tiltedProduct_symmetric (Q : Law) {lam : ℝ} (_hl₀ : 0 ≤ lam) (_hl₁ : lam < 1) (n : ℕ)
-    (σ : Equiv.Perm (Fin 3)) :
-    MeasurePreserving (fun m : Fin n → Message ↦ fun j ↦ permute σ (m j))
-      (tiltedProduct Q lam n) (tiltedProduct Q lam n) := by
-  apply measurePreserving_density
-    (measurePreserving_pi (fun _ ↦ Q.probability.toMeasure) (fun _ ↦ Q.probability.toMeasure)
-      (fun _ ↦ Q.symmetric σ))
-    (ENNReal.measurable_ofReal.comp (continuous_normalizer n lam).measurable)
-  intro m
-  change ENNReal.ofReal (normalizer (branchEdges lam (fun j ↦ permute σ (m j)))) = _
-  rw [normalizer_permute]
-  rfl
-
-def finiteProbability (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
-    ProbabilityMeasure Message := by
-  let ν : ProbabilityMeasure (Fin n → Message) := ⟨tiltedProduct Q lam n, tiltedProduct_isProbability Q hl₀ hl₁ n⟩
-  exact ν.map (branchOutput hl₀ hl₁)
-
-lemma integral_finiteProbability (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ)
-    {f : Message → ℝ} (hf : Measurable f) :
-    (∫ m, f m ∂(finiteProbability Q hl₀ hl₁ n).toMeasure) =
-      ∫ m : Fin n → Message, normalizer (branchEdges lam m)*f (branchOutput hl₀ hl₁ m)
-        ∂(Measure.pi fun _ ↦ Q.probability.toMeasure) := by
-  change (∫ m, f m ∂Measure.map (branchOutput hl₀ hl₁) (tiltedProduct Q lam n)) = _
-  rw [integral_map (continuous_branchOutput hl₀ hl₁ n).measurable.aemeasurable hf.aestronglyMeasurable,
-    integral_tiltedProduct Q hl₀ hl₁]
-
-lemma finiteProbability_balanced (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) (i : Fin 3) :
-    (∫ m, m.1 i ∂(finiteProbability Q hl₀ hl₁ n).toMeasure) = 1 := by
-  rw [integral_finiteProbability Q hl₀ hl₁ n (continuous_coordinate i).measurable]
-  have he (m : Fin n → Message) :
-      normalizer (branchEdges lam m)*(branchOutput hl₀ hl₁ m).1 i = branchProduct (branchEdges lam m) i := by
-    change normalizer (branchEdges lam m)*(branchProduct (branchEdges lam m) i/normalizer (branchEdges lam m)) = _
-    field_simp [ne_of_gt (normalizer_positive (branchEdges_positive hl₀ hl₁ m))]
-  simp_rw [he]
-  change (∫ m : Fin n → Message, ∏ j, edgeMessage lam (m j) i ∂(Measure.pi fun _ ↦ Q.probability.toMeasure)) = _
-  rw [integral_fintype_prod_eq_pow (fun m : Message ↦ edgeMessage lam m i), Q.edge_balanced]
-  simp
-
-lemma finiteProbability_symmetric (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ)
-    (σ : Equiv.Perm (Fin 3)) :
-    MeasurePreserving (permute σ) (finiteProbability Q hl₀ hl₁ n).toMeasure
-      (finiteProbability Q hl₀ hl₁ n).toMeasure := by
-  refine ⟨(continuous_permute σ).measurable, ?_⟩
-  change Measure.map (permute σ) (Measure.map (branchOutput hl₀ hl₁) (tiltedProduct Q lam n)) =
-    Measure.map (branchOutput hl₀ hl₁) (tiltedProduct Q lam n)
-  rw [Measure.map_map (continuous_permute σ).measurable (continuous_branchOutput hl₀ hl₁ n).measurable]
-  have he : permute σ ∘ branchOutput hl₀ hl₁ =
-      branchOutput hl₀ hl₁ ∘ (fun m : Fin n → Message ↦ fun j ↦ permute σ (m j)) := by
-    funext m; exact (branchOutput_permute hl₀ hl₁ σ m).symm
-  rw [he, ← Measure.map_map (continuous_branchOutput hl₀ hl₁ n).measurable
-    (tiltedProduct_symmetric Q hl₀ hl₁ n σ).measurable,
-    (tiltedProduct_symmetric Q hl₀ hl₁ n σ).map_eq]
-
-def finiteLaw (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) : Law where
-  probability := finiteProbability Q hl₀ hl₁ n
-  balanced := finiteProbability_balanced Q hl₀ hl₁ n
-  symmetric := finiteProbability_symmetric Q hl₀ hl₁ n
-
-end ThreeState.TreeClauses.Experiment
-
-end 
-
-noncomputable section
-open Set MeasureTheory
-open scoped BigOperators ENNReal
-namespace ThreeState.TreeClauses.Experiment
-open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
-
-variable {α : Type*}
-
-lemma channel_real {lam : ℝ} (hlam : Admissible lam) (i j : Spin) :
-    (channel lam hlam i j).toReal = channelWeight lam i j := by
-  exact ENNReal.toReal_ofReal (channelWeight_nonneg hlam i j)
-
-lemma edge_pmf_real {lam : ℝ} (hlam : Admissible lam) (p : Spin → PMF α) (i : Spin) (a : α) :
-    (((channel lam hlam i).bind p) a).toReal =
-      discreteWeight p a*edgeMessage lam (discreteMessage p a) i := by
-  rw [PMF.bind_apply, tsum_fintype,
-    ENNReal.toReal_sum (fun j _ ↦ ENNReal.mul_ne_top ((channel lam hlam i).apply_ne_top j) ((p j).apply_ne_top a))]
-  simp only [ENNReal.toReal_mul, channel_real]
-  have he : discreteWeight p a*edgeMessage lam (discreteMessage p a) i =
-      lam*(p i a).toReal+(1-lam)*discreteWeight p a := by
-    dsimp only [edgeMessage, edge, centered]
-    calc
-      _ = lam*(discreteWeight p a*(discreteMessage p a : Spin → ℝ) i) +
-          (1-lam)*discreteWeight p a := by ring
-      _ = _ := by rw [discrete_bayes]
-  rw [he]
-  fin_cases i <;> simp [Fin.sum_univ_three, channelWeight, discreteWeight, avg] <;> ring
-
-def discreteBranchLaw (p : Spin → PMF α) (lam : ℝ) (hlam : Admissible lam) (n : ℕ) (i : Spin) :
-    PMF (Fin n → α) := productPMF (fun _ ↦ (channel lam hlam i).bind p)
-
-lemma discreteBranchLaw_real (p : Spin → PMF α) {lam : ℝ} (hlam : Admissible lam)
-    (n : ℕ) (i : Spin) (v : Fin n → α) :
-    (discreteBranchLaw p lam hlam n i v).toReal =
-      (∏ j, discreteWeight p (v j))*branchProduct (branchEdges lam (fun j ↦ discreteMessage p (v j))) i := by
-  simp only [discreteBranchLaw, productPMF_apply, ENNReal.toReal_prod, edge_pmf_real,
-    Finset.prod_mul_distrib, branchProduct, branchEdges]
-
-lemma discreteBranchWeight (p : Spin → PMF α) {lam : ℝ} (hlam : Admissible lam)
-    (n : ℕ) (v : Fin n → α) :
-    discreteWeight (discreteBranchLaw p lam hlam n) v =
-      (∏ j, discreteWeight p (v j))*normalizer (branchEdges lam (fun j ↦ discreteMessage p (v j))) := by
-  unfold discreteWeight
-  simp only [discreteBranchLaw_real]
-  exact avg_mul _ _
-
-lemma discreteBranchMessage (p : Spin → PMF α) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
-    (hlam : Admissible lam) (n : ℕ) (v : Fin n → α) (hv : (∏ j, discreteWeight p (v j)) ≠ 0) :
-    discreteMessage (discreteBranchLaw p lam hlam n) v =
-      branchOutput hl₀ hl₁ (fun j ↦ discreteMessage p (v j)) := by
-  have hz := normalizer_positive (branchEdges_positive hl₀ hl₁ (fun j ↦ discreteMessage p (v j)))
-  have hw : discreteWeight (discreteBranchLaw p lam hlam n) v ≠ 0 := by
-    rw [discreteBranchWeight]; exact mul_ne_zero hv (ne_of_gt hz)
-  apply Subtype.ext
-  funext i
-  rw [discrete_coord hw, discreteBranchLaw_real, discreteBranchWeight]
-  change _ = branchProduct (branchEdges lam (fun j ↦ discreteMessage p (v j))) i /
-    normalizer (branchEdges lam (fun j ↦ discreteMessage p (v j)))
-  exact mul_div_mul_left _ _ hv
-
-lemma discreteBranch_integrand (p : Spin → PMF α) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
-    (hlam : Admissible lam) (n : ℕ) (v : Fin n → α) (f : Message → ℝ) :
-    discreteWeight (discreteBranchLaw p lam hlam n) v*f (discreteMessage (discreteBranchLaw p lam hlam n) v) =
-      (∏ j, discreteWeight p (v j))*
-        (normalizer (branchEdges lam (fun j ↦ discreteMessage p (v j)))*
-          f (branchOutput hl₀ hl₁ (fun j ↦ discreteMessage p (v j)))) := by
-  rw [discreteBranchWeight]
-  by_cases hv : (∏ j, discreteWeight p (v j)) = 0
-  · simp [hv]
-  · rw [discreteBranchMessage p hl₀ hl₁ hlam n v hv, mul_assoc]
-
-section Measures
-variable [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α]
-
-lemma discreteMessage_preserving (p : Spin → PMF α) :
-    MeasurePreserving (discreteMessage p) (discreteMarginal p).toMeasure (discreteProbability p).toMeasure :=
-  ⟨measurable_of_countable _, rfl⟩
-
-lemma integral_discreteBranchLaw (p : Spin → PMF α) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
-    (hlam : Admissible lam) (n : ℕ) {f : Message → ℝ} (hf : Continuous f) :
-    (∫ m, f m ∂(discreteProbability (discreteBranchLaw p lam hlam n)).toMeasure) =
-      ∫ m : Fin n → Message, normalizer (branchEdges lam m)*f (branchOutput hl₀ hl₁ m)
-        ∂(Measure.pi fun _ ↦ (discreteProbability p).toMeasure) := by
-  let g : (Fin n → Message) → ℝ := fun m ↦ normalizer (branchEdges lam m)*f (branchOutput hl₀ hl₁ m)
-  have hg : Continuous g := (continuous_normalizer n lam).mul (hf.comp (continuous_branchOutput hl₀ hl₁ n))
-  have hp := measurePreserving_pi (fun _ : Fin n ↦ (discreteMarginal p).toMeasure)
-    (fun _ : Fin n ↦ (discreteProbability p).toMeasure) (fun _ ↦ discreteMessage_preserving p)
-  conv_rhs => rw [← hp.map_eq, integral_map hp.measurable.aemeasurable hg.measurable.aestronglyMeasurable]
-  rw [integral_discreteProbability _ hf.measurable, ← productPMF_measure]
-  rw [PMF.integral_eq_tsum, PMF.integral_eq_tsum]
-  · apply tsum_congr
-    intro v
-    simp only [smul_eq_mul, productPMF_apply, ENNReal.toReal_prod, discreteMarginal_real]
-    exact discreteBranch_integrand p hl₀ hl₁ hlam n v f
-  · rw [productPMF_measure]
-    exact hp.integrable_comp_of_integrable (compact_integrable hg)
-  · exact (discreteMessage_preserving _).integrable_comp_of_integrable (compact_integrable hf)
-
-lemma probability_ext_continuous {X : Type*} [TopologicalSpace X] [MeasurableSpace X]
-    [BorelSpace X] [HasOuterApproxClosed X] {P Q : ProbabilityMeasure X}
-    (h : ∀ (f : X → ℝ), Continuous f → (∫ x, f x ∂P.toMeasure) = ∫ x, f x ∂Q.toMeasure) : P = Q := by
-  apply Subtype.ext
-  exact congrArg FiniteMeasure.toMeasure (FiniteMeasure.ext_of_forall_integral_eq
-    (μ := P.toFiniteMeasure) (ν := Q.toFiniteMeasure) (fun f ↦ h f f.continuous))
-
-lemma discreteBranchLaw_probability (p : Spin → PMF α) (Q : Law)
-    (hQ : Q.probability = discreteProbability p) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
-    (hlam : Admissible lam) (n : ℕ) :
-    discreteProbability (discreteBranchLaw p lam hlam n) = finiteProbability Q hl₀ hl₁ n := by
-  apply probability_ext_continuous
-  intro f hf
-  rw [integral_discreteBranchLaw p hl₀ hl₁ hlam n hf,
-    integral_finiteProbability Q hl₀ hl₁ n hf.measurable, hQ]
-
-end Measures
-
-end ThreeState.TreeClauses.Experiment
-
-end 
-
-noncomputable section
-open Set MeasureTheory
-namespace ThreeState.TreeClauses.Experiment
-open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
-
-lemma measurable_symEntropy : Measurable (fun m : Message ↦ symEntropy m.1) := by
-  have h (i : Fin 3) : Measurable (fun m : Message ↦ (m.1 i-1)*Real.log (m.1 i)) :=
-    ((continuous_coordinate i).measurable.sub measurable_const).mul
-      (Real.measurable_log.comp (continuous_coordinate i).measurable)
-  exact (((h 0).add (h 1)).add (h 2)).div_const 3 |>.div_const 2
-
-lemma measurable_info : Measurable (fun m : Message ↦ info m.1) := by
-  have h (i : Fin 3) : Measurable (fun m : Message ↦ m.1 i*Real.log (m.1 i)) :=
-    (continuous_coordinate i).measurable.mul (Real.measurable_log.comp (continuous_coordinate i).measurable)
-  exact (((h 0).add (h 1)).add (h 2)).div_const 3
-
-lemma measurable_correction : Measurable (fun m : Message ↦ correction m.1) := by
-  have hl (i : Fin 3) : Measurable (fun m : Message ↦ Real.log (m.1 i)) :=
-    Real.measurable_log.comp (continuous_coordinate i).measurable
-  have hu (i : Fin 3) : Measurable (fun m : Message ↦ logCentered m.1 i) :=
-    (hl i).sub ((((hl 0).add (hl 1)).add (hl 2)).div_const 3)
-  have h (i : Fin 3) : Measurable (fun m : Message ↦ m.1 i*logCentered m.1 i^2) :=
-    (continuous_coordinate i).measurable.mul ((hu i).pow_const 2)
-  exact (((h 0).add (h 1)).add (h 2)).div_const 3 |>.div_const 2
-
-lemma continuous_symEntropy_comp {X : Type*} [TopologicalSpace X] {f : X → Vec}
-    (hf : Continuous f) (hp : ∀ x i, 0 < f x i) : Continuous (fun x ↦ symEntropy (f x)) := by
-  unfold symEntropy
-  exact (continuous_avg.comp (continuous_pi (fun i ↦
-    (((continuous_apply i).comp hf).sub continuous_const).mul
-      (((continuous_apply i).comp hf).log (fun x ↦ ne_of_gt (hp x i)))))).div_const 2
-
-lemma continuous_correction_comp {X : Type*} [TopologicalSpace X] {f : X → Vec}
-    (hf : Continuous f) (hp : ∀ x i, 0 < f x i) : Continuous (fun x ↦ correction (f x)) := by
-  have hl (i : Fin 3) : Continuous (fun x ↦ Real.log (f x i)) :=
-    ((continuous_apply i).comp hf).log (fun x ↦ ne_of_gt (hp x i))
-  have hu (i : Fin 3) : Continuous (fun x ↦ logCentered (f x) i) :=
-    (hl i).sub (continuous_avg.comp (continuous_pi hl))
-  exact (continuous_avg.comp (continuous_pi (fun i ↦
-    ((continuous_apply i).comp hf).mul ((hu i).pow 2)))).div_const 2
-
-lemma continuous_info_comp {X : Type*} [TopologicalSpace X] {f : X → Vec}
-    (hf : Continuous f) : Continuous (fun x ↦ info (f x)) :=
-  continuous_avg.comp (continuous_pi (fun i ↦ Real.continuous_mul_log.comp ((continuous_apply i).comp hf)))
-
-lemma finite_integrable_of_comp (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ)
-    {f : Message → ℝ} (hf : Measurable f) (hc : Continuous (fun m : Fin n → Message ↦ f (branchOutput hl₀ hl₁ m))) :
-    Integrable f (finiteProbability Q hl₀ hl₁ n).toMeasure := by
-  have := tiltedProduct_isProbability Q hl₀ hl₁ n
-  change Integrable f (Measure.map (branchOutput hl₀ hl₁) (tiltedProduct Q lam n))
-  exact (integrable_map_measure hf.aestronglyMeasurable
-    (continuous_branchOutput hl₀ hl₁ n).measurable.aemeasurable).2 (compact_integrable hc)
-
-lemma finite_integrable_symEntropy (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
-    Integrable (fun m : Message ↦ symEntropy m.1) (finiteProbability Q hl₀ hl₁ n).toMeasure :=
-  finite_integrable_of_comp Q hl₀ hl₁ n measurable_symEntropy
-    (continuous_symEntropy_comp (continuous_subtype_val.comp (continuous_branchOutput hl₀ hl₁ n))
-      (fun m i ↦ (normalizedProduct_positive (branchEdges_positive hl₀ hl₁ m)).1 i))
-
-lemma finite_integrable_correction (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
-    Integrable (fun m : Message ↦ correction m.1) (finiteProbability Q hl₀ hl₁ n).toMeasure :=
-  finite_integrable_of_comp Q hl₀ hl₁ n measurable_correction
-    (continuous_correction_comp (continuous_subtype_val.comp (continuous_branchOutput hl₀ hl₁ n))
-      (fun m i ↦ (normalizedProduct_positive (branchEdges_positive hl₀ hl₁ m)).1 i))
-
-lemma measurableSet_positive : MeasurableSet {m : Message | PositiveMessage m.1} := by
-  have he : {m : Message | PositiveMessage m.1} = {m : Message | ∀ i, 0 < m.1 i} := by
-    ext m; exact ⟨fun h ↦ h.1, fun h ↦ ⟨h, m.2.2⟩⟩
-  rw [he]
-  simp only [ofPred_forall]
-  exact MeasurableSet.iInter (fun i ↦ measurableSet_lt measurable_const (continuous_coordinate i).measurable)
-
-lemma finite_positive (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
-    ∀ᵐ m ∂(finiteProbability Q hl₀ hl₁ n).toMeasure, PositiveMessage m.1 := by
-  change ∀ᵐ m ∂Measure.map (branchOutput hl₀ hl₁) (tiltedProduct Q lam n), PositiveMessage m.1
-  rw [ae_map_iff (continuous_branchOutput hl₀ hl₁ n).measurable.aemeasurable measurableSet_positive]
-  exact Filter.Eventually.of_forall (fun m ↦ normalizedProduct_positive (branchEdges_positive hl₀ hl₁ m))
-
-lemma finiteLaw_symEntropy (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
-    (∫ m, symEntropy m.1 ∂(finiteLaw Q hl₀ hl₁ n).probability.toMeasure) =
-      n*(∫ m, symEntropy (edgeMessage lam m) ∂Q.probability.toMeasure) := by
-  change (∫ m, symEntropy m.1 ∂(finiteProbability Q hl₀ hl₁ n).toMeasure) = _
-  rw [integral_finiteProbability Q hl₀ hl₁ n measurable_symEntropy]
-  exact finite_symEntropy Q hl₀ hl₁ n
-
-lemma finiteLaw_correction (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
-    (∫ m, correction m.1 ∂(finiteLaw Q hl₀ hl₁ n).probability.toMeasure) =
-      n*(∫ m, correction (edgeMessage lam m) ∂Q.probability.toMeasure)+
-      2*((n:ℝ)^2-n)*(∫ m, symEntropy (edgeMessage lam m) ∂Q.probability.toMeasure)^2 := by
-  change (∫ m, correction m.1 ∂(finiteProbability Q hl₀ hl₁ n).toMeasure) = _
-  rw [integral_finiteProbability Q hl₀ hl₁ n measurable_correction]
-  exact finite_correction Q hl₀ hl₁ n
-
-lemma finiteLaw_info (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
-    (∫ m, info m.1 ∂(finiteLaw Q hl₀ hl₁ n).probability.toMeasure) =
-      n*(∫ m, info (edgeMessage lam m) ∂Q.probability.toMeasure)-correlationEntropy Q lam n := by
-  change (∫ m, info m.1 ∂(finiteProbability Q hl₀ hl₁ n).toMeasure) = _
-  rw [integral_finiteProbability Q hl₀ hl₁ n measurable_info]
-  exact finite_info Q hl₀ hl₁ n
-
-end ThreeState.TreeClauses.Experiment
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeStateClauses.Basic
+import OAI.Probability.ThreeStateClauses.DiscreteLaw
+import OAI.Probability.ThreeStateClauses.Products
+import OAI.Probability.ThreeStateClauses.BranchMoments
+
+namespace OAI
+
+/-! Finite Update. -/
+
+open scoped BigOperators ENNReal NNReal Topology
+open Filter
+noncomputable section
+open Set MeasureTheory
+open scoped BigOperators
+namespace ThreeState.TreeClauses.Experiment
+open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
+
+def branchOutput {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) {n : ℕ}
+    (m : Fin n → Message) : Message :=
+  ⟨normalizedProduct (branchEdges lam m),
+    (fun i ↦ (normalizedProduct_positive (branchEdges_positive hl₀ hl₁ m)).1 i |>.le),
+    (normalizedProduct_positive (branchEdges_positive hl₀ hl₁ m)).2⟩
+
+lemma continuous_branchOutput {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
+    Continuous (branchOutput hl₀ hl₁ (n := n)) := by
+  apply Continuous.subtype_mk
+  exact continuous_pi (fun i ↦ (continuous_branchProduct lam i).div (continuous_normalizer n lam)
+    (fun m ↦ ne_of_gt (normalizer_positive (branchEdges_positive hl₀ hl₁ m))))
+
+def tiltedProduct (Q : Law) (lam : ℝ) (n : ℕ) : Measure (Fin n → Message) :=
+  (Measure.pi fun _ ↦ Q.probability.toMeasure).withDensity
+    (fun m ↦ ENNReal.ofReal (normalizer (branchEdges lam m)))
+
+lemma tiltedProduct_isProbability (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
+    IsProbabilityMeasure (tiltedProduct Q lam n) := by
+  constructor
+  rw [tiltedProduct, withDensity_apply _ MeasurableSet.univ, setLIntegral_univ,
+    ← ofReal_integral_eq_lintegral_ofReal (compact_integrable (continuous_normalizer n lam))
+      (Filter.Eventually.of_forall (fun m ↦ (normalizer_positive (branchEdges_positive hl₀ hl₁ m)).le)),
+    integral_normalizer]
+  norm_num
+
+lemma integral_tiltedProduct (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ)
+    (f : (Fin n → Message) → ℝ) :
+    (∫ m, f m ∂tiltedProduct Q lam n) =
+      ∫ m, normalizer (branchEdges lam m)*f m ∂(Measure.pi fun _ ↦ Q.probability.toMeasure) := by
+  rw [tiltedProduct, integral_withDensity_eq_integral_toReal_smul
+    (show Measurable (fun m : Fin n → Message ↦ ENNReal.ofReal (normalizer (branchEdges lam m))) from
+      ENNReal.measurable_ofReal.comp (continuous_normalizer n lam).measurable)
+    (Filter.Eventually.of_forall (fun _ ↦ ENNReal.ofReal_lt_top))]
+  congr 1; funext m
+  rw [ENNReal.toReal_ofReal (normalizer_positive (branchEdges_positive hl₀ hl₁ m)).le]
+  rfl
+
+lemma measurePreserving_density {X : Type*} [MeasurableSpace X] {μ : Measure X}
+    {T : X → X} (hT : MeasurePreserving T μ μ) {w : X → ENNReal} (hw : Measurable w)
+    (hinv : ∀ x, w (T x) = w x) :
+    MeasurePreserving T (μ.withDensity w) (μ.withDensity w) := by
+  refine ⟨hT.measurable, Measure.ext (fun s hs ↦ ?_)⟩
+  rw [Measure.map_apply hT.measurable hs, withDensity_apply _ (hT.measurable hs), withDensity_apply _ hs]
+  have h := hT.setLIntegral_comp_preimage hs hw
+  simpa only [hinv] using h
+
+lemma branchProduct_permute {n : ℕ} (lam : ℝ) (σ : Equiv.Perm (Fin 3))
+    (m : Fin n → Message) (i : Fin 3) :
+    branchProduct (branchEdges lam (fun j ↦ permute σ (m j))) i =
+      branchProduct (branchEdges lam m) (σ i) := rfl
+
+lemma normalizer_permute {n : ℕ} (lam : ℝ) (σ : Equiv.Perm (Fin 3)) (m : Fin n → Message) :
+    normalizer (branchEdges lam (fun j ↦ permute σ (m j))) = normalizer (branchEdges lam m) := by
+  unfold normalizer
+  change avg (fun i ↦ branchProduct (branchEdges lam m) (σ i)) = _
+  exact avg_permute_vec σ (branchProduct (branchEdges lam m))
+
+lemma branchOutput_permute {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) {n : ℕ}
+    (σ : Equiv.Perm (Fin 3)) (m : Fin n → Message) :
+    branchOutput hl₀ hl₁ (fun j ↦ permute σ (m j)) = permute σ (branchOutput hl₀ hl₁ m) := by
+  apply Subtype.ext; funext i
+  change branchProduct (branchEdges lam (fun j ↦ permute σ (m j))) i /
+    normalizer (branchEdges lam (fun j ↦ permute σ (m j))) =
+    branchProduct (branchEdges lam m) (σ i) / normalizer (branchEdges lam m)
+  rw [branchProduct_permute, normalizer_permute]
+
+lemma tiltedProduct_symmetric (Q : Law) {lam : ℝ} (_hl₀ : 0 ≤ lam) (_hl₁ : lam < 1) (n : ℕ)
+    (σ : Equiv.Perm (Fin 3)) :
+    MeasurePreserving (fun m : Fin n → Message ↦ fun j ↦ permute σ (m j))
+      (tiltedProduct Q lam n) (tiltedProduct Q lam n) := by
+  apply measurePreserving_density
+    (measurePreserving_pi (fun _ ↦ Q.probability.toMeasure) (fun _ ↦ Q.probability.toMeasure)
+      (fun _ ↦ Q.symmetric σ))
+    (ENNReal.measurable_ofReal.comp (continuous_normalizer n lam).measurable)
+  intro m
+  change ENNReal.ofReal (normalizer (branchEdges lam (fun j ↦ permute σ (m j)))) = _
+  rw [normalizer_permute]
+  rfl
+
+def finiteProbability (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
+    ProbabilityMeasure Message := by
+  let ν : ProbabilityMeasure (Fin n → Message) := ⟨tiltedProduct Q lam n, tiltedProduct_isProbability Q hl₀ hl₁ n⟩
+  exact ν.map (branchOutput hl₀ hl₁)
+
+lemma integral_finiteProbability (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ)
+    {f : Message → ℝ} (hf : Measurable f) :
+    (∫ m, f m ∂(finiteProbability Q hl₀ hl₁ n).toMeasure) =
+      ∫ m : Fin n → Message, normalizer (branchEdges lam m)*f (branchOutput hl₀ hl₁ m)
+        ∂(Measure.pi fun _ ↦ Q.probability.toMeasure) := by
+  change (∫ m, f m ∂Measure.map (branchOutput hl₀ hl₁) (tiltedProduct Q lam n)) = _
+  rw [integral_map (continuous_branchOutput hl₀ hl₁ n).measurable.aemeasurable hf.aestronglyMeasurable,
+    integral_tiltedProduct Q hl₀ hl₁]
+
+lemma finiteProbability_balanced (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) (i : Fin 3) :
+    (∫ m, m.1 i ∂(finiteProbability Q hl₀ hl₁ n).toMeasure) = 1 := by
+  rw [integral_finiteProbability Q hl₀ hl₁ n (continuous_coordinate i).measurable]
+  have he (m : Fin n → Message) :
+      normalizer (branchEdges lam m)*(branchOutput hl₀ hl₁ m).1 i = branchProduct (branchEdges lam m) i := by
+    change normalizer (branchEdges lam m)*(branchProduct (branchEdges lam m) i/normalizer (branchEdges lam m)) = _
+    field_simp [ne_of_gt (normalizer_positive (branchEdges_positive hl₀ hl₁ m))]
+  simp_rw [he]
+  change (∫ m : Fin n → Message, ∏ j, edgeMessage lam (m j) i ∂(Measure.pi fun _ ↦ Q.probability.toMeasure)) = _
+  rw [integral_fintype_prod_eq_pow (fun m : Message ↦ edgeMessage lam m i), Q.edge_balanced]
+  simp
+
+lemma finiteProbability_symmetric (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ)
+    (σ : Equiv.Perm (Fin 3)) :
+    MeasurePreserving (permute σ) (finiteProbability Q hl₀ hl₁ n).toMeasure
+      (finiteProbability Q hl₀ hl₁ n).toMeasure := by
+  refine ⟨(continuous_permute σ).measurable, ?_⟩
+  change Measure.map (permute σ) (Measure.map (branchOutput hl₀ hl₁) (tiltedProduct Q lam n)) =
+    Measure.map (branchOutput hl₀ hl₁) (tiltedProduct Q lam n)
+  rw [Measure.map_map (continuous_permute σ).measurable (continuous_branchOutput hl₀ hl₁ n).measurable]
+  have he : permute σ ∘ branchOutput hl₀ hl₁ =
+      branchOutput hl₀ hl₁ ∘ (fun m : Fin n → Message ↦ fun j ↦ permute σ (m j)) := by
+    funext m; exact (branchOutput_permute hl₀ hl₁ σ m).symm
+  rw [he, ← Measure.map_map (continuous_branchOutput hl₀ hl₁ n).measurable
+    (tiltedProduct_symmetric Q hl₀ hl₁ n σ).measurable,
+    (tiltedProduct_symmetric Q hl₀ hl₁ n σ).map_eq]
+
+def finiteLaw (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) : Law where
+  probability := finiteProbability Q hl₀ hl₁ n
+  balanced := finiteProbability_balanced Q hl₀ hl₁ n
+  symmetric := finiteProbability_symmetric Q hl₀ hl₁ n
+
+end ThreeState.TreeClauses.Experiment
+
+end 
+
+noncomputable section
+open Set MeasureTheory
+open scoped BigOperators ENNReal
+namespace ThreeState.TreeClauses.Experiment
+open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
+
+variable {α : Type*}
+
+lemma channel_real {lam : ℝ} (hlam : Admissible lam) (i j : Spin) :
+    (channel lam hlam i j).toReal = channelWeight lam i j := by
+  exact ENNReal.toReal_ofReal (channelWeight_nonneg hlam i j)
+
+lemma edge_pmf_real {lam : ℝ} (hlam : Admissible lam) (p : Spin → PMF α) (i : Spin) (a : α) :
+    (((channel lam hlam i).bind p) a).toReal =
+      discreteWeight p a*edgeMessage lam (discreteMessage p a) i := by
+  rw [PMF.bind_apply, tsum_fintype,
+    ENNReal.toReal_sum (fun j _ ↦ ENNReal.mul_ne_top ((channel lam hlam i).apply_ne_top j) ((p j).apply_ne_top a))]
+  simp only [ENNReal.toReal_mul, channel_real]
+  have he : discreteWeight p a*edgeMessage lam (discreteMessage p a) i =
+      lam*(p i a).toReal+(1-lam)*discreteWeight p a := by
+    dsimp only [edgeMessage, edge, centered]
+    calc
+      _ = lam*(discreteWeight p a*(discreteMessage p a : Spin → ℝ) i) +
+          (1-lam)*discreteWeight p a := by ring
+      _ = _ := by rw [discrete_bayes]
+  rw [he]
+  fin_cases i <;> simp [Fin.sum_univ_three, channelWeight, discreteWeight, avg] <;> ring
+
+def discreteBranchLaw (p : Spin → PMF α) (lam : ℝ) (hlam : Admissible lam) (n : ℕ) (i : Spin) :
+    PMF (Fin n → α) := productPMF (fun _ ↦ (channel lam hlam i).bind p)
+
+lemma discreteBranchLaw_real (p : Spin → PMF α) {lam : ℝ} (hlam : Admissible lam)
+    (n : ℕ) (i : Spin) (v : Fin n → α) :
+    (discreteBranchLaw p lam hlam n i v).toReal =
+      (∏ j, discreteWeight p (v j))*branchProduct (branchEdges lam (fun j ↦ discreteMessage p (v j))) i := by
+  simp only [discreteBranchLaw, productPMF_apply, ENNReal.toReal_prod, edge_pmf_real,
+    Finset.prod_mul_distrib, branchProduct, branchEdges]
+
+lemma discreteBranchWeight (p : Spin → PMF α) {lam : ℝ} (hlam : Admissible lam)
+    (n : ℕ) (v : Fin n → α) :
+    discreteWeight (discreteBranchLaw p lam hlam n) v =
+      (∏ j, discreteWeight p (v j))*normalizer (branchEdges lam (fun j ↦ discreteMessage p (v j))) := by
+  unfold discreteWeight
+  simp only [discreteBranchLaw_real]
+  exact avg_mul _ _
+
+lemma discreteBranchMessage (p : Spin → PMF α) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
+    (hlam : Admissible lam) (n : ℕ) (v : Fin n → α) (hv : (∏ j, discreteWeight p (v j)) ≠ 0) :
+    discreteMessage (discreteBranchLaw p lam hlam n) v =
+      branchOutput hl₀ hl₁ (fun j ↦ discreteMessage p (v j)) := by
+  have hz := normalizer_positive (branchEdges_positive hl₀ hl₁ (fun j ↦ discreteMessage p (v j)))
+  have hw : discreteWeight (discreteBranchLaw p lam hlam n) v ≠ 0 := by
+    rw [discreteBranchWeight]; exact mul_ne_zero hv (ne_of_gt hz)
+  apply Subtype.ext
+  funext i
+  rw [discrete_coord hw, discreteBranchLaw_real, discreteBranchWeight]
+  change _ = branchProduct (branchEdges lam (fun j ↦ discreteMessage p (v j))) i /
+    normalizer (branchEdges lam (fun j ↦ discreteMessage p (v j)))
+  exact mul_div_mul_left _ _ hv
+
+lemma discreteBranch_integrand (p : Spin → PMF α) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
+    (hlam : Admissible lam) (n : ℕ) (v : Fin n → α) (f : Message → ℝ) :
+    discreteWeight (discreteBranchLaw p lam hlam n) v*f (discreteMessage (discreteBranchLaw p lam hlam n) v) =
+      (∏ j, discreteWeight p (v j))*
+        (normalizer (branchEdges lam (fun j ↦ discreteMessage p (v j)))*
+          f (branchOutput hl₀ hl₁ (fun j ↦ discreteMessage p (v j)))) := by
+  rw [discreteBranchWeight]
+  by_cases hv : (∏ j, discreteWeight p (v j)) = 0
+  · simp [hv]
+  · rw [discreteBranchMessage p hl₀ hl₁ hlam n v hv, mul_assoc]
+
+section Measures
+variable [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α]
+
+lemma discreteMessage_preserving (p : Spin → PMF α) :
+    MeasurePreserving (discreteMessage p) (discreteMarginal p).toMeasure (discreteProbability p).toMeasure :=
+  ⟨measurable_of_countable _, rfl⟩
+
+lemma integral_discreteBranchLaw (p : Spin → PMF α) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
+    (hlam : Admissible lam) (n : ℕ) {f : Message → ℝ} (hf : Continuous f) :
+    (∫ m, f m ∂(discreteProbability (discreteBranchLaw p lam hlam n)).toMeasure) =
+      ∫ m : Fin n → Message, normalizer (branchEdges lam m)*f (branchOutput hl₀ hl₁ m)
+        ∂(Measure.pi fun _ ↦ (discreteProbability p).toMeasure) := by
+  let g : (Fin n → Message) → ℝ := fun m ↦ normalizer (branchEdges lam m)*f (branchOutput hl₀ hl₁ m)
+  have hg : Continuous g := (continuous_normalizer n lam).mul (hf.comp (continuous_branchOutput hl₀ hl₁ n))
+  have hp := measurePreserving_pi (fun _ : Fin n ↦ (discreteMarginal p).toMeasure)
+    (fun _ : Fin n ↦ (discreteProbability p).toMeasure) (fun _ ↦ discreteMessage_preserving p)
+  conv_rhs => rw [← hp.map_eq, integral_map hp.measurable.aemeasurable hg.measurable.aestronglyMeasurable]
+  rw [integral_discreteProbability _ hf.measurable, ← productPMF_measure]
+  rw [PMF.integral_eq_tsum, PMF.integral_eq_tsum]
+  · apply tsum_congr
+    intro v
+    simp only [smul_eq_mul, productPMF_apply, ENNReal.toReal_prod, discreteMarginal_real]
+    exact discreteBranch_integrand p hl₀ hl₁ hlam n v f
+  · rw [productPMF_measure]
+    exact hp.integrable_comp_of_integrable (compact_integrable hg)
+  · exact (discreteMessage_preserving _).integrable_comp_of_integrable (compact_integrable hf)
+
+lemma probability_ext_continuous {X : Type*} [TopologicalSpace X] [MeasurableSpace X]
+    [BorelSpace X] [HasOuterApproxClosed X] {P Q : ProbabilityMeasure X}
+    (h : ∀ (f : X → ℝ), Continuous f → (∫ x, f x ∂P.toMeasure) = ∫ x, f x ∂Q.toMeasure) : P = Q := by
+  apply Subtype.ext
+  exact congrArg FiniteMeasure.toMeasure (FiniteMeasure.ext_of_forall_integral_eq
+    (μ := P.toFiniteMeasure) (ν := Q.toFiniteMeasure) (fun f ↦ h f f.continuous))
+
+lemma discreteBranchLaw_probability (p : Spin → PMF α) (Q : Law)
+    (hQ : Q.probability = discreteProbability p) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
+    (hlam : Admissible lam) (n : ℕ) :
+    discreteProbability (discreteBranchLaw p lam hlam n) = finiteProbability Q hl₀ hl₁ n := by
+  apply probability_ext_continuous
+  intro f hf
+  rw [integral_discreteBranchLaw p hl₀ hl₁ hlam n hf,
+    integral_finiteProbability Q hl₀ hl₁ n hf.measurable, hQ]
+
+end Measures
+
+end ThreeState.TreeClauses.Experiment
+
+end 
+
+noncomputable section
+open Set MeasureTheory
+namespace ThreeState.TreeClauses.Experiment
+open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
+
+lemma measurable_symEntropy : Measurable (fun m : Message ↦ symEntropy m.1) := by
+  have h (i : Fin 3) : Measurable (fun m : Message ↦ (m.1 i-1)*Real.log (m.1 i)) :=
+    ((continuous_coordinate i).measurable.sub measurable_const).mul
+      (Real.measurable_log.comp (continuous_coordinate i).measurable)
+  exact (((h 0).add (h 1)).add (h 2)).div_const 3 |>.div_const 2
+
+lemma measurable_info : Measurable (fun m : Message ↦ info m.1) := by
+  have h (i : Fin 3) : Measurable (fun m : Message ↦ m.1 i*Real.log (m.1 i)) :=
+    (continuous_coordinate i).measurable.mul (Real.measurable_log.comp (continuous_coordinate i).measurable)
+  exact (((h 0).add (h 1)).add (h 2)).div_const 3
+
+lemma measurable_correction : Measurable (fun m : Message ↦ correction m.1) := by
+  have hl (i : Fin 3) : Measurable (fun m : Message ↦ Real.log (m.1 i)) :=
+    Real.measurable_log.comp (continuous_coordinate i).measurable
+  have hu (i : Fin 3) : Measurable (fun m : Message ↦ logCentered m.1 i) :=
+    (hl i).sub ((((hl 0).add (hl 1)).add (hl 2)).div_const 3)
+  have h (i : Fin 3) : Measurable (fun m : Message ↦ m.1 i*logCentered m.1 i^2) :=
+    (continuous_coordinate i).measurable.mul ((hu i).pow_const 2)
+  exact (((h 0).add (h 1)).add (h 2)).div_const 3 |>.div_const 2
+
+lemma continuous_symEntropy_comp {X : Type*} [TopologicalSpace X] {f : X → Vec}
+    (hf : Continuous f) (hp : ∀ x i, 0 < f x i) : Continuous (fun x ↦ symEntropy (f x)) := by
+  unfold symEntropy
+  exact (continuous_avg.comp (continuous_pi (fun i ↦
+    (((continuous_apply i).comp hf).sub continuous_const).mul
+      (((continuous_apply i).comp hf).log (fun x ↦ ne_of_gt (hp x i)))))).div_const 2
+
+lemma continuous_correction_comp {X : Type*} [TopologicalSpace X] {f : X → Vec}
+    (hf : Continuous f) (hp : ∀ x i, 0 < f x i) : Continuous (fun x ↦ correction (f x)) := by
+  have hl (i : Fin 3) : Continuous (fun x ↦ Real.log (f x i)) :=
+    ((continuous_apply i).comp hf).log (fun x ↦ ne_of_gt (hp x i))
+  have hu (i : Fin 3) : Continuous (fun x ↦ logCentered (f x) i) :=
+    (hl i).sub (continuous_avg.comp (continuous_pi hl))
+  exact (continuous_avg.comp (continuous_pi (fun i ↦
+    ((continuous_apply i).comp hf).mul ((hu i).pow 2)))).div_const 2
+
+lemma continuous_info_comp {X : Type*} [TopologicalSpace X] {f : X → Vec}
+    (hf : Continuous f) : Continuous (fun x ↦ info (f x)) :=
+  continuous_avg.comp (continuous_pi (fun i ↦ Real.continuous_mul_log.comp ((continuous_apply i).comp hf)))
+
+lemma finite_integrable_of_comp (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ)
+    {f : Message → ℝ} (hf : Measurable f) (hc : Continuous (fun m : Fin n → Message ↦ f (branchOutput hl₀ hl₁ m))) :
+    Integrable f (finiteProbability Q hl₀ hl₁ n).toMeasure := by
+  have := tiltedProduct_isProbability Q hl₀ hl₁ n
+  change Integrable f (Measure.map (branchOutput hl₀ hl₁) (tiltedProduct Q lam n))
+  exact (integrable_map_measure hf.aestronglyMeasurable
+    (continuous_branchOutput hl₀ hl₁ n).measurable.aemeasurable).2 (compact_integrable hc)
+
+lemma finite_integrable_symEntropy (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
+    Integrable (fun m : Message ↦ symEntropy m.1) (finiteProbability Q hl₀ hl₁ n).toMeasure :=
+  finite_integrable_of_comp Q hl₀ hl₁ n measurable_symEntropy
+    (continuous_symEntropy_comp (continuous_subtype_val.comp (continuous_branchOutput hl₀ hl₁ n))
+      (fun m i ↦ (normalizedProduct_positive (branchEdges_positive hl₀ hl₁ m)).1 i))
+
+lemma finite_integrable_correction (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
+    Integrable (fun m : Message ↦ correction m.1) (finiteProbability Q hl₀ hl₁ n).toMeasure :=
+  finite_integrable_of_comp Q hl₀ hl₁ n measurable_correction
+    (continuous_correction_comp (continuous_subtype_val.comp (continuous_branchOutput hl₀ hl₁ n))
+      (fun m i ↦ (normalizedProduct_positive (branchEdges_positive hl₀ hl₁ m)).1 i))
+
+lemma measurableSet_positive : MeasurableSet {m : Message | PositiveMessage m.1} := by
+  have he : {m : Message | PositiveMessage m.1} = {m : Message | ∀ i, 0 < m.1 i} := by
+    ext m; exact ⟨fun h ↦ h.1, fun h ↦ ⟨h, m.2.2⟩⟩
+  rw [he]
+  simp only [ofPred_forall]
+  exact MeasurableSet.iInter (fun i ↦ measurableSet_lt measurable_const (continuous_coordinate i).measurable)
+
+lemma finite_positive (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
+    ∀ᵐ m ∂(finiteProbability Q hl₀ hl₁ n).toMeasure, PositiveMessage m.1 := by
+  change ∀ᵐ m ∂Measure.map (branchOutput hl₀ hl₁) (tiltedProduct Q lam n), PositiveMessage m.1
+  rw [ae_map_iff (continuous_branchOutput hl₀ hl₁ n).measurable.aemeasurable measurableSet_positive]
+  exact Filter.Eventually.of_forall (fun m ↦ normalizedProduct_positive (branchEdges_positive hl₀ hl₁ m))
+
+lemma finiteLaw_symEntropy (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
+    (∫ m, symEntropy m.1 ∂(finiteLaw Q hl₀ hl₁ n).probability.toMeasure) =
+      n*(∫ m, symEntropy (edgeMessage lam m) ∂Q.probability.toMeasure) := by
+  change (∫ m, symEntropy m.1 ∂(finiteProbability Q hl₀ hl₁ n).toMeasure) = _
+  rw [integral_finiteProbability Q hl₀ hl₁ n measurable_symEntropy]
+  exact finite_symEntropy Q hl₀ hl₁ n
+
+lemma finiteLaw_correction (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
+    (∫ m, correction m.1 ∂(finiteLaw Q hl₀ hl₁ n).probability.toMeasure) =
+      n*(∫ m, correction (edgeMessage lam m) ∂Q.probability.toMeasure)+
+      2*((n:ℝ)^2-n)*(∫ m, symEntropy (edgeMessage lam m) ∂Q.probability.toMeasure)^2 := by
+  change (∫ m, correction m.1 ∂(finiteProbability Q hl₀ hl₁ n).toMeasure) = _
+  rw [integral_finiteProbability Q hl₀ hl₁ n measurable_correction]
+  exact finite_correction Q hl₀ hl₁ n
+
+lemma finiteLaw_info (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
+    (∫ m, info m.1 ∂(finiteLaw Q hl₀ hl₁ n).probability.toMeasure) =
+      n*(∫ m, info (edgeMessage lam m) ∂Q.probability.toMeasure)-correlationEntropy Q lam n := by
+  change (∫ m, info m.1 ∂(finiteProbability Q hl₀ hl₁ n).toMeasure) = _
+  rw [integral_finiteProbability Q hl₀ hl₁ n measurable_info]
+  exact finite_info Q hl₀ hl₁ n
+
+end ThreeState.TreeClauses.Experiment
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeStateClauses/InformationLoss.lean b/lean/OAI/Probability/ThreeStateClauses/InformationLoss.lean
index 86119ce05c3e53ab0cfcd9ab7d37cbae559a3f9c..d6d8eb786bd80adfee38e88f446304bbaf15505a 100644
--- a/lean/OAI/Probability/ThreeStateClauses/InformationLoss.lean
+++ b/lean/OAI/Probability/ThreeStateClauses/InformationLoss.lean
@@ -1,261 +1,261 @@
-import Mathlib
-import OAI.Probability.ThreeStateClauses.CompactLimits
-
-namespace OAI
-
-/-! Information Loss. -/
-
-open scoped BigOperators ENNReal NNReal Topology
-open Filter
-noncomputable section
-open scoped BigOperators
-namespace ThreeState.TreeClauses.Experiment
-
-variable {α β γ : Type}
-
-def multisetKernel (K : α → PMF β) (s : Multiset α) : PMF (Multiset β) :=
-  Multiset.recOn s (PMF.pure 0)
-    (fun a _ p ↦ (K a).bind (fun b ↦ p.map (Multiset.cons b))) (by
-      intro a a' s p
-      apply heq_of_eq
-      simp only [PMF.map_bind, PMF.map_comp]
-      rw [PMF.bind_comm (K a) (K a')]
-      congr 1
-      funext b'
-      congr 1
-      funext b
-      congr 1
-      funext t
-      exact Multiset.cons_swap _ _ _)
-
-@[simp] lemma multisetKernel_zero (K : α → PMF β) : multisetKernel K 0 = PMF.pure 0 := by
-  rw [multisetKernel, Multiset.recOn_0]
-
-@[simp] lemma multisetKernel_cons (K : α → PMF β) (a : α) (s : Multiset α) :
-    multisetKernel K (a ::ₘ s) = (K a).bind (fun b ↦ (multisetKernel K s).map (Multiset.cons b)) := by
-  rw [multisetKernel, Multiset.recOn_cons]
-  rfl
-
-lemma multisetKernel_iid (p : PMF α) (K : α → PMF β) (n : ℕ) :
-    ((iidList p n).map (fun xs : List α ↦ (xs : Multiset α))).bind (multisetKernel K) =
-      (iidList (p.bind K) n).map (fun xs : List β ↦ (xs : Multiset β)) := by
-  induction n with
-  | zero => simp [iidList, PMF.pure_map]
-  | succ n ih =>
-    simp only [iidList, PMF.map_bind, PMF.map_comp, PMF.bind_bind, PMF.bind_map]
-    congr 1
-    funext a
-    simp only [Function.comp_def, ← Multiset.cons_coe, multisetKernel_cons]
-    rw [PMF.bind_comm (iidList p n) (K a)]
-    congr 1
-    funext b
-    rw [← PMF.map_bind]
-    have hi := ih
-    rw [PMF.bind_map] at hi
-    dsimp only [Function.comp_def] at hi
-    rw [hi, PMF.map_comp]
-    rfl
-
-lemma observationLaw_succ (lam : ℝ) (hlam : Admissible lam) (offspring : PMF ℕ)
-    (ℓ : ℕ) (i : Spin) :
-    observationLaw lam hlam offspring (ℓ+1) i = offspring.bind (fun n ↦
-      (iidList ((channel lam hlam i).bind (observationLaw lam hlam offspring ℓ)) n).map
-        (fun xs : List (Observation ℓ) ↦ (xs : Multiset (Observation ℓ)))) := by
-  rw [observationLaw]
-  congr 1
-  funext n
-  exact PMF.map_id _
-
-def observationExpansion (lam : ℝ) (hlam : Admissible lam) (offspring : PMF ℕ) :
-    (ℓ : ℕ) → Observation ℓ → PMF (Observation (ℓ+1))
-  | 0, i => observationLaw lam hlam offspring 1 i
-  | ℓ+1, s => multisetKernel (observationExpansion lam hlam offspring ℓ) s
-
-lemma observationLaw_degrades (lam : ℝ) (hlam : Admissible lam) (offspring : PMF ℕ)
-    (ℓ : ℕ) (i : Spin) :
-    (observationLaw lam hlam offspring ℓ i).bind (observationExpansion lam hlam offspring ℓ) =
-      observationLaw lam hlam offspring (ℓ+1) i := by
-  induction ℓ generalizing i with
-  | zero => exact PMF.pure_bind _ _
-  | succ ℓ ih =>
-    rw [observationLaw_succ, observationLaw_succ]
-    change ((offspring.bind (fun n ↦ (iidList ((channel lam hlam i).bind (observationLaw lam hlam offspring ℓ)) n).map
-      (fun xs : List (Observation ℓ) ↦ (xs : Multiset (Observation ℓ))))).bind
-      (multisetKernel (observationExpansion lam hlam offspring ℓ))) = _
-    rw [PMF.bind_bind]
-    congr 1
-    funext n
-    rw [multisetKernel_iid]
-    congr 2
-    rw [PMF.bind_bind]
-    congr 1
-    funext j
-    exact ih j
-
-end ThreeState.TreeClauses.Experiment
-
-end 
-
-noncomputable section
-open Set MeasureTheory Filter
-open scoped BigOperators Topology
-namespace ThreeState.TreeClauses.Experiment
-open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
-
-lemma coord_center_abs_le_two (m : Message) (i : Spin) : |m.1 i-1| ≤ 2 :=
-  abs_le.mpr ⟨by linarith [coordinate_nonneg m i], by linarith [coordinate_le_three m i]⟩
-
-variable {α β : Type*} [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α]
-  [Countable β] [MeasurableSpace β] [MeasurableSingletonClass β]
-
-lemma discrete_degradation_abs (p : Spin → PMF α) (K : α → PMF β) (i : Spin) :
-    (∫ b, |(discreteMessage (fun j ↦ (p j).bind K) b).1 i-1|
-      ∂(discreteMarginal (fun j ↦ (p j).bind K)).toMeasure) ≤
-    ∫ a, |(discreteMessage p a).1 i-1| ∂(discreteMarginal p).toMeasure := by
-  let q := fun j ↦ (p j).bind K
-  let g : β → ℝ := fun b ↦ if 0 ≤ (discreteMessage q b).1 i-1 then 1 else -1
-  have hg (b : β) : |g b| ≤ 1 := by dsimp [g]; split_ifs <;> norm_num
-  have he (b : β) : ((discreteMessage q b).1 i-1)*g b = |(discreteMessage q b).1 i-1| := by
-    dsimp [g]; split_ifs with h
-    · simp [abs_of_nonneg h]
-    · simp [abs_of_neg (lt_of_not_ge h)]
-  have hb (z : α × β) : |((discreteMessage p z.1).1 i-1)*g z.2| ≤ 2 := by
-    rw [abs_mul]
-    exact (mul_le_mul (coord_center_abs_le_two _ _) (hg _) (abs_nonneg _) (by norm_num)).trans_eq (by norm_num)
-  have hi := discrete_degradation_centered p K i hg
-  have hid (z : α × β) : ((discreteMessage p z.1).1 i-(discreteMessage q z.2).1 i)*g z.2 =
-      ((discreteMessage p z.1).1 i-1)*g z.2-|(discreteMessage q z.2).1 i-1| := by
-    rw [← he]; ring
-  change (∫ z : α × β, ((discreteMessage p z.1).1 i-(discreteMessage q z.2).1 i)*g z.2 ∂_) = 0 at hi
-  simp_rw [hid] at hi
-  rw [integral_sub (pmf_bounded_integrable _ hb)
-    (pmf_bounded_integrable _ (fun z : α × β ↦ by simpa using coord_center_abs_le_two (discreteMessage q z.2) i))] at hi
-  have hle : (∫ z, ((discreteMessage p z.1).1 i-1)*g z.2 ∂(kernelJoint (discreteMarginal p) K).toMeasure) ≤
-      ∫ z, |(discreteMessage p z.1).1 i-1| ∂(kernelJoint (discreteMarginal p) K).toMeasure :=
-    integral_mono (μ := (kernelJoint (discreteMarginal p) K).toMeasure)
-    (pmf_bounded_integrable _ hb)
-    (pmf_bounded_integrable _ (fun z : α × β ↦ by simpa using coord_center_abs_le_two (discreteMessage p z.1) i))
-    (fun z : α × β ↦ (le_abs_self _).trans (by
-      rw [abs_mul]; exact mul_le_of_le_one_right (abs_nonneg _) (hg _)))
-  rw [integral_kernelJoint_fst (f := fun a ↦ |(discreteMessage p a).1 i-1|)] at hle
-  rw [integral_kernelJoint_snd (f := fun b ↦ |(discreteMessage q b).1 i-1|), ← discreteMarginal_bind] at hi
-  change _ - (∫ b, |(discreteMessage q b).1 i-1| ∂(discreteMarginal q).toMeasure) = 0 at hi
-  linarith
-
-lemma discrete_degradation_tv (p : Spin → PMF α) (K : α → PMF β) :
-    (∫ m, tvMessage m ∂(discreteProbability (fun i ↦ (p i).bind K)).toMeasure) ≤
-      ∫ m, tvMessage m ∂(discreteProbability p).toMeasure := by
-  rw [integral_discreteProbability _ continuous_tvMessage.measurable,
-    integral_discreteProbability _ continuous_tvMessage.measurable]
-  unfold tvMessage
-  rw [integral_div, integral_div, integral_avg_measure, integral_avg_measure]
-  · exact div_le_div_of_nonneg_right (by
-      dsimp [avg]
-      linarith [discrete_degradation_abs p K 0, discrete_degradation_abs p K 1,
-        discrete_degradation_abs p K 2]) (by norm_num)
-  all_goals intro i; exact pmf_bounded_integrable _ (fun a ↦ by simpa using coord_center_abs_le_two _ i)
-
-lemma advantage_antitone (lam : ℝ) (hlam : Admissible lam) (offspring : PMF ℕ) :
-    Antitone (advantage lam hlam offspring) := by
-  apply antitone_nat_of_succ_le
-  intro ℓ
-  simp only [advantage_eq_integral]
-  have h := discrete_degradation_tv (observationLaw lam hlam offspring ℓ)
-    (observationExpansion lam hlam offspring ℓ)
-  simpa only [observationLaw_degrades] using h
-
-lemma advantage_tendsto (lam : ℝ) (hlam : Admissible lam) (offspring : PMF ℕ) :
-    Tendsto (advantage lam hlam offspring) atTop (𝓝 (⨅ ℓ, advantage lam hlam offspring ℓ)) :=
-  tendsto_atTop_ciInf (advantage_antitone lam hlam offspring)
-    ⟨0, by rintro _ ⟨ℓ,rfl⟩; exact advantage_nonneg lam hlam offspring ℓ⟩
-
-end ThreeState.TreeClauses.Experiment
-
-end 
-
-noncomputable section
-open Set MeasureTheory
-open scoped BigOperators
-namespace ThreeState.TreeClauses.Experiment
-open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
-
-variable {α : Type*} [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α]
-
-lemma integral_discreteMarginal (p : Spin → PMF α) {f : α → ℝ} {B : ℝ} (hB : ∀ a, |f a| ≤ B) :
-    (∫ a, f a ∂(discreteMarginal p).toMeasure) =
-      ((∫ a, f a ∂(p 0).toMeasure)+(∫ a, f a ∂(p 1).toMeasure)+(∫ a, f a ∂(p 2).toMeasure))/3 := by
-  rw [discreteMarginal, integral_pmf_bind _ _ hB, PMF.integral_eq_sum]
-  simp only [Fin.sum_univ_three, uniformSpin_apply, ENNReal.toReal_div,
-    ENNReal.toReal_one, ENNReal.toReal_ofNat, smul_eq_mul]
-  ring
-
-lemma coord_sq_le_tv (m : Message) (i : Spin) : (m.1 i-1)^2 ≤ 12*tvMessage m := by
-  have h := mul_le_mul_of_nonneg_left (coord_center_abs_le_two m i) (abs_nonneg (m.1 i-1))
-  rw [← sq, sq_abs] at h
-  have hs : |m.1 i-1| ≤ ∑ j : Spin, |m.1 j-1| :=
-    Finset.single_le_sum (f := fun j : Spin ↦ |m.1 j-1|) (fun j _ ↦ abs_nonneg _) (Finset.mem_univ i)
-  simp only [Fin.sum_univ_three] at hs
-  dsimp [tvMessage, avg]
-  linarith
-
-omit [Countable α] [MeasurableSingletonClass α] in
-lemma integral_scaled_square (μ : Measure α) {v f : α → ℝ} (r : ℝ)
-    (hv : Integrable (fun a ↦ v a^2) μ) (hf : Integrable (fun a ↦ f a^2) μ)
-    (hc : Integrable (fun a ↦ v a*f a) μ) :
-    (∫ a, (v a-r*f a)^2 ∂μ) = (∫ a, v a^2 ∂μ)-2*r*(∫ a, v a*f a ∂μ)+r^2*(∫ a, f a^2 ∂μ) := by
-  have he (a : α) : (v a-r*f a)^2 = (v a^2-(2*r)*(v a*f a))+r^2*f a^2 := by ring
-  simp_rw [he]
-  rw [integral_add (f := fun a ↦ v a^2-(2*r)*(v a*f a)) (g := fun a ↦ r^2*f a^2)
-    (hv.sub (hc.const_mul (2*r))) (hf.const_mul (r^2))]
-  rw [integral_sub hv (hc.const_mul (2*r))]
-  rw [integral_const_mul, integral_const_mul]
-
-lemma estimator_tv_lower (p : Spin → PMF α) {f : α → ℝ} {B C : ℝ}
-    (hB : ∀ a, |f a| ≤ B) (hC : 0 < C)
-    (hzero : (∫ a, f a ∂(discreteMarginal p).toMeasure) = 0)
-    (hroot : (∫ a, f a ∂(p 0).toMeasure) = 2)
-    (hsecond : (∫ a, f a^2 ∂(discreteMarginal p).toMeasure) ≤ C) :
-    1/(3*C) ≤ ∫ m, tvMessage m ∂(discreteProbability p).toMeasure := by
-  let v : α → ℝ := fun a ↦ (discreteMessage p a).1 0-1
-  have hv (a : α) : |v a| ≤ 2 := coord_center_abs_le_two _ _
-  have hf₂ := pmf_bounded_integrable (discreteMarginal p) (fun a ↦ show |f a^2| ≤ B^2 by
-    rw [abs_of_nonneg (sq_nonneg _)]; nlinarith [hB a, abs_nonneg (f a), sq_abs (f a)])
-  have hv₂ := pmf_bounded_integrable (discreteMarginal p) (fun a ↦ show |v a^2| ≤ (2:ℝ)^2 by
-    rw [abs_of_nonneg (sq_nonneg _)]; nlinarith [hv a, abs_nonneg (v a), sq_abs (v a)])
-  have hvf := pmf_bounded_integrable (discreteMarginal p) (fun a ↦ show |v a*f a| ≤ 2*B by
-    rw [abs_mul]; exact mul_le_mul (hv a) (hB a) (abs_nonneg _) (by norm_num))
-  have hcross : (∫ a, v a*f a ∂(discreteMarginal p).toMeasure) = 2 := by
-    have he (a : α) : v a*f a = (discreteMessage p a).1 0*f a-f a := by dsimp [v]; ring
-    simp_rw [he]
-    rw [integral_sub (pmf_bounded_integrable _ (fun a ↦ show |(discreteMessage p a).1 0*f a| ≤ 3*B by
-      rw [abs_mul, abs_of_nonneg (coordinate_nonneg _ _)]; exact mul_le_mul (coordinate_le_three _ _) (hB a) (abs_nonneg _) (by norm_num)))
-      (pmf_bounded_integrable _ hB), discrete_bayes_integral p 0 hB, hroot, hzero, sub_zero]
-  let r := 2/C
-  have hr : r*C = 2 := div_mul_cancel₀ 2 hC.ne'
-  have h := MeasureTheory.integral_nonneg (μ := (discreteMarginal p).toMeasure)
-    (f := fun a ↦ (v a-r*f a)^2) (fun _ ↦ sq_nonneg _)
-  rw [integral_scaled_square _ r hv₂ hf₂ hvf, hcross] at h
-  have hb := mul_le_mul_of_nonneg_left hsecond (sq_nonneg r)
-  have hr2 : r^2*C = 2*r := by
-    have hh := congrArg (fun x : ℝ ↦ r*x) hr
-    nlinarith only [hh]
-  have hD : 4 ≤ C*(∫ a, v a^2 ∂(discreteMarginal p).toMeasure) := by
-    have hs : 2*r ≤ ∫ a, v a^2 ∂(discreteMarginal p).toMeasure := by
-      nlinarith only [h, hb, hr2]
-    have hh := mul_le_mul_of_nonneg_left hs hC.le
-    nlinarith only [hh, hr]
-  have hT := integral_mono hv₂ (pmf_bounded_integrable (discreteMarginal p)
-    (fun a ↦ show |12*tvMessage (discreteMessage p a)| ≤ 12 by
-      rw [abs_mul, abs_of_nonneg (by norm_num : (0:ℝ) ≤ 12), abs_of_nonneg (tvMessage_nonneg _)];
-      nlinarith [tvMessage_le_one (discreteMessage p a)]))
-    (fun a ↦ coord_sq_le_tv (discreteMessage p a) 0)
-  rw [integral_const_mul, ← integral_discreteProbability p continuous_tvMessage.measurable] at hT
-  apply (div_le_iff₀ (by positivity : 0 < 3*C)).2
-  have hh := mul_le_mul_of_nonneg_left hT hC.le
-  nlinarith only [hD, hh]
-
-end ThreeState.TreeClauses.Experiment
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeStateClauses.CompactLimits
+
+namespace OAI
+
+/-! Information Loss. -/
+
+open scoped BigOperators ENNReal NNReal Topology
+open Filter
+noncomputable section
+open scoped BigOperators
+namespace ThreeState.TreeClauses.Experiment
+
+variable {α β γ : Type}
+
+def multisetKernel (K : α → PMF β) (s : Multiset α) : PMF (Multiset β) :=
+  Multiset.recOn s (PMF.pure 0)
+    (fun a _ p ↦ (K a).bind (fun b ↦ p.map (Multiset.cons b))) (by
+      intro a a' s p
+      apply heq_of_eq
+      simp only [PMF.map_bind, PMF.map_comp]
+      rw [PMF.bind_comm (K a) (K a')]
+      congr 1
+      funext b'
+      congr 1
+      funext b
+      congr 1
+      funext t
+      exact Multiset.cons_swap _ _ _)
+
+@[simp] lemma multisetKernel_zero (K : α → PMF β) : multisetKernel K 0 = PMF.pure 0 := by
+  rw [multisetKernel, Multiset.recOn_0]
+
+@[simp] lemma multisetKernel_cons (K : α → PMF β) (a : α) (s : Multiset α) :
+    multisetKernel K (a ::ₘ s) = (K a).bind (fun b ↦ (multisetKernel K s).map (Multiset.cons b)) := by
+  rw [multisetKernel, Multiset.recOn_cons]
+  rfl
+
+lemma multisetKernel_iid (p : PMF α) (K : α → PMF β) (n : ℕ) :
+    ((iidList p n).map (fun xs : List α ↦ (xs : Multiset α))).bind (multisetKernel K) =
+      (iidList (p.bind K) n).map (fun xs : List β ↦ (xs : Multiset β)) := by
+  induction n with
+  | zero => simp [iidList, PMF.pure_map]
+  | succ n ih =>
+    simp only [iidList, PMF.map_bind, PMF.map_comp, PMF.bind_bind, PMF.bind_map]
+    congr 1
+    funext a
+    simp only [Function.comp_def, ← Multiset.cons_coe, multisetKernel_cons]
+    rw [PMF.bind_comm (iidList p n) (K a)]
+    congr 1
+    funext b
+    rw [← PMF.map_bind]
+    have hi := ih
+    rw [PMF.bind_map] at hi
+    dsimp only [Function.comp_def] at hi
+    rw [hi, PMF.map_comp]
+    rfl
+
+lemma observationLaw_succ (lam : ℝ) (hlam : Admissible lam) (offspring : PMF ℕ)
+    (ℓ : ℕ) (i : Spin) :
+    observationLaw lam hlam offspring (ℓ+1) i = offspring.bind (fun n ↦
+      (iidList ((channel lam hlam i).bind (observationLaw lam hlam offspring ℓ)) n).map
+        (fun xs : List (Observation ℓ) ↦ (xs : Multiset (Observation ℓ)))) := by
+  rw [observationLaw]
+  congr 1
+  funext n
+  exact PMF.map_id _
+
+def observationExpansion (lam : ℝ) (hlam : Admissible lam) (offspring : PMF ℕ) :
+    (ℓ : ℕ) → Observation ℓ → PMF (Observation (ℓ+1))
+  | 0, i => observationLaw lam hlam offspring 1 i
+  | ℓ+1, s => multisetKernel (observationExpansion lam hlam offspring ℓ) s
+
+lemma observationLaw_degrades (lam : ℝ) (hlam : Admissible lam) (offspring : PMF ℕ)
+    (ℓ : ℕ) (i : Spin) :
+    (observationLaw lam hlam offspring ℓ i).bind (observationExpansion lam hlam offspring ℓ) =
+      observationLaw lam hlam offspring (ℓ+1) i := by
+  induction ℓ generalizing i with
+  | zero => exact PMF.pure_bind _ _
+  | succ ℓ ih =>
+    rw [observationLaw_succ, observationLaw_succ]
+    change ((offspring.bind (fun n ↦ (iidList ((channel lam hlam i).bind (observationLaw lam hlam offspring ℓ)) n).map
+      (fun xs : List (Observation ℓ) ↦ (xs : Multiset (Observation ℓ))))).bind
+      (multisetKernel (observationExpansion lam hlam offspring ℓ))) = _
+    rw [PMF.bind_bind]
+    congr 1
+    funext n
+    rw [multisetKernel_iid]
+    congr 2
+    rw [PMF.bind_bind]
+    congr 1
+    funext j
+    exact ih j
+
+end ThreeState.TreeClauses.Experiment
+
+end 
+
+noncomputable section
+open Set MeasureTheory Filter
+open scoped BigOperators Topology
+namespace ThreeState.TreeClauses.Experiment
+open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
+
+lemma coord_center_abs_le_two (m : Message) (i : Spin) : |m.1 i-1| ≤ 2 :=
+  abs_le.mpr ⟨by linarith [coordinate_nonneg m i], by linarith [coordinate_le_three m i]⟩
+
+variable {α β : Type*} [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α]
+  [Countable β] [MeasurableSpace β] [MeasurableSingletonClass β]
+
+lemma discrete_degradation_abs (p : Spin → PMF α) (K : α → PMF β) (i : Spin) :
+    (∫ b, |(discreteMessage (fun j ↦ (p j).bind K) b).1 i-1|
+      ∂(discreteMarginal (fun j ↦ (p j).bind K)).toMeasure) ≤
+    ∫ a, |(discreteMessage p a).1 i-1| ∂(discreteMarginal p).toMeasure := by
+  let q := fun j ↦ (p j).bind K
+  let g : β → ℝ := fun b ↦ if 0 ≤ (discreteMessage q b).1 i-1 then 1 else -1
+  have hg (b : β) : |g b| ≤ 1 := by dsimp [g]; split_ifs <;> norm_num
+  have he (b : β) : ((discreteMessage q b).1 i-1)*g b = |(discreteMessage q b).1 i-1| := by
+    dsimp [g]; split_ifs with h
+    · simp [abs_of_nonneg h]
+    · simp [abs_of_neg (lt_of_not_ge h)]
+  have hb (z : α × β) : |((discreteMessage p z.1).1 i-1)*g z.2| ≤ 2 := by
+    rw [abs_mul]
+    exact (mul_le_mul (coord_center_abs_le_two _ _) (hg _) (abs_nonneg _) (by norm_num)).trans_eq (by norm_num)
+  have hi := discrete_degradation_centered p K i hg
+  have hid (z : α × β) : ((discreteMessage p z.1).1 i-(discreteMessage q z.2).1 i)*g z.2 =
+      ((discreteMessage p z.1).1 i-1)*g z.2-|(discreteMessage q z.2).1 i-1| := by
+    rw [← he]; ring
+  change (∫ z : α × β, ((discreteMessage p z.1).1 i-(discreteMessage q z.2).1 i)*g z.2 ∂_) = 0 at hi
+  simp_rw [hid] at hi
+  rw [integral_sub (pmf_bounded_integrable _ hb)
+    (pmf_bounded_integrable _ (fun z : α × β ↦ by simpa using coord_center_abs_le_two (discreteMessage q z.2) i))] at hi
+  have hle : (∫ z, ((discreteMessage p z.1).1 i-1)*g z.2 ∂(kernelJoint (discreteMarginal p) K).toMeasure) ≤
+      ∫ z, |(discreteMessage p z.1).1 i-1| ∂(kernelJoint (discreteMarginal p) K).toMeasure :=
+    integral_mono (μ := (kernelJoint (discreteMarginal p) K).toMeasure)
+    (pmf_bounded_integrable _ hb)
+    (pmf_bounded_integrable _ (fun z : α × β ↦ by simpa using coord_center_abs_le_two (discreteMessage p z.1) i))
+    (fun z : α × β ↦ (le_abs_self _).trans (by
+      rw [abs_mul]; exact mul_le_of_le_one_right (abs_nonneg _) (hg _)))
+  rw [integral_kernelJoint_fst (f := fun a ↦ |(discreteMessage p a).1 i-1|)] at hle
+  rw [integral_kernelJoint_snd (f := fun b ↦ |(discreteMessage q b).1 i-1|), ← discreteMarginal_bind] at hi
+  change _ - (∫ b, |(discreteMessage q b).1 i-1| ∂(discreteMarginal q).toMeasure) = 0 at hi
+  linarith
+
+lemma discrete_degradation_tv (p : Spin → PMF α) (K : α → PMF β) :
+    (∫ m, tvMessage m ∂(discreteProbability (fun i ↦ (p i).bind K)).toMeasure) ≤
+      ∫ m, tvMessage m ∂(discreteProbability p).toMeasure := by
+  rw [integral_discreteProbability _ continuous_tvMessage.measurable,
+    integral_discreteProbability _ continuous_tvMessage.measurable]
+  unfold tvMessage
+  rw [integral_div, integral_div, integral_avg_measure, integral_avg_measure]
+  · exact div_le_div_of_nonneg_right (by
+      dsimp [avg]
+      linarith [discrete_degradation_abs p K 0, discrete_degradation_abs p K 1,
+        discrete_degradation_abs p K 2]) (by norm_num)
+  all_goals intro i; exact pmf_bounded_integrable _ (fun a ↦ by simpa using coord_center_abs_le_two _ i)
+
+lemma advantage_antitone (lam : ℝ) (hlam : Admissible lam) (offspring : PMF ℕ) :
+    Antitone (advantage lam hlam offspring) := by
+  apply antitone_nat_of_succ_le
+  intro ℓ
+  simp only [advantage_eq_integral]
+  have h := discrete_degradation_tv (observationLaw lam hlam offspring ℓ)
+    (observationExpansion lam hlam offspring ℓ)
+  simpa only [observationLaw_degrades] using h
+
+lemma advantage_tendsto (lam : ℝ) (hlam : Admissible lam) (offspring : PMF ℕ) :
+    Tendsto (advantage lam hlam offspring) atTop (𝓝 (⨅ ℓ, advantage lam hlam offspring ℓ)) :=
+  tendsto_atTop_ciInf (advantage_antitone lam hlam offspring)
+    ⟨0, by rintro _ ⟨ℓ,rfl⟩; exact advantage_nonneg lam hlam offspring ℓ⟩
+
+end ThreeState.TreeClauses.Experiment
+
+end 
+
+noncomputable section
+open Set MeasureTheory
+open scoped BigOperators
+namespace ThreeState.TreeClauses.Experiment
+open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
+
+variable {α : Type*} [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α]
+
+lemma integral_discreteMarginal (p : Spin → PMF α) {f : α → ℝ} {B : ℝ} (hB : ∀ a, |f a| ≤ B) :
+    (∫ a, f a ∂(discreteMarginal p).toMeasure) =
+      ((∫ a, f a ∂(p 0).toMeasure)+(∫ a, f a ∂(p 1).toMeasure)+(∫ a, f a ∂(p 2).toMeasure))/3 := by
+  rw [discreteMarginal, integral_pmf_bind _ _ hB, PMF.integral_eq_sum]
+  simp only [Fin.sum_univ_three, uniformSpin_apply, ENNReal.toReal_div,
+    ENNReal.toReal_one, ENNReal.toReal_ofNat, smul_eq_mul]
+  ring
+
+lemma coord_sq_le_tv (m : Message) (i : Spin) : (m.1 i-1)^2 ≤ 12*tvMessage m := by
+  have h := mul_le_mul_of_nonneg_left (coord_center_abs_le_two m i) (abs_nonneg (m.1 i-1))
+  rw [← sq, sq_abs] at h
+  have hs : |m.1 i-1| ≤ ∑ j : Spin, |m.1 j-1| :=
+    Finset.single_le_sum (f := fun j : Spin ↦ |m.1 j-1|) (fun j _ ↦ abs_nonneg _) (Finset.mem_univ i)
+  simp only [Fin.sum_univ_three] at hs
+  dsimp [tvMessage, avg]
+  linarith
+
+omit [Countable α] [MeasurableSingletonClass α] in
+lemma integral_scaled_square (μ : Measure α) {v f : α → ℝ} (r : ℝ)
+    (hv : Integrable (fun a ↦ v a^2) μ) (hf : Integrable (fun a ↦ f a^2) μ)
+    (hc : Integrable (fun a ↦ v a*f a) μ) :
+    (∫ a, (v a-r*f a)^2 ∂μ) = (∫ a, v a^2 ∂μ)-2*r*(∫ a, v a*f a ∂μ)+r^2*(∫ a, f a^2 ∂μ) := by
+  have he (a : α) : (v a-r*f a)^2 = (v a^2-(2*r)*(v a*f a))+r^2*f a^2 := by ring
+  simp_rw [he]
+  rw [integral_add (f := fun a ↦ v a^2-(2*r)*(v a*f a)) (g := fun a ↦ r^2*f a^2)
+    (hv.sub (hc.const_mul (2*r))) (hf.const_mul (r^2))]
+  rw [integral_sub hv (hc.const_mul (2*r))]
+  rw [integral_const_mul, integral_const_mul]
+
+lemma estimator_tv_lower (p : Spin → PMF α) {f : α → ℝ} {B C : ℝ}
+    (hB : ∀ a, |f a| ≤ B) (hC : 0 < C)
+    (hzero : (∫ a, f a ∂(discreteMarginal p).toMeasure) = 0)
+    (hroot : (∫ a, f a ∂(p 0).toMeasure) = 2)
+    (hsecond : (∫ a, f a^2 ∂(discreteMarginal p).toMeasure) ≤ C) :
+    1/(3*C) ≤ ∫ m, tvMessage m ∂(discreteProbability p).toMeasure := by
+  let v : α → ℝ := fun a ↦ (discreteMessage p a).1 0-1
+  have hv (a : α) : |v a| ≤ 2 := coord_center_abs_le_two _ _
+  have hf₂ := pmf_bounded_integrable (discreteMarginal p) (fun a ↦ show |f a^2| ≤ B^2 by
+    rw [abs_of_nonneg (sq_nonneg _)]; nlinarith [hB a, abs_nonneg (f a), sq_abs (f a)])
+  have hv₂ := pmf_bounded_integrable (discreteMarginal p) (fun a ↦ show |v a^2| ≤ (2:ℝ)^2 by
+    rw [abs_of_nonneg (sq_nonneg _)]; nlinarith [hv a, abs_nonneg (v a), sq_abs (v a)])
+  have hvf := pmf_bounded_integrable (discreteMarginal p) (fun a ↦ show |v a*f a| ≤ 2*B by
+    rw [abs_mul]; exact mul_le_mul (hv a) (hB a) (abs_nonneg _) (by norm_num))
+  have hcross : (∫ a, v a*f a ∂(discreteMarginal p).toMeasure) = 2 := by
+    have he (a : α) : v a*f a = (discreteMessage p a).1 0*f a-f a := by dsimp [v]; ring
+    simp_rw [he]
+    rw [integral_sub (pmf_bounded_integrable _ (fun a ↦ show |(discreteMessage p a).1 0*f a| ≤ 3*B by
+      rw [abs_mul, abs_of_nonneg (coordinate_nonneg _ _)]; exact mul_le_mul (coordinate_le_three _ _) (hB a) (abs_nonneg _) (by norm_num)))
+      (pmf_bounded_integrable _ hB), discrete_bayes_integral p 0 hB, hroot, hzero, sub_zero]
+  let r := 2/C
+  have hr : r*C = 2 := div_mul_cancel₀ 2 hC.ne'
+  have h := MeasureTheory.integral_nonneg (μ := (discreteMarginal p).toMeasure)
+    (f := fun a ↦ (v a-r*f a)^2) (fun _ ↦ sq_nonneg _)
+  rw [integral_scaled_square _ r hv₂ hf₂ hvf, hcross] at h
+  have hb := mul_le_mul_of_nonneg_left hsecond (sq_nonneg r)
+  have hr2 : r^2*C = 2*r := by
+    have hh := congrArg (fun x : ℝ ↦ r*x) hr
+    nlinarith only [hh]
+  have hD : 4 ≤ C*(∫ a, v a^2 ∂(discreteMarginal p).toMeasure) := by
+    have hs : 2*r ≤ ∫ a, v a^2 ∂(discreteMarginal p).toMeasure := by
+      nlinarith only [h, hb, hr2]
+    have hh := mul_le_mul_of_nonneg_left hs hC.le
+    nlinarith only [hh, hr]
+  have hT := integral_mono hv₂ (pmf_bounded_integrable (discreteMarginal p)
+    (fun a ↦ show |12*tvMessage (discreteMessage p a)| ≤ 12 by
+      rw [abs_mul, abs_of_nonneg (by norm_num : (0:ℝ) ≤ 12), abs_of_nonneg (tvMessage_nonneg _)];
+      nlinarith [tvMessage_le_one (discreteMessage p a)]))
+    (fun a ↦ coord_sq_le_tv (discreteMessage p a) 0)
+  rw [integral_const_mul, ← integral_discreteProbability p continuous_tvMessage.measurable] at hT
+  apply (div_le_iff₀ (by positivity : 0 < 3*C)).2
+  have hh := mul_le_mul_of_nonneg_left hT hC.le
+  nlinarith only [hD, hh]
+
+end ThreeState.TreeClauses.Experiment
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeStateClauses/Main.lean b/lean/OAI/Probability/ThreeStateClauses/Main.lean
index 50afbe5a7060cc8cec8e968062cf5d8a17d17227..2ae82e84f729678813f3ed15db23cff7c9a1d283 100644
--- a/lean/OAI/Probability/ThreeStateClauses/Main.lean
+++ b/lean/OAI/Probability/ThreeStateClauses/Main.lean
@@ -1,41 +1,41 @@
-import Mathlib
-import OAI.Probability.ThreeStateClauses.Basic
-import OAI.Probability.ThreeStateClauses.RadialAlgebra
-import OAI.Probability.ThreeStateClauses.RadialBounds
-import OAI.Probability.ThreeStateClauses.RadialCalculus
-import OAI.Probability.ThreeStateClauses.PositiveMoments
-import OAI.Probability.ThreeStateClauses.PosteriorLaw
-import OAI.Probability.ThreeStateClauses.DiscreteLaw
-import OAI.Probability.ThreeStateClauses.Products
-import OAI.Probability.ThreeStateClauses.Degradation
-import OAI.Probability.ThreeStateClauses.RegularLaw
-import OAI.Probability.ThreeStateClauses.BranchMoments
-import OAI.Probability.ThreeStateClauses.FiniteUpdate
-import OAI.Probability.ThreeStateClauses.Correlation
-import OAI.Probability.ThreeStateClauses.Projection
-import OAI.Probability.ThreeStateClauses.BranchGrowth
-import OAI.Probability.ThreeStateClauses.CompactLimits
-import OAI.Probability.ThreeStateClauses.ScalarBounds
-import OAI.Probability.ThreeStateClauses.PositiveFixed
-import OAI.Probability.ThreeStateClauses.RegularCritical
-import OAI.Probability.ThreeStateClauses.RegularStatistics
-import OAI.Probability.ThreeStateClauses.InformationLoss
-import OAI.Probability.ThreeStateClauses.RegularReconstruction
-import OAI.Probability.ThreeStateClauses.ObservedMixtures
-import OAI.Probability.ThreeStateClauses.OffspringLaws
-import OAI.Probability.ThreeStateClauses.PoissonMoments
-import OAI.Probability.ThreeStateClauses.PoissonLaw
-import OAI.Probability.ThreeStateClauses.PoissonCalculus
-import OAI.Probability.ThreeStateClauses.PoissonFixed
-import OAI.Probability.ThreeStateClauses.CountableStatistics
-import OAI.Probability.ThreeStateClauses.PoissonReconstruction
-
-namespace OAI
-
-/-! Positive-channel nonreconstruction and both-sign supercritical reconstruction for regular and observed Poisson trees. -/
-
-open scoped BigOperators ENNReal NNReal Topology
-open Filter
-
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeStateClauses.Basic
+import OAI.Probability.ThreeStateClauses.RadialAlgebra
+import OAI.Probability.ThreeStateClauses.RadialBounds
+import OAI.Probability.ThreeStateClauses.RadialCalculus
+import OAI.Probability.ThreeStateClauses.PositiveMoments
+import OAI.Probability.ThreeStateClauses.PosteriorLaw
+import OAI.Probability.ThreeStateClauses.DiscreteLaw
+import OAI.Probability.ThreeStateClauses.Products
+import OAI.Probability.ThreeStateClauses.Degradation
+import OAI.Probability.ThreeStateClauses.RegularLaw
+import OAI.Probability.ThreeStateClauses.BranchMoments
+import OAI.Probability.ThreeStateClauses.FiniteUpdate
+import OAI.Probability.ThreeStateClauses.Correlation
+import OAI.Probability.ThreeStateClauses.Projection
+import OAI.Probability.ThreeStateClauses.BranchGrowth
+import OAI.Probability.ThreeStateClauses.CompactLimits
+import OAI.Probability.ThreeStateClauses.ScalarBounds
+import OAI.Probability.ThreeStateClauses.PositiveFixed
+import OAI.Probability.ThreeStateClauses.RegularCritical
+import OAI.Probability.ThreeStateClauses.RegularStatistics
+import OAI.Probability.ThreeStateClauses.InformationLoss
+import OAI.Probability.ThreeStateClauses.RegularReconstruction
+import OAI.Probability.ThreeStateClauses.ObservedMixtures
+import OAI.Probability.ThreeStateClauses.OffspringLaws
+import OAI.Probability.ThreeStateClauses.PoissonMoments
+import OAI.Probability.ThreeStateClauses.PoissonLaw
+import OAI.Probability.ThreeStateClauses.PoissonCalculus
+import OAI.Probability.ThreeStateClauses.PoissonFixed
+import OAI.Probability.ThreeStateClauses.CountableStatistics
+import OAI.Probability.ThreeStateClauses.PoissonReconstruction
+
+namespace OAI
+
+/-! Positive-channel nonreconstruction and both-sign supercritical reconstruction for regular and observed Poisson trees. -/
+
+open scoped BigOperators ENNReal NNReal Topology
+open Filter
+
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeStateClauses/ObservedMixtures.lean b/lean/OAI/Probability/ThreeStateClauses/ObservedMixtures.lean
index 683ca33490e33aa717a30b084ea4d4916d36620a..095aef591f762a4159951359f04f3864b05b6b6d 100644
--- a/lean/OAI/Probability/ThreeStateClauses/ObservedMixtures.lean
+++ b/lean/OAI/Probability/ThreeStateClauses/ObservedMixtures.lean
@@ -1,215 +1,215 @@
-import Mathlib
-import OAI.Probability.ThreeStateClauses.CompactLimits
-
-namespace OAI
-
-/-! Observed Mixtures. -/
-
-open scoped BigOperators ENNReal NNReal Topology
-open Filter
-noncomputable section
-open Set MeasureTheory Filter
-open scoped BigOperators Topology
-namespace ThreeState.TreeClauses.Experiment
-open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
-
-variable {X : Type*} [MeasurableSpace X]
-
-def mixtureProbability (ρ : PMF ℕ) (P : ℕ → ProbabilityMeasure X) : ProbabilityMeasure X :=
-  ⟨Measure.sum (fun n ↦ ρ n • (P n).toMeasure), ⟨by
-    rw [Measure.sum_apply _ MeasurableSet.univ]
-    simpa only [Measure.smul_apply, measure_univ, smul_eq_mul, mul_one] using ρ.tsum_coe⟩⟩
-
-lemma integral_mixtureProbability (ρ : PMF ℕ) (P : ℕ → ProbabilityMeasure X)
-    {f : X → ℝ} (hf : Integrable f (mixtureProbability ρ P).toMeasure) :
-    (∫ x, f x ∂(mixtureProbability ρ P).toMeasure) =
-      ∑' n, (ρ n).toReal*(∫ x, f x ∂(P n).toMeasure) := by
-  rw [show (mixtureProbability ρ P).toMeasure = Measure.sum (fun n ↦ ρ n • (P n).toMeasure) from rfl,
-    integral_sum_measure hf]
-  simp only [integral_smul_measure, smul_eq_mul]
-
-lemma pmf_hasSum_real (ρ : PMF ℕ) : HasSum (fun n ↦ (ρ n).toReal) 1 := by
-  have hs := ENNReal.summable_toReal ρ.tsum_coe_ne_top
-  convert hs.hasSum using 1
-  rw [← ENNReal.tsum_toReal_eq (fun n ↦ ρ.apply_ne_top n), ρ.tsum_coe]
-  norm_num
-
-lemma mixture_balanced (ρ : PMF ℕ) (Q : ℕ → Law) (i : Fin 3) :
-    (∫ m, m.1 i ∂(mixtureProbability ρ (fun n ↦ (Q n).probability)).toMeasure) = 1 := by
-  rw [integral_mixtureProbability _ _ (compact_integrable (continuous_coordinate i))]
-  simp only [Law.balanced, mul_one]
-  exact (pmf_hasSum_real ρ).tsum_eq
-
-lemma mixture_symmetric (ρ : PMF ℕ) (Q : ℕ → Law) (σ : Equiv.Perm (Fin 3)) :
-    MeasurePreserving (permute σ) (mixtureProbability ρ (fun n ↦ (Q n).probability)).toMeasure
-      (mixtureProbability ρ (fun n ↦ (Q n).probability)).toMeasure := by
-  refine ⟨(continuous_permute σ).measurable, ?_⟩
-  change Measure.map _ (Measure.sum _) = Measure.sum _
-  rw [Measure.map_sum (continuous_permute σ).measurable.aemeasurable]
-  congr 1; funext n
-  rw [Measure.map_smul _ (continuous_permute σ).measurable.aemeasurable,
-    (Q n).symmetric σ |>.map_eq]
-
-def mixtureLaw (ρ : PMF ℕ) (Q : ℕ → Law) : Law where
-  probability := mixtureProbability ρ (fun n ↦ (Q n).probability)
-  balanced := mixture_balanced ρ Q
-  symmetric := mixture_symmetric ρ Q
-
-def offspringLaw (ρ : PMF ℕ) (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) : Law :=
-  mixtureLaw ρ (finiteLaw Q hl₀ hl₁)
-
-lemma mixture_integrable (ρ : PMF ℕ) (P : ℕ → ProbabilityMeasure X) {f : X → ℝ}
-    (hf : ∀ n, Integrable f (P n).toMeasure)
-    (hs : Summable (fun n ↦ (ρ n).toReal*(∫ x, |f x| ∂(P n).toMeasure))) :
-    Integrable f (mixtureProbability ρ P).toMeasure := by
-  apply integrable_sum_measure
-  · intro n; exact (hf n).smul_measure (ρ.apply_ne_top n)
-  · simpa only [integral_smul_measure, smul_eq_mul, Real.norm_eq_abs] using hs
-
-variable [TopologicalSpace X] [CompactSpace X] [BorelSpace X]
-
-lemma mixtureProbability_tendsto {ι : Type*} {F : Filter ι} (ρ : PMF ℕ)
-    {P : ι → ℕ → ProbabilityMeasure X} {R : ℕ → ProbabilityMeasure X}
-    (hP : ∀ n, Tendsto (fun i ↦ P i n) F (𝓝 (R n))) :
-    Tendsto (fun i ↦ mixtureProbability ρ (P i)) F (𝓝 (mixtureProbability ρ R)) := by
-  apply ProbabilityMeasure.tendsto_iff_forall_integral_tendsto.mpr
-  intro f
-  simp_rw [integral_mixtureProbability _ _ (compact_integrable f.continuous)]
-  apply tendsto_tsum_of_dominated_convergence ((pmf_hasSum_real ρ).summable.mul_right ‖f‖)
-  · intro n
-    exact tendsto_const_nhds.mul ((probability_continuous_integral f.continuous).tendsto (R n) |>.comp (hP n))
-  · exact Filter.Eventually.of_forall (fun i n ↦ by
-      rw [norm_mul, Real.norm_of_nonneg ENNReal.toReal_nonneg]
-      apply mul_le_mul_of_nonneg_left _ ENNReal.toReal_nonneg
-      exact (norm_integral_le_integral_norm _).trans (by
-        exact (integral_mono (compact_integrable f.continuous.norm) (integrable_const _)
-          (fun x ↦ f.norm_coe_le_norm x)).trans_eq (by simp)))
-
-lemma offspringLaw_tendsto {ι : Type*} {F : Filter ι} (ρ : PMF ℕ) {Q : ι → Law} {R : Law}
-    {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
-    (hQ : Tendsto (fun i ↦ (Q i).probability) F (𝓝 R.probability)) :
-    Tendsto (fun i ↦ (offspringLaw ρ (Q i) hl₀ hl₁).probability) F
-      (𝓝 (offspringLaw ρ R hl₀ hl₁).probability) :=
-  mixtureProbability_tendsto ρ (fun n ↦ finiteProbability_tendsto hQ hl₀ hl₁ n)
-
-end ThreeState.TreeClauses.Experiment
-
-end 
-
-noncomputable section
-open Set MeasureTheory
-open scoped BigOperators ENNReal
-namespace ThreeState.TreeClauses.Experiment
-open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
-
-variable {α : ℕ → Type*}
-
-def sigmaPMF (ρ : PMF ℕ) (p : ∀ n, PMF (α n)) : PMF (Sigma α) :=
-  ⟨fun z ↦ ρ z.1*p z.1 z.2, ENNReal.summable.hasSum_iff.2 (by
-    rw [ENNReal.tsum_sigma']
-    simp only [ENNReal.tsum_mul_left]
-    have hs (n : ℕ) : (∑' a, p n a) = 1 := (p n).tsum_coe
-    simp only [hs, mul_one]
-    exact ρ.tsum_coe)⟩
-
-@[simp] lemma sigmaPMF_apply (ρ : PMF ℕ) (p : ∀ n, PMF (α n)) (n : ℕ) (a : α n) :
-    sigmaPMF ρ p ⟨n,a⟩ = ρ n*p n a := rfl
-
-lemma sigmaPMF_eq_bind (ρ : PMF ℕ) (p : ∀ n, PMF (α n)) :
-    sigmaPMF ρ p = ρ.bind (fun n ↦ (p n).map (Sigma.mk n)) := by
-  classical
-  apply PMF.ext
-  rintro ⟨n,a⟩
-  rw [PMF.bind_apply, tsum_eq_single n]
-  · simp [PMF.map_apply]
-  · intro m hm
-    have hn : n ≠ m := Ne.symm hm
-    simp [PMF.map_apply, hn]
-
-lemma sigmaPMF_bind {β : Type*} (ρ : PMF ℕ) (p : ∀ n, PMF (α n))
-    (K : Sigma α → PMF β) :
-    (sigmaPMF ρ p).bind K = ρ.bind (fun n ↦ (p n).bind (fun a ↦ K ⟨n,a⟩)) := by
-  rw [sigmaPMF_eq_bind, PMF.bind_bind]
-  simp only [PMF.bind_map, Function.comp_def]
-
-lemma sigmaPMF_map {β : Type*} (ρ : PMF ℕ) (p : ∀ n, PMF (α n))
-    (T : Sigma α → β) :
-    (sigmaPMF ρ p).map T = ρ.bind (fun n ↦ (p n).map (fun a ↦ T ⟨n,a⟩)) :=
-  sigmaPMF_bind ρ p (fun z ↦ PMF.pure (T z))
-
-lemma sigmaPMF_bind_components {β : ℕ → Type*} (ρ : PMF ℕ) (p : ∀ n, PMF (α n))
-    (K : ∀ n, α n → PMF (β n)) :
-    (sigmaPMF ρ p).bind (fun z ↦ (K z.1 z.2).map (Sigma.mk z.1)) =
-      sigmaPMF ρ (fun n ↦ (p n).bind (K n)) := by
-  rw [sigmaPMF_bind, sigmaPMF_eq_bind]
-  simp only [PMF.map_bind]
-
-lemma discreteSigma_marginal (ρ : PMF ℕ) (p : ∀ n, Spin → PMF (α n)) :
-    discreteMarginal (fun i ↦ sigmaPMF ρ (fun n ↦ p n i)) =
-      sigmaPMF ρ (fun n ↦ discreteMarginal (p n)) := by
-  apply PMF.ext
-  rintro ⟨n,a⟩
-  simp only [discreteMarginal_apply, sigmaPMF_apply, div_eq_mul_inv]
-  ring
-
-lemma discreteSigma_weight (ρ : PMF ℕ) (p : ∀ n, Spin → PMF (α n)) (n : ℕ) (a : α n) :
-    discreteWeight (fun i ↦ sigmaPMF ρ (fun n ↦ p n i)) ⟨n,a⟩ =
-      (ρ n).toReal*discreteWeight (p n) a := by
-  simp only [discreteWeight, sigmaPMF_apply, ENNReal.toReal_mul]
-  exact avg_mul _ _
-
-lemma discreteSigma_message (ρ : PMF ℕ) (p : ∀ n, Spin → PMF (α n))
-    (n : ℕ) (a : α n) (hρ : (ρ n).toReal ≠ 0) :
-    discreteMessage (fun i ↦ sigmaPMF ρ (fun n ↦ p n i)) ⟨n,a⟩ = discreteMessage (p n) a := by
-  by_cases ha : discreteWeight (p n) a = 0
-  · simp only [discreteMessage, discreteSigma_weight, ha, mul_zero, dite_true]
-  · apply Subtype.ext
-    funext i
-    rw [discrete_coord (by rw [discreteSigma_weight]; exact mul_ne_zero hρ ha),
-      discrete_coord ha, sigmaPMF_apply, ENNReal.toReal_mul, discreteSigma_weight]
-    exact mul_div_mul_left _ _ hρ
-
-section Measures
-variable [∀ n, Countable (α n)] [∀ n, MeasurableSpace (α n)]
-  [∀ n, MeasurableSingletonClass (α n)]
-
-instance sigmaMeasurableSingleton : MeasurableSingletonClass (Sigma α) := by
-  constructor
-  intro z
-  apply MeasurableSpace.measurableSet_iInf.mpr
-  intro n
-  change MeasurableSet ((Sigma.mk n) ⁻¹' {z})
-  exact Set.Countable.measurableSet (Set.to_countable _)
-
-lemma integral_sigmaPMF (ρ : PMF ℕ) (p : ∀ n, PMF (α n)) {f : Sigma α → ℝ} {B : ℝ}
-    (hf : ∀ a, |f a| ≤ B) :
-    (∫ a, f a ∂(sigmaPMF ρ p).toMeasure) =
-      ∑' n, (ρ n).toReal*(∫ a, f ⟨n,a⟩ ∂(p n).toMeasure) := by
-  rw [sigmaPMF_eq_bind, pmf_measure_bind, integral_sum_measure]
-  · simp only [integral_smul_measure, smul_eq_mul, integral_pmf_map]
-  · rw [← pmf_measure_bind]
-    exact pmf_bounded_integrable _ hf
-
-lemma discreteSigma_probability (ρ : PMF ℕ) (p : ∀ n, Spin → PMF (α n)) :
-    discreteProbability (fun i ↦ sigmaPMF ρ (fun n ↦ p n i)) =
-      mixtureProbability ρ (fun n ↦ discreteProbability (p n)) := by
-  apply probability_ext_continuous
-  intro f hf
-  obtain ⟨B,hB⟩ := isCompact_univ.exists_bound_of_continuousOn hf.continuousOn
-  have hbf (m : Message) : |f m| ≤ B := by simpa only [Real.norm_eq_abs] using hB m (mem_univ m)
-  rw [integral_discreteProbability _ hf.measurable, discreteSigma_marginal,
-    integral_sigmaPMF ρ _ (fun z ↦ hbf _),
-    integral_mixtureProbability _ _ (compact_integrable hf)]
-  apply tsum_congr
-  intro n
-  rw [integral_discreteProbability _ hf.measurable]
-  by_cases hρ : (ρ n).toReal = 0
-  · simp [hρ]
-  · simp_rw [discreteSigma_message ρ p n _ hρ]
-
-end Measures
-end ThreeState.TreeClauses.Experiment
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeStateClauses.CompactLimits
+
+namespace OAI
+
+/-! Observed Mixtures. -/
+
+open scoped BigOperators ENNReal NNReal Topology
+open Filter
+noncomputable section
+open Set MeasureTheory Filter
+open scoped BigOperators Topology
+namespace ThreeState.TreeClauses.Experiment
+open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
+
+variable {X : Type*} [MeasurableSpace X]
+
+def mixtureProbability (ρ : PMF ℕ) (P : ℕ → ProbabilityMeasure X) : ProbabilityMeasure X :=
+  ⟨Measure.sum (fun n ↦ ρ n • (P n).toMeasure), ⟨by
+    rw [Measure.sum_apply _ MeasurableSet.univ]
+    simpa only [Measure.smul_apply, measure_univ, smul_eq_mul, mul_one] using ρ.tsum_coe⟩⟩
+
+lemma integral_mixtureProbability (ρ : PMF ℕ) (P : ℕ → ProbabilityMeasure X)
+    {f : X → ℝ} (hf : Integrable f (mixtureProbability ρ P).toMeasure) :
+    (∫ x, f x ∂(mixtureProbability ρ P).toMeasure) =
+      ∑' n, (ρ n).toReal*(∫ x, f x ∂(P n).toMeasure) := by
+  rw [show (mixtureProbability ρ P).toMeasure = Measure.sum (fun n ↦ ρ n • (P n).toMeasure) from rfl,
+    integral_sum_measure hf]
+  simp only [integral_smul_measure, smul_eq_mul]
+
+lemma pmf_hasSum_real (ρ : PMF ℕ) : HasSum (fun n ↦ (ρ n).toReal) 1 := by
+  have hs := ENNReal.summable_toReal ρ.tsum_coe_ne_top
+  convert hs.hasSum using 1
+  rw [← ENNReal.tsum_toReal_eq (fun n ↦ ρ.apply_ne_top n), ρ.tsum_coe]
+  norm_num
+
+lemma mixture_balanced (ρ : PMF ℕ) (Q : ℕ → Law) (i : Fin 3) :
+    (∫ m, m.1 i ∂(mixtureProbability ρ (fun n ↦ (Q n).probability)).toMeasure) = 1 := by
+  rw [integral_mixtureProbability _ _ (compact_integrable (continuous_coordinate i))]
+  simp only [Law.balanced, mul_one]
+  exact (pmf_hasSum_real ρ).tsum_eq
+
+lemma mixture_symmetric (ρ : PMF ℕ) (Q : ℕ → Law) (σ : Equiv.Perm (Fin 3)) :
+    MeasurePreserving (permute σ) (mixtureProbability ρ (fun n ↦ (Q n).probability)).toMeasure
+      (mixtureProbability ρ (fun n ↦ (Q n).probability)).toMeasure := by
+  refine ⟨(continuous_permute σ).measurable, ?_⟩
+  change Measure.map _ (Measure.sum _) = Measure.sum _
+  rw [Measure.map_sum (continuous_permute σ).measurable.aemeasurable]
+  congr 1; funext n
+  rw [Measure.map_smul _ (continuous_permute σ).measurable.aemeasurable,
+    (Q n).symmetric σ |>.map_eq]
+
+def mixtureLaw (ρ : PMF ℕ) (Q : ℕ → Law) : Law where
+  probability := mixtureProbability ρ (fun n ↦ (Q n).probability)
+  balanced := mixture_balanced ρ Q
+  symmetric := mixture_symmetric ρ Q
+
+def offspringLaw (ρ : PMF ℕ) (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) : Law :=
+  mixtureLaw ρ (finiteLaw Q hl₀ hl₁)
+
+lemma mixture_integrable (ρ : PMF ℕ) (P : ℕ → ProbabilityMeasure X) {f : X → ℝ}
+    (hf : ∀ n, Integrable f (P n).toMeasure)
+    (hs : Summable (fun n ↦ (ρ n).toReal*(∫ x, |f x| ∂(P n).toMeasure))) :
+    Integrable f (mixtureProbability ρ P).toMeasure := by
+  apply integrable_sum_measure
+  · intro n; exact (hf n).smul_measure (ρ.apply_ne_top n)
+  · simpa only [integral_smul_measure, smul_eq_mul, Real.norm_eq_abs] using hs
+
+variable [TopologicalSpace X] [CompactSpace X] [BorelSpace X]
+
+lemma mixtureProbability_tendsto {ι : Type*} {F : Filter ι} (ρ : PMF ℕ)
+    {P : ι → ℕ → ProbabilityMeasure X} {R : ℕ → ProbabilityMeasure X}
+    (hP : ∀ n, Tendsto (fun i ↦ P i n) F (𝓝 (R n))) :
+    Tendsto (fun i ↦ mixtureProbability ρ (P i)) F (𝓝 (mixtureProbability ρ R)) := by
+  apply ProbabilityMeasure.tendsto_iff_forall_integral_tendsto.mpr
+  intro f
+  simp_rw [integral_mixtureProbability _ _ (compact_integrable f.continuous)]
+  apply tendsto_tsum_of_dominated_convergence ((pmf_hasSum_real ρ).summable.mul_right ‖f‖)
+  · intro n
+    exact tendsto_const_nhds.mul ((probability_continuous_integral f.continuous).tendsto (R n) |>.comp (hP n))
+  · exact Filter.Eventually.of_forall (fun i n ↦ by
+      rw [norm_mul, Real.norm_of_nonneg ENNReal.toReal_nonneg]
+      apply mul_le_mul_of_nonneg_left _ ENNReal.toReal_nonneg
+      exact (norm_integral_le_integral_norm _).trans (by
+        exact (integral_mono (compact_integrable f.continuous.norm) (integrable_const _)
+          (fun x ↦ f.norm_coe_le_norm x)).trans_eq (by simp)))
+
+lemma offspringLaw_tendsto {ι : Type*} {F : Filter ι} (ρ : PMF ℕ) {Q : ι → Law} {R : Law}
+    {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
+    (hQ : Tendsto (fun i ↦ (Q i).probability) F (𝓝 R.probability)) :
+    Tendsto (fun i ↦ (offspringLaw ρ (Q i) hl₀ hl₁).probability) F
+      (𝓝 (offspringLaw ρ R hl₀ hl₁).probability) :=
+  mixtureProbability_tendsto ρ (fun n ↦ finiteProbability_tendsto hQ hl₀ hl₁ n)
+
+end ThreeState.TreeClauses.Experiment
+
+end 
+
+noncomputable section
+open Set MeasureTheory
+open scoped BigOperators ENNReal
+namespace ThreeState.TreeClauses.Experiment
+open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
+
+variable {α : ℕ → Type*}
+
+def sigmaPMF (ρ : PMF ℕ) (p : ∀ n, PMF (α n)) : PMF (Sigma α) :=
+  ⟨fun z ↦ ρ z.1*p z.1 z.2, ENNReal.summable.hasSum_iff.2 (by
+    rw [ENNReal.tsum_sigma']
+    simp only [ENNReal.tsum_mul_left]
+    have hs (n : ℕ) : (∑' a, p n a) = 1 := (p n).tsum_coe
+    simp only [hs, mul_one]
+    exact ρ.tsum_coe)⟩
+
+@[simp] lemma sigmaPMF_apply (ρ : PMF ℕ) (p : ∀ n, PMF (α n)) (n : ℕ) (a : α n) :
+    sigmaPMF ρ p ⟨n,a⟩ = ρ n*p n a := rfl
+
+lemma sigmaPMF_eq_bind (ρ : PMF ℕ) (p : ∀ n, PMF (α n)) :
+    sigmaPMF ρ p = ρ.bind (fun n ↦ (p n).map (Sigma.mk n)) := by
+  classical
+  apply PMF.ext
+  rintro ⟨n,a⟩
+  rw [PMF.bind_apply, tsum_eq_single n]
+  · simp [PMF.map_apply]
+  · intro m hm
+    have hn : n ≠ m := Ne.symm hm
+    simp [PMF.map_apply, hn]
+
+lemma sigmaPMF_bind {β : Type*} (ρ : PMF ℕ) (p : ∀ n, PMF (α n))
+    (K : Sigma α → PMF β) :
+    (sigmaPMF ρ p).bind K = ρ.bind (fun n ↦ (p n).bind (fun a ↦ K ⟨n,a⟩)) := by
+  rw [sigmaPMF_eq_bind, PMF.bind_bind]
+  simp only [PMF.bind_map, Function.comp_def]
+
+lemma sigmaPMF_map {β : Type*} (ρ : PMF ℕ) (p : ∀ n, PMF (α n))
+    (T : Sigma α → β) :
+    (sigmaPMF ρ p).map T = ρ.bind (fun n ↦ (p n).map (fun a ↦ T ⟨n,a⟩)) :=
+  sigmaPMF_bind ρ p (fun z ↦ PMF.pure (T z))
+
+lemma sigmaPMF_bind_components {β : ℕ → Type*} (ρ : PMF ℕ) (p : ∀ n, PMF (α n))
+    (K : ∀ n, α n → PMF (β n)) :
+    (sigmaPMF ρ p).bind (fun z ↦ (K z.1 z.2).map (Sigma.mk z.1)) =
+      sigmaPMF ρ (fun n ↦ (p n).bind (K n)) := by
+  rw [sigmaPMF_bind, sigmaPMF_eq_bind]
+  simp only [PMF.map_bind]
+
+lemma discreteSigma_marginal (ρ : PMF ℕ) (p : ∀ n, Spin → PMF (α n)) :
+    discreteMarginal (fun i ↦ sigmaPMF ρ (fun n ↦ p n i)) =
+      sigmaPMF ρ (fun n ↦ discreteMarginal (p n)) := by
+  apply PMF.ext
+  rintro ⟨n,a⟩
+  simp only [discreteMarginal_apply, sigmaPMF_apply, div_eq_mul_inv]
+  ring
+
+lemma discreteSigma_weight (ρ : PMF ℕ) (p : ∀ n, Spin → PMF (α n)) (n : ℕ) (a : α n) :
+    discreteWeight (fun i ↦ sigmaPMF ρ (fun n ↦ p n i)) ⟨n,a⟩ =
+      (ρ n).toReal*discreteWeight (p n) a := by
+  simp only [discreteWeight, sigmaPMF_apply, ENNReal.toReal_mul]
+  exact avg_mul _ _
+
+lemma discreteSigma_message (ρ : PMF ℕ) (p : ∀ n, Spin → PMF (α n))
+    (n : ℕ) (a : α n) (hρ : (ρ n).toReal ≠ 0) :
+    discreteMessage (fun i ↦ sigmaPMF ρ (fun n ↦ p n i)) ⟨n,a⟩ = discreteMessage (p n) a := by
+  by_cases ha : discreteWeight (p n) a = 0
+  · simp only [discreteMessage, discreteSigma_weight, ha, mul_zero, dite_true]
+  · apply Subtype.ext
+    funext i
+    rw [discrete_coord (by rw [discreteSigma_weight]; exact mul_ne_zero hρ ha),
+      discrete_coord ha, sigmaPMF_apply, ENNReal.toReal_mul, discreteSigma_weight]
+    exact mul_div_mul_left _ _ hρ
+
+section Measures
+variable [∀ n, Countable (α n)] [∀ n, MeasurableSpace (α n)]
+  [∀ n, MeasurableSingletonClass (α n)]
+
+instance sigmaMeasurableSingleton : MeasurableSingletonClass (Sigma α) := by
+  constructor
+  intro z
+  apply MeasurableSpace.measurableSet_iInf.mpr
+  intro n
+  change MeasurableSet ((Sigma.mk n) ⁻¹' {z})
+  exact Set.Countable.measurableSet (Set.to_countable _)
+
+lemma integral_sigmaPMF (ρ : PMF ℕ) (p : ∀ n, PMF (α n)) {f : Sigma α → ℝ} {B : ℝ}
+    (hf : ∀ a, |f a| ≤ B) :
+    (∫ a, f a ∂(sigmaPMF ρ p).toMeasure) =
+      ∑' n, (ρ n).toReal*(∫ a, f ⟨n,a⟩ ∂(p n).toMeasure) := by
+  rw [sigmaPMF_eq_bind, pmf_measure_bind, integral_sum_measure]
+  · simp only [integral_smul_measure, smul_eq_mul, integral_pmf_map]
+  · rw [← pmf_measure_bind]
+    exact pmf_bounded_integrable _ hf
+
+lemma discreteSigma_probability (ρ : PMF ℕ) (p : ∀ n, Spin → PMF (α n)) :
+    discreteProbability (fun i ↦ sigmaPMF ρ (fun n ↦ p n i)) =
+      mixtureProbability ρ (fun n ↦ discreteProbability (p n)) := by
+  apply probability_ext_continuous
+  intro f hf
+  obtain ⟨B,hB⟩ := isCompact_univ.exists_bound_of_continuousOn hf.continuousOn
+  have hbf (m : Message) : |f m| ≤ B := by simpa only [Real.norm_eq_abs] using hB m (mem_univ m)
+  rw [integral_discreteProbability _ hf.measurable, discreteSigma_marginal,
+    integral_sigmaPMF ρ _ (fun z ↦ hbf _),
+    integral_mixtureProbability _ _ (compact_integrable hf)]
+  apply tsum_congr
+  intro n
+  rw [integral_discreteProbability _ hf.measurable]
+  by_cases hρ : (ρ n).toReal = 0
+  · simp [hρ]
+  · simp_rw [discreteSigma_message ρ p n _ hρ]
+
+end Measures
+end ThreeState.TreeClauses.Experiment
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeStateClauses/OffspringLaws.lean b/lean/OAI/Probability/ThreeStateClauses/OffspringLaws.lean
index 9a6ca26c1dae11d9de6e7c1355bb281b893963e3..f501f2de3e497aaf440a37d97671b5901e94d8ae 100644
--- a/lean/OAI/Probability/ThreeStateClauses/OffspringLaws.lean
+++ b/lean/OAI/Probability/ThreeStateClauses/OffspringLaws.lean
@@ -1,256 +1,256 @@
-import Mathlib
-import OAI.Probability.ThreeStateClauses.RegularCritical
-import OAI.Probability.ThreeStateClauses.ObservedMixtures
-
-namespace OAI
-
-/-! Offspring Laws. -/
-
-open scoped BigOperators ENNReal NNReal Topology
-open Filter
-noncomputable section
-open Set MeasureTheory
-open scoped BigOperators ENNReal
-namespace ThreeState.TreeClauses.Tree
-open ThreeState.TreeClauses.Experiment ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
-
-@[reducible] def OrderedObservation : ℕ → Type
-  | 0 => Spin
-  | ℓ+1 => Σ n : ℕ, Fin n → OrderedObservation ℓ
-
-instance orderedObservationCountable : (ℓ : ℕ) → Countable (OrderedObservation ℓ)
-  | 0 => inferInstanceAs (Countable Spin)
-  | ℓ+1 => letI := orderedObservationCountable ℓ
-           inferInstanceAs (Countable (Σ n : ℕ, Fin n → OrderedObservation ℓ))
-
-instance orderedObservationMeasurable : (ℓ : ℕ) → MeasurableSpace (OrderedObservation ℓ)
-  | 0 => inferInstanceAs (MeasurableSpace Spin)
-  | ℓ+1 => letI := orderedObservationMeasurable ℓ
-           inferInstanceAs (MeasurableSpace (Σ n : ℕ, Fin n → OrderedObservation ℓ))
-
-instance orderedObservationSingleton : (ℓ : ℕ) → MeasurableSingletonClass (OrderedObservation ℓ)
-  | 0 => inferInstanceAs (MeasurableSingletonClass Spin)
-  | ℓ+1 => letI := orderedObservationSingleton ℓ
-           inferInstanceAs (MeasurableSingletonClass (Σ n : ℕ, Fin n → OrderedObservation ℓ))
-
-def orderedLaw (ρ : PMF ℕ) (lam : ℝ) (hlam : Admissible lam) :
-    (ℓ : ℕ) → Spin → PMF (OrderedObservation ℓ)
-  | 0, i => PMF.pure i
-  | ℓ+1, i => sigmaPMF ρ (fun n ↦ productPMF (fun _ : Fin n ↦
-      (channel lam hlam i).bind (orderedLaw ρ lam hlam ℓ)))
-
-def forgetOrdered : (ℓ : ℕ) → OrderedObservation ℓ → Observation ℓ
-  | 0, i => i
-  | ℓ+1, ⟨_,v⟩ => (List.ofFn (fun j ↦ forgetOrdered ℓ (v j)) : Multiset (Observation ℓ))
-
-lemma orderedLaw_forget (ρ : PMF ℕ) (lam : ℝ) (hlam : Admissible lam) (ℓ : ℕ) (i : Spin) :
-    (orderedLaw ρ lam hlam ℓ i).map (forgetOrdered ℓ) = observationLaw lam hlam ρ ℓ i := by
-  induction ℓ generalizing i with
-  | zero => exact PMF.map_id _
-  | succ ℓ ih =>
-    rw [orderedLaw, sigmaPMF_map]
-    change ρ.bind _ = ρ.bind _
-    congr 1
-    funext n
-    change (productPMF (fun _ : Fin n ↦ (channel lam hlam i).bind (orderedLaw ρ lam hlam ℓ))).map
-      ((fun xs : List (Observation ℓ) ↦ (xs : Multiset (Observation ℓ))) ∘
-        List.ofFn ∘ (fun v j ↦ forgetOrdered ℓ (v j))) = _
-    rw [← PMF.map_comp, ← PMF.map_comp, productPMF_map]
-    simp_rw [PMF.map_bind, ih]
-    rw [productPMF_list]
-    exact (PMF.map_id ((iidList ((channel lam hlam i).bind
-      (observationLaw lam hlam ρ ℓ)) n).map (fun xs : List (Observation ℓ) ↦ (xs : Multiset (Observation ℓ))))).symm
-
-def orderedExpansion (ρ : PMF ℕ) (lam : ℝ) (hlam : Admissible lam) :
-    (ℓ : ℕ) → OrderedObservation ℓ → PMF (OrderedObservation (ℓ+1))
-  | 0, i => sigmaPMF ρ (fun n ↦ productPMF (fun _ : Fin n ↦ channel lam hlam i))
-  | ℓ+1, ⟨n,v⟩ => (productPMF (fun j ↦ orderedExpansion ρ lam hlam ℓ (v j))).map (Sigma.mk n)
-
-lemma orderedLaw_degrades (ρ : PMF ℕ) (lam : ℝ) (hlam : Admissible lam) (ℓ : ℕ) (i : Spin) :
-    (orderedLaw ρ lam hlam ℓ i).bind (orderedExpansion ρ lam hlam ℓ) =
-      orderedLaw ρ lam hlam (ℓ+1) i := by
-  induction ℓ generalizing i with
-  | zero =>
-    change (PMF.pure i).bind _ = sigmaPMF ρ (fun n ↦
-      productPMF (fun _ : Fin n ↦ (channel lam hlam i).bind PMF.pure))
-    rw [PMF.pure_bind]
-    simp only [PMF.bind_pure]
-    rfl
-  | succ ℓ ih =>
-    change (sigmaPMF ρ (fun n ↦ productPMF (fun _ : Fin n ↦
-      (channel lam hlam i).bind (orderedLaw ρ lam hlam ℓ)))).bind
-      (fun z ↦ (productPMF (fun j ↦ orderedExpansion ρ lam hlam ℓ (z.2 j))).map (Sigma.mk (β := fun n ↦ Fin n → OrderedObservation (ℓ+1)) z.1)) = _
-    rw [sigmaPMF_bind_components ρ _ (fun n (v : Fin n → OrderedObservation ℓ) ↦
-      productPMF (fun j ↦ orderedExpansion ρ lam hlam ℓ (v j)))]
-    simp only [productPMF_bind, PMF.bind_bind]
-    simp_rw [ih]
-    rfl
-
-def offspringPosteriorLaw (ρ : PMF ℕ) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) : ℕ → Law
-  | 0 => rootLaw
-  | ℓ+1 => offspringLaw ρ (offspringPosteriorLaw ρ hl₀ hl₁ ℓ) hl₀ hl₁
-
-lemma offspringPosteriorLaw_probability (ρ : PMF ℕ) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
-    (hlam : Admissible lam) (ℓ : ℕ) :
-    (offspringPosteriorLaw ρ hl₀ hl₁ ℓ).probability = discreteProbability (orderedLaw ρ lam hlam ℓ) := by
-  induction ℓ with
-  | zero => rfl
-  | succ ℓ ih =>
-    change mixtureProbability ρ (fun n ↦ finiteProbability (offspringPosteriorLaw ρ hl₀ hl₁ ℓ) hl₀ hl₁ n) =
-      discreteProbability (fun i ↦ sigmaPMF ρ (fun n ↦ discreteBranchLaw (orderedLaw ρ lam hlam ℓ) lam hlam n i))
-    rw [discreteSigma_probability]
-    congr 1
-    funext n
-    exact (discreteBranchLaw_probability _ _ ih hl₀ hl₁ hlam n).symm
-
-end ThreeState.TreeClauses.Tree
-
-end 
-
-noncomputable section
-open Set MeasureTheory
-open scoped BigOperators ENNReal
-namespace ThreeState.TreeClauses.Tree
-open ThreeState.TreeClauses.Experiment
-
-lemma sigmaPMF_map_components {α β : ℕ → Type*} (ρ : PMF ℕ) (p : ∀ n, PMF (α n))
-    (f : ∀ n, α n → β n) :
-    (sigmaPMF ρ p).map (fun z ↦ ⟨z.1,f z.1 z.2⟩) = sigmaPMF ρ (fun n ↦ (p n).map (f n)) := by
-  rw [sigmaPMF_map, sigmaPMF_eq_bind]
-  congr 1
-  funext n
-  rw [PMF.map_comp]
-  rfl
-
-def relabelOrdered (σ : Equiv.Perm Spin) : (ℓ : ℕ) → OrderedObservation ℓ → OrderedObservation ℓ
-  | 0, i => σ i
-  | ℓ+1, ⟨n,v⟩ => ⟨n,fun j ↦ relabelOrdered σ ℓ (v j)⟩
-
-lemma orderedLaw_permute (ρ : PMF ℕ) (lam : ℝ) (hlam : Admissible lam) (σ : Equiv.Perm Spin)
-    (ℓ : ℕ) (i : Spin) :
-    (orderedLaw ρ lam hlam ℓ i).map (relabelOrdered σ ℓ) = orderedLaw ρ lam hlam ℓ (σ i) := by
-  induction ℓ generalizing i with
-  | zero => exact PMF.pure_bind _ _
-  | succ ℓ ih =>
-    change (sigmaPMF ρ (fun n ↦ productPMF (fun _ : Fin n ↦
-      (channel lam hlam i).bind (orderedLaw ρ lam hlam ℓ)))).map
-      (fun z ↦ Sigma.mk (β := fun n ↦ Fin n → OrderedObservation ℓ) z.1
-        (fun j ↦ relabelOrdered σ ℓ (z.2 j))) = _
-    rw [sigmaPMF_map_components ρ _ (fun n (v : Fin n → OrderedObservation ℓ) j ↦ relabelOrdered σ ℓ (v j))]
-    simp_rw [productPMF_map, PMF.map_bind, ih]
-    have he : (channel lam hlam i).bind (fun j ↦ orderedLaw ρ lam hlam ℓ (σ j)) =
-        (channel lam hlam (σ i)).bind (orderedLaw ρ lam hlam ℓ) := by
-      rw [← channel_permute hlam σ i, PMF.bind_map]
-      rfl
-    simp_rw [he]
-    rfl
-
-def orderedNoise (c : ℝ) (hc : Admissible c) :
-    (ℓ : ℕ) → OrderedObservation ℓ → PMF (OrderedObservation ℓ)
-  | 0, i => PMF.pure i
-  | ℓ+1, ⟨n,v⟩ => (productPMF (fun j ↦
-      (channel c hc 0).bind (fun s ↦ (orderedNoise c hc ℓ (v j)).map
-        (relabelOrdered (Equiv.addRight s) ℓ)))).map (Sigma.mk n)
-
-lemma orderedLaw_noise (ρ : PMF ℕ) (lam c : ℝ) (hlam : Admissible lam) (hc : Admissible c)
-    (hprod : Admissible (lam*c)) (ℓ : ℕ) (i : Spin) :
-    (orderedLaw ρ lam hlam ℓ i).bind (orderedNoise c hc ℓ) = orderedLaw ρ (lam*c) hprod ℓ i := by
-  induction ℓ generalizing i with
-  | zero => exact PMF.pure_bind _ _
-  | succ ℓ ih =>
-    change (sigmaPMF ρ (fun n ↦ productPMF (fun _ : Fin n ↦
-      (channel lam hlam i).bind (orderedLaw ρ lam hlam ℓ)))).bind
-      (fun z ↦ (productPMF (fun j ↦ (channel c hc 0).bind (fun s ↦
-        (orderedNoise c hc ℓ (z.2 j)).map (relabelOrdered (Equiv.addRight s) ℓ)) )).map (Sigma.mk (β := fun n ↦ Fin n → OrderedObservation ℓ) z.1)) = _
-    rw [sigmaPMF_bind_components ρ _ (fun n (v : Fin n → OrderedObservation ℓ) ↦
-      productPMF (fun j ↦ (channel c hc 0).bind (fun s ↦
-        (orderedNoise c hc ℓ (v j)).map (relabelOrdered (Equiv.addRight s) ℓ))))]
-    change sigmaPMF ρ _ = sigmaPMF ρ _
-    congr 1
-    funext n
-    rw [productPMF_bind (fun _ : Fin n ↦ (channel lam hlam i).bind (orderedLaw ρ lam hlam ℓ))
-      (fun _ a ↦ (channel c hc 0).bind (fun s ↦
-        (orderedNoise c hc ℓ a).map (relabelOrdered (Equiv.addRight s) ℓ)))]
-    congr 1
-    funext j
-    rw [PMF.bind_bind]
-    have he (k : Spin) : (orderedLaw ρ lam hlam ℓ k).bind (fun a ↦ (channel c hc 0).bind
-        (fun s ↦ (orderedNoise c hc ℓ a).map (relabelOrdered (Equiv.addRight s) ℓ))) =
-        (channel c hc k).bind (orderedLaw ρ (lam*c) hprod ℓ) := by
-      rw [PMF.bind_comm]
-      simp_rw [← PMF.map_bind, ih, orderedLaw_permute]
-      change (channel c hc 0).bind (fun s ↦ orderedLaw ρ (lam*c) hprod ℓ (k+s)) = _
-      have hm := channel_permute hc (Equiv.addLeft k) 0
-      simp only [Equiv.coe_addLeft, add_zero] at hm
-      rw [← hm, PMF.bind_map]
-      rfl
-    simp_rw [he, ← PMF.bind_bind, channel_compose hlam hc hprod]
-
-lemma ordered_channel_mu_le (ρ : PMF ℕ) (lam c : ℝ) (hlam : Admissible lam) (hc : Admissible c)
-    (hprod : Admissible (lam*c)) (ℓ : ℕ) :
-    (∫ m, xMoment m ∂(discreteProbability (orderedLaw ρ (lam*c) hprod ℓ)).toMeasure) ≤
-      ∫ m, xMoment m ∂(discreteProbability (orderedLaw ρ lam hlam ℓ)).toMeasure := by
-  have h := discrete_degradation_mu (orderedLaw ρ lam hlam ℓ) (orderedNoise c hc ℓ)
-  change (∫ m, xMoment m ∂(discreteProbability (fun i ↦
-    (orderedLaw ρ lam hlam ℓ i).bind (orderedNoise c hc ℓ))).toMeasure) ≤ _ at h
-  simpa only [orderedLaw_noise ρ lam c hlam hc hprod] using h
-
-end ThreeState.TreeClauses.Tree
-
-end 
-
-noncomputable section
-open Set MeasureTheory Filter
-open scoped Topology
-namespace ThreeState.TreeClauses.Experiment
-open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
-
-lemma degrading_offspring_fixed_point (Q : ℕ → Law) (C : ℕ → ProbabilityMeasure (Message × Message))
-    (hfst : ∀ ℓ, (C ℓ).map Prod.fst = (Q ℓ).probability)
-    (hsnd : ∀ ℓ, (C ℓ).map Prod.snd = (Q (ℓ+1)).probability)
-    (hloss : ∀ ℓ, (∫ z, quadraticGap z ∂(C ℓ).toMeasure) = (Q ℓ).mu-(Q (ℓ+1)).mu)
-    {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (ρ : PMF ℕ)
-    (hrec : ∀ ℓ, (Q (ℓ+1)).probability = (offspringLaw ρ (Q ℓ) hl₀ hl₁).probability) :
-    ∃ R : Law, R.probability = (offspringLaw ρ R hl₀ hl₁).probability ∧
-      Tendsto (fun ℓ ↦ (Q ℓ).mu) atTop (𝓝 R.mu) := by
-  have hanti : Antitone (fun ℓ ↦ (Q ℓ).mu) := by
-    apply antitone_nat_of_succ_le
-    intro ℓ
-    have h := MeasureTheory.integral_nonneg (μ := (C ℓ).toMeasure) (f := quadraticGap) quadraticGap_nonneg
-    rw [hloss] at h
-    linarith
-  let a := ⨅ ℓ, (Q ℓ).mu
-  have hlim : Tendsto (fun ℓ ↦ (Q ℓ).mu) atTop (𝓝 a) :=
-    tendsto_atTop_ciInf hanti ⟨0, by rintro _ ⟨ℓ,rfl⟩; exact (Q ℓ).mu_nonneg⟩
-  have hdiff : Tendsto (fun ℓ ↦ (Q ℓ).mu-(Q (ℓ+1)).mu) atTop (𝓝 0) := by
-    simpa using hlim.sub (hlim.comp (tendsto_add_atTop_nat 1))
-  obtain ⟨D, hD⟩ := exists_clusterPt_of_compactSpace (Filter.map C atTop)
-  let F : Filter ℕ := Filter.comap C (𝓝 D) ⊓ atTop
-  have : F.NeBot := neBot_inf_comap_iff_map'.mpr hD
-  have hF : F ≤ atTop := inf_le_right
-  have hC : Tendsto C F (𝓝 D) := tendsto_comap.mono_left inf_le_left
-  have hgap : (∫ z, quadraticGap z ∂D.toMeasure) = 0 := by
-    have hg := (probability_continuous_integral continuous_quadraticGap).tendsto D |>.comp hC
-    simp only [Function.comp_def, hloss] at hg
-    exact tendsto_nhds_unique hg (hdiff.mono_left hF)
-  have hdiag := coupling_diagonal_of_loss_zero D hgap
-  have hP := (ProbabilityMeasure.continuous_map (continuous_fst : Continuous (Prod.fst : Message × Message → Message))).tendsto D |>.comp hC
-  simp only [Function.comp_def, hfst] at hP
-  let R : Law := limitLaw hP
-  have hS := (ProbabilityMeasure.continuous_map (continuous_snd : Continuous (Prod.snd : Message × Message → Message))).tendsto D |>.comp hC
-  simp only [Function.comp_def, hsnd] at hS
-  rw [← hdiag] at hS
-  have hR : R.probability = (offspringLaw ρ R hl₀ hl₁).probability := by
-    have hr := offspringLaw_tendsto (R := R) ρ hl₀ hl₁ hP
-    simp only [← hrec] at hr
-    exact tendsto_nhds_unique hS hr
-  have hm : a = R.mu := by
-    have hp := (probability_continuous_integral continuous_xMoment).tendsto R.probability |>.comp hP
-    exact tendsto_nhds_unique (hlim.mono_left hF) hp
-  exact ⟨R, hR, hm ▸ hlim⟩
-
-end ThreeState.TreeClauses.Experiment
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeStateClauses.RegularCritical
+import OAI.Probability.ThreeStateClauses.ObservedMixtures
+
+namespace OAI
+
+/-! Offspring Laws. -/
+
+open scoped BigOperators ENNReal NNReal Topology
+open Filter
+noncomputable section
+open Set MeasureTheory
+open scoped BigOperators ENNReal
+namespace ThreeState.TreeClauses.Tree
+open ThreeState.TreeClauses.Experiment ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
+
+@[reducible] def OrderedObservation : ℕ → Type
+  | 0 => Spin
+  | ℓ+1 => Σ n : ℕ, Fin n → OrderedObservation ℓ
+
+instance orderedObservationCountable : (ℓ : ℕ) → Countable (OrderedObservation ℓ)
+  | 0 => inferInstanceAs (Countable Spin)
+  | ℓ+1 => letI := orderedObservationCountable ℓ
+           inferInstanceAs (Countable (Σ n : ℕ, Fin n → OrderedObservation ℓ))
+
+instance orderedObservationMeasurable : (ℓ : ℕ) → MeasurableSpace (OrderedObservation ℓ)
+  | 0 => inferInstanceAs (MeasurableSpace Spin)
+  | ℓ+1 => letI := orderedObservationMeasurable ℓ
+           inferInstanceAs (MeasurableSpace (Σ n : ℕ, Fin n → OrderedObservation ℓ))
+
+instance orderedObservationSingleton : (ℓ : ℕ) → MeasurableSingletonClass (OrderedObservation ℓ)
+  | 0 => inferInstanceAs (MeasurableSingletonClass Spin)
+  | ℓ+1 => letI := orderedObservationSingleton ℓ
+           inferInstanceAs (MeasurableSingletonClass (Σ n : ℕ, Fin n → OrderedObservation ℓ))
+
+def orderedLaw (ρ : PMF ℕ) (lam : ℝ) (hlam : Admissible lam) :
+    (ℓ : ℕ) → Spin → PMF (OrderedObservation ℓ)
+  | 0, i => PMF.pure i
+  | ℓ+1, i => sigmaPMF ρ (fun n ↦ productPMF (fun _ : Fin n ↦
+      (channel lam hlam i).bind (orderedLaw ρ lam hlam ℓ)))
+
+def forgetOrdered : (ℓ : ℕ) → OrderedObservation ℓ → Observation ℓ
+  | 0, i => i
+  | ℓ+1, ⟨_,v⟩ => (List.ofFn (fun j ↦ forgetOrdered ℓ (v j)) : Multiset (Observation ℓ))
+
+lemma orderedLaw_forget (ρ : PMF ℕ) (lam : ℝ) (hlam : Admissible lam) (ℓ : ℕ) (i : Spin) :
+    (orderedLaw ρ lam hlam ℓ i).map (forgetOrdered ℓ) = observationLaw lam hlam ρ ℓ i := by
+  induction ℓ generalizing i with
+  | zero => exact PMF.map_id _
+  | succ ℓ ih =>
+    rw [orderedLaw, sigmaPMF_map]
+    change ρ.bind _ = ρ.bind _
+    congr 1
+    funext n
+    change (productPMF (fun _ : Fin n ↦ (channel lam hlam i).bind (orderedLaw ρ lam hlam ℓ))).map
+      ((fun xs : List (Observation ℓ) ↦ (xs : Multiset (Observation ℓ))) ∘
+        List.ofFn ∘ (fun v j ↦ forgetOrdered ℓ (v j))) = _
+    rw [← PMF.map_comp, ← PMF.map_comp, productPMF_map]
+    simp_rw [PMF.map_bind, ih]
+    rw [productPMF_list]
+    exact (PMF.map_id ((iidList ((channel lam hlam i).bind
+      (observationLaw lam hlam ρ ℓ)) n).map (fun xs : List (Observation ℓ) ↦ (xs : Multiset (Observation ℓ))))).symm
+
+def orderedExpansion (ρ : PMF ℕ) (lam : ℝ) (hlam : Admissible lam) :
+    (ℓ : ℕ) → OrderedObservation ℓ → PMF (OrderedObservation (ℓ+1))
+  | 0, i => sigmaPMF ρ (fun n ↦ productPMF (fun _ : Fin n ↦ channel lam hlam i))
+  | ℓ+1, ⟨n,v⟩ => (productPMF (fun j ↦ orderedExpansion ρ lam hlam ℓ (v j))).map (Sigma.mk n)
+
+lemma orderedLaw_degrades (ρ : PMF ℕ) (lam : ℝ) (hlam : Admissible lam) (ℓ : ℕ) (i : Spin) :
+    (orderedLaw ρ lam hlam ℓ i).bind (orderedExpansion ρ lam hlam ℓ) =
+      orderedLaw ρ lam hlam (ℓ+1) i := by
+  induction ℓ generalizing i with
+  | zero =>
+    change (PMF.pure i).bind _ = sigmaPMF ρ (fun n ↦
+      productPMF (fun _ : Fin n ↦ (channel lam hlam i).bind PMF.pure))
+    rw [PMF.pure_bind]
+    simp only [PMF.bind_pure]
+    rfl
+  | succ ℓ ih =>
+    change (sigmaPMF ρ (fun n ↦ productPMF (fun _ : Fin n ↦
+      (channel lam hlam i).bind (orderedLaw ρ lam hlam ℓ)))).bind
+      (fun z ↦ (productPMF (fun j ↦ orderedExpansion ρ lam hlam ℓ (z.2 j))).map (Sigma.mk (β := fun n ↦ Fin n → OrderedObservation (ℓ+1)) z.1)) = _
+    rw [sigmaPMF_bind_components ρ _ (fun n (v : Fin n → OrderedObservation ℓ) ↦
+      productPMF (fun j ↦ orderedExpansion ρ lam hlam ℓ (v j)))]
+    simp only [productPMF_bind, PMF.bind_bind]
+    simp_rw [ih]
+    rfl
+
+def offspringPosteriorLaw (ρ : PMF ℕ) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) : ℕ → Law
+  | 0 => rootLaw
+  | ℓ+1 => offspringLaw ρ (offspringPosteriorLaw ρ hl₀ hl₁ ℓ) hl₀ hl₁
+
+lemma offspringPosteriorLaw_probability (ρ : PMF ℕ) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
+    (hlam : Admissible lam) (ℓ : ℕ) :
+    (offspringPosteriorLaw ρ hl₀ hl₁ ℓ).probability = discreteProbability (orderedLaw ρ lam hlam ℓ) := by
+  induction ℓ with
+  | zero => rfl
+  | succ ℓ ih =>
+    change mixtureProbability ρ (fun n ↦ finiteProbability (offspringPosteriorLaw ρ hl₀ hl₁ ℓ) hl₀ hl₁ n) =
+      discreteProbability (fun i ↦ sigmaPMF ρ (fun n ↦ discreteBranchLaw (orderedLaw ρ lam hlam ℓ) lam hlam n i))
+    rw [discreteSigma_probability]
+    congr 1
+    funext n
+    exact (discreteBranchLaw_probability _ _ ih hl₀ hl₁ hlam n).symm
+
+end ThreeState.TreeClauses.Tree
+
+end 
+
+noncomputable section
+open Set MeasureTheory
+open scoped BigOperators ENNReal
+namespace ThreeState.TreeClauses.Tree
+open ThreeState.TreeClauses.Experiment
+
+lemma sigmaPMF_map_components {α β : ℕ → Type*} (ρ : PMF ℕ) (p : ∀ n, PMF (α n))
+    (f : ∀ n, α n → β n) :
+    (sigmaPMF ρ p).map (fun z ↦ ⟨z.1,f z.1 z.2⟩) = sigmaPMF ρ (fun n ↦ (p n).map (f n)) := by
+  rw [sigmaPMF_map, sigmaPMF_eq_bind]
+  congr 1
+  funext n
+  rw [PMF.map_comp]
+  rfl
+
+def relabelOrdered (σ : Equiv.Perm Spin) : (ℓ : ℕ) → OrderedObservation ℓ → OrderedObservation ℓ
+  | 0, i => σ i
+  | ℓ+1, ⟨n,v⟩ => ⟨n,fun j ↦ relabelOrdered σ ℓ (v j)⟩
+
+lemma orderedLaw_permute (ρ : PMF ℕ) (lam : ℝ) (hlam : Admissible lam) (σ : Equiv.Perm Spin)
+    (ℓ : ℕ) (i : Spin) :
+    (orderedLaw ρ lam hlam ℓ i).map (relabelOrdered σ ℓ) = orderedLaw ρ lam hlam ℓ (σ i) := by
+  induction ℓ generalizing i with
+  | zero => exact PMF.pure_bind _ _
+  | succ ℓ ih =>
+    change (sigmaPMF ρ (fun n ↦ productPMF (fun _ : Fin n ↦
+      (channel lam hlam i).bind (orderedLaw ρ lam hlam ℓ)))).map
+      (fun z ↦ Sigma.mk (β := fun n ↦ Fin n → OrderedObservation ℓ) z.1
+        (fun j ↦ relabelOrdered σ ℓ (z.2 j))) = _
+    rw [sigmaPMF_map_components ρ _ (fun n (v : Fin n → OrderedObservation ℓ) j ↦ relabelOrdered σ ℓ (v j))]
+    simp_rw [productPMF_map, PMF.map_bind, ih]
+    have he : (channel lam hlam i).bind (fun j ↦ orderedLaw ρ lam hlam ℓ (σ j)) =
+        (channel lam hlam (σ i)).bind (orderedLaw ρ lam hlam ℓ) := by
+      rw [← channel_permute hlam σ i, PMF.bind_map]
+      rfl
+    simp_rw [he]
+    rfl
+
+def orderedNoise (c : ℝ) (hc : Admissible c) :
+    (ℓ : ℕ) → OrderedObservation ℓ → PMF (OrderedObservation ℓ)
+  | 0, i => PMF.pure i
+  | ℓ+1, ⟨n,v⟩ => (productPMF (fun j ↦
+      (channel c hc 0).bind (fun s ↦ (orderedNoise c hc ℓ (v j)).map
+        (relabelOrdered (Equiv.addRight s) ℓ)))).map (Sigma.mk n)
+
+lemma orderedLaw_noise (ρ : PMF ℕ) (lam c : ℝ) (hlam : Admissible lam) (hc : Admissible c)
+    (hprod : Admissible (lam*c)) (ℓ : ℕ) (i : Spin) :
+    (orderedLaw ρ lam hlam ℓ i).bind (orderedNoise c hc ℓ) = orderedLaw ρ (lam*c) hprod ℓ i := by
+  induction ℓ generalizing i with
+  | zero => exact PMF.pure_bind _ _
+  | succ ℓ ih =>
+    change (sigmaPMF ρ (fun n ↦ productPMF (fun _ : Fin n ↦
+      (channel lam hlam i).bind (orderedLaw ρ lam hlam ℓ)))).bind
+      (fun z ↦ (productPMF (fun j ↦ (channel c hc 0).bind (fun s ↦
+        (orderedNoise c hc ℓ (z.2 j)).map (relabelOrdered (Equiv.addRight s) ℓ)) )).map (Sigma.mk (β := fun n ↦ Fin n → OrderedObservation ℓ) z.1)) = _
+    rw [sigmaPMF_bind_components ρ _ (fun n (v : Fin n → OrderedObservation ℓ) ↦
+      productPMF (fun j ↦ (channel c hc 0).bind (fun s ↦
+        (orderedNoise c hc ℓ (v j)).map (relabelOrdered (Equiv.addRight s) ℓ))))]
+    change sigmaPMF ρ _ = sigmaPMF ρ _
+    congr 1
+    funext n
+    rw [productPMF_bind (fun _ : Fin n ↦ (channel lam hlam i).bind (orderedLaw ρ lam hlam ℓ))
+      (fun _ a ↦ (channel c hc 0).bind (fun s ↦
+        (orderedNoise c hc ℓ a).map (relabelOrdered (Equiv.addRight s) ℓ)))]
+    congr 1
+    funext j
+    rw [PMF.bind_bind]
+    have he (k : Spin) : (orderedLaw ρ lam hlam ℓ k).bind (fun a ↦ (channel c hc 0).bind
+        (fun s ↦ (orderedNoise c hc ℓ a).map (relabelOrdered (Equiv.addRight s) ℓ))) =
+        (channel c hc k).bind (orderedLaw ρ (lam*c) hprod ℓ) := by
+      rw [PMF.bind_comm]
+      simp_rw [← PMF.map_bind, ih, orderedLaw_permute]
+      change (channel c hc 0).bind (fun s ↦ orderedLaw ρ (lam*c) hprod ℓ (k+s)) = _
+      have hm := channel_permute hc (Equiv.addLeft k) 0
+      simp only [Equiv.coe_addLeft, add_zero] at hm
+      rw [← hm, PMF.bind_map]
+      rfl
+    simp_rw [he, ← PMF.bind_bind, channel_compose hlam hc hprod]
+
+lemma ordered_channel_mu_le (ρ : PMF ℕ) (lam c : ℝ) (hlam : Admissible lam) (hc : Admissible c)
+    (hprod : Admissible (lam*c)) (ℓ : ℕ) :
+    (∫ m, xMoment m ∂(discreteProbability (orderedLaw ρ (lam*c) hprod ℓ)).toMeasure) ≤
+      ∫ m, xMoment m ∂(discreteProbability (orderedLaw ρ lam hlam ℓ)).toMeasure := by
+  have h := discrete_degradation_mu (orderedLaw ρ lam hlam ℓ) (orderedNoise c hc ℓ)
+  change (∫ m, xMoment m ∂(discreteProbability (fun i ↦
+    (orderedLaw ρ lam hlam ℓ i).bind (orderedNoise c hc ℓ))).toMeasure) ≤ _ at h
+  simpa only [orderedLaw_noise ρ lam c hlam hc hprod] using h
+
+end ThreeState.TreeClauses.Tree
+
+end 
+
+noncomputable section
+open Set MeasureTheory Filter
+open scoped Topology
+namespace ThreeState.TreeClauses.Experiment
+open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
+
+lemma degrading_offspring_fixed_point (Q : ℕ → Law) (C : ℕ → ProbabilityMeasure (Message × Message))
+    (hfst : ∀ ℓ, (C ℓ).map Prod.fst = (Q ℓ).probability)
+    (hsnd : ∀ ℓ, (C ℓ).map Prod.snd = (Q (ℓ+1)).probability)
+    (hloss : ∀ ℓ, (∫ z, quadraticGap z ∂(C ℓ).toMeasure) = (Q ℓ).mu-(Q (ℓ+1)).mu)
+    {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (ρ : PMF ℕ)
+    (hrec : ∀ ℓ, (Q (ℓ+1)).probability = (offspringLaw ρ (Q ℓ) hl₀ hl₁).probability) :
+    ∃ R : Law, R.probability = (offspringLaw ρ R hl₀ hl₁).probability ∧
+      Tendsto (fun ℓ ↦ (Q ℓ).mu) atTop (𝓝 R.mu) := by
+  have hanti : Antitone (fun ℓ ↦ (Q ℓ).mu) := by
+    apply antitone_nat_of_succ_le
+    intro ℓ
+    have h := MeasureTheory.integral_nonneg (μ := (C ℓ).toMeasure) (f := quadraticGap) quadraticGap_nonneg
+    rw [hloss] at h
+    linarith
+  let a := ⨅ ℓ, (Q ℓ).mu
+  have hlim : Tendsto (fun ℓ ↦ (Q ℓ).mu) atTop (𝓝 a) :=
+    tendsto_atTop_ciInf hanti ⟨0, by rintro _ ⟨ℓ,rfl⟩; exact (Q ℓ).mu_nonneg⟩
+  have hdiff : Tendsto (fun ℓ ↦ (Q ℓ).mu-(Q (ℓ+1)).mu) atTop (𝓝 0) := by
+    simpa using hlim.sub (hlim.comp (tendsto_add_atTop_nat 1))
+  obtain ⟨D, hD⟩ := exists_clusterPt_of_compactSpace (Filter.map C atTop)
+  let F : Filter ℕ := Filter.comap C (𝓝 D) ⊓ atTop
+  have : F.NeBot := neBot_inf_comap_iff_map'.mpr hD
+  have hF : F ≤ atTop := inf_le_right
+  have hC : Tendsto C F (𝓝 D) := tendsto_comap.mono_left inf_le_left
+  have hgap : (∫ z, quadraticGap z ∂D.toMeasure) = 0 := by
+    have hg := (probability_continuous_integral continuous_quadraticGap).tendsto D |>.comp hC
+    simp only [Function.comp_def, hloss] at hg
+    exact tendsto_nhds_unique hg (hdiff.mono_left hF)
+  have hdiag := coupling_diagonal_of_loss_zero D hgap
+  have hP := (ProbabilityMeasure.continuous_map (continuous_fst : Continuous (Prod.fst : Message × Message → Message))).tendsto D |>.comp hC
+  simp only [Function.comp_def, hfst] at hP
+  let R : Law := limitLaw hP
+  have hS := (ProbabilityMeasure.continuous_map (continuous_snd : Continuous (Prod.snd : Message × Message → Message))).tendsto D |>.comp hC
+  simp only [Function.comp_def, hsnd] at hS
+  rw [← hdiag] at hS
+  have hR : R.probability = (offspringLaw ρ R hl₀ hl₁).probability := by
+    have hr := offspringLaw_tendsto (R := R) ρ hl₀ hl₁ hP
+    simp only [← hrec] at hr
+    exact tendsto_nhds_unique hS hr
+  have hm : a = R.mu := by
+    have hp := (probability_continuous_integral continuous_xMoment).tendsto R.probability |>.comp hP
+    exact tendsto_nhds_unique (hlim.mono_left hF) hp
+  exact ⟨R, hR, hm ▸ hlim⟩
+
+end ThreeState.TreeClauses.Experiment
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeStateClauses/PoissonCalculus.lean b/lean/OAI/Probability/ThreeStateClauses/PoissonCalculus.lean
index 854395fa48a97e9004080a2df317bcf92ce0277e..69f261b2374e3711f75c20c68658ec22bf42eea6 100644
--- a/lean/OAI/Probability/ThreeStateClauses/PoissonCalculus.lean
+++ b/lean/OAI/Probability/ThreeStateClauses/PoissonCalculus.lean
@@ -1,545 +1,545 @@
-import Mathlib
-import OAI.Probability.ThreeStateClauses.PositiveFixed
-import OAI.Probability.ThreeStateClauses.PoissonLaw
-
-namespace OAI
-
-/-! Poisson Calculus. -/
-
-open scoped BigOperators ENNReal NNReal Topology
-open Filter
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped BigOperators NNReal
-namespace ThreeState.TreeClauses.Probability
-
-def exponentialAverage (f : ℕ → ℝ) (t : ℝ) : ℝ := ∑' n, t^n/(n.factorial:ℝ)*f n
-
-def poissonAverage (f : ℕ → ℝ) (t : ℝ) : ℝ := Real.exp (-t)*exponentialAverage f t
-
-lemma exponentialAverage_summable {f : ℕ → ℝ} {B : ℝ} (hB : ∀ n, |f n| ≤ B) (t : ℝ) :
-    Summable (fun n ↦ t^n/(n.factorial:ℝ)*f n) := by
-  apply Summable.of_norm_bounded ((Real.summable_pow_div_factorial |t|).mul_right B)
-  intro n
-  rw [Real.norm_eq_abs, abs_mul, abs_div, abs_pow, Nat.abs_cast]
-  exact mul_le_mul_of_nonneg_left (hB n) (by positivity)
-
-private def exponentialDerivative (f : ℕ → ℝ) : ℕ → ℝ → ℝ
-  | 0, _ => 0
-  | n+1, t => t^n/(n.factorial:ℝ)*f (n+1)
-
-private lemma exponentialDerivative_summable {f : ℕ → ℝ} {B : ℝ} (hB : ∀ n, |f n| ≤ B) (t : ℝ) :
-    Summable (fun n ↦ exponentialDerivative f n t) := by
-  apply (summable_nat_add_iff 1).mp
-  exact exponentialAverage_summable (fun n ↦ hB (n+1)) t
-
-private lemma exponentialDerivative_sum {f : ℕ → ℝ} {B : ℝ} (hB : ∀ n, |f n| ≤ B) (t : ℝ) :
-    (∑' n, exponentialDerivative f n t) = exponentialAverage (fun n ↦ f (n+1)) t := by
-  rw [← (exponentialDerivative_summable hB t).sum_add_tsum_nat_add 1]
-  simp [exponentialDerivative, exponentialAverage]
-
-lemma hasDerivAt_exponentialAverage {f : ℕ → ℝ} {B : ℝ} (hB : ∀ n, |f n| ≤ B) (t : ℝ) :
-    HasDerivAt (exponentialAverage f) (exponentialAverage (fun n ↦ f (n+1)) t) t := by
-  let R := |t|+1
-  have hR : 0 < R := by dsimp [R]; positivity
-  have ht : t ∈ Ioo (-R) R := by
-    constructor <;> dsimp [R] <;> linarith [le_abs_self t, neg_abs_le t]
-  let u : ℕ → ℝ := fun n ↦ exponentialDerivative (fun _ ↦ B) n R
-  have hu : Summable u := exponentialDerivative_summable (B := |B|) (fun _ ↦ le_refl _) R
-  have hd (n : ℕ) (y : ℝ) : HasDerivAt (fun z : ℝ ↦ z^n/(n.factorial:ℝ)*f n)
-      (exponentialDerivative f n y) y := by
-    cases n with
-    | zero => simpa [exponentialDerivative] using hasDerivAt_const y (f 0)
-    | succ n =>
-      convert (((hasDerivAt_id y).pow (n+1)).div_const ((n+1).factorial:ℝ)).mul_const (f (n+1)) using 1
-      all_goals try rfl
-      all_goals try simp only [exponentialDerivative, id_eq]
-      rw [Nat.factorial_succ, Nat.cast_mul]
-      simp only [Nat.add_sub_cancel, mul_one]
-      have hn : ((n+1:ℕ):ℝ) ≠ 0 := by positivity
-      field_simp
-  have hb (n : ℕ) (y : ℝ) (hy : y ∈ Ioo (-R) R) : ‖exponentialDerivative f n y‖ ≤ u n := by
-    have hyR : |y| ≤ R := (abs_le).2 ⟨le_of_lt hy.1, le_of_lt hy.2⟩
-    cases n with
-    | zero => simp [exponentialDerivative, u]
-    | succ n =>
-      dsimp [exponentialDerivative, u]
-      rw [abs_mul, abs_div, abs_pow, Nat.abs_cast]
-      exact mul_le_mul (div_le_div_of_nonneg_right (pow_le_pow_left₀ (abs_nonneg y) hyR n) (by positivity))
-        (hB (n+1)) (abs_nonneg _) (by positivity)
-  have hh := hasDerivAt_tsum_of_isPreconnected hu isOpen_Ioo (convex_Ioo (-R) R).isPreconnected
-    (fun n y _ ↦ hd n y) hb ht (exponentialAverage_summable hB t) ht
-  rw [exponentialDerivative_sum hB t] at hh
-  exact hh
-
-theorem hasDerivAt_poissonAverage {f : ℕ → ℝ} {B : ℝ} (hB : ∀ n, |f n| ≤ B) (t : ℝ) :
-    HasDerivAt (poissonAverage f)
-      (poissonAverage (fun n ↦ f (n+1)) t-poissonAverage f t) t := by
-  have hd := ((hasDerivAt_neg t).exp).mul (hasDerivAt_exponentialAverage hB t)
-  exact hd.congr_deriv (by unfold poissonAverage; ring)
-
-lemma poissonAverage_eq_tsum {f : ℕ → ℝ} {B : ℝ} (hB : ∀ n, |f n| ≤ B) (t : ℝ) :
-    poissonAverage f t = ∑' n, poissonWeight t n*f n := by
-  rw [poissonAverage, exponentialAverage, ← (exponentialAverage_summable hB t).tsum_mul_left]
-  apply tsum_congr; intro n; unfold poissonWeight; ring
-
-lemma poissonAverage_eq_integral {f : ℕ → ℝ} {B : ℝ} (hB : ∀ n, |f n| ≤ B) (t : ℝ≥0) :
-    poissonAverage f t = ∫ n, f n ∂(poissonPMF t).toMeasure := by
-  rw [PMF.integral_eq_tsum]
-  · simp only [poissonPMF_real, smul_eq_mul]
-    exact poissonAverage_eq_tsum hB t
-  · exact Integrable.of_bound (measurable_of_countable _).aestronglyMeasurable B
-      (Eventually.of_forall (fun n ↦ by simpa only [Real.norm_eq_abs] using hB n))
-
-lemma continuous_poissonAverage {f : ℕ → ℝ} {B : ℝ} (hB : ∀ n, |f n| ≤ B) :
-    Continuous (poissonAverage f) := continuous_iff_continuousAt.mpr
-  (fun t ↦ (hasDerivAt_poissonAverage hB t).continuousAt)
-
-lemma poissonAverage_zero (f : ℕ → ℝ) : poissonAverage f 0 = f 0 := by
-  rw [poissonAverage, neg_zero, Real.exp_zero, one_mul, exponentialAverage, tsum_eq_single 0]
-  · simp
-  · intro n hn
-    simp [zero_pow hn]
-
-end ThreeState.TreeClauses.Probability
-
-end 
-
-noncomputable section
-open Set MeasureTheory ProbabilityTheory
-open scoped NNReal
-namespace ThreeState.TreeClauses.Experiment
-open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive ThreeState.TreeClauses.Probability
-
-lemma offspring_positive (ρ : PMF ℕ) (Q : Law) {lam : ℝ}
-    (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) :
-    ∀ᵐ m ∂(offspringLaw ρ Q hl₀ hl₁).probability.toMeasure, PositiveMessage m.1 := by
-  apply Measure.ae_sum_iff.mpr
-  intro n
-  exact Measure.ae_smul_measure (finite_positive Q hl₀ hl₁ n) _
-
-lemma correction_nonneg (m : Message) : 0 ≤ correction m.1 := by
-  unfold correction
-  apply div_nonneg _ (by norm_num)
-  have h := avg_mono (fun i ↦ mul_nonneg (coordinate_nonneg m i) (sq_nonneg (logCentered m.1 i)))
-  simpa only [avg_const] using h
-
-private lemma poisson_mixture_integrable_nonneg (t : ℝ≥0) (P : ℕ → ProbabilityMeasure Message)
-    {f : Message → ℝ} (hi : ∀ n, Integrable f (P n).toMeasure)
-    (hp : ∀ n, ∀ᵐ m ∂(P n).toMeasure, 0 ≤ f m)
-    (hs : Summable (fun n ↦ poissonWeight t n*(∫ m, f m ∂(P n).toMeasure))) :
-    Integrable f (mixtureProbability (poissonPMF t) P).toMeasure := by
-  apply mixture_integrable _ P hi
-  simp only [poissonPMF_real]
-  apply hs.congr
-  intro n
-  congr 1
-  exact integral_congr_ae ((hp n).mono (fun m hm ↦ (abs_of_nonneg hm).symm))
-
-lemma poisson_integrable_symEntropy (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
-    (t : ℝ≥0) : Integrable (fun m : Message ↦ symEntropy m.1) (poissonLaw Q hl₀ hl₁ t).probability.toMeasure := by
-  apply poisson_mixture_integrable_nonneg t (fun n ↦ finiteProbability Q hl₀ hl₁ n)
-    (finite_integrable_symEntropy Q hl₀ hl₁)
-    (fun n ↦ (finite_positive Q hl₀ hl₁ n).mono (fun m hm ↦ symEntropy_nonneg hm))
-  have he (n : ℕ) : (∫ m, symEntropy m.1 ∂(finiteProbability Q hl₀ hl₁ n).toMeasure) =
-      (n:ℝ)*edgeJMean Q lam := finiteLaw_symEntropy Q hl₀ hl₁ n
-  simp_rw [he, ← mul_assoc]
-  exact (poissonWeight_mean t).summable.mul_right _
-
-lemma poisson_integrable_correction (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
-    (t : ℝ≥0) : Integrable (fun m : Message ↦ correction m.1) (poissonLaw Q hl₀ hl₁ t).probability.toMeasure := by
-  apply poisson_mixture_integrable_nonneg t (fun n ↦ finiteProbability Q hl₀ hl₁ n)
-    (finite_integrable_correction Q hl₀ hl₁)
-    (fun _ ↦ Filter.Eventually.of_forall correction_nonneg)
-  have h := ((poissonWeight_mean t).summable.mul_right
-    (∫ m, correction (edgeMessage lam m) ∂Q.probability.toMeasure)).add
-    ((poissonWeight_factorial_second t).summable.mul_right (2*(edgeJMean Q lam)^2))
-  apply h.congr
-  intro n
-  have he := finiteLaw_correction Q hl₀ hl₁ n
-  change (∫ m, correction m.1 ∂(finiteProbability Q hl₀ hl₁ n).toMeasure) = _ at he
-  rw [he]
-  unfold edgeJMean
-  ring
-
-lemma poissonLaw_symEntropy (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
-    (t : ℝ≥0) :
-    (∫ m, symEntropy m.1 ∂(poissonLaw Q hl₀ hl₁ t).probability.toMeasure) = (t:ℝ)*edgeJMean Q lam := by
-  rw [show (poissonLaw Q hl₀ hl₁ t).probability = mixtureProbability (poissonPMF t)
-    (fun n ↦ (finiteLaw Q hl₀ hl₁ n).probability) from rfl,
-    integral_mixtureProbability _ _ (poisson_integrable_symEntropy Q hl₀ hl₁ t)]
-  simp only [poissonPMF_real, finiteLaw_symEntropy]
-  exact ((poissonWeight_mean t).mul_right (edgeJMean Q lam)).tsum_eq |> fun h ↦ by
-    simpa only [mul_assoc, edgeJMean] using h
-
-lemma poissonLaw_correction (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
-    (t : ℝ≥0) :
-    (∫ m, correction m.1 ∂(poissonLaw Q hl₀ hl₁ t).probability.toMeasure) =
-      (t:ℝ)*(∫ m, correction (edgeMessage lam m) ∂Q.probability.toMeasure)+
-      2*(t:ℝ)^2*(edgeJMean Q lam)^2 := by
-  rw [show (poissonLaw Q hl₀ hl₁ t).probability = mixtureProbability (poissonPMF t)
-    (fun n ↦ (finiteLaw Q hl₀ hl₁ n).probability) from rfl,
-    integral_mixtureProbability _ _ (poisson_integrable_correction Q hl₀ hl₁ t)]
-  simp only [poissonPMF_real, finiteLaw_correction]
-  have h := ((poissonWeight_mean t).mul_right
-    (∫ m, correction (edgeMessage lam m) ∂Q.probability.toMeasure)).add
-    ((poissonWeight_factorial_second t).mul_right (2*(edgeJMean Q lam)^2))
-  convert h.tsum_eq using 1
-  · apply tsum_congr; intro n; unfold edgeJMean; ring
-  · ring
-
-def iMean (Q : Law) : ℝ := ∫ m, info m.1 ∂Q.probability.toMeasure
-
-def edgeIMean (Q : Law) (lam : ℝ) : ℝ := ∫ m, info (edgeMessage lam m) ∂Q.probability.toMeasure
-
-lemma bounded_iMean : ∃ B : ℝ, ∀ Q : Law, |iMean Q| ≤ B := by
-  have hc : Continuous (fun m : Message ↦ info m.1) := continuous_info_comp continuous_subtype_val
-  obtain ⟨B,hB⟩ := isCompact_univ.exists_bound_of_continuousOn hc.continuousOn
-  refine ⟨B, fun Q ↦ ?_⟩
-  apply (abs_integral_le_integral_abs).trans
-  exact (integral_mono (compact_integrable (continuous_info_comp continuous_subtype_val).abs)
-    (integrable_const B) (fun m ↦ by simpa only [Real.norm_eq_abs] using hB m (mem_univ m))).trans_eq (by simp)
-
-def poissonCorrelation (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (t : ℝ) : ℝ :=
-  t*edgeIMean Q lam-poissonAverage (fun n ↦ iMean (finiteLaw Q hl₀ hl₁ n)) t
-
-lemma poissonLaw_info (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (t : ℝ≥0) :
-    iMean (poissonLaw Q hl₀ hl₁ t) = (t:ℝ)*edgeIMean Q lam-poissonCorrelation Q hl₀ hl₁ t := by
-  obtain ⟨B,hB⟩ := bounded_iMean
-  have he : iMean (poissonLaw Q hl₀ hl₁ t) = poissonAverage (fun n ↦ iMean (finiteLaw Q hl₀ hl₁ n)) t := by
-    unfold iMean
-    rw [show (poissonLaw Q hl₀ hl₁ t).probability = mixtureProbability (poissonPMF t)
-      (fun n ↦ (finiteLaw Q hl₀ hl₁ n).probability) from rfl,
-      integral_mixtureProbability _ _ (compact_integrable (continuous_info_comp continuous_subtype_val))]
-    change _ = poissonAverage (fun n ↦ iMean (finiteLaw Q hl₀ hl₁ n)) t
-    rw [poissonAverage_eq_tsum (fun n ↦ hB (finiteLaw Q hl₀ hl₁ n))]
-    simp only [poissonPMF_real, iMean]
-  rw [he, poissonCorrelation]
-  ring
-
-lemma poissonCorrelation_continuous (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) :
-    Continuous (poissonCorrelation Q hl₀ hl₁) := by
-  obtain ⟨B,hB⟩ := bounded_iMean
-  exact (continuous_id.mul continuous_const).sub
-    (continuous_poissonAverage (fun n ↦ hB (finiteLaw Q hl₀ hl₁ n)))
-
-end ThreeState.TreeClauses.Experiment
-
-end 
-
-noncomputable section
-open Set MeasureTheory
-open scoped NNReal
-namespace ThreeState.TreeClauses.Positive
-open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Experiment
-
-lemma poisson_fixed_edgeJ (Q : Law) {lam : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1) (d : ℝ≥0)
-    (hcrit : (d:ℝ)*lam^2 = 1) (hfixed : Q.probability = (poissonLaw Q hl₀.le hl₁ d).probability) :
-    edgeJMean Q lam = lam^2*jMean Q := by
-  have h := poissonLaw_symEntropy Q hl₀.le hl₁ d
-  change (∫ m, symEntropy m.1 ∂((poissonLaw Q hl₀.le hl₁ d).probability).toMeasure) = _ at h
-  rw [← hfixed] at h
-  change jMean Q = (d:ℝ)*edgeJMean Q lam at h
-  rw [h]
-  calc
-    _ = ((d:ℝ)*lam^2)*edgeJMean Q lam := by rw [hcrit, one_mul]
-    _ = _ := by ring
-
-lemma functional_integral {α : Type*} [MeasurableSpace α] (μ : Measure α) (f : α → Vec)
-    (hI : Integrable (fun m ↦ info (f m)) μ)
-    (hJ : Integrable (fun m ↦ symEntropy (f m)) μ)
-    (hB : Integrable (fun m ↦ correction (f m)) μ) :
-    (∫ m, functional (f m) ∂μ) = 3*(∫ m, info (f m) ∂μ)-
-      2*(∫ m, symEntropy (f m) ∂μ)+(4/5)*(∫ m, correction (f m) ∂μ) := by
-  unfold functional
-  rw [integral_add (f := fun m ↦ 3*info (f m)-2*symEntropy (f m))
-    (g := fun m ↦ (4/5)*correction (f m)) ((hI.const_mul _).sub (hJ.const_mul _)) (hB.const_mul _),
-    integral_sub (f := fun m ↦ 3*info (f m)) (g := fun m ↦ 2*symEntropy (f m))
-      (hI.const_mul _) (hJ.const_mul _), integral_const_mul, integral_const_mul, integral_const_mul]
-
-lemma integrated_radial (Q : Law) {lam : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1)
-    (hp : ∀ᵐ m ∂Q.probability.toMeasure, PositiveMessage m.1)
-    (hJ : Integrable (fun m : Message ↦ symEntropy m.1) Q.probability.toMeasure)
-    (hB : Integrable (fun m : Message ↦ correction m.1) Q.probability.toMeasure) :
-    (1-lam^2)/2*Q.nu ≤ (∫ m, functional m.1 ∂Q.probability.toMeasure)-
-    (lam^2)⁻¹*(∫ m, functional (edgeMessage lam m) ∂Q.probability.toMeasure) := by
-  have hI : Integrable (fun m : Message ↦ info m.1) Q.probability.toMeasure :=
-    compact_integrable (continuous_info_comp continuous_subtype_val)
-  have hJe := edgeJ_integrable Q hl₀.le hl₁
-  have hIe : Integrable (fun m : Message ↦ info (edgeMessage lam m)) Q.probability.toMeasure :=
-    compact_integrable (continuous_info_comp (continuous_edgeMessage lam))
-  have hBe : Integrable (fun m : Message ↦ correction (edgeMessage lam m)) Q.probability.toMeasure :=
-    compact_integrable (continuous_correction_comp (continuous_edgeMessage lam)
-      (fun m i ↦ (edge_positive hl₀.le hl₁ m).1 i))
-  have hF : Integrable (fun m : Message ↦ functional m.1) Q.probability.toMeasure :=
-    (hI.const_mul 3 |>.sub (hJ.const_mul 2)).add (hB.const_mul (4/5))
-  have hFe : Integrable (fun m : Message ↦ functional (edgeMessage lam m)) Q.probability.toMeasure :=
-    (hIe.const_mul 3 |>.sub (hJe.const_mul 2)).add (hBe.const_mul (4/5))
-  have hrad := integral_mono_ae (μ := Q.probability.toMeasure)
-    (f := fun m : Message ↦ (1-lam^2)/2*xMoment m^2)
-    (g := fun m ↦ functional m.1-(lam^2)⁻¹*functional (edgeMessage lam m))
-    ((compact_integrable (continuous_xMoment.pow 2)).const_mul _) (hF.sub (hFe.const_mul _))
-    (hp.mono (fun m hm ↦ radial_inequality m.1 hm lam hl₀ hl₁))
-  rw [integral_const_mul, integral_sub hF (hFe.const_mul _), integral_const_mul] at hrad
-  exact hrad
-
-lemma poisson_fixed_entropy_radial (Q : Law) {lam : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1) (d : ℝ≥0)
-    (hcrit : (d:ℝ)*lam^2 = 1) (hfixed : Q.probability = (poissonLaw Q hl₀.le hl₁ d).probability) :
-    (1-lam^2)/2*Q.nu ≤ (8/5)*(jMean Q)^2-3*poissonCorrelation Q hl₀.le hl₁ d := by
-  have hp := offspring_positive (poissonPMF d) Q hl₀.le hl₁
-  have hJ := poisson_integrable_symEntropy Q hl₀.le hl₁ d
-  have hB := poisson_integrable_correction Q hl₀.le hl₁ d
-  change ∀ᵐ m ∂(poissonLaw Q hl₀.le hl₁ d).probability.toMeasure, PositiveMessage m.1 at hp
-  rw [← hfixed] at hp hJ hB
-  have hI : Integrable (fun m : Message ↦ info m.1) Q.probability.toMeasure :=
-    compact_integrable (continuous_info_comp continuous_subtype_val)
-  have hJe := edgeJ_integrable Q hl₀.le hl₁
-  have hIe : Integrable (fun m : Message ↦ info (edgeMessage lam m)) Q.probability.toMeasure :=
-    compact_integrable (continuous_info_comp (continuous_edgeMessage lam))
-  have hBe : Integrable (fun m : Message ↦ correction (edgeMessage lam m)) Q.probability.toMeasure :=
-    compact_integrable (continuous_correction_comp (continuous_edgeMessage lam) (fun m i ↦ (edge_positive hl₀.le hl₁ m).1 i))
-  have hrad := integrated_radial Q hl₀ hl₁ hp hJ hB
-  have hJid := poissonLaw_symEntropy Q hl₀.le hl₁ d
-  have hBid := poissonLaw_correction Q hl₀.le hl₁ d
-  have hIid := poissonLaw_info Q hl₀.le hl₁ d
-  change (∫ m, symEntropy m.1 ∂((poissonLaw Q hl₀.le hl₁ d).probability).toMeasure) = _ at hJid
-  change (∫ m, correction m.1 ∂((poissonLaw Q hl₀.le hl₁ d).probability).toMeasure) = _ at hBid
-  change (∫ m, info m.1 ∂((poissonLaw Q hl₀.le hl₁ d).probability).toMeasure) = _ at hIid
-  rw [← hfixed] at hJid hBid hIid
-  have hninv : (lam^2)⁻¹ = (d:ℝ) := by
-    apply mul_left_cancel₀ (ne_of_gt (sq_pos_of_pos hl₀))
-    rw [mul_inv_cancel₀ (ne_of_gt (sq_pos_of_pos hl₀))]
-    nlinarith [hcrit]
-  have hFid := functional_integral Q.probability.toMeasure (fun m : Message ↦ m.1) hI hJ hB
-  have hFeid := functional_integral Q.probability.toMeasure (edgeMessage lam) hIe hJe hBe
-  rw [hFid, hFeid, hninv, hIid, hBid] at hrad
-  change jMean Q = (d:ℝ)*edgeJMean Q lam at hJid
-  change _ ≤ 3*((d:ℝ)*(∫ m, info (edgeMessage lam m) ∂Q.probability.toMeasure)-poissonCorrelation Q hl₀.le hl₁ d)-
-    2*jMean Q+(4/5)*((d:ℝ)*(∫ m, correction (edgeMessage lam m) ∂Q.probability.toMeasure)+
-      2*((d:ℝ)^2)*(edgeJMean Q lam)^2)-
-    (d:ℝ)*(3*(∫ m, info (edgeMessage lam m) ∂Q.probability.toMeasure)-2*edgeJMean Q lam+
-      (4/5)*(∫ m, correction (edgeMessage lam m) ∂Q.probability.toMeasure)) at hrad
-  have he : ((d:ℝ)^2)*(edgeJMean Q lam)^2 = (jMean Q)^2 := by
-    rw [poisson_fixed_edgeJ Q hl₀ hl₁ d hcrit hfixed]
-    have hn : ((d:ℝ)^2)*lam^4 = 1 := by
-      nlinarith [sq_nonneg ((d:ℝ)*lam^2-1), hcrit]
-    calc
-      _ = (((d:ℝ)^2)*lam^4)*(jMean Q)^2 := by ring
-      _ = _ := by rw [hn]; ring
-  nlinarith [hJid, he]
-
-lemma poisson_fixed_moment_bounds (Q : Law) {lam : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1) (d : ℝ≥0)
-    (hcrit : (d:ℝ)*lam^2 = 1) (hfixed : Q.probability = (poissonLaw Q hl₀.le hl₁ d).probability) :
-    0 ≤ jMean Q ∧ jMean Q ≤ Q.mu-lam*qExact lam*Q.nu ∧ cExact lam*Q.nu ≤ Q.eta := by
-  have hp := offspring_positive (poissonPMF d) Q hl₀.le hl₁
-  have hJ := poisson_integrable_symEntropy Q hl₀.le hl₁ d
-  change ∀ᵐ m ∂(poissonLaw Q hl₀.le hl₁ d).probability.toMeasure, PositiveMessage m.1 at hp
-  rw [← hfixed] at hp hJ
-  exact expected_moment_bounds Q hl₀ hl₁ hp hJ (poisson_fixed_edgeJ Q hl₀ hl₁ d hcrit hfixed)
-
-end ThreeState.TreeClauses.Positive
-
-end 
-
-noncomputable section
-open Set MeasureTheory ProbabilityTheory
-open scoped NNReal
-namespace ThreeState.TreeClauses.Experiment
-open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive ThreeState.TreeClauses.Probability
-
-lemma finite_iMean_succ (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
-    iMean (finiteLaw Q hl₀ hl₁ (n+1)) = iMean (finiteLaw Q hl₀ hl₁ n)+edgeIMean Q lam-
-      pairCorrelation (finiteLaw Q hl₀ hl₁ n) Q lam := by
-  unfold iMean edgeIMean
-  rw [finiteLaw_info, finiteLaw_info, correlationEntropy_succ Q hl₀ hl₁, Nat.cast_add, Nat.cast_one]
-  ring
-
-lemma finite_iMean_zero (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) :
-    iMean (finiteLaw Q hl₀ hl₁ 0) = 0 := by
-  rw [iMean, finiteLaw_info]
-  norm_num [correlationEntropy, normalizer, branchProduct, avg]
-
-lemma poissonCorrelation_zero (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) :
-    poissonCorrelation Q hl₀ hl₁ 0 = 0 := by
-  simp [poissonCorrelation, poissonAverage_zero, finite_iMean_zero]
-
-def poissonSlope (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (t : ℝ) : ℝ :=
-  edgeIMean Q lam-(poissonAverage (fun n ↦ iMean (finiteLaw Q hl₀ hl₁ (n+1))) t-
-    poissonAverage (fun n ↦ iMean (finiteLaw Q hl₀ hl₁ n)) t)
-
-lemma hasDerivAt_poissonCorrelation (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (t : ℝ) :
-    HasDerivAt (poissonCorrelation Q hl₀ hl₁) (poissonSlope Q hl₀ hl₁ t) t := by
-  obtain ⟨B,hB⟩ := bounded_iMean
-  unfold poissonCorrelation poissonSlope
-  convert ((hasDerivAt_id t).mul_const (edgeIMean Q lam)).sub
-    (hasDerivAt_poissonAverage (fun n ↦ hB (finiteLaw Q hl₀ hl₁ n)) t) using 1 <;> first | rfl | simp only [one_mul]
-
-lemma continuous_poissonSlope (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) :
-    Continuous (poissonSlope Q hl₀ hl₁) := by
-  obtain ⟨B,hB⟩ := bounded_iMean
-  exact continuous_const.sub
-    ((continuous_poissonAverage (fun n ↦ hB (finiteLaw Q hl₀ hl₁ (n+1)))).sub
-      (continuous_poissonAverage (fun n ↦ hB (finiteLaw Q hl₀ hl₁ n))))
-
-lemma bounded_pairCorrelation (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) :
-    ∃ B : ℝ, ∀ R : Law, |pairCorrelation R Q lam| ≤ B := by
-  have hc := continuous_compact_integral (μ := Q.probability.toMeasure) (continuous_pairEntropy hl₀ hl₁)
-  obtain ⟨B,hB⟩ := isCompact_univ.exists_bound_of_continuousOn hc.continuousOn
-  refine ⟨B, fun R ↦ ?_⟩
-  apply (abs_integral_le_integral_abs).trans
-  exact (integral_mono (compact_integrable hc.abs) (integrable_const B)
-    (fun r ↦ by simpa only [Real.norm_eq_abs] using hB r (mem_univ r))).trans_eq (by simp)
-
-lemma poissonSlope_eq_pairCorrelation (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (t : ℝ≥0) :
-    poissonSlope Q hl₀ hl₁ t = pairCorrelation (poissonLaw Q hl₀ hl₁ t) Q lam := by
-  obtain ⟨B,hB⟩ := bounded_iMean
-  obtain ⟨C,hC⟩ := bounded_pairCorrelation Q hl₀ hl₁
-  have hi := pmf_bounded_integrable (poissonPMF t) (fun n ↦ hB (finiteLaw Q hl₀ hl₁ n))
-  have hi' := pmf_bounded_integrable (poissonPMF t) (fun n ↦ hB (finiteLaw Q hl₀ hl₁ (n+1)))
-  have hc := pmf_bounded_integrable (poissonPMF t) (fun n ↦ hC (finiteLaw Q hl₀ hl₁ n))
-  have he : pairCorrelation (poissonLaw Q hl₀ hl₁ t) Q lam =
-      ∫ n, pairCorrelation (finiteLaw Q hl₀ hl₁ n) Q lam ∂(poissonPMF t).toMeasure := by
-    change (∫ r, (∫ m, pairZ lam r m*Real.log (pairZ lam r m) ∂Q.probability.toMeasure)
-      ∂(mixtureProbability (poissonPMF t) (fun n ↦ (finiteLaw Q hl₀ hl₁ n).probability)).toMeasure) = _
-    rw [integral_mixtureProbability _ _ (compact_integrable (continuous_compact_integral (continuous_pairEntropy hl₀ hl₁))),
-      PMF.integral_eq_tsum _ _ hc]
-    rfl
-  rw [he, poissonSlope, poissonAverage_eq_integral (fun n ↦ hB (finiteLaw Q hl₀ hl₁ (n+1))),
-    poissonAverage_eq_integral (fun n ↦ hB (finiteLaw Q hl₀ hl₁ n))]
-  have he' := integral_congr_ae (μ := (poissonPMF t).toMeasure)
-    (Filter.Eventually.of_forall (finite_iMean_succ Q hl₀ hl₁))
-  rw [integral_sub (f := fun n ↦ iMean (finiteLaw Q hl₀ hl₁ n)+edgeIMean Q lam)
-    (g := fun n ↦ pairCorrelation (finiteLaw Q hl₀ hl₁ n) Q lam) (hi.add (integrable_const _)) hc,
-    integral_add (f := fun n ↦ iMean (finiteLaw Q hl₀ hl₁ n))
-      (g := fun _ ↦ edgeIMean Q lam) hi (integrable_const _)] at he'
-  simp only [integral_const, Measure.real, measure_univ, ENNReal.toReal_one, one_smul] at he'
-  linarith
-
-end ThreeState.TreeClauses.Experiment
-
-end 
-
-noncomputable section
-open Set MeasureTheory
-open scoped NNReal
-namespace ThreeState.TreeClauses.Positive
-open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Experiment
-
-lemma poisson_increment_bound (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
-    (d t : ℝ≥0) (htd : t ≤ d)
-    (hfixed : Q.probability = (poissonLaw Q hl₀ hl₁ d).probability) :
-    3*lam^4*Q.mu^2*(t:ℝ)/(1+(t:ℝ)*(lam^2*Q.mu))-
-      lam^4*Q.mu*Q.nu*Real.sqrt t ≤ 3*poissonSlope Q hl₀ hl₁ t := by
-  have hmu := Q.mu_nonneg
-  have hnuQ := Q.nu_nonneg
-  let L : ℝ := (t:ℝ)*(lam^2*Q.mu)/(1+(t:ℝ)*(lam^2*Q.mu))
-  have ht : 0 ≤ (t:ℝ) := t.coe_nonneg
-  have hd : 1 ≤ 1+(t:ℝ)*(lam^2*Q.mu) := by nlinarith only [mul_nonneg ht (mul_nonneg (sq_nonneg lam) hmu)]
-  have hL : 0 ≤ L := by dsimp [L]; positivity
-  have hproj := poisson_projection Q hl₀ hl₁ t
-  have hnu : (poissonLaw Q hl₀ hl₁ t).nu ≤ Q.nu := by
-    have hh := poisson_nu_mono Q hl₀ hl₁ htd
-    change _ ≤ ∫ m, xMoment m^2 ∂(poissonLaw Q hl₀ hl₁ d).probability.toMeasure at hh
-    rwa [← hfixed] at hh
-  have hc := correlation_truncated Q (poissonLaw Q hl₀ hl₁ t) hl₀ hl₁ hnu hL hproj
-  have hLu : L ≤ (t:ℝ)*(lam^2*Q.mu) := by
-    dsimp only [L]
-    exact div_le_self (by positivity) hd
-  have hs : Real.sqrt L ≤ lam*Real.sqrt Q.mu*Real.sqrt t := by
-    have he : (lam*Real.sqrt Q.mu*Real.sqrt t)^2 = (t:ℝ)*(lam^2*Q.mu) := by
-      calc
-        _ = lam^2*(Real.sqrt Q.mu)^2*(Real.sqrt t)^2 := by ring
-        _ = _ := by rw [Real.sq_sqrt Q.mu_nonneg, Real.sq_sqrt ht]; ring
-    have hp : 0 ≤ lam*Real.sqrt Q.mu*Real.sqrt t := by positivity
-    nlinarith only [he, hp, Real.sq_sqrt hL, Real.sqrt_nonneg L, hLu]
-  have hmul := mul_le_mul_of_nonneg_left hs (show 0 ≤ lam^3*Real.sqrt Q.mu*Q.nu by positivity)
-  have heq : (lam^3*Real.sqrt Q.mu*Q.nu)*(lam*Real.sqrt Q.mu*Real.sqrt t) =
-      lam^4*Q.mu*Q.nu*Real.sqrt t := by
-    calc
-      _ = lam^4*(Real.sqrt Q.mu)^2*Q.nu*Real.sqrt t := by ring
-      _ = _ := by rw [Real.sq_sqrt Q.mu_nonneg]
-  rw [heq] at hmul
-  have hmain : 3*lam^2*Q.mu*L = 3*lam^4*Q.mu^2*(t:ℝ)/(1+(t:ℝ)*(lam^2*Q.mu)) := by
-    dsimp only [L]; ring
-  rw [hmain, ← poissonSlope_eq_pairCorrelation] at hc
-  linarith only [hc, hmul]
-
-lemma sqrt_intensity_tangent {lam t : ℝ} (hl : 0 < lam) (ht : 0 ≤ t) :
-    Real.sqrt t ≤ (2*lam^2*t+1)/(2*lam*Real.sqrt 2) := by
-  have hs2 := Real.sq_sqrt (by norm_num : (0:ℝ) ≤ 2)
-  have hst := Real.sq_sqrt ht
-  have hs2p := Real.sqrt_pos.mpr (by norm_num : (0:ℝ) < 2)
-  apply (le_div_iff₀ (by positivity : 0 < 2*lam*Real.sqrt 2)).2
-  have he : (lam*Real.sqrt 2*Real.sqrt t)^2 = 2*lam^2*t := by
-    calc
-      _ = lam^2*(Real.sqrt 2)^2*(Real.sqrt t)^2 := by ring
-      _ = _ := by rw [hs2, hst]; ring
-  nlinarith only [sq_nonneg (lam*Real.sqrt 2*Real.sqrt t-1), he]
-
-theorem poisson_fixed_correlation_bound (Q : Law) {lam : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1)
-    (d : ℝ≥0) (hcrit : (d:ℝ)*lam^2 = 1)
-    (hfixed : Q.probability = (poissonLaw Q hl₀.le hl₁ d).probability) :
-    3*Q.mu^2/(2*(1+2*Q.mu/3))-lam/Real.sqrt 2*Q.mu*Q.nu ≤
-      3*poissonCorrelation Q hl₀.le hl₁ d := by
-  let q := 2*Q.mu/3
-  let β := lam^2*Q.mu
-  let A := 3*lam^4*Q.mu^2/(1+q)^2
-  let B := lam^3*Q.mu*Q.nu/(2*Real.sqrt 2)
-  let P : ℝ → ℝ := fun t ↦ A*((1+2*q)*t^2/2-β*t^3/3)-B*(lam^2*t^2+t)
-  let P' : ℝ → ℝ := fun t ↦ A*((1+2*q)*t-β*t^2)-B*(2*lam^2*t+1)
-  have hmu := Q.mu_nonneg
-  have hnu := Q.nu_nonneg
-  have hq : 0 ≤ q := by dsimp [q]; positivity
-  have hβ : 0 ≤ β := by dsimp [β]; positivity
-  have hA : 0 ≤ A := by dsimp [A]; positivity
-  have hB : 0 ≤ B := by dsimp [B]; positivity
-  have hsq : 0 < 1+q := by positivity
-  have hs2 : 0 < Real.sqrt 2 := Real.sqrt_pos.mpr (by norm_num)
-  have dP (t : ℝ) : HasDerivAt P (P' t) t := by
-    apply ((((((hasDerivAt_id t).pow 2).const_mul (1+2*q)).div_const 2 |>.sub
-      ((((hasDerivAt_id t).pow 3).const_mul β).div_const 3)).const_mul A).sub
-      (((((hasDerivAt_id t).pow 2).const_mul (lam^2)).add (hasDerivAt_id t)).const_mul B)).congr_deriv
-    dsimp [P']; norm_num only [Nat.cast_ofNat, Nat.reduceSub, pow_one]; ring
-  have hbound (t : ℝ) (ht : t ∈ Icc 0 (d:ℝ)) : P' t ≤ 3*poissonSlope Q hl₀.le hl₁ t := by
-    let u : ℝ≥0 := ⟨t,ht.1⟩
-    have hi := poisson_increment_bound Q hl₀.le hl₁ d u ht.2 hfixed
-    have hr := mul_le_mul_of_nonneg_left (reciprocal_tangent (mul_nonneg ht.1 hβ) hq)
-      (mul_nonneg hA ht.1)
-    have hr' : A*((1+2*q)*t-β*t^2) ≤ 3*lam^4*Q.mu^2*t/(1+t*β) := by
-      calc
-        _ = A*t*(1+2*q-t*β) := by ring
-        _ ≤ A*t*((1+q)^2/(1+t*β)) := hr
-        _ = _ := by dsimp only [A]; field_simp
-    have hs := mul_le_mul_of_nonneg_left (sqrt_intensity_tangent hl₀ ht.1)
-      (show 0 ≤ lam^4*Q.mu*Q.nu by positivity)
-    have hs' : lam^4*Q.mu*Q.nu*Real.sqrt t ≤ B*(2*lam^2*t+1) := by
-      calc
-        _ ≤ lam^4*Q.mu*Q.nu*((2*lam^2*t+1)/(2*lam*Real.sqrt 2)) := hs
-        _ = _ := by dsimp only [B]; field_simp
-    change 3*lam^4*Q.mu^2*t/(1+t*β)-lam^4*Q.mu*Q.nu*Real.sqrt t ≤
-      3*poissonSlope Q hl₀.le hl₁ t at hi
-    dsimp only [P']
-    linarith only [hr', hs', hi]
-  have hm : MonotoneOn (fun t ↦ 3*poissonCorrelation Q hl₀.le hl₁ t-P t) (Icc 0 (d:ℝ)) := by
-    apply monotoneOn_Icc_of_hasDerivAt_nonneg
-      (fun t _ ↦ ((hasDerivAt_poissonCorrelation Q hl₀.le hl₁ t).const_mul 3).sub (dP t))
-    intro t ht
-    exact sub_nonneg.mpr (hbound t ⟨ht.1.le,ht.2.le⟩)
-  have hh := hm ⟨le_rfl,d.coe_nonneg⟩ ⟨d.coe_nonneg,le_rfl⟩ d.coe_nonneg
-  have hz : P 0 = 0 := by dsimp [P]; ring
-  change 3*poissonCorrelation Q hl₀.le hl₁ 0-P 0 ≤ 3*poissonCorrelation Q hl₀.le hl₁ d-P d at hh
-  rw [poissonCorrelation_zero, hz] at hh
-  have hd : (d:ℝ) = 1/lam^2 := (eq_div_iff (ne_of_gt (sq_pos_of_pos hl₀))).2 hcrit
-  have hp : P d = 3*Q.mu^2/(2*(1+2*Q.mu/3))-lam/Real.sqrt 2*Q.mu*Q.nu := by
-    dsimp only [P,A,B,β,q]
-    rw [hd]
-    field_simp
-    ring
-  rw [hp] at hh
-  linarith only [hh]
-
-end ThreeState.TreeClauses.Positive
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeStateClauses.PositiveFixed
+import OAI.Probability.ThreeStateClauses.PoissonLaw
+
+namespace OAI
+
+/-! Poisson Calculus. -/
+
+open scoped BigOperators ENNReal NNReal Topology
+open Filter
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped BigOperators NNReal
+namespace ThreeState.TreeClauses.Probability
+
+def exponentialAverage (f : ℕ → ℝ) (t : ℝ) : ℝ := ∑' n, t^n/(n.factorial:ℝ)*f n
+
+def poissonAverage (f : ℕ → ℝ) (t : ℝ) : ℝ := Real.exp (-t)*exponentialAverage f t
+
+lemma exponentialAverage_summable {f : ℕ → ℝ} {B : ℝ} (hB : ∀ n, |f n| ≤ B) (t : ℝ) :
+    Summable (fun n ↦ t^n/(n.factorial:ℝ)*f n) := by
+  apply Summable.of_norm_bounded ((Real.summable_pow_div_factorial |t|).mul_right B)
+  intro n
+  rw [Real.norm_eq_abs, abs_mul, abs_div, abs_pow, Nat.abs_cast]
+  exact mul_le_mul_of_nonneg_left (hB n) (by positivity)
+
+private def exponentialDerivative (f : ℕ → ℝ) : ℕ → ℝ → ℝ
+  | 0, _ => 0
+  | n+1, t => t^n/(n.factorial:ℝ)*f (n+1)
+
+private lemma exponentialDerivative_summable {f : ℕ → ℝ} {B : ℝ} (hB : ∀ n, |f n| ≤ B) (t : ℝ) :
+    Summable (fun n ↦ exponentialDerivative f n t) := by
+  apply (summable_nat_add_iff 1).mp
+  exact exponentialAverage_summable (fun n ↦ hB (n+1)) t
+
+private lemma exponentialDerivative_sum {f : ℕ → ℝ} {B : ℝ} (hB : ∀ n, |f n| ≤ B) (t : ℝ) :
+    (∑' n, exponentialDerivative f n t) = exponentialAverage (fun n ↦ f (n+1)) t := by
+  rw [← (exponentialDerivative_summable hB t).sum_add_tsum_nat_add 1]
+  simp [exponentialDerivative, exponentialAverage]
+
+lemma hasDerivAt_exponentialAverage {f : ℕ → ℝ} {B : ℝ} (hB : ∀ n, |f n| ≤ B) (t : ℝ) :
+    HasDerivAt (exponentialAverage f) (exponentialAverage (fun n ↦ f (n+1)) t) t := by
+  let R := |t|+1
+  have hR : 0 < R := by dsimp [R]; positivity
+  have ht : t ∈ Ioo (-R) R := by
+    constructor <;> dsimp [R] <;> linarith [le_abs_self t, neg_abs_le t]
+  let u : ℕ → ℝ := fun n ↦ exponentialDerivative (fun _ ↦ B) n R
+  have hu : Summable u := exponentialDerivative_summable (B := |B|) (fun _ ↦ le_refl _) R
+  have hd (n : ℕ) (y : ℝ) : HasDerivAt (fun z : ℝ ↦ z^n/(n.factorial:ℝ)*f n)
+      (exponentialDerivative f n y) y := by
+    cases n with
+    | zero => simpa [exponentialDerivative] using hasDerivAt_const y (f 0)
+    | succ n =>
+      convert (((hasDerivAt_id y).pow (n+1)).div_const ((n+1).factorial:ℝ)).mul_const (f (n+1)) using 1
+      all_goals try rfl
+      all_goals try simp only [exponentialDerivative, id_eq]
+      rw [Nat.factorial_succ, Nat.cast_mul]
+      simp only [Nat.add_sub_cancel, mul_one]
+      have hn : ((n+1:ℕ):ℝ) ≠ 0 := by positivity
+      field_simp
+  have hb (n : ℕ) (y : ℝ) (hy : y ∈ Ioo (-R) R) : ‖exponentialDerivative f n y‖ ≤ u n := by
+    have hyR : |y| ≤ R := (abs_le).2 ⟨le_of_lt hy.1, le_of_lt hy.2⟩
+    cases n with
+    | zero => simp [exponentialDerivative, u]
+    | succ n =>
+      dsimp [exponentialDerivative, u]
+      rw [abs_mul, abs_div, abs_pow, Nat.abs_cast]
+      exact mul_le_mul (div_le_div_of_nonneg_right (pow_le_pow_left₀ (abs_nonneg y) hyR n) (by positivity))
+        (hB (n+1)) (abs_nonneg _) (by positivity)
+  have hh := hasDerivAt_tsum_of_isPreconnected hu isOpen_Ioo (convex_Ioo (-R) R).isPreconnected
+    (fun n y _ ↦ hd n y) hb ht (exponentialAverage_summable hB t) ht
+  rw [exponentialDerivative_sum hB t] at hh
+  exact hh
+
+theorem hasDerivAt_poissonAverage {f : ℕ → ℝ} {B : ℝ} (hB : ∀ n, |f n| ≤ B) (t : ℝ) :
+    HasDerivAt (poissonAverage f)
+      (poissonAverage (fun n ↦ f (n+1)) t-poissonAverage f t) t := by
+  have hd := ((hasDerivAt_neg t).exp).mul (hasDerivAt_exponentialAverage hB t)
+  exact hd.congr_deriv (by unfold poissonAverage; ring)
+
+lemma poissonAverage_eq_tsum {f : ℕ → ℝ} {B : ℝ} (hB : ∀ n, |f n| ≤ B) (t : ℝ) :
+    poissonAverage f t = ∑' n, poissonWeight t n*f n := by
+  rw [poissonAverage, exponentialAverage, ← (exponentialAverage_summable hB t).tsum_mul_left]
+  apply tsum_congr; intro n; unfold poissonWeight; ring
+
+lemma poissonAverage_eq_integral {f : ℕ → ℝ} {B : ℝ} (hB : ∀ n, |f n| ≤ B) (t : ℝ≥0) :
+    poissonAverage f t = ∫ n, f n ∂(poissonPMF t).toMeasure := by
+  rw [PMF.integral_eq_tsum]
+  · simp only [poissonPMF_real, smul_eq_mul]
+    exact poissonAverage_eq_tsum hB t
+  · exact Integrable.of_bound (measurable_of_countable _).aestronglyMeasurable B
+      (Eventually.of_forall (fun n ↦ by simpa only [Real.norm_eq_abs] using hB n))
+
+lemma continuous_poissonAverage {f : ℕ → ℝ} {B : ℝ} (hB : ∀ n, |f n| ≤ B) :
+    Continuous (poissonAverage f) := continuous_iff_continuousAt.mpr
+  (fun t ↦ (hasDerivAt_poissonAverage hB t).continuousAt)
+
+lemma poissonAverage_zero (f : ℕ → ℝ) : poissonAverage f 0 = f 0 := by
+  rw [poissonAverage, neg_zero, Real.exp_zero, one_mul, exponentialAverage, tsum_eq_single 0]
+  · simp
+  · intro n hn
+    simp [zero_pow hn]
+
+end ThreeState.TreeClauses.Probability
+
+end 
+
+noncomputable section
+open Set MeasureTheory ProbabilityTheory
+open scoped NNReal
+namespace ThreeState.TreeClauses.Experiment
+open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive ThreeState.TreeClauses.Probability
+
+lemma offspring_positive (ρ : PMF ℕ) (Q : Law) {lam : ℝ}
+    (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) :
+    ∀ᵐ m ∂(offspringLaw ρ Q hl₀ hl₁).probability.toMeasure, PositiveMessage m.1 := by
+  apply Measure.ae_sum_iff.mpr
+  intro n
+  exact Measure.ae_smul_measure (finite_positive Q hl₀ hl₁ n) _
+
+lemma correction_nonneg (m : Message) : 0 ≤ correction m.1 := by
+  unfold correction
+  apply div_nonneg _ (by norm_num)
+  have h := avg_mono (fun i ↦ mul_nonneg (coordinate_nonneg m i) (sq_nonneg (logCentered m.1 i)))
+  simpa only [avg_const] using h
+
+private lemma poisson_mixture_integrable_nonneg (t : ℝ≥0) (P : ℕ → ProbabilityMeasure Message)
+    {f : Message → ℝ} (hi : ∀ n, Integrable f (P n).toMeasure)
+    (hp : ∀ n, ∀ᵐ m ∂(P n).toMeasure, 0 ≤ f m)
+    (hs : Summable (fun n ↦ poissonWeight t n*(∫ m, f m ∂(P n).toMeasure))) :
+    Integrable f (mixtureProbability (poissonPMF t) P).toMeasure := by
+  apply mixture_integrable _ P hi
+  simp only [poissonPMF_real]
+  apply hs.congr
+  intro n
+  congr 1
+  exact integral_congr_ae ((hp n).mono (fun m hm ↦ (abs_of_nonneg hm).symm))
+
+lemma poisson_integrable_symEntropy (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
+    (t : ℝ≥0) : Integrable (fun m : Message ↦ symEntropy m.1) (poissonLaw Q hl₀ hl₁ t).probability.toMeasure := by
+  apply poisson_mixture_integrable_nonneg t (fun n ↦ finiteProbability Q hl₀ hl₁ n)
+    (finite_integrable_symEntropy Q hl₀ hl₁)
+    (fun n ↦ (finite_positive Q hl₀ hl₁ n).mono (fun m hm ↦ symEntropy_nonneg hm))
+  have he (n : ℕ) : (∫ m, symEntropy m.1 ∂(finiteProbability Q hl₀ hl₁ n).toMeasure) =
+      (n:ℝ)*edgeJMean Q lam := finiteLaw_symEntropy Q hl₀ hl₁ n
+  simp_rw [he, ← mul_assoc]
+  exact (poissonWeight_mean t).summable.mul_right _
+
+lemma poisson_integrable_correction (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
+    (t : ℝ≥0) : Integrable (fun m : Message ↦ correction m.1) (poissonLaw Q hl₀ hl₁ t).probability.toMeasure := by
+  apply poisson_mixture_integrable_nonneg t (fun n ↦ finiteProbability Q hl₀ hl₁ n)
+    (finite_integrable_correction Q hl₀ hl₁)
+    (fun _ ↦ Filter.Eventually.of_forall correction_nonneg)
+  have h := ((poissonWeight_mean t).summable.mul_right
+    (∫ m, correction (edgeMessage lam m) ∂Q.probability.toMeasure)).add
+    ((poissonWeight_factorial_second t).summable.mul_right (2*(edgeJMean Q lam)^2))
+  apply h.congr
+  intro n
+  have he := finiteLaw_correction Q hl₀ hl₁ n
+  change (∫ m, correction m.1 ∂(finiteProbability Q hl₀ hl₁ n).toMeasure) = _ at he
+  rw [he]
+  unfold edgeJMean
+  ring
+
+lemma poissonLaw_symEntropy (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
+    (t : ℝ≥0) :
+    (∫ m, symEntropy m.1 ∂(poissonLaw Q hl₀ hl₁ t).probability.toMeasure) = (t:ℝ)*edgeJMean Q lam := by
+  rw [show (poissonLaw Q hl₀ hl₁ t).probability = mixtureProbability (poissonPMF t)
+    (fun n ↦ (finiteLaw Q hl₀ hl₁ n).probability) from rfl,
+    integral_mixtureProbability _ _ (poisson_integrable_symEntropy Q hl₀ hl₁ t)]
+  simp only [poissonPMF_real, finiteLaw_symEntropy]
+  exact ((poissonWeight_mean t).mul_right (edgeJMean Q lam)).tsum_eq |> fun h ↦ by
+    simpa only [mul_assoc, edgeJMean] using h
+
+lemma poissonLaw_correction (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
+    (t : ℝ≥0) :
+    (∫ m, correction m.1 ∂(poissonLaw Q hl₀ hl₁ t).probability.toMeasure) =
+      (t:ℝ)*(∫ m, correction (edgeMessage lam m) ∂Q.probability.toMeasure)+
+      2*(t:ℝ)^2*(edgeJMean Q lam)^2 := by
+  rw [show (poissonLaw Q hl₀ hl₁ t).probability = mixtureProbability (poissonPMF t)
+    (fun n ↦ (finiteLaw Q hl₀ hl₁ n).probability) from rfl,
+    integral_mixtureProbability _ _ (poisson_integrable_correction Q hl₀ hl₁ t)]
+  simp only [poissonPMF_real, finiteLaw_correction]
+  have h := ((poissonWeight_mean t).mul_right
+    (∫ m, correction (edgeMessage lam m) ∂Q.probability.toMeasure)).add
+    ((poissonWeight_factorial_second t).mul_right (2*(edgeJMean Q lam)^2))
+  convert h.tsum_eq using 1
+  · apply tsum_congr; intro n; unfold edgeJMean; ring
+  · ring
+
+def iMean (Q : Law) : ℝ := ∫ m, info m.1 ∂Q.probability.toMeasure
+
+def edgeIMean (Q : Law) (lam : ℝ) : ℝ := ∫ m, info (edgeMessage lam m) ∂Q.probability.toMeasure
+
+lemma bounded_iMean : ∃ B : ℝ, ∀ Q : Law, |iMean Q| ≤ B := by
+  have hc : Continuous (fun m : Message ↦ info m.1) := continuous_info_comp continuous_subtype_val
+  obtain ⟨B,hB⟩ := isCompact_univ.exists_bound_of_continuousOn hc.continuousOn
+  refine ⟨B, fun Q ↦ ?_⟩
+  apply (abs_integral_le_integral_abs).trans
+  exact (integral_mono (compact_integrable (continuous_info_comp continuous_subtype_val).abs)
+    (integrable_const B) (fun m ↦ by simpa only [Real.norm_eq_abs] using hB m (mem_univ m))).trans_eq (by simp)
+
+def poissonCorrelation (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (t : ℝ) : ℝ :=
+  t*edgeIMean Q lam-poissonAverage (fun n ↦ iMean (finiteLaw Q hl₀ hl₁ n)) t
+
+lemma poissonLaw_info (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (t : ℝ≥0) :
+    iMean (poissonLaw Q hl₀ hl₁ t) = (t:ℝ)*edgeIMean Q lam-poissonCorrelation Q hl₀ hl₁ t := by
+  obtain ⟨B,hB⟩ := bounded_iMean
+  have he : iMean (poissonLaw Q hl₀ hl₁ t) = poissonAverage (fun n ↦ iMean (finiteLaw Q hl₀ hl₁ n)) t := by
+    unfold iMean
+    rw [show (poissonLaw Q hl₀ hl₁ t).probability = mixtureProbability (poissonPMF t)
+      (fun n ↦ (finiteLaw Q hl₀ hl₁ n).probability) from rfl,
+      integral_mixtureProbability _ _ (compact_integrable (continuous_info_comp continuous_subtype_val))]
+    change _ = poissonAverage (fun n ↦ iMean (finiteLaw Q hl₀ hl₁ n)) t
+    rw [poissonAverage_eq_tsum (fun n ↦ hB (finiteLaw Q hl₀ hl₁ n))]
+    simp only [poissonPMF_real, iMean]
+  rw [he, poissonCorrelation]
+  ring
+
+lemma poissonCorrelation_continuous (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) :
+    Continuous (poissonCorrelation Q hl₀ hl₁) := by
+  obtain ⟨B,hB⟩ := bounded_iMean
+  exact (continuous_id.mul continuous_const).sub
+    (continuous_poissonAverage (fun n ↦ hB (finiteLaw Q hl₀ hl₁ n)))
+
+end ThreeState.TreeClauses.Experiment
+
+end 
+
+noncomputable section
+open Set MeasureTheory
+open scoped NNReal
+namespace ThreeState.TreeClauses.Positive
+open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Experiment
+
+lemma poisson_fixed_edgeJ (Q : Law) {lam : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1) (d : ℝ≥0)
+    (hcrit : (d:ℝ)*lam^2 = 1) (hfixed : Q.probability = (poissonLaw Q hl₀.le hl₁ d).probability) :
+    edgeJMean Q lam = lam^2*jMean Q := by
+  have h := poissonLaw_symEntropy Q hl₀.le hl₁ d
+  change (∫ m, symEntropy m.1 ∂((poissonLaw Q hl₀.le hl₁ d).probability).toMeasure) = _ at h
+  rw [← hfixed] at h
+  change jMean Q = (d:ℝ)*edgeJMean Q lam at h
+  rw [h]
+  calc
+    _ = ((d:ℝ)*lam^2)*edgeJMean Q lam := by rw [hcrit, one_mul]
+    _ = _ := by ring
+
+lemma functional_integral {α : Type*} [MeasurableSpace α] (μ : Measure α) (f : α → Vec)
+    (hI : Integrable (fun m ↦ info (f m)) μ)
+    (hJ : Integrable (fun m ↦ symEntropy (f m)) μ)
+    (hB : Integrable (fun m ↦ correction (f m)) μ) :
+    (∫ m, functional (f m) ∂μ) = 3*(∫ m, info (f m) ∂μ)-
+      2*(∫ m, symEntropy (f m) ∂μ)+(4/5)*(∫ m, correction (f m) ∂μ) := by
+  unfold functional
+  rw [integral_add (f := fun m ↦ 3*info (f m)-2*symEntropy (f m))
+    (g := fun m ↦ (4/5)*correction (f m)) ((hI.const_mul _).sub (hJ.const_mul _)) (hB.const_mul _),
+    integral_sub (f := fun m ↦ 3*info (f m)) (g := fun m ↦ 2*symEntropy (f m))
+      (hI.const_mul _) (hJ.const_mul _), integral_const_mul, integral_const_mul, integral_const_mul]
+
+lemma integrated_radial (Q : Law) {lam : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1)
+    (hp : ∀ᵐ m ∂Q.probability.toMeasure, PositiveMessage m.1)
+    (hJ : Integrable (fun m : Message ↦ symEntropy m.1) Q.probability.toMeasure)
+    (hB : Integrable (fun m : Message ↦ correction m.1) Q.probability.toMeasure) :
+    (1-lam^2)/2*Q.nu ≤ (∫ m, functional m.1 ∂Q.probability.toMeasure)-
+    (lam^2)⁻¹*(∫ m, functional (edgeMessage lam m) ∂Q.probability.toMeasure) := by
+  have hI : Integrable (fun m : Message ↦ info m.1) Q.probability.toMeasure :=
+    compact_integrable (continuous_info_comp continuous_subtype_val)
+  have hJe := edgeJ_integrable Q hl₀.le hl₁
+  have hIe : Integrable (fun m : Message ↦ info (edgeMessage lam m)) Q.probability.toMeasure :=
+    compact_integrable (continuous_info_comp (continuous_edgeMessage lam))
+  have hBe : Integrable (fun m : Message ↦ correction (edgeMessage lam m)) Q.probability.toMeasure :=
+    compact_integrable (continuous_correction_comp (continuous_edgeMessage lam)
+      (fun m i ↦ (edge_positive hl₀.le hl₁ m).1 i))
+  have hF : Integrable (fun m : Message ↦ functional m.1) Q.probability.toMeasure :=
+    (hI.const_mul 3 |>.sub (hJ.const_mul 2)).add (hB.const_mul (4/5))
+  have hFe : Integrable (fun m : Message ↦ functional (edgeMessage lam m)) Q.probability.toMeasure :=
+    (hIe.const_mul 3 |>.sub (hJe.const_mul 2)).add (hBe.const_mul (4/5))
+  have hrad := integral_mono_ae (μ := Q.probability.toMeasure)
+    (f := fun m : Message ↦ (1-lam^2)/2*xMoment m^2)
+    (g := fun m ↦ functional m.1-(lam^2)⁻¹*functional (edgeMessage lam m))
+    ((compact_integrable (continuous_xMoment.pow 2)).const_mul _) (hF.sub (hFe.const_mul _))
+    (hp.mono (fun m hm ↦ radial_inequality m.1 hm lam hl₀ hl₁))
+  rw [integral_const_mul, integral_sub hF (hFe.const_mul _), integral_const_mul] at hrad
+  exact hrad
+
+lemma poisson_fixed_entropy_radial (Q : Law) {lam : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1) (d : ℝ≥0)
+    (hcrit : (d:ℝ)*lam^2 = 1) (hfixed : Q.probability = (poissonLaw Q hl₀.le hl₁ d).probability) :
+    (1-lam^2)/2*Q.nu ≤ (8/5)*(jMean Q)^2-3*poissonCorrelation Q hl₀.le hl₁ d := by
+  have hp := offspring_positive (poissonPMF d) Q hl₀.le hl₁
+  have hJ := poisson_integrable_symEntropy Q hl₀.le hl₁ d
+  have hB := poisson_integrable_correction Q hl₀.le hl₁ d
+  change ∀ᵐ m ∂(poissonLaw Q hl₀.le hl₁ d).probability.toMeasure, PositiveMessage m.1 at hp
+  rw [← hfixed] at hp hJ hB
+  have hI : Integrable (fun m : Message ↦ info m.1) Q.probability.toMeasure :=
+    compact_integrable (continuous_info_comp continuous_subtype_val)
+  have hJe := edgeJ_integrable Q hl₀.le hl₁
+  have hIe : Integrable (fun m : Message ↦ info (edgeMessage lam m)) Q.probability.toMeasure :=
+    compact_integrable (continuous_info_comp (continuous_edgeMessage lam))
+  have hBe : Integrable (fun m : Message ↦ correction (edgeMessage lam m)) Q.probability.toMeasure :=
+    compact_integrable (continuous_correction_comp (continuous_edgeMessage lam) (fun m i ↦ (edge_positive hl₀.le hl₁ m).1 i))
+  have hrad := integrated_radial Q hl₀ hl₁ hp hJ hB
+  have hJid := poissonLaw_symEntropy Q hl₀.le hl₁ d
+  have hBid := poissonLaw_correction Q hl₀.le hl₁ d
+  have hIid := poissonLaw_info Q hl₀.le hl₁ d
+  change (∫ m, symEntropy m.1 ∂((poissonLaw Q hl₀.le hl₁ d).probability).toMeasure) = _ at hJid
+  change (∫ m, correction m.1 ∂((poissonLaw Q hl₀.le hl₁ d).probability).toMeasure) = _ at hBid
+  change (∫ m, info m.1 ∂((poissonLaw Q hl₀.le hl₁ d).probability).toMeasure) = _ at hIid
+  rw [← hfixed] at hJid hBid hIid
+  have hninv : (lam^2)⁻¹ = (d:ℝ) := by
+    apply mul_left_cancel₀ (ne_of_gt (sq_pos_of_pos hl₀))
+    rw [mul_inv_cancel₀ (ne_of_gt (sq_pos_of_pos hl₀))]
+    nlinarith [hcrit]
+  have hFid := functional_integral Q.probability.toMeasure (fun m : Message ↦ m.1) hI hJ hB
+  have hFeid := functional_integral Q.probability.toMeasure (edgeMessage lam) hIe hJe hBe
+  rw [hFid, hFeid, hninv, hIid, hBid] at hrad
+  change jMean Q = (d:ℝ)*edgeJMean Q lam at hJid
+  change _ ≤ 3*((d:ℝ)*(∫ m, info (edgeMessage lam m) ∂Q.probability.toMeasure)-poissonCorrelation Q hl₀.le hl₁ d)-
+    2*jMean Q+(4/5)*((d:ℝ)*(∫ m, correction (edgeMessage lam m) ∂Q.probability.toMeasure)+
+      2*((d:ℝ)^2)*(edgeJMean Q lam)^2)-
+    (d:ℝ)*(3*(∫ m, info (edgeMessage lam m) ∂Q.probability.toMeasure)-2*edgeJMean Q lam+
+      (4/5)*(∫ m, correction (edgeMessage lam m) ∂Q.probability.toMeasure)) at hrad
+  have he : ((d:ℝ)^2)*(edgeJMean Q lam)^2 = (jMean Q)^2 := by
+    rw [poisson_fixed_edgeJ Q hl₀ hl₁ d hcrit hfixed]
+    have hn : ((d:ℝ)^2)*lam^4 = 1 := by
+      nlinarith [sq_nonneg ((d:ℝ)*lam^2-1), hcrit]
+    calc
+      _ = (((d:ℝ)^2)*lam^4)*(jMean Q)^2 := by ring
+      _ = _ := by rw [hn]; ring
+  nlinarith [hJid, he]
+
+lemma poisson_fixed_moment_bounds (Q : Law) {lam : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1) (d : ℝ≥0)
+    (hcrit : (d:ℝ)*lam^2 = 1) (hfixed : Q.probability = (poissonLaw Q hl₀.le hl₁ d).probability) :
+    0 ≤ jMean Q ∧ jMean Q ≤ Q.mu-lam*qExact lam*Q.nu ∧ cExact lam*Q.nu ≤ Q.eta := by
+  have hp := offspring_positive (poissonPMF d) Q hl₀.le hl₁
+  have hJ := poisson_integrable_symEntropy Q hl₀.le hl₁ d
+  change ∀ᵐ m ∂(poissonLaw Q hl₀.le hl₁ d).probability.toMeasure, PositiveMessage m.1 at hp
+  rw [← hfixed] at hp hJ
+  exact expected_moment_bounds Q hl₀ hl₁ hp hJ (poisson_fixed_edgeJ Q hl₀ hl₁ d hcrit hfixed)
+
+end ThreeState.TreeClauses.Positive
+
+end 
+
+noncomputable section
+open Set MeasureTheory ProbabilityTheory
+open scoped NNReal
+namespace ThreeState.TreeClauses.Experiment
+open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive ThreeState.TreeClauses.Probability
+
+lemma finite_iMean_succ (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
+    iMean (finiteLaw Q hl₀ hl₁ (n+1)) = iMean (finiteLaw Q hl₀ hl₁ n)+edgeIMean Q lam-
+      pairCorrelation (finiteLaw Q hl₀ hl₁ n) Q lam := by
+  unfold iMean edgeIMean
+  rw [finiteLaw_info, finiteLaw_info, correlationEntropy_succ Q hl₀ hl₁, Nat.cast_add, Nat.cast_one]
+  ring
+
+lemma finite_iMean_zero (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) :
+    iMean (finiteLaw Q hl₀ hl₁ 0) = 0 := by
+  rw [iMean, finiteLaw_info]
+  norm_num [correlationEntropy, normalizer, branchProduct, avg]
+
+lemma poissonCorrelation_zero (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) :
+    poissonCorrelation Q hl₀ hl₁ 0 = 0 := by
+  simp [poissonCorrelation, poissonAverage_zero, finite_iMean_zero]
+
+def poissonSlope (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (t : ℝ) : ℝ :=
+  edgeIMean Q lam-(poissonAverage (fun n ↦ iMean (finiteLaw Q hl₀ hl₁ (n+1))) t-
+    poissonAverage (fun n ↦ iMean (finiteLaw Q hl₀ hl₁ n)) t)
+
+lemma hasDerivAt_poissonCorrelation (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (t : ℝ) :
+    HasDerivAt (poissonCorrelation Q hl₀ hl₁) (poissonSlope Q hl₀ hl₁ t) t := by
+  obtain ⟨B,hB⟩ := bounded_iMean
+  unfold poissonCorrelation poissonSlope
+  convert ((hasDerivAt_id t).mul_const (edgeIMean Q lam)).sub
+    (hasDerivAt_poissonAverage (fun n ↦ hB (finiteLaw Q hl₀ hl₁ n)) t) using 1 <;> first | rfl | simp only [one_mul]
+
+lemma continuous_poissonSlope (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) :
+    Continuous (poissonSlope Q hl₀ hl₁) := by
+  obtain ⟨B,hB⟩ := bounded_iMean
+  exact continuous_const.sub
+    ((continuous_poissonAverage (fun n ↦ hB (finiteLaw Q hl₀ hl₁ (n+1)))).sub
+      (continuous_poissonAverage (fun n ↦ hB (finiteLaw Q hl₀ hl₁ n))))
+
+lemma bounded_pairCorrelation (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) :
+    ∃ B : ℝ, ∀ R : Law, |pairCorrelation R Q lam| ≤ B := by
+  have hc := continuous_compact_integral (μ := Q.probability.toMeasure) (continuous_pairEntropy hl₀ hl₁)
+  obtain ⟨B,hB⟩ := isCompact_univ.exists_bound_of_continuousOn hc.continuousOn
+  refine ⟨B, fun R ↦ ?_⟩
+  apply (abs_integral_le_integral_abs).trans
+  exact (integral_mono (compact_integrable hc.abs) (integrable_const B)
+    (fun r ↦ by simpa only [Real.norm_eq_abs] using hB r (mem_univ r))).trans_eq (by simp)
+
+lemma poissonSlope_eq_pairCorrelation (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (t : ℝ≥0) :
+    poissonSlope Q hl₀ hl₁ t = pairCorrelation (poissonLaw Q hl₀ hl₁ t) Q lam := by
+  obtain ⟨B,hB⟩ := bounded_iMean
+  obtain ⟨C,hC⟩ := bounded_pairCorrelation Q hl₀ hl₁
+  have hi := pmf_bounded_integrable (poissonPMF t) (fun n ↦ hB (finiteLaw Q hl₀ hl₁ n))
+  have hi' := pmf_bounded_integrable (poissonPMF t) (fun n ↦ hB (finiteLaw Q hl₀ hl₁ (n+1)))
+  have hc := pmf_bounded_integrable (poissonPMF t) (fun n ↦ hC (finiteLaw Q hl₀ hl₁ n))
+  have he : pairCorrelation (poissonLaw Q hl₀ hl₁ t) Q lam =
+      ∫ n, pairCorrelation (finiteLaw Q hl₀ hl₁ n) Q lam ∂(poissonPMF t).toMeasure := by
+    change (∫ r, (∫ m, pairZ lam r m*Real.log (pairZ lam r m) ∂Q.probability.toMeasure)
+      ∂(mixtureProbability (poissonPMF t) (fun n ↦ (finiteLaw Q hl₀ hl₁ n).probability)).toMeasure) = _
+    rw [integral_mixtureProbability _ _ (compact_integrable (continuous_compact_integral (continuous_pairEntropy hl₀ hl₁))),
+      PMF.integral_eq_tsum _ _ hc]
+    rfl
+  rw [he, poissonSlope, poissonAverage_eq_integral (fun n ↦ hB (finiteLaw Q hl₀ hl₁ (n+1))),
+    poissonAverage_eq_integral (fun n ↦ hB (finiteLaw Q hl₀ hl₁ n))]
+  have he' := integral_congr_ae (μ := (poissonPMF t).toMeasure)
+    (Filter.Eventually.of_forall (finite_iMean_succ Q hl₀ hl₁))
+  rw [integral_sub (f := fun n ↦ iMean (finiteLaw Q hl₀ hl₁ n)+edgeIMean Q lam)
+    (g := fun n ↦ pairCorrelation (finiteLaw Q hl₀ hl₁ n) Q lam) (hi.add (integrable_const _)) hc,
+    integral_add (f := fun n ↦ iMean (finiteLaw Q hl₀ hl₁ n))
+      (g := fun _ ↦ edgeIMean Q lam) hi (integrable_const _)] at he'
+  simp only [integral_const, Measure.real, measure_univ, ENNReal.toReal_one, one_smul] at he'
+  linarith
+
+end ThreeState.TreeClauses.Experiment
+
+end 
+
+noncomputable section
+open Set MeasureTheory
+open scoped NNReal
+namespace ThreeState.TreeClauses.Positive
+open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Experiment
+
+lemma poisson_increment_bound (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
+    (d t : ℝ≥0) (htd : t ≤ d)
+    (hfixed : Q.probability = (poissonLaw Q hl₀ hl₁ d).probability) :
+    3*lam^4*Q.mu^2*(t:ℝ)/(1+(t:ℝ)*(lam^2*Q.mu))-
+      lam^4*Q.mu*Q.nu*Real.sqrt t ≤ 3*poissonSlope Q hl₀ hl₁ t := by
+  have hmu := Q.mu_nonneg
+  have hnuQ := Q.nu_nonneg
+  let L : ℝ := (t:ℝ)*(lam^2*Q.mu)/(1+(t:ℝ)*(lam^2*Q.mu))
+  have ht : 0 ≤ (t:ℝ) := t.coe_nonneg
+  have hd : 1 ≤ 1+(t:ℝ)*(lam^2*Q.mu) := by nlinarith only [mul_nonneg ht (mul_nonneg (sq_nonneg lam) hmu)]
+  have hL : 0 ≤ L := by dsimp [L]; positivity
+  have hproj := poisson_projection Q hl₀ hl₁ t
+  have hnu : (poissonLaw Q hl₀ hl₁ t).nu ≤ Q.nu := by
+    have hh := poisson_nu_mono Q hl₀ hl₁ htd
+    change _ ≤ ∫ m, xMoment m^2 ∂(poissonLaw Q hl₀ hl₁ d).probability.toMeasure at hh
+    rwa [← hfixed] at hh
+  have hc := correlation_truncated Q (poissonLaw Q hl₀ hl₁ t) hl₀ hl₁ hnu hL hproj
+  have hLu : L ≤ (t:ℝ)*(lam^2*Q.mu) := by
+    dsimp only [L]
+    exact div_le_self (by positivity) hd
+  have hs : Real.sqrt L ≤ lam*Real.sqrt Q.mu*Real.sqrt t := by
+    have he : (lam*Real.sqrt Q.mu*Real.sqrt t)^2 = (t:ℝ)*(lam^2*Q.mu) := by
+      calc
+        _ = lam^2*(Real.sqrt Q.mu)^2*(Real.sqrt t)^2 := by ring
+        _ = _ := by rw [Real.sq_sqrt Q.mu_nonneg, Real.sq_sqrt ht]; ring
+    have hp : 0 ≤ lam*Real.sqrt Q.mu*Real.sqrt t := by positivity
+    nlinarith only [he, hp, Real.sq_sqrt hL, Real.sqrt_nonneg L, hLu]
+  have hmul := mul_le_mul_of_nonneg_left hs (show 0 ≤ lam^3*Real.sqrt Q.mu*Q.nu by positivity)
+  have heq : (lam^3*Real.sqrt Q.mu*Q.nu)*(lam*Real.sqrt Q.mu*Real.sqrt t) =
+      lam^4*Q.mu*Q.nu*Real.sqrt t := by
+    calc
+      _ = lam^4*(Real.sqrt Q.mu)^2*Q.nu*Real.sqrt t := by ring
+      _ = _ := by rw [Real.sq_sqrt Q.mu_nonneg]
+  rw [heq] at hmul
+  have hmain : 3*lam^2*Q.mu*L = 3*lam^4*Q.mu^2*(t:ℝ)/(1+(t:ℝ)*(lam^2*Q.mu)) := by
+    dsimp only [L]; ring
+  rw [hmain, ← poissonSlope_eq_pairCorrelation] at hc
+  linarith only [hc, hmul]
+
+lemma sqrt_intensity_tangent {lam t : ℝ} (hl : 0 < lam) (ht : 0 ≤ t) :
+    Real.sqrt t ≤ (2*lam^2*t+1)/(2*lam*Real.sqrt 2) := by
+  have hs2 := Real.sq_sqrt (by norm_num : (0:ℝ) ≤ 2)
+  have hst := Real.sq_sqrt ht
+  have hs2p := Real.sqrt_pos.mpr (by norm_num : (0:ℝ) < 2)
+  apply (le_div_iff₀ (by positivity : 0 < 2*lam*Real.sqrt 2)).2
+  have he : (lam*Real.sqrt 2*Real.sqrt t)^2 = 2*lam^2*t := by
+    calc
+      _ = lam^2*(Real.sqrt 2)^2*(Real.sqrt t)^2 := by ring
+      _ = _ := by rw [hs2, hst]; ring
+  nlinarith only [sq_nonneg (lam*Real.sqrt 2*Real.sqrt t-1), he]
+
+theorem poisson_fixed_correlation_bound (Q : Law) {lam : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1)
+    (d : ℝ≥0) (hcrit : (d:ℝ)*lam^2 = 1)
+    (hfixed : Q.probability = (poissonLaw Q hl₀.le hl₁ d).probability) :
+    3*Q.mu^2/(2*(1+2*Q.mu/3))-lam/Real.sqrt 2*Q.mu*Q.nu ≤
+      3*poissonCorrelation Q hl₀.le hl₁ d := by
+  let q := 2*Q.mu/3
+  let β := lam^2*Q.mu
+  let A := 3*lam^4*Q.mu^2/(1+q)^2
+  let B := lam^3*Q.mu*Q.nu/(2*Real.sqrt 2)
+  let P : ℝ → ℝ := fun t ↦ A*((1+2*q)*t^2/2-β*t^3/3)-B*(lam^2*t^2+t)
+  let P' : ℝ → ℝ := fun t ↦ A*((1+2*q)*t-β*t^2)-B*(2*lam^2*t+1)
+  have hmu := Q.mu_nonneg
+  have hnu := Q.nu_nonneg
+  have hq : 0 ≤ q := by dsimp [q]; positivity
+  have hβ : 0 ≤ β := by dsimp [β]; positivity
+  have hA : 0 ≤ A := by dsimp [A]; positivity
+  have hB : 0 ≤ B := by dsimp [B]; positivity
+  have hsq : 0 < 1+q := by positivity
+  have hs2 : 0 < Real.sqrt 2 := Real.sqrt_pos.mpr (by norm_num)
+  have dP (t : ℝ) : HasDerivAt P (P' t) t := by
+    apply ((((((hasDerivAt_id t).pow 2).const_mul (1+2*q)).div_const 2 |>.sub
+      ((((hasDerivAt_id t).pow 3).const_mul β).div_const 3)).const_mul A).sub
+      (((((hasDerivAt_id t).pow 2).const_mul (lam^2)).add (hasDerivAt_id t)).const_mul B)).congr_deriv
+    dsimp [P']; norm_num only [Nat.cast_ofNat, Nat.reduceSub, pow_one]; ring
+  have hbound (t : ℝ) (ht : t ∈ Icc 0 (d:ℝ)) : P' t ≤ 3*poissonSlope Q hl₀.le hl₁ t := by
+    let u : ℝ≥0 := ⟨t,ht.1⟩
+    have hi := poisson_increment_bound Q hl₀.le hl₁ d u ht.2 hfixed
+    have hr := mul_le_mul_of_nonneg_left (reciprocal_tangent (mul_nonneg ht.1 hβ) hq)
+      (mul_nonneg hA ht.1)
+    have hr' : A*((1+2*q)*t-β*t^2) ≤ 3*lam^4*Q.mu^2*t/(1+t*β) := by
+      calc
+        _ = A*t*(1+2*q-t*β) := by ring
+        _ ≤ A*t*((1+q)^2/(1+t*β)) := hr
+        _ = _ := by dsimp only [A]; field_simp
+    have hs := mul_le_mul_of_nonneg_left (sqrt_intensity_tangent hl₀ ht.1)
+      (show 0 ≤ lam^4*Q.mu*Q.nu by positivity)
+    have hs' : lam^4*Q.mu*Q.nu*Real.sqrt t ≤ B*(2*lam^2*t+1) := by
+      calc
+        _ ≤ lam^4*Q.mu*Q.nu*((2*lam^2*t+1)/(2*lam*Real.sqrt 2)) := hs
+        _ = _ := by dsimp only [B]; field_simp
+    change 3*lam^4*Q.mu^2*t/(1+t*β)-lam^4*Q.mu*Q.nu*Real.sqrt t ≤
+      3*poissonSlope Q hl₀.le hl₁ t at hi
+    dsimp only [P']
+    linarith only [hr', hs', hi]
+  have hm : MonotoneOn (fun t ↦ 3*poissonCorrelation Q hl₀.le hl₁ t-P t) (Icc 0 (d:ℝ)) := by
+    apply monotoneOn_Icc_of_hasDerivAt_nonneg
+      (fun t _ ↦ ((hasDerivAt_poissonCorrelation Q hl₀.le hl₁ t).const_mul 3).sub (dP t))
+    intro t ht
+    exact sub_nonneg.mpr (hbound t ⟨ht.1.le,ht.2.le⟩)
+  have hh := hm ⟨le_rfl,d.coe_nonneg⟩ ⟨d.coe_nonneg,le_rfl⟩ d.coe_nonneg
+  have hz : P 0 = 0 := by dsimp [P]; ring
+  change 3*poissonCorrelation Q hl₀.le hl₁ 0-P 0 ≤ 3*poissonCorrelation Q hl₀.le hl₁ d-P d at hh
+  rw [poissonCorrelation_zero, hz] at hh
+  have hd : (d:ℝ) = 1/lam^2 := (eq_div_iff (ne_of_gt (sq_pos_of_pos hl₀))).2 hcrit
+  have hp : P d = 3*Q.mu^2/(2*(1+2*Q.mu/3))-lam/Real.sqrt 2*Q.mu*Q.nu := by
+    dsimp only [P,A,B,β,q]
+    rw [hd]
+    field_simp
+    ring
+  rw [hp] at hh
+  linarith only [hh]
+
+end ThreeState.TreeClauses.Positive
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeStateClauses/PoissonFixed.lean b/lean/OAI/Probability/ThreeStateClauses/PoissonFixed.lean
index d48e42cddee5ffa72beeb4d086d98e47d391e56a..31fa4987dfe84cdfa34ac12d921d3f1f17ab437b 100644
--- a/lean/OAI/Probability/ThreeStateClauses/PoissonFixed.lean
+++ b/lean/OAI/Probability/ThreeStateClauses/PoissonFixed.lean
@@ -1,346 +1,346 @@
-import Mathlib
-import OAI.Probability.ThreeStateClauses.OffspringLaws
-import OAI.Probability.ThreeStateClauses.PoissonCalculus
-
-namespace OAI
-
-/-! Poisson Fixed. -/
-
-open scoped BigOperators ENNReal NNReal Topology
-open Filter
-noncomputable section
-open Set MeasureTheory ProbabilityTheory
-open scoped NNReal ENNReal BigOperators
-namespace ThreeState.TreeClauses.Experiment
-open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive ThreeState.TreeClauses.Probability
-
-def uniformMessage : Message := ⟨fun _ ↦ 1, by norm_num, by norm_num [avg]⟩
-
-lemma uniformMessage_x : xMoment uniformMessage = 0 := by norm_num [xMoment, momentX, centered, uniformMessage, avg]
-
-lemma branch_uniform (lam : ℝ) (n : ℕ) :
-    normalizer (branchEdges lam (fun _ : Fin n ↦ uniformMessage)) = 1 := by
-  norm_num [normalizer, branchEdges, branchProduct, edgeMessage, edge, centered, uniformMessage, avg]
-
-lemma branchOutput_uniform {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
-    branchOutput hl₀ hl₁ (fun _ : Fin n ↦ uniformMessage) = uniformMessage := by
-  apply Subtype.ext
-  funext i
-  norm_num [branchOutput, normalizedProduct, branch_uniform, branchProduct, branchEdges,
-    edgeMessage, edge, centered, uniformMessage, normalizer, avg]
-
-lemma finite_uniform_atom (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
-    (Q.probability.toMeasure {uniformMessage})^n ≤
-      (finiteProbability Q hl₀ hl₁ n).toMeasure {uniformMessage} := by
-  change _ ≤ (Measure.map (branchOutput hl₀ hl₁) (tiltedProduct Q lam n)) {uniformMessage}
-  rw [Measure.map_apply (continuous_branchOutput hl₀ hl₁ n).measurable (measurableSet_singleton _)]
-  have he : (tiltedProduct Q lam n) {fun _ : Fin n ↦ uniformMessage} =
-      (Q.probability.toMeasure {uniformMessage})^n := by
-    rw [tiltedProduct, withDensity_apply _ (measurableSet_singleton _), lintegral_singleton]
-    simp only [branch_uniform, ENNReal.ofReal_one, one_mul, Measure.pi_singleton,
-      Finset.prod_const, Finset.card_univ, Fintype.card_fin]
-  rw [← he]
-  apply measure_mono
-  intro m hm
-  rcases hm with rfl
-  exact branchOutput_uniform hl₀ hl₁ n
-
-lemma poisson_generating (t q : ℝ) :
-    HasSum (fun n ↦ poissonWeight t n*q^n) (Real.exp (t*(q-1))) := by
-  have hx : HasSum (fun n : ℕ ↦ (t*q)^n/(n.factorial:ℝ)) (Real.exp (t*q)) := by
-    rw [Real.exp_eq_exp_ℝ]
-    exact NormedSpace.expSeries_div_hasSum_exp (t*q)
-  have h := hx.mul_left (Real.exp (-t))
-  have he : Real.exp (-t)*Real.exp (t*q) = Real.exp (t*(q-1)) := by
-    rw [← Real.exp_add]; congr 1; ring
-  rw [he] at h
-  have hf : (fun n : ℕ ↦ poissonWeight t n*q^n) =
-      (fun n ↦ Real.exp (-t)*((t*q)^n/(n.factorial:ℝ))) := by
-    funext n
-    dsimp [poissonWeight]
-    rw [mul_pow]
-    ring
-  rw [hf]
-  exact h
-
-lemma poisson_uniform_atom (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (t : ℝ≥0) :
-    Real.exp ((t:ℝ)*((Q.probability.toMeasure {uniformMessage}).toReal-1)) ≤
-      ((poissonLaw Q hl₀ hl₁ t).probability.toMeasure {uniformMessage}).toReal := by
-  let q := (Q.probability.toMeasure {uniformMessage}).toReal
-  have hq : 0 ≤ q := ENNReal.toReal_nonneg
-  have he : ENNReal.ofReal q = Q.probability.toMeasure {uniformMessage} :=
-    ENNReal.ofReal_toReal (measure_ne_top _ _)
-  have h : ENNReal.ofReal (Real.exp ((t:ℝ)*(q-1))) ≤
-      (poissonLaw Q hl₀ hl₁ t).probability.toMeasure {uniformMessage} := by
-    rw [← (poisson_generating t q).tsum_eq, ENNReal.ofReal_tsum_of_nonneg
-      (fun n ↦ mul_nonneg (poissonWeight_nonneg t.coe_nonneg n) (pow_nonneg hq n))
-      (poisson_generating t q).summable]
-    change _ ≤ (Measure.sum (fun n ↦ (poissonPMF t) n • (finiteProbability Q hl₀ hl₁ n).toMeasure)) _
-    rw [Measure.sum_apply _ (measurableSet_singleton _)]
-    apply ENNReal.tsum_le_tsum
-    intro n
-    rw [ENNReal.ofReal_mul (poissonWeight_nonneg t.coe_nonneg n), ENNReal.ofReal_pow hq, he,
-      ← poissonPMF_real, ENNReal.ofReal_toReal ((poissonPMF t).apply_ne_top n)]
-    exact mul_le_mul' le_rfl (finite_uniform_atom Q hl₀ hl₁ n)
-  have hh := ENNReal.toReal_mono (measure_ne_top _ _) h
-  rwa [ENNReal.toReal_ofReal (Real.exp_pos _).le] at hh
-
-def Law.nonuniformMass (Q : Law) : ℝ := 1-(Q.probability.toMeasure {uniformMessage}).toReal
-
-lemma Law.nonuniformMass_nonneg (Q : Law) : 0 ≤ Q.nonuniformMass := by
-  apply sub_nonneg.mpr
-  exact (ENNReal.toReal_le_toReal (measure_ne_top _ _) ENNReal.one_ne_top).2 (prob_le_one)
-
-lemma Law.nonuniformMass_le_one (Q : Law) : Q.nonuniformMass ≤ 1 := by
-  unfold Law.nonuniformMass
-  exact sub_le_self _ ENNReal.toReal_nonneg
-
-lemma Law.nonuniform_second_moment (Q : Law) : Q.mu^2 ≤ Q.nonuniformMass*Q.nu := by
-  have hi := compact_integrable continuous_xMoment (μ := Q.probability.toMeasure)
-  have hi₂ := compact_integrable (continuous_xMoment.pow 2) (μ := Q.probability.toMeasure)
-  have hquad (a : ℝ) : 0 ≤ Q.nonuniformMass*(a*a)+(-2*Q.mu)*a+Q.nu := by
-    have h := integral_nonneg (μ := Q.probability.toMeasure.restrict {uniformMessage}ᶜ)
-      (f := fun m : Message ↦ (xMoment m-a)^2) (fun m ↦ sq_nonneg (xMoment m-a))
-    rw [setIntegral_compl (f := fun m : Message ↦ (xMoment m-a)^2) (measurableSet_singleton _)
-      (compact_integrable ((continuous_xMoment.sub continuous_const).pow 2)),
-      integral_singleton, uniformMessage_x] at h
-    have hf : (fun m : Message ↦ (xMoment m-a)^2) =
-        (fun m ↦ xMoment m^2-2*a*xMoment m+a^2) := by funext m; ring
-    rw [hf, integral_add (f := fun m ↦ xMoment m^2-2*a*xMoment m)
-      (g := fun _ ↦ a^2) (hi₂.sub (hi.const_mul _)) (integrable_const _),
-      integral_sub (f := fun m : Message ↦ xMoment m^2) (g := fun m ↦ 2*a*xMoment m) hi₂ (hi.const_mul _), integral_const_mul] at h
-    simp only [integral_const, measure_univ, Measure.real, ENNReal.toReal_one,
-      smul_eq_mul, one_mul, zero_sub, neg_sq] at h
-    change 0 ≤ Q.nu-2*a*Q.mu+a^2-(Q.probability.toMeasure {uniformMessage}).toReal*a^2 at h
-    dsimp only [Law.nonuniformMass]
-    nlinarith only [h]
-  have h := discrim_le_zero hquad
-  dsimp only [discrim] at h
-  nlinarith only [h]
-
-end ThreeState.TreeClauses.Experiment
-
-end 
-
-noncomputable section
-open Set
-namespace ThreeState.TreeClauses.Probability
-open ThreeState.TreeClauses.Radial
-
-lemma neg_log_one_sub_lower {s : ℝ} (hs₀ : 0 ≤ s) (hs₁ : s < 1) :
-    2*s/(2-s) ≤ -Real.log (1-s) := by
-  let f : ℝ → ℝ := fun t ↦ -Real.log (1-t)-2*t/(2-t)
-  let f' : ℝ → ℝ := fun t ↦ t^2/((1-t)*(2-t)^2)
-  have hd (t : ℝ) (ht : t ∈ Icc 0 s) : HasDerivAt f (f' t) t := by
-    have ht1 : 0 < 1-t := by linarith [ht.2]
-    have ht2 : 0 < 2-t := by linarith
-    apply (((((hasDerivAt_id t).const_sub 1).log (ne_of_gt ht1)).neg).sub
-      (((hasDerivAt_id t).const_mul 2).div ((hasDerivAt_id t).const_sub 2) (ne_of_gt ht2))).congr_deriv
-    dsimp only [f', id_eq]
-    field_simp
-    ring
-  have hm : MonotoneOn f (Icc 0 s) := by
-    apply monotoneOn_Icc_of_hasDerivAt_nonneg hd
-    intro t ht
-    dsimp only [f', id_eq]
-    have : 0 < 1-t := by linarith [ht.2]
-    positivity
-  have h := hm ⟨le_rfl,hs₀⟩ ⟨hs₀,le_rfl⟩ hs₀
-  dsimp only [f] at h
-  norm_num at h
-  linarith only [h]
-
-lemma poisson_subfixed_mass_bound {d s : ℝ} (hd : 1 < d) (hs₀ : 0 ≤ s)
-    (hs : Real.exp (-d*s) ≤ 1-s) : s ≤ 2*(1-1/d) := by
-  have hs₁ : s < 1 := by linarith [Real.exp_pos (-d*s)]
-  have hlog : -Real.log (1-s) ≤ d*s := by
-    have h := Real.log_le_log (Real.exp_pos (-d*s)) hs
-    rw [Real.log_exp] at h
-    linarith only [h]
-  have hl := neg_log_one_sub_lower hs₀ hs₁
-  have hden : 0 < 2-s := by linarith
-  have hmul := (div_le_iff₀ hden).mp (hl.trans hlog)
-  rcases eq_or_lt_of_le hs₀ with h | h
-  · have he : s = 0 := h.symm
-    rw [he]
-    have hdiv : 1/d < 1 := (div_lt_one (by linarith : 0 < d)).2 hd
-    linarith only [hdiv]
-  · have hh : 2 ≤ d*(2-s) := (mul_le_mul_iff_right₀ h).mp (by nlinarith only [hmul])
-    have he : 2*(1-1/d) = (2*d-2)/d := by field_simp
-    rw [he]
-    apply (le_div_iff₀ (by linarith : 0 < d)).2
-    nlinarith only [hh]
-
-end ThreeState.TreeClauses.Probability
-
-end 
-
-noncomputable section
-open Set MeasureTheory
-open scoped NNReal
-namespace ThreeState.TreeClauses.Positive
-open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Experiment ThreeState.TreeClauses.Probability
-
-lemma poisson_fixed_mass_bound (Q : Law) {lam : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1)
-    (d : ℝ≥0) (hd : 1 < (d:ℝ)) (hcrit : (d:ℝ)*lam^2 = 1)
-    (hfixed : Q.probability = (poissonLaw Q hl₀.le hl₁ d).probability) :
-    Q.nonuniformMass ≤ 2*(1-lam^2) := by
-  have ha := poisson_uniform_atom Q hl₀.le hl₁ d
-  rw [← hfixed] at ha
-  have hsub : Real.exp (-(d:ℝ)*Q.nonuniformMass) ≤ 1-Q.nonuniformMass := by
-    calc
-      _ = Real.exp ((d:ℝ)*((Q.probability.toMeasure {uniformMessage}).toReal-1)) := by
-        congr 1; dsimp only [Law.nonuniformMass]; ring
-      _ ≤ (Q.probability.toMeasure {uniformMessage}).toReal := ha
-      _ = _ := by dsimp only [Law.nonuniformMass]; ring
-  have hs := poisson_subfixed_mass_bound hd Q.nonuniformMass_nonneg hsub
-  have he : 1/(d:ℝ) = lam^2 := (div_eq_iff (by positivity : (d:ℝ) ≠ 0)).2 (by nlinarith only [hcrit])
-  rwa [he] at hs
-
-theorem poisson_fixed_mu_zero (Q : Law) {lam : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1)
-    (d : ℝ≥0) (hd : 1 < (d:ℝ)) (hcrit : (d:ℝ)*lam^2 = 1)
-    (hfixed : Q.probability = (poissonLaw Q hl₀.le hl₁ d).probability) : Q.mu = 0 := by
-  by_contra hmu0
-  have hmu : 0 < Q.mu := lt_of_le_of_ne Q.mu_nonneg (Ne.symm hmu0)
-  have hnu : 0 < Q.nu := (sq_pos_of_pos hmu).trans_le Q.second_moment_lower
-  have hmoment := Q.nonuniform_second_moment
-  have hs₀ : 0 < Q.nonuniformMass := by
-    by_contra h
-    have hh := mul_nonpos_of_nonpos_of_nonneg (le_of_not_gt h) hnu.le
-    nlinarith only [hmoment, hh, sq_pos_of_pos hmu]
-  have hinfo := poisson_fixed_moment_bounds Q hl₀ hl₁ d hcrit hfixed
-  have hcl := cExact_lower hl₀.le hl₁.le
-  have hc : 0 < cExact lam := (cLower_pos hl₀.le).trans_le hcl
-  have hetaNonneg : 0 ≤ Q.eta := (mul_nonneg hc.le hnu.le).trans hinfo.2.2
-  have hsq : (cExact lam*Q.nu)^2 ≤ Q.eta^2 :=
-    (sq_le_sq₀ (mul_nonneg hc.le hnu.le) hetaNonneg).2 hinfo.2.2
-  have hregion : cExact lam^2*Q.nu ≤ Q.mu := by
-    apply (mul_le_mul_iff_right₀ hnu).mp
-    nlinarith [Q.moment_discriminant]
-  have hmuC : Q.mu*cExact lam^2 ≤ Q.nonuniformMass := by
-    apply (mul_le_mul_iff_right₀ hmu).mp
-    have hh := mul_le_mul_of_nonneg_left hmoment (sq_nonneg (cExact lam))
-    have hh' := mul_le_mul_of_nonneg_left hregion hs₀.le
-    nlinarith only [hh, hh']
-  have hregionLower : Q.mu*cLower lam^2 ≤ Q.nonuniformMass := by
-    have hs : cLower lam^2 ≤ cExact lam^2 :=
-      (sq_le_sq₀ (cLower_pos hl₀.le).le hc.le).2 hcl
-    exact (mul_le_mul_of_nonneg_left hs hmu.le).trans hmuC
-  have hstrict := scalar_strict hl₀ hl₁ hmu.le hs₀ Q.nonuniformMass_le_one hregionLower
-    (Or.inr (poisson_fixed_mass_bound Q hl₀ hl₁ d hd hcrit hfixed))
-  have hJ := jMean_square_bound hl₀.le hl₁.le hmu.le hnu.le hinfo.1 hinfo.2.1 hregion
-  have hH := poisson_fixed_correlation_bound Q hl₀ hl₁ d hcrit hfixed
-  have hrad := poisson_fixed_entropy_radial Q hl₀ hl₁ d hcrit hfixed
-  have hH' : 3*Q.mu^2/(2*(1+2*Q.mu/3))-(1/Real.sqrt 2)*(lam*Q.mu*Q.nu) ≤
-      3*poissonCorrelation Q hl₀.le hl₁ d := by
-    convert hH using 1
-    ring
-  have hg := mul_le_mul_of_nonneg_right (le_of_lt strict_correlation_coefficient)
-    (show 0 ≤ lam*Q.mu*Q.nu by positivity)
-  have hb₀ : (1-lam^2)/2*Q.nu+3*Q.mu^2/(2*(1+2*Q.mu/3))+
-      (17/25)*lam*Q.mu*Q.nu ≤ (8/5)*Q.mu^2 := by
-    nlinarith only [hJ,hH',hrad,hg]
-  have hb : ((1-lam^2)/2+17*lam*Q.mu/25)*Q.nu+3*Q.mu^2/(2*(1+2*Q.mu/3)) ≤
-      (8/5)*Q.mu^2 := by nlinarith only [hb₀]
-  have halpha : 0 < 1-lam^2 := by nlinarith only [mul_pos (sub_pos.mpr hl₁) (by linarith : 0 < 1+lam)]
-  have hcoef : 0 ≤ ((1-lam^2)/2+17*lam*Q.mu/25)/Q.nonuniformMass := by positivity
-  have hsub := mul_le_mul_of_nonneg_left hmoment hcoef
-  have hnecessary : scalarT lam Q.mu Q.nonuniformMass*Q.mu^2 ≤ (8/5)*Q.mu^2 := by
-    calc
-      _ = (((1-lam^2)/2+17*lam*Q.mu/25)/Q.nonuniformMass)*Q.mu^2+
-          3*Q.mu^2/(2*(1+2*Q.mu/3)) := by unfold scalarT; ring
-      _ ≤ (((1-lam^2)/2+17*lam*Q.mu/25)/Q.nonuniformMass)*(Q.nonuniformMass*Q.nu)+
-          3*Q.mu^2/(2*(1+2*Q.mu/3)) := by linarith only [hsub]
-      _ = ((1-lam^2)/2+17*lam*Q.mu/25)*Q.nu+3*Q.mu^2/(2*(1+2*Q.mu/3)) := by
-        field_simp [ne_of_gt hs₀]
-      _ ≤ _ := hb
-  have hh := mul_lt_mul_of_pos_right hstrict (sq_pos_of_pos hmu)
-  linarith only [hh,hnecessary]
-
-end ThreeState.TreeClauses.Positive
-
-end 
-
-noncomputable section
-open Set MeasureTheory Filter
-open scoped Topology NNReal
-namespace ThreeState.TreeClauses.Tree
-open ThreeState.TreeClauses.Experiment ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
-
-lemma ordered_advantage_sq_moment (ρ : PMF ℕ) (lam : ℝ) (hlam : Admissible lam) (ℓ : ℕ) :
-    (advantage lam hlam ρ ℓ)^2 ≤
-      (∫ m, xMoment m ∂(discreteProbability (orderedLaw ρ lam hlam ℓ)).toMeasure)/2 := by
-  have h := discrete_degradation_mu (orderedLaw ρ lam hlam ℓ) (fun v ↦ PMF.pure (forgetOrdered ℓ v))
-  change (∫ m, xMoment m ∂(discreteProbability (fun i ↦
-    (orderedLaw ρ lam hlam ℓ i).map (forgetOrdered ℓ))).toMeasure) ≤ _ at h
-  simp only [orderedLaw_forget] at h
-  have ha := advantage_sq_bound lam hlam ρ ℓ
-  linarith
-
-lemma poisson_positive_critical_mu (d : ℝ≥0) (hd : 1 < (d:ℝ)) {lam : ℝ}
-    (hl₀ : 0 < lam) (hl₁ : lam < 1) (hlam : Admissible lam) (hcrit : (d:ℝ)*lam^2 = 1) :
-    Tendsto (fun ℓ ↦ (offspringPosteriorLaw (poissonPMF d) hl₀.le hl₁ ℓ).mu) atTop (𝓝 0) := by
-  let ρ := poissonPMF d
-  let Q := offspringPosteriorLaw ρ hl₀.le hl₁
-  let C := fun ℓ ↦ degradationCoupling (orderedLaw ρ lam hlam ℓ) (orderedExpansion ρ lam hlam ℓ)
-  have hf (ℓ : ℕ) : (C ℓ).map Prod.fst = (Q ℓ).probability := by
-    rw [degradationCoupling_fst, offspringPosteriorLaw_probability ρ hl₀.le hl₁ hlam ℓ]
-  have hs (ℓ : ℕ) : (C ℓ).map Prod.snd = (Q (ℓ+1)).probability := by
-    rw [degradationCoupling_snd]
-    have he : (fun i ↦ (orderedLaw ρ lam hlam ℓ i).bind (orderedExpansion ρ lam hlam ℓ)) =
-        orderedLaw ρ lam hlam (ℓ+1) := funext (orderedLaw_degrades ρ lam hlam ℓ)
-    rw [he]
-    exact (offspringPosteriorLaw_probability ρ hl₀.le hl₁ hlam (ℓ+1)).symm
-  have hl (ℓ : ℕ) : (∫ z, quadraticGap z ∂(C ℓ).toMeasure) = (Q ℓ).mu-(Q (ℓ+1)).mu := by
-    rw [degradationCoupling_loss]
-    have he : (fun i ↦ (orderedLaw ρ lam hlam ℓ i).bind (orderedExpansion ρ lam hlam ℓ)) =
-        orderedLaw ρ lam hlam (ℓ+1) := funext (orderedLaw_degrades ρ lam hlam ℓ)
-    rw [he]
-    simp only [Q, Law.mu, offspringPosteriorLaw_probability ρ hl₀.le hl₁ hlam]
-  obtain ⟨R,hR,hlim⟩ := degrading_offspring_fixed_point Q C hf hs hl hl₀.le hl₁ ρ (fun _ ↦ rfl)
-  rw [poisson_fixed_mu_zero R hl₀ hl₁ d hd hcrit hR] at hlim
-  exact hlim
-
-theorem poisson_positive_nonreconstruction (d : ℝ≥0) (hd : 1 < (d:ℝ))
-    (lam : ℝ) (hlam : Admissible lam) (hl₀ : 0 ≤ lam) (hKS : (d:ℝ)*lam^2 ≤ 1) :
-    Tendsto (advantage lam hlam (poissonPMF d)) atTop (𝓝 0) := by
-  have hdpos : 0 < (d:ℝ) := by linarith
-  let t : ℝ := 1/Real.sqrt d
-  have ht : 0 < t := by dsimp [t]; exact div_pos (by norm_num) (Real.sqrt_pos.mpr hdpos)
-  have htsq : (d:ℝ)*t^2 = 1 := by
-    dsimp [t]
-    rw [div_pow, one_pow, Real.sq_sqrt hdpos.le, mul_one_div_cancel (ne_of_gt hdpos)]
-  have htone : t < 1 := by
-    have h := mul_lt_mul_of_pos_right hd (sq_pos_of_pos ht)
-    nlinarith only [h, htsq, ht]
-  have htadm : Admissible t := ⟨by linarith, htone.le⟩
-  have hlt : lam ≤ t := by
-    have hs : lam^2 ≤ t^2 := (mul_le_mul_iff_right₀ hdpos).mp (by nlinarith only [hKS,htsq])
-    nlinarith only [hs,hl₀,ht]
-  let c := lam/t
-  have hcpos : 0 ≤ c := div_nonneg hl₀ ht.le
-  have hcone : c ≤ 1 := (div_le_one ht).2 hlt
-  have hcadm : Admissible c := ⟨by linarith, hcone⟩
-  have he : t*c = lam := by dsimp [c]; field_simp
-  have hprod : Admissible (t*c) := he.symm ▸ hlam
-  let ρ := poissonPMF d
-  have hm := poisson_positive_critical_mu d hd ht htone htadm htsq
-  have hu : Tendsto (fun ℓ ↦ Real.sqrt (offspringPosteriorLaw ρ ht.le htone ℓ).mu) atTop (𝓝 0) := by
-    simpa only [Real.sqrt_zero, Function.comp_def] using Real.continuous_sqrt.continuousAt.tendsto.comp hm
-  apply squeeze_zero (fun ℓ ↦ advantage_nonneg lam hlam ρ ℓ) _ hu
-  intro ℓ
-  apply (sq_le_sq₀ (advantage_nonneg lam hlam ρ ℓ) (Real.sqrt_nonneg _)).mp
-  rw [Real.sq_sqrt (offspringPosteriorLaw ρ ht.le htone ℓ).mu_nonneg]
-  have h := ordered_channel_mu_le ρ t c htadm hcadm hprod ℓ
-  have hw := ordered_advantage_sq_moment ρ lam hlam ℓ
-  have heq : orderedLaw ρ (t*c) hprod ℓ = orderedLaw ρ lam hlam ℓ := by congr 1
-  rw [heq] at h
-  rw [← offspringPosteriorLaw_probability ρ ht.le htone htadm ℓ] at h
-  change _ ≤ (offspringPosteriorLaw ρ ht.le htone ℓ).mu at h
-  linarith [(offspringPosteriorLaw ρ ht.le htone ℓ).mu_nonneg]
-
-end ThreeState.TreeClauses.Tree
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeStateClauses.OffspringLaws
+import OAI.Probability.ThreeStateClauses.PoissonCalculus
+
+namespace OAI
+
+/-! Poisson Fixed. -/
+
+open scoped BigOperators ENNReal NNReal Topology
+open Filter
+noncomputable section
+open Set MeasureTheory ProbabilityTheory
+open scoped NNReal ENNReal BigOperators
+namespace ThreeState.TreeClauses.Experiment
+open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive ThreeState.TreeClauses.Probability
+
+def uniformMessage : Message := ⟨fun _ ↦ 1, by norm_num, by norm_num [avg]⟩
+
+lemma uniformMessage_x : xMoment uniformMessage = 0 := by norm_num [xMoment, momentX, centered, uniformMessage, avg]
+
+lemma branch_uniform (lam : ℝ) (n : ℕ) :
+    normalizer (branchEdges lam (fun _ : Fin n ↦ uniformMessage)) = 1 := by
+  norm_num [normalizer, branchEdges, branchProduct, edgeMessage, edge, centered, uniformMessage, avg]
+
+lemma branchOutput_uniform {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
+    branchOutput hl₀ hl₁ (fun _ : Fin n ↦ uniformMessage) = uniformMessage := by
+  apply Subtype.ext
+  funext i
+  norm_num [branchOutput, normalizedProduct, branch_uniform, branchProduct, branchEdges,
+    edgeMessage, edge, centered, uniformMessage, normalizer, avg]
+
+lemma finite_uniform_atom (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
+    (Q.probability.toMeasure {uniformMessage})^n ≤
+      (finiteProbability Q hl₀ hl₁ n).toMeasure {uniformMessage} := by
+  change _ ≤ (Measure.map (branchOutput hl₀ hl₁) (tiltedProduct Q lam n)) {uniformMessage}
+  rw [Measure.map_apply (continuous_branchOutput hl₀ hl₁ n).measurable (measurableSet_singleton _)]
+  have he : (tiltedProduct Q lam n) {fun _ : Fin n ↦ uniformMessage} =
+      (Q.probability.toMeasure {uniformMessage})^n := by
+    rw [tiltedProduct, withDensity_apply _ (measurableSet_singleton _), lintegral_singleton]
+    simp only [branch_uniform, ENNReal.ofReal_one, one_mul, Measure.pi_singleton,
+      Finset.prod_const, Finset.card_univ, Fintype.card_fin]
+  rw [← he]
+  apply measure_mono
+  intro m hm
+  rcases hm with rfl
+  exact branchOutput_uniform hl₀ hl₁ n
+
+lemma poisson_generating (t q : ℝ) :
+    HasSum (fun n ↦ poissonWeight t n*q^n) (Real.exp (t*(q-1))) := by
+  have hx : HasSum (fun n : ℕ ↦ (t*q)^n/(n.factorial:ℝ)) (Real.exp (t*q)) := by
+    rw [Real.exp_eq_exp_ℝ]
+    exact NormedSpace.expSeries_div_hasSum_exp (t*q)
+  have h := hx.mul_left (Real.exp (-t))
+  have he : Real.exp (-t)*Real.exp (t*q) = Real.exp (t*(q-1)) := by
+    rw [← Real.exp_add]; congr 1; ring
+  rw [he] at h
+  have hf : (fun n : ℕ ↦ poissonWeight t n*q^n) =
+      (fun n ↦ Real.exp (-t)*((t*q)^n/(n.factorial:ℝ))) := by
+    funext n
+    dsimp [poissonWeight]
+    rw [mul_pow]
+    ring
+  rw [hf]
+  exact h
+
+lemma poisson_uniform_atom (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (t : ℝ≥0) :
+    Real.exp ((t:ℝ)*((Q.probability.toMeasure {uniformMessage}).toReal-1)) ≤
+      ((poissonLaw Q hl₀ hl₁ t).probability.toMeasure {uniformMessage}).toReal := by
+  let q := (Q.probability.toMeasure {uniformMessage}).toReal
+  have hq : 0 ≤ q := ENNReal.toReal_nonneg
+  have he : ENNReal.ofReal q = Q.probability.toMeasure {uniformMessage} :=
+    ENNReal.ofReal_toReal (measure_ne_top _ _)
+  have h : ENNReal.ofReal (Real.exp ((t:ℝ)*(q-1))) ≤
+      (poissonLaw Q hl₀ hl₁ t).probability.toMeasure {uniformMessage} := by
+    rw [← (poisson_generating t q).tsum_eq, ENNReal.ofReal_tsum_of_nonneg
+      (fun n ↦ mul_nonneg (poissonWeight_nonneg t.coe_nonneg n) (pow_nonneg hq n))
+      (poisson_generating t q).summable]
+    change _ ≤ (Measure.sum (fun n ↦ (poissonPMF t) n • (finiteProbability Q hl₀ hl₁ n).toMeasure)) _
+    rw [Measure.sum_apply _ (measurableSet_singleton _)]
+    apply ENNReal.tsum_le_tsum
+    intro n
+    rw [ENNReal.ofReal_mul (poissonWeight_nonneg t.coe_nonneg n), ENNReal.ofReal_pow hq, he,
+      ← poissonPMF_real, ENNReal.ofReal_toReal ((poissonPMF t).apply_ne_top n)]
+    exact mul_le_mul' le_rfl (finite_uniform_atom Q hl₀ hl₁ n)
+  have hh := ENNReal.toReal_mono (measure_ne_top _ _) h
+  rwa [ENNReal.toReal_ofReal (Real.exp_pos _).le] at hh
+
+def Law.nonuniformMass (Q : Law) : ℝ := 1-(Q.probability.toMeasure {uniformMessage}).toReal
+
+lemma Law.nonuniformMass_nonneg (Q : Law) : 0 ≤ Q.nonuniformMass := by
+  apply sub_nonneg.mpr
+  exact (ENNReal.toReal_le_toReal (measure_ne_top _ _) ENNReal.one_ne_top).2 (prob_le_one)
+
+lemma Law.nonuniformMass_le_one (Q : Law) : Q.nonuniformMass ≤ 1 := by
+  unfold Law.nonuniformMass
+  exact sub_le_self _ ENNReal.toReal_nonneg
+
+lemma Law.nonuniform_second_moment (Q : Law) : Q.mu^2 ≤ Q.nonuniformMass*Q.nu := by
+  have hi := compact_integrable continuous_xMoment (μ := Q.probability.toMeasure)
+  have hi₂ := compact_integrable (continuous_xMoment.pow 2) (μ := Q.probability.toMeasure)
+  have hquad (a : ℝ) : 0 ≤ Q.nonuniformMass*(a*a)+(-2*Q.mu)*a+Q.nu := by
+    have h := integral_nonneg (μ := Q.probability.toMeasure.restrict {uniformMessage}ᶜ)
+      (f := fun m : Message ↦ (xMoment m-a)^2) (fun m ↦ sq_nonneg (xMoment m-a))
+    rw [setIntegral_compl (f := fun m : Message ↦ (xMoment m-a)^2) (measurableSet_singleton _)
+      (compact_integrable ((continuous_xMoment.sub continuous_const).pow 2)),
+      integral_singleton, uniformMessage_x] at h
+    have hf : (fun m : Message ↦ (xMoment m-a)^2) =
+        (fun m ↦ xMoment m^2-2*a*xMoment m+a^2) := by funext m; ring
+    rw [hf, integral_add (f := fun m ↦ xMoment m^2-2*a*xMoment m)
+      (g := fun _ ↦ a^2) (hi₂.sub (hi.const_mul _)) (integrable_const _),
+      integral_sub (f := fun m : Message ↦ xMoment m^2) (g := fun m ↦ 2*a*xMoment m) hi₂ (hi.const_mul _), integral_const_mul] at h
+    simp only [integral_const, measure_univ, Measure.real, ENNReal.toReal_one,
+      smul_eq_mul, one_mul, zero_sub, neg_sq] at h
+    change 0 ≤ Q.nu-2*a*Q.mu+a^2-(Q.probability.toMeasure {uniformMessage}).toReal*a^2 at h
+    dsimp only [Law.nonuniformMass]
+    nlinarith only [h]
+  have h := discrim_le_zero hquad
+  dsimp only [discrim] at h
+  nlinarith only [h]
+
+end ThreeState.TreeClauses.Experiment
+
+end 
+
+noncomputable section
+open Set
+namespace ThreeState.TreeClauses.Probability
+open ThreeState.TreeClauses.Radial
+
+lemma neg_log_one_sub_lower {s : ℝ} (hs₀ : 0 ≤ s) (hs₁ : s < 1) :
+    2*s/(2-s) ≤ -Real.log (1-s) := by
+  let f : ℝ → ℝ := fun t ↦ -Real.log (1-t)-2*t/(2-t)
+  let f' : ℝ → ℝ := fun t ↦ t^2/((1-t)*(2-t)^2)
+  have hd (t : ℝ) (ht : t ∈ Icc 0 s) : HasDerivAt f (f' t) t := by
+    have ht1 : 0 < 1-t := by linarith [ht.2]
+    have ht2 : 0 < 2-t := by linarith
+    apply (((((hasDerivAt_id t).const_sub 1).log (ne_of_gt ht1)).neg).sub
+      (((hasDerivAt_id t).const_mul 2).div ((hasDerivAt_id t).const_sub 2) (ne_of_gt ht2))).congr_deriv
+    dsimp only [f', id_eq]
+    field_simp
+    ring
+  have hm : MonotoneOn f (Icc 0 s) := by
+    apply monotoneOn_Icc_of_hasDerivAt_nonneg hd
+    intro t ht
+    dsimp only [f', id_eq]
+    have : 0 < 1-t := by linarith [ht.2]
+    positivity
+  have h := hm ⟨le_rfl,hs₀⟩ ⟨hs₀,le_rfl⟩ hs₀
+  dsimp only [f] at h
+  norm_num at h
+  linarith only [h]
+
+lemma poisson_subfixed_mass_bound {d s : ℝ} (hd : 1 < d) (hs₀ : 0 ≤ s)
+    (hs : Real.exp (-d*s) ≤ 1-s) : s ≤ 2*(1-1/d) := by
+  have hs₁ : s < 1 := by linarith [Real.exp_pos (-d*s)]
+  have hlog : -Real.log (1-s) ≤ d*s := by
+    have h := Real.log_le_log (Real.exp_pos (-d*s)) hs
+    rw [Real.log_exp] at h
+    linarith only [h]
+  have hl := neg_log_one_sub_lower hs₀ hs₁
+  have hden : 0 < 2-s := by linarith
+  have hmul := (div_le_iff₀ hden).mp (hl.trans hlog)
+  rcases eq_or_lt_of_le hs₀ with h | h
+  · have he : s = 0 := h.symm
+    rw [he]
+    have hdiv : 1/d < 1 := (div_lt_one (by linarith : 0 < d)).2 hd
+    linarith only [hdiv]
+  · have hh : 2 ≤ d*(2-s) := (mul_le_mul_iff_right₀ h).mp (by nlinarith only [hmul])
+    have he : 2*(1-1/d) = (2*d-2)/d := by field_simp
+    rw [he]
+    apply (le_div_iff₀ (by linarith : 0 < d)).2
+    nlinarith only [hh]
+
+end ThreeState.TreeClauses.Probability
+
+end 
+
+noncomputable section
+open Set MeasureTheory
+open scoped NNReal
+namespace ThreeState.TreeClauses.Positive
+open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Experiment ThreeState.TreeClauses.Probability
+
+lemma poisson_fixed_mass_bound (Q : Law) {lam : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1)
+    (d : ℝ≥0) (hd : 1 < (d:ℝ)) (hcrit : (d:ℝ)*lam^2 = 1)
+    (hfixed : Q.probability = (poissonLaw Q hl₀.le hl₁ d).probability) :
+    Q.nonuniformMass ≤ 2*(1-lam^2) := by
+  have ha := poisson_uniform_atom Q hl₀.le hl₁ d
+  rw [← hfixed] at ha
+  have hsub : Real.exp (-(d:ℝ)*Q.nonuniformMass) ≤ 1-Q.nonuniformMass := by
+    calc
+      _ = Real.exp ((d:ℝ)*((Q.probability.toMeasure {uniformMessage}).toReal-1)) := by
+        congr 1; dsimp only [Law.nonuniformMass]; ring
+      _ ≤ (Q.probability.toMeasure {uniformMessage}).toReal := ha
+      _ = _ := by dsimp only [Law.nonuniformMass]; ring
+  have hs := poisson_subfixed_mass_bound hd Q.nonuniformMass_nonneg hsub
+  have he : 1/(d:ℝ) = lam^2 := (div_eq_iff (by positivity : (d:ℝ) ≠ 0)).2 (by nlinarith only [hcrit])
+  rwa [he] at hs
+
+theorem poisson_fixed_mu_zero (Q : Law) {lam : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1)
+    (d : ℝ≥0) (hd : 1 < (d:ℝ)) (hcrit : (d:ℝ)*lam^2 = 1)
+    (hfixed : Q.probability = (poissonLaw Q hl₀.le hl₁ d).probability) : Q.mu = 0 := by
+  by_contra hmu0
+  have hmu : 0 < Q.mu := lt_of_le_of_ne Q.mu_nonneg (Ne.symm hmu0)
+  have hnu : 0 < Q.nu := (sq_pos_of_pos hmu).trans_le Q.second_moment_lower
+  have hmoment := Q.nonuniform_second_moment
+  have hs₀ : 0 < Q.nonuniformMass := by
+    by_contra h
+    have hh := mul_nonpos_of_nonpos_of_nonneg (le_of_not_gt h) hnu.le
+    nlinarith only [hmoment, hh, sq_pos_of_pos hmu]
+  have hinfo := poisson_fixed_moment_bounds Q hl₀ hl₁ d hcrit hfixed
+  have hcl := cExact_lower hl₀.le hl₁.le
+  have hc : 0 < cExact lam := (cLower_pos hl₀.le).trans_le hcl
+  have hetaNonneg : 0 ≤ Q.eta := (mul_nonneg hc.le hnu.le).trans hinfo.2.2
+  have hsq : (cExact lam*Q.nu)^2 ≤ Q.eta^2 :=
+    (sq_le_sq₀ (mul_nonneg hc.le hnu.le) hetaNonneg).2 hinfo.2.2
+  have hregion : cExact lam^2*Q.nu ≤ Q.mu := by
+    apply (mul_le_mul_iff_right₀ hnu).mp
+    nlinarith [Q.moment_discriminant]
+  have hmuC : Q.mu*cExact lam^2 ≤ Q.nonuniformMass := by
+    apply (mul_le_mul_iff_right₀ hmu).mp
+    have hh := mul_le_mul_of_nonneg_left hmoment (sq_nonneg (cExact lam))
+    have hh' := mul_le_mul_of_nonneg_left hregion hs₀.le
+    nlinarith only [hh, hh']
+  have hregionLower : Q.mu*cLower lam^2 ≤ Q.nonuniformMass := by
+    have hs : cLower lam^2 ≤ cExact lam^2 :=
+      (sq_le_sq₀ (cLower_pos hl₀.le).le hc.le).2 hcl
+    exact (mul_le_mul_of_nonneg_left hs hmu.le).trans hmuC
+  have hstrict := scalar_strict hl₀ hl₁ hmu.le hs₀ Q.nonuniformMass_le_one hregionLower
+    (Or.inr (poisson_fixed_mass_bound Q hl₀ hl₁ d hd hcrit hfixed))
+  have hJ := jMean_square_bound hl₀.le hl₁.le hmu.le hnu.le hinfo.1 hinfo.2.1 hregion
+  have hH := poisson_fixed_correlation_bound Q hl₀ hl₁ d hcrit hfixed
+  have hrad := poisson_fixed_entropy_radial Q hl₀ hl₁ d hcrit hfixed
+  have hH' : 3*Q.mu^2/(2*(1+2*Q.mu/3))-(1/Real.sqrt 2)*(lam*Q.mu*Q.nu) ≤
+      3*poissonCorrelation Q hl₀.le hl₁ d := by
+    convert hH using 1
+    ring
+  have hg := mul_le_mul_of_nonneg_right (le_of_lt strict_correlation_coefficient)
+    (show 0 ≤ lam*Q.mu*Q.nu by positivity)
+  have hb₀ : (1-lam^2)/2*Q.nu+3*Q.mu^2/(2*(1+2*Q.mu/3))+
+      (17/25)*lam*Q.mu*Q.nu ≤ (8/5)*Q.mu^2 := by
+    nlinarith only [hJ,hH',hrad,hg]
+  have hb : ((1-lam^2)/2+17*lam*Q.mu/25)*Q.nu+3*Q.mu^2/(2*(1+2*Q.mu/3)) ≤
+      (8/5)*Q.mu^2 := by nlinarith only [hb₀]
+  have halpha : 0 < 1-lam^2 := by nlinarith only [mul_pos (sub_pos.mpr hl₁) (by linarith : 0 < 1+lam)]
+  have hcoef : 0 ≤ ((1-lam^2)/2+17*lam*Q.mu/25)/Q.nonuniformMass := by positivity
+  have hsub := mul_le_mul_of_nonneg_left hmoment hcoef
+  have hnecessary : scalarT lam Q.mu Q.nonuniformMass*Q.mu^2 ≤ (8/5)*Q.mu^2 := by
+    calc
+      _ = (((1-lam^2)/2+17*lam*Q.mu/25)/Q.nonuniformMass)*Q.mu^2+
+          3*Q.mu^2/(2*(1+2*Q.mu/3)) := by unfold scalarT; ring
+      _ ≤ (((1-lam^2)/2+17*lam*Q.mu/25)/Q.nonuniformMass)*(Q.nonuniformMass*Q.nu)+
+          3*Q.mu^2/(2*(1+2*Q.mu/3)) := by linarith only [hsub]
+      _ = ((1-lam^2)/2+17*lam*Q.mu/25)*Q.nu+3*Q.mu^2/(2*(1+2*Q.mu/3)) := by
+        field_simp [ne_of_gt hs₀]
+      _ ≤ _ := hb
+  have hh := mul_lt_mul_of_pos_right hstrict (sq_pos_of_pos hmu)
+  linarith only [hh,hnecessary]
+
+end ThreeState.TreeClauses.Positive
+
+end 
+
+noncomputable section
+open Set MeasureTheory Filter
+open scoped Topology NNReal
+namespace ThreeState.TreeClauses.Tree
+open ThreeState.TreeClauses.Experiment ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
+
+lemma ordered_advantage_sq_moment (ρ : PMF ℕ) (lam : ℝ) (hlam : Admissible lam) (ℓ : ℕ) :
+    (advantage lam hlam ρ ℓ)^2 ≤
+      (∫ m, xMoment m ∂(discreteProbability (orderedLaw ρ lam hlam ℓ)).toMeasure)/2 := by
+  have h := discrete_degradation_mu (orderedLaw ρ lam hlam ℓ) (fun v ↦ PMF.pure (forgetOrdered ℓ v))
+  change (∫ m, xMoment m ∂(discreteProbability (fun i ↦
+    (orderedLaw ρ lam hlam ℓ i).map (forgetOrdered ℓ))).toMeasure) ≤ _ at h
+  simp only [orderedLaw_forget] at h
+  have ha := advantage_sq_bound lam hlam ρ ℓ
+  linarith
+
+lemma poisson_positive_critical_mu (d : ℝ≥0) (hd : 1 < (d:ℝ)) {lam : ℝ}
+    (hl₀ : 0 < lam) (hl₁ : lam < 1) (hlam : Admissible lam) (hcrit : (d:ℝ)*lam^2 = 1) :
+    Tendsto (fun ℓ ↦ (offspringPosteriorLaw (poissonPMF d) hl₀.le hl₁ ℓ).mu) atTop (𝓝 0) := by
+  let ρ := poissonPMF d
+  let Q := offspringPosteriorLaw ρ hl₀.le hl₁
+  let C := fun ℓ ↦ degradationCoupling (orderedLaw ρ lam hlam ℓ) (orderedExpansion ρ lam hlam ℓ)
+  have hf (ℓ : ℕ) : (C ℓ).map Prod.fst = (Q ℓ).probability := by
+    rw [degradationCoupling_fst, offspringPosteriorLaw_probability ρ hl₀.le hl₁ hlam ℓ]
+  have hs (ℓ : ℕ) : (C ℓ).map Prod.snd = (Q (ℓ+1)).probability := by
+    rw [degradationCoupling_snd]
+    have he : (fun i ↦ (orderedLaw ρ lam hlam ℓ i).bind (orderedExpansion ρ lam hlam ℓ)) =
+        orderedLaw ρ lam hlam (ℓ+1) := funext (orderedLaw_degrades ρ lam hlam ℓ)
+    rw [he]
+    exact (offspringPosteriorLaw_probability ρ hl₀.le hl₁ hlam (ℓ+1)).symm
+  have hl (ℓ : ℕ) : (∫ z, quadraticGap z ∂(C ℓ).toMeasure) = (Q ℓ).mu-(Q (ℓ+1)).mu := by
+    rw [degradationCoupling_loss]
+    have he : (fun i ↦ (orderedLaw ρ lam hlam ℓ i).bind (orderedExpansion ρ lam hlam ℓ)) =
+        orderedLaw ρ lam hlam (ℓ+1) := funext (orderedLaw_degrades ρ lam hlam ℓ)
+    rw [he]
+    simp only [Q, Law.mu, offspringPosteriorLaw_probability ρ hl₀.le hl₁ hlam]
+  obtain ⟨R,hR,hlim⟩ := degrading_offspring_fixed_point Q C hf hs hl hl₀.le hl₁ ρ (fun _ ↦ rfl)
+  rw [poisson_fixed_mu_zero R hl₀ hl₁ d hd hcrit hR] at hlim
+  exact hlim
+
+theorem poisson_positive_nonreconstruction (d : ℝ≥0) (hd : 1 < (d:ℝ))
+    (lam : ℝ) (hlam : Admissible lam) (hl₀ : 0 ≤ lam) (hKS : (d:ℝ)*lam^2 ≤ 1) :
+    Tendsto (advantage lam hlam (poissonPMF d)) atTop (𝓝 0) := by
+  have hdpos : 0 < (d:ℝ) := by linarith
+  let t : ℝ := 1/Real.sqrt d
+  have ht : 0 < t := by dsimp [t]; exact div_pos (by norm_num) (Real.sqrt_pos.mpr hdpos)
+  have htsq : (d:ℝ)*t^2 = 1 := by
+    dsimp [t]
+    rw [div_pow, one_pow, Real.sq_sqrt hdpos.le, mul_one_div_cancel (ne_of_gt hdpos)]
+  have htone : t < 1 := by
+    have h := mul_lt_mul_of_pos_right hd (sq_pos_of_pos ht)
+    nlinarith only [h, htsq, ht]
+  have htadm : Admissible t := ⟨by linarith, htone.le⟩
+  have hlt : lam ≤ t := by
+    have hs : lam^2 ≤ t^2 := (mul_le_mul_iff_right₀ hdpos).mp (by nlinarith only [hKS,htsq])
+    nlinarith only [hs,hl₀,ht]
+  let c := lam/t
+  have hcpos : 0 ≤ c := div_nonneg hl₀ ht.le
+  have hcone : c ≤ 1 := (div_le_one ht).2 hlt
+  have hcadm : Admissible c := ⟨by linarith, hcone⟩
+  have he : t*c = lam := by dsimp [c]; field_simp
+  have hprod : Admissible (t*c) := he.symm ▸ hlam
+  let ρ := poissonPMF d
+  have hm := poisson_positive_critical_mu d hd ht htone htadm htsq
+  have hu : Tendsto (fun ℓ ↦ Real.sqrt (offspringPosteriorLaw ρ ht.le htone ℓ).mu) atTop (𝓝 0) := by
+    simpa only [Real.sqrt_zero, Function.comp_def] using Real.continuous_sqrt.continuousAt.tendsto.comp hm
+  apply squeeze_zero (fun ℓ ↦ advantage_nonneg lam hlam ρ ℓ) _ hu
+  intro ℓ
+  apply (sq_le_sq₀ (advantage_nonneg lam hlam ρ ℓ) (Real.sqrt_nonneg _)).mp
+  rw [Real.sq_sqrt (offspringPosteriorLaw ρ ht.le htone ℓ).mu_nonneg]
+  have h := ordered_channel_mu_le ρ t c htadm hcadm hprod ℓ
+  have hw := ordered_advantage_sq_moment ρ lam hlam ℓ
+  have heq : orderedLaw ρ (t*c) hprod ℓ = orderedLaw ρ lam hlam ℓ := by congr 1
+  rw [heq] at h
+  rw [← offspringPosteriorLaw_probability ρ ht.le htone htadm ℓ] at h
+  change _ ≤ (offspringPosteriorLaw ρ ht.le htone ℓ).mu at h
+  linarith [(offspringPosteriorLaw ρ ht.le htone ℓ).mu_nonneg]
+
+end ThreeState.TreeClauses.Tree
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeStateClauses/PoissonLaw.lean b/lean/OAI/Probability/ThreeStateClauses/PoissonLaw.lean
index bde325a6268509eae38590a1b4934d455a7e95a6..a256e603de4ccbacaf884da5f7e84211b35e0dd1 100644
--- a/lean/OAI/Probability/ThreeStateClauses/PoissonLaw.lean
+++ b/lean/OAI/Probability/ThreeStateClauses/PoissonLaw.lean
@@ -1,136 +1,136 @@
-import Mathlib
-import OAI.Probability.ThreeStateClauses.BranchGrowth
-import OAI.Probability.ThreeStateClauses.ObservedMixtures
-import OAI.Probability.ThreeStateClauses.PoissonMoments
-
-namespace OAI
-
-/-! Poisson Law. -/
-
-open scoped BigOperators ENNReal NNReal Topology
-open Filter
-noncomputable section
-open Set MeasureTheory ProbabilityTheory
-open scoped BigOperators NNReal
-namespace ThreeState.TreeClauses.Experiment
-open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive ThreeState.TreeClauses.Probability
-
-def poissonLaw (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (t : ℝ≥0) : Law :=
-  offspringLaw (poissonPMF t) Q hl₀ hl₁
-
-lemma mixture_mu (ρ : PMF ℕ) (Q : ℕ → Law) :
-    (mixtureLaw ρ Q).mu = ∫ n, (Q n).mu ∂ρ.toMeasure := by
-  change (∫ m, xMoment m ∂(mixtureProbability ρ (fun n ↦ (Q n).probability)).toMeasure) = _
-  rw [integral_mixtureProbability _ _ (compact_integrable continuous_xMoment),
-    PMF.integral_eq_tsum _ _ (pmf_bounded_integrable ρ (fun n ↦ show |(Q n).mu| ≤ 1 by
-      rw [abs_of_nonneg (Q n).mu_nonneg]; exact (Q n).mu_le_one))]
-  rfl
-
-lemma Law.nu_le_one (Q : Law) : Q.nu ≤ 1 := by
-  apply (integral_mono (compact_integrable (continuous_xMoment.pow 2)) (integrable_const (1:ℝ)) _).trans_eq
-    (by simp)
-  intro m
-  change xMoment m^2 ≤ 1
-  nlinarith [xMoment_nonneg m, xMoment_le_one m]
-
-lemma mixture_nu (ρ : PMF ℕ) (Q : ℕ → Law) :
-    (mixtureLaw ρ Q).nu = ∫ n, (Q n).nu ∂ρ.toMeasure := by
-  change (∫ m, xMoment m^2 ∂(mixtureProbability ρ (fun n ↦ (Q n).probability)).toMeasure) = _
-  rw [integral_mixtureProbability _ _ (f := fun m ↦ xMoment m^2) (compact_integrable (continuous_xMoment.pow 2)),
-    PMF.integral_eq_tsum _ _ (pmf_bounded_integrable ρ (fun n ↦ show |(Q n).nu| ≤ 1 by
-      rw [abs_of_nonneg (Q n).nu_nonneg]; exact (Q n).nu_le_one))]
-  rfl
-
-private lemma poisson_quadratic_integral (t : ℝ≥0) (b a : ℝ) (M : ℕ → ℝ)
-    (hM : Integrable M (poissonPMF t).toMeasure) :
-    (∫ n : ℕ, (2*n*b+2*((n:ℝ)^2-n)*b^2)*(a*a)+(-4*n*b)*a+2*M n
-      ∂(poissonPMF t).toMeasure) =
-    (2*t*b+2*(t:ℝ)^2*b^2)*(a*a)+(-4*t*b)*a+2*(∫ n, M n ∂(poissonPMF t).toMeasure) := by
-  have hn := poisson_integrable_first t
-  have hn₂ := poisson_integrable_second t
-  have he (n : ℕ) : (2*n*b+2*((n:ℝ)^2-n)*b^2)*(a*a)+(-4*n*b)*a+2*M n =
-      (2*b*(a*a)-4*b*a)*(n:ℝ)+(2*b^2*(a*a))*((n:ℝ)^2-n)+2*M n := by ring
-  let A : ℝ := 2*b*(a*a)-4*b*a
-  let B : ℝ := 2*b^2*(a*a)
-  have hA : Integrable (fun n : ℕ ↦ A*(n:ℝ)) (poissonPMF t).toMeasure := hn.const_mul A
-  have hB : Integrable (fun n : ℕ ↦ B*((n:ℝ)^2-n)) (poissonPMF t).toMeasure := (hn₂.sub hn).const_mul B
-  calc
-    _ = ∫ n : ℕ, (A*(n:ℝ)+B*((n:ℝ)^2-n))+2*M n ∂(poissonPMF t).toMeasure :=
-      integral_congr_ae (Filter.Eventually.of_forall he)
-    _ = (∫ n : ℕ, A*(n:ℝ)+B*((n:ℝ)^2-n) ∂(poissonPMF t).toMeasure)+
-        (∫ n, 2*M n ∂(poissonPMF t).toMeasure) := integral_add (hA.add hB) (hM.const_mul 2)
-    _ = A*(t:ℝ)+B*(t:ℝ)^2+2*(∫ n, M n ∂(poissonPMF t).toMeasure) := by
-      rw [integral_add hA hB, integral_const_mul, integral_const_mul, integral_const_mul,
-        poisson_integral_first, poisson_integral_factorial_second]
-    _ = _ := by dsimp only [A,B]; ring
-
-lemma poisson_projection_quadratic (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
-    (t : ℝ≥0) (a : ℝ) :
-    0 ≤ (2*t*(lam^2*Q.mu)+2*(t:ℝ)^2*(lam^2*Q.mu)^2)*(a*a)+
-      (-4*t*(lam^2*Q.mu))*a+2*(poissonLaw Q hl₀ hl₁ t).mu := by
-  have hmu := pmf_bounded_integrable (poissonPMF t)
-    (fun n ↦ show |(finiteLaw Q hl₀ hl₁ n).mu| ≤ 1 by
-      rw [abs_of_nonneg (finiteLaw Q hl₀ hl₁ n).mu_nonneg]; exact (finiteLaw Q hl₀ hl₁ n).mu_le_one)
-  have h := integral_nonneg (μ := (poissonPMF t).toMeasure)
-    (f := fun n : ℕ ↦ (2*n*(lam^2*Q.mu)+2*((n:ℝ)^2-n)*(lam^2*Q.mu)^2)*(a*a)+
-      (-4*n*(lam^2*Q.mu))*a+2*(finiteLaw Q hl₀ hl₁ n).mu)
-    (fun n ↦ finite_projection_quadratic Q hl₀ hl₁ n a)
-  rw [poisson_quadratic_integral t (lam^2*Q.mu) a _ hmu] at h
-  have hm : (∫ n, (finiteLaw Q hl₀ hl₁ n).mu ∂(poissonPMF t).toMeasure) =
-      (poissonLaw Q hl₀ hl₁ t).mu := (mixture_mu _ _).symm
-  rwa [hm] at h
-
-theorem poisson_projection (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (t : ℝ≥0) :
-    (t:ℝ)*(lam^2*Q.mu)/(1+(t:ℝ)*(lam^2*Q.mu)) ≤ (poissonLaw Q hl₀ hl₁ t).mu := by
-  let β := lam^2*Q.mu
-  have hb : 0 ≤ β := mul_nonneg (sq_nonneg _) Q.mu_nonneg
-  have ht : 0 ≤ (t:ℝ) := t.coe_nonneg
-  have hd : 0 < 1+(t:ℝ)*β := by positivity
-  have h := poisson_projection_quadratic Q hl₀ hl₁ t (1/(1+(t:ℝ)*β))
-  change 0 ≤ (2*t*β+2*(t:ℝ)^2*β^2)*(1/(1+(t:ℝ)*β)*(1/(1+(t:ℝ)*β)))+
-      (-4*t*β)*(1/(1+(t:ℝ)*β))+2*(poissonLaw Q hl₀ hl₁ t).mu at h
-  have he : (2*t*β+2*(t:ℝ)^2*β^2)*(1/(1+(t:ℝ)*β)*(1/(1+(t:ℝ)*β)))+
-      (-4*t*β)*(1/(1+(t:ℝ)*β)) = -2*((t:ℝ)*β/(1+(t:ℝ)*β)) := by
-    field_simp; ring
-  rw [he] at h
-  linarith
-
-lemma poisson_integral_mono_intensity {f : ℕ → ℝ} {B : ℝ} (hB : ∀ n, |f n| ≤ B)
-    (hf : Monotone f) {s t : ℝ≥0} (hst : s ≤ t) :
-    (∫ n, f n ∂(poissonPMF s).toMeasure) ≤ ∫ n, f n ∂(poissonPMF t).toMeasure := by
-  have hts : s+(t-s) = t := add_tsub_cancel_of_le hst
-  have hi : Integrable f (poissonMeasure s ∗ poissonMeasure (t-s)) := by
-    rw [poissonMeasure_conv_poissonMeasure, hts, ← poissonPMF_toMeasure]
-    exact pmf_bounded_integrable _ hB
-  rw [poissonPMF_toMeasure, poissonPMF_toMeasure, ← hts, ← poissonMeasure_conv_poissonMeasure,
-    integral_conv hi]
-  apply integral_mono (by rw [← poissonPMF_toMeasure]; exact pmf_bounded_integrable _ hB)
-  · apply Integrable.of_bound (measurable_of_countable _).aestronglyMeasurable B
-    exact Filter.Eventually.of_forall (fun n ↦ by
-      rw [Real.norm_eq_abs]
-      apply (abs_integral_le_integral_abs).trans
-      apply (integral_mono (by rw [← poissonPMF_toMeasure]; exact (pmf_bounded_integrable _ (fun m ↦ hB (n+m))).abs)
-        (integrable_const _) (fun m ↦ hB (n+m))).trans_eq
-      simp)
-  · intro n
-    calc
-      f n = ∫ _, f n ∂poissonMeasure (t-s) := by simp
-      _ ≤ _ := integral_mono (integrable_const _) (by
-        rw [← poissonPMF_toMeasure]; exact pmf_bounded_integrable _ (fun m ↦ hB (n+m)))
-        (fun m ↦ hf (Nat.le_add_right n m))
-
-lemma poisson_nu_mono (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
-    {s t : ℝ≥0} (hst : s ≤ t) :
-    (poissonLaw Q hl₀ hl₁ s).nu ≤ (poissonLaw Q hl₀ hl₁ t).nu := by
-  change (mixtureLaw _ _).nu ≤ (mixtureLaw _ _).nu
-  rw [mixture_nu, mixture_nu]
-  apply poisson_integral_mono_intensity (B := 1) _ (fun _ _ h ↦ finite_nu_mono Q hl₀ hl₁ h) hst
-  intro n
-  rw [abs_of_nonneg (finiteLaw Q hl₀ hl₁ n).nu_nonneg]
-  exact (finiteLaw Q hl₀ hl₁ n).nu_le_one
-
-end ThreeState.TreeClauses.Experiment
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeStateClauses.BranchGrowth
+import OAI.Probability.ThreeStateClauses.ObservedMixtures
+import OAI.Probability.ThreeStateClauses.PoissonMoments
+
+namespace OAI
+
+/-! Poisson Law. -/
+
+open scoped BigOperators ENNReal NNReal Topology
+open Filter
+noncomputable section
+open Set MeasureTheory ProbabilityTheory
+open scoped BigOperators NNReal
+namespace ThreeState.TreeClauses.Experiment
+open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive ThreeState.TreeClauses.Probability
+
+def poissonLaw (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (t : ℝ≥0) : Law :=
+  offspringLaw (poissonPMF t) Q hl₀ hl₁
+
+lemma mixture_mu (ρ : PMF ℕ) (Q : ℕ → Law) :
+    (mixtureLaw ρ Q).mu = ∫ n, (Q n).mu ∂ρ.toMeasure := by
+  change (∫ m, xMoment m ∂(mixtureProbability ρ (fun n ↦ (Q n).probability)).toMeasure) = _
+  rw [integral_mixtureProbability _ _ (compact_integrable continuous_xMoment),
+    PMF.integral_eq_tsum _ _ (pmf_bounded_integrable ρ (fun n ↦ show |(Q n).mu| ≤ 1 by
+      rw [abs_of_nonneg (Q n).mu_nonneg]; exact (Q n).mu_le_one))]
+  rfl
+
+lemma Law.nu_le_one (Q : Law) : Q.nu ≤ 1 := by
+  apply (integral_mono (compact_integrable (continuous_xMoment.pow 2)) (integrable_const (1:ℝ)) _).trans_eq
+    (by simp)
+  intro m
+  change xMoment m^2 ≤ 1
+  nlinarith [xMoment_nonneg m, xMoment_le_one m]
+
+lemma mixture_nu (ρ : PMF ℕ) (Q : ℕ → Law) :
+    (mixtureLaw ρ Q).nu = ∫ n, (Q n).nu ∂ρ.toMeasure := by
+  change (∫ m, xMoment m^2 ∂(mixtureProbability ρ (fun n ↦ (Q n).probability)).toMeasure) = _
+  rw [integral_mixtureProbability _ _ (f := fun m ↦ xMoment m^2) (compact_integrable (continuous_xMoment.pow 2)),
+    PMF.integral_eq_tsum _ _ (pmf_bounded_integrable ρ (fun n ↦ show |(Q n).nu| ≤ 1 by
+      rw [abs_of_nonneg (Q n).nu_nonneg]; exact (Q n).nu_le_one))]
+  rfl
+
+private lemma poisson_quadratic_integral (t : ℝ≥0) (b a : ℝ) (M : ℕ → ℝ)
+    (hM : Integrable M (poissonPMF t).toMeasure) :
+    (∫ n : ℕ, (2*n*b+2*((n:ℝ)^2-n)*b^2)*(a*a)+(-4*n*b)*a+2*M n
+      ∂(poissonPMF t).toMeasure) =
+    (2*t*b+2*(t:ℝ)^2*b^2)*(a*a)+(-4*t*b)*a+2*(∫ n, M n ∂(poissonPMF t).toMeasure) := by
+  have hn := poisson_integrable_first t
+  have hn₂ := poisson_integrable_second t
+  have he (n : ℕ) : (2*n*b+2*((n:ℝ)^2-n)*b^2)*(a*a)+(-4*n*b)*a+2*M n =
+      (2*b*(a*a)-4*b*a)*(n:ℝ)+(2*b^2*(a*a))*((n:ℝ)^2-n)+2*M n := by ring
+  let A : ℝ := 2*b*(a*a)-4*b*a
+  let B : ℝ := 2*b^2*(a*a)
+  have hA : Integrable (fun n : ℕ ↦ A*(n:ℝ)) (poissonPMF t).toMeasure := hn.const_mul A
+  have hB : Integrable (fun n : ℕ ↦ B*((n:ℝ)^2-n)) (poissonPMF t).toMeasure := (hn₂.sub hn).const_mul B
+  calc
+    _ = ∫ n : ℕ, (A*(n:ℝ)+B*((n:ℝ)^2-n))+2*M n ∂(poissonPMF t).toMeasure :=
+      integral_congr_ae (Filter.Eventually.of_forall he)
+    _ = (∫ n : ℕ, A*(n:ℝ)+B*((n:ℝ)^2-n) ∂(poissonPMF t).toMeasure)+
+        (∫ n, 2*M n ∂(poissonPMF t).toMeasure) := integral_add (hA.add hB) (hM.const_mul 2)
+    _ = A*(t:ℝ)+B*(t:ℝ)^2+2*(∫ n, M n ∂(poissonPMF t).toMeasure) := by
+      rw [integral_add hA hB, integral_const_mul, integral_const_mul, integral_const_mul,
+        poisson_integral_first, poisson_integral_factorial_second]
+    _ = _ := by dsimp only [A,B]; ring
+
+lemma poisson_projection_quadratic (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
+    (t : ℝ≥0) (a : ℝ) :
+    0 ≤ (2*t*(lam^2*Q.mu)+2*(t:ℝ)^2*(lam^2*Q.mu)^2)*(a*a)+
+      (-4*t*(lam^2*Q.mu))*a+2*(poissonLaw Q hl₀ hl₁ t).mu := by
+  have hmu := pmf_bounded_integrable (poissonPMF t)
+    (fun n ↦ show |(finiteLaw Q hl₀ hl₁ n).mu| ≤ 1 by
+      rw [abs_of_nonneg (finiteLaw Q hl₀ hl₁ n).mu_nonneg]; exact (finiteLaw Q hl₀ hl₁ n).mu_le_one)
+  have h := integral_nonneg (μ := (poissonPMF t).toMeasure)
+    (f := fun n : ℕ ↦ (2*n*(lam^2*Q.mu)+2*((n:ℝ)^2-n)*(lam^2*Q.mu)^2)*(a*a)+
+      (-4*n*(lam^2*Q.mu))*a+2*(finiteLaw Q hl₀ hl₁ n).mu)
+    (fun n ↦ finite_projection_quadratic Q hl₀ hl₁ n a)
+  rw [poisson_quadratic_integral t (lam^2*Q.mu) a _ hmu] at h
+  have hm : (∫ n, (finiteLaw Q hl₀ hl₁ n).mu ∂(poissonPMF t).toMeasure) =
+      (poissonLaw Q hl₀ hl₁ t).mu := (mixture_mu _ _).symm
+  rwa [hm] at h
+
+theorem poisson_projection (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (t : ℝ≥0) :
+    (t:ℝ)*(lam^2*Q.mu)/(1+(t:ℝ)*(lam^2*Q.mu)) ≤ (poissonLaw Q hl₀ hl₁ t).mu := by
+  let β := lam^2*Q.mu
+  have hb : 0 ≤ β := mul_nonneg (sq_nonneg _) Q.mu_nonneg
+  have ht : 0 ≤ (t:ℝ) := t.coe_nonneg
+  have hd : 0 < 1+(t:ℝ)*β := by positivity
+  have h := poisson_projection_quadratic Q hl₀ hl₁ t (1/(1+(t:ℝ)*β))
+  change 0 ≤ (2*t*β+2*(t:ℝ)^2*β^2)*(1/(1+(t:ℝ)*β)*(1/(1+(t:ℝ)*β)))+
+      (-4*t*β)*(1/(1+(t:ℝ)*β))+2*(poissonLaw Q hl₀ hl₁ t).mu at h
+  have he : (2*t*β+2*(t:ℝ)^2*β^2)*(1/(1+(t:ℝ)*β)*(1/(1+(t:ℝ)*β)))+
+      (-4*t*β)*(1/(1+(t:ℝ)*β)) = -2*((t:ℝ)*β/(1+(t:ℝ)*β)) := by
+    field_simp; ring
+  rw [he] at h
+  linarith
+
+lemma poisson_integral_mono_intensity {f : ℕ → ℝ} {B : ℝ} (hB : ∀ n, |f n| ≤ B)
+    (hf : Monotone f) {s t : ℝ≥0} (hst : s ≤ t) :
+    (∫ n, f n ∂(poissonPMF s).toMeasure) ≤ ∫ n, f n ∂(poissonPMF t).toMeasure := by
+  have hts : s+(t-s) = t := add_tsub_cancel_of_le hst
+  have hi : Integrable f (poissonMeasure s ∗ poissonMeasure (t-s)) := by
+    rw [poissonMeasure_conv_poissonMeasure, hts, ← poissonPMF_toMeasure]
+    exact pmf_bounded_integrable _ hB
+  rw [poissonPMF_toMeasure, poissonPMF_toMeasure, ← hts, ← poissonMeasure_conv_poissonMeasure,
+    integral_conv hi]
+  apply integral_mono (by rw [← poissonPMF_toMeasure]; exact pmf_bounded_integrable _ hB)
+  · apply Integrable.of_bound (measurable_of_countable _).aestronglyMeasurable B
+    exact Filter.Eventually.of_forall (fun n ↦ by
+      rw [Real.norm_eq_abs]
+      apply (abs_integral_le_integral_abs).trans
+      apply (integral_mono (by rw [← poissonPMF_toMeasure]; exact (pmf_bounded_integrable _ (fun m ↦ hB (n+m))).abs)
+        (integrable_const _) (fun m ↦ hB (n+m))).trans_eq
+      simp)
+  · intro n
+    calc
+      f n = ∫ _, f n ∂poissonMeasure (t-s) := by simp
+      _ ≤ _ := integral_mono (integrable_const _) (by
+        rw [← poissonPMF_toMeasure]; exact pmf_bounded_integrable _ (fun m ↦ hB (n+m)))
+        (fun m ↦ hf (Nat.le_add_right n m))
+
+lemma poisson_nu_mono (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
+    {s t : ℝ≥0} (hst : s ≤ t) :
+    (poissonLaw Q hl₀ hl₁ s).nu ≤ (poissonLaw Q hl₀ hl₁ t).nu := by
+  change (mixtureLaw _ _).nu ≤ (mixtureLaw _ _).nu
+  rw [mixture_nu, mixture_nu]
+  apply poisson_integral_mono_intensity (B := 1) _ (fun _ _ h ↦ finite_nu_mono Q hl₀ hl₁ h) hst
+  intro n
+  rw [abs_of_nonneg (finiteLaw Q hl₀ hl₁ n).nu_nonneg]
+  exact (finiteLaw Q hl₀ hl₁ n).nu_le_one
+
+end ThreeState.TreeClauses.Experiment
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeStateClauses/PoissonMoments.lean b/lean/OAI/Probability/ThreeStateClauses/PoissonMoments.lean
index d116d6a39108924a13b2e6ae43823be5d0aec6e5..c6ccfca4d36bb6e4d57d7d0d169a6f5bafb069e2 100644
--- a/lean/OAI/Probability/ThreeStateClauses/PoissonMoments.lean
+++ b/lean/OAI/Probability/ThreeStateClauses/PoissonMoments.lean
@@ -1,98 +1,98 @@
-import Mathlib
-
-namespace OAI
-
-/-! Poisson Moments. -/
-
-open scoped BigOperators ENNReal NNReal Topology
-open Filter
-noncomputable section
-open MeasureTheory ProbabilityTheory
-open scoped BigOperators NNReal
-namespace ThreeState.TreeClauses.Probability
-
-def poissonWeight (t : ℝ) (n : ℕ) : ℝ := Real.exp (-t)*t^n/(n.factorial:ℝ)
-
-lemma poissonWeight_nonneg {t : ℝ} (ht : 0 ≤ t) (n : ℕ) : 0 ≤ poissonWeight t n := by
-  unfold poissonWeight; positivity
-
-lemma poissonWeight_mass (t : ℝ≥0) : HasSum (poissonWeight t) 1 :=
-  hasSum_one_poissonMeasure t
-
-def _root_.OAI.ThreeState.TreeClauses.poissonPMF (rate : ℝ≥0) : PMF ℕ := by
-  refine ⟨fun count ↦ ENNReal.ofReal (poissonWeight rate count), ?_⟩
-  apply ENNReal.hasSum_coe.mpr
-  rw [← Real.toNNReal_one]
-  exact (poissonWeight_mass rate).toNNReal (poissonWeight_nonneg rate.coe_nonneg)
-
-lemma poissonWeight_succ (t : ℝ) (n : ℕ) :
-    poissonWeight t (n+1)*((n:ℝ)+1) = t*poissonWeight t n := by
-  unfold poissonWeight
-  rw [Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, pow_succ]
-  have hn : (n:ℝ)+1 ≠ 0 := by positivity
-  field_simp
-
-lemma poissonWeight_mean (t : ℝ≥0) : HasSum (fun n ↦ poissonWeight t n*(n:ℝ)) t := by
-  apply (hasSum_nat_add_iff' 1).mp
-  simp only [Finset.range_one, Finset.sum_singleton, Nat.cast_zero, mul_zero, sub_zero,
-    Nat.cast_add, Nat.cast_one, poissonWeight_succ]
-  simpa using (poissonWeight_mass t).mul_left (t:ℝ)
-
-lemma poissonWeight_factorial_second (t : ℝ≥0) :
-    HasSum (fun n ↦ poissonWeight t n*((n:ℝ)^2-n)) ((t:ℝ)^2) := by
-  apply (hasSum_nat_add_iff' 1).mp
-  simp only [Finset.range_one, Finset.sum_singleton, Nat.cast_zero, zero_pow (by decide : 2 ≠ 0),
-    sub_self, mul_zero, sub_zero]
-  have he (n : ℕ) : poissonWeight t (n+1)*(((n+1:ℕ):ℝ)^2-((n+1:ℕ):ℝ)) =
-      (t:ℝ)*(poissonWeight t n*(n:ℝ)) := by
-    push_cast
-    calc
-      _ = (poissonWeight t (n+1)*((n:ℝ)+1))*(n:ℝ) := by ring
-      _ = _ := by rw [poissonWeight_succ]; ring
-  simp_rw [he]
-  simpa [pow_two] using (poissonWeight_mean t).mul_left (t:ℝ)
-
-lemma poissonWeight_second (t : ℝ≥0) :
-    HasSum (fun n ↦ poissonWeight t n*(n:ℝ)^2) ((t:ℝ)^2+t) := by
-  have h := (poissonWeight_factorial_second t).add (poissonWeight_mean t)
-  convert h using 1
-  ext n; ring
-
-lemma poissonPMF_real (t : ℝ≥0) (n : ℕ) : (poissonPMF t n).toReal = poissonWeight t n := by
-  change (ENNReal.ofReal (poissonWeight t n)).toReal = _
-  exact ENNReal.toReal_ofReal (poissonWeight_nonneg t.coe_nonneg n)
-
-lemma poissonPMF_toMeasure (t : ℝ≥0) : (poissonPMF t).toMeasure = poissonMeasure t := by
-  apply Measure.ext_of_singleton
-  intro n
-  rw [PMF.toMeasure_apply_singleton, poissonMeasure_singleton]
-  · rfl
-  · exact measurableSet_singleton n
-
-lemma poisson_integrable_first (t : ℝ≥0) : Integrable (fun n : ℕ ↦ (n:ℝ)) (poissonPMF t).toMeasure := by
-  rw [poissonPMF_toMeasure, integrable_poissonMeasure_iff]
-  simpa only [Real.norm_eq_abs, Nat.abs_cast, smul_eq_mul, poissonWeight] using (poissonWeight_mean t).summable
-
-lemma poisson_integrable_second (t : ℝ≥0) : Integrable (fun n : ℕ ↦ (n:ℝ)^2) (poissonPMF t).toMeasure := by
-  rw [poissonPMF_toMeasure, integrable_poissonMeasure_iff]
-  simpa only [Real.norm_eq_abs, abs_pow, Nat.abs_cast, smul_eq_mul, poissonWeight] using (poissonWeight_second t).summable
-
-lemma poisson_integral_first (t : ℝ≥0) : (∫ n : ℕ, (n:ℝ) ∂(poissonPMF t).toMeasure) = t := by
-  rw [PMF.integral_eq_tsum _ _ (poisson_integrable_first t)]
-  simpa only [poissonPMF_real, smul_eq_mul] using (poissonWeight_mean t).tsum_eq
-
-lemma poisson_integral_second (t : ℝ≥0) : (∫ n : ℕ, (n:ℝ)^2 ∂(poissonPMF t).toMeasure) = (t:ℝ)^2+t := by
-  rw [PMF.integral_eq_tsum _ _ (poisson_integrable_second t)]
-  simpa only [poissonPMF_real, smul_eq_mul] using (poissonWeight_second t).tsum_eq
-
-lemma poisson_integral_factorial_second (t : ℝ≥0) :
-    (∫ n : ℕ, ((n:ℝ)^2-n) ∂(poissonPMF t).toMeasure) = (t:ℝ)^2 := by
-  rw [integral_sub (poisson_integrable_second t) (poisson_integrable_first t),
-    poisson_integral_second, poisson_integral_first]
-  ring
-
-end ThreeState.TreeClauses.Probability
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-! Poisson Moments. -/
+
+open scoped BigOperators ENNReal NNReal Topology
+open Filter
+noncomputable section
+open MeasureTheory ProbabilityTheory
+open scoped BigOperators NNReal
+namespace ThreeState.TreeClauses.Probability
+
+def poissonWeight (t : ℝ) (n : ℕ) : ℝ := Real.exp (-t)*t^n/(n.factorial:ℝ)
+
+lemma poissonWeight_nonneg {t : ℝ} (ht : 0 ≤ t) (n : ℕ) : 0 ≤ poissonWeight t n := by
+  unfold poissonWeight; positivity
+
+lemma poissonWeight_mass (t : ℝ≥0) : HasSum (poissonWeight t) 1 :=
+  hasSum_one_poissonMeasure t
+
+def _root_.OAI.ThreeState.TreeClauses.poissonPMF (rate : ℝ≥0) : PMF ℕ := by
+  refine ⟨fun count ↦ ENNReal.ofReal (poissonWeight rate count), ?_⟩
+  apply ENNReal.hasSum_coe.mpr
+  rw [← Real.toNNReal_one]
+  exact (poissonWeight_mass rate).toNNReal (poissonWeight_nonneg rate.coe_nonneg)
+
+lemma poissonWeight_succ (t : ℝ) (n : ℕ) :
+    poissonWeight t (n+1)*((n:ℝ)+1) = t*poissonWeight t n := by
+  unfold poissonWeight
+  rw [Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, pow_succ]
+  have hn : (n:ℝ)+1 ≠ 0 := by positivity
+  field_simp
+
+lemma poissonWeight_mean (t : ℝ≥0) : HasSum (fun n ↦ poissonWeight t n*(n:ℝ)) t := by
+  apply (hasSum_nat_add_iff' 1).mp
+  simp only [Finset.range_one, Finset.sum_singleton, Nat.cast_zero, mul_zero, sub_zero,
+    Nat.cast_add, Nat.cast_one, poissonWeight_succ]
+  simpa using (poissonWeight_mass t).mul_left (t:ℝ)
+
+lemma poissonWeight_factorial_second (t : ℝ≥0) :
+    HasSum (fun n ↦ poissonWeight t n*((n:ℝ)^2-n)) ((t:ℝ)^2) := by
+  apply (hasSum_nat_add_iff' 1).mp
+  simp only [Finset.range_one, Finset.sum_singleton, Nat.cast_zero, zero_pow (by decide : 2 ≠ 0),
+    sub_self, mul_zero, sub_zero]
+  have he (n : ℕ) : poissonWeight t (n+1)*(((n+1:ℕ):ℝ)^2-((n+1:ℕ):ℝ)) =
+      (t:ℝ)*(poissonWeight t n*(n:ℝ)) := by
+    push_cast
+    calc
+      _ = (poissonWeight t (n+1)*((n:ℝ)+1))*(n:ℝ) := by ring
+      _ = _ := by rw [poissonWeight_succ]; ring
+  simp_rw [he]
+  simpa [pow_two] using (poissonWeight_mean t).mul_left (t:ℝ)
+
+lemma poissonWeight_second (t : ℝ≥0) :
+    HasSum (fun n ↦ poissonWeight t n*(n:ℝ)^2) ((t:ℝ)^2+t) := by
+  have h := (poissonWeight_factorial_second t).add (poissonWeight_mean t)
+  convert h using 1
+  ext n; ring
+
+lemma poissonPMF_real (t : ℝ≥0) (n : ℕ) : (poissonPMF t n).toReal = poissonWeight t n := by
+  change (ENNReal.ofReal (poissonWeight t n)).toReal = _
+  exact ENNReal.toReal_ofReal (poissonWeight_nonneg t.coe_nonneg n)
+
+lemma poissonPMF_toMeasure (t : ℝ≥0) : (poissonPMF t).toMeasure = poissonMeasure t := by
+  apply Measure.ext_of_singleton
+  intro n
+  rw [PMF.toMeasure_apply_singleton, poissonMeasure_singleton]
+  · rfl
+  · exact measurableSet_singleton n
+
+lemma poisson_integrable_first (t : ℝ≥0) : Integrable (fun n : ℕ ↦ (n:ℝ)) (poissonPMF t).toMeasure := by
+  rw [poissonPMF_toMeasure, integrable_poissonMeasure_iff]
+  simpa only [Real.norm_eq_abs, Nat.abs_cast, smul_eq_mul, poissonWeight] using (poissonWeight_mean t).summable
+
+lemma poisson_integrable_second (t : ℝ≥0) : Integrable (fun n : ℕ ↦ (n:ℝ)^2) (poissonPMF t).toMeasure := by
+  rw [poissonPMF_toMeasure, integrable_poissonMeasure_iff]
+  simpa only [Real.norm_eq_abs, abs_pow, Nat.abs_cast, smul_eq_mul, poissonWeight] using (poissonWeight_second t).summable
+
+lemma poisson_integral_first (t : ℝ≥0) : (∫ n : ℕ, (n:ℝ) ∂(poissonPMF t).toMeasure) = t := by
+  rw [PMF.integral_eq_tsum _ _ (poisson_integrable_first t)]
+  simpa only [poissonPMF_real, smul_eq_mul] using (poissonWeight_mean t).tsum_eq
+
+lemma poisson_integral_second (t : ℝ≥0) : (∫ n : ℕ, (n:ℝ)^2 ∂(poissonPMF t).toMeasure) = (t:ℝ)^2+t := by
+  rw [PMF.integral_eq_tsum _ _ (poisson_integrable_second t)]
+  simpa only [poissonPMF_real, smul_eq_mul] using (poissonWeight_second t).tsum_eq
+
+lemma poisson_integral_factorial_second (t : ℝ≥0) :
+    (∫ n : ℕ, ((n:ℝ)^2-n) ∂(poissonPMF t).toMeasure) = (t:ℝ)^2 := by
+  rw [integral_sub (poisson_integrable_second t) (poisson_integrable_first t),
+    poisson_integral_second, poisson_integral_first]
+  ring
+
+end ThreeState.TreeClauses.Probability
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeStateClauses/PoissonReconstruction.lean b/lean/OAI/Probability/ThreeStateClauses/PoissonReconstruction.lean
index 1975986f0ed656d9cd312ba2adab42ba3e3e09cc..978ac414f30396966974e8cce0b585c3329410a9 100644
--- a/lean/OAI/Probability/ThreeStateClauses/PoissonReconstruction.lean
+++ b/lean/OAI/Probability/ThreeStateClauses/PoissonReconstruction.lean
@@ -1,203 +1,203 @@
-import Mathlib
-import OAI.Probability.ThreeStateClauses.RegularStatistics
-import OAI.Probability.ThreeStateClauses.OffspringLaws
-import OAI.Probability.ThreeStateClauses.PoissonMoments
-import OAI.Probability.ThreeStateClauses.CountableStatistics
-
-namespace OAI
-
-/-! Poisson Reconstruction. -/
-
-open scoped BigOperators ENNReal NNReal Topology
-open Filter
-noncomputable section
-open Set MeasureTheory
-open scoped BigOperators NNReal
-namespace ThreeState.TreeClauses.Tree
-open ThreeState.TreeClauses.Experiment ThreeState.TreeClauses.Probability
-
-def orderedStatistic (d lam : ℝ) : (ℓ : ℕ) → OrderedObservation ℓ → ℝ
-  | 0, i => spinScore i
-  | ℓ+1, ⟨_,v⟩ => (∑ j, orderedStatistic d lam ℓ (v j))/(d*lam)
-
-def observedStatistic (d lam : ℝ) : (ℓ : ℕ) → Observation ℓ → ℝ
-  | 0, i => spinScore i
-  | ℓ+1, s => (s.map (observedStatistic d lam ℓ)).sum/(d*lam)
-
-lemma observedStatistic_forget (d lam : ℝ) (ℓ : ℕ) (v : OrderedObservation ℓ) :
-    observedStatistic d lam ℓ (forgetOrdered ℓ v) = orderedStatistic d lam ℓ v := by
-  induction ℓ with
-  | zero => rfl
-  | succ ℓ ih =>
-    rcases v with ⟨n,v⟩
-    simp only [forgetOrdered, observedStatistic, orderedStatistic]
-    congr 1
-    simp only [Multiset.map_coe, Multiset.sum_coe, List.map_ofFn, Function.comp_def, ih, List.sum_ofFn]
-
-lemma poissonStatistic_square_integrable (d : ℝ≥0) (lam : ℝ) (hlam : Admissible lam)
-    (ℓ : ℕ) (i : Spin) :
-    Integrable (fun v ↦ orderedStatistic d lam ℓ v^2)
-      (orderedLaw (poissonPMF d) lam hlam ℓ i).toMeasure := by
-  induction ℓ generalizing i with
-  | zero => exact Integrable.of_finite
-  | succ ℓ ih =>
-    let p := (channel lam hlam i).bind (orderedLaw (poissonPMF d) lam hlam ℓ)
-    have hp : Integrable (fun v ↦ orderedStatistic d lam ℓ v^2) p.toMeasure :=
-      pmf_integrable_bind_finite _ _ ih
-    change Integrable (fun z : Σ n, Fin n → OrderedObservation ℓ ↦
-      ((∑ j, orderedStatistic d lam ℓ (z.2 j))/((d:ℝ)*lam))^2)
-      (sigmaPMF (poissonPMF d) (fun n ↦ productPMF (fun _ : Fin n ↦ p))).toMeasure
-    apply sigmaPMF_integrable
-    · intro n
-      simpa only [div_pow] using (product_square_sum_integrable p hp n).div_const (((d:ℝ)*lam)^2)
-    · change Integrable (fun n ↦ ∫ v : Fin n → OrderedObservation ℓ,
-        |((∑ j, orderedStatistic d lam ℓ (v j))/((d:ℝ)*lam))^2|
-        ∂(productPMF (fun _ : Fin n ↦ p)).toMeasure) (poissonPMF d).toMeasure
-      have hab (x : ℝ) : |x^2| = x^2 := abs_of_nonneg (sq_nonneg x)
-      simp_rw [hab, div_pow, integral_div,
-        integral_product_sq_sum_integrable p hp]
-      exact (((poisson_integrable_first d).mul_const _).add
-        (((poisson_integrable_second d).sub (poisson_integrable_first d)).mul_const _)).div_const _
-
-lemma poissonStatistic_mean (d : ℝ≥0) (hd : (d:ℝ) ≠ 0) (lam : ℝ) (hlam : Admissible lam)
-    (hl : lam ≠ 0) (ℓ : ℕ) (i : Spin) :
-    (∫ v, orderedStatistic d lam ℓ v
-      ∂(orderedLaw (poissonPMF d) lam hlam ℓ i).toMeasure) = spinScore i := by
-  induction ℓ generalizing i with
-  | zero =>
-    change (∫ v : Spin, spinScore v ∂(PMF.pure i).toMeasure) = spinScore i
-    rw [PMF.toMeasure_pure, integral_dirac]
-  | succ ℓ ih =>
-    let p := (channel lam hlam i).bind (orderedLaw (poissonPMF d) lam hlam ℓ)
-    have hp₂ : Integrable (fun v ↦ orderedStatistic d lam ℓ v^2) p.toMeasure :=
-      pmf_integrable_bind_finite _ _ (poissonStatistic_square_integrable d lam hlam ℓ)
-    have hp := pmf_integrable_of_square p hp₂
-    have hi := pmf_integrable_of_square _ (poissonStatistic_square_integrable d lam hlam (ℓ+1) i)
-    change (∫ z : Σ n, Fin n → OrderedObservation ℓ,
-      (∑ j, orderedStatistic d lam ℓ (z.2 j))/((d:ℝ)*lam)
-      ∂(sigmaPMF (poissonPMF d) (fun n ↦ productPMF (fun _ : Fin n ↦ p))).toMeasure) = _
-    change Integrable (fun z : Σ n, Fin n → OrderedObservation ℓ ↦
-      (∑ j, orderedStatistic d lam ℓ (z.2 j))/((d:ℝ)*lam))
-      (sigmaPMF (poissonPMF d) (fun n ↦ productPMF (fun _ : Fin n ↦ p))).toMeasure at hi
-    rw [integral_sigmaPMF_integrable _ _ hi]
-    simp_rw [integral_div, integral_product_sum_integrable p hp]
-    have hm : (∫ v, orderedStatistic d lam ℓ v ∂p.toMeasure) = lam*spinScore i := by
-      rw [integral_pmf_bind_integrable _ _ hp]
-      simp_rw [ih]
-      exact channel_spinScore lam hlam i
-    rw [hm, integral_mul_const, poisson_integral_first]
-    field_simp
-
-end ThreeState.TreeClauses.Tree
-
-end 
-
-noncomputable section
-open Set MeasureTheory Filter
-open scoped BigOperators Topology NNReal
-namespace ThreeState.TreeClauses.Tree
-open ThreeState.TreeClauses.Experiment ThreeState.TreeClauses.Probability
-
-def poissonKSConstant (d lam : ℝ) : ℝ := 4*(d*lam^2)/(d*lam^2-1)
-
-lemma poissonKSConstant_pos {d lam : ℝ} (hKS : 1 < d*lam^2) : 0 < poissonKSConstant d lam := by
-  unfold poissonKSConstant; positivity
-
-lemma poissonKSConstant_ge_four {d lam : ℝ} (hKS : 1 < d*lam^2) : 4 ≤ poissonKSConstant d lam := by
-  rw [poissonKSConstant, le_div_iff₀ (by linarith)]
-  linarith
-
-lemma poissonKSConstant_identity {d lam : ℝ} (hKS : 1 < d*lam^2) :
-    poissonKSConstant d lam*(d*lam^2-1) = 4*(d*lam^2) := by
-  rw [poissonKSConstant, div_mul_cancel₀ _ (by linarith)]
-
-lemma poissonStatistic_second (d : ℝ≥0) (lam : ℝ) (hlam : Admissible lam)
-    (hKS : 1 < (d:ℝ)*lam^2) (ℓ : ℕ) (i : Spin) :
-    (∫ v, orderedStatistic d lam ℓ v^2
-      ∂(orderedLaw (poissonPMF d) lam hlam ℓ i).toMeasure) ≤ poissonKSConstant d lam := by
-  have hd : 0 < (d:ℝ) := by
-    by_contra h
-    have he : (d:ℝ) = 0 := le_antisymm (not_lt.mp h) d.coe_nonneg
-    norm_num [he] at hKS
-  have hl : lam ≠ 0 := by intro h; norm_num [h] at hKS
-  have hden : 0 < ((d:ℝ)*lam)^2 := sq_pos_of_ne_zero (mul_ne_zero hd.ne' hl)
-  induction ℓ generalizing i with
-  | zero =>
-    change (∫ v : Spin, spinScore v^2 ∂(PMF.pure i).toMeasure) ≤ _
-    rw [PMF.toMeasure_pure, integral_dirac]
-    exact (spinScore_square i).trans (poissonKSConstant_ge_four hKS)
-  | succ ℓ ih =>
-    let p := (channel lam hlam i).bind (orderedLaw (poissonPMF d) lam hlam ℓ)
-    have hp₂ : Integrable (fun v ↦ orderedStatistic d lam ℓ v^2) p.toMeasure :=
-      pmf_integrable_bind_finite _ _ (poissonStatistic_square_integrable d lam hlam ℓ)
-    have hp := pmf_integrable_of_square p hp₂
-    have hmean : (∫ v, orderedStatistic d lam ℓ v ∂p.toMeasure) = lam*spinScore i := by
-      rw [integral_pmf_bind_integrable _ _ hp]
-      simp_rw [poissonStatistic_mean d hd.ne' lam hlam hl ℓ]
-      exact channel_spinScore lam hlam i
-    have hsecond : (∫ v, orderedStatistic d lam ℓ v^2 ∂p.toMeasure) ≤ poissonKSConstant d lam := by
-      rw [integral_pmf_bind_integrable _ _ hp₂]
-      exact (integral_mono Integrable.of_finite (integrable_const _) ih).trans_eq (by simp)
-    have hi := poissonStatistic_square_integrable d lam hlam (ℓ+1) i
-    change (∫ z : Σ n, Fin n → OrderedObservation ℓ,
-      ((∑ j, orderedStatistic d lam ℓ (z.2 j))/((d:ℝ)*lam))^2
-      ∂(sigmaPMF (poissonPMF d) (fun n ↦ productPMF (fun _ : Fin n ↦ p))).toMeasure) ≤ _
-    change Integrable (fun z : Σ n, Fin n → OrderedObservation ℓ ↦
-      ((∑ j, orderedStatistic d lam ℓ (z.2 j))/((d:ℝ)*lam))^2)
-      (sigmaPMF (poissonPMF d) (fun n ↦ productPMF (fun _ : Fin n ↦ p))).toMeasure at hi
-    rw [integral_sigmaPMF_integrable _ _ hi]
-    simp_rw [div_pow, integral_div, integral_product_sq_sum_integrable p hp₂]
-    rw [hmean, integral_add
-      (f := fun n : ℕ ↦ (n:ℝ)*(∫ v, orderedStatistic d lam ℓ v^2 ∂p.toMeasure))
-      (g := fun n : ℕ ↦ ((n:ℝ)^2-n)*(lam*spinScore i)^2)
-      ((poisson_integrable_first d).mul_const _)
-      (((poisson_integrable_second d).sub (poisson_integrable_first d)).mul_const _),
-      integral_mul_const, integral_mul_const, poisson_integral_first, poisson_integral_factorial_second]
-    apply (div_le_iff₀ hden).2
-    have hs := mul_le_mul_of_nonneg_left (spinScore_square i) (sq_nonneg ((d:ℝ)*lam))
-    have hm := mul_le_mul_of_nonneg_left hsecond hd.le
-    have hid := congrArg (fun x : ℝ ↦ x*(d:ℝ)) (poissonKSConstant_identity hKS)
-    nlinarith only [hs,hm,hid]
-
-lemma poisson_estimator_advantage (d : ℝ≥0) (lam : ℝ) (hlam : Admissible lam)
-    (hKS : 1 < (d:ℝ)*lam^2) (ℓ : ℕ) :
-    1/(3*poissonKSConstant d lam) ≤ advantage lam hlam (poissonPMF d) ℓ := by
-  have hd : (d:ℝ) ≠ 0 := by intro h; norm_num [h] at hKS
-  have hl : lam ≠ 0 := by intro h; norm_num [h] at hKS
-  let p := observationLaw lam hlam (poissonPMF d) ℓ
-  let f := observedStatistic d lam ℓ
-  have hInt (i : Spin) : Integrable (fun o ↦ f o^2) (p i).toMeasure := by
-    dsimp [p, f]
-    rw [← orderedLaw_forget _ lam hlam ℓ i, pmf_integrable_map_iff]
-    simpa only [observedStatistic_forget] using poissonStatistic_square_integrable d lam hlam ℓ i
-  have hmean (i : Spin) : (∫ o, f o ∂(p i).toMeasure) = spinScore i := by
-    dsimp [p, f]
-    rw [← orderedLaw_forget _ lam hlam ℓ i, integral_pmf_map]
-    simp_rw [observedStatistic_forget]
-    exact poissonStatistic_mean d hd lam hlam hl ℓ i
-  have hsecond (i : Spin) : (∫ o, f o^2 ∂(p i).toMeasure) ≤ poissonKSConstant d lam := by
-    dsimp [p, f]
-    rw [← orderedLaw_forget _ lam hlam ℓ i, integral_pmf_map]
-    simp_rw [observedStatistic_forget]
-    exact poissonStatistic_second d lam hlam hKS ℓ i
-  rw [advantage_eq_integral]
-  apply estimator_tv_lower_integrable (p := p) hInt (poissonKSConstant_pos hKS)
-  · rw [integral_discreteMarginal_integrable p (fun i ↦ pmf_integrable_of_square _ (hInt i)),
-      hmean 0, hmean 1, hmean 2]
-    norm_num [spinScore, show (1:Spin) ≠ 0 by decide, show (2:Spin) ≠ 0 by decide]
-  · exact hmean 0
-  · rw [integral_discreteMarginal_integrable p hInt]
-    linarith [hsecond 0, hsecond 1, hsecond 2]
-
-theorem poisson_reconstructs_above_KS (d : ℝ≥0) (lam : ℝ) (hlam : Admissible lam)
-    (hKS : 1 < (d:ℝ)*lam^2) : Reconstructs lam hlam (poissonPMF d) := by
-  refine ⟨⨅ ℓ, advantage lam hlam (poissonPMF d) ℓ, ?_, advantage_tendsto _ _ _⟩
-  have he : 1/(3*poissonKSConstant d lam) ≤ ⨅ ℓ, advantage lam hlam (poissonPMF d) ℓ := by
-    exact le_ciInf (fun ℓ ↦ poisson_estimator_advantage d lam hlam hKS ℓ)
-  exact (div_pos (by norm_num) (mul_pos (by norm_num) (poissonKSConstant_pos hKS))).trans_le he
-
-end ThreeState.TreeClauses.Tree
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeStateClauses.RegularStatistics
+import OAI.Probability.ThreeStateClauses.OffspringLaws
+import OAI.Probability.ThreeStateClauses.PoissonMoments
+import OAI.Probability.ThreeStateClauses.CountableStatistics
+
+namespace OAI
+
+/-! Poisson Reconstruction. -/
+
+open scoped BigOperators ENNReal NNReal Topology
+open Filter
+noncomputable section
+open Set MeasureTheory
+open scoped BigOperators NNReal
+namespace ThreeState.TreeClauses.Tree
+open ThreeState.TreeClauses.Experiment ThreeState.TreeClauses.Probability
+
+def orderedStatistic (d lam : ℝ) : (ℓ : ℕ) → OrderedObservation ℓ → ℝ
+  | 0, i => spinScore i
+  | ℓ+1, ⟨_,v⟩ => (∑ j, orderedStatistic d lam ℓ (v j))/(d*lam)
+
+def observedStatistic (d lam : ℝ) : (ℓ : ℕ) → Observation ℓ → ℝ
+  | 0, i => spinScore i
+  | ℓ+1, s => (s.map (observedStatistic d lam ℓ)).sum/(d*lam)
+
+lemma observedStatistic_forget (d lam : ℝ) (ℓ : ℕ) (v : OrderedObservation ℓ) :
+    observedStatistic d lam ℓ (forgetOrdered ℓ v) = orderedStatistic d lam ℓ v := by
+  induction ℓ with
+  | zero => rfl
+  | succ ℓ ih =>
+    rcases v with ⟨n,v⟩
+    simp only [forgetOrdered, observedStatistic, orderedStatistic]
+    congr 1
+    simp only [Multiset.map_coe, Multiset.sum_coe, List.map_ofFn, Function.comp_def, ih, List.sum_ofFn]
+
+lemma poissonStatistic_square_integrable (d : ℝ≥0) (lam : ℝ) (hlam : Admissible lam)
+    (ℓ : ℕ) (i : Spin) :
+    Integrable (fun v ↦ orderedStatistic d lam ℓ v^2)
+      (orderedLaw (poissonPMF d) lam hlam ℓ i).toMeasure := by
+  induction ℓ generalizing i with
+  | zero => exact Integrable.of_finite
+  | succ ℓ ih =>
+    let p := (channel lam hlam i).bind (orderedLaw (poissonPMF d) lam hlam ℓ)
+    have hp : Integrable (fun v ↦ orderedStatistic d lam ℓ v^2) p.toMeasure :=
+      pmf_integrable_bind_finite _ _ ih
+    change Integrable (fun z : Σ n, Fin n → OrderedObservation ℓ ↦
+      ((∑ j, orderedStatistic d lam ℓ (z.2 j))/((d:ℝ)*lam))^2)
+      (sigmaPMF (poissonPMF d) (fun n ↦ productPMF (fun _ : Fin n ↦ p))).toMeasure
+    apply sigmaPMF_integrable
+    · intro n
+      simpa only [div_pow] using (product_square_sum_integrable p hp n).div_const (((d:ℝ)*lam)^2)
+    · change Integrable (fun n ↦ ∫ v : Fin n → OrderedObservation ℓ,
+        |((∑ j, orderedStatistic d lam ℓ (v j))/((d:ℝ)*lam))^2|
+        ∂(productPMF (fun _ : Fin n ↦ p)).toMeasure) (poissonPMF d).toMeasure
+      have hab (x : ℝ) : |x^2| = x^2 := abs_of_nonneg (sq_nonneg x)
+      simp_rw [hab, div_pow, integral_div,
+        integral_product_sq_sum_integrable p hp]
+      exact (((poisson_integrable_first d).mul_const _).add
+        (((poisson_integrable_second d).sub (poisson_integrable_first d)).mul_const _)).div_const _
+
+lemma poissonStatistic_mean (d : ℝ≥0) (hd : (d:ℝ) ≠ 0) (lam : ℝ) (hlam : Admissible lam)
+    (hl : lam ≠ 0) (ℓ : ℕ) (i : Spin) :
+    (∫ v, orderedStatistic d lam ℓ v
+      ∂(orderedLaw (poissonPMF d) lam hlam ℓ i).toMeasure) = spinScore i := by
+  induction ℓ generalizing i with
+  | zero =>
+    change (∫ v : Spin, spinScore v ∂(PMF.pure i).toMeasure) = spinScore i
+    rw [PMF.toMeasure_pure, integral_dirac]
+  | succ ℓ ih =>
+    let p := (channel lam hlam i).bind (orderedLaw (poissonPMF d) lam hlam ℓ)
+    have hp₂ : Integrable (fun v ↦ orderedStatistic d lam ℓ v^2) p.toMeasure :=
+      pmf_integrable_bind_finite _ _ (poissonStatistic_square_integrable d lam hlam ℓ)
+    have hp := pmf_integrable_of_square p hp₂
+    have hi := pmf_integrable_of_square _ (poissonStatistic_square_integrable d lam hlam (ℓ+1) i)
+    change (∫ z : Σ n, Fin n → OrderedObservation ℓ,
+      (∑ j, orderedStatistic d lam ℓ (z.2 j))/((d:ℝ)*lam)
+      ∂(sigmaPMF (poissonPMF d) (fun n ↦ productPMF (fun _ : Fin n ↦ p))).toMeasure) = _
+    change Integrable (fun z : Σ n, Fin n → OrderedObservation ℓ ↦
+      (∑ j, orderedStatistic d lam ℓ (z.2 j))/((d:ℝ)*lam))
+      (sigmaPMF (poissonPMF d) (fun n ↦ productPMF (fun _ : Fin n ↦ p))).toMeasure at hi
+    rw [integral_sigmaPMF_integrable _ _ hi]
+    simp_rw [integral_div, integral_product_sum_integrable p hp]
+    have hm : (∫ v, orderedStatistic d lam ℓ v ∂p.toMeasure) = lam*spinScore i := by
+      rw [integral_pmf_bind_integrable _ _ hp]
+      simp_rw [ih]
+      exact channel_spinScore lam hlam i
+    rw [hm, integral_mul_const, poisson_integral_first]
+    field_simp
+
+end ThreeState.TreeClauses.Tree
+
+end 
+
+noncomputable section
+open Set MeasureTheory Filter
+open scoped BigOperators Topology NNReal
+namespace ThreeState.TreeClauses.Tree
+open ThreeState.TreeClauses.Experiment ThreeState.TreeClauses.Probability
+
+def poissonKSConstant (d lam : ℝ) : ℝ := 4*(d*lam^2)/(d*lam^2-1)
+
+lemma poissonKSConstant_pos {d lam : ℝ} (hKS : 1 < d*lam^2) : 0 < poissonKSConstant d lam := by
+  unfold poissonKSConstant; positivity
+
+lemma poissonKSConstant_ge_four {d lam : ℝ} (hKS : 1 < d*lam^2) : 4 ≤ poissonKSConstant d lam := by
+  rw [poissonKSConstant, le_div_iff₀ (by linarith)]
+  linarith
+
+lemma poissonKSConstant_identity {d lam : ℝ} (hKS : 1 < d*lam^2) :
+    poissonKSConstant d lam*(d*lam^2-1) = 4*(d*lam^2) := by
+  rw [poissonKSConstant, div_mul_cancel₀ _ (by linarith)]
+
+lemma poissonStatistic_second (d : ℝ≥0) (lam : ℝ) (hlam : Admissible lam)
+    (hKS : 1 < (d:ℝ)*lam^2) (ℓ : ℕ) (i : Spin) :
+    (∫ v, orderedStatistic d lam ℓ v^2
+      ∂(orderedLaw (poissonPMF d) lam hlam ℓ i).toMeasure) ≤ poissonKSConstant d lam := by
+  have hd : 0 < (d:ℝ) := by
+    by_contra h
+    have he : (d:ℝ) = 0 := le_antisymm (not_lt.mp h) d.coe_nonneg
+    norm_num [he] at hKS
+  have hl : lam ≠ 0 := by intro h; norm_num [h] at hKS
+  have hden : 0 < ((d:ℝ)*lam)^2 := sq_pos_of_ne_zero (mul_ne_zero hd.ne' hl)
+  induction ℓ generalizing i with
+  | zero =>
+    change (∫ v : Spin, spinScore v^2 ∂(PMF.pure i).toMeasure) ≤ _
+    rw [PMF.toMeasure_pure, integral_dirac]
+    exact (spinScore_square i).trans (poissonKSConstant_ge_four hKS)
+  | succ ℓ ih =>
+    let p := (channel lam hlam i).bind (orderedLaw (poissonPMF d) lam hlam ℓ)
+    have hp₂ : Integrable (fun v ↦ orderedStatistic d lam ℓ v^2) p.toMeasure :=
+      pmf_integrable_bind_finite _ _ (poissonStatistic_square_integrable d lam hlam ℓ)
+    have hp := pmf_integrable_of_square p hp₂
+    have hmean : (∫ v, orderedStatistic d lam ℓ v ∂p.toMeasure) = lam*spinScore i := by
+      rw [integral_pmf_bind_integrable _ _ hp]
+      simp_rw [poissonStatistic_mean d hd.ne' lam hlam hl ℓ]
+      exact channel_spinScore lam hlam i
+    have hsecond : (∫ v, orderedStatistic d lam ℓ v^2 ∂p.toMeasure) ≤ poissonKSConstant d lam := by
+      rw [integral_pmf_bind_integrable _ _ hp₂]
+      exact (integral_mono Integrable.of_finite (integrable_const _) ih).trans_eq (by simp)
+    have hi := poissonStatistic_square_integrable d lam hlam (ℓ+1) i
+    change (∫ z : Σ n, Fin n → OrderedObservation ℓ,
+      ((∑ j, orderedStatistic d lam ℓ (z.2 j))/((d:ℝ)*lam))^2
+      ∂(sigmaPMF (poissonPMF d) (fun n ↦ productPMF (fun _ : Fin n ↦ p))).toMeasure) ≤ _
+    change Integrable (fun z : Σ n, Fin n → OrderedObservation ℓ ↦
+      ((∑ j, orderedStatistic d lam ℓ (z.2 j))/((d:ℝ)*lam))^2)
+      (sigmaPMF (poissonPMF d) (fun n ↦ productPMF (fun _ : Fin n ↦ p))).toMeasure at hi
+    rw [integral_sigmaPMF_integrable _ _ hi]
+    simp_rw [div_pow, integral_div, integral_product_sq_sum_integrable p hp₂]
+    rw [hmean, integral_add
+      (f := fun n : ℕ ↦ (n:ℝ)*(∫ v, orderedStatistic d lam ℓ v^2 ∂p.toMeasure))
+      (g := fun n : ℕ ↦ ((n:ℝ)^2-n)*(lam*spinScore i)^2)
+      ((poisson_integrable_first d).mul_const _)
+      (((poisson_integrable_second d).sub (poisson_integrable_first d)).mul_const _),
+      integral_mul_const, integral_mul_const, poisson_integral_first, poisson_integral_factorial_second]
+    apply (div_le_iff₀ hden).2
+    have hs := mul_le_mul_of_nonneg_left (spinScore_square i) (sq_nonneg ((d:ℝ)*lam))
+    have hm := mul_le_mul_of_nonneg_left hsecond hd.le
+    have hid := congrArg (fun x : ℝ ↦ x*(d:ℝ)) (poissonKSConstant_identity hKS)
+    nlinarith only [hs,hm,hid]
+
+lemma poisson_estimator_advantage (d : ℝ≥0) (lam : ℝ) (hlam : Admissible lam)
+    (hKS : 1 < (d:ℝ)*lam^2) (ℓ : ℕ) :
+    1/(3*poissonKSConstant d lam) ≤ advantage lam hlam (poissonPMF d) ℓ := by
+  have hd : (d:ℝ) ≠ 0 := by intro h; norm_num [h] at hKS
+  have hl : lam ≠ 0 := by intro h; norm_num [h] at hKS
+  let p := observationLaw lam hlam (poissonPMF d) ℓ
+  let f := observedStatistic d lam ℓ
+  have hInt (i : Spin) : Integrable (fun o ↦ f o^2) (p i).toMeasure := by
+    dsimp [p, f]
+    rw [← orderedLaw_forget _ lam hlam ℓ i, pmf_integrable_map_iff]
+    simpa only [observedStatistic_forget] using poissonStatistic_square_integrable d lam hlam ℓ i
+  have hmean (i : Spin) : (∫ o, f o ∂(p i).toMeasure) = spinScore i := by
+    dsimp [p, f]
+    rw [← orderedLaw_forget _ lam hlam ℓ i, integral_pmf_map]
+    simp_rw [observedStatistic_forget]
+    exact poissonStatistic_mean d hd lam hlam hl ℓ i
+  have hsecond (i : Spin) : (∫ o, f o^2 ∂(p i).toMeasure) ≤ poissonKSConstant d lam := by
+    dsimp [p, f]
+    rw [← orderedLaw_forget _ lam hlam ℓ i, integral_pmf_map]
+    simp_rw [observedStatistic_forget]
+    exact poissonStatistic_second d lam hlam hKS ℓ i
+  rw [advantage_eq_integral]
+  apply estimator_tv_lower_integrable (p := p) hInt (poissonKSConstant_pos hKS)
+  · rw [integral_discreteMarginal_integrable p (fun i ↦ pmf_integrable_of_square _ (hInt i)),
+      hmean 0, hmean 1, hmean 2]
+    norm_num [spinScore, show (1:Spin) ≠ 0 by decide, show (2:Spin) ≠ 0 by decide]
+  · exact hmean 0
+  · rw [integral_discreteMarginal_integrable p hInt]
+    linarith [hsecond 0, hsecond 1, hsecond 2]
+
+theorem poisson_reconstructs_above_KS (d : ℝ≥0) (lam : ℝ) (hlam : Admissible lam)
+    (hKS : 1 < (d:ℝ)*lam^2) : Reconstructs lam hlam (poissonPMF d) := by
+  refine ⟨⨅ ℓ, advantage lam hlam (poissonPMF d) ℓ, ?_, advantage_tendsto _ _ _⟩
+  have he : 1/(3*poissonKSConstant d lam) ≤ ⨅ ℓ, advantage lam hlam (poissonPMF d) ℓ := by
+    exact le_ciInf (fun ℓ ↦ poisson_estimator_advantage d lam hlam hKS ℓ)
+  exact (div_pos (by norm_num) (mul_pos (by norm_num) (poissonKSConstant_pos hKS))).trans_le he
+
+end ThreeState.TreeClauses.Tree
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeStateClauses/PositiveFixed.lean b/lean/OAI/Probability/ThreeStateClauses/PositiveFixed.lean
index ddd52e8a033dcbe5761187374cf19a9e993c2686..9dd90458563da262e1d50122bf26ecd2c11a4291 100644
--- a/lean/OAI/Probability/ThreeStateClauses/PositiveFixed.lean
+++ b/lean/OAI/Probability/ThreeStateClauses/PositiveFixed.lean
@@ -1,605 +1,605 @@
-import Mathlib
-import OAI.Probability.ThreeStateClauses.BranchGrowth
-import OAI.Probability.ThreeStateClauses.ScalarBounds
-
-namespace OAI
-
-/-! Positive Fixed. -/
-
-open scoped BigOperators ENNReal NNReal Topology
-open Filter
-noncomputable section
-open Set MeasureTheory
-namespace ThreeState.TreeClauses.Positive
-open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Experiment
-
-def qExact (lam : ℝ) : ℝ := (37/27)/(2+lam)
-def cExact (lam : ℝ) : ℝ := 37/27+2*lam*qExact lam
-
-def jMean (Q : Law) : ℝ := ∫ m, symEntropy m.1 ∂Q.probability.toMeasure
-
-def edgeJMean (Q : Law) (lam : ℝ) : ℝ := ∫ m, symEntropy (edgeMessage lam m) ∂Q.probability.toMeasure
-
-lemma edgeJ_integrable (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) :
-    Integrable (fun m ↦ symEntropy (edgeMessage lam m)) Q.probability.toMeasure :=
-  compact_integrable (continuous_symEntropy_comp (continuous_edgeMessage lam)
-    (fun m i ↦ (edge_positive hl₀ hl₁ m).1 i))
-
-lemma residual_integrable (Q : Law) {lam : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1)
-    (hp : ∀ᵐ m ∂Q.probability.toMeasure, PositiveMessage m.1)
-    (hJ : Integrable (fun m : Message ↦ symEntropy m.1) Q.probability.toMeasure) :
-    Integrable (fun m : Message ↦ residual m.1 lam) Q.probability.toMeasure := by
-  have h := (((edgeJ_integrable Q hl₀.le hl₁).div_const (lam^2)).sub hJ).const_mul (2/(1-lam))
-  apply h.congr
-  filter_upwards [hp] with m hm
-  change 2/(1-lam)*(symEntropy (edge lam m.1)/lam^2-symEntropy m.1) = _
-  rw [symEntropy_edge_residual hm ⟨hl₀.le,hl₁.le⟩]
-  field_simp [ne_of_gt hl₀, ne_of_gt (sub_pos.mpr hl₁)]
-  ring
-
-lemma momentA_integrable (Q : Law)
-    (hp : ∀ᵐ m ∂Q.probability.toMeasure, PositiveMessage m.1)
-    (hJ : Integrable (fun m : Message ↦ symEntropy m.1) Q.probability.toMeasure) :
-    Integrable (fun m : Message ↦ momentA m.1) Q.probability.toMeasure := by
-  have h := ((compact_integrable continuous_yMoment).sub ((compact_integrable continuous_xMoment).const_mul 2)).add (hJ.const_mul 2)
-  apply h.congr
-  filter_upwards [hp] with m hm
-  exact (momentA_identity hm).symm
-
-lemma momentQ_integrable (Q : Law) {lam : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1)
-    (hp : ∀ᵐ m ∂Q.probability.toMeasure, PositiveMessage m.1)
-    (hJ : Integrable (fun m : Message ↦ symEntropy m.1) Q.probability.toMeasure) :
-    Integrable (fun m : Message ↦ momentQ m.1 lam) Q.probability.toMeasure := by
-  have hC := residual_integrable Q hl₀ hl₁ hp hJ
-  have h := (((compact_integrable continuous_xMoment).sub hJ).sub (hC.div_const 2)).div_const lam
-  apply h.congr
-  filter_upwards [hp] with m hm
-  have he := symEntropy_residual hm ⟨hl₀.le,hl₁.le⟩
-  apply (div_eq_iff (ne_of_gt hl₀)).2
-  change momentX (centered m.1)-symEntropy m.1-residual m.1 lam/2 = _
-  linarith
-
-lemma expected_moment_bounds (Q : Law) {lam : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1)
-    (hp : ∀ᵐ m ∂Q.probability.toMeasure, PositiveMessage m.1)
-    (hJ : Integrable (fun m : Message ↦ symEntropy m.1) Q.probability.toMeasure)
-    (hadd : edgeJMean Q lam = lam^2*jMean Q) :
-    0 ≤ jMean Q ∧ jMean Q ≤ Q.mu-lam*qExact lam*Q.nu ∧ cExact lam*Q.nu ≤ Q.eta := by
-  have hC := residual_integrable Q hl₀ hl₁ hp hJ
-  have hA := momentA_integrable Q hp hJ
-  have hQ := momentQ_integrable Q hl₀ hl₁ hp hJ
-  have hx := compact_integrable continuous_xMoment (μ := Q.probability.toMeasure)
-  have hx₂ := compact_integrable (continuous_xMoment.pow 2) (μ := Q.probability.toMeasure)
-  have hcx : (∫ m, residual m.1 lam ∂Q.probability.toMeasure) = 0 := by
-    have he := integral_congr_ae (hp.mono (fun m hm ↦ symEntropy_edge_residual hm ⟨hl₀.le,hl₁.le⟩))
-    change edgeJMean Q lam = (∫ m : Message, lam^2*(symEntropy m.1+(1-lam)*residual m.1 lam/2) ∂Q.probability.toMeasure) at he
-    rw [integral_const_mul,
-      integral_add (f := fun m : Message ↦ symEntropy m.1) (g := fun m ↦ (1-lam)*residual m.1 lam/2)
-        hJ ((hC.const_mul _).div_const _), integral_div, integral_const_mul] at he
-    change edgeJMean Q lam = lam^2*(jMean Q+(1-lam)*(∫ m, residual m.1 lam ∂Q.probability.toMeasure)/2) at he
-    have hc : 0 < lam^2*(1-lam) := mul_pos (sq_pos_of_pos hl₀) (sub_pos.mpr hl₁)
-    nlinarith [hadd]
-  have hjx : jMean Q = Q.mu-lam*(∫ m, momentQ m.1 lam ∂Q.probability.toMeasure) := by
-    have he := integral_congr_ae (hp.mono (fun m hm ↦ symEntropy_residual hm ⟨hl₀.le,hl₁.le⟩))
-    change jMean Q = (∫ m : Message, xMoment m-lam*momentQ m.1 lam-residual m.1 lam/2 ∂Q.probability.toMeasure) at he
-    rw [integral_sub (f := fun m : Message ↦ xMoment m-lam*momentQ m.1 lam)
-      (g := fun m ↦ residual m.1 lam/2) (hx.sub (hQ.const_mul _)) (hC.div_const _),
-      integral_sub (f := xMoment) (g := fun m ↦ lam*momentQ m.1 lam) hx (hQ.const_mul _),
-      integral_const_mul, integral_div, hcx] at he
-    simpa only [sub_zero, zero_div, jMean, Law.mu] using he
-  have hyx : Q.eta = (∫ m, momentA m.1 ∂Q.probability.toMeasure)+
-      2*lam*(∫ m, momentQ m.1 lam ∂Q.probability.toMeasure) := by
-    have he := integral_congr_ae (hp.mono (fun m hm ↦ momentY_residual hm ⟨hl₀.le,hl₁.le⟩))
-    rw [integral_add (f := fun m : Message ↦ momentA m.1+2*lam*momentQ m.1 lam)
-      (g := fun m ↦ residual m.1 lam) (hA.add (hQ.const_mul _)) hC,
-      integral_add (f := fun m : Message ↦ momentA m.1) (g := fun m ↦ 2*lam*momentQ m.1 lam) hA (hQ.const_mul _),
-      integral_const_mul, hcx, add_zero] at he
-    exact he
-  have hAb : (37/27)*Q.nu ≤ ∫ m, momentA m.1 ∂Q.probability.toMeasure := by
-    have h := integral_mono_ae (f := fun m : Message ↦ (37/27)*xMoment m^2)
-      (g := fun m ↦ momentA m.1) (hx₂.const_mul (37/27)) hA
-      (hp.mono (fun m hm ↦ momentA_lower hm))
-    rwa [integral_const_mul] at h
-  have hQb : qExact lam*Q.nu ≤ ∫ m, momentQ m.1 lam ∂Q.probability.toMeasure := by
-    have h := integral_mono_ae (f := fun m : Message ↦ momentA m.1/(2+lam))
-      (g := fun m ↦ momentQ m.1 lam) (hA.div_const _) hQ
-      (hp.mono (fun m hm ↦ momentQ_lower hm hl₀.le hl₁.le))
-    rw [integral_div] at h
-    have hden : 0 < 2+lam := by linarith
-    have hx := (div_le_div_iff_of_pos_right hden).2 hAb
-    have hh := hx.trans h
-    simpa only [qExact, div_mul_eq_mul_div] using hh
-  refine ⟨integral_nonneg_of_ae (hp.mono (fun m hm ↦ symEntropy_nonneg hm)), ?_, ?_⟩
-  · rw [hjx]; nlinarith [mul_nonneg hl₀.le (sub_nonneg.mpr hQb)]
-  · rw [hyx]
-    dsimp only [cExact]
-    nlinarith [mul_nonneg hl₀.le (sub_nonneg.mpr hQb)]
-
-theorem regular_fixed_moment_bounds (Q : Law) {lam : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1) (n : ℕ)
-    (hcrit : (n:ℝ)*lam^2 = 1) (hfixed : Q.probability = finiteProbability Q hl₀.le hl₁ n) :
-    0 ≤ jMean Q ∧ jMean Q ≤ Q.mu-lam*qExact lam*Q.nu ∧ cExact lam*Q.nu ≤ Q.eta := by
-  have hp := finite_positive Q hl₀.le hl₁ n
-  have hJ := finite_integrable_symEntropy Q hl₀.le hl₁ n
-  rw [← hfixed] at hp hJ
-  apply expected_moment_bounds Q hl₀ hl₁ hp hJ
-  have he := finiteLaw_symEntropy Q hl₀.le hl₁ n
-  change (∫ m, symEntropy m.1 ∂(finiteProbability Q hl₀.le hl₁ n).toMeasure) = (n:ℝ)*edgeJMean Q lam at he
-  rw [← hfixed] at he
-  change jMean Q = (n:ℝ)*edgeJMean Q lam at he
-  calc
-    edgeJMean Q lam = ((n:ℝ)*lam^2)*edgeJMean Q lam := by rw [hcrit]; ring
-    _ = lam^2*((n:ℝ)*edgeJMean Q lam) := by ring
-    _ = lam^2*jMean Q := by rw [← he]
-
-end ThreeState.TreeClauses.Positive
-
-end 
-
-noncomputable section
-open scoped BigOperators
-namespace ThreeState.TreeClauses.Positive
-
-lemma branch_sum_first (k : ℕ) :
-    (∑ j ∈ Finset.range k, ((j:ℝ)+1)) = (k:ℝ)*((k:ℝ)+1)/2 := by
-  induction k with
-  | zero => simp
-  | succ k ih =>
-    rw [Finset.sum_range_succ, ih]
-    push_cast
-    ring
-
-lemma branch_sum_second (k : ℕ) :
-    (∑ j ∈ Finset.range k, ((j:ℝ)+1)*(j:ℝ)) = (k:ℝ)*((k:ℝ)+1)*((k:ℝ)-1)/3 := by
-  induction k with
-  | zero => simp
-  | succ k ih =>
-    rw [Finset.sum_range_succ, ih]
-    push_cast
-    ring
-
-lemma reciprocal_tangent {a q : ℝ} (ha : 0 ≤ a) (_hq : 0 ≤ q) :
-    1+2*q-a ≤ (1+q)^2/(1+a) := by
-  apply (le_div_iff₀ (by positivity : 0 < 1+a)).2
-  nlinarith [sq_nonneg (a-q)]
-
-lemma weighted_branch_jensen {lam mu : ℝ} (hmu : 0 ≤ mu) (k : ℕ)
-    (hcrit : ((k:ℝ)+1)*lam^2 = 1) :
-    (k:ℝ)*((k:ℝ)+1)/(2*(1+2*mu/3)) ≤
-      ∑ j ∈ Finset.range k, ((j:ℝ)+1)/(1+(j:ℝ)*(lam^2*mu)) := by
-  let q := 2*mu/3
-  let beta := lam^2*mu
-  have hq : 0 ≤ q := by dsimp [q]; positivity
-  have hb : 0 ≤ beta := by dsimp [beta]; positivity
-  have hh := Finset.sum_le_sum (s := Finset.range k) (fun j _ ↦
-    mul_le_mul_of_nonneg_right (reciprocal_tangent (mul_nonneg (Nat.cast_nonneg j) hb) hq)
-      (show (0:ℝ) ≤ (j:ℝ)+1 by positivity))
-  have hleft : (∑ j ∈ Finset.range k, (1+2*q-(j:ℝ)*beta)*((j:ℝ)+1)) =
-      (1+2*q)*((k:ℝ)*((k:ℝ)+1)/2)-beta*((k:ℝ)*((k:ℝ)+1)*((k:ℝ)-1)/3) := by
-    simp_rw [sub_mul]
-    rw [Finset.sum_sub_distrib, ← Finset.mul_sum]
-    have hid : (∑ j ∈ Finset.range k, (j:ℝ)*beta*((j:ℝ)+1)) =
-        beta*∑ j ∈ Finset.range k, ((j:ℝ)+1)*(j:ℝ) := by
-      rw [Finset.mul_sum]; apply Finset.sum_congr rfl; intro j _; ring
-    rw [hid, branch_sum_first, branch_sum_second]
-  have hright : (∑ j ∈ Finset.range k, ((1+q)^2/(1+(j:ℝ)*beta))*((j:ℝ)+1)) =
-      (1+q)^2*∑ j ∈ Finset.range k, ((j:ℝ)+1)/(1+(j:ℝ)*beta) := by
-    rw [Finset.mul_sum]; apply Finset.sum_congr rfl; intro j _; ring
-  rw [hleft, hright] at hh
-  have hbB : beta*((k:ℝ)*((k:ℝ)+1)*((k:ℝ)-1)/3) ≤ q*((k:ℝ)*((k:ℝ)+1)/2) := by
-    have he : beta*((k:ℝ)*((k:ℝ)+1)*((k:ℝ)-1)/3) = mu*(k:ℝ)*((k:ℝ)-1)/3 := by
-      dsimp [beta]
-      linear_combination (mu*(k:ℝ)*((k:ℝ)-1)/3)*hcrit
-    rw [he]
-    dsimp [q]
-    nlinarith [mul_nonneg hmu (Nat.cast_nonneg k)]
-  have hd : 0 < 1+q := by positivity
-  have hh' : (k:ℝ)*((k:ℝ)+1)/2 ≤
-      (1+q)*(∑ j ∈ Finset.range k, ((j:ℝ)+1)/(1+(j:ℝ)*beta)) := by
-    apply le_of_mul_le_mul_left (a := 1+q) _ hd
-    nlinarith
-  apply (div_le_iff₀ (show 0 < 2*(1+2*mu/3) by positivity)).2
-  dsimp [q, beta] at hh'
-  nlinarith
-
-lemma sqrt_branch_sum {lam : ℝ} (hl : 0 < lam) (k : ℕ)
-    (hcrit : ((k:ℝ)+1)*lam^2 = 1) :
-    lam^4*(∑ j ∈ Finset.range k, Real.sqrt ((j:ℝ)+1)) ≤
-      (1-lam^2)*lam/Real.sqrt 2 := by
-  have hs2 := Real.sq_sqrt (by norm_num : (0:ℝ) ≤ 2)
-  have hs2p := Real.sqrt_pos.mpr (by norm_num : (0:ℝ) < 2)
-  have hl2 : 0 < lam^2 := sq_pos_of_pos hl
-  have hp := Finset.sum_le_sum (s := Finset.range k) (fun j _ ↦
-    sq_nonneg (Real.sqrt ((j:ℝ)+1)-1/(lam*Real.sqrt 2)))
-  have hpoint (j : ℕ) :
-      2*lam*Real.sqrt 2*Real.sqrt ((j:ℝ)+1) ≤ 2*lam^2*((j:ℝ)+1)+1 := by
-    have hh := sq_nonneg (lam*Real.sqrt 2*Real.sqrt ((j:ℝ)+1)-1)
-    have he : (lam*Real.sqrt 2*Real.sqrt ((j:ℝ)+1))^2 = 2*lam^2*((j:ℝ)+1) := by
-      calc
-        _ = lam^2*(Real.sqrt 2)^2*(Real.sqrt ((j:ℝ)+1))^2 := by ring
-        _ = _ := by rw [hs2, Real.sq_sqrt (by positivity)]; ring
-    nlinarith
-  have hh := Finset.sum_le_sum (s := Finset.range k) (fun j _ ↦ hpoint j)
-  rw [← Finset.mul_sum, Finset.sum_add_distrib, ← Finset.mul_sum, branch_sum_first] at hh
-  simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_one] at hh
-  have he : 2*lam^2*((k:ℝ)*((k:ℝ)+1)/2)+(k:ℝ) = 2*(k:ℝ) := by
-    linear_combination (k:ℝ)*hcrit
-  rw [he] at hh
-  have he' : (k:ℝ)*lam^2 = 1-lam^2 := by nlinarith [hcrit]
-  apply (le_div_iff₀ hs2p).2
-  have hm := mul_le_mul_of_nonneg_left hh (show 0 ≤ lam^3/2 by positivity)
-  nlinarith
-
-end ThreeState.TreeClauses.Positive
-
-end 
-
-noncomputable section
-open Set MeasureTheory
-open scoped BigOperators
-namespace ThreeState.TreeClauses.Positive
-open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Experiment
-
-lemma truncated_quadratic {a b t L C : ℝ} (ha : 0 ≤ a) (_hb : 0 ≤ b)
-    (hL : 0 ≤ L) (ht : L ≤ t) (hC : 0 ≤ C)
-    (hbound : a*t-b*Real.sqrt t ≤ C) : a*L-b*Real.sqrt L ≤ C := by
-  have hsL := Real.sqrt_nonneg L
-  have hst := Real.sqrt_nonneg t
-  have hsqL := Real.sq_sqrt hL
-  have hsqt := Real.sq_sqrt (hL.trans ht)
-  have hsle := Real.sqrt_le_sqrt ht
-  by_cases h : a*Real.sqrt L-b ≤ 0
-  · have hh := mul_nonpos_of_nonneg_of_nonpos hsL h
-    nlinarith
-  · have hh := mul_nonneg (sub_nonneg.mpr hsle) (le_of_lt (lt_of_not_ge h))
-    have hh' := mul_nonneg (mul_nonneg ha (sub_nonneg.mpr hsle)) hst
-    nlinarith
-
-lemma product_cubic_bound (Q R : Law) (hnu : R.nu ≤ Q.nu) :
-    Q.eta*R.eta ≤ Real.sqrt Q.mu*Q.nu*Real.sqrt R.mu := by
-  have hmu := Q.mu_nonneg
-  have hnuQ := Q.nu_nonneg
-  have hR : R.eta^2 ≤ R.mu*Q.nu :=
-    R.moment_discriminant.trans (mul_le_mul_of_nonneg_left hnu R.mu_nonneg)
-  have hprod := mul_le_mul Q.moment_discriminant hR (sq_nonneg R.eta)
-    (mul_nonneg Q.mu_nonneg Q.nu_nonneg)
-  have hs : (Real.sqrt Q.mu*Q.nu*Real.sqrt R.mu)^2 = (Q.mu*Q.nu)*(R.mu*Q.nu) := by
-    calc
-      _ = (Real.sqrt Q.mu)^2*Q.nu^2*(Real.sqrt R.mu)^2 := by ring
-      _ = _ := by rw [Real.sq_sqrt Q.mu_nonneg, Real.sq_sqrt R.mu_nonneg]; ring
-  have hp : 0 ≤ Real.sqrt Q.mu*Q.nu*Real.sqrt R.mu := by positivity
-  nlinarith [sq_nonneg (Q.eta*R.eta-Real.sqrt Q.mu*Q.nu*Real.sqrt R.mu)]
-
-lemma correlation_truncated (Q R : Law) {lam L : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
-    (hnu : R.nu ≤ Q.nu) (hL : 0 ≤ L) (hproj : L ≤ R.mu) :
-    3*lam^2*Q.mu*L-lam^3*Real.sqrt Q.mu*Q.nu*Real.sqrt L ≤ 3*pairCorrelation R Q lam := by
-  have hmu := Q.mu_nonneg
-  have hnuQ := Q.nu_nonneg
-  have hi := pairCorrelation_increment R Q hl₀ hl₁
-  have he := product_cubic_bound Q R hnu
-  have hh : 3*lam^2*Q.mu*R.mu-(lam^3*Real.sqrt Q.mu*Q.nu)*Real.sqrt R.mu ≤
-      3*pairCorrelation R Q lam := by
-    nlinarith [mul_nonneg (pow_nonneg hl₀ 3) (sub_nonneg.mpr he)]
-  exact truncated_quadratic (by positivity) (by positivity) hL hproj
-    (mul_nonneg (by norm_num) (pairCorrelation_nonneg R Q hl₀ hl₁)) hh
-
-lemma finite_increment_bound (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
-    (N j : ℕ) (hj : 1 ≤ j) (hjN : j ≤ N)
-    (hfixed : Q.probability = finiteProbability Q hl₀ hl₁ N) :
-    3*lam^4*Q.mu^2*(j:ℝ)/(1+((j:ℝ)-1)*(lam^2*Q.mu))-
-      lam^4*Q.mu*Q.nu*Real.sqrt j ≤ 3*pairCorrelation (finiteLaw Q hl₀ hl₁ j) Q lam := by
-  have hmu := Q.mu_nonneg
-  have hnuQ := Q.nu_nonneg
-  let L : ℝ := (j:ℝ)*(lam^2*Q.mu)/(1+((j:ℝ)-1)*(lam^2*Q.mu))
-  have hj' : (1:ℝ) ≤ j := by exact_mod_cast hj
-  have hd : 1 ≤ 1+((j:ℝ)-1)*(lam^2*Q.mu) := by
-    nlinarith [mul_nonneg (sub_nonneg.mpr hj') (mul_nonneg (sq_nonneg lam) Q.mu_nonneg)]
-  have hL : 0 ≤ L := by dsimp [L]; positivity
-  have hproj := finite_projection Q hl₀ hl₁ j hj
-  have hnu : (finiteLaw Q hl₀ hl₁ j).nu ≤ Q.nu := by
-    have hh := finite_nu_mono Q hl₀ hl₁ hjN
-    change _ ≤ ∫ m, xMoment m^2 ∂(finiteProbability Q hl₀ hl₁ N).toMeasure at hh
-    rw [← hfixed] at hh
-    exact hh
-  have hc := correlation_truncated Q (finiteLaw Q hl₀ hl₁ j) hl₀ hl₁ hnu hL hproj
-  have hLu : L ≤ (j:ℝ)*(lam^2*Q.mu) := by
-    dsimp only [L]
-    exact div_le_self (by positivity) hd
-  have hs : Real.sqrt L ≤ lam*Real.sqrt Q.mu*Real.sqrt j := by
-    have he : (lam*Real.sqrt Q.mu*Real.sqrt j)^2 = (j:ℝ)*(lam^2*Q.mu) := by
-      calc
-        _ = lam^2*(Real.sqrt Q.mu)^2*(Real.sqrt j)^2 := by ring
-        _ = _ := by rw [Real.sq_sqrt Q.mu_nonneg, Real.sq_sqrt (Nat.cast_nonneg j)]; ring
-    have hp : 0 ≤ lam*Real.sqrt Q.mu*Real.sqrt j := by positivity
-    nlinarith [Real.sq_sqrt hL, Real.sqrt_nonneg L]
-  have hmul := mul_le_mul_of_nonneg_left hs (show 0 ≤ lam^3*Real.sqrt Q.mu*Q.nu by positivity)
-  have heq : (lam^3*Real.sqrt Q.mu*Q.nu)*(lam*Real.sqrt Q.mu*Real.sqrt j) =
-      lam^4*Q.mu*Q.nu*Real.sqrt j := by
-    calc
-      _ = lam^4*(Real.sqrt Q.mu)^2*Q.nu*Real.sqrt j := by ring
-      _ = _ := by rw [Real.sq_sqrt Q.mu_nonneg]
-  rw [heq] at hmul
-  have hmain : 3*lam^2*Q.mu*L = 3*lam^4*Q.mu^2*(j:ℝ)/(1+((j:ℝ)-1)*(lam^2*Q.mu)) := by
-    dsimp only [L]; ring
-  rw [hmain] at hc
-  linarith
-
-lemma finite_correlation_sum (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
-    (N : ℕ) (hfixed : Q.probability = finiteProbability Q hl₀ hl₁ N)
-    (k : ℕ) (hk : k+1 ≤ N) :
-    (∑ j ∈ Finset.range k,
-      (3*lam^4*Q.mu^2*((j:ℝ)+1)/(1+(j:ℝ)*(lam^2*Q.mu))-
-        lam^4*Q.mu*Q.nu*Real.sqrt ((j:ℝ)+1))) ≤ 3*correlationEntropy Q lam (k+1) := by
-  induction k with
-  | zero =>
-    simpa using mul_nonneg (by norm_num : (0:ℝ) ≤ 3) (correlationEntropy_nonneg Q hl₀ hl₁ 1)
-  | succ k ih =>
-    have hprev := ih (by omega)
-    have hi := finite_increment_bound Q hl₀ hl₁ N (k+1) (by omega) (by omega) hfixed
-    simp only [Nat.cast_add, Nat.cast_one, add_sub_cancel_right] at hi
-    rw [Finset.sum_range_succ, correlationEntropy_succ Q hl₀ hl₁ (k+1)]
-    linarith
-
-theorem regular_fixed_correlation_bounds (Q : Law) {lam : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1)
-    (n : ℕ) (hn : 2 ≤ n) (hcrit : (n:ℝ)*lam^2 = 1)
-    (hfixed : Q.probability = finiteProbability Q hl₀.le hl₁ n) :
-    3*Q.mu^2/(2*(1+2*Q.mu/3))-lam/Real.sqrt 2*Q.mu*Q.nu ≤
-      3*correlationEntropy Q lam n/(1-lam^2) := by
-  obtain ⟨k,rfl⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : n ≠ 0)
-  have hc : ((k:ℝ)+1)*lam^2 = 1 := by simpa using hcrit
-  have halpha : 0 < 1-lam^2 := by nlinarith [mul_pos (sub_pos.mpr hl₁) (by linarith : 0 < 1+lam)]
-  have hmu := Q.mu_nonneg
-  have hnu := Q.nu_nonneg
-  have hh := finite_correlation_sum Q hl₀.le hl₁ (k+1) hfixed k (by omega)
-  have hid : (∑ j ∈ Finset.range k,
-      (3*lam^4*Q.mu^2*((j:ℝ)+1)/(1+(j:ℝ)*(lam^2*Q.mu))-
-        lam^4*Q.mu*Q.nu*Real.sqrt ((j:ℝ)+1))) =
-      (3*lam^4*Q.mu^2)*(∑ j ∈ Finset.range k, ((j:ℝ)+1)/(1+(j:ℝ)*(lam^2*Q.mu)))-
-      (Q.mu*Q.nu)*(lam^4*∑ j ∈ Finset.range k, Real.sqrt ((j:ℝ)+1)) := by
-    rw [Finset.sum_sub_distrib, Finset.mul_sum, Finset.mul_sum, Finset.mul_sum]
-    congr 1 <;> apply Finset.sum_congr rfl <;> intro j _ <;> ring
-  rw [hid] at hh
-  have hJ := mul_le_mul_of_nonneg_left (weighted_branch_jensen hmu k hc)
-    (show 0 ≤ 3*lam^4*Q.mu^2 by positivity)
-  have hS := mul_le_mul_of_nonneg_left (sqrt_branch_sum hl₀ k hc) (mul_nonneg hmu hnu)
-  have he : lam^4*(k:ℝ)*((k:ℝ)+1) = 1-lam^2 := by
-    calc
-      _ = ((k:ℝ)*lam^2)*(((k:ℝ)+1)*lam^2) := by ring
-      _ = 1-lam^2 := by rw [hc]; nlinarith [hc]
-  have heJ : 3*lam^4*Q.mu^2*((k:ℝ)*((k:ℝ)+1)/(2*(1+2*Q.mu/3))) =
-      (1-lam^2)*(3*Q.mu^2/(2*(1+2*Q.mu/3))) := by
-    calc
-      _ = (lam^4*(k:ℝ)*((k:ℝ)+1))*(3*Q.mu^2/(2*(1+2*Q.mu/3))) := by ring
-      _ = _ := by rw [he]
-  rw [heJ] at hJ
-  apply (le_div_iff₀ halpha).2
-  calc
-    _ = (1-lam^2)*(3*Q.mu^2/(2*(1+2*Q.mu/3)))-Q.mu*Q.nu*((1-lam^2)*lam/Real.sqrt 2) := by ring
-    _ ≤ 3*correlationEntropy Q lam (k+1) := by linarith
-
-end ThreeState.TreeClauses.Positive
-
-end 
-
-noncomputable section
-open Set MeasureTheory
-namespace ThreeState.TreeClauses.Positive
-open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Experiment
-
-lemma fixed_edgeJ (Q : Law) {lam : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1) (n : ℕ)
-    (hcrit : (n:ℝ)*lam^2 = 1) (hfixed : Q.probability = finiteProbability Q hl₀.le hl₁ n) :
-    edgeJMean Q lam = lam^2*jMean Q := by
-  have h := finiteLaw_symEntropy Q hl₀.le hl₁ n
-  change (∫ m, symEntropy m.1 ∂(finiteProbability Q hl₀.le hl₁ n).toMeasure) = _ at h
-  rw [← hfixed] at h
-  change jMean Q = (n:ℝ)*edgeJMean Q lam at h
-  rw [h]
-  calc
-    _ = ((n:ℝ)*lam^2)*edgeJMean Q lam := by rw [hcrit, one_mul]
-    _ = _ := by ring
-
-lemma regular_fixed_entropy_radial (Q : Law) {lam : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1) (n : ℕ)
-    (hcrit : (n:ℝ)*lam^2 = 1) (hfixed : Q.probability = finiteProbability Q hl₀.le hl₁ n) :
-    (1-lam^2)/2*Q.nu ≤ (8/5)*(1-lam^2)*(jMean Q)^2-3*correlationEntropy Q lam n := by
-  have hp := finite_positive Q hl₀.le hl₁ n
-  have hJ := finite_integrable_symEntropy Q hl₀.le hl₁ n
-  have hB := finite_integrable_correction Q hl₀.le hl₁ n
-  rw [← hfixed] at hp hJ hB
-  have hI : Integrable (fun m : Message ↦ info m.1) Q.probability.toMeasure :=
-    compact_integrable (continuous_info_comp continuous_subtype_val)
-  have hJe := edgeJ_integrable Q hl₀.le hl₁
-  have hIe : Integrable (fun m : Message ↦ info (edgeMessage lam m)) Q.probability.toMeasure :=
-    compact_integrable (continuous_info_comp (continuous_edgeMessage lam))
-  have hBe : Integrable (fun m : Message ↦ correction (edgeMessage lam m)) Q.probability.toMeasure :=
-    compact_integrable (continuous_correction_comp (continuous_edgeMessage lam) (fun m i ↦ (edge_positive hl₀.le hl₁ m).1 i))
-  have hF : Integrable (fun m : Message ↦ functional m.1) Q.probability.toMeasure :=
-    (hI.const_mul 3 |>.sub (hJ.const_mul 2)).add (hB.const_mul (4/5))
-  have hFe : Integrable (fun m : Message ↦ functional (edgeMessage lam m)) Q.probability.toMeasure :=
-    (hIe.const_mul 3 |>.sub (hJe.const_mul 2)).add (hBe.const_mul (4/5))
-  have hrad := integral_mono_ae (μ := Q.probability.toMeasure)
-    (f := fun m : Message ↦ (1-lam^2)/2*xMoment m^2)
-    (g := fun m ↦ functional m.1-(lam^2)⁻¹*functional (edgeMessage lam m))
-    ((compact_integrable (continuous_xMoment.pow 2)).const_mul _) (hF.sub (hFe.const_mul _))
-    (hp.mono (fun m hm ↦ radial_inequality m.1 hm lam hl₀ hl₁))
-  rw [integral_const_mul, integral_sub hF (hFe.const_mul _), integral_const_mul] at hrad
-  change (1-lam^2)/2*Q.nu ≤ _ at hrad
-  have hJid := finiteLaw_symEntropy Q hl₀.le hl₁ n
-  have hBid := finiteLaw_correction Q hl₀.le hl₁ n
-  have hIid := finiteLaw_info Q hl₀.le hl₁ n
-  change (∫ m, symEntropy m.1 ∂(finiteProbability Q hl₀.le hl₁ n).toMeasure) = _ at hJid
-  change (∫ m, correction m.1 ∂(finiteProbability Q hl₀.le hl₁ n).toMeasure) = _ at hBid
-  change (∫ m, info m.1 ∂(finiteProbability Q hl₀.le hl₁ n).toMeasure) = _ at hIid
-  rw [← hfixed] at hJid hBid hIid
-  have hninv : (lam^2)⁻¹ = (n:ℝ) := by
-    apply mul_left_cancel₀ (ne_of_gt (sq_pos_of_pos hl₀))
-    rw [mul_inv_cancel₀ (ne_of_gt (sq_pos_of_pos hl₀))]
-    nlinarith [hcrit]
-  have hFid : (∫ m, functional m.1 ∂Q.probability.toMeasure) =
-      3*(∫ m, info m.1 ∂Q.probability.toMeasure)-2*jMean Q+
-      (4/5)*(∫ m, correction m.1 ∂Q.probability.toMeasure) := by
-    unfold functional
-    rw [integral_add (f := fun m : Message ↦ 3*info m.1-2*symEntropy m.1)
-      (g := fun m ↦ (4/5)*correction m.1) ((hI.const_mul _).sub (hJ.const_mul _)) (hB.const_mul _),
-      integral_sub (f := fun m : Message ↦ 3*info m.1) (g := fun m ↦ 2*symEntropy m.1)
-        (hI.const_mul _) (hJ.const_mul _), integral_const_mul, integral_const_mul, integral_const_mul]
-    rfl
-  have hFeid : (∫ m, functional (edgeMessage lam m) ∂Q.probability.toMeasure) =
-      3*(∫ m, info (edgeMessage lam m) ∂Q.probability.toMeasure)-2*edgeJMean Q lam+
-      (4/5)*(∫ m, correction (edgeMessage lam m) ∂Q.probability.toMeasure) := by
-    unfold functional
-    rw [integral_add (f := fun m : Message ↦ 3*info (edgeMessage lam m)-2*symEntropy (edgeMessage lam m))
-      (g := fun m ↦ (4/5)*correction (edgeMessage lam m)) ((hIe.const_mul _).sub (hJe.const_mul _)) (hBe.const_mul _),
-      integral_sub (f := fun m : Message ↦ 3*info (edgeMessage lam m)) (g := fun m ↦ 2*symEntropy (edgeMessage lam m))
-        (hIe.const_mul _) (hJe.const_mul _), integral_const_mul, integral_const_mul, integral_const_mul]
-    rfl
-  rw [hFid, hFeid, hninv, hIid, hBid] at hrad
-  change jMean Q = (n:ℝ)*edgeJMean Q lam at hJid
-  change _ ≤ 3*((n:ℝ)*(∫ m, info (edgeMessage lam m) ∂Q.probability.toMeasure)-correlationEntropy Q lam n)-
-    2*jMean Q+(4/5)*((n:ℝ)*(∫ m, correction (edgeMessage lam m) ∂Q.probability.toMeasure)+
-      2*((n:ℝ)^2-n)*(edgeJMean Q lam)^2)-
-    (n:ℝ)*(3*(∫ m, info (edgeMessage lam m) ∂Q.probability.toMeasure)-2*edgeJMean Q lam+
-      (4/5)*(∫ m, correction (edgeMessage lam m) ∂Q.probability.toMeasure)) at hrad
-  have he : ((n:ℝ)^2-n)*(edgeJMean Q lam)^2 = (1-lam^2)*(jMean Q)^2 := by
-    rw [fixed_edgeJ Q hl₀ hl₁ n hcrit hfixed]
-    have hn : ((n:ℝ)^2-n)*lam^4 = 1-lam^2 := by
-      nlinarith [sq_nonneg ((n:ℝ)*lam^2-1), hcrit]
-    calc
-      _ = (((n:ℝ)^2-n)*lam^4)*(jMean Q)^2 := by ring
-      _ = _ := by rw [hn]
-  nlinarith [hJid, he]
-
-end ThreeState.TreeClauses.Positive
-
-end 
-
-noncomputable section
-namespace ThreeState.TreeClauses.Positive
-
-lemma qExact_lower {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam ≤ 1) :
-    (37/81:ℝ) ≤ qExact lam := by
-  unfold qExact
-  apply (le_div_iff₀ (by positivity : 0 < 2+lam)).2
-  linarith
-
-lemma cExact_lower {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam ≤ 1) :
-    cLower lam ≤ cExact lam := by
-  have hq := qExact_lower hl₀ hl₁
-  unfold cExact cLower
-  nlinarith [mul_nonneg hl₀ (sub_nonneg.mpr hq)]
-
-lemma qExact_coefficient {lam : ℝ} :
-    8*(37/27)*lam*qExact lam ≤ cExact lam^2 := by
-  unfold cExact
-  nlinarith [sq_nonneg ((37/27:ℝ)-2*lam*qExact lam)]
-
-lemma jMean_square_bound {lam mu nu J : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam ≤ 1)
-    (hmu : 0 ≤ mu) (hnu : 0 ≤ nu) (hJ : 0 ≤ J) (hupper : J ≤ mu-lam*qExact lam*nu)
-    (hregion : cExact lam^2*nu ≤ mu) :
-    (8/5)*J^2 ≤ (8/5)*mu^2-(113/81)*lam*mu*nu := by
-  let t := lam*qExact lam*nu
-  have hq := qExact_lower hl₀ hl₁
-  have ht : 0 ≤ t := by dsimp [t]; positivity
-  have hc := mul_le_mul_of_nonneg_right (qExact_coefficient (lam := lam)) hnu
-  have htu : 8*(37/27)*t ≤ mu := by dsimp [t]; nlinarith
-  have hJsq : J^2 ≤ (mu-t)^2 := by
-    have hu : J ≤ mu-t := hupper
-    nlinarith
-  have htt : t^2 ≤ (27/296)*mu*t := by
-    have hh := mul_le_mul_of_nonneg_right htu ht
-    nlinarith
-  have htl : (37/81)*lam*nu ≤ t := by
-    dsimp [t]
-    nlinarith [mul_nonneg (mul_nonneg hl₀ hnu) (sub_nonneg.mpr hq)]
-  have hm := mul_le_mul_of_nonneg_left htl hmu
-  nlinarith
-
-lemma strict_correlation_coefficient : (1:ℝ)/Real.sqrt 2+17/25 < 113/81 := by
-  have hs := Real.sq_sqrt (by norm_num : (0:ℝ) ≤ 2)
-  have hp := Real.sqrt_pos.mpr (by norm_num : (0:ℝ) < 2)
-  have hh : (2025:ℝ)/1448 < Real.sqrt 2 := by nlinarith
-  have hfrac : (1:ℝ)/Real.sqrt 2 < 1448/2025 := (div_lt_iff₀ hp).2 (by nlinarith)
-  linarith
-
-end ThreeState.TreeClauses.Positive
-
-end 
-
-noncomputable section
-open Set MeasureTheory
-namespace ThreeState.TreeClauses.Positive
-open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Experiment
-
-theorem regular_fixed_mu_zero (Q : Law) {lam : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1)
-    (n : ℕ) (hn : 2 ≤ n) (hcrit : (n:ℝ)*lam^2 = 1)
-    (hfixed : Q.probability = finiteProbability Q hl₀.le hl₁ n) : Q.mu = 0 := by
-  by_contra hmu0
-  have hmu : 0 < Q.mu := lt_of_le_of_ne Q.mu_nonneg (Ne.symm hmu0)
-  have hnu : 0 < Q.nu := (sq_pos_of_pos hmu).trans_le Q.second_moment_lower
-  have hinfo := regular_fixed_moment_bounds Q hl₀ hl₁ n hcrit hfixed
-  have hcl := cExact_lower hl₀.le hl₁.le
-  have hc : 0 < cExact lam := (cLower_pos hl₀.le).trans_le hcl
-  have hetaNonneg : 0 ≤ Q.eta := (mul_nonneg hc.le hnu.le).trans hinfo.2.2
-  have hsq : (cExact lam*Q.nu)^2 ≤ Q.eta^2 :=
-    (sq_le_sq₀ (mul_nonneg hc.le hnu.le) hetaNonneg).2 hinfo.2.2
-  have hregion : cExact lam^2*Q.nu ≤ Q.mu := by
-    apply (mul_le_mul_iff_right₀ hnu).mp
-    nlinarith [Q.moment_discriminant]
-  have hmuC : Q.mu*cExact lam^2 ≤ 1 := by
-    apply (mul_le_mul_iff_right₀ hmu).mp
-    have hh := mul_le_mul_of_nonneg_left Q.second_moment_lower (sq_nonneg (cExact lam))
-    nlinarith
-  have hregionLower : Q.mu*cLower lam^2 ≤ 1 := by
-    have hs : cLower lam^2 ≤ cExact lam^2 :=
-      (sq_le_sq₀ (cLower_pos hl₀.le).le hc.le).2 hcl
-    exact (mul_le_mul_of_nonneg_left hs hmu.le).trans hmuC
-  have hlam : lam^2 ≤ 1/2 := by
-    have hn' : (2:ℝ) ≤ n := by exact_mod_cast hn
-    have hh := mul_le_mul_of_nonneg_right hn' (sq_nonneg lam)
-    nlinarith [hcrit]
-  have hstrict := scalar_strict hl₀ hl₁ hmu.le (by norm_num : (0:ℝ) < 1)
-    (le_refl (1:ℝ)) hregionLower (Or.inl hlam)
-  have hJ := jMean_square_bound hl₀.le hl₁.le hmu.le hnu.le hinfo.1 hinfo.2.1 hregion
-  have hH := regular_fixed_correlation_bounds Q hl₀ hl₁ n hn hcrit hfixed
-  have hrad := regular_fixed_entropy_radial Q hl₀ hl₁ n hcrit hfixed
-  have halpha : 0 < 1-lam^2 := by nlinarith
-  have hent : Q.nu/2+3*correlationEntropy Q lam n/(1-lam^2) ≤ (8/5)*(jMean Q)^2 := by
-    calc
-      _ = ((1-lam^2)/2*Q.nu+3*correlationEntropy Q lam n)/(1-lam^2) := by
-        field_simp [ne_of_gt halpha]
-      _ ≤ _ := (div_le_iff₀ halpha).2 (by nlinarith)
-  have hH' : 3*Q.mu^2/(2*(1+2*Q.mu/3))-(1/Real.sqrt 2)*(lam*Q.mu*Q.nu) ≤
-      3*correlationEntropy Q lam n/(1-lam^2) := by
-    convert hH using 1
-    ring
-  have hg := mul_le_mul_of_nonneg_right (le_of_lt strict_correlation_coefficient)
-    (show 0 ≤ lam*Q.mu*Q.nu by positivity)
-  have hb : Q.nu/2+3*Q.mu^2/(2*(1+2*Q.mu/3))+(17/25)*lam*Q.mu*Q.nu ≤ (8/5)*Q.mu^2 := by
-    nlinarith
-  have hcoef : 0 ≤ (1-lam^2)/2+17*lam*Q.mu/25 := by positivity
-  have hsub := mul_le_mul_of_nonneg_left Q.second_moment_lower hcoef
-  have hnecessary : scalarT lam Q.mu 1*Q.mu^2 ≤ (8/5)*Q.mu^2 := by
-    calc
-      _ = ((1-lam^2)/2+17*lam*Q.mu/25)*Q.mu^2+3*Q.mu^2/(2*(1+2*Q.mu/3)) := by
-        unfold scalarT; ring
-      _ ≤ ((1-lam^2)/2+17*lam*Q.mu/25)*Q.nu+3*Q.mu^2/(2*(1+2*Q.mu/3)) := by linarith
-      _ ≤ _ := by nlinarith [mul_nonneg (sq_nonneg lam) hnu.le]
-  have hh := mul_lt_mul_of_pos_right hstrict (sq_pos_of_pos hmu)
-  linarith
-
-lemma message_uniform_of_x_zero {m : Message} (h : xMoment m = 0) : m.1 = fun _ ↦ 1 := by
-  dsimp [xMoment, momentX, centered, avg] at h
-  funext i
-  fin_cases i <;> dsimp <;> nlinarith [sq_nonneg (m.1 0-1), sq_nonneg (m.1 1-1), sq_nonneg (m.1 2-1)]
-
-theorem regular_fixed_uniform (Q : Law) {lam : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1)
-    (n : ℕ) (hn : 2 ≤ n) (hcrit : (n:ℝ)*lam^2 = 1)
-    (hfixed : Q.probability = finiteProbability Q hl₀.le hl₁ n) :
-    ∀ᵐ m ∂Q.probability.toMeasure, m.1 = fun _ ↦ 1 := by
-  have hzero := regular_fixed_mu_zero Q hl₀ hl₁ n hn hcrit hfixed
-  have he : ∀ᵐ m ∂Q.probability.toMeasure, xMoment m = 0 :=
-    (integral_eq_zero_iff_of_nonneg xMoment_nonneg (compact_integrable continuous_xMoment)).1 hzero
-  exact he.mono (fun _ hm ↦ message_uniform_of_x_zero hm)
-
-end ThreeState.TreeClauses.Positive
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeStateClauses.BranchGrowth
+import OAI.Probability.ThreeStateClauses.ScalarBounds
+
+namespace OAI
+
+/-! Positive Fixed. -/
+
+open scoped BigOperators ENNReal NNReal Topology
+open Filter
+noncomputable section
+open Set MeasureTheory
+namespace ThreeState.TreeClauses.Positive
+open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Experiment
+
+def qExact (lam : ℝ) : ℝ := (37/27)/(2+lam)
+def cExact (lam : ℝ) : ℝ := 37/27+2*lam*qExact lam
+
+def jMean (Q : Law) : ℝ := ∫ m, symEntropy m.1 ∂Q.probability.toMeasure
+
+def edgeJMean (Q : Law) (lam : ℝ) : ℝ := ∫ m, symEntropy (edgeMessage lam m) ∂Q.probability.toMeasure
+
+lemma edgeJ_integrable (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) :
+    Integrable (fun m ↦ symEntropy (edgeMessage lam m)) Q.probability.toMeasure :=
+  compact_integrable (continuous_symEntropy_comp (continuous_edgeMessage lam)
+    (fun m i ↦ (edge_positive hl₀ hl₁ m).1 i))
+
+lemma residual_integrable (Q : Law) {lam : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1)
+    (hp : ∀ᵐ m ∂Q.probability.toMeasure, PositiveMessage m.1)
+    (hJ : Integrable (fun m : Message ↦ symEntropy m.1) Q.probability.toMeasure) :
+    Integrable (fun m : Message ↦ residual m.1 lam) Q.probability.toMeasure := by
+  have h := (((edgeJ_integrable Q hl₀.le hl₁).div_const (lam^2)).sub hJ).const_mul (2/(1-lam))
+  apply h.congr
+  filter_upwards [hp] with m hm
+  change 2/(1-lam)*(symEntropy (edge lam m.1)/lam^2-symEntropy m.1) = _
+  rw [symEntropy_edge_residual hm ⟨hl₀.le,hl₁.le⟩]
+  field_simp [ne_of_gt hl₀, ne_of_gt (sub_pos.mpr hl₁)]
+  ring
+
+lemma momentA_integrable (Q : Law)
+    (hp : ∀ᵐ m ∂Q.probability.toMeasure, PositiveMessage m.1)
+    (hJ : Integrable (fun m : Message ↦ symEntropy m.1) Q.probability.toMeasure) :
+    Integrable (fun m : Message ↦ momentA m.1) Q.probability.toMeasure := by
+  have h := ((compact_integrable continuous_yMoment).sub ((compact_integrable continuous_xMoment).const_mul 2)).add (hJ.const_mul 2)
+  apply h.congr
+  filter_upwards [hp] with m hm
+  exact (momentA_identity hm).symm
+
+lemma momentQ_integrable (Q : Law) {lam : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1)
+    (hp : ∀ᵐ m ∂Q.probability.toMeasure, PositiveMessage m.1)
+    (hJ : Integrable (fun m : Message ↦ symEntropy m.1) Q.probability.toMeasure) :
+    Integrable (fun m : Message ↦ momentQ m.1 lam) Q.probability.toMeasure := by
+  have hC := residual_integrable Q hl₀ hl₁ hp hJ
+  have h := (((compact_integrable continuous_xMoment).sub hJ).sub (hC.div_const 2)).div_const lam
+  apply h.congr
+  filter_upwards [hp] with m hm
+  have he := symEntropy_residual hm ⟨hl₀.le,hl₁.le⟩
+  apply (div_eq_iff (ne_of_gt hl₀)).2
+  change momentX (centered m.1)-symEntropy m.1-residual m.1 lam/2 = _
+  linarith
+
+lemma expected_moment_bounds (Q : Law) {lam : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1)
+    (hp : ∀ᵐ m ∂Q.probability.toMeasure, PositiveMessage m.1)
+    (hJ : Integrable (fun m : Message ↦ symEntropy m.1) Q.probability.toMeasure)
+    (hadd : edgeJMean Q lam = lam^2*jMean Q) :
+    0 ≤ jMean Q ∧ jMean Q ≤ Q.mu-lam*qExact lam*Q.nu ∧ cExact lam*Q.nu ≤ Q.eta := by
+  have hC := residual_integrable Q hl₀ hl₁ hp hJ
+  have hA := momentA_integrable Q hp hJ
+  have hQ := momentQ_integrable Q hl₀ hl₁ hp hJ
+  have hx := compact_integrable continuous_xMoment (μ := Q.probability.toMeasure)
+  have hx₂ := compact_integrable (continuous_xMoment.pow 2) (μ := Q.probability.toMeasure)
+  have hcx : (∫ m, residual m.1 lam ∂Q.probability.toMeasure) = 0 := by
+    have he := integral_congr_ae (hp.mono (fun m hm ↦ symEntropy_edge_residual hm ⟨hl₀.le,hl₁.le⟩))
+    change edgeJMean Q lam = (∫ m : Message, lam^2*(symEntropy m.1+(1-lam)*residual m.1 lam/2) ∂Q.probability.toMeasure) at he
+    rw [integral_const_mul,
+      integral_add (f := fun m : Message ↦ symEntropy m.1) (g := fun m ↦ (1-lam)*residual m.1 lam/2)
+        hJ ((hC.const_mul _).div_const _), integral_div, integral_const_mul] at he
+    change edgeJMean Q lam = lam^2*(jMean Q+(1-lam)*(∫ m, residual m.1 lam ∂Q.probability.toMeasure)/2) at he
+    have hc : 0 < lam^2*(1-lam) := mul_pos (sq_pos_of_pos hl₀) (sub_pos.mpr hl₁)
+    nlinarith [hadd]
+  have hjx : jMean Q = Q.mu-lam*(∫ m, momentQ m.1 lam ∂Q.probability.toMeasure) := by
+    have he := integral_congr_ae (hp.mono (fun m hm ↦ symEntropy_residual hm ⟨hl₀.le,hl₁.le⟩))
+    change jMean Q = (∫ m : Message, xMoment m-lam*momentQ m.1 lam-residual m.1 lam/2 ∂Q.probability.toMeasure) at he
+    rw [integral_sub (f := fun m : Message ↦ xMoment m-lam*momentQ m.1 lam)
+      (g := fun m ↦ residual m.1 lam/2) (hx.sub (hQ.const_mul _)) (hC.div_const _),
+      integral_sub (f := xMoment) (g := fun m ↦ lam*momentQ m.1 lam) hx (hQ.const_mul _),
+      integral_const_mul, integral_div, hcx] at he
+    simpa only [sub_zero, zero_div, jMean, Law.mu] using he
+  have hyx : Q.eta = (∫ m, momentA m.1 ∂Q.probability.toMeasure)+
+      2*lam*(∫ m, momentQ m.1 lam ∂Q.probability.toMeasure) := by
+    have he := integral_congr_ae (hp.mono (fun m hm ↦ momentY_residual hm ⟨hl₀.le,hl₁.le⟩))
+    rw [integral_add (f := fun m : Message ↦ momentA m.1+2*lam*momentQ m.1 lam)
+      (g := fun m ↦ residual m.1 lam) (hA.add (hQ.const_mul _)) hC,
+      integral_add (f := fun m : Message ↦ momentA m.1) (g := fun m ↦ 2*lam*momentQ m.1 lam) hA (hQ.const_mul _),
+      integral_const_mul, hcx, add_zero] at he
+    exact he
+  have hAb : (37/27)*Q.nu ≤ ∫ m, momentA m.1 ∂Q.probability.toMeasure := by
+    have h := integral_mono_ae (f := fun m : Message ↦ (37/27)*xMoment m^2)
+      (g := fun m ↦ momentA m.1) (hx₂.const_mul (37/27)) hA
+      (hp.mono (fun m hm ↦ momentA_lower hm))
+    rwa [integral_const_mul] at h
+  have hQb : qExact lam*Q.nu ≤ ∫ m, momentQ m.1 lam ∂Q.probability.toMeasure := by
+    have h := integral_mono_ae (f := fun m : Message ↦ momentA m.1/(2+lam))
+      (g := fun m ↦ momentQ m.1 lam) (hA.div_const _) hQ
+      (hp.mono (fun m hm ↦ momentQ_lower hm hl₀.le hl₁.le))
+    rw [integral_div] at h
+    have hden : 0 < 2+lam := by linarith
+    have hx := (div_le_div_iff_of_pos_right hden).2 hAb
+    have hh := hx.trans h
+    simpa only [qExact, div_mul_eq_mul_div] using hh
+  refine ⟨integral_nonneg_of_ae (hp.mono (fun m hm ↦ symEntropy_nonneg hm)), ?_, ?_⟩
+  · rw [hjx]; nlinarith [mul_nonneg hl₀.le (sub_nonneg.mpr hQb)]
+  · rw [hyx]
+    dsimp only [cExact]
+    nlinarith [mul_nonneg hl₀.le (sub_nonneg.mpr hQb)]
+
+theorem regular_fixed_moment_bounds (Q : Law) {lam : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1) (n : ℕ)
+    (hcrit : (n:ℝ)*lam^2 = 1) (hfixed : Q.probability = finiteProbability Q hl₀.le hl₁ n) :
+    0 ≤ jMean Q ∧ jMean Q ≤ Q.mu-lam*qExact lam*Q.nu ∧ cExact lam*Q.nu ≤ Q.eta := by
+  have hp := finite_positive Q hl₀.le hl₁ n
+  have hJ := finite_integrable_symEntropy Q hl₀.le hl₁ n
+  rw [← hfixed] at hp hJ
+  apply expected_moment_bounds Q hl₀ hl₁ hp hJ
+  have he := finiteLaw_symEntropy Q hl₀.le hl₁ n
+  change (∫ m, symEntropy m.1 ∂(finiteProbability Q hl₀.le hl₁ n).toMeasure) = (n:ℝ)*edgeJMean Q lam at he
+  rw [← hfixed] at he
+  change jMean Q = (n:ℝ)*edgeJMean Q lam at he
+  calc
+    edgeJMean Q lam = ((n:ℝ)*lam^2)*edgeJMean Q lam := by rw [hcrit]; ring
+    _ = lam^2*((n:ℝ)*edgeJMean Q lam) := by ring
+    _ = lam^2*jMean Q := by rw [← he]
+
+end ThreeState.TreeClauses.Positive
+
+end 
+
+noncomputable section
+open scoped BigOperators
+namespace ThreeState.TreeClauses.Positive
+
+lemma branch_sum_first (k : ℕ) :
+    (∑ j ∈ Finset.range k, ((j:ℝ)+1)) = (k:ℝ)*((k:ℝ)+1)/2 := by
+  induction k with
+  | zero => simp
+  | succ k ih =>
+    rw [Finset.sum_range_succ, ih]
+    push_cast
+    ring
+
+lemma branch_sum_second (k : ℕ) :
+    (∑ j ∈ Finset.range k, ((j:ℝ)+1)*(j:ℝ)) = (k:ℝ)*((k:ℝ)+1)*((k:ℝ)-1)/3 := by
+  induction k with
+  | zero => simp
+  | succ k ih =>
+    rw [Finset.sum_range_succ, ih]
+    push_cast
+    ring
+
+lemma reciprocal_tangent {a q : ℝ} (ha : 0 ≤ a) (_hq : 0 ≤ q) :
+    1+2*q-a ≤ (1+q)^2/(1+a) := by
+  apply (le_div_iff₀ (by positivity : 0 < 1+a)).2
+  nlinarith [sq_nonneg (a-q)]
+
+lemma weighted_branch_jensen {lam mu : ℝ} (hmu : 0 ≤ mu) (k : ℕ)
+    (hcrit : ((k:ℝ)+1)*lam^2 = 1) :
+    (k:ℝ)*((k:ℝ)+1)/(2*(1+2*mu/3)) ≤
+      ∑ j ∈ Finset.range k, ((j:ℝ)+1)/(1+(j:ℝ)*(lam^2*mu)) := by
+  let q := 2*mu/3
+  let beta := lam^2*mu
+  have hq : 0 ≤ q := by dsimp [q]; positivity
+  have hb : 0 ≤ beta := by dsimp [beta]; positivity
+  have hh := Finset.sum_le_sum (s := Finset.range k) (fun j _ ↦
+    mul_le_mul_of_nonneg_right (reciprocal_tangent (mul_nonneg (Nat.cast_nonneg j) hb) hq)
+      (show (0:ℝ) ≤ (j:ℝ)+1 by positivity))
+  have hleft : (∑ j ∈ Finset.range k, (1+2*q-(j:ℝ)*beta)*((j:ℝ)+1)) =
+      (1+2*q)*((k:ℝ)*((k:ℝ)+1)/2)-beta*((k:ℝ)*((k:ℝ)+1)*((k:ℝ)-1)/3) := by
+    simp_rw [sub_mul]
+    rw [Finset.sum_sub_distrib, ← Finset.mul_sum]
+    have hid : (∑ j ∈ Finset.range k, (j:ℝ)*beta*((j:ℝ)+1)) =
+        beta*∑ j ∈ Finset.range k, ((j:ℝ)+1)*(j:ℝ) := by
+      rw [Finset.mul_sum]; apply Finset.sum_congr rfl; intro j _; ring
+    rw [hid, branch_sum_first, branch_sum_second]
+  have hright : (∑ j ∈ Finset.range k, ((1+q)^2/(1+(j:ℝ)*beta))*((j:ℝ)+1)) =
+      (1+q)^2*∑ j ∈ Finset.range k, ((j:ℝ)+1)/(1+(j:ℝ)*beta) := by
+    rw [Finset.mul_sum]; apply Finset.sum_congr rfl; intro j _; ring
+  rw [hleft, hright] at hh
+  have hbB : beta*((k:ℝ)*((k:ℝ)+1)*((k:ℝ)-1)/3) ≤ q*((k:ℝ)*((k:ℝ)+1)/2) := by
+    have he : beta*((k:ℝ)*((k:ℝ)+1)*((k:ℝ)-1)/3) = mu*(k:ℝ)*((k:ℝ)-1)/3 := by
+      dsimp [beta]
+      linear_combination (mu*(k:ℝ)*((k:ℝ)-1)/3)*hcrit
+    rw [he]
+    dsimp [q]
+    nlinarith [mul_nonneg hmu (Nat.cast_nonneg k)]
+  have hd : 0 < 1+q := by positivity
+  have hh' : (k:ℝ)*((k:ℝ)+1)/2 ≤
+      (1+q)*(∑ j ∈ Finset.range k, ((j:ℝ)+1)/(1+(j:ℝ)*beta)) := by
+    apply le_of_mul_le_mul_left (a := 1+q) _ hd
+    nlinarith
+  apply (div_le_iff₀ (show 0 < 2*(1+2*mu/3) by positivity)).2
+  dsimp [q, beta] at hh'
+  nlinarith
+
+lemma sqrt_branch_sum {lam : ℝ} (hl : 0 < lam) (k : ℕ)
+    (hcrit : ((k:ℝ)+1)*lam^2 = 1) :
+    lam^4*(∑ j ∈ Finset.range k, Real.sqrt ((j:ℝ)+1)) ≤
+      (1-lam^2)*lam/Real.sqrt 2 := by
+  have hs2 := Real.sq_sqrt (by norm_num : (0:ℝ) ≤ 2)
+  have hs2p := Real.sqrt_pos.mpr (by norm_num : (0:ℝ) < 2)
+  have hl2 : 0 < lam^2 := sq_pos_of_pos hl
+  have hp := Finset.sum_le_sum (s := Finset.range k) (fun j _ ↦
+    sq_nonneg (Real.sqrt ((j:ℝ)+1)-1/(lam*Real.sqrt 2)))
+  have hpoint (j : ℕ) :
+      2*lam*Real.sqrt 2*Real.sqrt ((j:ℝ)+1) ≤ 2*lam^2*((j:ℝ)+1)+1 := by
+    have hh := sq_nonneg (lam*Real.sqrt 2*Real.sqrt ((j:ℝ)+1)-1)
+    have he : (lam*Real.sqrt 2*Real.sqrt ((j:ℝ)+1))^2 = 2*lam^2*((j:ℝ)+1) := by
+      calc
+        _ = lam^2*(Real.sqrt 2)^2*(Real.sqrt ((j:ℝ)+1))^2 := by ring
+        _ = _ := by rw [hs2, Real.sq_sqrt (by positivity)]; ring
+    nlinarith
+  have hh := Finset.sum_le_sum (s := Finset.range k) (fun j _ ↦ hpoint j)
+  rw [← Finset.mul_sum, Finset.sum_add_distrib, ← Finset.mul_sum, branch_sum_first] at hh
+  simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_one] at hh
+  have he : 2*lam^2*((k:ℝ)*((k:ℝ)+1)/2)+(k:ℝ) = 2*(k:ℝ) := by
+    linear_combination (k:ℝ)*hcrit
+  rw [he] at hh
+  have he' : (k:ℝ)*lam^2 = 1-lam^2 := by nlinarith [hcrit]
+  apply (le_div_iff₀ hs2p).2
+  have hm := mul_le_mul_of_nonneg_left hh (show 0 ≤ lam^3/2 by positivity)
+  nlinarith
+
+end ThreeState.TreeClauses.Positive
+
+end 
+
+noncomputable section
+open Set MeasureTheory
+open scoped BigOperators
+namespace ThreeState.TreeClauses.Positive
+open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Experiment
+
+lemma truncated_quadratic {a b t L C : ℝ} (ha : 0 ≤ a) (_hb : 0 ≤ b)
+    (hL : 0 ≤ L) (ht : L ≤ t) (hC : 0 ≤ C)
+    (hbound : a*t-b*Real.sqrt t ≤ C) : a*L-b*Real.sqrt L ≤ C := by
+  have hsL := Real.sqrt_nonneg L
+  have hst := Real.sqrt_nonneg t
+  have hsqL := Real.sq_sqrt hL
+  have hsqt := Real.sq_sqrt (hL.trans ht)
+  have hsle := Real.sqrt_le_sqrt ht
+  by_cases h : a*Real.sqrt L-b ≤ 0
+  · have hh := mul_nonpos_of_nonneg_of_nonpos hsL h
+    nlinarith
+  · have hh := mul_nonneg (sub_nonneg.mpr hsle) (le_of_lt (lt_of_not_ge h))
+    have hh' := mul_nonneg (mul_nonneg ha (sub_nonneg.mpr hsle)) hst
+    nlinarith
+
+lemma product_cubic_bound (Q R : Law) (hnu : R.nu ≤ Q.nu) :
+    Q.eta*R.eta ≤ Real.sqrt Q.mu*Q.nu*Real.sqrt R.mu := by
+  have hmu := Q.mu_nonneg
+  have hnuQ := Q.nu_nonneg
+  have hR : R.eta^2 ≤ R.mu*Q.nu :=
+    R.moment_discriminant.trans (mul_le_mul_of_nonneg_left hnu R.mu_nonneg)
+  have hprod := mul_le_mul Q.moment_discriminant hR (sq_nonneg R.eta)
+    (mul_nonneg Q.mu_nonneg Q.nu_nonneg)
+  have hs : (Real.sqrt Q.mu*Q.nu*Real.sqrt R.mu)^2 = (Q.mu*Q.nu)*(R.mu*Q.nu) := by
+    calc
+      _ = (Real.sqrt Q.mu)^2*Q.nu^2*(Real.sqrt R.mu)^2 := by ring
+      _ = _ := by rw [Real.sq_sqrt Q.mu_nonneg, Real.sq_sqrt R.mu_nonneg]; ring
+  have hp : 0 ≤ Real.sqrt Q.mu*Q.nu*Real.sqrt R.mu := by positivity
+  nlinarith [sq_nonneg (Q.eta*R.eta-Real.sqrt Q.mu*Q.nu*Real.sqrt R.mu)]
+
+lemma correlation_truncated (Q R : Law) {lam L : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
+    (hnu : R.nu ≤ Q.nu) (hL : 0 ≤ L) (hproj : L ≤ R.mu) :
+    3*lam^2*Q.mu*L-lam^3*Real.sqrt Q.mu*Q.nu*Real.sqrt L ≤ 3*pairCorrelation R Q lam := by
+  have hmu := Q.mu_nonneg
+  have hnuQ := Q.nu_nonneg
+  have hi := pairCorrelation_increment R Q hl₀ hl₁
+  have he := product_cubic_bound Q R hnu
+  have hh : 3*lam^2*Q.mu*R.mu-(lam^3*Real.sqrt Q.mu*Q.nu)*Real.sqrt R.mu ≤
+      3*pairCorrelation R Q lam := by
+    nlinarith [mul_nonneg (pow_nonneg hl₀ 3) (sub_nonneg.mpr he)]
+  exact truncated_quadratic (by positivity) (by positivity) hL hproj
+    (mul_nonneg (by norm_num) (pairCorrelation_nonneg R Q hl₀ hl₁)) hh
+
+lemma finite_increment_bound (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
+    (N j : ℕ) (hj : 1 ≤ j) (hjN : j ≤ N)
+    (hfixed : Q.probability = finiteProbability Q hl₀ hl₁ N) :
+    3*lam^4*Q.mu^2*(j:ℝ)/(1+((j:ℝ)-1)*(lam^2*Q.mu))-
+      lam^4*Q.mu*Q.nu*Real.sqrt j ≤ 3*pairCorrelation (finiteLaw Q hl₀ hl₁ j) Q lam := by
+  have hmu := Q.mu_nonneg
+  have hnuQ := Q.nu_nonneg
+  let L : ℝ := (j:ℝ)*(lam^2*Q.mu)/(1+((j:ℝ)-1)*(lam^2*Q.mu))
+  have hj' : (1:ℝ) ≤ j := by exact_mod_cast hj
+  have hd : 1 ≤ 1+((j:ℝ)-1)*(lam^2*Q.mu) := by
+    nlinarith [mul_nonneg (sub_nonneg.mpr hj') (mul_nonneg (sq_nonneg lam) Q.mu_nonneg)]
+  have hL : 0 ≤ L := by dsimp [L]; positivity
+  have hproj := finite_projection Q hl₀ hl₁ j hj
+  have hnu : (finiteLaw Q hl₀ hl₁ j).nu ≤ Q.nu := by
+    have hh := finite_nu_mono Q hl₀ hl₁ hjN
+    change _ ≤ ∫ m, xMoment m^2 ∂(finiteProbability Q hl₀ hl₁ N).toMeasure at hh
+    rw [← hfixed] at hh
+    exact hh
+  have hc := correlation_truncated Q (finiteLaw Q hl₀ hl₁ j) hl₀ hl₁ hnu hL hproj
+  have hLu : L ≤ (j:ℝ)*(lam^2*Q.mu) := by
+    dsimp only [L]
+    exact div_le_self (by positivity) hd
+  have hs : Real.sqrt L ≤ lam*Real.sqrt Q.mu*Real.sqrt j := by
+    have he : (lam*Real.sqrt Q.mu*Real.sqrt j)^2 = (j:ℝ)*(lam^2*Q.mu) := by
+      calc
+        _ = lam^2*(Real.sqrt Q.mu)^2*(Real.sqrt j)^2 := by ring
+        _ = _ := by rw [Real.sq_sqrt Q.mu_nonneg, Real.sq_sqrt (Nat.cast_nonneg j)]; ring
+    have hp : 0 ≤ lam*Real.sqrt Q.mu*Real.sqrt j := by positivity
+    nlinarith [Real.sq_sqrt hL, Real.sqrt_nonneg L]
+  have hmul := mul_le_mul_of_nonneg_left hs (show 0 ≤ lam^3*Real.sqrt Q.mu*Q.nu by positivity)
+  have heq : (lam^3*Real.sqrt Q.mu*Q.nu)*(lam*Real.sqrt Q.mu*Real.sqrt j) =
+      lam^4*Q.mu*Q.nu*Real.sqrt j := by
+    calc
+      _ = lam^4*(Real.sqrt Q.mu)^2*Q.nu*Real.sqrt j := by ring
+      _ = _ := by rw [Real.sq_sqrt Q.mu_nonneg]
+  rw [heq] at hmul
+  have hmain : 3*lam^2*Q.mu*L = 3*lam^4*Q.mu^2*(j:ℝ)/(1+((j:ℝ)-1)*(lam^2*Q.mu)) := by
+    dsimp only [L]; ring
+  rw [hmain] at hc
+  linarith
+
+lemma finite_correlation_sum (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
+    (N : ℕ) (hfixed : Q.probability = finiteProbability Q hl₀ hl₁ N)
+    (k : ℕ) (hk : k+1 ≤ N) :
+    (∑ j ∈ Finset.range k,
+      (3*lam^4*Q.mu^2*((j:ℝ)+1)/(1+(j:ℝ)*(lam^2*Q.mu))-
+        lam^4*Q.mu*Q.nu*Real.sqrt ((j:ℝ)+1))) ≤ 3*correlationEntropy Q lam (k+1) := by
+  induction k with
+  | zero =>
+    simpa using mul_nonneg (by norm_num : (0:ℝ) ≤ 3) (correlationEntropy_nonneg Q hl₀ hl₁ 1)
+  | succ k ih =>
+    have hprev := ih (by omega)
+    have hi := finite_increment_bound Q hl₀ hl₁ N (k+1) (by omega) (by omega) hfixed
+    simp only [Nat.cast_add, Nat.cast_one, add_sub_cancel_right] at hi
+    rw [Finset.sum_range_succ, correlationEntropy_succ Q hl₀ hl₁ (k+1)]
+    linarith
+
+theorem regular_fixed_correlation_bounds (Q : Law) {lam : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1)
+    (n : ℕ) (hn : 2 ≤ n) (hcrit : (n:ℝ)*lam^2 = 1)
+    (hfixed : Q.probability = finiteProbability Q hl₀.le hl₁ n) :
+    3*Q.mu^2/(2*(1+2*Q.mu/3))-lam/Real.sqrt 2*Q.mu*Q.nu ≤
+      3*correlationEntropy Q lam n/(1-lam^2) := by
+  obtain ⟨k,rfl⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : n ≠ 0)
+  have hc : ((k:ℝ)+1)*lam^2 = 1 := by simpa using hcrit
+  have halpha : 0 < 1-lam^2 := by nlinarith [mul_pos (sub_pos.mpr hl₁) (by linarith : 0 < 1+lam)]
+  have hmu := Q.mu_nonneg
+  have hnu := Q.nu_nonneg
+  have hh := finite_correlation_sum Q hl₀.le hl₁ (k+1) hfixed k (by omega)
+  have hid : (∑ j ∈ Finset.range k,
+      (3*lam^4*Q.mu^2*((j:ℝ)+1)/(1+(j:ℝ)*(lam^2*Q.mu))-
+        lam^4*Q.mu*Q.nu*Real.sqrt ((j:ℝ)+1))) =
+      (3*lam^4*Q.mu^2)*(∑ j ∈ Finset.range k, ((j:ℝ)+1)/(1+(j:ℝ)*(lam^2*Q.mu)))-
+      (Q.mu*Q.nu)*(lam^4*∑ j ∈ Finset.range k, Real.sqrt ((j:ℝ)+1)) := by
+    rw [Finset.sum_sub_distrib, Finset.mul_sum, Finset.mul_sum, Finset.mul_sum]
+    congr 1 <;> apply Finset.sum_congr rfl <;> intro j _ <;> ring
+  rw [hid] at hh
+  have hJ := mul_le_mul_of_nonneg_left (weighted_branch_jensen hmu k hc)
+    (show 0 ≤ 3*lam^4*Q.mu^2 by positivity)
+  have hS := mul_le_mul_of_nonneg_left (sqrt_branch_sum hl₀ k hc) (mul_nonneg hmu hnu)
+  have he : lam^4*(k:ℝ)*((k:ℝ)+1) = 1-lam^2 := by
+    calc
+      _ = ((k:ℝ)*lam^2)*(((k:ℝ)+1)*lam^2) := by ring
+      _ = 1-lam^2 := by rw [hc]; nlinarith [hc]
+  have heJ : 3*lam^4*Q.mu^2*((k:ℝ)*((k:ℝ)+1)/(2*(1+2*Q.mu/3))) =
+      (1-lam^2)*(3*Q.mu^2/(2*(1+2*Q.mu/3))) := by
+    calc
+      _ = (lam^4*(k:ℝ)*((k:ℝ)+1))*(3*Q.mu^2/(2*(1+2*Q.mu/3))) := by ring
+      _ = _ := by rw [he]
+  rw [heJ] at hJ
+  apply (le_div_iff₀ halpha).2
+  calc
+    _ = (1-lam^2)*(3*Q.mu^2/(2*(1+2*Q.mu/3)))-Q.mu*Q.nu*((1-lam^2)*lam/Real.sqrt 2) := by ring
+    _ ≤ 3*correlationEntropy Q lam (k+1) := by linarith
+
+end ThreeState.TreeClauses.Positive
+
+end 
+
+noncomputable section
+open Set MeasureTheory
+namespace ThreeState.TreeClauses.Positive
+open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Experiment
+
+lemma fixed_edgeJ (Q : Law) {lam : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1) (n : ℕ)
+    (hcrit : (n:ℝ)*lam^2 = 1) (hfixed : Q.probability = finiteProbability Q hl₀.le hl₁ n) :
+    edgeJMean Q lam = lam^2*jMean Q := by
+  have h := finiteLaw_symEntropy Q hl₀.le hl₁ n
+  change (∫ m, symEntropy m.1 ∂(finiteProbability Q hl₀.le hl₁ n).toMeasure) = _ at h
+  rw [← hfixed] at h
+  change jMean Q = (n:ℝ)*edgeJMean Q lam at h
+  rw [h]
+  calc
+    _ = ((n:ℝ)*lam^2)*edgeJMean Q lam := by rw [hcrit, one_mul]
+    _ = _ := by ring
+
+lemma regular_fixed_entropy_radial (Q : Law) {lam : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1) (n : ℕ)
+    (hcrit : (n:ℝ)*lam^2 = 1) (hfixed : Q.probability = finiteProbability Q hl₀.le hl₁ n) :
+    (1-lam^2)/2*Q.nu ≤ (8/5)*(1-lam^2)*(jMean Q)^2-3*correlationEntropy Q lam n := by
+  have hp := finite_positive Q hl₀.le hl₁ n
+  have hJ := finite_integrable_symEntropy Q hl₀.le hl₁ n
+  have hB := finite_integrable_correction Q hl₀.le hl₁ n
+  rw [← hfixed] at hp hJ hB
+  have hI : Integrable (fun m : Message ↦ info m.1) Q.probability.toMeasure :=
+    compact_integrable (continuous_info_comp continuous_subtype_val)
+  have hJe := edgeJ_integrable Q hl₀.le hl₁
+  have hIe : Integrable (fun m : Message ↦ info (edgeMessage lam m)) Q.probability.toMeasure :=
+    compact_integrable (continuous_info_comp (continuous_edgeMessage lam))
+  have hBe : Integrable (fun m : Message ↦ correction (edgeMessage lam m)) Q.probability.toMeasure :=
+    compact_integrable (continuous_correction_comp (continuous_edgeMessage lam) (fun m i ↦ (edge_positive hl₀.le hl₁ m).1 i))
+  have hF : Integrable (fun m : Message ↦ functional m.1) Q.probability.toMeasure :=
+    (hI.const_mul 3 |>.sub (hJ.const_mul 2)).add (hB.const_mul (4/5))
+  have hFe : Integrable (fun m : Message ↦ functional (edgeMessage lam m)) Q.probability.toMeasure :=
+    (hIe.const_mul 3 |>.sub (hJe.const_mul 2)).add (hBe.const_mul (4/5))
+  have hrad := integral_mono_ae (μ := Q.probability.toMeasure)
+    (f := fun m : Message ↦ (1-lam^2)/2*xMoment m^2)
+    (g := fun m ↦ functional m.1-(lam^2)⁻¹*functional (edgeMessage lam m))
+    ((compact_integrable (continuous_xMoment.pow 2)).const_mul _) (hF.sub (hFe.const_mul _))
+    (hp.mono (fun m hm ↦ radial_inequality m.1 hm lam hl₀ hl₁))
+  rw [integral_const_mul, integral_sub hF (hFe.const_mul _), integral_const_mul] at hrad
+  change (1-lam^2)/2*Q.nu ≤ _ at hrad
+  have hJid := finiteLaw_symEntropy Q hl₀.le hl₁ n
+  have hBid := finiteLaw_correction Q hl₀.le hl₁ n
+  have hIid := finiteLaw_info Q hl₀.le hl₁ n
+  change (∫ m, symEntropy m.1 ∂(finiteProbability Q hl₀.le hl₁ n).toMeasure) = _ at hJid
+  change (∫ m, correction m.1 ∂(finiteProbability Q hl₀.le hl₁ n).toMeasure) = _ at hBid
+  change (∫ m, info m.1 ∂(finiteProbability Q hl₀.le hl₁ n).toMeasure) = _ at hIid
+  rw [← hfixed] at hJid hBid hIid
+  have hninv : (lam^2)⁻¹ = (n:ℝ) := by
+    apply mul_left_cancel₀ (ne_of_gt (sq_pos_of_pos hl₀))
+    rw [mul_inv_cancel₀ (ne_of_gt (sq_pos_of_pos hl₀))]
+    nlinarith [hcrit]
+  have hFid : (∫ m, functional m.1 ∂Q.probability.toMeasure) =
+      3*(∫ m, info m.1 ∂Q.probability.toMeasure)-2*jMean Q+
+      (4/5)*(∫ m, correction m.1 ∂Q.probability.toMeasure) := by
+    unfold functional
+    rw [integral_add (f := fun m : Message ↦ 3*info m.1-2*symEntropy m.1)
+      (g := fun m ↦ (4/5)*correction m.1) ((hI.const_mul _).sub (hJ.const_mul _)) (hB.const_mul _),
+      integral_sub (f := fun m : Message ↦ 3*info m.1) (g := fun m ↦ 2*symEntropy m.1)
+        (hI.const_mul _) (hJ.const_mul _), integral_const_mul, integral_const_mul, integral_const_mul]
+    rfl
+  have hFeid : (∫ m, functional (edgeMessage lam m) ∂Q.probability.toMeasure) =
+      3*(∫ m, info (edgeMessage lam m) ∂Q.probability.toMeasure)-2*edgeJMean Q lam+
+      (4/5)*(∫ m, correction (edgeMessage lam m) ∂Q.probability.toMeasure) := by
+    unfold functional
+    rw [integral_add (f := fun m : Message ↦ 3*info (edgeMessage lam m)-2*symEntropy (edgeMessage lam m))
+      (g := fun m ↦ (4/5)*correction (edgeMessage lam m)) ((hIe.const_mul _).sub (hJe.const_mul _)) (hBe.const_mul _),
+      integral_sub (f := fun m : Message ↦ 3*info (edgeMessage lam m)) (g := fun m ↦ 2*symEntropy (edgeMessage lam m))
+        (hIe.const_mul _) (hJe.const_mul _), integral_const_mul, integral_const_mul, integral_const_mul]
+    rfl
+  rw [hFid, hFeid, hninv, hIid, hBid] at hrad
+  change jMean Q = (n:ℝ)*edgeJMean Q lam at hJid
+  change _ ≤ 3*((n:ℝ)*(∫ m, info (edgeMessage lam m) ∂Q.probability.toMeasure)-correlationEntropy Q lam n)-
+    2*jMean Q+(4/5)*((n:ℝ)*(∫ m, correction (edgeMessage lam m) ∂Q.probability.toMeasure)+
+      2*((n:ℝ)^2-n)*(edgeJMean Q lam)^2)-
+    (n:ℝ)*(3*(∫ m, info (edgeMessage lam m) ∂Q.probability.toMeasure)-2*edgeJMean Q lam+
+      (4/5)*(∫ m, correction (edgeMessage lam m) ∂Q.probability.toMeasure)) at hrad
+  have he : ((n:ℝ)^2-n)*(edgeJMean Q lam)^2 = (1-lam^2)*(jMean Q)^2 := by
+    rw [fixed_edgeJ Q hl₀ hl₁ n hcrit hfixed]
+    have hn : ((n:ℝ)^2-n)*lam^4 = 1-lam^2 := by
+      nlinarith [sq_nonneg ((n:ℝ)*lam^2-1), hcrit]
+    calc
+      _ = (((n:ℝ)^2-n)*lam^4)*(jMean Q)^2 := by ring
+      _ = _ := by rw [hn]
+  nlinarith [hJid, he]
+
+end ThreeState.TreeClauses.Positive
+
+end 
+
+noncomputable section
+namespace ThreeState.TreeClauses.Positive
+
+lemma qExact_lower {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam ≤ 1) :
+    (37/81:ℝ) ≤ qExact lam := by
+  unfold qExact
+  apply (le_div_iff₀ (by positivity : 0 < 2+lam)).2
+  linarith
+
+lemma cExact_lower {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam ≤ 1) :
+    cLower lam ≤ cExact lam := by
+  have hq := qExact_lower hl₀ hl₁
+  unfold cExact cLower
+  nlinarith [mul_nonneg hl₀ (sub_nonneg.mpr hq)]
+
+lemma qExact_coefficient {lam : ℝ} :
+    8*(37/27)*lam*qExact lam ≤ cExact lam^2 := by
+  unfold cExact
+  nlinarith [sq_nonneg ((37/27:ℝ)-2*lam*qExact lam)]
+
+lemma jMean_square_bound {lam mu nu J : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam ≤ 1)
+    (hmu : 0 ≤ mu) (hnu : 0 ≤ nu) (hJ : 0 ≤ J) (hupper : J ≤ mu-lam*qExact lam*nu)
+    (hregion : cExact lam^2*nu ≤ mu) :
+    (8/5)*J^2 ≤ (8/5)*mu^2-(113/81)*lam*mu*nu := by
+  let t := lam*qExact lam*nu
+  have hq := qExact_lower hl₀ hl₁
+  have ht : 0 ≤ t := by dsimp [t]; positivity
+  have hc := mul_le_mul_of_nonneg_right (qExact_coefficient (lam := lam)) hnu
+  have htu : 8*(37/27)*t ≤ mu := by dsimp [t]; nlinarith
+  have hJsq : J^2 ≤ (mu-t)^2 := by
+    have hu : J ≤ mu-t := hupper
+    nlinarith
+  have htt : t^2 ≤ (27/296)*mu*t := by
+    have hh := mul_le_mul_of_nonneg_right htu ht
+    nlinarith
+  have htl : (37/81)*lam*nu ≤ t := by
+    dsimp [t]
+    nlinarith [mul_nonneg (mul_nonneg hl₀ hnu) (sub_nonneg.mpr hq)]
+  have hm := mul_le_mul_of_nonneg_left htl hmu
+  nlinarith
+
+lemma strict_correlation_coefficient : (1:ℝ)/Real.sqrt 2+17/25 < 113/81 := by
+  have hs := Real.sq_sqrt (by norm_num : (0:ℝ) ≤ 2)
+  have hp := Real.sqrt_pos.mpr (by norm_num : (0:ℝ) < 2)
+  have hh : (2025:ℝ)/1448 < Real.sqrt 2 := by nlinarith
+  have hfrac : (1:ℝ)/Real.sqrt 2 < 1448/2025 := (div_lt_iff₀ hp).2 (by nlinarith)
+  linarith
+
+end ThreeState.TreeClauses.Positive
+
+end 
+
+noncomputable section
+open Set MeasureTheory
+namespace ThreeState.TreeClauses.Positive
+open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Experiment
+
+theorem regular_fixed_mu_zero (Q : Law) {lam : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1)
+    (n : ℕ) (hn : 2 ≤ n) (hcrit : (n:ℝ)*lam^2 = 1)
+    (hfixed : Q.probability = finiteProbability Q hl₀.le hl₁ n) : Q.mu = 0 := by
+  by_contra hmu0
+  have hmu : 0 < Q.mu := lt_of_le_of_ne Q.mu_nonneg (Ne.symm hmu0)
+  have hnu : 0 < Q.nu := (sq_pos_of_pos hmu).trans_le Q.second_moment_lower
+  have hinfo := regular_fixed_moment_bounds Q hl₀ hl₁ n hcrit hfixed
+  have hcl := cExact_lower hl₀.le hl₁.le
+  have hc : 0 < cExact lam := (cLower_pos hl₀.le).trans_le hcl
+  have hetaNonneg : 0 ≤ Q.eta := (mul_nonneg hc.le hnu.le).trans hinfo.2.2
+  have hsq : (cExact lam*Q.nu)^2 ≤ Q.eta^2 :=
+    (sq_le_sq₀ (mul_nonneg hc.le hnu.le) hetaNonneg).2 hinfo.2.2
+  have hregion : cExact lam^2*Q.nu ≤ Q.mu := by
+    apply (mul_le_mul_iff_right₀ hnu).mp
+    nlinarith [Q.moment_discriminant]
+  have hmuC : Q.mu*cExact lam^2 ≤ 1 := by
+    apply (mul_le_mul_iff_right₀ hmu).mp
+    have hh := mul_le_mul_of_nonneg_left Q.second_moment_lower (sq_nonneg (cExact lam))
+    nlinarith
+  have hregionLower : Q.mu*cLower lam^2 ≤ 1 := by
+    have hs : cLower lam^2 ≤ cExact lam^2 :=
+      (sq_le_sq₀ (cLower_pos hl₀.le).le hc.le).2 hcl
+    exact (mul_le_mul_of_nonneg_left hs hmu.le).trans hmuC
+  have hlam : lam^2 ≤ 1/2 := by
+    have hn' : (2:ℝ) ≤ n := by exact_mod_cast hn
+    have hh := mul_le_mul_of_nonneg_right hn' (sq_nonneg lam)
+    nlinarith [hcrit]
+  have hstrict := scalar_strict hl₀ hl₁ hmu.le (by norm_num : (0:ℝ) < 1)
+    (le_refl (1:ℝ)) hregionLower (Or.inl hlam)
+  have hJ := jMean_square_bound hl₀.le hl₁.le hmu.le hnu.le hinfo.1 hinfo.2.1 hregion
+  have hH := regular_fixed_correlation_bounds Q hl₀ hl₁ n hn hcrit hfixed
+  have hrad := regular_fixed_entropy_radial Q hl₀ hl₁ n hcrit hfixed
+  have halpha : 0 < 1-lam^2 := by nlinarith
+  have hent : Q.nu/2+3*correlationEntropy Q lam n/(1-lam^2) ≤ (8/5)*(jMean Q)^2 := by
+    calc
+      _ = ((1-lam^2)/2*Q.nu+3*correlationEntropy Q lam n)/(1-lam^2) := by
+        field_simp [ne_of_gt halpha]
+      _ ≤ _ := (div_le_iff₀ halpha).2 (by nlinarith)
+  have hH' : 3*Q.mu^2/(2*(1+2*Q.mu/3))-(1/Real.sqrt 2)*(lam*Q.mu*Q.nu) ≤
+      3*correlationEntropy Q lam n/(1-lam^2) := by
+    convert hH using 1
+    ring
+  have hg := mul_le_mul_of_nonneg_right (le_of_lt strict_correlation_coefficient)
+    (show 0 ≤ lam*Q.mu*Q.nu by positivity)
+  have hb : Q.nu/2+3*Q.mu^2/(2*(1+2*Q.mu/3))+(17/25)*lam*Q.mu*Q.nu ≤ (8/5)*Q.mu^2 := by
+    nlinarith
+  have hcoef : 0 ≤ (1-lam^2)/2+17*lam*Q.mu/25 := by positivity
+  have hsub := mul_le_mul_of_nonneg_left Q.second_moment_lower hcoef
+  have hnecessary : scalarT lam Q.mu 1*Q.mu^2 ≤ (8/5)*Q.mu^2 := by
+    calc
+      _ = ((1-lam^2)/2+17*lam*Q.mu/25)*Q.mu^2+3*Q.mu^2/(2*(1+2*Q.mu/3)) := by
+        unfold scalarT; ring
+      _ ≤ ((1-lam^2)/2+17*lam*Q.mu/25)*Q.nu+3*Q.mu^2/(2*(1+2*Q.mu/3)) := by linarith
+      _ ≤ _ := by nlinarith [mul_nonneg (sq_nonneg lam) hnu.le]
+  have hh := mul_lt_mul_of_pos_right hstrict (sq_pos_of_pos hmu)
+  linarith
+
+lemma message_uniform_of_x_zero {m : Message} (h : xMoment m = 0) : m.1 = fun _ ↦ 1 := by
+  dsimp [xMoment, momentX, centered, avg] at h
+  funext i
+  fin_cases i <;> dsimp <;> nlinarith [sq_nonneg (m.1 0-1), sq_nonneg (m.1 1-1), sq_nonneg (m.1 2-1)]
+
+theorem regular_fixed_uniform (Q : Law) {lam : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1)
+    (n : ℕ) (hn : 2 ≤ n) (hcrit : (n:ℝ)*lam^2 = 1)
+    (hfixed : Q.probability = finiteProbability Q hl₀.le hl₁ n) :
+    ∀ᵐ m ∂Q.probability.toMeasure, m.1 = fun _ ↦ 1 := by
+  have hzero := regular_fixed_mu_zero Q hl₀ hl₁ n hn hcrit hfixed
+  have he : ∀ᵐ m ∂Q.probability.toMeasure, xMoment m = 0 :=
+    (integral_eq_zero_iff_of_nonneg xMoment_nonneg (compact_integrable continuous_xMoment)).1 hzero
+  exact he.mono (fun _ hm ↦ message_uniform_of_x_zero hm)
+
+end ThreeState.TreeClauses.Positive
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeStateClauses/PositiveMoments.lean b/lean/OAI/Probability/ThreeStateClauses/PositiveMoments.lean
index fbc9e0ecef525f162de12bf4dd2f7ac401f12bd4..5e0f876ee879d0ced37e519d9f389beadb9d2fe4 100644
--- a/lean/OAI/Probability/ThreeStateClauses/PositiveMoments.lean
+++ b/lean/OAI/Probability/ThreeStateClauses/PositiveMoments.lean
@@ -1,429 +1,429 @@
-import Mathlib
-import OAI.Probability.ThreeStateClauses.RadialBounds
-
-namespace OAI
-
-/-! Positive Moments. -/
-
-open scoped BigOperators ENNReal NNReal Topology
-open Filter
-noncomputable section
-open Set MeasureTheory
-namespace ThreeState.TreeClauses.Positive
-open Radial
-
-def fourthResolvent (v : Vec) (t : ℝ) : ℝ := avg (fun i ↦ v i^4/(1+t*v i))
-
-def fourthEdgeResolvent (v : Vec) (lam t : ℝ) : ℝ :=
-  avg (fun i ↦ v i^4/((1+t*v i)*(1+t*lam*v i)))
-
-def momentA (m : Vec) : ℝ := ∫ t in (0:ℝ)..1, t^2*fourthResolvent (centered m) t
-
-def momentQ (m : Vec) (lam : ℝ) : ℝ :=
-  (1/2)*∫ t in (0:ℝ)..1, t^2*fourthEdgeResolvent (centered m) lam t
-
-lemma fourthMoment {v : Vec} (hv : avg v = 0) :
-    avg (fun i ↦ v i^4) = 6*momentX v^2 := by
-  simp only [momentX, avg]; rw [third_eq hv]; ring
-
-lemma fourthResolvent_eq {v : Vec} (hv : avg v = 0) {t : ℝ}
-    (hp : ∀ i, 1+t*v i ≠ 0) (hD : denom v t ≠ 0) :
-    fourthResolvent v t =
-      (6*momentX v^2-10*t*momentX v*momentY v+4*t^2*momentY v^2)/denom v t := by
-  have he (i : Fin 3) : v i^4/(1+t*v i) =
-      (v i^3*(v i-t*v i^2+2*t^2*momentY v))/denom v t := by
-    calc
-      _ = v i^3*(v i/(1+t*v i)) := by ring
-      _ = _ := by rw [rational_identity hv t i (hp i) hD]; ring
-  simp only [fourthResolvent, he, avg_div]
-  congr 1
-  simp only [momentX, momentY, avg]
-  rw [third_eq hv]; ring
-
-lemma fraction_lower {z k : ℝ} (hz₀ : 0 ≤ z) (hz₁ : z < 1)
-    (_hk₀ : -1 ≤ k) (hk₁ : k ≤ 1) (hD : 0 < 1-3*z^2+2*z^3*k) :
-    4+(64/9)*(z-1/2)^2 ≤ (6-10*z*k+4*z^2*k^2)/(1-3*z^2+2*z^3*k) := by
-  have hB : 0 < 1+z-2*z^2 := by nlinarith [mul_pos (by linarith : 0 < 1-z) (by linarith : 0 < 1+2*z)]
-  have hBtop : 1+z-2*z^2 ≤ 9/8 := by nlinarith [sq_nonneg (z-1/4)]
-  have hnum : 6-10*z+4*z^2 ≤ 6-10*z*k+4*z^2*k^2 := by
-    have hfac : 0 ≤ 10-4*z*(k+1) := by nlinarith [mul_nonneg hz₀ (by linarith : 0 ≤ 1-k)]
-    have := mul_nonneg (mul_nonneg hz₀ (by linarith : 0 ≤ 1-k)) hfac
-    nlinarith
-  have hnum0 : 0 ≤ 6-10*z+4*z^2 := by
-    nlinarith [mul_nonneg (by linarith : 0 ≤ 1-z) (by linarith : 0 ≤ 6-4*z)]
-  have hden : 1-3*z^2+2*z^3*k ≤ 1-3*z^2+2*z^3 := by
-    nlinarith [mul_nonneg (pow_nonneg hz₀ 3) (by linarith : 0 ≤ 1-k)]
-  have hdenpos : 0 < 1-3*z^2+2*z^3 := lt_of_lt_of_le hD hden
-  have heq : (6-10*z+4*z^2)/(1-3*z^2+2*z^3) = (6-4*z)/(1+z-2*z^2) := by
-    apply (div_eq_div_iff (ne_of_gt hdenpos) (ne_of_gt hB)).2
-    ring
-  have hbase : 4+(64/9)*(z-1/2)^2 ≤ (6-4*z)/(1+z-2*z^2) := by
-    apply (le_div_iff₀ hB).2
-    have := mul_nonneg (sq_nonneg (z-1/2)) (sub_nonneg.mpr hBtop)
-    nlinarith
-  calc
-    _ ≤ (6-4*z)/(1+z-2*z^2) := hbase
-    _ = (6-10*z+4*z^2)/(1-3*z^2+2*z^3) := heq.symm
-    _ ≤ (6-10*z+4*z^2)/(1-3*z^2+2*z^3*k) :=
-      div_le_div_of_nonneg_left hnum0 hD hden
-    _ ≤ _ := (div_le_div_iff_of_pos_right hD).2 hnum
-
-lemma fourthResolvent_nonneg {m : Vec} (hm : PositiveMessage m)
-    {t : ℝ} (ht : t ∈ Icc (0:ℝ) 1) : 0 ≤ fourthResolvent (centered m) t := by
-  have hi (i : Fin 3) : 0 ≤ centered m i^4/(1+t*centered m i) :=
-    div_nonneg (by positivity) (edge_pos hm ht i).le
-  unfold fourthResolvent avg
-  exact div_nonneg (add_nonneg (add_nonneg (hi 0) (hi 1)) (hi 2)) (by norm_num)
-
-lemma fourthResolvent_lower {m : Vec} (hm : PositiveMessage m) {t : ℝ}
-    (ht : t ∈ Icc (0:ℝ) 1) :
-    (4+(64/9)*(t*Real.sqrt (momentX (centered m))-1/2)^2)*momentX (centered m)^2 ≤
-      fourthResolvent (centered m) t := by
-  let v := centered m
-  let x := momentX v
-  let y := momentY v
-  have hx₀ : 0 ≤ x := momentX_nonneg v
-  have hx₁ : x < 1 := momentX_lt_one hm
-  by_cases hx : x = 0
-  · have hn := fourthResolvent_nonneg hm ht
-    change (4+(64/9)*(t*Real.sqrt x-1/2)^2)*x^2 ≤ fourthResolvent v t
-    simpa [hx] using hn
-  have hxpos : 0 < x := lt_of_le_of_ne hx₀ (Ne.symm hx)
-  let r := Real.sqrt x
-  have hrpos : 0 < r := Real.sqrt_pos.mpr hxpos
-  have hr2 : r^2 = x := Real.sq_sqrt hx₀
-  have hr₁ : r < 1 := by nlinarith
-  let k := y/r^3
-  have hyr : y = k*r^3 := (div_mul_cancel₀ _ (pow_ne_zero 3 (ne_of_gt hrpos))).symm
-  have hk2 : k^2 ≤ 1 := by
-    have hxy : y^2 ≤ x^3 := moment_discriminant (avg_centered hm)
-    have heq : k^2 = y^2/r^6 := by dsimp [k]; ring
-    rw [heq, div_le_iff₀ (pow_pos hrpos 6)]
-    rw [← hr2] at hxy
-    nlinarith [hxy]
-  have hk₀ : -1 ≤ k := by nlinarith [sq_nonneg (k+1)]
-  have hk₁ : k ≤ 1 := by nlinarith [sq_nonneg (k-1)]
-  have hz₀ : 0 ≤ t*r := mul_nonneg ht.1 hrpos.le
-  have hz₁ : t*r < 1 := by nlinarith [ht.2]
-  have hD := denom_pos hm ht
-  have hDeq : denom v t = 1-3*(t*r)^2+2*(t*r)^3*k := by
-    dsimp only [denom]
-    change 1-3*t^2*x+2*t^3*y = _
-    rw [← hr2, hyr]; ring
-  have hD' : 0 < 1-3*(t*r)^2+2*(t*r)^3*k := by rw [← hDeq]; exact hD
-  have heq : fourthResolvent v t = x^2*((6-10*(t*r)*k+4*(t*r)^2*k^2)/(1-3*(t*r)^2+2*(t*r)^3*k)) := by
-    rw [fourthResolvent_eq (avg_centered hm) (fun i ↦ ne_of_gt (edge_pos hm ht i)) (ne_of_gt hD), hDeq]
-    change (6*x^2-10*t*x*y+4*t^2*y^2)/_ = _
-    rw [← hr2, hyr]; ring
-  change (4+(64/9)*(t*r-1/2)^2)*x^2 ≤ fourthResolvent v t
-  rw [heq, mul_comm (x^2)]
-  exact mul_le_mul_of_nonneg_right (fraction_lower hz₀ hz₁ hk₀ hk₁ hD') (sq_nonneg x)
-
-lemma fourthResolvent_continuousOn {m : Vec} (hm : PositiveMessage m) :
-    ContinuousOn (fourthResolvent (centered m)) (Icc (0:ℝ) 1) := by
-  have hi (i : Fin 3) : ContinuousOn (fun t : ℝ ↦ centered m i^4/(1+t*centered m i)) (Icc 0 1) :=
-    continuousOn_const.div (by fun_prop) (fun t ht ↦ ne_of_gt (edge_pos hm ht i))
-  exact (((hi 0).add (hi 1)).add (hi 2)).div_const 3
-
-lemma lower_integral (r x : ℝ) :
-    (∫ t in (0:ℝ)..1, t^2*((4+(64/9)*(t*r-1/2)^2)*x^2)) =
-      (37/27+(64/45)*(r-5/8)^2)*x^2 := by
-  conv_lhs => arg 1; ext t; ring_nf
-  simp (disch := apply Continuous.intervalIntegrable; fun_prop) only
-    [intervalIntegral.integral_add, intervalIntegral.integral_mul_const, integral_pow]
-  norm_num
-  ring
-
-theorem momentA_lower {m : Vec} (hm : PositiveMessage m) :
-    (37/27)*momentX (centered m)^2 ≤ momentA m := by
-  have hcr : ContinuousOn (fun t : ℝ ↦ t^2*fourthResolvent (centered m) t) (Icc 0 1) :=
-    (by fun_prop : ContinuousOn (fun t : ℝ ↦ t^2) (Icc 0 1)).mul (fourthResolvent_continuousOn hm)
-  have hcl : Continuous (fun t : ℝ ↦ t^2*((4+(64/9)*(t*Real.sqrt (momentX (centered m))-1/2)^2)*momentX (centered m)^2)) := by fun_prop
-  have hi := intervalIntegral.integral_mono_on (μ := volume) (by norm_num : (0:ℝ) ≤ 1)
-    (hcl.intervalIntegrable 0 1) (hcr.intervalIntegrable_of_Icc (by norm_num))
-    (fun t ht ↦ mul_le_mul_of_nonneg_left (fourthResolvent_lower hm ht) (sq_nonneg t))
-  rw [lower_integral] at hi
-  have hn : 0 ≤ (Real.sqrt (momentX (centered m))-5/8)^2*momentX (centered m)^2 := by positivity
-  dsimp only [momentA]
-  linarith
-
-lemma avg_mono {a b : Vec} (h : ∀ i, a i ≤ b i) : avg a ≤ avg b := by
-  unfold avg
-  exact div_le_div_of_nonneg_right (add_le_add (add_le_add (h 0) (h 1)) (h 2)) (by norm_num)
-
-lemma fourthResolvent_four {m : Vec} (hm : PositiveMessage m) {t : ℝ}
-    (ht : t ∈ Icc (0:ℝ) 1) : 4*momentX (centered m)^2 ≤ fourthResolvent (centered m) t := by
-  have h := fourthResolvent_lower hm ht
-  nlinarith [mul_nonneg (sq_nonneg (t*Real.sqrt (momentX (centered m))-1/2))
-    (sq_nonneg (momentX (centered m)))]
-
-lemma fourthEdgeResolvent_lower {m : Vec} (hm : PositiveMessage m) {lam t : ℝ}
-    (hl₀ : 0 ≤ lam) (hl₁ : lam ≤ 1) (ht : t ∈ Icc (0:ℝ) 1) :
-    (2/(2+lam))*fourthResolvent (centered m) t ≤ fourthEdgeResolvent (centered m) lam t := by
-  let v := centered m
-  let a : Vec := fun i ↦ v i^4/(1+t*v i)
-  let b : Vec := fun i ↦ 1+t*lam*v i
-  let c : ℝ := 1+lam/2
-  have hc : 0 < c := by dsimp [c]; linarith
-  have htl : t*lam ∈ Icc (0:ℝ) 1 := ⟨mul_nonneg ht.1 hl₀, by nlinarith [ht.2, mul_le_mul_of_nonneg_left hl₁ ht.1]⟩
-  have hb (i : Fin 3) : 0 < b i := by dsimp [b]; exact edge_pos hm htl i
-  have ha (i : Fin 3) : 0 ≤ a i := div_nonneg (by positivity) (edge_pos hm ht i).le
-  have hab (i : Fin 3) : a i*b i = (1-lam)*a i+lam*v i^4 := by
-    dsimp only [a, b]
-    have hd : 1+t*v i ≠ 0 := ne_of_gt (edge_pos hm ht i)
-    calc
-      _ = (v i^4/(1+t*v i))*((1-lam)+lam*(1+t*v i)) := by ring
-      _ = (1-lam)*(v i^4/(1+t*v i))+lam*((v i^4/(1+t*v i))*(1+t*v i)) := by ring
-      _ = _ := by rw [div_mul_cancel₀ _ hd]
-  have habavg : avg (fun i ↦ a i*b i) = (1-lam)*fourthResolvent v t+6*lam*momentX v^2 := by
-    simp_rw [hab]
-    rw [avg_add, avg_mul, avg_mul, fourthMoment (avg_centered hm)]
-    change (1-lam)*fourthResolvent v t+lam*(6*momentX v^2) = _
-    ring
-  have havga : avg a = fourthResolvent v t := rfl
-  have havgab : avg (fun i ↦ a i/b i) = fourthEdgeResolvent v lam t := by
-    simp only [a, b, fourthEdgeResolvent, div_div]
-  have htangent (i : Fin 3) : a i*(2*c-b i) ≤ c^2*(a i/b i) := by
-    rw [← mul_div_assoc]
-    apply (le_div_iff₀ (hb i)).2
-    nlinarith [mul_nonneg (ha i) (sq_nonneg (b i-c))]
-  have hs : 2*c*fourthResolvent v t-((1-lam)*fourthResolvent v t+6*lam*momentX v^2) ≤
-      c^2*fourthEdgeResolvent v lam t := by
-    calc
-      _ = avg (fun i ↦ a i*(2*c-b i)) := by
-        rw [← habavg, ← havga]
-        simp only [avg]; ring
-      _ ≤ avg (fun i ↦ c^2*(a i/b i)) := avg_mono htangent
-      _ = _ := by rw [avg_mul, havgab]
-  have hfour := fourthResolvent_four hm ht
-  have hh : 0 ≤ lam*((3/2)*fourthResolvent v t-6*momentX v^2) :=
-    mul_nonneg hl₀ (by linarith)
-  have hd : 0 ≤ c*(c*fourthEdgeResolvent v lam t-fourthResolvent v t) := by
-    dsimp only [c] at hs ⊢
-    nlinarith
-  have hh' := (mul_nonneg_iff_of_pos_left hc).mp hd
-  have hfinal : fourthResolvent v t/c ≤ fourthEdgeResolvent v lam t := by
-    apply (div_le_iff₀ hc).2
-    nlinarith
-  have heq : (2/(2+lam))*fourthResolvent v t = fourthResolvent v t/c := by
-    dsimp [c]
-    field_simp
-  rw [heq]
-  exact hfinal
-
-lemma fourthEdgeResolvent_continuousOn {m : Vec} (hm : PositiveMessage m) {lam : ℝ}
-    (hl₀ : 0 ≤ lam) (hl₁ : lam ≤ 1) :
-    ContinuousOn (fourthEdgeResolvent (centered m) lam) (Icc (0:ℝ) 1) := by
-  have hi (i : Fin 3) : ContinuousOn
-      (fun t : ℝ ↦ centered m i^4/((1+t*centered m i)*(1+t*lam*centered m i))) (Icc 0 1) := by
-    apply continuousOn_const.div (by fun_prop)
-    intro t ht
-    have htl : t*lam ∈ Icc (0:ℝ) 1 := ⟨mul_nonneg ht.1 hl₀, by nlinarith [ht.2, mul_le_mul_of_nonneg_left hl₁ ht.1]⟩
-    exact ne_of_gt (mul_pos (edge_pos hm ht i) (edge_pos hm htl i))
-  exact (((hi 0).add (hi 1)).add (hi 2)).div_const 3
-
-theorem momentQ_lower {m : Vec} (hm : PositiveMessage m) {lam : ℝ}
-    (hl₀ : 0 ≤ lam) (hl₁ : lam ≤ 1) : momentA m/(2+lam) ≤ momentQ m lam := by
-  have hcr : ContinuousOn (fun t : ℝ ↦ t^2*fourthEdgeResolvent (centered m) lam t) (Icc 0 1) :=
-    (by fun_prop : ContinuousOn (fun t : ℝ ↦ t^2) (Icc 0 1)).mul (fourthEdgeResolvent_continuousOn hm hl₀ hl₁)
-  have hcl : ContinuousOn (fun t : ℝ ↦ t^2*((2/(2+lam))*fourthResolvent (centered m) t)) (Icc 0 1) :=
-    (by fun_prop : ContinuousOn (fun t : ℝ ↦ t^2) (Icc 0 1)).mul
-      (continuousOn_const.mul (fourthResolvent_continuousOn hm))
-  have hi := intervalIntegral.integral_mono_on (μ := volume) (by norm_num : (0:ℝ) ≤ 1)
-    (hcl.intervalIntegrable_of_Icc (by norm_num)) (hcr.intervalIntegrable_of_Icc (by norm_num))
-    (fun t ht ↦ mul_le_mul_of_nonneg_left (fourthEdgeResolvent_lower hm hl₀ hl₁ ht) (sq_nonneg t))
-  have he : (∫ t in (0:ℝ)..1, t^2*((2/(2+lam))*fourthResolvent (centered m) t)) =
-      (2/(2+lam))*momentA m := by
-    rw [momentA, ← intervalIntegral.integral_const_mul]
-    congr 1; funext t; ring
-  rw [he] at hi
-  have heq : (2/(2+lam))*momentA m = 2*(momentA m/(2+lam)) := by ring
-  rw [heq] at hi
-  dsimp only [momentQ]
-  linarith
-
-end ThreeState.TreeClauses.Positive
-
-end 
-
-noncomputable section
-open Set MeasureTheory
-namespace ThreeState.TreeClauses.Positive
-open Radial
-
-def residual (m : Vec) (lam : ℝ) : ℝ :=
-  ∫ t in (0:ℝ)..1, t*avg (fun i ↦ centered m i^3 /
-    ((1+t*centered m i)*(1+t*lam*centered m i)))
-
-lemma avg_integrable {f : ℝ → Vec} (h : ∀ i, IntervalIntegrable (fun t ↦ f t i) volume 0 1) :
-    IntervalIntegrable (fun t ↦ avg (f t)) volume 0 1 :=
-  (((h 0).add (h 1)).add (h 2)).div_const 3
-
-lemma avg_continuousOn {f : ℝ → Vec} {S : Set ℝ} (h : ∀ i, ContinuousOn (fun t ↦ f t i) S) :
-    ContinuousOn (fun t ↦ avg (f t)) S :=
-  (((h 0).add (h 1)).add (h 2)).div_const 3
-
-lemma edge_message {m : Vec} (hm : PositiveMessage m) {lam : ℝ} (hl : lam ∈ Icc (0:ℝ) 1) :
-    PositiveMessage (edge lam m) := by
-  refine ⟨fun i ↦ edge_pos hm hl i, ?_⟩
-  change avg (fun i ↦ 1+lam*centered m i) = 1
-  rw [avg_add, avg_const, avg_mul]
-  rw [avg_centered hm]; ring
-
-lemma centered_edge (m : Vec) (lam : ℝ) : centered (edge lam m) = fun i ↦ lam*centered m i := by
-  funext i; simp [centered, edge]
-
-lemma symEntropy_integral {m : Vec} (hm : PositiveMessage m) :
-    symEntropy m = (1/2)*∫ t in (0:ℝ)..1, avg (fun i ↦ centered m i^2/(1+t*centered m i)) := by
-  have hi (i : Fin 3) : IntervalIntegrable (fun t : ℝ ↦ centered m i^2/(1+t*centered m i)) volume 0 1 :=
-    (continuousOn_const.div (by fun_prop : ContinuousOn (fun t : ℝ ↦ 1+t*centered m i) (Icc 0 1))
-      (fun t ht ↦ ne_of_gt (edge_pos hm ht i))).intervalIntegrable_of_Icc (by norm_num)
-  rw [integral_avg hi]
-  have he (i : Fin 3) : (∫ t in (0:ℝ)..1, centered m i^2/(1+t*centered m i)) =
-      centered m i*Real.log (m i) := by
-    rw [← integral_log_edge hm i, ← intervalIntegral.integral_const_mul]
-    congr 1; funext t; ring
-  simp_rw [he]
-  simp only [symEntropy, centered, avg]
-  ring
-
-lemma symEntropy_nonneg {m : Vec} (hm : PositiveMessage m) : 0 ≤ symEntropy m := by
-  have hi (i : Fin 3) : 0 ≤ (m i-1)*Real.log (m i) := by
-    rcases le_total 1 (m i) with h | h
-    · exact mul_nonneg (sub_nonneg.mpr h) (Real.log_nonneg h)
-    · exact mul_nonneg_of_nonpos_of_nonpos (sub_nonpos.mpr h) (Real.log_nonpos (hm.1 i).le h)
-  dsimp only [symEntropy, avg]
-  exact div_nonneg (div_nonneg (add_nonneg (add_nonneg (hi 0) (hi 1)) (hi 2)) (by norm_num)) (by norm_num)
-
-lemma residual_continuousOn {m : Vec} (hm : PositiveMessage m) {lam : ℝ}
-    (hl : lam ∈ Icc (0:ℝ) 1) : ContinuousOn
-      (fun t : ℝ ↦ t*avg (fun i ↦ centered m i^3/((1+t*centered m i)*(1+t*lam*centered m i)))) (Icc 0 1) := by
-  apply continuousOn_id.mul
-  apply avg_continuousOn
-  intro i
-  apply continuousOn_const.div (by fun_prop)
-  intro t ht
-  have htl : t*lam ∈ Icc (0:ℝ) 1 := ⟨mul_nonneg ht.1 hl.1, by nlinarith [ht.2, mul_le_mul_of_nonneg_left hl.2 ht.1]⟩
-  exact ne_of_gt (mul_pos (edge_pos hm ht i) (edge_pos hm htl i))
-
-lemma symEntropy_residual {m : Vec} (hm : PositiveMessage m) {lam : ℝ}
-    (hl : lam ∈ Icc (0:ℝ) 1) :
-    symEntropy m = momentX (centered m)-lam*momentQ m lam-residual m lam/2 := by
-  let v := centered m
-  have hi : IntervalIntegrable (fun t : ℝ ↦ avg (fun i ↦ v i^2/(1+t*v i))) volume 0 1 := by
-    apply (avg_continuousOn (fun i ↦ continuousOn_const.div (by fun_prop) (fun t ht ↦ ne_of_gt (edge_pos hm ht i)))).intervalIntegrable_of_Icc
-    norm_num
-  have hq : IntervalIntegrable (fun t : ℝ ↦ t^2*fourthEdgeResolvent v lam t) volume 0 1 :=
-    ((by fun_prop : ContinuousOn (fun t : ℝ ↦ t^2) (Icc 0 1)).mul
-      (fourthEdgeResolvent_continuousOn hm hl.1 hl.2)).intervalIntegrable_of_Icc (by norm_num)
-  have hr : IntervalIntegrable (fun t : ℝ ↦ t*avg (fun i ↦ centered m i^3/((1+t*centered m i)*(1+t*lam*centered m i)))) volume 0 1 :=
-    (residual_continuousOn hm hl).intervalIntegrable_of_Icc (by norm_num)
-  have hp (t : ℝ) (ht : t ∈ Icc (0:ℝ) 1) : avg (fun i ↦ v i^2/(1+t*v i)) =
-      2*momentX v-lam*(t^2*fourthEdgeResolvent v lam t)-
-      t*avg (fun i ↦ v i^3/((1+t*v i)*(1+t*lam*v i))) := by
-    have he (i : Fin 3) : v i^2/(1+t*v i) = v i^2-lam*t^2*(v i^4/((1+t*v i)*(1+t*lam*v i)))-
-        t*(v i^3/((1+t*v i)*(1+t*lam*v i))) := by
-      have htlam : t*lam ∈ Icc (0:ℝ) 1 := ⟨mul_nonneg ht.1 hl.1, by nlinarith [ht.2, mul_le_mul_of_nonneg_left hl.2 ht.1]⟩
-      have h₁ : 1+t*v i ≠ 0 := ne_of_gt (edge_pos hm ht i)
-      have h₂ : 1+t*lam*v i ≠ 0 := ne_of_gt (edge_pos hm htlam i)
-      have h₁' : 1+v i*t ≠ 0 := by simpa only [mul_comm] using h₁
-      have h₂' : 1+v i*t*lam ≠ 0 := by convert h₂ using 1; ring
-      field_simp [h₁', h₂']
-      ring
-    simp_rw [he]
-    rw [avg_sub, avg_sub, avg_mul, avg_mul]
-    dsimp only [momentX, fourthEdgeResolvent]
-    ring
-  rw [symEntropy_integral hm]
-  have hconvert := intervalIntegral.integral_congr (μ := volume) (a := (0:ℝ)) (b := 1) (fun t ht ↦ hp t (by simpa using ht))
-  rw [hconvert, intervalIntegral.integral_sub ((intervalIntegrable_const).sub (hq.const_mul lam)) hr,
-    intervalIntegral.integral_sub intervalIntegrable_const (hq.const_mul lam),
-    intervalIntegral.integral_const, intervalIntegral.integral_const_mul]
-  dsimp only [momentQ, residual]
-  ring
-
-lemma powerResolvent_continuousOn {m : Vec} (hm : PositiveMessage m) (n : ℕ)
-    {lam : ℝ} (hl : lam ∈ Icc (0:ℝ) 1) :
-    ContinuousOn (fun t : ℝ ↦ avg (fun i ↦ centered m i^n/(1+t*lam*centered m i))) (Icc 0 1) := by
-  apply avg_continuousOn
-  intro i
-  apply continuousOn_const.div (by fun_prop)
-  intro t ht
-  have htl : t*lam ∈ Icc (0:ℝ) 1 := ⟨mul_nonneg ht.1 hl.1, by nlinarith [ht.2, mul_le_mul_of_nonneg_left hl.2 ht.1]⟩
-  exact ne_of_gt (edge_pos hm htl i)
-
-lemma edge_symEntropy_integral {m : Vec} (hm : PositiveMessage m) {lam : ℝ}
-    (hl : lam ∈ Icc (0:ℝ) 1) :
-    symEntropy (edge lam m) = (lam^2/2)*∫ t in (0:ℝ)..1,
-      avg (fun i ↦ centered m i^2/(1+t*lam*centered m i)) := by
-  rw [symEntropy_integral (edge_message hm hl), centered_edge]
-  have he (t : ℝ) : avg (fun i ↦ (lam*centered m i)^2/(1+t*(lam*centered m i))) =
-      lam^2*avg (fun i ↦ centered m i^2/(1+t*lam*centered m i)) := by
-    dsimp only [avg]; ring
-  simp_rw [he]
-  rw [intervalIntegral.integral_const_mul]
-  ring
-
-lemma symEntropy_edge_residual {m : Vec} (hm : PositiveMessage m) {lam : ℝ}
-    (hl : lam ∈ Icc (0:ℝ) 1) :
-    symEntropy (edge lam m) = lam^2*(symEntropy m+(1-lam)*residual m lam/2) := by
-  let v := centered m
-  have hi : IntervalIntegrable (fun t : ℝ ↦ avg (fun i ↦ v i^2/(1+t*v i))) volume 0 1 := by
-    simpa using (powerResolvent_continuousOn hm 2 (by norm_num : (1:ℝ) ∈ Icc 0 1)).intervalIntegrable_of_Icc (μ := volume) (by norm_num)
-  have hr : IntervalIntegrable (fun t : ℝ ↦ t*avg (fun i ↦ v i^3/((1+t*v i)*(1+t*lam*v i)))) volume 0 1 :=
-    (residual_continuousOn hm hl).intervalIntegrable_of_Icc (by norm_num)
-  have hp (t : ℝ) (ht : t ∈ Icc (0:ℝ) 1) : avg (fun i ↦ v i^2/(1+t*lam*v i)) =
-      avg (fun i ↦ v i^2/(1+t*v i))+(1-lam)*(t*avg (fun i ↦ v i^3/((1+t*v i)*(1+t*lam*v i)))) := by
-    have he (i : Fin 3) : v i^2/(1+t*lam*v i) = v i^2/(1+t*v i)+
-        (1-lam)*t*(v i^3/((1+t*v i)*(1+t*lam*v i))) := by
-      have htlam : t*lam ∈ Icc (0:ℝ) 1 := ⟨mul_nonneg ht.1 hl.1, by nlinarith [ht.2, mul_le_mul_of_nonneg_left hl.2 ht.1]⟩
-      have h₁ : 1+v i*t ≠ 0 := by simpa only [mul_comm] using ne_of_gt (edge_pos hm ht i)
-      have h₂ : 1+v i*t*lam ≠ 0 := by convert ne_of_gt (edge_pos hm htlam i) using 1; ring
-      field_simp [h₁, h₂]
-      ring
-    simp_rw [he]
-    rw [avg_add, avg_mul]
-    ring
-  rw [edge_symEntropy_integral hm hl, symEntropy_integral hm]
-  have he := intervalIntegral.integral_congr (μ := volume) (a := (0:ℝ)) (b := 1) (fun t ht ↦ hp t (by simpa using ht))
-  rw [he, intervalIntegral.integral_add hi (hr.const_mul (1-lam)), intervalIntegral.integral_const_mul]
-  dsimp only [residual]
-  ring
-
-lemma momentA_identity {m : Vec} (hm : PositiveMessage m) :
-    momentA m = momentY (centered m)-2*momentX (centered m)+2*symEntropy m := by
-  let v := centered m
-  have hi : IntervalIntegrable (fun t : ℝ ↦ avg (fun i ↦ v i^2/(1+t*v i))) volume 0 1 := by
-    simpa using (powerResolvent_continuousOn hm 2 (by norm_num : (1:ℝ) ∈ Icc 0 1)).intervalIntegrable_of_Icc (μ := volume) (by norm_num)
-  have ha : IntervalIntegrable (fun t : ℝ ↦ t^2*fourthResolvent v t) volume 0 1 :=
-    ((by fun_prop : ContinuousOn (fun t : ℝ ↦ t^2) (Icc 0 1)).mul
-      (fourthResolvent_continuousOn hm)).intervalIntegrable_of_Icc (by norm_num)
-  have hp (t : ℝ) (ht : t ∈ Icc (0:ℝ) 1) : avg (fun i ↦ v i^2/(1+t*v i)) =
-      2*momentX v-t*(2*momentY v)+t^2*fourthResolvent v t := by
-    have he (i : Fin 3) : v i^2/(1+t*v i) = v i^2-t*v i^3+t^2*(v i^4/(1+t*v i)) := by
-      have h₁ : 1+v i*t ≠ 0 := by simpa only [mul_comm] using ne_of_gt (edge_pos hm ht i)
-      field_simp [h₁]
-      ring
-    simp_rw [he]
-    rw [avg_add, avg_sub, avg_mul, avg_mul]
-    dsimp only [momentX, momentY, fourthResolvent]
-    ring
-  have he := intervalIntegral.integral_congr (μ := volume) (a := (0:ℝ)) (b := 1) (fun t ht ↦ hp t (by simpa using ht))
-  rw [symEntropy_integral hm, he]
-  rw [intervalIntegral.integral_add (by apply Continuous.intervalIntegrable; fun_prop) ha,
-    intervalIntegral.integral_sub intervalIntegrable_const (by apply Continuous.intervalIntegrable; fun_prop),
-    intervalIntegral.integral_const, intervalIntegral.integral_mul_const, integral_id]
-  dsimp only [momentA]
-  ring
-
-lemma momentY_residual {m : Vec} (hm : PositiveMessage m) {lam : ℝ}
-    (hl : lam ∈ Icc (0:ℝ) 1) :
-    momentY (centered m) = momentA m+2*lam*momentQ m lam+residual m lam := by
-  rw [momentA_identity hm, symEntropy_residual hm hl]
-  ring
-
-end ThreeState.TreeClauses.Positive
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeStateClauses.RadialBounds
+
+namespace OAI
+
+/-! Positive Moments. -/
+
+open scoped BigOperators ENNReal NNReal Topology
+open Filter
+noncomputable section
+open Set MeasureTheory
+namespace ThreeState.TreeClauses.Positive
+open Radial
+
+def fourthResolvent (v : Vec) (t : ℝ) : ℝ := avg (fun i ↦ v i^4/(1+t*v i))
+
+def fourthEdgeResolvent (v : Vec) (lam t : ℝ) : ℝ :=
+  avg (fun i ↦ v i^4/((1+t*v i)*(1+t*lam*v i)))
+
+def momentA (m : Vec) : ℝ := ∫ t in (0:ℝ)..1, t^2*fourthResolvent (centered m) t
+
+def momentQ (m : Vec) (lam : ℝ) : ℝ :=
+  (1/2)*∫ t in (0:ℝ)..1, t^2*fourthEdgeResolvent (centered m) lam t
+
+lemma fourthMoment {v : Vec} (hv : avg v = 0) :
+    avg (fun i ↦ v i^4) = 6*momentX v^2 := by
+  simp only [momentX, avg]; rw [third_eq hv]; ring
+
+lemma fourthResolvent_eq {v : Vec} (hv : avg v = 0) {t : ℝ}
+    (hp : ∀ i, 1+t*v i ≠ 0) (hD : denom v t ≠ 0) :
+    fourthResolvent v t =
+      (6*momentX v^2-10*t*momentX v*momentY v+4*t^2*momentY v^2)/denom v t := by
+  have he (i : Fin 3) : v i^4/(1+t*v i) =
+      (v i^3*(v i-t*v i^2+2*t^2*momentY v))/denom v t := by
+    calc
+      _ = v i^3*(v i/(1+t*v i)) := by ring
+      _ = _ := by rw [rational_identity hv t i (hp i) hD]; ring
+  simp only [fourthResolvent, he, avg_div]
+  congr 1
+  simp only [momentX, momentY, avg]
+  rw [third_eq hv]; ring
+
+lemma fraction_lower {z k : ℝ} (hz₀ : 0 ≤ z) (hz₁ : z < 1)
+    (_hk₀ : -1 ≤ k) (hk₁ : k ≤ 1) (hD : 0 < 1-3*z^2+2*z^3*k) :
+    4+(64/9)*(z-1/2)^2 ≤ (6-10*z*k+4*z^2*k^2)/(1-3*z^2+2*z^3*k) := by
+  have hB : 0 < 1+z-2*z^2 := by nlinarith [mul_pos (by linarith : 0 < 1-z) (by linarith : 0 < 1+2*z)]
+  have hBtop : 1+z-2*z^2 ≤ 9/8 := by nlinarith [sq_nonneg (z-1/4)]
+  have hnum : 6-10*z+4*z^2 ≤ 6-10*z*k+4*z^2*k^2 := by
+    have hfac : 0 ≤ 10-4*z*(k+1) := by nlinarith [mul_nonneg hz₀ (by linarith : 0 ≤ 1-k)]
+    have := mul_nonneg (mul_nonneg hz₀ (by linarith : 0 ≤ 1-k)) hfac
+    nlinarith
+  have hnum0 : 0 ≤ 6-10*z+4*z^2 := by
+    nlinarith [mul_nonneg (by linarith : 0 ≤ 1-z) (by linarith : 0 ≤ 6-4*z)]
+  have hden : 1-3*z^2+2*z^3*k ≤ 1-3*z^2+2*z^3 := by
+    nlinarith [mul_nonneg (pow_nonneg hz₀ 3) (by linarith : 0 ≤ 1-k)]
+  have hdenpos : 0 < 1-3*z^2+2*z^3 := lt_of_lt_of_le hD hden
+  have heq : (6-10*z+4*z^2)/(1-3*z^2+2*z^3) = (6-4*z)/(1+z-2*z^2) := by
+    apply (div_eq_div_iff (ne_of_gt hdenpos) (ne_of_gt hB)).2
+    ring
+  have hbase : 4+(64/9)*(z-1/2)^2 ≤ (6-4*z)/(1+z-2*z^2) := by
+    apply (le_div_iff₀ hB).2
+    have := mul_nonneg (sq_nonneg (z-1/2)) (sub_nonneg.mpr hBtop)
+    nlinarith
+  calc
+    _ ≤ (6-4*z)/(1+z-2*z^2) := hbase
+    _ = (6-10*z+4*z^2)/(1-3*z^2+2*z^3) := heq.symm
+    _ ≤ (6-10*z+4*z^2)/(1-3*z^2+2*z^3*k) :=
+      div_le_div_of_nonneg_left hnum0 hD hden
+    _ ≤ _ := (div_le_div_iff_of_pos_right hD).2 hnum
+
+lemma fourthResolvent_nonneg {m : Vec} (hm : PositiveMessage m)
+    {t : ℝ} (ht : t ∈ Icc (0:ℝ) 1) : 0 ≤ fourthResolvent (centered m) t := by
+  have hi (i : Fin 3) : 0 ≤ centered m i^4/(1+t*centered m i) :=
+    div_nonneg (by positivity) (edge_pos hm ht i).le
+  unfold fourthResolvent avg
+  exact div_nonneg (add_nonneg (add_nonneg (hi 0) (hi 1)) (hi 2)) (by norm_num)
+
+lemma fourthResolvent_lower {m : Vec} (hm : PositiveMessage m) {t : ℝ}
+    (ht : t ∈ Icc (0:ℝ) 1) :
+    (4+(64/9)*(t*Real.sqrt (momentX (centered m))-1/2)^2)*momentX (centered m)^2 ≤
+      fourthResolvent (centered m) t := by
+  let v := centered m
+  let x := momentX v
+  let y := momentY v
+  have hx₀ : 0 ≤ x := momentX_nonneg v
+  have hx₁ : x < 1 := momentX_lt_one hm
+  by_cases hx : x = 0
+  · have hn := fourthResolvent_nonneg hm ht
+    change (4+(64/9)*(t*Real.sqrt x-1/2)^2)*x^2 ≤ fourthResolvent v t
+    simpa [hx] using hn
+  have hxpos : 0 < x := lt_of_le_of_ne hx₀ (Ne.symm hx)
+  let r := Real.sqrt x
+  have hrpos : 0 < r := Real.sqrt_pos.mpr hxpos
+  have hr2 : r^2 = x := Real.sq_sqrt hx₀
+  have hr₁ : r < 1 := by nlinarith
+  let k := y/r^3
+  have hyr : y = k*r^3 := (div_mul_cancel₀ _ (pow_ne_zero 3 (ne_of_gt hrpos))).symm
+  have hk2 : k^2 ≤ 1 := by
+    have hxy : y^2 ≤ x^3 := moment_discriminant (avg_centered hm)
+    have heq : k^2 = y^2/r^6 := by dsimp [k]; ring
+    rw [heq, div_le_iff₀ (pow_pos hrpos 6)]
+    rw [← hr2] at hxy
+    nlinarith [hxy]
+  have hk₀ : -1 ≤ k := by nlinarith [sq_nonneg (k+1)]
+  have hk₁ : k ≤ 1 := by nlinarith [sq_nonneg (k-1)]
+  have hz₀ : 0 ≤ t*r := mul_nonneg ht.1 hrpos.le
+  have hz₁ : t*r < 1 := by nlinarith [ht.2]
+  have hD := denom_pos hm ht
+  have hDeq : denom v t = 1-3*(t*r)^2+2*(t*r)^3*k := by
+    dsimp only [denom]
+    change 1-3*t^2*x+2*t^3*y = _
+    rw [← hr2, hyr]; ring
+  have hD' : 0 < 1-3*(t*r)^2+2*(t*r)^3*k := by rw [← hDeq]; exact hD
+  have heq : fourthResolvent v t = x^2*((6-10*(t*r)*k+4*(t*r)^2*k^2)/(1-3*(t*r)^2+2*(t*r)^3*k)) := by
+    rw [fourthResolvent_eq (avg_centered hm) (fun i ↦ ne_of_gt (edge_pos hm ht i)) (ne_of_gt hD), hDeq]
+    change (6*x^2-10*t*x*y+4*t^2*y^2)/_ = _
+    rw [← hr2, hyr]; ring
+  change (4+(64/9)*(t*r-1/2)^2)*x^2 ≤ fourthResolvent v t
+  rw [heq, mul_comm (x^2)]
+  exact mul_le_mul_of_nonneg_right (fraction_lower hz₀ hz₁ hk₀ hk₁ hD') (sq_nonneg x)
+
+lemma fourthResolvent_continuousOn {m : Vec} (hm : PositiveMessage m) :
+    ContinuousOn (fourthResolvent (centered m)) (Icc (0:ℝ) 1) := by
+  have hi (i : Fin 3) : ContinuousOn (fun t : ℝ ↦ centered m i^4/(1+t*centered m i)) (Icc 0 1) :=
+    continuousOn_const.div (by fun_prop) (fun t ht ↦ ne_of_gt (edge_pos hm ht i))
+  exact (((hi 0).add (hi 1)).add (hi 2)).div_const 3
+
+lemma lower_integral (r x : ℝ) :
+    (∫ t in (0:ℝ)..1, t^2*((4+(64/9)*(t*r-1/2)^2)*x^2)) =
+      (37/27+(64/45)*(r-5/8)^2)*x^2 := by
+  conv_lhs => arg 1; ext t; ring_nf
+  simp (disch := apply Continuous.intervalIntegrable; fun_prop) only
+    [intervalIntegral.integral_add, intervalIntegral.integral_mul_const, integral_pow]
+  norm_num
+  ring
+
+theorem momentA_lower {m : Vec} (hm : PositiveMessage m) :
+    (37/27)*momentX (centered m)^2 ≤ momentA m := by
+  have hcr : ContinuousOn (fun t : ℝ ↦ t^2*fourthResolvent (centered m) t) (Icc 0 1) :=
+    (by fun_prop : ContinuousOn (fun t : ℝ ↦ t^2) (Icc 0 1)).mul (fourthResolvent_continuousOn hm)
+  have hcl : Continuous (fun t : ℝ ↦ t^2*((4+(64/9)*(t*Real.sqrt (momentX (centered m))-1/2)^2)*momentX (centered m)^2)) := by fun_prop
+  have hi := intervalIntegral.integral_mono_on (μ := volume) (by norm_num : (0:ℝ) ≤ 1)
+    (hcl.intervalIntegrable 0 1) (hcr.intervalIntegrable_of_Icc (by norm_num))
+    (fun t ht ↦ mul_le_mul_of_nonneg_left (fourthResolvent_lower hm ht) (sq_nonneg t))
+  rw [lower_integral] at hi
+  have hn : 0 ≤ (Real.sqrt (momentX (centered m))-5/8)^2*momentX (centered m)^2 := by positivity
+  dsimp only [momentA]
+  linarith
+
+lemma avg_mono {a b : Vec} (h : ∀ i, a i ≤ b i) : avg a ≤ avg b := by
+  unfold avg
+  exact div_le_div_of_nonneg_right (add_le_add (add_le_add (h 0) (h 1)) (h 2)) (by norm_num)
+
+lemma fourthResolvent_four {m : Vec} (hm : PositiveMessage m) {t : ℝ}
+    (ht : t ∈ Icc (0:ℝ) 1) : 4*momentX (centered m)^2 ≤ fourthResolvent (centered m) t := by
+  have h := fourthResolvent_lower hm ht
+  nlinarith [mul_nonneg (sq_nonneg (t*Real.sqrt (momentX (centered m))-1/2))
+    (sq_nonneg (momentX (centered m)))]
+
+lemma fourthEdgeResolvent_lower {m : Vec} (hm : PositiveMessage m) {lam t : ℝ}
+    (hl₀ : 0 ≤ lam) (hl₁ : lam ≤ 1) (ht : t ∈ Icc (0:ℝ) 1) :
+    (2/(2+lam))*fourthResolvent (centered m) t ≤ fourthEdgeResolvent (centered m) lam t := by
+  let v := centered m
+  let a : Vec := fun i ↦ v i^4/(1+t*v i)
+  let b : Vec := fun i ↦ 1+t*lam*v i
+  let c : ℝ := 1+lam/2
+  have hc : 0 < c := by dsimp [c]; linarith
+  have htl : t*lam ∈ Icc (0:ℝ) 1 := ⟨mul_nonneg ht.1 hl₀, by nlinarith [ht.2, mul_le_mul_of_nonneg_left hl₁ ht.1]⟩
+  have hb (i : Fin 3) : 0 < b i := by dsimp [b]; exact edge_pos hm htl i
+  have ha (i : Fin 3) : 0 ≤ a i := div_nonneg (by positivity) (edge_pos hm ht i).le
+  have hab (i : Fin 3) : a i*b i = (1-lam)*a i+lam*v i^4 := by
+    dsimp only [a, b]
+    have hd : 1+t*v i ≠ 0 := ne_of_gt (edge_pos hm ht i)
+    calc
+      _ = (v i^4/(1+t*v i))*((1-lam)+lam*(1+t*v i)) := by ring
+      _ = (1-lam)*(v i^4/(1+t*v i))+lam*((v i^4/(1+t*v i))*(1+t*v i)) := by ring
+      _ = _ := by rw [div_mul_cancel₀ _ hd]
+  have habavg : avg (fun i ↦ a i*b i) = (1-lam)*fourthResolvent v t+6*lam*momentX v^2 := by
+    simp_rw [hab]
+    rw [avg_add, avg_mul, avg_mul, fourthMoment (avg_centered hm)]
+    change (1-lam)*fourthResolvent v t+lam*(6*momentX v^2) = _
+    ring
+  have havga : avg a = fourthResolvent v t := rfl
+  have havgab : avg (fun i ↦ a i/b i) = fourthEdgeResolvent v lam t := by
+    simp only [a, b, fourthEdgeResolvent, div_div]
+  have htangent (i : Fin 3) : a i*(2*c-b i) ≤ c^2*(a i/b i) := by
+    rw [← mul_div_assoc]
+    apply (le_div_iff₀ (hb i)).2
+    nlinarith [mul_nonneg (ha i) (sq_nonneg (b i-c))]
+  have hs : 2*c*fourthResolvent v t-((1-lam)*fourthResolvent v t+6*lam*momentX v^2) ≤
+      c^2*fourthEdgeResolvent v lam t := by
+    calc
+      _ = avg (fun i ↦ a i*(2*c-b i)) := by
+        rw [← habavg, ← havga]
+        simp only [avg]; ring
+      _ ≤ avg (fun i ↦ c^2*(a i/b i)) := avg_mono htangent
+      _ = _ := by rw [avg_mul, havgab]
+  have hfour := fourthResolvent_four hm ht
+  have hh : 0 ≤ lam*((3/2)*fourthResolvent v t-6*momentX v^2) :=
+    mul_nonneg hl₀ (by linarith)
+  have hd : 0 ≤ c*(c*fourthEdgeResolvent v lam t-fourthResolvent v t) := by
+    dsimp only [c] at hs ⊢
+    nlinarith
+  have hh' := (mul_nonneg_iff_of_pos_left hc).mp hd
+  have hfinal : fourthResolvent v t/c ≤ fourthEdgeResolvent v lam t := by
+    apply (div_le_iff₀ hc).2
+    nlinarith
+  have heq : (2/(2+lam))*fourthResolvent v t = fourthResolvent v t/c := by
+    dsimp [c]
+    field_simp
+  rw [heq]
+  exact hfinal
+
+lemma fourthEdgeResolvent_continuousOn {m : Vec} (hm : PositiveMessage m) {lam : ℝ}
+    (hl₀ : 0 ≤ lam) (hl₁ : lam ≤ 1) :
+    ContinuousOn (fourthEdgeResolvent (centered m) lam) (Icc (0:ℝ) 1) := by
+  have hi (i : Fin 3) : ContinuousOn
+      (fun t : ℝ ↦ centered m i^4/((1+t*centered m i)*(1+t*lam*centered m i))) (Icc 0 1) := by
+    apply continuousOn_const.div (by fun_prop)
+    intro t ht
+    have htl : t*lam ∈ Icc (0:ℝ) 1 := ⟨mul_nonneg ht.1 hl₀, by nlinarith [ht.2, mul_le_mul_of_nonneg_left hl₁ ht.1]⟩
+    exact ne_of_gt (mul_pos (edge_pos hm ht i) (edge_pos hm htl i))
+  exact (((hi 0).add (hi 1)).add (hi 2)).div_const 3
+
+theorem momentQ_lower {m : Vec} (hm : PositiveMessage m) {lam : ℝ}
+    (hl₀ : 0 ≤ lam) (hl₁ : lam ≤ 1) : momentA m/(2+lam) ≤ momentQ m lam := by
+  have hcr : ContinuousOn (fun t : ℝ ↦ t^2*fourthEdgeResolvent (centered m) lam t) (Icc 0 1) :=
+    (by fun_prop : ContinuousOn (fun t : ℝ ↦ t^2) (Icc 0 1)).mul (fourthEdgeResolvent_continuousOn hm hl₀ hl₁)
+  have hcl : ContinuousOn (fun t : ℝ ↦ t^2*((2/(2+lam))*fourthResolvent (centered m) t)) (Icc 0 1) :=
+    (by fun_prop : ContinuousOn (fun t : ℝ ↦ t^2) (Icc 0 1)).mul
+      (continuousOn_const.mul (fourthResolvent_continuousOn hm))
+  have hi := intervalIntegral.integral_mono_on (μ := volume) (by norm_num : (0:ℝ) ≤ 1)
+    (hcl.intervalIntegrable_of_Icc (by norm_num)) (hcr.intervalIntegrable_of_Icc (by norm_num))
+    (fun t ht ↦ mul_le_mul_of_nonneg_left (fourthEdgeResolvent_lower hm hl₀ hl₁ ht) (sq_nonneg t))
+  have he : (∫ t in (0:ℝ)..1, t^2*((2/(2+lam))*fourthResolvent (centered m) t)) =
+      (2/(2+lam))*momentA m := by
+    rw [momentA, ← intervalIntegral.integral_const_mul]
+    congr 1; funext t; ring
+  rw [he] at hi
+  have heq : (2/(2+lam))*momentA m = 2*(momentA m/(2+lam)) := by ring
+  rw [heq] at hi
+  dsimp only [momentQ]
+  linarith
+
+end ThreeState.TreeClauses.Positive
+
+end 
+
+noncomputable section
+open Set MeasureTheory
+namespace ThreeState.TreeClauses.Positive
+open Radial
+
+def residual (m : Vec) (lam : ℝ) : ℝ :=
+  ∫ t in (0:ℝ)..1, t*avg (fun i ↦ centered m i^3 /
+    ((1+t*centered m i)*(1+t*lam*centered m i)))
+
+lemma avg_integrable {f : ℝ → Vec} (h : ∀ i, IntervalIntegrable (fun t ↦ f t i) volume 0 1) :
+    IntervalIntegrable (fun t ↦ avg (f t)) volume 0 1 :=
+  (((h 0).add (h 1)).add (h 2)).div_const 3
+
+lemma avg_continuousOn {f : ℝ → Vec} {S : Set ℝ} (h : ∀ i, ContinuousOn (fun t ↦ f t i) S) :
+    ContinuousOn (fun t ↦ avg (f t)) S :=
+  (((h 0).add (h 1)).add (h 2)).div_const 3
+
+lemma edge_message {m : Vec} (hm : PositiveMessage m) {lam : ℝ} (hl : lam ∈ Icc (0:ℝ) 1) :
+    PositiveMessage (edge lam m) := by
+  refine ⟨fun i ↦ edge_pos hm hl i, ?_⟩
+  change avg (fun i ↦ 1+lam*centered m i) = 1
+  rw [avg_add, avg_const, avg_mul]
+  rw [avg_centered hm]; ring
+
+lemma centered_edge (m : Vec) (lam : ℝ) : centered (edge lam m) = fun i ↦ lam*centered m i := by
+  funext i; simp [centered, edge]
+
+lemma symEntropy_integral {m : Vec} (hm : PositiveMessage m) :
+    symEntropy m = (1/2)*∫ t in (0:ℝ)..1, avg (fun i ↦ centered m i^2/(1+t*centered m i)) := by
+  have hi (i : Fin 3) : IntervalIntegrable (fun t : ℝ ↦ centered m i^2/(1+t*centered m i)) volume 0 1 :=
+    (continuousOn_const.div (by fun_prop : ContinuousOn (fun t : ℝ ↦ 1+t*centered m i) (Icc 0 1))
+      (fun t ht ↦ ne_of_gt (edge_pos hm ht i))).intervalIntegrable_of_Icc (by norm_num)
+  rw [integral_avg hi]
+  have he (i : Fin 3) : (∫ t in (0:ℝ)..1, centered m i^2/(1+t*centered m i)) =
+      centered m i*Real.log (m i) := by
+    rw [← integral_log_edge hm i, ← intervalIntegral.integral_const_mul]
+    congr 1; funext t; ring
+  simp_rw [he]
+  simp only [symEntropy, centered, avg]
+  ring
+
+lemma symEntropy_nonneg {m : Vec} (hm : PositiveMessage m) : 0 ≤ symEntropy m := by
+  have hi (i : Fin 3) : 0 ≤ (m i-1)*Real.log (m i) := by
+    rcases le_total 1 (m i) with h | h
+    · exact mul_nonneg (sub_nonneg.mpr h) (Real.log_nonneg h)
+    · exact mul_nonneg_of_nonpos_of_nonpos (sub_nonpos.mpr h) (Real.log_nonpos (hm.1 i).le h)
+  dsimp only [symEntropy, avg]
+  exact div_nonneg (div_nonneg (add_nonneg (add_nonneg (hi 0) (hi 1)) (hi 2)) (by norm_num)) (by norm_num)
+
+lemma residual_continuousOn {m : Vec} (hm : PositiveMessage m) {lam : ℝ}
+    (hl : lam ∈ Icc (0:ℝ) 1) : ContinuousOn
+      (fun t : ℝ ↦ t*avg (fun i ↦ centered m i^3/((1+t*centered m i)*(1+t*lam*centered m i)))) (Icc 0 1) := by
+  apply continuousOn_id.mul
+  apply avg_continuousOn
+  intro i
+  apply continuousOn_const.div (by fun_prop)
+  intro t ht
+  have htl : t*lam ∈ Icc (0:ℝ) 1 := ⟨mul_nonneg ht.1 hl.1, by nlinarith [ht.2, mul_le_mul_of_nonneg_left hl.2 ht.1]⟩
+  exact ne_of_gt (mul_pos (edge_pos hm ht i) (edge_pos hm htl i))
+
+lemma symEntropy_residual {m : Vec} (hm : PositiveMessage m) {lam : ℝ}
+    (hl : lam ∈ Icc (0:ℝ) 1) :
+    symEntropy m = momentX (centered m)-lam*momentQ m lam-residual m lam/2 := by
+  let v := centered m
+  have hi : IntervalIntegrable (fun t : ℝ ↦ avg (fun i ↦ v i^2/(1+t*v i))) volume 0 1 := by
+    apply (avg_continuousOn (fun i ↦ continuousOn_const.div (by fun_prop) (fun t ht ↦ ne_of_gt (edge_pos hm ht i)))).intervalIntegrable_of_Icc
+    norm_num
+  have hq : IntervalIntegrable (fun t : ℝ ↦ t^2*fourthEdgeResolvent v lam t) volume 0 1 :=
+    ((by fun_prop : ContinuousOn (fun t : ℝ ↦ t^2) (Icc 0 1)).mul
+      (fourthEdgeResolvent_continuousOn hm hl.1 hl.2)).intervalIntegrable_of_Icc (by norm_num)
+  have hr : IntervalIntegrable (fun t : ℝ ↦ t*avg (fun i ↦ centered m i^3/((1+t*centered m i)*(1+t*lam*centered m i)))) volume 0 1 :=
+    (residual_continuousOn hm hl).intervalIntegrable_of_Icc (by norm_num)
+  have hp (t : ℝ) (ht : t ∈ Icc (0:ℝ) 1) : avg (fun i ↦ v i^2/(1+t*v i)) =
+      2*momentX v-lam*(t^2*fourthEdgeResolvent v lam t)-
+      t*avg (fun i ↦ v i^3/((1+t*v i)*(1+t*lam*v i))) := by
+    have he (i : Fin 3) : v i^2/(1+t*v i) = v i^2-lam*t^2*(v i^4/((1+t*v i)*(1+t*lam*v i)))-
+        t*(v i^3/((1+t*v i)*(1+t*lam*v i))) := by
+      have htlam : t*lam ∈ Icc (0:ℝ) 1 := ⟨mul_nonneg ht.1 hl.1, by nlinarith [ht.2, mul_le_mul_of_nonneg_left hl.2 ht.1]⟩
+      have h₁ : 1+t*v i ≠ 0 := ne_of_gt (edge_pos hm ht i)
+      have h₂ : 1+t*lam*v i ≠ 0 := ne_of_gt (edge_pos hm htlam i)
+      have h₁' : 1+v i*t ≠ 0 := by simpa only [mul_comm] using h₁
+      have h₂' : 1+v i*t*lam ≠ 0 := by convert h₂ using 1; ring
+      field_simp [h₁', h₂']
+      ring
+    simp_rw [he]
+    rw [avg_sub, avg_sub, avg_mul, avg_mul]
+    dsimp only [momentX, fourthEdgeResolvent]
+    ring
+  rw [symEntropy_integral hm]
+  have hconvert := intervalIntegral.integral_congr (μ := volume) (a := (0:ℝ)) (b := 1) (fun t ht ↦ hp t (by simpa using ht))
+  rw [hconvert, intervalIntegral.integral_sub ((intervalIntegrable_const).sub (hq.const_mul lam)) hr,
+    intervalIntegral.integral_sub intervalIntegrable_const (hq.const_mul lam),
+    intervalIntegral.integral_const, intervalIntegral.integral_const_mul]
+  dsimp only [momentQ, residual]
+  ring
+
+lemma powerResolvent_continuousOn {m : Vec} (hm : PositiveMessage m) (n : ℕ)
+    {lam : ℝ} (hl : lam ∈ Icc (0:ℝ) 1) :
+    ContinuousOn (fun t : ℝ ↦ avg (fun i ↦ centered m i^n/(1+t*lam*centered m i))) (Icc 0 1) := by
+  apply avg_continuousOn
+  intro i
+  apply continuousOn_const.div (by fun_prop)
+  intro t ht
+  have htl : t*lam ∈ Icc (0:ℝ) 1 := ⟨mul_nonneg ht.1 hl.1, by nlinarith [ht.2, mul_le_mul_of_nonneg_left hl.2 ht.1]⟩
+  exact ne_of_gt (edge_pos hm htl i)
+
+lemma edge_symEntropy_integral {m : Vec} (hm : PositiveMessage m) {lam : ℝ}
+    (hl : lam ∈ Icc (0:ℝ) 1) :
+    symEntropy (edge lam m) = (lam^2/2)*∫ t in (0:ℝ)..1,
+      avg (fun i ↦ centered m i^2/(1+t*lam*centered m i)) := by
+  rw [symEntropy_integral (edge_message hm hl), centered_edge]
+  have he (t : ℝ) : avg (fun i ↦ (lam*centered m i)^2/(1+t*(lam*centered m i))) =
+      lam^2*avg (fun i ↦ centered m i^2/(1+t*lam*centered m i)) := by
+    dsimp only [avg]; ring
+  simp_rw [he]
+  rw [intervalIntegral.integral_const_mul]
+  ring
+
+lemma symEntropy_edge_residual {m : Vec} (hm : PositiveMessage m) {lam : ℝ}
+    (hl : lam ∈ Icc (0:ℝ) 1) :
+    symEntropy (edge lam m) = lam^2*(symEntropy m+(1-lam)*residual m lam/2) := by
+  let v := centered m
+  have hi : IntervalIntegrable (fun t : ℝ ↦ avg (fun i ↦ v i^2/(1+t*v i))) volume 0 1 := by
+    simpa using (powerResolvent_continuousOn hm 2 (by norm_num : (1:ℝ) ∈ Icc 0 1)).intervalIntegrable_of_Icc (μ := volume) (by norm_num)
+  have hr : IntervalIntegrable (fun t : ℝ ↦ t*avg (fun i ↦ v i^3/((1+t*v i)*(1+t*lam*v i)))) volume 0 1 :=
+    (residual_continuousOn hm hl).intervalIntegrable_of_Icc (by norm_num)
+  have hp (t : ℝ) (ht : t ∈ Icc (0:ℝ) 1) : avg (fun i ↦ v i^2/(1+t*lam*v i)) =
+      avg (fun i ↦ v i^2/(1+t*v i))+(1-lam)*(t*avg (fun i ↦ v i^3/((1+t*v i)*(1+t*lam*v i)))) := by
+    have he (i : Fin 3) : v i^2/(1+t*lam*v i) = v i^2/(1+t*v i)+
+        (1-lam)*t*(v i^3/((1+t*v i)*(1+t*lam*v i))) := by
+      have htlam : t*lam ∈ Icc (0:ℝ) 1 := ⟨mul_nonneg ht.1 hl.1, by nlinarith [ht.2, mul_le_mul_of_nonneg_left hl.2 ht.1]⟩
+      have h₁ : 1+v i*t ≠ 0 := by simpa only [mul_comm] using ne_of_gt (edge_pos hm ht i)
+      have h₂ : 1+v i*t*lam ≠ 0 := by convert ne_of_gt (edge_pos hm htlam i) using 1; ring
+      field_simp [h₁, h₂]
+      ring
+    simp_rw [he]
+    rw [avg_add, avg_mul]
+    ring
+  rw [edge_symEntropy_integral hm hl, symEntropy_integral hm]
+  have he := intervalIntegral.integral_congr (μ := volume) (a := (0:ℝ)) (b := 1) (fun t ht ↦ hp t (by simpa using ht))
+  rw [he, intervalIntegral.integral_add hi (hr.const_mul (1-lam)), intervalIntegral.integral_const_mul]
+  dsimp only [residual]
+  ring
+
+lemma momentA_identity {m : Vec} (hm : PositiveMessage m) :
+    momentA m = momentY (centered m)-2*momentX (centered m)+2*symEntropy m := by
+  let v := centered m
+  have hi : IntervalIntegrable (fun t : ℝ ↦ avg (fun i ↦ v i^2/(1+t*v i))) volume 0 1 := by
+    simpa using (powerResolvent_continuousOn hm 2 (by norm_num : (1:ℝ) ∈ Icc 0 1)).intervalIntegrable_of_Icc (μ := volume) (by norm_num)
+  have ha : IntervalIntegrable (fun t : ℝ ↦ t^2*fourthResolvent v t) volume 0 1 :=
+    ((by fun_prop : ContinuousOn (fun t : ℝ ↦ t^2) (Icc 0 1)).mul
+      (fourthResolvent_continuousOn hm)).intervalIntegrable_of_Icc (by norm_num)
+  have hp (t : ℝ) (ht : t ∈ Icc (0:ℝ) 1) : avg (fun i ↦ v i^2/(1+t*v i)) =
+      2*momentX v-t*(2*momentY v)+t^2*fourthResolvent v t := by
+    have he (i : Fin 3) : v i^2/(1+t*v i) = v i^2-t*v i^3+t^2*(v i^4/(1+t*v i)) := by
+      have h₁ : 1+v i*t ≠ 0 := by simpa only [mul_comm] using ne_of_gt (edge_pos hm ht i)
+      field_simp [h₁]
+      ring
+    simp_rw [he]
+    rw [avg_add, avg_sub, avg_mul, avg_mul]
+    dsimp only [momentX, momentY, fourthResolvent]
+    ring
+  have he := intervalIntegral.integral_congr (μ := volume) (a := (0:ℝ)) (b := 1) (fun t ht ↦ hp t (by simpa using ht))
+  rw [symEntropy_integral hm, he]
+  rw [intervalIntegral.integral_add (by apply Continuous.intervalIntegrable; fun_prop) ha,
+    intervalIntegral.integral_sub intervalIntegrable_const (by apply Continuous.intervalIntegrable; fun_prop),
+    intervalIntegral.integral_const, intervalIntegral.integral_mul_const, integral_id]
+  dsimp only [momentA]
+  ring
+
+lemma momentY_residual {m : Vec} (hm : PositiveMessage m) {lam : ℝ}
+    (hl : lam ∈ Icc (0:ℝ) 1) :
+    momentY (centered m) = momentA m+2*lam*momentQ m lam+residual m lam := by
+  rw [momentA_identity hm, symEntropy_residual hm hl]
+  ring
+
+end ThreeState.TreeClauses.Positive
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeStateClauses/PosteriorLaw.lean b/lean/OAI/Probability/ThreeStateClauses/PosteriorLaw.lean
index 0dd10f7adf9cf64ab038068c0542cc0a90312358..d68b235ffbd57e233b72f90c39efe091ebbfe11d 100644
--- a/lean/OAI/Probability/ThreeStateClauses/PosteriorLaw.lean
+++ b/lean/OAI/Probability/ThreeStateClauses/PosteriorLaw.lean
@@ -1,292 +1,292 @@
-import Mathlib
-import OAI.Probability.ThreeStateClauses.PositiveMoments
-
-namespace OAI
-
-/-! Posterior Law. -/
-
-open scoped BigOperators ENNReal NNReal Topology
-open Filter
-noncomputable section
-open Set MeasureTheory
-namespace ThreeState.TreeClauses.Experiment
-open Radial ThreeState.TreeClauses.Positive
-
-def messageSet : Set Vec := {m | (∀ i, 0 ≤ m i) ∧ avg m = 1}
-abbrev Message := messageSet
-
-lemma coordinate_nonneg (m : Message) (i : Fin 3) : 0 ≤ m.1 i := m.2.1 i
-lemma coordinate_le_three (m : Message) (i : Fin 3) : m.1 i ≤ 3 := by
-  have hs := m.2.2
-  dsimp only [avg] at hs
-  have h₀ := m.2.1 0
-  have h₁ := m.2.1 1
-  have h₂ := m.2.1 2
-  fin_cases i <;> dsimp at * <;> linarith
-
-lemma continuous_avg : Continuous (avg : Vec → ℝ) :=
-  (((continuous_apply 0).add (continuous_apply 1)).add (continuous_apply 2)).div_const 3
-
-lemma isClosed_messageSet : IsClosed messageSet := by
-  have h : IsClosed {m : Vec | ∀ i, 0 ≤ m i} := by
-    simp only [ofPred_forall]
-    exact isClosed_iInter (fun i ↦ isClosed_le continuous_const (continuous_apply i))
-  exact h.inter (isClosed_eq continuous_avg continuous_const)
-
-lemma isCompact_messageSet : IsCompact messageSet :=
-  (isCompact_Icc (a := fun _ : Fin 3 ↦ (0:ℝ)) (b := fun _ : Fin 3 ↦ (3:ℝ))).of_isClosed_subset
-    isClosed_messageSet (fun m hm ↦ ⟨hm.1, fun i ↦ coordinate_le_three ⟨m, hm⟩ i⟩)
-
-instance : CompactSpace Message := isCompact_iff_compactSpace.mp isCompact_messageSet
-
-lemma continuous_coordinate (i : Fin 3) : Continuous (fun m : Message ↦ m.1 i) :=
-  (continuous_apply i).comp continuous_subtype_val
-
-lemma integral_avg_measure {X : Type*} [MeasurableSpace X] {ν : Measure X} {f : X → Vec}
-    (hf : ∀ i, Integrable (fun x ↦ f x i) ν) :
-    (∫ x, avg (f x) ∂ν) = avg (fun i ↦ ∫ x, f x i ∂ν) := by
-  simp only [avg, integral_div]
-  rw [integral_add (f := fun x ↦ f x 0+f x 1) (g := fun x ↦ f x 2) ((hf 0).add (hf 1)) (hf 2),
-    integral_add (f := fun x ↦ f x 0) (g := fun x ↦ f x 1) (hf 0) (hf 1)]
-
-lemma integrable_avg {X : Type*} [MeasurableSpace X] {ν : Measure X} {f : X → Vec}
-    (hf : ∀ i, Integrable (fun x ↦ f x i) ν) : Integrable (fun x ↦ avg (f x)) ν :=
-  (((hf 0).add (hf 1)).add (hf 2)).div_const 3
-
-def permute (σ : Equiv.Perm (Fin 3)) (m : Message) : Message :=
-  ⟨fun i ↦ m.1 (σ i), fun i ↦ m.2.1 (σ i), by
-    have h : avg (fun i ↦ m.1 (σ i)) = avg m.1 := by
-      have he := Equiv.sum_comp σ m.1
-      simpa only [Fin.sum_univ_three, avg, div_left_inj' (by norm_num : (3:ℝ) ≠ 0)] using he
-    rw [h]; exact m.2.2⟩
-
-lemma continuous_permute (σ : Equiv.Perm (Fin 3)) : Continuous (permute σ) := by
-  apply Continuous.subtype_mk
-  exact continuous_pi (fun i ↦ continuous_coordinate (σ i))
-
-structure Law where
-  probability : ProbabilityMeasure Message
-  balanced : ∀ i, ∫ m, m.1 i ∂probability.toMeasure = 1
-  symmetric : ∀ σ, MeasurePreserving (permute σ) probability.toMeasure probability.toMeasure
-
-def edgeMessage (lam : ℝ) (m : Message) : Vec := edge lam m.1
-
-lemma edge_lower {lam : ℝ} (hl : 0 ≤ lam) (m : Message) (i : Fin 3) :
-    1-lam ≤ edgeMessage lam m i := by
-  dsimp [edgeMessage, edge]
-  nlinarith [mul_nonneg hl (coordinate_nonneg m i)]
-
-lemma edge_upper {lam : ℝ} (hl : 0 ≤ lam) (m : Message) (i : Fin 3) :
-    edgeMessage lam m i ≤ 1+2*lam := by
-  dsimp [edgeMessage, edge]
-  nlinarith [mul_nonneg hl (sub_nonneg.mpr (coordinate_le_three m i))]
-
-lemma edge_positive {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (m : Message) :
-    PositiveMessage (edgeMessage lam m) := by
-  refine ⟨fun i ↦ lt_of_lt_of_le (sub_pos.mpr hl₁) (edge_lower hl₀ m i), ?_⟩
-  have hc : avg (centered m.1) = 0 := by
-    change avg (fun i ↦ m.1 i-1) = 0
-    rw [avg_sub, avg_const, m.2.2]; ring
-  change avg (fun i ↦ 1+lam*centered m.1 i) = 1
-  rw [avg_add, avg_const, avg_mul, hc]; ring
-
-lemma continuous_edgeMessage (lam : ℝ) : Continuous (edgeMessage lam) := by
-  apply continuous_pi
-  intro i
-  exact continuous_const.add (continuous_const.mul ((continuous_coordinate i).sub continuous_const))
-
-lemma continuous_edge_log {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (i : Fin 3) :
-    Continuous (fun m : Message ↦ Real.log (edgeMessage lam m i)) :=
-  ((continuous_apply i).comp (continuous_edgeMessage lam)).log
-    (fun m ↦ ne_of_gt ((edge_positive hl₀ hl₁ m).1 i))
-
-end ThreeState.TreeClauses.Experiment
-
-end 
-
-noncomputable section
-open Set MeasureTheory
-namespace ThreeState.TreeClauses.Experiment
-open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
-
-lemma avg_permute_vec (σ : Equiv.Perm (Fin 3)) (v : Vec) :
-    avg (fun i ↦ v (σ i)) = avg v := by
-  have h := Equiv.sum_comp σ v
-  simpa only [Fin.sum_univ_three, avg, div_left_inj' (by norm_num : (3:ℝ) ≠ 0)] using h
-
-def permuteMeasurableEquiv (σ : Equiv.Perm (Fin 3)) : Message ≃ᵐ Message where
-  toFun := permute σ
-  invFun := permute σ.symm
-  left_inv m := by apply Subtype.ext; funext i; simp [permute]
-  right_inv m := by apply Subtype.ext; funext i; simp [permute]
-  measurable_toFun := (continuous_permute σ).measurable
-  measurable_invFun := (continuous_permute σ.symm).measurable
-
-lemma compact_integrable {X : Type*} [MeasurableSpace X] [TopologicalSpace X]
-    [BorelSpace X] [CompactSpace X] {μ : Measure X} [IsFiniteMeasure μ]
-    {f : X → ℝ} (hf : Continuous f) : Integrable f μ :=
-  hf.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace f)
-
-lemma Law.integral_equivariant (Q : Law) {f : Message → Vec}
-    (hf : ∀ i, Integrable (fun m ↦ f m i) Q.probability.toMeasure)
-    (hs : ∀ σ m i, f (permute σ m) i = f m (σ i)) (i : Fin 3) :
-    (∫ m, f m i ∂Q.probability.toMeasure) = ∫ m, avg (f m) ∂Q.probability.toMeasure := by
-  have he (j : Fin 3) : (∫ m, f m j ∂Q.probability.toMeasure) = ∫ m, f m i ∂Q.probability.toMeasure := by
-    let σ := Equiv.swap i j
-    have hp : MeasurePreserving (permuteMeasurableEquiv σ) Q.probability.toMeasure Q.probability.toMeasure := Q.symmetric σ
-    have h := hp.integral_comp' (fun m ↦ f m i)
-    change (∫ m, f (permute σ m) i ∂Q.probability.toMeasure) = _ at h
-    simp_rw [hs] at h
-    simpa only [σ, Equiv.swap_apply_left] using h
-  rw [integral_avg_measure hf]
-  dsimp only [avg]
-  rw [he 0, he 1, he 2]
-  ring
-
-lemma continuous_edge_centeredLog {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (i : Fin 3) :
-    Continuous (fun m : Message ↦ logCentered (edgeMessage lam m) i) := by
-  apply (continuous_edge_log hl₀ hl₁ i).sub
-  exact continuous_avg.comp (continuous_pi (fun j ↦ continuous_edge_log hl₀ hl₁ j))
-
-lemma edge_permute (lam : ℝ) (σ : Equiv.Perm (Fin 3)) (m : Message) (i : Fin 3) :
-    edgeMessage lam (permute σ m) i = edgeMessage lam m (σ i) := rfl
-
-lemma edge_logCentered_permute (lam : ℝ) (σ : Equiv.Perm (Fin 3)) (m : Message) (i : Fin 3) :
-    logCentered (edgeMessage lam (permute σ m)) i = logCentered (edgeMessage lam m) (σ i) := by
-  unfold logCentered logAverage
-  simp_rw [edge_permute]
-  rw [avg_permute_vec σ (fun i ↦ Real.log (edgeMessage lam m i))]
-
-lemma Law.edge_balanced (Q : Law) (lam : ℝ) (i : Fin 3) :
-    (∫ m, edgeMessage lam m i ∂Q.probability.toMeasure) = 1 := by
-  have hi : Integrable (fun m : Message ↦ m.1 i) Q.probability.toMeasure := compact_integrable (continuous_coordinate i)
-  change (∫ m : Message, 1+lam*(m.1 i-1) ∂Q.probability.toMeasure) = 1
-  rw [integral_add (f := fun _ : Message ↦ (1:ℝ)) (g := fun m ↦ lam*(m.1 i-1))
-    (integrable_const _) ((hi.sub (integrable_const _)).const_mul _),
-    integral_const_mul, integral_sub (f := fun m : Message ↦ m.1 i) (g := fun _ ↦ (1:ℝ)) hi (integrable_const _), Q.balanced]
-  simp
-
-lemma symEntropy_centeredLog' {m : Vec} (hm : avg m = 1) :
-    avg (fun i ↦ m i*logCentered m i) = 2*symEntropy m := by
-  have he : avg (fun i ↦ m i*logAverage m) = logAverage m := by
-    calc
-      _ = logAverage m*avg m := by dsimp only [avg]; ring
-      _ = _ := by rw [hm]; ring
-  simp only [logCentered, mul_sub, avg_sub, he]
-  dsimp only [symEntropy, logAverage, avg]
-  ring
-
-lemma Law.edge_centeredLog_mean (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (i : Fin 3) :
-    (∫ m, edgeMessage lam m i*logCentered (edgeMessage lam m) i ∂Q.probability.toMeasure) =
-      2*(∫ m, symEntropy (edgeMessage lam m) ∂Q.probability.toMeasure) := by
-  rw [Q.integral_equivariant (f := fun m i ↦ edgeMessage lam m i*logCentered (edgeMessage lam m) i)
-    (fun i ↦ compact_integrable (((continuous_apply i).comp (continuous_edgeMessage lam)).mul
-    (continuous_edge_centeredLog hl₀ hl₁ i))) (fun σ m i ↦ by rw [edge_permute, edge_logCentered_permute]) i]
-  simp_rw [symEntropy_centeredLog' (edge_positive hl₀ hl₁ _).2]
-  rw [integral_const_mul]
-
-lemma Law.edge_centeredLog_sq (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (i : Fin 3) :
-    (∫ m, edgeMessage lam m i*logCentered (edgeMessage lam m) i^2 ∂Q.probability.toMeasure) =
-      2*(∫ m, correction (edgeMessage lam m) ∂Q.probability.toMeasure) := by
-  rw [Q.integral_equivariant (f := fun m i ↦ edgeMessage lam m i*logCentered (edgeMessage lam m) i^2)
-    (fun i ↦ compact_integrable (((continuous_apply i).comp (continuous_edgeMessage lam)).mul
-    ((continuous_edge_centeredLog hl₀ hl₁ i).pow 2))) (fun σ m i ↦ by rw [edge_permute, edge_logCentered_permute]) i]
-  rw [← integral_const_mul]
-  congr 1; funext m; simp only [correction]; ring
-
-end ThreeState.TreeClauses.Experiment
-
-end 
-
-noncomputable section
-open Set MeasureTheory
-namespace ThreeState.TreeClauses.Experiment
-open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
-
-def xMoment (m : Message) : ℝ := momentX (centered m.1)
-def yMoment (m : Message) : ℝ := momentY (centered m.1)
-
-lemma message_centered_avg (m : Message) : avg (centered m.1) = 0 := by
-  change avg (fun i ↦ m.1 i-1) = 0
-  rw [avg_sub, avg_const, m.2.2]; ring
-
-lemma continuous_xMoment : Continuous xMoment := by
-  unfold xMoment momentX centered
-  exact (continuous_avg.comp (continuous_pi (fun i ↦ ((continuous_coordinate i).sub continuous_const).pow 2))).div_const 2
-
-lemma continuous_yMoment : Continuous yMoment := by
-  unfold yMoment momentY centered
-  exact (continuous_avg.comp (continuous_pi (fun i ↦ ((continuous_coordinate i).sub continuous_const).pow 3))).div_const 2
-
-lemma xMoment_nonneg (m : Message) : 0 ≤ xMoment m := momentX_nonneg _
-
-lemma xMoment_le_one (m : Message) : xMoment m ≤ 1 := by
-  have h (i : Fin 3) : (m.1 i)^2 ≤ 3*m.1 i := by
-    nlinarith [mul_nonneg (coordinate_nonneg m i) (sub_nonneg.mpr (coordinate_le_three m i))]
-  have hs := m.2.2
-  dsimp only [avg] at hs
-  dsimp only [xMoment, momentX, centered, avg]
-  nlinarith [h 0, h 1, h 2]
-
-lemma yMoment_discriminant (m : Message) : yMoment m^2 ≤ xMoment m^3 :=
-  moment_discriminant (message_centered_avg m)
-
-lemma moment_quadratic_nonneg {x y a : ℝ} (hx : 0 ≤ x) (hy : y^2 ≤ x^3) :
-    0 ≤ x*a^2+2*y*a+x^2 := by
-  rcases eq_or_lt_of_le hx with h | h
-  · have hx0 : x = 0 := h.symm
-    have hy0 : y = 0 := by rw [hx0] at hy; nlinarith [sq_nonneg y]
-    simp [hx0,hy0]
-  · have hh : 0 ≤ x*(x*a^2+2*y*a+x^2) := by nlinarith [sq_nonneg (x*a+y)]
-    exact nonneg_of_mul_nonneg_right hh h
-
-def Law.mu (Q : Law) := ∫ m, xMoment m ∂Q.probability.toMeasure
-def Law.nu (Q : Law) := ∫ m, xMoment m^2 ∂Q.probability.toMeasure
-def Law.eta (Q : Law) := ∫ m, yMoment m ∂Q.probability.toMeasure
-
-lemma Law.mu_nonneg (Q : Law) : 0 ≤ Q.mu := integral_nonneg xMoment_nonneg
-lemma Law.nu_nonneg (Q : Law) : 0 ≤ Q.nu := integral_nonneg (fun _ ↦ sq_nonneg _)
-
-lemma Law.mu_le_one (Q : Law) : Q.mu ≤ 1 := by
-  have h := integral_mono (μ := Q.probability.toMeasure)
-    (compact_integrable continuous_xMoment) (integrable_const (1:ℝ)) xMoment_le_one
-  simpa [Law.mu] using h
-
-lemma Law.second_moment_lower (Q : Law) : Q.mu^2 ≤ Q.nu := by
-  have hi := compact_integrable continuous_xMoment (μ := Q.probability.toMeasure)
-  have hi₂ := compact_integrable (continuous_xMoment.pow 2) (μ := Q.probability.toMeasure)
-  have h := integral_nonneg (μ := Q.probability.toMeasure) (f := fun m : Message ↦ (xMoment m-Q.mu)^2)
-    (fun m ↦ sq_nonneg (xMoment m-Q.mu))
-  have hfunc : (fun m : Message ↦ (xMoment m-Q.mu)^2) =
-      (fun m ↦ xMoment m^2 - 2*Q.mu*xMoment m + Q.mu^2) := by funext m; ring
-  rw [hfunc, integral_add (f := fun m ↦ xMoment m^2-2*Q.mu*xMoment m)
-    (g := fun _ ↦ Q.mu^2) (hi₂.sub (hi.const_mul _)) (integrable_const _),
-    integral_sub (f := fun m ↦ xMoment m^2) (g := fun m ↦ 2*Q.mu*xMoment m) hi₂ (hi.const_mul (2*Q.mu)), integral_const_mul] at h
-  simp only [integral_const, measure_univ, Measure.real, ENNReal.toReal_one, smul_eq_mul, one_mul] at h
-  change 0 ≤ Q.nu-2*Q.mu*Q.mu+Q.mu^2 at h
-  linarith
-
-lemma Law.moment_discriminant (Q : Law) : Q.eta^2 ≤ Q.mu*Q.nu := by
-  have hi := compact_integrable continuous_xMoment (μ := Q.probability.toMeasure)
-  have hi₂ := compact_integrable (continuous_xMoment.pow 2) (μ := Q.probability.toMeasure)
-  have hy := compact_integrable continuous_yMoment (μ := Q.probability.toMeasure)
-  have hquad (a : ℝ) : 0 ≤ Q.mu*(a*a)+(2*Q.eta)*a+Q.nu := by
-    have h := integral_nonneg (μ := Q.probability.toMeasure)
-      (f := fun m : Message ↦ xMoment m*a^2+2*yMoment m*a+xMoment m^2)
-      (fun m : Message ↦ moment_quadratic_nonneg (a := a) (xMoment_nonneg m) (yMoment_discriminant m))
-    rw [integral_add (f := fun m ↦ xMoment m*a^2+2*yMoment m*a) (g := fun m ↦ xMoment m^2)
-      ((hi.mul_const _).add ((hy.const_mul _).mul_const _)) hi₂,
-      integral_add (f := fun m ↦ xMoment m*a^2) (g := fun m ↦ 2*yMoment m*a)
-        (hi.mul_const _) ((hy.const_mul _).mul_const _),
-      integral_mul_const, integral_mul_const, integral_const_mul] at h
-    change 0 ≤ Q.mu*a^2+2*Q.eta*a+Q.nu at h
-    nlinarith
-  have h := discrim_le_zero hquad
-  dsimp only [discrim] at h
-  nlinarith
-
-end ThreeState.TreeClauses.Experiment
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeStateClauses.PositiveMoments
+
+namespace OAI
+
+/-! Posterior Law. -/
+
+open scoped BigOperators ENNReal NNReal Topology
+open Filter
+noncomputable section
+open Set MeasureTheory
+namespace ThreeState.TreeClauses.Experiment
+open Radial ThreeState.TreeClauses.Positive
+
+def messageSet : Set Vec := {m | (∀ i, 0 ≤ m i) ∧ avg m = 1}
+abbrev Message := messageSet
+
+lemma coordinate_nonneg (m : Message) (i : Fin 3) : 0 ≤ m.1 i := m.2.1 i
+lemma coordinate_le_three (m : Message) (i : Fin 3) : m.1 i ≤ 3 := by
+  have hs := m.2.2
+  dsimp only [avg] at hs
+  have h₀ := m.2.1 0
+  have h₁ := m.2.1 1
+  have h₂ := m.2.1 2
+  fin_cases i <;> dsimp at * <;> linarith
+
+lemma continuous_avg : Continuous (avg : Vec → ℝ) :=
+  (((continuous_apply 0).add (continuous_apply 1)).add (continuous_apply 2)).div_const 3
+
+lemma isClosed_messageSet : IsClosed messageSet := by
+  have h : IsClosed {m : Vec | ∀ i, 0 ≤ m i} := by
+    simp only [ofPred_forall]
+    exact isClosed_iInter (fun i ↦ isClosed_le continuous_const (continuous_apply i))
+  exact h.inter (isClosed_eq continuous_avg continuous_const)
+
+lemma isCompact_messageSet : IsCompact messageSet :=
+  (isCompact_Icc (a := fun _ : Fin 3 ↦ (0:ℝ)) (b := fun _ : Fin 3 ↦ (3:ℝ))).of_isClosed_subset
+    isClosed_messageSet (fun m hm ↦ ⟨hm.1, fun i ↦ coordinate_le_three ⟨m, hm⟩ i⟩)
+
+instance : CompactSpace Message := isCompact_iff_compactSpace.mp isCompact_messageSet
+
+lemma continuous_coordinate (i : Fin 3) : Continuous (fun m : Message ↦ m.1 i) :=
+  (continuous_apply i).comp continuous_subtype_val
+
+lemma integral_avg_measure {X : Type*} [MeasurableSpace X] {ν : Measure X} {f : X → Vec}
+    (hf : ∀ i, Integrable (fun x ↦ f x i) ν) :
+    (∫ x, avg (f x) ∂ν) = avg (fun i ↦ ∫ x, f x i ∂ν) := by
+  simp only [avg, integral_div]
+  rw [integral_add (f := fun x ↦ f x 0+f x 1) (g := fun x ↦ f x 2) ((hf 0).add (hf 1)) (hf 2),
+    integral_add (f := fun x ↦ f x 0) (g := fun x ↦ f x 1) (hf 0) (hf 1)]
+
+lemma integrable_avg {X : Type*} [MeasurableSpace X] {ν : Measure X} {f : X → Vec}
+    (hf : ∀ i, Integrable (fun x ↦ f x i) ν) : Integrable (fun x ↦ avg (f x)) ν :=
+  (((hf 0).add (hf 1)).add (hf 2)).div_const 3
+
+def permute (σ : Equiv.Perm (Fin 3)) (m : Message) : Message :=
+  ⟨fun i ↦ m.1 (σ i), fun i ↦ m.2.1 (σ i), by
+    have h : avg (fun i ↦ m.1 (σ i)) = avg m.1 := by
+      have he := Equiv.sum_comp σ m.1
+      simpa only [Fin.sum_univ_three, avg, div_left_inj' (by norm_num : (3:ℝ) ≠ 0)] using he
+    rw [h]; exact m.2.2⟩
+
+lemma continuous_permute (σ : Equiv.Perm (Fin 3)) : Continuous (permute σ) := by
+  apply Continuous.subtype_mk
+  exact continuous_pi (fun i ↦ continuous_coordinate (σ i))
+
+structure Law where
+  probability : ProbabilityMeasure Message
+  balanced : ∀ i, ∫ m, m.1 i ∂probability.toMeasure = 1
+  symmetric : ∀ σ, MeasurePreserving (permute σ) probability.toMeasure probability.toMeasure
+
+def edgeMessage (lam : ℝ) (m : Message) : Vec := edge lam m.1
+
+lemma edge_lower {lam : ℝ} (hl : 0 ≤ lam) (m : Message) (i : Fin 3) :
+    1-lam ≤ edgeMessage lam m i := by
+  dsimp [edgeMessage, edge]
+  nlinarith [mul_nonneg hl (coordinate_nonneg m i)]
+
+lemma edge_upper {lam : ℝ} (hl : 0 ≤ lam) (m : Message) (i : Fin 3) :
+    edgeMessage lam m i ≤ 1+2*lam := by
+  dsimp [edgeMessage, edge]
+  nlinarith [mul_nonneg hl (sub_nonneg.mpr (coordinate_le_three m i))]
+
+lemma edge_positive {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (m : Message) :
+    PositiveMessage (edgeMessage lam m) := by
+  refine ⟨fun i ↦ lt_of_lt_of_le (sub_pos.mpr hl₁) (edge_lower hl₀ m i), ?_⟩
+  have hc : avg (centered m.1) = 0 := by
+    change avg (fun i ↦ m.1 i-1) = 0
+    rw [avg_sub, avg_const, m.2.2]; ring
+  change avg (fun i ↦ 1+lam*centered m.1 i) = 1
+  rw [avg_add, avg_const, avg_mul, hc]; ring
+
+lemma continuous_edgeMessage (lam : ℝ) : Continuous (edgeMessage lam) := by
+  apply continuous_pi
+  intro i
+  exact continuous_const.add (continuous_const.mul ((continuous_coordinate i).sub continuous_const))
+
+lemma continuous_edge_log {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (i : Fin 3) :
+    Continuous (fun m : Message ↦ Real.log (edgeMessage lam m i)) :=
+  ((continuous_apply i).comp (continuous_edgeMessage lam)).log
+    (fun m ↦ ne_of_gt ((edge_positive hl₀ hl₁ m).1 i))
+
+end ThreeState.TreeClauses.Experiment
+
+end 
+
+noncomputable section
+open Set MeasureTheory
+namespace ThreeState.TreeClauses.Experiment
+open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
+
+lemma avg_permute_vec (σ : Equiv.Perm (Fin 3)) (v : Vec) :
+    avg (fun i ↦ v (σ i)) = avg v := by
+  have h := Equiv.sum_comp σ v
+  simpa only [Fin.sum_univ_three, avg, div_left_inj' (by norm_num : (3:ℝ) ≠ 0)] using h
+
+def permuteMeasurableEquiv (σ : Equiv.Perm (Fin 3)) : Message ≃ᵐ Message where
+  toFun := permute σ
+  invFun := permute σ.symm
+  left_inv m := by apply Subtype.ext; funext i; simp [permute]
+  right_inv m := by apply Subtype.ext; funext i; simp [permute]
+  measurable_toFun := (continuous_permute σ).measurable
+  measurable_invFun := (continuous_permute σ.symm).measurable
+
+lemma compact_integrable {X : Type*} [MeasurableSpace X] [TopologicalSpace X]
+    [BorelSpace X] [CompactSpace X] {μ : Measure X} [IsFiniteMeasure μ]
+    {f : X → ℝ} (hf : Continuous f) : Integrable f μ :=
+  hf.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace f)
+
+lemma Law.integral_equivariant (Q : Law) {f : Message → Vec}
+    (hf : ∀ i, Integrable (fun m ↦ f m i) Q.probability.toMeasure)
+    (hs : ∀ σ m i, f (permute σ m) i = f m (σ i)) (i : Fin 3) :
+    (∫ m, f m i ∂Q.probability.toMeasure) = ∫ m, avg (f m) ∂Q.probability.toMeasure := by
+  have he (j : Fin 3) : (∫ m, f m j ∂Q.probability.toMeasure) = ∫ m, f m i ∂Q.probability.toMeasure := by
+    let σ := Equiv.swap i j
+    have hp : MeasurePreserving (permuteMeasurableEquiv σ) Q.probability.toMeasure Q.probability.toMeasure := Q.symmetric σ
+    have h := hp.integral_comp' (fun m ↦ f m i)
+    change (∫ m, f (permute σ m) i ∂Q.probability.toMeasure) = _ at h
+    simp_rw [hs] at h
+    simpa only [σ, Equiv.swap_apply_left] using h
+  rw [integral_avg_measure hf]
+  dsimp only [avg]
+  rw [he 0, he 1, he 2]
+  ring
+
+lemma continuous_edge_centeredLog {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (i : Fin 3) :
+    Continuous (fun m : Message ↦ logCentered (edgeMessage lam m) i) := by
+  apply (continuous_edge_log hl₀ hl₁ i).sub
+  exact continuous_avg.comp (continuous_pi (fun j ↦ continuous_edge_log hl₀ hl₁ j))
+
+lemma edge_permute (lam : ℝ) (σ : Equiv.Perm (Fin 3)) (m : Message) (i : Fin 3) :
+    edgeMessage lam (permute σ m) i = edgeMessage lam m (σ i) := rfl
+
+lemma edge_logCentered_permute (lam : ℝ) (σ : Equiv.Perm (Fin 3)) (m : Message) (i : Fin 3) :
+    logCentered (edgeMessage lam (permute σ m)) i = logCentered (edgeMessage lam m) (σ i) := by
+  unfold logCentered logAverage
+  simp_rw [edge_permute]
+  rw [avg_permute_vec σ (fun i ↦ Real.log (edgeMessage lam m i))]
+
+lemma Law.edge_balanced (Q : Law) (lam : ℝ) (i : Fin 3) :
+    (∫ m, edgeMessage lam m i ∂Q.probability.toMeasure) = 1 := by
+  have hi : Integrable (fun m : Message ↦ m.1 i) Q.probability.toMeasure := compact_integrable (continuous_coordinate i)
+  change (∫ m : Message, 1+lam*(m.1 i-1) ∂Q.probability.toMeasure) = 1
+  rw [integral_add (f := fun _ : Message ↦ (1:ℝ)) (g := fun m ↦ lam*(m.1 i-1))
+    (integrable_const _) ((hi.sub (integrable_const _)).const_mul _),
+    integral_const_mul, integral_sub (f := fun m : Message ↦ m.1 i) (g := fun _ ↦ (1:ℝ)) hi (integrable_const _), Q.balanced]
+  simp
+
+lemma symEntropy_centeredLog' {m : Vec} (hm : avg m = 1) :
+    avg (fun i ↦ m i*logCentered m i) = 2*symEntropy m := by
+  have he : avg (fun i ↦ m i*logAverage m) = logAverage m := by
+    calc
+      _ = logAverage m*avg m := by dsimp only [avg]; ring
+      _ = _ := by rw [hm]; ring
+  simp only [logCentered, mul_sub, avg_sub, he]
+  dsimp only [symEntropy, logAverage, avg]
+  ring
+
+lemma Law.edge_centeredLog_mean (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (i : Fin 3) :
+    (∫ m, edgeMessage lam m i*logCentered (edgeMessage lam m) i ∂Q.probability.toMeasure) =
+      2*(∫ m, symEntropy (edgeMessage lam m) ∂Q.probability.toMeasure) := by
+  rw [Q.integral_equivariant (f := fun m i ↦ edgeMessage lam m i*logCentered (edgeMessage lam m) i)
+    (fun i ↦ compact_integrable (((continuous_apply i).comp (continuous_edgeMessage lam)).mul
+    (continuous_edge_centeredLog hl₀ hl₁ i))) (fun σ m i ↦ by rw [edge_permute, edge_logCentered_permute]) i]
+  simp_rw [symEntropy_centeredLog' (edge_positive hl₀ hl₁ _).2]
+  rw [integral_const_mul]
+
+lemma Law.edge_centeredLog_sq (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (i : Fin 3) :
+    (∫ m, edgeMessage lam m i*logCentered (edgeMessage lam m) i^2 ∂Q.probability.toMeasure) =
+      2*(∫ m, correction (edgeMessage lam m) ∂Q.probability.toMeasure) := by
+  rw [Q.integral_equivariant (f := fun m i ↦ edgeMessage lam m i*logCentered (edgeMessage lam m) i^2)
+    (fun i ↦ compact_integrable (((continuous_apply i).comp (continuous_edgeMessage lam)).mul
+    ((continuous_edge_centeredLog hl₀ hl₁ i).pow 2))) (fun σ m i ↦ by rw [edge_permute, edge_logCentered_permute]) i]
+  rw [← integral_const_mul]
+  congr 1; funext m; simp only [correction]; ring
+
+end ThreeState.TreeClauses.Experiment
+
+end 
+
+noncomputable section
+open Set MeasureTheory
+namespace ThreeState.TreeClauses.Experiment
+open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
+
+def xMoment (m : Message) : ℝ := momentX (centered m.1)
+def yMoment (m : Message) : ℝ := momentY (centered m.1)
+
+lemma message_centered_avg (m : Message) : avg (centered m.1) = 0 := by
+  change avg (fun i ↦ m.1 i-1) = 0
+  rw [avg_sub, avg_const, m.2.2]; ring
+
+lemma continuous_xMoment : Continuous xMoment := by
+  unfold xMoment momentX centered
+  exact (continuous_avg.comp (continuous_pi (fun i ↦ ((continuous_coordinate i).sub continuous_const).pow 2))).div_const 2
+
+lemma continuous_yMoment : Continuous yMoment := by
+  unfold yMoment momentY centered
+  exact (continuous_avg.comp (continuous_pi (fun i ↦ ((continuous_coordinate i).sub continuous_const).pow 3))).div_const 2
+
+lemma xMoment_nonneg (m : Message) : 0 ≤ xMoment m := momentX_nonneg _
+
+lemma xMoment_le_one (m : Message) : xMoment m ≤ 1 := by
+  have h (i : Fin 3) : (m.1 i)^2 ≤ 3*m.1 i := by
+    nlinarith [mul_nonneg (coordinate_nonneg m i) (sub_nonneg.mpr (coordinate_le_three m i))]
+  have hs := m.2.2
+  dsimp only [avg] at hs
+  dsimp only [xMoment, momentX, centered, avg]
+  nlinarith [h 0, h 1, h 2]
+
+lemma yMoment_discriminant (m : Message) : yMoment m^2 ≤ xMoment m^3 :=
+  moment_discriminant (message_centered_avg m)
+
+lemma moment_quadratic_nonneg {x y a : ℝ} (hx : 0 ≤ x) (hy : y^2 ≤ x^3) :
+    0 ≤ x*a^2+2*y*a+x^2 := by
+  rcases eq_or_lt_of_le hx with h | h
+  · have hx0 : x = 0 := h.symm
+    have hy0 : y = 0 := by rw [hx0] at hy; nlinarith [sq_nonneg y]
+    simp [hx0,hy0]
+  · have hh : 0 ≤ x*(x*a^2+2*y*a+x^2) := by nlinarith [sq_nonneg (x*a+y)]
+    exact nonneg_of_mul_nonneg_right hh h
+
+def Law.mu (Q : Law) := ∫ m, xMoment m ∂Q.probability.toMeasure
+def Law.nu (Q : Law) := ∫ m, xMoment m^2 ∂Q.probability.toMeasure
+def Law.eta (Q : Law) := ∫ m, yMoment m ∂Q.probability.toMeasure
+
+lemma Law.mu_nonneg (Q : Law) : 0 ≤ Q.mu := integral_nonneg xMoment_nonneg
+lemma Law.nu_nonneg (Q : Law) : 0 ≤ Q.nu := integral_nonneg (fun _ ↦ sq_nonneg _)
+
+lemma Law.mu_le_one (Q : Law) : Q.mu ≤ 1 := by
+  have h := integral_mono (μ := Q.probability.toMeasure)
+    (compact_integrable continuous_xMoment) (integrable_const (1:ℝ)) xMoment_le_one
+  simpa [Law.mu] using h
+
+lemma Law.second_moment_lower (Q : Law) : Q.mu^2 ≤ Q.nu := by
+  have hi := compact_integrable continuous_xMoment (μ := Q.probability.toMeasure)
+  have hi₂ := compact_integrable (continuous_xMoment.pow 2) (μ := Q.probability.toMeasure)
+  have h := integral_nonneg (μ := Q.probability.toMeasure) (f := fun m : Message ↦ (xMoment m-Q.mu)^2)
+    (fun m ↦ sq_nonneg (xMoment m-Q.mu))
+  have hfunc : (fun m : Message ↦ (xMoment m-Q.mu)^2) =
+      (fun m ↦ xMoment m^2 - 2*Q.mu*xMoment m + Q.mu^2) := by funext m; ring
+  rw [hfunc, integral_add (f := fun m ↦ xMoment m^2-2*Q.mu*xMoment m)
+    (g := fun _ ↦ Q.mu^2) (hi₂.sub (hi.const_mul _)) (integrable_const _),
+    integral_sub (f := fun m ↦ xMoment m^2) (g := fun m ↦ 2*Q.mu*xMoment m) hi₂ (hi.const_mul (2*Q.mu)), integral_const_mul] at h
+  simp only [integral_const, measure_univ, Measure.real, ENNReal.toReal_one, smul_eq_mul, one_mul] at h
+  change 0 ≤ Q.nu-2*Q.mu*Q.mu+Q.mu^2 at h
+  linarith
+
+lemma Law.moment_discriminant (Q : Law) : Q.eta^2 ≤ Q.mu*Q.nu := by
+  have hi := compact_integrable continuous_xMoment (μ := Q.probability.toMeasure)
+  have hi₂ := compact_integrable (continuous_xMoment.pow 2) (μ := Q.probability.toMeasure)
+  have hy := compact_integrable continuous_yMoment (μ := Q.probability.toMeasure)
+  have hquad (a : ℝ) : 0 ≤ Q.mu*(a*a)+(2*Q.eta)*a+Q.nu := by
+    have h := integral_nonneg (μ := Q.probability.toMeasure)
+      (f := fun m : Message ↦ xMoment m*a^2+2*yMoment m*a+xMoment m^2)
+      (fun m : Message ↦ moment_quadratic_nonneg (a := a) (xMoment_nonneg m) (yMoment_discriminant m))
+    rw [integral_add (f := fun m ↦ xMoment m*a^2+2*yMoment m*a) (g := fun m ↦ xMoment m^2)
+      ((hi.mul_const _).add ((hy.const_mul _).mul_const _)) hi₂,
+      integral_add (f := fun m ↦ xMoment m*a^2) (g := fun m ↦ 2*yMoment m*a)
+        (hi.mul_const _) ((hy.const_mul _).mul_const _),
+      integral_mul_const, integral_mul_const, integral_const_mul] at h
+    change 0 ≤ Q.mu*a^2+2*Q.eta*a+Q.nu at h
+    nlinarith
+  have h := discrim_le_zero hquad
+  dsimp only [discrim] at h
+  nlinarith
+
+end ThreeState.TreeClauses.Experiment
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeStateClauses/Products.lean b/lean/OAI/Probability/ThreeStateClauses/Products.lean
index 0b6996fc2d65afd4a4ca2f2782cae5bba7d7c90d..81b1550431a0df88cda0b4e1d75d7d48c3eb3244 100644
--- a/lean/OAI/Probability/ThreeStateClauses/Products.lean
+++ b/lean/OAI/Probability/ThreeStateClauses/Products.lean
@@ -1,100 +1,100 @@
-import Mathlib
-
-namespace OAI
-
-/-! Products. -/
-
-open scoped BigOperators ENNReal NNReal Topology
-open Filter
-noncomputable section
-open Set MeasureTheory
-open scoped BigOperators ENNReal
-namespace ThreeState.TreeClauses.Experiment
-
-variable {α β : Type*}
-
-lemma ennreal_tsum_fin_product (n : ℕ) (f : Fin n → α → ℝ≥0∞) :
-    (∑' v : Fin n → α, ∏ j, f j (v j)) = ∏ j, ∑' a, f j a := by
-  induction n with
-  | zero => simp
-  | succ n ih =>
-    rw [← (Fin.consEquiv (fun _ : Fin (n+1) ↦ α)).tsum_eq]
-    change (∑' z : α × (Fin n → α), ∏ j, f j (Fin.cons (α := fun _ : Fin (n+1) ↦ α) z.1 z.2 j)) = _
-    simp_rw [Fin.prod_univ_succ, Fin.cons_zero, Fin.cons_succ]
-    rw [ENNReal.tsum_prod' (f := fun z : α × (Fin n → α) ↦ f 0 z.1*∏ j, f j.succ (z.2 j))]
-    simp_rw [ENNReal.tsum_mul_left, ih, ENNReal.tsum_mul_right]
-
-def productPMF {n : ℕ} (p : Fin n → PMF α) : PMF (Fin n → α) :=
-  ⟨fun v ↦ ∏ j, p j (v j), ENNReal.summable.hasSum_iff.2 (by
-    rw [ennreal_tsum_fin_product]
-    simp only [PMF.tsum_coe, Finset.prod_const_one])⟩
-
-@[simp] lemma productPMF_apply {n : ℕ} (p : Fin n → PMF α) (v : Fin n → α) :
-    productPMF p v = ∏ j, p j (v j) := rfl
-
-lemma productPMF_pure {n : ℕ} (v : Fin n → α) :
-    productPMF (fun j ↦ PMF.pure (v j)) = PMF.pure v := by
-  classical
-  apply PMF.ext
-  intro w
-  by_cases h : w = v
-  · subst w; simp
-  · have hn : ∃ j, w j ≠ v j := by simpa only [funext_iff, not_forall] using h
-    obtain ⟨j,hj⟩ := hn
-    simp only [productPMF_apply, PMF.pure_apply, ite_eq_right h]
-    exact Finset.prod_eq_zero (Finset.mem_univ j) (ite_eq_right hj)
-
-lemma productPMF_bind {n : ℕ} (p : Fin n → PMF α) (K : Fin n → α → PMF β) :
-    (productPMF p).bind (fun v ↦ productPMF (fun j ↦ K j (v j))) =
-      productPMF (fun j ↦ (p j).bind (K j)) := by
-  apply PMF.ext
-  intro w
-  simp only [PMF.bind_apply, productPMF_apply, ← Finset.prod_mul_distrib]
-  exact ennreal_tsum_fin_product n (fun j a ↦ p j a*K j a (w j))
-
-lemma productPMF_map {n : ℕ} (p : Fin n → PMF α) (T : Fin n → α → β) :
-    (productPMF p).map (fun v j ↦ T j (v j)) = productPMF (fun j ↦ (p j).map (T j)) := by
-  change (productPMF p).bind (fun v ↦ PMF.pure (fun j ↦ T j (v j))) = _
-  simp_rw [← productPMF_pure]
-  exact productPMF_bind p (fun j a ↦ PMF.pure (T j a))
-
-lemma productPMF_measure [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α]
-    {n : ℕ} (p : Fin n → PMF α) :
-    (productPMF p).toMeasure = Measure.pi (fun j ↦ (p j).toMeasure) := by
-  have h : productPMF p = (Measure.pi (fun j ↦ (p j).toMeasure)).toPMF := by
-    apply PMF.ext
-    intro v
-    simp [productPMF_apply, Measure.toPMF_apply, Measure.pi_singleton, PMF.toMeasure_apply_singleton]
-  rw [h, Measure.toPMF_toMeasure]
-
-lemma productPMF_cons {n : ℕ} (p : Fin (n+1) → PMF α) :
-    productPMF p = (p 0).bind (fun a ↦ (productPMF (fun j ↦ p j.succ)).map (Fin.cons a)) := by
-  classical
-  apply PMF.ext
-  intro v
-  rw [PMF.bind_apply, tsum_eq_single (v 0)]
-  · rw [PMF.map_apply, tsum_eq_single (Fin.tail v)]
-    · simp [productPMF_apply, Fin.prod_univ_succ, Fin.tail]
-    · intro w hw
-      have hn : v ≠ Fin.cons (v 0) w := by
-        intro he
-        have := congrArg Fin.tail he
-        simp only [Fin.tail_cons] at this
-        exact hw this.symm
-      simp [hn]
-  · intro a ha
-    have he : (productPMF (fun j ↦ p j.succ)).map (Fin.cons a) v = 0 := by
-      rw [PMF.map_apply]
-      apply ENNReal.tsum_eq_zero.mpr
-      intro w
-      have hn : v ≠ Fin.cons a w := by
-        intro h
-        exact ha (congrFun h 0).symm
-      simp [hn]
-    rw [he, mul_zero]
-
-end ThreeState.TreeClauses.Experiment
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-! Products. -/
+
+open scoped BigOperators ENNReal NNReal Topology
+open Filter
+noncomputable section
+open Set MeasureTheory
+open scoped BigOperators ENNReal
+namespace ThreeState.TreeClauses.Experiment
+
+variable {α β : Type*}
+
+lemma ennreal_tsum_fin_product (n : ℕ) (f : Fin n → α → ℝ≥0∞) :
+    (∑' v : Fin n → α, ∏ j, f j (v j)) = ∏ j, ∑' a, f j a := by
+  induction n with
+  | zero => simp
+  | succ n ih =>
+    rw [← (Fin.consEquiv (fun _ : Fin (n+1) ↦ α)).tsum_eq]
+    change (∑' z : α × (Fin n → α), ∏ j, f j (Fin.cons (α := fun _ : Fin (n+1) ↦ α) z.1 z.2 j)) = _
+    simp_rw [Fin.prod_univ_succ, Fin.cons_zero, Fin.cons_succ]
+    rw [ENNReal.tsum_prod' (f := fun z : α × (Fin n → α) ↦ f 0 z.1*∏ j, f j.succ (z.2 j))]
+    simp_rw [ENNReal.tsum_mul_left, ih, ENNReal.tsum_mul_right]
+
+def productPMF {n : ℕ} (p : Fin n → PMF α) : PMF (Fin n → α) :=
+  ⟨fun v ↦ ∏ j, p j (v j), ENNReal.summable.hasSum_iff.2 (by
+    rw [ennreal_tsum_fin_product]
+    simp only [PMF.tsum_coe, Finset.prod_const_one])⟩
+
+@[simp] lemma productPMF_apply {n : ℕ} (p : Fin n → PMF α) (v : Fin n → α) :
+    productPMF p v = ∏ j, p j (v j) := rfl
+
+lemma productPMF_pure {n : ℕ} (v : Fin n → α) :
+    productPMF (fun j ↦ PMF.pure (v j)) = PMF.pure v := by
+  classical
+  apply PMF.ext
+  intro w
+  by_cases h : w = v
+  · subst w; simp
+  · have hn : ∃ j, w j ≠ v j := by simpa only [funext_iff, not_forall] using h
+    obtain ⟨j,hj⟩ := hn
+    simp only [productPMF_apply, PMF.pure_apply, ite_eq_right h]
+    exact Finset.prod_eq_zero (Finset.mem_univ j) (ite_eq_right hj)
+
+lemma productPMF_bind {n : ℕ} (p : Fin n → PMF α) (K : Fin n → α → PMF β) :
+    (productPMF p).bind (fun v ↦ productPMF (fun j ↦ K j (v j))) =
+      productPMF (fun j ↦ (p j).bind (K j)) := by
+  apply PMF.ext
+  intro w
+  simp only [PMF.bind_apply, productPMF_apply, ← Finset.prod_mul_distrib]
+  exact ennreal_tsum_fin_product n (fun j a ↦ p j a*K j a (w j))
+
+lemma productPMF_map {n : ℕ} (p : Fin n → PMF α) (T : Fin n → α → β) :
+    (productPMF p).map (fun v j ↦ T j (v j)) = productPMF (fun j ↦ (p j).map (T j)) := by
+  change (productPMF p).bind (fun v ↦ PMF.pure (fun j ↦ T j (v j))) = _
+  simp_rw [← productPMF_pure]
+  exact productPMF_bind p (fun j a ↦ PMF.pure (T j a))
+
+lemma productPMF_measure [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α]
+    {n : ℕ} (p : Fin n → PMF α) :
+    (productPMF p).toMeasure = Measure.pi (fun j ↦ (p j).toMeasure) := by
+  have h : productPMF p = (Measure.pi (fun j ↦ (p j).toMeasure)).toPMF := by
+    apply PMF.ext
+    intro v
+    simp [productPMF_apply, Measure.toPMF_apply, Measure.pi_singleton, PMF.toMeasure_apply_singleton]
+  rw [h, Measure.toPMF_toMeasure]
+
+lemma productPMF_cons {n : ℕ} (p : Fin (n+1) → PMF α) :
+    productPMF p = (p 0).bind (fun a ↦ (productPMF (fun j ↦ p j.succ)).map (Fin.cons a)) := by
+  classical
+  apply PMF.ext
+  intro v
+  rw [PMF.bind_apply, tsum_eq_single (v 0)]
+  · rw [PMF.map_apply, tsum_eq_single (Fin.tail v)]
+    · simp [productPMF_apply, Fin.prod_univ_succ, Fin.tail]
+    · intro w hw
+      have hn : v ≠ Fin.cons (v 0) w := by
+        intro he
+        have := congrArg Fin.tail he
+        simp only [Fin.tail_cons] at this
+        exact hw this.symm
+      simp [hn]
+  · intro a ha
+    have he : (productPMF (fun j ↦ p j.succ)).map (Fin.cons a) v = 0 := by
+      rw [PMF.map_apply]
+      apply ENNReal.tsum_eq_zero.mpr
+      intro w
+      have hn : v ≠ Fin.cons a w := by
+        intro h
+        exact ha (congrFun h 0).symm
+      simp [hn]
+    rw [he, mul_zero]
+
+end ThreeState.TreeClauses.Experiment
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeStateClauses/Projection.lean b/lean/OAI/Probability/ThreeStateClauses/Projection.lean
index 9f8131aa8454ff2184ee453a75d68d0d1ef31ac8..acc05d7c87ae142f7a775e27211d303840d0617a 100644
--- a/lean/OAI/Probability/ThreeStateClauses/Projection.lean
+++ b/lean/OAI/Probability/ThreeStateClauses/Projection.lean
@@ -1,271 +1,271 @@
-import Mathlib
-import OAI.Probability.ThreeStateClauses.FiniteUpdate
-import OAI.Probability.ThreeStateClauses.Correlation
-
-namespace OAI
-
-/-! Projection. -/
-
-open scoped BigOperators ENNReal NNReal Topology
-open Filter
-noncomputable section
-open Set MeasureTheory
-open scoped BigOperators
-namespace ThreeState.TreeClauses.Experiment
-open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
-
-def edgeCentered (lam : ℝ) (m : Message) (i : Fin 3) : ℝ := edgeMessage lam m i-1
-
-lemma continuous_edgeCentered (lam : ℝ) (i : Fin 3) :
-    Continuous (fun m : Message ↦ edgeCentered lam m i) :=
-  ((continuous_apply i).comp (continuous_edgeMessage lam)).sub continuous_const
-
-lemma Law.edgeCentered_mean (Q : Law) (lam : ℝ) (i : Fin 3) :
-    (∫ m, edgeCentered lam m i ∂Q.probability.toMeasure) = 0 := by
-  unfold edgeCentered
-  rw [integral_sub (f := fun m ↦ edgeMessage lam m i) (g := fun _ ↦ (1:ℝ))
-    (compact_integrable ((continuous_apply i).comp (continuous_edgeMessage lam))) (integrable_const _),
-    Q.edge_balanced]
-  simp
-
-lemma Law.edgeCentered_second (Q : Law) (lam : ℝ) (i : Fin 3) :
-    (∫ m, edgeCentered lam m i^2 ∂Q.probability.toMeasure) = 2*lam^2*Q.mu := by
-  rw [Q.integral_equivariant (f := fun m i ↦ edgeCentered lam m i^2)
-    (fun i ↦ compact_integrable ((continuous_edgeCentered lam i).pow 2))
-    (fun _ _ _ ↦ rfl)]
-  have he (m : Message) : avg (fun i ↦ edgeCentered lam m i^2) = 2*lam^2*xMoment m := by
-    have hh := xMoment_edge lam m
-    change (avg (fun i ↦ edgeCentered lam m i^2))/2 = _ at hh
-    linarith
-  simp_rw [he]
-  rw [integral_const_mul]
-  rfl
-
-lemma Law.edgeCentered_tilt_diag (Q : Law) (lam : ℝ) (i : Fin 3) :
-    (∫ m, edgeMessage lam m i*edgeCentered lam m i ∂Q.probability.toMeasure) = 2*lam^2*Q.mu := by
-  have he (m : Message) : edgeMessage lam m i*edgeCentered lam m i =
-      edgeCentered lam m i+edgeCentered lam m i^2 := by unfold edgeCentered; ring
-  simp_rw [he]
-  rw [integral_add (f := fun m ↦ edgeCentered lam m i) (g := fun m ↦ edgeCentered lam m i^2)
-    (compact_integrable (continuous_edgeCentered lam i))
-    (compact_integrable ((continuous_edgeCentered lam i).pow 2)),
-    Q.edgeCentered_mean, Q.edgeCentered_second, zero_add]
-
-lemma Law.edgeCentered_tilt_mean (Q : Law) (lam : ℝ) (i : Fin 3) :
-    (∫ m, edgeMessage lam m i*edgeCentered lam m 0 ∂Q.probability.toMeasure) =
-      if i=0 then 2*(lam^2*Q.mu) else -(lam^2*Q.mu) := by
-  have hi (i : Fin 3) : Integrable (fun m ↦ edgeMessage lam m i*edgeCentered lam m 0) Q.probability.toMeasure :=
-    compact_integrable (((continuous_apply i).comp (continuous_edgeMessage lam)).mul (continuous_edgeCentered lam 0))
-  have he : (∫ m, edgeMessage lam m 1*edgeCentered lam m 0 ∂Q.probability.toMeasure) =
-      ∫ m, edgeMessage lam m 2*edgeCentered lam m 0 ∂Q.probability.toMeasure := by
-    have hh := (show MeasurePreserving (permuteMeasurableEquiv (Equiv.swap 1 2))
-      Q.probability.toMeasure Q.probability.toMeasure from Q.symmetric _).integral_comp'
-        (fun m ↦ edgeMessage lam m 1*edgeCentered lam m 0)
-    simpa [permuteMeasurableEquiv, edgeCentered, edge_permute, Equiv.swap_apply_def] using hh.symm
-  have hsum : avg (fun i ↦ ∫ m, edgeMessage lam m i*edgeCentered lam m 0 ∂Q.probability.toMeasure) = 0 := by
-    rw [← integral_avg_measure hi]
-    have hh (m : Message) : avg (fun i ↦ edgeMessage lam m i*edgeCentered lam m 0) = edgeCentered lam m 0 := by
-      have hav : avg (edgeMessage lam m) = 1 := by
-        dsimp only [edgeMessage, edge, avg]
-        have hm := m.2.2
-        dsimp only [avg] at hm
-        linear_combination lam*hm
-      calc
-        _ = avg (edgeMessage lam m)*edgeCentered lam m 0 := by dsimp [avg]; ring
-        _ = _ := by rw [hav, one_mul]
-    simp_rw [hh]
-    exact Q.edgeCentered_mean lam 0
-  dsimp only [avg] at hsum
-  rw [Q.edgeCentered_tilt_diag lam 0, he] at hsum
-  fin_cases i
-  · simp [Q.edgeCentered_tilt_diag]; ring
-  · change (∫ m, edgeMessage lam m 1*edgeCentered lam m 0 ∂Q.probability.toMeasure) = -(lam^2*Q.mu)
-    rw [he]; linarith
-  · change (∫ m, edgeMessage lam m 2*edgeCentered lam m 0 ∂Q.probability.toMeasure) = -(lam^2*Q.mu)
-    linarith
-
-def branchSum (lam : ℝ) {n : ℕ} (m : Fin n → Message) : ℝ := ∑ j, edgeCentered lam (m j) 0
-
-lemma continuous_branchSum (lam : ℝ) (n : ℕ) : Continuous (branchSum lam (n := n)) :=
-  continuous_finsetSum _ (fun j _ ↦ (continuous_edgeCentered lam 0).comp (continuous_apply j))
-
-lemma integral_branchProduct_sum (Q : Law) (lam : ℝ) (n : ℕ) (i : Fin 3) :
-    (∫ m : Fin n → Message, branchProduct (branchEdges lam m) i*branchSum lam m
-      ∂(Measure.pi fun _ ↦ Q.probability.toMeasure)) =
-      n*(if i=0 then 2*(lam^2*Q.mu) else -(lam^2*Q.mu)) := by
-  have hh := integral_weighted_sum (ι := Fin n)
-    (compact_integrable ((continuous_apply i).comp (continuous_edgeMessage lam)))
-    (compact_integrable (((continuous_apply i).comp (continuous_edgeMessage lam)).mul
-      (continuous_edgeCentered lam 0))) (Q.edge_balanced lam i)
-  simp only [Function.comp_apply] at hh
-  rw [Q.edgeCentered_tilt_mean] at hh
-  simpa only [weightProduct, branchProduct, branchEdges, branchSum, Fintype.card_fin, Function.comp_apply] using hh
-
-lemma integral_normalizer_sum (Q : Law) (lam : ℝ) (n : ℕ) :
-    (∫ m : Fin n → Message, normalizer (branchEdges lam m)*branchSum lam m
-      ∂(Measure.pi fun _ ↦ Q.probability.toMeasure)) = 0 := by
-  have he (m : Fin n → Message) : normalizer (branchEdges lam m)*branchSum lam m =
-      avg (fun i ↦ branchProduct (branchEdges lam m) i*branchSum lam m) := by dsimp [normalizer, avg]; ring
-  simp_rw [he]
-  rw [integral_avg_measure (f := fun m : Fin n → Message ↦ fun i ↦ branchProduct (branchEdges lam m) i*branchSum lam m)
-    (fun i ↦ compact_integrable ((continuous_branchProduct lam i).mul (continuous_branchSum lam n)))]
-  simp_rw [integral_branchProduct_sum]
-  norm_num [avg, show (2 : Fin 3) ≠ 0 from by decide]
-  ring
-
-lemma integral_normalizer_sq_sum (Q : Law) (lam : ℝ) (n : ℕ) :
-    (∫ m : Fin n → Message, normalizer (branchEdges lam m)*branchSum lam m^2
-      ∂(Measure.pi fun _ ↦ Q.probability.toMeasure)) =
-      2*n*(lam^2*Q.mu)+2*((n:ℝ)^2-n)*(lam^2*Q.mu)^2 := by
-  have he (m : Fin n → Message) : normalizer (branchEdges lam m)*branchSum lam m^2 =
-      avg (fun i ↦ branchProduct (branchEdges lam m) i*branchSum lam m^2) := by dsimp [normalizer, avg]; ring
-  simp_rw [he]
-  rw [integral_avg_measure (f := fun m : Fin n → Message ↦ fun i ↦ branchProduct (branchEdges lam m) i*branchSum lam m^2)
-    (fun i ↦ compact_integrable ((continuous_branchProduct lam i).mul ((continuous_branchSum lam n).pow 2)))]
-  have hh (i : Fin 3) : (∫ m : Fin n → Message, branchProduct (branchEdges lam m) i*branchSum lam m^2
-      ∂(Measure.pi fun _ ↦ Q.probability.toMeasure)) =
-      n*(∫ m, edgeMessage lam m i*edgeCentered lam m 0^2 ∂Q.probability.toMeasure)+
-      ((n:ℝ)^2-n)*(if i=0 then 2*(lam^2*Q.mu) else -(lam^2*Q.mu))^2 := by
-    have h := integral_weighted_sq_sum (ι := Fin n)
-      (compact_integrable ((continuous_apply i).comp (continuous_edgeMessage lam)))
-      (compact_integrable (((continuous_apply i).comp (continuous_edgeMessage lam)).mul
-        (continuous_edgeCentered lam 0)))
-      (compact_integrable (((continuous_apply i).comp (continuous_edgeMessage lam)).mul
-        ((continuous_edgeCentered lam 0).pow 2))) (Q.edge_balanced lam i)
-    simp only [Function.comp_apply] at h
-    rw [Q.edgeCentered_tilt_mean] at h
-    simpa only [weightProduct, branchProduct, branchEdges, branchSum, Fintype.card_fin, Function.comp_apply] using h
-  simp_rw [hh]
-  rw [avg_add, avg_mul]
-  have hmean : avg (fun i ↦ ∫ m, edgeMessage lam m i*edgeCentered lam m 0^2 ∂Q.probability.toMeasure) =
-      2*lam^2*Q.mu := by
-    rw [← integral_avg_measure (f := fun m : Message ↦ fun i ↦ edgeMessage lam m i*edgeCentered lam m 0^2)
-      (fun i ↦ compact_integrable (((continuous_apply i).comp (continuous_edgeMessage lam)).mul
-      ((continuous_edgeCentered lam 0).pow 2)))]
-    have hh (m : Message) : avg (fun i ↦ edgeMessage lam m i*edgeCentered lam m 0^2) = edgeCentered lam m 0^2 := by
-      have hm := m.2.2
-      dsimp only [edgeMessage, edge, edgeCentered, avg] at *
-      linear_combination (1+lam*(m.1 0-1)-1)^2*lam*hm
-    simp_rw [hh]
-    exact Q.edgeCentered_second lam 0
-  rw [hmean]
-  norm_num [avg, show (2 : Fin 3) ≠ 0 from by decide]
-  ring
-
-lemma Law.centered_second (Q : Law) (i : Fin 3) :
-    (∫ m, (m.1 i-1)^2 ∂Q.probability.toMeasure) = 2*Q.mu := by
-  simpa [edgeCentered, edgeMessage, edge] using Q.edgeCentered_second 1 i
-
-lemma integral_posterior_sum (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
-    (∫ m : Fin n → Message, ((branchOutput hl₀ hl₁ m).1 0-1)*branchSum lam m
-      ∂tiltedProduct Q lam n) = 2*n*(lam^2*Q.mu) := by
-  rw [integral_tiltedProduct Q hl₀ hl₁]
-  have he (m : Fin n → Message) :
-      normalizer (branchEdges lam m)*(((branchOutput hl₀ hl₁ m).1 0-1)*branchSum lam m) =
-      branchProduct (branchEdges lam m) 0*branchSum lam m-normalizer (branchEdges lam m)*branchSum lam m := by
-    change normalizer (branchEdges lam m)*((branchProduct (branchEdges lam m) 0/normalizer (branchEdges lam m)-1)*branchSum lam m) = _
-    field_simp [ne_of_gt (normalizer_positive (branchEdges_positive hl₀ hl₁ m))]
-  simp_rw [he]
-  rw [integral_sub (f := fun m : Fin n → Message ↦ branchProduct (branchEdges lam m) 0*branchSum lam m)
-    (g := fun m ↦ normalizer (branchEdges lam m)*branchSum lam m)
-    (compact_integrable ((continuous_branchProduct lam 0).mul (continuous_branchSum lam n)))
-    (compact_integrable ((continuous_normalizer n lam).mul (continuous_branchSum lam n))),
-    integral_branchProduct_sum, integral_normalizer_sum]
-  simp; ring
-
-lemma finite_projection_quadratic (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
-    (n : ℕ) (a : ℝ) :
-    0 ≤ (2*n*(lam^2*Q.mu)+2*((n:ℝ)^2-n)*(lam^2*Q.mu)^2)*(a*a)+
-      (-4*n*(lam^2*Q.mu))*a+2*(finiteLaw Q hl₀ hl₁ n).mu := by
-  let beta := lam^2*Q.mu
-  have : IsProbabilityMeasure (tiltedProduct Q lam n) := tiltedProduct_isProbability Q hl₀ hl₁ n
-  let v : (Fin n → Message) → ℝ := fun m ↦ (branchOutput hl₀ hl₁ m).1 0-1
-  have hv : Continuous v := ((continuous_coordinate 0).comp (continuous_branchOutput hl₀ hl₁ n)).sub continuous_const
-  have hv₂ : (∫ m, v m^2 ∂tiltedProduct Q lam n) = 2*(finiteLaw Q hl₀ hl₁ n).mu := by
-    have hh := (finiteLaw Q hl₀ hl₁ n).centered_second 0
-    change (∫ m, (m.1 0-1)^2 ∂(finiteProbability Q hl₀ hl₁ n).toMeasure) = _ at hh
-    change (∫ m, (m.1 0-1)^2 ∂Measure.map (branchOutput hl₀ hl₁) (tiltedProduct Q lam n)) = _ at hh
-    rw [integral_map (f := fun m : Message ↦ (m.1 0-1)^2) (continuous_branchOutput hl₀ hl₁ n).measurable.aemeasurable
-      (((continuous_coordinate 0).sub continuous_const).pow 2).measurable.aestronglyMeasurable] at hh
-    exact hh
-  have hT₂ : (∫ m, branchSum lam m^2 ∂tiltedProduct Q lam n) =
-      2*n*beta+2*((n:ℝ)^2-n)*beta^2 := by
-    rw [integral_tiltedProduct Q hl₀ hl₁]
-    exact integral_normalizer_sq_sum Q lam n
-  have hvT : (∫ m, v m*branchSum lam m ∂tiltedProduct Q lam n) = 2*n*beta :=
-    integral_posterior_sum Q hl₀ hl₁ n
-  show 0 ≤ (2*n*beta+2*((n:ℝ)^2-n)*beta^2)*(a*a)+
-      (-4*n*beta)*a+2*(finiteLaw Q hl₀ hl₁ n).mu
-  have hh := integral_nonneg (μ := tiltedProduct Q lam n)
-    (f := fun m : Fin n → Message ↦ (v m-a*branchSum lam m)^2) (fun m ↦ sq_nonneg (v m-a*branchSum lam m))
-  have he (m : Fin n → Message) : (v m-a*branchSum lam m)^2 =
-      v m^2-2*a*(v m*branchSum lam m)+a^2*branchSum lam m^2 := by ring
-  simp_rw [he] at hh
-  rw [integral_add (f := fun m ↦ v m^2-2*a*(v m*branchSum lam m))
-    (g := fun m ↦ a^2*branchSum lam m^2)
-    ((compact_integrable (hv.pow 2)).sub ((compact_integrable (hv.mul (continuous_branchSum lam n))).const_mul _))
-    ((compact_integrable ((continuous_branchSum lam n).pow 2)).const_mul _),
-    integral_sub (f := fun m ↦ v m^2) (g := fun m ↦ 2*a*(v m*branchSum lam m))
-      (compact_integrable (hv.pow 2)) ((compact_integrable (hv.mul (continuous_branchSum lam n))).const_mul _),
-    integral_const_mul, integral_const_mul, hv₂, hvT, hT₂] at hh
-  nlinarith [hh]
-
-theorem finite_projection (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
-    (n : ℕ) (hn : 1 ≤ n) :
-    (n : ℝ)*(lam^2*Q.mu)/(1+((n:ℝ)-1)*(lam^2*Q.mu)) ≤ (finiteLaw Q hl₀ hl₁ n).mu := by
-  let beta := lam^2*Q.mu
-  have hb : 0 ≤ beta := mul_nonneg (sq_nonneg _) Q.mu_nonneg
-  have hn' : (1:ℝ) ≤ n := by exact_mod_cast hn
-  have hd : 0 < 1+((n:ℝ)-1)*beta := by nlinarith [mul_nonneg (sub_nonneg.mpr hn') hb]
-  have : IsProbabilityMeasure (tiltedProduct Q lam n) := tiltedProduct_isProbability Q hl₀ hl₁ n
-  let v : (Fin n → Message) → ℝ := fun m ↦ (branchOutput hl₀ hl₁ m).1 0-1
-  have hv : Continuous v := ((continuous_coordinate 0).comp (continuous_branchOutput hl₀ hl₁ n)).sub continuous_const
-  have hv₂ : (∫ m, v m^2 ∂tiltedProduct Q lam n) = 2*(finiteLaw Q hl₀ hl₁ n).mu := by
-    have hh := (finiteLaw Q hl₀ hl₁ n).centered_second 0
-    change (∫ m, (m.1 0-1)^2 ∂(finiteProbability Q hl₀ hl₁ n).toMeasure) = _ at hh
-    change (∫ m, (m.1 0-1)^2 ∂Measure.map (branchOutput hl₀ hl₁) (tiltedProduct Q lam n)) = _ at hh
-    rw [integral_map (f := fun m : Message ↦ (m.1 0-1)^2) (continuous_branchOutput hl₀ hl₁ n).measurable.aemeasurable
-      (((continuous_coordinate 0).sub continuous_const).pow 2).measurable.aestronglyMeasurable] at hh
-    exact hh
-  have hT₂ : (∫ m, branchSum lam m^2 ∂tiltedProduct Q lam n) =
-      2*n*beta+2*((n:ℝ)^2-n)*beta^2 := by
-    rw [integral_tiltedProduct Q hl₀ hl₁]
-    exact integral_normalizer_sq_sum Q lam n
-  have hvT : (∫ m, v m*branchSum lam m ∂tiltedProduct Q lam n) = 2*n*beta :=
-    integral_posterior_sum Q hl₀ hl₁ n
-  have hquad (a : ℝ) : 0 ≤ (2*n*beta+2*((n:ℝ)^2-n)*beta^2)*(a*a)+
-      (-4*n*beta)*a+2*(finiteLaw Q hl₀ hl₁ n).mu := by
-    have hh := integral_nonneg (μ := tiltedProduct Q lam n)
-      (f := fun m : Fin n → Message ↦ (v m-a*branchSum lam m)^2) (fun m ↦ sq_nonneg (v m-a*branchSum lam m))
-    have he (m : Fin n → Message) : (v m-a*branchSum lam m)^2 =
-        v m^2-2*a*(v m*branchSum lam m)+a^2*branchSum lam m^2 := by ring
-    simp_rw [he] at hh
-    rw [integral_add (f := fun m ↦ v m^2-2*a*(v m*branchSum lam m))
-      (g := fun m ↦ a^2*branchSum lam m^2)
-      ((compact_integrable (hv.pow 2)).sub ((compact_integrable (hv.mul (continuous_branchSum lam n))).const_mul _))
-      ((compact_integrable ((continuous_branchSum lam n).pow 2)).const_mul _),
-      integral_sub (f := fun m ↦ v m^2) (g := fun m ↦ 2*a*(v m*branchSum lam m))
-        (compact_integrable (hv.pow 2)) ((compact_integrable (hv.mul (continuous_branchSum lam n))).const_mul _),
-      integral_const_mul, integral_const_mul, hv₂, hvT, hT₂] at hh
-    nlinarith [hh]
-  have hh := discrim_le_zero hquad
-  dsimp only [discrim] at hh
-  have hcs : ((n:ℝ)*beta)^2 ≤ (finiteLaw Q hl₀ hl₁ n).mu*((n:ℝ)*beta+((n:ℝ)^2-n)*beta^2) := by nlinarith [hh]
-  rcases eq_or_lt_of_le hb with hb0 | hbpos
-  · have hb0' : beta = 0 := hb0.symm
-    change (n:ℝ)*beta/(1+((n:ℝ)-1)*beta) ≤ _
-    simp only [hb0', mul_zero, add_zero, zero_div]
-    exact (finiteLaw Q hl₀ hl₁ n).mu_nonneg
-  · have hnb : 0 < (n:ℝ)*beta := mul_pos (lt_of_lt_of_le zero_lt_one hn') hbpos
-    apply (div_le_iff₀ hd).2
-    apply le_of_mul_le_mul_left (a := (n:ℝ)*beta) (b := (n:ℝ)*beta) (c := (finiteLaw Q hl₀ hl₁ n).mu*(1+((n:ℝ)-1)*beta)) _ hnb
-    calc
-      (n:ℝ)*beta*((n:ℝ)*beta) = ((n:ℝ)*beta)^2 := by ring
-      _ ≤ _ := hcs
-      _ = _ := by ring
-
-end ThreeState.TreeClauses.Experiment
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeStateClauses.FiniteUpdate
+import OAI.Probability.ThreeStateClauses.Correlation
+
+namespace OAI
+
+/-! Projection. -/
+
+open scoped BigOperators ENNReal NNReal Topology
+open Filter
+noncomputable section
+open Set MeasureTheory
+open scoped BigOperators
+namespace ThreeState.TreeClauses.Experiment
+open ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
+
+def edgeCentered (lam : ℝ) (m : Message) (i : Fin 3) : ℝ := edgeMessage lam m i-1
+
+lemma continuous_edgeCentered (lam : ℝ) (i : Fin 3) :
+    Continuous (fun m : Message ↦ edgeCentered lam m i) :=
+  ((continuous_apply i).comp (continuous_edgeMessage lam)).sub continuous_const
+
+lemma Law.edgeCentered_mean (Q : Law) (lam : ℝ) (i : Fin 3) :
+    (∫ m, edgeCentered lam m i ∂Q.probability.toMeasure) = 0 := by
+  unfold edgeCentered
+  rw [integral_sub (f := fun m ↦ edgeMessage lam m i) (g := fun _ ↦ (1:ℝ))
+    (compact_integrable ((continuous_apply i).comp (continuous_edgeMessage lam))) (integrable_const _),
+    Q.edge_balanced]
+  simp
+
+lemma Law.edgeCentered_second (Q : Law) (lam : ℝ) (i : Fin 3) :
+    (∫ m, edgeCentered lam m i^2 ∂Q.probability.toMeasure) = 2*lam^2*Q.mu := by
+  rw [Q.integral_equivariant (f := fun m i ↦ edgeCentered lam m i^2)
+    (fun i ↦ compact_integrable ((continuous_edgeCentered lam i).pow 2))
+    (fun _ _ _ ↦ rfl)]
+  have he (m : Message) : avg (fun i ↦ edgeCentered lam m i^2) = 2*lam^2*xMoment m := by
+    have hh := xMoment_edge lam m
+    change (avg (fun i ↦ edgeCentered lam m i^2))/2 = _ at hh
+    linarith
+  simp_rw [he]
+  rw [integral_const_mul]
+  rfl
+
+lemma Law.edgeCentered_tilt_diag (Q : Law) (lam : ℝ) (i : Fin 3) :
+    (∫ m, edgeMessage lam m i*edgeCentered lam m i ∂Q.probability.toMeasure) = 2*lam^2*Q.mu := by
+  have he (m : Message) : edgeMessage lam m i*edgeCentered lam m i =
+      edgeCentered lam m i+edgeCentered lam m i^2 := by unfold edgeCentered; ring
+  simp_rw [he]
+  rw [integral_add (f := fun m ↦ edgeCentered lam m i) (g := fun m ↦ edgeCentered lam m i^2)
+    (compact_integrable (continuous_edgeCentered lam i))
+    (compact_integrable ((continuous_edgeCentered lam i).pow 2)),
+    Q.edgeCentered_mean, Q.edgeCentered_second, zero_add]
+
+lemma Law.edgeCentered_tilt_mean (Q : Law) (lam : ℝ) (i : Fin 3) :
+    (∫ m, edgeMessage lam m i*edgeCentered lam m 0 ∂Q.probability.toMeasure) =
+      if i=0 then 2*(lam^2*Q.mu) else -(lam^2*Q.mu) := by
+  have hi (i : Fin 3) : Integrable (fun m ↦ edgeMessage lam m i*edgeCentered lam m 0) Q.probability.toMeasure :=
+    compact_integrable (((continuous_apply i).comp (continuous_edgeMessage lam)).mul (continuous_edgeCentered lam 0))
+  have he : (∫ m, edgeMessage lam m 1*edgeCentered lam m 0 ∂Q.probability.toMeasure) =
+      ∫ m, edgeMessage lam m 2*edgeCentered lam m 0 ∂Q.probability.toMeasure := by
+    have hh := (show MeasurePreserving (permuteMeasurableEquiv (Equiv.swap 1 2))
+      Q.probability.toMeasure Q.probability.toMeasure from Q.symmetric _).integral_comp'
+        (fun m ↦ edgeMessage lam m 1*edgeCentered lam m 0)
+    simpa [permuteMeasurableEquiv, edgeCentered, edge_permute, Equiv.swap_apply_def] using hh.symm
+  have hsum : avg (fun i ↦ ∫ m, edgeMessage lam m i*edgeCentered lam m 0 ∂Q.probability.toMeasure) = 0 := by
+    rw [← integral_avg_measure hi]
+    have hh (m : Message) : avg (fun i ↦ edgeMessage lam m i*edgeCentered lam m 0) = edgeCentered lam m 0 := by
+      have hav : avg (edgeMessage lam m) = 1 := by
+        dsimp only [edgeMessage, edge, avg]
+        have hm := m.2.2
+        dsimp only [avg] at hm
+        linear_combination lam*hm
+      calc
+        _ = avg (edgeMessage lam m)*edgeCentered lam m 0 := by dsimp [avg]; ring
+        _ = _ := by rw [hav, one_mul]
+    simp_rw [hh]
+    exact Q.edgeCentered_mean lam 0
+  dsimp only [avg] at hsum
+  rw [Q.edgeCentered_tilt_diag lam 0, he] at hsum
+  fin_cases i
+  · simp [Q.edgeCentered_tilt_diag]; ring
+  · change (∫ m, edgeMessage lam m 1*edgeCentered lam m 0 ∂Q.probability.toMeasure) = -(lam^2*Q.mu)
+    rw [he]; linarith
+  · change (∫ m, edgeMessage lam m 2*edgeCentered lam m 0 ∂Q.probability.toMeasure) = -(lam^2*Q.mu)
+    linarith
+
+def branchSum (lam : ℝ) {n : ℕ} (m : Fin n → Message) : ℝ := ∑ j, edgeCentered lam (m j) 0
+
+lemma continuous_branchSum (lam : ℝ) (n : ℕ) : Continuous (branchSum lam (n := n)) :=
+  continuous_finsetSum _ (fun j _ ↦ (continuous_edgeCentered lam 0).comp (continuous_apply j))
+
+lemma integral_branchProduct_sum (Q : Law) (lam : ℝ) (n : ℕ) (i : Fin 3) :
+    (∫ m : Fin n → Message, branchProduct (branchEdges lam m) i*branchSum lam m
+      ∂(Measure.pi fun _ ↦ Q.probability.toMeasure)) =
+      n*(if i=0 then 2*(lam^2*Q.mu) else -(lam^2*Q.mu)) := by
+  have hh := integral_weighted_sum (ι := Fin n)
+    (compact_integrable ((continuous_apply i).comp (continuous_edgeMessage lam)))
+    (compact_integrable (((continuous_apply i).comp (continuous_edgeMessage lam)).mul
+      (continuous_edgeCentered lam 0))) (Q.edge_balanced lam i)
+  simp only [Function.comp_apply] at hh
+  rw [Q.edgeCentered_tilt_mean] at hh
+  simpa only [weightProduct, branchProduct, branchEdges, branchSum, Fintype.card_fin, Function.comp_apply] using hh
+
+lemma integral_normalizer_sum (Q : Law) (lam : ℝ) (n : ℕ) :
+    (∫ m : Fin n → Message, normalizer (branchEdges lam m)*branchSum lam m
+      ∂(Measure.pi fun _ ↦ Q.probability.toMeasure)) = 0 := by
+  have he (m : Fin n → Message) : normalizer (branchEdges lam m)*branchSum lam m =
+      avg (fun i ↦ branchProduct (branchEdges lam m) i*branchSum lam m) := by dsimp [normalizer, avg]; ring
+  simp_rw [he]
+  rw [integral_avg_measure (f := fun m : Fin n → Message ↦ fun i ↦ branchProduct (branchEdges lam m) i*branchSum lam m)
+    (fun i ↦ compact_integrable ((continuous_branchProduct lam i).mul (continuous_branchSum lam n)))]
+  simp_rw [integral_branchProduct_sum]
+  norm_num [avg, show (2 : Fin 3) ≠ 0 from by decide]
+  ring
+
+lemma integral_normalizer_sq_sum (Q : Law) (lam : ℝ) (n : ℕ) :
+    (∫ m : Fin n → Message, normalizer (branchEdges lam m)*branchSum lam m^2
+      ∂(Measure.pi fun _ ↦ Q.probability.toMeasure)) =
+      2*n*(lam^2*Q.mu)+2*((n:ℝ)^2-n)*(lam^2*Q.mu)^2 := by
+  have he (m : Fin n → Message) : normalizer (branchEdges lam m)*branchSum lam m^2 =
+      avg (fun i ↦ branchProduct (branchEdges lam m) i*branchSum lam m^2) := by dsimp [normalizer, avg]; ring
+  simp_rw [he]
+  rw [integral_avg_measure (f := fun m : Fin n → Message ↦ fun i ↦ branchProduct (branchEdges lam m) i*branchSum lam m^2)
+    (fun i ↦ compact_integrable ((continuous_branchProduct lam i).mul ((continuous_branchSum lam n).pow 2)))]
+  have hh (i : Fin 3) : (∫ m : Fin n → Message, branchProduct (branchEdges lam m) i*branchSum lam m^2
+      ∂(Measure.pi fun _ ↦ Q.probability.toMeasure)) =
+      n*(∫ m, edgeMessage lam m i*edgeCentered lam m 0^2 ∂Q.probability.toMeasure)+
+      ((n:ℝ)^2-n)*(if i=0 then 2*(lam^2*Q.mu) else -(lam^2*Q.mu))^2 := by
+    have h := integral_weighted_sq_sum (ι := Fin n)
+      (compact_integrable ((continuous_apply i).comp (continuous_edgeMessage lam)))
+      (compact_integrable (((continuous_apply i).comp (continuous_edgeMessage lam)).mul
+        (continuous_edgeCentered lam 0)))
+      (compact_integrable (((continuous_apply i).comp (continuous_edgeMessage lam)).mul
+        ((continuous_edgeCentered lam 0).pow 2))) (Q.edge_balanced lam i)
+    simp only [Function.comp_apply] at h
+    rw [Q.edgeCentered_tilt_mean] at h
+    simpa only [weightProduct, branchProduct, branchEdges, branchSum, Fintype.card_fin, Function.comp_apply] using h
+  simp_rw [hh]
+  rw [avg_add, avg_mul]
+  have hmean : avg (fun i ↦ ∫ m, edgeMessage lam m i*edgeCentered lam m 0^2 ∂Q.probability.toMeasure) =
+      2*lam^2*Q.mu := by
+    rw [← integral_avg_measure (f := fun m : Message ↦ fun i ↦ edgeMessage lam m i*edgeCentered lam m 0^2)
+      (fun i ↦ compact_integrable (((continuous_apply i).comp (continuous_edgeMessage lam)).mul
+      ((continuous_edgeCentered lam 0).pow 2)))]
+    have hh (m : Message) : avg (fun i ↦ edgeMessage lam m i*edgeCentered lam m 0^2) = edgeCentered lam m 0^2 := by
+      have hm := m.2.2
+      dsimp only [edgeMessage, edge, edgeCentered, avg] at *
+      linear_combination (1+lam*(m.1 0-1)-1)^2*lam*hm
+    simp_rw [hh]
+    exact Q.edgeCentered_second lam 0
+  rw [hmean]
+  norm_num [avg, show (2 : Fin 3) ≠ 0 from by decide]
+  ring
+
+lemma Law.centered_second (Q : Law) (i : Fin 3) :
+    (∫ m, (m.1 i-1)^2 ∂Q.probability.toMeasure) = 2*Q.mu := by
+  simpa [edgeCentered, edgeMessage, edge] using Q.edgeCentered_second 1 i
+
+lemma integral_posterior_sum (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1) (n : ℕ) :
+    (∫ m : Fin n → Message, ((branchOutput hl₀ hl₁ m).1 0-1)*branchSum lam m
+      ∂tiltedProduct Q lam n) = 2*n*(lam^2*Q.mu) := by
+  rw [integral_tiltedProduct Q hl₀ hl₁]
+  have he (m : Fin n → Message) :
+      normalizer (branchEdges lam m)*(((branchOutput hl₀ hl₁ m).1 0-1)*branchSum lam m) =
+      branchProduct (branchEdges lam m) 0*branchSum lam m-normalizer (branchEdges lam m)*branchSum lam m := by
+    change normalizer (branchEdges lam m)*((branchProduct (branchEdges lam m) 0/normalizer (branchEdges lam m)-1)*branchSum lam m) = _
+    field_simp [ne_of_gt (normalizer_positive (branchEdges_positive hl₀ hl₁ m))]
+  simp_rw [he]
+  rw [integral_sub (f := fun m : Fin n → Message ↦ branchProduct (branchEdges lam m) 0*branchSum lam m)
+    (g := fun m ↦ normalizer (branchEdges lam m)*branchSum lam m)
+    (compact_integrable ((continuous_branchProduct lam 0).mul (continuous_branchSum lam n)))
+    (compact_integrable ((continuous_normalizer n lam).mul (continuous_branchSum lam n))),
+    integral_branchProduct_sum, integral_normalizer_sum]
+  simp; ring
+
+lemma finite_projection_quadratic (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
+    (n : ℕ) (a : ℝ) :
+    0 ≤ (2*n*(lam^2*Q.mu)+2*((n:ℝ)^2-n)*(lam^2*Q.mu)^2)*(a*a)+
+      (-4*n*(lam^2*Q.mu))*a+2*(finiteLaw Q hl₀ hl₁ n).mu := by
+  let beta := lam^2*Q.mu
+  have : IsProbabilityMeasure (tiltedProduct Q lam n) := tiltedProduct_isProbability Q hl₀ hl₁ n
+  let v : (Fin n → Message) → ℝ := fun m ↦ (branchOutput hl₀ hl₁ m).1 0-1
+  have hv : Continuous v := ((continuous_coordinate 0).comp (continuous_branchOutput hl₀ hl₁ n)).sub continuous_const
+  have hv₂ : (∫ m, v m^2 ∂tiltedProduct Q lam n) = 2*(finiteLaw Q hl₀ hl₁ n).mu := by
+    have hh := (finiteLaw Q hl₀ hl₁ n).centered_second 0
+    change (∫ m, (m.1 0-1)^2 ∂(finiteProbability Q hl₀ hl₁ n).toMeasure) = _ at hh
+    change (∫ m, (m.1 0-1)^2 ∂Measure.map (branchOutput hl₀ hl₁) (tiltedProduct Q lam n)) = _ at hh
+    rw [integral_map (f := fun m : Message ↦ (m.1 0-1)^2) (continuous_branchOutput hl₀ hl₁ n).measurable.aemeasurable
+      (((continuous_coordinate 0).sub continuous_const).pow 2).measurable.aestronglyMeasurable] at hh
+    exact hh
+  have hT₂ : (∫ m, branchSum lam m^2 ∂tiltedProduct Q lam n) =
+      2*n*beta+2*((n:ℝ)^2-n)*beta^2 := by
+    rw [integral_tiltedProduct Q hl₀ hl₁]
+    exact integral_normalizer_sq_sum Q lam n
+  have hvT : (∫ m, v m*branchSum lam m ∂tiltedProduct Q lam n) = 2*n*beta :=
+    integral_posterior_sum Q hl₀ hl₁ n
+  show 0 ≤ (2*n*beta+2*((n:ℝ)^2-n)*beta^2)*(a*a)+
+      (-4*n*beta)*a+2*(finiteLaw Q hl₀ hl₁ n).mu
+  have hh := integral_nonneg (μ := tiltedProduct Q lam n)
+    (f := fun m : Fin n → Message ↦ (v m-a*branchSum lam m)^2) (fun m ↦ sq_nonneg (v m-a*branchSum lam m))
+  have he (m : Fin n → Message) : (v m-a*branchSum lam m)^2 =
+      v m^2-2*a*(v m*branchSum lam m)+a^2*branchSum lam m^2 := by ring
+  simp_rw [he] at hh
+  rw [integral_add (f := fun m ↦ v m^2-2*a*(v m*branchSum lam m))
+    (g := fun m ↦ a^2*branchSum lam m^2)
+    ((compact_integrable (hv.pow 2)).sub ((compact_integrable (hv.mul (continuous_branchSum lam n))).const_mul _))
+    ((compact_integrable ((continuous_branchSum lam n).pow 2)).const_mul _),
+    integral_sub (f := fun m ↦ v m^2) (g := fun m ↦ 2*a*(v m*branchSum lam m))
+      (compact_integrable (hv.pow 2)) ((compact_integrable (hv.mul (continuous_branchSum lam n))).const_mul _),
+    integral_const_mul, integral_const_mul, hv₂, hvT, hT₂] at hh
+  nlinarith [hh]
+
+theorem finite_projection (Q : Law) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
+    (n : ℕ) (hn : 1 ≤ n) :
+    (n : ℝ)*(lam^2*Q.mu)/(1+((n:ℝ)-1)*(lam^2*Q.mu)) ≤ (finiteLaw Q hl₀ hl₁ n).mu := by
+  let beta := lam^2*Q.mu
+  have hb : 0 ≤ beta := mul_nonneg (sq_nonneg _) Q.mu_nonneg
+  have hn' : (1:ℝ) ≤ n := by exact_mod_cast hn
+  have hd : 0 < 1+((n:ℝ)-1)*beta := by nlinarith [mul_nonneg (sub_nonneg.mpr hn') hb]
+  have : IsProbabilityMeasure (tiltedProduct Q lam n) := tiltedProduct_isProbability Q hl₀ hl₁ n
+  let v : (Fin n → Message) → ℝ := fun m ↦ (branchOutput hl₀ hl₁ m).1 0-1
+  have hv : Continuous v := ((continuous_coordinate 0).comp (continuous_branchOutput hl₀ hl₁ n)).sub continuous_const
+  have hv₂ : (∫ m, v m^2 ∂tiltedProduct Q lam n) = 2*(finiteLaw Q hl₀ hl₁ n).mu := by
+    have hh := (finiteLaw Q hl₀ hl₁ n).centered_second 0
+    change (∫ m, (m.1 0-1)^2 ∂(finiteProbability Q hl₀ hl₁ n).toMeasure) = _ at hh
+    change (∫ m, (m.1 0-1)^2 ∂Measure.map (branchOutput hl₀ hl₁) (tiltedProduct Q lam n)) = _ at hh
+    rw [integral_map (f := fun m : Message ↦ (m.1 0-1)^2) (continuous_branchOutput hl₀ hl₁ n).measurable.aemeasurable
+      (((continuous_coordinate 0).sub continuous_const).pow 2).measurable.aestronglyMeasurable] at hh
+    exact hh
+  have hT₂ : (∫ m, branchSum lam m^2 ∂tiltedProduct Q lam n) =
+      2*n*beta+2*((n:ℝ)^2-n)*beta^2 := by
+    rw [integral_tiltedProduct Q hl₀ hl₁]
+    exact integral_normalizer_sq_sum Q lam n
+  have hvT : (∫ m, v m*branchSum lam m ∂tiltedProduct Q lam n) = 2*n*beta :=
+    integral_posterior_sum Q hl₀ hl₁ n
+  have hquad (a : ℝ) : 0 ≤ (2*n*beta+2*((n:ℝ)^2-n)*beta^2)*(a*a)+
+      (-4*n*beta)*a+2*(finiteLaw Q hl₀ hl₁ n).mu := by
+    have hh := integral_nonneg (μ := tiltedProduct Q lam n)
+      (f := fun m : Fin n → Message ↦ (v m-a*branchSum lam m)^2) (fun m ↦ sq_nonneg (v m-a*branchSum lam m))
+    have he (m : Fin n → Message) : (v m-a*branchSum lam m)^2 =
+        v m^2-2*a*(v m*branchSum lam m)+a^2*branchSum lam m^2 := by ring
+    simp_rw [he] at hh
+    rw [integral_add (f := fun m ↦ v m^2-2*a*(v m*branchSum lam m))
+      (g := fun m ↦ a^2*branchSum lam m^2)
+      ((compact_integrable (hv.pow 2)).sub ((compact_integrable (hv.mul (continuous_branchSum lam n))).const_mul _))
+      ((compact_integrable ((continuous_branchSum lam n).pow 2)).const_mul _),
+      integral_sub (f := fun m ↦ v m^2) (g := fun m ↦ 2*a*(v m*branchSum lam m))
+        (compact_integrable (hv.pow 2)) ((compact_integrable (hv.mul (continuous_branchSum lam n))).const_mul _),
+      integral_const_mul, integral_const_mul, hv₂, hvT, hT₂] at hh
+    nlinarith [hh]
+  have hh := discrim_le_zero hquad
+  dsimp only [discrim] at hh
+  have hcs : ((n:ℝ)*beta)^2 ≤ (finiteLaw Q hl₀ hl₁ n).mu*((n:ℝ)*beta+((n:ℝ)^2-n)*beta^2) := by nlinarith [hh]
+  rcases eq_or_lt_of_le hb with hb0 | hbpos
+  · have hb0' : beta = 0 := hb0.symm
+    change (n:ℝ)*beta/(1+((n:ℝ)-1)*beta) ≤ _
+    simp only [hb0', mul_zero, add_zero, zero_div]
+    exact (finiteLaw Q hl₀ hl₁ n).mu_nonneg
+  · have hnb : 0 < (n:ℝ)*beta := mul_pos (lt_of_lt_of_le zero_lt_one hn') hbpos
+    apply (div_le_iff₀ hd).2
+    apply le_of_mul_le_mul_left (a := (n:ℝ)*beta) (b := (n:ℝ)*beta) (c := (finiteLaw Q hl₀ hl₁ n).mu*(1+((n:ℝ)-1)*beta)) _ hnb
+    calc
+      (n:ℝ)*beta*((n:ℝ)*beta) = ((n:ℝ)*beta)^2 := by ring
+      _ ≤ _ := hcs
+      _ = _ := by ring
+
+end ThreeState.TreeClauses.Experiment
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeStateClauses/RadialAlgebra.lean b/lean/OAI/Probability/ThreeStateClauses/RadialAlgebra.lean
index 2d42f24b3e3acd3ff40813487b0b6f3cebcf339b..e22fe422ec56f09921877cc5f6dac80523fa3156 100644
--- a/lean/OAI/Probability/ThreeStateClauses/RadialAlgebra.lean
+++ b/lean/OAI/Probability/ThreeStateClauses/RadialAlgebra.lean
@@ -1,161 +1,161 @@
-import Mathlib
-
-namespace OAI
-
-/-! Radial Algebra. -/
-
-open scoped BigOperators ENNReal NNReal Topology
-open Filter
-noncomputable section
-namespace ThreeState.TreeClauses.Radial
-
-abbrev Vec := Fin 3 → ℝ
-
-def avg (w : Vec) : ℝ := (w 0 + w 1 + w 2) / 3
-
-def PositiveMessage (m : Vec) : Prop := (∀ i, 0 < m i) ∧ avg m = 1
-
-def centered (m : Vec) : Vec := fun i ↦ m i - 1
-
-def momentX (v : Vec) : ℝ := avg (fun i ↦ v i ^ 2) / 2
-
-def momentY (v : Vec) : ℝ := avg (fun i ↦ v i ^ 3) / 2
-
-def logAverage (m : Vec) : ℝ := avg (fun i ↦ Real.log (m i))
-
-def logCentered (m : Vec) : Vec := fun i ↦ Real.log (m i) - logAverage m
-
-def info (m : Vec) : ℝ := avg (fun i ↦ m i * Real.log (m i))
-
-def symEntropy (m : Vec) : ℝ := avg (fun i ↦ (m i - 1) * Real.log (m i)) / 2
-
-def correction (m : Vec) : ℝ := avg (fun i ↦ m i * logCentered m i ^ 2) / 2
-
-def functional (m : Vec) : ℝ := 3 * info m - 2 * symEntropy m + (4 / 5) * correction m
-
-def edge (t : ℝ) (m : Vec) : Vec := fun i ↦ 1 + t * (m i - 1)
-
-def RadialInequality : Prop := ∀ (m : Vec), PositiveMessage m → ∀ (lam : ℝ),
-  0 < lam → lam < 1 →
-  (1 - lam ^ 2) / 2 * momentX (centered m) ^ 2 ≤
-    functional m - (lam ^ 2)⁻¹ * functional (edge lam m)
-
-lemma avg_add (v w : Vec) : avg (fun i ↦ v i + w i) = avg v + avg w := by
-  simp only [avg]; ring
-
-lemma avg_sub (v w : Vec) : avg (fun i ↦ v i - w i) = avg v - avg w := by
-  simp only [avg]; ring
-
-lemma avg_mul (a : ℝ) (v : Vec) : avg (fun i ↦ a * v i) = a * avg v := by
-  simp only [avg]; ring
-
-lemma avg_const (a : ℝ) : avg (fun _ ↦ a) = a := by simp only [avg]; ring
-
-lemma avg_centered {m : Vec} (hm : PositiveMessage m) : avg (centered m) = 0 := by
-  have h := hm.2
-  dsimp [centered, avg] at *
-  linarith
-
-lemma info_eq (m : Vec) : info m = logAverage m + 2 * symEntropy m := by
-  simp only [info, logAverage, symEntropy, avg]; ring
-
-lemma functional_eq (m : Vec) :
-    functional m = 3 * logAverage m + 4 * symEntropy m + (4 / 5) * correction m := by
-  rw [functional, info_eq]; ring
-
-lemma third_eq {v : Vec} (hv : avg v = 0) : v 2 = -v 0 - v 1 := by
-  simp only [avg] at hv; linarith
-
-lemma momentX_nonneg (v : Vec) : 0 ≤ momentX v := by
-  unfold momentX avg; positivity
-
-lemma cubic {v : Vec} (hv : avg v = 0) (i : Fin 3) :
-    v i ^ 3 = 3 * momentX v * v i + 2 * momentY v := by
-  have hc := third_eq hv
-  fin_cases i
-  · change v 0 ^ 3 = 3 * momentX v * v 0 + 2 * momentY v
-    simp only [momentX, momentY, avg]; rw [hc]; ring
-  · change v 1 ^ 3 = 3 * momentX v * v 1 + 2 * momentY v
-    simp only [momentX, momentY, avg]; rw [hc]; ring
-  · change v 2 ^ 3 = 3 * momentX v * v 2 + 2 * momentY v
-    simp only [momentX, momentY, avg]; rw [hc]; ring
-
-lemma product_eq {v : Vec} (hv : avg v = 0) (t : ℝ) :
-    (1 + t * v 0) * (1 + t * v 1) * (1 + t * v 2) =
-      1 - 3 * t ^ 2 * momentX v + 2 * t ^ 3 * momentY v := by
-  simp only [momentX, momentY, avg]; rw [third_eq hv]; ring
-
-lemma discriminant {v : Vec} (hv : avg v = 0) :
-    (v 0 - v 1) ^ 2 * (v 0 - v 2) ^ 2 * (v 1 - v 2) ^ 2 =
-      108 * (momentX v ^ 3 - momentY v ^ 2) := by
-  simp only [momentX, momentY, avg]; rw [third_eq hv]; ring
-
-lemma moment_discriminant {v : Vec} (hv : avg v = 0) :
-    momentY v ^ 2 ≤ momentX v ^ 3 := by
-  have h : 0 ≤ (v 0 - v 1) ^ 2 * (v 0 - v 2) ^ 2 * (v 1 - v 2) ^ 2 := by positivity
-  rw [discriminant hv] at h; linarith
-
-lemma momentX_lt_one {m : Vec} (hm : PositiveMessage m) : momentX (centered m) < 1 := by
-  have hs : m 0 + m 1 + m 2 = 3 := by simpa only [avg, div_eq_iff (by norm_num : (3:ℝ) ≠ 0), one_mul] using hm.2
-  have h01 := mul_pos (hm.1 0) (hm.1 1)
-  have h02 := mul_pos (hm.1 0) (hm.1 2)
-  have h12 := mul_pos (hm.1 1) (hm.1 2)
-  dsimp [momentX, avg, centered]
-  nlinarith [sq_nonneg (m 0 + m 1 + m 2 - 3)]
-
-lemma denominator_pos {m : Vec} (hm : PositiveMessage m) (t : ℝ)
-    (ht : t ∈ Set.Icc (0 : ℝ) 1) :
-    0 < 1 - 3 * t ^ 2 * momentX (centered m) + 2 * t ^ 3 * momentY (centered m) := by
-  rw [← product_eq (avg_centered hm)]
-  have hi (i : Fin 3) : 0 < 1 + t * centered m i := by
-    dsimp [centered]
-    by_cases h : t = 0
-    · subst t; norm_num
-    · have ht0 : 0 < t := lt_of_le_of_ne ht.1 (Ne.symm h)
-      have := mul_pos ht0 (hm.1 i)
-      nlinarith [ht.2]
-  exact mul_pos (mul_pos (hi 0) (hi 1)) (hi 2)
-
-def hpoly (v : Vec) : Vec := fun i ↦ 1 - v i + v i ^ 2 - 3 * momentX v
-
-def Hpoly (j : ℕ) (v : Vec) : ℝ := avg (fun i ↦ hpoly v i ^ j)
-
-lemma Hpoly_one {v : Vec} (hv : avg v = 0) : Hpoly 1 v = 1 - momentX v := by
-  simp only [Hpoly, hpoly, momentX, avg, pow_one]; rw [third_eq hv]; ring
-
-lemma Hpoly_two {v : Vec} (hv : avg v = 0) :
-    Hpoly 2 v = 3 * momentX v ^ 2 - 4 * momentY v + 1 := by
-  simp only [Hpoly, hpoly, momentX, momentY, avg]; rw [third_eq hv]; ring
-
-lemma Hpoly_three {v : Vec} (hv : avg v = 0) :
-    Hpoly 3 v = -9 * momentX v ^ 3 + 9 * momentX v ^ 2 +
-      6 * momentX v * momentY v + 3 * momentX v + 4 * momentY v ^ 2 - 14 * momentY v + 1 := by
-  simp only [Hpoly, hpoly, momentX, momentY, avg]; rw [third_eq hv]; ring
-
-def polyA (x y : ℝ) : ℝ :=
-  3*x^4 + 12*x^3 + x^2 + (-6*x^3 - 28*x^2 - 2*x)*y + (26*x - 2)*y^2 - 4*y^3
-
-def polyC (x y : ℝ) : ℝ :=
-  6*x^4 - 2*x^3 + (-13*x^3 - 10*x^2 - x)*y + (10*x^2 + 22*x + 4)*y^2 - 16*y^3
-
-def polyN (x y t : ℝ) : ℝ := (24/5) * (polyA x y + t * polyC x y)
-
-def polyL (x y : ℝ) : ℝ :=
-  1500*x^5 - 1020*x^4 + 240*x^3 + 168*y^4 +
-  (-480*x^2 - 756*x - 52)*y^3 +
-  (2114*x^3 + 710*x^2 + 48*x - 80)*y^2 +
-  (-3093*x^4 + 736*x^3 - 35*x^2)*y
-
-lemma polyN_one (x y : ℝ) :
-    polyN x y 1 = (24/5) * (x-y) * (9*x-10*y+1) * (x^2+x-2*y) := by
-  simp only [polyN, polyA, polyC]; ring
-
-lemma polyN_affine (x y t : ℝ) :
-    polyN x y t = (1-t)*polyN x y 0 + t*polyN x y 1 := by
-  simp only [polyN]; ring
-
-end ThreeState.TreeClauses.Radial
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-! Radial Algebra. -/
+
+open scoped BigOperators ENNReal NNReal Topology
+open Filter
+noncomputable section
+namespace ThreeState.TreeClauses.Radial
+
+abbrev Vec := Fin 3 → ℝ
+
+def avg (w : Vec) : ℝ := (w 0 + w 1 + w 2) / 3
+
+def PositiveMessage (m : Vec) : Prop := (∀ i, 0 < m i) ∧ avg m = 1
+
+def centered (m : Vec) : Vec := fun i ↦ m i - 1
+
+def momentX (v : Vec) : ℝ := avg (fun i ↦ v i ^ 2) / 2
+
+def momentY (v : Vec) : ℝ := avg (fun i ↦ v i ^ 3) / 2
+
+def logAverage (m : Vec) : ℝ := avg (fun i ↦ Real.log (m i))
+
+def logCentered (m : Vec) : Vec := fun i ↦ Real.log (m i) - logAverage m
+
+def info (m : Vec) : ℝ := avg (fun i ↦ m i * Real.log (m i))
+
+def symEntropy (m : Vec) : ℝ := avg (fun i ↦ (m i - 1) * Real.log (m i)) / 2
+
+def correction (m : Vec) : ℝ := avg (fun i ↦ m i * logCentered m i ^ 2) / 2
+
+def functional (m : Vec) : ℝ := 3 * info m - 2 * symEntropy m + (4 / 5) * correction m
+
+def edge (t : ℝ) (m : Vec) : Vec := fun i ↦ 1 + t * (m i - 1)
+
+def RadialInequality : Prop := ∀ (m : Vec), PositiveMessage m → ∀ (lam : ℝ),
+  0 < lam → lam < 1 →
+  (1 - lam ^ 2) / 2 * momentX (centered m) ^ 2 ≤
+    functional m - (lam ^ 2)⁻¹ * functional (edge lam m)
+
+lemma avg_add (v w : Vec) : avg (fun i ↦ v i + w i) = avg v + avg w := by
+  simp only [avg]; ring
+
+lemma avg_sub (v w : Vec) : avg (fun i ↦ v i - w i) = avg v - avg w := by
+  simp only [avg]; ring
+
+lemma avg_mul (a : ℝ) (v : Vec) : avg (fun i ↦ a * v i) = a * avg v := by
+  simp only [avg]; ring
+
+lemma avg_const (a : ℝ) : avg (fun _ ↦ a) = a := by simp only [avg]; ring
+
+lemma avg_centered {m : Vec} (hm : PositiveMessage m) : avg (centered m) = 0 := by
+  have h := hm.2
+  dsimp [centered, avg] at *
+  linarith
+
+lemma info_eq (m : Vec) : info m = logAverage m + 2 * symEntropy m := by
+  simp only [info, logAverage, symEntropy, avg]; ring
+
+lemma functional_eq (m : Vec) :
+    functional m = 3 * logAverage m + 4 * symEntropy m + (4 / 5) * correction m := by
+  rw [functional, info_eq]; ring
+
+lemma third_eq {v : Vec} (hv : avg v = 0) : v 2 = -v 0 - v 1 := by
+  simp only [avg] at hv; linarith
+
+lemma momentX_nonneg (v : Vec) : 0 ≤ momentX v := by
+  unfold momentX avg; positivity
+
+lemma cubic {v : Vec} (hv : avg v = 0) (i : Fin 3) :
+    v i ^ 3 = 3 * momentX v * v i + 2 * momentY v := by
+  have hc := third_eq hv
+  fin_cases i
+  · change v 0 ^ 3 = 3 * momentX v * v 0 + 2 * momentY v
+    simp only [momentX, momentY, avg]; rw [hc]; ring
+  · change v 1 ^ 3 = 3 * momentX v * v 1 + 2 * momentY v
+    simp only [momentX, momentY, avg]; rw [hc]; ring
+  · change v 2 ^ 3 = 3 * momentX v * v 2 + 2 * momentY v
+    simp only [momentX, momentY, avg]; rw [hc]; ring
+
+lemma product_eq {v : Vec} (hv : avg v = 0) (t : ℝ) :
+    (1 + t * v 0) * (1 + t * v 1) * (1 + t * v 2) =
+      1 - 3 * t ^ 2 * momentX v + 2 * t ^ 3 * momentY v := by
+  simp only [momentX, momentY, avg]; rw [third_eq hv]; ring
+
+lemma discriminant {v : Vec} (hv : avg v = 0) :
+    (v 0 - v 1) ^ 2 * (v 0 - v 2) ^ 2 * (v 1 - v 2) ^ 2 =
+      108 * (momentX v ^ 3 - momentY v ^ 2) := by
+  simp only [momentX, momentY, avg]; rw [third_eq hv]; ring
+
+lemma moment_discriminant {v : Vec} (hv : avg v = 0) :
+    momentY v ^ 2 ≤ momentX v ^ 3 := by
+  have h : 0 ≤ (v 0 - v 1) ^ 2 * (v 0 - v 2) ^ 2 * (v 1 - v 2) ^ 2 := by positivity
+  rw [discriminant hv] at h; linarith
+
+lemma momentX_lt_one {m : Vec} (hm : PositiveMessage m) : momentX (centered m) < 1 := by
+  have hs : m 0 + m 1 + m 2 = 3 := by simpa only [avg, div_eq_iff (by norm_num : (3:ℝ) ≠ 0), one_mul] using hm.2
+  have h01 := mul_pos (hm.1 0) (hm.1 1)
+  have h02 := mul_pos (hm.1 0) (hm.1 2)
+  have h12 := mul_pos (hm.1 1) (hm.1 2)
+  dsimp [momentX, avg, centered]
+  nlinarith [sq_nonneg (m 0 + m 1 + m 2 - 3)]
+
+lemma denominator_pos {m : Vec} (hm : PositiveMessage m) (t : ℝ)
+    (ht : t ∈ Set.Icc (0 : ℝ) 1) :
+    0 < 1 - 3 * t ^ 2 * momentX (centered m) + 2 * t ^ 3 * momentY (centered m) := by
+  rw [← product_eq (avg_centered hm)]
+  have hi (i : Fin 3) : 0 < 1 + t * centered m i := by
+    dsimp [centered]
+    by_cases h : t = 0
+    · subst t; norm_num
+    · have ht0 : 0 < t := lt_of_le_of_ne ht.1 (Ne.symm h)
+      have := mul_pos ht0 (hm.1 i)
+      nlinarith [ht.2]
+  exact mul_pos (mul_pos (hi 0) (hi 1)) (hi 2)
+
+def hpoly (v : Vec) : Vec := fun i ↦ 1 - v i + v i ^ 2 - 3 * momentX v
+
+def Hpoly (j : ℕ) (v : Vec) : ℝ := avg (fun i ↦ hpoly v i ^ j)
+
+lemma Hpoly_one {v : Vec} (hv : avg v = 0) : Hpoly 1 v = 1 - momentX v := by
+  simp only [Hpoly, hpoly, momentX, avg, pow_one]; rw [third_eq hv]; ring
+
+lemma Hpoly_two {v : Vec} (hv : avg v = 0) :
+    Hpoly 2 v = 3 * momentX v ^ 2 - 4 * momentY v + 1 := by
+  simp only [Hpoly, hpoly, momentX, momentY, avg]; rw [third_eq hv]; ring
+
+lemma Hpoly_three {v : Vec} (hv : avg v = 0) :
+    Hpoly 3 v = -9 * momentX v ^ 3 + 9 * momentX v ^ 2 +
+      6 * momentX v * momentY v + 3 * momentX v + 4 * momentY v ^ 2 - 14 * momentY v + 1 := by
+  simp only [Hpoly, hpoly, momentX, momentY, avg]; rw [third_eq hv]; ring
+
+def polyA (x y : ℝ) : ℝ :=
+  3*x^4 + 12*x^3 + x^2 + (-6*x^3 - 28*x^2 - 2*x)*y + (26*x - 2)*y^2 - 4*y^3
+
+def polyC (x y : ℝ) : ℝ :=
+  6*x^4 - 2*x^3 + (-13*x^3 - 10*x^2 - x)*y + (10*x^2 + 22*x + 4)*y^2 - 16*y^3
+
+def polyN (x y t : ℝ) : ℝ := (24/5) * (polyA x y + t * polyC x y)
+
+def polyL (x y : ℝ) : ℝ :=
+  1500*x^5 - 1020*x^4 + 240*x^3 + 168*y^4 +
+  (-480*x^2 - 756*x - 52)*y^3 +
+  (2114*x^3 + 710*x^2 + 48*x - 80)*y^2 +
+  (-3093*x^4 + 736*x^3 - 35*x^2)*y
+
+lemma polyN_one (x y : ℝ) :
+    polyN x y 1 = (24/5) * (x-y) * (9*x-10*y+1) * (x^2+x-2*y) := by
+  simp only [polyN, polyA, polyC]; ring
+
+lemma polyN_affine (x y t : ℝ) :
+    polyN x y t = (1-t)*polyN x y 0 + t*polyN x y 1 := by
+  simp only [polyN]; ring
+
+end ThreeState.TreeClauses.Radial
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeStateClauses/RadialBounds.lean b/lean/OAI/Probability/ThreeStateClauses/RadialBounds.lean
index 61febc731cf771e987e211d4ce7c5e3e1a4b59b8..d28b74f63401ce748d8da207cfe2f9a1b739040a 100644
--- a/lean/OAI/Probability/ThreeStateClauses/RadialBounds.lean
+++ b/lean/OAI/Probability/ThreeStateClauses/RadialBounds.lean
@@ -1,466 +1,466 @@
-import Mathlib
-import OAI.Probability.ThreeStateClauses.RadialAlgebra
-
-namespace OAI
-
-/-! Radial Bounds. -/
-
-open scoped BigOperators ENNReal NNReal Topology
-open Filter
-noncomputable section
-namespace ThreeState.TreeClauses.Radial
-
-def f₀ (r : ℝ) : ℝ := 480*r^4 - 842*r^3 - 117*r^2 + 445*r + 160
-
-def f₁ (r : ℝ) : ℝ :=
-  336*r^6 - 1008*r^5 + 1096*r^4 - 896*r^3 + 63*r^2 + 685*r + 240
-
-def f₂ (r : ℝ) : ℝ := 440*r^4 - 1104*r^3 + 368*r^2 + 413*r + 210
-
-lemma f₀_certificate (p : ℝ) (hp : 0 ≤ p) :
-    4*(1+p)^4 * f₀ ((p-1/2)/(1+p)) =
-      504*p^4 + 4386*p^3 + 12267*p^2 - 1161*p + 174 := by
-  have h : 1+p ≠ 0 := by positivity
-  dsimp [f₀]; field_simp; ring
-
-lemma f₁_certificate (p : ℝ) (hp : 0 ≤ p) :
-    4*(1+p)^6 * f₁ ((p-1/2)/(1+p)) =
-      2064*p^6 + 15486*p^5 + 36669*p^4 + 31368*p^3 +
-        20184*p^2 - 2724*p + 522 := by
-  have h : 1+p ≠ 0 := by positivity
-  dsimp [f₁]; field_simp; ring
-
-lemma f₂_certificate (p : ℝ) (hp : 0 ≤ p) :
-    4*(1+p)^4 * f₂ ((p-1/2)/(1+p)) =
-      1308*p^4 + 7650*p^3 + 12366*p^2 - 1842*p + 1044 := by
-  have h : 1+p ≠ 0 := by positivity
-  dsimp [f₂]; field_simp; ring
-
-lemma f_positive (r : ℝ) (hr₀ : -(1/2:ℝ) ≤ r) (hr₁ : r ≤ 1) :
-    0 < f₀ r ∧ 0 < f₁ r ∧ 0 < f₂ r := by
-  rcases hr₁.eq_or_lt with hr | hr
-  · subst r; norm_num [f₀, f₁, f₂]
-  let p := (r+1/2)/(1-r)
-  have hp : 0 ≤ p := div_nonneg (by linarith) (by linarith)
-  have hp1 : 0 < 1+p := by positivity
-  have hrp : r = (p-1/2)/(1+p) := by
-    have heq : p*(1-r) = r+1/2 := div_mul_cancel₀ _ (by linarith : 1-r ≠ 0)
-    apply (eq_div_iff (ne_of_gt hp1)).2
-    nlinarith [heq]
-  have h₀ : 0 < 12267*p^2 - 1161*p + 174 := by
-    nlinarith [sq_nonneg (p - (1161 / (2*12267) : ℝ))]
-  have h₁ : 0 < 20184*p^2 - 2724*p + 522 := by
-    nlinarith [sq_nonneg (p - (2724 / (2*20184) : ℝ))]
-  have h₂ : 0 < 12366*p^2 - 1842*p + 1044 := by
-    nlinarith [sq_nonneg (p - (1842 / (2*12366) : ℝ))]
-  rw [hrp]
-  constructor
-  · have ht : 0 ≤ 504*p^4 + 4386*p^3 := by positivity
-    have hc := f₀_certificate p hp
-    exact (mul_pos_iff_of_pos_left (by positivity : 0 < 4*(1+p)^4)).1 (by linarith)
-  constructor
-  · have ht : 0 ≤ 2064*p^6 + 15486*p^5 + 36669*p^4 + 31368*p^3 := by positivity
-    have hc := f₁_certificate p hp
-    exact (mul_pos_iff_of_pos_left (by positivity : 0 < 4*(1+p)^6)).1 (by linarith)
-  · have ht : 0 ≤ 1308*p^4 + 7650*p^3 := by positivity
-    have hc := f₂_certificate p hp
-    exact (mul_pos_iff_of_pos_left (by positivity : 0 < 4*(1+p)^4)).1 (by linarith)
-
-def a₀ (r : ℝ) : ℝ := (1-r)^3*(3-2*r)^2*(10*r^2+r+1)
-
-def a₁ (r : ℝ) : ℝ := 2*(1-r)^3*(3-2*r)*(9+5*r+20*r^2-4*r^3)
-
-def a₂ (r : ℝ) : ℝ := 48*(1-r)^3
-
-def b₀ (r : ℝ) : ℝ := r^2*(1-r)^3*(3-2*r)^3*f₀ r
-
-def b₁ (r : ℝ) : ℝ := (1-r)^3*(3-2*r)^2*f₁ r
-
-def b₂ (r : ℝ) : ℝ := 2*(1-r)^3*(3-2*r)*f₂ r
-
-def b₃ (r : ℝ) : ℝ := 24*(1-r)^3*(25-4*r^2-10*r)
-
-lemma a_nonneg (r : ℝ) (hr₀ : -(1/2:ℝ) ≤ r) (hr₁ : r ≤ 1) :
-    0 ≤ a₀ r ∧ 0 ≤ a₁ r ∧ 0 ≤ a₂ r := by
-  have h1 : 0 ≤ 1-r := by linarith
-  have h3 : 0 ≤ 3-2*r := by linarith
-  have hq : 0 ≤ 10*r^2+r+1 := by nlinarith [sq_nonneg (r+1/20)]
-  have hc : 0 ≤ 9+5*r+20*r^2-4*r^3 := by
-    have h : 0 ≤ r^2*(20-4*r) := mul_nonneg (sq_nonneg r) (by linarith)
-    nlinarith
-  dsimp [a₀, a₁, a₂]
-  exact ⟨by positivity, by positivity, by positivity⟩
-
-lemma b_nonneg (r : ℝ) (hr₀ : -(1/2:ℝ) ≤ r) (hr₁ : r ≤ 1) :
-    0 ≤ b₀ r ∧ 0 ≤ b₁ r ∧ 0 ≤ b₂ r ∧ 0 ≤ b₃ r := by
-  have h1 : 0 ≤ 1-r := by linarith
-  have h3 : 0 ≤ 3-2*r := by linarith
-  obtain ⟨hf₀, hf₁, hf₂⟩ := f_positive r hr₀ hr₁
-  have hq : 0 ≤ 25-4*r^2-10*r := by
-    have h : 0 ≤ (1-r)*(r+1/2) := mul_nonneg h1 (by linarith)
-    nlinarith
-  dsimp [b₀, b₁, b₂, b₃]
-  exact ⟨by positivity, by positivity, by positivity, by positivity⟩
-
-def normalizedA (x r : ℝ) : ℝ :=
-  (3-6*r)*x^2 + (12-28*r+26*r^2-4*r^3)*x + 1-2*r-2*r^2
-
-def normalizedL (x r : ℝ) : ℝ :=
-  (1500-3093*r+2114*r^2-480*r^3)*x^3 +
-  (168*r^4-756*r^3+710*r^2+736*r-1020)*x^2 +
-  (240-35*r+48*r^2-52*r^3)*x - 80*r^2
-
-lemma normalizeA (x r : ℝ) : polyA x (r*x) = x^2*normalizedA x r := by
-  dsimp [polyA, normalizedA]; ring
-
-lemma normalizeL (x r : ℝ) : polyL x (r*x) = x^2*normalizedL x r := by
-  dsimp [polyL, normalizedL]; ring
-
-def paramX (r z : ℝ) : ℝ := (r^2+z/(3-2*r))/(1+z)
-
-lemma a_certificate (r z : ℝ) (hr : r ≤ 1) (hz : 0 ≤ z) :
-    (1+z)^2*(3-2*r)^2*normalizedA (paramX r z) r =
-      a₀ r + a₁ r*z + a₂ r*z^2 := by
-  have h1 : 1+z ≠ 0 := by positivity
-  have h3 : 3-2*r ≠ 0 := by linarith
-  dsimp [paramX, normalizedA, a₀, a₁, a₂]
-  field_simp
-  ring
-
-lemma b_certificate (r z : ℝ) (hr : r ≤ 1) (hz : 0 ≤ z) :
-    (1+z)^3*(3-2*r)^3*normalizedL (paramX r z) r =
-      b₀ r + b₁ r*z + b₂ r*z^2 + b₃ r*z^3 := by
-  have h1 : 1+z ≠ 0 := by positivity
-  have h3 : 3-2*r ≠ 0 := by linarith
-  dsimp [paramX, normalizedL, b₀, b₁, b₂, b₃, f₀, f₁, f₂]
-  field_simp
-  ring
-
-lemma param_polynomial_nonneg (r z : ℝ) (hr₀ : -(1/2:ℝ) ≤ r)
-    (hr₁ : r ≤ 1) (hz : 0 ≤ z) :
-    0 ≤ polyA (paramX r z) (r*paramX r z) ∧
-    0 ≤ polyL (paramX r z) (r*paramX r z) := by
-  have h1 : 0 < 1+z := by positivity
-  have h3 : 0 < 3-2*r := by linarith
-  obtain ⟨ha₀, ha₁, ha₂⟩ := a_nonneg r hr₀ hr₁
-  obtain ⟨hb₀, hb₁, hb₂, hb₃⟩ := b_nonneg r hr₀ hr₁
-  constructor
-  · rw [normalizeA]
-    apply mul_nonneg (sq_nonneg _)
-    apply (nonneg_of_mul_nonneg_left _ (by positivity : 0 < (1+z)^2*(3-2*r)^2))
-    rw [mul_comm, a_certificate r z hr₁ hz]
-    positivity
-  · rw [normalizeL]
-    apply mul_nonneg (sq_nonneg _)
-    apply (nonneg_of_mul_nonneg_left _ (by positivity : 0 < (1+z)^3*(3-2*r)^3))
-    rw [mul_comm, b_certificate r z hr₁ hz]
-    positivity
-
-lemma domain_parametrization {x y : ℝ} (hx₀ : 0 < x) (hx₁ : x < 1)
-    (hxy : y^2 ≤ x^3) (hD : 0 < 1-3*x+2*y) :
-    ∃ r z : ℝ, -(1/2:ℝ) ≤ r ∧ r < 1 ∧ 0 ≤ z ∧
-      x = paramX r z ∧ y = r*x := by
-  let r := y/x
-  have hyr : y = r*x := by dsimp [r]; field_simp
-  have hrx : r^2 ≤ x := by
-    dsimp [r]; rw [div_pow]
-    apply (div_le_iff₀ (pow_pos hx₀ 2)).2
-    nlinarith
-  have hr₁ : r < 1 := by nlinarith [sq_nonneg (r-1)]
-  have h3 : 0 < 3-2*r := by linarith
-  have hD' : 0 < 1-(3-2*r)*x := by rw [hyr] at hD; nlinarith
-  have hfactor : 0 < (1-r)^2*(2*r+1) := by
-    nlinarith [mul_nonneg h3.le (sub_nonneg.mpr hrx)]
-  have hr₀ : -(1/2:ℝ) ≤ r := by
-    have hsq : 0 < (1-r)^2 := sq_pos_of_pos (by linarith)
-    have := (mul_pos_iff_of_pos_left hsq).1 hfactor
-    linarith
-  have hden : 0 < 1/(3-2*r)-x := by
-    apply sub_pos.mpr
-    apply (lt_div_iff₀ h3).2
-    nlinarith
-  let z := (x-r^2)/(1/(3-2*r)-x)
-  have hz : 0 ≤ z := div_nonneg (sub_nonneg.mpr hrx) hden.le
-  have hz1 : 0 < 1+z := by positivity
-  have hzEq : z*(1/(3-2*r)-x) = x-r^2 := div_mul_cancel₀ _ (ne_of_gt hden)
-  refine ⟨r, z, hr₀, hr₁, hz, ?_, hyr⟩
-  dsimp only [paramX]
-  apply (eq_div_iff (ne_of_gt hz1)).2
-  simp only [div_eq_mul_inv, one_mul] at hzEq ⊢
-  nlinarith
-
-lemma polyN_one_nonneg {x y : ℝ} (hx₀ : 0 ≤ x) (hx₁ : x ≤ 1)
-    (hxy : y^2 ≤ x^3) : 0 ≤ polyN x y 1 := by
-  have hx3 : x^3 ≤ x^2 := by
-    nlinarith [mul_nonneg (sq_nonneg x) (sub_nonneg.mpr hx₁)]
-  have hyx : y ≤ x := by nlinarith [sq_nonneg (y-x)]
-  have hlast : 0 ≤ x^2+x-2*y := by
-    by_contra h
-    have hminus : 0 < 2*y-(x^2+x) := by linarith
-    have hplus : 0 < 2*y+(x^2+x) := by nlinarith
-    have hp := mul_pos hminus hplus
-    have hsq : 0 ≤ x^2*(x-1)^2 := by positivity
-    nlinarith
-  rw [polyN_one]
-  have hfirst : 0 ≤ x-y := by linarith
-  have hsecond : 0 ≤ 9*x-10*y+1 := by linarith
-  positivity
-
-lemma polynomial_goal {x y : ℝ} (hx₀ : 0 ≤ x) (hx₁ : x < 1)
-    (hxy : y^2 ≤ x^3) (hD : 0 < 1-3*x+2*y) :
-    0 ≤ polyN x y 0 ∧ 0 ≤ polyN x y 1 ∧ 0 ≤ polyL x y := by
-  rcases eq_or_lt_of_le hx₀ with hx | hx
-  · have hy : y = 0 := by rw [← hx] at hxy; nlinarith [sq_nonneg y]
-    rw [← hx, hy]; norm_num [polyN, polyA, polyC, polyL]
-  · obtain ⟨r,z,hr₀,hr₁,hz,hxp,hyr⟩ := domain_parametrization hx hx₁ hxy hD
-    have hs := param_polynomial_nonneg r z hr₀ hr₁.le hz
-    rw [← hxp, ← hyr] at hs
-    refine ⟨?_, polyN_one_nonneg hx₀ hx₁.le hxy, hs.2⟩
-    dsimp only [polyN]
-    nlinarith [hs.1]
-
-lemma polyN_nonneg {x y : ℝ} (hx₀ : 0 ≤ x) (hx₁ : x < 1)
-    (hxy : y^2 ≤ x^3) (hD : 0 < 1-3*x+2*y)
-    {t : ℝ} (ht : t ∈ Set.Icc (0:ℝ) 1) : 0 ≤ polyN x y t := by
-  obtain ⟨h0,h1,_⟩ := polynomial_goal hx₀ hx₁ hxy hD
-  rw [polyN_affine]
-  exact add_nonneg (mul_nonneg (sub_nonneg.mpr ht.2) h0) (mul_nonneg ht.1 h1)
-
-end ThreeState.TreeClauses.Radial
-
-end 
-
-noncomputable section
-open MeasureTheory Set
-open scoped Interval
-namespace ThreeState.TreeClauses.Radial
-
-def denom (v : Vec) (t : ℝ) : ℝ := 1 - 3*t^2*momentX v + 2*t^3*momentY v
-
-def kpoly (v : Vec) (t : ℝ) : Vec := fun i ↦ v i - t * (v i^2 - 2*momentX v)
-
-def invMoment (m : Vec) (j : ℕ) : ℝ := avg (fun i ↦ (m i)⁻¹ ^ j)
-
-def logInvMoment (m : Vec) (j : ℕ) : ℝ :=
-  avg (fun i ↦ logCentered m i * (m i)⁻¹ ^ j)
-
-def logMoment (m : Vec) : ℝ := avg (fun i ↦ m i * logCentered m i)
-
-def radialG (m : Vec) : ℝ :=
-  6 - 12 * invMoment m 1 + 8 * invMoment m 2 - 2 * invMoment m 3 +
-  (4/5) * (logInvMoment m 2 - 3*logInvMoment m 1 +
-    (2*invMoment m 3 - 3*invMoment m 2)*logMoment m +
-    3*(invMoment m 1-1)*(invMoment m 2-invMoment m 1))
-
-lemma edge_pos {m : Vec} (hm : PositiveMessage m) {t : ℝ} (ht : t ∈ Icc (0:ℝ) 1)
-    (i : Fin 3) : 0 < 1+t*centered m i := by
-  by_cases h : t = 0
-  · subst t; norm_num
-  · have ht0 : 0 < t := lt_of_le_of_ne ht.1 (Ne.symm h)
-    have := mul_pos ht0 (hm.1 i)
-    dsimp only [centered]; nlinarith [ht.2]
-
-lemma denom_pos {m : Vec} (hm : PositiveMessage m) {t : ℝ} (ht : t ∈ Icc (0:ℝ) 1) :
-    0 < denom (centered m) t := denominator_pos hm t ht
-
-lemma rational_identity {v : Vec} (hv : avg v = 0) (t : ℝ) (i : Fin 3)
-    (hi : 1+t*v i ≠ 0) (hD : denom v t ≠ 0) :
-    v i / (1+t*v i) = (v i - t*v i^2 + 2*t^2*momentY v) / denom v t := by
-  apply (div_eq_div_iff hi hD).2
-  dsimp only [denom]
-  linear_combination t^2 * cubic hv i
-
-lemma rational_centered {v : Vec} (hv : avg v = 0) (t : ℝ)
-    (hi : ∀ i, 1+t*v i ≠ 0) (hD : denom v t ≠ 0) (i : Fin 3) :
-    v i / (1+t*v i) - avg (fun j ↦ v j / (1+t*v j)) = kpoly v t i / denom v t := by
-  simp_rw [rational_identity hv t _ (hi _) hD]
-  dsimp only [avg, kpoly, momentX, momentY]
-  have hs := third_eq hv
-  field_simp
-  rw [hs]
-  ring
-
-lemma hasDerivAt_log_edge {v : Vec} {t : ℝ} (i : Fin 3) (hi : 1+t*v i ≠ 0) :
-    HasDerivAt (fun s : ℝ ↦ Real.log (1+s*v i)) (v i/(1+t*v i)) t := by
-  convert ((hasDerivAt_const t (1:ℝ)).add ((hasDerivAt_id t).mul_const (v i))).log hi using 1 <;> simp
-
-lemma integral_log_edge {m : Vec} (hm : PositiveMessage m) (i : Fin 3) :
-    (∫ t in (0:ℝ)..1, centered m i / (1+t*centered m i)) = Real.log (m i) := by
-  have hc : ContinuousOn (fun t : ℝ ↦ centered m i / (1+t*centered m i)) (Icc 0 1) := by
-    apply continuousOn_const.div (by fun_prop)
-    intro t ht; exact ne_of_gt (edge_pos hm ht i)
-  have hint : IntervalIntegrable (fun t : ℝ ↦ centered m i / (1+t*centered m i)) volume 0 1 := by
-    apply ContinuousOn.intervalIntegrable
-    simpa only [uIcc_of_le (by norm_num : (0:ℝ) ≤ 1)] using hc
-  have h := intervalIntegral.integral_eq_sub_of_hasDerivAt
-    (fun t (ht : t ∈ uIcc (0:ℝ) 1) ↦
-      hasDerivAt_log_edge i (ne_of_gt (edge_pos hm (by simpa using ht) i))) hint
-  simpa [centered] using h
-
-lemma integral_avg {f : ℝ → Vec}
-    (hf : ∀ i, IntervalIntegrable (fun t ↦ f t i) volume (0:ℝ) 1) :
-    (∫ t in (0:ℝ)..1, avg (f t)) = avg (fun i ↦ ∫ t in (0:ℝ)..1, f t i) := by
-  simp only [avg, intervalIntegral.integral_div,
-    intervalIntegral.integral_add (hf 0) (hf 1),
-    intervalIntegral.integral_add ((hf 0).add (hf 1)) (hf 2)]
-
-lemma integral_k {m : Vec} (hm : PositiveMessage m) (i : Fin 3) :
-    (∫ t in (0:ℝ)..1, kpoly (centered m) t i / denom (centered m) t) = logCentered m i := by
-  have hint (j : Fin 3) : IntervalIntegrable
-      (fun t : ℝ ↦ centered m j/(1+t*centered m j)) volume 0 1 := by
-    apply ContinuousOn.intervalIntegrable
-    rw [uIcc_of_le (by norm_num : (0:ℝ) ≤ 1)]
-    exact continuousOn_const.div (by fun_prop) (fun t ht ↦ ne_of_gt (edge_pos hm ht j))
-  have havg : IntervalIntegrable
-      (fun t : ℝ ↦ avg (fun j ↦ centered m j/(1+t*centered m j))) volume 0 1 := by
-    exact (((hint 0).add (hint 1)).add (hint 2)).div_const 3
-  calc
-    _ = ∫ t in (0:ℝ)..1, centered m i/(1+t*centered m i) -
-        avg (fun j ↦ centered m j/(1+t*centered m j)) := by
-      apply intervalIntegral.integral_congr
-      intro t ht
-      symm
-      exact rational_centered (avg_centered hm) t
-        (fun j ↦ ne_of_gt (edge_pos hm (by simpa using ht) j))
-        (ne_of_gt (denom_pos hm (by simpa using ht))) i
-    _ = _ := by
-      rw [intervalIntegral.integral_sub (hint i) havg, integral_log_edge hm,
-        integral_avg hint]
-      simp only [integral_log_edge hm, logCentered, logAverage]
-
-lemma avg_div (v : Vec) (a : ℝ) : avg (fun i ↦ v i / a) = avg v / a := by
-  simp only [avg]; ring
-
-lemma avg_logCentered (m : Vec) : avg (logCentered m) = 0 := by
-  simp only [logCentered, logAverage, avg]; ring
-
-lemma h_inverse {m : Vec} (hm : PositiveMessage m) (i : Fin 3) :
-    (m i)⁻¹ = hpoly (centered m) i / denom (centered m) 1 := by
-  have hc := cubic (avg_centered hm) i
-  have hD := ne_of_gt (denom_pos hm (by norm_num : (1:ℝ) ∈ Icc 0 1))
-  apply (eq_div_iff hD).2
-  apply (mul_left_cancel₀ (ne_of_gt (hm.1 i)))
-  rw [← mul_assoc, mul_inv_cancel₀ (ne_of_gt (hm.1 i)), one_mul]
-  dsimp only [denom, hpoly, centered] at *
-  norm_num only [one_pow, mul_one, one_mul]
-  linear_combination -hc
-
-def polynomialR (v : Vec) : ℝ :=
-  (6-12*momentX v^2)*denom v 1^3 - 12*Hpoly 1 v*denom v 1^2 +
-  8*Hpoly 2 v*denom v 1 - 2*Hpoly 3 v +
-  (12/5)*(Hpoly 1 v-denom v 1)*(Hpoly 2 v-Hpoly 1 v*denom v 1)
-
-def logWeight (v : Vec) : Vec := fun i ↦
-  hpoly v i^2*denom v 1 - 3*hpoly v i*denom v 1^2 +
-  (2*Hpoly 3 v-3*Hpoly 2 v*denom v 1)*v i
-
-lemma polynomialN_representation {v : Vec} (hv : avg v = 0) (t : ℝ) :
-    (4/5)*avg (fun i ↦ kpoly v t i * logWeight v i) =
-      polyN (momentX v) (momentY v) t := by
-  simp only [logWeight, Hpoly, kpoly, hpoly, denom, polyN, polyA, polyC,
-    momentX, momentY, avg]
-  rw [third_eq hv]
-  ring
-
-lemma G_algebra {m : Vec} (hm : PositiveMessage m) :
-    denom (centered m) 1^3 * (radialG m-12*momentX (centered m)^2) =
-      polynomialR (centered m) + (4/5)*avg (fun i ↦ logCentered m i * logWeight (centered m) i) := by
-  have hD := ne_of_gt (denom_pos hm (by norm_num : (1:ℝ) ∈ Icc 0 1))
-  simp only [radialG, invMoment, logInvMoment, logMoment]
-  simp_rw [h_inverse hm]
-  simp only [polynomialR, logWeight, Hpoly, avg, logCentered, logAverage, centered]
-  field_simp
-  ring
-
-lemma integrable_k {m : Vec} (hm : PositiveMessage m) (i : Fin 3) :
-    IntervalIntegrable (fun t ↦ kpoly (centered m) t i / denom (centered m) t) volume (0:ℝ) 1 := by
-  apply ContinuousOn.intervalIntegrable
-  rw [uIcc_of_le (by norm_num : (0:ℝ) ≤ 1)]
-  apply ContinuousOn.div (by unfold kpoly; fun_prop) (by unfold denom; fun_prop)
-  intro t ht; exact ne_of_gt (denom_pos hm ht)
-
-lemma integral_N {m : Vec} (hm : PositiveMessage m) :
-    (∫ t in (0:ℝ)..1, polyN (momentX (centered m)) (momentY (centered m)) t /
-      denom (centered m) t) =
-      (4/5)*avg (fun i ↦ logCentered m i * logWeight (centered m) i) := by
-  have heq (t : ℝ) : polyN (momentX (centered m)) (momentY (centered m)) t /
-      denom (centered m) t = (4/5)*avg (fun i ↦
-        (kpoly (centered m) t i / denom (centered m) t) * logWeight (centered m) i) := by
-    rw [← polynomialN_representation (avg_centered hm)]
-    simp only [avg]; ring
-  simp_rw [heq]
-  rw [intervalIntegral.integral_const_mul,
-    integral_avg (fun i ↦ (integrable_k hm i).mul_const _)]
-  simp only [intervalIntegral.integral_mul_const, integral_k hm]
-
-lemma RN_representation {m : Vec} (hm : PositiveMessage m) :
-    denom (centered m) 1^3 * (radialG m-12*momentX (centered m)^2) =
-      polynomialR (centered m) +
-      (∫ t in (0:ℝ)..1, polyN (momentX (centered m)) (momentY (centered m)) t /
-        denom (centered m) t) := by rw [integral_N hm, G_algebra hm]
-
-lemma polynomial_integral (x y : ℝ) :
-    (∫ t in (0:ℝ)..1, polyN x y t * (2-(1-3*t^2*x+2*t^3*y))) =
-      (24/5)*(polyA x y*(1+x-y/2)+polyC x y*(1/2+3*x/4-2*y/5)) := by
-  simp only [polyN]
-  conv_lhs => arg 1; ext t; ring_nf
-  simp (disch := apply Continuous.intervalIntegrable; fun_prop) only
-    [intervalIntegral.integral_add,
-      intervalIntegral.integral_mul_const, intervalIntegral.integral_const_mul,
-      integral_pow, intervalIntegral.integral_const]
-  norm_num
-  ring
-
-lemma L_representation {v : Vec} (hv : avg v = 0) :
-    polynomialR v + (∫ t in (0:ℝ)..1, polyN (momentX v) (momentY v) t * (2-denom v t)) =
-      (6/25)*polyL (momentX v) (momentY v) := by
-  simp only [denom]
-  rw [polynomial_integral]
-  simp only [polynomialR, denom, Hpoly_one hv, Hpoly_two hv, Hpoly_three hv,
-    polyL, polyA, polyC]
-  ring
-
-theorem radialG_bound {m : Vec} (hm : PositiveMessage m) :
-    12*momentX (centered m)^2 ≤ radialG m := by
-  let v := centered m
-  have hx₀ := momentX_nonneg v
-  have hx₁ := momentX_lt_one hm
-  have hxy := moment_discriminant (avg_centered hm)
-  have hD : 0 < 1-3*momentX v+2*momentY v := by
-    simpa [v, denom] using denom_pos hm (by norm_num : (1:ℝ) ∈ Icc 0 1)
-  have hN (t : ℝ) (ht : t ∈ Icc (0:ℝ) 1) : 0 ≤ polyN (momentX v) (momentY v) t :=
-    polyN_nonneg hx₀ hx₁ hxy hD ht
-  have hL := (polynomial_goal hx₀ hx₁ hxy hD).2.2
-  have hcN : Continuous (fun t : ℝ ↦ polyN (momentX v) (momentY v) t) := by
-    unfold polyN; fun_prop
-  have hcl : Continuous (fun t : ℝ ↦ polyN (momentX v) (momentY v) t * (2-denom v t)) := by
-    unfold denom; fun_prop
-  have hcr : ContinuousOn (fun t : ℝ ↦ polyN (momentX v) (momentY v) t / denom v t)
-      (Icc 0 1) := by
-    exact hcN.continuousOn.div (by unfold denom; fun_prop)
-      (fun t ht ↦ ne_of_gt (denom_pos hm ht))
-  have hir : IntervalIntegrable (fun t : ℝ ↦ polyN (momentX v) (momentY v) t / denom v t) volume 0 1 :=
-    hcr.intervalIntegrable_of_Icc (by norm_num : (0:ℝ) ≤ 1)
-  have hi : (∫ t in (0:ℝ)..1, polyN (momentX v) (momentY v) t * (2-denom v t)) ≤
-      (∫ t in (0:ℝ)..1, polyN (momentX v) (momentY v) t / denom v t) := by
-    apply intervalIntegral.integral_mono_on (by norm_num) (hcl.intervalIntegrable _ _) hir
-    intro t ht
-    have hd := denom_pos hm ht
-    have hh : 2-denom v t ≤ 1/denom v t := by
-      apply (le_div_iff₀ hd).2
-      nlinarith [sq_nonneg (1-denom v t)]
-    simpa only [mul_one_div] using mul_le_mul_of_nonneg_left hh (hN t ht)
-  have hp : 0 ≤ denom v 1^3 * (radialG m-12*momentX v^2) := by
-    rw [RN_representation hm]
-    have he := L_representation (avg_centered hm)
-    dsimp only [v] at hi hL ⊢
-    linarith
-  have hd : 0 < denom v 1^3 := pow_pos (denom_pos hm (by norm_num)) 3
-  have := (mul_nonneg_iff_of_pos_left hd).1 hp
-  linarith
-
-end ThreeState.TreeClauses.Radial
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeStateClauses.RadialAlgebra
+
+namespace OAI
+
+/-! Radial Bounds. -/
+
+open scoped BigOperators ENNReal NNReal Topology
+open Filter
+noncomputable section
+namespace ThreeState.TreeClauses.Radial
+
+def f₀ (r : ℝ) : ℝ := 480*r^4 - 842*r^3 - 117*r^2 + 445*r + 160
+
+def f₁ (r : ℝ) : ℝ :=
+  336*r^6 - 1008*r^5 + 1096*r^4 - 896*r^3 + 63*r^2 + 685*r + 240
+
+def f₂ (r : ℝ) : ℝ := 440*r^4 - 1104*r^3 + 368*r^2 + 413*r + 210
+
+lemma f₀_certificate (p : ℝ) (hp : 0 ≤ p) :
+    4*(1+p)^4 * f₀ ((p-1/2)/(1+p)) =
+      504*p^4 + 4386*p^3 + 12267*p^2 - 1161*p + 174 := by
+  have h : 1+p ≠ 0 := by positivity
+  dsimp [f₀]; field_simp; ring
+
+lemma f₁_certificate (p : ℝ) (hp : 0 ≤ p) :
+    4*(1+p)^6 * f₁ ((p-1/2)/(1+p)) =
+      2064*p^6 + 15486*p^5 + 36669*p^4 + 31368*p^3 +
+        20184*p^2 - 2724*p + 522 := by
+  have h : 1+p ≠ 0 := by positivity
+  dsimp [f₁]; field_simp; ring
+
+lemma f₂_certificate (p : ℝ) (hp : 0 ≤ p) :
+    4*(1+p)^4 * f₂ ((p-1/2)/(1+p)) =
+      1308*p^4 + 7650*p^3 + 12366*p^2 - 1842*p + 1044 := by
+  have h : 1+p ≠ 0 := by positivity
+  dsimp [f₂]; field_simp; ring
+
+lemma f_positive (r : ℝ) (hr₀ : -(1/2:ℝ) ≤ r) (hr₁ : r ≤ 1) :
+    0 < f₀ r ∧ 0 < f₁ r ∧ 0 < f₂ r := by
+  rcases hr₁.eq_or_lt with hr | hr
+  · subst r; norm_num [f₀, f₁, f₂]
+  let p := (r+1/2)/(1-r)
+  have hp : 0 ≤ p := div_nonneg (by linarith) (by linarith)
+  have hp1 : 0 < 1+p := by positivity
+  have hrp : r = (p-1/2)/(1+p) := by
+    have heq : p*(1-r) = r+1/2 := div_mul_cancel₀ _ (by linarith : 1-r ≠ 0)
+    apply (eq_div_iff (ne_of_gt hp1)).2
+    nlinarith [heq]
+  have h₀ : 0 < 12267*p^2 - 1161*p + 174 := by
+    nlinarith [sq_nonneg (p - (1161 / (2*12267) : ℝ))]
+  have h₁ : 0 < 20184*p^2 - 2724*p + 522 := by
+    nlinarith [sq_nonneg (p - (2724 / (2*20184) : ℝ))]
+  have h₂ : 0 < 12366*p^2 - 1842*p + 1044 := by
+    nlinarith [sq_nonneg (p - (1842 / (2*12366) : ℝ))]
+  rw [hrp]
+  constructor
+  · have ht : 0 ≤ 504*p^4 + 4386*p^3 := by positivity
+    have hc := f₀_certificate p hp
+    exact (mul_pos_iff_of_pos_left (by positivity : 0 < 4*(1+p)^4)).1 (by linarith)
+  constructor
+  · have ht : 0 ≤ 2064*p^6 + 15486*p^5 + 36669*p^4 + 31368*p^3 := by positivity
+    have hc := f₁_certificate p hp
+    exact (mul_pos_iff_of_pos_left (by positivity : 0 < 4*(1+p)^6)).1 (by linarith)
+  · have ht : 0 ≤ 1308*p^4 + 7650*p^3 := by positivity
+    have hc := f₂_certificate p hp
+    exact (mul_pos_iff_of_pos_left (by positivity : 0 < 4*(1+p)^4)).1 (by linarith)
+
+def a₀ (r : ℝ) : ℝ := (1-r)^3*(3-2*r)^2*(10*r^2+r+1)
+
+def a₁ (r : ℝ) : ℝ := 2*(1-r)^3*(3-2*r)*(9+5*r+20*r^2-4*r^3)
+
+def a₂ (r : ℝ) : ℝ := 48*(1-r)^3
+
+def b₀ (r : ℝ) : ℝ := r^2*(1-r)^3*(3-2*r)^3*f₀ r
+
+def b₁ (r : ℝ) : ℝ := (1-r)^3*(3-2*r)^2*f₁ r
+
+def b₂ (r : ℝ) : ℝ := 2*(1-r)^3*(3-2*r)*f₂ r
+
+def b₃ (r : ℝ) : ℝ := 24*(1-r)^3*(25-4*r^2-10*r)
+
+lemma a_nonneg (r : ℝ) (hr₀ : -(1/2:ℝ) ≤ r) (hr₁ : r ≤ 1) :
+    0 ≤ a₀ r ∧ 0 ≤ a₁ r ∧ 0 ≤ a₂ r := by
+  have h1 : 0 ≤ 1-r := by linarith
+  have h3 : 0 ≤ 3-2*r := by linarith
+  have hq : 0 ≤ 10*r^2+r+1 := by nlinarith [sq_nonneg (r+1/20)]
+  have hc : 0 ≤ 9+5*r+20*r^2-4*r^3 := by
+    have h : 0 ≤ r^2*(20-4*r) := mul_nonneg (sq_nonneg r) (by linarith)
+    nlinarith
+  dsimp [a₀, a₁, a₂]
+  exact ⟨by positivity, by positivity, by positivity⟩
+
+lemma b_nonneg (r : ℝ) (hr₀ : -(1/2:ℝ) ≤ r) (hr₁ : r ≤ 1) :
+    0 ≤ b₀ r ∧ 0 ≤ b₁ r ∧ 0 ≤ b₂ r ∧ 0 ≤ b₃ r := by
+  have h1 : 0 ≤ 1-r := by linarith
+  have h3 : 0 ≤ 3-2*r := by linarith
+  obtain ⟨hf₀, hf₁, hf₂⟩ := f_positive r hr₀ hr₁
+  have hq : 0 ≤ 25-4*r^2-10*r := by
+    have h : 0 ≤ (1-r)*(r+1/2) := mul_nonneg h1 (by linarith)
+    nlinarith
+  dsimp [b₀, b₁, b₂, b₃]
+  exact ⟨by positivity, by positivity, by positivity, by positivity⟩
+
+def normalizedA (x r : ℝ) : ℝ :=
+  (3-6*r)*x^2 + (12-28*r+26*r^2-4*r^3)*x + 1-2*r-2*r^2
+
+def normalizedL (x r : ℝ) : ℝ :=
+  (1500-3093*r+2114*r^2-480*r^3)*x^3 +
+  (168*r^4-756*r^3+710*r^2+736*r-1020)*x^2 +
+  (240-35*r+48*r^2-52*r^3)*x - 80*r^2
+
+lemma normalizeA (x r : ℝ) : polyA x (r*x) = x^2*normalizedA x r := by
+  dsimp [polyA, normalizedA]; ring
+
+lemma normalizeL (x r : ℝ) : polyL x (r*x) = x^2*normalizedL x r := by
+  dsimp [polyL, normalizedL]; ring
+
+def paramX (r z : ℝ) : ℝ := (r^2+z/(3-2*r))/(1+z)
+
+lemma a_certificate (r z : ℝ) (hr : r ≤ 1) (hz : 0 ≤ z) :
+    (1+z)^2*(3-2*r)^2*normalizedA (paramX r z) r =
+      a₀ r + a₁ r*z + a₂ r*z^2 := by
+  have h1 : 1+z ≠ 0 := by positivity
+  have h3 : 3-2*r ≠ 0 := by linarith
+  dsimp [paramX, normalizedA, a₀, a₁, a₂]
+  field_simp
+  ring
+
+lemma b_certificate (r z : ℝ) (hr : r ≤ 1) (hz : 0 ≤ z) :
+    (1+z)^3*(3-2*r)^3*normalizedL (paramX r z) r =
+      b₀ r + b₁ r*z + b₂ r*z^2 + b₃ r*z^3 := by
+  have h1 : 1+z ≠ 0 := by positivity
+  have h3 : 3-2*r ≠ 0 := by linarith
+  dsimp [paramX, normalizedL, b₀, b₁, b₂, b₃, f₀, f₁, f₂]
+  field_simp
+  ring
+
+lemma param_polynomial_nonneg (r z : ℝ) (hr₀ : -(1/2:ℝ) ≤ r)
+    (hr₁ : r ≤ 1) (hz : 0 ≤ z) :
+    0 ≤ polyA (paramX r z) (r*paramX r z) ∧
+    0 ≤ polyL (paramX r z) (r*paramX r z) := by
+  have h1 : 0 < 1+z := by positivity
+  have h3 : 0 < 3-2*r := by linarith
+  obtain ⟨ha₀, ha₁, ha₂⟩ := a_nonneg r hr₀ hr₁
+  obtain ⟨hb₀, hb₁, hb₂, hb₃⟩ := b_nonneg r hr₀ hr₁
+  constructor
+  · rw [normalizeA]
+    apply mul_nonneg (sq_nonneg _)
+    apply (nonneg_of_mul_nonneg_left _ (by positivity : 0 < (1+z)^2*(3-2*r)^2))
+    rw [mul_comm, a_certificate r z hr₁ hz]
+    positivity
+  · rw [normalizeL]
+    apply mul_nonneg (sq_nonneg _)
+    apply (nonneg_of_mul_nonneg_left _ (by positivity : 0 < (1+z)^3*(3-2*r)^3))
+    rw [mul_comm, b_certificate r z hr₁ hz]
+    positivity
+
+lemma domain_parametrization {x y : ℝ} (hx₀ : 0 < x) (hx₁ : x < 1)
+    (hxy : y^2 ≤ x^3) (hD : 0 < 1-3*x+2*y) :
+    ∃ r z : ℝ, -(1/2:ℝ) ≤ r ∧ r < 1 ∧ 0 ≤ z ∧
+      x = paramX r z ∧ y = r*x := by
+  let r := y/x
+  have hyr : y = r*x := by dsimp [r]; field_simp
+  have hrx : r^2 ≤ x := by
+    dsimp [r]; rw [div_pow]
+    apply (div_le_iff₀ (pow_pos hx₀ 2)).2
+    nlinarith
+  have hr₁ : r < 1 := by nlinarith [sq_nonneg (r-1)]
+  have h3 : 0 < 3-2*r := by linarith
+  have hD' : 0 < 1-(3-2*r)*x := by rw [hyr] at hD; nlinarith
+  have hfactor : 0 < (1-r)^2*(2*r+1) := by
+    nlinarith [mul_nonneg h3.le (sub_nonneg.mpr hrx)]
+  have hr₀ : -(1/2:ℝ) ≤ r := by
+    have hsq : 0 < (1-r)^2 := sq_pos_of_pos (by linarith)
+    have := (mul_pos_iff_of_pos_left hsq).1 hfactor
+    linarith
+  have hden : 0 < 1/(3-2*r)-x := by
+    apply sub_pos.mpr
+    apply (lt_div_iff₀ h3).2
+    nlinarith
+  let z := (x-r^2)/(1/(3-2*r)-x)
+  have hz : 0 ≤ z := div_nonneg (sub_nonneg.mpr hrx) hden.le
+  have hz1 : 0 < 1+z := by positivity
+  have hzEq : z*(1/(3-2*r)-x) = x-r^2 := div_mul_cancel₀ _ (ne_of_gt hden)
+  refine ⟨r, z, hr₀, hr₁, hz, ?_, hyr⟩
+  dsimp only [paramX]
+  apply (eq_div_iff (ne_of_gt hz1)).2
+  simp only [div_eq_mul_inv, one_mul] at hzEq ⊢
+  nlinarith
+
+lemma polyN_one_nonneg {x y : ℝ} (hx₀ : 0 ≤ x) (hx₁ : x ≤ 1)
+    (hxy : y^2 ≤ x^3) : 0 ≤ polyN x y 1 := by
+  have hx3 : x^3 ≤ x^2 := by
+    nlinarith [mul_nonneg (sq_nonneg x) (sub_nonneg.mpr hx₁)]
+  have hyx : y ≤ x := by nlinarith [sq_nonneg (y-x)]
+  have hlast : 0 ≤ x^2+x-2*y := by
+    by_contra h
+    have hminus : 0 < 2*y-(x^2+x) := by linarith
+    have hplus : 0 < 2*y+(x^2+x) := by nlinarith
+    have hp := mul_pos hminus hplus
+    have hsq : 0 ≤ x^2*(x-1)^2 := by positivity
+    nlinarith
+  rw [polyN_one]
+  have hfirst : 0 ≤ x-y := by linarith
+  have hsecond : 0 ≤ 9*x-10*y+1 := by linarith
+  positivity
+
+lemma polynomial_goal {x y : ℝ} (hx₀ : 0 ≤ x) (hx₁ : x < 1)
+    (hxy : y^2 ≤ x^3) (hD : 0 < 1-3*x+2*y) :
+    0 ≤ polyN x y 0 ∧ 0 ≤ polyN x y 1 ∧ 0 ≤ polyL x y := by
+  rcases eq_or_lt_of_le hx₀ with hx | hx
+  · have hy : y = 0 := by rw [← hx] at hxy; nlinarith [sq_nonneg y]
+    rw [← hx, hy]; norm_num [polyN, polyA, polyC, polyL]
+  · obtain ⟨r,z,hr₀,hr₁,hz,hxp,hyr⟩ := domain_parametrization hx hx₁ hxy hD
+    have hs := param_polynomial_nonneg r z hr₀ hr₁.le hz
+    rw [← hxp, ← hyr] at hs
+    refine ⟨?_, polyN_one_nonneg hx₀ hx₁.le hxy, hs.2⟩
+    dsimp only [polyN]
+    nlinarith [hs.1]
+
+lemma polyN_nonneg {x y : ℝ} (hx₀ : 0 ≤ x) (hx₁ : x < 1)
+    (hxy : y^2 ≤ x^3) (hD : 0 < 1-3*x+2*y)
+    {t : ℝ} (ht : t ∈ Set.Icc (0:ℝ) 1) : 0 ≤ polyN x y t := by
+  obtain ⟨h0,h1,_⟩ := polynomial_goal hx₀ hx₁ hxy hD
+  rw [polyN_affine]
+  exact add_nonneg (mul_nonneg (sub_nonneg.mpr ht.2) h0) (mul_nonneg ht.1 h1)
+
+end ThreeState.TreeClauses.Radial
+
+end 
+
+noncomputable section
+open MeasureTheory Set
+open scoped Interval
+namespace ThreeState.TreeClauses.Radial
+
+def denom (v : Vec) (t : ℝ) : ℝ := 1 - 3*t^2*momentX v + 2*t^3*momentY v
+
+def kpoly (v : Vec) (t : ℝ) : Vec := fun i ↦ v i - t * (v i^2 - 2*momentX v)
+
+def invMoment (m : Vec) (j : ℕ) : ℝ := avg (fun i ↦ (m i)⁻¹ ^ j)
+
+def logInvMoment (m : Vec) (j : ℕ) : ℝ :=
+  avg (fun i ↦ logCentered m i * (m i)⁻¹ ^ j)
+
+def logMoment (m : Vec) : ℝ := avg (fun i ↦ m i * logCentered m i)
+
+def radialG (m : Vec) : ℝ :=
+  6 - 12 * invMoment m 1 + 8 * invMoment m 2 - 2 * invMoment m 3 +
+  (4/5) * (logInvMoment m 2 - 3*logInvMoment m 1 +
+    (2*invMoment m 3 - 3*invMoment m 2)*logMoment m +
+    3*(invMoment m 1-1)*(invMoment m 2-invMoment m 1))
+
+lemma edge_pos {m : Vec} (hm : PositiveMessage m) {t : ℝ} (ht : t ∈ Icc (0:ℝ) 1)
+    (i : Fin 3) : 0 < 1+t*centered m i := by
+  by_cases h : t = 0
+  · subst t; norm_num
+  · have ht0 : 0 < t := lt_of_le_of_ne ht.1 (Ne.symm h)
+    have := mul_pos ht0 (hm.1 i)
+    dsimp only [centered]; nlinarith [ht.2]
+
+lemma denom_pos {m : Vec} (hm : PositiveMessage m) {t : ℝ} (ht : t ∈ Icc (0:ℝ) 1) :
+    0 < denom (centered m) t := denominator_pos hm t ht
+
+lemma rational_identity {v : Vec} (hv : avg v = 0) (t : ℝ) (i : Fin 3)
+    (hi : 1+t*v i ≠ 0) (hD : denom v t ≠ 0) :
+    v i / (1+t*v i) = (v i - t*v i^2 + 2*t^2*momentY v) / denom v t := by
+  apply (div_eq_div_iff hi hD).2
+  dsimp only [denom]
+  linear_combination t^2 * cubic hv i
+
+lemma rational_centered {v : Vec} (hv : avg v = 0) (t : ℝ)
+    (hi : ∀ i, 1+t*v i ≠ 0) (hD : denom v t ≠ 0) (i : Fin 3) :
+    v i / (1+t*v i) - avg (fun j ↦ v j / (1+t*v j)) = kpoly v t i / denom v t := by
+  simp_rw [rational_identity hv t _ (hi _) hD]
+  dsimp only [avg, kpoly, momentX, momentY]
+  have hs := third_eq hv
+  field_simp
+  rw [hs]
+  ring
+
+lemma hasDerivAt_log_edge {v : Vec} {t : ℝ} (i : Fin 3) (hi : 1+t*v i ≠ 0) :
+    HasDerivAt (fun s : ℝ ↦ Real.log (1+s*v i)) (v i/(1+t*v i)) t := by
+  convert ((hasDerivAt_const t (1:ℝ)).add ((hasDerivAt_id t).mul_const (v i))).log hi using 1 <;> simp
+
+lemma integral_log_edge {m : Vec} (hm : PositiveMessage m) (i : Fin 3) :
+    (∫ t in (0:ℝ)..1, centered m i / (1+t*centered m i)) = Real.log (m i) := by
+  have hc : ContinuousOn (fun t : ℝ ↦ centered m i / (1+t*centered m i)) (Icc 0 1) := by
+    apply continuousOn_const.div (by fun_prop)
+    intro t ht; exact ne_of_gt (edge_pos hm ht i)
+  have hint : IntervalIntegrable (fun t : ℝ ↦ centered m i / (1+t*centered m i)) volume 0 1 := by
+    apply ContinuousOn.intervalIntegrable
+    simpa only [uIcc_of_le (by norm_num : (0:ℝ) ≤ 1)] using hc
+  have h := intervalIntegral.integral_eq_sub_of_hasDerivAt
+    (fun t (ht : t ∈ uIcc (0:ℝ) 1) ↦
+      hasDerivAt_log_edge i (ne_of_gt (edge_pos hm (by simpa using ht) i))) hint
+  simpa [centered] using h
+
+lemma integral_avg {f : ℝ → Vec}
+    (hf : ∀ i, IntervalIntegrable (fun t ↦ f t i) volume (0:ℝ) 1) :
+    (∫ t in (0:ℝ)..1, avg (f t)) = avg (fun i ↦ ∫ t in (0:ℝ)..1, f t i) := by
+  simp only [avg, intervalIntegral.integral_div,
+    intervalIntegral.integral_add (hf 0) (hf 1),
+    intervalIntegral.integral_add ((hf 0).add (hf 1)) (hf 2)]
+
+lemma integral_k {m : Vec} (hm : PositiveMessage m) (i : Fin 3) :
+    (∫ t in (0:ℝ)..1, kpoly (centered m) t i / denom (centered m) t) = logCentered m i := by
+  have hint (j : Fin 3) : IntervalIntegrable
+      (fun t : ℝ ↦ centered m j/(1+t*centered m j)) volume 0 1 := by
+    apply ContinuousOn.intervalIntegrable
+    rw [uIcc_of_le (by norm_num : (0:ℝ) ≤ 1)]
+    exact continuousOn_const.div (by fun_prop) (fun t ht ↦ ne_of_gt (edge_pos hm ht j))
+  have havg : IntervalIntegrable
+      (fun t : ℝ ↦ avg (fun j ↦ centered m j/(1+t*centered m j))) volume 0 1 := by
+    exact (((hint 0).add (hint 1)).add (hint 2)).div_const 3
+  calc
+    _ = ∫ t in (0:ℝ)..1, centered m i/(1+t*centered m i) -
+        avg (fun j ↦ centered m j/(1+t*centered m j)) := by
+      apply intervalIntegral.integral_congr
+      intro t ht
+      symm
+      exact rational_centered (avg_centered hm) t
+        (fun j ↦ ne_of_gt (edge_pos hm (by simpa using ht) j))
+        (ne_of_gt (denom_pos hm (by simpa using ht))) i
+    _ = _ := by
+      rw [intervalIntegral.integral_sub (hint i) havg, integral_log_edge hm,
+        integral_avg hint]
+      simp only [integral_log_edge hm, logCentered, logAverage]
+
+lemma avg_div (v : Vec) (a : ℝ) : avg (fun i ↦ v i / a) = avg v / a := by
+  simp only [avg]; ring
+
+lemma avg_logCentered (m : Vec) : avg (logCentered m) = 0 := by
+  simp only [logCentered, logAverage, avg]; ring
+
+lemma h_inverse {m : Vec} (hm : PositiveMessage m) (i : Fin 3) :
+    (m i)⁻¹ = hpoly (centered m) i / denom (centered m) 1 := by
+  have hc := cubic (avg_centered hm) i
+  have hD := ne_of_gt (denom_pos hm (by norm_num : (1:ℝ) ∈ Icc 0 1))
+  apply (eq_div_iff hD).2
+  apply (mul_left_cancel₀ (ne_of_gt (hm.1 i)))
+  rw [← mul_assoc, mul_inv_cancel₀ (ne_of_gt (hm.1 i)), one_mul]
+  dsimp only [denom, hpoly, centered] at *
+  norm_num only [one_pow, mul_one, one_mul]
+  linear_combination -hc
+
+def polynomialR (v : Vec) : ℝ :=
+  (6-12*momentX v^2)*denom v 1^3 - 12*Hpoly 1 v*denom v 1^2 +
+  8*Hpoly 2 v*denom v 1 - 2*Hpoly 3 v +
+  (12/5)*(Hpoly 1 v-denom v 1)*(Hpoly 2 v-Hpoly 1 v*denom v 1)
+
+def logWeight (v : Vec) : Vec := fun i ↦
+  hpoly v i^2*denom v 1 - 3*hpoly v i*denom v 1^2 +
+  (2*Hpoly 3 v-3*Hpoly 2 v*denom v 1)*v i
+
+lemma polynomialN_representation {v : Vec} (hv : avg v = 0) (t : ℝ) :
+    (4/5)*avg (fun i ↦ kpoly v t i * logWeight v i) =
+      polyN (momentX v) (momentY v) t := by
+  simp only [logWeight, Hpoly, kpoly, hpoly, denom, polyN, polyA, polyC,
+    momentX, momentY, avg]
+  rw [third_eq hv]
+  ring
+
+lemma G_algebra {m : Vec} (hm : PositiveMessage m) :
+    denom (centered m) 1^3 * (radialG m-12*momentX (centered m)^2) =
+      polynomialR (centered m) + (4/5)*avg (fun i ↦ logCentered m i * logWeight (centered m) i) := by
+  have hD := ne_of_gt (denom_pos hm (by norm_num : (1:ℝ) ∈ Icc 0 1))
+  simp only [radialG, invMoment, logInvMoment, logMoment]
+  simp_rw [h_inverse hm]
+  simp only [polynomialR, logWeight, Hpoly, avg, logCentered, logAverage, centered]
+  field_simp
+  ring
+
+lemma integrable_k {m : Vec} (hm : PositiveMessage m) (i : Fin 3) :
+    IntervalIntegrable (fun t ↦ kpoly (centered m) t i / denom (centered m) t) volume (0:ℝ) 1 := by
+  apply ContinuousOn.intervalIntegrable
+  rw [uIcc_of_le (by norm_num : (0:ℝ) ≤ 1)]
+  apply ContinuousOn.div (by unfold kpoly; fun_prop) (by unfold denom; fun_prop)
+  intro t ht; exact ne_of_gt (denom_pos hm ht)
+
+lemma integral_N {m : Vec} (hm : PositiveMessage m) :
+    (∫ t in (0:ℝ)..1, polyN (momentX (centered m)) (momentY (centered m)) t /
+      denom (centered m) t) =
+      (4/5)*avg (fun i ↦ logCentered m i * logWeight (centered m) i) := by
+  have heq (t : ℝ) : polyN (momentX (centered m)) (momentY (centered m)) t /
+      denom (centered m) t = (4/5)*avg (fun i ↦
+        (kpoly (centered m) t i / denom (centered m) t) * logWeight (centered m) i) := by
+    rw [← polynomialN_representation (avg_centered hm)]
+    simp only [avg]; ring
+  simp_rw [heq]
+  rw [intervalIntegral.integral_const_mul,
+    integral_avg (fun i ↦ (integrable_k hm i).mul_const _)]
+  simp only [intervalIntegral.integral_mul_const, integral_k hm]
+
+lemma RN_representation {m : Vec} (hm : PositiveMessage m) :
+    denom (centered m) 1^3 * (radialG m-12*momentX (centered m)^2) =
+      polynomialR (centered m) +
+      (∫ t in (0:ℝ)..1, polyN (momentX (centered m)) (momentY (centered m)) t /
+        denom (centered m) t) := by rw [integral_N hm, G_algebra hm]
+
+lemma polynomial_integral (x y : ℝ) :
+    (∫ t in (0:ℝ)..1, polyN x y t * (2-(1-3*t^2*x+2*t^3*y))) =
+      (24/5)*(polyA x y*(1+x-y/2)+polyC x y*(1/2+3*x/4-2*y/5)) := by
+  simp only [polyN]
+  conv_lhs => arg 1; ext t; ring_nf
+  simp (disch := apply Continuous.intervalIntegrable; fun_prop) only
+    [intervalIntegral.integral_add,
+      intervalIntegral.integral_mul_const, intervalIntegral.integral_const_mul,
+      integral_pow, intervalIntegral.integral_const]
+  norm_num
+  ring
+
+lemma L_representation {v : Vec} (hv : avg v = 0) :
+    polynomialR v + (∫ t in (0:ℝ)..1, polyN (momentX v) (momentY v) t * (2-denom v t)) =
+      (6/25)*polyL (momentX v) (momentY v) := by
+  simp only [denom]
+  rw [polynomial_integral]
+  simp only [polynomialR, denom, Hpoly_one hv, Hpoly_two hv, Hpoly_three hv,
+    polyL, polyA, polyC]
+  ring
+
+theorem radialG_bound {m : Vec} (hm : PositiveMessage m) :
+    12*momentX (centered m)^2 ≤ radialG m := by
+  let v := centered m
+  have hx₀ := momentX_nonneg v
+  have hx₁ := momentX_lt_one hm
+  have hxy := moment_discriminant (avg_centered hm)
+  have hD : 0 < 1-3*momentX v+2*momentY v := by
+    simpa [v, denom] using denom_pos hm (by norm_num : (1:ℝ) ∈ Icc 0 1)
+  have hN (t : ℝ) (ht : t ∈ Icc (0:ℝ) 1) : 0 ≤ polyN (momentX v) (momentY v) t :=
+    polyN_nonneg hx₀ hx₁ hxy hD ht
+  have hL := (polynomial_goal hx₀ hx₁ hxy hD).2.2
+  have hcN : Continuous (fun t : ℝ ↦ polyN (momentX v) (momentY v) t) := by
+    unfold polyN; fun_prop
+  have hcl : Continuous (fun t : ℝ ↦ polyN (momentX v) (momentY v) t * (2-denom v t)) := by
+    unfold denom; fun_prop
+  have hcr : ContinuousOn (fun t : ℝ ↦ polyN (momentX v) (momentY v) t / denom v t)
+      (Icc 0 1) := by
+    exact hcN.continuousOn.div (by unfold denom; fun_prop)
+      (fun t ht ↦ ne_of_gt (denom_pos hm ht))
+  have hir : IntervalIntegrable (fun t : ℝ ↦ polyN (momentX v) (momentY v) t / denom v t) volume 0 1 :=
+    hcr.intervalIntegrable_of_Icc (by norm_num : (0:ℝ) ≤ 1)
+  have hi : (∫ t in (0:ℝ)..1, polyN (momentX v) (momentY v) t * (2-denom v t)) ≤
+      (∫ t in (0:ℝ)..1, polyN (momentX v) (momentY v) t / denom v t) := by
+    apply intervalIntegral.integral_mono_on (by norm_num) (hcl.intervalIntegrable _ _) hir
+    intro t ht
+    have hd := denom_pos hm ht
+    have hh : 2-denom v t ≤ 1/denom v t := by
+      apply (le_div_iff₀ hd).2
+      nlinarith [sq_nonneg (1-denom v t)]
+    simpa only [mul_one_div] using mul_le_mul_of_nonneg_left hh (hN t ht)
+  have hp : 0 ≤ denom v 1^3 * (radialG m-12*momentX v^2) := by
+    rw [RN_representation hm]
+    have he := L_representation (avg_centered hm)
+    dsimp only [v] at hi hL ⊢
+    linarith
+  have hd : 0 < denom v 1^3 := pow_pos (denom_pos hm (by norm_num)) 3
+  have := (mul_nonneg_iff_of_pos_left hd).1 hp
+  linarith
+
+end ThreeState.TreeClauses.Radial
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeStateClauses/RadialCalculus.lean b/lean/OAI/Probability/ThreeStateClauses/RadialCalculus.lean
index 62485675debf17db854a8bdbb5b62b11a8f50cc3..e052ece837d0bf5f998ead7fa973add02a561724 100644
--- a/lean/OAI/Probability/ThreeStateClauses/RadialCalculus.lean
+++ b/lean/OAI/Probability/ThreeStateClauses/RadialCalculus.lean
@@ -1,378 +1,378 @@
-import Mathlib
-import OAI.Probability.ThreeStateClauses.RadialBounds
-
-namespace OAI
-
-/-! Radial Calculus. -/
-
-open scoped BigOperators ENNReal NNReal Topology
-open Filter
-noncomputable section
-open Set
-namespace ThreeState.TreeClauses.Radial
-
-def rayG₀ (v : Vec) (t : ℝ) : ℝ := avg (fun i ↦ Real.log (1+t*v i))
-def rayG₁ (v : Vec) (t : ℝ) : ℝ := avg (fun i ↦ v i/(1+t*v i))
-def rayG₂ (v : Vec) (t : ℝ) : ℝ := avg (fun i ↦ -v i^2/(1+t*v i)^2)
-def rayG₃ (v : Vec) (t : ℝ) : ℝ := avg (fun i ↦ 2*v i^3/(1+t*v i)^3)
-
-def rayI₀ (v : Vec) (t : ℝ) : ℝ := avg (fun i ↦ (1+t*v i)*Real.log (1+t*v i))
-def rayI₁ (v : Vec) (t : ℝ) : ℝ := avg (fun i ↦ v i*Real.log (1+t*v i))
-def rayI₂ (v : Vec) (t : ℝ) : ℝ := avg (fun i ↦ v i^2/(1+t*v i))
-def rayI₃ (v : Vec) (t : ℝ) : ℝ := avg (fun i ↦ -v i^3/(1+t*v i)^2)
-
-def rayH₀ (v : Vec) (t : ℝ) : ℝ := avg (fun i ↦ (1+t*v i)*Real.log (1+t*v i)^2)
-def rayH₁ (v : Vec) (t : ℝ) : ℝ := avg (fun i ↦ v i*(Real.log (1+t*v i)^2+2*Real.log (1+t*v i)))
-def rayH₂ (v : Vec) (t : ℝ) : ℝ := avg (fun i ↦ (2*v i^2/(1+t*v i))*(Real.log (1+t*v i)+1))
-def rayH₃ (v : Vec) (t : ℝ) : ℝ := avg (fun i ↦ -2*v i^3*Real.log (1+t*v i)/(1+t*v i)^2)
-
-lemma hasDerivAt_avg {f : ℝ → Vec} {df : Vec} {t : ℝ}
-    (hf : ∀ i, HasDerivAt (fun s ↦ f s i) (df i) t) :
-    HasDerivAt (fun s ↦ avg (f s)) (avg df) t :=
-  (((hf 0).add (hf 1)).add (hf 2)).div_const 3
-
-lemma hasDerivAt_ray_coord (v : Vec) (t : ℝ) (i : Fin 3) :
-    HasDerivAt (fun s : ℝ ↦ 1+s*v i) (v i) t := by
-  simpa using (hasDerivAt_mul_const (v i)).const_add (1:ℝ)
-
-lemma hasDerivAt_rayG₀ {v : Vec} {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
-    HasDerivAt (rayG₀ v) (rayG₁ v t) t :=
-  hasDerivAt_avg (fun i ↦ hasDerivAt_log_edge i (hp i))
-
-lemma hasDerivAt_rayG₁ {v : Vec} {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
-    HasDerivAt (rayG₁ v) (rayG₂ v t) t := by
-  apply hasDerivAt_avg
-  intro i
-  apply ((hasDerivAt_const t (v i)).div (hasDerivAt_ray_coord v t i) (hp i)).congr_deriv
-  simp only [zero_mul, zero_sub]; ring
-
-lemma hasDerivAt_rayG₂ {v : Vec} {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
-    HasDerivAt (rayG₂ v) (rayG₃ v t) t := by
-  apply hasDerivAt_avg
-  intro i
-  apply ((hasDerivAt_const t (-v i^2)).div ((hasDerivAt_ray_coord v t i).pow 2) (pow_ne_zero 2 (hp i))).congr_deriv
-  dsimp
-  field_simp [hp i]; ring
-
-lemma hasDerivAt_rayI₀ {v : Vec} (hv : avg v = 0) {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
-    HasDerivAt (rayI₀ v) (rayI₁ v t) t := by
-  have hd := hasDerivAt_avg (fun i ↦
-    (hasDerivAt_ray_coord v t i).mul (hasDerivAt_log_edge i (hp i)))
-  apply hd.congr_deriv
-  simp only [rayI₁, avg] at *
-  have he (i : Fin 3) : (1+t*v i)*(v i/(1+t*v i)) = v i := by field_simp [hp i]
-  simp_rw [he]
-  linarith
-
-lemma hasDerivAt_rayI₁ {v : Vec} {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
-    HasDerivAt (rayI₁ v) (rayI₂ v t) t := by
-  apply hasDerivAt_avg
-  intro i
-  apply ((hasDerivAt_log_edge i (hp i)).const_mul (v i)).congr_deriv
-  ring
-
-lemma hasDerivAt_rayI₂ {v : Vec} {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
-    HasDerivAt (rayI₂ v) (rayI₃ v t) t := by
-  apply hasDerivAt_avg
-  intro i
-  apply ((hasDerivAt_const t (v i^2)).div (hasDerivAt_ray_coord v t i) (hp i)).congr_deriv
-  simp only [zero_mul, zero_sub]; ring
-
-lemma hasDerivAt_rayH₀ {v : Vec} {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
-    HasDerivAt (rayH₀ v) (rayH₁ v t) t := by
-  apply hasDerivAt_avg
-  intro i
-  apply ((hasDerivAt_ray_coord v t i).mul ((hasDerivAt_log_edge i (hp i)).pow 2)).congr_deriv
-  dsimp
-  have he : (1+t*v i)*(v i/(1+t*v i)) = v i := by field_simp [hp i]
-  calc
-    _ = v i*Real.log (1+t*v i)^2 + 2*Real.log (1+t*v i)*((1+t*v i)*(v i/(1+t*v i))) := by ring
-    _ = _ := by rw [he]; ring
-
-lemma hasDerivAt_rayH₁ {v : Vec} {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
-    HasDerivAt (rayH₁ v) (rayH₂ v t) t := by
-  apply hasDerivAt_avg
-  intro i
-  have hl := hasDerivAt_log_edge i (hp i)
-  apply (((hl.pow 2).add (hl.const_mul 2)).const_mul (v i)).congr_deriv
-  norm_num only [Nat.cast_ofNat, Nat.reduceSub, pow_one]; ring
-
-lemma hasDerivAt_rayH₂ {v : Vec} {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
-    HasDerivAt (rayH₂ v) (rayH₃ v t) t := by
-  apply hasDerivAt_avg
-  intro i
-  have hl := hasDerivAt_log_edge i (hp i)
-  have ha := hasDerivAt_ray_coord v t i
-  apply (((hasDerivAt_const t (2*v i^2)).div ha (hp i)).mul (hl.add_const 1)).congr_deriv
-  simp only [Pi.div_apply]
-  field_simp [hp i]; ring
-
-def rayF₀ (v : Vec) (t : ℝ) : ℝ :=
-  rayG₀ v t + 2*rayI₀ v t + (2/5)*rayH₀ v t - (4/5)*rayG₀ v t*rayI₀ v t + (2/5)*rayG₀ v t^2
-
-def rayF₁ (v : Vec) (t : ℝ) : ℝ :=
-  rayG₁ v t + 2*rayI₁ v t + (2/5)*rayH₁ v t -
-  (4/5)*(rayG₁ v t*rayI₀ v t+rayG₀ v t*rayI₁ v t) + (4/5)*rayG₀ v t*rayG₁ v t
-
-def rayF₂ (v : Vec) (t : ℝ) : ℝ :=
-  rayG₂ v t + 2*rayI₂ v t + (2/5)*rayH₂ v t -
-  (4/5)*(rayG₂ v t*rayI₀ v t+2*rayG₁ v t*rayI₁ v t+rayG₀ v t*rayI₂ v t) +
-  (4/5)*(rayG₁ v t^2+rayG₀ v t*rayG₂ v t)
-
-def rayF₃ (v : Vec) (t : ℝ) : ℝ :=
-  rayG₃ v t + 2*rayI₃ v t + (2/5)*rayH₃ v t -
-  (4/5)*(rayG₃ v t*rayI₀ v t+3*rayG₂ v t*rayI₁ v t+
-    3*rayG₁ v t*rayI₂ v t+rayG₀ v t*rayI₃ v t) +
-  (4/5)*(3*rayG₁ v t*rayG₂ v t+rayG₀ v t*rayG₃ v t)
-
-lemma hasDerivAt_rayF₀ {v : Vec} (hv : avg v = 0) {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
-    HasDerivAt (rayF₀ v) (rayF₁ v t) t := by
-  have hg := hasDerivAt_rayG₀ hp
-  have hi := hasDerivAt_rayI₀ hv hp
-  have hh := hasDerivAt_rayH₀ hp
-  apply (((hg.add (hi.const_mul 2)).add (hh.const_mul (2/5))).sub
-    ((hg.const_mul (4/5)).mul hi) |>.add ((hg.pow 2).const_mul (2/5))).congr_deriv
-  dsimp only [rayF₁, Pi.pow_apply]
-  norm_num only [Nat.cast_ofNat, Nat.reduceSub, pow_one]
-  ring
-
-lemma hasDerivAt_rayF₁ {v : Vec} (hv : avg v = 0) {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
-    HasDerivAt (rayF₁ v) (rayF₂ v t) t := by
-  have hg₀ := hasDerivAt_rayG₀ hp
-  have hg₁ := hasDerivAt_rayG₁ hp
-  have hi₀ := hasDerivAt_rayI₀ hv hp
-  have hi₁ := hasDerivAt_rayI₁ hp
-  have hh₁ := hasDerivAt_rayH₁ hp
-  apply (((hg₁.add (hi₁.const_mul 2)).add (hh₁.const_mul (2/5))).sub
-    (((hg₁.mul hi₀).add (hg₀.mul hi₁)).const_mul (4/5)) |>.add
-      ((hg₀.const_mul (4/5)).mul hg₁)).congr_deriv
-  dsimp only [rayF₂, Pi.add_apply]
-  ring
-
-lemma hasDerivAt_rayF₂ {v : Vec} (hv : avg v = 0) {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
-    HasDerivAt (rayF₂ v) (rayF₃ v t) t := by
-  have hg₀ := hasDerivAt_rayG₀ hp
-  have hg₁ := hasDerivAt_rayG₁ hp
-  have hg₂ := hasDerivAt_rayG₂ hp
-  have hi₀ := hasDerivAt_rayI₀ hv hp
-  have hi₁ := hasDerivAt_rayI₁ hp
-  have hi₂ := hasDerivAt_rayI₂ hp
-  have hh₂ := hasDerivAt_rayH₂ hp
-  apply (((hg₂.add (hi₂.const_mul 2)).add (hh₂.const_mul (2/5))).sub
-    ((((hg₂.mul hi₀).add ((hg₁.const_mul 2).mul hi₁)).add (hg₀.mul hi₂)).const_mul (4/5)) |>.add
-      (((hg₁.pow 2).add (hg₀.mul hg₂)).const_mul (4/5))).congr_deriv
-  dsimp only [rayF₃, Pi.add_apply]
-  norm_num only [Nat.cast_ofNat, Nat.reduceSub, pow_one]
-  ring
-
-lemma rayF₀_eq_functional {v : Vec} (hv : avg v = 0) (t : ℝ) :
-    rayF₀ v t = functional (fun i ↦ 1+t*v i) := by
-  rw [functional_eq]
-  simp only [rayF₀, rayG₀, rayI₀, rayH₀, logAverage, symEntropy, correction, logCentered, avg]
-  rw [third_eq hv]
-  ring
-
-lemma rayF₀_zero (v : Vec) : rayF₀ v 0 = 0 := by
-  norm_num [rayF₀, rayG₀, rayI₀, rayH₀, avg]
-
-lemma rayF₁_zero {v : Vec} (hv : avg v = 0) : rayF₁ v 0 = 0 := by
-  simpa [rayF₁, rayG₀, rayG₁, rayI₀, rayI₁, rayH₁, avg] using hv
-
-lemma rayG₁_scaled {v : Vec} {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
-    t*rayG₁ v t = 1-invMoment (fun i ↦ 1+t*v i) 1 := by
-  have he (i : Fin 3) : t*(v i/(1+t*v i)) = 1-(1+t*v i)⁻¹ := by
-    field_simp [hp i]; ring
-  rw [rayG₁, ← avg_mul]
-  simp_rw [he]
-  simp only [avg, invMoment, pow_one]; ring
-
-lemma rayG₂_scaled {v : Vec} {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
-    t^2*rayG₂ v t = -1+2*invMoment (fun i ↦ 1+t*v i) 1-invMoment (fun i ↦ 1+t*v i) 2 := by
-  have he (i : Fin 3) : t^2*(-v i^2/(1+t*v i)^2) = -1+2*(1+t*v i)⁻¹-(1+t*v i)⁻¹^2 := by
-    field_simp [hp i]; ring
-  rw [rayG₂, ← avg_mul]
-  simp_rw [he]
-  simp only [avg, invMoment, pow_one]; ring
-
-lemma rayG₃_scaled {v : Vec} {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
-    t^3*rayG₃ v t = 2-6*invMoment (fun i ↦ 1+t*v i) 1+
-      6*invMoment (fun i ↦ 1+t*v i) 2-2*invMoment (fun i ↦ 1+t*v i) 3 := by
-  have he (i : Fin 3) : t^3*(2*v i^3/(1+t*v i)^3) =
-      2-6*(1+t*v i)⁻¹+6*(1+t*v i)⁻¹^2-2*(1+t*v i)⁻¹^3 := by
-    field_simp [hp i]; ring
-  rw [rayG₃, ← avg_mul]
-  simp_rw [he]
-  simp only [avg, invMoment, pow_one]; ring
-
-lemma rayI₁_scaled (v : Vec) (t : ℝ) : t*rayI₁ v t = rayI₀ v t-rayG₀ v t := by
-  simp only [rayI₁, rayI₀, rayG₀, avg]; ring
-
-lemma rayI₂_scaled {v : Vec} (hv : avg v = 0) {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
-    t^2*rayI₂ v t = invMoment (fun i ↦ 1+t*v i) 1-1 := by
-  have he (i : Fin 3) : t^2*(v i^2/(1+t*v i)) = (1+t*v i)-2+(1+t*v i)⁻¹ := by
-    field_simp [hp i]; ring
-  rw [rayI₂, ← avg_mul]
-  simp_rw [he]
-  simp only [avg, invMoment, pow_one]
-  rw [third_eq hv]; ring
-
-lemma rayI₃_scaled {v : Vec} (hv : avg v = 0) {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
-    t^3*rayI₃ v t = 2-3*invMoment (fun i ↦ 1+t*v i) 1+invMoment (fun i ↦ 1+t*v i) 2 := by
-  have he (i : Fin 3) : t^3*(-v i^3/(1+t*v i)^2) =
-      -(1+t*v i)+3-3*(1+t*v i)⁻¹+(1+t*v i)⁻¹^2 := by
-    field_simp [hp i]; ring
-  rw [rayI₃, ← avg_mul]
-  simp_rw [he]
-  simp only [avg, invMoment, pow_one]
-  rw [third_eq hv]; ring
-
-lemma rayH₃_scaled {v : Vec} {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
-    t^3*rayH₃ v t = -2*(rayI₀ v t-3*rayG₀ v t+
-      3*(logInvMoment (fun i ↦ 1+t*v i) 1+rayG₀ v t*invMoment (fun i ↦ 1+t*v i) 1)-
-      (logInvMoment (fun i ↦ 1+t*v i) 2+rayG₀ v t*invMoment (fun i ↦ 1+t*v i) 2)) := by
-  have he (i : Fin 3) : t^3*(-2*v i^3*Real.log (1+t*v i)/(1+t*v i)^2) =
-      -2*((1+t*v i)*Real.log (1+t*v i)-3*Real.log (1+t*v i)+
-        3*Real.log (1+t*v i)*(1+t*v i)⁻¹-Real.log (1+t*v i)*(1+t*v i)⁻¹^2) := by
-    field_simp [hp i]; ring
-  rw [rayH₃, ← avg_mul]
-  simp_rw [he]
-  simp only [avg, rayI₀, rayG₀, logInvMoment, logCentered, logAverage, invMoment, pow_one]
-  ring
-
-lemma logMoment_ray {v : Vec} (hv : avg v = 0) (t : ℝ) :
-    logMoment (fun i ↦ 1+t*v i) = rayI₀ v t-rayG₀ v t := by
-  simp only [logMoment, rayI₀, rayG₀, logCentered, logAverage, avg]
-  rw [third_eq hv]; ring
-
-lemma rayF₃_scaled {v : Vec} (hv : avg v = 0) {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
-    t^3*rayF₃ v t = radialG (fun i ↦ 1+t*v i) := by
-  calc
-    _ = t^3*rayG₃ v t + 2*(t^3*rayI₃ v t) + (2/5)*(t^3*rayH₃ v t) -
-      (4/5)*((t^3*rayG₃ v t)*rayI₀ v t+3*(t^2*rayG₂ v t)*(t*rayI₁ v t)+
-        3*(t*rayG₁ v t)*(t^2*rayI₂ v t)+rayG₀ v t*(t^3*rayI₃ v t)) +
-      (4/5)*(3*(t*rayG₁ v t)*(t^2*rayG₂ v t)+rayG₀ v t*(t^3*rayG₃ v t)) := by
-      unfold rayF₃; ring
-    _ = _ := by
-      rw [rayG₁_scaled hp, rayG₂_scaled hp, rayG₃_scaled hp,
-        rayI₁_scaled, rayI₂_scaled hv hp, rayI₃_scaled hv hp, rayH₃_scaled hp]
-      rw [radialG, logMoment_ray hv]
-      ring
-
-end ThreeState.TreeClauses.Radial
-
-end 
-
-noncomputable section
-open Set
-namespace ThreeState.TreeClauses.Radial
-
-lemma monotoneOn_Icc_of_hasDerivAt_nonneg {a b : ℝ} {f f' : ℝ → ℝ}
-    (hf : ∀ t ∈ Icc a b, HasDerivAt f (f' t) t)
-    (hn : ∀ t ∈ Ioo a b, 0 ≤ f' t) : MonotoneOn f (Icc a b) := by
-  apply monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc a b)
-    (fun t ht ↦ (hf t ht).continuousAt.continuousWithinAt)
-  · intro t ht
-    rw [interior_Icc] at ht
-    exact (hf t ⟨ht.1.le, ht.2.le⟩).hasDerivWithinAt
-  · simpa only [interior_Icc] using hn
-
-lemma radial_calculus {f f₁ f₂ f₃ : ℝ → ℝ} {x : ℝ}
-    (h₀ : ∀ t ∈ Icc (0:ℝ) 1, HasDerivAt f (f₁ t) t)
-    (h₁ : ∀ t ∈ Icc (0:ℝ) 1, HasDerivAt f₁ (f₂ t) t)
-    (h₂ : ∀ t ∈ Icc (0:ℝ) 1, HasDerivAt f₂ (f₃ t) t)
-    (hf0 : f 0 = 0) (hf1 : f₁ 0 = 0)
-    (hb : ∀ t ∈ Ioo (0:ℝ) 1, 12*t*x^2 ≤ f₃ t)
-    {lam : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1) :
-    (1-lam^2)/2*x^2 ≤ f 1-(lam^2)⁻¹*f lam := by
-  let Y : ℝ → ℝ := fun t ↦ t*f₁ t-2*f t-t^4*x^2
-  let Y₁ : ℝ → ℝ := fun t ↦ t*f₂ t-f₁ t-4*t^3*x^2
-  let Y₂ : ℝ → ℝ := fun t ↦ t*f₃ t-12*t^2*x^2
-  have dY (t : ℝ) (ht : t ∈ Icc (0:ℝ) 1) : HasDerivAt Y (Y₁ t) t := by
-    apply ((((hasDerivAt_id t).mul (h₁ t ht)).sub ((h₀ t ht).const_mul 2)).sub
-      (((hasDerivAt_id t).pow 4).mul_const (x^2))).congr_deriv
-    dsimp [Y₁]; ring
-  have dY₁ (t : ℝ) (ht : t ∈ Icc (0:ℝ) 1) : HasDerivAt Y₁ (Y₂ t) t := by
-    apply ((((hasDerivAt_id t).mul (h₂ t ht)).sub (h₁ t ht)).sub
-      ((((hasDerivAt_id t).pow 3).const_mul 4).mul_const (x^2))).congr_deriv
-    dsimp [Y₂]; ring
-  have hm₁ : MonotoneOn Y₁ (Icc (0:ℝ) 1) := by
-    apply monotoneOn_Icc_of_hasDerivAt_nonneg dY₁
-    intro t ht
-    have h := mul_le_mul_of_nonneg_left (hb t ht) ht.1.le
-    dsimp only [Y₂]; nlinarith
-  have hn₁ (t : ℝ) (ht : t ∈ Icc (0:ℝ) 1) : 0 ≤ Y₁ t := by
-    have h := hm₁ (by norm_num : (0:ℝ) ∈ Icc 0 1) ht ht.1
-    simpa [Y₁, hf1] using h
-  have hm : MonotoneOn Y (Icc (0:ℝ) 1) :=
-    monotoneOn_Icc_of_hasDerivAt_nonneg dY (fun t ht ↦ hn₁ t ⟨ht.1.le, ht.2.le⟩)
-  have hn (t : ℝ) (ht : t ∈ Icc (0:ℝ) 1) : 0 ≤ Y t := by
-    have h := hm (by norm_num : (0:ℝ) ∈ Icc 0 1) ht ht.1
-    simpa [Y, hf0] using h
-  let q : ℝ → ℝ := fun t ↦ (f t-(1/2)*t^4*x^2)/t^2
-  have dq (t : ℝ) (ht : t ∈ Icc lam 1) : HasDerivAt q (Y t/t^3) t := by
-    have ht0 : 0 < t := lt_of_lt_of_le hl₀ ht.1
-    have hti : t ∈ Icc (0:ℝ) 1 := ⟨ht0.le, ht.2⟩
-    apply (((h₀ t hti).sub ((((hasDerivAt_id t).pow 4).const_mul (1/2)).mul_const (x^2))).div
-      ((hasDerivAt_id t).pow 2) (pow_ne_zero 2 (ne_of_gt ht0))).congr_deriv
-    dsimp [Y]
-    norm_num only [Nat.cast_ofNat, Nat.reduceSub, pow_one]
-    field_simp [ne_of_gt ht0]; ring
-  have hmq : MonotoneOn q (Icc lam 1) := by
-    apply monotoneOn_Icc_of_hasDerivAt_nonneg dq
-    intro t ht
-    exact div_nonneg (hn t ⟨(lt_trans hl₀ ht.1).le, ht.2.le⟩)
-      (pow_nonneg (lt_trans hl₀ ht.1).le 3)
-  have hr := hmq (by exact ⟨le_rfl, hl₁.le⟩ : lam ∈ Icc lam 1)
-    (by exact ⟨hl₁.le, le_rfl⟩ : (1:ℝ) ∈ Icc lam 1) hl₁.le
-  have hr' : f lam/lam^2-lam^2/2*x^2 ≤ f 1-(1/2)*x^2 := by
-    calc
-      f lam/lam^2-lam^2/2*x^2 = q lam := by
-        dsimp [q]; field_simp [ne_of_gt hl₀]
-      _ ≤ q 1 := hr
-      _ = _ := by norm_num [q]
-  rw [inv_mul_eq_div]
-  linarith
-
-lemma positive_edge {m : Vec} (hm : PositiveMessage m) {t : ℝ} (ht : t ∈ Icc (0:ℝ) 1) :
-    PositiveMessage (edge t m) := by
-  refine ⟨fun i ↦ edge_pos hm ht i, ?_⟩
-  change avg (fun i ↦ 1+t*centered m i) = 1
-  rw [avg_add, avg_const, avg_mul, avg_centered hm]
-  ring
-
-lemma momentX_edge (m : Vec) (t : ℝ) :
-    momentX (centered (edge t m)) = t^2*momentX (centered m) := by
-  simp only [momentX, centered, edge, avg]; ring
-
-lemma rayF₃_lower {m : Vec} (hm : PositiveMessage m) {t : ℝ} (ht : t ∈ Ioo (0:ℝ) 1) :
-    12*t*momentX (centered m)^2 ≤ rayF₃ (centered m) t := by
-  have h := radialG_bound (positive_edge hm ⟨ht.1.le, ht.2.le⟩)
-  have hp (i : Fin 3) : 1+t*centered m i ≠ 0 :=
-    ne_of_gt (edge_pos hm ⟨ht.1.le, ht.2.le⟩ i)
-  rw [momentX_edge] at h
-  have he : t^3*rayF₃ (centered m) t = radialG (edge t m) :=
-    rayF₃_scaled (avg_centered hm) hp
-  rw [← he] at h
-  have h' : 0 ≤ t^3*(rayF₃ (centered m) t-12*t*momentX (centered m)^2) := by nlinarith
-  exact sub_nonneg.mp ((mul_nonneg_iff_of_pos_left (pow_pos ht.1 3)).mp h')
-
-theorem radial_inequality : RadialInequality := by
-  intro m hm lam hl₀ hl₁
-  have hp (t : ℝ) (ht : t ∈ Icc (0:ℝ) 1) (i : Fin 3) : 1+t*centered m i ≠ 0 :=
-    ne_of_gt (edge_pos hm ht i)
-  have h := radial_calculus
-    (fun t ht ↦ hasDerivAt_rayF₀ (avg_centered hm) (hp t ht))
-    (fun t ht ↦ hasDerivAt_rayF₁ (avg_centered hm) (hp t ht))
-    (fun t ht ↦ hasDerivAt_rayF₂ (avg_centered hm) (hp t ht))
-    (rayF₀_zero (centered m)) (rayF₁_zero (avg_centered hm))
-    (fun _ ht ↦ rayF₃_lower hm ht) hl₀ hl₁
-  rw [rayF₀_eq_functional (avg_centered hm), rayF₀_eq_functional (avg_centered hm)] at h
-  have he1 : (fun i ↦ 1+(1:ℝ)*centered m i) = m := by funext i; simp [centered]
-  rw [he1] at h
-  exact h
-
-end ThreeState.TreeClauses.Radial
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeStateClauses.RadialBounds
+
+namespace OAI
+
+/-! Radial Calculus. -/
+
+open scoped BigOperators ENNReal NNReal Topology
+open Filter
+noncomputable section
+open Set
+namespace ThreeState.TreeClauses.Radial
+
+def rayG₀ (v : Vec) (t : ℝ) : ℝ := avg (fun i ↦ Real.log (1+t*v i))
+def rayG₁ (v : Vec) (t : ℝ) : ℝ := avg (fun i ↦ v i/(1+t*v i))
+def rayG₂ (v : Vec) (t : ℝ) : ℝ := avg (fun i ↦ -v i^2/(1+t*v i)^2)
+def rayG₃ (v : Vec) (t : ℝ) : ℝ := avg (fun i ↦ 2*v i^3/(1+t*v i)^3)
+
+def rayI₀ (v : Vec) (t : ℝ) : ℝ := avg (fun i ↦ (1+t*v i)*Real.log (1+t*v i))
+def rayI₁ (v : Vec) (t : ℝ) : ℝ := avg (fun i ↦ v i*Real.log (1+t*v i))
+def rayI₂ (v : Vec) (t : ℝ) : ℝ := avg (fun i ↦ v i^2/(1+t*v i))
+def rayI₃ (v : Vec) (t : ℝ) : ℝ := avg (fun i ↦ -v i^3/(1+t*v i)^2)
+
+def rayH₀ (v : Vec) (t : ℝ) : ℝ := avg (fun i ↦ (1+t*v i)*Real.log (1+t*v i)^2)
+def rayH₁ (v : Vec) (t : ℝ) : ℝ := avg (fun i ↦ v i*(Real.log (1+t*v i)^2+2*Real.log (1+t*v i)))
+def rayH₂ (v : Vec) (t : ℝ) : ℝ := avg (fun i ↦ (2*v i^2/(1+t*v i))*(Real.log (1+t*v i)+1))
+def rayH₃ (v : Vec) (t : ℝ) : ℝ := avg (fun i ↦ -2*v i^3*Real.log (1+t*v i)/(1+t*v i)^2)
+
+lemma hasDerivAt_avg {f : ℝ → Vec} {df : Vec} {t : ℝ}
+    (hf : ∀ i, HasDerivAt (fun s ↦ f s i) (df i) t) :
+    HasDerivAt (fun s ↦ avg (f s)) (avg df) t :=
+  (((hf 0).add (hf 1)).add (hf 2)).div_const 3
+
+lemma hasDerivAt_ray_coord (v : Vec) (t : ℝ) (i : Fin 3) :
+    HasDerivAt (fun s : ℝ ↦ 1+s*v i) (v i) t := by
+  simpa using (hasDerivAt_mul_const (v i)).const_add (1:ℝ)
+
+lemma hasDerivAt_rayG₀ {v : Vec} {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
+    HasDerivAt (rayG₀ v) (rayG₁ v t) t :=
+  hasDerivAt_avg (fun i ↦ hasDerivAt_log_edge i (hp i))
+
+lemma hasDerivAt_rayG₁ {v : Vec} {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
+    HasDerivAt (rayG₁ v) (rayG₂ v t) t := by
+  apply hasDerivAt_avg
+  intro i
+  apply ((hasDerivAt_const t (v i)).div (hasDerivAt_ray_coord v t i) (hp i)).congr_deriv
+  simp only [zero_mul, zero_sub]; ring
+
+lemma hasDerivAt_rayG₂ {v : Vec} {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
+    HasDerivAt (rayG₂ v) (rayG₃ v t) t := by
+  apply hasDerivAt_avg
+  intro i
+  apply ((hasDerivAt_const t (-v i^2)).div ((hasDerivAt_ray_coord v t i).pow 2) (pow_ne_zero 2 (hp i))).congr_deriv
+  dsimp
+  field_simp [hp i]; ring
+
+lemma hasDerivAt_rayI₀ {v : Vec} (hv : avg v = 0) {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
+    HasDerivAt (rayI₀ v) (rayI₁ v t) t := by
+  have hd := hasDerivAt_avg (fun i ↦
+    (hasDerivAt_ray_coord v t i).mul (hasDerivAt_log_edge i (hp i)))
+  apply hd.congr_deriv
+  simp only [rayI₁, avg] at *
+  have he (i : Fin 3) : (1+t*v i)*(v i/(1+t*v i)) = v i := by field_simp [hp i]
+  simp_rw [he]
+  linarith
+
+lemma hasDerivAt_rayI₁ {v : Vec} {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
+    HasDerivAt (rayI₁ v) (rayI₂ v t) t := by
+  apply hasDerivAt_avg
+  intro i
+  apply ((hasDerivAt_log_edge i (hp i)).const_mul (v i)).congr_deriv
+  ring
+
+lemma hasDerivAt_rayI₂ {v : Vec} {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
+    HasDerivAt (rayI₂ v) (rayI₃ v t) t := by
+  apply hasDerivAt_avg
+  intro i
+  apply ((hasDerivAt_const t (v i^2)).div (hasDerivAt_ray_coord v t i) (hp i)).congr_deriv
+  simp only [zero_mul, zero_sub]; ring
+
+lemma hasDerivAt_rayH₀ {v : Vec} {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
+    HasDerivAt (rayH₀ v) (rayH₁ v t) t := by
+  apply hasDerivAt_avg
+  intro i
+  apply ((hasDerivAt_ray_coord v t i).mul ((hasDerivAt_log_edge i (hp i)).pow 2)).congr_deriv
+  dsimp
+  have he : (1+t*v i)*(v i/(1+t*v i)) = v i := by field_simp [hp i]
+  calc
+    _ = v i*Real.log (1+t*v i)^2 + 2*Real.log (1+t*v i)*((1+t*v i)*(v i/(1+t*v i))) := by ring
+    _ = _ := by rw [he]; ring
+
+lemma hasDerivAt_rayH₁ {v : Vec} {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
+    HasDerivAt (rayH₁ v) (rayH₂ v t) t := by
+  apply hasDerivAt_avg
+  intro i
+  have hl := hasDerivAt_log_edge i (hp i)
+  apply (((hl.pow 2).add (hl.const_mul 2)).const_mul (v i)).congr_deriv
+  norm_num only [Nat.cast_ofNat, Nat.reduceSub, pow_one]; ring
+
+lemma hasDerivAt_rayH₂ {v : Vec} {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
+    HasDerivAt (rayH₂ v) (rayH₃ v t) t := by
+  apply hasDerivAt_avg
+  intro i
+  have hl := hasDerivAt_log_edge i (hp i)
+  have ha := hasDerivAt_ray_coord v t i
+  apply (((hasDerivAt_const t (2*v i^2)).div ha (hp i)).mul (hl.add_const 1)).congr_deriv
+  simp only [Pi.div_apply]
+  field_simp [hp i]; ring
+
+def rayF₀ (v : Vec) (t : ℝ) : ℝ :=
+  rayG₀ v t + 2*rayI₀ v t + (2/5)*rayH₀ v t - (4/5)*rayG₀ v t*rayI₀ v t + (2/5)*rayG₀ v t^2
+
+def rayF₁ (v : Vec) (t : ℝ) : ℝ :=
+  rayG₁ v t + 2*rayI₁ v t + (2/5)*rayH₁ v t -
+  (4/5)*(rayG₁ v t*rayI₀ v t+rayG₀ v t*rayI₁ v t) + (4/5)*rayG₀ v t*rayG₁ v t
+
+def rayF₂ (v : Vec) (t : ℝ) : ℝ :=
+  rayG₂ v t + 2*rayI₂ v t + (2/5)*rayH₂ v t -
+  (4/5)*(rayG₂ v t*rayI₀ v t+2*rayG₁ v t*rayI₁ v t+rayG₀ v t*rayI₂ v t) +
+  (4/5)*(rayG₁ v t^2+rayG₀ v t*rayG₂ v t)
+
+def rayF₃ (v : Vec) (t : ℝ) : ℝ :=
+  rayG₃ v t + 2*rayI₃ v t + (2/5)*rayH₃ v t -
+  (4/5)*(rayG₃ v t*rayI₀ v t+3*rayG₂ v t*rayI₁ v t+
+    3*rayG₁ v t*rayI₂ v t+rayG₀ v t*rayI₃ v t) +
+  (4/5)*(3*rayG₁ v t*rayG₂ v t+rayG₀ v t*rayG₃ v t)
+
+lemma hasDerivAt_rayF₀ {v : Vec} (hv : avg v = 0) {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
+    HasDerivAt (rayF₀ v) (rayF₁ v t) t := by
+  have hg := hasDerivAt_rayG₀ hp
+  have hi := hasDerivAt_rayI₀ hv hp
+  have hh := hasDerivAt_rayH₀ hp
+  apply (((hg.add (hi.const_mul 2)).add (hh.const_mul (2/5))).sub
+    ((hg.const_mul (4/5)).mul hi) |>.add ((hg.pow 2).const_mul (2/5))).congr_deriv
+  dsimp only [rayF₁, Pi.pow_apply]
+  norm_num only [Nat.cast_ofNat, Nat.reduceSub, pow_one]
+  ring
+
+lemma hasDerivAt_rayF₁ {v : Vec} (hv : avg v = 0) {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
+    HasDerivAt (rayF₁ v) (rayF₂ v t) t := by
+  have hg₀ := hasDerivAt_rayG₀ hp
+  have hg₁ := hasDerivAt_rayG₁ hp
+  have hi₀ := hasDerivAt_rayI₀ hv hp
+  have hi₁ := hasDerivAt_rayI₁ hp
+  have hh₁ := hasDerivAt_rayH₁ hp
+  apply (((hg₁.add (hi₁.const_mul 2)).add (hh₁.const_mul (2/5))).sub
+    (((hg₁.mul hi₀).add (hg₀.mul hi₁)).const_mul (4/5)) |>.add
+      ((hg₀.const_mul (4/5)).mul hg₁)).congr_deriv
+  dsimp only [rayF₂, Pi.add_apply]
+  ring
+
+lemma hasDerivAt_rayF₂ {v : Vec} (hv : avg v = 0) {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
+    HasDerivAt (rayF₂ v) (rayF₃ v t) t := by
+  have hg₀ := hasDerivAt_rayG₀ hp
+  have hg₁ := hasDerivAt_rayG₁ hp
+  have hg₂ := hasDerivAt_rayG₂ hp
+  have hi₀ := hasDerivAt_rayI₀ hv hp
+  have hi₁ := hasDerivAt_rayI₁ hp
+  have hi₂ := hasDerivAt_rayI₂ hp
+  have hh₂ := hasDerivAt_rayH₂ hp
+  apply (((hg₂.add (hi₂.const_mul 2)).add (hh₂.const_mul (2/5))).sub
+    ((((hg₂.mul hi₀).add ((hg₁.const_mul 2).mul hi₁)).add (hg₀.mul hi₂)).const_mul (4/5)) |>.add
+      (((hg₁.pow 2).add (hg₀.mul hg₂)).const_mul (4/5))).congr_deriv
+  dsimp only [rayF₃, Pi.add_apply]
+  norm_num only [Nat.cast_ofNat, Nat.reduceSub, pow_one]
+  ring
+
+lemma rayF₀_eq_functional {v : Vec} (hv : avg v = 0) (t : ℝ) :
+    rayF₀ v t = functional (fun i ↦ 1+t*v i) := by
+  rw [functional_eq]
+  simp only [rayF₀, rayG₀, rayI₀, rayH₀, logAverage, symEntropy, correction, logCentered, avg]
+  rw [third_eq hv]
+  ring
+
+lemma rayF₀_zero (v : Vec) : rayF₀ v 0 = 0 := by
+  norm_num [rayF₀, rayG₀, rayI₀, rayH₀, avg]
+
+lemma rayF₁_zero {v : Vec} (hv : avg v = 0) : rayF₁ v 0 = 0 := by
+  simpa [rayF₁, rayG₀, rayG₁, rayI₀, rayI₁, rayH₁, avg] using hv
+
+lemma rayG₁_scaled {v : Vec} {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
+    t*rayG₁ v t = 1-invMoment (fun i ↦ 1+t*v i) 1 := by
+  have he (i : Fin 3) : t*(v i/(1+t*v i)) = 1-(1+t*v i)⁻¹ := by
+    field_simp [hp i]; ring
+  rw [rayG₁, ← avg_mul]
+  simp_rw [he]
+  simp only [avg, invMoment, pow_one]; ring
+
+lemma rayG₂_scaled {v : Vec} {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
+    t^2*rayG₂ v t = -1+2*invMoment (fun i ↦ 1+t*v i) 1-invMoment (fun i ↦ 1+t*v i) 2 := by
+  have he (i : Fin 3) : t^2*(-v i^2/(1+t*v i)^2) = -1+2*(1+t*v i)⁻¹-(1+t*v i)⁻¹^2 := by
+    field_simp [hp i]; ring
+  rw [rayG₂, ← avg_mul]
+  simp_rw [he]
+  simp only [avg, invMoment, pow_one]; ring
+
+lemma rayG₃_scaled {v : Vec} {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
+    t^3*rayG₃ v t = 2-6*invMoment (fun i ↦ 1+t*v i) 1+
+      6*invMoment (fun i ↦ 1+t*v i) 2-2*invMoment (fun i ↦ 1+t*v i) 3 := by
+  have he (i : Fin 3) : t^3*(2*v i^3/(1+t*v i)^3) =
+      2-6*(1+t*v i)⁻¹+6*(1+t*v i)⁻¹^2-2*(1+t*v i)⁻¹^3 := by
+    field_simp [hp i]; ring
+  rw [rayG₃, ← avg_mul]
+  simp_rw [he]
+  simp only [avg, invMoment, pow_one]; ring
+
+lemma rayI₁_scaled (v : Vec) (t : ℝ) : t*rayI₁ v t = rayI₀ v t-rayG₀ v t := by
+  simp only [rayI₁, rayI₀, rayG₀, avg]; ring
+
+lemma rayI₂_scaled {v : Vec} (hv : avg v = 0) {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
+    t^2*rayI₂ v t = invMoment (fun i ↦ 1+t*v i) 1-1 := by
+  have he (i : Fin 3) : t^2*(v i^2/(1+t*v i)) = (1+t*v i)-2+(1+t*v i)⁻¹ := by
+    field_simp [hp i]; ring
+  rw [rayI₂, ← avg_mul]
+  simp_rw [he]
+  simp only [avg, invMoment, pow_one]
+  rw [third_eq hv]; ring
+
+lemma rayI₃_scaled {v : Vec} (hv : avg v = 0) {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
+    t^3*rayI₃ v t = 2-3*invMoment (fun i ↦ 1+t*v i) 1+invMoment (fun i ↦ 1+t*v i) 2 := by
+  have he (i : Fin 3) : t^3*(-v i^3/(1+t*v i)^2) =
+      -(1+t*v i)+3-3*(1+t*v i)⁻¹+(1+t*v i)⁻¹^2 := by
+    field_simp [hp i]; ring
+  rw [rayI₃, ← avg_mul]
+  simp_rw [he]
+  simp only [avg, invMoment, pow_one]
+  rw [third_eq hv]; ring
+
+lemma rayH₃_scaled {v : Vec} {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
+    t^3*rayH₃ v t = -2*(rayI₀ v t-3*rayG₀ v t+
+      3*(logInvMoment (fun i ↦ 1+t*v i) 1+rayG₀ v t*invMoment (fun i ↦ 1+t*v i) 1)-
+      (logInvMoment (fun i ↦ 1+t*v i) 2+rayG₀ v t*invMoment (fun i ↦ 1+t*v i) 2)) := by
+  have he (i : Fin 3) : t^3*(-2*v i^3*Real.log (1+t*v i)/(1+t*v i)^2) =
+      -2*((1+t*v i)*Real.log (1+t*v i)-3*Real.log (1+t*v i)+
+        3*Real.log (1+t*v i)*(1+t*v i)⁻¹-Real.log (1+t*v i)*(1+t*v i)⁻¹^2) := by
+    field_simp [hp i]; ring
+  rw [rayH₃, ← avg_mul]
+  simp_rw [he]
+  simp only [avg, rayI₀, rayG₀, logInvMoment, logCentered, logAverage, invMoment, pow_one]
+  ring
+
+lemma logMoment_ray {v : Vec} (hv : avg v = 0) (t : ℝ) :
+    logMoment (fun i ↦ 1+t*v i) = rayI₀ v t-rayG₀ v t := by
+  simp only [logMoment, rayI₀, rayG₀, logCentered, logAverage, avg]
+  rw [third_eq hv]; ring
+
+lemma rayF₃_scaled {v : Vec} (hv : avg v = 0) {t : ℝ} (hp : ∀ i, 1+t*v i ≠ 0) :
+    t^3*rayF₃ v t = radialG (fun i ↦ 1+t*v i) := by
+  calc
+    _ = t^3*rayG₃ v t + 2*(t^3*rayI₃ v t) + (2/5)*(t^3*rayH₃ v t) -
+      (4/5)*((t^3*rayG₃ v t)*rayI₀ v t+3*(t^2*rayG₂ v t)*(t*rayI₁ v t)+
+        3*(t*rayG₁ v t)*(t^2*rayI₂ v t)+rayG₀ v t*(t^3*rayI₃ v t)) +
+      (4/5)*(3*(t*rayG₁ v t)*(t^2*rayG₂ v t)+rayG₀ v t*(t^3*rayG₃ v t)) := by
+      unfold rayF₃; ring
+    _ = _ := by
+      rw [rayG₁_scaled hp, rayG₂_scaled hp, rayG₃_scaled hp,
+        rayI₁_scaled, rayI₂_scaled hv hp, rayI₃_scaled hv hp, rayH₃_scaled hp]
+      rw [radialG, logMoment_ray hv]
+      ring
+
+end ThreeState.TreeClauses.Radial
+
+end 
+
+noncomputable section
+open Set
+namespace ThreeState.TreeClauses.Radial
+
+lemma monotoneOn_Icc_of_hasDerivAt_nonneg {a b : ℝ} {f f' : ℝ → ℝ}
+    (hf : ∀ t ∈ Icc a b, HasDerivAt f (f' t) t)
+    (hn : ∀ t ∈ Ioo a b, 0 ≤ f' t) : MonotoneOn f (Icc a b) := by
+  apply monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc a b)
+    (fun t ht ↦ (hf t ht).continuousAt.continuousWithinAt)
+  · intro t ht
+    rw [interior_Icc] at ht
+    exact (hf t ⟨ht.1.le, ht.2.le⟩).hasDerivWithinAt
+  · simpa only [interior_Icc] using hn
+
+lemma radial_calculus {f f₁ f₂ f₃ : ℝ → ℝ} {x : ℝ}
+    (h₀ : ∀ t ∈ Icc (0:ℝ) 1, HasDerivAt f (f₁ t) t)
+    (h₁ : ∀ t ∈ Icc (0:ℝ) 1, HasDerivAt f₁ (f₂ t) t)
+    (h₂ : ∀ t ∈ Icc (0:ℝ) 1, HasDerivAt f₂ (f₃ t) t)
+    (hf0 : f 0 = 0) (hf1 : f₁ 0 = 0)
+    (hb : ∀ t ∈ Ioo (0:ℝ) 1, 12*t*x^2 ≤ f₃ t)
+    {lam : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1) :
+    (1-lam^2)/2*x^2 ≤ f 1-(lam^2)⁻¹*f lam := by
+  let Y : ℝ → ℝ := fun t ↦ t*f₁ t-2*f t-t^4*x^2
+  let Y₁ : ℝ → ℝ := fun t ↦ t*f₂ t-f₁ t-4*t^3*x^2
+  let Y₂ : ℝ → ℝ := fun t ↦ t*f₃ t-12*t^2*x^2
+  have dY (t : ℝ) (ht : t ∈ Icc (0:ℝ) 1) : HasDerivAt Y (Y₁ t) t := by
+    apply ((((hasDerivAt_id t).mul (h₁ t ht)).sub ((h₀ t ht).const_mul 2)).sub
+      (((hasDerivAt_id t).pow 4).mul_const (x^2))).congr_deriv
+    dsimp [Y₁]; ring
+  have dY₁ (t : ℝ) (ht : t ∈ Icc (0:ℝ) 1) : HasDerivAt Y₁ (Y₂ t) t := by
+    apply ((((hasDerivAt_id t).mul (h₂ t ht)).sub (h₁ t ht)).sub
+      ((((hasDerivAt_id t).pow 3).const_mul 4).mul_const (x^2))).congr_deriv
+    dsimp [Y₂]; ring
+  have hm₁ : MonotoneOn Y₁ (Icc (0:ℝ) 1) := by
+    apply monotoneOn_Icc_of_hasDerivAt_nonneg dY₁
+    intro t ht
+    have h := mul_le_mul_of_nonneg_left (hb t ht) ht.1.le
+    dsimp only [Y₂]; nlinarith
+  have hn₁ (t : ℝ) (ht : t ∈ Icc (0:ℝ) 1) : 0 ≤ Y₁ t := by
+    have h := hm₁ (by norm_num : (0:ℝ) ∈ Icc 0 1) ht ht.1
+    simpa [Y₁, hf1] using h
+  have hm : MonotoneOn Y (Icc (0:ℝ) 1) :=
+    monotoneOn_Icc_of_hasDerivAt_nonneg dY (fun t ht ↦ hn₁ t ⟨ht.1.le, ht.2.le⟩)
+  have hn (t : ℝ) (ht : t ∈ Icc (0:ℝ) 1) : 0 ≤ Y t := by
+    have h := hm (by norm_num : (0:ℝ) ∈ Icc 0 1) ht ht.1
+    simpa [Y, hf0] using h
+  let q : ℝ → ℝ := fun t ↦ (f t-(1/2)*t^4*x^2)/t^2
+  have dq (t : ℝ) (ht : t ∈ Icc lam 1) : HasDerivAt q (Y t/t^3) t := by
+    have ht0 : 0 < t := lt_of_lt_of_le hl₀ ht.1
+    have hti : t ∈ Icc (0:ℝ) 1 := ⟨ht0.le, ht.2⟩
+    apply (((h₀ t hti).sub ((((hasDerivAt_id t).pow 4).const_mul (1/2)).mul_const (x^2))).div
+      ((hasDerivAt_id t).pow 2) (pow_ne_zero 2 (ne_of_gt ht0))).congr_deriv
+    dsimp [Y]
+    norm_num only [Nat.cast_ofNat, Nat.reduceSub, pow_one]
+    field_simp [ne_of_gt ht0]; ring
+  have hmq : MonotoneOn q (Icc lam 1) := by
+    apply monotoneOn_Icc_of_hasDerivAt_nonneg dq
+    intro t ht
+    exact div_nonneg (hn t ⟨(lt_trans hl₀ ht.1).le, ht.2.le⟩)
+      (pow_nonneg (lt_trans hl₀ ht.1).le 3)
+  have hr := hmq (by exact ⟨le_rfl, hl₁.le⟩ : lam ∈ Icc lam 1)
+    (by exact ⟨hl₁.le, le_rfl⟩ : (1:ℝ) ∈ Icc lam 1) hl₁.le
+  have hr' : f lam/lam^2-lam^2/2*x^2 ≤ f 1-(1/2)*x^2 := by
+    calc
+      f lam/lam^2-lam^2/2*x^2 = q lam := by
+        dsimp [q]; field_simp [ne_of_gt hl₀]
+      _ ≤ q 1 := hr
+      _ = _ := by norm_num [q]
+  rw [inv_mul_eq_div]
+  linarith
+
+lemma positive_edge {m : Vec} (hm : PositiveMessage m) {t : ℝ} (ht : t ∈ Icc (0:ℝ) 1) :
+    PositiveMessage (edge t m) := by
+  refine ⟨fun i ↦ edge_pos hm ht i, ?_⟩
+  change avg (fun i ↦ 1+t*centered m i) = 1
+  rw [avg_add, avg_const, avg_mul, avg_centered hm]
+  ring
+
+lemma momentX_edge (m : Vec) (t : ℝ) :
+    momentX (centered (edge t m)) = t^2*momentX (centered m) := by
+  simp only [momentX, centered, edge, avg]; ring
+
+lemma rayF₃_lower {m : Vec} (hm : PositiveMessage m) {t : ℝ} (ht : t ∈ Ioo (0:ℝ) 1) :
+    12*t*momentX (centered m)^2 ≤ rayF₃ (centered m) t := by
+  have h := radialG_bound (positive_edge hm ⟨ht.1.le, ht.2.le⟩)
+  have hp (i : Fin 3) : 1+t*centered m i ≠ 0 :=
+    ne_of_gt (edge_pos hm ⟨ht.1.le, ht.2.le⟩ i)
+  rw [momentX_edge] at h
+  have he : t^3*rayF₃ (centered m) t = radialG (edge t m) :=
+    rayF₃_scaled (avg_centered hm) hp
+  rw [← he] at h
+  have h' : 0 ≤ t^3*(rayF₃ (centered m) t-12*t*momentX (centered m)^2) := by nlinarith
+  exact sub_nonneg.mp ((mul_nonneg_iff_of_pos_left (pow_pos ht.1 3)).mp h')
+
+theorem radial_inequality : RadialInequality := by
+  intro m hm lam hl₀ hl₁
+  have hp (t : ℝ) (ht : t ∈ Icc (0:ℝ) 1) (i : Fin 3) : 1+t*centered m i ≠ 0 :=
+    ne_of_gt (edge_pos hm ht i)
+  have h := radial_calculus
+    (fun t ht ↦ hasDerivAt_rayF₀ (avg_centered hm) (hp t ht))
+    (fun t ht ↦ hasDerivAt_rayF₁ (avg_centered hm) (hp t ht))
+    (fun t ht ↦ hasDerivAt_rayF₂ (avg_centered hm) (hp t ht))
+    (rayF₀_zero (centered m)) (rayF₁_zero (avg_centered hm))
+    (fun _ ht ↦ rayF₃_lower hm ht) hl₀ hl₁
+  rw [rayF₀_eq_functional (avg_centered hm), rayF₀_eq_functional (avg_centered hm)] at h
+  have he1 : (fun i ↦ 1+(1:ℝ)*centered m i) = m := by funext i; simp [centered]
+  rw [he1] at h
+  exact h
+
+end ThreeState.TreeClauses.Radial
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeStateClauses/RegularCritical.lean b/lean/OAI/Probability/ThreeStateClauses/RegularCritical.lean
index 7a5ca724758dc6f123cb4389cf3e9ef37969cd8c..4f9ea571d61432794323550b192de015bcb0ebec 100644
--- a/lean/OAI/Probability/ThreeStateClauses/RegularCritical.lean
+++ b/lean/OAI/Probability/ThreeStateClauses/RegularCritical.lean
@@ -1,249 +1,249 @@
-import Mathlib
-import OAI.Probability.ThreeStateClauses.CompactLimits
-import OAI.Probability.ThreeStateClauses.PositiveFixed
-
-namespace OAI
-
-/-! Regular Critical. -/
-
-open scoped BigOperators ENNReal NNReal Topology
-open Filter
-noncomputable section
-open Set MeasureTheory Filter
-open scoped Topology
-namespace ThreeState.TreeClauses.Tree
-open ThreeState.TreeClauses.Experiment ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
-
-lemma regular_positive_critical_mu (b : ℕ) (hb : 2 ≤ b) {lam : ℝ}
-    (hl₀ : 0 < lam) (hl₁ : lam < 1) (hlam : Admissible lam) (hcrit : (b:ℝ)*lam^2 = 1) :
-    Tendsto (fun ℓ ↦ (regularPosteriorLaw b hl₀.le hl₁ ℓ).mu) atTop (𝓝 0) := by
-  let Q := regularPosteriorLaw b hl₀.le hl₁
-  let C := fun ℓ ↦ degradationCoupling (regularLaw b lam hlam ℓ) (regularExpansion b lam hlam ℓ)
-  have hf (ℓ : ℕ) : (C ℓ).map Prod.fst = (Q ℓ).probability := by
-    rw [degradationCoupling_fst, regularPosteriorLaw_probability b hl₀.le hl₁ hlam ℓ]
-  have hs (ℓ : ℕ) : (C ℓ).map Prod.snd = (Q (ℓ+1)).probability := by
-    rw [degradationCoupling_snd]
-    simp only [regularLaw_degrades]
-    exact (regularPosteriorLaw_probability b hl₀.le hl₁ hlam (ℓ+1)).symm
-  have hl (ℓ : ℕ) : (∫ z, quadraticGap z ∂(C ℓ).toMeasure) = (Q ℓ).mu-(Q (ℓ+1)).mu := by
-    rw [degradationCoupling_loss]
-    simp only [regularLaw_degrades]
-    simp only [Q, Law.mu, regularPosteriorLaw_probability b hl₀.le hl₁ hlam]
-  obtain ⟨R,hR,hlim⟩ := degrading_finite_fixed_point Q C hf hs hl hl₀.le hl₁ b (fun _ ↦ rfl)
-  rw [regular_fixed_mu_zero R hl₀ hl₁ b hb hcrit hR] at hlim
-  exact hlim
-
-lemma regular_advantage_sq_bound (b : ℕ) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
-    (hlam : Admissible lam) (ℓ : ℕ) :
-    (advantage lam hlam (PMF.pure b) ℓ)^2 ≤ (regularPosteriorLaw b hl₀ hl₁ ℓ).mu/2 := by
-  have h := discrete_degradation_mu (regularLaw b lam hlam ℓ) (fun v ↦ PMF.pure (forgetRegular b ℓ v))
-  change (∫ m, xMoment m ∂(discreteProbability (fun i ↦ (regularLaw b lam hlam ℓ i).map (forgetRegular b ℓ))).toMeasure) ≤ _ at h
-  simp only [regularLaw_forget] at h
-  have ha := advantage_sq_bound lam hlam (PMF.pure b) ℓ
-  unfold Law.mu
-  rw [regularPosteriorLaw_probability b hl₀ hl₁ hlam ℓ]
-  linarith
-
-theorem regular_positive_critical_nonreconstruction (b : ℕ) (hb : 2 ≤ b)
-    (lam : ℝ) (hlam : Admissible lam) (hl₀ : 0 < lam) (hcrit : (b:ℝ)*lam^2 = 1) :
-    Tendsto (advantage lam hlam (PMF.pure b)) atTop (𝓝 0) := by
-  have hb' : (2:ℝ) ≤ b := by exact_mod_cast hb
-  have hl₁ : lam < 1 := by
-    have h := mul_le_mul_of_nonneg_right hb' (sq_nonneg lam)
-    nlinarith
-  have hm := regular_positive_critical_mu b hb hl₀ hl₁ hlam hcrit
-  have hu : Tendsto (fun ℓ ↦ Real.sqrt (regularPosteriorLaw b hl₀.le hl₁ ℓ).mu) atTop (𝓝 0) := by
-    simpa only [Real.sqrt_zero, Function.comp_def] using Real.continuous_sqrt.continuousAt.tendsto.comp hm
-  apply squeeze_zero (fun ℓ ↦ advantage_nonneg lam hlam (PMF.pure b) ℓ) _ hu
-  intro ℓ
-  apply (sq_le_sq₀ (advantage_nonneg lam hlam (PMF.pure b) ℓ) (Real.sqrt_nonneg _)).mp
-  rw [Real.sq_sqrt (regularPosteriorLaw b hl₀.le hl₁ ℓ).mu_nonneg]
-  have h := regular_advantage_sq_bound b hl₀.le hl₁ hlam ℓ
-  linarith [(regularPosteriorLaw b hl₀.le hl₁ ℓ).mu_nonneg]
-
-end ThreeState.TreeClauses.Tree
-
-end 
-
-noncomputable section
-open Set MeasureTheory
-open scoped BigOperators ENNReal
-namespace ThreeState.TreeClauses.Tree
-open ThreeState.TreeClauses.Experiment
-
-lemma channelWeight_alt (lam : ℝ) (i j : Spin) :
-    channelWeight lam i j = lam*(if i = j then 1 else 0)+(1-lam)/3 := by
-  unfold channelWeight
-  split_ifs <;> ring
-
-lemma channelWeight_compose (lam c : ℝ) (i j : Spin) :
-    (∑ k : Spin, channelWeight lam i k*channelWeight c k j) = channelWeight (lam*c) i j := by
-  classical
-  simp only [channelWeight_alt, add_mul, mul_add, Finset.sum_add_distrib,
-    ← Finset.mul_sum, mul_assoc]
-  by_cases h : i = j
-  · subst j
-    simp
-    ring
-  · simp [h]
-    ring
-
-lemma channel_compose {lam c : ℝ} (hlam : Admissible lam) (hc : Admissible c)
-    (hprod : Admissible (lam*c)) (i : Spin) :
-    (channel lam hlam i).bind (channel c hc) = channel (lam*c) hprod i := by
-  apply PMF.ext
-  intro j
-  apply (ENNReal.toReal_eq_toReal_iff' ((channel lam hlam i).bind (channel c hc) |>.apply_ne_top j)
-    ((channel (lam*c) hprod i).apply_ne_top j)).mp
-  rw [PMF.bind_apply, tsum_fintype,
-    ENNReal.toReal_sum (fun k _ ↦ ENNReal.mul_ne_top ((channel lam hlam i).apply_ne_top k)
-      ((channel c hc k).apply_ne_top j))]
-  simp only [ENNReal.toReal_mul, channel_real]
-  exact channelWeight_compose lam c i j
-
-lemma channel_permute {lam : ℝ} (hlam : Admissible lam) (σ : Equiv.Perm Spin) (i : Spin) :
-    (channel lam hlam i).map σ = channel lam hlam (σ i) := by
-  classical
-  apply PMF.ext
-  intro j
-  rw [PMF.map_apply, tsum_eq_single (σ.symm j)]
-  · simp only [σ.apply_symm_apply, ↓reduceIte]
-    change ENNReal.ofReal (channelWeight lam i (σ.symm j)) =
-      ENNReal.ofReal (channelWeight lam (σ i) j)
-    congr 1
-    unfold channelWeight
-    simp only [Equiv.eq_symm_apply]
-  · intro a ha
-    have hne : j ≠ σ a := by
-      intro h
-      exact ha (σ.injective (h.symm.trans (σ.apply_symm_apply j).symm))
-    simp only [hne, ↓reduceIte]
-
-def relabelRegular (b : ℕ) (σ : Equiv.Perm Spin) : (ℓ : ℕ) → RegularObservation b ℓ → RegularObservation b ℓ
-  | 0, i => σ i
-  | ℓ+1, v => fun j ↦ relabelRegular b σ ℓ (v j)
-
-lemma regularLaw_permute (b : ℕ) (lam : ℝ) (hlam : Admissible lam) (σ : Equiv.Perm Spin)
-    (ℓ : ℕ) (i : Spin) :
-    (regularLaw b lam hlam ℓ i).map (relabelRegular b σ ℓ) = regularLaw b lam hlam ℓ (σ i) := by
-  induction ℓ generalizing i with
-  | zero => exact PMF.pure_bind _ _
-  | succ ℓ ih =>
-    change (productPMF (fun _ : Fin b ↦ (channel lam hlam i).bind (regularLaw b lam hlam ℓ))).map
-      (fun v j ↦ relabelRegular b σ ℓ (v j)) = _
-    rw [productPMF_map]
-    simp_rw [PMF.map_bind, ih]
-    have he : (channel lam hlam i).bind (fun j ↦ regularLaw b lam hlam ℓ (σ j)) =
-        (channel lam hlam (σ i)).bind (regularLaw b lam hlam ℓ) := by
-      rw [← channel_permute hlam σ i, PMF.bind_map]
-      rfl
-    simp_rw [he]
-    rfl
-
-def channelNoise (b : ℕ) (c : ℝ) (hc : Admissible c) :
-    (ℓ : ℕ) → RegularObservation b ℓ → PMF (RegularObservation b ℓ)
-  | 0, i => PMF.pure i
-  | ℓ+1, v => productPMF (fun j ↦
-      (channel c hc 0).bind (fun s ↦ (channelNoise b c hc ℓ (v j)).map
-        (relabelRegular b (Equiv.addRight s) ℓ)))
-
-lemma regularLaw_noise (b : ℕ) (lam c : ℝ) (hlam : Admissible lam) (hc : Admissible c)
-    (hprod : Admissible (lam*c)) (ℓ : ℕ) (i : Spin) :
-    (regularLaw b lam hlam ℓ i).bind (channelNoise b c hc ℓ) =
-      regularLaw b (lam*c) hprod ℓ i := by
-  induction ℓ generalizing i with
-  | zero => exact PMF.pure_bind _ _
-  | succ ℓ ih =>
-    change (productPMF (fun _ : Fin b ↦ (channel lam hlam i).bind (regularLaw b lam hlam ℓ))).bind
-      (fun v ↦ productPMF (fun j ↦ (channel c hc 0).bind (fun s ↦
-        (channelNoise b c hc ℓ (v j)).map (relabelRegular b (Equiv.addRight s) ℓ)))) =
-      productPMF (fun _ : Fin b ↦ (channel (lam*c) hprod i).bind (regularLaw b (lam*c) hprod ℓ))
-    rw [productPMF_bind (fun _ : Fin b ↦ (channel lam hlam i).bind (regularLaw b lam hlam ℓ))
-      (fun _ a ↦ (channel c hc 0).bind (fun s ↦
-        (channelNoise b c hc ℓ a).map (relabelRegular b (Equiv.addRight s) ℓ)))]
-    congr 1
-    funext j
-    rw [PMF.bind_bind]
-    have he (k : Spin) : (regularLaw b lam hlam ℓ k).bind (fun a ↦ (channel c hc 0).bind
-        (fun s ↦ (channelNoise b c hc ℓ a).map (relabelRegular b (Equiv.addRight s) ℓ))) =
-        (channel c hc k).bind (regularLaw b (lam*c) hprod ℓ) := by
-      rw [PMF.bind_comm]
-      simp_rw [← PMF.map_bind, ih, regularLaw_permute]
-      change (channel c hc 0).bind (fun s ↦ regularLaw b (lam*c) hprod ℓ (k+s)) = _
-      have hm := channel_permute hc (Equiv.addLeft k) 0
-      simp only [Equiv.coe_addLeft, add_zero] at hm
-      rw [← hm, PMF.bind_map]
-      rfl
-    simp_rw [he, ← PMF.bind_bind, channel_compose hlam hc hprod]
-
-end ThreeState.TreeClauses.Tree
-
-end 
-
-noncomputable section
-open Set MeasureTheory Filter
-open scoped Topology
-namespace ThreeState.TreeClauses.Tree
-open ThreeState.TreeClauses.Experiment ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
-
-lemma regular_advantage_sq_moment (b : ℕ) (lam : ℝ) (hlam : Admissible lam) (ℓ : ℕ) :
-    (advantage lam hlam (PMF.pure b) ℓ)^2 ≤
-      (∫ m, xMoment m ∂(discreteProbability (regularLaw b lam hlam ℓ)).toMeasure)/2 := by
-  have h := discrete_degradation_mu (regularLaw b lam hlam ℓ) (fun v ↦ PMF.pure (forgetRegular b ℓ v))
-  change (∫ m, xMoment m ∂(discreteProbability (fun i ↦
-    (regularLaw b lam hlam ℓ i).map (forgetRegular b ℓ))).toMeasure) ≤ _ at h
-  simp only [regularLaw_forget] at h
-  have ha := advantage_sq_bound lam hlam (PMF.pure b) ℓ
-  linarith
-
-lemma regular_channel_mu_le (b : ℕ) (lam c : ℝ) (hlam : Admissible lam)
-    (hc : Admissible c) (hprod : Admissible (lam*c)) (ℓ : ℕ) :
-    (∫ m, xMoment m ∂(discreteProbability (regularLaw b (lam*c) hprod ℓ)).toMeasure) ≤
-      ∫ m, xMoment m ∂(discreteProbability (regularLaw b lam hlam ℓ)).toMeasure := by
-  have h := discrete_degradation_mu (regularLaw b lam hlam ℓ) (channelNoise b c hc ℓ)
-  simpa only [regularLaw_noise b lam c hlam hc hprod] using h
-
-theorem regular_positive_nonreconstruction (b : ℕ) (hb : 2 ≤ b)
-    (lam : ℝ) (hlam : Admissible lam) (hl₀ : 0 ≤ lam) (hKS : (b:ℝ)*lam^2 ≤ 1) :
-    Tendsto (advantage lam hlam (PMF.pure b)) atTop (𝓝 0) := by
-  have hb' : (2:ℝ) ≤ b := by exact_mod_cast hb
-  have hbpos : (0:ℝ) < b := by linarith
-  let t : ℝ := 1/Real.sqrt b
-  have ht : 0 < t := by dsimp [t]; exact div_pos (by norm_num) (Real.sqrt_pos.mpr hbpos)
-  have htsq : (b:ℝ)*t^2 = 1 := by
-    dsimp [t]
-    rw [div_pow, one_pow, Real.sq_sqrt hbpos.le, mul_one_div_cancel (ne_of_gt hbpos)]
-  have htone : t < 1 := by
-    have h := mul_le_mul_of_nonneg_right hb' (sq_nonneg t)
-    nlinarith
-  have htadm : Admissible t := ⟨by linarith, htone.le⟩
-  have hlt : lam ≤ t := by
-    have hs : lam^2 ≤ t^2 := (mul_le_mul_iff_right₀ hbpos).mp (by nlinarith [hKS, htsq])
-    nlinarith
-  let c := lam/t
-  have hcpos : 0 ≤ c := div_nonneg hl₀ ht.le
-  have hcone : c ≤ 1 := (div_le_one ht).2 hlt
-  have hcadm : Admissible c := ⟨by linarith, hcone⟩
-  have he : t*c = lam := by dsimp [c]; field_simp
-  have hprod : Admissible (t*c) := he.symm ▸ hlam
-  have hm := regular_positive_critical_mu b hb ht htone htadm htsq
-  have hu : Tendsto (fun ℓ ↦ Real.sqrt (regularPosteriorLaw b ht.le htone ℓ).mu) atTop (𝓝 0) := by
-    simpa only [Real.sqrt_zero, Function.comp_def] using Real.continuous_sqrt.continuousAt.tendsto.comp hm
-  apply squeeze_zero (fun ℓ ↦ advantage_nonneg lam hlam (PMF.pure b) ℓ) _ hu
-  intro ℓ
-  apply (sq_le_sq₀ (advantage_nonneg lam hlam (PMF.pure b) ℓ) (Real.sqrt_nonneg _)).mp
-  rw [Real.sq_sqrt (regularPosteriorLaw b ht.le htone ℓ).mu_nonneg]
-  have h := regular_channel_mu_le b t c htadm hcadm hprod ℓ
-  have hw := regular_advantage_sq_moment b lam hlam ℓ
-  have heq : regularLaw b (t*c) hprod ℓ = regularLaw b lam hlam ℓ := by
-    congr 1
-  rw [heq] at h
-  rw [← regularPosteriorLaw_probability b ht.le htone htadm ℓ] at h
-  change _ ≤ (regularPosteriorLaw b ht.le htone ℓ).mu at h
-  linarith [(regularPosteriorLaw b ht.le htone ℓ).mu_nonneg]
-
-end ThreeState.TreeClauses.Tree
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeStateClauses.CompactLimits
+import OAI.Probability.ThreeStateClauses.PositiveFixed
+
+namespace OAI
+
+/-! Regular Critical. -/
+
+open scoped BigOperators ENNReal NNReal Topology
+open Filter
+noncomputable section
+open Set MeasureTheory Filter
+open scoped Topology
+namespace ThreeState.TreeClauses.Tree
+open ThreeState.TreeClauses.Experiment ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
+
+lemma regular_positive_critical_mu (b : ℕ) (hb : 2 ≤ b) {lam : ℝ}
+    (hl₀ : 0 < lam) (hl₁ : lam < 1) (hlam : Admissible lam) (hcrit : (b:ℝ)*lam^2 = 1) :
+    Tendsto (fun ℓ ↦ (regularPosteriorLaw b hl₀.le hl₁ ℓ).mu) atTop (𝓝 0) := by
+  let Q := regularPosteriorLaw b hl₀.le hl₁
+  let C := fun ℓ ↦ degradationCoupling (regularLaw b lam hlam ℓ) (regularExpansion b lam hlam ℓ)
+  have hf (ℓ : ℕ) : (C ℓ).map Prod.fst = (Q ℓ).probability := by
+    rw [degradationCoupling_fst, regularPosteriorLaw_probability b hl₀.le hl₁ hlam ℓ]
+  have hs (ℓ : ℕ) : (C ℓ).map Prod.snd = (Q (ℓ+1)).probability := by
+    rw [degradationCoupling_snd]
+    simp only [regularLaw_degrades]
+    exact (regularPosteriorLaw_probability b hl₀.le hl₁ hlam (ℓ+1)).symm
+  have hl (ℓ : ℕ) : (∫ z, quadraticGap z ∂(C ℓ).toMeasure) = (Q ℓ).mu-(Q (ℓ+1)).mu := by
+    rw [degradationCoupling_loss]
+    simp only [regularLaw_degrades]
+    simp only [Q, Law.mu, regularPosteriorLaw_probability b hl₀.le hl₁ hlam]
+  obtain ⟨R,hR,hlim⟩ := degrading_finite_fixed_point Q C hf hs hl hl₀.le hl₁ b (fun _ ↦ rfl)
+  rw [regular_fixed_mu_zero R hl₀ hl₁ b hb hcrit hR] at hlim
+  exact hlim
+
+lemma regular_advantage_sq_bound (b : ℕ) {lam : ℝ} (hl₀ : 0 ≤ lam) (hl₁ : lam < 1)
+    (hlam : Admissible lam) (ℓ : ℕ) :
+    (advantage lam hlam (PMF.pure b) ℓ)^2 ≤ (regularPosteriorLaw b hl₀ hl₁ ℓ).mu/2 := by
+  have h := discrete_degradation_mu (regularLaw b lam hlam ℓ) (fun v ↦ PMF.pure (forgetRegular b ℓ v))
+  change (∫ m, xMoment m ∂(discreteProbability (fun i ↦ (regularLaw b lam hlam ℓ i).map (forgetRegular b ℓ))).toMeasure) ≤ _ at h
+  simp only [regularLaw_forget] at h
+  have ha := advantage_sq_bound lam hlam (PMF.pure b) ℓ
+  unfold Law.mu
+  rw [regularPosteriorLaw_probability b hl₀ hl₁ hlam ℓ]
+  linarith
+
+theorem regular_positive_critical_nonreconstruction (b : ℕ) (hb : 2 ≤ b)
+    (lam : ℝ) (hlam : Admissible lam) (hl₀ : 0 < lam) (hcrit : (b:ℝ)*lam^2 = 1) :
+    Tendsto (advantage lam hlam (PMF.pure b)) atTop (𝓝 0) := by
+  have hb' : (2:ℝ) ≤ b := by exact_mod_cast hb
+  have hl₁ : lam < 1 := by
+    have h := mul_le_mul_of_nonneg_right hb' (sq_nonneg lam)
+    nlinarith
+  have hm := regular_positive_critical_mu b hb hl₀ hl₁ hlam hcrit
+  have hu : Tendsto (fun ℓ ↦ Real.sqrt (regularPosteriorLaw b hl₀.le hl₁ ℓ).mu) atTop (𝓝 0) := by
+    simpa only [Real.sqrt_zero, Function.comp_def] using Real.continuous_sqrt.continuousAt.tendsto.comp hm
+  apply squeeze_zero (fun ℓ ↦ advantage_nonneg lam hlam (PMF.pure b) ℓ) _ hu
+  intro ℓ
+  apply (sq_le_sq₀ (advantage_nonneg lam hlam (PMF.pure b) ℓ) (Real.sqrt_nonneg _)).mp
+  rw [Real.sq_sqrt (regularPosteriorLaw b hl₀.le hl₁ ℓ).mu_nonneg]
+  have h := regular_advantage_sq_bound b hl₀.le hl₁ hlam ℓ
+  linarith [(regularPosteriorLaw b hl₀.le hl₁ ℓ).mu_nonneg]
+
+end ThreeState.TreeClauses.Tree
+
+end 
+
+noncomputable section
+open Set MeasureTheory
+open scoped BigOperators ENNReal
+namespace ThreeState.TreeClauses.Tree
+open ThreeState.TreeClauses.Experiment
+
+lemma channelWeight_alt (lam : ℝ) (i j : Spin) :
+    channelWeight lam i j = lam*(if i = j then 1 else 0)+(1-lam)/3 := by
+  unfold channelWeight
+  split_ifs <;> ring
+
+lemma channelWeight_compose (lam c : ℝ) (i j : Spin) :
+    (∑ k : Spin, channelWeight lam i k*channelWeight c k j) = channelWeight (lam*c) i j := by
+  classical
+  simp only [channelWeight_alt, add_mul, mul_add, Finset.sum_add_distrib,
+    ← Finset.mul_sum, mul_assoc]
+  by_cases h : i = j
+  · subst j
+    simp
+    ring
+  · simp [h]
+    ring
+
+lemma channel_compose {lam c : ℝ} (hlam : Admissible lam) (hc : Admissible c)
+    (hprod : Admissible (lam*c)) (i : Spin) :
+    (channel lam hlam i).bind (channel c hc) = channel (lam*c) hprod i := by
+  apply PMF.ext
+  intro j
+  apply (ENNReal.toReal_eq_toReal_iff' ((channel lam hlam i).bind (channel c hc) |>.apply_ne_top j)
+    ((channel (lam*c) hprod i).apply_ne_top j)).mp
+  rw [PMF.bind_apply, tsum_fintype,
+    ENNReal.toReal_sum (fun k _ ↦ ENNReal.mul_ne_top ((channel lam hlam i).apply_ne_top k)
+      ((channel c hc k).apply_ne_top j))]
+  simp only [ENNReal.toReal_mul, channel_real]
+  exact channelWeight_compose lam c i j
+
+lemma channel_permute {lam : ℝ} (hlam : Admissible lam) (σ : Equiv.Perm Spin) (i : Spin) :
+    (channel lam hlam i).map σ = channel lam hlam (σ i) := by
+  classical
+  apply PMF.ext
+  intro j
+  rw [PMF.map_apply, tsum_eq_single (σ.symm j)]
+  · simp only [σ.apply_symm_apply, ↓reduceIte]
+    change ENNReal.ofReal (channelWeight lam i (σ.symm j)) =
+      ENNReal.ofReal (channelWeight lam (σ i) j)
+    congr 1
+    unfold channelWeight
+    simp only [Equiv.eq_symm_apply]
+  · intro a ha
+    have hne : j ≠ σ a := by
+      intro h
+      exact ha (σ.injective (h.symm.trans (σ.apply_symm_apply j).symm))
+    simp only [hne, ↓reduceIte]
+
+def relabelRegular (b : ℕ) (σ : Equiv.Perm Spin) : (ℓ : ℕ) → RegularObservation b ℓ → RegularObservation b ℓ
+  | 0, i => σ i
+  | ℓ+1, v => fun j ↦ relabelRegular b σ ℓ (v j)
+
+lemma regularLaw_permute (b : ℕ) (lam : ℝ) (hlam : Admissible lam) (σ : Equiv.Perm Spin)
+    (ℓ : ℕ) (i : Spin) :
+    (regularLaw b lam hlam ℓ i).map (relabelRegular b σ ℓ) = regularLaw b lam hlam ℓ (σ i) := by
+  induction ℓ generalizing i with
+  | zero => exact PMF.pure_bind _ _
+  | succ ℓ ih =>
+    change (productPMF (fun _ : Fin b ↦ (channel lam hlam i).bind (regularLaw b lam hlam ℓ))).map
+      (fun v j ↦ relabelRegular b σ ℓ (v j)) = _
+    rw [productPMF_map]
+    simp_rw [PMF.map_bind, ih]
+    have he : (channel lam hlam i).bind (fun j ↦ regularLaw b lam hlam ℓ (σ j)) =
+        (channel lam hlam (σ i)).bind (regularLaw b lam hlam ℓ) := by
+      rw [← channel_permute hlam σ i, PMF.bind_map]
+      rfl
+    simp_rw [he]
+    rfl
+
+def channelNoise (b : ℕ) (c : ℝ) (hc : Admissible c) :
+    (ℓ : ℕ) → RegularObservation b ℓ → PMF (RegularObservation b ℓ)
+  | 0, i => PMF.pure i
+  | ℓ+1, v => productPMF (fun j ↦
+      (channel c hc 0).bind (fun s ↦ (channelNoise b c hc ℓ (v j)).map
+        (relabelRegular b (Equiv.addRight s) ℓ)))
+
+lemma regularLaw_noise (b : ℕ) (lam c : ℝ) (hlam : Admissible lam) (hc : Admissible c)
+    (hprod : Admissible (lam*c)) (ℓ : ℕ) (i : Spin) :
+    (regularLaw b lam hlam ℓ i).bind (channelNoise b c hc ℓ) =
+      regularLaw b (lam*c) hprod ℓ i := by
+  induction ℓ generalizing i with
+  | zero => exact PMF.pure_bind _ _
+  | succ ℓ ih =>
+    change (productPMF (fun _ : Fin b ↦ (channel lam hlam i).bind (regularLaw b lam hlam ℓ))).bind
+      (fun v ↦ productPMF (fun j ↦ (channel c hc 0).bind (fun s ↦
+        (channelNoise b c hc ℓ (v j)).map (relabelRegular b (Equiv.addRight s) ℓ)))) =
+      productPMF (fun _ : Fin b ↦ (channel (lam*c) hprod i).bind (regularLaw b (lam*c) hprod ℓ))
+    rw [productPMF_bind (fun _ : Fin b ↦ (channel lam hlam i).bind (regularLaw b lam hlam ℓ))
+      (fun _ a ↦ (channel c hc 0).bind (fun s ↦
+        (channelNoise b c hc ℓ a).map (relabelRegular b (Equiv.addRight s) ℓ)))]
+    congr 1
+    funext j
+    rw [PMF.bind_bind]
+    have he (k : Spin) : (regularLaw b lam hlam ℓ k).bind (fun a ↦ (channel c hc 0).bind
+        (fun s ↦ (channelNoise b c hc ℓ a).map (relabelRegular b (Equiv.addRight s) ℓ))) =
+        (channel c hc k).bind (regularLaw b (lam*c) hprod ℓ) := by
+      rw [PMF.bind_comm]
+      simp_rw [← PMF.map_bind, ih, regularLaw_permute]
+      change (channel c hc 0).bind (fun s ↦ regularLaw b (lam*c) hprod ℓ (k+s)) = _
+      have hm := channel_permute hc (Equiv.addLeft k) 0
+      simp only [Equiv.coe_addLeft, add_zero] at hm
+      rw [← hm, PMF.bind_map]
+      rfl
+    simp_rw [he, ← PMF.bind_bind, channel_compose hlam hc hprod]
+
+end ThreeState.TreeClauses.Tree
+
+end 
+
+noncomputable section
+open Set MeasureTheory Filter
+open scoped Topology
+namespace ThreeState.TreeClauses.Tree
+open ThreeState.TreeClauses.Experiment ThreeState.TreeClauses.Radial ThreeState.TreeClauses.Positive
+
+lemma regular_advantage_sq_moment (b : ℕ) (lam : ℝ) (hlam : Admissible lam) (ℓ : ℕ) :
+    (advantage lam hlam (PMF.pure b) ℓ)^2 ≤
+      (∫ m, xMoment m ∂(discreteProbability (regularLaw b lam hlam ℓ)).toMeasure)/2 := by
+  have h := discrete_degradation_mu (regularLaw b lam hlam ℓ) (fun v ↦ PMF.pure (forgetRegular b ℓ v))
+  change (∫ m, xMoment m ∂(discreteProbability (fun i ↦
+    (regularLaw b lam hlam ℓ i).map (forgetRegular b ℓ))).toMeasure) ≤ _ at h
+  simp only [regularLaw_forget] at h
+  have ha := advantage_sq_bound lam hlam (PMF.pure b) ℓ
+  linarith
+
+lemma regular_channel_mu_le (b : ℕ) (lam c : ℝ) (hlam : Admissible lam)
+    (hc : Admissible c) (hprod : Admissible (lam*c)) (ℓ : ℕ) :
+    (∫ m, xMoment m ∂(discreteProbability (regularLaw b (lam*c) hprod ℓ)).toMeasure) ≤
+      ∫ m, xMoment m ∂(discreteProbability (regularLaw b lam hlam ℓ)).toMeasure := by
+  have h := discrete_degradation_mu (regularLaw b lam hlam ℓ) (channelNoise b c hc ℓ)
+  simpa only [regularLaw_noise b lam c hlam hc hprod] using h
+
+theorem regular_positive_nonreconstruction (b : ℕ) (hb : 2 ≤ b)
+    (lam : ℝ) (hlam : Admissible lam) (hl₀ : 0 ≤ lam) (hKS : (b:ℝ)*lam^2 ≤ 1) :
+    Tendsto (advantage lam hlam (PMF.pure b)) atTop (𝓝 0) := by
+  have hb' : (2:ℝ) ≤ b := by exact_mod_cast hb
+  have hbpos : (0:ℝ) < b := by linarith
+  let t : ℝ := 1/Real.sqrt b
+  have ht : 0 < t := by dsimp [t]; exact div_pos (by norm_num) (Real.sqrt_pos.mpr hbpos)
+  have htsq : (b:ℝ)*t^2 = 1 := by
+    dsimp [t]
+    rw [div_pow, one_pow, Real.sq_sqrt hbpos.le, mul_one_div_cancel (ne_of_gt hbpos)]
+  have htone : t < 1 := by
+    have h := mul_le_mul_of_nonneg_right hb' (sq_nonneg t)
+    nlinarith
+  have htadm : Admissible t := ⟨by linarith, htone.le⟩
+  have hlt : lam ≤ t := by
+    have hs : lam^2 ≤ t^2 := (mul_le_mul_iff_right₀ hbpos).mp (by nlinarith [hKS, htsq])
+    nlinarith
+  let c := lam/t
+  have hcpos : 0 ≤ c := div_nonneg hl₀ ht.le
+  have hcone : c ≤ 1 := (div_le_one ht).2 hlt
+  have hcadm : Admissible c := ⟨by linarith, hcone⟩
+  have he : t*c = lam := by dsimp [c]; field_simp
+  have hprod : Admissible (t*c) := he.symm ▸ hlam
+  have hm := regular_positive_critical_mu b hb ht htone htadm htsq
+  have hu : Tendsto (fun ℓ ↦ Real.sqrt (regularPosteriorLaw b ht.le htone ℓ).mu) atTop (𝓝 0) := by
+    simpa only [Real.sqrt_zero, Function.comp_def] using Real.continuous_sqrt.continuousAt.tendsto.comp hm
+  apply squeeze_zero (fun ℓ ↦ advantage_nonneg lam hlam (PMF.pure b) ℓ) _ hu
+  intro ℓ
+  apply (sq_le_sq₀ (advantage_nonneg lam hlam (PMF.pure b) ℓ) (Real.sqrt_nonneg _)).mp
+  rw [Real.sq_sqrt (regularPosteriorLaw b ht.le htone ℓ).mu_nonneg]
+  have h := regular_channel_mu_le b t c htadm hcadm hprod ℓ
+  have hw := regular_advantage_sq_moment b lam hlam ℓ
+  have heq : regularLaw b (t*c) hprod ℓ = regularLaw b lam hlam ℓ := by
+    congr 1
+  rw [heq] at h
+  rw [← regularPosteriorLaw_probability b ht.le htone htadm ℓ] at h
+  change _ ≤ (regularPosteriorLaw b ht.le htone ℓ).mu at h
+  linarith [(regularPosteriorLaw b ht.le htone ℓ).mu_nonneg]
+
+end ThreeState.TreeClauses.Tree
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeStateClauses/RegularLaw.lean b/lean/OAI/Probability/ThreeStateClauses/RegularLaw.lean
index d280d02a48b2bc6498192277fbdf1889193fcf40..abfa01af07382a31f70668a216d208f179d8486f 100644
--- a/lean/OAI/Probability/ThreeStateClauses/RegularLaw.lean
+++ b/lean/OAI/Probability/ThreeStateClauses/RegularLaw.lean
@@ -1,114 +1,114 @@
-import Mathlib
-import OAI.Probability.ThreeStateClauses.Basic
-import OAI.Probability.ThreeStateClauses.PosteriorLaw
-import OAI.Probability.ThreeStateClauses.Products
-
-namespace OAI
-
-/-! Regular Law. -/
-
-open scoped BigOperators ENNReal NNReal Topology
-open Filter
-noncomputable section
-open Set MeasureTheory
-open scoped BigOperators ENNReal
-namespace ThreeState.TreeClauses.Tree
-open ThreeState.TreeClauses.Experiment
-
-def RegularObservation (b : ℕ) : ℕ → Type
-  | 0 => Spin
-  | n+1 => Fin b → RegularObservation b n
-
-instance regularObservationFintype (b ℓ : ℕ) : Fintype (RegularObservation b ℓ) := by
-  induction ℓ with
-  | zero => exact inferInstanceAs (Fintype Spin)
-  | succ ℓ ih => letI := ih; exact inferInstanceAs (Fintype (Fin b → RegularObservation b ℓ))
-
-instance regularObservationMeasurable (b : ℕ) : (ℓ : ℕ) → MeasurableSpace (RegularObservation b ℓ)
-  | 0 => inferInstanceAs (MeasurableSpace Spin)
-  | ℓ+1 => letI := regularObservationMeasurable b ℓ
-           inferInstanceAs (MeasurableSpace (Fin b → RegularObservation b ℓ))
-instance regularObservationSingleton (b : ℕ) : (ℓ : ℕ) → MeasurableSingletonClass (RegularObservation b ℓ)
-  | 0 => inferInstanceAs (MeasurableSingletonClass Spin)
-  | ℓ+1 => letI := regularObservationSingleton b ℓ
-           inferInstanceAs (MeasurableSingletonClass (Fin b → RegularObservation b ℓ))
-
-instance observationCountable (ℓ : ℕ) : Countable (Observation ℓ) := by
-  induction ℓ with
-  | zero => exact inferInstanceAs (Countable Spin)
-  | succ ℓ ih => let := ih; exact inferInstanceAs (Countable (Multiset (Observation ℓ)))
-
-instance observationMeasurable (ℓ : ℕ) : MeasurableSpace (Observation ℓ) := ⊤
-instance observationSingleton (ℓ : ℕ) : MeasurableSingletonClass (Observation ℓ) :=
-  ⟨fun _ ↦ MeasurableSpace.measurableSet_top⟩
-
-def regularLaw (b : ℕ) (lam : ℝ) (hlam : Admissible lam) :
-    (ℓ : ℕ) → Spin → PMF (RegularObservation b ℓ)
-  | 0, i => PMF.pure i
-  | ℓ+1, i => productPMF (fun _ : Fin b ↦ (channel lam hlam i).bind (regularLaw b lam hlam ℓ))
-
-def forgetRegular (b : ℕ) : (ℓ : ℕ) → RegularObservation b ℓ → Observation ℓ
-  | 0, i => i
-  | ℓ+1, v => (List.ofFn (fun j ↦ forgetRegular b ℓ (v j)) : Multiset (Observation ℓ))
-
-lemma productPMF_list {α : Type} (p : PMF α) (n : ℕ) :
-    (productPMF (fun _ : Fin n ↦ p)).map List.ofFn = iidList p n := by
-  induction n with
-  | zero =>
-    have h : productPMF (fun _ : Fin 0 ↦ p) = PMF.pure (fun i : Fin 0 ↦ i.elim0) := by
-      apply PMF.ext
-      intro v
-      simp [PMF.pure_apply, Subsingleton.elim v (fun i : Fin 0 ↦ i.elim0)]
-    rw [h, PMF.pure_map]
-    rfl
-  | succ n ih =>
-    rw [productPMF_cons, PMF.map_bind]
-    change (p.bind (fun a ↦ _)) = p.bind (fun a ↦ _)
-    congr 1
-    funext a
-    rw [PMF.map_comp]
-    have he : (List.ofFn ∘ Fin.cons (n := n) (α := fun _ ↦ α) a : (Fin n → α) → List α) = List.cons a ∘ List.ofFn := by
-      funext v; exact List.ofFn_cons a v
-    rw [he, ← PMF.map_comp, ih]
-
-lemma regularLaw_forget (b : ℕ) (lam : ℝ) (hlam : Admissible lam) (ℓ : ℕ) (i : Spin) :
-    (regularLaw b lam hlam ℓ i).map (forgetRegular b ℓ) =
-      observationLaw lam hlam (PMF.pure b) ℓ i := by
-  induction ℓ generalizing i with
-  | zero => exact PMF.map_id _
-  | succ ℓ ih =>
-    simp only [observationLaw, PMF.pure_bind]
-    change (productPMF (fun _ : Fin b ↦ (channel lam hlam i).bind (regularLaw b lam hlam ℓ))).map
-      ((fun xs : List (Observation ℓ) ↦ (xs : Multiset (Observation ℓ))) ∘
-        List.ofFn ∘ (fun v j ↦ forgetRegular b ℓ (v j))) = _
-    rw [← PMF.map_comp, ← PMF.map_comp, productPMF_map]
-    simp_rw [PMF.map_bind, ih]
-    rw [productPMF_list]
-    exact (PMF.map_id _).symm
-
-def regularExpansion (b : ℕ) (lam : ℝ) (hlam : Admissible lam) :
-    (ℓ : ℕ) → RegularObservation b ℓ → PMF (RegularObservation b (ℓ+1))
-  | 0, i => productPMF (fun _ : Fin b ↦ channel lam hlam i)
-  | ℓ+1, v => productPMF (fun j : Fin b ↦ regularExpansion b lam hlam ℓ (v j))
-
-lemma regularLaw_degrades (b : ℕ) (lam : ℝ) (hlam : Admissible lam) (ℓ : ℕ) (i : Spin) :
-    (regularLaw b lam hlam ℓ i).bind (regularExpansion b lam hlam ℓ) =
-      regularLaw b lam hlam (ℓ+1) i := by
-  induction ℓ generalizing i with
-  | zero =>
-    change (PMF.pure i).bind (fun j ↦ productPMF (fun _ : Fin b ↦ channel lam hlam j)) =
-      productPMF (fun _ : Fin b ↦ (channel lam hlam i).bind PMF.pure)
-    rw [PMF.pure_bind, PMF.bind_pure]
-  | succ ℓ ih =>
-    change (productPMF (fun _ : Fin b ↦ (channel lam hlam i).bind (regularLaw b lam hlam ℓ))).bind
-      (fun v ↦ productPMF (fun j ↦ regularExpansion b lam hlam ℓ (v j))) = _
-    rw [productPMF_bind]
-    simp only [PMF.bind_bind]
-    simp_rw [ih]
-    rfl
-
-end ThreeState.TreeClauses.Tree
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeStateClauses.Basic
+import OAI.Probability.ThreeStateClauses.PosteriorLaw
+import OAI.Probability.ThreeStateClauses.Products
+
+namespace OAI
+
+/-! Regular Law. -/
+
+open scoped BigOperators ENNReal NNReal Topology
+open Filter
+noncomputable section
+open Set MeasureTheory
+open scoped BigOperators ENNReal
+namespace ThreeState.TreeClauses.Tree
+open ThreeState.TreeClauses.Experiment
+
+def RegularObservation (b : ℕ) : ℕ → Type
+  | 0 => Spin
+  | n+1 => Fin b → RegularObservation b n
+
+instance regularObservationFintype (b ℓ : ℕ) : Fintype (RegularObservation b ℓ) := by
+  induction ℓ with
+  | zero => exact inferInstanceAs (Fintype Spin)
+  | succ ℓ ih => letI := ih; exact inferInstanceAs (Fintype (Fin b → RegularObservation b ℓ))
+
+instance regularObservationMeasurable (b : ℕ) : (ℓ : ℕ) → MeasurableSpace (RegularObservation b ℓ)
+  | 0 => inferInstanceAs (MeasurableSpace Spin)
+  | ℓ+1 => letI := regularObservationMeasurable b ℓ
+           inferInstanceAs (MeasurableSpace (Fin b → RegularObservation b ℓ))
+instance regularObservationSingleton (b : ℕ) : (ℓ : ℕ) → MeasurableSingletonClass (RegularObservation b ℓ)
+  | 0 => inferInstanceAs (MeasurableSingletonClass Spin)
+  | ℓ+1 => letI := regularObservationSingleton b ℓ
+           inferInstanceAs (MeasurableSingletonClass (Fin b → RegularObservation b ℓ))
+
+instance observationCountable (ℓ : ℕ) : Countable (Observation ℓ) := by
+  induction ℓ with
+  | zero => exact inferInstanceAs (Countable Spin)
+  | succ ℓ ih => let := ih; exact inferInstanceAs (Countable (Multiset (Observation ℓ)))
+
+instance observationMeasurable (ℓ : ℕ) : MeasurableSpace (Observation ℓ) := ⊤
+instance observationSingleton (ℓ : ℕ) : MeasurableSingletonClass (Observation ℓ) :=
+  ⟨fun _ ↦ MeasurableSpace.measurableSet_top⟩
+
+def regularLaw (b : ℕ) (lam : ℝ) (hlam : Admissible lam) :
+    (ℓ : ℕ) → Spin → PMF (RegularObservation b ℓ)
+  | 0, i => PMF.pure i
+  | ℓ+1, i => productPMF (fun _ : Fin b ↦ (channel lam hlam i).bind (regularLaw b lam hlam ℓ))
+
+def forgetRegular (b : ℕ) : (ℓ : ℕ) → RegularObservation b ℓ → Observation ℓ
+  | 0, i => i
+  | ℓ+1, v => (List.ofFn (fun j ↦ forgetRegular b ℓ (v j)) : Multiset (Observation ℓ))
+
+lemma productPMF_list {α : Type} (p : PMF α) (n : ℕ) :
+    (productPMF (fun _ : Fin n ↦ p)).map List.ofFn = iidList p n := by
+  induction n with
+  | zero =>
+    have h : productPMF (fun _ : Fin 0 ↦ p) = PMF.pure (fun i : Fin 0 ↦ i.elim0) := by
+      apply PMF.ext
+      intro v
+      simp [PMF.pure_apply, Subsingleton.elim v (fun i : Fin 0 ↦ i.elim0)]
+    rw [h, PMF.pure_map]
+    rfl
+  | succ n ih =>
+    rw [productPMF_cons, PMF.map_bind]
+    change (p.bind (fun a ↦ _)) = p.bind (fun a ↦ _)
+    congr 1
+    funext a
+    rw [PMF.map_comp]
+    have he : (List.ofFn ∘ Fin.cons (n := n) (α := fun _ ↦ α) a : (Fin n → α) → List α) = List.cons a ∘ List.ofFn := by
+      funext v; exact List.ofFn_cons a v
+    rw [he, ← PMF.map_comp, ih]
+
+lemma regularLaw_forget (b : ℕ) (lam : ℝ) (hlam : Admissible lam) (ℓ : ℕ) (i : Spin) :
+    (regularLaw b lam hlam ℓ i).map (forgetRegular b ℓ) =
+      observationLaw lam hlam (PMF.pure b) ℓ i := by
+  induction ℓ generalizing i with
+  | zero => exact PMF.map_id _
+  | succ ℓ ih =>
+    simp only [observationLaw, PMF.pure_bind]
+    change (productPMF (fun _ : Fin b ↦ (channel lam hlam i).bind (regularLaw b lam hlam ℓ))).map
+      ((fun xs : List (Observation ℓ) ↦ (xs : Multiset (Observation ℓ))) ∘
+        List.ofFn ∘ (fun v j ↦ forgetRegular b ℓ (v j))) = _
+    rw [← PMF.map_comp, ← PMF.map_comp, productPMF_map]
+    simp_rw [PMF.map_bind, ih]
+    rw [productPMF_list]
+    exact (PMF.map_id _).symm
+
+def regularExpansion (b : ℕ) (lam : ℝ) (hlam : Admissible lam) :
+    (ℓ : ℕ) → RegularObservation b ℓ → PMF (RegularObservation b (ℓ+1))
+  | 0, i => productPMF (fun _ : Fin b ↦ channel lam hlam i)
+  | ℓ+1, v => productPMF (fun j : Fin b ↦ regularExpansion b lam hlam ℓ (v j))
+
+lemma regularLaw_degrades (b : ℕ) (lam : ℝ) (hlam : Admissible lam) (ℓ : ℕ) (i : Spin) :
+    (regularLaw b lam hlam ℓ i).bind (regularExpansion b lam hlam ℓ) =
+      regularLaw b lam hlam (ℓ+1) i := by
+  induction ℓ generalizing i with
+  | zero =>
+    change (PMF.pure i).bind (fun j ↦ productPMF (fun _ : Fin b ↦ channel lam hlam j)) =
+      productPMF (fun _ : Fin b ↦ (channel lam hlam i).bind PMF.pure)
+    rw [PMF.pure_bind, PMF.bind_pure]
+  | succ ℓ ih =>
+    change (productPMF (fun _ : Fin b ↦ (channel lam hlam i).bind (regularLaw b lam hlam ℓ))).bind
+      (fun v ↦ productPMF (fun j ↦ regularExpansion b lam hlam ℓ (v j))) = _
+    rw [productPMF_bind]
+    simp only [PMF.bind_bind]
+    simp_rw [ih]
+    rfl
+
+end ThreeState.TreeClauses.Tree
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeStateClauses/RegularReconstruction.lean b/lean/OAI/Probability/ThreeStateClauses/RegularReconstruction.lean
index c73846b4cb98c84bec6e75dd79f1fc2430249575..7105569e95c30848214c890091bce4a7f23782cc 100644
--- a/lean/OAI/Probability/ThreeStateClauses/RegularReconstruction.lean
+++ b/lean/OAI/Probability/ThreeStateClauses/RegularReconstruction.lean
@@ -1,114 +1,114 @@
-import Mathlib
-import OAI.Probability.ThreeStateClauses.RegularStatistics
-import OAI.Probability.ThreeStateClauses.InformationLoss
-
-namespace OAI
-
-/-! Regular Reconstruction. -/
-
-open scoped BigOperators ENNReal NNReal Topology
-open Filter
-noncomputable section
-open Set MeasureTheory Filter
-open scoped BigOperators Topology
-namespace ThreeState.TreeClauses.Tree
-open ThreeState.TreeClauses.Experiment
-
-def ksConstant (b : ℕ) (lam : ℝ) : ℝ := 4*((b:ℝ)*lam^2)/((b:ℝ)*lam^2-1)
-
-lemma ksConstant_pos {b : ℕ} {lam : ℝ} (hKS : 1 < (b:ℝ)*lam^2) : 0 < ksConstant b lam := by
-  unfold ksConstant; positivity
-
-lemma ksConstant_ge_four {b : ℕ} {lam : ℝ} (hKS : 1 < (b:ℝ)*lam^2) : 4 ≤ ksConstant b lam := by
-  rw [ksConstant, le_div_iff₀ (by linarith)]
-  linarith
-
-lemma ksConstant_identity {b : ℕ} {lam : ℝ} (hKS : 1 < (b:ℝ)*lam^2) :
-    ksConstant b lam*((b:ℝ)*lam^2-1) = 4*((b:ℝ)*lam^2) := by
-  rw [ksConstant, div_mul_cancel₀ _ (by linarith)]
-
-lemma regularStatistic_second (b : ℕ) (lam : ℝ) (hlam : Admissible lam)
-    (hKS : 1 < (b:ℝ)*lam^2) (ℓ : ℕ) (i : Spin) :
-    (∫ v, regularStatistic b lam ℓ v^2 ∂(regularLaw b lam hlam ℓ i).toMeasure) ≤ ksConstant b lam := by
-  have hb : 0 < b := by
-    by_contra h
-    have hb0 : b = 0 := by omega
-    norm_num [hb0] at hKS
-  have hb' : (0:ℝ) < b := Nat.cast_pos.mpr hb
-  have hl : lam ≠ 0 := by intro h; norm_num [h] at hKS
-  have hden : 0 < ((b:ℝ)*lam)^2 := sq_pos_of_ne_zero (mul_ne_zero (ne_of_gt hb') hl)
-  have hC := ksConstant_pos hKS
-  induction ℓ generalizing i with
-  | zero =>
-    change (∫ v : Spin, spinScore v^2 ∂(PMF.pure i).toMeasure) ≤ _
-    rw [PMF.toMeasure_pure, integral_dirac]
-    exact (spinScore_square i).trans (ksConstant_ge_four hKS)
-  | succ ℓ ih =>
-    have hmean : (∫ v, regularStatistic b lam ℓ v
-        ∂((channel lam hlam i).bind (regularLaw b lam hlam ℓ)).toMeasure) = lam*spinScore i := by
-      rw [integral_pmf_bind_finite]
-      simp_rw [regularStatistic_mean b hb lam hlam hl ℓ]
-      exact channel_spinScore lam hlam i
-    have hsecond : (∫ v, regularStatistic b lam ℓ v^2
-        ∂((channel lam hlam i).bind (regularLaw b lam hlam ℓ)).toMeasure) ≤ ksConstant b lam := by
-      rw [integral_pmf_bind_finite]
-      exact (integral_mono Integrable.of_finite (integrable_const _) (ih)).trans_eq (by simp)
-    change (∫ v : Fin b → RegularObservation b ℓ,
-      ((∑ j, regularStatistic b lam ℓ (v j))/((b:ℝ)*lam))^2
-      ∂(productPMF (fun _ ↦ (channel lam hlam i).bind (regularLaw b lam hlam ℓ))).toMeasure) ≤ _
-    simp_rw [div_pow]
-    rw [integral_div, integral_product_sq_sum, hmean]
-    apply (div_le_iff₀ hden).2
-    have hb1 : (1:ℝ) ≤ b := by exact_mod_cast hb
-    have hn : 0 ≤ (b:ℝ)^2-b := by nlinarith
-    have hs := mul_le_mul_of_nonneg_left (spinScore_square i) (sq_nonneg lam)
-    have hp := mul_le_mul_of_nonneg_left hs hn
-    have hi := congrArg (fun x : ℝ ↦ x*(b:ℝ)) (ksConstant_identity hKS)
-    have hm := mul_le_mul_of_nonneg_left hsecond hb'.le
-    nlinarith [mul_nonneg hb'.le (sq_nonneg lam)]
-
-lemma regular_estimator_advantage (b : ℕ) (lam : ℝ) (hlam : Admissible lam)
-    (hKS : 1 < (b:ℝ)*lam^2) (ℓ : ℕ) :
-    1/(3*ksConstant b lam) ≤ advantage lam hlam (PMF.pure b) ℓ := by
-  have hb : 0 < b := by
-    by_contra h
-    have hb0 : b = 0 := by omega
-    norm_num [hb0] at hKS
-  have hl : lam ≠ 0 := by intro h; norm_num [h] at hKS
-  let p := observationLaw lam hlam (PMF.pure b) ℓ
-  let f := observedRegularStatistic b lam ℓ
-  obtain ⟨B,hB0,hB⟩ := observedRegularStatistic_bound b lam ℓ
-  have hmean (i : Spin) : (∫ o, f o ∂(p i).toMeasure) = spinScore i := by
-    dsimp [p, f]
-    rw [← regularLaw_forget b lam hlam ℓ i, integral_pmf_map]
-    simp_rw [observedRegularStatistic_forget]
-    exact regularStatistic_mean b hb lam hlam hl ℓ i
-  have hsecond (i : Spin) : (∫ o, f o^2 ∂(p i).toMeasure) ≤ ksConstant b lam := by
-    dsimp [p, f]
-    rw [← regularLaw_forget b lam hlam ℓ i, integral_pmf_map]
-    simp_rw [observedRegularStatistic_forget]
-    exact regularStatistic_second b lam hlam hKS ℓ i
-  rw [advantage_eq_integral]
-  apply estimator_tv_lower (p := p) hB (ksConstant_pos hKS)
-  · rw [integral_discreteMarginal p hB, hmean 0, hmean 1, hmean 2]
-    norm_num [spinScore, show (1:Spin) ≠ 0 by decide, show (2:Spin) ≠ 0 by decide]
-  · exact hmean 0
-  · have hB₂ (o : Observation ℓ) : |f o^2| ≤ B^2 := by
-      rw [abs_of_nonneg (sq_nonneg _)]
-      have hh : |f o| ≤ B := hB o
-      nlinarith [hh, sq_abs (f o), abs_nonneg (f o)]
-    rw [integral_discreteMarginal p hB₂]
-    linarith [hsecond 0, hsecond 1, hsecond 2]
-
-theorem regular_reconstructs_above_KS (b : ℕ) (lam : ℝ) (hlam : Admissible lam)
-    (hKS : 1 < (b:ℝ)*lam^2) : Reconstructs lam hlam (PMF.pure b) := by
-  refine ⟨⨅ ℓ, advantage lam hlam (PMF.pure b) ℓ, ?_, advantage_tendsto _ _ _⟩
-  have he : 1/(3*ksConstant b lam) ≤ ⨅ ℓ, advantage lam hlam (PMF.pure b) ℓ := by
-    exact le_ciInf (fun ℓ ↦ regular_estimator_advantage b lam hlam hKS ℓ)
-  exact (div_pos (by norm_num) (mul_pos (by norm_num) (ksConstant_pos hKS))).trans_le he
-
-end ThreeState.TreeClauses.Tree
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeStateClauses.RegularStatistics
+import OAI.Probability.ThreeStateClauses.InformationLoss
+
+namespace OAI
+
+/-! Regular Reconstruction. -/
+
+open scoped BigOperators ENNReal NNReal Topology
+open Filter
+noncomputable section
+open Set MeasureTheory Filter
+open scoped BigOperators Topology
+namespace ThreeState.TreeClauses.Tree
+open ThreeState.TreeClauses.Experiment
+
+def ksConstant (b : ℕ) (lam : ℝ) : ℝ := 4*((b:ℝ)*lam^2)/((b:ℝ)*lam^2-1)
+
+lemma ksConstant_pos {b : ℕ} {lam : ℝ} (hKS : 1 < (b:ℝ)*lam^2) : 0 < ksConstant b lam := by
+  unfold ksConstant; positivity
+
+lemma ksConstant_ge_four {b : ℕ} {lam : ℝ} (hKS : 1 < (b:ℝ)*lam^2) : 4 ≤ ksConstant b lam := by
+  rw [ksConstant, le_div_iff₀ (by linarith)]
+  linarith
+
+lemma ksConstant_identity {b : ℕ} {lam : ℝ} (hKS : 1 < (b:ℝ)*lam^2) :
+    ksConstant b lam*((b:ℝ)*lam^2-1) = 4*((b:ℝ)*lam^2) := by
+  rw [ksConstant, div_mul_cancel₀ _ (by linarith)]
+
+lemma regularStatistic_second (b : ℕ) (lam : ℝ) (hlam : Admissible lam)
+    (hKS : 1 < (b:ℝ)*lam^2) (ℓ : ℕ) (i : Spin) :
+    (∫ v, regularStatistic b lam ℓ v^2 ∂(regularLaw b lam hlam ℓ i).toMeasure) ≤ ksConstant b lam := by
+  have hb : 0 < b := by
+    by_contra h
+    have hb0 : b = 0 := by omega
+    norm_num [hb0] at hKS
+  have hb' : (0:ℝ) < b := Nat.cast_pos.mpr hb
+  have hl : lam ≠ 0 := by intro h; norm_num [h] at hKS
+  have hden : 0 < ((b:ℝ)*lam)^2 := sq_pos_of_ne_zero (mul_ne_zero (ne_of_gt hb') hl)
+  have hC := ksConstant_pos hKS
+  induction ℓ generalizing i with
+  | zero =>
+    change (∫ v : Spin, spinScore v^2 ∂(PMF.pure i).toMeasure) ≤ _
+    rw [PMF.toMeasure_pure, integral_dirac]
+    exact (spinScore_square i).trans (ksConstant_ge_four hKS)
+  | succ ℓ ih =>
+    have hmean : (∫ v, regularStatistic b lam ℓ v
+        ∂((channel lam hlam i).bind (regularLaw b lam hlam ℓ)).toMeasure) = lam*spinScore i := by
+      rw [integral_pmf_bind_finite]
+      simp_rw [regularStatistic_mean b hb lam hlam hl ℓ]
+      exact channel_spinScore lam hlam i
+    have hsecond : (∫ v, regularStatistic b lam ℓ v^2
+        ∂((channel lam hlam i).bind (regularLaw b lam hlam ℓ)).toMeasure) ≤ ksConstant b lam := by
+      rw [integral_pmf_bind_finite]
+      exact (integral_mono Integrable.of_finite (integrable_const _) (ih)).trans_eq (by simp)
+    change (∫ v : Fin b → RegularObservation b ℓ,
+      ((∑ j, regularStatistic b lam ℓ (v j))/((b:ℝ)*lam))^2
+      ∂(productPMF (fun _ ↦ (channel lam hlam i).bind (regularLaw b lam hlam ℓ))).toMeasure) ≤ _
+    simp_rw [div_pow]
+    rw [integral_div, integral_product_sq_sum, hmean]
+    apply (div_le_iff₀ hden).2
+    have hb1 : (1:ℝ) ≤ b := by exact_mod_cast hb
+    have hn : 0 ≤ (b:ℝ)^2-b := by nlinarith
+    have hs := mul_le_mul_of_nonneg_left (spinScore_square i) (sq_nonneg lam)
+    have hp := mul_le_mul_of_nonneg_left hs hn
+    have hi := congrArg (fun x : ℝ ↦ x*(b:ℝ)) (ksConstant_identity hKS)
+    have hm := mul_le_mul_of_nonneg_left hsecond hb'.le
+    nlinarith [mul_nonneg hb'.le (sq_nonneg lam)]
+
+lemma regular_estimator_advantage (b : ℕ) (lam : ℝ) (hlam : Admissible lam)
+    (hKS : 1 < (b:ℝ)*lam^2) (ℓ : ℕ) :
+    1/(3*ksConstant b lam) ≤ advantage lam hlam (PMF.pure b) ℓ := by
+  have hb : 0 < b := by
+    by_contra h
+    have hb0 : b = 0 := by omega
+    norm_num [hb0] at hKS
+  have hl : lam ≠ 0 := by intro h; norm_num [h] at hKS
+  let p := observationLaw lam hlam (PMF.pure b) ℓ
+  let f := observedRegularStatistic b lam ℓ
+  obtain ⟨B,hB0,hB⟩ := observedRegularStatistic_bound b lam ℓ
+  have hmean (i : Spin) : (∫ o, f o ∂(p i).toMeasure) = spinScore i := by
+    dsimp [p, f]
+    rw [← regularLaw_forget b lam hlam ℓ i, integral_pmf_map]
+    simp_rw [observedRegularStatistic_forget]
+    exact regularStatistic_mean b hb lam hlam hl ℓ i
+  have hsecond (i : Spin) : (∫ o, f o^2 ∂(p i).toMeasure) ≤ ksConstant b lam := by
+    dsimp [p, f]
+    rw [← regularLaw_forget b lam hlam ℓ i, integral_pmf_map]
+    simp_rw [observedRegularStatistic_forget]
+    exact regularStatistic_second b lam hlam hKS ℓ i
+  rw [advantage_eq_integral]
+  apply estimator_tv_lower (p := p) hB (ksConstant_pos hKS)
+  · rw [integral_discreteMarginal p hB, hmean 0, hmean 1, hmean 2]
+    norm_num [spinScore, show (1:Spin) ≠ 0 by decide, show (2:Spin) ≠ 0 by decide]
+  · exact hmean 0
+  · have hB₂ (o : Observation ℓ) : |f o^2| ≤ B^2 := by
+      rw [abs_of_nonneg (sq_nonneg _)]
+      have hh : |f o| ≤ B := hB o
+      nlinarith [hh, sq_abs (f o), abs_nonneg (f o)]
+    rw [integral_discreteMarginal p hB₂]
+    linarith [hsecond 0, hsecond 1, hsecond 2]
+
+theorem regular_reconstructs_above_KS (b : ℕ) (lam : ℝ) (hlam : Admissible lam)
+    (hKS : 1 < (b:ℝ)*lam^2) : Reconstructs lam hlam (PMF.pure b) := by
+  refine ⟨⨅ ℓ, advantage lam hlam (PMF.pure b) ℓ, ?_, advantage_tendsto _ _ _⟩
+  have he : 1/(3*ksConstant b lam) ≤ ⨅ ℓ, advantage lam hlam (PMF.pure b) ℓ := by
+    exact le_ciInf (fun ℓ ↦ regular_estimator_advantage b lam hlam hKS ℓ)
+  exact (div_pos (by norm_num) (mul_pos (by norm_num) (ksConstant_pos hKS))).trans_le he
+
+end ThreeState.TreeClauses.Tree
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeStateClauses/RegularStatistics.lean b/lean/OAI/Probability/ThreeStateClauses/RegularStatistics.lean
index 3175a4b149578ef99d802ffa0c57d21cc727e490..0603ffa833ff4ea6c270984c5efd433ae1681f12 100644
--- a/lean/OAI/Probability/ThreeStateClauses/RegularStatistics.lean
+++ b/lean/OAI/Probability/ThreeStateClauses/RegularStatistics.lean
@@ -1,132 +1,132 @@
-import Mathlib
-import OAI.Probability.ThreeStateClauses.RegularCritical
-
-namespace OAI
-
-/-! Regular Statistics. -/
-
-open scoped BigOperators ENNReal NNReal Topology
-open Filter
-noncomputable section
-open Set MeasureTheory
-open scoped BigOperators
-namespace ThreeState.TreeClauses.Experiment
-
-variable {α β : Type*} [Fintype α] [MeasurableSpace α] [MeasurableSingletonClass α]
-  [Fintype β] [MeasurableSpace β] [MeasurableSingletonClass β]
-
-omit [MeasurableSpace α] [MeasurableSingletonClass α] in
-lemma finite_abs_bound (f : α → ℝ) : ∃ C : ℝ, ∀ a, |f a| ≤ C := by
-  obtain ⟨C,hC⟩ := (Set.finite_range (fun a ↦ |f a|)).bddAbove
-  exact ⟨C, fun a ↦ hC ⟨a,rfl⟩⟩
-
-lemma integral_pmf_bind_finite (p : PMF α) (K : α → PMF β) (f : β → ℝ) :
-    (∫ b, f b ∂(p.bind K).toMeasure) = ∫ a, (∫ b, f b ∂(K a).toMeasure) ∂p.toMeasure := by
-  obtain ⟨C,hC⟩ := finite_abs_bound f
-  exact integral_pmf_bind p K hC
-
-lemma integral_product_sum (p : PMF α) (f : α → ℝ) (n : ℕ) :
-    (∫ v : Fin n → α, (∑ j, f (v j)) ∂(productPMF (fun _ ↦ p)).toMeasure) =
-      (n:ℝ)*(∫ a, f a ∂p.toMeasure) := by
-  rw [productPMF_measure]
-  simpa [weightProduct] using integral_weighted_sum (ι := Fin n) (a := fun _ : α ↦ (1:ℝ))
-    (u := f) (integrable_const 1) Integrable.of_finite (by simp)
-
-lemma integral_product_sq_sum (p : PMF α) (f : α → ℝ) (n : ℕ) :
-    (∫ v : Fin n → α, (∑ j, f (v j))^2 ∂(productPMF (fun _ ↦ p)).toMeasure) =
-      (n:ℝ)*(∫ a, f a^2 ∂p.toMeasure)+((n:ℝ)^2-n)*(∫ a, f a ∂p.toMeasure)^2 := by
-  rw [productPMF_measure]
-  simpa [weightProduct] using integral_weighted_sq_sum (ι := Fin n) (a := fun _ : α ↦ (1:ℝ))
-    (u := f) (integrable_const 1) Integrable.of_finite Integrable.of_finite (by simp)
-
-end ThreeState.TreeClauses.Experiment
-
-end 
-
-noncomputable section
-open Set MeasureTheory
-open scoped BigOperators
-namespace ThreeState.TreeClauses.Tree
-open ThreeState.TreeClauses.Experiment
-
-def spinScore (i : Spin) : ℝ := if i=0 then 2 else -1
-
-lemma spinScore_abs (i : Spin) : |spinScore i| ≤ 2 := by
-  unfold spinScore; split_ifs <;> norm_num
-
-lemma spinScore_square (i : Spin) : spinScore i^2 ≤ 4 := by
-  unfold spinScore; split_ifs <;> norm_num
-
-lemma channel_spinScore (lam : ℝ) (hlam : Admissible lam) (i : Spin) :
-    (∫ j, spinScore j ∂(channel lam hlam i).toMeasure) = lam*spinScore i := by
-  rw [PMF.integral_eq_sum]
-  have he (j : Spin) : (channel lam hlam i j).toReal = channelWeight lam i j :=
-    ENNReal.toReal_ofReal (channelWeight_nonneg hlam i j)
-  simp only [smul_eq_mul, he]
-  simp_rw [channelWeight_alt, add_mul, Finset.sum_add_distrib]
-  have hs : ∑ j : Spin, spinScore j = 0 := by
-    norm_num [Fin.sum_univ_three, spinScore, show (1:Spin) ≠ 0 by decide, show (2:Spin) ≠ 0 by decide]
-  simp only [mul_ite, mul_one, mul_zero, ite_mul, zero_mul, Finset.sum_ite_eq,
-    Finset.mem_univ, ite_true, ← Finset.mul_sum, hs, mul_zero, add_zero]
-
-def regularStatistic (b : ℕ) (lam : ℝ) : (ℓ : ℕ) → RegularObservation b ℓ → ℝ
-  | 0, i => spinScore i
-  | ℓ+1, v => (∑ j : Fin b, regularStatistic b lam ℓ (v j))/((b:ℝ)*lam)
-
-def observedRegularStatistic (b : ℕ) (lam : ℝ) : (ℓ : ℕ) → Observation ℓ → ℝ
-  | 0, i => spinScore i
-  | ℓ+1, s => if s.card = b then (s.map (observedRegularStatistic b lam ℓ)).sum/((b:ℝ)*lam) else 0
-
-lemma abs_multiset_sum_bound {α : Type*} (f : α → ℝ) (C : ℝ) (hf : ∀ a, |f a| ≤ C)
-    (s : Multiset α) : |(s.map f).sum| ≤ (s.card:ℝ)*C := by
-  induction s using Multiset.induction_on with
-  | empty => simp
-  | @cons a s ih =>
-    simp only [Multiset.map_cons, Multiset.sum_cons, Multiset.card_cons, Nat.cast_add, Nat.cast_one]
-    exact (abs_add_le _ _).trans (by linarith [hf a])
-
-lemma observedRegularStatistic_bound (b : ℕ) (lam : ℝ) (ℓ : ℕ) :
-    ∃ C : ℝ, 0 ≤ C ∧ ∀ o, |observedRegularStatistic b lam ℓ o| ≤ C := by
-  induction ℓ with
-  | zero => exact ⟨2, by norm_num, spinScore_abs⟩
-  | succ ℓ ih =>
-    obtain ⟨C,hC,hbound⟩ := ih
-    refine ⟨(b:ℝ)*C/|((b:ℝ)*lam)|, by positivity, ?_⟩
-    intro s
-    change |if s.card=b then _ else 0| ≤ _
-    split_ifs with h
-    · rw [abs_div]
-      exact div_le_div_of_nonneg_right (by simpa only [h] using abs_multiset_sum_bound _ C hbound s) (abs_nonneg _)
-    · simp only [abs_zero]; positivity
-
-lemma observedRegularStatistic_forget (b : ℕ) (lam : ℝ) (ℓ : ℕ)
-    (v : RegularObservation b ℓ) :
-    observedRegularStatistic b lam ℓ (forgetRegular b ℓ v) = regularStatistic b lam ℓ v := by
-  induction ℓ with
-  | zero => rfl
-  | succ ℓ ih =>
-    simp only [forgetRegular, observedRegularStatistic, Multiset.coe_card, List.length_ofFn, ite_true, regularStatistic]
-    congr 1
-    simp only [Multiset.map_coe, Multiset.sum_coe, List.map_ofFn, Function.comp_def, ih, List.sum_ofFn]
-
-lemma regularStatistic_mean (b : ℕ) (hb : 0 < b) (lam : ℝ) (hlam : Admissible lam)
-    (hl : lam ≠ 0) (ℓ : ℕ) (i : Spin) :
-    (∫ v, regularStatistic b lam ℓ v ∂(regularLaw b lam hlam ℓ i).toMeasure) = spinScore i := by
-  have hb' : (b:ℝ) ≠ 0 := by exact_mod_cast (Nat.ne_of_gt hb)
-  induction ℓ generalizing i with
-  | zero =>
-    change (∫ v : Spin, spinScore v ∂(PMF.pure i).toMeasure) = spinScore i
-    rw [PMF.toMeasure_pure, integral_dirac]
-  | succ ℓ ih =>
-    change (∫ v, (∑ j : Fin b, regularStatistic b lam ℓ (v j))/((b:ℝ)*lam)
-      ∂(productPMF (fun _ ↦ (channel lam hlam i).bind (regularLaw b lam hlam ℓ))).toMeasure) = _
-    rw [integral_div, integral_product_sum, integral_pmf_bind_finite]
-    simp_rw [ih]
-    rw [channel_spinScore]
-    field_simp
-
-end ThreeState.TreeClauses.Tree
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.ThreeStateClauses.RegularCritical
+
+namespace OAI
+
+/-! Regular Statistics. -/
+
+open scoped BigOperators ENNReal NNReal Topology
+open Filter
+noncomputable section
+open Set MeasureTheory
+open scoped BigOperators
+namespace ThreeState.TreeClauses.Experiment
+
+variable {α β : Type*} [Fintype α] [MeasurableSpace α] [MeasurableSingletonClass α]
+  [Fintype β] [MeasurableSpace β] [MeasurableSingletonClass β]
+
+omit [MeasurableSpace α] [MeasurableSingletonClass α] in
+lemma finite_abs_bound (f : α → ℝ) : ∃ C : ℝ, ∀ a, |f a| ≤ C := by
+  obtain ⟨C,hC⟩ := (Set.finite_range (fun a ↦ |f a|)).bddAbove
+  exact ⟨C, fun a ↦ hC ⟨a,rfl⟩⟩
+
+lemma integral_pmf_bind_finite (p : PMF α) (K : α → PMF β) (f : β → ℝ) :
+    (∫ b, f b ∂(p.bind K).toMeasure) = ∫ a, (∫ b, f b ∂(K a).toMeasure) ∂p.toMeasure := by
+  obtain ⟨C,hC⟩ := finite_abs_bound f
+  exact integral_pmf_bind p K hC
+
+lemma integral_product_sum (p : PMF α) (f : α → ℝ) (n : ℕ) :
+    (∫ v : Fin n → α, (∑ j, f (v j)) ∂(productPMF (fun _ ↦ p)).toMeasure) =
+      (n:ℝ)*(∫ a, f a ∂p.toMeasure) := by
+  rw [productPMF_measure]
+  simpa [weightProduct] using integral_weighted_sum (ι := Fin n) (a := fun _ : α ↦ (1:ℝ))
+    (u := f) (integrable_const 1) Integrable.of_finite (by simp)
+
+lemma integral_product_sq_sum (p : PMF α) (f : α → ℝ) (n : ℕ) :
+    (∫ v : Fin n → α, (∑ j, f (v j))^2 ∂(productPMF (fun _ ↦ p)).toMeasure) =
+      (n:ℝ)*(∫ a, f a^2 ∂p.toMeasure)+((n:ℝ)^2-n)*(∫ a, f a ∂p.toMeasure)^2 := by
+  rw [productPMF_measure]
+  simpa [weightProduct] using integral_weighted_sq_sum (ι := Fin n) (a := fun _ : α ↦ (1:ℝ))
+    (u := f) (integrable_const 1) Integrable.of_finite Integrable.of_finite (by simp)
+
+end ThreeState.TreeClauses.Experiment
+
+end 
+
+noncomputable section
+open Set MeasureTheory
+open scoped BigOperators
+namespace ThreeState.TreeClauses.Tree
+open ThreeState.TreeClauses.Experiment
+
+def spinScore (i : Spin) : ℝ := if i=0 then 2 else -1
+
+lemma spinScore_abs (i : Spin) : |spinScore i| ≤ 2 := by
+  unfold spinScore; split_ifs <;> norm_num
+
+lemma spinScore_square (i : Spin) : spinScore i^2 ≤ 4 := by
+  unfold spinScore; split_ifs <;> norm_num
+
+lemma channel_spinScore (lam : ℝ) (hlam : Admissible lam) (i : Spin) :
+    (∫ j, spinScore j ∂(channel lam hlam i).toMeasure) = lam*spinScore i := by
+  rw [PMF.integral_eq_sum]
+  have he (j : Spin) : (channel lam hlam i j).toReal = channelWeight lam i j :=
+    ENNReal.toReal_ofReal (channelWeight_nonneg hlam i j)
+  simp only [smul_eq_mul, he]
+  simp_rw [channelWeight_alt, add_mul, Finset.sum_add_distrib]
+  have hs : ∑ j : Spin, spinScore j = 0 := by
+    norm_num [Fin.sum_univ_three, spinScore, show (1:Spin) ≠ 0 by decide, show (2:Spin) ≠ 0 by decide]
+  simp only [mul_ite, mul_one, mul_zero, ite_mul, zero_mul, Finset.sum_ite_eq,
+    Finset.mem_univ, ite_true, ← Finset.mul_sum, hs, mul_zero, add_zero]
+
+def regularStatistic (b : ℕ) (lam : ℝ) : (ℓ : ℕ) → RegularObservation b ℓ → ℝ
+  | 0, i => spinScore i
+  | ℓ+1, v => (∑ j : Fin b, regularStatistic b lam ℓ (v j))/((b:ℝ)*lam)
+
+def observedRegularStatistic (b : ℕ) (lam : ℝ) : (ℓ : ℕ) → Observation ℓ → ℝ
+  | 0, i => spinScore i
+  | ℓ+1, s => if s.card = b then (s.map (observedRegularStatistic b lam ℓ)).sum/((b:ℝ)*lam) else 0
+
+lemma abs_multiset_sum_bound {α : Type*} (f : α → ℝ) (C : ℝ) (hf : ∀ a, |f a| ≤ C)
+    (s : Multiset α) : |(s.map f).sum| ≤ (s.card:ℝ)*C := by
+  induction s using Multiset.induction_on with
+  | empty => simp
+  | @cons a s ih =>
+    simp only [Multiset.map_cons, Multiset.sum_cons, Multiset.card_cons, Nat.cast_add, Nat.cast_one]
+    exact (abs_add_le _ _).trans (by linarith [hf a])
+
+lemma observedRegularStatistic_bound (b : ℕ) (lam : ℝ) (ℓ : ℕ) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ o, |observedRegularStatistic b lam ℓ o| ≤ C := by
+  induction ℓ with
+  | zero => exact ⟨2, by norm_num, spinScore_abs⟩
+  | succ ℓ ih =>
+    obtain ⟨C,hC,hbound⟩ := ih
+    refine ⟨(b:ℝ)*C/|((b:ℝ)*lam)|, by positivity, ?_⟩
+    intro s
+    change |if s.card=b then _ else 0| ≤ _
+    split_ifs with h
+    · rw [abs_div]
+      exact div_le_div_of_nonneg_right (by simpa only [h] using abs_multiset_sum_bound _ C hbound s) (abs_nonneg _)
+    · simp only [abs_zero]; positivity
+
+lemma observedRegularStatistic_forget (b : ℕ) (lam : ℝ) (ℓ : ℕ)
+    (v : RegularObservation b ℓ) :
+    observedRegularStatistic b lam ℓ (forgetRegular b ℓ v) = regularStatistic b lam ℓ v := by
+  induction ℓ with
+  | zero => rfl
+  | succ ℓ ih =>
+    simp only [forgetRegular, observedRegularStatistic, Multiset.coe_card, List.length_ofFn, ite_true, regularStatistic]
+    congr 1
+    simp only [Multiset.map_coe, Multiset.sum_coe, List.map_ofFn, Function.comp_def, ih, List.sum_ofFn]
+
+lemma regularStatistic_mean (b : ℕ) (hb : 0 < b) (lam : ℝ) (hlam : Admissible lam)
+    (hl : lam ≠ 0) (ℓ : ℕ) (i : Spin) :
+    (∫ v, regularStatistic b lam ℓ v ∂(regularLaw b lam hlam ℓ i).toMeasure) = spinScore i := by
+  have hb' : (b:ℝ) ≠ 0 := by exact_mod_cast (Nat.ne_of_gt hb)
+  induction ℓ generalizing i with
+  | zero =>
+    change (∫ v : Spin, spinScore v ∂(PMF.pure i).toMeasure) = spinScore i
+    rw [PMF.toMeasure_pure, integral_dirac]
+  | succ ℓ ih =>
+    change (∫ v, (∑ j : Fin b, regularStatistic b lam ℓ (v j))/((b:ℝ)*lam)
+      ∂(productPMF (fun _ ↦ (channel lam hlam i).bind (regularLaw b lam hlam ℓ))).toMeasure) = _
+    rw [integral_div, integral_product_sum, integral_pmf_bind_finite]
+    simp_rw [ih]
+    rw [channel_spinScore]
+    field_simp
+
+end ThreeState.TreeClauses.Tree
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/ThreeStateClauses/ScalarBounds.lean b/lean/OAI/Probability/ThreeStateClauses/ScalarBounds.lean
index 0bb2d75eeb42cadaead99fb26dea345f2831931b..13d53e3b413e9181d3a5ca411e402814ae1bae65 100644
--- a/lean/OAI/Probability/ThreeStateClauses/ScalarBounds.lean
+++ b/lean/OAI/Probability/ThreeStateClauses/ScalarBounds.lean
@@ -1,115 +1,115 @@
-import Mathlib
-
-namespace OAI
-
-/-! Scalar Bounds. -/
-
-open scoped BigOperators ENNReal NNReal Topology
-open Filter
-noncomputable section
-namespace ThreeState.TreeClauses.Positive
-
-def cLower (lam : ℝ) : ℝ := (37/27)*(1+2*lam/3)
-def scalarT (lam mu s : ℝ) : ℝ :=
-  (1-lam^2)/(2*s)+3/(2*(1+2*mu/3))+(17*lam*mu)/(25*s)
-
-lemma cLower_pos {lam : ℝ} (hl : 0 ≤ lam) : 0 < cLower lam := by
-  dsimp [cLower]; positivity
-
-lemma small_polynomial {lam : ℝ} (h₀ : 0 ≤ lam) (h₁ : lam ≤ 3/10) :
-    (2/5-lam^2/2)*cLower lam^2 > (2/3)*(11/10+lam^2/2) := by
-  let u := (10/3)*lam
-  have hu₀ : 0 ≤ u := by dsimp [u]; positivity
-  have hu₁ : u ≤ 1 := by dsimp [u]; linarith
-  have heq : (2/5-lam^2/2)*cLower lam^2-(2/3)*(11/10+lam^2/2) =
-      (65000*(1-u)^4+1355200*u*(1-u)^3+3367745*u^2*(1-u)^2+
-        2806680*u^3*(1-u)+716814*u^4)/3645000 := by dsimp [u, cLower]; ring
-  have hp : 0 < (2/5-lam^2/2)*cLower lam^2-(2/3)*(11/10+lam^2/2) := by
-    rw [heq]
-    rcases lt_or_eq_of_le hu₁ with h | h
-    · have : 0 < 1-u := sub_pos.mpr h
-      positivity
-    · rw [h]; norm_num
-  linarith
-
-lemma middle_polynomial {lam : ℝ} (h₀ : 3/10 ≤ lam) (h₁ : lam ≤ 71/100) :
-    (2/5-lam^2/2)*cLower lam^2 > 1-17*lam/25 := by
-  let u := (100/41)*(lam-3/10)
-  have hu₀ : 0 ≤ u := by dsimp [u]; positivity
-  have hu₁ : u ≤ 1 := by dsimp [u]; linarith
-  have heq : (2/5-lam^2/2)*cLower lam^2-(1-17*lam/25) =
-      (53796960000*(1-u)^4+340999128000*u*(1-u)^3+592285169900*u^2*(1-u)^2+
-        337132599220*u^3*(1-u)+28181130511*u^4)/328050000000 := by dsimp [u, cLower]; ring
-  have hp : 0 < (2/5-lam^2/2)*cLower lam^2-(1-17*lam/25) := by
-    rw [heq]
-    rcases lt_or_eq_of_le hu₁ with h | h
-    · have : 0 < 1-u := sub_pos.mpr h
-      positivity
-    · rw [h]; norm_num
-  linarith
-
-lemma large_polynomial {lam : ℝ} (h : 7/10 ≤ lam) :
-    (3/20)*cLower lam^2 > 1-17*lam/25 := by
-  have heq : (3/20)*cLower lam^2-(1-17*lam/25) =
-      44801/546750+(67297/54675)*(lam-7/10)+(1369/10935)*(lam-7/10)^2 := by
-    dsimp [cLower]; ring
-  have hpos : 0 < (3/20)*cLower lam^2-(1-17*lam/25) := by
-    rw [heq]; positivity
-  linarith
-
-lemma substitute_moment_region {C B D mu : ℝ} (hC : 0 < C) (hD : 0 ≤ D)
-    (hmu : mu*C^2 ≤ 1) (hBD : D < B*C^2) : D*mu < B := by
-  have h := mul_le_mul_of_nonneg_left hmu hD
-  have hprod : (D*mu)*C^2 < B*C^2 := by nlinarith
-  exact (mul_lt_mul_iff_left₀ (sq_pos_of_pos hC)).mp hprod
-
-lemma reciprocal_lower {mu : ℝ} (hmu : 0 ≤ mu) :
-    3/2-mu ≤ 3/(2*(1+2*mu/3)) := by
-  apply (le_div_iff₀ (by positivity : 0 < 2*(1+2*mu/3))).2
-  nlinarith [sq_nonneg mu]
-
-theorem scalar_strict {lam mu s : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1)
-    (hmu : 0 ≤ mu) (hs₀ : 0 < s) (hs₁ : s ≤ 1) (hregion : mu*cLower lam^2 ≤ s)
-    (hmodel : lam^2 ≤ 1/2 ∨ s ≤ 2*(1-lam^2)) : 8/5 < scalarT lam mu s := by
-  have hC := cLower_pos hl₀.le
-  have hmuC : mu*cLower lam^2 ≤ 1 := hregion.trans hs₁
-  have hlam : 0 < 1-lam^2 := by nlinarith [mul_pos (sub_pos.mpr hl₁) (by linarith : 0 < 1+lam)]
-  have hfirst : (1-lam^2)/2 ≤ (1-lam^2)/(2*s) :=
-    div_le_div_of_nonneg_left hlam.le (by positivity) (by nlinarith)
-  have hlast : 17*lam*mu/25 ≤ 17*lam*mu/(25*s) :=
-    div_le_div_of_nonneg_left (by positivity) (by positivity) (by nlinarith)
-  have hrecip := reciprocal_lower hmu
-  by_cases hsmall : lam ≤ 3/10
-  · have hp := small_polynomial hl₀.le hsmall
-    have hD : 0 ≤ (2/3)*(11/10+lam^2/2) := by positivity
-    have hsub := substitute_moment_region hC hD hmuC hp
-    have hden : 0 < 2*(1+2*mu/3) := by positivity
-    have hfrac : 11/10+lam^2/2 < 3/(2*(1+2*mu/3)) := by
-      apply (lt_div_iff₀ hden).2
-      nlinarith
-    dsimp [scalarT]
-    have : 0 ≤ 17*lam*mu/25 := by positivity
-    linarith
-  · have hlow : 3/10 ≤ lam := le_of_not_ge hsmall
-    have hD : 0 ≤ 1-17*lam/25 := by linarith
-    rcases le_or_gt (lam^2) (1/2) with hmid | hbig
-    · have hhigh : lam ≤ 71/100 := by nlinarith
-      have hp := middle_polynomial hlow hhigh
-      have hsub := substitute_moment_region hC hD hmuC hp
-      dsimp [scalarT]
-      nlinarith
-    · have hsurvive : s ≤ 2*(1-lam^2) := hmodel.resolve_left (not_le.mpr hbig)
-      have hquarter : 1/4 ≤ (1-lam^2)/(2*s) := by
-        apply (le_div_iff₀ (by positivity : 0 < 2*s)).2
-        linarith
-      have hlarge : 7/10 ≤ lam := by nlinarith
-      have hp := large_polynomial hlarge
-      have hsub := substitute_moment_region hC hD hmuC hp
-      dsimp [scalarT]
-      nlinarith
-
-end ThreeState.TreeClauses.Positive
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-! Scalar Bounds. -/
+
+open scoped BigOperators ENNReal NNReal Topology
+open Filter
+noncomputable section
+namespace ThreeState.TreeClauses.Positive
+
+def cLower (lam : ℝ) : ℝ := (37/27)*(1+2*lam/3)
+def scalarT (lam mu s : ℝ) : ℝ :=
+  (1-lam^2)/(2*s)+3/(2*(1+2*mu/3))+(17*lam*mu)/(25*s)
+
+lemma cLower_pos {lam : ℝ} (hl : 0 ≤ lam) : 0 < cLower lam := by
+  dsimp [cLower]; positivity
+
+lemma small_polynomial {lam : ℝ} (h₀ : 0 ≤ lam) (h₁ : lam ≤ 3/10) :
+    (2/5-lam^2/2)*cLower lam^2 > (2/3)*(11/10+lam^2/2) := by
+  let u := (10/3)*lam
+  have hu₀ : 0 ≤ u := by dsimp [u]; positivity
+  have hu₁ : u ≤ 1 := by dsimp [u]; linarith
+  have heq : (2/5-lam^2/2)*cLower lam^2-(2/3)*(11/10+lam^2/2) =
+      (65000*(1-u)^4+1355200*u*(1-u)^3+3367745*u^2*(1-u)^2+
+        2806680*u^3*(1-u)+716814*u^4)/3645000 := by dsimp [u, cLower]; ring
+  have hp : 0 < (2/5-lam^2/2)*cLower lam^2-(2/3)*(11/10+lam^2/2) := by
+    rw [heq]
+    rcases lt_or_eq_of_le hu₁ with h | h
+    · have : 0 < 1-u := sub_pos.mpr h
+      positivity
+    · rw [h]; norm_num
+  linarith
+
+lemma middle_polynomial {lam : ℝ} (h₀ : 3/10 ≤ lam) (h₁ : lam ≤ 71/100) :
+    (2/5-lam^2/2)*cLower lam^2 > 1-17*lam/25 := by
+  let u := (100/41)*(lam-3/10)
+  have hu₀ : 0 ≤ u := by dsimp [u]; positivity
+  have hu₁ : u ≤ 1 := by dsimp [u]; linarith
+  have heq : (2/5-lam^2/2)*cLower lam^2-(1-17*lam/25) =
+      (53796960000*(1-u)^4+340999128000*u*(1-u)^3+592285169900*u^2*(1-u)^2+
+        337132599220*u^3*(1-u)+28181130511*u^4)/328050000000 := by dsimp [u, cLower]; ring
+  have hp : 0 < (2/5-lam^2/2)*cLower lam^2-(1-17*lam/25) := by
+    rw [heq]
+    rcases lt_or_eq_of_le hu₁ with h | h
+    · have : 0 < 1-u := sub_pos.mpr h
+      positivity
+    · rw [h]; norm_num
+  linarith
+
+lemma large_polynomial {lam : ℝ} (h : 7/10 ≤ lam) :
+    (3/20)*cLower lam^2 > 1-17*lam/25 := by
+  have heq : (3/20)*cLower lam^2-(1-17*lam/25) =
+      44801/546750+(67297/54675)*(lam-7/10)+(1369/10935)*(lam-7/10)^2 := by
+    dsimp [cLower]; ring
+  have hpos : 0 < (3/20)*cLower lam^2-(1-17*lam/25) := by
+    rw [heq]; positivity
+  linarith
+
+lemma substitute_moment_region {C B D mu : ℝ} (hC : 0 < C) (hD : 0 ≤ D)
+    (hmu : mu*C^2 ≤ 1) (hBD : D < B*C^2) : D*mu < B := by
+  have h := mul_le_mul_of_nonneg_left hmu hD
+  have hprod : (D*mu)*C^2 < B*C^2 := by nlinarith
+  exact (mul_lt_mul_iff_left₀ (sq_pos_of_pos hC)).mp hprod
+
+lemma reciprocal_lower {mu : ℝ} (hmu : 0 ≤ mu) :
+    3/2-mu ≤ 3/(2*(1+2*mu/3)) := by
+  apply (le_div_iff₀ (by positivity : 0 < 2*(1+2*mu/3))).2
+  nlinarith [sq_nonneg mu]
+
+theorem scalar_strict {lam mu s : ℝ} (hl₀ : 0 < lam) (hl₁ : lam < 1)
+    (hmu : 0 ≤ mu) (hs₀ : 0 < s) (hs₁ : s ≤ 1) (hregion : mu*cLower lam^2 ≤ s)
+    (hmodel : lam^2 ≤ 1/2 ∨ s ≤ 2*(1-lam^2)) : 8/5 < scalarT lam mu s := by
+  have hC := cLower_pos hl₀.le
+  have hmuC : mu*cLower lam^2 ≤ 1 := hregion.trans hs₁
+  have hlam : 0 < 1-lam^2 := by nlinarith [mul_pos (sub_pos.mpr hl₁) (by linarith : 0 < 1+lam)]
+  have hfirst : (1-lam^2)/2 ≤ (1-lam^2)/(2*s) :=
+    div_le_div_of_nonneg_left hlam.le (by positivity) (by nlinarith)
+  have hlast : 17*lam*mu/25 ≤ 17*lam*mu/(25*s) :=
+    div_le_div_of_nonneg_left (by positivity) (by positivity) (by nlinarith)
+  have hrecip := reciprocal_lower hmu
+  by_cases hsmall : lam ≤ 3/10
+  · have hp := small_polynomial hl₀.le hsmall
+    have hD : 0 ≤ (2/3)*(11/10+lam^2/2) := by positivity
+    have hsub := substitute_moment_region hC hD hmuC hp
+    have hden : 0 < 2*(1+2*mu/3) := by positivity
+    have hfrac : 11/10+lam^2/2 < 3/(2*(1+2*mu/3)) := by
+      apply (lt_div_iff₀ hden).2
+      nlinarith
+    dsimp [scalarT]
+    have : 0 ≤ 17*lam*mu/25 := by positivity
+    linarith
+  · have hlow : 3/10 ≤ lam := le_of_not_ge hsmall
+    have hD : 0 ≤ 1-17*lam/25 := by linarith
+    rcases le_or_gt (lam^2) (1/2) with hmid | hbig
+    · have hhigh : lam ≤ 71/100 := by nlinarith
+      have hp := middle_polynomial hlow hhigh
+      have hsub := substitute_moment_region hC hD hmuC hp
+      dsimp [scalarT]
+      nlinarith
+    · have hsurvive : s ≤ 2*(1-lam^2) := hmodel.resolve_left (not_le.mpr hbig)
+      have hquarter : 1/4 ≤ (1-lam^2)/(2*s) := by
+        apply (le_div_iff₀ (by positivity : 0 < 2*s)).2
+        linarith
+      have hlarge : 7/10 ≤ lam := by nlinarith
+      have hp := large_polynomial hlarge
+      have hsub := substitute_moment_region hC hD hmuC hp
+      dsimp [scalarT]
+      nlinarith
+
+end ThreeState.TreeClauses.Positive
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/TraceReconstruction.lean b/lean/OAI/Probability/TraceReconstruction.lean
index c4667feec965913829a9d2e7c66ce34a63035699..a4f3f0f32540d93a4267678de4330fdacb8e6978 100644
--- a/lean/OAI/Probability/TraceReconstruction.lean
+++ b/lean/OAI/Probability/TraceReconstruction.lean
@@ -1,427 +1,427 @@
-import OAI.Probability.TraceReconstruction.Induction
-import OAI.Probability.TraceReconstruction.Testing
-
-namespace OAI
-
-/-!
-# Bounds
-
-Restored construction and final scales; quantitative sample lower bound and both fixed-channel superpolynomial endpoints.
--/
-
-open scoped ENNReal
-noncomputable section
-
-open Filter
-open scoped Topology
-
-
-noncomputable section
-namespace TraceReconstruction
-open Filter
-open scoped Topology Classical
-
-/-- The exact source error is negligible compared to the squared score energy. -/
-theorem source_final_error_eventually (B : ℕ) (ζ : ℝ) (hB : 2≤B) (hζ : ζ∈Set.Ioo 0 1)
-    {n q : ℕ→ℝ} (hn : Tendsto n atTop atTop) (hq : ∀t,q t∈Set.Ioc 0 (1/2:ℝ))
-    (hH : Tendsto (fun t=>(q t)^3*Real.log (n t)) atTop atTop) :
-    ∀ᶠt in atTop,∀ (U : ConcreteLevelPair (q t) ⟨(hq t).1.le,by linarith [(hq t).2]⟩ (n t)
-      ((n t)^(-100*(totalScoreExponent B (sourceDepth B ζ ((q t)^3*Real.log (n t)))+B+1)))
-      (totalScoreExponent B (sourceDepth B ζ ((q t)^3*Real.log (n t))))
-      (initialOverlapLower (q t))
-      ((sourceOverlapConstant (q t))^(2*B*sourceDepth B ζ ((q t)^3*Real.log (n t))+2))
-      (sourceDepth B ζ ((q t)^3*Real.log (n t)))
-      (sourceHalfMoment B (sourceDepth B ζ ((q t)^3*Real.log (n t))) (n t)
-        (sourceDepth B ζ ((q t)^3*Real.log (n t))))),
-      U.error≤(n t)^(-2*totalScoreExponent B (sourceDepth B ζ ((q t)^3*Real.log (n t)))) := by
-  have ht := SourceExpUpper.uniform (fun D _=>(D:ℝ)+1) SourcePolyUpper.depth.expUpper
-    B ζ 1 hB hζ zero_lt_one hn hq hH
-  filter_upwards [ht,hn.eventually_ge_atTop 2] with t ht hn2 U
-  let D := sourceDepth B ζ ((q t)^3*Real.log (n t))
-  let d := totalScoreExponent B D
-  have hn0 : 0<n t := by linarith only [hn2]
-  have hn1 : 1≤n t := by linarith only [hn2]
-  have hD : (D:ℝ)+1≤n t := by
-    have hh := ht D
-    rw [sourceTargetScale_top,one_mul] at hh
-    exact hh.trans (by simpa only [Real.rpow_one] using
-      Real.rpow_le_rpow_of_exponent_le hn1 (sourceEpsilon_le_one B D))
-  have hd : 11/5≤d := (totalScoreExponent_bounds B D hB).1
-  calc
-    U.error ≤ ((D:ℝ)+1)*(n t)^(-100*(d+B+1))*n t := U.error_le hn0 (Real.rpow_nonneg hn0.le _)
-    _ ≤ (n t)*(n t)^(-100*(d+B+1))*n t := by gcongr
-    _ = (n t)^(-100*(d+B+1)+2) := by
-      rw [Real.rpow_add hn0,Real.rpow_two]
-      ring
-    _ ≤ (n t)^(-2*d) := Real.rpow_le_rpow_of_exponent_le hn1 (by linarith [Nat.cast_nonneg B (α:=ℝ)])
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open Filter
-open scoped Topology Classical
-
-theorem exists_source_slope {c : ℝ} (hc : c<1/(4*Real.log 2)) :
-    ∃B : ℕ,2≤B ∧ ∃ζ : ℝ,ζ∈Set.Ioo 0 1 ∧ c<finalSlope B ζ := by
-  have hn : Tendsto (fun B : ℕ=>(B:ℝ)) atTop atTop := tendsto_natCast_atTop_atTop
-  have hn1 : Tendsto (fun B : ℕ=>(B:ℝ)+1) atTop atTop :=
-    tendsto_atTop_mono (fun B=>by linarith) hn
-  have h1 := hn1.const_div_atTop (1:ℝ)
-  have h2 := hn.const_div_atTop (3/10:ℝ)
-  have hlim := ((tendsto_const_nhds (x:=(1:ℝ))).sub h1).mul
-    ((tendsto_const_nhds (x:=(1/4:ℝ))).sub h2) |>.div_const (Real.log 2)
-  have hlim' : Tendsto (fun B : ℕ=>(1-1/((B:ℝ)+1))*((1/4:ℝ)-(3/10:ℝ)/(B:ℝ))/Real.log 2)
-      atTop (𝓝 (1/(4*Real.log 2))) := by
-    convert hlim using 1
-    ring_nf
-  obtain ⟨B,hB,hβ⟩ := ((eventually_ge_atTop 2).and (hlim'.eventually_const_lt hc)).exists
-  refine ⟨B,hB,1/((B:ℝ)+1),⟨by positivity,?_⟩,?_⟩
-  · have hBr : (2:ℝ)≤B := by exact_mod_cast hB
-    exact (div_lt_one (by positivity)).mpr (by linarith)
-  · have hBr : (B:ℝ)≠0 := by exact_mod_cast (show B≠0 by omega)
-    have he : finalSlope B (1/((B:ℝ)+1))=
-        (1-1/((B:ℝ)+1))*((1/4:ℝ)-(3/10:ℝ)/(B:ℝ))/Real.log 2 := by
-      unfold finalSlope blockGain
-      field_simp
-      ring
-    rw [he]
-    exact hβ
-
-/-- Constant-size subdivision and larger deletion preserve the joint regime.
- The factor 16 includes the factor eight of the working deletion cube. -/
-theorem subdivided_working_scale {n q : ℕ→ℝ} (hn : Tendsto n atTop atTop)
-    (hq : ∀t,q t∈Set.Ioo 0 1)
-    (hH : Tendsto (fun t=>(q t)^3*Real.log (n t)) atTop atTop)
-    {G : ℝ} (hG : 0<G) :
-    Tendsto (fun t=>n t/G) atTop atTop ∧
-    Tendsto (fun t=>(workingDeletion (q t))^3*Real.log (n t/G)) atTop atTop ∧
-    ∀ᶠt in atTop,(q t)^3*Real.log (n t)/16 ≤
-      (workingDeletion (q t))^3*Real.log (n t/G) := by
-  have hN := hn.atTop_div_const hG
-  have hl := Real.tendsto_log_atTop.comp hn
-  have hh : ∀ᶠt in atTop,(q t)^3*Real.log (n t)/16 ≤
-      (workingDeletion (q t))^3*Real.log (n t/G) := by
-    filter_upwards [hn.eventually_ge_atTop 1,hl.eventually_ge_atTop (max 0 (2*Real.log G))] with t hn1 hl
-    have hn0 : 0<n t := by linarith only [hn1]
-    have hl0 : 0≤Real.log (n t) := (le_max_left _ _).trans hl
-    have hlg : 2*Real.log G≤Real.log (n t) := (le_max_right _ _).trans hl
-    have hhalf : Real.log (n t)/2≤Real.log (n t/G) := by
-      rw [Real.log_div hn0.ne' hG.ne']
-      linarith only [hlg]
-    have hb := (workingScale_bounds (hq t) hl0).1
-    have hm := mul_le_mul_of_nonneg_left hhalf (pow_nonneg (workingDeletion_bounds (hq t)).1.1.le 3)
-    nlinarith only [hb,hm]
-  exact ⟨hN,tendsto_atTop_mono' atTop hh (hH.atTop_div_const (by norm_num : (0:ℝ)<16)),hh⟩
-
-theorem workingDeletion_Ioc {q : ℝ} (hq : q∈Set.Ioo 0 1) :
-    workingDeletion q ∈Set.Ioc 0 (1/2:ℝ) :=
-  ⟨(workingDeletion_bounds hq).1.1,min_le_right _ _⟩
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open Filter
-open scoped Topology Classical
-
-/-- Full sharp slope restoration after fixed gadget packing. Both deletion
- probability and length may vary; the working channel is only internal. -/
-theorem source_restored_exponent_margin (B : ℕ) (ζ c G a : ℝ)
-    (hB : 2≤B) (hζ : ζ∈Set.Ioo 0 1) (hc : 0<c) (hα : c<finalSlope B ζ)
-    (hG : 0<G) (ha : 0<a) {n q : ℕ→ℝ} (hn : Tendsto n atTop atTop)
-    (hq : ∀t,q t∈Set.Ioo 0 1)
-    (hH : Tendsto (fun t=>(q t)^3*Real.log (n t)) atTop atTop) :
-    ∀ᶠt in atTop,(n t)^(c*Real.log ((q t)^3*Real.log (n t))) ≤
-      a*(n t/G)^(totalScoreExponent B
-        (sourceDepth B ζ ((workingDeletion (q t))^3*Real.log (n t/G)))) := by
-  let α := finalSlope B ζ
-  have hα0 : 0<α := hc.trans hα
-  obtain ⟨ρ,hρc,hρ1⟩ := exists_between ((div_lt_one hα0).mpr hα)
-  have hρ0 : 0<ρ := (div_pos hc hα0).trans hρc
-  have hρc' : c<ρ*α := (div_lt_iff₀ hα0).mp hρc
-  let H := fun t=>(q t)^3*Real.log (n t)
-  let J := fun t=>(workingDeletion (q t))^3*Real.log (n t/G)
-  let d := fun t=>totalScoreExponent B (sourceDepth B ζ (J t))
-  let K := blockGain B+(warmupBlocks B:ℝ)*blockGain B+α*Real.log 16
-  have hsc := subdivided_working_scale hn hq hH hG
-  have hdepth := sourceDepth_tendsto B ζ hB hζ.2 hsc.2.1
-  have hd : ∀ᶠt in atTop,(ρ*α)*Real.log (H t)-ρ*K≤ρ*d t := by
-    filter_upwards [hsc.2.2,hH.eventually_gt_atTop 0,
-      hdepth.eventually_ge_atTop (warmupBlocks B)] with t hj hp hD
-    have hJ : 0<J t := lt_of_lt_of_le (div_pos hp (by norm_num)) hj
-    have hlog : Real.log (H t)-Real.log 16≤Real.log (J t) := by
-      have hh := Real.log_le_log (div_pos hp (by norm_num)) hj
-      simpa only [Real.log_div (ne_of_gt hp) (by norm_num : (16:ℝ)≠0)] using hh
-    have hs := (totalScoreExponent_slope_lower B ζ (J t) hB hD).le
-    have hm := mul_le_mul_of_nonneg_left hlog hα0.le
-    have hbase : α*Real.log (H t)-K≤d t := by
-      dsimp only [K,d,α] at *
-      linarith only [hs,hm]
-    have hh := mul_le_mul_of_nonneg_left hbase hρ0.le
-    nlinarith only [hh]
-  have hex := eventually_exponent_margin hn hH hρc' ha hd
-  have hsub := ((PowerRate.base hn).div_const hn hG).lower hn hρ1
-  filter_upwards [hex,hsub,hn.eventually_gt_atTop 0] with t he hs hp
-  apply he.trans
-  apply mul_le_mul_of_nonneg_left _ ha.le
-  rw [Real.rpow_mul hp.le]
-  exact Real.rpow_le_rpow (Real.rpow_nonneg hp.le _) hs
-    (by have hh := (totalScoreExponent_bounds B (sourceDepth B ζ (J t)) hB).1; dsimp [d]; linarith only [hh])
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open Filter
-open scoped Topology Classical ENNReal
-
-theorem sample_tv_small {s G N d E M : ℝ} (hs : s∈Set.Ioc 0 1)
-    (hG : 0<G) (hN : 1 ≤ N) (hd : 0 ≤ d) (hM : 0 ≤ M)
-    (hE : E ≤ N^(-2*d)) (hbudget : M ≤ (s^2/(64*G))*N^d) :
-    Real.sqrt (M*G*N^(-d))+2*M*G*E ≤ s/4 := by
-  have hN0 : 0<N := by linarith only [hN]
-  have hV0 : 0 ≤ N^(-d) := Real.rpow_nonneg hN0.le _
-  have hV1 : N^(-d) ≤ 1 := by
-    simpa only [Real.rpow_zero] using Real.rpow_le_rpow_of_exponent_le hN (neg_nonpos.mpr hd)
-  have hVV : N^(-2*d)=(N^(-d))^2 := by
-    rw [←Real.rpow_natCast,←Real.rpow_mul hN0.le]
-    norm_num only [Nat.cast_ofNat]
-    congr 1
-    ring
-  have hprod : N^d*N^(-d)=1 := by rw [←Real.rpow_add hN0,add_neg_cancel,Real.rpow_zero]
-  have hMV : M*G*N^(-d) ≤ s^2/64 := by
-    calc
-      _  ≤  ((s^2/(64*G))*N^d)*G*N^(-d) := by gcongr
-      _ = (s^2/(64*G)*G)*(N^d*N^(-d)) := by ring
-      _ = s^2/64 := by rw [hprod]; field_simp
-  have hsq : Real.sqrt (M*G*N^(-d)) ≤ s/8 := by
-    apply (Real.sqrt_le_left (div_nonneg hs.1.le (by norm_num) : 0 ≤ s/8)).mpr
-    nlinarith only [hMV]
-  have hE' : E ≤ N^(-d) := by
-    rw [hVV] at hE
-    nlinarith only [hE,mul_le_mul_of_nonneg_left hV1 hV0]
-  have he := mul_le_mul_of_nonneg_left hE' (show 0 ≤ 2*M*G by positivity)
-  have hss := mul_nonneg hs.1.le (sub_nonneg.mpr hs.2)
-  nlinarith only [hsq,he,hMV,hss]
-
-/-- Source Theorem 1: all fixed positive success levels, not only binary
- testing above one half, with complete independent traces and known channel. -/
-theorem quantitative_sample_lower_bound
-    (s c : ℝ) (hs : s ∈ Set.Ioc 0 1)
-    (hc : c ∈ Set.Ioo 0 (1 / (4 * Real.log 2)))
-    (n : ℕ → ℕ) (q : ℕ → ℝ)
-    (hn : Tendsto n atTop atTop)
-    (hq : ∀ k, q k ∈ Set.Ioo 0 1)
-    (hscale : Tendsto (fun k => (q k) ^ 3 * Real.log (n k)) atTop atTop) :
-    ∀ᶠ k in atTop,
-      ENNReal.ofReal ((n k : ℝ) ^ (c * Real.log ((q k) ^ 3 * Real.log (n k))))  ≤ 
-        sampleComplexity (n k) (q k) s := by
-  obtain ⟨B,hB,ζ,hζ,hα⟩ := exists_source_slope hc.2
-  obtain ⟨g,hg⟩ := exists_nat_gt (max 1 (2/s))
-  have hg1 : 1<(g:ℝ) := (le_max_left _ _).trans_lt hg
-  have hg0 : 0<(g:ℝ) := by linarith only [hg1]
-  have hgs : 1/(2:ℝ)^g<s/2 := by
-    have hp : (g:ℝ)<(2:ℝ)^g := by
-      clear hg hg1 hg0
-      induction g with
-      | zero => norm_num
-      | succ g ih =>
-        push_cast
-        rw [pow_succ]
-        have hg : (0:ℝ) ≤ g := Nat.cast_nonneg g
-        have hp : (1:ℝ) ≤ 2^g := one_le_pow₀ (by norm_num)
-        nlinarith only [ih,hg,hp]
-    have hbig : 2/s<(2:ℝ)^g := ((le_max_right _ _).trans_lt hg).trans hp
-    have hh := (div_lt_iff₀ hs.1).mp hbig
-    apply (div_lt_iff₀ (by positivity : (0:ℝ)<2^g)).mpr
-    nlinarith only [hh]
-  have hnr : Tendsto (fun t=>(n t:ℝ)) atTop atTop := tendsto_natCast_atTop_atTop.comp hn
-  let N := fun t=>(n t:ℝ)/(g:ℝ)
-  let r := fun t=>workingDeletion (q t)
-  have hr (t) : r t∈Set.Ioc 0 (1/2:ℝ) := workingDeletion_Ioc (hq t)
-  have hsc := subdivided_working_scale hnr hq hscale hg0
-  have hpair := source_final_pair_eventually B ζ hB hζ hsc.1 hr hsc.2.1
-  have herr := source_final_error_eventually B ζ hB hζ hsc.1 hr hsc.2.1
-  have hmargin := source_restored_exponent_margin B ζ c (g:ℝ) (s^2/(64*g))
-    hB hζ hc.1 hα hg0 (div_pos (sq_pos_of_pos hs.1) (mul_pos (by norm_num) hg0)) hnr hq hscale
-  filter_upwards [hpair,herr,hmargin,hsc.1.eventually_ge_atTop 2] with t hp he hm hn2
-  obtain ⟨U⟩ := hp
-  have hN1 : 1 ≤ N t := by dsimp [N]; linarith only [hn2]
-  have hN2 : 1<N t := by dsimp [N]; linarith only [hn2]
-  have hk := (sourceHalfMoment_bounds B
-    (sourceDepth B ζ ((r t)^3*Real.log (N t)))
-    (sourceDepth B ζ ((r t)^3*Real.log (N t))) hB hN2).1
-  obtain ⟨W,hW,hTV⟩ := U.restored_family (by have hh := (hr t).2; dsimp [r] at *; linarith only [hh])
-    ⟨(hq t).1.le,(hq t).2.le⟩ (workingDeletion_bounds (hq t)).2.2 hk g (n t)
-    (by field_simp; exact le_rfl)
-  apply le_sInf
-  rintro z ⟨m,hbudget,rfl⟩
-  rw [← ENNReal.ofReal_natCast]
-  apply ENNReal.ofReal_le_ofReal
-  by_contra hnot
-  have hm' : (m:ℝ) ≤ (s^2/(64*g))*(N t)^(
-      totalScoreExponent B (sourceDepth B ζ ((r t)^3*Real.log (N t)))) :=
-    (le_of_lt (lt_of_not_ge hnot)).trans hm
-  have hclose (θ : Fin g→Bool) := (hTV θ m).trans
-    (sample_tv_small hs hg0 hN1
-      (by have hd := (totalScoreExponent_bounds B (sourceDepth B ζ ((r t)^3*Real.log (N t))) hB).1; linarith only [hd])
-      (Nat.cast_nonneg m) (he U) hm')
-  have hbad := family_testing_bound ⟨(hq t).1.le,(hq t).2.le⟩ W hW hclose hbudget
-  linarith only [hbad,hgs,hs.1]
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open Filter
-open scoped Topology Classical ENNReal
-
-namespace FiniteLaw
-variable {α : Type*} [Fintype α]
-theorem tv_product_one (P Q : FiniteLaw α) : (P.product 1).tv (Q.product 1)=P.tv Q := by
-  rw [show 1=0+1 from rfl,tv_product_succ]
-  simp only [tv,tensor,product,Fintype.sum_prod_type,Fintype.prod_empty,mul_one,Fintype.sum_unique]
-end FiniteLaw
-
-theorem totalScoreExponent_two (D : ℕ) : totalScoreExponent 2 D=11/5+(D:ℝ)/5 := by
-  simp only [totalScoreExponent,ite_self,blockGain_two,Finset.sum_const,Finset.card_range,nsmul_eq_mul]
-  ring
-
-theorem minTraceTV_nonneg_of_pair {n : ℕ} (q : ℝ) (x y : Word n) (hxy : x≠y) :
-    0 ≤ minTraceTV n q := by
-  apply le_csInf (show {t | ∃x y : Word n,x≠y ∧ t=traceTV q x y}.Nonempty from
-    ⟨_,x,y,hxy,rfl⟩)
-  rintro b ⟨u,v,_,rfl⟩
-  exact traceTV_nonneg q u v
-
-theorem minTraceTV_le_pair {n : ℕ} (q : ℝ) (x y : Word n) (hxy : x≠y) :
-    minTraceTV n q ≤ traceTV q x y := by
-  apply csInf_le (show BddBelow {t | ∃x y : Word n,x≠y ∧ t=traceTV q x y} from ?_) ⟨x,y,hxy,rfl⟩
-  exact ⟨0,by rintro b ⟨u,v,_,rfl⟩; exact traceTV_nonneg q u v⟩
-
-theorem one_trace_power_bound {N d C R : ℝ} (hN : 1 ≤ N) (hC : 0<C)
-    (hd : 2*(C+1) ≤ d) (hR : R ≤ Real.sqrt (N^(-d))+2*N^(-2*d)) :
-    N^C*R ≤ 3/N := by
-  have hN0 : 0<N := by linarith only [hN]
-  have he : (N^(-(C+1)))^2=N^(-2*(C+1)) := by
-    rw [←Real.rpow_natCast,←Real.rpow_mul hN0.le]
-    norm_num only [Nat.cast_ofNat]
-    congr 1
-    ring
-  have hs : Real.sqrt (N^(-d)) ≤ N^(-(C+1)) := by
-    apply (Real.sqrt_le_left (Real.rpow_nonneg hN0.le _)).mpr
-    rw [he]
-    exact Real.rpow_le_rpow_of_exponent_le hN (by linarith only [hd])
-  have herr : N^(-2*d) ≤ N^(-(C+1)) :=
-    Real.rpow_le_rpow_of_exponent_le hN (by linarith only [hd,hC])
-  calc
-    N^C*R ≤ N^C*(3*N^(-(C+1))) :=
-      mul_le_mul_of_nonneg_left (by linarith only [hR,hs,herr]) (Real.rpow_nonneg hN0.le _)
-    _ = 3/N := by
-      rw [mul_left_comm,←Real.rpow_add hN0]
-      rw [show C+ -(C+1)=(-1:ℝ) by ring,Real.rpow_neg_one]
-      ring
-
-/-- Source Theorem 2, first conclusion, over all integer lengths and at the
- same fixed channel for every power. -/
-theorem superpolynomial_one_trace
-    (q : ℝ) (hq : q ∈ Set.Ioo 0 1) (C : ℝ) (hC : 0<C) :
-    Tendsto (fun n : ℕ=>(n:ℝ)^C*minTraceTV n q) atTop (𝓝 0) := by
-  let r := workingDeletion q
-  have hr : r∈Set.Ioc 0 (1/2:ℝ) := workingDeletion_Ioc hq
-  have hn : Tendsto (fun n : ℕ=>(n:ℝ)) atTop atTop := tendsto_natCast_atTop_atTop
-  have hJ : Tendsto (fun n : ℕ=>r^3*Real.log (n:ℝ)) atTop atTop :=
-    (Real.tendsto_log_atTop.comp hn).const_mul_atTop (pow_pos hr.1 3)
-  have hpair := source_final_pair_eventually 2 (1/2) (by omega) (by norm_num) hn (fun _=>hr) hJ
-  have herr := source_final_error_eventually 2 (1/2) (by omega) (by norm_num) hn (fun _=>hr) hJ
-  obtain ⟨D0,hD0⟩ := exists_nat_ge (5*(2*(C+1)))
-  have hdepth := (sourceDepth_tendsto 2 (1/2) (by omega) (by norm_num) hJ).eventually_ge_atTop D0
-  have hb : ∀ᶠ n : ℕ in atTop,0 ≤ (n:ℝ)^C*minTraceTV n q ∧
-      (n:ℝ)^C*minTraceTV n q ≤ 3/(n:ℝ) := by
-    filter_upwards [hpair,herr,hdepth,hn.eventually_ge_atTop 2] with n hp he hD hn2
-    obtain ⟨U⟩ := hp
-    have hn1 : 1<(n:ℝ) := by linarith only [hn2]
-    have hk := (sourceHalfMoment_bounds 2
-      (sourceDepth 2 (1/2) (r^3*Real.log (n:ℝ)))
-      (sourceDepth 2 (1/2) (r^3*Real.log (n:ℝ))) (by omega) hn1).1
-    obtain ⟨x,y,hxy,hTV⟩ := U.restored_words (by linarith only [hr.2])
-      ⟨hq.1.le,hq.2.le⟩ (workingDeletion_bounds hq).2.2 hk n le_rfl
-    have ht := hTV 1
-    rw [FiniteLaw.tv_product_one] at ht
-    rw [completeTraceLaw_tv q ⟨hq.1.le,hq.2.le⟩ x y] at ht
-    norm_num only [Nat.cast_one,one_mul,mul_one] at ht
-    have hd : 2*(C+1) ≤ totalScoreExponent 2 (sourceDepth 2 (1/2) (r^3*Real.log (n:ℝ))) := by
-      rw [totalScoreExponent_two]
-      have hDD : (D0:ℝ) ≤ (sourceDepth 2 (1/2) (r^3*Real.log (n:ℝ)):ℝ) := by exact_mod_cast hD
-      linarith only [hD0,hDD]
-    refine ⟨mul_nonneg (Real.rpow_nonneg (Nat.cast_nonneg n) _) (minTraceTV_nonneg_of_pair q x y hxy),?_⟩
-    apply (mul_le_mul_of_nonneg_left (minTraceTV_le_pair q x y hxy)
-      (Real.rpow_nonneg (Nat.cast_nonneg n) C)).trans
-    apply one_trace_power_bound hn1.le hC hd
-    linarith only [ht,he U]
-  exact squeeze_zero' (hb.mono fun _ h=>h.1) (hb.mono fun _ h=>h.2) (hn.const_div_atTop 3)
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open Filter
-open scoped Topology Classical ENNReal
-
-/-- Source Theorem 2's sample conclusion at the same fixed deletion channel. -/
-theorem superpolynomial_sample_complexity
-    (q : ℝ) (hq : q ∈ Set.Ioo 0 1) (s : ℝ) (hs : s ∈ Set.Ioc 0 1)
-    (C : ℝ) (hC : 0 < C) :
-    Tendsto
-      (fun n : ℕ => sampleComplexity n q s / ENNReal.ofReal ((n : ℝ) ^ C))
-      atTop (𝓝 ⊤) := by
-  let c : ℝ := 1/(8*Real.log 2)
-  have hl : 0 < Real.log 2 := Real.log_pos (by norm_num)
-  have hc0 : 0 < c := by dsimp [c]; positivity
-  have hc : c ∈ Set.Ioo 0 (1/(4*Real.log 2)) := by
-    refine ⟨hc0,?_⟩
-    dsimp [c]
-    apply one_div_lt_one_div_of_lt
-    · positivity
-    · nlinarith only [hl]
-  have hn : Tendsto (fun n : ℕ => (n:ℝ)) atTop atTop := tendsto_natCast_atTop_atTop
-  have hH : Tendsto (fun n : ℕ => q^3*Real.log (n:ℝ)) atTop atTop :=
-    (Real.tendsto_log_atTop.comp hn).const_mul_atTop (pow_pos hq.1 3)
-  have hb := quantitative_sample_lower_bound s c hs hc id (fun _ => q) tendsto_id (fun _ => hq) hH
-  have he := ((Real.tendsto_log_atTop.comp hH).const_mul_atTop hc0).eventually_ge_atTop (2*C+1)
-  have hbound : ∀ᶠ n : ℕ in atTop,(n:ℝ≥0∞) ≤
-      sampleComplexity n q s / ENNReal.ofReal ((n:ℝ)^C) := by
-    filter_upwards [hb,he,hn.eventually_gt_atTop 1] with n hb he hn1
-    dsimp only [Function.comp_apply] at he
-    have hn0 : 0 < (n:ℝ) := zero_lt_one.trans hn1
-    have hpow : (n:ℝ)^(C+1) ≤ (n:ℝ)^(c*Real.log (q^3*Real.log (n:ℝ))) :=
-      Real.rpow_le_rpow_of_exponent_le hn1.le (by linarith only [he,hC])
-    have hnC : 0 < (n:ℝ)^C := Real.rpow_pos_of_pos hn0 C
-    have hdiv : ENNReal.ofReal ((n:ℝ)^(C+1)) /
-        ENNReal.ofReal ((n:ℝ)^C) = (n:ℝ≥0∞) := by
-      rw [← ENNReal.ofReal_div_of_pos hnC,Real.rpow_add hn0,Real.rpow_one]
-      rw [mul_div_cancel_left₀ _ hnC.ne',ENNReal.ofReal_natCast]
-    rw [← hdiv]
-    exact ENNReal.div_le_div_right ((ENNReal.ofReal_le_ofReal hpow).trans hb) _
-  apply ENNReal.tendsto_nhds_top
-  intro k
-  filter_upwards [hbound,eventually_gt_atTop k] with n hn hk
-  exact (show (k:ℝ≥0∞) < (n:ℝ≥0∞) by exact_mod_cast hk).trans_le hn
-
-end TraceReconstruction
-
-end
-end
-end
-end
-end
-end
-end
-
-end OAI
+import OAI.Probability.TraceReconstruction.Induction
+import OAI.Probability.TraceReconstruction.Testing
+
+namespace OAI
+
+/-!
+# Bounds
+
+Restored construction and final scales; quantitative sample lower bound and both fixed-channel superpolynomial endpoints.
+-/
+
+open scoped ENNReal
+noncomputable section
+
+open Filter
+open scoped Topology
+
+
+noncomputable section
+namespace TraceReconstruction
+open Filter
+open scoped Topology Classical
+
+/-- The exact source error is negligible compared to the squared score energy. -/
+theorem source_final_error_eventually (B : ℕ) (ζ : ℝ) (hB : 2≤B) (hζ : ζ∈Set.Ioo 0 1)
+    {n q : ℕ→ℝ} (hn : Tendsto n atTop atTop) (hq : ∀t,q t∈Set.Ioc 0 (1/2:ℝ))
+    (hH : Tendsto (fun t=>(q t)^3*Real.log (n t)) atTop atTop) :
+    ∀ᶠt in atTop,∀ (U : ConcreteLevelPair (q t) ⟨(hq t).1.le,by linarith [(hq t).2]⟩ (n t)
+      ((n t)^(-100*(totalScoreExponent B (sourceDepth B ζ ((q t)^3*Real.log (n t)))+B+1)))
+      (totalScoreExponent B (sourceDepth B ζ ((q t)^3*Real.log (n t))))
+      (initialOverlapLower (q t))
+      ((sourceOverlapConstant (q t))^(2*B*sourceDepth B ζ ((q t)^3*Real.log (n t))+2))
+      (sourceDepth B ζ ((q t)^3*Real.log (n t)))
+      (sourceHalfMoment B (sourceDepth B ζ ((q t)^3*Real.log (n t))) (n t)
+        (sourceDepth B ζ ((q t)^3*Real.log (n t))))),
+      U.error≤(n t)^(-2*totalScoreExponent B (sourceDepth B ζ ((q t)^3*Real.log (n t)))) := by
+  have ht := SourceExpUpper.uniform (fun D _=>(D:ℝ)+1) SourcePolyUpper.depth.expUpper
+    B ζ 1 hB hζ zero_lt_one hn hq hH
+  filter_upwards [ht,hn.eventually_ge_atTop 2] with t ht hn2 U
+  let D := sourceDepth B ζ ((q t)^3*Real.log (n t))
+  let d := totalScoreExponent B D
+  have hn0 : 0<n t := by linarith only [hn2]
+  have hn1 : 1≤n t := by linarith only [hn2]
+  have hD : (D:ℝ)+1≤n t := by
+    have hh := ht D
+    rw [sourceTargetScale_top,one_mul] at hh
+    exact hh.trans (by simpa only [Real.rpow_one] using
+      Real.rpow_le_rpow_of_exponent_le hn1 (sourceEpsilon_le_one B D))
+  have hd : 11/5≤d := (totalScoreExponent_bounds B D hB).1
+  calc
+    U.error ≤ ((D:ℝ)+1)*(n t)^(-100*(d+B+1))*n t := U.error_le hn0 (Real.rpow_nonneg hn0.le _)
+    _ ≤ (n t)*(n t)^(-100*(d+B+1))*n t := by gcongr
+    _ = (n t)^(-100*(d+B+1)+2) := by
+      rw [Real.rpow_add hn0,Real.rpow_two]
+      ring
+    _ ≤ (n t)^(-2*d) := Real.rpow_le_rpow_of_exponent_le hn1 (by linarith [Nat.cast_nonneg B (α:=ℝ)])
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open Filter
+open scoped Topology Classical
+
+theorem exists_source_slope {c : ℝ} (hc : c<1/(4*Real.log 2)) :
+    ∃B : ℕ,2≤B ∧ ∃ζ : ℝ,ζ∈Set.Ioo 0 1 ∧ c<finalSlope B ζ := by
+  have hn : Tendsto (fun B : ℕ=>(B:ℝ)) atTop atTop := tendsto_natCast_atTop_atTop
+  have hn1 : Tendsto (fun B : ℕ=>(B:ℝ)+1) atTop atTop :=
+    tendsto_atTop_mono (fun B=>by linarith) hn
+  have h1 := hn1.const_div_atTop (1:ℝ)
+  have h2 := hn.const_div_atTop (3/10:ℝ)
+  have hlim := ((tendsto_const_nhds (x:=(1:ℝ))).sub h1).mul
+    ((tendsto_const_nhds (x:=(1/4:ℝ))).sub h2) |>.div_const (Real.log 2)
+  have hlim' : Tendsto (fun B : ℕ=>(1-1/((B:ℝ)+1))*((1/4:ℝ)-(3/10:ℝ)/(B:ℝ))/Real.log 2)
+      atTop (𝓝 (1/(4*Real.log 2))) := by
+    convert hlim using 1
+    ring_nf
+  obtain ⟨B,hB,hβ⟩ := ((eventually_ge_atTop 2).and (hlim'.eventually_const_lt hc)).exists
+  refine ⟨B,hB,1/((B:ℝ)+1),⟨by positivity,?_⟩,?_⟩
+  · have hBr : (2:ℝ)≤B := by exact_mod_cast hB
+    exact (div_lt_one (by positivity)).mpr (by linarith)
+  · have hBr : (B:ℝ)≠0 := by exact_mod_cast (show B≠0 by omega)
+    have he : finalSlope B (1/((B:ℝ)+1))=
+        (1-1/((B:ℝ)+1))*((1/4:ℝ)-(3/10:ℝ)/(B:ℝ))/Real.log 2 := by
+      unfold finalSlope blockGain
+      field_simp
+      ring
+    rw [he]
+    exact hβ
+
+/-- Constant-size subdivision and larger deletion preserve the joint regime.
+ The factor 16 includes the factor eight of the working deletion cube. -/
+theorem subdivided_working_scale {n q : ℕ→ℝ} (hn : Tendsto n atTop atTop)
+    (hq : ∀t,q t∈Set.Ioo 0 1)
+    (hH : Tendsto (fun t=>(q t)^3*Real.log (n t)) atTop atTop)
+    {G : ℝ} (hG : 0<G) :
+    Tendsto (fun t=>n t/G) atTop atTop ∧
+    Tendsto (fun t=>(workingDeletion (q t))^3*Real.log (n t/G)) atTop atTop ∧
+    ∀ᶠt in atTop,(q t)^3*Real.log (n t)/16 ≤
+      (workingDeletion (q t))^3*Real.log (n t/G) := by
+  have hN := hn.atTop_div_const hG
+  have hl := Real.tendsto_log_atTop.comp hn
+  have hh : ∀ᶠt in atTop,(q t)^3*Real.log (n t)/16 ≤
+      (workingDeletion (q t))^3*Real.log (n t/G) := by
+    filter_upwards [hn.eventually_ge_atTop 1,hl.eventually_ge_atTop (max 0 (2*Real.log G))] with t hn1 hl
+    have hn0 : 0<n t := by linarith only [hn1]
+    have hl0 : 0≤Real.log (n t) := (le_max_left _ _).trans hl
+    have hlg : 2*Real.log G≤Real.log (n t) := (le_max_right _ _).trans hl
+    have hhalf : Real.log (n t)/2≤Real.log (n t/G) := by
+      rw [Real.log_div hn0.ne' hG.ne']
+      linarith only [hlg]
+    have hb := (workingScale_bounds (hq t) hl0).1
+    have hm := mul_le_mul_of_nonneg_left hhalf (pow_nonneg (workingDeletion_bounds (hq t)).1.1.le 3)
+    nlinarith only [hb,hm]
+  exact ⟨hN,tendsto_atTop_mono' atTop hh (hH.atTop_div_const (by norm_num : (0:ℝ)<16)),hh⟩
+
+theorem workingDeletion_Ioc {q : ℝ} (hq : q∈Set.Ioo 0 1) :
+    workingDeletion q ∈Set.Ioc 0 (1/2:ℝ) :=
+  ⟨(workingDeletion_bounds hq).1.1,min_le_right _ _⟩
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open Filter
+open scoped Topology Classical
+
+/-- Full sharp slope restoration after fixed gadget packing. Both deletion
+ probability and length may vary; the working channel is only internal. -/
+theorem source_restored_exponent_margin (B : ℕ) (ζ c G a : ℝ)
+    (hB : 2≤B) (hζ : ζ∈Set.Ioo 0 1) (hc : 0<c) (hα : c<finalSlope B ζ)
+    (hG : 0<G) (ha : 0<a) {n q : ℕ→ℝ} (hn : Tendsto n atTop atTop)
+    (hq : ∀t,q t∈Set.Ioo 0 1)
+    (hH : Tendsto (fun t=>(q t)^3*Real.log (n t)) atTop atTop) :
+    ∀ᶠt in atTop,(n t)^(c*Real.log ((q t)^3*Real.log (n t))) ≤
+      a*(n t/G)^(totalScoreExponent B
+        (sourceDepth B ζ ((workingDeletion (q t))^3*Real.log (n t/G)))) := by
+  let α := finalSlope B ζ
+  have hα0 : 0<α := hc.trans hα
+  obtain ⟨ρ,hρc,hρ1⟩ := exists_between ((div_lt_one hα0).mpr hα)
+  have hρ0 : 0<ρ := (div_pos hc hα0).trans hρc
+  have hρc' : c<ρ*α := (div_lt_iff₀ hα0).mp hρc
+  let H := fun t=>(q t)^3*Real.log (n t)
+  let J := fun t=>(workingDeletion (q t))^3*Real.log (n t/G)
+  let d := fun t=>totalScoreExponent B (sourceDepth B ζ (J t))
+  let K := blockGain B+(warmupBlocks B:ℝ)*blockGain B+α*Real.log 16
+  have hsc := subdivided_working_scale hn hq hH hG
+  have hdepth := sourceDepth_tendsto B ζ hB hζ.2 hsc.2.1
+  have hd : ∀ᶠt in atTop,(ρ*α)*Real.log (H t)-ρ*K≤ρ*d t := by
+    filter_upwards [hsc.2.2,hH.eventually_gt_atTop 0,
+      hdepth.eventually_ge_atTop (warmupBlocks B)] with t hj hp hD
+    have hJ : 0<J t := lt_of_lt_of_le (div_pos hp (by norm_num)) hj
+    have hlog : Real.log (H t)-Real.log 16≤Real.log (J t) := by
+      have hh := Real.log_le_log (div_pos hp (by norm_num)) hj
+      simpa only [Real.log_div (ne_of_gt hp) (by norm_num : (16:ℝ)≠0)] using hh
+    have hs := (totalScoreExponent_slope_lower B ζ (J t) hB hD).le
+    have hm := mul_le_mul_of_nonneg_left hlog hα0.le
+    have hbase : α*Real.log (H t)-K≤d t := by
+      dsimp only [K,d,α] at *
+      linarith only [hs,hm]
+    have hh := mul_le_mul_of_nonneg_left hbase hρ0.le
+    nlinarith only [hh]
+  have hex := eventually_exponent_margin hn hH hρc' ha hd
+  have hsub := ((PowerRate.base hn).div_const hn hG).lower hn hρ1
+  filter_upwards [hex,hsub,hn.eventually_gt_atTop 0] with t he hs hp
+  apply he.trans
+  apply mul_le_mul_of_nonneg_left _ ha.le
+  rw [Real.rpow_mul hp.le]
+  exact Real.rpow_le_rpow (Real.rpow_nonneg hp.le _) hs
+    (by have hh := (totalScoreExponent_bounds B (sourceDepth B ζ (J t)) hB).1; dsimp [d]; linarith only [hh])
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open Filter
+open scoped Topology Classical ENNReal
+
+theorem sample_tv_small {s G N d E M : ℝ} (hs : s∈Set.Ioc 0 1)
+    (hG : 0<G) (hN : 1 ≤ N) (hd : 0 ≤ d) (hM : 0 ≤ M)
+    (hE : E ≤ N^(-2*d)) (hbudget : M ≤ (s^2/(64*G))*N^d) :
+    Real.sqrt (M*G*N^(-d))+2*M*G*E ≤ s/4 := by
+  have hN0 : 0<N := by linarith only [hN]
+  have hV0 : 0 ≤ N^(-d) := Real.rpow_nonneg hN0.le _
+  have hV1 : N^(-d) ≤ 1 := by
+    simpa only [Real.rpow_zero] using Real.rpow_le_rpow_of_exponent_le hN (neg_nonpos.mpr hd)
+  have hVV : N^(-2*d)=(N^(-d))^2 := by
+    rw [←Real.rpow_natCast,←Real.rpow_mul hN0.le]
+    norm_num only [Nat.cast_ofNat]
+    congr 1
+    ring
+  have hprod : N^d*N^(-d)=1 := by rw [←Real.rpow_add hN0,add_neg_cancel,Real.rpow_zero]
+  have hMV : M*G*N^(-d) ≤ s^2/64 := by
+    calc
+      _  ≤  ((s^2/(64*G))*N^d)*G*N^(-d) := by gcongr
+      _ = (s^2/(64*G)*G)*(N^d*N^(-d)) := by ring
+      _ = s^2/64 := by rw [hprod]; field_simp
+  have hsq : Real.sqrt (M*G*N^(-d)) ≤ s/8 := by
+    apply (Real.sqrt_le_left (div_nonneg hs.1.le (by norm_num) : 0 ≤ s/8)).mpr
+    nlinarith only [hMV]
+  have hE' : E ≤ N^(-d) := by
+    rw [hVV] at hE
+    nlinarith only [hE,mul_le_mul_of_nonneg_left hV1 hV0]
+  have he := mul_le_mul_of_nonneg_left hE' (show 0 ≤ 2*M*G by positivity)
+  have hss := mul_nonneg hs.1.le (sub_nonneg.mpr hs.2)
+  nlinarith only [hsq,he,hMV,hss]
+
+/-- Source Theorem 1: all fixed positive success levels, not only binary
+ testing above one half, with complete independent traces and known channel. -/
+theorem quantitative_sample_lower_bound
+    (s c : ℝ) (hs : s ∈ Set.Ioc 0 1)
+    (hc : c ∈ Set.Ioo 0 (1 / (4 * Real.log 2)))
+    (n : ℕ → ℕ) (q : ℕ → ℝ)
+    (hn : Tendsto n atTop atTop)
+    (hq : ∀ k, q k ∈ Set.Ioo 0 1)
+    (hscale : Tendsto (fun k => (q k) ^ 3 * Real.log (n k)) atTop atTop) :
+    ∀ᶠ k in atTop,
+      ENNReal.ofReal ((n k : ℝ) ^ (c * Real.log ((q k) ^ 3 * Real.log (n k))))  ≤ 
+        sampleComplexity (n k) (q k) s := by
+  obtain ⟨B,hB,ζ,hζ,hα⟩ := exists_source_slope hc.2
+  obtain ⟨g,hg⟩ := exists_nat_gt (max 1 (2/s))
+  have hg1 : 1<(g:ℝ) := (le_max_left _ _).trans_lt hg
+  have hg0 : 0<(g:ℝ) := by linarith only [hg1]
+  have hgs : 1/(2:ℝ)^g<s/2 := by
+    have hp : (g:ℝ)<(2:ℝ)^g := by
+      clear hg hg1 hg0
+      induction g with
+      | zero => norm_num
+      | succ g ih =>
+        push_cast
+        rw [pow_succ]
+        have hg : (0:ℝ) ≤ g := Nat.cast_nonneg g
+        have hp : (1:ℝ) ≤ 2^g := one_le_pow₀ (by norm_num)
+        nlinarith only [ih,hg,hp]
+    have hbig : 2/s<(2:ℝ)^g := ((le_max_right _ _).trans_lt hg).trans hp
+    have hh := (div_lt_iff₀ hs.1).mp hbig
+    apply (div_lt_iff₀ (by positivity : (0:ℝ)<2^g)).mpr
+    nlinarith only [hh]
+  have hnr : Tendsto (fun t=>(n t:ℝ)) atTop atTop := tendsto_natCast_atTop_atTop.comp hn
+  let N := fun t=>(n t:ℝ)/(g:ℝ)
+  let r := fun t=>workingDeletion (q t)
+  have hr (t) : r t∈Set.Ioc 0 (1/2:ℝ) := workingDeletion_Ioc (hq t)
+  have hsc := subdivided_working_scale hnr hq hscale hg0
+  have hpair := source_final_pair_eventually B ζ hB hζ hsc.1 hr hsc.2.1
+  have herr := source_final_error_eventually B ζ hB hζ hsc.1 hr hsc.2.1
+  have hmargin := source_restored_exponent_margin B ζ c (g:ℝ) (s^2/(64*g))
+    hB hζ hc.1 hα hg0 (div_pos (sq_pos_of_pos hs.1) (mul_pos (by norm_num) hg0)) hnr hq hscale
+  filter_upwards [hpair,herr,hmargin,hsc.1.eventually_ge_atTop 2] with t hp he hm hn2
+  obtain ⟨U⟩ := hp
+  have hN1 : 1 ≤ N t := by dsimp [N]; linarith only [hn2]
+  have hN2 : 1<N t := by dsimp [N]; linarith only [hn2]
+  have hk := (sourceHalfMoment_bounds B
+    (sourceDepth B ζ ((r t)^3*Real.log (N t)))
+    (sourceDepth B ζ ((r t)^3*Real.log (N t))) hB hN2).1
+  obtain ⟨W,hW,hTV⟩ := U.restored_family (by have hh := (hr t).2; dsimp [r] at *; linarith only [hh])
+    ⟨(hq t).1.le,(hq t).2.le⟩ (workingDeletion_bounds (hq t)).2.2 hk g (n t)
+    (by field_simp; exact le_rfl)
+  apply le_sInf
+  rintro z ⟨m,hbudget,rfl⟩
+  rw [← ENNReal.ofReal_natCast]
+  apply ENNReal.ofReal_le_ofReal
+  by_contra hnot
+  have hm' : (m:ℝ) ≤ (s^2/(64*g))*(N t)^(
+      totalScoreExponent B (sourceDepth B ζ ((r t)^3*Real.log (N t)))) :=
+    (le_of_lt (lt_of_not_ge hnot)).trans hm
+  have hclose (θ : Fin g→Bool) := (hTV θ m).trans
+    (sample_tv_small hs hg0 hN1
+      (by have hd := (totalScoreExponent_bounds B (sourceDepth B ζ ((r t)^3*Real.log (N t))) hB).1; linarith only [hd])
+      (Nat.cast_nonneg m) (he U) hm')
+  have hbad := family_testing_bound ⟨(hq t).1.le,(hq t).2.le⟩ W hW hclose hbudget
+  linarith only [hbad,hgs,hs.1]
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open Filter
+open scoped Topology Classical ENNReal
+
+namespace FiniteLaw
+variable {α : Type*} [Fintype α]
+theorem tv_product_one (P Q : FiniteLaw α) : (P.product 1).tv (Q.product 1)=P.tv Q := by
+  rw [show 1=0+1 from rfl,tv_product_succ]
+  simp only [tv,tensor,product,Fintype.sum_prod_type,Fintype.prod_empty,mul_one,Fintype.sum_unique]
+end FiniteLaw
+
+theorem totalScoreExponent_two (D : ℕ) : totalScoreExponent 2 D=11/5+(D:ℝ)/5 := by
+  simp only [totalScoreExponent,ite_self,blockGain_two,Finset.sum_const,Finset.card_range,nsmul_eq_mul]
+  ring
+
+theorem minTraceTV_nonneg_of_pair {n : ℕ} (q : ℝ) (x y : Word n) (hxy : x≠y) :
+    0 ≤ minTraceTV n q := by
+  apply le_csInf (show {t | ∃x y : Word n,x≠y ∧ t=traceTV q x y}.Nonempty from
+    ⟨_,x,y,hxy,rfl⟩)
+  rintro b ⟨u,v,_,rfl⟩
+  exact traceTV_nonneg q u v
+
+theorem minTraceTV_le_pair {n : ℕ} (q : ℝ) (x y : Word n) (hxy : x≠y) :
+    minTraceTV n q ≤ traceTV q x y := by
+  apply csInf_le (show BddBelow {t | ∃x y : Word n,x≠y ∧ t=traceTV q x y} from ?_) ⟨x,y,hxy,rfl⟩
+  exact ⟨0,by rintro b ⟨u,v,_,rfl⟩; exact traceTV_nonneg q u v⟩
+
+theorem one_trace_power_bound {N d C R : ℝ} (hN : 1 ≤ N) (hC : 0<C)
+    (hd : 2*(C+1) ≤ d) (hR : R ≤ Real.sqrt (N^(-d))+2*N^(-2*d)) :
+    N^C*R ≤ 3/N := by
+  have hN0 : 0<N := by linarith only [hN]
+  have he : (N^(-(C+1)))^2=N^(-2*(C+1)) := by
+    rw [←Real.rpow_natCast,←Real.rpow_mul hN0.le]
+    norm_num only [Nat.cast_ofNat]
+    congr 1
+    ring
+  have hs : Real.sqrt (N^(-d)) ≤ N^(-(C+1)) := by
+    apply (Real.sqrt_le_left (Real.rpow_nonneg hN0.le _)).mpr
+    rw [he]
+    exact Real.rpow_le_rpow_of_exponent_le hN (by linarith only [hd])
+  have herr : N^(-2*d) ≤ N^(-(C+1)) :=
+    Real.rpow_le_rpow_of_exponent_le hN (by linarith only [hd,hC])
+  calc
+    N^C*R ≤ N^C*(3*N^(-(C+1))) :=
+      mul_le_mul_of_nonneg_left (by linarith only [hR,hs,herr]) (Real.rpow_nonneg hN0.le _)
+    _ = 3/N := by
+      rw [mul_left_comm,←Real.rpow_add hN0]
+      rw [show C+ -(C+1)=(-1:ℝ) by ring,Real.rpow_neg_one]
+      ring
+
+/-- Source Theorem 2, first conclusion, over all integer lengths and at the
+ same fixed channel for every power. -/
+theorem superpolynomial_one_trace
+    (q : ℝ) (hq : q ∈ Set.Ioo 0 1) (C : ℝ) (hC : 0<C) :
+    Tendsto (fun n : ℕ=>(n:ℝ)^C*minTraceTV n q) atTop (𝓝 0) := by
+  let r := workingDeletion q
+  have hr : r∈Set.Ioc 0 (1/2:ℝ) := workingDeletion_Ioc hq
+  have hn : Tendsto (fun n : ℕ=>(n:ℝ)) atTop atTop := tendsto_natCast_atTop_atTop
+  have hJ : Tendsto (fun n : ℕ=>r^3*Real.log (n:ℝ)) atTop atTop :=
+    (Real.tendsto_log_atTop.comp hn).const_mul_atTop (pow_pos hr.1 3)
+  have hpair := source_final_pair_eventually 2 (1/2) (by omega) (by norm_num) hn (fun _=>hr) hJ
+  have herr := source_final_error_eventually 2 (1/2) (by omega) (by norm_num) hn (fun _=>hr) hJ
+  obtain ⟨D0,hD0⟩ := exists_nat_ge (5*(2*(C+1)))
+  have hdepth := (sourceDepth_tendsto 2 (1/2) (by omega) (by norm_num) hJ).eventually_ge_atTop D0
+  have hb : ∀ᶠ n : ℕ in atTop,0 ≤ (n:ℝ)^C*minTraceTV n q ∧
+      (n:ℝ)^C*minTraceTV n q ≤ 3/(n:ℝ) := by
+    filter_upwards [hpair,herr,hdepth,hn.eventually_ge_atTop 2] with n hp he hD hn2
+    obtain ⟨U⟩ := hp
+    have hn1 : 1<(n:ℝ) := by linarith only [hn2]
+    have hk := (sourceHalfMoment_bounds 2
+      (sourceDepth 2 (1/2) (r^3*Real.log (n:ℝ)))
+      (sourceDepth 2 (1/2) (r^3*Real.log (n:ℝ))) (by omega) hn1).1
+    obtain ⟨x,y,hxy,hTV⟩ := U.restored_words (by linarith only [hr.2])
+      ⟨hq.1.le,hq.2.le⟩ (workingDeletion_bounds hq).2.2 hk n le_rfl
+    have ht := hTV 1
+    rw [FiniteLaw.tv_product_one] at ht
+    rw [completeTraceLaw_tv q ⟨hq.1.le,hq.2.le⟩ x y] at ht
+    norm_num only [Nat.cast_one,one_mul,mul_one] at ht
+    have hd : 2*(C+1) ≤ totalScoreExponent 2 (sourceDepth 2 (1/2) (r^3*Real.log (n:ℝ))) := by
+      rw [totalScoreExponent_two]
+      have hDD : (D0:ℝ) ≤ (sourceDepth 2 (1/2) (r^3*Real.log (n:ℝ)):ℝ) := by exact_mod_cast hD
+      linarith only [hD0,hDD]
+    refine ⟨mul_nonneg (Real.rpow_nonneg (Nat.cast_nonneg n) _) (minTraceTV_nonneg_of_pair q x y hxy),?_⟩
+    apply (mul_le_mul_of_nonneg_left (minTraceTV_le_pair q x y hxy)
+      (Real.rpow_nonneg (Nat.cast_nonneg n) C)).trans
+    apply one_trace_power_bound hn1.le hC hd
+    linarith only [ht,he U]
+  exact squeeze_zero' (hb.mono fun _ h=>h.1) (hb.mono fun _ h=>h.2) (hn.const_div_atTop 3)
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open Filter
+open scoped Topology Classical ENNReal
+
+/-- Source Theorem 2's sample conclusion at the same fixed deletion channel. -/
+theorem superpolynomial_sample_complexity
+    (q : ℝ) (hq : q ∈ Set.Ioo 0 1) (s : ℝ) (hs : s ∈ Set.Ioc 0 1)
+    (C : ℝ) (hC : 0 < C) :
+    Tendsto
+      (fun n : ℕ => sampleComplexity n q s / ENNReal.ofReal ((n : ℝ) ^ C))
+      atTop (𝓝 ⊤) := by
+  let c : ℝ := 1/(8*Real.log 2)
+  have hl : 0 < Real.log 2 := Real.log_pos (by norm_num)
+  have hc0 : 0 < c := by dsimp [c]; positivity
+  have hc : c ∈ Set.Ioo 0 (1/(4*Real.log 2)) := by
+    refine ⟨hc0,?_⟩
+    dsimp [c]
+    apply one_div_lt_one_div_of_lt
+    · positivity
+    · nlinarith only [hl]
+  have hn : Tendsto (fun n : ℕ => (n:ℝ)) atTop atTop := tendsto_natCast_atTop_atTop
+  have hH : Tendsto (fun n : ℕ => q^3*Real.log (n:ℝ)) atTop atTop :=
+    (Real.tendsto_log_atTop.comp hn).const_mul_atTop (pow_pos hq.1 3)
+  have hb := quantitative_sample_lower_bound s c hs hc id (fun _ => q) tendsto_id (fun _ => hq) hH
+  have he := ((Real.tendsto_log_atTop.comp hH).const_mul_atTop hc0).eventually_ge_atTop (2*C+1)
+  have hbound : ∀ᶠ n : ℕ in atTop,(n:ℝ≥0∞) ≤
+      sampleComplexity n q s / ENNReal.ofReal ((n:ℝ)^C) := by
+    filter_upwards [hb,he,hn.eventually_gt_atTop 1] with n hb he hn1
+    dsimp only [Function.comp_apply] at he
+    have hn0 : 0 < (n:ℝ) := zero_lt_one.trans hn1
+    have hpow : (n:ℝ)^(C+1) ≤ (n:ℝ)^(c*Real.log (q^3*Real.log (n:ℝ))) :=
+      Real.rpow_le_rpow_of_exponent_le hn1.le (by linarith only [he,hC])
+    have hnC : 0 < (n:ℝ)^C := Real.rpow_pos_of_pos hn0 C
+    have hdiv : ENNReal.ofReal ((n:ℝ)^(C+1)) /
+        ENNReal.ofReal ((n:ℝ)^C) = (n:ℝ≥0∞) := by
+      rw [← ENNReal.ofReal_div_of_pos hnC,Real.rpow_add hn0,Real.rpow_one]
+      rw [mul_div_cancel_left₀ _ hnC.ne',ENNReal.ofReal_natCast]
+    rw [← hdiv]
+    exact ENNReal.div_le_div_right ((ENNReal.ofReal_le_ofReal hpow).trans hb) _
+  apply ENNReal.tendsto_nhds_top
+  intro k
+  filter_upwards [hbound,eventually_gt_atTop k] with n hn hk
+  exact (show (k:ℝ≥0∞) < (n:ℝ≥0∞) by exact_mod_cast hk).trans_le hn
+
+end TraceReconstruction
+
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/TraceReconstruction/Amplification.lean b/lean/OAI/Probability/TraceReconstruction/Amplification.lean
index 2ba8dc8e8c98d0c8dc6734b7f40b112cad2c178b..a85f2d14c9b837bbc4dc5bb74a0683db18f61c6e 100644
--- a/lean/OAI/Probability/TraceReconstruction/Amplification.lean
+++ b/lean/OAI/Probability/TraceReconstruction/Amplification.lean
@@ -1,828 +1,828 @@
-import OAI.Probability.TraceReconstruction.SourceOracle
-
-namespace OAI
-
-/-!
-# Amplification
-
-Oracle products, mixed powers, padding, warmup and nested template Hellinger estimates, overlap, and block assembly.
--/
-
-open scoped ENNReal
-noncomputable section
-
-open Filter
-open scoped Topology
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- The literal product/concatenation decoder charges only coordinates at
- which the two deterministic input templates differ. -/
-theorem OracleTracePair.mixed_tv {q E V : ℝ} {x y : List Bool} {k n : ℕ}
-    (C : OracleTracePair q x y k E V) (hq : q∈Set.Icc 0 1) (hk : 1≤k)
-    (τ σ : Fin n → Bool) (M : ℕ)
-    (hτ : (coordinateWord (fun i => if τ i then y else x)).length≤M)
-    (hσ : (coordinateWord (fun i => if σ i then y else x)).length≤M) :
-    (listTraceLaw q hq (coordinateWord (fun i => if τ i then y else x)) M hτ).tv
-      (listTraceLaw q hq (coordinateWord (fun i => if σ i then y else x)) M hσ) ≤
-      Real.sqrt ((∑i,if τ i=σ i then (0:ℝ) else 1)*V)+2*(n:ℝ)*E := by
-  let := C.finiteOutcome
-  have ha (ρ : Fin n → Bool) : ApproximatesTrace q
-      (FiniteLaw.coordinateProduct (fun i => if ρ i then C.second else C.first))
-      (fun z => coordinateWord (fun i => C.decode (z i)))
-      (coordinateWord (fun i => if ρ i then y else x)) ((n:ℝ)*E) := by
-    have H := coordinateProduct_approximates hq
-      (fun i => if ρ i then C.second else C.first) (fun _ => C.decode)
-      (fun i => if ρ i then y else x) (fun _ => E) (fun i => ?_)
-    · simpa only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul] using H
-    · cases ρ i <;> simp only [Bool.false_eq_true,ite_false,ite_true]
-      · exact C.first_simulation
-      · exact C.second_simulation
-  have H := approximations_trace_tv hq _ _ _ _ _ _ _ M hτ hσ (ha τ) (ha σ)
-  have htv := FiniteLaw.tv_mixedProduct_le C.first C.second C.score C.density k hk C.moment τ σ
-  linarith only [H,htv]
-
-theorem coordinateWord_constant (w : List Bool) (n : ℕ) :
-    coordinateWord (fun _ : Fin n => w)=pureAtomRun w n := by
-  induction n with
-  | zero => simp [coordinateWord,pureAtomRun]
-  | succ n ih =>
-    rw [coordinateWord_succ,ih]
-    simp only [pureAtomRun,List.replicate_succ,List.flatten_cons]
-
-/-- Tensorization of a genuine approximate oracle, on actual word powers. -/
-theorem OracleTracePair.power_tv {q E V : ℝ} {x y : List Bool} {k : ℕ}
-    (C : OracleTracePair q x y k E V) (hq : q∈Set.Icc 0 1) (hk : 1≤k)
-    (n M : ℕ) (hx : (pureAtomRun x n).length≤M) (hy : (pureAtomRun y n).length≤M) :
-    (listTraceLaw q hq (pureAtomRun x n) M hx).tv
-      (listTraceLaw q hq (pureAtomRun y n) M hy) ≤
-      Real.sqrt ((n:ℝ)*V)+2*(n:ℝ)*E := by
-  have H := C.mixed_tv hq hk (fun _ : Fin n => false) (fun _ => true) M
-    (by simpa only [Bool.false_eq_true,ite_false,coordinateWord_constant] using hx)
-    (by simpa only [ite_true,coordinateWord_constant] using hy)
-  simpa only [Bool.false_eq_true,ite_false,ite_true,coordinateWord_constant,
-    Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,mul_one] using H
-
-/-- Genuine lifting for complete traces, with no auxiliary overlap hypothesis. -/
-theorem listTraceOverlap_lift {F k R : ℕ} {C p q : ℝ}
-    (hk : 1≤k) (hR : 1≤R) (hp : p∈Set.Ioc 0 (1/8)) (hq : q∈Set.Icc 0 1)
-    (hC : 1073741824≤p*C) (hF : C*k*R≤F) (w : List Bool)
-    (hov : p≤listTraceOverlap q hq (pureAtomRun w k) (pureAtomRun w (k+1))) :
-    (2/3:ℝ)≤listTraceOverlap q hq (pureAtomRun w (F*R)) (pureAtomRun w (F*R+R)) := by
-  let M := max (pureAtomRun w k).length (pureAtomRun w (k+1)).length
-  have H := adjacent_power_lift hk hR hp hq hC hF
-    (listTraceLaw q hq (pureAtomRun w k) M (le_max_left _ _))
-    (listTraceLaw q hq (pureAtomRun w (k+1)) M (le_max_right _ _)) Subtype.val w
-    (listTraceLaw_simulates q hq _ M _) (listTraceLaw_simulates q hq _ M _) hov
-  obtain ⟨β,fin,P,Q,g,hP,hQ,hO⟩ := H
-  let := fin
-  exact hO.trans (simulations_overlap_le hq P Q g _ _ hP hQ)
-
-theorem pureAtomRun_mul (w : List Bool) (n m : ℕ) :
-    pureAtomRun (pureAtomRun w n) m=pureAtomRun w (m*n) := by
-  induction m with
-  | zero => simp [pureAtomRun]
-  | succ m ih =>
-    have H : pureAtomRun (pureAtomRun w n) (m+1)=
-        pureAtomRun w n++pureAtomRun (pureAtomRun w n) m := by
-      simp [pureAtomRun,List.replicate_succ]
-    rw [H,ih,←pureAtomRun_add]
-    congr 1
-    ring
-
-/-- Quantitative overlap transfer used for the initial pure comparison and
- the small-dimensional warmup refinement. All three laws are ordinary traces. -/
-theorem OracleTracePair.lifted_power_overlap {q E V p C₀ : ℝ} {w z : List Bool}
-    {k h P F : ℕ} (O : OracleTracePair q (pureAtomRun w P) z k E V)
-    (hq : q∈Set.Icc 0 1) (hk : 1≤k) (hh : 1≤h) (hP : 1≤P)
-    (hp : p∈Set.Ioc 0 (1/8)) (hC : 1073741824≤p*C₀) (hF : C₀*h*P≤F)
-    (hov : p≤listTraceOverlap q hq (pureAtomRun w h) (pureAtomRun w (h+1))) :
-    (2/3:ℝ)-(Real.sqrt ((F:ℝ)*V)+2*(F:ℝ)*E)-
-      (Real.sqrt (((F+1:ℕ):ℝ)*V)+2*((F+1:ℕ):ℝ)*E) ≤
-      listTraceOverlap q hq (pureAtomRun z F) (pureAtomRun z (F+1)) := by
-  let a := pureAtomRun (pureAtomRun w P) F
-  let b := pureAtomRun (pureAtomRun w P) (F+1)
-  let x := pureAtomRun z F
-  let y := pureAtomRun z (F+1)
-  let M := max (max a.length b.length) (max x.length y.length)
-  have ha : a.length≤M := (le_max_left _ _).trans (le_max_left _ _)
-  have hb : b.length≤M := (le_max_right _ _).trans (le_max_left _ _)
-  have hx : x.length≤M := (le_max_left _ _).trans (le_max_right _ _)
-  have hy : y.length≤M := (le_max_right _ _).trans (le_max_right _ _)
-  have ht1 := O.power_tv hq hk F M ha hx
-  have ht2 := O.power_tv hq hk (F+1) M hb hy
-  have hol := listTraceOverlap_lift hh hP hp hq hC hF w hov
-  have he : F*P+P=(F+1)*P := by ring
-  have heA : pureAtomRun w (F*P)=a := (pureAtomRun_mul w P F).symm
-  have heB : pureAtomRun w (F*P+P)=b := by
-    rw [he]; exact (pureAtomRun_mul w P (F+1)).symm
-  rw [heA,heB] at hol
-  rw [listTraceOverlap_ambient hq _ _ M ha hb,FiniteLaw.overlap_eq_one_sub_tv] at hol
-  rw [listTraceOverlap_ambient hq _ _ M hx hy,FiniteLaw.overlap_eq_one_sub_tv]
-  have H := (FiniteLaw.tv_triangle (listTraceLaw q hq x M hx) (listTraceLaw q hq a M ha)
-    (listTraceLaw q hq y M hy)).trans
-    (add_le_add le_rfl (FiniteLaw.tv_triangle (listTraceLaw q hq a M ha)
-      (listTraceLaw q hq b M hb) (listTraceLaw q hq y M hy)))
-  rw [FiniteLaw.tv_symm (listTraceLaw q hq x M hx) (listTraceLaw q hq a M ha)] at H
-  linarith only [H,ht1,ht2,hol]
-
-end TraceReconstruction
-
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-namespace FiniteLaw
-variable {α : Type*} [Fintype α]
-
-theorem hellingerSq_mixedProduct_le {n : ℕ} (P Q : FiniteLaw α) {V : ℝ}
-    (hH : P.hellingerSq Q ≤ V) (τ σ : Fin n → Bool) :
-    (coordinateProduct (fun i => if τ i then Q else P)).hellingerSq
-      (coordinateProduct (fun i => if σ i then Q else P)) ≤
-      (∑i,if τ i=σ i then (0:ℝ) else 1)*V := by
-  apply (hellingerSq_coordinateProduct_le _ _).trans
-  rw [Finset.sum_mul]
-  apply Finset.sum_le_sum
-  intro i _
-  cases τ i <;> cases σ i <;> simp only [Bool.false_eq_true,Bool.true_eq_false,
-    ite_false,ite_true,zero_mul,one_mul,hellingerSq_self,le_refl]
-  · exact hH
-  · rw [hellingerSq_symm]; exact hH
-
-end FiniteLaw
-
-theorem ApproximatesTrace.power {α : Type*} [Fintype α] {q E : ℝ} {P : FiniteLaw α}
-    {g : α → List Bool} {w : List Bool} (H : ApproximatesTrace q P g w E)
-    (hq : q∈Set.Icc 0 1) (F : ℕ) :
-    ApproximatesTrace q (P.product F) (fun z => coordinateWord (fun i => g (z i)))
-      (pureAtomRun w F) ((F:ℝ)*E) := by
-  have h := coordinateProduct_approximates hq (fun _ : Fin F => P) (fun _ => g)
-    (fun _ => w) (fun _ => E) (fun _ => H)
-  have he : FiniteLaw.coordinateProduct (fun _ : Fin F => P)=P.product F := rfl
-  rw [he,coordinateWord_constant] at h
-  simpa only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul] using h
-
-/-- Repeated mixed templates pay only for the differing leaves in squared
- Hellinger, while approximate leaf simulation is charged linearly. -/
-theorem OracleTracePair.mixed_power_tv {q E V : ℝ} {x y : List Bool} {k n : ℕ}
-    (C : OracleTracePair q x y k E V) (hq : q∈Set.Icc 0 1) (hk : 1 ≤ k)
-    (τ σ : Fin n → Bool) (F M : ℕ)
-    (hτ : (pureAtomRun (coordinateWord (fun i => if τ i then y else x)) F).length ≤ M)
-    (hσ : (pureAtomRun (coordinateWord (fun i => if σ i then y else x)) F).length ≤ M) :
-    (listTraceLaw q hq (pureAtomRun (coordinateWord (fun i => if τ i then y else x)) F) M hτ).tv
-      (listTraceLaw q hq (pureAtomRun (coordinateWord (fun i => if σ i then y else x)) F) M hσ) ≤
-      Real.sqrt ((F:ℝ)*(∑i,if τ i=σ i then (0:ℝ) else 1)*V)+2*(F:ℝ)*(n:ℝ)*E := by
-  let := C.finiteOutcome
-  let A (ρ : Fin n → Bool) := FiniteLaw.coordinateProduct (fun i => if ρ i then C.second else C.first)
-  let g (z : Fin n → C.Outcome) := coordinateWord (fun i => C.decode (z i))
-  have ha (ρ : Fin n → Bool) : ApproximatesTrace q (A ρ) g
-      (coordinateWord (fun i => if ρ i then y else x)) ((n:ℝ)*E) := by
-    have H := coordinateProduct_approximates hq
-      (fun i => if ρ i then C.second else C.first) (fun _ => C.decode)
-      (fun i => if ρ i then y else x) (fun _ => E) (fun i => ?_)
-    · simpa only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul] using H
-    · cases ρ i <;> simp only [Bool.false_eq_true,ite_false,ite_true]
-      · exact C.first_simulation
-      · exact C.second_simulation
-  have hH : C.first.hellingerSq C.second ≤ V :=
-    (C.first.hellingerSq_le_score C.second C.score C.density).trans
-      ((C.first.secondMoment_le_evenMomentSq C.score k hk).trans C.moment)
-  have hm := FiniteLaw.hellingerSq_mixedProduct_le C.first C.second hH τ σ
-  have hp := (FiniteLaw.hellingerSq_product_le (A τ) (A σ) F).trans
-    (mul_le_mul_of_nonneg_left hm (Nat.cast_nonneg F))
-  have htv := (FiniteLaw.tv_le_sqrt_hellingerSq ((A τ).product F) ((A σ).product F)).trans
-    (Real.sqrt_le_sqrt hp)
-  have H := approximations_trace_tv hq _ _ _ _ _ _ _ M hτ hσ
-    ((ha τ).power hq F) ((ha σ).power hq F)
-  rw [←mul_assoc] at htv
-  linarith only [H,htv]
-
-end TraceReconstruction
-
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- An exact common prefix and suffix are observed as ordinary independent
- complete traces. They neither change the score moment nor cost simulation
- error. This is the source's deterministic residual convention. -/
-def OracleTracePair.pad {q E V : ℝ} {x y : List Bool} {k : ℕ}
-    (C : OracleTracePair q x y k E V) (hq : q∈Set.Icc 0 1) (u v : List Bool) :
-    OracleTracePair q (u++x++v) (u++y++v) k E V := by
-  let := C.finiteOutcome
-  let P := listTraceLaw q hq u u.length le_rfl
-  let Q := listTraceLaw q hq v v.length le_rfl
-  refine {
-    Outcome := (BoundedTrace u.length × C.Outcome) × BoundedTrace v.length
-    finiteOutcome := inferInstance
-    first := (P.prod C.first).prod Q
-    second := (P.prod C.second).prod Q
-    decode := fun z => z.1.1.val++C.decode z.1.2++z.2.val
-    score := fun z => C.score z.1.2
-    density := by
-      intro z
-      change (P.mass z.1.1*C.second.mass z.1.2)*Q.mass z.2=_
-      rw [C.density]
-      change _=(P.mass z.1.1*C.first.mass z.1.2)*Q.mass z.2*(1+C.score z.1.2)
-      ring
-    likelihood_nonneg := fun z => C.likelihood_nonneg z.1.2
-    centered := by
-      simp only [FiniteLaw.expect_prod,FiniteLaw.expect_const,C.centered]
-    moment := by
-      simpa only [FiniteLaw.evenMomentSq,FiniteLaw.expect_prod,FiniteLaw.expect_const] using C.moment
-    first_simulation := by
-      have H := ((listTraceLaw_simulates q hq u u.length le_rfl).approximate.append hq
-        C.first_simulation).append hq (listTraceLaw_simulates q hq v v.length le_rfl).approximate
-      simpa only [zero_add,add_zero] using H
-    second_simulation := by
-      have H := ((listTraceLaw_simulates q hq u u.length le_rfl).approximate.append hq
-        C.second_simulation).append hq (listTraceLaw_simulates q hq v v.length le_rfl).approximate
-      simpa only [zero_add,add_zero] using H }
-
-/-- Literal template equality under deterministic pure residual padding. -/
-theorem complementaryWords_pad (L P R a len : ℕ) (sign : Fin len → Bool)
-    (w : Bool → List Bool) (τ : Bool) (hfit : a+len ≤ P) :
-    complementaryWords (L+P+R) (L+a) len sign w τ =
-      pureAtomRun (w false) L++complementaryWords P a len sign w τ++pureAtomRun (w false) R := by
-  unfold complementaryWords
-  rw [atomString_three]
-  have hp (i : Fin L) : templateType (L+P+R) (L+a) len sign τ ⟨i.val,by omega⟩=false := by
-    have hi := i.isLt
-    simp only [templateType,dite_eq_right (by omega : ¬(L+a ≤ i.val ∧ i.val < L+a+len))]
-  have hm (i : Fin P) : templateType (L+P+R) (L+a) len sign τ ⟨L+i.val,by omega⟩=
-      templateType P a len sign τ i := by
-    unfold templateType
-    split_ifs with h h' h'
-    · congr 2
-      apply Fin.ext
-      simp only
-      omega
-    · simp only at h
-      omega
-    · simp only at h
-      omega
-    · rfl
-  have hr (i : Fin R) : templateType (L+P+R) (L+a) len sign τ ⟨L+P+i.val,by omega⟩=false := by
-    simp only [templateType,dite_eq_right (by omega : ¬(L+a ≤ L+P+i.val ∧ L+P+i.val < L+a+len))]
-  simp_rw [hp,hm,hr]
-  change coordinateWord (fun _ : Fin L => w false)++_++coordinateWord (fun _ : Fin R => w false)=_
-  rw [coordinateWord_constant,coordinateWord_constant]
-
-def centeredLeftResidual (N P : ℕ) : ℕ := N/2-1-(P/2-1)
-def centeredRightResidual (N P : ℕ) : ℕ := N-(centeredLeftResidual N P+P)
-
-theorem centeredResidual_geometry {N P : ℕ} (hP : 2 ≤ P) (hN : P ≤ N) :
-    centeredLeftResidual N P+P+centeredRightResidual N P=N ∧
-    centeredLeftResidual N P+(P/2-1)=N/2-1 := by
-  simp only [centeredRightResidual,centeredLeftResidual]
-  omega
-
-/-- The auxiliary one-window oracle is centered at the same actual source
- motif after appending pure residuals, also for odd ambient atom counts. -/
-theorem centeredComplementaryWords_pad (N P s : ℕ) (w : Bool → List Bool) (τ : Bool)
-    (hP : 2 ≤ P) (hN : P ≤ N) (hfit : P/2-1+2^s ≤ P) :
-    complementaryWords N (N/2-1) (2^s) (prouhetBool s) w τ =
-      pureAtomRun (w false) (centeredLeftResidual N P)++
-      complementaryWords P (P/2-1) (2^s) (prouhetBool s) w τ++
-      pureAtomRun (w false) (centeredRightResidual N P) := by
-  have h := complementaryWords_pad (centeredLeftResidual N P) P (centeredRightResidual N P)
-    (P/2-1) (2^s) (prouhetBool s) w τ hfit
-  rw [(centeredResidual_geometry hP hN).1,(centeredResidual_geometry hP hN).2] at h
-  exact h
-
-end TraceReconstruction
-
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- The source's special one-window warmup observation. The ambient word
- need not have the root's arithmetic divisibility: exact pure residuals
- restore its literal center, at no additional statistical error. -/
-theorem actual_one_window_oracle {q E V : ℝ} {x y : List Bool} {k : ℕ}
-    (C : OracleTracePair q x y k E V) (hq : q∈Set.Icc 0 1) (hE : 0 ≤ E)
-    (s N T t h R : ℕ) (hT : 24 ≤ T) (he : Even T)
-    (hMotif : 24*2^s ≤ T) (hroot : 4*(t+1) ≤ T)
-    (hfit : t*(2*h+1)+T ≤ N)
-    (ω a η L : ℝ) (ha : 0 < a) (hL : 0 < L) (hω : ω∈Set.Ioo 0 1)
-    (hov : ω ≤ listTraceOverlap q hq (pureAtomRun x h) (pureAtomRun x (h+1)))
-    (hk : 1 ≤ k) (hV : V ≤ L^2)
-    (hscale : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount (fun _ => T) s 0 ≤ 1/2)
-    (hη : hiddenSmoothingBound t s ω ≤ η)
-    (herror : sourceClippingCost (fun _ => T) (fun _ => t) s 0 k ω a L ≤ 1/2) :
-    Nonempty (OracleTracePair q
-      (complementaryWords N (N/2-1) (2^s) (prouhetBool s) (fun τ => if τ then y else x) false)
-      (complementaryWords N (N/2-1) (2^s) (prouhetBool s) (fun τ => if τ then y else x) true) k
-      (sourceClippingCost (fun _ => T) (fun _ => t) s 0 k ω a L+(T:ℝ)*E)
-      (64*sourceSharpEnergyBudget (fun _ => T) (fun _ => t) s 0 R ω η a ((6*(2*k:ℕ):ℝ)*L))) := by
-  let := C.finiteOutcome
-  let p := t*(2*h+1)+T
-  let atom : Bool → List Bool := fun τ => if τ then y else x
-  have H : SourceLayoutMargins s (fun _ => p) (fun _ => T) (fun _ => t) (fun _ => h) 0 := by
-    constructor <;> intro j hj <;> omega
-  have hm : C.first.expect (fun z => |C.score z|^(2*k)) ≤ L^(2*k) := by
-    have hv := C.first.expect_even_pow_le_of_moment C.score k hk (C.moment.trans hV)
-    simpa only [pow_mul] using hv
-  obtain ⟨W⟩ := actual_source_oracle hq hE s (fun _ => p) (fun _ => T) (fun _ => t)
-    (fun _ => h) atom 0 R k H (fun _ => by omega) rfl
-    (fun j hj => by omega) (fun _ _ => hT) (fun _ _ => he) (fun _ _ => hMotif) hroot
-    C.first C.second C.decode C.score C.density C.likelihood_nonneg
-    C.first_simulation C.second_simulation ω a η L ha hL hω
-    (fun j hj => by
-      have hj' : j=0 := by omega
-      subst j
-      exact hov)
-    hk hm hscale (fun _ _ => hη) herror
-  have hp : 2 ≤ p := by dsimp [p]; omega
-  have hmp : p/2-1+2^s ≤ p := by dsimp [p] at *; omega
-  let U := pureAtomRun x (centeredLeftResidual N p)
-  let B := pureAtomRun x (centeredRightResidual N p)
-  have heq (τ : Bool) := centeredComplementaryWords_pad N p s atom τ hp hfit hmp
-  have HH := W.pad hq U B
-  simp only [sourceBlockWord,sourceFitTail,add_zero] at HH
-  change OracleTracePair q
-    (U++complementaryWords p (p/2-1) (2^s) (prouhetBool s) atom false++B)
-    (U++complementaryWords p (p/2-1) (2^s) (prouhetBool s) atom true++B) k _ _ at HH
-  have hleft : U++complementaryWords p (p/2-1) (2^s) (prouhetBool s) atom false++B =
-      complementaryWords N (N/2-1) (2^s) (prouhetBool s) atom false := (heq false).symm
-  have hright : U++complementaryWords p (p/2-1) (2^s) (prouhetBool s) atom true++B =
-      complementaryWords N (N/2-1) (2^s) (prouhetBool s) atom true := (heq true).symm
-  rw [hleft,hright] at HH
-  have hc : SourceLeafCount (fun _ => T) 1=T := by
-    calc
-      _ = Fintype.card (Fin (T*1)) := Fintype.card_congr finProdFinEquiv
-      _ = T := by simp
-  simp only [hc] at HH
-  exact ⟨HH⟩
-
-end TraceReconstruction
-
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem sourceLeafCount_one (T : ℕ → ℕ) : SourceLeafCount T 1=T 0 := by
-  calc
-    _ = Fintype.card (Fin (T 0*1)) := Fintype.card_congr finProdFinEquiv
-    _ = T 0 := by simp
-
-/-- Exact two-degree cutoff for the one-window warmup refinement. In
- particular the remaining geometric term is quadratic in the old energy. -/
-theorem sourceSharpEnergyBudget_one_window (T n : ℕ → ℕ) (s : ℕ) (ω η a w : ℝ) :
-    sourceSharpEnergyBudget T n s 0 2 ω η a w =
-      36*(T 0:ℝ)*w^2*(η^2+a^2)+4*(w^2*2^s)^2/(1-w^2*2^s) := by
-  simp only [sourceSharpEnergyBudget,Finset.sum_range_succ,Finset.sum_range_zero]
-  norm_num [sourceSharpDegreeBudget,sourceSparseCount,sourceLeafCount_one,Nat.choose_one_right]
-  ring
-
-theorem sourceSharpEnergyBudget_warmup (T n : ℕ → ℕ) (s : ℕ) (ω η w : ℝ)
-    (h : w^2*2^s ≤ 1/2) :
-    sourceSharpEnergyBudget T n s 0 2 ω η η w ≤
-      72*(T 0:ℝ)*w^2*η^2+8*(w^2*2^s)^2 := by
-  rw [sourceSharpEnergyBudget_one_window]
-  have hd : 0 < 1-w^2*2^s := by linarith
-  have hb : 4*(w^2*2^s)^2/(1-w^2*2^s) ≤ 8*(w^2*2^s)^2 := by
-    apply (div_le_iff₀ hd).mpr
-    have hh := mul_le_mul_of_nonneg_left h (sq_nonneg (w^2*2^s))
-    nlinarith only [hh]
-  nlinarith only [hb]
-
-end TraceReconstruction
-
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- The number of differing labels of complementary templates is bounded
- by the actual motif length, without any assumption that it fills the word. -/
-theorem templateType_difference_count (N a L : ℕ) (sign : Fin L → Bool) (τ σ : Bool) :
-    (∑i : Fin N,if templateType N a L sign τ i=templateType N a L sign σ i then (0:ℝ) else 1) ≤ L := by
-  let S : Finset (Fin N) := Finset.univ.filter (fun i => a ≤ i.val ∧ i.val < a+L)
-  have hi (i : S) : a ≤ i.val.val ∧ i.val.val < a+L := (Finset.mem_filter.mp i.property).2
-  let f : S → Fin L := fun i => ⟨i.val.val-a,by have h := hi i; omega⟩
-  have hf : Function.Injective f := by
-    intro i j hij
-    have h := congrArg Fin.val hij
-    have h₁ := hi i
-    have h₂ := hi j
-    dsimp [f] at h
-    apply Subtype.ext
-    apply Fin.ext
-    omega
-  have hc : S.card ≤ L := by
-    simpa only [Fintype.card_coe,Fintype.card_fin] using Fintype.card_le_of_injective f hf
-  calc
-    _ ≤ ∑i : Fin N,if i∈S then (1:ℝ) else 0 := by
-      apply Finset.sum_le_sum
-      intro i _
-      by_cases h : i∈S
-      · rw [ite_eq_left h]; split_ifs <;> norm_num
-      · have h' : ¬(a ≤ i.val ∧ i.val < a+L) := by simpa only [S,Finset.mem_filter,Finset.mem_univ,true_and] using h
-        simp [templateType,h',h]
-    _ = (S.card:ℝ) := by simp
-    _ ≤ L := by exact_mod_cast hc
-
-/-- Actual paired finite observations with a common complete-trace decoder
- and a squared Hellinger bound. This auxiliary structure is constructed from
- the source score oracle and closed under literal template substitution. -/
-structure HellingerTracePair (q : ℝ) (x y : List Bool) (E V : ℝ) where
-  Outcome : Type
-  finiteOutcome : Fintype Outcome
-  first : @FiniteLaw Outcome finiteOutcome
-  second : @FiniteLaw Outcome finiteOutcome
-  decode : Outcome → List Bool
-  first_simulation : @ApproximatesTrace Outcome finiteOutcome q first decode x E
-  second_simulation : @ApproximatesTrace Outcome finiteOutcome q second decode y E
-  bound : @FiniteLaw.hellingerSq Outcome finiteOutcome first second ≤ V
-
-def OracleTracePair.hellinger {q E V : ℝ} {x y : List Bool} {k : ℕ}
-    (C : OracleTracePair q x y k E V) (hk : 1 ≤ k) : HellingerTracePair q x y E V := by
-  let := C.finiteOutcome
-  exact {
-    Outcome := C.Outcome
-    finiteOutcome := C.finiteOutcome
-    first := C.first
-    second := C.second
-    decode := C.decode
-    first_simulation := C.first_simulation
-    second_simulation := C.second_simulation
-    bound := (C.first.hellingerSq_le_score C.second C.score C.density).trans
-      ((C.first.secondMoment_le_evenMomentSq C.score k hk).trans C.moment) }
-
-def HellingerTracePair.mix {q E V : ℝ} {x y : List Bool} (C : HellingerTracePair q x y E V)
-    (hq : q∈Set.Icc 0 1) {N : ℕ} (τ σ : Fin N → Bool) :
-    HellingerTracePair q (coordinateWord (fun i => if τ i then y else x))
-      (coordinateWord (fun i => if σ i then y else x)) ((N:ℝ)*E)
-      ((∑i,if τ i=σ i then (0:ℝ) else 1)*V) := by
-  let := C.finiteOutcome
-  have ha (ρ : Fin N → Bool) : ApproximatesTrace q
-      (FiniteLaw.coordinateProduct (fun i => if ρ i then C.second else C.first))
-      (fun z => coordinateWord (fun i => C.decode (z i)))
-      (coordinateWord (fun i => if ρ i then y else x)) ((N:ℝ)*E) := by
-    have H := coordinateProduct_approximates hq
-      (fun i => if ρ i then C.second else C.first) (fun _ => C.decode)
-      (fun i => if ρ i then y else x) (fun _ => E) (fun i => ?_)
-    · simpa only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul] using H
-    · cases ρ i <;> simp only [Bool.false_eq_true,ite_false,ite_true]
-      · exact C.first_simulation
-      · exact C.second_simulation
-  exact {
-    Outcome := Fin N → C.Outcome
-    finiteOutcome := inferInstance
-    first := FiniteLaw.coordinateProduct (fun i => if τ i then C.second else C.first)
-    second := FiniteLaw.coordinateProduct (fun i => if σ i then C.second else C.first)
-    decode := fun z => coordinateWord (fun i => C.decode (z i))
-    first_simulation := ha τ
-    second_simulation := ha σ
-    bound := FiniteLaw.hellingerSq_mixedProduct_le C.first C.second C.bound τ σ }
-
-def HellingerTracePair.weaken {q E V W : ℝ} {x y : List Bool}
-    (C : HellingerTracePair q x y E V) (h : V ≤ W) : HellingerTracePair q x y E W :=
-  { C with bound := C.bound.trans h }
-
-def HellingerTracePair.template {q E V : ℝ} {x y : List Bool} (C : HellingerTracePair q x y E V)
-    (hq : q∈Set.Icc 0 1) (N a L : ℕ) (sign : Fin L → Bool) :
-    HellingerTracePair q
-      (complementaryWords N a L sign (fun τ => if τ then y else x) false)
-      (complementaryWords N a L sign (fun τ => if τ then y else x) true) ((N:ℝ)*E) ((L:ℝ)*V) := by
-  let := C.finiteOutcome
-  let D := C.mix hq (templateType N a L sign false) (templateType N a L sign true)
-  have hV : 0 ≤ V := (FiniteLaw.hellingerSq_nonneg _ _).trans C.bound
-  exact D.weaken (mul_le_mul_of_nonneg_right (templateType_difference_count N a L sign false true) hV)
-
-theorem HellingerTracePair.power_tv {q E V : ℝ} {x y : List Bool}
-    (C : HellingerTracePair q x y E V) (hq : q∈Set.Icc 0 1)
-    (F M : ℕ) (hx : (pureAtomRun x F).length ≤ M) (hy : (pureAtomRun y F).length ≤ M) :
-    (listTraceLaw q hq (pureAtomRun x F) M hx).tv
-      (listTraceLaw q hq (pureAtomRun y F) M hy) ≤ Real.sqrt ((F:ℝ)*V)+2*(F:ℝ)*E := by
-  let := C.finiteOutcome
-  have htv := (FiniteLaw.tv_le_sqrt_hellingerSq (C.first.product F) (C.second.product F)).trans
-    (Real.sqrt_le_sqrt ((FiniteLaw.hellingerSq_product_le C.first C.second F).trans
-      (mul_le_mul_of_nonneg_left C.bound (Nat.cast_nonneg F))))
-  have H := approximations_trace_tv hq _ _ _ _ _ _ _ M hx hy
-    (C.first_simulation.power hq F) (C.second_simulation.power hq F)
-  linarith only [H,htv]
-
-end TraceReconstruction
-
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem templateType_one_count (N a L : ℕ) (sign : Fin L → Bool) (τ : Bool) :
-    (∑i : Fin N,if false=templateType N a L sign τ i then (0:ℝ) else 1) ≤ L := by
-  apply le_trans _ (templateType_difference_count N a L sign false true)
-  apply Finset.sum_le_sum
-  intro i _
-  by_cases h : a ≤ i.val ∧ i.val < a+L
-  · simp only [templateType,dite_eq_left h]
-    cases hs : sign ⟨i.val-a,by omega⟩ <;> cases τ <;> norm_num
-  · simp [templateType,h]
-
-def HellingerTracePair.pure_template {q E V : ℝ} {x y : List Bool}
-    (C : HellingerTracePair q x y E V) (hq : q∈Set.Icc 0 1)
-    (N a L : ℕ) (sign : Fin L → Bool) (τ : Bool) :
-    HellingerTracePair q (pureAtomRun x N)
-      (complementaryWords N a L sign (fun υ => if υ then y else x) τ) ((N:ℝ)*E) ((L:ℝ)*V) := by
-  let := C.finiteOutcome
-  let D := C.mix hq (fun _ : Fin N => false) (templateType N a L sign τ)
-  have hV : 0 ≤ V := (FiniteLaw.hellingerSq_nonneg _ _).trans C.bound
-  have W := D.weaken (mul_le_mul_of_nonneg_right (templateType_one_count N a L sign τ) hV)
-  simp only [Bool.false_eq_true,ite_false,coordinateWord_constant] at W
-  exact W
-
-theorem bool_word_eta (w : Bool → List Bool) : (fun τ => if τ then w true else w false)=w := by
-  funext τ
-  cases τ <;> rfl
-
-/-- Literal dense substitution of the old auxiliary laws, used only for
- overlap comparison. Statistical cost is paid only for differing motif
- leaves, not the filler atoms. -/
-def sourceDensePair {q E V : ℝ} (hq : q∈Set.Icc 0 1) (s : ℕ) (P : ℕ → ℕ)
-    (atom : Bool → List Bool) (C : HellingerTracePair q (atom false) (atom true) E V) :
-    (j : ℕ) → HellingerTracePair q (sourceBlockWord s P atom j false)
-      (sourceBlockWord s P atom j true) ((sourceBlockLength P 1 j:ℝ)*E) (((2^s:ℕ):ℝ)^j*V)
-  | 0 => by simpa only [sourceBlockWord,sourceBlockLength,Nat.cast_one,pow_zero,one_mul] using C
-  | j+1 => by
-      have D := (sourceDensePair hq s P atom C j).template hq (P j) (P j/2-1) (2^s) (prouhetBool s)
-      rw [bool_word_eta] at D
-      simpa only [sourceBlockWord,sourceBlockLength,Nat.cast_mul,pow_succ,
-        mul_assoc,mul_left_comm,mul_comm] using D
-
-theorem HellingerTracePair.lifted_power_overlap {q E V p C₀ : ℝ} {w z : List Bool}
-    {h P F : ℕ} (O : HellingerTracePair q (pureAtomRun w P) z E V)
-    (hq : q∈Set.Icc 0 1) (hh : 1 ≤ h) (hP : 1 ≤ P)
-    (hp : p∈Set.Ioc 0 (1/8)) (hC : 1073741824 ≤ p*C₀) (hF : C₀*h*P ≤ F)
-    (hov : p ≤ listTraceOverlap q hq (pureAtomRun w h) (pureAtomRun w (h+1))) :
-    (2/3:ℝ)-(Real.sqrt ((F:ℝ)*V)+2*(F:ℝ)*E)-
-      (Real.sqrt (((F+1:ℕ):ℝ)*V)+2*((F+1:ℕ):ℝ)*E) ≤
-      listTraceOverlap q hq (pureAtomRun z F) (pureAtomRun z (F+1)) := by
-  let a := pureAtomRun (pureAtomRun w P) F
-  let b := pureAtomRun (pureAtomRun w P) (F+1)
-  let x := pureAtomRun z F
-  let y := pureAtomRun z (F+1)
-  let M := max (max a.length b.length) (max x.length y.length)
-  have ha : a.length ≤ M := (le_max_left _ _).trans (le_max_left _ _)
-  have hb : b.length ≤ M := (le_max_right _ _).trans (le_max_left _ _)
-  have hx : x.length ≤ M := (le_max_left _ _).trans (le_max_right _ _)
-  have hy : y.length ≤ M := (le_max_right _ _).trans (le_max_right _ _)
-  have ht1 := O.power_tv hq F M ha hx
-  have ht2 := O.power_tv hq (F+1) M hb hy
-  have hol := listTraceOverlap_lift hh hP hp hq hC hF w hov
-  have he : F*P+P=(F+1)*P := by ring
-  have heA : pureAtomRun w (F*P)=a := (pureAtomRun_mul w P F).symm
-  have heB : pureAtomRun w (F*P+P)=b := by
-    rw [he]; exact (pureAtomRun_mul w P (F+1)).symm
-  rw [heA,heB] at hol
-  rw [listTraceOverlap_ambient hq _ _ M ha hb,FiniteLaw.overlap_eq_one_sub_tv] at hol
-  rw [listTraceOverlap_ambient hq _ _ M hx hy,FiniteLaw.overlap_eq_one_sub_tv]
-  have H := (FiniteLaw.tv_triangle (listTraceLaw q hq x M hx) (listTraceLaw q hq a M ha)
-    (listTraceLaw q hq y M hy)).trans
-    (add_le_add le_rfl (FiniteLaw.tv_triangle (listTraceLaw q hq a M ha)
-      (listTraceLaw q hq b M hb) (listTraceLaw q hq y M hy)))
-  rw [FiniteLaw.tv_symm (listTraceLaw q hq x M hx) (listTraceLaw q hq a M ha)] at H
-  linarith only [H,ht1,ht2,hol]
-
-theorem HellingerTracePair.lifted_overlap_quarter {q E V p C₀ : ℝ} {w z : List Bool}
-    {h P F : ℕ} (O : HellingerTracePair q (pureAtomRun w P) z E V)
-    (hq : q∈Set.Icc 0 1) (hE : 0 ≤ E) (hh : 1 ≤ h) (hP : 1 ≤ P)
-    (hp : p∈Set.Ioc 0 (1/8)) (hC : 1073741824 ≤ p*C₀) (hF : C₀*h*P ≤ F)
-    (hov : p ≤ listTraceOverlap q hq (pureAtomRun w h) (pureAtomRun w (h+1)))
-    (hsmall : Real.sqrt (((F+1:ℕ):ℝ)*V)+2*((F+1:ℕ):ℝ)*E ≤ 1/4) :
-    p ≤ listTraceOverlap q hq (pureAtomRun z F) (pureAtomRun z (F+1)) := by
-  let := O.finiteOutcome
-  have hV : 0 ≤ V := (FiniteLaw.hellingerSq_nonneg _ _).trans O.bound
-  have ht := O.lifted_power_overlap hq hh hP hp hC hF hov
-  have hm : Real.sqrt ((F:ℝ)*V)+2*(F:ℝ)*E ≤
-      Real.sqrt (((F+1:ℕ):ℝ)*V)+2*((F+1:ℕ):ℝ)*E := by
-    apply add_le_add
-    · apply Real.sqrt_le_sqrt
-      exact mul_le_mul_of_nonneg_right (by norm_num) hV
-    · exact mul_le_mul_of_nonneg_right (by norm_num) hE
-  linarith only [hm,hsmall,ht,hp.2]
-
-/-- Finite nested overlap induction at the main (post-warmup) dimensions.
- Every closeness premise is an explicit scalar estimate on the old oracle;
- the repeated-word overlaps themselves are proved, not assumed. -/
-theorem source_nested_overlap {q E V p C₀ : ℝ} (hq : q∈Set.Icc 0 1)
-    (hE : 0 ≤ E) (s : ℕ) (P h : ℕ → ℕ) (atom : Bool → List Bool)
-    (C : HellingerTracePair q (atom false) (atom true) E V) (b : ℕ)
-    (hp : p∈Set.Ioc 0 (1/8)) (hC : 1073741824 ≤ p*C₀)
-    (hcount : ∀j ≤ b,1 ≤ h j) (hP : ∀j < b,1 ≤ P j)
-    (hlift : ∀j < b,C₀*h j*P j ≤ h (j+1))
-    (hsmall : ∀j < b,
-      Real.sqrt (((h (j+1)+1:ℕ):ℝ)*(((2^s:ℕ):ℝ)^(j+1)*V))+
-        2*((h (j+1)+1:ℕ):ℝ)*((sourceBlockLength P 1 (j+1):ℝ)*E) ≤ 1/4)
-    (hbase : p ≤ listTraceOverlap q hq (pureAtomRun (atom false) (h 0))
-      (pureAtomRun (atom false) (h 0+1))) :
-    ∀j ≤ b,p ≤ listTraceOverlap q hq (pureAtomRun (sourceBlockWord s P atom j false) (h j))
-      (pureAtomRun (sourceBlockWord s P atom j false) (h j+1)) := by
-  intro j hj
-  induction j with
-  | zero => exact hbase
-  | succ j ih =>
-    have hjb : j < b := by omega
-    have D := (sourceDensePair hq s P atom C j).pure_template hq (P j) (P j/2-1) (2^s) (prouhetBool s) false
-    rw [bool_word_eta] at D
-    have H := D.lifted_overlap_quarter hq (by positivity) (hcount j (by omega)) (hP j hjb)
-      hp hC (hlift j hjb) (ih (by omega)) ?_
-    · exact H
-    · simpa only [sourceBlockLength,Nat.cast_mul,pow_succ,mul_assoc,mul_left_comm,mul_comm]
-        using hsmall j hjb
-
-end TraceReconstruction
-
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-
-theorem sourceFitTail_nonneg (T n : ℕ → ℕ) (b : ℕ) : 0 ≤ sourceFitTail T n b := by
-  induction b with
-  | zero => rfl
-  | succ b ih =>
-    cases b with
-    | zero => rfl
-    | succ b => exact mul_nonneg (Nat.cast_nonneg _) (add_nonneg (by positivity) ih)
-
-theorem sourceEffectiveHiddenTail_nonneg (T n : ℕ → ℕ) (b : ℕ) (ω : ℝ) :
-    0 ≤ sourceEffectiveHiddenTail T n b ω := by
-  apply Finset.sum_nonneg
-  intro v _
-  split_ifs <;> positivity
-
-theorem sourceClippingCost_nonneg (T n : ℕ → ℕ) (s b k : ℕ) (ω a L : ℝ) (ha : 0 ≤ a) :
-    0 ≤ sourceClippingCost T n s b k ω a L := by
-  exact add_nonneg (add_nonneg (sourceEffectiveHiddenTail_nonneg _ _ _ _)
-    (div_nonneg (sourceFitTail_nonneg _ _ _) ha)) (Real.sqrt_nonneg _)
-
-/-- The two-dimensional warmup uses the actual one-window cancellation
- oracle at its second nested overlap step. No unproved side overlap is
- inserted into the recursive source observation. -/
-theorem source_warmup_nested_overlap {q E V p C₀ : ℝ} (hq : q∈Set.Icc 0 1)
-    (hE : 0 ≤ E) (s k : ℕ) (P h : ℕ → ℕ) (atom : Bool → List Bool)
-    (C : OracleTracePair q (atom false) (atom true) k E V)
-    (hp : p∈Set.Ioc 0 (1/8)) (hC : 1073741824 ≤ p*C₀)
-    (hcount : ∀j ≤ 1,1 ≤ h j) (hP : ∀j < 2,1 ≤ P j)
-    (hlift : ∀j < 2,C₀*h j*P j ≤ h (j+1))
-    (hfirst : Real.sqrt (((h 1+1:ℕ):ℝ)*((2^s:ℕ):ℝ)*V)+
-      2*((h 1+1:ℕ):ℝ)*((P 0:ℝ)*E) ≤ 1/4)
-    (hbase : p ≤ listTraceOverlap q hq (pureAtomRun (atom false) (h 0))
-      (pureAtomRun (atom false) (h 0+1)))
-    (T t : ℕ) (hT : 24 ≤ T) (he : Even T) (hMotif : 24*2^s ≤ T)
-    (hroot : 4*(t+1) ≤ T) (hfit : t*(2*h 0+1)+T ≤ P 0)
-    (a η L : ℝ) (ha : 0 < a) (hL : 0 < L) (hk : 1 ≤ k) (hV : V ≤ L^2)
-    (hscale : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount (fun _ => T) s 0 ≤ 1/2)
-    (hη : hiddenSmoothingBound t s p ≤ η)
-    (herror : sourceClippingCost (fun _ => T) (fun _ => t) s 0 k p a L ≤ 1/2)
-    (hsecond :
-      Real.sqrt (((h 2+1:ℕ):ℝ)*((2^s:ℕ):ℝ)*
-        (64*sourceSharpEnergyBudget (fun _ => T) (fun _ => t) s 0 2 p η a ((6*(2*k:ℕ):ℝ)*L)))+
-      2*((h 2+1:ℕ):ℝ)*((P 1:ℝ)*
-        (sourceClippingCost (fun _ => T) (fun _ => t) s 0 k p a L+(T:ℝ)*E)) ≤ 1/4) :
-    ∀j ≤ 2,p ≤ listTraceOverlap q hq (pureAtomRun (sourceBlockWord s P atom j false) (h j))
-      (pureAtomRun (sourceBlockWord s P atom j false) (h j+1)) := by
-  have hp' : p∈Set.Ioo 0 1 := ⟨hp.1,lt_of_le_of_lt hp.2 (by norm_num)⟩
-  have D := (C.hellinger hk).pure_template hq (P 0) (P 0/2-1) (2^s) (prouhetBool s) false
-  rw [bool_word_eta] at D
-  have h1 := D.lifted_overlap_quarter hq (by positivity) (hcount 0 (by omega))
-    (hP 0 (by omega)) hp hC (hlift 0 (by omega)) hbase ?_
-  · obtain ⟨W⟩ := actual_one_window_oracle C hq hE s (P 0) T t (h 0) 2 hT he hMotif
-      hroot hfit p a η L ha hL hp' hbase hk hV hscale hη herror
-    rw [bool_word_eta] at W
-    have W' := (W.hellinger hk).pure_template hq (P 1) (P 1/2-1) (2^s) (prouhetBool s) false
-    have heword : (fun τ => if τ then
-        complementaryWords (P 0) (P 0/2-1) (2^s) (prouhetBool s) atom true else
-        complementaryWords (P 0) (P 0/2-1) (2^s) (prouhetBool s) atom false) =
-        sourceBlockWord s P atom 1 := bool_word_eta _
-    rw [heword] at W'
-    have hE' : 0 ≤ sourceClippingCost (fun _ => T) (fun _ => t) s 0 k p a L+(T:ℝ)*E := by
-      exact add_nonneg (sourceClippingCost_nonneg _ _ _ _ _ _ _ _ ha.le) (by positivity)
-    have h2 := W'.lifted_overlap_quarter hq (mul_nonneg (Nat.cast_nonneg _) hE')
-      (hcount 1 le_rfl) (hP 1 (by omega)) hp hC (hlift 1 (by omega)) h1 ?_
-    · intro j hj
-      interval_cases j
-      · exact hbase
-      · exact h1
-      · exact h2
-    · simpa only [mul_assoc] using hsecond
-  · simpa only [mul_assoc] using hfirst
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem OracleTracePair.error_nonneg {q E V : ℝ} {x y : List Bool} {k : ℕ}
-    (C : OracleTracePair q x y k E V) : 0 ≤ E := by
-  let := C.finiteOutcome
-  have h := C.first_simulation (fun _ => 0) (fun _ => ⟨le_rfl,zero_le_one⟩)
-  exact (abs_nonneg _).trans h
-
-/-- Finite main-block production: actual oracle, complete trace projection,
- and all side overlaps follow from arithmetic budgets and the OLD invariant.
- In particular none of the new side laws or their overlap are hypotheses. -/
-theorem actual_main_block {q E V ω C₀ : ℝ} (hq : q∈Set.Icc 0 1)
-    (s : ℕ) (P T n h : ℕ → ℕ) (atom : Bool → List Bool) (b R k : ℕ)
-    (C : OracleTracePair q (atom false) (atom true) k E V)
-    (H : SourceLayoutMargins s P T n h b)
-    (hTpos : ∀j,0 < T j) (hP : P b=n b*(2*h b+1)+T b)
-    (hn : ∀j < b,0 < n j) (hT : ∀j ≤ b,24 ≤ T j) (he : ∀j ≤ b,Even (T j))
-    (hMotif : ∀j ≤ b,24*2^s ≤ T j) (hroot : 4*(n b+1) ≤ T b)
-    (a η L : ℝ) (ha : 0 < a) (hL : 0 < L) (hω : ω∈Set.Ioc 0 (1/8))
-    (hk : 1 ≤ k) (hV : V ≤ L^2)
-    (hscale : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b ≤ 1/2)
-    (hη : ∀j < b+1,hiddenSmoothingBound (n j) s ω ≤ η)
-    (herror : sourceClippingCost T n s b k ω a L ≤ 1/2)
-    (hC : 1073741824 ≤ ω*C₀) (hcount : ∀j ≤ b+1,1 ≤ h j)
-    (hpos : ∀j < b+1,1 ≤ P j)
-    (hlift : ∀j < b+1,C₀*h j*P j ≤ h (j+1))
-    (hsmall : ∀j < b+1,
-      Real.sqrt (((h (j+1)+1:ℕ):ℝ)*(((2^s:ℕ):ℝ)^(j+1)*V))+
-        2*((h (j+1)+1:ℕ):ℝ)*((sourceBlockLength P 1 (j+1):ℝ)*E) ≤ 1/4)
-    (hbase : ω ≤ listTraceOverlap q hq (pureAtomRun (atom false) (h 0))
-      (pureAtomRun (atom false) (h 0+1))) :
-    Nonempty (OracleTracePair q (sourceBlockWord s P atom (b+1) false)
-      (sourceBlockWord s P atom (b+1) true) k
-      (sourceClippingCost T n s b k ω a L+(SourceLeafCount T (b+1):ℝ)*E+sourceFitTail T n (b+1))
-      (64*sourceSharpEnergyBudget T n s b R ω η a ((6*(2*k:ℕ):ℝ)*L))) ∧
-    (∀j ≤ b+1,ω ≤ listTraceOverlap q hq
-      (pureAtomRun (sourceBlockWord s P atom j false) (h j))
-      (pureAtomRun (sourceBlockWord s P atom j false) (h j+1))) := by
-  let := C.finiteOutcome
-  have hov := source_nested_overlap hq C.error_nonneg s P h atom (C.hellinger hk) (b+1)
-    hω hC hcount hpos hlift hsmall hbase
-  refine ⟨?_,hov⟩
-  have hm : C.first.expect (fun z => |C.score z|^(2*k)) ≤ L^(2*k) := by
-    have H := C.first.expect_even_pow_le_of_moment C.score k hk (C.moment.trans hV)
-    simpa only [pow_mul] using H
-  exact actual_source_oracle hq C.error_nonneg s P T n h atom b R k H hTpos hP hn hT he hMotif hroot
-    C.first C.second C.decode C.score C.density C.likelihood_nonneg C.first_simulation C.second_simulation
-    ω a η L ha hL ⟨hω.1,lt_of_le_of_lt hω.2 (by norm_num)⟩
-    (fun j hj => hov j (by omega)) hk hm hscale hη herror
-
-/-- Conversion of a produced genuine oracle into the manuscript invariant.
- The scalar bounds here are exactly those to be discharged by scale margins. -/
-def OracleTracePair.toConcrete {q E V N Δ d ω F : ℝ} {x y : List Bool} {k i h : ℕ}
-    (C : OracleTracePair q x y k E V) (hq : q∈Set.Icc 0 1)
-    (hlen : x.length=y.length) (hne : x≠y) (hlo : N/2 ≤ (x.length:ℝ))
-    (hhi : (x.length:ℝ) ≤ N) (herror : E/(x.length:ℝ) ≤ ((i:ℝ)+1)*Δ)
-    (hV : V ≤ N^(-d)) (hhlo : x.length ≤ h) (hhhi : (h:ℝ) ≤ F*(x.length:ℝ))
-    (hov : ω ≤ listTraceOverlap q hq (pureAtomRun x h) (pureAtomRun x (h+1))) :
-    ConcreteLevelPair q hq N Δ d ω F i k := {
-  left := x
-  right := y
-  equalLength := hlen
-  different := hne
-  length_lower := hlo
-  length_upper := hhi
-  Outcome := C.Outcome
-  finiteOutcome := C.finiteOutcome
-  firstLaw := C.first
-  secondLaw := C.second
-  decode := C.decode
-  error := E
-  error_nonneg := C.error_nonneg
-  normalized_error := herror
-  first_simulation := C.first_simulation
-  second_simulation := C.second_simulation
-  score := C.score
-  density := C.density
-  score_bound := C.moment.trans hV
-  overlapCount := h
-  overlapCount_lower := hhlo
-  overlapCount_upper := hhhi
-  power_overlap := hov }
-
-end TraceReconstruction
-
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-
-end OAI
+import OAI.Probability.TraceReconstruction.SourceOracle
+
+namespace OAI
+
+/-!
+# Amplification
+
+Oracle products, mixed powers, padding, warmup and nested template Hellinger estimates, overlap, and block assembly.
+-/
+
+open scoped ENNReal
+noncomputable section
+
+open Filter
+open scoped Topology
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- The literal product/concatenation decoder charges only coordinates at
+ which the two deterministic input templates differ. -/
+theorem OracleTracePair.mixed_tv {q E V : ℝ} {x y : List Bool} {k n : ℕ}
+    (C : OracleTracePair q x y k E V) (hq : q∈Set.Icc 0 1) (hk : 1≤k)
+    (τ σ : Fin n → Bool) (M : ℕ)
+    (hτ : (coordinateWord (fun i => if τ i then y else x)).length≤M)
+    (hσ : (coordinateWord (fun i => if σ i then y else x)).length≤M) :
+    (listTraceLaw q hq (coordinateWord (fun i => if τ i then y else x)) M hτ).tv
+      (listTraceLaw q hq (coordinateWord (fun i => if σ i then y else x)) M hσ) ≤
+      Real.sqrt ((∑i,if τ i=σ i then (0:ℝ) else 1)*V)+2*(n:ℝ)*E := by
+  let := C.finiteOutcome
+  have ha (ρ : Fin n → Bool) : ApproximatesTrace q
+      (FiniteLaw.coordinateProduct (fun i => if ρ i then C.second else C.first))
+      (fun z => coordinateWord (fun i => C.decode (z i)))
+      (coordinateWord (fun i => if ρ i then y else x)) ((n:ℝ)*E) := by
+    have H := coordinateProduct_approximates hq
+      (fun i => if ρ i then C.second else C.first) (fun _ => C.decode)
+      (fun i => if ρ i then y else x) (fun _ => E) (fun i => ?_)
+    · simpa only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul] using H
+    · cases ρ i <;> simp only [Bool.false_eq_true,ite_false,ite_true]
+      · exact C.first_simulation
+      · exact C.second_simulation
+  have H := approximations_trace_tv hq _ _ _ _ _ _ _ M hτ hσ (ha τ) (ha σ)
+  have htv := FiniteLaw.tv_mixedProduct_le C.first C.second C.score C.density k hk C.moment τ σ
+  linarith only [H,htv]
+
+theorem coordinateWord_constant (w : List Bool) (n : ℕ) :
+    coordinateWord (fun _ : Fin n => w)=pureAtomRun w n := by
+  induction n with
+  | zero => simp [coordinateWord,pureAtomRun]
+  | succ n ih =>
+    rw [coordinateWord_succ,ih]
+    simp only [pureAtomRun,List.replicate_succ,List.flatten_cons]
+
+/-- Tensorization of a genuine approximate oracle, on actual word powers. -/
+theorem OracleTracePair.power_tv {q E V : ℝ} {x y : List Bool} {k : ℕ}
+    (C : OracleTracePair q x y k E V) (hq : q∈Set.Icc 0 1) (hk : 1≤k)
+    (n M : ℕ) (hx : (pureAtomRun x n).length≤M) (hy : (pureAtomRun y n).length≤M) :
+    (listTraceLaw q hq (pureAtomRun x n) M hx).tv
+      (listTraceLaw q hq (pureAtomRun y n) M hy) ≤
+      Real.sqrt ((n:ℝ)*V)+2*(n:ℝ)*E := by
+  have H := C.mixed_tv hq hk (fun _ : Fin n => false) (fun _ => true) M
+    (by simpa only [Bool.false_eq_true,ite_false,coordinateWord_constant] using hx)
+    (by simpa only [ite_true,coordinateWord_constant] using hy)
+  simpa only [Bool.false_eq_true,ite_false,ite_true,coordinateWord_constant,
+    Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,mul_one] using H
+
+/-- Genuine lifting for complete traces, with no auxiliary overlap hypothesis. -/
+theorem listTraceOverlap_lift {F k R : ℕ} {C p q : ℝ}
+    (hk : 1≤k) (hR : 1≤R) (hp : p∈Set.Ioc 0 (1/8)) (hq : q∈Set.Icc 0 1)
+    (hC : 1073741824≤p*C) (hF : C*k*R≤F) (w : List Bool)
+    (hov : p≤listTraceOverlap q hq (pureAtomRun w k) (pureAtomRun w (k+1))) :
+    (2/3:ℝ)≤listTraceOverlap q hq (pureAtomRun w (F*R)) (pureAtomRun w (F*R+R)) := by
+  let M := max (pureAtomRun w k).length (pureAtomRun w (k+1)).length
+  have H := adjacent_power_lift hk hR hp hq hC hF
+    (listTraceLaw q hq (pureAtomRun w k) M (le_max_left _ _))
+    (listTraceLaw q hq (pureAtomRun w (k+1)) M (le_max_right _ _)) Subtype.val w
+    (listTraceLaw_simulates q hq _ M _) (listTraceLaw_simulates q hq _ M _) hov
+  obtain ⟨β,fin,P,Q,g,hP,hQ,hO⟩ := H
+  let := fin
+  exact hO.trans (simulations_overlap_le hq P Q g _ _ hP hQ)
+
+theorem pureAtomRun_mul (w : List Bool) (n m : ℕ) :
+    pureAtomRun (pureAtomRun w n) m=pureAtomRun w (m*n) := by
+  induction m with
+  | zero => simp [pureAtomRun]
+  | succ m ih =>
+    have H : pureAtomRun (pureAtomRun w n) (m+1)=
+        pureAtomRun w n++pureAtomRun (pureAtomRun w n) m := by
+      simp [pureAtomRun,List.replicate_succ]
+    rw [H,ih,←pureAtomRun_add]
+    congr 1
+    ring
+
+/-- Quantitative overlap transfer used for the initial pure comparison and
+ the small-dimensional warmup refinement. All three laws are ordinary traces. -/
+theorem OracleTracePair.lifted_power_overlap {q E V p C₀ : ℝ} {w z : List Bool}
+    {k h P F : ℕ} (O : OracleTracePair q (pureAtomRun w P) z k E V)
+    (hq : q∈Set.Icc 0 1) (hk : 1≤k) (hh : 1≤h) (hP : 1≤P)
+    (hp : p∈Set.Ioc 0 (1/8)) (hC : 1073741824≤p*C₀) (hF : C₀*h*P≤F)
+    (hov : p≤listTraceOverlap q hq (pureAtomRun w h) (pureAtomRun w (h+1))) :
+    (2/3:ℝ)-(Real.sqrt ((F:ℝ)*V)+2*(F:ℝ)*E)-
+      (Real.sqrt (((F+1:ℕ):ℝ)*V)+2*((F+1:ℕ):ℝ)*E) ≤
+      listTraceOverlap q hq (pureAtomRun z F) (pureAtomRun z (F+1)) := by
+  let a := pureAtomRun (pureAtomRun w P) F
+  let b := pureAtomRun (pureAtomRun w P) (F+1)
+  let x := pureAtomRun z F
+  let y := pureAtomRun z (F+1)
+  let M := max (max a.length b.length) (max x.length y.length)
+  have ha : a.length≤M := (le_max_left _ _).trans (le_max_left _ _)
+  have hb : b.length≤M := (le_max_right _ _).trans (le_max_left _ _)
+  have hx : x.length≤M := (le_max_left _ _).trans (le_max_right _ _)
+  have hy : y.length≤M := (le_max_right _ _).trans (le_max_right _ _)
+  have ht1 := O.power_tv hq hk F M ha hx
+  have ht2 := O.power_tv hq hk (F+1) M hb hy
+  have hol := listTraceOverlap_lift hh hP hp hq hC hF w hov
+  have he : F*P+P=(F+1)*P := by ring
+  have heA : pureAtomRun w (F*P)=a := (pureAtomRun_mul w P F).symm
+  have heB : pureAtomRun w (F*P+P)=b := by
+    rw [he]; exact (pureAtomRun_mul w P (F+1)).symm
+  rw [heA,heB] at hol
+  rw [listTraceOverlap_ambient hq _ _ M ha hb,FiniteLaw.overlap_eq_one_sub_tv] at hol
+  rw [listTraceOverlap_ambient hq _ _ M hx hy,FiniteLaw.overlap_eq_one_sub_tv]
+  have H := (FiniteLaw.tv_triangle (listTraceLaw q hq x M hx) (listTraceLaw q hq a M ha)
+    (listTraceLaw q hq y M hy)).trans
+    (add_le_add le_rfl (FiniteLaw.tv_triangle (listTraceLaw q hq a M ha)
+      (listTraceLaw q hq b M hb) (listTraceLaw q hq y M hy)))
+  rw [FiniteLaw.tv_symm (listTraceLaw q hq x M hx) (listTraceLaw q hq a M ha)] at H
+  linarith only [H,ht1,ht2,hol]
+
+end TraceReconstruction
+
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+namespace FiniteLaw
+variable {α : Type*} [Fintype α]
+
+theorem hellingerSq_mixedProduct_le {n : ℕ} (P Q : FiniteLaw α) {V : ℝ}
+    (hH : P.hellingerSq Q ≤ V) (τ σ : Fin n → Bool) :
+    (coordinateProduct (fun i => if τ i then Q else P)).hellingerSq
+      (coordinateProduct (fun i => if σ i then Q else P)) ≤
+      (∑i,if τ i=σ i then (0:ℝ) else 1)*V := by
+  apply (hellingerSq_coordinateProduct_le _ _).trans
+  rw [Finset.sum_mul]
+  apply Finset.sum_le_sum
+  intro i _
+  cases τ i <;> cases σ i <;> simp only [Bool.false_eq_true,Bool.true_eq_false,
+    ite_false,ite_true,zero_mul,one_mul,hellingerSq_self,le_refl]
+  · exact hH
+  · rw [hellingerSq_symm]; exact hH
+
+end FiniteLaw
+
+theorem ApproximatesTrace.power {α : Type*} [Fintype α] {q E : ℝ} {P : FiniteLaw α}
+    {g : α → List Bool} {w : List Bool} (H : ApproximatesTrace q P g w E)
+    (hq : q∈Set.Icc 0 1) (F : ℕ) :
+    ApproximatesTrace q (P.product F) (fun z => coordinateWord (fun i => g (z i)))
+      (pureAtomRun w F) ((F:ℝ)*E) := by
+  have h := coordinateProduct_approximates hq (fun _ : Fin F => P) (fun _ => g)
+    (fun _ => w) (fun _ => E) (fun _ => H)
+  have he : FiniteLaw.coordinateProduct (fun _ : Fin F => P)=P.product F := rfl
+  rw [he,coordinateWord_constant] at h
+  simpa only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul] using h
+
+/-- Repeated mixed templates pay only for the differing leaves in squared
+ Hellinger, while approximate leaf simulation is charged linearly. -/
+theorem OracleTracePair.mixed_power_tv {q E V : ℝ} {x y : List Bool} {k n : ℕ}
+    (C : OracleTracePair q x y k E V) (hq : q∈Set.Icc 0 1) (hk : 1 ≤ k)
+    (τ σ : Fin n → Bool) (F M : ℕ)
+    (hτ : (pureAtomRun (coordinateWord (fun i => if τ i then y else x)) F).length ≤ M)
+    (hσ : (pureAtomRun (coordinateWord (fun i => if σ i then y else x)) F).length ≤ M) :
+    (listTraceLaw q hq (pureAtomRun (coordinateWord (fun i => if τ i then y else x)) F) M hτ).tv
+      (listTraceLaw q hq (pureAtomRun (coordinateWord (fun i => if σ i then y else x)) F) M hσ) ≤
+      Real.sqrt ((F:ℝ)*(∑i,if τ i=σ i then (0:ℝ) else 1)*V)+2*(F:ℝ)*(n:ℝ)*E := by
+  let := C.finiteOutcome
+  let A (ρ : Fin n → Bool) := FiniteLaw.coordinateProduct (fun i => if ρ i then C.second else C.first)
+  let g (z : Fin n → C.Outcome) := coordinateWord (fun i => C.decode (z i))
+  have ha (ρ : Fin n → Bool) : ApproximatesTrace q (A ρ) g
+      (coordinateWord (fun i => if ρ i then y else x)) ((n:ℝ)*E) := by
+    have H := coordinateProduct_approximates hq
+      (fun i => if ρ i then C.second else C.first) (fun _ => C.decode)
+      (fun i => if ρ i then y else x) (fun _ => E) (fun i => ?_)
+    · simpa only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul] using H
+    · cases ρ i <;> simp only [Bool.false_eq_true,ite_false,ite_true]
+      · exact C.first_simulation
+      · exact C.second_simulation
+  have hH : C.first.hellingerSq C.second ≤ V :=
+    (C.first.hellingerSq_le_score C.second C.score C.density).trans
+      ((C.first.secondMoment_le_evenMomentSq C.score k hk).trans C.moment)
+  have hm := FiniteLaw.hellingerSq_mixedProduct_le C.first C.second hH τ σ
+  have hp := (FiniteLaw.hellingerSq_product_le (A τ) (A σ) F).trans
+    (mul_le_mul_of_nonneg_left hm (Nat.cast_nonneg F))
+  have htv := (FiniteLaw.tv_le_sqrt_hellingerSq ((A τ).product F) ((A σ).product F)).trans
+    (Real.sqrt_le_sqrt hp)
+  have H := approximations_trace_tv hq _ _ _ _ _ _ _ M hτ hσ
+    ((ha τ).power hq F) ((ha σ).power hq F)
+  rw [←mul_assoc] at htv
+  linarith only [H,htv]
+
+end TraceReconstruction
+
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- An exact common prefix and suffix are observed as ordinary independent
+ complete traces. They neither change the score moment nor cost simulation
+ error. This is the source's deterministic residual convention. -/
+def OracleTracePair.pad {q E V : ℝ} {x y : List Bool} {k : ℕ}
+    (C : OracleTracePair q x y k E V) (hq : q∈Set.Icc 0 1) (u v : List Bool) :
+    OracleTracePair q (u++x++v) (u++y++v) k E V := by
+  let := C.finiteOutcome
+  let P := listTraceLaw q hq u u.length le_rfl
+  let Q := listTraceLaw q hq v v.length le_rfl
+  refine {
+    Outcome := (BoundedTrace u.length × C.Outcome) × BoundedTrace v.length
+    finiteOutcome := inferInstance
+    first := (P.prod C.first).prod Q
+    second := (P.prod C.second).prod Q
+    decode := fun z => z.1.1.val++C.decode z.1.2++z.2.val
+    score := fun z => C.score z.1.2
+    density := by
+      intro z
+      change (P.mass z.1.1*C.second.mass z.1.2)*Q.mass z.2=_
+      rw [C.density]
+      change _=(P.mass z.1.1*C.first.mass z.1.2)*Q.mass z.2*(1+C.score z.1.2)
+      ring
+    likelihood_nonneg := fun z => C.likelihood_nonneg z.1.2
+    centered := by
+      simp only [FiniteLaw.expect_prod,FiniteLaw.expect_const,C.centered]
+    moment := by
+      simpa only [FiniteLaw.evenMomentSq,FiniteLaw.expect_prod,FiniteLaw.expect_const] using C.moment
+    first_simulation := by
+      have H := ((listTraceLaw_simulates q hq u u.length le_rfl).approximate.append hq
+        C.first_simulation).append hq (listTraceLaw_simulates q hq v v.length le_rfl).approximate
+      simpa only [zero_add,add_zero] using H
+    second_simulation := by
+      have H := ((listTraceLaw_simulates q hq u u.length le_rfl).approximate.append hq
+        C.second_simulation).append hq (listTraceLaw_simulates q hq v v.length le_rfl).approximate
+      simpa only [zero_add,add_zero] using H }
+
+/-- Literal template equality under deterministic pure residual padding. -/
+theorem complementaryWords_pad (L P R a len : ℕ) (sign : Fin len → Bool)
+    (w : Bool → List Bool) (τ : Bool) (hfit : a+len ≤ P) :
+    complementaryWords (L+P+R) (L+a) len sign w τ =
+      pureAtomRun (w false) L++complementaryWords P a len sign w τ++pureAtomRun (w false) R := by
+  unfold complementaryWords
+  rw [atomString_three]
+  have hp (i : Fin L) : templateType (L+P+R) (L+a) len sign τ ⟨i.val,by omega⟩=false := by
+    have hi := i.isLt
+    simp only [templateType,dite_eq_right (by omega : ¬(L+a ≤ i.val ∧ i.val < L+a+len))]
+  have hm (i : Fin P) : templateType (L+P+R) (L+a) len sign τ ⟨L+i.val,by omega⟩=
+      templateType P a len sign τ i := by
+    unfold templateType
+    split_ifs with h h' h'
+    · congr 2
+      apply Fin.ext
+      simp only
+      omega
+    · simp only at h
+      omega
+    · simp only at h
+      omega
+    · rfl
+  have hr (i : Fin R) : templateType (L+P+R) (L+a) len sign τ ⟨L+P+i.val,by omega⟩=false := by
+    simp only [templateType,dite_eq_right (by omega : ¬(L+a ≤ L+P+i.val ∧ L+P+i.val < L+a+len))]
+  simp_rw [hp,hm,hr]
+  change coordinateWord (fun _ : Fin L => w false)++_++coordinateWord (fun _ : Fin R => w false)=_
+  rw [coordinateWord_constant,coordinateWord_constant]
+
+def centeredLeftResidual (N P : ℕ) : ℕ := N/2-1-(P/2-1)
+def centeredRightResidual (N P : ℕ) : ℕ := N-(centeredLeftResidual N P+P)
+
+theorem centeredResidual_geometry {N P : ℕ} (hP : 2 ≤ P) (hN : P ≤ N) :
+    centeredLeftResidual N P+P+centeredRightResidual N P=N ∧
+    centeredLeftResidual N P+(P/2-1)=N/2-1 := by
+  simp only [centeredRightResidual,centeredLeftResidual]
+  omega
+
+/-- The auxiliary one-window oracle is centered at the same actual source
+ motif after appending pure residuals, also for odd ambient atom counts. -/
+theorem centeredComplementaryWords_pad (N P s : ℕ) (w : Bool → List Bool) (τ : Bool)
+    (hP : 2 ≤ P) (hN : P ≤ N) (hfit : P/2-1+2^s ≤ P) :
+    complementaryWords N (N/2-1) (2^s) (prouhetBool s) w τ =
+      pureAtomRun (w false) (centeredLeftResidual N P)++
+      complementaryWords P (P/2-1) (2^s) (prouhetBool s) w τ++
+      pureAtomRun (w false) (centeredRightResidual N P) := by
+  have h := complementaryWords_pad (centeredLeftResidual N P) P (centeredRightResidual N P)
+    (P/2-1) (2^s) (prouhetBool s) w τ hfit
+  rw [(centeredResidual_geometry hP hN).1,(centeredResidual_geometry hP hN).2] at h
+  exact h
+
+end TraceReconstruction
+
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- The source's special one-window warmup observation. The ambient word
+ need not have the root's arithmetic divisibility: exact pure residuals
+ restore its literal center, at no additional statistical error. -/
+theorem actual_one_window_oracle {q E V : ℝ} {x y : List Bool} {k : ℕ}
+    (C : OracleTracePair q x y k E V) (hq : q∈Set.Icc 0 1) (hE : 0 ≤ E)
+    (s N T t h R : ℕ) (hT : 24 ≤ T) (he : Even T)
+    (hMotif : 24*2^s ≤ T) (hroot : 4*(t+1) ≤ T)
+    (hfit : t*(2*h+1)+T ≤ N)
+    (ω a η L : ℝ) (ha : 0 < a) (hL : 0 < L) (hω : ω∈Set.Ioo 0 1)
+    (hov : ω ≤ listTraceOverlap q hq (pureAtomRun x h) (pureAtomRun x (h+1)))
+    (hk : 1 ≤ k) (hV : V ≤ L^2)
+    (hscale : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount (fun _ => T) s 0 ≤ 1/2)
+    (hη : hiddenSmoothingBound t s ω ≤ η)
+    (herror : sourceClippingCost (fun _ => T) (fun _ => t) s 0 k ω a L ≤ 1/2) :
+    Nonempty (OracleTracePair q
+      (complementaryWords N (N/2-1) (2^s) (prouhetBool s) (fun τ => if τ then y else x) false)
+      (complementaryWords N (N/2-1) (2^s) (prouhetBool s) (fun τ => if τ then y else x) true) k
+      (sourceClippingCost (fun _ => T) (fun _ => t) s 0 k ω a L+(T:ℝ)*E)
+      (64*sourceSharpEnergyBudget (fun _ => T) (fun _ => t) s 0 R ω η a ((6*(2*k:ℕ):ℝ)*L))) := by
+  let := C.finiteOutcome
+  let p := t*(2*h+1)+T
+  let atom : Bool → List Bool := fun τ => if τ then y else x
+  have H : SourceLayoutMargins s (fun _ => p) (fun _ => T) (fun _ => t) (fun _ => h) 0 := by
+    constructor <;> intro j hj <;> omega
+  have hm : C.first.expect (fun z => |C.score z|^(2*k)) ≤ L^(2*k) := by
+    have hv := C.first.expect_even_pow_le_of_moment C.score k hk (C.moment.trans hV)
+    simpa only [pow_mul] using hv
+  obtain ⟨W⟩ := actual_source_oracle hq hE s (fun _ => p) (fun _ => T) (fun _ => t)
+    (fun _ => h) atom 0 R k H (fun _ => by omega) rfl
+    (fun j hj => by omega) (fun _ _ => hT) (fun _ _ => he) (fun _ _ => hMotif) hroot
+    C.first C.second C.decode C.score C.density C.likelihood_nonneg
+    C.first_simulation C.second_simulation ω a η L ha hL hω
+    (fun j hj => by
+      have hj' : j=0 := by omega
+      subst j
+      exact hov)
+    hk hm hscale (fun _ _ => hη) herror
+  have hp : 2 ≤ p := by dsimp [p]; omega
+  have hmp : p/2-1+2^s ≤ p := by dsimp [p] at *; omega
+  let U := pureAtomRun x (centeredLeftResidual N p)
+  let B := pureAtomRun x (centeredRightResidual N p)
+  have heq (τ : Bool) := centeredComplementaryWords_pad N p s atom τ hp hfit hmp
+  have HH := W.pad hq U B
+  simp only [sourceBlockWord,sourceFitTail,add_zero] at HH
+  change OracleTracePair q
+    (U++complementaryWords p (p/2-1) (2^s) (prouhetBool s) atom false++B)
+    (U++complementaryWords p (p/2-1) (2^s) (prouhetBool s) atom true++B) k _ _ at HH
+  have hleft : U++complementaryWords p (p/2-1) (2^s) (prouhetBool s) atom false++B =
+      complementaryWords N (N/2-1) (2^s) (prouhetBool s) atom false := (heq false).symm
+  have hright : U++complementaryWords p (p/2-1) (2^s) (prouhetBool s) atom true++B =
+      complementaryWords N (N/2-1) (2^s) (prouhetBool s) atom true := (heq true).symm
+  rw [hleft,hright] at HH
+  have hc : SourceLeafCount (fun _ => T) 1=T := by
+    calc
+      _ = Fintype.card (Fin (T*1)) := Fintype.card_congr finProdFinEquiv
+      _ = T := by simp
+  simp only [hc] at HH
+  exact ⟨HH⟩
+
+end TraceReconstruction
+
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem sourceLeafCount_one (T : ℕ → ℕ) : SourceLeafCount T 1=T 0 := by
+  calc
+    _ = Fintype.card (Fin (T 0*1)) := Fintype.card_congr finProdFinEquiv
+    _ = T 0 := by simp
+
+/-- Exact two-degree cutoff for the one-window warmup refinement. In
+ particular the remaining geometric term is quadratic in the old energy. -/
+theorem sourceSharpEnergyBudget_one_window (T n : ℕ → ℕ) (s : ℕ) (ω η a w : ℝ) :
+    sourceSharpEnergyBudget T n s 0 2 ω η a w =
+      36*(T 0:ℝ)*w^2*(η^2+a^2)+4*(w^2*2^s)^2/(1-w^2*2^s) := by
+  simp only [sourceSharpEnergyBudget,Finset.sum_range_succ,Finset.sum_range_zero]
+  norm_num [sourceSharpDegreeBudget,sourceSparseCount,sourceLeafCount_one,Nat.choose_one_right]
+  ring
+
+theorem sourceSharpEnergyBudget_warmup (T n : ℕ → ℕ) (s : ℕ) (ω η w : ℝ)
+    (h : w^2*2^s ≤ 1/2) :
+    sourceSharpEnergyBudget T n s 0 2 ω η η w ≤
+      72*(T 0:ℝ)*w^2*η^2+8*(w^2*2^s)^2 := by
+  rw [sourceSharpEnergyBudget_one_window]
+  have hd : 0 < 1-w^2*2^s := by linarith
+  have hb : 4*(w^2*2^s)^2/(1-w^2*2^s) ≤ 8*(w^2*2^s)^2 := by
+    apply (div_le_iff₀ hd).mpr
+    have hh := mul_le_mul_of_nonneg_left h (sq_nonneg (w^2*2^s))
+    nlinarith only [hh]
+  nlinarith only [hb]
+
+end TraceReconstruction
+
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- The number of differing labels of complementary templates is bounded
+ by the actual motif length, without any assumption that it fills the word. -/
+theorem templateType_difference_count (N a L : ℕ) (sign : Fin L → Bool) (τ σ : Bool) :
+    (∑i : Fin N,if templateType N a L sign τ i=templateType N a L sign σ i then (0:ℝ) else 1) ≤ L := by
+  let S : Finset (Fin N) := Finset.univ.filter (fun i => a ≤ i.val ∧ i.val < a+L)
+  have hi (i : S) : a ≤ i.val.val ∧ i.val.val < a+L := (Finset.mem_filter.mp i.property).2
+  let f : S → Fin L := fun i => ⟨i.val.val-a,by have h := hi i; omega⟩
+  have hf : Function.Injective f := by
+    intro i j hij
+    have h := congrArg Fin.val hij
+    have h₁ := hi i
+    have h₂ := hi j
+    dsimp [f] at h
+    apply Subtype.ext
+    apply Fin.ext
+    omega
+  have hc : S.card ≤ L := by
+    simpa only [Fintype.card_coe,Fintype.card_fin] using Fintype.card_le_of_injective f hf
+  calc
+    _ ≤ ∑i : Fin N,if i∈S then (1:ℝ) else 0 := by
+      apply Finset.sum_le_sum
+      intro i _
+      by_cases h : i∈S
+      · rw [ite_eq_left h]; split_ifs <;> norm_num
+      · have h' : ¬(a ≤ i.val ∧ i.val < a+L) := by simpa only [S,Finset.mem_filter,Finset.mem_univ,true_and] using h
+        simp [templateType,h',h]
+    _ = (S.card:ℝ) := by simp
+    _ ≤ L := by exact_mod_cast hc
+
+/-- Actual paired finite observations with a common complete-trace decoder
+ and a squared Hellinger bound. This auxiliary structure is constructed from
+ the source score oracle and closed under literal template substitution. -/
+structure HellingerTracePair (q : ℝ) (x y : List Bool) (E V : ℝ) where
+  Outcome : Type
+  finiteOutcome : Fintype Outcome
+  first : @FiniteLaw Outcome finiteOutcome
+  second : @FiniteLaw Outcome finiteOutcome
+  decode : Outcome → List Bool
+  first_simulation : @ApproximatesTrace Outcome finiteOutcome q first decode x E
+  second_simulation : @ApproximatesTrace Outcome finiteOutcome q second decode y E
+  bound : @FiniteLaw.hellingerSq Outcome finiteOutcome first second ≤ V
+
+def OracleTracePair.hellinger {q E V : ℝ} {x y : List Bool} {k : ℕ}
+    (C : OracleTracePair q x y k E V) (hk : 1 ≤ k) : HellingerTracePair q x y E V := by
+  let := C.finiteOutcome
+  exact {
+    Outcome := C.Outcome
+    finiteOutcome := C.finiteOutcome
+    first := C.first
+    second := C.second
+    decode := C.decode
+    first_simulation := C.first_simulation
+    second_simulation := C.second_simulation
+    bound := (C.first.hellingerSq_le_score C.second C.score C.density).trans
+      ((C.first.secondMoment_le_evenMomentSq C.score k hk).trans C.moment) }
+
+def HellingerTracePair.mix {q E V : ℝ} {x y : List Bool} (C : HellingerTracePair q x y E V)
+    (hq : q∈Set.Icc 0 1) {N : ℕ} (τ σ : Fin N → Bool) :
+    HellingerTracePair q (coordinateWord (fun i => if τ i then y else x))
+      (coordinateWord (fun i => if σ i then y else x)) ((N:ℝ)*E)
+      ((∑i,if τ i=σ i then (0:ℝ) else 1)*V) := by
+  let := C.finiteOutcome
+  have ha (ρ : Fin N → Bool) : ApproximatesTrace q
+      (FiniteLaw.coordinateProduct (fun i => if ρ i then C.second else C.first))
+      (fun z => coordinateWord (fun i => C.decode (z i)))
+      (coordinateWord (fun i => if ρ i then y else x)) ((N:ℝ)*E) := by
+    have H := coordinateProduct_approximates hq
+      (fun i => if ρ i then C.second else C.first) (fun _ => C.decode)
+      (fun i => if ρ i then y else x) (fun _ => E) (fun i => ?_)
+    · simpa only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul] using H
+    · cases ρ i <;> simp only [Bool.false_eq_true,ite_false,ite_true]
+      · exact C.first_simulation
+      · exact C.second_simulation
+  exact {
+    Outcome := Fin N → C.Outcome
+    finiteOutcome := inferInstance
+    first := FiniteLaw.coordinateProduct (fun i => if τ i then C.second else C.first)
+    second := FiniteLaw.coordinateProduct (fun i => if σ i then C.second else C.first)
+    decode := fun z => coordinateWord (fun i => C.decode (z i))
+    first_simulation := ha τ
+    second_simulation := ha σ
+    bound := FiniteLaw.hellingerSq_mixedProduct_le C.first C.second C.bound τ σ }
+
+def HellingerTracePair.weaken {q E V W : ℝ} {x y : List Bool}
+    (C : HellingerTracePair q x y E V) (h : V ≤ W) : HellingerTracePair q x y E W :=
+  { C with bound := C.bound.trans h }
+
+def HellingerTracePair.template {q E V : ℝ} {x y : List Bool} (C : HellingerTracePair q x y E V)
+    (hq : q∈Set.Icc 0 1) (N a L : ℕ) (sign : Fin L → Bool) :
+    HellingerTracePair q
+      (complementaryWords N a L sign (fun τ => if τ then y else x) false)
+      (complementaryWords N a L sign (fun τ => if τ then y else x) true) ((N:ℝ)*E) ((L:ℝ)*V) := by
+  let := C.finiteOutcome
+  let D := C.mix hq (templateType N a L sign false) (templateType N a L sign true)
+  have hV : 0 ≤ V := (FiniteLaw.hellingerSq_nonneg _ _).trans C.bound
+  exact D.weaken (mul_le_mul_of_nonneg_right (templateType_difference_count N a L sign false true) hV)
+
+theorem HellingerTracePair.power_tv {q E V : ℝ} {x y : List Bool}
+    (C : HellingerTracePair q x y E V) (hq : q∈Set.Icc 0 1)
+    (F M : ℕ) (hx : (pureAtomRun x F).length ≤ M) (hy : (pureAtomRun y F).length ≤ M) :
+    (listTraceLaw q hq (pureAtomRun x F) M hx).tv
+      (listTraceLaw q hq (pureAtomRun y F) M hy) ≤ Real.sqrt ((F:ℝ)*V)+2*(F:ℝ)*E := by
+  let := C.finiteOutcome
+  have htv := (FiniteLaw.tv_le_sqrt_hellingerSq (C.first.product F) (C.second.product F)).trans
+    (Real.sqrt_le_sqrt ((FiniteLaw.hellingerSq_product_le C.first C.second F).trans
+      (mul_le_mul_of_nonneg_left C.bound (Nat.cast_nonneg F))))
+  have H := approximations_trace_tv hq _ _ _ _ _ _ _ M hx hy
+    (C.first_simulation.power hq F) (C.second_simulation.power hq F)
+  linarith only [H,htv]
+
+end TraceReconstruction
+
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem templateType_one_count (N a L : ℕ) (sign : Fin L → Bool) (τ : Bool) :
+    (∑i : Fin N,if false=templateType N a L sign τ i then (0:ℝ) else 1) ≤ L := by
+  apply le_trans _ (templateType_difference_count N a L sign false true)
+  apply Finset.sum_le_sum
+  intro i _
+  by_cases h : a ≤ i.val ∧ i.val < a+L
+  · simp only [templateType,dite_eq_left h]
+    cases hs : sign ⟨i.val-a,by omega⟩ <;> cases τ <;> norm_num
+  · simp [templateType,h]
+
+def HellingerTracePair.pure_template {q E V : ℝ} {x y : List Bool}
+    (C : HellingerTracePair q x y E V) (hq : q∈Set.Icc 0 1)
+    (N a L : ℕ) (sign : Fin L → Bool) (τ : Bool) :
+    HellingerTracePair q (pureAtomRun x N)
+      (complementaryWords N a L sign (fun υ => if υ then y else x) τ) ((N:ℝ)*E) ((L:ℝ)*V) := by
+  let := C.finiteOutcome
+  let D := C.mix hq (fun _ : Fin N => false) (templateType N a L sign τ)
+  have hV : 0 ≤ V := (FiniteLaw.hellingerSq_nonneg _ _).trans C.bound
+  have W := D.weaken (mul_le_mul_of_nonneg_right (templateType_one_count N a L sign τ) hV)
+  simp only [Bool.false_eq_true,ite_false,coordinateWord_constant] at W
+  exact W
+
+theorem bool_word_eta (w : Bool → List Bool) : (fun τ => if τ then w true else w false)=w := by
+  funext τ
+  cases τ <;> rfl
+
+/-- Literal dense substitution of the old auxiliary laws, used only for
+ overlap comparison. Statistical cost is paid only for differing motif
+ leaves, not the filler atoms. -/
+def sourceDensePair {q E V : ℝ} (hq : q∈Set.Icc 0 1) (s : ℕ) (P : ℕ → ℕ)
+    (atom : Bool → List Bool) (C : HellingerTracePair q (atom false) (atom true) E V) :
+    (j : ℕ) → HellingerTracePair q (sourceBlockWord s P atom j false)
+      (sourceBlockWord s P atom j true) ((sourceBlockLength P 1 j:ℝ)*E) (((2^s:ℕ):ℝ)^j*V)
+  | 0 => by simpa only [sourceBlockWord,sourceBlockLength,Nat.cast_one,pow_zero,one_mul] using C
+  | j+1 => by
+      have D := (sourceDensePair hq s P atom C j).template hq (P j) (P j/2-1) (2^s) (prouhetBool s)
+      rw [bool_word_eta] at D
+      simpa only [sourceBlockWord,sourceBlockLength,Nat.cast_mul,pow_succ,
+        mul_assoc,mul_left_comm,mul_comm] using D
+
+theorem HellingerTracePair.lifted_power_overlap {q E V p C₀ : ℝ} {w z : List Bool}
+    {h P F : ℕ} (O : HellingerTracePair q (pureAtomRun w P) z E V)
+    (hq : q∈Set.Icc 0 1) (hh : 1 ≤ h) (hP : 1 ≤ P)
+    (hp : p∈Set.Ioc 0 (1/8)) (hC : 1073741824 ≤ p*C₀) (hF : C₀*h*P ≤ F)
+    (hov : p ≤ listTraceOverlap q hq (pureAtomRun w h) (pureAtomRun w (h+1))) :
+    (2/3:ℝ)-(Real.sqrt ((F:ℝ)*V)+2*(F:ℝ)*E)-
+      (Real.sqrt (((F+1:ℕ):ℝ)*V)+2*((F+1:ℕ):ℝ)*E) ≤
+      listTraceOverlap q hq (pureAtomRun z F) (pureAtomRun z (F+1)) := by
+  let a := pureAtomRun (pureAtomRun w P) F
+  let b := pureAtomRun (pureAtomRun w P) (F+1)
+  let x := pureAtomRun z F
+  let y := pureAtomRun z (F+1)
+  let M := max (max a.length b.length) (max x.length y.length)
+  have ha : a.length ≤ M := (le_max_left _ _).trans (le_max_left _ _)
+  have hb : b.length ≤ M := (le_max_right _ _).trans (le_max_left _ _)
+  have hx : x.length ≤ M := (le_max_left _ _).trans (le_max_right _ _)
+  have hy : y.length ≤ M := (le_max_right _ _).trans (le_max_right _ _)
+  have ht1 := O.power_tv hq F M ha hx
+  have ht2 := O.power_tv hq (F+1) M hb hy
+  have hol := listTraceOverlap_lift hh hP hp hq hC hF w hov
+  have he : F*P+P=(F+1)*P := by ring
+  have heA : pureAtomRun w (F*P)=a := (pureAtomRun_mul w P F).symm
+  have heB : pureAtomRun w (F*P+P)=b := by
+    rw [he]; exact (pureAtomRun_mul w P (F+1)).symm
+  rw [heA,heB] at hol
+  rw [listTraceOverlap_ambient hq _ _ M ha hb,FiniteLaw.overlap_eq_one_sub_tv] at hol
+  rw [listTraceOverlap_ambient hq _ _ M hx hy,FiniteLaw.overlap_eq_one_sub_tv]
+  have H := (FiniteLaw.tv_triangle (listTraceLaw q hq x M hx) (listTraceLaw q hq a M ha)
+    (listTraceLaw q hq y M hy)).trans
+    (add_le_add le_rfl (FiniteLaw.tv_triangle (listTraceLaw q hq a M ha)
+      (listTraceLaw q hq b M hb) (listTraceLaw q hq y M hy)))
+  rw [FiniteLaw.tv_symm (listTraceLaw q hq x M hx) (listTraceLaw q hq a M ha)] at H
+  linarith only [H,ht1,ht2,hol]
+
+theorem HellingerTracePair.lifted_overlap_quarter {q E V p C₀ : ℝ} {w z : List Bool}
+    {h P F : ℕ} (O : HellingerTracePair q (pureAtomRun w P) z E V)
+    (hq : q∈Set.Icc 0 1) (hE : 0 ≤ E) (hh : 1 ≤ h) (hP : 1 ≤ P)
+    (hp : p∈Set.Ioc 0 (1/8)) (hC : 1073741824 ≤ p*C₀) (hF : C₀*h*P ≤ F)
+    (hov : p ≤ listTraceOverlap q hq (pureAtomRun w h) (pureAtomRun w (h+1)))
+    (hsmall : Real.sqrt (((F+1:ℕ):ℝ)*V)+2*((F+1:ℕ):ℝ)*E ≤ 1/4) :
+    p ≤ listTraceOverlap q hq (pureAtomRun z F) (pureAtomRun z (F+1)) := by
+  let := O.finiteOutcome
+  have hV : 0 ≤ V := (FiniteLaw.hellingerSq_nonneg _ _).trans O.bound
+  have ht := O.lifted_power_overlap hq hh hP hp hC hF hov
+  have hm : Real.sqrt ((F:ℝ)*V)+2*(F:ℝ)*E ≤
+      Real.sqrt (((F+1:ℕ):ℝ)*V)+2*((F+1:ℕ):ℝ)*E := by
+    apply add_le_add
+    · apply Real.sqrt_le_sqrt
+      exact mul_le_mul_of_nonneg_right (by norm_num) hV
+    · exact mul_le_mul_of_nonneg_right (by norm_num) hE
+  linarith only [hm,hsmall,ht,hp.2]
+
+/-- Finite nested overlap induction at the main (post-warmup) dimensions.
+ Every closeness premise is an explicit scalar estimate on the old oracle;
+ the repeated-word overlaps themselves are proved, not assumed. -/
+theorem source_nested_overlap {q E V p C₀ : ℝ} (hq : q∈Set.Icc 0 1)
+    (hE : 0 ≤ E) (s : ℕ) (P h : ℕ → ℕ) (atom : Bool → List Bool)
+    (C : HellingerTracePair q (atom false) (atom true) E V) (b : ℕ)
+    (hp : p∈Set.Ioc 0 (1/8)) (hC : 1073741824 ≤ p*C₀)
+    (hcount : ∀j ≤ b,1 ≤ h j) (hP : ∀j < b,1 ≤ P j)
+    (hlift : ∀j < b,C₀*h j*P j ≤ h (j+1))
+    (hsmall : ∀j < b,
+      Real.sqrt (((h (j+1)+1:ℕ):ℝ)*(((2^s:ℕ):ℝ)^(j+1)*V))+
+        2*((h (j+1)+1:ℕ):ℝ)*((sourceBlockLength P 1 (j+1):ℝ)*E) ≤ 1/4)
+    (hbase : p ≤ listTraceOverlap q hq (pureAtomRun (atom false) (h 0))
+      (pureAtomRun (atom false) (h 0+1))) :
+    ∀j ≤ b,p ≤ listTraceOverlap q hq (pureAtomRun (sourceBlockWord s P atom j false) (h j))
+      (pureAtomRun (sourceBlockWord s P atom j false) (h j+1)) := by
+  intro j hj
+  induction j with
+  | zero => exact hbase
+  | succ j ih =>
+    have hjb : j < b := by omega
+    have D := (sourceDensePair hq s P atom C j).pure_template hq (P j) (P j/2-1) (2^s) (prouhetBool s) false
+    rw [bool_word_eta] at D
+    have H := D.lifted_overlap_quarter hq (by positivity) (hcount j (by omega)) (hP j hjb)
+      hp hC (hlift j hjb) (ih (by omega)) ?_
+    · exact H
+    · simpa only [sourceBlockLength,Nat.cast_mul,pow_succ,mul_assoc,mul_left_comm,mul_comm]
+        using hsmall j hjb
+
+end TraceReconstruction
+
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+
+theorem sourceFitTail_nonneg (T n : ℕ → ℕ) (b : ℕ) : 0 ≤ sourceFitTail T n b := by
+  induction b with
+  | zero => rfl
+  | succ b ih =>
+    cases b with
+    | zero => rfl
+    | succ b => exact mul_nonneg (Nat.cast_nonneg _) (add_nonneg (by positivity) ih)
+
+theorem sourceEffectiveHiddenTail_nonneg (T n : ℕ → ℕ) (b : ℕ) (ω : ℝ) :
+    0 ≤ sourceEffectiveHiddenTail T n b ω := by
+  apply Finset.sum_nonneg
+  intro v _
+  split_ifs <;> positivity
+
+theorem sourceClippingCost_nonneg (T n : ℕ → ℕ) (s b k : ℕ) (ω a L : ℝ) (ha : 0 ≤ a) :
+    0 ≤ sourceClippingCost T n s b k ω a L := by
+  exact add_nonneg (add_nonneg (sourceEffectiveHiddenTail_nonneg _ _ _ _)
+    (div_nonneg (sourceFitTail_nonneg _ _ _) ha)) (Real.sqrt_nonneg _)
+
+/-- The two-dimensional warmup uses the actual one-window cancellation
+ oracle at its second nested overlap step. No unproved side overlap is
+ inserted into the recursive source observation. -/
+theorem source_warmup_nested_overlap {q E V p C₀ : ℝ} (hq : q∈Set.Icc 0 1)
+    (hE : 0 ≤ E) (s k : ℕ) (P h : ℕ → ℕ) (atom : Bool → List Bool)
+    (C : OracleTracePair q (atom false) (atom true) k E V)
+    (hp : p∈Set.Ioc 0 (1/8)) (hC : 1073741824 ≤ p*C₀)
+    (hcount : ∀j ≤ 1,1 ≤ h j) (hP : ∀j < 2,1 ≤ P j)
+    (hlift : ∀j < 2,C₀*h j*P j ≤ h (j+1))
+    (hfirst : Real.sqrt (((h 1+1:ℕ):ℝ)*((2^s:ℕ):ℝ)*V)+
+      2*((h 1+1:ℕ):ℝ)*((P 0:ℝ)*E) ≤ 1/4)
+    (hbase : p ≤ listTraceOverlap q hq (pureAtomRun (atom false) (h 0))
+      (pureAtomRun (atom false) (h 0+1)))
+    (T t : ℕ) (hT : 24 ≤ T) (he : Even T) (hMotif : 24*2^s ≤ T)
+    (hroot : 4*(t+1) ≤ T) (hfit : t*(2*h 0+1)+T ≤ P 0)
+    (a η L : ℝ) (ha : 0 < a) (hL : 0 < L) (hk : 1 ≤ k) (hV : V ≤ L^2)
+    (hscale : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount (fun _ => T) s 0 ≤ 1/2)
+    (hη : hiddenSmoothingBound t s p ≤ η)
+    (herror : sourceClippingCost (fun _ => T) (fun _ => t) s 0 k p a L ≤ 1/2)
+    (hsecond :
+      Real.sqrt (((h 2+1:ℕ):ℝ)*((2^s:ℕ):ℝ)*
+        (64*sourceSharpEnergyBudget (fun _ => T) (fun _ => t) s 0 2 p η a ((6*(2*k:ℕ):ℝ)*L)))+
+      2*((h 2+1:ℕ):ℝ)*((P 1:ℝ)*
+        (sourceClippingCost (fun _ => T) (fun _ => t) s 0 k p a L+(T:ℝ)*E)) ≤ 1/4) :
+    ∀j ≤ 2,p ≤ listTraceOverlap q hq (pureAtomRun (sourceBlockWord s P atom j false) (h j))
+      (pureAtomRun (sourceBlockWord s P atom j false) (h j+1)) := by
+  have hp' : p∈Set.Ioo 0 1 := ⟨hp.1,lt_of_le_of_lt hp.2 (by norm_num)⟩
+  have D := (C.hellinger hk).pure_template hq (P 0) (P 0/2-1) (2^s) (prouhetBool s) false
+  rw [bool_word_eta] at D
+  have h1 := D.lifted_overlap_quarter hq (by positivity) (hcount 0 (by omega))
+    (hP 0 (by omega)) hp hC (hlift 0 (by omega)) hbase ?_
+  · obtain ⟨W⟩ := actual_one_window_oracle C hq hE s (P 0) T t (h 0) 2 hT he hMotif
+      hroot hfit p a η L ha hL hp' hbase hk hV hscale hη herror
+    rw [bool_word_eta] at W
+    have W' := (W.hellinger hk).pure_template hq (P 1) (P 1/2-1) (2^s) (prouhetBool s) false
+    have heword : (fun τ => if τ then
+        complementaryWords (P 0) (P 0/2-1) (2^s) (prouhetBool s) atom true else
+        complementaryWords (P 0) (P 0/2-1) (2^s) (prouhetBool s) atom false) =
+        sourceBlockWord s P atom 1 := bool_word_eta _
+    rw [heword] at W'
+    have hE' : 0 ≤ sourceClippingCost (fun _ => T) (fun _ => t) s 0 k p a L+(T:ℝ)*E := by
+      exact add_nonneg (sourceClippingCost_nonneg _ _ _ _ _ _ _ _ ha.le) (by positivity)
+    have h2 := W'.lifted_overlap_quarter hq (mul_nonneg (Nat.cast_nonneg _) hE')
+      (hcount 1 le_rfl) (hP 1 (by omega)) hp hC (hlift 1 (by omega)) h1 ?_
+    · intro j hj
+      interval_cases j
+      · exact hbase
+      · exact h1
+      · exact h2
+    · simpa only [mul_assoc] using hsecond
+  · simpa only [mul_assoc] using hfirst
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem OracleTracePair.error_nonneg {q E V : ℝ} {x y : List Bool} {k : ℕ}
+    (C : OracleTracePair q x y k E V) : 0 ≤ E := by
+  let := C.finiteOutcome
+  have h := C.first_simulation (fun _ => 0) (fun _ => ⟨le_rfl,zero_le_one⟩)
+  exact (abs_nonneg _).trans h
+
+/-- Finite main-block production: actual oracle, complete trace projection,
+ and all side overlaps follow from arithmetic budgets and the OLD invariant.
+ In particular none of the new side laws or their overlap are hypotheses. -/
+theorem actual_main_block {q E V ω C₀ : ℝ} (hq : q∈Set.Icc 0 1)
+    (s : ℕ) (P T n h : ℕ → ℕ) (atom : Bool → List Bool) (b R k : ℕ)
+    (C : OracleTracePair q (atom false) (atom true) k E V)
+    (H : SourceLayoutMargins s P T n h b)
+    (hTpos : ∀j,0 < T j) (hP : P b=n b*(2*h b+1)+T b)
+    (hn : ∀j < b,0 < n j) (hT : ∀j ≤ b,24 ≤ T j) (he : ∀j ≤ b,Even (T j))
+    (hMotif : ∀j ≤ b,24*2^s ≤ T j) (hroot : 4*(n b+1) ≤ T b)
+    (a η L : ℝ) (ha : 0 < a) (hL : 0 < L) (hω : ω∈Set.Ioc 0 (1/8))
+    (hk : 1 ≤ k) (hV : V ≤ L^2)
+    (hscale : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b ≤ 1/2)
+    (hη : ∀j < b+1,hiddenSmoothingBound (n j) s ω ≤ η)
+    (herror : sourceClippingCost T n s b k ω a L ≤ 1/2)
+    (hC : 1073741824 ≤ ω*C₀) (hcount : ∀j ≤ b+1,1 ≤ h j)
+    (hpos : ∀j < b+1,1 ≤ P j)
+    (hlift : ∀j < b+1,C₀*h j*P j ≤ h (j+1))
+    (hsmall : ∀j < b+1,
+      Real.sqrt (((h (j+1)+1:ℕ):ℝ)*(((2^s:ℕ):ℝ)^(j+1)*V))+
+        2*((h (j+1)+1:ℕ):ℝ)*((sourceBlockLength P 1 (j+1):ℝ)*E) ≤ 1/4)
+    (hbase : ω ≤ listTraceOverlap q hq (pureAtomRun (atom false) (h 0))
+      (pureAtomRun (atom false) (h 0+1))) :
+    Nonempty (OracleTracePair q (sourceBlockWord s P atom (b+1) false)
+      (sourceBlockWord s P atom (b+1) true) k
+      (sourceClippingCost T n s b k ω a L+(SourceLeafCount T (b+1):ℝ)*E+sourceFitTail T n (b+1))
+      (64*sourceSharpEnergyBudget T n s b R ω η a ((6*(2*k:ℕ):ℝ)*L))) ∧
+    (∀j ≤ b+1,ω ≤ listTraceOverlap q hq
+      (pureAtomRun (sourceBlockWord s P atom j false) (h j))
+      (pureAtomRun (sourceBlockWord s P atom j false) (h j+1))) := by
+  let := C.finiteOutcome
+  have hov := source_nested_overlap hq C.error_nonneg s P h atom (C.hellinger hk) (b+1)
+    hω hC hcount hpos hlift hsmall hbase
+  refine ⟨?_,hov⟩
+  have hm : C.first.expect (fun z => |C.score z|^(2*k)) ≤ L^(2*k) := by
+    have H := C.first.expect_even_pow_le_of_moment C.score k hk (C.moment.trans hV)
+    simpa only [pow_mul] using H
+  exact actual_source_oracle hq C.error_nonneg s P T n h atom b R k H hTpos hP hn hT he hMotif hroot
+    C.first C.second C.decode C.score C.density C.likelihood_nonneg C.first_simulation C.second_simulation
+    ω a η L ha hL ⟨hω.1,lt_of_le_of_lt hω.2 (by norm_num)⟩
+    (fun j hj => hov j (by omega)) hk hm hscale hη herror
+
+/-- Conversion of a produced genuine oracle into the manuscript invariant.
+ The scalar bounds here are exactly those to be discharged by scale margins. -/
+def OracleTracePair.toConcrete {q E V N Δ d ω F : ℝ} {x y : List Bool} {k i h : ℕ}
+    (C : OracleTracePair q x y k E V) (hq : q∈Set.Icc 0 1)
+    (hlen : x.length=y.length) (hne : x≠y) (hlo : N/2 ≤ (x.length:ℝ))
+    (hhi : (x.length:ℝ) ≤ N) (herror : E/(x.length:ℝ) ≤ ((i:ℝ)+1)*Δ)
+    (hV : V ≤ N^(-d)) (hhlo : x.length ≤ h) (hhhi : (h:ℝ) ≤ F*(x.length:ℝ))
+    (hov : ω ≤ listTraceOverlap q hq (pureAtomRun x h) (pureAtomRun x (h+1))) :
+    ConcreteLevelPair q hq N Δ d ω F i k := {
+  left := x
+  right := y
+  equalLength := hlen
+  different := hne
+  length_lower := hlo
+  length_upper := hhi
+  Outcome := C.Outcome
+  finiteOutcome := C.finiteOutcome
+  firstLaw := C.first
+  secondLaw := C.second
+  decode := C.decode
+  error := E
+  error_nonneg := C.error_nonneg
+  normalized_error := herror
+  first_simulation := C.first_simulation
+  second_simulation := C.second_simulation
+  score := C.score
+  density := C.density
+  score_bound := C.moment.trans hV
+  overlapCount := h
+  overlapCount_lower := hhlo
+  overlapCount_upper := hhhi
+  power_overlap := hov }
+
+end TraceReconstruction
+
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/TraceReconstruction/Anchors.lean b/lean/OAI/Probability/TraceReconstruction/Anchors.lean
index 3a21e1c9468c37a72d19bfea35f25cad358fd47a..0dd20dc0b673d40694df52a8ec2f642dca04d522 100644
--- a/lean/OAI/Probability/TraceReconstruction/Anchors.lean
+++ b/lean/OAI/Probability/TraceReconstruction/Anchors.lean
@@ -1,1229 +1,1229 @@
-import OAI.Probability.TraceReconstruction.LeafOracle
-
-namespace OAI
-
-/-!
-# Anchors
-
-Forced anchors, extra-leaf closeness, and correlated group square-sum estimates.
--/
-
-open scoped ENNReal
-noncomputable section
-
-open Filter
-open scoped Topology
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-open FiniteLaw
-
-namespace FiniteLaw
-/-- Two distinct actual child coordinates are independent. This does not
-assert independence of their shared parent-global jitter. -/
-theorem coordinateProduct_expect_two {α : Type*} [Fintype α] {n : ℕ}
-    (P : Fin n → FiniteLaw α) (a c : Fin n) (hne : a≠c) (f h : α → ℝ) :
-    (coordinateProduct P).expect (fun x => f (x a)*h (x c)) = (P a).expect f*(P c).expect h := by
-  let F : Fin n → α → ℝ := fun j x => if j=a then f x else if j=c then h x else 1
-  have hp : ∀x : Fin n → α, (∏j,F j (x j)) = f (x a)*h (x c) := by
-    intro x
-    rw [Finset.prod_eq_mul a c hne]
-    · simp [F,hne.symm]
-    · intro j hj hjs
-      simp [F,hjs.1,hjs.2]
-    · simp
-    · simp
-  rw [show (fun x : Fin n → α => f (x a)*h (x c)) = (fun x => ∏j,F j (x j)) by
-    funext x; exact (hp x).symm,coordinateProduct_expect_product]
-  rw [Finset.prod_eq_mul a c hne]
-  · simp [F,hne.symm]
-  · intro j hj hjs
-    simp [F,hjs.1,hjs.2,expect_const]
-  · simp
-  · simp
-
-/-- Exact local/global/children factorization with the parent-global draw
-outside the product of child expectations. -/
-theorem expect_prod_two {α β γ : Type*} [Fintype α] [Fintype β] [Fintype γ]
-    {n : ℕ} (P : FiniteLaw α) (Q : FiniteLaw β) (R : Fin n → FiniteLaw γ)
-    (a c : Fin n) (hne : a≠c) (f : α → ℝ) (h k : β → γ → ℝ) :
-    (P.prod (Q.prod (coordinateProduct R))).expect (fun x =>
-      f x.1 * (h x.2.1 (x.2.2 a)*k x.2.1 (x.2.2 c))) =
-      P.expect f * Q.expect (fun z => (R a).expect (h z)*(R c).expect (k z)) := by
-  simp_rw [expect_prod,expect_const_mul,coordinateProduct_expect_two R a c hne]
-  rw [expect_mul_const]
-end FiniteLaw
-
-/-- A concrete binary selection of labels in the complete index tree.
-Unlike a free abstract shape this retains every actual column label. -/
-def BinaryWindowLabels (T : ℕ → ℕ) : ℕ → Type
-  | 0 => PUnit
-  | b+1 => Fin (T b) × Fin (T b) × BinaryWindowLabels T b × BinaryWindowLabels T b
-
-noncomputable instance binaryWindowLabelsFintype (T : ℕ → ℕ) :
-    (b : ℕ) → Fintype (BinaryWindowLabels T b)
-  | 0 => inferInstanceAs (Fintype PUnit)
-  | b+1 => by
-      letI := binaryWindowLabelsFintype T b
-      exact inferInstanceAs (Fintype (Fin (T b) × Fin (T b) × BinaryWindowLabels T b × BinaryWindowLabels T b))
-
-/-- The forced children are distinct at every split. Central restrictions
-are separate deterministic predicates, not conditioning events. -/
-def BinaryWindowLabels.distinct (T : ℕ → ℕ) : (b : ℕ) → BinaryWindowLabels T b → Prop
-  | 0, _ => True
-  | b+1, p => p.1≠p.2.1 ∧ distinct T b p.2.2.1 ∧ distinct T b p.2.2.2
-
-/-- Product of the first-child unsigned anchor at every forced nonleaf,
-evaluated using the literal complete-tree orientation assignment. -/
-def binaryAnchorKernel (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ) :
-    (b : ℕ) → BinaryWindowLabels T b → WindowAssignments T n g Bool b → ℤ → ℝ
-  | 0, _, _, _ => 1
-  | b+1, p, bits, incoming =>
-      motifIndicator s ((p.1.val:ℤ)-center b+(∑i,(bitIndex (bits.1 i):ℤ))+incoming) *
-      (binaryAnchorKernel T n g s center globalShift b p.2.2.1 (bits.2.2 p.1)
-        (globalShift b bits.2.1 p.1) *
-       binaryAnchorKernel T n g s center globalShift b p.2.2.2 (bits.2.2 p.2.1)
-        (globalShift b bits.2.1 p.2.1))
-
-def binaryAnchorAverage (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (b : ℕ) (p : BinaryWindowLabels T b) (d : WindowAssignments T n g (Option Bool) b)
-    (incoming : ℤ) : ℝ :=
-  (windowPosteriorLaw T n g b d).expect (fun bits =>
-    binaryAnchorKernel T n g s center globalShift b p bits incoming)
-
-theorem binaryAnchorAverage_zero (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (p : BinaryWindowLabels T 0) (d : WindowAssignments T n g (Option Bool) 0) (incoming : ℤ) :
-    binaryAnchorAverage T n g s center globalShift 0 p d incoming = 1 := by
-  apply expect_const
-
-theorem binaryAnchorAverage_succ (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (b : ℕ) (p : BinaryWindowLabels T (b+1)) (d : WindowAssignments T n g (Option Bool) (b+1))
-    (incoming : ℤ) (hd : p.1≠p.2.1) :
-    binaryAnchorAverage T n g s center globalShift (b+1) p d incoming =
-      (posteriorArray d.1).expect (fun loc =>
-        motifIndicator s ((p.1.val:ℤ)-center b+(∑i,(bitIndex (loc i):ℤ))+incoming)) *
-      (posteriorArray d.2.1).expect (fun glob =>
-        binaryAnchorAverage T n g s center globalShift b p.2.2.1 (d.2.2 p.1)
-          (globalShift b glob p.1) *
-        binaryAnchorAverage T n g s center globalShift b p.2.2.2 (d.2.2 p.2.1)
-          (globalShift b glob p.2.1)) := by
-  exact FiniteLaw.expect_prod_two (posteriorArray d.1) (posteriorArray d.2.1)
-    (fun i => windowPosteriorLaw T n g b (d.2.2 i)) p.1 p.2.1 hd
-    (fun loc => motifIndicator s ((p.1.val:ℤ)-center b+(∑i,(bitIndex (loc i):ℤ))+incoming))
-    (fun glob bits => binaryAnchorKernel T n g s center globalShift b p.2.2.1 bits
-      (globalShift b glob p.1))
-    (fun glob bits => binaryAnchorKernel T n g s center globalShift b p.2.2.2 bits
-      (globalShift b glob p.2.1))
-
-/-- Local unsigned anchor averaged against its actual known/hidden flags. -/
-def localAnchorAverage {m : ℕ} (d : Fin m → Option Bool) (s : ℕ) (z : ℤ) : ℝ :=
-  (posteriorArray d).expect (fun bits => motifIndicator s (z+∑i,(bitIndex (bits i):ℤ)))
-
-theorem localAnchorAverage_kernel {m : ℕ} (d : Fin m → Option Bool) (s : ℕ) (z : ℤ) :
-    localAnchorAverage d s z = windowKernel (∑i,hiddenTrial (d i)) (2^s)
-      (z+∑i,(visibleOnes (d i):ℤ)) := by
-  have h := posterior_unsigned_motif d s z
-  unfold localAnchorAverage posteriorArray coordinateProduct expect
-  convert h using 1
-  congr 2
-  exact Subsingleton.elim _ _
-
-/-- Product of all lower sibling-group factors. Each group still averages
-its own single common global draw, not two independent global draws. -/
-def binaryAnchorBelow (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ) :
-    (b : ℕ) → BinaryWindowLabels T (b+1) → WindowAssignments T n g (Option Bool) (b+1) → ℝ
-  | 0, _, _ => 1
-  | b+1, p, d =>
-      (posteriorArray d.2.1).expect (fun glob =>
-        localAnchorAverage (d.2.2 p.1).1 s
-          ((p.2.2.1.1.val:ℤ)-center b+globalShift (b+1) glob p.1) *
-        localAnchorAverage (d.2.2 p.2.1).1 s
-          ((p.2.2.2.1.val:ℤ)-center b+globalShift (b+1) glob p.2.1)) *
-      (binaryAnchorBelow T n g s center globalShift b p.2.2.1 (d.2.2 p.1) *
-       binaryAnchorBelow T n g s center globalShift b p.2.2.2 (d.2.2 p.2.1))
-
-/-- Exact factorization of the forced anchor expectation into its root
-kernel and all correlated sibling groups, at the real metadata/labels. -/
-theorem binaryAnchorAverage_factor (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (b : ℕ) (p : BinaryWindowLabels T (b+1)) (d : WindowAssignments T n g (Option Bool) (b+1))
-    (incoming : ℤ) (hd : BinaryWindowLabels.distinct T (b+1) p) :
-    binaryAnchorAverage T n g s center globalShift (b+1) p d incoming =
-      localAnchorAverage d.1 s ((p.1.val:ℤ)-center b+incoming) *
-        binaryAnchorBelow T n g s center globalShift b p d := by
-  induction b generalizing incoming with
-  | zero =>
-    rw [binaryAnchorAverage_succ T n g s center globalShift _ p d incoming hd.1]
-    simp only [binaryAnchorAverage_zero,expect_const,binaryAnchorBelow,mul_one]
-    unfold localAnchorAverage
-    congr 1
-    funext loc
-    congr 1
-    omega
-  | succ b ih =>
-    have hl := fun incoming => ih p.2.2.1 (d.2.2 p.1) incoming hd.2.1
-    have hr := fun incoming => ih p.2.2.2 (d.2.2 p.2.1) incoming hd.2.2
-    have hroot : (posteriorArray d.1).expect (fun loc =>
-        motifIndicator s ((p.1.val:ℤ)-center (b+1)+(∑i,(bitIndex (loc i):ℤ))+incoming)) =
-        localAnchorAverage d.1 s ((p.1.val:ℤ)-center (b+1)+incoming) := by
-      apply congrArg (FiniteLaw.expect (posteriorArray d.1))
-      funext loc
-      congr 1
-      omega
-    have hh : (posteriorArray d.2.1).expect (fun glob =>
-        binaryAnchorAverage T n g s center globalShift (b+1) p.2.2.1 (d.2.2 p.1)
-          (globalShift (b+1) glob p.1) *
-        binaryAnchorAverage T n g s center globalShift (b+1) p.2.2.2 (d.2.2 p.2.1)
-          (globalShift (b+1) glob p.2.1)) =
-        binaryAnchorBelow T n g s center globalShift (b+1) p d := by
-      calc
-        _ = (posteriorArray d.2.1).expect (fun glob =>
-          (localAnchorAverage (d.2.2 p.1).1 s
-            ((p.2.2.1.1.val:ℤ)-center b+globalShift (b+1) glob p.1) *
-          localAnchorAverage (d.2.2 p.2.1).1 s
-            ((p.2.2.2.1.val:ℤ)-center b+globalShift (b+1) glob p.2.1)) *
-          (binaryAnchorBelow T n g s center globalShift b p.2.2.1 (d.2.2 p.1) *
-           binaryAnchorBelow T n g s center globalShift b p.2.2.2 (d.2.2 p.2.1))) := by
-            apply congrArg (FiniteLaw.expect (posteriorArray d.2.1))
-            funext glob
-            exact (congrArg₂ (·*·) (hl _) (hr _)).trans (by ring)
-        _ = _ := expect_mul_const _ _ _
-    exact (binaryAnchorAverage_succ T n g s center globalShift _ p d incoming hd.1).trans
-      (congrArg₂ (·*·) hroot hh)
-
-/-- One plateau group's local means are the genuine shared-global joint
-anchor, with exactly the affine shift appearing in the layout. -/
-theorem localAnchor_pair_plateau {mg m₁ m₂ : ℕ}
-    (dg : Fin mg → Option Bool) (d₁ : Fin m₁ → Option Bool) (d₂ : Fin m₂ → Option Bool)
-    (s : ℕ) (z₁ z₂ shift : ℤ) :
-    (posteriorArray dg).expect (fun glob =>
-      localAnchorAverage d₁ s (z₁+(∑i,(bitIndex (glob i):ℤ))-shift) *
-      localAnchorAverage d₂ s (z₂+(∑i,(bitIndex (glob i):ℤ))-shift)) =
-    posteriorJointAnchors dg d₁ d₂ s (z₁-shift) (z₂-shift) := by
-  unfold localAnchorAverage posteriorArray coordinateProduct expect posteriorJointAnchors
-  have he : ∀a b c d : ℤ,(a+b-c)+d = (a-c)+b+d := by intros; omega
-  simp_rw [he,mul_assoc]
-  congr 2
-  · exact Subsingleton.elim _ _
-  · funext bg
-    congr 1
-    congr 1
-    · congr 2 ; exact Subsingleton.elim _ _
-    · congr 2 ; exact Subsingleton.elim _ _
-
-end TraceReconstruction
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-open FiniteLaw
-
-/-- A role-selected forced subtree remembers the actual complete-tree labels,
-not merely its abstract binary shape. -/
-def forcedWindowLabels (T : ℕ → ℕ) : (b : ℕ) →
-    (S : Finset (WindowLeaf T (b+1))) → (a : OccupiedWindowChoice T b S) →
-    (occupiedWindowTerm T b S a).Forced → BinaryWindowLabels T (b+1)
-  | 0, S, _, .node i j _ _ _ _ _ =>
-      ((S.equivFin.symm i).val.1,(S.equivFin.symm j).val.1,PUnit.unit,PUnit.unit)
-  | b+1, S, a, .node i j _ _ _ left right =>
-      (((selectedChildren S).equivFin.symm i).val,
-       ((selectedChildren S).equivFin.symm j).val,
-       forcedWindowLabels T b (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val)
-         (a.1 i) left,
-       forcedWindowLabels T b (selectedChildLeaves S ((selectedChildren S).equivFin.symm j).val)
-         (a.1 j) right)
-
-theorem forcedWindowLabels_distinct (T : ℕ → ℕ) (b : ℕ)
-    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S)
-    (F : (occupiedWindowTerm T b S a).Forced) :
-    BinaryWindowLabels.distinct T (b+1) (forcedWindowLabels T b S a F) := by
-  induction b with
-  | zero =>
-    cases F with
-    | node i j hne hi hj left right =>
-      refine ⟨?_,trivial,trivial⟩
-      intro he
-      apply hne
-      apply S.equivFin.symm.injective
-      apply Subtype.ext
-      exact Prod.ext he (Subsingleton.elim (α := Fin 1) _ _)
-  | succ b ih =>
-    cases F with
-    | node i j hne hi hj left right =>
-      refine ⟨?_,ih _ _ left,ih _ _ right⟩
-      intro he
-      exact hne ((selectedChildren S).equivFin.symm.injective (Subtype.ext he))
-
-theorem motifIndicator_nonneg (s : ℕ) (z : ℤ) : 0 ≤ motifIndicator s z := by
-  unfold motifIndicator
-  split_ifs <;> norm_num
-
-theorem motifIndicator_le_one (s : ℕ) (z : ℤ) : motifIndicator s z≤1 := by
-  unfold motifIndicator
-  split_ifs <;> norm_num
-
-theorem binaryAnchorKernel_nonneg (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (b : ℕ) (p : BinaryWindowLabels T b) (bits : WindowAssignments T n g Bool b)
-    (incoming : ℤ) : 0≤binaryAnchorKernel T n g s center globalShift b p bits incoming := by
-  induction b generalizing incoming with
-  | zero => exact zero_le_one
-  | succ b ih =>
-    exact mul_nonneg (motifIndicator_nonneg _ _) (mul_nonneg (ih _ _ _) (ih _ _ _))
-
-theorem bottomRoleWeight_abs_le_indicator (χ σ : ℝ) (h : IsMotifValue χ σ) (a : Fin 3) :
-    |bottomRoleWeight χ σ a|≤χ := by
-  rcases h with ⟨rfl,rfl⟩ | ⟨rfl,rfl|rfl⟩ <;>
-    fin_cases a <;> norm_num [bottomRoleWeight]
-
-theorem internalRoleWeight_active_abs_le_indicator (odd : Prop) [Decidable odd]
-    (χ σ : ℝ) (h : IsMotifValue χ σ) (a : Fin 3) (ha : a≠0) :
-    |internalRoleWeight odd χ σ a|≤χ := by
-  rcases h with ⟨rfl,rfl⟩ | ⟨rfl,rfl|rfl⟩ <;>
-    fin_cases a <;> by_cases ho : odd <;> norm_num [internalRoleWeight,roleWeight,ho] at *
-
-theorem occupiedRootWeight_bottom_anchor (T : ℕ → ℕ) (s : ℕ)
-    (S : Finset (WindowLeaf T 1)) (a : OccupiedWindowChoice T 0 S) (X : ℤ) (i : Fin S.card) :
-    |occupiedRootWeight T s 0 S a X|≤
-      motifIndicator s (((S.equivFin.symm i).val.1.val:ℤ)-X) := by
-  exact (abs_prod_le_coordinate _ i (fun j _ =>
-    bottomRoleWeight_abs_le_one _ _ (prouhet_motif_value _ _) _)).trans
-      (bottomRoleWeight_abs_le_indicator _ _ (prouhet_motif_value _ _) _)
-
-theorem occupiedRootWeight_internal_anchor (T : ℕ → ℕ) (s b : ℕ)
-    (S : Finset (WindowLeaf T (b+1+1))) (a : OccupiedWindowChoice T (b+1) S) (X : ℤ)
-    (i : Fin (selectedChildren S).card) (hi : a.2 i≠0) :
-    |occupiedRootWeight T s (b+1) S a X|≤
-      motifIndicator s ((((selectedChildren S).equivFin.symm i).val.val:ℤ)-X) := by
-  exact (abs_prod_le_coordinate _ i (fun j _ =>
-    internalRoleWeight_abs_le_one _ _ _ (prouhet_motif_value _ _) _)).trans
-      (internalRoleWeight_active_abs_le_indicator _ _ _ (prouhet_motif_value _ _) _ hi)
-
-theorem abs_prod_le_two {ι : Type*} [Fintype ι] (f : ι → ℝ) (a c : ι) (hne : a≠c)
-    (h : ∀j,j≠a → j≠c → |f j|≤1) : |∏j,f j|≤|f a| * |f c| := by
-  rw [Finset.abs_prod]
-  calc
-    _ ≤ ∏j, if j=a then |f a| else if j=c then |f c| else 1 := by
-      apply Finset.prod_le_prod₀ (fun _ _ => abs_nonneg _)
-      intro j hj
-      by_cases hja : j=a
-      · simp [hja]
-      · by_cases hjc : j=c
-        · simp [hjc,hne.symm]
-        · simpa only [ite_eq_right hja,ite_eq_right hjc] using h j hja hjc
-    _ = _ := by
-      rw [Finset.prod_eq_mul a c hne]
-      · simp [hne.symm]
-      · intro j hj hjs
-        simp [hjs.1,hjs.2]
-      · simp
-      · simp
-
-/-- The real role term is pointwise dominated by its selected unsigned
-anchors. No independence assumption is used in this domination. -/
-theorem sourceRoleTerm_forced_anchor (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (b : ℕ) (bits : WindowAssignments T n g Bool (b+1)) (incoming : ℤ)
-    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S)
-    (F : (occupiedWindowTerm T b S a).Forced) :
-    |sourceRoleTerm T n g s center globalShift b bits incoming S a|≤
-      binaryAnchorKernel T n g s center globalShift (b+1) (forcedWindowLabels T b S a F) bits incoming := by
-  induction b generalizing incoming with
-  | zero =>
-    cases F with
-    | node i j hne hi hj left right =>
-      rw [sourceRoleTerm_zero]
-      change _ ≤ motifIndicator s (((S.equivFin.symm i).val.1.val:ℤ)-center 0+
-        (∑i,(bitIndex (bits.1 i):ℤ))+incoming)*(1*1)
-      rw [mul_one,mul_one]
-      convert occupiedRootWeight_bottom_anchor T s S a
-        (center 0-(∑i,(bitIndex (bits.1 i):ℤ))-incoming) i using 1 ; congr 1 ; omega
-  | succ b ih =>
-    cases F with
-    | node i j hne hi hj left right =>
-      rw [sourceRoleTerm_succ,abs_mul]
-      have hroot := occupiedRootWeight_internal_anchor T s b S a
-        (center (b+1)-(∑i,(bitIndex (bits.1 i):ℤ))-incoming) i hi
-      have hprod := abs_prod_le_two (fun k : Fin (selectedChildren S).card =>
-        sourceRoleTerm T n g s center globalShift b
-          (bits.2.2 ((selectedChildren S).equivFin.symm k).val)
-          (globalShift (b+1) bits.2.1 ((selectedChildren S).equivFin.symm k).val)
-          (selectedChildLeaves S ((selectedChildren S).equivFin.symm k).val) (a.1 k)) i j hne
-          (fun k _ _ => occupiedWindowWeight_abs_le_one T s _ b [] _ _)
-      have hl := ih (bits.2.2 ((selectedChildren S).equivFin.symm i).val)
-        (globalShift (b+1) bits.2.1 ((selectedChildren S).equivFin.symm i).val) _ _ left
-      have hr := ih (bits.2.2 ((selectedChildren S).equivFin.symm j).val)
-        (globalShift (b+1) bits.2.1 ((selectedChildren S).equivFin.symm j).val) _ _ right
-      have hp := mul_le_mul hl hr (abs_nonneg _) (binaryAnchorKernel_nonneg T n g s center globalShift _ _ _ _)
-      have hh := mul_le_mul hroot (hprod.trans hp) (abs_nonneg _) (motifIndicator_nonneg _ _)
-      apply hh.trans_eq
-      change motifIndicator s _ * _ = motifIndicator s
-        ((((selectedChildren S).equivFin.symm i).val.val:ℤ)-center (b+1)+
-          (∑k,(bitIndex (bits.1 k):ℤ))+incoming) * _
-      congr 2
-      omega
-
-/-- Therefore the coefficient average is bounded by the actual forced
-anchor average, with the true posterior law and shared jitter retained. -/
-theorem sourceRoleAverage_forced_anchor (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ)
-    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S)
-    (F : (occupiedWindowTerm T b S a).Forced) :
-    |sourceRoleAverage T n g s center globalShift b d incoming S a|≤
-      binaryAnchorAverage T n g s center globalShift (b+1) (forcedWindowLabels T b S a F) d incoming := by
-  exact ((windowPosteriorLaw T n g (b+1) d).abs_expect_le _).trans
-    ((windowPosteriorLaw T n g (b+1) d).expect_mono (fun bits =>
-      sourceRoleTerm_forced_anchor T n g s center globalShift b bits incoming S a F))
-
-theorem sourceRoleTerm_valid (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (b : ℕ) (bits : WindowAssignments T n g Bool (b+1)) (incoming : ℤ)
-    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S)
-    (h : sourceRoleTerm T n g s center globalShift b bits incoming S a ≠ 0) :
-    (occupiedWindowTerm T b S a).valid := by
-  have hh := (occupiedWindowShape T s
-    (sourceTreeAnchors T n g center globalShift (b+1) bits incoming) b [] S).term_valid
-    (occupiedWindowChoiceEquiv T s
-      (sourceTreeAnchors T n g center globalShift (b+1) bits incoming) b [] S a) h
-  simpa only [occupiedWindowTerm_eq] using hh
-
-theorem sourceRoleAverage_valid (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ)
-    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S)
-    (h : sourceRoleAverage T n g s center globalShift b d incoming S a ≠ 0) :
-    (occupiedWindowTerm T b S a).valid := by
-  by_contra hn
-  have hh : ∀bits, sourceRoleTerm T n g s center globalShift b bits incoming S a=0 := by
-    intro bits
-    by_contra hz
-    exact hn (sourceRoleTerm_valid T n g s center globalShift b bits incoming S a hz)
-  apply h
-  change (windowPosteriorLaw T n g (b+1) d).expect _ = 0
-  rw [show (fun bits => sourceRoleTerm T n g s center globalShift b bits incoming S a) =
-    (fun _ => 0) from funext hh,expect_const]
-
-/-- The source's two alternatives now apply to actual posterior coefficient
-terms: signed singleton smoothing, or domination by a genuine forced tree. -/
-theorem sourceRoleAverage_alternative (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ)
-    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S)
-    (η : ℝ) (hη : 0≤η) (hcap : MetadataSmoothingCap T n g s η (b+1) d)
-    (ho : (occupiedWindowTerm T b S a).isOdd) :
-    |sourceRoleAverage T n g s center globalShift b d incoming S a|≤η ∨
-    ∃F : (occupiedWindowTerm T b S a).Forced,
-      BinaryWindowLabels.distinct T (b+1) (forcedWindowLabels T b S a F) ∧
-      |sourceRoleAverage T n g s center globalShift b d incoming S a|≤
-        binaryAnchorAverage T n g s center globalShift (b+1) (forcedWindowLabels T b S a F) d incoming := by
-  by_cases hz : sourceRoleAverage T n g s center globalShift b d incoming S a=0
-  · exact Or.inl (by simpa only [hz,abs_zero] using hη)
-  by_cases hn : (occupiedWindowTerm T b S a).noSingleton
-  · obtain ⟨F⟩ := OccupiedParityTree.forced_nonempty (occupiedWindowTerm T b S a)
-      (sourceRoleAverage_valid T n g s center globalShift b d incoming S a hz) hn ho
-    exact Or.inr ⟨F,forcedWindowLabels_distinct T b S a F,
-      sourceRoleAverage_forced_anchor T n g s center globalShift b d incoming S a F⟩
-  · exact Or.inl (sourceRoleAverage_singleton T n g s center globalShift b d incoming S a η hcap hn)
-
-end TraceReconstruction
-
-
-namespace TraceReconstruction
-open scoped Classical
-open FiniteLaw
-
-/-- The unsigned bottom constraint at one literal complete-tree leaf. -/
-def sourceBottomIndicator (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ) :
-    (b : ℕ) → WindowAssignments T n g Bool (b+1) → ℤ → WindowLeaf T (b+1) → ℝ
-  | 0, bits, incoming, v =>
-      motifIndicator s ((v.1.val:ℤ)-center 0+(∑i,(bitIndex (bits.1 i):ℤ))+incoming)
-  | b+1, bits, _, v => sourceBottomIndicator T n g s center globalShift b
-      (bits.2.2 v.1) (globalShift (b+1) bits.2.1 v.1) v.2
-
-def sourceLeafProduct (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (b : ℕ) (bits : WindowAssignments T n g Bool (b+1)) (incoming : ℤ)
-    (S : Finset (WindowLeaf T (b+1))) : ℝ :=
-  ∏v∈S, sourceBottomIndicator T n g s center globalShift b bits incoming v
-
-theorem sourceBottomIndicator_zero_one (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (b : ℕ) (bits : WindowAssignments T n g Bool (b+1)) (incoming : ℤ)
-    (v : WindowLeaf T (b+1)) :
-    sourceBottomIndicator T n g s center globalShift b bits incoming v=0 ∨
-    sourceBottomIndicator T n g s center globalShift b bits incoming v=1 := by
-  induction b generalizing incoming with
-  | zero => unfold sourceBottomIndicator motifIndicator; split_ifs <;> simp
-  | succ b ih => exact ih _ _ _
-
-theorem prod_zero_one {ι : Type*} (S : Finset ι) (f : ι → ℝ)
-    (h : ∀i∈S,f i=0 ∨ f i=1) : (∏i∈S,f i)=0 ∨ (∏i∈S,f i)=1 := by
-  by_cases hz : ∃i∈S,f i=0
-  · obtain ⟨i,hi,hz⟩ := hz
-    exact Or.inl (Finset.prod_eq_zero hi hz)
-  · right
-    apply Finset.prod_eq_one
-    intro i hi
-    exact (h i hi).resolve_left (fun he => hz ⟨i,hi,he⟩)
-
-theorem sourceLeafProduct_zero_one (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (b : ℕ) (bits : WindowAssignments T n g Bool (b+1)) (incoming : ℤ)
-    (S : Finset (WindowLeaf T (b+1))) :
-    sourceLeafProduct T n g s center globalShift b bits incoming S=0 ∨
-    sourceLeafProduct T n g s center globalShift b bits incoming S=1 :=
-  prod_zero_one S _ (fun v _ => sourceBottomIndicator_zero_one T n g s center globalShift b bits incoming v)
-
-theorem binaryAnchorKernel_zero_one (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (b : ℕ) (p : BinaryWindowLabels T b) (bits : WindowAssignments T n g Bool b)
-    (incoming : ℤ) :
-    binaryAnchorKernel T n g s center globalShift b p bits incoming=0 ∨
-    binaryAnchorKernel T n g s center globalShift b p bits incoming=1 := by
-  induction b generalizing incoming with
-  | zero => exact Or.inr rfl
-  | succ b ih =>
-    have hl := ih p.2.2.1 (bits.2.2 p.1) (globalShift b bits.2.1 p.1)
-    have hr := ih p.2.2.2 (bits.2.2 p.2.1) (globalShift b bits.2.1 p.2.1)
-    change (_*(_*_)=0) ∨ (_*(_*_)=1)
-    rcases hl with hl|hl <;> rcases hr with hr|hr <;> rw [hl,hr] <;>
-      unfold motifIndicator <;> split_ifs <;> norm_num
-
-theorem sourceLeafProduct_succ (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (b : ℕ) (bits : WindowAssignments T n g Bool (b+1+1)) (incoming : ℤ)
-    (S : Finset (WindowLeaf T (b+1+1))) :
-    sourceLeafProduct T n g s center globalShift (b+1) bits incoming S =
-      ∏i : Fin (selectedChildren S).card,
-        sourceLeafProduct T n g s center globalShift b
-          (bits.2.2 ((selectedChildren S).equivFin.symm i).val)
-          (globalShift (b+1) bits.2.1 ((selectedChildren S).equivFin.symm i).val)
-          (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val) := by
-  symm
-  exact (selected_prod_index (selectedChildren S) (fun h =>
-    ∏v∈selectedChildLeaves S h,
-      sourceBottomIndicator T n g s center globalShift b (bits.2.2 h)
-        (globalShift (b+1) bits.2.1 h) v)).trans
-      (selected_prod_children S (fun v =>
-        sourceBottomIndicator T n g s center globalShift (b+1) bits incoming v))
-
-/-- Every selected leaf is active in the bottom expansion, including those
-outside the forced tree. Their absolute constraints are not lost. -/
-theorem sourceRoleTerm_all_leaf_constraints (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (b : ℕ) (bits : WindowAssignments T n g Bool (b+1)) (incoming : ℤ)
-    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S) :
-    |sourceRoleTerm T n g s center globalShift b bits incoming S a|≤
-      sourceLeafProduct T n g s center globalShift b bits incoming S := by
-  induction b generalizing incoming with
-  | zero =>
-    rw [sourceRoleTerm_zero]
-    change |∏i : Fin S.card, _|≤_
-    rw [Finset.abs_prod]
-    calc
-      _ ≤ ∏i : Fin S.card, motifIndicator s
-          (((S.equivFin.symm i).val.1.val:ℤ)-(center 0-(∑i,(bitIndex (bits.1 i):ℤ))-incoming)) :=
-        Finset.prod_le_prod₀ (fun _ _ => abs_nonneg _)
-          (fun i _ => bottomRoleWeight_abs_le_indicator _ _ (prouhet_motif_value _ _) _)
-      _ = ∏v∈S, motifIndicator s ((v.1.val:ℤ)-
-          (center 0-(∑i,(bitIndex (bits.1 i):ℤ))-incoming)) :=
-        selected_prod_index S (fun v : WindowLeaf T 1 => motifIndicator s
-          ((v.1.val:ℤ)-(center 0-(∑i,(bitIndex (bits.1 i):ℤ))-incoming)))
-      _ = _ := by
-        apply Finset.prod_congr rfl
-        intro v hv
-        change motifIndicator s _ = motifIndicator s _
-        congr 1
-        omega
-  | succ b ih =>
-    rw [sourceRoleTerm_succ,abs_mul,sourceLeafProduct_succ]
-    apply (mul_le_mul_of_nonneg_right
-      (occupiedRootWeight_abs_le_one T s (b+1) S a _) (abs_nonneg _)).trans
-    rw [one_mul,Finset.abs_prod]
-    apply Finset.prod_le_prod₀ (fun _ _ => abs_nonneg _)
-    intro i hi
-    exact ih _ _ _ _
-
-theorem sourceLeafProduct_antitone (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (b : ℕ) (bits : WindowAssignments T n g Bool (b+1)) (incoming : ℤ)
-    (E S : Finset (WindowLeaf T (b+1))) (hES : E⊆S) :
-    sourceLeafProduct T n g s center globalShift b bits incoming S≤
-      sourceLeafProduct T n g s center globalShift b bits incoming E := by
-  apply Finset.prod_le_prod_of_subset_of_le_one₀ hES
-  · intro v hv
-    rcases sourceBottomIndicator_zero_one T n g s center globalShift b bits incoming v with h|h <;> norm_num [h]
-  · intro v hv he
-    rcases sourceBottomIndicator_zero_one T n g s center globalShift b bits incoming v with h|h <;> norm_num [h]
-
-theorem le_mul_of_zero_one {x a b : ℝ} (ha : a=0 ∨ a=1) (hb : b=0 ∨ b=1)
-    (hxa : x≤a) (hxb : x≤b) : x≤a*b := by
-  rcases ha with rfl|rfl <;> rcases hb with rfl|rfl <;> norm_num at * <;> assumption
-
-/-- Actual forced anchor AND extra-leaf constraints in one bit expectation.
-In particular E can be precisely the nonforced leaves; keeping it arbitrary
-here permits the later label summation without any conditioning gates. -/
-theorem sourceRoleAverage_forced_extra (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ)
-    (S E : Finset (WindowLeaf T (b+1))) (hES : E⊆S) (a : OccupiedWindowChoice T b S)
-    (F : (occupiedWindowTerm T b S a).Forced) :
-    |sourceRoleAverage T n g s center globalShift b d incoming S a|≤
-      (windowPosteriorLaw T n g (b+1) d).expect (fun bits =>
-        binaryAnchorKernel T n g s center globalShift (b+1) (forcedWindowLabels T b S a F) bits incoming *
-        sourceLeafProduct T n g s center globalShift b bits incoming E) := by
-  apply ((windowPosteriorLaw T n g (b+1) d).abs_expect_le _).trans
-  apply FiniteLaw.expect_mono
-  intro bits
-  exact le_mul_of_zero_one
-    (binaryAnchorKernel_zero_one T n g s center globalShift _ _ bits incoming)
-    (sourceLeafProduct_zero_one T n g s center globalShift b bits incoming E)
-    (sourceRoleTerm_forced_anchor T n g s center globalShift b bits incoming S a F)
-    ((sourceRoleTerm_all_leaf_constraints T n g s center globalShift b bits incoming S a).trans
-      (sourceLeafProduct_antitone T n g s center globalShift b bits incoming E S hES))
-
-end TraceReconstruction
-
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-open FiniteLaw
-
-/-- Deterministic closeness of each forced split; it is a restriction on
-labels, never an additional conditioning event for the random bits. -/
-def BinaryWindowLabels.close (T : ℕ → ℕ) (s : ℕ) : (b : ℕ) → BinaryWindowLabels T b → Prop
-  | 0, _ => True
-  | b+1, p => |(p.1.val:ℤ)-p.2.1.val|<(2^s:ℕ) ∧ close T s b p.2.2.1 ∧ close T s b p.2.2.2
-
-theorem motifIndicator_pair_close (s : ℕ) (a c X : ℤ)
-    (ha : motifIndicator s (a-X)≠0) (hc : motifIndicator s (c-X)≠0) :
-    |a-c|<(2^s:ℕ) := by
-  have hsa : 0≤a-X ∧ a-X<(2^s:ℕ) := by
-    by_contra h
-    exact ha (by simp only [motifIndicator,ite_eq_right h])
-  have hsc : 0≤c-X ∧ c-X<(2^s:ℕ) := by
-    by_contra h
-    exact hc (by simp only [motifIndicator,ite_eq_right h])
-  exact abs_lt.mpr ⟨by omega,by omega⟩
-
-theorem nonzero_of_abs_le {x y : ℝ} (h : x≠0) (hle : |x|≤y) : y≠0 := by
-  intro hy
-  rw [hy] at hle
-  exact h (abs_eq_zero.mp (le_antisymm hle (abs_nonneg x)))
-
-/-- A nonzero literal role multiplier enforces all close-child restrictions
-of every forced subtree chosen from its fixed role pattern. -/
-theorem sourceRoleTerm_forced_close (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (b : ℕ) (bits : WindowAssignments T n g Bool (b+1)) (incoming : ℤ)
-    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S)
-    (F : (occupiedWindowTerm T b S a).Forced)
-    (h : sourceRoleTerm T n g s center globalShift b bits incoming S a≠0) :
-    BinaryWindowLabels.close T s (b+1) (forcedWindowLabels T b S a F) := by
-  induction b generalizing incoming with
-  | zero =>
-    cases F with
-    | node i j hne hi hj left right =>
-      rw [sourceRoleTerm_zero] at h
-      change |((S.equivFin.symm i).val.1.val:ℤ)-(S.equivFin.symm j).val.1.val|<(2^s:ℕ) ∧ True ∧ True
-      refine ⟨?_,trivial,trivial⟩
-      exact motifIndicator_pair_close s _ _ _
-        (nonzero_of_abs_le h (occupiedRootWeight_bottom_anchor T s S a _ i))
-        (nonzero_of_abs_le h (occupiedRootWeight_bottom_anchor T s S a _ j))
-  | succ b ih =>
-    cases F with
-    | node i j hne hi hj left right =>
-      rw [sourceRoleTerm_succ] at h
-      have hm := mul_ne_zero_iff.mp h
-      have hc := Finset.prod_ne_zero_iff.mp hm.2
-      change |(((selectedChildren S).equivFin.symm i).val.val:ℤ)-
-        ((selectedChildren S).equivFin.symm j).val.val|<(2^s:ℕ) ∧
-        BinaryWindowLabels.close T s (b+1)
-          (forcedWindowLabels T b (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val)
-            (a.1 i) left) ∧
-        BinaryWindowLabels.close T s (b+1)
-          (forcedWindowLabels T b (selectedChildLeaves S ((selectedChildren S).equivFin.symm j).val)
-            (a.1 j) right)
-      refine ⟨?_,ih _ _ _ _ left (hc i (Finset.mem_univ i)),
-        ih _ _ _ _ right (hc j (Finset.mem_univ j))⟩
-      exact motifIndicator_pair_close s _ _ _
-        (nonzero_of_abs_le hm.1 (occupiedRootWeight_internal_anchor T s b S a _ i hi))
-        (nonzero_of_abs_le hm.1 (occupiedRootWeight_internal_anchor T s b S a _ j hj))
-
-/-- Closeness remains a pure label condition after averaging, without
-conditioning on it. -/
-theorem sourceRoleAverage_forced_close (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ)
-    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S)
-    (F : (occupiedWindowTerm T b S a).Forced)
-    (h : sourceRoleAverage T n g s center globalShift b d incoming S a≠0) :
-    BinaryWindowLabels.close T s (b+1) (forcedWindowLabels T b S a F) := by
-  by_contra hn
-  have hh : ∀bits, sourceRoleTerm T n g s center globalShift b bits incoming S a=0 := by
-    intro bits
-    by_contra hz
-    exact hn (sourceRoleTerm_forced_close T n g s center globalShift b bits incoming S a F hz)
-  apply h
-  change (windowPosteriorLaw T n g (b+1) d).expect _ = 0
-  rw [show (fun bits => sourceRoleTerm T n g s center globalShift b bits incoming S a) =
-    (fun _ => 0) from funext hh,expect_const]
-
-end TraceReconstruction
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-open FiniteLaw
-
-/-- The literal finite child labels embedded in the integer line. -/
-def finIntegerLabels (N : ℕ) : Finset ℤ := Finset.univ.image (fun i : Fin N => (i.val:ℤ))
-
-theorem sum_finIntegerLabels (N : ℕ) (f : ℤ → ℝ) :
-    (∑a∈finIntegerLabels N,f a)=∑i : Fin N,f i.val := by
-  rw [finIntegerLabels,Finset.sum_image]
-  intro a ha b hb he
-  apply Fin.ext
-  change (a.val:ℤ)=(b.val:ℤ) at he
-  exact_mod_cast he
-
-theorem sum_close_finIntegerLabels (N L : ℕ) (a : ℤ) (f : ℤ → ℝ) :
-    (∑c∈closeLabels (finIntegerLabels N) L a,f c) =
-      ∑i : Fin N, if |(i.val:ℤ)-a|<(L:ℤ) then f i.val else 0 := by
-  rw [closeLabels,Finset.sum_filter,sum_finIntegerLabels]
-
-/-- Any deterministic plateau/distinctness restriction can be retained in
-child-pair summation. Only closeness is used to upper bound its cost. -/
-theorem localAnchor_finite_energy {m N : ℕ} (d : Fin m → Option Bool) (s : ℕ)
-    (c : ℤ) (R : Fin N → Fin N → Prop) [DecidableRel R]
-    (hR : ∀a b,R a b → |(b.val:ℤ)-a.val|<(2^s:ℕ)) :
-    (∑a : Fin N, ∑b : Fin N, if R a b then localAnchorAverage d s ((a.val:ℤ)-c)^2 else 0) ≤
-      2*((2^s:ℕ):ℝ)^3*binomialBound (∑i,hiddenTrial (d i)) := by
-  have H := root_anchor_energy (∑i,hiddenTrial (d i)) (2^s)
-    (finIntegerLabels N) (finIntegerLabels N) (c-(∑i,(visibleOnes (d i):ℤ)))
-  rw [sum_finIntegerLabels] at H
-  simp_rw [sum_close_finIntegerLabels] at H
-  calc
-    _ ≤ ∑a : Fin N, ∑b : Fin N,
-        if |(b.val:ℤ)-a.val|<(2^s:ℕ) then
-          windowKernel (∑i,hiddenTrial (d i)) (2^s)
-            ((a.val:ℤ)-(c-∑i,(visibleOnes (d i):ℤ)))^2 else 0 := by
-      apply Finset.sum_le_sum
-      intro a ha
-      apply Finset.sum_le_sum
-      intro b hb
-      by_cases h : R a b
-      · rw [ite_eq_left h,ite_eq_left (hR a b h),localAnchorAverage_kernel]
-        convert le_refl _ using 1 ; congr 2 ; omega
-      · rw [ite_eq_right h]
-        split_ifs <;> positivity
-    _ ≤ _ := H
-
-/-- Actual shared-jitter energy over two finite sibling pairs. Parent-label
-plateau restrictions are discharged before applying this lemma; arbitrary
-restrictions on these child pairs are kept, not conditioned on. -/
-theorem posteriorJoint_finite_energy {mg m₁ m₂ N : ℕ}
-    (dg : Fin mg → Option Bool) (d₁ : Fin m₁ → Option Bool) (d₂ : Fin m₂ → Option Bool)
-    (s : ℕ) (c₁ c₂ : ℤ) (R : Fin N → Fin N → Prop) [DecidableRel R]
-    (hR : ∀a b,R a b → |(b.val:ℤ)-a.val|<(2^s:ℕ)) :
-    (∑a : Fin N, ∑c : Fin N, ∑b : Fin N, ∑d : Fin N,
-      if R a c ∧ R b d then
-        posteriorJointAnchors dg d₁ d₂ s ((a.val:ℤ)-c₁) ((b.val:ℤ)-c₂)^2 else 0) ≤
-       4*((2^s:ℕ):ℝ)^6*binomialBound (∑i,hiddenTrial (dg i))*binomialBound (∑i,hiddenTrial (d₂ i)) := by
-  have H := posterior_sibling_anchor_energy dg d₁ d₂ s
-    (finIntegerLabels N) (finIntegerLabels N) (finIntegerLabels N) (finIntegerLabels N) c₁ c₂
-  simp_rw [sum_finIntegerLabels,sum_close_finIntegerLabels] at H
-  calc
-    _ = ∑a : Fin N, ∑b : Fin N, ∑c : Fin N, ∑d : Fin N,
-      if R a c ∧ R b d then
-        posteriorJointAnchors dg d₁ d₂ s ((a.val:ℤ)-c₁) ((b.val:ℤ)-c₂)^2 else 0 := by
-      apply Finset.sum_congr rfl
-      intro a ha
-      rw [Finset.sum_comm]
-    _ ≤ ∑a : Fin N, ∑b : Fin N, ∑c : Fin N,
-      if |(c.val:ℤ)-a.val|<(2^s:ℕ) then
-        ∑d : Fin N, if |(d.val:ℤ)-b.val|<(2^s:ℕ) then
-          posteriorJointAnchors dg d₁ d₂ s ((a.val:ℤ)-c₁) ((b.val:ℤ)-c₂)^2 else 0
-      else 0 := by
-      apply Finset.sum_le_sum
-      intro a ha
-      apply Finset.sum_le_sum
-      intro b hb
-      apply Finset.sum_le_sum
-      intro c hc
-      by_cases h : |(c.val:ℤ)-a.val|<(2^s:ℕ)
-      · rw [ite_eq_left h]
-        apply Finset.sum_le_sum
-        intro d hd
-        by_cases hr : R a c ∧ R b d
-        · rw [ite_eq_left hr,ite_eq_left (hR b d hr.2)]
-        · rw [ite_eq_right hr]
-          split_ifs <;> positivity
-      · rw [ite_eq_right h]
-        apply le_of_eq
-        apply Finset.sum_eq_zero
-        intro d hd
-        exact ite_eq_right (fun hr => h (hR a c hr.1))
-    _ ≤ _ := H
-
-end TraceReconstruction
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-open FiniteLaw
-
-/-- The actual finite central domain, distinctness and close-label restriction.
-It is purely deterministic and is never added to the orientation posterior. -/
-def anchorPairAllowed (T : ℕ → ℕ) (A : (b : ℕ) → Finset (Fin (T b))) (s b : ℕ)
-    (a c : Fin (T b)) : Prop := a∈A b ∧ c∈A b ∧ a≠c ∧ |(c.val:ℤ)-a.val|<(2^s:ℕ)
-
-/-- The source global plateau identity is required pointwise for bits, not
-just in distribution and not after conditioning on successful layout. -/
-def HasGlobalPlateau (T g : ℕ → ℕ) (A : (b : ℕ) → Finset (Fin (T b))) (shift : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ) : Prop :=
-  ∀b glob u, u∈A b → globalShift b glob u=(∑i,(bitIndex (glob i):ℤ))-shift b
-
-/-- Numerical binomial caps extracted from the literal known/hidden metadata.
-In applications these are proved using the source hidden-trial lower bounds. -/
-def MetadataBinomialCaps (T n g : ℕ → ℕ) (B C : ℕ → ℝ) :
-    (b : ℕ) → WindowAssignments T n g (Option Bool) b → Prop
-  | 0, _ => True
-  | b+1, d => binomialBound (∑i, hiddenTrial (d.1 i)) ≤ B b ∧
-      binomialBound (∑i, hiddenTrial (d.2.1 i)) ≤ C b ∧
-      ∀u, MetadataBinomialCaps T n g B C b (d.2.2 u)
-
-/-- One correlated sibling-group anchor, using the actual child metadata
-and their parent's one shared global orientation collection. -/
-def binarySiblingAnchor (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1+1))
-    (u v : Fin (T (b+1))) (a c : Fin (T b)) : ℝ :=
-  (posteriorArray d.2.1).expect (fun glob =>
-    localAnchorAverage (d.2.2 u).1 s ((a.val:ℤ)-center b+globalShift (b+1) glob u) *
-    localAnchorAverage (d.2.2 v).1 s ((c.val:ℤ)-center b+globalShift (b+1) glob v))
-
-theorem binarySiblingAnchor_plateau (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (A : (b : ℕ) → Finset (Fin (T b))) (shift : ℕ → ℤ)
-    (hplat : HasGlobalPlateau T g A shift globalShift)
-    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1+1))
-    (u v : Fin (T (b+1))) (hu : u∈A (b+1)) (hv : v∈A (b+1)) (a c : Fin (T b)) :
-    binarySiblingAnchor T n g s center globalShift b d u v a c =
-      posteriorJointAnchors d.2.1 (d.2.2 u).1 (d.2.2 v).1 s
-        ((a.val:ℤ)-(center b+shift (b+1))) ((c.val:ℤ)-(center b+shift (b+1))) := by
-  calc
-    _ = (posteriorArray d.2.1).expect (fun glob =>
-        localAnchorAverage (d.2.2 u).1 s ((a.val:ℤ)-center b+(∑i,(bitIndex (glob i):ℤ))-shift (b+1)) *
-        localAnchorAverage (d.2.2 v).1 s ((c.val:ℤ)-center b+(∑i,(bitIndex (glob i):ℤ))-shift (b+1))) := by
-      apply congrArg (FiniteLaw.expect (posteriorArray d.2.1))
-      funext glob
-      have hu' := hplat (b+1) glob u hu
-      have hv' := hplat (b+1) glob v hv
-      rw [hu',hv']
-      congr 2 <;> omega
-    _ = _ := by
-      rw [localAnchor_pair_plateau]
-      congr 1 <;> omega
-
-/-- Exact dynamic sum of squared group anchors over the allowed finite
-labels below a fixed forced split. Distinct groups retain their own metadata. -/
-def binaryDescendantEnergy (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (A : (b : ℕ) → Finset (Fin (T b))) :
-    (b : ℕ) → WindowAssignments T n g (Option Bool) (b+1) → Fin (T b) → Fin (T b) → ℝ
-  | 0, _, _, _ => 1
-  | b+1, d, u, v =>
-      ∑a : Fin (T b),∑c : Fin (T b),∑e : Fin (T b),∑f : Fin (T b),
-        if anchorPairAllowed T A s b a c ∧ anchorPairAllowed T A s b e f then
-          binarySiblingAnchor T n g s center globalShift b d u v a e ^ 2 *
-            binaryDescendantEnergy T n g s center globalShift A b (d.2.2 u) a c *
-            binaryDescendantEnergy T n g s center globalShift A b (d.2.2 v) e f
-        else 0
-
-theorem binaryDescendantEnergy_nonneg (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (A : (b : ℕ) → Finset (Fin (T b))) (b : ℕ)
-    (d : WindowAssignments T n g (Option Bool) (b+1)) (u v : Fin (T b)) :
-    0≤binaryDescendantEnergy T n g s center globalShift A b d u v := by
-  induction b with
-  | zero => exact zero_le_one
-  | succ b ih =>
-    apply Finset.sum_nonneg
-    intro a ha
-    apply Finset.sum_nonneg
-    intro c hc
-    apply Finset.sum_nonneg
-    intro e he
-    apply Finset.sum_nonneg
-    intro f hf
-    split_ifs
-    · exact mul_nonneg (mul_nonneg (sq_nonneg _) (ih _ _ _)) (ih _ _ _)
-    · exact le_refl _
-
-/-- True common-jitter energy at one group, with its exact finite domains. -/
-theorem binarySiblingAnchor_energy (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (A : (b : ℕ) → Finset (Fin (T b))) (shift : ℕ → ℤ)
-    (hplat : HasGlobalPlateau T g A shift globalShift)
-    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1+1))
-    (u v : Fin (T (b+1))) (hu : u∈A (b+1)) (hv : v∈A (b+1)) :
-    (∑a : Fin (T b),∑c : Fin (T b),∑e : Fin (T b),∑f : Fin (T b),
-        if anchorPairAllowed T A s b a c ∧ anchorPairAllowed T A s b e f then
-          binarySiblingAnchor T n g s center globalShift b d u v a e ^ 2 else 0) ≤
-      4*((2^s:ℕ):ℝ)^6*binomialBound (∑i,hiddenTrial (d.2.1 i))*
-        binomialBound (∑i,hiddenTrial ((d.2.2 v).1 i)) := by
-  simp_rw [binarySiblingAnchor_plateau T n g s center globalShift A shift hplat b d u v hu hv]
-  exact posteriorJoint_finite_energy d.2.1 (d.2.2 u).1 (d.2.2 v).1 s
-    (center b+shift (b+1)) (center b+shift (b+1)) (anchorPairAllowed T A s b)
-    (fun a c h => h.2.2.2)
-
-/-- The complete bottom-up forced-label energy estimate. Its only numerical
-inputs are actual metadata hidden-count binomial caps. The group bound
-keeps one shared global kernel until AFTER squaring. -/
-theorem binaryDescendantEnergy_le (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (A : (b : ℕ) → Finset (Fin (T b))) (shift : ℕ → ℤ)
-    (hplat : HasGlobalPlateau T g A shift globalShift)
-    (B C : ℕ → ℝ) (hB : ∀b,0≤B b) (hC : ∀b,0≤C b)
-    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1))
-    (hcap : MetadataBinomialCaps T n g B C (b+1) d)
-    (u v : Fin (T b)) (hu : u∈A b) (hv : v∈A b) :
-    binaryDescendantEnergy T n g s center globalShift A b d u v ≤
-      binaryTreeBudget (fun j => 4*((2^s:ℕ):ℝ)^6*C (j+1)*B j) b := by
-  let K : ℕ → ℝ := fun j => 4*((2^s:ℕ):ℝ)^6*C (j+1)*B j
-  have hK : ∀j,0≤K j := fun j => mul_nonneg (mul_nonneg (by positivity) (hC _)) (hB _)
-  induction b with
-  | zero => exact le_refl _
-  | succ b ih =>
-    have hbudget := binaryTreeBudget_nonneg K hK b
-    have hgroup := binarySiblingAnchor_energy T n g s center globalShift A shift hplat b d u v hu hv
-    have hnum : 4*((2^s:ℕ):ℝ)^6*binomialBound (∑i,hiddenTrial (d.2.1 i))*
-        binomialBound (∑i,hiddenTrial ((d.2.2 v).1 i)) ≤ K b := by
-      exact mul_le_mul
-        (mul_le_mul_of_nonneg_left hcap.2.1 (by positivity))
-        (hcap.2.2 v).1 (by unfold binomialBound; positivity)
-        (mul_nonneg (by positivity) (hC _))
-    calc
-      _ ≤ (∑a : Fin (T b),∑c : Fin (T b),∑e : Fin (T b),∑f : Fin (T b),
-        if anchorPairAllowed T A s b a c ∧ anchorPairAllowed T A s b e f then
-          binarySiblingAnchor T n g s center globalShift b d u v a e ^ 2 else 0) *
-          (binaryTreeBudget K b)^2 := by
-        simp only [binaryDescendantEnergy,Finset.sum_mul]
-        apply Finset.sum_le_sum
-        intro a ha
-        apply Finset.sum_le_sum
-        intro c hc
-        apply Finset.sum_le_sum
-        intro e he
-        apply Finset.sum_le_sum
-        intro f hf
-        split_ifs with hr
-        · have hl := ih (d.2.2 u) (hcap.2.2 u) a c hr.1.1 hr.1.2.1
-          have hh := ih (d.2.2 v) (hcap.2.2 v) e f hr.2.1 hr.2.2.1
-          have hm := mul_le_mul hl hh
-            (binaryDescendantEnergy_nonneg T n g s center globalShift A b (d.2.2 v) e f) hbudget
-          have hf := mul_le_mul_of_nonneg_left hm
-            (sq_nonneg (binarySiblingAnchor T n g s center globalShift b d u v a e))
-          simpa only [K,pow_two,mul_assoc] using hf
-        · simp only [zero_mul,le_refl]
-      _ ≤ K b*(binaryTreeBudget K b)^2 :=
-        mul_le_mul_of_nonneg_right (hgroup.trans hnum) (sq_nonneg _)
-      _ = _ := rfl
-
-end TraceReconstruction
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-open FiniteLaw
-
-def BinaryWindowLabels.allowed (T : ℕ → ℕ) (A : (b : ℕ) → Finset (Fin (T b))) (s : ℕ) :
-    (b : ℕ) → BinaryWindowLabels T b → Prop
-  | 0, _ => True
-  | b+1, p => anchorPairAllowed T A s b p.1 p.2.1 ∧
-      allowed T A s b p.2.2.1 ∧ allowed T A s b p.2.2.2
-
-theorem BinaryWindowLabels.allowed_distinct (T : ℕ → ℕ) (A : (b : ℕ) → Finset (Fin (T b)))
-    (s b : ℕ) (p : BinaryWindowLabels T b) (hp : allowed T A s b p) : distinct T b p := by
-  induction b with
-  | zero => trivial
-  | succ b ih => exact ⟨hp.1.2.2.1,ih _ hp.2.1,ih _ hp.2.2⟩
-
-/-- Regroup the two real top child labels, without changing the actual
-finite label type or declaration order. -/
-def joinBinaryLabels {T : ℕ → ℕ} {b : ℕ} (a : Fin (T b) × Fin (T b))
-    (p : BinaryWindowLabels T b × BinaryWindowLabels T b) : BinaryWindowLabels T (b+1) :=
-  (a.1,a.2,p.1,p.2)
-
-theorem sum_binary_labels (T : ℕ → ℕ) (b : ℕ) (f : BinaryWindowLabels T (b+1) → ℝ) :
-    (∑p, f p) = ∑a : Fin (T b) × Fin (T b),
-      ∑p : BinaryWindowLabels T b × BinaryWindowLabels T b, f (joinBinaryLabels a p) := by
-  let e : BinaryWindowLabels T (b+1) ≃
-      (Fin (T b) × Fin (T b)) × (BinaryWindowLabels T b × BinaryWindowLabels T b) :=
-    { toFun := fun p => ((p.1,p.2.1),(p.2.2.1,p.2.2.2))
-      invFun := fun z => joinBinaryLabels z.1 z.2
-      left_inv := fun _ => rfl
-      right_inv := fun _ => rfl }
-  exact (Fintype.sum_equiv e f (fun z => f (joinBinaryLabels z.1 z.2)) (fun _ => rfl)).trans
-    (Fintype.sum_prod_type _)
-
-/-- Exact finite tensor summation after preserving both deterministically
-allowed label domains. No probabilistic independence is asserted here. -/
-theorem sum_pair_tensor {α β σ τ : Type*} [Fintype α] [Fintype β] [Fintype σ] [Fintype τ]
-    (R : α → Prop) (S : β → Prop) (P : σ → Prop) (Q : τ → Prop)
-    [DecidablePred R] [DecidablePred S] [DecidablePred P] [DecidablePred Q]
-    (K : α → β → ℝ) (f : α → σ → ℝ) (h : β → τ → ℝ) :
-    (∑a,∑x,∑b,∑y, if R a ∧ P x ∧ S b ∧ Q y then K a b*f a x*h b y else 0) =
-      ∑a,∑b,if R a ∧ S b then K a b*(∑x,if P x then f a x else 0)*
-        (∑y,if Q y then h b y else 0) else 0 := by
-  apply Finset.sum_congr rfl
-  intro a ha
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro b hb
-  by_cases hr : R a
-  · by_cases hs : S b
-    · rw [ite_eq_left ⟨hr,hs⟩]
-      simp only [Finset.sum_mul,Finset.mul_sum]
-      conv_rhs => rw [Finset.sum_comm]
-      apply Finset.sum_congr rfl
-      intro x hx
-      apply Finset.sum_congr rfl
-      intro y hy
-      by_cases hp : P x <;> by_cases hq : Q y <;> simp only [hr,hs,hp,hq,and_self,
-        false_and,and_false,ite_true,ite_false,mul_zero,zero_mul]
-    · simp only [hs,and_false,false_and,ite_false,Finset.sum_const_zero]
-  · simp only [hr,false_and,ite_false,Finset.sum_const_zero]
-
-/-- Literal sum over all descendant labels is the recursively compiled
-sibling-group energy. This is the bridge from the actual anchor average to
-bottom-up summation; no coefficient estimate is supplied as a hypothesis. -/
-theorem binaryAnchorBelow_sum (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (A : (b : ℕ) → Finset (Fin (T b))) (b : ℕ)
-    (d : WindowAssignments T n g (Option Bool) (b+1)) (u v : Fin (T b)) :
-    (∑p : BinaryWindowLabels T b × BinaryWindowLabels T b,
-      if BinaryWindowLabels.allowed T A s b p.1 ∧ BinaryWindowLabels.allowed T A s b p.2 then
-        binaryAnchorBelow T n g s center globalShift b (joinBinaryLabels (u,v) p) d ^ 2 else 0) =
-      binaryDescendantEnergy T n g s center globalShift A b d u v := by
-  induction b with
-  | zero =>
-    have hcard : Fintype.card (BinaryWindowLabels T 0) = 1 := rfl
-    simp only [BinaryWindowLabels.allowed,binaryAnchorBelow,binaryDescendantEnergy,
-      and_self,ite_true,one_pow,Finset.sum_const,Finset.card_univ,Fintype.card_prod,hcard,nsmul_eq_mul,
-      mul_one,Nat.cast_one]
-  | succ b ih =>
-    rw [Fintype.sum_prod_type,sum_binary_labels]
-    simp_rw [sum_binary_labels]
-    have hpoint (a c : Fin (T b) × Fin (T b))
-        (p q : BinaryWindowLabels T b × BinaryWindowLabels T b) :
-        (if BinaryWindowLabels.allowed T A s (b+1) (joinBinaryLabels a p) ∧
-          BinaryWindowLabels.allowed T A s (b+1) (joinBinaryLabels c q) then
-          binaryAnchorBelow T n g s center globalShift (b+1)
-            (joinBinaryLabels (u,v) (joinBinaryLabels a p,joinBinaryLabels c q)) d ^ 2 else 0) =
-        (if anchorPairAllowed T A s b a.1 a.2 ∧
-          (BinaryWindowLabels.allowed T A s b p.1 ∧ BinaryWindowLabels.allowed T A s b p.2) ∧
-          anchorPairAllowed T A s b c.1 c.2 ∧
-          (BinaryWindowLabels.allowed T A s b q.1 ∧ BinaryWindowLabels.allowed T A s b q.2) then
-          binarySiblingAnchor T n g s center globalShift b d u v a.1 c.1 ^ 2 *
-            binaryAnchorBelow T n g s center globalShift b (joinBinaryLabels a p) (d.2.2 u) ^ 2 *
-            binaryAnchorBelow T n g s center globalShift b (joinBinaryLabels c q) (d.2.2 v) ^ 2 else 0) := by
-      simp only [BinaryWindowLabels.allowed,joinBinaryLabels,and_assoc,binaryAnchorBelow,binarySiblingAnchor]
-      split_ifs <;> ring
-    simp_rw [hpoint]
-    rw [sum_pair_tensor]
-    simp_rw [ih]
-    change _ = ∑a : Fin (T b),∑c : Fin (T b),∑e : Fin (T b),∑f : Fin (T b), _
-    simp only [Fintype.sum_prod_type]
-
-/-- The square-sum of actual anchor averages over all admissible forced
-indices, including the root local smoothing cost exactly once. -/
-theorem binaryAnchorAverage_sum (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (A : (b : ℕ) → Finset (Fin (T b))) (b : ℕ)
-    (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ) :
-    (∑p : BinaryWindowLabels T (b+1), if BinaryWindowLabels.allowed T A s (b+1) p then
-      binaryAnchorAverage T n g s center globalShift (b+1) p d incoming ^ 2 else 0) =
-      ∑u : Fin (T b),∑v : Fin (T b), if anchorPairAllowed T A s b u v then
-        localAnchorAverage d.1 s ((u.val:ℤ)-center b+incoming)^2 *
-          binaryDescendantEnergy T n g s center globalShift A b d u v else 0 := by
-  rw [sum_binary_labels,Fintype.sum_prod_type]
-  apply Finset.sum_congr rfl
-  intro u hu
-  apply Finset.sum_congr rfl
-  intro v hv
-  by_cases hr : anchorPairAllowed T A s b u v
-  · rw [ite_eq_left hr,←binaryAnchorBelow_sum,Finset.mul_sum]
-    apply Finset.sum_congr rfl
-    intro p hp
-    by_cases hsub : BinaryWindowLabels.allowed T A s b p.1 ∧ BinaryWindowLabels.allowed T A s b p.2
-    · have ha : BinaryWindowLabels.allowed T A s (b+1) (joinBinaryLabels (u,v) p) := ⟨hr,hsub⟩
-      rw [ite_eq_left ha,ite_eq_left hsub,binaryAnchorAverage_factor T n g s center globalShift _ _ _ _
-        (BinaryWindowLabels.allowed_distinct T A s _ _ ha),mul_pow]
-      rfl
-    · have ha : ¬BinaryWindowLabels.allowed T A s (b+1) (joinBinaryLabels (u,v) p) := fun h => hsub h.2
-      rw [ite_eq_right ha,ite_eq_right hsub,mul_zero]
-  · rw [ite_eq_right hr]
-    apply Finset.sum_eq_zero
-    intro p hp
-    exact ite_eq_right (fun h => hr h.1)
-
-end TraceReconstruction
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-open FiniteLaw
-
-/-- Full finite forced-anchor squared coefficient budget, with the source
-power of one local root cost and one correlated global/local cost per group.
-This conclusion is about literal posterior averages and actual labels. -/
-theorem binaryAnchorAverage_energy (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (A : (b : ℕ) → Finset (Fin (T b))) (shift : ℕ → ℤ)
-    (hplat : HasGlobalPlateau T g A shift globalShift)
-    (B C : ℕ → ℝ) (hB : ∀b,0≤B b) (hC : ∀b,0≤C b)
-    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1))
-    (hcap : MetadataBinomialCaps T n g B C (b+1) d) (incoming : ℤ) :
-    (∑p : BinaryWindowLabels T (b+1), if BinaryWindowLabels.allowed T A s (b+1) p then
-      binaryAnchorAverage T n g s center globalShift (b+1) p d incoming ^ 2 else 0) ≤
-        (2*((2^s:ℕ):ℝ)^3*B b) *
-          binaryTreeBudget (fun j => 4*((2^s:ℕ):ℝ)^6*C (j+1)*B j) b := by
-  let K : ℕ → ℝ := fun j => 4*((2^s:ℕ):ℝ)^6*C (j+1)*B j
-  have hK : ∀j,0≤K j := fun j => mul_nonneg (mul_nonneg (by positivity) (hC _)) (hB _)
-  have hbudget := binaryTreeBudget_nonneg K hK b
-  have hroot := localAnchor_finite_energy d.1 s (center b-incoming)
-    (anchorPairAllowed T A s b) (fun a c h => h.2.2.2)
-  have he (u : Fin (T b)) : (u.val:ℤ)-(center b-incoming)=(u.val:ℤ)-center b+incoming := by omega
-  simp_rw [he] at hroot
-  rw [binaryAnchorAverage_sum]
-  calc
-    _ ≤ (∑u : Fin (T b),∑v : Fin (T b),if anchorPairAllowed T A s b u v then
-        localAnchorAverage d.1 s ((u.val:ℤ)-center b+incoming)^2 else 0) * binaryTreeBudget K b := by
-      simp only [Finset.sum_mul]
-      apply Finset.sum_le_sum
-      intro u hu
-      apply Finset.sum_le_sum
-      intro v hv
-      split_ifs with hr
-      · exact mul_le_mul_of_nonneg_left
-          (binaryDescendantEnergy_le T n g s center globalShift A shift hplat B C hB hC b d hcap
-            u v hr.1 hr.2.1) (sq_nonneg _)
-      · simp only [zero_mul,le_refl]
-    _ ≤ (2*((2^s:ℕ):ℝ)^3*B b)*binaryTreeBudget K b := by
-      apply mul_le_mul_of_nonneg_right _ hbudget
-      exact hroot.trans (mul_le_mul_of_nonneg_left hcap.1 (by positivity))
-
-end TraceReconstruction
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-open FiniteLaw
-
-/-- A literal unsigned length-L translated motif admits at most L child
-labels in any finite window, even when the intended motif does not fit. -/
-theorem motifIndicator_finite_sum (s N : ℕ) (X : ℤ) :
-    (∑h : Fin N,motifIndicator s ((h.val:ℤ)-X)) ≤ (2^s:ℕ) := by
-  let A := Finset.univ.filter (fun h : Fin N => 0≤(h.val:ℤ)-X ∧ (h.val:ℤ)-X<(2^s:ℕ))
-  have hc : A.card≤2^s := by
-    have H := Finset.card_le_card_of_injOn (s:=A) (t:=Finset.range (2^s))
-      (fun h : Fin N => ((h.val:ℤ)-X).toNat) ?_ ?_
-    · simpa only [Finset.card_range] using H
-    · intro h hh
-      have h' := (Finset.mem_filter.mp hh).2
-      apply Finset.mem_range.mpr
-      dsimp only
-      omega
-    · intro h hh j hj he
-      have h' := (Finset.mem_filter.mp hh).2
-      have j' := (Finset.mem_filter.mp hj).2
-      apply Fin.ext
-      dsimp only at he
-      omega
-  have he : (∑h : Fin N,motifIndicator s ((h.val:ℤ)-X)) = (A.card:ℝ) := by
-    simp only [motifIndicator,A,Finset.sum_ite,Finset.sum_const_zero,add_zero,
-      Finset.sum_const,nsmul_eq_mul,mul_one]
-  rw [he]
-  exact_mod_cast hc
-
-/-- The complete bits law is arbitrary. All extra-leaf offsets may be
-mutually correlated and correlated with the retained forced-anchor kernel.
-Only the extra leaf label variables themselves are summed here. -/
-def extraLabelAverage {Ω : Type*} [Fintype Ω] (P : FiniteLaw Ω)
-    (s N e : ℕ) (f : Ω → ℝ) (X : Ω → Fin e → ℤ) (h : Fin e → Fin N) : ℝ :=
-  P.expect (fun ω => f ω*∏j,motifIndicator s (((h j).val:ℤ)-X ω j))
-
-theorem extraLabelAverage_nonneg {Ω : Type*} [Fintype Ω] (P : FiniteLaw Ω)
-    (s N e : ℕ) (f : Ω → ℝ) (hf : ∀ω,0≤f ω) (X : Ω → Fin e → ℤ)
-    (h : Fin e → Fin N) : 0≤extraLabelAverage P s N e f X h := by
-  apply Finset.sum_nonneg
-  intro ω hω
-  apply mul_nonneg (P.nonneg ω)
-  exact mul_nonneg (hf ω) (Finset.prod_nonneg (fun j _ => motifIndicator_nonneg _ _))
-
-/-- Source the correlated-anchor estimate, extra leaf l1 cost before squaring. No free
-T_1 cost and no invalid decoupling of global displacements enter. -/
-theorem extraLabelAverage_sum {Ω : Type*} [Fintype Ω] (P : FiniteLaw Ω)
-    (s N e : ℕ) (f : Ω → ℝ) (hf : ∀ω,0≤f ω) (X : Ω → Fin e → ℤ) :
-    (∑h : Fin e → Fin N,extraLabelAverage P s N e f X h) ≤
-      ((2^s:ℕ):ℝ)^e*P.expect f := by
-  unfold extraLabelAverage expect
-  rw [Finset.sum_comm]
-  simp_rw [←mul_assoc,←Finset.mul_sum]
-  calc
-    _ ≤ ∑ω,P.mass ω*f ω*(((2^s:ℕ):ℝ)^e) := by
-      apply Finset.sum_le_sum
-      intro ω hω
-      apply mul_le_mul_of_nonneg_left _ (mul_nonneg (P.nonneg _) (hf ω))
-      have he := Fintype.prod_sum (fun (j : Fin e) (h : Fin N) => motifIndicator s ((h.val:ℤ)-X ω j))
-      rw [←he]
-      calc
-        _ ≤ ∏_j : Fin e,((2^s:ℕ):ℝ) := by
-          apply Finset.prod_le_prod₀
-          · intro j hj
-            exact Finset.sum_nonneg (fun h hh => motifIndicator_nonneg _ _)
-          · intro j hj
-            exact motifIndicator_finite_sum s N (X ω j)
-        _ = _ := by simp
-    _ = _ := by rw [←Finset.sum_mul]; ring
-
-/-- The squared extra-leaf cost follows from nonnegativity AFTER the whole
-shared-bit average, exactly as in the source (not from independence). -/
-theorem extraLabelAverage_energy {Ω : Type*} [Fintype Ω] (P : FiniteLaw Ω)
-    (s N e : ℕ) (f : Ω → ℝ) (hf : ∀ω,0≤f ω) (X : Ω → Fin e → ℤ) :
-    (∑h : Fin e → Fin N,extraLabelAverage P s N e f X h ^ 2) ≤
-      ((2^s:ℕ):ℝ)^(2*e)*(P.expect f)^2 := by
-  have hn (h : Fin e → Fin N) := extraLabelAverage_nonneg P s N e f hf X h
-  calc
-    _ ≤ (∑h : Fin e → Fin N,extraLabelAverage P s N e f X h)^2 :=
-      Finset.sum_sq_le_sq_sum_of_nonneg (fun h _ => hn h)
-    _ ≤ (((2^s:ℕ):ℝ)^e*P.expect f)^2 := by
-      apply pow_le_pow_left₀ (Finset.sum_nonneg (fun h _ => hn h))
-      exact extraLabelAverage_sum P s N e f hf X
-    _ = _ := by rw [mul_pow,←pow_mul]; congr 2; omega
-end TraceReconstruction
-
-end
-end
-end
-end
-end
-end
-end
-end
-end
-
-end OAI
+import OAI.Probability.TraceReconstruction.LeafOracle
+
+namespace OAI
+
+/-!
+# Anchors
+
+Forced anchors, extra-leaf closeness, and correlated group square-sum estimates.
+-/
+
+open scoped ENNReal
+noncomputable section
+
+open Filter
+open scoped Topology
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+open FiniteLaw
+
+namespace FiniteLaw
+/-- Two distinct actual child coordinates are independent. This does not
+assert independence of their shared parent-global jitter. -/
+theorem coordinateProduct_expect_two {α : Type*} [Fintype α] {n : ℕ}
+    (P : Fin n → FiniteLaw α) (a c : Fin n) (hne : a≠c) (f h : α → ℝ) :
+    (coordinateProduct P).expect (fun x => f (x a)*h (x c)) = (P a).expect f*(P c).expect h := by
+  let F : Fin n → α → ℝ := fun j x => if j=a then f x else if j=c then h x else 1
+  have hp : ∀x : Fin n → α, (∏j,F j (x j)) = f (x a)*h (x c) := by
+    intro x
+    rw [Finset.prod_eq_mul a c hne]
+    · simp [F,hne.symm]
+    · intro j hj hjs
+      simp [F,hjs.1,hjs.2]
+    · simp
+    · simp
+  rw [show (fun x : Fin n → α => f (x a)*h (x c)) = (fun x => ∏j,F j (x j)) by
+    funext x; exact (hp x).symm,coordinateProduct_expect_product]
+  rw [Finset.prod_eq_mul a c hne]
+  · simp [F,hne.symm]
+  · intro j hj hjs
+    simp [F,hjs.1,hjs.2,expect_const]
+  · simp
+  · simp
+
+/-- Exact local/global/children factorization with the parent-global draw
+outside the product of child expectations. -/
+theorem expect_prod_two {α β γ : Type*} [Fintype α] [Fintype β] [Fintype γ]
+    {n : ℕ} (P : FiniteLaw α) (Q : FiniteLaw β) (R : Fin n → FiniteLaw γ)
+    (a c : Fin n) (hne : a≠c) (f : α → ℝ) (h k : β → γ → ℝ) :
+    (P.prod (Q.prod (coordinateProduct R))).expect (fun x =>
+      f x.1 * (h x.2.1 (x.2.2 a)*k x.2.1 (x.2.2 c))) =
+      P.expect f * Q.expect (fun z => (R a).expect (h z)*(R c).expect (k z)) := by
+  simp_rw [expect_prod,expect_const_mul,coordinateProduct_expect_two R a c hne]
+  rw [expect_mul_const]
+end FiniteLaw
+
+/-- A concrete binary selection of labels in the complete index tree.
+Unlike a free abstract shape this retains every actual column label. -/
+def BinaryWindowLabels (T : ℕ → ℕ) : ℕ → Type
+  | 0 => PUnit
+  | b+1 => Fin (T b) × Fin (T b) × BinaryWindowLabels T b × BinaryWindowLabels T b
+
+noncomputable instance binaryWindowLabelsFintype (T : ℕ → ℕ) :
+    (b : ℕ) → Fintype (BinaryWindowLabels T b)
+  | 0 => inferInstanceAs (Fintype PUnit)
+  | b+1 => by
+      letI := binaryWindowLabelsFintype T b
+      exact inferInstanceAs (Fintype (Fin (T b) × Fin (T b) × BinaryWindowLabels T b × BinaryWindowLabels T b))
+
+/-- The forced children are distinct at every split. Central restrictions
+are separate deterministic predicates, not conditioning events. -/
+def BinaryWindowLabels.distinct (T : ℕ → ℕ) : (b : ℕ) → BinaryWindowLabels T b → Prop
+  | 0, _ => True
+  | b+1, p => p.1≠p.2.1 ∧ distinct T b p.2.2.1 ∧ distinct T b p.2.2.2
+
+/-- Product of the first-child unsigned anchor at every forced nonleaf,
+evaluated using the literal complete-tree orientation assignment. -/
+def binaryAnchorKernel (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ) :
+    (b : ℕ) → BinaryWindowLabels T b → WindowAssignments T n g Bool b → ℤ → ℝ
+  | 0, _, _, _ => 1
+  | b+1, p, bits, incoming =>
+      motifIndicator s ((p.1.val:ℤ)-center b+(∑i,(bitIndex (bits.1 i):ℤ))+incoming) *
+      (binaryAnchorKernel T n g s center globalShift b p.2.2.1 (bits.2.2 p.1)
+        (globalShift b bits.2.1 p.1) *
+       binaryAnchorKernel T n g s center globalShift b p.2.2.2 (bits.2.2 p.2.1)
+        (globalShift b bits.2.1 p.2.1))
+
+def binaryAnchorAverage (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (b : ℕ) (p : BinaryWindowLabels T b) (d : WindowAssignments T n g (Option Bool) b)
+    (incoming : ℤ) : ℝ :=
+  (windowPosteriorLaw T n g b d).expect (fun bits =>
+    binaryAnchorKernel T n g s center globalShift b p bits incoming)
+
+theorem binaryAnchorAverage_zero (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (p : BinaryWindowLabels T 0) (d : WindowAssignments T n g (Option Bool) 0) (incoming : ℤ) :
+    binaryAnchorAverage T n g s center globalShift 0 p d incoming = 1 := by
+  apply expect_const
+
+theorem binaryAnchorAverage_succ (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (b : ℕ) (p : BinaryWindowLabels T (b+1)) (d : WindowAssignments T n g (Option Bool) (b+1))
+    (incoming : ℤ) (hd : p.1≠p.2.1) :
+    binaryAnchorAverage T n g s center globalShift (b+1) p d incoming =
+      (posteriorArray d.1).expect (fun loc =>
+        motifIndicator s ((p.1.val:ℤ)-center b+(∑i,(bitIndex (loc i):ℤ))+incoming)) *
+      (posteriorArray d.2.1).expect (fun glob =>
+        binaryAnchorAverage T n g s center globalShift b p.2.2.1 (d.2.2 p.1)
+          (globalShift b glob p.1) *
+        binaryAnchorAverage T n g s center globalShift b p.2.2.2 (d.2.2 p.2.1)
+          (globalShift b glob p.2.1)) := by
+  exact FiniteLaw.expect_prod_two (posteriorArray d.1) (posteriorArray d.2.1)
+    (fun i => windowPosteriorLaw T n g b (d.2.2 i)) p.1 p.2.1 hd
+    (fun loc => motifIndicator s ((p.1.val:ℤ)-center b+(∑i,(bitIndex (loc i):ℤ))+incoming))
+    (fun glob bits => binaryAnchorKernel T n g s center globalShift b p.2.2.1 bits
+      (globalShift b glob p.1))
+    (fun glob bits => binaryAnchorKernel T n g s center globalShift b p.2.2.2 bits
+      (globalShift b glob p.2.1))
+
+/-- Local unsigned anchor averaged against its actual known/hidden flags. -/
+def localAnchorAverage {m : ℕ} (d : Fin m → Option Bool) (s : ℕ) (z : ℤ) : ℝ :=
+  (posteriorArray d).expect (fun bits => motifIndicator s (z+∑i,(bitIndex (bits i):ℤ)))
+
+theorem localAnchorAverage_kernel {m : ℕ} (d : Fin m → Option Bool) (s : ℕ) (z : ℤ) :
+    localAnchorAverage d s z = windowKernel (∑i,hiddenTrial (d i)) (2^s)
+      (z+∑i,(visibleOnes (d i):ℤ)) := by
+  have h := posterior_unsigned_motif d s z
+  unfold localAnchorAverage posteriorArray coordinateProduct expect
+  convert h using 1
+  congr 2
+  exact Subsingleton.elim _ _
+
+/-- Product of all lower sibling-group factors. Each group still averages
+its own single common global draw, not two independent global draws. -/
+def binaryAnchorBelow (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ) :
+    (b : ℕ) → BinaryWindowLabels T (b+1) → WindowAssignments T n g (Option Bool) (b+1) → ℝ
+  | 0, _, _ => 1
+  | b+1, p, d =>
+      (posteriorArray d.2.1).expect (fun glob =>
+        localAnchorAverage (d.2.2 p.1).1 s
+          ((p.2.2.1.1.val:ℤ)-center b+globalShift (b+1) glob p.1) *
+        localAnchorAverage (d.2.2 p.2.1).1 s
+          ((p.2.2.2.1.val:ℤ)-center b+globalShift (b+1) glob p.2.1)) *
+      (binaryAnchorBelow T n g s center globalShift b p.2.2.1 (d.2.2 p.1) *
+       binaryAnchorBelow T n g s center globalShift b p.2.2.2 (d.2.2 p.2.1))
+
+/-- Exact factorization of the forced anchor expectation into its root
+kernel and all correlated sibling groups, at the real metadata/labels. -/
+theorem binaryAnchorAverage_factor (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (b : ℕ) (p : BinaryWindowLabels T (b+1)) (d : WindowAssignments T n g (Option Bool) (b+1))
+    (incoming : ℤ) (hd : BinaryWindowLabels.distinct T (b+1) p) :
+    binaryAnchorAverage T n g s center globalShift (b+1) p d incoming =
+      localAnchorAverage d.1 s ((p.1.val:ℤ)-center b+incoming) *
+        binaryAnchorBelow T n g s center globalShift b p d := by
+  induction b generalizing incoming with
+  | zero =>
+    rw [binaryAnchorAverage_succ T n g s center globalShift _ p d incoming hd.1]
+    simp only [binaryAnchorAverage_zero,expect_const,binaryAnchorBelow,mul_one]
+    unfold localAnchorAverage
+    congr 1
+    funext loc
+    congr 1
+    omega
+  | succ b ih =>
+    have hl := fun incoming => ih p.2.2.1 (d.2.2 p.1) incoming hd.2.1
+    have hr := fun incoming => ih p.2.2.2 (d.2.2 p.2.1) incoming hd.2.2
+    have hroot : (posteriorArray d.1).expect (fun loc =>
+        motifIndicator s ((p.1.val:ℤ)-center (b+1)+(∑i,(bitIndex (loc i):ℤ))+incoming)) =
+        localAnchorAverage d.1 s ((p.1.val:ℤ)-center (b+1)+incoming) := by
+      apply congrArg (FiniteLaw.expect (posteriorArray d.1))
+      funext loc
+      congr 1
+      omega
+    have hh : (posteriorArray d.2.1).expect (fun glob =>
+        binaryAnchorAverage T n g s center globalShift (b+1) p.2.2.1 (d.2.2 p.1)
+          (globalShift (b+1) glob p.1) *
+        binaryAnchorAverage T n g s center globalShift (b+1) p.2.2.2 (d.2.2 p.2.1)
+          (globalShift (b+1) glob p.2.1)) =
+        binaryAnchorBelow T n g s center globalShift (b+1) p d := by
+      calc
+        _ = (posteriorArray d.2.1).expect (fun glob =>
+          (localAnchorAverage (d.2.2 p.1).1 s
+            ((p.2.2.1.1.val:ℤ)-center b+globalShift (b+1) glob p.1) *
+          localAnchorAverage (d.2.2 p.2.1).1 s
+            ((p.2.2.2.1.val:ℤ)-center b+globalShift (b+1) glob p.2.1)) *
+          (binaryAnchorBelow T n g s center globalShift b p.2.2.1 (d.2.2 p.1) *
+           binaryAnchorBelow T n g s center globalShift b p.2.2.2 (d.2.2 p.2.1))) := by
+            apply congrArg (FiniteLaw.expect (posteriorArray d.2.1))
+            funext glob
+            exact (congrArg₂ (·*·) (hl _) (hr _)).trans (by ring)
+        _ = _ := expect_mul_const _ _ _
+    exact (binaryAnchorAverage_succ T n g s center globalShift _ p d incoming hd.1).trans
+      (congrArg₂ (·*·) hroot hh)
+
+/-- One plateau group's local means are the genuine shared-global joint
+anchor, with exactly the affine shift appearing in the layout. -/
+theorem localAnchor_pair_plateau {mg m₁ m₂ : ℕ}
+    (dg : Fin mg → Option Bool) (d₁ : Fin m₁ → Option Bool) (d₂ : Fin m₂ → Option Bool)
+    (s : ℕ) (z₁ z₂ shift : ℤ) :
+    (posteriorArray dg).expect (fun glob =>
+      localAnchorAverage d₁ s (z₁+(∑i,(bitIndex (glob i):ℤ))-shift) *
+      localAnchorAverage d₂ s (z₂+(∑i,(bitIndex (glob i):ℤ))-shift)) =
+    posteriorJointAnchors dg d₁ d₂ s (z₁-shift) (z₂-shift) := by
+  unfold localAnchorAverage posteriorArray coordinateProduct expect posteriorJointAnchors
+  have he : ∀a b c d : ℤ,(a+b-c)+d = (a-c)+b+d := by intros; omega
+  simp_rw [he,mul_assoc]
+  congr 2
+  · exact Subsingleton.elim _ _
+  · funext bg
+    congr 1
+    congr 1
+    · congr 2 ; exact Subsingleton.elim _ _
+    · congr 2 ; exact Subsingleton.elim _ _
+
+end TraceReconstruction
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+open FiniteLaw
+
+/-- A role-selected forced subtree remembers the actual complete-tree labels,
+not merely its abstract binary shape. -/
+def forcedWindowLabels (T : ℕ → ℕ) : (b : ℕ) →
+    (S : Finset (WindowLeaf T (b+1))) → (a : OccupiedWindowChoice T b S) →
+    (occupiedWindowTerm T b S a).Forced → BinaryWindowLabels T (b+1)
+  | 0, S, _, .node i j _ _ _ _ _ =>
+      ((S.equivFin.symm i).val.1,(S.equivFin.symm j).val.1,PUnit.unit,PUnit.unit)
+  | b+1, S, a, .node i j _ _ _ left right =>
+      (((selectedChildren S).equivFin.symm i).val,
+       ((selectedChildren S).equivFin.symm j).val,
+       forcedWindowLabels T b (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val)
+         (a.1 i) left,
+       forcedWindowLabels T b (selectedChildLeaves S ((selectedChildren S).equivFin.symm j).val)
+         (a.1 j) right)
+
+theorem forcedWindowLabels_distinct (T : ℕ → ℕ) (b : ℕ)
+    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S)
+    (F : (occupiedWindowTerm T b S a).Forced) :
+    BinaryWindowLabels.distinct T (b+1) (forcedWindowLabels T b S a F) := by
+  induction b with
+  | zero =>
+    cases F with
+    | node i j hne hi hj left right =>
+      refine ⟨?_,trivial,trivial⟩
+      intro he
+      apply hne
+      apply S.equivFin.symm.injective
+      apply Subtype.ext
+      exact Prod.ext he (Subsingleton.elim (α := Fin 1) _ _)
+  | succ b ih =>
+    cases F with
+    | node i j hne hi hj left right =>
+      refine ⟨?_,ih _ _ left,ih _ _ right⟩
+      intro he
+      exact hne ((selectedChildren S).equivFin.symm.injective (Subtype.ext he))
+
+theorem motifIndicator_nonneg (s : ℕ) (z : ℤ) : 0 ≤ motifIndicator s z := by
+  unfold motifIndicator
+  split_ifs <;> norm_num
+
+theorem motifIndicator_le_one (s : ℕ) (z : ℤ) : motifIndicator s z≤1 := by
+  unfold motifIndicator
+  split_ifs <;> norm_num
+
+theorem binaryAnchorKernel_nonneg (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (b : ℕ) (p : BinaryWindowLabels T b) (bits : WindowAssignments T n g Bool b)
+    (incoming : ℤ) : 0≤binaryAnchorKernel T n g s center globalShift b p bits incoming := by
+  induction b generalizing incoming with
+  | zero => exact zero_le_one
+  | succ b ih =>
+    exact mul_nonneg (motifIndicator_nonneg _ _) (mul_nonneg (ih _ _ _) (ih _ _ _))
+
+theorem bottomRoleWeight_abs_le_indicator (χ σ : ℝ) (h : IsMotifValue χ σ) (a : Fin 3) :
+    |bottomRoleWeight χ σ a|≤χ := by
+  rcases h with ⟨rfl,rfl⟩ | ⟨rfl,rfl|rfl⟩ <;>
+    fin_cases a <;> norm_num [bottomRoleWeight]
+
+theorem internalRoleWeight_active_abs_le_indicator (odd : Prop) [Decidable odd]
+    (χ σ : ℝ) (h : IsMotifValue χ σ) (a : Fin 3) (ha : a≠0) :
+    |internalRoleWeight odd χ σ a|≤χ := by
+  rcases h with ⟨rfl,rfl⟩ | ⟨rfl,rfl|rfl⟩ <;>
+    fin_cases a <;> by_cases ho : odd <;> norm_num [internalRoleWeight,roleWeight,ho] at *
+
+theorem occupiedRootWeight_bottom_anchor (T : ℕ → ℕ) (s : ℕ)
+    (S : Finset (WindowLeaf T 1)) (a : OccupiedWindowChoice T 0 S) (X : ℤ) (i : Fin S.card) :
+    |occupiedRootWeight T s 0 S a X|≤
+      motifIndicator s (((S.equivFin.symm i).val.1.val:ℤ)-X) := by
+  exact (abs_prod_le_coordinate _ i (fun j _ =>
+    bottomRoleWeight_abs_le_one _ _ (prouhet_motif_value _ _) _)).trans
+      (bottomRoleWeight_abs_le_indicator _ _ (prouhet_motif_value _ _) _)
+
+theorem occupiedRootWeight_internal_anchor (T : ℕ → ℕ) (s b : ℕ)
+    (S : Finset (WindowLeaf T (b+1+1))) (a : OccupiedWindowChoice T (b+1) S) (X : ℤ)
+    (i : Fin (selectedChildren S).card) (hi : a.2 i≠0) :
+    |occupiedRootWeight T s (b+1) S a X|≤
+      motifIndicator s ((((selectedChildren S).equivFin.symm i).val.val:ℤ)-X) := by
+  exact (abs_prod_le_coordinate _ i (fun j _ =>
+    internalRoleWeight_abs_le_one _ _ _ (prouhet_motif_value _ _) _)).trans
+      (internalRoleWeight_active_abs_le_indicator _ _ _ (prouhet_motif_value _ _) _ hi)
+
+theorem abs_prod_le_two {ι : Type*} [Fintype ι] (f : ι → ℝ) (a c : ι) (hne : a≠c)
+    (h : ∀j,j≠a → j≠c → |f j|≤1) : |∏j,f j|≤|f a| * |f c| := by
+  rw [Finset.abs_prod]
+  calc
+    _ ≤ ∏j, if j=a then |f a| else if j=c then |f c| else 1 := by
+      apply Finset.prod_le_prod₀ (fun _ _ => abs_nonneg _)
+      intro j hj
+      by_cases hja : j=a
+      · simp [hja]
+      · by_cases hjc : j=c
+        · simp [hjc,hne.symm]
+        · simpa only [ite_eq_right hja,ite_eq_right hjc] using h j hja hjc
+    _ = _ := by
+      rw [Finset.prod_eq_mul a c hne]
+      · simp [hne.symm]
+      · intro j hj hjs
+        simp [hjs.1,hjs.2]
+      · simp
+      · simp
+
+/-- The real role term is pointwise dominated by its selected unsigned
+anchors. No independence assumption is used in this domination. -/
+theorem sourceRoleTerm_forced_anchor (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (b : ℕ) (bits : WindowAssignments T n g Bool (b+1)) (incoming : ℤ)
+    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S)
+    (F : (occupiedWindowTerm T b S a).Forced) :
+    |sourceRoleTerm T n g s center globalShift b bits incoming S a|≤
+      binaryAnchorKernel T n g s center globalShift (b+1) (forcedWindowLabels T b S a F) bits incoming := by
+  induction b generalizing incoming with
+  | zero =>
+    cases F with
+    | node i j hne hi hj left right =>
+      rw [sourceRoleTerm_zero]
+      change _ ≤ motifIndicator s (((S.equivFin.symm i).val.1.val:ℤ)-center 0+
+        (∑i,(bitIndex (bits.1 i):ℤ))+incoming)*(1*1)
+      rw [mul_one,mul_one]
+      convert occupiedRootWeight_bottom_anchor T s S a
+        (center 0-(∑i,(bitIndex (bits.1 i):ℤ))-incoming) i using 1 ; congr 1 ; omega
+  | succ b ih =>
+    cases F with
+    | node i j hne hi hj left right =>
+      rw [sourceRoleTerm_succ,abs_mul]
+      have hroot := occupiedRootWeight_internal_anchor T s b S a
+        (center (b+1)-(∑i,(bitIndex (bits.1 i):ℤ))-incoming) i hi
+      have hprod := abs_prod_le_two (fun k : Fin (selectedChildren S).card =>
+        sourceRoleTerm T n g s center globalShift b
+          (bits.2.2 ((selectedChildren S).equivFin.symm k).val)
+          (globalShift (b+1) bits.2.1 ((selectedChildren S).equivFin.symm k).val)
+          (selectedChildLeaves S ((selectedChildren S).equivFin.symm k).val) (a.1 k)) i j hne
+          (fun k _ _ => occupiedWindowWeight_abs_le_one T s _ b [] _ _)
+      have hl := ih (bits.2.2 ((selectedChildren S).equivFin.symm i).val)
+        (globalShift (b+1) bits.2.1 ((selectedChildren S).equivFin.symm i).val) _ _ left
+      have hr := ih (bits.2.2 ((selectedChildren S).equivFin.symm j).val)
+        (globalShift (b+1) bits.2.1 ((selectedChildren S).equivFin.symm j).val) _ _ right
+      have hp := mul_le_mul hl hr (abs_nonneg _) (binaryAnchorKernel_nonneg T n g s center globalShift _ _ _ _)
+      have hh := mul_le_mul hroot (hprod.trans hp) (abs_nonneg _) (motifIndicator_nonneg _ _)
+      apply hh.trans_eq
+      change motifIndicator s _ * _ = motifIndicator s
+        ((((selectedChildren S).equivFin.symm i).val.val:ℤ)-center (b+1)+
+          (∑k,(bitIndex (bits.1 k):ℤ))+incoming) * _
+      congr 2
+      omega
+
+/-- Therefore the coefficient average is bounded by the actual forced
+anchor average, with the true posterior law and shared jitter retained. -/
+theorem sourceRoleAverage_forced_anchor (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ)
+    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S)
+    (F : (occupiedWindowTerm T b S a).Forced) :
+    |sourceRoleAverage T n g s center globalShift b d incoming S a|≤
+      binaryAnchorAverage T n g s center globalShift (b+1) (forcedWindowLabels T b S a F) d incoming := by
+  exact ((windowPosteriorLaw T n g (b+1) d).abs_expect_le _).trans
+    ((windowPosteriorLaw T n g (b+1) d).expect_mono (fun bits =>
+      sourceRoleTerm_forced_anchor T n g s center globalShift b bits incoming S a F))
+
+theorem sourceRoleTerm_valid (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (b : ℕ) (bits : WindowAssignments T n g Bool (b+1)) (incoming : ℤ)
+    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S)
+    (h : sourceRoleTerm T n g s center globalShift b bits incoming S a ≠ 0) :
+    (occupiedWindowTerm T b S a).valid := by
+  have hh := (occupiedWindowShape T s
+    (sourceTreeAnchors T n g center globalShift (b+1) bits incoming) b [] S).term_valid
+    (occupiedWindowChoiceEquiv T s
+      (sourceTreeAnchors T n g center globalShift (b+1) bits incoming) b [] S a) h
+  simpa only [occupiedWindowTerm_eq] using hh
+
+theorem sourceRoleAverage_valid (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ)
+    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S)
+    (h : sourceRoleAverage T n g s center globalShift b d incoming S a ≠ 0) :
+    (occupiedWindowTerm T b S a).valid := by
+  by_contra hn
+  have hh : ∀bits, sourceRoleTerm T n g s center globalShift b bits incoming S a=0 := by
+    intro bits
+    by_contra hz
+    exact hn (sourceRoleTerm_valid T n g s center globalShift b bits incoming S a hz)
+  apply h
+  change (windowPosteriorLaw T n g (b+1) d).expect _ = 0
+  rw [show (fun bits => sourceRoleTerm T n g s center globalShift b bits incoming S a) =
+    (fun _ => 0) from funext hh,expect_const]
+
+/-- The source's two alternatives now apply to actual posterior coefficient
+terms: signed singleton smoothing, or domination by a genuine forced tree. -/
+theorem sourceRoleAverage_alternative (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ)
+    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S)
+    (η : ℝ) (hη : 0≤η) (hcap : MetadataSmoothingCap T n g s η (b+1) d)
+    (ho : (occupiedWindowTerm T b S a).isOdd) :
+    |sourceRoleAverage T n g s center globalShift b d incoming S a|≤η ∨
+    ∃F : (occupiedWindowTerm T b S a).Forced,
+      BinaryWindowLabels.distinct T (b+1) (forcedWindowLabels T b S a F) ∧
+      |sourceRoleAverage T n g s center globalShift b d incoming S a|≤
+        binaryAnchorAverage T n g s center globalShift (b+1) (forcedWindowLabels T b S a F) d incoming := by
+  by_cases hz : sourceRoleAverage T n g s center globalShift b d incoming S a=0
+  · exact Or.inl (by simpa only [hz,abs_zero] using hη)
+  by_cases hn : (occupiedWindowTerm T b S a).noSingleton
+  · obtain ⟨F⟩ := OccupiedParityTree.forced_nonempty (occupiedWindowTerm T b S a)
+      (sourceRoleAverage_valid T n g s center globalShift b d incoming S a hz) hn ho
+    exact Or.inr ⟨F,forcedWindowLabels_distinct T b S a F,
+      sourceRoleAverage_forced_anchor T n g s center globalShift b d incoming S a F⟩
+  · exact Or.inl (sourceRoleAverage_singleton T n g s center globalShift b d incoming S a η hcap hn)
+
+end TraceReconstruction
+
+
+namespace TraceReconstruction
+open scoped Classical
+open FiniteLaw
+
+/-- The unsigned bottom constraint at one literal complete-tree leaf. -/
+def sourceBottomIndicator (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ) :
+    (b : ℕ) → WindowAssignments T n g Bool (b+1) → ℤ → WindowLeaf T (b+1) → ℝ
+  | 0, bits, incoming, v =>
+      motifIndicator s ((v.1.val:ℤ)-center 0+(∑i,(bitIndex (bits.1 i):ℤ))+incoming)
+  | b+1, bits, _, v => sourceBottomIndicator T n g s center globalShift b
+      (bits.2.2 v.1) (globalShift (b+1) bits.2.1 v.1) v.2
+
+def sourceLeafProduct (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (b : ℕ) (bits : WindowAssignments T n g Bool (b+1)) (incoming : ℤ)
+    (S : Finset (WindowLeaf T (b+1))) : ℝ :=
+  ∏v∈S, sourceBottomIndicator T n g s center globalShift b bits incoming v
+
+theorem sourceBottomIndicator_zero_one (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (b : ℕ) (bits : WindowAssignments T n g Bool (b+1)) (incoming : ℤ)
+    (v : WindowLeaf T (b+1)) :
+    sourceBottomIndicator T n g s center globalShift b bits incoming v=0 ∨
+    sourceBottomIndicator T n g s center globalShift b bits incoming v=1 := by
+  induction b generalizing incoming with
+  | zero => unfold sourceBottomIndicator motifIndicator; split_ifs <;> simp
+  | succ b ih => exact ih _ _ _
+
+theorem prod_zero_one {ι : Type*} (S : Finset ι) (f : ι → ℝ)
+    (h : ∀i∈S,f i=0 ∨ f i=1) : (∏i∈S,f i)=0 ∨ (∏i∈S,f i)=1 := by
+  by_cases hz : ∃i∈S,f i=0
+  · obtain ⟨i,hi,hz⟩ := hz
+    exact Or.inl (Finset.prod_eq_zero hi hz)
+  · right
+    apply Finset.prod_eq_one
+    intro i hi
+    exact (h i hi).resolve_left (fun he => hz ⟨i,hi,he⟩)
+
+theorem sourceLeafProduct_zero_one (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (b : ℕ) (bits : WindowAssignments T n g Bool (b+1)) (incoming : ℤ)
+    (S : Finset (WindowLeaf T (b+1))) :
+    sourceLeafProduct T n g s center globalShift b bits incoming S=0 ∨
+    sourceLeafProduct T n g s center globalShift b bits incoming S=1 :=
+  prod_zero_one S _ (fun v _ => sourceBottomIndicator_zero_one T n g s center globalShift b bits incoming v)
+
+theorem binaryAnchorKernel_zero_one (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (b : ℕ) (p : BinaryWindowLabels T b) (bits : WindowAssignments T n g Bool b)
+    (incoming : ℤ) :
+    binaryAnchorKernel T n g s center globalShift b p bits incoming=0 ∨
+    binaryAnchorKernel T n g s center globalShift b p bits incoming=1 := by
+  induction b generalizing incoming with
+  | zero => exact Or.inr rfl
+  | succ b ih =>
+    have hl := ih p.2.2.1 (bits.2.2 p.1) (globalShift b bits.2.1 p.1)
+    have hr := ih p.2.2.2 (bits.2.2 p.2.1) (globalShift b bits.2.1 p.2.1)
+    change (_*(_*_)=0) ∨ (_*(_*_)=1)
+    rcases hl with hl|hl <;> rcases hr with hr|hr <;> rw [hl,hr] <;>
+      unfold motifIndicator <;> split_ifs <;> norm_num
+
+theorem sourceLeafProduct_succ (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (b : ℕ) (bits : WindowAssignments T n g Bool (b+1+1)) (incoming : ℤ)
+    (S : Finset (WindowLeaf T (b+1+1))) :
+    sourceLeafProduct T n g s center globalShift (b+1) bits incoming S =
+      ∏i : Fin (selectedChildren S).card,
+        sourceLeafProduct T n g s center globalShift b
+          (bits.2.2 ((selectedChildren S).equivFin.symm i).val)
+          (globalShift (b+1) bits.2.1 ((selectedChildren S).equivFin.symm i).val)
+          (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val) := by
+  symm
+  exact (selected_prod_index (selectedChildren S) (fun h =>
+    ∏v∈selectedChildLeaves S h,
+      sourceBottomIndicator T n g s center globalShift b (bits.2.2 h)
+        (globalShift (b+1) bits.2.1 h) v)).trans
+      (selected_prod_children S (fun v =>
+        sourceBottomIndicator T n g s center globalShift (b+1) bits incoming v))
+
+/-- Every selected leaf is active in the bottom expansion, including those
+outside the forced tree. Their absolute constraints are not lost. -/
+theorem sourceRoleTerm_all_leaf_constraints (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (b : ℕ) (bits : WindowAssignments T n g Bool (b+1)) (incoming : ℤ)
+    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S) :
+    |sourceRoleTerm T n g s center globalShift b bits incoming S a|≤
+      sourceLeafProduct T n g s center globalShift b bits incoming S := by
+  induction b generalizing incoming with
+  | zero =>
+    rw [sourceRoleTerm_zero]
+    change |∏i : Fin S.card, _|≤_
+    rw [Finset.abs_prod]
+    calc
+      _ ≤ ∏i : Fin S.card, motifIndicator s
+          (((S.equivFin.symm i).val.1.val:ℤ)-(center 0-(∑i,(bitIndex (bits.1 i):ℤ))-incoming)) :=
+        Finset.prod_le_prod₀ (fun _ _ => abs_nonneg _)
+          (fun i _ => bottomRoleWeight_abs_le_indicator _ _ (prouhet_motif_value _ _) _)
+      _ = ∏v∈S, motifIndicator s ((v.1.val:ℤ)-
+          (center 0-(∑i,(bitIndex (bits.1 i):ℤ))-incoming)) :=
+        selected_prod_index S (fun v : WindowLeaf T 1 => motifIndicator s
+          ((v.1.val:ℤ)-(center 0-(∑i,(bitIndex (bits.1 i):ℤ))-incoming)))
+      _ = _ := by
+        apply Finset.prod_congr rfl
+        intro v hv
+        change motifIndicator s _ = motifIndicator s _
+        congr 1
+        omega
+  | succ b ih =>
+    rw [sourceRoleTerm_succ,abs_mul,sourceLeafProduct_succ]
+    apply (mul_le_mul_of_nonneg_right
+      (occupiedRootWeight_abs_le_one T s (b+1) S a _) (abs_nonneg _)).trans
+    rw [one_mul,Finset.abs_prod]
+    apply Finset.prod_le_prod₀ (fun _ _ => abs_nonneg _)
+    intro i hi
+    exact ih _ _ _ _
+
+theorem sourceLeafProduct_antitone (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (b : ℕ) (bits : WindowAssignments T n g Bool (b+1)) (incoming : ℤ)
+    (E S : Finset (WindowLeaf T (b+1))) (hES : E⊆S) :
+    sourceLeafProduct T n g s center globalShift b bits incoming S≤
+      sourceLeafProduct T n g s center globalShift b bits incoming E := by
+  apply Finset.prod_le_prod_of_subset_of_le_one₀ hES
+  · intro v hv
+    rcases sourceBottomIndicator_zero_one T n g s center globalShift b bits incoming v with h|h <;> norm_num [h]
+  · intro v hv he
+    rcases sourceBottomIndicator_zero_one T n g s center globalShift b bits incoming v with h|h <;> norm_num [h]
+
+theorem le_mul_of_zero_one {x a b : ℝ} (ha : a=0 ∨ a=1) (hb : b=0 ∨ b=1)
+    (hxa : x≤a) (hxb : x≤b) : x≤a*b := by
+  rcases ha with rfl|rfl <;> rcases hb with rfl|rfl <;> norm_num at * <;> assumption
+
+/-- Actual forced anchor AND extra-leaf constraints in one bit expectation.
+In particular E can be precisely the nonforced leaves; keeping it arbitrary
+here permits the later label summation without any conditioning gates. -/
+theorem sourceRoleAverage_forced_extra (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ)
+    (S E : Finset (WindowLeaf T (b+1))) (hES : E⊆S) (a : OccupiedWindowChoice T b S)
+    (F : (occupiedWindowTerm T b S a).Forced) :
+    |sourceRoleAverage T n g s center globalShift b d incoming S a|≤
+      (windowPosteriorLaw T n g (b+1) d).expect (fun bits =>
+        binaryAnchorKernel T n g s center globalShift (b+1) (forcedWindowLabels T b S a F) bits incoming *
+        sourceLeafProduct T n g s center globalShift b bits incoming E) := by
+  apply ((windowPosteriorLaw T n g (b+1) d).abs_expect_le _).trans
+  apply FiniteLaw.expect_mono
+  intro bits
+  exact le_mul_of_zero_one
+    (binaryAnchorKernel_zero_one T n g s center globalShift _ _ bits incoming)
+    (sourceLeafProduct_zero_one T n g s center globalShift b bits incoming E)
+    (sourceRoleTerm_forced_anchor T n g s center globalShift b bits incoming S a F)
+    ((sourceRoleTerm_all_leaf_constraints T n g s center globalShift b bits incoming S a).trans
+      (sourceLeafProduct_antitone T n g s center globalShift b bits incoming E S hES))
+
+end TraceReconstruction
+
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+open FiniteLaw
+
+/-- Deterministic closeness of each forced split; it is a restriction on
+labels, never an additional conditioning event for the random bits. -/
+def BinaryWindowLabels.close (T : ℕ → ℕ) (s : ℕ) : (b : ℕ) → BinaryWindowLabels T b → Prop
+  | 0, _ => True
+  | b+1, p => |(p.1.val:ℤ)-p.2.1.val|<(2^s:ℕ) ∧ close T s b p.2.2.1 ∧ close T s b p.2.2.2
+
+theorem motifIndicator_pair_close (s : ℕ) (a c X : ℤ)
+    (ha : motifIndicator s (a-X)≠0) (hc : motifIndicator s (c-X)≠0) :
+    |a-c|<(2^s:ℕ) := by
+  have hsa : 0≤a-X ∧ a-X<(2^s:ℕ) := by
+    by_contra h
+    exact ha (by simp only [motifIndicator,ite_eq_right h])
+  have hsc : 0≤c-X ∧ c-X<(2^s:ℕ) := by
+    by_contra h
+    exact hc (by simp only [motifIndicator,ite_eq_right h])
+  exact abs_lt.mpr ⟨by omega,by omega⟩
+
+theorem nonzero_of_abs_le {x y : ℝ} (h : x≠0) (hle : |x|≤y) : y≠0 := by
+  intro hy
+  rw [hy] at hle
+  exact h (abs_eq_zero.mp (le_antisymm hle (abs_nonneg x)))
+
+/-- A nonzero literal role multiplier enforces all close-child restrictions
+of every forced subtree chosen from its fixed role pattern. -/
+theorem sourceRoleTerm_forced_close (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (b : ℕ) (bits : WindowAssignments T n g Bool (b+1)) (incoming : ℤ)
+    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S)
+    (F : (occupiedWindowTerm T b S a).Forced)
+    (h : sourceRoleTerm T n g s center globalShift b bits incoming S a≠0) :
+    BinaryWindowLabels.close T s (b+1) (forcedWindowLabels T b S a F) := by
+  induction b generalizing incoming with
+  | zero =>
+    cases F with
+    | node i j hne hi hj left right =>
+      rw [sourceRoleTerm_zero] at h
+      change |((S.equivFin.symm i).val.1.val:ℤ)-(S.equivFin.symm j).val.1.val|<(2^s:ℕ) ∧ True ∧ True
+      refine ⟨?_,trivial,trivial⟩
+      exact motifIndicator_pair_close s _ _ _
+        (nonzero_of_abs_le h (occupiedRootWeight_bottom_anchor T s S a _ i))
+        (nonzero_of_abs_le h (occupiedRootWeight_bottom_anchor T s S a _ j))
+  | succ b ih =>
+    cases F with
+    | node i j hne hi hj left right =>
+      rw [sourceRoleTerm_succ] at h
+      have hm := mul_ne_zero_iff.mp h
+      have hc := Finset.prod_ne_zero_iff.mp hm.2
+      change |(((selectedChildren S).equivFin.symm i).val.val:ℤ)-
+        ((selectedChildren S).equivFin.symm j).val.val|<(2^s:ℕ) ∧
+        BinaryWindowLabels.close T s (b+1)
+          (forcedWindowLabels T b (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val)
+            (a.1 i) left) ∧
+        BinaryWindowLabels.close T s (b+1)
+          (forcedWindowLabels T b (selectedChildLeaves S ((selectedChildren S).equivFin.symm j).val)
+            (a.1 j) right)
+      refine ⟨?_,ih _ _ _ _ left (hc i (Finset.mem_univ i)),
+        ih _ _ _ _ right (hc j (Finset.mem_univ j))⟩
+      exact motifIndicator_pair_close s _ _ _
+        (nonzero_of_abs_le hm.1 (occupiedRootWeight_internal_anchor T s b S a _ i hi))
+        (nonzero_of_abs_le hm.1 (occupiedRootWeight_internal_anchor T s b S a _ j hj))
+
+/-- Closeness remains a pure label condition after averaging, without
+conditioning on it. -/
+theorem sourceRoleAverage_forced_close (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ)
+    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S)
+    (F : (occupiedWindowTerm T b S a).Forced)
+    (h : sourceRoleAverage T n g s center globalShift b d incoming S a≠0) :
+    BinaryWindowLabels.close T s (b+1) (forcedWindowLabels T b S a F) := by
+  by_contra hn
+  have hh : ∀bits, sourceRoleTerm T n g s center globalShift b bits incoming S a=0 := by
+    intro bits
+    by_contra hz
+    exact hn (sourceRoleTerm_forced_close T n g s center globalShift b bits incoming S a F hz)
+  apply h
+  change (windowPosteriorLaw T n g (b+1) d).expect _ = 0
+  rw [show (fun bits => sourceRoleTerm T n g s center globalShift b bits incoming S a) =
+    (fun _ => 0) from funext hh,expect_const]
+
+end TraceReconstruction
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+open FiniteLaw
+
+/-- The literal finite child labels embedded in the integer line. -/
+def finIntegerLabels (N : ℕ) : Finset ℤ := Finset.univ.image (fun i : Fin N => (i.val:ℤ))
+
+theorem sum_finIntegerLabels (N : ℕ) (f : ℤ → ℝ) :
+    (∑a∈finIntegerLabels N,f a)=∑i : Fin N,f i.val := by
+  rw [finIntegerLabels,Finset.sum_image]
+  intro a ha b hb he
+  apply Fin.ext
+  change (a.val:ℤ)=(b.val:ℤ) at he
+  exact_mod_cast he
+
+theorem sum_close_finIntegerLabels (N L : ℕ) (a : ℤ) (f : ℤ → ℝ) :
+    (∑c∈closeLabels (finIntegerLabels N) L a,f c) =
+      ∑i : Fin N, if |(i.val:ℤ)-a|<(L:ℤ) then f i.val else 0 := by
+  rw [closeLabels,Finset.sum_filter,sum_finIntegerLabels]
+
+/-- Any deterministic plateau/distinctness restriction can be retained in
+child-pair summation. Only closeness is used to upper bound its cost. -/
+theorem localAnchor_finite_energy {m N : ℕ} (d : Fin m → Option Bool) (s : ℕ)
+    (c : ℤ) (R : Fin N → Fin N → Prop) [DecidableRel R]
+    (hR : ∀a b,R a b → |(b.val:ℤ)-a.val|<(2^s:ℕ)) :
+    (∑a : Fin N, ∑b : Fin N, if R a b then localAnchorAverage d s ((a.val:ℤ)-c)^2 else 0) ≤
+      2*((2^s:ℕ):ℝ)^3*binomialBound (∑i,hiddenTrial (d i)) := by
+  have H := root_anchor_energy (∑i,hiddenTrial (d i)) (2^s)
+    (finIntegerLabels N) (finIntegerLabels N) (c-(∑i,(visibleOnes (d i):ℤ)))
+  rw [sum_finIntegerLabels] at H
+  simp_rw [sum_close_finIntegerLabels] at H
+  calc
+    _ ≤ ∑a : Fin N, ∑b : Fin N,
+        if |(b.val:ℤ)-a.val|<(2^s:ℕ) then
+          windowKernel (∑i,hiddenTrial (d i)) (2^s)
+            ((a.val:ℤ)-(c-∑i,(visibleOnes (d i):ℤ)))^2 else 0 := by
+      apply Finset.sum_le_sum
+      intro a ha
+      apply Finset.sum_le_sum
+      intro b hb
+      by_cases h : R a b
+      · rw [ite_eq_left h,ite_eq_left (hR a b h),localAnchorAverage_kernel]
+        convert le_refl _ using 1 ; congr 2 ; omega
+      · rw [ite_eq_right h]
+        split_ifs <;> positivity
+    _ ≤ _ := H
+
+/-- Actual shared-jitter energy over two finite sibling pairs. Parent-label
+plateau restrictions are discharged before applying this lemma; arbitrary
+restrictions on these child pairs are kept, not conditioned on. -/
+theorem posteriorJoint_finite_energy {mg m₁ m₂ N : ℕ}
+    (dg : Fin mg → Option Bool) (d₁ : Fin m₁ → Option Bool) (d₂ : Fin m₂ → Option Bool)
+    (s : ℕ) (c₁ c₂ : ℤ) (R : Fin N → Fin N → Prop) [DecidableRel R]
+    (hR : ∀a b,R a b → |(b.val:ℤ)-a.val|<(2^s:ℕ)) :
+    (∑a : Fin N, ∑c : Fin N, ∑b : Fin N, ∑d : Fin N,
+      if R a c ∧ R b d then
+        posteriorJointAnchors dg d₁ d₂ s ((a.val:ℤ)-c₁) ((b.val:ℤ)-c₂)^2 else 0) ≤
+       4*((2^s:ℕ):ℝ)^6*binomialBound (∑i,hiddenTrial (dg i))*binomialBound (∑i,hiddenTrial (d₂ i)) := by
+  have H := posterior_sibling_anchor_energy dg d₁ d₂ s
+    (finIntegerLabels N) (finIntegerLabels N) (finIntegerLabels N) (finIntegerLabels N) c₁ c₂
+  simp_rw [sum_finIntegerLabels,sum_close_finIntegerLabels] at H
+  calc
+    _ = ∑a : Fin N, ∑b : Fin N, ∑c : Fin N, ∑d : Fin N,
+      if R a c ∧ R b d then
+        posteriorJointAnchors dg d₁ d₂ s ((a.val:ℤ)-c₁) ((b.val:ℤ)-c₂)^2 else 0 := by
+      apply Finset.sum_congr rfl
+      intro a ha
+      rw [Finset.sum_comm]
+    _ ≤ ∑a : Fin N, ∑b : Fin N, ∑c : Fin N,
+      if |(c.val:ℤ)-a.val|<(2^s:ℕ) then
+        ∑d : Fin N, if |(d.val:ℤ)-b.val|<(2^s:ℕ) then
+          posteriorJointAnchors dg d₁ d₂ s ((a.val:ℤ)-c₁) ((b.val:ℤ)-c₂)^2 else 0
+      else 0 := by
+      apply Finset.sum_le_sum
+      intro a ha
+      apply Finset.sum_le_sum
+      intro b hb
+      apply Finset.sum_le_sum
+      intro c hc
+      by_cases h : |(c.val:ℤ)-a.val|<(2^s:ℕ)
+      · rw [ite_eq_left h]
+        apply Finset.sum_le_sum
+        intro d hd
+        by_cases hr : R a c ∧ R b d
+        · rw [ite_eq_left hr,ite_eq_left (hR b d hr.2)]
+        · rw [ite_eq_right hr]
+          split_ifs <;> positivity
+      · rw [ite_eq_right h]
+        apply le_of_eq
+        apply Finset.sum_eq_zero
+        intro d hd
+        exact ite_eq_right (fun hr => h (hR a c hr.1))
+    _ ≤ _ := H
+
+end TraceReconstruction
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+open FiniteLaw
+
+/-- The actual finite central domain, distinctness and close-label restriction.
+It is purely deterministic and is never added to the orientation posterior. -/
+def anchorPairAllowed (T : ℕ → ℕ) (A : (b : ℕ) → Finset (Fin (T b))) (s b : ℕ)
+    (a c : Fin (T b)) : Prop := a∈A b ∧ c∈A b ∧ a≠c ∧ |(c.val:ℤ)-a.val|<(2^s:ℕ)
+
+/-- The source global plateau identity is required pointwise for bits, not
+just in distribution and not after conditioning on successful layout. -/
+def HasGlobalPlateau (T g : ℕ → ℕ) (A : (b : ℕ) → Finset (Fin (T b))) (shift : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ) : Prop :=
+  ∀b glob u, u∈A b → globalShift b glob u=(∑i,(bitIndex (glob i):ℤ))-shift b
+
+/-- Numerical binomial caps extracted from the literal known/hidden metadata.
+In applications these are proved using the source hidden-trial lower bounds. -/
+def MetadataBinomialCaps (T n g : ℕ → ℕ) (B C : ℕ → ℝ) :
+    (b : ℕ) → WindowAssignments T n g (Option Bool) b → Prop
+  | 0, _ => True
+  | b+1, d => binomialBound (∑i, hiddenTrial (d.1 i)) ≤ B b ∧
+      binomialBound (∑i, hiddenTrial (d.2.1 i)) ≤ C b ∧
+      ∀u, MetadataBinomialCaps T n g B C b (d.2.2 u)
+
+/-- One correlated sibling-group anchor, using the actual child metadata
+and their parent's one shared global orientation collection. -/
+def binarySiblingAnchor (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1+1))
+    (u v : Fin (T (b+1))) (a c : Fin (T b)) : ℝ :=
+  (posteriorArray d.2.1).expect (fun glob =>
+    localAnchorAverage (d.2.2 u).1 s ((a.val:ℤ)-center b+globalShift (b+1) glob u) *
+    localAnchorAverage (d.2.2 v).1 s ((c.val:ℤ)-center b+globalShift (b+1) glob v))
+
+theorem binarySiblingAnchor_plateau (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (A : (b : ℕ) → Finset (Fin (T b))) (shift : ℕ → ℤ)
+    (hplat : HasGlobalPlateau T g A shift globalShift)
+    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1+1))
+    (u v : Fin (T (b+1))) (hu : u∈A (b+1)) (hv : v∈A (b+1)) (a c : Fin (T b)) :
+    binarySiblingAnchor T n g s center globalShift b d u v a c =
+      posteriorJointAnchors d.2.1 (d.2.2 u).1 (d.2.2 v).1 s
+        ((a.val:ℤ)-(center b+shift (b+1))) ((c.val:ℤ)-(center b+shift (b+1))) := by
+  calc
+    _ = (posteriorArray d.2.1).expect (fun glob =>
+        localAnchorAverage (d.2.2 u).1 s ((a.val:ℤ)-center b+(∑i,(bitIndex (glob i):ℤ))-shift (b+1)) *
+        localAnchorAverage (d.2.2 v).1 s ((c.val:ℤ)-center b+(∑i,(bitIndex (glob i):ℤ))-shift (b+1))) := by
+      apply congrArg (FiniteLaw.expect (posteriorArray d.2.1))
+      funext glob
+      have hu' := hplat (b+1) glob u hu
+      have hv' := hplat (b+1) glob v hv
+      rw [hu',hv']
+      congr 2 <;> omega
+    _ = _ := by
+      rw [localAnchor_pair_plateau]
+      congr 1 <;> omega
+
+/-- Exact dynamic sum of squared group anchors over the allowed finite
+labels below a fixed forced split. Distinct groups retain their own metadata. -/
+def binaryDescendantEnergy (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (A : (b : ℕ) → Finset (Fin (T b))) :
+    (b : ℕ) → WindowAssignments T n g (Option Bool) (b+1) → Fin (T b) → Fin (T b) → ℝ
+  | 0, _, _, _ => 1
+  | b+1, d, u, v =>
+      ∑a : Fin (T b),∑c : Fin (T b),∑e : Fin (T b),∑f : Fin (T b),
+        if anchorPairAllowed T A s b a c ∧ anchorPairAllowed T A s b e f then
+          binarySiblingAnchor T n g s center globalShift b d u v a e ^ 2 *
+            binaryDescendantEnergy T n g s center globalShift A b (d.2.2 u) a c *
+            binaryDescendantEnergy T n g s center globalShift A b (d.2.2 v) e f
+        else 0
+
+theorem binaryDescendantEnergy_nonneg (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (A : (b : ℕ) → Finset (Fin (T b))) (b : ℕ)
+    (d : WindowAssignments T n g (Option Bool) (b+1)) (u v : Fin (T b)) :
+    0≤binaryDescendantEnergy T n g s center globalShift A b d u v := by
+  induction b with
+  | zero => exact zero_le_one
+  | succ b ih =>
+    apply Finset.sum_nonneg
+    intro a ha
+    apply Finset.sum_nonneg
+    intro c hc
+    apply Finset.sum_nonneg
+    intro e he
+    apply Finset.sum_nonneg
+    intro f hf
+    split_ifs
+    · exact mul_nonneg (mul_nonneg (sq_nonneg _) (ih _ _ _)) (ih _ _ _)
+    · exact le_refl _
+
+/-- True common-jitter energy at one group, with its exact finite domains. -/
+theorem binarySiblingAnchor_energy (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (A : (b : ℕ) → Finset (Fin (T b))) (shift : ℕ → ℤ)
+    (hplat : HasGlobalPlateau T g A shift globalShift)
+    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1+1))
+    (u v : Fin (T (b+1))) (hu : u∈A (b+1)) (hv : v∈A (b+1)) :
+    (∑a : Fin (T b),∑c : Fin (T b),∑e : Fin (T b),∑f : Fin (T b),
+        if anchorPairAllowed T A s b a c ∧ anchorPairAllowed T A s b e f then
+          binarySiblingAnchor T n g s center globalShift b d u v a e ^ 2 else 0) ≤
+      4*((2^s:ℕ):ℝ)^6*binomialBound (∑i,hiddenTrial (d.2.1 i))*
+        binomialBound (∑i,hiddenTrial ((d.2.2 v).1 i)) := by
+  simp_rw [binarySiblingAnchor_plateau T n g s center globalShift A shift hplat b d u v hu hv]
+  exact posteriorJoint_finite_energy d.2.1 (d.2.2 u).1 (d.2.2 v).1 s
+    (center b+shift (b+1)) (center b+shift (b+1)) (anchorPairAllowed T A s b)
+    (fun a c h => h.2.2.2)
+
+/-- The complete bottom-up forced-label energy estimate. Its only numerical
+inputs are actual metadata hidden-count binomial caps. The group bound
+keeps one shared global kernel until AFTER squaring. -/
+theorem binaryDescendantEnergy_le (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (A : (b : ℕ) → Finset (Fin (T b))) (shift : ℕ → ℤ)
+    (hplat : HasGlobalPlateau T g A shift globalShift)
+    (B C : ℕ → ℝ) (hB : ∀b,0≤B b) (hC : ∀b,0≤C b)
+    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1))
+    (hcap : MetadataBinomialCaps T n g B C (b+1) d)
+    (u v : Fin (T b)) (hu : u∈A b) (hv : v∈A b) :
+    binaryDescendantEnergy T n g s center globalShift A b d u v ≤
+      binaryTreeBudget (fun j => 4*((2^s:ℕ):ℝ)^6*C (j+1)*B j) b := by
+  let K : ℕ → ℝ := fun j => 4*((2^s:ℕ):ℝ)^6*C (j+1)*B j
+  have hK : ∀j,0≤K j := fun j => mul_nonneg (mul_nonneg (by positivity) (hC _)) (hB _)
+  induction b with
+  | zero => exact le_refl _
+  | succ b ih =>
+    have hbudget := binaryTreeBudget_nonneg K hK b
+    have hgroup := binarySiblingAnchor_energy T n g s center globalShift A shift hplat b d u v hu hv
+    have hnum : 4*((2^s:ℕ):ℝ)^6*binomialBound (∑i,hiddenTrial (d.2.1 i))*
+        binomialBound (∑i,hiddenTrial ((d.2.2 v).1 i)) ≤ K b := by
+      exact mul_le_mul
+        (mul_le_mul_of_nonneg_left hcap.2.1 (by positivity))
+        (hcap.2.2 v).1 (by unfold binomialBound; positivity)
+        (mul_nonneg (by positivity) (hC _))
+    calc
+      _ ≤ (∑a : Fin (T b),∑c : Fin (T b),∑e : Fin (T b),∑f : Fin (T b),
+        if anchorPairAllowed T A s b a c ∧ anchorPairAllowed T A s b e f then
+          binarySiblingAnchor T n g s center globalShift b d u v a e ^ 2 else 0) *
+          (binaryTreeBudget K b)^2 := by
+        simp only [binaryDescendantEnergy,Finset.sum_mul]
+        apply Finset.sum_le_sum
+        intro a ha
+        apply Finset.sum_le_sum
+        intro c hc
+        apply Finset.sum_le_sum
+        intro e he
+        apply Finset.sum_le_sum
+        intro f hf
+        split_ifs with hr
+        · have hl := ih (d.2.2 u) (hcap.2.2 u) a c hr.1.1 hr.1.2.1
+          have hh := ih (d.2.2 v) (hcap.2.2 v) e f hr.2.1 hr.2.2.1
+          have hm := mul_le_mul hl hh
+            (binaryDescendantEnergy_nonneg T n g s center globalShift A b (d.2.2 v) e f) hbudget
+          have hf := mul_le_mul_of_nonneg_left hm
+            (sq_nonneg (binarySiblingAnchor T n g s center globalShift b d u v a e))
+          simpa only [K,pow_two,mul_assoc] using hf
+        · simp only [zero_mul,le_refl]
+      _ ≤ K b*(binaryTreeBudget K b)^2 :=
+        mul_le_mul_of_nonneg_right (hgroup.trans hnum) (sq_nonneg _)
+      _ = _ := rfl
+
+end TraceReconstruction
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+open FiniteLaw
+
+def BinaryWindowLabels.allowed (T : ℕ → ℕ) (A : (b : ℕ) → Finset (Fin (T b))) (s : ℕ) :
+    (b : ℕ) → BinaryWindowLabels T b → Prop
+  | 0, _ => True
+  | b+1, p => anchorPairAllowed T A s b p.1 p.2.1 ∧
+      allowed T A s b p.2.2.1 ∧ allowed T A s b p.2.2.2
+
+theorem BinaryWindowLabels.allowed_distinct (T : ℕ → ℕ) (A : (b : ℕ) → Finset (Fin (T b)))
+    (s b : ℕ) (p : BinaryWindowLabels T b) (hp : allowed T A s b p) : distinct T b p := by
+  induction b with
+  | zero => trivial
+  | succ b ih => exact ⟨hp.1.2.2.1,ih _ hp.2.1,ih _ hp.2.2⟩
+
+/-- Regroup the two real top child labels, without changing the actual
+finite label type or declaration order. -/
+def joinBinaryLabels {T : ℕ → ℕ} {b : ℕ} (a : Fin (T b) × Fin (T b))
+    (p : BinaryWindowLabels T b × BinaryWindowLabels T b) : BinaryWindowLabels T (b+1) :=
+  (a.1,a.2,p.1,p.2)
+
+theorem sum_binary_labels (T : ℕ → ℕ) (b : ℕ) (f : BinaryWindowLabels T (b+1) → ℝ) :
+    (∑p, f p) = ∑a : Fin (T b) × Fin (T b),
+      ∑p : BinaryWindowLabels T b × BinaryWindowLabels T b, f (joinBinaryLabels a p) := by
+  let e : BinaryWindowLabels T (b+1) ≃
+      (Fin (T b) × Fin (T b)) × (BinaryWindowLabels T b × BinaryWindowLabels T b) :=
+    { toFun := fun p => ((p.1,p.2.1),(p.2.2.1,p.2.2.2))
+      invFun := fun z => joinBinaryLabels z.1 z.2
+      left_inv := fun _ => rfl
+      right_inv := fun _ => rfl }
+  exact (Fintype.sum_equiv e f (fun z => f (joinBinaryLabels z.1 z.2)) (fun _ => rfl)).trans
+    (Fintype.sum_prod_type _)
+
+/-- Exact finite tensor summation after preserving both deterministically
+allowed label domains. No probabilistic independence is asserted here. -/
+theorem sum_pair_tensor {α β σ τ : Type*} [Fintype α] [Fintype β] [Fintype σ] [Fintype τ]
+    (R : α → Prop) (S : β → Prop) (P : σ → Prop) (Q : τ → Prop)
+    [DecidablePred R] [DecidablePred S] [DecidablePred P] [DecidablePred Q]
+    (K : α → β → ℝ) (f : α → σ → ℝ) (h : β → τ → ℝ) :
+    (∑a,∑x,∑b,∑y, if R a ∧ P x ∧ S b ∧ Q y then K a b*f a x*h b y else 0) =
+      ∑a,∑b,if R a ∧ S b then K a b*(∑x,if P x then f a x else 0)*
+        (∑y,if Q y then h b y else 0) else 0 := by
+  apply Finset.sum_congr rfl
+  intro a ha
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro b hb
+  by_cases hr : R a
+  · by_cases hs : S b
+    · rw [ite_eq_left ⟨hr,hs⟩]
+      simp only [Finset.sum_mul,Finset.mul_sum]
+      conv_rhs => rw [Finset.sum_comm]
+      apply Finset.sum_congr rfl
+      intro x hx
+      apply Finset.sum_congr rfl
+      intro y hy
+      by_cases hp : P x <;> by_cases hq : Q y <;> simp only [hr,hs,hp,hq,and_self,
+        false_and,and_false,ite_true,ite_false,mul_zero,zero_mul]
+    · simp only [hs,and_false,false_and,ite_false,Finset.sum_const_zero]
+  · simp only [hr,false_and,ite_false,Finset.sum_const_zero]
+
+/-- Literal sum over all descendant labels is the recursively compiled
+sibling-group energy. This is the bridge from the actual anchor average to
+bottom-up summation; no coefficient estimate is supplied as a hypothesis. -/
+theorem binaryAnchorBelow_sum (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (A : (b : ℕ) → Finset (Fin (T b))) (b : ℕ)
+    (d : WindowAssignments T n g (Option Bool) (b+1)) (u v : Fin (T b)) :
+    (∑p : BinaryWindowLabels T b × BinaryWindowLabels T b,
+      if BinaryWindowLabels.allowed T A s b p.1 ∧ BinaryWindowLabels.allowed T A s b p.2 then
+        binaryAnchorBelow T n g s center globalShift b (joinBinaryLabels (u,v) p) d ^ 2 else 0) =
+      binaryDescendantEnergy T n g s center globalShift A b d u v := by
+  induction b with
+  | zero =>
+    have hcard : Fintype.card (BinaryWindowLabels T 0) = 1 := rfl
+    simp only [BinaryWindowLabels.allowed,binaryAnchorBelow,binaryDescendantEnergy,
+      and_self,ite_true,one_pow,Finset.sum_const,Finset.card_univ,Fintype.card_prod,hcard,nsmul_eq_mul,
+      mul_one,Nat.cast_one]
+  | succ b ih =>
+    rw [Fintype.sum_prod_type,sum_binary_labels]
+    simp_rw [sum_binary_labels]
+    have hpoint (a c : Fin (T b) × Fin (T b))
+        (p q : BinaryWindowLabels T b × BinaryWindowLabels T b) :
+        (if BinaryWindowLabels.allowed T A s (b+1) (joinBinaryLabels a p) ∧
+          BinaryWindowLabels.allowed T A s (b+1) (joinBinaryLabels c q) then
+          binaryAnchorBelow T n g s center globalShift (b+1)
+            (joinBinaryLabels (u,v) (joinBinaryLabels a p,joinBinaryLabels c q)) d ^ 2 else 0) =
+        (if anchorPairAllowed T A s b a.1 a.2 ∧
+          (BinaryWindowLabels.allowed T A s b p.1 ∧ BinaryWindowLabels.allowed T A s b p.2) ∧
+          anchorPairAllowed T A s b c.1 c.2 ∧
+          (BinaryWindowLabels.allowed T A s b q.1 ∧ BinaryWindowLabels.allowed T A s b q.2) then
+          binarySiblingAnchor T n g s center globalShift b d u v a.1 c.1 ^ 2 *
+            binaryAnchorBelow T n g s center globalShift b (joinBinaryLabels a p) (d.2.2 u) ^ 2 *
+            binaryAnchorBelow T n g s center globalShift b (joinBinaryLabels c q) (d.2.2 v) ^ 2 else 0) := by
+      simp only [BinaryWindowLabels.allowed,joinBinaryLabels,and_assoc,binaryAnchorBelow,binarySiblingAnchor]
+      split_ifs <;> ring
+    simp_rw [hpoint]
+    rw [sum_pair_tensor]
+    simp_rw [ih]
+    change _ = ∑a : Fin (T b),∑c : Fin (T b),∑e : Fin (T b),∑f : Fin (T b), _
+    simp only [Fintype.sum_prod_type]
+
+/-- The square-sum of actual anchor averages over all admissible forced
+indices, including the root local smoothing cost exactly once. -/
+theorem binaryAnchorAverage_sum (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (A : (b : ℕ) → Finset (Fin (T b))) (b : ℕ)
+    (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ) :
+    (∑p : BinaryWindowLabels T (b+1), if BinaryWindowLabels.allowed T A s (b+1) p then
+      binaryAnchorAverage T n g s center globalShift (b+1) p d incoming ^ 2 else 0) =
+      ∑u : Fin (T b),∑v : Fin (T b), if anchorPairAllowed T A s b u v then
+        localAnchorAverage d.1 s ((u.val:ℤ)-center b+incoming)^2 *
+          binaryDescendantEnergy T n g s center globalShift A b d u v else 0 := by
+  rw [sum_binary_labels,Fintype.sum_prod_type]
+  apply Finset.sum_congr rfl
+  intro u hu
+  apply Finset.sum_congr rfl
+  intro v hv
+  by_cases hr : anchorPairAllowed T A s b u v
+  · rw [ite_eq_left hr,←binaryAnchorBelow_sum,Finset.mul_sum]
+    apply Finset.sum_congr rfl
+    intro p hp
+    by_cases hsub : BinaryWindowLabels.allowed T A s b p.1 ∧ BinaryWindowLabels.allowed T A s b p.2
+    · have ha : BinaryWindowLabels.allowed T A s (b+1) (joinBinaryLabels (u,v) p) := ⟨hr,hsub⟩
+      rw [ite_eq_left ha,ite_eq_left hsub,binaryAnchorAverage_factor T n g s center globalShift _ _ _ _
+        (BinaryWindowLabels.allowed_distinct T A s _ _ ha),mul_pow]
+      rfl
+    · have ha : ¬BinaryWindowLabels.allowed T A s (b+1) (joinBinaryLabels (u,v) p) := fun h => hsub h.2
+      rw [ite_eq_right ha,ite_eq_right hsub,mul_zero]
+  · rw [ite_eq_right hr]
+    apply Finset.sum_eq_zero
+    intro p hp
+    exact ite_eq_right (fun h => hr h.1)
+
+end TraceReconstruction
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+open FiniteLaw
+
+/-- Full finite forced-anchor squared coefficient budget, with the source
+power of one local root cost and one correlated global/local cost per group.
+This conclusion is about literal posterior averages and actual labels. -/
+theorem binaryAnchorAverage_energy (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (A : (b : ℕ) → Finset (Fin (T b))) (shift : ℕ → ℤ)
+    (hplat : HasGlobalPlateau T g A shift globalShift)
+    (B C : ℕ → ℝ) (hB : ∀b,0≤B b) (hC : ∀b,0≤C b)
+    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1))
+    (hcap : MetadataBinomialCaps T n g B C (b+1) d) (incoming : ℤ) :
+    (∑p : BinaryWindowLabels T (b+1), if BinaryWindowLabels.allowed T A s (b+1) p then
+      binaryAnchorAverage T n g s center globalShift (b+1) p d incoming ^ 2 else 0) ≤
+        (2*((2^s:ℕ):ℝ)^3*B b) *
+          binaryTreeBudget (fun j => 4*((2^s:ℕ):ℝ)^6*C (j+1)*B j) b := by
+  let K : ℕ → ℝ := fun j => 4*((2^s:ℕ):ℝ)^6*C (j+1)*B j
+  have hK : ∀j,0≤K j := fun j => mul_nonneg (mul_nonneg (by positivity) (hC _)) (hB _)
+  have hbudget := binaryTreeBudget_nonneg K hK b
+  have hroot := localAnchor_finite_energy d.1 s (center b-incoming)
+    (anchorPairAllowed T A s b) (fun a c h => h.2.2.2)
+  have he (u : Fin (T b)) : (u.val:ℤ)-(center b-incoming)=(u.val:ℤ)-center b+incoming := by omega
+  simp_rw [he] at hroot
+  rw [binaryAnchorAverage_sum]
+  calc
+    _ ≤ (∑u : Fin (T b),∑v : Fin (T b),if anchorPairAllowed T A s b u v then
+        localAnchorAverage d.1 s ((u.val:ℤ)-center b+incoming)^2 else 0) * binaryTreeBudget K b := by
+      simp only [Finset.sum_mul]
+      apply Finset.sum_le_sum
+      intro u hu
+      apply Finset.sum_le_sum
+      intro v hv
+      split_ifs with hr
+      · exact mul_le_mul_of_nonneg_left
+          (binaryDescendantEnergy_le T n g s center globalShift A shift hplat B C hB hC b d hcap
+            u v hr.1 hr.2.1) (sq_nonneg _)
+      · simp only [zero_mul,le_refl]
+    _ ≤ (2*((2^s:ℕ):ℝ)^3*B b)*binaryTreeBudget K b := by
+      apply mul_le_mul_of_nonneg_right _ hbudget
+      exact hroot.trans (mul_le_mul_of_nonneg_left hcap.1 (by positivity))
+
+end TraceReconstruction
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+open FiniteLaw
+
+/-- A literal unsigned length-L translated motif admits at most L child
+labels in any finite window, even when the intended motif does not fit. -/
+theorem motifIndicator_finite_sum (s N : ℕ) (X : ℤ) :
+    (∑h : Fin N,motifIndicator s ((h.val:ℤ)-X)) ≤ (2^s:ℕ) := by
+  let A := Finset.univ.filter (fun h : Fin N => 0≤(h.val:ℤ)-X ∧ (h.val:ℤ)-X<(2^s:ℕ))
+  have hc : A.card≤2^s := by
+    have H := Finset.card_le_card_of_injOn (s:=A) (t:=Finset.range (2^s))
+      (fun h : Fin N => ((h.val:ℤ)-X).toNat) ?_ ?_
+    · simpa only [Finset.card_range] using H
+    · intro h hh
+      have h' := (Finset.mem_filter.mp hh).2
+      apply Finset.mem_range.mpr
+      dsimp only
+      omega
+    · intro h hh j hj he
+      have h' := (Finset.mem_filter.mp hh).2
+      have j' := (Finset.mem_filter.mp hj).2
+      apply Fin.ext
+      dsimp only at he
+      omega
+  have he : (∑h : Fin N,motifIndicator s ((h.val:ℤ)-X)) = (A.card:ℝ) := by
+    simp only [motifIndicator,A,Finset.sum_ite,Finset.sum_const_zero,add_zero,
+      Finset.sum_const,nsmul_eq_mul,mul_one]
+  rw [he]
+  exact_mod_cast hc
+
+/-- The complete bits law is arbitrary. All extra-leaf offsets may be
+mutually correlated and correlated with the retained forced-anchor kernel.
+Only the extra leaf label variables themselves are summed here. -/
+def extraLabelAverage {Ω : Type*} [Fintype Ω] (P : FiniteLaw Ω)
+    (s N e : ℕ) (f : Ω → ℝ) (X : Ω → Fin e → ℤ) (h : Fin e → Fin N) : ℝ :=
+  P.expect (fun ω => f ω*∏j,motifIndicator s (((h j).val:ℤ)-X ω j))
+
+theorem extraLabelAverage_nonneg {Ω : Type*} [Fintype Ω] (P : FiniteLaw Ω)
+    (s N e : ℕ) (f : Ω → ℝ) (hf : ∀ω,0≤f ω) (X : Ω → Fin e → ℤ)
+    (h : Fin e → Fin N) : 0≤extraLabelAverage P s N e f X h := by
+  apply Finset.sum_nonneg
+  intro ω hω
+  apply mul_nonneg (P.nonneg ω)
+  exact mul_nonneg (hf ω) (Finset.prod_nonneg (fun j _ => motifIndicator_nonneg _ _))
+
+/-- Source the correlated-anchor estimate, extra leaf l1 cost before squaring. No free
+T_1 cost and no invalid decoupling of global displacements enter. -/
+theorem extraLabelAverage_sum {Ω : Type*} [Fintype Ω] (P : FiniteLaw Ω)
+    (s N e : ℕ) (f : Ω → ℝ) (hf : ∀ω,0≤f ω) (X : Ω → Fin e → ℤ) :
+    (∑h : Fin e → Fin N,extraLabelAverage P s N e f X h) ≤
+      ((2^s:ℕ):ℝ)^e*P.expect f := by
+  unfold extraLabelAverage expect
+  rw [Finset.sum_comm]
+  simp_rw [←mul_assoc,←Finset.mul_sum]
+  calc
+    _ ≤ ∑ω,P.mass ω*f ω*(((2^s:ℕ):ℝ)^e) := by
+      apply Finset.sum_le_sum
+      intro ω hω
+      apply mul_le_mul_of_nonneg_left _ (mul_nonneg (P.nonneg _) (hf ω))
+      have he := Fintype.prod_sum (fun (j : Fin e) (h : Fin N) => motifIndicator s ((h.val:ℤ)-X ω j))
+      rw [←he]
+      calc
+        _ ≤ ∏_j : Fin e,((2^s:ℕ):ℝ) := by
+          apply Finset.prod_le_prod₀
+          · intro j hj
+            exact Finset.sum_nonneg (fun h hh => motifIndicator_nonneg _ _)
+          · intro j hj
+            exact motifIndicator_finite_sum s N (X ω j)
+        _ = _ := by simp
+    _ = _ := by rw [←Finset.sum_mul]; ring
+
+/-- The squared extra-leaf cost follows from nonnegativity AFTER the whole
+shared-bit average, exactly as in the source (not from independence). -/
+theorem extraLabelAverage_energy {Ω : Type*} [Fintype Ω] (P : FiniteLaw Ω)
+    (s N e : ℕ) (f : Ω → ℝ) (hf : ∀ω,0≤f ω) (X : Ω → Fin e → ℤ) :
+    (∑h : Fin e → Fin N,extraLabelAverage P s N e f X h ^ 2) ≤
+      ((2^s:ℕ):ℝ)^(2*e)*(P.expect f)^2 := by
+  have hn (h : Fin e → Fin N) := extraLabelAverage_nonneg P s N e f hf X h
+  calc
+    _ ≤ (∑h : Fin e → Fin N,extraLabelAverage P s N e f X h)^2 :=
+      Finset.sum_sq_le_sq_sum_of_nonneg (fun h _ => hn h)
+    _ ≤ (((2^s:ℕ):ℝ)^e*P.expect f)^2 := by
+      apply pow_le_pow_left₀ (Finset.sum_nonneg (fun h _ => hn h))
+      exact extraLabelAverage_sum P s N e f hf X
+    _ = _ := by rw [mul_pow,←pow_mul]; congr 2; omega
+end TraceReconstruction
+
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/TraceReconstruction/Binomial.lean b/lean/OAI/Probability/TraceReconstruction/Binomial.lean
index c193be4ffb18064330940b5badbe89dcd1149ac8..3ef5f8270e8d705982e78fdcd87a883251b74e1a 100644
--- a/lean/OAI/Probability/TraceReconstruction/Binomial.lean
+++ b/lean/OAI/Probability/TraceReconstruction/Binomial.lean
@@ -1,2111 +1,2111 @@
-import Mathlib
-
-namespace OAI
-
-/-!
-# Binomial
-
-Fair and biased binomial laws, generating series, differences, concentration, Gaussian overlap, hidden counts, and local lower bounds.
--/
-
-open scoped ENNReal
-noncomputable section
-
-namespace TraceReconstruction
-
-/-! ## Quantitative fair-binomial anti-concentration
-
-A direct inductive bound, avoiding an unformalized appeal to Stirling.
-It is one of the estimates needed for signed binomial smoothing and
-for the local/global anchor kernels.
--/
-
-/-- Fair-binomial mass, extended by zero beyond its natural support. -/
-def fairBinomial (n k : ℕ) : ℝ := (n.choose k : ℝ) / 2 ^ n
-
-theorem fairBinomial_nonneg (n k : ℕ) : 0 ≤ fairBinomial n k := by
-  unfold fairBinomial
-  positivity
-
-theorem centralBinom_square_bound (n : ℕ) :
-    ((n : ℝ) + 1) * (n.centralBinom : ℝ) ^ 2 ≤ (16 : ℝ) ^ n := by
-  induction n with
-  | zero => norm_num
-  | succ n ih =>
-    have hn : (0 : ℝ) < n + 1 := by positivity
-    have hr : ((n : ℝ) + 1) * ((n + 1).centralBinom : ℝ) =
-        2 * (2 * n + 1) * (n.centralBinom : ℝ) := by
-      exact_mod_cast Nat.succ_mul_centralBinom_succ n
-    have hc : 4 * (2 * (n : ℝ) + 1) ^ 2 * (n + 2) ≤
-        16 * (n + 1) ^ 3 := by nlinarith [sq_nonneg (n : ℝ)]
-    have hh : ((n : ℝ) + 1) ^ 2 *
-        ((n + 2) * ((n + 1).centralBinom : ℝ) ^ 2) ≤
-        ((n : ℝ) + 1) ^ 2 * (16 * 16 ^ n) := by
-      calc
-        _ = (n + 2) * (((n : ℝ) + 1) * ((n + 1).centralBinom : ℝ)) ^ 2 := by ring
-        _ = (4 * (2 * (n : ℝ) + 1) ^ 2 * (n + 2)) *
-            (n.centralBinom : ℝ) ^ 2 := by rw [hr]; ring
-        _ ≤ (16 * ((n : ℝ) + 1) ^ 3) * (n.centralBinom : ℝ) ^ 2 :=
-          mul_le_mul_of_nonneg_right hc (sq_nonneg _)
-        _ = (16 * ((n : ℝ) + 1) ^ 2) *
-            (((n : ℝ) + 1) * (n.centralBinom : ℝ) ^ 2) := by ring
-        _ ≤ (16 * ((n : ℝ) + 1) ^ 2) * 16 ^ n :=
-          mul_le_mul_of_nonneg_left ih (by positivity)
-        _ = _ := by ring
-    have := (mul_le_mul_iff_right₀ (sq_pos_of_pos hn)).mp hh
-    rw [pow_succ (16 : ℝ) n]
-    convert this using 1 <;> push_cast <;> first | rfl | ring
-
-theorem choose_odd_le_twice_central (n k : ℕ) :
-    (2 * n + 1).choose k ≤ 2 * n.centralBinom := by
-  cases k with
-  | zero => simp only [Nat.choose_zero_right]; have := Nat.centralBinom_pos n; omega
-  | succ k =>
-    rw [Nat.choose_succ_succ]
-    have h₁ := Nat.choose_le_centralBinom k n
-    have h₂ := Nat.choose_le_centralBinom (k + 1) n
-    simpa [Nat.succ_eq_add_one, two_mul] using Nat.add_le_add h₁ h₂
-
-theorem fairBinomial_square_bound (n k : ℕ) :
-    ((n : ℝ) + 1) * fairBinomial n k ^ 2 ≤ 2 := by
-  have hcases : n % 2 = 0 ∨ n % 2 = 1 := Nat.mod_two_eq_zero_or_one n
-  rcases hcases with he | ho
-  · obtain ⟨m, rfl⟩ : ∃ m, n = 2 * m := ⟨n / 2, by omega⟩
-    have hc : ((2 * m).choose k : ℝ) ≤ (m.centralBinom : ℝ) :=
-      by exact_mod_cast Nat.choose_le_centralBinom k m
-    have hcs : ((2 * m).choose k : ℝ) ^ 2 ≤ (m.centralBinom : ℝ) ^ 2 :=
-      pow_le_pow_left₀ (by positivity) hc 2
-    have ht := centralBinom_square_bound m
-    have hp : ((2 : ℝ) ^ (2 * m)) ^ 2 = 16 ^ m := by
-      rw [← pow_mul, show 2 * m * 2 = 4 * m by omega, pow_mul]
-      norm_num
-    unfold fairBinomial
-    rw [div_pow, ← mul_div_assoc, div_le_iff₀ (by positivity : (0 : ℝ) < (2 ^ (2 * m)) ^ 2)]
-    rw [hp]
-    push_cast
-    nlinarith [mul_le_mul_of_nonneg_left hcs (show (0 : ℝ) ≤ m + 1 by positivity),
-      sq_nonneg ((2 * m).choose k : ℝ)]
-  · obtain ⟨m, rfl⟩ : ∃ m, n = 2 * m + 1 := ⟨n / 2, by omega⟩
-    have hc : ((2 * m + 1).choose k : ℝ) ≤ 2 * (m.centralBinom : ℝ) :=
-      by exact_mod_cast choose_odd_le_twice_central m k
-    have hcs : ((2 * m + 1).choose k : ℝ) ^ 2 ≤ 4 * (m.centralBinom : ℝ) ^ 2 := by
-      have := pow_le_pow_left₀ (by positivity : (0 : ℝ) ≤ ((2 * m + 1).choose k : ℝ)) hc 2
-      nlinarith
-    have ht := centralBinom_square_bound m
-    have hp : ((2 : ℝ) ^ (2 * m + 1)) ^ 2 = 4 * 16 ^ m := by
-      rw [pow_add, mul_pow, ← pow_mul, show 2 * m * 2 = 4 * m by omega, pow_mul]
-      norm_num
-      ring
-    unfold fairBinomial
-    rw [div_pow, ← mul_div_assoc, div_le_iff₀ (by positivity : (0 : ℝ) < (2 ^ (2 * m + 1)) ^ 2)]
-    rw [hp]
-    push_cast
-    nlinarith [mul_le_mul_of_nonneg_left hcs (show (0 : ℝ) ≤ m + 1 by positivity)]
-
-
-/-! ## Discrete convolution estimates for the hidden displacement kernels -/
-
-abbrev IntSeries := AddMonoidAlgebra ℝ ℤ
-
-def seriesL1 (f : IntSeries) : ℝ := ∑' z : ℤ, |f.coeff z|
-def seriesL2Sq (f : IntSeries) : ℝ := ∑' z : ℤ, f.coeff z ^ 2
-
-theorem series_summable (f : IntSeries) (φ : ℝ → ℝ) (hφ : φ 0 = 0) :
-    Summable (fun z : ℤ => φ (f.coeff z)) :=
-  summable_of_hasFiniteSupport (f.coeff.hasFiniteSupport.fun_comp hφ)
-
-theorem seriesL1_nonneg (f : IntSeries) : 0 ≤ seriesL1 f :=
-  tsum_nonneg fun _ => abs_nonneg _
-
-theorem seriesL2Sq_nonneg (f : IntSeries) : 0 ≤ seriesL2Sq f :=
-  tsum_nonneg fun _ => sq_nonneg _
-
-theorem seriesL1_eq_sum (f : IntSeries) :
-    seriesL1 f = ∑ z ∈ f.coeff.support, |f.coeff z| := by
-  apply tsum_eq_sum
-  intro z hz
-  simp [Finsupp.notMem_support_iff.mp hz]
-
-theorem seriesL2Sq_eq_sum (f : IntSeries) :
-    seriesL2Sq f = ∑ z ∈ f.coeff.support, f.coeff z ^ 2 := by
-  apply tsum_eq_sum
-  intro z hz
-  simp [Finsupp.notMem_support_iff.mp hz]
-
-theorem series_summable_shift (f : IntSeries) (φ : ℝ → ℝ)
-    (hφ : φ 0 = 0) (a : ℤ) :
-    Summable (fun z : ℤ => φ (f.coeff (a + z))) := by
-  exact (Equiv.addLeft a).summable_iff.mpr (series_summable f φ hφ)
-
-theorem series_sum_shift (f : IntSeries) (φ : ℝ → ℝ) (a : ℤ) :
-    (∑' z : ℤ, φ (f.coeff (a + z))) = ∑' z : ℤ, φ (f.coeff z) :=
-  (Equiv.addLeft a).tsum_eq (fun z => φ (f.coeff z))
-
-theorem series_coeff_mul_abs_le (f g : IntSeries) (z : ℤ) :
-    |(f * g).coeff z| ≤ ∑ a ∈ f.coeff.support, |f.coeff a| * |g.coeff (-a + z)| := by
-  rw [AddMonoidAlgebra.coeff_mul_apply_left, Finsupp.sum]
-  simpa only [abs_mul] using Finset.abs_sum_le_sum_abs
-    (fun a => f.coeff a * g.coeff (-a + z)) f.coeff.support
-
-theorem seriesL1_mul_le (f g : IntSeries) :
-    seriesL1 (f * g) ≤ seriesL1 f * seriesL1 g := by
-  have hh (a : ℤ) : Summable (fun z : ℤ => |f.coeff a| * |g.coeff (-a + z)|) :=
-    (series_summable_shift g abs (by simp) (-a)).mul_left _
-  calc
-    seriesL1 (f * g) ≤ ∑' z : ℤ, ∑ a ∈ f.coeff.support,
-        |f.coeff a| * |g.coeff (-a + z)| :=
-      Summable.tsum_le_tsum (series_coeff_mul_abs_le f g) (series_summable (f * g) abs (by simp))
-        (summable_sum fun a _ => hh a)
-    _ = ∑ a ∈ f.coeff.support, ∑' z : ℤ, |f.coeff a| * |g.coeff (-a + z)| :=
-      Summable.tsum_finsetSum fun a _ => hh a
-    _ = (∑ a ∈ f.coeff.support, |f.coeff a|) * seriesL1 g := by
-      simp_rw [tsum_mul_left, series_sum_shift]
-      exact (Finset.sum_mul _ _ _).symm
-    _ = seriesL1 f * seriesL1 g := by rw [seriesL1_eq_sum f]
-
-theorem series_coeff_mul_sq_le (f g : IntSeries) (z : ℤ) :
-    (f * g).coeff z ^ 2 ≤ seriesL1 f *
-      ∑ a ∈ f.coeff.support, |f.coeff a| * (g.coeff (-a + z)) ^ 2 := by
-  rw [AddMonoidAlgebra.coeff_mul_apply_left, Finsupp.sum, seriesL1_eq_sum]
-  apply Finset.sum_sq_le_sum_mul_sum_of_sq_le_mul
-  · intro a _; exact abs_nonneg _
-  · intro a _; positivity
-  · intro a _
-    rw [mul_pow, ← mul_assoc, ← sq, sq_abs]
-
-theorem seriesL2Sq_mul_le (f g : IntSeries) :
-    seriesL2Sq (f * g) ≤ seriesL1 f ^ 2 * seriesL2Sq g := by
-  have hh (a : ℤ) : Summable (fun z : ℤ => |f.coeff a| * (g.coeff (-a + z)) ^ 2) :=
-    (series_summable_shift g (fun x => x ^ 2) (by simp) (-a)).mul_left _
-  calc
-    seriesL2Sq (f * g) ≤ ∑' z : ℤ, seriesL1 f *
-        ∑ a ∈ f.coeff.support, |f.coeff a| * (g.coeff (-a + z)) ^ 2 :=
-      Summable.tsum_le_tsum (series_coeff_mul_sq_le f g)
-        (series_summable (f * g) (fun x => x ^ 2) (by simp))
-        ((summable_sum fun a _ => hh a).mul_left _)
-    _ = seriesL1 f * ∑ a ∈ f.coeff.support,
-        ∑' z : ℤ, |f.coeff a| * (g.coeff (-a + z)) ^ 2 := by
-      rw [tsum_mul_left, Summable.tsum_finsetSum fun a _ => hh a]
-    _ = seriesL1 f ^ 2 * seriesL2Sq g := by
-      simp_rw [tsum_mul_left, series_sum_shift g (fun x => x ^ 2)]
-      rw [← Finset.sum_mul, ← seriesL1_eq_sum]
-      unfold seriesL2Sq
-      ring
-
-theorem series_coeff_mul_bound (f g : IntSeries) (M : ℝ)
-    (hg : ∀ z, |g.coeff z| ≤ M) (z : ℤ) :
-    |(f * g).coeff z| ≤ seriesL1 f * M := by
-  calc
-    _ ≤ ∑ a ∈ f.coeff.support, |f.coeff a| * |g.coeff (-a + z)| :=
-      series_coeff_mul_abs_le f g z
-    _ ≤ ∑ a ∈ f.coeff.support, |f.coeff a| * M :=
-      Finset.sum_le_sum fun a _ => mul_le_mul_of_nonneg_left (hg _) (abs_nonneg _)
-    _ = seriesL1 f * M := by rw [← Finset.sum_mul, ← seriesL1_eq_sum]
-
-/-! The probability generating series of the fair binomial kernel. -/
-def fairSeries (n : ℕ) : IntSeries :=
-  (AddMonoidAlgebra.single 1 (1 / 2) + AddMonoidAlgebra.single 0 (1 / 2)) ^ n
-
-theorem fairSeries_add (m n : ℕ) : fairSeries (m + n) = fairSeries m * fairSeries n :=
-  pow_add _ m n
-
-theorem fairSeries_sum (n : ℕ) :
-    fairSeries n = ∑ k ∈ Finset.range (n + 1),
-      AddMonoidAlgebra.single (k : ℤ) (fairBinomial n k) := by
-  unfold fairSeries
-  rw [add_pow]
-  apply Finset.sum_congr rfl
-  intro k hk
-  have hkn : k ≤ n := by simpa only [Finset.mem_range, Nat.lt_succ_iff] using hk
-  rw [AddMonoidAlgebra.single_pow, AddMonoidAlgebra.single_pow,
-    AddMonoidAlgebra.natCast_def, AddMonoidAlgebra.single_mul_single,
-    AddMonoidAlgebra.single_mul_single]
-  simp only [nsmul_eq_mul, mul_one, mul_zero, add_zero]
-  congr 1
-  rw [← pow_add, Nat.add_sub_of_le hkn]
-  unfold fairBinomial
-  rw [div_pow]
-  simp [div_eq_mul_inv, mul_comm]
-
-theorem fairSeries_coeff_nat (n k : ℕ) :
-    (fairSeries n).coeff (k : ℤ) = fairBinomial n k := by
-  rw [fairSeries_sum, AddMonoidAlgebra.coeff_sum, Finsupp.finsetSum_apply]
-  by_cases hk : k ≤ n
-  · rw [Finset.sum_eq_single k]
-    · simp only [AddMonoidAlgebra.coeff_single, Finsupp.single_apply, ite_true]
-    · intro b _ hb
-      simp only [AddMonoidAlgebra.coeff_single, Finsupp.single_apply, Nat.cast_inj, ite_eq_right hb]
-    · intro h
-      exact (h (Finset.mem_range.mpr (by omega))).elim
-  · rw [Finset.sum_eq_zero]
-    · simp [fairBinomial, Nat.choose_eq_zero_of_lt (by omega : n < k)]
-    · intro b hb
-      have hbk : b ≠ k := by simp only [Finset.mem_range] at hb; omega
-      simp only [AddMonoidAlgebra.coeff_single, Finsupp.single_apply, Nat.cast_inj, ite_eq_right hbk]
-
-theorem fairSeries_coeff_neg (n : ℕ) (z : ℤ) (hz : z < 0) :
-    (fairSeries n).coeff z = 0 := by
-  rw [fairSeries_sum, AddMonoidAlgebra.coeff_sum, Finsupp.finsetSum_apply]
-  apply Finset.sum_eq_zero
-  intro b _
-  have h : (b : ℤ) ≠ z := by omega
-  simp only [AddMonoidAlgebra.coeff_single, Finsupp.single_apply, ite_eq_right h]
-
-theorem seriesL1_eq_sum_range (f : IntSeries) (n : ℕ)
-    (hneg : ∀ z < 0, f.coeff z = 0)
-    (hhigh : ∀ k : ℕ, n < k → f.coeff k = 0) :
-    seriesL1 f = ∑ k ∈ Finset.range (n + 1), |f.coeff (k : ℤ)| := by
-  let e : ℕ ↪ ℤ := ⟨Nat.cast, Nat.cast_injective⟩
-  calc
-    seriesL1 f = ∑ z ∈ (Finset.range (n + 1)).map e, |f.coeff z| := by
-      apply tsum_eq_sum
-      intro z hz
-      by_cases h : 0 ≤ z
-      · have heq : ((z.toNat : ℕ) : ℤ) = z := Int.toNat_of_nonneg h
-        have hh : n < z.toNat := by
-          by_contra hn
-          apply hz
-          exact Finset.mem_map.mpr ⟨z.toNat, Finset.mem_range.mpr (by omega), heq⟩
-        rw [← heq, hhigh _ hh, abs_zero]
-      · rw [hneg z (by omega), abs_zero]
-    _ = _ := Finset.sum_map _ e _
-
-def firstDifference : IntSeries :=
-  AddMonoidAlgebra.single 0 1 - AddMonoidAlgebra.single 1 1
-
-theorem firstDifference_coeff (f : IntSeries) (z : ℤ) :
-    (firstDifference * f).coeff z = f.coeff z - f.coeff (z - 1) := by
-  rw [firstDifference, sub_mul, AddMonoidAlgebra.coeff_sub, Finsupp.sub_apply,
-    mul_comm _ f, AddMonoidAlgebra.coeff_mul_single_apply,
-    mul_comm _ f, AddMonoidAlgebra.coeff_mul_single_apply]
-  simp only [neg_zero, add_zero, mul_one, sub_eq_add_neg]
-
-theorem fairSeries_difference_coeff_zero (n : ℕ) :
-    (firstDifference * fairSeries n).coeff 0 = fairBinomial n 0 := by
-  rw [firstDifference_coeff]
-  norm_num only [zero_sub]
-  rw [fairSeries_coeff_neg n (-1) (by norm_num), sub_zero]
-  exact fairSeries_coeff_nat n 0
-
-theorem fairSeries_difference_coeff_succ (n k : ℕ) :
-    (firstDifference * fairSeries n).coeff (↑(k + 1) : ℤ) =
-      fairBinomial n (k + 1) - fairBinomial n k := by
-  rw [firstDifference_coeff, fairSeries_coeff_nat]
-  simp only [Nat.cast_add, Nat.cast_one, add_sub_cancel_right, fairSeries_coeff_nat]
-
-theorem fairBinomial_unimodal_left (n k : ℕ) (hk : k < n / 2) :
-    fairBinomial n k ≤ fairBinomial n (k + 1) := by
-  apply div_le_div_of_nonneg_right _ (by positivity)
-  exact_mod_cast Nat.choose_le_succ_of_lt_half_left hk
-
-theorem fairBinomial_unimodal_right (n k : ℕ) (hk : n / 2 ≤ k) :
-    fairBinomial n (k + 1) ≤ fairBinomial n k := by
-  apply div_le_div_of_nonneg_right _ (by positivity)
-  have hnk : n - k ≤ k + 1 := by omega
-  have hmul := Nat.mul_le_mul_left (n.choose k) hnk
-  rw [← Nat.choose_succ_right_eq] at hmul
-  exact_mod_cast Nat.le_of_mul_le_mul_right hmul (by omega : 0 < k + 1)
-
-theorem fairBinomial_variation (n : ℕ) :
-    (∑ k ∈ Finset.range (n + 1), |fairBinomial n (k + 1) - fairBinomial n k|) =
-      2 * fairBinomial n (n / 2) - fairBinomial n 0 := by
-  have hmid : n / 2 ≤ n + 1 := by omega
-  rw [← Finset.sum_range_add_sum_Ico _ hmid]
-  have hl : (∑ k ∈ Finset.range (n / 2),
-      |fairBinomial n (k + 1) - fairBinomial n k|) =
-      fairBinomial n (n / 2) - fairBinomial n 0 := by
-    rw [← Finset.sum_range_sub (fairBinomial n) (n / 2)]
-    apply Finset.sum_congr rfl
-    intro k hk
-    exact abs_of_nonneg (sub_nonneg.mpr
-      (fairBinomial_unimodal_left n k (Finset.mem_range.mp hk)))
-  have hr : (∑ k ∈ Finset.Ico (n / 2) (n + 1),
-      |fairBinomial n (k + 1) - fairBinomial n k|) =
-      fairBinomial n (n / 2) - fairBinomial n (n + 1) := by
-    calc
-      _ = ∑ k ∈ Finset.Ico (n / 2) (n + 1),
-          -(fairBinomial n (k + 1) - fairBinomial n k) := by
-        apply Finset.sum_congr rfl
-        intro k hk
-        exact abs_of_nonpos (sub_nonpos.mpr
-          (fairBinomial_unimodal_right n k (Finset.mem_Ico.mp hk).1))
-      _ = _ := by rw [Finset.sum_neg_distrib, Finset.sum_Ico_sub _ hmid]; ring
-  rw [hl, hr]
-  have hzero : fairBinomial n (n + 1) = 0 := by simp [fairBinomial]
-  rw [hzero]
-  ring
-
-theorem fairSeries_difference_L1 (n : ℕ) :
-    seriesL1 (firstDifference * fairSeries n) = 2 * fairBinomial n (n / 2) := by
-  rw [seriesL1_eq_sum_range _ (n + 1)]
-  · rw [Finset.sum_range_succ', Nat.cast_zero, fairSeries_difference_coeff_zero,
-      abs_of_nonneg (fairBinomial_nonneg n 0)]
-    simp_rw [fairSeries_difference_coeff_succ]
-    rw [fairBinomial_variation]
-    ring
-  · intro z hz
-    rw [firstDifference_coeff, fairSeries_coeff_neg n z hz,
-      fairSeries_coeff_neg n (z - 1) (by omega), sub_zero]
-  · intro k hk
-    have hkpos : 0 < k := by omega
-    obtain ⟨v, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (Nat.ne_of_gt hkpos)
-    rw [fairSeries_difference_coeff_succ]
-    simp [fairBinomial, Nat.choose_eq_zero_of_lt (by omega : n < v),
-      Nat.choose_eq_zero_of_lt (by omega : n < v + 1)]
-
-def binomialBound (n : ℕ) : ℝ := Real.sqrt (2 / ((n : ℝ) + 1))
-
-theorem binomialBound_nonneg (n : ℕ) : 0 ≤ binomialBound n := Real.sqrt_nonneg _
-
-theorem fairBinomial_le_bound (n k : ℕ) : fairBinomial n k ≤ binomialBound n := by
-  apply Real.le_sqrt_of_sq_le
-  apply (le_div_iff₀ (by positivity : (0 : ℝ) < n + 1)).mpr
-  simpa only [mul_comm] using fairBinomial_square_bound n k
-
-theorem fairSeries_nonneg (n : ℕ) (z : ℤ) : 0 ≤ (fairSeries n).coeff z := by
-  by_cases hz : 0 ≤ z
-  · rw [← Int.toNat_of_nonneg hz, fairSeries_coeff_nat]
-    exact fairBinomial_nonneg _ _
-  · rw [fairSeries_coeff_neg n z (by omega)]
-
-theorem fairSeries_coeff_bound (n : ℕ) (z : ℤ) :
-    |(fairSeries n).coeff z| ≤ binomialBound n := by
-  by_cases hz : 0 ≤ z
-  · rw [← Int.toNat_of_nonneg hz, fairSeries_coeff_nat,
-      abs_of_nonneg (fairBinomial_nonneg _ _)]
-    exact fairBinomial_le_bound _ _
-  · rw [fairSeries_coeff_neg n z (by omega), abs_zero]
-    exact binomialBound_nonneg n
-
-theorem fairSeries_L1 (n : ℕ) : seriesL1 (fairSeries n) = 1 := by
-  rw [seriesL1_eq_sum_range _ n (fairSeries_coeff_neg n)]
-  · simp_rw [fairSeries_coeff_nat, abs_of_nonneg (fairBinomial_nonneg _ _)]
-    unfold fairBinomial
-    rw [← Finset.sum_div]
-    have hsum : (∑ k ∈ Finset.range (n + 1), (n.choose k : ℝ)) = (2 : ℝ) ^ n := by
-      exact_mod_cast Nat.sum_range_choose n
-    rw [hsum, div_self (by positivity)]
-  · intro k hk
-    rw [fairSeries_coeff_nat]
-    simp [fairBinomial, Nat.choose_eq_zero_of_lt hk]
-
-theorem seriesL2Sq_le_L1_mul_bound (f : IntSeries) (M : ℝ)
-    (hf : ∀ z, |f.coeff z| ≤ M) : seriesL2Sq f ≤ seriesL1 f * M := by
-  calc
-    _ ≤ ∑' z : ℤ, |f.coeff z| * M := by
-      apply Summable.tsum_le_tsum
-      · intro z
-        calc
-          f.coeff z ^ 2 = |f.coeff z| * |f.coeff z| := by rw [← sq, sq_abs]
-          _ ≤ _ := mul_le_mul_of_nonneg_left (hf z) (abs_nonneg _)
-      · exact series_summable f (fun x => x ^ 2) (by simp)
-      · exact (series_summable f abs (by simp)).mul_right M
-    _ = _ := tsum_mul_right
-
-theorem fairSeries_L2Sq_bound (n : ℕ) : seriesL2Sq (fairSeries n) ≤ binomialBound n := by
-  simpa only [fairSeries_L1, one_mul] using
-    seriesL2Sq_le_L1_mul_bound (fairSeries n) (binomialBound n) (fairSeries_coeff_bound n)
-
-theorem fairSeries_difference_L1_bound (n : ℕ) :
-    seriesL1 (firstDifference * fairSeries n) ≤ 2 * binomialBound n := by
-  rw [fairSeries_difference_L1]
-  exact mul_le_mul_of_nonneg_left (fairBinomial_le_bound n (n / 2)) (by norm_num)
-
-theorem seriesL1_single (z : ℤ) (r : ℝ) :
-    seriesL1 (AddMonoidAlgebra.single z r) = |r| := by
-  unfold seriesL1
-  simp only [AddMonoidAlgebra.coeff_single, Finsupp.single_apply]
-  simp only [apply_ite abs, abs_zero]
-  exact tsum_ite_eq' z (fun _ : ℤ => |r|)
-
-theorem seriesL1_one : seriesL1 (1 : IntSeries) = 1 := by
-  change seriesL1 (AddMonoidAlgebra.single 0 1) = 1
-  rw [seriesL1_single, abs_one]
-
-theorem seriesL1_pow_le (f : IntSeries) (s : ℕ) : seriesL1 (f ^ s) ≤ seriesL1 f ^ s := by
-  induction s with
-  | zero => simp only [pow_zero, seriesL1_one, le_refl]
-  | succ s ih =>
-    rw [pow_succ, pow_succ]
-    exact (seriesL1_mul_le _ _).trans
-      (mul_le_mul_of_nonneg_right ih (seriesL1_nonneg f))
-
-theorem fairSeries_mul_power (n s : ℕ) :
-    firstDifference ^ s * fairSeries ((s + 1) * n) =
-      (firstDifference * fairSeries n) ^ s * fairSeries n := by
-  simp only [fairSeries, add_mul, one_mul, pow_add, pow_mul, mul_pow]
-  ring
-
-/-- Quantitative higher-order smoothing, with n trials assigned to each
-    difference and one further group reserved for anti-concentration. -/
-theorem binomial_difference_coeff_bound (n s : ℕ) (z : ℤ) :
-    |(firstDifference ^ s * fairSeries ((s + 1) * n)).coeff z| ≤
-      (2 * binomialBound n) ^ s * binomialBound n := by
-  rw [fairSeries_mul_power]
-  apply (series_coeff_mul_bound _ _ _ (fairSeries_coeff_bound n) z).trans
-  apply mul_le_mul_of_nonneg_right _ (binomialBound_nonneg n)
-  exact (seriesL1_pow_le _ _).trans
-    (pow_le_pow_left₀ (seriesL1_nonneg _) (fairSeries_difference_L1_bound n) s)
-
-theorem binomial_difference_L2Sq_bound (n s : ℕ) :
-    seriesL2Sq (firstDifference ^ s * fairSeries ((s + 1) * n)) ≤
-      ((2 * binomialBound n) ^ s) ^ 2 * binomialBound n := by
-  rw [fairSeries_mul_power]
-  apply (seriesL2Sq_mul_le _ _).trans
-  apply mul_le_mul
-  · apply pow_le_pow_left₀ (seriesL1_nonneg _)
-    exact (seriesL1_pow_le _ _).trans
-      (pow_le_pow_left₀ (seriesL1_nonneg _) (fairSeries_difference_L1_bound n) s)
-  · exact fairSeries_L2Sq_bound n
-  · exact seriesL2Sq_nonneg _
-  · positivity
-
-theorem seriesL1_add_le (f g : IntSeries) :
-    seriesL1 (f + g) ≤ seriesL1 f + seriesL1 g := by
-  calc
-    _ ≤ ∑' z : ℤ, (|f.coeff z| + |g.coeff z|) := by
-      apply Summable.tsum_le_tsum
-      · intro z
-        simpa only [AddMonoidAlgebra.coeff_add, Finsupp.add_apply] using
-          abs_add_le (f.coeff z) (g.coeff z)
-      · exact series_summable (f + g) abs (by simp)
-      · exact (series_summable f abs (by simp)).add (series_summable g abs (by simp))
-    _ = _ := (series_summable f abs (by simp)).tsum_add (series_summable g abs (by simp))
-
-theorem seriesL1_zero : seriesL1 (0 : IntSeries) = 0 := by
-  simp [seriesL1]
-
-theorem seriesL1_sum_le {ι : Type*} (A : Finset ι) (f : ι → IntSeries) :
-    seriesL1 (∑ i ∈ A, f i) ≤ ∑ i ∈ A, seriesL1 (f i) := by
-  classical
-  induction A using Finset.induction_on with
-  | empty => simp only [Finset.sum_empty, seriesL1_zero, le_refl]
-  | @insert a A ha ih =>
-    rw [Finset.sum_insert ha, Finset.sum_insert ha]
-    exact (seriesL1_add_le _ _).trans (add_le_add le_rfl ih)
-
-theorem seriesL1_prod_le {ι : Type*} (A : Finset ι) (f : ι → IntSeries) :
-    seriesL1 (∏ i ∈ A, f i) ≤ ∏ i ∈ A, seriesL1 (f i) := by
-  classical
-  induction A using Finset.induction_on with
-  | empty => simp only [Finset.prod_empty, seriesL1_one, le_refl]
-  | @insert a A ha ih =>
-    rw [Finset.prod_insert ha, Finset.prod_insert ha]
-    exact (seriesL1_mul_le _ _).trans (mul_le_mul_of_nonneg_left ih (seriesL1_nonneg _))
-
-def geomSeries (k : ℕ) : IntSeries := ∑ a ∈ Finset.range k, (AddMonoidAlgebra.single 1 1) ^ a
-
-def prouhetSeries (s : ℕ) : IntSeries :=
-  ∏ v ∈ Finset.range s, (1 - (AddMonoidAlgebra.single 1 1) ^ (2 ^ v))
-
-def prouhetQuotient (s : ℕ) : IntSeries :=
-  ∏ v ∈ Finset.range s, geomSeries (2 ^ v)
-
-theorem geomSeries_difference (k : ℕ) :
-    geomSeries k * firstDifference = 1 - (AddMonoidAlgebra.single 1 1 : IntSeries) ^ k := by
-  exact geom_sum_mul_neg _ _
-
-theorem geomSeries_L1_le (k : ℕ) : seriesL1 (geomSeries k) ≤ k := by
-  apply (seriesL1_sum_le _ _).trans
-  simp only [AddMonoidAlgebra.single_pow, one_pow, seriesL1_single, abs_one,
-    Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_one, le_refl]
-
-theorem prouhetSeries_factor (s : ℕ) :
-    prouhetSeries s = prouhetQuotient s * firstDifference ^ s := by
-  unfold prouhetSeries prouhetQuotient
-  simp_rw [← geomSeries_difference]
-  rw [Finset.prod_mul_distrib]
-  simp only [Finset.prod_const, Finset.card_range]
-
-theorem prouhetQuotient_L1_le (s : ℕ) :
-    seriesL1 (prouhetQuotient s) ≤ (2 : ℝ) ^ (s * s) := by
-  calc
-    _ ≤ ∏ v ∈ Finset.range s, seriesL1 (geomSeries (2 ^ v)) := seriesL1_prod_le _ _
-    _ ≤ ∏ _v ∈ Finset.range s, ((2 : ℝ) ^ s) := by
-      apply Finset.prod_le_prod₀
-      · intro v _; exact seriesL1_nonneg _
-      · intro v hv
-        apply (geomSeries_L1_le _).trans
-        push_cast
-        apply pow_le_pow_right₀ (by norm_num : (1 : ℝ) ≤ 2)
-        exact (Finset.mem_range.mp hv).le
-    _ = _ := by simp only [Finset.prod_const, Finset.card_range, ← pow_mul]
-
-/-- Prouhet smoothing in an explicit finite-trial form.  The constants
-    2^(s*s) and (2 sqrt(2/(n+1)))^s are exp(O(s^2)) after assigning
-    n = floor(g/(s+1)) trials to each group.  No Fourier theorem or
-    probabilistic estimate is assumed. -/
-theorem prouhet_binomial_coeff_bound (n s : ℕ) (z : ℤ) :
-    |(prouhetSeries s * fairSeries ((s + 1) * n)).coeff z| ≤
-      (2 : ℝ) ^ (s * s) * ((2 * binomialBound n) ^ s * binomialBound n) := by
-  rw [prouhetSeries_factor, mul_assoc]
-  apply (series_coeff_mul_bound _ _ _ (binomial_difference_coeff_bound n s) z).trans
-  exact mul_le_mul_of_nonneg_right (prouhetQuotient_L1_le s)
-    (by positivity [binomialBound_nonneg n])
-
-theorem prouhet_binomial_L2Sq_bound (n s : ℕ) :
-    seriesL2Sq (prouhetSeries s * fairSeries ((s + 1) * n)) ≤
-      ((2 : ℝ) ^ (s * s)) ^ 2 * (((2 * binomialBound n) ^ s) ^ 2 * binomialBound n) := by
-  rw [prouhetSeries_factor, mul_assoc]
-  apply (seriesL2Sq_mul_le _ _).trans
-  apply mul_le_mul
-  · exact pow_le_pow_left₀ (seriesL1_nonneg _) (prouhetQuotient_L1_le s) 2
-  · exact binomial_difference_L2Sq_bound n s
-  · exact seriesL2Sq_nonneg _
-  · positivity
-
-/-- Arbitrary trial counts are obtained by convolving with the independent
-    unused trials; this cannot enlarge the coefficient or squared-L2 bounds. -/
-theorem prouhet_binomial_coeff_bound_remainder (n s r : ℕ) (z : ℤ) :
-    |(prouhetSeries s * fairSeries ((s + 1) * n + r)).coeff z| ≤
-      (2 : ℝ) ^ (s * s) * ((2 * binomialBound n) ^ s * binomialBound n) := by
-  rw [fairSeries_add, ← mul_assoc, mul_comm (prouhetSeries s * _)]
-  simpa only [fairSeries_L1, one_mul] using
-    series_coeff_mul_bound (fairSeries r) (prouhetSeries s * fairSeries ((s + 1) * n))
-      _ (prouhet_binomial_coeff_bound n s) z
-
-theorem prouhet_binomial_L2Sq_bound_remainder (n s r : ℕ) :
-    seriesL2Sq (prouhetSeries s * fairSeries ((s + 1) * n + r)) ≤
-      ((2 : ℝ) ^ (s * s)) ^ 2 * (((2 * binomialBound n) ^ s) ^ 2 * binomialBound n) := by
-  rw [fairSeries_add, ← mul_assoc, mul_comm (prouhetSeries s * _)]
-  calc
-    _ ≤ seriesL1 (fairSeries r) ^ 2 *
-        seriesL2Sq (prouhetSeries s * fairSeries ((s + 1) * n)) := seriesL2Sq_mul_le _ _
-    _ ≤ _ := by simpa only [fairSeries_L1, one_pow, one_mul] using
-        prouhet_binomial_L2Sq_bound n s
-
-theorem signed_binomial_smoothing (g s : ℕ) :
-    (∀ z : ℤ, |(prouhetSeries s * fairSeries g).coeff z| ≤
-      (2 : ℝ) ^ (s * s) * ((2 * binomialBound (g / (s + 1))) ^ s *
-        binomialBound (g / (s + 1)))) ∧
-    (seriesL2Sq (prouhetSeries s * fairSeries g) ≤
-      ((2 : ℝ) ^ (s * s)) ^ 2 * (((2 * binomialBound (g / (s + 1))) ^ s) ^ 2 *
-        binomialBound (g / (s + 1)))) := by
-  have hg : (s + 1) * (g / (s + 1)) + g % (s + 1) = g := Nat.div_add_mod g (s + 1)
-  constructor
-  · intro z
-    simpa only [hg] using prouhet_binomial_coeff_bound_remainder (g / (s + 1)) s
-      (g % (s + 1)) z
-  · simpa only [hg] using prouhet_binomial_L2Sq_bound_remainder (g / (s + 1)) s
-      (g % (s + 1))
-
-theorem binomialBound_quotient_le (g s : ℕ) :
-    binomialBound (g / (s + 1)) ≤
-      Real.sqrt (2 * ((s : ℝ) + 1) / ((g : ℝ) + 1)) := by
-  apply Real.sqrt_le_sqrt
-  have hn : g + 1 ≤ (g / (s + 1) + 1) * (s + 1) := by
-    have hmod := Nat.mod_lt g (by omega : 0 < s + 1)
-    have hdiv := Nat.div_add_mod g (s + 1)
-    nlinarith
-  have hnr : (g : ℝ) + 1 ≤ ((g / (s + 1) : ℕ) + (1 : ℝ)) * ((s : ℝ) + 1) := by
-    exact_mod_cast hn
-  apply (div_le_div_iff₀ (by positivity : (0 : ℝ) < (g / (s + 1) : ℕ) + 1)
-    (by positivity : (0 : ℝ) < g + 1)).mpr
-  nlinarith
-
-theorem prouhetSeries_succ_coeff (s : ℕ) (z : ℤ) :
-    (prouhetSeries (s + 1)).coeff z =
-    (prouhetSeries s).coeff z - (prouhetSeries s).coeff (z - (2 ^ s : ℕ)) := by
-  rw [prouhetSeries, Finset.prod_range_succ, mul_sub, mul_one]
-  change (prouhetSeries s - prouhetSeries s *
-    (AddMonoidAlgebra.single 1 1 : IntSeries) ^ (2 ^ s)).coeff z = _
-  rw [AddMonoidAlgebra.coeff_sub, Finsupp.sub_apply]
-  simp only [AddMonoidAlgebra.single_pow, one_pow, nsmul_eq_mul, mul_one,
-    AddMonoidAlgebra.coeff_mul_single_apply, sub_eq_add_neg]
-
-/-- The exact generating polynomial from the manuscript has unit signed
-    coefficients at every index 0,...,2^s-1, and no other coefficients. -/
-theorem prouhetSeries_coeff_spec (s : ℕ) :
-    (∀ z : ℤ, (z < 0 ∨ (2 ^ s : ℕ) ≤ z) → (prouhetSeries s).coeff z = 0) ∧
-    (∀ z : ℤ, 0 ≤ z → z < (2 ^ s : ℕ) → |(prouhetSeries s).coeff z| = 1) := by
-  induction s with
-  | zero =>
-    have hcoeff (z : ℤ) : (prouhetSeries 0).coeff z = if 0 = z then 1 else 0 := by
-      change (AddMonoidAlgebra.single 0 (1 : ℝ)).coeff z = _
-      exact Finsupp.single_apply
-    constructor
-    · intro z hz
-      rw [hcoeff, ite_eq_right]
-      norm_num only [pow_zero, Nat.cast_one] at hz
-      omega
-    · intro z h0 h1
-      have hz : z = 0 := by norm_num only [pow_zero, Nat.cast_one] at h1; omega
-      subst hz
-      rw [hcoeff, ite_eq_left rfl, abs_one]
-  | succ s ih =>
-    have hp : (0 : ℤ) < (2 ^ s : ℕ) := by positivity
-    have hd : ((2 ^ (s + 1) : ℕ) : ℤ) = 2 * (2 ^ s : ℕ) := by push_cast; ring
-    constructor
-    · intro z hz
-      rw [prouhetSeries_succ_coeff]
-      rcases hz with hz | hz
-      · rw [ih.1 z (Or.inl hz), ih.1 _ (Or.inl (by omega)), sub_zero]
-      · rw [hd] at hz
-        rw [ih.1 z (Or.inr (by omega)), ih.1 _ (Or.inr (by omega)), sub_zero]
-    · intro z h0 h1
-      rw [hd] at h1
-      rw [prouhetSeries_succ_coeff]
-      by_cases hz : z < (2 ^ s : ℕ)
-      · rw [ih.1 (z - (2 ^ s : ℕ)) (Or.inl (by omega)), sub_zero]
-        exact ih.2 z h0 hz
-      · rw [ih.1 z (Or.inr (by omega)), zero_sub, abs_neg]
-        exact ih.2 _ (by omega) (by omega)
-
-
-/-! ## Nonfair anti-concentration without an assumed Stirling estimate -/
-
-def binomialMass (n : ℕ) (p : ℝ) (k : ℕ) : ℝ :=
-  (n.choose k : ℝ) * p ^ k * (1 - p) ^ (n - k)
-
-theorem binomialMass_nonneg (n k : ℕ) {p : ℝ} (hp : p ∈ Set.Icc 0 1) :
-    0 ≤ binomialMass n p k := by
-  unfold binomialMass
-  positivity [hp.1, sub_nonneg.mpr hp.2]
-
-theorem sum_binomialMass (n : ℕ) (p : ℝ) :
-    ∑ k ∈ Finset.range (n + 1), binomialMass n p k = 1 := by
-  have h := add_pow p (1 - p) n
-  rw [show p + (1 - p) = 1 by ring, one_pow] at h
-  rw [h]
-  apply Finset.sum_congr rfl
-  intro k _
-  unfold binomialMass
-  ring
-
-theorem binomialMass_harmonic_step (n k : ℕ) (p : ℝ) :
-    ((n : ℝ) + 1) * p * (binomialMass n p k / ((k : ℝ) + 1)) =
-      binomialMass (n + 1) p (k + 1) := by
-  have hc : ((n : ℝ) + 1) * (n.choose k : ℝ) =
-      ((n + 1).choose (k + 1) : ℝ) * ((k : ℝ) + 1) := by
-    exact_mod_cast Nat.add_one_mul_choose_eq n k
-  unfold binomialMass
-  rw [Nat.add_sub_add_right, pow_succ]
-  field_simp
-  nlinarith [congrArg (fun x : ℝ => x * p ^ k * (1 - p) ^ (n - k) * p) hc]
-
-theorem binomialMass_harmonic_identity (n : ℕ) (p : ℝ) :
-    ((n : ℝ) + 1) * p *
-      (∑ k ∈ Finset.range (n + 1), binomialMass n p k / ((k : ℝ) + 1)) =
-    1 - (1 - p) ^ (n + 1) := by
-  rw [Finset.mul_sum]
-  simp_rw [binomialMass_harmonic_step]
-  have h := sum_binomialMass (n + 1) p
-  rw [Finset.sum_range_succ'] at h
-  simp only [binomialMass, Nat.choose_zero_right, Nat.cast_one, pow_zero,
-    mul_one, one_mul, Nat.sub_zero] at h ⊢
-  linarith
-
-theorem binomialMass_harmonic_le (n : ℕ) {p : ℝ} (hp : p ∈ Set.Ioc 0 1) :
-    (∑ k ∈ Finset.range (n + 1), binomialMass n p k / ((k : ℝ) + 1)) ≤
-      1 / (((n : ℝ) + 1) * p) := by
-  apply (le_div_iff₀ (mul_pos (by positivity) hp.1)).mpr
-  rw [mul_comm, binomialMass_harmonic_identity]
-  have h := pow_nonneg (sub_nonneg.mpr hp.2) (n + 1)
-  linarith
-
-/-- The weighted Cauchy--Schwarz step for mixing the fair estimates. -/
-theorem weighted_sum_sq_le {ι : Type*} (S : Finset ι) (w f : ι → ℝ)
-    (hw : ∀ i, 0 ≤ w i) :
-    (∑ i ∈ S, w i * f i) ^ 2 ≤ (∑ i ∈ S, w i) * ∑ i ∈ S, w i * f i ^ 2 := by
-  have heq (i : ι) : w i * f i = Real.sqrt (w i) * (Real.sqrt (w i) * f i) := by
-    rw [← mul_assoc, ← sq, Real.sq_sqrt (hw i)]
-  simp_rw [heq]
-  apply (Finset.sum_mul_sq_le_sq_mul_sq S (fun i => Real.sqrt (w i))
-    (fun i => Real.sqrt (w i) * f i)).trans_eq
-  simp only [mul_pow, Real.sq_sqrt (hw _)]
-
-theorem mixed_binomialBound_le (n : ℕ) {p : ℝ} (hp : p ∈ Set.Ioc 0 1) :
-    (∑ k ∈ Finset.range (n + 1), binomialMass n p k * binomialBound k) ≤
-      Real.sqrt (2 / (((n : ℝ) + 1) * p)) := by
-  have hw (k : ℕ) := binomialMass_nonneg n k (show p ∈ Set.Icc 0 1 from ⟨hp.1.le, hp.2⟩)
-  have hs := weighted_sum_sq_le (Finset.range (n + 1)) (binomialMass n p)
-    binomialBound hw
-  rw [sum_binomialMass, one_mul] at hs
-  have hb (k : ℕ) : binomialBound k ^ 2 = 2 / ((k : ℝ) + 1) :=
-    Real.sq_sqrt (by positivity)
-  simp_rw [hb] at hs
-  have heq : (∑ k ∈ Finset.range (n + 1), binomialMass n p k * (2 / ((k : ℝ) + 1))) =
-      2 * ∑ k ∈ Finset.range (n + 1), binomialMass n p k / ((k : ℝ) + 1) := by
-    rw [Finset.mul_sum]
-    apply Finset.sum_congr rfl
-    intro k _
-    ring
-  rw [heq] at hs
-  have hh := binomialMass_harmonic_le n hp
-  have hp0 := hp.1
-  have hbound : (∑ k ∈ Finset.range (n + 1), binomialMass n p k * binomialBound k) ^ 2 ≤
-      2 / (((n : ℝ) + 1) * p) := by
-    calc
-      _ ≤ 2 * ∑ k ∈ Finset.range (n + 1), binomialMass n p k / ((k : ℝ) + 1) := hs
-      _ ≤ 2 * (1 / (((n : ℝ) + 1) * p)) := mul_le_mul_of_nonneg_left hh (by norm_num)
-      _ = _ := by ring
-  have hsr := Real.sq_sqrt (show (0 : ℝ) ≤ 2 / (((n : ℝ) + 1) * p) by positivity)
-  have hpos := Real.sqrt_nonneg (2 / (((n : ℝ) + 1) * p))
-  nlinarith only [hbound, hsr, hpos]
-
-
-def mixedFairSeries (n : ℕ) (p : ℝ) : IntSeries :=
-  (p • fairSeries 1 + (1 - p) • (1 : IntSeries)) ^ n
-
-def biasedSeries (n : ℕ) (p : ℝ) : IntSeries :=
-  (AddMonoidAlgebra.single 1 p + AddMonoidAlgebra.single 0 (1 - p)) ^ n
-
-theorem mixedFairSeries_expansion (n : ℕ) (p : ℝ) :
-    mixedFairSeries n p =
-      ∑ k ∈ Finset.range (n + 1), binomialMass n p k • fairSeries k := by
-  rw [mixedFairSeries, add_pow]
-  apply Finset.sum_congr rfl
-  intro k _
-  simp only [smul_pow, one_pow, smul_mul_smul, mul_one, fairSeries, pow_one]
-  rw [mul_comm, ← nsmul_eq_mul, ← Nat.cast_smul_eq_nsmul ℝ, smul_smul]
-  congr 1
-  unfold binomialMass
-  ring
-
-theorem mixedFairSeries_eq_biased (n : ℕ) (p : ℝ) :
-    mixedFairSeries n p = biasedSeries n (p / 2) := by
-  unfold mixedFairSeries biasedSeries
-  congr 1
-  ext z
-  simp only [fairSeries, pow_one, smul_add, AddMonoidAlgebra.one_def, AddMonoidAlgebra.coeff_add,
-    Finsupp.add_apply, AddMonoidAlgebra.coeff_smul, Finsupp.smul_apply,
-    smul_eq_mul, AddMonoidAlgebra.coeff_single, Finsupp.single_apply]
-  split_ifs <;> first | omega | ring
-
-theorem mixedFairSeries_coeff_bound (n : ℕ) {p : ℝ} (hp : p ∈ Set.Ioc 0 1) (z : ℤ) :
-    |(mixedFairSeries n p).coeff z| ≤ Real.sqrt (2 / (((n : ℝ) + 1) * p)) := by
-  rw [mixedFairSeries_expansion, AddMonoidAlgebra.coeff_sum, Finsupp.finsetSum_apply]
-  simp only [AddMonoidAlgebra.coeff_smul, Finsupp.smul_apply, smul_eq_mul]
-  calc
-    _ ≤ ∑ k ∈ Finset.range (n + 1), |binomialMass n p k * (fairSeries k).coeff z| :=
-      Finset.abs_sum_le_sum_abs _ _
-    _ ≤ ∑ k ∈ Finset.range (n + 1), binomialMass n p k * binomialBound k := by
-      apply Finset.sum_le_sum
-      intro k _
-      have hw := binomialMass_nonneg n k (show p ∈ Set.Icc 0 1 from ⟨hp.1.le, hp.2⟩)
-      rw [abs_mul, abs_of_nonneg hw]
-      exact mul_le_mul_of_nonneg_left (fairSeries_coeff_bound k z) hw
-    _ ≤ _ := mixed_binomialBound_le n hp
-
-/-- At success probabilities at most 1/2, this gives the sharp variance
-    scale up to an absolute constant, using only the fair-coin estimate. -/
-theorem biasedSeries_coeff_bound_low (n : ℕ) {p : ℝ}
-    (hp : p ∈ Set.Ioc 0 (1 / 2)) (z : ℤ) :
-    |(biasedSeries n p).coeff z| ≤ Real.sqrt (1 / (((n : ℝ) + 1) * p)) := by
-  have h2p : 2 * p ∈ Set.Ioc 0 1 := ⟨by linarith [hp.1], by linarith [hp.2]⟩
-  have h := mixedFairSeries_coeff_bound n h2p z
-  rw [mixedFairSeries_eq_biased, show 2 * p / 2 = p by ring] at h
-  have heq : 2 / (((n : ℝ) + 1) * (2 * p)) = 1 / (((n : ℝ) + 1) * p) := by
-    field_simp
-  rwa [heq] at h
-
-
-theorem biasedSeries_sum (n : ℕ) (p : ℝ) :
-    biasedSeries n p = ∑ k ∈ Finset.range (n + 1),
-      AddMonoidAlgebra.single (k : ℤ) (binomialMass n p k) := by
-  rw [biasedSeries, add_pow]
-  apply Finset.sum_congr rfl
-  intro k _
-  rw [AddMonoidAlgebra.single_pow, AddMonoidAlgebra.single_pow,
-    AddMonoidAlgebra.natCast_def, AddMonoidAlgebra.single_mul_single,
-    AddMonoidAlgebra.single_mul_single]
-  simp only [nsmul_eq_mul, mul_one, mul_zero, add_zero]
-  congr 1
-  unfold binomialMass
-  ring
-
-theorem biasedSeries_coeff_nat (n k : ℕ) (p : ℝ) :
-    (biasedSeries n p).coeff (k : ℤ) = binomialMass n p k := by
-  rw [biasedSeries_sum, AddMonoidAlgebra.coeff_sum, Finsupp.finsetSum_apply]
-  by_cases hk : k ≤ n
-  · rw [Finset.sum_eq_single k]
-    · simp only [AddMonoidAlgebra.coeff_single, Finsupp.single_apply, ite_true]
-    · intro b _ hb
-      simp only [AddMonoidAlgebra.coeff_single, Finsupp.single_apply, Nat.cast_inj, ite_eq_right hb]
-    · intro h
-      exact (h (Finset.mem_range.mpr (by omega))).elim
-  · rw [Finset.sum_eq_zero]
-    · simp [binomialMass, Nat.choose_eq_zero_of_lt (by omega : n < k)]
-    · intro b hb
-      have hbk : b ≠ k := by simp only [Finset.mem_range] at hb; omega
-      simp only [AddMonoidAlgebra.coeff_single, Finsupp.single_apply, Nat.cast_inj, ite_eq_right hbk]
-
-theorem biasedSeries_coeff_neg (n : ℕ) (p : ℝ) (z : ℤ) (hz : z < 0) :
-    (biasedSeries n p).coeff z = 0 := by
-  rw [biasedSeries_sum, AddMonoidAlgebra.coeff_sum, Finsupp.finsetSum_apply]
-  apply Finset.sum_eq_zero
-  intro b _
-  have h : (b : ℤ) ≠ z := by omega
-  simp only [AddMonoidAlgebra.coeff_single, Finsupp.single_apply, ite_eq_right h]
-
-theorem biasedSeries_coeff_high (n : ℕ) (p : ℝ) (z : ℤ) (hz : (n : ℤ) < z) :
-    (biasedSeries n p).coeff z = 0 := by
-  rw [biasedSeries_sum, AddMonoidAlgebra.coeff_sum, Finsupp.finsetSum_apply]
-  apply Finset.sum_eq_zero
-  intro b hb
-  have h : (b : ℤ) ≠ z := by simp only [Finset.mem_range] at hb; omega
-  simp only [AddMonoidAlgebra.coeff_single, Finsupp.single_apply, ite_eq_right h]
-
-theorem binomialMass_symm (n k : ℕ) (p : ℝ) (hk : k ≤ n) :
-    binomialMass n (1 - p) (n - k) = binomialMass n p k := by
-  unfold binomialMass
-  rw [Nat.choose_symm hk, Nat.sub_sub_self hk, show 1 - (1 - p) = p by ring]
-  ring
-
-theorem biasedSeries_coeff_reflect (n : ℕ) (p : ℝ) (z : ℤ) :
-    (biasedSeries n (1 - p)).coeff ((n : ℤ) - z) = (biasedSeries n p).coeff z := by
-  by_cases hz : z < 0
-  · rw [biasedSeries_coeff_neg n p z hz,
-      biasedSeries_coeff_high n (1 - p) _ (by omega)]
-  by_cases hz' : (n : ℤ) < z
-  · rw [biasedSeries_coeff_high n p z hz',
-      biasedSeries_coeff_neg n (1 - p) _ (by omega)]
-  have h0 : 0 ≤ z := by omega
-  have hk : z.toNat ≤ n := by omega
-  have hzN : (z.toNat : ℤ) = z := Int.toNat_of_nonneg h0
-  have hsub : (n : ℤ) - z = ((n - z.toNat : ℕ) : ℤ) := by omega
-  rw [hsub, ← hzN, biasedSeries_coeff_nat, biasedSeries_coeff_nat]
-  exact binomialMass_symm n z.toNat p hk
-
-/-- Uniform nonfair-binomial anti-concentration.  This is the absolute
-    variance-scale upper bound used in the manuscript's rank kernels. -/
-theorem biasedSeries_coeff_bound (n : ℕ) {p : ℝ}
-    (hp : p ∈ Set.Ioo 0 1) (z : ℤ) :
-    |(biasedSeries n p).coeff z| ≤
-      Real.sqrt (1 / (((n : ℝ) + 1) * p * (1 - p))) := by
-  by_cases hlow : p ≤ 1 / 2
-  · apply (biasedSeries_coeff_bound_low n ⟨hp.1, hlow⟩ z).trans
-    apply Real.sqrt_le_sqrt
-    apply one_div_le_one_div_of_le
-    · exact mul_pos (mul_pos (by positivity) hp.1) (sub_pos.mpr hp.2)
-    · have hn : (0 : ℝ) ≤ n + 1 := by positivity
-      nlinarith [mul_nonneg hn (sq_nonneg p)]
-  · have hq : 1 - p ∈ Set.Ioc 0 (1 / 2) := ⟨by linarith [hp.2], by linarith⟩
-    rw [← biasedSeries_coeff_reflect n p z]
-    apply (biasedSeries_coeff_bound_low n hq ((n : ℤ) - z)).trans
-    apply Real.sqrt_le_sqrt
-    apply one_div_le_one_div_of_le
-    · exact mul_pos (mul_pos (by positivity) hp.1) (sub_pos.mpr hp.2)
-    · nlinarith [mul_nonneg (show (0 : ℝ) ≤ n + 1 by positivity)
-        (mul_nonneg (sub_nonneg.mpr hp.2.le) (sub_nonneg.mpr hp.2.le))]
-
-
-theorem binomialMass_mul_index (n k : ℕ) (p : ℝ) :
-    binomialMass (n + 1) p (k + 1) * ((k : ℝ) + 1) =
-      ((n : ℝ) + 1) * p * binomialMass n p k := by
-  have h := binomialMass_harmonic_step n k p
-  have hk : ((k : ℝ) + 1) ≠ 0 := by positivity
-  rw [← h]
-  field_simp
-
-theorem binomialMass_index_sum (n : ℕ) (p : ℝ) (F : ℕ → ℝ) :
-    (∑ k ∈ Finset.range (n + 2), binomialMass (n + 1) p k * (k : ℝ) * F k) =
-      ((n : ℝ) + 1) * p *
-        ∑ k ∈ Finset.range (n + 1), binomialMass n p k * F (k + 1) := by
-  rw [show n + 2 = (n + 1) + 1 by omega, Finset.sum_range_succ']
-  simp only [Nat.cast_zero, mul_zero, zero_mul, add_zero, Nat.cast_add, Nat.cast_one]
-  simp_rw [binomialMass_mul_index]
-  rw [Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro k _
-  ring
-
-theorem binomialMass_mean (n : ℕ) (p : ℝ) :
-    ∑ k ∈ Finset.range (n + 1), binomialMass n p k * (k : ℝ) = (n : ℝ) * p := by
-  cases n with
-  | zero => simp
-  | succ n =>
-    simpa only [mul_one, sum_binomialMass, Nat.cast_add, Nat.cast_one] using
-      binomialMass_index_sum n p (fun _ => 1)
-
-theorem binomialMass_second_moment (n : ℕ) (p : ℝ) :
-    ∑ k ∈ Finset.range (n + 1), binomialMass n p k * (k : ℝ) ^ 2 =
-      (n : ℝ) * ((n : ℝ) - 1) * p ^ 2 + (n : ℝ) * p := by
-  cases n with
-  | zero => simp
-  | succ n =>
-    have h := binomialMass_index_sum n p (fun k => k)
-    simp only [Nat.cast_add, Nat.cast_one] at h ⊢
-    simp_rw [mul_add, mul_one, Finset.sum_add_distrib, binomialMass_mean,
-      sum_binomialMass] at h
-    calc
-      _ = ∑ k ∈ Finset.range (n + 2), binomialMass (n + 1) p k * (k : ℝ) * (k : ℝ) := by
-        apply Finset.sum_congr rfl
-        intro k _
-        ring
-      _ = _ := by rw [h]; ring
-
-theorem binomialMass_variance (n : ℕ) (p : ℝ) :
-    (∑ k ∈ Finset.range (n + 1), binomialMass n p k * ((k : ℝ) - (n : ℝ) * p) ^ 2) =
-      (n : ℝ) * p * (1 - p) := by
-  calc
-    _ = (∑ k ∈ Finset.range (n + 1), binomialMass n p k * (k : ℝ) ^ 2) -
-        2 * (n : ℝ) * p * (∑ k ∈ Finset.range (n + 1), binomialMass n p k * (k : ℝ)) +
-        ((n : ℝ) * p) ^ 2 * (∑ k ∈ Finset.range (n + 1), binomialMass n p k) := by
-      rw [Finset.mul_sum, Finset.mul_sum, ← Finset.sum_sub_distrib, ← Finset.sum_add_distrib]
-      apply Finset.sum_congr rfl
-      intro k _
-      ring
-    _ = _ := by rw [binomialMass_second_moment, binomialMass_mean, sum_binomialMass]; ring
-
-theorem binomialMass_adjacent (n k : ℕ) (p : ℝ) (hk : k < n) :
-    ((k : ℝ) + 1) * (1 - p) * binomialMass n p (k + 1) =
-      ((n : ℝ) - k) * p * binomialMass n p k := by
-  have hc : (n.choose (k + 1) : ℝ) * ((k : ℝ) + 1) =
-      (n.choose k : ℝ) * ((n : ℝ) - (k : ℝ)) := by
-    simpa only [Nat.cast_mul, Nat.cast_add, Nat.cast_one, Nat.cast_sub hk.le] using
-      congrArg (fun t : ℕ => (t : ℝ)) (Nat.choose_succ_right_eq n k)
-  have hnk : n - k = (n - (k + 1)) + 1 := by omega
-  unfold binomialMass
-  rw [hnk, pow_succ, pow_succ]
-  linear_combination p ^ k * p * (1 - p) ^ (n - (k + 1)) * (1 - p) * hc
-
-theorem binomialMass_le_mode (n l k : ℕ) {p : ℝ} (hp : p ∈ Set.Ioo 0 1)
-    (hl : l ≤ n) (hmean : (n : ℝ) * p = l) :
-    binomialMass n p k ≤ binomialMass n p l := by
-  have h0 (j : ℕ) := binomialMass_nonneg n j ⟨hp.1.le, hp.2.le⟩
-  have hleft (j : ℕ) (hj : j < l) : binomialMass n p j ≤ binomialMass n p (j + 1) := by
-    have hrec := binomialMass_adjacent n j p (by omega)
-    have hjr : (j : ℝ) + 1 ≤ l := by exact_mod_cast hj
-    have hcoef : (0 : ℝ) < ((j : ℝ) + 1) * (1 - p) := by
-      exact mul_pos (by positivity) (sub_pos.mpr hp.2)
-    have hdiff : ((j : ℝ) + 1) * (1 - p) ≤ ((n : ℝ) - j) * p := by
-      nlinarith [hp.1]
-    have := mul_le_mul_of_nonneg_right hdiff (h0 j)
-    nlinarith only [hrec, this, hcoef]
-  have hright (j : ℕ) (hj : l ≤ j) : binomialMass n p (j + 1) ≤ binomialMass n p j := by
-    by_cases hjn : j < n
-    · have hrec := binomialMass_adjacent n j p hjn
-      have hjr : (l : ℝ) ≤ j := by exact_mod_cast hj
-      have hcoef : (0 : ℝ) < ((j : ℝ) + 1) * (1 - p) := by
-        exact mul_pos (by positivity) (sub_pos.mpr hp.2)
-      have hdiff : ((n : ℝ) - j) * p ≤ ((j : ℝ) + 1) * (1 - p) := by
-        nlinarith [hp.2]
-      have := mul_le_mul_of_nonneg_right hdiff (h0 j)
-      nlinarith only [hrec, this, hcoef]
-    · have hjn' : n < j + 1 := by omega
-      simp only [binomialMass, Nat.choose_eq_zero_of_lt hjn', Nat.cast_zero, zero_mul]
-      exact h0 j
-  rcases le_total k l with hk | hk
-  · have haux : ∀ b : ℕ, k ≤ b → b ≤ l → binomialMass n p k ≤ binomialMass n p b := by
-      intro b hkb
-      induction b, hkb using Nat.le_induction with
-      | base => intro _; exact le_rfl
-      | succ b hb ih =>
-        intro hbl
-        exact (ih (by omega)).trans (hleft b (by omega))
-    exact haux l hk le_rfl
-  · induction k, hk using Nat.le_induction with
-    | base => exact le_rfl
-    | succ j hj ih => exact (hright j hj).trans ih
-
-
-/-- A discrete law cannot have a very small maximum when its variance is bounded.
-The explicit constant is deliberately loose, avoiding any use of Stirling. -/
-theorem discrete_mode_lower (n l : ℕ) (f : ℕ → ℝ) (M v : ℝ)
-    (hf : ∀ k, 0 ≤ f k) (hsum : ∑ k ∈ Finset.range (n + 1), f k = 1)
-    (hmax : ∀ k, f k ≤ M)
-    (hvar : ∑ k ∈ Finset.range (n + 1), f k * ((k : ℝ) - l) ^ 2 = v)
-    (hv : 1 ≤ v) : 1 / (12 * Real.sqrt v) ≤ M := by
-  classical
-  let r : ℕ := ⌈2 * Real.sqrt v⌉₊
-  have hv0 : 0 ≤ v := by linarith
-  have hs : 1 ≤ Real.sqrt v := Real.one_le_sqrt.mpr hv
-  have hs2 := Real.sq_sqrt hv0
-  have hrlo : 2 * Real.sqrt v ≤ (r : ℝ) := Nat.le_ceil _
-  have hrhi : (r : ℝ) < 2 * Real.sqrt v + 1 := Nat.ceil_lt_add_one (by positivity)
-  have hrpos : (0 : ℝ) < r := by linarith
-  have hM : 0 ≤ M := (hf 0).trans (hmax 0)
-  let S := (Finset.range (n + 1)).filter (fun k => l - r ≤ k ∧ k ≤ l + r)
-  have hcard : S.card ≤ 2 * r + 1 := by
-    have hsub : S ⊆ Finset.Icc (l - r) (l + r) := by
-      intro k hk
-      exact Finset.mem_Icc.mpr (Finset.mem_filter.mp hk).2
-    have := Finset.card_le_card hsub
-    rw [Nat.card_Icc] at this
-    omega
-  have hout (k : ℕ) (hk : ¬(l - r ≤ k ∧ k ≤ l + r)) :
-      (r : ℝ) ^ 2 ≤ ((k : ℝ) - l) ^ 2 := by
-    by_cases hleft : k < l - r
-    · have hnat : k + r < l := by omega
-      have hreal : (k : ℝ) + r < l := by exact_mod_cast hnat
-      nlinarith [sq_nonneg ((l : ℝ) - k - r)]
-    · have hnat : l + r < k := by omega
-      have hreal : (l : ℝ) + r < k := by exact_mod_cast hnat
-      nlinarith [sq_nonneg ((k : ℝ) - l - r)]
-  have hpoint (k : ℕ) : (r : ℝ) ^ 2 * f k ≤
-      (if l - r ≤ k ∧ k ≤ l + r then (r : ℝ) ^ 2 * M else 0) +
-        f k * ((k : ℝ) - l) ^ 2 := by
-    split_ifs with hk
-    · have := mul_le_mul_of_nonneg_left (hmax k) (sq_nonneg (r : ℝ))
-      have := mul_nonneg (hf k) (sq_nonneg ((k : ℝ) - l))
-      linarith
-    · simpa only [zero_add, mul_comm] using mul_le_mul_of_nonneg_left (hout k hk) (hf k)
-  have htotal := Finset.sum_le_sum (fun k (_ : k ∈ Finset.range (n + 1)) => hpoint k)
-  rw [← Finset.mul_sum, hsum, mul_one, Finset.sum_add_distrib, hvar,
-    ← Finset.sum_filter] at htotal
-  change (r : ℝ) ^ 2 ≤ ∑ k ∈ S, (r : ℝ) ^ 2 * M + v at htotal
-  rw [Finset.sum_const, nsmul_eq_mul] at htotal
-  have hcardR : (S.card : ℝ) ≤ 2 * r + 1 := by exact_mod_cast hcard
-  have hbound := mul_le_mul_of_nonneg_right hcardR (mul_nonneg (sq_nonneg (r : ℝ)) hM)
-  have htotal' : (r : ℝ) ^ 2 ≤ (2 * r + 1) * ((r : ℝ) ^ 2 * M) + v := by
-    linarith
-  have hrv : 4 * v ≤ (r : ℝ) ^ 2 := by nlinarith [sq_nonneg ((r : ℝ) - 2 * Real.sqrt v)]
-  have hcancel : (3 : ℝ) / 4 ≤ (2 * r + 1) * M := by
-    have hrr : (0 : ℝ) < (r : ℝ) ^ 2 := sq_pos_of_pos hrpos
-    nlinarith only [htotal', hrv, hrr]
-  have hsize : 2 * (r : ℝ) + 1 ≤ 7 * Real.sqrt v := by linarith
-  have hfinal := mul_le_mul_of_nonneg_right hsize hM
-  apply (div_le_iff₀ (by positivity : 0 < 12 * Real.sqrt v)).mpr
-  nlinarith only [hcancel, hfinal]
-
-theorem binomialMass_mean_lower (n l : ℕ) {p : ℝ} (hp : p ∈ Set.Ioo 0 1)
-    (hl : l ≤ n) (hmean : (n : ℝ) * p = l) (hv : 1 ≤ (n : ℝ) * p * (1 - p)) :
-    1 / (12 * Real.sqrt ((n : ℝ) * p * (1 - p))) ≤ binomialMass n p l := by
-  apply discrete_mode_lower n l (binomialMass n p)
-    (binomialMass n p l) ((n : ℝ) * p * (1 - p))
-  · intro k; exact binomialMass_nonneg n k ⟨hp.1.le, hp.2.le⟩
-  · exact sum_binomialMass n p
-  · intro k; exact binomialMass_le_mode n l k hp hl hmean
-  · rw [← hmean]; exact binomialMass_variance n p
-  · exact hv
-
-
-theorem binomialMass_mean_inverse_sq (n l : ℕ) {p : ℝ} (hp : p ∈ Set.Ioo 0 1)
-    (hl : l ≤ n) (hmean : (n : ℝ) * p = l) (hv : 1 ≤ (n : ℝ) * p * (1 - p)) :
-    (binomialMass n p l)⁻¹ ^ 2 ≤ 144 * ((n : ℝ) * p * (1 - p)) := by
-  have hs : 0 < Real.sqrt ((n : ℝ) * p * (1 - p)) :=
-    Real.sqrt_pos.mpr (by linarith)
-  have hlow := binomialMass_mean_lower n l hp hl hmean hv
-  have hpos : 0 < binomialMass n p l := lt_of_lt_of_le (by positivity) hlow
-  have hi : (binomialMass n p l)⁻¹ ≤ 12 * Real.sqrt ((n : ℝ) * p * (1 - p)) := by
-    rw [← one_div]
-    apply (div_le_iff₀ hpos).mpr
-    have := (div_le_iff₀ (by positivity)).mp hlow
-    nlinarith only [this]
-  have his := sq_le_sq₀ (by positivity) (by positivity) |>.mpr hi
-  have hs2 := Real.sq_sqrt (show 0 ≤ (n : ℝ) * p * (1 - p) by linarith)
-  nlinarith only [his, hs2]
-
-theorem seriesL1_smul (a : ℝ) (f : IntSeries) :
-    seriesL1 (a • f) = |a| * seriesL1 f := by
-  unfold seriesL1
-  simp only [AddMonoidAlgebra.coeff_smul, Finsupp.smul_apply, smul_eq_mul, abs_mul]
-  exact tsum_mul_left
-
-theorem mixedFairSeries_difference_L1_bound (n : ℕ) {p : ℝ} (hp : p ∈ Set.Ioc 0 1) :
-    seriesL1 (firstDifference * mixedFairSeries n p) ≤
-      2 * Real.sqrt (2 / (((n : ℝ) + 1) * p)) := by
-  rw [mixedFairSeries_expansion, Finset.mul_sum]
-  simp_rw [mul_smul_comm]
-  calc
-    _ ≤ ∑ k ∈ Finset.range (n + 1),
-        seriesL1 (binomialMass n p k • (firstDifference * fairSeries k)) := seriesL1_sum_le _ _
-    _ ≤ ∑ k ∈ Finset.range (n + 1), binomialMass n p k * (2 * binomialBound k) := by
-      apply Finset.sum_le_sum
-      intro k _
-      have hw := binomialMass_nonneg n k ⟨hp.1.le, hp.2⟩
-      rw [seriesL1_smul, abs_of_nonneg hw]
-      exact mul_le_mul_of_nonneg_left (fairSeries_difference_L1_bound k) hw
-    _ = 2 * ∑ k ∈ Finset.range (n + 1), binomialMass n p k * binomialBound k := by
-      rw [Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro k _; ring
-    _ ≤ _ := mul_le_mul_of_nonneg_left (mixed_binomialBound_le n hp) (by norm_num)
-
-theorem biasedSeries_difference_L1_bound_low (n : ℕ) {p : ℝ}
-    (hp : p ∈ Set.Ioc 0 (1 / 2)) :
-    seriesL1 (firstDifference * biasedSeries n p) ≤
-      2 * Real.sqrt (1 / (((n : ℝ) + 1) * p)) := by
-  have h := mixedFairSeries_difference_L1_bound n
-    (show 2 * p ∈ Set.Ioc 0 1 from ⟨by linarith [hp.1], by linarith [hp.2]⟩)
-  rw [mixedFairSeries_eq_biased] at h
-  have hp2 : 2 * p / 2 = p := by ring
-  have heq : 2 / (((n : ℝ) + 1) * (2 * p)) = 1 / (((n : ℝ) + 1) * p) := by
-    have hp0 : p ≠ 0 := ne_of_gt hp.1
-    have hn0 : ((n : ℝ) + 1) ≠ 0 := by positivity
-    field_simp
-  rwa [hp2, heq] at h
-
-theorem biasedSeries_difference_reflect (n : ℕ) (p : ℝ) (z : ℤ) :
-    (firstDifference * biasedSeries n (1 - p)).coeff z =
-      -(firstDifference * biasedSeries n p).coeff ((n : ℤ) + 1 - z) := by
-  rw [firstDifference_coeff, firstDifference_coeff]
-  have h1 := biasedSeries_coeff_reflect n p ((n : ℤ) - z)
-  have h2 := biasedSeries_coeff_reflect n p ((n : ℤ) - (z - 1))
-  have hz1 : (n : ℤ) - ((n : ℤ) - z) = z := by ring
-  have hz2 : (n : ℤ) - ((n : ℤ) - (z - 1)) = z - 1 := by ring
-  rw [hz1] at h1
-  rw [hz2] at h2
-  rw [h1, h2]
-  have he1 : (n : ℤ) - (z - 1) = n + 1 - z := by ring
-  have he2 : (n : ℤ) + 1 - z - 1 = n - z := by ring
-  rw [he1, he2]
-  ring
-
-theorem biasedSeries_difference_L1_reflect (n : ℕ) (p : ℝ) :
-    seriesL1 (firstDifference * biasedSeries n (1 - p)) =
-      seriesL1 (firstDifference * biasedSeries n p) := by
-  unfold seriesL1
-  simp_rw [biasedSeries_difference_reflect, abs_neg]
-  exact (Equiv.subLeft ((n : ℤ) + 1)).tsum_eq
-    (fun z : ℤ => |(firstDifference * biasedSeries n p).coeff z|)
-
-theorem biasedSeries_difference_L1_bound (n : ℕ) {p : ℝ} (hp : p ∈ Set.Ioo 0 1) :
-    seriesL1 (firstDifference * biasedSeries n p) ≤
-      2 * Real.sqrt (1 / (((n : ℝ) + 1) * p * (1 - p))) := by
-  by_cases hlow : p ≤ 1 / 2
-  · apply (biasedSeries_difference_L1_bound_low n ⟨hp.1, hlow⟩).trans
-    apply mul_le_mul_of_nonneg_left _ (by norm_num)
-    apply Real.sqrt_le_sqrt
-    apply one_div_le_one_div_of_le
-    · exact mul_pos (mul_pos (by positivity) hp.1) (sub_pos.mpr hp.2)
-    · nlinarith [mul_nonneg (show (0 : ℝ) ≤ n + 1 by positivity) (sq_nonneg p)]
-  · have hq : 1 - p ∈ Set.Ioc 0 (1 / 2) := ⟨by linarith [hp.2], by linarith⟩
-    rw [← biasedSeries_difference_L1_reflect n p]
-    apply (biasedSeries_difference_L1_bound_low n hq).trans
-    apply mul_le_mul_of_nonneg_left _ (by norm_num)
-    apply Real.sqrt_le_sqrt
-    apply one_div_le_one_div_of_le
-    · exact mul_pos (mul_pos (by positivity) hp.1) (sub_pos.mpr hp.2)
-    · nlinarith [mul_nonneg (show (0 : ℝ) ≤ n + 1 by positivity)
-        (mul_nonneg (sub_nonneg.mpr hp.2.le) (sub_nonneg.mpr hp.2.le))]
-
-
-def biasedBinomialBound (n : ℕ) (p : ℝ) : ℝ :=
-  Real.sqrt (1 / (((n : ℝ) + 1) * p * (1 - p)))
-
-theorem biasedSeries_L1 (n : ℕ) {p : ℝ} (hp : p ∈ Set.Icc 0 1) :
-    seriesL1 (biasedSeries n p) = 1 := by
-  rw [seriesL1_eq_sum_range _ n (biasedSeries_coeff_neg n p)]
-  · simp_rw [biasedSeries_coeff_nat, abs_of_nonneg (binomialMass_nonneg _ _ hp)]
-    exact sum_binomialMass n p
-  · intro k hk
-    exact biasedSeries_coeff_high n p k (by exact_mod_cast hk)
-
-theorem biasedSeries_L2Sq_bound (n : ℕ) {p : ℝ} (hp : p ∈ Set.Ioo 0 1) :
-    seriesL2Sq (biasedSeries n p) ≤ biasedBinomialBound n p := by
-  have := seriesL2Sq_le_L1_mul_bound (biasedSeries n p) _ (biasedSeries_coeff_bound n hp)
-  simpa only [biasedSeries_L1 n ⟨hp.1.le, hp.2.le⟩, one_mul, biasedBinomialBound] using this
-
-theorem biasedSeries_add (m n : ℕ) (p : ℝ) :
-    biasedSeries (m + n) p = biasedSeries m p * biasedSeries n p := by
-  exact pow_add _ _ _
-
-theorem biasedSeries_mul_power (n s : ℕ) (p : ℝ) :
-    firstDifference ^ s * biasedSeries ((s + 1) * n) p =
-      (firstDifference * biasedSeries n p) ^ s * biasedSeries n p := by
-  simp only [biasedSeries, add_mul, one_mul, pow_add, pow_mul, mul_pow]
-  ring
-
-theorem biased_difference_coeff_bound (n s : ℕ) {p : ℝ} (hp : p ∈ Set.Ioo 0 1) (z : ℤ) :
-    |(firstDifference ^ s * biasedSeries ((s + 1) * n) p).coeff z| ≤
-      (2 * biasedBinomialBound n p) ^ s * biasedBinomialBound n p := by
-  rw [biasedSeries_mul_power]
-  apply (series_coeff_mul_bound _ _ _ (biasedSeries_coeff_bound n hp) z).trans
-  apply mul_le_mul_of_nonneg_right _ (Real.sqrt_nonneg _)
-  exact (seriesL1_pow_le _ _).trans
-    (pow_le_pow_left₀ (seriesL1_nonneg _) (biasedSeries_difference_L1_bound n hp) s)
-
-theorem biased_difference_L2Sq_bound (n s : ℕ) {p : ℝ} (hp : p ∈ Set.Ioo 0 1) :
-    seriesL2Sq (firstDifference ^ s * biasedSeries ((s + 1) * n) p) ≤
-      ((2 * biasedBinomialBound n p) ^ s) ^ 2 * biasedBinomialBound n p := by
-  rw [biasedSeries_mul_power]
-  apply (seriesL2Sq_mul_le _ _).trans
-  apply mul_le_mul
-  · apply pow_le_pow_left₀ (seriesL1_nonneg _)
-    exact (seriesL1_pow_le _ _).trans
-      (pow_le_pow_left₀ (seriesL1_nonneg _) (biasedSeries_difference_L1_bound n hp) s)
-  · exact biasedSeries_L2Sq_bound n hp
-  · exact seriesL2Sq_nonneg _
-  · exact sq_nonneg _
-
-theorem biased_difference_coeff_bound_remainder (n s r : ℕ) {p : ℝ}
-    (hp : p ∈ Set.Ioo 0 1) (z : ℤ) :
-    |(firstDifference ^ s * biasedSeries ((s + 1) * n + r) p).coeff z| ≤
-      (2 * biasedBinomialBound n p) ^ s * biasedBinomialBound n p := by
-  rw [biasedSeries_add, ← mul_assoc, mul_comm (firstDifference ^ s * _)]
-  simpa only [biasedSeries_L1 r ⟨hp.1.le, hp.2.le⟩, one_mul] using
-    series_coeff_mul_bound (biasedSeries r p) (firstDifference ^ s * biasedSeries ((s + 1) * n) p)
-      _ (biased_difference_coeff_bound n s hp) z
-
-theorem biased_difference_L2Sq_bound_remainder (n s r : ℕ) {p : ℝ}
-    (hp : p ∈ Set.Ioo 0 1) :
-    seriesL2Sq (firstDifference ^ s * biasedSeries ((s + 1) * n + r) p) ≤
-      ((2 * biasedBinomialBound n p) ^ s) ^ 2 * biasedBinomialBound n p := by
-  rw [biasedSeries_add, ← mul_assoc, mul_comm (firstDifference ^ s * _)]
-  calc
-    _ ≤ seriesL1 (biasedSeries r p) ^ 2 *
-        seriesL2Sq (firstDifference ^ s * biasedSeries ((s + 1) * n) p) := seriesL2Sq_mul_le _ _
-    _ ≤ _ := by simpa only [biasedSeries_L1 r ⟨hp.1.le, hp.2.le⟩, one_pow, one_mul] using
-        biased_difference_L2Sq_bound n s hp
-
-/-- All fixed-order nonfair binomial differences, at the exact variance scale.
-This is the analytic estimate needed for the initial conditional rank kernels. -/
-theorem biased_difference_smoothing (g s : ℕ) {p : ℝ} (hp : p ∈ Set.Ioo 0 1) :
-    (∀ z : ℤ, |(firstDifference ^ s * biasedSeries g p).coeff z| ≤
-      (2 * biasedBinomialBound (g / (s + 1)) p) ^ s *
-        biasedBinomialBound (g / (s + 1)) p) ∧
-    (seriesL2Sq (firstDifference ^ s * biasedSeries g p) ≤
-      ((2 * biasedBinomialBound (g / (s + 1)) p) ^ s) ^ 2 *
-        biasedBinomialBound (g / (s + 1)) p) := by
-  have hg : (s + 1) * (g / (s + 1)) + g % (s + 1) = g := Nat.div_add_mod g (s + 1)
-  constructor
-  · intro z
-    simpa only [hg] using biased_difference_coeff_bound_remainder (g / (s + 1)) s
-      (g % (s + 1)) hp z
-  · simpa only [hg] using biased_difference_L2Sq_bound_remainder (g / (s + 1)) s
-      (g % (s + 1)) hp
-
-theorem biasedBinomialBound_quotient_le (g s : ℕ) {p : ℝ} (hp : p ∈ Set.Ioo 0 1) :
-    biasedBinomialBound (g / (s + 1)) p ≤
-      Real.sqrt (((s : ℝ) + 1) / (((g : ℝ) + 1) * p * (1 - p))) := by
-  unfold biasedBinomialBound
-  apply Real.sqrt_le_sqrt
-  have hpq : 0 < p * (1 - p) := mul_pos hp.1 (sub_pos.mpr hp.2)
-  have hnat : g + 1 ≤ (s + 1) * (g / (s + 1) + 1) := by
-    have hdiv := Nat.div_add_mod g (s + 1)
-    have hmod : g % (s + 1) < s + 1 := Nat.mod_lt g (by omega)
-    calc
-      g + 1 = (s + 1) * (g / (s + 1)) + g % (s + 1) + 1 := by omega
-      _ ≤ (s + 1) * (g / (s + 1)) + (s + 1) := by omega
-      _ = _ := by ring
-  have hreal : (g : ℝ) + 1 ≤ ((s : ℝ) + 1) * ((g / (s + 1) : ℕ) + 1 : ℝ) := by
-    exact_mod_cast hnat
-  have hp0 := hp.1
-  have hq0 := sub_pos.mpr hp.2
-  apply (div_le_div_iff₀ (by positivity) (by positivity)).mpr
-  nlinarith only [mul_le_mul_of_nonneg_right hreal hpq.le]
-
-/-! ## Gaussian overlap by exponential tilting of the exact binomial law -/
-
-theorem binomialMass_tilt (n k : ℕ) {p a : ℝ} (hp : p ∈ Set.Ioo 0 1)
-    (ha : 0 < a) :
-    binomialMass n p k * a ^ k =
-      (1 - p + p * a) ^ n * binomialMass n (p * a / (1 - p + p * a)) k := by
-  by_cases hk : k ≤ n
-  · have hA : 0 < 1 - p + p * a := by positivity [sub_pos.mpr hp.2, hp.1]
-    have hsub : 1 - p * a / (1 - p + p * a) = (1 - p) / (1 - p + p * a) := by
-      field_simp; ring
-    unfold binomialMass
-    rw [hsub, div_pow, div_pow, mul_pow]
-    have hpow : (1 - p + p * a) ^ n =
-        (1 - p + p * a) ^ k * (1 - p + p * a) ^ (n - k) := by
-      rw [← pow_add, Nat.add_sub_of_le hk]
-    rw [hpow]
-    field_simp
-  · simp [binomialMass, Nat.choose_eq_zero_of_lt (by omega : n < k)]
-
-theorem binomialMass_tilt_product (g h k : ℕ) {p a : ℝ} (hp : p ∈ Set.Ioo 0 1)
-    (ha : 0 < a) :
-    binomialMass g p k * binomialMass h p k =
-      ((1 - p + p * a) ^ g * (1 - p + p / a) ^ h) *
-        (binomialMass g (p * a / (1 - p + p * a)) k *
-          binomialMass h ((p / a) / (1 - p + p / a)) k) := by
-  have h1 := binomialMass_tilt g k hp ha
-  have h2 := binomialMass_tilt h k hp (inv_pos.mpr ha)
-  have hid := congrArg₂ (· * ·) h1 h2
-  simp only [inv_pow, ← div_eq_mul_inv] at hid
-  have ha0 : a ^ k ≠ 0 := pow_ne_zero _ ha.ne'
-  calc
-    _ = (binomialMass g p k * a ^ k) * (binomialMass h p k / a ^ k) := by field_simp
-    _ = _ := by convert hid using 1; ring
-
-/-- Tilting on opposite sides retains an anti-concentration factor as well
-as the Gaussian penalty for different numbers of trials. -/
-theorem binomial_overlap_tilt_bound (g h : ℕ) {p a : ℝ}
-    (hp : p ∈ Set.Ioo 0 1) (ha : 0 < a) :
-    (∑ k ∈ Finset.range (h + 1), binomialMass g p k * binomialMass h p k) ≤
-      ((1 - p + p * a) ^ g * (1 - p + p / a) ^ h) *
-        biasedBinomialBound g (p * a / (1 - p + p * a)) := by
-  have hA : 0 < 1 - p + p * a := by positivity [sub_pos.mpr hp.2, hp.1]
-  have hB : 0 < 1 - p + p / a := by positivity [sub_pos.mpr hp.2, hp.1]
-  have hpA : p * a / (1 - p + p * a) ∈ Set.Ioo (0 : ℝ) 1 := by
-    constructor
-    · positivity [hp.1]
-    · apply (div_lt_one hA).mpr; linarith [hp.2]
-  have hpB : (p / a) / (1 - p + p / a) ∈ Set.Ioo (0 : ℝ) 1 := by
-    constructor
-    · positivity [hp.1]
-    · apply (div_lt_one hB).mpr; linarith [hp.2]
-  simp_rw [binomialMass_tilt_product g h _ hp ha, ← Finset.mul_sum]
-  apply mul_le_mul_of_nonneg_left _ (by positivity)
-  calc
-    _ ≤ ∑ k ∈ Finset.range (h + 1),
-        biasedBinomialBound g (p * a / (1 - p + p * a)) *
-          binomialMass h ((p / a) / (1 - p + p / a)) k := by
-      apply Finset.sum_le_sum
-      intro k _
-      apply mul_le_mul_of_nonneg_right _ (binomialMass_nonneg _ _ ⟨hpB.1.le, hpB.2.le⟩)
-      have hb := biasedSeries_coeff_bound g hpA (k : ℤ)
-      rw [biasedSeries_coeff_nat, abs_of_nonneg (binomialMass_nonneg _ _ ⟨hpA.1.le, hpA.2.le⟩)] at hb
-      exact hb
-    _ = _ := by rw [← Finset.mul_sum, sum_binomialMass, mul_one]
-
-
-theorem binomial_tilt_variance {p t : ℝ} (hp : p ∈ Set.Ioo 0 1)
-    (ht : t ∈ Set.Icc 0 (1 / 4 : ℝ)) :
-    p * (1 - p) / 2 ≤
-      (p * (1 - t) / (1 - p + p * (1 - t))) *
-        (1 - p * (1 - t) / (1 - p + p * (1 - t))) := by
-  have hA : 0 < 1 - p + p * (1 - t) := by
-    have := hp.1; have := hp.2; have := ht.2
-    positivity [sub_pos.mpr hp.2, show (0 : ℝ) < 1 - t by linarith]
-  have hA1 : 1 - p + p * (1 - t) ≤ 1 := by nlinarith [mul_nonneg hp.1.le ht.1]
-  have hAsq : (1 - p + p * (1 - t)) ^ 2 ≤ 1 := by nlinarith
-  have hpq : 0 < p * (1 - p) := mul_pos hp.1 (sub_pos.mpr hp.2)
-  have heq : (p * (1 - t) / (1 - p + p * (1 - t))) *
-        (1 - p * (1 - t) / (1 - p + p * (1 - t))) =
-      p * (1 - p) * (1 - t) / (1 - p + p * (1 - t)) ^ 2 := by
-    field_simp; ring
-  rw [heq]
-  apply (le_div_iff₀ (sq_pos_of_pos hA)).mpr
-  have hh := mul_le_mul_of_nonneg_left hAsq hpq.le
-  nlinarith [ht.2]
-
-theorem binomial_tilt_normalizer (g h : ℕ) {p : ℝ}
-    (hp : p ∈ Set.Ico (1 / 2 : ℝ) 1) (hhg : h ≤ g) (hS : 0 < g + h) :
-    let t : ℝ := ((g : ℝ) - h) / (4 * ((g : ℝ) + h))
-    ((1 - p + p * (1 - t)) ^ g * (1 - p + p / (1 - t)) ^ h) ≤
-      Real.exp (-((g : ℝ) - h) ^ 2 / (16 * ((g : ℝ) + h))) := by
-  dsimp only
-  set t : ℝ := ((g : ℝ) - h) / (4 * ((g : ℝ) + h)) with htdef
-  have hSr : (0 : ℝ) < (g : ℝ) + h := by exact_mod_cast hS
-  have hgr : (h : ℝ) ≤ g := by exact_mod_cast hhg
-  have hd : (0 : ℝ) ≤ (g : ℝ) - h := sub_nonneg.mpr hgr
-  have ht0 : 0 ≤ t := div_nonneg hd (by positivity)
-  have ht1 : t ≤ 1 / 4 := by
-    apply (div_le_iff₀ (by positivity : (0 : ℝ) < 4 * ((g : ℝ) + h))).mpr
-    nlinarith [show (0 : ℝ) ≤ h by positivity]
-  have htlt : 0 < 1 - t := by linarith
-  have hp0 : 0 < p := by linarith [hp.1]
-  have hA : 0 < 1 - p + p * (1 - t) := by positivity [sub_pos.mpr hp.2]
-  have hB : 0 < 1 - p + p / (1 - t) := by positivity [sub_pos.mpr hp.2]
-  have hlogA : Real.log (1 - p + p * (1 - t)) ≤ -p * t := by
-    have := Real.log_le_sub_one_of_pos hA
-    nlinarith only [this]
-  have hlogB : Real.log (1 - p + p / (1 - t)) ≤ p * t / (1 - t) := by
-    have := Real.log_le_sub_one_of_pos hB
-    have heq : 1 - p + p / (1 - t) - 1 = p * t / (1 - t) := by field_simp; ring
-    rwa [heq] at this
-  have hlog : Real.log ((1 - p + p * (1 - t)) ^ g * (1 - p + p / (1 - t)) ^ h) ≤
-      -(p * ((g : ℝ) - h) * t) + 2 * p * h * t ^ 2 := by
-    rw [Real.log_mul (pow_pos hA _).ne' (pow_pos hB _).ne', Real.log_pow, Real.log_pow]
-    have hb : (h : ℝ) * (p * t / (1 - t)) ≤ (h : ℝ) * (p * t + 2 * p * t ^ 2) := by
-      apply mul_le_mul_of_nonneg_left _ (by positivity)
-      apply (div_le_iff₀ htlt).mpr
-      have hh := mul_nonneg (mul_nonneg hp0.le (sq_nonneg t)) (show 0 ≤ 1 - 2 * t by linarith)
-      nlinarith only [hh]
-    have ha := mul_le_mul_of_nonneg_left hlogA (show (0 : ℝ) ≤ g by positivity)
-    have hb' := mul_le_mul_of_nonneg_left hlogB (show (0 : ℝ) ≤ h by positivity)
-    linarith
-  have hht : 4 * (h : ℝ) * t ≤ (g : ℝ) - h := by
-    rw [htdef, ← mul_div_assoc]
-    apply (div_le_iff₀ (by positivity : (0 : ℝ) < 4 * ((g : ℝ) + h))).mpr
-    have hm := mul_nonneg (show (0 : ℝ) ≤ g by positivity) hd
-    nlinarith only [hm]
-  have hhalf : -(p * ((g : ℝ) - h) * t) + 2 * p * h * t ^ 2 ≤
-      -(p * ((g : ℝ) - h) * t) / 2 := by
-    have hm := mul_le_mul_of_nonneg_right hht (mul_nonneg hp0.le ht0)
-    nlinarith only [hm]
-  have hlast : -(p * ((g : ℝ) - h) * t) / 2 ≤
-      -((g : ℝ) - h) ^ 2 / (16 * ((g : ℝ) + h)) := by
-    rw [htdef]
-    apply (div_le_div_iff₀ (by norm_num : (0 : ℝ) < 2) (by positivity)).mpr
-    field_simp
-    have hm := mul_nonneg (show 0 ≤ 2 * p - 1 by linarith [hp.1]) (sq_nonneg ((g : ℝ) - h))
-    nlinarith only [hm]
-  have hexp := Real.exp_le_exp.mpr (hlog.trans (hhalf.trans hlast))
-  rwa [Real.exp_log (mul_pos (pow_pos hA _) (pow_pos hB _))] at hexp
-
-
-/-- The Gaussian cross-overlap estimate used by the conditional rank-kernel
-square-sum argument. The constant is absolute, also as p tends to one. -/
-theorem binomial_gaussian_overlap (g h : ℕ) {p : ℝ}
-    (hp : p ∈ Set.Ico (1 / 2 : ℝ) 1) (hhg : h ≤ g) (hS : 0 < g + h) :
-    (∑ k ∈ Finset.range (h + 1), binomialMass g p k * binomialMass h p k) ≤
-      Real.sqrt (2 / (((g : ℝ) + 1) * p * (1 - p))) *
-        Real.exp (-((g : ℝ) - h) ^ 2 / (16 * ((g : ℝ) + h))) := by
-  let t : ℝ := ((g : ℝ) - h) / (4 * ((g : ℝ) + h))
-  have hSr : (0 : ℝ) < (g : ℝ) + h := by exact_mod_cast hS
-  have hgr : (h : ℝ) ≤ g := by exact_mod_cast hhg
-  have ht0 : 0 ≤ t := div_nonneg (sub_nonneg.mpr hgr) (by positivity)
-  have ht1 : t ≤ 1 / 4 := by
-    apply (div_le_iff₀ (by positivity : (0 : ℝ) < 4 * ((g : ℝ) + h))).mpr
-    nlinarith [show (0 : ℝ) ≤ h by positivity]
-  have htlt : 0 < 1 - t := by linarith
-  have hp0 : 0 < p := by linarith [hp.1]
-  have hp' : p ∈ Set.Ioo (0 : ℝ) 1 := ⟨hp0, hp.2⟩
-  have hA : 0 < 1 - p + p * (1 - t) := by positivity [sub_pos.mpr hp.2]
-  have hB : 0 < 1 - p + p / (1 - t) := by positivity [sub_pos.mpr hp.2]
-  have hv := binomial_tilt_variance hp' ⟨ht0, ht1⟩
-  have hpq : 0 < p * (1 - p) := mul_pos hp0 (sub_pos.mpr hp.2)
-  have hx : 0 < (p * (1 - t) / (1 - p + p * (1 - t))) *
-      (1 - p * (1 - t) / (1 - p + p * (1 - t))) := lt_of_lt_of_le (by positivity) hv
-  have hb : biasedBinomialBound g (p * (1 - t) / (1 - p + p * (1 - t))) ≤
-      Real.sqrt (2 / (((g : ℝ) + 1) * p * (1 - p))) := by
-    apply Real.sqrt_le_sqrt
-    have hg : (0 : ℝ) < (g : ℝ) + 1 := by positivity
-    apply (div_le_div_iff₀ (by nlinarith only [mul_pos hg hx]) (by nlinarith only [mul_pos hg hpq])).mpr
-    have hm := mul_le_mul_of_nonneg_left hv hg.le
-    nlinarith only [hm]
-  have hn := binomial_tilt_normalizer g h hp hhg hS
-  change ((1 - p + p * (1 - t)) ^ g * (1 - p + p / (1 - t)) ^ h) ≤ _ at hn
-  have he := binomial_overlap_tilt_bound g h hp' htlt
-  apply he.trans
-  have hbound := mul_le_mul hn hb (Real.sqrt_nonneg _) (Real.exp_nonneg _)
-  simpa only [mul_comm] using hbound
-
-
-theorem sqrt_times_gaussian_le {x : ℝ} (hx : 0 ≤ x) :
-    Real.sqrt (x + 1) * Real.exp (-x / 96) ≤ 16 := by
-  have he : Real.exp (-x / 96) ^ 2 = 1 / Real.exp (x / 48) := by
-    rw [pow_two, ← Real.exp_add]
-    rw [show -x / 96 + -x / 96 = -(x / 48) by ring, Real.exp_neg, one_div]
-  have hs : (Real.sqrt (x + 1) * Real.exp (-x / 96)) ^ 2 ≤ 256 := by
-    rw [mul_pow, Real.sq_sqrt (by linarith), he, mul_one_div]
-    apply (div_le_iff₀ (Real.exp_pos _)).mpr
-    have hh := Real.add_one_le_exp (x / 48)
-    nlinarith only [hh, hx]
-  nlinarith only [hs]
-
-theorem sqrt_recip_product_le {a b : ℝ} (ha : 0 < a) (hb : 0 < b) (hab : a ≤ 2 * b) :
-    Real.sqrt (1 / a) * Real.sqrt (1 / b) ≤ 2 / a := by
-  rw [← Real.sqrt_mul (by positivity), Real.sqrt_le_iff]
-  refine ⟨by positivity, ?_⟩
-  rw [div_pow]
-  apply (le_div_iff₀ (sq_pos_of_pos ha)).mpr
-  field_simp
-  nlinarith
-
-theorem binomialMass_le_bound (n k : ℕ) {p : ℝ} (hp : p ∈ Set.Ioo 0 1) :
-    binomialMass n p k ≤ biasedBinomialBound n p := by
-  have h := biasedSeries_coeff_bound n hp (k : ℤ)
-  rw [biasedSeries_coeff_nat, abs_of_nonneg (binomialMass_nonneg _ _ ⟨hp.1.le, hp.2.le⟩)] at h
-  exact h
-
-/-- A uniform product-supremum estimate, including very unequal trial
-counts. This justifies eliminating a largest gap in the rank kernel. -/
-theorem binomial_product_sup (g h k : ℕ) {p : ℝ}
-    (hp : p ∈ Set.Ico (1 / 2 : ℝ) 1) :
-    binomialMass g p k * binomialMass h p k ≤
-      32 / (((g : ℝ) + 1) * p * (1 - p)) := by
-  have hp0 : 0 < p := by linarith [hp.1]
-  have hp' : p ∈ Set.Ioo (0 : ℝ) 1 := ⟨hp0, hp.2⟩
-  have hpq : 0 < p * (1 - p) := mul_pos hp0 (sub_pos.mpr hp.2)
-  have hpq1 : p * (1 - p) ≤ 1 := by nlinarith only [sq_nonneg (p - 1 / 2)]
-  have hden : 0 < ((g : ℝ) + 1) * p * (1 - p) := by nlinarith only [mul_pos (show (0 : ℝ) < (g : ℝ) + 1 by positivity) hpq]
-  by_cases hnear : g ≤ 2 * h
-  · have hdenh : 0 < ((h : ℝ) + 1) * p * (1 - p) := by nlinarith only [mul_pos (show (0 : ℝ) < (h : ℝ) + 1 by positivity) hpq]
-    have hgh : ((g : ℝ) + 1) * p * (1 - p) ≤ 2 * (((h : ℝ) + 1) * p * (1 - p)) := by
-      have hcast : (g : ℝ) ≤ 2 * h := by exact_mod_cast hnear
-      have hm := mul_le_mul_of_nonneg_right (show (g : ℝ) + 1 ≤ 2 * ((h : ℝ) + 1) by linarith) hpq.le
-      nlinarith only [hm]
-    calc
-      _ ≤ biasedBinomialBound g p * biasedBinomialBound h p :=
-        mul_le_mul (binomialMass_le_bound g k hp') (binomialMass_le_bound h k hp')
-          (binomialMass_nonneg _ _ ⟨hp0.le, hp.2.le⟩) (Real.sqrt_nonneg _)
-      _ ≤ 2 / (((g : ℝ) + 1) * p * (1 - p)) := sqrt_recip_product_le hden hdenh hgh
-      _ ≤ _ := div_le_div_of_nonneg_right (by norm_num) hden.le
-  · by_cases hk : k ≤ h
-    · have hhg : h ≤ g := by omega
-      have hS : 0 < g + h := by omega
-      have hsum : binomialMass g p k * binomialMass h p k ≤
-          ∑ j ∈ Finset.range (h + 1), binomialMass g p j * binomialMass h p j := by
-        exact Finset.single_le_sum (s := Finset.range (h + 1)) (a := k) (fun j _ => mul_nonneg
-          (binomialMass_nonneg g j ⟨hp0.le, hp.2.le⟩)
-          (binomialMass_nonneg h j ⟨hp0.le, hp.2.le⟩)) (Finset.mem_range.mpr (by omega))
-      have hgauss := binomial_gaussian_overlap g h hp hhg hS
-      have hSr : (0 : ℝ) < (g : ℝ) + h := by exact_mod_cast hS
-      have hfar : 2 * (h : ℝ) < g := by exact_mod_cast (show 2 * h < g by omega)
-      have hexp : Real.exp (-((g : ℝ) - h) ^ 2 / (16 * ((g : ℝ) + h))) ≤ Real.exp (-(g : ℝ) / 96) := by
-        apply Real.exp_le_exp.mpr
-        apply (div_le_div_iff₀ (by positivity : (0 : ℝ) < 16 * ((g : ℝ) + h)) (by norm_num : (0 : ℝ) < 96)).mpr
-        have hn := mul_nonneg (show 0 ≤ (g : ℝ) - 2 * h by linarith) (show 0 ≤ 5 * (g : ℝ) - 3 * h by linarith)
-        nlinarith only [hn, sq_nonneg (h : ℝ)]
-      have hcoeff : Real.sqrt (2 / (((g : ℝ) + 1) * p * (1 - p))) ≤
-          (2 / (((g : ℝ) + 1) * p * (1 - p))) * Real.sqrt ((g : ℝ) + 1) := by
-        rw [Real.sqrt_le_iff]
-        refine ⟨by positivity, ?_⟩
-        rw [mul_pow, Real.sq_sqrt (by positivity), div_pow]
-        have hgpos : (0 : ℝ) < (g : ℝ) + 1 := by positivity
-        have hqpos : 0 < 1 - p := sub_pos.mpr hp.2
-        field_simp
-        have hm := mul_le_mul_of_nonneg_left hpq1 (show (0 : ℝ) ≤ (g : ℝ) + 1 by positivity)
-        nlinarith only [hm, hpq1]
-      calc
-        _ ≤ Real.sqrt (2 / (((g : ℝ) + 1) * p * (1 - p))) *
-            Real.exp (-((g : ℝ) - h) ^ 2 / (16 * ((g : ℝ) + h))) := hsum.trans hgauss
-        _ ≤ ((2 / (((g : ℝ) + 1) * p * (1 - p))) * Real.sqrt ((g : ℝ) + 1)) *
-            Real.exp (-(g : ℝ) / 96) := mul_le_mul hcoeff hexp (Real.exp_nonneg _) (by positivity)
-        _ ≤ (2 / (((g : ℝ) + 1) * p * (1 - p))) * 16 := by
-          rw [mul_assoc]
-          exact mul_le_mul_of_nonneg_left (sqrt_times_gaussian_le (by positivity)) (by positivity)
-        _ = _ := by ring
-    · simp [binomialMass, Nat.choose_eq_zero_of_lt (by omega : h < k)]
-      positivity
-
-end TraceReconstruction
-
-namespace TraceReconstruction
-
-/-- Exact moment generating function, including both endpoint channels. -/
-theorem binomialMass_mgf (n : ℕ) (p t : ℝ) :
-    (∑ k ∈ Finset.range (n + 1), binomialMass n p k * Real.exp (t * k)) =
-      (1 - p + p * Real.exp t) ^ n := by
-  rw [add_comm (1 - p), add_pow]
-  apply Finset.sum_congr rfl
-  intro k _
-  rw [show t * (k : ℝ) = (k : ℝ) * t by ring, Real.exp_nat_mul, mul_pow]
-  unfold binomialMass
-  ring
-
-theorem binomialMass_mgf_le (n : ℕ) {p : ℝ} (hp : p ∈ Set.Icc 0 1) (t : ℝ) :
-    (∑ k ∈ Finset.range (n + 1), binomialMass n p k * Real.exp (t * k)) ≤
-      Real.exp ((n : ℝ) * p * (Real.exp t - 1)) := by
-  rw [binomialMass_mgf]
-  have hn : 0 ≤ 1 - p + p * Real.exp t := by positivity [sub_nonneg.mpr hp.2, hp.1]
-  have he : 1 - p + p * Real.exp t ≤ Real.exp (p * (Real.exp t - 1)) := by
-    nlinarith only [Real.add_one_le_exp (p * (Real.exp t - 1))]
-  calc
-    _ ≤ Real.exp (p * (Real.exp t - 1)) ^ n := pow_le_pow_left₀ hn he n
-    _ = _ := by rw [← Real.exp_nat_mul]; congr 1; ring
-
-/-- Exponential Markov bound for a finite nonnegative measure. -/
-theorem finite_exp_markov {α : Type*} (S : Finset α) (w X : α → ℝ)
-    (hw : ∀ a ∈ S, 0 ≤ w a) (a t : ℝ) (ht : 0 ≤ t) :
-    (∑ z ∈ S with a ≤ X z, w z) ≤
-      Real.exp (-t * a) * ∑ z ∈ S, w z * Real.exp (t * X z) := by
-  classical
-  calc
-    _ = ∑ z ∈ S with a ≤ X z, Real.exp (-t * a) * (w z * Real.exp (t * a)) := by
-      apply Finset.sum_congr rfl
-      intro z _
-      rw [mul_left_comm, ← Real.exp_add]
-      simp
-    _ ≤ ∑ z ∈ S with a ≤ X z, Real.exp (-t * a) * (w z * Real.exp (t * X z)) := by
-      apply Finset.sum_le_sum
-      intro z hz
-      have h := Finset.mem_filter.mp hz
-      exact mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_left
-        (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left h.2 ht)) (hw z h.1))
-        (Real.exp_nonneg _)
-    _ ≤ ∑ z ∈ S, Real.exp (-t * a) * (w z * Real.exp (t * X z)) := by
-      apply Finset.sum_le_sum_of_subset_of_nonneg (Finset.filter_subset _ _)
-      intro z hz _
-      positivity [hw z hz]
-    _ = _ := by rw [Finset.mul_sum]
-
-/-- A uniform multiplicative lower tail, sufficient for hidden-flag counts. -/
-theorem binomialMass_lower_half (n : ℕ) {p : ℝ} (hp : p ∈ Set.Icc 0 1) :
-    (∑ k ∈ (Finset.range (n + 1)).filter (fun k : ℕ => (k : ℝ) ≤ n * p / 2),
-      binomialMass n p k) ≤ Real.exp (-(n : ℝ) * p / 12) := by
-  have he : Real.exp (-(1 / 2 : ℝ)) ≤ 2 / 3 := by
-    rw [Real.exp_neg]
-    apply (inv_le_iff_one_le_mul₀ (Real.exp_pos _)).mpr
-    nlinarith only [Real.add_one_le_exp (1 / 2 : ℝ)]
-  have hmark := finite_exp_markov (Finset.range (n + 1)) (binomialMass n p)
-    (fun k => -(k : ℝ)) (fun k _ => binomialMass_nonneg n k hp)
-    (-(n : ℝ) * p / 2) (1 / 2) (by norm_num)
-  have hf : (Finset.range (n + 1)).filter (fun k : ℕ => -(n : ℝ) * p / 2 ≤ -(k : ℝ)) =
-      (Finset.range (n + 1)).filter (fun k : ℕ => (k : ℝ) ≤ n * p / 2) := by
-    ext k
-    simp only [Finset.mem_filter]
-    constructor <;> rintro ⟨h, h'⟩ <;> exact ⟨h, by linarith⟩
-  rw [hf] at hmark
-  calc
-    _ ≤ Real.exp (-(1 / 2 : ℝ) * (-(n : ℝ) * p / 2)) *
-        (∑ k ∈ Finset.range (n + 1), binomialMass n p k * Real.exp (-(1 / 2 : ℝ) * k)) := by
-      convert hmark using 1
-      congr 2
-      funext k
-      congr 2
-      ring
-    _ ≤ Real.exp (-(1 / 2 : ℝ) * (-(n : ℝ) * p / 2)) *
-        Real.exp ((n : ℝ) * p * (Real.exp (-(1 / 2 : ℝ)) - 1)) := by
-      gcongr
-      exact binomialMass_mgf_le n hp _
-    _ = Real.exp ((n : ℝ) * p * (Real.exp (-(1 / 2 : ℝ)) - 3 / 4)) := by
-      rw [← Real.exp_add]
-      congr 1
-      ring
-    _ ≤ _ := by
-      apply Real.exp_le_exp.mpr
-      nlinarith [mul_le_mul_of_nonneg_left he (mul_nonneg (Nat.cast_nonneg n) hp.1)]
-
-end TraceReconstruction
-
-namespace TraceReconstruction
-theorem fairBinomial_centered_mgf (n : ℕ) (t : ℝ) :
-    (∑ k ∈ Finset.range (n + 1), binomialMass n (1 / 2) k *
-      Real.exp (t * ((k : ℝ) - n / 2))) ≤ Real.exp ((n : ℝ) * t ^ 2 / 8) := by
-  have he : (1 / 2 : ℝ) + Real.exp t / 2 = Real.exp (t / 2) * Real.cosh (t / 2) := by
-    rw [Real.cosh_eq, ← mul_div_assoc, mul_add, ← Real.exp_add, ← Real.exp_add]
-    rw [show t / 2 + t / 2 = t by ring, show t / 2 + -(t / 2) = 0 by ring, Real.exp_zero]
-    ring
-  have hm : (∑ k ∈ Finset.range (n + 1), binomialMass n (1 / 2) k *
-      Real.exp (t * ((k : ℝ) - n / 2))) =
-      Real.exp (-t * n / 2) * (1 - 1 / 2 + (1 / 2) * Real.exp t) ^ n := by
-    rw [← binomialMass_mgf, Finset.mul_sum]
-    apply Finset.sum_congr rfl
-    intro k _
-    rw [mul_left_comm, ← Real.exp_add]
-    congr 2
-    ring
-  rw [hm, show (1 - 1 / 2 + (1 / 2) * Real.exp t : ℝ) = 1 / 2 + Real.exp t / 2 by ring, he, mul_pow]
-  have hb : Real.cosh (t / 2) ^ n ≤ Real.exp (t ^ 2 / 8) ^ n := by
-    apply pow_le_pow_left₀ (le_of_lt (Real.cosh_pos _))
-    convert Real.cosh_le_exp_half_sq (t / 2) using 1; congr 1; ring
-  calc
-    _ ≤ Real.exp (-t * n / 2) * (Real.exp (t / 2) ^ n * Real.exp (t ^ 2 / 8) ^ n) := by gcongr
-    _ = _ := by
-      simp only [← Real.exp_nat_mul, ← Real.exp_add]
-      congr 1
-      ring
-
-theorem fairBinomial_upper_tail (n : ℕ) (hn : 0 < n) (a : ℝ) (ha : 0 ≤ a) :
-    (∑ k ∈ (Finset.range (n + 1)).filter (fun k : ℕ => a ≤ (k : ℝ) - n / 2),
-      binomialMass n (1 / 2) k) ≤ Real.exp (-2 * a ^ 2 / n) := by
-  have hnR : (0 : ℝ) < n := by exact_mod_cast hn
-  have h := finite_exp_markov (Finset.range (n + 1)) (binomialMass n (1 / 2))
-    (fun k => (k : ℝ) - n / 2) (fun k _ => binomialMass_nonneg n k (by norm_num))
-    a (4 * a / n) (by positivity)
-  calc
-    _ ≤ Real.exp (-(4 * a / n) * a) *
-      (∑ k ∈ Finset.range (n + 1), binomialMass n (1 / 2) k *
-        Real.exp ((4 * a / n) * ((k : ℝ) - n / 2))) := h
-    _ ≤ Real.exp (-(4 * a / n) * a) * Real.exp ((n : ℝ) * (4 * a / n) ^ 2 / 8) := by
-      gcongr
-      exact fairBinomial_centered_mgf n _
-    _ = _ := by
-      rw [← Real.exp_add]
-      congr 1
-      field_simp [ne_of_gt hnR]
-      ring
-end TraceReconstruction
-
-namespace TraceReconstruction
-def hypergeomMass (V M m v : ℕ) : ℝ :=
-  if v ≤ m then (V.choose v : ℝ) * (M.choose (m-v) : ℝ) / ((V+M).choose m : ℝ) else 0
-
-theorem hypergeomMass_nonneg (V M m v : ℕ) : 0 ≤ hypergeomMass V M m v := by
-  unfold hypergeomMass
-  split <;> positivity
-
-theorem hypergeomMass_eq_binomial (V M m v : ℕ) {p : ℝ}
-    (hp : p ∈ Set.Ioo 0 1) (hm : m ≤ V+M) (hv : v ≤ m) :
-    hypergeomMass V M m v = binomialMass V p v * binomialMass M p (m-v) /
-      binomialMass (V+M) p m := by
-  unfold hypergeomMass
-  rw [ite_eq_left hv]
-  by_cases hvV : v ≤ V
-  · by_cases hkM : m-v ≤ M
-    · unfold binomialMass
-      have hc : (0 : ℝ) < (V+M).choose m := by exact_mod_cast Nat.choose_pos hm
-      have hp1 := hp.1
-      have hq1 := sub_pos.mpr hp.2
-      have he : V-v+(M-(m-v)) = V+M-m := by omega
-      have he' : v+(m-v) = m := by omega
-      apply (div_eq_div_iff (ne_of_gt hc) (by positivity)).mpr
-      calc
-        _ = ((V.choose v : ℝ) * (M.choose (m-v) : ℝ) * ((V+M).choose m : ℝ)) *
-            (p^m * (1-p)^(V+M-m)) := by ring
-        _ = _ := by
-          rw [show p^m = p^v * p^(m-v) by rw [← pow_add, he'],
-            show (1-p)^(V+M-m) = (1-p)^(V-v)*(1-p)^(M-(m-v)) by rw [← pow_add, he]]
-          ring
-    · rw [Nat.choose_eq_zero_of_lt (by omega : M < m-v)]
-      simp [binomialMass, Nat.choose_eq_zero_of_lt (by omega : M < m-v)]
-  · rw [Nat.choose_eq_zero_of_lt (by omega : V < v)]
-    simp [binomialMass, Nat.choose_eq_zero_of_lt (by omega : V < v)]
-
-theorem binomialMass_sq_upper (n k : ℕ) {p : ℝ} (hp : p ∈ Set.Ioo 0 1) :
-    binomialMass n p k ^ 2 ≤ 1 / (((n : ℝ)+1)*p*(1-p)) := by
-  have hp1 := hp.1
-  have hq1 := sub_pos.mpr hp.2
-  have h := biasedSeries_coeff_bound n hp (k : ℤ)
-  rw [biasedSeries_coeff_nat, abs_of_nonneg (binomialMass_nonneg n k ⟨hp.1.le,hp.2.le⟩)] at h
-  have hs := sq_le_sq₀ (binomialMass_nonneg n k ⟨hp.1.le,hp.2.le⟩)
-    (Real.sqrt_nonneg _) |>.mpr h
-  rwa [Real.sq_sqrt (by positivity)] at hs
-
-theorem hypergeomMass_sq_upper (V M m v : ℕ) {p : ℝ}
-    (hp : p ∈ Set.Ioo 0 1) (hm : m ≤ V+M) (hmean : (m : ℝ) = (V+M : ℕ)*p)
-    (hvar : 1 ≤ (V+M : ℕ)*p*(1-p)) :
-    hypergeomMass V M m v ^ 2 ≤
-      144 * (V+M : ℕ) / (((V : ℝ)+1)*((M : ℝ)+1)*p*(1-p)) := by
-  have hp1 := hp.1
-  have hq1 := sub_pos.mpr hp.2
-  by_cases hv : v ≤ m
-  · rw [hypergeomMass_eq_binomial V M m v hp hm hv, div_eq_mul_inv, mul_pow, mul_pow]
-    have hV := binomialMass_sq_upper V v hp
-    have hM := binomialMass_sq_upper M (m-v) hp
-    have hN := binomialMass_mean_inverse_sq (V+M) m hp hm hmean.symm hvar
-    have hvm := mul_le_mul hV hM (sq_nonneg _) (by positivity)
-    have h := mul_le_mul hvm hN (sq_nonneg _) (by positivity)
-    apply h.trans
-    have hp0 : p ≠ 0 := ne_of_gt hp.1
-    have hq0 : 1-p ≠ 0 := ne_of_gt (sub_pos.mpr hp.2)
-    have hV0 : (V : ℝ)+1 ≠ 0 := by positivity
-    have hM0 : (M : ℝ)+1 ≠ 0 := by positivity
-    apply le_of_eq
-    field_simp
-  · simp only [hypergeomMass, ite_eq_right hv, zero_pow (by decide : 2 ≠ 0)]
-    positivity
-
-theorem sum_hypergeomMass (V M m : ℕ) (hm : m ≤ V+M) :
-    ∑ v ∈ Finset.range (m+1), hypergeomMass V M m v = 1 := by
-  have hc : (0 : ℝ) < (V+M).choose m := by exact_mod_cast Nat.choose_pos hm
-  have hv := Nat.add_choose_eq V M m
-  rw [Finset.Nat.sum_antidiagonal_eq_sum_range_succ_mk] at hv
-  simp only [hypergeomMass]
-  have he : (∑ v ∈ Finset.range (m+1),
-      if v ≤ m then (V.choose v : ℝ)*(M.choose (m-v) : ℝ)/((V+M).choose m : ℝ) else 0) =
-    (∑ v ∈ Finset.range (m+1), (V.choose v : ℝ)*(M.choose (m-v) : ℝ))/((V+M).choose m : ℝ) := by
-    rw [Finset.sum_div]
-    apply Finset.sum_congr rfl
-    intro v hv'
-    rw [ite_eq_left (by simp only [Finset.mem_range] at hv'; omega)]
-  rw [he]
-  have he' : (∑ v ∈ Finset.range (m+1), (V.choose v : ℝ)*(M.choose (m-v) : ℝ)) =
-      ((V+M).choose m : ℝ) := by exact_mod_cast hv.symm
-  rw [he', div_self (ne_of_gt hc)]
-
-theorem hypergeomMass_high (V M m v : ℕ) (hv : V < v) : hypergeomMass V M m v = 0 := by
-  unfold hypergeomMass
-  rw [Nat.choose_eq_zero_of_lt hv]
-  simp
-
-/-- Adjacent-mass identity for the hidden count. The central regime has full support
-on 0,...,M; this is exactly the regime used in adjacent-power lifting. -/
-theorem hypergeom_hidden_adjacent (V M m k : ℕ) (hM : M ≤ m) (hV : m ≤ V) (hk : k < M) :
-    (((k:ℝ)+1)*((V:ℝ)-m+k+1)) * hypergeomMass M V m (k+1) =
-    (((M:ℝ)-k)*((m:ℝ)-k)) * hypergeomMass M V m k := by
-  have hkM : k ≤ M := by omega
-  have hkm : k ≤ m := by omega
-  have hkm' : k+1 ≤ m := by omega
-  have hv : m-(k+1) ≤ V := by omega
-  have h1 := Nat.choose_succ_right_eq M k
-  have h2 := Nat.choose_succ_right_eq V (m-(k+1))
-  rw [show m-(k+1)+1=m-k by omega] at h2
-  have h1' : (M.choose (k+1):ℝ)*((k:ℝ)+1) = (M.choose k:ℝ)*((M:ℝ)-k) := by
-    exact_mod_cast h1
-  have h2' : (V.choose (m-k):ℝ)*((m:ℝ)-k) =
-      (V.choose (m-(k+1)):ℝ)*((V:ℝ)-m+k+1) := by
-    have hvsub : V-(m-(k+1)) = V-m+k+1 := by omega
-    rw [hvsub] at h2
-    exact_mod_cast h2
-  unfold hypergeomMass
-  rw [ite_eq_left hkm', ite_eq_left hkm]
-  have hmul := congrArg (fun z : ℝ => z*(V.choose (m-(k+1)) : ℝ)*((V:ℝ)-m+k+1)) h1'
-  have hmul' := congrArg (fun z : ℝ => z*(M.choose k : ℝ)*((M:ℝ)-k)) h2'
-  have he : (((k:ℝ)+1)*((V:ℝ)-m+k+1)) *
-        ((M.choose (k+1):ℝ)*(V.choose (m-(k+1)):ℝ)) =
-      (((M:ℝ)-k)*((m:ℝ)-k))*((M.choose k:ℝ)*(V.choose (m-k):ℝ)) := by
-    nlinarith only [hmul, hmul']
-  simpa only [mul_div_assoc] using congrArg (fun z : ℝ => z/((M+V).choose m:ℝ)) he
-
-def hypergeomHiddenMode (V M m : ℕ) : ℕ := (M+1)*(m+1)/(V+M+2)
-
-theorem hypergeomHiddenMode_le (V M m : ℕ) (hV : m ≤ V) :
-    hypergeomHiddenMode V M m ≤ M := by
-  unfold hypergeomHiddenMode
-  have h1 : (M+1)*(m+1) < (M+1)*(V+M+2) := Nat.mul_lt_mul_of_pos_left (by omega) (by omega)
-  have hh : (M+1)*(m+1)/(V+M+2) < M+1 := by
-    apply (Nat.div_lt_iff_lt_mul (by omega)).mpr
-    exact h1
-  omega
-
-theorem hypergeom_hidden_unimodal_left (V M m k : ℕ) (hM : M ≤ m) (hV : m ≤ V)
-    (hk : k < hypergeomHiddenMode V M m) :
-    hypergeomMass M V m k ≤ hypergeomMass M V m (k+1) := by
-  have hl := hypergeomHiddenMode_le V M m hV
-  have hkM : k < M := by omega
-  have hrec := hypergeom_hidden_adjacent V M m k hM hV hkM
-  have hc : (k+1)*(V+M+2) ≤ (M+1)*(m+1) := by
-    have hh : k+1 ≤ (M+1)*(m+1)/(V+M+2) := hk
-    exact (Nat.le_div_iff_mul_le (by omega)).mp hh
-  have hc' : ((k:ℝ)+1)*((V:ℝ)+M+2) ≤ ((M:ℝ)+1)*((m:ℝ)+1) := by exact_mod_cast hc
-  have hh : ((k:ℝ)+1)*((V:ℝ)-m+k+1) ≤ ((M:ℝ)-k)*((m:ℝ)-k) := by nlinarith
-  have hm := mul_le_mul_of_nonneg_right hh (hypergeomMass_nonneg M V m k)
-  have hV' : (m:ℝ) ≤ V := by exact_mod_cast hV
-  have ha : 0 < ((k:ℝ)+1)*((V:ℝ)-m+k+1) := by positivity
-  exact (mul_le_mul_iff_right₀ ha).mp (by nlinarith only [hm, hrec])
-
-theorem hypergeom_hidden_unimodal_right (V M m k : ℕ) (hM : M ≤ m) (hV : m ≤ V)
-    (hk : hypergeomHiddenMode V M m ≤ k) :
-    hypergeomMass M V m (k+1) ≤ hypergeomMass M V m k := by
-  by_cases hkM : k < M
-  · have hrec := hypergeom_hidden_adjacent V M m k hM hV hkM
-    have hc : (M+1)*(m+1) < (k+1)*(V+M+2) := by
-      apply (Nat.div_lt_iff_lt_mul (by omega)).mp
-      exact Nat.lt_succ_of_le hk
-    have hc' : ((M:ℝ)+1)*((m:ℝ)+1) < ((k:ℝ)+1)*((V:ℝ)+M+2) := by exact_mod_cast hc
-    have hh : ((M:ℝ)-k)*((m:ℝ)-k) ≤ ((k:ℝ)+1)*((V:ℝ)-m+k+1) := by nlinarith
-    have hm := mul_le_mul_of_nonneg_right hh (hypergeomMass_nonneg M V m k)
-    have hV' : (m:ℝ) ≤ V := by exact_mod_cast hV
-    have ha : 0 < ((k:ℝ)+1)*((V:ℝ)-m+k+1) := by positivity
-    exact (mul_le_mul_iff_right₀ ha).mp (by nlinarith only [hm, hrec])
-  · rw [hypergeomMass_high M V m (k+1) (by omega)]
-    exact hypergeomMass_nonneg M V m k
-
-/-- Exact total variation of a zero-extended unimodal sequence. -/
-theorem unimodal_variation (f : ℕ → ℝ) (n l : ℕ) (hl : l ≤ n+1)
-    (hleft : ∀ k < l, f k ≤ f (k+1))
-    (hright : ∀ k, l ≤ k → f (k+1) ≤ f k) (hz : f (n+1)=0) :
-    f 0 + ∑ k ∈ Finset.range (n+1), |f (k+1)-f k| = 2*f l := by
-  rw [← Finset.sum_range_add_sum_Ico _ hl]
-  have h1 : (∑ k ∈ Finset.range l, |f (k+1)-f k|) = f l-f 0 := by
-    rw [← Finset.sum_range_sub f l]
-    apply Finset.sum_congr rfl
-    intro k hk
-    exact abs_of_nonneg (sub_nonneg.mpr (hleft k (Finset.mem_range.mp hk)))
-  have h2 : (∑ k ∈ Finset.Ico l (n+1), |f (k+1)-f k|) = f l-f (n+1) := by
-    calc
-      _ = ∑ k ∈ Finset.Ico l (n+1), -(f (k+1)-f k) := by
-        apply Finset.sum_congr rfl
-        intro k hk
-        exact abs_of_nonpos (sub_nonpos.mpr (hright k (Finset.mem_Ico.mp hk).1))
-      _ = _ := by rw [Finset.sum_neg_distrib, Finset.sum_Ico_sub _ hl]; ring
-  rw [h1,h2,hz]
-  ring
-
-theorem hypergeom_hidden_variation (V M m : ℕ) (hM : M ≤ m) (hV : m ≤ V) :
-    hypergeomMass M V m 0 +
-      ∑ k ∈ Finset.range (M+1), |hypergeomMass M V m (k+1)-hypergeomMass M V m k| =
-      2*hypergeomMass M V m (hypergeomHiddenMode V M m) := by
-  apply unimodal_variation _ M (hypergeomHiddenMode V M m)
-    ((hypergeomHiddenMode_le V M m hV).trans (by omega))
-    (fun k hk => hypergeom_hidden_unimodal_left V M m k hM hV hk)
-    (fun k hk => hypergeom_hidden_unimodal_right V M m k hM hV hk)
-  exact hypergeomMass_high M V m (M+1) (by omega)
-theorem choose_pred_weight (n r : ℕ) (hn : 0 < n) (hr : r ≤ n) :
-    (n:ℝ)*((n-1).choose r : ℝ) = ((n:ℝ)-r)*(n.choose r : ℝ) := by
-  have h := Nat.choose_mul_succ_eq (n-1) r
-  rw [Nat.sub_add_cancel hn] at h
-  have h' : ((n-1).choose r : ℝ)*(n:ℝ) = (n.choose r : ℝ)*((n:ℝ)-r) := by
-    exact_mod_cast h
-  nlinarith only [h']
-
-theorem choose_pred_weight_succ (n r : ℕ) (hn : 0 < n) :
-    (n:ℝ)*((n-1).choose r : ℝ) = ((r:ℝ)+1)*(n.choose (r+1) : ℝ) := by
-  have h := Nat.add_one_mul_choose_eq (n-1) r
-  rw [Nat.sub_add_cancel hn] at h
-  exact_mod_cast h.trans (Nat.mul_comm _ _)
-
-def hypergeomHiddenFlux (V M m k : ℕ) : ℝ :=
-  (V:ℝ)/(V+M)*hypergeomMass M (V-1) m k
-
-theorem hypergeomHiddenFlux_formula (V M m k : ℕ) (hM : M ≤ m) (hV : m < V) (hk : k ≤ M) :
-    hypergeomHiddenFlux V M m k =
-      ((V:ℝ)-m+k)/((V:ℝ)+M-m)*hypergeomMass M V m k := by
-  have hN : 0 < V+M := by omega
-  have hk' : k ≤ m := by omega
-  have hr : m-k ≤ V := by omega
-  have hmN : m ≤ V+M := by omega
-  have hpred : m ≤ V+M-1 := by omega
-  have hVN : (0:ℝ) < V+M := by exact_mod_cast hN
-  have hd : (0:ℝ) < (V:ℝ)+M-m := by
-    have hr : (m:ℝ) < (V:ℝ)+M := by exact_mod_cast (show m < V+M by omega)
-    linarith
-  have hCN : (0:ℝ) < (V+M).choose m := by exact_mod_cast Nat.choose_pos hmN
-  have hCP : (0:ℝ) < (V+M-1).choose m := by exact_mod_cast Nat.choose_pos hpred
-  have hcw := choose_pred_weight (V+M) m hN hmN
-  have hcv := choose_pred_weight V (m-k) (by omega) hr
-  push_cast [Nat.cast_sub hk'] at hcv
-  push_cast at hcw
-  unfold hypergeomHiddenFlux hypergeomMass
-  rw [ite_eq_left hk', ite_eq_left hk', show M+(V-1)=V+M-1 by omega]
-  have hden : ((V:ℝ)+M)*((V+M-1).choose m:ℝ) =
-      ((V:ℝ)+M-m)*((V+M).choose m:ℝ) := hcw
-  rw [show M+V=V+M by omega]
-  calc
-    _ = (M.choose k:ℝ)*((V:ℝ)*((V-1).choose (m-k):ℝ)) /
-        (((V:ℝ)+M)*((V+M-1).choose m:ℝ)) := by field_simp
-    _ = _ := by rw [hden, hcv]; field_simp; ring
-
-theorem hypergeomHiddenFlux_high (V M m k : ℕ) (hk : M < k) :
-    hypergeomHiddenFlux V M m k = 0 := by
-  simp only [hypergeomHiddenFlux, hypergeomMass_high M (V-1) m k hk, mul_zero]
-
-theorem hypergeomHiddenFlux_boundary (V M m : ℕ) (hM : M ≤ m) (hV : m < V) :
-    hypergeomHiddenFlux V M m M = hypergeomMass M V m M := by
-  rw [hypergeomHiddenFlux_formula V M m M hM hV le_rfl]
-  have hd : (0:ℝ) < (V:ℝ)+M-m := by
-    have hr : (m:ℝ) < (V:ℝ)+M := by exact_mod_cast (show m < V+M by omega)
-    linarith
-  have he : (V:ℝ)-m+M = (V:ℝ)+M-m := by ring
-  rw [he, div_self hd.ne', one_mul]
-
-theorem hypergeom_adjacent_hidden_difference (V M m k : ℕ)
-    (hM : M ≤ m) (hV : m < V) (hk : k ≤ M) :
-    hypergeomMass M V (m+1) (k+1) - hypergeomMass M V m k =
-      hypergeomHiddenFlux V M m (k+1) - hypergeomHiddenFlux V M m k := by
-  by_cases hkM : k = M
-  · subst k
-    rw [hypergeomMass_high M V (m+1) (M+1) (by omega),
-      hypergeomHiddenFlux_high V M m (M+1) (by omega), hypergeomHiddenFlux_boundary V M m hM hV]
-  have hklt : k < M := by omega
-  have hkm : k+1 ≤ m := by omega
-  have hkm' : k ≤ m := by omega
-  have hN : 0 < V+M := by omega
-  have hmN : m ≤ V+M := by omega
-  have hmN' : m+1 ≤ V+M := by omega
-  have hpred : m ≤ V+M-1 := by omega
-  have hN0 : (0:ℝ) < (V:ℝ)+M := by exact_mod_cast hN
-  have hCN : (0:ℝ) < (V+M).choose m := by exact_mod_cast Nat.choose_pos hmN
-  have hCN' : (0:ℝ) < (V+M).choose (m+1) := by exact_mod_cast Nat.choose_pos hmN'
-  have hCP : (0:ℝ) < (V+M-1).choose m := by exact_mod_cast Nat.choose_pos hpred
-  have hcw := choose_pred_weight (V+M) m hN hmN
-  have hcws := choose_pred_weight_succ (V+M) m hN
-  have hcv := choose_pred_weight V (m-k) (by omega) (by omega)
-  have hcvs := choose_pred_weight_succ V (m-(k+1)) (by omega)
-  rw [show m-(k+1)+1=m-k by omega] at hcvs
-  have hMrec := Nat.choose_succ_right_eq M k
-  have hMrec' : (M.choose (k+1):ℝ)*((k:ℝ)+1) = (M.choose k:ℝ)*((M:ℝ)-k) := by
-    exact_mod_cast hMrec
-  push_cast at hcw hcws
-  push_cast [Nat.cast_sub hkm'] at hcv
-  push_cast [Nat.cast_sub hkm] at hcvs
-  unfold hypergeomHiddenFlux hypergeomMass
-  rw [ite_eq_left (by omega : k+1 ≤ m+1), ite_eq_left hkm', ite_eq_left hkm, ite_eq_left hkm',
-    show m+1-(k+1)=m-k by omega, show M+V=V+M by omega,
-    show M+(V-1)=V+M-1 by omega]
-  have hpold : (M.choose k:ℝ)*(V.choose (m-k):ℝ)/( (V+M).choose m:ℝ) =
-      ((V:ℝ)+M-m)*(M.choose k:ℝ)*(V.choose (m-k):ℝ) /
-      (((V:ℝ)+M)*((V+M-1).choose m:ℝ)) := by
-    apply (div_eq_div_iff hCN.ne' (mul_pos hN0 hCP).ne').mpr
-    rw [hcw]; ring
-  have hpnew : (M.choose (k+1):ℝ)*(V.choose (m-k):ℝ)/( (V+M).choose (m+1):ℝ) =
-      ((m:ℝ)+1)*(M.choose (k+1):ℝ)*(V.choose (m-k):ℝ) /
-      (((V:ℝ)+M)*((V+M-1).choose m:ℝ)) := by
-    apply (div_eq_div_iff hCN'.ne' (mul_pos hN0 hCP).ne').mpr
-    rw [hcws]; ring
-  rw [hpold,hpnew]
-  have hF (j : ℕ) : (V:ℝ)/((V:ℝ)+M)*((M.choose j:ℝ)*((V-1).choose (m-j):ℝ)/
-      ((V+M-1).choose m:ℝ)) = (M.choose j:ℝ)*((V:ℝ)*((V-1).choose (m-j):ℝ)) /
-      (((V:ℝ)+M)*((V+M-1).choose m:ℝ)) := by field_simp
-  rw [hF, hF, hcv, hcvs]
-  rw [← sub_div, ← sub_div]
-  congr 1
-  have hh := congrArg (fun z : ℝ => z*(V.choose (m-k):ℝ)) hMrec'
-  nlinarith only [hh]
-
-
-theorem hypergeom_next_zero (V M m : ℕ) (hV : m < V) :
-    hypergeomMass M V (m+1) 0 = hypergeomHiddenFlux V M m 0 := by
-  have hN : 0 < V+M := by omega
-  have hN0 : (0:ℝ) < (V:ℝ)+M := by exact_mod_cast hN
-  have hCN : (0:ℝ) < (V+M).choose (m+1) := by exact_mod_cast Nat.choose_pos (show m+1 ≤ V+M by omega)
-  have hCP : (0:ℝ) < (V+M-1).choose m := by exact_mod_cast Nat.choose_pos (show m ≤ V+M-1 by omega)
-  have hcw := choose_pred_weight_succ (V+M) m hN
-  have hcv := choose_pred_weight_succ V m (by omega)
-  push_cast at hcw hcv
-  unfold hypergeomHiddenFlux hypergeomMass
-  simp only [Nat.zero_le, ite_true, Nat.choose_zero_right, Nat.sub_zero, Nat.cast_one, one_mul]
-  rw [show M+V=V+M by omega, show M+(V-1)=V+M-1 by omega]
-  rw [div_mul_div_comm]
-  apply (div_eq_div_iff hCN.ne' (mul_pos hN0 hCP).ne').mpr
-  have hh := congrArg (fun z : ℝ => z*(V.choose (m+1):ℝ)) hcw
-  have hh' := congrArg (fun z : ℝ => z*((V+M).choose (m+1):ℝ)) hcv
-  nlinarith only [hh,hh']
-
-/-- Exact adjacent visible-count TV, expressed with complementary hidden counts.
-The shift by one is essential: the two totals are different. -/
-def adjacentHypergeomTV (V M m : ℕ) : ℝ :=
-  (1/2:ℝ)*(hypergeomMass M V (m+1) 0 +
-    ∑ k ∈ Finset.range (M+1), |hypergeomMass M V (m+1) (k+1)-hypergeomMass M V m k|)
-
-theorem adjacentHypergeomTV_eq_flux (V M m : ℕ) (hM : M ≤ m) (hV : m < V) :
-    adjacentHypergeomTV V M m =
-      hypergeomHiddenFlux V M m (hypergeomHiddenMode (V-1) M m) := by
-  unfold adjacentHypergeomTV
-  rw [hypergeom_next_zero V M m hV]
-  have he : (∑ k ∈ Finset.range (M+1),
-      |hypergeomMass M V (m+1) (k+1)-hypergeomMass M V m k|) =
-      ∑ k ∈ Finset.range (M+1), |hypergeomHiddenFlux V M m (k+1)-hypergeomHiddenFlux V M m k| := by
-    apply Finset.sum_congr rfl
-    intro k hk
-    rw [hypergeom_adjacent_hidden_difference V M m k hM hV (by simpa only [Finset.mem_range, Nat.lt_succ_iff] using hk)]
-  rw [he]
-  unfold hypergeomHiddenFlux
-  simp_rw [← mul_sub, abs_mul, abs_of_nonneg (by positivity : (0:ℝ) ≤ (V:ℝ)/(V+M))]
-  rw [← Finset.mul_sum, ← mul_add, hypergeom_hidden_variation (V-1) M m hM (by omega)]
-  ring
-
-theorem adjacentHypergeomTV_le_mass (V M m : ℕ) (hM : M ≤ m) (hV : m < V) :
-    adjacentHypergeomTV V M m ≤ hypergeomMass M V m (hypergeomHiddenMode (V-1) M m) := by
-  rw [adjacentHypergeomTV_eq_flux V M m hM hV,
-    hypergeomHiddenFlux_formula V M m _ hM hV (hypergeomHiddenMode_le (V-1) M m (by omega))]
-  have hk := hypergeomHiddenMode_le (V-1) M m (by omega : m ≤ V-1)
-  have hk' : (hypergeomHiddenMode (V-1) M m:ℝ) ≤ M := by exact_mod_cast hk
-  have hm' : (m:ℝ) < V := by exact_mod_cast hV
-  have hd : (0:ℝ) < (V:ℝ)+M-m := by linarith [Nat.cast_nonneg (α := ℝ) M]
-  apply mul_le_of_le_one_left (hypergeomMass_nonneg _ _ _ _)
-  apply (div_le_one hd).mpr
-  linarith
-
-end TraceReconstruction
-
-namespace TraceReconstruction
-def bernoulliRelativeEntropy (θ p : ℝ) : ℝ :=
-  θ*Real.log (θ/p)+(1-θ)*Real.log ((1-θ)/(1-p))
-
-theorem bernoulliRelativeEntropy_upper {θ p : ℝ}
-    (hθ : θ ∈ Set.Ioo 0 1) (hp : p ∈ Set.Ioo 0 1) :
-    bernoulliRelativeEntropy θ p ≤ (θ-p)^2/(p*(1-p)) := by
-  have h1 := mul_le_mul_of_nonneg_left (Real.log_le_sub_one_of_pos (div_pos hθ.1 hp.1)) hθ.1.le
-  have h2 := mul_le_mul_of_nonneg_left (Real.log_le_sub_one_of_pos
-    (div_pos (sub_pos.mpr hθ.2) (sub_pos.mpr hp.2))) (sub_nonneg.mpr hθ.2.le)
-  have he : θ*(θ/p-1)+(1-θ)*((1-θ)/(1-p)-1) = (θ-p)^2/(p*(1-p)) := by
-    field_simp [hp.1.ne', (sub_pos.mpr hp.2).ne']; ring
-  unfold bernoulliRelativeEntropy
-  rw [← he]
-  linarith
-
-theorem binomialMass_change_probability (n k : ℕ) {θ p : ℝ}
-    (hθ : θ ∈ Set.Ioo 0 1) (hp : p ∈ Set.Ioo 0 1) :
-    binomialMass n p k = binomialMass n θ k *
-      Real.exp ((k:ℝ)*Real.log (p/θ)+(n-k:ℕ)*Real.log ((1-p)/(1-θ))) := by
-  rw [Real.exp_add, Real.exp_nat_mul, Real.exp_nat_mul,
-    Real.exp_log (div_pos hp.1 hθ.1),
-    Real.exp_log (div_pos (sub_pos.mpr hp.2) (sub_pos.mpr hθ.2))]
-  unfold binomialMass
-  simp only [div_pow]
-  field_simp [hθ.1.ne', (sub_pos.mpr hθ.2).ne']
-
-theorem binomialMass_mean_change_probability (n k : ℕ) {θ p : ℝ}
-    (hθ : θ ∈ Set.Ioo 0 1) (hp : p ∈ Set.Ioo 0 1)
-    (hk : k ≤ n) (hmean : (n:ℝ)*θ = k) :
-    binomialMass n p k = binomialMass n θ k * Real.exp (-(n:ℝ)*bernoulliRelativeEntropy θ p) := by
-  rw [binomialMass_change_probability n k hθ hp]
-  congr 2
-  rw [Nat.cast_sub hk, ← hmean]
-  unfold bernoulliRelativeEntropy
-  rw [Real.log_div hp.1.ne' hθ.1.ne', Real.log_div (sub_pos.mpr hp.2).ne' (sub_pos.mpr hθ.2).ne',
-    Real.log_div hθ.1.ne' hp.1.ne', Real.log_div (sub_pos.mpr hθ.2).ne' (sub_pos.mpr hp.2).ne']
-  ring
-
-/-- Uniform local binomial lower bound without Stirling. It applies at a
-count `k` by changing to the channel with mean exactly `k`. -/
-theorem binomialMass_local_lower (n k : ℕ) {θ p : ℝ}
-    (hθ : θ ∈ Set.Ioo 0 1) (hp : p ∈ Set.Ioo 0 1)
-    (hk : k ≤ n) (hmean : (n:ℝ)*θ = k) (hv : 1 ≤ (n:ℝ)*θ*(1-θ)) :
-    Real.exp (-(n:ℝ)*(θ-p)^2/(p*(1-p))) /
-      (12*Real.sqrt ((n:ℝ)*θ*(1-θ))) ≤ binomialMass n p k := by
-  rw [binomialMass_mean_change_probability n k hθ hp hk hmean]
-  have hlow := binomialMass_mean_lower n k hθ hk hmean hv
-  have hexp : Real.exp (-(n:ℝ)*(θ-p)^2/(p*(1-p))) ≤
-      Real.exp (-(n:ℝ)*bernoulliRelativeEntropy θ p) := by
-    apply Real.exp_le_exp.mpr
-    have := mul_le_mul_of_nonpos_left (bernoulliRelativeEntropy_upper hθ hp) (neg_nonpos.mpr (Nat.cast_nonneg n))
-    simpa only [mul_div_assoc] using this
-  have hh := mul_le_mul hlow hexp (Real.exp_pos _).le
-    (binomialMass_nonneg n k ⟨hθ.1.le,hθ.2.le⟩)
-  simpa only [one_div, div_eq_mul_inv, mul_comm, one_mul] using hh
-
-end TraceReconstruction
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-!
+# Binomial
+
+Fair and biased binomial laws, generating series, differences, concentration, Gaussian overlap, hidden counts, and local lower bounds.
+-/
+
+open scoped ENNReal
+noncomputable section
+
+namespace TraceReconstruction
+
+/-! ## Quantitative fair-binomial anti-concentration
+
+A direct inductive bound, avoiding an unformalized appeal to Stirling.
+It is one of the estimates needed for signed binomial smoothing and
+for the local/global anchor kernels.
+-/
+
+/-- Fair-binomial mass, extended by zero beyond its natural support. -/
+def fairBinomial (n k : ℕ) : ℝ := (n.choose k : ℝ) / 2 ^ n
+
+theorem fairBinomial_nonneg (n k : ℕ) : 0 ≤ fairBinomial n k := by
+  unfold fairBinomial
+  positivity
+
+theorem centralBinom_square_bound (n : ℕ) :
+    ((n : ℝ) + 1) * (n.centralBinom : ℝ) ^ 2 ≤ (16 : ℝ) ^ n := by
+  induction n with
+  | zero => norm_num
+  | succ n ih =>
+    have hn : (0 : ℝ) < n + 1 := by positivity
+    have hr : ((n : ℝ) + 1) * ((n + 1).centralBinom : ℝ) =
+        2 * (2 * n + 1) * (n.centralBinom : ℝ) := by
+      exact_mod_cast Nat.succ_mul_centralBinom_succ n
+    have hc : 4 * (2 * (n : ℝ) + 1) ^ 2 * (n + 2) ≤
+        16 * (n + 1) ^ 3 := by nlinarith [sq_nonneg (n : ℝ)]
+    have hh : ((n : ℝ) + 1) ^ 2 *
+        ((n + 2) * ((n + 1).centralBinom : ℝ) ^ 2) ≤
+        ((n : ℝ) + 1) ^ 2 * (16 * 16 ^ n) := by
+      calc
+        _ = (n + 2) * (((n : ℝ) + 1) * ((n + 1).centralBinom : ℝ)) ^ 2 := by ring
+        _ = (4 * (2 * (n : ℝ) + 1) ^ 2 * (n + 2)) *
+            (n.centralBinom : ℝ) ^ 2 := by rw [hr]; ring
+        _ ≤ (16 * ((n : ℝ) + 1) ^ 3) * (n.centralBinom : ℝ) ^ 2 :=
+          mul_le_mul_of_nonneg_right hc (sq_nonneg _)
+        _ = (16 * ((n : ℝ) + 1) ^ 2) *
+            (((n : ℝ) + 1) * (n.centralBinom : ℝ) ^ 2) := by ring
+        _ ≤ (16 * ((n : ℝ) + 1) ^ 2) * 16 ^ n :=
+          mul_le_mul_of_nonneg_left ih (by positivity)
+        _ = _ := by ring
+    have := (mul_le_mul_iff_right₀ (sq_pos_of_pos hn)).mp hh
+    rw [pow_succ (16 : ℝ) n]
+    convert this using 1 <;> push_cast <;> first | rfl | ring
+
+theorem choose_odd_le_twice_central (n k : ℕ) :
+    (2 * n + 1).choose k ≤ 2 * n.centralBinom := by
+  cases k with
+  | zero => simp only [Nat.choose_zero_right]; have := Nat.centralBinom_pos n; omega
+  | succ k =>
+    rw [Nat.choose_succ_succ]
+    have h₁ := Nat.choose_le_centralBinom k n
+    have h₂ := Nat.choose_le_centralBinom (k + 1) n
+    simpa [Nat.succ_eq_add_one, two_mul] using Nat.add_le_add h₁ h₂
+
+theorem fairBinomial_square_bound (n k : ℕ) :
+    ((n : ℝ) + 1) * fairBinomial n k ^ 2 ≤ 2 := by
+  have hcases : n % 2 = 0 ∨ n % 2 = 1 := Nat.mod_two_eq_zero_or_one n
+  rcases hcases with he | ho
+  · obtain ⟨m, rfl⟩ : ∃ m, n = 2 * m := ⟨n / 2, by omega⟩
+    have hc : ((2 * m).choose k : ℝ) ≤ (m.centralBinom : ℝ) :=
+      by exact_mod_cast Nat.choose_le_centralBinom k m
+    have hcs : ((2 * m).choose k : ℝ) ^ 2 ≤ (m.centralBinom : ℝ) ^ 2 :=
+      pow_le_pow_left₀ (by positivity) hc 2
+    have ht := centralBinom_square_bound m
+    have hp : ((2 : ℝ) ^ (2 * m)) ^ 2 = 16 ^ m := by
+      rw [← pow_mul, show 2 * m * 2 = 4 * m by omega, pow_mul]
+      norm_num
+    unfold fairBinomial
+    rw [div_pow, ← mul_div_assoc, div_le_iff₀ (by positivity : (0 : ℝ) < (2 ^ (2 * m)) ^ 2)]
+    rw [hp]
+    push_cast
+    nlinarith [mul_le_mul_of_nonneg_left hcs (show (0 : ℝ) ≤ m + 1 by positivity),
+      sq_nonneg ((2 * m).choose k : ℝ)]
+  · obtain ⟨m, rfl⟩ : ∃ m, n = 2 * m + 1 := ⟨n / 2, by omega⟩
+    have hc : ((2 * m + 1).choose k : ℝ) ≤ 2 * (m.centralBinom : ℝ) :=
+      by exact_mod_cast choose_odd_le_twice_central m k
+    have hcs : ((2 * m + 1).choose k : ℝ) ^ 2 ≤ 4 * (m.centralBinom : ℝ) ^ 2 := by
+      have := pow_le_pow_left₀ (by positivity : (0 : ℝ) ≤ ((2 * m + 1).choose k : ℝ)) hc 2
+      nlinarith
+    have ht := centralBinom_square_bound m
+    have hp : ((2 : ℝ) ^ (2 * m + 1)) ^ 2 = 4 * 16 ^ m := by
+      rw [pow_add, mul_pow, ← pow_mul, show 2 * m * 2 = 4 * m by omega, pow_mul]
+      norm_num
+      ring
+    unfold fairBinomial
+    rw [div_pow, ← mul_div_assoc, div_le_iff₀ (by positivity : (0 : ℝ) < (2 ^ (2 * m + 1)) ^ 2)]
+    rw [hp]
+    push_cast
+    nlinarith [mul_le_mul_of_nonneg_left hcs (show (0 : ℝ) ≤ m + 1 by positivity)]
+
+
+/-! ## Discrete convolution estimates for the hidden displacement kernels -/
+
+abbrev IntSeries := AddMonoidAlgebra ℝ ℤ
+
+def seriesL1 (f : IntSeries) : ℝ := ∑' z : ℤ, |f.coeff z|
+def seriesL2Sq (f : IntSeries) : ℝ := ∑' z : ℤ, f.coeff z ^ 2
+
+theorem series_summable (f : IntSeries) (φ : ℝ → ℝ) (hφ : φ 0 = 0) :
+    Summable (fun z : ℤ => φ (f.coeff z)) :=
+  summable_of_hasFiniteSupport (f.coeff.hasFiniteSupport.fun_comp hφ)
+
+theorem seriesL1_nonneg (f : IntSeries) : 0 ≤ seriesL1 f :=
+  tsum_nonneg fun _ => abs_nonneg _
+
+theorem seriesL2Sq_nonneg (f : IntSeries) : 0 ≤ seriesL2Sq f :=
+  tsum_nonneg fun _ => sq_nonneg _
+
+theorem seriesL1_eq_sum (f : IntSeries) :
+    seriesL1 f = ∑ z ∈ f.coeff.support, |f.coeff z| := by
+  apply tsum_eq_sum
+  intro z hz
+  simp [Finsupp.notMem_support_iff.mp hz]
+
+theorem seriesL2Sq_eq_sum (f : IntSeries) :
+    seriesL2Sq f = ∑ z ∈ f.coeff.support, f.coeff z ^ 2 := by
+  apply tsum_eq_sum
+  intro z hz
+  simp [Finsupp.notMem_support_iff.mp hz]
+
+theorem series_summable_shift (f : IntSeries) (φ : ℝ → ℝ)
+    (hφ : φ 0 = 0) (a : ℤ) :
+    Summable (fun z : ℤ => φ (f.coeff (a + z))) := by
+  exact (Equiv.addLeft a).summable_iff.mpr (series_summable f φ hφ)
+
+theorem series_sum_shift (f : IntSeries) (φ : ℝ → ℝ) (a : ℤ) :
+    (∑' z : ℤ, φ (f.coeff (a + z))) = ∑' z : ℤ, φ (f.coeff z) :=
+  (Equiv.addLeft a).tsum_eq (fun z => φ (f.coeff z))
+
+theorem series_coeff_mul_abs_le (f g : IntSeries) (z : ℤ) :
+    |(f * g).coeff z| ≤ ∑ a ∈ f.coeff.support, |f.coeff a| * |g.coeff (-a + z)| := by
+  rw [AddMonoidAlgebra.coeff_mul_apply_left, Finsupp.sum]
+  simpa only [abs_mul] using Finset.abs_sum_le_sum_abs
+    (fun a => f.coeff a * g.coeff (-a + z)) f.coeff.support
+
+theorem seriesL1_mul_le (f g : IntSeries) :
+    seriesL1 (f * g) ≤ seriesL1 f * seriesL1 g := by
+  have hh (a : ℤ) : Summable (fun z : ℤ => |f.coeff a| * |g.coeff (-a + z)|) :=
+    (series_summable_shift g abs (by simp) (-a)).mul_left _
+  calc
+    seriesL1 (f * g) ≤ ∑' z : ℤ, ∑ a ∈ f.coeff.support,
+        |f.coeff a| * |g.coeff (-a + z)| :=
+      Summable.tsum_le_tsum (series_coeff_mul_abs_le f g) (series_summable (f * g) abs (by simp))
+        (summable_sum fun a _ => hh a)
+    _ = ∑ a ∈ f.coeff.support, ∑' z : ℤ, |f.coeff a| * |g.coeff (-a + z)| :=
+      Summable.tsum_finsetSum fun a _ => hh a
+    _ = (∑ a ∈ f.coeff.support, |f.coeff a|) * seriesL1 g := by
+      simp_rw [tsum_mul_left, series_sum_shift]
+      exact (Finset.sum_mul _ _ _).symm
+    _ = seriesL1 f * seriesL1 g := by rw [seriesL1_eq_sum f]
+
+theorem series_coeff_mul_sq_le (f g : IntSeries) (z : ℤ) :
+    (f * g).coeff z ^ 2 ≤ seriesL1 f *
+      ∑ a ∈ f.coeff.support, |f.coeff a| * (g.coeff (-a + z)) ^ 2 := by
+  rw [AddMonoidAlgebra.coeff_mul_apply_left, Finsupp.sum, seriesL1_eq_sum]
+  apply Finset.sum_sq_le_sum_mul_sum_of_sq_le_mul
+  · intro a _; exact abs_nonneg _
+  · intro a _; positivity
+  · intro a _
+    rw [mul_pow, ← mul_assoc, ← sq, sq_abs]
+
+theorem seriesL2Sq_mul_le (f g : IntSeries) :
+    seriesL2Sq (f * g) ≤ seriesL1 f ^ 2 * seriesL2Sq g := by
+  have hh (a : ℤ) : Summable (fun z : ℤ => |f.coeff a| * (g.coeff (-a + z)) ^ 2) :=
+    (series_summable_shift g (fun x => x ^ 2) (by simp) (-a)).mul_left _
+  calc
+    seriesL2Sq (f * g) ≤ ∑' z : ℤ, seriesL1 f *
+        ∑ a ∈ f.coeff.support, |f.coeff a| * (g.coeff (-a + z)) ^ 2 :=
+      Summable.tsum_le_tsum (series_coeff_mul_sq_le f g)
+        (series_summable (f * g) (fun x => x ^ 2) (by simp))
+        ((summable_sum fun a _ => hh a).mul_left _)
+    _ = seriesL1 f * ∑ a ∈ f.coeff.support,
+        ∑' z : ℤ, |f.coeff a| * (g.coeff (-a + z)) ^ 2 := by
+      rw [tsum_mul_left, Summable.tsum_finsetSum fun a _ => hh a]
+    _ = seriesL1 f ^ 2 * seriesL2Sq g := by
+      simp_rw [tsum_mul_left, series_sum_shift g (fun x => x ^ 2)]
+      rw [← Finset.sum_mul, ← seriesL1_eq_sum]
+      unfold seriesL2Sq
+      ring
+
+theorem series_coeff_mul_bound (f g : IntSeries) (M : ℝ)
+    (hg : ∀ z, |g.coeff z| ≤ M) (z : ℤ) :
+    |(f * g).coeff z| ≤ seriesL1 f * M := by
+  calc
+    _ ≤ ∑ a ∈ f.coeff.support, |f.coeff a| * |g.coeff (-a + z)| :=
+      series_coeff_mul_abs_le f g z
+    _ ≤ ∑ a ∈ f.coeff.support, |f.coeff a| * M :=
+      Finset.sum_le_sum fun a _ => mul_le_mul_of_nonneg_left (hg _) (abs_nonneg _)
+    _ = seriesL1 f * M := by rw [← Finset.sum_mul, ← seriesL1_eq_sum]
+
+/-! The probability generating series of the fair binomial kernel. -/
+def fairSeries (n : ℕ) : IntSeries :=
+  (AddMonoidAlgebra.single 1 (1 / 2) + AddMonoidAlgebra.single 0 (1 / 2)) ^ n
+
+theorem fairSeries_add (m n : ℕ) : fairSeries (m + n) = fairSeries m * fairSeries n :=
+  pow_add _ m n
+
+theorem fairSeries_sum (n : ℕ) :
+    fairSeries n = ∑ k ∈ Finset.range (n + 1),
+      AddMonoidAlgebra.single (k : ℤ) (fairBinomial n k) := by
+  unfold fairSeries
+  rw [add_pow]
+  apply Finset.sum_congr rfl
+  intro k hk
+  have hkn : k ≤ n := by simpa only [Finset.mem_range, Nat.lt_succ_iff] using hk
+  rw [AddMonoidAlgebra.single_pow, AddMonoidAlgebra.single_pow,
+    AddMonoidAlgebra.natCast_def, AddMonoidAlgebra.single_mul_single,
+    AddMonoidAlgebra.single_mul_single]
+  simp only [nsmul_eq_mul, mul_one, mul_zero, add_zero]
+  congr 1
+  rw [← pow_add, Nat.add_sub_of_le hkn]
+  unfold fairBinomial
+  rw [div_pow]
+  simp [div_eq_mul_inv, mul_comm]
+
+theorem fairSeries_coeff_nat (n k : ℕ) :
+    (fairSeries n).coeff (k : ℤ) = fairBinomial n k := by
+  rw [fairSeries_sum, AddMonoidAlgebra.coeff_sum, Finsupp.finsetSum_apply]
+  by_cases hk : k ≤ n
+  · rw [Finset.sum_eq_single k]
+    · simp only [AddMonoidAlgebra.coeff_single, Finsupp.single_apply, ite_true]
+    · intro b _ hb
+      simp only [AddMonoidAlgebra.coeff_single, Finsupp.single_apply, Nat.cast_inj, ite_eq_right hb]
+    · intro h
+      exact (h (Finset.mem_range.mpr (by omega))).elim
+  · rw [Finset.sum_eq_zero]
+    · simp [fairBinomial, Nat.choose_eq_zero_of_lt (by omega : n < k)]
+    · intro b hb
+      have hbk : b ≠ k := by simp only [Finset.mem_range] at hb; omega
+      simp only [AddMonoidAlgebra.coeff_single, Finsupp.single_apply, Nat.cast_inj, ite_eq_right hbk]
+
+theorem fairSeries_coeff_neg (n : ℕ) (z : ℤ) (hz : z < 0) :
+    (fairSeries n).coeff z = 0 := by
+  rw [fairSeries_sum, AddMonoidAlgebra.coeff_sum, Finsupp.finsetSum_apply]
+  apply Finset.sum_eq_zero
+  intro b _
+  have h : (b : ℤ) ≠ z := by omega
+  simp only [AddMonoidAlgebra.coeff_single, Finsupp.single_apply, ite_eq_right h]
+
+theorem seriesL1_eq_sum_range (f : IntSeries) (n : ℕ)
+    (hneg : ∀ z < 0, f.coeff z = 0)
+    (hhigh : ∀ k : ℕ, n < k → f.coeff k = 0) :
+    seriesL1 f = ∑ k ∈ Finset.range (n + 1), |f.coeff (k : ℤ)| := by
+  let e : ℕ ↪ ℤ := ⟨Nat.cast, Nat.cast_injective⟩
+  calc
+    seriesL1 f = ∑ z ∈ (Finset.range (n + 1)).map e, |f.coeff z| := by
+      apply tsum_eq_sum
+      intro z hz
+      by_cases h : 0 ≤ z
+      · have heq : ((z.toNat : ℕ) : ℤ) = z := Int.toNat_of_nonneg h
+        have hh : n < z.toNat := by
+          by_contra hn
+          apply hz
+          exact Finset.mem_map.mpr ⟨z.toNat, Finset.mem_range.mpr (by omega), heq⟩
+        rw [← heq, hhigh _ hh, abs_zero]
+      · rw [hneg z (by omega), abs_zero]
+    _ = _ := Finset.sum_map _ e _
+
+def firstDifference : IntSeries :=
+  AddMonoidAlgebra.single 0 1 - AddMonoidAlgebra.single 1 1
+
+theorem firstDifference_coeff (f : IntSeries) (z : ℤ) :
+    (firstDifference * f).coeff z = f.coeff z - f.coeff (z - 1) := by
+  rw [firstDifference, sub_mul, AddMonoidAlgebra.coeff_sub, Finsupp.sub_apply,
+    mul_comm _ f, AddMonoidAlgebra.coeff_mul_single_apply,
+    mul_comm _ f, AddMonoidAlgebra.coeff_mul_single_apply]
+  simp only [neg_zero, add_zero, mul_one, sub_eq_add_neg]
+
+theorem fairSeries_difference_coeff_zero (n : ℕ) :
+    (firstDifference * fairSeries n).coeff 0 = fairBinomial n 0 := by
+  rw [firstDifference_coeff]
+  norm_num only [zero_sub]
+  rw [fairSeries_coeff_neg n (-1) (by norm_num), sub_zero]
+  exact fairSeries_coeff_nat n 0
+
+theorem fairSeries_difference_coeff_succ (n k : ℕ) :
+    (firstDifference * fairSeries n).coeff (↑(k + 1) : ℤ) =
+      fairBinomial n (k + 1) - fairBinomial n k := by
+  rw [firstDifference_coeff, fairSeries_coeff_nat]
+  simp only [Nat.cast_add, Nat.cast_one, add_sub_cancel_right, fairSeries_coeff_nat]
+
+theorem fairBinomial_unimodal_left (n k : ℕ) (hk : k < n / 2) :
+    fairBinomial n k ≤ fairBinomial n (k + 1) := by
+  apply div_le_div_of_nonneg_right _ (by positivity)
+  exact_mod_cast Nat.choose_le_succ_of_lt_half_left hk
+
+theorem fairBinomial_unimodal_right (n k : ℕ) (hk : n / 2 ≤ k) :
+    fairBinomial n (k + 1) ≤ fairBinomial n k := by
+  apply div_le_div_of_nonneg_right _ (by positivity)
+  have hnk : n - k ≤ k + 1 := by omega
+  have hmul := Nat.mul_le_mul_left (n.choose k) hnk
+  rw [← Nat.choose_succ_right_eq] at hmul
+  exact_mod_cast Nat.le_of_mul_le_mul_right hmul (by omega : 0 < k + 1)
+
+theorem fairBinomial_variation (n : ℕ) :
+    (∑ k ∈ Finset.range (n + 1), |fairBinomial n (k + 1) - fairBinomial n k|) =
+      2 * fairBinomial n (n / 2) - fairBinomial n 0 := by
+  have hmid : n / 2 ≤ n + 1 := by omega
+  rw [← Finset.sum_range_add_sum_Ico _ hmid]
+  have hl : (∑ k ∈ Finset.range (n / 2),
+      |fairBinomial n (k + 1) - fairBinomial n k|) =
+      fairBinomial n (n / 2) - fairBinomial n 0 := by
+    rw [← Finset.sum_range_sub (fairBinomial n) (n / 2)]
+    apply Finset.sum_congr rfl
+    intro k hk
+    exact abs_of_nonneg (sub_nonneg.mpr
+      (fairBinomial_unimodal_left n k (Finset.mem_range.mp hk)))
+  have hr : (∑ k ∈ Finset.Ico (n / 2) (n + 1),
+      |fairBinomial n (k + 1) - fairBinomial n k|) =
+      fairBinomial n (n / 2) - fairBinomial n (n + 1) := by
+    calc
+      _ = ∑ k ∈ Finset.Ico (n / 2) (n + 1),
+          -(fairBinomial n (k + 1) - fairBinomial n k) := by
+        apply Finset.sum_congr rfl
+        intro k hk
+        exact abs_of_nonpos (sub_nonpos.mpr
+          (fairBinomial_unimodal_right n k (Finset.mem_Ico.mp hk).1))
+      _ = _ := by rw [Finset.sum_neg_distrib, Finset.sum_Ico_sub _ hmid]; ring
+  rw [hl, hr]
+  have hzero : fairBinomial n (n + 1) = 0 := by simp [fairBinomial]
+  rw [hzero]
+  ring
+
+theorem fairSeries_difference_L1 (n : ℕ) :
+    seriesL1 (firstDifference * fairSeries n) = 2 * fairBinomial n (n / 2) := by
+  rw [seriesL1_eq_sum_range _ (n + 1)]
+  · rw [Finset.sum_range_succ', Nat.cast_zero, fairSeries_difference_coeff_zero,
+      abs_of_nonneg (fairBinomial_nonneg n 0)]
+    simp_rw [fairSeries_difference_coeff_succ]
+    rw [fairBinomial_variation]
+    ring
+  · intro z hz
+    rw [firstDifference_coeff, fairSeries_coeff_neg n z hz,
+      fairSeries_coeff_neg n (z - 1) (by omega), sub_zero]
+  · intro k hk
+    have hkpos : 0 < k := by omega
+    obtain ⟨v, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (Nat.ne_of_gt hkpos)
+    rw [fairSeries_difference_coeff_succ]
+    simp [fairBinomial, Nat.choose_eq_zero_of_lt (by omega : n < v),
+      Nat.choose_eq_zero_of_lt (by omega : n < v + 1)]
+
+def binomialBound (n : ℕ) : ℝ := Real.sqrt (2 / ((n : ℝ) + 1))
+
+theorem binomialBound_nonneg (n : ℕ) : 0 ≤ binomialBound n := Real.sqrt_nonneg _
+
+theorem fairBinomial_le_bound (n k : ℕ) : fairBinomial n k ≤ binomialBound n := by
+  apply Real.le_sqrt_of_sq_le
+  apply (le_div_iff₀ (by positivity : (0 : ℝ) < n + 1)).mpr
+  simpa only [mul_comm] using fairBinomial_square_bound n k
+
+theorem fairSeries_nonneg (n : ℕ) (z : ℤ) : 0 ≤ (fairSeries n).coeff z := by
+  by_cases hz : 0 ≤ z
+  · rw [← Int.toNat_of_nonneg hz, fairSeries_coeff_nat]
+    exact fairBinomial_nonneg _ _
+  · rw [fairSeries_coeff_neg n z (by omega)]
+
+theorem fairSeries_coeff_bound (n : ℕ) (z : ℤ) :
+    |(fairSeries n).coeff z| ≤ binomialBound n := by
+  by_cases hz : 0 ≤ z
+  · rw [← Int.toNat_of_nonneg hz, fairSeries_coeff_nat,
+      abs_of_nonneg (fairBinomial_nonneg _ _)]
+    exact fairBinomial_le_bound _ _
+  · rw [fairSeries_coeff_neg n z (by omega), abs_zero]
+    exact binomialBound_nonneg n
+
+theorem fairSeries_L1 (n : ℕ) : seriesL1 (fairSeries n) = 1 := by
+  rw [seriesL1_eq_sum_range _ n (fairSeries_coeff_neg n)]
+  · simp_rw [fairSeries_coeff_nat, abs_of_nonneg (fairBinomial_nonneg _ _)]
+    unfold fairBinomial
+    rw [← Finset.sum_div]
+    have hsum : (∑ k ∈ Finset.range (n + 1), (n.choose k : ℝ)) = (2 : ℝ) ^ n := by
+      exact_mod_cast Nat.sum_range_choose n
+    rw [hsum, div_self (by positivity)]
+  · intro k hk
+    rw [fairSeries_coeff_nat]
+    simp [fairBinomial, Nat.choose_eq_zero_of_lt hk]
+
+theorem seriesL2Sq_le_L1_mul_bound (f : IntSeries) (M : ℝ)
+    (hf : ∀ z, |f.coeff z| ≤ M) : seriesL2Sq f ≤ seriesL1 f * M := by
+  calc
+    _ ≤ ∑' z : ℤ, |f.coeff z| * M := by
+      apply Summable.tsum_le_tsum
+      · intro z
+        calc
+          f.coeff z ^ 2 = |f.coeff z| * |f.coeff z| := by rw [← sq, sq_abs]
+          _ ≤ _ := mul_le_mul_of_nonneg_left (hf z) (abs_nonneg _)
+      · exact series_summable f (fun x => x ^ 2) (by simp)
+      · exact (series_summable f abs (by simp)).mul_right M
+    _ = _ := tsum_mul_right
+
+theorem fairSeries_L2Sq_bound (n : ℕ) : seriesL2Sq (fairSeries n) ≤ binomialBound n := by
+  simpa only [fairSeries_L1, one_mul] using
+    seriesL2Sq_le_L1_mul_bound (fairSeries n) (binomialBound n) (fairSeries_coeff_bound n)
+
+theorem fairSeries_difference_L1_bound (n : ℕ) :
+    seriesL1 (firstDifference * fairSeries n) ≤ 2 * binomialBound n := by
+  rw [fairSeries_difference_L1]
+  exact mul_le_mul_of_nonneg_left (fairBinomial_le_bound n (n / 2)) (by norm_num)
+
+theorem seriesL1_single (z : ℤ) (r : ℝ) :
+    seriesL1 (AddMonoidAlgebra.single z r) = |r| := by
+  unfold seriesL1
+  simp only [AddMonoidAlgebra.coeff_single, Finsupp.single_apply]
+  simp only [apply_ite abs, abs_zero]
+  exact tsum_ite_eq' z (fun _ : ℤ => |r|)
+
+theorem seriesL1_one : seriesL1 (1 : IntSeries) = 1 := by
+  change seriesL1 (AddMonoidAlgebra.single 0 1) = 1
+  rw [seriesL1_single, abs_one]
+
+theorem seriesL1_pow_le (f : IntSeries) (s : ℕ) : seriesL1 (f ^ s) ≤ seriesL1 f ^ s := by
+  induction s with
+  | zero => simp only [pow_zero, seriesL1_one, le_refl]
+  | succ s ih =>
+    rw [pow_succ, pow_succ]
+    exact (seriesL1_mul_le _ _).trans
+      (mul_le_mul_of_nonneg_right ih (seriesL1_nonneg f))
+
+theorem fairSeries_mul_power (n s : ℕ) :
+    firstDifference ^ s * fairSeries ((s + 1) * n) =
+      (firstDifference * fairSeries n) ^ s * fairSeries n := by
+  simp only [fairSeries, add_mul, one_mul, pow_add, pow_mul, mul_pow]
+  ring
+
+/-- Quantitative higher-order smoothing, with n trials assigned to each
+    difference and one further group reserved for anti-concentration. -/
+theorem binomial_difference_coeff_bound (n s : ℕ) (z : ℤ) :
+    |(firstDifference ^ s * fairSeries ((s + 1) * n)).coeff z| ≤
+      (2 * binomialBound n) ^ s * binomialBound n := by
+  rw [fairSeries_mul_power]
+  apply (series_coeff_mul_bound _ _ _ (fairSeries_coeff_bound n) z).trans
+  apply mul_le_mul_of_nonneg_right _ (binomialBound_nonneg n)
+  exact (seriesL1_pow_le _ _).trans
+    (pow_le_pow_left₀ (seriesL1_nonneg _) (fairSeries_difference_L1_bound n) s)
+
+theorem binomial_difference_L2Sq_bound (n s : ℕ) :
+    seriesL2Sq (firstDifference ^ s * fairSeries ((s + 1) * n)) ≤
+      ((2 * binomialBound n) ^ s) ^ 2 * binomialBound n := by
+  rw [fairSeries_mul_power]
+  apply (seriesL2Sq_mul_le _ _).trans
+  apply mul_le_mul
+  · apply pow_le_pow_left₀ (seriesL1_nonneg _)
+    exact (seriesL1_pow_le _ _).trans
+      (pow_le_pow_left₀ (seriesL1_nonneg _) (fairSeries_difference_L1_bound n) s)
+  · exact fairSeries_L2Sq_bound n
+  · exact seriesL2Sq_nonneg _
+  · positivity
+
+theorem seriesL1_add_le (f g : IntSeries) :
+    seriesL1 (f + g) ≤ seriesL1 f + seriesL1 g := by
+  calc
+    _ ≤ ∑' z : ℤ, (|f.coeff z| + |g.coeff z|) := by
+      apply Summable.tsum_le_tsum
+      · intro z
+        simpa only [AddMonoidAlgebra.coeff_add, Finsupp.add_apply] using
+          abs_add_le (f.coeff z) (g.coeff z)
+      · exact series_summable (f + g) abs (by simp)
+      · exact (series_summable f abs (by simp)).add (series_summable g abs (by simp))
+    _ = _ := (series_summable f abs (by simp)).tsum_add (series_summable g abs (by simp))
+
+theorem seriesL1_zero : seriesL1 (0 : IntSeries) = 0 := by
+  simp [seriesL1]
+
+theorem seriesL1_sum_le {ι : Type*} (A : Finset ι) (f : ι → IntSeries) :
+    seriesL1 (∑ i ∈ A, f i) ≤ ∑ i ∈ A, seriesL1 (f i) := by
+  classical
+  induction A using Finset.induction_on with
+  | empty => simp only [Finset.sum_empty, seriesL1_zero, le_refl]
+  | @insert a A ha ih =>
+    rw [Finset.sum_insert ha, Finset.sum_insert ha]
+    exact (seriesL1_add_le _ _).trans (add_le_add le_rfl ih)
+
+theorem seriesL1_prod_le {ι : Type*} (A : Finset ι) (f : ι → IntSeries) :
+    seriesL1 (∏ i ∈ A, f i) ≤ ∏ i ∈ A, seriesL1 (f i) := by
+  classical
+  induction A using Finset.induction_on with
+  | empty => simp only [Finset.prod_empty, seriesL1_one, le_refl]
+  | @insert a A ha ih =>
+    rw [Finset.prod_insert ha, Finset.prod_insert ha]
+    exact (seriesL1_mul_le _ _).trans (mul_le_mul_of_nonneg_left ih (seriesL1_nonneg _))
+
+def geomSeries (k : ℕ) : IntSeries := ∑ a ∈ Finset.range k, (AddMonoidAlgebra.single 1 1) ^ a
+
+def prouhetSeries (s : ℕ) : IntSeries :=
+  ∏ v ∈ Finset.range s, (1 - (AddMonoidAlgebra.single 1 1) ^ (2 ^ v))
+
+def prouhetQuotient (s : ℕ) : IntSeries :=
+  ∏ v ∈ Finset.range s, geomSeries (2 ^ v)
+
+theorem geomSeries_difference (k : ℕ) :
+    geomSeries k * firstDifference = 1 - (AddMonoidAlgebra.single 1 1 : IntSeries) ^ k := by
+  exact geom_sum_mul_neg _ _
+
+theorem geomSeries_L1_le (k : ℕ) : seriesL1 (geomSeries k) ≤ k := by
+  apply (seriesL1_sum_le _ _).trans
+  simp only [AddMonoidAlgebra.single_pow, one_pow, seriesL1_single, abs_one,
+    Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_one, le_refl]
+
+theorem prouhetSeries_factor (s : ℕ) :
+    prouhetSeries s = prouhetQuotient s * firstDifference ^ s := by
+  unfold prouhetSeries prouhetQuotient
+  simp_rw [← geomSeries_difference]
+  rw [Finset.prod_mul_distrib]
+  simp only [Finset.prod_const, Finset.card_range]
+
+theorem prouhetQuotient_L1_le (s : ℕ) :
+    seriesL1 (prouhetQuotient s) ≤ (2 : ℝ) ^ (s * s) := by
+  calc
+    _ ≤ ∏ v ∈ Finset.range s, seriesL1 (geomSeries (2 ^ v)) := seriesL1_prod_le _ _
+    _ ≤ ∏ _v ∈ Finset.range s, ((2 : ℝ) ^ s) := by
+      apply Finset.prod_le_prod₀
+      · intro v _; exact seriesL1_nonneg _
+      · intro v hv
+        apply (geomSeries_L1_le _).trans
+        push_cast
+        apply pow_le_pow_right₀ (by norm_num : (1 : ℝ) ≤ 2)
+        exact (Finset.mem_range.mp hv).le
+    _ = _ := by simp only [Finset.prod_const, Finset.card_range, ← pow_mul]
+
+/-- Prouhet smoothing in an explicit finite-trial form.  The constants
+    2^(s*s) and (2 sqrt(2/(n+1)))^s are exp(O(s^2)) after assigning
+    n = floor(g/(s+1)) trials to each group.  No Fourier theorem or
+    probabilistic estimate is assumed. -/
+theorem prouhet_binomial_coeff_bound (n s : ℕ) (z : ℤ) :
+    |(prouhetSeries s * fairSeries ((s + 1) * n)).coeff z| ≤
+      (2 : ℝ) ^ (s * s) * ((2 * binomialBound n) ^ s * binomialBound n) := by
+  rw [prouhetSeries_factor, mul_assoc]
+  apply (series_coeff_mul_bound _ _ _ (binomial_difference_coeff_bound n s) z).trans
+  exact mul_le_mul_of_nonneg_right (prouhetQuotient_L1_le s)
+    (by positivity [binomialBound_nonneg n])
+
+theorem prouhet_binomial_L2Sq_bound (n s : ℕ) :
+    seriesL2Sq (prouhetSeries s * fairSeries ((s + 1) * n)) ≤
+      ((2 : ℝ) ^ (s * s)) ^ 2 * (((2 * binomialBound n) ^ s) ^ 2 * binomialBound n) := by
+  rw [prouhetSeries_factor, mul_assoc]
+  apply (seriesL2Sq_mul_le _ _).trans
+  apply mul_le_mul
+  · exact pow_le_pow_left₀ (seriesL1_nonneg _) (prouhetQuotient_L1_le s) 2
+  · exact binomial_difference_L2Sq_bound n s
+  · exact seriesL2Sq_nonneg _
+  · positivity
+
+/-- Arbitrary trial counts are obtained by convolving with the independent
+    unused trials; this cannot enlarge the coefficient or squared-L2 bounds. -/
+theorem prouhet_binomial_coeff_bound_remainder (n s r : ℕ) (z : ℤ) :
+    |(prouhetSeries s * fairSeries ((s + 1) * n + r)).coeff z| ≤
+      (2 : ℝ) ^ (s * s) * ((2 * binomialBound n) ^ s * binomialBound n) := by
+  rw [fairSeries_add, ← mul_assoc, mul_comm (prouhetSeries s * _)]
+  simpa only [fairSeries_L1, one_mul] using
+    series_coeff_mul_bound (fairSeries r) (prouhetSeries s * fairSeries ((s + 1) * n))
+      _ (prouhet_binomial_coeff_bound n s) z
+
+theorem prouhet_binomial_L2Sq_bound_remainder (n s r : ℕ) :
+    seriesL2Sq (prouhetSeries s * fairSeries ((s + 1) * n + r)) ≤
+      ((2 : ℝ) ^ (s * s)) ^ 2 * (((2 * binomialBound n) ^ s) ^ 2 * binomialBound n) := by
+  rw [fairSeries_add, ← mul_assoc, mul_comm (prouhetSeries s * _)]
+  calc
+    _ ≤ seriesL1 (fairSeries r) ^ 2 *
+        seriesL2Sq (prouhetSeries s * fairSeries ((s + 1) * n)) := seriesL2Sq_mul_le _ _
+    _ ≤ _ := by simpa only [fairSeries_L1, one_pow, one_mul] using
+        prouhet_binomial_L2Sq_bound n s
+
+theorem signed_binomial_smoothing (g s : ℕ) :
+    (∀ z : ℤ, |(prouhetSeries s * fairSeries g).coeff z| ≤
+      (2 : ℝ) ^ (s * s) * ((2 * binomialBound (g / (s + 1))) ^ s *
+        binomialBound (g / (s + 1)))) ∧
+    (seriesL2Sq (prouhetSeries s * fairSeries g) ≤
+      ((2 : ℝ) ^ (s * s)) ^ 2 * (((2 * binomialBound (g / (s + 1))) ^ s) ^ 2 *
+        binomialBound (g / (s + 1)))) := by
+  have hg : (s + 1) * (g / (s + 1)) + g % (s + 1) = g := Nat.div_add_mod g (s + 1)
+  constructor
+  · intro z
+    simpa only [hg] using prouhet_binomial_coeff_bound_remainder (g / (s + 1)) s
+      (g % (s + 1)) z
+  · simpa only [hg] using prouhet_binomial_L2Sq_bound_remainder (g / (s + 1)) s
+      (g % (s + 1))
+
+theorem binomialBound_quotient_le (g s : ℕ) :
+    binomialBound (g / (s + 1)) ≤
+      Real.sqrt (2 * ((s : ℝ) + 1) / ((g : ℝ) + 1)) := by
+  apply Real.sqrt_le_sqrt
+  have hn : g + 1 ≤ (g / (s + 1) + 1) * (s + 1) := by
+    have hmod := Nat.mod_lt g (by omega : 0 < s + 1)
+    have hdiv := Nat.div_add_mod g (s + 1)
+    nlinarith
+  have hnr : (g : ℝ) + 1 ≤ ((g / (s + 1) : ℕ) + (1 : ℝ)) * ((s : ℝ) + 1) := by
+    exact_mod_cast hn
+  apply (div_le_div_iff₀ (by positivity : (0 : ℝ) < (g / (s + 1) : ℕ) + 1)
+    (by positivity : (0 : ℝ) < g + 1)).mpr
+  nlinarith
+
+theorem prouhetSeries_succ_coeff (s : ℕ) (z : ℤ) :
+    (prouhetSeries (s + 1)).coeff z =
+    (prouhetSeries s).coeff z - (prouhetSeries s).coeff (z - (2 ^ s : ℕ)) := by
+  rw [prouhetSeries, Finset.prod_range_succ, mul_sub, mul_one]
+  change (prouhetSeries s - prouhetSeries s *
+    (AddMonoidAlgebra.single 1 1 : IntSeries) ^ (2 ^ s)).coeff z = _
+  rw [AddMonoidAlgebra.coeff_sub, Finsupp.sub_apply]
+  simp only [AddMonoidAlgebra.single_pow, one_pow, nsmul_eq_mul, mul_one,
+    AddMonoidAlgebra.coeff_mul_single_apply, sub_eq_add_neg]
+
+/-- The exact generating polynomial from the manuscript has unit signed
+    coefficients at every index 0,...,2^s-1, and no other coefficients. -/
+theorem prouhetSeries_coeff_spec (s : ℕ) :
+    (∀ z : ℤ, (z < 0 ∨ (2 ^ s : ℕ) ≤ z) → (prouhetSeries s).coeff z = 0) ∧
+    (∀ z : ℤ, 0 ≤ z → z < (2 ^ s : ℕ) → |(prouhetSeries s).coeff z| = 1) := by
+  induction s with
+  | zero =>
+    have hcoeff (z : ℤ) : (prouhetSeries 0).coeff z = if 0 = z then 1 else 0 := by
+      change (AddMonoidAlgebra.single 0 (1 : ℝ)).coeff z = _
+      exact Finsupp.single_apply
+    constructor
+    · intro z hz
+      rw [hcoeff, ite_eq_right]
+      norm_num only [pow_zero, Nat.cast_one] at hz
+      omega
+    · intro z h0 h1
+      have hz : z = 0 := by norm_num only [pow_zero, Nat.cast_one] at h1; omega
+      subst hz
+      rw [hcoeff, ite_eq_left rfl, abs_one]
+  | succ s ih =>
+    have hp : (0 : ℤ) < (2 ^ s : ℕ) := by positivity
+    have hd : ((2 ^ (s + 1) : ℕ) : ℤ) = 2 * (2 ^ s : ℕ) := by push_cast; ring
+    constructor
+    · intro z hz
+      rw [prouhetSeries_succ_coeff]
+      rcases hz with hz | hz
+      · rw [ih.1 z (Or.inl hz), ih.1 _ (Or.inl (by omega)), sub_zero]
+      · rw [hd] at hz
+        rw [ih.1 z (Or.inr (by omega)), ih.1 _ (Or.inr (by omega)), sub_zero]
+    · intro z h0 h1
+      rw [hd] at h1
+      rw [prouhetSeries_succ_coeff]
+      by_cases hz : z < (2 ^ s : ℕ)
+      · rw [ih.1 (z - (2 ^ s : ℕ)) (Or.inl (by omega)), sub_zero]
+        exact ih.2 z h0 hz
+      · rw [ih.1 z (Or.inr (by omega)), zero_sub, abs_neg]
+        exact ih.2 _ (by omega) (by omega)
+
+
+/-! ## Nonfair anti-concentration without an assumed Stirling estimate -/
+
+def binomialMass (n : ℕ) (p : ℝ) (k : ℕ) : ℝ :=
+  (n.choose k : ℝ) * p ^ k * (1 - p) ^ (n - k)
+
+theorem binomialMass_nonneg (n k : ℕ) {p : ℝ} (hp : p ∈ Set.Icc 0 1) :
+    0 ≤ binomialMass n p k := by
+  unfold binomialMass
+  positivity [hp.1, sub_nonneg.mpr hp.2]
+
+theorem sum_binomialMass (n : ℕ) (p : ℝ) :
+    ∑ k ∈ Finset.range (n + 1), binomialMass n p k = 1 := by
+  have h := add_pow p (1 - p) n
+  rw [show p + (1 - p) = 1 by ring, one_pow] at h
+  rw [h]
+  apply Finset.sum_congr rfl
+  intro k _
+  unfold binomialMass
+  ring
+
+theorem binomialMass_harmonic_step (n k : ℕ) (p : ℝ) :
+    ((n : ℝ) + 1) * p * (binomialMass n p k / ((k : ℝ) + 1)) =
+      binomialMass (n + 1) p (k + 1) := by
+  have hc : ((n : ℝ) + 1) * (n.choose k : ℝ) =
+      ((n + 1).choose (k + 1) : ℝ) * ((k : ℝ) + 1) := by
+    exact_mod_cast Nat.add_one_mul_choose_eq n k
+  unfold binomialMass
+  rw [Nat.add_sub_add_right, pow_succ]
+  field_simp
+  nlinarith [congrArg (fun x : ℝ => x * p ^ k * (1 - p) ^ (n - k) * p) hc]
+
+theorem binomialMass_harmonic_identity (n : ℕ) (p : ℝ) :
+    ((n : ℝ) + 1) * p *
+      (∑ k ∈ Finset.range (n + 1), binomialMass n p k / ((k : ℝ) + 1)) =
+    1 - (1 - p) ^ (n + 1) := by
+  rw [Finset.mul_sum]
+  simp_rw [binomialMass_harmonic_step]
+  have h := sum_binomialMass (n + 1) p
+  rw [Finset.sum_range_succ'] at h
+  simp only [binomialMass, Nat.choose_zero_right, Nat.cast_one, pow_zero,
+    mul_one, one_mul, Nat.sub_zero] at h ⊢
+  linarith
+
+theorem binomialMass_harmonic_le (n : ℕ) {p : ℝ} (hp : p ∈ Set.Ioc 0 1) :
+    (∑ k ∈ Finset.range (n + 1), binomialMass n p k / ((k : ℝ) + 1)) ≤
+      1 / (((n : ℝ) + 1) * p) := by
+  apply (le_div_iff₀ (mul_pos (by positivity) hp.1)).mpr
+  rw [mul_comm, binomialMass_harmonic_identity]
+  have h := pow_nonneg (sub_nonneg.mpr hp.2) (n + 1)
+  linarith
+
+/-- The weighted Cauchy--Schwarz step for mixing the fair estimates. -/
+theorem weighted_sum_sq_le {ι : Type*} (S : Finset ι) (w f : ι → ℝ)
+    (hw : ∀ i, 0 ≤ w i) :
+    (∑ i ∈ S, w i * f i) ^ 2 ≤ (∑ i ∈ S, w i) * ∑ i ∈ S, w i * f i ^ 2 := by
+  have heq (i : ι) : w i * f i = Real.sqrt (w i) * (Real.sqrt (w i) * f i) := by
+    rw [← mul_assoc, ← sq, Real.sq_sqrt (hw i)]
+  simp_rw [heq]
+  apply (Finset.sum_mul_sq_le_sq_mul_sq S (fun i => Real.sqrt (w i))
+    (fun i => Real.sqrt (w i) * f i)).trans_eq
+  simp only [mul_pow, Real.sq_sqrt (hw _)]
+
+theorem mixed_binomialBound_le (n : ℕ) {p : ℝ} (hp : p ∈ Set.Ioc 0 1) :
+    (∑ k ∈ Finset.range (n + 1), binomialMass n p k * binomialBound k) ≤
+      Real.sqrt (2 / (((n : ℝ) + 1) * p)) := by
+  have hw (k : ℕ) := binomialMass_nonneg n k (show p ∈ Set.Icc 0 1 from ⟨hp.1.le, hp.2⟩)
+  have hs := weighted_sum_sq_le (Finset.range (n + 1)) (binomialMass n p)
+    binomialBound hw
+  rw [sum_binomialMass, one_mul] at hs
+  have hb (k : ℕ) : binomialBound k ^ 2 = 2 / ((k : ℝ) + 1) :=
+    Real.sq_sqrt (by positivity)
+  simp_rw [hb] at hs
+  have heq : (∑ k ∈ Finset.range (n + 1), binomialMass n p k * (2 / ((k : ℝ) + 1))) =
+      2 * ∑ k ∈ Finset.range (n + 1), binomialMass n p k / ((k : ℝ) + 1) := by
+    rw [Finset.mul_sum]
+    apply Finset.sum_congr rfl
+    intro k _
+    ring
+  rw [heq] at hs
+  have hh := binomialMass_harmonic_le n hp
+  have hp0 := hp.1
+  have hbound : (∑ k ∈ Finset.range (n + 1), binomialMass n p k * binomialBound k) ^ 2 ≤
+      2 / (((n : ℝ) + 1) * p) := by
+    calc
+      _ ≤ 2 * ∑ k ∈ Finset.range (n + 1), binomialMass n p k / ((k : ℝ) + 1) := hs
+      _ ≤ 2 * (1 / (((n : ℝ) + 1) * p)) := mul_le_mul_of_nonneg_left hh (by norm_num)
+      _ = _ := by ring
+  have hsr := Real.sq_sqrt (show (0 : ℝ) ≤ 2 / (((n : ℝ) + 1) * p) by positivity)
+  have hpos := Real.sqrt_nonneg (2 / (((n : ℝ) + 1) * p))
+  nlinarith only [hbound, hsr, hpos]
+
+
+def mixedFairSeries (n : ℕ) (p : ℝ) : IntSeries :=
+  (p • fairSeries 1 + (1 - p) • (1 : IntSeries)) ^ n
+
+def biasedSeries (n : ℕ) (p : ℝ) : IntSeries :=
+  (AddMonoidAlgebra.single 1 p + AddMonoidAlgebra.single 0 (1 - p)) ^ n
+
+theorem mixedFairSeries_expansion (n : ℕ) (p : ℝ) :
+    mixedFairSeries n p =
+      ∑ k ∈ Finset.range (n + 1), binomialMass n p k • fairSeries k := by
+  rw [mixedFairSeries, add_pow]
+  apply Finset.sum_congr rfl
+  intro k _
+  simp only [smul_pow, one_pow, smul_mul_smul, mul_one, fairSeries, pow_one]
+  rw [mul_comm, ← nsmul_eq_mul, ← Nat.cast_smul_eq_nsmul ℝ, smul_smul]
+  congr 1
+  unfold binomialMass
+  ring
+
+theorem mixedFairSeries_eq_biased (n : ℕ) (p : ℝ) :
+    mixedFairSeries n p = biasedSeries n (p / 2) := by
+  unfold mixedFairSeries biasedSeries
+  congr 1
+  ext z
+  simp only [fairSeries, pow_one, smul_add, AddMonoidAlgebra.one_def, AddMonoidAlgebra.coeff_add,
+    Finsupp.add_apply, AddMonoidAlgebra.coeff_smul, Finsupp.smul_apply,
+    smul_eq_mul, AddMonoidAlgebra.coeff_single, Finsupp.single_apply]
+  split_ifs <;> first | omega | ring
+
+theorem mixedFairSeries_coeff_bound (n : ℕ) {p : ℝ} (hp : p ∈ Set.Ioc 0 1) (z : ℤ) :
+    |(mixedFairSeries n p).coeff z| ≤ Real.sqrt (2 / (((n : ℝ) + 1) * p)) := by
+  rw [mixedFairSeries_expansion, AddMonoidAlgebra.coeff_sum, Finsupp.finsetSum_apply]
+  simp only [AddMonoidAlgebra.coeff_smul, Finsupp.smul_apply, smul_eq_mul]
+  calc
+    _ ≤ ∑ k ∈ Finset.range (n + 1), |binomialMass n p k * (fairSeries k).coeff z| :=
+      Finset.abs_sum_le_sum_abs _ _
+    _ ≤ ∑ k ∈ Finset.range (n + 1), binomialMass n p k * binomialBound k := by
+      apply Finset.sum_le_sum
+      intro k _
+      have hw := binomialMass_nonneg n k (show p ∈ Set.Icc 0 1 from ⟨hp.1.le, hp.2⟩)
+      rw [abs_mul, abs_of_nonneg hw]
+      exact mul_le_mul_of_nonneg_left (fairSeries_coeff_bound k z) hw
+    _ ≤ _ := mixed_binomialBound_le n hp
+
+/-- At success probabilities at most 1/2, this gives the sharp variance
+    scale up to an absolute constant, using only the fair-coin estimate. -/
+theorem biasedSeries_coeff_bound_low (n : ℕ) {p : ℝ}
+    (hp : p ∈ Set.Ioc 0 (1 / 2)) (z : ℤ) :
+    |(biasedSeries n p).coeff z| ≤ Real.sqrt (1 / (((n : ℝ) + 1) * p)) := by
+  have h2p : 2 * p ∈ Set.Ioc 0 1 := ⟨by linarith [hp.1], by linarith [hp.2]⟩
+  have h := mixedFairSeries_coeff_bound n h2p z
+  rw [mixedFairSeries_eq_biased, show 2 * p / 2 = p by ring] at h
+  have heq : 2 / (((n : ℝ) + 1) * (2 * p)) = 1 / (((n : ℝ) + 1) * p) := by
+    field_simp
+  rwa [heq] at h
+
+
+theorem biasedSeries_sum (n : ℕ) (p : ℝ) :
+    biasedSeries n p = ∑ k ∈ Finset.range (n + 1),
+      AddMonoidAlgebra.single (k : ℤ) (binomialMass n p k) := by
+  rw [biasedSeries, add_pow]
+  apply Finset.sum_congr rfl
+  intro k _
+  rw [AddMonoidAlgebra.single_pow, AddMonoidAlgebra.single_pow,
+    AddMonoidAlgebra.natCast_def, AddMonoidAlgebra.single_mul_single,
+    AddMonoidAlgebra.single_mul_single]
+  simp only [nsmul_eq_mul, mul_one, mul_zero, add_zero]
+  congr 1
+  unfold binomialMass
+  ring
+
+theorem biasedSeries_coeff_nat (n k : ℕ) (p : ℝ) :
+    (biasedSeries n p).coeff (k : ℤ) = binomialMass n p k := by
+  rw [biasedSeries_sum, AddMonoidAlgebra.coeff_sum, Finsupp.finsetSum_apply]
+  by_cases hk : k ≤ n
+  · rw [Finset.sum_eq_single k]
+    · simp only [AddMonoidAlgebra.coeff_single, Finsupp.single_apply, ite_true]
+    · intro b _ hb
+      simp only [AddMonoidAlgebra.coeff_single, Finsupp.single_apply, Nat.cast_inj, ite_eq_right hb]
+    · intro h
+      exact (h (Finset.mem_range.mpr (by omega))).elim
+  · rw [Finset.sum_eq_zero]
+    · simp [binomialMass, Nat.choose_eq_zero_of_lt (by omega : n < k)]
+    · intro b hb
+      have hbk : b ≠ k := by simp only [Finset.mem_range] at hb; omega
+      simp only [AddMonoidAlgebra.coeff_single, Finsupp.single_apply, Nat.cast_inj, ite_eq_right hbk]
+
+theorem biasedSeries_coeff_neg (n : ℕ) (p : ℝ) (z : ℤ) (hz : z < 0) :
+    (biasedSeries n p).coeff z = 0 := by
+  rw [biasedSeries_sum, AddMonoidAlgebra.coeff_sum, Finsupp.finsetSum_apply]
+  apply Finset.sum_eq_zero
+  intro b _
+  have h : (b : ℤ) ≠ z := by omega
+  simp only [AddMonoidAlgebra.coeff_single, Finsupp.single_apply, ite_eq_right h]
+
+theorem biasedSeries_coeff_high (n : ℕ) (p : ℝ) (z : ℤ) (hz : (n : ℤ) < z) :
+    (biasedSeries n p).coeff z = 0 := by
+  rw [biasedSeries_sum, AddMonoidAlgebra.coeff_sum, Finsupp.finsetSum_apply]
+  apply Finset.sum_eq_zero
+  intro b hb
+  have h : (b : ℤ) ≠ z := by simp only [Finset.mem_range] at hb; omega
+  simp only [AddMonoidAlgebra.coeff_single, Finsupp.single_apply, ite_eq_right h]
+
+theorem binomialMass_symm (n k : ℕ) (p : ℝ) (hk : k ≤ n) :
+    binomialMass n (1 - p) (n - k) = binomialMass n p k := by
+  unfold binomialMass
+  rw [Nat.choose_symm hk, Nat.sub_sub_self hk, show 1 - (1 - p) = p by ring]
+  ring
+
+theorem biasedSeries_coeff_reflect (n : ℕ) (p : ℝ) (z : ℤ) :
+    (biasedSeries n (1 - p)).coeff ((n : ℤ) - z) = (biasedSeries n p).coeff z := by
+  by_cases hz : z < 0
+  · rw [biasedSeries_coeff_neg n p z hz,
+      biasedSeries_coeff_high n (1 - p) _ (by omega)]
+  by_cases hz' : (n : ℤ) < z
+  · rw [biasedSeries_coeff_high n p z hz',
+      biasedSeries_coeff_neg n (1 - p) _ (by omega)]
+  have h0 : 0 ≤ z := by omega
+  have hk : z.toNat ≤ n := by omega
+  have hzN : (z.toNat : ℤ) = z := Int.toNat_of_nonneg h0
+  have hsub : (n : ℤ) - z = ((n - z.toNat : ℕ) : ℤ) := by omega
+  rw [hsub, ← hzN, biasedSeries_coeff_nat, biasedSeries_coeff_nat]
+  exact binomialMass_symm n z.toNat p hk
+
+/-- Uniform nonfair-binomial anti-concentration.  This is the absolute
+    variance-scale upper bound used in the manuscript's rank kernels. -/
+theorem biasedSeries_coeff_bound (n : ℕ) {p : ℝ}
+    (hp : p ∈ Set.Ioo 0 1) (z : ℤ) :
+    |(biasedSeries n p).coeff z| ≤
+      Real.sqrt (1 / (((n : ℝ) + 1) * p * (1 - p))) := by
+  by_cases hlow : p ≤ 1 / 2
+  · apply (biasedSeries_coeff_bound_low n ⟨hp.1, hlow⟩ z).trans
+    apply Real.sqrt_le_sqrt
+    apply one_div_le_one_div_of_le
+    · exact mul_pos (mul_pos (by positivity) hp.1) (sub_pos.mpr hp.2)
+    · have hn : (0 : ℝ) ≤ n + 1 := by positivity
+      nlinarith [mul_nonneg hn (sq_nonneg p)]
+  · have hq : 1 - p ∈ Set.Ioc 0 (1 / 2) := ⟨by linarith [hp.2], by linarith⟩
+    rw [← biasedSeries_coeff_reflect n p z]
+    apply (biasedSeries_coeff_bound_low n hq ((n : ℤ) - z)).trans
+    apply Real.sqrt_le_sqrt
+    apply one_div_le_one_div_of_le
+    · exact mul_pos (mul_pos (by positivity) hp.1) (sub_pos.mpr hp.2)
+    · nlinarith [mul_nonneg (show (0 : ℝ) ≤ n + 1 by positivity)
+        (mul_nonneg (sub_nonneg.mpr hp.2.le) (sub_nonneg.mpr hp.2.le))]
+
+
+theorem binomialMass_mul_index (n k : ℕ) (p : ℝ) :
+    binomialMass (n + 1) p (k + 1) * ((k : ℝ) + 1) =
+      ((n : ℝ) + 1) * p * binomialMass n p k := by
+  have h := binomialMass_harmonic_step n k p
+  have hk : ((k : ℝ) + 1) ≠ 0 := by positivity
+  rw [← h]
+  field_simp
+
+theorem binomialMass_index_sum (n : ℕ) (p : ℝ) (F : ℕ → ℝ) :
+    (∑ k ∈ Finset.range (n + 2), binomialMass (n + 1) p k * (k : ℝ) * F k) =
+      ((n : ℝ) + 1) * p *
+        ∑ k ∈ Finset.range (n + 1), binomialMass n p k * F (k + 1) := by
+  rw [show n + 2 = (n + 1) + 1 by omega, Finset.sum_range_succ']
+  simp only [Nat.cast_zero, mul_zero, zero_mul, add_zero, Nat.cast_add, Nat.cast_one]
+  simp_rw [binomialMass_mul_index]
+  rw [Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro k _
+  ring
+
+theorem binomialMass_mean (n : ℕ) (p : ℝ) :
+    ∑ k ∈ Finset.range (n + 1), binomialMass n p k * (k : ℝ) = (n : ℝ) * p := by
+  cases n with
+  | zero => simp
+  | succ n =>
+    simpa only [mul_one, sum_binomialMass, Nat.cast_add, Nat.cast_one] using
+      binomialMass_index_sum n p (fun _ => 1)
+
+theorem binomialMass_second_moment (n : ℕ) (p : ℝ) :
+    ∑ k ∈ Finset.range (n + 1), binomialMass n p k * (k : ℝ) ^ 2 =
+      (n : ℝ) * ((n : ℝ) - 1) * p ^ 2 + (n : ℝ) * p := by
+  cases n with
+  | zero => simp
+  | succ n =>
+    have h := binomialMass_index_sum n p (fun k => k)
+    simp only [Nat.cast_add, Nat.cast_one] at h ⊢
+    simp_rw [mul_add, mul_one, Finset.sum_add_distrib, binomialMass_mean,
+      sum_binomialMass] at h
+    calc
+      _ = ∑ k ∈ Finset.range (n + 2), binomialMass (n + 1) p k * (k : ℝ) * (k : ℝ) := by
+        apply Finset.sum_congr rfl
+        intro k _
+        ring
+      _ = _ := by rw [h]; ring
+
+theorem binomialMass_variance (n : ℕ) (p : ℝ) :
+    (∑ k ∈ Finset.range (n + 1), binomialMass n p k * ((k : ℝ) - (n : ℝ) * p) ^ 2) =
+      (n : ℝ) * p * (1 - p) := by
+  calc
+    _ = (∑ k ∈ Finset.range (n + 1), binomialMass n p k * (k : ℝ) ^ 2) -
+        2 * (n : ℝ) * p * (∑ k ∈ Finset.range (n + 1), binomialMass n p k * (k : ℝ)) +
+        ((n : ℝ) * p) ^ 2 * (∑ k ∈ Finset.range (n + 1), binomialMass n p k) := by
+      rw [Finset.mul_sum, Finset.mul_sum, ← Finset.sum_sub_distrib, ← Finset.sum_add_distrib]
+      apply Finset.sum_congr rfl
+      intro k _
+      ring
+    _ = _ := by rw [binomialMass_second_moment, binomialMass_mean, sum_binomialMass]; ring
+
+theorem binomialMass_adjacent (n k : ℕ) (p : ℝ) (hk : k < n) :
+    ((k : ℝ) + 1) * (1 - p) * binomialMass n p (k + 1) =
+      ((n : ℝ) - k) * p * binomialMass n p k := by
+  have hc : (n.choose (k + 1) : ℝ) * ((k : ℝ) + 1) =
+      (n.choose k : ℝ) * ((n : ℝ) - (k : ℝ)) := by
+    simpa only [Nat.cast_mul, Nat.cast_add, Nat.cast_one, Nat.cast_sub hk.le] using
+      congrArg (fun t : ℕ => (t : ℝ)) (Nat.choose_succ_right_eq n k)
+  have hnk : n - k = (n - (k + 1)) + 1 := by omega
+  unfold binomialMass
+  rw [hnk, pow_succ, pow_succ]
+  linear_combination p ^ k * p * (1 - p) ^ (n - (k + 1)) * (1 - p) * hc
+
+theorem binomialMass_le_mode (n l k : ℕ) {p : ℝ} (hp : p ∈ Set.Ioo 0 1)
+    (hl : l ≤ n) (hmean : (n : ℝ) * p = l) :
+    binomialMass n p k ≤ binomialMass n p l := by
+  have h0 (j : ℕ) := binomialMass_nonneg n j ⟨hp.1.le, hp.2.le⟩
+  have hleft (j : ℕ) (hj : j < l) : binomialMass n p j ≤ binomialMass n p (j + 1) := by
+    have hrec := binomialMass_adjacent n j p (by omega)
+    have hjr : (j : ℝ) + 1 ≤ l := by exact_mod_cast hj
+    have hcoef : (0 : ℝ) < ((j : ℝ) + 1) * (1 - p) := by
+      exact mul_pos (by positivity) (sub_pos.mpr hp.2)
+    have hdiff : ((j : ℝ) + 1) * (1 - p) ≤ ((n : ℝ) - j) * p := by
+      nlinarith [hp.1]
+    have := mul_le_mul_of_nonneg_right hdiff (h0 j)
+    nlinarith only [hrec, this, hcoef]
+  have hright (j : ℕ) (hj : l ≤ j) : binomialMass n p (j + 1) ≤ binomialMass n p j := by
+    by_cases hjn : j < n
+    · have hrec := binomialMass_adjacent n j p hjn
+      have hjr : (l : ℝ) ≤ j := by exact_mod_cast hj
+      have hcoef : (0 : ℝ) < ((j : ℝ) + 1) * (1 - p) := by
+        exact mul_pos (by positivity) (sub_pos.mpr hp.2)
+      have hdiff : ((n : ℝ) - j) * p ≤ ((j : ℝ) + 1) * (1 - p) := by
+        nlinarith [hp.2]
+      have := mul_le_mul_of_nonneg_right hdiff (h0 j)
+      nlinarith only [hrec, this, hcoef]
+    · have hjn' : n < j + 1 := by omega
+      simp only [binomialMass, Nat.choose_eq_zero_of_lt hjn', Nat.cast_zero, zero_mul]
+      exact h0 j
+  rcases le_total k l with hk | hk
+  · have haux : ∀ b : ℕ, k ≤ b → b ≤ l → binomialMass n p k ≤ binomialMass n p b := by
+      intro b hkb
+      induction b, hkb using Nat.le_induction with
+      | base => intro _; exact le_rfl
+      | succ b hb ih =>
+        intro hbl
+        exact (ih (by omega)).trans (hleft b (by omega))
+    exact haux l hk le_rfl
+  · induction k, hk using Nat.le_induction with
+    | base => exact le_rfl
+    | succ j hj ih => exact (hright j hj).trans ih
+
+
+/-- A discrete law cannot have a very small maximum when its variance is bounded.
+The explicit constant is deliberately loose, avoiding any use of Stirling. -/
+theorem discrete_mode_lower (n l : ℕ) (f : ℕ → ℝ) (M v : ℝ)
+    (hf : ∀ k, 0 ≤ f k) (hsum : ∑ k ∈ Finset.range (n + 1), f k = 1)
+    (hmax : ∀ k, f k ≤ M)
+    (hvar : ∑ k ∈ Finset.range (n + 1), f k * ((k : ℝ) - l) ^ 2 = v)
+    (hv : 1 ≤ v) : 1 / (12 * Real.sqrt v) ≤ M := by
+  classical
+  let r : ℕ := ⌈2 * Real.sqrt v⌉₊
+  have hv0 : 0 ≤ v := by linarith
+  have hs : 1 ≤ Real.sqrt v := Real.one_le_sqrt.mpr hv
+  have hs2 := Real.sq_sqrt hv0
+  have hrlo : 2 * Real.sqrt v ≤ (r : ℝ) := Nat.le_ceil _
+  have hrhi : (r : ℝ) < 2 * Real.sqrt v + 1 := Nat.ceil_lt_add_one (by positivity)
+  have hrpos : (0 : ℝ) < r := by linarith
+  have hM : 0 ≤ M := (hf 0).trans (hmax 0)
+  let S := (Finset.range (n + 1)).filter (fun k => l - r ≤ k ∧ k ≤ l + r)
+  have hcard : S.card ≤ 2 * r + 1 := by
+    have hsub : S ⊆ Finset.Icc (l - r) (l + r) := by
+      intro k hk
+      exact Finset.mem_Icc.mpr (Finset.mem_filter.mp hk).2
+    have := Finset.card_le_card hsub
+    rw [Nat.card_Icc] at this
+    omega
+  have hout (k : ℕ) (hk : ¬(l - r ≤ k ∧ k ≤ l + r)) :
+      (r : ℝ) ^ 2 ≤ ((k : ℝ) - l) ^ 2 := by
+    by_cases hleft : k < l - r
+    · have hnat : k + r < l := by omega
+      have hreal : (k : ℝ) + r < l := by exact_mod_cast hnat
+      nlinarith [sq_nonneg ((l : ℝ) - k - r)]
+    · have hnat : l + r < k := by omega
+      have hreal : (l : ℝ) + r < k := by exact_mod_cast hnat
+      nlinarith [sq_nonneg ((k : ℝ) - l - r)]
+  have hpoint (k : ℕ) : (r : ℝ) ^ 2 * f k ≤
+      (if l - r ≤ k ∧ k ≤ l + r then (r : ℝ) ^ 2 * M else 0) +
+        f k * ((k : ℝ) - l) ^ 2 := by
+    split_ifs with hk
+    · have := mul_le_mul_of_nonneg_left (hmax k) (sq_nonneg (r : ℝ))
+      have := mul_nonneg (hf k) (sq_nonneg ((k : ℝ) - l))
+      linarith
+    · simpa only [zero_add, mul_comm] using mul_le_mul_of_nonneg_left (hout k hk) (hf k)
+  have htotal := Finset.sum_le_sum (fun k (_ : k ∈ Finset.range (n + 1)) => hpoint k)
+  rw [← Finset.mul_sum, hsum, mul_one, Finset.sum_add_distrib, hvar,
+    ← Finset.sum_filter] at htotal
+  change (r : ℝ) ^ 2 ≤ ∑ k ∈ S, (r : ℝ) ^ 2 * M + v at htotal
+  rw [Finset.sum_const, nsmul_eq_mul] at htotal
+  have hcardR : (S.card : ℝ) ≤ 2 * r + 1 := by exact_mod_cast hcard
+  have hbound := mul_le_mul_of_nonneg_right hcardR (mul_nonneg (sq_nonneg (r : ℝ)) hM)
+  have htotal' : (r : ℝ) ^ 2 ≤ (2 * r + 1) * ((r : ℝ) ^ 2 * M) + v := by
+    linarith
+  have hrv : 4 * v ≤ (r : ℝ) ^ 2 := by nlinarith [sq_nonneg ((r : ℝ) - 2 * Real.sqrt v)]
+  have hcancel : (3 : ℝ) / 4 ≤ (2 * r + 1) * M := by
+    have hrr : (0 : ℝ) < (r : ℝ) ^ 2 := sq_pos_of_pos hrpos
+    nlinarith only [htotal', hrv, hrr]
+  have hsize : 2 * (r : ℝ) + 1 ≤ 7 * Real.sqrt v := by linarith
+  have hfinal := mul_le_mul_of_nonneg_right hsize hM
+  apply (div_le_iff₀ (by positivity : 0 < 12 * Real.sqrt v)).mpr
+  nlinarith only [hcancel, hfinal]
+
+theorem binomialMass_mean_lower (n l : ℕ) {p : ℝ} (hp : p ∈ Set.Ioo 0 1)
+    (hl : l ≤ n) (hmean : (n : ℝ) * p = l) (hv : 1 ≤ (n : ℝ) * p * (1 - p)) :
+    1 / (12 * Real.sqrt ((n : ℝ) * p * (1 - p))) ≤ binomialMass n p l := by
+  apply discrete_mode_lower n l (binomialMass n p)
+    (binomialMass n p l) ((n : ℝ) * p * (1 - p))
+  · intro k; exact binomialMass_nonneg n k ⟨hp.1.le, hp.2.le⟩
+  · exact sum_binomialMass n p
+  · intro k; exact binomialMass_le_mode n l k hp hl hmean
+  · rw [← hmean]; exact binomialMass_variance n p
+  · exact hv
+
+
+theorem binomialMass_mean_inverse_sq (n l : ℕ) {p : ℝ} (hp : p ∈ Set.Ioo 0 1)
+    (hl : l ≤ n) (hmean : (n : ℝ) * p = l) (hv : 1 ≤ (n : ℝ) * p * (1 - p)) :
+    (binomialMass n p l)⁻¹ ^ 2 ≤ 144 * ((n : ℝ) * p * (1 - p)) := by
+  have hs : 0 < Real.sqrt ((n : ℝ) * p * (1 - p)) :=
+    Real.sqrt_pos.mpr (by linarith)
+  have hlow := binomialMass_mean_lower n l hp hl hmean hv
+  have hpos : 0 < binomialMass n p l := lt_of_lt_of_le (by positivity) hlow
+  have hi : (binomialMass n p l)⁻¹ ≤ 12 * Real.sqrt ((n : ℝ) * p * (1 - p)) := by
+    rw [← one_div]
+    apply (div_le_iff₀ hpos).mpr
+    have := (div_le_iff₀ (by positivity)).mp hlow
+    nlinarith only [this]
+  have his := sq_le_sq₀ (by positivity) (by positivity) |>.mpr hi
+  have hs2 := Real.sq_sqrt (show 0 ≤ (n : ℝ) * p * (1 - p) by linarith)
+  nlinarith only [his, hs2]
+
+theorem seriesL1_smul (a : ℝ) (f : IntSeries) :
+    seriesL1 (a • f) = |a| * seriesL1 f := by
+  unfold seriesL1
+  simp only [AddMonoidAlgebra.coeff_smul, Finsupp.smul_apply, smul_eq_mul, abs_mul]
+  exact tsum_mul_left
+
+theorem mixedFairSeries_difference_L1_bound (n : ℕ) {p : ℝ} (hp : p ∈ Set.Ioc 0 1) :
+    seriesL1 (firstDifference * mixedFairSeries n p) ≤
+      2 * Real.sqrt (2 / (((n : ℝ) + 1) * p)) := by
+  rw [mixedFairSeries_expansion, Finset.mul_sum]
+  simp_rw [mul_smul_comm]
+  calc
+    _ ≤ ∑ k ∈ Finset.range (n + 1),
+        seriesL1 (binomialMass n p k • (firstDifference * fairSeries k)) := seriesL1_sum_le _ _
+    _ ≤ ∑ k ∈ Finset.range (n + 1), binomialMass n p k * (2 * binomialBound k) := by
+      apply Finset.sum_le_sum
+      intro k _
+      have hw := binomialMass_nonneg n k ⟨hp.1.le, hp.2⟩
+      rw [seriesL1_smul, abs_of_nonneg hw]
+      exact mul_le_mul_of_nonneg_left (fairSeries_difference_L1_bound k) hw
+    _ = 2 * ∑ k ∈ Finset.range (n + 1), binomialMass n p k * binomialBound k := by
+      rw [Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro k _; ring
+    _ ≤ _ := mul_le_mul_of_nonneg_left (mixed_binomialBound_le n hp) (by norm_num)
+
+theorem biasedSeries_difference_L1_bound_low (n : ℕ) {p : ℝ}
+    (hp : p ∈ Set.Ioc 0 (1 / 2)) :
+    seriesL1 (firstDifference * biasedSeries n p) ≤
+      2 * Real.sqrt (1 / (((n : ℝ) + 1) * p)) := by
+  have h := mixedFairSeries_difference_L1_bound n
+    (show 2 * p ∈ Set.Ioc 0 1 from ⟨by linarith [hp.1], by linarith [hp.2]⟩)
+  rw [mixedFairSeries_eq_biased] at h
+  have hp2 : 2 * p / 2 = p := by ring
+  have heq : 2 / (((n : ℝ) + 1) * (2 * p)) = 1 / (((n : ℝ) + 1) * p) := by
+    have hp0 : p ≠ 0 := ne_of_gt hp.1
+    have hn0 : ((n : ℝ) + 1) ≠ 0 := by positivity
+    field_simp
+  rwa [hp2, heq] at h
+
+theorem biasedSeries_difference_reflect (n : ℕ) (p : ℝ) (z : ℤ) :
+    (firstDifference * biasedSeries n (1 - p)).coeff z =
+      -(firstDifference * biasedSeries n p).coeff ((n : ℤ) + 1 - z) := by
+  rw [firstDifference_coeff, firstDifference_coeff]
+  have h1 := biasedSeries_coeff_reflect n p ((n : ℤ) - z)
+  have h2 := biasedSeries_coeff_reflect n p ((n : ℤ) - (z - 1))
+  have hz1 : (n : ℤ) - ((n : ℤ) - z) = z := by ring
+  have hz2 : (n : ℤ) - ((n : ℤ) - (z - 1)) = z - 1 := by ring
+  rw [hz1] at h1
+  rw [hz2] at h2
+  rw [h1, h2]
+  have he1 : (n : ℤ) - (z - 1) = n + 1 - z := by ring
+  have he2 : (n : ℤ) + 1 - z - 1 = n - z := by ring
+  rw [he1, he2]
+  ring
+
+theorem biasedSeries_difference_L1_reflect (n : ℕ) (p : ℝ) :
+    seriesL1 (firstDifference * biasedSeries n (1 - p)) =
+      seriesL1 (firstDifference * biasedSeries n p) := by
+  unfold seriesL1
+  simp_rw [biasedSeries_difference_reflect, abs_neg]
+  exact (Equiv.subLeft ((n : ℤ) + 1)).tsum_eq
+    (fun z : ℤ => |(firstDifference * biasedSeries n p).coeff z|)
+
+theorem biasedSeries_difference_L1_bound (n : ℕ) {p : ℝ} (hp : p ∈ Set.Ioo 0 1) :
+    seriesL1 (firstDifference * biasedSeries n p) ≤
+      2 * Real.sqrt (1 / (((n : ℝ) + 1) * p * (1 - p))) := by
+  by_cases hlow : p ≤ 1 / 2
+  · apply (biasedSeries_difference_L1_bound_low n ⟨hp.1, hlow⟩).trans
+    apply mul_le_mul_of_nonneg_left _ (by norm_num)
+    apply Real.sqrt_le_sqrt
+    apply one_div_le_one_div_of_le
+    · exact mul_pos (mul_pos (by positivity) hp.1) (sub_pos.mpr hp.2)
+    · nlinarith [mul_nonneg (show (0 : ℝ) ≤ n + 1 by positivity) (sq_nonneg p)]
+  · have hq : 1 - p ∈ Set.Ioc 0 (1 / 2) := ⟨by linarith [hp.2], by linarith⟩
+    rw [← biasedSeries_difference_L1_reflect n p]
+    apply (biasedSeries_difference_L1_bound_low n hq).trans
+    apply mul_le_mul_of_nonneg_left _ (by norm_num)
+    apply Real.sqrt_le_sqrt
+    apply one_div_le_one_div_of_le
+    · exact mul_pos (mul_pos (by positivity) hp.1) (sub_pos.mpr hp.2)
+    · nlinarith [mul_nonneg (show (0 : ℝ) ≤ n + 1 by positivity)
+        (mul_nonneg (sub_nonneg.mpr hp.2.le) (sub_nonneg.mpr hp.2.le))]
+
+
+def biasedBinomialBound (n : ℕ) (p : ℝ) : ℝ :=
+  Real.sqrt (1 / (((n : ℝ) + 1) * p * (1 - p)))
+
+theorem biasedSeries_L1 (n : ℕ) {p : ℝ} (hp : p ∈ Set.Icc 0 1) :
+    seriesL1 (biasedSeries n p) = 1 := by
+  rw [seriesL1_eq_sum_range _ n (biasedSeries_coeff_neg n p)]
+  · simp_rw [biasedSeries_coeff_nat, abs_of_nonneg (binomialMass_nonneg _ _ hp)]
+    exact sum_binomialMass n p
+  · intro k hk
+    exact biasedSeries_coeff_high n p k (by exact_mod_cast hk)
+
+theorem biasedSeries_L2Sq_bound (n : ℕ) {p : ℝ} (hp : p ∈ Set.Ioo 0 1) :
+    seriesL2Sq (biasedSeries n p) ≤ biasedBinomialBound n p := by
+  have := seriesL2Sq_le_L1_mul_bound (biasedSeries n p) _ (biasedSeries_coeff_bound n hp)
+  simpa only [biasedSeries_L1 n ⟨hp.1.le, hp.2.le⟩, one_mul, biasedBinomialBound] using this
+
+theorem biasedSeries_add (m n : ℕ) (p : ℝ) :
+    biasedSeries (m + n) p = biasedSeries m p * biasedSeries n p := by
+  exact pow_add _ _ _
+
+theorem biasedSeries_mul_power (n s : ℕ) (p : ℝ) :
+    firstDifference ^ s * biasedSeries ((s + 1) * n) p =
+      (firstDifference * biasedSeries n p) ^ s * biasedSeries n p := by
+  simp only [biasedSeries, add_mul, one_mul, pow_add, pow_mul, mul_pow]
+  ring
+
+theorem biased_difference_coeff_bound (n s : ℕ) {p : ℝ} (hp : p ∈ Set.Ioo 0 1) (z : ℤ) :
+    |(firstDifference ^ s * biasedSeries ((s + 1) * n) p).coeff z| ≤
+      (2 * biasedBinomialBound n p) ^ s * biasedBinomialBound n p := by
+  rw [biasedSeries_mul_power]
+  apply (series_coeff_mul_bound _ _ _ (biasedSeries_coeff_bound n hp) z).trans
+  apply mul_le_mul_of_nonneg_right _ (Real.sqrt_nonneg _)
+  exact (seriesL1_pow_le _ _).trans
+    (pow_le_pow_left₀ (seriesL1_nonneg _) (biasedSeries_difference_L1_bound n hp) s)
+
+theorem biased_difference_L2Sq_bound (n s : ℕ) {p : ℝ} (hp : p ∈ Set.Ioo 0 1) :
+    seriesL2Sq (firstDifference ^ s * biasedSeries ((s + 1) * n) p) ≤
+      ((2 * biasedBinomialBound n p) ^ s) ^ 2 * biasedBinomialBound n p := by
+  rw [biasedSeries_mul_power]
+  apply (seriesL2Sq_mul_le _ _).trans
+  apply mul_le_mul
+  · apply pow_le_pow_left₀ (seriesL1_nonneg _)
+    exact (seriesL1_pow_le _ _).trans
+      (pow_le_pow_left₀ (seriesL1_nonneg _) (biasedSeries_difference_L1_bound n hp) s)
+  · exact biasedSeries_L2Sq_bound n hp
+  · exact seriesL2Sq_nonneg _
+  · exact sq_nonneg _
+
+theorem biased_difference_coeff_bound_remainder (n s r : ℕ) {p : ℝ}
+    (hp : p ∈ Set.Ioo 0 1) (z : ℤ) :
+    |(firstDifference ^ s * biasedSeries ((s + 1) * n + r) p).coeff z| ≤
+      (2 * biasedBinomialBound n p) ^ s * biasedBinomialBound n p := by
+  rw [biasedSeries_add, ← mul_assoc, mul_comm (firstDifference ^ s * _)]
+  simpa only [biasedSeries_L1 r ⟨hp.1.le, hp.2.le⟩, one_mul] using
+    series_coeff_mul_bound (biasedSeries r p) (firstDifference ^ s * biasedSeries ((s + 1) * n) p)
+      _ (biased_difference_coeff_bound n s hp) z
+
+theorem biased_difference_L2Sq_bound_remainder (n s r : ℕ) {p : ℝ}
+    (hp : p ∈ Set.Ioo 0 1) :
+    seriesL2Sq (firstDifference ^ s * biasedSeries ((s + 1) * n + r) p) ≤
+      ((2 * biasedBinomialBound n p) ^ s) ^ 2 * biasedBinomialBound n p := by
+  rw [biasedSeries_add, ← mul_assoc, mul_comm (firstDifference ^ s * _)]
+  calc
+    _ ≤ seriesL1 (biasedSeries r p) ^ 2 *
+        seriesL2Sq (firstDifference ^ s * biasedSeries ((s + 1) * n) p) := seriesL2Sq_mul_le _ _
+    _ ≤ _ := by simpa only [biasedSeries_L1 r ⟨hp.1.le, hp.2.le⟩, one_pow, one_mul] using
+        biased_difference_L2Sq_bound n s hp
+
+/-- All fixed-order nonfair binomial differences, at the exact variance scale.
+This is the analytic estimate needed for the initial conditional rank kernels. -/
+theorem biased_difference_smoothing (g s : ℕ) {p : ℝ} (hp : p ∈ Set.Ioo 0 1) :
+    (∀ z : ℤ, |(firstDifference ^ s * biasedSeries g p).coeff z| ≤
+      (2 * biasedBinomialBound (g / (s + 1)) p) ^ s *
+        biasedBinomialBound (g / (s + 1)) p) ∧
+    (seriesL2Sq (firstDifference ^ s * biasedSeries g p) ≤
+      ((2 * biasedBinomialBound (g / (s + 1)) p) ^ s) ^ 2 *
+        biasedBinomialBound (g / (s + 1)) p) := by
+  have hg : (s + 1) * (g / (s + 1)) + g % (s + 1) = g := Nat.div_add_mod g (s + 1)
+  constructor
+  · intro z
+    simpa only [hg] using biased_difference_coeff_bound_remainder (g / (s + 1)) s
+      (g % (s + 1)) hp z
+  · simpa only [hg] using biased_difference_L2Sq_bound_remainder (g / (s + 1)) s
+      (g % (s + 1)) hp
+
+theorem biasedBinomialBound_quotient_le (g s : ℕ) {p : ℝ} (hp : p ∈ Set.Ioo 0 1) :
+    biasedBinomialBound (g / (s + 1)) p ≤
+      Real.sqrt (((s : ℝ) + 1) / (((g : ℝ) + 1) * p * (1 - p))) := by
+  unfold biasedBinomialBound
+  apply Real.sqrt_le_sqrt
+  have hpq : 0 < p * (1 - p) := mul_pos hp.1 (sub_pos.mpr hp.2)
+  have hnat : g + 1 ≤ (s + 1) * (g / (s + 1) + 1) := by
+    have hdiv := Nat.div_add_mod g (s + 1)
+    have hmod : g % (s + 1) < s + 1 := Nat.mod_lt g (by omega)
+    calc
+      g + 1 = (s + 1) * (g / (s + 1)) + g % (s + 1) + 1 := by omega
+      _ ≤ (s + 1) * (g / (s + 1)) + (s + 1) := by omega
+      _ = _ := by ring
+  have hreal : (g : ℝ) + 1 ≤ ((s : ℝ) + 1) * ((g / (s + 1) : ℕ) + 1 : ℝ) := by
+    exact_mod_cast hnat
+  have hp0 := hp.1
+  have hq0 := sub_pos.mpr hp.2
+  apply (div_le_div_iff₀ (by positivity) (by positivity)).mpr
+  nlinarith only [mul_le_mul_of_nonneg_right hreal hpq.le]
+
+/-! ## Gaussian overlap by exponential tilting of the exact binomial law -/
+
+theorem binomialMass_tilt (n k : ℕ) {p a : ℝ} (hp : p ∈ Set.Ioo 0 1)
+    (ha : 0 < a) :
+    binomialMass n p k * a ^ k =
+      (1 - p + p * a) ^ n * binomialMass n (p * a / (1 - p + p * a)) k := by
+  by_cases hk : k ≤ n
+  · have hA : 0 < 1 - p + p * a := by positivity [sub_pos.mpr hp.2, hp.1]
+    have hsub : 1 - p * a / (1 - p + p * a) = (1 - p) / (1 - p + p * a) := by
+      field_simp; ring
+    unfold binomialMass
+    rw [hsub, div_pow, div_pow, mul_pow]
+    have hpow : (1 - p + p * a) ^ n =
+        (1 - p + p * a) ^ k * (1 - p + p * a) ^ (n - k) := by
+      rw [← pow_add, Nat.add_sub_of_le hk]
+    rw [hpow]
+    field_simp
+  · simp [binomialMass, Nat.choose_eq_zero_of_lt (by omega : n < k)]
+
+theorem binomialMass_tilt_product (g h k : ℕ) {p a : ℝ} (hp : p ∈ Set.Ioo 0 1)
+    (ha : 0 < a) :
+    binomialMass g p k * binomialMass h p k =
+      ((1 - p + p * a) ^ g * (1 - p + p / a) ^ h) *
+        (binomialMass g (p * a / (1 - p + p * a)) k *
+          binomialMass h ((p / a) / (1 - p + p / a)) k) := by
+  have h1 := binomialMass_tilt g k hp ha
+  have h2 := binomialMass_tilt h k hp (inv_pos.mpr ha)
+  have hid := congrArg₂ (· * ·) h1 h2
+  simp only [inv_pow, ← div_eq_mul_inv] at hid
+  have ha0 : a ^ k ≠ 0 := pow_ne_zero _ ha.ne'
+  calc
+    _ = (binomialMass g p k * a ^ k) * (binomialMass h p k / a ^ k) := by field_simp
+    _ = _ := by convert hid using 1; ring
+
+/-- Tilting on opposite sides retains an anti-concentration factor as well
+as the Gaussian penalty for different numbers of trials. -/
+theorem binomial_overlap_tilt_bound (g h : ℕ) {p a : ℝ}
+    (hp : p ∈ Set.Ioo 0 1) (ha : 0 < a) :
+    (∑ k ∈ Finset.range (h + 1), binomialMass g p k * binomialMass h p k) ≤
+      ((1 - p + p * a) ^ g * (1 - p + p / a) ^ h) *
+        biasedBinomialBound g (p * a / (1 - p + p * a)) := by
+  have hA : 0 < 1 - p + p * a := by positivity [sub_pos.mpr hp.2, hp.1]
+  have hB : 0 < 1 - p + p / a := by positivity [sub_pos.mpr hp.2, hp.1]
+  have hpA : p * a / (1 - p + p * a) ∈ Set.Ioo (0 : ℝ) 1 := by
+    constructor
+    · positivity [hp.1]
+    · apply (div_lt_one hA).mpr; linarith [hp.2]
+  have hpB : (p / a) / (1 - p + p / a) ∈ Set.Ioo (0 : ℝ) 1 := by
+    constructor
+    · positivity [hp.1]
+    · apply (div_lt_one hB).mpr; linarith [hp.2]
+  simp_rw [binomialMass_tilt_product g h _ hp ha, ← Finset.mul_sum]
+  apply mul_le_mul_of_nonneg_left _ (by positivity)
+  calc
+    _ ≤ ∑ k ∈ Finset.range (h + 1),
+        biasedBinomialBound g (p * a / (1 - p + p * a)) *
+          binomialMass h ((p / a) / (1 - p + p / a)) k := by
+      apply Finset.sum_le_sum
+      intro k _
+      apply mul_le_mul_of_nonneg_right _ (binomialMass_nonneg _ _ ⟨hpB.1.le, hpB.2.le⟩)
+      have hb := biasedSeries_coeff_bound g hpA (k : ℤ)
+      rw [biasedSeries_coeff_nat, abs_of_nonneg (binomialMass_nonneg _ _ ⟨hpA.1.le, hpA.2.le⟩)] at hb
+      exact hb
+    _ = _ := by rw [← Finset.mul_sum, sum_binomialMass, mul_one]
+
+
+theorem binomial_tilt_variance {p t : ℝ} (hp : p ∈ Set.Ioo 0 1)
+    (ht : t ∈ Set.Icc 0 (1 / 4 : ℝ)) :
+    p * (1 - p) / 2 ≤
+      (p * (1 - t) / (1 - p + p * (1 - t))) *
+        (1 - p * (1 - t) / (1 - p + p * (1 - t))) := by
+  have hA : 0 < 1 - p + p * (1 - t) := by
+    have := hp.1; have := hp.2; have := ht.2
+    positivity [sub_pos.mpr hp.2, show (0 : ℝ) < 1 - t by linarith]
+  have hA1 : 1 - p + p * (1 - t) ≤ 1 := by nlinarith [mul_nonneg hp.1.le ht.1]
+  have hAsq : (1 - p + p * (1 - t)) ^ 2 ≤ 1 := by nlinarith
+  have hpq : 0 < p * (1 - p) := mul_pos hp.1 (sub_pos.mpr hp.2)
+  have heq : (p * (1 - t) / (1 - p + p * (1 - t))) *
+        (1 - p * (1 - t) / (1 - p + p * (1 - t))) =
+      p * (1 - p) * (1 - t) / (1 - p + p * (1 - t)) ^ 2 := by
+    field_simp; ring
+  rw [heq]
+  apply (le_div_iff₀ (sq_pos_of_pos hA)).mpr
+  have hh := mul_le_mul_of_nonneg_left hAsq hpq.le
+  nlinarith [ht.2]
+
+theorem binomial_tilt_normalizer (g h : ℕ) {p : ℝ}
+    (hp : p ∈ Set.Ico (1 / 2 : ℝ) 1) (hhg : h ≤ g) (hS : 0 < g + h) :
+    let t : ℝ := ((g : ℝ) - h) / (4 * ((g : ℝ) + h))
+    ((1 - p + p * (1 - t)) ^ g * (1 - p + p / (1 - t)) ^ h) ≤
+      Real.exp (-((g : ℝ) - h) ^ 2 / (16 * ((g : ℝ) + h))) := by
+  dsimp only
+  set t : ℝ := ((g : ℝ) - h) / (4 * ((g : ℝ) + h)) with htdef
+  have hSr : (0 : ℝ) < (g : ℝ) + h := by exact_mod_cast hS
+  have hgr : (h : ℝ) ≤ g := by exact_mod_cast hhg
+  have hd : (0 : ℝ) ≤ (g : ℝ) - h := sub_nonneg.mpr hgr
+  have ht0 : 0 ≤ t := div_nonneg hd (by positivity)
+  have ht1 : t ≤ 1 / 4 := by
+    apply (div_le_iff₀ (by positivity : (0 : ℝ) < 4 * ((g : ℝ) + h))).mpr
+    nlinarith [show (0 : ℝ) ≤ h by positivity]
+  have htlt : 0 < 1 - t := by linarith
+  have hp0 : 0 < p := by linarith [hp.1]
+  have hA : 0 < 1 - p + p * (1 - t) := by positivity [sub_pos.mpr hp.2]
+  have hB : 0 < 1 - p + p / (1 - t) := by positivity [sub_pos.mpr hp.2]
+  have hlogA : Real.log (1 - p + p * (1 - t)) ≤ -p * t := by
+    have := Real.log_le_sub_one_of_pos hA
+    nlinarith only [this]
+  have hlogB : Real.log (1 - p + p / (1 - t)) ≤ p * t / (1 - t) := by
+    have := Real.log_le_sub_one_of_pos hB
+    have heq : 1 - p + p / (1 - t) - 1 = p * t / (1 - t) := by field_simp; ring
+    rwa [heq] at this
+  have hlog : Real.log ((1 - p + p * (1 - t)) ^ g * (1 - p + p / (1 - t)) ^ h) ≤
+      -(p * ((g : ℝ) - h) * t) + 2 * p * h * t ^ 2 := by
+    rw [Real.log_mul (pow_pos hA _).ne' (pow_pos hB _).ne', Real.log_pow, Real.log_pow]
+    have hb : (h : ℝ) * (p * t / (1 - t)) ≤ (h : ℝ) * (p * t + 2 * p * t ^ 2) := by
+      apply mul_le_mul_of_nonneg_left _ (by positivity)
+      apply (div_le_iff₀ htlt).mpr
+      have hh := mul_nonneg (mul_nonneg hp0.le (sq_nonneg t)) (show 0 ≤ 1 - 2 * t by linarith)
+      nlinarith only [hh]
+    have ha := mul_le_mul_of_nonneg_left hlogA (show (0 : ℝ) ≤ g by positivity)
+    have hb' := mul_le_mul_of_nonneg_left hlogB (show (0 : ℝ) ≤ h by positivity)
+    linarith
+  have hht : 4 * (h : ℝ) * t ≤ (g : ℝ) - h := by
+    rw [htdef, ← mul_div_assoc]
+    apply (div_le_iff₀ (by positivity : (0 : ℝ) < 4 * ((g : ℝ) + h))).mpr
+    have hm := mul_nonneg (show (0 : ℝ) ≤ g by positivity) hd
+    nlinarith only [hm]
+  have hhalf : -(p * ((g : ℝ) - h) * t) + 2 * p * h * t ^ 2 ≤
+      -(p * ((g : ℝ) - h) * t) / 2 := by
+    have hm := mul_le_mul_of_nonneg_right hht (mul_nonneg hp0.le ht0)
+    nlinarith only [hm]
+  have hlast : -(p * ((g : ℝ) - h) * t) / 2 ≤
+      -((g : ℝ) - h) ^ 2 / (16 * ((g : ℝ) + h)) := by
+    rw [htdef]
+    apply (div_le_div_iff₀ (by norm_num : (0 : ℝ) < 2) (by positivity)).mpr
+    field_simp
+    have hm := mul_nonneg (show 0 ≤ 2 * p - 1 by linarith [hp.1]) (sq_nonneg ((g : ℝ) - h))
+    nlinarith only [hm]
+  have hexp := Real.exp_le_exp.mpr (hlog.trans (hhalf.trans hlast))
+  rwa [Real.exp_log (mul_pos (pow_pos hA _) (pow_pos hB _))] at hexp
+
+
+/-- The Gaussian cross-overlap estimate used by the conditional rank-kernel
+square-sum argument. The constant is absolute, also as p tends to one. -/
+theorem binomial_gaussian_overlap (g h : ℕ) {p : ℝ}
+    (hp : p ∈ Set.Ico (1 / 2 : ℝ) 1) (hhg : h ≤ g) (hS : 0 < g + h) :
+    (∑ k ∈ Finset.range (h + 1), binomialMass g p k * binomialMass h p k) ≤
+      Real.sqrt (2 / (((g : ℝ) + 1) * p * (1 - p))) *
+        Real.exp (-((g : ℝ) - h) ^ 2 / (16 * ((g : ℝ) + h))) := by
+  let t : ℝ := ((g : ℝ) - h) / (4 * ((g : ℝ) + h))
+  have hSr : (0 : ℝ) < (g : ℝ) + h := by exact_mod_cast hS
+  have hgr : (h : ℝ) ≤ g := by exact_mod_cast hhg
+  have ht0 : 0 ≤ t := div_nonneg (sub_nonneg.mpr hgr) (by positivity)
+  have ht1 : t ≤ 1 / 4 := by
+    apply (div_le_iff₀ (by positivity : (0 : ℝ) < 4 * ((g : ℝ) + h))).mpr
+    nlinarith [show (0 : ℝ) ≤ h by positivity]
+  have htlt : 0 < 1 - t := by linarith
+  have hp0 : 0 < p := by linarith [hp.1]
+  have hp' : p ∈ Set.Ioo (0 : ℝ) 1 := ⟨hp0, hp.2⟩
+  have hA : 0 < 1 - p + p * (1 - t) := by positivity [sub_pos.mpr hp.2]
+  have hB : 0 < 1 - p + p / (1 - t) := by positivity [sub_pos.mpr hp.2]
+  have hv := binomial_tilt_variance hp' ⟨ht0, ht1⟩
+  have hpq : 0 < p * (1 - p) := mul_pos hp0 (sub_pos.mpr hp.2)
+  have hx : 0 < (p * (1 - t) / (1 - p + p * (1 - t))) *
+      (1 - p * (1 - t) / (1 - p + p * (1 - t))) := lt_of_lt_of_le (by positivity) hv
+  have hb : biasedBinomialBound g (p * (1 - t) / (1 - p + p * (1 - t))) ≤
+      Real.sqrt (2 / (((g : ℝ) + 1) * p * (1 - p))) := by
+    apply Real.sqrt_le_sqrt
+    have hg : (0 : ℝ) < (g : ℝ) + 1 := by positivity
+    apply (div_le_div_iff₀ (by nlinarith only [mul_pos hg hx]) (by nlinarith only [mul_pos hg hpq])).mpr
+    have hm := mul_le_mul_of_nonneg_left hv hg.le
+    nlinarith only [hm]
+  have hn := binomial_tilt_normalizer g h hp hhg hS
+  change ((1 - p + p * (1 - t)) ^ g * (1 - p + p / (1 - t)) ^ h) ≤ _ at hn
+  have he := binomial_overlap_tilt_bound g h hp' htlt
+  apply he.trans
+  have hbound := mul_le_mul hn hb (Real.sqrt_nonneg _) (Real.exp_nonneg _)
+  simpa only [mul_comm] using hbound
+
+
+theorem sqrt_times_gaussian_le {x : ℝ} (hx : 0 ≤ x) :
+    Real.sqrt (x + 1) * Real.exp (-x / 96) ≤ 16 := by
+  have he : Real.exp (-x / 96) ^ 2 = 1 / Real.exp (x / 48) := by
+    rw [pow_two, ← Real.exp_add]
+    rw [show -x / 96 + -x / 96 = -(x / 48) by ring, Real.exp_neg, one_div]
+  have hs : (Real.sqrt (x + 1) * Real.exp (-x / 96)) ^ 2 ≤ 256 := by
+    rw [mul_pow, Real.sq_sqrt (by linarith), he, mul_one_div]
+    apply (div_le_iff₀ (Real.exp_pos _)).mpr
+    have hh := Real.add_one_le_exp (x / 48)
+    nlinarith only [hh, hx]
+  nlinarith only [hs]
+
+theorem sqrt_recip_product_le {a b : ℝ} (ha : 0 < a) (hb : 0 < b) (hab : a ≤ 2 * b) :
+    Real.sqrt (1 / a) * Real.sqrt (1 / b) ≤ 2 / a := by
+  rw [← Real.sqrt_mul (by positivity), Real.sqrt_le_iff]
+  refine ⟨by positivity, ?_⟩
+  rw [div_pow]
+  apply (le_div_iff₀ (sq_pos_of_pos ha)).mpr
+  field_simp
+  nlinarith
+
+theorem binomialMass_le_bound (n k : ℕ) {p : ℝ} (hp : p ∈ Set.Ioo 0 1) :
+    binomialMass n p k ≤ biasedBinomialBound n p := by
+  have h := biasedSeries_coeff_bound n hp (k : ℤ)
+  rw [biasedSeries_coeff_nat, abs_of_nonneg (binomialMass_nonneg _ _ ⟨hp.1.le, hp.2.le⟩)] at h
+  exact h
+
+/-- A uniform product-supremum estimate, including very unequal trial
+counts. This justifies eliminating a largest gap in the rank kernel. -/
+theorem binomial_product_sup (g h k : ℕ) {p : ℝ}
+    (hp : p ∈ Set.Ico (1 / 2 : ℝ) 1) :
+    binomialMass g p k * binomialMass h p k ≤
+      32 / (((g : ℝ) + 1) * p * (1 - p)) := by
+  have hp0 : 0 < p := by linarith [hp.1]
+  have hp' : p ∈ Set.Ioo (0 : ℝ) 1 := ⟨hp0, hp.2⟩
+  have hpq : 0 < p * (1 - p) := mul_pos hp0 (sub_pos.mpr hp.2)
+  have hpq1 : p * (1 - p) ≤ 1 := by nlinarith only [sq_nonneg (p - 1 / 2)]
+  have hden : 0 < ((g : ℝ) + 1) * p * (1 - p) := by nlinarith only [mul_pos (show (0 : ℝ) < (g : ℝ) + 1 by positivity) hpq]
+  by_cases hnear : g ≤ 2 * h
+  · have hdenh : 0 < ((h : ℝ) + 1) * p * (1 - p) := by nlinarith only [mul_pos (show (0 : ℝ) < (h : ℝ) + 1 by positivity) hpq]
+    have hgh : ((g : ℝ) + 1) * p * (1 - p) ≤ 2 * (((h : ℝ) + 1) * p * (1 - p)) := by
+      have hcast : (g : ℝ) ≤ 2 * h := by exact_mod_cast hnear
+      have hm := mul_le_mul_of_nonneg_right (show (g : ℝ) + 1 ≤ 2 * ((h : ℝ) + 1) by linarith) hpq.le
+      nlinarith only [hm]
+    calc
+      _ ≤ biasedBinomialBound g p * biasedBinomialBound h p :=
+        mul_le_mul (binomialMass_le_bound g k hp') (binomialMass_le_bound h k hp')
+          (binomialMass_nonneg _ _ ⟨hp0.le, hp.2.le⟩) (Real.sqrt_nonneg _)
+      _ ≤ 2 / (((g : ℝ) + 1) * p * (1 - p)) := sqrt_recip_product_le hden hdenh hgh
+      _ ≤ _ := div_le_div_of_nonneg_right (by norm_num) hden.le
+  · by_cases hk : k ≤ h
+    · have hhg : h ≤ g := by omega
+      have hS : 0 < g + h := by omega
+      have hsum : binomialMass g p k * binomialMass h p k ≤
+          ∑ j ∈ Finset.range (h + 1), binomialMass g p j * binomialMass h p j := by
+        exact Finset.single_le_sum (s := Finset.range (h + 1)) (a := k) (fun j _ => mul_nonneg
+          (binomialMass_nonneg g j ⟨hp0.le, hp.2.le⟩)
+          (binomialMass_nonneg h j ⟨hp0.le, hp.2.le⟩)) (Finset.mem_range.mpr (by omega))
+      have hgauss := binomial_gaussian_overlap g h hp hhg hS
+      have hSr : (0 : ℝ) < (g : ℝ) + h := by exact_mod_cast hS
+      have hfar : 2 * (h : ℝ) < g := by exact_mod_cast (show 2 * h < g by omega)
+      have hexp : Real.exp (-((g : ℝ) - h) ^ 2 / (16 * ((g : ℝ) + h))) ≤ Real.exp (-(g : ℝ) / 96) := by
+        apply Real.exp_le_exp.mpr
+        apply (div_le_div_iff₀ (by positivity : (0 : ℝ) < 16 * ((g : ℝ) + h)) (by norm_num : (0 : ℝ) < 96)).mpr
+        have hn := mul_nonneg (show 0 ≤ (g : ℝ) - 2 * h by linarith) (show 0 ≤ 5 * (g : ℝ) - 3 * h by linarith)
+        nlinarith only [hn, sq_nonneg (h : ℝ)]
+      have hcoeff : Real.sqrt (2 / (((g : ℝ) + 1) * p * (1 - p))) ≤
+          (2 / (((g : ℝ) + 1) * p * (1 - p))) * Real.sqrt ((g : ℝ) + 1) := by
+        rw [Real.sqrt_le_iff]
+        refine ⟨by positivity, ?_⟩
+        rw [mul_pow, Real.sq_sqrt (by positivity), div_pow]
+        have hgpos : (0 : ℝ) < (g : ℝ) + 1 := by positivity
+        have hqpos : 0 < 1 - p := sub_pos.mpr hp.2
+        field_simp
+        have hm := mul_le_mul_of_nonneg_left hpq1 (show (0 : ℝ) ≤ (g : ℝ) + 1 by positivity)
+        nlinarith only [hm, hpq1]
+      calc
+        _ ≤ Real.sqrt (2 / (((g : ℝ) + 1) * p * (1 - p))) *
+            Real.exp (-((g : ℝ) - h) ^ 2 / (16 * ((g : ℝ) + h))) := hsum.trans hgauss
+        _ ≤ ((2 / (((g : ℝ) + 1) * p * (1 - p))) * Real.sqrt ((g : ℝ) + 1)) *
+            Real.exp (-(g : ℝ) / 96) := mul_le_mul hcoeff hexp (Real.exp_nonneg _) (by positivity)
+        _ ≤ (2 / (((g : ℝ) + 1) * p * (1 - p))) * 16 := by
+          rw [mul_assoc]
+          exact mul_le_mul_of_nonneg_left (sqrt_times_gaussian_le (by positivity)) (by positivity)
+        _ = _ := by ring
+    · simp [binomialMass, Nat.choose_eq_zero_of_lt (by omega : h < k)]
+      positivity
+
+end TraceReconstruction
+
+namespace TraceReconstruction
+
+/-- Exact moment generating function, including both endpoint channels. -/
+theorem binomialMass_mgf (n : ℕ) (p t : ℝ) :
+    (∑ k ∈ Finset.range (n + 1), binomialMass n p k * Real.exp (t * k)) =
+      (1 - p + p * Real.exp t) ^ n := by
+  rw [add_comm (1 - p), add_pow]
+  apply Finset.sum_congr rfl
+  intro k _
+  rw [show t * (k : ℝ) = (k : ℝ) * t by ring, Real.exp_nat_mul, mul_pow]
+  unfold binomialMass
+  ring
+
+theorem binomialMass_mgf_le (n : ℕ) {p : ℝ} (hp : p ∈ Set.Icc 0 1) (t : ℝ) :
+    (∑ k ∈ Finset.range (n + 1), binomialMass n p k * Real.exp (t * k)) ≤
+      Real.exp ((n : ℝ) * p * (Real.exp t - 1)) := by
+  rw [binomialMass_mgf]
+  have hn : 0 ≤ 1 - p + p * Real.exp t := by positivity [sub_nonneg.mpr hp.2, hp.1]
+  have he : 1 - p + p * Real.exp t ≤ Real.exp (p * (Real.exp t - 1)) := by
+    nlinarith only [Real.add_one_le_exp (p * (Real.exp t - 1))]
+  calc
+    _ ≤ Real.exp (p * (Real.exp t - 1)) ^ n := pow_le_pow_left₀ hn he n
+    _ = _ := by rw [← Real.exp_nat_mul]; congr 1; ring
+
+/-- Exponential Markov bound for a finite nonnegative measure. -/
+theorem finite_exp_markov {α : Type*} (S : Finset α) (w X : α → ℝ)
+    (hw : ∀ a ∈ S, 0 ≤ w a) (a t : ℝ) (ht : 0 ≤ t) :
+    (∑ z ∈ S with a ≤ X z, w z) ≤
+      Real.exp (-t * a) * ∑ z ∈ S, w z * Real.exp (t * X z) := by
+  classical
+  calc
+    _ = ∑ z ∈ S with a ≤ X z, Real.exp (-t * a) * (w z * Real.exp (t * a)) := by
+      apply Finset.sum_congr rfl
+      intro z _
+      rw [mul_left_comm, ← Real.exp_add]
+      simp
+    _ ≤ ∑ z ∈ S with a ≤ X z, Real.exp (-t * a) * (w z * Real.exp (t * X z)) := by
+      apply Finset.sum_le_sum
+      intro z hz
+      have h := Finset.mem_filter.mp hz
+      exact mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_left
+        (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left h.2 ht)) (hw z h.1))
+        (Real.exp_nonneg _)
+    _ ≤ ∑ z ∈ S, Real.exp (-t * a) * (w z * Real.exp (t * X z)) := by
+      apply Finset.sum_le_sum_of_subset_of_nonneg (Finset.filter_subset _ _)
+      intro z hz _
+      positivity [hw z hz]
+    _ = _ := by rw [Finset.mul_sum]
+
+/-- A uniform multiplicative lower tail, sufficient for hidden-flag counts. -/
+theorem binomialMass_lower_half (n : ℕ) {p : ℝ} (hp : p ∈ Set.Icc 0 1) :
+    (∑ k ∈ (Finset.range (n + 1)).filter (fun k : ℕ => (k : ℝ) ≤ n * p / 2),
+      binomialMass n p k) ≤ Real.exp (-(n : ℝ) * p / 12) := by
+  have he : Real.exp (-(1 / 2 : ℝ)) ≤ 2 / 3 := by
+    rw [Real.exp_neg]
+    apply (inv_le_iff_one_le_mul₀ (Real.exp_pos _)).mpr
+    nlinarith only [Real.add_one_le_exp (1 / 2 : ℝ)]
+  have hmark := finite_exp_markov (Finset.range (n + 1)) (binomialMass n p)
+    (fun k => -(k : ℝ)) (fun k _ => binomialMass_nonneg n k hp)
+    (-(n : ℝ) * p / 2) (1 / 2) (by norm_num)
+  have hf : (Finset.range (n + 1)).filter (fun k : ℕ => -(n : ℝ) * p / 2 ≤ -(k : ℝ)) =
+      (Finset.range (n + 1)).filter (fun k : ℕ => (k : ℝ) ≤ n * p / 2) := by
+    ext k
+    simp only [Finset.mem_filter]
+    constructor <;> rintro ⟨h, h'⟩ <;> exact ⟨h, by linarith⟩
+  rw [hf] at hmark
+  calc
+    _ ≤ Real.exp (-(1 / 2 : ℝ) * (-(n : ℝ) * p / 2)) *
+        (∑ k ∈ Finset.range (n + 1), binomialMass n p k * Real.exp (-(1 / 2 : ℝ) * k)) := by
+      convert hmark using 1
+      congr 2
+      funext k
+      congr 2
+      ring
+    _ ≤ Real.exp (-(1 / 2 : ℝ) * (-(n : ℝ) * p / 2)) *
+        Real.exp ((n : ℝ) * p * (Real.exp (-(1 / 2 : ℝ)) - 1)) := by
+      gcongr
+      exact binomialMass_mgf_le n hp _
+    _ = Real.exp ((n : ℝ) * p * (Real.exp (-(1 / 2 : ℝ)) - 3 / 4)) := by
+      rw [← Real.exp_add]
+      congr 1
+      ring
+    _ ≤ _ := by
+      apply Real.exp_le_exp.mpr
+      nlinarith [mul_le_mul_of_nonneg_left he (mul_nonneg (Nat.cast_nonneg n) hp.1)]
+
+end TraceReconstruction
+
+namespace TraceReconstruction
+theorem fairBinomial_centered_mgf (n : ℕ) (t : ℝ) :
+    (∑ k ∈ Finset.range (n + 1), binomialMass n (1 / 2) k *
+      Real.exp (t * ((k : ℝ) - n / 2))) ≤ Real.exp ((n : ℝ) * t ^ 2 / 8) := by
+  have he : (1 / 2 : ℝ) + Real.exp t / 2 = Real.exp (t / 2) * Real.cosh (t / 2) := by
+    rw [Real.cosh_eq, ← mul_div_assoc, mul_add, ← Real.exp_add, ← Real.exp_add]
+    rw [show t / 2 + t / 2 = t by ring, show t / 2 + -(t / 2) = 0 by ring, Real.exp_zero]
+    ring
+  have hm : (∑ k ∈ Finset.range (n + 1), binomialMass n (1 / 2) k *
+      Real.exp (t * ((k : ℝ) - n / 2))) =
+      Real.exp (-t * n / 2) * (1 - 1 / 2 + (1 / 2) * Real.exp t) ^ n := by
+    rw [← binomialMass_mgf, Finset.mul_sum]
+    apply Finset.sum_congr rfl
+    intro k _
+    rw [mul_left_comm, ← Real.exp_add]
+    congr 2
+    ring
+  rw [hm, show (1 - 1 / 2 + (1 / 2) * Real.exp t : ℝ) = 1 / 2 + Real.exp t / 2 by ring, he, mul_pow]
+  have hb : Real.cosh (t / 2) ^ n ≤ Real.exp (t ^ 2 / 8) ^ n := by
+    apply pow_le_pow_left₀ (le_of_lt (Real.cosh_pos _))
+    convert Real.cosh_le_exp_half_sq (t / 2) using 1; congr 1; ring
+  calc
+    _ ≤ Real.exp (-t * n / 2) * (Real.exp (t / 2) ^ n * Real.exp (t ^ 2 / 8) ^ n) := by gcongr
+    _ = _ := by
+      simp only [← Real.exp_nat_mul, ← Real.exp_add]
+      congr 1
+      ring
+
+theorem fairBinomial_upper_tail (n : ℕ) (hn : 0 < n) (a : ℝ) (ha : 0 ≤ a) :
+    (∑ k ∈ (Finset.range (n + 1)).filter (fun k : ℕ => a ≤ (k : ℝ) - n / 2),
+      binomialMass n (1 / 2) k) ≤ Real.exp (-2 * a ^ 2 / n) := by
+  have hnR : (0 : ℝ) < n := by exact_mod_cast hn
+  have h := finite_exp_markov (Finset.range (n + 1)) (binomialMass n (1 / 2))
+    (fun k => (k : ℝ) - n / 2) (fun k _ => binomialMass_nonneg n k (by norm_num))
+    a (4 * a / n) (by positivity)
+  calc
+    _ ≤ Real.exp (-(4 * a / n) * a) *
+      (∑ k ∈ Finset.range (n + 1), binomialMass n (1 / 2) k *
+        Real.exp ((4 * a / n) * ((k : ℝ) - n / 2))) := h
+    _ ≤ Real.exp (-(4 * a / n) * a) * Real.exp ((n : ℝ) * (4 * a / n) ^ 2 / 8) := by
+      gcongr
+      exact fairBinomial_centered_mgf n _
+    _ = _ := by
+      rw [← Real.exp_add]
+      congr 1
+      field_simp [ne_of_gt hnR]
+      ring
+end TraceReconstruction
+
+namespace TraceReconstruction
+def hypergeomMass (V M m v : ℕ) : ℝ :=
+  if v ≤ m then (V.choose v : ℝ) * (M.choose (m-v) : ℝ) / ((V+M).choose m : ℝ) else 0
+
+theorem hypergeomMass_nonneg (V M m v : ℕ) : 0 ≤ hypergeomMass V M m v := by
+  unfold hypergeomMass
+  split <;> positivity
+
+theorem hypergeomMass_eq_binomial (V M m v : ℕ) {p : ℝ}
+    (hp : p ∈ Set.Ioo 0 1) (hm : m ≤ V+M) (hv : v ≤ m) :
+    hypergeomMass V M m v = binomialMass V p v * binomialMass M p (m-v) /
+      binomialMass (V+M) p m := by
+  unfold hypergeomMass
+  rw [ite_eq_left hv]
+  by_cases hvV : v ≤ V
+  · by_cases hkM : m-v ≤ M
+    · unfold binomialMass
+      have hc : (0 : ℝ) < (V+M).choose m := by exact_mod_cast Nat.choose_pos hm
+      have hp1 := hp.1
+      have hq1 := sub_pos.mpr hp.2
+      have he : V-v+(M-(m-v)) = V+M-m := by omega
+      have he' : v+(m-v) = m := by omega
+      apply (div_eq_div_iff (ne_of_gt hc) (by positivity)).mpr
+      calc
+        _ = ((V.choose v : ℝ) * (M.choose (m-v) : ℝ) * ((V+M).choose m : ℝ)) *
+            (p^m * (1-p)^(V+M-m)) := by ring
+        _ = _ := by
+          rw [show p^m = p^v * p^(m-v) by rw [← pow_add, he'],
+            show (1-p)^(V+M-m) = (1-p)^(V-v)*(1-p)^(M-(m-v)) by rw [← pow_add, he]]
+          ring
+    · rw [Nat.choose_eq_zero_of_lt (by omega : M < m-v)]
+      simp [binomialMass, Nat.choose_eq_zero_of_lt (by omega : M < m-v)]
+  · rw [Nat.choose_eq_zero_of_lt (by omega : V < v)]
+    simp [binomialMass, Nat.choose_eq_zero_of_lt (by omega : V < v)]
+
+theorem binomialMass_sq_upper (n k : ℕ) {p : ℝ} (hp : p ∈ Set.Ioo 0 1) :
+    binomialMass n p k ^ 2 ≤ 1 / (((n : ℝ)+1)*p*(1-p)) := by
+  have hp1 := hp.1
+  have hq1 := sub_pos.mpr hp.2
+  have h := biasedSeries_coeff_bound n hp (k : ℤ)
+  rw [biasedSeries_coeff_nat, abs_of_nonneg (binomialMass_nonneg n k ⟨hp.1.le,hp.2.le⟩)] at h
+  have hs := sq_le_sq₀ (binomialMass_nonneg n k ⟨hp.1.le,hp.2.le⟩)
+    (Real.sqrt_nonneg _) |>.mpr h
+  rwa [Real.sq_sqrt (by positivity)] at hs
+
+theorem hypergeomMass_sq_upper (V M m v : ℕ) {p : ℝ}
+    (hp : p ∈ Set.Ioo 0 1) (hm : m ≤ V+M) (hmean : (m : ℝ) = (V+M : ℕ)*p)
+    (hvar : 1 ≤ (V+M : ℕ)*p*(1-p)) :
+    hypergeomMass V M m v ^ 2 ≤
+      144 * (V+M : ℕ) / (((V : ℝ)+1)*((M : ℝ)+1)*p*(1-p)) := by
+  have hp1 := hp.1
+  have hq1 := sub_pos.mpr hp.2
+  by_cases hv : v ≤ m
+  · rw [hypergeomMass_eq_binomial V M m v hp hm hv, div_eq_mul_inv, mul_pow, mul_pow]
+    have hV := binomialMass_sq_upper V v hp
+    have hM := binomialMass_sq_upper M (m-v) hp
+    have hN := binomialMass_mean_inverse_sq (V+M) m hp hm hmean.symm hvar
+    have hvm := mul_le_mul hV hM (sq_nonneg _) (by positivity)
+    have h := mul_le_mul hvm hN (sq_nonneg _) (by positivity)
+    apply h.trans
+    have hp0 : p ≠ 0 := ne_of_gt hp.1
+    have hq0 : 1-p ≠ 0 := ne_of_gt (sub_pos.mpr hp.2)
+    have hV0 : (V : ℝ)+1 ≠ 0 := by positivity
+    have hM0 : (M : ℝ)+1 ≠ 0 := by positivity
+    apply le_of_eq
+    field_simp
+  · simp only [hypergeomMass, ite_eq_right hv, zero_pow (by decide : 2 ≠ 0)]
+    positivity
+
+theorem sum_hypergeomMass (V M m : ℕ) (hm : m ≤ V+M) :
+    ∑ v ∈ Finset.range (m+1), hypergeomMass V M m v = 1 := by
+  have hc : (0 : ℝ) < (V+M).choose m := by exact_mod_cast Nat.choose_pos hm
+  have hv := Nat.add_choose_eq V M m
+  rw [Finset.Nat.sum_antidiagonal_eq_sum_range_succ_mk] at hv
+  simp only [hypergeomMass]
+  have he : (∑ v ∈ Finset.range (m+1),
+      if v ≤ m then (V.choose v : ℝ)*(M.choose (m-v) : ℝ)/((V+M).choose m : ℝ) else 0) =
+    (∑ v ∈ Finset.range (m+1), (V.choose v : ℝ)*(M.choose (m-v) : ℝ))/((V+M).choose m : ℝ) := by
+    rw [Finset.sum_div]
+    apply Finset.sum_congr rfl
+    intro v hv'
+    rw [ite_eq_left (by simp only [Finset.mem_range] at hv'; omega)]
+  rw [he]
+  have he' : (∑ v ∈ Finset.range (m+1), (V.choose v : ℝ)*(M.choose (m-v) : ℝ)) =
+      ((V+M).choose m : ℝ) := by exact_mod_cast hv.symm
+  rw [he', div_self (ne_of_gt hc)]
+
+theorem hypergeomMass_high (V M m v : ℕ) (hv : V < v) : hypergeomMass V M m v = 0 := by
+  unfold hypergeomMass
+  rw [Nat.choose_eq_zero_of_lt hv]
+  simp
+
+/-- Adjacent-mass identity for the hidden count. The central regime has full support
+on 0,...,M; this is exactly the regime used in adjacent-power lifting. -/
+theorem hypergeom_hidden_adjacent (V M m k : ℕ) (hM : M ≤ m) (hV : m ≤ V) (hk : k < M) :
+    (((k:ℝ)+1)*((V:ℝ)-m+k+1)) * hypergeomMass M V m (k+1) =
+    (((M:ℝ)-k)*((m:ℝ)-k)) * hypergeomMass M V m k := by
+  have hkM : k ≤ M := by omega
+  have hkm : k ≤ m := by omega
+  have hkm' : k+1 ≤ m := by omega
+  have hv : m-(k+1) ≤ V := by omega
+  have h1 := Nat.choose_succ_right_eq M k
+  have h2 := Nat.choose_succ_right_eq V (m-(k+1))
+  rw [show m-(k+1)+1=m-k by omega] at h2
+  have h1' : (M.choose (k+1):ℝ)*((k:ℝ)+1) = (M.choose k:ℝ)*((M:ℝ)-k) := by
+    exact_mod_cast h1
+  have h2' : (V.choose (m-k):ℝ)*((m:ℝ)-k) =
+      (V.choose (m-(k+1)):ℝ)*((V:ℝ)-m+k+1) := by
+    have hvsub : V-(m-(k+1)) = V-m+k+1 := by omega
+    rw [hvsub] at h2
+    exact_mod_cast h2
+  unfold hypergeomMass
+  rw [ite_eq_left hkm', ite_eq_left hkm]
+  have hmul := congrArg (fun z : ℝ => z*(V.choose (m-(k+1)) : ℝ)*((V:ℝ)-m+k+1)) h1'
+  have hmul' := congrArg (fun z : ℝ => z*(M.choose k : ℝ)*((M:ℝ)-k)) h2'
+  have he : (((k:ℝ)+1)*((V:ℝ)-m+k+1)) *
+        ((M.choose (k+1):ℝ)*(V.choose (m-(k+1)):ℝ)) =
+      (((M:ℝ)-k)*((m:ℝ)-k))*((M.choose k:ℝ)*(V.choose (m-k):ℝ)) := by
+    nlinarith only [hmul, hmul']
+  simpa only [mul_div_assoc] using congrArg (fun z : ℝ => z/((M+V).choose m:ℝ)) he
+
+def hypergeomHiddenMode (V M m : ℕ) : ℕ := (M+1)*(m+1)/(V+M+2)
+
+theorem hypergeomHiddenMode_le (V M m : ℕ) (hV : m ≤ V) :
+    hypergeomHiddenMode V M m ≤ M := by
+  unfold hypergeomHiddenMode
+  have h1 : (M+1)*(m+1) < (M+1)*(V+M+2) := Nat.mul_lt_mul_of_pos_left (by omega) (by omega)
+  have hh : (M+1)*(m+1)/(V+M+2) < M+1 := by
+    apply (Nat.div_lt_iff_lt_mul (by omega)).mpr
+    exact h1
+  omega
+
+theorem hypergeom_hidden_unimodal_left (V M m k : ℕ) (hM : M ≤ m) (hV : m ≤ V)
+    (hk : k < hypergeomHiddenMode V M m) :
+    hypergeomMass M V m k ≤ hypergeomMass M V m (k+1) := by
+  have hl := hypergeomHiddenMode_le V M m hV
+  have hkM : k < M := by omega
+  have hrec := hypergeom_hidden_adjacent V M m k hM hV hkM
+  have hc : (k+1)*(V+M+2) ≤ (M+1)*(m+1) := by
+    have hh : k+1 ≤ (M+1)*(m+1)/(V+M+2) := hk
+    exact (Nat.le_div_iff_mul_le (by omega)).mp hh
+  have hc' : ((k:ℝ)+1)*((V:ℝ)+M+2) ≤ ((M:ℝ)+1)*((m:ℝ)+1) := by exact_mod_cast hc
+  have hh : ((k:ℝ)+1)*((V:ℝ)-m+k+1) ≤ ((M:ℝ)-k)*((m:ℝ)-k) := by nlinarith
+  have hm := mul_le_mul_of_nonneg_right hh (hypergeomMass_nonneg M V m k)
+  have hV' : (m:ℝ) ≤ V := by exact_mod_cast hV
+  have ha : 0 < ((k:ℝ)+1)*((V:ℝ)-m+k+1) := by positivity
+  exact (mul_le_mul_iff_right₀ ha).mp (by nlinarith only [hm, hrec])
+
+theorem hypergeom_hidden_unimodal_right (V M m k : ℕ) (hM : M ≤ m) (hV : m ≤ V)
+    (hk : hypergeomHiddenMode V M m ≤ k) :
+    hypergeomMass M V m (k+1) ≤ hypergeomMass M V m k := by
+  by_cases hkM : k < M
+  · have hrec := hypergeom_hidden_adjacent V M m k hM hV hkM
+    have hc : (M+1)*(m+1) < (k+1)*(V+M+2) := by
+      apply (Nat.div_lt_iff_lt_mul (by omega)).mp
+      exact Nat.lt_succ_of_le hk
+    have hc' : ((M:ℝ)+1)*((m:ℝ)+1) < ((k:ℝ)+1)*((V:ℝ)+M+2) := by exact_mod_cast hc
+    have hh : ((M:ℝ)-k)*((m:ℝ)-k) ≤ ((k:ℝ)+1)*((V:ℝ)-m+k+1) := by nlinarith
+    have hm := mul_le_mul_of_nonneg_right hh (hypergeomMass_nonneg M V m k)
+    have hV' : (m:ℝ) ≤ V := by exact_mod_cast hV
+    have ha : 0 < ((k:ℝ)+1)*((V:ℝ)-m+k+1) := by positivity
+    exact (mul_le_mul_iff_right₀ ha).mp (by nlinarith only [hm, hrec])
+  · rw [hypergeomMass_high M V m (k+1) (by omega)]
+    exact hypergeomMass_nonneg M V m k
+
+/-- Exact total variation of a zero-extended unimodal sequence. -/
+theorem unimodal_variation (f : ℕ → ℝ) (n l : ℕ) (hl : l ≤ n+1)
+    (hleft : ∀ k < l, f k ≤ f (k+1))
+    (hright : ∀ k, l ≤ k → f (k+1) ≤ f k) (hz : f (n+1)=0) :
+    f 0 + ∑ k ∈ Finset.range (n+1), |f (k+1)-f k| = 2*f l := by
+  rw [← Finset.sum_range_add_sum_Ico _ hl]
+  have h1 : (∑ k ∈ Finset.range l, |f (k+1)-f k|) = f l-f 0 := by
+    rw [← Finset.sum_range_sub f l]
+    apply Finset.sum_congr rfl
+    intro k hk
+    exact abs_of_nonneg (sub_nonneg.mpr (hleft k (Finset.mem_range.mp hk)))
+  have h2 : (∑ k ∈ Finset.Ico l (n+1), |f (k+1)-f k|) = f l-f (n+1) := by
+    calc
+      _ = ∑ k ∈ Finset.Ico l (n+1), -(f (k+1)-f k) := by
+        apply Finset.sum_congr rfl
+        intro k hk
+        exact abs_of_nonpos (sub_nonpos.mpr (hright k (Finset.mem_Ico.mp hk).1))
+      _ = _ := by rw [Finset.sum_neg_distrib, Finset.sum_Ico_sub _ hl]; ring
+  rw [h1,h2,hz]
+  ring
+
+theorem hypergeom_hidden_variation (V M m : ℕ) (hM : M ≤ m) (hV : m ≤ V) :
+    hypergeomMass M V m 0 +
+      ∑ k ∈ Finset.range (M+1), |hypergeomMass M V m (k+1)-hypergeomMass M V m k| =
+      2*hypergeomMass M V m (hypergeomHiddenMode V M m) := by
+  apply unimodal_variation _ M (hypergeomHiddenMode V M m)
+    ((hypergeomHiddenMode_le V M m hV).trans (by omega))
+    (fun k hk => hypergeom_hidden_unimodal_left V M m k hM hV hk)
+    (fun k hk => hypergeom_hidden_unimodal_right V M m k hM hV hk)
+  exact hypergeomMass_high M V m (M+1) (by omega)
+theorem choose_pred_weight (n r : ℕ) (hn : 0 < n) (hr : r ≤ n) :
+    (n:ℝ)*((n-1).choose r : ℝ) = ((n:ℝ)-r)*(n.choose r : ℝ) := by
+  have h := Nat.choose_mul_succ_eq (n-1) r
+  rw [Nat.sub_add_cancel hn] at h
+  have h' : ((n-1).choose r : ℝ)*(n:ℝ) = (n.choose r : ℝ)*((n:ℝ)-r) := by
+    exact_mod_cast h
+  nlinarith only [h']
+
+theorem choose_pred_weight_succ (n r : ℕ) (hn : 0 < n) :
+    (n:ℝ)*((n-1).choose r : ℝ) = ((r:ℝ)+1)*(n.choose (r+1) : ℝ) := by
+  have h := Nat.add_one_mul_choose_eq (n-1) r
+  rw [Nat.sub_add_cancel hn] at h
+  exact_mod_cast h.trans (Nat.mul_comm _ _)
+
+def hypergeomHiddenFlux (V M m k : ℕ) : ℝ :=
+  (V:ℝ)/(V+M)*hypergeomMass M (V-1) m k
+
+theorem hypergeomHiddenFlux_formula (V M m k : ℕ) (hM : M ≤ m) (hV : m < V) (hk : k ≤ M) :
+    hypergeomHiddenFlux V M m k =
+      ((V:ℝ)-m+k)/((V:ℝ)+M-m)*hypergeomMass M V m k := by
+  have hN : 0 < V+M := by omega
+  have hk' : k ≤ m := by omega
+  have hr : m-k ≤ V := by omega
+  have hmN : m ≤ V+M := by omega
+  have hpred : m ≤ V+M-1 := by omega
+  have hVN : (0:ℝ) < V+M := by exact_mod_cast hN
+  have hd : (0:ℝ) < (V:ℝ)+M-m := by
+    have hr : (m:ℝ) < (V:ℝ)+M := by exact_mod_cast (show m < V+M by omega)
+    linarith
+  have hCN : (0:ℝ) < (V+M).choose m := by exact_mod_cast Nat.choose_pos hmN
+  have hCP : (0:ℝ) < (V+M-1).choose m := by exact_mod_cast Nat.choose_pos hpred
+  have hcw := choose_pred_weight (V+M) m hN hmN
+  have hcv := choose_pred_weight V (m-k) (by omega) hr
+  push_cast [Nat.cast_sub hk'] at hcv
+  push_cast at hcw
+  unfold hypergeomHiddenFlux hypergeomMass
+  rw [ite_eq_left hk', ite_eq_left hk', show M+(V-1)=V+M-1 by omega]
+  have hden : ((V:ℝ)+M)*((V+M-1).choose m:ℝ) =
+      ((V:ℝ)+M-m)*((V+M).choose m:ℝ) := hcw
+  rw [show M+V=V+M by omega]
+  calc
+    _ = (M.choose k:ℝ)*((V:ℝ)*((V-1).choose (m-k):ℝ)) /
+        (((V:ℝ)+M)*((V+M-1).choose m:ℝ)) := by field_simp
+    _ = _ := by rw [hden, hcv]; field_simp; ring
+
+theorem hypergeomHiddenFlux_high (V M m k : ℕ) (hk : M < k) :
+    hypergeomHiddenFlux V M m k = 0 := by
+  simp only [hypergeomHiddenFlux, hypergeomMass_high M (V-1) m k hk, mul_zero]
+
+theorem hypergeomHiddenFlux_boundary (V M m : ℕ) (hM : M ≤ m) (hV : m < V) :
+    hypergeomHiddenFlux V M m M = hypergeomMass M V m M := by
+  rw [hypergeomHiddenFlux_formula V M m M hM hV le_rfl]
+  have hd : (0:ℝ) < (V:ℝ)+M-m := by
+    have hr : (m:ℝ) < (V:ℝ)+M := by exact_mod_cast (show m < V+M by omega)
+    linarith
+  have he : (V:ℝ)-m+M = (V:ℝ)+M-m := by ring
+  rw [he, div_self hd.ne', one_mul]
+
+theorem hypergeom_adjacent_hidden_difference (V M m k : ℕ)
+    (hM : M ≤ m) (hV : m < V) (hk : k ≤ M) :
+    hypergeomMass M V (m+1) (k+1) - hypergeomMass M V m k =
+      hypergeomHiddenFlux V M m (k+1) - hypergeomHiddenFlux V M m k := by
+  by_cases hkM : k = M
+  · subst k
+    rw [hypergeomMass_high M V (m+1) (M+1) (by omega),
+      hypergeomHiddenFlux_high V M m (M+1) (by omega), hypergeomHiddenFlux_boundary V M m hM hV]
+  have hklt : k < M := by omega
+  have hkm : k+1 ≤ m := by omega
+  have hkm' : k ≤ m := by omega
+  have hN : 0 < V+M := by omega
+  have hmN : m ≤ V+M := by omega
+  have hmN' : m+1 ≤ V+M := by omega
+  have hpred : m ≤ V+M-1 := by omega
+  have hN0 : (0:ℝ) < (V:ℝ)+M := by exact_mod_cast hN
+  have hCN : (0:ℝ) < (V+M).choose m := by exact_mod_cast Nat.choose_pos hmN
+  have hCN' : (0:ℝ) < (V+M).choose (m+1) := by exact_mod_cast Nat.choose_pos hmN'
+  have hCP : (0:ℝ) < (V+M-1).choose m := by exact_mod_cast Nat.choose_pos hpred
+  have hcw := choose_pred_weight (V+M) m hN hmN
+  have hcws := choose_pred_weight_succ (V+M) m hN
+  have hcv := choose_pred_weight V (m-k) (by omega) (by omega)
+  have hcvs := choose_pred_weight_succ V (m-(k+1)) (by omega)
+  rw [show m-(k+1)+1=m-k by omega] at hcvs
+  have hMrec := Nat.choose_succ_right_eq M k
+  have hMrec' : (M.choose (k+1):ℝ)*((k:ℝ)+1) = (M.choose k:ℝ)*((M:ℝ)-k) := by
+    exact_mod_cast hMrec
+  push_cast at hcw hcws
+  push_cast [Nat.cast_sub hkm'] at hcv
+  push_cast [Nat.cast_sub hkm] at hcvs
+  unfold hypergeomHiddenFlux hypergeomMass
+  rw [ite_eq_left (by omega : k+1 ≤ m+1), ite_eq_left hkm', ite_eq_left hkm, ite_eq_left hkm',
+    show m+1-(k+1)=m-k by omega, show M+V=V+M by omega,
+    show M+(V-1)=V+M-1 by omega]
+  have hpold : (M.choose k:ℝ)*(V.choose (m-k):ℝ)/( (V+M).choose m:ℝ) =
+      ((V:ℝ)+M-m)*(M.choose k:ℝ)*(V.choose (m-k):ℝ) /
+      (((V:ℝ)+M)*((V+M-1).choose m:ℝ)) := by
+    apply (div_eq_div_iff hCN.ne' (mul_pos hN0 hCP).ne').mpr
+    rw [hcw]; ring
+  have hpnew : (M.choose (k+1):ℝ)*(V.choose (m-k):ℝ)/( (V+M).choose (m+1):ℝ) =
+      ((m:ℝ)+1)*(M.choose (k+1):ℝ)*(V.choose (m-k):ℝ) /
+      (((V:ℝ)+M)*((V+M-1).choose m:ℝ)) := by
+    apply (div_eq_div_iff hCN'.ne' (mul_pos hN0 hCP).ne').mpr
+    rw [hcws]; ring
+  rw [hpold,hpnew]
+  have hF (j : ℕ) : (V:ℝ)/((V:ℝ)+M)*((M.choose j:ℝ)*((V-1).choose (m-j):ℝ)/
+      ((V+M-1).choose m:ℝ)) = (M.choose j:ℝ)*((V:ℝ)*((V-1).choose (m-j):ℝ)) /
+      (((V:ℝ)+M)*((V+M-1).choose m:ℝ)) := by field_simp
+  rw [hF, hF, hcv, hcvs]
+  rw [← sub_div, ← sub_div]
+  congr 1
+  have hh := congrArg (fun z : ℝ => z*(V.choose (m-k):ℝ)) hMrec'
+  nlinarith only [hh]
+
+
+theorem hypergeom_next_zero (V M m : ℕ) (hV : m < V) :
+    hypergeomMass M V (m+1) 0 = hypergeomHiddenFlux V M m 0 := by
+  have hN : 0 < V+M := by omega
+  have hN0 : (0:ℝ) < (V:ℝ)+M := by exact_mod_cast hN
+  have hCN : (0:ℝ) < (V+M).choose (m+1) := by exact_mod_cast Nat.choose_pos (show m+1 ≤ V+M by omega)
+  have hCP : (0:ℝ) < (V+M-1).choose m := by exact_mod_cast Nat.choose_pos (show m ≤ V+M-1 by omega)
+  have hcw := choose_pred_weight_succ (V+M) m hN
+  have hcv := choose_pred_weight_succ V m (by omega)
+  push_cast at hcw hcv
+  unfold hypergeomHiddenFlux hypergeomMass
+  simp only [Nat.zero_le, ite_true, Nat.choose_zero_right, Nat.sub_zero, Nat.cast_one, one_mul]
+  rw [show M+V=V+M by omega, show M+(V-1)=V+M-1 by omega]
+  rw [div_mul_div_comm]
+  apply (div_eq_div_iff hCN.ne' (mul_pos hN0 hCP).ne').mpr
+  have hh := congrArg (fun z : ℝ => z*(V.choose (m+1):ℝ)) hcw
+  have hh' := congrArg (fun z : ℝ => z*((V+M).choose (m+1):ℝ)) hcv
+  nlinarith only [hh,hh']
+
+/-- Exact adjacent visible-count TV, expressed with complementary hidden counts.
+The shift by one is essential: the two totals are different. -/
+def adjacentHypergeomTV (V M m : ℕ) : ℝ :=
+  (1/2:ℝ)*(hypergeomMass M V (m+1) 0 +
+    ∑ k ∈ Finset.range (M+1), |hypergeomMass M V (m+1) (k+1)-hypergeomMass M V m k|)
+
+theorem adjacentHypergeomTV_eq_flux (V M m : ℕ) (hM : M ≤ m) (hV : m < V) :
+    adjacentHypergeomTV V M m =
+      hypergeomHiddenFlux V M m (hypergeomHiddenMode (V-1) M m) := by
+  unfold adjacentHypergeomTV
+  rw [hypergeom_next_zero V M m hV]
+  have he : (∑ k ∈ Finset.range (M+1),
+      |hypergeomMass M V (m+1) (k+1)-hypergeomMass M V m k|) =
+      ∑ k ∈ Finset.range (M+1), |hypergeomHiddenFlux V M m (k+1)-hypergeomHiddenFlux V M m k| := by
+    apply Finset.sum_congr rfl
+    intro k hk
+    rw [hypergeom_adjacent_hidden_difference V M m k hM hV (by simpa only [Finset.mem_range, Nat.lt_succ_iff] using hk)]
+  rw [he]
+  unfold hypergeomHiddenFlux
+  simp_rw [← mul_sub, abs_mul, abs_of_nonneg (by positivity : (0:ℝ) ≤ (V:ℝ)/(V+M))]
+  rw [← Finset.mul_sum, ← mul_add, hypergeom_hidden_variation (V-1) M m hM (by omega)]
+  ring
+
+theorem adjacentHypergeomTV_le_mass (V M m : ℕ) (hM : M ≤ m) (hV : m < V) :
+    adjacentHypergeomTV V M m ≤ hypergeomMass M V m (hypergeomHiddenMode (V-1) M m) := by
+  rw [adjacentHypergeomTV_eq_flux V M m hM hV,
+    hypergeomHiddenFlux_formula V M m _ hM hV (hypergeomHiddenMode_le (V-1) M m (by omega))]
+  have hk := hypergeomHiddenMode_le (V-1) M m (by omega : m ≤ V-1)
+  have hk' : (hypergeomHiddenMode (V-1) M m:ℝ) ≤ M := by exact_mod_cast hk
+  have hm' : (m:ℝ) < V := by exact_mod_cast hV
+  have hd : (0:ℝ) < (V:ℝ)+M-m := by linarith [Nat.cast_nonneg (α := ℝ) M]
+  apply mul_le_of_le_one_left (hypergeomMass_nonneg _ _ _ _)
+  apply (div_le_one hd).mpr
+  linarith
+
+end TraceReconstruction
+
+namespace TraceReconstruction
+def bernoulliRelativeEntropy (θ p : ℝ) : ℝ :=
+  θ*Real.log (θ/p)+(1-θ)*Real.log ((1-θ)/(1-p))
+
+theorem bernoulliRelativeEntropy_upper {θ p : ℝ}
+    (hθ : θ ∈ Set.Ioo 0 1) (hp : p ∈ Set.Ioo 0 1) :
+    bernoulliRelativeEntropy θ p ≤ (θ-p)^2/(p*(1-p)) := by
+  have h1 := mul_le_mul_of_nonneg_left (Real.log_le_sub_one_of_pos (div_pos hθ.1 hp.1)) hθ.1.le
+  have h2 := mul_le_mul_of_nonneg_left (Real.log_le_sub_one_of_pos
+    (div_pos (sub_pos.mpr hθ.2) (sub_pos.mpr hp.2))) (sub_nonneg.mpr hθ.2.le)
+  have he : θ*(θ/p-1)+(1-θ)*((1-θ)/(1-p)-1) = (θ-p)^2/(p*(1-p)) := by
+    field_simp [hp.1.ne', (sub_pos.mpr hp.2).ne']; ring
+  unfold bernoulliRelativeEntropy
+  rw [← he]
+  linarith
+
+theorem binomialMass_change_probability (n k : ℕ) {θ p : ℝ}
+    (hθ : θ ∈ Set.Ioo 0 1) (hp : p ∈ Set.Ioo 0 1) :
+    binomialMass n p k = binomialMass n θ k *
+      Real.exp ((k:ℝ)*Real.log (p/θ)+(n-k:ℕ)*Real.log ((1-p)/(1-θ))) := by
+  rw [Real.exp_add, Real.exp_nat_mul, Real.exp_nat_mul,
+    Real.exp_log (div_pos hp.1 hθ.1),
+    Real.exp_log (div_pos (sub_pos.mpr hp.2) (sub_pos.mpr hθ.2))]
+  unfold binomialMass
+  simp only [div_pow]
+  field_simp [hθ.1.ne', (sub_pos.mpr hθ.2).ne']
+
+theorem binomialMass_mean_change_probability (n k : ℕ) {θ p : ℝ}
+    (hθ : θ ∈ Set.Ioo 0 1) (hp : p ∈ Set.Ioo 0 1)
+    (hk : k ≤ n) (hmean : (n:ℝ)*θ = k) :
+    binomialMass n p k = binomialMass n θ k * Real.exp (-(n:ℝ)*bernoulliRelativeEntropy θ p) := by
+  rw [binomialMass_change_probability n k hθ hp]
+  congr 2
+  rw [Nat.cast_sub hk, ← hmean]
+  unfold bernoulliRelativeEntropy
+  rw [Real.log_div hp.1.ne' hθ.1.ne', Real.log_div (sub_pos.mpr hp.2).ne' (sub_pos.mpr hθ.2).ne',
+    Real.log_div hθ.1.ne' hp.1.ne', Real.log_div (sub_pos.mpr hθ.2).ne' (sub_pos.mpr hp.2).ne']
+  ring
+
+/-- Uniform local binomial lower bound without Stirling. It applies at a
+count `k` by changing to the channel with mean exactly `k`. -/
+theorem binomialMass_local_lower (n k : ℕ) {θ p : ℝ}
+    (hθ : θ ∈ Set.Ioo 0 1) (hp : p ∈ Set.Ioo 0 1)
+    (hk : k ≤ n) (hmean : (n:ℝ)*θ = k) (hv : 1 ≤ (n:ℝ)*θ*(1-θ)) :
+    Real.exp (-(n:ℝ)*(θ-p)^2/(p*(1-p))) /
+      (12*Real.sqrt ((n:ℝ)*θ*(1-θ))) ≤ binomialMass n p k := by
+  rw [binomialMass_mean_change_probability n k hθ hp hk hmean]
+  have hlow := binomialMass_mean_lower n k hθ hk hmean hv
+  have hexp : Real.exp (-(n:ℝ)*(θ-p)^2/(p*(1-p))) ≤
+      Real.exp (-(n:ℝ)*bernoulliRelativeEntropy θ p) := by
+    apply Real.exp_le_exp.mpr
+    have := mul_le_mul_of_nonpos_left (bernoulliRelativeEntropy_upper hθ hp) (neg_nonpos.mpr (Nat.cast_nonneg n))
+    simpa only [mul_div_assoc] using this
+  have hh := mul_le_mul hlow hexp (Real.exp_pos _).le
+    (binomialMass_nonneg n k ⟨hθ.1.le,hθ.2.le⟩)
+  simpa only [one_div, div_eq_mul_inv, mul_comm, one_mul] using hh
+
+end TraceReconstruction
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/TraceReconstruction/Differences.lean b/lean/OAI/Probability/TraceReconstruction/Differences.lean
index ef019921e16e82b37a4a64b1061cf0c315ce9155..f405c02734a7fd98aa9701bbe01582ae8fcbfffb 100644
--- a/lean/OAI/Probability/TraceReconstruction/Differences.lean
+++ b/lean/OAI/Probability/TraceReconstruction/Differences.lean
@@ -1,2475 +1,2475 @@
-import OAI.Probability.TraceReconstruction.Model
-import OAI.Probability.TraceReconstruction.Parity
-import OAI.Probability.TraceReconstruction.Ranks
-
-namespace OAI
-
-/-!
-# Differences
-
-Tensor gap differences, positional shifts, two-gap elimination, central clipping, disjoint rank regions, and weighted period sums.
--/
-
-open scoped ENNReal
-noncomputable section
-
-open Filter
-open scoped Topology
-
-namespace TraceReconstruction
-
-/-- Forward length differences of the actual binomial probability
-polynomial, before imposing the conditional rank total. -/
-def binomialLengthDifference (g s : ℕ) (p : ℝ) : IntSeries :=
-  (biasedSeries 1 p - 1)^s * biasedSeries g p
-
-/-- Increasing the number of trials corresponds exactly to multiplying
-by the Bernoulli step polynomial. -/
-theorem binomialLengthDifference_succ (g s : ℕ) (p : ℝ) :
-    binomialLengthDifference g (s+1) p =
-      binomialLengthDifference (g+1) s p - binomialLengthDifference g s p := by
-  unfold binomialLengthDifference
-  rw [biasedSeries_add]
-  ring
-
-theorem biasedSeries_step_sub_one (p : ℝ) :
-    biasedSeries 1 p - 1 = (-p) • firstDifference := by
-  unfold biasedSeries firstDifference
-  rw [pow_one,smul_sub,AddMonoidAlgebra.smul_single,AddMonoidAlgebra.smul_single]
-  simp only [smul_eq_mul,mul_one]
-  have h1 : (1 : IntSeries) = AddMonoidAlgebra.single 0 1 := rfl
-  rw [h1]
-  simp only [AddMonoidAlgebra.single_sub]
-  ext z
-  simp only [AddMonoidAlgebra.coeff_sub,AddMonoidAlgebra.coeff_add,
-    Finsupp.sub_apply,Finsupp.add_apply,AddMonoidAlgebra.coeff_single,Finsupp.single_apply]
-  split_ifs <;> ring
-
-theorem binomialLengthDifference_eq (g s : ℕ) (p : ℝ) :
-    binomialLengthDifference g s p =
-      (-p)^s • (firstDifference^s * biasedSeries g p) := by
-  unfold binomialLengthDifference
-  rw [biasedSeries_step_sub_one,smul_pow,smul_mul_assoc]
-
-theorem seriesL2Sq_smul (c : ℝ) (f : IntSeries) :
-    seriesL2Sq (c • f) = c^2 * seriesL2Sq f := by
-  unfold seriesL2Sq
-  simp only [AddMonoidAlgebra.coeff_smul,Finsupp.smul_apply,smul_eq_mul,mul_pow]
-  exact tsum_mul_left
-
-/-- Pointwise and square-sum bounds for every forward length difference.
-The quotient form keeps every order and every channel factor explicit. -/
-theorem binomialLengthDifference_bounds (g s : ℕ) {p : ℝ} (hp : p ∈ Set.Ioo 0 1) :
-    (∀ z : ℤ, |(binomialLengthDifference g s p).coeff z| ≤
-      (2*biasedBinomialBound (g/(s+1)) p)^s * biasedBinomialBound (g/(s+1)) p) ∧
-    (seriesL2Sq (binomialLengthDifference g s p) ≤
-      ((2*biasedBinomialBound (g/(s+1)) p)^s)^2 * biasedBinomialBound (g/(s+1)) p) := by
-  have hpow : |(-p)^s| ≤ 1 := by
-    rw [abs_pow,abs_neg,abs_of_pos hp.1]
-    exact pow_le_one₀ hp.1.le hp.2.le
-  have hs := biased_difference_smoothing g s hp
-  rw [binomialLengthDifference_eq]
-  constructor
-  · intro z
-    simp only [AddMonoidAlgebra.coeff_smul,Finsupp.smul_apply,smul_eq_mul,abs_mul]
-    exact (mul_le_mul_of_nonneg_right hpow (abs_nonneg _)).trans (by simpa using hs.1 z)
-  · rw [seriesL2Sq_smul]
-    have h : ((-p)^s) ^ 2 ≤ 1 := by nlinarith [sq_abs ((-p)^s),abs_nonneg ((-p)^s)]
-    exact (mul_le_mul_of_nonneg_right h (seriesL2Sq_nonneg _)).trans (by simpa using hs.2)
-
-/-- Explicit fixed-order smoothing constant; no asymptotic hypothesis is hidden. -/
-def lengthDifferenceConstant (s : ℕ) (p : ℝ) : ℝ :=
-  2 * Real.sqrt (((s:ℝ)+1)/(p*(1-p)))
-
-theorem lengthDifferenceConstant_pos (s : ℕ) {p : ℝ} (hp : p ∈ Set.Ioo 0 1) :
-    0 < lengthDifferenceConstant s p := by
-  unfold lengthDifferenceConstant
-  positivity [hp.1,sub_pos.mpr hp.2]
-
-theorem biasedBinomialBound_length_le (g s : ℕ) {p : ℝ} (hp : p ∈ Set.Ioo 0 1) :
-    2*biasedBinomialBound (g/(s+1)) p ≤
-      lengthDifferenceConstant s p / Real.sqrt ((g:ℝ)+1) := by
-  calc
-    _ ≤ 2*Real.sqrt (((s:ℝ)+1)/(((g:ℝ)+1)*p*(1-p))) :=
-      mul_le_mul_of_nonneg_left (biasedBinomialBound_quotient_le g s hp) (by norm_num)
-    _ = _ := by
-      unfold lengthDifferenceConstant
-      rw [show ((s:ℝ)+1)/(((g:ℝ)+1)*p*(1-p)) =
-        (((s:ℝ)+1)/(p*(1-p)))/((g:ℝ)+1) by field_simp]
-      rw [Real.sqrt_div (by positivity [hp.1,sub_pos.mpr hp.2])]
-      ring
-
-/-- Squared supremum and squared L2 smoothing estimates in a form suitable
-for exact constrained rank profiles. -/
-theorem binomialLengthDifference_numeric (g s : ℕ) {p : ℝ} (hp : p ∈ Set.Ioo 0 1) :
-    (∀ z : ℤ, (binomialLengthDifference g s p).coeff z ^ 2 ≤
-      (lengthDifferenceConstant s p)^(2*s+2) / ((g:ℝ)+1)^(s+1)) ∧
-    (seriesL2Sq (binomialLengthDifference g s p) ≤
-      (lengthDifferenceConstant s p)^(2*s+1) /
-        (((g:ℝ)+1)^s * Real.sqrt ((g:ℝ)+1))) := by
-  have hb := binomialLengthDifference_bounds g s hp
-  have hc := biasedBinomialBound_length_le g s hp
-  have hB : 0 ≤ biasedBinomialBound (g/(s+1)) p := Real.sqrt_nonneg _
-  have hC : 0 < lengthDifferenceConstant s p := lengthDifferenceConstant_pos s hp
-  have hg : 0 < Real.sqrt ((g:ℝ)+1) := Real.sqrt_pos.mpr (by positivity)
-  have hB' : biasedBinomialBound (g/(s+1)) p ≤
-      lengthDifferenceConstant s p / Real.sqrt ((g:ℝ)+1) := by linarith
-  have hs := pow_le_pow_left₀ (by positivity : 0 ≤ 2*biasedBinomialBound (g/(s+1)) p) hc s
-  constructor
-  · intro z
-    have h := (hb.1 z).trans (mul_le_mul hs hB' hB (by positivity))
-    rw [← pow_succ] at h
-    have hsq := pow_le_pow_left₀ (abs_nonneg _) h 2
-    rw [sq_abs] at hsq
-    refine hsq.trans_eq ?_
-    rw [← pow_mul,div_pow]
-    have he : (s+1)*2=2*s+2 := by omega
-    rw [he]
-    congr 1
-    rw [show 2*s+2=2*(s+1) by omega,pow_mul,Real.sq_sqrt (by positivity)]
-  · have hsq := pow_le_pow_left₀ (by positivity :
-        0 ≤ (2*biasedBinomialBound (g/(s+1)) p)^s) hs 2
-    refine hb.2.trans ((mul_le_mul hsq hB' hB (by positivity)).trans_eq ?_)
-    rw [← pow_mul,← pow_succ,div_pow]
-    congr 1
-    · congr 1; omega
-    · rw [pow_succ,show s*2=2*s by omega,pow_mul,Real.sq_sqrt (by positivity)]
-
-end TraceReconstruction
-
-namespace TraceReconstruction
-
-theorem series_finite_sum_sq_le {α : Type*} [Fintype α]
-    (f : IntSeries) (z : α → ℤ) (hz : Function.Injective z) :
-    (∑ a, f.coeff (z a)^2) ≤ seriesL2Sq f := by
-  simpa only [tsum_fintype,seriesL2Sq] using
-    Summable.tsum_le_tsum_of_inj z hz (fun _ _ => sq_nonneg _)
-      (fun _ => le_rfl) (hasSum_fintype _).summable (series_summable f (fun x => x^2) (by simp))
-
-/-- Product coefficients of signed gap series, including an arbitrary
-translation of each observed rank gap. -/
-def signedRankProduct {l r : ℕ} (f : Fin (r+1) → IntSeries)
-    (shift : Fin (r+1) → ℤ) (v : IncreasingTuple l r) : ℝ :=
-  ∏ i, (f i).coeff ((tupleGap v.val i : ℤ)-shift i)
-
-/-- Largest-gap elimination also holds for signed finite-difference series.
-The fixed sum of the observed gaps supplies the missing square-root factor;
-translations are unrestricted and the estimate contains no omitted ranks. -/
-theorem signedRankProduct_square_sum_le {l r : ℕ}
-    (f : Fin (r+1) → IntSeries) (shift : Fin (r+1) → ℤ)
-    (e : Fin (r+1)) (M : ℝ) (hM : 0 ≤ M)
-    (hbound : ∀ z : ℤ, (f e).coeff z ^ 2 ≤ M) :
-    (∑ v : IncreasingTuple l r, signedRankProduct f shift v ^ 2) ≤
-      M * ∏ i : Fin r, seriesL2Sq (f (Equiv.swap 0 e i.succ)) := by
-  classical
-  let σ : Equiv.Perm (Fin (r+1)) := Equiv.swap 0 e
-  let F : Fin (r+1) → Fin (l-r+1) → ℝ :=
-    fun i k => (f (σ i)).coeff ((k.val:ℤ)-shift (σ i)) ^ 2
-  have hF : ∀ i k, 0 ≤ F i k := fun _ _ => sq_nonneg _
-  have hpoint (v : IncreasingTuple l r) :
-      signedRankProduct f shift v ^ 2 = ∏ i, F i (rankGapProfile σ v i) := by
-    unfold signedRankProduct
-    rw [← Finset.prod_pow]
-    dsimp [F,rankGapProfile]
-    exact (Equiv.prod_comp σ _).symm
-  simp_rw [hpoint]
-  calc
-    _ ≤ ∑ z : Fin (r+1) → Fin (l-r+1),
-        if ∑ i, (z i).val = l-r then ∏ i, F i (z i) else 0 := by
-      exact sum_injective_profile_le (rankGapProfile σ) (rankGapProfile_injective σ)
-        (fun z => ∑ i, (z i).val = l-r) (rankGapProfile_sum σ)
-        (fun z => ∏ i, F i (z i)) (fun _ => Finset.prod_nonneg (fun _ _ => hF _ _))
-    _ ≤ M * ∏ i : Fin r, ∑ k : Fin (l-r+1), F i.succ k :=
-      constrained_sum_product_le r (l-r) F hF M hM (by
-        intro k
-        simpa only [F,σ,Equiv.swap_apply_left] using hbound ((k.val:ℤ)-shift e))
-    _ ≤ _ := by
-      apply mul_le_mul_of_nonneg_left _ hM
-      apply Finset.prod_le_prod₀ (fun _ _ => Finset.sum_nonneg (fun _ _ => hF _ _))
-      intro i _
-      dsimp only [F]
-      apply series_finite_sum_sq_le
-      intro a b h
-      exact Fin.ext (Int.ofNat_inj.mp (sub_left_injective h))
-
-/-- Finite mixtures of signed rank products can be estimated termwise.
-This is the exact Cauchy--Schwarz form used when expanding positional
-finite-difference operators into gap derivatives. -/
-theorem rank_mixture_square_sum_le {α V : Type*} [Fintype α] [Fintype V]
-    (c : α → ℝ) (F : α → V → ℝ) (B : α → ℝ)
-    (hB : ∀ a, (∑ v, F a v ^ 2) ≤ B a) :
-    (∑ v, (∑ a, c a * F a v)^2) ≤ (∑ a, |c a|) * ∑ a, |c a| * B a := by
-  have hp (v : V) : (∑ a, c a*F a v)^2 ≤
-      (∑ a, |c a|) * ∑ a, |c a| * F a v^2 := by
-    apply Finset.sum_sq_le_sum_mul_sum_of_sq_le_mul Finset.univ
-      (fun _ _ => abs_nonneg _) (fun _ _ => mul_nonneg (abs_nonneg _) (sq_nonneg _))
-    intro a _
-    rw [mul_pow,← mul_assoc,← sq, sq_abs]
-  calc
-    _ ≤ ∑ v, (∑ a, |c a|) * ∑ a, |c a| * F a v^2 :=
-      Finset.sum_le_sum (fun v _ => hp v)
-    _ = (∑ a, |c a|) * ∑ a, |c a| * ∑ v, F a v^2 := by
-      rw [← Finset.mul_sum,Finset.sum_comm]
-      simp_rw [← Finset.mul_sum]
-    _ ≤ _ := by
-      apply mul_le_mul_of_nonneg_left _ (Finset.sum_nonneg (fun _ _ => abs_nonneg _))
-      apply Finset.sum_le_sum
-      intro a _
-      exact mul_le_mul_of_nonneg_left (hB a) (abs_nonneg _)
-
-end TraceReconstruction
-
-namespace TraceReconstruction
-
-/-- The tensor product of finitely supported one-dimensional coefficient
-series, represented on its literal vector-valued exponents. -/
-def tensorSeries {ι : Type*} [Fintype ι] (f : ι → IntSeries) :
-    AddMonoidAlgebra ℝ (ι → ℤ) :=
-by
-  classical
-  exact AddMonoidAlgebra.ofCoeff (Finsupp.onFinset (Fintype.piFinset (fun i => (f i).coeff.support))
-    (fun z => ∏ i, (f i).coeff (z i)) (by
-      classical
-      intro z hz
-      simp only [Fintype.mem_piFinset,Finsupp.mem_support_iff]
-      exact fun i => (Finset.prod_ne_zero_iff.mp hz) i (Finset.mem_univ i)))
-
-theorem tensorSeries_coeff {ι : Type*} [Fintype ι] (f : ι → IntSeries) (z : ι → ℤ) :
-    (tensorSeries f).coeff z = ∏ i, (f i).coeff (z i) := rfl
-
-theorem tensorSeries_support {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (f : ι → IntSeries) :
-    (tensorSeries f).coeff.support = Fintype.piFinset (fun i => (f i).coeff.support) := by
-  classical
-  ext z
-  simp [tensorSeries_coeff,Finset.prod_ne_zero_iff]
-
-/-- Literal tensor multiplication is coordinatewise convolution. -/
-theorem tensorSeries_mul {ι : Type*} [Fintype ι] (f g : ι → IntSeries) :
-    tensorSeries (fun i => f i*g i) = tensorSeries f * tensorSeries g := by
-  classical
-  ext z
-  rw [tensorSeries_coeff,AddMonoidAlgebra.coeff_mul_apply_left,Finsupp.sum,tensorSeries_support]
-  simp_rw [tensorSeries_coeff,← Finset.prod_mul_distrib]
-  simp only [Pi.add_apply, Pi.neg_apply]
-  rw [← Finset.prod_univ_sum (fun i => (f i).coeff.support)
-    (fun i j => (f i).coeff j * (g i).coeff (-j + z i))]
-  apply Finset.prod_congr rfl
-  intro i _
-  rw [AddMonoidAlgebra.coeff_mul_apply_left,Finsupp.sum]
-
-theorem tensorSeries_one {ι : Type*} [Fintype ι] :
-    tensorSeries (fun _ : ι => (1 : IntSeries)) = 1 := by
-  classical
-  ext z
-  rw [tensorSeries_coeff]
-  change (∏ i, Finsupp.single 0 (1:ℝ) (z i)) = Finsupp.single 0 (1:ℝ) z
-  simp only [Finsupp.single_apply]
-  by_cases hz : z = 0
-  · subst z; simp
-  · have he : ∃ i, z i ≠ 0 := by simpa [funext_iff] using hz
-    obtain ⟨i,hi⟩ := he
-    rw [ite_eq_right (Ne.symm hz)]
-    exact Finset.prod_eq_zero (Finset.mem_univ i) (by simp [Ne.symm hi])
-
-
-/-- Embedding one integral exponent into its own gap coordinate. -/
-def coordinateExponent {ι : Type*} [DecidableEq ι] (i : ι) : ℤ →+ (ι → ℤ) where
-  toFun := fun a => Pi.single (M := fun _ => ℤ) i a
-  map_zero' := Pi.single_zero i
-  map_add' := fun a b => by
-    ext j
-    by_cases h : j=i
-    · subst j; simp
-    · simp [Pi.single_eq_of_ne h]
-
-def coordinateSeriesHom {ι : Type*} [DecidableEq ι] (i : ι) :
-    IntSeries →+* AddMonoidAlgebra ℝ (ι → ℤ) :=
-  AddMonoidAlgebra.mapDomainRingHom ℝ (coordinateExponent i)
-
-theorem coordinateExponent_injective {ι : Type*} [DecidableEq ι] (i : ι) :
-    Function.Injective (coordinateExponent i) := by
-  intro a b h
-  simpa only [coordinateExponent,AddMonoidHom.coe_mk,ZeroHom.coe_mk,Pi.single_eq_same]
-    using congrFun h i
-
-theorem tensorSeries_coordinate {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (i : ι) (f : IntSeries) :
-    tensorSeries (fun j => if j=i then f else 1) = coordinateSeriesHom i f := by
-  classical
-  ext z
-  rw [tensorSeries_coeff]
-  change (∏ j, (if j=i then f else (1:IntSeries)).coeff (z j)) =
-    (Finsupp.mapDomain (coordinateExponent i) f.coeff) z
-  by_cases hz : ∀ j, j ≠ i → z j = 0
-  · have he : coordinateExponent i (z i) = z := by
-      funext j
-      by_cases hj : j=i
-      · subst j; simp [coordinateExponent]
-      · simp [coordinateExponent,Pi.single_eq_of_ne hj,hz j hj]
-    rw [← he,Finsupp.mapDomain_apply_of_injective (coordinateExponent_injective i)]
-    rw [Finset.prod_eq_single i]
-    · simp [coordinateExponent]
-    · intro j _ hj
-      simp [hj,coordinateExponent]
-    · simp
-  · push Not at hz
-    obtain ⟨j,hji,hj⟩ := hz
-    have hn : z ∉ Set.range (coordinateExponent i) := by
-      rintro ⟨a,ha⟩
-      have h := congrFun ha j
-      simp [coordinateExponent,Pi.single_eq_of_ne hji] at h
-      exact hj h.symm
-    rw [Finsupp.mapDomain_of_notMem_range _ _ hn]
-    apply Finset.prod_eq_zero (Finset.mem_univ j)
-    simp only [ite_eq_right hji]
-    change Finsupp.single 0 (1:ℝ) (z j) = 0
-    simp [Ne.symm hj]
-
-theorem tensorSeries_prod {ι α : Type*} [Fintype ι]
-    (s : Finset α) (f : α → ι → IntSeries) :
-    tensorSeries (fun i => ∏ a ∈ s, f a i) = ∏ a ∈ s, tensorSeries (f a) := by
-  classical
-  induction s using Finset.induction_on with
-  | empty => simp [tensorSeries_one]
-  | @insert a s ha ih =>
-    simp only [Finset.prod_insert ha]
-    rw [tensorSeries_mul,ih]
-
-/-- The literal product of the coordinate embeddings has precisely the
-independent product coefficients, with no external tensor-product axioms. -/
-theorem tensorSeries_eq_prod {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (f : ι → IntSeries) :
-    tensorSeries f = ∏ i, coordinateSeriesHom i (f i) := by
-  have he : f = fun j => ∏ i, if j=i then f i else 1 := by
-    funext j
-    simp
-  conv_lhs => rw [he,tensorSeries_prod]
-  apply Finset.prod_congr rfl
-  intro i _
-  exact tensorSeries_coordinate i (f i)
-
-end TraceReconstruction
-
-namespace TraceReconstruction
-
-/-- Explicit fixed-order gap derivative bound, uniformly over translated
-rank restrictions and the choice of the eliminated gap. -/
-theorem rank_gap_derivatives_bound {l r : ℕ}
-    (g s : Fin (r+1) → ℕ) (shift : Fin (r+1) → ℤ)
-    {p : ℝ} (hp : p ∈ Set.Ioo 0 1) (e : Fin (r+1)) :
-    (∑ v : IncreasingTuple l r,
-      signedRankProduct (fun i => binomialLengthDifference (g i) (s i) p) shift v ^ 2) ≤
-    (lengthDifferenceConstant (s e) p)^(2*s e+2) / ((g e:ℝ)+1)^(s e+1) *
-      ∏ i : Fin r,
-        (lengthDifferenceConstant (s (Equiv.swap 0 e i.succ)) p)^(2*s (Equiv.swap 0 e i.succ)+1) /
-        (((g (Equiv.swap 0 e i.succ):ℝ)+1)^(s (Equiv.swap 0 e i.succ)) *
-          Real.sqrt ((g (Equiv.swap 0 e i.succ):ℝ)+1)) := by
-  have hC := (lengthDifferenceConstant_pos (s e) hp).le
-  refine (signedRankProduct_square_sum_le _ shift e _ (by positivity)
-    (binomialLengthDifference_numeric (g e) (s e) hp).1).trans ?_
-  apply mul_le_mul_of_nonneg_left _ (by positivity)
-  apply Finset.prod_le_prod₀ (fun _ _ => seriesL2Sq_nonneg _)
-  intro i _
-  exact (binomialLengthDifference_numeric _ _ hp).2
-
-/-- Increasing an order increases only its explicit smoothing constant. -/
-theorem lengthDifferenceConstant_mono {s t : ℕ} (hst : s ≤ t)
-    {p : ℝ} (hp : p ∈ Set.Ioo 0 1) :
-    lengthDifferenceConstant s p ≤ lengthDifferenceConstant t p := by
-  unfold lengthDifferenceConstant
-  apply mul_le_mul_of_nonneg_left _ (by norm_num)
-  apply Real.sqrt_le_sqrt
-  exact div_le_div_of_nonneg_right (by exact_mod_cast Nat.succ_le_succ hst)
-    (mul_pos hp.1 (sub_pos.mpr hp.2)).le
-
-end TraceReconstruction
-
-namespace TraceReconstruction
-
-inductive GapDifference
-  | unchanged | first | second
-  deriving DecidableEq
-
-def GapDifference.order : GapDifference → ℕ
-  | .unchanged => 0 | .first => 1 | .second => 2
-
-def adjacentGap {r : ℕ} (h : Fin r) (b : Bool) : Fin (r+1) :=
-  if b then h.succ else h.castSucc
-
-def gapTermWeight (k : GapDifference) (a : Bool × Bool) : ℝ :=
-  match k with
-  | .unchanged => 1/4
-  | .first => (if a.1 then -1 else 1)/2
-  | .second => (if a.1 then -1 else 1)*(if a.2 then -1 else 1)/2
-
-def gapTermOrder {r : ℕ} (h : Fin r) (k : GapDifference) (a : Bool × Bool)
-    (i : Fin (r+1)) : ℕ :=
-  match k with
-  | .unchanged => 0
-  | .first => if i = adjacentGap h a.1 then 1 else 0
-  | .second => (if i = adjacentGap h a.1 then 1 else 0) +
-      (if i = adjacentGap h a.2 then 1 else 0)
-
-def gapTermExtra {r : ℕ} (h : Fin r) (k : GapDifference) (a : Bool × Bool)
-    (i : Fin (r+1)) : ℕ :=
-  match k with
-  | .unchanged => (if i = h.castSucc then 1 else 0) + (if i = h.succ then 1 else 0)
-  | .first => if i = adjacentGap h a.2 then 1 else 0
-  | .second => 0
-
-/-- Polynomial of a centered positional operator after removing from
-both incident gaps the one unit needed for either direction of movement. -/
-def gapOperator {r : ℕ} (h : Fin r) (k : GapDifference) (p : ℝ) :
-    AddMonoidAlgebra ℝ (Fin (r+1) → ℤ) :=
-  let L := coordinateSeriesHom h.castSucc (biasedSeries 1 p)
-  let R := coordinateSeriesHom h.succ (biasedSeries 1 p)
-  match k with
-  | .unchanged => L*R
-  | .first => (1/2:ℝ) • (L^2-R^2)
-  | .second => (1/2:ℝ) • (L^2+R^2-2*L*R)
-
-theorem tensorSeries_two_coordinates {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (a b : ι) (f g : IntSeries) :
-    tensorSeries (fun i => f^(if i=a then 1 else 0) * g^(if i=b then 1 else 0)) =
-      coordinateSeriesHom a f * coordinateSeriesHom b g := by
-  rw [tensorSeries_mul]
-  have h (c : ι) (u : IntSeries) :
-      (fun i => u^(if i=c then 1 else 0)) = (fun i => if i=c then u else 1) := by
-    funext i
-    split <;> simp
-  rw [h,h,tensorSeries_coordinate,tensorSeries_coordinate]
-
-theorem gapLocalTerm_factor {r : ℕ} (h : Fin r) (k : GapDifference)
-    (a : Bool × Bool) (p : ℝ) :
-    tensorSeries (fun i => (biasedSeries 1 p-1)^gapTermOrder h k a i *
-      (biasedSeries 1 p)^gapTermExtra h k a i) =
-      match k with
-      | .unchanged => coordinateSeriesHom h.castSucc (biasedSeries 1 p) *
-          coordinateSeriesHom h.succ (biasedSeries 1 p)
-      | .first => (coordinateSeriesHom (adjacentGap h a.1) (biasedSeries 1 p)-1) *
-          coordinateSeriesHom (adjacentGap h a.2) (biasedSeries 1 p)
-      | .second => (coordinateSeriesHom (adjacentGap h a.1) (biasedSeries 1 p)-1) *
-          (coordinateSeriesHom (adjacentGap h a.2) (biasedSeries 1 p)-1) := by
-  cases k <;> simp only [gapTermOrder,gapTermExtra,pow_zero,pow_add,one_mul,mul_one]
-  all_goals rw [tensorSeries_two_coordinates]
-  all_goals simp only [map_sub,map_one]
-
-/-- Exact four-term assignment of positional differences to the adjacent
-binomial gap lengths, including all signs and all bounded shifts. -/
-theorem gapOperator_expansion {r : ℕ} (h : Fin r) (k : GapDifference) (p : ℝ) :
-    gapOperator h k p = ∑ a : Bool × Bool, gapTermWeight k a •
-      tensorSeries (fun i => (biasedSeries 1 p-1)^gapTermOrder h k a i *
-        (biasedSeries 1 p)^gapTermExtra h k a i) := by
-  simp_rw [gapLocalTerm_factor]
-  cases k <;> simp only [gapOperator,gapTermWeight,Fintype.sum_prod_type,
-    Fintype.sum_bool,adjacentGap,Bool.false_eq_true,ite_false,ite_true]
-  · module
-  all_goals simp only [Algebra.smul_def]
-  all_goals norm_num
-  all_goals ring
-
-theorem gapTermWeight_l1 (k : GapDifference) :
-    (∑ a : Bool × Bool, |gapTermWeight k a|) ≤ 2 := by
-  cases k <;> norm_num [gapTermWeight,Fintype.sum_prod_type,Fintype.sum_bool]
-
-theorem gapTermOrder_total {r : ℕ} (h : Fin r) (k : GapDifference) (a : Bool × Bool) :
-    (∑ i, gapTermOrder h k a i) = k.order := by
-  cases k <;> simp [gapTermOrder,GapDifference.order,Finset.sum_add_distrib]
-
-/-- Exact global expansion inside the unconstrained vector of gap counts. -/
-theorem gapOperators_expansion {r : ℕ} (g : Fin (r+1) → ℕ)
-    (k : Fin r → GapDifference) (p : ℝ) :
-    tensorSeries (fun i => biasedSeries (g i) p) * (∏ h, gapOperator h (k h) p) =
-      ∑ a : Fin r → Bool × Bool, (∏ h, gapTermWeight (k h) (a h)) •
-        tensorSeries (fun i => binomialLengthDifference
-          (g i + ∑ h, gapTermExtra h (k h) (a h) i)
-          (∑ h, gapTermOrder h (k h) (a h) i) p) := by
-  classical
-  simp_rw [gapOperator_expansion]
-  rw [Finset.prod_univ_sum]
-  simp only [Fintype.piFinset_univ]
-  rw [Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro a _
-  rw [Finset.prod_smul, mul_smul_comm, ← tensorSeries_prod, ← tensorSeries_mul]
-  congr 2
-  funext i
-  simp only [binomialLengthDifference,biasedSeries,
-    Finset.prod_mul_distrib,Finset.prod_pow_eq_pow_sum]
-  ring
-
-theorem gapOperators_weight_l1 {r : ℕ} (k : Fin r → GapDifference) :
-    (∑ a : Fin r → Bool × Bool, |∏ h, gapTermWeight (k h) (a h)|) ≤ (2:ℝ)^r := by
-  simp_rw [Finset.abs_prod]
-  rw [← Fintype.prod_sum (fun h a => |gapTermWeight (k h) a|)]
-  calc
-    _ ≤ ∏ _h : Fin r, (2:ℝ) := by
-      apply Finset.prod_le_prod₀ (fun _ _ => Finset.sum_nonneg fun _ _ => abs_nonneg _)
-      intro h _
-      exact gapTermWeight_l1 (k h)
-    _ = _ := by simp
-
-end TraceReconstruction
-
-
-
-noncomputable section
-namespace TraceReconstruction
-
-theorem prod_eliminate_fin {α : Type*} [CommMonoid α] {r : ℕ}
-    (f : Fin (r+1) → α) (e : Fin (r+1)) :
-    f e * (∏ i : Fin r, f (Equiv.swap 0 e i.succ)) = ∏ i, f i := by
-  calc
-    _ = ∏ i : Fin (r+1), f (Equiv.swap 0 e i) := by
-      rw [Fin.prod_univ_succ,Equiv.swap_apply_left]
-    _ = _ := Equiv.prod_comp (Equiv.swap 0 e) f
-
-/-- Uniform smoothing with the individual gap scale retained, and a common
-lower gap scale reserved for every derivative order. -/
-theorem lengthDifference_uniform (g b s : ℕ) {p C G : ℝ}
-    (hp : p ∈ Set.Ioo 0 1) (hC : lengthDifferenceConstant s p ≤ C)
-    (hG : 0 < G) (hb : b ≤ g) (hGg : G ≤ (g:ℝ)+1) :
-    (∀ z : ℤ, (binomialLengthDifference g s p).coeff z^2 ≤
-      (C/Real.sqrt ((b:ℝ)+1)) * (C^(2*s+1)/(G^s*Real.sqrt ((b:ℝ)+1)))) ∧
-    seriesL2Sq (binomialLengthDifference g s p) ≤
-      C^(2*s+1)/(G^s*Real.sqrt ((b:ℝ)+1)) := by
-  have hC0 : 0 < C := (lengthDifferenceConstant_pos s hp).trans_le hC
-  have hb0 : 0 < (b:ℝ)+1 := by positivity
-  have hg0 : 0 < (g:ℝ)+1 := by positivity
-  have hb' : (b:ℝ)+1 ≤ (g:ℝ)+1 := by exact_mod_cast Nat.add_le_add_right hb 1
-  have hpow : G^s ≤ ((g:ℝ)+1)^s := pow_le_pow_left₀ hG.le hGg s
-  have hx := binomialLengthDifference_numeric g s hp
-  constructor
-  · intro z
-    refine (hx.1 z).trans ?_
-    have hc := pow_le_pow_left₀ (lengthDifferenceConstant_pos s hp).le hC (2*s+2)
-    have hd : G^s*((b:ℝ)+1) ≤ ((g:ℝ)+1)^(s+1) := by
-      rw [pow_succ]
-      exact mul_le_mul hpow hb' hb0.le (by positivity)
-    calc
-      _ ≤ C^(2*s+2)/(G^s*((b:ℝ)+1)) := by
-        apply div_le_div₀ (by positivity) hc (by positivity) hd
-      _ = _ := by
-        have hsq := Real.sq_sqrt hb0.le
-        have hs : Real.sqrt ((b:ℝ)+1) ≠ 0 := (Real.sqrt_pos.mpr hb0).ne'
-        rw [show 2*s+2=(2*s+1)+1 by omega,pow_succ]
-        field_simp
-        nlinarith only [hsq]
-  · refine hx.2.trans ?_
-    apply div_le_div₀ (by positivity)
-      (pow_le_pow_left₀ (lengthDifferenceConstant_pos s hp).le hC (2*s+1))
-      (by positivity)
-    exact mul_le_mul hpow (Real.sqrt_le_sqrt hb') (Real.sqrt_nonneg _) (by positivity)
-
-/-- A single translated signed rank product, with every smoothing scale
-and the largest-gap constraint exposed. -/
-theorem rank_derivatives_uniform {l r : ℕ}
-    (g b s : Fin (r+1) → ℕ) (shift : Fin (r+1) → ℤ)
-    {p C G : ℝ} (hp : p ∈ Set.Ioo 0 1)
-    (hC : ∀ i, lengthDifferenceConstant (s i) p ≤ C) (hG : 0 < G)
-    (hb : ∀ i, b i ≤ g i) (hGg : ∀ i, G ≤ (g i:ℝ)+1) (e : Fin (r+1)) :
-    (∑ v : IncreasingTuple l r,
-      signedRankProduct (fun i => binomialLengthDifference (g i) (s i) p) shift v ^ 2) ≤
-      C^(2*(∑ i, s i)+r+2) /
-        (G^(∑ i, s i) * Real.sqrt ((b e:ℝ)+1) * ∏ i, Real.sqrt ((b i:ℝ)+1)) := by
-  let F (i : Fin (r+1)) : ℝ := C^(2*s i+1)/(G^(s i)*Real.sqrt ((b i:ℝ)+1))
-  have hC0 : 0 < C := (lengthDifferenceConstant_pos (s e) hp).trans_le (hC e)
-  have hF : ∀ i, 0 ≤ F i := fun i => by dsimp [F]; positivity
-  have hbound (i : Fin (r+1)) := lengthDifference_uniform (g i) (b i) (s i) hp (hC i) hG (hb i) (hGg i)
-  calc
-    _ ≤ (C/Real.sqrt ((b e:ℝ)+1)*F e) *
-        ∏ i : Fin r, seriesL2Sq (binomialLengthDifference (g (Equiv.swap 0 e i.succ))
-          (s (Equiv.swap 0 e i.succ)) p) := by
-      apply signedRankProduct_square_sum_le _ shift e _ (by positivity [hF e])
-      exact (hbound e).1
-    _ ≤ (C/Real.sqrt ((b e:ℝ)+1)*F e) * ∏ i : Fin r, F (Equiv.swap 0 e i.succ) := by
-      apply mul_le_mul_of_nonneg_left _ (by positivity [hF e])
-      apply Finset.prod_le_prod₀ (fun _ _ => seriesL2Sq_nonneg _)
-      intro i _
-      exact (hbound _).2
-    _ = (C/Real.sqrt ((b e:ℝ)+1))*∏ i, F i := by
-      rw [mul_assoc,prod_eliminate_fin]
-    _ = _ := by
-      simp only [F,Finset.prod_div_distrib,Finset.prod_mul_distrib,
-        Finset.prod_pow_eq_pow_sum,Finset.sum_add_distrib,← Finset.mul_sum,
-        Finset.sum_const,Finset.card_univ,Fintype.card_fin,smul_eq_mul,mul_one]
-      rw [show 2*(∑ i, s i)+r+2=(2*(∑ i, s i)+(r+1))+1 by omega,pow_succ]
-      ring
-
-end TraceReconstruction
-
-
-
-noncomputable section
-namespace TraceReconstruction
-
-/-- Enough vacant slots for independent one-step displacements of every tag. -/
-def SafeTaggedTuple {n r : ℕ} (p : Fin r → Fin n) : Prop :=
-  (∀ i, 1 ≤ (p i).val ∧ (p i).val+1 < n) ∧
-  ∀ i j, i < j → (p i).val+3 ≤ (p j).val
-
-def shiftedTag {n r : ℕ} (p : Fin r → Fin n) (hp : SafeTaggedTuple p)
-    (a : Fin r → Fin 3) (i : Fin r) : Fin n :=
-  ⟨(p i).val+(a i).val-1, by
-    have h := hp.1 i
-    have ha := (a i).isLt
-    omega⟩
-
-theorem shiftedTag_strictMono {n r : ℕ} (p : Fin r → Fin n) (hp : SafeTaggedTuple p)
-    (a : Fin r → Fin 3) : StrictMono (shiftedTag p hp a) := by
-  intro i j hij
-  change (p i).val+(a i).val-1 < (p j).val+(a j).val-1
-  have h := hp.2 i j hij
-  have hi := hp.1 i
-  have hj := hp.1 j
-  have hai := (a i).isLt
-  omega
-
-def gapIncidence (r : ℕ) (i : Fin (r+1)) : ℕ :=
-  Fin.cons (α := fun _ => ℕ) 0 (fun _ : Fin r => 1) i + Fin.snoc (α := fun _ => ℕ) (fun _ : Fin r => 1) 0 i
-
-def shiftGapExtra {r : ℕ} (a : Fin r → Fin 3) (i : Fin (r+1)) : ℕ :=
-  Fin.cons (α := fun _ => ℕ) 0 (fun h => 2-(a h).val) i + Fin.snoc (α := fun _ => ℕ) (fun h => (a h).val) 0 i
-
-/-- Literal adjacent-gap allocation of the units used by a displacement.
-Value zero means translate left, one leave fixed, two translate right. -/
-def localShiftExtra {r : ℕ} (h : Fin r) (a : Fin 3) (i : Fin (r+1)) : ℕ :=
-  (if i=h.castSucc then a.val else 0) + (if i=h.succ then 2-a.val else 0)
-
-theorem sum_castSucc_indicator {r : ℕ} (f : Fin r → ℕ) (i : Fin (r+1)) :
-    (∑ h, if i=h.castSucc then f h else 0) = Fin.snoc (α := fun _ => ℕ) f 0 i := by
-  induction i using Fin.lastCases with
-  | last => simp [eq_comm]
-  | cast h => simp
-
-theorem sum_succ_indicator {r : ℕ} (f : Fin r → ℕ) (i : Fin (r+1)) :
-    (∑ h, if i=h.succ then f h else 0) = Fin.cons (α := fun _ => ℕ) 0 f i := by
-  induction i using Fin.cases with
-  | zero => simp [eq_comm]
-  | succ h => simp
-
-theorem shiftGapExtra_eq_sum {r : ℕ} (a : Fin r → Fin 3) (i : Fin (r+1)) :
-    shiftGapExtra a i = ∑ h, localShiftExtra h (a h) i := by
-  simp only [localShiftExtra,Finset.sum_add_distrib,sum_castSucc_indicator,
-    sum_succ_indicator,shiftGapExtra]
-  omega
-
-theorem gapIncidence_le {n r : ℕ} (p : Fin r → Fin n) (hp : SafeTaggedTuple p)
-    (i : Fin (r+1)) : gapIncidence r i ≤ tupleGap p i := by
-  cases r with
-  | zero =>
-    have hi : i=0 := Fin.eq_zero i
-    subst i
-    simp [gapIncidence,Fin.snoc_zero]
-  | succ r =>
-    induction i using Fin.cases with
-    | zero => simpa only [gapIncidence,Fin.cons_zero,zero_add,Fin.snoc_apply_zero,
-        tupleGap_zero] using (hp.1 0).1
-    | succ i =>
-      induction i using Fin.lastCases with
-      | last =>
-        simp only [gapIncidence,Fin.succ_last,Fin.cons_last,Fin.snoc_last,add_zero,
-          tupleGap_last]
-        have h := (hp.1 (Fin.last r)).2
-        omega
-      | cast i =>
-        simp only [gapIncidence,Fin.cons_succ]
-        rw [Fin.succ_castSucc,Fin.snoc_castSucc]
-        rw [← Fin.succ_castSucc,tupleGap_middle]
-        have h := hp.2 i.castSucc i.succ (by simp)
-        omega
-
-/-- The exact geometric identity relating translated tag positions to
-nonnegative binomial-gap lengths. -/
-theorem tupleGap_shifted {n r : ℕ} (p : Fin r → Fin n) (hp : SafeTaggedTuple p)
-    (a : Fin r → Fin 3) (i : Fin (r+1)) :
-    tupleGap (shiftedTag p hp a) i =
-      tupleGap p i - gapIncidence r i + shiftGapExtra a i := by
-  cases r with
-  | zero =>
-    have hi : i=0 := Fin.eq_zero i
-    subst i
-    simp [tupleGap,tupleLower,tupleUpper,gapIncidence,shiftGapExtra,Fin.snoc_zero]
-  | succ r =>
-    induction i using Fin.cases with
-    | zero =>
-      simp only [tupleGap_zero,gapIncidence,shiftGapExtra,Fin.cons_zero,zero_add,
-        Fin.snoc_apply_zero,shiftedTag]
-      have h := (hp.1 0).1
-      omega
-    | succ i =>
-      induction i using Fin.lastCases with
-      | last =>
-        simp only [gapIncidence,shiftGapExtra,Fin.succ_last,Fin.cons_last,
-          Fin.snoc_last,add_zero,tupleGap_last,shiftedTag]
-        have h := hp.1 (Fin.last r)
-        have ha := (a (Fin.last r)).isLt
-        omega
-      | cast i =>
-        simp only [gapIncidence,shiftGapExtra,Fin.cons_succ]
-        rw [Fin.succ_castSucc,Fin.snoc_castSucc,Fin.snoc_castSucc]
-        rw [← Fin.succ_castSucc,tupleGap_middle,tupleGap_middle]
-        change (p i.succ).val+(a i.succ).val-1-((p i.castSucc).val+(a i.castSucc).val-1+1) = _
-        have hi := hp.1 i.castSucc
-        have hj := hp.1 i.succ
-        have hij := hp.2 i.castSucc i.succ (by simp)
-        have hai := (a i.castSucc).isLt
-        have haj := (a i.succ).isLt
-        omega
-
-
-/-- Coefficients of the unchanged, centered first, and centered second
-one-step positional operators. -/
-def positionalWeight (k : GapDifference) (a : Fin 3) : ℝ :=
-  match k with
-  | .unchanged => if a=1 then 1 else 0
-  | .first => if a=0 then -1/2 else if a=2 then 1/2 else 0
-  | .second => if a=1 then -1 else 1/2
-
-theorem localShift_factor {r : ℕ} (h : Fin r) (a : Fin 3) (p : ℝ) :
-    tensorSeries (fun i => (biasedSeries 1 p)^localShiftExtra h a i) =
-      coordinateSeriesHom h.castSucc (biasedSeries 1 p)^a.val *
-        coordinateSeriesHom h.succ (biasedSeries 1 p)^(2-a.val) := by
-  have hpow (u : IntSeries) (t : ℕ) (c i : Fin (r+1)) :
-      u^(if i=c then t else 0) = (u^t)^(if i=c then 1 else 0) := by
-    split <;> simp
-  have he : (fun i => (biasedSeries 1 p)^localShiftExtra h a i) =
-      (fun i => ((biasedSeries 1 p)^a.val)^(if i=h.castSucc then 1 else 0) *
-        ((biasedSeries 1 p)^(2-a.val))^(if i=h.succ then 1 else 0)) := by
-    funext i
-    rw [localShiftExtra,pow_add,hpow _ _ h.castSucc i,hpow _ _ h.succ i]
-  rw [he,tensorSeries_two_coordinates,map_pow,map_pow]
-
-theorem gapOperator_positional_expansion {r : ℕ} (h : Fin r) (k : GapDifference)
-    (p : ℝ) :
-    gapOperator h k p = ∑ a : Fin 3, positionalWeight k a •
-      tensorSeries (fun i => (biasedSeries 1 p)^localShiftExtra h a i) := by
-  simp_rw [localShift_factor]
-  rw [Fin.sum_univ_three]
-  cases k <;> norm_num [positionalWeight,gapOperator,Fin.ext_iff]
-  all_goals try simp only [two_mul,add_mul]
-  all_goals module
-
-/-- Literal centered differences of all rank kernels are coefficients of
-the adjacent-gap operator product, with no formal negative trial lengths. -/
-theorem shifted_gap_series_expansion {n r : ℕ} (p : Fin r → Fin n)
-    (hp : SafeTaggedTuple p) (k : Fin r → GapDifference) (θ : ℝ) :
-    (∑ a : Fin r → Fin 3, (∏ h, positionalWeight (k h) (a h)) •
-      tensorSeries (fun i => biasedSeries (tupleGap (shiftedTag p hp a) i) θ)) =
-    tensorSeries (fun i => biasedSeries (tupleGap p i-gapIncidence r i) θ) *
-      ∏ h, gapOperator h (k h) θ := by
-  classical
-  simp_rw [gapOperator_positional_expansion]
-  rw [Finset.prod_univ_sum]
-  simp only [Fintype.piFinset_univ]
-  rw [Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro a _
-  rw [Finset.prod_smul,mul_smul_comm,← tensorSeries_prod,← tensorSeries_mul]
-  congr 2
-  funext i
-  rw [tupleGap_shifted,shiftGapExtra_eq_sum]
-  simp only [biasedSeries,pow_one,Finset.prod_pow_eq_pow_sum,pow_add]
-
-
-/-- Simultaneous literal positional operators applied to the actual
-conditional retained-subset kernel. -/
-def positionalRankDifference {n l r : ℕ} (hl : l ≤ n) (p : Fin r → Fin n)
-    (hp : SafeTaggedTuple p) (k : Fin r → GapDifference) (v : IncreasingTuple l r) : ℝ :=
-  ∑ a : Fin r → Fin 3, (∏ h, positionalWeight (k h) (a h)) *
-    subsetRankKernel hl (shiftedTag p hp a) v.val
-
-theorem positionalRankDifference_expansion {n l r : ℕ} (hl : l ≤ n)
-    (p : Fin r → Fin n) (hp : SafeTaggedTuple p) (k : Fin r → GapDifference)
-    {θ : ℝ} (hθ : θ ∈ Set.Ioo 0 1) (v : IncreasingTuple l r) :
-    positionalRankDifference hl p hp k v = θ^r / binomialMass n θ l *
-      ∑ a : Fin r → Bool × Bool, (∏ h, gapTermWeight (k h) (a h)) *
-        signedRankProduct (fun i => binomialLengthDifference
-          (tupleGap p i-gapIncidence r i + ∑ h, gapTermExtra h (k h) (a h) i)
-          (∑ h, gapTermOrder h (k h) (a h) i) θ) (fun _ => 0) v := by
-  have h := shifted_gap_series_expansion p hp k θ
-  rw [gapOperators_expansion] at h
-  have hh := congrArg (fun f : AddMonoidAlgebra ℝ (Fin (r+1) → ℤ) =>
-    f.coeff (fun i => (tupleGap v.val i:ℤ))) h
-  simp only [AddMonoidAlgebra.coeff_sum,Finsupp.finsetSum_apply,
-    AddMonoidAlgebra.coeff_smul,Finsupp.smul_apply,smul_eq_mul,tensorSeries_coeff,
-    biasedSeries_coeff_nat] at hh
-  unfold positionalRankDifference
-  simp_rw [subsetRankKernel_binomial_formula hl _ (shiftedTag_strictMono p hp _) _ v.property θ hθ]
-  calc
-    _ = θ^r / binomialMass n θ l *
-        ∑ a : Fin r → Fin 3, (∏ h, positionalWeight (k h) (a h)) *
-          ∏ i, binomialMass (tupleGap (shiftedTag p hp a) i) θ (tupleGap v.val i) := by
-      rw [Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro a _
-      ring
-    _ = _ := by
-      exact congrArg (fun t => θ^r / binomialMass n θ l * t) (by
-        simpa only [signedRankProduct,sub_zero] using hh)
-
-/-- The total derivative order is the source value |D|+2|V|. -/
-theorem gap_global_order {r : ℕ} (k : Fin r → GapDifference) (a : Fin r → Bool × Bool) :
-    (∑ i : Fin (r+1), ∑ h, gapTermOrder h (k h) (a h) i) = ∑ h, (k h).order := by
-  rw [Finset.sum_comm]
-  simp_rw [gapTermOrder_total]
-
-end TraceReconstruction
-
-
-
-noncomputable section
-namespace TraceReconstruction
-
-theorem gapTermOrder_sum_le_global {r : ℕ} (k : Fin r → GapDifference)
-    (a : Fin r → Bool × Bool) (i : Fin (r+1)) :
-    (∑ h, gapTermOrder h (k h) (a h) i) ≤ ∑ h, (k h).order := by
-  rw [← gap_global_order k a]
-  exact Finset.single_le_sum (f := fun j : Fin (r+1) => ∑ h, gapTermOrder h (k h) (a h) j)
-    (fun _ _ => Nat.zero_le _) (Finset.mem_univ i)
-
-/-- Literal mixed centered positional differences of a conditional rank
-kernel, with a single explicit constant depending only on the total order
-and the Bernoulli parameter. Every gap scale remains available for summing
-over the period tuples in the signed initial construction. -/
-theorem positionalRankDifference_square_sum_le {n l r : ℕ} (hl : l ≤ n)
-    (p : Fin r → Fin n) (hp : SafeTaggedTuple p) (k : Fin r → GapDifference)
-    {θ G : ℝ} (hθ : θ ∈ Set.Ioo 0 1) (hG : 0 < G)
-    (hgap : ∀ i, G ≤ (tupleGap p i-gapIncidence r i : ℕ)+1)
-    (e : Fin (r+1)) :
-    (∑ v : IncreasingTuple l r, positionalRankDifference hl p hp k v ^ 2) ≤
-      (θ^r / binomialMass n θ l)^2 * (2:ℝ)^(2*r) *
-        (lengthDifferenceConstant (∑ h, (k h).order) θ)^(2*(∑ h, (k h).order)+r+2) /
-      (G^(∑ h, (k h).order) * Real.sqrt ((tupleGap p e-gapIncidence r e:ℕ)+1) *
-        ∏ i, Real.sqrt ((tupleGap p i-gapIncidence r i:ℕ)+1)) := by
-  let d := ∑ h, (k h).order
-  let C := lengthDifferenceConstant d θ
-  let b (i : Fin (r+1)) := tupleGap p i-gapIncidence r i
-  let B := C^(2*d+r+2) / (G^d*Real.sqrt ((b e:ℝ)+1)*∏ i,Real.sqrt ((b i:ℝ)+1))
-  let c (a : Fin r → Bool × Bool) := ∏ h, gapTermWeight (k h) (a h)
-  let F (a : Fin r → Bool × Bool) (v : IncreasingTuple l r) :=
-    signedRankProduct (fun i => binomialLengthDifference
-      (b i+∑ h, gapTermExtra h (k h) (a h) i)
-      (∑ h, gapTermOrder h (k h) (a h) i) θ) (fun _ => 0) v
-  have hC : 0 < C := lengthDifferenceConstant_pos d hθ
-  have hB0 : 0 ≤ B := by dsimp [B]; positivity
-  have hB (a : Fin r → Bool × Bool) : (∑ v, F a v^2) ≤ B := by
-    have hd (i : Fin (r+1)) : lengthDifferenceConstant (∑ h, gapTermOrder h (k h) (a h) i) θ ≤ C :=
-      lengthDifferenceConstant_mono (gapTermOrder_sum_le_global k a i) hθ
-    have h := rank_derivatives_uniform (l := l)
-      (fun i => b i+∑ h, gapTermExtra h (k h) (a h) i) b
-      (fun i => ∑ h, gapTermOrder h (k h) (a h) i) (fun _ => 0)
-      hθ hd hG (fun _ => Nat.le_add_right _ _) (by
-        intro i
-        exact (hgap i).trans (by exact_mod_cast Nat.add_le_add_right (Nat.le_add_right (b i) _) 1)) e
-    simpa only [F,B,d,gap_global_order] using h
-  have hmix := rank_mixture_square_sum_le c F (fun _ => B) hB
-  have hM : (∑ a, |c a|) ≤ (2:ℝ)^r := gapOperators_weight_l1 k
-  have hM0 : 0 ≤ ∑ a, |c a| := Finset.sum_nonneg fun _ _ => abs_nonneg _
-  have hmix' : (∑ v, (∑ a, c a*F a v)^2) ≤ (2:ℝ)^(2*r)*B := by
-    refine hmix.trans ?_
-    rw [← Finset.sum_mul,← mul_assoc]
-    have hs : (∑ a, |c a|)^2 ≤ ((2:ℝ)^r)^2 := (sq_le_sq₀ hM0 (by positivity)).mpr hM
-    calc
-      _ = (∑ a, |c a|)^2*B := by ring
-      _ ≤ ((2:ℝ)^r)^2*B := mul_le_mul_of_nonneg_right hs hB0
-      _ = _ := by rw [← pow_mul,Nat.mul_comm r 2]
-  simp_rw [positionalRankDifference_expansion hl p hp k hθ,mul_pow]
-  rw [← Finset.mul_sum]
-  calc
-    _ ≤ (θ^r / binomialMass n θ l)^2 * ((2:ℝ)^(2*r)*B) :=
-      mul_le_mul_of_nonneg_left hmix' (sq_nonneg _)
-    _ = _ := by dsimp [B,C,d,b]; ring
-
-end TraceReconstruction
-
-
-
-noncomputable section
-namespace TraceReconstruction
-
-theorem gapIncidence_le_two (r : ℕ) (i : Fin (r+1)) : gapIncidence r i ≤ 2 := by
-  have h₁ : Fin.cons (α := fun _ => ℕ) 0 (fun _ : Fin r => 1) i ≤ 1 := by
-    induction i using Fin.cases <;> simp
-  have h₂ : Fin.snoc (α := fun _ => ℕ) (fun _ : Fin r => 1) 0 i ≤ 1 := by
-    induction i using Fin.lastCases <;> simp
-  exact Nat.add_le_add h₁ h₂
-
-/-- Constant middle-period tags admit every independent left/unchanged/right
-choice without violating slot order or either end boundary. -/
-theorem constant_period_safe {u m r : ℕ} (off : Fin m)
-    (hoff₁ : 1 ≤ off.val) (hoff₂ : off.val+1 < m) (hm : 3 ≤ m)
-    (j : Fin r → Fin u) (hj : StrictMono j) :
-    SafeTaggedTuple (periodSlot (fun _ => off) ∘ j) := by
-  constructor
-  · intro i
-    change 1 ≤ (j i).val*m+off.val ∧ (j i).val*m+off.val+1 < u*m
-    constructor
-    · omega
-    · have h := Nat.mul_le_mul_right m (Nat.succ_le_of_lt (j i).isLt)
-      rw [Nat.succ_mul] at h
-      omega
-  · intro i h hih
-    change (j i).val*m+off.val+3 ≤ (j h).val*m+off.val
-    have h := Nat.mul_le_mul_right m (Nat.succ_le_iff.mpr (hj hih))
-    rw [Nat.succ_mul] at h
-    omega
-
-/-- Removing both incident displacement units still leaves a fixed
-fraction of each actual coarse gap. Doubled coarse gaps are used to keep
-all combinatorial labels integer-valued. -/
-theorem period_derivative_base_lower {u m r : ℕ} (hm : 32 ≤ m) (hr : 0 < r)
-    (off : Fin u → Fin m)
-    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
-    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2)
-    (j : Fin r → Fin u) (hj : StrictMono j)
-    (hsafe : SafeTaggedTuple (periodSlot off ∘ j)) (i : Fin (r+1)) :
-    (m:ℝ)*coarsePeriodGap j i/4 ≤
-      ((tupleGap (periodSlot off ∘ j) i-gapIncidence r i:ℕ):ℝ)+1 := by
-  have he := abs_le.mp (period_gap_rounding hr off hoff hdiff j hj i)
-  have hs := gapIncidence_le (periodSlot off ∘ j) hsafe i
-  have hs₂ : (gapIncidence r i:ℝ) ≤ 2 := by exact_mod_cast gapIncidence_le_two r i
-  have hc : 1 ≤ (coarsePeriodGap j i:ℝ) := by exact_mod_cast coarsePeriodGap_pos j i
-  have hm' : (32:ℝ) ≤ m := by exact_mod_cast hm
-  rw [Nat.cast_sub hs]
-  nlinarith
-
-theorem period_derivative_common_lower {u m r : ℕ} (hm : 32 ≤ m) (hr : 0 < r)
-    (off : Fin u → Fin m)
-    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
-    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2)
-    (j : Fin r → Fin u) (hj : StrictMono j)
-    (hsafe : SafeTaggedTuple (periodSlot off ∘ j)) (i : Fin (r+1)) :
-    (m:ℝ)/4 ≤ ((tupleGap (periodSlot off ∘ j) i-gapIncidence r i:ℕ):ℝ)+1 := by
-  refine le_trans ?_ (period_derivative_base_lower hm hr off hoff hdiff j hj hsafe i)
-  have hc : 1 ≤ (coarsePeriodGap j i:ℝ) := by exact_mod_cast coarsePeriodGap_pos j i
-  nlinarith [show (0:ℝ) ≤ m by positivity]
-
-/-- A largest coarse gap supplies the additional square-root improvement
-coming from the fixed sum of the observed ranks. -/
-theorem period_derivative_largest_lower {u m r : ℕ} (hm : 32 ≤ m) (hr : 0 < r)
-    (off : Fin u → Fin m)
-    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
-    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2)
-    (j : Fin r → Fin u) (hj : StrictMono j)
-    (hsafe : SafeTaggedTuple (periodSlot off ∘ j)) (e : Fin (r+1))
-    (he : 2*u ≤ (r+1)*coarsePeriodGap j e) :
-    (u:ℝ)*m/(2*(r+1:ℝ)) ≤
-      ((tupleGap (periodSlot off ∘ j) e-gapIncidence r e:ℕ):ℝ)+1 := by
-  refine le_trans ?_ (period_derivative_base_lower hm hr off hoff hdiff j hj hsafe e)
-  have he' : 2*(u:ℝ) ≤ (r+1:ℝ)*coarsePeriodGap j e := by exact_mod_cast he
-  apply (div_le_iff₀ (by positivity)).mpr
-  have hh := mul_le_mul_of_nonneg_right he' (show (0:ℝ) ≤ m by positivity)
-  nlinarith only [hh]
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-
-theorem binomial_rank_normalization {n l r : ℕ} {θ : ℝ}
-    (hθ : θ ∈ Set.Ioo 0 1) (hl : l ≤ n)
-    (hmean : (n:ℝ)*θ=l) (hv : 1 ≤ (n:ℝ)*θ*(1-θ)) :
-    (θ^r/binomialMass n θ l)^2 ≤ 144*(n:ℝ) := by
-  have hpow : (θ^r)^2 ≤ 1 := pow_le_one₀ (pow_nonneg hθ.1.le r) (pow_le_one₀ hθ.1.le hθ.2.le) 
-  calc
-    _ = (θ^r)^2*(binomialMass n θ l)⁻¹^2 := by rw [div_eq_mul_inv,mul_pow]
-    _ ≤ (binomialMass n θ l)⁻¹^2 := mul_le_of_le_one_left (sq_nonneg _) hpow
-    _ ≤ 144*((n:ℝ)*θ*(1-θ)) := binomialMass_mean_inverse_sq n l hθ hl hmean hv
-    _ ≤ _ := by
-      have h : θ*(1-θ) ≤ 1 := by nlinarith [sq_nonneg θ]
-      have hh := mul_le_mul_of_nonneg_left h (show 0 ≤ (n:ℝ) by positivity)
-      nlinarith only [hh]
-
-/-- The exact mean-count version, with arbitrary proven positive lower
-scales for all gaps; no asymptotic or analytic estimate is assumed. -/
-theorem positionalRankDifference_scale_bound {n l r : ℕ} (hl : l ≤ n)
-    (p : Fin r → Fin n) (hp : SafeTaggedTuple p) (k : Fin r → GapDifference)
-    {θ G H : ℝ} (hθ : θ ∈ Set.Ioo 0 1) (hG : 0 < G) (hH : 0 < H)
-    (hmean : (n:ℝ)*θ=l) (hv : 1 ≤ (n:ℝ)*θ*(1-θ))
-    (B : Fin (r+1) → ℝ) (hB : ∀ i, 0 < B i)
-    (hgap : ∀ i, G ≤ (tupleGap p i-gapIncidence r i:ℕ)+1)
-    (hscale : ∀ i, B i ≤ (tupleGap p i-gapIncidence r i:ℕ)+1)
-    (e : Fin (r+1)) (hlarge : H ≤ Real.sqrt ((tupleGap p e-gapIncidence r e:ℕ)+1)) :
-    (∑ v : IncreasingTuple l r, positionalRankDifference hl p hp k v ^ 2) ≤
-      144*(n:ℝ)*(2:ℝ)^(2*r)*
-        (lengthDifferenceConstant (∑ h, (k h).order) θ)^(2*(∑ h, (k h).order)+r+2) /
-      (G^(∑ h, (k h).order)*H*∏ i, Real.sqrt (B i)) := by
-  refine (positionalRankDifference_square_sum_le hl p hp k hθ hG hgap e).trans ?_
-  have hC := (lengthDifferenceConstant_pos (∑ h, (k h).order) hθ).le
-  apply div_le_div₀ (by positivity)
-  · exact mul_le_mul_of_nonneg_right
-      (mul_le_mul_of_nonneg_right (binomial_rank_normalization hθ hl hmean hv) (by positivity))
-      (pow_nonneg hC _)
-  · exact mul_pos (mul_pos (pow_pos hG _) hH)
-      (Finset.prod_pos (fun i _ => Real.sqrt_pos.mpr (hB i)))
-  · apply mul_le_mul
-    · exact mul_le_mul_of_nonneg_left hlarge (by positivity)
-    · exact Finset.prod_le_prod₀ (fun i _ => (Real.sqrt_nonneg (B i)))
-        (fun i _ => Real.sqrt_le_sqrt (hscale i))
-    · exact Finset.prod_nonneg (fun _ _ => Real.sqrt_nonneg _)
-    · positivity
-
-/-- Explicit source rank-difference prefactor. It depends on the period
-count, slot length, derivative order and Bernoulli parameter, not the
-selected period tuple. -/
-def periodDifferencePrefactor (u m r d : ℕ) (θ : ℝ) : ℝ :=
-  144*((u*m:ℕ):ℝ)*(2:ℝ)^(2*r)*(lengthDifferenceConstant d θ)^(2*d+r+2) /
-    (((m:ℝ)/4)^d*Real.sqrt ((u:ℝ)*m/(2*(r+1:ℝ)))*
-      (Real.sqrt ((m:ℝ)/4))^(r+1))
-
-/-- The finite-difference estimate for actual translated tagged slots in
-periodic words. The tuple dependence is exactly the product of inverse
-square roots of its doubled coarse gaps. -/
-theorem period_positionalRankDifference_bound {u m l r : ℕ}
-    (hm : 32 ≤ m) (hr : 0 < r) (hl : l ≤ u*m)
-    (off : Fin u → Fin m)
-    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
-    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2)
-    (j : IncreasingTuple u r) (hsafe : SafeTaggedTuple (periodSlot off ∘ j.val))
-    (k : Fin r → GapDifference) {θ : ℝ} (hθ : θ ∈ Set.Ioo 0 1)
-    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ)) :
-    (∑ v : IncreasingTuple l r,
-      positionalRankDifference hl (periodSlot off ∘ j.val) hsafe k v ^ 2) ≤
-      periodDifferencePrefactor u m r (∑ h, (k h).order) θ *
-        ∏ i, (Real.sqrt (coarsePeriodGap j.val i))⁻¹ := by
-  have hu : 0 < u := lt_of_le_of_lt (Nat.zero_le (j.val ⟨0,hr⟩).val) (j.val ⟨0,hr⟩).isLt
-  have hm0 : (0:ℝ) < m := by exact_mod_cast (show 0 < m by omega)
-  have hu0 : (0:ℝ) < u := by exact_mod_cast hu
-  obtain ⟨e,he⟩ := exists_large_period_gap hr j
-  have hb (i : Fin (r+1)) := period_derivative_base_lower hm hr off hoff hdiff j.val j.property hsafe i
-  have hg (i : Fin (r+1)) := period_derivative_common_lower hm hr off hoff hdiff j.val j.property hsafe i
-  have hc (i : Fin (r+1)) : (0:ℝ) < coarsePeriodGap j.val i := by exact_mod_cast coarsePeriodGap_pos j.val i
-  have h := positionalRankDifference_scale_bound hl (periodSlot off ∘ j.val) hsafe k hθ
-    (G := (m:ℝ)/4) (H := Real.sqrt ((u:ℝ)*m/(2*(r+1:ℝ)))) (by positivity) (by positivity)
-    hmean hv (fun i => (m:ℝ)*coarsePeriodGap j.val i/4) (fun i => by positivity [hc i]) hg hb e
-    (Real.sqrt_le_sqrt (period_derivative_largest_lower hm hr off hoff hdiff j.val j.property hsafe e he))
-  refine h.trans_eq ?_
-  have hs (i : Fin (r+1)) : Real.sqrt ((m:ℝ)*coarsePeriodGap j.val i/4) =
-      Real.sqrt ((m:ℝ)/4)*Real.sqrt (coarsePeriodGap j.val i) := by
-    rw [show (m:ℝ)*coarsePeriodGap j.val i/4=((m:ℝ)/4)*coarsePeriodGap j.val i by ring,
-      Real.sqrt_mul (by positivity)]
-  simp_rw [hs]
-  rw [Finset.prod_mul_distrib]
-  simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fin]
-  rw [periodDifferencePrefactor,Finset.prod_inv_distrib]
-  ring
-
-end TraceReconstruction
-
-open Filter
-open scoped Topology
-
-noncomputable section
-namespace TraceReconstruction
-
-/-- Two independent exact sum constraints determine a missing coordinate on
- each side of a fixed cut. This is the second-gap elimination needed when
- a period label has been fixed. -/
-theorem two_sums_determine {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (A : Finset ι) {e d : ι} (he : e∈A) (hd : d∉A)
-    (x y : ι → ℕ) (hA : ∑ i∈A, x i = ∑ i∈A, y i)
-    (ht : ∑ i, x i = ∑ i, y i)
-    (hxy : ∀ i, i≠e → i≠d → x i=y i) : x=y := by
-  have hs : (∑ i∈A.erase e, x i) = ∑ i∈A.erase e, y i := by
-    apply Finset.sum_congr rfl
-    intro i hi
-    have hi' := Finset.mem_erase.mp hi
-    apply hxy i hi'.1
-    intro hid
-    exact hd (hid ▸ hi'.2)
-  have hxe : x e=y e := by
-    have hx := Finset.sum_erase_add A x he
-    have hy := Finset.sum_erase_add A y he
-    omega
-  have hs' : (∑ i∈Finset.univ.erase d, x i) = ∑ i∈Finset.univ.erase d, y i := by
-    apply Finset.sum_congr rfl
-    intro i hi
-    by_cases hie : i=e
-    · simpa only [hie] using hxe
-    · exact hxy i hie (Finset.mem_erase.mp hi).1
-  have hxd : x d=y d := by
-    have hx := Finset.sum_erase_add Finset.univ x (Finset.mem_univ d)
-    have hy := Finset.sum_erase_add Finset.univ y (Finset.mem_univ d)
-    omega
-  funext i
-  by_cases hie : i=e
-  · simpa only [hie] using hxe
-  by_cases hid : i=d
-  · simpa only [hid] using hxd
-  exact hxy i hie hid
-
-abbrev RemainingCoordinates {ι : Type*} (e d : ι) := {i : ι // i≠e ∧ i≠d}
-
-/-- Restricting an injective gap profile to all but the two eliminated
- coordinates is still injective on two fixed-sum fibers. -/
-theorem two_sums_restrict_injective {α ι : Type*} [Fintype ι] [DecidableEq ι]
-    {U L V : ℕ} (A : Finset ι) {e d : ι} (he : e∈A) (hd : d∉A)
-    (p : α → ι → Fin (U+1)) (hp : Function.Injective p)
-    (hA : ∀ a, ∑ i∈A, (p a i).val=L) (ht : ∀ a, ∑ i, (p a i).val=V) :
-    Function.Injective (fun a => fun i : RemainingCoordinates e d => p a i.val) := by
-  intro a b hab
-  apply hp
-  have hv : (fun i => (p a i).val) = (fun i => (p b i).val) := by
-    apply two_sums_determine A he hd
-    · rw [hA,hA]
-    · rw [ht,ht]
-    · intro i hie hid
-      exact congrArg Fin.val (congrFun hab ⟨i,hie,hid⟩)
-  funext i
-  exact Fin.ext (congrFun hv i)
-
-/-- Once the two coordinates are eliminated all remaining gap weights sum
- independently; any additional constraints may be dropped. -/
-theorem two_sums_product_bound {α ι : Type*} [Fintype α] [Fintype ι] [DecidableEq ι]
-    {U L V : ℕ} (A : Finset ι) {e d : ι} (he : e∈A) (hd : d∉A)
-    (p : α → ι → Fin (U+1)) (hp : Function.Injective p)
-    (hA : ∀ a, ∑ i∈A, (p a i).val=L) (ht : ∀ a, ∑ i, (p a i).val=V)
-    (w : ι → Fin (U+1) → ℝ) (hw : ∀ i k, 0≤w i k) :
-    (∑ a, ∏ i : RemainingCoordinates e d, w i.val (p a i.val)) ≤
-      ∏ i : RemainingCoordinates e d, ∑ k, w i.val k := by
-  classical
-  have hh := sum_injective_profile_le
-    (fun a => fun i : RemainingCoordinates e d => p a i.val)
-    (two_sums_restrict_injective A he hd p hp hA ht)
-    (fun _ => True) (fun _ => trivial)
-    (fun δ => ∏ i : RemainingCoordinates e d, w i.val (δ i))
-    (fun δ => Finset.prod_nonneg (fun i _ => hw i.val (δ i)))
-  simp only [ite_true] at hh
-  exact hh.trans_eq (by rw [Finset.prod_univ_sum]; simp only [Fintype.piFinset_univ])
-
-/-- The sum of doubled coarse gaps to a specified period label is exact,
- including the half endpoint cells in the source convention. -/
-theorem coarsePeriodGap_prefix {u r : ℕ} (j : Fin r → Fin u) (hj : StrictMono j)
-    (i : Fin r) : (∑ g∈Finset.Iic i.castSucc, coarsePeriodGap j g) = 2*(j i).val+1 := by
-  classical
-  have hg := tupleGap_prefix j hj i
-  have hz : (0 : Fin (r+1)) ∈ Finset.Iic i.castSucc := by simp
-  have hl : ∀ g∈Finset.Iic i.castSucc, g≠Fin.last r := by
-    intro g hg he
-    have hle := Finset.mem_Iic.mp hg
-    have hval : g.val ≤ i.val := hle
-    have he' := congrArg Fin.val he
-    simp only [Fin.val_last] at he'
-    omega
-  have hc : (∑ g∈Finset.Iic i.castSucc,
-      if g=0 ∨ g=Fin.last r then 1 else 2) = 2*i.val+1 := by
-    have hsub : (∑ g∈(Finset.Iic i.castSucc).erase 0,
-        if g=0 ∨ g=Fin.last r then 1 else 2) = 2*i.val := by
-      have hc : ((Finset.Iic i.castSucc).erase 0).card = i.val := by
-        rw [Finset.card_erase_of_mem hz,Fin.card_Iic]
-        simp
-      calc
-        _ = ∑ _g∈(Finset.Iic i.castSucc).erase 0, 2 := by
-          apply Finset.sum_congr rfl
-          intro g hg
-          rw [ite_eq_right (not_or.mpr ⟨(Finset.mem_erase.mp hg).1,
-            hl g (Finset.mem_erase.mp hg).2⟩)]
-        _ = _ := by simp [hc,Nat.mul_comm]
-    have hsum := Finset.sum_erase_add (Finset.Iic i.castSucc)
-      (fun g => if g=0 ∨ g=Fin.last r then 1 else 2) hz
-    rw [hsub] at hsum
-    simpa using hsum.symm
-  simp only [coarsePeriodGap,Finset.sum_add_distrib,← Finset.mul_sum]
-  omega
-
-/-- The full inverse-square-root sum, including the zero cell (whose
- value is zero), has the same elementary integral bound. -/
-theorem fin_sum_inverse_sqrt (U : ℕ) :
-    (∑ k : Fin (U+1), (Real.sqrt (k.val:ℝ))⁻¹) ≤ 2*Real.sqrt U := by
-  rw [Fin.sum_univ_succ]
-  simp only [Fin.val_zero, Nat.cast_zero, Real.sqrt_zero, inv_zero,zero_add,Fin.val_succ]
-  have h := sum_inverse_sqrt U
-  rw [Finset.sum_range] at h
-  simpa only [Nat.cast_add,Nat.cast_one] using h
-
-theorem prod_two_coordinates {ι : Type*} [Fintype ι] [DecidableEq ι]
-    {e d : ι} (hed : e≠d) (f : ι → ℝ) :
-    (∏ i, f i) = f e*f d*∏ i : RemainingCoordinates e d, f i.val := by
-  have hd : d∈Finset.univ.erase e := by simp [hed.symm]
-  have h1 := Finset.prod_erase_mul Finset.univ f (Finset.mem_univ e)
-  have h2 := Finset.prod_erase_mul (Finset.univ.erase e) f hd
-  have hrem : (∏ i∈(Finset.univ.erase e).erase d, f i) =
-      ∏ i : RemainingCoordinates e d, f i.val :=
-    Finset.prod_subtype _ (by intro i; simp [and_comm]) f
-  rw [← h1,← h2,hrem]
-  ring
-
-theorem inverse_sqrt_pair {x y R : ℝ} (hR : 0<R) (hx : R≤x) (hy : R≤y) :
-    (Real.sqrt x)⁻¹*(Real.sqrt y)⁻¹ ≤ R⁻¹ := by
-  have hs : 0<Real.sqrt R := Real.sqrt_pos.mpr hR
-  have hix := (inv_le_inv₀ (hs.trans_le (Real.sqrt_le_sqrt hx)) hs).mpr (Real.sqrt_le_sqrt hx)
-  have hiy := (inv_le_inv₀ (hs.trans_le (Real.sqrt_le_sqrt hy)) hs).mpr (Real.sqrt_le_sqrt hy)
-  calc
-    _ ≤ (Real.sqrt R)⁻¹*(Real.sqrt R)⁻¹ := mul_le_mul hix hiy (by positivity) (by positivity)
-    _ = _ := by rw [← mul_inv,Real.mul_self_sqrt hR.le]
-
-/-- Explicit two-gap summation on a fixed-sum fiber. Both large gaps
- contribute R^(-1), rather than losing a free label factor. -/
-theorem two_large_gaps_inverse_sum {α ι : Type*} [Fintype α] [Fintype ι] [DecidableEq ι]
-    {U L V : ℕ} (A : Finset ι) {e d : ι} (he : e∈A) (hd : d∉A)
-    (p : α → ι → Fin (U+1)) (hp : Function.Injective p)
-    (hA : ∀ a, ∑ i∈A, (p a i).val=L) (ht : ∀ a, ∑ i, (p a i).val=V)
-    {R : ℝ} (hR : 0<R) (heR : ∀ a, R≤(p a e).val) (hdR : ∀ a, R≤(p a d).val) :
-    (∑ a, ∏ i, (Real.sqrt ((p a i).val:ℝ))⁻¹) ≤
-      R⁻¹*(2*Real.sqrt U)^Fintype.card (RemainingCoordinates e d) := by
-  classical
-  have hed : e≠d := by intro h; exact hd (h ▸ he)
-  have hpoint (a) := inverse_sqrt_pair hR (heR a) (hdR a)
-  calc
-    _ ≤ ∑ a, R⁻¹*∏ i : RemainingCoordinates e d, (Real.sqrt ((p a i.val).val:ℝ))⁻¹ := by
-      apply Finset.sum_le_sum
-      intro a _
-      rw [prod_two_coordinates hed]
-      exact mul_le_mul_of_nonneg_right (hpoint a) (by positivity)
-    _ = R⁻¹*∑ a, ∏ i : RemainingCoordinates e d, (Real.sqrt ((p a i.val).val:ℝ))⁻¹ := by
-      rw [Finset.mul_sum]
-    _ ≤ R⁻¹*∏ _i : RemainingCoordinates e d, ∑ k : Fin (U+1), (Real.sqrt (k.val:ℝ))⁻¹ := by
-      apply mul_le_mul_of_nonneg_left _ (inv_nonneg.mpr hR.le)
-      exact two_sums_product_bound A he hd p hp hA ht
-        (fun _ k => (Real.sqrt (k.val:ℝ))⁻¹) (fun _ _ => by positivity)
-    _ ≤ R⁻¹*∏ _i : RemainingCoordinates e d, (2*Real.sqrt U) := by
-      apply mul_le_mul_of_nonneg_left _ (inv_nonneg.mpr hR.le)
-      exact Finset.prod_le_prod₀ (fun _ _ => by positivity) (fun _ _ => fin_sum_inverse_sqrt U)
-    _ = _ := by simp
-
-/-- The fixed involved label is a genuine geometric fiber, not an extra
- analytic hypothesis on the rank kernel. -/
-def FixedPeriodTuple (u r : ℕ) (a : Fin r) (h : Fin u) :=
-  {j : IncreasingTuple u r // j.val a=h}
-
-noncomputable instance fixedPeriodTupleFintype (u r : ℕ) (a : Fin r) (h : Fin u) :
-    Fintype (FixedPeriodTuple u r a h) := by
-  unfold FixedPeriodTuple
-  exact Fintype.ofFinite _
-
-def fixedPeriodProfile {u r : ℕ} (a : Fin r) (h : Fin u) (j : FixedPeriodTuple u r a h) :
-    Fin (r+1) → Fin (2*u+1) := fun i =>
-  ⟨coarsePeriodGap j.val.val i, Nat.lt_succ_of_le
-    (coarsePeriodGap_le (Nat.zero_lt_of_lt a.isLt) j.val.val j.val.property i)⟩
-
-theorem fixedPeriodProfile_injective {u r : ℕ} (a : Fin r) (h : Fin u) :
-    Function.Injective (fixedPeriodProfile a h) := by
-  intro j k hjk
-  apply Subtype.ext
-  apply Subtype.ext
-  apply coarsePeriodGap_injective j.val.property k.val.property
-  funext i
-  exact congrArg Fin.val (congrFun hjk i)
-
-theorem fixedPeriodProfile_prefix {u r : ℕ} (a : Fin r) (h : Fin u)
-    (j : FixedPeriodTuple u r a h) :
-    (∑ i∈Finset.Iic a.castSucc, (fixedPeriodProfile a h j i).val)=2*h.val+1 := by
-  change (∑ i∈Finset.Iic a.castSucc, coarsePeriodGap j.val.val i)=_
-  rw [coarsePeriodGap_prefix _ j.val.property,j.property]
-
-theorem fixedPeriodProfile_total {u r : ℕ} (a : Fin r) (h : Fin u)
-    (j : FixedPeriodTuple u r a h) :
-    (∑ i, (fixedPeriodProfile a h j i).val)=2*u :=
-  coarsePeriodGap_sum (Nat.zero_lt_of_lt a.isLt) j.val.val j.val.property
-
-end TraceReconstruction
-noncomputable section
-namespace TraceReconstruction
-
-/-- An elementary centered MGF bound valid on both sides of the mean,
- with constants independent of the Bernoulli parameter. -/
-theorem binomialMass_centered_mgf (n : ℕ) {p t : ℝ}
-    (hp : p∈Set.Icc 0 1) (ht : |t|≤1) :
-    (∑ k∈Finset.range (n+1), binomialMass n p k*Real.exp (t*((k:ℝ)-n*p))) ≤
-      Real.exp ((n:ℝ)*t^2) := by
-  have he := (abs_le.mp (Real.abs_exp_sub_one_sub_id_le ht)).2
-  have hn : 0≤(n:ℝ)*p := mul_nonneg (Nat.cast_nonneg _) hp.1
-  have hn' : (n:ℝ)*p≤n := mul_le_of_le_one_right (Nat.cast_nonneg _) hp.2
-  calc
-    _ = Real.exp (-t*(n:ℝ)*p)*
-        ∑ k∈Finset.range (n+1), binomialMass n p k*Real.exp (t*k) := by
-      rw [Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro k _
-      rw [mul_left_comm,← Real.exp_add]
-      congr 2
-      ring
-    _ ≤ Real.exp (-t*(n:ℝ)*p)*Real.exp ((n:ℝ)*p*(Real.exp t-1)) :=
-      mul_le_mul_of_nonneg_left (binomialMass_mgf_le n hp t) (Real.exp_nonneg _)
-    _ ≤ Real.exp ((n:ℝ)*t^2) := by
-      rw [← Real.exp_add]
-      apply Real.exp_le_exp.mpr
-      have he' := mul_le_mul_of_nonneg_left he hn
-      have hs := mul_le_mul_of_nonneg_right hn' (sq_nonneg t)
-      nlinarith only [he',hs]
-
-/-- Either signed deviation has the same Chernoff estimate. -/
-theorem binomialMass_signed_tail (n : ℕ) {p t a ε : ℝ}
-    (hp : p∈Set.Icc 0 1) (ht : t∈Set.Icc 0 1) (hε : ε=1 ∨ ε= -1) :
-    (∑ k∈(Finset.range (n+1)).filter (fun k : ℕ => a≤ε*((k:ℝ)-n*p)), binomialMass n p k) ≤
-      Real.exp (-t*a+(n:ℝ)*t^2) := by
-  have heps : |t*ε|≤1 := by rcases hε with rfl | rfl <;> simpa [abs_of_nonneg ht.1] using ht.2
-  have hm := finite_exp_markov (Finset.range (n+1)) (binomialMass n p)
-    (fun k : ℕ => ε*((k:ℝ)-n*p)) (fun k _ => binomialMass_nonneg n k hp) a t ht.1
-  calc
-    _ ≤ Real.exp (-t*a)*∑ k∈Finset.range (n+1),
-        binomialMass n p k*Real.exp ((t*ε)*((k:ℝ)-n*p)) := by
-      simpa only [mul_assoc] using hm
-    _ ≤ Real.exp (-t*a)*Real.exp ((n:ℝ)*(t*ε)^2) :=
-      mul_le_mul_of_nonneg_left (binomialMass_centered_mgf n hp heps) (Real.exp_nonneg _)
-    _ = _ := by rcases hε with rfl | rfl <;> simp [Real.exp_add]
-
-/-- A uniform two-sided tail at any finite cutoff. Unlike a quoted
- concentration theorem, this follows here from the finite binomial MGF. -/
-theorem binomialMass_abs_tail (n : ℕ) {p t a : ℝ}
-    (hp : p∈Set.Icc 0 1) (ht : t∈Set.Icc 0 1) :
-    (∑ k∈(Finset.range (n+1)).filter (fun k : ℕ => a≤|(k:ℝ)-n*p|), binomialMass n p k) ≤
-      2*Real.exp (-t*a+(n:ℝ)*t^2) := by
-  have hpoint (k : ℕ) : (if a≤|(k:ℝ)-n*p| then binomialMass n p k else 0) ≤
-      (if a≤(k:ℝ)-n*p then binomialMass n p k else 0)+
-      (if a≤-((k:ℝ)-n*p) then binomialMass n p k else 0) := by
-    have hn := binomialMass_nonneg n k hp
-    by_cases h : a≤|(k:ℝ)-n*p|
-    · rw [ite_eq_left h]
-      have hh := le_abs.mp h
-      rcases hh with h1 | h2
-      · rw [ite_eq_left h1]
-        split_ifs <;> linarith
-      · rw [ite_eq_left h2]
-        split_ifs <;> linarith
-    · rw [ite_eq_right h]
-      split_ifs <;> linarith
-  have hh := Finset.sum_le_sum (s := Finset.range (n+1)) (fun k _ => hpoint k)
-  rw [Finset.sum_add_distrib,← Finset.sum_filter,← Finset.sum_filter,← Finset.sum_filter] at hh
-  have h1 := binomialMass_signed_tail n (a:=a) hp ht (Or.inl rfl : (1:ℝ)=1 ∨ (1:ℝ)= -1)
-  have h2 := binomialMass_signed_tail n (a:=a) hp ht (Or.inr rfl : (-1:ℝ)=1 ∨ (-1:ℝ)= -1)
-  simp only [one_mul,neg_one_mul] at h1 h2
-  linarith
-
-end TraceReconstruction
-noncomputable section
-namespace TraceReconstruction
-
-/-- Cardinality bookkeeping for the two eliminated gap coordinates. -/
-theorem remainingCoordinates_card {ι : Type*} [Fintype ι] [DecidableEq ι]
-    {e d : ι} (hed : e≠d) :
-    Fintype.card (RemainingCoordinates e d)=Fintype.card ι-2 := by
-  classical
-  have hset : (Finset.univ.filter (fun i : ι => i≠e ∧ i≠d)) = (Finset.univ.erase e).erase d := by
-    ext i
-    simp [and_comm]
-  rw [Fintype.card_subtype, hset]
-  rw [Finset.card_erase_of_mem (by simp [hed.symm]),
-    Finset.card_erase_of_mem (Finset.mem_univ e)]
-  simp only [Finset.card_univ]
-  omega
-
-/-- A finite sum's largest entry on a nonempty set controls its average,
- with the full number of coordinates in place of the subset cardinality. -/
-theorem exists_large_in_subset {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (A : Finset ι) (hA : A.Nonempty) (x : ι → ℝ) (hx : ∀ i, 0≤x i)
-    {R : ℝ} (hR : R*Fintype.card ι ≤ ∑ i∈A, x i) :
-    ∃ e∈A, R≤x e := by
-  obtain ⟨e,he,hm⟩ := A.exists_max_image x hA
-  refine ⟨e,he,?_⟩
-  have hc : (A.card:ℝ)≤Fintype.card ι := by exact_mod_cast Finset.card_le_univ A
-  have hcard : (0:ℝ)<Fintype.card ι := by
-    exact_mod_cast Fintype.card_pos_iff.mpr ⟨e⟩
-  have hs : (∑ i∈A, x i) ≤ (Fintype.card ι:ℝ)*x e := by
-    calc
-      _ ≤ ∑ _i∈A, x e := Finset.sum_le_sum (fun i hi => hm i hi)
-      _ = (A.card:ℝ)*x e := by simp
-      _ ≤ _ := mul_le_mul_of_nonneg_right hc (hx e)
-  nlinarith
-
-/-- Covering the fixed-sum profile by its selected largest coordinates
- costs only the square of its (fixed) dimension. -/
-theorem two_sided_large_gaps_sum {α ι : Type*} [Fintype α] [Fintype ι] [DecidableEq ι]
-    {U L V : ℕ} (A : Finset ι)
-    (p : α → ι → Fin (U+1)) (hp : Function.Injective p)
-    (hA : ∀ a, ∑ i∈A, (p a i).val=L) (ht : ∀ a, ∑ i, (p a i).val=V)
-    {R : ℝ} (hR : 0<R)
-    (hleft : ∀ a, ∃ e∈A, R≤(p a e).val)
-    (hright : ∀ a, ∃ d∉A, R≤(p a d).val) :
-    (∑ a, ∏ i, (Real.sqrt ((p a i).val:ℝ))⁻¹) ≤
-      (Fintype.card ι:ℝ)^2*R⁻¹*(2*Real.sqrt U)^(Fintype.card ι-2) := by
-  classical
-  choose e he heR using hleft
-  choose d hd hdR using hright
-  let f (a : α) := (e a,d a)
-  let W (a : α) := ∏ i, (Real.sqrt ((p a i).val:ℝ))⁻¹
-  have hfiber (z : ι × ι) :
-      (∑ a : {a // f a=z}, W a.val) ≤ R⁻¹*(2*Real.sqrt U)^(Fintype.card ι-2) := by
-    by_cases hne : Nonempty {a // f a=z}
-    · obtain ⟨a⟩ := hne
-      have ha := congrArg Prod.fst a.property
-      have ha' := congrArg Prod.snd a.property
-      have hz1 : z.1∈A := by simpa only [f] using ha ▸ he a.val
-      have hz2 : z.2∉A := by simpa only [f] using ha' ▸ hd a.val
-      have hzd : z.1≠z.2 := by intro h; exact hz2 (h ▸ hz1)
-      have hh := two_large_gaps_inverse_sum A hz1 hz2
-        (fun a : {a // f a=z} => p a.val) (hp.comp Subtype.val_injective)
-        (fun a => hA a.val) (fun a => ht a.val) hR
-        (fun a => by have h := congrArg Prod.fst a.property; simpa only [f] using h ▸ heR a.val)
-        (fun a => by have h := congrArg Prod.snd a.property; simpa only [f] using h ▸ hdR a.val)
-      simpa only [W,remainingCoordinates_card hzd] using hh
-    · have : IsEmpty {a // f a=z} := not_nonempty_iff.mp hne
-      simp only [Fintype.sum_empty]
-      positivity
-  calc
-    _ = ∑ z : ι × ι, ∑ a : {a // f a=z}, W a.val := by
-      exact (Fintype.sum_fiberwise f W).symm
-    _ ≤ ∑ _z : ι × ι, R⁻¹*(2*Real.sqrt U)^(Fintype.card ι-2) :=
-      Finset.sum_le_sum (fun z _ => hfiber z)
-    _ = _ := by simp [pow_two,mul_assoc]
-
-/-- The fixed central label leaves a macroscopic sum on both sides. -/
-theorem fixedPeriodProfile_central_sum {u r : ℕ} (a : Fin r) (h : Fin u)
-    (hleft : (u:ℝ)/3≤2*h.val+1)
-    (hright : (u:ℝ)/3≤2*u-(2*h.val+1)) :
-    (∑ j : FixedPeriodTuple u r a h,
-      ∏ i, (Real.sqrt (coarsePeriodGap j.val.val i))⁻¹) ≤
-      ((r+1:ℕ):ℝ)^2 * ((u:ℝ)/(3*(r+1:ℝ)))⁻¹ *
-        (2*Real.sqrt (2*u:ℕ))^(r-1) := by
-  classical
-  have hu : (0:ℝ)<u := by exact_mod_cast (Nat.zero_lt_of_lt h.isLt)
-  let A := Finset.Iic a.castSucc
-  let p := fixedPeriodProfile a h
-  have hR : 0<(u:ℝ)/(3*(r+1:ℝ)) := by positivity
-  have hL (j : FixedPeriodTuple u r a h) :
-      ∃ e∈A, (u:ℝ)/(3*(r+1:ℝ))≤(p j e).val := by
-    apply exists_large_in_subset A ⟨0, by simp [A]⟩ (fun i => (p j i).val)
-      (fun _ => Nat.cast_nonneg _)
-    have hs := fixedPeriodProfile_prefix a h j
-    have hs' : (∑ i∈A, ((p j i).val:ℝ))=2*h.val+1 := by exact_mod_cast hs
-    rw [hs']
-    simp only [Fintype.card_fin, Nat.cast_add, Nat.cast_one]
-    convert hleft using 1; field_simp
-  have hD (j : FixedPeriodTuple u r a h) :
-      ∃ d∉A, (u:ℝ)/(3*(r+1:ℝ))≤(p j d).val := by
-    have ha : a.val<r := a.isLt
-    have hlast : Fin.last r∈Aᶜ := by simp [A]
-    obtain ⟨d,hd,hdR⟩ := exists_large_in_subset Aᶜ ⟨Fin.last r,hlast⟩
-      (fun i => ((p j i).val:ℝ)) (fun _ => Nat.cast_nonneg _) (R := (u:ℝ)/(3*(r+1:ℝ))) (by
-        have hs := fixedPeriodProfile_prefix a h j
-        have ht := fixedPeriodProfile_total a h j
-        have hs' : (∑ i∈A, ((p j i).val:ℝ))=2*h.val+1 := by exact_mod_cast hs
-        have ht' : (∑ i, ((p j i).val:ℝ))=2*u := by exact_mod_cast ht
-        have hc := Finset.sum_compl_add_sum A (fun i => ((p j i).val:ℝ))
-        have hsum : (∑ i∈Aᶜ, ((p j i).val:ℝ))=2*u-(2*h.val+1) := by linarith
-        rw [hsum]
-        simp only [Fintype.card_fin, Nat.cast_add, Nat.cast_one]
-        convert hright using 1; field_simp)
-    exact ⟨d,Finset.mem_compl.mp hd,hdR⟩
-  have hh := two_sided_large_gaps_sum A p (fixedPeriodProfile_injective a h)
-    (fixedPeriodProfile_prefix a h) (fixedPeriodProfile_total a h) hR hL hD
-  simpa only [p,fixedPeriodProfile,Fin.val_mk,Fintype.card_fin,show r+1-2=r-1 by omega] using hh
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-
-/-- Any finite cutoff only removes nonnegative mass from the full binomial
- support, even if the cutoff extends beyond that support. -/
-theorem binomialMass_cutoff_filter_le (n U : ℕ) {p : ℝ} (hp : p∈Set.Icc 0 1)
-    (P : ℕ → Prop) [DecidablePred P] :
-    (∑ k : Fin (U+1), if P k.val then binomialMass n p k.val else 0) ≤
-      ∑ k∈(Finset.range (n+1)).filter P, binomialMass n p k := by
-  rw [Fin.sum_univ_eq_sum_range (fun k : ℕ => if P k then binomialMass n p k else 0) (U+1),Finset.sum_filter]
-  by_cases hU : U≤n
-  · apply Finset.sum_le_sum_of_subset_of_nonneg (Finset.range_mono (by omega))
-    intro k _ _
-    split_ifs
-    · exact binomialMass_nonneg n k hp
-    · rfl
-  · apply le_of_eq
-    symm
-    apply Finset.sum_subset (Finset.range_mono (by omega))
-    intro k _ hk
-    have hnk : n<k := by simp only [Finset.mem_range] at hk; omega
-    simp [binomialMass,Nat.choose_eq_zero_of_lt hnk]
-
-/-- A single-coordinate event for any injective family of profiles is
- bounded by the corresponding full product marginal. -/
-theorem profile_coordinate_event_le {α ι κ : Type*} [Fintype α] [Fintype ι] [Fintype κ]
-    [DecidableEq ι] (p : α → ι → κ) (hp : Function.Injective p)
-    (w : ι → κ → ℝ) (hw : ∀ i k, 0≤w i k)
-    (hs : ∀ i, ∑ k, w i k≤1) (e : ι) (P : κ → Prop) [DecidablePred P]
-    {T : ℝ} (_hT : 0≤T) (ht : (∑ k, if P k then w e k else 0)≤T) :
-    (∑ a, if P (p a e) then ∏ i, w i (p a i) else 0) ≤ T := by
-  classical
-  let w' (i : ι) (k : κ) := if i=e then (if P k then w i k else 0) else w i k
-  have hw' (i k) : 0≤w' i k := by
-    dsimp [w']
-    split_ifs
-    · exact hw i k
-    · exact le_rfl
-    · exact hw i k
-  have hprod (δ : ι → κ) : (∏ i, w' i (δ i)) =
-      if P (δ e) then ∏ i, w i (δ i) else 0 := by
-    by_cases h : P (δ e)
-    · rw [ite_eq_left h]
-      apply Finset.prod_congr rfl
-      intro i _
-      by_cases hi : i=e
-      · subst i; simp [w',h]
-      · simp [w',hi]
-    · rw [ite_eq_right h]
-      exact Finset.prod_eq_zero (Finset.mem_univ e) (by simp [w',h])
-  have hsum (i) : (∑ k, w' i k)≤ if i=e then T else 1 := by
-    by_cases hi : i=e
-    · subst i; simpa only [w',ite_true] using ht
-    · simpa only [w',ite_eq_right hi] using hs i
-  calc
-    _ = ∑ a, ∏ i, w' i (p a i) := by simp_rw [hprod]
-    _ ≤ ∑ δ : ι → κ, ∏ i, w' i (δ i) := by
-      have hh := sum_injective_profile_le p hp (fun _ => True) (fun _ => trivial)
-        (fun δ => ∏ i, w' i (δ i)) (fun δ => Finset.prod_nonneg (fun i _ => hw' i (δ i)))
-      simpa only [ite_true] using hh
-    _ = ∏ i, ∑ k, w' i k := (Fintype.prod_sum _).symm
-    _ ≤ ∏ i : ι, if i=e then T else 1 :=
-      Finset.prod_le_prod₀ (fun i _ => Finset.sum_nonneg (fun k _ => hw' i k)) (fun i _ => hsum i)
-    _ = T := by simp
-
-/-- Exponentially small clipping loss for a single gap of the actual
- conditional retained-subset rank kernel. -/
-theorem subsetRankKernel_gap_tail {n l r : ℕ} (hl : l≤n)
-    (p : Fin r → Fin n) (hp : StrictMono p) (e : Fin (r+1))
-    {θ t a : ℝ} (hθ : θ∈Set.Ioo 0 1) (ht : t∈Set.Icc 0 1) :
-    (∑ v : IncreasingTuple l r,
-      if a≤|(tupleGap v.val e:ℝ)-(tupleGap p e:ℝ)*θ|
-      then subsetRankKernel hl p v.val else 0) ≤
-      θ^r / binomialMass n θ l *
-        (2*Real.exp (-t*a+(tupleGap p e:ℝ)*t^2)) := by
-  classical
-  have hθ' : θ∈Set.Icc 0 1 := ⟨hθ.1.le,hθ.2.le⟩
-  let prof := rankGapProfile (n := l) (r := r) (Equiv.refl _)
-  let w (i : Fin (r+1)) (k : Fin (l-r+1)) := binomialMass (tupleGap p i) θ k.val
-  have hw (i k) : 0≤w i k := binomialMass_nonneg _ _ hθ'
-  have hs (i) : (∑ k, w i k)≤1 := by
-    have hh := binomialMass_cutoff_filter_le (tupleGap p i) (l-r) hθ' (fun _ => True)
-    simpa only [ite_true,Finset.filter_true,sum_binomialMass] using hh
-  have ht' : (∑ k : Fin (l-r+1),
-      if a≤|(k.val:ℝ)-(tupleGap p e:ℝ)*θ| then w e k else 0) ≤
-      2*Real.exp (-t*a+(tupleGap p e:ℝ)*t^2) := by
-    exact (binomialMass_cutoff_filter_le (tupleGap p e) (l-r) hθ'
-      (fun k : ℕ => a≤|(k:ℝ)-(tupleGap p e:ℝ)*θ|)).trans
-        (binomialMass_abs_tail _ hθ' ht)
-  have hh := profile_coordinate_event_le prof (rankGapProfile_injective _) w hw hs e
-    (fun k => a≤|(k.val:ℝ)-(tupleGap p e:ℝ)*θ|) (by positivity) ht'
-  have hfac : 0≤θ^r / binomialMass n θ l :=
-    div_nonneg (pow_nonneg hθ.1.le _) (binomialMass_nonneg _ _ hθ')
-  calc
-    _ = θ^r / binomialMass n θ l *
-        ∑ v : IncreasingTuple l r,
-          if a≤|(tupleGap v.val e:ℝ)-(tupleGap p e:ℝ)*θ|
-          then ∏ i, binomialMass (tupleGap p i) θ (tupleGap v.val i) else 0 := by
-      rw [Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro v _
-      split_ifs
-      · exact subsetRankKernel_binomial_formula hl p hp v.val v.property θ hθ
-      · simp
-    _ ≤ _ := mul_le_mul_of_nonneg_left (by
-      simpa only [prof,rankGapProfile,Equiv.refl_apply,w] using hh) hfac
-
-/-- A coordinatewise rank region has a union-bound clipping estimate. -/
-theorem subsetRankKernel_outside_region {n l r : ℕ} (hl : l≤n)
-    (p : Fin r → Fin n) (hp : StrictMono p)
-    {θ t a : ℝ} (hθ : θ∈Set.Ioo 0 1) (ht : t∈Set.Icc 0 1) :
-    (∑ v : IncreasingTuple l r,
-      if ∃ i, a≤|(tupleGap v.val i:ℝ)-(tupleGap p i:ℝ)*θ|
-      then subsetRankKernel hl p v.val else 0) ≤
-      ((r+1:ℕ):ℝ)*θ^r / binomialMass n θ l *
-        (2*Real.exp (-t*a+(n:ℝ)*t^2)) := by
-  classical
-  have hθ' : θ∈Set.Icc 0 1 := ⟨hθ.1.le,hθ.2.le⟩
-  have hpoint (v : IncreasingTuple l r) :
-      (if ∃ i, a≤|(tupleGap v.val i:ℝ)-(tupleGap p i:ℝ)*θ|
-      then subsetRankKernel hl p v.val else 0) ≤
-      ∑ i, if a≤|(tupleGap v.val i:ℝ)-(tupleGap p i:ℝ)*θ|
-        then subsetRankKernel hl p v.val else 0 := by
-    have hn (i) : 0≤(if a≤|(tupleGap v.val i:ℝ)-(tupleGap p i:ℝ)*θ|
-        then subsetRankKernel hl p v.val else 0) := by
-      split_ifs
-      · exact subsetRankKernel_nonneg hl p v.val
-      · rfl
-    split_ifs with h
-    · obtain ⟨e,he⟩ := h
-      have hh := Finset.single_le_sum (fun i _ => hn i) (Finset.mem_univ e)
-      simpa only [ite_eq_left he] using hh
-    · exact Finset.sum_nonneg (fun i _ => hn i)
-  calc
-    _ ≤ ∑ v : IncreasingTuple l r, ∑ i,
-        if a≤|(tupleGap v.val i:ℝ)-(tupleGap p i:ℝ)*θ|
-        then subsetRankKernel hl p v.val else 0 := Finset.sum_le_sum (fun v _ => hpoint v)
-    _ = ∑ i, ∑ v : IncreasingTuple l r,
-        if a≤|(tupleGap v.val i:ℝ)-(tupleGap p i:ℝ)*θ|
-        then subsetRankKernel hl p v.val else 0 := Finset.sum_comm
-    _ ≤ ∑ _i : Fin (r+1), θ^r / binomialMass n θ l *
-        (2*Real.exp (-t*a+(n:ℝ)*t^2)) := by
-      apply Finset.sum_le_sum
-      intro i _
-      apply (subsetRankKernel_gap_tail hl p hp i hθ ht).trans
-      apply mul_le_mul_of_nonneg_left _ (div_nonneg (pow_nonneg hθ.1.le _) (binomialMass_nonneg _ _ hθ'))
-      apply mul_le_mul_of_nonneg_left _ (by norm_num)
-      apply Real.exp_le_exp.mpr
-      have hgap : tupleGap p i≤n := (tupleGap_le_remainder ⟨p,hp⟩ i).trans (Nat.sub_le _ _)
-      have hg : (tupleGap p i:ℝ)≤n := by exact_mod_cast hgap
-      have hh := mul_le_mul_of_nonneg_right hg (sq_nonneg t)
-      linarith
-    _ = _ := by simp; ring
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-
-/-- An inverse square root at an integer, extended by zero at zero,
- never exceeds one. -/
-theorem inverse_sqrt_nat_le_one (k : ℕ) : (Real.sqrt (k:ℝ))⁻¹≤1 := by
-  by_cases hk : k=0
-  · simp [hk]
-  have hk' : (1:ℝ)≤k := by exact_mod_cast (Nat.one_le_iff_ne_zero.mpr hk)
-  have hs : 1≤Real.sqrt (k:ℝ) := by simpa using Real.sqrt_le_sqrt hk'
-  exact (inv_le_one₀ (by positivity : 0<Real.sqrt (k:ℝ))).mpr hs
-
-/-- A bounded gap costs a constant depending only on that bound, rather
- than an additional square root of the ambient period count. -/
-theorem fin_bounded_inverse_sqrt (U M : ℕ) :
-    (∑ k : Fin (U+1), if k.val≤M then (Real.sqrt (k.val:ℝ))⁻¹ else 0) ≤ M+1 := by
-  classical
-  calc
-    _ ≤ ∑ k : Fin (U+1), if k.val≤M then (1:ℝ) else 0 := by
-      apply Finset.sum_le_sum
-      intro k _
-      split_ifs
-      · exact inverse_sqrt_nat_le_one k.val
-      · rfl
-    _ = ((Finset.univ.filter (fun k : Fin (U+1) => k.val≤M)).card:ℝ) := by simp
-    _ ≤ M+1 := by
-      have hh : (Finset.univ.filter (fun k : Fin (U+1) => k.val≤M)).card≤M+1 := by
-        rw [← Finset.card_range (M+1)]
-        apply Finset.card_le_card_of_injOn Fin.val
-          (fun k hk => by
-            have hh := (Finset.mem_filter.mp hk).2
-            exact Finset.mem_range.mpr (Nat.lt_succ_of_le hh))
-        intro k _ j _ hkj; exact Fin.ext hkj
-      exact_mod_cast hh
-
-theorem prod_one_special {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (e : ι) (A B : ℝ) :
-    (∏ i : ι, if i=e then A else B)=A*B^(Fintype.card ι-1) := by
-  rw [← Finset.mul_prod_erase Finset.univ (fun i => if i=e then A else B) (Finset.mem_univ e)]
-  simp only [ite_true]
-  have hh : (∏ i∈Finset.univ.erase e, if i=e then A else B)=
-      ∏ _i∈Finset.univ.erase e, B := by
-    apply Finset.prod_congr rfl
-    intro i hi
-    exact ite_eq_right (Finset.mem_erase.mp hi).1
-  rw [hh]
-  simp
-
-/-- Refined two-gap elimination when a third gap has bounded size. -/
-theorem two_large_bounded_gap_sum {α ι : Type*} [Fintype α] [Fintype ι] [DecidableEq ι]
-    {U L V M : ℕ} (A : Finset ι) {e d c : ι} (he : e∈A) (hd : d∉A)
-    (hce : c≠e) (hcd : c≠d)
-    (p : α → ι → Fin (U+1)) (hp : Function.Injective p)
-    (hA : ∀ a, ∑ i∈A, (p a i).val=L) (ht : ∀ a, ∑ i, (p a i).val=V)
-    {R : ℝ} (hR : 0<R) (heR : ∀ a, R≤(p a e).val) (hdR : ∀ a, R≤(p a d).val)
-    (hcM : ∀ a, (p a c).val≤M) :
-    (∑ a, ∏ i, (Real.sqrt ((p a i).val:ℝ))⁻¹) ≤
-      R⁻¹*(M+1)*(2*Real.sqrt U)^(Fintype.card ι-3) := by
-  classical
-  have hed : e≠d := by intro h; exact hd (h ▸ he)
-  let c' : RemainingCoordinates e d := ⟨c,hce,hcd⟩
-  let w (i : ι) (k : Fin (U+1)) :=
-    if i=c then (if k.val≤M then (Real.sqrt (k.val:ℝ))⁻¹ else 0)
-    else (Real.sqrt (k.val:ℝ))⁻¹
-  have hw (i k) : 0≤w i k := by dsimp [w]; split_ifs <;> positivity
-  have hwP (a i) : w i (p a i)=(Real.sqrt ((p a i).val:ℝ))⁻¹ := by
-    by_cases hi : i=c
-    · subst i; simp only [w,ite_true,ite_eq_left (hcM a)]
-    · simp only [w,ite_eq_right hi]
-  have hwS (i : RemainingCoordinates e d) : (∑ k, w i.val k)≤
-      if i=c' then (M+1:ℝ) else 2*Real.sqrt U := by
-    by_cases hi : i=c'
-    · subst i
-      simpa only [c',w,ite_true] using fin_bounded_inverse_sqrt U M
-    · have hic : i.val≠c := by intro h; apply hi; exact Subtype.ext h
-      simpa only [w,ite_eq_right hi,ite_eq_right hic] using fin_sum_inverse_sqrt U
-  calc
-    _ ≤ ∑ a, R⁻¹*∏ i : RemainingCoordinates e d, w i.val (p a i.val) := by
-      apply Finset.sum_le_sum
-      intro a _
-      simp_rw [hwP]
-      rw [prod_two_coordinates hed]
-      exact mul_le_mul_of_nonneg_right (inverse_sqrt_pair hR (heR a) (hdR a)) (by positivity)
-    _ = R⁻¹*∑ a, ∏ i : RemainingCoordinates e d, w i.val (p a i.val) := by rw [Finset.mul_sum]
-    _ ≤ R⁻¹*∏ i : RemainingCoordinates e d, ∑ k, w i.val k :=
-      mul_le_mul_of_nonneg_left (two_sums_product_bound A he hd p hp hA ht w hw) (inv_nonneg.mpr hR.le)
-    _ ≤ R⁻¹*∏ i : RemainingCoordinates e d, if i=c' then (M+1:ℝ) else 2*Real.sqrt U := by
-      apply mul_le_mul_of_nonneg_left _ (inv_nonneg.mpr hR.le)
-      exact Finset.prod_le_prod₀ (fun i _ => Finset.sum_nonneg (fun k _ => hw i.val k)) (fun i _ => hwS i)
-    _ = _ := by
-      rw [prod_one_special,remainingCoordinates_card hed]
-      simp only [Nat.sub_sub]
-      ring
-
-/-- A small internal gap cannot be either selected macroscopic gap.
- The bound retains the extra factor u^(-1/2) needed for nonlinear terms. -/
-theorem two_sided_bounded_gaps_sum {α ι : Type*} [Fintype α] [Fintype ι] [DecidableEq ι]
-    {U L V M : ℕ} (A : Finset ι) (c : ι)
-    (p : α → ι → Fin (U+1)) (hp : Function.Injective p)
-    (hA : ∀ a, ∑ i∈A, (p a i).val=L) (ht : ∀ a, ∑ i, (p a i).val=V)
-    {R : ℝ} (hR : 0<R) (hMR : (M:ℝ)<R)
-    (hleft : ∀ a, ∃ e∈A, R≤(p a e).val)
-    (hright : ∀ a, ∃ d∉A, R≤(p a d).val)
-    (hcM : ∀ a, (p a c).val≤M) :
-    (∑ a, ∏ i, (Real.sqrt ((p a i).val:ℝ))⁻¹) ≤
-      (Fintype.card ι:ℝ)^2*R⁻¹*(M+1)*(2*Real.sqrt U)^(Fintype.card ι-3) := by
-  classical
-  choose e he heR using hleft
-  choose d hd hdR using hright
-  let f (a : α) := (e a,d a)
-  let W (a : α) := ∏ i, (Real.sqrt ((p a i).val:ℝ))⁻¹
-  have hfiber (z : ι × ι) :
-      (∑ a : {a // f a=z}, W a.val) ≤ R⁻¹*(M+1)*(2*Real.sqrt U)^(Fintype.card ι-3) := by
-    by_cases hne : Nonempty {a // f a=z}
-    · obtain ⟨a⟩ := hne
-      have ha := congrArg Prod.fst a.property
-      have ha' := congrArg Prod.snd a.property
-      have hz1 : z.1∈A := by simpa only [f] using ha ▸ he a.val
-      have hz2 : z.2∉A := by simpa only [f] using ha' ▸ hd a.val
-      have hzR : R≤(p a.val z.1).val := by simpa only [f] using ha ▸ heR a.val
-      have hzR' : R≤(p a.val z.2).val := by simpa only [f] using ha' ▸ hdR a.val
-      have hc : ((p a.val c).val:ℝ)≤M := by exact_mod_cast hcM a.val
-      have hce : c≠z.1 := by intro h; rw [h] at hc; linarith
-      have hcd : c≠z.2 := by intro h; rw [h] at hc; linarith
-      exact two_large_bounded_gap_sum A hz1 hz2 hce hcd
-        (fun a : {a // f a=z} => p a.val) (hp.comp Subtype.val_injective)
-        (fun a => hA a.val) (fun a => ht a.val) hR
-        (fun a => by have h := congrArg Prod.fst a.property; simpa only [f] using h ▸ heR a.val)
-        (fun a => by have h := congrArg Prod.snd a.property; simpa only [f] using h ▸ hdR a.val)
-        (fun a => hcM a.val)
-    · have : IsEmpty {a // f a=z} := not_nonempty_iff.mp hne
-      simp only [Fintype.sum_empty]
-      positivity
-  calc
-    _ = ∑ z : ι × ι, ∑ a : {a // f a=z}, W a.val := (Fintype.sum_fiberwise f W).symm
-    _ ≤ ∑ _z : ι × ι, R⁻¹*(M+1)*(2*Real.sqrt U)^(Fintype.card ι-3) :=
-      Finset.sum_le_sum (fun z _ => hfiber z)
-    _ = _ := by simp [pow_two,mul_assoc]
-
-end TraceReconstruction
-
-
-noncomputable section
-open scoped Classical
-namespace TraceReconstruction
-
-/-- The literal rank-clipping rectangles used in the manuscript, with
- doubled coarse gaps to avoid half-integer natural indices. Only the
- first and internal gaps are needed to determine the period tuple. -/
-def PeriodRankRegion {u l r : ℕ} (m : ℕ) (θ : ℝ)
-    (j : IncreasingTuple u r) (v : IncreasingTuple l r) : Prop :=
-  ∀ i : Fin r, |(tupleGap v.val i.castSucc:ℝ)-
-    θ*m*(coarsePeriodGap j.val i.castSucc:ℝ)/2| < (m:ℝ)/8
-
-/-- Enough accuracy to distinguish two integer gap lengths. -/
-theorem nearby_gap_centers_equal {x y : ℕ} {z θ m : ℝ}
-    (hθ : 1 / 2 ≤ θ) (hm : 0 < m)
-    (hx : |z - θ * m * x / 2| < m / 8) (hy : |z - θ * m * y / 2| < m / 8) : x = y := by
-  have hx' := abs_lt.mp hx
-  have hy' := abs_lt.mp hy
-  have hscale : m / 4 ≤ θ * m / 2 := by nlinarith
-  by_contra hxy
-  obtain h | h := lt_or_gt_of_ne hxy
-  · have h' : (x:ℝ)+1≤y := by exact_mod_cast h
-    have hh := mul_le_mul_of_nonneg_left h' (by positivity : 0 ≤ θ * m / 2)
-    nlinarith
-  · have h' : (y:ℝ)+1≤x := by exact_mod_cast h
-    have hh := mul_le_mul_of_nonneg_left h' (by positivity : 0 ≤ θ * m / 2)
-    nlinarith
-
-/-- Regions for distinct tagged period tuples are genuinely disjoint,
- before any coefficient or orientation averaging. -/
-theorem periodRankRegion_unique {u l r m : ℕ} (hm : 0 < m) {θ : ℝ} (hθ : 1 / 2 ≤ θ)
-    {j k : IncreasingTuple u r} {v : IncreasingTuple l r}
-    (hj : PeriodRankRegion m θ j v) (hk : PeriodRankRegion m θ k v) : j=k := by
-  apply Subtype.ext
-  by_cases hr : r=0
-  · subst r; exact Subsingleton.elim _ _
-  have hg (i : Fin r) : coarsePeriodGap j.val i.castSucc=coarsePeriodGap k.val i.castSucc :=
-    nearby_gap_centers_equal hθ (by exact_mod_cast hm) (hj i) (hk i)
-  have hsum := coarsePeriodGap_sum (Nat.pos_of_ne_zero hr) j.val j.property
-  have hsum' := coarsePeriodGap_sum (Nat.pos_of_ne_zero hr) k.val k.property
-  have hgSum : (∑ i : Fin r, coarsePeriodGap j.val i.castSucc)=
-      ∑ i : Fin r, coarsePeriodGap k.val i.castSucc := Finset.sum_congr rfl (fun i _ => hg i)
-  rw [Fin.sum_univ_castSucc] at hsum hsum'
-  have hlast : coarsePeriodGap j.val (Fin.last r)=coarsePeriodGap k.val (Fin.last r) := by omega
-  apply coarsePeriodGap_injective j.property k.property
-  funext i
-  refine Fin.lastCases hlast (fun i => hg i) i
-
-/-- Orthogonality of a finite family with pointwise disjoint supports. -/
-theorem disjoint_support_sum_sq {α β : Type*} [Fintype α] [Fintype β]
-    (f : α → β → ℝ) (hf : ∀ a b, a≠b → ∀ v, f a v*f b v=0) :
-    (∑ v, (∑ a, f a v)^2) = ∑ a, ∑ v, f a v^2 := by
-  classical
-  have hp (v) : (∑ a, f a v)^2=∑ a, f a v^2 := by
-    rw [pow_two,Finset.sum_mul]
-    apply Finset.sum_congr rfl
-    intro a _
-    rw [Finset.mul_sum]
-    rw [Finset.sum_eq_single a]
-    · ring
-    · intro b _ hba
-      exact hf a b hba.symm v
-    · intro ha
-      exact False.elim (ha (Finset.mem_univ a))
-  simp_rw [hp]
-  exact Finset.sum_comm
-
-/-- Squared norm of the clipped sum is at most the sum of the original
- squared norms, with arbitrary signed multipliers on each tuple. -/
-theorem period_clipped_square_sum {u l r m : ℕ} (hm : 0 < m) {θ : ℝ} (hθ : 1 / 2 ≤ θ)
-    (W : IncreasingTuple u r → IncreasingTuple l r → ℝ)
-    (a : IncreasingTuple u r → ℝ) :
-    (∑ v, (∑ j, if PeriodRankRegion m θ j v then a j*W j v else 0)^2) ≤
-      ∑ j, a j^2*∑ v, W j v^2 := by
-  classical
-  rw [disjoint_support_sum_sq _ (by
-    intro j k hjk v
-    by_cases hj : PeriodRankRegion m θ j v
-    · have hk : ¬PeriodRankRegion m θ k v := fun hk => hjk (periodRankRegion_unique hm hθ hj hk)
-      simp only [ite_eq_left hj,ite_eq_right hk,mul_zero]
-    · simp only [ite_eq_right hj,zero_mul])]
-  apply Finset.sum_le_sum
-  intro j _
-  rw [Finset.mul_sum]
-  apply Finset.sum_le_sum
-  intro v _
-  split_ifs
-  · exact le_of_eq (mul_pow _ _ _)
-  · simp only [zero_pow (by norm_num : 2 ≠ 0)]
-    exact mul_nonneg (sq_nonneg _) (sq_nonneg _)
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-instance : Fintype GapDifference :=
-  ⟨{.unchanged,.first,.second}, by intro k; cases k <;> simp⟩
-
-theorem sum_gapDifference (f : GapDifference → ℝ) :
-    (∑ k, f k)=f .unchanged+f .first+f .second := by
-  change (∑ k ∈ ({.unchanged,.first,.second} : Finset GapDifference), f k)=_
-  simp [Finset.sum_insert]; ring
-
-/-- The sign-dependent displacement in {-1,0,1}, represented by Fin 3. -/
-def motifShift (χ β τ : ℝ) : Fin 3 :=
-  if χ=0 then 1 else if β*τ=1 then 2 else 0
-
-/-- Coefficients of 1 + χ V + τ β D. -/
-def motifDifferenceWeight (χ β τ : ℝ) : GapDifference → ℝ
-  | .unchanged => 1 | .first => τ*β | .second => χ
-
-/-- The centered operator expansion is an exact displacement, not an
- approximation of the trace kernel. -/
-theorem motif_positional_identity {χ β τ : ℝ} (hβ : IsMotifValue χ β)
-    (hτ : IsTypeSign τ) (a : Fin 3) :
-    (∑ k, motifDifferenceWeight χ β τ k * positionalWeight k a)=
-      if a=motifShift χ β τ then 1 else 0 := by
-  rw [sum_gapDifference]
-  rcases hβ with ⟨rfl,rfl⟩ | ⟨rfl,rfl | rfl⟩ <;>
-  rcases hτ with rfl | rfl <;> fin_cases a <;>
-    norm_num [motifDifferenceWeight,motifShift,positionalWeight,Fin.ext_iff]
-
-theorem product_function_delta {ι κ : Type*} [Fintype ι] [DecidableEq κ]
-    (a b : ι → κ) :
-    (∏ i, if a i=b i then (1:ℝ) else 0)=if a=b then 1 else 0 := by
-  classical
-  by_cases h : a=b
-  · subst a; simp
-  · rw [ite_eq_right h]
-    obtain ⟨i,hi⟩ := Function.ne_iff.mp h
-    exact Finset.prod_eq_zero (Finset.mem_univ i) (ite_eq_right hi)
-
-/-- Tensorization of the exact local shift identity, with an arbitrary
- array W (in particular any literal retained-subset rank kernel). -/
-theorem motif_shift_expansion {r : ℕ} (χ β : Fin r → ℝ) (τ : ℝ)
-    (hβ : ∀ h, IsMotifValue (χ h) (β h)) (hτ : IsTypeSign τ)
-    (W : (Fin r → Fin 3) → ℝ) :
-    W (fun h => motifShift (χ h) (β h) τ)=
-      ∑ k : Fin r → GapDifference,
-        (∏ h, motifDifferenceWeight (χ h) (β h) τ (k h))*
-        (∑ a : Fin r → Fin 3, (∏ h, positionalWeight (k h) (a h))*W a) := by
-  classical
-  symm
-  simp_rw [Finset.mul_sum]
-  rw [Finset.sum_comm]
-  calc
-    _ = ∑ a, (∑ k : Fin r → GapDifference,
-        ∏ h, motifDifferenceWeight (χ h) (β h) τ (k h)*positionalWeight (k h) (a h))*W a := by
-      apply Finset.sum_congr rfl
-      intro a _
-      rw [Finset.sum_mul]
-      apply Finset.sum_congr rfl
-      intro k _
-      rw [Finset.prod_mul_distrib]
-      ring
-    _ = ∑ a, (∏ h, if a h=motifShift (χ h) (β h) τ then (1:ℝ) else 0)*W a := by
-      apply Finset.sum_congr rfl
-      intro a _
-      rw [← Fintype.prod_sum (fun h k => motifDifferenceWeight (χ h) (β h) τ k * positionalWeight k (a h))]
-      simp_rw [motif_positional_identity (hβ _) hτ]
-    _ = _ := by simp_rw [product_function_delta]; simp
-
-def firstDifferenceCount {r : ℕ} (k : Fin r → GapDifference) : ℕ :=
-  (Finset.univ.filter fun h => k h = .first).card
-
-def secondDifferenceCount {r : ℕ} (k : Fin r → GapDifference) : ℕ :=
-  (Finset.univ.filter fun h => k h = .second).card
-
-/-- Every factor of the hypothesis sign comes precisely from a D operator. -/
-theorem motifDifferenceWeight_product {r : ℕ} (χ β : Fin r → ℝ) (τ : ℝ)
-    (k : Fin r → GapDifference) :
-    (∏ h, motifDifferenceWeight (χ h) (β h) τ (k h))=
-      τ^firstDifferenceCount k *∏ h, motifDifferenceWeight (χ h) (β h) 1 (k h) := by
-  classical
-  have hw (h) : motifDifferenceWeight (χ h) (β h) τ (k h)=
-      (if k h=.first then τ else 1)*motifDifferenceWeight (χ h) (β h) 1 (k h) := by
-    cases k h <;> simp [motifDifferenceWeight]
-  simp_rw [hw]
-  rw [Finset.prod_mul_distrib]
-  congr 1
-  rw [← Finset.prod_filter]
-  simp [firstDifferenceCount]
-
-/-- The signed difference retains exactly the patterns having an odd
- number of centered first differences, with the source's factor two. -/
-theorem motif_signed_shift_expansion {r : ℕ} (χ β : Fin r → ℝ)
-    (hβ : ∀ h, IsMotifValue (χ h) (β h)) (W : (Fin r → Fin 3) → ℝ) :
-    W (fun h => motifShift (χ h) (β h) 1)-
-      W (fun h => motifShift (χ h) (β h) (-1))=
-      ∑ k : Fin r → GapDifference,
-        (if Odd (firstDifferenceCount k) then
-          2*∏ h, motifDifferenceWeight (χ h) (β h) 1 (k h) else 0)*
-        (∑ a : Fin r → Fin 3, (∏ h, positionalWeight (k h) (a h))*W a) := by
-  rw [motif_shift_expansion χ β 1 hβ (Or.inl rfl) W,
-    motif_shift_expansion χ β (-1) hβ (Or.inr rfl) W,← Finset.sum_sub_distrib]
-  apply Finset.sum_congr rfl
-  intro k _
-  rw [motifDifferenceWeight_product χ β 1 k,motifDifferenceWeight_product χ β (-1) k]
-  rw [sign_pow_parity (Or.inr rfl : IsTypeSign (-1))]
-  simp only [one_pow,one_mul]
-  split_ifs <;> ring
-
-theorem difference_order_eq_counts {r : ℕ} (k : Fin r → GapDifference) :
-    (∑ h, (k h).order)=firstDifferenceCount k+2*secondDifferenceCount k := by
-  classical
-  have hd (h) : (k h).order=(if k h=.first then 1 else 0)+
-      2*(if k h=.second then 1 else 0) := by cases k h <;> simp [GapDifference.order]
-  simp_rw [hd]
-  rw [Finset.sum_add_distrib,← Finset.mul_sum]
-  simp [firstDifferenceCount,secondDifferenceCount]
-
-/-- The only signed pattern of order below three is the singleton D term. -/
-theorem odd_pattern_linear_or_order_three {r : ℕ} (k : Fin r → GapDifference)
-    (hk : Odd (firstDifferenceCount k)) :
-    (firstDifferenceCount k=1 ∧ secondDifferenceCount k=0) ∨
-      3≤∑ h, (k h).order := by
-  rw [difference_order_eq_counts]
-  obtain ⟨n,hn⟩ := hk
-  omega
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem shiftGapExtra_le {r : ℕ} (a : Fin r → Fin 3) (i : Fin (r+1)) :
-    shiftGapExtra a i  ≤  2*gapIncidence r i := by
-  have hl : Fin.cons (α := fun _ =>  ℕ) 0 (fun h =>  2-(a h).val) i  ≤ 
-      2*Fin.cons (α := fun _ =>  ℕ) 0 (fun _ : Fin r =>  1) i := by
-    induction i using Fin.cases with
-    | zero =>  simp
-    | succ i =>  simp
-  have hr : Fin.snoc (α := fun _ =>  ℕ) (fun h =>  (a h).val) 0 i  ≤ 
-      2*Fin.snoc (α := fun _ =>  ℕ) (fun _ : Fin r =>  1) 0 i := by
-    induction i using Fin.lastCases with
-    | last =>  simp
-    | cast i =>  simpa using Nat.le_of_lt_succ (a i).isLt
-  dsimp [shiftGapExtra,gapIncidence]
-  omega
-
-/-- A unit positional shift changes each actual rank gap by at most two. -/
-theorem tupleGap_shifted_close {n r : ℕ} (p : Fin r → Fin n)
-    (hp : SafeTaggedTuple p) (a : Fin r → Fin 3) (i : Fin (r+1)) :
-    |(tupleGap (shiftedTag p hp a) i:ℝ)-(tupleGap p i:ℝ)| ≤ 2 := by
-  rw [tupleGap_shifted,Nat.cast_add,Nat.cast_sub (gapIncidence_le p hp i)]
-  have hi : (gapIncidence r i:ℝ) ≤ 2 := by exact_mod_cast gapIncidence_le_two r i
-  have he : (shiftGapExtra a i:ℝ) ≤ 2*(gapIncidence r i:ℝ) := by exact_mod_cast shiftGapExtra_le a i
-  have h0 : (0:ℝ) ≤ shiftGapExtra a i := by positivity
-  apply abs_le.mpr
-  constructor  <;>  linarith
-
-theorem period_shifted_gap_rounding {u m r : ℕ} (hr : 0 < r)
-    (off : Fin u → Fin m)
-    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
-    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2)
-    (j : IncreasingTuple u r) (hsafe : SafeTaggedTuple (periodSlot off ∘ j.val))
-    (a : Fin r → Fin 3) (i : Fin (r+1)) :
-    |(tupleGap (shiftedTag (periodSlot off ∘ j.val) hsafe a) i:ℝ)-
-      (m:ℝ)*coarsePeriodGap j.val i/2| ≤ 6 := by
-  have ht := abs_sub_le (tupleGap (shiftedTag (periodSlot off ∘ j.val) hsafe a) i:ℝ)
-    (tupleGap (periodSlot off ∘ j.val) i:ℝ) ((m:ℝ)*coarsePeriodGap j.val i/2)
-  have h1 := tupleGap_shifted_close (periodSlot off ∘ j.val) hsafe a i
-  have h2 := period_gap_rounding hr off hoff hdiff j.val j.property i
-  linarith
-
-theorem coarse_outside_implies_gap_tail {z g c θ m : ℝ}
-    (hθ : θ ∈ Set.Icc 0 1) (hm : 128 ≤ m) (hg : |g-m*c/2| ≤ 6)
-    (hz : m/8 ≤ |z-θ*m*c/2|) : m/16 ≤ |z-g*θ| := by
-  have he : |g*θ-θ*m*c/2| ≤ 6 := by
-    have h := mul_le_mul_of_nonneg_right hg hθ.1
-    rw [show g*θ-θ*m*c/2=(g-m*c/2)*θ by ring,abs_mul,abs_of_nonneg hθ.1]
-    exact h.trans (by nlinarith [hθ.2])
-  have ht := abs_sub_le z (g*θ) (θ*m*c/2)
-  linarith
-
-/-- The manuscript's rank rectangles lose exponentially little mass for
- every bounded shift of a tagged tuple. -/
-theorem period_shifted_outside_region {u m l r : ℕ}
-    (hu : 0 < u) (hm : 128 ≤ m) (hr : 0 < r) (hl : l ≤ u*m)
-    (off : Fin u → Fin m)
-    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
-    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2)
-    (j : IncreasingTuple u r) (hsafe : SafeTaggedTuple (periodSlot off ∘ j.val))
-    (a : Fin r → Fin 3) {θ : ℝ} (hθ : θ ∈ Set.Ioo 0 1) :
-    (∑ v : IncreasingTuple l r,
-      if PeriodRankRegion m θ j v then 0 else
-        subsetRankKernel hl (shiftedTag (periodSlot off ∘ j.val) hsafe a) v.val)  ≤ 
-      ((r+1:ℕ):ℝ)*θ^r /binomialMass (u*m) θ l *
-        (2*Real.exp (-(m:ℝ)/(1024*u))) := by
-  let p := shiftedTag (periodSlot off ∘ j.val) hsafe a
-  have hu' : (0:ℝ) < u := by exact_mod_cast hu
-  have ht : (1/(32*(u:ℝ))) ∈ Set.Icc 0 1 := by
-    constructor
-    · positivity
-    · apply (div_le_one (by positivity)).mpr
-      have : (1:ℝ) ≤ u := by exact_mod_cast hu
-      linarith
-  have hx : -(1/(32*(u:ℝ)))*((m:ℝ)/16)+((u*m:ℕ):ℝ)*(1/(32*(u:ℝ)))^2 =
-      -(m:ℝ)/(1024*u) := by push_cast; field_simp; ring
-  calc
-    _  ≤  ∑ v : IncreasingTuple l r,
-      if ∃ i, (m:ℝ)/16 ≤ |(tupleGap v.val i:ℝ)-(tupleGap p i:ℝ)*θ|
-      then subsetRankKernel hl p v.val else 0 := by
-      apply Finset.sum_le_sum
-      intro v _
-      by_cases hv : PeriodRankRegion m θ j v
-      · simp only [ite_eq_left hv]
-        split_ifs
-        · exact subsetRankKernel_nonneg hl p v.val
-        · rfl
-      · simp only [PeriodRankRegion,not_forall,not_lt] at hv
-        obtain ⟨i,hi⟩ := hv
-        have he : ∃ i', (m:ℝ)/16 ≤ |(tupleGap v.val i':ℝ)-(tupleGap p i':ℝ)*θ| :=
-          ⟨i.castSucc,coarse_outside_implies_gap_tail ⟨hθ.1.le,hθ.2.le⟩
-            (by exact_mod_cast hm) (period_shifted_gap_rounding hr off hoff hdiff j hsafe a i.castSucc) hi⟩
-        rw [ite_eq_left he]
-        split_ifs
-        · exact subsetRankKernel_nonneg hl p v.val
-        · rfl
-    _  ≤  _ := by
-      have hh := subsetRankKernel_outside_region hl p (shiftedTag_strictMono _ hsafe a) hθ ht (a := (m:ℝ)/16)
-      simpa only [hx] using hh
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- The literal support indicator of the length-2^s signed block. -/
-def motifIndicator (s : ℕ) (z : ℤ) : ℝ :=
-  if 0 ≤ z ∧ z < (2^s : ℕ) then 1 else 0
-
-theorem prouhet_motif_value (s : ℕ) (z : ℤ) :
-    IsMotifValue (motifIndicator s z) ((prouhetSeries s).coeff z) := by
-  by_cases hz : 0 ≤ z ∧ z < (2^s : ℕ)
-  · right
-    refine ⟨by simp only [motifIndicator,ite_eq_left hz], ?_⟩
-    have hv := (prouhetSeries_coeff_spec s).2 z hz.1 hz.2
-    exact (abs_eq (by norm_num : (0:ℝ) ≤ 1)).mp hv
-  · left
-    refine ⟨by simp only [motifIndicator,ite_eq_right hz], ?_⟩
-    apply (prouhetSeries_coeff_spec s).1 z
-    omega
-
-theorem motif_weight_abs_le (χ β : ℝ) (h : IsMotifValue χ β) (k : GapDifference) :
-    |motifDifferenceWeight χ β 1 k| ≤ 1 := by
-  rcases h with ⟨rfl,rfl⟩ | ⟨rfl,rfl | rfl⟩ <;> cases k <;>
-    norm_num [motifDifferenceWeight]
-
-theorem motif_weight_abs_le_indicator (χ β : ℝ) (h : IsMotifValue χ β)
-    (k : GapDifference) (hk : k ≠ .unchanged) :
-    |motifDifferenceWeight χ β 1 k| ≤ χ := by
-  rcases h with ⟨rfl,rfl⟩ | ⟨rfl,rfl | rfl⟩ <;> cases k <;>
-    norm_num [motifDifferenceWeight] at *
-
-/-- A kernel-pattern multiplier averaged over the actual finitely supported
- anchor law. No independence of its involved positions is asserted. -/
-def motifPatternMultiplier {r : ℕ} (π : IntSeries) (s : ℕ)
-    (j : Fin r → ℤ) (k : Fin r → GapDifference) : ℝ :=
-  ∑ X ∈ π.coeff.support, π.coeff X *
-    ∏ h, motifDifferenceWeight (motifIndicator s (j h-X))
-      ((prouhetSeries s).coeff (j h-X)) 1 (k h)
-
-def linearPattern {r : ℕ} (h : Fin r) (a : Fin r) : GapDifference :=
-  if a=h then .first else .unchanged
-
-theorem motif_linear_multiplier {r : ℕ} (π : IntSeries) (s : ℕ)
-    (j : Fin r → ℤ) (h : Fin r) :
-    motifPatternMultiplier π s j (linearPattern h)=
-      (prouhetSeries s * π).coeff (j h) := by
-  classical
-  rw [mul_comm,AddMonoidAlgebra.coeff_mul_apply_left,Finsupp.sum]
-  apply Finset.sum_congr rfl
-  intro X _
-  congr 1
-  have hw (a : Fin r) : motifDifferenceWeight (motifIndicator s (j a-X))
-      ((prouhetSeries s).coeff (j a-X)) 1 (linearPattern h a)=
-      if a=h then (prouhetSeries s).coeff (j h-X) else 1 := by
-    by_cases ha : a=h <;> simp [linearPattern,ha,motifDifferenceWeight]
-  simp_rw [hw]
-  simp [sub_eq_neg_add]
-
-theorem motifPatternMultiplier_abs_le {r : ℕ} (π : IntSeries)
-    (hπ : ∀ X, 0 ≤ π.coeff X) (s : ℕ) (j : Fin r → ℤ)
-    (k : Fin r → GapDifference) (h : Fin r) (hk : k h ≠ .unchanged) :
-    |motifPatternMultiplier π s j k| ≤
-      ∑ X ∈ π.coeff.support, π.coeff X*motifIndicator s (j h-X) := by
-  classical
-  apply (Finset.abs_sum_le_sum_abs _ _).trans
-  apply Finset.sum_le_sum
-  intro X _
-  rw [abs_mul,abs_of_nonneg (hπ X),Finset.abs_prod]
-  apply mul_le_mul_of_nonneg_left _ (hπ X)
-  have hi (a) := motif_weight_abs_le _ _ (prouhet_motif_value s (j a-X)) (k a)
-  have hh := motif_weight_abs_le_indicator _ _ (prouhet_motif_value s (j h-X)) (k h) hk
-  calc
-    _ ≤ ∏ a : Fin r, if a=h then motifIndicator s (j h-X) else 1 := by
-      apply Finset.prod_le_prod₀
-      · intro a _; positivity
-      · intro a _
-        by_cases ha : a=h
-        · subst a; simpa using hh
-        · simpa [ha] using hi a
-    _ = _ := by simp
-
-/-- A nonzero averaged pattern requires every involved tagged position to
- lie in one common translate of the motif; correlations are retained. -/
-theorem motif_multiplier_nonzero_support {r : ℕ} (π : IntSeries) (s : ℕ)
-    (j : Fin r → ℤ) (k : Fin r → GapDifference)
-    (hm : motifPatternMultiplier π s j k ≠ 0) :
-    ∃ X ∈ π.coeff.support, ∀ h, k h ≠ .unchanged →
-      0 ≤ j h-X ∧ j h-X < (2^s : ℕ) := by
-  classical
-  obtain ⟨X,hX,hv⟩ := Finset.exists_ne_zero_of_sum_ne_zero hm
-  refine ⟨X,hX,?_⟩
-  intro h hk
-  by_contra hh
-  apply hv
-  apply mul_eq_zero_of_right
-  apply Finset.prod_eq_zero (Finset.mem_univ h)
-  have hi : motifIndicator s (j h-X)=0 := by simp only [motifIndicator,ite_eq_right hh]
-  have hw := motif_weight_abs_le_indicator _ _ (prouhet_motif_value s (j h-X)) (k h) hk
-  exact abs_nonpos_iff.mp (hw.trans_eq hi)
-
-theorem motif_multiplier_bounded_separation {r : ℕ} (π : IntSeries) (s : ℕ)
-    (j : Fin r → ℤ) (k : Fin r → GapDifference)
-    (hm : motifPatternMultiplier π s j k ≠ 0)
-    (a b : Fin r) (ha : k a ≠ .unchanged) (hb : k b ≠ .unchanged) :
-    |j a-j b| < (2^s : ℕ) := by
-  obtain ⟨X,_,hX⟩ := motif_multiplier_nonzero_support π s j k hm
-  have h1 := hX a ha
-  have h2 := hX b hb
-  exact abs_lt.mpr ⟨by omega,by omega⟩
-
-/-- Mixing commutes with the exact signed positional expansion. -/
-theorem mixed_signed_shift_expansion {r : ℕ} (π : IntSeries) (s : ℕ)
-    (j : Fin r → ℤ) (W : (Fin r → Fin 3) → ℝ) :
-    (∑ X ∈ π.coeff.support, π.coeff X *
-      (W (fun h => motifShift (motifIndicator s (j h-X)) ((prouhetSeries s).coeff (j h-X)) 1)-
-       W (fun h => motifShift (motifIndicator s (j h-X)) ((prouhetSeries s).coeff (j h-X)) (-1))))=
-      ∑ k : Fin r → GapDifference,
-        (if Odd (firstDifferenceCount k) then 2*motifPatternMultiplier π s j k else 0)*
-        (∑ a : Fin r → Fin 3, (∏ h, positionalWeight (k h) (a h))*W a) := by
-  calc
-    _ = ∑ X ∈ π.coeff.support, ∑ k : Fin r → GapDifference,
-        π.coeff X*((if Odd (firstDifferenceCount k) then
-          2*∏ h, motifDifferenceWeight (motifIndicator s (j h-X))
-            ((prouhetSeries s).coeff (j h-X)) 1 (k h) else 0)*
-          (∑ a : Fin r → Fin 3, (∏ h, positionalWeight (k h) (a h))*W a)) := by
-      apply Finset.sum_congr rfl
-      intro X _
-      rw [motif_signed_shift_expansion _ _ (fun h => prouhet_motif_value s (j h-X)) W,
-        Finset.mul_sum]
-    _ = _ := by
-      rw [Finset.sum_comm]
-      apply Finset.sum_congr rfl
-      intro k _
-      by_cases hk : Odd (firstDifferenceCount k)
-      · simp only [ite_eq_left hk,motifPatternMultiplier]
-        calc
-          _ = (∑ X ∈ π.coeff.support, 2*(π.coeff X *
-                ∏ h, motifDifferenceWeight (motifIndicator s (j h-X))
-                  ((prouhetSeries s).coeff (j h-X)) 1 (k h))) *
-              (∑ a : Fin r → Fin 3, (∏ h, positionalWeight (k h) (a h))*W a) := by
-            rw [Finset.sum_mul]
-            apply Finset.sum_congr rfl
-            intro X _; ring
-          _ = _ := by rw [← Finset.mul_sum]
-      · simp only [ite_eq_right hk,zero_mul,mul_zero,Finset.sum_const_zero]
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- A general weighted fiber sum, including multipliers which vanish on
- inadmissible anchor labels. -/
-theorem weighted_fiber_sum_le {α β : Type*} [Fintype α] [Fintype β]
-    (f : α → β) (a : α → ℝ) (w : β → ℝ) (P : α → ℝ) (C : ℝ)
-    (hP : ∀ x, 0 ≤ P x) (ha : ∀ x, a x^2 ≤ w (f x)^2)
-    (hC : ∀ y, w y ≠ 0 → (∑ x : {x // f x=y}, P x.val) ≤ C) :
-    (∑ x, a x^2*P x) ≤ C*∑ y, w y^2 := by
-  classical
-  calc
-    _ ≤ ∑ x, w (f x)^2*P x :=
-      Finset.sum_le_sum fun x _ => mul_le_mul_of_nonneg_right (ha x) (hP x)
-    _ = ∑ y, w y^2*∑ x : {x // f x=y}, P x.val := by
-      rw [← Fintype.sum_fiberwise f (fun x => w (f x)^2*P x)]
-      apply Finset.sum_congr rfl
-      intro y _
-      rw [Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro x _
-      rw [x.property]
-    _ ≤ ∑ y, w y^2*C := by
-      apply Finset.sum_le_sum
-      intro y _
-      by_cases hy : w y=0
-      · simp [hy]
-      · exact mul_le_mul_of_nonneg_left (hC y hy) (sq_nonneg _)
-    _ = _ := by rw [← Finset.sum_mul]; ring
-
-/-- Explicit central-anchor summation constant for doubled period gaps. -/
-def periodCentralSumConstant (u r : ℕ) : ℝ :=
-  ((r+1:ℕ):ℝ)^2*((u:ℝ)/(3*(r+1:ℝ)))⁻¹*(2*Real.sqrt (2*u:ℕ))^(r-1)
-
-theorem weighted_period_gap_sum {u r : ℕ} (a : Fin r)
-    (A : IncreasingTuple u r → ℝ) (w : Fin u → ℝ)
-    (hA : ∀ j, A j^2 ≤ w (j.val a)^2)
-    (hw : ∀ h, w h ≠ 0 → (u:ℝ)/3 ≤ 2*h.val+1 ∧ (u:ℝ)/3 ≤ 2*u-(2*h.val+1)) :
-    (∑ j : IncreasingTuple u r, A j^2*∏ i, (Real.sqrt (coarsePeriodGap j.val i))⁻¹) ≤
-      periodCentralSumConstant u r *∑ h, w h^2 := by
-  apply weighted_fiber_sum_le (fun j : IncreasingTuple u r => j.val a) A w
-    (fun j => ∏ i, (Real.sqrt (coarsePeriodGap j.val i))⁻¹)
-    (periodCentralSumConstant u r) (fun _ => by positivity) hA
-  intro h hh
-  calc
-    _ = ∑ j : FixedPeriodTuple u r a h,
-        ∏ i, (Real.sqrt (coarsePeriodGap j.val.val i))⁻¹ :=
-      by
-        congr 1
-        exact congrArg (fun i : Fintype (FixedPeriodTuple u r a h) => @Finset.univ _ i)
-          (Subsingleton.elim _ _)
-    _ ≤ _ := fixedPeriodProfile_central_sum a h (hw h hh).1 (hw h hh).2
-
-/-- The two largest gaps exist even after arbitrary restrictions on the
- tuple, provided its anchor label is central. -/
-theorem fixedPeriodProfile_large_sides {u r : ℕ} (a : Fin r) (h : Fin u)
-    (hleft : (u:ℝ)/3 ≤ 2*h.val+1)
-    (hright : (u:ℝ)/3 ≤ 2*u-(2*h.val+1)) (j : FixedPeriodTuple u r a h) :
-    (∃ e ∈ Finset.Iic a.castSucc, (u:ℝ)/(3*(r+1:ℝ)) ≤ (fixedPeriodProfile a h j e).val) ∧
-    (∃ d ∉ Finset.Iic a.castSucc, (u:ℝ)/(3*(r+1:ℝ)) ≤ (fixedPeriodProfile a h j d).val) := by
-  classical
-  let A := Finset.Iic a.castSucc
-  let p := fixedPeriodProfile a h
-  have hL : ∃ e∈A, (u:ℝ)/(3*(r+1:ℝ)) ≤ (p j e).val := by
-    apply exists_large_in_subset A ⟨0, by simp [A]⟩ (fun i => (p j i).val)
-      (fun _ => Nat.cast_nonneg _)
-    have hs := fixedPeriodProfile_prefix a h j
-    have hs' : (∑ i∈A, ((p j i).val:ℝ))=2*h.val+1 := by exact_mod_cast hs
-    rw [hs']
-    simp only [Fintype.card_fin, Nat.cast_add, Nat.cast_one]
-    convert hleft using 1; field_simp
-  have hD : ∃ d∉A, (u:ℝ)/(3*(r+1:ℝ)) ≤ (p j d).val := by
-    have ha : a.val<r := a.isLt
-    have hlast : Fin.last r∈Aᶜ := by simp [A]
-    obtain ⟨d,hd,hdR⟩ := exists_large_in_subset Aᶜ ⟨Fin.last r,hlast⟩
-      (fun i => ((p j i).val:ℝ)) (fun _ => Nat.cast_nonneg _) (R := (u:ℝ)/(3*(r+1:ℝ))) (by
-        have hs := fixedPeriodProfile_prefix a h j
-        have ht := fixedPeriodProfile_total a h j
-        have hs' : (∑ i∈A, ((p j i).val:ℝ))=2*h.val+1 := by exact_mod_cast hs
-        have ht' : (∑ i, ((p j i).val:ℝ))=2*u := by exact_mod_cast ht
-        have hc := Finset.sum_compl_add_sum A (fun i => ((p j i).val:ℝ))
-        have hsum : (∑ i∈Aᶜ, ((p j i).val:ℝ))=2*u-(2*h.val+1) := by linarith
-        rw [hsum]
-        simp only [Fintype.card_fin, Nat.cast_add, Nat.cast_one]
-        convert hright using 1; field_simp)
-    exact ⟨d,Finset.mem_compl.mp hd,hdR⟩
-  exact ⟨hL,hD⟩
-
-/-- Source's bounded gap refinement at a prescribed central label. -/
-theorem fixedPeriodProfile_bounded_central_sum {u r M : ℕ} (a : Fin r) (h : Fin u)
-    (c : Fin (r+1)) (hleft : (u:ℝ)/3 ≤ 2*h.val+1)
-    (hright : (u:ℝ)/3 ≤ 2*u-(2*h.val+1)) (hM : (M:ℝ) < (u:ℝ)/(3*(r+1:ℝ))) :
-    (∑ j : FixedPeriodTuple u r a h,
-      if coarsePeriodGap j.val.val c ≤ M then ∏ i, (Real.sqrt (coarsePeriodGap j.val.val i))⁻¹ else 0) ≤
-      ((r+1:ℕ):ℝ)^2*((u:ℝ)/(3*(r+1:ℝ)))⁻¹*(M+1)*(2*Real.sqrt (2*u:ℕ))^(r-2) := by
-  classical
-  let α := {j : FixedPeriodTuple u r a h // coarsePeriodGap j.val.val c ≤ M}
-  let p (j : α) := fixedPeriodProfile a h j.val
-  have hu : (0:ℝ) < u := by exact_mod_cast (Nat.zero_lt_of_lt h.isLt)
-  have hR : 0 < (u:ℝ)/(3*(r+1:ℝ)) := by positivity
-  have hs := two_sided_bounded_gaps_sum (Finset.Iic a.castSucc) c p
-    ((fixedPeriodProfile_injective a h).comp Subtype.val_injective)
-    (fun j => fixedPeriodProfile_prefix a h j.val)
-    (fun j => fixedPeriodProfile_total a h j.val) hR hM
-    (fun j => (fixedPeriodProfile_large_sides a h hleft hright j.val).1)
-    (fun j => (fixedPeriodProfile_large_sides a h hleft hright j.val).2)
-    (fun j => j.property)
-  have he : (∑ j : FixedPeriodTuple u r a h,
-      if coarsePeriodGap j.val.val c ≤ M then ∏ i, (Real.sqrt (coarsePeriodGap j.val.val i))⁻¹ else 0)=
-      ∑ j : α, ∏ i, (Real.sqrt (coarsePeriodGap j.val.val.val i))⁻¹ := by
-    rw [← Finset.sum_filter]
-    exact Finset.sum_subtype _ (by intro j; simp) _
-  rw [he]
-  simpa only [α,p,fixedPeriodProfile,Fin.val_mk,Fintype.card_fin,
-    show r+1-3=r-2 by omega] using hs
-
-def periodBoundedSumConstant (u r M : ℕ) : ℝ :=
-  ((r+1:ℕ):ℝ)^2*((u:ℝ)/(3*(r+1:ℝ)))⁻¹*(M+1)*(2*Real.sqrt (2*u:ℕ))^(r-2)
-
-theorem weighted_period_bounded_gap_sum {u r M : ℕ} (a : Fin r) (c : Fin (r+1))
-    (A : IncreasingTuple u r → ℝ) (w : Fin u → ℝ)
-    (hA : ∀ j, A j^2 ≤ w (j.val a)^2)
-    (hsupport : ∀ j, A j ≠ 0 → coarsePeriodGap j.val c ≤ M)
-    (hw : ∀ h, w h ≠ 0 → (u:ℝ)/3 ≤ 2*h.val+1 ∧ (u:ℝ)/3 ≤ 2*u-(2*h.val+1))
-    (hM : (M:ℝ) < (u:ℝ)/(3*(r+1:ℝ))) :
-    (∑ j : IncreasingTuple u r, A j^2*∏ i, (Real.sqrt (coarsePeriodGap j.val i))⁻¹) ≤
-      periodBoundedSumConstant u r M *∑ h, w h^2 := by
-  have hp (j : IncreasingTuple u r) :
-      A j^2*(∏ i, (Real.sqrt (coarsePeriodGap j.val i))⁻¹)=
-      A j^2*(if coarsePeriodGap j.val c ≤ M then ∏ i, (Real.sqrt (coarsePeriodGap j.val i))⁻¹ else 0) := by
-    by_cases hj : A j=0
-    · simp [hj]
-    · rw [ite_eq_left (hsupport j hj)]
-  simp_rw [hp]
-  apply weighted_fiber_sum_le (fun j : IncreasingTuple u r => j.val a) A w
-    (fun j => if coarsePeriodGap j.val c ≤ M then ∏ i, (Real.sqrt (coarsePeriodGap j.val i))⁻¹ else 0)
-    (periodBoundedSumConstant u r M) (fun _ => by split_ifs <;> positivity) hA
-  intro h hh
-  calc
-    _ = ∑ j : FixedPeriodTuple u r a h,
-        if coarsePeriodGap j.val.val c ≤ M then ∏ i, (Real.sqrt (coarsePeriodGap j.val.val i))⁻¹ else 0 :=
-      by
-        congr 1
-        exact congrArg (fun i : Fintype (FixedPeriodTuple u r a h) => @Finset.univ _ i)
-          (Subsingleton.elim _ _)
-    _ ≤ _ := fixedPeriodProfile_bounded_central_sum a h c (hw h hh).1 (hw h hh).2 hM
-
-end TraceReconstruction
-
-
-namespace TraceReconstruction
-
-/-- A displaced child window as an actual interval of the deterministic
- parent template. `t` is the realized count of atoms before the window. -/
-def childWindowIndex {P T : ℕ} (t : ℕ) (hfit : t+T ≤ P) (h : Fin T) : Fin P :=
-  ⟨t+h.val, by omega⟩
-
-theorem templateType_inside_window {P T a L t : ℕ}
-    (sign : Fin L → Bool) (τ : Bool) (hfit : t+T ≤ P)
-    (hleft : t ≤ a) (h : Fin T) :
-    templateType P a L sign τ (childWindowIndex t hfit h)=
-      templateType T (a-t) L sign τ h := by
-  have he : (a ≤ (childWindowIndex t hfit h).val ∧
-      (childWindowIndex t hfit h).val < a+L) ↔
-      a-t ≤ h.val ∧ h.val < a-t+L := by dsimp [childWindowIndex]; omega
-  unfold templateType
-  split_ifs with h1 h2 h2
-  · congr 2
-    apply Fin.ext
-    dsimp [childWindowIndex]
-    omega
-  · exact False.elim (h2 (he.mp h1))
-  · exact False.elim (h1 (he.mpr h2))
-  · rfl
-
-/-- When the motif fits its assigned window, every atom outside that
- window has the default pure type, including pieces crossing nominal cuts. -/
-theorem templateType_outside_window {P T a L t : ℕ}
-    (sign : Fin L → Bool) (τ : Bool) (hleft : t ≤ a) (hright : a+L ≤ t+T)
-    (i : Fin P) (hi : i.val < t ∨ t+T ≤ i.val) :
-    templateType P a L sign τ i=false := by
-  have he : ¬(a ≤ i.val ∧ i.val < a+L) := by omega
-  simp [templateType,he]
-
-/-- A literal centered window remains within its own nominal parent unit,
- and contains the motif, whenever its jitter has the source fit bound.
- This elementary statement is independent of the distribution of the jitter. -/
-theorem centered_window_fit (P T L a A t : ℕ) (Y : ℤ)
-    (hP : 2*T ≤ P) (hA : 2*A=P-T) (ha : 2*a=P)
-    (hL : 8*L ≤ T) (hY : |8*Y| ≤ (T:ℤ)) (ht : (t:ℤ)=A+Y) :
-    t+T ≤ P ∧ t ≤ a ∧ a+L ≤ t+T := by
-  have hy := abs_le.mp hY
-  omega
-
-/-- Flattened deterministic parent type, before any random partition. -/
-def parentGroupType {R P L : ℕ} (a : ℕ) (sign : Fin L → Bool)
-    (τ : Fin R → Bool) (u : Fin R) (h : Fin P) : Bool :=
-  templateType P a L sign (τ u) h
-
-theorem parentGroup_window_types {R P T L : ℕ} (a : ℕ) (sign : Fin L → Bool)
-    (τ : Fin R → Bool) (t : Fin R → ℕ)
-    (hfit : ∀ u, t u+T ≤ P) (hleft : ∀ u, t u ≤ a)
-    (u : Fin R) (h : Fin T) :
-    parentGroupType a sign τ u (childWindowIndex (t u) (hfit u) h)=
-      templateType T (a-t u) L sign (τ u) h :=
-  templateType_inside_window sign (τ u) (hfit u) (hleft u) h
-
-end TraceReconstruction
-
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-
-end OAI
+import OAI.Probability.TraceReconstruction.Model
+import OAI.Probability.TraceReconstruction.Parity
+import OAI.Probability.TraceReconstruction.Ranks
+
+namespace OAI
+
+/-!
+# Differences
+
+Tensor gap differences, positional shifts, two-gap elimination, central clipping, disjoint rank regions, and weighted period sums.
+-/
+
+open scoped ENNReal
+noncomputable section
+
+open Filter
+open scoped Topology
+
+namespace TraceReconstruction
+
+/-- Forward length differences of the actual binomial probability
+polynomial, before imposing the conditional rank total. -/
+def binomialLengthDifference (g s : ℕ) (p : ℝ) : IntSeries :=
+  (biasedSeries 1 p - 1)^s * biasedSeries g p
+
+/-- Increasing the number of trials corresponds exactly to multiplying
+by the Bernoulli step polynomial. -/
+theorem binomialLengthDifference_succ (g s : ℕ) (p : ℝ) :
+    binomialLengthDifference g (s+1) p =
+      binomialLengthDifference (g+1) s p - binomialLengthDifference g s p := by
+  unfold binomialLengthDifference
+  rw [biasedSeries_add]
+  ring
+
+theorem biasedSeries_step_sub_one (p : ℝ) :
+    biasedSeries 1 p - 1 = (-p) • firstDifference := by
+  unfold biasedSeries firstDifference
+  rw [pow_one,smul_sub,AddMonoidAlgebra.smul_single,AddMonoidAlgebra.smul_single]
+  simp only [smul_eq_mul,mul_one]
+  have h1 : (1 : IntSeries) = AddMonoidAlgebra.single 0 1 := rfl
+  rw [h1]
+  simp only [AddMonoidAlgebra.single_sub]
+  ext z
+  simp only [AddMonoidAlgebra.coeff_sub,AddMonoidAlgebra.coeff_add,
+    Finsupp.sub_apply,Finsupp.add_apply,AddMonoidAlgebra.coeff_single,Finsupp.single_apply]
+  split_ifs <;> ring
+
+theorem binomialLengthDifference_eq (g s : ℕ) (p : ℝ) :
+    binomialLengthDifference g s p =
+      (-p)^s • (firstDifference^s * biasedSeries g p) := by
+  unfold binomialLengthDifference
+  rw [biasedSeries_step_sub_one,smul_pow,smul_mul_assoc]
+
+theorem seriesL2Sq_smul (c : ℝ) (f : IntSeries) :
+    seriesL2Sq (c • f) = c^2 * seriesL2Sq f := by
+  unfold seriesL2Sq
+  simp only [AddMonoidAlgebra.coeff_smul,Finsupp.smul_apply,smul_eq_mul,mul_pow]
+  exact tsum_mul_left
+
+/-- Pointwise and square-sum bounds for every forward length difference.
+The quotient form keeps every order and every channel factor explicit. -/
+theorem binomialLengthDifference_bounds (g s : ℕ) {p : ℝ} (hp : p ∈ Set.Ioo 0 1) :
+    (∀ z : ℤ, |(binomialLengthDifference g s p).coeff z| ≤
+      (2*biasedBinomialBound (g/(s+1)) p)^s * biasedBinomialBound (g/(s+1)) p) ∧
+    (seriesL2Sq (binomialLengthDifference g s p) ≤
+      ((2*biasedBinomialBound (g/(s+1)) p)^s)^2 * biasedBinomialBound (g/(s+1)) p) := by
+  have hpow : |(-p)^s| ≤ 1 := by
+    rw [abs_pow,abs_neg,abs_of_pos hp.1]
+    exact pow_le_one₀ hp.1.le hp.2.le
+  have hs := biased_difference_smoothing g s hp
+  rw [binomialLengthDifference_eq]
+  constructor
+  · intro z
+    simp only [AddMonoidAlgebra.coeff_smul,Finsupp.smul_apply,smul_eq_mul,abs_mul]
+    exact (mul_le_mul_of_nonneg_right hpow (abs_nonneg _)).trans (by simpa using hs.1 z)
+  · rw [seriesL2Sq_smul]
+    have h : ((-p)^s) ^ 2 ≤ 1 := by nlinarith [sq_abs ((-p)^s),abs_nonneg ((-p)^s)]
+    exact (mul_le_mul_of_nonneg_right h (seriesL2Sq_nonneg _)).trans (by simpa using hs.2)
+
+/-- Explicit fixed-order smoothing constant; no asymptotic hypothesis is hidden. -/
+def lengthDifferenceConstant (s : ℕ) (p : ℝ) : ℝ :=
+  2 * Real.sqrt (((s:ℝ)+1)/(p*(1-p)))
+
+theorem lengthDifferenceConstant_pos (s : ℕ) {p : ℝ} (hp : p ∈ Set.Ioo 0 1) :
+    0 < lengthDifferenceConstant s p := by
+  unfold lengthDifferenceConstant
+  positivity [hp.1,sub_pos.mpr hp.2]
+
+theorem biasedBinomialBound_length_le (g s : ℕ) {p : ℝ} (hp : p ∈ Set.Ioo 0 1) :
+    2*biasedBinomialBound (g/(s+1)) p ≤
+      lengthDifferenceConstant s p / Real.sqrt ((g:ℝ)+1) := by
+  calc
+    _ ≤ 2*Real.sqrt (((s:ℝ)+1)/(((g:ℝ)+1)*p*(1-p))) :=
+      mul_le_mul_of_nonneg_left (biasedBinomialBound_quotient_le g s hp) (by norm_num)
+    _ = _ := by
+      unfold lengthDifferenceConstant
+      rw [show ((s:ℝ)+1)/(((g:ℝ)+1)*p*(1-p)) =
+        (((s:ℝ)+1)/(p*(1-p)))/((g:ℝ)+1) by field_simp]
+      rw [Real.sqrt_div (by positivity [hp.1,sub_pos.mpr hp.2])]
+      ring
+
+/-- Squared supremum and squared L2 smoothing estimates in a form suitable
+for exact constrained rank profiles. -/
+theorem binomialLengthDifference_numeric (g s : ℕ) {p : ℝ} (hp : p ∈ Set.Ioo 0 1) :
+    (∀ z : ℤ, (binomialLengthDifference g s p).coeff z ^ 2 ≤
+      (lengthDifferenceConstant s p)^(2*s+2) / ((g:ℝ)+1)^(s+1)) ∧
+    (seriesL2Sq (binomialLengthDifference g s p) ≤
+      (lengthDifferenceConstant s p)^(2*s+1) /
+        (((g:ℝ)+1)^s * Real.sqrt ((g:ℝ)+1))) := by
+  have hb := binomialLengthDifference_bounds g s hp
+  have hc := biasedBinomialBound_length_le g s hp
+  have hB : 0 ≤ biasedBinomialBound (g/(s+1)) p := Real.sqrt_nonneg _
+  have hC : 0 < lengthDifferenceConstant s p := lengthDifferenceConstant_pos s hp
+  have hg : 0 < Real.sqrt ((g:ℝ)+1) := Real.sqrt_pos.mpr (by positivity)
+  have hB' : biasedBinomialBound (g/(s+1)) p ≤
+      lengthDifferenceConstant s p / Real.sqrt ((g:ℝ)+1) := by linarith
+  have hs := pow_le_pow_left₀ (by positivity : 0 ≤ 2*biasedBinomialBound (g/(s+1)) p) hc s
+  constructor
+  · intro z
+    have h := (hb.1 z).trans (mul_le_mul hs hB' hB (by positivity))
+    rw [← pow_succ] at h
+    have hsq := pow_le_pow_left₀ (abs_nonneg _) h 2
+    rw [sq_abs] at hsq
+    refine hsq.trans_eq ?_
+    rw [← pow_mul,div_pow]
+    have he : (s+1)*2=2*s+2 := by omega
+    rw [he]
+    congr 1
+    rw [show 2*s+2=2*(s+1) by omega,pow_mul,Real.sq_sqrt (by positivity)]
+  · have hsq := pow_le_pow_left₀ (by positivity :
+        0 ≤ (2*biasedBinomialBound (g/(s+1)) p)^s) hs 2
+    refine hb.2.trans ((mul_le_mul hsq hB' hB (by positivity)).trans_eq ?_)
+    rw [← pow_mul,← pow_succ,div_pow]
+    congr 1
+    · congr 1; omega
+    · rw [pow_succ,show s*2=2*s by omega,pow_mul,Real.sq_sqrt (by positivity)]
+
+end TraceReconstruction
+
+namespace TraceReconstruction
+
+theorem series_finite_sum_sq_le {α : Type*} [Fintype α]
+    (f : IntSeries) (z : α → ℤ) (hz : Function.Injective z) :
+    (∑ a, f.coeff (z a)^2) ≤ seriesL2Sq f := by
+  simpa only [tsum_fintype,seriesL2Sq] using
+    Summable.tsum_le_tsum_of_inj z hz (fun _ _ => sq_nonneg _)
+      (fun _ => le_rfl) (hasSum_fintype _).summable (series_summable f (fun x => x^2) (by simp))
+
+/-- Product coefficients of signed gap series, including an arbitrary
+translation of each observed rank gap. -/
+def signedRankProduct {l r : ℕ} (f : Fin (r+1) → IntSeries)
+    (shift : Fin (r+1) → ℤ) (v : IncreasingTuple l r) : ℝ :=
+  ∏ i, (f i).coeff ((tupleGap v.val i : ℤ)-shift i)
+
+/-- Largest-gap elimination also holds for signed finite-difference series.
+The fixed sum of the observed gaps supplies the missing square-root factor;
+translations are unrestricted and the estimate contains no omitted ranks. -/
+theorem signedRankProduct_square_sum_le {l r : ℕ}
+    (f : Fin (r+1) → IntSeries) (shift : Fin (r+1) → ℤ)
+    (e : Fin (r+1)) (M : ℝ) (hM : 0 ≤ M)
+    (hbound : ∀ z : ℤ, (f e).coeff z ^ 2 ≤ M) :
+    (∑ v : IncreasingTuple l r, signedRankProduct f shift v ^ 2) ≤
+      M * ∏ i : Fin r, seriesL2Sq (f (Equiv.swap 0 e i.succ)) := by
+  classical
+  let σ : Equiv.Perm (Fin (r+1)) := Equiv.swap 0 e
+  let F : Fin (r+1) → Fin (l-r+1) → ℝ :=
+    fun i k => (f (σ i)).coeff ((k.val:ℤ)-shift (σ i)) ^ 2
+  have hF : ∀ i k, 0 ≤ F i k := fun _ _ => sq_nonneg _
+  have hpoint (v : IncreasingTuple l r) :
+      signedRankProduct f shift v ^ 2 = ∏ i, F i (rankGapProfile σ v i) := by
+    unfold signedRankProduct
+    rw [← Finset.prod_pow]
+    dsimp [F,rankGapProfile]
+    exact (Equiv.prod_comp σ _).symm
+  simp_rw [hpoint]
+  calc
+    _ ≤ ∑ z : Fin (r+1) → Fin (l-r+1),
+        if ∑ i, (z i).val = l-r then ∏ i, F i (z i) else 0 := by
+      exact sum_injective_profile_le (rankGapProfile σ) (rankGapProfile_injective σ)
+        (fun z => ∑ i, (z i).val = l-r) (rankGapProfile_sum σ)
+        (fun z => ∏ i, F i (z i)) (fun _ => Finset.prod_nonneg (fun _ _ => hF _ _))
+    _ ≤ M * ∏ i : Fin r, ∑ k : Fin (l-r+1), F i.succ k :=
+      constrained_sum_product_le r (l-r) F hF M hM (by
+        intro k
+        simpa only [F,σ,Equiv.swap_apply_left] using hbound ((k.val:ℤ)-shift e))
+    _ ≤ _ := by
+      apply mul_le_mul_of_nonneg_left _ hM
+      apply Finset.prod_le_prod₀ (fun _ _ => Finset.sum_nonneg (fun _ _ => hF _ _))
+      intro i _
+      dsimp only [F]
+      apply series_finite_sum_sq_le
+      intro a b h
+      exact Fin.ext (Int.ofNat_inj.mp (sub_left_injective h))
+
+/-- Finite mixtures of signed rank products can be estimated termwise.
+This is the exact Cauchy--Schwarz form used when expanding positional
+finite-difference operators into gap derivatives. -/
+theorem rank_mixture_square_sum_le {α V : Type*} [Fintype α] [Fintype V]
+    (c : α → ℝ) (F : α → V → ℝ) (B : α → ℝ)
+    (hB : ∀ a, (∑ v, F a v ^ 2) ≤ B a) :
+    (∑ v, (∑ a, c a * F a v)^2) ≤ (∑ a, |c a|) * ∑ a, |c a| * B a := by
+  have hp (v : V) : (∑ a, c a*F a v)^2 ≤
+      (∑ a, |c a|) * ∑ a, |c a| * F a v^2 := by
+    apply Finset.sum_sq_le_sum_mul_sum_of_sq_le_mul Finset.univ
+      (fun _ _ => abs_nonneg _) (fun _ _ => mul_nonneg (abs_nonneg _) (sq_nonneg _))
+    intro a _
+    rw [mul_pow,← mul_assoc,← sq, sq_abs]
+  calc
+    _ ≤ ∑ v, (∑ a, |c a|) * ∑ a, |c a| * F a v^2 :=
+      Finset.sum_le_sum (fun v _ => hp v)
+    _ = (∑ a, |c a|) * ∑ a, |c a| * ∑ v, F a v^2 := by
+      rw [← Finset.mul_sum,Finset.sum_comm]
+      simp_rw [← Finset.mul_sum]
+    _ ≤ _ := by
+      apply mul_le_mul_of_nonneg_left _ (Finset.sum_nonneg (fun _ _ => abs_nonneg _))
+      apply Finset.sum_le_sum
+      intro a _
+      exact mul_le_mul_of_nonneg_left (hB a) (abs_nonneg _)
+
+end TraceReconstruction
+
+namespace TraceReconstruction
+
+/-- The tensor product of finitely supported one-dimensional coefficient
+series, represented on its literal vector-valued exponents. -/
+def tensorSeries {ι : Type*} [Fintype ι] (f : ι → IntSeries) :
+    AddMonoidAlgebra ℝ (ι → ℤ) :=
+by
+  classical
+  exact AddMonoidAlgebra.ofCoeff (Finsupp.onFinset (Fintype.piFinset (fun i => (f i).coeff.support))
+    (fun z => ∏ i, (f i).coeff (z i)) (by
+      classical
+      intro z hz
+      simp only [Fintype.mem_piFinset,Finsupp.mem_support_iff]
+      exact fun i => (Finset.prod_ne_zero_iff.mp hz) i (Finset.mem_univ i)))
+
+theorem tensorSeries_coeff {ι : Type*} [Fintype ι] (f : ι → IntSeries) (z : ι → ℤ) :
+    (tensorSeries f).coeff z = ∏ i, (f i).coeff (z i) := rfl
+
+theorem tensorSeries_support {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (f : ι → IntSeries) :
+    (tensorSeries f).coeff.support = Fintype.piFinset (fun i => (f i).coeff.support) := by
+  classical
+  ext z
+  simp [tensorSeries_coeff,Finset.prod_ne_zero_iff]
+
+/-- Literal tensor multiplication is coordinatewise convolution. -/
+theorem tensorSeries_mul {ι : Type*} [Fintype ι] (f g : ι → IntSeries) :
+    tensorSeries (fun i => f i*g i) = tensorSeries f * tensorSeries g := by
+  classical
+  ext z
+  rw [tensorSeries_coeff,AddMonoidAlgebra.coeff_mul_apply_left,Finsupp.sum,tensorSeries_support]
+  simp_rw [tensorSeries_coeff,← Finset.prod_mul_distrib]
+  simp only [Pi.add_apply, Pi.neg_apply]
+  rw [← Finset.prod_univ_sum (fun i => (f i).coeff.support)
+    (fun i j => (f i).coeff j * (g i).coeff (-j + z i))]
+  apply Finset.prod_congr rfl
+  intro i _
+  rw [AddMonoidAlgebra.coeff_mul_apply_left,Finsupp.sum]
+
+theorem tensorSeries_one {ι : Type*} [Fintype ι] :
+    tensorSeries (fun _ : ι => (1 : IntSeries)) = 1 := by
+  classical
+  ext z
+  rw [tensorSeries_coeff]
+  change (∏ i, Finsupp.single 0 (1:ℝ) (z i)) = Finsupp.single 0 (1:ℝ) z
+  simp only [Finsupp.single_apply]
+  by_cases hz : z = 0
+  · subst z; simp
+  · have he : ∃ i, z i ≠ 0 := by simpa [funext_iff] using hz
+    obtain ⟨i,hi⟩ := he
+    rw [ite_eq_right (Ne.symm hz)]
+    exact Finset.prod_eq_zero (Finset.mem_univ i) (by simp [Ne.symm hi])
+
+
+/-- Embedding one integral exponent into its own gap coordinate. -/
+def coordinateExponent {ι : Type*} [DecidableEq ι] (i : ι) : ℤ →+ (ι → ℤ) where
+  toFun := fun a => Pi.single (M := fun _ => ℤ) i a
+  map_zero' := Pi.single_zero i
+  map_add' := fun a b => by
+    ext j
+    by_cases h : j=i
+    · subst j; simp
+    · simp [Pi.single_eq_of_ne h]
+
+def coordinateSeriesHom {ι : Type*} [DecidableEq ι] (i : ι) :
+    IntSeries →+* AddMonoidAlgebra ℝ (ι → ℤ) :=
+  AddMonoidAlgebra.mapDomainRingHom ℝ (coordinateExponent i)
+
+theorem coordinateExponent_injective {ι : Type*} [DecidableEq ι] (i : ι) :
+    Function.Injective (coordinateExponent i) := by
+  intro a b h
+  simpa only [coordinateExponent,AddMonoidHom.coe_mk,ZeroHom.coe_mk,Pi.single_eq_same]
+    using congrFun h i
+
+theorem tensorSeries_coordinate {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (i : ι) (f : IntSeries) :
+    tensorSeries (fun j => if j=i then f else 1) = coordinateSeriesHom i f := by
+  classical
+  ext z
+  rw [tensorSeries_coeff]
+  change (∏ j, (if j=i then f else (1:IntSeries)).coeff (z j)) =
+    (Finsupp.mapDomain (coordinateExponent i) f.coeff) z
+  by_cases hz : ∀ j, j ≠ i → z j = 0
+  · have he : coordinateExponent i (z i) = z := by
+      funext j
+      by_cases hj : j=i
+      · subst j; simp [coordinateExponent]
+      · simp [coordinateExponent,Pi.single_eq_of_ne hj,hz j hj]
+    rw [← he,Finsupp.mapDomain_apply_of_injective (coordinateExponent_injective i)]
+    rw [Finset.prod_eq_single i]
+    · simp [coordinateExponent]
+    · intro j _ hj
+      simp [hj,coordinateExponent]
+    · simp
+  · push Not at hz
+    obtain ⟨j,hji,hj⟩ := hz
+    have hn : z ∉ Set.range (coordinateExponent i) := by
+      rintro ⟨a,ha⟩
+      have h := congrFun ha j
+      simp [coordinateExponent,Pi.single_eq_of_ne hji] at h
+      exact hj h.symm
+    rw [Finsupp.mapDomain_of_notMem_range _ _ hn]
+    apply Finset.prod_eq_zero (Finset.mem_univ j)
+    simp only [ite_eq_right hji]
+    change Finsupp.single 0 (1:ℝ) (z j) = 0
+    simp [Ne.symm hj]
+
+theorem tensorSeries_prod {ι α : Type*} [Fintype ι]
+    (s : Finset α) (f : α → ι → IntSeries) :
+    tensorSeries (fun i => ∏ a ∈ s, f a i) = ∏ a ∈ s, tensorSeries (f a) := by
+  classical
+  induction s using Finset.induction_on with
+  | empty => simp [tensorSeries_one]
+  | @insert a s ha ih =>
+    simp only [Finset.prod_insert ha]
+    rw [tensorSeries_mul,ih]
+
+/-- The literal product of the coordinate embeddings has precisely the
+independent product coefficients, with no external tensor-product axioms. -/
+theorem tensorSeries_eq_prod {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (f : ι → IntSeries) :
+    tensorSeries f = ∏ i, coordinateSeriesHom i (f i) := by
+  have he : f = fun j => ∏ i, if j=i then f i else 1 := by
+    funext j
+    simp
+  conv_lhs => rw [he,tensorSeries_prod]
+  apply Finset.prod_congr rfl
+  intro i _
+  exact tensorSeries_coordinate i (f i)
+
+end TraceReconstruction
+
+namespace TraceReconstruction
+
+/-- Explicit fixed-order gap derivative bound, uniformly over translated
+rank restrictions and the choice of the eliminated gap. -/
+theorem rank_gap_derivatives_bound {l r : ℕ}
+    (g s : Fin (r+1) → ℕ) (shift : Fin (r+1) → ℤ)
+    {p : ℝ} (hp : p ∈ Set.Ioo 0 1) (e : Fin (r+1)) :
+    (∑ v : IncreasingTuple l r,
+      signedRankProduct (fun i => binomialLengthDifference (g i) (s i) p) shift v ^ 2) ≤
+    (lengthDifferenceConstant (s e) p)^(2*s e+2) / ((g e:ℝ)+1)^(s e+1) *
+      ∏ i : Fin r,
+        (lengthDifferenceConstant (s (Equiv.swap 0 e i.succ)) p)^(2*s (Equiv.swap 0 e i.succ)+1) /
+        (((g (Equiv.swap 0 e i.succ):ℝ)+1)^(s (Equiv.swap 0 e i.succ)) *
+          Real.sqrt ((g (Equiv.swap 0 e i.succ):ℝ)+1)) := by
+  have hC := (lengthDifferenceConstant_pos (s e) hp).le
+  refine (signedRankProduct_square_sum_le _ shift e _ (by positivity)
+    (binomialLengthDifference_numeric (g e) (s e) hp).1).trans ?_
+  apply mul_le_mul_of_nonneg_left _ (by positivity)
+  apply Finset.prod_le_prod₀ (fun _ _ => seriesL2Sq_nonneg _)
+  intro i _
+  exact (binomialLengthDifference_numeric _ _ hp).2
+
+/-- Increasing an order increases only its explicit smoothing constant. -/
+theorem lengthDifferenceConstant_mono {s t : ℕ} (hst : s ≤ t)
+    {p : ℝ} (hp : p ∈ Set.Ioo 0 1) :
+    lengthDifferenceConstant s p ≤ lengthDifferenceConstant t p := by
+  unfold lengthDifferenceConstant
+  apply mul_le_mul_of_nonneg_left _ (by norm_num)
+  apply Real.sqrt_le_sqrt
+  exact div_le_div_of_nonneg_right (by exact_mod_cast Nat.succ_le_succ hst)
+    (mul_pos hp.1 (sub_pos.mpr hp.2)).le
+
+end TraceReconstruction
+
+namespace TraceReconstruction
+
+inductive GapDifference
+  | unchanged | first | second
+  deriving DecidableEq
+
+def GapDifference.order : GapDifference → ℕ
+  | .unchanged => 0 | .first => 1 | .second => 2
+
+def adjacentGap {r : ℕ} (h : Fin r) (b : Bool) : Fin (r+1) :=
+  if b then h.succ else h.castSucc
+
+def gapTermWeight (k : GapDifference) (a : Bool × Bool) : ℝ :=
+  match k with
+  | .unchanged => 1/4
+  | .first => (if a.1 then -1 else 1)/2
+  | .second => (if a.1 then -1 else 1)*(if a.2 then -1 else 1)/2
+
+def gapTermOrder {r : ℕ} (h : Fin r) (k : GapDifference) (a : Bool × Bool)
+    (i : Fin (r+1)) : ℕ :=
+  match k with
+  | .unchanged => 0
+  | .first => if i = adjacentGap h a.1 then 1 else 0
+  | .second => (if i = adjacentGap h a.1 then 1 else 0) +
+      (if i = adjacentGap h a.2 then 1 else 0)
+
+def gapTermExtra {r : ℕ} (h : Fin r) (k : GapDifference) (a : Bool × Bool)
+    (i : Fin (r+1)) : ℕ :=
+  match k with
+  | .unchanged => (if i = h.castSucc then 1 else 0) + (if i = h.succ then 1 else 0)
+  | .first => if i = adjacentGap h a.2 then 1 else 0
+  | .second => 0
+
+/-- Polynomial of a centered positional operator after removing from
+both incident gaps the one unit needed for either direction of movement. -/
+def gapOperator {r : ℕ} (h : Fin r) (k : GapDifference) (p : ℝ) :
+    AddMonoidAlgebra ℝ (Fin (r+1) → ℤ) :=
+  let L := coordinateSeriesHom h.castSucc (biasedSeries 1 p)
+  let R := coordinateSeriesHom h.succ (biasedSeries 1 p)
+  match k with
+  | .unchanged => L*R
+  | .first => (1/2:ℝ) • (L^2-R^2)
+  | .second => (1/2:ℝ) • (L^2+R^2-2*L*R)
+
+theorem tensorSeries_two_coordinates {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (a b : ι) (f g : IntSeries) :
+    tensorSeries (fun i => f^(if i=a then 1 else 0) * g^(if i=b then 1 else 0)) =
+      coordinateSeriesHom a f * coordinateSeriesHom b g := by
+  rw [tensorSeries_mul]
+  have h (c : ι) (u : IntSeries) :
+      (fun i => u^(if i=c then 1 else 0)) = (fun i => if i=c then u else 1) := by
+    funext i
+    split <;> simp
+  rw [h,h,tensorSeries_coordinate,tensorSeries_coordinate]
+
+theorem gapLocalTerm_factor {r : ℕ} (h : Fin r) (k : GapDifference)
+    (a : Bool × Bool) (p : ℝ) :
+    tensorSeries (fun i => (biasedSeries 1 p-1)^gapTermOrder h k a i *
+      (biasedSeries 1 p)^gapTermExtra h k a i) =
+      match k with
+      | .unchanged => coordinateSeriesHom h.castSucc (biasedSeries 1 p) *
+          coordinateSeriesHom h.succ (biasedSeries 1 p)
+      | .first => (coordinateSeriesHom (adjacentGap h a.1) (biasedSeries 1 p)-1) *
+          coordinateSeriesHom (adjacentGap h a.2) (biasedSeries 1 p)
+      | .second => (coordinateSeriesHom (adjacentGap h a.1) (biasedSeries 1 p)-1) *
+          (coordinateSeriesHom (adjacentGap h a.2) (biasedSeries 1 p)-1) := by
+  cases k <;> simp only [gapTermOrder,gapTermExtra,pow_zero,pow_add,one_mul,mul_one]
+  all_goals rw [tensorSeries_two_coordinates]
+  all_goals simp only [map_sub,map_one]
+
+/-- Exact four-term assignment of positional differences to the adjacent
+binomial gap lengths, including all signs and all bounded shifts. -/
+theorem gapOperator_expansion {r : ℕ} (h : Fin r) (k : GapDifference) (p : ℝ) :
+    gapOperator h k p = ∑ a : Bool × Bool, gapTermWeight k a •
+      tensorSeries (fun i => (biasedSeries 1 p-1)^gapTermOrder h k a i *
+        (biasedSeries 1 p)^gapTermExtra h k a i) := by
+  simp_rw [gapLocalTerm_factor]
+  cases k <;> simp only [gapOperator,gapTermWeight,Fintype.sum_prod_type,
+    Fintype.sum_bool,adjacentGap,Bool.false_eq_true,ite_false,ite_true]
+  · module
+  all_goals simp only [Algebra.smul_def]
+  all_goals norm_num
+  all_goals ring
+
+theorem gapTermWeight_l1 (k : GapDifference) :
+    (∑ a : Bool × Bool, |gapTermWeight k a|) ≤ 2 := by
+  cases k <;> norm_num [gapTermWeight,Fintype.sum_prod_type,Fintype.sum_bool]
+
+theorem gapTermOrder_total {r : ℕ} (h : Fin r) (k : GapDifference) (a : Bool × Bool) :
+    (∑ i, gapTermOrder h k a i) = k.order := by
+  cases k <;> simp [gapTermOrder,GapDifference.order,Finset.sum_add_distrib]
+
+/-- Exact global expansion inside the unconstrained vector of gap counts. -/
+theorem gapOperators_expansion {r : ℕ} (g : Fin (r+1) → ℕ)
+    (k : Fin r → GapDifference) (p : ℝ) :
+    tensorSeries (fun i => biasedSeries (g i) p) * (∏ h, gapOperator h (k h) p) =
+      ∑ a : Fin r → Bool × Bool, (∏ h, gapTermWeight (k h) (a h)) •
+        tensorSeries (fun i => binomialLengthDifference
+          (g i + ∑ h, gapTermExtra h (k h) (a h) i)
+          (∑ h, gapTermOrder h (k h) (a h) i) p) := by
+  classical
+  simp_rw [gapOperator_expansion]
+  rw [Finset.prod_univ_sum]
+  simp only [Fintype.piFinset_univ]
+  rw [Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro a _
+  rw [Finset.prod_smul, mul_smul_comm, ← tensorSeries_prod, ← tensorSeries_mul]
+  congr 2
+  funext i
+  simp only [binomialLengthDifference,biasedSeries,
+    Finset.prod_mul_distrib,Finset.prod_pow_eq_pow_sum]
+  ring
+
+theorem gapOperators_weight_l1 {r : ℕ} (k : Fin r → GapDifference) :
+    (∑ a : Fin r → Bool × Bool, |∏ h, gapTermWeight (k h) (a h)|) ≤ (2:ℝ)^r := by
+  simp_rw [Finset.abs_prod]
+  rw [← Fintype.prod_sum (fun h a => |gapTermWeight (k h) a|)]
+  calc
+    _ ≤ ∏ _h : Fin r, (2:ℝ) := by
+      apply Finset.prod_le_prod₀ (fun _ _ => Finset.sum_nonneg fun _ _ => abs_nonneg _)
+      intro h _
+      exact gapTermWeight_l1 (k h)
+    _ = _ := by simp
+
+end TraceReconstruction
+
+
+
+noncomputable section
+namespace TraceReconstruction
+
+theorem prod_eliminate_fin {α : Type*} [CommMonoid α] {r : ℕ}
+    (f : Fin (r+1) → α) (e : Fin (r+1)) :
+    f e * (∏ i : Fin r, f (Equiv.swap 0 e i.succ)) = ∏ i, f i := by
+  calc
+    _ = ∏ i : Fin (r+1), f (Equiv.swap 0 e i) := by
+      rw [Fin.prod_univ_succ,Equiv.swap_apply_left]
+    _ = _ := Equiv.prod_comp (Equiv.swap 0 e) f
+
+/-- Uniform smoothing with the individual gap scale retained, and a common
+lower gap scale reserved for every derivative order. -/
+theorem lengthDifference_uniform (g b s : ℕ) {p C G : ℝ}
+    (hp : p ∈ Set.Ioo 0 1) (hC : lengthDifferenceConstant s p ≤ C)
+    (hG : 0 < G) (hb : b ≤ g) (hGg : G ≤ (g:ℝ)+1) :
+    (∀ z : ℤ, (binomialLengthDifference g s p).coeff z^2 ≤
+      (C/Real.sqrt ((b:ℝ)+1)) * (C^(2*s+1)/(G^s*Real.sqrt ((b:ℝ)+1)))) ∧
+    seriesL2Sq (binomialLengthDifference g s p) ≤
+      C^(2*s+1)/(G^s*Real.sqrt ((b:ℝ)+1)) := by
+  have hC0 : 0 < C := (lengthDifferenceConstant_pos s hp).trans_le hC
+  have hb0 : 0 < (b:ℝ)+1 := by positivity
+  have hg0 : 0 < (g:ℝ)+1 := by positivity
+  have hb' : (b:ℝ)+1 ≤ (g:ℝ)+1 := by exact_mod_cast Nat.add_le_add_right hb 1
+  have hpow : G^s ≤ ((g:ℝ)+1)^s := pow_le_pow_left₀ hG.le hGg s
+  have hx := binomialLengthDifference_numeric g s hp
+  constructor
+  · intro z
+    refine (hx.1 z).trans ?_
+    have hc := pow_le_pow_left₀ (lengthDifferenceConstant_pos s hp).le hC (2*s+2)
+    have hd : G^s*((b:ℝ)+1) ≤ ((g:ℝ)+1)^(s+1) := by
+      rw [pow_succ]
+      exact mul_le_mul hpow hb' hb0.le (by positivity)
+    calc
+      _ ≤ C^(2*s+2)/(G^s*((b:ℝ)+1)) := by
+        apply div_le_div₀ (by positivity) hc (by positivity) hd
+      _ = _ := by
+        have hsq := Real.sq_sqrt hb0.le
+        have hs : Real.sqrt ((b:ℝ)+1) ≠ 0 := (Real.sqrt_pos.mpr hb0).ne'
+        rw [show 2*s+2=(2*s+1)+1 by omega,pow_succ]
+        field_simp
+        nlinarith only [hsq]
+  · refine hx.2.trans ?_
+    apply div_le_div₀ (by positivity)
+      (pow_le_pow_left₀ (lengthDifferenceConstant_pos s hp).le hC (2*s+1))
+      (by positivity)
+    exact mul_le_mul hpow (Real.sqrt_le_sqrt hb') (Real.sqrt_nonneg _) (by positivity)
+
+/-- A single translated signed rank product, with every smoothing scale
+and the largest-gap constraint exposed. -/
+theorem rank_derivatives_uniform {l r : ℕ}
+    (g b s : Fin (r+1) → ℕ) (shift : Fin (r+1) → ℤ)
+    {p C G : ℝ} (hp : p ∈ Set.Ioo 0 1)
+    (hC : ∀ i, lengthDifferenceConstant (s i) p ≤ C) (hG : 0 < G)
+    (hb : ∀ i, b i ≤ g i) (hGg : ∀ i, G ≤ (g i:ℝ)+1) (e : Fin (r+1)) :
+    (∑ v : IncreasingTuple l r,
+      signedRankProduct (fun i => binomialLengthDifference (g i) (s i) p) shift v ^ 2) ≤
+      C^(2*(∑ i, s i)+r+2) /
+        (G^(∑ i, s i) * Real.sqrt ((b e:ℝ)+1) * ∏ i, Real.sqrt ((b i:ℝ)+1)) := by
+  let F (i : Fin (r+1)) : ℝ := C^(2*s i+1)/(G^(s i)*Real.sqrt ((b i:ℝ)+1))
+  have hC0 : 0 < C := (lengthDifferenceConstant_pos (s e) hp).trans_le (hC e)
+  have hF : ∀ i, 0 ≤ F i := fun i => by dsimp [F]; positivity
+  have hbound (i : Fin (r+1)) := lengthDifference_uniform (g i) (b i) (s i) hp (hC i) hG (hb i) (hGg i)
+  calc
+    _ ≤ (C/Real.sqrt ((b e:ℝ)+1)*F e) *
+        ∏ i : Fin r, seriesL2Sq (binomialLengthDifference (g (Equiv.swap 0 e i.succ))
+          (s (Equiv.swap 0 e i.succ)) p) := by
+      apply signedRankProduct_square_sum_le _ shift e _ (by positivity [hF e])
+      exact (hbound e).1
+    _ ≤ (C/Real.sqrt ((b e:ℝ)+1)*F e) * ∏ i : Fin r, F (Equiv.swap 0 e i.succ) := by
+      apply mul_le_mul_of_nonneg_left _ (by positivity [hF e])
+      apply Finset.prod_le_prod₀ (fun _ _ => seriesL2Sq_nonneg _)
+      intro i _
+      exact (hbound _).2
+    _ = (C/Real.sqrt ((b e:ℝ)+1))*∏ i, F i := by
+      rw [mul_assoc,prod_eliminate_fin]
+    _ = _ := by
+      simp only [F,Finset.prod_div_distrib,Finset.prod_mul_distrib,
+        Finset.prod_pow_eq_pow_sum,Finset.sum_add_distrib,← Finset.mul_sum,
+        Finset.sum_const,Finset.card_univ,Fintype.card_fin,smul_eq_mul,mul_one]
+      rw [show 2*(∑ i, s i)+r+2=(2*(∑ i, s i)+(r+1))+1 by omega,pow_succ]
+      ring
+
+end TraceReconstruction
+
+
+
+noncomputable section
+namespace TraceReconstruction
+
+/-- Enough vacant slots for independent one-step displacements of every tag. -/
+def SafeTaggedTuple {n r : ℕ} (p : Fin r → Fin n) : Prop :=
+  (∀ i, 1 ≤ (p i).val ∧ (p i).val+1 < n) ∧
+  ∀ i j, i < j → (p i).val+3 ≤ (p j).val
+
+def shiftedTag {n r : ℕ} (p : Fin r → Fin n) (hp : SafeTaggedTuple p)
+    (a : Fin r → Fin 3) (i : Fin r) : Fin n :=
+  ⟨(p i).val+(a i).val-1, by
+    have h := hp.1 i
+    have ha := (a i).isLt
+    omega⟩
+
+theorem shiftedTag_strictMono {n r : ℕ} (p : Fin r → Fin n) (hp : SafeTaggedTuple p)
+    (a : Fin r → Fin 3) : StrictMono (shiftedTag p hp a) := by
+  intro i j hij
+  change (p i).val+(a i).val-1 < (p j).val+(a j).val-1
+  have h := hp.2 i j hij
+  have hi := hp.1 i
+  have hj := hp.1 j
+  have hai := (a i).isLt
+  omega
+
+def gapIncidence (r : ℕ) (i : Fin (r+1)) : ℕ :=
+  Fin.cons (α := fun _ => ℕ) 0 (fun _ : Fin r => 1) i + Fin.snoc (α := fun _ => ℕ) (fun _ : Fin r => 1) 0 i
+
+def shiftGapExtra {r : ℕ} (a : Fin r → Fin 3) (i : Fin (r+1)) : ℕ :=
+  Fin.cons (α := fun _ => ℕ) 0 (fun h => 2-(a h).val) i + Fin.snoc (α := fun _ => ℕ) (fun h => (a h).val) 0 i
+
+/-- Literal adjacent-gap allocation of the units used by a displacement.
+Value zero means translate left, one leave fixed, two translate right. -/
+def localShiftExtra {r : ℕ} (h : Fin r) (a : Fin 3) (i : Fin (r+1)) : ℕ :=
+  (if i=h.castSucc then a.val else 0) + (if i=h.succ then 2-a.val else 0)
+
+theorem sum_castSucc_indicator {r : ℕ} (f : Fin r → ℕ) (i : Fin (r+1)) :
+    (∑ h, if i=h.castSucc then f h else 0) = Fin.snoc (α := fun _ => ℕ) f 0 i := by
+  induction i using Fin.lastCases with
+  | last => simp [eq_comm]
+  | cast h => simp
+
+theorem sum_succ_indicator {r : ℕ} (f : Fin r → ℕ) (i : Fin (r+1)) :
+    (∑ h, if i=h.succ then f h else 0) = Fin.cons (α := fun _ => ℕ) 0 f i := by
+  induction i using Fin.cases with
+  | zero => simp [eq_comm]
+  | succ h => simp
+
+theorem shiftGapExtra_eq_sum {r : ℕ} (a : Fin r → Fin 3) (i : Fin (r+1)) :
+    shiftGapExtra a i = ∑ h, localShiftExtra h (a h) i := by
+  simp only [localShiftExtra,Finset.sum_add_distrib,sum_castSucc_indicator,
+    sum_succ_indicator,shiftGapExtra]
+  omega
+
+theorem gapIncidence_le {n r : ℕ} (p : Fin r → Fin n) (hp : SafeTaggedTuple p)
+    (i : Fin (r+1)) : gapIncidence r i ≤ tupleGap p i := by
+  cases r with
+  | zero =>
+    have hi : i=0 := Fin.eq_zero i
+    subst i
+    simp [gapIncidence,Fin.snoc_zero]
+  | succ r =>
+    induction i using Fin.cases with
+    | zero => simpa only [gapIncidence,Fin.cons_zero,zero_add,Fin.snoc_apply_zero,
+        tupleGap_zero] using (hp.1 0).1
+    | succ i =>
+      induction i using Fin.lastCases with
+      | last =>
+        simp only [gapIncidence,Fin.succ_last,Fin.cons_last,Fin.snoc_last,add_zero,
+          tupleGap_last]
+        have h := (hp.1 (Fin.last r)).2
+        omega
+      | cast i =>
+        simp only [gapIncidence,Fin.cons_succ]
+        rw [Fin.succ_castSucc,Fin.snoc_castSucc]
+        rw [← Fin.succ_castSucc,tupleGap_middle]
+        have h := hp.2 i.castSucc i.succ (by simp)
+        omega
+
+/-- The exact geometric identity relating translated tag positions to
+nonnegative binomial-gap lengths. -/
+theorem tupleGap_shifted {n r : ℕ} (p : Fin r → Fin n) (hp : SafeTaggedTuple p)
+    (a : Fin r → Fin 3) (i : Fin (r+1)) :
+    tupleGap (shiftedTag p hp a) i =
+      tupleGap p i - gapIncidence r i + shiftGapExtra a i := by
+  cases r with
+  | zero =>
+    have hi : i=0 := Fin.eq_zero i
+    subst i
+    simp [tupleGap,tupleLower,tupleUpper,gapIncidence,shiftGapExtra,Fin.snoc_zero]
+  | succ r =>
+    induction i using Fin.cases with
+    | zero =>
+      simp only [tupleGap_zero,gapIncidence,shiftGapExtra,Fin.cons_zero,zero_add,
+        Fin.snoc_apply_zero,shiftedTag]
+      have h := (hp.1 0).1
+      omega
+    | succ i =>
+      induction i using Fin.lastCases with
+      | last =>
+        simp only [gapIncidence,shiftGapExtra,Fin.succ_last,Fin.cons_last,
+          Fin.snoc_last,add_zero,tupleGap_last,shiftedTag]
+        have h := hp.1 (Fin.last r)
+        have ha := (a (Fin.last r)).isLt
+        omega
+      | cast i =>
+        simp only [gapIncidence,shiftGapExtra,Fin.cons_succ]
+        rw [Fin.succ_castSucc,Fin.snoc_castSucc,Fin.snoc_castSucc]
+        rw [← Fin.succ_castSucc,tupleGap_middle,tupleGap_middle]
+        change (p i.succ).val+(a i.succ).val-1-((p i.castSucc).val+(a i.castSucc).val-1+1) = _
+        have hi := hp.1 i.castSucc
+        have hj := hp.1 i.succ
+        have hij := hp.2 i.castSucc i.succ (by simp)
+        have hai := (a i.castSucc).isLt
+        have haj := (a i.succ).isLt
+        omega
+
+
+/-- Coefficients of the unchanged, centered first, and centered second
+one-step positional operators. -/
+def positionalWeight (k : GapDifference) (a : Fin 3) : ℝ :=
+  match k with
+  | .unchanged => if a=1 then 1 else 0
+  | .first => if a=0 then -1/2 else if a=2 then 1/2 else 0
+  | .second => if a=1 then -1 else 1/2
+
+theorem localShift_factor {r : ℕ} (h : Fin r) (a : Fin 3) (p : ℝ) :
+    tensorSeries (fun i => (biasedSeries 1 p)^localShiftExtra h a i) =
+      coordinateSeriesHom h.castSucc (biasedSeries 1 p)^a.val *
+        coordinateSeriesHom h.succ (biasedSeries 1 p)^(2-a.val) := by
+  have hpow (u : IntSeries) (t : ℕ) (c i : Fin (r+1)) :
+      u^(if i=c then t else 0) = (u^t)^(if i=c then 1 else 0) := by
+    split <;> simp
+  have he : (fun i => (biasedSeries 1 p)^localShiftExtra h a i) =
+      (fun i => ((biasedSeries 1 p)^a.val)^(if i=h.castSucc then 1 else 0) *
+        ((biasedSeries 1 p)^(2-a.val))^(if i=h.succ then 1 else 0)) := by
+    funext i
+    rw [localShiftExtra,pow_add,hpow _ _ h.castSucc i,hpow _ _ h.succ i]
+  rw [he,tensorSeries_two_coordinates,map_pow,map_pow]
+
+theorem gapOperator_positional_expansion {r : ℕ} (h : Fin r) (k : GapDifference)
+    (p : ℝ) :
+    gapOperator h k p = ∑ a : Fin 3, positionalWeight k a •
+      tensorSeries (fun i => (biasedSeries 1 p)^localShiftExtra h a i) := by
+  simp_rw [localShift_factor]
+  rw [Fin.sum_univ_three]
+  cases k <;> norm_num [positionalWeight,gapOperator,Fin.ext_iff]
+  all_goals try simp only [two_mul,add_mul]
+  all_goals module
+
+/-- Literal centered differences of all rank kernels are coefficients of
+the adjacent-gap operator product, with no formal negative trial lengths. -/
+theorem shifted_gap_series_expansion {n r : ℕ} (p : Fin r → Fin n)
+    (hp : SafeTaggedTuple p) (k : Fin r → GapDifference) (θ : ℝ) :
+    (∑ a : Fin r → Fin 3, (∏ h, positionalWeight (k h) (a h)) •
+      tensorSeries (fun i => biasedSeries (tupleGap (shiftedTag p hp a) i) θ)) =
+    tensorSeries (fun i => biasedSeries (tupleGap p i-gapIncidence r i) θ) *
+      ∏ h, gapOperator h (k h) θ := by
+  classical
+  simp_rw [gapOperator_positional_expansion]
+  rw [Finset.prod_univ_sum]
+  simp only [Fintype.piFinset_univ]
+  rw [Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro a _
+  rw [Finset.prod_smul,mul_smul_comm,← tensorSeries_prod,← tensorSeries_mul]
+  congr 2
+  funext i
+  rw [tupleGap_shifted,shiftGapExtra_eq_sum]
+  simp only [biasedSeries,pow_one,Finset.prod_pow_eq_pow_sum,pow_add]
+
+
+/-- Simultaneous literal positional operators applied to the actual
+conditional retained-subset kernel. -/
+def positionalRankDifference {n l r : ℕ} (hl : l ≤ n) (p : Fin r → Fin n)
+    (hp : SafeTaggedTuple p) (k : Fin r → GapDifference) (v : IncreasingTuple l r) : ℝ :=
+  ∑ a : Fin r → Fin 3, (∏ h, positionalWeight (k h) (a h)) *
+    subsetRankKernel hl (shiftedTag p hp a) v.val
+
+theorem positionalRankDifference_expansion {n l r : ℕ} (hl : l ≤ n)
+    (p : Fin r → Fin n) (hp : SafeTaggedTuple p) (k : Fin r → GapDifference)
+    {θ : ℝ} (hθ : θ ∈ Set.Ioo 0 1) (v : IncreasingTuple l r) :
+    positionalRankDifference hl p hp k v = θ^r / binomialMass n θ l *
+      ∑ a : Fin r → Bool × Bool, (∏ h, gapTermWeight (k h) (a h)) *
+        signedRankProduct (fun i => binomialLengthDifference
+          (tupleGap p i-gapIncidence r i + ∑ h, gapTermExtra h (k h) (a h) i)
+          (∑ h, gapTermOrder h (k h) (a h) i) θ) (fun _ => 0) v := by
+  have h := shifted_gap_series_expansion p hp k θ
+  rw [gapOperators_expansion] at h
+  have hh := congrArg (fun f : AddMonoidAlgebra ℝ (Fin (r+1) → ℤ) =>
+    f.coeff (fun i => (tupleGap v.val i:ℤ))) h
+  simp only [AddMonoidAlgebra.coeff_sum,Finsupp.finsetSum_apply,
+    AddMonoidAlgebra.coeff_smul,Finsupp.smul_apply,smul_eq_mul,tensorSeries_coeff,
+    biasedSeries_coeff_nat] at hh
+  unfold positionalRankDifference
+  simp_rw [subsetRankKernel_binomial_formula hl _ (shiftedTag_strictMono p hp _) _ v.property θ hθ]
+  calc
+    _ = θ^r / binomialMass n θ l *
+        ∑ a : Fin r → Fin 3, (∏ h, positionalWeight (k h) (a h)) *
+          ∏ i, binomialMass (tupleGap (shiftedTag p hp a) i) θ (tupleGap v.val i) := by
+      rw [Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro a _
+      ring
+    _ = _ := by
+      exact congrArg (fun t => θ^r / binomialMass n θ l * t) (by
+        simpa only [signedRankProduct,sub_zero] using hh)
+
+/-- The total derivative order is the source value |D|+2|V|. -/
+theorem gap_global_order {r : ℕ} (k : Fin r → GapDifference) (a : Fin r → Bool × Bool) :
+    (∑ i : Fin (r+1), ∑ h, gapTermOrder h (k h) (a h) i) = ∑ h, (k h).order := by
+  rw [Finset.sum_comm]
+  simp_rw [gapTermOrder_total]
+
+end TraceReconstruction
+
+
+
+noncomputable section
+namespace TraceReconstruction
+
+theorem gapTermOrder_sum_le_global {r : ℕ} (k : Fin r → GapDifference)
+    (a : Fin r → Bool × Bool) (i : Fin (r+1)) :
+    (∑ h, gapTermOrder h (k h) (a h) i) ≤ ∑ h, (k h).order := by
+  rw [← gap_global_order k a]
+  exact Finset.single_le_sum (f := fun j : Fin (r+1) => ∑ h, gapTermOrder h (k h) (a h) j)
+    (fun _ _ => Nat.zero_le _) (Finset.mem_univ i)
+
+/-- Literal mixed centered positional differences of a conditional rank
+kernel, with a single explicit constant depending only on the total order
+and the Bernoulli parameter. Every gap scale remains available for summing
+over the period tuples in the signed initial construction. -/
+theorem positionalRankDifference_square_sum_le {n l r : ℕ} (hl : l ≤ n)
+    (p : Fin r → Fin n) (hp : SafeTaggedTuple p) (k : Fin r → GapDifference)
+    {θ G : ℝ} (hθ : θ ∈ Set.Ioo 0 1) (hG : 0 < G)
+    (hgap : ∀ i, G ≤ (tupleGap p i-gapIncidence r i : ℕ)+1)
+    (e : Fin (r+1)) :
+    (∑ v : IncreasingTuple l r, positionalRankDifference hl p hp k v ^ 2) ≤
+      (θ^r / binomialMass n θ l)^2 * (2:ℝ)^(2*r) *
+        (lengthDifferenceConstant (∑ h, (k h).order) θ)^(2*(∑ h, (k h).order)+r+2) /
+      (G^(∑ h, (k h).order) * Real.sqrt ((tupleGap p e-gapIncidence r e:ℕ)+1) *
+        ∏ i, Real.sqrt ((tupleGap p i-gapIncidence r i:ℕ)+1)) := by
+  let d := ∑ h, (k h).order
+  let C := lengthDifferenceConstant d θ
+  let b (i : Fin (r+1)) := tupleGap p i-gapIncidence r i
+  let B := C^(2*d+r+2) / (G^d*Real.sqrt ((b e:ℝ)+1)*∏ i,Real.sqrt ((b i:ℝ)+1))
+  let c (a : Fin r → Bool × Bool) := ∏ h, gapTermWeight (k h) (a h)
+  let F (a : Fin r → Bool × Bool) (v : IncreasingTuple l r) :=
+    signedRankProduct (fun i => binomialLengthDifference
+      (b i+∑ h, gapTermExtra h (k h) (a h) i)
+      (∑ h, gapTermOrder h (k h) (a h) i) θ) (fun _ => 0) v
+  have hC : 0 < C := lengthDifferenceConstant_pos d hθ
+  have hB0 : 0 ≤ B := by dsimp [B]; positivity
+  have hB (a : Fin r → Bool × Bool) : (∑ v, F a v^2) ≤ B := by
+    have hd (i : Fin (r+1)) : lengthDifferenceConstant (∑ h, gapTermOrder h (k h) (a h) i) θ ≤ C :=
+      lengthDifferenceConstant_mono (gapTermOrder_sum_le_global k a i) hθ
+    have h := rank_derivatives_uniform (l := l)
+      (fun i => b i+∑ h, gapTermExtra h (k h) (a h) i) b
+      (fun i => ∑ h, gapTermOrder h (k h) (a h) i) (fun _ => 0)
+      hθ hd hG (fun _ => Nat.le_add_right _ _) (by
+        intro i
+        exact (hgap i).trans (by exact_mod_cast Nat.add_le_add_right (Nat.le_add_right (b i) _) 1)) e
+    simpa only [F,B,d,gap_global_order] using h
+  have hmix := rank_mixture_square_sum_le c F (fun _ => B) hB
+  have hM : (∑ a, |c a|) ≤ (2:ℝ)^r := gapOperators_weight_l1 k
+  have hM0 : 0 ≤ ∑ a, |c a| := Finset.sum_nonneg fun _ _ => abs_nonneg _
+  have hmix' : (∑ v, (∑ a, c a*F a v)^2) ≤ (2:ℝ)^(2*r)*B := by
+    refine hmix.trans ?_
+    rw [← Finset.sum_mul,← mul_assoc]
+    have hs : (∑ a, |c a|)^2 ≤ ((2:ℝ)^r)^2 := (sq_le_sq₀ hM0 (by positivity)).mpr hM
+    calc
+      _ = (∑ a, |c a|)^2*B := by ring
+      _ ≤ ((2:ℝ)^r)^2*B := mul_le_mul_of_nonneg_right hs hB0
+      _ = _ := by rw [← pow_mul,Nat.mul_comm r 2]
+  simp_rw [positionalRankDifference_expansion hl p hp k hθ,mul_pow]
+  rw [← Finset.mul_sum]
+  calc
+    _ ≤ (θ^r / binomialMass n θ l)^2 * ((2:ℝ)^(2*r)*B) :=
+      mul_le_mul_of_nonneg_left hmix' (sq_nonneg _)
+    _ = _ := by dsimp [B,C,d,b]; ring
+
+end TraceReconstruction
+
+
+
+noncomputable section
+namespace TraceReconstruction
+
+theorem gapIncidence_le_two (r : ℕ) (i : Fin (r+1)) : gapIncidence r i ≤ 2 := by
+  have h₁ : Fin.cons (α := fun _ => ℕ) 0 (fun _ : Fin r => 1) i ≤ 1 := by
+    induction i using Fin.cases <;> simp
+  have h₂ : Fin.snoc (α := fun _ => ℕ) (fun _ : Fin r => 1) 0 i ≤ 1 := by
+    induction i using Fin.lastCases <;> simp
+  exact Nat.add_le_add h₁ h₂
+
+/-- Constant middle-period tags admit every independent left/unchanged/right
+choice without violating slot order or either end boundary. -/
+theorem constant_period_safe {u m r : ℕ} (off : Fin m)
+    (hoff₁ : 1 ≤ off.val) (hoff₂ : off.val+1 < m) (hm : 3 ≤ m)
+    (j : Fin r → Fin u) (hj : StrictMono j) :
+    SafeTaggedTuple (periodSlot (fun _ => off) ∘ j) := by
+  constructor
+  · intro i
+    change 1 ≤ (j i).val*m+off.val ∧ (j i).val*m+off.val+1 < u*m
+    constructor
+    · omega
+    · have h := Nat.mul_le_mul_right m (Nat.succ_le_of_lt (j i).isLt)
+      rw [Nat.succ_mul] at h
+      omega
+  · intro i h hih
+    change (j i).val*m+off.val+3 ≤ (j h).val*m+off.val
+    have h := Nat.mul_le_mul_right m (Nat.succ_le_iff.mpr (hj hih))
+    rw [Nat.succ_mul] at h
+    omega
+
+/-- Removing both incident displacement units still leaves a fixed
+fraction of each actual coarse gap. Doubled coarse gaps are used to keep
+all combinatorial labels integer-valued. -/
+theorem period_derivative_base_lower {u m r : ℕ} (hm : 32 ≤ m) (hr : 0 < r)
+    (off : Fin u → Fin m)
+    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
+    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2)
+    (j : Fin r → Fin u) (hj : StrictMono j)
+    (hsafe : SafeTaggedTuple (periodSlot off ∘ j)) (i : Fin (r+1)) :
+    (m:ℝ)*coarsePeriodGap j i/4 ≤
+      ((tupleGap (periodSlot off ∘ j) i-gapIncidence r i:ℕ):ℝ)+1 := by
+  have he := abs_le.mp (period_gap_rounding hr off hoff hdiff j hj i)
+  have hs := gapIncidence_le (periodSlot off ∘ j) hsafe i
+  have hs₂ : (gapIncidence r i:ℝ) ≤ 2 := by exact_mod_cast gapIncidence_le_two r i
+  have hc : 1 ≤ (coarsePeriodGap j i:ℝ) := by exact_mod_cast coarsePeriodGap_pos j i
+  have hm' : (32:ℝ) ≤ m := by exact_mod_cast hm
+  rw [Nat.cast_sub hs]
+  nlinarith
+
+theorem period_derivative_common_lower {u m r : ℕ} (hm : 32 ≤ m) (hr : 0 < r)
+    (off : Fin u → Fin m)
+    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
+    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2)
+    (j : Fin r → Fin u) (hj : StrictMono j)
+    (hsafe : SafeTaggedTuple (periodSlot off ∘ j)) (i : Fin (r+1)) :
+    (m:ℝ)/4 ≤ ((tupleGap (periodSlot off ∘ j) i-gapIncidence r i:ℕ):ℝ)+1 := by
+  refine le_trans ?_ (period_derivative_base_lower hm hr off hoff hdiff j hj hsafe i)
+  have hc : 1 ≤ (coarsePeriodGap j i:ℝ) := by exact_mod_cast coarsePeriodGap_pos j i
+  nlinarith [show (0:ℝ) ≤ m by positivity]
+
+/-- A largest coarse gap supplies the additional square-root improvement
+coming from the fixed sum of the observed ranks. -/
+theorem period_derivative_largest_lower {u m r : ℕ} (hm : 32 ≤ m) (hr : 0 < r)
+    (off : Fin u → Fin m)
+    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
+    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2)
+    (j : Fin r → Fin u) (hj : StrictMono j)
+    (hsafe : SafeTaggedTuple (periodSlot off ∘ j)) (e : Fin (r+1))
+    (he : 2*u ≤ (r+1)*coarsePeriodGap j e) :
+    (u:ℝ)*m/(2*(r+1:ℝ)) ≤
+      ((tupleGap (periodSlot off ∘ j) e-gapIncidence r e:ℕ):ℝ)+1 := by
+  refine le_trans ?_ (period_derivative_base_lower hm hr off hoff hdiff j hj hsafe e)
+  have he' : 2*(u:ℝ) ≤ (r+1:ℝ)*coarsePeriodGap j e := by exact_mod_cast he
+  apply (div_le_iff₀ (by positivity)).mpr
+  have hh := mul_le_mul_of_nonneg_right he' (show (0:ℝ) ≤ m by positivity)
+  nlinarith only [hh]
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+
+theorem binomial_rank_normalization {n l r : ℕ} {θ : ℝ}
+    (hθ : θ ∈ Set.Ioo 0 1) (hl : l ≤ n)
+    (hmean : (n:ℝ)*θ=l) (hv : 1 ≤ (n:ℝ)*θ*(1-θ)) :
+    (θ^r/binomialMass n θ l)^2 ≤ 144*(n:ℝ) := by
+  have hpow : (θ^r)^2 ≤ 1 := pow_le_one₀ (pow_nonneg hθ.1.le r) (pow_le_one₀ hθ.1.le hθ.2.le) 
+  calc
+    _ = (θ^r)^2*(binomialMass n θ l)⁻¹^2 := by rw [div_eq_mul_inv,mul_pow]
+    _ ≤ (binomialMass n θ l)⁻¹^2 := mul_le_of_le_one_left (sq_nonneg _) hpow
+    _ ≤ 144*((n:ℝ)*θ*(1-θ)) := binomialMass_mean_inverse_sq n l hθ hl hmean hv
+    _ ≤ _ := by
+      have h : θ*(1-θ) ≤ 1 := by nlinarith [sq_nonneg θ]
+      have hh := mul_le_mul_of_nonneg_left h (show 0 ≤ (n:ℝ) by positivity)
+      nlinarith only [hh]
+
+/-- The exact mean-count version, with arbitrary proven positive lower
+scales for all gaps; no asymptotic or analytic estimate is assumed. -/
+theorem positionalRankDifference_scale_bound {n l r : ℕ} (hl : l ≤ n)
+    (p : Fin r → Fin n) (hp : SafeTaggedTuple p) (k : Fin r → GapDifference)
+    {θ G H : ℝ} (hθ : θ ∈ Set.Ioo 0 1) (hG : 0 < G) (hH : 0 < H)
+    (hmean : (n:ℝ)*θ=l) (hv : 1 ≤ (n:ℝ)*θ*(1-θ))
+    (B : Fin (r+1) → ℝ) (hB : ∀ i, 0 < B i)
+    (hgap : ∀ i, G ≤ (tupleGap p i-gapIncidence r i:ℕ)+1)
+    (hscale : ∀ i, B i ≤ (tupleGap p i-gapIncidence r i:ℕ)+1)
+    (e : Fin (r+1)) (hlarge : H ≤ Real.sqrt ((tupleGap p e-gapIncidence r e:ℕ)+1)) :
+    (∑ v : IncreasingTuple l r, positionalRankDifference hl p hp k v ^ 2) ≤
+      144*(n:ℝ)*(2:ℝ)^(2*r)*
+        (lengthDifferenceConstant (∑ h, (k h).order) θ)^(2*(∑ h, (k h).order)+r+2) /
+      (G^(∑ h, (k h).order)*H*∏ i, Real.sqrt (B i)) := by
+  refine (positionalRankDifference_square_sum_le hl p hp k hθ hG hgap e).trans ?_
+  have hC := (lengthDifferenceConstant_pos (∑ h, (k h).order) hθ).le
+  apply div_le_div₀ (by positivity)
+  · exact mul_le_mul_of_nonneg_right
+      (mul_le_mul_of_nonneg_right (binomial_rank_normalization hθ hl hmean hv) (by positivity))
+      (pow_nonneg hC _)
+  · exact mul_pos (mul_pos (pow_pos hG _) hH)
+      (Finset.prod_pos (fun i _ => Real.sqrt_pos.mpr (hB i)))
+  · apply mul_le_mul
+    · exact mul_le_mul_of_nonneg_left hlarge (by positivity)
+    · exact Finset.prod_le_prod₀ (fun i _ => (Real.sqrt_nonneg (B i)))
+        (fun i _ => Real.sqrt_le_sqrt (hscale i))
+    · exact Finset.prod_nonneg (fun _ _ => Real.sqrt_nonneg _)
+    · positivity
+
+/-- Explicit source rank-difference prefactor. It depends on the period
+count, slot length, derivative order and Bernoulli parameter, not the
+selected period tuple. -/
+def periodDifferencePrefactor (u m r d : ℕ) (θ : ℝ) : ℝ :=
+  144*((u*m:ℕ):ℝ)*(2:ℝ)^(2*r)*(lengthDifferenceConstant d θ)^(2*d+r+2) /
+    (((m:ℝ)/4)^d*Real.sqrt ((u:ℝ)*m/(2*(r+1:ℝ)))*
+      (Real.sqrt ((m:ℝ)/4))^(r+1))
+
+/-- The finite-difference estimate for actual translated tagged slots in
+periodic words. The tuple dependence is exactly the product of inverse
+square roots of its doubled coarse gaps. -/
+theorem period_positionalRankDifference_bound {u m l r : ℕ}
+    (hm : 32 ≤ m) (hr : 0 < r) (hl : l ≤ u*m)
+    (off : Fin u → Fin m)
+    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
+    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2)
+    (j : IncreasingTuple u r) (hsafe : SafeTaggedTuple (periodSlot off ∘ j.val))
+    (k : Fin r → GapDifference) {θ : ℝ} (hθ : θ ∈ Set.Ioo 0 1)
+    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ)) :
+    (∑ v : IncreasingTuple l r,
+      positionalRankDifference hl (periodSlot off ∘ j.val) hsafe k v ^ 2) ≤
+      periodDifferencePrefactor u m r (∑ h, (k h).order) θ *
+        ∏ i, (Real.sqrt (coarsePeriodGap j.val i))⁻¹ := by
+  have hu : 0 < u := lt_of_le_of_lt (Nat.zero_le (j.val ⟨0,hr⟩).val) (j.val ⟨0,hr⟩).isLt
+  have hm0 : (0:ℝ) < m := by exact_mod_cast (show 0 < m by omega)
+  have hu0 : (0:ℝ) < u := by exact_mod_cast hu
+  obtain ⟨e,he⟩ := exists_large_period_gap hr j
+  have hb (i : Fin (r+1)) := period_derivative_base_lower hm hr off hoff hdiff j.val j.property hsafe i
+  have hg (i : Fin (r+1)) := period_derivative_common_lower hm hr off hoff hdiff j.val j.property hsafe i
+  have hc (i : Fin (r+1)) : (0:ℝ) < coarsePeriodGap j.val i := by exact_mod_cast coarsePeriodGap_pos j.val i
+  have h := positionalRankDifference_scale_bound hl (periodSlot off ∘ j.val) hsafe k hθ
+    (G := (m:ℝ)/4) (H := Real.sqrt ((u:ℝ)*m/(2*(r+1:ℝ)))) (by positivity) (by positivity)
+    hmean hv (fun i => (m:ℝ)*coarsePeriodGap j.val i/4) (fun i => by positivity [hc i]) hg hb e
+    (Real.sqrt_le_sqrt (period_derivative_largest_lower hm hr off hoff hdiff j.val j.property hsafe e he))
+  refine h.trans_eq ?_
+  have hs (i : Fin (r+1)) : Real.sqrt ((m:ℝ)*coarsePeriodGap j.val i/4) =
+      Real.sqrt ((m:ℝ)/4)*Real.sqrt (coarsePeriodGap j.val i) := by
+    rw [show (m:ℝ)*coarsePeriodGap j.val i/4=((m:ℝ)/4)*coarsePeriodGap j.val i by ring,
+      Real.sqrt_mul (by positivity)]
+  simp_rw [hs]
+  rw [Finset.prod_mul_distrib]
+  simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fin]
+  rw [periodDifferencePrefactor,Finset.prod_inv_distrib]
+  ring
+
+end TraceReconstruction
+
+open Filter
+open scoped Topology
+
+noncomputable section
+namespace TraceReconstruction
+
+/-- Two independent exact sum constraints determine a missing coordinate on
+ each side of a fixed cut. This is the second-gap elimination needed when
+ a period label has been fixed. -/
+theorem two_sums_determine {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (A : Finset ι) {e d : ι} (he : e∈A) (hd : d∉A)
+    (x y : ι → ℕ) (hA : ∑ i∈A, x i = ∑ i∈A, y i)
+    (ht : ∑ i, x i = ∑ i, y i)
+    (hxy : ∀ i, i≠e → i≠d → x i=y i) : x=y := by
+  have hs : (∑ i∈A.erase e, x i) = ∑ i∈A.erase e, y i := by
+    apply Finset.sum_congr rfl
+    intro i hi
+    have hi' := Finset.mem_erase.mp hi
+    apply hxy i hi'.1
+    intro hid
+    exact hd (hid ▸ hi'.2)
+  have hxe : x e=y e := by
+    have hx := Finset.sum_erase_add A x he
+    have hy := Finset.sum_erase_add A y he
+    omega
+  have hs' : (∑ i∈Finset.univ.erase d, x i) = ∑ i∈Finset.univ.erase d, y i := by
+    apply Finset.sum_congr rfl
+    intro i hi
+    by_cases hie : i=e
+    · simpa only [hie] using hxe
+    · exact hxy i hie (Finset.mem_erase.mp hi).1
+  have hxd : x d=y d := by
+    have hx := Finset.sum_erase_add Finset.univ x (Finset.mem_univ d)
+    have hy := Finset.sum_erase_add Finset.univ y (Finset.mem_univ d)
+    omega
+  funext i
+  by_cases hie : i=e
+  · simpa only [hie] using hxe
+  by_cases hid : i=d
+  · simpa only [hid] using hxd
+  exact hxy i hie hid
+
+abbrev RemainingCoordinates {ι : Type*} (e d : ι) := {i : ι // i≠e ∧ i≠d}
+
+/-- Restricting an injective gap profile to all but the two eliminated
+ coordinates is still injective on two fixed-sum fibers. -/
+theorem two_sums_restrict_injective {α ι : Type*} [Fintype ι] [DecidableEq ι]
+    {U L V : ℕ} (A : Finset ι) {e d : ι} (he : e∈A) (hd : d∉A)
+    (p : α → ι → Fin (U+1)) (hp : Function.Injective p)
+    (hA : ∀ a, ∑ i∈A, (p a i).val=L) (ht : ∀ a, ∑ i, (p a i).val=V) :
+    Function.Injective (fun a => fun i : RemainingCoordinates e d => p a i.val) := by
+  intro a b hab
+  apply hp
+  have hv : (fun i => (p a i).val) = (fun i => (p b i).val) := by
+    apply two_sums_determine A he hd
+    · rw [hA,hA]
+    · rw [ht,ht]
+    · intro i hie hid
+      exact congrArg Fin.val (congrFun hab ⟨i,hie,hid⟩)
+  funext i
+  exact Fin.ext (congrFun hv i)
+
+/-- Once the two coordinates are eliminated all remaining gap weights sum
+ independently; any additional constraints may be dropped. -/
+theorem two_sums_product_bound {α ι : Type*} [Fintype α] [Fintype ι] [DecidableEq ι]
+    {U L V : ℕ} (A : Finset ι) {e d : ι} (he : e∈A) (hd : d∉A)
+    (p : α → ι → Fin (U+1)) (hp : Function.Injective p)
+    (hA : ∀ a, ∑ i∈A, (p a i).val=L) (ht : ∀ a, ∑ i, (p a i).val=V)
+    (w : ι → Fin (U+1) → ℝ) (hw : ∀ i k, 0≤w i k) :
+    (∑ a, ∏ i : RemainingCoordinates e d, w i.val (p a i.val)) ≤
+      ∏ i : RemainingCoordinates e d, ∑ k, w i.val k := by
+  classical
+  have hh := sum_injective_profile_le
+    (fun a => fun i : RemainingCoordinates e d => p a i.val)
+    (two_sums_restrict_injective A he hd p hp hA ht)
+    (fun _ => True) (fun _ => trivial)
+    (fun δ => ∏ i : RemainingCoordinates e d, w i.val (δ i))
+    (fun δ => Finset.prod_nonneg (fun i _ => hw i.val (δ i)))
+  simp only [ite_true] at hh
+  exact hh.trans_eq (by rw [Finset.prod_univ_sum]; simp only [Fintype.piFinset_univ])
+
+/-- The sum of doubled coarse gaps to a specified period label is exact,
+ including the half endpoint cells in the source convention. -/
+theorem coarsePeriodGap_prefix {u r : ℕ} (j : Fin r → Fin u) (hj : StrictMono j)
+    (i : Fin r) : (∑ g∈Finset.Iic i.castSucc, coarsePeriodGap j g) = 2*(j i).val+1 := by
+  classical
+  have hg := tupleGap_prefix j hj i
+  have hz : (0 : Fin (r+1)) ∈ Finset.Iic i.castSucc := by simp
+  have hl : ∀ g∈Finset.Iic i.castSucc, g≠Fin.last r := by
+    intro g hg he
+    have hle := Finset.mem_Iic.mp hg
+    have hval : g.val ≤ i.val := hle
+    have he' := congrArg Fin.val he
+    simp only [Fin.val_last] at he'
+    omega
+  have hc : (∑ g∈Finset.Iic i.castSucc,
+      if g=0 ∨ g=Fin.last r then 1 else 2) = 2*i.val+1 := by
+    have hsub : (∑ g∈(Finset.Iic i.castSucc).erase 0,
+        if g=0 ∨ g=Fin.last r then 1 else 2) = 2*i.val := by
+      have hc : ((Finset.Iic i.castSucc).erase 0).card = i.val := by
+        rw [Finset.card_erase_of_mem hz,Fin.card_Iic]
+        simp
+      calc
+        _ = ∑ _g∈(Finset.Iic i.castSucc).erase 0, 2 := by
+          apply Finset.sum_congr rfl
+          intro g hg
+          rw [ite_eq_right (not_or.mpr ⟨(Finset.mem_erase.mp hg).1,
+            hl g (Finset.mem_erase.mp hg).2⟩)]
+        _ = _ := by simp [hc,Nat.mul_comm]
+    have hsum := Finset.sum_erase_add (Finset.Iic i.castSucc)
+      (fun g => if g=0 ∨ g=Fin.last r then 1 else 2) hz
+    rw [hsub] at hsum
+    simpa using hsum.symm
+  simp only [coarsePeriodGap,Finset.sum_add_distrib,← Finset.mul_sum]
+  omega
+
+/-- The full inverse-square-root sum, including the zero cell (whose
+ value is zero), has the same elementary integral bound. -/
+theorem fin_sum_inverse_sqrt (U : ℕ) :
+    (∑ k : Fin (U+1), (Real.sqrt (k.val:ℝ))⁻¹) ≤ 2*Real.sqrt U := by
+  rw [Fin.sum_univ_succ]
+  simp only [Fin.val_zero, Nat.cast_zero, Real.sqrt_zero, inv_zero,zero_add,Fin.val_succ]
+  have h := sum_inverse_sqrt U
+  rw [Finset.sum_range] at h
+  simpa only [Nat.cast_add,Nat.cast_one] using h
+
+theorem prod_two_coordinates {ι : Type*} [Fintype ι] [DecidableEq ι]
+    {e d : ι} (hed : e≠d) (f : ι → ℝ) :
+    (∏ i, f i) = f e*f d*∏ i : RemainingCoordinates e d, f i.val := by
+  have hd : d∈Finset.univ.erase e := by simp [hed.symm]
+  have h1 := Finset.prod_erase_mul Finset.univ f (Finset.mem_univ e)
+  have h2 := Finset.prod_erase_mul (Finset.univ.erase e) f hd
+  have hrem : (∏ i∈(Finset.univ.erase e).erase d, f i) =
+      ∏ i : RemainingCoordinates e d, f i.val :=
+    Finset.prod_subtype _ (by intro i; simp [and_comm]) f
+  rw [← h1,← h2,hrem]
+  ring
+
+theorem inverse_sqrt_pair {x y R : ℝ} (hR : 0<R) (hx : R≤x) (hy : R≤y) :
+    (Real.sqrt x)⁻¹*(Real.sqrt y)⁻¹ ≤ R⁻¹ := by
+  have hs : 0<Real.sqrt R := Real.sqrt_pos.mpr hR
+  have hix := (inv_le_inv₀ (hs.trans_le (Real.sqrt_le_sqrt hx)) hs).mpr (Real.sqrt_le_sqrt hx)
+  have hiy := (inv_le_inv₀ (hs.trans_le (Real.sqrt_le_sqrt hy)) hs).mpr (Real.sqrt_le_sqrt hy)
+  calc
+    _ ≤ (Real.sqrt R)⁻¹*(Real.sqrt R)⁻¹ := mul_le_mul hix hiy (by positivity) (by positivity)
+    _ = _ := by rw [← mul_inv,Real.mul_self_sqrt hR.le]
+
+/-- Explicit two-gap summation on a fixed-sum fiber. Both large gaps
+ contribute R^(-1), rather than losing a free label factor. -/
+theorem two_large_gaps_inverse_sum {α ι : Type*} [Fintype α] [Fintype ι] [DecidableEq ι]
+    {U L V : ℕ} (A : Finset ι) {e d : ι} (he : e∈A) (hd : d∉A)
+    (p : α → ι → Fin (U+1)) (hp : Function.Injective p)
+    (hA : ∀ a, ∑ i∈A, (p a i).val=L) (ht : ∀ a, ∑ i, (p a i).val=V)
+    {R : ℝ} (hR : 0<R) (heR : ∀ a, R≤(p a e).val) (hdR : ∀ a, R≤(p a d).val) :
+    (∑ a, ∏ i, (Real.sqrt ((p a i).val:ℝ))⁻¹) ≤
+      R⁻¹*(2*Real.sqrt U)^Fintype.card (RemainingCoordinates e d) := by
+  classical
+  have hed : e≠d := by intro h; exact hd (h ▸ he)
+  have hpoint (a) := inverse_sqrt_pair hR (heR a) (hdR a)
+  calc
+    _ ≤ ∑ a, R⁻¹*∏ i : RemainingCoordinates e d, (Real.sqrt ((p a i.val).val:ℝ))⁻¹ := by
+      apply Finset.sum_le_sum
+      intro a _
+      rw [prod_two_coordinates hed]
+      exact mul_le_mul_of_nonneg_right (hpoint a) (by positivity)
+    _ = R⁻¹*∑ a, ∏ i : RemainingCoordinates e d, (Real.sqrt ((p a i.val).val:ℝ))⁻¹ := by
+      rw [Finset.mul_sum]
+    _ ≤ R⁻¹*∏ _i : RemainingCoordinates e d, ∑ k : Fin (U+1), (Real.sqrt (k.val:ℝ))⁻¹ := by
+      apply mul_le_mul_of_nonneg_left _ (inv_nonneg.mpr hR.le)
+      exact two_sums_product_bound A he hd p hp hA ht
+        (fun _ k => (Real.sqrt (k.val:ℝ))⁻¹) (fun _ _ => by positivity)
+    _ ≤ R⁻¹*∏ _i : RemainingCoordinates e d, (2*Real.sqrt U) := by
+      apply mul_le_mul_of_nonneg_left _ (inv_nonneg.mpr hR.le)
+      exact Finset.prod_le_prod₀ (fun _ _ => by positivity) (fun _ _ => fin_sum_inverse_sqrt U)
+    _ = _ := by simp
+
+/-- The fixed involved label is a genuine geometric fiber, not an extra
+ analytic hypothesis on the rank kernel. -/
+def FixedPeriodTuple (u r : ℕ) (a : Fin r) (h : Fin u) :=
+  {j : IncreasingTuple u r // j.val a=h}
+
+noncomputable instance fixedPeriodTupleFintype (u r : ℕ) (a : Fin r) (h : Fin u) :
+    Fintype (FixedPeriodTuple u r a h) := by
+  unfold FixedPeriodTuple
+  exact Fintype.ofFinite _
+
+def fixedPeriodProfile {u r : ℕ} (a : Fin r) (h : Fin u) (j : FixedPeriodTuple u r a h) :
+    Fin (r+1) → Fin (2*u+1) := fun i =>
+  ⟨coarsePeriodGap j.val.val i, Nat.lt_succ_of_le
+    (coarsePeriodGap_le (Nat.zero_lt_of_lt a.isLt) j.val.val j.val.property i)⟩
+
+theorem fixedPeriodProfile_injective {u r : ℕ} (a : Fin r) (h : Fin u) :
+    Function.Injective (fixedPeriodProfile a h) := by
+  intro j k hjk
+  apply Subtype.ext
+  apply Subtype.ext
+  apply coarsePeriodGap_injective j.val.property k.val.property
+  funext i
+  exact congrArg Fin.val (congrFun hjk i)
+
+theorem fixedPeriodProfile_prefix {u r : ℕ} (a : Fin r) (h : Fin u)
+    (j : FixedPeriodTuple u r a h) :
+    (∑ i∈Finset.Iic a.castSucc, (fixedPeriodProfile a h j i).val)=2*h.val+1 := by
+  change (∑ i∈Finset.Iic a.castSucc, coarsePeriodGap j.val.val i)=_
+  rw [coarsePeriodGap_prefix _ j.val.property,j.property]
+
+theorem fixedPeriodProfile_total {u r : ℕ} (a : Fin r) (h : Fin u)
+    (j : FixedPeriodTuple u r a h) :
+    (∑ i, (fixedPeriodProfile a h j i).val)=2*u :=
+  coarsePeriodGap_sum (Nat.zero_lt_of_lt a.isLt) j.val.val j.val.property
+
+end TraceReconstruction
+noncomputable section
+namespace TraceReconstruction
+
+/-- An elementary centered MGF bound valid on both sides of the mean,
+ with constants independent of the Bernoulli parameter. -/
+theorem binomialMass_centered_mgf (n : ℕ) {p t : ℝ}
+    (hp : p∈Set.Icc 0 1) (ht : |t|≤1) :
+    (∑ k∈Finset.range (n+1), binomialMass n p k*Real.exp (t*((k:ℝ)-n*p))) ≤
+      Real.exp ((n:ℝ)*t^2) := by
+  have he := (abs_le.mp (Real.abs_exp_sub_one_sub_id_le ht)).2
+  have hn : 0≤(n:ℝ)*p := mul_nonneg (Nat.cast_nonneg _) hp.1
+  have hn' : (n:ℝ)*p≤n := mul_le_of_le_one_right (Nat.cast_nonneg _) hp.2
+  calc
+    _ = Real.exp (-t*(n:ℝ)*p)*
+        ∑ k∈Finset.range (n+1), binomialMass n p k*Real.exp (t*k) := by
+      rw [Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro k _
+      rw [mul_left_comm,← Real.exp_add]
+      congr 2
+      ring
+    _ ≤ Real.exp (-t*(n:ℝ)*p)*Real.exp ((n:ℝ)*p*(Real.exp t-1)) :=
+      mul_le_mul_of_nonneg_left (binomialMass_mgf_le n hp t) (Real.exp_nonneg _)
+    _ ≤ Real.exp ((n:ℝ)*t^2) := by
+      rw [← Real.exp_add]
+      apply Real.exp_le_exp.mpr
+      have he' := mul_le_mul_of_nonneg_left he hn
+      have hs := mul_le_mul_of_nonneg_right hn' (sq_nonneg t)
+      nlinarith only [he',hs]
+
+/-- Either signed deviation has the same Chernoff estimate. -/
+theorem binomialMass_signed_tail (n : ℕ) {p t a ε : ℝ}
+    (hp : p∈Set.Icc 0 1) (ht : t∈Set.Icc 0 1) (hε : ε=1 ∨ ε= -1) :
+    (∑ k∈(Finset.range (n+1)).filter (fun k : ℕ => a≤ε*((k:ℝ)-n*p)), binomialMass n p k) ≤
+      Real.exp (-t*a+(n:ℝ)*t^2) := by
+  have heps : |t*ε|≤1 := by rcases hε with rfl | rfl <;> simpa [abs_of_nonneg ht.1] using ht.2
+  have hm := finite_exp_markov (Finset.range (n+1)) (binomialMass n p)
+    (fun k : ℕ => ε*((k:ℝ)-n*p)) (fun k _ => binomialMass_nonneg n k hp) a t ht.1
+  calc
+    _ ≤ Real.exp (-t*a)*∑ k∈Finset.range (n+1),
+        binomialMass n p k*Real.exp ((t*ε)*((k:ℝ)-n*p)) := by
+      simpa only [mul_assoc] using hm
+    _ ≤ Real.exp (-t*a)*Real.exp ((n:ℝ)*(t*ε)^2) :=
+      mul_le_mul_of_nonneg_left (binomialMass_centered_mgf n hp heps) (Real.exp_nonneg _)
+    _ = _ := by rcases hε with rfl | rfl <;> simp [Real.exp_add]
+
+/-- A uniform two-sided tail at any finite cutoff. Unlike a quoted
+ concentration theorem, this follows here from the finite binomial MGF. -/
+theorem binomialMass_abs_tail (n : ℕ) {p t a : ℝ}
+    (hp : p∈Set.Icc 0 1) (ht : t∈Set.Icc 0 1) :
+    (∑ k∈(Finset.range (n+1)).filter (fun k : ℕ => a≤|(k:ℝ)-n*p|), binomialMass n p k) ≤
+      2*Real.exp (-t*a+(n:ℝ)*t^2) := by
+  have hpoint (k : ℕ) : (if a≤|(k:ℝ)-n*p| then binomialMass n p k else 0) ≤
+      (if a≤(k:ℝ)-n*p then binomialMass n p k else 0)+
+      (if a≤-((k:ℝ)-n*p) then binomialMass n p k else 0) := by
+    have hn := binomialMass_nonneg n k hp
+    by_cases h : a≤|(k:ℝ)-n*p|
+    · rw [ite_eq_left h]
+      have hh := le_abs.mp h
+      rcases hh with h1 | h2
+      · rw [ite_eq_left h1]
+        split_ifs <;> linarith
+      · rw [ite_eq_left h2]
+        split_ifs <;> linarith
+    · rw [ite_eq_right h]
+      split_ifs <;> linarith
+  have hh := Finset.sum_le_sum (s := Finset.range (n+1)) (fun k _ => hpoint k)
+  rw [Finset.sum_add_distrib,← Finset.sum_filter,← Finset.sum_filter,← Finset.sum_filter] at hh
+  have h1 := binomialMass_signed_tail n (a:=a) hp ht (Or.inl rfl : (1:ℝ)=1 ∨ (1:ℝ)= -1)
+  have h2 := binomialMass_signed_tail n (a:=a) hp ht (Or.inr rfl : (-1:ℝ)=1 ∨ (-1:ℝ)= -1)
+  simp only [one_mul,neg_one_mul] at h1 h2
+  linarith
+
+end TraceReconstruction
+noncomputable section
+namespace TraceReconstruction
+
+/-- Cardinality bookkeeping for the two eliminated gap coordinates. -/
+theorem remainingCoordinates_card {ι : Type*} [Fintype ι] [DecidableEq ι]
+    {e d : ι} (hed : e≠d) :
+    Fintype.card (RemainingCoordinates e d)=Fintype.card ι-2 := by
+  classical
+  have hset : (Finset.univ.filter (fun i : ι => i≠e ∧ i≠d)) = (Finset.univ.erase e).erase d := by
+    ext i
+    simp [and_comm]
+  rw [Fintype.card_subtype, hset]
+  rw [Finset.card_erase_of_mem (by simp [hed.symm]),
+    Finset.card_erase_of_mem (Finset.mem_univ e)]
+  simp only [Finset.card_univ]
+  omega
+
+/-- A finite sum's largest entry on a nonempty set controls its average,
+ with the full number of coordinates in place of the subset cardinality. -/
+theorem exists_large_in_subset {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (A : Finset ι) (hA : A.Nonempty) (x : ι → ℝ) (hx : ∀ i, 0≤x i)
+    {R : ℝ} (hR : R*Fintype.card ι ≤ ∑ i∈A, x i) :
+    ∃ e∈A, R≤x e := by
+  obtain ⟨e,he,hm⟩ := A.exists_max_image x hA
+  refine ⟨e,he,?_⟩
+  have hc : (A.card:ℝ)≤Fintype.card ι := by exact_mod_cast Finset.card_le_univ A
+  have hcard : (0:ℝ)<Fintype.card ι := by
+    exact_mod_cast Fintype.card_pos_iff.mpr ⟨e⟩
+  have hs : (∑ i∈A, x i) ≤ (Fintype.card ι:ℝ)*x e := by
+    calc
+      _ ≤ ∑ _i∈A, x e := Finset.sum_le_sum (fun i hi => hm i hi)
+      _ = (A.card:ℝ)*x e := by simp
+      _ ≤ _ := mul_le_mul_of_nonneg_right hc (hx e)
+  nlinarith
+
+/-- Covering the fixed-sum profile by its selected largest coordinates
+ costs only the square of its (fixed) dimension. -/
+theorem two_sided_large_gaps_sum {α ι : Type*} [Fintype α] [Fintype ι] [DecidableEq ι]
+    {U L V : ℕ} (A : Finset ι)
+    (p : α → ι → Fin (U+1)) (hp : Function.Injective p)
+    (hA : ∀ a, ∑ i∈A, (p a i).val=L) (ht : ∀ a, ∑ i, (p a i).val=V)
+    {R : ℝ} (hR : 0<R)
+    (hleft : ∀ a, ∃ e∈A, R≤(p a e).val)
+    (hright : ∀ a, ∃ d∉A, R≤(p a d).val) :
+    (∑ a, ∏ i, (Real.sqrt ((p a i).val:ℝ))⁻¹) ≤
+      (Fintype.card ι:ℝ)^2*R⁻¹*(2*Real.sqrt U)^(Fintype.card ι-2) := by
+  classical
+  choose e he heR using hleft
+  choose d hd hdR using hright
+  let f (a : α) := (e a,d a)
+  let W (a : α) := ∏ i, (Real.sqrt ((p a i).val:ℝ))⁻¹
+  have hfiber (z : ι × ι) :
+      (∑ a : {a // f a=z}, W a.val) ≤ R⁻¹*(2*Real.sqrt U)^(Fintype.card ι-2) := by
+    by_cases hne : Nonempty {a // f a=z}
+    · obtain ⟨a⟩ := hne
+      have ha := congrArg Prod.fst a.property
+      have ha' := congrArg Prod.snd a.property
+      have hz1 : z.1∈A := by simpa only [f] using ha ▸ he a.val
+      have hz2 : z.2∉A := by simpa only [f] using ha' ▸ hd a.val
+      have hzd : z.1≠z.2 := by intro h; exact hz2 (h ▸ hz1)
+      have hh := two_large_gaps_inverse_sum A hz1 hz2
+        (fun a : {a // f a=z} => p a.val) (hp.comp Subtype.val_injective)
+        (fun a => hA a.val) (fun a => ht a.val) hR
+        (fun a => by have h := congrArg Prod.fst a.property; simpa only [f] using h ▸ heR a.val)
+        (fun a => by have h := congrArg Prod.snd a.property; simpa only [f] using h ▸ hdR a.val)
+      simpa only [W,remainingCoordinates_card hzd] using hh
+    · have : IsEmpty {a // f a=z} := not_nonempty_iff.mp hne
+      simp only [Fintype.sum_empty]
+      positivity
+  calc
+    _ = ∑ z : ι × ι, ∑ a : {a // f a=z}, W a.val := by
+      exact (Fintype.sum_fiberwise f W).symm
+    _ ≤ ∑ _z : ι × ι, R⁻¹*(2*Real.sqrt U)^(Fintype.card ι-2) :=
+      Finset.sum_le_sum (fun z _ => hfiber z)
+    _ = _ := by simp [pow_two,mul_assoc]
+
+/-- The fixed central label leaves a macroscopic sum on both sides. -/
+theorem fixedPeriodProfile_central_sum {u r : ℕ} (a : Fin r) (h : Fin u)
+    (hleft : (u:ℝ)/3≤2*h.val+1)
+    (hright : (u:ℝ)/3≤2*u-(2*h.val+1)) :
+    (∑ j : FixedPeriodTuple u r a h,
+      ∏ i, (Real.sqrt (coarsePeriodGap j.val.val i))⁻¹) ≤
+      ((r+1:ℕ):ℝ)^2 * ((u:ℝ)/(3*(r+1:ℝ)))⁻¹ *
+        (2*Real.sqrt (2*u:ℕ))^(r-1) := by
+  classical
+  have hu : (0:ℝ)<u := by exact_mod_cast (Nat.zero_lt_of_lt h.isLt)
+  let A := Finset.Iic a.castSucc
+  let p := fixedPeriodProfile a h
+  have hR : 0<(u:ℝ)/(3*(r+1:ℝ)) := by positivity
+  have hL (j : FixedPeriodTuple u r a h) :
+      ∃ e∈A, (u:ℝ)/(3*(r+1:ℝ))≤(p j e).val := by
+    apply exists_large_in_subset A ⟨0, by simp [A]⟩ (fun i => (p j i).val)
+      (fun _ => Nat.cast_nonneg _)
+    have hs := fixedPeriodProfile_prefix a h j
+    have hs' : (∑ i∈A, ((p j i).val:ℝ))=2*h.val+1 := by exact_mod_cast hs
+    rw [hs']
+    simp only [Fintype.card_fin, Nat.cast_add, Nat.cast_one]
+    convert hleft using 1; field_simp
+  have hD (j : FixedPeriodTuple u r a h) :
+      ∃ d∉A, (u:ℝ)/(3*(r+1:ℝ))≤(p j d).val := by
+    have ha : a.val<r := a.isLt
+    have hlast : Fin.last r∈Aᶜ := by simp [A]
+    obtain ⟨d,hd,hdR⟩ := exists_large_in_subset Aᶜ ⟨Fin.last r,hlast⟩
+      (fun i => ((p j i).val:ℝ)) (fun _ => Nat.cast_nonneg _) (R := (u:ℝ)/(3*(r+1:ℝ))) (by
+        have hs := fixedPeriodProfile_prefix a h j
+        have ht := fixedPeriodProfile_total a h j
+        have hs' : (∑ i∈A, ((p j i).val:ℝ))=2*h.val+1 := by exact_mod_cast hs
+        have ht' : (∑ i, ((p j i).val:ℝ))=2*u := by exact_mod_cast ht
+        have hc := Finset.sum_compl_add_sum A (fun i => ((p j i).val:ℝ))
+        have hsum : (∑ i∈Aᶜ, ((p j i).val:ℝ))=2*u-(2*h.val+1) := by linarith
+        rw [hsum]
+        simp only [Fintype.card_fin, Nat.cast_add, Nat.cast_one]
+        convert hright using 1; field_simp)
+    exact ⟨d,Finset.mem_compl.mp hd,hdR⟩
+  have hh := two_sided_large_gaps_sum A p (fixedPeriodProfile_injective a h)
+    (fixedPeriodProfile_prefix a h) (fixedPeriodProfile_total a h) hR hL hD
+  simpa only [p,fixedPeriodProfile,Fin.val_mk,Fintype.card_fin,show r+1-2=r-1 by omega] using hh
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+
+/-- Any finite cutoff only removes nonnegative mass from the full binomial
+ support, even if the cutoff extends beyond that support. -/
+theorem binomialMass_cutoff_filter_le (n U : ℕ) {p : ℝ} (hp : p∈Set.Icc 0 1)
+    (P : ℕ → Prop) [DecidablePred P] :
+    (∑ k : Fin (U+1), if P k.val then binomialMass n p k.val else 0) ≤
+      ∑ k∈(Finset.range (n+1)).filter P, binomialMass n p k := by
+  rw [Fin.sum_univ_eq_sum_range (fun k : ℕ => if P k then binomialMass n p k else 0) (U+1),Finset.sum_filter]
+  by_cases hU : U≤n
+  · apply Finset.sum_le_sum_of_subset_of_nonneg (Finset.range_mono (by omega))
+    intro k _ _
+    split_ifs
+    · exact binomialMass_nonneg n k hp
+    · rfl
+  · apply le_of_eq
+    symm
+    apply Finset.sum_subset (Finset.range_mono (by omega))
+    intro k _ hk
+    have hnk : n<k := by simp only [Finset.mem_range] at hk; omega
+    simp [binomialMass,Nat.choose_eq_zero_of_lt hnk]
+
+/-- A single-coordinate event for any injective family of profiles is
+ bounded by the corresponding full product marginal. -/
+theorem profile_coordinate_event_le {α ι κ : Type*} [Fintype α] [Fintype ι] [Fintype κ]
+    [DecidableEq ι] (p : α → ι → κ) (hp : Function.Injective p)
+    (w : ι → κ → ℝ) (hw : ∀ i k, 0≤w i k)
+    (hs : ∀ i, ∑ k, w i k≤1) (e : ι) (P : κ → Prop) [DecidablePred P]
+    {T : ℝ} (_hT : 0≤T) (ht : (∑ k, if P k then w e k else 0)≤T) :
+    (∑ a, if P (p a e) then ∏ i, w i (p a i) else 0) ≤ T := by
+  classical
+  let w' (i : ι) (k : κ) := if i=e then (if P k then w i k else 0) else w i k
+  have hw' (i k) : 0≤w' i k := by
+    dsimp [w']
+    split_ifs
+    · exact hw i k
+    · exact le_rfl
+    · exact hw i k
+  have hprod (δ : ι → κ) : (∏ i, w' i (δ i)) =
+      if P (δ e) then ∏ i, w i (δ i) else 0 := by
+    by_cases h : P (δ e)
+    · rw [ite_eq_left h]
+      apply Finset.prod_congr rfl
+      intro i _
+      by_cases hi : i=e
+      · subst i; simp [w',h]
+      · simp [w',hi]
+    · rw [ite_eq_right h]
+      exact Finset.prod_eq_zero (Finset.mem_univ e) (by simp [w',h])
+  have hsum (i) : (∑ k, w' i k)≤ if i=e then T else 1 := by
+    by_cases hi : i=e
+    · subst i; simpa only [w',ite_true] using ht
+    · simpa only [w',ite_eq_right hi] using hs i
+  calc
+    _ = ∑ a, ∏ i, w' i (p a i) := by simp_rw [hprod]
+    _ ≤ ∑ δ : ι → κ, ∏ i, w' i (δ i) := by
+      have hh := sum_injective_profile_le p hp (fun _ => True) (fun _ => trivial)
+        (fun δ => ∏ i, w' i (δ i)) (fun δ => Finset.prod_nonneg (fun i _ => hw' i (δ i)))
+      simpa only [ite_true] using hh
+    _ = ∏ i, ∑ k, w' i k := (Fintype.prod_sum _).symm
+    _ ≤ ∏ i : ι, if i=e then T else 1 :=
+      Finset.prod_le_prod₀ (fun i _ => Finset.sum_nonneg (fun k _ => hw' i k)) (fun i _ => hsum i)
+    _ = T := by simp
+
+/-- Exponentially small clipping loss for a single gap of the actual
+ conditional retained-subset rank kernel. -/
+theorem subsetRankKernel_gap_tail {n l r : ℕ} (hl : l≤n)
+    (p : Fin r → Fin n) (hp : StrictMono p) (e : Fin (r+1))
+    {θ t a : ℝ} (hθ : θ∈Set.Ioo 0 1) (ht : t∈Set.Icc 0 1) :
+    (∑ v : IncreasingTuple l r,
+      if a≤|(tupleGap v.val e:ℝ)-(tupleGap p e:ℝ)*θ|
+      then subsetRankKernel hl p v.val else 0) ≤
+      θ^r / binomialMass n θ l *
+        (2*Real.exp (-t*a+(tupleGap p e:ℝ)*t^2)) := by
+  classical
+  have hθ' : θ∈Set.Icc 0 1 := ⟨hθ.1.le,hθ.2.le⟩
+  let prof := rankGapProfile (n := l) (r := r) (Equiv.refl _)
+  let w (i : Fin (r+1)) (k : Fin (l-r+1)) := binomialMass (tupleGap p i) θ k.val
+  have hw (i k) : 0≤w i k := binomialMass_nonneg _ _ hθ'
+  have hs (i) : (∑ k, w i k)≤1 := by
+    have hh := binomialMass_cutoff_filter_le (tupleGap p i) (l-r) hθ' (fun _ => True)
+    simpa only [ite_true,Finset.filter_true,sum_binomialMass] using hh
+  have ht' : (∑ k : Fin (l-r+1),
+      if a≤|(k.val:ℝ)-(tupleGap p e:ℝ)*θ| then w e k else 0) ≤
+      2*Real.exp (-t*a+(tupleGap p e:ℝ)*t^2) := by
+    exact (binomialMass_cutoff_filter_le (tupleGap p e) (l-r) hθ'
+      (fun k : ℕ => a≤|(k:ℝ)-(tupleGap p e:ℝ)*θ|)).trans
+        (binomialMass_abs_tail _ hθ' ht)
+  have hh := profile_coordinate_event_le prof (rankGapProfile_injective _) w hw hs e
+    (fun k => a≤|(k.val:ℝ)-(tupleGap p e:ℝ)*θ|) (by positivity) ht'
+  have hfac : 0≤θ^r / binomialMass n θ l :=
+    div_nonneg (pow_nonneg hθ.1.le _) (binomialMass_nonneg _ _ hθ')
+  calc
+    _ = θ^r / binomialMass n θ l *
+        ∑ v : IncreasingTuple l r,
+          if a≤|(tupleGap v.val e:ℝ)-(tupleGap p e:ℝ)*θ|
+          then ∏ i, binomialMass (tupleGap p i) θ (tupleGap v.val i) else 0 := by
+      rw [Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro v _
+      split_ifs
+      · exact subsetRankKernel_binomial_formula hl p hp v.val v.property θ hθ
+      · simp
+    _ ≤ _ := mul_le_mul_of_nonneg_left (by
+      simpa only [prof,rankGapProfile,Equiv.refl_apply,w] using hh) hfac
+
+/-- A coordinatewise rank region has a union-bound clipping estimate. -/
+theorem subsetRankKernel_outside_region {n l r : ℕ} (hl : l≤n)
+    (p : Fin r → Fin n) (hp : StrictMono p)
+    {θ t a : ℝ} (hθ : θ∈Set.Ioo 0 1) (ht : t∈Set.Icc 0 1) :
+    (∑ v : IncreasingTuple l r,
+      if ∃ i, a≤|(tupleGap v.val i:ℝ)-(tupleGap p i:ℝ)*θ|
+      then subsetRankKernel hl p v.val else 0) ≤
+      ((r+1:ℕ):ℝ)*θ^r / binomialMass n θ l *
+        (2*Real.exp (-t*a+(n:ℝ)*t^2)) := by
+  classical
+  have hθ' : θ∈Set.Icc 0 1 := ⟨hθ.1.le,hθ.2.le⟩
+  have hpoint (v : IncreasingTuple l r) :
+      (if ∃ i, a≤|(tupleGap v.val i:ℝ)-(tupleGap p i:ℝ)*θ|
+      then subsetRankKernel hl p v.val else 0) ≤
+      ∑ i, if a≤|(tupleGap v.val i:ℝ)-(tupleGap p i:ℝ)*θ|
+        then subsetRankKernel hl p v.val else 0 := by
+    have hn (i) : 0≤(if a≤|(tupleGap v.val i:ℝ)-(tupleGap p i:ℝ)*θ|
+        then subsetRankKernel hl p v.val else 0) := by
+      split_ifs
+      · exact subsetRankKernel_nonneg hl p v.val
+      · rfl
+    split_ifs with h
+    · obtain ⟨e,he⟩ := h
+      have hh := Finset.single_le_sum (fun i _ => hn i) (Finset.mem_univ e)
+      simpa only [ite_eq_left he] using hh
+    · exact Finset.sum_nonneg (fun i _ => hn i)
+  calc
+    _ ≤ ∑ v : IncreasingTuple l r, ∑ i,
+        if a≤|(tupleGap v.val i:ℝ)-(tupleGap p i:ℝ)*θ|
+        then subsetRankKernel hl p v.val else 0 := Finset.sum_le_sum (fun v _ => hpoint v)
+    _ = ∑ i, ∑ v : IncreasingTuple l r,
+        if a≤|(tupleGap v.val i:ℝ)-(tupleGap p i:ℝ)*θ|
+        then subsetRankKernel hl p v.val else 0 := Finset.sum_comm
+    _ ≤ ∑ _i : Fin (r+1), θ^r / binomialMass n θ l *
+        (2*Real.exp (-t*a+(n:ℝ)*t^2)) := by
+      apply Finset.sum_le_sum
+      intro i _
+      apply (subsetRankKernel_gap_tail hl p hp i hθ ht).trans
+      apply mul_le_mul_of_nonneg_left _ (div_nonneg (pow_nonneg hθ.1.le _) (binomialMass_nonneg _ _ hθ'))
+      apply mul_le_mul_of_nonneg_left _ (by norm_num)
+      apply Real.exp_le_exp.mpr
+      have hgap : tupleGap p i≤n := (tupleGap_le_remainder ⟨p,hp⟩ i).trans (Nat.sub_le _ _)
+      have hg : (tupleGap p i:ℝ)≤n := by exact_mod_cast hgap
+      have hh := mul_le_mul_of_nonneg_right hg (sq_nonneg t)
+      linarith
+    _ = _ := by simp; ring
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+
+/-- An inverse square root at an integer, extended by zero at zero,
+ never exceeds one. -/
+theorem inverse_sqrt_nat_le_one (k : ℕ) : (Real.sqrt (k:ℝ))⁻¹≤1 := by
+  by_cases hk : k=0
+  · simp [hk]
+  have hk' : (1:ℝ)≤k := by exact_mod_cast (Nat.one_le_iff_ne_zero.mpr hk)
+  have hs : 1≤Real.sqrt (k:ℝ) := by simpa using Real.sqrt_le_sqrt hk'
+  exact (inv_le_one₀ (by positivity : 0<Real.sqrt (k:ℝ))).mpr hs
+
+/-- A bounded gap costs a constant depending only on that bound, rather
+ than an additional square root of the ambient period count. -/
+theorem fin_bounded_inverse_sqrt (U M : ℕ) :
+    (∑ k : Fin (U+1), if k.val≤M then (Real.sqrt (k.val:ℝ))⁻¹ else 0) ≤ M+1 := by
+  classical
+  calc
+    _ ≤ ∑ k : Fin (U+1), if k.val≤M then (1:ℝ) else 0 := by
+      apply Finset.sum_le_sum
+      intro k _
+      split_ifs
+      · exact inverse_sqrt_nat_le_one k.val
+      · rfl
+    _ = ((Finset.univ.filter (fun k : Fin (U+1) => k.val≤M)).card:ℝ) := by simp
+    _ ≤ M+1 := by
+      have hh : (Finset.univ.filter (fun k : Fin (U+1) => k.val≤M)).card≤M+1 := by
+        rw [← Finset.card_range (M+1)]
+        apply Finset.card_le_card_of_injOn Fin.val
+          (fun k hk => by
+            have hh := (Finset.mem_filter.mp hk).2
+            exact Finset.mem_range.mpr (Nat.lt_succ_of_le hh))
+        intro k _ j _ hkj; exact Fin.ext hkj
+      exact_mod_cast hh
+
+theorem prod_one_special {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (e : ι) (A B : ℝ) :
+    (∏ i : ι, if i=e then A else B)=A*B^(Fintype.card ι-1) := by
+  rw [← Finset.mul_prod_erase Finset.univ (fun i => if i=e then A else B) (Finset.mem_univ e)]
+  simp only [ite_true]
+  have hh : (∏ i∈Finset.univ.erase e, if i=e then A else B)=
+      ∏ _i∈Finset.univ.erase e, B := by
+    apply Finset.prod_congr rfl
+    intro i hi
+    exact ite_eq_right (Finset.mem_erase.mp hi).1
+  rw [hh]
+  simp
+
+/-- Refined two-gap elimination when a third gap has bounded size. -/
+theorem two_large_bounded_gap_sum {α ι : Type*} [Fintype α] [Fintype ι] [DecidableEq ι]
+    {U L V M : ℕ} (A : Finset ι) {e d c : ι} (he : e∈A) (hd : d∉A)
+    (hce : c≠e) (hcd : c≠d)
+    (p : α → ι → Fin (U+1)) (hp : Function.Injective p)
+    (hA : ∀ a, ∑ i∈A, (p a i).val=L) (ht : ∀ a, ∑ i, (p a i).val=V)
+    {R : ℝ} (hR : 0<R) (heR : ∀ a, R≤(p a e).val) (hdR : ∀ a, R≤(p a d).val)
+    (hcM : ∀ a, (p a c).val≤M) :
+    (∑ a, ∏ i, (Real.sqrt ((p a i).val:ℝ))⁻¹) ≤
+      R⁻¹*(M+1)*(2*Real.sqrt U)^(Fintype.card ι-3) := by
+  classical
+  have hed : e≠d := by intro h; exact hd (h ▸ he)
+  let c' : RemainingCoordinates e d := ⟨c,hce,hcd⟩
+  let w (i : ι) (k : Fin (U+1)) :=
+    if i=c then (if k.val≤M then (Real.sqrt (k.val:ℝ))⁻¹ else 0)
+    else (Real.sqrt (k.val:ℝ))⁻¹
+  have hw (i k) : 0≤w i k := by dsimp [w]; split_ifs <;> positivity
+  have hwP (a i) : w i (p a i)=(Real.sqrt ((p a i).val:ℝ))⁻¹ := by
+    by_cases hi : i=c
+    · subst i; simp only [w,ite_true,ite_eq_left (hcM a)]
+    · simp only [w,ite_eq_right hi]
+  have hwS (i : RemainingCoordinates e d) : (∑ k, w i.val k)≤
+      if i=c' then (M+1:ℝ) else 2*Real.sqrt U := by
+    by_cases hi : i=c'
+    · subst i
+      simpa only [c',w,ite_true] using fin_bounded_inverse_sqrt U M
+    · have hic : i.val≠c := by intro h; apply hi; exact Subtype.ext h
+      simpa only [w,ite_eq_right hi,ite_eq_right hic] using fin_sum_inverse_sqrt U
+  calc
+    _ ≤ ∑ a, R⁻¹*∏ i : RemainingCoordinates e d, w i.val (p a i.val) := by
+      apply Finset.sum_le_sum
+      intro a _
+      simp_rw [hwP]
+      rw [prod_two_coordinates hed]
+      exact mul_le_mul_of_nonneg_right (inverse_sqrt_pair hR (heR a) (hdR a)) (by positivity)
+    _ = R⁻¹*∑ a, ∏ i : RemainingCoordinates e d, w i.val (p a i.val) := by rw [Finset.mul_sum]
+    _ ≤ R⁻¹*∏ i : RemainingCoordinates e d, ∑ k, w i.val k :=
+      mul_le_mul_of_nonneg_left (two_sums_product_bound A he hd p hp hA ht w hw) (inv_nonneg.mpr hR.le)
+    _ ≤ R⁻¹*∏ i : RemainingCoordinates e d, if i=c' then (M+1:ℝ) else 2*Real.sqrt U := by
+      apply mul_le_mul_of_nonneg_left _ (inv_nonneg.mpr hR.le)
+      exact Finset.prod_le_prod₀ (fun i _ => Finset.sum_nonneg (fun k _ => hw i.val k)) (fun i _ => hwS i)
+    _ = _ := by
+      rw [prod_one_special,remainingCoordinates_card hed]
+      simp only [Nat.sub_sub]
+      ring
+
+/-- A small internal gap cannot be either selected macroscopic gap.
+ The bound retains the extra factor u^(-1/2) needed for nonlinear terms. -/
+theorem two_sided_bounded_gaps_sum {α ι : Type*} [Fintype α] [Fintype ι] [DecidableEq ι]
+    {U L V M : ℕ} (A : Finset ι) (c : ι)
+    (p : α → ι → Fin (U+1)) (hp : Function.Injective p)
+    (hA : ∀ a, ∑ i∈A, (p a i).val=L) (ht : ∀ a, ∑ i, (p a i).val=V)
+    {R : ℝ} (hR : 0<R) (hMR : (M:ℝ)<R)
+    (hleft : ∀ a, ∃ e∈A, R≤(p a e).val)
+    (hright : ∀ a, ∃ d∉A, R≤(p a d).val)
+    (hcM : ∀ a, (p a c).val≤M) :
+    (∑ a, ∏ i, (Real.sqrt ((p a i).val:ℝ))⁻¹) ≤
+      (Fintype.card ι:ℝ)^2*R⁻¹*(M+1)*(2*Real.sqrt U)^(Fintype.card ι-3) := by
+  classical
+  choose e he heR using hleft
+  choose d hd hdR using hright
+  let f (a : α) := (e a,d a)
+  let W (a : α) := ∏ i, (Real.sqrt ((p a i).val:ℝ))⁻¹
+  have hfiber (z : ι × ι) :
+      (∑ a : {a // f a=z}, W a.val) ≤ R⁻¹*(M+1)*(2*Real.sqrt U)^(Fintype.card ι-3) := by
+    by_cases hne : Nonempty {a // f a=z}
+    · obtain ⟨a⟩ := hne
+      have ha := congrArg Prod.fst a.property
+      have ha' := congrArg Prod.snd a.property
+      have hz1 : z.1∈A := by simpa only [f] using ha ▸ he a.val
+      have hz2 : z.2∉A := by simpa only [f] using ha' ▸ hd a.val
+      have hzR : R≤(p a.val z.1).val := by simpa only [f] using ha ▸ heR a.val
+      have hzR' : R≤(p a.val z.2).val := by simpa only [f] using ha' ▸ hdR a.val
+      have hc : ((p a.val c).val:ℝ)≤M := by exact_mod_cast hcM a.val
+      have hce : c≠z.1 := by intro h; rw [h] at hc; linarith
+      have hcd : c≠z.2 := by intro h; rw [h] at hc; linarith
+      exact two_large_bounded_gap_sum A hz1 hz2 hce hcd
+        (fun a : {a // f a=z} => p a.val) (hp.comp Subtype.val_injective)
+        (fun a => hA a.val) (fun a => ht a.val) hR
+        (fun a => by have h := congrArg Prod.fst a.property; simpa only [f] using h ▸ heR a.val)
+        (fun a => by have h := congrArg Prod.snd a.property; simpa only [f] using h ▸ hdR a.val)
+        (fun a => hcM a.val)
+    · have : IsEmpty {a // f a=z} := not_nonempty_iff.mp hne
+      simp only [Fintype.sum_empty]
+      positivity
+  calc
+    _ = ∑ z : ι × ι, ∑ a : {a // f a=z}, W a.val := (Fintype.sum_fiberwise f W).symm
+    _ ≤ ∑ _z : ι × ι, R⁻¹*(M+1)*(2*Real.sqrt U)^(Fintype.card ι-3) :=
+      Finset.sum_le_sum (fun z _ => hfiber z)
+    _ = _ := by simp [pow_two,mul_assoc]
+
+end TraceReconstruction
+
+
+noncomputable section
+open scoped Classical
+namespace TraceReconstruction
+
+/-- The literal rank-clipping rectangles used in the manuscript, with
+ doubled coarse gaps to avoid half-integer natural indices. Only the
+ first and internal gaps are needed to determine the period tuple. -/
+def PeriodRankRegion {u l r : ℕ} (m : ℕ) (θ : ℝ)
+    (j : IncreasingTuple u r) (v : IncreasingTuple l r) : Prop :=
+  ∀ i : Fin r, |(tupleGap v.val i.castSucc:ℝ)-
+    θ*m*(coarsePeriodGap j.val i.castSucc:ℝ)/2| < (m:ℝ)/8
+
+/-- Enough accuracy to distinguish two integer gap lengths. -/
+theorem nearby_gap_centers_equal {x y : ℕ} {z θ m : ℝ}
+    (hθ : 1 / 2 ≤ θ) (hm : 0 < m)
+    (hx : |z - θ * m * x / 2| < m / 8) (hy : |z - θ * m * y / 2| < m / 8) : x = y := by
+  have hx' := abs_lt.mp hx
+  have hy' := abs_lt.mp hy
+  have hscale : m / 4 ≤ θ * m / 2 := by nlinarith
+  by_contra hxy
+  obtain h | h := lt_or_gt_of_ne hxy
+  · have h' : (x:ℝ)+1≤y := by exact_mod_cast h
+    have hh := mul_le_mul_of_nonneg_left h' (by positivity : 0 ≤ θ * m / 2)
+    nlinarith
+  · have h' : (y:ℝ)+1≤x := by exact_mod_cast h
+    have hh := mul_le_mul_of_nonneg_left h' (by positivity : 0 ≤ θ * m / 2)
+    nlinarith
+
+/-- Regions for distinct tagged period tuples are genuinely disjoint,
+ before any coefficient or orientation averaging. -/
+theorem periodRankRegion_unique {u l r m : ℕ} (hm : 0 < m) {θ : ℝ} (hθ : 1 / 2 ≤ θ)
+    {j k : IncreasingTuple u r} {v : IncreasingTuple l r}
+    (hj : PeriodRankRegion m θ j v) (hk : PeriodRankRegion m θ k v) : j=k := by
+  apply Subtype.ext
+  by_cases hr : r=0
+  · subst r; exact Subsingleton.elim _ _
+  have hg (i : Fin r) : coarsePeriodGap j.val i.castSucc=coarsePeriodGap k.val i.castSucc :=
+    nearby_gap_centers_equal hθ (by exact_mod_cast hm) (hj i) (hk i)
+  have hsum := coarsePeriodGap_sum (Nat.pos_of_ne_zero hr) j.val j.property
+  have hsum' := coarsePeriodGap_sum (Nat.pos_of_ne_zero hr) k.val k.property
+  have hgSum : (∑ i : Fin r, coarsePeriodGap j.val i.castSucc)=
+      ∑ i : Fin r, coarsePeriodGap k.val i.castSucc := Finset.sum_congr rfl (fun i _ => hg i)
+  rw [Fin.sum_univ_castSucc] at hsum hsum'
+  have hlast : coarsePeriodGap j.val (Fin.last r)=coarsePeriodGap k.val (Fin.last r) := by omega
+  apply coarsePeriodGap_injective j.property k.property
+  funext i
+  refine Fin.lastCases hlast (fun i => hg i) i
+
+/-- Orthogonality of a finite family with pointwise disjoint supports. -/
+theorem disjoint_support_sum_sq {α β : Type*} [Fintype α] [Fintype β]
+    (f : α → β → ℝ) (hf : ∀ a b, a≠b → ∀ v, f a v*f b v=0) :
+    (∑ v, (∑ a, f a v)^2) = ∑ a, ∑ v, f a v^2 := by
+  classical
+  have hp (v) : (∑ a, f a v)^2=∑ a, f a v^2 := by
+    rw [pow_two,Finset.sum_mul]
+    apply Finset.sum_congr rfl
+    intro a _
+    rw [Finset.mul_sum]
+    rw [Finset.sum_eq_single a]
+    · ring
+    · intro b _ hba
+      exact hf a b hba.symm v
+    · intro ha
+      exact False.elim (ha (Finset.mem_univ a))
+  simp_rw [hp]
+  exact Finset.sum_comm
+
+/-- Squared norm of the clipped sum is at most the sum of the original
+ squared norms, with arbitrary signed multipliers on each tuple. -/
+theorem period_clipped_square_sum {u l r m : ℕ} (hm : 0 < m) {θ : ℝ} (hθ : 1 / 2 ≤ θ)
+    (W : IncreasingTuple u r → IncreasingTuple l r → ℝ)
+    (a : IncreasingTuple u r → ℝ) :
+    (∑ v, (∑ j, if PeriodRankRegion m θ j v then a j*W j v else 0)^2) ≤
+      ∑ j, a j^2*∑ v, W j v^2 := by
+  classical
+  rw [disjoint_support_sum_sq _ (by
+    intro j k hjk v
+    by_cases hj : PeriodRankRegion m θ j v
+    · have hk : ¬PeriodRankRegion m θ k v := fun hk => hjk (periodRankRegion_unique hm hθ hj hk)
+      simp only [ite_eq_left hj,ite_eq_right hk,mul_zero]
+    · simp only [ite_eq_right hj,zero_mul])]
+  apply Finset.sum_le_sum
+  intro j _
+  rw [Finset.mul_sum]
+  apply Finset.sum_le_sum
+  intro v _
+  split_ifs
+  · exact le_of_eq (mul_pow _ _ _)
+  · simp only [zero_pow (by norm_num : 2 ≠ 0)]
+    exact mul_nonneg (sq_nonneg _) (sq_nonneg _)
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+instance : Fintype GapDifference :=
+  ⟨{.unchanged,.first,.second}, by intro k; cases k <;> simp⟩
+
+theorem sum_gapDifference (f : GapDifference → ℝ) :
+    (∑ k, f k)=f .unchanged+f .first+f .second := by
+  change (∑ k ∈ ({.unchanged,.first,.second} : Finset GapDifference), f k)=_
+  simp [Finset.sum_insert]; ring
+
+/-- The sign-dependent displacement in {-1,0,1}, represented by Fin 3. -/
+def motifShift (χ β τ : ℝ) : Fin 3 :=
+  if χ=0 then 1 else if β*τ=1 then 2 else 0
+
+/-- Coefficients of 1 + χ V + τ β D. -/
+def motifDifferenceWeight (χ β τ : ℝ) : GapDifference → ℝ
+  | .unchanged => 1 | .first => τ*β | .second => χ
+
+/-- The centered operator expansion is an exact displacement, not an
+ approximation of the trace kernel. -/
+theorem motif_positional_identity {χ β τ : ℝ} (hβ : IsMotifValue χ β)
+    (hτ : IsTypeSign τ) (a : Fin 3) :
+    (∑ k, motifDifferenceWeight χ β τ k * positionalWeight k a)=
+      if a=motifShift χ β τ then 1 else 0 := by
+  rw [sum_gapDifference]
+  rcases hβ with ⟨rfl,rfl⟩ | ⟨rfl,rfl | rfl⟩ <;>
+  rcases hτ with rfl | rfl <;> fin_cases a <;>
+    norm_num [motifDifferenceWeight,motifShift,positionalWeight,Fin.ext_iff]
+
+theorem product_function_delta {ι κ : Type*} [Fintype ι] [DecidableEq κ]
+    (a b : ι → κ) :
+    (∏ i, if a i=b i then (1:ℝ) else 0)=if a=b then 1 else 0 := by
+  classical
+  by_cases h : a=b
+  · subst a; simp
+  · rw [ite_eq_right h]
+    obtain ⟨i,hi⟩ := Function.ne_iff.mp h
+    exact Finset.prod_eq_zero (Finset.mem_univ i) (ite_eq_right hi)
+
+/-- Tensorization of the exact local shift identity, with an arbitrary
+ array W (in particular any literal retained-subset rank kernel). -/
+theorem motif_shift_expansion {r : ℕ} (χ β : Fin r → ℝ) (τ : ℝ)
+    (hβ : ∀ h, IsMotifValue (χ h) (β h)) (hτ : IsTypeSign τ)
+    (W : (Fin r → Fin 3) → ℝ) :
+    W (fun h => motifShift (χ h) (β h) τ)=
+      ∑ k : Fin r → GapDifference,
+        (∏ h, motifDifferenceWeight (χ h) (β h) τ (k h))*
+        (∑ a : Fin r → Fin 3, (∏ h, positionalWeight (k h) (a h))*W a) := by
+  classical
+  symm
+  simp_rw [Finset.mul_sum]
+  rw [Finset.sum_comm]
+  calc
+    _ = ∑ a, (∑ k : Fin r → GapDifference,
+        ∏ h, motifDifferenceWeight (χ h) (β h) τ (k h)*positionalWeight (k h) (a h))*W a := by
+      apply Finset.sum_congr rfl
+      intro a _
+      rw [Finset.sum_mul]
+      apply Finset.sum_congr rfl
+      intro k _
+      rw [Finset.prod_mul_distrib]
+      ring
+    _ = ∑ a, (∏ h, if a h=motifShift (χ h) (β h) τ then (1:ℝ) else 0)*W a := by
+      apply Finset.sum_congr rfl
+      intro a _
+      rw [← Fintype.prod_sum (fun h k => motifDifferenceWeight (χ h) (β h) τ k * positionalWeight k (a h))]
+      simp_rw [motif_positional_identity (hβ _) hτ]
+    _ = _ := by simp_rw [product_function_delta]; simp
+
+def firstDifferenceCount {r : ℕ} (k : Fin r → GapDifference) : ℕ :=
+  (Finset.univ.filter fun h => k h = .first).card
+
+def secondDifferenceCount {r : ℕ} (k : Fin r → GapDifference) : ℕ :=
+  (Finset.univ.filter fun h => k h = .second).card
+
+/-- Every factor of the hypothesis sign comes precisely from a D operator. -/
+theorem motifDifferenceWeight_product {r : ℕ} (χ β : Fin r → ℝ) (τ : ℝ)
+    (k : Fin r → GapDifference) :
+    (∏ h, motifDifferenceWeight (χ h) (β h) τ (k h))=
+      τ^firstDifferenceCount k *∏ h, motifDifferenceWeight (χ h) (β h) 1 (k h) := by
+  classical
+  have hw (h) : motifDifferenceWeight (χ h) (β h) τ (k h)=
+      (if k h=.first then τ else 1)*motifDifferenceWeight (χ h) (β h) 1 (k h) := by
+    cases k h <;> simp [motifDifferenceWeight]
+  simp_rw [hw]
+  rw [Finset.prod_mul_distrib]
+  congr 1
+  rw [← Finset.prod_filter]
+  simp [firstDifferenceCount]
+
+/-- The signed difference retains exactly the patterns having an odd
+ number of centered first differences, with the source's factor two. -/
+theorem motif_signed_shift_expansion {r : ℕ} (χ β : Fin r → ℝ)
+    (hβ : ∀ h, IsMotifValue (χ h) (β h)) (W : (Fin r → Fin 3) → ℝ) :
+    W (fun h => motifShift (χ h) (β h) 1)-
+      W (fun h => motifShift (χ h) (β h) (-1))=
+      ∑ k : Fin r → GapDifference,
+        (if Odd (firstDifferenceCount k) then
+          2*∏ h, motifDifferenceWeight (χ h) (β h) 1 (k h) else 0)*
+        (∑ a : Fin r → Fin 3, (∏ h, positionalWeight (k h) (a h))*W a) := by
+  rw [motif_shift_expansion χ β 1 hβ (Or.inl rfl) W,
+    motif_shift_expansion χ β (-1) hβ (Or.inr rfl) W,← Finset.sum_sub_distrib]
+  apply Finset.sum_congr rfl
+  intro k _
+  rw [motifDifferenceWeight_product χ β 1 k,motifDifferenceWeight_product χ β (-1) k]
+  rw [sign_pow_parity (Or.inr rfl : IsTypeSign (-1))]
+  simp only [one_pow,one_mul]
+  split_ifs <;> ring
+
+theorem difference_order_eq_counts {r : ℕ} (k : Fin r → GapDifference) :
+    (∑ h, (k h).order)=firstDifferenceCount k+2*secondDifferenceCount k := by
+  classical
+  have hd (h) : (k h).order=(if k h=.first then 1 else 0)+
+      2*(if k h=.second then 1 else 0) := by cases k h <;> simp [GapDifference.order]
+  simp_rw [hd]
+  rw [Finset.sum_add_distrib,← Finset.mul_sum]
+  simp [firstDifferenceCount,secondDifferenceCount]
+
+/-- The only signed pattern of order below three is the singleton D term. -/
+theorem odd_pattern_linear_or_order_three {r : ℕ} (k : Fin r → GapDifference)
+    (hk : Odd (firstDifferenceCount k)) :
+    (firstDifferenceCount k=1 ∧ secondDifferenceCount k=0) ∨
+      3≤∑ h, (k h).order := by
+  rw [difference_order_eq_counts]
+  obtain ⟨n,hn⟩ := hk
+  omega
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem shiftGapExtra_le {r : ℕ} (a : Fin r → Fin 3) (i : Fin (r+1)) :
+    shiftGapExtra a i  ≤  2*gapIncidence r i := by
+  have hl : Fin.cons (α := fun _ =>  ℕ) 0 (fun h =>  2-(a h).val) i  ≤ 
+      2*Fin.cons (α := fun _ =>  ℕ) 0 (fun _ : Fin r =>  1) i := by
+    induction i using Fin.cases with
+    | zero =>  simp
+    | succ i =>  simp
+  have hr : Fin.snoc (α := fun _ =>  ℕ) (fun h =>  (a h).val) 0 i  ≤ 
+      2*Fin.snoc (α := fun _ =>  ℕ) (fun _ : Fin r =>  1) 0 i := by
+    induction i using Fin.lastCases with
+    | last =>  simp
+    | cast i =>  simpa using Nat.le_of_lt_succ (a i).isLt
+  dsimp [shiftGapExtra,gapIncidence]
+  omega
+
+/-- A unit positional shift changes each actual rank gap by at most two. -/
+theorem tupleGap_shifted_close {n r : ℕ} (p : Fin r → Fin n)
+    (hp : SafeTaggedTuple p) (a : Fin r → Fin 3) (i : Fin (r+1)) :
+    |(tupleGap (shiftedTag p hp a) i:ℝ)-(tupleGap p i:ℝ)| ≤ 2 := by
+  rw [tupleGap_shifted,Nat.cast_add,Nat.cast_sub (gapIncidence_le p hp i)]
+  have hi : (gapIncidence r i:ℝ) ≤ 2 := by exact_mod_cast gapIncidence_le_two r i
+  have he : (shiftGapExtra a i:ℝ) ≤ 2*(gapIncidence r i:ℝ) := by exact_mod_cast shiftGapExtra_le a i
+  have h0 : (0:ℝ) ≤ shiftGapExtra a i := by positivity
+  apply abs_le.mpr
+  constructor  <;>  linarith
+
+theorem period_shifted_gap_rounding {u m r : ℕ} (hr : 0 < r)
+    (off : Fin u → Fin m)
+    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
+    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2)
+    (j : IncreasingTuple u r) (hsafe : SafeTaggedTuple (periodSlot off ∘ j.val))
+    (a : Fin r → Fin 3) (i : Fin (r+1)) :
+    |(tupleGap (shiftedTag (periodSlot off ∘ j.val) hsafe a) i:ℝ)-
+      (m:ℝ)*coarsePeriodGap j.val i/2| ≤ 6 := by
+  have ht := abs_sub_le (tupleGap (shiftedTag (periodSlot off ∘ j.val) hsafe a) i:ℝ)
+    (tupleGap (periodSlot off ∘ j.val) i:ℝ) ((m:ℝ)*coarsePeriodGap j.val i/2)
+  have h1 := tupleGap_shifted_close (periodSlot off ∘ j.val) hsafe a i
+  have h2 := period_gap_rounding hr off hoff hdiff j.val j.property i
+  linarith
+
+theorem coarse_outside_implies_gap_tail {z g c θ m : ℝ}
+    (hθ : θ ∈ Set.Icc 0 1) (hm : 128 ≤ m) (hg : |g-m*c/2| ≤ 6)
+    (hz : m/8 ≤ |z-θ*m*c/2|) : m/16 ≤ |z-g*θ| := by
+  have he : |g*θ-θ*m*c/2| ≤ 6 := by
+    have h := mul_le_mul_of_nonneg_right hg hθ.1
+    rw [show g*θ-θ*m*c/2=(g-m*c/2)*θ by ring,abs_mul,abs_of_nonneg hθ.1]
+    exact h.trans (by nlinarith [hθ.2])
+  have ht := abs_sub_le z (g*θ) (θ*m*c/2)
+  linarith
+
+/-- The manuscript's rank rectangles lose exponentially little mass for
+ every bounded shift of a tagged tuple. -/
+theorem period_shifted_outside_region {u m l r : ℕ}
+    (hu : 0 < u) (hm : 128 ≤ m) (hr : 0 < r) (hl : l ≤ u*m)
+    (off : Fin u → Fin m)
+    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
+    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2)
+    (j : IncreasingTuple u r) (hsafe : SafeTaggedTuple (periodSlot off ∘ j.val))
+    (a : Fin r → Fin 3) {θ : ℝ} (hθ : θ ∈ Set.Ioo 0 1) :
+    (∑ v : IncreasingTuple l r,
+      if PeriodRankRegion m θ j v then 0 else
+        subsetRankKernel hl (shiftedTag (periodSlot off ∘ j.val) hsafe a) v.val)  ≤ 
+      ((r+1:ℕ):ℝ)*θ^r /binomialMass (u*m) θ l *
+        (2*Real.exp (-(m:ℝ)/(1024*u))) := by
+  let p := shiftedTag (periodSlot off ∘ j.val) hsafe a
+  have hu' : (0:ℝ) < u := by exact_mod_cast hu
+  have ht : (1/(32*(u:ℝ))) ∈ Set.Icc 0 1 := by
+    constructor
+    · positivity
+    · apply (div_le_one (by positivity)).mpr
+      have : (1:ℝ) ≤ u := by exact_mod_cast hu
+      linarith
+  have hx : -(1/(32*(u:ℝ)))*((m:ℝ)/16)+((u*m:ℕ):ℝ)*(1/(32*(u:ℝ)))^2 =
+      -(m:ℝ)/(1024*u) := by push_cast; field_simp; ring
+  calc
+    _  ≤  ∑ v : IncreasingTuple l r,
+      if ∃ i, (m:ℝ)/16 ≤ |(tupleGap v.val i:ℝ)-(tupleGap p i:ℝ)*θ|
+      then subsetRankKernel hl p v.val else 0 := by
+      apply Finset.sum_le_sum
+      intro v _
+      by_cases hv : PeriodRankRegion m θ j v
+      · simp only [ite_eq_left hv]
+        split_ifs
+        · exact subsetRankKernel_nonneg hl p v.val
+        · rfl
+      · simp only [PeriodRankRegion,not_forall,not_lt] at hv
+        obtain ⟨i,hi⟩ := hv
+        have he : ∃ i', (m:ℝ)/16 ≤ |(tupleGap v.val i':ℝ)-(tupleGap p i':ℝ)*θ| :=
+          ⟨i.castSucc,coarse_outside_implies_gap_tail ⟨hθ.1.le,hθ.2.le⟩
+            (by exact_mod_cast hm) (period_shifted_gap_rounding hr off hoff hdiff j hsafe a i.castSucc) hi⟩
+        rw [ite_eq_left he]
+        split_ifs
+        · exact subsetRankKernel_nonneg hl p v.val
+        · rfl
+    _  ≤  _ := by
+      have hh := subsetRankKernel_outside_region hl p (shiftedTag_strictMono _ hsafe a) hθ ht (a := (m:ℝ)/16)
+      simpa only [hx] using hh
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- The literal support indicator of the length-2^s signed block. -/
+def motifIndicator (s : ℕ) (z : ℤ) : ℝ :=
+  if 0 ≤ z ∧ z < (2^s : ℕ) then 1 else 0
+
+theorem prouhet_motif_value (s : ℕ) (z : ℤ) :
+    IsMotifValue (motifIndicator s z) ((prouhetSeries s).coeff z) := by
+  by_cases hz : 0 ≤ z ∧ z < (2^s : ℕ)
+  · right
+    refine ⟨by simp only [motifIndicator,ite_eq_left hz], ?_⟩
+    have hv := (prouhetSeries_coeff_spec s).2 z hz.1 hz.2
+    exact (abs_eq (by norm_num : (0:ℝ) ≤ 1)).mp hv
+  · left
+    refine ⟨by simp only [motifIndicator,ite_eq_right hz], ?_⟩
+    apply (prouhetSeries_coeff_spec s).1 z
+    omega
+
+theorem motif_weight_abs_le (χ β : ℝ) (h : IsMotifValue χ β) (k : GapDifference) :
+    |motifDifferenceWeight χ β 1 k| ≤ 1 := by
+  rcases h with ⟨rfl,rfl⟩ | ⟨rfl,rfl | rfl⟩ <;> cases k <;>
+    norm_num [motifDifferenceWeight]
+
+theorem motif_weight_abs_le_indicator (χ β : ℝ) (h : IsMotifValue χ β)
+    (k : GapDifference) (hk : k ≠ .unchanged) :
+    |motifDifferenceWeight χ β 1 k| ≤ χ := by
+  rcases h with ⟨rfl,rfl⟩ | ⟨rfl,rfl | rfl⟩ <;> cases k <;>
+    norm_num [motifDifferenceWeight] at *
+
+/-- A kernel-pattern multiplier averaged over the actual finitely supported
+ anchor law. No independence of its involved positions is asserted. -/
+def motifPatternMultiplier {r : ℕ} (π : IntSeries) (s : ℕ)
+    (j : Fin r → ℤ) (k : Fin r → GapDifference) : ℝ :=
+  ∑ X ∈ π.coeff.support, π.coeff X *
+    ∏ h, motifDifferenceWeight (motifIndicator s (j h-X))
+      ((prouhetSeries s).coeff (j h-X)) 1 (k h)
+
+def linearPattern {r : ℕ} (h : Fin r) (a : Fin r) : GapDifference :=
+  if a=h then .first else .unchanged
+
+theorem motif_linear_multiplier {r : ℕ} (π : IntSeries) (s : ℕ)
+    (j : Fin r → ℤ) (h : Fin r) :
+    motifPatternMultiplier π s j (linearPattern h)=
+      (prouhetSeries s * π).coeff (j h) := by
+  classical
+  rw [mul_comm,AddMonoidAlgebra.coeff_mul_apply_left,Finsupp.sum]
+  apply Finset.sum_congr rfl
+  intro X _
+  congr 1
+  have hw (a : Fin r) : motifDifferenceWeight (motifIndicator s (j a-X))
+      ((prouhetSeries s).coeff (j a-X)) 1 (linearPattern h a)=
+      if a=h then (prouhetSeries s).coeff (j h-X) else 1 := by
+    by_cases ha : a=h <;> simp [linearPattern,ha,motifDifferenceWeight]
+  simp_rw [hw]
+  simp [sub_eq_neg_add]
+
+theorem motifPatternMultiplier_abs_le {r : ℕ} (π : IntSeries)
+    (hπ : ∀ X, 0 ≤ π.coeff X) (s : ℕ) (j : Fin r → ℤ)
+    (k : Fin r → GapDifference) (h : Fin r) (hk : k h ≠ .unchanged) :
+    |motifPatternMultiplier π s j k| ≤
+      ∑ X ∈ π.coeff.support, π.coeff X*motifIndicator s (j h-X) := by
+  classical
+  apply (Finset.abs_sum_le_sum_abs _ _).trans
+  apply Finset.sum_le_sum
+  intro X _
+  rw [abs_mul,abs_of_nonneg (hπ X),Finset.abs_prod]
+  apply mul_le_mul_of_nonneg_left _ (hπ X)
+  have hi (a) := motif_weight_abs_le _ _ (prouhet_motif_value s (j a-X)) (k a)
+  have hh := motif_weight_abs_le_indicator _ _ (prouhet_motif_value s (j h-X)) (k h) hk
+  calc
+    _ ≤ ∏ a : Fin r, if a=h then motifIndicator s (j h-X) else 1 := by
+      apply Finset.prod_le_prod₀
+      · intro a _; positivity
+      · intro a _
+        by_cases ha : a=h
+        · subst a; simpa using hh
+        · simpa [ha] using hi a
+    _ = _ := by simp
+
+/-- A nonzero averaged pattern requires every involved tagged position to
+ lie in one common translate of the motif; correlations are retained. -/
+theorem motif_multiplier_nonzero_support {r : ℕ} (π : IntSeries) (s : ℕ)
+    (j : Fin r → ℤ) (k : Fin r → GapDifference)
+    (hm : motifPatternMultiplier π s j k ≠ 0) :
+    ∃ X ∈ π.coeff.support, ∀ h, k h ≠ .unchanged →
+      0 ≤ j h-X ∧ j h-X < (2^s : ℕ) := by
+  classical
+  obtain ⟨X,hX,hv⟩ := Finset.exists_ne_zero_of_sum_ne_zero hm
+  refine ⟨X,hX,?_⟩
+  intro h hk
+  by_contra hh
+  apply hv
+  apply mul_eq_zero_of_right
+  apply Finset.prod_eq_zero (Finset.mem_univ h)
+  have hi : motifIndicator s (j h-X)=0 := by simp only [motifIndicator,ite_eq_right hh]
+  have hw := motif_weight_abs_le_indicator _ _ (prouhet_motif_value s (j h-X)) (k h) hk
+  exact abs_nonpos_iff.mp (hw.trans_eq hi)
+
+theorem motif_multiplier_bounded_separation {r : ℕ} (π : IntSeries) (s : ℕ)
+    (j : Fin r → ℤ) (k : Fin r → GapDifference)
+    (hm : motifPatternMultiplier π s j k ≠ 0)
+    (a b : Fin r) (ha : k a ≠ .unchanged) (hb : k b ≠ .unchanged) :
+    |j a-j b| < (2^s : ℕ) := by
+  obtain ⟨X,_,hX⟩ := motif_multiplier_nonzero_support π s j k hm
+  have h1 := hX a ha
+  have h2 := hX b hb
+  exact abs_lt.mpr ⟨by omega,by omega⟩
+
+/-- Mixing commutes with the exact signed positional expansion. -/
+theorem mixed_signed_shift_expansion {r : ℕ} (π : IntSeries) (s : ℕ)
+    (j : Fin r → ℤ) (W : (Fin r → Fin 3) → ℝ) :
+    (∑ X ∈ π.coeff.support, π.coeff X *
+      (W (fun h => motifShift (motifIndicator s (j h-X)) ((prouhetSeries s).coeff (j h-X)) 1)-
+       W (fun h => motifShift (motifIndicator s (j h-X)) ((prouhetSeries s).coeff (j h-X)) (-1))))=
+      ∑ k : Fin r → GapDifference,
+        (if Odd (firstDifferenceCount k) then 2*motifPatternMultiplier π s j k else 0)*
+        (∑ a : Fin r → Fin 3, (∏ h, positionalWeight (k h) (a h))*W a) := by
+  calc
+    _ = ∑ X ∈ π.coeff.support, ∑ k : Fin r → GapDifference,
+        π.coeff X*((if Odd (firstDifferenceCount k) then
+          2*∏ h, motifDifferenceWeight (motifIndicator s (j h-X))
+            ((prouhetSeries s).coeff (j h-X)) 1 (k h) else 0)*
+          (∑ a : Fin r → Fin 3, (∏ h, positionalWeight (k h) (a h))*W a)) := by
+      apply Finset.sum_congr rfl
+      intro X _
+      rw [motif_signed_shift_expansion _ _ (fun h => prouhet_motif_value s (j h-X)) W,
+        Finset.mul_sum]
+    _ = _ := by
+      rw [Finset.sum_comm]
+      apply Finset.sum_congr rfl
+      intro k _
+      by_cases hk : Odd (firstDifferenceCount k)
+      · simp only [ite_eq_left hk,motifPatternMultiplier]
+        calc
+          _ = (∑ X ∈ π.coeff.support, 2*(π.coeff X *
+                ∏ h, motifDifferenceWeight (motifIndicator s (j h-X))
+                  ((prouhetSeries s).coeff (j h-X)) 1 (k h))) *
+              (∑ a : Fin r → Fin 3, (∏ h, positionalWeight (k h) (a h))*W a) := by
+            rw [Finset.sum_mul]
+            apply Finset.sum_congr rfl
+            intro X _; ring
+          _ = _ := by rw [← Finset.mul_sum]
+      · simp only [ite_eq_right hk,zero_mul,mul_zero,Finset.sum_const_zero]
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- A general weighted fiber sum, including multipliers which vanish on
+ inadmissible anchor labels. -/
+theorem weighted_fiber_sum_le {α β : Type*} [Fintype α] [Fintype β]
+    (f : α → β) (a : α → ℝ) (w : β → ℝ) (P : α → ℝ) (C : ℝ)
+    (hP : ∀ x, 0 ≤ P x) (ha : ∀ x, a x^2 ≤ w (f x)^2)
+    (hC : ∀ y, w y ≠ 0 → (∑ x : {x // f x=y}, P x.val) ≤ C) :
+    (∑ x, a x^2*P x) ≤ C*∑ y, w y^2 := by
+  classical
+  calc
+    _ ≤ ∑ x, w (f x)^2*P x :=
+      Finset.sum_le_sum fun x _ => mul_le_mul_of_nonneg_right (ha x) (hP x)
+    _ = ∑ y, w y^2*∑ x : {x // f x=y}, P x.val := by
+      rw [← Fintype.sum_fiberwise f (fun x => w (f x)^2*P x)]
+      apply Finset.sum_congr rfl
+      intro y _
+      rw [Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro x _
+      rw [x.property]
+    _ ≤ ∑ y, w y^2*C := by
+      apply Finset.sum_le_sum
+      intro y _
+      by_cases hy : w y=0
+      · simp [hy]
+      · exact mul_le_mul_of_nonneg_left (hC y hy) (sq_nonneg _)
+    _ = _ := by rw [← Finset.sum_mul]; ring
+
+/-- Explicit central-anchor summation constant for doubled period gaps. -/
+def periodCentralSumConstant (u r : ℕ) : ℝ :=
+  ((r+1:ℕ):ℝ)^2*((u:ℝ)/(3*(r+1:ℝ)))⁻¹*(2*Real.sqrt (2*u:ℕ))^(r-1)
+
+theorem weighted_period_gap_sum {u r : ℕ} (a : Fin r)
+    (A : IncreasingTuple u r → ℝ) (w : Fin u → ℝ)
+    (hA : ∀ j, A j^2 ≤ w (j.val a)^2)
+    (hw : ∀ h, w h ≠ 0 → (u:ℝ)/3 ≤ 2*h.val+1 ∧ (u:ℝ)/3 ≤ 2*u-(2*h.val+1)) :
+    (∑ j : IncreasingTuple u r, A j^2*∏ i, (Real.sqrt (coarsePeriodGap j.val i))⁻¹) ≤
+      periodCentralSumConstant u r *∑ h, w h^2 := by
+  apply weighted_fiber_sum_le (fun j : IncreasingTuple u r => j.val a) A w
+    (fun j => ∏ i, (Real.sqrt (coarsePeriodGap j.val i))⁻¹)
+    (periodCentralSumConstant u r) (fun _ => by positivity) hA
+  intro h hh
+  calc
+    _ = ∑ j : FixedPeriodTuple u r a h,
+        ∏ i, (Real.sqrt (coarsePeriodGap j.val.val i))⁻¹ :=
+      by
+        congr 1
+        exact congrArg (fun i : Fintype (FixedPeriodTuple u r a h) => @Finset.univ _ i)
+          (Subsingleton.elim _ _)
+    _ ≤ _ := fixedPeriodProfile_central_sum a h (hw h hh).1 (hw h hh).2
+
+/-- The two largest gaps exist even after arbitrary restrictions on the
+ tuple, provided its anchor label is central. -/
+theorem fixedPeriodProfile_large_sides {u r : ℕ} (a : Fin r) (h : Fin u)
+    (hleft : (u:ℝ)/3 ≤ 2*h.val+1)
+    (hright : (u:ℝ)/3 ≤ 2*u-(2*h.val+1)) (j : FixedPeriodTuple u r a h) :
+    (∃ e ∈ Finset.Iic a.castSucc, (u:ℝ)/(3*(r+1:ℝ)) ≤ (fixedPeriodProfile a h j e).val) ∧
+    (∃ d ∉ Finset.Iic a.castSucc, (u:ℝ)/(3*(r+1:ℝ)) ≤ (fixedPeriodProfile a h j d).val) := by
+  classical
+  let A := Finset.Iic a.castSucc
+  let p := fixedPeriodProfile a h
+  have hL : ∃ e∈A, (u:ℝ)/(3*(r+1:ℝ)) ≤ (p j e).val := by
+    apply exists_large_in_subset A ⟨0, by simp [A]⟩ (fun i => (p j i).val)
+      (fun _ => Nat.cast_nonneg _)
+    have hs := fixedPeriodProfile_prefix a h j
+    have hs' : (∑ i∈A, ((p j i).val:ℝ))=2*h.val+1 := by exact_mod_cast hs
+    rw [hs']
+    simp only [Fintype.card_fin, Nat.cast_add, Nat.cast_one]
+    convert hleft using 1; field_simp
+  have hD : ∃ d∉A, (u:ℝ)/(3*(r+1:ℝ)) ≤ (p j d).val := by
+    have ha : a.val<r := a.isLt
+    have hlast : Fin.last r∈Aᶜ := by simp [A]
+    obtain ⟨d,hd,hdR⟩ := exists_large_in_subset Aᶜ ⟨Fin.last r,hlast⟩
+      (fun i => ((p j i).val:ℝ)) (fun _ => Nat.cast_nonneg _) (R := (u:ℝ)/(3*(r+1:ℝ))) (by
+        have hs := fixedPeriodProfile_prefix a h j
+        have ht := fixedPeriodProfile_total a h j
+        have hs' : (∑ i∈A, ((p j i).val:ℝ))=2*h.val+1 := by exact_mod_cast hs
+        have ht' : (∑ i, ((p j i).val:ℝ))=2*u := by exact_mod_cast ht
+        have hc := Finset.sum_compl_add_sum A (fun i => ((p j i).val:ℝ))
+        have hsum : (∑ i∈Aᶜ, ((p j i).val:ℝ))=2*u-(2*h.val+1) := by linarith
+        rw [hsum]
+        simp only [Fintype.card_fin, Nat.cast_add, Nat.cast_one]
+        convert hright using 1; field_simp)
+    exact ⟨d,Finset.mem_compl.mp hd,hdR⟩
+  exact ⟨hL,hD⟩
+
+/-- Source's bounded gap refinement at a prescribed central label. -/
+theorem fixedPeriodProfile_bounded_central_sum {u r M : ℕ} (a : Fin r) (h : Fin u)
+    (c : Fin (r+1)) (hleft : (u:ℝ)/3 ≤ 2*h.val+1)
+    (hright : (u:ℝ)/3 ≤ 2*u-(2*h.val+1)) (hM : (M:ℝ) < (u:ℝ)/(3*(r+1:ℝ))) :
+    (∑ j : FixedPeriodTuple u r a h,
+      if coarsePeriodGap j.val.val c ≤ M then ∏ i, (Real.sqrt (coarsePeriodGap j.val.val i))⁻¹ else 0) ≤
+      ((r+1:ℕ):ℝ)^2*((u:ℝ)/(3*(r+1:ℝ)))⁻¹*(M+1)*(2*Real.sqrt (2*u:ℕ))^(r-2) := by
+  classical
+  let α := {j : FixedPeriodTuple u r a h // coarsePeriodGap j.val.val c ≤ M}
+  let p (j : α) := fixedPeriodProfile a h j.val
+  have hu : (0:ℝ) < u := by exact_mod_cast (Nat.zero_lt_of_lt h.isLt)
+  have hR : 0 < (u:ℝ)/(3*(r+1:ℝ)) := by positivity
+  have hs := two_sided_bounded_gaps_sum (Finset.Iic a.castSucc) c p
+    ((fixedPeriodProfile_injective a h).comp Subtype.val_injective)
+    (fun j => fixedPeriodProfile_prefix a h j.val)
+    (fun j => fixedPeriodProfile_total a h j.val) hR hM
+    (fun j => (fixedPeriodProfile_large_sides a h hleft hright j.val).1)
+    (fun j => (fixedPeriodProfile_large_sides a h hleft hright j.val).2)
+    (fun j => j.property)
+  have he : (∑ j : FixedPeriodTuple u r a h,
+      if coarsePeriodGap j.val.val c ≤ M then ∏ i, (Real.sqrt (coarsePeriodGap j.val.val i))⁻¹ else 0)=
+      ∑ j : α, ∏ i, (Real.sqrt (coarsePeriodGap j.val.val.val i))⁻¹ := by
+    rw [← Finset.sum_filter]
+    exact Finset.sum_subtype _ (by intro j; simp) _
+  rw [he]
+  simpa only [α,p,fixedPeriodProfile,Fin.val_mk,Fintype.card_fin,
+    show r+1-3=r-2 by omega] using hs
+
+def periodBoundedSumConstant (u r M : ℕ) : ℝ :=
+  ((r+1:ℕ):ℝ)^2*((u:ℝ)/(3*(r+1:ℝ)))⁻¹*(M+1)*(2*Real.sqrt (2*u:ℕ))^(r-2)
+
+theorem weighted_period_bounded_gap_sum {u r M : ℕ} (a : Fin r) (c : Fin (r+1))
+    (A : IncreasingTuple u r → ℝ) (w : Fin u → ℝ)
+    (hA : ∀ j, A j^2 ≤ w (j.val a)^2)
+    (hsupport : ∀ j, A j ≠ 0 → coarsePeriodGap j.val c ≤ M)
+    (hw : ∀ h, w h ≠ 0 → (u:ℝ)/3 ≤ 2*h.val+1 ∧ (u:ℝ)/3 ≤ 2*u-(2*h.val+1))
+    (hM : (M:ℝ) < (u:ℝ)/(3*(r+1:ℝ))) :
+    (∑ j : IncreasingTuple u r, A j^2*∏ i, (Real.sqrt (coarsePeriodGap j.val i))⁻¹) ≤
+      periodBoundedSumConstant u r M *∑ h, w h^2 := by
+  have hp (j : IncreasingTuple u r) :
+      A j^2*(∏ i, (Real.sqrt (coarsePeriodGap j.val i))⁻¹)=
+      A j^2*(if coarsePeriodGap j.val c ≤ M then ∏ i, (Real.sqrt (coarsePeriodGap j.val i))⁻¹ else 0) := by
+    by_cases hj : A j=0
+    · simp [hj]
+    · rw [ite_eq_left (hsupport j hj)]
+  simp_rw [hp]
+  apply weighted_fiber_sum_le (fun j : IncreasingTuple u r => j.val a) A w
+    (fun j => if coarsePeriodGap j.val c ≤ M then ∏ i, (Real.sqrt (coarsePeriodGap j.val i))⁻¹ else 0)
+    (periodBoundedSumConstant u r M) (fun _ => by split_ifs <;> positivity) hA
+  intro h hh
+  calc
+    _ = ∑ j : FixedPeriodTuple u r a h,
+        if coarsePeriodGap j.val.val c ≤ M then ∏ i, (Real.sqrt (coarsePeriodGap j.val.val i))⁻¹ else 0 :=
+      by
+        congr 1
+        exact congrArg (fun i : Fintype (FixedPeriodTuple u r a h) => @Finset.univ _ i)
+          (Subsingleton.elim _ _)
+    _ ≤ _ := fixedPeriodProfile_bounded_central_sum a h c (hw h hh).1 (hw h hh).2 hM
+
+end TraceReconstruction
+
+
+namespace TraceReconstruction
+
+/-- A displaced child window as an actual interval of the deterministic
+ parent template. `t` is the realized count of atoms before the window. -/
+def childWindowIndex {P T : ℕ} (t : ℕ) (hfit : t+T ≤ P) (h : Fin T) : Fin P :=
+  ⟨t+h.val, by omega⟩
+
+theorem templateType_inside_window {P T a L t : ℕ}
+    (sign : Fin L → Bool) (τ : Bool) (hfit : t+T ≤ P)
+    (hleft : t ≤ a) (h : Fin T) :
+    templateType P a L sign τ (childWindowIndex t hfit h)=
+      templateType T (a-t) L sign τ h := by
+  have he : (a ≤ (childWindowIndex t hfit h).val ∧
+      (childWindowIndex t hfit h).val < a+L) ↔
+      a-t ≤ h.val ∧ h.val < a-t+L := by dsimp [childWindowIndex]; omega
+  unfold templateType
+  split_ifs with h1 h2 h2
+  · congr 2
+    apply Fin.ext
+    dsimp [childWindowIndex]
+    omega
+  · exact False.elim (h2 (he.mp h1))
+  · exact False.elim (h1 (he.mpr h2))
+  · rfl
+
+/-- When the motif fits its assigned window, every atom outside that
+ window has the default pure type, including pieces crossing nominal cuts. -/
+theorem templateType_outside_window {P T a L t : ℕ}
+    (sign : Fin L → Bool) (τ : Bool) (hleft : t ≤ a) (hright : a+L ≤ t+T)
+    (i : Fin P) (hi : i.val < t ∨ t+T ≤ i.val) :
+    templateType P a L sign τ i=false := by
+  have he : ¬(a ≤ i.val ∧ i.val < a+L) := by omega
+  simp [templateType,he]
+
+/-- A literal centered window remains within its own nominal parent unit,
+ and contains the motif, whenever its jitter has the source fit bound.
+ This elementary statement is independent of the distribution of the jitter. -/
+theorem centered_window_fit (P T L a A t : ℕ) (Y : ℤ)
+    (hP : 2*T ≤ P) (hA : 2*A=P-T) (ha : 2*a=P)
+    (hL : 8*L ≤ T) (hY : |8*Y| ≤ (T:ℤ)) (ht : (t:ℤ)=A+Y) :
+    t+T ≤ P ∧ t ≤ a ∧ a+L ≤ t+T := by
+  have hy := abs_le.mp hY
+  omega
+
+/-- Flattened deterministic parent type, before any random partition. -/
+def parentGroupType {R P L : ℕ} (a : ℕ) (sign : Fin L → Bool)
+    (τ : Fin R → Bool) (u : Fin R) (h : Fin P) : Bool :=
+  templateType P a L sign (τ u) h
+
+theorem parentGroup_window_types {R P T L : ℕ} (a : ℕ) (sign : Fin L → Bool)
+    (τ : Fin R → Bool) (t : Fin R → ℕ)
+    (hfit : ∀ u, t u+T ≤ P) (hleft : ∀ u, t u ≤ a)
+    (u : Fin R) (h : Fin T) :
+    parentGroupType a sign τ u (childWindowIndex (t u) (hfit u) h)=
+      templateType T (a-t u) L sign (τ u) h :=
+  templateType_inside_window sign (τ u) (hfit u) (hleft u) h
+
+end TraceReconstruction
+
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/TraceReconstruction/Energy.lean b/lean/OAI/Probability/TraceReconstruction/Energy.lean
index b15f1d9c4c63077d2402b8d19c3ef59a860ae50c..d9e5fdc1443974446846b430df0ec47cd311e7f0 100644
--- a/lean/OAI/Probability/TraceReconstruction/Energy.lean
+++ b/lean/OAI/Probability/TraceReconstruction/Energy.lean
@@ -1,881 +1,881 @@
-import OAI.Probability.TraceReconstruction.Shapes
-
-namespace OAI
-
-/-!
-# Energy
-
-Shaped envelopes, actual coefficient aggregation, role counts, central contributions, and full coefficient energy.
--/
-
-open scoped ENNReal
-noncomputable section
-
-open Filter
-open scoped Topology
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-open FiniteLaw
-namespace OccupiedParityTree
-
-/-- Actual unsigned coefficient envelope on one ordered occupied shape.
-No bits are resampled for the additional leaves. -/
-def Forced.envelope (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) (x : t.Labels T)
-    (bits : WindowAssignments T n g Bool b) (incoming : ℤ) : ℝ :=
-  binaryAnchorKernel T n g s center globalShift b (F.labels T x) bits incoming *
-    ∏e : F.ExtraLeafEdge, motifIndicator s ((x e.val.val).val-
-      sourceTreeAnchors T n g center globalShift b bits incoming (e.val.val.parentPath T x))
-
-def Forced.averagedEnvelope (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) (x : t.Labels T)
-    (d : WindowAssignments T n g (Option Bool) b) (incoming : ℤ) : ℝ :=
-  (windowPosteriorLaw T n g b d).expect
-    (fun bits => F.envelope T n g s center globalShift x bits incoming)
-
-/-- Offsets for all extra leaf constraints, with only their labels replaced
-by zero. The location of each constraint is not changed by that replacement. -/
-def Forced.extraOffsets (T n g : ℕ → ℕ) (hT : 0<T 0) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced)
-    (p : BinaryWindowLabels T b) (z : F.InteriorLabels T)
-    (bits : WindowAssignments T n g Bool b) (incoming : ℤ) (e : F.ExtraLeafEdge) : ℤ :=
-  sourceTreeAnchors T n g center globalShift b bits incoming
-    (e.val.val.parentPath T ((F.fullLabelsEquiv T).symm (p,(fun _ => ⟨0,hT⟩),z)))
-
-theorem Forced.averagedEnvelope_split (T n g : ℕ → ℕ) (hT : 0<T 0) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced)
-    (p : BinaryWindowLabels T b) (h : F.ExtraLeafEdge → Fin (T 0)) (z : F.InteriorLabels T)
-    (d : WindowAssignments T n g (Option Bool) b) (incoming : ℤ) :
-    F.averagedEnvelope T n g s center globalShift ((F.fullLabelsEquiv T).symm (p,h,z)) d incoming =
-      indexedExtraAverage (windowPosteriorLaw T n g b d) s (T 0)
-        (fun bits => binaryAnchorKernel T n g s center globalShift b p bits incoming)
-        (fun bits e => F.extraOffsets T n g hT center globalShift p z bits incoming e) h := by
-  let x := (F.fullLabelsEquiv T).symm (p,h,z)
-  have hx : F.fullLabelsEquiv T x=(p,h,z) := Equiv.apply_symm_apply _ _
-  have hp : F.labels T x=p := by rw [←F.fullLabelsEquiv_fst,hx]
-  apply congrArg (FiniteLaw.expect (windowPosteriorLaw T n g b d))
-  funext bits
-  change F.envelope T n g s center globalShift x bits incoming = _
-  unfold Forced.envelope
-  rw [hp]
-  congr 1
-  apply Finset.prod_congr rfl
-  intro e he
-  have hv := congrArg (fun a => (a.2.1 e).val) hx
-  simp only [Forced.fullLabelsEquiv_leaf_value] at hv
-  rw [hv]
-  unfold Forced.extraOffsets
-  congr 2
-  exact congrArg (sourceTreeAnchors T n g center globalShift b bits incoming)
-    (F.parentPath_extraLeaf_invariant T p h (fun _ => ⟨0,hT⟩) z e.val.val)
-
-/-- Summing every actual numeric labeling of a fixed shape costs exactly
-one l1-squared factor per extra leaf and one factor per extra interior label.
-The forced-anchor energy remains a single genuine correlated posterior. -/
-theorem Forced.envelope_energy (T n g : ℕ → ℕ) (hT : 0<T 0) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced)
-    (R : BinaryWindowLabels T b → Prop) [DecidablePred R]
-    (d : WindowAssignments T n g (Option Bool) b) (incoming : ℤ) :
-    (∑x : t.Labels T, if R (F.labels T x) then
-      (F.averagedEnvelope T n g s center globalShift x d incoming)^2 else 0) ≤
-      (Fintype.card (F.InteriorLabels T):ℝ) * ((2^s:ℕ):ℝ)^(2*Fintype.card F.ExtraLeafEdge) *
-        ∑p : BinaryWindowLabels T b, if R p then
-          (binaryAnchorAverage T n g s center globalShift b p d incoming)^2 else 0 := by
-  let L := ((2^s:ℕ):ℝ)^(2*Fintype.card F.ExtraLeafEdge)
-  have hs := (F.fullLabelsEquiv T).symm.sum_comp (fun x : t.Labels T =>
-    if R (F.labels T x) then (F.averagedEnvelope T n g s center globalShift x d incoming)^2 else 0)
-  rw [←hs]
-  simp only [Fintype.sum_prod_type]
-  calc
-    _ ≤ ∑p : BinaryWindowLabels T b, ∑z : F.InteriorLabels T,
-        if R p then L*(binaryAnchorAverage T n g s center globalShift b p d incoming)^2 else 0 := by
-      apply Finset.sum_le_sum
-      intro p hp
-      rw [Finset.sum_comm]
-      apply Finset.sum_le_sum
-      intro z hz
-      have he (h : F.ExtraLeafEdge → Fin (T 0)) :
-          F.labels T ((F.fullLabelsEquiv T).symm (p,h,z))=p := by
-        rw [←F.fullLabelsEquiv_fst,Equiv.apply_symm_apply]
-      simp_rw [he]
-      by_cases hR : R p
-      · simp only [ite_eq_left hR]
-        simp_rw [F.averagedEnvelope_split T n g hT s center globalShift]
-        exact indexedExtraAverage_energy (windowPosteriorLaw T n g b d) s (T 0)
-          (fun bits => binaryAnchorKernel T n g s center globalShift b p bits incoming)
-          (fun bits => binaryAnchorKernel_nonneg T n g s center globalShift b p bits incoming)
-          (fun bits e => F.extraOffsets T n g hT center globalShift p z bits incoming e)
-      · simp only [ite_eq_right hR,Finset.sum_const_zero,le_refl]
-    _ = _ := by
-      simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
-      rw [Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro p hp
-      by_cases hr : R p <;> simp [hr,L,mul_assoc]
-/-- Connected forced-shape coefficient-envelope estimate with the true
-common-global/local sibling energy, and no independent-sibling substitute. -/
-theorem Forced.envelope_energy_root (T n g : ℕ → ℕ) (hT : 0<T 0) (s : ℕ)
-    (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (A : (b : ℕ) → Finset (Fin (T b))) (shift : ℕ → ℤ)
-    (hplat : HasGlobalPlateau T g A shift globalShift)
-    (B C : ℕ → ℝ) (hB : ∀b,0≤B b) (hC : ∀b,0≤C b)
-    (b : ℕ) (t : OccupiedParityTree (b+1)) (F : t.Forced)
-    (d : WindowAssignments T n g (Option Bool) (b+1))
-    (hcap : MetadataBinomialCaps T n g B C (b+1) d) (incoming : ℤ) :
-    (∑x : t.Labels T, if BinaryWindowLabels.allowed T A s (b+1) (F.labels T x) then
-      (F.averagedEnvelope T n g s center globalShift x d incoming)^2 else 0) ≤
-      ((Fintype.card (F.InteriorLabels T):ℝ) * ((2^s:ℕ):ℝ)^(2*Fintype.card F.ExtraLeafEdge)) *
-        ((2*((2^s:ℕ):ℝ)^3*B b) *
-          binaryTreeBudget (fun j => 4*((2^s:ℕ):ℝ)^6*C (j+1)*B j) b) := by
-  apply (F.envelope_energy T n g hT s center globalShift
-    (BinaryWindowLabels.allowed T A s (b+1)) d incoming).trans
-  exact mul_le_mul_of_nonneg_left
-    (binaryAnchorAverage_energy T n g s center globalShift A shift hplat B C hB hC b d hcap incoming)
-    (by positivity)
-
-end OccupiedParityTree
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-namespace OccupiedParityTree
-
-/-- The actual full-tree leaf address of a bottom edge, including the edge's
-own numeric label. -/
-def Edge.bottomAddress (T : ℕ → ℕ) : {b : ℕ} → {t : OccupiedParityTree b} →
-    (x : t.Labels T) → (e : t.Edge) → e.height=0 → WindowLeaf T b
-  | _, .leaf, _, e, _ => e.elim
-  | b+1, .node k children roles, x, Sum.inl i, he => by
-      change b=0 at he
-      subst b
-      exact (x (Sum.inl i),(⟨0,by decide⟩ : Fin 1))
-  | b+1, .node k children roles, x, Sum.inr z, he =>
-      (x (Sum.inl z.1), Edge.bottomAddress T (x.child T z.1) z.2 he)
-
-theorem Edge.bottomAddress_mem (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b}
-    (x : t.Labels T) (e : t.Edge) (he : e.height=0) :
-    e.bottomAddress T x he ∈ t.labelledLeaves T x := by
-  induction t with
-  | leaf => exact e.elim
-  | @node b k children roles ih =>
-    cases e with
-    | inl i =>
-      change b=0 at he
-      subst b
-      apply Finset.mem_biUnion.mpr
-      refine ⟨i,Finset.mem_univ i,?_⟩
-      apply Finset.mem_image.mpr
-      refine ⟨(⟨0,by decide⟩ : Fin 1),?_,rfl⟩
-      have hz (u : OccupiedParityTree 0) (y : u.Labels T) :
-          u.labelledLeaves T y = {(⟨0,by decide⟩ : Fin 1)} := by cases u; rfl
-      rw [hz]
-      exact Finset.mem_singleton_self _
-    | inr z =>
-      apply Finset.mem_biUnion.mpr
-      refine ⟨z.1,Finset.mem_univ _,?_⟩
-      exact Finset.mem_image.mpr ⟨z.2.bottomAddress T (x.child T z.1) he, ih z.1 _ _ he,rfl⟩
-
-/-- Leaf constraints read from the real shape label address are exactly the
-ones in the genuine complete-tree latent law. -/
-theorem Edge.bottomIndicator_eq (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (b : ℕ) (t : OccupiedParityTree (b+1)) (x : t.Labels T)
-    (e : t.Edge) (he : e.height=0)
-    (bits : WindowAssignments T n g Bool (b+1)) (incoming : ℤ) :
-    sourceBottomIndicator T n g s center globalShift b bits incoming (e.bottomAddress T x he) =
-      motifIndicator s ((x e).val-sourceTreeAnchors T n g center globalShift (b+1) bits incoming
-        (e.parentPath T x)) := by
-  induction b generalizing incoming with
-  | zero =>
-    cases t with
-    | node k children roles =>
-      cases e with
-      | inl i =>
-        change motifIndicator s _ = motifIndicator s _
-        congr 1
-        change ((x (Sum.inl i)).val:ℤ)-center 0+(∑i,(bitIndex (bits.1 i):ℤ))+incoming =
-          ((x (Sum.inl i)).val:ℤ)-(center 0-(∑i,(bitIndex (bits.1 i):ℤ))-incoming)
-        omega
-      | inr z => have hh := z.2.height_lt; omega
-  | succ b ih =>
-    cases t with
-    | node k children roles =>
-      cases e with
-      | inl i => change b+1=0 at he; omega
-      | inr z =>
-        change sourceBottomIndicator T n g s center globalShift b
-          (bits.2.2 (x (Sum.inl z.1))) (globalShift (b+1) bits.2.1 (x (Sum.inl z.1)))
-          (z.2.bottomAddress T (x.child T z.1) he) = motifIndicator s ((x.child T z.1 z.2).val-
-          sourceTreeAnchors T n g center globalShift (b+1+1) bits incoming
-            ((x (Sum.inl z.1)).val::z.2.parentPath T (x.child T z.1)))
-        rw [sourceTreeAnchors_child T n g center globalShift (b+1) bits incoming
-          (x (Sum.inl z.1)) (z.2.parentPath T (x.child T z.1))]
-        exact ih _ _ _ he _ _
-
-end OccupiedParityTree
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-open FiniteLaw
-
-/-- Taking repeated constraints from a zero-one product can only enlarge it.
-The map need not be injective. This keeps the real leaf addresses, including
-coincident labels in the ambient enlarged summation. -/
-theorem product_zero_one_pull_le {α β : Type*} [Fintype β]
-    (S : Finset α) (f : α → ℝ) (hf : ∀a∈S,f a=0 ∨ f a=1)
-    (g : β → α) (hg : ∀b,g b∈S) : (∏a∈S,f a)≤∏b,f (g b) := by
-  rcases prod_zero_one S f hf with hz | ho
-  · rw [hz]
-    exact Finset.prod_nonneg (fun b _ => by rcases hf (g b) (hg b) with h|h <;> simp [h])
-  · have h (b : β) : f (g b)=1 := by
-      rcases hf (g b) (hg b) with h|h
-      · have hz := Finset.prod_eq_zero (hg b) h
-        rw [ho] at hz
-        norm_num at hz
-      · exact h
-    rw [ho]
-    simp only [h,Finset.prod_const_one,le_refl]
-
-/-- Every actual role coefficient is bounded by its actual forced shape
-and extra-leaf envelope before averaging over the single whole latent law. -/
-theorem sourceRoleTerm_envelope (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (b : ℕ) (bits : WindowAssignments T n g Bool (b+1)) (incoming : ℤ)
-    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S)
-    (F : (occupiedWindowTerm T b S a).Forced) :
-    |sourceRoleTerm T n g s center globalShift b bits incoming S a|≤
-      F.envelope T n g s center globalShift (occupiedTermLabels T b S a) bits incoming := by
-  let x := occupiedTermLabels T b S a
-  let f := sourceBottomIndicator T n g s center globalShift b bits incoming
-  let addr : F.ExtraLeafEdge → WindowLeaf T (b+1) :=
-    fun e => e.val.val.bottomAddress T x e.property
-  have hm (e : F.ExtraLeafEdge) : addr e∈S := by
-    rw [←occupiedTermLabels_leaves T b S a]
-    exact e.val.val.bottomAddress_mem T x e.property
-  have he (e : F.ExtraLeafEdge) : f (addr e)=motifIndicator s ((x e.val.val).val-
-      sourceTreeAnchors T n g center globalShift (b+1) bits incoming (e.val.val.parentPath T x)) :=
-    e.val.val.bottomIndicator_eq T n g s center globalShift b _ x e.property bits incoming
-  have hle := product_zero_one_pull_le S f (fun v hv =>
-    sourceBottomIndicator_zero_one T n g s center globalShift b bits incoming v) addr hm
-  simp_rw [he] at hle
-  apply le_mul_of_zero_one
-    (binaryAnchorKernel_zero_one T n g s center globalShift _ _ bits incoming)
-    (prod_zero_one Finset.univ _ (fun e _ => by unfold motifIndicator; split_ifs <;> simp))
-  · simpa only [occupiedTermLabels_forced] using
-      sourceRoleTerm_forced_anchor T n g s center globalShift b bits incoming S a F
-  · exact (sourceRoleTerm_all_leaf_constraints T n g s center globalShift b bits incoming S a).trans hle
-
-/-- The averaged coefficient uses one exact metadata fiber. No fit event,
-plateau event or leaf-emission observation is conditioned on here. -/
-theorem sourceRoleAverage_envelope (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ)
-    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S)
-    (F : (occupiedWindowTerm T b S a).Forced) :
-    |sourceRoleAverage T n g s center globalShift b d incoming S a|≤
-      F.averagedEnvelope T n g s center globalShift (occupiedTermLabels T b S a) d incoming := by
-  apply ((windowPosteriorLaw T n g (b+1) d).abs_expect_le _).trans
-  exact FiniteLaw.expect_mono _ (fun bits =>
-    sourceRoleTerm_envelope T n g s center globalShift b bits incoming S a F)
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-open FiniteLaw
-open OccupiedParityTree
-
-theorem occupiedWindowTerm_leafCount (T : ℕ → ℕ) (b : ℕ)
-    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S) :
-    (occupiedWindowTerm T b S a).leafCount=S.card := by
-  rw [←occupiedWindowTerm_eq T 0 (fun _ => 0) b [] S a]
-  exact occupiedWindowShape_leafCount T 0 (fun _ => 0) b [] S _
-
-theorem occupiedWindowTerm_occupied (T : ℕ → ℕ) (b : ℕ)
-    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S) (hS : S.Nonempty) :
-    (occupiedWindowTerm T b S a).occupied := by
-  rw [←occupiedWindowTerm_eq T 0 (fun _ => 0) b [] S a]
-  exact occupiedWindowShape_occupied T 0 (fun _ => 0) b [] S hS _
-
-/-- Only the unlabeled ordered role shape is selected here; no child label
-or hidden orientation enters the forced choice. -/
-abbrev DegreeForcedShape (b r : ℕ) :=
-  {t : OccupiedParityTree b // t.occupied ∧ t.leafCount=r ∧ Nonempty t.Forced}
-
-instance degreeForcedShapeFinite (b r : ℕ) : Finite (DegreeForcedShape b r) := by
-  let := finite_bounded_occupied_shapes b r
-  exact Finite.of_injective
-    (fun t : DegreeForcedShape b r =>
-      (⟨t.val,t.property.1,t.property.2.1.le⟩ : {t : OccupiedParityTree b // t.occupied ∧ t.leafCount≤r}))
-    (fun _ _ h => Subtype.ext (congrArg
-      (fun q : {t : OccupiedParityTree b // t.occupied ∧ t.leafCount≤r} => q.val) h))
-
-noncomputable instance degreeForcedShapeFintype (b r : ℕ) : Fintype (DegreeForcedShape b r) :=
-  Fintype.ofFinite _
-
-def DegreeForcedShape.forced {b r : ℕ} (t : DegreeForcedShape b r) : t.val.Forced :=
-  Classical.choice t.property.2.2
-
-theorem card_degreeForcedShape_le (b r : ℕ) :
-    Fintype.card (DegreeForcedShape b r) ≤ shapeRoleConstant r b := by
-  let f : DegreeForcedShape b r → FiniteRoleShape r b := fun t =>
-    Classical.choose (exists_finiteRoleShape t.val r t.property.1 t.property.2.1.le)
-  apply Fintype.card_le_of_injective f
-  intro t u he
-  apply Subtype.ext
-  have ht := Classical.choose_spec (exists_finiteRoleShape t.val r t.property.1 t.property.2.1.le)
-  have hu := Classical.choose_spec (exists_finiteRoleShape u.val r u.property.1 u.property.2.1.le)
-  exact ht.symm.trans ((congrArg (FiniteRoleShape.toTree r b) he).trans hu)
-
-abbrev SourceRoleIndex (T : ℕ → ℕ) (b : ℕ) :=
-  (S : Finset (WindowLeaf T (b+1))) × OccupiedWindowChoice T b S
-
-abbrev DegreeForcedRole (T : ℕ → ℕ) (b r : ℕ) :=
-  {w : SourceRoleIndex T b // w.1.card=r ∧ Nonempty (occupiedWindowTerm T b w.1 w.2).Forced}
-
-def DegreeForcedRole.shape (T : ℕ → ℕ) (b r : ℕ) (w : DegreeForcedRole T b r) :
-    DegreeForcedShape (b+1) r := by
-  refine ⟨occupiedWindowTerm T b w.val.1 w.val.2,?_,?_,w.property.2⟩
-  · apply occupiedWindowTerm_occupied
-    apply Finset.card_pos.mp
-    obtain ⟨F⟩ := w.property.2
-    have hh := F.leaf_lower
-    rw [occupiedWindowTerm_leafCount] at hh
-    exact (Nat.pow_pos (by decide : (0:ℕ)<2)).trans_le hh
-  · exact (occupiedWindowTerm_leafCount T b w.val.1 w.val.2).trans w.property.1
-
-def DegreeForcedRole.encoding (T : ℕ → ℕ) (b r : ℕ) (w : DegreeForcedRole T b r) :
-    (t : DegreeForcedShape (b+1) r) × t.val.Labels T :=
-  ⟨w.shape T b r,occupiedTermLabels T b w.val.1 w.val.2⟩
-
-theorem DegreeForcedRole.encoding_injective (T : ℕ → ℕ) (b r : ℕ) :
-    Function.Injective (DegreeForcedRole.encoding T b r) := by
-  intro w v he
-  apply Subtype.ext
-  apply occupiedTermEncoding_injective T b
-  exact congrArg (fun z : (t : DegreeForcedShape (b+1) r) × t.val.Labels T =>
-    (⟨z.1.val,z.2⟩ : (t : OccupiedParityTree (b+1)) × t.Labels T)) he
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-open FiniteLaw
-open OccupiedParityTree
-
-/-- A degree-r sum over the genuine pairs (selected leaves, roles),
-restricted only by deterministic allowed forced labels. -/
-def degreeForcedRoleEnergy (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (A : (b : ℕ) → Finset (Fin (T b))) (b r : ℕ)
-    (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ) : ℝ :=
-  ∑w : DegreeForcedRole T b r,
-    if BinaryWindowLabels.allowed T A s (b+1)
-      ((w.shape T b r).forced.labels T (occupiedTermLabels T b w.val.1 w.val.2)) then
-      (sourceRoleAverage T n g s center globalShift b d incoming w.val.1 w.val.2)^2 else 0
-
-/-- The actual coefficient indices inject into shape+numeric-labels. Thus
-summing the enlarged envelope really bounds the original coefficient sum,
-with no identification of different selected monomials. -/
-theorem degreeForcedRoleEnergy_le_envelopes (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (A : (b : ℕ) → Finset (Fin (T b))) (b r : ℕ)
-    (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ) :
-    degreeForcedRoleEnergy T n g s center globalShift A b r d incoming ≤
-      ∑t : DegreeForcedShape (b+1) r, ∑x : t.val.Labels T,
-        if BinaryWindowLabels.allowed T A s (b+1) (t.forced.labels T x) then
-          (t.forced.averagedEnvelope T n g s center globalShift x d incoming)^2 else 0 := by
-  let f := fun z : (t : DegreeForcedShape (b+1) r) × t.val.Labels T =>
-    if BinaryWindowLabels.allowed T A s (b+1) (z.1.forced.labels T z.2) then
-      (z.1.forced.averagedEnvelope T n g s center globalShift z.2 d incoming)^2 else 0
-  have hf : ∀z,0≤f z := by intro z; dsimp [f]; split_ifs <;> positivity
-  calc
-    _ ≤ ∑w : DegreeForcedRole T b r,f (w.encoding T b r) := by
-      apply Finset.sum_le_sum
-      intro w hw
-      dsimp only [f,DegreeForcedRole.encoding]
-      by_cases h : BinaryWindowLabels.allowed T A s (b+1)
-          ((w.shape T b r).forced.labels T (occupiedTermLabels T b w.val.1 w.val.2))
-      · simp only [ite_eq_left h]
-        have hh := sourceRoleAverage_envelope T n g s center globalShift b d incoming
-          w.val.1 w.val.2 (w.shape T b r).forced
-        rw [←sq_abs (sourceRoleAverage T n g s center globalShift b d incoming w.val.1 w.val.2)]
-        exact (sq_le_sq₀ (abs_nonneg _) ((abs_nonneg _).trans hh)).mpr hh
-      · simp only [ite_eq_right h,le_refl]
-    _ ≤ ∑z,f z := by
-      simpa only [ite_true] using sum_injective_profile_le
-        (DegreeForcedRole.encoding T b r) (DegreeForcedRole.encoding_injective T b r)
-        (fun _ => True) (fun _ => trivial) f hf
-    _ = _ := Fintype.sum_sigma _
-
-/-- Uniform arithmetic interior-label budget of a degree-r depth-(b+1)
-shape. This includes no factor for a bottom label. -/
-def interiorLabelBudget (T : ℕ → ℕ) (b r : ℕ) : ℝ :=
-  ((∏j : Fin (b+1),if j.val=0 then 1 else T j.val : ℕ):ℝ)^(r-2^(b+1))
-
-theorem degreeForcedRoleEnergy_bound (T n g : ℕ → ℕ) (hT : ∀j,0<T j) (s : ℕ)
-    (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (A : (b : ℕ) → Finset (Fin (T b))) (shift : ℕ → ℤ)
-    (hplat : HasGlobalPlateau T g A shift globalShift)
-    (B C : ℕ → ℝ) (hB : ∀b,0≤B b) (hC : ∀b,0≤C b)
-    (b r : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1))
-    (hcap : MetadataBinomialCaps T n g B C (b+1) d) (incoming : ℤ) :
-    degreeForcedRoleEnergy T n g s center globalShift A b r d incoming ≤
-      (shapeRoleConstant r (b+1):ℝ) * interiorLabelBudget T b r *
-        ((2^s:ℕ):ℝ)^(2*(r-2^(b+1))) *
-        ((2*((2^s:ℕ):ℝ)^3*B b) *
-          binaryTreeBudget (fun j => 4*((2^s:ℕ):ℝ)^6*C (j+1)*B j) b) := by
-  let K := interiorLabelBudget T b r * ((2^s:ℕ):ℝ)^(2*(r-2^(b+1))) *
-    ((2*((2^s:ℕ):ℝ)^3*B b) * binaryTreeBudget (fun j => 4*((2^s:ℕ):ℝ)^6*C (j+1)*B j) b)
-  have hbud : 0≤binaryTreeBudget (fun j => 4*((2^s:ℕ):ℝ)^6*C (j+1)*B j) b := by
-    exact binaryTreeBudget_nonneg _ (fun j =>
-      mul_nonneg (mul_nonneg (by positivity) (hC (j+1))) (hB j)) b
-  have hb := hB b
-  have hK : 0≤K := by dsimp [K,interiorLabelBudget]; positivity
-  calc
-    _ ≤ ∑t : DegreeForcedShape (b+1) r, ∑x : t.val.Labels T,
-        if BinaryWindowLabels.allowed T A s (b+1) (t.forced.labels T x) then
-          (t.forced.averagedEnvelope T n g s center globalShift x d incoming)^2 else 0 :=
-      degreeForcedRoleEnergy_le_envelopes T n g s center globalShift A b r d incoming
-    _ ≤ ∑_t : DegreeForcedShape (b+1) r,K := by
-      apply Finset.sum_le_sum
-      intro t ht
-      apply (t.forced.envelope_energy_root T n g (hT 0) s center globalShift A shift hplat
-        B C hB hC b t.val d hcap incoming).trans
-      have hi := t.forced.card_interiorLabels_le T hT t.property.1
-      rw [t.property.2.1] at hi
-      have hi' : (Fintype.card (t.forced.InteriorLabels T):ℝ) ≤ interiorLabelBudget T b r := by
-        dsimp [interiorLabelBudget]
-        exact_mod_cast hi
-      rw [t.forced.card_extraLeafEdge,t.property.2.1]
-      exact mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hi' (by positivity)) (by positivity)
-    _ = (Fintype.card (DegreeForcedShape (b+1) r):ℝ)*K := by
-      rw [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
-    _ ≤ (shapeRoleConstant r (b+1):ℝ)*K := by
-      apply mul_le_mul_of_nonneg_right _ hK
-      exact_mod_cast card_degreeForcedShape_le (b+1) r
-    _ = _ := by dsimp [K]; ring
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- Total role multiplicity for one actual selected monomial. Every level
-has at most the number of selected leaves; no ambient-label factor occurs. -/
-theorem occupiedWindowChoice_card_le (T : ℕ → ℕ) (b : ℕ)
-    (S : Finset (WindowLeaf T (b+1))) :
-    Fintype.card (OccupiedWindowChoice T b S) ≤ 3^((b+1)*S.card) := by
-  induction b with
-  | zero =>
-    change Fintype.card (Fin S.card → Fin 3) ≤ _
-    simp
-  | succ b ih =>
-    change Fintype.card (((i : Fin (selectedChildren S).card) →
-      OccupiedWindowChoice T b (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val)) ×
-      (Fin (selectedChildren S).card → Fin 3)) ≤ _
-    rw [Fintype.card_prod,Fintype.card_pi,Fintype.card_fun,Fintype.card_fin,Fintype.card_fin]
-    have hprod := Finset.prod_le_prod (s:=Finset.univ) (fun i _ =>
-      ih (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val))
-    have hs : (∑i : Fin (selectedChildren S).card,
-        (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val).card)=S.card :=
-      (selected_sum_index (selectedChildren S) (fun h => (selectedChildLeaves S h).card)).trans
-        (selected_children_card S)
-    have hc : (selectedChildren S).card≤S.card := Finset.card_image_le
-    calc
-      _ ≤ (∏i : Fin (selectedChildren S).card,
-          3^((b+1)*(selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val).card)) *
-          3^(selectedChildren S).card := Nat.mul_le_mul_right _ hprod
-      _ = 3^((b+1)*S.card+(selectedChildren S).card) := by
-        rw [Finset.prod_pow_eq_pow_sum,←Finset.mul_sum,hs,←pow_add]
-      _ ≤ _ := Nat.pow_le_pow_right (by decide) (by simp only [Nat.add_mul,Nat.one_mul]; omega)
-
-abbrev DegreeLeafSubset (T : ℕ → ℕ) (b r : ℕ) :=
-  {S : Finset (WindowLeaf T (b+1)) // S.card=r}
-
-theorem degreeLeafSubset_card (T : ℕ → ℕ) (b r : ℕ) :
-    Fintype.card (DegreeLeafSubset T b r)=(Fintype.card (WindowLeaf T (b+1))).choose r := by
-  have h := Finset.card_powersetCard r (Finset.univ : Finset (WindowLeaf T (b+1)))
-  have he : (Finset.univ.powersetCard r : Finset (Finset (WindowLeaf T (b+1))))=(Finset.univ.filter (fun S : Finset (WindowLeaf T (b+1)) => S.card=r)) := by
-    ext S
-    simp only [Finset.mem_powersetCard,Finset.subset_univ,true_and,Finset.mem_filter,Finset.mem_univ]
-  rw [he,Finset.card_univ] at h
-  rw [Fintype.card_subtype]
-  exact h
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-open FiniteLaw
-
-/-- All actual motif support lies in the chosen central interval at every
-vertex of the full window tree. This is a predicate of the complete bits,
-not an event on which any posterior law is conditioned. -/
-def SourceCentral (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (A : (b : ℕ) → Finset (Fin (T b))) :
-    (b : ℕ) → WindowAssignments T n g Bool b → ℤ → Prop
-  | 0, _, _ => True
-  | b+1, bits, incoming =>
-      (∀h : Fin (T b), motifIndicator s ((h.val:ℤ)-center b+
-        (∑i,(bitIndex (bits.1 i):ℤ))+incoming)≠0 → h∈A b) ∧
-      ∀h, SourceCentral T n g s center globalShift A b (bits.2.2 h) (globalShift b bits.2.1 h)
-
-def BinaryWindowLabels.central (T : ℕ → ℕ) (A : (b : ℕ) → Finset (Fin (T b))) :
-    (b : ℕ) → BinaryWindowLabels T b → Prop
-  | 0, _ => True
-  | b+1, p => p.1∈A b ∧ p.2.1∈A b ∧ central T A b p.2.2.1 ∧ central T A b p.2.2.2
-
-theorem BinaryWindowLabels.allowed_of_central (T : ℕ → ℕ)
-    (A : (b : ℕ) → Finset (Fin (T b))) (s b : ℕ) (p : BinaryWindowLabels T b)
-    (hc : central T A b p) (hd : distinct T b p) (hs : close T s b p) :
-    allowed T A s b p := by
-  induction b with
-  | zero => trivial
-  | succ b ih =>
-    exact ⟨⟨hc.1,hc.2.1,hd.1,by simpa only [abs_sub_comm] using hs.1⟩,
-      ih _ hc.2.2.1 hd.2.1 hs.2.1,ih _ hc.2.2.2 hd.2.2 hs.2.2⟩
-
-theorem sourceRoleTerm_forced_central (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (A : (b : ℕ) → Finset (Fin (T b)))
-    (b : ℕ) (bits : WindowAssignments T n g Bool (b+1)) (incoming : ℤ)
-    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S)
-    (F : (occupiedWindowTerm T b S a).Forced)
-    (hc : SourceCentral T n g s center globalShift A (b+1) bits incoming)
-    (hz : sourceRoleTerm T n g s center globalShift b bits incoming S a≠0) :
-    BinaryWindowLabels.central T A (b+1) (forcedWindowLabels T b S a F) := by
-  induction b generalizing incoming with
-  | zero =>
-    cases F with
-    | node i j hne hi hj left right =>
-      rw [sourceRoleTerm_zero] at hz
-      change (S.equivFin.symm i).val.1 ∈ A 0 ∧ (S.equivFin.symm j).val.1 ∈ A 0 ∧ True ∧ True
-      refine ⟨hc.1 _ ?_,hc.1 _ ?_,trivial,trivial⟩
-      · convert nonzero_of_abs_le hz (occupiedRootWeight_bottom_anchor T s S a (center 0-(∑k,(bitIndex (bits.1 k):ℤ))-incoming) i) using 1
-        congr 1; ring
-      · convert nonzero_of_abs_le hz (occupiedRootWeight_bottom_anchor T s S a (center 0-(∑k,(bitIndex (bits.1 k):ℤ))-incoming) j) using 1
-        congr 1; ring
-  | succ b ih =>
-    cases F with
-    | node i j hne hi hj left right =>
-      rw [sourceRoleTerm_succ] at hz
-      have hm := mul_ne_zero_iff.mp hz
-      have hp := Finset.prod_ne_zero_iff.mp hm.2
-      change ((selectedChildren S).equivFin.symm i).val ∈ A (b+1) ∧
-        ((selectedChildren S).equivFin.symm j).val ∈ A (b+1) ∧
-        BinaryWindowLabels.central T A (b+1)
-          (forcedWindowLabels T b (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val)
-            (a.1 i) left) ∧
-        BinaryWindowLabels.central T A (b+1)
-          (forcedWindowLabels T b (selectedChildLeaves S ((selectedChildren S).equivFin.symm j).val)
-            (a.1 j) right)
-      refine ⟨hc.1 _ ?_,hc.1 _ ?_,
-        ih _ _ _ _ left (hc.2 _) (hp i (Finset.mem_univ i)),
-        ih _ _ _ _ right (hc.2 _) (hp j (Finset.mem_univ j))⟩
-      · convert nonzero_of_abs_le hm.1 (occupiedRootWeight_internal_anchor T s b S a _ i hi) using 1
-        congr 1; ring
-      · convert nonzero_of_abs_le hm.1 (occupiedRootWeight_internal_anchor T s b S a _ j hj) using 1
-        congr 1; ring
-
-/-- The discarded noncentral role contributions are bounded by the actual
-conditional probability of a failed full-tree layout. Typical metadata may
-bound this mass; the averaging distribution itself remains unconditional. -/
-def sourceNoncentralMass (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (A : (b : ℕ) → Finset (Fin (T b))) (b : ℕ)
-    (d : WindowAssignments T n g (Option Bool) b) (incoming : ℤ) : ℝ :=
-  (windowPosteriorLaw T n g b d).expect
-    (fun bits => if SourceCentral T n g s center globalShift A b bits incoming then 0 else 1)
-
-theorem sourceNoncentralMass_nonneg (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (A : (b : ℕ) → Finset (Fin (T b))) (b : ℕ)
-    (d : WindowAssignments T n g (Option Bool) b) (incoming : ℤ) :
-    0 ≤ sourceNoncentralMass T n g s center globalShift A b d incoming := by
-  exact Finset.sum_nonneg (fun bits _ => mul_nonneg
-    ((windowPosteriorLaw T n g b d).nonneg bits) (by dsimp only; split_ifs <;> norm_num))
-
-theorem sourceRoleAverage_not_allowed (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (A : (b : ℕ) → Finset (Fin (T b)))
-    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ)
-    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S)
-    (F : (occupiedWindowTerm T b S a).Forced)
-    (h : ¬BinaryWindowLabels.allowed T A s (b+1) (forcedWindowLabels T b S a F)) :
-    |sourceRoleAverage T n g s center globalShift b d incoming S a|≤
-      sourceNoncentralMass T n g s center globalShift A (b+1) d incoming := by
-  apply ((windowPosteriorLaw T n g (b+1) d).abs_expect_le _).trans
-  apply FiniteLaw.expect_mono
-  intro bits
-  by_cases hc : SourceCentral T n g s center globalShift A (b+1) bits incoming
-  · rw [ite_eq_left hc]
-    have hz : sourceRoleTerm T n g s center globalShift b bits incoming S a=0 := by
-      by_contra hn
-      exact h (BinaryWindowLabels.allowed_of_central T A s (b+1) _
-        (sourceRoleTerm_forced_central T n g s center globalShift A b bits incoming S a F hc hn)
-        (forcedWindowLabels_distinct T b S a F)
-        (sourceRoleTerm_forced_close T n g s center globalShift b bits incoming S a F hn))
-    rw [hz,abs_zero]
-  · rw [ite_eq_right hc]
-    exact occupiedWindowWeight_abs_le_one T s _ b [] S a
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-open FiniteLaw
-
-/-- Extension by zero commutes with the finite subtype sum, including an
-empty subtype and predicates depending on dependent role indices. -/
-theorem sum_subtype_zero_extend {α : Type*} [Fintype α] (p : α → Prop) [DecidablePred p]
-    (f : {x // p x} → ℝ) :
-    (∑x, if h : p x then f ⟨x,h⟩ else 0) = ∑x,f x := by
-  have h := Fintype.sum_subtype_add_sum_subtype p
-    (fun x => if h : p x then f ⟨x,h⟩ else 0)
-  calc
-    _ = (∑x : {x // p x}, if h : p x.val then f ⟨x.val,h⟩ else 0) +
-        ∑x : {x // ¬p x}, if h : p x.val then f ⟨x.val,h⟩ else 0 := h.symm
-    _ = (∑x,f x)+0 := congrArg₂ (·+·)
-      (Finset.sum_congr rfl (fun x _ => dite_eq_left x.property))
-      (Finset.sum_eq_zero (fun x _ => dite_eq_right x.property))
-    _ = _ := add_zero _
-
-/-- The coefficient is defined directly from the two literal leaf-type
-patterns under the common complete posterior bit tree. -/
-def sourceCoefficient (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ)
-    (S : Finset (WindowLeaf T (b+1))) : ℝ :=
-  (windowPosteriorLaw T n g (b+1) d).expect (fun bits =>
-    ∏v∈S,(1-sourceTreeLeafSign T n g s center globalShift (b+1) bits incoming 1 v)/2) -
-  (windowPosteriorLaw T n g (b+1) d).expect (fun bits =>
-    ∏v∈S,(1-sourceTreeLeafSign T n g s center globalShift (b+1) bits incoming (-1) v)/2)
-
-theorem sourceCoefficient_expansion (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ)
-    (S : Finset (WindowLeaf T (b+1))) :
-    sourceCoefficient T n g s center globalShift b d incoming S =
-      2*∑a : OccupiedWindowChoice T b S,
-        if (occupiedWindowTerm T b S a).isOdd then
-          sourceRoleAverage T n g s center globalShift b d incoming S a else 0 := by
-  unfold sourceCoefficient
-  simp_rw [sourceTreeLeafSign_eq]
-  exact occupiedWindow_averaged_difference _ T s _ b [] S
-
-def sourceGoodRoleSquare (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (A : (b : ℕ) → Finset (Fin (T b))) (b r : ℕ)
-    (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ)
-    (w : SourceRoleIndex T b) : ℝ :=
-  if h : w.1.card=r ∧ Nonempty (occupiedWindowTerm T b w.1 w.2).Forced then
-    let u : DegreeForcedRole T b r := ⟨w,h⟩
-    if BinaryWindowLabels.allowed T A s (b+1)
-      ((u.shape T b r).forced.labels T (occupiedTermLabels T b w.1 w.2)) then
-      (sourceRoleAverage T n g s center globalShift b d incoming w.1 w.2)^2 else 0
-  else 0
-
-theorem sourceGoodRoleSquare_nonneg (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (A : (b : ℕ) → Finset (Fin (T b))) (b r : ℕ)
-    (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ)
-    (w : SourceRoleIndex T b) :
-    0 ≤ sourceGoodRoleSquare T n g s center globalShift A b r d incoming w := by
-  unfold sourceGoodRoleSquare
-  split_ifs <;> positivity
-
-theorem sourceGoodRoleSquare_sum (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (A : (b : ℕ) → Finset (Fin (T b))) (b r : ℕ)
-    (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ) :
-    (∑w : SourceRoleIndex T b, sourceGoodRoleSquare T n g s center globalShift A b r d incoming w)=
-      degreeForcedRoleEnergy T n g s center globalShift A b r d incoming := by
-  unfold sourceGoodRoleSquare degreeForcedRoleEnergy
-  exact sum_subtype_zero_extend
-    (fun w : SourceRoleIndex T b => w.1.card=r ∧ Nonempty (occupiedWindowTerm T b w.1 w.2).Forced)
-    (fun w : DegreeForcedRole T b r => if BinaryWindowLabels.allowed T A s (b+1)
-      ((w.shape T b r).forced.labels T (occupiedTermLabels T b w.val.1 w.val.2)) then
-      (sourceRoleAverage T n g s center globalShift b d incoming w.val.1 w.val.2)^2 else 0)
-
-/-- Every actual odd role term is either smoothed at an actual singleton,
-discarded by the literal central-layout failure event, or retained in the
-forced-tree sum. There is no unproved coefficient-bound assumption. -/
-theorem sourceRoleSquare_split (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (A : (b : ℕ) → Finset (Fin (T b))) (b r : ℕ)
-    (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ)
-    (S : Finset (WindowLeaf T (b+1))) (hS : S.card=r) (a : OccupiedWindowChoice T b S)
-    (η : ℝ) (hcap : MetadataSmoothingCap T n g s η (b+1) d) :
-    (if (occupiedWindowTerm T b S a).isOdd then
-      sourceRoleAverage T n g s center globalShift b d incoming S a else 0)^2 ≤
-      η^2+(sourceNoncentralMass T n g s center globalShift A (b+1) d incoming)^2+
-        sourceGoodRoleSquare T n g s center globalShift A b r d incoming ⟨S,a⟩ := by
-  have hg := sourceGoodRoleSquare_nonneg T n g s center globalShift A b r d incoming ⟨S,a⟩
-  by_cases ho : (occupiedWindowTerm T b S a).isOdd
-  swap
-  · rw [ite_eq_right ho]; nlinarith [sq_nonneg η, sq_nonneg (sourceNoncentralMass T n g s center globalShift A (b+1) d incoming)]
-  rw [ite_eq_left ho]
-  by_cases hz : sourceRoleAverage T n g s center globalShift b d incoming S a=0
-  · rw [hz]; nlinarith [sq_nonneg η, sq_nonneg (sourceNoncentralMass T n g s center globalShift A (b+1) d incoming)]
-  by_cases hn : (occupiedWindowTerm T b S a).noSingleton
-  swap
-  · have hs := sourceRoleAverage_singleton T n g s center globalShift b d incoming S a η hcap hn
-    have hs2 := (sq_le_sq₀ (abs_nonneg _) ((abs_nonneg _).trans hs)).mpr hs
-    rw [sq_abs] at hs2
-    nlinarith [sq_nonneg (sourceNoncentralMass T n g s center globalShift A (b+1) d incoming)]
-  have hf := OccupiedParityTree.forced_nonempty (occupiedWindowTerm T b S a)
-    (sourceRoleAverage_valid T n g s center globalShift b d incoming S a hz) hn ho
-  let w : DegreeForcedRole T b r := ⟨⟨S,a⟩,hS,hf⟩
-  let F : (occupiedWindowTerm T b S a).Forced := (w.shape T b r).forced
-  change _ ≤ _+ (if h : S.card=r ∧ Nonempty (occupiedWindowTerm T b S a).Forced then _ else 0)
-  rw [dite_eq_left ⟨hS,hf⟩]
-  change _ ≤ _ + (if BinaryWindowLabels.allowed T A s (b+1)
-    (F.labels T (occupiedTermLabels T b S a)) then _ else 0)
-  rw [occupiedTermLabels_forced T b S a F]
-  by_cases ha : BinaryWindowLabels.allowed T A s (b+1) (forcedWindowLabels T b S a F)
-  · rw [ite_eq_left ha]; nlinarith [sq_nonneg η,
-      sq_nonneg (sourceNoncentralMass T n g s center globalShift A (b+1) d incoming)]
-  · rw [ite_eq_right ha]
-    have hs := sourceRoleAverage_not_allowed T n g s center globalShift A b d incoming S a F ha
-    have hs2 := (sq_le_sq₀ (abs_nonneg _) (sourceNoncentralMass_nonneg T n g s center globalShift A (b+1) d incoming)).mpr hs
-    rw [sq_abs] at hs2
-    nlinarith [sq_nonneg η]
-
-end TraceReconstruction
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-open FiniteLaw
-
-theorem sourceGoodRoleSquare_degree_sum (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (A : (b : ℕ) → Finset (Fin (T b))) (b r : ℕ)
-    (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ) :
-    (∑S : DegreeLeafSubset T b r, ∑a : OccupiedWindowChoice T b S.val,
-      sourceGoodRoleSquare T n g s center globalShift A b r d incoming ⟨S.val,a⟩)=
-      degreeForcedRoleEnergy T n g s center globalShift A b r d incoming := by
-  rw [←sourceGoodRoleSquare_sum,Fintype.sum_sigma]
-  symm
-  calc
-    _ = ∑S : Finset (WindowLeaf T (b+1)), if h : S.card=r then
-        ∑a : OccupiedWindowChoice T b S,
-          sourceGoodRoleSquare T n g s center globalShift A b r d incoming ⟨S,a⟩ else 0 := by
-      apply Finset.sum_congr rfl
-      intro S _
-      by_cases h : S.card=r
-      · rw [dite_eq_left h]
-      · simp [sourceGoodRoleSquare,h]
-    _ = _ := sum_subtype_zero_extend
-      (fun S : Finset (WindowLeaf T (b+1)) => S.card=r)
-      (fun S : DegreeLeafSubset T b r => ∑a : OccupiedWindowChoice T b S.val,
-        sourceGoodRoleSquare T n g s center globalShift A b r d incoming ⟨S.val,a⟩)
-
-theorem sourceCoefficient_square_bound (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (A : (b : ℕ) → Finset (Fin (T b))) (b r : ℕ)
-    (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ)
-    (S : Finset (WindowLeaf T (b+1))) (hS : S.card=r)
-    (η : ℝ) (hcap : MetadataSmoothingCap T n g s η (b+1) d) :
-    (sourceCoefficient T n g s center globalShift b d incoming S)^2 ≤
-      4*(3^((b+1)*r):ℕ)*
-        ((3^((b+1)*r):ℕ)*(η^2+
-          (sourceNoncentralMass T n g s center globalShift A (b+1) d incoming)^2)+
-          ∑a : OccupiedWindowChoice T b S,
-            sourceGoodRoleSquare T n g s center globalShift A b r d incoming ⟨S,a⟩) := by
-  let f : OccupiedWindowChoice T b S → ℝ := fun a =>
-    if (occupiedWindowTerm T b S a).isOdd then
-      sourceRoleAverage T n g s center globalShift b d incoming S a else 0
-  let M : ℝ := (3^((b+1)*r):ℕ)
-  let E : ℝ := η^2+(sourceNoncentralMass T n g s center globalShift A (b+1) d incoming)^2
-  let G := fun a => sourceGoodRoleSquare T n g s center globalShift A b r d incoming ⟨S,a⟩
-  have hE : 0 ≤ E := add_nonneg (sq_nonneg _) (sq_nonneg _)
-  have hM : 0 ≤ M := Nat.cast_nonneg _
-  have hcard : (Fintype.card (OccupiedWindowChoice T b S):ℝ) ≤ M := by
-    dsimp only [M]
-    exact_mod_cast (hS ▸ occupiedWindowChoice_card_le T b S)
-  have hcs : (∑a,f a)^2 ≤ (Fintype.card (OccupiedWindowChoice T b S):ℝ)*∑a,(f a)^2 := by
-    simpa using Finset.sum_mul_sq_le_sq_mul_sq Finset.univ (fun _ => (1:ℝ)) f
-  have hs : (∑a,(f a)^2) ≤ M*E+∑a,G a := by
-    calc
-      _ ≤ ∑a : OccupiedWindowChoice T b S,(E+G a) :=
-        Finset.sum_le_sum (fun a _ => sourceRoleSquare_split T n g s center globalShift A b r d incoming S hS a η hcap)
-      _ = (Fintype.card (OccupiedWindowChoice T b S):ℝ)*E+∑a,G a := by simp [Finset.sum_add_distrib]
-      _ ≤ M*E+∑a,G a := add_le_add (mul_le_mul_of_nonneg_right hcard hE) le_rfl
-  calc
-    _ = 4*(∑a,f a)^2 := by rw [sourceCoefficient_expansion]; dsimp only [f]; ring
-    _ ≤ 4*((Fintype.card (OccupiedWindowChoice T b S):ℝ)*∑a,(f a)^2) := by linarith
-    _ ≤ 4*(M*∑a,(f a)^2) := mul_le_mul_of_nonneg_left
-      (mul_le_mul_of_nonneg_right hcard (Finset.sum_nonneg (fun _ _ => sq_nonneg _))) (by norm_num)
-    _ ≤ 4*(M*(M*E+∑a,G a)) := mul_le_mul_of_nonneg_left
-      (mul_le_mul_of_nonneg_left hs hM) (by norm_num)
-    _ = _ := by dsimp only [M,E,G]; ring
-
-/-- The full degree coefficient sum, including every selected monomial,
-singletons, and off-center labels. This is the finite, unabsorbed version
-of the manuscript's low-degree coefficient estimate. -/
-theorem sourceCoefficient_degree_energy (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (A : (b : ℕ) → Finset (Fin (T b))) (b r : ℕ)
-    (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ)
-    (η : ℝ) (hcap : MetadataSmoothingCap T n g s η (b+1) d) :
-    (∑S : DegreeLeafSubset T b r,
-      (sourceCoefficient T n g s center globalShift b d incoming S.val)^2) ≤
-      4*(3^((b+1)*r):ℕ)*
-        (((Fintype.card (WindowLeaf T (b+1))).choose r:ℕ)*(3^((b+1)*r):ℕ)*
-          (η^2+(sourceNoncentralMass T n g s center globalShift A (b+1) d incoming)^2)+
-          degreeForcedRoleEnergy T n g s center globalShift A b r d incoming) := by
-  calc
-    _ ≤ ∑S : DegreeLeafSubset T b r,
-        4*(3^((b+1)*r):ℕ)*
-          ((3^((b+1)*r):ℕ)*(η^2+
-            (sourceNoncentralMass T n g s center globalShift A (b+1) d incoming)^2)+
-            ∑a : OccupiedWindowChoice T b S.val,
-              sourceGoodRoleSquare T n g s center globalShift A b r d incoming ⟨S.val,a⟩) :=
-      Finset.sum_le_sum (fun S _ => sourceCoefficient_square_bound T n g s center globalShift A b r d incoming S.val S.property η hcap)
-    _ = _ := by
-      rw [←Finset.mul_sum,Finset.sum_add_distrib,sourceGoodRoleSquare_degree_sum]
-      simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,degreeLeafSubset_card]
-      ring
-
-end TraceReconstruction
-
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-
-end OAI
+import OAI.Probability.TraceReconstruction.Shapes
+
+namespace OAI
+
+/-!
+# Energy
+
+Shaped envelopes, actual coefficient aggregation, role counts, central contributions, and full coefficient energy.
+-/
+
+open scoped ENNReal
+noncomputable section
+
+open Filter
+open scoped Topology
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+open FiniteLaw
+namespace OccupiedParityTree
+
+/-- Actual unsigned coefficient envelope on one ordered occupied shape.
+No bits are resampled for the additional leaves. -/
+def Forced.envelope (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) (x : t.Labels T)
+    (bits : WindowAssignments T n g Bool b) (incoming : ℤ) : ℝ :=
+  binaryAnchorKernel T n g s center globalShift b (F.labels T x) bits incoming *
+    ∏e : F.ExtraLeafEdge, motifIndicator s ((x e.val.val).val-
+      sourceTreeAnchors T n g center globalShift b bits incoming (e.val.val.parentPath T x))
+
+def Forced.averagedEnvelope (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) (x : t.Labels T)
+    (d : WindowAssignments T n g (Option Bool) b) (incoming : ℤ) : ℝ :=
+  (windowPosteriorLaw T n g b d).expect
+    (fun bits => F.envelope T n g s center globalShift x bits incoming)
+
+/-- Offsets for all extra leaf constraints, with only their labels replaced
+by zero. The location of each constraint is not changed by that replacement. -/
+def Forced.extraOffsets (T n g : ℕ → ℕ) (hT : 0<T 0) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced)
+    (p : BinaryWindowLabels T b) (z : F.InteriorLabels T)
+    (bits : WindowAssignments T n g Bool b) (incoming : ℤ) (e : F.ExtraLeafEdge) : ℤ :=
+  sourceTreeAnchors T n g center globalShift b bits incoming
+    (e.val.val.parentPath T ((F.fullLabelsEquiv T).symm (p,(fun _ => ⟨0,hT⟩),z)))
+
+theorem Forced.averagedEnvelope_split (T n g : ℕ → ℕ) (hT : 0<T 0) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced)
+    (p : BinaryWindowLabels T b) (h : F.ExtraLeafEdge → Fin (T 0)) (z : F.InteriorLabels T)
+    (d : WindowAssignments T n g (Option Bool) b) (incoming : ℤ) :
+    F.averagedEnvelope T n g s center globalShift ((F.fullLabelsEquiv T).symm (p,h,z)) d incoming =
+      indexedExtraAverage (windowPosteriorLaw T n g b d) s (T 0)
+        (fun bits => binaryAnchorKernel T n g s center globalShift b p bits incoming)
+        (fun bits e => F.extraOffsets T n g hT center globalShift p z bits incoming e) h := by
+  let x := (F.fullLabelsEquiv T).symm (p,h,z)
+  have hx : F.fullLabelsEquiv T x=(p,h,z) := Equiv.apply_symm_apply _ _
+  have hp : F.labels T x=p := by rw [←F.fullLabelsEquiv_fst,hx]
+  apply congrArg (FiniteLaw.expect (windowPosteriorLaw T n g b d))
+  funext bits
+  change F.envelope T n g s center globalShift x bits incoming = _
+  unfold Forced.envelope
+  rw [hp]
+  congr 1
+  apply Finset.prod_congr rfl
+  intro e he
+  have hv := congrArg (fun a => (a.2.1 e).val) hx
+  simp only [Forced.fullLabelsEquiv_leaf_value] at hv
+  rw [hv]
+  unfold Forced.extraOffsets
+  congr 2
+  exact congrArg (sourceTreeAnchors T n g center globalShift b bits incoming)
+    (F.parentPath_extraLeaf_invariant T p h (fun _ => ⟨0,hT⟩) z e.val.val)
+
+/-- Summing every actual numeric labeling of a fixed shape costs exactly
+one l1-squared factor per extra leaf and one factor per extra interior label.
+The forced-anchor energy remains a single genuine correlated posterior. -/
+theorem Forced.envelope_energy (T n g : ℕ → ℕ) (hT : 0<T 0) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced)
+    (R : BinaryWindowLabels T b → Prop) [DecidablePred R]
+    (d : WindowAssignments T n g (Option Bool) b) (incoming : ℤ) :
+    (∑x : t.Labels T, if R (F.labels T x) then
+      (F.averagedEnvelope T n g s center globalShift x d incoming)^2 else 0) ≤
+      (Fintype.card (F.InteriorLabels T):ℝ) * ((2^s:ℕ):ℝ)^(2*Fintype.card F.ExtraLeafEdge) *
+        ∑p : BinaryWindowLabels T b, if R p then
+          (binaryAnchorAverage T n g s center globalShift b p d incoming)^2 else 0 := by
+  let L := ((2^s:ℕ):ℝ)^(2*Fintype.card F.ExtraLeafEdge)
+  have hs := (F.fullLabelsEquiv T).symm.sum_comp (fun x : t.Labels T =>
+    if R (F.labels T x) then (F.averagedEnvelope T n g s center globalShift x d incoming)^2 else 0)
+  rw [←hs]
+  simp only [Fintype.sum_prod_type]
+  calc
+    _ ≤ ∑p : BinaryWindowLabels T b, ∑z : F.InteriorLabels T,
+        if R p then L*(binaryAnchorAverage T n g s center globalShift b p d incoming)^2 else 0 := by
+      apply Finset.sum_le_sum
+      intro p hp
+      rw [Finset.sum_comm]
+      apply Finset.sum_le_sum
+      intro z hz
+      have he (h : F.ExtraLeafEdge → Fin (T 0)) :
+          F.labels T ((F.fullLabelsEquiv T).symm (p,h,z))=p := by
+        rw [←F.fullLabelsEquiv_fst,Equiv.apply_symm_apply]
+      simp_rw [he]
+      by_cases hR : R p
+      · simp only [ite_eq_left hR]
+        simp_rw [F.averagedEnvelope_split T n g hT s center globalShift]
+        exact indexedExtraAverage_energy (windowPosteriorLaw T n g b d) s (T 0)
+          (fun bits => binaryAnchorKernel T n g s center globalShift b p bits incoming)
+          (fun bits => binaryAnchorKernel_nonneg T n g s center globalShift b p bits incoming)
+          (fun bits e => F.extraOffsets T n g hT center globalShift p z bits incoming e)
+      · simp only [ite_eq_right hR,Finset.sum_const_zero,le_refl]
+    _ = _ := by
+      simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
+      rw [Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro p hp
+      by_cases hr : R p <;> simp [hr,L,mul_assoc]
+/-- Connected forced-shape coefficient-envelope estimate with the true
+common-global/local sibling energy, and no independent-sibling substitute. -/
+theorem Forced.envelope_energy_root (T n g : ℕ → ℕ) (hT : 0<T 0) (s : ℕ)
+    (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (A : (b : ℕ) → Finset (Fin (T b))) (shift : ℕ → ℤ)
+    (hplat : HasGlobalPlateau T g A shift globalShift)
+    (B C : ℕ → ℝ) (hB : ∀b,0≤B b) (hC : ∀b,0≤C b)
+    (b : ℕ) (t : OccupiedParityTree (b+1)) (F : t.Forced)
+    (d : WindowAssignments T n g (Option Bool) (b+1))
+    (hcap : MetadataBinomialCaps T n g B C (b+1) d) (incoming : ℤ) :
+    (∑x : t.Labels T, if BinaryWindowLabels.allowed T A s (b+1) (F.labels T x) then
+      (F.averagedEnvelope T n g s center globalShift x d incoming)^2 else 0) ≤
+      ((Fintype.card (F.InteriorLabels T):ℝ) * ((2^s:ℕ):ℝ)^(2*Fintype.card F.ExtraLeafEdge)) *
+        ((2*((2^s:ℕ):ℝ)^3*B b) *
+          binaryTreeBudget (fun j => 4*((2^s:ℕ):ℝ)^6*C (j+1)*B j) b) := by
+  apply (F.envelope_energy T n g hT s center globalShift
+    (BinaryWindowLabels.allowed T A s (b+1)) d incoming).trans
+  exact mul_le_mul_of_nonneg_left
+    (binaryAnchorAverage_energy T n g s center globalShift A shift hplat B C hB hC b d hcap incoming)
+    (by positivity)
+
+end OccupiedParityTree
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+namespace OccupiedParityTree
+
+/-- The actual full-tree leaf address of a bottom edge, including the edge's
+own numeric label. -/
+def Edge.bottomAddress (T : ℕ → ℕ) : {b : ℕ} → {t : OccupiedParityTree b} →
+    (x : t.Labels T) → (e : t.Edge) → e.height=0 → WindowLeaf T b
+  | _, .leaf, _, e, _ => e.elim
+  | b+1, .node k children roles, x, Sum.inl i, he => by
+      change b=0 at he
+      subst b
+      exact (x (Sum.inl i),(⟨0,by decide⟩ : Fin 1))
+  | b+1, .node k children roles, x, Sum.inr z, he =>
+      (x (Sum.inl z.1), Edge.bottomAddress T (x.child T z.1) z.2 he)
+
+theorem Edge.bottomAddress_mem (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b}
+    (x : t.Labels T) (e : t.Edge) (he : e.height=0) :
+    e.bottomAddress T x he ∈ t.labelledLeaves T x := by
+  induction t with
+  | leaf => exact e.elim
+  | @node b k children roles ih =>
+    cases e with
+    | inl i =>
+      change b=0 at he
+      subst b
+      apply Finset.mem_biUnion.mpr
+      refine ⟨i,Finset.mem_univ i,?_⟩
+      apply Finset.mem_image.mpr
+      refine ⟨(⟨0,by decide⟩ : Fin 1),?_,rfl⟩
+      have hz (u : OccupiedParityTree 0) (y : u.Labels T) :
+          u.labelledLeaves T y = {(⟨0,by decide⟩ : Fin 1)} := by cases u; rfl
+      rw [hz]
+      exact Finset.mem_singleton_self _
+    | inr z =>
+      apply Finset.mem_biUnion.mpr
+      refine ⟨z.1,Finset.mem_univ _,?_⟩
+      exact Finset.mem_image.mpr ⟨z.2.bottomAddress T (x.child T z.1) he, ih z.1 _ _ he,rfl⟩
+
+/-- Leaf constraints read from the real shape label address are exactly the
+ones in the genuine complete-tree latent law. -/
+theorem Edge.bottomIndicator_eq (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (b : ℕ) (t : OccupiedParityTree (b+1)) (x : t.Labels T)
+    (e : t.Edge) (he : e.height=0)
+    (bits : WindowAssignments T n g Bool (b+1)) (incoming : ℤ) :
+    sourceBottomIndicator T n g s center globalShift b bits incoming (e.bottomAddress T x he) =
+      motifIndicator s ((x e).val-sourceTreeAnchors T n g center globalShift (b+1) bits incoming
+        (e.parentPath T x)) := by
+  induction b generalizing incoming with
+  | zero =>
+    cases t with
+    | node k children roles =>
+      cases e with
+      | inl i =>
+        change motifIndicator s _ = motifIndicator s _
+        congr 1
+        change ((x (Sum.inl i)).val:ℤ)-center 0+(∑i,(bitIndex (bits.1 i):ℤ))+incoming =
+          ((x (Sum.inl i)).val:ℤ)-(center 0-(∑i,(bitIndex (bits.1 i):ℤ))-incoming)
+        omega
+      | inr z => have hh := z.2.height_lt; omega
+  | succ b ih =>
+    cases t with
+    | node k children roles =>
+      cases e with
+      | inl i => change b+1=0 at he; omega
+      | inr z =>
+        change sourceBottomIndicator T n g s center globalShift b
+          (bits.2.2 (x (Sum.inl z.1))) (globalShift (b+1) bits.2.1 (x (Sum.inl z.1)))
+          (z.2.bottomAddress T (x.child T z.1) he) = motifIndicator s ((x.child T z.1 z.2).val-
+          sourceTreeAnchors T n g center globalShift (b+1+1) bits incoming
+            ((x (Sum.inl z.1)).val::z.2.parentPath T (x.child T z.1)))
+        rw [sourceTreeAnchors_child T n g center globalShift (b+1) bits incoming
+          (x (Sum.inl z.1)) (z.2.parentPath T (x.child T z.1))]
+        exact ih _ _ _ he _ _
+
+end OccupiedParityTree
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+open FiniteLaw
+
+/-- Taking repeated constraints from a zero-one product can only enlarge it.
+The map need not be injective. This keeps the real leaf addresses, including
+coincident labels in the ambient enlarged summation. -/
+theorem product_zero_one_pull_le {α β : Type*} [Fintype β]
+    (S : Finset α) (f : α → ℝ) (hf : ∀a∈S,f a=0 ∨ f a=1)
+    (g : β → α) (hg : ∀b,g b∈S) : (∏a∈S,f a)≤∏b,f (g b) := by
+  rcases prod_zero_one S f hf with hz | ho
+  · rw [hz]
+    exact Finset.prod_nonneg (fun b _ => by rcases hf (g b) (hg b) with h|h <;> simp [h])
+  · have h (b : β) : f (g b)=1 := by
+      rcases hf (g b) (hg b) with h|h
+      · have hz := Finset.prod_eq_zero (hg b) h
+        rw [ho] at hz
+        norm_num at hz
+      · exact h
+    rw [ho]
+    simp only [h,Finset.prod_const_one,le_refl]
+
+/-- Every actual role coefficient is bounded by its actual forced shape
+and extra-leaf envelope before averaging over the single whole latent law. -/
+theorem sourceRoleTerm_envelope (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (b : ℕ) (bits : WindowAssignments T n g Bool (b+1)) (incoming : ℤ)
+    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S)
+    (F : (occupiedWindowTerm T b S a).Forced) :
+    |sourceRoleTerm T n g s center globalShift b bits incoming S a|≤
+      F.envelope T n g s center globalShift (occupiedTermLabels T b S a) bits incoming := by
+  let x := occupiedTermLabels T b S a
+  let f := sourceBottomIndicator T n g s center globalShift b bits incoming
+  let addr : F.ExtraLeafEdge → WindowLeaf T (b+1) :=
+    fun e => e.val.val.bottomAddress T x e.property
+  have hm (e : F.ExtraLeafEdge) : addr e∈S := by
+    rw [←occupiedTermLabels_leaves T b S a]
+    exact e.val.val.bottomAddress_mem T x e.property
+  have he (e : F.ExtraLeafEdge) : f (addr e)=motifIndicator s ((x e.val.val).val-
+      sourceTreeAnchors T n g center globalShift (b+1) bits incoming (e.val.val.parentPath T x)) :=
+    e.val.val.bottomIndicator_eq T n g s center globalShift b _ x e.property bits incoming
+  have hle := product_zero_one_pull_le S f (fun v hv =>
+    sourceBottomIndicator_zero_one T n g s center globalShift b bits incoming v) addr hm
+  simp_rw [he] at hle
+  apply le_mul_of_zero_one
+    (binaryAnchorKernel_zero_one T n g s center globalShift _ _ bits incoming)
+    (prod_zero_one Finset.univ _ (fun e _ => by unfold motifIndicator; split_ifs <;> simp))
+  · simpa only [occupiedTermLabels_forced] using
+      sourceRoleTerm_forced_anchor T n g s center globalShift b bits incoming S a F
+  · exact (sourceRoleTerm_all_leaf_constraints T n g s center globalShift b bits incoming S a).trans hle
+
+/-- The averaged coefficient uses one exact metadata fiber. No fit event,
+plateau event or leaf-emission observation is conditioned on here. -/
+theorem sourceRoleAverage_envelope (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ)
+    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S)
+    (F : (occupiedWindowTerm T b S a).Forced) :
+    |sourceRoleAverage T n g s center globalShift b d incoming S a|≤
+      F.averagedEnvelope T n g s center globalShift (occupiedTermLabels T b S a) d incoming := by
+  apply ((windowPosteriorLaw T n g (b+1) d).abs_expect_le _).trans
+  exact FiniteLaw.expect_mono _ (fun bits =>
+    sourceRoleTerm_envelope T n g s center globalShift b bits incoming S a F)
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+open FiniteLaw
+open OccupiedParityTree
+
+theorem occupiedWindowTerm_leafCount (T : ℕ → ℕ) (b : ℕ)
+    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S) :
+    (occupiedWindowTerm T b S a).leafCount=S.card := by
+  rw [←occupiedWindowTerm_eq T 0 (fun _ => 0) b [] S a]
+  exact occupiedWindowShape_leafCount T 0 (fun _ => 0) b [] S _
+
+theorem occupiedWindowTerm_occupied (T : ℕ → ℕ) (b : ℕ)
+    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S) (hS : S.Nonempty) :
+    (occupiedWindowTerm T b S a).occupied := by
+  rw [←occupiedWindowTerm_eq T 0 (fun _ => 0) b [] S a]
+  exact occupiedWindowShape_occupied T 0 (fun _ => 0) b [] S hS _
+
+/-- Only the unlabeled ordered role shape is selected here; no child label
+or hidden orientation enters the forced choice. -/
+abbrev DegreeForcedShape (b r : ℕ) :=
+  {t : OccupiedParityTree b // t.occupied ∧ t.leafCount=r ∧ Nonempty t.Forced}
+
+instance degreeForcedShapeFinite (b r : ℕ) : Finite (DegreeForcedShape b r) := by
+  let := finite_bounded_occupied_shapes b r
+  exact Finite.of_injective
+    (fun t : DegreeForcedShape b r =>
+      (⟨t.val,t.property.1,t.property.2.1.le⟩ : {t : OccupiedParityTree b // t.occupied ∧ t.leafCount≤r}))
+    (fun _ _ h => Subtype.ext (congrArg
+      (fun q : {t : OccupiedParityTree b // t.occupied ∧ t.leafCount≤r} => q.val) h))
+
+noncomputable instance degreeForcedShapeFintype (b r : ℕ) : Fintype (DegreeForcedShape b r) :=
+  Fintype.ofFinite _
+
+def DegreeForcedShape.forced {b r : ℕ} (t : DegreeForcedShape b r) : t.val.Forced :=
+  Classical.choice t.property.2.2
+
+theorem card_degreeForcedShape_le (b r : ℕ) :
+    Fintype.card (DegreeForcedShape b r) ≤ shapeRoleConstant r b := by
+  let f : DegreeForcedShape b r → FiniteRoleShape r b := fun t =>
+    Classical.choose (exists_finiteRoleShape t.val r t.property.1 t.property.2.1.le)
+  apply Fintype.card_le_of_injective f
+  intro t u he
+  apply Subtype.ext
+  have ht := Classical.choose_spec (exists_finiteRoleShape t.val r t.property.1 t.property.2.1.le)
+  have hu := Classical.choose_spec (exists_finiteRoleShape u.val r u.property.1 u.property.2.1.le)
+  exact ht.symm.trans ((congrArg (FiniteRoleShape.toTree r b) he).trans hu)
+
+abbrev SourceRoleIndex (T : ℕ → ℕ) (b : ℕ) :=
+  (S : Finset (WindowLeaf T (b+1))) × OccupiedWindowChoice T b S
+
+abbrev DegreeForcedRole (T : ℕ → ℕ) (b r : ℕ) :=
+  {w : SourceRoleIndex T b // w.1.card=r ∧ Nonempty (occupiedWindowTerm T b w.1 w.2).Forced}
+
+def DegreeForcedRole.shape (T : ℕ → ℕ) (b r : ℕ) (w : DegreeForcedRole T b r) :
+    DegreeForcedShape (b+1) r := by
+  refine ⟨occupiedWindowTerm T b w.val.1 w.val.2,?_,?_,w.property.2⟩
+  · apply occupiedWindowTerm_occupied
+    apply Finset.card_pos.mp
+    obtain ⟨F⟩ := w.property.2
+    have hh := F.leaf_lower
+    rw [occupiedWindowTerm_leafCount] at hh
+    exact (Nat.pow_pos (by decide : (0:ℕ)<2)).trans_le hh
+  · exact (occupiedWindowTerm_leafCount T b w.val.1 w.val.2).trans w.property.1
+
+def DegreeForcedRole.encoding (T : ℕ → ℕ) (b r : ℕ) (w : DegreeForcedRole T b r) :
+    (t : DegreeForcedShape (b+1) r) × t.val.Labels T :=
+  ⟨w.shape T b r,occupiedTermLabels T b w.val.1 w.val.2⟩
+
+theorem DegreeForcedRole.encoding_injective (T : ℕ → ℕ) (b r : ℕ) :
+    Function.Injective (DegreeForcedRole.encoding T b r) := by
+  intro w v he
+  apply Subtype.ext
+  apply occupiedTermEncoding_injective T b
+  exact congrArg (fun z : (t : DegreeForcedShape (b+1) r) × t.val.Labels T =>
+    (⟨z.1.val,z.2⟩ : (t : OccupiedParityTree (b+1)) × t.Labels T)) he
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+open FiniteLaw
+open OccupiedParityTree
+
+/-- A degree-r sum over the genuine pairs (selected leaves, roles),
+restricted only by deterministic allowed forced labels. -/
+def degreeForcedRoleEnergy (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (A : (b : ℕ) → Finset (Fin (T b))) (b r : ℕ)
+    (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ) : ℝ :=
+  ∑w : DegreeForcedRole T b r,
+    if BinaryWindowLabels.allowed T A s (b+1)
+      ((w.shape T b r).forced.labels T (occupiedTermLabels T b w.val.1 w.val.2)) then
+      (sourceRoleAverage T n g s center globalShift b d incoming w.val.1 w.val.2)^2 else 0
+
+/-- The actual coefficient indices inject into shape+numeric-labels. Thus
+summing the enlarged envelope really bounds the original coefficient sum,
+with no identification of different selected monomials. -/
+theorem degreeForcedRoleEnergy_le_envelopes (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (A : (b : ℕ) → Finset (Fin (T b))) (b r : ℕ)
+    (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ) :
+    degreeForcedRoleEnergy T n g s center globalShift A b r d incoming ≤
+      ∑t : DegreeForcedShape (b+1) r, ∑x : t.val.Labels T,
+        if BinaryWindowLabels.allowed T A s (b+1) (t.forced.labels T x) then
+          (t.forced.averagedEnvelope T n g s center globalShift x d incoming)^2 else 0 := by
+  let f := fun z : (t : DegreeForcedShape (b+1) r) × t.val.Labels T =>
+    if BinaryWindowLabels.allowed T A s (b+1) (z.1.forced.labels T z.2) then
+      (z.1.forced.averagedEnvelope T n g s center globalShift z.2 d incoming)^2 else 0
+  have hf : ∀z,0≤f z := by intro z; dsimp [f]; split_ifs <;> positivity
+  calc
+    _ ≤ ∑w : DegreeForcedRole T b r,f (w.encoding T b r) := by
+      apply Finset.sum_le_sum
+      intro w hw
+      dsimp only [f,DegreeForcedRole.encoding]
+      by_cases h : BinaryWindowLabels.allowed T A s (b+1)
+          ((w.shape T b r).forced.labels T (occupiedTermLabels T b w.val.1 w.val.2))
+      · simp only [ite_eq_left h]
+        have hh := sourceRoleAverage_envelope T n g s center globalShift b d incoming
+          w.val.1 w.val.2 (w.shape T b r).forced
+        rw [←sq_abs (sourceRoleAverage T n g s center globalShift b d incoming w.val.1 w.val.2)]
+        exact (sq_le_sq₀ (abs_nonneg _) ((abs_nonneg _).trans hh)).mpr hh
+      · simp only [ite_eq_right h,le_refl]
+    _ ≤ ∑z,f z := by
+      simpa only [ite_true] using sum_injective_profile_le
+        (DegreeForcedRole.encoding T b r) (DegreeForcedRole.encoding_injective T b r)
+        (fun _ => True) (fun _ => trivial) f hf
+    _ = _ := Fintype.sum_sigma _
+
+/-- Uniform arithmetic interior-label budget of a degree-r depth-(b+1)
+shape. This includes no factor for a bottom label. -/
+def interiorLabelBudget (T : ℕ → ℕ) (b r : ℕ) : ℝ :=
+  ((∏j : Fin (b+1),if j.val=0 then 1 else T j.val : ℕ):ℝ)^(r-2^(b+1))
+
+theorem degreeForcedRoleEnergy_bound (T n g : ℕ → ℕ) (hT : ∀j,0<T j) (s : ℕ)
+    (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (A : (b : ℕ) → Finset (Fin (T b))) (shift : ℕ → ℤ)
+    (hplat : HasGlobalPlateau T g A shift globalShift)
+    (B C : ℕ → ℝ) (hB : ∀b,0≤B b) (hC : ∀b,0≤C b)
+    (b r : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1))
+    (hcap : MetadataBinomialCaps T n g B C (b+1) d) (incoming : ℤ) :
+    degreeForcedRoleEnergy T n g s center globalShift A b r d incoming ≤
+      (shapeRoleConstant r (b+1):ℝ) * interiorLabelBudget T b r *
+        ((2^s:ℕ):ℝ)^(2*(r-2^(b+1))) *
+        ((2*((2^s:ℕ):ℝ)^3*B b) *
+          binaryTreeBudget (fun j => 4*((2^s:ℕ):ℝ)^6*C (j+1)*B j) b) := by
+  let K := interiorLabelBudget T b r * ((2^s:ℕ):ℝ)^(2*(r-2^(b+1))) *
+    ((2*((2^s:ℕ):ℝ)^3*B b) * binaryTreeBudget (fun j => 4*((2^s:ℕ):ℝ)^6*C (j+1)*B j) b)
+  have hbud : 0≤binaryTreeBudget (fun j => 4*((2^s:ℕ):ℝ)^6*C (j+1)*B j) b := by
+    exact binaryTreeBudget_nonneg _ (fun j =>
+      mul_nonneg (mul_nonneg (by positivity) (hC (j+1))) (hB j)) b
+  have hb := hB b
+  have hK : 0≤K := by dsimp [K,interiorLabelBudget]; positivity
+  calc
+    _ ≤ ∑t : DegreeForcedShape (b+1) r, ∑x : t.val.Labels T,
+        if BinaryWindowLabels.allowed T A s (b+1) (t.forced.labels T x) then
+          (t.forced.averagedEnvelope T n g s center globalShift x d incoming)^2 else 0 :=
+      degreeForcedRoleEnergy_le_envelopes T n g s center globalShift A b r d incoming
+    _ ≤ ∑_t : DegreeForcedShape (b+1) r,K := by
+      apply Finset.sum_le_sum
+      intro t ht
+      apply (t.forced.envelope_energy_root T n g (hT 0) s center globalShift A shift hplat
+        B C hB hC b t.val d hcap incoming).trans
+      have hi := t.forced.card_interiorLabels_le T hT t.property.1
+      rw [t.property.2.1] at hi
+      have hi' : (Fintype.card (t.forced.InteriorLabels T):ℝ) ≤ interiorLabelBudget T b r := by
+        dsimp [interiorLabelBudget]
+        exact_mod_cast hi
+      rw [t.forced.card_extraLeafEdge,t.property.2.1]
+      exact mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hi' (by positivity)) (by positivity)
+    _ = (Fintype.card (DegreeForcedShape (b+1) r):ℝ)*K := by
+      rw [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
+    _ ≤ (shapeRoleConstant r (b+1):ℝ)*K := by
+      apply mul_le_mul_of_nonneg_right _ hK
+      exact_mod_cast card_degreeForcedShape_le (b+1) r
+    _ = _ := by dsimp [K]; ring
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- Total role multiplicity for one actual selected monomial. Every level
+has at most the number of selected leaves; no ambient-label factor occurs. -/
+theorem occupiedWindowChoice_card_le (T : ℕ → ℕ) (b : ℕ)
+    (S : Finset (WindowLeaf T (b+1))) :
+    Fintype.card (OccupiedWindowChoice T b S) ≤ 3^((b+1)*S.card) := by
+  induction b with
+  | zero =>
+    change Fintype.card (Fin S.card → Fin 3) ≤ _
+    simp
+  | succ b ih =>
+    change Fintype.card (((i : Fin (selectedChildren S).card) →
+      OccupiedWindowChoice T b (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val)) ×
+      (Fin (selectedChildren S).card → Fin 3)) ≤ _
+    rw [Fintype.card_prod,Fintype.card_pi,Fintype.card_fun,Fintype.card_fin,Fintype.card_fin]
+    have hprod := Finset.prod_le_prod (s:=Finset.univ) (fun i _ =>
+      ih (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val))
+    have hs : (∑i : Fin (selectedChildren S).card,
+        (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val).card)=S.card :=
+      (selected_sum_index (selectedChildren S) (fun h => (selectedChildLeaves S h).card)).trans
+        (selected_children_card S)
+    have hc : (selectedChildren S).card≤S.card := Finset.card_image_le
+    calc
+      _ ≤ (∏i : Fin (selectedChildren S).card,
+          3^((b+1)*(selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val).card)) *
+          3^(selectedChildren S).card := Nat.mul_le_mul_right _ hprod
+      _ = 3^((b+1)*S.card+(selectedChildren S).card) := by
+        rw [Finset.prod_pow_eq_pow_sum,←Finset.mul_sum,hs,←pow_add]
+      _ ≤ _ := Nat.pow_le_pow_right (by decide) (by simp only [Nat.add_mul,Nat.one_mul]; omega)
+
+abbrev DegreeLeafSubset (T : ℕ → ℕ) (b r : ℕ) :=
+  {S : Finset (WindowLeaf T (b+1)) // S.card=r}
+
+theorem degreeLeafSubset_card (T : ℕ → ℕ) (b r : ℕ) :
+    Fintype.card (DegreeLeafSubset T b r)=(Fintype.card (WindowLeaf T (b+1))).choose r := by
+  have h := Finset.card_powersetCard r (Finset.univ : Finset (WindowLeaf T (b+1)))
+  have he : (Finset.univ.powersetCard r : Finset (Finset (WindowLeaf T (b+1))))=(Finset.univ.filter (fun S : Finset (WindowLeaf T (b+1)) => S.card=r)) := by
+    ext S
+    simp only [Finset.mem_powersetCard,Finset.subset_univ,true_and,Finset.mem_filter,Finset.mem_univ]
+  rw [he,Finset.card_univ] at h
+  rw [Fintype.card_subtype]
+  exact h
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+open FiniteLaw
+
+/-- All actual motif support lies in the chosen central interval at every
+vertex of the full window tree. This is a predicate of the complete bits,
+not an event on which any posterior law is conditioned. -/
+def SourceCentral (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (A : (b : ℕ) → Finset (Fin (T b))) :
+    (b : ℕ) → WindowAssignments T n g Bool b → ℤ → Prop
+  | 0, _, _ => True
+  | b+1, bits, incoming =>
+      (∀h : Fin (T b), motifIndicator s ((h.val:ℤ)-center b+
+        (∑i,(bitIndex (bits.1 i):ℤ))+incoming)≠0 → h∈A b) ∧
+      ∀h, SourceCentral T n g s center globalShift A b (bits.2.2 h) (globalShift b bits.2.1 h)
+
+def BinaryWindowLabels.central (T : ℕ → ℕ) (A : (b : ℕ) → Finset (Fin (T b))) :
+    (b : ℕ) → BinaryWindowLabels T b → Prop
+  | 0, _ => True
+  | b+1, p => p.1∈A b ∧ p.2.1∈A b ∧ central T A b p.2.2.1 ∧ central T A b p.2.2.2
+
+theorem BinaryWindowLabels.allowed_of_central (T : ℕ → ℕ)
+    (A : (b : ℕ) → Finset (Fin (T b))) (s b : ℕ) (p : BinaryWindowLabels T b)
+    (hc : central T A b p) (hd : distinct T b p) (hs : close T s b p) :
+    allowed T A s b p := by
+  induction b with
+  | zero => trivial
+  | succ b ih =>
+    exact ⟨⟨hc.1,hc.2.1,hd.1,by simpa only [abs_sub_comm] using hs.1⟩,
+      ih _ hc.2.2.1 hd.2.1 hs.2.1,ih _ hc.2.2.2 hd.2.2 hs.2.2⟩
+
+theorem sourceRoleTerm_forced_central (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (A : (b : ℕ) → Finset (Fin (T b)))
+    (b : ℕ) (bits : WindowAssignments T n g Bool (b+1)) (incoming : ℤ)
+    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S)
+    (F : (occupiedWindowTerm T b S a).Forced)
+    (hc : SourceCentral T n g s center globalShift A (b+1) bits incoming)
+    (hz : sourceRoleTerm T n g s center globalShift b bits incoming S a≠0) :
+    BinaryWindowLabels.central T A (b+1) (forcedWindowLabels T b S a F) := by
+  induction b generalizing incoming with
+  | zero =>
+    cases F with
+    | node i j hne hi hj left right =>
+      rw [sourceRoleTerm_zero] at hz
+      change (S.equivFin.symm i).val.1 ∈ A 0 ∧ (S.equivFin.symm j).val.1 ∈ A 0 ∧ True ∧ True
+      refine ⟨hc.1 _ ?_,hc.1 _ ?_,trivial,trivial⟩
+      · convert nonzero_of_abs_le hz (occupiedRootWeight_bottom_anchor T s S a (center 0-(∑k,(bitIndex (bits.1 k):ℤ))-incoming) i) using 1
+        congr 1; ring
+      · convert nonzero_of_abs_le hz (occupiedRootWeight_bottom_anchor T s S a (center 0-(∑k,(bitIndex (bits.1 k):ℤ))-incoming) j) using 1
+        congr 1; ring
+  | succ b ih =>
+    cases F with
+    | node i j hne hi hj left right =>
+      rw [sourceRoleTerm_succ] at hz
+      have hm := mul_ne_zero_iff.mp hz
+      have hp := Finset.prod_ne_zero_iff.mp hm.2
+      change ((selectedChildren S).equivFin.symm i).val ∈ A (b+1) ∧
+        ((selectedChildren S).equivFin.symm j).val ∈ A (b+1) ∧
+        BinaryWindowLabels.central T A (b+1)
+          (forcedWindowLabels T b (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val)
+            (a.1 i) left) ∧
+        BinaryWindowLabels.central T A (b+1)
+          (forcedWindowLabels T b (selectedChildLeaves S ((selectedChildren S).equivFin.symm j).val)
+            (a.1 j) right)
+      refine ⟨hc.1 _ ?_,hc.1 _ ?_,
+        ih _ _ _ _ left (hc.2 _) (hp i (Finset.mem_univ i)),
+        ih _ _ _ _ right (hc.2 _) (hp j (Finset.mem_univ j))⟩
+      · convert nonzero_of_abs_le hm.1 (occupiedRootWeight_internal_anchor T s b S a _ i hi) using 1
+        congr 1; ring
+      · convert nonzero_of_abs_le hm.1 (occupiedRootWeight_internal_anchor T s b S a _ j hj) using 1
+        congr 1; ring
+
+/-- The discarded noncentral role contributions are bounded by the actual
+conditional probability of a failed full-tree layout. Typical metadata may
+bound this mass; the averaging distribution itself remains unconditional. -/
+def sourceNoncentralMass (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (A : (b : ℕ) → Finset (Fin (T b))) (b : ℕ)
+    (d : WindowAssignments T n g (Option Bool) b) (incoming : ℤ) : ℝ :=
+  (windowPosteriorLaw T n g b d).expect
+    (fun bits => if SourceCentral T n g s center globalShift A b bits incoming then 0 else 1)
+
+theorem sourceNoncentralMass_nonneg (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (A : (b : ℕ) → Finset (Fin (T b))) (b : ℕ)
+    (d : WindowAssignments T n g (Option Bool) b) (incoming : ℤ) :
+    0 ≤ sourceNoncentralMass T n g s center globalShift A b d incoming := by
+  exact Finset.sum_nonneg (fun bits _ => mul_nonneg
+    ((windowPosteriorLaw T n g b d).nonneg bits) (by dsimp only; split_ifs <;> norm_num))
+
+theorem sourceRoleAverage_not_allowed (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (A : (b : ℕ) → Finset (Fin (T b)))
+    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ)
+    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S)
+    (F : (occupiedWindowTerm T b S a).Forced)
+    (h : ¬BinaryWindowLabels.allowed T A s (b+1) (forcedWindowLabels T b S a F)) :
+    |sourceRoleAverage T n g s center globalShift b d incoming S a|≤
+      sourceNoncentralMass T n g s center globalShift A (b+1) d incoming := by
+  apply ((windowPosteriorLaw T n g (b+1) d).abs_expect_le _).trans
+  apply FiniteLaw.expect_mono
+  intro bits
+  by_cases hc : SourceCentral T n g s center globalShift A (b+1) bits incoming
+  · rw [ite_eq_left hc]
+    have hz : sourceRoleTerm T n g s center globalShift b bits incoming S a=0 := by
+      by_contra hn
+      exact h (BinaryWindowLabels.allowed_of_central T A s (b+1) _
+        (sourceRoleTerm_forced_central T n g s center globalShift A b bits incoming S a F hc hn)
+        (forcedWindowLabels_distinct T b S a F)
+        (sourceRoleTerm_forced_close T n g s center globalShift b bits incoming S a F hn))
+    rw [hz,abs_zero]
+  · rw [ite_eq_right hc]
+    exact occupiedWindowWeight_abs_le_one T s _ b [] S a
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+open FiniteLaw
+
+/-- Extension by zero commutes with the finite subtype sum, including an
+empty subtype and predicates depending on dependent role indices. -/
+theorem sum_subtype_zero_extend {α : Type*} [Fintype α] (p : α → Prop) [DecidablePred p]
+    (f : {x // p x} → ℝ) :
+    (∑x, if h : p x then f ⟨x,h⟩ else 0) = ∑x,f x := by
+  have h := Fintype.sum_subtype_add_sum_subtype p
+    (fun x => if h : p x then f ⟨x,h⟩ else 0)
+  calc
+    _ = (∑x : {x // p x}, if h : p x.val then f ⟨x.val,h⟩ else 0) +
+        ∑x : {x // ¬p x}, if h : p x.val then f ⟨x.val,h⟩ else 0 := h.symm
+    _ = (∑x,f x)+0 := congrArg₂ (·+·)
+      (Finset.sum_congr rfl (fun x _ => dite_eq_left x.property))
+      (Finset.sum_eq_zero (fun x _ => dite_eq_right x.property))
+    _ = _ := add_zero _
+
+/-- The coefficient is defined directly from the two literal leaf-type
+patterns under the common complete posterior bit tree. -/
+def sourceCoefficient (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ)
+    (S : Finset (WindowLeaf T (b+1))) : ℝ :=
+  (windowPosteriorLaw T n g (b+1) d).expect (fun bits =>
+    ∏v∈S,(1-sourceTreeLeafSign T n g s center globalShift (b+1) bits incoming 1 v)/2) -
+  (windowPosteriorLaw T n g (b+1) d).expect (fun bits =>
+    ∏v∈S,(1-sourceTreeLeafSign T n g s center globalShift (b+1) bits incoming (-1) v)/2)
+
+theorem sourceCoefficient_expansion (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ)
+    (S : Finset (WindowLeaf T (b+1))) :
+    sourceCoefficient T n g s center globalShift b d incoming S =
+      2*∑a : OccupiedWindowChoice T b S,
+        if (occupiedWindowTerm T b S a).isOdd then
+          sourceRoleAverage T n g s center globalShift b d incoming S a else 0 := by
+  unfold sourceCoefficient
+  simp_rw [sourceTreeLeafSign_eq]
+  exact occupiedWindow_averaged_difference _ T s _ b [] S
+
+def sourceGoodRoleSquare (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (A : (b : ℕ) → Finset (Fin (T b))) (b r : ℕ)
+    (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ)
+    (w : SourceRoleIndex T b) : ℝ :=
+  if h : w.1.card=r ∧ Nonempty (occupiedWindowTerm T b w.1 w.2).Forced then
+    let u : DegreeForcedRole T b r := ⟨w,h⟩
+    if BinaryWindowLabels.allowed T A s (b+1)
+      ((u.shape T b r).forced.labels T (occupiedTermLabels T b w.1 w.2)) then
+      (sourceRoleAverage T n g s center globalShift b d incoming w.1 w.2)^2 else 0
+  else 0
+
+theorem sourceGoodRoleSquare_nonneg (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (A : (b : ℕ) → Finset (Fin (T b))) (b r : ℕ)
+    (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ)
+    (w : SourceRoleIndex T b) :
+    0 ≤ sourceGoodRoleSquare T n g s center globalShift A b r d incoming w := by
+  unfold sourceGoodRoleSquare
+  split_ifs <;> positivity
+
+theorem sourceGoodRoleSquare_sum (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (A : (b : ℕ) → Finset (Fin (T b))) (b r : ℕ)
+    (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ) :
+    (∑w : SourceRoleIndex T b, sourceGoodRoleSquare T n g s center globalShift A b r d incoming w)=
+      degreeForcedRoleEnergy T n g s center globalShift A b r d incoming := by
+  unfold sourceGoodRoleSquare degreeForcedRoleEnergy
+  exact sum_subtype_zero_extend
+    (fun w : SourceRoleIndex T b => w.1.card=r ∧ Nonempty (occupiedWindowTerm T b w.1 w.2).Forced)
+    (fun w : DegreeForcedRole T b r => if BinaryWindowLabels.allowed T A s (b+1)
+      ((w.shape T b r).forced.labels T (occupiedTermLabels T b w.val.1 w.val.2)) then
+      (sourceRoleAverage T n g s center globalShift b d incoming w.val.1 w.val.2)^2 else 0)
+
+/-- Every actual odd role term is either smoothed at an actual singleton,
+discarded by the literal central-layout failure event, or retained in the
+forced-tree sum. There is no unproved coefficient-bound assumption. -/
+theorem sourceRoleSquare_split (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (A : (b : ℕ) → Finset (Fin (T b))) (b r : ℕ)
+    (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ)
+    (S : Finset (WindowLeaf T (b+1))) (hS : S.card=r) (a : OccupiedWindowChoice T b S)
+    (η : ℝ) (hcap : MetadataSmoothingCap T n g s η (b+1) d) :
+    (if (occupiedWindowTerm T b S a).isOdd then
+      sourceRoleAverage T n g s center globalShift b d incoming S a else 0)^2 ≤
+      η^2+(sourceNoncentralMass T n g s center globalShift A (b+1) d incoming)^2+
+        sourceGoodRoleSquare T n g s center globalShift A b r d incoming ⟨S,a⟩ := by
+  have hg := sourceGoodRoleSquare_nonneg T n g s center globalShift A b r d incoming ⟨S,a⟩
+  by_cases ho : (occupiedWindowTerm T b S a).isOdd
+  swap
+  · rw [ite_eq_right ho]; nlinarith [sq_nonneg η, sq_nonneg (sourceNoncentralMass T n g s center globalShift A (b+1) d incoming)]
+  rw [ite_eq_left ho]
+  by_cases hz : sourceRoleAverage T n g s center globalShift b d incoming S a=0
+  · rw [hz]; nlinarith [sq_nonneg η, sq_nonneg (sourceNoncentralMass T n g s center globalShift A (b+1) d incoming)]
+  by_cases hn : (occupiedWindowTerm T b S a).noSingleton
+  swap
+  · have hs := sourceRoleAverage_singleton T n g s center globalShift b d incoming S a η hcap hn
+    have hs2 := (sq_le_sq₀ (abs_nonneg _) ((abs_nonneg _).trans hs)).mpr hs
+    rw [sq_abs] at hs2
+    nlinarith [sq_nonneg (sourceNoncentralMass T n g s center globalShift A (b+1) d incoming)]
+  have hf := OccupiedParityTree.forced_nonempty (occupiedWindowTerm T b S a)
+    (sourceRoleAverage_valid T n g s center globalShift b d incoming S a hz) hn ho
+  let w : DegreeForcedRole T b r := ⟨⟨S,a⟩,hS,hf⟩
+  let F : (occupiedWindowTerm T b S a).Forced := (w.shape T b r).forced
+  change _ ≤ _+ (if h : S.card=r ∧ Nonempty (occupiedWindowTerm T b S a).Forced then _ else 0)
+  rw [dite_eq_left ⟨hS,hf⟩]
+  change _ ≤ _ + (if BinaryWindowLabels.allowed T A s (b+1)
+    (F.labels T (occupiedTermLabels T b S a)) then _ else 0)
+  rw [occupiedTermLabels_forced T b S a F]
+  by_cases ha : BinaryWindowLabels.allowed T A s (b+1) (forcedWindowLabels T b S a F)
+  · rw [ite_eq_left ha]; nlinarith [sq_nonneg η,
+      sq_nonneg (sourceNoncentralMass T n g s center globalShift A (b+1) d incoming)]
+  · rw [ite_eq_right ha]
+    have hs := sourceRoleAverage_not_allowed T n g s center globalShift A b d incoming S a F ha
+    have hs2 := (sq_le_sq₀ (abs_nonneg _) (sourceNoncentralMass_nonneg T n g s center globalShift A (b+1) d incoming)).mpr hs
+    rw [sq_abs] at hs2
+    nlinarith [sq_nonneg η]
+
+end TraceReconstruction
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+open FiniteLaw
+
+theorem sourceGoodRoleSquare_degree_sum (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (A : (b : ℕ) → Finset (Fin (T b))) (b r : ℕ)
+    (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ) :
+    (∑S : DegreeLeafSubset T b r, ∑a : OccupiedWindowChoice T b S.val,
+      sourceGoodRoleSquare T n g s center globalShift A b r d incoming ⟨S.val,a⟩)=
+      degreeForcedRoleEnergy T n g s center globalShift A b r d incoming := by
+  rw [←sourceGoodRoleSquare_sum,Fintype.sum_sigma]
+  symm
+  calc
+    _ = ∑S : Finset (WindowLeaf T (b+1)), if h : S.card=r then
+        ∑a : OccupiedWindowChoice T b S,
+          sourceGoodRoleSquare T n g s center globalShift A b r d incoming ⟨S,a⟩ else 0 := by
+      apply Finset.sum_congr rfl
+      intro S _
+      by_cases h : S.card=r
+      · rw [dite_eq_left h]
+      · simp [sourceGoodRoleSquare,h]
+    _ = _ := sum_subtype_zero_extend
+      (fun S : Finset (WindowLeaf T (b+1)) => S.card=r)
+      (fun S : DegreeLeafSubset T b r => ∑a : OccupiedWindowChoice T b S.val,
+        sourceGoodRoleSquare T n g s center globalShift A b r d incoming ⟨S.val,a⟩)
+
+theorem sourceCoefficient_square_bound (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (A : (b : ℕ) → Finset (Fin (T b))) (b r : ℕ)
+    (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ)
+    (S : Finset (WindowLeaf T (b+1))) (hS : S.card=r)
+    (η : ℝ) (hcap : MetadataSmoothingCap T n g s η (b+1) d) :
+    (sourceCoefficient T n g s center globalShift b d incoming S)^2 ≤
+      4*(3^((b+1)*r):ℕ)*
+        ((3^((b+1)*r):ℕ)*(η^2+
+          (sourceNoncentralMass T n g s center globalShift A (b+1) d incoming)^2)+
+          ∑a : OccupiedWindowChoice T b S,
+            sourceGoodRoleSquare T n g s center globalShift A b r d incoming ⟨S,a⟩) := by
+  let f : OccupiedWindowChoice T b S → ℝ := fun a =>
+    if (occupiedWindowTerm T b S a).isOdd then
+      sourceRoleAverage T n g s center globalShift b d incoming S a else 0
+  let M : ℝ := (3^((b+1)*r):ℕ)
+  let E : ℝ := η^2+(sourceNoncentralMass T n g s center globalShift A (b+1) d incoming)^2
+  let G := fun a => sourceGoodRoleSquare T n g s center globalShift A b r d incoming ⟨S,a⟩
+  have hE : 0 ≤ E := add_nonneg (sq_nonneg _) (sq_nonneg _)
+  have hM : 0 ≤ M := Nat.cast_nonneg _
+  have hcard : (Fintype.card (OccupiedWindowChoice T b S):ℝ) ≤ M := by
+    dsimp only [M]
+    exact_mod_cast (hS ▸ occupiedWindowChoice_card_le T b S)
+  have hcs : (∑a,f a)^2 ≤ (Fintype.card (OccupiedWindowChoice T b S):ℝ)*∑a,(f a)^2 := by
+    simpa using Finset.sum_mul_sq_le_sq_mul_sq Finset.univ (fun _ => (1:ℝ)) f
+  have hs : (∑a,(f a)^2) ≤ M*E+∑a,G a := by
+    calc
+      _ ≤ ∑a : OccupiedWindowChoice T b S,(E+G a) :=
+        Finset.sum_le_sum (fun a _ => sourceRoleSquare_split T n g s center globalShift A b r d incoming S hS a η hcap)
+      _ = (Fintype.card (OccupiedWindowChoice T b S):ℝ)*E+∑a,G a := by simp [Finset.sum_add_distrib]
+      _ ≤ M*E+∑a,G a := add_le_add (mul_le_mul_of_nonneg_right hcard hE) le_rfl
+  calc
+    _ = 4*(∑a,f a)^2 := by rw [sourceCoefficient_expansion]; dsimp only [f]; ring
+    _ ≤ 4*((Fintype.card (OccupiedWindowChoice T b S):ℝ)*∑a,(f a)^2) := by linarith
+    _ ≤ 4*(M*∑a,(f a)^2) := mul_le_mul_of_nonneg_left
+      (mul_le_mul_of_nonneg_right hcard (Finset.sum_nonneg (fun _ _ => sq_nonneg _))) (by norm_num)
+    _ ≤ 4*(M*(M*E+∑a,G a)) := mul_le_mul_of_nonneg_left
+      (mul_le_mul_of_nonneg_left hs hM) (by norm_num)
+    _ = _ := by dsimp only [M,E,G]; ring
+
+/-- The full degree coefficient sum, including every selected monomial,
+singletons, and off-center labels. This is the finite, unabsorbed version
+of the manuscript's low-degree coefficient estimate. -/
+theorem sourceCoefficient_degree_energy (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (A : (b : ℕ) → Finset (Fin (T b))) (b r : ℕ)
+    (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ)
+    (η : ℝ) (hcap : MetadataSmoothingCap T n g s η (b+1) d) :
+    (∑S : DegreeLeafSubset T b r,
+      (sourceCoefficient T n g s center globalShift b d incoming S.val)^2) ≤
+      4*(3^((b+1)*r):ℕ)*
+        (((Fintype.card (WindowLeaf T (b+1))).choose r:ℕ)*(3^((b+1)*r):ℕ)*
+          (η^2+(sourceNoncentralMass T n g s center globalShift A (b+1) d incoming)^2)+
+          degreeForcedRoleEnergy T n g s center globalShift A b r d incoming) := by
+  calc
+    _ ≤ ∑S : DegreeLeafSubset T b r,
+        4*(3^((b+1)*r):ℕ)*
+          ((3^((b+1)*r):ℕ)*(η^2+
+            (sourceNoncentralMass T n g s center globalShift A (b+1) d incoming)^2)+
+            ∑a : OccupiedWindowChoice T b S.val,
+              sourceGoodRoleSquare T n g s center globalShift A b r d incoming ⟨S.val,a⟩) :=
+      Finset.sum_le_sum (fun S _ => sourceCoefficient_square_bound T n g s center globalShift A b r d incoming S.val S.property η hcap)
+    _ = _ := by
+      rw [←Finset.mul_sum,Finset.sum_add_distrib,sourceGoodRoleSquare_degree_sum]
+      simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,degreeLeafSubset_card]
+      ring
+
+end TraceReconstruction
+
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/TraceReconstruction/ErrorRates.lean b/lean/OAI/Probability/TraceReconstruction/ErrorRates.lean
index f35a9e9f17881044a45750992f5cd965e052f188..fc5a6bb40d3468bb1a2882a6f04f9db4494b81a3 100644
--- a/lean/OAI/Probability/TraceReconstruction/ErrorRates.lean
+++ b/lean/OAI/Probability/TraceReconstruction/ErrorRates.lean
@@ -1,967 +1,967 @@
-import OAI.Probability.TraceReconstruction.LayoutRates
-
-namespace OAI
-
-/-!
-# ErrorRates
-
-Realization of block caps, sparse scales, sharp and smoothing budgets, tree tails, fit errors, and uniform polynomial/exponential bounds.
--/
-
-open scoped ENNReal
-noncomputable section
-
-open Filter
-open scoped Topology
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem sourceWindowSize_le (N : ℝ) (b j : ℕ) (ε : ℝ) (hN : 0 ≤ N) :
-    (sourceWindowSize N b ε j:ℝ) ≤ N^(scaleKappa b (j+1) ε) := by
-  have hh := Nat.floor_le (div_nonneg (Real.rpow_nonneg hN (scaleKappa b (j+1) ε)) (by norm_num : (0:ℝ) ≤ 4))
-  unfold sourceWindowSize
-  push_cast
-  linarith only [hh]
-
--- We use only positive target scales below.
-theorem SourceArithmeticBounds.scheduled_trials {C N F ε : ℝ} {s b ell k : ℕ}
-    (H : SourceArithmeticBounds C N F ε s b ell k) (hN : 0 < N) (hC : 0 < C) (hF : 0 < F)
-    (j : ℕ) (hj : j+1 < b) :
-    N^(scaleKappa b (j+2) ε+ε)/(4800*(C^(2*b)*F)) ≤
-      (sourceScheduledTrials (sourceBlockWindows N b ε) (sourceArithmeticTrials C N b ε ell k) (j+1):ℝ) := by
-  let T := sourceBlockWindows N b ε
-  let n := sourceArithmeticTrials C N b ε ell k
-  have htbig := (H.windows (j+1) hj).1
-  have hsize : (T (j+1):ℝ) ≤ N^(scaleKappa b (j+2) ε) := by
-    dsimp [T]
-    rw [sourceBlockWindows,ite_eq_left hj]
-    have hh := Nat.floor_le (div_nonneg (Real.rpow_nonneg hN.le (scaleKappa b (j+2) ε)) (by norm_num : (0:ℝ) ≤ 4))
-    simp only [sourceWindowSize,Nat.cast_mul,Nat.cast_ofNat]
-    linarith only [hh]
-  have hx : 8 ≤ N^(scaleKappa b (j+2) ε) := by
-    have hh : (24:ℝ) ≤ T (j+1) := by exact_mod_cast htbig
-    linarith only [hh,hsize]
-  have hlo := (sourceWindowSize_bounds (b:=b) (j:=j+1) hx).1
-  have hn := (H.trials j (by omega)).1
-  have hdvd : 4 ∣ T (j+1) := by
-    dsimp [T]
-    rw [sourceBlockWindows,ite_eq_left hj,sourceWindowSize]
-    exact dvd_mul_right _ _
-  have htlo : N^(scaleKappa b (j+2) ε)/8 ≤ ((T (j+1)/4:ℕ):ℝ) := by
-    rw [Nat.cast_div hdvd (by norm_num : (4:ℝ)≠0)]
-    have hh : (sourceWindowSize N b ε (j+1):ℝ)=(T (j+1):ℝ) := by simp [T,sourceBlockWindows,hj]
-    rw [hh] at hlo
-    linarith only [hlo]
-  rw [sourceScheduledTrials,Nat.cast_mul,Real.rpow_add hN]
-  calc
-    _ = (N^(scaleKappa b (j+2) ε)/8)*(N^ε/(600*(C^(2*b)*F))) := by ring
-    _ ≤ _ := mul_le_mul htlo hn (by positivity) (by positivity)
-
-theorem SourceArithmeticBounds.binomial_caps {C N F ε ω : ℝ} {s b ell k : ℕ}
-    (H : SourceArithmeticBounds C N F ε s b ell k) (hN : 0 < N)
-    (hC : 1 ≤ C) (hF : 1 ≤ F) (hω : 0 < ω) :
-    (∀j<b,hiddenBinomialCap (sourceArithmeticTrials C N b ε ell k j) ω ≤
-       (200*(C^(2*b)*F)/ω)*N^(-ε/2)) ∧
-    (∀j,j+1<b →hiddenBinomialCap
-      (sourceScheduledTrials (sourceBlockWindows N b ε) (sourceArithmeticTrials C N b ε ell k) (j+1)) ω ≤
-        (200*(C^(2*b)*F)/ω)*N^(-(scaleKappa b (j+2) ε+ε)/2)) := by
-  let G := C^(2*b)*F
-  have hG : 1 ≤ G := one_le_mul_of_one_le_of_one_le (one_le_pow₀ hC) hF
-  have hcoef : 4*(4800*G) ≤ (200*G/ω)^2*ω^2 := by
-    have hident : (200*G/ω)^2*ω^2=40000*G^2 := by field_simp; ring
-    rw [hident]
-    nlinarith only [hG,sq_nonneg (G-1)]
-  constructor
-  · intro j hj
-    apply hiddenBinomialCap_rpow hN (by positivity : 0 < 600*G) (by positivity) (H.trials j hj).1
-    dsimp [G] at *
-    have hnon : 0 ≤ C^(2*b)*F := by positivity
-    linarith only [hcoef,hnon]
-  · intro j hj
-    exact hiddenBinomialCap_rpow hN (by positivity : 0 < 4800*G) (by positivity)
-      (H.scheduled_trials hN (by linarith only [hC]) (by linarith only [hF]) j hj) hcoef
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem sourceSparseCount_scale (N : ℝ) (b s : ℕ) (ε : ℝ) (hN : 0 < N) :
-    (sourceSparseCount (sourceBlockWindows N b ε) s (b-1):ℝ) ≤
-      ((2^s:ℕ):ℝ)*N^(sourceInteriorExponent b ε) := by
-  unfold sourceSparseCount sourceInteriorExponent
-  push_cast
-  apply mul_le_mul_of_nonneg_left _ (by positivity)
-  rw [Real.rpow_sum_of_pos hN]
-  apply Finset.prod_le_prod₀
-  · intro j _
-    positivity
-  · intro j hj
-    have hj' := Finset.mem_range.mp hj
-    rw [sourceBlockWindows,ite_eq_left (by omega)]
-    exact sourceWindowSize_le N b (j+1) ε hN.le
-
-theorem sourceLeafCount_succ (T : ℕ → ℕ) (j : ℕ) :
-    SourceLeafCount T (j+1)=T j*SourceLeafCount T j := by
-  exact (Fintype.card_prod (α:=Fin (T j)) (β:=WindowLeaf T j)).trans (by rw [Fintype.card_fin])
-
-theorem sourceLeafCount_le_length {P T : ℕ → ℕ} {ell b : ℕ} (hell : 1 ≤ ell)
-    (hPT : ∀j<b,T j ≤ P j) : SourceLeafCount T b ≤ sourceBlockLength P ell b := by
-  induction b with
-  | zero =>
-    change Fintype.card (Fin 1) ≤ ell
-    simpa using hell
-  | succ b ih =>
-    rw [sourceLeafCount_succ,sourceBlockLength]
-    exact Nat.mul_le_mul (hPT b (by omega)) (ih (fun j hj=>hPT j (by omega)))
-
-theorem sourceArithmeticLength_eq_block (C N : ℝ) (b : ℕ) (ε : ℝ) (ell k j : ℕ) :
-    sourceArithmeticLength C N b ε ell k j=
-      sourceBlockLength (sourceArithmeticP C N b ε ell k) ell j := by
-  induction j with
-  | zero => rfl
-  | succ j ih => rw [sourceArithmeticLength_step,sourceBlockLength,ih,mul_comm]
-
-theorem SourceArithmeticBounds.leaf_bound {C N F ε : ℝ} {s b ell k : ℕ}
-    (H : SourceArithmeticBounds C N F ε s b ell k) (hN : 0 < N) :
-    (SourceLeafCount (sourceBlockWindows N b ε) b:ℝ) ≤ N := by
-  have hell : 1 ≤ ell := by
-    have hh := (H.lengths 0 (Nat.zero_le _)).1
-    rw [show sourceArithmeticLength C N b ε ell k 0=ell by rfl] at hh
-    have hp : 0 < N^(scaleA b 0 ε)/2 := by positivity
-    have he : (0:ℝ)<ell := hp.trans_le hh
-    have he' : 0<ell := by exact_mod_cast he
-    omega
-  have hh := sourceLeafCount_le_length (P:=sourceArithmeticP C N b ε ell k) hell
-    (fun j hj=>Nat.le_trans (by omega : sourceBlockWindows N b ε j ≤ 2*sourceBlockWindows N b ε j)
-      (H.windows j hj).2.2.1)
-  have hh' : (SourceLeafCount (sourceBlockWindows N b ε) b:ℝ) ≤
-      sourceArithmeticLength C N b ε ell k b := by
-    rw [sourceArithmeticLength_eq_block]
-    exact_mod_cast hh
-  apply hh'.trans
-  simpa only [scaleA_top,Real.rpow_one] using (H.lengths b le_rfl).2
-
-/-- Full sparse degree, with the excluded bottom window and the exact
- source gap. This is the high-degree geometric gain, before constants. -/
-theorem source_sparse_weight_bound (N ε d W w : ℝ) (b s : ℕ) (hN : 0 < N) (hW : 0 ≤ W)
-    (hw : w^2 ≤ W*N^(-scaleA b 0 ε*d)) :
-    w^2*(sourceSparseCount (sourceBlockWindows N b ε) s (b-1):ℝ) ≤
-      W*((2^s:ℕ):ℝ)*N^(-(scaleA b 0 ε*d-sourceInteriorExponent b ε)) := by
-  have hr : N^(-scaleA b 0 ε*d)*N^(sourceInteriorExponent b ε)=
-      N^(-(scaleA b 0 ε*d-sourceInteriorExponent b ε)) := by
-    rw [←Real.rpow_add hN]
-    congr 1
-    ring
-  calc
-    _ ≤ (W*N^(-scaleA b 0 ε*d))*(((2^s:ℕ):ℝ)*N^(sourceInteriorExponent b ε)) :=
-      mul_le_mul hw (sourceSparseCount_scale N b s ε hN) (by positivity) (by positivity)
-    _ = (W*((2^s:ℕ):ℝ))*(N^(-scaleA b 0 ε*d)*N^(sourceInteriorExponent b ε)) := by ring
-    _ = _ := by rw [hr]
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-def sourceBadRoleFactor (b R : ℕ) : ℝ :=
-  ∑r∈Finset.range R,4*((3^((b+1)*r):ℕ):ℝ)^2
-
-def sourceForcedRoleFactor (b R : ℕ) : ℝ :=
-  ∑r∈Finset.range R,4*((3^((b+1)*r):ℕ):ℝ)*(OccupiedParityTree.shapeRoleConstant r (b+1):ℝ)
-
-/-- Explicit finite sharp energy bound. The two undesirable pattern classes
- are suppressed by eta and a, and the genuine binary forced skeleton pays
- its full source exponent. The high tail uses the sparse, not dense count. -/
-theorem source_sharp_numeric (T n : ℕ→ℕ) (s b R : ℕ) (ω η a w N ε d K W Q gap : ℝ)
-    (hN : 1 ≤ N) (hK : 0 ≤ K) (hW : 0 ≤ W) (_hQ : 0 ≤ Q)
-    (hleaves : (SourceLeafCount T (b+1):ℝ) ≤ N) (hwone : w^2 ≤ 1)
-    (hw : w^2 ≤ W*N^(-scaleA (b+1) 0 ε*d))
-    (hsmall : w^2*((2^s:ℕ):ℝ)*(sourceSparseCount T s b:ℝ) ≤ 1)
-    (hhalf : w^2*(sourceSparseCount T s b:ℝ) ≤ 1/2)
-    (hsparse : w^2*(sourceSparseCount T s b:ℝ) ≤ Q*N^(-gap))
-    (hn : ∀j≤b,hiddenBinomialCap (n j) ω ≤ K*N^(-ε/2))
-    (hg : ∀j<b,hiddenBinomialCap (sourceScheduledTrials T n (j+1)) ω ≤
-      K*N^(-(scaleKappa (b+1) (j+2) ε+ε)/2)) :
-    sourceSharpEnergyBudget T n s b R ω η a w ≤
-      sourceBadRoleFactor b R*N^R*(η^2+a^2)+
-      sourceForcedRoleFactor b R*W^(2^(b+1))*sourceSkeletonPrefactor s b K*N^(-forcedDecay (b+1) ε d)+
-      8*Q^R*N^(-gap*(R:ℝ)) := by
-  have hN0 : 0 < N := by linarith only [hN]
-  let A := W^(2^(b+1))*sourceSkeletonPrefactor s b K*N^(-forcedDecay (b+1) ε d)
-  have hA : 0 ≤ A := by dsimp [A]; positivity [sourceSkeletonPrefactor_nonneg s b hK]
-  have hlow (r : ℕ) (hr : r∈Finset.range R) :
-      w^(2*r)*(if r=0 then 0 else sourceSharpDegreeBudget T n s b r ω η a) ≤
-        4*((3^((b+1)*r):ℕ):ℝ)^2*N^R*(η^2+a^2)+
-          (4*((3^((b+1)*r):ℕ):ℝ)*(OccupiedParityTree.shapeRoleConstant r (b+1):ℝ))*A := by
-    have hr' : r ≤ R := (Finset.mem_range.mp hr).le
-    have hw0 : 0 ≤ w^(2*r) := by rw [pow_mul]; exact pow_nonneg (sq_nonneg w) r
-    have hw1 : w^(2*r) ≤ 1 := by rw [pow_mul]; exact pow_le_one₀ (sq_nonneg w) hwone
-    have hc : ((SourceLeafCount T (b+1)).choose r:ℝ) ≤ N^R := by
-      have hh : ((SourceLeafCount T (b+1)).choose r:ℝ) ≤ (SourceLeafCount T (b+1):ℝ)^r := by
-        exact_mod_cast Nat.choose_le_pow (SourceLeafCount T (b+1)) r
-      exact hh.trans ((pow_le_pow_left₀ (by positivity) hleaves r).trans (pow_le_pow_right₀ hN hr'))
-    have hbad : w^(2*r)*(((SourceLeafCount T (b+1)).choose r:ℝ)) ≤ N^R := by
-      apply (mul_le_mul hw1 hc (by positivity) (by norm_num : (0:ℝ) ≤ 1)).trans_eq
-      exact one_mul _
-    by_cases hr0 : r=0
-    · rw [ite_eq_left hr0,mul_zero]
-      positivity
-    rw [ite_eq_right hr0,sourceSharpDegreeBudget,mul_add]
-    have hB : w^(2*r)*(4*((3^((b+1)*r):ℕ):ℝ)*
-        (((SourceLeafCount T (b+1)).choose r:ℝ)*((3^((b+1)*r):ℕ):ℝ)*(η^2+a^2))) ≤
-        4*((3^((b+1)*r):ℕ):ℝ)^2*N^R*(η^2+a^2) := by
-      have hh := mul_le_mul_of_nonneg_right hbad
-        (show 0 ≤ 4*((3^((b+1)*r):ℕ):ℝ)^2*(η^2+a^2) by positivity)
-      nlinarith only [hh]
-    rw [mul_add]
-    apply add_le_add hB
-    by_cases hrf : 2^(b+1) ≤ r
-    · rw [ite_eq_left hrf]
-      have hh := mul_le_mul_of_nonneg_left
-        (source_forced_degree_decay T n s b r ω N ε d K W w hN0 hK hW hrf hsmall hw hn hg)
-        (show 0 ≤ 4*((3^((b+1)*r):ℕ):ℝ) by positivity)
-      dsimp [A]
-      nlinarith only [hh]
-    · rw [ite_eq_right hrf,mul_zero,mul_zero]
-      positivity
-  have hsum := Finset.sum_le_sum hlow
-  rw [Finset.sum_add_distrib] at hsum
-  have hbadSum : (∑r∈Finset.range R,4*((3^((b+1)*r):ℕ):ℝ)^2*N^R*(η^2+a^2))=
-      sourceBadRoleFactor b R*N^R*(η^2+a^2) := by simp only [sourceBadRoleFactor,Finset.sum_mul]
-  have hforceSum : (∑r∈Finset.range R,(4*((3^((b+1)*r):ℕ):ℝ)*(OccupiedParityTree.shapeRoleConstant r (b+1):ℝ))*A)=
-      sourceForcedRoleFactor b R*W^(2^(b+1))*sourceSkeletonPrefactor s b K*N^(-forcedDecay (b+1) ε d) := by
-    rw [←Finset.sum_mul]
-    dsimp [sourceForcedRoleFactor,A]
-    ring
-  rw [hbadSum,hforceSum] at hsum
-  apply add_le_add hsum
-  have hx : 0 ≤ w^2*(sourceSparseCount T s b:ℝ) := by positivity
-  have htail : 4*(w^2*(sourceSparseCount T s b:ℝ))^R/(1-w^2*(sourceSparseCount T s b:ℝ)) ≤
-      8*(w^2*(sourceSparseCount T s b:ℝ))^R := by
-    apply (div_le_iff₀ (by linarith only [hhalf])).mpr
-    have hh := mul_le_mul_of_nonneg_right hhalf (pow_nonneg hx R)
-    nlinarith only [hh]
-  apply htail.trans
-  have hrp : (N^(-gap))^R=N^(-gap*(R:ℝ)) := by rw [←Real.rpow_natCast,←Real.rpow_mul hN0.le]
-  have hh := mul_le_mul_of_nonneg_left (pow_le_pow_left₀ hx hsparse R) (by norm_num : (0:ℝ)≤8)
-  simpa only [mul_pow,hrp,mul_assoc] using hh
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-def sourceSmoothingPrefactor (s : ℕ) (K : ℝ) : ℝ :=
-  (2:ℝ)^(s*s+s)*(((s:ℝ)+1)*K)^(s+1)
-
-theorem hiddenSmoothingBound_scale {N e K ω : ℝ} {m s : ℕ}
-    (hN : 0 < N) (hK : 0 ≤ K)
-    (hm : hiddenBinomialCap m ω ≤ K*N^(-e/2)) :
-    hiddenSmoothingBound m s ω ≤ sourceSmoothingPrefactor s K*N^(-((s:ℝ)+1)*e/2) := by
-  have hs : 1 ≤ (s:ℝ)+1 := by linarith [Nat.cast_nonneg s (α:=ℝ)]
-  have hs0 : 0 ≤ (s:ℝ)+1 := by positivity
-  have heq : 4*((s:ℝ)+1)/((m:ℝ)*ω^2+2)=((s:ℝ)+1)*(4/((m:ℝ)*ω^2+2)) := by ring
-  have hh : Real.sqrt (4*((s:ℝ)+1)/((m:ℝ)*ω^2+2)) ≤ ((s:ℝ)+1)*K*N^(-e/2) := by
-    rw [heq,Real.sqrt_mul hs0]
-    have hsq : Real.sqrt ((s:ℝ)+1) ≤ (s:ℝ)+1 := by
-      apply (Real.sqrt_le_iff).mpr
-      exact ⟨hs0,by nlinarith only [hs,sq_nonneg ((s:ℝ))]⟩
-    have h := mul_le_mul hsq hm (Real.sqrt_nonneg _) hs0
-    simpa only [hiddenBinomialCap,mul_assoc] using h
-  have hleft := mul_le_mul
-    (pow_le_pow_left₀ (by positivity) (mul_le_mul_of_nonneg_left hh (by norm_num : (0:ℝ)≤2)) s)
-    hh (Real.sqrt_nonneg _) (by positivity)
-  apply (mul_le_mul_of_nonneg_left hleft (by positivity : (0:ℝ)≤2^(s*s))).trans_eq
-  have hr : (N^(-e/2))^(s+1)=N^(-((s:ℝ)+1)*e/2) := by
-    rw [←Real.rpow_natCast,←Real.rpow_mul hN.le]
-    push_cast
-    congr 1
-    ring
-  dsimp [sourceSmoothingPrefactor]
-  rw [pow_add,pow_succ (((s:ℝ)+1)*K)]
-  simp only [mul_pow]
-  have he : (N^(-e/2))^s*N^(-e/2)=N^(-((s:ℝ)+1)*e/2) := by rw [←pow_succ,hr]
-  calc
-    _ = (2:ℝ)^(s*s)*2^s*((s:ℝ)+1)^s*K^s*(((s:ℝ)+1)*K)*
-      ((N^(-e/2))^s*N^(-e/2)) := by ring
-    _ = _ := by rw [he]; ring
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem sourceLeafCount_positive {T : ℕ → ℕ} {b : ℕ} (hT : ∀j<b,1 ≤ T j) :
-    1 ≤ SourceLeafCount T b := by
-  induction b with
-  | zero => change 1 ≤ Fintype.card (Fin 1); simp
-  | succ b ih =>
-    rw [sourceLeafCount_succ]
-    exact one_le_mul_of_one_le_of_one_le (hT b (by omega)) (ih (fun j hj=>hT j (by omega)))
-
-theorem sourceFitTail_simple {T n : ℕ → ℕ} {b : ℕ} {v : ℝ}
-    (hT : ∀j<b,1 ≤ T j)
-    (hv : ∀j,j+1<b → v ≤ 2*((T j:ℝ)/8)^2/(((T (j+1):ℝ)+1)*(n j:ℝ))) :
-    sourceFitTail T n b ≤ 2*(b:ℝ)*(SourceLeafCount T b:ℝ)*Real.exp (-v) := by
-  induction b using Nat.twoStepInduction with
-  | zero => simp [sourceFitTail]
-  | one => simp only [sourceFitTail]; positivity
-  | more b _ ih =>
-    have hi := ih (fun j hj=>hT j (by omega)) (fun j hj=>hv j (by omega))
-    have hleaf : (1:ℝ) ≤ SourceLeafCount T (b+1) := by
-      exact_mod_cast sourceLeafCount_positive (fun j hj=>hT j (show j<b+2 by omega))
-    have he := Real.exp_le_exp.mpr (neg_le_neg (hv b (by omega)))
-    rw [sourceFitTail,sourceLeafCount_succ,Nat.cast_mul]
-    have hmul := mul_le_mul_of_nonneg_left (add_le_add (mul_le_mul_of_nonneg_left he (by norm_num : (0:ℝ)≤2)) hi)
-      (show 0 ≤ (T (b+1):ℝ) by positivity)
-    have hlast := mul_le_mul_of_nonneg_left hleaf (show 0 ≤ (T (b+1):ℝ)*Real.exp (-v) by positivity)
-    push_cast at hmul ⊢
-    have hex : -(2*((T b:ℝ)/8)^2/(((T (b+1):ℝ)+1)*(n b:ℝ))) =
-        -2*((T b:ℝ)/8)^2/(((T (b+1):ℝ)+1)*(n b:ℝ)) := by ring
-    rw [hex] at hmul
-    nlinarith only [hmul,hlast]
-
-theorem windowCollection_card_bound {T : ℕ → ℕ} {b : ℕ} (hT : ∀j<b,1 ≤ T j) :
-    Fintype.card (WindowCollection T b) ≤ 2*b*SourceLeafCount T b := by
-  induction b with
-  | zero => change Fintype.card PEmpty ≤ _; simp
-  | succ b ih =>
-    have hi := ih (fun j hj=>hT j (by omega))
-    have hp := sourceLeafCount_positive (b:=b) (fun j hj=>hT j (show j<b+1 by omega))
-    have ht := hT b (by omega)
-    have hc : Fintype.card (WindowCollection T (b+1))=2+T b*Fintype.card (WindowCollection T b) := by
-      change Fintype.card (Bool ⊕ (Fin (T b) × WindowCollection T b))=_
-      simp only [Fintype.card_sum,Fintype.card_bool,Fintype.card_prod,Fintype.card_fin]
-    rw [hc,sourceLeafCount_succ]
-    have hm := Nat.mul_le_mul_left (T b) hi
-    have hh : 1 ≤ T b*SourceLeafCount T b := one_le_mul_of_one_le_of_one_le ht hp
-    nlinarith only [hm,hh]
-
-theorem collectionCount_lower {T n g : ℕ→ℕ} {b : ℕ} {M : ℝ}
-    (hn : ∀j<b,M ≤ (n j:ℝ)) (hg : ∀j<b,g j=0 ∨ M ≤ (g j:ℝ))
-    (v : WindowCollection T b) :
-    collectionCount T n g b v=0 ∨ M ≤ (collectionCount T n g b v:ℝ) := by
-  induction b with
-  | zero => exact PEmpty.elim v
-  | succ b ih =>
-    rcases v with a | ⟨u,v⟩
-    · cases a
-      · exact Or.inr (hn b (by omega))
-      · exact hg b (by omega)
-    · exact ih (fun j hj=>hn j (by omega)) (fun j hj=>hg j (by omega)) v
-
-theorem sourceEffectiveHiddenTail_simple {T n : ℕ→ℕ} {b : ℕ} {M ω : ℝ}
-    (hT : ∀j<b,1 ≤ T j) (hn : ∀j<b,M ≤ (n j:ℝ))
-    (hg : ∀j<b,sourceScheduledTrials T n j=0 ∨ M ≤ (sourceScheduledTrials T n j:ℝ)) :
-    sourceEffectiveHiddenTail T n b ω ≤ 2*(b:ℝ)*(SourceLeafCount T b:ℝ)*Real.exp (-M*ω^4/16) := by
-  unfold sourceEffectiveHiddenTail
-  have hh (v : WindowCollection T b) :
-      (if collectionCount T n (sourceScheduledTrials T n) b v=0 then 0 else
-        Real.exp (-(collectionCount T n (sourceScheduledTrials T n) b v:ℝ)*ω^4/16)) ≤ Real.exp (-M*ω^4/16) := by
-    rcases collectionCount_lower hn hg v with hz | hm
-    · rw [ite_eq_left hz]; positivity
-    · split_ifs
-      · positivity
-      · apply Real.exp_le_exp.mpr
-        have hh := mul_le_mul_of_nonneg_right hm (pow_nonneg (sq_nonneg ω) 2)
-        nlinarith only [hh]
-  apply (Finset.sum_le_sum (fun v _=>hh v)).trans
-  simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
-  apply mul_le_mul_of_nonneg_right _ (Real.exp_pos _).le
-  exact_mod_cast windowCollection_card_bound hT
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- The literal centering fluctuation has the source epsilon exponent.
- This does not treat nested window starts as independent. -/
-theorem SourceArithmeticBounds.fit_exponent {C N F ε : ℝ} {s b ell k j : ℕ}
-    (H : SourceArithmeticBounds C N F ε s b ell k) (hC : 0<C) (hF : 0<F) (hN : 1 ≤ N) (he : 0 ≤ ε)
-    (hκ : ∀v<b,3*ε ≤ scaleKappa b (v+1) ε) (hj : j+1<b) :
-    N^ε/512 ≤ 2*((sourceBlockWindows N b ε j:ℝ)/8)^2 /
-      (((sourceBlockWindows N b ε (j+1):ℝ)+1)*(sourceArithmeticTrials C N b ε ell k j:ℝ)) := by
-  have hN0 : 0<N := by linarith only [hN]
-  have hpresent := (H.windows j (by omega)).1
-  have hwin : 24 ≤ sourceWindowSize N b ε j := by
-    simpa only [sourceBlockWindows,ite_eq_left (by omega : j<b)] using hpresent
-  have hlarge : 8 ≤ N^(scaleKappa b (j+1) ε) := by
-    have hh : (24:ℝ) ≤ sourceWindowSize N b ε j := by exact_mod_cast hwin
-    exact (by norm_num : (8:ℝ)≤24).trans (hh.trans (sourceWindowSize_le N b j ε hN0.le))
-  have hl : N^(scaleKappa b (j+1) ε)/2 ≤ (sourceBlockWindows N b ε j:ℝ) := by
-    simpa only [sourceBlockWindows,ite_eq_left (by omega : j<b)] using (sourceWindowSize_bounds hlarge).1
-  have hu : (sourceBlockWindows N b ε (j+1):ℝ)+1 ≤ 2*N^(scaleKappa b (j+2) ε) := by
-    have hh := sourceWindowSize_le N b (j+1) ε hN0.le
-    have he' : 0 ≤ scaleKappa b (j+2) ε := by linarith only [hκ (j+1) hj,he]
-    have ho := Real.one_le_rpow hN he'
-    simpa only [sourceBlockWindows,ite_eq_left hj] using (show (sourceWindowSize N b ε (j+1):ℝ)+1 ≤ 2*N^(scaleKappa b (j+2) ε) by linarith only [hh,ho])
-  have htr := (H.trials j (by omega)).2
-  have hd : ((sourceBlockWindows N b ε (j+1):ℝ)+1)*sourceArithmeticTrials C N b ε ell k j ≤
-      4*N^(scaleKappa b (j+2) ε)*N^ε := by
-    have hh := mul_le_mul hu htr (by positivity) (by positivity)
-    nlinarith only [hh]
-  have htrial : 0 < (sourceArithmeticTrials C N b ε ell k j:ℝ) :=
-    (show 0 < N^ε/(600*(C^(2*b)*F)) by positivity).trans_le (H.trials j (by omega)).1
-  have hdpos : 0 < ((sourceBlockWindows N b ε (j+1):ℝ)+1)*sourceArithmeticTrials C N b ε ell k j := by positivity
-  apply (le_div_iff₀ hdpos).mpr
-  have hmul := mul_le_mul_of_nonneg_left hd (show 0 ≤ N^ε/512 by positivity)
-  have hsquare := pow_le_pow_left₀ (show 0 ≤ N^(scaleKappa b (j+1) ε)/2 by positivity) hl 2
-  have hpow : N^ε*N^(scaleKappa b (j+2) ε)*N^ε=(N^(scaleKappa b (j+1) ε))^2 := by
-    rw [←Real.rpow_add hN0,←Real.rpow_add hN0,←Real.rpow_natCast,←Real.rpow_mul hN0.le]
-    norm_num only [Nat.cast_ofNat]
-    congr 1
-    have hx := scaleKappa_step b (j+1) ε (by omega)
-    linarith only [hx]
-  nlinarith only [hmul,hsquare,hpow]
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open Filter
-open scoped Topology Classical
-
-/-- A convenience for the manuscript's fixed-B polynomial losses. This
- predicate is only used to prove bounds on the literal finite constants. -/
-def SourcePolyUpper (f : ℕ→ℝ) : Prop :=
-  ∃ C : ℝ,0 ≤ C ∧ ∃ h : ℕ,∀D,f D ≤ C*((D:ℝ)+1)^h
-
-theorem SourcePolyUpper.const (c : ℝ) : SourcePolyUpper (fun _=>c) := by
-  refine ⟨|c|,abs_nonneg _,0,?_⟩
-  intro D
-  simpa using le_abs_self c
-
-theorem SourcePolyUpper.depth : SourcePolyUpper (fun D=>(D:ℝ)+1) := by
-  exact ⟨1,zero_le_one,1,by simp⟩
-
-theorem SourcePolyUpper.mono {f g : ℕ→ℝ} (hg : SourcePolyUpper g) (hf : ∀D,f D ≤ g D) :
-    SourcePolyUpper f := by
-  obtain ⟨C,hC,h,H⟩ := hg
-  exact ⟨C,hC,h,fun D=>(hf D).trans (H D)⟩
-
-theorem SourcePolyUpper.add {f g : ℕ→ℝ} (hf : SourcePolyUpper f) (hg : SourcePolyUpper g) :
-    SourcePolyUpper (fun D=>f D+g D) := by
-  obtain ⟨C,hC,h,H⟩ := hf
-  obtain ⟨K,hK,k,G⟩ := hg
-  refine ⟨C+K,add_nonneg hC hK,h+k,?_⟩
-  intro D
-  have hd : 1 ≤ (D:ℝ)+1 := by linarith [Nat.cast_nonneg D (α:=ℝ)]
-  have hh := (H D).trans (mul_le_mul_of_nonneg_left (pow_le_pow_right₀ hd (by omega : h ≤ h+k)) hC)
-  have hk := (G D).trans (mul_le_mul_of_nonneg_left (pow_le_pow_right₀ hd (by omega : k ≤ h+k)) hK)
-  nlinarith only [hh,hk]
-
-theorem SourcePolyUpper.mul {f g : ℕ→ℝ} (hf : SourcePolyUpper f) (hg : SourcePolyUpper g)
-    (hf0 : ∀D,0 ≤ f D) (hg0 : ∀D,0 ≤ g D) : SourcePolyUpper (fun D=>f D*g D) := by
-  obtain ⟨C,hC,h,H⟩ := hf
-  obtain ⟨K,hK,k,G⟩ := hg
-  refine ⟨C*K,mul_nonneg hC hK,h+k,?_⟩
-  intro D
-  calc
-    f D*g D ≤ (C*((D:ℝ)+1)^h)*(K*((D:ℝ)+1)^k) :=
-      mul_le_mul (H D) (G D) (hg0 D) ((hf0 D).trans (H D))
-    _ = _ := by rw [pow_add]; ring
-
-theorem SourcePolyUpper.pow {f : ℕ→ℝ} (hf : SourcePolyUpper f) (hf0 : ∀D,0 ≤ f D) (r : ℕ) :
-    SourcePolyUpper (fun D=>(f D)^r) := by
-  obtain ⟨C,hC,h,H⟩ := hf
-  refine ⟨C^r,pow_nonneg hC _,h*r,?_⟩
-  intro D
-  simpa only [mul_pow,pow_mul] using pow_le_pow_left₀ (hf0 D) (H D) r
-
-theorem sourceEpsilon_inverse_poly (B : ℕ) :
-    SourcePolyUpper (fun D=>(sourceEpsilon B D)⁻¹) := by
-  refine ⟨(2:ℝ)^(4*B+100),by positivity,1,?_⟩
-  intro D
-  simp only [sourceEpsilon,one_div,inv_inv,pow_one]
-  exact le_rfl
-
-theorem totalScoreExponent_poly (B : ℕ) (hB : 2 ≤ B) :
-    SourcePolyUpper (totalScoreExponent B) := by
-  apply (SourcePolyUpper.const 3 |>.add
-    (SourcePolyUpper.depth.mul (SourcePolyUpper.const (B:ℝ)) (fun D=>by positivity) (fun _=>by positivity))).mono
-  intro D
-  have hh := (totalScoreExponent_bounds B D hB).2
-  nlinarith only [hh,Nat.cast_nonneg B (α:=ℝ)]
-
-theorem SourcePolyUpper.ceil {f : ℕ→ℝ} (hf : SourcePolyUpper f) (hf0 : ∀D,0 ≤ f D) :
-    SourcePolyUpper (fun D=>(⌈f D⌉₊:ℝ)) := by
-  apply (hf.add (SourcePolyUpper.const 1)).mono
-  intro D
-  exact (Nat.ceil_lt_add_one (hf0 D)).le
-
-theorem sourceCancellationOrder_poly (B : ℕ) (hB : 2 ≤ B) :
-    SourcePolyUpper (fun D=>(sourceCancellationOrder B D:ℝ)) := by
-  let f := fun D=>totalScoreExponent B D+(sourceDegreeCutoff B:ℝ)+B+10
-  have hf : SourcePolyUpper f := (((totalScoreExponent_poly B hB).add
-    (SourcePolyUpper.const (sourceDegreeCutoff B:ℝ))).add (SourcePolyUpper.const (B:ℝ))).add
-      (SourcePolyUpper.const 10)
-  have hf0 (D) : 0 ≤ f D := by
-    have hd := (totalScoreExponent_bounds B D hB).1
-    have hr : 0 ≤ (sourceDegreeCutoff B:ℝ) := Nat.cast_nonneg _
-    dsimp [f]
-    linarith only [hd,hr,Nat.cast_nonneg B (α:=ℝ)]
-  have hh := ((SourcePolyUpper.const 100).mul hf (fun _=>by norm_num) hf0).mul
-    (sourceEpsilon_inverse_poly B) (fun D=>mul_nonneg (by norm_num) (hf0 D))
-      (fun D=>(inv_pos.mpr (sourceEpsilon_pos B D)).le)
-  have hc := hh.ceil (fun D=>mul_nonneg (mul_nonneg (by norm_num) (hf0 D))
-    (inv_pos.mpr (sourceEpsilon_pos B D)).le)
-  simpa only [sourceCancellationOrder,f,div_eq_mul_inv] using hc
-
-theorem sourceCancellationOrder_large (B D : ℕ) :
-    100*(totalScoreExponent B D+(sourceDegreeCutoff B:ℝ)+B+10) ≤
-      (sourceCancellationOrder B D:ℝ)*sourceEpsilon B D := by
-  exact (div_le_iff₀ (sourceEpsilon_pos B D)).mp (Nat.le_ceil _)
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open Filter
-open scoped Topology Classical
-
-/-- Literal finite-block factors are subpower uniformly over the joint
- deletion regime when their logarithms grow polynomially in the depth. -/
-def SourceExpUpper (f : ℕ→ℝ→ℝ) : Prop :=
-  ∃ C : ℝ,0 ≤ C ∧ ∃ h : ℕ,∀D q,q∈Set.Ioc 0 (1/2:ℝ) →
-    f D q ≤ Real.exp (C*((D:ℝ)+1)^h*q^(-3:ℤ))
-
-theorem SourcePolyUpper.expUpper {f : ℕ→ℝ} (hf : SourcePolyUpper f) :
-    SourceExpUpper (fun D _=>f D) := by
-  obtain ⟨C,hC,h,H⟩ := hf
-  refine ⟨C,hC,h,?_⟩
-  intro D q hq
-  have hΛ := deletion_inverse_cube_ge_one ⟨hq.1,by linarith only [hq.2]⟩
-  have hmul := mul_le_mul_of_nonneg_left hΛ (show 0 ≤ C*((D:ℝ)+1)^h by positivity)
-  exact (H D).trans ((by linarith only [hmul] : C*((D:ℝ)+1)^h ≤ C*((D:ℝ)+1)^h*q^(-3:ℤ)+1).trans
-    (Real.add_one_le_exp _))
-
-theorem SourceExpUpper.const (c : ℝ) : SourceExpUpper (fun _ _=>c) :=
-  (SourcePolyUpper.const c).expUpper
-
-theorem SourceExpUpper.mono {f g : ℕ→ℝ→ℝ} (hg : SourceExpUpper g)
-    (hf : ∀D q,q∈Set.Ioc 0 (1/2:ℝ) → f D q ≤ g D q) : SourceExpUpper f := by
-  obtain ⟨C,hC,h,H⟩ := hg
-  exact ⟨C,hC,h,fun D q hq=>(hf D q hq).trans (H D q hq)⟩
-
-theorem SourceExpUpper.mul {f g : ℕ→ℝ→ℝ} (hf : SourceExpUpper f) (hg : SourceExpUpper g)
-    (hf0 : ∀D q,q∈Set.Ioc 0 (1/2:ℝ) → 0 ≤ f D q)
-    (hg0 : ∀D q,q∈Set.Ioc 0 (1/2:ℝ) → 0 ≤ g D q) :
-    SourceExpUpper (fun D q=>f D q*g D q) := by
-  obtain ⟨C,hC,h,H⟩ := hf
-  obtain ⟨K,hK,k,G⟩ := hg
-  refine ⟨C+K,add_nonneg hC hK,h+k,?_⟩
-  intro D q hq
-  have hΛ : 0 ≤ q^(-3:ℤ) := (zpow_pos hq.1 _).le
-  have hd : 1 ≤ (D:ℝ)+1 := by linarith [Nat.cast_nonneg D (α:=ℝ)]
-  have hh := mul_le_mul_of_nonneg_right
-    (mul_le_mul_of_nonneg_left (pow_le_pow_right₀ hd (by omega : h≤h+k)) hC) hΛ
-  have hk := mul_le_mul_of_nonneg_right
-    (mul_le_mul_of_nonneg_left (pow_le_pow_right₀ hd (by omega : k≤h+k)) hK) hΛ
-  calc
-    _ ≤ Real.exp (C*((D:ℝ)+1)^h*q^(-3:ℤ))*Real.exp (K*((D:ℝ)+1)^k*q^(-3:ℤ)) :=
-      mul_le_mul (H D q hq) (G D q hq) (hg0 D q hq) ((hf0 D q hq).trans (H D q hq))
-    _ ≤ _ := by rw [←Real.exp_add]; apply Real.exp_le_exp.mpr; nlinarith only [hh,hk]
-
-theorem SourceExpUpper.pow_variably {f : ℕ→ℝ→ℝ} {r : ℕ→ℕ} (hf : SourceExpUpper f)
-    (hf0 : ∀D q,q∈Set.Ioc 0 (1/2:ℝ) → 0 ≤ f D q)
-    (hr : SourcePolyUpper (fun D=>(r D:ℝ))) : SourceExpUpper (fun D q=>(f D q)^(r D)) := by
-  obtain ⟨C,hC,h,H⟩ := hf
-  obtain ⟨K,hK,k,G⟩ := hr
-  refine ⟨C*K,mul_nonneg hC hK,h+k,?_⟩
-  intro D q hq
-  have hΛ : 0≤q^(-3:ℤ) := (zpow_pos hq.1 _).le
-  have hh := mul_le_mul_of_nonneg_right (G D) (show 0≤C*((D:ℝ)+1)^h*q^(-3:ℤ) by positivity)
-  calc
-    _ ≤ (Real.exp (C*((D:ℝ)+1)^h*q^(-3:ℤ)))^(r D) := pow_le_pow_left₀ (hf0 D q hq) (H D q hq) _
-    _ = Real.exp ((r D:ℝ)*(C*((D:ℝ)+1)^h*q^(-3:ℤ))) := (Real.exp_nat_mul _ _).symm
-    _ ≤ _ := by apply Real.exp_le_exp.mpr; rw [pow_add]; nlinarith only [hh]
-
-theorem SourceExpUpper.pow {f : ℕ→ℝ→ℝ} (hf : SourceExpUpper f)
-    (hf0 : ∀D q,q∈Set.Ioc 0 (1/2:ℝ) → 0 ≤ f D q) (r : ℕ) :
-    SourceExpUpper (fun D q=>(f D q)^r) :=
-  hf.pow_variably hf0 (SourcePolyUpper.const (r:ℝ))
-
-theorem sourceOverlapConstant_expUpper : SourceExpUpper (fun _ q=>sourceOverlapConstant q) := by
-  refine ⟨sourceChannelLogConstant,sourceChannelLogConstant_pos.le,0,?_⟩
-  intro D q hq
-  simp only [pow_zero,mul_one]
-  apply (Real.log_le_iff_le_exp (sourceOverlapConstant_pos hq.1)).mp
-  exact sourceOverlapConstant_log_bound ⟨hq.1,by linarith only [hq.2]⟩
-
-theorem initialOverlapLower_inverse_expUpper : SourceExpUpper (fun _ q=>(initialOverlapLower q)⁻¹) := by
-  refine ⟨1536+(32+3*primitivePeriodConstant),by unfold primitivePeriodConstant; positivity,0,?_⟩
-  intro D q hq
-  have hp := initialOverlapLower_pos q
-  have hΛ := deletion_inverse_cube_ge_one ⟨hq.1,by linarith only [hq.2]⟩
-  rw [←Real.log_le_iff_le_exp (inv_pos.mpr hp),Real.log_inv,initialOverlapLower,
-    Real.log_div (Real.exp_pos _).ne' (by norm_num),Real.log_exp]
-  simp only [pow_zero,mul_one,zpow_neg,zpow_ofNat] at hΛ ⊢
-  have hh := Real.log_le_self (by norm_num : (0:ℝ)≤1536)
-  have hmul := mul_le_mul_of_nonneg_left hΛ (by norm_num : (0:ℝ)≤1536)
-  simp only [div_eq_mul_inv] at *
-  nlinarith only [hh,hmul]
-
-/-- All indices are inside the same physical n,q instance. -/
-theorem SourceExpUpper.uniform (f : ℕ→ℝ→ℝ) (hf : SourceExpUpper f)
-    (B : ℕ) (ζ δ : ℝ) (hB : 2 ≤ B) (hζ : ζ∈Set.Ioo 0 1) (hδ : 0<δ)
-    {n q : ℕ→ℝ} (hn : Tendsto n atTop atTop) (hq : ∀t,q t∈Set.Ioc 0 (1/2:ℝ))
-    (hH : Tendsto (fun t=>(q t)^3*Real.log (n t)) atTop atTop) :
-    ∀ᶠt in atTop,∀i,f (sourceDepth B ζ ((q t)^3*Real.log (n t))) (q t) ≤
-      (sourceTargetScale B (sourceDepth B ζ ((q t)^3*Real.log (n t))) (n t) i)^(
-        δ*sourceEpsilon B (sourceDepth B ζ ((q t)^3*Real.log (n t)))) := by
-  obtain ⟨C,hC,h,H⟩ := hf
-  have hrel (t) : Real.log (n t)=(q t)^(-3:ℤ)*((q t)^3*Real.log (n t)) := by
-    rw [zpow_neg,zpow_ofNat]
-    field_simp [(hq t).1.ne']
-  filter_upwards [source_uniform_exp_absorption B h ζ C δ hB hζ hC hδ hn hH
-    (fun t=>zpow_pos (hq t).1 _) hrel] with t ht i
-  exact (H _ _ (hq t)).trans (ht i)
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open Filter
-open scoped Topology Classical
-
-def sourceMomentEnvelope (B D : ℕ) : ℝ :=
-  ((10^8:ℝ)*2^(10*B)*((B:ℝ)+4)*2+1)*((D:ℝ)+1)*2^(B*D)
-
-def sourceCapEnvelope (B D : ℕ) (q : ℝ) : ℝ :=
-  200*(sourceOverlapConstant q)^(2*B*(D+1)+2)*(initialOverlapLower q)⁻¹
-
-def sourceWeightEnvelope (B D : ℕ) : ℝ := 144*(sourceMomentEnvelope B D)^2
-
-theorem sourceMomentEnvelope_pos (B D : ℕ) : 0 < sourceMomentEnvelope B D := by
-  unfold sourceMomentEnvelope
-  positivity
-
-theorem sourceCapEnvelope_pos (B D : ℕ) {q : ℝ} (hq : 0<q) : 0 < sourceCapEnvelope B D q := by
-  unfold sourceCapEnvelope
-  positivity [sourceOverlapConstant_pos hq,initialOverlapLower_pos q]
-
-theorem sourceMomentEnvelope_expUpper (B : ℕ) : SourceExpUpper (fun D _=>sourceMomentEnvelope B D) := by
-  have hr : SourcePolyUpper (fun D=>((B*D:ℕ):ℝ)) := by
-    apply ((SourcePolyUpper.const (B:ℝ)).mul SourcePolyUpper.depth (fun _=>by positivity) (fun _=>by positivity)).mono
-    intro D
-    push_cast
-    nlinarith only [Nat.cast_nonneg B (α:=ℝ)]
-  have hpow := (SourceExpUpper.const 2).pow_variably (fun _ _ _=>by norm_num) hr
-  have hp := ((SourcePolyUpper.const ((10^8:ℝ)*2^(10*B)*((B:ℝ)+4)*2+1)).mul
-    SourcePolyUpper.depth (fun _=>by positivity) (fun _=>by positivity)).expUpper
-  exact hp.mul hpow (fun _ _ _=>by positivity) (fun _ _ _=>by positivity)
-
-theorem sourceCapEnvelope_expUpper (B : ℕ) : SourceExpUpper (sourceCapEnvelope B) := by
-  have hr : SourcePolyUpper (fun D=>((2*B*(D+1)+2:ℕ):ℝ)) := by
-    have hp := ((SourcePolyUpper.const ((2*B:ℕ):ℝ)).mul SourcePolyUpper.depth
-      (fun _=>by positivity) (fun _=>by positivity)).add (SourcePolyUpper.const 2)
-    simpa only [Nat.cast_mul,Nat.cast_add,Nat.cast_one,Nat.cast_ofNat] using hp
-  have hc := sourceOverlapConstant_expUpper.pow_variably (fun _ q hq=>(sourceOverlapConstant_pos hq.1).le) hr
-  exact ((SourceExpUpper.const 200).mul hc (fun _ _ _=>by norm_num)
-    (fun _ q hq=>pow_nonneg (sourceOverlapConstant_pos hq.1).le _)).mul initialOverlapLower_inverse_expUpper
-    (fun _ q hq=>by positivity [sourceOverlapConstant_pos hq.1])
-    (fun _ q _=>(inv_pos.mpr (initialOverlapLower_pos q)).le)
-
-theorem sourceWeightEnvelope_expUpper (B : ℕ) : SourceExpUpper (fun D _=>sourceWeightEnvelope B D) := by
-  exact (SourceExpUpper.const 144).mul
-    ((sourceMomentEnvelope_expUpper B).pow (fun D _ _=>(sourceMomentEnvelope_pos B D).le) 2)
-    (fun _ _ _=>by norm_num) (fun _ _ _=>sq_nonneg _)
-
-theorem sourceMotif_expUpper (B : ℕ) (hB : 2 ≤ B) :
-    SourceExpUpper (fun D _=>(2:ℝ)^(sourceCancellationOrder B D)) :=
-  (SourceExpUpper.const 2).pow_variably (fun _ _ _=>by norm_num) (sourceCancellationOrder_poly B hB)
-
-theorem sourceSmoothingPrefactor_expUpper (B : ℕ) (hB : 2 ≤ B) :
-    SourceExpUpper (fun D q=>sourceSmoothingPrefactor (sourceCancellationOrder B D) (sourceCapEnvelope B D q)) := by
-  let s := sourceCancellationOrder B
-  have hs := sourceCancellationOrder_poly B hB
-  have hs0 (D) : 0 ≤ (s D:ℝ) := Nat.cast_nonneg _
-  have hp : SourcePolyUpper (fun D=>((s D*s D+s D:ℕ):ℝ)) := by
-    simpa only [Nat.cast_add,Nat.cast_mul] using (hs.mul hs hs0 hs0).add hs
-  have ht : SourcePolyUpper (fun D=>((s D+1:ℕ):ℝ)) := by
-    simpa only [Nat.cast_add,Nat.cast_one] using hs.add (SourcePolyUpper.const 1)
-  have hone := (SourceExpUpper.const 2).pow_variably (fun _ _ _=>by norm_num) hp
-  have htwo := ((ht.expUpper).mul (sourceCapEnvelope_expUpper B)
-    (fun _ _ _=>Nat.cast_nonneg _) (fun D _ hq=>(sourceCapEnvelope_pos B D hq.1).le)).pow_variably
-      (fun D q hq=>mul_nonneg (Nat.cast_nonneg _) (sourceCapEnvelope_pos B D hq.1).le) ht
-  have H := hone.mul htwo (fun _ _ _=>by positivity)
-    (fun D q hq=>pow_nonneg (mul_nonneg (Nat.cast_nonneg _) (sourceCapEnvelope_pos B D hq.1).le) _)
-  simpa only [sourceSmoothingPrefactor,s,Nat.cast_add,Nat.cast_one] using H
-
-theorem sourceSkeletonPrefactor_expUpper (B b : ℕ) (hB : 2 ≤ B) :
-    SourceExpUpper (fun D q=>sourceSkeletonPrefactor (sourceCancellationOrder B D) b (sourceCapEnvelope B D q)) := by
-  have hL := sourceMotif_expUpper B hB
-  have hK := sourceCapEnvelope_expUpper B
-  have hA := ((SourceExpUpper.const 2).mul (hL.pow (fun _ _ _=>by positivity) 3)
-    (fun _ _ _=>by norm_num) (fun _ _ _=>by positivity)).mul hK
-      (fun _ _ _=>by positivity) (fun D _ hq=>(sourceCapEnvelope_pos B D hq.1).le)
-  have hF := ((SourceExpUpper.const 4).mul (hL.pow (fun _ _ _=>by positivity) 6)
-    (fun _ _ _=>by norm_num) (fun _ _ _=>by positivity)).mul
-      (hK.pow (fun D _ hq=>(sourceCapEnvelope_pos B D hq.1).le) 2)
-      (fun _ _ _=>by positivity) (fun _ _ _=>sq_nonneg _)
-  have H := hA.mul (hF.pow (fun _ _ _=>by positivity) (2^b-1))
-    (fun D q hq=>by positivity [sourceCapEnvelope_pos B D hq.1]) (fun _ _ _=>by positivity)
-  simpa only [sourceSkeletonPrefactor,binaryTreeBudget_const,Nat.cast_pow,Nat.cast_ofNat] using H
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem SourceArithmeticBounds.fit_tail {C N F ε : ℝ} {s b ell k : ℕ}
-    (H : SourceArithmeticBounds C N F ε s b ell k) (hC : 0<C) (hF : 0<F)
-    (hN : 1≤N) (he : 0≤ε) (hκ : ∀v<b,3*ε ≤ scaleKappa b (v+1) ε) :
-    sourceFitTail (sourceBlockWindows N b ε) (sourceArithmeticTrials C N b ε ell k) b ≤
-      2*(b:ℝ)*N*Real.exp (-(N^ε/512)) := by
-  have hN0 : 0<N := by linarith only [hN]
-  apply (sourceFitTail_simple (fun j hj=>(by omega : 1≤24).trans (H.windows j hj).1)
-    (fun j hj=>H.fit_exponent hC hF hN he hκ hj)).trans
-  have hh := mul_le_mul_of_nonneg_left (H.leaf_bound hN0) (show 0≤2*(b:ℝ) by positivity)
-  exact mul_le_mul_of_nonneg_right hh (Real.exp_pos _).le
-
-theorem SourceArithmeticBounds.metadata_tail {C N F ε ω : ℝ} {s b ell k : ℕ}
-    (H : SourceArithmeticBounds C N F ε s b ell k) (hN : 0<N) :
-    sourceEffectiveHiddenTail (sourceBlockWindows N b ε) (sourceArithmeticTrials C N b ε ell k) b ω ≤
-      2*(b:ℝ)*N*Real.exp (-(N^ε/(600*(C^(2*b)*F)))*ω^4/16) := by
-  apply (sourceEffectiveHiddenTail_simple (fun j hj=>(by omega : 1≤24).trans (H.windows j hj).1)
-    (fun j hj=>(H.trials j hj).1) ?_).trans
-  · exact mul_le_mul_of_nonneg_right
-      (mul_le_mul_of_nonneg_left (H.leaf_bound hN) (show 0≤2*(b:ℝ) by positivity)) (Real.exp_pos _).le
-  · intro j hj
-    cases j with
-    | zero => exact Or.inl rfl
-    | succ j =>
-      right
-      rw [sourceScheduledTrials,Nat.cast_mul]
-      have ht := (H.windows (j+1) hj).1
-      have hd : 1 ≤ sourceBlockWindows N b ε (j+1)/4 := by omega
-      have hd' : (1:ℝ) ≤ (sourceBlockWindows N b ε (j+1)/4:ℕ) := by exact_mod_cast hd
-      exact (H.trials j (by omega)).1.trans
-        (le_mul_of_one_le_left (Nat.cast_nonneg _) hd')
-
-/-- Quantitative clipping without any default-limit increase or limiting
- interchange: the power is the finite even moment actually used by the oracle. -/
-theorem source_clipping_moment_power (T : ℕ→ℕ) (s b k : ℕ) (N gap L : ℝ)
-    (hN : 0<N)
-    (hhalf : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b ≤ 1/2)
-    (hscale : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b ≤ N^(-gap/2)/8) :
-    Real.sqrt (6*(4*sourceCenteredBudget T s b k L)^k) ≤ 3*N^(-gap*(k:ℝ)/4) := by
-  let x := ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b
-  have hx : 0≤x := by dsimp [x]; positivity
-  have hden : 0<1-x := by dsimp [x]; linarith only [hhalf]
-  have hc : 4*sourceCenteredBudget T s b k L ≤ N^(-gap/2) := by
-    have hh : x/(1-x) ≤ 2*x := by
-      apply (div_le_iff₀ hden).mpr
-      have hxhalf : x≤1/2 := hhalf
-      nlinarith only [mul_le_mul_of_nonneg_left hxhalf hx]
-    change 4*(x/(1-x)) ≤ _
-    nlinarith only [hh,hscale]
-  have hc0 : 0 ≤ 4*sourceCenteredBudget T s b k L :=
-    mul_nonneg (by norm_num) (sourceCenteredBudget_nonneg T s b k L (by linarith only [hhalf]))
-  have hh := pow_le_pow_left₀ hc0 hc k
-  have hr : (N^(-gap/2))^k=(N^(-gap*(k:ℝ)/4))^2 := by
-    rw [←Real.rpow_natCast,←Real.rpow_mul hN.le,←Real.rpow_natCast,←Real.rpow_mul hN.le]
-    norm_num only [Nat.cast_ofNat]
-    congr 1
-    ring
-  rw [hr] at hh
-  apply (Real.sqrt_le_left (by positivity : 0≤3*N^(-gap*(k:ℝ)/4))).mpr
-  nlinarith only [hh,sq_nonneg (N^(-gap*(k:ℝ)/4))]
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open Filter
-open scoped Topology Classical
-
-theorem SourceExpUpper.add {f g : ℕ→ℝ→ℝ} (hf : SourceExpUpper f) (hg : SourceExpUpper g) :
-    SourceExpUpper (fun D q=>f D q+g D q) := by
-  obtain ⟨C,hC,h,H⟩ := hf
-  obtain ⟨K,hK,k,G⟩ := hg
-  refine ⟨C+K+1,by positivity,h+k,?_⟩
-  intro D q hq
-  have hΛ := deletion_inverse_cube_ge_one ⟨hq.1,by linarith only [hq.2]⟩
-  have hd : 1≤(D:ℝ)+1 := by linarith [Nat.cast_nonneg D (α:=ℝ)]
-  have hh := mul_le_mul_of_nonneg_right
-    (mul_le_mul_of_nonneg_left (pow_le_pow_right₀ hd (by omega : h≤h+k)) hC) (by linarith only [hΛ] : 0≤q^(-3:ℤ))
-  have hk := mul_le_mul_of_nonneg_right
-    (mul_le_mul_of_nonneg_left (pow_le_pow_right₀ hd (by omega : k≤h+k)) hK) (by linarith only [hΛ] : 0≤q^(-3:ℤ))
-  have hz : 1≤((D:ℝ)+1)^(h+k)*q^(-3:ℤ) := one_le_mul_of_one_le_of_one_le (one_le_pow₀ hd) hΛ
-  have hc0 : 0≤C*((D:ℝ)+1)^(h+k)*q^(-3:ℤ) := by positivity
-  have hk0 : 0≤K*((D:ℝ)+1)^(h+k)*q^(-3:ℤ) := by positivity
-  have he1 : f D q ≤ Real.exp ((C+K)*((D:ℝ)+1)^(h+k)*q^(-3:ℤ)) :=
-    (H D q hq).trans (Real.exp_le_exp.mpr (by nlinarith only [hh,hk0]))
-  have he2 : g D q ≤ Real.exp ((C+K)*((D:ℝ)+1)^(h+k)*q^(-3:ℤ)) :=
-    (G D q hq).trans (Real.exp_le_exp.mpr (by nlinarith only [hk,hc0]))
-  have he := Real.add_one_le_exp (((D:ℝ)+1)^(h+k)*q^(-3:ℤ))
-  have he' := mul_le_mul_of_nonneg_left (show 2≤Real.exp (((D:ℝ)+1)^(h+k)*q^(-3:ℤ)) by linarith only [he,hz])
-    (Real.exp_pos ((C+K)*((D:ℝ)+1)^(h+k)*q^(-3:ℤ))).le
-  rw [←Real.exp_add] at he'
-  convert (show f D q+g D q ≤ Real.exp ((C+K)*((D:ℝ)+1)^(h+k)*q^(-3:ℤ)+((D:ℝ)+1)^(h+k)*q^(-3:ℤ)) by linarith only [he1,he2,he']) using 1
-  congr 1
-  ring
-
-def sourceScaleInverseEnvelope (B D : ℕ) : ℝ := 2*(2:ℝ)^(B*D)
-
-theorem sourceScaleInverseEnvelope_expUpper (B : ℕ) : SourceExpUpper (fun D _=>sourceScaleInverseEnvelope B D) := by
-  have hr : SourcePolyUpper (fun D=>((B*D:ℕ):ℝ)) := by
-    apply ((SourcePolyUpper.const (B:ℝ)).mul SourcePolyUpper.depth (fun _=>by positivity) (fun _=>by positivity)).mono
-    intro D
-    push_cast
-    nlinarith only [Nat.cast_nonneg B (α:=ℝ)]
-  exact (SourceExpUpper.const 2).mul ((SourceExpUpper.const 2).pow_variably (fun _ _ _=>by norm_num) hr)
-    (fun _ _ _=>by norm_num) (fun _ _ _=>by positivity)
-
-theorem source_log_scale_lower (B D i : ℕ) {n : ℝ} (hB : 2≤B) (hn : 1<n) :
-    Real.log n ≤ sourceScaleInverseEnvelope B D*Real.log (sourceTargetScale B D n i) := by
-  have hh := mul_le_mul_of_nonneg_right (sourceScaleProduct_lower B D i hB) (Real.log_pos hn).le
-  rw [sourceTargetScale,Real.log_rpow (by linarith only [hn] : 0<n)]
-  dsimp [sourceScaleInverseEnvelope]
-  have hp : 0<(2:ℝ)*2^(B*D) := by positivity
-  have H := mul_le_mul_of_nonneg_left hh hp.le
-  field_simp at H
-  nlinarith only [H]
-
-theorem SourceExpUpper.uniform_log {f : ℕ→ℝ→ℝ} (hf : SourceExpUpper f)
-    (hf0 : ∀D q,q∈Set.Ioc 0 (1/2:ℝ) → 0≤f D q)
-    (B : ℕ) (ζ δ : ℝ) (hB : 2≤B) (hζ : ζ∈Set.Ioo 0 1) (hδ : 0<δ)
-    {n q : ℕ→ℝ} (hn : Tendsto n atTop atTop) (hq : ∀t,q t∈Set.Ioc 0 (1/2:ℝ))
-    (hH : Tendsto (fun t=>(q t)^3*Real.log (n t)) atTop atTop) :
-    ∀ᶠt in atTop,∀i,f (sourceDepth B ζ ((q t)^3*Real.log (n t))) (q t)*Real.log (n t) ≤
-      (sourceTargetScale B (sourceDepth B ζ ((q t)^3*Real.log (n t))) (n t) i)^
-        (δ*sourceEpsilon B (sourceDepth B ζ ((q t)^3*Real.log (n t)))) := by
-  let g := fun D q=>f D q*sourceScaleInverseEnvelope B D*(2/δ)*(sourceEpsilon B D)⁻¹
-  have hg : SourceExpUpper g :=
-    ((hf.mul (sourceScaleInverseEnvelope_expUpper B) hf0 (fun _ _ _=>by unfold sourceScaleInverseEnvelope; positivity)).mul
-      (SourceExpUpper.const (2/δ)) (fun D q hq=>by unfold sourceScaleInverseEnvelope; positivity [hf0 D q hq]) (fun _ _ _=>by positivity)).mul
-        (sourceEpsilon_inverse_poly B).expUpper (fun D q hq=>by unfold sourceScaleInverseEnvelope; positivity [hf0 D q hq])
-          (fun D _ _=>(inv_pos.mpr (sourceEpsilon_pos B D)).le)
-  have hu := SourceExpUpper.uniform g hg B ζ (δ/2) hB hζ (by positivity) hn hq hH
-  filter_upwards [hu,hn.eventually_gt_atTop 1] with t ht hnt
-  intro i
-  let D := sourceDepth B ζ ((q t)^3*Real.log (n t))
-  let N := sourceTargetScale B D (n t) i
-  let ε := sourceEpsilon B D
-  have he : 0<ε := sourceEpsilon_pos B D
-  have hN : 0<N := Real.rpow_pos_of_pos (by linarith only [hnt]) _
-  have hl := source_log_scale_lower B D i hB hnt
-  have hb := Real.log_le_rpow_div hN.le (show 0<δ*ε/2 by positivity)
-  have hmul := mul_le_mul_of_nonneg_left hb
-    (show 0≤f D (q t)*sourceScaleInverseEnvelope B D by unfold sourceScaleInverseEnvelope; positivity [hf0 D _ (hq t)])
-  have hg' := mul_le_mul_of_nonneg_right (ht i) (Real.rpow_pos_of_pos hN (δ*ε/2)).le
-  have hr : N^((δ/2)*ε)*N^(δ*ε/2)=N^(δ*ε) := by rw [←Real.rpow_add hN]; congr 1; ring
-  change g D (q t)*N^(δ*ε/2) ≤ N^((δ/2)*ε)*N^(δ*ε/2) at hg'
-  rw [hr] at hg'
-  have heq : f D (q t)*sourceScaleInverseEnvelope B D*(N^(δ*ε/2)/(δ*ε/2))=g D (q t)*N^(δ*ε/2) := by
-    dsimp [g,ε]
-    field_simp
-  have hfirst := mul_le_mul_of_nonneg_left hl (hf0 D (q t) (hq t))
-  change f D (q t)*Real.log (n t) ≤ N^(δ*ε)
-  rw [heq] at hmul
-  nlinarith only [hfirst,hmul,hg']
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open Filter
-open scoped Topology Classical
-
-/-- Uniform exponential domination at every working scale, with the
- full depth and the varying known deletion probability retained. -/
-theorem SourceExpUpper.uniform_tail {f g h : ℕ→ℝ→ℝ}
-    (hf : SourceExpUpper f) (hg : SourceExpUpper g) (hh : SourceExpUpper h)
-    (hf0 : ∀D q,q∈Set.Ioc 0 (1/2:ℝ) → 0<f D q)
-    (_hg0 : ∀D q,q∈Set.Ioc 0 (1/2:ℝ) → 0≤g D q)
-    (hh0 : ∀D q,q∈Set.Ioc 0 (1/2:ℝ) → 0≤h D q)
-    (B : ℕ) (ζ : ℝ) (hB : 2≤B) (hζ : ζ∈Set.Ioo 0 1)
-    {n q : ℕ→ℝ} (hn : Tendsto n atTop atTop) (hq : ∀t,q t∈Set.Ioc 0 (1/2:ℝ))
-    (hH : Tendsto (fun t=>(q t)^3*Real.log (n t)) atTop atTop) :
-    ∀ᶠt in atTop,∀i,
-      g (sourceDepth B ζ ((q t)^3*Real.log (n t))) (q t)*
-        Real.exp (-(sourceTargetScale B (sourceDepth B ζ ((q t)^3*Real.log (n t))) (n t) i)^
-          (sourceEpsilon B (sourceDepth B ζ ((q t)^3*Real.log (n t))))/
-          f (sourceDepth B ζ ((q t)^3*Real.log (n t))) (q t)) ≤
-      (n t)^(-(h (sourceDepth B ζ ((q t)^3*Real.log (n t))) (q t))) := by
-  have hfu := SourceExpUpper.uniform f hf B ζ (1/4) hB hζ (by norm_num) hn hq hH
-  have hgu := SourceExpUpper.uniform g hg B ζ (1/4) hB hζ (by norm_num) hn hq hH
-  have hhu := hh.uniform_log hh0 B ζ (1/4) hB hζ (by norm_num) hn hq hH
-  have hcu := SourceExpUpper.uniform _ (SourceExpUpper.const 2) B ζ (1/4) hB hζ (by norm_num) hn hq hH
-  filter_upwards [hfu,hgu,hhu,hcu,hn.eventually_gt_atTop 1] with t hft hgt hht hct hnt
-  intro i
-  let D := sourceDepth B ζ ((q t)^3*Real.log (n t))
-  let N := sourceTargetScale B D (n t) i
-  let ε := sourceEpsilon B D
-  let X := N^((1/4)*ε)
-  have hN : 0<N := Real.rpow_pos_of_pos (by linarith only [hnt]) _
-  have hx : 0<X := Real.rpow_pos_of_pos hN _
-  have hfD := hf0 D (q t) (hq t)
-  have hfX : f D (q t)≤X := hft i
-  have hgX : g D (q t)≤X := hgt i
-  have hhX : h D (q t)*Real.log (n t)≤X := hht i
-  have hX2 : 2≤X := hct i
-  have hpow : N^ε=X^4 := by
-    dsimp [X]
-    rw [←Real.rpow_natCast,←Real.rpow_mul hN.le]
-    norm_num only [Nat.cast_ofNat]
-    congr 1
-    ring
-  have hsize : 2*X*f D (q t) ≤ X^4 := by
-    have h1 := mul_le_mul_of_nonneg_left hfX (show 0≤2*X by positivity)
-    have h2 : 2*X^2 ≤ X^4 := by nlinarith only [hX2,sq_nonneg (X^2-2),sq_nonneg (X-2)]
-    nlinarith only [h1,h2]
-  have hden : 2*X ≤ N^ε/f D (q t) := (le_div_iff₀ hfD).mpr (by rw [hpow]; exact hsize)
-  have hexp : Real.exp (g D (q t)-N^ε/f D (q t)) ≤ (n t)^(-h D (q t)) := by
-    rw [Real.rpow_def_of_pos (by linarith only [hnt] : 0<n t)]
-    exact Real.exp_le_exp.mpr (by nlinarith only [hden,hgX,hhX])
-  apply le_trans _ hexp
-  rw [Real.exp_sub]
-  have he := mul_le_mul_of_nonneg_right
-    (show g D (q t) ≤ Real.exp (g D (q t)) by linarith only [Real.add_one_le_exp (g D (q t))])
-    (Real.exp_pos (-N^ε/f D (q t))).le
-  have hnexp : -N^ε/f D (q t) = -(N^ε/f D (q t)) := by ring
-  rw [hnexp, Real.exp_neg] at he
-  change g D (q t)*Real.exp (-N^ε/f D (q t)) ≤ _
-  rw [hnexp, Real.exp_neg]
-  simpa only [div_eq_mul_inv] using he
-
-end TraceReconstruction
-
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-
-end OAI
+import OAI.Probability.TraceReconstruction.LayoutRates
+
+namespace OAI
+
+/-!
+# ErrorRates
+
+Realization of block caps, sparse scales, sharp and smoothing budgets, tree tails, fit errors, and uniform polynomial/exponential bounds.
+-/
+
+open scoped ENNReal
+noncomputable section
+
+open Filter
+open scoped Topology
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem sourceWindowSize_le (N : ℝ) (b j : ℕ) (ε : ℝ) (hN : 0 ≤ N) :
+    (sourceWindowSize N b ε j:ℝ) ≤ N^(scaleKappa b (j+1) ε) := by
+  have hh := Nat.floor_le (div_nonneg (Real.rpow_nonneg hN (scaleKappa b (j+1) ε)) (by norm_num : (0:ℝ) ≤ 4))
+  unfold sourceWindowSize
+  push_cast
+  linarith only [hh]
+
+-- We use only positive target scales below.
+theorem SourceArithmeticBounds.scheduled_trials {C N F ε : ℝ} {s b ell k : ℕ}
+    (H : SourceArithmeticBounds C N F ε s b ell k) (hN : 0 < N) (hC : 0 < C) (hF : 0 < F)
+    (j : ℕ) (hj : j+1 < b) :
+    N^(scaleKappa b (j+2) ε+ε)/(4800*(C^(2*b)*F)) ≤
+      (sourceScheduledTrials (sourceBlockWindows N b ε) (sourceArithmeticTrials C N b ε ell k) (j+1):ℝ) := by
+  let T := sourceBlockWindows N b ε
+  let n := sourceArithmeticTrials C N b ε ell k
+  have htbig := (H.windows (j+1) hj).1
+  have hsize : (T (j+1):ℝ) ≤ N^(scaleKappa b (j+2) ε) := by
+    dsimp [T]
+    rw [sourceBlockWindows,ite_eq_left hj]
+    have hh := Nat.floor_le (div_nonneg (Real.rpow_nonneg hN.le (scaleKappa b (j+2) ε)) (by norm_num : (0:ℝ) ≤ 4))
+    simp only [sourceWindowSize,Nat.cast_mul,Nat.cast_ofNat]
+    linarith only [hh]
+  have hx : 8 ≤ N^(scaleKappa b (j+2) ε) := by
+    have hh : (24:ℝ) ≤ T (j+1) := by exact_mod_cast htbig
+    linarith only [hh,hsize]
+  have hlo := (sourceWindowSize_bounds (b:=b) (j:=j+1) hx).1
+  have hn := (H.trials j (by omega)).1
+  have hdvd : 4 ∣ T (j+1) := by
+    dsimp [T]
+    rw [sourceBlockWindows,ite_eq_left hj,sourceWindowSize]
+    exact dvd_mul_right _ _
+  have htlo : N^(scaleKappa b (j+2) ε)/8 ≤ ((T (j+1)/4:ℕ):ℝ) := by
+    rw [Nat.cast_div hdvd (by norm_num : (4:ℝ)≠0)]
+    have hh : (sourceWindowSize N b ε (j+1):ℝ)=(T (j+1):ℝ) := by simp [T,sourceBlockWindows,hj]
+    rw [hh] at hlo
+    linarith only [hlo]
+  rw [sourceScheduledTrials,Nat.cast_mul,Real.rpow_add hN]
+  calc
+    _ = (N^(scaleKappa b (j+2) ε)/8)*(N^ε/(600*(C^(2*b)*F))) := by ring
+    _ ≤ _ := mul_le_mul htlo hn (by positivity) (by positivity)
+
+theorem SourceArithmeticBounds.binomial_caps {C N F ε ω : ℝ} {s b ell k : ℕ}
+    (H : SourceArithmeticBounds C N F ε s b ell k) (hN : 0 < N)
+    (hC : 1 ≤ C) (hF : 1 ≤ F) (hω : 0 < ω) :
+    (∀j<b,hiddenBinomialCap (sourceArithmeticTrials C N b ε ell k j) ω ≤
+       (200*(C^(2*b)*F)/ω)*N^(-ε/2)) ∧
+    (∀j,j+1<b →hiddenBinomialCap
+      (sourceScheduledTrials (sourceBlockWindows N b ε) (sourceArithmeticTrials C N b ε ell k) (j+1)) ω ≤
+        (200*(C^(2*b)*F)/ω)*N^(-(scaleKappa b (j+2) ε+ε)/2)) := by
+  let G := C^(2*b)*F
+  have hG : 1 ≤ G := one_le_mul_of_one_le_of_one_le (one_le_pow₀ hC) hF
+  have hcoef : 4*(4800*G) ≤ (200*G/ω)^2*ω^2 := by
+    have hident : (200*G/ω)^2*ω^2=40000*G^2 := by field_simp; ring
+    rw [hident]
+    nlinarith only [hG,sq_nonneg (G-1)]
+  constructor
+  · intro j hj
+    apply hiddenBinomialCap_rpow hN (by positivity : 0 < 600*G) (by positivity) (H.trials j hj).1
+    dsimp [G] at *
+    have hnon : 0 ≤ C^(2*b)*F := by positivity
+    linarith only [hcoef,hnon]
+  · intro j hj
+    exact hiddenBinomialCap_rpow hN (by positivity : 0 < 4800*G) (by positivity)
+      (H.scheduled_trials hN (by linarith only [hC]) (by linarith only [hF]) j hj) hcoef
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem sourceSparseCount_scale (N : ℝ) (b s : ℕ) (ε : ℝ) (hN : 0 < N) :
+    (sourceSparseCount (sourceBlockWindows N b ε) s (b-1):ℝ) ≤
+      ((2^s:ℕ):ℝ)*N^(sourceInteriorExponent b ε) := by
+  unfold sourceSparseCount sourceInteriorExponent
+  push_cast
+  apply mul_le_mul_of_nonneg_left _ (by positivity)
+  rw [Real.rpow_sum_of_pos hN]
+  apply Finset.prod_le_prod₀
+  · intro j _
+    positivity
+  · intro j hj
+    have hj' := Finset.mem_range.mp hj
+    rw [sourceBlockWindows,ite_eq_left (by omega)]
+    exact sourceWindowSize_le N b (j+1) ε hN.le
+
+theorem sourceLeafCount_succ (T : ℕ → ℕ) (j : ℕ) :
+    SourceLeafCount T (j+1)=T j*SourceLeafCount T j := by
+  exact (Fintype.card_prod (α:=Fin (T j)) (β:=WindowLeaf T j)).trans (by rw [Fintype.card_fin])
+
+theorem sourceLeafCount_le_length {P T : ℕ → ℕ} {ell b : ℕ} (hell : 1 ≤ ell)
+    (hPT : ∀j<b,T j ≤ P j) : SourceLeafCount T b ≤ sourceBlockLength P ell b := by
+  induction b with
+  | zero =>
+    change Fintype.card (Fin 1) ≤ ell
+    simpa using hell
+  | succ b ih =>
+    rw [sourceLeafCount_succ,sourceBlockLength]
+    exact Nat.mul_le_mul (hPT b (by omega)) (ih (fun j hj=>hPT j (by omega)))
+
+theorem sourceArithmeticLength_eq_block (C N : ℝ) (b : ℕ) (ε : ℝ) (ell k j : ℕ) :
+    sourceArithmeticLength C N b ε ell k j=
+      sourceBlockLength (sourceArithmeticP C N b ε ell k) ell j := by
+  induction j with
+  | zero => rfl
+  | succ j ih => rw [sourceArithmeticLength_step,sourceBlockLength,ih,mul_comm]
+
+theorem SourceArithmeticBounds.leaf_bound {C N F ε : ℝ} {s b ell k : ℕ}
+    (H : SourceArithmeticBounds C N F ε s b ell k) (hN : 0 < N) :
+    (SourceLeafCount (sourceBlockWindows N b ε) b:ℝ) ≤ N := by
+  have hell : 1 ≤ ell := by
+    have hh := (H.lengths 0 (Nat.zero_le _)).1
+    rw [show sourceArithmeticLength C N b ε ell k 0=ell by rfl] at hh
+    have hp : 0 < N^(scaleA b 0 ε)/2 := by positivity
+    have he : (0:ℝ)<ell := hp.trans_le hh
+    have he' : 0<ell := by exact_mod_cast he
+    omega
+  have hh := sourceLeafCount_le_length (P:=sourceArithmeticP C N b ε ell k) hell
+    (fun j hj=>Nat.le_trans (by omega : sourceBlockWindows N b ε j ≤ 2*sourceBlockWindows N b ε j)
+      (H.windows j hj).2.2.1)
+  have hh' : (SourceLeafCount (sourceBlockWindows N b ε) b:ℝ) ≤
+      sourceArithmeticLength C N b ε ell k b := by
+    rw [sourceArithmeticLength_eq_block]
+    exact_mod_cast hh
+  apply hh'.trans
+  simpa only [scaleA_top,Real.rpow_one] using (H.lengths b le_rfl).2
+
+/-- Full sparse degree, with the excluded bottom window and the exact
+ source gap. This is the high-degree geometric gain, before constants. -/
+theorem source_sparse_weight_bound (N ε d W w : ℝ) (b s : ℕ) (hN : 0 < N) (hW : 0 ≤ W)
+    (hw : w^2 ≤ W*N^(-scaleA b 0 ε*d)) :
+    w^2*(sourceSparseCount (sourceBlockWindows N b ε) s (b-1):ℝ) ≤
+      W*((2^s:ℕ):ℝ)*N^(-(scaleA b 0 ε*d-sourceInteriorExponent b ε)) := by
+  have hr : N^(-scaleA b 0 ε*d)*N^(sourceInteriorExponent b ε)=
+      N^(-(scaleA b 0 ε*d-sourceInteriorExponent b ε)) := by
+    rw [←Real.rpow_add hN]
+    congr 1
+    ring
+  calc
+    _ ≤ (W*N^(-scaleA b 0 ε*d))*(((2^s:ℕ):ℝ)*N^(sourceInteriorExponent b ε)) :=
+      mul_le_mul hw (sourceSparseCount_scale N b s ε hN) (by positivity) (by positivity)
+    _ = (W*((2^s:ℕ):ℝ))*(N^(-scaleA b 0 ε*d)*N^(sourceInteriorExponent b ε)) := by ring
+    _ = _ := by rw [hr]
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+def sourceBadRoleFactor (b R : ℕ) : ℝ :=
+  ∑r∈Finset.range R,4*((3^((b+1)*r):ℕ):ℝ)^2
+
+def sourceForcedRoleFactor (b R : ℕ) : ℝ :=
+  ∑r∈Finset.range R,4*((3^((b+1)*r):ℕ):ℝ)*(OccupiedParityTree.shapeRoleConstant r (b+1):ℝ)
+
+/-- Explicit finite sharp energy bound. The two undesirable pattern classes
+ are suppressed by eta and a, and the genuine binary forced skeleton pays
+ its full source exponent. The high tail uses the sparse, not dense count. -/
+theorem source_sharp_numeric (T n : ℕ→ℕ) (s b R : ℕ) (ω η a w N ε d K W Q gap : ℝ)
+    (hN : 1 ≤ N) (hK : 0 ≤ K) (hW : 0 ≤ W) (_hQ : 0 ≤ Q)
+    (hleaves : (SourceLeafCount T (b+1):ℝ) ≤ N) (hwone : w^2 ≤ 1)
+    (hw : w^2 ≤ W*N^(-scaleA (b+1) 0 ε*d))
+    (hsmall : w^2*((2^s:ℕ):ℝ)*(sourceSparseCount T s b:ℝ) ≤ 1)
+    (hhalf : w^2*(sourceSparseCount T s b:ℝ) ≤ 1/2)
+    (hsparse : w^2*(sourceSparseCount T s b:ℝ) ≤ Q*N^(-gap))
+    (hn : ∀j≤b,hiddenBinomialCap (n j) ω ≤ K*N^(-ε/2))
+    (hg : ∀j<b,hiddenBinomialCap (sourceScheduledTrials T n (j+1)) ω ≤
+      K*N^(-(scaleKappa (b+1) (j+2) ε+ε)/2)) :
+    sourceSharpEnergyBudget T n s b R ω η a w ≤
+      sourceBadRoleFactor b R*N^R*(η^2+a^2)+
+      sourceForcedRoleFactor b R*W^(2^(b+1))*sourceSkeletonPrefactor s b K*N^(-forcedDecay (b+1) ε d)+
+      8*Q^R*N^(-gap*(R:ℝ)) := by
+  have hN0 : 0 < N := by linarith only [hN]
+  let A := W^(2^(b+1))*sourceSkeletonPrefactor s b K*N^(-forcedDecay (b+1) ε d)
+  have hA : 0 ≤ A := by dsimp [A]; positivity [sourceSkeletonPrefactor_nonneg s b hK]
+  have hlow (r : ℕ) (hr : r∈Finset.range R) :
+      w^(2*r)*(if r=0 then 0 else sourceSharpDegreeBudget T n s b r ω η a) ≤
+        4*((3^((b+1)*r):ℕ):ℝ)^2*N^R*(η^2+a^2)+
+          (4*((3^((b+1)*r):ℕ):ℝ)*(OccupiedParityTree.shapeRoleConstant r (b+1):ℝ))*A := by
+    have hr' : r ≤ R := (Finset.mem_range.mp hr).le
+    have hw0 : 0 ≤ w^(2*r) := by rw [pow_mul]; exact pow_nonneg (sq_nonneg w) r
+    have hw1 : w^(2*r) ≤ 1 := by rw [pow_mul]; exact pow_le_one₀ (sq_nonneg w) hwone
+    have hc : ((SourceLeafCount T (b+1)).choose r:ℝ) ≤ N^R := by
+      have hh : ((SourceLeafCount T (b+1)).choose r:ℝ) ≤ (SourceLeafCount T (b+1):ℝ)^r := by
+        exact_mod_cast Nat.choose_le_pow (SourceLeafCount T (b+1)) r
+      exact hh.trans ((pow_le_pow_left₀ (by positivity) hleaves r).trans (pow_le_pow_right₀ hN hr'))
+    have hbad : w^(2*r)*(((SourceLeafCount T (b+1)).choose r:ℝ)) ≤ N^R := by
+      apply (mul_le_mul hw1 hc (by positivity) (by norm_num : (0:ℝ) ≤ 1)).trans_eq
+      exact one_mul _
+    by_cases hr0 : r=0
+    · rw [ite_eq_left hr0,mul_zero]
+      positivity
+    rw [ite_eq_right hr0,sourceSharpDegreeBudget,mul_add]
+    have hB : w^(2*r)*(4*((3^((b+1)*r):ℕ):ℝ)*
+        (((SourceLeafCount T (b+1)).choose r:ℝ)*((3^((b+1)*r):ℕ):ℝ)*(η^2+a^2))) ≤
+        4*((3^((b+1)*r):ℕ):ℝ)^2*N^R*(η^2+a^2) := by
+      have hh := mul_le_mul_of_nonneg_right hbad
+        (show 0 ≤ 4*((3^((b+1)*r):ℕ):ℝ)^2*(η^2+a^2) by positivity)
+      nlinarith only [hh]
+    rw [mul_add]
+    apply add_le_add hB
+    by_cases hrf : 2^(b+1) ≤ r
+    · rw [ite_eq_left hrf]
+      have hh := mul_le_mul_of_nonneg_left
+        (source_forced_degree_decay T n s b r ω N ε d K W w hN0 hK hW hrf hsmall hw hn hg)
+        (show 0 ≤ 4*((3^((b+1)*r):ℕ):ℝ) by positivity)
+      dsimp [A]
+      nlinarith only [hh]
+    · rw [ite_eq_right hrf,mul_zero,mul_zero]
+      positivity
+  have hsum := Finset.sum_le_sum hlow
+  rw [Finset.sum_add_distrib] at hsum
+  have hbadSum : (∑r∈Finset.range R,4*((3^((b+1)*r):ℕ):ℝ)^2*N^R*(η^2+a^2))=
+      sourceBadRoleFactor b R*N^R*(η^2+a^2) := by simp only [sourceBadRoleFactor,Finset.sum_mul]
+  have hforceSum : (∑r∈Finset.range R,(4*((3^((b+1)*r):ℕ):ℝ)*(OccupiedParityTree.shapeRoleConstant r (b+1):ℝ))*A)=
+      sourceForcedRoleFactor b R*W^(2^(b+1))*sourceSkeletonPrefactor s b K*N^(-forcedDecay (b+1) ε d) := by
+    rw [←Finset.sum_mul]
+    dsimp [sourceForcedRoleFactor,A]
+    ring
+  rw [hbadSum,hforceSum] at hsum
+  apply add_le_add hsum
+  have hx : 0 ≤ w^2*(sourceSparseCount T s b:ℝ) := by positivity
+  have htail : 4*(w^2*(sourceSparseCount T s b:ℝ))^R/(1-w^2*(sourceSparseCount T s b:ℝ)) ≤
+      8*(w^2*(sourceSparseCount T s b:ℝ))^R := by
+    apply (div_le_iff₀ (by linarith only [hhalf])).mpr
+    have hh := mul_le_mul_of_nonneg_right hhalf (pow_nonneg hx R)
+    nlinarith only [hh]
+  apply htail.trans
+  have hrp : (N^(-gap))^R=N^(-gap*(R:ℝ)) := by rw [←Real.rpow_natCast,←Real.rpow_mul hN0.le]
+  have hh := mul_le_mul_of_nonneg_left (pow_le_pow_left₀ hx hsparse R) (by norm_num : (0:ℝ)≤8)
+  simpa only [mul_pow,hrp,mul_assoc] using hh
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+def sourceSmoothingPrefactor (s : ℕ) (K : ℝ) : ℝ :=
+  (2:ℝ)^(s*s+s)*(((s:ℝ)+1)*K)^(s+1)
+
+theorem hiddenSmoothingBound_scale {N e K ω : ℝ} {m s : ℕ}
+    (hN : 0 < N) (hK : 0 ≤ K)
+    (hm : hiddenBinomialCap m ω ≤ K*N^(-e/2)) :
+    hiddenSmoothingBound m s ω ≤ sourceSmoothingPrefactor s K*N^(-((s:ℝ)+1)*e/2) := by
+  have hs : 1 ≤ (s:ℝ)+1 := by linarith [Nat.cast_nonneg s (α:=ℝ)]
+  have hs0 : 0 ≤ (s:ℝ)+1 := by positivity
+  have heq : 4*((s:ℝ)+1)/((m:ℝ)*ω^2+2)=((s:ℝ)+1)*(4/((m:ℝ)*ω^2+2)) := by ring
+  have hh : Real.sqrt (4*((s:ℝ)+1)/((m:ℝ)*ω^2+2)) ≤ ((s:ℝ)+1)*K*N^(-e/2) := by
+    rw [heq,Real.sqrt_mul hs0]
+    have hsq : Real.sqrt ((s:ℝ)+1) ≤ (s:ℝ)+1 := by
+      apply (Real.sqrt_le_iff).mpr
+      exact ⟨hs0,by nlinarith only [hs,sq_nonneg ((s:ℝ))]⟩
+    have h := mul_le_mul hsq hm (Real.sqrt_nonneg _) hs0
+    simpa only [hiddenBinomialCap,mul_assoc] using h
+  have hleft := mul_le_mul
+    (pow_le_pow_left₀ (by positivity) (mul_le_mul_of_nonneg_left hh (by norm_num : (0:ℝ)≤2)) s)
+    hh (Real.sqrt_nonneg _) (by positivity)
+  apply (mul_le_mul_of_nonneg_left hleft (by positivity : (0:ℝ)≤2^(s*s))).trans_eq
+  have hr : (N^(-e/2))^(s+1)=N^(-((s:ℝ)+1)*e/2) := by
+    rw [←Real.rpow_natCast,←Real.rpow_mul hN.le]
+    push_cast
+    congr 1
+    ring
+  dsimp [sourceSmoothingPrefactor]
+  rw [pow_add,pow_succ (((s:ℝ)+1)*K)]
+  simp only [mul_pow]
+  have he : (N^(-e/2))^s*N^(-e/2)=N^(-((s:ℝ)+1)*e/2) := by rw [←pow_succ,hr]
+  calc
+    _ = (2:ℝ)^(s*s)*2^s*((s:ℝ)+1)^s*K^s*(((s:ℝ)+1)*K)*
+      ((N^(-e/2))^s*N^(-e/2)) := by ring
+    _ = _ := by rw [he]; ring
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem sourceLeafCount_positive {T : ℕ → ℕ} {b : ℕ} (hT : ∀j<b,1 ≤ T j) :
+    1 ≤ SourceLeafCount T b := by
+  induction b with
+  | zero => change 1 ≤ Fintype.card (Fin 1); simp
+  | succ b ih =>
+    rw [sourceLeafCount_succ]
+    exact one_le_mul_of_one_le_of_one_le (hT b (by omega)) (ih (fun j hj=>hT j (by omega)))
+
+theorem sourceFitTail_simple {T n : ℕ → ℕ} {b : ℕ} {v : ℝ}
+    (hT : ∀j<b,1 ≤ T j)
+    (hv : ∀j,j+1<b → v ≤ 2*((T j:ℝ)/8)^2/(((T (j+1):ℝ)+1)*(n j:ℝ))) :
+    sourceFitTail T n b ≤ 2*(b:ℝ)*(SourceLeafCount T b:ℝ)*Real.exp (-v) := by
+  induction b using Nat.twoStepInduction with
+  | zero => simp [sourceFitTail]
+  | one => simp only [sourceFitTail]; positivity
+  | more b _ ih =>
+    have hi := ih (fun j hj=>hT j (by omega)) (fun j hj=>hv j (by omega))
+    have hleaf : (1:ℝ) ≤ SourceLeafCount T (b+1) := by
+      exact_mod_cast sourceLeafCount_positive (fun j hj=>hT j (show j<b+2 by omega))
+    have he := Real.exp_le_exp.mpr (neg_le_neg (hv b (by omega)))
+    rw [sourceFitTail,sourceLeafCount_succ,Nat.cast_mul]
+    have hmul := mul_le_mul_of_nonneg_left (add_le_add (mul_le_mul_of_nonneg_left he (by norm_num : (0:ℝ)≤2)) hi)
+      (show 0 ≤ (T (b+1):ℝ) by positivity)
+    have hlast := mul_le_mul_of_nonneg_left hleaf (show 0 ≤ (T (b+1):ℝ)*Real.exp (-v) by positivity)
+    push_cast at hmul ⊢
+    have hex : -(2*((T b:ℝ)/8)^2/(((T (b+1):ℝ)+1)*(n b:ℝ))) =
+        -2*((T b:ℝ)/8)^2/(((T (b+1):ℝ)+1)*(n b:ℝ)) := by ring
+    rw [hex] at hmul
+    nlinarith only [hmul,hlast]
+
+theorem windowCollection_card_bound {T : ℕ → ℕ} {b : ℕ} (hT : ∀j<b,1 ≤ T j) :
+    Fintype.card (WindowCollection T b) ≤ 2*b*SourceLeafCount T b := by
+  induction b with
+  | zero => change Fintype.card PEmpty ≤ _; simp
+  | succ b ih =>
+    have hi := ih (fun j hj=>hT j (by omega))
+    have hp := sourceLeafCount_positive (b:=b) (fun j hj=>hT j (show j<b+1 by omega))
+    have ht := hT b (by omega)
+    have hc : Fintype.card (WindowCollection T (b+1))=2+T b*Fintype.card (WindowCollection T b) := by
+      change Fintype.card (Bool ⊕ (Fin (T b) × WindowCollection T b))=_
+      simp only [Fintype.card_sum,Fintype.card_bool,Fintype.card_prod,Fintype.card_fin]
+    rw [hc,sourceLeafCount_succ]
+    have hm := Nat.mul_le_mul_left (T b) hi
+    have hh : 1 ≤ T b*SourceLeafCount T b := one_le_mul_of_one_le_of_one_le ht hp
+    nlinarith only [hm,hh]
+
+theorem collectionCount_lower {T n g : ℕ→ℕ} {b : ℕ} {M : ℝ}
+    (hn : ∀j<b,M ≤ (n j:ℝ)) (hg : ∀j<b,g j=0 ∨ M ≤ (g j:ℝ))
+    (v : WindowCollection T b) :
+    collectionCount T n g b v=0 ∨ M ≤ (collectionCount T n g b v:ℝ) := by
+  induction b with
+  | zero => exact PEmpty.elim v
+  | succ b ih =>
+    rcases v with a | ⟨u,v⟩
+    · cases a
+      · exact Or.inr (hn b (by omega))
+      · exact hg b (by omega)
+    · exact ih (fun j hj=>hn j (by omega)) (fun j hj=>hg j (by omega)) v
+
+theorem sourceEffectiveHiddenTail_simple {T n : ℕ→ℕ} {b : ℕ} {M ω : ℝ}
+    (hT : ∀j<b,1 ≤ T j) (hn : ∀j<b,M ≤ (n j:ℝ))
+    (hg : ∀j<b,sourceScheduledTrials T n j=0 ∨ M ≤ (sourceScheduledTrials T n j:ℝ)) :
+    sourceEffectiveHiddenTail T n b ω ≤ 2*(b:ℝ)*(SourceLeafCount T b:ℝ)*Real.exp (-M*ω^4/16) := by
+  unfold sourceEffectiveHiddenTail
+  have hh (v : WindowCollection T b) :
+      (if collectionCount T n (sourceScheduledTrials T n) b v=0 then 0 else
+        Real.exp (-(collectionCount T n (sourceScheduledTrials T n) b v:ℝ)*ω^4/16)) ≤ Real.exp (-M*ω^4/16) := by
+    rcases collectionCount_lower hn hg v with hz | hm
+    · rw [ite_eq_left hz]; positivity
+    · split_ifs
+      · positivity
+      · apply Real.exp_le_exp.mpr
+        have hh := mul_le_mul_of_nonneg_right hm (pow_nonneg (sq_nonneg ω) 2)
+        nlinarith only [hh]
+  apply (Finset.sum_le_sum (fun v _=>hh v)).trans
+  simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
+  apply mul_le_mul_of_nonneg_right _ (Real.exp_pos _).le
+  exact_mod_cast windowCollection_card_bound hT
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- The literal centering fluctuation has the source epsilon exponent.
+ This does not treat nested window starts as independent. -/
+theorem SourceArithmeticBounds.fit_exponent {C N F ε : ℝ} {s b ell k j : ℕ}
+    (H : SourceArithmeticBounds C N F ε s b ell k) (hC : 0<C) (hF : 0<F) (hN : 1 ≤ N) (he : 0 ≤ ε)
+    (hκ : ∀v<b,3*ε ≤ scaleKappa b (v+1) ε) (hj : j+1<b) :
+    N^ε/512 ≤ 2*((sourceBlockWindows N b ε j:ℝ)/8)^2 /
+      (((sourceBlockWindows N b ε (j+1):ℝ)+1)*(sourceArithmeticTrials C N b ε ell k j:ℝ)) := by
+  have hN0 : 0<N := by linarith only [hN]
+  have hpresent := (H.windows j (by omega)).1
+  have hwin : 24 ≤ sourceWindowSize N b ε j := by
+    simpa only [sourceBlockWindows,ite_eq_left (by omega : j<b)] using hpresent
+  have hlarge : 8 ≤ N^(scaleKappa b (j+1) ε) := by
+    have hh : (24:ℝ) ≤ sourceWindowSize N b ε j := by exact_mod_cast hwin
+    exact (by norm_num : (8:ℝ)≤24).trans (hh.trans (sourceWindowSize_le N b j ε hN0.le))
+  have hl : N^(scaleKappa b (j+1) ε)/2 ≤ (sourceBlockWindows N b ε j:ℝ) := by
+    simpa only [sourceBlockWindows,ite_eq_left (by omega : j<b)] using (sourceWindowSize_bounds hlarge).1
+  have hu : (sourceBlockWindows N b ε (j+1):ℝ)+1 ≤ 2*N^(scaleKappa b (j+2) ε) := by
+    have hh := sourceWindowSize_le N b (j+1) ε hN0.le
+    have he' : 0 ≤ scaleKappa b (j+2) ε := by linarith only [hκ (j+1) hj,he]
+    have ho := Real.one_le_rpow hN he'
+    simpa only [sourceBlockWindows,ite_eq_left hj] using (show (sourceWindowSize N b ε (j+1):ℝ)+1 ≤ 2*N^(scaleKappa b (j+2) ε) by linarith only [hh,ho])
+  have htr := (H.trials j (by omega)).2
+  have hd : ((sourceBlockWindows N b ε (j+1):ℝ)+1)*sourceArithmeticTrials C N b ε ell k j ≤
+      4*N^(scaleKappa b (j+2) ε)*N^ε := by
+    have hh := mul_le_mul hu htr (by positivity) (by positivity)
+    nlinarith only [hh]
+  have htrial : 0 < (sourceArithmeticTrials C N b ε ell k j:ℝ) :=
+    (show 0 < N^ε/(600*(C^(2*b)*F)) by positivity).trans_le (H.trials j (by omega)).1
+  have hdpos : 0 < ((sourceBlockWindows N b ε (j+1):ℝ)+1)*sourceArithmeticTrials C N b ε ell k j := by positivity
+  apply (le_div_iff₀ hdpos).mpr
+  have hmul := mul_le_mul_of_nonneg_left hd (show 0 ≤ N^ε/512 by positivity)
+  have hsquare := pow_le_pow_left₀ (show 0 ≤ N^(scaleKappa b (j+1) ε)/2 by positivity) hl 2
+  have hpow : N^ε*N^(scaleKappa b (j+2) ε)*N^ε=(N^(scaleKappa b (j+1) ε))^2 := by
+    rw [←Real.rpow_add hN0,←Real.rpow_add hN0,←Real.rpow_natCast,←Real.rpow_mul hN0.le]
+    norm_num only [Nat.cast_ofNat]
+    congr 1
+    have hx := scaleKappa_step b (j+1) ε (by omega)
+    linarith only [hx]
+  nlinarith only [hmul,hsquare,hpow]
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open Filter
+open scoped Topology Classical
+
+/-- A convenience for the manuscript's fixed-B polynomial losses. This
+ predicate is only used to prove bounds on the literal finite constants. -/
+def SourcePolyUpper (f : ℕ→ℝ) : Prop :=
+  ∃ C : ℝ,0 ≤ C ∧ ∃ h : ℕ,∀D,f D ≤ C*((D:ℝ)+1)^h
+
+theorem SourcePolyUpper.const (c : ℝ) : SourcePolyUpper (fun _=>c) := by
+  refine ⟨|c|,abs_nonneg _,0,?_⟩
+  intro D
+  simpa using le_abs_self c
+
+theorem SourcePolyUpper.depth : SourcePolyUpper (fun D=>(D:ℝ)+1) := by
+  exact ⟨1,zero_le_one,1,by simp⟩
+
+theorem SourcePolyUpper.mono {f g : ℕ→ℝ} (hg : SourcePolyUpper g) (hf : ∀D,f D ≤ g D) :
+    SourcePolyUpper f := by
+  obtain ⟨C,hC,h,H⟩ := hg
+  exact ⟨C,hC,h,fun D=>(hf D).trans (H D)⟩
+
+theorem SourcePolyUpper.add {f g : ℕ→ℝ} (hf : SourcePolyUpper f) (hg : SourcePolyUpper g) :
+    SourcePolyUpper (fun D=>f D+g D) := by
+  obtain ⟨C,hC,h,H⟩ := hf
+  obtain ⟨K,hK,k,G⟩ := hg
+  refine ⟨C+K,add_nonneg hC hK,h+k,?_⟩
+  intro D
+  have hd : 1 ≤ (D:ℝ)+1 := by linarith [Nat.cast_nonneg D (α:=ℝ)]
+  have hh := (H D).trans (mul_le_mul_of_nonneg_left (pow_le_pow_right₀ hd (by omega : h ≤ h+k)) hC)
+  have hk := (G D).trans (mul_le_mul_of_nonneg_left (pow_le_pow_right₀ hd (by omega : k ≤ h+k)) hK)
+  nlinarith only [hh,hk]
+
+theorem SourcePolyUpper.mul {f g : ℕ→ℝ} (hf : SourcePolyUpper f) (hg : SourcePolyUpper g)
+    (hf0 : ∀D,0 ≤ f D) (hg0 : ∀D,0 ≤ g D) : SourcePolyUpper (fun D=>f D*g D) := by
+  obtain ⟨C,hC,h,H⟩ := hf
+  obtain ⟨K,hK,k,G⟩ := hg
+  refine ⟨C*K,mul_nonneg hC hK,h+k,?_⟩
+  intro D
+  calc
+    f D*g D ≤ (C*((D:ℝ)+1)^h)*(K*((D:ℝ)+1)^k) :=
+      mul_le_mul (H D) (G D) (hg0 D) ((hf0 D).trans (H D))
+    _ = _ := by rw [pow_add]; ring
+
+theorem SourcePolyUpper.pow {f : ℕ→ℝ} (hf : SourcePolyUpper f) (hf0 : ∀D,0 ≤ f D) (r : ℕ) :
+    SourcePolyUpper (fun D=>(f D)^r) := by
+  obtain ⟨C,hC,h,H⟩ := hf
+  refine ⟨C^r,pow_nonneg hC _,h*r,?_⟩
+  intro D
+  simpa only [mul_pow,pow_mul] using pow_le_pow_left₀ (hf0 D) (H D) r
+
+theorem sourceEpsilon_inverse_poly (B : ℕ) :
+    SourcePolyUpper (fun D=>(sourceEpsilon B D)⁻¹) := by
+  refine ⟨(2:ℝ)^(4*B+100),by positivity,1,?_⟩
+  intro D
+  simp only [sourceEpsilon,one_div,inv_inv,pow_one]
+  exact le_rfl
+
+theorem totalScoreExponent_poly (B : ℕ) (hB : 2 ≤ B) :
+    SourcePolyUpper (totalScoreExponent B) := by
+  apply (SourcePolyUpper.const 3 |>.add
+    (SourcePolyUpper.depth.mul (SourcePolyUpper.const (B:ℝ)) (fun D=>by positivity) (fun _=>by positivity))).mono
+  intro D
+  have hh := (totalScoreExponent_bounds B D hB).2
+  nlinarith only [hh,Nat.cast_nonneg B (α:=ℝ)]
+
+theorem SourcePolyUpper.ceil {f : ℕ→ℝ} (hf : SourcePolyUpper f) (hf0 : ∀D,0 ≤ f D) :
+    SourcePolyUpper (fun D=>(⌈f D⌉₊:ℝ)) := by
+  apply (hf.add (SourcePolyUpper.const 1)).mono
+  intro D
+  exact (Nat.ceil_lt_add_one (hf0 D)).le
+
+theorem sourceCancellationOrder_poly (B : ℕ) (hB : 2 ≤ B) :
+    SourcePolyUpper (fun D=>(sourceCancellationOrder B D:ℝ)) := by
+  let f := fun D=>totalScoreExponent B D+(sourceDegreeCutoff B:ℝ)+B+10
+  have hf : SourcePolyUpper f := (((totalScoreExponent_poly B hB).add
+    (SourcePolyUpper.const (sourceDegreeCutoff B:ℝ))).add (SourcePolyUpper.const (B:ℝ))).add
+      (SourcePolyUpper.const 10)
+  have hf0 (D) : 0 ≤ f D := by
+    have hd := (totalScoreExponent_bounds B D hB).1
+    have hr : 0 ≤ (sourceDegreeCutoff B:ℝ) := Nat.cast_nonneg _
+    dsimp [f]
+    linarith only [hd,hr,Nat.cast_nonneg B (α:=ℝ)]
+  have hh := ((SourcePolyUpper.const 100).mul hf (fun _=>by norm_num) hf0).mul
+    (sourceEpsilon_inverse_poly B) (fun D=>mul_nonneg (by norm_num) (hf0 D))
+      (fun D=>(inv_pos.mpr (sourceEpsilon_pos B D)).le)
+  have hc := hh.ceil (fun D=>mul_nonneg (mul_nonneg (by norm_num) (hf0 D))
+    (inv_pos.mpr (sourceEpsilon_pos B D)).le)
+  simpa only [sourceCancellationOrder,f,div_eq_mul_inv] using hc
+
+theorem sourceCancellationOrder_large (B D : ℕ) :
+    100*(totalScoreExponent B D+(sourceDegreeCutoff B:ℝ)+B+10) ≤
+      (sourceCancellationOrder B D:ℝ)*sourceEpsilon B D := by
+  exact (div_le_iff₀ (sourceEpsilon_pos B D)).mp (Nat.le_ceil _)
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open Filter
+open scoped Topology Classical
+
+/-- Literal finite-block factors are subpower uniformly over the joint
+ deletion regime when their logarithms grow polynomially in the depth. -/
+def SourceExpUpper (f : ℕ→ℝ→ℝ) : Prop :=
+  ∃ C : ℝ,0 ≤ C ∧ ∃ h : ℕ,∀D q,q∈Set.Ioc 0 (1/2:ℝ) →
+    f D q ≤ Real.exp (C*((D:ℝ)+1)^h*q^(-3:ℤ))
+
+theorem SourcePolyUpper.expUpper {f : ℕ→ℝ} (hf : SourcePolyUpper f) :
+    SourceExpUpper (fun D _=>f D) := by
+  obtain ⟨C,hC,h,H⟩ := hf
+  refine ⟨C,hC,h,?_⟩
+  intro D q hq
+  have hΛ := deletion_inverse_cube_ge_one ⟨hq.1,by linarith only [hq.2]⟩
+  have hmul := mul_le_mul_of_nonneg_left hΛ (show 0 ≤ C*((D:ℝ)+1)^h by positivity)
+  exact (H D).trans ((by linarith only [hmul] : C*((D:ℝ)+1)^h ≤ C*((D:ℝ)+1)^h*q^(-3:ℤ)+1).trans
+    (Real.add_one_le_exp _))
+
+theorem SourceExpUpper.const (c : ℝ) : SourceExpUpper (fun _ _=>c) :=
+  (SourcePolyUpper.const c).expUpper
+
+theorem SourceExpUpper.mono {f g : ℕ→ℝ→ℝ} (hg : SourceExpUpper g)
+    (hf : ∀D q,q∈Set.Ioc 0 (1/2:ℝ) → f D q ≤ g D q) : SourceExpUpper f := by
+  obtain ⟨C,hC,h,H⟩ := hg
+  exact ⟨C,hC,h,fun D q hq=>(hf D q hq).trans (H D q hq)⟩
+
+theorem SourceExpUpper.mul {f g : ℕ→ℝ→ℝ} (hf : SourceExpUpper f) (hg : SourceExpUpper g)
+    (hf0 : ∀D q,q∈Set.Ioc 0 (1/2:ℝ) → 0 ≤ f D q)
+    (hg0 : ∀D q,q∈Set.Ioc 0 (1/2:ℝ) → 0 ≤ g D q) :
+    SourceExpUpper (fun D q=>f D q*g D q) := by
+  obtain ⟨C,hC,h,H⟩ := hf
+  obtain ⟨K,hK,k,G⟩ := hg
+  refine ⟨C+K,add_nonneg hC hK,h+k,?_⟩
+  intro D q hq
+  have hΛ : 0 ≤ q^(-3:ℤ) := (zpow_pos hq.1 _).le
+  have hd : 1 ≤ (D:ℝ)+1 := by linarith [Nat.cast_nonneg D (α:=ℝ)]
+  have hh := mul_le_mul_of_nonneg_right
+    (mul_le_mul_of_nonneg_left (pow_le_pow_right₀ hd (by omega : h≤h+k)) hC) hΛ
+  have hk := mul_le_mul_of_nonneg_right
+    (mul_le_mul_of_nonneg_left (pow_le_pow_right₀ hd (by omega : k≤h+k)) hK) hΛ
+  calc
+    _ ≤ Real.exp (C*((D:ℝ)+1)^h*q^(-3:ℤ))*Real.exp (K*((D:ℝ)+1)^k*q^(-3:ℤ)) :=
+      mul_le_mul (H D q hq) (G D q hq) (hg0 D q hq) ((hf0 D q hq).trans (H D q hq))
+    _ ≤ _ := by rw [←Real.exp_add]; apply Real.exp_le_exp.mpr; nlinarith only [hh,hk]
+
+theorem SourceExpUpper.pow_variably {f : ℕ→ℝ→ℝ} {r : ℕ→ℕ} (hf : SourceExpUpper f)
+    (hf0 : ∀D q,q∈Set.Ioc 0 (1/2:ℝ) → 0 ≤ f D q)
+    (hr : SourcePolyUpper (fun D=>(r D:ℝ))) : SourceExpUpper (fun D q=>(f D q)^(r D)) := by
+  obtain ⟨C,hC,h,H⟩ := hf
+  obtain ⟨K,hK,k,G⟩ := hr
+  refine ⟨C*K,mul_nonneg hC hK,h+k,?_⟩
+  intro D q hq
+  have hΛ : 0≤q^(-3:ℤ) := (zpow_pos hq.1 _).le
+  have hh := mul_le_mul_of_nonneg_right (G D) (show 0≤C*((D:ℝ)+1)^h*q^(-3:ℤ) by positivity)
+  calc
+    _ ≤ (Real.exp (C*((D:ℝ)+1)^h*q^(-3:ℤ)))^(r D) := pow_le_pow_left₀ (hf0 D q hq) (H D q hq) _
+    _ = Real.exp ((r D:ℝ)*(C*((D:ℝ)+1)^h*q^(-3:ℤ))) := (Real.exp_nat_mul _ _).symm
+    _ ≤ _ := by apply Real.exp_le_exp.mpr; rw [pow_add]; nlinarith only [hh]
+
+theorem SourceExpUpper.pow {f : ℕ→ℝ→ℝ} (hf : SourceExpUpper f)
+    (hf0 : ∀D q,q∈Set.Ioc 0 (1/2:ℝ) → 0 ≤ f D q) (r : ℕ) :
+    SourceExpUpper (fun D q=>(f D q)^r) :=
+  hf.pow_variably hf0 (SourcePolyUpper.const (r:ℝ))
+
+theorem sourceOverlapConstant_expUpper : SourceExpUpper (fun _ q=>sourceOverlapConstant q) := by
+  refine ⟨sourceChannelLogConstant,sourceChannelLogConstant_pos.le,0,?_⟩
+  intro D q hq
+  simp only [pow_zero,mul_one]
+  apply (Real.log_le_iff_le_exp (sourceOverlapConstant_pos hq.1)).mp
+  exact sourceOverlapConstant_log_bound ⟨hq.1,by linarith only [hq.2]⟩
+
+theorem initialOverlapLower_inverse_expUpper : SourceExpUpper (fun _ q=>(initialOverlapLower q)⁻¹) := by
+  refine ⟨1536+(32+3*primitivePeriodConstant),by unfold primitivePeriodConstant; positivity,0,?_⟩
+  intro D q hq
+  have hp := initialOverlapLower_pos q
+  have hΛ := deletion_inverse_cube_ge_one ⟨hq.1,by linarith only [hq.2]⟩
+  rw [←Real.log_le_iff_le_exp (inv_pos.mpr hp),Real.log_inv,initialOverlapLower,
+    Real.log_div (Real.exp_pos _).ne' (by norm_num),Real.log_exp]
+  simp only [pow_zero,mul_one,zpow_neg,zpow_ofNat] at hΛ ⊢
+  have hh := Real.log_le_self (by norm_num : (0:ℝ)≤1536)
+  have hmul := mul_le_mul_of_nonneg_left hΛ (by norm_num : (0:ℝ)≤1536)
+  simp only [div_eq_mul_inv] at *
+  nlinarith only [hh,hmul]
+
+/-- All indices are inside the same physical n,q instance. -/
+theorem SourceExpUpper.uniform (f : ℕ→ℝ→ℝ) (hf : SourceExpUpper f)
+    (B : ℕ) (ζ δ : ℝ) (hB : 2 ≤ B) (hζ : ζ∈Set.Ioo 0 1) (hδ : 0<δ)
+    {n q : ℕ→ℝ} (hn : Tendsto n atTop atTop) (hq : ∀t,q t∈Set.Ioc 0 (1/2:ℝ))
+    (hH : Tendsto (fun t=>(q t)^3*Real.log (n t)) atTop atTop) :
+    ∀ᶠt in atTop,∀i,f (sourceDepth B ζ ((q t)^3*Real.log (n t))) (q t) ≤
+      (sourceTargetScale B (sourceDepth B ζ ((q t)^3*Real.log (n t))) (n t) i)^(
+        δ*sourceEpsilon B (sourceDepth B ζ ((q t)^3*Real.log (n t)))) := by
+  obtain ⟨C,hC,h,H⟩ := hf
+  have hrel (t) : Real.log (n t)=(q t)^(-3:ℤ)*((q t)^3*Real.log (n t)) := by
+    rw [zpow_neg,zpow_ofNat]
+    field_simp [(hq t).1.ne']
+  filter_upwards [source_uniform_exp_absorption B h ζ C δ hB hζ hC hδ hn hH
+    (fun t=>zpow_pos (hq t).1 _) hrel] with t ht i
+  exact (H _ _ (hq t)).trans (ht i)
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open Filter
+open scoped Topology Classical
+
+def sourceMomentEnvelope (B D : ℕ) : ℝ :=
+  ((10^8:ℝ)*2^(10*B)*((B:ℝ)+4)*2+1)*((D:ℝ)+1)*2^(B*D)
+
+def sourceCapEnvelope (B D : ℕ) (q : ℝ) : ℝ :=
+  200*(sourceOverlapConstant q)^(2*B*(D+1)+2)*(initialOverlapLower q)⁻¹
+
+def sourceWeightEnvelope (B D : ℕ) : ℝ := 144*(sourceMomentEnvelope B D)^2
+
+theorem sourceMomentEnvelope_pos (B D : ℕ) : 0 < sourceMomentEnvelope B D := by
+  unfold sourceMomentEnvelope
+  positivity
+
+theorem sourceCapEnvelope_pos (B D : ℕ) {q : ℝ} (hq : 0<q) : 0 < sourceCapEnvelope B D q := by
+  unfold sourceCapEnvelope
+  positivity [sourceOverlapConstant_pos hq,initialOverlapLower_pos q]
+
+theorem sourceMomentEnvelope_expUpper (B : ℕ) : SourceExpUpper (fun D _=>sourceMomentEnvelope B D) := by
+  have hr : SourcePolyUpper (fun D=>((B*D:ℕ):ℝ)) := by
+    apply ((SourcePolyUpper.const (B:ℝ)).mul SourcePolyUpper.depth (fun _=>by positivity) (fun _=>by positivity)).mono
+    intro D
+    push_cast
+    nlinarith only [Nat.cast_nonneg B (α:=ℝ)]
+  have hpow := (SourceExpUpper.const 2).pow_variably (fun _ _ _=>by norm_num) hr
+  have hp := ((SourcePolyUpper.const ((10^8:ℝ)*2^(10*B)*((B:ℝ)+4)*2+1)).mul
+    SourcePolyUpper.depth (fun _=>by positivity) (fun _=>by positivity)).expUpper
+  exact hp.mul hpow (fun _ _ _=>by positivity) (fun _ _ _=>by positivity)
+
+theorem sourceCapEnvelope_expUpper (B : ℕ) : SourceExpUpper (sourceCapEnvelope B) := by
+  have hr : SourcePolyUpper (fun D=>((2*B*(D+1)+2:ℕ):ℝ)) := by
+    have hp := ((SourcePolyUpper.const ((2*B:ℕ):ℝ)).mul SourcePolyUpper.depth
+      (fun _=>by positivity) (fun _=>by positivity)).add (SourcePolyUpper.const 2)
+    simpa only [Nat.cast_mul,Nat.cast_add,Nat.cast_one,Nat.cast_ofNat] using hp
+  have hc := sourceOverlapConstant_expUpper.pow_variably (fun _ q hq=>(sourceOverlapConstant_pos hq.1).le) hr
+  exact ((SourceExpUpper.const 200).mul hc (fun _ _ _=>by norm_num)
+    (fun _ q hq=>pow_nonneg (sourceOverlapConstant_pos hq.1).le _)).mul initialOverlapLower_inverse_expUpper
+    (fun _ q hq=>by positivity [sourceOverlapConstant_pos hq.1])
+    (fun _ q _=>(inv_pos.mpr (initialOverlapLower_pos q)).le)
+
+theorem sourceWeightEnvelope_expUpper (B : ℕ) : SourceExpUpper (fun D _=>sourceWeightEnvelope B D) := by
+  exact (SourceExpUpper.const 144).mul
+    ((sourceMomentEnvelope_expUpper B).pow (fun D _ _=>(sourceMomentEnvelope_pos B D).le) 2)
+    (fun _ _ _=>by norm_num) (fun _ _ _=>sq_nonneg _)
+
+theorem sourceMotif_expUpper (B : ℕ) (hB : 2 ≤ B) :
+    SourceExpUpper (fun D _=>(2:ℝ)^(sourceCancellationOrder B D)) :=
+  (SourceExpUpper.const 2).pow_variably (fun _ _ _=>by norm_num) (sourceCancellationOrder_poly B hB)
+
+theorem sourceSmoothingPrefactor_expUpper (B : ℕ) (hB : 2 ≤ B) :
+    SourceExpUpper (fun D q=>sourceSmoothingPrefactor (sourceCancellationOrder B D) (sourceCapEnvelope B D q)) := by
+  let s := sourceCancellationOrder B
+  have hs := sourceCancellationOrder_poly B hB
+  have hs0 (D) : 0 ≤ (s D:ℝ) := Nat.cast_nonneg _
+  have hp : SourcePolyUpper (fun D=>((s D*s D+s D:ℕ):ℝ)) := by
+    simpa only [Nat.cast_add,Nat.cast_mul] using (hs.mul hs hs0 hs0).add hs
+  have ht : SourcePolyUpper (fun D=>((s D+1:ℕ):ℝ)) := by
+    simpa only [Nat.cast_add,Nat.cast_one] using hs.add (SourcePolyUpper.const 1)
+  have hone := (SourceExpUpper.const 2).pow_variably (fun _ _ _=>by norm_num) hp
+  have htwo := ((ht.expUpper).mul (sourceCapEnvelope_expUpper B)
+    (fun _ _ _=>Nat.cast_nonneg _) (fun D _ hq=>(sourceCapEnvelope_pos B D hq.1).le)).pow_variably
+      (fun D q hq=>mul_nonneg (Nat.cast_nonneg _) (sourceCapEnvelope_pos B D hq.1).le) ht
+  have H := hone.mul htwo (fun _ _ _=>by positivity)
+    (fun D q hq=>pow_nonneg (mul_nonneg (Nat.cast_nonneg _) (sourceCapEnvelope_pos B D hq.1).le) _)
+  simpa only [sourceSmoothingPrefactor,s,Nat.cast_add,Nat.cast_one] using H
+
+theorem sourceSkeletonPrefactor_expUpper (B b : ℕ) (hB : 2 ≤ B) :
+    SourceExpUpper (fun D q=>sourceSkeletonPrefactor (sourceCancellationOrder B D) b (sourceCapEnvelope B D q)) := by
+  have hL := sourceMotif_expUpper B hB
+  have hK := sourceCapEnvelope_expUpper B
+  have hA := ((SourceExpUpper.const 2).mul (hL.pow (fun _ _ _=>by positivity) 3)
+    (fun _ _ _=>by norm_num) (fun _ _ _=>by positivity)).mul hK
+      (fun _ _ _=>by positivity) (fun D _ hq=>(sourceCapEnvelope_pos B D hq.1).le)
+  have hF := ((SourceExpUpper.const 4).mul (hL.pow (fun _ _ _=>by positivity) 6)
+    (fun _ _ _=>by norm_num) (fun _ _ _=>by positivity)).mul
+      (hK.pow (fun D _ hq=>(sourceCapEnvelope_pos B D hq.1).le) 2)
+      (fun _ _ _=>by positivity) (fun _ _ _=>sq_nonneg _)
+  have H := hA.mul (hF.pow (fun _ _ _=>by positivity) (2^b-1))
+    (fun D q hq=>by positivity [sourceCapEnvelope_pos B D hq.1]) (fun _ _ _=>by positivity)
+  simpa only [sourceSkeletonPrefactor,binaryTreeBudget_const,Nat.cast_pow,Nat.cast_ofNat] using H
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem SourceArithmeticBounds.fit_tail {C N F ε : ℝ} {s b ell k : ℕ}
+    (H : SourceArithmeticBounds C N F ε s b ell k) (hC : 0<C) (hF : 0<F)
+    (hN : 1≤N) (he : 0≤ε) (hκ : ∀v<b,3*ε ≤ scaleKappa b (v+1) ε) :
+    sourceFitTail (sourceBlockWindows N b ε) (sourceArithmeticTrials C N b ε ell k) b ≤
+      2*(b:ℝ)*N*Real.exp (-(N^ε/512)) := by
+  have hN0 : 0<N := by linarith only [hN]
+  apply (sourceFitTail_simple (fun j hj=>(by omega : 1≤24).trans (H.windows j hj).1)
+    (fun j hj=>H.fit_exponent hC hF hN he hκ hj)).trans
+  have hh := mul_le_mul_of_nonneg_left (H.leaf_bound hN0) (show 0≤2*(b:ℝ) by positivity)
+  exact mul_le_mul_of_nonneg_right hh (Real.exp_pos _).le
+
+theorem SourceArithmeticBounds.metadata_tail {C N F ε ω : ℝ} {s b ell k : ℕ}
+    (H : SourceArithmeticBounds C N F ε s b ell k) (hN : 0<N) :
+    sourceEffectiveHiddenTail (sourceBlockWindows N b ε) (sourceArithmeticTrials C N b ε ell k) b ω ≤
+      2*(b:ℝ)*N*Real.exp (-(N^ε/(600*(C^(2*b)*F)))*ω^4/16) := by
+  apply (sourceEffectiveHiddenTail_simple (fun j hj=>(by omega : 1≤24).trans (H.windows j hj).1)
+    (fun j hj=>(H.trials j hj).1) ?_).trans
+  · exact mul_le_mul_of_nonneg_right
+      (mul_le_mul_of_nonneg_left (H.leaf_bound hN) (show 0≤2*(b:ℝ) by positivity)) (Real.exp_pos _).le
+  · intro j hj
+    cases j with
+    | zero => exact Or.inl rfl
+    | succ j =>
+      right
+      rw [sourceScheduledTrials,Nat.cast_mul]
+      have ht := (H.windows (j+1) hj).1
+      have hd : 1 ≤ sourceBlockWindows N b ε (j+1)/4 := by omega
+      have hd' : (1:ℝ) ≤ (sourceBlockWindows N b ε (j+1)/4:ℕ) := by exact_mod_cast hd
+      exact (H.trials j (by omega)).1.trans
+        (le_mul_of_one_le_left (Nat.cast_nonneg _) hd')
+
+/-- Quantitative clipping without any default-limit increase or limiting
+ interchange: the power is the finite even moment actually used by the oracle. -/
+theorem source_clipping_moment_power (T : ℕ→ℕ) (s b k : ℕ) (N gap L : ℝ)
+    (hN : 0<N)
+    (hhalf : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b ≤ 1/2)
+    (hscale : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b ≤ N^(-gap/2)/8) :
+    Real.sqrt (6*(4*sourceCenteredBudget T s b k L)^k) ≤ 3*N^(-gap*(k:ℝ)/4) := by
+  let x := ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b
+  have hx : 0≤x := by dsimp [x]; positivity
+  have hden : 0<1-x := by dsimp [x]; linarith only [hhalf]
+  have hc : 4*sourceCenteredBudget T s b k L ≤ N^(-gap/2) := by
+    have hh : x/(1-x) ≤ 2*x := by
+      apply (div_le_iff₀ hden).mpr
+      have hxhalf : x≤1/2 := hhalf
+      nlinarith only [mul_le_mul_of_nonneg_left hxhalf hx]
+    change 4*(x/(1-x)) ≤ _
+    nlinarith only [hh,hscale]
+  have hc0 : 0 ≤ 4*sourceCenteredBudget T s b k L :=
+    mul_nonneg (by norm_num) (sourceCenteredBudget_nonneg T s b k L (by linarith only [hhalf]))
+  have hh := pow_le_pow_left₀ hc0 hc k
+  have hr : (N^(-gap/2))^k=(N^(-gap*(k:ℝ)/4))^2 := by
+    rw [←Real.rpow_natCast,←Real.rpow_mul hN.le,←Real.rpow_natCast,←Real.rpow_mul hN.le]
+    norm_num only [Nat.cast_ofNat]
+    congr 1
+    ring
+  rw [hr] at hh
+  apply (Real.sqrt_le_left (by positivity : 0≤3*N^(-gap*(k:ℝ)/4))).mpr
+  nlinarith only [hh,sq_nonneg (N^(-gap*(k:ℝ)/4))]
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open Filter
+open scoped Topology Classical
+
+theorem SourceExpUpper.add {f g : ℕ→ℝ→ℝ} (hf : SourceExpUpper f) (hg : SourceExpUpper g) :
+    SourceExpUpper (fun D q=>f D q+g D q) := by
+  obtain ⟨C,hC,h,H⟩ := hf
+  obtain ⟨K,hK,k,G⟩ := hg
+  refine ⟨C+K+1,by positivity,h+k,?_⟩
+  intro D q hq
+  have hΛ := deletion_inverse_cube_ge_one ⟨hq.1,by linarith only [hq.2]⟩
+  have hd : 1≤(D:ℝ)+1 := by linarith [Nat.cast_nonneg D (α:=ℝ)]
+  have hh := mul_le_mul_of_nonneg_right
+    (mul_le_mul_of_nonneg_left (pow_le_pow_right₀ hd (by omega : h≤h+k)) hC) (by linarith only [hΛ] : 0≤q^(-3:ℤ))
+  have hk := mul_le_mul_of_nonneg_right
+    (mul_le_mul_of_nonneg_left (pow_le_pow_right₀ hd (by omega : k≤h+k)) hK) (by linarith only [hΛ] : 0≤q^(-3:ℤ))
+  have hz : 1≤((D:ℝ)+1)^(h+k)*q^(-3:ℤ) := one_le_mul_of_one_le_of_one_le (one_le_pow₀ hd) hΛ
+  have hc0 : 0≤C*((D:ℝ)+1)^(h+k)*q^(-3:ℤ) := by positivity
+  have hk0 : 0≤K*((D:ℝ)+1)^(h+k)*q^(-3:ℤ) := by positivity
+  have he1 : f D q ≤ Real.exp ((C+K)*((D:ℝ)+1)^(h+k)*q^(-3:ℤ)) :=
+    (H D q hq).trans (Real.exp_le_exp.mpr (by nlinarith only [hh,hk0]))
+  have he2 : g D q ≤ Real.exp ((C+K)*((D:ℝ)+1)^(h+k)*q^(-3:ℤ)) :=
+    (G D q hq).trans (Real.exp_le_exp.mpr (by nlinarith only [hk,hc0]))
+  have he := Real.add_one_le_exp (((D:ℝ)+1)^(h+k)*q^(-3:ℤ))
+  have he' := mul_le_mul_of_nonneg_left (show 2≤Real.exp (((D:ℝ)+1)^(h+k)*q^(-3:ℤ)) by linarith only [he,hz])
+    (Real.exp_pos ((C+K)*((D:ℝ)+1)^(h+k)*q^(-3:ℤ))).le
+  rw [←Real.exp_add] at he'
+  convert (show f D q+g D q ≤ Real.exp ((C+K)*((D:ℝ)+1)^(h+k)*q^(-3:ℤ)+((D:ℝ)+1)^(h+k)*q^(-3:ℤ)) by linarith only [he1,he2,he']) using 1
+  congr 1
+  ring
+
+def sourceScaleInverseEnvelope (B D : ℕ) : ℝ := 2*(2:ℝ)^(B*D)
+
+theorem sourceScaleInverseEnvelope_expUpper (B : ℕ) : SourceExpUpper (fun D _=>sourceScaleInverseEnvelope B D) := by
+  have hr : SourcePolyUpper (fun D=>((B*D:ℕ):ℝ)) := by
+    apply ((SourcePolyUpper.const (B:ℝ)).mul SourcePolyUpper.depth (fun _=>by positivity) (fun _=>by positivity)).mono
+    intro D
+    push_cast
+    nlinarith only [Nat.cast_nonneg B (α:=ℝ)]
+  exact (SourceExpUpper.const 2).mul ((SourceExpUpper.const 2).pow_variably (fun _ _ _=>by norm_num) hr)
+    (fun _ _ _=>by norm_num) (fun _ _ _=>by positivity)
+
+theorem source_log_scale_lower (B D i : ℕ) {n : ℝ} (hB : 2≤B) (hn : 1<n) :
+    Real.log n ≤ sourceScaleInverseEnvelope B D*Real.log (sourceTargetScale B D n i) := by
+  have hh := mul_le_mul_of_nonneg_right (sourceScaleProduct_lower B D i hB) (Real.log_pos hn).le
+  rw [sourceTargetScale,Real.log_rpow (by linarith only [hn] : 0<n)]
+  dsimp [sourceScaleInverseEnvelope]
+  have hp : 0<(2:ℝ)*2^(B*D) := by positivity
+  have H := mul_le_mul_of_nonneg_left hh hp.le
+  field_simp at H
+  nlinarith only [H]
+
+theorem SourceExpUpper.uniform_log {f : ℕ→ℝ→ℝ} (hf : SourceExpUpper f)
+    (hf0 : ∀D q,q∈Set.Ioc 0 (1/2:ℝ) → 0≤f D q)
+    (B : ℕ) (ζ δ : ℝ) (hB : 2≤B) (hζ : ζ∈Set.Ioo 0 1) (hδ : 0<δ)
+    {n q : ℕ→ℝ} (hn : Tendsto n atTop atTop) (hq : ∀t,q t∈Set.Ioc 0 (1/2:ℝ))
+    (hH : Tendsto (fun t=>(q t)^3*Real.log (n t)) atTop atTop) :
+    ∀ᶠt in atTop,∀i,f (sourceDepth B ζ ((q t)^3*Real.log (n t))) (q t)*Real.log (n t) ≤
+      (sourceTargetScale B (sourceDepth B ζ ((q t)^3*Real.log (n t))) (n t) i)^
+        (δ*sourceEpsilon B (sourceDepth B ζ ((q t)^3*Real.log (n t)))) := by
+  let g := fun D q=>f D q*sourceScaleInverseEnvelope B D*(2/δ)*(sourceEpsilon B D)⁻¹
+  have hg : SourceExpUpper g :=
+    ((hf.mul (sourceScaleInverseEnvelope_expUpper B) hf0 (fun _ _ _=>by unfold sourceScaleInverseEnvelope; positivity)).mul
+      (SourceExpUpper.const (2/δ)) (fun D q hq=>by unfold sourceScaleInverseEnvelope; positivity [hf0 D q hq]) (fun _ _ _=>by positivity)).mul
+        (sourceEpsilon_inverse_poly B).expUpper (fun D q hq=>by unfold sourceScaleInverseEnvelope; positivity [hf0 D q hq])
+          (fun D _ _=>(inv_pos.mpr (sourceEpsilon_pos B D)).le)
+  have hu := SourceExpUpper.uniform g hg B ζ (δ/2) hB hζ (by positivity) hn hq hH
+  filter_upwards [hu,hn.eventually_gt_atTop 1] with t ht hnt
+  intro i
+  let D := sourceDepth B ζ ((q t)^3*Real.log (n t))
+  let N := sourceTargetScale B D (n t) i
+  let ε := sourceEpsilon B D
+  have he : 0<ε := sourceEpsilon_pos B D
+  have hN : 0<N := Real.rpow_pos_of_pos (by linarith only [hnt]) _
+  have hl := source_log_scale_lower B D i hB hnt
+  have hb := Real.log_le_rpow_div hN.le (show 0<δ*ε/2 by positivity)
+  have hmul := mul_le_mul_of_nonneg_left hb
+    (show 0≤f D (q t)*sourceScaleInverseEnvelope B D by unfold sourceScaleInverseEnvelope; positivity [hf0 D _ (hq t)])
+  have hg' := mul_le_mul_of_nonneg_right (ht i) (Real.rpow_pos_of_pos hN (δ*ε/2)).le
+  have hr : N^((δ/2)*ε)*N^(δ*ε/2)=N^(δ*ε) := by rw [←Real.rpow_add hN]; congr 1; ring
+  change g D (q t)*N^(δ*ε/2) ≤ N^((δ/2)*ε)*N^(δ*ε/2) at hg'
+  rw [hr] at hg'
+  have heq : f D (q t)*sourceScaleInverseEnvelope B D*(N^(δ*ε/2)/(δ*ε/2))=g D (q t)*N^(δ*ε/2) := by
+    dsimp [g,ε]
+    field_simp
+  have hfirst := mul_le_mul_of_nonneg_left hl (hf0 D (q t) (hq t))
+  change f D (q t)*Real.log (n t) ≤ N^(δ*ε)
+  rw [heq] at hmul
+  nlinarith only [hfirst,hmul,hg']
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open Filter
+open scoped Topology Classical
+
+/-- Uniform exponential domination at every working scale, with the
+ full depth and the varying known deletion probability retained. -/
+theorem SourceExpUpper.uniform_tail {f g h : ℕ→ℝ→ℝ}
+    (hf : SourceExpUpper f) (hg : SourceExpUpper g) (hh : SourceExpUpper h)
+    (hf0 : ∀D q,q∈Set.Ioc 0 (1/2:ℝ) → 0<f D q)
+    (_hg0 : ∀D q,q∈Set.Ioc 0 (1/2:ℝ) → 0≤g D q)
+    (hh0 : ∀D q,q∈Set.Ioc 0 (1/2:ℝ) → 0≤h D q)
+    (B : ℕ) (ζ : ℝ) (hB : 2≤B) (hζ : ζ∈Set.Ioo 0 1)
+    {n q : ℕ→ℝ} (hn : Tendsto n atTop atTop) (hq : ∀t,q t∈Set.Ioc 0 (1/2:ℝ))
+    (hH : Tendsto (fun t=>(q t)^3*Real.log (n t)) atTop atTop) :
+    ∀ᶠt in atTop,∀i,
+      g (sourceDepth B ζ ((q t)^3*Real.log (n t))) (q t)*
+        Real.exp (-(sourceTargetScale B (sourceDepth B ζ ((q t)^3*Real.log (n t))) (n t) i)^
+          (sourceEpsilon B (sourceDepth B ζ ((q t)^3*Real.log (n t))))/
+          f (sourceDepth B ζ ((q t)^3*Real.log (n t))) (q t)) ≤
+      (n t)^(-(h (sourceDepth B ζ ((q t)^3*Real.log (n t))) (q t))) := by
+  have hfu := SourceExpUpper.uniform f hf B ζ (1/4) hB hζ (by norm_num) hn hq hH
+  have hgu := SourceExpUpper.uniform g hg B ζ (1/4) hB hζ (by norm_num) hn hq hH
+  have hhu := hh.uniform_log hh0 B ζ (1/4) hB hζ (by norm_num) hn hq hH
+  have hcu := SourceExpUpper.uniform _ (SourceExpUpper.const 2) B ζ (1/4) hB hζ (by norm_num) hn hq hH
+  filter_upwards [hfu,hgu,hhu,hcu,hn.eventually_gt_atTop 1] with t hft hgt hht hct hnt
+  intro i
+  let D := sourceDepth B ζ ((q t)^3*Real.log (n t))
+  let N := sourceTargetScale B D (n t) i
+  let ε := sourceEpsilon B D
+  let X := N^((1/4)*ε)
+  have hN : 0<N := Real.rpow_pos_of_pos (by linarith only [hnt]) _
+  have hx : 0<X := Real.rpow_pos_of_pos hN _
+  have hfD := hf0 D (q t) (hq t)
+  have hfX : f D (q t)≤X := hft i
+  have hgX : g D (q t)≤X := hgt i
+  have hhX : h D (q t)*Real.log (n t)≤X := hht i
+  have hX2 : 2≤X := hct i
+  have hpow : N^ε=X^4 := by
+    dsimp [X]
+    rw [←Real.rpow_natCast,←Real.rpow_mul hN.le]
+    norm_num only [Nat.cast_ofNat]
+    congr 1
+    ring
+  have hsize : 2*X*f D (q t) ≤ X^4 := by
+    have h1 := mul_le_mul_of_nonneg_left hfX (show 0≤2*X by positivity)
+    have h2 : 2*X^2 ≤ X^4 := by nlinarith only [hX2,sq_nonneg (X^2-2),sq_nonneg (X-2)]
+    nlinarith only [h1,h2]
+  have hden : 2*X ≤ N^ε/f D (q t) := (le_div_iff₀ hfD).mpr (by rw [hpow]; exact hsize)
+  have hexp : Real.exp (g D (q t)-N^ε/f D (q t)) ≤ (n t)^(-h D (q t)) := by
+    rw [Real.rpow_def_of_pos (by linarith only [hnt] : 0<n t)]
+    exact Real.exp_le_exp.mpr (by nlinarith only [hden,hgX,hhX])
+  apply le_trans _ hexp
+  rw [Real.exp_sub]
+  have he := mul_le_mul_of_nonneg_right
+    (show g D (q t) ≤ Real.exp (g D (q t)) by linarith only [Real.add_one_le_exp (g D (q t))])
+    (Real.exp_pos (-N^ε/f D (q t))).le
+  have hnexp : -N^ε/f D (q t) = -(N^ε/f D (q t)) := by ring
+  rw [hnexp, Real.exp_neg] at he
+  change g D (q t)*Real.exp (-N^ε/f D (q t)) ≤ _
+  rw [hnexp, Real.exp_neg]
+  simpa only [div_eq_mul_inv] using he
+
+end TraceReconstruction
+
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/TraceReconstruction/FiniteLaw.lean b/lean/OAI/Probability/TraceReconstruction/FiniteLaw.lean
index 8872ef594ec98bb4fe8684d0a87abb974c248e63..65f9f6cd47cc1db19506300b64ef52704e89d392 100644
--- a/lean/OAI/Probability/TraceReconstruction/FiniteLaw.lean
+++ b/lean/OAI/Probability/TraceReconstruction/FiniteLaw.lean
@@ -1,1487 +1,1487 @@
-import Mathlib
-
-namespace OAI
-
-/-!
-# FiniteLaw
-
-Finite laws, common-reference total variation, Hellinger affinity, overlap, scalar moments, hypercontractivity, clipping, and leaf densities.
--/
-
-open scoped ENNReal
-noncomputable section
-
-noncomputable section
-namespace TraceReconstruction
-
-/-- Finite probability laws used for the exact statistical transfer. -/
-structure FiniteLaw (α : Type*) [Fintype α] where
-  mass : α → ℝ
-  nonneg : ∀ a, 0 ≤ mass a
-  total : ∑ a, mass a = 1
-
-namespace FiniteLaw
-variable {α β : Type*} [Fintype α] [Fintype β]
-
-def tv (P Q : FiniteLaw α) : ℝ := (1 / 2 : ℝ) * ∑ a, |P.mass a - Q.mass a|
-def hellingerSq (P Q : FiniteLaw α) : ℝ :=
-  ∑ a, (Real.sqrt (P.mass a) - Real.sqrt (Q.mass a)) ^ 2
-def affinity (P Q : FiniteLaw α) : ℝ :=
-  ∑ a, Real.sqrt (P.mass a) * Real.sqrt (Q.mass a)
-
-theorem affinity_nonneg (P Q : FiniteLaw α) : 0 ≤ affinity P Q := by
-  apply Finset.sum_nonneg
-  intro a _; positivity
-
-theorem hellingerSq_nonneg (P Q : FiniteLaw α) : 0 ≤ hellingerSq P Q := by
-  apply Finset.sum_nonneg
-  intro a _; positivity
-
-theorem hellingerSq_eq (P Q : FiniteLaw α) :
-    hellingerSq P Q = 2 - 2 * affinity P Q := by
-  unfold hellingerSq affinity
-  calc
-    _ = (∑ a, P.mass a) + (∑ a, Q.mass a) - 2 * ∑ a,
-        Real.sqrt (P.mass a) * Real.sqrt (Q.mass a) := by
-      rw [Finset.mul_sum, ← Finset.sum_add_distrib, ← Finset.sum_sub_distrib]
-      apply Finset.sum_congr rfl
-      intro a _
-      have hp := Real.sq_sqrt (P.nonneg a)
-      have hq := Real.sq_sqrt (Q.nonneg a)
-      nlinarith only [hp, hq]
-    _ = _ := by rw [P.total, Q.total]; ring
-
-theorem affinity_le_one (P Q : FiniteLaw α) : affinity P Q ≤ 1 := by
-  have h := hellingerSq_nonneg P Q
-  rw [hellingerSq_eq] at h
-  linarith
-
-theorem tv_nonneg (P Q : FiniteLaw α) : 0 ≤ tv P Q := by unfold tv; positivity
-
-theorem tv_sq_le_hellingerSq (P Q : FiniteLaw α) :
-    tv P Q ^ 2 ≤ hellingerSq P Q := by
-  have hpoint (a : α) : |P.mass a - Q.mass a| ^ 2 ≤
-      (Real.sqrt (P.mass a) - Real.sqrt (Q.mass a)) ^ 2 *
-        (Real.sqrt (P.mass a) + Real.sqrt (Q.mass a)) ^ 2 := by
-    rw [sq_abs]
-    have hp := Real.sq_sqrt (P.nonneg a)
-    have hq := Real.sq_sqrt (Q.nonneg a)
-    nlinarith only [hp, hq, sq_nonneg (P.mass a - Q.mass a)]
-  have h := Finset.sum_sq_le_sum_mul_sum_of_sq_le_mul
-    (s := Finset.univ)
-    (r := fun a => |P.mass a - Q.mass a|)
-    (f := fun a => (Real.sqrt (P.mass a) - Real.sqrt (Q.mass a)) ^ 2)
-    (g := fun a => (Real.sqrt (P.mass a) + Real.sqrt (Q.mass a)) ^ 2)
-    (fun a _ => sq_nonneg _) (fun a _ => sq_nonneg _) (fun a _ => hpoint a)
-  have hplus : (∑ a, (Real.sqrt (P.mass a) + Real.sqrt (Q.mass a)) ^ 2) =
-      2 + 2 * affinity P Q := by
-    unfold affinity
-    calc
-      _ = (∑ a, P.mass a) + (∑ a, Q.mass a) + 2 * ∑ a,
-          Real.sqrt (P.mass a) * Real.sqrt (Q.mass a) := by
-        rw [Finset.mul_sum, ← Finset.sum_add_distrib, ← Finset.sum_add_distrib]
-        apply Finset.sum_congr rfl
-        intro a _
-        have hp := Real.sq_sqrt (P.nonneg a)
-        have hq := Real.sq_sqrt (Q.nonneg a)
-        nlinarith only [hp, hq]
-      _ = _ := by rw [P.total, Q.total]; ring
-  rw [hplus] at h
-  change (∑ a, |P.mass a - Q.mass a|) ^ 2 ≤ hellingerSq P Q * (2 + 2 * affinity P Q) at h
-  have hbound := mul_le_mul_of_nonneg_left (affinity_le_one P Q) (hellingerSq_nonneg P Q)
-  unfold tv
-  nlinarith only [h, hbound]
-
-theorem tv_le_sqrt_hellingerSq (P Q : FiniteLaw α) :
-    tv P Q ≤ Real.sqrt (hellingerSq P Q) := by
-  have h := tv_sq_le_hellingerSq P Q
-  have hs := Real.sq_sqrt (hellingerSq_nonneg P Q)
-  have hn := Real.sqrt_nonneg (hellingerSq P Q)
-  nlinarith only [h, hs, hn]
-
-def product (P : FiniteLaw α) (m : ℕ) : FiniteLaw (Fin m → α) where
-  mass a := ∏ j, P.mass (a j)
-  nonneg a := Finset.prod_nonneg fun j _ => P.nonneg (a j)
-  total := by
-    rw [← Fintype.prod_sum]
-    simp [P.total]
-
-theorem affinity_product (P Q : FiniteLaw α) (m : ℕ) :
-    affinity (P.product m) (Q.product m) = affinity P Q ^ m := by
-  unfold affinity product
-  simp only
-  have hexp (a : Fin m → α) :
-      Real.sqrt (∏ j, P.mass (a j)) * Real.sqrt (∏ j, Q.mass (a j)) =
-        ∏ j, (Real.sqrt (P.mass (a j)) * Real.sqrt (Q.mass (a j))) := by
-    rw [Real.sqrt_prod _ (fun j _ => P.nonneg (a j)),
-      Real.sqrt_prod _ (fun j _ => Q.nonneg (a j)), Finset.prod_mul_distrib]
-  simp_rw [hexp]
-  rw [← Fintype.prod_sum (fun (_ : Fin m) a => Real.sqrt (P.mass a) * Real.sqrt (Q.mass a))]
-  simp
-
-theorem hellingerSq_product_le (P Q : FiniteLaw α) (m : ℕ) :
-    hellingerSq (P.product m) (Q.product m) ≤ (m : ℝ) * hellingerSq P Q := by
-  rw [hellingerSq_eq, affinity_product, hellingerSq_eq]
-  have ha := affinity_nonneg P Q
-  have hb := affinity_le_one P Q
-  have hpow : 1 - affinity P Q ^ m ≤ (m : ℝ) * (1 - affinity P Q) := by
-    induction m with
-    | zero => simp
-    | succ m ih =>
-      rw [pow_succ, Nat.cast_add, Nat.cast_one]
-      have hmul := mul_le_mul_of_nonneg_right ih ha
-      have hm : (0 : ℝ) ≤ (m : ℝ) * (1 - affinity P Q) := by positivity
-      have hmon := mul_le_mul_of_nonneg_left hb hm
-      nlinarith only [hmul, hmon]
-  linarith
-
-
-theorem sqrt_sub_one_sq_le {t : ℝ} (ht : 0 ≤ t) :
-    (Real.sqrt t - 1) ^ 2 ≤ (t - 1) ^ 2 := by
-  have hs2 := Real.sq_sqrt ht
-  have hprod := mul_nonneg (Real.sqrt_nonneg t)
-    (mul_nonneg (show 0 ≤ Real.sqrt t + 2 by positivity) (sq_nonneg (Real.sqrt t - 1)))
-  conv_rhs => rw [← hs2]
-  nlinarith only [hprod]
-
-theorem hellingerSq_le_score (P Q : FiniteLaw α) (Z : α → ℝ)
-    (hZ : ∀ a, Q.mass a = P.mass a * (1 + Z a)) :
-    hellingerSq P Q ≤ ∑ a, P.mass a * Z a ^ 2 := by
-  apply Finset.sum_le_sum
-  intro a _
-  by_cases hp0 : P.mass a = 0
-  · have hq0 : Q.mass a = 0 := by rw [hZ, hp0, zero_mul]
-    simp [hp0, hq0]
-  have hp : 0 < P.mass a := lt_of_le_of_ne (P.nonneg a) (Ne.symm hp0)
-  have ht : 0 ≤ 1 + Z a := by
-    have hq := Q.nonneg a
-    rw [hZ] at hq
-    exact nonneg_of_mul_nonneg_right hq hp
-  have hs := sqrt_sub_one_sq_le ht
-  have hmul := mul_le_mul_of_nonneg_left hs (P.nonneg a)
-  have hp2 := Real.sq_sqrt (P.nonneg a)
-  rw [hZ, Real.sqrt_mul (P.nonneg a)]
-  calc
-    _ = P.mass a * (Real.sqrt (1 + Z a) - 1) ^ 2 := by
-      calc
-        _ = Real.sqrt (P.mass a) ^ 2 * (Real.sqrt (1 + Z a) - 1) ^ 2 := by ring
-        _ = _ := by rw [hp2]
-    _ ≤ P.mass a * ((1 + Z a) - 1) ^ 2 := hmul
-    _ = _ := by ring
-
-theorem tv_product_le_sqrt_score (P Q : FiniteLaw α) (Z : α → ℝ)
-    (hZ : ∀ a, Q.mass a = P.mass a * (1 + Z a)) (m : ℕ) :
-    tv (P.product m) (Q.product m) ≤ Real.sqrt ((m : ℝ) * ∑ a, P.mass a * Z a ^ 2) := by
-  apply (tv_le_sqrt_hellingerSq _ _).trans
-  apply Real.sqrt_le_sqrt
-  exact (hellingerSq_product_le P Q m).trans
-    (mul_le_mul_of_nonneg_left (hellingerSq_le_score P Q Z hZ) (by positivity))
-
-def bind (P : FiniteLaw α) (K : α → FiniteLaw β) : FiniteLaw β where
-  mass b := ∑ a, P.mass a * (K a).mass b
-  nonneg b := Finset.sum_nonneg fun a _ => mul_nonneg (P.nonneg a) ((K a).nonneg b)
-  total := by
-    rw [Finset.sum_comm]
-    simp_rw [← Finset.mul_sum, (K _).total, mul_one]
-    exact P.total
-
-theorem tv_bind_le (P Q : FiniteLaw α) (K : α → FiniteLaw β) :
-    tv (P.bind K) (Q.bind K) ≤ tv P Q := by
-  unfold tv bind
-  simp only
-  apply mul_le_mul_of_nonneg_left _ (by norm_num)
-  calc
-    _ = ∑ b, |∑ a, (P.mass a - Q.mass a) * (K a).mass b| := by
-      apply Finset.sum_congr rfl
-      intro b _
-      congr 1
-      rw [← Finset.sum_sub_distrib]
-      apply Finset.sum_congr rfl
-      intro a _; ring
-    _ ≤ ∑ b, ∑ a, |(P.mass a - Q.mass a) * (K a).mass b| :=
-      Finset.sum_le_sum fun b _ => Finset.abs_sum_le_sum_abs _ _
-    _ = ∑ a, |P.mass a - Q.mass a| := by
-      rw [Finset.sum_comm]
-      apply Finset.sum_congr rfl
-      intro a _
-      simp_rw [abs_mul, abs_of_nonneg ((K a).nonneg _), ← Finset.mul_sum, (K a).total, mul_one]
-
-theorem tv_triangle (P Q R : FiniteLaw α) : tv P R ≤ tv P Q + tv Q R := by
-  unfold tv
-  rw [← mul_add, ← Finset.sum_add_distrib]
-  apply mul_le_mul_of_nonneg_left _ (by norm_num)
-  apply Finset.sum_le_sum
-  intro a _
-  exact abs_sub_le _ _ _
-
-theorem tv_symm (P Q : FiniteLaw α) : tv P Q = tv Q P := by
-  unfold tv
-  congr 1
-  apply Finset.sum_congr rfl
-  intro a _; exact abs_sub_comm _ _
-
-theorem expectation_diff_le_tv (P Q : FiniteLaw α) (f : α → ℝ)
-    (hf : ∀ a, f a ∈ Set.Icc 0 1) :
-    (∑ a, P.mass a * f a) - (∑ a, Q.mass a * f a) ≤ tv P Q := by
-  have hz : (∑ a, (P.mass a - Q.mass a)) = 0 := by
-    rw [Finset.sum_sub_distrib, P.total, Q.total, sub_self]
-  have h (a : α) : (P.mass a - Q.mass a) * f a ≤
-      (|P.mass a - Q.mass a| + (P.mass a - Q.mass a)) / 2 := by
-    by_cases hd : 0 ≤ P.mass a - Q.mass a
-    · rw [abs_of_nonneg hd]
-      have := mul_le_mul_of_nonneg_left (hf a).2 hd
-      linarith
-    · rw [abs_of_neg (lt_of_not_ge hd)]
-      have := mul_nonpos_of_nonpos_of_nonneg (le_of_not_ge hd) (hf a).1
-      linarith
-  calc
-    _ = ∑ a, (P.mass a - Q.mass a) * f a := by
-      rw [← Finset.sum_sub_distrib]
-      apply Finset.sum_congr rfl
-      intro a _; ring
-    _ ≤ ∑ a, (|P.mass a - Q.mass a| + (P.mass a - Q.mass a)) / 2 :=
-      Finset.sum_le_sum fun a _ => h a
-    _ = tv P Q := by
-      rw [← Finset.sum_div, Finset.sum_add_distrib, hz, add_zero]
-      unfold tv
-      ring
-
-theorem binary_testing_bound (P Q : FiniteLaw α) (f : α → ℝ)
-    (hf : ∀ a, f a ∈ Set.Icc 0 1) :
-    ((∑ a, P.mass a * f a) + (∑ a, Q.mass a * (1 - f a))) / 2 ≤
-      (1 + tv P Q) / 2 := by
-  have h := expectation_diff_le_tv P Q f hf
-  have heq : (∑ a, Q.mass a * (1 - f a)) = 1 - ∑ a, Q.mass a * f a := by
-    simp_rw [mul_sub, mul_one, Finset.sum_sub_distrib, Q.total]
-  rw [heq]
-  linarith
-
-
-theorem sum_max_eq (P Q : FiniteLaw α) :
-    (∑ a, max (P.mass a) (Q.mass a)) = 1 + tv P Q := by
-  have hpoint (a : α) : max (P.mass a) (Q.mass a) =
-      (P.mass a + Q.mass a + |P.mass a - Q.mass a|) / 2 := by
-    by_cases h : P.mass a ≤ Q.mass a
-    · rw [max_eq_right h, abs_of_nonpos (sub_nonpos.mpr h)]; ring
-    · rw [max_eq_left (le_of_not_ge h), abs_of_nonneg (sub_nonneg.mpr (le_of_not_ge h))]; ring
-  simp_rw [hpoint]
-  rw [← Finset.sum_div, Finset.sum_add_distrib, Finset.sum_add_distrib, P.total, Q.total]
-  unfold tv
-  ring
-
-/-- Joint exact recovery of independent binary choices, allowing arbitrary
-randomized estimators. No coordinatewise decision rule is assumed. -/
-theorem independent_binary_testing (P Q : FiniteLaw α) (g : ℕ)
-    (A : (Fin g → α) → (Fin g → Bool) → ℝ)
-    (hA0 : ∀ z θ, 0 ≤ A z θ) (hA1 : ∀ z, ∑ θ, A z θ = 1) :
-    (1 / (2 : ℝ) ^ g) * ∑ θ : Fin g → Bool, ∑ z : Fin g → α,
-      (∏ j, (if θ j then P.mass (z j) else Q.mass (z j))) * A z θ ≤
-        ((1 + tv P Q) / 2) ^ g := by
-  classical
-  have hpoint (z : Fin g → α) (θ : Fin g → Bool) :
-      (∏ j, (if θ j then P.mass (z j) else Q.mass (z j))) ≤
-        ∏ j, max (P.mass (z j)) (Q.mass (z j)) := by
-    apply Finset.prod_le_prod₀
-    · intro j _; split <;> first | exact P.nonneg _ | exact Q.nonneg _
-    · intro j _; split <;> first | exact le_max_left _ _ | exact le_max_right _ _
-  have hsum (z : Fin g → α) :
-      (∑ θ : Fin g → Bool,
-        (∏ j, (if θ j then P.mass (z j) else Q.mass (z j))) * A z θ) ≤
-          ∏ j, max (P.mass (z j)) (Q.mass (z j)) := by
-    calc
-      _ ≤ ∑ θ : Fin g → Bool,
-          (∏ j, max (P.mass (z j)) (Q.mass (z j))) * A z θ := by
-        apply Finset.sum_le_sum
-        intro θ _
-        exact mul_le_mul_of_nonneg_right (hpoint z θ) (hA0 z θ)
-      _ = _ := by rw [← Finset.mul_sum, hA1, mul_one]
-  rw [Finset.sum_comm]
-  calc
-    _ ≤ (1 / (2 : ℝ) ^ g) * ∑ z : Fin g → α, ∏ j,
-        max (P.mass (z j)) (Q.mass (z j)) :=
-      mul_le_mul_of_nonneg_left (Finset.sum_le_sum fun z _ => hsum z) (by positivity)
-    _ = (1 / (2 : ℝ) ^ g) * (∑ a, max (P.mass a) (Q.mass a)) ^ g := by
-      rw [← Fintype.prod_sum (fun (_ : Fin g) a => max (P.mass a) (Q.mass a))]
-      simp
-    _ = _ := by rw [sum_max_eq, div_pow]; ring
-
-def tensor (P : FiniteLaw α) (R : FiniteLaw β) : FiniteLaw (α × β) where
-  mass z := P.mass z.1 * R.mass z.2
-  nonneg z := mul_nonneg (P.nonneg z.1) (R.nonneg z.2)
-  total := by
-    rw [Fintype.sum_prod_type]
-    simp_rw [← Finset.mul_sum, R.total, mul_one]
-    exact P.total
-
-theorem tv_tensor_le (P Q : FiniteLaw α) (R S : FiniteLaw β) :
-    tv (P.tensor R) (Q.tensor S) ≤ tv P Q + tv R S := by
-  unfold tv tensor
-  simp only
-  rw [Fintype.sum_prod_type, ← mul_add]
-  apply mul_le_mul_of_nonneg_left _ (by norm_num)
-  have hpoint (a : α) (b : β) :
-      |P.mass a * R.mass b - Q.mass a * S.mass b| ≤
-        |P.mass a - Q.mass a| * R.mass b + Q.mass a * |R.mass b - S.mass b| := by
-    calc
-      _ = |(P.mass a - Q.mass a) * R.mass b + Q.mass a * (R.mass b - S.mass b)| := by
-        congr 1; ring
-      _ ≤ |(P.mass a - Q.mass a) * R.mass b| + |Q.mass a * (R.mass b - S.mass b)| :=
-        abs_add_le _ _
-      _ = _ := by rw [abs_mul, abs_mul, abs_of_nonneg (R.nonneg b), abs_of_nonneg (Q.nonneg a)]
-  calc
-    _ ≤ ∑ a, ∑ b, (|P.mass a - Q.mass a| * R.mass b + Q.mass a * |R.mass b - S.mass b|) := by
-      apply Finset.sum_le_sum
-      intro a _
-      exact Finset.sum_le_sum fun b _ => hpoint a b
-    _ = _ := by
-      simp_rw [Finset.sum_add_distrib, ← Finset.mul_sum, R.total, mul_one]
-      rw [← Finset.sum_mul, Q.total, one_mul]
-
-theorem tv_product_succ (P Q : FiniteLaw α) (m : ℕ) :
-    tv (P.product (m + 1)) (Q.product (m + 1)) =
-      tv (P.tensor (P.product m)) (Q.tensor (Q.product m)) := by
-  unfold tv product tensor
-  simp only
-  congr 1
-  rw [← (Fin.consEquiv (fun _ : Fin (m + 1) => α)).sum_comp]
-  apply Finset.sum_congr rfl
-  intro a _
-  simp only [Fin.prod_univ_succ, Fin.consEquiv_apply, Fin.cons_zero, Fin.cons_succ]
-
-theorem tv_product_le (P Q : FiniteLaw α) (m : ℕ) :
-    tv (P.product m) (Q.product m) ≤ (m : ℝ) * tv P Q := by
-  induction m with
-  | zero => simp [tv, product]
-  | succ m ih =>
-    rw [tv_product_succ]
-    apply (tv_tensor_le P Q (P.product m) (Q.product m)).trans
-    simpa only [Nat.cast_add, Nat.cast_one, add_mul, one_mul, add_comm] using
-      add_le_add_left ih (tv P Q)
-
-
-theorem affinity_bind_ge (P Q : FiniteLaw α) (K : α → FiniteLaw β) :
-    affinity P Q ≤ affinity (P.bind K) (Q.bind K) := by
-  have hpoint (b : β) :
-      (∑ a, Real.sqrt (P.mass a) * Real.sqrt (Q.mass a) * (K a).mass b) ≤
-        Real.sqrt ((P.bind K).mass b) * Real.sqrt ((Q.bind K).mass b) := by
-    have hcs := Finset.sum_sq_le_sum_mul_sum_of_sq_le_mul (s := Finset.univ)
-      (r := fun a => Real.sqrt (P.mass a) * Real.sqrt (Q.mass a) * (K a).mass b)
-      (f := fun a => P.mass a * (K a).mass b) (g := fun a => Q.mass a * (K a).mass b)
-      (fun a _ => mul_nonneg (P.nonneg a) ((K a).nonneg b))
-      (fun a _ => mul_nonneg (Q.nonneg a) ((K a).nonneg b)) (by
-        intro a _
-        rw [mul_pow, mul_pow, Real.sq_sqrt (P.nonneg a), Real.sq_sqrt (Q.nonneg a)]
-        nlinarith only [sq_nonneg ((K a).mass b)])
-    have hp2 := Real.sq_sqrt ((P.bind K).nonneg b)
-    have hq2 := Real.sq_sqrt ((Q.bind K).nonneg b)
-    have hsq : (Real.sqrt ((P.bind K).mass b) * Real.sqrt ((Q.bind K).mass b)) ^ 2 =
-        (P.bind K).mass b * (Q.bind K).mass b := by rw [mul_pow, hp2, hq2]
-    change (∑ a, Real.sqrt (P.mass a) * Real.sqrt (Q.mass a) * (K a).mass b) ^ 2 ≤
-      (P.bind K).mass b * (Q.bind K).mass b at hcs
-    have hn : 0 ≤ Real.sqrt ((P.bind K).mass b) * Real.sqrt ((Q.bind K).mass b) := by positivity
-    nlinarith only [hcs, hsq, hn]
-  calc
-    affinity P Q = ∑ b, ∑ a, Real.sqrt (P.mass a) * Real.sqrt (Q.mass a) * (K a).mass b := by
-      rw [Finset.sum_comm]
-      simp_rw [← Finset.mul_sum, (K _).total, mul_one]
-      rfl
-    _ ≤ _ := Finset.sum_le_sum fun b _ => hpoint b
-
-theorem hellingerSq_bind_le (P Q : FiniteLaw α) (K : α → FiniteLaw β) :
-    hellingerSq (P.bind K) (Q.bind K) ≤ hellingerSq P Q := by
-  rw [hellingerSq_eq, hellingerSq_eq]
-  linarith [affinity_bind_ge P Q K]
-
-/-- Restoration of approximate oracle simulations before taking independent
-complete observations; the simulation error is charged once per sample. -/
-theorem approximate_score_transfer (P Q : FiniteLaw α) (K : α → FiniteLaw β)
-    (R S : FiniteLaw β) (Z : α → ℝ)
-    (hZ : ∀ a, Q.mass a = P.mass a * (1 + Z a)) (E : ℝ)
-    (hR : tv R (P.bind K) ≤ E) (hS : tv S (Q.bind K) ≤ E) (m : ℕ) :
-    tv (R.product m) (S.product m) ≤
-      Real.sqrt ((m : ℝ) * ∑ a, P.mass a * Z a ^ 2) + 2 * m * E := by
-  have hmid : tv ((P.bind K).product m) ((Q.bind K).product m) ≤
-      Real.sqrt ((m : ℝ) * ∑ a, P.mass a * Z a ^ 2) := by
-    apply (tv_le_sqrt_hellingerSq _ _).trans
-    apply Real.sqrt_le_sqrt
-    apply (hellingerSq_product_le _ _ m).trans
-    exact mul_le_mul_of_nonneg_left
-      ((hellingerSq_bind_le P Q K).trans (hellingerSq_le_score P Q Z hZ)) (by positivity)
-  have hr := (tv_product_le R (P.bind K) m).trans
-    (mul_le_mul_of_nonneg_left hR (by positivity : (0 : ℝ) ≤ m))
-  have hs := (tv_product_le S (Q.bind K) m).trans
-    (mul_le_mul_of_nonneg_left hS (by positivity : (0 : ℝ) ≤ m))
-  rw [tv_symm (S.product m)] at hs
-  have htri1 := tv_triangle (R.product m) ((P.bind K).product m) (S.product m)
-  have htri2 := tv_triangle ((P.bind K).product m) ((Q.bind K).product m) (S.product m)
-  linarith
-
-def overlap (P Q : FiniteLaw α) : ℝ := ∑ a, min (P.mass a) (Q.mass a)
-
-theorem overlap_eq_one_sub_tv (P Q : FiniteLaw α) : overlap P Q = 1 - tv P Q := by
-  have hm : overlap P Q + (∑ a, max (P.mass a) (Q.mass a)) = 2 := by
-    rw [overlap, ← Finset.sum_add_distrib]
-    simp_rw [min_add_max]
-    rw [Finset.sum_add_distrib, P.total, Q.total]
-    norm_num
-  rw [sum_max_eq] at hm
-  linarith
-
-theorem overlap_nonneg (P Q : FiniteLaw α) : 0 ≤ overlap P Q := by
-  apply Finset.sum_nonneg
-  intro a _
-  exact le_min (P.nonneg a) (Q.nonneg a)
-
-def commonPart (P Q : FiniteLaw α) (h : 0 < overlap P Q) : FiniteLaw α where
-  mass a := min (P.mass a) (Q.mass a) / overlap P Q
-  nonneg a := div_nonneg (le_min (P.nonneg a) (Q.nonneg a)) h.le
-  total := by rw [← Finset.sum_div]; exact div_self h.ne'
-
-theorem commonPart_le_left (P Q : FiniteLaw α) (h : 0 < overlap P Q)
-    {ω : ℝ} (hω : ω ≤ overlap P Q) (a : α) :
-    ω * (commonPart P Q h).mass a ≤ P.mass a := by
-  change ω * (min (P.mass a) (Q.mass a) / overlap P Q) ≤ _
-  calc
-    _ ≤ overlap P Q * (min (P.mass a) (Q.mass a) / overlap P Q) :=
-      mul_le_mul_of_nonneg_right hω (div_nonneg (le_min (P.nonneg a) (Q.nonneg a)) h.le)
-    _ = min (P.mass a) (Q.mass a) := by field_simp
-    _ ≤ _ := min_le_left _ _
-
-def remainderLaw (P H : FiniteLaw α) (ω : ℝ) (hω : ω < 1)
-    (hdom : ∀ a, ω * H.mass a ≤ P.mass a) : FiniteLaw α where
-  mass a := (P.mass a - ω * H.mass a) / (1 - ω)
-  nonneg a := div_nonneg (sub_nonneg.mpr (hdom a)) (sub_pos.mpr hω).le
-  total := by
-    rw [← Finset.sum_div, Finset.sum_sub_distrib, ← Finset.mul_sum, P.total, H.total, mul_one]
-    exact div_self (sub_pos.mpr hω).ne'
-
-theorem common_part_decomposition (P Q : FiniteLaw α) {ω : ℝ}
-    (hω : ω ∈ Set.Ioo 0 1) (hov : ω ≤ overlap P Q) :
-    ∃ H P' Q' : FiniteLaw α,
-      (∀ a, P.mass a = ω * H.mass a + (1 - ω) * P'.mass a) ∧
-      (∀ a, Q.mass a = ω * H.mass a + (1 - ω) * Q'.mass a) := by
-  have h : 0 < overlap P Q := lt_of_lt_of_le hω.1 hov
-  let H := commonPart P Q h
-  have hP : ∀ a, ω * H.mass a ≤ P.mass a := commonPart_le_left P Q h hov
-  have hQ : ∀ a, ω * H.mass a ≤ Q.mass a := by
-    intro a
-    change ω * (min (P.mass a) (Q.mass a) / overlap P Q) ≤ _
-    calc
-      _ ≤ overlap P Q * (min (P.mass a) (Q.mass a) / overlap P Q) :=
-        mul_le_mul_of_nonneg_right hov (div_nonneg (le_min (P.nonneg a) (Q.nonneg a)) h.le)
-      _ = min (P.mass a) (Q.mass a) := by field_simp
-      _ ≤ _ := min_le_right _ _
-  refine ⟨H, remainderLaw P H ω hω.2 hP, remainderLaw Q H ω hω.2 hQ, ?_, ?_⟩
-  · intro a
-    change P.mass a = ω * H.mass a + (1 - ω) * ((P.mass a - ω * H.mass a) / (1 - ω))
-    field_simp [(sub_pos.mpr hω.2).ne']
-    ring
-  · intro a
-    change Q.mass a = ω * H.mass a + (1 - ω) * ((Q.mass a - ω * H.mass a) / (1 - ω))
-    field_simp [(sub_pos.mpr hω.2).ne']
-    ring
-
-end FiniteLaw
-end TraceReconstruction
-
-namespace TraceReconstruction.FiniteLaw
-variable {α β : Type*} [Fintype α] [Fintype β]
-
-def prod (P : FiniteLaw α) (Q : FiniteLaw β) : FiniteLaw (α × β) where
-  mass x := P.mass x.1 * Q.mass x.2
-  nonneg x := mul_nonneg (P.nonneg _) (Q.nonneg _)
-  total := by
-    rw [Fintype.sum_prod_type]
-    simp_rw [← Finset.mul_sum, Q.total, mul_one]
-    exact P.total
-
-def expect (P : FiniteLaw α) (f : α → ℝ) : ℝ := ∑ a, P.mass a * f a
-
-theorem expect_const (P : FiniteLaw α) (c : ℝ) : P.expect (fun _ => c) = c := by
-  unfold expect
-  rw [← Finset.sum_mul, P.total, one_mul]
-
-theorem expect_mono (P : FiniteLaw α) {f g : α → ℝ} (h : ∀ a, f a ≤ g a) :
-    P.expect f ≤ P.expect g := Finset.sum_le_sum fun a _ =>
-      mul_le_mul_of_nonneg_left (h a) (P.nonneg a)
-
-theorem expect_prod (P : FiniteLaw α) (Q : FiniteLaw β) (f : α × β → ℝ) :
-    (P.prod Q).expect f = P.expect (fun a => Q.expect (fun b => f (a, b))) := by
-  unfold expect prod
-  rw [Fintype.sum_prod_type]
-  simp_rw [Finset.mul_sum, mul_assoc]
-
-/-- Association is only used for increasing likelihood-threshold events. -/
-def Associated [Preorder α] (P : FiniteLaw α) : Prop :=
-  ∀ f g : α → ℝ, Monotone f → Monotone g →
-    P.expect f * P.expect g ≤ P.expect (fun a => f a * g a)
-
-theorem associated_linearOrder [LinearOrder α] (P : FiniteLaw α) : P.Associated := by
-  intro f g hf hg
-  have hn (a b : α) : 0 ≤ (f a - f b) * (g a - g b) := by
-    rcases le_total a b with hab | hba
-    · exact mul_nonneg_of_nonpos_of_nonpos (sub_nonpos.mpr (hf hab))
-        (sub_nonpos.mpr (hg hab))
-    · exact mul_nonneg (sub_nonneg.mpr (hf hba)) (sub_nonneg.mpr (hg hba))
-  have h := Finset.sum_nonneg (s := Finset.univ) (fun a _ =>
-    Finset.sum_nonneg (s := Finset.univ) (fun b _ => mul_nonneg (mul_nonneg (P.nonneg a)
-      (P.nonneg b)) (hn a b)))
-  have hid : (∑ a, ∑ b, P.mass a * P.mass b * ((f a - f b) * (g a - g b))) =
-      2 * (P.expect (fun a => f a * g a) - P.expect f * P.expect g) := by
-    have hp (a b : α) : P.mass a * P.mass b * ((f a - f b) * (g a - g b)) =
-        (P.mass a * (f a * g a)) * P.mass b + P.mass a * (P.mass b * (f b * g b)) -
-        (P.mass a * f a) * (P.mass b * g b) - (P.mass a * g a) * (P.mass b * f b) := by ring
-    simp_rw [hp, Finset.sum_sub_distrib, Finset.sum_add_distrib,
-      ← Finset.mul_sum, ← Finset.sum_mul, P.total, one_mul, mul_one]
-    unfold expect
-    ring
-  rw [hid] at h
-  linarith
-
-theorem Associated.prod [Preorder α] [Preorder β] {P : FiniteLaw α} {Q : FiniteLaw β}
-    (hP : P.Associated) (hQ : Q.Associated) : (P.prod Q).Associated := by
-  intro f g hf hg
-  rw [expect_prod, expect_prod, expect_prod]
-  calc
-    _ ≤ P.expect (fun a => Q.expect (fun b => f (a, b)) * Q.expect (fun b => g (a, b))) := by
-      apply hP
-      · intro a a' ha
-        exact Q.expect_mono (fun b => hf ⟨ha, le_rfl⟩)
-      · intro a a' ha
-        exact Q.expect_mono (fun b => hg ⟨ha, le_rfl⟩)
-    _ ≤ _ := by
-      apply P.expect_mono
-      intro a
-      apply hQ
-      · intro b b' hb; exact hf ⟨le_rfl, hb⟩
-      · intro b b' hb; exact hg ⟨le_rfl, hb⟩
-
-theorem product_succ_expect (P : FiniteLaw α) (n : ℕ) (f : (Fin (n + 1) → α) → ℝ) :
-    (P.product (n + 1)).expect f =
-      P.expect (fun a => (P.product n).expect (fun b => f (Fin.cons a b))) := by
-  unfold expect product
-  rw [← (Fin.consEquiv (fun _ : Fin (n + 1) => α)).sum_comp, Fintype.sum_prod_type]
-  simp_rw [Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro a _
-  apply Finset.sum_congr rfl
-  intro b _
-  change (∏ j, P.mass ((Fin.cons a b : Fin (n + 1) → α) j)) * f (Fin.cons a b) = _
-  rw [Fin.prod_univ_succ]
-  simp only [Fin.cons_zero, Fin.cons_succ]
-  ring
-
-theorem associated_product [Preorder α] (P : FiniteLaw α) (hP : P.Associated) (n : ℕ) :
-    (P.product n).Associated := by
-  induction n with
-  | zero =>
-    intro f g _ _
-    simp [expect, product]
-  | succ n ih =>
-    intro f g hf hg
-    rw [product_succ_expect, product_succ_expect, product_succ_expect]
-    have hcons {a a' : α} {b b' : Fin n → α} (ha : a ≤ a') (hb : b ≤ b') :
-        (Fin.cons a b : Fin (n + 1) → α) ≤ (Fin.cons a' b' : Fin (n + 1) → α) := by
-      intro j
-      refine Fin.cases ?_ (fun i => ?_) j
-      · exact ha
-      · exact hb i
-    calc
-      _ ≤ P.expect (fun a => (P.product n).expect (fun b => f (Fin.cons a b)) *
-          (P.product n).expect (fun b => g (Fin.cons a b))) := by
-        apply hP
-        · intro a a' ha
-          exact (P.product n).expect_mono (fun b => hf (hcons ha le_rfl))
-        · intro a a' ha
-          exact (P.product n).expect_mono (fun b => hg (hcons ha le_rfl))
-      _ ≤ _ := by
-        apply P.expect_mono
-        intro a
-        apply ih
-        · intro b b' hb; exact hf (hcons le_rfl hb)
-        · intro b b' hb; exact hg (hcons le_rfl hb)
-
-/-- An elementary Paley--Zygmund bound at threshold 1/2, for a density. -/
-theorem density_threshold_mass (P : FiniteLaw α) (f : α → ℝ)
-    (hmean : P.expect f = 1) :
-    (1 : ℝ) / 4 ≤
-      (P.expect (fun a => if 1 / 2 ≤ f a then 1 else 0)) * P.expect (fun a => f a ^ 2) := by
-  classical
-  let S : Finset α := Finset.univ.filter (fun a => 1 / 2 ≤ f a)
-  have hsmall : (∑ a ∈ Finset.univ \ S, P.mass a * f a) ≤ 1 / 2 := by
-    calc
-      _ ≤ ∑ a ∈ Finset.univ \ S, P.mass a * (1 / 2) := by
-        apply Finset.sum_le_sum
-        intro a ha
-        have hnot : ¬ 1 / 2 ≤ f a := by simpa [S] using (Finset.mem_sdiff.mp ha).2
-        exact mul_le_mul_of_nonneg_left (le_of_lt (lt_of_not_ge hnot)) (P.nonneg a)
-      _ ≤ ∑ a, P.mass a * (1 / 2) := by
-        apply Finset.sum_le_sum_of_subset_of_nonneg (Finset.sdiff_subset)
-        intro a _ _; positivity [P.nonneg a]
-      _ = 1 / 2 := P.expect_const _
-  have hlarge : 1 / 2 ≤ ∑ a ∈ S, P.mass a * f a := by
-    have hsplit := Finset.sum_sdiff (Finset.subset_univ S) (f := fun a => P.mass a * f a)
-    change (∑ a, P.mass a * f a) = 1 at hmean
-    rw [hmean] at hsplit
-    linarith
-  have hc := Finset.sum_sq_le_sum_mul_sum_of_sq_le_mul (s := S)
-    (r := fun a => P.mass a * f a) (f := P.mass) (g := fun a => P.mass a * f a ^ 2)
-    (fun a _ => P.nonneg a) (fun a _ => by positivity [P.nonneg a])
-    (fun a _ => by nlinarith)
-  have ht : (∑ a ∈ S, P.mass a * f a ^ 2) ≤ P.expect (fun a => f a ^ 2) := by
-    apply Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ S)
-    intro a _ _; positivity [P.nonneg a]
-  have hmass : P.expect (fun a => if 1 / 2 ≤ f a then 1 else 0) = ∑ a ∈ S, P.mass a := by
-    unfold expect S
-    simp_rw [mul_ite, mul_one, mul_zero]
-    rw [Finset.sum_filter]
-  rw [hmass]
-  have hm := mul_le_mul_of_nonneg_left ht (Finset.sum_nonneg (s := S) (fun a _ => P.nonneg a))
-  nlinarith
-
-/-- The common-density overlap used for consecutive pure primitive runs. -/
-theorem increasing_density_overlap [Preorder α] (P : FiniteLaw α) (hP : P.Associated)
-    (f g : α → ℝ) (hf : ∀ a, 0 ≤ f a) (hg : ∀ a, 0 ≤ g a)
-    (hfm : Monotone f) (hgm : Monotone g)
-    (hf1 : P.expect f = 1) (hg1 : P.expect g = 1) :
-    (1 : ℝ) / 32 ≤ P.expect (fun a => min (f a) (g a)) *
-      P.expect (fun a => f a ^ 2) * P.expect (fun a => g a ^ 2) := by
-  let F : α → ℝ := fun a => if 1 / 2 ≤ f a then 1 else 0
-  let G : α → ℝ := fun a => if 1 / 2 ≤ g a then 1 else 0
-  have hF : Monotone F := by
-    intro a b hab
-    dsimp [F]
-    split_ifs with ha hb hb
-    · rfl
-    · exact False.elim (hb (ha.trans (hfm hab)))
-    · norm_num
-    · rfl
-  have hG : Monotone G := by
-    intro a b hab
-    dsimp [G]
-    split_ifs with ha hb hb
-    · rfl
-    · exact False.elim (hb (ha.trans (hgm hab)))
-    · norm_num
-    · rfl
-  have hcorr := hP F G hF hG
-  have hpoint : ∀ a, (1 / 2 : ℝ) * (F a * G a) ≤ min (f a) (g a) := by
-    intro a
-    dsimp [F, G]
-    split_ifs <;> simp only [mul_one, mul_zero]
-    · exact le_min ‹_› ‹_›
-    all_goals exact le_min (hf a) (hg a)
-  have hint := P.expect_mono hpoint
-  have heq : P.expect (fun a => (1 / 2 : ℝ) * (F a * G a)) =
-      (1 / 2 : ℝ) * P.expect (fun a => F a * G a) := by
-    unfold expect
-    rw [Finset.mul_sum]
-    apply Finset.sum_congr rfl
-    intro a _; ring
-  rw [heq] at hint
-  have hfm2 : 0 ≤ P.expect (fun a => f a ^ 2) :=
-    Finset.sum_nonneg (fun a _ => mul_nonneg (P.nonneg a) (sq_nonneg _))
-  have hgm2 : 0 ≤ P.expect (fun a => g a ^ 2) :=
-    Finset.sum_nonneg (fun a _ => mul_nonneg (P.nonneg a) (sq_nonneg _))
-  have hff := density_threshold_mass P f hf1
-  have hgg := density_threshold_mass P g hg1
-  change 1 / 4 ≤ P.expect F * P.expect (fun a => f a ^ 2) at hff
-  change 1 / 4 ≤ P.expect G * P.expect (fun a => g a ^ 2) at hgg
-  have hmul := mul_le_mul hff hgg (by norm_num : (0 : ℝ) ≤ 1 / 4) (by linarith :
-    0 ≤ P.expect F * P.expect (fun a => f a ^ 2))
-  have hcomb : (1 / 2 : ℝ) * (P.expect F * P.expect G) ≤
-      P.expect (fun a => min (f a) (g a)) := by linarith
-  have hmul2 := mul_le_mul_of_nonneg_right hcomb (mul_nonneg hfm2 hgm2)
-  nlinarith only [hmul, hmul2]
-
-end TraceReconstruction.FiniteLaw
-
-namespace TraceReconstruction.FiniteLaw
-variable {α : Type*} [Fintype α]
-/-- Orientation `false` uses P,Q in chronological order; true reverses them. -/
-def orientedProd (P Q : FiniteLaw α) (b : Bool) : FiniteLaw (α × α) :=
-  if b then Q.prod P else P.prod Q
-
-def pairedHiddenMass (H : FiniteLaw α) (ω : ℝ) (z : α × α) : ℝ :=
-  ω ^ 2 * H.mass z.1 * H.mass z.2
-
-theorem sum_pairedHiddenMass (H : FiniteLaw α) (ω : ℝ) :
-    (∑ z, pairedHiddenMass H ω z) = ω ^ 2 := by
-  unfold pairedHiddenMass
-  rw [Fintype.sum_prod_type]
-  simp_rw [← Finset.mul_sum, H.total, mul_one]
-  rw [← Finset.mul_sum, H.total, mul_one]
-
-theorem pairedHiddenMass_le (P Q H : FiniteLaw α) {ω : ℝ} (hω : 0 ≤ ω)
-    (hP : ∀ a, ω * H.mass a ≤ P.mass a) (hQ : ∀ a, ω * H.mass a ≤ Q.mass a)
-    (b : Bool) (z : α × α) : pairedHiddenMass H ω z ≤ (orientedProd P Q b).mass z := by
-  have hm (R S : FiniteLaw α) (hR : ∀ a, ω * H.mass a ≤ R.mass a)
-      (hS : ∀ a, ω * H.mass a ≤ S.mass a) :
-      pairedHiddenMass H ω z ≤ R.mass z.1 * S.mass z.2 := by
-    calc
-      _ = (ω * H.mass z.1) * (ω * H.mass z.2) := by unfold pairedHiddenMass; ring
-      _ ≤ _ := mul_le_mul (hR z.1) (hS z.2)
-        (mul_nonneg hω (H.nonneg _)) (R.nonneg _)
-  cases b
-  · exact hm P Q hP hQ
-  · exact hm Q P hQ hP
-
-/-- None is the joint hidden flag; a visible flag reports the true orientation. -/
-def pairedFlagMass (P Q H : FiniteLaw α) (ω : ℝ) (b : Bool)
-    (z : Option Bool × (α × α)) : ℝ :=
-  match z.1 with
-  | none => pairedHiddenMass H ω z.2
-  | some c => if c = b then (orientedProd P Q b).mass z.2 - pairedHiddenMass H ω z.2 else 0
-
-theorem pairedFlagMass_nonneg (P Q H : FiniteLaw α) {ω : ℝ} (hω : 0 ≤ ω)
-    (hP : ∀ a, ω * H.mass a ≤ P.mass a) (hQ : ∀ a, ω * H.mass a ≤ Q.mass a)
-    (b : Bool) (z : Option Bool × (α × α)) : 0 ≤ pairedFlagMass P Q H ω b z := by
-  unfold pairedFlagMass
-  split
-  · unfold pairedHiddenMass
-    positivity [H.nonneg z.2.1, H.nonneg z.2.2]
-  · split_ifs
-    · exact sub_nonneg.mpr (pairedHiddenMass_le P Q H hω hP hQ b z.2)
-    · rfl
-
-theorem pairedFlagMass_sum (P Q H : FiniteLaw α) (ω : ℝ) (b : Bool) :
-    (∑ z, pairedFlagMass P Q H ω b z) = 1 := by
-  rw [Fintype.sum_prod_type, Finset.sum_comm]
-  have hp (z : α × α) : (∑ c : Option Bool, pairedFlagMass P Q H ω b (c, z)) =
-      (orientedProd P Q b).mass z := by
-    cases b <;> simp [pairedFlagMass, Fintype.sum_option ]
-  simp_rw [hp]
-  exact (orientedProd P Q b).total
-
-def pairedFlagLaw (P Q H : FiniteLaw α) {ω : ℝ} (hω : 0 ≤ ω)
-    (hP : ∀ a, ω * H.mass a ≤ P.mass a) (hQ : ∀ a, ω * H.mass a ≤ Q.mass a)
-    (b : Bool) : FiniteLaw (Option Bool × (α × α)) :=
-  ⟨pairedFlagMass P Q H ω b, pairedFlagMass_nonneg P Q H hω hP hQ b,
-    pairedFlagMass_sum P Q H ω b⟩
-
-/-- Forgetting the report recovers conditional independence for each orientation. -/
-theorem pairedFlag_simulation (P Q H : FiniteLaw α) (ω : ℝ) (b : Bool) (z : α × α) :
-    (∑ c : Option Bool, pairedFlagMass P Q H ω b (c, z)) =
-      (orientedProd P Q b).mass z := by
-  cases b <;> simp [pairedFlagMass, Fintype.sum_option ]
-
-/-- Hidden reports carry exactly zero information about the fair orientation,
-even after both strings have been observed. -/
-theorem pairedFlag_hidden_joint (P Q H : FiniteLaw α) (ω : ℝ) (b : Bool) (z : α × α) :
-    (1 / 2 : ℝ) * pairedFlagMass P Q H ω b (none, z) =
-      (1 / 2 : ℝ) * (ω ^ 2 * (H.prod H).mass z) := by
-  unfold pairedFlagMass pairedHiddenMass prod
-  ring
-
-theorem pairedFlag_hidden_probability (P Q H : FiniteLaw α) (ω : ℝ) (b : Bool) :
-    (∑ z, pairedFlagMass P Q H ω b (none, z)) = ω ^ 2 := sum_pairedHiddenMass H ω
-
-end TraceReconstruction.FiniteLaw
-
-namespace TraceReconstruction.FiniteLaw
-variable {α : Type*} [Fintype α]
-theorem small_geometric_sum (n : ℕ) {x : ℝ} (hx : x ∈ Set.Icc 0 (1 / 2)) :
-    (∑ j ∈ Finset.range n, x ^ j) ≤ 2 := by
-  have he := geom_sum_mul x n
-  have hn : 0 ≤ ∑ j ∈ Finset.range n, x ^ j :=
-    Finset.sum_nonneg (fun j _ => pow_nonneg hx.1 j)
-  have hp := pow_nonneg hx.1 n
-  nlinarith [mul_le_mul_of_nonneg_left hx.2 hn]
-
-/-- Elementary tail of a binomial polynomial after cancelling orders zero and one. -/
-theorem small_binomial_tail (n : ℕ) {x : ℝ} (hx : 0 ≤ x) (hsmall : (n : ℝ) * x ≤ 1 / 2) :
-    (∑ j ∈ Finset.range (n - 1), (n.choose (j + 2) : ℝ) * x ^ (j + 2)) ≤
-      2 * ((n : ℝ) * x) ^ 2 := by
-  calc
-    _ ≤ ∑ j ∈ Finset.range (n - 1), ((n : ℝ) * x) ^ 2 * ((n : ℝ) * x) ^ j := by
-      apply Finset.sum_le_sum
-      intro j _
-      have hc : (n.choose (j + 2) : ℝ) ≤ (n : ℝ) ^ (j + 2) := by
-        exact_mod_cast Nat.choose_le_pow n (j + 2)
-      calc
-        _ ≤ (n : ℝ) ^ (j + 2) * x ^ (j + 2) :=
-          mul_le_mul_of_nonneg_right hc (pow_nonneg hx _)
-        _ = _ := by rw [← mul_pow, pow_add]; ring
-    _ = ((n : ℝ) * x) ^ 2 * ∑ j ∈ Finset.range (n - 1), ((n : ℝ) * x) ^ j := by
-      rw [Finset.mul_sum]
-    _ ≤ _ := by
-      have h := small_geometric_sum (n - 1) ⟨mul_nonneg (Nat.cast_nonneg n) hx, hsmall⟩
-      nlinarith [mul_le_mul_of_nonneg_left h (sq_nonneg ((n : ℝ) * x))]
-
-theorem expect_pow_expansion (P : FiniteLaw α) (Z : α → ℝ) (u : ℝ) (t : ℕ) :
-    P.expect (fun a => (1 + u * Z a) ^ t) =
-      ∑ j ∈ Finset.range (t + 1), (t.choose j : ℝ) * u ^ j * P.expect (fun a => Z a ^ j) := by
-  unfold expect
-  simp_rw [add_comm (1 : ℝ), add_pow, one_pow, mul_one, mul_pow, Finset.mul_sum]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro j _
-  apply Finset.sum_congr rfl
-  intro a _
-  ring
-
-theorem expect_small_affine_pow (P : FiniteLaw α) (Z : α → ℝ) (t : ℕ) (ht : 2 ≤ t)
-    (hcenter : P.expect Z = 0)
-    (hmom : ∀ j ≤ t, |P.expect (fun a => Z a ^ j)| ≤ 2)
-    {u : ℝ} (hu : (t : ℝ) * |u| ≤ 1 / 2) :
-    P.expect (fun a => (1 + u * Z a) ^ t) ≤ 1 + 4 * ((t : ℝ) * u) ^ 2 := by
-  rw [expect_pow_expansion]
-  have ht' : t + 1 = 2 + (t - 1) := by omega
-  rw [ht', Finset.sum_range_add]
-  have hz : P.expect (fun _ : α => (1 : ℝ)) = 1 := by simp [expect, P.total]
-  simp only [Finset.sum_range_succ, Finset.sum_range_zero, zero_add,
-    Nat.choose_zero_right, Nat.cast_one, pow_zero, mul_one, one_mul,
-    Nat.choose_one_right, pow_one]
-  rw [hz, hcenter]
-  simp only [mul_zero, add_zero]
-  have hb : (∑ j ∈ Finset.range (t - 1), (t.choose (2 + j) : ℝ) * u ^ (2 + j) *
-      P.expect (fun a => Z a ^ (2 + j))) ≤
-      2 * ∑ j ∈ Finset.range (t - 1), (t.choose (j + 2) : ℝ) * |u| ^ (j + 2) := by
-    rw [Finset.mul_sum]
-    apply Finset.sum_le_sum
-    intro j hj
-    have hjt : 2 + j ≤ t := by have := Finset.mem_range.mp hj; omega
-    have hm := hmom (2 + j) hjt
-    calc
-      _ ≤ |(t.choose (2 + j) : ℝ) * u ^ (2 + j) * P.expect (fun a => Z a ^ (2 + j))| :=
-        le_abs_self _
-      _ ≤ (t.choose (2 + j) : ℝ) * |u| ^ (2 + j) * 2 := by
-        rw [abs_mul, abs_mul, abs_of_nonneg (Nat.cast_nonneg _), abs_pow]
-        exact mul_le_mul_of_nonneg_left hm (by positivity)
-      _ = _ := by rw [Nat.add_comm 2 j]; ring
-  have htail := small_binomial_tail t (abs_nonneg u) hu
-  have he : ((t : ℝ) * |u|) ^ 2 = ((t : ℝ) * u) ^ 2 := by rw [mul_pow, mul_pow, sq_abs]
-  rw [he] at htail
-  linarith
-
-/-- Only the centered first moment must vanish; lower absolute moments
-are obtained from the one available source moment, without boundedness. -/
-theorem expect_lower_moment_le_two (P : FiniteLaw α) (Z : α → ℝ) (t : ℕ)
-    (ht : P.expect (fun a => |Z a| ^ t) ≤ 1) (j : ℕ) (hj : j ≤ t) :
-    |P.expect (fun a => Z a ^ j)| ≤ 2 := by
-  have hpoint (a : α) : |Z a| ^ j ≤ 1 + |Z a| ^ t := by
-    by_cases hx : |Z a| ≤ 1
-    · have h := pow_le_one₀ (n := j) (abs_nonneg _) hx
-      nlinarith [pow_nonneg (abs_nonneg (Z a)) t]
-    · have h := pow_le_pow_right₀ (le_of_lt (lt_of_not_ge hx)) hj
-      linarith
-  calc
-    _ ≤ ∑ a, |P.mass a * Z a ^ j| := Finset.abs_sum_le_sum_abs _ _
-    _ = ∑ a, P.mass a * |Z a| ^ j := by
-      apply Finset.sum_congr rfl
-      intro a _
-      rw [abs_mul, abs_of_nonneg (P.nonneg a), abs_pow]
-    _ ≤ ∑ a, P.mass a * (1 + |Z a| ^ t) := by
-      apply Finset.sum_le_sum
-      intro a _
-      exact mul_le_mul_of_nonneg_left (hpoint a) (P.nonneg a)
-    _ = 1 + P.expect (fun a => |Z a| ^ t) := by
-      simp_rw [mul_add, mul_one, Finset.sum_add_distrib, P.total]
-      rfl
-    _ ≤ 2 := by linarith
-
-theorem expect_affine_pow_le (P : FiniteLaw α) (Z : α → ℝ) (t : ℕ)
-    (hmom : ∀ j ≤ t, |P.expect (fun a => Z a ^ j)| ≤ 2) (u : ℝ) :
-    P.expect (fun a => (1 + u * Z a) ^ t) ≤ 2 * (1 + |u|) ^ t := by
-  rw [expect_pow_expansion, add_comm (1 : ℝ), add_pow, Finset.mul_sum]
-  apply Finset.sum_le_sum
-  intro j hj
-  have h := hmom j (by have := Finset.mem_range.mp hj; omega)
-  calc
-    _ ≤ |(t.choose j : ℝ) * u ^ j * P.expect (fun a => Z a ^ j)| := le_abs_self _
-    _ ≤ (t.choose j : ℝ) * |u| ^ j * 2 := by
-      rw [abs_mul, abs_mul, abs_of_nonneg (Nat.cast_nonneg _), abs_pow]
-      exact mul_le_mul_of_nonneg_left h (by positivity)
-    _ = _ := by simp only [one_pow, mul_one]; ring
-
-/-- Scalar two-smoothness at even moment order; its constant is linear
-in that order after taking the square root, as required for tensorization. -/
-theorem scalar_even_moment_unit (P : FiniteLaw α) (Z : α → ℝ) (k : ℕ) (hk : 1 ≤ k)
-    (hcenter : P.expect Z = 0) (ht : P.expect (fun a => |Z a| ^ (2 * k)) ≤ 1) (u : ℝ) :
-    P.expect (fun a => (1 + u * Z a) ^ (2 * k)) ≤
-      (1 + (6 * (2 * k : ℕ) : ℝ) ^ 2 * u ^ 2) ^ k := by
-  have hmom := expect_lower_moment_le_two P Z (2 * k) ht
-  have hkR : (1 : ℝ) ≤ k := by exact_mod_cast hk
-  have htR : (1 : ℝ) ≤ (2 * k : ℕ) := by exact_mod_cast (show 1 ≤ 2 * k by omega)
-  have hbase : 1 ≤ 1 + (6 * (2 * k : ℕ) : ℝ) ^ 2 * u ^ 2 := by nlinarith [mul_nonneg (sq_nonneg (6 * ((2 * k : ℕ) : ℝ))) (sq_nonneg u)]
-  by_cases hu : ((2 * k : ℕ) : ℝ) * |u| ≤ 1 / 2
-  · have hs := expect_small_affine_pow P Z (2 * k) (by omega) hcenter hmom hu
-    have hb : 1 + 4 * (((2 * k : ℕ) : ℝ) * u) ^ 2 ≤
-        1 + (6 * (2 * k : ℕ) : ℝ) ^ 2 * u ^ 2 := by nlinarith [sq_nonneg (((2 * k : ℕ) : ℝ) * u)]
-    exact hs.trans (hb.trans (le_self_pow₀ hbase (by omega)))
-  · have hu' : 1 / 2 ≤ ((2 * k : ℕ) : ℝ) * |u| := le_of_lt (lt_of_not_ge hu)
-    have hsum : 1 + |u| ≤ 3 * ((2 * k : ℕ) : ℝ) * |u| := by
-      have hh := mul_le_mul_of_nonneg_right htR (abs_nonneg u)
-      nlinarith
-    have hc : (2 : ℝ) ≤ (2 : ℝ) ^ (2 * k) := le_self_pow₀ (by norm_num) (by omega)
-    calc
-      _ ≤ 2 * (1 + |u|) ^ (2 * k) := expect_affine_pow_le P Z (2 * k) hmom u
-      _ ≤ 2 * (3 * ((2 * k : ℕ) : ℝ) * |u|) ^ (2 * k) := by gcongr
-      _ ≤ 2 ^ (2 * k) * (3 * ((2 * k : ℕ) : ℝ) * |u|) ^ (2 * k) := by
-        exact mul_le_mul_of_nonneg_right hc (by positivity)
-      _ = ((6 * (2 * k : ℕ) : ℝ) ^ 2 * u ^ 2) ^ k := by
-        rw [← mul_pow, pow_mul]
-        congr 1
-        simp only [mul_pow, sq_abs]
-        ring
-      _ ≤ _ := pow_le_pow_left₀ (by positivity) (by linarith) k
-
-/-- Homogeneous version of the scalar even-moment inequality. -/
-theorem scalar_even_moment (P : FiniteLaw α) (Z : α → ℝ) (k : ℕ) (hk : 1 ≤ k)
-    (hcenter : P.expect Z = 0) (ht : P.expect (fun a => |Z a| ^ (2 * k)) ≤ 1) (b v : ℝ) :
-    P.expect (fun a => (b + v * Z a) ^ (2 * k)) ≤
-      (b ^ 2 + (6 * (2 * k : ℕ) : ℝ) ^ 2 * v ^ 2) ^ k := by
-  have heven (a : α) : |Z a| ^ (2 * k) = Z a ^ (2 * k) := by
-    rw [pow_mul, sq_abs, ← pow_mul]
-  have hmoment : P.expect (fun a => Z a ^ (2 * k)) ≤ 1 := by
-    simpa only [heven] using ht
-  have hc : 1 ≤ (6 * (2 * k : ℕ) : ℝ) ^ 2 := by
-    have hn : (1 : ℝ) ≤ (2 * k : ℕ) := by exact_mod_cast (show 1 ≤ 2 * k by omega)
-    nlinarith
-  by_cases hb : b = 0
-  · subst b
-    simp only [zero_add, zero_pow (by omega : 2 ≠ 0)]
-    have he : P.expect (fun a => (v * Z a) ^ (2 * k)) =
-        v ^ (2 * k) * P.expect (fun a => Z a ^ (2 * k)) := by
-      simp only [expect, mul_pow, Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro a _
-      ring
-    rw [he]
-    calc
-      _ ≤ v ^ (2 * k) := by
-        have hv : 0 ≤ v ^ (2 * k) := by rw [pow_mul]; positivity
-        nlinarith [mul_le_mul_of_nonneg_left hmoment hv]
-      _ = (v ^ 2) ^ k := pow_mul v 2 k
-      _ ≤ _ := pow_le_pow_left₀ (sq_nonneg v) (by nlinarith [mul_le_mul_of_nonneg_right hc (sq_nonneg v)]) k
-  · have hs := scalar_even_moment_unit P Z k hk hcenter ht (v / b)
-    have hbpow : 0 ≤ b ^ (2 * k) := by rw [pow_mul]; positivity
-    have he : P.expect (fun a => (b + v * Z a) ^ (2 * k)) =
-        b ^ (2 * k) * P.expect (fun a => (1 + v / b * Z a) ^ (2 * k)) := by
-      unfold expect
-      rw [Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro a _
-      have h : b + v * Z a = b * (1 + v / b * Z a) := by field_simp [hb]
-      dsimp only
-      rw [h, mul_pow]
-      ring
-    rw [he]
-    calc
-      _ ≤ b ^ (2 * k) * (1 + (6 * (2 * k : ℕ) : ℝ) ^ 2 * (v / b) ^ 2) ^ k :=
-        mul_le_mul_of_nonneg_left hs hbpow
-      _ = _ := by
-        rw [pow_mul, ← mul_pow]
-        congr 1
-        field_simp [hb]
-
-theorem weighted_minkowski_bound (P : FiniteLaw α) (k : ℕ) (hk : 1 ≤ k)
-    (f g : α → ℝ) (hf : ∀ a, 0 ≤ f a) (hg : ∀ a, 0 ≤ g a)
-    (A B : ℝ) (hA : 0 ≤ A) (hB : 0 ≤ B)
-    (hfa : P.expect (fun a => f a ^ k) ≤ A ^ k)
-    (hgb : P.expect (fun a => g a ^ k) ≤ B ^ k) :
-    P.expect (fun a => (f a + g a) ^ k) ≤ (A + B) ^ k := by
-  have hkR : (1 : ℝ) ≤ k := by exact_mod_cast hk
-  have hk0 : (k : ℝ) ≠ 0 := by positivity
-  let w : α → ℝ := fun a => P.mass a ^ (1 / (k : ℝ))
-  have hw (a : α) : 0 ≤ w a := Real.rpow_nonneg (P.nonneg a) _
-  have hwk (a : α) : w a ^ k = P.mass a := by
-    dsimp [w]
-    rw [← Real.rpow_natCast, ← Real.rpow_mul (P.nonneg a), one_div_mul_cancel hk0, Real.rpow_one]
-  have hsingle (F : α → ℝ) (hF : ∀ a, 0 ≤ F a) :
-      (∑ a, |w a * F a| ^ (k : ℝ)) = P.expect (fun a => F a ^ k) := by
-    apply Finset.sum_congr rfl
-    intro a _
-    rw [abs_of_nonneg (mul_nonneg (hw a) (hF a)), Real.rpow_natCast, mul_pow, hwk]
-  have hsum : (∑ a, |w a * f a + w a * g a| ^ (k : ℝ)) =
-      P.expect (fun a => (f a + g a) ^ k) := by
-    simp_rw [← mul_add]
-    exact hsingle _ (fun a => add_nonneg (hf a) (hg a))
-  have hm := Real.Lp_add_le Finset.univ (fun a => w a * f a) (fun a => w a * g a) hkR
-  rw [hsum, hsingle f hf, hsingle g hg] at hm
-  have hroots (F : α → ℝ) (hF : ∀ a, 0 ≤ F a) (C : ℝ) (hC : 0 ≤ C)
-      (hFC : P.expect (fun a => F a ^ k) ≤ C ^ k) :
-      (P.expect (fun a => F a ^ k)) ^ (1 / (k : ℝ)) ≤ C := by
-    calc
-      _ ≤ (C ^ k) ^ (1 / (k : ℝ)) :=
-        Real.rpow_le_rpow (Finset.sum_nonneg (fun a _ => mul_nonneg (P.nonneg a) (pow_nonneg (hF a) k))) hFC (by positivity)
-      _ = C := by
-        rw [← Real.rpow_natCast, one_div, Real.rpow_rpow_inv hC hk0]
-  have hroot : (P.expect (fun a => (f a + g a) ^ k)) ^ (1 / (k : ℝ)) ≤ A + B :=
-    hm.trans (add_le_add (hroots f hf A hA hfa) (hroots g hg B hB hgb))
-  have hexp : 0 ≤ P.expect (fun a => (f a + g a) ^ k) :=
-    Finset.sum_nonneg (fun a _ => mul_nonneg (P.nonneg a) (pow_nonneg (add_nonneg (hf a) (hg a)) k))
-  have hp := pow_le_pow_left₀ (Real.rpow_nonneg hexp _) hroot k
-  rw [← Real.rpow_natCast, ← Real.rpow_mul hexp, one_div_mul_cancel hk0, Real.rpow_one] at hp
-  exact hp
-theorem product_succ_expect_tail (P : FiniteLaw α) (n : ℕ) (f : (Fin (n + 1) → α) → ℝ) :
-    (P.product (n + 1)).expect f =
-      (P.product n).expect (fun b => P.expect (fun a => f (Fin.cons a b))) := by
-  rw [product_succ_expect]
-  unfold expect
-  simp_rw [Finset.mul_sum]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro b _
-  apply Finset.sum_congr rfl
-  intro a _
-  ring
-
-end TraceReconstruction.FiniteLaw
-namespace TraceReconstruction
-variable {α : Type*}
-def multilinear {n : ℕ} (A : (Fin n → Bool) → ℝ) (Z : α → ℝ) (x : Fin n → α) : ℝ :=
-  ∑ σ, A σ * ∏ j, if σ j then Z (x j) else 1
-
-def multilinearEnergy {n : ℕ} (A : (Fin n → Bool) → ℝ) (R : ℝ) : ℝ :=
-  ∑ σ, A σ ^ 2 * ∏ j, if σ j then R ^ 2 else 1
-
-theorem multilinearEnergy_nonneg {n : ℕ} (A : (Fin n → Bool) → ℝ) (R : ℝ) :
-    0 ≤ multilinearEnergy A R := by
-  apply Finset.sum_nonneg
-  intro σ _
-  apply mul_nonneg (sq_nonneg _)
-  apply Finset.prod_nonneg
-  intro j _
-  split <;> positivity
-
-theorem multilinear_cons {n : ℕ} (A : (Fin (n + 1) → Bool) → ℝ) (Z : α → ℝ)
-    (x : Fin n → α) (a : α) :
-    multilinear A Z (Fin.cons a x) =
-      multilinear (fun σ => A (Fin.cons false σ)) Z x +
-      Z a * multilinear (fun σ => A (Fin.cons true σ)) Z x := by
-  unfold multilinear
-  rw [← (Fin.consEquiv (fun _ : Fin (n + 1) => Bool)).sum_comp, Fintype.sum_prod_type]
-  simp only [Fin.consEquiv, Equiv.coe_fn_mk, Fintype.sum_bool, Fin.prod_univ_succ, Fin.cons_zero, Fin.cons_succ,
-    Bool.false_eq_true, ↓reduceIte, one_mul]
-  rw [Finset.mul_sum]
-  rw [add_comm]
-  congr 1
-  apply Finset.sum_congr rfl
-  intro σ _
-  ring
-
-theorem multilinearEnergy_cons {n : ℕ} (A : (Fin (n + 1) → Bool) → ℝ) (R : ℝ) :
-    multilinearEnergy A R =
-      multilinearEnergy (fun σ => A (Fin.cons false σ)) R +
-      R ^ 2 * multilinearEnergy (fun σ => A (Fin.cons true σ)) R := by
-  unfold multilinearEnergy
-  rw [← (Fin.consEquiv (fun _ : Fin (n + 1) => Bool)).sum_comp, Fintype.sum_prod_type]
-  simp only [Fin.consEquiv, Equiv.coe_fn_mk, Fintype.sum_bool, Fin.prod_univ_succ, Fin.cons_zero, Fin.cons_succ,
-    Bool.false_eq_true, ↓reduceIte, one_mul]
-  rw [Finset.mul_sum]
-  rw [add_comm]
-  congr 1
-  apply Finset.sum_congr rfl
-  intro σ _
-  ring
-
-/-- The finite-product moment inequality used for every score expansion.
-The order-cost is linear per degree, and no boundedness assumption is needed. -/
-theorem multilinear_even_moment {α : Type*} [Fintype α] (P : FiniteLaw α)
-    (Z : α → ℝ) (k : ℕ) (hk : 1 ≤ k) (hcenter : P.expect Z = 0)
-    (ht : P.expect (fun a => |Z a| ^ (2 * k)) ≤ 1)
-    (n : ℕ) (A : (Fin n → Bool) → ℝ) :
-    (P.product n).expect (fun x => multilinear A Z x ^ (2 * k)) ≤
-      (multilinearEnergy A (6 * (2 * k : ℕ))) ^ k := by
-  induction n with
-  | zero =>
-    simp only [FiniteLaw.expect, FiniteLaw.product, multilinear, multilinearEnergy,
-      Fintype.prod_empty, mul_one, one_mul]
-    simp only [Finset.univ_unique, Finset.sum_singleton]
-    exact le_of_eq (pow_mul _ 2 k)
-  | succ n ih =>
-    let A0 : (Fin n → Bool) → ℝ := fun σ => A (Fin.cons false σ)
-    let A1 : (Fin n → Bool) → ℝ := fun σ => A (Fin.cons true σ)
-    let R : ℝ := 6 * (2 * k : ℕ)
-    let F0 : (Fin n → α) → ℝ := fun x => multilinear A0 Z x
-    let F1 : (Fin n → α) → ℝ := fun x => multilinear A1 Z x
-    have h0 := ih A0
-    have h1 := ih A1
-    have hA0 := multilinearEnergy_nonneg A0 R
-    have hA1 := multilinearEnergy_nonneg A1 R
-    rw [FiniteLaw.product_succ_expect_tail, multilinearEnergy_cons]
-    calc
-      _ ≤ (P.product n).expect (fun x => (F0 x ^ 2 + R ^ 2 * F1 x ^ 2) ^ k) := by
-        apply FiniteLaw.expect_mono
-        intro x
-        simp_rw [multilinear_cons]
-        have hs := FiniteLaw.scalar_even_moment P Z k hk hcenter ht (F0 x) (F1 x)
-        simpa only [F0, F1, A0, A1, R, mul_comm] using hs
-      _ ≤ (multilinearEnergy A0 R + R ^ 2 * multilinearEnergy A1 R) ^ k := by
-        apply FiniteLaw.weighted_minkowski_bound (P.product n) k hk
-          (fun x => F0 x ^ 2) (fun x => R ^ 2 * F1 x ^ 2)
-          (fun x => sq_nonneg _) (fun x => mul_nonneg (sq_nonneg _) (sq_nonneg _))
-          (multilinearEnergy A0 R) (R ^ 2 * multilinearEnergy A1 R) hA0
-          (mul_nonneg (sq_nonneg _) hA1)
-        · simpa only [← pow_mul] using h0
-        · have he : (P.product n).expect (fun x => (R ^ 2 * F1 x ^ 2) ^ k) =
-              (R ^ 2) ^ k * (P.product n).expect (fun x => F1 x ^ (2 * k)) := by
-            unfold FiniteLaw.expect
-            simp only [mul_pow, ← pow_mul, Finset.mul_sum]
-            apply Finset.sum_congr rfl
-            intro x _
-            ring
-          rw [he, mul_pow (R ^ 2) (multilinearEnergy A1 R)]
-          exact mul_le_mul_of_nonneg_left h1 (pow_nonneg (sq_nonneg R) k)
-      _ = _ := rfl
-
-def monomialDegree {n : ℕ} (σ : Fin n → Bool) : ℕ :=
-  (Finset.univ.filter (fun j => σ j = true)).card
-
-theorem monomial_weight {n : ℕ} (σ : Fin n → Bool) (R : ℝ) :
-    (∏ j, if σ j then R ^ 2 else 1) = R ^ (2 * monomialDegree σ) := by
-  rw [← Finset.prod_filter]
-  simp only [Finset.prod_const, ← pow_mul, monomialDegree]
-
-theorem multilinearEnergy_homogeneous {n : ℕ} (A : (Fin n → Bool) → ℝ)
-    (r : ℕ) (hA : ∀ σ, monomialDegree σ ≠ r → A σ = 0) (R : ℝ) :
-    multilinearEnergy A R = R ^ (2 * r) * ∑ σ, A σ ^ 2 := by
-  unfold multilinearEnergy
-  rw [Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro σ _
-  rw [monomial_weight]
-  by_cases hd : monomialDegree σ = r
-  · rw [hd, mul_comm]
-  · rw [hA σ hd, zero_pow (by omega : 2 ≠ 0), zero_mul, mul_zero]
-
-theorem homogeneous_even_moment {α : Type*} [Fintype α] (P : FiniteLaw α)
-    (Z : α → ℝ) (k : ℕ) (hk : 1 ≤ k) (hcenter : P.expect Z = 0)
-    (ht : P.expect (fun a => |Z a| ^ (2 * k)) ≤ 1)
-    (n r : ℕ) (A : (Fin n → Bool) → ℝ)
-    (hA : ∀ σ, monomialDegree σ ≠ r → A σ = 0) :
-    (P.product n).expect (fun x => multilinear A Z x ^ (2 * k)) ≤
-      ((6 * (2 * k : ℕ) : ℝ) ^ (2 * r) * ∑ σ, A σ ^ 2) ^ k := by
-  have h := multilinear_even_moment P Z k hk hcenter ht n A
-  rw [multilinearEnergy_homogeneous A r hA] at h
-  exact h
-
-/-- Squared L^(2k)-norm for a finite law, expressed without measure coercions. -/
-def FiniteLaw.evenMomentSq {α : Type*} [Fintype α] (P : FiniteLaw α)
-    (f : α → ℝ) (k : ℕ) : ℝ :=
-  (P.expect (fun a => |f a| ^ (2 * k))) ^ (1 / (k : ℝ))
-
-theorem homogeneous_moment_sq {α : Type*} [Fintype α] (P : FiniteLaw α)
-    (Z : α → ℝ) (k : ℕ) (hk : 1 ≤ k) (hcenter : P.expect Z = 0)
-    (ht : P.expect (fun a => |Z a| ^ (2 * k)) ≤ 1)
-    (n r : ℕ) (A : (Fin n → Bool) → ℝ)
-    (hA : ∀ σ, monomialDegree σ ≠ r → A σ = 0) :
-    (P.product n).evenMomentSq (multilinear A Z) k ≤
-      (6 * (2 * k : ℕ) : ℝ) ^ (2 * r) * ∑ σ, A σ ^ 2 := by
-  have h := homogeneous_even_moment P Z k hk hcenter ht n r A hA
-  have he (x : Fin n → α) : |multilinear A Z x| ^ (2 * k) =
-      multilinear A Z x ^ (2 * k) := by rw [pow_mul, sq_abs, ← pow_mul]
-  unfold FiniteLaw.evenMomentSq
-  simp_rw [he]
-  have hn : 0 ≤ (P.product n).expect (fun x => multilinear A Z x ^ (2 * k)) := by
-    apply Finset.sum_nonneg
-    intro x _
-    apply mul_nonneg ((P.product n).nonneg _)
-    change 0 ≤ multilinear A Z x ^ (2 * k)
-    rw [pow_mul]
-    positivity
-  have hb : 0 ≤ (6 * (2 * k : ℕ) : ℝ) ^ (2 * r) * ∑ σ, A σ ^ 2 := by positivity
-  calc
-    _ ≤ (((6 * (2 * k : ℕ) : ℝ) ^ (2 * r) * ∑ σ, A σ ^ 2) ^ k) ^ (1 / (k : ℝ)) :=
-      Real.rpow_le_rpow hn h (by positivity)
-    _ = _ := by rw [← Real.rpow_natCast, one_div, Real.rpow_rpow_inv hb (by positivity)]
-end TraceReconstruction
-
-namespace TraceReconstruction.FiniteLaw
-variable {α : Type*} [Fintype α]
-def eventMass (P : FiniteLaw α) (A : Finset α) : ℝ := ∑ a ∈ A, P.mass a
-
-theorem eventMass_nonneg (P : FiniteLaw α) (A : Finset α) : 0 ≤ P.eventMass A :=
-  Finset.sum_nonneg (fun a _ => P.nonneg a)
-
-theorem eventMass_le_one (P : FiniteLaw α) (A : Finset α) : P.eventMass A ≤ 1 := by
-  rw [← P.total]
-  exact Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _)
-    (fun a _ _ => P.nonneg a)
-
-def condition (P : FiniteLaw α) (A : Finset α) (hA : 0 < P.eventMass A) : FiniteLaw α := by
-  classical
-  exact {
-    mass := fun a => if a ∈ A then P.mass a / P.eventMass A else 0
-    nonneg := fun a => by split <;> positivity [P.nonneg a]
-    total := by
-      rw [← Finset.sum_filter]
-      simp only [Finset.filter_mem_eq_inter, Finset.univ_inter]
-      rw [← Finset.sum_div]
-      exact div_self (ne_of_gt hA) }
-
-theorem overlap_condition (P : FiniteLaw α) (A : Finset α) (hA : 0 < P.eventMass A) :
-    overlap P (condition P A hA) = P.eventMass A := by
-  classical
-  have he (a : α) : min (P.mass a) ((condition P A hA).mass a) =
-      if a ∈ A then P.mass a else 0 := by
-    by_cases ha : a ∈ A
-    · simp only [condition, ha, ↓reduceIte]
-      apply min_eq_left
-      apply (le_div_iff₀ hA).mpr
-      have h := mul_le_mul_of_nonneg_left (P.eventMass_le_one A) (P.nonneg a)
-      simpa only [mul_one] using h
-    · simp only [condition, ha, ↓reduceIte, min_eq_right (P.nonneg a)]
-  unfold overlap
-  simp_rw [he]
-  rw [← Finset.sum_filter]
-  simp only [Finset.filter_mem_eq_inter, Finset.univ_inter, eventMass]
-
-theorem tv_condition (P : FiniteLaw α) (A : Finset α) (hA : 0 < P.eventMass A) :
-    tv P (condition P A hA) = 1 - P.eventMass A := by
-  have he := P.overlap_condition A hA
-  rw [overlap_eq_one_sub_tv] at he
-  linarith
-
-/-- Densities relative to one finite reference, preserving the exact normalization. -/
-def withDensity (P : FiniteLaw α) (f : α → ℝ) (hf : ∀ a, 0 ≤ f a)
-    (hmean : P.expect f = 1) : FiniteLaw α where
-  mass a := P.mass a * f a
-  nonneg a := mul_nonneg (P.nonneg a) (hf a)
-  total := hmean
-
-theorem density_bad_mass_sq (P : FiniteLaw α) (f : α → ℝ) (A : Finset α) :
-    (∑ a ∈ A, P.mass a * f a) ^ 2 ≤
-      (P.eventMass A) * P.expect (fun a => f a ^ 2) := by
-  have h := Finset.sum_mul_sq_le_sq_mul_sq A (fun a => Real.sqrt (P.mass a))
-    (fun a => Real.sqrt (P.mass a) * f a)
-  simp only [mul_pow, Real.sq_sqrt (P.nonneg _)] at h
-  have he : (∑ a ∈ A, Real.sqrt (P.mass a) * (Real.sqrt (P.mass a) * f a)) =
-      ∑ a ∈ A, P.mass a * f a := by
-    apply Finset.sum_congr rfl
-    intro a _
-    rw [← mul_assoc, ← pow_two, Real.sq_sqrt (P.nonneg a)]
-  rw [he] at h
-  apply h.trans
-  apply mul_le_mul_of_nonneg_left _ (P.eventMass_nonneg A)
-  exact Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _) (fun a _ _ =>
-    mul_nonneg (P.nonneg a) (sq_nonneg _))
-
-/-- Loss from the small-reference-density event by the exact available moment. -/
-theorem low_density_reference_mass (P : FiniteLaw α) (f : α → ℝ) (t : ℕ)
-    (A : Finset α) (hA : ∀ a ∈ A, f a < 1 / 2) :
-    P.eventMass A ≤ (2 : ℝ) ^ t * P.expect (fun a => |f a - 1| ^ t) := by
-  have hlocal (a : α) (ha : a ∈ A) : 1 ≤ (2 : ℝ) ^ t * |f a - 1| ^ t := by
-    rw [← mul_pow]
-    apply one_le_pow₀
-    have hh := neg_le_abs (f a - 1)
-    linarith [hA a ha]
-  calc
-    _ ≤ ∑ a ∈ A, (2 : ℝ) ^ t * (P.mass a * |f a - 1| ^ t) := by
-      apply Finset.sum_le_sum
-      intro a ha
-      nlinarith [mul_le_mul_of_nonneg_left (hlocal a ha) (P.nonneg a)]
-    _ ≤ ∑ a, (2 : ℝ) ^ t * (P.mass a * |f a - 1| ^ t) := by
-      exact Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _) (fun a _ _ => by positivity [P.nonneg a])
-    _ = _ := by rw [← Finset.mul_sum]; rfl
-
-/-- Exact common-event likelihood ratio: no lower bound is imposed off the event. -/
-theorem conditioned_density_identity (R : FiniteLaw α) (f g : α → ℝ)
-    (hf : ∀ a, 0 ≤ f a) (hg : ∀ a, 0 ≤ g a)
-    (hfm : R.expect f = 1) (hgm : R.expect g = 1)
-    (A : Finset α) (hAf : ∀ a ∈ A, 0 < f a)
-    (hP : 0 < (R.withDensity f hf hfm).eventMass A)
-    (hQ : 0 < (R.withDensity g hg hgm).eventMass A) (a : α) :
-    (condition (R.withDensity g hg hgm) A hQ).mass a =
-      (condition (R.withDensity f hf hfm) A hP).mass a *
-      ((R.withDensity f hf hfm).eventMass A /
-        (R.withDensity g hg hgm).eventMass A * (g a / f a)) := by
-  classical
-  by_cases ha : a ∈ A
-  · simp only [condition, ha, ↓reduceIte]
-    change R.mass a * g a / (R.withDensity g hg hgm).eventMass A =
-      R.mass a * f a / (R.withDensity f hf hfm).eventMass A *
-        ((R.withDensity f hf hfm).eventMass A /
-          (R.withDensity g hg hgm).eventMass A * (g a / f a))
-    field_simp [ne_of_gt hP, ne_of_gt hQ, ne_of_gt (hAf a ha)]
-  · simp only [condition, ha, ↓reduceIte, zero_mul]
-
-/-- The raw likelihood score on a common clipping event, at every integer moment. -/
-theorem clipped_raw_moment (R : FiniteLaw α) (f g : α → ℝ)
-    (hf : ∀ a, 0 ≤ f a) (hfm : R.expect f = 1)
-    (A : Finset α) (hAf : ∀ a ∈ A, 1 / 2 ≤ f a)
-    (hP : 0 < (R.withDensity f hf hfm).eventMass A) (k : ℕ) :
-    (condition (R.withDensity f hf hfm) A hP).expect
-        (fun a => |g a / f a - 1| ^ (k + 1)) ≤
-      (2 : ℝ) ^ k / (R.withDensity f hf hfm).eventMass A *
-        R.expect (fun a => |g a - f a| ^ (k + 1)) := by
-  classical
-  have hpoint (a : α) (ha : a ∈ A) :
-      f a * |g a / f a - 1| ^ (k + 1) ≤
-        (2 : ℝ) ^ k * |g a - f a| ^ (k + 1) := by
-    have hp : 0 < f a := lt_of_lt_of_le (by norm_num) (hAf a ha)
-    rw [div_sub_one (ne_of_gt hp), abs_div, abs_of_pos hp]
-    have hi : 0 ≤ 1 / f a := by positivity
-    have hi2 : 1 / f a ≤ 2 := (div_le_iff₀ hp).mpr (by linarith [hAf a ha])
-    calc
-      _ = |g a - f a| ^ (k + 1) * (1 / f a) ^ k := by
-        rw [div_pow, div_pow, one_pow, pow_succ (f a)]
-        field_simp [ne_of_gt hp]
-      _ ≤ _ := by
-        simpa only [mul_comm] using mul_le_mul_of_nonneg_left
-          (pow_le_pow_left₀ hi hi2 k) (show 0 ≤ |g a - f a| ^ (k + 1) by positivity)
-  unfold expect
-  calc
-    _ = ∑ a ∈ A, (R.mass a * (f a * |g a / f a - 1| ^ (k + 1))) /
-          (R.withDensity f hf hfm).eventMass A := by
-      simp only [condition, withDensity]
-      simp_rw [ite_mul, zero_mul]
-      rw [← Finset.sum_filter]
-      simp only [Finset.filter_mem_eq_inter, Finset.univ_inter]
-      apply Finset.sum_congr rfl
-      intro a _
-      ring
-    _ ≤ ∑ a ∈ A, ((2 : ℝ) ^ k * (R.mass a * |g a - f a| ^ (k + 1))) /
-          (R.withDensity f hf hfm).eventMass A := by
-      apply Finset.sum_le_sum
-      intro a ha
-      apply div_le_div_of_nonneg_right _ hP.le
-      nlinarith [mul_le_mul_of_nonneg_left (hpoint a ha) (R.nonneg a)]
-    _ ≤ ∑ a, ((2 : ℝ) ^ k * (R.mass a * |g a - f a| ^ (k + 1))) /
-          (R.withDensity f hf hfm).eventMass A := by
-      apply Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _)
-      intro a _ _
-      positivity [R.nonneg a]
-    _ = _ := by rw [← Finset.sum_div, ← Finset.mul_sum]; ring
-
-/-- Normalizing the second conditioned law costs only its scalar normalization error. -/
-theorem score_rescaling_moment (P : FiniteLaw α) (u : α → ℝ)
-    (c : ℝ) (hc : 0 ≤ c) (k : ℕ) :
-    P.expect (fun a => |c * u a - 1| ^ (k + 1)) ≤
-      2 ^ k * (c ^ (k + 1) * P.expect (fun a => |u a - 1| ^ (k + 1)) +
-        |c - 1| ^ (k + 1)) := by
-  have hb (a : α) : |c * u a - 1| ^ (k + 1) ≤
-      2 ^ k * (c ^ (k + 1) * |u a - 1| ^ (k + 1) + |c - 1| ^ (k + 1)) := by
-    have he : c * u a - 1 = c * (u a - 1) + (c - 1) := by ring
-    rw [he]
-    calc
-      _ ≤ (|c * (u a - 1)| + |c - 1|) ^ (k + 1) :=
-        pow_le_pow_left₀ (abs_nonneg _) (abs_add_le _ _) _
-      _ ≤ _ := by
-        have h := add_pow_le (abs_nonneg (c * (u a - 1))) (abs_nonneg (c - 1)) (k + 1)
-        simpa only [Nat.add_sub_cancel, abs_mul, abs_of_nonneg hc, mul_pow] using h
-  unfold expect
-  calc
-    _ ≤ ∑ a, P.mass a * (2 ^ k * (c ^ (k + 1) * |u a - 1| ^ (k + 1) +
-        |c - 1| ^ (k + 1))) := Finset.sum_le_sum (fun a _ =>
-          mul_le_mul_of_nonneg_left (hb a) (P.nonneg a))
-    _ = 2 ^ k * (c ^ (k + 1) * (∑ a, P.mass a * |u a - 1| ^ (k + 1)) +
-        |c - 1| ^ (k + 1) * ∑ a, P.mass a) := by
-      simp only [mul_add, Finset.sum_add_distrib, Finset.mul_sum]
-      congr 1 <;> apply Finset.sum_congr rfl <;> intro a _ <;> ring
-    _ = _ := by rw [P.total, mul_one]
-end TraceReconstruction.FiniteLaw
-
-namespace TraceReconstruction
-namespace FiniteLaw
-variable {α β : Type*} [Fintype α] [Fintype β]
-def coordinateProduct {n : ℕ} (P : Fin n → FiniteLaw α) : FiniteLaw (Fin n → α) where
-  mass x := ∏ i, (P i).mass (x i)
-  nonneg x := Finset.prod_nonneg (fun i _ => (P i).nonneg (x i))
-  total := by rw [← Fintype.prod_sum]; simp only [FiniteLaw.total, Finset.prod_const_one]
-
-def mixture (R : FiniteLaw β) (P : β → FiniteLaw α) : FiniteLaw α where
-  mass a := ∑ b, R.mass b * (P b).mass a
-  nonneg a := Finset.sum_nonneg (fun b _ => mul_nonneg (R.nonneg b) ((P b).nonneg a))
-  total := by
-    rw [Finset.sum_comm]
-    simp_rw [← Finset.mul_sum, FiniteLaw.total, mul_one]
-    exact R.total
-end FiniteLaw
-variable {α Ω : Type*} [Fintype Ω]
-
-def selectionCoefficient {n : ℕ} (θ σ : Fin n → Bool) : ℝ :=
-  ∏ i, if σ i then (if θ i then 1 else 0) else 1
-
-def selectionDensity {n : ℕ} (θ : Fin n → Bool) (Z : α → ℝ) (x : Fin n → α) : ℝ :=
-  ∏ i, if θ i then 1 + Z (x i) else 1
-
-def averagedCoefficient {n : ℕ} (R : FiniteLaw Ω) (θ : Ω → Fin n → Bool)
-    (σ : Fin n → Bool) : ℝ := R.expect (fun ω => selectionCoefficient (θ ω) σ)
-
-def averagedDensity {n : ℕ} (R : FiniteLaw Ω) (θ : Ω → Fin n → Bool)
-    (Z : α → ℝ) (x : Fin n → α) : ℝ := R.expect (fun ω => selectionDensity (θ ω) Z x)
-
-theorem selectionDensity_expansion {n : ℕ} (θ : Fin n → Bool) (Z : α → ℝ)
-    (x : Fin n → α) : selectionDensity θ Z x = multilinear (selectionCoefficient θ) Z x := by
-  unfold selectionDensity multilinear selectionCoefficient
-  simp_rw [← Finset.prod_mul_distrib]
-  rw [← Fintype.prod_sum (fun (i : Fin n) (b : Bool) =>
-    (if b then (if θ i then (1 : ℝ) else 0) else 1) * (if b then Z (x i) else 1))]
-  apply Finset.prod_congr rfl
-  intro i _
-  cases θ i <;> simp [add_comm]
-
-theorem averagedDensity_expansion {n : ℕ} (R : FiniteLaw Ω) (θ : Ω → Fin n → Bool)
-    (Z : α → ℝ) (x : Fin n → α) :
-    averagedDensity R θ Z x = multilinear (averagedCoefficient R θ) Z x := by
-  unfold averagedDensity averagedCoefficient
-  simp_rw [selectionDensity_expansion]
-  unfold multilinear FiniteLaw.expect
-  simp_rw [Finset.mul_sum, Finset.sum_mul]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro σ _
-  apply Finset.sum_congr rfl
-  intro ω _
-  ring
-
-theorem averagedDensity_difference {n : ℕ} (R : FiniteLaw Ω)
-    (θ φ : Ω → Fin n → Bool) (Z : α → ℝ) (x : Fin n → α) :
-    averagedDensity R θ Z x - averagedDensity R φ Z x =
-      multilinear (fun σ => averagedCoefficient R θ σ - averagedCoefficient R φ σ) Z x := by
-  rw [averagedDensity_expansion, averagedDensity_expansion]
-  simp only [multilinear, sub_mul, Finset.sum_sub_distrib]
-
-theorem coefficient_nonneg {n : ℕ} (R : FiniteLaw Ω) (θ : Ω → Fin n → Bool)
-    (σ : Fin n → Bool) : 0 ≤ averagedCoefficient R θ σ := by
-  apply Finset.sum_nonneg
-  intro ω _
-  apply mul_nonneg (R.nonneg ω)
-  apply Finset.prod_nonneg
-  intro i _
-  cases σ i <;> cases θ ω i <;> norm_num
-
-theorem coefficient_le_one {n : ℕ} (R : FiniteLaw Ω) (θ : Ω → Fin n → Bool)
-    (σ : Fin n → Bool) : averagedCoefficient R θ σ ≤ 1 := by
-  rw [← R.total]
-  apply Finset.sum_le_sum
-  intro ω _
-  have h : selectionCoefficient (θ ω) σ ≤ 1 := by
-    apply Finset.prod_le_one₀
-    · intro i _; cases σ i <;> cases θ ω i <;> norm_num
-    · intro i _; cases σ i <;> cases θ ω i <;> norm_num
-  simpa only [mul_one] using mul_le_mul_of_nonneg_left h (R.nonneg ω)
-
-/-- The coefficient for the empty subset is exactly one, before or after averaging. -/
-theorem averagedCoefficient_empty {n : ℕ} (R : FiniteLaw Ω) (θ : Ω → Fin n → Bool) :
-    averagedCoefficient R θ (fun _ => false) = 1 := by
-  simp [averagedCoefficient, selectionCoefficient, FiniteLaw.expect, R.total]
-
-/-- Exact density of the conditionally independent leaf oracle used in the source. -/
-theorem leaf_oracle_density [Fintype α] {n : ℕ} (P Q : FiniteLaw α) (Z : α → ℝ)
-    (hscore : ∀ a, Q.mass a = P.mass a * (1 + Z a))
-    (R : FiniteLaw Ω) (θ : Ω → Fin n → Bool) (x : Fin n → α) :
-    (R.mixture (fun ω => FiniteLaw.coordinateProduct (fun i => if θ ω i then Q else P))).mass x =
-      (∏ i, P.mass (x i)) * averagedDensity R θ Z x := by
-  unfold FiniteLaw.mixture FiniteLaw.coordinateProduct averagedDensity FiniteLaw.expect
-  rw [Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro ω _
-  have he : (∏ i, (if θ ω i then Q else P).mass (x i)) =
-      (∏ i, P.mass (x i)) * selectionDensity (θ ω) Z x := by
-    rw [selectionDensity, ← Finset.prod_mul_distrib]
-    apply Finset.prod_congr rfl
-    intro i _
-    cases θ ω i <;> simp only [Bool.false_eq_true, ↓reduceIte, mul_one, hscore]
-  change R.mass ω * (∏ i, (if θ ω i then Q else P).mass (x i)) =
-    (∏ i, P.mass (x i)) * (R.mass ω * selectionDensity (θ ω) Z x)
-  rw [he]
-  ring
-end TraceReconstruction
-
-end
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-!
+# FiniteLaw
+
+Finite laws, common-reference total variation, Hellinger affinity, overlap, scalar moments, hypercontractivity, clipping, and leaf densities.
+-/
+
+open scoped ENNReal
+noncomputable section
+
+noncomputable section
+namespace TraceReconstruction
+
+/-- Finite probability laws used for the exact statistical transfer. -/
+structure FiniteLaw (α : Type*) [Fintype α] where
+  mass : α → ℝ
+  nonneg : ∀ a, 0 ≤ mass a
+  total : ∑ a, mass a = 1
+
+namespace FiniteLaw
+variable {α β : Type*} [Fintype α] [Fintype β]
+
+def tv (P Q : FiniteLaw α) : ℝ := (1 / 2 : ℝ) * ∑ a, |P.mass a - Q.mass a|
+def hellingerSq (P Q : FiniteLaw α) : ℝ :=
+  ∑ a, (Real.sqrt (P.mass a) - Real.sqrt (Q.mass a)) ^ 2
+def affinity (P Q : FiniteLaw α) : ℝ :=
+  ∑ a, Real.sqrt (P.mass a) * Real.sqrt (Q.mass a)
+
+theorem affinity_nonneg (P Q : FiniteLaw α) : 0 ≤ affinity P Q := by
+  apply Finset.sum_nonneg
+  intro a _; positivity
+
+theorem hellingerSq_nonneg (P Q : FiniteLaw α) : 0 ≤ hellingerSq P Q := by
+  apply Finset.sum_nonneg
+  intro a _; positivity
+
+theorem hellingerSq_eq (P Q : FiniteLaw α) :
+    hellingerSq P Q = 2 - 2 * affinity P Q := by
+  unfold hellingerSq affinity
+  calc
+    _ = (∑ a, P.mass a) + (∑ a, Q.mass a) - 2 * ∑ a,
+        Real.sqrt (P.mass a) * Real.sqrt (Q.mass a) := by
+      rw [Finset.mul_sum, ← Finset.sum_add_distrib, ← Finset.sum_sub_distrib]
+      apply Finset.sum_congr rfl
+      intro a _
+      have hp := Real.sq_sqrt (P.nonneg a)
+      have hq := Real.sq_sqrt (Q.nonneg a)
+      nlinarith only [hp, hq]
+    _ = _ := by rw [P.total, Q.total]; ring
+
+theorem affinity_le_one (P Q : FiniteLaw α) : affinity P Q ≤ 1 := by
+  have h := hellingerSq_nonneg P Q
+  rw [hellingerSq_eq] at h
+  linarith
+
+theorem tv_nonneg (P Q : FiniteLaw α) : 0 ≤ tv P Q := by unfold tv; positivity
+
+theorem tv_sq_le_hellingerSq (P Q : FiniteLaw α) :
+    tv P Q ^ 2 ≤ hellingerSq P Q := by
+  have hpoint (a : α) : |P.mass a - Q.mass a| ^ 2 ≤
+      (Real.sqrt (P.mass a) - Real.sqrt (Q.mass a)) ^ 2 *
+        (Real.sqrt (P.mass a) + Real.sqrt (Q.mass a)) ^ 2 := by
+    rw [sq_abs]
+    have hp := Real.sq_sqrt (P.nonneg a)
+    have hq := Real.sq_sqrt (Q.nonneg a)
+    nlinarith only [hp, hq, sq_nonneg (P.mass a - Q.mass a)]
+  have h := Finset.sum_sq_le_sum_mul_sum_of_sq_le_mul
+    (s := Finset.univ)
+    (r := fun a => |P.mass a - Q.mass a|)
+    (f := fun a => (Real.sqrt (P.mass a) - Real.sqrt (Q.mass a)) ^ 2)
+    (g := fun a => (Real.sqrt (P.mass a) + Real.sqrt (Q.mass a)) ^ 2)
+    (fun a _ => sq_nonneg _) (fun a _ => sq_nonneg _) (fun a _ => hpoint a)
+  have hplus : (∑ a, (Real.sqrt (P.mass a) + Real.sqrt (Q.mass a)) ^ 2) =
+      2 + 2 * affinity P Q := by
+    unfold affinity
+    calc
+      _ = (∑ a, P.mass a) + (∑ a, Q.mass a) + 2 * ∑ a,
+          Real.sqrt (P.mass a) * Real.sqrt (Q.mass a) := by
+        rw [Finset.mul_sum, ← Finset.sum_add_distrib, ← Finset.sum_add_distrib]
+        apply Finset.sum_congr rfl
+        intro a _
+        have hp := Real.sq_sqrt (P.nonneg a)
+        have hq := Real.sq_sqrt (Q.nonneg a)
+        nlinarith only [hp, hq]
+      _ = _ := by rw [P.total, Q.total]; ring
+  rw [hplus] at h
+  change (∑ a, |P.mass a - Q.mass a|) ^ 2 ≤ hellingerSq P Q * (2 + 2 * affinity P Q) at h
+  have hbound := mul_le_mul_of_nonneg_left (affinity_le_one P Q) (hellingerSq_nonneg P Q)
+  unfold tv
+  nlinarith only [h, hbound]
+
+theorem tv_le_sqrt_hellingerSq (P Q : FiniteLaw α) :
+    tv P Q ≤ Real.sqrt (hellingerSq P Q) := by
+  have h := tv_sq_le_hellingerSq P Q
+  have hs := Real.sq_sqrt (hellingerSq_nonneg P Q)
+  have hn := Real.sqrt_nonneg (hellingerSq P Q)
+  nlinarith only [h, hs, hn]
+
+def product (P : FiniteLaw α) (m : ℕ) : FiniteLaw (Fin m → α) where
+  mass a := ∏ j, P.mass (a j)
+  nonneg a := Finset.prod_nonneg fun j _ => P.nonneg (a j)
+  total := by
+    rw [← Fintype.prod_sum]
+    simp [P.total]
+
+theorem affinity_product (P Q : FiniteLaw α) (m : ℕ) :
+    affinity (P.product m) (Q.product m) = affinity P Q ^ m := by
+  unfold affinity product
+  simp only
+  have hexp (a : Fin m → α) :
+      Real.sqrt (∏ j, P.mass (a j)) * Real.sqrt (∏ j, Q.mass (a j)) =
+        ∏ j, (Real.sqrt (P.mass (a j)) * Real.sqrt (Q.mass (a j))) := by
+    rw [Real.sqrt_prod _ (fun j _ => P.nonneg (a j)),
+      Real.sqrt_prod _ (fun j _ => Q.nonneg (a j)), Finset.prod_mul_distrib]
+  simp_rw [hexp]
+  rw [← Fintype.prod_sum (fun (_ : Fin m) a => Real.sqrt (P.mass a) * Real.sqrt (Q.mass a))]
+  simp
+
+theorem hellingerSq_product_le (P Q : FiniteLaw α) (m : ℕ) :
+    hellingerSq (P.product m) (Q.product m) ≤ (m : ℝ) * hellingerSq P Q := by
+  rw [hellingerSq_eq, affinity_product, hellingerSq_eq]
+  have ha := affinity_nonneg P Q
+  have hb := affinity_le_one P Q
+  have hpow : 1 - affinity P Q ^ m ≤ (m : ℝ) * (1 - affinity P Q) := by
+    induction m with
+    | zero => simp
+    | succ m ih =>
+      rw [pow_succ, Nat.cast_add, Nat.cast_one]
+      have hmul := mul_le_mul_of_nonneg_right ih ha
+      have hm : (0 : ℝ) ≤ (m : ℝ) * (1 - affinity P Q) := by positivity
+      have hmon := mul_le_mul_of_nonneg_left hb hm
+      nlinarith only [hmul, hmon]
+  linarith
+
+
+theorem sqrt_sub_one_sq_le {t : ℝ} (ht : 0 ≤ t) :
+    (Real.sqrt t - 1) ^ 2 ≤ (t - 1) ^ 2 := by
+  have hs2 := Real.sq_sqrt ht
+  have hprod := mul_nonneg (Real.sqrt_nonneg t)
+    (mul_nonneg (show 0 ≤ Real.sqrt t + 2 by positivity) (sq_nonneg (Real.sqrt t - 1)))
+  conv_rhs => rw [← hs2]
+  nlinarith only [hprod]
+
+theorem hellingerSq_le_score (P Q : FiniteLaw α) (Z : α → ℝ)
+    (hZ : ∀ a, Q.mass a = P.mass a * (1 + Z a)) :
+    hellingerSq P Q ≤ ∑ a, P.mass a * Z a ^ 2 := by
+  apply Finset.sum_le_sum
+  intro a _
+  by_cases hp0 : P.mass a = 0
+  · have hq0 : Q.mass a = 0 := by rw [hZ, hp0, zero_mul]
+    simp [hp0, hq0]
+  have hp : 0 < P.mass a := lt_of_le_of_ne (P.nonneg a) (Ne.symm hp0)
+  have ht : 0 ≤ 1 + Z a := by
+    have hq := Q.nonneg a
+    rw [hZ] at hq
+    exact nonneg_of_mul_nonneg_right hq hp
+  have hs := sqrt_sub_one_sq_le ht
+  have hmul := mul_le_mul_of_nonneg_left hs (P.nonneg a)
+  have hp2 := Real.sq_sqrt (P.nonneg a)
+  rw [hZ, Real.sqrt_mul (P.nonneg a)]
+  calc
+    _ = P.mass a * (Real.sqrt (1 + Z a) - 1) ^ 2 := by
+      calc
+        _ = Real.sqrt (P.mass a) ^ 2 * (Real.sqrt (1 + Z a) - 1) ^ 2 := by ring
+        _ = _ := by rw [hp2]
+    _ ≤ P.mass a * ((1 + Z a) - 1) ^ 2 := hmul
+    _ = _ := by ring
+
+theorem tv_product_le_sqrt_score (P Q : FiniteLaw α) (Z : α → ℝ)
+    (hZ : ∀ a, Q.mass a = P.mass a * (1 + Z a)) (m : ℕ) :
+    tv (P.product m) (Q.product m) ≤ Real.sqrt ((m : ℝ) * ∑ a, P.mass a * Z a ^ 2) := by
+  apply (tv_le_sqrt_hellingerSq _ _).trans
+  apply Real.sqrt_le_sqrt
+  exact (hellingerSq_product_le P Q m).trans
+    (mul_le_mul_of_nonneg_left (hellingerSq_le_score P Q Z hZ) (by positivity))
+
+def bind (P : FiniteLaw α) (K : α → FiniteLaw β) : FiniteLaw β where
+  mass b := ∑ a, P.mass a * (K a).mass b
+  nonneg b := Finset.sum_nonneg fun a _ => mul_nonneg (P.nonneg a) ((K a).nonneg b)
+  total := by
+    rw [Finset.sum_comm]
+    simp_rw [← Finset.mul_sum, (K _).total, mul_one]
+    exact P.total
+
+theorem tv_bind_le (P Q : FiniteLaw α) (K : α → FiniteLaw β) :
+    tv (P.bind K) (Q.bind K) ≤ tv P Q := by
+  unfold tv bind
+  simp only
+  apply mul_le_mul_of_nonneg_left _ (by norm_num)
+  calc
+    _ = ∑ b, |∑ a, (P.mass a - Q.mass a) * (K a).mass b| := by
+      apply Finset.sum_congr rfl
+      intro b _
+      congr 1
+      rw [← Finset.sum_sub_distrib]
+      apply Finset.sum_congr rfl
+      intro a _; ring
+    _ ≤ ∑ b, ∑ a, |(P.mass a - Q.mass a) * (K a).mass b| :=
+      Finset.sum_le_sum fun b _ => Finset.abs_sum_le_sum_abs _ _
+    _ = ∑ a, |P.mass a - Q.mass a| := by
+      rw [Finset.sum_comm]
+      apply Finset.sum_congr rfl
+      intro a _
+      simp_rw [abs_mul, abs_of_nonneg ((K a).nonneg _), ← Finset.mul_sum, (K a).total, mul_one]
+
+theorem tv_triangle (P Q R : FiniteLaw α) : tv P R ≤ tv P Q + tv Q R := by
+  unfold tv
+  rw [← mul_add, ← Finset.sum_add_distrib]
+  apply mul_le_mul_of_nonneg_left _ (by norm_num)
+  apply Finset.sum_le_sum
+  intro a _
+  exact abs_sub_le _ _ _
+
+theorem tv_symm (P Q : FiniteLaw α) : tv P Q = tv Q P := by
+  unfold tv
+  congr 1
+  apply Finset.sum_congr rfl
+  intro a _; exact abs_sub_comm _ _
+
+theorem expectation_diff_le_tv (P Q : FiniteLaw α) (f : α → ℝ)
+    (hf : ∀ a, f a ∈ Set.Icc 0 1) :
+    (∑ a, P.mass a * f a) - (∑ a, Q.mass a * f a) ≤ tv P Q := by
+  have hz : (∑ a, (P.mass a - Q.mass a)) = 0 := by
+    rw [Finset.sum_sub_distrib, P.total, Q.total, sub_self]
+  have h (a : α) : (P.mass a - Q.mass a) * f a ≤
+      (|P.mass a - Q.mass a| + (P.mass a - Q.mass a)) / 2 := by
+    by_cases hd : 0 ≤ P.mass a - Q.mass a
+    · rw [abs_of_nonneg hd]
+      have := mul_le_mul_of_nonneg_left (hf a).2 hd
+      linarith
+    · rw [abs_of_neg (lt_of_not_ge hd)]
+      have := mul_nonpos_of_nonpos_of_nonneg (le_of_not_ge hd) (hf a).1
+      linarith
+  calc
+    _ = ∑ a, (P.mass a - Q.mass a) * f a := by
+      rw [← Finset.sum_sub_distrib]
+      apply Finset.sum_congr rfl
+      intro a _; ring
+    _ ≤ ∑ a, (|P.mass a - Q.mass a| + (P.mass a - Q.mass a)) / 2 :=
+      Finset.sum_le_sum fun a _ => h a
+    _ = tv P Q := by
+      rw [← Finset.sum_div, Finset.sum_add_distrib, hz, add_zero]
+      unfold tv
+      ring
+
+theorem binary_testing_bound (P Q : FiniteLaw α) (f : α → ℝ)
+    (hf : ∀ a, f a ∈ Set.Icc 0 1) :
+    ((∑ a, P.mass a * f a) + (∑ a, Q.mass a * (1 - f a))) / 2 ≤
+      (1 + tv P Q) / 2 := by
+  have h := expectation_diff_le_tv P Q f hf
+  have heq : (∑ a, Q.mass a * (1 - f a)) = 1 - ∑ a, Q.mass a * f a := by
+    simp_rw [mul_sub, mul_one, Finset.sum_sub_distrib, Q.total]
+  rw [heq]
+  linarith
+
+
+theorem sum_max_eq (P Q : FiniteLaw α) :
+    (∑ a, max (P.mass a) (Q.mass a)) = 1 + tv P Q := by
+  have hpoint (a : α) : max (P.mass a) (Q.mass a) =
+      (P.mass a + Q.mass a + |P.mass a - Q.mass a|) / 2 := by
+    by_cases h : P.mass a ≤ Q.mass a
+    · rw [max_eq_right h, abs_of_nonpos (sub_nonpos.mpr h)]; ring
+    · rw [max_eq_left (le_of_not_ge h), abs_of_nonneg (sub_nonneg.mpr (le_of_not_ge h))]; ring
+  simp_rw [hpoint]
+  rw [← Finset.sum_div, Finset.sum_add_distrib, Finset.sum_add_distrib, P.total, Q.total]
+  unfold tv
+  ring
+
+/-- Joint exact recovery of independent binary choices, allowing arbitrary
+randomized estimators. No coordinatewise decision rule is assumed. -/
+theorem independent_binary_testing (P Q : FiniteLaw α) (g : ℕ)
+    (A : (Fin g → α) → (Fin g → Bool) → ℝ)
+    (hA0 : ∀ z θ, 0 ≤ A z θ) (hA1 : ∀ z, ∑ θ, A z θ = 1) :
+    (1 / (2 : ℝ) ^ g) * ∑ θ : Fin g → Bool, ∑ z : Fin g → α,
+      (∏ j, (if θ j then P.mass (z j) else Q.mass (z j))) * A z θ ≤
+        ((1 + tv P Q) / 2) ^ g := by
+  classical
+  have hpoint (z : Fin g → α) (θ : Fin g → Bool) :
+      (∏ j, (if θ j then P.mass (z j) else Q.mass (z j))) ≤
+        ∏ j, max (P.mass (z j)) (Q.mass (z j)) := by
+    apply Finset.prod_le_prod₀
+    · intro j _; split <;> first | exact P.nonneg _ | exact Q.nonneg _
+    · intro j _; split <;> first | exact le_max_left _ _ | exact le_max_right _ _
+  have hsum (z : Fin g → α) :
+      (∑ θ : Fin g → Bool,
+        (∏ j, (if θ j then P.mass (z j) else Q.mass (z j))) * A z θ) ≤
+          ∏ j, max (P.mass (z j)) (Q.mass (z j)) := by
+    calc
+      _ ≤ ∑ θ : Fin g → Bool,
+          (∏ j, max (P.mass (z j)) (Q.mass (z j))) * A z θ := by
+        apply Finset.sum_le_sum
+        intro θ _
+        exact mul_le_mul_of_nonneg_right (hpoint z θ) (hA0 z θ)
+      _ = _ := by rw [← Finset.mul_sum, hA1, mul_one]
+  rw [Finset.sum_comm]
+  calc
+    _ ≤ (1 / (2 : ℝ) ^ g) * ∑ z : Fin g → α, ∏ j,
+        max (P.mass (z j)) (Q.mass (z j)) :=
+      mul_le_mul_of_nonneg_left (Finset.sum_le_sum fun z _ => hsum z) (by positivity)
+    _ = (1 / (2 : ℝ) ^ g) * (∑ a, max (P.mass a) (Q.mass a)) ^ g := by
+      rw [← Fintype.prod_sum (fun (_ : Fin g) a => max (P.mass a) (Q.mass a))]
+      simp
+    _ = _ := by rw [sum_max_eq, div_pow]; ring
+
+def tensor (P : FiniteLaw α) (R : FiniteLaw β) : FiniteLaw (α × β) where
+  mass z := P.mass z.1 * R.mass z.2
+  nonneg z := mul_nonneg (P.nonneg z.1) (R.nonneg z.2)
+  total := by
+    rw [Fintype.sum_prod_type]
+    simp_rw [← Finset.mul_sum, R.total, mul_one]
+    exact P.total
+
+theorem tv_tensor_le (P Q : FiniteLaw α) (R S : FiniteLaw β) :
+    tv (P.tensor R) (Q.tensor S) ≤ tv P Q + tv R S := by
+  unfold tv tensor
+  simp only
+  rw [Fintype.sum_prod_type, ← mul_add]
+  apply mul_le_mul_of_nonneg_left _ (by norm_num)
+  have hpoint (a : α) (b : β) :
+      |P.mass a * R.mass b - Q.mass a * S.mass b| ≤
+        |P.mass a - Q.mass a| * R.mass b + Q.mass a * |R.mass b - S.mass b| := by
+    calc
+      _ = |(P.mass a - Q.mass a) * R.mass b + Q.mass a * (R.mass b - S.mass b)| := by
+        congr 1; ring
+      _ ≤ |(P.mass a - Q.mass a) * R.mass b| + |Q.mass a * (R.mass b - S.mass b)| :=
+        abs_add_le _ _
+      _ = _ := by rw [abs_mul, abs_mul, abs_of_nonneg (R.nonneg b), abs_of_nonneg (Q.nonneg a)]
+  calc
+    _ ≤ ∑ a, ∑ b, (|P.mass a - Q.mass a| * R.mass b + Q.mass a * |R.mass b - S.mass b|) := by
+      apply Finset.sum_le_sum
+      intro a _
+      exact Finset.sum_le_sum fun b _ => hpoint a b
+    _ = _ := by
+      simp_rw [Finset.sum_add_distrib, ← Finset.mul_sum, R.total, mul_one]
+      rw [← Finset.sum_mul, Q.total, one_mul]
+
+theorem tv_product_succ (P Q : FiniteLaw α) (m : ℕ) :
+    tv (P.product (m + 1)) (Q.product (m + 1)) =
+      tv (P.tensor (P.product m)) (Q.tensor (Q.product m)) := by
+  unfold tv product tensor
+  simp only
+  congr 1
+  rw [← (Fin.consEquiv (fun _ : Fin (m + 1) => α)).sum_comp]
+  apply Finset.sum_congr rfl
+  intro a _
+  simp only [Fin.prod_univ_succ, Fin.consEquiv_apply, Fin.cons_zero, Fin.cons_succ]
+
+theorem tv_product_le (P Q : FiniteLaw α) (m : ℕ) :
+    tv (P.product m) (Q.product m) ≤ (m : ℝ) * tv P Q := by
+  induction m with
+  | zero => simp [tv, product]
+  | succ m ih =>
+    rw [tv_product_succ]
+    apply (tv_tensor_le P Q (P.product m) (Q.product m)).trans
+    simpa only [Nat.cast_add, Nat.cast_one, add_mul, one_mul, add_comm] using
+      add_le_add_left ih (tv P Q)
+
+
+theorem affinity_bind_ge (P Q : FiniteLaw α) (K : α → FiniteLaw β) :
+    affinity P Q ≤ affinity (P.bind K) (Q.bind K) := by
+  have hpoint (b : β) :
+      (∑ a, Real.sqrt (P.mass a) * Real.sqrt (Q.mass a) * (K a).mass b) ≤
+        Real.sqrt ((P.bind K).mass b) * Real.sqrt ((Q.bind K).mass b) := by
+    have hcs := Finset.sum_sq_le_sum_mul_sum_of_sq_le_mul (s := Finset.univ)
+      (r := fun a => Real.sqrt (P.mass a) * Real.sqrt (Q.mass a) * (K a).mass b)
+      (f := fun a => P.mass a * (K a).mass b) (g := fun a => Q.mass a * (K a).mass b)
+      (fun a _ => mul_nonneg (P.nonneg a) ((K a).nonneg b))
+      (fun a _ => mul_nonneg (Q.nonneg a) ((K a).nonneg b)) (by
+        intro a _
+        rw [mul_pow, mul_pow, Real.sq_sqrt (P.nonneg a), Real.sq_sqrt (Q.nonneg a)]
+        nlinarith only [sq_nonneg ((K a).mass b)])
+    have hp2 := Real.sq_sqrt ((P.bind K).nonneg b)
+    have hq2 := Real.sq_sqrt ((Q.bind K).nonneg b)
+    have hsq : (Real.sqrt ((P.bind K).mass b) * Real.sqrt ((Q.bind K).mass b)) ^ 2 =
+        (P.bind K).mass b * (Q.bind K).mass b := by rw [mul_pow, hp2, hq2]
+    change (∑ a, Real.sqrt (P.mass a) * Real.sqrt (Q.mass a) * (K a).mass b) ^ 2 ≤
+      (P.bind K).mass b * (Q.bind K).mass b at hcs
+    have hn : 0 ≤ Real.sqrt ((P.bind K).mass b) * Real.sqrt ((Q.bind K).mass b) := by positivity
+    nlinarith only [hcs, hsq, hn]
+  calc
+    affinity P Q = ∑ b, ∑ a, Real.sqrt (P.mass a) * Real.sqrt (Q.mass a) * (K a).mass b := by
+      rw [Finset.sum_comm]
+      simp_rw [← Finset.mul_sum, (K _).total, mul_one]
+      rfl
+    _ ≤ _ := Finset.sum_le_sum fun b _ => hpoint b
+
+theorem hellingerSq_bind_le (P Q : FiniteLaw α) (K : α → FiniteLaw β) :
+    hellingerSq (P.bind K) (Q.bind K) ≤ hellingerSq P Q := by
+  rw [hellingerSq_eq, hellingerSq_eq]
+  linarith [affinity_bind_ge P Q K]
+
+/-- Restoration of approximate oracle simulations before taking independent
+complete observations; the simulation error is charged once per sample. -/
+theorem approximate_score_transfer (P Q : FiniteLaw α) (K : α → FiniteLaw β)
+    (R S : FiniteLaw β) (Z : α → ℝ)
+    (hZ : ∀ a, Q.mass a = P.mass a * (1 + Z a)) (E : ℝ)
+    (hR : tv R (P.bind K) ≤ E) (hS : tv S (Q.bind K) ≤ E) (m : ℕ) :
+    tv (R.product m) (S.product m) ≤
+      Real.sqrt ((m : ℝ) * ∑ a, P.mass a * Z a ^ 2) + 2 * m * E := by
+  have hmid : tv ((P.bind K).product m) ((Q.bind K).product m) ≤
+      Real.sqrt ((m : ℝ) * ∑ a, P.mass a * Z a ^ 2) := by
+    apply (tv_le_sqrt_hellingerSq _ _).trans
+    apply Real.sqrt_le_sqrt
+    apply (hellingerSq_product_le _ _ m).trans
+    exact mul_le_mul_of_nonneg_left
+      ((hellingerSq_bind_le P Q K).trans (hellingerSq_le_score P Q Z hZ)) (by positivity)
+  have hr := (tv_product_le R (P.bind K) m).trans
+    (mul_le_mul_of_nonneg_left hR (by positivity : (0 : ℝ) ≤ m))
+  have hs := (tv_product_le S (Q.bind K) m).trans
+    (mul_le_mul_of_nonneg_left hS (by positivity : (0 : ℝ) ≤ m))
+  rw [tv_symm (S.product m)] at hs
+  have htri1 := tv_triangle (R.product m) ((P.bind K).product m) (S.product m)
+  have htri2 := tv_triangle ((P.bind K).product m) ((Q.bind K).product m) (S.product m)
+  linarith
+
+def overlap (P Q : FiniteLaw α) : ℝ := ∑ a, min (P.mass a) (Q.mass a)
+
+theorem overlap_eq_one_sub_tv (P Q : FiniteLaw α) : overlap P Q = 1 - tv P Q := by
+  have hm : overlap P Q + (∑ a, max (P.mass a) (Q.mass a)) = 2 := by
+    rw [overlap, ← Finset.sum_add_distrib]
+    simp_rw [min_add_max]
+    rw [Finset.sum_add_distrib, P.total, Q.total]
+    norm_num
+  rw [sum_max_eq] at hm
+  linarith
+
+theorem overlap_nonneg (P Q : FiniteLaw α) : 0 ≤ overlap P Q := by
+  apply Finset.sum_nonneg
+  intro a _
+  exact le_min (P.nonneg a) (Q.nonneg a)
+
+def commonPart (P Q : FiniteLaw α) (h : 0 < overlap P Q) : FiniteLaw α where
+  mass a := min (P.mass a) (Q.mass a) / overlap P Q
+  nonneg a := div_nonneg (le_min (P.nonneg a) (Q.nonneg a)) h.le
+  total := by rw [← Finset.sum_div]; exact div_self h.ne'
+
+theorem commonPart_le_left (P Q : FiniteLaw α) (h : 0 < overlap P Q)
+    {ω : ℝ} (hω : ω ≤ overlap P Q) (a : α) :
+    ω * (commonPart P Q h).mass a ≤ P.mass a := by
+  change ω * (min (P.mass a) (Q.mass a) / overlap P Q) ≤ _
+  calc
+    _ ≤ overlap P Q * (min (P.mass a) (Q.mass a) / overlap P Q) :=
+      mul_le_mul_of_nonneg_right hω (div_nonneg (le_min (P.nonneg a) (Q.nonneg a)) h.le)
+    _ = min (P.mass a) (Q.mass a) := by field_simp
+    _ ≤ _ := min_le_left _ _
+
+def remainderLaw (P H : FiniteLaw α) (ω : ℝ) (hω : ω < 1)
+    (hdom : ∀ a, ω * H.mass a ≤ P.mass a) : FiniteLaw α where
+  mass a := (P.mass a - ω * H.mass a) / (1 - ω)
+  nonneg a := div_nonneg (sub_nonneg.mpr (hdom a)) (sub_pos.mpr hω).le
+  total := by
+    rw [← Finset.sum_div, Finset.sum_sub_distrib, ← Finset.mul_sum, P.total, H.total, mul_one]
+    exact div_self (sub_pos.mpr hω).ne'
+
+theorem common_part_decomposition (P Q : FiniteLaw α) {ω : ℝ}
+    (hω : ω ∈ Set.Ioo 0 1) (hov : ω ≤ overlap P Q) :
+    ∃ H P' Q' : FiniteLaw α,
+      (∀ a, P.mass a = ω * H.mass a + (1 - ω) * P'.mass a) ∧
+      (∀ a, Q.mass a = ω * H.mass a + (1 - ω) * Q'.mass a) := by
+  have h : 0 < overlap P Q := lt_of_lt_of_le hω.1 hov
+  let H := commonPart P Q h
+  have hP : ∀ a, ω * H.mass a ≤ P.mass a := commonPart_le_left P Q h hov
+  have hQ : ∀ a, ω * H.mass a ≤ Q.mass a := by
+    intro a
+    change ω * (min (P.mass a) (Q.mass a) / overlap P Q) ≤ _
+    calc
+      _ ≤ overlap P Q * (min (P.mass a) (Q.mass a) / overlap P Q) :=
+        mul_le_mul_of_nonneg_right hov (div_nonneg (le_min (P.nonneg a) (Q.nonneg a)) h.le)
+      _ = min (P.mass a) (Q.mass a) := by field_simp
+      _ ≤ _ := min_le_right _ _
+  refine ⟨H, remainderLaw P H ω hω.2 hP, remainderLaw Q H ω hω.2 hQ, ?_, ?_⟩
+  · intro a
+    change P.mass a = ω * H.mass a + (1 - ω) * ((P.mass a - ω * H.mass a) / (1 - ω))
+    field_simp [(sub_pos.mpr hω.2).ne']
+    ring
+  · intro a
+    change Q.mass a = ω * H.mass a + (1 - ω) * ((Q.mass a - ω * H.mass a) / (1 - ω))
+    field_simp [(sub_pos.mpr hω.2).ne']
+    ring
+
+end FiniteLaw
+end TraceReconstruction
+
+namespace TraceReconstruction.FiniteLaw
+variable {α β : Type*} [Fintype α] [Fintype β]
+
+def prod (P : FiniteLaw α) (Q : FiniteLaw β) : FiniteLaw (α × β) where
+  mass x := P.mass x.1 * Q.mass x.2
+  nonneg x := mul_nonneg (P.nonneg _) (Q.nonneg _)
+  total := by
+    rw [Fintype.sum_prod_type]
+    simp_rw [← Finset.mul_sum, Q.total, mul_one]
+    exact P.total
+
+def expect (P : FiniteLaw α) (f : α → ℝ) : ℝ := ∑ a, P.mass a * f a
+
+theorem expect_const (P : FiniteLaw α) (c : ℝ) : P.expect (fun _ => c) = c := by
+  unfold expect
+  rw [← Finset.sum_mul, P.total, one_mul]
+
+theorem expect_mono (P : FiniteLaw α) {f g : α → ℝ} (h : ∀ a, f a ≤ g a) :
+    P.expect f ≤ P.expect g := Finset.sum_le_sum fun a _ =>
+      mul_le_mul_of_nonneg_left (h a) (P.nonneg a)
+
+theorem expect_prod (P : FiniteLaw α) (Q : FiniteLaw β) (f : α × β → ℝ) :
+    (P.prod Q).expect f = P.expect (fun a => Q.expect (fun b => f (a, b))) := by
+  unfold expect prod
+  rw [Fintype.sum_prod_type]
+  simp_rw [Finset.mul_sum, mul_assoc]
+
+/-- Association is only used for increasing likelihood-threshold events. -/
+def Associated [Preorder α] (P : FiniteLaw α) : Prop :=
+  ∀ f g : α → ℝ, Monotone f → Monotone g →
+    P.expect f * P.expect g ≤ P.expect (fun a => f a * g a)
+
+theorem associated_linearOrder [LinearOrder α] (P : FiniteLaw α) : P.Associated := by
+  intro f g hf hg
+  have hn (a b : α) : 0 ≤ (f a - f b) * (g a - g b) := by
+    rcases le_total a b with hab | hba
+    · exact mul_nonneg_of_nonpos_of_nonpos (sub_nonpos.mpr (hf hab))
+        (sub_nonpos.mpr (hg hab))
+    · exact mul_nonneg (sub_nonneg.mpr (hf hba)) (sub_nonneg.mpr (hg hba))
+  have h := Finset.sum_nonneg (s := Finset.univ) (fun a _ =>
+    Finset.sum_nonneg (s := Finset.univ) (fun b _ => mul_nonneg (mul_nonneg (P.nonneg a)
+      (P.nonneg b)) (hn a b)))
+  have hid : (∑ a, ∑ b, P.mass a * P.mass b * ((f a - f b) * (g a - g b))) =
+      2 * (P.expect (fun a => f a * g a) - P.expect f * P.expect g) := by
+    have hp (a b : α) : P.mass a * P.mass b * ((f a - f b) * (g a - g b)) =
+        (P.mass a * (f a * g a)) * P.mass b + P.mass a * (P.mass b * (f b * g b)) -
+        (P.mass a * f a) * (P.mass b * g b) - (P.mass a * g a) * (P.mass b * f b) := by ring
+    simp_rw [hp, Finset.sum_sub_distrib, Finset.sum_add_distrib,
+      ← Finset.mul_sum, ← Finset.sum_mul, P.total, one_mul, mul_one]
+    unfold expect
+    ring
+  rw [hid] at h
+  linarith
+
+theorem Associated.prod [Preorder α] [Preorder β] {P : FiniteLaw α} {Q : FiniteLaw β}
+    (hP : P.Associated) (hQ : Q.Associated) : (P.prod Q).Associated := by
+  intro f g hf hg
+  rw [expect_prod, expect_prod, expect_prod]
+  calc
+    _ ≤ P.expect (fun a => Q.expect (fun b => f (a, b)) * Q.expect (fun b => g (a, b))) := by
+      apply hP
+      · intro a a' ha
+        exact Q.expect_mono (fun b => hf ⟨ha, le_rfl⟩)
+      · intro a a' ha
+        exact Q.expect_mono (fun b => hg ⟨ha, le_rfl⟩)
+    _ ≤ _ := by
+      apply P.expect_mono
+      intro a
+      apply hQ
+      · intro b b' hb; exact hf ⟨le_rfl, hb⟩
+      · intro b b' hb; exact hg ⟨le_rfl, hb⟩
+
+theorem product_succ_expect (P : FiniteLaw α) (n : ℕ) (f : (Fin (n + 1) → α) → ℝ) :
+    (P.product (n + 1)).expect f =
+      P.expect (fun a => (P.product n).expect (fun b => f (Fin.cons a b))) := by
+  unfold expect product
+  rw [← (Fin.consEquiv (fun _ : Fin (n + 1) => α)).sum_comp, Fintype.sum_prod_type]
+  simp_rw [Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro a _
+  apply Finset.sum_congr rfl
+  intro b _
+  change (∏ j, P.mass ((Fin.cons a b : Fin (n + 1) → α) j)) * f (Fin.cons a b) = _
+  rw [Fin.prod_univ_succ]
+  simp only [Fin.cons_zero, Fin.cons_succ]
+  ring
+
+theorem associated_product [Preorder α] (P : FiniteLaw α) (hP : P.Associated) (n : ℕ) :
+    (P.product n).Associated := by
+  induction n with
+  | zero =>
+    intro f g _ _
+    simp [expect, product]
+  | succ n ih =>
+    intro f g hf hg
+    rw [product_succ_expect, product_succ_expect, product_succ_expect]
+    have hcons {a a' : α} {b b' : Fin n → α} (ha : a ≤ a') (hb : b ≤ b') :
+        (Fin.cons a b : Fin (n + 1) → α) ≤ (Fin.cons a' b' : Fin (n + 1) → α) := by
+      intro j
+      refine Fin.cases ?_ (fun i => ?_) j
+      · exact ha
+      · exact hb i
+    calc
+      _ ≤ P.expect (fun a => (P.product n).expect (fun b => f (Fin.cons a b)) *
+          (P.product n).expect (fun b => g (Fin.cons a b))) := by
+        apply hP
+        · intro a a' ha
+          exact (P.product n).expect_mono (fun b => hf (hcons ha le_rfl))
+        · intro a a' ha
+          exact (P.product n).expect_mono (fun b => hg (hcons ha le_rfl))
+      _ ≤ _ := by
+        apply P.expect_mono
+        intro a
+        apply ih
+        · intro b b' hb; exact hf (hcons le_rfl hb)
+        · intro b b' hb; exact hg (hcons le_rfl hb)
+
+/-- An elementary Paley--Zygmund bound at threshold 1/2, for a density. -/
+theorem density_threshold_mass (P : FiniteLaw α) (f : α → ℝ)
+    (hmean : P.expect f = 1) :
+    (1 : ℝ) / 4 ≤
+      (P.expect (fun a => if 1 / 2 ≤ f a then 1 else 0)) * P.expect (fun a => f a ^ 2) := by
+  classical
+  let S : Finset α := Finset.univ.filter (fun a => 1 / 2 ≤ f a)
+  have hsmall : (∑ a ∈ Finset.univ \ S, P.mass a * f a) ≤ 1 / 2 := by
+    calc
+      _ ≤ ∑ a ∈ Finset.univ \ S, P.mass a * (1 / 2) := by
+        apply Finset.sum_le_sum
+        intro a ha
+        have hnot : ¬ 1 / 2 ≤ f a := by simpa [S] using (Finset.mem_sdiff.mp ha).2
+        exact mul_le_mul_of_nonneg_left (le_of_lt (lt_of_not_ge hnot)) (P.nonneg a)
+      _ ≤ ∑ a, P.mass a * (1 / 2) := by
+        apply Finset.sum_le_sum_of_subset_of_nonneg (Finset.sdiff_subset)
+        intro a _ _; positivity [P.nonneg a]
+      _ = 1 / 2 := P.expect_const _
+  have hlarge : 1 / 2 ≤ ∑ a ∈ S, P.mass a * f a := by
+    have hsplit := Finset.sum_sdiff (Finset.subset_univ S) (f := fun a => P.mass a * f a)
+    change (∑ a, P.mass a * f a) = 1 at hmean
+    rw [hmean] at hsplit
+    linarith
+  have hc := Finset.sum_sq_le_sum_mul_sum_of_sq_le_mul (s := S)
+    (r := fun a => P.mass a * f a) (f := P.mass) (g := fun a => P.mass a * f a ^ 2)
+    (fun a _ => P.nonneg a) (fun a _ => by positivity [P.nonneg a])
+    (fun a _ => by nlinarith)
+  have ht : (∑ a ∈ S, P.mass a * f a ^ 2) ≤ P.expect (fun a => f a ^ 2) := by
+    apply Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ S)
+    intro a _ _; positivity [P.nonneg a]
+  have hmass : P.expect (fun a => if 1 / 2 ≤ f a then 1 else 0) = ∑ a ∈ S, P.mass a := by
+    unfold expect S
+    simp_rw [mul_ite, mul_one, mul_zero]
+    rw [Finset.sum_filter]
+  rw [hmass]
+  have hm := mul_le_mul_of_nonneg_left ht (Finset.sum_nonneg (s := S) (fun a _ => P.nonneg a))
+  nlinarith
+
+/-- The common-density overlap used for consecutive pure primitive runs. -/
+theorem increasing_density_overlap [Preorder α] (P : FiniteLaw α) (hP : P.Associated)
+    (f g : α → ℝ) (hf : ∀ a, 0 ≤ f a) (hg : ∀ a, 0 ≤ g a)
+    (hfm : Monotone f) (hgm : Monotone g)
+    (hf1 : P.expect f = 1) (hg1 : P.expect g = 1) :
+    (1 : ℝ) / 32 ≤ P.expect (fun a => min (f a) (g a)) *
+      P.expect (fun a => f a ^ 2) * P.expect (fun a => g a ^ 2) := by
+  let F : α → ℝ := fun a => if 1 / 2 ≤ f a then 1 else 0
+  let G : α → ℝ := fun a => if 1 / 2 ≤ g a then 1 else 0
+  have hF : Monotone F := by
+    intro a b hab
+    dsimp [F]
+    split_ifs with ha hb hb
+    · rfl
+    · exact False.elim (hb (ha.trans (hfm hab)))
+    · norm_num
+    · rfl
+  have hG : Monotone G := by
+    intro a b hab
+    dsimp [G]
+    split_ifs with ha hb hb
+    · rfl
+    · exact False.elim (hb (ha.trans (hgm hab)))
+    · norm_num
+    · rfl
+  have hcorr := hP F G hF hG
+  have hpoint : ∀ a, (1 / 2 : ℝ) * (F a * G a) ≤ min (f a) (g a) := by
+    intro a
+    dsimp [F, G]
+    split_ifs <;> simp only [mul_one, mul_zero]
+    · exact le_min ‹_› ‹_›
+    all_goals exact le_min (hf a) (hg a)
+  have hint := P.expect_mono hpoint
+  have heq : P.expect (fun a => (1 / 2 : ℝ) * (F a * G a)) =
+      (1 / 2 : ℝ) * P.expect (fun a => F a * G a) := by
+    unfold expect
+    rw [Finset.mul_sum]
+    apply Finset.sum_congr rfl
+    intro a _; ring
+  rw [heq] at hint
+  have hfm2 : 0 ≤ P.expect (fun a => f a ^ 2) :=
+    Finset.sum_nonneg (fun a _ => mul_nonneg (P.nonneg a) (sq_nonneg _))
+  have hgm2 : 0 ≤ P.expect (fun a => g a ^ 2) :=
+    Finset.sum_nonneg (fun a _ => mul_nonneg (P.nonneg a) (sq_nonneg _))
+  have hff := density_threshold_mass P f hf1
+  have hgg := density_threshold_mass P g hg1
+  change 1 / 4 ≤ P.expect F * P.expect (fun a => f a ^ 2) at hff
+  change 1 / 4 ≤ P.expect G * P.expect (fun a => g a ^ 2) at hgg
+  have hmul := mul_le_mul hff hgg (by norm_num : (0 : ℝ) ≤ 1 / 4) (by linarith :
+    0 ≤ P.expect F * P.expect (fun a => f a ^ 2))
+  have hcomb : (1 / 2 : ℝ) * (P.expect F * P.expect G) ≤
+      P.expect (fun a => min (f a) (g a)) := by linarith
+  have hmul2 := mul_le_mul_of_nonneg_right hcomb (mul_nonneg hfm2 hgm2)
+  nlinarith only [hmul, hmul2]
+
+end TraceReconstruction.FiniteLaw
+
+namespace TraceReconstruction.FiniteLaw
+variable {α : Type*} [Fintype α]
+/-- Orientation `false` uses P,Q in chronological order; true reverses them. -/
+def orientedProd (P Q : FiniteLaw α) (b : Bool) : FiniteLaw (α × α) :=
+  if b then Q.prod P else P.prod Q
+
+def pairedHiddenMass (H : FiniteLaw α) (ω : ℝ) (z : α × α) : ℝ :=
+  ω ^ 2 * H.mass z.1 * H.mass z.2
+
+theorem sum_pairedHiddenMass (H : FiniteLaw α) (ω : ℝ) :
+    (∑ z, pairedHiddenMass H ω z) = ω ^ 2 := by
+  unfold pairedHiddenMass
+  rw [Fintype.sum_prod_type]
+  simp_rw [← Finset.mul_sum, H.total, mul_one]
+  rw [← Finset.mul_sum, H.total, mul_one]
+
+theorem pairedHiddenMass_le (P Q H : FiniteLaw α) {ω : ℝ} (hω : 0 ≤ ω)
+    (hP : ∀ a, ω * H.mass a ≤ P.mass a) (hQ : ∀ a, ω * H.mass a ≤ Q.mass a)
+    (b : Bool) (z : α × α) : pairedHiddenMass H ω z ≤ (orientedProd P Q b).mass z := by
+  have hm (R S : FiniteLaw α) (hR : ∀ a, ω * H.mass a ≤ R.mass a)
+      (hS : ∀ a, ω * H.mass a ≤ S.mass a) :
+      pairedHiddenMass H ω z ≤ R.mass z.1 * S.mass z.2 := by
+    calc
+      _ = (ω * H.mass z.1) * (ω * H.mass z.2) := by unfold pairedHiddenMass; ring
+      _ ≤ _ := mul_le_mul (hR z.1) (hS z.2)
+        (mul_nonneg hω (H.nonneg _)) (R.nonneg _)
+  cases b
+  · exact hm P Q hP hQ
+  · exact hm Q P hQ hP
+
+/-- None is the joint hidden flag; a visible flag reports the true orientation. -/
+def pairedFlagMass (P Q H : FiniteLaw α) (ω : ℝ) (b : Bool)
+    (z : Option Bool × (α × α)) : ℝ :=
+  match z.1 with
+  | none => pairedHiddenMass H ω z.2
+  | some c => if c = b then (orientedProd P Q b).mass z.2 - pairedHiddenMass H ω z.2 else 0
+
+theorem pairedFlagMass_nonneg (P Q H : FiniteLaw α) {ω : ℝ} (hω : 0 ≤ ω)
+    (hP : ∀ a, ω * H.mass a ≤ P.mass a) (hQ : ∀ a, ω * H.mass a ≤ Q.mass a)
+    (b : Bool) (z : Option Bool × (α × α)) : 0 ≤ pairedFlagMass P Q H ω b z := by
+  unfold pairedFlagMass
+  split
+  · unfold pairedHiddenMass
+    positivity [H.nonneg z.2.1, H.nonneg z.2.2]
+  · split_ifs
+    · exact sub_nonneg.mpr (pairedHiddenMass_le P Q H hω hP hQ b z.2)
+    · rfl
+
+theorem pairedFlagMass_sum (P Q H : FiniteLaw α) (ω : ℝ) (b : Bool) :
+    (∑ z, pairedFlagMass P Q H ω b z) = 1 := by
+  rw [Fintype.sum_prod_type, Finset.sum_comm]
+  have hp (z : α × α) : (∑ c : Option Bool, pairedFlagMass P Q H ω b (c, z)) =
+      (orientedProd P Q b).mass z := by
+    cases b <;> simp [pairedFlagMass, Fintype.sum_option ]
+  simp_rw [hp]
+  exact (orientedProd P Q b).total
+
+def pairedFlagLaw (P Q H : FiniteLaw α) {ω : ℝ} (hω : 0 ≤ ω)
+    (hP : ∀ a, ω * H.mass a ≤ P.mass a) (hQ : ∀ a, ω * H.mass a ≤ Q.mass a)
+    (b : Bool) : FiniteLaw (Option Bool × (α × α)) :=
+  ⟨pairedFlagMass P Q H ω b, pairedFlagMass_nonneg P Q H hω hP hQ b,
+    pairedFlagMass_sum P Q H ω b⟩
+
+/-- Forgetting the report recovers conditional independence for each orientation. -/
+theorem pairedFlag_simulation (P Q H : FiniteLaw α) (ω : ℝ) (b : Bool) (z : α × α) :
+    (∑ c : Option Bool, pairedFlagMass P Q H ω b (c, z)) =
+      (orientedProd P Q b).mass z := by
+  cases b <;> simp [pairedFlagMass, Fintype.sum_option ]
+
+/-- Hidden reports carry exactly zero information about the fair orientation,
+even after both strings have been observed. -/
+theorem pairedFlag_hidden_joint (P Q H : FiniteLaw α) (ω : ℝ) (b : Bool) (z : α × α) :
+    (1 / 2 : ℝ) * pairedFlagMass P Q H ω b (none, z) =
+      (1 / 2 : ℝ) * (ω ^ 2 * (H.prod H).mass z) := by
+  unfold pairedFlagMass pairedHiddenMass prod
+  ring
+
+theorem pairedFlag_hidden_probability (P Q H : FiniteLaw α) (ω : ℝ) (b : Bool) :
+    (∑ z, pairedFlagMass P Q H ω b (none, z)) = ω ^ 2 := sum_pairedHiddenMass H ω
+
+end TraceReconstruction.FiniteLaw
+
+namespace TraceReconstruction.FiniteLaw
+variable {α : Type*} [Fintype α]
+theorem small_geometric_sum (n : ℕ) {x : ℝ} (hx : x ∈ Set.Icc 0 (1 / 2)) :
+    (∑ j ∈ Finset.range n, x ^ j) ≤ 2 := by
+  have he := geom_sum_mul x n
+  have hn : 0 ≤ ∑ j ∈ Finset.range n, x ^ j :=
+    Finset.sum_nonneg (fun j _ => pow_nonneg hx.1 j)
+  have hp := pow_nonneg hx.1 n
+  nlinarith [mul_le_mul_of_nonneg_left hx.2 hn]
+
+/-- Elementary tail of a binomial polynomial after cancelling orders zero and one. -/
+theorem small_binomial_tail (n : ℕ) {x : ℝ} (hx : 0 ≤ x) (hsmall : (n : ℝ) * x ≤ 1 / 2) :
+    (∑ j ∈ Finset.range (n - 1), (n.choose (j + 2) : ℝ) * x ^ (j + 2)) ≤
+      2 * ((n : ℝ) * x) ^ 2 := by
+  calc
+    _ ≤ ∑ j ∈ Finset.range (n - 1), ((n : ℝ) * x) ^ 2 * ((n : ℝ) * x) ^ j := by
+      apply Finset.sum_le_sum
+      intro j _
+      have hc : (n.choose (j + 2) : ℝ) ≤ (n : ℝ) ^ (j + 2) := by
+        exact_mod_cast Nat.choose_le_pow n (j + 2)
+      calc
+        _ ≤ (n : ℝ) ^ (j + 2) * x ^ (j + 2) :=
+          mul_le_mul_of_nonneg_right hc (pow_nonneg hx _)
+        _ = _ := by rw [← mul_pow, pow_add]; ring
+    _ = ((n : ℝ) * x) ^ 2 * ∑ j ∈ Finset.range (n - 1), ((n : ℝ) * x) ^ j := by
+      rw [Finset.mul_sum]
+    _ ≤ _ := by
+      have h := small_geometric_sum (n - 1) ⟨mul_nonneg (Nat.cast_nonneg n) hx, hsmall⟩
+      nlinarith [mul_le_mul_of_nonneg_left h (sq_nonneg ((n : ℝ) * x))]
+
+theorem expect_pow_expansion (P : FiniteLaw α) (Z : α → ℝ) (u : ℝ) (t : ℕ) :
+    P.expect (fun a => (1 + u * Z a) ^ t) =
+      ∑ j ∈ Finset.range (t + 1), (t.choose j : ℝ) * u ^ j * P.expect (fun a => Z a ^ j) := by
+  unfold expect
+  simp_rw [add_comm (1 : ℝ), add_pow, one_pow, mul_one, mul_pow, Finset.mul_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro j _
+  apply Finset.sum_congr rfl
+  intro a _
+  ring
+
+theorem expect_small_affine_pow (P : FiniteLaw α) (Z : α → ℝ) (t : ℕ) (ht : 2 ≤ t)
+    (hcenter : P.expect Z = 0)
+    (hmom : ∀ j ≤ t, |P.expect (fun a => Z a ^ j)| ≤ 2)
+    {u : ℝ} (hu : (t : ℝ) * |u| ≤ 1 / 2) :
+    P.expect (fun a => (1 + u * Z a) ^ t) ≤ 1 + 4 * ((t : ℝ) * u) ^ 2 := by
+  rw [expect_pow_expansion]
+  have ht' : t + 1 = 2 + (t - 1) := by omega
+  rw [ht', Finset.sum_range_add]
+  have hz : P.expect (fun _ : α => (1 : ℝ)) = 1 := by simp [expect, P.total]
+  simp only [Finset.sum_range_succ, Finset.sum_range_zero, zero_add,
+    Nat.choose_zero_right, Nat.cast_one, pow_zero, mul_one, one_mul,
+    Nat.choose_one_right, pow_one]
+  rw [hz, hcenter]
+  simp only [mul_zero, add_zero]
+  have hb : (∑ j ∈ Finset.range (t - 1), (t.choose (2 + j) : ℝ) * u ^ (2 + j) *
+      P.expect (fun a => Z a ^ (2 + j))) ≤
+      2 * ∑ j ∈ Finset.range (t - 1), (t.choose (j + 2) : ℝ) * |u| ^ (j + 2) := by
+    rw [Finset.mul_sum]
+    apply Finset.sum_le_sum
+    intro j hj
+    have hjt : 2 + j ≤ t := by have := Finset.mem_range.mp hj; omega
+    have hm := hmom (2 + j) hjt
+    calc
+      _ ≤ |(t.choose (2 + j) : ℝ) * u ^ (2 + j) * P.expect (fun a => Z a ^ (2 + j))| :=
+        le_abs_self _
+      _ ≤ (t.choose (2 + j) : ℝ) * |u| ^ (2 + j) * 2 := by
+        rw [abs_mul, abs_mul, abs_of_nonneg (Nat.cast_nonneg _), abs_pow]
+        exact mul_le_mul_of_nonneg_left hm (by positivity)
+      _ = _ := by rw [Nat.add_comm 2 j]; ring
+  have htail := small_binomial_tail t (abs_nonneg u) hu
+  have he : ((t : ℝ) * |u|) ^ 2 = ((t : ℝ) * u) ^ 2 := by rw [mul_pow, mul_pow, sq_abs]
+  rw [he] at htail
+  linarith
+
+/-- Only the centered first moment must vanish; lower absolute moments
+are obtained from the one available source moment, without boundedness. -/
+theorem expect_lower_moment_le_two (P : FiniteLaw α) (Z : α → ℝ) (t : ℕ)
+    (ht : P.expect (fun a => |Z a| ^ t) ≤ 1) (j : ℕ) (hj : j ≤ t) :
+    |P.expect (fun a => Z a ^ j)| ≤ 2 := by
+  have hpoint (a : α) : |Z a| ^ j ≤ 1 + |Z a| ^ t := by
+    by_cases hx : |Z a| ≤ 1
+    · have h := pow_le_one₀ (n := j) (abs_nonneg _) hx
+      nlinarith [pow_nonneg (abs_nonneg (Z a)) t]
+    · have h := pow_le_pow_right₀ (le_of_lt (lt_of_not_ge hx)) hj
+      linarith
+  calc
+    _ ≤ ∑ a, |P.mass a * Z a ^ j| := Finset.abs_sum_le_sum_abs _ _
+    _ = ∑ a, P.mass a * |Z a| ^ j := by
+      apply Finset.sum_congr rfl
+      intro a _
+      rw [abs_mul, abs_of_nonneg (P.nonneg a), abs_pow]
+    _ ≤ ∑ a, P.mass a * (1 + |Z a| ^ t) := by
+      apply Finset.sum_le_sum
+      intro a _
+      exact mul_le_mul_of_nonneg_left (hpoint a) (P.nonneg a)
+    _ = 1 + P.expect (fun a => |Z a| ^ t) := by
+      simp_rw [mul_add, mul_one, Finset.sum_add_distrib, P.total]
+      rfl
+    _ ≤ 2 := by linarith
+
+theorem expect_affine_pow_le (P : FiniteLaw α) (Z : α → ℝ) (t : ℕ)
+    (hmom : ∀ j ≤ t, |P.expect (fun a => Z a ^ j)| ≤ 2) (u : ℝ) :
+    P.expect (fun a => (1 + u * Z a) ^ t) ≤ 2 * (1 + |u|) ^ t := by
+  rw [expect_pow_expansion, add_comm (1 : ℝ), add_pow, Finset.mul_sum]
+  apply Finset.sum_le_sum
+  intro j hj
+  have h := hmom j (by have := Finset.mem_range.mp hj; omega)
+  calc
+    _ ≤ |(t.choose j : ℝ) * u ^ j * P.expect (fun a => Z a ^ j)| := le_abs_self _
+    _ ≤ (t.choose j : ℝ) * |u| ^ j * 2 := by
+      rw [abs_mul, abs_mul, abs_of_nonneg (Nat.cast_nonneg _), abs_pow]
+      exact mul_le_mul_of_nonneg_left h (by positivity)
+    _ = _ := by simp only [one_pow, mul_one]; ring
+
+/-- Scalar two-smoothness at even moment order; its constant is linear
+in that order after taking the square root, as required for tensorization. -/
+theorem scalar_even_moment_unit (P : FiniteLaw α) (Z : α → ℝ) (k : ℕ) (hk : 1 ≤ k)
+    (hcenter : P.expect Z = 0) (ht : P.expect (fun a => |Z a| ^ (2 * k)) ≤ 1) (u : ℝ) :
+    P.expect (fun a => (1 + u * Z a) ^ (2 * k)) ≤
+      (1 + (6 * (2 * k : ℕ) : ℝ) ^ 2 * u ^ 2) ^ k := by
+  have hmom := expect_lower_moment_le_two P Z (2 * k) ht
+  have hkR : (1 : ℝ) ≤ k := by exact_mod_cast hk
+  have htR : (1 : ℝ) ≤ (2 * k : ℕ) := by exact_mod_cast (show 1 ≤ 2 * k by omega)
+  have hbase : 1 ≤ 1 + (6 * (2 * k : ℕ) : ℝ) ^ 2 * u ^ 2 := by nlinarith [mul_nonneg (sq_nonneg (6 * ((2 * k : ℕ) : ℝ))) (sq_nonneg u)]
+  by_cases hu : ((2 * k : ℕ) : ℝ) * |u| ≤ 1 / 2
+  · have hs := expect_small_affine_pow P Z (2 * k) (by omega) hcenter hmom hu
+    have hb : 1 + 4 * (((2 * k : ℕ) : ℝ) * u) ^ 2 ≤
+        1 + (6 * (2 * k : ℕ) : ℝ) ^ 2 * u ^ 2 := by nlinarith [sq_nonneg (((2 * k : ℕ) : ℝ) * u)]
+    exact hs.trans (hb.trans (le_self_pow₀ hbase (by omega)))
+  · have hu' : 1 / 2 ≤ ((2 * k : ℕ) : ℝ) * |u| := le_of_lt (lt_of_not_ge hu)
+    have hsum : 1 + |u| ≤ 3 * ((2 * k : ℕ) : ℝ) * |u| := by
+      have hh := mul_le_mul_of_nonneg_right htR (abs_nonneg u)
+      nlinarith
+    have hc : (2 : ℝ) ≤ (2 : ℝ) ^ (2 * k) := le_self_pow₀ (by norm_num) (by omega)
+    calc
+      _ ≤ 2 * (1 + |u|) ^ (2 * k) := expect_affine_pow_le P Z (2 * k) hmom u
+      _ ≤ 2 * (3 * ((2 * k : ℕ) : ℝ) * |u|) ^ (2 * k) := by gcongr
+      _ ≤ 2 ^ (2 * k) * (3 * ((2 * k : ℕ) : ℝ) * |u|) ^ (2 * k) := by
+        exact mul_le_mul_of_nonneg_right hc (by positivity)
+      _ = ((6 * (2 * k : ℕ) : ℝ) ^ 2 * u ^ 2) ^ k := by
+        rw [← mul_pow, pow_mul]
+        congr 1
+        simp only [mul_pow, sq_abs]
+        ring
+      _ ≤ _ := pow_le_pow_left₀ (by positivity) (by linarith) k
+
+/-- Homogeneous version of the scalar even-moment inequality. -/
+theorem scalar_even_moment (P : FiniteLaw α) (Z : α → ℝ) (k : ℕ) (hk : 1 ≤ k)
+    (hcenter : P.expect Z = 0) (ht : P.expect (fun a => |Z a| ^ (2 * k)) ≤ 1) (b v : ℝ) :
+    P.expect (fun a => (b + v * Z a) ^ (2 * k)) ≤
+      (b ^ 2 + (6 * (2 * k : ℕ) : ℝ) ^ 2 * v ^ 2) ^ k := by
+  have heven (a : α) : |Z a| ^ (2 * k) = Z a ^ (2 * k) := by
+    rw [pow_mul, sq_abs, ← pow_mul]
+  have hmoment : P.expect (fun a => Z a ^ (2 * k)) ≤ 1 := by
+    simpa only [heven] using ht
+  have hc : 1 ≤ (6 * (2 * k : ℕ) : ℝ) ^ 2 := by
+    have hn : (1 : ℝ) ≤ (2 * k : ℕ) := by exact_mod_cast (show 1 ≤ 2 * k by omega)
+    nlinarith
+  by_cases hb : b = 0
+  · subst b
+    simp only [zero_add, zero_pow (by omega : 2 ≠ 0)]
+    have he : P.expect (fun a => (v * Z a) ^ (2 * k)) =
+        v ^ (2 * k) * P.expect (fun a => Z a ^ (2 * k)) := by
+      simp only [expect, mul_pow, Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro a _
+      ring
+    rw [he]
+    calc
+      _ ≤ v ^ (2 * k) := by
+        have hv : 0 ≤ v ^ (2 * k) := by rw [pow_mul]; positivity
+        nlinarith [mul_le_mul_of_nonneg_left hmoment hv]
+      _ = (v ^ 2) ^ k := pow_mul v 2 k
+      _ ≤ _ := pow_le_pow_left₀ (sq_nonneg v) (by nlinarith [mul_le_mul_of_nonneg_right hc (sq_nonneg v)]) k
+  · have hs := scalar_even_moment_unit P Z k hk hcenter ht (v / b)
+    have hbpow : 0 ≤ b ^ (2 * k) := by rw [pow_mul]; positivity
+    have he : P.expect (fun a => (b + v * Z a) ^ (2 * k)) =
+        b ^ (2 * k) * P.expect (fun a => (1 + v / b * Z a) ^ (2 * k)) := by
+      unfold expect
+      rw [Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro a _
+      have h : b + v * Z a = b * (1 + v / b * Z a) := by field_simp [hb]
+      dsimp only
+      rw [h, mul_pow]
+      ring
+    rw [he]
+    calc
+      _ ≤ b ^ (2 * k) * (1 + (6 * (2 * k : ℕ) : ℝ) ^ 2 * (v / b) ^ 2) ^ k :=
+        mul_le_mul_of_nonneg_left hs hbpow
+      _ = _ := by
+        rw [pow_mul, ← mul_pow]
+        congr 1
+        field_simp [hb]
+
+theorem weighted_minkowski_bound (P : FiniteLaw α) (k : ℕ) (hk : 1 ≤ k)
+    (f g : α → ℝ) (hf : ∀ a, 0 ≤ f a) (hg : ∀ a, 0 ≤ g a)
+    (A B : ℝ) (hA : 0 ≤ A) (hB : 0 ≤ B)
+    (hfa : P.expect (fun a => f a ^ k) ≤ A ^ k)
+    (hgb : P.expect (fun a => g a ^ k) ≤ B ^ k) :
+    P.expect (fun a => (f a + g a) ^ k) ≤ (A + B) ^ k := by
+  have hkR : (1 : ℝ) ≤ k := by exact_mod_cast hk
+  have hk0 : (k : ℝ) ≠ 0 := by positivity
+  let w : α → ℝ := fun a => P.mass a ^ (1 / (k : ℝ))
+  have hw (a : α) : 0 ≤ w a := Real.rpow_nonneg (P.nonneg a) _
+  have hwk (a : α) : w a ^ k = P.mass a := by
+    dsimp [w]
+    rw [← Real.rpow_natCast, ← Real.rpow_mul (P.nonneg a), one_div_mul_cancel hk0, Real.rpow_one]
+  have hsingle (F : α → ℝ) (hF : ∀ a, 0 ≤ F a) :
+      (∑ a, |w a * F a| ^ (k : ℝ)) = P.expect (fun a => F a ^ k) := by
+    apply Finset.sum_congr rfl
+    intro a _
+    rw [abs_of_nonneg (mul_nonneg (hw a) (hF a)), Real.rpow_natCast, mul_pow, hwk]
+  have hsum : (∑ a, |w a * f a + w a * g a| ^ (k : ℝ)) =
+      P.expect (fun a => (f a + g a) ^ k) := by
+    simp_rw [← mul_add]
+    exact hsingle _ (fun a => add_nonneg (hf a) (hg a))
+  have hm := Real.Lp_add_le Finset.univ (fun a => w a * f a) (fun a => w a * g a) hkR
+  rw [hsum, hsingle f hf, hsingle g hg] at hm
+  have hroots (F : α → ℝ) (hF : ∀ a, 0 ≤ F a) (C : ℝ) (hC : 0 ≤ C)
+      (hFC : P.expect (fun a => F a ^ k) ≤ C ^ k) :
+      (P.expect (fun a => F a ^ k)) ^ (1 / (k : ℝ)) ≤ C := by
+    calc
+      _ ≤ (C ^ k) ^ (1 / (k : ℝ)) :=
+        Real.rpow_le_rpow (Finset.sum_nonneg (fun a _ => mul_nonneg (P.nonneg a) (pow_nonneg (hF a) k))) hFC (by positivity)
+      _ = C := by
+        rw [← Real.rpow_natCast, one_div, Real.rpow_rpow_inv hC hk0]
+  have hroot : (P.expect (fun a => (f a + g a) ^ k)) ^ (1 / (k : ℝ)) ≤ A + B :=
+    hm.trans (add_le_add (hroots f hf A hA hfa) (hroots g hg B hB hgb))
+  have hexp : 0 ≤ P.expect (fun a => (f a + g a) ^ k) :=
+    Finset.sum_nonneg (fun a _ => mul_nonneg (P.nonneg a) (pow_nonneg (add_nonneg (hf a) (hg a)) k))
+  have hp := pow_le_pow_left₀ (Real.rpow_nonneg hexp _) hroot k
+  rw [← Real.rpow_natCast, ← Real.rpow_mul hexp, one_div_mul_cancel hk0, Real.rpow_one] at hp
+  exact hp
+theorem product_succ_expect_tail (P : FiniteLaw α) (n : ℕ) (f : (Fin (n + 1) → α) → ℝ) :
+    (P.product (n + 1)).expect f =
+      (P.product n).expect (fun b => P.expect (fun a => f (Fin.cons a b))) := by
+  rw [product_succ_expect]
+  unfold expect
+  simp_rw [Finset.mul_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro b _
+  apply Finset.sum_congr rfl
+  intro a _
+  ring
+
+end TraceReconstruction.FiniteLaw
+namespace TraceReconstruction
+variable {α : Type*}
+def multilinear {n : ℕ} (A : (Fin n → Bool) → ℝ) (Z : α → ℝ) (x : Fin n → α) : ℝ :=
+  ∑ σ, A σ * ∏ j, if σ j then Z (x j) else 1
+
+def multilinearEnergy {n : ℕ} (A : (Fin n → Bool) → ℝ) (R : ℝ) : ℝ :=
+  ∑ σ, A σ ^ 2 * ∏ j, if σ j then R ^ 2 else 1
+
+theorem multilinearEnergy_nonneg {n : ℕ} (A : (Fin n → Bool) → ℝ) (R : ℝ) :
+    0 ≤ multilinearEnergy A R := by
+  apply Finset.sum_nonneg
+  intro σ _
+  apply mul_nonneg (sq_nonneg _)
+  apply Finset.prod_nonneg
+  intro j _
+  split <;> positivity
+
+theorem multilinear_cons {n : ℕ} (A : (Fin (n + 1) → Bool) → ℝ) (Z : α → ℝ)
+    (x : Fin n → α) (a : α) :
+    multilinear A Z (Fin.cons a x) =
+      multilinear (fun σ => A (Fin.cons false σ)) Z x +
+      Z a * multilinear (fun σ => A (Fin.cons true σ)) Z x := by
+  unfold multilinear
+  rw [← (Fin.consEquiv (fun _ : Fin (n + 1) => Bool)).sum_comp, Fintype.sum_prod_type]
+  simp only [Fin.consEquiv, Equiv.coe_fn_mk, Fintype.sum_bool, Fin.prod_univ_succ, Fin.cons_zero, Fin.cons_succ,
+    Bool.false_eq_true, ↓reduceIte, one_mul]
+  rw [Finset.mul_sum]
+  rw [add_comm]
+  congr 1
+  apply Finset.sum_congr rfl
+  intro σ _
+  ring
+
+theorem multilinearEnergy_cons {n : ℕ} (A : (Fin (n + 1) → Bool) → ℝ) (R : ℝ) :
+    multilinearEnergy A R =
+      multilinearEnergy (fun σ => A (Fin.cons false σ)) R +
+      R ^ 2 * multilinearEnergy (fun σ => A (Fin.cons true σ)) R := by
+  unfold multilinearEnergy
+  rw [← (Fin.consEquiv (fun _ : Fin (n + 1) => Bool)).sum_comp, Fintype.sum_prod_type]
+  simp only [Fin.consEquiv, Equiv.coe_fn_mk, Fintype.sum_bool, Fin.prod_univ_succ, Fin.cons_zero, Fin.cons_succ,
+    Bool.false_eq_true, ↓reduceIte, one_mul]
+  rw [Finset.mul_sum]
+  rw [add_comm]
+  congr 1
+  apply Finset.sum_congr rfl
+  intro σ _
+  ring
+
+/-- The finite-product moment inequality used for every score expansion.
+The order-cost is linear per degree, and no boundedness assumption is needed. -/
+theorem multilinear_even_moment {α : Type*} [Fintype α] (P : FiniteLaw α)
+    (Z : α → ℝ) (k : ℕ) (hk : 1 ≤ k) (hcenter : P.expect Z = 0)
+    (ht : P.expect (fun a => |Z a| ^ (2 * k)) ≤ 1)
+    (n : ℕ) (A : (Fin n → Bool) → ℝ) :
+    (P.product n).expect (fun x => multilinear A Z x ^ (2 * k)) ≤
+      (multilinearEnergy A (6 * (2 * k : ℕ))) ^ k := by
+  induction n with
+  | zero =>
+    simp only [FiniteLaw.expect, FiniteLaw.product, multilinear, multilinearEnergy,
+      Fintype.prod_empty, mul_one, one_mul]
+    simp only [Finset.univ_unique, Finset.sum_singleton]
+    exact le_of_eq (pow_mul _ 2 k)
+  | succ n ih =>
+    let A0 : (Fin n → Bool) → ℝ := fun σ => A (Fin.cons false σ)
+    let A1 : (Fin n → Bool) → ℝ := fun σ => A (Fin.cons true σ)
+    let R : ℝ := 6 * (2 * k : ℕ)
+    let F0 : (Fin n → α) → ℝ := fun x => multilinear A0 Z x
+    let F1 : (Fin n → α) → ℝ := fun x => multilinear A1 Z x
+    have h0 := ih A0
+    have h1 := ih A1
+    have hA0 := multilinearEnergy_nonneg A0 R
+    have hA1 := multilinearEnergy_nonneg A1 R
+    rw [FiniteLaw.product_succ_expect_tail, multilinearEnergy_cons]
+    calc
+      _ ≤ (P.product n).expect (fun x => (F0 x ^ 2 + R ^ 2 * F1 x ^ 2) ^ k) := by
+        apply FiniteLaw.expect_mono
+        intro x
+        simp_rw [multilinear_cons]
+        have hs := FiniteLaw.scalar_even_moment P Z k hk hcenter ht (F0 x) (F1 x)
+        simpa only [F0, F1, A0, A1, R, mul_comm] using hs
+      _ ≤ (multilinearEnergy A0 R + R ^ 2 * multilinearEnergy A1 R) ^ k := by
+        apply FiniteLaw.weighted_minkowski_bound (P.product n) k hk
+          (fun x => F0 x ^ 2) (fun x => R ^ 2 * F1 x ^ 2)
+          (fun x => sq_nonneg _) (fun x => mul_nonneg (sq_nonneg _) (sq_nonneg _))
+          (multilinearEnergy A0 R) (R ^ 2 * multilinearEnergy A1 R) hA0
+          (mul_nonneg (sq_nonneg _) hA1)
+        · simpa only [← pow_mul] using h0
+        · have he : (P.product n).expect (fun x => (R ^ 2 * F1 x ^ 2) ^ k) =
+              (R ^ 2) ^ k * (P.product n).expect (fun x => F1 x ^ (2 * k)) := by
+            unfold FiniteLaw.expect
+            simp only [mul_pow, ← pow_mul, Finset.mul_sum]
+            apply Finset.sum_congr rfl
+            intro x _
+            ring
+          rw [he, mul_pow (R ^ 2) (multilinearEnergy A1 R)]
+          exact mul_le_mul_of_nonneg_left h1 (pow_nonneg (sq_nonneg R) k)
+      _ = _ := rfl
+
+def monomialDegree {n : ℕ} (σ : Fin n → Bool) : ℕ :=
+  (Finset.univ.filter (fun j => σ j = true)).card
+
+theorem monomial_weight {n : ℕ} (σ : Fin n → Bool) (R : ℝ) :
+    (∏ j, if σ j then R ^ 2 else 1) = R ^ (2 * monomialDegree σ) := by
+  rw [← Finset.prod_filter]
+  simp only [Finset.prod_const, ← pow_mul, monomialDegree]
+
+theorem multilinearEnergy_homogeneous {n : ℕ} (A : (Fin n → Bool) → ℝ)
+    (r : ℕ) (hA : ∀ σ, monomialDegree σ ≠ r → A σ = 0) (R : ℝ) :
+    multilinearEnergy A R = R ^ (2 * r) * ∑ σ, A σ ^ 2 := by
+  unfold multilinearEnergy
+  rw [Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro σ _
+  rw [monomial_weight]
+  by_cases hd : monomialDegree σ = r
+  · rw [hd, mul_comm]
+  · rw [hA σ hd, zero_pow (by omega : 2 ≠ 0), zero_mul, mul_zero]
+
+theorem homogeneous_even_moment {α : Type*} [Fintype α] (P : FiniteLaw α)
+    (Z : α → ℝ) (k : ℕ) (hk : 1 ≤ k) (hcenter : P.expect Z = 0)
+    (ht : P.expect (fun a => |Z a| ^ (2 * k)) ≤ 1)
+    (n r : ℕ) (A : (Fin n → Bool) → ℝ)
+    (hA : ∀ σ, monomialDegree σ ≠ r → A σ = 0) :
+    (P.product n).expect (fun x => multilinear A Z x ^ (2 * k)) ≤
+      ((6 * (2 * k : ℕ) : ℝ) ^ (2 * r) * ∑ σ, A σ ^ 2) ^ k := by
+  have h := multilinear_even_moment P Z k hk hcenter ht n A
+  rw [multilinearEnergy_homogeneous A r hA] at h
+  exact h
+
+/-- Squared L^(2k)-norm for a finite law, expressed without measure coercions. -/
+def FiniteLaw.evenMomentSq {α : Type*} [Fintype α] (P : FiniteLaw α)
+    (f : α → ℝ) (k : ℕ) : ℝ :=
+  (P.expect (fun a => |f a| ^ (2 * k))) ^ (1 / (k : ℝ))
+
+theorem homogeneous_moment_sq {α : Type*} [Fintype α] (P : FiniteLaw α)
+    (Z : α → ℝ) (k : ℕ) (hk : 1 ≤ k) (hcenter : P.expect Z = 0)
+    (ht : P.expect (fun a => |Z a| ^ (2 * k)) ≤ 1)
+    (n r : ℕ) (A : (Fin n → Bool) → ℝ)
+    (hA : ∀ σ, monomialDegree σ ≠ r → A σ = 0) :
+    (P.product n).evenMomentSq (multilinear A Z) k ≤
+      (6 * (2 * k : ℕ) : ℝ) ^ (2 * r) * ∑ σ, A σ ^ 2 := by
+  have h := homogeneous_even_moment P Z k hk hcenter ht n r A hA
+  have he (x : Fin n → α) : |multilinear A Z x| ^ (2 * k) =
+      multilinear A Z x ^ (2 * k) := by rw [pow_mul, sq_abs, ← pow_mul]
+  unfold FiniteLaw.evenMomentSq
+  simp_rw [he]
+  have hn : 0 ≤ (P.product n).expect (fun x => multilinear A Z x ^ (2 * k)) := by
+    apply Finset.sum_nonneg
+    intro x _
+    apply mul_nonneg ((P.product n).nonneg _)
+    change 0 ≤ multilinear A Z x ^ (2 * k)
+    rw [pow_mul]
+    positivity
+  have hb : 0 ≤ (6 * (2 * k : ℕ) : ℝ) ^ (2 * r) * ∑ σ, A σ ^ 2 := by positivity
+  calc
+    _ ≤ (((6 * (2 * k : ℕ) : ℝ) ^ (2 * r) * ∑ σ, A σ ^ 2) ^ k) ^ (1 / (k : ℝ)) :=
+      Real.rpow_le_rpow hn h (by positivity)
+    _ = _ := by rw [← Real.rpow_natCast, one_div, Real.rpow_rpow_inv hb (by positivity)]
+end TraceReconstruction
+
+namespace TraceReconstruction.FiniteLaw
+variable {α : Type*} [Fintype α]
+def eventMass (P : FiniteLaw α) (A : Finset α) : ℝ := ∑ a ∈ A, P.mass a
+
+theorem eventMass_nonneg (P : FiniteLaw α) (A : Finset α) : 0 ≤ P.eventMass A :=
+  Finset.sum_nonneg (fun a _ => P.nonneg a)
+
+theorem eventMass_le_one (P : FiniteLaw α) (A : Finset α) : P.eventMass A ≤ 1 := by
+  rw [← P.total]
+  exact Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _)
+    (fun a _ _ => P.nonneg a)
+
+def condition (P : FiniteLaw α) (A : Finset α) (hA : 0 < P.eventMass A) : FiniteLaw α := by
+  classical
+  exact {
+    mass := fun a => if a ∈ A then P.mass a / P.eventMass A else 0
+    nonneg := fun a => by split <;> positivity [P.nonneg a]
+    total := by
+      rw [← Finset.sum_filter]
+      simp only [Finset.filter_mem_eq_inter, Finset.univ_inter]
+      rw [← Finset.sum_div]
+      exact div_self (ne_of_gt hA) }
+
+theorem overlap_condition (P : FiniteLaw α) (A : Finset α) (hA : 0 < P.eventMass A) :
+    overlap P (condition P A hA) = P.eventMass A := by
+  classical
+  have he (a : α) : min (P.mass a) ((condition P A hA).mass a) =
+      if a ∈ A then P.mass a else 0 := by
+    by_cases ha : a ∈ A
+    · simp only [condition, ha, ↓reduceIte]
+      apply min_eq_left
+      apply (le_div_iff₀ hA).mpr
+      have h := mul_le_mul_of_nonneg_left (P.eventMass_le_one A) (P.nonneg a)
+      simpa only [mul_one] using h
+    · simp only [condition, ha, ↓reduceIte, min_eq_right (P.nonneg a)]
+  unfold overlap
+  simp_rw [he]
+  rw [← Finset.sum_filter]
+  simp only [Finset.filter_mem_eq_inter, Finset.univ_inter, eventMass]
+
+theorem tv_condition (P : FiniteLaw α) (A : Finset α) (hA : 0 < P.eventMass A) :
+    tv P (condition P A hA) = 1 - P.eventMass A := by
+  have he := P.overlap_condition A hA
+  rw [overlap_eq_one_sub_tv] at he
+  linarith
+
+/-- Densities relative to one finite reference, preserving the exact normalization. -/
+def withDensity (P : FiniteLaw α) (f : α → ℝ) (hf : ∀ a, 0 ≤ f a)
+    (hmean : P.expect f = 1) : FiniteLaw α where
+  mass a := P.mass a * f a
+  nonneg a := mul_nonneg (P.nonneg a) (hf a)
+  total := hmean
+
+theorem density_bad_mass_sq (P : FiniteLaw α) (f : α → ℝ) (A : Finset α) :
+    (∑ a ∈ A, P.mass a * f a) ^ 2 ≤
+      (P.eventMass A) * P.expect (fun a => f a ^ 2) := by
+  have h := Finset.sum_mul_sq_le_sq_mul_sq A (fun a => Real.sqrt (P.mass a))
+    (fun a => Real.sqrt (P.mass a) * f a)
+  simp only [mul_pow, Real.sq_sqrt (P.nonneg _)] at h
+  have he : (∑ a ∈ A, Real.sqrt (P.mass a) * (Real.sqrt (P.mass a) * f a)) =
+      ∑ a ∈ A, P.mass a * f a := by
+    apply Finset.sum_congr rfl
+    intro a _
+    rw [← mul_assoc, ← pow_two, Real.sq_sqrt (P.nonneg a)]
+  rw [he] at h
+  apply h.trans
+  apply mul_le_mul_of_nonneg_left _ (P.eventMass_nonneg A)
+  exact Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _) (fun a _ _ =>
+    mul_nonneg (P.nonneg a) (sq_nonneg _))
+
+/-- Loss from the small-reference-density event by the exact available moment. -/
+theorem low_density_reference_mass (P : FiniteLaw α) (f : α → ℝ) (t : ℕ)
+    (A : Finset α) (hA : ∀ a ∈ A, f a < 1 / 2) :
+    P.eventMass A ≤ (2 : ℝ) ^ t * P.expect (fun a => |f a - 1| ^ t) := by
+  have hlocal (a : α) (ha : a ∈ A) : 1 ≤ (2 : ℝ) ^ t * |f a - 1| ^ t := by
+    rw [← mul_pow]
+    apply one_le_pow₀
+    have hh := neg_le_abs (f a - 1)
+    linarith [hA a ha]
+  calc
+    _ ≤ ∑ a ∈ A, (2 : ℝ) ^ t * (P.mass a * |f a - 1| ^ t) := by
+      apply Finset.sum_le_sum
+      intro a ha
+      nlinarith [mul_le_mul_of_nonneg_left (hlocal a ha) (P.nonneg a)]
+    _ ≤ ∑ a, (2 : ℝ) ^ t * (P.mass a * |f a - 1| ^ t) := by
+      exact Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _) (fun a _ _ => by positivity [P.nonneg a])
+    _ = _ := by rw [← Finset.mul_sum]; rfl
+
+/-- Exact common-event likelihood ratio: no lower bound is imposed off the event. -/
+theorem conditioned_density_identity (R : FiniteLaw α) (f g : α → ℝ)
+    (hf : ∀ a, 0 ≤ f a) (hg : ∀ a, 0 ≤ g a)
+    (hfm : R.expect f = 1) (hgm : R.expect g = 1)
+    (A : Finset α) (hAf : ∀ a ∈ A, 0 < f a)
+    (hP : 0 < (R.withDensity f hf hfm).eventMass A)
+    (hQ : 0 < (R.withDensity g hg hgm).eventMass A) (a : α) :
+    (condition (R.withDensity g hg hgm) A hQ).mass a =
+      (condition (R.withDensity f hf hfm) A hP).mass a *
+      ((R.withDensity f hf hfm).eventMass A /
+        (R.withDensity g hg hgm).eventMass A * (g a / f a)) := by
+  classical
+  by_cases ha : a ∈ A
+  · simp only [condition, ha, ↓reduceIte]
+    change R.mass a * g a / (R.withDensity g hg hgm).eventMass A =
+      R.mass a * f a / (R.withDensity f hf hfm).eventMass A *
+        ((R.withDensity f hf hfm).eventMass A /
+          (R.withDensity g hg hgm).eventMass A * (g a / f a))
+    field_simp [ne_of_gt hP, ne_of_gt hQ, ne_of_gt (hAf a ha)]
+  · simp only [condition, ha, ↓reduceIte, zero_mul]
+
+/-- The raw likelihood score on a common clipping event, at every integer moment. -/
+theorem clipped_raw_moment (R : FiniteLaw α) (f g : α → ℝ)
+    (hf : ∀ a, 0 ≤ f a) (hfm : R.expect f = 1)
+    (A : Finset α) (hAf : ∀ a ∈ A, 1 / 2 ≤ f a)
+    (hP : 0 < (R.withDensity f hf hfm).eventMass A) (k : ℕ) :
+    (condition (R.withDensity f hf hfm) A hP).expect
+        (fun a => |g a / f a - 1| ^ (k + 1)) ≤
+      (2 : ℝ) ^ k / (R.withDensity f hf hfm).eventMass A *
+        R.expect (fun a => |g a - f a| ^ (k + 1)) := by
+  classical
+  have hpoint (a : α) (ha : a ∈ A) :
+      f a * |g a / f a - 1| ^ (k + 1) ≤
+        (2 : ℝ) ^ k * |g a - f a| ^ (k + 1) := by
+    have hp : 0 < f a := lt_of_lt_of_le (by norm_num) (hAf a ha)
+    rw [div_sub_one (ne_of_gt hp), abs_div, abs_of_pos hp]
+    have hi : 0 ≤ 1 / f a := by positivity
+    have hi2 : 1 / f a ≤ 2 := (div_le_iff₀ hp).mpr (by linarith [hAf a ha])
+    calc
+      _ = |g a - f a| ^ (k + 1) * (1 / f a) ^ k := by
+        rw [div_pow, div_pow, one_pow, pow_succ (f a)]
+        field_simp [ne_of_gt hp]
+      _ ≤ _ := by
+        simpa only [mul_comm] using mul_le_mul_of_nonneg_left
+          (pow_le_pow_left₀ hi hi2 k) (show 0 ≤ |g a - f a| ^ (k + 1) by positivity)
+  unfold expect
+  calc
+    _ = ∑ a ∈ A, (R.mass a * (f a * |g a / f a - 1| ^ (k + 1))) /
+          (R.withDensity f hf hfm).eventMass A := by
+      simp only [condition, withDensity]
+      simp_rw [ite_mul, zero_mul]
+      rw [← Finset.sum_filter]
+      simp only [Finset.filter_mem_eq_inter, Finset.univ_inter]
+      apply Finset.sum_congr rfl
+      intro a _
+      ring
+    _ ≤ ∑ a ∈ A, ((2 : ℝ) ^ k * (R.mass a * |g a - f a| ^ (k + 1))) /
+          (R.withDensity f hf hfm).eventMass A := by
+      apply Finset.sum_le_sum
+      intro a ha
+      apply div_le_div_of_nonneg_right _ hP.le
+      nlinarith [mul_le_mul_of_nonneg_left (hpoint a ha) (R.nonneg a)]
+    _ ≤ ∑ a, ((2 : ℝ) ^ k * (R.mass a * |g a - f a| ^ (k + 1))) /
+          (R.withDensity f hf hfm).eventMass A := by
+      apply Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _)
+      intro a _ _
+      positivity [R.nonneg a]
+    _ = _ := by rw [← Finset.sum_div, ← Finset.mul_sum]; ring
+
+/-- Normalizing the second conditioned law costs only its scalar normalization error. -/
+theorem score_rescaling_moment (P : FiniteLaw α) (u : α → ℝ)
+    (c : ℝ) (hc : 0 ≤ c) (k : ℕ) :
+    P.expect (fun a => |c * u a - 1| ^ (k + 1)) ≤
+      2 ^ k * (c ^ (k + 1) * P.expect (fun a => |u a - 1| ^ (k + 1)) +
+        |c - 1| ^ (k + 1)) := by
+  have hb (a : α) : |c * u a - 1| ^ (k + 1) ≤
+      2 ^ k * (c ^ (k + 1) * |u a - 1| ^ (k + 1) + |c - 1| ^ (k + 1)) := by
+    have he : c * u a - 1 = c * (u a - 1) + (c - 1) := by ring
+    rw [he]
+    calc
+      _ ≤ (|c * (u a - 1)| + |c - 1|) ^ (k + 1) :=
+        pow_le_pow_left₀ (abs_nonneg _) (abs_add_le _ _) _
+      _ ≤ _ := by
+        have h := add_pow_le (abs_nonneg (c * (u a - 1))) (abs_nonneg (c - 1)) (k + 1)
+        simpa only [Nat.add_sub_cancel, abs_mul, abs_of_nonneg hc, mul_pow] using h
+  unfold expect
+  calc
+    _ ≤ ∑ a, P.mass a * (2 ^ k * (c ^ (k + 1) * |u a - 1| ^ (k + 1) +
+        |c - 1| ^ (k + 1))) := Finset.sum_le_sum (fun a _ =>
+          mul_le_mul_of_nonneg_left (hb a) (P.nonneg a))
+    _ = 2 ^ k * (c ^ (k + 1) * (∑ a, P.mass a * |u a - 1| ^ (k + 1)) +
+        |c - 1| ^ (k + 1) * ∑ a, P.mass a) := by
+      simp only [mul_add, Finset.sum_add_distrib, Finset.mul_sum]
+      congr 1 <;> apply Finset.sum_congr rfl <;> intro a _ <;> ring
+    _ = _ := by rw [P.total, mul_one]
+end TraceReconstruction.FiniteLaw
+
+namespace TraceReconstruction
+namespace FiniteLaw
+variable {α β : Type*} [Fintype α] [Fintype β]
+def coordinateProduct {n : ℕ} (P : Fin n → FiniteLaw α) : FiniteLaw (Fin n → α) where
+  mass x := ∏ i, (P i).mass (x i)
+  nonneg x := Finset.prod_nonneg (fun i _ => (P i).nonneg (x i))
+  total := by rw [← Fintype.prod_sum]; simp only [FiniteLaw.total, Finset.prod_const_one]
+
+def mixture (R : FiniteLaw β) (P : β → FiniteLaw α) : FiniteLaw α where
+  mass a := ∑ b, R.mass b * (P b).mass a
+  nonneg a := Finset.sum_nonneg (fun b _ => mul_nonneg (R.nonneg b) ((P b).nonneg a))
+  total := by
+    rw [Finset.sum_comm]
+    simp_rw [← Finset.mul_sum, FiniteLaw.total, mul_one]
+    exact R.total
+end FiniteLaw
+variable {α Ω : Type*} [Fintype Ω]
+
+def selectionCoefficient {n : ℕ} (θ σ : Fin n → Bool) : ℝ :=
+  ∏ i, if σ i then (if θ i then 1 else 0) else 1
+
+def selectionDensity {n : ℕ} (θ : Fin n → Bool) (Z : α → ℝ) (x : Fin n → α) : ℝ :=
+  ∏ i, if θ i then 1 + Z (x i) else 1
+
+def averagedCoefficient {n : ℕ} (R : FiniteLaw Ω) (θ : Ω → Fin n → Bool)
+    (σ : Fin n → Bool) : ℝ := R.expect (fun ω => selectionCoefficient (θ ω) σ)
+
+def averagedDensity {n : ℕ} (R : FiniteLaw Ω) (θ : Ω → Fin n → Bool)
+    (Z : α → ℝ) (x : Fin n → α) : ℝ := R.expect (fun ω => selectionDensity (θ ω) Z x)
+
+theorem selectionDensity_expansion {n : ℕ} (θ : Fin n → Bool) (Z : α → ℝ)
+    (x : Fin n → α) : selectionDensity θ Z x = multilinear (selectionCoefficient θ) Z x := by
+  unfold selectionDensity multilinear selectionCoefficient
+  simp_rw [← Finset.prod_mul_distrib]
+  rw [← Fintype.prod_sum (fun (i : Fin n) (b : Bool) =>
+    (if b then (if θ i then (1 : ℝ) else 0) else 1) * (if b then Z (x i) else 1))]
+  apply Finset.prod_congr rfl
+  intro i _
+  cases θ i <;> simp [add_comm]
+
+theorem averagedDensity_expansion {n : ℕ} (R : FiniteLaw Ω) (θ : Ω → Fin n → Bool)
+    (Z : α → ℝ) (x : Fin n → α) :
+    averagedDensity R θ Z x = multilinear (averagedCoefficient R θ) Z x := by
+  unfold averagedDensity averagedCoefficient
+  simp_rw [selectionDensity_expansion]
+  unfold multilinear FiniteLaw.expect
+  simp_rw [Finset.mul_sum, Finset.sum_mul]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro σ _
+  apply Finset.sum_congr rfl
+  intro ω _
+  ring
+
+theorem averagedDensity_difference {n : ℕ} (R : FiniteLaw Ω)
+    (θ φ : Ω → Fin n → Bool) (Z : α → ℝ) (x : Fin n → α) :
+    averagedDensity R θ Z x - averagedDensity R φ Z x =
+      multilinear (fun σ => averagedCoefficient R θ σ - averagedCoefficient R φ σ) Z x := by
+  rw [averagedDensity_expansion, averagedDensity_expansion]
+  simp only [multilinear, sub_mul, Finset.sum_sub_distrib]
+
+theorem coefficient_nonneg {n : ℕ} (R : FiniteLaw Ω) (θ : Ω → Fin n → Bool)
+    (σ : Fin n → Bool) : 0 ≤ averagedCoefficient R θ σ := by
+  apply Finset.sum_nonneg
+  intro ω _
+  apply mul_nonneg (R.nonneg ω)
+  apply Finset.prod_nonneg
+  intro i _
+  cases σ i <;> cases θ ω i <;> norm_num
+
+theorem coefficient_le_one {n : ℕ} (R : FiniteLaw Ω) (θ : Ω → Fin n → Bool)
+    (σ : Fin n → Bool) : averagedCoefficient R θ σ ≤ 1 := by
+  rw [← R.total]
+  apply Finset.sum_le_sum
+  intro ω _
+  have h : selectionCoefficient (θ ω) σ ≤ 1 := by
+    apply Finset.prod_le_one₀
+    · intro i _; cases σ i <;> cases θ ω i <;> norm_num
+    · intro i _; cases σ i <;> cases θ ω i <;> norm_num
+  simpa only [mul_one] using mul_le_mul_of_nonneg_left h (R.nonneg ω)
+
+/-- The coefficient for the empty subset is exactly one, before or after averaging. -/
+theorem averagedCoefficient_empty {n : ℕ} (R : FiniteLaw Ω) (θ : Ω → Fin n → Bool) :
+    averagedCoefficient R θ (fun _ => false) = 1 := by
+  simp [averagedCoefficient, selectionCoefficient, FiniteLaw.expect, R.total]
+
+/-- Exact density of the conditionally independent leaf oracle used in the source. -/
+theorem leaf_oracle_density [Fintype α] {n : ℕ} (P Q : FiniteLaw α) (Z : α → ℝ)
+    (hscore : ∀ a, Q.mass a = P.mass a * (1 + Z a))
+    (R : FiniteLaw Ω) (θ : Ω → Fin n → Bool) (x : Fin n → α) :
+    (R.mixture (fun ω => FiniteLaw.coordinateProduct (fun i => if θ ω i then Q else P))).mass x =
+      (∏ i, P.mass (x i)) * averagedDensity R θ Z x := by
+  unfold FiniteLaw.mixture FiniteLaw.coordinateProduct averagedDensity FiniteLaw.expect
+  rw [Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro ω _
+  have he : (∏ i, (if θ ω i then Q else P).mass (x i)) =
+      (∏ i, P.mass (x i)) * selectionDensity (θ ω) Z x := by
+    rw [selectionDensity, ← Finset.prod_mul_distrib]
+    apply Finset.prod_congr rfl
+    intro i _
+    cases θ ω i <;> simp only [Bool.false_eq_true, ↓reduceIte, mul_one, hscore]
+  change R.mass ω * (∏ i, (if θ ω i then Q else P).mass (x i)) =
+    (∏ i, P.mass (x i)) * (R.mass ω * selectionDensity (θ ω) Z x)
+  rw [he]
+  ring
+end TraceReconstruction
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/TraceReconstruction/Induction.lean b/lean/OAI/Probability/TraceReconstruction/Induction.lean
index 06710f824b35d86fa1e0e38d4319e6d23f2aff49..933f46fdee0b606ec611e3c92a631077d7dd1d18 100644
--- a/lean/OAI/Probability/TraceReconstruction/Induction.lean
+++ b/lean/OAI/Probability/TraceReconstruction/Induction.lean
@@ -1,708 +1,708 @@
-import OAI.Probability.TraceReconstruction.Amplification
-import OAI.Probability.TraceReconstruction.Propagation
-
-namespace OAI
-
-/-!
-# Induction
-
-Concrete level invariants, metadata prefixes, score/error budgets, overlap, the induction step, and the complete construction.
--/
-
-open scoped ENNReal
-noncomputable section
-
-open Filter
-open scoped Topology
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- The literal induction invariant, including the depth-dependent moment,
- actual complete-trace simulator, distinct words, length, and power overlap. -/
-abbrev SourceLevelPair (B D : ℕ) (n q : ℝ) (hq : q∈Set.Icc 0 1) (i : ℕ) :=
-  ConcreteLevelPair q hq (sourceTargetScale B D n i)
-    (n^(-100*(totalScoreExponent B D+B+1))) (totalScoreExponent B i)
-    (initialOverlapLower q) ((sourceOverlapConstant q)^(2*B*i+2)) i (sourceHalfMoment B D n i)
-
-theorem source_level_geometry {B D i : ℕ} {n q : ℝ} {hq : q∈Set.Icc 0 1}
-    (U : SourceLevelPair B D n q hq i) (hB : 2 ≤ B) (hi : i<D)
-    (hn : 1 ≤ n) (hq' : q∈Set.Ioc 0 (1/2:ℝ))
-    (H : SourceNumericMargins B D q (sourceTargetScale B D n (i+1))) :
-    SourceArithmeticBounds (sourceOverlapConstant q) (sourceTargetScale B D n (i+1))
-      ((sourceOverlapConstant q)^(2*B*i+2)) (sourceEpsilon B D)
-      (sourceCancellationOrder B D) (sourceDimension B i) U.left.length U.overlapCount := by
-  let N := sourceTargetScale B D n (i+1)
-  let b := sourceDimension B i
-  let C := sourceOverlapConstant q
-  have hC : 2 ≤ C := sourceOverlapConstant_ge_two hq'
-  have hb := sourceDimension_bounds B i hB
-  have hN := (sourceTargetScale_bounds B D (i+1) hB hn).1
-  have he := sourceEpsilon_pos B D
-  have hG := source_local_envelope B D i b hq' (Nat.le_of_lt hi) hb.2
-  apply source_arithmetic_geometry hC hN (one_le_pow₀ (by linarith only [hC])) he.le
-    (fun j _=>sourceA_nonneg B D b j hb.2 (by omega))
-    (fun j hj=>sourceKappa_three_epsilon B D b (j+1) hb.2 (by omega) (by omega))
-  · have hh := mul_le_mul_of_nonneg_left hG (show (0:ℝ) ≤ 1200 by norm_num)
-    exact (by nlinarith only [hh] : 1200*C^(2*b)*C^(2*B*i+2) ≤ 1200*sourceCapEnvelope B D q).trans H.geometry
-  · simpa only [Nat.cast_pow,Nat.cast_ofNat] using H.motif
-  · simpa only [sourceTargetScale_step B D i n (by linarith only [hn]) hi] using U.length_lower
-  · simpa only [sourceTargetScale_step B D i n (by linarith only [hn]) hi] using U.length_upper
-  · exact U.overlapCount_lower
-  · exact U.overlapCount_upper
-
-theorem SourceArithmeticBounds.copies_positive {C N F ε : ℝ} {s b ell k : ℕ}
-    (H : SourceArithmeticBounds C N F ε s b ell k) (hN : 0<N) {j : ℕ} (hj : j<b) :
-    1 ≤ sourceArithmeticP C N b ε ell k j := by
-  have hh := (H.copies j hj).1
-  have hp : (0:ℝ)<N^(scaleA b j ε)*N^ε/2 := by positivity
-  have hp' : (0:ℝ)<sourceArithmeticP C N b ε ell k j := hp.trans_le hh
-  have hp'' : 0<sourceArithmeticP C N b ε ell k j := by exact_mod_cast hp'
-  omega
-
-theorem SourceArithmeticBounds.trials_positive {C N F ε : ℝ} {s b ell k : ℕ}
-    (H : SourceArithmeticBounds C N F ε s b ell k) (hN : 0<N) (hC : 0<C) (hF : 0<F)
-    {j : ℕ} (hj : j<b) : 0<sourceArithmeticTrials C N b ε ell k j := by
-  have hh := (H.trials j hj).1
-  have hp : (0:ℝ)<N^ε/(600*(C^(2*b)*F)) := by positivity
-  exact_mod_cast hp.trans_le hh
-
-theorem source_level_exponent_margins (B D i : ℕ) (hB : 2 ≤ B) (hi : i ≤ D) :
-    let ε := sourceEpsilon B D
-    let b := sourceDimension B i
-    let e := scaleA b 0 ε*totalScoreExponent B i
-    (i<warmupBlocks B → b=2 ∧ 27/50 ≤ e ∧ ε+scaleA b 1 ε ≤ e ∧ ε+1-2*e ≤ 0) ∧
-    (warmupBlocks B ≤ i → ∀j ≤ b,ε+scaleA b j ε ≤ e) := by
-  let ε := sourceEpsilon B D
-  have he := sourceEpsilon_pos B D
-  have he1 : ε ≤ 1/100 := by
-    have hh := sourceEpsilon_gap_small B D
-    have hg := sourceGapFloor_le_one B
-    dsimp [ε]
-    linarith only [hh,hg]
-  have hd := (totalScoreExponent_bounds B i hB).1
-  have hd' : totalScoreExponent B i ≤ 3+(B:ℝ)*D := by
-    simpa only [mul_comm] using (totalScoreExponent_mono B hB hi).trans (totalScoreExponent_bounds B D hB).2
-  constructor
-  · intro hw
-    have hb : sourceDimension B i=2 := ite_eq_left hw
-    have hh := source_warmup_score_margin B D hB hd hd'
-    simp only [hb]
-    exact ⟨True.intro,hh.1,by linarith only [hh.1,hh.2,he1],by linarith only [hh.1,he1]⟩
-  · intro hw j hj
-    have hb : sourceDimension B i=B := ite_eq_right (by omega)
-    rw [hb] at hj ⊢
-    have hlarge := totalScoreExponent_after_warmup B i hB hw
-    have hh := source_main_score_margin B D hB (by linarith only [hlarge]) hd'
-    have ha : scaleA B j ε ≤ 1 := by
-      exact (scaleA_mono_index B j B ε he.le hj).trans (by rw [scaleA_top])
-    have heps : ε ≤ 1/(2:ℝ)^B := by
-      have hs := sourceEpsilon_small B D
-      have hm := mul_le_mul_of_nonneg_right (show (1:ℝ) ≤ D+1 by have : 0 ≤ (D:ℝ) := Nat.cast_nonneg D; linarith only [this]) (show 0 ≤ (2:ℝ)^B*ε by positivity)
-      apply (le_div_iff₀ (by positivity)).mpr
-      dsimp [ε] at *
-      nlinarith only [hs,hm]
-    linarith only [hh,ha,heps]
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem sourceEffectiveHiddenTail_one (T n : ℕ→ℕ) (ω : ℝ) :
-    sourceEffectiveHiddenTail T n 1 ω = if n 0=0 then 0 else Real.exp (-(n 0:ℝ)*ω^4/16) := by
-  unfold sourceEffectiveHiddenTail
-  change (∑v : Bool ⊕ (Fin (T 0) × PEmpty),_)=_
-  simp only [Fintype.sum_sum_type,Finset.univ_eq_empty,Finset.sum_empty,
-    add_zero,Fintype.sum_bool,collectionCount,sourceScheduledTrials,
-    ite_true,zero_add]
-  rfl
-
-theorem sourceEffectiveHiddenTail_step (T n : ℕ→ℕ) (b : ℕ) (ω : ℝ) :
-    sourceEffectiveHiddenTail T n (b+1) ω =
-      (if n b=0 then 0 else Real.exp (-(n b:ℝ)*ω^4/16))+
-      (if sourceScheduledTrials T n b=0 then 0 else Real.exp (-(sourceScheduledTrials T n b:ℝ)*ω^4/16))+
-      (T b:ℝ)*sourceEffectiveHiddenTail T n b ω := by
-  unfold sourceEffectiveHiddenTail
-  change (∑v : Bool ⊕ (Fin (T b) × WindowCollection T b),_)=_
-  simp only [Fintype.sum_sum_type,Fintype.sum_prod_type,Fintype.sum_bool,collectionCount]
-  let f := fun v : ℕ => if v=0 then (0:ℝ) else Real.exp (-(v:ℝ)*ω^4/16)
-  change (f (sourceScheduledTrials T n b)+f (n b)+(∑ _ : Fin (T b),
-    ∑ v : WindowCollection T b,f (collectionCount T n (sourceScheduledTrials T n) b v))) =
-    f (n b)+f (sourceScheduledTrials T n b)+(T b:ℝ)*
-    (∑ v : WindowCollection T b,f (collectionCount T n (sourceScheduledTrials T n) b v))
-  rw [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]
-  ring
-
-theorem sourceEffectiveHiddenTail_two_ge_one (T n : ℕ→ℕ) (ω : ℝ) (hT : 1 ≤ T 1) :
-    sourceEffectiveHiddenTail (fun _=>T 0) (fun _=>n 0) 1 ω ≤ sourceEffectiveHiddenTail T n 2 ω := by
-  rw [sourceEffectiveHiddenTail_step T n 1 ω,sourceEffectiveHiddenTail_one,
-    sourceEffectiveHiddenTail_one]
-  have hnonneg (v : ℕ) : 0 ≤ (if v=0 then 0 else Real.exp (-(v:ℝ)*ω^4/16)) := by
-    split_ifs <;> positivity
-  have ht : (1:ℝ) ≤ T 1 := by exact_mod_cast hT
-  have hh := le_mul_of_one_le_left (hnonneg (n 0)) ht
-  linarith only [hh,hnonneg (n 1),hnonneg (sourceScheduledTrials T n 1)]
-
-theorem sourceSparseCount_motif_le (T : ℕ→ℕ) (s b : ℕ) (hT : ∀j<b,1 ≤ T (j+1)) :
-    (2:ℝ)^s ≤ (sourceSparseCount T s b:ℝ) := by
-  have hp : 1 ≤ ∏j∈Finset.range b,T (j+1) := Finset.one_le_prod (fun j hj=>hT j (Finset.mem_range.mp hj))
-  have hh : 2^s ≤ sourceSparseCount T s b := by
-    unfold sourceSparseCount
-    exact Nat.le_mul_of_pos_right _ hp
-  exact_mod_cast hh
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem source_level_tail_budget {B D i : ℕ} {n q : ℝ} {hq : q∈Set.Icc 0 1}
-    (U : SourceLevelPair B D n q hq i) (hB : 2 ≤ B) (hi : i<D) (hn : 4 ≤ n)
-    (hq' : q∈Set.Ioc 0 (1/2:ℝ))
-    (H : SourceNumericMargins B D q (sourceTargetScale B D n (i+1)))
-    (htail : 2*(B:ℝ)*Real.exp
-      (-(sourceTargetScale B D n (i+1))^(sourceEpsilon B D)/sourceTailDenominator B D q) ≤
-        n^(-sourceTailExponent B D)) :
-    let N := sourceTargetScale B D n (i+1)
-    let b := sourceDimension B i
-    let ε := sourceEpsilon B D
-    let T := sourceBlockWindows N b ε
-    let t := sourceArithmeticTrials (sourceOverlapConstant q) N b ε U.left.length U.overlapCount
-    sourceEffectiveHiddenTail T t b (initialOverlapLower q) ≤ n^(-200*(totalScoreExponent B D+B+1)) ∧
-    sourceFitTail T t b/N^(-20*sourceWorkSize B D) ≤ n^(-200*(totalScoreExponent B D+B+1)) ∧
-    sourceFitTail T t b ≤ n^(-200*(totalScoreExponent B D+B+1)) := by
-  have hn1 : 1 ≤ n := by linarith only [hn]
-  have H' := source_level_geometry U hB hi hn1 hq' H
-  have hb := sourceDimension_bounds B i hB
-  have hN := sourceTargetScale_bounds B D (i+1) hB hn1
-  have hC := sourceOverlapConstant_pos hq'.1
-  have hω := initialOverlapLower_pos q
-  have hG := source_local_envelope B D i (sourceDimension B i) hq' (Nat.le_of_lt hi) hb.2
-  have hf : 9600*((sourceOverlapConstant q)^(2*sourceDimension B i)*
-      (sourceOverlapConstant q)^(2*B*i+2))/(initialOverlapLower q)^4 ≤ sourceTailDenominator B D q := by
-    have hh := mul_le_mul_of_nonneg_right hG (show 0 ≤ 9600*(initialOverlapLower q)⁻¹^4 by positivity)
-    unfold sourceTailDenominator
-    rw [div_eq_mul_inv,←inv_pow]
-    nlinarith only [hh]
-  exact source_exponential_error_budget H' hB hb.2 hC (by positivity) hω hn1 hN.1 hN.2
-    (by
-      have hh : 0 ≤ 9600*sourceCapEnvelope B D q*(initialOverlapLower q)⁻¹^4 := by positivity [sourceCapEnvelope_pos B D hq'.1]
-      unfold sourceTailDenominator
-      linarith only [hh]) hf
-    (by simpa only [neg_div] using htail)
-
-theorem source_level_all_budgets {B D i : ℕ} {n q : ℝ} {hq : q∈Set.Icc 0 1}
-    (U : SourceLevelPair B D n q hq i) (hB : 2 ≤ B) (hi : i<D) (hn : 4 ≤ n)
-    (hq' : q∈Set.Ioc 0 (1/2:ℝ))
-    (H : SourceNumericMargins B D q (sourceTargetScale B D n (i+1)))
-    (htail : 2*(B:ℝ)*Real.exp
-      (-(sourceTargetScale B D n (i+1))^(sourceEpsilon B D)/sourceTailDenominator B D q) ≤
-        n^(-sourceTailExponent B D)) :
-    let N := sourceTargetScale B D n (i+1)
-    let b := sourceDimension B i
-    let ε := sourceEpsilon B D
-    let s := sourceCancellationOrder B D
-    let T := sourceBlockWindows N b ε
-    let t := sourceArithmeticTrials (sourceOverlapConstant q) N b ε U.left.length U.overlapCount
-    let k := sourceHalfMoment B D n (i+1)
-    let L := N^(-scaleA b 0 ε*totalScoreExponent B i/2)
-    let a := N^(-20*sourceWorkSize B D)
-    let ω := initialOverlapLower q
-    let gap := (totalScoreExponent B i+1)*sourceGapFloor B
-    (((6*(2*k:ℕ):ℝ)*L)^2*(sourceSparseCount T s (b-1):ℝ) ≤ N^(-gap/2)/8) ∧
-    (((6*(2*k:ℕ):ℝ)*L)^2*(sourceSparseCount T s (b-1):ℝ) ≤ 1/2) ∧
-    (64*sourceSharpEnergyBudget T t s (b-1) (sourceDegreeCutoff B) ω a a ((6*(2*k:ℕ):ℝ)*L) ≤
-       N^(-totalScoreExponent B (i+1))) ∧
-    (∀j<b,hiddenSmoothingBound (t j) s ω ≤ a) ∧
-    sourceClippingCost T t s (b-1) k ω a L ≤ 1/2 ∧
-    sourceClippingCost T t s (b-1) k ω a L+sourceFitTail T t b ≤
-      n^(-100*(totalScoreExponent B D+B+1)) := by
-  have hn1 : 1 ≤ n := by linarith only [hn]
-  have hn2 : 1<n := by linarith only [hn]
-  have hC : 1 ≤ sourceOverlapConstant q := (by norm_num : (1:ℝ) ≤ 2).trans (sourceOverlapConstant_ge_two hq')
-  have H' := source_level_geometry U hB hi hn1 hq' H
-  have hb := sourceDimension_bounds B i hB
-  have hb1 : sourceDimension B i-1+1=sourceDimension B i := by omega
-  have hN := sourceTargetScale_bounds B D (i+1) hB hn1
-  have he := source_level_energy B D i U.left.length U.overlapCount (sourceHalfMoment B D n (i+1))
-    (sourceOverlapConstant q) (sourceTargetScale B D n (i+1)) ((sourceOverlapConstant q)^(2*B*i+2))
-    (initialOverlapLower q) (sourceCapEnvelope B D q) hB hi hN.1 hC (one_le_pow₀ hC) (initialOverlapLower_pos q)
-    H' (sourceHalfMoment_bounds B D (i+1) hB hn2).2
-    (source_cap_envelope B D i (sourceDimension B i) hq' (Nat.le_of_lt hi) hb.2)
-    (by simpa only [Nat.cast_pow,Nat.cast_ofNat] using H.sparse)
-    (H.bad _ (by omega)) (by simpa only [hb1] using H.force (sourceDimension B i-1) (by omega)) H.tail H.smoothing
-  have ht := source_level_tail_budget U hB hi hn hq' H htail
-  have hg : sourceGapFloor B ≤ (totalScoreExponent B i+1)*sourceGapFloor B := by
-    exact le_mul_of_one_le_left (sourceGapFloor_pos B).le
-      (by linarith only [(totalScoreExponent_bounds B i hB).1])
-  have hc := source_level_clipping B D (i+1) (sourceCancellationOrder B D) (sourceDimension B i-1)
-    (sourceBlockWindows (sourceTargetScale B D n (i+1)) (sourceDimension B i) (sourceEpsilon B D))
-    (sourceArithmeticTrials (sourceOverlapConstant q) (sourceTargetScale B D n (i+1))
-      (sourceDimension B i) (sourceEpsilon B D) U.left.length U.overlapCount)
-    n (initialOverlapLower q) ((sourceTargetScale B D n (i+1))^(-20*sourceWorkSize B D))
-    ((totalScoreExponent B i+1)*sourceGapFloor B)
-    ((sourceTargetScale B D n (i+1))^(-scaleA (sourceDimension B i) 0 (sourceEpsilon B D)*totalScoreExponent B i/2))
-    hB hn hg (by simpa only [hb1] using ht.1) (by simpa only [hb1] using ht.2.1)
-    (by simpa only [hb1] using ht.2.2) he.2.1 he.1
-  exact ⟨he.1,he.2.1,he.2.2.1,he.2.2.2,hc.1,by simpa only [hb1] using hc.2⟩
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem source_old_variance_scale (B D i : ℕ) {n : ℝ} (hn : 0<n) (hi : i<D) :
-    (sourceTargetScale B D n i)^(-totalScoreExponent B i)=
-      (sourceTargetScale B D n (i+1))^(-scaleA (sourceDimension B i) 0 (sourceEpsilon B D)*totalScoreExponent B i) := by
-  rw [sourceTargetScale_step B D i n hn.le hi,←Real.rpow_mul (show 0 ≤ sourceTargetScale B D n (i+1) from (Real.rpow_pos_of_pos hn _).le)]
-  congr 1
-  ring
-
-theorem source_old_variance_square (B D i : ℕ) {n : ℝ} (hn : 0<n) (hi : i<D) :
-    (sourceTargetScale B D n i)^(-totalScoreExponent B i)=
-      ((sourceTargetScale B D n (i+1))^(-scaleA (sourceDimension B i) 0 (sourceEpsilon B D)*totalScoreExponent B i/2))^2 := by
-  rw [source_old_variance_scale B D i hn hi,←Real.rpow_natCast,←Real.rpow_mul (show 0 ≤ sourceTargetScale B D n (i+1) from (Real.rpow_pos_of_pos hn _).le)]
-  congr 1
-  norm_num
-
-theorem source_level_dense_overlap {B D i j : ℕ} {n q : ℝ} {hq : q∈Set.Icc 0 1}
-    (U : SourceLevelPair B D n q hq i) (hB : 2 ≤ B) (hi : i<D) (hn : 4 ≤ n)
-    (hq' : q∈Set.Ioc 0 (1/2:ℝ))
-    (H : SourceNumericMargins B D q (sourceTargetScale B D n (i+1)))
-    (hj : j ≤ sourceDimension B i)
-    (hgap : sourceEpsilon B D+scaleA (sourceDimension B i) j (sourceEpsilon B D) ≤
-      scaleA (sourceDimension B i) 0 (sourceEpsilon B D)*totalScoreExponent B i) :
-    let N := sourceTargetScale B D n (i+1)
-    let b := sourceDimension B i
-    let ε := sourceEpsilon B D
-    let s := sourceCancellationOrder B D
-    let P := sourceArithmeticP (sourceOverlapConstant q) N b ε U.left.length U.overlapCount
-    let h := sourceArithmeticCount (sourceOverlapConstant q) N b ε U.left.length U.overlapCount
-    Real.sqrt (((h j+1:ℕ):ℝ)*(((2^s:ℕ):ℝ)^j*(sourceTargetScale B D n i)^(-totalScoreExponent B i)))+
-      2*((h j+1:ℕ):ℝ)*((sourceBlockLength P 1 j:ℝ)*U.error) ≤ 1/4 := by
-  have hn1 : 1 ≤ n := by linarith only [hn]
-  have hn0 : 0<n := by linarith only [hn]
-  have H' := source_level_geometry U hB hi hn1 hq' H
-  have hb := sourceDimension_bounds B i hB
-  have he := sourceEpsilon_pos B D
-  have he1 : sourceEpsilon B D ≤ 1 := by
-    linarith only [sourceEpsilon_gap_small B D,sourceGapFloor_le_one B]
-  have hN := sourceTargetScale_bounds B D (i+1) hB hn1
-  have hC : 1 ≤ sourceOverlapConstant q := (by norm_num : (1:ℝ) ≤ 2).trans (sourceOverlapConstant_ge_two hq')
-  have hE : U.error ≤ ((i:ℝ)+1)*n^(-100*(totalScoreExponent B D+B+1))*U.left.length := by
-    apply (div_le_iff₀ ?_).mp U.normalized_error
-    exact (show 0<sourceTargetScale B D n i/2 by positivity [Real.rpow_pos_of_pos hn0 (sourceScaleProduct B D i)]).trans_le U.length_lower
-  apply source_literal_dense_overlap (V:=(sourceTargetScale B D n i)^(-totalScoreExponent B i)) (e:=scaleA (sourceDimension B i) 0 (sourceEpsilon B D)*totalScoreExponent B i) (W:=sourceWeightEnvelope B D) (L:=(2:ℝ)^(sourceCancellationOrder B D))
-    H' hB hb.2 hC (pow_nonneg (by linarith only [hC]) _) hn1 hN.1 hN.2 he.le he1
-    (sourceCapEnvelope_pos B D hq'.1).le (sourceWeightEnvelope_nonneg B D) (one_le_pow₀ (by norm_num)) rfl
-    (source_local_envelope B D i (sourceDimension B i) hq' (Nat.le_of_lt hi) hb.2) (Nat.le_of_lt hi) hj
-    (sourceA_nonneg B D _ j hb.2 (by omega))
-    ((scaleA_mono_index _ j _ _ he.le hj).trans (by rw [scaleA_top])) hE rfl
-    (by simpa only [neg_mul] using (source_old_variance_scale B D i hn0 hi).le) hgap H.error H.overlap
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- The second warmup lift is produced from the one-window oracle, whose
- quadratic energy gain is essential before the main-block regime. -/
-theorem source_level_warmup_second {B D i : ℕ} {n q : ℝ} {hq : q∈Set.Icc 0 1}
-    (U : SourceLevelPair B D n q hq i) (hB : 2 ≤ B) (hi : i<D) (hn : 4 ≤ n)
-    (hq' : q∈Set.Ioc 0 (1/2:ℝ)) (hiw : i<warmupBlocks B)
-    (H : SourceNumericMargins B D q (sourceTargetScale B D n (i+1)))
-    (htail : 2*(B:ℝ)*Real.exp
-      (-(sourceTargetScale B D n (i+1))^(sourceEpsilon B D)/sourceTailDenominator B D q) ≤
-        n^(-sourceTailExponent B D)) :
-    let N := sourceTargetScale B D n (i+1)
-    let ε := sourceEpsilon B D
-    let s := sourceCancellationOrder B D
-    let T := sourceBlockWindows N 2 ε
-    let t := sourceArithmeticTrials (sourceOverlapConstant q) N 2 ε U.left.length U.overlapCount
-    let P := sourceArithmeticP (sourceOverlapConstant q) N 2 ε U.left.length U.overlapCount
-    let h := sourceArithmeticCount (sourceOverlapConstant q) N 2 ε U.left.length U.overlapCount
-    let k := sourceHalfMoment B D n (i+1)
-    let L := N^(-scaleA 2 0 ε*totalScoreExponent B i/2)
-    let a := N^(-20*sourceWorkSize B D)
-    let ω := initialOverlapLower q
-    let cost := sourceClippingCost (fun _=>T 0) (fun _=>t 0) s 0 k ω a L
-    (((6*(2*k:ℕ):ℝ)*L)^2*(2:ℝ)^s ≤ 1/2) ∧ cost ≤ 1/2 ∧
-    Real.sqrt (((h 2+1:ℕ):ℝ)*((2^s:ℕ):ℝ)*
-      (64*sourceSharpEnergyBudget (fun _=>T 0) (fun _=>t 0) s 0 2 ω a a ((6*(2*k:ℕ):ℝ)*L)))+
-      2*((h 2+1:ℕ):ℝ)*((P 1:ℝ)*(cost+(T 0:ℝ)*U.error)) ≤ 1/4 := by
-  let N := sourceTargetScale B D n (i+1)
-  let ε := sourceEpsilon B D
-  let s := sourceCancellationOrder B D
-  let C := sourceOverlapConstant q
-  let F := C^(2*B*i+2)
-  let K := sourceCapEnvelope B D q
-  let W := sourceWeightEnvelope B D
-  let T := sourceBlockWindows N 2 ε
-  let t := sourceArithmeticTrials C N 2 ε U.left.length U.overlapCount
-  let P := sourceArithmeticP C N 2 ε U.left.length U.overlapCount
-  let h := sourceArithmeticCount C N 2 ε U.left.length U.overlapCount
-  let k := sourceHalfMoment B D n (i+1)
-  let e := scaleA 2 0 ε*totalScoreExponent B i
-  let L := N^(-scaleA 2 0 ε*totalScoreExponent B i/2)
-  let a := N^(-20*sourceWorkSize B D)
-  let ω := initialOverlapLower q
-  let w := (6*(2*k:ℕ):ℝ)*L
-  let gap := (totalScoreExponent B i+1)*sourceGapFloor B
-  let Δ := n^(-100*(totalScoreExponent B D+B+1))
-  have hb : sourceDimension B i=2 := ite_eq_left hiw
-  have hn1 : 1 ≤ n := by linarith only [hn]
-  have hn0 : 0<n := by linarith only [hn]
-  have hN := sourceTargetScale_bounds B D (i+1) hB hn1
-  have hN0 : 0<N := by dsimp [N]; linarith only [hN.1]
-  have hC : 1 ≤ C := (by norm_num : (1:ℝ) ≤ 2).trans (sourceOverlapConstant_ge_two hq')
-  have hC0 : 0<C := by linarith only [hC]
-  have hF : 0<F := by positivity
-  have hK : 0 ≤ K := (sourceCapEnvelope_pos B D hq'.1).le
-  have hW : 0 ≤ W := sourceWeightEnvelope_nonneg B D
-  have he := sourceEpsilon_pos B D
-  have he1 : ε ≤ 1 := by
-    dsimp [ε]
-    linarith only [sourceEpsilon_gap_small B D,sourceGapFloor_le_one B]
-  have H' := source_level_geometry U hB hi hn1 hq' H
-  rw [hb] at H'
-  have Hnum := source_level_all_budgets U hB hi hn hq' H htail
-  simp only [hb,show (2:ℕ)-1=1 by rfl] at Hnum
-  have ht := source_level_tail_budget U hB hi hn hq' H htail
-  simp only [hb] at ht
-  have hmot : (2:ℝ)^s ≤ (sourceSparseCount T s 1:ℝ) :=
-    sourceSparseCount_motif_le T s 1 (fun j _=>Nat.succ_le_of_lt (H'.window_positive (j+1)))
-  have hs := mul_le_mul_of_nonneg_left hmot (sq_nonneg w)
-  have hs1 : w^2*(2:ℝ)^s ≤ 1/2 := hs.trans Hnum.2.1
-  have hs2 : w^2*(2:ℝ)^s ≤ N^(-gap/2)/8 := hs.trans Hnum.1
-  have hm : sourceEffectiveHiddenTail (fun _=>T 0) (fun _=>t 0) 1 ω ≤
-      n^(-200*(totalScoreExponent B D+B+1)) :=
-    (sourceEffectiveHiddenTail_two_ge_one T t ω (Nat.succ_le_of_lt (H'.window_positive 1))).trans ht.1
-  have hg : sourceGapFloor B ≤ gap := le_mul_of_one_le_left (sourceGapFloor_pos B).le
-    (by linarith only [(totalScoreExponent_bounds B i hB).1])
-  have hc := source_one_window_clipping B D (i+1) s (T 0) (t 0) n ω a gap L hB hn hg hm hs1 hs2
-  have hG : C^4*F ≤ K := by
-    simpa only [hb,show (2:ℕ)*2=4 by rfl] using
-      source_local_envelope B D i (sourceDimension B i) hq' (Nat.le_of_lt hi) (sourceDimension_bounds B i hB).2
-  have hE : U.error ≤ ((i:ℝ)+1)*Δ*U.left.length := by
-    apply (div_le_iff₀ ?_).mp U.normalized_error
-    exact (show 0<sourceTargetScale B D n i/2 by positivity [Real.rpow_pos_of_pos hn0 (sourceScaleProduct B D i)]).trans_le U.length_lower
-  have herr := source_warmup_error_budget H' hB hC hF.le hn1 hN.1 hN.2 he.le he1 hK hG
-    (Nat.le_of_lt hi) hE U.error_nonneg rfl hc.2 H.error
-  have hcount : ((h 2+1:ℕ):ℝ) ≤ (K+1)*N := by
-    have hh := H'.count_bound hC hF.le hN.1 (j:=2) le_rfl (by rw [scaleA_top]; norm_num)
-    have hh' := mul_le_mul_of_nonneg_right (add_le_add hG (show (1:ℝ) ≤ 1 from le_rfl)) hN0.le
-    simp only [scaleA_top,Real.rpow_one,show (2:ℕ)*2=4 by rfl] at hh
-    exact hh.trans hh'
-  have hwindow : (T 0:ℝ) ≤ N := by
-    have hp : T 0 ≤ P 0 := by have hh := (H'.windows 0 (by omega)).2.2.1; change 2*T 0 ≤ P 0 at hh; omega
-    exact (show (T 0:ℝ) ≤ P 0 by exact_mod_cast hp).trans
-      (H'.copies_le_target hN.1 (by omega) ((scaleA_mono_index 2 1 2 ε he.le (by omega)).trans (by rw [scaleA_top])))
-  have hw : w^2 ≤ W*N^(-e) := by
-    rw [source_weight_square hN0 (show L=N^(-scaleA 2 0 ε*totalScoreExponent B i/2) from rfl)]
-    have hh := mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (Nat.cast_nonneg k)
-      (sourceHalfMoment_bounds B D (i+1) hB (by linarith only [hn])).2 2) (by norm_num : (0:ℝ) ≤ 144)
-    simpa only [W,sourceWeightEnvelope,sourceMomentEnvelope,e,neg_mul] using
-      mul_le_mul_of_nonneg_right hh (Real.rpow_pos_of_pos hN0 (-e)).le
-  have henergy := mul_le_mul_of_nonneg_left
-    (sourceSharpEnergyBudget_warmup (fun _=>T 0) (fun _=>t 0) s ω a w hs1) (by norm_num : (0:ℝ) ≤ 64)
-  have hpref := source_overlap_prefactors hK hW (one_le_pow₀ (by norm_num : (1:ℝ) ≤ 2)) hB (j:=0) (by omega) H.overlap
-  have hexp := (source_level_exponent_margins B D i hB (Nat.le_of_lt hi)).1 hiw
-  rw [hb] at hexp
-  have hM : 1 ≤ sourceWorkSize B D := by
-    unfold sourceWorkSize
-    linarith only [(totalScoreExponent_bounds B D hB).1,Nat.cast_nonneg (sourceDegreeCutoff B) (α:=ℝ),Nat.cast_nonneg B (α:=ℝ)]
-  have hv := source_warmup_variance_budget hN.1 hK hW (show 0 ≤ (2:ℝ)^s by positivity)
-    (Nat.cast_nonneg (T 0)) (Nat.cast_nonneg (h 2+1)) hcount hwindow hw henergy hpref.2.1 hpref.2.2
-    (by change ε+2-e-40*sourceWorkSize B D ≤ 0; have hloe : 27/50 ≤ e := hexp.2.1; linarith only [he1,hloe,hM])
-    hexp.2.2.2
-  refine ⟨hs1,hc.1,?_⟩
-  have hh := overlap_sqrt_budget hv herr
-  simpa only [N,ε,s,C,F,K,W,T,t,P,h,k,e,L,a,ω,w,Nat.cast_pow,Nat.cast_ofNat,mul_assoc] using hh
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem source_level_overlap {B D i : ℕ} {n q : ℝ} {hq : q∈Set.Icc 0 1}
-    (U : SourceLevelPair B D n q hq i) (hB : 2 ≤ B) (hi : i<D) (hn : 4 ≤ n)
-    (hq' : q∈Set.Ioc 0 (1/2:ℝ))
-    (H : SourceNumericMargins B D q (sourceTargetScale B D n (i+1)))
-    (htail : 2*(B:ℝ)*Real.exp
-      (-(sourceTargetScale B D n (i+1))^(sourceEpsilon B D)/sourceTailDenominator B D q) ≤
-        n^(-sourceTailExponent B D)) :
-    let N := sourceTargetScale B D n (i+1)
-    let b := sourceDimension B i
-    let ε := sourceEpsilon B D
-    let s := sourceCancellationOrder B D
-    let P := sourceArithmeticP (sourceOverlapConstant q) N b ε U.left.length U.overlapCount
-    let h := sourceArithmeticCount (sourceOverlapConstant q) N b ε U.left.length U.overlapCount
-    let atom := fun (τ : Bool)=>if τ then U.right else U.left
-    ∀j ≤ b, initialOverlapLower q ≤ listTraceOverlap q hq
-      (pureAtomRun (sourceBlockWord s P atom j false) (h j))
-      (pureAtomRun (sourceBlockWord s P atom j false) (h j+1)) := by
-  let N := sourceTargetScale B D n (i+1)
-  let b := sourceDimension B i
-  let ε := sourceEpsilon B D
-  let s := sourceCancellationOrder B D
-  let P := sourceArithmeticP (sourceOverlapConstant q) N b ε U.left.length U.overlapCount
-  let h := sourceArithmeticCount (sourceOverlapConstant q) N b ε U.left.length U.overlapCount
-  let T := sourceBlockWindows N b ε
-  let t := sourceArithmeticTrials (sourceOverlapConstant q) N b ε U.left.length U.overlapCount
-  let atom := fun (τ : Bool)=>if τ then U.right else U.left
-  let k := sourceHalfMoment B D n (i+1)
-  let L := N^(-scaleA b 0 ε*totalScoreExponent B i/2)
-  let a := N^(-20*sourceWorkSize B D)
-  have hn1 : 1 ≤ n := by linarith only [hn]
-  have hn2 : 1 < n := by linarith only [hn]
-  have hN : 0<N := (by norm_num : (0:ℝ)<1).trans_le (sourceTargetScale_bounds B D (i+1) hB hn1).1
-  have H' := source_level_geometry U hB hi hn1 hq' H
-  have hb := sourceDimension_bounds B i hB
-  have hk : 1 ≤ k := (sourceHalfMoment_bounds B D (i+1) hB hn2).1
-  let O := U.oracle.restrict hk (sourceHalfMoment_step B D i hB hi hn2)
-  have hC := sourceOverlapConstant_large hq'
-  have hp : initialOverlapLower q∈Set.Ioc 0 (1/8:ℝ) :=
-    ⟨initialOverlapLower_pos q,(initialOverlapLower_lt_eighth hq'.1).le⟩
-  have hc (j : ℕ) (hj : j ≤ b) : 1 ≤ h j :=
-    (H'.length_positive hN hj).trans (H'.counts j hj).1
-  have hP (j : ℕ) (hj : j<b) : 1 ≤ P j := H'.copies_positive hN hj
-  have hlift (j : ℕ) (_hj : j<b) : sourceOverlapConstant q*h j*P j ≤ h (j+1) :=
-    sourceArithmetic_count_lift _ _ _ _ _ _ _
-  have hbase : initialOverlapLower q ≤ listTraceOverlap q hq
-      (pureAtomRun (atom false) (h 0)) (pureAtomRun (atom false) (h 0+1)) := U.power_overlap
-  by_cases hiw : i<warmupBlocks B
-  · have hb2 : sourceDimension B i=2 := ite_eq_left hiw
-    have hb' : b=2 := hb2
-    have hw := source_level_warmup_second U hB hi hn hq' hiw H htail
-    have hnum := source_level_all_budgets U hB hi hn hq' H htail
-    have hexp := (source_level_exponent_margins B D i hB (Nat.le_of_lt hi)).1 hiw
-    have hfirst := source_level_dense_overlap U hB hi hn hq' H (j:=1) (by omega) hexp.2.2.1
-    have hfirst' : Real.sqrt (((h 1+1:ℕ):ℝ)*((2^s:ℕ):ℝ)*(sourceTargetScale B D n i)^(-totalScoreExponent B i))+
-        2*((h 1+1:ℕ):ℝ)*((P 0:ℝ)*U.error) ≤ 1/4 := by
-      simpa only [pow_one,sourceBlockLength,mul_one,mul_assoc] using hfirst
-    have hev : Even (T 0) := by
-      change Even (sourceBlockWindows N b ε 0)
-      rw [sourceBlockWindows,ite_eq_left (by omega : 0<b)]
-      exact sourceWindowSize_even _ _ _ _
-    have hfit : t 0*(2*h 0+1)+T 0 ≤ P 0 := by
-      dsimp only [t]
-      rw [sourceArithmetic_inner_trials (sourceOverlapConstant q) N b ε U.left.length U.overlapCount 0 (by omega : 0+1≠b)]
-      have hh := sourceLocalTrials_budget (P 0) (h 0)
-      have ht := (H'.windows 0 (by omega)).2.2.1
-      change 2*T 0 ≤ P 0 at ht
-      rw [mul_assoc] at hh
-      change sourceLocalTrials (P 0) (h 0)*(2*h 0+1)+T 0 ≤ P 0
-      omega
-    have hs : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount (fun _=>T 0) s 0 ≤ 1/2 := by
-      simpa only [L,N,b,ε,k,s,T,t,a,hb2,sourceSparseCount,Finset.range_zero,Finset.prod_empty,Nat.mul_one,Nat.cast_pow,Nat.cast_ofNat] using hw.1
-    have hclip : sourceClippingCost (fun _=>T 0) (fun _=>t 0) s 0 k (initialOverlapLower q) a L ≤ 1/2 := by
-      simpa only [L,N,b,ε,k,s,T,t,a,P,h,hb2] using hw.2.1
-    have hv : (sourceTargetScale B D n i)^(-totalScoreExponent B i) ≤ L^2 := by
-      rw [source_old_variance_square B D i (by linarith only [hn]) hi]
-    have hh := source_warmup_nested_overlap hq U.error_nonneg s k P h atom O hp hC
-      (fun j hj=>hc j (by omega)) (fun j hj=>hP j (by omega)) (fun j hj=>hlift j (by omega))
-      hfirst' hbase (T 0) (t 0) (H'.windows 0 (by omega)).1 hev (H'.windows 0 (by omega)).2.1
-      (H'.windows 0 (by omega)).2.2.2 hfit a a L (by positivity) (by positivity) hk hv hs
-      (hnum.2.2.2.1 0 (by omega)) hclip (by simpa only [L,N,b,ε,k,s,T,t,a,P,h,hb2] using hw.2.2)
-    change ∀j : ℕ,j≤b → initialOverlapLower q ≤ listTraceOverlap q hq
-      (pureAtomRun (sourceBlockWord s P atom j false) (h j))
-      (pureAtomRun (sourceBlockWord s P atom j false) (h j+1))
-    intro j hj
-    exact hh j (by change j≤b at hj; omega)
-  · apply source_nested_overlap hq U.error_nonneg s P h atom (O.hellinger hk) b hp hC hc hP hlift _ hbase
-    intro j hj
-    exact source_level_dense_overlap U hB hi hn hq' H (by omega)
-      ((source_level_exponent_margins B D i hB (Nat.le_of_lt hi)).2 (by omega) (j+1) (by omega))
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- The exact manuscript block is now a producer: no new level invariant or
- overlap is assumed. All hypotheses besides the old actual pair are literal
- scalar schedule margins. -/
-theorem source_level_step {B D i : ℕ} {n q : ℝ} {hq : q∈Set.Icc 0 1}
-    (U : SourceLevelPair B D n q hq i) (hB : 2 ≤ B) (hi : i<D) (hn : 4 ≤ n)
-    (hq' : q∈Set.Ioc 0 (1/2:ℝ))
-    (H : SourceNumericMargins B D q (sourceTargetScale B D n (i+1)))
-    (htail : 2*(B:ℝ)*Real.exp
-      (-(sourceTargetScale B D n (i+1))^(sourceEpsilon B D)/sourceTailDenominator B D q) ≤
-        n^(-sourceTailExponent B D)) :
-    Nonempty (SourceLevelPair B D n q hq (i+1)) := by
-  let N := sourceTargetScale B D n (i+1)
-  let b := sourceDimension B i
-  let ε := sourceEpsilon B D
-  let s := sourceCancellationOrder B D
-  let C := sourceOverlapConstant q
-  let P := sourceArithmeticP C N b ε U.left.length U.overlapCount
-  let h := sourceArithmeticCount C N b ε U.left.length U.overlapCount
-  let T := sourceBlockWindows N b ε
-  let t := sourceArithmeticTrials C N b ε U.left.length U.overlapCount
-  let atom := fun (τ : Bool)=>if τ then U.right else U.left
-  let k := sourceHalfMoment B D n (i+1)
-  let L := N^(-scaleA b 0 ε*totalScoreExponent B i/2)
-  let a := N^(-20*sourceWorkSize B D)
-  let ω := initialOverlapLower q
-  let Δ := n^(-100*(totalScoreExponent B D+B+1))
-  have hn1 : 1 ≤ n := by linarith only [hn]
-  have hn2 : 1 < n := by linarith only [hn]
-  have hn0 : 0<n := by linarith only [hn]
-  have hN : 0<N := (by norm_num : (0:ℝ)<1).trans_le (sourceTargetScale_bounds B D (i+1) hB hn1).1
-  have hC : 1 ≤ C := (by norm_num : (1:ℝ)≤2).trans (sourceOverlapConstant_ge_two hq')
-  have H' := source_level_geometry U hB hi hn1 hq' H
-  have hb := sourceDimension_bounds B i hB
-  have hb1 : b-1+1=b := by dsimp [b]; omega
-  have hk : 1 ≤ k := (sourceHalfMoment_bounds B D (i+1) hB hn2).1
-  let O := U.oracle.restrict hk (sourceHalfMoment_step B D i hB hi hn2)
-  let := O.finiteOutcome
-  have hnum := source_level_all_budgets U hB hi hn hq' H htail
-  have hov := source_level_overlap U hB hi hn hq' H htail
-  have hv : (sourceTargetScale B D n i)^(-totalScoreExponent B i) ≤ L^2 := by
-    rw [source_old_variance_square B D i hn0 hi]
-  have hm : O.first.expect (fun z=>|O.score z|^(2*k)) ≤ L^(2*k) := by
-    have hh := O.first.expect_even_pow_le_of_moment O.score k hk (O.moment.trans hv)
-    simpa only [pow_mul] using hh
-  have he (j : ℕ) (hj : j ≤ b-1) : Even (T j) := by
-    change Even (sourceBlockWindows N b ε j)
-    rw [sourceBlockWindows,ite_eq_left (by omega : j<b)]
-    exact sourceWindowSize_even _ _ _ _
-  obtain ⟨W⟩ := actual_source_oracle hq U.error_nonneg s P T t h atom (b-1) (sourceDegreeCutoff B) k
-    H'.layout H'.window_positive (by
-      simpa only [P,t,h,T,sourceBlockWindows,ite_eq_left (show b-1<b by omega)] using
-        sourceArithmetic_root_copies C N b ε U.left.length U.overlapCount (by omega))
-    (fun j hj=>H'.trials_positive hN (by linarith only [hC]) (by positivity) (by omega))
-    (fun j hj=>(H'.windows j (by omega)).1) he (fun j hj=>(H'.windows j (by omega)).2.1)
-    (H'.windows (b-1) (by omega)).2.2.2 O.first O.second O.decode O.score O.density O.likelihood_nonneg
-    O.first_simulation O.second_simulation ω a a L (by positivity) (by positivity)
-    ⟨initialOverlapLower_pos q,lt_of_lt_of_le (initialOverlapLower_lt_eighth hq'.1) (by norm_num)⟩
-    (fun j hj=>hov j (by omega)) hk hm hnum.2.1
-    (fun j hj=>hnum.2.2.2.1 j (by omega)) hnum.2.2.2.2.1
-  rw [hb1] at W
-  have hlen (τ : Bool) : (sourceBlockWord s P atom b τ).length=sourceArithmeticLength C N b ε U.left.length U.overlapCount b := by
-    rw [sourceArithmeticLength_eq_block]
-    apply sourceBlockWord_length
-    intro σ
-    cases σ
-    · rfl
-    · exact U.equalLength.symm
-  have hinj : Function.Injective atom := by
-    intro τ σ hτσ
-    cases τ <;> cases σ
-    · rfl
-    · exact False.elim (U.different hτσ)
-    · exact False.elim (U.different hτσ.symm)
-    · rfl
-  have hne : sourceBlockWord s P atom b false≠sourceBlockWord s P atom b true := by
-    intro hh
-    have hh' := sourceBlockWord_injective s P atom U.left.length (by
-      intro τ; cases τ; rfl; exact U.equalLength.symm) hinj b
-      (fun j hj=>Nat.lt_of_lt_of_le (by omega : 0<1) (H'.copies_positive hN hj)) hh
-    contradiction
-  have hlength := H'.lengths b le_rfl
-  change N^(scaleA b b ε)/2 ≤ (sourceArithmeticLength C N b ε U.left.length U.overlapCount b:ℝ) ∧
-    (sourceArithmeticLength C N b ε U.left.length U.overlapCount b:ℝ) ≤ N^(scaleA b b ε) at hlength
-  simp only [scaleA_top,Real.rpow_one] at hlength
-  have hE : U.error ≤ ((i:ℝ)+1)*Δ*U.left.length := by
-    apply (div_le_iff₀ ?_).mp U.normalized_error
-    exact (show 0<sourceTargetScale B D n i/2 by positivity [Real.rpow_pos_of_pos hn0 (sourceScaleProduct B D i)]).trans_le U.length_lower
-  have hleaf : SourceLeafCount T b*U.left.length ≤ sourceArithmeticLength C N b ε U.left.length U.overlapCount b := by
-    rw [sourceArithmeticLength_eq_block,sourceBlockLength_factor]
-    rw [Nat.mul_comm]
-    apply Nat.mul_le_mul_left
-    apply sourceLeafCount_le_length (by omega : 1≤1)
-    intro j hj
-    have hh := (H'.windows j hj).2.2.1
-    change 2*T j ≤ P j at hh
-    change T j ≤ P j
-    omega
-  have herror := normalized_block_error (H'.length_positive hN le_rfl) hE
-    (show 0≤Δ by positivity) hleaf hnum.2.2.2.2.2
-  have hfac : C^(2*b)*C^(2*B*i+2) ≤ C^(2*B*(i+1)+2) := by
-    rw [←pow_add]
-    apply pow_le_pow_right₀ hC
-    dsimp [b]
-    nlinarith only [hb.2]
-  refine ⟨W.toConcrete hq ((hlen false).trans (hlen true).symm) hne ?_ ?_ ?_ hnum.2.2.1 ?_ ?_ (hov b le_rfl)⟩
-  · rw [hlen]; exact hlength.1
-  · rw [hlen]; exact hlength.2
-  · rw [hlen]
-    simpa only [Δ,Nat.cast_add,Nat.cast_one,add_assoc,show (1:ℝ)+1=2 by norm_num] using herror
-  · rw [hlen]; exact (H'.counts b le_rfl).1
-  · rw [hlen]
-    exact (H'.counts b le_rfl).2.trans (mul_le_mul_of_nonneg_right hfac (Nat.cast_nonneg _))
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open Filter
-open scoped Topology Classical
-
-/-- Full source construction at every scheduled level, uniformly in the
- joint deletion regime. No level witnesses are assumed. -/
-theorem source_all_levels_eventually (B : ℕ) (ζ : ℝ) (hB : 2 ≤ B)
-    (hζ : ζ∈Set.Ioo 0 1) {n q : ℕ→ℝ} (hn : Tendsto n atTop atTop)
-    (hq : ∀t,q t∈Set.Ioc 0 (1/2:ℝ))
-    (hH : Tendsto (fun t=>(q t)^3*Real.log (n t)) atTop atTop) :
-    ∀ᶠt in atTop,∀i ≤ sourceDepth B ζ ((q t)^3*Real.log (n t)),
-      Nonempty (SourceLevelPair B (sourceDepth B ζ ((q t)^3*Real.log (n t)))
-        (n t) (q t) ⟨(hq t).1.le,by linarith [(hq t).2]⟩ i) := by
-  filter_upwards [source_initial_at_smallest_scale B ζ hB hζ hn hq hH,
-    source_numeric_margins_eventually B ζ hB hζ hn hq hH,
-    source_tails_eventually B ζ hB hζ hn hq hH,hn.eventually_ge_atTop 4] with t h0 hm ht hn4
-  intro i hi
-  induction i with
-  | zero => simpa only [SourceLevelPair,totalScoreExponent,Finset.range_zero,
-      Finset.sum_empty,add_zero,Nat.mul_zero,Nat.zero_add] using h0
-  | succ i ih =>
-    obtain ⟨U⟩ := ih (by omega)
-    exact source_level_step U hB (by omega) hn4 (hq t) (hm (i+1)) (ht (i+1))
-
-/-- The final source pair has actual length in [n/2,n], an exact common
- complete-trace simulator, prescribed high moment and inductively obtained
- power overlap. This is the construction theorem, before statistical transfer. -/
-theorem source_final_pair_eventually (B : ℕ) (ζ : ℝ) (hB : 2 ≤ B)
-    (hζ : ζ∈Set.Ioo 0 1) {n q : ℕ→ℝ} (hn : Tendsto n atTop atTop)
-    (hq : ∀t,q t∈Set.Ioc 0 (1/2:ℝ))
-    (hH : Tendsto (fun t=>(q t)^3*Real.log (n t)) atTop atTop) :
-    ∀ᶠt in atTop,Nonempty (ConcreteLevelPair (q t) ⟨(hq t).1.le,by linarith [(hq t).2]⟩ (n t)
-      ((n t)^(-100*(totalScoreExponent B (sourceDepth B ζ ((q t)^3*Real.log (n t)))+B+1)))
-      (totalScoreExponent B (sourceDepth B ζ ((q t)^3*Real.log (n t))))
-      (initialOverlapLower (q t))
-      ((sourceOverlapConstant (q t))^(2*B*sourceDepth B ζ ((q t)^3*Real.log (n t))+2))
-      (sourceDepth B ζ ((q t)^3*Real.log (n t)))
-      (sourceHalfMoment B (sourceDepth B ζ ((q t)^3*Real.log (n t))) (n t)
-        (sourceDepth B ζ ((q t)^3*Real.log (n t))))) := by
-  filter_upwards [source_all_levels_eventually B ζ hB hζ hn hq hH] with t ht
-  simpa only [SourceLevelPair,sourceTargetScale_top] using ht _ le_rfl
-
-end TraceReconstruction
-
-end
-end
-end
-end
-end
-end
-end
-end
-end
-
-end OAI
+import OAI.Probability.TraceReconstruction.Amplification
+import OAI.Probability.TraceReconstruction.Propagation
+
+namespace OAI
+
+/-!
+# Induction
+
+Concrete level invariants, metadata prefixes, score/error budgets, overlap, the induction step, and the complete construction.
+-/
+
+open scoped ENNReal
+noncomputable section
+
+open Filter
+open scoped Topology
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- The literal induction invariant, including the depth-dependent moment,
+ actual complete-trace simulator, distinct words, length, and power overlap. -/
+abbrev SourceLevelPair (B D : ℕ) (n q : ℝ) (hq : q∈Set.Icc 0 1) (i : ℕ) :=
+  ConcreteLevelPair q hq (sourceTargetScale B D n i)
+    (n^(-100*(totalScoreExponent B D+B+1))) (totalScoreExponent B i)
+    (initialOverlapLower q) ((sourceOverlapConstant q)^(2*B*i+2)) i (sourceHalfMoment B D n i)
+
+theorem source_level_geometry {B D i : ℕ} {n q : ℝ} {hq : q∈Set.Icc 0 1}
+    (U : SourceLevelPair B D n q hq i) (hB : 2 ≤ B) (hi : i<D)
+    (hn : 1 ≤ n) (hq' : q∈Set.Ioc 0 (1/2:ℝ))
+    (H : SourceNumericMargins B D q (sourceTargetScale B D n (i+1))) :
+    SourceArithmeticBounds (sourceOverlapConstant q) (sourceTargetScale B D n (i+1))
+      ((sourceOverlapConstant q)^(2*B*i+2)) (sourceEpsilon B D)
+      (sourceCancellationOrder B D) (sourceDimension B i) U.left.length U.overlapCount := by
+  let N := sourceTargetScale B D n (i+1)
+  let b := sourceDimension B i
+  let C := sourceOverlapConstant q
+  have hC : 2 ≤ C := sourceOverlapConstant_ge_two hq'
+  have hb := sourceDimension_bounds B i hB
+  have hN := (sourceTargetScale_bounds B D (i+1) hB hn).1
+  have he := sourceEpsilon_pos B D
+  have hG := source_local_envelope B D i b hq' (Nat.le_of_lt hi) hb.2
+  apply source_arithmetic_geometry hC hN (one_le_pow₀ (by linarith only [hC])) he.le
+    (fun j _=>sourceA_nonneg B D b j hb.2 (by omega))
+    (fun j hj=>sourceKappa_three_epsilon B D b (j+1) hb.2 (by omega) (by omega))
+  · have hh := mul_le_mul_of_nonneg_left hG (show (0:ℝ) ≤ 1200 by norm_num)
+    exact (by nlinarith only [hh] : 1200*C^(2*b)*C^(2*B*i+2) ≤ 1200*sourceCapEnvelope B D q).trans H.geometry
+  · simpa only [Nat.cast_pow,Nat.cast_ofNat] using H.motif
+  · simpa only [sourceTargetScale_step B D i n (by linarith only [hn]) hi] using U.length_lower
+  · simpa only [sourceTargetScale_step B D i n (by linarith only [hn]) hi] using U.length_upper
+  · exact U.overlapCount_lower
+  · exact U.overlapCount_upper
+
+theorem SourceArithmeticBounds.copies_positive {C N F ε : ℝ} {s b ell k : ℕ}
+    (H : SourceArithmeticBounds C N F ε s b ell k) (hN : 0<N) {j : ℕ} (hj : j<b) :
+    1 ≤ sourceArithmeticP C N b ε ell k j := by
+  have hh := (H.copies j hj).1
+  have hp : (0:ℝ)<N^(scaleA b j ε)*N^ε/2 := by positivity
+  have hp' : (0:ℝ)<sourceArithmeticP C N b ε ell k j := hp.trans_le hh
+  have hp'' : 0<sourceArithmeticP C N b ε ell k j := by exact_mod_cast hp'
+  omega
+
+theorem SourceArithmeticBounds.trials_positive {C N F ε : ℝ} {s b ell k : ℕ}
+    (H : SourceArithmeticBounds C N F ε s b ell k) (hN : 0<N) (hC : 0<C) (hF : 0<F)
+    {j : ℕ} (hj : j<b) : 0<sourceArithmeticTrials C N b ε ell k j := by
+  have hh := (H.trials j hj).1
+  have hp : (0:ℝ)<N^ε/(600*(C^(2*b)*F)) := by positivity
+  exact_mod_cast hp.trans_le hh
+
+theorem source_level_exponent_margins (B D i : ℕ) (hB : 2 ≤ B) (hi : i ≤ D) :
+    let ε := sourceEpsilon B D
+    let b := sourceDimension B i
+    let e := scaleA b 0 ε*totalScoreExponent B i
+    (i<warmupBlocks B → b=2 ∧ 27/50 ≤ e ∧ ε+scaleA b 1 ε ≤ e ∧ ε+1-2*e ≤ 0) ∧
+    (warmupBlocks B ≤ i → ∀j ≤ b,ε+scaleA b j ε ≤ e) := by
+  let ε := sourceEpsilon B D
+  have he := sourceEpsilon_pos B D
+  have he1 : ε ≤ 1/100 := by
+    have hh := sourceEpsilon_gap_small B D
+    have hg := sourceGapFloor_le_one B
+    dsimp [ε]
+    linarith only [hh,hg]
+  have hd := (totalScoreExponent_bounds B i hB).1
+  have hd' : totalScoreExponent B i ≤ 3+(B:ℝ)*D := by
+    simpa only [mul_comm] using (totalScoreExponent_mono B hB hi).trans (totalScoreExponent_bounds B D hB).2
+  constructor
+  · intro hw
+    have hb : sourceDimension B i=2 := ite_eq_left hw
+    have hh := source_warmup_score_margin B D hB hd hd'
+    simp only [hb]
+    exact ⟨True.intro,hh.1,by linarith only [hh.1,hh.2,he1],by linarith only [hh.1,he1]⟩
+  · intro hw j hj
+    have hb : sourceDimension B i=B := ite_eq_right (by omega)
+    rw [hb] at hj ⊢
+    have hlarge := totalScoreExponent_after_warmup B i hB hw
+    have hh := source_main_score_margin B D hB (by linarith only [hlarge]) hd'
+    have ha : scaleA B j ε ≤ 1 := by
+      exact (scaleA_mono_index B j B ε he.le hj).trans (by rw [scaleA_top])
+    have heps : ε ≤ 1/(2:ℝ)^B := by
+      have hs := sourceEpsilon_small B D
+      have hm := mul_le_mul_of_nonneg_right (show (1:ℝ) ≤ D+1 by have : 0 ≤ (D:ℝ) := Nat.cast_nonneg D; linarith only [this]) (show 0 ≤ (2:ℝ)^B*ε by positivity)
+      apply (le_div_iff₀ (by positivity)).mpr
+      dsimp [ε] at *
+      nlinarith only [hs,hm]
+    linarith only [hh,ha,heps]
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem sourceEffectiveHiddenTail_one (T n : ℕ→ℕ) (ω : ℝ) :
+    sourceEffectiveHiddenTail T n 1 ω = if n 0=0 then 0 else Real.exp (-(n 0:ℝ)*ω^4/16) := by
+  unfold sourceEffectiveHiddenTail
+  change (∑v : Bool ⊕ (Fin (T 0) × PEmpty),_)=_
+  simp only [Fintype.sum_sum_type,Finset.univ_eq_empty,Finset.sum_empty,
+    add_zero,Fintype.sum_bool,collectionCount,sourceScheduledTrials,
+    ite_true,zero_add]
+  rfl
+
+theorem sourceEffectiveHiddenTail_step (T n : ℕ→ℕ) (b : ℕ) (ω : ℝ) :
+    sourceEffectiveHiddenTail T n (b+1) ω =
+      (if n b=0 then 0 else Real.exp (-(n b:ℝ)*ω^4/16))+
+      (if sourceScheduledTrials T n b=0 then 0 else Real.exp (-(sourceScheduledTrials T n b:ℝ)*ω^4/16))+
+      (T b:ℝ)*sourceEffectiveHiddenTail T n b ω := by
+  unfold sourceEffectiveHiddenTail
+  change (∑v : Bool ⊕ (Fin (T b) × WindowCollection T b),_)=_
+  simp only [Fintype.sum_sum_type,Fintype.sum_prod_type,Fintype.sum_bool,collectionCount]
+  let f := fun v : ℕ => if v=0 then (0:ℝ) else Real.exp (-(v:ℝ)*ω^4/16)
+  change (f (sourceScheduledTrials T n b)+f (n b)+(∑ _ : Fin (T b),
+    ∑ v : WindowCollection T b,f (collectionCount T n (sourceScheduledTrials T n) b v))) =
+    f (n b)+f (sourceScheduledTrials T n b)+(T b:ℝ)*
+    (∑ v : WindowCollection T b,f (collectionCount T n (sourceScheduledTrials T n) b v))
+  rw [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]
+  ring
+
+theorem sourceEffectiveHiddenTail_two_ge_one (T n : ℕ→ℕ) (ω : ℝ) (hT : 1 ≤ T 1) :
+    sourceEffectiveHiddenTail (fun _=>T 0) (fun _=>n 0) 1 ω ≤ sourceEffectiveHiddenTail T n 2 ω := by
+  rw [sourceEffectiveHiddenTail_step T n 1 ω,sourceEffectiveHiddenTail_one,
+    sourceEffectiveHiddenTail_one]
+  have hnonneg (v : ℕ) : 0 ≤ (if v=0 then 0 else Real.exp (-(v:ℝ)*ω^4/16)) := by
+    split_ifs <;> positivity
+  have ht : (1:ℝ) ≤ T 1 := by exact_mod_cast hT
+  have hh := le_mul_of_one_le_left (hnonneg (n 0)) ht
+  linarith only [hh,hnonneg (n 1),hnonneg (sourceScheduledTrials T n 1)]
+
+theorem sourceSparseCount_motif_le (T : ℕ→ℕ) (s b : ℕ) (hT : ∀j<b,1 ≤ T (j+1)) :
+    (2:ℝ)^s ≤ (sourceSparseCount T s b:ℝ) := by
+  have hp : 1 ≤ ∏j∈Finset.range b,T (j+1) := Finset.one_le_prod (fun j hj=>hT j (Finset.mem_range.mp hj))
+  have hh : 2^s ≤ sourceSparseCount T s b := by
+    unfold sourceSparseCount
+    exact Nat.le_mul_of_pos_right _ hp
+  exact_mod_cast hh
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem source_level_tail_budget {B D i : ℕ} {n q : ℝ} {hq : q∈Set.Icc 0 1}
+    (U : SourceLevelPair B D n q hq i) (hB : 2 ≤ B) (hi : i<D) (hn : 4 ≤ n)
+    (hq' : q∈Set.Ioc 0 (1/2:ℝ))
+    (H : SourceNumericMargins B D q (sourceTargetScale B D n (i+1)))
+    (htail : 2*(B:ℝ)*Real.exp
+      (-(sourceTargetScale B D n (i+1))^(sourceEpsilon B D)/sourceTailDenominator B D q) ≤
+        n^(-sourceTailExponent B D)) :
+    let N := sourceTargetScale B D n (i+1)
+    let b := sourceDimension B i
+    let ε := sourceEpsilon B D
+    let T := sourceBlockWindows N b ε
+    let t := sourceArithmeticTrials (sourceOverlapConstant q) N b ε U.left.length U.overlapCount
+    sourceEffectiveHiddenTail T t b (initialOverlapLower q) ≤ n^(-200*(totalScoreExponent B D+B+1)) ∧
+    sourceFitTail T t b/N^(-20*sourceWorkSize B D) ≤ n^(-200*(totalScoreExponent B D+B+1)) ∧
+    sourceFitTail T t b ≤ n^(-200*(totalScoreExponent B D+B+1)) := by
+  have hn1 : 1 ≤ n := by linarith only [hn]
+  have H' := source_level_geometry U hB hi hn1 hq' H
+  have hb := sourceDimension_bounds B i hB
+  have hN := sourceTargetScale_bounds B D (i+1) hB hn1
+  have hC := sourceOverlapConstant_pos hq'.1
+  have hω := initialOverlapLower_pos q
+  have hG := source_local_envelope B D i (sourceDimension B i) hq' (Nat.le_of_lt hi) hb.2
+  have hf : 9600*((sourceOverlapConstant q)^(2*sourceDimension B i)*
+      (sourceOverlapConstant q)^(2*B*i+2))/(initialOverlapLower q)^4 ≤ sourceTailDenominator B D q := by
+    have hh := mul_le_mul_of_nonneg_right hG (show 0 ≤ 9600*(initialOverlapLower q)⁻¹^4 by positivity)
+    unfold sourceTailDenominator
+    rw [div_eq_mul_inv,←inv_pow]
+    nlinarith only [hh]
+  exact source_exponential_error_budget H' hB hb.2 hC (by positivity) hω hn1 hN.1 hN.2
+    (by
+      have hh : 0 ≤ 9600*sourceCapEnvelope B D q*(initialOverlapLower q)⁻¹^4 := by positivity [sourceCapEnvelope_pos B D hq'.1]
+      unfold sourceTailDenominator
+      linarith only [hh]) hf
+    (by simpa only [neg_div] using htail)
+
+theorem source_level_all_budgets {B D i : ℕ} {n q : ℝ} {hq : q∈Set.Icc 0 1}
+    (U : SourceLevelPair B D n q hq i) (hB : 2 ≤ B) (hi : i<D) (hn : 4 ≤ n)
+    (hq' : q∈Set.Ioc 0 (1/2:ℝ))
+    (H : SourceNumericMargins B D q (sourceTargetScale B D n (i+1)))
+    (htail : 2*(B:ℝ)*Real.exp
+      (-(sourceTargetScale B D n (i+1))^(sourceEpsilon B D)/sourceTailDenominator B D q) ≤
+        n^(-sourceTailExponent B D)) :
+    let N := sourceTargetScale B D n (i+1)
+    let b := sourceDimension B i
+    let ε := sourceEpsilon B D
+    let s := sourceCancellationOrder B D
+    let T := sourceBlockWindows N b ε
+    let t := sourceArithmeticTrials (sourceOverlapConstant q) N b ε U.left.length U.overlapCount
+    let k := sourceHalfMoment B D n (i+1)
+    let L := N^(-scaleA b 0 ε*totalScoreExponent B i/2)
+    let a := N^(-20*sourceWorkSize B D)
+    let ω := initialOverlapLower q
+    let gap := (totalScoreExponent B i+1)*sourceGapFloor B
+    (((6*(2*k:ℕ):ℝ)*L)^2*(sourceSparseCount T s (b-1):ℝ) ≤ N^(-gap/2)/8) ∧
+    (((6*(2*k:ℕ):ℝ)*L)^2*(sourceSparseCount T s (b-1):ℝ) ≤ 1/2) ∧
+    (64*sourceSharpEnergyBudget T t s (b-1) (sourceDegreeCutoff B) ω a a ((6*(2*k:ℕ):ℝ)*L) ≤
+       N^(-totalScoreExponent B (i+1))) ∧
+    (∀j<b,hiddenSmoothingBound (t j) s ω ≤ a) ∧
+    sourceClippingCost T t s (b-1) k ω a L ≤ 1/2 ∧
+    sourceClippingCost T t s (b-1) k ω a L+sourceFitTail T t b ≤
+      n^(-100*(totalScoreExponent B D+B+1)) := by
+  have hn1 : 1 ≤ n := by linarith only [hn]
+  have hn2 : 1<n := by linarith only [hn]
+  have hC : 1 ≤ sourceOverlapConstant q := (by norm_num : (1:ℝ) ≤ 2).trans (sourceOverlapConstant_ge_two hq')
+  have H' := source_level_geometry U hB hi hn1 hq' H
+  have hb := sourceDimension_bounds B i hB
+  have hb1 : sourceDimension B i-1+1=sourceDimension B i := by omega
+  have hN := sourceTargetScale_bounds B D (i+1) hB hn1
+  have he := source_level_energy B D i U.left.length U.overlapCount (sourceHalfMoment B D n (i+1))
+    (sourceOverlapConstant q) (sourceTargetScale B D n (i+1)) ((sourceOverlapConstant q)^(2*B*i+2))
+    (initialOverlapLower q) (sourceCapEnvelope B D q) hB hi hN.1 hC (one_le_pow₀ hC) (initialOverlapLower_pos q)
+    H' (sourceHalfMoment_bounds B D (i+1) hB hn2).2
+    (source_cap_envelope B D i (sourceDimension B i) hq' (Nat.le_of_lt hi) hb.2)
+    (by simpa only [Nat.cast_pow,Nat.cast_ofNat] using H.sparse)
+    (H.bad _ (by omega)) (by simpa only [hb1] using H.force (sourceDimension B i-1) (by omega)) H.tail H.smoothing
+  have ht := source_level_tail_budget U hB hi hn hq' H htail
+  have hg : sourceGapFloor B ≤ (totalScoreExponent B i+1)*sourceGapFloor B := by
+    exact le_mul_of_one_le_left (sourceGapFloor_pos B).le
+      (by linarith only [(totalScoreExponent_bounds B i hB).1])
+  have hc := source_level_clipping B D (i+1) (sourceCancellationOrder B D) (sourceDimension B i-1)
+    (sourceBlockWindows (sourceTargetScale B D n (i+1)) (sourceDimension B i) (sourceEpsilon B D))
+    (sourceArithmeticTrials (sourceOverlapConstant q) (sourceTargetScale B D n (i+1))
+      (sourceDimension B i) (sourceEpsilon B D) U.left.length U.overlapCount)
+    n (initialOverlapLower q) ((sourceTargetScale B D n (i+1))^(-20*sourceWorkSize B D))
+    ((totalScoreExponent B i+1)*sourceGapFloor B)
+    ((sourceTargetScale B D n (i+1))^(-scaleA (sourceDimension B i) 0 (sourceEpsilon B D)*totalScoreExponent B i/2))
+    hB hn hg (by simpa only [hb1] using ht.1) (by simpa only [hb1] using ht.2.1)
+    (by simpa only [hb1] using ht.2.2) he.2.1 he.1
+  exact ⟨he.1,he.2.1,he.2.2.1,he.2.2.2,hc.1,by simpa only [hb1] using hc.2⟩
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem source_old_variance_scale (B D i : ℕ) {n : ℝ} (hn : 0<n) (hi : i<D) :
+    (sourceTargetScale B D n i)^(-totalScoreExponent B i)=
+      (sourceTargetScale B D n (i+1))^(-scaleA (sourceDimension B i) 0 (sourceEpsilon B D)*totalScoreExponent B i) := by
+  rw [sourceTargetScale_step B D i n hn.le hi,←Real.rpow_mul (show 0 ≤ sourceTargetScale B D n (i+1) from (Real.rpow_pos_of_pos hn _).le)]
+  congr 1
+  ring
+
+theorem source_old_variance_square (B D i : ℕ) {n : ℝ} (hn : 0<n) (hi : i<D) :
+    (sourceTargetScale B D n i)^(-totalScoreExponent B i)=
+      ((sourceTargetScale B D n (i+1))^(-scaleA (sourceDimension B i) 0 (sourceEpsilon B D)*totalScoreExponent B i/2))^2 := by
+  rw [source_old_variance_scale B D i hn hi,←Real.rpow_natCast,←Real.rpow_mul (show 0 ≤ sourceTargetScale B D n (i+1) from (Real.rpow_pos_of_pos hn _).le)]
+  congr 1
+  norm_num
+
+theorem source_level_dense_overlap {B D i j : ℕ} {n q : ℝ} {hq : q∈Set.Icc 0 1}
+    (U : SourceLevelPair B D n q hq i) (hB : 2 ≤ B) (hi : i<D) (hn : 4 ≤ n)
+    (hq' : q∈Set.Ioc 0 (1/2:ℝ))
+    (H : SourceNumericMargins B D q (sourceTargetScale B D n (i+1)))
+    (hj : j ≤ sourceDimension B i)
+    (hgap : sourceEpsilon B D+scaleA (sourceDimension B i) j (sourceEpsilon B D) ≤
+      scaleA (sourceDimension B i) 0 (sourceEpsilon B D)*totalScoreExponent B i) :
+    let N := sourceTargetScale B D n (i+1)
+    let b := sourceDimension B i
+    let ε := sourceEpsilon B D
+    let s := sourceCancellationOrder B D
+    let P := sourceArithmeticP (sourceOverlapConstant q) N b ε U.left.length U.overlapCount
+    let h := sourceArithmeticCount (sourceOverlapConstant q) N b ε U.left.length U.overlapCount
+    Real.sqrt (((h j+1:ℕ):ℝ)*(((2^s:ℕ):ℝ)^j*(sourceTargetScale B D n i)^(-totalScoreExponent B i)))+
+      2*((h j+1:ℕ):ℝ)*((sourceBlockLength P 1 j:ℝ)*U.error) ≤ 1/4 := by
+  have hn1 : 1 ≤ n := by linarith only [hn]
+  have hn0 : 0<n := by linarith only [hn]
+  have H' := source_level_geometry U hB hi hn1 hq' H
+  have hb := sourceDimension_bounds B i hB
+  have he := sourceEpsilon_pos B D
+  have he1 : sourceEpsilon B D ≤ 1 := by
+    linarith only [sourceEpsilon_gap_small B D,sourceGapFloor_le_one B]
+  have hN := sourceTargetScale_bounds B D (i+1) hB hn1
+  have hC : 1 ≤ sourceOverlapConstant q := (by norm_num : (1:ℝ) ≤ 2).trans (sourceOverlapConstant_ge_two hq')
+  have hE : U.error ≤ ((i:ℝ)+1)*n^(-100*(totalScoreExponent B D+B+1))*U.left.length := by
+    apply (div_le_iff₀ ?_).mp U.normalized_error
+    exact (show 0<sourceTargetScale B D n i/2 by positivity [Real.rpow_pos_of_pos hn0 (sourceScaleProduct B D i)]).trans_le U.length_lower
+  apply source_literal_dense_overlap (V:=(sourceTargetScale B D n i)^(-totalScoreExponent B i)) (e:=scaleA (sourceDimension B i) 0 (sourceEpsilon B D)*totalScoreExponent B i) (W:=sourceWeightEnvelope B D) (L:=(2:ℝ)^(sourceCancellationOrder B D))
+    H' hB hb.2 hC (pow_nonneg (by linarith only [hC]) _) hn1 hN.1 hN.2 he.le he1
+    (sourceCapEnvelope_pos B D hq'.1).le (sourceWeightEnvelope_nonneg B D) (one_le_pow₀ (by norm_num)) rfl
+    (source_local_envelope B D i (sourceDimension B i) hq' (Nat.le_of_lt hi) hb.2) (Nat.le_of_lt hi) hj
+    (sourceA_nonneg B D _ j hb.2 (by omega))
+    ((scaleA_mono_index _ j _ _ he.le hj).trans (by rw [scaleA_top])) hE rfl
+    (by simpa only [neg_mul] using (source_old_variance_scale B D i hn0 hi).le) hgap H.error H.overlap
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- The second warmup lift is produced from the one-window oracle, whose
+ quadratic energy gain is essential before the main-block regime. -/
+theorem source_level_warmup_second {B D i : ℕ} {n q : ℝ} {hq : q∈Set.Icc 0 1}
+    (U : SourceLevelPair B D n q hq i) (hB : 2 ≤ B) (hi : i<D) (hn : 4 ≤ n)
+    (hq' : q∈Set.Ioc 0 (1/2:ℝ)) (hiw : i<warmupBlocks B)
+    (H : SourceNumericMargins B D q (sourceTargetScale B D n (i+1)))
+    (htail : 2*(B:ℝ)*Real.exp
+      (-(sourceTargetScale B D n (i+1))^(sourceEpsilon B D)/sourceTailDenominator B D q) ≤
+        n^(-sourceTailExponent B D)) :
+    let N := sourceTargetScale B D n (i+1)
+    let ε := sourceEpsilon B D
+    let s := sourceCancellationOrder B D
+    let T := sourceBlockWindows N 2 ε
+    let t := sourceArithmeticTrials (sourceOverlapConstant q) N 2 ε U.left.length U.overlapCount
+    let P := sourceArithmeticP (sourceOverlapConstant q) N 2 ε U.left.length U.overlapCount
+    let h := sourceArithmeticCount (sourceOverlapConstant q) N 2 ε U.left.length U.overlapCount
+    let k := sourceHalfMoment B D n (i+1)
+    let L := N^(-scaleA 2 0 ε*totalScoreExponent B i/2)
+    let a := N^(-20*sourceWorkSize B D)
+    let ω := initialOverlapLower q
+    let cost := sourceClippingCost (fun _=>T 0) (fun _=>t 0) s 0 k ω a L
+    (((6*(2*k:ℕ):ℝ)*L)^2*(2:ℝ)^s ≤ 1/2) ∧ cost ≤ 1/2 ∧
+    Real.sqrt (((h 2+1:ℕ):ℝ)*((2^s:ℕ):ℝ)*
+      (64*sourceSharpEnergyBudget (fun _=>T 0) (fun _=>t 0) s 0 2 ω a a ((6*(2*k:ℕ):ℝ)*L)))+
+      2*((h 2+1:ℕ):ℝ)*((P 1:ℝ)*(cost+(T 0:ℝ)*U.error)) ≤ 1/4 := by
+  let N := sourceTargetScale B D n (i+1)
+  let ε := sourceEpsilon B D
+  let s := sourceCancellationOrder B D
+  let C := sourceOverlapConstant q
+  let F := C^(2*B*i+2)
+  let K := sourceCapEnvelope B D q
+  let W := sourceWeightEnvelope B D
+  let T := sourceBlockWindows N 2 ε
+  let t := sourceArithmeticTrials C N 2 ε U.left.length U.overlapCount
+  let P := sourceArithmeticP C N 2 ε U.left.length U.overlapCount
+  let h := sourceArithmeticCount C N 2 ε U.left.length U.overlapCount
+  let k := sourceHalfMoment B D n (i+1)
+  let e := scaleA 2 0 ε*totalScoreExponent B i
+  let L := N^(-scaleA 2 0 ε*totalScoreExponent B i/2)
+  let a := N^(-20*sourceWorkSize B D)
+  let ω := initialOverlapLower q
+  let w := (6*(2*k:ℕ):ℝ)*L
+  let gap := (totalScoreExponent B i+1)*sourceGapFloor B
+  let Δ := n^(-100*(totalScoreExponent B D+B+1))
+  have hb : sourceDimension B i=2 := ite_eq_left hiw
+  have hn1 : 1 ≤ n := by linarith only [hn]
+  have hn0 : 0<n := by linarith only [hn]
+  have hN := sourceTargetScale_bounds B D (i+1) hB hn1
+  have hN0 : 0<N := by dsimp [N]; linarith only [hN.1]
+  have hC : 1 ≤ C := (by norm_num : (1:ℝ) ≤ 2).trans (sourceOverlapConstant_ge_two hq')
+  have hC0 : 0<C := by linarith only [hC]
+  have hF : 0<F := by positivity
+  have hK : 0 ≤ K := (sourceCapEnvelope_pos B D hq'.1).le
+  have hW : 0 ≤ W := sourceWeightEnvelope_nonneg B D
+  have he := sourceEpsilon_pos B D
+  have he1 : ε ≤ 1 := by
+    dsimp [ε]
+    linarith only [sourceEpsilon_gap_small B D,sourceGapFloor_le_one B]
+  have H' := source_level_geometry U hB hi hn1 hq' H
+  rw [hb] at H'
+  have Hnum := source_level_all_budgets U hB hi hn hq' H htail
+  simp only [hb,show (2:ℕ)-1=1 by rfl] at Hnum
+  have ht := source_level_tail_budget U hB hi hn hq' H htail
+  simp only [hb] at ht
+  have hmot : (2:ℝ)^s ≤ (sourceSparseCount T s 1:ℝ) :=
+    sourceSparseCount_motif_le T s 1 (fun j _=>Nat.succ_le_of_lt (H'.window_positive (j+1)))
+  have hs := mul_le_mul_of_nonneg_left hmot (sq_nonneg w)
+  have hs1 : w^2*(2:ℝ)^s ≤ 1/2 := hs.trans Hnum.2.1
+  have hs2 : w^2*(2:ℝ)^s ≤ N^(-gap/2)/8 := hs.trans Hnum.1
+  have hm : sourceEffectiveHiddenTail (fun _=>T 0) (fun _=>t 0) 1 ω ≤
+      n^(-200*(totalScoreExponent B D+B+1)) :=
+    (sourceEffectiveHiddenTail_two_ge_one T t ω (Nat.succ_le_of_lt (H'.window_positive 1))).trans ht.1
+  have hg : sourceGapFloor B ≤ gap := le_mul_of_one_le_left (sourceGapFloor_pos B).le
+    (by linarith only [(totalScoreExponent_bounds B i hB).1])
+  have hc := source_one_window_clipping B D (i+1) s (T 0) (t 0) n ω a gap L hB hn hg hm hs1 hs2
+  have hG : C^4*F ≤ K := by
+    simpa only [hb,show (2:ℕ)*2=4 by rfl] using
+      source_local_envelope B D i (sourceDimension B i) hq' (Nat.le_of_lt hi) (sourceDimension_bounds B i hB).2
+  have hE : U.error ≤ ((i:ℝ)+1)*Δ*U.left.length := by
+    apply (div_le_iff₀ ?_).mp U.normalized_error
+    exact (show 0<sourceTargetScale B D n i/2 by positivity [Real.rpow_pos_of_pos hn0 (sourceScaleProduct B D i)]).trans_le U.length_lower
+  have herr := source_warmup_error_budget H' hB hC hF.le hn1 hN.1 hN.2 he.le he1 hK hG
+    (Nat.le_of_lt hi) hE U.error_nonneg rfl hc.2 H.error
+  have hcount : ((h 2+1:ℕ):ℝ) ≤ (K+1)*N := by
+    have hh := H'.count_bound hC hF.le hN.1 (j:=2) le_rfl (by rw [scaleA_top]; norm_num)
+    have hh' := mul_le_mul_of_nonneg_right (add_le_add hG (show (1:ℝ) ≤ 1 from le_rfl)) hN0.le
+    simp only [scaleA_top,Real.rpow_one,show (2:ℕ)*2=4 by rfl] at hh
+    exact hh.trans hh'
+  have hwindow : (T 0:ℝ) ≤ N := by
+    have hp : T 0 ≤ P 0 := by have hh := (H'.windows 0 (by omega)).2.2.1; change 2*T 0 ≤ P 0 at hh; omega
+    exact (show (T 0:ℝ) ≤ P 0 by exact_mod_cast hp).trans
+      (H'.copies_le_target hN.1 (by omega) ((scaleA_mono_index 2 1 2 ε he.le (by omega)).trans (by rw [scaleA_top])))
+  have hw : w^2 ≤ W*N^(-e) := by
+    rw [source_weight_square hN0 (show L=N^(-scaleA 2 0 ε*totalScoreExponent B i/2) from rfl)]
+    have hh := mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (Nat.cast_nonneg k)
+      (sourceHalfMoment_bounds B D (i+1) hB (by linarith only [hn])).2 2) (by norm_num : (0:ℝ) ≤ 144)
+    simpa only [W,sourceWeightEnvelope,sourceMomentEnvelope,e,neg_mul] using
+      mul_le_mul_of_nonneg_right hh (Real.rpow_pos_of_pos hN0 (-e)).le
+  have henergy := mul_le_mul_of_nonneg_left
+    (sourceSharpEnergyBudget_warmup (fun _=>T 0) (fun _=>t 0) s ω a w hs1) (by norm_num : (0:ℝ) ≤ 64)
+  have hpref := source_overlap_prefactors hK hW (one_le_pow₀ (by norm_num : (1:ℝ) ≤ 2)) hB (j:=0) (by omega) H.overlap
+  have hexp := (source_level_exponent_margins B D i hB (Nat.le_of_lt hi)).1 hiw
+  rw [hb] at hexp
+  have hM : 1 ≤ sourceWorkSize B D := by
+    unfold sourceWorkSize
+    linarith only [(totalScoreExponent_bounds B D hB).1,Nat.cast_nonneg (sourceDegreeCutoff B) (α:=ℝ),Nat.cast_nonneg B (α:=ℝ)]
+  have hv := source_warmup_variance_budget hN.1 hK hW (show 0 ≤ (2:ℝ)^s by positivity)
+    (Nat.cast_nonneg (T 0)) (Nat.cast_nonneg (h 2+1)) hcount hwindow hw henergy hpref.2.1 hpref.2.2
+    (by change ε+2-e-40*sourceWorkSize B D ≤ 0; have hloe : 27/50 ≤ e := hexp.2.1; linarith only [he1,hloe,hM])
+    hexp.2.2.2
+  refine ⟨hs1,hc.1,?_⟩
+  have hh := overlap_sqrt_budget hv herr
+  simpa only [N,ε,s,C,F,K,W,T,t,P,h,k,e,L,a,ω,w,Nat.cast_pow,Nat.cast_ofNat,mul_assoc] using hh
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem source_level_overlap {B D i : ℕ} {n q : ℝ} {hq : q∈Set.Icc 0 1}
+    (U : SourceLevelPair B D n q hq i) (hB : 2 ≤ B) (hi : i<D) (hn : 4 ≤ n)
+    (hq' : q∈Set.Ioc 0 (1/2:ℝ))
+    (H : SourceNumericMargins B D q (sourceTargetScale B D n (i+1)))
+    (htail : 2*(B:ℝ)*Real.exp
+      (-(sourceTargetScale B D n (i+1))^(sourceEpsilon B D)/sourceTailDenominator B D q) ≤
+        n^(-sourceTailExponent B D)) :
+    let N := sourceTargetScale B D n (i+1)
+    let b := sourceDimension B i
+    let ε := sourceEpsilon B D
+    let s := sourceCancellationOrder B D
+    let P := sourceArithmeticP (sourceOverlapConstant q) N b ε U.left.length U.overlapCount
+    let h := sourceArithmeticCount (sourceOverlapConstant q) N b ε U.left.length U.overlapCount
+    let atom := fun (τ : Bool)=>if τ then U.right else U.left
+    ∀j ≤ b, initialOverlapLower q ≤ listTraceOverlap q hq
+      (pureAtomRun (sourceBlockWord s P atom j false) (h j))
+      (pureAtomRun (sourceBlockWord s P atom j false) (h j+1)) := by
+  let N := sourceTargetScale B D n (i+1)
+  let b := sourceDimension B i
+  let ε := sourceEpsilon B D
+  let s := sourceCancellationOrder B D
+  let P := sourceArithmeticP (sourceOverlapConstant q) N b ε U.left.length U.overlapCount
+  let h := sourceArithmeticCount (sourceOverlapConstant q) N b ε U.left.length U.overlapCount
+  let T := sourceBlockWindows N b ε
+  let t := sourceArithmeticTrials (sourceOverlapConstant q) N b ε U.left.length U.overlapCount
+  let atom := fun (τ : Bool)=>if τ then U.right else U.left
+  let k := sourceHalfMoment B D n (i+1)
+  let L := N^(-scaleA b 0 ε*totalScoreExponent B i/2)
+  let a := N^(-20*sourceWorkSize B D)
+  have hn1 : 1 ≤ n := by linarith only [hn]
+  have hn2 : 1 < n := by linarith only [hn]
+  have hN : 0<N := (by norm_num : (0:ℝ)<1).trans_le (sourceTargetScale_bounds B D (i+1) hB hn1).1
+  have H' := source_level_geometry U hB hi hn1 hq' H
+  have hb := sourceDimension_bounds B i hB
+  have hk : 1 ≤ k := (sourceHalfMoment_bounds B D (i+1) hB hn2).1
+  let O := U.oracle.restrict hk (sourceHalfMoment_step B D i hB hi hn2)
+  have hC := sourceOverlapConstant_large hq'
+  have hp : initialOverlapLower q∈Set.Ioc 0 (1/8:ℝ) :=
+    ⟨initialOverlapLower_pos q,(initialOverlapLower_lt_eighth hq'.1).le⟩
+  have hc (j : ℕ) (hj : j ≤ b) : 1 ≤ h j :=
+    (H'.length_positive hN hj).trans (H'.counts j hj).1
+  have hP (j : ℕ) (hj : j<b) : 1 ≤ P j := H'.copies_positive hN hj
+  have hlift (j : ℕ) (_hj : j<b) : sourceOverlapConstant q*h j*P j ≤ h (j+1) :=
+    sourceArithmetic_count_lift _ _ _ _ _ _ _
+  have hbase : initialOverlapLower q ≤ listTraceOverlap q hq
+      (pureAtomRun (atom false) (h 0)) (pureAtomRun (atom false) (h 0+1)) := U.power_overlap
+  by_cases hiw : i<warmupBlocks B
+  · have hb2 : sourceDimension B i=2 := ite_eq_left hiw
+    have hb' : b=2 := hb2
+    have hw := source_level_warmup_second U hB hi hn hq' hiw H htail
+    have hnum := source_level_all_budgets U hB hi hn hq' H htail
+    have hexp := (source_level_exponent_margins B D i hB (Nat.le_of_lt hi)).1 hiw
+    have hfirst := source_level_dense_overlap U hB hi hn hq' H (j:=1) (by omega) hexp.2.2.1
+    have hfirst' : Real.sqrt (((h 1+1:ℕ):ℝ)*((2^s:ℕ):ℝ)*(sourceTargetScale B D n i)^(-totalScoreExponent B i))+
+        2*((h 1+1:ℕ):ℝ)*((P 0:ℝ)*U.error) ≤ 1/4 := by
+      simpa only [pow_one,sourceBlockLength,mul_one,mul_assoc] using hfirst
+    have hev : Even (T 0) := by
+      change Even (sourceBlockWindows N b ε 0)
+      rw [sourceBlockWindows,ite_eq_left (by omega : 0<b)]
+      exact sourceWindowSize_even _ _ _ _
+    have hfit : t 0*(2*h 0+1)+T 0 ≤ P 0 := by
+      dsimp only [t]
+      rw [sourceArithmetic_inner_trials (sourceOverlapConstant q) N b ε U.left.length U.overlapCount 0 (by omega : 0+1≠b)]
+      have hh := sourceLocalTrials_budget (P 0) (h 0)
+      have ht := (H'.windows 0 (by omega)).2.2.1
+      change 2*T 0 ≤ P 0 at ht
+      rw [mul_assoc] at hh
+      change sourceLocalTrials (P 0) (h 0)*(2*h 0+1)+T 0 ≤ P 0
+      omega
+    have hs : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount (fun _=>T 0) s 0 ≤ 1/2 := by
+      simpa only [L,N,b,ε,k,s,T,t,a,hb2,sourceSparseCount,Finset.range_zero,Finset.prod_empty,Nat.mul_one,Nat.cast_pow,Nat.cast_ofNat] using hw.1
+    have hclip : sourceClippingCost (fun _=>T 0) (fun _=>t 0) s 0 k (initialOverlapLower q) a L ≤ 1/2 := by
+      simpa only [L,N,b,ε,k,s,T,t,a,P,h,hb2] using hw.2.1
+    have hv : (sourceTargetScale B D n i)^(-totalScoreExponent B i) ≤ L^2 := by
+      rw [source_old_variance_square B D i (by linarith only [hn]) hi]
+    have hh := source_warmup_nested_overlap hq U.error_nonneg s k P h atom O hp hC
+      (fun j hj=>hc j (by omega)) (fun j hj=>hP j (by omega)) (fun j hj=>hlift j (by omega))
+      hfirst' hbase (T 0) (t 0) (H'.windows 0 (by omega)).1 hev (H'.windows 0 (by omega)).2.1
+      (H'.windows 0 (by omega)).2.2.2 hfit a a L (by positivity) (by positivity) hk hv hs
+      (hnum.2.2.2.1 0 (by omega)) hclip (by simpa only [L,N,b,ε,k,s,T,t,a,P,h,hb2] using hw.2.2)
+    change ∀j : ℕ,j≤b → initialOverlapLower q ≤ listTraceOverlap q hq
+      (pureAtomRun (sourceBlockWord s P atom j false) (h j))
+      (pureAtomRun (sourceBlockWord s P atom j false) (h j+1))
+    intro j hj
+    exact hh j (by change j≤b at hj; omega)
+  · apply source_nested_overlap hq U.error_nonneg s P h atom (O.hellinger hk) b hp hC hc hP hlift _ hbase
+    intro j hj
+    exact source_level_dense_overlap U hB hi hn hq' H (by omega)
+      ((source_level_exponent_margins B D i hB (Nat.le_of_lt hi)).2 (by omega) (j+1) (by omega))
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- The exact manuscript block is now a producer: no new level invariant or
+ overlap is assumed. All hypotheses besides the old actual pair are literal
+ scalar schedule margins. -/
+theorem source_level_step {B D i : ℕ} {n q : ℝ} {hq : q∈Set.Icc 0 1}
+    (U : SourceLevelPair B D n q hq i) (hB : 2 ≤ B) (hi : i<D) (hn : 4 ≤ n)
+    (hq' : q∈Set.Ioc 0 (1/2:ℝ))
+    (H : SourceNumericMargins B D q (sourceTargetScale B D n (i+1)))
+    (htail : 2*(B:ℝ)*Real.exp
+      (-(sourceTargetScale B D n (i+1))^(sourceEpsilon B D)/sourceTailDenominator B D q) ≤
+        n^(-sourceTailExponent B D)) :
+    Nonempty (SourceLevelPair B D n q hq (i+1)) := by
+  let N := sourceTargetScale B D n (i+1)
+  let b := sourceDimension B i
+  let ε := sourceEpsilon B D
+  let s := sourceCancellationOrder B D
+  let C := sourceOverlapConstant q
+  let P := sourceArithmeticP C N b ε U.left.length U.overlapCount
+  let h := sourceArithmeticCount C N b ε U.left.length U.overlapCount
+  let T := sourceBlockWindows N b ε
+  let t := sourceArithmeticTrials C N b ε U.left.length U.overlapCount
+  let atom := fun (τ : Bool)=>if τ then U.right else U.left
+  let k := sourceHalfMoment B D n (i+1)
+  let L := N^(-scaleA b 0 ε*totalScoreExponent B i/2)
+  let a := N^(-20*sourceWorkSize B D)
+  let ω := initialOverlapLower q
+  let Δ := n^(-100*(totalScoreExponent B D+B+1))
+  have hn1 : 1 ≤ n := by linarith only [hn]
+  have hn2 : 1 < n := by linarith only [hn]
+  have hn0 : 0<n := by linarith only [hn]
+  have hN : 0<N := (by norm_num : (0:ℝ)<1).trans_le (sourceTargetScale_bounds B D (i+1) hB hn1).1
+  have hC : 1 ≤ C := (by norm_num : (1:ℝ)≤2).trans (sourceOverlapConstant_ge_two hq')
+  have H' := source_level_geometry U hB hi hn1 hq' H
+  have hb := sourceDimension_bounds B i hB
+  have hb1 : b-1+1=b := by dsimp [b]; omega
+  have hk : 1 ≤ k := (sourceHalfMoment_bounds B D (i+1) hB hn2).1
+  let O := U.oracle.restrict hk (sourceHalfMoment_step B D i hB hi hn2)
+  let := O.finiteOutcome
+  have hnum := source_level_all_budgets U hB hi hn hq' H htail
+  have hov := source_level_overlap U hB hi hn hq' H htail
+  have hv : (sourceTargetScale B D n i)^(-totalScoreExponent B i) ≤ L^2 := by
+    rw [source_old_variance_square B D i hn0 hi]
+  have hm : O.first.expect (fun z=>|O.score z|^(2*k)) ≤ L^(2*k) := by
+    have hh := O.first.expect_even_pow_le_of_moment O.score k hk (O.moment.trans hv)
+    simpa only [pow_mul] using hh
+  have he (j : ℕ) (hj : j ≤ b-1) : Even (T j) := by
+    change Even (sourceBlockWindows N b ε j)
+    rw [sourceBlockWindows,ite_eq_left (by omega : j<b)]
+    exact sourceWindowSize_even _ _ _ _
+  obtain ⟨W⟩ := actual_source_oracle hq U.error_nonneg s P T t h atom (b-1) (sourceDegreeCutoff B) k
+    H'.layout H'.window_positive (by
+      simpa only [P,t,h,T,sourceBlockWindows,ite_eq_left (show b-1<b by omega)] using
+        sourceArithmetic_root_copies C N b ε U.left.length U.overlapCount (by omega))
+    (fun j hj=>H'.trials_positive hN (by linarith only [hC]) (by positivity) (by omega))
+    (fun j hj=>(H'.windows j (by omega)).1) he (fun j hj=>(H'.windows j (by omega)).2.1)
+    (H'.windows (b-1) (by omega)).2.2.2 O.first O.second O.decode O.score O.density O.likelihood_nonneg
+    O.first_simulation O.second_simulation ω a a L (by positivity) (by positivity)
+    ⟨initialOverlapLower_pos q,lt_of_lt_of_le (initialOverlapLower_lt_eighth hq'.1) (by norm_num)⟩
+    (fun j hj=>hov j (by omega)) hk hm hnum.2.1
+    (fun j hj=>hnum.2.2.2.1 j (by omega)) hnum.2.2.2.2.1
+  rw [hb1] at W
+  have hlen (τ : Bool) : (sourceBlockWord s P atom b τ).length=sourceArithmeticLength C N b ε U.left.length U.overlapCount b := by
+    rw [sourceArithmeticLength_eq_block]
+    apply sourceBlockWord_length
+    intro σ
+    cases σ
+    · rfl
+    · exact U.equalLength.symm
+  have hinj : Function.Injective atom := by
+    intro τ σ hτσ
+    cases τ <;> cases σ
+    · rfl
+    · exact False.elim (U.different hτσ)
+    · exact False.elim (U.different hτσ.symm)
+    · rfl
+  have hne : sourceBlockWord s P atom b false≠sourceBlockWord s P atom b true := by
+    intro hh
+    have hh' := sourceBlockWord_injective s P atom U.left.length (by
+      intro τ; cases τ; rfl; exact U.equalLength.symm) hinj b
+      (fun j hj=>Nat.lt_of_lt_of_le (by omega : 0<1) (H'.copies_positive hN hj)) hh
+    contradiction
+  have hlength := H'.lengths b le_rfl
+  change N^(scaleA b b ε)/2 ≤ (sourceArithmeticLength C N b ε U.left.length U.overlapCount b:ℝ) ∧
+    (sourceArithmeticLength C N b ε U.left.length U.overlapCount b:ℝ) ≤ N^(scaleA b b ε) at hlength
+  simp only [scaleA_top,Real.rpow_one] at hlength
+  have hE : U.error ≤ ((i:ℝ)+1)*Δ*U.left.length := by
+    apply (div_le_iff₀ ?_).mp U.normalized_error
+    exact (show 0<sourceTargetScale B D n i/2 by positivity [Real.rpow_pos_of_pos hn0 (sourceScaleProduct B D i)]).trans_le U.length_lower
+  have hleaf : SourceLeafCount T b*U.left.length ≤ sourceArithmeticLength C N b ε U.left.length U.overlapCount b := by
+    rw [sourceArithmeticLength_eq_block,sourceBlockLength_factor]
+    rw [Nat.mul_comm]
+    apply Nat.mul_le_mul_left
+    apply sourceLeafCount_le_length (by omega : 1≤1)
+    intro j hj
+    have hh := (H'.windows j hj).2.2.1
+    change 2*T j ≤ P j at hh
+    change T j ≤ P j
+    omega
+  have herror := normalized_block_error (H'.length_positive hN le_rfl) hE
+    (show 0≤Δ by positivity) hleaf hnum.2.2.2.2.2
+  have hfac : C^(2*b)*C^(2*B*i+2) ≤ C^(2*B*(i+1)+2) := by
+    rw [←pow_add]
+    apply pow_le_pow_right₀ hC
+    dsimp [b]
+    nlinarith only [hb.2]
+  refine ⟨W.toConcrete hq ((hlen false).trans (hlen true).symm) hne ?_ ?_ ?_ hnum.2.2.1 ?_ ?_ (hov b le_rfl)⟩
+  · rw [hlen]; exact hlength.1
+  · rw [hlen]; exact hlength.2
+  · rw [hlen]
+    simpa only [Δ,Nat.cast_add,Nat.cast_one,add_assoc,show (1:ℝ)+1=2 by norm_num] using herror
+  · rw [hlen]; exact (H'.counts b le_rfl).1
+  · rw [hlen]
+    exact (H'.counts b le_rfl).2.trans (mul_le_mul_of_nonneg_right hfac (Nat.cast_nonneg _))
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open Filter
+open scoped Topology Classical
+
+/-- Full source construction at every scheduled level, uniformly in the
+ joint deletion regime. No level witnesses are assumed. -/
+theorem source_all_levels_eventually (B : ℕ) (ζ : ℝ) (hB : 2 ≤ B)
+    (hζ : ζ∈Set.Ioo 0 1) {n q : ℕ→ℝ} (hn : Tendsto n atTop atTop)
+    (hq : ∀t,q t∈Set.Ioc 0 (1/2:ℝ))
+    (hH : Tendsto (fun t=>(q t)^3*Real.log (n t)) atTop atTop) :
+    ∀ᶠt in atTop,∀i ≤ sourceDepth B ζ ((q t)^3*Real.log (n t)),
+      Nonempty (SourceLevelPair B (sourceDepth B ζ ((q t)^3*Real.log (n t)))
+        (n t) (q t) ⟨(hq t).1.le,by linarith [(hq t).2]⟩ i) := by
+  filter_upwards [source_initial_at_smallest_scale B ζ hB hζ hn hq hH,
+    source_numeric_margins_eventually B ζ hB hζ hn hq hH,
+    source_tails_eventually B ζ hB hζ hn hq hH,hn.eventually_ge_atTop 4] with t h0 hm ht hn4
+  intro i hi
+  induction i with
+  | zero => simpa only [SourceLevelPair,totalScoreExponent,Finset.range_zero,
+      Finset.sum_empty,add_zero,Nat.mul_zero,Nat.zero_add] using h0
+  | succ i ih =>
+    obtain ⟨U⟩ := ih (by omega)
+    exact source_level_step U hB (by omega) hn4 (hq t) (hm (i+1)) (ht (i+1))
+
+/-- The final source pair has actual length in [n/2,n], an exact common
+ complete-trace simulator, prescribed high moment and inductively obtained
+ power overlap. This is the construction theorem, before statistical transfer. -/
+theorem source_final_pair_eventually (B : ℕ) (ζ : ℝ) (hB : 2 ≤ B)
+    (hζ : ζ∈Set.Ioo 0 1) {n q : ℕ→ℝ} (hn : Tendsto n atTop atTop)
+    (hq : ∀t,q t∈Set.Ioc 0 (1/2:ℝ))
+    (hH : Tendsto (fun t=>(q t)^3*Real.log (n t)) atTop atTop) :
+    ∀ᶠt in atTop,Nonempty (ConcreteLevelPair (q t) ⟨(hq t).1.le,by linarith [(hq t).2]⟩ (n t)
+      ((n t)^(-100*(totalScoreExponent B (sourceDepth B ζ ((q t)^3*Real.log (n t)))+B+1)))
+      (totalScoreExponent B (sourceDepth B ζ ((q t)^3*Real.log (n t))))
+      (initialOverlapLower (q t))
+      ((sourceOverlapConstant (q t))^(2*B*sourceDepth B ζ ((q t)^3*Real.log (n t))+2))
+      (sourceDepth B ζ ((q t)^3*Real.log (n t)))
+      (sourceHalfMoment B (sourceDepth B ζ ((q t)^3*Real.log (n t))) (n t)
+        (sourceDepth B ζ ((q t)^3*Real.log (n t))))) := by
+  filter_upwards [source_all_levels_eventually B ζ hB hζ hn hq hH] with t ht
+  simpa only [SourceLevelPair,sourceTargetScale_top] using ht _ le_rfl
+
+end TraceReconstruction
+
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/TraceReconstruction/InitialDensity.lean b/lean/OAI/Probability/TraceReconstruction/InitialDensity.lean
index e4a00fd9bf1cff04b1171f9f860349a8c6033695..d539477dc41182ecab2f550d1cf9000f2f7d091f 100644
--- a/lean/OAI/Probability/TraceReconstruction/InitialDensity.lean
+++ b/lean/OAI/Probability/TraceReconstruction/InitialDensity.lean
@@ -1,527 +1,527 @@
-import OAI.Probability.TraceReconstruction.Ranks
-
-namespace OAI
-
-/-!
-# InitialDensity
-
-Defect monomials, orthogonality, normalized initial densities, and common mass.
--/
-
-open scoped ENNReal
-noncomputable section
-
-open Filter
-open scoped Topology
-
-noncomputable section
-namespace TraceReconstruction
-open FiniteLaw
-
-/-- A monotone tuple of defect slots is uniquely identified by the
-retained ranks whenever all these ranks carry defects. -/
-theorem exists_unique_defect_tuple {n l u r : ℕ}
-    (d : Fin u → Fin n) (hd : StrictMono d) (v : IncreasingTuple l r)
-    (S : FixedSubset (Fin n) l)
-    (h : ∀ i, slotAtRank S (v.val i) ∈ Set.range d) :
-    ∃! j : IncreasingTuple u r, taggedRankEvent (d ∘ j.val) v.val S := by
-  classical
-  choose j hj using h
-  have hm : StrictMono j := by
-    intro a b hab
-    apply hd.lt_iff_lt.mp
-    rw [hj a,hj b]
-    exact slotAtRank_strictMono S (v.property hab)
-  refine ⟨⟨j,hm⟩,?_,?_⟩
-  · intro i
-    exact (hj i).symm
-  · intro k hk
-    apply Subtype.ext
-    funext i
-    apply hd.injective
-    exact (hk i).symm.trans (hj i).symm
-
-/-- The degree coefficient is the sum over actual tagged defect tuples,
-not a separately postulated kernel array. -/
-theorem defect_tuple_indicator {n l u r : ℕ}
-    (d : Fin u → Fin n) (hd : StrictMono d) (v : IncreasingTuple l r)
-    (S : FixedSubset (Fin n) l) :
-    (if ∀ i, slotAtRank S (v.val i) ∈ Set.range d then (1:ℝ) else 0) =
-      ∑ j : IncreasingTuple u r, if taggedRankEvent (d ∘ j.val) v.val S then 1 else 0 := by
-  classical
-  by_cases h : ∀ i, slotAtRank S (v.val i) ∈ Set.range d
-  · obtain ⟨j,hj,hu⟩ := exists_unique_defect_tuple d hd v S h
-    rw [ite_eq_left h]
-    symm
-    rw [Finset.sum_eq_single j]
-    · exact ite_eq_left hj
-    · intro k _ hkj
-      have hk : ¬taggedRankEvent (d ∘ k.val) v.val S := by
-        intro hk
-        exact hkj (hu k hk)
-      exact ite_eq_right hk
-    · simp
-  · rw [ite_eq_right h]
-    symm
-    apply Finset.sum_eq_zero
-    intro j _
-    apply ite_eq_right
-    intro hj
-    apply h
-    intro i
-    exact ⟨j.val i,(hj i).symm⟩
-
-def rankDefectCoefficient {n l u r : ℕ} (hl : l ≤ n) (d : Fin u → Fin n)
-    (v : IncreasingTuple l r) : ℝ := by
-  classical
-  exact (uniformSubsetLaw l (by simpa using hl)).expect (fun S =>
-    if ∀ i, slotAtRank S (v.val i) ∈ Set.range d then 1 else 0)
-
-theorem rankDefectCoefficient_eq {n l u r : ℕ} (hl : l ≤ n)
-    (d : Fin u → Fin n) (hd : StrictMono d) (v : IncreasingTuple l r) :
-    rankDefectCoefficient hl d v =
-      ∑ j : IncreasingTuple u r, subsetRankKernel hl (d ∘ j.val) v.val := by
-  classical
-  unfold rankDefectCoefficient FiniteLaw.expect subsetRankKernel
-  simp_rw [defect_tuple_indicator d hd v, Finset.mul_sum]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro j _
-  apply Finset.sum_congr rfl
-  intro S _
-  split_ifs <;> simp
-
-/-- Bool-mask form of the selected-rank coefficient, in the exact
-multilinear density expansion. -/
-theorem selected_defect_coefficient {n l : ℕ} (D : Finset (Fin n))
-    (S : FixedSubset (Fin n) l) (σ : Fin l → Bool) :
-    selectionCoefficient (defectAtRank D S) σ =
-      if ∀ i, σ i = true → slotAtRank S i ∈ D then (1:ℝ) else 0 := by
-  classical
-  unfold selectionCoefficient defectAtRank
-  by_cases h : ∀ i, σ i = true → slotAtRank S i ∈ D
-  · rw [ite_eq_left h]
-    apply Finset.prod_eq_one
-    intro i _
-    cases hs : σ i
-    · simp
-    · simp [h i hs]
-  · rw [ite_eq_right h]
-    push Not at h
-    obtain ⟨i,hi,hni⟩ := h
-    apply Finset.prod_eq_zero (Finset.mem_univ i)
-    simp [hi,hni]
-
-def rankMask {l r : ℕ} (v : IncreasingTuple l r) : Fin l → Bool :=
-  fun i => decide (i ∈ Set.range v.val)
-
-theorem averaged_rank_coefficient {n l u r : ℕ} (hl : l ≤ n)
-    (d : Fin u → Fin n) (v : IncreasingTuple l r) :
-    averagedCoefficient (uniformSubsetLaw l (by simpa using hl))
-      (defectAtRank (Finset.univ.image d)) (rankMask v) =
-      rankDefectCoefficient hl d v := by
-  classical
-  unfold averagedCoefficient rankDefectCoefficient FiniteLaw.expect
-  apply Finset.sum_congr rfl
-  intro S _
-  congr 1
-  dsimp only
-  rw [selected_defect_coefficient]
-  congr 1
-  simp only [rankMask,decide_eq_true_eq,Finset.mem_image,Finset.mem_univ,true_and,
-    Set.mem_range]
-  apply propext
-  constructor
-  · intro h i
-    exact h (v.val i) ⟨i,rfl⟩
-  · rintro h i ⟨j,rfl⟩
-    exact h j
-
-end TraceReconstruction
-
-namespace TraceReconstruction
-open FiniteLaw
-
-/-- Canonical increasing tuple associated with a selected set of ranks. -/
-def tupleOfMask {l r : ℕ} (σ : Fin l → Bool) (hσ : monomialDegree σ = r) :
-    IncreasingTuple l r := by
-  classical
-  let S : Finset (Fin l) := Finset.univ.filter (fun i => σ i = true)
-  have hS : S.card = r := hσ
-  exact ⟨fun i => ((S.orderIsoOfFin hS) i).val, (S.orderIsoOfFin hS).strictMono⟩
-
-theorem rankMask_tupleOfMask {l r : ℕ} (σ : Fin l → Bool) (hσ : monomialDegree σ = r) :
-    rankMask (tupleOfMask σ hσ) = σ := by
-  classical
-  funext i
-  have he : i ∈ Set.range (tupleOfMask σ hσ).val ↔ σ i = true := by
-    constructor
-    · rintro ⟨j,rfl⟩
-      exact (Finset.mem_filter.mp (((Finset.univ.filter (fun k => σ k = true)).orderIsoOfFin hσ) j).property).2
-    · intro hi
-      let z : ↥(Finset.univ.filter (fun k => σ k = true)) := ⟨i,by simp [hi]⟩
-      refine ⟨((Finset.univ.filter (fun k => σ k = true)).orderIsoOfFin hσ).symm z,?_⟩
-      exact congrArg Subtype.val (((Finset.univ.filter (fun k => σ k = true)).orderIsoOfFin hσ).apply_symm_apply z)
-  simp only [rankMask,he]
-  exact Bool.decide_eq_true
-
-theorem tupleOfMask_injective {l r : ℕ} : Function.Injective
-    (fun σ : {σ : Fin l → Bool // monomialDegree σ = r} => tupleOfMask σ.val σ.property) := by
-  intro σ τ h
-  apply Subtype.ext
-  have he := congrArg rankMask h
-  simpa only [rankMask_tupleOfMask] using he
-
-/-- Changing from Boolean monomial labels to ordered rank tuples loses no
-mass; an injection suffices for the upper bounds. -/
-theorem sum_degree_coeff_sq_le {l r : ℕ} (A : (Fin l → Bool) → ℝ) :
-    (∑ σ : Fin l → Bool, if monomialDegree σ = r then A σ ^ 2 else 0) ≤
-      ∑ v : IncreasingTuple l r, A (rankMask v)^2 := by
-  classical
-  rw [← Finset.sum_filter]
-  have he : (∑ σ ∈ Finset.univ.filter (fun σ : Fin l → Bool => monomialDegree σ = r), A σ ^ 2) =
-      ∑ σ : {σ : Fin l → Bool // monomialDegree σ = r}, A σ.val ^ 2 := by
-    exact Finset.sum_subtype _ (by intro σ; simp) _
-  rw [he]
-  have h := sum_injective_profile_le
-    (fun σ : {σ : Fin l → Bool // monomialDegree σ = r} => tupleOfMask σ.val σ.property)
-    tupleOfMask_injective (fun _ => True) (fun _ => True.intro)
-    (fun v => A (rankMask v)^2) (fun _ => sq_nonneg _)
-  simpa only [rankMask_tupleOfMask,ite_true] using h
-
-end TraceReconstruction
-
-noncomputable section
-namespace TraceReconstruction
-open FiniteLaw
-
-/-- Exact orthogonality of the multilinear expansion; this avoids any
-loss in the channel power in consecutive-run overlap. -/
-theorem multilinear_second_moment {α : Type*} [Fintype α] (P : FiniteLaw α)
-    (Z : α → ℝ) (hcenter : P.expect Z = 0) (R : ℝ)
-    (hvar : P.expect (fun a => Z a^2) = R^2) (n : ℕ)
-    (A : (Fin n → Bool) → ℝ) :
-    (P.product n).expect (fun x => multilinear A Z x ^ 2) =
-      multilinearEnergy A R := by
-  induction n with
-  | zero => simp [FiniteLaw.expect,FiniteLaw.product,multilinear,multilinearEnergy]
-  | succ n ih =>
-    rw [product_succ_expect_tail]
-    simp_rw [multilinear_cons]
-    have he (x : Fin n → α) :
-        P.expect (fun a => (multilinear (fun σ => A (Fin.cons false σ)) Z x +
-            Z a * multilinear (fun σ => A (Fin.cons true σ)) Z x)^2) =
-          multilinear (fun σ => A (Fin.cons false σ)) Z x ^ 2 +
-            R^2 * multilinear (fun σ => A (Fin.cons true σ)) Z x ^ 2 := by
-      have hp (a : α) :
-          P.mass a * (multilinear (fun σ => A (Fin.cons false σ)) Z x +
-            Z a * multilinear (fun σ => A (Fin.cons true σ)) Z x)^2 =
-          P.mass a * (multilinear (fun σ => A (Fin.cons false σ)) Z x)^2 +
-            2 * multilinear (fun σ => A (Fin.cons false σ)) Z x *
-              multilinear (fun σ => A (Fin.cons true σ)) Z x * (P.mass a * Z a) +
-            (multilinear (fun σ => A (Fin.cons true σ)) Z x)^2 * (P.mass a * Z a^2) := by ring
-      unfold FiniteLaw.expect
-      simp_rw [hp,Finset.sum_add_distrib,← Finset.sum_mul,← Finset.mul_sum]
-      simp only [FiniteLaw.expect] at hcenter hvar
-      rw [P.total,hcenter,hvar]
-      ring
-    simp_rw [he]
-    rw [multilinearEnergy_cons]
-    have hadd (f g : (Fin n → α) → ℝ) :
-        (P.product n).expect (fun x => f x + R^2*g x) =
-          (P.product n).expect f + R^2*(P.product n).expect g := by
-      unfold FiniteLaw.expect
-      simp_rw [mul_add,Finset.sum_add_distrib]
-      rw [Finset.mul_sum]
-      congr 1
-      apply Finset.sum_congr rfl
-      intro x _
-      ring
-    rw [hadd,ih,ih]
-
-/-- Association requires only a total preorder, allowing tied likelihoods. -/
-theorem FiniteLaw.associated_totalPreorder {α : Type*} [Fintype α] [Preorder α]
-    (P : FiniteLaw α) (htotal : ∀ a b : α, a ≤ b ∨ b ≤ a) : P.Associated := by
-  intro f g hf hg
-  have hn (a b : α) : 0 ≤ (f a - f b) * (g a - g b) := by
-    rcases htotal a b with hab | hba
-    · exact mul_nonneg_of_nonpos_of_nonpos (sub_nonpos.mpr (hf hab))
-        (sub_nonpos.mpr (hg hab))
-    · exact mul_nonneg (sub_nonneg.mpr (hf hba)) (sub_nonneg.mpr (hg hba))
-  have h := Finset.sum_nonneg (s := Finset.univ) (fun a _ =>
-    Finset.sum_nonneg (s := Finset.univ) (fun b _ => mul_nonneg (mul_nonneg (P.nonneg a)
-      (P.nonneg b)) (hn a b)))
-  have hid : (∑ a, ∑ b, P.mass a * P.mass b * ((f a - f b) * (g a - g b))) =
-      2 * (P.expect (fun a => f a * g a) - P.expect f * P.expect g) := by
-    have hp (a b : α) : P.mass a * P.mass b * ((f a - f b) * (g a - g b)) =
-        (P.mass a * (f a * g a)) * P.mass b + P.mass a * (P.mass b * (f b * g b)) -
-        (P.mass a * f a) * (P.mass b * g b) - (P.mass a * g a) * (P.mass b * f b) := by ring
-    simp_rw [hp, Finset.sum_sub_distrib, Finset.sum_add_distrib,
-      ← Finset.mul_sum, ← Finset.sum_mul, P.total, one_mul, mul_one]
-    unfold FiniteLaw.expect
-    ring
-  rw [hid] at h
-  linarith
-
-/-- Conditional average densities are nonnegative when the coordinate
-likelihoods are nonnegative. -/
-theorem averagedDensity_nonneg {α Ω : Type*} [Fintype Ω] {n : ℕ}
-    (R : FiniteLaw Ω) (θ : Ω → Fin n → Bool) (Z : α → ℝ)
-    (hZ : ∀ a, 0 ≤ 1+Z a) (x : Fin n → α) :
-    0 ≤ averagedDensity R θ Z x := by
-  apply Finset.sum_nonneg
-  intro ω _
-  apply mul_nonneg (R.nonneg ω)
-  apply Finset.prod_nonneg
-  intro i _
-  split_ifs
-  · exact hZ _
-  · norm_num
-
-theorem averagedDensity_mono {α Ω : Type*} [Preorder α] [Fintype Ω] {n : ℕ}
-    (R : FiniteLaw Ω) (θ : Ω → Fin n → Bool) (Z : α → ℝ)
-    (hZ : ∀ a, 0 ≤ 1+Z a) (hZm : Monotone Z) :
-    Monotone (averagedDensity R θ Z) := by
-  intro x y hxy
-  apply R.expect_mono
-  intro ω
-  apply Finset.prod_le_prod₀
-  · intro i _
-    split_ifs
-    · exact hZ _
-    · norm_num
-  · intro i _
-    split_ifs
-    · linarith [hZm (hxy i)]
-    · rfl
-
-/-- Every independent-score likelihood product is normalized. -/
-theorem selectionDensity_mean {α : Type*} [Fintype α] (P : FiniteLaw α)
-    (Z : α → ℝ) (hZ : P.expect Z = 0) {n : ℕ} (θ : Fin n → Bool) :
-    (P.product n).expect (selectionDensity θ Z) = 1 := by
-  unfold FiniteLaw.expect FiniteLaw.product selectionDensity
-  simp_rw [← Finset.prod_mul_distrib]
-  rw [← Fintype.prod_sum (fun (i : Fin n) (a : α) => P.mass a * (if θ i then 1+Z a else 1))]
-  apply Finset.prod_eq_one
-  intro i _
-  cases θ i
-  · simp only [Bool.false_eq_true,↓reduceIte,mul_one,P.total]
-  · simp only [↓reduceIte,mul_add,mul_one,Finset.sum_add_distrib,P.total]
-    exact add_eq_left.mpr hZ
-
-theorem averagedDensity_mean {α Ω : Type*} [Fintype α] [Fintype Ω]
-    (P : FiniteLaw α) (R : FiniteLaw Ω) (Z : α → ℝ) (hZ : P.expect Z = 0)
-    {n : ℕ} (θ : Ω → Fin n → Bool) :
-    (P.product n).expect (averagedDensity R θ Z) = 1 := by
-  unfold averagedDensity FiniteLaw.expect
-  simp_rw [Finset.mul_sum]
-  rw [Finset.sum_comm]
-  have he (ω : Ω) : (∑ x : Fin n → α, (P.product n).mass x * (R.mass ω * selectionDensity (θ ω) Z x)) =
-      R.mass ω * (P.product n).expect (selectionDensity (θ ω) Z) := by
-    rw [FiniteLaw.expect,Finset.mul_sum]
-    apply Finset.sum_congr rfl
-    intro x _
-    ring
-  simp_rw [he,selectionDensity_mean P Z hZ,mul_one]
-  exact R.total
-
-end TraceReconstruction
-
-namespace TraceReconstruction
-open FiniteLaw
-
-/-- Exact decomposition of the coefficient energy by its degree. -/
-theorem multilinearEnergy_degrees {n : ℕ} (A : (Fin n → Bool) → ℝ) (R : ℝ) :
-    multilinearEnergy A R = ∑ r ∈ Finset.range (n+1),
-      R^(2*r) * ∑ σ : Fin n → Bool, if monomialDegree σ = r then A σ^2 else 0 := by
-  classical
-  unfold multilinearEnergy
-  simp_rw [monomial_weight]
-  simp_rw [Finset.mul_sum]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro σ _
-  have hd : monomialDegree σ < n+1 := by
-    have h := Finset.card_filter_le (s := Finset.univ) (p := fun i : Fin n => σ i = true)
-    simpa [monomialDegree] using Nat.lt_succ_of_le h
-  rw [Finset.sum_eq_single (monomialDegree σ)]
-  · simp [mul_comm]
-  · intro r _ hr
-    simp [Ne.symm hr]
-  · simp [Finset.mem_range,hd]
-
-/-- A zero-degree selection is precisely the empty selection. -/
-theorem monomialDegree_eq_zero_iff {n : ℕ} (σ : Fin n → Bool) :
-    monomialDegree σ = 0 ↔ σ = fun _ => false := by
-  classical
-  constructor
-  · intro h
-    have he : Finset.univ.filter (fun i => σ i = true) = ∅ := Finset.card_eq_zero.mp h
-    funext i
-    have hn : σ i ≠ true := by
-      intro hi
-      have : i ∈ Finset.univ.filter (fun j => σ j = true) := by simp [hi]
-      rw [he] at this
-      exact Finset.notMem_empty i this
-    exact Bool.eq_false_iff.mpr hn
-  · rintro rfl
-    simp [monomialDegree]
-
-theorem averaged_degree_zero {Ω : Type*} [Fintype Ω] {n : ℕ}
-    (R : FiniteLaw Ω) (θ : Ω → Fin n → Bool) :
-    (∑ σ : Fin n → Bool, if monomialDegree σ = 0 then averagedCoefficient R θ σ ^ 2 else 0) = 1 := by
-  classical
-  simp_rw [monomialDegree_eq_zero_iff]
-  simp [averagedCoefficient_empty]
-
-/-- There is no increasing selection with more entries than its ambient set. -/
-theorem increasingTuple_isEmpty {n r : ℕ} (h : n < r) : IsEmpty (IncreasingTuple n r) := by
-  refine ⟨fun v => ?_⟩
-  have hc := Fintype.card_le_of_injective v.val v.property.injective
-  simp only [Fintype.card_fin] at hc
-  omega
-
-/-- The finite law of uniformly retained slots has no coefficients beyond
-its number of actual defect slots. -/
-theorem defect_degree_bound {n l u r : ℕ} (hl : l ≤ n)
-    (d : Fin u → Fin n) (hd : StrictMono d) (hru : u < r) :
-    (∑ σ : Fin l → Bool, if monomialDegree σ = r then
-      averagedCoefficient (uniformSubsetLaw l (by simpa using hl))
-        (defectAtRank (Finset.univ.image d)) σ ^ 2 else 0) = 0 := by
-  classical
-  apply le_antisymm
-  · refine (sum_degree_coeff_sq_le (r:=r) (averagedCoefficient (uniformSubsetLaw l (by simpa using hl))
-        (defectAtRank (Finset.univ.image d)))).trans ?_
-    simp_rw [averaged_rank_coefficient hl d,rankDefectCoefficient_eq hl d hd]
-    have := increasingTuple_isEmpty hru
-    simp
-  · apply Finset.sum_nonneg
-    intro σ _
-    split_ifs <;> positivity
-
-/-- Coefficient version of the conditional-rank square-sum estimate. -/
-theorem period_degree_square_sum {u m l r : ℕ} (hl : l ≤ u*m) (hr : 0 < r)
-    (hru : r ≤ u) (hm : 32 ≤ m) (hum : u ≤ m) {δ θ : ℝ}
-    (hδ : δ ∈ Set.Ioc 0 (1/2:ℝ))
-    (hθ : θ ∈ Set.Icc (1-δ^2) (1-δ^2/4))
-    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
-    (off : Fin u → Fin m)
-    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
-    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2) :
-    (∑ σ : Fin l → Bool, if monomialDegree σ = r then
-      averagedCoefficient (uniformSubsetLaw l (by simpa using hl))
-        (defectAtRank (Finset.univ.image (periodSlot off))) σ ^ 2 else 0) ≤
-      ((3*rankSchurConstant/δ)*Real.sqrt ((u:ℝ)/((m:ℝ)*r)))^r := by
-  refine (sum_degree_coeff_sq_le (r:=r) (averagedCoefficient (uniformSubsetLaw l (by simpa using hl))
-    (defectAtRank (Finset.univ.image (periodSlot off))))).trans ?_
-  simp_rw [averaged_rank_coefficient hl (periodSlot off),
-    rankDefectCoefficient_eq hl (periodSlot off) (periodSlot_strictMono off)]
-  exact period_rank_square_sum_delta hl hr hru hm hum hδ hθ hmean hv off hoff hdiff
-
-/-- The full coefficient energy is controlled by the subgaussian degree
-series, uniformly in the number of retained coordinates. -/
-theorem period_density_energy {u m l : ℕ} (hl : l ≤ u*m)
-    (hm : 32 ≤ m) (hum : u ≤ m) {δ θ : ℝ}
-    (hδ : δ ∈ Set.Ioc 0 (1/2:ℝ))
-    (hθ : θ ∈ Set.Icc (1-δ^2) (1-δ^2/4))
-    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
-    (off : Fin u → Fin m)
-    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
-    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2) (R : ℝ) :
-    multilinearEnergy
-      (averagedCoefficient (uniformSubsetLaw l (by simpa using hl))
-        (defectAtRank (Finset.univ.image (periodSlot off)))) R ≤
-      ∑ r ∈ Finset.range (l+1),
-        (R^2*(3*rankSchurConstant/δ)*Real.sqrt ((u:ℝ)/m))^r / (Real.sqrt (r:ℝ))^r := by
-  rw [multilinearEnergy_degrees]
-  apply Finset.sum_le_sum
-  intro r _
-  by_cases hr : r = 0
-  · subst r
-    rw [averaged_degree_zero]
-    norm_num
-  · by_cases hru : r ≤ u
-    · have hr0 : 0 < r := Nat.pos_of_ne_zero hr
-      have hb := period_degree_square_sum hl hr0 hru hm hum hδ hθ hmean hv off hoff hdiff
-      calc
-        _ ≤ R^(2*r)*((3*rankSchurConstant/δ)*Real.sqrt ((u:ℝ)/((m:ℝ)*r)))^r :=
-          mul_le_mul_of_nonneg_left hb (by rw [pow_mul]; positivity)
-        _ = _ := by
-          have hm0 : (0:ℝ) < m := by exact_mod_cast (show 0 < m by omega)
-          have hr0 : (0:ℝ) < r := by exact_mod_cast hr0
-          rw [show (u:ℝ)/((m:ℝ)*r)=((u:ℝ)/m)/r by field_simp]
-          rw [Real.sqrt_div (by positivity),pow_mul,← mul_pow,← div_pow]
-          congr 1
-          ring
-    · rw [defect_degree_bound hl (periodSlot off) (periodSlot_strictMono off) (by omega),mul_zero]
-      have := hδ.1
-      positivity [rankSchurConstant_pos]
-
-/-- A convenient exponential majorant for the full nonnegative degree series. -/
-theorem degree_series_le_exp (a : ℝ) (N : ℕ) :
-    (∑ r ∈ Finset.range N, a^r / (Real.sqrt (r:ℝ))^r) ≤ 2*Real.exp (a^2) := by
-  have hb := degree_series_bound a N
-  have hp := Real.exp_pos (a^2)
-  have he : Real.exp (2*a^2) = (Real.exp (a^2))^2 := by
-    rw [show 2*a^2=a^2+a^2 by ring,Real.exp_add]
-    ring
-  rw [he] at hb
-  nlinarith [sq_nonneg ((∑ r ∈ Finset.range N, a^r / (Real.sqrt (r:ℝ))^r)+2*Real.exp (a^2))]
-
-/-- Actual conditional initial-oracle density, with its exact second moment. -/
-theorem period_density_second_moment {α : Type*} [Fintype α]
-    (P : FiniteLaw α) (Z : α → ℝ) (hZ : P.expect Z = 0)
-    {u m l : ℕ} (hl : l ≤ u*m)
-    (hm : 32 ≤ m) (hum : u ≤ m) {δ θ : ℝ}
-    (hδ : δ ∈ Set.Ioc 0 (1/2:ℝ))
-    (hθ : θ ∈ Set.Icc (1-δ^2) (1-δ^2/4))
-    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
-    (off : Fin u → Fin m)
-    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
-    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2) :
-    (P.product l).expect (fun x =>
-      averagedDensity (uniformSubsetLaw l (by simpa using hl))
-        (defectAtRank (Finset.univ.image (periodSlot off))) Z x ^ 2) ≤
-      2*Real.exp ((P.expect (fun a => Z a^2))^2 * (3*rankSchurConstant/δ)^2 * ((u:ℝ)/m)) := by
-  have hv0 : 0 ≤ P.expect (fun a => Z a^2) :=
-    Finset.sum_nonneg (fun a _ => mul_nonneg (P.nonneg a) (sq_nonneg _))
-  simp_rw [averagedDensity_expansion]
-  rw [multilinear_second_moment P Z hZ (Real.sqrt (P.expect (fun a => Z a^2)))
-    (Real.sq_sqrt hv0).symm]
-  refine (period_density_energy hl hm hum hδ hθ hmean hv off hoff hdiff _).trans ?_
-  apply (degree_series_le_exp _ (l+1)).trans
-  rw [Real.sq_sqrt hv0]
-  have he : (P.expect (fun a => Z a^2)*(3*rankSchurConstant/δ)*Real.sqrt ((u:ℝ)/m))^2 =
-      (P.expect (fun a => Z a^2))^2*(3*rankSchurConstant/δ)^2*((u:ℝ)/m) := by
-    rw [mul_pow,mul_pow,Real.sq_sqrt (by positivity)]
-  rw [he]
-
-/-- The source association argument holds for any two conditional mixture
-experiments using the same independent primitive marks, even when their
-retained-subset sample spaces have different original sizes. -/
-theorem averagedDensity_overlap {α Ω Ψ : Type*}
-    [Fintype α] [Fintype Ω] [Fintype Ψ]
-    (P : FiniteLaw α) (R : FiniteLaw Ω) (S : FiniteLaw Ψ)
-    (Z : α → ℝ) (hZ : P.expect Z = 0) (hZpos : ∀ a, 0 ≤ 1+Z a)
-    {l : ℕ} (f : Ω → Fin l → Bool) (g : Ψ → Fin l → Bool) :
-    (1:ℝ)/32 ≤
-      (P.product l).expect (fun x => min (averagedDensity R f Z x) (averagedDensity S g Z x)) *
-      (P.product l).expect (fun x => averagedDensity R f Z x ^ 2) *
-      (P.product l).expect (fun x => averagedDensity S g Z x ^ 2) := by
-  let : Preorder α := Preorder.lift Z
-  have htotal : ∀ a b : α, a ≤ b ∨ b ≤ a := fun a b => le_total (Z a) (Z b)
-  have hm : Monotone Z := fun _ _ h => h
-  exact increasing_density_overlap (P.product l)
-    (associated_product P (P.associated_totalPreorder htotal) l)
-    (averagedDensity R f Z) (averagedDensity S g Z)
-    (averagedDensity_nonneg R f Z hZpos) (averagedDensity_nonneg S g Z hZpos)
-    (averagedDensity_mono R f Z hZpos hm) (averagedDensity_mono S g Z hZpos hm)
-    (averagedDensity_mean P R Z hZ f) (averagedDensity_mean P S Z hZ g)
-
-end TraceReconstruction
-
-end
-end
-end
-
-end OAI
+import OAI.Probability.TraceReconstruction.Ranks
+
+namespace OAI
+
+/-!
+# InitialDensity
+
+Defect monomials, orthogonality, normalized initial densities, and common mass.
+-/
+
+open scoped ENNReal
+noncomputable section
+
+open Filter
+open scoped Topology
+
+noncomputable section
+namespace TraceReconstruction
+open FiniteLaw
+
+/-- A monotone tuple of defect slots is uniquely identified by the
+retained ranks whenever all these ranks carry defects. -/
+theorem exists_unique_defect_tuple {n l u r : ℕ}
+    (d : Fin u → Fin n) (hd : StrictMono d) (v : IncreasingTuple l r)
+    (S : FixedSubset (Fin n) l)
+    (h : ∀ i, slotAtRank S (v.val i) ∈ Set.range d) :
+    ∃! j : IncreasingTuple u r, taggedRankEvent (d ∘ j.val) v.val S := by
+  classical
+  choose j hj using h
+  have hm : StrictMono j := by
+    intro a b hab
+    apply hd.lt_iff_lt.mp
+    rw [hj a,hj b]
+    exact slotAtRank_strictMono S (v.property hab)
+  refine ⟨⟨j,hm⟩,?_,?_⟩
+  · intro i
+    exact (hj i).symm
+  · intro k hk
+    apply Subtype.ext
+    funext i
+    apply hd.injective
+    exact (hk i).symm.trans (hj i).symm
+
+/-- The degree coefficient is the sum over actual tagged defect tuples,
+not a separately postulated kernel array. -/
+theorem defect_tuple_indicator {n l u r : ℕ}
+    (d : Fin u → Fin n) (hd : StrictMono d) (v : IncreasingTuple l r)
+    (S : FixedSubset (Fin n) l) :
+    (if ∀ i, slotAtRank S (v.val i) ∈ Set.range d then (1:ℝ) else 0) =
+      ∑ j : IncreasingTuple u r, if taggedRankEvent (d ∘ j.val) v.val S then 1 else 0 := by
+  classical
+  by_cases h : ∀ i, slotAtRank S (v.val i) ∈ Set.range d
+  · obtain ⟨j,hj,hu⟩ := exists_unique_defect_tuple d hd v S h
+    rw [ite_eq_left h]
+    symm
+    rw [Finset.sum_eq_single j]
+    · exact ite_eq_left hj
+    · intro k _ hkj
+      have hk : ¬taggedRankEvent (d ∘ k.val) v.val S := by
+        intro hk
+        exact hkj (hu k hk)
+      exact ite_eq_right hk
+    · simp
+  · rw [ite_eq_right h]
+    symm
+    apply Finset.sum_eq_zero
+    intro j _
+    apply ite_eq_right
+    intro hj
+    apply h
+    intro i
+    exact ⟨j.val i,(hj i).symm⟩
+
+def rankDefectCoefficient {n l u r : ℕ} (hl : l ≤ n) (d : Fin u → Fin n)
+    (v : IncreasingTuple l r) : ℝ := by
+  classical
+  exact (uniformSubsetLaw l (by simpa using hl)).expect (fun S =>
+    if ∀ i, slotAtRank S (v.val i) ∈ Set.range d then 1 else 0)
+
+theorem rankDefectCoefficient_eq {n l u r : ℕ} (hl : l ≤ n)
+    (d : Fin u → Fin n) (hd : StrictMono d) (v : IncreasingTuple l r) :
+    rankDefectCoefficient hl d v =
+      ∑ j : IncreasingTuple u r, subsetRankKernel hl (d ∘ j.val) v.val := by
+  classical
+  unfold rankDefectCoefficient FiniteLaw.expect subsetRankKernel
+  simp_rw [defect_tuple_indicator d hd v, Finset.mul_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro j _
+  apply Finset.sum_congr rfl
+  intro S _
+  split_ifs <;> simp
+
+/-- Bool-mask form of the selected-rank coefficient, in the exact
+multilinear density expansion. -/
+theorem selected_defect_coefficient {n l : ℕ} (D : Finset (Fin n))
+    (S : FixedSubset (Fin n) l) (σ : Fin l → Bool) :
+    selectionCoefficient (defectAtRank D S) σ =
+      if ∀ i, σ i = true → slotAtRank S i ∈ D then (1:ℝ) else 0 := by
+  classical
+  unfold selectionCoefficient defectAtRank
+  by_cases h : ∀ i, σ i = true → slotAtRank S i ∈ D
+  · rw [ite_eq_left h]
+    apply Finset.prod_eq_one
+    intro i _
+    cases hs : σ i
+    · simp
+    · simp [h i hs]
+  · rw [ite_eq_right h]
+    push Not at h
+    obtain ⟨i,hi,hni⟩ := h
+    apply Finset.prod_eq_zero (Finset.mem_univ i)
+    simp [hi,hni]
+
+def rankMask {l r : ℕ} (v : IncreasingTuple l r) : Fin l → Bool :=
+  fun i => decide (i ∈ Set.range v.val)
+
+theorem averaged_rank_coefficient {n l u r : ℕ} (hl : l ≤ n)
+    (d : Fin u → Fin n) (v : IncreasingTuple l r) :
+    averagedCoefficient (uniformSubsetLaw l (by simpa using hl))
+      (defectAtRank (Finset.univ.image d)) (rankMask v) =
+      rankDefectCoefficient hl d v := by
+  classical
+  unfold averagedCoefficient rankDefectCoefficient FiniteLaw.expect
+  apply Finset.sum_congr rfl
+  intro S _
+  congr 1
+  dsimp only
+  rw [selected_defect_coefficient]
+  congr 1
+  simp only [rankMask,decide_eq_true_eq,Finset.mem_image,Finset.mem_univ,true_and,
+    Set.mem_range]
+  apply propext
+  constructor
+  · intro h i
+    exact h (v.val i) ⟨i,rfl⟩
+  · rintro h i ⟨j,rfl⟩
+    exact h j
+
+end TraceReconstruction
+
+namespace TraceReconstruction
+open FiniteLaw
+
+/-- Canonical increasing tuple associated with a selected set of ranks. -/
+def tupleOfMask {l r : ℕ} (σ : Fin l → Bool) (hσ : monomialDegree σ = r) :
+    IncreasingTuple l r := by
+  classical
+  let S : Finset (Fin l) := Finset.univ.filter (fun i => σ i = true)
+  have hS : S.card = r := hσ
+  exact ⟨fun i => ((S.orderIsoOfFin hS) i).val, (S.orderIsoOfFin hS).strictMono⟩
+
+theorem rankMask_tupleOfMask {l r : ℕ} (σ : Fin l → Bool) (hσ : monomialDegree σ = r) :
+    rankMask (tupleOfMask σ hσ) = σ := by
+  classical
+  funext i
+  have he : i ∈ Set.range (tupleOfMask σ hσ).val ↔ σ i = true := by
+    constructor
+    · rintro ⟨j,rfl⟩
+      exact (Finset.mem_filter.mp (((Finset.univ.filter (fun k => σ k = true)).orderIsoOfFin hσ) j).property).2
+    · intro hi
+      let z : ↥(Finset.univ.filter (fun k => σ k = true)) := ⟨i,by simp [hi]⟩
+      refine ⟨((Finset.univ.filter (fun k => σ k = true)).orderIsoOfFin hσ).symm z,?_⟩
+      exact congrArg Subtype.val (((Finset.univ.filter (fun k => σ k = true)).orderIsoOfFin hσ).apply_symm_apply z)
+  simp only [rankMask,he]
+  exact Bool.decide_eq_true
+
+theorem tupleOfMask_injective {l r : ℕ} : Function.Injective
+    (fun σ : {σ : Fin l → Bool // monomialDegree σ = r} => tupleOfMask σ.val σ.property) := by
+  intro σ τ h
+  apply Subtype.ext
+  have he := congrArg rankMask h
+  simpa only [rankMask_tupleOfMask] using he
+
+/-- Changing from Boolean monomial labels to ordered rank tuples loses no
+mass; an injection suffices for the upper bounds. -/
+theorem sum_degree_coeff_sq_le {l r : ℕ} (A : (Fin l → Bool) → ℝ) :
+    (∑ σ : Fin l → Bool, if monomialDegree σ = r then A σ ^ 2 else 0) ≤
+      ∑ v : IncreasingTuple l r, A (rankMask v)^2 := by
+  classical
+  rw [← Finset.sum_filter]
+  have he : (∑ σ ∈ Finset.univ.filter (fun σ : Fin l → Bool => monomialDegree σ = r), A σ ^ 2) =
+      ∑ σ : {σ : Fin l → Bool // monomialDegree σ = r}, A σ.val ^ 2 := by
+    exact Finset.sum_subtype _ (by intro σ; simp) _
+  rw [he]
+  have h := sum_injective_profile_le
+    (fun σ : {σ : Fin l → Bool // monomialDegree σ = r} => tupleOfMask σ.val σ.property)
+    tupleOfMask_injective (fun _ => True) (fun _ => True.intro)
+    (fun v => A (rankMask v)^2) (fun _ => sq_nonneg _)
+  simpa only [rankMask_tupleOfMask,ite_true] using h
+
+end TraceReconstruction
+
+noncomputable section
+namespace TraceReconstruction
+open FiniteLaw
+
+/-- Exact orthogonality of the multilinear expansion; this avoids any
+loss in the channel power in consecutive-run overlap. -/
+theorem multilinear_second_moment {α : Type*} [Fintype α] (P : FiniteLaw α)
+    (Z : α → ℝ) (hcenter : P.expect Z = 0) (R : ℝ)
+    (hvar : P.expect (fun a => Z a^2) = R^2) (n : ℕ)
+    (A : (Fin n → Bool) → ℝ) :
+    (P.product n).expect (fun x => multilinear A Z x ^ 2) =
+      multilinearEnergy A R := by
+  induction n with
+  | zero => simp [FiniteLaw.expect,FiniteLaw.product,multilinear,multilinearEnergy]
+  | succ n ih =>
+    rw [product_succ_expect_tail]
+    simp_rw [multilinear_cons]
+    have he (x : Fin n → α) :
+        P.expect (fun a => (multilinear (fun σ => A (Fin.cons false σ)) Z x +
+            Z a * multilinear (fun σ => A (Fin.cons true σ)) Z x)^2) =
+          multilinear (fun σ => A (Fin.cons false σ)) Z x ^ 2 +
+            R^2 * multilinear (fun σ => A (Fin.cons true σ)) Z x ^ 2 := by
+      have hp (a : α) :
+          P.mass a * (multilinear (fun σ => A (Fin.cons false σ)) Z x +
+            Z a * multilinear (fun σ => A (Fin.cons true σ)) Z x)^2 =
+          P.mass a * (multilinear (fun σ => A (Fin.cons false σ)) Z x)^2 +
+            2 * multilinear (fun σ => A (Fin.cons false σ)) Z x *
+              multilinear (fun σ => A (Fin.cons true σ)) Z x * (P.mass a * Z a) +
+            (multilinear (fun σ => A (Fin.cons true σ)) Z x)^2 * (P.mass a * Z a^2) := by ring
+      unfold FiniteLaw.expect
+      simp_rw [hp,Finset.sum_add_distrib,← Finset.sum_mul,← Finset.mul_sum]
+      simp only [FiniteLaw.expect] at hcenter hvar
+      rw [P.total,hcenter,hvar]
+      ring
+    simp_rw [he]
+    rw [multilinearEnergy_cons]
+    have hadd (f g : (Fin n → α) → ℝ) :
+        (P.product n).expect (fun x => f x + R^2*g x) =
+          (P.product n).expect f + R^2*(P.product n).expect g := by
+      unfold FiniteLaw.expect
+      simp_rw [mul_add,Finset.sum_add_distrib]
+      rw [Finset.mul_sum]
+      congr 1
+      apply Finset.sum_congr rfl
+      intro x _
+      ring
+    rw [hadd,ih,ih]
+
+/-- Association requires only a total preorder, allowing tied likelihoods. -/
+theorem FiniteLaw.associated_totalPreorder {α : Type*} [Fintype α] [Preorder α]
+    (P : FiniteLaw α) (htotal : ∀ a b : α, a ≤ b ∨ b ≤ a) : P.Associated := by
+  intro f g hf hg
+  have hn (a b : α) : 0 ≤ (f a - f b) * (g a - g b) := by
+    rcases htotal a b with hab | hba
+    · exact mul_nonneg_of_nonpos_of_nonpos (sub_nonpos.mpr (hf hab))
+        (sub_nonpos.mpr (hg hab))
+    · exact mul_nonneg (sub_nonneg.mpr (hf hba)) (sub_nonneg.mpr (hg hba))
+  have h := Finset.sum_nonneg (s := Finset.univ) (fun a _ =>
+    Finset.sum_nonneg (s := Finset.univ) (fun b _ => mul_nonneg (mul_nonneg (P.nonneg a)
+      (P.nonneg b)) (hn a b)))
+  have hid : (∑ a, ∑ b, P.mass a * P.mass b * ((f a - f b) * (g a - g b))) =
+      2 * (P.expect (fun a => f a * g a) - P.expect f * P.expect g) := by
+    have hp (a b : α) : P.mass a * P.mass b * ((f a - f b) * (g a - g b)) =
+        (P.mass a * (f a * g a)) * P.mass b + P.mass a * (P.mass b * (f b * g b)) -
+        (P.mass a * f a) * (P.mass b * g b) - (P.mass a * g a) * (P.mass b * f b) := by ring
+    simp_rw [hp, Finset.sum_sub_distrib, Finset.sum_add_distrib,
+      ← Finset.mul_sum, ← Finset.sum_mul, P.total, one_mul, mul_one]
+    unfold FiniteLaw.expect
+    ring
+  rw [hid] at h
+  linarith
+
+/-- Conditional average densities are nonnegative when the coordinate
+likelihoods are nonnegative. -/
+theorem averagedDensity_nonneg {α Ω : Type*} [Fintype Ω] {n : ℕ}
+    (R : FiniteLaw Ω) (θ : Ω → Fin n → Bool) (Z : α → ℝ)
+    (hZ : ∀ a, 0 ≤ 1+Z a) (x : Fin n → α) :
+    0 ≤ averagedDensity R θ Z x := by
+  apply Finset.sum_nonneg
+  intro ω _
+  apply mul_nonneg (R.nonneg ω)
+  apply Finset.prod_nonneg
+  intro i _
+  split_ifs
+  · exact hZ _
+  · norm_num
+
+theorem averagedDensity_mono {α Ω : Type*} [Preorder α] [Fintype Ω] {n : ℕ}
+    (R : FiniteLaw Ω) (θ : Ω → Fin n → Bool) (Z : α → ℝ)
+    (hZ : ∀ a, 0 ≤ 1+Z a) (hZm : Monotone Z) :
+    Monotone (averagedDensity R θ Z) := by
+  intro x y hxy
+  apply R.expect_mono
+  intro ω
+  apply Finset.prod_le_prod₀
+  · intro i _
+    split_ifs
+    · exact hZ _
+    · norm_num
+  · intro i _
+    split_ifs
+    · linarith [hZm (hxy i)]
+    · rfl
+
+/-- Every independent-score likelihood product is normalized. -/
+theorem selectionDensity_mean {α : Type*} [Fintype α] (P : FiniteLaw α)
+    (Z : α → ℝ) (hZ : P.expect Z = 0) {n : ℕ} (θ : Fin n → Bool) :
+    (P.product n).expect (selectionDensity θ Z) = 1 := by
+  unfold FiniteLaw.expect FiniteLaw.product selectionDensity
+  simp_rw [← Finset.prod_mul_distrib]
+  rw [← Fintype.prod_sum (fun (i : Fin n) (a : α) => P.mass a * (if θ i then 1+Z a else 1))]
+  apply Finset.prod_eq_one
+  intro i _
+  cases θ i
+  · simp only [Bool.false_eq_true,↓reduceIte,mul_one,P.total]
+  · simp only [↓reduceIte,mul_add,mul_one,Finset.sum_add_distrib,P.total]
+    exact add_eq_left.mpr hZ
+
+theorem averagedDensity_mean {α Ω : Type*} [Fintype α] [Fintype Ω]
+    (P : FiniteLaw α) (R : FiniteLaw Ω) (Z : α → ℝ) (hZ : P.expect Z = 0)
+    {n : ℕ} (θ : Ω → Fin n → Bool) :
+    (P.product n).expect (averagedDensity R θ Z) = 1 := by
+  unfold averagedDensity FiniteLaw.expect
+  simp_rw [Finset.mul_sum]
+  rw [Finset.sum_comm]
+  have he (ω : Ω) : (∑ x : Fin n → α, (P.product n).mass x * (R.mass ω * selectionDensity (θ ω) Z x)) =
+      R.mass ω * (P.product n).expect (selectionDensity (θ ω) Z) := by
+    rw [FiniteLaw.expect,Finset.mul_sum]
+    apply Finset.sum_congr rfl
+    intro x _
+    ring
+  simp_rw [he,selectionDensity_mean P Z hZ,mul_one]
+  exact R.total
+
+end TraceReconstruction
+
+namespace TraceReconstruction
+open FiniteLaw
+
+/-- Exact decomposition of the coefficient energy by its degree. -/
+theorem multilinearEnergy_degrees {n : ℕ} (A : (Fin n → Bool) → ℝ) (R : ℝ) :
+    multilinearEnergy A R = ∑ r ∈ Finset.range (n+1),
+      R^(2*r) * ∑ σ : Fin n → Bool, if monomialDegree σ = r then A σ^2 else 0 := by
+  classical
+  unfold multilinearEnergy
+  simp_rw [monomial_weight]
+  simp_rw [Finset.mul_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro σ _
+  have hd : monomialDegree σ < n+1 := by
+    have h := Finset.card_filter_le (s := Finset.univ) (p := fun i : Fin n => σ i = true)
+    simpa [monomialDegree] using Nat.lt_succ_of_le h
+  rw [Finset.sum_eq_single (monomialDegree σ)]
+  · simp [mul_comm]
+  · intro r _ hr
+    simp [Ne.symm hr]
+  · simp [Finset.mem_range,hd]
+
+/-- A zero-degree selection is precisely the empty selection. -/
+theorem monomialDegree_eq_zero_iff {n : ℕ} (σ : Fin n → Bool) :
+    monomialDegree σ = 0 ↔ σ = fun _ => false := by
+  classical
+  constructor
+  · intro h
+    have he : Finset.univ.filter (fun i => σ i = true) = ∅ := Finset.card_eq_zero.mp h
+    funext i
+    have hn : σ i ≠ true := by
+      intro hi
+      have : i ∈ Finset.univ.filter (fun j => σ j = true) := by simp [hi]
+      rw [he] at this
+      exact Finset.notMem_empty i this
+    exact Bool.eq_false_iff.mpr hn
+  · rintro rfl
+    simp [monomialDegree]
+
+theorem averaged_degree_zero {Ω : Type*} [Fintype Ω] {n : ℕ}
+    (R : FiniteLaw Ω) (θ : Ω → Fin n → Bool) :
+    (∑ σ : Fin n → Bool, if monomialDegree σ = 0 then averagedCoefficient R θ σ ^ 2 else 0) = 1 := by
+  classical
+  simp_rw [monomialDegree_eq_zero_iff]
+  simp [averagedCoefficient_empty]
+
+/-- There is no increasing selection with more entries than its ambient set. -/
+theorem increasingTuple_isEmpty {n r : ℕ} (h : n < r) : IsEmpty (IncreasingTuple n r) := by
+  refine ⟨fun v => ?_⟩
+  have hc := Fintype.card_le_of_injective v.val v.property.injective
+  simp only [Fintype.card_fin] at hc
+  omega
+
+/-- The finite law of uniformly retained slots has no coefficients beyond
+its number of actual defect slots. -/
+theorem defect_degree_bound {n l u r : ℕ} (hl : l ≤ n)
+    (d : Fin u → Fin n) (hd : StrictMono d) (hru : u < r) :
+    (∑ σ : Fin l → Bool, if monomialDegree σ = r then
+      averagedCoefficient (uniformSubsetLaw l (by simpa using hl))
+        (defectAtRank (Finset.univ.image d)) σ ^ 2 else 0) = 0 := by
+  classical
+  apply le_antisymm
+  · refine (sum_degree_coeff_sq_le (r:=r) (averagedCoefficient (uniformSubsetLaw l (by simpa using hl))
+        (defectAtRank (Finset.univ.image d)))).trans ?_
+    simp_rw [averaged_rank_coefficient hl d,rankDefectCoefficient_eq hl d hd]
+    have := increasingTuple_isEmpty hru
+    simp
+  · apply Finset.sum_nonneg
+    intro σ _
+    split_ifs <;> positivity
+
+/-- Coefficient version of the conditional-rank square-sum estimate. -/
+theorem period_degree_square_sum {u m l r : ℕ} (hl : l ≤ u*m) (hr : 0 < r)
+    (hru : r ≤ u) (hm : 32 ≤ m) (hum : u ≤ m) {δ θ : ℝ}
+    (hδ : δ ∈ Set.Ioc 0 (1/2:ℝ))
+    (hθ : θ ∈ Set.Icc (1-δ^2) (1-δ^2/4))
+    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
+    (off : Fin u → Fin m)
+    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
+    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2) :
+    (∑ σ : Fin l → Bool, if monomialDegree σ = r then
+      averagedCoefficient (uniformSubsetLaw l (by simpa using hl))
+        (defectAtRank (Finset.univ.image (periodSlot off))) σ ^ 2 else 0) ≤
+      ((3*rankSchurConstant/δ)*Real.sqrt ((u:ℝ)/((m:ℝ)*r)))^r := by
+  refine (sum_degree_coeff_sq_le (r:=r) (averagedCoefficient (uniformSubsetLaw l (by simpa using hl))
+    (defectAtRank (Finset.univ.image (periodSlot off))))).trans ?_
+  simp_rw [averaged_rank_coefficient hl (periodSlot off),
+    rankDefectCoefficient_eq hl (periodSlot off) (periodSlot_strictMono off)]
+  exact period_rank_square_sum_delta hl hr hru hm hum hδ hθ hmean hv off hoff hdiff
+
+/-- The full coefficient energy is controlled by the subgaussian degree
+series, uniformly in the number of retained coordinates. -/
+theorem period_density_energy {u m l : ℕ} (hl : l ≤ u*m)
+    (hm : 32 ≤ m) (hum : u ≤ m) {δ θ : ℝ}
+    (hδ : δ ∈ Set.Ioc 0 (1/2:ℝ))
+    (hθ : θ ∈ Set.Icc (1-δ^2) (1-δ^2/4))
+    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
+    (off : Fin u → Fin m)
+    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
+    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2) (R : ℝ) :
+    multilinearEnergy
+      (averagedCoefficient (uniformSubsetLaw l (by simpa using hl))
+        (defectAtRank (Finset.univ.image (periodSlot off)))) R ≤
+      ∑ r ∈ Finset.range (l+1),
+        (R^2*(3*rankSchurConstant/δ)*Real.sqrt ((u:ℝ)/m))^r / (Real.sqrt (r:ℝ))^r := by
+  rw [multilinearEnergy_degrees]
+  apply Finset.sum_le_sum
+  intro r _
+  by_cases hr : r = 0
+  · subst r
+    rw [averaged_degree_zero]
+    norm_num
+  · by_cases hru : r ≤ u
+    · have hr0 : 0 < r := Nat.pos_of_ne_zero hr
+      have hb := period_degree_square_sum hl hr0 hru hm hum hδ hθ hmean hv off hoff hdiff
+      calc
+        _ ≤ R^(2*r)*((3*rankSchurConstant/δ)*Real.sqrt ((u:ℝ)/((m:ℝ)*r)))^r :=
+          mul_le_mul_of_nonneg_left hb (by rw [pow_mul]; positivity)
+        _ = _ := by
+          have hm0 : (0:ℝ) < m := by exact_mod_cast (show 0 < m by omega)
+          have hr0 : (0:ℝ) < r := by exact_mod_cast hr0
+          rw [show (u:ℝ)/((m:ℝ)*r)=((u:ℝ)/m)/r by field_simp]
+          rw [Real.sqrt_div (by positivity),pow_mul,← mul_pow,← div_pow]
+          congr 1
+          ring
+    · rw [defect_degree_bound hl (periodSlot off) (periodSlot_strictMono off) (by omega),mul_zero]
+      have := hδ.1
+      positivity [rankSchurConstant_pos]
+
+/-- A convenient exponential majorant for the full nonnegative degree series. -/
+theorem degree_series_le_exp (a : ℝ) (N : ℕ) :
+    (∑ r ∈ Finset.range N, a^r / (Real.sqrt (r:ℝ))^r) ≤ 2*Real.exp (a^2) := by
+  have hb := degree_series_bound a N
+  have hp := Real.exp_pos (a^2)
+  have he : Real.exp (2*a^2) = (Real.exp (a^2))^2 := by
+    rw [show 2*a^2=a^2+a^2 by ring,Real.exp_add]
+    ring
+  rw [he] at hb
+  nlinarith [sq_nonneg ((∑ r ∈ Finset.range N, a^r / (Real.sqrt (r:ℝ))^r)+2*Real.exp (a^2))]
+
+/-- Actual conditional initial-oracle density, with its exact second moment. -/
+theorem period_density_second_moment {α : Type*} [Fintype α]
+    (P : FiniteLaw α) (Z : α → ℝ) (hZ : P.expect Z = 0)
+    {u m l : ℕ} (hl : l ≤ u*m)
+    (hm : 32 ≤ m) (hum : u ≤ m) {δ θ : ℝ}
+    (hδ : δ ∈ Set.Ioc 0 (1/2:ℝ))
+    (hθ : θ ∈ Set.Icc (1-δ^2) (1-δ^2/4))
+    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
+    (off : Fin u → Fin m)
+    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
+    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2) :
+    (P.product l).expect (fun x =>
+      averagedDensity (uniformSubsetLaw l (by simpa using hl))
+        (defectAtRank (Finset.univ.image (periodSlot off))) Z x ^ 2) ≤
+      2*Real.exp ((P.expect (fun a => Z a^2))^2 * (3*rankSchurConstant/δ)^2 * ((u:ℝ)/m)) := by
+  have hv0 : 0 ≤ P.expect (fun a => Z a^2) :=
+    Finset.sum_nonneg (fun a _ => mul_nonneg (P.nonneg a) (sq_nonneg _))
+  simp_rw [averagedDensity_expansion]
+  rw [multilinear_second_moment P Z hZ (Real.sqrt (P.expect (fun a => Z a^2)))
+    (Real.sq_sqrt hv0).symm]
+  refine (period_density_energy hl hm hum hδ hθ hmean hv off hoff hdiff _).trans ?_
+  apply (degree_series_le_exp _ (l+1)).trans
+  rw [Real.sq_sqrt hv0]
+  have he : (P.expect (fun a => Z a^2)*(3*rankSchurConstant/δ)*Real.sqrt ((u:ℝ)/m))^2 =
+      (P.expect (fun a => Z a^2))^2*(3*rankSchurConstant/δ)^2*((u:ℝ)/m) := by
+    rw [mul_pow,mul_pow,Real.sq_sqrt (by positivity)]
+  rw [he]
+
+/-- The source association argument holds for any two conditional mixture
+experiments using the same independent primitive marks, even when their
+retained-subset sample spaces have different original sizes. -/
+theorem averagedDensity_overlap {α Ω Ψ : Type*}
+    [Fintype α] [Fintype Ω] [Fintype Ψ]
+    (P : FiniteLaw α) (R : FiniteLaw Ω) (S : FiniteLaw Ψ)
+    (Z : α → ℝ) (hZ : P.expect Z = 0) (hZpos : ∀ a, 0 ≤ 1+Z a)
+    {l : ℕ} (f : Ω → Fin l → Bool) (g : Ψ → Fin l → Bool) :
+    (1:ℝ)/32 ≤
+      (P.product l).expect (fun x => min (averagedDensity R f Z x) (averagedDensity S g Z x)) *
+      (P.product l).expect (fun x => averagedDensity R f Z x ^ 2) *
+      (P.product l).expect (fun x => averagedDensity S g Z x ^ 2) := by
+  let : Preorder α := Preorder.lift Z
+  have htotal : ∀ a b : α, a ≤ b ∨ b ≤ a := fun a b => le_total (Z a) (Z b)
+  have hm : Monotone Z := fun _ _ h => h
+  exact increasing_density_overlap (P.product l)
+    (associated_product P (P.associated_totalPreorder htotal) l)
+    (averagedDensity R f Z) (averagedDensity S g Z)
+    (averagedDensity_nonneg R f Z hZpos) (averagedDensity_nonneg S g Z hZpos)
+    (averagedDensity_mono R f Z hZpos hm) (averagedDensity_mono S g Z hZpos hm)
+    (averagedDensity_mean P R Z hZ f) (averagedDensity_mean P S Z hZ g)
+
+end TraceReconstruction
+
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/TraceReconstruction/InitialMoments.lean b/lean/OAI/Probability/TraceReconstruction/InitialMoments.lean
index 4b08127b02ebfd66aca682d325b07fe9048d27fa..2dab9878f760774cdaf4d296c8f75fe00be29bfe 100644
--- a/lean/OAI/Probability/TraceReconstruction/InitialMoments.lean
+++ b/lean/OAI/Probability/TraceReconstruction/InitialMoments.lean
@@ -1,1215 +1,1215 @@
-import OAI.Probability.TraceReconstruction.Patterns
-
-namespace OAI
-
-/-!
-# InitialMoments
-
-Initial coefficient tails, primitive conditioning, mark-list oracle densities, pure norm estimates, and moment energies.
--/
-
-open scoped ENNReal
-noncomputable section
-
-open Filter
-open scoped Topology
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem signedPeriodDefects_comp {u m r : ℕ} (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
-    (s : ℕ) (X : ℤ) (τ : ℝ) (j : IncreasingTuple u r) :
-    signedPeriodDefects off hsafe s X τ ∘ j.val =
-      shiftedTag (periodSlot (fun _ => off) ∘ j.val) (hsafe.comp j.val j.property)
-        (fun i => motifShift (motifIndicator s ((j.val i).val-X))
-          ((prouhetSeries s).coeff ((j.val i).val-X)) τ) := by
-  funext i; apply Fin.ext; rfl
-
-def signedTupleKernel {u m l r : ℕ} (hl : l ≤ u*m) (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
-    (π : IntSeries) (s : ℕ) (j : IncreasingTuple u r) (v : IncreasingTuple l r) : ℝ :=
-  ∑X∈π.coeff.support, π.coeff X*
-    (subsetRankKernel hl (signedPeriodDefects off hsafe s X 1 ∘ j.val) v.val -
-     subsetRankKernel hl (signedPeriodDefects off hsafe s X (-1) ∘ j.val) v.val)
-
-theorem signedRankCoefficient_tuple_sum {u m l r : ℕ} (hl : l ≤ u*m) (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off))) (π : IntSeries) (s : ℕ)
-    (v : IncreasingTuple l r) :
-    signedRankCoefficient hl off hsafe π s v=∑j, signedTupleKernel hl off hsafe π s j v := by
-  unfold signedRankCoefficient signedTupleKernel
-  simp_rw [rankDefectCoefficient_eq _ _ (signedPeriodDefects_strictMono _ _ _ _ _),
-    ← Finset.sum_sub_distrib,Finset.mul_sum]
-  exact Finset.sum_comm
-
-theorem signedTupleKernel_expansion {u m l r : ℕ} (hl : l ≤ u*m) (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off))) (π : IntSeries) (s : ℕ)
-    (j : IncreasingTuple u r) (v : IncreasingTuple l r) :
-    signedTupleKernel hl off hsafe π s j v = ∑k : Fin r → GapDifference,
-      if Odd (firstDifferenceCount k) then
-        2*motifPatternMultiplier π s (fun i => (j.val i).val) k *
-          positionalRankDifference hl (periodSlot (fun _ => off) ∘ j.val)
-            (hsafe.comp j.val j.property) k v else 0 := by
-  unfold signedTupleKernel
-  simp_rw [signedPeriodDefects_comp]
-  have H := mixed_signed_shift_expansion π s (fun i => (j.val i).val)
-    (fun a => subsetRankKernel hl
-      (shiftedTag (periodSlot (fun _ => off) ∘ j.val) (hsafe.comp j.val j.property) a) v.val)
-  simpa only [ite_mul,zero_mul,positionalRankDifference] using H
-
-def clippedSignedRankCoefficient {u m l r : ℕ} (hl : l ≤ u*m) (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
-    (π : IntSeries) (s : ℕ) (θ : ℝ) (v : IncreasingTuple l r) : ℝ :=
-  ∑j, if PeriodRankRegion m θ j v then signedTupleKernel hl off hsafe π s j v else 0
-
-def clippedPatternCoefficient {u m l r : ℕ} (hl : l ≤ u*m) (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
-    (π : IntSeries) (s : ℕ) (θ : ℝ) (k : Fin r → GapDifference) (v : IncreasingTuple l r) : ℝ :=
-  ∑j, if PeriodRankRegion m θ j v then
-    motifPatternMultiplier π s (fun i => (j.val i).val) k *
-      positionalRankDifference hl (periodSlot (fun _ => off) ∘ j.val)
-        (hsafe.comp j.val j.property) k v else 0
-
-theorem clippedSignedRankCoefficient_expansion {u m l r : ℕ} (hl : l ≤ u*m) (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
-    (π : IntSeries) (s : ℕ) (θ : ℝ) (v : IncreasingTuple l r) :
-    clippedSignedRankCoefficient hl off hsafe π s θ v =
-      2*∑k : Fin r → GapDifference, if Odd (firstDifferenceCount k) then
-        clippedPatternCoefficient hl off hsafe π s θ k v else 0 := by
-  unfold clippedSignedRankCoefficient clippedPatternCoefficient
-  have he (j : IncreasingTuple u r) :
-      (if PeriodRankRegion m θ j v then signedTupleKernel hl off hsafe π s j v else 0) =
-      ∑k : Fin r → GapDifference, if Odd (firstDifferenceCount k) then
-        (if PeriodRankRegion m θ j v then
-          2*motifPatternMultiplier π s (fun i => (j.val i).val) k *
-            positionalRankDifference hl (periodSlot (fun _ => off) ∘ j.val)
-              (hsafe.comp j.val j.property) k v else 0) else 0 := by
-    by_cases hv : PeriodRankRegion m θ j v
-    · simp only [ite_eq_left hv, signedTupleKernel_expansion]
-    · simp [hv]
-  simp_rw [he]
-  rw [Finset.sum_comm,Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro k _
-  by_cases hk : Odd (firstDifferenceCount k)
-  · simp only [ite_eq_left hk,Finset.mul_sum]
-    apply Finset.sum_congr rfl
-    intro j _
-    split_ifs <;> ring
-  · simp [hk]
-
-/-- Removing coarse rank rectangles loses an explicitly exponentially
- small amount in the actual coefficient, before any finite differences. -/
-theorem signedTupleKernel_tail {u m l r : ℕ}
-    (hu : 0 < u) (hm : 128 ≤ m) (hr : 0 < r) (hl : l ≤ u*m)
-    (off : Fin m) (hoff : |(off.val:ℝ)-(m:ℝ)/2|≤3)
-    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
-    (π : IntSeries) (hπ : ∀X, 0≤π.coeff X) (hπsum : ∑X∈π.coeff.support, π.coeff X=1)
-    (s : ℕ) (j : IncreasingTuple u r) {θ : ℝ} (hθ : θ∈Set.Ioo 0 1) :
-    (∑v : IncreasingTuple l r, if PeriodRankRegion m θ j v then 0 else
-      |signedTupleKernel hl off hsafe π s j v|) ≤
-      2*(((r+1:ℕ):ℝ)*θ^r/binomialMass (u*m) θ l * (2*Real.exp (-(m:ℝ)/(1024*u)))) := by
-  let K (X : ℤ) (τ : ℝ) (v : IncreasingTuple l r) :=
-    subsetRankKernel hl (signedPeriodDefects off hsafe s X τ ∘ j.val) v.val
-  let B := ((r+1:ℕ):ℝ)*θ^r/binomialMass (u*m) θ l * (2*Real.exp (-(m:ℝ)/(1024*u)))
-  have hK (X : ℤ) (τ : ℝ) : (∑v : IncreasingTuple l r,
-      if PeriodRankRegion m θ j v then 0 else K X τ v)≤B := by
-    simp_rw [K,signedPeriodDefects_comp]
-    exact period_shifted_outside_region hu hm hr hl (fun _ => off) (fun _ => hoff)
-      (by intro _ _; simp) j (hsafe.comp j.val j.property) _ hθ
-  have hnon (X : ℤ) (τ : ℝ) (v : IncreasingTuple l r) : 0≤K X τ v :=
-    subsetRankKernel_nonneg _ _ _
-  calc
-    _ ≤ ∑v : IncreasingTuple l r, ∑X∈π.coeff.support, π.coeff X*
-        (if PeriodRankRegion m θ j v then 0 else K X 1 v+K X (-1) v) := by
-      apply Finset.sum_le_sum
-      intro v _
-      by_cases hv : PeriodRankRegion m θ j v
-      · simp [hv]
-      · simp only [ite_eq_right hv]
-        refine (Finset.abs_sum_le_sum_abs _ _).trans ?_
-        apply Finset.sum_le_sum
-        intro X _
-        rw [abs_mul,abs_of_nonneg (hπ X)]
-        apply mul_le_mul_of_nonneg_left _ (hπ X)
-        exact (abs_sub _ _).trans_eq (by rw [abs_of_nonneg (hnon X 1 v),abs_of_nonneg (hnon X (-1) v)])
-    _ = ∑X∈π.coeff.support, π.coeff X*((∑v : IncreasingTuple l r,
-        if PeriodRankRegion m θ j v then 0 else K X 1 v)+
-        (∑v : IncreasingTuple l r, if PeriodRankRegion m θ j v then 0 else K X (-1) v)) := by
-      rw [Finset.sum_comm]
-      apply Finset.sum_congr rfl
-      intro X _
-      rw [← Finset.mul_sum,← Finset.sum_add_distrib]
-      congr 1; apply Finset.sum_congr rfl
-      intro v _; split_ifs <;> ring
-    _ ≤ ∑X∈π.coeff.support, π.coeff X*(B+B) := by
-      apply Finset.sum_le_sum
-      intro X _
-      exact mul_le_mul_of_nonneg_left (add_le_add (hK X 1) (hK X (-1))) (hπ X)
-    _ = _ := by rw [← Finset.sum_mul,hπsum]; dsimp [B]; ring
-
-end TraceReconstruction
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem motif_pattern_common_short_gap {r : ℕ} (k : Fin r → GapDifference)
-    (hk : Odd (firstDifferenceCount k)) (hnl : ¬∃a, k=linearPattern a) :
-    ∃ a : Fin r, k a≠.unchanged ∧ ∃c : Fin (r+1), ∀ {u s : ℕ} (π : IntSeries)
-      (j : IncreasingTuple u r),
-      motifPatternMultiplier π s (fun h => (j.val h).val) k≠0 →
-        coarsePeriodGap j.val c ≤ 2*(2^s) := by
-  obtain ⟨a,b,hab,ha,hb⟩ := odd_pattern_pair k hk hnl
-  cases r with
-  | zero => exact Fin.elim0 a
-  | succ r =>
-    have har : a.val < r := by have := b.isLt; exact lt_of_lt_of_le hab (Nat.le_of_lt_succ b.isLt)
-    let i : Fin r := ⟨a.val,har⟩
-    have hi : i.castSucc=a := by ext; rfl
-    refine ⟨a,ha,i.castSucc.succ,?_⟩
-    intro u s π j hj
-    apply motif_pattern_bounded_gap π s j k i b
-    · simpa only [hi] using hab
-    · simpa only [hi] using ha
-    · exact hb
-    · exact hj
-
-theorem clipped_linear_pattern_norm {u m l r : ℕ}
-    (hm : 32 ≤ m) (hr : 0 < r) (hl : l ≤ u*m)
-    (off : Fin m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 3)
-    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
-    (π : IntSeries) (s : ℕ) (hcentral : CentralMotifSupport u s π)
-    {θ : ℝ} (hθ : θ∈Set.Ioo 0 1) (hhalf : 1/2 ≤ θ)
-    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ)) (a : Fin r) :
-    (∑v, (clippedPatternCoefficient hl off hsafe π s θ (linearPattern a) v)^2)  ≤ 
-      periodDifferencePrefactor u m r 1 θ * periodCentralSumConstant u r *
-        seriesL2Sq (prouhetSeries s*π) := by
-  have hb := period_pattern_norm hm hr hl (fun _ => off) (fun _ => hoff)
-    (by intro _ _; simp) (fun j => hsafe.comp j.val j.property) (linearPattern a)
-    hθ hhalf hmean hv a
-    (fun j => motifPatternMultiplier π s (fun i => (j.val i).val) (linearPattern a))
-    (fun h => (prouhetSeries s*π).coeff h.val)
-    (by intro j; rw [motif_linear_multiplier])
-    (prouhet_convolution_central π hcentral)
-  rw [linearPattern_order] at hb
-  apply hb.trans
-  apply mul_le_mul_of_nonneg_left
-    (seriesL2Sq_partial _ (fun h : Fin u => (h.val:ℤ)) (by intro a b h; apply Fin.ext; change (a.val:ℤ)=b.val at h; exact_mod_cast h))
-  unfold periodDifferencePrefactor lengthDifferenceConstant periodCentralSumConstant
-  positivity
-
-theorem clipped_nonlinear_pattern_norm {u m l r : ℕ}
-    (hm : 32 ≤ m) (hr : 0 < r) (hl : l ≤ u*m)
-    (off : Fin m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 3)
-    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
-    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (s : ℕ) (hcentral : CentralMotifSupport u s π)
-    {θ : ℝ} (hθ : θ∈Set.Ioo 0 1) (hhalf : 1/2 ≤ θ)
-    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
-    (k : Fin r → GapDifference) (hk : Odd (firstDifferenceCount k)) (hnl : ¬∃a, k=linearPattern a)
-    (hM : ((2*(2^s):ℕ):ℝ) < (u:ℝ)/(3*(r+1:ℝ))) :
-    (∑v, (clippedPatternCoefficient hl off hsafe π s θ k v)^2)  ≤ 
-      periodDifferencePrefactor u m r (∑h, (k h).order) θ *
-        periodBoundedSumConstant u r (2*(2^s)) * seriesL2Sq (geomSeries (2^s)*π) := by
-  obtain ⟨a,ha,c,hc⟩ := motif_pattern_common_short_gap k hk hnl
-  have hb := period_pattern_bounded_norm hm hr hl (fun _ => off) (fun _ => hoff)
-    (by intro _ _; simp) (fun j => hsafe.comp j.val j.property) k
-    hθ hhalf hmean hv a c
-    (fun j => motifPatternMultiplier π s (fun i => (j.val i).val) k)
-    (fun h => (geomSeries (2^s)*π).coeff h.val)
-    (fun j => motifPatternMultiplier_sq_le π hπ s _ k a ha)
-    (fun j hj => hc π j hj) (unsigned_convolution_central π hcentral) hM
-  apply hb.trans
-  apply mul_le_mul_of_nonneg_left
-    (seriesL2Sq_partial _ (fun h : Fin u => (h.val:ℤ)) (by intro a b h; apply Fin.ext; change (a.val:ℤ)=b.val at h; exact_mod_cast h))
-  unfold periodDifferencePrefactor lengthDifferenceConstant periodBoundedSumConstant
-  positivity
-
-def signedPatternBound (u m r s : ℕ) (π : IntSeries) (θ : ℝ)
-    (k : Fin r → GapDifference) : ℝ :=
-  if ∃a, k=linearPattern a then
-    periodDifferencePrefactor u m r 1 θ * periodCentralSumConstant u r * seriesL2Sq (prouhetSeries s*π)
-  else periodDifferencePrefactor u m r (∑h, (k h).order) θ *
-    periodBoundedSumConstant u r (2*(2^s)) * seriesL2Sq (geomSeries (2^s)*π)
-
-theorem sum_square_card_bound {α β : Type*} [Fintype α] [Fintype β] (f : α → β → ℝ) :
-    (∑ b, (∑a, f a b)^2)  ≤  (Fintype.card α:ℝ)*∑a, ∑b, f a b^2 := by
-  calc
-    _  ≤  ∑b, (Fintype.card α:ℝ)*∑a, f a b^2 := by
-      apply Finset.sum_le_sum
-      intro b _
-      have h := Finset.sum_sq_le_sum_mul_sum_of_sq_le_mul Finset.univ
-        (r := fun a => f a b) (f := fun _ => (1:ℝ)) (g := fun a => f a b^2)
-        (by intro _ _; norm_num) (by intro _ _; positivity) (by intro _ _; simp)
-      simpa using h
-    _ = _ := by rw [← Finset.mul_sum,Finset.sum_comm]
-
-theorem clipped_signed_initial_norm {u m l r : ℕ}
-    (hm : 32 ≤ m) (hr : 0 < r) (hl : l ≤ u*m)
-    (off : Fin m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 3)
-    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
-    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (s : ℕ) (hcentral : CentralMotifSupport u s π)
-    {θ : ℝ} (hθ : θ∈Set.Ioo 0 1) (hhalf : 1/2 ≤ θ)
-    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
-    (hM : ((2*(2^s):ℕ):ℝ) < (u:ℝ)/(3*(r+1:ℝ))) :
-    (∑v : IncreasingTuple l r, (clippedSignedRankCoefficient hl off hsafe π s θ v)^2)  ≤ 
-      4*(Fintype.card (Fin r → GapDifference):ℝ)*
-        ∑k : Fin r → GapDifference, if Odd (firstDifferenceCount k) then
-          signedPatternBound u m r s π θ k else 0 := by
-  simp_rw [clippedSignedRankCoefficient_expansion,mul_pow,show (2:ℝ)^2=4 by norm_num]
-  rw [← Finset.mul_sum]
-  calc
-    _  ≤  4*((Fintype.card (Fin r → GapDifference):ℝ)*∑k : Fin r → GapDifference, ∑v : IncreasingTuple l r,
-        (if Odd (firstDifferenceCount k) then clippedPatternCoefficient hl off hsafe π s θ k v else 0)^2) := by
-      exact mul_le_mul_of_nonneg_left (sum_square_card_bound (fun (k : Fin r → GapDifference) (v : IncreasingTuple l r) =>
-        if Odd (firstDifferenceCount k) then clippedPatternCoefficient hl off hsafe π s θ k v else 0)) (by norm_num)
-    _  ≤  _ := by
-      rw [← mul_assoc]
-      apply mul_le_mul_of_nonneg_left _ (by positivity)
-      apply Finset.sum_le_sum
-      intro k _
-      by_cases hk : Odd (firstDifferenceCount k)
-      · simp only [ite_eq_left hk]
-        unfold signedPatternBound
-        split_ifs with hlin
-        · obtain ⟨a,rfl⟩ := hlin
-          exact clipped_linear_pattern_norm hm hr hl off hoff hsafe π s hcentral hθ hhalf hmean hv a
-        · exact clipped_nonlinear_pattern_norm hm hr hl off hoff hsafe π hπ s hcentral
-            hθ hhalf hmean hv k hk hlin hM
-      · simp [hk]
-
-end TraceReconstruction
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem signedRankCoefficient_tail {u m l r : ℕ}
-    (hu : 0 < u) (hm : 128 ≤ m) (hr : 0 < r) (hl : l ≤ u*m)
-    (off : Fin m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 3)
-    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
-    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support, π.coeff X=1)
-    (s : ℕ) {θ : ℝ} (hθ : θ∈Set.Ioo 0 1) :
-    (∑v : IncreasingTuple l r, |signedRankCoefficient hl off hsafe π s v-
-        clippedSignedRankCoefficient hl off hsafe π s θ v|) ≤
-      (Fintype.card (IncreasingTuple u r):ℝ)*
-      (2*(((r+1:ℕ):ℝ)*θ^r/binomialMass (u*m) θ l * (2*Real.exp (-(m:ℝ)/(1024*u))))) := by
-  simp_rw [signedRankCoefficient_tuple_sum,clippedSignedRankCoefficient,← Finset.sum_sub_distrib]
-  calc
-    _ ≤ ∑v : IncreasingTuple l r, ∑j : IncreasingTuple u r,
-        if PeriodRankRegion m θ j v then 0 else |signedTupleKernel hl off hsafe π s j v| := by
-      apply Finset.sum_le_sum
-      intro v _
-      refine (Finset.abs_sum_le_sum_abs _ _).trans_eq ?_
-      apply Finset.sum_congr rfl
-      intro j _
-      split_ifs <;> simp
-    _ = ∑j : IncreasingTuple u r, ∑v : IncreasingTuple l r,
-        if PeriodRankRegion m θ j v then 0 else |signedTupleKernel hl off hsafe π s j v| := Finset.sum_comm
-    _ ≤ _ := by
-      calc
-        _ ≤ ∑j : IncreasingTuple u r,
-            2*(((r+1:ℕ):ℝ)*θ^r/binomialMass (u*m) θ l * (2*Real.exp (-(m:ℝ)/(1024*u)))) :=
-          Finset.sum_le_sum (fun j _ => signedTupleKernel_tail hu hm hr hl off hoff hsafe π hπ hπsum s j hθ)
-        _ = _ := by simp
-
-/-- A globally discarded l1 tail is also negligible in coefficient energy.
- This does not assume pointwise cancellation on the discarded ranks. -/
-theorem finite_sum_sq_le_abs_sum_sq {α : Type*} [Fintype α] (f : α → ℝ) :
-    (∑a, f a^2) ≤ (∑a, |f a|)^2 := by
-  have h := Finset.sum_sq_le_sq_sum_of_nonneg (s := Finset.univ) (f := fun a => |f a|)
-    (by intro a _; exact abs_nonneg _)
-  simpa only [sq_abs] using h
-
-theorem initial_signed_energy {u m l r : ℕ}
-    (hu : 0 < u) (hm : 128 ≤ m) (hr : 0 < r) (hl : l ≤ u*m)
-    (off : Fin m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 3)
-    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
-    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support, π.coeff X=1)
-    (s : ℕ) (hcentral : CentralMotifSupport u s π)
-    {θ : ℝ} (hθ : θ∈Set.Ioo 0 1) (hhalf : 1/2 ≤ θ)
-    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
-    (hM : ((2*(2^s):ℕ):ℝ) < (u:ℝ)/(3*(r+1:ℝ))) :
-    (∑v : IncreasingTuple l r, (signedRankCoefficient hl off hsafe π s v)^2) ≤
-      8*(Fintype.card (Fin r → GapDifference):ℝ)*
-        (∑k : Fin r → GapDifference, if Odd (firstDifferenceCount k) then
-          signedPatternBound u m r s π θ k else 0) +
-      2*((Fintype.card (IncreasingTuple u r):ℝ)*
-        (2*(((r+1:ℕ):ℝ)*θ^r/binomialMass (u*m) θ l * (2*Real.exp (-(m:ℝ)/(1024*u))))))^2 := by
-  let A : IncreasingTuple l r → ℝ := signedRankCoefficient hl off hsafe π s
-  let C : IncreasingTuple l r → ℝ := clippedSignedRankCoefficient hl off hsafe π s θ
-  have hn := clipped_signed_initial_norm (by omega) hr hl off hoff hsafe π hπ s hcentral hθ hhalf hmean hv hM
-  have ht := signedRankCoefficient_tail hu hm hr hl off hoff hsafe π hπ hπsum s hθ
-  have hE := (finite_sum_sq_le_abs_sum_sq (fun v : IncreasingTuple l r => A v-C v)).trans
-    (pow_le_pow_left₀ (Finset.sum_nonneg (fun v _ => abs_nonneg (A v-C v))) ht 2)
-  have hsum : (∑v : IncreasingTuple l r, A v^2) ≤
-      2*(∑v : IncreasingTuple l r, C v^2)+2*(∑v : IncreasingTuple l r, (A v-C v)^2) := by
-    simp only [Finset.mul_sum,← Finset.sum_add_distrib]
-    apply Finset.sum_le_sum
-    intro v _
-    nlinarith [sq_nonneg (A v-2*C v)]
-  calc
-    _ ≤ 2*(∑v : IncreasingTuple l r, C v^2)+2*(∑v : IncreasingTuple l r, (A v-C v)^2) := hsum
-    _ ≤ _ := by dsimp [C] at *; linarith
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- Exact complete-trace expectation, recursively on the actual binary word. -/
-def listDeletionExpectation (q : ℝ) : List Bool → (List Bool → ℝ) → ℝ
-  | [], F => F []
-  | b::w, F => q*listDeletionExpectation q w F +
-      (1-q)*listDeletionExpectation q w (fun t => F (b::t))
-
-theorem traceOfMask_cons {n : ℕ} (x : Word (n+1)) (b : Bool) (a : Mask n) :
-    traceOfMask x (Fin.cons b a)=
-      if b then x 0::traceOfMask (fun i => x i.succ) a
-      else traceOfMask (fun i => x i.succ) a := by
-  rw [trace_filterMap,List.ofFn_succ,trace_filterMap]
-  cases b <;> simp
-
-theorem maskWeight_cons {n : ℕ} (q : ℝ) (b : Bool) (a : Mask n) :
-    maskWeight q (Fin.cons b a)=(if b then 1-q else q)*maskWeight q a := by
-  simp [maskWeight,Fin.prod_univ_succ]
-
-theorem listDeletionExpectation_ofFn {n : ℕ} (q : ℝ) (x : Word n) (F : List Bool → ℝ) :
-    listDeletionExpectation q (List.ofFn x) F =
-      ∑a : Mask n, maskWeight q a*F (traceOfMask x a) := by
-  induction n generalizing F with
-  | zero => simp [listDeletionExpectation,maskWeight,traceOfMask]
-  | succ n ih =>
-    rw [List.ofFn_succ,listDeletionExpectation,ih,ih]
-    symm
-    rw [← (Fin.consEquiv (fun _ : Fin (n+1) => Bool)).sum_comp,Fintype.sum_prod_type]
-    change (∑b : Bool, ∑a : Mask n, maskWeight q (Fin.cons b a)*F (traceOfMask x (Fin.cons b a)))=_
-    simp_rw [maskWeight_cons,traceOfMask_cons]
-    simp only [Bool.false_eq_true,ite_false,ite_true,Fintype.sum_bool]
-    simp only [mul_assoc,← Finset.mul_sum]
-    ring
-
-theorem listDeletionExpectation_append (q : ℝ) (x y : List Bool) (F : List Bool → ℝ) :
-    listDeletionExpectation q (x++y) F =
-      listDeletionExpectation q x (fun t => listDeletionExpectation q y (fun u => F (t++u))) := by
-  induction x generalizing F with
-  | nil => simp [listDeletionExpectation]
-  | cons b x ih => simp only [List.cons_append,listDeletionExpectation,ih]
-
-/-- The primitive alphabet consists of the literal strings 01 and 1. -/
-def primitiveBlock (b : Bool) : List Bool := if b then [true] else [false,true]
-
-theorem primitiveBlock_expectation (q : ℝ) (b : Bool) (F : List Bool → ℝ) :
-    listDeletionExpectation q (primitiveBlock b) F =
-      ∑z : PrimitiveMark, (if b then defectPrimitiveMass q z else ordinaryPrimitiveMass q z)*F (primitiveOutput z) := by
-  cases b <;> simp [primitiveBlock,listDeletionExpectation,primitiveOutput,
-    defectPrimitiveMass,ordinaryPrimitiveMass,Fin.sum_univ_succ]
-  ring
-
-theorem primitiveBlock_erasure_expectation {q : ℝ} (hq : q∈Set.Ioo 0 1)
-    (b : Bool) (F : List Bool → ℝ) :
-    listDeletionExpectation q (primitiveBlock b) F =
-      primitiveErasure q * F []+(1-primitiveErasure q)*
-        (primitiveMarkLaw q hq b).expect (fun z => F (primitiveOutput z)) := by
-  rw [primitiveBlock_expectation]
-  simp_rw [← primitiveMark_simulation hq]
-  simp only [add_mul,Finset.sum_add_distrib]
-  simp only [ite_mul,zero_mul,Finset.sum_ite_eq',Finset.mem_univ,ite_true,
-    primitiveOutput,Matrix.cons_val_zero]
-  simp only [FiniteLaw.expect,primitiveMarkLaw,mul_assoc,← Finset.mul_sum]
-
-/-- Independent silent block erasures and conditionally emitted primitive marks.
- An emitted empty mark is retained in the list and differs from a silent erasure. -/
-def silentMarkExpectation (q : ℝ) : List Bool → (List PrimitiveMark → ℝ) → ℝ
-  | [], F => F []
-  | b::bs, F => primitiveErasure q*silentMarkExpectation q bs F +
-      (1-primitiveErasure q)*∑z : PrimitiveMark, primitiveMarkMass q b z*
-        silentMarkExpectation q bs (fun ys => F (z::ys))
-
-def joinPrimitiveMarks (ys : List PrimitiveMark) : List Bool := (ys.map primitiveOutput).flatten
-
-theorem silentMarkExpectation_linear (q : ℝ) (bs : List Bool)
-    (F G : List PrimitiveMark → ℝ) (a b : ℝ) :
-    silentMarkExpectation q bs (fun ys => a*F ys+b*G ys)=
-      a*silentMarkExpectation q bs F+b*silentMarkExpectation q bs G := by
-  induction bs generalizing F G with
-  | nil => rfl
-  | cons x xs ih =>
-    simp only [silentMarkExpectation,ih,mul_add,Finset.sum_add_distrib]
-    have hf : (∑ z : PrimitiveMark, primitiveMarkMass q x z *
-        (a*silentMarkExpectation q xs (fun ys => F (z::ys)))) =
-        a*∑ z : PrimitiveMark, primitiveMarkMass q x z * silentMarkExpectation q xs (fun ys => F (z::ys)) := by
-      rw [Finset.mul_sum]; apply Finset.sum_congr rfl; intro z _; ring
-    have hg : (∑ z : PrimitiveMark, primitiveMarkMass q x z *
-        (b*silentMarkExpectation q xs (fun ys => G (z::ys)))) =
-        b*∑ z : PrimitiveMark, primitiveMarkMass q x z * silentMarkExpectation q xs (fun ys => G (z::ys)) := by
-      rw [Finset.mul_sum]; apply Finset.sum_congr rfl; intro z _; ring
-    rw [hf,hg]; ring
-
-theorem silentMarkExpectation_sum {ι : Type*} [Fintype ι] (q : ℝ) (bs : List Bool)
-    (F : ι → List PrimitiveMark → ℝ) :
-    silentMarkExpectation q bs (fun ys => ∑i, F i ys)=∑i, silentMarkExpectation q bs (F i) := by
-  induction bs generalizing F with
-  | nil => rfl
-  | cons x xs ih =>
-    simp only [silentMarkExpectation,ih,Finset.mul_sum,Finset.sum_add_distrib]
-    congr 1
-    rw [Finset.sum_comm]
-
-theorem silentMarkExpectation_scale (q : ℝ) (bs : List Bool)
-    (F : List PrimitiveMark → ℝ) (a : ℝ) :
-    silentMarkExpectation q bs (fun ys => a*F ys)=a*silentMarkExpectation q bs F := by
-  have h := silentMarkExpectation_linear q bs F F a 0
-  simpa using h
-
-/-- The common deterministic mark-joining simulator is exact for arbitrary
- concatenations of primitive blocks; block boundaries are not reported. -/
-theorem primitive_run_simulation {q : ℝ} (hq : q∈Set.Ioo 0 1)
-    (bs : List Bool) (F : List Bool → ℝ) :
-    listDeletionExpectation q ((bs.map primitiveBlock).flatten) F =
-      silentMarkExpectation q bs (fun ys => F (joinPrimitiveMarks ys)) := by
-  induction bs generalizing F with
-  | nil => rfl
-  | cons b bs ih =>
-    simp only [List.map_cons,List.flatten_cons]
-    rw [listDeletionExpectation_append,primitiveBlock_erasure_expectation hq]
-    simp only [List.nil_append,ih,FiniteLaw.expect,primitiveMarkLaw]
-    simp only [silentMarkExpectation]
-    rfl
-
-end TraceReconstruction
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- Every emitted mark is counted, including the empty primitive output. -/
-def emittedMarkExpectation (q : ℝ) : List Bool → (List PrimitiveMark → ℝ) → ℝ
-  | [], F => F []
-  | b::bs, F => ∑z : PrimitiveMark, primitiveMarkMass q b z*
-      emittedMarkExpectation q bs (fun ys => F (z::ys))
-
-theorem emittedMarkExpectation_ofFn {n : ℕ} (q : ℝ) (θ : Fin n → Bool)
-    (F : List PrimitiveMark → ℝ) :
-    emittedMarkExpectation q (List.ofFn θ) F=
-      ∑x : Fin n → PrimitiveMark, (∏i, primitiveMarkMass q (θ i) (x i))*F (List.ofFn x) := by
-  induction n generalizing F with
-  | zero => simp [emittedMarkExpectation]
-  | succ n ih =>
-    rw [List.ofFn_succ,emittedMarkExpectation]
-    simp_rw [ih,Finset.mul_sum]
-    rw [← (Fin.consEquiv (fun _ : Fin (n+1) => PrimitiveMark)).sum_comp,Fintype.sum_prod_type]
-    apply Finset.sum_congr rfl
-    intro z _
-    apply Finset.sum_congr rfl
-    intro x _
-    simp [Fin.consEquiv,Fin.prod_univ_succ,List.ofFn_succ,mul_assoc]
-
-/-- Conditioning the independent erasures on the return count chooses
- a uniformly random subset of the original block slots. -/
-theorem silentMarkExpectation_mask_mixture {n : ℕ} (q : ℝ) (θ : Fin n → Bool)
-    (F : List PrimitiveMark → ℝ) :
-    silentMarkExpectation q (List.ofFn θ) F=
-      ∑a : Mask n, maskWeight (primitiveErasure q) a*
-        emittedMarkExpectation q (traceOfMask θ a) F := by
-  induction n generalizing F with
-  | zero => simp [silentMarkExpectation,emittedMarkExpectation,maskWeight,traceOfMask]
-  | succ n ih =>
-    rw [List.ofFn_succ,silentMarkExpectation]
-    simp_rw [ih]
-    symm
-    rw [← (Fin.consEquiv (fun _ : Fin (n+1) => Bool)).sum_comp,Fintype.sum_prod_type]
-    change (∑b : Bool, ∑a : Mask n, maskWeight (primitiveErasure q) (Fin.cons b a)*
-      emittedMarkExpectation q (traceOfMask θ (Fin.cons b a)) F)=_
-    simp_rw [maskWeight_cons,traceOfMask_cons]
-    simp only [Fintype.sum_bool,ite_true,Bool.false_eq_true,ite_false,emittedMarkExpectation]
-    simp_rw [mul_assoc,Finset.mul_sum]
-    rw [Finset.sum_comm,add_comm]
-    congr 1
-    apply Finset.sum_congr rfl
-    intro z _
-    apply Finset.sum_congr rfl
-    intro a _
-    ring
-
-def maskSubset {n : ℕ} (a : Mask n) : Finset (Fin n) := Finset.univ.filter fun i => a i
-
-def maskSubsetEquiv (n : ℕ) : Mask n ≃ Finset (Fin n) where
-  toFun := maskSubset
-  invFun S i := decide (i∈S)
-  left_inv a := by funext i; simp [maskSubset]
-  right_inv S := by ext i; simp [maskSubset]
-
-theorem maskWeight_card {n : ℕ} (q : ℝ) (a : Mask n) :
-    maskWeight q a = (1-q)^(maskSubset a).card*q^(n-(maskSubset a).card) := by
-  unfold maskWeight maskSubset
-  rw [Finset.prod_ite]
-  simp only [Finset.prod_const]
-  congr 1
-  have h := Finset.card_filter_add_card_filter_not (s := (Finset.univ : Finset (Fin n))) (p := fun i => a i)
-  rw [Finset.card_univ,Fintype.card_fin] at h
-  rw [show (Finset.univ.filter fun i : Fin n => ¬a i=true).card=
-      n-(Finset.univ.filter fun i : Fin n => a i=true).card by omega]
-
-theorem slotAtRank_ofFn {n l : ℕ} (S : FixedSubset (Fin n) l) :
-    List.ofFn (slotAtRank S)=S.val.sort := by
-  apply List.ext_getElem
-  · simp [S.property]
-  · intro i hi hj
-    simp only [List.getElem_ofFn]
-    rfl
-
-theorem filtered_finRange_eq_sort {n : ℕ} (S : Finset (Fin n)) :
-    ((List.finRange n).filter fun i => decide (i∈S))=S.sort := by
-  apply ((List.sortedLT_finRange n).pairwise.filter _).eq_of_mem_iff S.sortedLT_sort.pairwise
-  intro a
-  simp
-
-theorem selected_types_ofFn {n l : ℕ} (θ : Fin n → Bool) (S : FixedSubset (Fin n) l) :
-    traceOfMask θ (fun i => decide (i∈S.val))=List.ofFn (fun v => θ (slotAtRank S v)) := by
-  rw [traceOfMask,filtered_finRange_eq_sort,← slotAtRank_ofFn S,List.map_ofFn]
-  rfl
-
-theorem silentMarkExpectation_subset_mixture {n : ℕ} (q : ℝ) (θ : Fin n → Bool)
-    (F : List PrimitiveMark → ℝ) :
-    silentMarkExpectation q (List.ofFn θ) F=
-      ∑S : Finset (Fin n), subsetWeight (1-primitiveErasure q) S*
-        emittedMarkExpectation q (traceOfMask θ (fun i => decide (i∈S))) F := by
-  rw [silentMarkExpectation_mask_mixture,← (maskSubsetEquiv n).symm.sum_comp]
-  apply Finset.sum_congr rfl
-  intro S _
-  rw [maskWeight_card]
-  change (1-primitiveErasure q)^(maskSubsetEquiv n ((maskSubsetEquiv n).symm S)).card *
-      (primitiveErasure q)^(n-(maskSubsetEquiv n ((maskSubsetEquiv n).symm S)).card)*
-      emittedMarkExpectation q (traceOfMask θ (fun i => decide (i∈S))) F=_
-  rw [Equiv.apply_symm_apply]
-  simp [subsetWeight]
-
-def cardinalSubsetEquiv (n : ℕ) : Finset (Fin n) ≃ Σl : Fin (n+1), FixedSubset (Fin n) l.val where
-  toFun S := ⟨⟨S.card, by have := Finset.card_le_univ S; simpa using Nat.lt_succ_of_le this⟩,⟨S,rfl⟩⟩
-  invFun x := x.2.val
-  left_inv _ := rfl
-  right_inv x := by
-    rcases x with ⟨⟨l,hl⟩,⟨S,hS⟩⟩
-    dsimp
-    change S.card=l at hS
-    subst l
-    rfl
-
-theorem subsetWeight_count_factor {n l : ℕ} (hl : l ≤ n) (p : ℝ) (S : FixedSubset (Fin n) l) :
-    subsetWeight p S.val=binomialMass n p l*(uniformSubsetLaw l (by simpa using hl)).mass S := by
-  have hc : (n.choose l:ℝ)≠0 := by exact_mod_cast (Nat.choose_pos hl).ne'
-  simp only [subsetWeight,binomialMass,uniformSubsetLaw,S.property,Fintype.card_fin]
-  field_simp
-
-/-- The literal conditional-list oracle used by the rank coefficients. -/
-def conditionalPrimitiveLaw {n l : ℕ} (hl : l ≤ n) (q : ℝ) (hq : q∈Set.Ioo 0 1)
-    (θ : Fin n → Bool) : FiniteLaw (Fin l → PrimitiveMark) :=
-  (uniformSubsetLaw l (by simpa using hl)).mixture fun S =>
-    FiniteLaw.coordinateProduct (fun v => primitiveMarkLaw q hq (θ (slotAtRank S v)))
-
-theorem silentMarkExpectation_count_mixture {n : ℕ} (q : ℝ) (hq : q∈Set.Ioo 0 1)
-    (θ : Fin n → Bool) (F : List PrimitiveMark → ℝ) :
-    silentMarkExpectation q (List.ofFn θ) F=
-      ∑l : Fin (n+1), binomialMass n (1-primitiveErasure q) l.val*
-        (conditionalPrimitiveLaw (Nat.le_of_lt_succ l.isLt) q hq θ).expect (fun x => F (List.ofFn x)) := by
-  rw [silentMarkExpectation_subset_mixture,← (cardinalSubsetEquiv n).symm.sum_comp,Fintype.sum_sigma]
-  apply Finset.sum_congr rfl
-  intro l _
-  change (∑S : FixedSubset (Fin n) l.val, subsetWeight (1-primitiveErasure q) S.val*
-      emittedMarkExpectation q (traceOfMask θ (fun i => decide (i∈S.val))) F)=_
-  calc
-    _ = ∑S : FixedSubset (Fin n) l.val, binomialMass n (1-primitiveErasure q) l.val*
-        (uniformSubsetLaw l.val (by simpa using Nat.le_of_lt_succ l.isLt)).mass S*
-        (∑x : Fin l.val → PrimitiveMark, (∏v, primitiveMarkMass q (θ (slotAtRank S v)) (x v))*F (List.ofFn x)) := by
-      apply Finset.sum_congr rfl
-      intro S _
-      rw [subsetWeight_count_factor (Nat.le_of_lt_succ l.isLt),selected_types_ofFn θ S,emittedMarkExpectation_ofFn]
-    _ = _ := by
-      unfold conditionalPrimitiveLaw FiniteLaw.expect FiniteLaw.mixture FiniteLaw.coordinateProduct primitiveMarkLaw
-      simp_rw [Finset.sum_mul,Finset.mul_sum]
-      rw [Finset.sum_comm]
-      apply Finset.sum_congr rfl
-      intro x _
-      apply Finset.sum_congr rfl
-      intro S _
-      ring
-
-end TraceReconstruction
-
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- Density of the actual uniform-subset primitive oracle, over the ordinary
- mark product law. This is exactly the channel defined by silent erasures. -/
-theorem conditionalPrimitiveLaw_density {n l : ℕ} (hl : l ≤ n)
-    {q : ℝ} (hq : q∈Set.Ioo 0 1) (θ : Fin n → Bool) (x : Fin l → PrimitiveMark) :
-    (conditionalPrimitiveLaw hl q hq θ).mass x =
-      ((primitiveMarkLaw q hq false).product l).mass x *
-        averagedDensity (uniformSubsetLaw l (by simpa using hl))
-          (fun S v => θ (slotAtRank S v)) (primitiveScore q) x := by
-  have he : (fun S : FixedSubset (Fin n) l =>
-      FiniteLaw.coordinateProduct (fun v => primitiveMarkLaw q hq (θ (slotAtRank S v)))) =
-      (fun S : FixedSubset (Fin n) l =>
-      FiniteLaw.coordinateProduct (fun v => if θ (slotAtRank S v) then
-        primitiveMarkLaw q hq true else primitiveMarkLaw q hq false)) := by
-    funext S
-    congr 1
-    funext v
-    cases θ (slotAtRank S v) <;> rfl
-  unfold conditionalPrimitiveLaw
-  rw [he]
-  exact leaf_oracle_density (primitiveMarkLaw q hq false) (primitiveMarkLaw q hq true)
-    (primitiveScore q) (primitive_score_identity hq)
-    (uniformSubsetLaw l (by simpa using hl)) (fun S v => θ (slotAtRank S v)) x
-
-def defectTypes {n u : ℕ} (d : Fin u → Fin n) (i : Fin n) : Bool := decide (i∈Set.range d)
-
-theorem defectTypes_atRank {n l u : ℕ} (d : Fin u → Fin n) (S : FixedSubset (Fin n) l) :
-    (fun v => defectTypes d (slotAtRank S v))=defectAtRank (Finset.univ.image d) S := by
-  funext v
-  simp [defectTypes,defectAtRank,Set.mem_range]
-
-theorem conditionalDefectLaw_density {n l u : ℕ} (hl : l ≤ n)
-    {q : ℝ} (hq : q∈Set.Ioo 0 1) (d : Fin u → Fin n) (x : Fin l → PrimitiveMark) :
-    (conditionalPrimitiveLaw hl q hq (defectTypes d)).mass x =
-      ((primitiveMarkLaw q hq false).product l).mass x *
-        multilinear (averagedCoefficient (uniformSubsetLaw l (by simpa using hl))
-          (defectAtRank (Finset.univ.image d))) (primitiveScore q) x := by
-  rw [conditionalPrimitiveLaw_density,show (fun S v => defectTypes d (slotAtRank S v))=
-      defectAtRank (Finset.univ.image d) by funext S; exact defectTypes_atRank d S,
-      averagedDensity_expansion]
-
-/-- The finite-support law of the hidden motif anchor, not an additional
- distributional assumption or a changed channel parameter. -/
-def seriesProbabilityLaw (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X)
-    (hπsum : ∑X∈π.coeff.support, π.coeff X=1) : FiniteLaw ↥π.coeff.support where
-  mass X := π.coeff X.val
-  nonneg X := hπ X.val
-  total := by simpa only [Finset.univ_eq_attach,Finset.sum_attach] using hπsum
-
-def mixedMotifCoefficient {u m l : ℕ} (hl : l ≤ u*m) (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
-    (π : IntSeries) (s : ℕ) (τ : ℝ) (σ : Fin l → Bool) : ℝ :=
-  ∑X∈π.coeff.support, π.coeff X*
-    averagedCoefficient (uniformSubsetLaw l (by simpa using hl))
-      (defectAtRank (Finset.univ.image (signedPeriodDefects off hsafe s X τ))) σ
-
-def conditionalMotifLaw {u m l : ℕ} (hl : l ≤ u*m) (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
-    {q : ℝ} (hq : q∈Set.Ioo 0 1) (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X)
-    (hπsum : ∑X∈π.coeff.support, π.coeff X=1) (s : ℕ) (τ : ℝ) :
-    FiniteLaw (Fin l → PrimitiveMark) :=
-  (seriesProbabilityLaw π hπ hπsum).mixture fun X =>
-    conditionalPrimitiveLaw hl q hq (defectTypes (signedPeriodDefects off hsafe s X.val τ))
-
-theorem conditionalMotifLaw_density {u m l : ℕ} (hl : l ≤ u*m) (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
-    {q : ℝ} (hq : q∈Set.Ioo 0 1) (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X)
-    (hπsum : ∑X∈π.coeff.support, π.coeff X=1) (s : ℕ) (τ : ℝ) (x : Fin l → PrimitiveMark) :
-    (conditionalMotifLaw hl off hsafe hq π hπ hπsum s τ).mass x=
-      ((primitiveMarkLaw q hq false).product l).mass x *
-        multilinear (mixedMotifCoefficient hl off hsafe π s τ) (primitiveScore q) x := by
-  unfold conditionalMotifLaw FiniteLaw.mixture seriesProbabilityLaw
-  simp_rw [conditionalDefectLaw_density]
-  let F : ℤ → ℝ := fun X => π.coeff X *
-    (((primitiveMarkLaw q hq false).product l).mass x *
-      multilinear (averagedCoefficient (uniformSubsetLaw l (by simpa using hl))
-        (defectAtRank (Finset.univ.image (signedPeriodDefects off hsafe s X τ))))
-        (primitiveScore q) x)
-  change (∑ X : ↥π.coeff.support, F X.val) = _
-  rw [Finset.sum_coe_sort]
-  dsimp only [F]
-  unfold mixedMotifCoefficient multilinear
-  simp_rw [Finset.mul_sum,Finset.sum_mul]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro σ _
-  rw [Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro X _
-  ring
-
-theorem mixedMotifCoefficient_rank_difference {u m l r : ℕ} (hl : l ≤ u*m) (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
-    (π : IntSeries) (s : ℕ) (v : IncreasingTuple l r) :
-    mixedMotifCoefficient hl off hsafe π s 1 (rankMask v)-
-      mixedMotifCoefficient hl off hsafe π s (-1) (rankMask v)=
-        signedRankCoefficient hl off hsafe π s v := by
-  unfold mixedMotifCoefficient signedRankCoefficient
-  simp_rw [averaged_rank_coefficient hl _ v]
-  rw [← Finset.sum_sub_distrib]
-  apply Finset.sum_congr rfl
-  intro X _
-  ring
-
-end TraceReconstruction
-
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-def unchangedPattern (r : ℕ) : Fin r → GapDifference := fun _ => .unchanged
-
-theorem unchanged_positional_value {r : ℕ} (W : (Fin r → Fin 3) → ℝ) :
-    (∑ a : Fin r → Fin 3, (∏ h, positionalWeight (unchangedPattern r h) (a h))*W a)=
-      W (fun _ => 1) := by
-  simp only [unchangedPattern,positionalWeight]
-  simp_rw [product_function_delta]
-  simp
-
-/-- Comparing a signed displaced motif with the literal centered pure run
- retains every nonconstant positional pattern, not just odd ones. -/
-theorem motif_pure_shift_expansion {r : ℕ} (χ β : Fin r → ℝ)
-    (hβ : ∀h, IsMotifValue (χ h) (β h)) (W : (Fin r → Fin 3) → ℝ) :
-    W (fun h => motifShift (χ h) (β h) 1)-W (fun _ => 1)=
-      ∑ k : Fin r → GapDifference, if k=unchangedPattern r then 0 else
-        (∏h, motifDifferenceWeight (χ h) (β h) 1 (k h))*
-          (∑a : Fin r → Fin 3, (∏h, positionalWeight (k h) (a h))*W a) := by
-  rw [motif_shift_expansion χ β 1 hβ (Or.inl rfl) W]
-  have he : W (fun _ => 1)=∑k : Fin r → GapDifference,
-      if k=unchangedPattern r then
-        (∑a : Fin r → Fin 3, (∏h, positionalWeight (k h) (a h))*W a) else 0 := by
-    simp [unchanged_positional_value]
-  rw [he,← Finset.sum_sub_distrib]
-  apply Finset.sum_congr rfl
-  intro k _
-  by_cases hk : k=unchangedPattern r
-  · subst k
-    simp [unchangedPattern,motifDifferenceWeight]
-  · simp [hk]
-
-theorem mixed_pure_shift_expansion {r : ℕ} (π : IntSeries) (s : ℕ)
-    (j : Fin r → ℤ) (W : (Fin r → Fin 3) → ℝ) :
-    (∑X∈π.coeff.support, π.coeff X *
-      (W (fun h => motifShift (motifIndicator s (j h-X)) ((prouhetSeries s).coeff (j h-X)) 1)-
-       W (fun _ => 1)))=
-      ∑k : Fin r → GapDifference, if k=unchangedPattern r then 0 else
-        motifPatternMultiplier π s j k *
-          (∑a : Fin r → Fin 3, (∏h, positionalWeight (k h) (a h))*W a) := by
-  simp_rw [motif_pure_shift_expansion _ _ (fun h => prouhet_motif_value s (j h-_)) W,
-    Finset.mul_sum]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro k _
-  by_cases hk : k=unchangedPattern r
-  · simp [hk]
-  · simp only [ite_eq_right hk,motifPatternMultiplier,Finset.sum_mul,mul_assoc]
-    simp_rw [Finset.mul_sum]
-    exact Finset.sum_comm
-
-theorem pattern_order_zero {r : ℕ} (k : Fin r → GapDifference)
-    (hk : (∑h, (k h).order)=0) : k=unchangedPattern r := by
-  funext h
-  have hh := Finset.single_le_sum (fun i (_ : i∈(Finset.univ : Finset (Fin r))) =>
-    Nat.zero_le (k i).order) (Finset.mem_univ h)
-  rw [hk] at hh
-  have hz : (k h).order=0 := by omega
-  cases he : k h <;> simp_all [GapDifference.order,unchangedPattern]
-
-theorem nonconstant_pattern_linear_or_order_two {r : ℕ} (k : Fin r → GapDifference)
-    (hk : k≠unchangedPattern r) : (∃h, k=linearPattern h) ∨ 2≤∑h,(k h).order := by
-  by_cases hlin : ∃h, k=linearPattern h
-  · exact Or.inl hlin
-  right
-  by_contra hsmall
-  have horder : (∑h,(k h).order)=1 := by
-    have hn : (∑h,(k h).order)≠0 := fun he => hk (pattern_order_zero k he)
-    omega
-  have hc := difference_order_eq_counts k
-  have hfirst : firstDifferenceCount k=1 := by omega
-  have hsecond : secondDifferenceCount k=0 := by omega
-  exact hlin (first_second_counts_linear k hfirst hsecond)
-
-
-/-- A nonconstant motif pattern always has an actual involved position. -/
-theorem nonconstant_pattern_position {r : ℕ} (k : Fin r → GapDifference)
-    (hk : k≠unchangedPattern r) : ∃a, k a≠.unchanged := by
-  by_contra h
-  push Not at h
-  exact hk (funext h)
-
-theorem clipped_pure_nonlinear_pattern_norm {u m l r : ℕ}
-    (hm : 32 ≤ m) (hr : 0 < r) (hl : l ≤ u*m)
-    (off : Fin m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 3)
-    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
-    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (s : ℕ) (hcentral : CentralMotifSupport u s π)
-    {θ : ℝ} (hθ : θ∈Set.Ioo 0 1) (hhalf : 1/2 ≤ θ)
-    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
-    (k : Fin r → GapDifference) (hk : k≠unchangedPattern r) :
-    (∑v : IncreasingTuple l r, (clippedPatternCoefficient hl off hsafe π s θ k v)^2) ≤
-      periodDifferencePrefactor u m r (∑h, (k h).order) θ *
-        periodCentralSumConstant u r * seriesL2Sq (geomSeries (2^s)*π) := by
-  obtain ⟨a,ha⟩ := nonconstant_pattern_position k hk
-  have hb := period_pattern_norm hm hr hl (fun _ => off) (fun _ => hoff)
-    (by intro _ _; simp) (fun j => hsafe.comp j.val j.property) k
-    hθ hhalf hmean hv a
-    (fun j => motifPatternMultiplier π s (fun i => (j.val i).val) k)
-    (fun h => (geomSeries (2^s)*π).coeff h.val)
-    (fun j => motifPatternMultiplier_sq_le π hπ s _ k a ha)
-    (unsigned_convolution_central π hcentral)
-  apply hb.trans
-  apply mul_le_mul_of_nonneg_left
-    (seriesL2Sq_partial _ (fun h : Fin u => (h.val:ℤ)) (by intro a b h; apply Fin.ext; change (a.val:ℤ)=b.val at h; exact_mod_cast h))
-  unfold periodDifferencePrefactor lengthDifferenceConstant periodCentralSumConstant
-  positivity
-
-def purePatternBound (u m r s : ℕ) (π : IntSeries) (θ : ℝ)
-    (k : Fin r → GapDifference) : ℝ :=
-  if ∃a, k=linearPattern a then
-    periodDifferencePrefactor u m r 1 θ * periodCentralSumConstant u r * seriesL2Sq (prouhetSeries s*π)
-  else periodDifferencePrefactor u m r (∑h, (k h).order) θ *
-    periodCentralSumConstant u r * seriesL2Sq (geomSeries (2^s)*π)
-
-/-- Complete clipped energy, comparing the signed motif to a pure run. -/
-theorem clipped_pure_initial_norm {u m l r : ℕ}
-    (hm : 32 ≤ m) (hr : 0 < r) (hl : l ≤ u*m)
-    (off : Fin m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 3)
-    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
-    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (s : ℕ) (hcentral : CentralMotifSupport u s π)
-    {θ : ℝ} (hθ : θ∈Set.Ioo 0 1) (hhalf : 1/2 ≤ θ)
-    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ)) :
-    (∑v : IncreasingTuple l r, (∑k : Fin r → GapDifference,
-      if k=unchangedPattern r then 0 else clippedPatternCoefficient hl off hsafe π s θ k v)^2) ≤
-      (Fintype.card (Fin r → GapDifference):ℝ)*
-        ∑k : Fin r → GapDifference, if k=unchangedPattern r then 0 else
-          purePatternBound u m r s π θ k := by
-  apply (sum_square_card_bound (fun (k : Fin r → GapDifference) (v : IncreasingTuple l r) =>
-    if k=unchangedPattern r then 0 else clippedPatternCoefficient hl off hsafe π s θ k v)).trans
-  apply mul_le_mul_of_nonneg_left _ (by positivity)
-  apply Finset.sum_le_sum
-  intro k _
-  by_cases hk : k=unchangedPattern r
-  · simp [hk]
-  · simp only [ite_eq_right hk]
-    unfold purePatternBound
-    split_ifs with hlin
-    · obtain ⟨a,rfl⟩ := hlin
-      exact clipped_linear_pattern_norm hm hr hl off hoff hsafe π s hcentral hθ hhalf hmean hv a
-    · exact clipped_pure_nonlinear_pattern_norm hm hr hl off hoff hsafe π hπ s hcentral
-        hθ hhalf hmean hv k hk
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem shiftedTag_center {n r : ℕ} (p : Fin r → Fin n) (hp : SafeTaggedTuple p) :
-    shiftedTag p hp (fun _ => 1)=p := by
-  funext i
-  apply Fin.ext
-  simp [shiftedTag]
-
-def pureRankCoefficient {u m l r : ℕ} (hl : l ≤ u*m) (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
-    (π : IntSeries) (s : ℕ) (v : IncreasingTuple l r) : ℝ :=
-  ∑X∈π.coeff.support, π.coeff X *
-    (rankDefectCoefficient hl (signedPeriodDefects off hsafe s X 1) v -
-     rankDefectCoefficient hl (periodSlot (fun _ => off)) v)
-
-def pureTupleKernel {u m l r : ℕ} (hl : l ≤ u*m) (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
-    (π : IntSeries) (s : ℕ) (j : IncreasingTuple u r) (v : IncreasingTuple l r) : ℝ :=
-  ∑X∈π.coeff.support, π.coeff X *
-    (subsetRankKernel hl (signedPeriodDefects off hsafe s X 1 ∘ j.val) v.val -
-     subsetRankKernel hl (periodSlot (fun _ => off) ∘ j.val) v.val)
-
-theorem pureRankCoefficient_tuple_sum {u m l r : ℕ} (hl : l ≤ u*m) (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off))) (π : IntSeries) (s : ℕ)
-    (v : IncreasingTuple l r) :
-    pureRankCoefficient hl off hsafe π s v=∑j, pureTupleKernel hl off hsafe π s j v := by
-  unfold pureRankCoefficient pureTupleKernel
-  simp_rw [rankDefectCoefficient_eq _ _ (signedPeriodDefects_strictMono _ _ _ _ _),
-    rankDefectCoefficient_eq _ _ (periodSlot_strictMono _),
-    ← Finset.sum_sub_distrib,Finset.mul_sum]
-  exact Finset.sum_comm
-
-theorem pureTupleKernel_expansion {u m l r : ℕ} (hl : l ≤ u*m) (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off))) (π : IntSeries) (s : ℕ)
-    (j : IncreasingTuple u r) (v : IncreasingTuple l r) :
-    pureTupleKernel hl off hsafe π s j v = ∑k : Fin r → GapDifference,
-      if k=unchangedPattern r then 0 else
-        motifPatternMultiplier π s (fun i => (j.val i).val) k *
-          positionalRankDifference hl (periodSlot (fun _ => off) ∘ j.val)
-            (hsafe.comp j.val j.property) k v := by
-  unfold pureTupleKernel
-  simp_rw [signedPeriodDefects_comp]
-  have H := mixed_pure_shift_expansion π s (fun i => (j.val i).val)
-    (fun a => subsetRankKernel hl
-      (shiftedTag (periodSlot (fun _ => off) ∘ j.val) (hsafe.comp j.val j.property) a) v.val)
-  simpa only [shiftedTag_center,positionalRankDifference] using H
-
-def clippedPureRankCoefficient {u m l r : ℕ} (hl : l ≤ u*m) (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
-    (π : IntSeries) (s : ℕ) (θ : ℝ) (v : IncreasingTuple l r) : ℝ :=
-  ∑j, if PeriodRankRegion m θ j v then pureTupleKernel hl off hsafe π s j v else 0
-
-theorem clippedPureRankCoefficient_expansion {u m l r : ℕ} (hl : l ≤ u*m) (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
-    (π : IntSeries) (s : ℕ) (θ : ℝ) (v : IncreasingTuple l r) :
-    clippedPureRankCoefficient hl off hsafe π s θ v =
-      ∑k : Fin r → GapDifference, if k=unchangedPattern r then 0 else
-        clippedPatternCoefficient hl off hsafe π s θ k v := by
-  unfold clippedPureRankCoefficient clippedPatternCoefficient
-  simp_rw [pureTupleKernel_expansion]
-  have he (j : IncreasingTuple u r) :
-      (if PeriodRankRegion m θ j v then ∑k : Fin r → GapDifference,
-        if k=unchangedPattern r then 0 else
-          motifPatternMultiplier π s (fun i => (j.val i).val) k *
-            positionalRankDifference hl (periodSlot (fun _ => off) ∘ j.val)
-              (hsafe.comp j.val j.property) k v else 0)=
-      ∑k : Fin r → GapDifference, if k=unchangedPattern r then 0 else
-        (if PeriodRankRegion m θ j v then
-          motifPatternMultiplier π s (fun i => (j.val i).val) k *
-            positionalRankDifference hl (periodSlot (fun _ => off) ∘ j.val)
-              (hsafe.comp j.val j.property) k v else 0) := by
-    by_cases hv : PeriodRankRegion m θ j v <;> simp [hv]
-  simp_rw [he]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro k _
-  by_cases hk : k=unchangedPattern r <;> simp [hk]
-
-theorem pureTupleKernel_tail {u m l r : ℕ}
-    (hu : 0 < u) (hm : 128 ≤ m) (hr : 0 < r) (hl : l ≤ u*m)
-    (off : Fin m) (hoff : |(off.val:ℝ)-(m:ℝ)/2|≤3)
-    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
-    (π : IntSeries) (hπ : ∀X, 0≤π.coeff X) (hπsum : ∑X∈π.coeff.support, π.coeff X=1)
-    (s : ℕ) (j : IncreasingTuple u r) {θ : ℝ} (hθ : θ∈Set.Ioo 0 1) :
-    (∑v : IncreasingTuple l r, if PeriodRankRegion m θ j v then 0 else
-      |pureTupleKernel hl off hsafe π s j v|) ≤
-      2*(((r+1:ℕ):ℝ)*θ^r/binomialMass (u*m) θ l * (2*Real.exp (-(m:ℝ)/(1024*u)))) := by
-  let K (X : ℤ) (τ : ℝ) (v : IncreasingTuple l r) :=
-    subsetRankKernel hl (if τ=1 then signedPeriodDefects off hsafe s X 1 ∘ j.val
-      else periodSlot (fun _ => off) ∘ j.val) v.val
-  let B := ((r+1:ℕ):ℝ)*θ^r/binomialMass (u*m) θ l * (2*Real.exp (-(m:ℝ)/(1024*u)))
-  have hK (X : ℤ) (τ : ℝ) : (∑v : IncreasingTuple l r,
-      if PeriodRankRegion m θ j v then 0 else K X τ v)≤B := by
-    dsimp [K]
-    split_ifs
-    · rw [signedPeriodDefects_comp]
-      exact period_shifted_outside_region hu hm hr hl (fun _ => off) (fun _ => hoff)
-        (by intro _ _; simp) j (hsafe.comp j.val j.property) _ hθ
-    · have H := period_shifted_outside_region hu hm hr hl (fun _ => off) (fun _ => hoff)
-        (by intro _ _; simp) j (hsafe.comp j.val j.property) (fun _ => 1) hθ
-      simpa only [shiftedTag_center] using H
-  have hnon (X : ℤ) (τ : ℝ) (v : IncreasingTuple l r) : 0≤K X τ v :=
-    subsetRankKernel_nonneg _ _ _
-  calc
-    _ ≤ ∑v : IncreasingTuple l r, ∑X∈π.coeff.support, π.coeff X*
-        (if PeriodRankRegion m θ j v then 0 else K X 1 v+K X (-1) v) := by
-      apply Finset.sum_le_sum
-      intro v _
-      by_cases hv : PeriodRankRegion m θ j v
-      · simp [hv]
-      · simp only [ite_eq_right hv]
-        have he : pureTupleKernel hl off hsafe π s j v=
-            ∑X∈π.coeff.support, π.coeff X*(K X 1 v-K X (-1) v) := by
-          norm_num [pureTupleKernel,K]
-        rw [he]
-        refine (Finset.abs_sum_le_sum_abs _ _).trans ?_
-        apply Finset.sum_le_sum
-        intro X _
-        rw [abs_mul,abs_of_nonneg (hπ X)]
-        apply mul_le_mul_of_nonneg_left _ (hπ X)
-        exact (abs_sub _ _).trans_eq (by rw [abs_of_nonneg (hnon X 1 v),abs_of_nonneg (hnon X (-1) v)])
-    _ = ∑X∈π.coeff.support, π.coeff X*((∑v : IncreasingTuple l r,
-        if PeriodRankRegion m θ j v then 0 else K X 1 v)+
-        (∑v : IncreasingTuple l r, if PeriodRankRegion m θ j v then 0 else K X (-1) v)) := by
-      rw [Finset.sum_comm]
-      apply Finset.sum_congr rfl
-      intro X _
-      rw [← Finset.mul_sum,← Finset.sum_add_distrib]
-      congr 1; apply Finset.sum_congr rfl
-      intro v _; split_ifs <;> ring
-    _ ≤ ∑X∈π.coeff.support, π.coeff X*(B+B) := by
-      apply Finset.sum_le_sum
-      intro X _
-      exact mul_le_mul_of_nonneg_left (add_le_add (hK X 1) (hK X (-1))) (hπ X)
-    _ = _ := by rw [← Finset.sum_mul,hπsum]; dsimp [B]; ring
-
-
-theorem pureRankCoefficient_tail {u m l r : ℕ}
-    (hu : 0 < u) (hm : 128 ≤ m) (hr : 0 < r) (hl : l ≤ u*m)
-    (off : Fin m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 3)
-    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
-    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support, π.coeff X=1)
-    (s : ℕ) {θ : ℝ} (hθ : θ∈Set.Ioo 0 1) :
-    (∑v : IncreasingTuple l r, |pureRankCoefficient hl off hsafe π s v-
-        clippedPureRankCoefficient hl off hsafe π s θ v|) ≤
-      (Fintype.card (IncreasingTuple u r):ℝ)*
-      (2*(((r+1:ℕ):ℝ)*θ^r/binomialMass (u*m) θ l * (2*Real.exp (-(m:ℝ)/(1024*u))))) := by
-  simp_rw [pureRankCoefficient_tuple_sum,clippedPureRankCoefficient,← Finset.sum_sub_distrib]
-  calc
-    _ ≤ ∑v : IncreasingTuple l r, ∑j : IncreasingTuple u r,
-        if PeriodRankRegion m θ j v then 0 else |pureTupleKernel hl off hsafe π s j v| := by
-      apply Finset.sum_le_sum
-      intro v _
-      refine (Finset.abs_sum_le_sum_abs _ _).trans_eq ?_
-      apply Finset.sum_congr rfl
-      intro j _
-      split_ifs <;> simp
-    _ = ∑j : IncreasingTuple u r, ∑v : IncreasingTuple l r,
-        if PeriodRankRegion m θ j v then 0 else |pureTupleKernel hl off hsafe π s j v| := Finset.sum_comm
-    _ ≤ _ := by
-      calc
-        _ ≤ ∑j : IncreasingTuple u r,
-            2*(((r+1:ℕ):ℝ)*θ^r/binomialMass (u*m) θ l * (2*Real.exp (-(m:ℝ)/(1024*u)))) :=
-          Finset.sum_le_sum (fun j _ => pureTupleKernel_tail hu hm hr hl off hoff hsafe π hπ hπsum s j hθ)
-        _ = _ := by simp
-
-theorem initial_pure_energy {u m l r : ℕ}
-    (hu : 0 < u) (hm : 128 ≤ m) (hr : 0 < r) (hl : l ≤ u*m)
-    (off : Fin m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 3)
-    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
-    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support, π.coeff X=1)
-    (s : ℕ) (hcentral : CentralMotifSupport u s π)
-    {θ : ℝ} (hθ : θ∈Set.Ioo 0 1) (hhalf : 1/2 ≤ θ)
-    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
-    :
-    (∑v : IncreasingTuple l r, (pureRankCoefficient hl off hsafe π s v)^2) ≤
-      2*(Fintype.card (Fin r → GapDifference):ℝ)*
-        (∑k : Fin r → GapDifference, if k=unchangedPattern r then 0 else
-          purePatternBound u m r s π θ k) +
-      2*((Fintype.card (IncreasingTuple u r):ℝ)*
-        (2*(((r+1:ℕ):ℝ)*θ^r/binomialMass (u*m) θ l * (2*Real.exp (-(m:ℝ)/(1024*u))))))^2 := by
-  let A : IncreasingTuple l r → ℝ := pureRankCoefficient hl off hsafe π s
-  let C : IncreasingTuple l r → ℝ := clippedPureRankCoefficient hl off hsafe π s θ
-  have hn := clipped_pure_initial_norm (by omega) hr hl off hoff hsafe π hπ s hcentral hθ hhalf hmean hv
-  simp_rw [← clippedPureRankCoefficient_expansion] at hn
-  have ht := pureRankCoefficient_tail hu hm hr hl off hoff hsafe π hπ hπsum s hθ
-  have hE := (finite_sum_sq_le_abs_sum_sq (fun v : IncreasingTuple l r => A v-C v)).trans
-    (pow_le_pow_left₀ (Finset.sum_nonneg (fun v _ => abs_nonneg (A v-C v))) ht 2)
-  have hsum : (∑v : IncreasingTuple l r, A v^2) ≤
-      2*(∑v : IncreasingTuple l r, C v^2)+2*(∑v : IncreasingTuple l r, (A v-C v)^2) := by
-    simp only [Finset.mul_sum,← Finset.sum_add_distrib]
-    apply Finset.sum_le_sum
-    intro v _
-    nlinarith [sq_nonneg (A v-2*C v)]
-  calc
-    _ ≤ 2*(∑v : IncreasingTuple l r, C v^2)+2*(∑v : IncreasingTuple l r, (A v-C v)^2) := hsum
-    _ ≤ _ := by dsimp [C] at *; linarith
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-def scaleCoefficient {n : ℕ} (A : (Fin n → Bool) → ℝ) (L : ℝ) (σ : Fin n → Bool) : ℝ :=
-  A σ * ∏ i, if σ i then L else 1
-
-theorem multilinear_scale {α : Type*} {n : ℕ} (A : (Fin n → Bool) → ℝ)
-    (Z : α → ℝ) {L : ℝ} (hL : L≠0) (x : Fin n → α) :
-    multilinear (scaleCoefficient A L) (fun a => Z a/L) x=multilinear A Z x := by
-  unfold multilinear scaleCoefficient
-  apply Finset.sum_congr rfl
-  intro σ _
-  rw [mul_assoc,← Finset.prod_mul_distrib]
-  congr 1
-  apply Finset.prod_congr rfl
-  intro i _
-  cases σ i <;> simp
-  exact mul_div_cancel₀ _ hL
-
-theorem multilinearEnergy_scale {n : ℕ} (A : (Fin n → Bool) → ℝ) (L R : ℝ) :
-    multilinearEnergy (scaleCoefficient A L) R=multilinearEnergy A (R*L) := by
-  unfold multilinearEnergy scaleCoefficient
-  apply Finset.sum_congr rfl
-  intro σ _
-  rw [mul_pow,← Finset.prod_pow,mul_assoc,← Finset.prod_mul_distrib]
-  congr 1
-  apply Finset.prod_congr rfl
-  intro i _
-  cases σ i <;> simp
-  ring
-
-/-- The exact norm-of-input factor in the conditional leaf expansion. -/
-theorem multilinear_moment_scaled {α : Type*} [Fintype α] (P : FiniteLaw α)
-    (Z : α → ℝ) (k : ℕ) (hk : 1 ≤ k) (hcenter : P.expect Z=0)
-    {L : ℝ} (hL : 0<L) (hMoment : P.expect (fun a => |Z a|^(2*k))≤L^(2*k))
-    (n : ℕ) (A : (Fin n → Bool) → ℝ) :
-    (P.product n).expect (fun x => |multilinear A Z x|^(2*k)) ≤
-      (multilinearEnergy A ((6*(2*k:ℕ):ℝ)*L))^k := by
-  have hcenter' : P.expect (fun a => Z a/L)=0 := by
-    unfold FiniteLaw.expect
-    simp_rw [← mul_div_assoc]
-    rw [← Finset.sum_div]
-    change P.expect Z / L = 0
-    rw [hcenter,zero_div]
-  have hmoment' : P.expect (fun a => |Z a/L|^(2*k))≤1 := by
-    simp_rw [abs_div,abs_of_pos hL,div_pow]
-    unfold FiniteLaw.expect
-    simp_rw [← mul_div_assoc]
-    rw [← Finset.sum_div]
-    apply (div_le_one (by positivity)).mpr hMoment
-  have H := multilinear_even_moment P (fun a => Z a/L) k hk hcenter' hmoment' n (scaleCoefficient A L)
-  simp_rw [multilinear_scale A Z hL.ne',multilinearEnergy_scale] at H
-  simpa only [pow_mul,sq_abs] using H
-
-theorem multilinear_moment_sq_scaled {α : Type*} [Fintype α] (P : FiniteLaw α)
-    (Z : α → ℝ) (k : ℕ) (hk : 1 ≤ k) (hcenter : P.expect Z=0)
-    {L : ℝ} (hL : 0<L) (hMoment : P.expect (fun a => |Z a|^(2*k))≤L^(2*k))
-    (n : ℕ) (A : (Fin n → Bool) → ℝ) :
-    (P.product n).evenMomentSq (multilinear A Z) k ≤
-      multilinearEnergy A ((6*(2*k:ℕ):ℝ)*L) := by
-  have H := multilinear_moment_scaled P Z k hk hcenter hL hMoment n A
-  have hn : 0≤(P.product n).expect (fun x => |multilinear A Z x|^(2*k)) := by
-    exact Finset.sum_nonneg (fun x _ => mul_nonneg ((P.product n).nonneg x) (by positivity))
-  unfold FiniteLaw.evenMomentSq
-  calc
-    _ ≤ (multilinearEnergy A ((6*(2*k:ℕ):ℝ)*L)^k)^(1/(k:ℝ)) :=
-      Real.rpow_le_rpow hn H (by positivity)
-    _ = _ := by
-      rw [← Real.rpow_natCast,one_div,Real.rpow_rpow_inv (multilinearEnergy_nonneg _ _) (by positivity)]
-
-theorem multilinearEnergy_mixture_le {Ω : Type*} [Fintype Ω] {n : ℕ}
-    (P : FiniteLaw Ω) (A : Ω → (Fin n → Bool) → ℝ) (R : ℝ) :
-    multilinearEnergy (fun σ => P.expect (fun ω => A ω σ)) R ≤
-      P.expect (fun ω => multilinearEnergy (A ω) R) := by
-  unfold multilinearEnergy FiniteLaw.expect
-  simp_rw [Finset.mul_sum]
-  rw [Finset.sum_comm]
-  apply Finset.sum_le_sum
-  intro σ _
-  have H := weighted_sum_sq_le Finset.univ P.mass (fun ω => A ω σ) (fun ω => P.nonneg ω)
-  rw [P.total,one_mul] at H
-  calc
-    _ ≤ (∑ω, P.mass ω*A ω σ^2)*(∏i,if σ i then R^2 else 1) :=
-      mul_le_mul_of_nonneg_right H (Finset.prod_nonneg (fun i _ => by split_ifs <;> positivity))
-    _ = _ := by rw [Finset.sum_mul]; apply Finset.sum_congr rfl; intro ω _; ring
-
-theorem multilinearEnergy_sub_le {n : ℕ} (A B : (Fin n → Bool) → ℝ) (R : ℝ) :
-    multilinearEnergy (fun σ => A σ-B σ) R ≤
-      2*multilinearEnergy A R+2*multilinearEnergy B R := by
-  unfold multilinearEnergy
-  simp_rw [Finset.mul_sum,← Finset.sum_add_distrib]
-  apply Finset.sum_le_sum
-  intro σ _
-  have h := mul_le_mul_of_nonneg_right (show (A σ-B σ)^2≤2*A σ^2+2*B σ^2 by nlinarith [sq_nonneg (A σ+B σ)])
-    (Finset.prod_nonneg (s := Finset.univ) (f := fun i => if σ i then R^2 else 1) (fun i _ => by split_ifs <;> positivity))
-  nlinarith only [h]
-
-end TraceReconstruction
-
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-
-end OAI
+import OAI.Probability.TraceReconstruction.Patterns
+
+namespace OAI
+
+/-!
+# InitialMoments
+
+Initial coefficient tails, primitive conditioning, mark-list oracle densities, pure norm estimates, and moment energies.
+-/
+
+open scoped ENNReal
+noncomputable section
+
+open Filter
+open scoped Topology
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem signedPeriodDefects_comp {u m r : ℕ} (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
+    (s : ℕ) (X : ℤ) (τ : ℝ) (j : IncreasingTuple u r) :
+    signedPeriodDefects off hsafe s X τ ∘ j.val =
+      shiftedTag (periodSlot (fun _ => off) ∘ j.val) (hsafe.comp j.val j.property)
+        (fun i => motifShift (motifIndicator s ((j.val i).val-X))
+          ((prouhetSeries s).coeff ((j.val i).val-X)) τ) := by
+  funext i; apply Fin.ext; rfl
+
+def signedTupleKernel {u m l r : ℕ} (hl : l ≤ u*m) (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
+    (π : IntSeries) (s : ℕ) (j : IncreasingTuple u r) (v : IncreasingTuple l r) : ℝ :=
+  ∑X∈π.coeff.support, π.coeff X*
+    (subsetRankKernel hl (signedPeriodDefects off hsafe s X 1 ∘ j.val) v.val -
+     subsetRankKernel hl (signedPeriodDefects off hsafe s X (-1) ∘ j.val) v.val)
+
+theorem signedRankCoefficient_tuple_sum {u m l r : ℕ} (hl : l ≤ u*m) (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off))) (π : IntSeries) (s : ℕ)
+    (v : IncreasingTuple l r) :
+    signedRankCoefficient hl off hsafe π s v=∑j, signedTupleKernel hl off hsafe π s j v := by
+  unfold signedRankCoefficient signedTupleKernel
+  simp_rw [rankDefectCoefficient_eq _ _ (signedPeriodDefects_strictMono _ _ _ _ _),
+    ← Finset.sum_sub_distrib,Finset.mul_sum]
+  exact Finset.sum_comm
+
+theorem signedTupleKernel_expansion {u m l r : ℕ} (hl : l ≤ u*m) (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off))) (π : IntSeries) (s : ℕ)
+    (j : IncreasingTuple u r) (v : IncreasingTuple l r) :
+    signedTupleKernel hl off hsafe π s j v = ∑k : Fin r → GapDifference,
+      if Odd (firstDifferenceCount k) then
+        2*motifPatternMultiplier π s (fun i => (j.val i).val) k *
+          positionalRankDifference hl (periodSlot (fun _ => off) ∘ j.val)
+            (hsafe.comp j.val j.property) k v else 0 := by
+  unfold signedTupleKernel
+  simp_rw [signedPeriodDefects_comp]
+  have H := mixed_signed_shift_expansion π s (fun i => (j.val i).val)
+    (fun a => subsetRankKernel hl
+      (shiftedTag (periodSlot (fun _ => off) ∘ j.val) (hsafe.comp j.val j.property) a) v.val)
+  simpa only [ite_mul,zero_mul,positionalRankDifference] using H
+
+def clippedSignedRankCoefficient {u m l r : ℕ} (hl : l ≤ u*m) (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
+    (π : IntSeries) (s : ℕ) (θ : ℝ) (v : IncreasingTuple l r) : ℝ :=
+  ∑j, if PeriodRankRegion m θ j v then signedTupleKernel hl off hsafe π s j v else 0
+
+def clippedPatternCoefficient {u m l r : ℕ} (hl : l ≤ u*m) (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
+    (π : IntSeries) (s : ℕ) (θ : ℝ) (k : Fin r → GapDifference) (v : IncreasingTuple l r) : ℝ :=
+  ∑j, if PeriodRankRegion m θ j v then
+    motifPatternMultiplier π s (fun i => (j.val i).val) k *
+      positionalRankDifference hl (periodSlot (fun _ => off) ∘ j.val)
+        (hsafe.comp j.val j.property) k v else 0
+
+theorem clippedSignedRankCoefficient_expansion {u m l r : ℕ} (hl : l ≤ u*m) (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
+    (π : IntSeries) (s : ℕ) (θ : ℝ) (v : IncreasingTuple l r) :
+    clippedSignedRankCoefficient hl off hsafe π s θ v =
+      2*∑k : Fin r → GapDifference, if Odd (firstDifferenceCount k) then
+        clippedPatternCoefficient hl off hsafe π s θ k v else 0 := by
+  unfold clippedSignedRankCoefficient clippedPatternCoefficient
+  have he (j : IncreasingTuple u r) :
+      (if PeriodRankRegion m θ j v then signedTupleKernel hl off hsafe π s j v else 0) =
+      ∑k : Fin r → GapDifference, if Odd (firstDifferenceCount k) then
+        (if PeriodRankRegion m θ j v then
+          2*motifPatternMultiplier π s (fun i => (j.val i).val) k *
+            positionalRankDifference hl (periodSlot (fun _ => off) ∘ j.val)
+              (hsafe.comp j.val j.property) k v else 0) else 0 := by
+    by_cases hv : PeriodRankRegion m θ j v
+    · simp only [ite_eq_left hv, signedTupleKernel_expansion]
+    · simp [hv]
+  simp_rw [he]
+  rw [Finset.sum_comm,Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro k _
+  by_cases hk : Odd (firstDifferenceCount k)
+  · simp only [ite_eq_left hk,Finset.mul_sum]
+    apply Finset.sum_congr rfl
+    intro j _
+    split_ifs <;> ring
+  · simp [hk]
+
+/-- Removing coarse rank rectangles loses an explicitly exponentially
+ small amount in the actual coefficient, before any finite differences. -/
+theorem signedTupleKernel_tail {u m l r : ℕ}
+    (hu : 0 < u) (hm : 128 ≤ m) (hr : 0 < r) (hl : l ≤ u*m)
+    (off : Fin m) (hoff : |(off.val:ℝ)-(m:ℝ)/2|≤3)
+    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
+    (π : IntSeries) (hπ : ∀X, 0≤π.coeff X) (hπsum : ∑X∈π.coeff.support, π.coeff X=1)
+    (s : ℕ) (j : IncreasingTuple u r) {θ : ℝ} (hθ : θ∈Set.Ioo 0 1) :
+    (∑v : IncreasingTuple l r, if PeriodRankRegion m θ j v then 0 else
+      |signedTupleKernel hl off hsafe π s j v|) ≤
+      2*(((r+1:ℕ):ℝ)*θ^r/binomialMass (u*m) θ l * (2*Real.exp (-(m:ℝ)/(1024*u)))) := by
+  let K (X : ℤ) (τ : ℝ) (v : IncreasingTuple l r) :=
+    subsetRankKernel hl (signedPeriodDefects off hsafe s X τ ∘ j.val) v.val
+  let B := ((r+1:ℕ):ℝ)*θ^r/binomialMass (u*m) θ l * (2*Real.exp (-(m:ℝ)/(1024*u)))
+  have hK (X : ℤ) (τ : ℝ) : (∑v : IncreasingTuple l r,
+      if PeriodRankRegion m θ j v then 0 else K X τ v)≤B := by
+    simp_rw [K,signedPeriodDefects_comp]
+    exact period_shifted_outside_region hu hm hr hl (fun _ => off) (fun _ => hoff)
+      (by intro _ _; simp) j (hsafe.comp j.val j.property) _ hθ
+  have hnon (X : ℤ) (τ : ℝ) (v : IncreasingTuple l r) : 0≤K X τ v :=
+    subsetRankKernel_nonneg _ _ _
+  calc
+    _ ≤ ∑v : IncreasingTuple l r, ∑X∈π.coeff.support, π.coeff X*
+        (if PeriodRankRegion m θ j v then 0 else K X 1 v+K X (-1) v) := by
+      apply Finset.sum_le_sum
+      intro v _
+      by_cases hv : PeriodRankRegion m θ j v
+      · simp [hv]
+      · simp only [ite_eq_right hv]
+        refine (Finset.abs_sum_le_sum_abs _ _).trans ?_
+        apply Finset.sum_le_sum
+        intro X _
+        rw [abs_mul,abs_of_nonneg (hπ X)]
+        apply mul_le_mul_of_nonneg_left _ (hπ X)
+        exact (abs_sub _ _).trans_eq (by rw [abs_of_nonneg (hnon X 1 v),abs_of_nonneg (hnon X (-1) v)])
+    _ = ∑X∈π.coeff.support, π.coeff X*((∑v : IncreasingTuple l r,
+        if PeriodRankRegion m θ j v then 0 else K X 1 v)+
+        (∑v : IncreasingTuple l r, if PeriodRankRegion m θ j v then 0 else K X (-1) v)) := by
+      rw [Finset.sum_comm]
+      apply Finset.sum_congr rfl
+      intro X _
+      rw [← Finset.mul_sum,← Finset.sum_add_distrib]
+      congr 1; apply Finset.sum_congr rfl
+      intro v _; split_ifs <;> ring
+    _ ≤ ∑X∈π.coeff.support, π.coeff X*(B+B) := by
+      apply Finset.sum_le_sum
+      intro X _
+      exact mul_le_mul_of_nonneg_left (add_le_add (hK X 1) (hK X (-1))) (hπ X)
+    _ = _ := by rw [← Finset.sum_mul,hπsum]; dsimp [B]; ring
+
+end TraceReconstruction
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem motif_pattern_common_short_gap {r : ℕ} (k : Fin r → GapDifference)
+    (hk : Odd (firstDifferenceCount k)) (hnl : ¬∃a, k=linearPattern a) :
+    ∃ a : Fin r, k a≠.unchanged ∧ ∃c : Fin (r+1), ∀ {u s : ℕ} (π : IntSeries)
+      (j : IncreasingTuple u r),
+      motifPatternMultiplier π s (fun h => (j.val h).val) k≠0 →
+        coarsePeriodGap j.val c ≤ 2*(2^s) := by
+  obtain ⟨a,b,hab,ha,hb⟩ := odd_pattern_pair k hk hnl
+  cases r with
+  | zero => exact Fin.elim0 a
+  | succ r =>
+    have har : a.val < r := by have := b.isLt; exact lt_of_lt_of_le hab (Nat.le_of_lt_succ b.isLt)
+    let i : Fin r := ⟨a.val,har⟩
+    have hi : i.castSucc=a := by ext; rfl
+    refine ⟨a,ha,i.castSucc.succ,?_⟩
+    intro u s π j hj
+    apply motif_pattern_bounded_gap π s j k i b
+    · simpa only [hi] using hab
+    · simpa only [hi] using ha
+    · exact hb
+    · exact hj
+
+theorem clipped_linear_pattern_norm {u m l r : ℕ}
+    (hm : 32 ≤ m) (hr : 0 < r) (hl : l ≤ u*m)
+    (off : Fin m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 3)
+    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
+    (π : IntSeries) (s : ℕ) (hcentral : CentralMotifSupport u s π)
+    {θ : ℝ} (hθ : θ∈Set.Ioo 0 1) (hhalf : 1/2 ≤ θ)
+    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ)) (a : Fin r) :
+    (∑v, (clippedPatternCoefficient hl off hsafe π s θ (linearPattern a) v)^2)  ≤ 
+      periodDifferencePrefactor u m r 1 θ * periodCentralSumConstant u r *
+        seriesL2Sq (prouhetSeries s*π) := by
+  have hb := period_pattern_norm hm hr hl (fun _ => off) (fun _ => hoff)
+    (by intro _ _; simp) (fun j => hsafe.comp j.val j.property) (linearPattern a)
+    hθ hhalf hmean hv a
+    (fun j => motifPatternMultiplier π s (fun i => (j.val i).val) (linearPattern a))
+    (fun h => (prouhetSeries s*π).coeff h.val)
+    (by intro j; rw [motif_linear_multiplier])
+    (prouhet_convolution_central π hcentral)
+  rw [linearPattern_order] at hb
+  apply hb.trans
+  apply mul_le_mul_of_nonneg_left
+    (seriesL2Sq_partial _ (fun h : Fin u => (h.val:ℤ)) (by intro a b h; apply Fin.ext; change (a.val:ℤ)=b.val at h; exact_mod_cast h))
+  unfold periodDifferencePrefactor lengthDifferenceConstant periodCentralSumConstant
+  positivity
+
+theorem clipped_nonlinear_pattern_norm {u m l r : ℕ}
+    (hm : 32 ≤ m) (hr : 0 < r) (hl : l ≤ u*m)
+    (off : Fin m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 3)
+    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
+    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (s : ℕ) (hcentral : CentralMotifSupport u s π)
+    {θ : ℝ} (hθ : θ∈Set.Ioo 0 1) (hhalf : 1/2 ≤ θ)
+    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
+    (k : Fin r → GapDifference) (hk : Odd (firstDifferenceCount k)) (hnl : ¬∃a, k=linearPattern a)
+    (hM : ((2*(2^s):ℕ):ℝ) < (u:ℝ)/(3*(r+1:ℝ))) :
+    (∑v, (clippedPatternCoefficient hl off hsafe π s θ k v)^2)  ≤ 
+      periodDifferencePrefactor u m r (∑h, (k h).order) θ *
+        periodBoundedSumConstant u r (2*(2^s)) * seriesL2Sq (geomSeries (2^s)*π) := by
+  obtain ⟨a,ha,c,hc⟩ := motif_pattern_common_short_gap k hk hnl
+  have hb := period_pattern_bounded_norm hm hr hl (fun _ => off) (fun _ => hoff)
+    (by intro _ _; simp) (fun j => hsafe.comp j.val j.property) k
+    hθ hhalf hmean hv a c
+    (fun j => motifPatternMultiplier π s (fun i => (j.val i).val) k)
+    (fun h => (geomSeries (2^s)*π).coeff h.val)
+    (fun j => motifPatternMultiplier_sq_le π hπ s _ k a ha)
+    (fun j hj => hc π j hj) (unsigned_convolution_central π hcentral) hM
+  apply hb.trans
+  apply mul_le_mul_of_nonneg_left
+    (seriesL2Sq_partial _ (fun h : Fin u => (h.val:ℤ)) (by intro a b h; apply Fin.ext; change (a.val:ℤ)=b.val at h; exact_mod_cast h))
+  unfold periodDifferencePrefactor lengthDifferenceConstant periodBoundedSumConstant
+  positivity
+
+def signedPatternBound (u m r s : ℕ) (π : IntSeries) (θ : ℝ)
+    (k : Fin r → GapDifference) : ℝ :=
+  if ∃a, k=linearPattern a then
+    periodDifferencePrefactor u m r 1 θ * periodCentralSumConstant u r * seriesL2Sq (prouhetSeries s*π)
+  else periodDifferencePrefactor u m r (∑h, (k h).order) θ *
+    periodBoundedSumConstant u r (2*(2^s)) * seriesL2Sq (geomSeries (2^s)*π)
+
+theorem sum_square_card_bound {α β : Type*} [Fintype α] [Fintype β] (f : α → β → ℝ) :
+    (∑ b, (∑a, f a b)^2)  ≤  (Fintype.card α:ℝ)*∑a, ∑b, f a b^2 := by
+  calc
+    _  ≤  ∑b, (Fintype.card α:ℝ)*∑a, f a b^2 := by
+      apply Finset.sum_le_sum
+      intro b _
+      have h := Finset.sum_sq_le_sum_mul_sum_of_sq_le_mul Finset.univ
+        (r := fun a => f a b) (f := fun _ => (1:ℝ)) (g := fun a => f a b^2)
+        (by intro _ _; norm_num) (by intro _ _; positivity) (by intro _ _; simp)
+      simpa using h
+    _ = _ := by rw [← Finset.mul_sum,Finset.sum_comm]
+
+theorem clipped_signed_initial_norm {u m l r : ℕ}
+    (hm : 32 ≤ m) (hr : 0 < r) (hl : l ≤ u*m)
+    (off : Fin m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 3)
+    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
+    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (s : ℕ) (hcentral : CentralMotifSupport u s π)
+    {θ : ℝ} (hθ : θ∈Set.Ioo 0 1) (hhalf : 1/2 ≤ θ)
+    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
+    (hM : ((2*(2^s):ℕ):ℝ) < (u:ℝ)/(3*(r+1:ℝ))) :
+    (∑v : IncreasingTuple l r, (clippedSignedRankCoefficient hl off hsafe π s θ v)^2)  ≤ 
+      4*(Fintype.card (Fin r → GapDifference):ℝ)*
+        ∑k : Fin r → GapDifference, if Odd (firstDifferenceCount k) then
+          signedPatternBound u m r s π θ k else 0 := by
+  simp_rw [clippedSignedRankCoefficient_expansion,mul_pow,show (2:ℝ)^2=4 by norm_num]
+  rw [← Finset.mul_sum]
+  calc
+    _  ≤  4*((Fintype.card (Fin r → GapDifference):ℝ)*∑k : Fin r → GapDifference, ∑v : IncreasingTuple l r,
+        (if Odd (firstDifferenceCount k) then clippedPatternCoefficient hl off hsafe π s θ k v else 0)^2) := by
+      exact mul_le_mul_of_nonneg_left (sum_square_card_bound (fun (k : Fin r → GapDifference) (v : IncreasingTuple l r) =>
+        if Odd (firstDifferenceCount k) then clippedPatternCoefficient hl off hsafe π s θ k v else 0)) (by norm_num)
+    _  ≤  _ := by
+      rw [← mul_assoc]
+      apply mul_le_mul_of_nonneg_left _ (by positivity)
+      apply Finset.sum_le_sum
+      intro k _
+      by_cases hk : Odd (firstDifferenceCount k)
+      · simp only [ite_eq_left hk]
+        unfold signedPatternBound
+        split_ifs with hlin
+        · obtain ⟨a,rfl⟩ := hlin
+          exact clipped_linear_pattern_norm hm hr hl off hoff hsafe π s hcentral hθ hhalf hmean hv a
+        · exact clipped_nonlinear_pattern_norm hm hr hl off hoff hsafe π hπ s hcentral
+            hθ hhalf hmean hv k hk hlin hM
+      · simp [hk]
+
+end TraceReconstruction
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem signedRankCoefficient_tail {u m l r : ℕ}
+    (hu : 0 < u) (hm : 128 ≤ m) (hr : 0 < r) (hl : l ≤ u*m)
+    (off : Fin m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 3)
+    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
+    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support, π.coeff X=1)
+    (s : ℕ) {θ : ℝ} (hθ : θ∈Set.Ioo 0 1) :
+    (∑v : IncreasingTuple l r, |signedRankCoefficient hl off hsafe π s v-
+        clippedSignedRankCoefficient hl off hsafe π s θ v|) ≤
+      (Fintype.card (IncreasingTuple u r):ℝ)*
+      (2*(((r+1:ℕ):ℝ)*θ^r/binomialMass (u*m) θ l * (2*Real.exp (-(m:ℝ)/(1024*u))))) := by
+  simp_rw [signedRankCoefficient_tuple_sum,clippedSignedRankCoefficient,← Finset.sum_sub_distrib]
+  calc
+    _ ≤ ∑v : IncreasingTuple l r, ∑j : IncreasingTuple u r,
+        if PeriodRankRegion m θ j v then 0 else |signedTupleKernel hl off hsafe π s j v| := by
+      apply Finset.sum_le_sum
+      intro v _
+      refine (Finset.abs_sum_le_sum_abs _ _).trans_eq ?_
+      apply Finset.sum_congr rfl
+      intro j _
+      split_ifs <;> simp
+    _ = ∑j : IncreasingTuple u r, ∑v : IncreasingTuple l r,
+        if PeriodRankRegion m θ j v then 0 else |signedTupleKernel hl off hsafe π s j v| := Finset.sum_comm
+    _ ≤ _ := by
+      calc
+        _ ≤ ∑j : IncreasingTuple u r,
+            2*(((r+1:ℕ):ℝ)*θ^r/binomialMass (u*m) θ l * (2*Real.exp (-(m:ℝ)/(1024*u)))) :=
+          Finset.sum_le_sum (fun j _ => signedTupleKernel_tail hu hm hr hl off hoff hsafe π hπ hπsum s j hθ)
+        _ = _ := by simp
+
+/-- A globally discarded l1 tail is also negligible in coefficient energy.
+ This does not assume pointwise cancellation on the discarded ranks. -/
+theorem finite_sum_sq_le_abs_sum_sq {α : Type*} [Fintype α] (f : α → ℝ) :
+    (∑a, f a^2) ≤ (∑a, |f a|)^2 := by
+  have h := Finset.sum_sq_le_sq_sum_of_nonneg (s := Finset.univ) (f := fun a => |f a|)
+    (by intro a _; exact abs_nonneg _)
+  simpa only [sq_abs] using h
+
+theorem initial_signed_energy {u m l r : ℕ}
+    (hu : 0 < u) (hm : 128 ≤ m) (hr : 0 < r) (hl : l ≤ u*m)
+    (off : Fin m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 3)
+    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
+    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support, π.coeff X=1)
+    (s : ℕ) (hcentral : CentralMotifSupport u s π)
+    {θ : ℝ} (hθ : θ∈Set.Ioo 0 1) (hhalf : 1/2 ≤ θ)
+    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
+    (hM : ((2*(2^s):ℕ):ℝ) < (u:ℝ)/(3*(r+1:ℝ))) :
+    (∑v : IncreasingTuple l r, (signedRankCoefficient hl off hsafe π s v)^2) ≤
+      8*(Fintype.card (Fin r → GapDifference):ℝ)*
+        (∑k : Fin r → GapDifference, if Odd (firstDifferenceCount k) then
+          signedPatternBound u m r s π θ k else 0) +
+      2*((Fintype.card (IncreasingTuple u r):ℝ)*
+        (2*(((r+1:ℕ):ℝ)*θ^r/binomialMass (u*m) θ l * (2*Real.exp (-(m:ℝ)/(1024*u))))))^2 := by
+  let A : IncreasingTuple l r → ℝ := signedRankCoefficient hl off hsafe π s
+  let C : IncreasingTuple l r → ℝ := clippedSignedRankCoefficient hl off hsafe π s θ
+  have hn := clipped_signed_initial_norm (by omega) hr hl off hoff hsafe π hπ s hcentral hθ hhalf hmean hv hM
+  have ht := signedRankCoefficient_tail hu hm hr hl off hoff hsafe π hπ hπsum s hθ
+  have hE := (finite_sum_sq_le_abs_sum_sq (fun v : IncreasingTuple l r => A v-C v)).trans
+    (pow_le_pow_left₀ (Finset.sum_nonneg (fun v _ => abs_nonneg (A v-C v))) ht 2)
+  have hsum : (∑v : IncreasingTuple l r, A v^2) ≤
+      2*(∑v : IncreasingTuple l r, C v^2)+2*(∑v : IncreasingTuple l r, (A v-C v)^2) := by
+    simp only [Finset.mul_sum,← Finset.sum_add_distrib]
+    apply Finset.sum_le_sum
+    intro v _
+    nlinarith [sq_nonneg (A v-2*C v)]
+  calc
+    _ ≤ 2*(∑v : IncreasingTuple l r, C v^2)+2*(∑v : IncreasingTuple l r, (A v-C v)^2) := hsum
+    _ ≤ _ := by dsimp [C] at *; linarith
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- Exact complete-trace expectation, recursively on the actual binary word. -/
+def listDeletionExpectation (q : ℝ) : List Bool → (List Bool → ℝ) → ℝ
+  | [], F => F []
+  | b::w, F => q*listDeletionExpectation q w F +
+      (1-q)*listDeletionExpectation q w (fun t => F (b::t))
+
+theorem traceOfMask_cons {n : ℕ} (x : Word (n+1)) (b : Bool) (a : Mask n) :
+    traceOfMask x (Fin.cons b a)=
+      if b then x 0::traceOfMask (fun i => x i.succ) a
+      else traceOfMask (fun i => x i.succ) a := by
+  rw [trace_filterMap,List.ofFn_succ,trace_filterMap]
+  cases b <;> simp
+
+theorem maskWeight_cons {n : ℕ} (q : ℝ) (b : Bool) (a : Mask n) :
+    maskWeight q (Fin.cons b a)=(if b then 1-q else q)*maskWeight q a := by
+  simp [maskWeight,Fin.prod_univ_succ]
+
+theorem listDeletionExpectation_ofFn {n : ℕ} (q : ℝ) (x : Word n) (F : List Bool → ℝ) :
+    listDeletionExpectation q (List.ofFn x) F =
+      ∑a : Mask n, maskWeight q a*F (traceOfMask x a) := by
+  induction n generalizing F with
+  | zero => simp [listDeletionExpectation,maskWeight,traceOfMask]
+  | succ n ih =>
+    rw [List.ofFn_succ,listDeletionExpectation,ih,ih]
+    symm
+    rw [← (Fin.consEquiv (fun _ : Fin (n+1) => Bool)).sum_comp,Fintype.sum_prod_type]
+    change (∑b : Bool, ∑a : Mask n, maskWeight q (Fin.cons b a)*F (traceOfMask x (Fin.cons b a)))=_
+    simp_rw [maskWeight_cons,traceOfMask_cons]
+    simp only [Bool.false_eq_true,ite_false,ite_true,Fintype.sum_bool]
+    simp only [mul_assoc,← Finset.mul_sum]
+    ring
+
+theorem listDeletionExpectation_append (q : ℝ) (x y : List Bool) (F : List Bool → ℝ) :
+    listDeletionExpectation q (x++y) F =
+      listDeletionExpectation q x (fun t => listDeletionExpectation q y (fun u => F (t++u))) := by
+  induction x generalizing F with
+  | nil => simp [listDeletionExpectation]
+  | cons b x ih => simp only [List.cons_append,listDeletionExpectation,ih]
+
+/-- The primitive alphabet consists of the literal strings 01 and 1. -/
+def primitiveBlock (b : Bool) : List Bool := if b then [true] else [false,true]
+
+theorem primitiveBlock_expectation (q : ℝ) (b : Bool) (F : List Bool → ℝ) :
+    listDeletionExpectation q (primitiveBlock b) F =
+      ∑z : PrimitiveMark, (if b then defectPrimitiveMass q z else ordinaryPrimitiveMass q z)*F (primitiveOutput z) := by
+  cases b <;> simp [primitiveBlock,listDeletionExpectation,primitiveOutput,
+    defectPrimitiveMass,ordinaryPrimitiveMass,Fin.sum_univ_succ]
+  ring
+
+theorem primitiveBlock_erasure_expectation {q : ℝ} (hq : q∈Set.Ioo 0 1)
+    (b : Bool) (F : List Bool → ℝ) :
+    listDeletionExpectation q (primitiveBlock b) F =
+      primitiveErasure q * F []+(1-primitiveErasure q)*
+        (primitiveMarkLaw q hq b).expect (fun z => F (primitiveOutput z)) := by
+  rw [primitiveBlock_expectation]
+  simp_rw [← primitiveMark_simulation hq]
+  simp only [add_mul,Finset.sum_add_distrib]
+  simp only [ite_mul,zero_mul,Finset.sum_ite_eq',Finset.mem_univ,ite_true,
+    primitiveOutput,Matrix.cons_val_zero]
+  simp only [FiniteLaw.expect,primitiveMarkLaw,mul_assoc,← Finset.mul_sum]
+
+/-- Independent silent block erasures and conditionally emitted primitive marks.
+ An emitted empty mark is retained in the list and differs from a silent erasure. -/
+def silentMarkExpectation (q : ℝ) : List Bool → (List PrimitiveMark → ℝ) → ℝ
+  | [], F => F []
+  | b::bs, F => primitiveErasure q*silentMarkExpectation q bs F +
+      (1-primitiveErasure q)*∑z : PrimitiveMark, primitiveMarkMass q b z*
+        silentMarkExpectation q bs (fun ys => F (z::ys))
+
+def joinPrimitiveMarks (ys : List PrimitiveMark) : List Bool := (ys.map primitiveOutput).flatten
+
+theorem silentMarkExpectation_linear (q : ℝ) (bs : List Bool)
+    (F G : List PrimitiveMark → ℝ) (a b : ℝ) :
+    silentMarkExpectation q bs (fun ys => a*F ys+b*G ys)=
+      a*silentMarkExpectation q bs F+b*silentMarkExpectation q bs G := by
+  induction bs generalizing F G with
+  | nil => rfl
+  | cons x xs ih =>
+    simp only [silentMarkExpectation,ih,mul_add,Finset.sum_add_distrib]
+    have hf : (∑ z : PrimitiveMark, primitiveMarkMass q x z *
+        (a*silentMarkExpectation q xs (fun ys => F (z::ys)))) =
+        a*∑ z : PrimitiveMark, primitiveMarkMass q x z * silentMarkExpectation q xs (fun ys => F (z::ys)) := by
+      rw [Finset.mul_sum]; apply Finset.sum_congr rfl; intro z _; ring
+    have hg : (∑ z : PrimitiveMark, primitiveMarkMass q x z *
+        (b*silentMarkExpectation q xs (fun ys => G (z::ys)))) =
+        b*∑ z : PrimitiveMark, primitiveMarkMass q x z * silentMarkExpectation q xs (fun ys => G (z::ys)) := by
+      rw [Finset.mul_sum]; apply Finset.sum_congr rfl; intro z _; ring
+    rw [hf,hg]; ring
+
+theorem silentMarkExpectation_sum {ι : Type*} [Fintype ι] (q : ℝ) (bs : List Bool)
+    (F : ι → List PrimitiveMark → ℝ) :
+    silentMarkExpectation q bs (fun ys => ∑i, F i ys)=∑i, silentMarkExpectation q bs (F i) := by
+  induction bs generalizing F with
+  | nil => rfl
+  | cons x xs ih =>
+    simp only [silentMarkExpectation,ih,Finset.mul_sum,Finset.sum_add_distrib]
+    congr 1
+    rw [Finset.sum_comm]
+
+theorem silentMarkExpectation_scale (q : ℝ) (bs : List Bool)
+    (F : List PrimitiveMark → ℝ) (a : ℝ) :
+    silentMarkExpectation q bs (fun ys => a*F ys)=a*silentMarkExpectation q bs F := by
+  have h := silentMarkExpectation_linear q bs F F a 0
+  simpa using h
+
+/-- The common deterministic mark-joining simulator is exact for arbitrary
+ concatenations of primitive blocks; block boundaries are not reported. -/
+theorem primitive_run_simulation {q : ℝ} (hq : q∈Set.Ioo 0 1)
+    (bs : List Bool) (F : List Bool → ℝ) :
+    listDeletionExpectation q ((bs.map primitiveBlock).flatten) F =
+      silentMarkExpectation q bs (fun ys => F (joinPrimitiveMarks ys)) := by
+  induction bs generalizing F with
+  | nil => rfl
+  | cons b bs ih =>
+    simp only [List.map_cons,List.flatten_cons]
+    rw [listDeletionExpectation_append,primitiveBlock_erasure_expectation hq]
+    simp only [List.nil_append,ih,FiniteLaw.expect,primitiveMarkLaw]
+    simp only [silentMarkExpectation]
+    rfl
+
+end TraceReconstruction
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- Every emitted mark is counted, including the empty primitive output. -/
+def emittedMarkExpectation (q : ℝ) : List Bool → (List PrimitiveMark → ℝ) → ℝ
+  | [], F => F []
+  | b::bs, F => ∑z : PrimitiveMark, primitiveMarkMass q b z*
+      emittedMarkExpectation q bs (fun ys => F (z::ys))
+
+theorem emittedMarkExpectation_ofFn {n : ℕ} (q : ℝ) (θ : Fin n → Bool)
+    (F : List PrimitiveMark → ℝ) :
+    emittedMarkExpectation q (List.ofFn θ) F=
+      ∑x : Fin n → PrimitiveMark, (∏i, primitiveMarkMass q (θ i) (x i))*F (List.ofFn x) := by
+  induction n generalizing F with
+  | zero => simp [emittedMarkExpectation]
+  | succ n ih =>
+    rw [List.ofFn_succ,emittedMarkExpectation]
+    simp_rw [ih,Finset.mul_sum]
+    rw [← (Fin.consEquiv (fun _ : Fin (n+1) => PrimitiveMark)).sum_comp,Fintype.sum_prod_type]
+    apply Finset.sum_congr rfl
+    intro z _
+    apply Finset.sum_congr rfl
+    intro x _
+    simp [Fin.consEquiv,Fin.prod_univ_succ,List.ofFn_succ,mul_assoc]
+
+/-- Conditioning the independent erasures on the return count chooses
+ a uniformly random subset of the original block slots. -/
+theorem silentMarkExpectation_mask_mixture {n : ℕ} (q : ℝ) (θ : Fin n → Bool)
+    (F : List PrimitiveMark → ℝ) :
+    silentMarkExpectation q (List.ofFn θ) F=
+      ∑a : Mask n, maskWeight (primitiveErasure q) a*
+        emittedMarkExpectation q (traceOfMask θ a) F := by
+  induction n generalizing F with
+  | zero => simp [silentMarkExpectation,emittedMarkExpectation,maskWeight,traceOfMask]
+  | succ n ih =>
+    rw [List.ofFn_succ,silentMarkExpectation]
+    simp_rw [ih]
+    symm
+    rw [← (Fin.consEquiv (fun _ : Fin (n+1) => Bool)).sum_comp,Fintype.sum_prod_type]
+    change (∑b : Bool, ∑a : Mask n, maskWeight (primitiveErasure q) (Fin.cons b a)*
+      emittedMarkExpectation q (traceOfMask θ (Fin.cons b a)) F)=_
+    simp_rw [maskWeight_cons,traceOfMask_cons]
+    simp only [Fintype.sum_bool,ite_true,Bool.false_eq_true,ite_false,emittedMarkExpectation]
+    simp_rw [mul_assoc,Finset.mul_sum]
+    rw [Finset.sum_comm,add_comm]
+    congr 1
+    apply Finset.sum_congr rfl
+    intro z _
+    apply Finset.sum_congr rfl
+    intro a _
+    ring
+
+def maskSubset {n : ℕ} (a : Mask n) : Finset (Fin n) := Finset.univ.filter fun i => a i
+
+def maskSubsetEquiv (n : ℕ) : Mask n ≃ Finset (Fin n) where
+  toFun := maskSubset
+  invFun S i := decide (i∈S)
+  left_inv a := by funext i; simp [maskSubset]
+  right_inv S := by ext i; simp [maskSubset]
+
+theorem maskWeight_card {n : ℕ} (q : ℝ) (a : Mask n) :
+    maskWeight q a = (1-q)^(maskSubset a).card*q^(n-(maskSubset a).card) := by
+  unfold maskWeight maskSubset
+  rw [Finset.prod_ite]
+  simp only [Finset.prod_const]
+  congr 1
+  have h := Finset.card_filter_add_card_filter_not (s := (Finset.univ : Finset (Fin n))) (p := fun i => a i)
+  rw [Finset.card_univ,Fintype.card_fin] at h
+  rw [show (Finset.univ.filter fun i : Fin n => ¬a i=true).card=
+      n-(Finset.univ.filter fun i : Fin n => a i=true).card by omega]
+
+theorem slotAtRank_ofFn {n l : ℕ} (S : FixedSubset (Fin n) l) :
+    List.ofFn (slotAtRank S)=S.val.sort := by
+  apply List.ext_getElem
+  · simp [S.property]
+  · intro i hi hj
+    simp only [List.getElem_ofFn]
+    rfl
+
+theorem filtered_finRange_eq_sort {n : ℕ} (S : Finset (Fin n)) :
+    ((List.finRange n).filter fun i => decide (i∈S))=S.sort := by
+  apply ((List.sortedLT_finRange n).pairwise.filter _).eq_of_mem_iff S.sortedLT_sort.pairwise
+  intro a
+  simp
+
+theorem selected_types_ofFn {n l : ℕ} (θ : Fin n → Bool) (S : FixedSubset (Fin n) l) :
+    traceOfMask θ (fun i => decide (i∈S.val))=List.ofFn (fun v => θ (slotAtRank S v)) := by
+  rw [traceOfMask,filtered_finRange_eq_sort,← slotAtRank_ofFn S,List.map_ofFn]
+  rfl
+
+theorem silentMarkExpectation_subset_mixture {n : ℕ} (q : ℝ) (θ : Fin n → Bool)
+    (F : List PrimitiveMark → ℝ) :
+    silentMarkExpectation q (List.ofFn θ) F=
+      ∑S : Finset (Fin n), subsetWeight (1-primitiveErasure q) S*
+        emittedMarkExpectation q (traceOfMask θ (fun i => decide (i∈S))) F := by
+  rw [silentMarkExpectation_mask_mixture,← (maskSubsetEquiv n).symm.sum_comp]
+  apply Finset.sum_congr rfl
+  intro S _
+  rw [maskWeight_card]
+  change (1-primitiveErasure q)^(maskSubsetEquiv n ((maskSubsetEquiv n).symm S)).card *
+      (primitiveErasure q)^(n-(maskSubsetEquiv n ((maskSubsetEquiv n).symm S)).card)*
+      emittedMarkExpectation q (traceOfMask θ (fun i => decide (i∈S))) F=_
+  rw [Equiv.apply_symm_apply]
+  simp [subsetWeight]
+
+def cardinalSubsetEquiv (n : ℕ) : Finset (Fin n) ≃ Σl : Fin (n+1), FixedSubset (Fin n) l.val where
+  toFun S := ⟨⟨S.card, by have := Finset.card_le_univ S; simpa using Nat.lt_succ_of_le this⟩,⟨S,rfl⟩⟩
+  invFun x := x.2.val
+  left_inv _ := rfl
+  right_inv x := by
+    rcases x with ⟨⟨l,hl⟩,⟨S,hS⟩⟩
+    dsimp
+    change S.card=l at hS
+    subst l
+    rfl
+
+theorem subsetWeight_count_factor {n l : ℕ} (hl : l ≤ n) (p : ℝ) (S : FixedSubset (Fin n) l) :
+    subsetWeight p S.val=binomialMass n p l*(uniformSubsetLaw l (by simpa using hl)).mass S := by
+  have hc : (n.choose l:ℝ)≠0 := by exact_mod_cast (Nat.choose_pos hl).ne'
+  simp only [subsetWeight,binomialMass,uniformSubsetLaw,S.property,Fintype.card_fin]
+  field_simp
+
+/-- The literal conditional-list oracle used by the rank coefficients. -/
+def conditionalPrimitiveLaw {n l : ℕ} (hl : l ≤ n) (q : ℝ) (hq : q∈Set.Ioo 0 1)
+    (θ : Fin n → Bool) : FiniteLaw (Fin l → PrimitiveMark) :=
+  (uniformSubsetLaw l (by simpa using hl)).mixture fun S =>
+    FiniteLaw.coordinateProduct (fun v => primitiveMarkLaw q hq (θ (slotAtRank S v)))
+
+theorem silentMarkExpectation_count_mixture {n : ℕ} (q : ℝ) (hq : q∈Set.Ioo 0 1)
+    (θ : Fin n → Bool) (F : List PrimitiveMark → ℝ) :
+    silentMarkExpectation q (List.ofFn θ) F=
+      ∑l : Fin (n+1), binomialMass n (1-primitiveErasure q) l.val*
+        (conditionalPrimitiveLaw (Nat.le_of_lt_succ l.isLt) q hq θ).expect (fun x => F (List.ofFn x)) := by
+  rw [silentMarkExpectation_subset_mixture,← (cardinalSubsetEquiv n).symm.sum_comp,Fintype.sum_sigma]
+  apply Finset.sum_congr rfl
+  intro l _
+  change (∑S : FixedSubset (Fin n) l.val, subsetWeight (1-primitiveErasure q) S.val*
+      emittedMarkExpectation q (traceOfMask θ (fun i => decide (i∈S.val))) F)=_
+  calc
+    _ = ∑S : FixedSubset (Fin n) l.val, binomialMass n (1-primitiveErasure q) l.val*
+        (uniformSubsetLaw l.val (by simpa using Nat.le_of_lt_succ l.isLt)).mass S*
+        (∑x : Fin l.val → PrimitiveMark, (∏v, primitiveMarkMass q (θ (slotAtRank S v)) (x v))*F (List.ofFn x)) := by
+      apply Finset.sum_congr rfl
+      intro S _
+      rw [subsetWeight_count_factor (Nat.le_of_lt_succ l.isLt),selected_types_ofFn θ S,emittedMarkExpectation_ofFn]
+    _ = _ := by
+      unfold conditionalPrimitiveLaw FiniteLaw.expect FiniteLaw.mixture FiniteLaw.coordinateProduct primitiveMarkLaw
+      simp_rw [Finset.sum_mul,Finset.mul_sum]
+      rw [Finset.sum_comm]
+      apply Finset.sum_congr rfl
+      intro x _
+      apply Finset.sum_congr rfl
+      intro S _
+      ring
+
+end TraceReconstruction
+
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- Density of the actual uniform-subset primitive oracle, over the ordinary
+ mark product law. This is exactly the channel defined by silent erasures. -/
+theorem conditionalPrimitiveLaw_density {n l : ℕ} (hl : l ≤ n)
+    {q : ℝ} (hq : q∈Set.Ioo 0 1) (θ : Fin n → Bool) (x : Fin l → PrimitiveMark) :
+    (conditionalPrimitiveLaw hl q hq θ).mass x =
+      ((primitiveMarkLaw q hq false).product l).mass x *
+        averagedDensity (uniformSubsetLaw l (by simpa using hl))
+          (fun S v => θ (slotAtRank S v)) (primitiveScore q) x := by
+  have he : (fun S : FixedSubset (Fin n) l =>
+      FiniteLaw.coordinateProduct (fun v => primitiveMarkLaw q hq (θ (slotAtRank S v)))) =
+      (fun S : FixedSubset (Fin n) l =>
+      FiniteLaw.coordinateProduct (fun v => if θ (slotAtRank S v) then
+        primitiveMarkLaw q hq true else primitiveMarkLaw q hq false)) := by
+    funext S
+    congr 1
+    funext v
+    cases θ (slotAtRank S v) <;> rfl
+  unfold conditionalPrimitiveLaw
+  rw [he]
+  exact leaf_oracle_density (primitiveMarkLaw q hq false) (primitiveMarkLaw q hq true)
+    (primitiveScore q) (primitive_score_identity hq)
+    (uniformSubsetLaw l (by simpa using hl)) (fun S v => θ (slotAtRank S v)) x
+
+def defectTypes {n u : ℕ} (d : Fin u → Fin n) (i : Fin n) : Bool := decide (i∈Set.range d)
+
+theorem defectTypes_atRank {n l u : ℕ} (d : Fin u → Fin n) (S : FixedSubset (Fin n) l) :
+    (fun v => defectTypes d (slotAtRank S v))=defectAtRank (Finset.univ.image d) S := by
+  funext v
+  simp [defectTypes,defectAtRank,Set.mem_range]
+
+theorem conditionalDefectLaw_density {n l u : ℕ} (hl : l ≤ n)
+    {q : ℝ} (hq : q∈Set.Ioo 0 1) (d : Fin u → Fin n) (x : Fin l → PrimitiveMark) :
+    (conditionalPrimitiveLaw hl q hq (defectTypes d)).mass x =
+      ((primitiveMarkLaw q hq false).product l).mass x *
+        multilinear (averagedCoefficient (uniformSubsetLaw l (by simpa using hl))
+          (defectAtRank (Finset.univ.image d))) (primitiveScore q) x := by
+  rw [conditionalPrimitiveLaw_density,show (fun S v => defectTypes d (slotAtRank S v))=
+      defectAtRank (Finset.univ.image d) by funext S; exact defectTypes_atRank d S,
+      averagedDensity_expansion]
+
+/-- The finite-support law of the hidden motif anchor, not an additional
+ distributional assumption or a changed channel parameter. -/
+def seriesProbabilityLaw (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X)
+    (hπsum : ∑X∈π.coeff.support, π.coeff X=1) : FiniteLaw ↥π.coeff.support where
+  mass X := π.coeff X.val
+  nonneg X := hπ X.val
+  total := by simpa only [Finset.univ_eq_attach,Finset.sum_attach] using hπsum
+
+def mixedMotifCoefficient {u m l : ℕ} (hl : l ≤ u*m) (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
+    (π : IntSeries) (s : ℕ) (τ : ℝ) (σ : Fin l → Bool) : ℝ :=
+  ∑X∈π.coeff.support, π.coeff X*
+    averagedCoefficient (uniformSubsetLaw l (by simpa using hl))
+      (defectAtRank (Finset.univ.image (signedPeriodDefects off hsafe s X τ))) σ
+
+def conditionalMotifLaw {u m l : ℕ} (hl : l ≤ u*m) (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
+    {q : ℝ} (hq : q∈Set.Ioo 0 1) (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X)
+    (hπsum : ∑X∈π.coeff.support, π.coeff X=1) (s : ℕ) (τ : ℝ) :
+    FiniteLaw (Fin l → PrimitiveMark) :=
+  (seriesProbabilityLaw π hπ hπsum).mixture fun X =>
+    conditionalPrimitiveLaw hl q hq (defectTypes (signedPeriodDefects off hsafe s X.val τ))
+
+theorem conditionalMotifLaw_density {u m l : ℕ} (hl : l ≤ u*m) (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
+    {q : ℝ} (hq : q∈Set.Ioo 0 1) (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X)
+    (hπsum : ∑X∈π.coeff.support, π.coeff X=1) (s : ℕ) (τ : ℝ) (x : Fin l → PrimitiveMark) :
+    (conditionalMotifLaw hl off hsafe hq π hπ hπsum s τ).mass x=
+      ((primitiveMarkLaw q hq false).product l).mass x *
+        multilinear (mixedMotifCoefficient hl off hsafe π s τ) (primitiveScore q) x := by
+  unfold conditionalMotifLaw FiniteLaw.mixture seriesProbabilityLaw
+  simp_rw [conditionalDefectLaw_density]
+  let F : ℤ → ℝ := fun X => π.coeff X *
+    (((primitiveMarkLaw q hq false).product l).mass x *
+      multilinear (averagedCoefficient (uniformSubsetLaw l (by simpa using hl))
+        (defectAtRank (Finset.univ.image (signedPeriodDefects off hsafe s X τ))))
+        (primitiveScore q) x)
+  change (∑ X : ↥π.coeff.support, F X.val) = _
+  rw [Finset.sum_coe_sort]
+  dsimp only [F]
+  unfold mixedMotifCoefficient multilinear
+  simp_rw [Finset.mul_sum,Finset.sum_mul]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro σ _
+  rw [Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro X _
+  ring
+
+theorem mixedMotifCoefficient_rank_difference {u m l r : ℕ} (hl : l ≤ u*m) (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
+    (π : IntSeries) (s : ℕ) (v : IncreasingTuple l r) :
+    mixedMotifCoefficient hl off hsafe π s 1 (rankMask v)-
+      mixedMotifCoefficient hl off hsafe π s (-1) (rankMask v)=
+        signedRankCoefficient hl off hsafe π s v := by
+  unfold mixedMotifCoefficient signedRankCoefficient
+  simp_rw [averaged_rank_coefficient hl _ v]
+  rw [← Finset.sum_sub_distrib]
+  apply Finset.sum_congr rfl
+  intro X _
+  ring
+
+end TraceReconstruction
+
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+def unchangedPattern (r : ℕ) : Fin r → GapDifference := fun _ => .unchanged
+
+theorem unchanged_positional_value {r : ℕ} (W : (Fin r → Fin 3) → ℝ) :
+    (∑ a : Fin r → Fin 3, (∏ h, positionalWeight (unchangedPattern r h) (a h))*W a)=
+      W (fun _ => 1) := by
+  simp only [unchangedPattern,positionalWeight]
+  simp_rw [product_function_delta]
+  simp
+
+/-- Comparing a signed displaced motif with the literal centered pure run
+ retains every nonconstant positional pattern, not just odd ones. -/
+theorem motif_pure_shift_expansion {r : ℕ} (χ β : Fin r → ℝ)
+    (hβ : ∀h, IsMotifValue (χ h) (β h)) (W : (Fin r → Fin 3) → ℝ) :
+    W (fun h => motifShift (χ h) (β h) 1)-W (fun _ => 1)=
+      ∑ k : Fin r → GapDifference, if k=unchangedPattern r then 0 else
+        (∏h, motifDifferenceWeight (χ h) (β h) 1 (k h))*
+          (∑a : Fin r → Fin 3, (∏h, positionalWeight (k h) (a h))*W a) := by
+  rw [motif_shift_expansion χ β 1 hβ (Or.inl rfl) W]
+  have he : W (fun _ => 1)=∑k : Fin r → GapDifference,
+      if k=unchangedPattern r then
+        (∑a : Fin r → Fin 3, (∏h, positionalWeight (k h) (a h))*W a) else 0 := by
+    simp [unchanged_positional_value]
+  rw [he,← Finset.sum_sub_distrib]
+  apply Finset.sum_congr rfl
+  intro k _
+  by_cases hk : k=unchangedPattern r
+  · subst k
+    simp [unchangedPattern,motifDifferenceWeight]
+  · simp [hk]
+
+theorem mixed_pure_shift_expansion {r : ℕ} (π : IntSeries) (s : ℕ)
+    (j : Fin r → ℤ) (W : (Fin r → Fin 3) → ℝ) :
+    (∑X∈π.coeff.support, π.coeff X *
+      (W (fun h => motifShift (motifIndicator s (j h-X)) ((prouhetSeries s).coeff (j h-X)) 1)-
+       W (fun _ => 1)))=
+      ∑k : Fin r → GapDifference, if k=unchangedPattern r then 0 else
+        motifPatternMultiplier π s j k *
+          (∑a : Fin r → Fin 3, (∏h, positionalWeight (k h) (a h))*W a) := by
+  simp_rw [motif_pure_shift_expansion _ _ (fun h => prouhet_motif_value s (j h-_)) W,
+    Finset.mul_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro k _
+  by_cases hk : k=unchangedPattern r
+  · simp [hk]
+  · simp only [ite_eq_right hk,motifPatternMultiplier,Finset.sum_mul,mul_assoc]
+    simp_rw [Finset.mul_sum]
+    exact Finset.sum_comm
+
+theorem pattern_order_zero {r : ℕ} (k : Fin r → GapDifference)
+    (hk : (∑h, (k h).order)=0) : k=unchangedPattern r := by
+  funext h
+  have hh := Finset.single_le_sum (fun i (_ : i∈(Finset.univ : Finset (Fin r))) =>
+    Nat.zero_le (k i).order) (Finset.mem_univ h)
+  rw [hk] at hh
+  have hz : (k h).order=0 := by omega
+  cases he : k h <;> simp_all [GapDifference.order,unchangedPattern]
+
+theorem nonconstant_pattern_linear_or_order_two {r : ℕ} (k : Fin r → GapDifference)
+    (hk : k≠unchangedPattern r) : (∃h, k=linearPattern h) ∨ 2≤∑h,(k h).order := by
+  by_cases hlin : ∃h, k=linearPattern h
+  · exact Or.inl hlin
+  right
+  by_contra hsmall
+  have horder : (∑h,(k h).order)=1 := by
+    have hn : (∑h,(k h).order)≠0 := fun he => hk (pattern_order_zero k he)
+    omega
+  have hc := difference_order_eq_counts k
+  have hfirst : firstDifferenceCount k=1 := by omega
+  have hsecond : secondDifferenceCount k=0 := by omega
+  exact hlin (first_second_counts_linear k hfirst hsecond)
+
+
+/-- A nonconstant motif pattern always has an actual involved position. -/
+theorem nonconstant_pattern_position {r : ℕ} (k : Fin r → GapDifference)
+    (hk : k≠unchangedPattern r) : ∃a, k a≠.unchanged := by
+  by_contra h
+  push Not at h
+  exact hk (funext h)
+
+theorem clipped_pure_nonlinear_pattern_norm {u m l r : ℕ}
+    (hm : 32 ≤ m) (hr : 0 < r) (hl : l ≤ u*m)
+    (off : Fin m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 3)
+    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
+    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (s : ℕ) (hcentral : CentralMotifSupport u s π)
+    {θ : ℝ} (hθ : θ∈Set.Ioo 0 1) (hhalf : 1/2 ≤ θ)
+    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
+    (k : Fin r → GapDifference) (hk : k≠unchangedPattern r) :
+    (∑v : IncreasingTuple l r, (clippedPatternCoefficient hl off hsafe π s θ k v)^2) ≤
+      periodDifferencePrefactor u m r (∑h, (k h).order) θ *
+        periodCentralSumConstant u r * seriesL2Sq (geomSeries (2^s)*π) := by
+  obtain ⟨a,ha⟩ := nonconstant_pattern_position k hk
+  have hb := period_pattern_norm hm hr hl (fun _ => off) (fun _ => hoff)
+    (by intro _ _; simp) (fun j => hsafe.comp j.val j.property) k
+    hθ hhalf hmean hv a
+    (fun j => motifPatternMultiplier π s (fun i => (j.val i).val) k)
+    (fun h => (geomSeries (2^s)*π).coeff h.val)
+    (fun j => motifPatternMultiplier_sq_le π hπ s _ k a ha)
+    (unsigned_convolution_central π hcentral)
+  apply hb.trans
+  apply mul_le_mul_of_nonneg_left
+    (seriesL2Sq_partial _ (fun h : Fin u => (h.val:ℤ)) (by intro a b h; apply Fin.ext; change (a.val:ℤ)=b.val at h; exact_mod_cast h))
+  unfold periodDifferencePrefactor lengthDifferenceConstant periodCentralSumConstant
+  positivity
+
+def purePatternBound (u m r s : ℕ) (π : IntSeries) (θ : ℝ)
+    (k : Fin r → GapDifference) : ℝ :=
+  if ∃a, k=linearPattern a then
+    periodDifferencePrefactor u m r 1 θ * periodCentralSumConstant u r * seriesL2Sq (prouhetSeries s*π)
+  else periodDifferencePrefactor u m r (∑h, (k h).order) θ *
+    periodCentralSumConstant u r * seriesL2Sq (geomSeries (2^s)*π)
+
+/-- Complete clipped energy, comparing the signed motif to a pure run. -/
+theorem clipped_pure_initial_norm {u m l r : ℕ}
+    (hm : 32 ≤ m) (hr : 0 < r) (hl : l ≤ u*m)
+    (off : Fin m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 3)
+    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
+    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (s : ℕ) (hcentral : CentralMotifSupport u s π)
+    {θ : ℝ} (hθ : θ∈Set.Ioo 0 1) (hhalf : 1/2 ≤ θ)
+    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ)) :
+    (∑v : IncreasingTuple l r, (∑k : Fin r → GapDifference,
+      if k=unchangedPattern r then 0 else clippedPatternCoefficient hl off hsafe π s θ k v)^2) ≤
+      (Fintype.card (Fin r → GapDifference):ℝ)*
+        ∑k : Fin r → GapDifference, if k=unchangedPattern r then 0 else
+          purePatternBound u m r s π θ k := by
+  apply (sum_square_card_bound (fun (k : Fin r → GapDifference) (v : IncreasingTuple l r) =>
+    if k=unchangedPattern r then 0 else clippedPatternCoefficient hl off hsafe π s θ k v)).trans
+  apply mul_le_mul_of_nonneg_left _ (by positivity)
+  apply Finset.sum_le_sum
+  intro k _
+  by_cases hk : k=unchangedPattern r
+  · simp [hk]
+  · simp only [ite_eq_right hk]
+    unfold purePatternBound
+    split_ifs with hlin
+    · obtain ⟨a,rfl⟩ := hlin
+      exact clipped_linear_pattern_norm hm hr hl off hoff hsafe π s hcentral hθ hhalf hmean hv a
+    · exact clipped_pure_nonlinear_pattern_norm hm hr hl off hoff hsafe π hπ s hcentral
+        hθ hhalf hmean hv k hk
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem shiftedTag_center {n r : ℕ} (p : Fin r → Fin n) (hp : SafeTaggedTuple p) :
+    shiftedTag p hp (fun _ => 1)=p := by
+  funext i
+  apply Fin.ext
+  simp [shiftedTag]
+
+def pureRankCoefficient {u m l r : ℕ} (hl : l ≤ u*m) (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
+    (π : IntSeries) (s : ℕ) (v : IncreasingTuple l r) : ℝ :=
+  ∑X∈π.coeff.support, π.coeff X *
+    (rankDefectCoefficient hl (signedPeriodDefects off hsafe s X 1) v -
+     rankDefectCoefficient hl (periodSlot (fun _ => off)) v)
+
+def pureTupleKernel {u m l r : ℕ} (hl : l ≤ u*m) (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
+    (π : IntSeries) (s : ℕ) (j : IncreasingTuple u r) (v : IncreasingTuple l r) : ℝ :=
+  ∑X∈π.coeff.support, π.coeff X *
+    (subsetRankKernel hl (signedPeriodDefects off hsafe s X 1 ∘ j.val) v.val -
+     subsetRankKernel hl (periodSlot (fun _ => off) ∘ j.val) v.val)
+
+theorem pureRankCoefficient_tuple_sum {u m l r : ℕ} (hl : l ≤ u*m) (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off))) (π : IntSeries) (s : ℕ)
+    (v : IncreasingTuple l r) :
+    pureRankCoefficient hl off hsafe π s v=∑j, pureTupleKernel hl off hsafe π s j v := by
+  unfold pureRankCoefficient pureTupleKernel
+  simp_rw [rankDefectCoefficient_eq _ _ (signedPeriodDefects_strictMono _ _ _ _ _),
+    rankDefectCoefficient_eq _ _ (periodSlot_strictMono _),
+    ← Finset.sum_sub_distrib,Finset.mul_sum]
+  exact Finset.sum_comm
+
+theorem pureTupleKernel_expansion {u m l r : ℕ} (hl : l ≤ u*m) (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off))) (π : IntSeries) (s : ℕ)
+    (j : IncreasingTuple u r) (v : IncreasingTuple l r) :
+    pureTupleKernel hl off hsafe π s j v = ∑k : Fin r → GapDifference,
+      if k=unchangedPattern r then 0 else
+        motifPatternMultiplier π s (fun i => (j.val i).val) k *
+          positionalRankDifference hl (periodSlot (fun _ => off) ∘ j.val)
+            (hsafe.comp j.val j.property) k v := by
+  unfold pureTupleKernel
+  simp_rw [signedPeriodDefects_comp]
+  have H := mixed_pure_shift_expansion π s (fun i => (j.val i).val)
+    (fun a => subsetRankKernel hl
+      (shiftedTag (periodSlot (fun _ => off) ∘ j.val) (hsafe.comp j.val j.property) a) v.val)
+  simpa only [shiftedTag_center,positionalRankDifference] using H
+
+def clippedPureRankCoefficient {u m l r : ℕ} (hl : l ≤ u*m) (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
+    (π : IntSeries) (s : ℕ) (θ : ℝ) (v : IncreasingTuple l r) : ℝ :=
+  ∑j, if PeriodRankRegion m θ j v then pureTupleKernel hl off hsafe π s j v else 0
+
+theorem clippedPureRankCoefficient_expansion {u m l r : ℕ} (hl : l ≤ u*m) (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
+    (π : IntSeries) (s : ℕ) (θ : ℝ) (v : IncreasingTuple l r) :
+    clippedPureRankCoefficient hl off hsafe π s θ v =
+      ∑k : Fin r → GapDifference, if k=unchangedPattern r then 0 else
+        clippedPatternCoefficient hl off hsafe π s θ k v := by
+  unfold clippedPureRankCoefficient clippedPatternCoefficient
+  simp_rw [pureTupleKernel_expansion]
+  have he (j : IncreasingTuple u r) :
+      (if PeriodRankRegion m θ j v then ∑k : Fin r → GapDifference,
+        if k=unchangedPattern r then 0 else
+          motifPatternMultiplier π s (fun i => (j.val i).val) k *
+            positionalRankDifference hl (periodSlot (fun _ => off) ∘ j.val)
+              (hsafe.comp j.val j.property) k v else 0)=
+      ∑k : Fin r → GapDifference, if k=unchangedPattern r then 0 else
+        (if PeriodRankRegion m θ j v then
+          motifPatternMultiplier π s (fun i => (j.val i).val) k *
+            positionalRankDifference hl (periodSlot (fun _ => off) ∘ j.val)
+              (hsafe.comp j.val j.property) k v else 0) := by
+    by_cases hv : PeriodRankRegion m θ j v <;> simp [hv]
+  simp_rw [he]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro k _
+  by_cases hk : k=unchangedPattern r <;> simp [hk]
+
+theorem pureTupleKernel_tail {u m l r : ℕ}
+    (hu : 0 < u) (hm : 128 ≤ m) (hr : 0 < r) (hl : l ≤ u*m)
+    (off : Fin m) (hoff : |(off.val:ℝ)-(m:ℝ)/2|≤3)
+    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
+    (π : IntSeries) (hπ : ∀X, 0≤π.coeff X) (hπsum : ∑X∈π.coeff.support, π.coeff X=1)
+    (s : ℕ) (j : IncreasingTuple u r) {θ : ℝ} (hθ : θ∈Set.Ioo 0 1) :
+    (∑v : IncreasingTuple l r, if PeriodRankRegion m θ j v then 0 else
+      |pureTupleKernel hl off hsafe π s j v|) ≤
+      2*(((r+1:ℕ):ℝ)*θ^r/binomialMass (u*m) θ l * (2*Real.exp (-(m:ℝ)/(1024*u)))) := by
+  let K (X : ℤ) (τ : ℝ) (v : IncreasingTuple l r) :=
+    subsetRankKernel hl (if τ=1 then signedPeriodDefects off hsafe s X 1 ∘ j.val
+      else periodSlot (fun _ => off) ∘ j.val) v.val
+  let B := ((r+1:ℕ):ℝ)*θ^r/binomialMass (u*m) θ l * (2*Real.exp (-(m:ℝ)/(1024*u)))
+  have hK (X : ℤ) (τ : ℝ) : (∑v : IncreasingTuple l r,
+      if PeriodRankRegion m θ j v then 0 else K X τ v)≤B := by
+    dsimp [K]
+    split_ifs
+    · rw [signedPeriodDefects_comp]
+      exact period_shifted_outside_region hu hm hr hl (fun _ => off) (fun _ => hoff)
+        (by intro _ _; simp) j (hsafe.comp j.val j.property) _ hθ
+    · have H := period_shifted_outside_region hu hm hr hl (fun _ => off) (fun _ => hoff)
+        (by intro _ _; simp) j (hsafe.comp j.val j.property) (fun _ => 1) hθ
+      simpa only [shiftedTag_center] using H
+  have hnon (X : ℤ) (τ : ℝ) (v : IncreasingTuple l r) : 0≤K X τ v :=
+    subsetRankKernel_nonneg _ _ _
+  calc
+    _ ≤ ∑v : IncreasingTuple l r, ∑X∈π.coeff.support, π.coeff X*
+        (if PeriodRankRegion m θ j v then 0 else K X 1 v+K X (-1) v) := by
+      apply Finset.sum_le_sum
+      intro v _
+      by_cases hv : PeriodRankRegion m θ j v
+      · simp [hv]
+      · simp only [ite_eq_right hv]
+        have he : pureTupleKernel hl off hsafe π s j v=
+            ∑X∈π.coeff.support, π.coeff X*(K X 1 v-K X (-1) v) := by
+          norm_num [pureTupleKernel,K]
+        rw [he]
+        refine (Finset.abs_sum_le_sum_abs _ _).trans ?_
+        apply Finset.sum_le_sum
+        intro X _
+        rw [abs_mul,abs_of_nonneg (hπ X)]
+        apply mul_le_mul_of_nonneg_left _ (hπ X)
+        exact (abs_sub _ _).trans_eq (by rw [abs_of_nonneg (hnon X 1 v),abs_of_nonneg (hnon X (-1) v)])
+    _ = ∑X∈π.coeff.support, π.coeff X*((∑v : IncreasingTuple l r,
+        if PeriodRankRegion m θ j v then 0 else K X 1 v)+
+        (∑v : IncreasingTuple l r, if PeriodRankRegion m θ j v then 0 else K X (-1) v)) := by
+      rw [Finset.sum_comm]
+      apply Finset.sum_congr rfl
+      intro X _
+      rw [← Finset.mul_sum,← Finset.sum_add_distrib]
+      congr 1; apply Finset.sum_congr rfl
+      intro v _; split_ifs <;> ring
+    _ ≤ ∑X∈π.coeff.support, π.coeff X*(B+B) := by
+      apply Finset.sum_le_sum
+      intro X _
+      exact mul_le_mul_of_nonneg_left (add_le_add (hK X 1) (hK X (-1))) (hπ X)
+    _ = _ := by rw [← Finset.sum_mul,hπsum]; dsimp [B]; ring
+
+
+theorem pureRankCoefficient_tail {u m l r : ℕ}
+    (hu : 0 < u) (hm : 128 ≤ m) (hr : 0 < r) (hl : l ≤ u*m)
+    (off : Fin m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 3)
+    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
+    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support, π.coeff X=1)
+    (s : ℕ) {θ : ℝ} (hθ : θ∈Set.Ioo 0 1) :
+    (∑v : IncreasingTuple l r, |pureRankCoefficient hl off hsafe π s v-
+        clippedPureRankCoefficient hl off hsafe π s θ v|) ≤
+      (Fintype.card (IncreasingTuple u r):ℝ)*
+      (2*(((r+1:ℕ):ℝ)*θ^r/binomialMass (u*m) θ l * (2*Real.exp (-(m:ℝ)/(1024*u))))) := by
+  simp_rw [pureRankCoefficient_tuple_sum,clippedPureRankCoefficient,← Finset.sum_sub_distrib]
+  calc
+    _ ≤ ∑v : IncreasingTuple l r, ∑j : IncreasingTuple u r,
+        if PeriodRankRegion m θ j v then 0 else |pureTupleKernel hl off hsafe π s j v| := by
+      apply Finset.sum_le_sum
+      intro v _
+      refine (Finset.abs_sum_le_sum_abs _ _).trans_eq ?_
+      apply Finset.sum_congr rfl
+      intro j _
+      split_ifs <;> simp
+    _ = ∑j : IncreasingTuple u r, ∑v : IncreasingTuple l r,
+        if PeriodRankRegion m θ j v then 0 else |pureTupleKernel hl off hsafe π s j v| := Finset.sum_comm
+    _ ≤ _ := by
+      calc
+        _ ≤ ∑j : IncreasingTuple u r,
+            2*(((r+1:ℕ):ℝ)*θ^r/binomialMass (u*m) θ l * (2*Real.exp (-(m:ℝ)/(1024*u)))) :=
+          Finset.sum_le_sum (fun j _ => pureTupleKernel_tail hu hm hr hl off hoff hsafe π hπ hπsum s j hθ)
+        _ = _ := by simp
+
+theorem initial_pure_energy {u m l r : ℕ}
+    (hu : 0 < u) (hm : 128 ≤ m) (hr : 0 < r) (hl : l ≤ u*m)
+    (off : Fin m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 3)
+    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
+    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support, π.coeff X=1)
+    (s : ℕ) (hcentral : CentralMotifSupport u s π)
+    {θ : ℝ} (hθ : θ∈Set.Ioo 0 1) (hhalf : 1/2 ≤ θ)
+    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
+    :
+    (∑v : IncreasingTuple l r, (pureRankCoefficient hl off hsafe π s v)^2) ≤
+      2*(Fintype.card (Fin r → GapDifference):ℝ)*
+        (∑k : Fin r → GapDifference, if k=unchangedPattern r then 0 else
+          purePatternBound u m r s π θ k) +
+      2*((Fintype.card (IncreasingTuple u r):ℝ)*
+        (2*(((r+1:ℕ):ℝ)*θ^r/binomialMass (u*m) θ l * (2*Real.exp (-(m:ℝ)/(1024*u))))))^2 := by
+  let A : IncreasingTuple l r → ℝ := pureRankCoefficient hl off hsafe π s
+  let C : IncreasingTuple l r → ℝ := clippedPureRankCoefficient hl off hsafe π s θ
+  have hn := clipped_pure_initial_norm (by omega) hr hl off hoff hsafe π hπ s hcentral hθ hhalf hmean hv
+  simp_rw [← clippedPureRankCoefficient_expansion] at hn
+  have ht := pureRankCoefficient_tail hu hm hr hl off hoff hsafe π hπ hπsum s hθ
+  have hE := (finite_sum_sq_le_abs_sum_sq (fun v : IncreasingTuple l r => A v-C v)).trans
+    (pow_le_pow_left₀ (Finset.sum_nonneg (fun v _ => abs_nonneg (A v-C v))) ht 2)
+  have hsum : (∑v : IncreasingTuple l r, A v^2) ≤
+      2*(∑v : IncreasingTuple l r, C v^2)+2*(∑v : IncreasingTuple l r, (A v-C v)^2) := by
+    simp only [Finset.mul_sum,← Finset.sum_add_distrib]
+    apply Finset.sum_le_sum
+    intro v _
+    nlinarith [sq_nonneg (A v-2*C v)]
+  calc
+    _ ≤ 2*(∑v : IncreasingTuple l r, C v^2)+2*(∑v : IncreasingTuple l r, (A v-C v)^2) := hsum
+    _ ≤ _ := by dsimp [C] at *; linarith
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+def scaleCoefficient {n : ℕ} (A : (Fin n → Bool) → ℝ) (L : ℝ) (σ : Fin n → Bool) : ℝ :=
+  A σ * ∏ i, if σ i then L else 1
+
+theorem multilinear_scale {α : Type*} {n : ℕ} (A : (Fin n → Bool) → ℝ)
+    (Z : α → ℝ) {L : ℝ} (hL : L≠0) (x : Fin n → α) :
+    multilinear (scaleCoefficient A L) (fun a => Z a/L) x=multilinear A Z x := by
+  unfold multilinear scaleCoefficient
+  apply Finset.sum_congr rfl
+  intro σ _
+  rw [mul_assoc,← Finset.prod_mul_distrib]
+  congr 1
+  apply Finset.prod_congr rfl
+  intro i _
+  cases σ i <;> simp
+  exact mul_div_cancel₀ _ hL
+
+theorem multilinearEnergy_scale {n : ℕ} (A : (Fin n → Bool) → ℝ) (L R : ℝ) :
+    multilinearEnergy (scaleCoefficient A L) R=multilinearEnergy A (R*L) := by
+  unfold multilinearEnergy scaleCoefficient
+  apply Finset.sum_congr rfl
+  intro σ _
+  rw [mul_pow,← Finset.prod_pow,mul_assoc,← Finset.prod_mul_distrib]
+  congr 1
+  apply Finset.prod_congr rfl
+  intro i _
+  cases σ i <;> simp
+  ring
+
+/-- The exact norm-of-input factor in the conditional leaf expansion. -/
+theorem multilinear_moment_scaled {α : Type*} [Fintype α] (P : FiniteLaw α)
+    (Z : α → ℝ) (k : ℕ) (hk : 1 ≤ k) (hcenter : P.expect Z=0)
+    {L : ℝ} (hL : 0<L) (hMoment : P.expect (fun a => |Z a|^(2*k))≤L^(2*k))
+    (n : ℕ) (A : (Fin n → Bool) → ℝ) :
+    (P.product n).expect (fun x => |multilinear A Z x|^(2*k)) ≤
+      (multilinearEnergy A ((6*(2*k:ℕ):ℝ)*L))^k := by
+  have hcenter' : P.expect (fun a => Z a/L)=0 := by
+    unfold FiniteLaw.expect
+    simp_rw [← mul_div_assoc]
+    rw [← Finset.sum_div]
+    change P.expect Z / L = 0
+    rw [hcenter,zero_div]
+  have hmoment' : P.expect (fun a => |Z a/L|^(2*k))≤1 := by
+    simp_rw [abs_div,abs_of_pos hL,div_pow]
+    unfold FiniteLaw.expect
+    simp_rw [← mul_div_assoc]
+    rw [← Finset.sum_div]
+    apply (div_le_one (by positivity)).mpr hMoment
+  have H := multilinear_even_moment P (fun a => Z a/L) k hk hcenter' hmoment' n (scaleCoefficient A L)
+  simp_rw [multilinear_scale A Z hL.ne',multilinearEnergy_scale] at H
+  simpa only [pow_mul,sq_abs] using H
+
+theorem multilinear_moment_sq_scaled {α : Type*} [Fintype α] (P : FiniteLaw α)
+    (Z : α → ℝ) (k : ℕ) (hk : 1 ≤ k) (hcenter : P.expect Z=0)
+    {L : ℝ} (hL : 0<L) (hMoment : P.expect (fun a => |Z a|^(2*k))≤L^(2*k))
+    (n : ℕ) (A : (Fin n → Bool) → ℝ) :
+    (P.product n).evenMomentSq (multilinear A Z) k ≤
+      multilinearEnergy A ((6*(2*k:ℕ):ℝ)*L) := by
+  have H := multilinear_moment_scaled P Z k hk hcenter hL hMoment n A
+  have hn : 0≤(P.product n).expect (fun x => |multilinear A Z x|^(2*k)) := by
+    exact Finset.sum_nonneg (fun x _ => mul_nonneg ((P.product n).nonneg x) (by positivity))
+  unfold FiniteLaw.evenMomentSq
+  calc
+    _ ≤ (multilinearEnergy A ((6*(2*k:ℕ):ℝ)*L)^k)^(1/(k:ℝ)) :=
+      Real.rpow_le_rpow hn H (by positivity)
+    _ = _ := by
+      rw [← Real.rpow_natCast,one_div,Real.rpow_rpow_inv (multilinearEnergy_nonneg _ _) (by positivity)]
+
+theorem multilinearEnergy_mixture_le {Ω : Type*} [Fintype Ω] {n : ℕ}
+    (P : FiniteLaw Ω) (A : Ω → (Fin n → Bool) → ℝ) (R : ℝ) :
+    multilinearEnergy (fun σ => P.expect (fun ω => A ω σ)) R ≤
+      P.expect (fun ω => multilinearEnergy (A ω) R) := by
+  unfold multilinearEnergy FiniteLaw.expect
+  simp_rw [Finset.mul_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_le_sum
+  intro σ _
+  have H := weighted_sum_sq_le Finset.univ P.mass (fun ω => A ω σ) (fun ω => P.nonneg ω)
+  rw [P.total,one_mul] at H
+  calc
+    _ ≤ (∑ω, P.mass ω*A ω σ^2)*(∏i,if σ i then R^2 else 1) :=
+      mul_le_mul_of_nonneg_right H (Finset.prod_nonneg (fun i _ => by split_ifs <;> positivity))
+    _ = _ := by rw [Finset.sum_mul]; apply Finset.sum_congr rfl; intro ω _; ring
+
+theorem multilinearEnergy_sub_le {n : ℕ} (A B : (Fin n → Bool) → ℝ) (R : ℝ) :
+    multilinearEnergy (fun σ => A σ-B σ) R ≤
+      2*multilinearEnergy A R+2*multilinearEnergy B R := by
+  unfold multilinearEnergy
+  simp_rw [Finset.mul_sum,← Finset.sum_add_distrib]
+  apply Finset.sum_le_sum
+  intro σ _
+  have h := mul_le_mul_of_nonneg_right (show (A σ-B σ)^2≤2*A σ^2+2*B σ^2 by nlinarith [sq_nonneg (A σ+B σ)])
+    (Finset.prod_nonneg (s := Finset.univ) (f := fun i => if σ i then R^2 else 1) (fun i _ => by split_ifs <;> positivity))
+  nlinarith only [h]
+
+end TraceReconstruction
+
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/TraceReconstruction/InitialOracle.lean b/lean/OAI/Probability/TraceReconstruction/InitialOracle.lean
index f8309fb24a51fb465e6be69109f0a5e681eeb834..53ce412c20a6de6bcebbeae30e74d2e88d5abbb7 100644
--- a/lean/OAI/Probability/TraceReconstruction/InitialOracle.lean
+++ b/lean/OAI/Probability/TraceReconstruction/InitialOracle.lean
@@ -1,1718 +1,1718 @@
-import OAI.Probability.TraceReconstruction.SourceOracle
-
-namespace OAI
-
-/-!
-# InitialOracle
-
-Actual initial scores and anchors, count partitions and typicality, clipped metadata, common decoding, sharp hidden-count energy, tails, and budget caps.
--/
-
-open scoped ENNReal
-noncomputable section
-
-open Filter
-open scoped Topology
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-open FiniteLaw
-
-def initialMomentRatio (u m : ℕ) (q w : ℝ) : ℝ :=
-  w^2*(3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/m)
-
-def initialSignedBudget (u m l R s : ℕ) (π : IntSeries) (θ q w : ℝ) : ℝ :=
-  (∑r∈Finset.range R,w^(2*r)*signedInitialDegreeBound u m r s π θ l)+
-    4*(initialMomentRatio u m q w)^R/(1-initialMomentRatio u m q w)
-
-theorem initialSignedBudget_nonneg (u m l R s : ℕ) (π : IntSeries) (θ q w : ℝ)
-    (hq : 0 ≤ q) (ha : initialMomentRatio u m q w < 1) :
-    0 ≤ initialSignedBudget u m l R s π θ q w := by
-  apply add_nonneg
-  · exact Finset.sum_nonneg (fun r _ => mul_nonneg (by rw [pow_mul]; positivity)
-      (signedInitialDegreeBound_nonneg _ _ _ _ _ _ _))
-  · apply div_nonneg _ (sub_nonneg.mpr ha.le)
-    exact mul_nonneg (by norm_num) (pow_nonneg (by unfold initialMomentRatio; positivity [rankSchurConstant_pos]) _)
-
-/-- The signed score numerator in the *actual conditional mark-list laws*,
- with the entire finite cancellation energy and geometric tail retained. -/
-theorem initial_signed_moment {u m l R : ℕ} (hu : 0 < u) (hm : 128 ≤ m)
-    (hum : u ≤ m) (hR : 0 < R) (hl : l ≤ u*m) (off : Fin m)
-    (hi : 1 ≤ off.val ∧ off.val+1 < m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 2)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
-    (π : IntSeries) (hπ : ∀X,0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
-    (s k : ℕ) (hcentral : CentralMotifSupport u s π) (hk : 1 ≤ k)
-    {q θ w : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
-    (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
-    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
-    (hM : ((2*(2^s):ℕ):ℝ) < (u:ℝ)/(3*(R+1:ℝ)))
-    (hw : w=((6*(2*k:ℕ):ℝ)*(2/q))) (ha : initialMomentRatio u m q w < 1) :
-    ((primitiveMarkLaw q ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩ false).product l).evenMomentSq
-      (fun x => initialMotifDensity hl off hsafe π s 1 q x-initialMotifDensity hl off hsafe π s (-1) q x) k  ≤ 
-        initialSignedBudget u m l R s π θ q w := by
-  have hq' : q∈Set.Ioo 0 1 := ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩
-  have he : (fun x => initialMotifDensity hl off hsafe π s 1 q x-initialMotifDensity hl off hsafe π s (-1) q x)=
-      multilinear (fun σ => mixedMotifCoefficient hl off hsafe π s 1 σ-mixedMotifCoefficient hl off hsafe π s (-1) σ)
-        (primitiveScore q) := by funext x; exact (multilinear_sub _ _ _ _).symm
-  rw [he]
-  apply (multilinear_moment_sq_scaled (primitiveMarkLaw q hq' false) (primitiveScore q) k hk
-    (primitive_score_centered hq') (div_pos (by norm_num) hq.1) (primitive_score_evenMoment hq' k) l _).trans
-  rw [←hw]
-  exact signed_initial_full_energy hu hm hum hR hl off hi hoff hsafe π hπ hπsum s hcentral hq hθ hmean hv hM
-    (by rw [hw]; positivity [hq.1]) ha
-
-/-- Quantitative normalized common clipping of the level-zero conditional
- pair. This theorem has only the literal count/geometric inequalities of the
- initial construction, not a signed-density or oracle conclusion premise. -/
-theorem initial_conditional_clipped_exists {u m l R : ℕ} (hu : 0 < u) (hm : 128 ≤ m)
-    (hum : u ≤ m) (hR : 0 < R) (hl : l ≤ u*m) (off : Fin m)
-    (hi : 1 ≤ off.val ∧ off.val+1 < m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 2)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
-    (π : IntSeries) (hπ : ∀X,0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
-    (s k : ℕ) (hcentral : CentralMotifSupport u s π) (hk : 1 ≤ k)
-    {q θ w : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
-    (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
-    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
-    (hM : ((2*(2^s):ℕ):ℝ) < (u:ℝ)/(3*(R+1:ℝ)))
-    (hw : w=((6*(2*k:ℕ):ℝ)*(2/q))) (hscale : initialMomentRatio u m q w ≤ 1/2)
-    (hclip : Real.sqrt (6*(4*(initialMomentRatio u m q w/(1-initialMomentRatio u m q w)))^k) ≤ 1/2) :
-    Nonempty (ClippedOracle
-      (conditionalMotifLaw hl off hsafe ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩ π hπ hπsum s (-1))
-      (conditionalMotifLaw hl off hsafe ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩ π hπ hπsum s 1) k
-      (Real.sqrt (6*(4*(initialMomentRatio u m q w/(1-initialMomentRatio u m q w)))^k))
-      (64*initialSignedBudget u m l R s π θ q w)) := by
-  have hq' : q∈Set.Ioo 0 1 := ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩
-  let A := (primitiveMarkLaw q hq' false).product l
-  let f := initialMotifDensity hl off hsafe π s (-1) q
-  let g := initialMotifDensity hl off hsafe π s 1 q
-  let U := initialMomentRatio u m q w/(1-initialMomentRatio u m q w)
-  have ha : initialMomentRatio u m q w < 1 := by linarith
-  have hz : 0 ≤ initialMomentRatio u m q w := by unfold initialMomentRatio; positivity [rankSchurConstant_pos,hq.1]
-  have hU : 0 ≤ U := div_nonneg hz (sub_nonneg.mpr ha.le)
-  have hU1 : U ≤ 1 := (div_le_iff₀ (sub_pos.mpr ha)).mpr (by linarith)
-  have hmom (τ : ℝ) : A.expect (fun x => |initialMotifDensity hl off hsafe π s τ q x-1|^(2*k)) ≤ U^k :=
-    A.expect_even_pow_le_of_moment _ k hk (initial_centered_moment hl (by omega) hum off hi hoff hsafe π hπ hπsum s k hk τ hq hθ hmean hv hw ha)
-  have hV := initialSignedBudget_nonneg u m l R s π θ q w hq.1.le ha
-  have hsig := A.expect_even_pow_le_of_moment _ k hk
-    (initial_signed_moment hu hm hum hR hl off hi hoff hsafe π hπ hπsum s k hcentral hk hq hθ hmean hv hM hw ha)
-  rw [initialMotifLaw_withDensity hl off hsafe π hπ hπsum s (-1) hq',
-    initialMotifLaw_withDensity hl off hsafe π hπ hπsum s 1 hq']
-  have H := A.exists_clippedOracle f g
-    (initialMotifDensity_nonneg hl off hsafe π hπ hπsum s (-1) hq')
-    (initialMotifDensity_nonneg hl off hsafe π hπ hπsum s 1 hq')
-    (initialMotifDensity_mean hl off hsafe π hπ hπsum s (-1) hq')
-    (initialMotifDensity_mean hl off hsafe π hπ hπsum s 1 hq')
-    Finset.univ 0 (by simp [eventMass]) (by simp [eventMass]) k hk U hU hU1 (hmom (-1)) (hmom 1)
-    (initialSignedBudget u m l R s π θ q w) hV (by simpa only [Finset.mem_univ,ite_true] using hsig)
-    (by simpa only [zero_add] using hclip)
-  simpa only [zero_add] using H
-
-end TraceReconstruction
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-namespace FiniteLaw
-variable {α : Type*} [Fintype α]
-
-theorem evenMomentSq_le_of_expect (P : FiniteLaw α) (F : α → ℝ) (k : ℕ) (hk : 1≤k)
-    {V : ℝ} (hV : 0≤V) (h : P.expect (fun a => |F a|^(2*k))≤V^k) :
-    P.evenMomentSq F k≤V := by
-  have H := Real.rpow_le_rpow (P.expect_nonneg _ (fun _ => by positivity)) h (by positivity : 0≤(k:ℝ)⁻¹)
-  rw [Real.pow_rpow_inv_natCast hV (by omega)] at H
-  simpa only [FiniteLaw.evenMomentSq,one_div] using H
-
-theorem tv_sigma_eq {ι : Type*} [Fintype ι] {β : ι → Type*} [∀i,Fintype (β i)]
-    (M : FiniteLaw ι) (P Q : ∀i,FiniteLaw (β i)) :
-    (M.sigma P).tv (M.sigma Q)=M.expect (fun i => (P i).tv (Q i)) := by
-  simp only [tv,sigma,expect,Fintype.sum_sigma,←mul_sub,abs_mul,abs_of_nonneg (M.nonneg _)]
-  rw [Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro i _
-  rw [←Finset.mul_sum]
-  ring
-
-end FiniteLaw
-namespace ClippedOracle
-variable {α : Type*} [Fintype α] {P Q : FiniteLaw α} {k : ℕ} {e V E W : ℝ}
-
-def weaken (C : ClippedOracle P Q k e V) (he : e≤E) (hV : V≤W) : ClippedOracle P Q k E W where
-  first := C.first
-  second := C.second
-  score := C.score
-  density := C.density
-  likelihood_nonneg := C.likelihood_nonneg
-  centered := C.centered
-  moment := C.moment.trans hV
-  first_tv := C.first_tv.trans he
-  second_tv := C.second_tv.trans he
-
-def trivial (P Q : FiniteLaw α) (k : ℕ) (hk : 1≤k) : ClippedOracle P Q k 1 0 where
-  first := P
-  second := P
-  score := fun _ => 0
-  density := fun _ => by ring
-  likelihood_nonneg := fun _ => by norm_num
-  centered := by simp [FiniteLaw.expect]
-  moment := by
-    apply P.evenMomentSq_le_of_expect _ k hk (by norm_num)
-    simp [FiniteLaw.expect,zero_pow (by omega : 2*k≠0)]
-  first_tv := by simp [FiniteLaw.tv]
-  second_tv := FiniteLaw.tv_le_one _ _
-
-/-- Reported metadata is retained, so the squared moment of a normalized
- score assembles without a mixing penalty. No conditioning of the original
- latent metadata distribution occurs. -/
-def sigma {ι : Type*} [Fintype ι] {β : ι → Type*} [∀i,Fintype (β i)]
-    (M : FiniteLaw ι) (P Q : ∀i,FiniteLaw (β i)) (e : ι → ℝ)
-    (k : ℕ) (hk : 1≤k) (V : ℝ) (hV : 0≤V)
-    (C : ∀i,ClippedOracle (P i) (Q i) k (e i) V) :
-    ClippedOracle (M.sigma P) (M.sigma Q) k (M.expect e) V where
-  first := M.sigma (fun i => (C i).first)
-  second := M.sigma (fun i => (C i).second)
-  score := fun a => (C a.1).score a.2
-  density := fun a => by dsimp only [FiniteLaw.sigma]; rw [(C a.1).density]; ring
-  likelihood_nonneg := fun a => (C a.1).likelihood_nonneg a.2
-  centered := by
-    rw [FiniteLaw.expect_sigma]
-    simp_rw [(C _).centered]
-    simp [FiniteLaw.expect]
-  moment := by
-    apply FiniteLaw.evenMomentSq_le_of_expect _ _ k hk hV
-    rw [FiniteLaw.expect_sigma]
-    calc
-      _ ≤ M.expect (fun _ => V^k) := M.expect_mono (fun i =>
-        (C i).first.expect_even_pow_le_of_moment _ k hk (C i).moment)
-      _ = _ := M.expect_const _
-  first_tv := by
-    rw [FiniteLaw.tv_sigma_eq]
-    exact M.expect_mono (fun i => (C i).first_tv)
-  second_tv := by
-    rw [FiniteLaw.tv_sigma_eq]
-    exact M.expect_mono (fun i => (C i).second_tv)
-
-/-- Replace bad conditional observations by a common law rather than making
- the original metadata law conditional on typicality. The exact cost is
- the true metadata failure mass plus the good conditional clipping cost. -/
-theorem assemble_good {ι : Type*} [Fintype ι] {β : ι → Type*} [∀i,Fintype (β i)]
-    (M : FiniteLaw ι) (P Q : ∀i,FiniteLaw (β i)) (Good : ι → Prop)
-    (k : ℕ) (hk : 1≤k) (e V : ℝ) (he : 0≤e) (hV : 0≤V)
-    (C : ∀i,Good i → Nonempty (ClippedOracle (P i) (Q i) k e V)) :
-    Nonempty (ClippedOracle (M.sigma P) (M.sigma Q) k
-      (e+M.expect (fun i => if Good i then 0 else 1)) V) := by
-  let err := fun i => if Good i then e else 1
-  have hi (i) : Nonempty (ClippedOracle (P i) (Q i) k (err i) V) := by
-    by_cases h : Good i
-    · simpa only [err,ite_eq_left h] using C i h
-    · exact ⟨(trivial (P i) (Q i) k hk).weaken (by simp [err,h]) hV⟩
-  let D := sigma M P Q err k hk V hV (fun i => Classical.choice (hi i))
-  refine ⟨D.weaken ?_ le_rfl⟩
-  calc
-    M.expect err ≤ M.expect (fun i => e+(if Good i then 0 else 1)) := M.expect_mono (fun i => by
-      dsimp only [err]; split_ifs <;> linarith)
-    _ = _ := by rw [M.expect_add,M.expect_const]
-
-end ClippedOracle
-end TraceReconstruction
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-open FiniteLaw
-
-theorem translated_fair_eval (g : ℕ) (c : ℤ) (F : ℤ → ℝ) :
-    (∑X∈(translateSeries c (fairSeries g)).coeff.support,
-      (translateSeries c (fairSeries g)).coeff X*F X)=
-    ∑k∈Finset.range (g+1),fairBinomial g k*F (c+k) := by
-  have H := congrArg (fun p : IntSeries => Finsupp.linearCombination ℝ F p.coeff)
-    (congrArg (translateSeries c) (fairSeries_sum g))
-  simp only [translateSeries,Finset.mul_sum,AddMonoidAlgebra.single_mul_single,one_mul,
-    AddMonoidAlgebra.coeff_sum,AddMonoidAlgebra.coeff_single,map_sum,
-    Finsupp.linearCombination_single,smul_eq_mul] at H
-  exact H
-
-/-- The literal initial anchor after conditioning the paired side outputs. -/
-def posteriorAnchorSeries {ι : Type*} [Fintype ι] (c : ℤ) (d : ι → Option Bool) : IntSeries :=
-  translateSeries (c-(∑i,(visibleOnes (d i):ℤ))-(∑i,hiddenTrial (d i)))
-    (fairSeries (∑i,hiddenTrial (d i)))
-
-theorem posterior_anchor_average {ι : Type*} [Fintype ι]
-    (c : ℤ) (d : ι → Option Bool) (F : ℤ → ℝ) :
-    (∑b : ι → Bool, (∏i,(flagPosterior (d i)).mass (b i))*
-      F (c-∑i,(bitIndex (b i):ℤ)))=
-    ∑X∈(posteriorAnchorSeries c d).coeff.support,(posteriorAnchorSeries c d).coeff X*F X := by
-  rw [posterior_count_average_reflected d (fun z => F (c-z)),posteriorAnchorSeries,translated_fair_eval]
-  apply Finset.sum_congr rfl
-  intro k _
-  congr 2
-  ring
-
-theorem posteriorAnchorSeries_nonneg {ι : Type*} [Fintype ι]
-    (c : ℤ) (d : ι → Option Bool) (X : ℤ) : 0 ≤ (posteriorAnchorSeries c d).coeff X :=
-  translated_binomial_nonneg _ _ _
-
-theorem posteriorAnchorSeries_total {ι : Type*} [Fintype ι]
-    (c : ℤ) (d : ι → Option Bool) :
-    ∑X∈(posteriorAnchorSeries c d).coeff.support,(posteriorAnchorSeries c d).coeff X=1 := by
-  have H := posterior_anchor_average c d (fun _ => 1)
-  simp only [mul_one] at H
-  rw [←H]
-  exact (FiniteLaw.indexedProduct (fun i => flagPosterior (d i))).total
-
-theorem posteriorAnchorSeries_support {ι : Type*} [Fintype ι]
-    (c : ℤ) (d : ι → Option Bool) (X : ℤ) (hX : X∈(posteriorAnchorSeries c d).coeff.support) :
-    c-Fintype.card ι ≤ X ∧ X ≤ c := by
-  have H := translated_binomial_support (∑i,hiddenTrial (d i))
-    (c-(∑i,(visibleOnes (d i):ℤ))-(∑i,hiddenTrial (d i))) X
-    (Finsupp.mem_support_iff.mp hX)
-  have hz : 0 ≤ ∑i,(visibleOnes (d i):ℤ) := Finset.sum_nonneg (fun _ _ => by positivity)
-  have hh : (∑i,(visibleOnes (d i):ℤ))+(∑i,hiddenTrial (d i)) ≤ Fintype.card ι := by
-    rw [Nat.cast_sum,←Finset.sum_add_distrib]
-    calc
-      _ ≤ ∑_i : ι,(1:ℤ) := Finset.sum_le_sum (fun i _ => by
-        cases h:d i with
-        | none => simp [visibleOnes,hiddenTrial]
-        | some b => cases b <;> simp [visibleOnes,hiddenTrial])
-      _ = _ := by simp
-  omega
-
-theorem posteriorAnchorSeries_central {ι : Type*} [Fintype ι]
-    (u s : ℕ) (c : ℤ) (d : ι → Option Bool)
-    (hleft : (u:ℝ)/3 ≤ 2*((c:ℝ)-Fintype.card ι)+1)
-    (hright : (u:ℝ)/3 ≤ 2*u-(2*((c:ℝ)+(2:ℝ)^s)+1)) :
-    CentralMotifSupport u s (posteriorAnchorSeries c d) := by
-  intro X hX
-  have H := posteriorAnchorSeries_support c d X hX
-  have H₁ : (c:ℝ)-Fintype.card ι ≤ X := by exact_mod_cast H.1
-  have H₂ : (X:ℝ) ≤ c := by exact_mod_cast H.2
-  constructor <;> linarith
-
-namespace PairedMechanisms
-variable {Side α : Type*} [Fintype Side] [Fintype α] {J : ℕ}
-
-def emissionOracle (M : PairedMechanisms Side J) (K : (Fin J → Bool) → FiniteLaw α) :
-    FiniteLaw (Reports Side J × α) :=
-  M.reports.joint (fun z => (orientationsGiven z).mixture K)
-
-/-- Bayes factorization is valid for arbitrary conditional finite emissions,
- including the dependent-length initial primitive-mark list. -/
-theorem emission_factorization (M : PairedMechanisms Side J) (K : (Fin J → Bool) → FiniteLaw α) :
-    (fairOrientations J).mixture (fun b => (M.reportsGiven b).prod (K b))=M.emissionOracle K := by
-  apply FiniteLaw.ext
-  funext zx
-  rcases zx with ⟨z,x⟩
-  change (∑b,(∏i : Fin J,(1/2:ℝ))*
-    ((∏i,FiniteLaw.pairedFlagMass (M.first i) (M.second i) (M.common i)
-      (M.hiddenWeight i) (b i) (z i))*(K b).mass x))=
-    (∏i,FiniteLaw.pairedReportMass (M.first i) (M.second i) (M.common i)
-      (M.hiddenWeight i) (z i))*(∑b,(∏i,(flagPosterior (z i).1).mass (b i))*(K b).mass x)
-  rw [Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro b _
-  rw [←mul_assoc,←Finset.prod_mul_distrib,FiniteLaw.pairedFlags_product_bayes,mul_assoc]
-
-end PairedMechanisms
-
-theorem conditionalMotifLaw_posterior {u m l J : ℕ} (hl : l ≤ u*m) (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
-    {q : ℝ} (hq : q∈Set.Ioo 0 1) (c : ℤ) (d : Fin J → Option Bool) (s : ℕ) (τ : ℝ) :
-    (FiniteLaw.coordinateProduct (fun i => flagPosterior (d i))).mixture
-      (fun b => conditionalPrimitiveLaw hl q hq
-        (defectTypes (signedPeriodDefects off hsafe s (c-∑i,(bitIndex (b i):ℤ)) τ)))=
-    conditionalMotifLaw hl off hsafe hq (posteriorAnchorSeries c d)
-      (posteriorAnchorSeries_nonneg c d) (posteriorAnchorSeries_total c d) s τ := by
-  apply FiniteLaw.ext
-  funext x
-  change (∑b : Fin J → Bool,(∏i : Fin J,(flagPosterior (d i)).mass (b i))*
-    (conditionalPrimitiveLaw hl q hq (defectTypes (signedPeriodDefects off hsafe s
-      (c-∑i,(bitIndex (b i):ℤ)) τ))).mass x)=_
-  classical
-  have H := posterior_anchor_average c d (fun X =>
-    (conditionalPrimitiveLaw hl q hq (defectTypes (signedPeriodDefects off hsafe s X τ))).mass x)
-  calc
-    _ = ∑X∈(posteriorAnchorSeries c d).coeff.support,
-        (posteriorAnchorSeries c d).coeff X*
-          (conditionalPrimitiveLaw hl q hq (defectTypes (signedPeriodDefects off hsafe s X τ))).mass x := by
-      convert H using 1
-      congr 1
-      ext b
-      simp
-    _ = _ := (Finset.sum_coe_sort _ _).symm
-
-
-/-- Initial observations retain the central count with its original binomial law. -/
-def countedMotifLaw {u m : ℕ} (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
-    {q : ℝ} (hq : q∈Set.Ioo 0 1) (π : IntSeries) (hπ : ∀X,0 ≤ π.coeff X)
-    (hπsum : ∑X∈π.coeff.support,π.coeff X=1) (s : ℕ) (τ : ℝ) :
-    FiniteLaw (BoundedMarks (u*m)) :=
-  (boundedCountLaw (le_refl (u*m)) (1-primitiveErasure q) (primitiveRetention_mem hq)).sigma
-    (fun l => conditionalMotifLaw (by omega) off hsafe hq π hπ hπsum s τ)
-
-theorem countedMotifLaw_posterior {u m J : ℕ} (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
-    {q : ℝ} (hq : q∈Set.Ioo 0 1) (c : ℤ) (d : Fin J → Option Bool) (s : ℕ) (τ : ℝ) :
-    (FiniteLaw.coordinateProduct (fun i => flagPosterior (d i))).mixture
-      (fun b => countedPrimitiveLaw (le_refl (u*m)) hq
-        (defectTypes (signedPeriodDefects off hsafe s (c-∑i,(bitIndex (b i):ℤ)) τ)))=
-    countedMotifLaw off hsafe hq (posteriorAnchorSeries c d)
-      (posteriorAnchorSeries_nonneg c d) (posteriorAnchorSeries_total c d) s τ := by
-  apply FiniteLaw.ext
-  funext x
-  rcases x with ⟨l,x⟩
-  have hl : l.val ≤ u*m := by omega
-  have H := congrArg (fun P : FiniteLaw (Fin l.val → PrimitiveMark) => P.mass x)
-    (conditionalMotifLaw_posterior hl off hsafe hq c d s τ)
-  change (∑b, _ * ((boundedCountLaw _ _ _).mass l *
-    (boundedConditionalPrimitiveLaw hq _ l).mass x))=_
-  simp only [boundedConditionalPrimitiveLaw,dite_eq_left hl]
-  change _ = (boundedCountLaw _ _ _).mass l * _
-  rw [←H]
-  simp only [FiniteLaw.mixture,Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro b _
-  ring
-
-
-end TraceReconstruction
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- The actual displaced primitive slot in a period. -/
-def motifOffset {m : ℕ} (off : Fin m) (hi : 1 ≤ off.val ∧ off.val+1 < m)
-    (s : ℕ) (X : ℤ) (τ : ℝ) (j : ℤ) : Fin m :=
-  ⟨off.val+(motifShift (motifIndicator s (j-X)) ((prouhetSeries s).coeff (j-X)) τ).val-1, by
-    have H := (motifShift (motifIndicator s (j-X)) ((prouhetSeries s).coeff (j-X)) τ).isLt
-    omega⟩
-
-theorem signedPeriodDefects_periodSlot {u m : ℕ} (off : Fin m)
-    (hi : 1 ≤ off.val ∧ off.val+1 < m)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
-    (s : ℕ) (X : ℤ) (τ : ℝ) :
-    signedPeriodDefects off hsafe s X τ =
-      periodSlot (fun j => motifOffset off hi s X τ j.val) := by
-  funext j
-  apply Fin.ext
-  simp only [signedPeriodDefects,shiftedTag,periodSlot,motifOffset]
-  omega
-
-def motifPeriodWord {m : ℕ} (off : Fin m) (hi : 1 ≤ off.val ∧ off.val+1 < m)
-    (u s : ℕ) (X : ℤ) (τ : ℝ) : List Bool :=
-  periodString m u (fun j => motifOffset off hi s X τ j.val)
-
-theorem signed_primitive_word {u m : ℕ} (off : Fin m)
-    (hi : 1 ≤ off.val ∧ off.val+1 < m)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
-    (s : ℕ) (X : ℤ) (τ : ℝ) :
-    (((List.ofFn (defectTypes (signedPeriodDefects off hsafe s X τ))).map primitiveBlock).flatten)=
-    motifPeriodWord off hi u s X τ := by
-  rw [signedPeriodDefects_periodSlot off hi,periodic_primitive_word]
-  rfl
-
-theorem motifOffset_outside {m : ℕ} (off : Fin m) (hi : 1 ≤ off.val ∧ off.val+1 < m)
-    (s : ℕ) (X : ℤ) (τ : ℝ) (j : ℤ) (hj : j<X ∨ X+(2^s:ℕ) ≤ j) :
-    motifOffset off hi s X τ j=off := by
-  apply Fin.ext
-  have h : ¬(0 ≤ j-X ∧ j-X<(2^s:ℕ)) := by omega
-  simp only [motifOffset,motifIndicator,ite_eq_right h,motifShift,ite_true]
-  norm_num
-
-theorem motifOffset_translate {m : ℕ} (off : Fin m) (hi : 1 ≤ off.val ∧ off.val+1 < m)
-    (s : ℕ) (X t : ℤ) (τ : ℝ) (j : ℤ) :
-    motifOffset off hi s X τ (t+j)=motifOffset off hi s (X-t) τ j := by
-  have h : t+j-X=j-(X-t) := by ring
-  simp only [motifOffset,h]
-
-theorem periodString_three {m : ℕ} (n k z : ℕ) (off : Fin (n+k+z) → Fin m) :
-    periodString m (n+k+z) off =
-      periodString m n (fun i => off ⟨i.val,by omega⟩) ++
-      periodString m k (fun i => off ⟨n+i.val,by omega⟩) ++
-      periodString m z (fun i => off ⟨n+k+i.val,by omega⟩) := by
-  simp only [periodString,List.ofFn_add,List.flatten_append]
-  rfl
-
-/-- Every randomized initial partition is a partition of the same literal
-period string, with no fit conditioning or change of channel. -/
-theorem motifPeriodWord_window {m : ℕ} (off : Fin m) (hi : 1 ≤ off.val ∧ off.val+1 < m)
-    (t K z s : ℕ) (a : ℤ) (τ : ℝ) (hleft : (t:ℤ) ≤ a) (hright : a+(2^s:ℕ) ≤ t+K) :
-    motifPeriodWord off hi (t+K+z) s a τ =
-      pureAtomRun (initialPeriod m off.val) t ++ motifPeriodWord off hi K s (a-t) τ ++
-      pureAtomRun (initialPeriod m off.val) z := by
-  unfold motifPeriodWord
-  rw [periodString_three]
-  congr 2
-  · rw [←periodString_constant off]
-    congr 1
-    funext i
-    exact motifOffset_outside off hi s a τ i.val (Or.inl (by have H:=i.isLt; omega))
-  · congr 1
-    funext i
-    rw [Nat.cast_add,motifOffset_translate]
-  · rw [←periodString_constant off]
-    congr 1
-    funext i
-    exact motifOffset_outside off hi s a τ (t+K+i.val) (Or.inr (by
-      have H : (0:ℤ) ≤ i.val := by positivity
-      omega))
-
-theorem motifPeriodWord_length {m : ℕ} (off : Fin m) (hi : 1 ≤ off.val ∧ off.val+1 < m)
-    (u s : ℕ) (X : ℤ) (τ : ℝ) :
-    (motifPeriodWord off hi u s X τ).length=u*(2*m-1) := periodString_length _ _ _
-
-end TraceReconstruction
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-namespace FiniteLaw
-variable {α β : Type*} [Fintype α] [Fintype β]
-
-theorem coordinateProduct_expect_pairs {n : ℕ} (P : Fin n → FiniteLaw α)
-    (Q : Fin n → FiniteLaw β) (F : (Fin n → α×β) → ℝ) :
-    (coordinateProduct (fun i => (P i).prod (Q i))).expect F=
-      (coordinateProduct P).expect (fun a =>
-        (coordinateProduct Q).expect (fun b => F (fun i => (a i,b i)))) := by
-  induction n with
-  | zero =>
-    have H (a : Fin 0 → α×β) : a = fun i => Fin.elim0 i := by funext i; exact Fin.elim0 i
-    have hF : F=fun _ => F (fun i => Fin.elim0 i) := by
-      funext a
-      exact congrArg F (H a)
-    rw [hF]
-    simp only [expect_const]
-  | succ n ih =>
-    rw [expect_coordinateProduct_succ,expect_prod,expect_coordinateProduct_succ]
-    congr 1
-    funext a
-    simp_rw [ih,expect_coordinateProduct_succ Q]
-    rw [expect_comm]
-    congr 1
-    funext x
-    congr 1
-    funext b
-    congr 1
-    funext y
-    congr 1
-    funext i
-    refine Fin.cases ?_ (fun j => ?_) i <;> simp
-
-end FiniteLaw
-
-/-- Three chronological streams, with side pairs independent across slots.
-This forgets flags but does not separate any marks inside the central list. -/
-def decodePairedEmission {Side α : Type*} {J : ℕ}
-    (side : Side → List Bool) (central : α → List Bool) (z : Reports Side J × α) : List Bool :=
-  coordinateWord (fun i => side (z.1 i).2.1) ++ central z.2 ++
-    coordinateWord (fun i => side (z.1 i).2.2)
-
-namespace PairedMechanisms
-variable {Side α : Type*} [Fintype Side] [Fintype α] {J : ℕ}
-
-theorem reportsGiven_paired_simulates (M : PairedMechanisms Side J) (b : Fin J → Bool)
-    {q : ℝ} (side : Side → List Bool) (central : α → List Bool) (K : FiniteLaw α)
-    (U V : Fin J → List Bool) (w : List Bool)
-    (hfirst : ∀i,SimulatesTrace q (M.first i) side (U i))
-    (hsecond : ∀i,SimulatesTrace q (M.second i) side (V i))
-    (hK : SimulatesTrace q K central w) :
-    SimulatesTrace q ((M.reportsGiven b).prod K) (decodePairedEmission side central)
-      (coordinateWord (fun i => if b i then V i else U i) ++ w ++
-        coordinateWord (fun i => if b i then U i else V i)) := by
-  let P : Fin J → FiniteLaw Side := fun i => if b i then M.second i else M.first i
-  let Q : Fin J → FiniteLaw Side := fun i => if b i then M.first i else M.second i
-  have HP : ∀i,SimulatesTrace q (P i) side (if b i then V i else U i) := by
-    intro i
-    dsimp [P]
-    cases h:b i
-    · exact hfirst i
-    · exact hsecond i
-  have HQ : ∀i,SimulatesTrace q (Q i) side (if b i then U i else V i) := by
-    intro i
-    dsimp [Q]
-    cases h:b i
-    · exact hsecond i
-    · exact hfirst i
-  have H := ((coordinateProduct_simulates P side _ HP).append hK).append
-    (coordinateProduct_simulates Q side _ HQ)
-  intro F
-  rw [←H F,FiniteLaw.expect_prod,FiniteLaw.expect_prod]
-  change (M.reportsGiven b).expect (fun z => K.expect (fun x =>
-    F (coordinateWord (fun i => side (z i).2.1) ++ central x ++
-      coordinateWord (fun i => side (z i).2.2))))=_
-  have hf := congrArg (fun L : FiniteLaw (Fin J → Side×Side) => L.expect (fun z =>
-    K.expect (fun x => F (coordinateWord (fun i => side (z i).1) ++ central x ++
-      coordinateWord (fun i => side (z i).2))))) (M.reportsGiven_forget b)
-  rw [FiniteLaw.expect_map] at hf
-  simp only [Function.comp_def] at hf
-  rw [hf]
-  have he : (fun i => FiniteLaw.orientedProd (M.first i) (M.second i) (b i))=
-      fun i => (P i).prod (Q i) := by
-    funext i
-    cases h:b i <;> simp [FiniteLaw.orientedProd,P,Q,h]
-  rw [he,FiniteLaw.coordinateProduct_expect_pairs]
-  simp only [FiniteLaw.expect_prod]
-  congr 1
-  funext a
-  rw [FiniteLaw.expect_comm]
-
-end PairedMechanisms
-end TraceReconstruction
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- Exact initial paired prefix and complementary suffix sizes. -/
-def initialPrefixPeriods {J : ℕ} (k : ℕ) (b : Fin J → Bool) : ℕ :=
-  ∑ i : Fin J, (k+bitIndex (b i))
-def initialSuffixPeriods {J : ℕ} (k : ℕ) (b : Fin J → Bool) : ℕ :=
-  ∑ i : Fin J, (k+(1-bitIndex (b i)))
-
-theorem initialPrefixPeriods_eval {J : ℕ} (k : ℕ) (b : Fin J → Bool) :
-    initialPrefixPeriods k b=J*k+∑i,bitIndex (b i) := by
-  simp [initialPrefixPeriods,Finset.sum_add_distrib]
-
-theorem initialPairPeriods_total {J : ℕ} (k : ℕ) (b : Fin J → Bool) :
-    initialPrefixPeriods k b+initialSuffixPeriods k b=J*(2*k+1) := by
-  unfold initialPrefixPeriods initialSuffixPeriods
-  rw [←Finset.sum_add_distrib]
-  have H (i : Fin J) : (k+bitIndex (b i))+(k+(1-bitIndex (b i)))=2*k+1 := by
-    cases h:b i <;> simp [bitIndex] <;> omega
-  simp only [H,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,Nat.cast_id]
-
-theorem initialPrefixPeriods_bounds {J : ℕ} (k : ℕ) (b : Fin J → Bool) :
-    J*k ≤ initialPrefixPeriods k b ∧ initialPrefixPeriods k b ≤ J*k+J := by
-  rw [initialPrefixPeriods_eval]
-  have H : (∑i,bitIndex (b i)) ≤ J := by
-    calc
-      _ ≤ ∑_i : Fin J,1 := Finset.sum_le_sum (fun i _ => by cases b i <;> simp [bitIndex])
-      _ = _ := by simp
-  omega
-
-theorem initial_paired_oracle_simulates {Side : Type*} [Fintype Side] {J K m : ℕ}
-    (M : PairedMechanisms Side J) (k s : ℕ) (a : ℤ) (τ : ℝ)
-    (off : Fin m) (hi : 1 ≤ off.val ∧ off.val+1 < m)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=K) (fun _ => off)))
-    {q : ℝ} (hq : q∈Set.Ioo 0 1) (side : Side → List Bool)
-    (hfirst : ∀i,SimulatesTrace q (M.first i) side (pureAtomRun (initialPeriod m off.val) k))
-    (hsecond : ∀i,SimulatesTrace q (M.second i) side (pureAtomRun (initialPeriod m off.val) (k+1)))
-    (ha : (J*k+J:ℕ) ≤ a) (ha' : a+(2^s:ℕ) ≤ (J*k+K:ℕ)) :
-    SimulatesTrace q
-      (M.emissionOracle (fun b => countedPrimitiveLaw (le_refl (K*m)) hq
-        (defectTypes (signedPeriodDefects off hsafe s
-          (a-(J*k:ℕ)-∑i,(bitIndex (b i):ℤ)) τ))))
-      (decodePairedEmission side decodeBoundedMarks)
-      (motifPeriodWord off hi (J*(2*k+1)+K) s a τ) := by
-  rw [←M.emission_factorization]
-  intro F
-  rw [FiniteLaw.expect_mixture]
-  have H (b : Fin J → Bool) :
-      ((M.reportsGiven b).prod (countedPrimitiveLaw (le_refl (K*m)) hq
-        (defectTypes (signedPeriodDefects off hsafe s
-          (a-(J*k:ℕ)-∑i,(bitIndex (b i):ℤ)) τ)))).expect
-        (fun z => F (decodePairedEmission side decodeBoundedMarks z))=
-      listDeletionExpectation q (motifPeriodWord off hi (J*(2*k+1)+K) s a τ) F := by
-    let t := initialPrefixPeriods k b
-    let z := initialSuffixPeriods k b
-    have ht := initialPrefixPeriods_bounds k b
-    have htot : t+K+z=J*(2*k+1)+K := by
-      have H:=initialPairPeriods_total k b
-      dsimp [t,z]
-      omega
-    have hanchor : a-(J*k:ℕ)-∑i,(bitIndex (b i):ℤ)=a-t := by
-      dsimp [t]
-      rw [initialPrefixPeriods_eval]
-      push_cast
-      ring
-    have hleft : (t:ℤ) ≤ a := by
-      have H : (t:ℤ) ≤ (J*k+J:ℕ) := by exact_mod_cast ht.2
-      omega
-    have hright : a+(2^s:ℕ) ≤ (t:ℤ)+K := by
-      have H : ((J*k:ℕ):ℤ) ≤ t := by exact_mod_cast ht.1
-      simp only [Nat.cast_add] at ha'
-      omega
-    have hp : coordinateWord (fun i => if b i then
-        pureAtomRun (initialPeriod m off.val) (k+1) else pureAtomRun (initialPeriod m off.val) k)=
-        pureAtomRun (initialPeriod m off.val) t := by
-      have h (i : Fin J) : (if b i then pureAtomRun (initialPeriod m off.val) (k+1) else
-          pureAtomRun (initialPeriod m off.val) k)=pureAtomRun (initialPeriod m off.val) (k+bitIndex (b i)) := by
-        cases b i <;> simp [bitIndex]
-      simp_rw [h]
-      exact coordinateWord_pureAtomRun _ _
-    have hz : coordinateWord (fun i => if b i then
-        pureAtomRun (initialPeriod m off.val) k else pureAtomRun (initialPeriod m off.val) (k+1))=
-        pureAtomRun (initialPeriod m off.val) z := by
-      have h (i : Fin J) : (if b i then pureAtomRun (initialPeriod m off.val) k else
-          pureAtomRun (initialPeriod m off.val) (k+1))=pureAtomRun (initialPeriod m off.val) (k+(1-bitIndex (b i))) := by
-        cases b i <;> simp [bitIndex]
-      simp_rw [h]
-      exact coordinateWord_pureAtomRun _ _
-    have hc := countedPrimitiveLaw_simulates (le_refl (K*m)) hq
-      (defectTypes (signedPeriodDefects off hsafe s (a-(J*k:ℕ)-∑i,(bitIndex (b i):ℤ)) τ))
-    rw [signed_primitive_word off hi] at hc
-    have hs := M.reportsGiven_paired_simulates b side decodeBoundedMarks _
-      (fun _ => pureAtomRun (initialPeriod m off.val) k)
-      (fun _ => pureAtomRun (initialPeriod m off.val) (k+1)) _ hfirst hsecond hc
-    have hword : coordinateWord (fun i => if b i then
-        pureAtomRun (initialPeriod m off.val) (k+1) else pureAtomRun (initialPeriod m off.val) k) ++
-        motifPeriodWord off hi K s (a-(J*k:ℕ)-∑i,(bitIndex (b i):ℤ)) τ ++
-        coordinateWord (fun i => if b i then pureAtomRun (initialPeriod m off.val) k else
-          pureAtomRun (initialPeriod m off.val) (k+1))=
-        motifPeriodWord off hi (J*(2*k+1)+K) s a τ := by
-      rw [hp,hz,hanchor,←motifPeriodWord_window off hi t K z s a τ hleft hright,htot]
-    rw [hword] at hs
-    exact hs F
-  simp only [H,FiniteLaw.expect_const]
-
-end TraceReconstruction
-
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-abbrev InitialMetadata (Side : Type*) (J N : ℕ) := Reports Side J × Fin (N+1)
-abbrev InitialObservation (Side : Type*) (J N : ℕ) :=
-  Σz : InitialMetadata Side J N, Fin z.2.val → PrimitiveMark
-
-def initialMetadataLaw {Side : Type*} [Fintype Side] {J : ℕ}
-    (M : PairedMechanisms Side J) (N : ℕ) {q : ℝ} (hq : q∈Set.Ioo 0 1) :
-    FiniteLaw (InitialMetadata Side J N) :=
-  M.reports.prod (boundedCountLaw (le_refl N) (1-primitiveErasure q) (primitiveRetention_mem hq))
-
-def initialConditionalLaw {Side : Type*} [Fintype Side] {J K m : ℕ}
-    (off : Fin m) (hsafe : SafeTaggedTuple (periodSlot (u:=K) (fun _ => off)))
-    {q : ℝ} (hq : q∈Set.Ioo 0 1) (c : ℤ) (s : ℕ) (τ : ℝ)
-    (z : InitialMetadata Side J (K*m)) : FiniteLaw (Fin z.2.val → PrimitiveMark) :=
-  conditionalMotifLaw (by have H:=z.2.isLt; omega) off hsafe hq
-    (posteriorAnchorSeries c (fun i => (z.1 i).1))
-    (posteriorAnchorSeries_nonneg c _) (posteriorAnchorSeries_total c _) s τ
-
-def initialSigmaLaw {Side : Type*} [Fintype Side] {J K m : ℕ}
-    (M : PairedMechanisms Side J) (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=K) (fun _ => off)))
-    {q : ℝ} (hq : q∈Set.Ioo 0 1) (c : ℤ) (s : ℕ) (τ : ℝ) :
-    FiniteLaw (InitialObservation Side J (K*m)) :=
-  (initialMetadataLaw M (K*m) hq).sigma (initialConditionalLaw off hsafe hq c s τ)
-
-def decodeInitialObservation {Side : Type*} {J N : ℕ} (side : Side → List Bool)
-    (z : InitialObservation Side J N) : List Bool :=
-  decodePairedEmission side decodeBoundedMarks (z.1.1,⟨z.1.2,z.2⟩)
-
-theorem initialSigmaLaw_expect {Side : Type*} [Fintype Side] {J K m : ℕ}
-    (M : PairedMechanisms Side J) (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=K) (fun _ => off)))
-    {q : ℝ} (hq : q∈Set.Ioo 0 1) (c : ℤ) (s : ℕ) (τ : ℝ)
-    (F : Reports Side J × BoundedMarks (K*m) → ℝ) :
-    (initialSigmaLaw M off hsafe hq c s τ).expect (fun z => F (z.1.1,⟨z.1.2,z.2⟩))=
-    (M.emissionOracle (fun b => countedPrimitiveLaw (le_refl (K*m)) hq
-      (defectTypes (signedPeriodDefects off hsafe s (c-∑i,(bitIndex (b i):ℤ)) τ)))).expect F := by
-  unfold PairedMechanisms.emissionOracle
-  rw [FiniteLaw.expect_joint]
-  have H (z : Reports Side J) : (PairedMechanisms.orientationsGiven z).mixture
-      (fun b => countedPrimitiveLaw (le_refl (K*m)) hq
-        (defectTypes (signedPeriodDefects off hsafe s (c-∑i,(bitIndex (b i):ℤ)) τ)))=
-      countedMotifLaw off hsafe hq (posteriorAnchorSeries c (fun i => (z i).1))
-        (posteriorAnchorSeries_nonneg c _) (posteriorAnchorSeries_total c _) s τ :=
-    countedMotifLaw_posterior off hsafe hq c (fun i => (z i).1) s τ
-  simp_rw [H]
-  simp only [initialSigmaLaw,initialMetadataLaw,FiniteLaw.expect_sigma,FiniteLaw.expect_prod,
-    countedMotifLaw,initialConditionalLaw]
-
-theorem initial_sigma_simulates {Side : Type*} [Fintype Side] {J K m : ℕ}
-    (M : PairedMechanisms Side J) (k s : ℕ) (a : ℤ) (τ : ℝ)
-    (off : Fin m) (hi : 1  ≤  off.val ∧ off.val+1  <  m)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=K) (fun _ => off)))
-    {q : ℝ} (hq : q∈Set.Ioo 0 1) (side : Side → List Bool)
-    (hfirst : ∀i,SimulatesTrace q (M.first i) side (pureAtomRun (initialPeriod m off.val) k))
-    (hsecond : ∀i,SimulatesTrace q (M.second i) side (pureAtomRun (initialPeriod m off.val) (k+1)))
-    (ha : (J*k+J:ℕ)  ≤  a) (ha' : a+(2^s:ℕ)  ≤  (J*k+K:ℕ)) :
-    SimulatesTrace q (initialSigmaLaw M off hsafe hq (a-(J*k:ℕ)) s τ)
-      (decodeInitialObservation side) (motifPeriodWord off hi (J*(2*k+1)+K) s a τ) := by
-  intro F
-  unfold decodeInitialObservation
-  rw [initialSigmaLaw_expect M off hsafe hq (a-(J*k:ℕ)) s τ (fun z => F (decodePairedEmission side decodeBoundedMarks z))]
-  exact initial_paired_oracle_simulates M k s a τ off hi hsafe hq side hfirst hsecond ha ha' F
-
-/-- This is the literal count-typicality condition; it does not change the
- central count's original binomial law. -/
-def initialCountAdmissible (N : ℕ) (q : ℝ) (l : Fin (N+1)) : Prop :=
-  (l.val:ℝ)/(N:ℝ)∈Set.Icc (1-q^2) (1-q^2/4) ∧
-    1  ≤  (N:ℝ)*((l.val:ℝ)/(N:ℝ))*(1-(l.val:ℝ)/(N:ℝ))
-
-def initialMetadataGood {Side : Type*} {J N : ℕ} (q ω : ℝ)
-    (z : InitialMetadata Side J N) : Prop :=
-  initialCountAdmissible N q z.2 ∧ (J:ℝ)*ω^2/2  ≤  ∑i,(hiddenTrial (z.1 i).1:ℝ)
-
-/-- The genuine initial metadata oracle is clipped by conditioning each
- admissible fiber only. Every bad report/count keeps its original mass and
- is replaced by a common law, so all discarded mass is charged. -/
-theorem initial_sigma_clipped_exists {Side : Type*} [Fintype Side] {J K m R : ℕ}
-    (M : PairedMechanisms Side J) (hK : 0 < K) (hm : 128 ≤ m) (hKm : K ≤ m)
-    (hR : 0 < R) (off : Fin m) (hi : 1 ≤ off.val ∧ off.val+1 < m)
-    (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 2)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=K) (fun _ => off)))
-    (c : ℤ) (s k : ℕ) (hk : 1 ≤ k)
-    (hleft : (K:ℝ)/3 ≤ 2*((c:ℝ)-J)+1)
-    (hright : (K:ℝ)/3 ≤ 2*K-(2*((c:ℝ)+(2:ℝ)^s)+1))
-    {q ω w V : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
-    (hM : ((2*(2^s):ℕ):ℝ) < (K:ℝ)/(3*(R+1:ℝ)))
-    (hw : w=((6*(2*k:ℕ):ℝ)*(2/q)))
-    (hscale : initialMomentRatio K m q w ≤ 1/2)
-    (hclip : Real.sqrt (6*(4*(initialMomentRatio K m q w/(1-initialMomentRatio K m q w)))^k) ≤ 1/2)
-    (hV : 0 ≤ V)
-    (hbudget : ∀z : InitialMetadata Side J (K*m),initialMetadataGood q ω z →
-      64*initialSignedBudget K m z.2.val R s (posteriorAnchorSeries c (fun i => (z.1 i).1))
-        ((z.2.val:ℝ)/(K*m:ℕ)) q w ≤ V) :
-    Nonempty (ClippedOracle
-      (initialSigmaLaw M off hsafe ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩ c s (-1))
-      (initialSigmaLaw M off hsafe ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩ c s 1) k
-      (Real.sqrt (6*(4*(initialMomentRatio K m q w/(1-initialMomentRatio K m q w)))^k)+
-        (initialMetadataLaw M (K*m) ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩).expect
-          (fun z => if initialMetadataGood q ω z then 0 else 1)) V) := by
-  apply ClippedOracle.assemble_good _ _ _ (initialMetadataGood q ω) k hk _ V (Real.sqrt_nonneg _) hV
-  intro z hz
-  have hmean : ((K*m:ℕ):ℝ)*((z.2.val:ℝ)/(K*m:ℕ))=z.2.val := by
-    exact mul_div_cancel₀ _ (by exact_mod_cast (Nat.mul_ne_zero (by omega : K≠0) (by omega : m≠0)))
-  have H := initial_conditional_clipped_exists hK hm hKm hR (by have H:=z.2.isLt; omega) off hi hoff hsafe
-    (posteriorAnchorSeries c (fun i => (z.1 i).1)) (posteriorAnchorSeries_nonneg c _)
-    (posteriorAnchorSeries_total c _) s k
-    (posteriorAnchorSeries_central K s c _ (by simpa using hleft) hright) hk hq hz.1.1 hmean hz.1.2 hM hw hscale hclip
-  exact ⟨(Classical.choice H).weaken le_rfl (hbudget z hz)⟩
-
-end TraceReconstruction
-
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-def purePeriodCoefficient {u m l : ℕ} (hl : l ≤ u*m) (off : Fin m) : (Fin l → Bool) → ℝ :=
-  averagedCoefficient (uniformSubsetLaw l (by simpa using hl))
-    (defectAtRank (Finset.univ.image (periodSlot (u:=u) (fun _ => off))))
-
-theorem mixedMotifCoefficient_pure_rank_difference {u m l r : ℕ} (hl : l ≤ u*m) (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
-    (π : IntSeries) (hπsum : ∑X∈π.coeff.support,π.coeff X=1) (s : ℕ) (v : IncreasingTuple l r) :
-    mixedMotifCoefficient hl off hsafe π s 1 (rankMask v)-purePeriodCoefficient hl off (rankMask v)=
-      pureRankCoefficient hl off hsafe π s v := by
-  unfold mixedMotifCoefficient purePeriodCoefficient pureRankCoefficient
-  simp_rw [averaged_rank_coefficient hl _ v,mul_sub,Finset.sum_sub_distrib,←Finset.sum_mul,hπsum,one_mul]
-
-/-- The pure comparison retains all low-degree cancellation, not just its
- total defect density. -/
-def initialPureDegreeBound (u m r s : ℕ) (π : IntSeries) (θ : ℝ) (l : ℕ) : ℝ :=
-  if r=0 then 0 else
-    2*(Fintype.card (Fin r → GapDifference):ℝ)*
-      (∑k : Fin r → GapDifference, if k=unchangedPattern r then 0 else purePatternBound u m r s π θ k)+
-    2*((Fintype.card (IncreasingTuple u r):ℝ)*
-      (2*(((r+1:ℕ):ℝ)*θ^r/binomialMass (u*m) θ l*(2*Real.exp (-(m:ℝ)/(1024*u))))))^2
-
-theorem initialPureDegreeBound_nonneg (u m r s : ℕ) (π : IntSeries) (θ : ℝ) (l : ℕ) :
-    0 ≤ initialPureDegreeBound u m r s π θ l := by
-  unfold initialPureDegreeBound
-  split_ifs
-  · exact le_rfl
-  · apply add_nonneg _ (by positivity)
-    apply mul_nonneg (by positivity)
-    apply Finset.sum_nonneg
-    intro k _
-    split_ifs
-    · exact le_rfl
-    · unfold purePatternBound
-      split_ifs  <;> apply mul_nonneg _ (seriesL2Sq_nonneg _)
-      all_goals dsimp only [periodDifferencePrefactor,lengthDifferenceConstant,periodCentralSumConstant]
-      all_goals positivity
-
-theorem pureMotif_degree_zero {u m l : ℕ} (hl : l ≤ u*m) (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
-    (π : IntSeries) (hπsum : ∑X∈π.coeff.support,π.coeff X=1) (s : ℕ) :
-    (∑σ : Fin l → Bool,if monomialDegree σ=0 then
-      (mixedMotifCoefficient hl off hsafe π s 1 σ-purePeriodCoefficient hl off σ)^2 else 0)=0 := by
-  simp_rw [monomialDegree_eq_zero_iff]
-  simp [mixedMotifCoefficient_empty hl off hsafe π hπsum,purePeriodCoefficient,averagedCoefficient_empty]
-
-theorem pureMotif_degree_cancellation {u m l r : ℕ}
-    (hu : 0 < u) (hm : 128 ≤ m) (hr : 0 < r) (hl : l ≤ u*m)
-    (off : Fin m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 3)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
-    (π : IntSeries) (hπ : ∀X,0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
-    (s : ℕ) (hcentral : CentralMotifSupport u s π)
-    {θ : ℝ} (hθ : θ∈Set.Ioo 0 1) (hhalf : 1/2 ≤ θ)
-    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ)) :
-    (∑σ : Fin l → Bool,if monomialDegree σ=r then
-      (mixedMotifCoefficient hl off hsafe π s 1 σ-purePeriodCoefficient hl off σ)^2 else 0) ≤ 
-        initialPureDegreeBound u m r s π θ l := by
-  refine (sum_degree_coeff_sq_le _).trans ?_
-  simp_rw [mixedMotifCoefficient_pure_rank_difference hl off hsafe π hπsum s]
-  simpa only [initialPureDegreeBound,ite_eq_right (by omega : r ≠ 0)] using
-    initial_pure_energy hu hm hr hl off hoff hsafe π hπ hπsum s hcentral hθ hhalf hmean hv
-
-theorem purePeriod_degree_geometric {u m l r : ℕ} (hl : l ≤ u*m) (hr : 0 < r)
-    (hm : 32 ≤ m) (hum : u ≤ m) (off : Fin m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 3)
-    {q θ : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
-    (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
-    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ)) (w : ℝ) :
-    w^(2*r)*(∑σ : Fin l → Bool,if monomialDegree σ=r then (purePeriodCoefficient hl off σ)^2 else 0) ≤ 
-      (initialMomentRatio u m q w)^r := by
-  have hw0 : 0 ≤ w^(2*r) := by rw [pow_mul]; positivity
-  have hp : 0 ≤ 3*rankSchurConstant/q := by positivity [rankSchurConstant_pos,hq.1]
-  by_cases hru : r ≤ u
-  · have hb := period_degree_square_sum hl hr hru hm hum hq hθ hmean hv (fun _ => off)
-      (fun _ => hoff) (fun _ _ => by simp)
-    have hsqrt : Real.sqrt ((u:ℝ)/((m:ℝ)*r)) ≤ Real.sqrt ((u:ℝ)/m) := by
-      apply Real.sqrt_le_sqrt
-      apply div_le_div_of_nonneg_left (by positivity) (by exact_mod_cast (show 0 < m by omega))
-      have hr1 : (1:ℝ) ≤ r := by exact_mod_cast hr
-      nlinarith
-    calc
-      _  ≤  w^(2*r)*((3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/((m:ℝ)*r)))^r :=
-        mul_le_mul_of_nonneg_left hb hw0
-      _  ≤  w^(2*r)*((3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/m))^r := by gcongr
-      _ = _ := by unfold initialMomentRatio; rw [pow_mul,mul_pow,mul_pow]; ring
-  · have H := defect_degree_bound hl (periodSlot (u:=u) (fun _ => off)) (periodSlot_strictMono _) (by omega : u < r)
-    change _=0 at H
-    change w^(2*r)*_ ≤ _
-    rw [show (∑σ : Fin l → Bool,if monomialDegree σ=r then (purePeriodCoefficient hl off σ)^2 else 0)=0 from H,mul_zero]
-    exact pow_nonneg (by unfold initialMomentRatio; positivity [rankSchurConstant_pos,hq.1]) _
-
-theorem pureMotif_degree_geometric {u m l r : ℕ} (hl : l ≤ u*m) (hr : 0 < r)
-    (hm : 32 ≤ m) (hum : u ≤ m) (off : Fin m)
-    (hi : 1 ≤ off.val ∧ off.val+1 < m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 2)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
-    (π : IntSeries) (hπ : ∀X,0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
-    (s : ℕ) {q θ : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
-    (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
-    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ)) (w : ℝ) :
-    w^(2*r)*(∑σ : Fin l → Bool,if monomialDegree σ=r then
-      (mixedMotifCoefficient hl off hsafe π s 1 σ-purePeriodCoefficient hl off σ)^2 else 0) ≤ 
-        4*(initialMomentRatio u m q w)^r := by
-  have H := mul_le_mul_of_nonneg_left (degree_square_sub_le
-    (mixedMotifCoefficient hl off hsafe π s 1) (purePeriodCoefficient hl off) r)
-    (by rw [pow_mul]; positivity : 0 ≤ w^(2*r))
-  have h1 := mixedMotif_degree_geometric hl hr hm hum off hi hoff hsafe π hπ hπsum s 1 hq hθ hmean hv w
-  have h2 := purePeriod_degree_geometric hl hr hm hum off (hoff.trans (by norm_num)) hq hθ hmean hv w
-  change _ ≤ initialMomentRatio u m q w ^r at h1
-  nlinarith only [H,h1,h2]
-
-def initialPureBudget (u m l R s : ℕ) (π : IntSeries) (θ q w : ℝ) : ℝ :=
-  (∑r∈Finset.range R,w^(2*r)*initialPureDegreeBound u m r s π θ l)+
-    4*(initialMomentRatio u m q w)^R/(1-initialMomentRatio u m q w)
-
-theorem pure_initial_full_energy {u m l R : ℕ} (hu : 0 < u) (hm : 128 ≤ m)
-    (hum : u ≤ m) (hR : 0 < R) (hl : l ≤ u*m) (off : Fin m)
-    (hi : 1 ≤ off.val ∧ off.val+1 < m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 2)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
-    (π : IntSeries) (hπ : ∀X,0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
-    (s : ℕ) (hcentral : CentralMotifSupport u s π)
-    {q θ w : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
-    (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
-    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
-    (hw : 0 ≤ w) (ha : initialMomentRatio u m q w < 1) :
-    multilinearEnergy (fun σ => mixedMotifCoefficient hl off hsafe π s 1 σ-purePeriodCoefficient hl off σ) w ≤ 
-      initialPureBudget u m l R s π θ q w := by
-  have hq2 : q^2 ≤ (1/4:ℝ) := by
-    calc q^2 ≤ (1/2:ℝ)^2 := pow_le_pow_left₀ hq.1.le hq.2 2
-         _ = _ := by norm_num
-  have hθ0 : θ∈Set.Ioo 0 1 := by constructor  <;> nlinarith [hq.1,hθ.1,hθ.2]
-  have hhalf : 1/2 ≤ θ := by nlinarith [hθ.1]
-  apply multilinearEnergy_low_high _ w _ 4 hw (by unfold initialMomentRatio; positivity [rankSchurConstant_pos,hq.1]) ha
-    (by norm_num) (initialPureDegreeBound u m · s π θ l) _ _
-    (fun r => initialPureDegreeBound_nonneg _ _ _ _ _ _ _)
-  · intro r _hr
-    by_cases hr0 : r=0
-    · subst r
-      rw [pureMotif_degree_zero hl off hsafe π hπsum s]
-      exact initialPureDegreeBound_nonneg _ _ _ _ _ _ _
-    · exact pureMotif_degree_cancellation hu hm (Nat.pos_of_ne_zero hr0) hl off (hoff.trans (by norm_num))
-        hsafe π hπ hπsum s hcentral hθ0 hhalf hmean hv
-  · intro r hr
-    exact pureMotif_degree_geometric hl (by omega) (by omega) hum off hi hoff hsafe π hπ hπsum s hq hθ hmean hv w
-
-end TraceReconstruction
-
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-def purePeriodDensity {u m l : ℕ} (hl : l ≤ u*m) (off : Fin m) (q : ℝ) :
-    (Fin l → PrimitiveMark) → ℝ := multilinear (purePeriodCoefficient hl off) (primitiveScore q)
-
-theorem purePeriodDensity_nonneg {u m l : ℕ} (hl : l ≤ u*m) (off : Fin m)
-    {q : ℝ} (hq : q∈Set.Ioo 0 1) (x : Fin l → PrimitiveMark) :
-    0 ≤ purePeriodDensity hl off q x := by
-  have H := (conditionalPrimitiveLaw hl q hq (defectTypes (periodSlot (u:=u) (fun _ => off)))).nonneg x
-  rw [conditionalDefectLaw_density] at H
-  exact nonneg_of_mul_nonneg_right H (primitive_product_mass_pos hq x)
-
-theorem purePeriodDensity_total {u m l : ℕ} (hl : l ≤ u*m) (off : Fin m)
-    {q : ℝ} (hq : q∈Set.Ioo 0 1) :
-    ((primitiveMarkLaw q hq false).product l).expect (purePeriodDensity hl off q)=1 := by
-  have H := (conditionalPrimitiveLaw hl q hq (defectTypes (periodSlot (u:=u) (fun _ => off)))).total
-  simpa only [conditionalDefectLaw_density,FiniteLaw.expect,purePeriodDensity,purePeriodCoefficient] using H
-
-theorem purePeriodLaw_withDensity {u m l : ℕ} (hl : l ≤ u*m) (off : Fin m)
-    {q : ℝ} (hq : q∈Set.Ioo 0 1) :
-    conditionalPrimitiveLaw hl q hq (defectTypes (periodSlot (u:=u) (fun _ => off)))=
-      ((primitiveMarkLaw q hq false).product l).withDensity (purePeriodDensity hl off q)
-        (purePeriodDensity_nonneg hl off hq) (purePeriodDensity_total hl off hq) := by
-  apply FiniteLaw.ext
-  funext x
-  exact conditionalDefectLaw_density hl hq _ x
-
-theorem purePeriod_centered_energy {u m l : ℕ} (hl : l ≤ u*m) (hm : 32 ≤ m) (hum : u ≤ m)
-    (off : Fin m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 3)
-    {q θ w : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
-    (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
-    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
-    (hw : 0 ≤ w) (ha : initialMomentRatio u m q w < 1) :
-    multilinearEnergy (nonconstantCoefficient (purePeriodCoefficient hl off)) w ≤
-      initialMomentRatio u m q w/(1-initialMomentRatio u m q w) := by
-  have H := multilinearEnergy_low_high (R:=1)
-    (nonconstantCoefficient (purePeriodCoefficient hl off)) w _ 1 hw
-    (by unfold initialMomentRatio; positivity [rankSchurConstant_pos,hq.1]) ha (by norm_num) (fun _ => 0)
-    (fun r hr => by
-      have he : r=0 := by omega
-      subst r
-      rw [nonconstant_degree_zero])
-    (fun r hr => by
-      rw [nonconstant_degree_pos _ (by omega)]
-      simpa only [one_mul] using purePeriod_degree_geometric hl (by omega) hm hum off hoff hq hθ hmean hv w)
-    (fun _ => le_rfl)
-  simpa using H
-
-theorem purePeriod_centered_moment {u m l : ℕ} (hl : l ≤ u*m) (hm : 32 ≤ m) (hum : u ≤ m)
-    (off : Fin m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 3) (k : ℕ) (hk : 1 ≤ k)
-    {q θ w : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
-    (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
-    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
-    (hw : w=((6*(2*k:ℕ):ℝ)*(2/q))) (ha : initialMomentRatio u m q w < 1) :
-    ((primitiveMarkLaw q ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩ false).product l).evenMomentSq
-      (fun x => purePeriodDensity hl off q x-1) k ≤
-      initialMomentRatio u m q w/(1-initialMomentRatio u m q w) := by
-  have hq' : q∈Set.Ioo 0 1 := ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩
-  have he : (fun x => purePeriodDensity hl off q x-1)=
-      multilinear (nonconstantCoefficient (purePeriodCoefficient hl off)) (primitiveScore q) := by
-    funext x
-    rw [nonconstant_multilinear]
-    simp only [purePeriodCoefficient,averagedCoefficient_empty,purePeriodDensity]
-  rw [he]
-  apply (multilinear_moment_sq_scaled (primitiveMarkLaw q hq' false) (primitiveScore q) k hk
-    (primitive_score_centered hq') (div_pos (by norm_num) hq.1) (primitive_score_evenMoment hq' k) l _).trans
-  rw [←hw]
-  exact purePeriod_centered_energy hl hm hum off hoff hq hθ hmean hv (by rw [hw]; positivity [hq.1]) ha
-
-theorem initial_pure_moment {u m l R : ℕ} (hu : 0 < u) (hm : 128 ≤ m)
-    (hum : u ≤ m) (hR : 0 < R) (hl : l ≤ u*m) (off : Fin m)
-    (hi : 1 ≤ off.val ∧ off.val+1 < m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 2)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
-    (π : IntSeries) (hπ : ∀X,0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
-    (s k : ℕ) (hcentral : CentralMotifSupport u s π) (hk : 1 ≤ k)
-    {q θ w : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
-    (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
-    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
-    (hw : w=((6*(2*k:ℕ):ℝ)*(2/q))) (ha : initialMomentRatio u m q w < 1) :
-    ((primitiveMarkLaw q ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩ false).product l).evenMomentSq
-      (fun x => initialMotifDensity hl off hsafe π s 1 q x-purePeriodDensity hl off q x) k ≤
-        initialPureBudget u m l R s π θ q w := by
-  have hq' : q∈Set.Ioo 0 1 := ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩
-  have he : (fun x => initialMotifDensity hl off hsafe π s 1 q x-purePeriodDensity hl off q x)=
-      multilinear (fun σ => mixedMotifCoefficient hl off hsafe π s 1 σ-purePeriodCoefficient hl off σ)
-        (primitiveScore q) := by funext x; exact (multilinear_sub _ _ _ _).symm
-  rw [he]
-  apply (multilinear_moment_sq_scaled (primitiveMarkLaw q hq' false) (primitiveScore q) k hk
-    (primitive_score_centered hq') (div_pos (by norm_num) hq.1) (primitive_score_evenMoment hq' k) l _).trans
-  rw [←hw]
-  exact pure_initial_full_energy hu hm hum hR hl off hi hoff hsafe π hπ hπsum s hcentral hq hθ hmean hv
-    (by rw [hw]; positivity [hq.1]) ha
-
-theorem initial_pure_conditional_clipped_exists {u m l R : ℕ} (hu : 0 < u) (hm : 128 ≤ m)
-    (hum : u ≤ m) (hR : 0 < R) (hl : l ≤ u*m) (off : Fin m)
-    (hi : 1 ≤ off.val ∧ off.val+1 < m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 2)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
-    (π : IntSeries) (hπ : ∀X,0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
-    (s k : ℕ) (hcentral : CentralMotifSupport u s π) (hk : 1 ≤ k)
-    {q θ w : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
-    (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
-    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
-    (hw : w=((6*(2*k:ℕ):ℝ)*(2/q))) (hscale : initialMomentRatio u m q w ≤ 1/2)
-    (hclip : Real.sqrt (6*(4*(initialMomentRatio u m q w/(1-initialMomentRatio u m q w)))^k) ≤ 1/2) :
-    Nonempty (ClippedOracle
-      (conditionalPrimitiveLaw hl q ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩
-        (defectTypes (periodSlot (u:=u) (fun _ => off))))
-      (conditionalMotifLaw hl off hsafe ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩ π hπ hπsum s 1) k
-      (Real.sqrt (6*(4*(initialMomentRatio u m q w/(1-initialMomentRatio u m q w)))^k))
-      (64*initialPureBudget u m l R s π θ q w)) := by
-  have hq' : q∈Set.Ioo 0 1 := ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩
-  let A := (primitiveMarkLaw q hq' false).product l
-  let f := purePeriodDensity hl off q
-  let g := initialMotifDensity hl off hsafe π s 1 q
-  let U := initialMomentRatio u m q w/(1-initialMomentRatio u m q w)
-  have ha : initialMomentRatio u m q w < 1 := by linarith
-  have hz : 0 ≤ initialMomentRatio u m q w := by unfold initialMomentRatio; positivity [rankSchurConstant_pos,hq.1]
-  have hU : 0 ≤ U := div_nonneg hz (sub_nonneg.mpr ha.le)
-  have hU1 : U ≤ 1 := (div_le_iff₀ (sub_pos.mpr ha)).mpr (by linarith)
-  have hf : A.expect (fun x => |f x-1|^(2*k)) ≤ U^k :=
-    A.expect_even_pow_le_of_moment _ k hk (purePeriod_centered_moment hl (by omega) hum off
-      (hoff.trans (by norm_num)) k hk hq hθ hmean hv hw ha)
-  have hg : A.expect (fun x => |g x-1|^(2*k)) ≤ U^k :=
-    A.expect_even_pow_le_of_moment _ k hk (initial_centered_moment hl (by omega) hum off hi hoff hsafe π hπ hπsum s k hk 1 hq hθ hmean hv hw ha)
-  have hsig := initial_pure_moment hu hm hum hR hl off hi hoff hsafe π hπ hπsum s k hcentral hk hq hθ hmean hv hw ha
-  have hV := (A.evenMomentSq_nonneg (fun x => g x-f x) k).trans hsig
-  have hsig' := A.expect_even_pow_le_of_moment _ k hk hsig
-  rw [purePeriodLaw_withDensity hl off hq',initialMotifLaw_withDensity hl off hsafe π hπ hπsum s 1 hq']
-  have H := A.exists_clippedOracle f g
-    (purePeriodDensity_nonneg hl off hq') (initialMotifDensity_nonneg hl off hsafe π hπ hπsum s 1 hq')
-    (purePeriodDensity_total hl off hq') (initialMotifDensity_mean hl off hsafe π hπ hπsum s 1 hq')
-    Finset.univ 0 (by simp [FiniteLaw.eventMass]) (by simp [FiniteLaw.eventMass]) k hk U hU hU1 hf hg
-    (initialPureBudget u m l R s π θ q w) hV (by simpa only [Finset.mem_univ,ite_true] using hsig')
-    (by simpa only [zero_add] using hclip)
-  simpa only [zero_add] using H
-
-end TraceReconstruction
-
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- A common finite decoder transfers normalized clipping to actual complete
- traces. Neither interface is permitted to assume the main lower bound. -/
-theorem ClippedOracle.tracePair {α : Type} [Fintype α] {P Q : FiniteLaw α}
-    {q e V : ℝ} {x y : List Bool} {k : ℕ} (f : α → List Bool)
-    (C : ClippedOracle P Q k e V) (hP : SimulatesTrace q P f x)
-    (hQ : SimulatesTrace q Q f y) : Nonempty (OracleTracePair q x y k e V) := by
-  refine ⟨{
-    Outcome := α
-    finiteOutcome := inferInstance
-    first := C.first
-    second := C.second
-    decode := f
-    score := C.score
-    density := C.density
-    likelihood_nonneg := C.likelihood_nonneg
-    centered := C.centered
-    moment := C.moment
-    first_simulation := ?_
-    second_simulation := ?_
-  }⟩
-  · simpa using hP.approximate.of_tv C.first_tv
-  · simpa using hQ.approximate.of_tv C.second_tv
-
-def initialPureConditionalLaw {Side : Type*} {J K m : ℕ} (off : Fin m)
-    {q : ℝ} (hq : q∈Set.Ioo 0 1) (z : InitialMetadata Side J (K*m)) :
-    FiniteLaw (Fin z.2.val → PrimitiveMark) :=
-  conditionalPrimitiveLaw (by have H:=z.2.isLt; omega) q hq
-    (defectTypes (periodSlot (u:=K) (fun _ => off)))
-
-def initialPureSigmaLaw {Side : Type*} [Fintype Side] {J K m : ℕ}
-    (M : PairedMechanisms Side J) (off : Fin m) {q : ℝ} (hq : q∈Set.Ioo 0 1) :
-    FiniteLaw (InitialObservation Side J (K*m)) :=
-  (initialMetadataLaw M (K*m) hq).sigma (initialPureConditionalLaw off hq)
-
-theorem initialPureSigmaLaw_expect {Side : Type*} [Fintype Side] {J K m : ℕ}
-    (M : PairedMechanisms Side J) (off : Fin m) {q : ℝ} (hq : q∈Set.Ioo 0 1)
-    (F : Reports Side J × BoundedMarks (K*m) → ℝ) :
-    (initialPureSigmaLaw M off hq).expect (fun z => F (z.1.1,⟨z.1.2,z.2⟩))=
-    (M.emissionOracle (fun _ => countedPrimitiveLaw (le_refl (K*m)) hq
-      (defectTypes (periodSlot (u:=K) (fun _ => off))))).expect F := by
-  simp only [initialPureSigmaLaw,initialMetadataLaw,FiniteLaw.expect_sigma,
-    FiniteLaw.expect_prod,PairedMechanisms.emissionOracle,FiniteLaw.expect_joint,
-    FiniteLaw.expect_mixture,FiniteLaw.expect_const,initialPureConditionalLaw,
-    countedPrimitiveLaw,FiniteLaw.expect_sigma]
-  have hl (l : Fin (K*m+1)) : l.val ≤ K*m := Nat.le_of_lt_succ l.isLt
-  simp only [boundedConditionalPrimitiveLaw,hl,dite_eq_left]
-
-theorem initial_pure_sigma_simulates {Side : Type*} [Fintype Side] {J K m : ℕ}
-    (M : PairedMechanisms Side J) (k : ℕ) (off : Fin m)
-    {q : ℝ} (hq : q∈Set.Ioo 0 1) (side : Side → List Bool)
-    (hfirst : ∀i,SimulatesTrace q (M.first i) side (pureAtomRun (initialPeriod m off.val) k))
-    (hsecond : ∀i,SimulatesTrace q (M.second i) side (pureAtomRun (initialPeriod m off.val) (k+1))) :
-    SimulatesTrace q (initialPureSigmaLaw (K:=K) M off hq) (decodeInitialObservation side)
-      (pureAtomRun (initialPeriod m off.val) (J*(2*k+1)+K)) := by
-  intro F
-  unfold decodeInitialObservation
-  rw [initialPureSigmaLaw_expect M off hq (fun z => F (decodePairedEmission side decodeBoundedMarks z)),
-    ←M.emission_factorization,FiniteLaw.expect_mixture]
-  have H (b : Fin J → Bool) :
-      ((M.reportsGiven b).prod (countedPrimitiveLaw (le_refl (K*m)) hq
-        (defectTypes (periodSlot (u:=K) (fun _ => off))))).expect
-        (fun z => F (decodePairedEmission side decodeBoundedMarks z))=
-      listDeletionExpectation q (pureAtomRun (initialPeriod m off.val) (J*(2*k+1)+K)) F := by
-    have hc := countedPrimitiveLaw_simulates (le_refl (K*m)) hq (defectTypes (periodSlot (u:=K) (fun _ => off)))
-    rw [periodic_primitive_word,periodString_constant] at hc
-    have hs := M.reportsGiven_paired_simulates b side decodeBoundedMarks _
-      (fun _ => pureAtomRun (initialPeriod m off.val) k)
-      (fun _ => pureAtomRun (initialPeriod m off.val) (k+1)) _ hfirst hsecond hc
-    have hp : coordinateWord (fun i => if b i then pureAtomRun (initialPeriod m off.val) (k+1)
-        else pureAtomRun (initialPeriod m off.val) k)=
-        pureAtomRun (initialPeriod m off.val) (initialPrefixPeriods k b) := by
-      have h (i : Fin J) : (if b i then pureAtomRun (initialPeriod m off.val) (k+1) else
-          pureAtomRun (initialPeriod m off.val) k)=pureAtomRun (initialPeriod m off.val) (k+bitIndex (b i)) := by
-        cases b i <;> simp [bitIndex]
-      simp_rw [h]
-      exact coordinateWord_pureAtomRun _ _
-    have hz : coordinateWord (fun i => if b i then pureAtomRun (initialPeriod m off.val) k
-        else pureAtomRun (initialPeriod m off.val) (k+1))=
-        pureAtomRun (initialPeriod m off.val) (initialSuffixPeriods k b) := by
-      have h (i : Fin J) : (if b i then pureAtomRun (initialPeriod m off.val) k else
-          pureAtomRun (initialPeriod m off.val) (k+1))=pureAtomRun (initialPeriod m off.val) (k+(1-bitIndex (b i))) := by
-        cases b i <;> simp [bitIndex]
-      simp_rw [h]
-      exact coordinateWord_pureAtomRun _ _
-    rw [hp,hz,←pureAtomRun_add,←pureAtomRun_add] at hs
-    have ht : initialPrefixPeriods k b+K+initialSuffixPeriods k b=J*(2*k+1)+K := by
-      have ht:=initialPairPeriods_total k b
-      omega
-    rw [ht] at hs
-    exact hs F
-  simp only [H,FiniteLaw.expect_const]
-
-theorem initial_pure_sigma_clipped_exists {Side : Type*} [Fintype Side] {J K m R : ℕ}
-    (M : PairedMechanisms Side J) (hK : 0 < K) (hm : 128 ≤ m) (hKm : K ≤ m)
-    (hR : 0 < R) (off : Fin m) (hi : 1 ≤ off.val ∧ off.val+1 < m)
-    (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 2)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=K) (fun _ => off)))
-    (c : ℤ) (s k : ℕ) (hk : 1 ≤ k)
-    (hleft : (K:ℝ)/3 ≤ 2*((c:ℝ)-J)+1)
-    (hright : (K:ℝ)/3 ≤ 2*K-(2*((c:ℝ)+(2:ℝ)^s)+1))
-    {q ω w V : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
-    (hw : w=((6*(2*k:ℕ):ℝ)*(2/q)))
-    (hscale : initialMomentRatio K m q w ≤ 1/2)
-    (hclip : Real.sqrt (6*(4*(initialMomentRatio K m q w/(1-initialMomentRatio K m q w)))^k) ≤ 1/2)
-    (hV : 0 ≤ V)
-    (hbudget : ∀z : InitialMetadata Side J (K*m),initialMetadataGood q ω z →
-      64*initialPureBudget K m z.2.val R s (posteriorAnchorSeries c (fun i => (z.1 i).1))
-        ((z.2.val:ℝ)/(K*m:ℕ)) q w ≤ V) :
-    Nonempty (ClippedOracle
-      (initialPureSigmaLaw M off ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩)
-      (initialSigmaLaw M off hsafe ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩ c s 1) k
-      (Real.sqrt (6*(4*(initialMomentRatio K m q w/(1-initialMomentRatio K m q w)))^k)+
-        (initialMetadataLaw M (K*m) ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩).expect
-          (fun z => if initialMetadataGood q ω z then 0 else 1)) V) := by
-  apply ClippedOracle.assemble_good _ _ _ (initialMetadataGood q ω) k hk _ V (Real.sqrt_nonneg _) hV
-  intro z hz
-  have hmean : ((K*m:ℕ):ℝ)*((z.2.val:ℝ)/(K*m:ℕ))=z.2.val := by
-    exact mul_div_cancel₀ _ (by exact_mod_cast (Nat.mul_ne_zero (by omega : K≠0) (by omega : m≠0)))
-  have H := initial_pure_conditional_clipped_exists hK hm hKm hR (by have H:=z.2.isLt; omega) off hi hoff hsafe
-    (posteriorAnchorSeries c (fun i => (z.1 i).1)) (posteriorAnchorSeries_nonneg c _)
-    (posteriorAnchorSeries_total c _) s k
-    (posteriorAnchorSeries_central K s c _ (by simpa using hleft) hright) hk hq hz.1.1 hmean hz.1.2 hw hscale hclip
-  exact ⟨(Classical.choice H).weaken le_rfl (hbudget z hz)⟩
-
-end TraceReconstruction
-
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- The half-mean hidden-count event yields the correct L², not supremum,
- exponent J^{-s-1/2} in the initial Prouhet cancellation. -/
-def hiddenSignedL2Bound (J s : ℕ) (ω : ℝ) : ℝ :=
-  ((2:ℝ)^(s*s))^2*((2*Real.sqrt (4*((s:ℝ)+1)/((J:ℝ)*ω^2+2)))^s)^2*
-    Real.sqrt (4*((s:ℝ)+1)/((J:ℝ)*ω^2+2))
-
-def hiddenUnsignedL2Bound (J s : ℕ) (ω : ℝ) : ℝ :=
-  ((2:ℝ)^s)^2*hiddenBinomialCap J ω
-
-theorem hiddenSignedL2Bound_nonneg (J s : ℕ) (ω : ℝ) : 0≤hiddenSignedL2Bound J s ω := by
-  unfold hiddenSignedL2Bound; positivity
-
-theorem hiddenUnsignedL2Bound_nonneg (J s : ℕ) (ω : ℝ) : 0≤hiddenUnsignedL2Bound J s ω := by
-  exact mul_nonneg (sq_nonneg _) (hiddenBinomialCap_nonneg _ _)
-
-theorem hidden_quotient_bound (J G s : ℕ) (ω : ℝ) (h : (J:ℝ)*ω^2/2≤G) :
-    binomialBound (G/(s+1))≤Real.sqrt (4*((s:ℝ)+1)/((J:ℝ)*ω^2+2)) := by
-  apply (binomialBound_quotient_le G s).trans
-  apply Real.sqrt_le_sqrt
-  apply (div_le_div_iff₀ (by positivity : (0:ℝ)<(G:ℝ)+1)
-    (by positivity : (0:ℝ)<(J:ℝ)*ω^2+2)).mpr
-  have hh := mul_le_mul_of_nonneg_left h (by positivity : (0:ℝ)≤4*((s:ℝ)+1))
-  nlinarith
-
-theorem posteriorAnchorSeries_signed_L2 {J : ℕ} (d : Fin J→Option Bool) (c : ℤ) (s : ℕ) (ω : ℝ)
-    (h : (J:ℝ)*ω^2/2≤∑i,(hiddenTrial (d i):ℝ)) :
-    seriesL2Sq (prouhetSeries s*posteriorAnchorSeries c d)≤hiddenSignedL2Bound J s ω := by
-  have hn : (J:ℝ)*ω^2/2≤((∑i,hiddenTrial (d i):ℕ):ℝ) := by simpa only [Nat.cast_sum] using h
-  have hb := hidden_quotient_bound J (∑i,hiddenTrial (d i)) s ω hn
-  apply (translated_binomial_smoothing (∑i,hiddenTrial (d i)) s _).trans
-  unfold hiddenSignedL2Bound
-  rw [mul_assoc]
-  apply mul_le_mul_of_nonneg_left _ (by positivity)
-  apply mul_le_mul
-  · apply pow_le_pow_left₀ (by unfold binomialBound; positivity)
-    exact pow_le_pow_left₀ (by unfold binomialBound; positivity) (mul_le_mul_of_nonneg_left hb (by norm_num)) _
-  · exact hb
-  · exact Real.sqrt_nonneg _
-  · positivity
-
-theorem posteriorAnchorSeries_unsigned_L2 {J : ℕ} (d : Fin J→Option Bool) (c : ℤ) (s : ℕ) (ω : ℝ)
-    (h : (J:ℝ)*ω^2/2≤∑i,(hiddenTrial (d i):ℝ)) :
-    seriesL2Sq (geomSeries (2^s)*posteriorAnchorSeries c d)≤hiddenUnsignedL2Bound J s ω := by
-  have hn : (J:ℝ)*ω^2/2≤((∑i,hiddenTrial (d i):ℕ):ℝ) := by simpa only [Nat.cast_sum] using h
-  calc
-    _ ≤ seriesL1 (geomSeries (2^s))^2*seriesL2Sq (posteriorAnchorSeries c d) := seriesL2Sq_mul_le _ _
-    _ ≤ ((2:ℝ)^s)^2*hiddenBinomialCap J ω := by
-      apply mul_le_mul
-      · exact pow_le_pow_left₀ (seriesL1_nonneg _) (by exact_mod_cast geomSeries_L1_le (2^s)) _
-      · rw [posteriorAnchorSeries,translateSeries_L2Sq]
-        exact (fairSeries_L2Sq_bound _).trans (hidden_binomial_bound J _ ω hn)
-      · exact seriesL2Sq_nonneg _
-      · positivity
-
-end TraceReconstruction
-
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-def initialDifferenceCap (d : ℕ) (q : ℝ) : ℝ :=
-  2*Real.sqrt (8*((d:ℝ)+1)/q^2)
-
-theorem initial_theta_variance {q θ : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
-    (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4)) :
-    θ∈Set.Ioo 0 1 ∧ 1/2≤θ ∧ q^2/8≤θ*(1-θ) := by
-  have hq2 : q^2≤1/4 := by nlinarith [hq.1,hq.2]
-  have hs : 0<q^2 := sq_pos_of_pos hq.1
-  have hh : 1/2≤θ := by linarith [hθ.1]
-  refine ⟨⟨by linarith,by linarith [hθ.2]⟩,hh,?_⟩
-  have H := mul_le_mul hh (show q^2/4≤1-θ by linarith [hθ.2])
-    (by positivity : (0:ℝ)≤q^2/4) (by linarith : 0≤θ)
-  nlinarith
-
-theorem initialDifferenceCap_bound (d : ℕ) {q θ : ℝ}
-    (hq : q∈Set.Ioc 0 (1/2:ℝ)) (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4)) :
-    lengthDifferenceConstant d θ≤ initialDifferenceCap d q := by
-  have ht := initial_theta_variance hq hθ
-  unfold lengthDifferenceConstant initialDifferenceCap
-  apply mul_le_mul_of_nonneg_left _ (by norm_num)
-  apply Real.sqrt_le_sqrt
-  apply (div_le_div_iff₀ (mul_pos ht.1.1 (sub_pos.mpr ht.1.2)) (sq_pos_of_pos hq.1)).mpr
-  have H := mul_le_mul_of_nonneg_left ht.2.2 (by positivity : (0:ℝ)≤8*((d:ℝ)+1))
-  nlinarith
-
-/-- All q dependence of a fixed rank-difference estimate is exposed here.
- The powers of u and m are retained exactly, including both large-gap gains. -/
-def initialPrefactorCap (u m r d : ℕ) (q : ℝ) : ℝ :=
-  144*((u*m:ℕ):ℝ)*(2:ℝ)^(2*r)*(initialDifferenceCap d q)^(2*d+r+2) /
-    (((m:ℝ)/4)^d*Real.sqrt ((u:ℝ)*m/(2*(r+1:ℝ)))*
-      (Real.sqrt ((m:ℝ)/4))^(r+1))
-
-theorem initialPrefactorCap_nonneg (u m r d : ℕ) (q : ℝ) : 0≤ initialPrefactorCap u m r d q := by
-  unfold initialPrefactorCap initialDifferenceCap
-  positivity
-
-theorem initial_prefactor_bound (u m r d : ℕ) {q θ : ℝ}
-    (hq : q∈Set.Ioc 0 (1/2:ℝ)) (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4)) :
-    periodDifferencePrefactor u m r d θ≤ initialPrefactorCap u m r d q := by
-  unfold periodDifferencePrefactor initialPrefactorCap
-  apply div_le_div_of_nonneg_right _ (by positivity)
-  apply mul_le_mul_of_nonneg_left _ (by positivity)
-  exact pow_le_pow_left₀ (by unfold lengthDifferenceConstant; positivity) (initialDifferenceCap_bound d hq hθ) _
-
-theorem binomial_mean_inverse_le_sqrt {n l : ℕ} {θ : ℝ}
-    (hθ : θ∈Set.Ioo 0 1) (hl : l≤n) (hmean : (n:ℝ)*θ=l)
-    (hv : 1≤(n:ℝ)*θ*(1-θ)) :
-    (binomialMass n θ l)⁻¹≤12*Real.sqrt (n:ℝ) := by
-  have h0 : 0≤(binomialMass n θ l)⁻¹ := inv_nonneg.mpr (binomialMass_nonneg n l ⟨hθ.1.le,hθ.2.le⟩)
-  have hi := binomialMass_mean_inverse_sq n l hθ hl hmean hv
-  have hv' : (n:ℝ)*θ*(1-θ)≤n := by
-    have h := mul_le_mul_of_nonneg_left (show θ*(1-θ)≤1 by nlinarith [sq_nonneg θ]) (Nat.cast_nonneg (α:=ℝ) n)
-    nlinarith
-  have hs := Real.sq_sqrt (Nat.cast_nonneg (α:=ℝ) n)
-  nlinarith [Real.sqrt_nonneg (n:ℝ)]
-
-def initialRankTailCap (u m r : ℕ) : ℝ :=
-  2*((Fintype.card (IncreasingTuple u r):ℝ)*
-    (48*((r+1:ℕ):ℝ)*Real.sqrt (u*m:ℕ)*Real.exp (-(m:ℝ)/(1024*u))))^2
-
-theorem initial_rank_tail_bound {u m l : ℕ} (r : ℕ) {θ : ℝ}
-    (hθ : θ∈Set.Ioo 0 1) (hl : l≤u*m) (hmean : ((u*m:ℕ):ℝ)*θ=l)
-    (hv : 1≤((u*m:ℕ):ℝ)*θ*(1-θ)) :
-    2*((Fintype.card (IncreasingTuple u r):ℝ)*
-      (2*(((r+1:ℕ):ℝ)*θ^r/binomialMass (u*m) θ l*(2*Real.exp (-(m:ℝ)/(1024*u))))))^2 ≤
-    initialRankTailCap u m r := by
-  have hp : θ^r≤1 := pow_le_one₀ hθ.1.le hθ.2.le
-  have hi := binomial_mean_inverse_le_sqrt hθ hl hmean hv
-  have hn : 0≤(binomialMass (u*m) θ l)⁻¹ := inv_nonneg.mpr (binomialMass_nonneg _ _ ⟨hθ.1.le,hθ.2.le⟩)
-  unfold initialRankTailCap
-  apply mul_le_mul_of_nonneg_left _ (by norm_num)
-  have hb := binomialMass_nonneg (u*m) l ⟨hθ.1.le,hθ.2.le⟩
-  apply pow_le_pow_left₀ (by
-    exact mul_nonneg (Nat.cast_nonneg _) (mul_nonneg (by norm_num)
-      (mul_nonneg (div_nonneg (mul_nonneg (Nat.cast_nonneg _) (pow_nonneg hθ.1.le _)) hb)
-        (by positivity))))
-  apply mul_le_mul_of_nonneg_left _ (by positivity)
-  have H := mul_le_mul hp hi hn (by norm_num : (0:ℝ)≤1)
-  have HH := mul_le_mul_of_nonneg_left H (by positivity : (0:ℝ)≤4*((r+1:ℕ):ℝ)*Real.exp (-(m:ℝ)/(1024*u)))
-  simp only [div_eq_mul_inv,one_mul] at *
-  nlinarith only [HH]
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- The central binomial typical interval also guarantees the rank-kernel
- variance condition, rather than imposing it by conditioning. -/
-theorem initialCountAdmissible_of_near {N : ℕ} {q : ℝ}
-    (hq : q∈Set.Ioc 0 (1/2:ℝ)) (hscale : 8≤(N:ℝ)*q^2) (l : Fin (N+1))
-    (hnear : |(l.val:ℝ)-(N:ℝ)*(1-q^2/2)|<(N:ℝ)*q^2/4) :
-    initialCountAdmissible N q l := by
-  have hN : (0:ℝ)<N := by
-    have h0 : (0:ℝ)≤N := Nat.cast_nonneg N
-    by_contra H
-    have hn : (N:ℝ)=0 := le_antisymm (le_of_not_gt H) h0
-    rw [hn,zero_mul] at hscale
-    norm_num at hscale
-  have hq2 : q^2≤1/4 := by nlinarith [hq.1,hq.2]
-  let θ : ℝ := (l.val:ℝ)/(N:ℝ)
-  have he : (N:ℝ)*θ=l.val := mul_div_cancel₀ _ hN.ne'
-  have hθ : θ∈Set.Icc (1-q^2) (1-q^2/4) := by
-    have hn := abs_lt.mp hnear
-    constructor
-    · apply le_of_lt
-      apply (mul_lt_mul_iff_right₀ hN).mp
-      nlinarith [sq_nonneg q]
-    · apply le_of_lt
-      apply (mul_lt_mul_iff_right₀ hN).mp
-      nlinarith
-  have hhalf : 3/4≤θ := by linarith [hθ.1]
-  have hlow : q^2/4≤1-θ := by linarith [hθ.2]
-  have hp := mul_le_mul ((mul_le_mul_iff_right₀ hN).mpr hhalf) hlow
-    (by positivity : (0:ℝ)≤q^2/4) (mul_nonneg hN.le (by linarith : 0≤θ))
-  refine ⟨hθ,?_⟩
-  change 1≤(N:ℝ)*θ*(1-θ)
-  nlinarith
-
-/-- An explicit unconditional tail for the actual count law. Its q⁴ exponent
- is more than enough under the source's q⁻¹=N^{o(1)} standing regime. -/
-theorem initial_count_bad_mass {N : ℕ} {q : ℝ}
-    (hq : q∈Set.Ioc 0 (1/2:ℝ)) (hscale : 8≤(N:ℝ)*q^2) :
-    (boundedCountLaw (le_refl N) (1-primitiveErasure q)
-      (primitiveRetention_mem ⟨hq.1,by linarith [hq.2]⟩)).expect
-      (fun l => if initialCountAdmissible N q l then 0 else 1) ≤
-    2*Real.exp (-(N:ℝ)*q^4/64) := by
-  let p := 1-primitiveErasure q
-  have hp : p∈Set.Icc 0 1 := primitiveRetention_mem ⟨hq.1,by linarith [hq.2]⟩
-  have ht : q^2/8∈Set.Icc (0:ℝ) 1 := ⟨by positivity,by nlinarith [hq.1,hq.2]⟩
-  have H := binomialMass_abs_tail N (a:=(N:ℝ)*q^2/4) hp ht
-  have he : -(q^2/8)*((N:ℝ)*q^2/4)+(N:ℝ)*(q^2/8)^2= -(N:ℝ)*q^4/64 := by ring
-  rw [he] at H
-  apply le_trans _ H
-  rw [Finset.sum_filter,←Fin.sum_univ_eq_sum_range]
-  apply Finset.sum_le_sum
-  intro l _
-  change binomialMass N p l.val*(if initialCountAdmissible N q l then 0 else 1)≤_
-  have hn := binomialMass_nonneg N l.val hp
-  by_cases hl : initialCountAdmissible N q l
-  · rw [ite_eq_left hl,mul_zero]
-    split_ifs <;> positivity
-  · rw [ite_eq_right hl,mul_one,ite_eq_left]
-    exact le_of_not_gt (fun h => hl (initialCountAdmissible_of_near hq hscale l (by simpa [p,primitiveErasure] using h)))
-
-/-- The true bad metadata mass is charged without changing any conditional
- law or requiring an unproved concentration conclusion. -/
-theorem initial_metadata_bad_mass {Side : Type*} [Fintype Side] {J N : ℕ}
-    (M : PairedMechanisms Side J) {q ω : ℝ}
-    (hq : q∈Set.Ioc 0 (1/2:ℝ)) (hscale : 8≤(N:ℝ)*q^2)
-    (hw : ∀i,M.hiddenWeight i=ω) (hp : ω^2∈Set.Icc 0 1) :
-    (initialMetadataLaw M N ⟨hq.1,by linarith [hq.2]⟩).expect
-      (fun z => if initialMetadataGood q ω z then 0 else 1) ≤
-    2*Real.exp (-(N:ℝ)*q^4/64)+Real.exp (-(J:ℝ)*ω^4/16) := by
-  calc
-    _ ≤ (initialMetadataLaw M N ⟨hq.1,by linarith [hq.2]⟩).expect
-        (fun z => (if initialCountAdmissible N q z.2 then 0 else 1)+
-          (if (PairedMechanisms.hiddenCount z.1:ℝ)<(J:ℝ)*ω^2/2 then 1 else 0)) := by
-      apply FiniteLaw.expect_mono
-      intro z
-      have he : (PairedMechanisms.hiddenCount z.1:ℝ)=∑i,(hiddenTrial (z.1 i).1:ℝ) := by
-        simp only [PairedMechanisms.hiddenCount,Nat.cast_sum]
-      unfold initialMetadataGood
-      rw [←he]
-      split_ifs <;> simp_all
-    _ ≤ _ := by
-      simp only [FiniteLaw.expect_add,initialMetadataLaw,FiniteLaw.expect_prod]
-      apply add_le_add
-      · exact (M.reports.expect_mono (fun _ => initial_count_bad_mass hq hscale)).trans (by rw [FiniteLaw.expect_const])
-      · convert M.hiddenCount_small hw hp using 1
-        congr 1
-        funext a
-        split_ifs <;> simp [FiniteLaw.expect,FiniteLaw.total]
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-def initialSignedPatternCap (J u m r s : ℕ) (q ω : ℝ) (k : Fin r → GapDifference) : ℝ :=
-  if ∃a,k=linearPattern a then
-    initialPrefactorCap u m r 1 q*periodCentralSumConstant u r*hiddenSignedL2Bound J s ω
-  else initialPrefactorCap u m r (∑h,(k h).order) q*
-    periodBoundedSumConstant u r (2*(2^s))*hiddenUnsignedL2Bound J s ω
-
-def initialPurePatternCap (J u m r s : ℕ) (q ω : ℝ) (k : Fin r → GapDifference) : ℝ :=
-  if ∃a,k=linearPattern a then
-    initialPrefactorCap u m r 1 q*periodCentralSumConstant u r*hiddenSignedL2Bound J s ω
-  else initialPrefactorCap u m r (∑h,(k h).order) q*
-    periodCentralSumConstant u r*hiddenUnsignedL2Bound J s ω
-
-theorem initial_signed_pattern_cap {J : ℕ} (d : Fin J → Option Bool) (c : ℤ)
-    (u m r s : ℕ) (k : Fin r → GapDifference) {q θ ω : ℝ}
-    (hq : q∈Set.Ioc 0 (1/2:ℝ)) (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
-    (hh : (J:ℝ)*ω^2/2≤∑i,(hiddenTrial (d i):ℝ)) :
-    signedPatternBound u m r s (posteriorAnchorSeries c d) θ k ≤
-      initialSignedPatternCap J u m r s q ω k := by
-  unfold signedPatternBound initialSignedPatternCap
-  split_ifs
-  · apply mul_le_mul
-    · exact mul_le_mul_of_nonneg_right (initial_prefactor_bound u m r 1 hq hθ)
-        (by unfold periodCentralSumConstant; positivity)
-    · exact posteriorAnchorSeries_signed_L2 d c s ω hh
-    · exact seriesL2Sq_nonneg _
-    · exact mul_nonneg (initialPrefactorCap_nonneg _ _ _ _ _) (by unfold periodCentralSumConstant; positivity)
-  · apply mul_le_mul
-    · exact mul_le_mul_of_nonneg_right (initial_prefactor_bound u m r _ hq hθ)
-        (by unfold periodBoundedSumConstant; positivity)
-    · exact posteriorAnchorSeries_unsigned_L2 d c s ω hh
-    · exact seriesL2Sq_nonneg _
-    · exact mul_nonneg (initialPrefactorCap_nonneg _ _ _ _ _) (by unfold periodBoundedSumConstant; positivity)
-
-theorem initial_pure_pattern_cap {J : ℕ} (d : Fin J → Option Bool) (c : ℤ)
-    (u m r s : ℕ) (k : Fin r → GapDifference) {q θ ω : ℝ}
-    (hq : q∈Set.Ioc 0 (1/2:ℝ)) (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
-    (hh : (J:ℝ)*ω^2/2≤∑i,(hiddenTrial (d i):ℝ)) :
-    purePatternBound u m r s (posteriorAnchorSeries c d) θ k ≤
-      initialPurePatternCap J u m r s q ω k := by
-  unfold purePatternBound initialPurePatternCap
-  split_ifs
-  · apply mul_le_mul
-    · exact mul_le_mul_of_nonneg_right (initial_prefactor_bound u m r 1 hq hθ)
-        (by unfold periodCentralSumConstant; positivity)
-    · exact posteriorAnchorSeries_signed_L2 d c s ω hh
-    · exact seriesL2Sq_nonneg _
-    · exact mul_nonneg (initialPrefactorCap_nonneg _ _ _ _ _) (by unfold periodCentralSumConstant; positivity)
-  · apply mul_le_mul
-    · exact mul_le_mul_of_nonneg_right (initial_prefactor_bound u m r _ hq hθ)
-        (by unfold periodCentralSumConstant; positivity)
-    · exact posteriorAnchorSeries_unsigned_L2 d c s ω hh
-    · exact seriesL2Sq_nonneg _
-    · exact mul_nonneg (initialPrefactorCap_nonneg _ _ _ _ _) (by unfold periodCentralSumConstant; positivity)
-
-def initialSignedDegreeCap (J u m r s : ℕ) (q ω : ℝ) : ℝ :=
-  if r=0 then 0 else
-    8*(Fintype.card (Fin r → GapDifference):ℝ)*
-      (∑k : Fin r → GapDifference,if Odd (firstDifferenceCount k) then
-        initialSignedPatternCap J u m r s q ω k else 0)+initialRankTailCap u m r
-
-def initialPureDegreeCap (J u m r s : ℕ) (q ω : ℝ) : ℝ :=
-  if r=0 then 0 else
-    2*(Fintype.card (Fin r → GapDifference):ℝ)*
-      (∑k : Fin r → GapDifference,if k=unchangedPattern r then 0 else
-        initialPurePatternCap J u m r s q ω k)+initialRankTailCap u m r
-
-theorem initial_signed_degree_cap {J u m l : ℕ} (d : Fin J → Option Bool) (c : ℤ) (r s : ℕ)
-    {q θ ω : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ)) (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
-    (hl : l≤u*m) (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1≤((u*m:ℕ):ℝ)*θ*(1-θ))
-    (hh : (J:ℝ)*ω^2/2≤∑i,(hiddenTrial (d i):ℝ)) :
-    signedInitialDegreeBound u m r s (posteriorAnchorSeries c d) θ l ≤
-      initialSignedDegreeCap J u m r s q ω := by
-  unfold signedInitialDegreeBound initialSignedDegreeCap
-  split_ifs
-  · exact le_rfl
-  · apply add_le_add
-    · apply mul_le_mul_of_nonneg_left _ (by positivity)
-      apply Finset.sum_le_sum
-      intro k _
-      split_ifs
-      · exact initial_signed_pattern_cap d c u m r s k hq hθ hh
-      · exact le_rfl
-    · exact initial_rank_tail_bound r (initial_theta_variance hq hθ).1 hl hmean hv
-
-theorem initial_pure_degree_cap {J u m l : ℕ} (d : Fin J → Option Bool) (c : ℤ) (r s : ℕ)
-    {q θ ω : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ)) (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
-    (hl : l≤u*m) (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1≤((u*m:ℕ):ℝ)*θ*(1-θ))
-    (hh : (J:ℝ)*ω^2/2≤∑i,(hiddenTrial (d i):ℝ)) :
-    initialPureDegreeBound u m r s (posteriorAnchorSeries c d) θ l ≤
-      initialPureDegreeCap J u m r s q ω := by
-  unfold initialPureDegreeBound initialPureDegreeCap
-  split_ifs
-  · exact le_rfl
-  · apply add_le_add
-    · apply mul_le_mul_of_nonneg_left _ (by positivity)
-      apply Finset.sum_le_sum
-      intro k _
-      split_ifs
-      · exact le_rfl
-      · exact initial_pure_pattern_cap d c u m r s k hq hθ hh
-    · exact initial_rank_tail_bound r (initial_theta_variance hq hθ).1 hl hmean hv
-
-/-- Scalar budgets independent of every actual report and count. The finite
- signed-pattern restriction is retained to preserve its three-difference gain. -/
-def initialSignedCap (J u m R s : ℕ) (q ω w : ℝ) : ℝ :=
-  (∑r∈Finset.range R,w^(2*r)*initialSignedDegreeCap J u m r s q ω)+
-    4*(initialMomentRatio u m q w)^R/(1-initialMomentRatio u m q w)
-
-def initialPureCap (J u m R s : ℕ) (q ω w : ℝ) : ℝ :=
-  (∑r∈Finset.range R,w^(2*r)*initialPureDegreeCap J u m r s q ω)+
-    4*(initialMomentRatio u m q w)^R/(1-initialMomentRatio u m q w)
-
-theorem initial_signed_budget_cap {J u m l : ℕ} (d : Fin J → Option Bool) (c : ℤ) (R s : ℕ)
-    {q θ ω w : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ)) (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
-    (hl : l≤u*m) (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1≤((u*m:ℕ):ℝ)*θ*(1-θ))
-    (hh : (J:ℝ)*ω^2/2≤∑i,(hiddenTrial (d i):ℝ)) :
-    initialSignedBudget u m l R s (posteriorAnchorSeries c d) θ q w ≤
-      initialSignedCap J u m R s q ω w := by
-  unfold initialSignedBudget initialSignedCap
-  apply add_le_add _ le_rfl
-  apply Finset.sum_le_sum
-  intro r _
-  exact mul_le_mul_of_nonneg_left (initial_signed_degree_cap d c r s hq hθ hl hmean hv hh)
-    (by rw [pow_mul]; positivity)
-
-theorem initial_pure_budget_cap {J u m l : ℕ} (d : Fin J → Option Bool) (c : ℤ) (R s : ℕ)
-    {q θ ω w : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ)) (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
-    (hl : l≤u*m) (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1≤((u*m:ℕ):ℝ)*θ*(1-θ))
-    (hh : (J:ℝ)*ω^2/2≤∑i,(hiddenTrial (d i):ℝ)) :
-    initialPureBudget u m l R s (posteriorAnchorSeries c d) θ q w ≤
-      initialPureCap J u m R s q ω w := by
-  unfold initialPureBudget initialPureCap
-  apply add_le_add _ le_rfl
-  apply Finset.sum_le_sum
-  intro r _
-  exact mul_le_mul_of_nonneg_left (initial_pure_degree_cap d c r s hq hθ hl hmean hv hh)
-    (by rw [pow_mul]; positivity)
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-def OracleTracePair.weaken {q E V E' V' : ℝ} {x y : List Bool} {k : ℕ}
-    (C : OracleTracePair q x y k E V) (hE : E ≤ E') (hV : V ≤ V') :
-    OracleTracePair q x y k E' V' :=
-  { C with
-    first_simulation := fun F hF => (C.first_simulation F hF).trans hE
-    second_simulation := fun F hF => (C.second_simulation F hF).trans hE
-    moment := C.moment.trans hV }
-
-def initialErrorCap (J K m k : ℕ) (q ω w : ℝ) : ℝ :=
-  Real.sqrt (6*(4*(initialMomentRatio K m q w/(1-initialMomentRatio K m q w)))^k)+
-    (2*Real.exp (-((K*m:ℕ):ℝ)*q^4/64)+Real.exp (-(J:ℝ)*ω^4/16))
-
-/-- Both required initial comparisons are genuine approximate complete-trace
- oracles. The metadata budgets have been eliminated: only explicit finite
- scalar bounds and literal period geometry remain to be instantiated. -/
-theorem actual_initial_oracles {J K m R : ℕ}
-    (hK : 0 < K) (hm : 128 ≤ m) (hKm : K ≤ m) (hR : 0 < R)
-    (off : Fin m) (hi : 1 ≤ off.val ∧ off.val+1 < m)
-    (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 2)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=K) (fun _ => off)))
-    (a : ℤ) (s k : ℕ) (hk : 1 ≤ k) {q w V W : ℝ}
-    (hq : q∈Set.Ioc 0 (1/2:ℝ))
-    (hseed : 16 ≤ q^3*(m:ℝ)) (hcount : 8 ≤ ((K*m:ℕ):ℝ)*q^2)
-    (ha : (J*initialSideCount q m+J:ℕ) ≤ a)
-    (ha' : a+(2^s:ℕ) ≤ (J*initialSideCount q m+K:ℕ))
-    (hleft : (K:ℝ)/3 ≤ 2*((a-(J*initialSideCount q m:ℕ):ℤ)-J)+1)
-    (hright : (K:ℝ)/3 ≤ 2*K-(2*((a-(J*initialSideCount q m:ℕ):ℤ)+(2:ℝ)^s)+1))
-    (hM : ((2*(2^s):ℕ):ℝ) < (K:ℝ)/(3*(R+1:ℝ)))
-    (hw : w=((6*(2*k:ℕ):ℝ)*(2/q)))
-    (hscale : initialMomentRatio K m q w ≤ 1/2)
-    (hclip : Real.sqrt (6*(4*(initialMomentRatio K m q w/(1-initialMomentRatio K m q w)))^k) ≤ 1/2)
-    (hV : 0 ≤ V) (hW : 0 ≤ W)
-    (hSV : 64*initialSignedCap J K m R s q (initialOverlapLower q) w ≤ V)
-    (hPW : 64*initialPureCap J K m R s q (initialOverlapLower q) w ≤ W) :
-    Nonempty (OracleTracePair q
-      (motifPeriodWord off hi (J*(2*initialSideCount q m+1)+K) s a (-1))
-      (motifPeriodWord off hi (J*(2*initialSideCount q m+1)+K) s a 1) k
-      (initialErrorCap J K m k q (initialOverlapLower q) w) V) ∧
-    Nonempty (OracleTracePair q
-      (pureAtomRun (initialPeriod m off.val) (J*(2*initialSideCount q m+1)+K))
-      (motifPeriodWord off hi (J*(2*initialSideCount q m+1)+K) s a 1) k
-      (initialErrorCap J K m k q (initialOverlapLower q) w) W) := by
-  let t := initialSideCount q m
-  let ω := initialOverlapLower q
-  let c : ℤ := a-(J*t:ℕ)
-  have hω : 0 < ω := initialOverlapLower_pos q
-  have hω1 : ω < 1/8 := initialOverlapLower_lt_eighth hq.1
-  have hqi : q∈Set.Ioo 0 1 := ⟨hq.1,by linarith [hq.2]⟩
-  have hqc : q∈Set.Icc 0 1 := ⟨hq.1.le,hqi.2.le⟩
-  let u := pureAtomRun (initialPeriod m off.val) t
-  let v := pureAtomRun (initialPeriod m off.val) (t+1)
-  let N := max u.length v.length
-  have hu : u.length ≤ N := le_max_left _ _
-  have hv : v.length ≤ N := le_max_right _ _
-  have hov : ω ≤ listTraceOverlap q hqc u v := initial_pure_trace_overlap hq (by omega)
-    hseed off (by linarith)
-  let M : PairedMechanisms (BoundedTrace N) J := tracePairedMechanisms hqc hω
-    (fun _ => u) (fun _ => v) N (fun _ => hu) (fun _ => hv) (fun _ => hov)
-  have hfirst (i : Fin J) : SimulatesTrace q (M.first i) Subtype.val u :=
-    listTraceLaw_simulates q hqc u N hu
-  have hsecond (i : Fin J) : SimulatesTrace q (M.second i) Subtype.val v :=
-    listTraceLaw_simulates q hqc v N hv
-  have htail := initial_metadata_bad_mass M hq hcount (fun _ => rfl)
-    (show ω^2∈Set.Icc (0:ℝ) 1 from ⟨sq_nonneg _,by nlinarith⟩)
-  have hsigned (z : InitialMetadata (BoundedTrace N) J (K*m))
-      (hz : initialMetadataGood q ω z) :
-      64*initialSignedBudget K m z.2.val R s (posteriorAnchorSeries c (fun i => (z.1 i).1))
-        ((z.2.val:ℝ)/(K*m:ℕ)) q w ≤ V := by
-    apply le_trans _ hSV
-    apply mul_le_mul_of_nonneg_left _ (by norm_num)
-    exact initial_signed_budget_cap _ c R s hq hz.1.1 (by have H:=z.2.isLt; omega)
-      (mul_div_cancel₀ _ (by exact_mod_cast (Nat.mul_ne_zero (by omega : K≠0) (by omega : m≠0))))
-      hz.1.2 hz.2
-  have hpure (z : InitialMetadata (BoundedTrace N) J (K*m))
-      (hz : initialMetadataGood q ω z) :
-      64*initialPureBudget K m z.2.val R s (posteriorAnchorSeries c (fun i => (z.1 i).1))
-        ((z.2.val:ℝ)/(K*m:ℕ)) q w ≤ W := by
-    apply le_trans _ hPW
-    apply mul_le_mul_of_nonneg_left _ (by norm_num)
-    exact initial_pure_budget_cap _ c R s hq hz.1.1 (by have H:=z.2.isLt; omega)
-      (mul_div_cancel₀ _ (by exact_mod_cast (Nat.mul_ne_zero (by omega : K≠0) (by omega : m≠0))))
-      hz.1.2 hz.2
-  have hsim (τ : ℝ) := initial_sigma_simulates M t s a τ off hi hsafe hqi
-    Subtype.val hfirst hsecond ha ha'
-  have hp := initial_pure_sigma_simulates (K:=K) M t off hqi Subtype.val hfirst hsecond
-  obtain ⟨S⟩ := initial_sigma_clipped_exists M hK hm hKm hR off hi hoff hsafe c s k hk
-    hleft hright hq hM hw hscale hclip hV hsigned
-  obtain ⟨P⟩ := initial_pure_sigma_clipped_exists M hK hm hKm hR off hi hoff hsafe c s k hk
-    hleft hright hq hw hscale hclip hW hpure
-  constructor
-  · obtain ⟨O⟩ := S.tracePair (decodeInitialObservation Subtype.val) (hsim (-1)) (hsim 1)
-    exact ⟨O.weaken (add_le_add le_rfl htail) le_rfl⟩
-  · obtain ⟨O⟩ := P.tracePair (decodeInitialObservation Subtype.val) hp (hsim 1)
-    exact ⟨O.weaken (add_le_add le_rfl htail) le_rfl⟩
-
-end TraceReconstruction
-
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-
-end OAI
+import OAI.Probability.TraceReconstruction.SourceOracle
+
+namespace OAI
+
+/-!
+# InitialOracle
+
+Actual initial scores and anchors, count partitions and typicality, clipped metadata, common decoding, sharp hidden-count energy, tails, and budget caps.
+-/
+
+open scoped ENNReal
+noncomputable section
+
+open Filter
+open scoped Topology
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+open FiniteLaw
+
+def initialMomentRatio (u m : ℕ) (q w : ℝ) : ℝ :=
+  w^2*(3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/m)
+
+def initialSignedBudget (u m l R s : ℕ) (π : IntSeries) (θ q w : ℝ) : ℝ :=
+  (∑r∈Finset.range R,w^(2*r)*signedInitialDegreeBound u m r s π θ l)+
+    4*(initialMomentRatio u m q w)^R/(1-initialMomentRatio u m q w)
+
+theorem initialSignedBudget_nonneg (u m l R s : ℕ) (π : IntSeries) (θ q w : ℝ)
+    (hq : 0 ≤ q) (ha : initialMomentRatio u m q w < 1) :
+    0 ≤ initialSignedBudget u m l R s π θ q w := by
+  apply add_nonneg
+  · exact Finset.sum_nonneg (fun r _ => mul_nonneg (by rw [pow_mul]; positivity)
+      (signedInitialDegreeBound_nonneg _ _ _ _ _ _ _))
+  · apply div_nonneg _ (sub_nonneg.mpr ha.le)
+    exact mul_nonneg (by norm_num) (pow_nonneg (by unfold initialMomentRatio; positivity [rankSchurConstant_pos]) _)
+
+/-- The signed score numerator in the *actual conditional mark-list laws*,
+ with the entire finite cancellation energy and geometric tail retained. -/
+theorem initial_signed_moment {u m l R : ℕ} (hu : 0 < u) (hm : 128 ≤ m)
+    (hum : u ≤ m) (hR : 0 < R) (hl : l ≤ u*m) (off : Fin m)
+    (hi : 1 ≤ off.val ∧ off.val+1 < m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 2)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
+    (π : IntSeries) (hπ : ∀X,0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
+    (s k : ℕ) (hcentral : CentralMotifSupport u s π) (hk : 1 ≤ k)
+    {q θ w : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
+    (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
+    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
+    (hM : ((2*(2^s):ℕ):ℝ) < (u:ℝ)/(3*(R+1:ℝ)))
+    (hw : w=((6*(2*k:ℕ):ℝ)*(2/q))) (ha : initialMomentRatio u m q w < 1) :
+    ((primitiveMarkLaw q ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩ false).product l).evenMomentSq
+      (fun x => initialMotifDensity hl off hsafe π s 1 q x-initialMotifDensity hl off hsafe π s (-1) q x) k  ≤ 
+        initialSignedBudget u m l R s π θ q w := by
+  have hq' : q∈Set.Ioo 0 1 := ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩
+  have he : (fun x => initialMotifDensity hl off hsafe π s 1 q x-initialMotifDensity hl off hsafe π s (-1) q x)=
+      multilinear (fun σ => mixedMotifCoefficient hl off hsafe π s 1 σ-mixedMotifCoefficient hl off hsafe π s (-1) σ)
+        (primitiveScore q) := by funext x; exact (multilinear_sub _ _ _ _).symm
+  rw [he]
+  apply (multilinear_moment_sq_scaled (primitiveMarkLaw q hq' false) (primitiveScore q) k hk
+    (primitive_score_centered hq') (div_pos (by norm_num) hq.1) (primitive_score_evenMoment hq' k) l _).trans
+  rw [←hw]
+  exact signed_initial_full_energy hu hm hum hR hl off hi hoff hsafe π hπ hπsum s hcentral hq hθ hmean hv hM
+    (by rw [hw]; positivity [hq.1]) ha
+
+/-- Quantitative normalized common clipping of the level-zero conditional
+ pair. This theorem has only the literal count/geometric inequalities of the
+ initial construction, not a signed-density or oracle conclusion premise. -/
+theorem initial_conditional_clipped_exists {u m l R : ℕ} (hu : 0 < u) (hm : 128 ≤ m)
+    (hum : u ≤ m) (hR : 0 < R) (hl : l ≤ u*m) (off : Fin m)
+    (hi : 1 ≤ off.val ∧ off.val+1 < m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 2)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
+    (π : IntSeries) (hπ : ∀X,0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
+    (s k : ℕ) (hcentral : CentralMotifSupport u s π) (hk : 1 ≤ k)
+    {q θ w : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
+    (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
+    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
+    (hM : ((2*(2^s):ℕ):ℝ) < (u:ℝ)/(3*(R+1:ℝ)))
+    (hw : w=((6*(2*k:ℕ):ℝ)*(2/q))) (hscale : initialMomentRatio u m q w ≤ 1/2)
+    (hclip : Real.sqrt (6*(4*(initialMomentRatio u m q w/(1-initialMomentRatio u m q w)))^k) ≤ 1/2) :
+    Nonempty (ClippedOracle
+      (conditionalMotifLaw hl off hsafe ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩ π hπ hπsum s (-1))
+      (conditionalMotifLaw hl off hsafe ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩ π hπ hπsum s 1) k
+      (Real.sqrt (6*(4*(initialMomentRatio u m q w/(1-initialMomentRatio u m q w)))^k))
+      (64*initialSignedBudget u m l R s π θ q w)) := by
+  have hq' : q∈Set.Ioo 0 1 := ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩
+  let A := (primitiveMarkLaw q hq' false).product l
+  let f := initialMotifDensity hl off hsafe π s (-1) q
+  let g := initialMotifDensity hl off hsafe π s 1 q
+  let U := initialMomentRatio u m q w/(1-initialMomentRatio u m q w)
+  have ha : initialMomentRatio u m q w < 1 := by linarith
+  have hz : 0 ≤ initialMomentRatio u m q w := by unfold initialMomentRatio; positivity [rankSchurConstant_pos,hq.1]
+  have hU : 0 ≤ U := div_nonneg hz (sub_nonneg.mpr ha.le)
+  have hU1 : U ≤ 1 := (div_le_iff₀ (sub_pos.mpr ha)).mpr (by linarith)
+  have hmom (τ : ℝ) : A.expect (fun x => |initialMotifDensity hl off hsafe π s τ q x-1|^(2*k)) ≤ U^k :=
+    A.expect_even_pow_le_of_moment _ k hk (initial_centered_moment hl (by omega) hum off hi hoff hsafe π hπ hπsum s k hk τ hq hθ hmean hv hw ha)
+  have hV := initialSignedBudget_nonneg u m l R s π θ q w hq.1.le ha
+  have hsig := A.expect_even_pow_le_of_moment _ k hk
+    (initial_signed_moment hu hm hum hR hl off hi hoff hsafe π hπ hπsum s k hcentral hk hq hθ hmean hv hM hw ha)
+  rw [initialMotifLaw_withDensity hl off hsafe π hπ hπsum s (-1) hq',
+    initialMotifLaw_withDensity hl off hsafe π hπ hπsum s 1 hq']
+  have H := A.exists_clippedOracle f g
+    (initialMotifDensity_nonneg hl off hsafe π hπ hπsum s (-1) hq')
+    (initialMotifDensity_nonneg hl off hsafe π hπ hπsum s 1 hq')
+    (initialMotifDensity_mean hl off hsafe π hπ hπsum s (-1) hq')
+    (initialMotifDensity_mean hl off hsafe π hπ hπsum s 1 hq')
+    Finset.univ 0 (by simp [eventMass]) (by simp [eventMass]) k hk U hU hU1 (hmom (-1)) (hmom 1)
+    (initialSignedBudget u m l R s π θ q w) hV (by simpa only [Finset.mem_univ,ite_true] using hsig)
+    (by simpa only [zero_add] using hclip)
+  simpa only [zero_add] using H
+
+end TraceReconstruction
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+namespace FiniteLaw
+variable {α : Type*} [Fintype α]
+
+theorem evenMomentSq_le_of_expect (P : FiniteLaw α) (F : α → ℝ) (k : ℕ) (hk : 1≤k)
+    {V : ℝ} (hV : 0≤V) (h : P.expect (fun a => |F a|^(2*k))≤V^k) :
+    P.evenMomentSq F k≤V := by
+  have H := Real.rpow_le_rpow (P.expect_nonneg _ (fun _ => by positivity)) h (by positivity : 0≤(k:ℝ)⁻¹)
+  rw [Real.pow_rpow_inv_natCast hV (by omega)] at H
+  simpa only [FiniteLaw.evenMomentSq,one_div] using H
+
+theorem tv_sigma_eq {ι : Type*} [Fintype ι] {β : ι → Type*} [∀i,Fintype (β i)]
+    (M : FiniteLaw ι) (P Q : ∀i,FiniteLaw (β i)) :
+    (M.sigma P).tv (M.sigma Q)=M.expect (fun i => (P i).tv (Q i)) := by
+  simp only [tv,sigma,expect,Fintype.sum_sigma,←mul_sub,abs_mul,abs_of_nonneg (M.nonneg _)]
+  rw [Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro i _
+  rw [←Finset.mul_sum]
+  ring
+
+end FiniteLaw
+namespace ClippedOracle
+variable {α : Type*} [Fintype α] {P Q : FiniteLaw α} {k : ℕ} {e V E W : ℝ}
+
+def weaken (C : ClippedOracle P Q k e V) (he : e≤E) (hV : V≤W) : ClippedOracle P Q k E W where
+  first := C.first
+  second := C.second
+  score := C.score
+  density := C.density
+  likelihood_nonneg := C.likelihood_nonneg
+  centered := C.centered
+  moment := C.moment.trans hV
+  first_tv := C.first_tv.trans he
+  second_tv := C.second_tv.trans he
+
+def trivial (P Q : FiniteLaw α) (k : ℕ) (hk : 1≤k) : ClippedOracle P Q k 1 0 where
+  first := P
+  second := P
+  score := fun _ => 0
+  density := fun _ => by ring
+  likelihood_nonneg := fun _ => by norm_num
+  centered := by simp [FiniteLaw.expect]
+  moment := by
+    apply P.evenMomentSq_le_of_expect _ k hk (by norm_num)
+    simp [FiniteLaw.expect,zero_pow (by omega : 2*k≠0)]
+  first_tv := by simp [FiniteLaw.tv]
+  second_tv := FiniteLaw.tv_le_one _ _
+
+/-- Reported metadata is retained, so the squared moment of a normalized
+ score assembles without a mixing penalty. No conditioning of the original
+ latent metadata distribution occurs. -/
+def sigma {ι : Type*} [Fintype ι] {β : ι → Type*} [∀i,Fintype (β i)]
+    (M : FiniteLaw ι) (P Q : ∀i,FiniteLaw (β i)) (e : ι → ℝ)
+    (k : ℕ) (hk : 1≤k) (V : ℝ) (hV : 0≤V)
+    (C : ∀i,ClippedOracle (P i) (Q i) k (e i) V) :
+    ClippedOracle (M.sigma P) (M.sigma Q) k (M.expect e) V where
+  first := M.sigma (fun i => (C i).first)
+  second := M.sigma (fun i => (C i).second)
+  score := fun a => (C a.1).score a.2
+  density := fun a => by dsimp only [FiniteLaw.sigma]; rw [(C a.1).density]; ring
+  likelihood_nonneg := fun a => (C a.1).likelihood_nonneg a.2
+  centered := by
+    rw [FiniteLaw.expect_sigma]
+    simp_rw [(C _).centered]
+    simp [FiniteLaw.expect]
+  moment := by
+    apply FiniteLaw.evenMomentSq_le_of_expect _ _ k hk hV
+    rw [FiniteLaw.expect_sigma]
+    calc
+      _ ≤ M.expect (fun _ => V^k) := M.expect_mono (fun i =>
+        (C i).first.expect_even_pow_le_of_moment _ k hk (C i).moment)
+      _ = _ := M.expect_const _
+  first_tv := by
+    rw [FiniteLaw.tv_sigma_eq]
+    exact M.expect_mono (fun i => (C i).first_tv)
+  second_tv := by
+    rw [FiniteLaw.tv_sigma_eq]
+    exact M.expect_mono (fun i => (C i).second_tv)
+
+/-- Replace bad conditional observations by a common law rather than making
+ the original metadata law conditional on typicality. The exact cost is
+ the true metadata failure mass plus the good conditional clipping cost. -/
+theorem assemble_good {ι : Type*} [Fintype ι] {β : ι → Type*} [∀i,Fintype (β i)]
+    (M : FiniteLaw ι) (P Q : ∀i,FiniteLaw (β i)) (Good : ι → Prop)
+    (k : ℕ) (hk : 1≤k) (e V : ℝ) (he : 0≤e) (hV : 0≤V)
+    (C : ∀i,Good i → Nonempty (ClippedOracle (P i) (Q i) k e V)) :
+    Nonempty (ClippedOracle (M.sigma P) (M.sigma Q) k
+      (e+M.expect (fun i => if Good i then 0 else 1)) V) := by
+  let err := fun i => if Good i then e else 1
+  have hi (i) : Nonempty (ClippedOracle (P i) (Q i) k (err i) V) := by
+    by_cases h : Good i
+    · simpa only [err,ite_eq_left h] using C i h
+    · exact ⟨(trivial (P i) (Q i) k hk).weaken (by simp [err,h]) hV⟩
+  let D := sigma M P Q err k hk V hV (fun i => Classical.choice (hi i))
+  refine ⟨D.weaken ?_ le_rfl⟩
+  calc
+    M.expect err ≤ M.expect (fun i => e+(if Good i then 0 else 1)) := M.expect_mono (fun i => by
+      dsimp only [err]; split_ifs <;> linarith)
+    _ = _ := by rw [M.expect_add,M.expect_const]
+
+end ClippedOracle
+end TraceReconstruction
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+open FiniteLaw
+
+theorem translated_fair_eval (g : ℕ) (c : ℤ) (F : ℤ → ℝ) :
+    (∑X∈(translateSeries c (fairSeries g)).coeff.support,
+      (translateSeries c (fairSeries g)).coeff X*F X)=
+    ∑k∈Finset.range (g+1),fairBinomial g k*F (c+k) := by
+  have H := congrArg (fun p : IntSeries => Finsupp.linearCombination ℝ F p.coeff)
+    (congrArg (translateSeries c) (fairSeries_sum g))
+  simp only [translateSeries,Finset.mul_sum,AddMonoidAlgebra.single_mul_single,one_mul,
+    AddMonoidAlgebra.coeff_sum,AddMonoidAlgebra.coeff_single,map_sum,
+    Finsupp.linearCombination_single,smul_eq_mul] at H
+  exact H
+
+/-- The literal initial anchor after conditioning the paired side outputs. -/
+def posteriorAnchorSeries {ι : Type*} [Fintype ι] (c : ℤ) (d : ι → Option Bool) : IntSeries :=
+  translateSeries (c-(∑i,(visibleOnes (d i):ℤ))-(∑i,hiddenTrial (d i)))
+    (fairSeries (∑i,hiddenTrial (d i)))
+
+theorem posterior_anchor_average {ι : Type*} [Fintype ι]
+    (c : ℤ) (d : ι → Option Bool) (F : ℤ → ℝ) :
+    (∑b : ι → Bool, (∏i,(flagPosterior (d i)).mass (b i))*
+      F (c-∑i,(bitIndex (b i):ℤ)))=
+    ∑X∈(posteriorAnchorSeries c d).coeff.support,(posteriorAnchorSeries c d).coeff X*F X := by
+  rw [posterior_count_average_reflected d (fun z => F (c-z)),posteriorAnchorSeries,translated_fair_eval]
+  apply Finset.sum_congr rfl
+  intro k _
+  congr 2
+  ring
+
+theorem posteriorAnchorSeries_nonneg {ι : Type*} [Fintype ι]
+    (c : ℤ) (d : ι → Option Bool) (X : ℤ) : 0 ≤ (posteriorAnchorSeries c d).coeff X :=
+  translated_binomial_nonneg _ _ _
+
+theorem posteriorAnchorSeries_total {ι : Type*} [Fintype ι]
+    (c : ℤ) (d : ι → Option Bool) :
+    ∑X∈(posteriorAnchorSeries c d).coeff.support,(posteriorAnchorSeries c d).coeff X=1 := by
+  have H := posterior_anchor_average c d (fun _ => 1)
+  simp only [mul_one] at H
+  rw [←H]
+  exact (FiniteLaw.indexedProduct (fun i => flagPosterior (d i))).total
+
+theorem posteriorAnchorSeries_support {ι : Type*} [Fintype ι]
+    (c : ℤ) (d : ι → Option Bool) (X : ℤ) (hX : X∈(posteriorAnchorSeries c d).coeff.support) :
+    c-Fintype.card ι ≤ X ∧ X ≤ c := by
+  have H := translated_binomial_support (∑i,hiddenTrial (d i))
+    (c-(∑i,(visibleOnes (d i):ℤ))-(∑i,hiddenTrial (d i))) X
+    (Finsupp.mem_support_iff.mp hX)
+  have hz : 0 ≤ ∑i,(visibleOnes (d i):ℤ) := Finset.sum_nonneg (fun _ _ => by positivity)
+  have hh : (∑i,(visibleOnes (d i):ℤ))+(∑i,hiddenTrial (d i)) ≤ Fintype.card ι := by
+    rw [Nat.cast_sum,←Finset.sum_add_distrib]
+    calc
+      _ ≤ ∑_i : ι,(1:ℤ) := Finset.sum_le_sum (fun i _ => by
+        cases h:d i with
+        | none => simp [visibleOnes,hiddenTrial]
+        | some b => cases b <;> simp [visibleOnes,hiddenTrial])
+      _ = _ := by simp
+  omega
+
+theorem posteriorAnchorSeries_central {ι : Type*} [Fintype ι]
+    (u s : ℕ) (c : ℤ) (d : ι → Option Bool)
+    (hleft : (u:ℝ)/3 ≤ 2*((c:ℝ)-Fintype.card ι)+1)
+    (hright : (u:ℝ)/3 ≤ 2*u-(2*((c:ℝ)+(2:ℝ)^s)+1)) :
+    CentralMotifSupport u s (posteriorAnchorSeries c d) := by
+  intro X hX
+  have H := posteriorAnchorSeries_support c d X hX
+  have H₁ : (c:ℝ)-Fintype.card ι ≤ X := by exact_mod_cast H.1
+  have H₂ : (X:ℝ) ≤ c := by exact_mod_cast H.2
+  constructor <;> linarith
+
+namespace PairedMechanisms
+variable {Side α : Type*} [Fintype Side] [Fintype α] {J : ℕ}
+
+def emissionOracle (M : PairedMechanisms Side J) (K : (Fin J → Bool) → FiniteLaw α) :
+    FiniteLaw (Reports Side J × α) :=
+  M.reports.joint (fun z => (orientationsGiven z).mixture K)
+
+/-- Bayes factorization is valid for arbitrary conditional finite emissions,
+ including the dependent-length initial primitive-mark list. -/
+theorem emission_factorization (M : PairedMechanisms Side J) (K : (Fin J → Bool) → FiniteLaw α) :
+    (fairOrientations J).mixture (fun b => (M.reportsGiven b).prod (K b))=M.emissionOracle K := by
+  apply FiniteLaw.ext
+  funext zx
+  rcases zx with ⟨z,x⟩
+  change (∑b,(∏i : Fin J,(1/2:ℝ))*
+    ((∏i,FiniteLaw.pairedFlagMass (M.first i) (M.second i) (M.common i)
+      (M.hiddenWeight i) (b i) (z i))*(K b).mass x))=
+    (∏i,FiniteLaw.pairedReportMass (M.first i) (M.second i) (M.common i)
+      (M.hiddenWeight i) (z i))*(∑b,(∏i,(flagPosterior (z i).1).mass (b i))*(K b).mass x)
+  rw [Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro b _
+  rw [←mul_assoc,←Finset.prod_mul_distrib,FiniteLaw.pairedFlags_product_bayes,mul_assoc]
+
+end PairedMechanisms
+
+theorem conditionalMotifLaw_posterior {u m l J : ℕ} (hl : l ≤ u*m) (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
+    {q : ℝ} (hq : q∈Set.Ioo 0 1) (c : ℤ) (d : Fin J → Option Bool) (s : ℕ) (τ : ℝ) :
+    (FiniteLaw.coordinateProduct (fun i => flagPosterior (d i))).mixture
+      (fun b => conditionalPrimitiveLaw hl q hq
+        (defectTypes (signedPeriodDefects off hsafe s (c-∑i,(bitIndex (b i):ℤ)) τ)))=
+    conditionalMotifLaw hl off hsafe hq (posteriorAnchorSeries c d)
+      (posteriorAnchorSeries_nonneg c d) (posteriorAnchorSeries_total c d) s τ := by
+  apply FiniteLaw.ext
+  funext x
+  change (∑b : Fin J → Bool,(∏i : Fin J,(flagPosterior (d i)).mass (b i))*
+    (conditionalPrimitiveLaw hl q hq (defectTypes (signedPeriodDefects off hsafe s
+      (c-∑i,(bitIndex (b i):ℤ)) τ))).mass x)=_
+  classical
+  have H := posterior_anchor_average c d (fun X =>
+    (conditionalPrimitiveLaw hl q hq (defectTypes (signedPeriodDefects off hsafe s X τ))).mass x)
+  calc
+    _ = ∑X∈(posteriorAnchorSeries c d).coeff.support,
+        (posteriorAnchorSeries c d).coeff X*
+          (conditionalPrimitiveLaw hl q hq (defectTypes (signedPeriodDefects off hsafe s X τ))).mass x := by
+      convert H using 1
+      congr 1
+      ext b
+      simp
+    _ = _ := (Finset.sum_coe_sort _ _).symm
+
+
+/-- Initial observations retain the central count with its original binomial law. -/
+def countedMotifLaw {u m : ℕ} (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
+    {q : ℝ} (hq : q∈Set.Ioo 0 1) (π : IntSeries) (hπ : ∀X,0 ≤ π.coeff X)
+    (hπsum : ∑X∈π.coeff.support,π.coeff X=1) (s : ℕ) (τ : ℝ) :
+    FiniteLaw (BoundedMarks (u*m)) :=
+  (boundedCountLaw (le_refl (u*m)) (1-primitiveErasure q) (primitiveRetention_mem hq)).sigma
+    (fun l => conditionalMotifLaw (by omega) off hsafe hq π hπ hπsum s τ)
+
+theorem countedMotifLaw_posterior {u m J : ℕ} (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
+    {q : ℝ} (hq : q∈Set.Ioo 0 1) (c : ℤ) (d : Fin J → Option Bool) (s : ℕ) (τ : ℝ) :
+    (FiniteLaw.coordinateProduct (fun i => flagPosterior (d i))).mixture
+      (fun b => countedPrimitiveLaw (le_refl (u*m)) hq
+        (defectTypes (signedPeriodDefects off hsafe s (c-∑i,(bitIndex (b i):ℤ)) τ)))=
+    countedMotifLaw off hsafe hq (posteriorAnchorSeries c d)
+      (posteriorAnchorSeries_nonneg c d) (posteriorAnchorSeries_total c d) s τ := by
+  apply FiniteLaw.ext
+  funext x
+  rcases x with ⟨l,x⟩
+  have hl : l.val ≤ u*m := by omega
+  have H := congrArg (fun P : FiniteLaw (Fin l.val → PrimitiveMark) => P.mass x)
+    (conditionalMotifLaw_posterior hl off hsafe hq c d s τ)
+  change (∑b, _ * ((boundedCountLaw _ _ _).mass l *
+    (boundedConditionalPrimitiveLaw hq _ l).mass x))=_
+  simp only [boundedConditionalPrimitiveLaw,dite_eq_left hl]
+  change _ = (boundedCountLaw _ _ _).mass l * _
+  rw [←H]
+  simp only [FiniteLaw.mixture,Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro b _
+  ring
+
+
+end TraceReconstruction
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- The actual displaced primitive slot in a period. -/
+def motifOffset {m : ℕ} (off : Fin m) (hi : 1 ≤ off.val ∧ off.val+1 < m)
+    (s : ℕ) (X : ℤ) (τ : ℝ) (j : ℤ) : Fin m :=
+  ⟨off.val+(motifShift (motifIndicator s (j-X)) ((prouhetSeries s).coeff (j-X)) τ).val-1, by
+    have H := (motifShift (motifIndicator s (j-X)) ((prouhetSeries s).coeff (j-X)) τ).isLt
+    omega⟩
+
+theorem signedPeriodDefects_periodSlot {u m : ℕ} (off : Fin m)
+    (hi : 1 ≤ off.val ∧ off.val+1 < m)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
+    (s : ℕ) (X : ℤ) (τ : ℝ) :
+    signedPeriodDefects off hsafe s X τ =
+      periodSlot (fun j => motifOffset off hi s X τ j.val) := by
+  funext j
+  apply Fin.ext
+  simp only [signedPeriodDefects,shiftedTag,periodSlot,motifOffset]
+  omega
+
+def motifPeriodWord {m : ℕ} (off : Fin m) (hi : 1 ≤ off.val ∧ off.val+1 < m)
+    (u s : ℕ) (X : ℤ) (τ : ℝ) : List Bool :=
+  periodString m u (fun j => motifOffset off hi s X τ j.val)
+
+theorem signed_primitive_word {u m : ℕ} (off : Fin m)
+    (hi : 1 ≤ off.val ∧ off.val+1 < m)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
+    (s : ℕ) (X : ℤ) (τ : ℝ) :
+    (((List.ofFn (defectTypes (signedPeriodDefects off hsafe s X τ))).map primitiveBlock).flatten)=
+    motifPeriodWord off hi u s X τ := by
+  rw [signedPeriodDefects_periodSlot off hi,periodic_primitive_word]
+  rfl
+
+theorem motifOffset_outside {m : ℕ} (off : Fin m) (hi : 1 ≤ off.val ∧ off.val+1 < m)
+    (s : ℕ) (X : ℤ) (τ : ℝ) (j : ℤ) (hj : j<X ∨ X+(2^s:ℕ) ≤ j) :
+    motifOffset off hi s X τ j=off := by
+  apply Fin.ext
+  have h : ¬(0 ≤ j-X ∧ j-X<(2^s:ℕ)) := by omega
+  simp only [motifOffset,motifIndicator,ite_eq_right h,motifShift,ite_true]
+  norm_num
+
+theorem motifOffset_translate {m : ℕ} (off : Fin m) (hi : 1 ≤ off.val ∧ off.val+1 < m)
+    (s : ℕ) (X t : ℤ) (τ : ℝ) (j : ℤ) :
+    motifOffset off hi s X τ (t+j)=motifOffset off hi s (X-t) τ j := by
+  have h : t+j-X=j-(X-t) := by ring
+  simp only [motifOffset,h]
+
+theorem periodString_three {m : ℕ} (n k z : ℕ) (off : Fin (n+k+z) → Fin m) :
+    periodString m (n+k+z) off =
+      periodString m n (fun i => off ⟨i.val,by omega⟩) ++
+      periodString m k (fun i => off ⟨n+i.val,by omega⟩) ++
+      periodString m z (fun i => off ⟨n+k+i.val,by omega⟩) := by
+  simp only [periodString,List.ofFn_add,List.flatten_append]
+  rfl
+
+/-- Every randomized initial partition is a partition of the same literal
+period string, with no fit conditioning or change of channel. -/
+theorem motifPeriodWord_window {m : ℕ} (off : Fin m) (hi : 1 ≤ off.val ∧ off.val+1 < m)
+    (t K z s : ℕ) (a : ℤ) (τ : ℝ) (hleft : (t:ℤ) ≤ a) (hright : a+(2^s:ℕ) ≤ t+K) :
+    motifPeriodWord off hi (t+K+z) s a τ =
+      pureAtomRun (initialPeriod m off.val) t ++ motifPeriodWord off hi K s (a-t) τ ++
+      pureAtomRun (initialPeriod m off.val) z := by
+  unfold motifPeriodWord
+  rw [periodString_three]
+  congr 2
+  · rw [←periodString_constant off]
+    congr 1
+    funext i
+    exact motifOffset_outside off hi s a τ i.val (Or.inl (by have H:=i.isLt; omega))
+  · congr 1
+    funext i
+    rw [Nat.cast_add,motifOffset_translate]
+  · rw [←periodString_constant off]
+    congr 1
+    funext i
+    exact motifOffset_outside off hi s a τ (t+K+i.val) (Or.inr (by
+      have H : (0:ℤ) ≤ i.val := by positivity
+      omega))
+
+theorem motifPeriodWord_length {m : ℕ} (off : Fin m) (hi : 1 ≤ off.val ∧ off.val+1 < m)
+    (u s : ℕ) (X : ℤ) (τ : ℝ) :
+    (motifPeriodWord off hi u s X τ).length=u*(2*m-1) := periodString_length _ _ _
+
+end TraceReconstruction
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+namespace FiniteLaw
+variable {α β : Type*} [Fintype α] [Fintype β]
+
+theorem coordinateProduct_expect_pairs {n : ℕ} (P : Fin n → FiniteLaw α)
+    (Q : Fin n → FiniteLaw β) (F : (Fin n → α×β) → ℝ) :
+    (coordinateProduct (fun i => (P i).prod (Q i))).expect F=
+      (coordinateProduct P).expect (fun a =>
+        (coordinateProduct Q).expect (fun b => F (fun i => (a i,b i)))) := by
+  induction n with
+  | zero =>
+    have H (a : Fin 0 → α×β) : a = fun i => Fin.elim0 i := by funext i; exact Fin.elim0 i
+    have hF : F=fun _ => F (fun i => Fin.elim0 i) := by
+      funext a
+      exact congrArg F (H a)
+    rw [hF]
+    simp only [expect_const]
+  | succ n ih =>
+    rw [expect_coordinateProduct_succ,expect_prod,expect_coordinateProduct_succ]
+    congr 1
+    funext a
+    simp_rw [ih,expect_coordinateProduct_succ Q]
+    rw [expect_comm]
+    congr 1
+    funext x
+    congr 1
+    funext b
+    congr 1
+    funext y
+    congr 1
+    funext i
+    refine Fin.cases ?_ (fun j => ?_) i <;> simp
+
+end FiniteLaw
+
+/-- Three chronological streams, with side pairs independent across slots.
+This forgets flags but does not separate any marks inside the central list. -/
+def decodePairedEmission {Side α : Type*} {J : ℕ}
+    (side : Side → List Bool) (central : α → List Bool) (z : Reports Side J × α) : List Bool :=
+  coordinateWord (fun i => side (z.1 i).2.1) ++ central z.2 ++
+    coordinateWord (fun i => side (z.1 i).2.2)
+
+namespace PairedMechanisms
+variable {Side α : Type*} [Fintype Side] [Fintype α] {J : ℕ}
+
+theorem reportsGiven_paired_simulates (M : PairedMechanisms Side J) (b : Fin J → Bool)
+    {q : ℝ} (side : Side → List Bool) (central : α → List Bool) (K : FiniteLaw α)
+    (U V : Fin J → List Bool) (w : List Bool)
+    (hfirst : ∀i,SimulatesTrace q (M.first i) side (U i))
+    (hsecond : ∀i,SimulatesTrace q (M.second i) side (V i))
+    (hK : SimulatesTrace q K central w) :
+    SimulatesTrace q ((M.reportsGiven b).prod K) (decodePairedEmission side central)
+      (coordinateWord (fun i => if b i then V i else U i) ++ w ++
+        coordinateWord (fun i => if b i then U i else V i)) := by
+  let P : Fin J → FiniteLaw Side := fun i => if b i then M.second i else M.first i
+  let Q : Fin J → FiniteLaw Side := fun i => if b i then M.first i else M.second i
+  have HP : ∀i,SimulatesTrace q (P i) side (if b i then V i else U i) := by
+    intro i
+    dsimp [P]
+    cases h:b i
+    · exact hfirst i
+    · exact hsecond i
+  have HQ : ∀i,SimulatesTrace q (Q i) side (if b i then U i else V i) := by
+    intro i
+    dsimp [Q]
+    cases h:b i
+    · exact hsecond i
+    · exact hfirst i
+  have H := ((coordinateProduct_simulates P side _ HP).append hK).append
+    (coordinateProduct_simulates Q side _ HQ)
+  intro F
+  rw [←H F,FiniteLaw.expect_prod,FiniteLaw.expect_prod]
+  change (M.reportsGiven b).expect (fun z => K.expect (fun x =>
+    F (coordinateWord (fun i => side (z i).2.1) ++ central x ++
+      coordinateWord (fun i => side (z i).2.2))))=_
+  have hf := congrArg (fun L : FiniteLaw (Fin J → Side×Side) => L.expect (fun z =>
+    K.expect (fun x => F (coordinateWord (fun i => side (z i).1) ++ central x ++
+      coordinateWord (fun i => side (z i).2))))) (M.reportsGiven_forget b)
+  rw [FiniteLaw.expect_map] at hf
+  simp only [Function.comp_def] at hf
+  rw [hf]
+  have he : (fun i => FiniteLaw.orientedProd (M.first i) (M.second i) (b i))=
+      fun i => (P i).prod (Q i) := by
+    funext i
+    cases h:b i <;> simp [FiniteLaw.orientedProd,P,Q,h]
+  rw [he,FiniteLaw.coordinateProduct_expect_pairs]
+  simp only [FiniteLaw.expect_prod]
+  congr 1
+  funext a
+  rw [FiniteLaw.expect_comm]
+
+end PairedMechanisms
+end TraceReconstruction
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- Exact initial paired prefix and complementary suffix sizes. -/
+def initialPrefixPeriods {J : ℕ} (k : ℕ) (b : Fin J → Bool) : ℕ :=
+  ∑ i : Fin J, (k+bitIndex (b i))
+def initialSuffixPeriods {J : ℕ} (k : ℕ) (b : Fin J → Bool) : ℕ :=
+  ∑ i : Fin J, (k+(1-bitIndex (b i)))
+
+theorem initialPrefixPeriods_eval {J : ℕ} (k : ℕ) (b : Fin J → Bool) :
+    initialPrefixPeriods k b=J*k+∑i,bitIndex (b i) := by
+  simp [initialPrefixPeriods,Finset.sum_add_distrib]
+
+theorem initialPairPeriods_total {J : ℕ} (k : ℕ) (b : Fin J → Bool) :
+    initialPrefixPeriods k b+initialSuffixPeriods k b=J*(2*k+1) := by
+  unfold initialPrefixPeriods initialSuffixPeriods
+  rw [←Finset.sum_add_distrib]
+  have H (i : Fin J) : (k+bitIndex (b i))+(k+(1-bitIndex (b i)))=2*k+1 := by
+    cases h:b i <;> simp [bitIndex] <;> omega
+  simp only [H,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,Nat.cast_id]
+
+theorem initialPrefixPeriods_bounds {J : ℕ} (k : ℕ) (b : Fin J → Bool) :
+    J*k ≤ initialPrefixPeriods k b ∧ initialPrefixPeriods k b ≤ J*k+J := by
+  rw [initialPrefixPeriods_eval]
+  have H : (∑i,bitIndex (b i)) ≤ J := by
+    calc
+      _ ≤ ∑_i : Fin J,1 := Finset.sum_le_sum (fun i _ => by cases b i <;> simp [bitIndex])
+      _ = _ := by simp
+  omega
+
+theorem initial_paired_oracle_simulates {Side : Type*} [Fintype Side] {J K m : ℕ}
+    (M : PairedMechanisms Side J) (k s : ℕ) (a : ℤ) (τ : ℝ)
+    (off : Fin m) (hi : 1 ≤ off.val ∧ off.val+1 < m)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=K) (fun _ => off)))
+    {q : ℝ} (hq : q∈Set.Ioo 0 1) (side : Side → List Bool)
+    (hfirst : ∀i,SimulatesTrace q (M.first i) side (pureAtomRun (initialPeriod m off.val) k))
+    (hsecond : ∀i,SimulatesTrace q (M.second i) side (pureAtomRun (initialPeriod m off.val) (k+1)))
+    (ha : (J*k+J:ℕ) ≤ a) (ha' : a+(2^s:ℕ) ≤ (J*k+K:ℕ)) :
+    SimulatesTrace q
+      (M.emissionOracle (fun b => countedPrimitiveLaw (le_refl (K*m)) hq
+        (defectTypes (signedPeriodDefects off hsafe s
+          (a-(J*k:ℕ)-∑i,(bitIndex (b i):ℤ)) τ))))
+      (decodePairedEmission side decodeBoundedMarks)
+      (motifPeriodWord off hi (J*(2*k+1)+K) s a τ) := by
+  rw [←M.emission_factorization]
+  intro F
+  rw [FiniteLaw.expect_mixture]
+  have H (b : Fin J → Bool) :
+      ((M.reportsGiven b).prod (countedPrimitiveLaw (le_refl (K*m)) hq
+        (defectTypes (signedPeriodDefects off hsafe s
+          (a-(J*k:ℕ)-∑i,(bitIndex (b i):ℤ)) τ)))).expect
+        (fun z => F (decodePairedEmission side decodeBoundedMarks z))=
+      listDeletionExpectation q (motifPeriodWord off hi (J*(2*k+1)+K) s a τ) F := by
+    let t := initialPrefixPeriods k b
+    let z := initialSuffixPeriods k b
+    have ht := initialPrefixPeriods_bounds k b
+    have htot : t+K+z=J*(2*k+1)+K := by
+      have H:=initialPairPeriods_total k b
+      dsimp [t,z]
+      omega
+    have hanchor : a-(J*k:ℕ)-∑i,(bitIndex (b i):ℤ)=a-t := by
+      dsimp [t]
+      rw [initialPrefixPeriods_eval]
+      push_cast
+      ring
+    have hleft : (t:ℤ) ≤ a := by
+      have H : (t:ℤ) ≤ (J*k+J:ℕ) := by exact_mod_cast ht.2
+      omega
+    have hright : a+(2^s:ℕ) ≤ (t:ℤ)+K := by
+      have H : ((J*k:ℕ):ℤ) ≤ t := by exact_mod_cast ht.1
+      simp only [Nat.cast_add] at ha'
+      omega
+    have hp : coordinateWord (fun i => if b i then
+        pureAtomRun (initialPeriod m off.val) (k+1) else pureAtomRun (initialPeriod m off.val) k)=
+        pureAtomRun (initialPeriod m off.val) t := by
+      have h (i : Fin J) : (if b i then pureAtomRun (initialPeriod m off.val) (k+1) else
+          pureAtomRun (initialPeriod m off.val) k)=pureAtomRun (initialPeriod m off.val) (k+bitIndex (b i)) := by
+        cases b i <;> simp [bitIndex]
+      simp_rw [h]
+      exact coordinateWord_pureAtomRun _ _
+    have hz : coordinateWord (fun i => if b i then
+        pureAtomRun (initialPeriod m off.val) k else pureAtomRun (initialPeriod m off.val) (k+1))=
+        pureAtomRun (initialPeriod m off.val) z := by
+      have h (i : Fin J) : (if b i then pureAtomRun (initialPeriod m off.val) k else
+          pureAtomRun (initialPeriod m off.val) (k+1))=pureAtomRun (initialPeriod m off.val) (k+(1-bitIndex (b i))) := by
+        cases b i <;> simp [bitIndex]
+      simp_rw [h]
+      exact coordinateWord_pureAtomRun _ _
+    have hc := countedPrimitiveLaw_simulates (le_refl (K*m)) hq
+      (defectTypes (signedPeriodDefects off hsafe s (a-(J*k:ℕ)-∑i,(bitIndex (b i):ℤ)) τ))
+    rw [signed_primitive_word off hi] at hc
+    have hs := M.reportsGiven_paired_simulates b side decodeBoundedMarks _
+      (fun _ => pureAtomRun (initialPeriod m off.val) k)
+      (fun _ => pureAtomRun (initialPeriod m off.val) (k+1)) _ hfirst hsecond hc
+    have hword : coordinateWord (fun i => if b i then
+        pureAtomRun (initialPeriod m off.val) (k+1) else pureAtomRun (initialPeriod m off.val) k) ++
+        motifPeriodWord off hi K s (a-(J*k:ℕ)-∑i,(bitIndex (b i):ℤ)) τ ++
+        coordinateWord (fun i => if b i then pureAtomRun (initialPeriod m off.val) k else
+          pureAtomRun (initialPeriod m off.val) (k+1))=
+        motifPeriodWord off hi (J*(2*k+1)+K) s a τ := by
+      rw [hp,hz,hanchor,←motifPeriodWord_window off hi t K z s a τ hleft hright,htot]
+    rw [hword] at hs
+    exact hs F
+  simp only [H,FiniteLaw.expect_const]
+
+end TraceReconstruction
+
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+abbrev InitialMetadata (Side : Type*) (J N : ℕ) := Reports Side J × Fin (N+1)
+abbrev InitialObservation (Side : Type*) (J N : ℕ) :=
+  Σz : InitialMetadata Side J N, Fin z.2.val → PrimitiveMark
+
+def initialMetadataLaw {Side : Type*} [Fintype Side] {J : ℕ}
+    (M : PairedMechanisms Side J) (N : ℕ) {q : ℝ} (hq : q∈Set.Ioo 0 1) :
+    FiniteLaw (InitialMetadata Side J N) :=
+  M.reports.prod (boundedCountLaw (le_refl N) (1-primitiveErasure q) (primitiveRetention_mem hq))
+
+def initialConditionalLaw {Side : Type*} [Fintype Side] {J K m : ℕ}
+    (off : Fin m) (hsafe : SafeTaggedTuple (periodSlot (u:=K) (fun _ => off)))
+    {q : ℝ} (hq : q∈Set.Ioo 0 1) (c : ℤ) (s : ℕ) (τ : ℝ)
+    (z : InitialMetadata Side J (K*m)) : FiniteLaw (Fin z.2.val → PrimitiveMark) :=
+  conditionalMotifLaw (by have H:=z.2.isLt; omega) off hsafe hq
+    (posteriorAnchorSeries c (fun i => (z.1 i).1))
+    (posteriorAnchorSeries_nonneg c _) (posteriorAnchorSeries_total c _) s τ
+
+def initialSigmaLaw {Side : Type*} [Fintype Side] {J K m : ℕ}
+    (M : PairedMechanisms Side J) (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=K) (fun _ => off)))
+    {q : ℝ} (hq : q∈Set.Ioo 0 1) (c : ℤ) (s : ℕ) (τ : ℝ) :
+    FiniteLaw (InitialObservation Side J (K*m)) :=
+  (initialMetadataLaw M (K*m) hq).sigma (initialConditionalLaw off hsafe hq c s τ)
+
+def decodeInitialObservation {Side : Type*} {J N : ℕ} (side : Side → List Bool)
+    (z : InitialObservation Side J N) : List Bool :=
+  decodePairedEmission side decodeBoundedMarks (z.1.1,⟨z.1.2,z.2⟩)
+
+theorem initialSigmaLaw_expect {Side : Type*} [Fintype Side] {J K m : ℕ}
+    (M : PairedMechanisms Side J) (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=K) (fun _ => off)))
+    {q : ℝ} (hq : q∈Set.Ioo 0 1) (c : ℤ) (s : ℕ) (τ : ℝ)
+    (F : Reports Side J × BoundedMarks (K*m) → ℝ) :
+    (initialSigmaLaw M off hsafe hq c s τ).expect (fun z => F (z.1.1,⟨z.1.2,z.2⟩))=
+    (M.emissionOracle (fun b => countedPrimitiveLaw (le_refl (K*m)) hq
+      (defectTypes (signedPeriodDefects off hsafe s (c-∑i,(bitIndex (b i):ℤ)) τ)))).expect F := by
+  unfold PairedMechanisms.emissionOracle
+  rw [FiniteLaw.expect_joint]
+  have H (z : Reports Side J) : (PairedMechanisms.orientationsGiven z).mixture
+      (fun b => countedPrimitiveLaw (le_refl (K*m)) hq
+        (defectTypes (signedPeriodDefects off hsafe s (c-∑i,(bitIndex (b i):ℤ)) τ)))=
+      countedMotifLaw off hsafe hq (posteriorAnchorSeries c (fun i => (z i).1))
+        (posteriorAnchorSeries_nonneg c _) (posteriorAnchorSeries_total c _) s τ :=
+    countedMotifLaw_posterior off hsafe hq c (fun i => (z i).1) s τ
+  simp_rw [H]
+  simp only [initialSigmaLaw,initialMetadataLaw,FiniteLaw.expect_sigma,FiniteLaw.expect_prod,
+    countedMotifLaw,initialConditionalLaw]
+
+theorem initial_sigma_simulates {Side : Type*} [Fintype Side] {J K m : ℕ}
+    (M : PairedMechanisms Side J) (k s : ℕ) (a : ℤ) (τ : ℝ)
+    (off : Fin m) (hi : 1  ≤  off.val ∧ off.val+1  <  m)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=K) (fun _ => off)))
+    {q : ℝ} (hq : q∈Set.Ioo 0 1) (side : Side → List Bool)
+    (hfirst : ∀i,SimulatesTrace q (M.first i) side (pureAtomRun (initialPeriod m off.val) k))
+    (hsecond : ∀i,SimulatesTrace q (M.second i) side (pureAtomRun (initialPeriod m off.val) (k+1)))
+    (ha : (J*k+J:ℕ)  ≤  a) (ha' : a+(2^s:ℕ)  ≤  (J*k+K:ℕ)) :
+    SimulatesTrace q (initialSigmaLaw M off hsafe hq (a-(J*k:ℕ)) s τ)
+      (decodeInitialObservation side) (motifPeriodWord off hi (J*(2*k+1)+K) s a τ) := by
+  intro F
+  unfold decodeInitialObservation
+  rw [initialSigmaLaw_expect M off hsafe hq (a-(J*k:ℕ)) s τ (fun z => F (decodePairedEmission side decodeBoundedMarks z))]
+  exact initial_paired_oracle_simulates M k s a τ off hi hsafe hq side hfirst hsecond ha ha' F
+
+/-- This is the literal count-typicality condition; it does not change the
+ central count's original binomial law. -/
+def initialCountAdmissible (N : ℕ) (q : ℝ) (l : Fin (N+1)) : Prop :=
+  (l.val:ℝ)/(N:ℝ)∈Set.Icc (1-q^2) (1-q^2/4) ∧
+    1  ≤  (N:ℝ)*((l.val:ℝ)/(N:ℝ))*(1-(l.val:ℝ)/(N:ℝ))
+
+def initialMetadataGood {Side : Type*} {J N : ℕ} (q ω : ℝ)
+    (z : InitialMetadata Side J N) : Prop :=
+  initialCountAdmissible N q z.2 ∧ (J:ℝ)*ω^2/2  ≤  ∑i,(hiddenTrial (z.1 i).1:ℝ)
+
+/-- The genuine initial metadata oracle is clipped by conditioning each
+ admissible fiber only. Every bad report/count keeps its original mass and
+ is replaced by a common law, so all discarded mass is charged. -/
+theorem initial_sigma_clipped_exists {Side : Type*} [Fintype Side] {J K m R : ℕ}
+    (M : PairedMechanisms Side J) (hK : 0 < K) (hm : 128 ≤ m) (hKm : K ≤ m)
+    (hR : 0 < R) (off : Fin m) (hi : 1 ≤ off.val ∧ off.val+1 < m)
+    (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 2)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=K) (fun _ => off)))
+    (c : ℤ) (s k : ℕ) (hk : 1 ≤ k)
+    (hleft : (K:ℝ)/3 ≤ 2*((c:ℝ)-J)+1)
+    (hright : (K:ℝ)/3 ≤ 2*K-(2*((c:ℝ)+(2:ℝ)^s)+1))
+    {q ω w V : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
+    (hM : ((2*(2^s):ℕ):ℝ) < (K:ℝ)/(3*(R+1:ℝ)))
+    (hw : w=((6*(2*k:ℕ):ℝ)*(2/q)))
+    (hscale : initialMomentRatio K m q w ≤ 1/2)
+    (hclip : Real.sqrt (6*(4*(initialMomentRatio K m q w/(1-initialMomentRatio K m q w)))^k) ≤ 1/2)
+    (hV : 0 ≤ V)
+    (hbudget : ∀z : InitialMetadata Side J (K*m),initialMetadataGood q ω z →
+      64*initialSignedBudget K m z.2.val R s (posteriorAnchorSeries c (fun i => (z.1 i).1))
+        ((z.2.val:ℝ)/(K*m:ℕ)) q w ≤ V) :
+    Nonempty (ClippedOracle
+      (initialSigmaLaw M off hsafe ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩ c s (-1))
+      (initialSigmaLaw M off hsafe ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩ c s 1) k
+      (Real.sqrt (6*(4*(initialMomentRatio K m q w/(1-initialMomentRatio K m q w)))^k)+
+        (initialMetadataLaw M (K*m) ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩).expect
+          (fun z => if initialMetadataGood q ω z then 0 else 1)) V) := by
+  apply ClippedOracle.assemble_good _ _ _ (initialMetadataGood q ω) k hk _ V (Real.sqrt_nonneg _) hV
+  intro z hz
+  have hmean : ((K*m:ℕ):ℝ)*((z.2.val:ℝ)/(K*m:ℕ))=z.2.val := by
+    exact mul_div_cancel₀ _ (by exact_mod_cast (Nat.mul_ne_zero (by omega : K≠0) (by omega : m≠0)))
+  have H := initial_conditional_clipped_exists hK hm hKm hR (by have H:=z.2.isLt; omega) off hi hoff hsafe
+    (posteriorAnchorSeries c (fun i => (z.1 i).1)) (posteriorAnchorSeries_nonneg c _)
+    (posteriorAnchorSeries_total c _) s k
+    (posteriorAnchorSeries_central K s c _ (by simpa using hleft) hright) hk hq hz.1.1 hmean hz.1.2 hM hw hscale hclip
+  exact ⟨(Classical.choice H).weaken le_rfl (hbudget z hz)⟩
+
+end TraceReconstruction
+
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+def purePeriodCoefficient {u m l : ℕ} (hl : l ≤ u*m) (off : Fin m) : (Fin l → Bool) → ℝ :=
+  averagedCoefficient (uniformSubsetLaw l (by simpa using hl))
+    (defectAtRank (Finset.univ.image (periodSlot (u:=u) (fun _ => off))))
+
+theorem mixedMotifCoefficient_pure_rank_difference {u m l r : ℕ} (hl : l ≤ u*m) (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
+    (π : IntSeries) (hπsum : ∑X∈π.coeff.support,π.coeff X=1) (s : ℕ) (v : IncreasingTuple l r) :
+    mixedMotifCoefficient hl off hsafe π s 1 (rankMask v)-purePeriodCoefficient hl off (rankMask v)=
+      pureRankCoefficient hl off hsafe π s v := by
+  unfold mixedMotifCoefficient purePeriodCoefficient pureRankCoefficient
+  simp_rw [averaged_rank_coefficient hl _ v,mul_sub,Finset.sum_sub_distrib,←Finset.sum_mul,hπsum,one_mul]
+
+/-- The pure comparison retains all low-degree cancellation, not just its
+ total defect density. -/
+def initialPureDegreeBound (u m r s : ℕ) (π : IntSeries) (θ : ℝ) (l : ℕ) : ℝ :=
+  if r=0 then 0 else
+    2*(Fintype.card (Fin r → GapDifference):ℝ)*
+      (∑k : Fin r → GapDifference, if k=unchangedPattern r then 0 else purePatternBound u m r s π θ k)+
+    2*((Fintype.card (IncreasingTuple u r):ℝ)*
+      (2*(((r+1:ℕ):ℝ)*θ^r/binomialMass (u*m) θ l*(2*Real.exp (-(m:ℝ)/(1024*u))))))^2
+
+theorem initialPureDegreeBound_nonneg (u m r s : ℕ) (π : IntSeries) (θ : ℝ) (l : ℕ) :
+    0 ≤ initialPureDegreeBound u m r s π θ l := by
+  unfold initialPureDegreeBound
+  split_ifs
+  · exact le_rfl
+  · apply add_nonneg _ (by positivity)
+    apply mul_nonneg (by positivity)
+    apply Finset.sum_nonneg
+    intro k _
+    split_ifs
+    · exact le_rfl
+    · unfold purePatternBound
+      split_ifs  <;> apply mul_nonneg _ (seriesL2Sq_nonneg _)
+      all_goals dsimp only [periodDifferencePrefactor,lengthDifferenceConstant,periodCentralSumConstant]
+      all_goals positivity
+
+theorem pureMotif_degree_zero {u m l : ℕ} (hl : l ≤ u*m) (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
+    (π : IntSeries) (hπsum : ∑X∈π.coeff.support,π.coeff X=1) (s : ℕ) :
+    (∑σ : Fin l → Bool,if monomialDegree σ=0 then
+      (mixedMotifCoefficient hl off hsafe π s 1 σ-purePeriodCoefficient hl off σ)^2 else 0)=0 := by
+  simp_rw [monomialDegree_eq_zero_iff]
+  simp [mixedMotifCoefficient_empty hl off hsafe π hπsum,purePeriodCoefficient,averagedCoefficient_empty]
+
+theorem pureMotif_degree_cancellation {u m l r : ℕ}
+    (hu : 0 < u) (hm : 128 ≤ m) (hr : 0 < r) (hl : l ≤ u*m)
+    (off : Fin m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 3)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
+    (π : IntSeries) (hπ : ∀X,0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
+    (s : ℕ) (hcentral : CentralMotifSupport u s π)
+    {θ : ℝ} (hθ : θ∈Set.Ioo 0 1) (hhalf : 1/2 ≤ θ)
+    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ)) :
+    (∑σ : Fin l → Bool,if monomialDegree σ=r then
+      (mixedMotifCoefficient hl off hsafe π s 1 σ-purePeriodCoefficient hl off σ)^2 else 0) ≤ 
+        initialPureDegreeBound u m r s π θ l := by
+  refine (sum_degree_coeff_sq_le _).trans ?_
+  simp_rw [mixedMotifCoefficient_pure_rank_difference hl off hsafe π hπsum s]
+  simpa only [initialPureDegreeBound,ite_eq_right (by omega : r ≠ 0)] using
+    initial_pure_energy hu hm hr hl off hoff hsafe π hπ hπsum s hcentral hθ hhalf hmean hv
+
+theorem purePeriod_degree_geometric {u m l r : ℕ} (hl : l ≤ u*m) (hr : 0 < r)
+    (hm : 32 ≤ m) (hum : u ≤ m) (off : Fin m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 3)
+    {q θ : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
+    (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
+    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ)) (w : ℝ) :
+    w^(2*r)*(∑σ : Fin l → Bool,if monomialDegree σ=r then (purePeriodCoefficient hl off σ)^2 else 0) ≤ 
+      (initialMomentRatio u m q w)^r := by
+  have hw0 : 0 ≤ w^(2*r) := by rw [pow_mul]; positivity
+  have hp : 0 ≤ 3*rankSchurConstant/q := by positivity [rankSchurConstant_pos,hq.1]
+  by_cases hru : r ≤ u
+  · have hb := period_degree_square_sum hl hr hru hm hum hq hθ hmean hv (fun _ => off)
+      (fun _ => hoff) (fun _ _ => by simp)
+    have hsqrt : Real.sqrt ((u:ℝ)/((m:ℝ)*r)) ≤ Real.sqrt ((u:ℝ)/m) := by
+      apply Real.sqrt_le_sqrt
+      apply div_le_div_of_nonneg_left (by positivity) (by exact_mod_cast (show 0 < m by omega))
+      have hr1 : (1:ℝ) ≤ r := by exact_mod_cast hr
+      nlinarith
+    calc
+      _  ≤  w^(2*r)*((3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/((m:ℝ)*r)))^r :=
+        mul_le_mul_of_nonneg_left hb hw0
+      _  ≤  w^(2*r)*((3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/m))^r := by gcongr
+      _ = _ := by unfold initialMomentRatio; rw [pow_mul,mul_pow,mul_pow]; ring
+  · have H := defect_degree_bound hl (periodSlot (u:=u) (fun _ => off)) (periodSlot_strictMono _) (by omega : u < r)
+    change _=0 at H
+    change w^(2*r)*_ ≤ _
+    rw [show (∑σ : Fin l → Bool,if monomialDegree σ=r then (purePeriodCoefficient hl off σ)^2 else 0)=0 from H,mul_zero]
+    exact pow_nonneg (by unfold initialMomentRatio; positivity [rankSchurConstant_pos,hq.1]) _
+
+theorem pureMotif_degree_geometric {u m l r : ℕ} (hl : l ≤ u*m) (hr : 0 < r)
+    (hm : 32 ≤ m) (hum : u ≤ m) (off : Fin m)
+    (hi : 1 ≤ off.val ∧ off.val+1 < m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 2)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
+    (π : IntSeries) (hπ : ∀X,0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
+    (s : ℕ) {q θ : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
+    (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
+    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ)) (w : ℝ) :
+    w^(2*r)*(∑σ : Fin l → Bool,if monomialDegree σ=r then
+      (mixedMotifCoefficient hl off hsafe π s 1 σ-purePeriodCoefficient hl off σ)^2 else 0) ≤ 
+        4*(initialMomentRatio u m q w)^r := by
+  have H := mul_le_mul_of_nonneg_left (degree_square_sub_le
+    (mixedMotifCoefficient hl off hsafe π s 1) (purePeriodCoefficient hl off) r)
+    (by rw [pow_mul]; positivity : 0 ≤ w^(2*r))
+  have h1 := mixedMotif_degree_geometric hl hr hm hum off hi hoff hsafe π hπ hπsum s 1 hq hθ hmean hv w
+  have h2 := purePeriod_degree_geometric hl hr hm hum off (hoff.trans (by norm_num)) hq hθ hmean hv w
+  change _ ≤ initialMomentRatio u m q w ^r at h1
+  nlinarith only [H,h1,h2]
+
+def initialPureBudget (u m l R s : ℕ) (π : IntSeries) (θ q w : ℝ) : ℝ :=
+  (∑r∈Finset.range R,w^(2*r)*initialPureDegreeBound u m r s π θ l)+
+    4*(initialMomentRatio u m q w)^R/(1-initialMomentRatio u m q w)
+
+theorem pure_initial_full_energy {u m l R : ℕ} (hu : 0 < u) (hm : 128 ≤ m)
+    (hum : u ≤ m) (hR : 0 < R) (hl : l ≤ u*m) (off : Fin m)
+    (hi : 1 ≤ off.val ∧ off.val+1 < m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 2)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
+    (π : IntSeries) (hπ : ∀X,0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
+    (s : ℕ) (hcentral : CentralMotifSupport u s π)
+    {q θ w : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
+    (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
+    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
+    (hw : 0 ≤ w) (ha : initialMomentRatio u m q w < 1) :
+    multilinearEnergy (fun σ => mixedMotifCoefficient hl off hsafe π s 1 σ-purePeriodCoefficient hl off σ) w ≤ 
+      initialPureBudget u m l R s π θ q w := by
+  have hq2 : q^2 ≤ (1/4:ℝ) := by
+    calc q^2 ≤ (1/2:ℝ)^2 := pow_le_pow_left₀ hq.1.le hq.2 2
+         _ = _ := by norm_num
+  have hθ0 : θ∈Set.Ioo 0 1 := by constructor  <;> nlinarith [hq.1,hθ.1,hθ.2]
+  have hhalf : 1/2 ≤ θ := by nlinarith [hθ.1]
+  apply multilinearEnergy_low_high _ w _ 4 hw (by unfold initialMomentRatio; positivity [rankSchurConstant_pos,hq.1]) ha
+    (by norm_num) (initialPureDegreeBound u m · s π θ l) _ _
+    (fun r => initialPureDegreeBound_nonneg _ _ _ _ _ _ _)
+  · intro r _hr
+    by_cases hr0 : r=0
+    · subst r
+      rw [pureMotif_degree_zero hl off hsafe π hπsum s]
+      exact initialPureDegreeBound_nonneg _ _ _ _ _ _ _
+    · exact pureMotif_degree_cancellation hu hm (Nat.pos_of_ne_zero hr0) hl off (hoff.trans (by norm_num))
+        hsafe π hπ hπsum s hcentral hθ0 hhalf hmean hv
+  · intro r hr
+    exact pureMotif_degree_geometric hl (by omega) (by omega) hum off hi hoff hsafe π hπ hπsum s hq hθ hmean hv w
+
+end TraceReconstruction
+
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+def purePeriodDensity {u m l : ℕ} (hl : l ≤ u*m) (off : Fin m) (q : ℝ) :
+    (Fin l → PrimitiveMark) → ℝ := multilinear (purePeriodCoefficient hl off) (primitiveScore q)
+
+theorem purePeriodDensity_nonneg {u m l : ℕ} (hl : l ≤ u*m) (off : Fin m)
+    {q : ℝ} (hq : q∈Set.Ioo 0 1) (x : Fin l → PrimitiveMark) :
+    0 ≤ purePeriodDensity hl off q x := by
+  have H := (conditionalPrimitiveLaw hl q hq (defectTypes (periodSlot (u:=u) (fun _ => off)))).nonneg x
+  rw [conditionalDefectLaw_density] at H
+  exact nonneg_of_mul_nonneg_right H (primitive_product_mass_pos hq x)
+
+theorem purePeriodDensity_total {u m l : ℕ} (hl : l ≤ u*m) (off : Fin m)
+    {q : ℝ} (hq : q∈Set.Ioo 0 1) :
+    ((primitiveMarkLaw q hq false).product l).expect (purePeriodDensity hl off q)=1 := by
+  have H := (conditionalPrimitiveLaw hl q hq (defectTypes (periodSlot (u:=u) (fun _ => off)))).total
+  simpa only [conditionalDefectLaw_density,FiniteLaw.expect,purePeriodDensity,purePeriodCoefficient] using H
+
+theorem purePeriodLaw_withDensity {u m l : ℕ} (hl : l ≤ u*m) (off : Fin m)
+    {q : ℝ} (hq : q∈Set.Ioo 0 1) :
+    conditionalPrimitiveLaw hl q hq (defectTypes (periodSlot (u:=u) (fun _ => off)))=
+      ((primitiveMarkLaw q hq false).product l).withDensity (purePeriodDensity hl off q)
+        (purePeriodDensity_nonneg hl off hq) (purePeriodDensity_total hl off hq) := by
+  apply FiniteLaw.ext
+  funext x
+  exact conditionalDefectLaw_density hl hq _ x
+
+theorem purePeriod_centered_energy {u m l : ℕ} (hl : l ≤ u*m) (hm : 32 ≤ m) (hum : u ≤ m)
+    (off : Fin m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 3)
+    {q θ w : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
+    (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
+    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
+    (hw : 0 ≤ w) (ha : initialMomentRatio u m q w < 1) :
+    multilinearEnergy (nonconstantCoefficient (purePeriodCoefficient hl off)) w ≤
+      initialMomentRatio u m q w/(1-initialMomentRatio u m q w) := by
+  have H := multilinearEnergy_low_high (R:=1)
+    (nonconstantCoefficient (purePeriodCoefficient hl off)) w _ 1 hw
+    (by unfold initialMomentRatio; positivity [rankSchurConstant_pos,hq.1]) ha (by norm_num) (fun _ => 0)
+    (fun r hr => by
+      have he : r=0 := by omega
+      subst r
+      rw [nonconstant_degree_zero])
+    (fun r hr => by
+      rw [nonconstant_degree_pos _ (by omega)]
+      simpa only [one_mul] using purePeriod_degree_geometric hl (by omega) hm hum off hoff hq hθ hmean hv w)
+    (fun _ => le_rfl)
+  simpa using H
+
+theorem purePeriod_centered_moment {u m l : ℕ} (hl : l ≤ u*m) (hm : 32 ≤ m) (hum : u ≤ m)
+    (off : Fin m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 3) (k : ℕ) (hk : 1 ≤ k)
+    {q θ w : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
+    (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
+    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
+    (hw : w=((6*(2*k:ℕ):ℝ)*(2/q))) (ha : initialMomentRatio u m q w < 1) :
+    ((primitiveMarkLaw q ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩ false).product l).evenMomentSq
+      (fun x => purePeriodDensity hl off q x-1) k ≤
+      initialMomentRatio u m q w/(1-initialMomentRatio u m q w) := by
+  have hq' : q∈Set.Ioo 0 1 := ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩
+  have he : (fun x => purePeriodDensity hl off q x-1)=
+      multilinear (nonconstantCoefficient (purePeriodCoefficient hl off)) (primitiveScore q) := by
+    funext x
+    rw [nonconstant_multilinear]
+    simp only [purePeriodCoefficient,averagedCoefficient_empty,purePeriodDensity]
+  rw [he]
+  apply (multilinear_moment_sq_scaled (primitiveMarkLaw q hq' false) (primitiveScore q) k hk
+    (primitive_score_centered hq') (div_pos (by norm_num) hq.1) (primitive_score_evenMoment hq' k) l _).trans
+  rw [←hw]
+  exact purePeriod_centered_energy hl hm hum off hoff hq hθ hmean hv (by rw [hw]; positivity [hq.1]) ha
+
+theorem initial_pure_moment {u m l R : ℕ} (hu : 0 < u) (hm : 128 ≤ m)
+    (hum : u ≤ m) (hR : 0 < R) (hl : l ≤ u*m) (off : Fin m)
+    (hi : 1 ≤ off.val ∧ off.val+1 < m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 2)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
+    (π : IntSeries) (hπ : ∀X,0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
+    (s k : ℕ) (hcentral : CentralMotifSupport u s π) (hk : 1 ≤ k)
+    {q θ w : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
+    (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
+    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
+    (hw : w=((6*(2*k:ℕ):ℝ)*(2/q))) (ha : initialMomentRatio u m q w < 1) :
+    ((primitiveMarkLaw q ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩ false).product l).evenMomentSq
+      (fun x => initialMotifDensity hl off hsafe π s 1 q x-purePeriodDensity hl off q x) k ≤
+        initialPureBudget u m l R s π θ q w := by
+  have hq' : q∈Set.Ioo 0 1 := ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩
+  have he : (fun x => initialMotifDensity hl off hsafe π s 1 q x-purePeriodDensity hl off q x)=
+      multilinear (fun σ => mixedMotifCoefficient hl off hsafe π s 1 σ-purePeriodCoefficient hl off σ)
+        (primitiveScore q) := by funext x; exact (multilinear_sub _ _ _ _).symm
+  rw [he]
+  apply (multilinear_moment_sq_scaled (primitiveMarkLaw q hq' false) (primitiveScore q) k hk
+    (primitive_score_centered hq') (div_pos (by norm_num) hq.1) (primitive_score_evenMoment hq' k) l _).trans
+  rw [←hw]
+  exact pure_initial_full_energy hu hm hum hR hl off hi hoff hsafe π hπ hπsum s hcentral hq hθ hmean hv
+    (by rw [hw]; positivity [hq.1]) ha
+
+theorem initial_pure_conditional_clipped_exists {u m l R : ℕ} (hu : 0 < u) (hm : 128 ≤ m)
+    (hum : u ≤ m) (hR : 0 < R) (hl : l ≤ u*m) (off : Fin m)
+    (hi : 1 ≤ off.val ∧ off.val+1 < m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 2)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
+    (π : IntSeries) (hπ : ∀X,0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
+    (s k : ℕ) (hcentral : CentralMotifSupport u s π) (hk : 1 ≤ k)
+    {q θ w : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
+    (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
+    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
+    (hw : w=((6*(2*k:ℕ):ℝ)*(2/q))) (hscale : initialMomentRatio u m q w ≤ 1/2)
+    (hclip : Real.sqrt (6*(4*(initialMomentRatio u m q w/(1-initialMomentRatio u m q w)))^k) ≤ 1/2) :
+    Nonempty (ClippedOracle
+      (conditionalPrimitiveLaw hl q ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩
+        (defectTypes (periodSlot (u:=u) (fun _ => off))))
+      (conditionalMotifLaw hl off hsafe ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩ π hπ hπsum s 1) k
+      (Real.sqrt (6*(4*(initialMomentRatio u m q w/(1-initialMomentRatio u m q w)))^k))
+      (64*initialPureBudget u m l R s π θ q w)) := by
+  have hq' : q∈Set.Ioo 0 1 := ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩
+  let A := (primitiveMarkLaw q hq' false).product l
+  let f := purePeriodDensity hl off q
+  let g := initialMotifDensity hl off hsafe π s 1 q
+  let U := initialMomentRatio u m q w/(1-initialMomentRatio u m q w)
+  have ha : initialMomentRatio u m q w < 1 := by linarith
+  have hz : 0 ≤ initialMomentRatio u m q w := by unfold initialMomentRatio; positivity [rankSchurConstant_pos,hq.1]
+  have hU : 0 ≤ U := div_nonneg hz (sub_nonneg.mpr ha.le)
+  have hU1 : U ≤ 1 := (div_le_iff₀ (sub_pos.mpr ha)).mpr (by linarith)
+  have hf : A.expect (fun x => |f x-1|^(2*k)) ≤ U^k :=
+    A.expect_even_pow_le_of_moment _ k hk (purePeriod_centered_moment hl (by omega) hum off
+      (hoff.trans (by norm_num)) k hk hq hθ hmean hv hw ha)
+  have hg : A.expect (fun x => |g x-1|^(2*k)) ≤ U^k :=
+    A.expect_even_pow_le_of_moment _ k hk (initial_centered_moment hl (by omega) hum off hi hoff hsafe π hπ hπsum s k hk 1 hq hθ hmean hv hw ha)
+  have hsig := initial_pure_moment hu hm hum hR hl off hi hoff hsafe π hπ hπsum s k hcentral hk hq hθ hmean hv hw ha
+  have hV := (A.evenMomentSq_nonneg (fun x => g x-f x) k).trans hsig
+  have hsig' := A.expect_even_pow_le_of_moment _ k hk hsig
+  rw [purePeriodLaw_withDensity hl off hq',initialMotifLaw_withDensity hl off hsafe π hπ hπsum s 1 hq']
+  have H := A.exists_clippedOracle f g
+    (purePeriodDensity_nonneg hl off hq') (initialMotifDensity_nonneg hl off hsafe π hπ hπsum s 1 hq')
+    (purePeriodDensity_total hl off hq') (initialMotifDensity_mean hl off hsafe π hπ hπsum s 1 hq')
+    Finset.univ 0 (by simp [FiniteLaw.eventMass]) (by simp [FiniteLaw.eventMass]) k hk U hU hU1 hf hg
+    (initialPureBudget u m l R s π θ q w) hV (by simpa only [Finset.mem_univ,ite_true] using hsig')
+    (by simpa only [zero_add] using hclip)
+  simpa only [zero_add] using H
+
+end TraceReconstruction
+
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- A common finite decoder transfers normalized clipping to actual complete
+ traces. Neither interface is permitted to assume the main lower bound. -/
+theorem ClippedOracle.tracePair {α : Type} [Fintype α] {P Q : FiniteLaw α}
+    {q e V : ℝ} {x y : List Bool} {k : ℕ} (f : α → List Bool)
+    (C : ClippedOracle P Q k e V) (hP : SimulatesTrace q P f x)
+    (hQ : SimulatesTrace q Q f y) : Nonempty (OracleTracePair q x y k e V) := by
+  refine ⟨{
+    Outcome := α
+    finiteOutcome := inferInstance
+    first := C.first
+    second := C.second
+    decode := f
+    score := C.score
+    density := C.density
+    likelihood_nonneg := C.likelihood_nonneg
+    centered := C.centered
+    moment := C.moment
+    first_simulation := ?_
+    second_simulation := ?_
+  }⟩
+  · simpa using hP.approximate.of_tv C.first_tv
+  · simpa using hQ.approximate.of_tv C.second_tv
+
+def initialPureConditionalLaw {Side : Type*} {J K m : ℕ} (off : Fin m)
+    {q : ℝ} (hq : q∈Set.Ioo 0 1) (z : InitialMetadata Side J (K*m)) :
+    FiniteLaw (Fin z.2.val → PrimitiveMark) :=
+  conditionalPrimitiveLaw (by have H:=z.2.isLt; omega) q hq
+    (defectTypes (periodSlot (u:=K) (fun _ => off)))
+
+def initialPureSigmaLaw {Side : Type*} [Fintype Side] {J K m : ℕ}
+    (M : PairedMechanisms Side J) (off : Fin m) {q : ℝ} (hq : q∈Set.Ioo 0 1) :
+    FiniteLaw (InitialObservation Side J (K*m)) :=
+  (initialMetadataLaw M (K*m) hq).sigma (initialPureConditionalLaw off hq)
+
+theorem initialPureSigmaLaw_expect {Side : Type*} [Fintype Side] {J K m : ℕ}
+    (M : PairedMechanisms Side J) (off : Fin m) {q : ℝ} (hq : q∈Set.Ioo 0 1)
+    (F : Reports Side J × BoundedMarks (K*m) → ℝ) :
+    (initialPureSigmaLaw M off hq).expect (fun z => F (z.1.1,⟨z.1.2,z.2⟩))=
+    (M.emissionOracle (fun _ => countedPrimitiveLaw (le_refl (K*m)) hq
+      (defectTypes (periodSlot (u:=K) (fun _ => off))))).expect F := by
+  simp only [initialPureSigmaLaw,initialMetadataLaw,FiniteLaw.expect_sigma,
+    FiniteLaw.expect_prod,PairedMechanisms.emissionOracle,FiniteLaw.expect_joint,
+    FiniteLaw.expect_mixture,FiniteLaw.expect_const,initialPureConditionalLaw,
+    countedPrimitiveLaw,FiniteLaw.expect_sigma]
+  have hl (l : Fin (K*m+1)) : l.val ≤ K*m := Nat.le_of_lt_succ l.isLt
+  simp only [boundedConditionalPrimitiveLaw,hl,dite_eq_left]
+
+theorem initial_pure_sigma_simulates {Side : Type*} [Fintype Side] {J K m : ℕ}
+    (M : PairedMechanisms Side J) (k : ℕ) (off : Fin m)
+    {q : ℝ} (hq : q∈Set.Ioo 0 1) (side : Side → List Bool)
+    (hfirst : ∀i,SimulatesTrace q (M.first i) side (pureAtomRun (initialPeriod m off.val) k))
+    (hsecond : ∀i,SimulatesTrace q (M.second i) side (pureAtomRun (initialPeriod m off.val) (k+1))) :
+    SimulatesTrace q (initialPureSigmaLaw (K:=K) M off hq) (decodeInitialObservation side)
+      (pureAtomRun (initialPeriod m off.val) (J*(2*k+1)+K)) := by
+  intro F
+  unfold decodeInitialObservation
+  rw [initialPureSigmaLaw_expect M off hq (fun z => F (decodePairedEmission side decodeBoundedMarks z)),
+    ←M.emission_factorization,FiniteLaw.expect_mixture]
+  have H (b : Fin J → Bool) :
+      ((M.reportsGiven b).prod (countedPrimitiveLaw (le_refl (K*m)) hq
+        (defectTypes (periodSlot (u:=K) (fun _ => off))))).expect
+        (fun z => F (decodePairedEmission side decodeBoundedMarks z))=
+      listDeletionExpectation q (pureAtomRun (initialPeriod m off.val) (J*(2*k+1)+K)) F := by
+    have hc := countedPrimitiveLaw_simulates (le_refl (K*m)) hq (defectTypes (periodSlot (u:=K) (fun _ => off)))
+    rw [periodic_primitive_word,periodString_constant] at hc
+    have hs := M.reportsGiven_paired_simulates b side decodeBoundedMarks _
+      (fun _ => pureAtomRun (initialPeriod m off.val) k)
+      (fun _ => pureAtomRun (initialPeriod m off.val) (k+1)) _ hfirst hsecond hc
+    have hp : coordinateWord (fun i => if b i then pureAtomRun (initialPeriod m off.val) (k+1)
+        else pureAtomRun (initialPeriod m off.val) k)=
+        pureAtomRun (initialPeriod m off.val) (initialPrefixPeriods k b) := by
+      have h (i : Fin J) : (if b i then pureAtomRun (initialPeriod m off.val) (k+1) else
+          pureAtomRun (initialPeriod m off.val) k)=pureAtomRun (initialPeriod m off.val) (k+bitIndex (b i)) := by
+        cases b i <;> simp [bitIndex]
+      simp_rw [h]
+      exact coordinateWord_pureAtomRun _ _
+    have hz : coordinateWord (fun i => if b i then pureAtomRun (initialPeriod m off.val) k
+        else pureAtomRun (initialPeriod m off.val) (k+1))=
+        pureAtomRun (initialPeriod m off.val) (initialSuffixPeriods k b) := by
+      have h (i : Fin J) : (if b i then pureAtomRun (initialPeriod m off.val) k else
+          pureAtomRun (initialPeriod m off.val) (k+1))=pureAtomRun (initialPeriod m off.val) (k+(1-bitIndex (b i))) := by
+        cases b i <;> simp [bitIndex]
+      simp_rw [h]
+      exact coordinateWord_pureAtomRun _ _
+    rw [hp,hz,←pureAtomRun_add,←pureAtomRun_add] at hs
+    have ht : initialPrefixPeriods k b+K+initialSuffixPeriods k b=J*(2*k+1)+K := by
+      have ht:=initialPairPeriods_total k b
+      omega
+    rw [ht] at hs
+    exact hs F
+  simp only [H,FiniteLaw.expect_const]
+
+theorem initial_pure_sigma_clipped_exists {Side : Type*} [Fintype Side] {J K m R : ℕ}
+    (M : PairedMechanisms Side J) (hK : 0 < K) (hm : 128 ≤ m) (hKm : K ≤ m)
+    (hR : 0 < R) (off : Fin m) (hi : 1 ≤ off.val ∧ off.val+1 < m)
+    (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 2)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=K) (fun _ => off)))
+    (c : ℤ) (s k : ℕ) (hk : 1 ≤ k)
+    (hleft : (K:ℝ)/3 ≤ 2*((c:ℝ)-J)+1)
+    (hright : (K:ℝ)/3 ≤ 2*K-(2*((c:ℝ)+(2:ℝ)^s)+1))
+    {q ω w V : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
+    (hw : w=((6*(2*k:ℕ):ℝ)*(2/q)))
+    (hscale : initialMomentRatio K m q w ≤ 1/2)
+    (hclip : Real.sqrt (6*(4*(initialMomentRatio K m q w/(1-initialMomentRatio K m q w)))^k) ≤ 1/2)
+    (hV : 0 ≤ V)
+    (hbudget : ∀z : InitialMetadata Side J (K*m),initialMetadataGood q ω z →
+      64*initialPureBudget K m z.2.val R s (posteriorAnchorSeries c (fun i => (z.1 i).1))
+        ((z.2.val:ℝ)/(K*m:ℕ)) q w ≤ V) :
+    Nonempty (ClippedOracle
+      (initialPureSigmaLaw M off ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩)
+      (initialSigmaLaw M off hsafe ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩ c s 1) k
+      (Real.sqrt (6*(4*(initialMomentRatio K m q w/(1-initialMomentRatio K m q w)))^k)+
+        (initialMetadataLaw M (K*m) ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩).expect
+          (fun z => if initialMetadataGood q ω z then 0 else 1)) V) := by
+  apply ClippedOracle.assemble_good _ _ _ (initialMetadataGood q ω) k hk _ V (Real.sqrt_nonneg _) hV
+  intro z hz
+  have hmean : ((K*m:ℕ):ℝ)*((z.2.val:ℝ)/(K*m:ℕ))=z.2.val := by
+    exact mul_div_cancel₀ _ (by exact_mod_cast (Nat.mul_ne_zero (by omega : K≠0) (by omega : m≠0)))
+  have H := initial_pure_conditional_clipped_exists hK hm hKm hR (by have H:=z.2.isLt; omega) off hi hoff hsafe
+    (posteriorAnchorSeries c (fun i => (z.1 i).1)) (posteriorAnchorSeries_nonneg c _)
+    (posteriorAnchorSeries_total c _) s k
+    (posteriorAnchorSeries_central K s c _ (by simpa using hleft) hright) hk hq hz.1.1 hmean hz.1.2 hw hscale hclip
+  exact ⟨(Classical.choice H).weaken le_rfl (hbudget z hz)⟩
+
+end TraceReconstruction
+
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- The half-mean hidden-count event yields the correct L², not supremum,
+ exponent J^{-s-1/2} in the initial Prouhet cancellation. -/
+def hiddenSignedL2Bound (J s : ℕ) (ω : ℝ) : ℝ :=
+  ((2:ℝ)^(s*s))^2*((2*Real.sqrt (4*((s:ℝ)+1)/((J:ℝ)*ω^2+2)))^s)^2*
+    Real.sqrt (4*((s:ℝ)+1)/((J:ℝ)*ω^2+2))
+
+def hiddenUnsignedL2Bound (J s : ℕ) (ω : ℝ) : ℝ :=
+  ((2:ℝ)^s)^2*hiddenBinomialCap J ω
+
+theorem hiddenSignedL2Bound_nonneg (J s : ℕ) (ω : ℝ) : 0≤hiddenSignedL2Bound J s ω := by
+  unfold hiddenSignedL2Bound; positivity
+
+theorem hiddenUnsignedL2Bound_nonneg (J s : ℕ) (ω : ℝ) : 0≤hiddenUnsignedL2Bound J s ω := by
+  exact mul_nonneg (sq_nonneg _) (hiddenBinomialCap_nonneg _ _)
+
+theorem hidden_quotient_bound (J G s : ℕ) (ω : ℝ) (h : (J:ℝ)*ω^2/2≤G) :
+    binomialBound (G/(s+1))≤Real.sqrt (4*((s:ℝ)+1)/((J:ℝ)*ω^2+2)) := by
+  apply (binomialBound_quotient_le G s).trans
+  apply Real.sqrt_le_sqrt
+  apply (div_le_div_iff₀ (by positivity : (0:ℝ)<(G:ℝ)+1)
+    (by positivity : (0:ℝ)<(J:ℝ)*ω^2+2)).mpr
+  have hh := mul_le_mul_of_nonneg_left h (by positivity : (0:ℝ)≤4*((s:ℝ)+1))
+  nlinarith
+
+theorem posteriorAnchorSeries_signed_L2 {J : ℕ} (d : Fin J→Option Bool) (c : ℤ) (s : ℕ) (ω : ℝ)
+    (h : (J:ℝ)*ω^2/2≤∑i,(hiddenTrial (d i):ℝ)) :
+    seriesL2Sq (prouhetSeries s*posteriorAnchorSeries c d)≤hiddenSignedL2Bound J s ω := by
+  have hn : (J:ℝ)*ω^2/2≤((∑i,hiddenTrial (d i):ℕ):ℝ) := by simpa only [Nat.cast_sum] using h
+  have hb := hidden_quotient_bound J (∑i,hiddenTrial (d i)) s ω hn
+  apply (translated_binomial_smoothing (∑i,hiddenTrial (d i)) s _).trans
+  unfold hiddenSignedL2Bound
+  rw [mul_assoc]
+  apply mul_le_mul_of_nonneg_left _ (by positivity)
+  apply mul_le_mul
+  · apply pow_le_pow_left₀ (by unfold binomialBound; positivity)
+    exact pow_le_pow_left₀ (by unfold binomialBound; positivity) (mul_le_mul_of_nonneg_left hb (by norm_num)) _
+  · exact hb
+  · exact Real.sqrt_nonneg _
+  · positivity
+
+theorem posteriorAnchorSeries_unsigned_L2 {J : ℕ} (d : Fin J→Option Bool) (c : ℤ) (s : ℕ) (ω : ℝ)
+    (h : (J:ℝ)*ω^2/2≤∑i,(hiddenTrial (d i):ℝ)) :
+    seriesL2Sq (geomSeries (2^s)*posteriorAnchorSeries c d)≤hiddenUnsignedL2Bound J s ω := by
+  have hn : (J:ℝ)*ω^2/2≤((∑i,hiddenTrial (d i):ℕ):ℝ) := by simpa only [Nat.cast_sum] using h
+  calc
+    _ ≤ seriesL1 (geomSeries (2^s))^2*seriesL2Sq (posteriorAnchorSeries c d) := seriesL2Sq_mul_le _ _
+    _ ≤ ((2:ℝ)^s)^2*hiddenBinomialCap J ω := by
+      apply mul_le_mul
+      · exact pow_le_pow_left₀ (seriesL1_nonneg _) (by exact_mod_cast geomSeries_L1_le (2^s)) _
+      · rw [posteriorAnchorSeries,translateSeries_L2Sq]
+        exact (fairSeries_L2Sq_bound _).trans (hidden_binomial_bound J _ ω hn)
+      · exact seriesL2Sq_nonneg _
+      · positivity
+
+end TraceReconstruction
+
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+def initialDifferenceCap (d : ℕ) (q : ℝ) : ℝ :=
+  2*Real.sqrt (8*((d:ℝ)+1)/q^2)
+
+theorem initial_theta_variance {q θ : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
+    (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4)) :
+    θ∈Set.Ioo 0 1 ∧ 1/2≤θ ∧ q^2/8≤θ*(1-θ) := by
+  have hq2 : q^2≤1/4 := by nlinarith [hq.1,hq.2]
+  have hs : 0<q^2 := sq_pos_of_pos hq.1
+  have hh : 1/2≤θ := by linarith [hθ.1]
+  refine ⟨⟨by linarith,by linarith [hθ.2]⟩,hh,?_⟩
+  have H := mul_le_mul hh (show q^2/4≤1-θ by linarith [hθ.2])
+    (by positivity : (0:ℝ)≤q^2/4) (by linarith : 0≤θ)
+  nlinarith
+
+theorem initialDifferenceCap_bound (d : ℕ) {q θ : ℝ}
+    (hq : q∈Set.Ioc 0 (1/2:ℝ)) (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4)) :
+    lengthDifferenceConstant d θ≤ initialDifferenceCap d q := by
+  have ht := initial_theta_variance hq hθ
+  unfold lengthDifferenceConstant initialDifferenceCap
+  apply mul_le_mul_of_nonneg_left _ (by norm_num)
+  apply Real.sqrt_le_sqrt
+  apply (div_le_div_iff₀ (mul_pos ht.1.1 (sub_pos.mpr ht.1.2)) (sq_pos_of_pos hq.1)).mpr
+  have H := mul_le_mul_of_nonneg_left ht.2.2 (by positivity : (0:ℝ)≤8*((d:ℝ)+1))
+  nlinarith
+
+/-- All q dependence of a fixed rank-difference estimate is exposed here.
+ The powers of u and m are retained exactly, including both large-gap gains. -/
+def initialPrefactorCap (u m r d : ℕ) (q : ℝ) : ℝ :=
+  144*((u*m:ℕ):ℝ)*(2:ℝ)^(2*r)*(initialDifferenceCap d q)^(2*d+r+2) /
+    (((m:ℝ)/4)^d*Real.sqrt ((u:ℝ)*m/(2*(r+1:ℝ)))*
+      (Real.sqrt ((m:ℝ)/4))^(r+1))
+
+theorem initialPrefactorCap_nonneg (u m r d : ℕ) (q : ℝ) : 0≤ initialPrefactorCap u m r d q := by
+  unfold initialPrefactorCap initialDifferenceCap
+  positivity
+
+theorem initial_prefactor_bound (u m r d : ℕ) {q θ : ℝ}
+    (hq : q∈Set.Ioc 0 (1/2:ℝ)) (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4)) :
+    periodDifferencePrefactor u m r d θ≤ initialPrefactorCap u m r d q := by
+  unfold periodDifferencePrefactor initialPrefactorCap
+  apply div_le_div_of_nonneg_right _ (by positivity)
+  apply mul_le_mul_of_nonneg_left _ (by positivity)
+  exact pow_le_pow_left₀ (by unfold lengthDifferenceConstant; positivity) (initialDifferenceCap_bound d hq hθ) _
+
+theorem binomial_mean_inverse_le_sqrt {n l : ℕ} {θ : ℝ}
+    (hθ : θ∈Set.Ioo 0 1) (hl : l≤n) (hmean : (n:ℝ)*θ=l)
+    (hv : 1≤(n:ℝ)*θ*(1-θ)) :
+    (binomialMass n θ l)⁻¹≤12*Real.sqrt (n:ℝ) := by
+  have h0 : 0≤(binomialMass n θ l)⁻¹ := inv_nonneg.mpr (binomialMass_nonneg n l ⟨hθ.1.le,hθ.2.le⟩)
+  have hi := binomialMass_mean_inverse_sq n l hθ hl hmean hv
+  have hv' : (n:ℝ)*θ*(1-θ)≤n := by
+    have h := mul_le_mul_of_nonneg_left (show θ*(1-θ)≤1 by nlinarith [sq_nonneg θ]) (Nat.cast_nonneg (α:=ℝ) n)
+    nlinarith
+  have hs := Real.sq_sqrt (Nat.cast_nonneg (α:=ℝ) n)
+  nlinarith [Real.sqrt_nonneg (n:ℝ)]
+
+def initialRankTailCap (u m r : ℕ) : ℝ :=
+  2*((Fintype.card (IncreasingTuple u r):ℝ)*
+    (48*((r+1:ℕ):ℝ)*Real.sqrt (u*m:ℕ)*Real.exp (-(m:ℝ)/(1024*u))))^2
+
+theorem initial_rank_tail_bound {u m l : ℕ} (r : ℕ) {θ : ℝ}
+    (hθ : θ∈Set.Ioo 0 1) (hl : l≤u*m) (hmean : ((u*m:ℕ):ℝ)*θ=l)
+    (hv : 1≤((u*m:ℕ):ℝ)*θ*(1-θ)) :
+    2*((Fintype.card (IncreasingTuple u r):ℝ)*
+      (2*(((r+1:ℕ):ℝ)*θ^r/binomialMass (u*m) θ l*(2*Real.exp (-(m:ℝ)/(1024*u))))))^2 ≤
+    initialRankTailCap u m r := by
+  have hp : θ^r≤1 := pow_le_one₀ hθ.1.le hθ.2.le
+  have hi := binomial_mean_inverse_le_sqrt hθ hl hmean hv
+  have hn : 0≤(binomialMass (u*m) θ l)⁻¹ := inv_nonneg.mpr (binomialMass_nonneg _ _ ⟨hθ.1.le,hθ.2.le⟩)
+  unfold initialRankTailCap
+  apply mul_le_mul_of_nonneg_left _ (by norm_num)
+  have hb := binomialMass_nonneg (u*m) l ⟨hθ.1.le,hθ.2.le⟩
+  apply pow_le_pow_left₀ (by
+    exact mul_nonneg (Nat.cast_nonneg _) (mul_nonneg (by norm_num)
+      (mul_nonneg (div_nonneg (mul_nonneg (Nat.cast_nonneg _) (pow_nonneg hθ.1.le _)) hb)
+        (by positivity))))
+  apply mul_le_mul_of_nonneg_left _ (by positivity)
+  have H := mul_le_mul hp hi hn (by norm_num : (0:ℝ)≤1)
+  have HH := mul_le_mul_of_nonneg_left H (by positivity : (0:ℝ)≤4*((r+1:ℕ):ℝ)*Real.exp (-(m:ℝ)/(1024*u)))
+  simp only [div_eq_mul_inv,one_mul] at *
+  nlinarith only [HH]
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- The central binomial typical interval also guarantees the rank-kernel
+ variance condition, rather than imposing it by conditioning. -/
+theorem initialCountAdmissible_of_near {N : ℕ} {q : ℝ}
+    (hq : q∈Set.Ioc 0 (1/2:ℝ)) (hscale : 8≤(N:ℝ)*q^2) (l : Fin (N+1))
+    (hnear : |(l.val:ℝ)-(N:ℝ)*(1-q^2/2)|<(N:ℝ)*q^2/4) :
+    initialCountAdmissible N q l := by
+  have hN : (0:ℝ)<N := by
+    have h0 : (0:ℝ)≤N := Nat.cast_nonneg N
+    by_contra H
+    have hn : (N:ℝ)=0 := le_antisymm (le_of_not_gt H) h0
+    rw [hn,zero_mul] at hscale
+    norm_num at hscale
+  have hq2 : q^2≤1/4 := by nlinarith [hq.1,hq.2]
+  let θ : ℝ := (l.val:ℝ)/(N:ℝ)
+  have he : (N:ℝ)*θ=l.val := mul_div_cancel₀ _ hN.ne'
+  have hθ : θ∈Set.Icc (1-q^2) (1-q^2/4) := by
+    have hn := abs_lt.mp hnear
+    constructor
+    · apply le_of_lt
+      apply (mul_lt_mul_iff_right₀ hN).mp
+      nlinarith [sq_nonneg q]
+    · apply le_of_lt
+      apply (mul_lt_mul_iff_right₀ hN).mp
+      nlinarith
+  have hhalf : 3/4≤θ := by linarith [hθ.1]
+  have hlow : q^2/4≤1-θ := by linarith [hθ.2]
+  have hp := mul_le_mul ((mul_le_mul_iff_right₀ hN).mpr hhalf) hlow
+    (by positivity : (0:ℝ)≤q^2/4) (mul_nonneg hN.le (by linarith : 0≤θ))
+  refine ⟨hθ,?_⟩
+  change 1≤(N:ℝ)*θ*(1-θ)
+  nlinarith
+
+/-- An explicit unconditional tail for the actual count law. Its q⁴ exponent
+ is more than enough under the source's q⁻¹=N^{o(1)} standing regime. -/
+theorem initial_count_bad_mass {N : ℕ} {q : ℝ}
+    (hq : q∈Set.Ioc 0 (1/2:ℝ)) (hscale : 8≤(N:ℝ)*q^2) :
+    (boundedCountLaw (le_refl N) (1-primitiveErasure q)
+      (primitiveRetention_mem ⟨hq.1,by linarith [hq.2]⟩)).expect
+      (fun l => if initialCountAdmissible N q l then 0 else 1) ≤
+    2*Real.exp (-(N:ℝ)*q^4/64) := by
+  let p := 1-primitiveErasure q
+  have hp : p∈Set.Icc 0 1 := primitiveRetention_mem ⟨hq.1,by linarith [hq.2]⟩
+  have ht : q^2/8∈Set.Icc (0:ℝ) 1 := ⟨by positivity,by nlinarith [hq.1,hq.2]⟩
+  have H := binomialMass_abs_tail N (a:=(N:ℝ)*q^2/4) hp ht
+  have he : -(q^2/8)*((N:ℝ)*q^2/4)+(N:ℝ)*(q^2/8)^2= -(N:ℝ)*q^4/64 := by ring
+  rw [he] at H
+  apply le_trans _ H
+  rw [Finset.sum_filter,←Fin.sum_univ_eq_sum_range]
+  apply Finset.sum_le_sum
+  intro l _
+  change binomialMass N p l.val*(if initialCountAdmissible N q l then 0 else 1)≤_
+  have hn := binomialMass_nonneg N l.val hp
+  by_cases hl : initialCountAdmissible N q l
+  · rw [ite_eq_left hl,mul_zero]
+    split_ifs <;> positivity
+  · rw [ite_eq_right hl,mul_one,ite_eq_left]
+    exact le_of_not_gt (fun h => hl (initialCountAdmissible_of_near hq hscale l (by simpa [p,primitiveErasure] using h)))
+
+/-- The true bad metadata mass is charged without changing any conditional
+ law or requiring an unproved concentration conclusion. -/
+theorem initial_metadata_bad_mass {Side : Type*} [Fintype Side] {J N : ℕ}
+    (M : PairedMechanisms Side J) {q ω : ℝ}
+    (hq : q∈Set.Ioc 0 (1/2:ℝ)) (hscale : 8≤(N:ℝ)*q^2)
+    (hw : ∀i,M.hiddenWeight i=ω) (hp : ω^2∈Set.Icc 0 1) :
+    (initialMetadataLaw M N ⟨hq.1,by linarith [hq.2]⟩).expect
+      (fun z => if initialMetadataGood q ω z then 0 else 1) ≤
+    2*Real.exp (-(N:ℝ)*q^4/64)+Real.exp (-(J:ℝ)*ω^4/16) := by
+  calc
+    _ ≤ (initialMetadataLaw M N ⟨hq.1,by linarith [hq.2]⟩).expect
+        (fun z => (if initialCountAdmissible N q z.2 then 0 else 1)+
+          (if (PairedMechanisms.hiddenCount z.1:ℝ)<(J:ℝ)*ω^2/2 then 1 else 0)) := by
+      apply FiniteLaw.expect_mono
+      intro z
+      have he : (PairedMechanisms.hiddenCount z.1:ℝ)=∑i,(hiddenTrial (z.1 i).1:ℝ) := by
+        simp only [PairedMechanisms.hiddenCount,Nat.cast_sum]
+      unfold initialMetadataGood
+      rw [←he]
+      split_ifs <;> simp_all
+    _ ≤ _ := by
+      simp only [FiniteLaw.expect_add,initialMetadataLaw,FiniteLaw.expect_prod]
+      apply add_le_add
+      · exact (M.reports.expect_mono (fun _ => initial_count_bad_mass hq hscale)).trans (by rw [FiniteLaw.expect_const])
+      · convert M.hiddenCount_small hw hp using 1
+        congr 1
+        funext a
+        split_ifs <;> simp [FiniteLaw.expect,FiniteLaw.total]
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+def initialSignedPatternCap (J u m r s : ℕ) (q ω : ℝ) (k : Fin r → GapDifference) : ℝ :=
+  if ∃a,k=linearPattern a then
+    initialPrefactorCap u m r 1 q*periodCentralSumConstant u r*hiddenSignedL2Bound J s ω
+  else initialPrefactorCap u m r (∑h,(k h).order) q*
+    periodBoundedSumConstant u r (2*(2^s))*hiddenUnsignedL2Bound J s ω
+
+def initialPurePatternCap (J u m r s : ℕ) (q ω : ℝ) (k : Fin r → GapDifference) : ℝ :=
+  if ∃a,k=linearPattern a then
+    initialPrefactorCap u m r 1 q*periodCentralSumConstant u r*hiddenSignedL2Bound J s ω
+  else initialPrefactorCap u m r (∑h,(k h).order) q*
+    periodCentralSumConstant u r*hiddenUnsignedL2Bound J s ω
+
+theorem initial_signed_pattern_cap {J : ℕ} (d : Fin J → Option Bool) (c : ℤ)
+    (u m r s : ℕ) (k : Fin r → GapDifference) {q θ ω : ℝ}
+    (hq : q∈Set.Ioc 0 (1/2:ℝ)) (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
+    (hh : (J:ℝ)*ω^2/2≤∑i,(hiddenTrial (d i):ℝ)) :
+    signedPatternBound u m r s (posteriorAnchorSeries c d) θ k ≤
+      initialSignedPatternCap J u m r s q ω k := by
+  unfold signedPatternBound initialSignedPatternCap
+  split_ifs
+  · apply mul_le_mul
+    · exact mul_le_mul_of_nonneg_right (initial_prefactor_bound u m r 1 hq hθ)
+        (by unfold periodCentralSumConstant; positivity)
+    · exact posteriorAnchorSeries_signed_L2 d c s ω hh
+    · exact seriesL2Sq_nonneg _
+    · exact mul_nonneg (initialPrefactorCap_nonneg _ _ _ _ _) (by unfold periodCentralSumConstant; positivity)
+  · apply mul_le_mul
+    · exact mul_le_mul_of_nonneg_right (initial_prefactor_bound u m r _ hq hθ)
+        (by unfold periodBoundedSumConstant; positivity)
+    · exact posteriorAnchorSeries_unsigned_L2 d c s ω hh
+    · exact seriesL2Sq_nonneg _
+    · exact mul_nonneg (initialPrefactorCap_nonneg _ _ _ _ _) (by unfold periodBoundedSumConstant; positivity)
+
+theorem initial_pure_pattern_cap {J : ℕ} (d : Fin J → Option Bool) (c : ℤ)
+    (u m r s : ℕ) (k : Fin r → GapDifference) {q θ ω : ℝ}
+    (hq : q∈Set.Ioc 0 (1/2:ℝ)) (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
+    (hh : (J:ℝ)*ω^2/2≤∑i,(hiddenTrial (d i):ℝ)) :
+    purePatternBound u m r s (posteriorAnchorSeries c d) θ k ≤
+      initialPurePatternCap J u m r s q ω k := by
+  unfold purePatternBound initialPurePatternCap
+  split_ifs
+  · apply mul_le_mul
+    · exact mul_le_mul_of_nonneg_right (initial_prefactor_bound u m r 1 hq hθ)
+        (by unfold periodCentralSumConstant; positivity)
+    · exact posteriorAnchorSeries_signed_L2 d c s ω hh
+    · exact seriesL2Sq_nonneg _
+    · exact mul_nonneg (initialPrefactorCap_nonneg _ _ _ _ _) (by unfold periodCentralSumConstant; positivity)
+  · apply mul_le_mul
+    · exact mul_le_mul_of_nonneg_right (initial_prefactor_bound u m r _ hq hθ)
+        (by unfold periodCentralSumConstant; positivity)
+    · exact posteriorAnchorSeries_unsigned_L2 d c s ω hh
+    · exact seriesL2Sq_nonneg _
+    · exact mul_nonneg (initialPrefactorCap_nonneg _ _ _ _ _) (by unfold periodCentralSumConstant; positivity)
+
+def initialSignedDegreeCap (J u m r s : ℕ) (q ω : ℝ) : ℝ :=
+  if r=0 then 0 else
+    8*(Fintype.card (Fin r → GapDifference):ℝ)*
+      (∑k : Fin r → GapDifference,if Odd (firstDifferenceCount k) then
+        initialSignedPatternCap J u m r s q ω k else 0)+initialRankTailCap u m r
+
+def initialPureDegreeCap (J u m r s : ℕ) (q ω : ℝ) : ℝ :=
+  if r=0 then 0 else
+    2*(Fintype.card (Fin r → GapDifference):ℝ)*
+      (∑k : Fin r → GapDifference,if k=unchangedPattern r then 0 else
+        initialPurePatternCap J u m r s q ω k)+initialRankTailCap u m r
+
+theorem initial_signed_degree_cap {J u m l : ℕ} (d : Fin J → Option Bool) (c : ℤ) (r s : ℕ)
+    {q θ ω : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ)) (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
+    (hl : l≤u*m) (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1≤((u*m:ℕ):ℝ)*θ*(1-θ))
+    (hh : (J:ℝ)*ω^2/2≤∑i,(hiddenTrial (d i):ℝ)) :
+    signedInitialDegreeBound u m r s (posteriorAnchorSeries c d) θ l ≤
+      initialSignedDegreeCap J u m r s q ω := by
+  unfold signedInitialDegreeBound initialSignedDegreeCap
+  split_ifs
+  · exact le_rfl
+  · apply add_le_add
+    · apply mul_le_mul_of_nonneg_left _ (by positivity)
+      apply Finset.sum_le_sum
+      intro k _
+      split_ifs
+      · exact initial_signed_pattern_cap d c u m r s k hq hθ hh
+      · exact le_rfl
+    · exact initial_rank_tail_bound r (initial_theta_variance hq hθ).1 hl hmean hv
+
+theorem initial_pure_degree_cap {J u m l : ℕ} (d : Fin J → Option Bool) (c : ℤ) (r s : ℕ)
+    {q θ ω : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ)) (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
+    (hl : l≤u*m) (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1≤((u*m:ℕ):ℝ)*θ*(1-θ))
+    (hh : (J:ℝ)*ω^2/2≤∑i,(hiddenTrial (d i):ℝ)) :
+    initialPureDegreeBound u m r s (posteriorAnchorSeries c d) θ l ≤
+      initialPureDegreeCap J u m r s q ω := by
+  unfold initialPureDegreeBound initialPureDegreeCap
+  split_ifs
+  · exact le_rfl
+  · apply add_le_add
+    · apply mul_le_mul_of_nonneg_left _ (by positivity)
+      apply Finset.sum_le_sum
+      intro k _
+      split_ifs
+      · exact le_rfl
+      · exact initial_pure_pattern_cap d c u m r s k hq hθ hh
+    · exact initial_rank_tail_bound r (initial_theta_variance hq hθ).1 hl hmean hv
+
+/-- Scalar budgets independent of every actual report and count. The finite
+ signed-pattern restriction is retained to preserve its three-difference gain. -/
+def initialSignedCap (J u m R s : ℕ) (q ω w : ℝ) : ℝ :=
+  (∑r∈Finset.range R,w^(2*r)*initialSignedDegreeCap J u m r s q ω)+
+    4*(initialMomentRatio u m q w)^R/(1-initialMomentRatio u m q w)
+
+def initialPureCap (J u m R s : ℕ) (q ω w : ℝ) : ℝ :=
+  (∑r∈Finset.range R,w^(2*r)*initialPureDegreeCap J u m r s q ω)+
+    4*(initialMomentRatio u m q w)^R/(1-initialMomentRatio u m q w)
+
+theorem initial_signed_budget_cap {J u m l : ℕ} (d : Fin J → Option Bool) (c : ℤ) (R s : ℕ)
+    {q θ ω w : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ)) (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
+    (hl : l≤u*m) (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1≤((u*m:ℕ):ℝ)*θ*(1-θ))
+    (hh : (J:ℝ)*ω^2/2≤∑i,(hiddenTrial (d i):ℝ)) :
+    initialSignedBudget u m l R s (posteriorAnchorSeries c d) θ q w ≤
+      initialSignedCap J u m R s q ω w := by
+  unfold initialSignedBudget initialSignedCap
+  apply add_le_add _ le_rfl
+  apply Finset.sum_le_sum
+  intro r _
+  exact mul_le_mul_of_nonneg_left (initial_signed_degree_cap d c r s hq hθ hl hmean hv hh)
+    (by rw [pow_mul]; positivity)
+
+theorem initial_pure_budget_cap {J u m l : ℕ} (d : Fin J → Option Bool) (c : ℤ) (R s : ℕ)
+    {q θ ω w : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ)) (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
+    (hl : l≤u*m) (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1≤((u*m:ℕ):ℝ)*θ*(1-θ))
+    (hh : (J:ℝ)*ω^2/2≤∑i,(hiddenTrial (d i):ℝ)) :
+    initialPureBudget u m l R s (posteriorAnchorSeries c d) θ q w ≤
+      initialPureCap J u m R s q ω w := by
+  unfold initialPureBudget initialPureCap
+  apply add_le_add _ le_rfl
+  apply Finset.sum_le_sum
+  intro r _
+  exact mul_le_mul_of_nonneg_left (initial_pure_degree_cap d c r s hq hθ hl hmean hv hh)
+    (by rw [pow_mul]; positivity)
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+def OracleTracePair.weaken {q E V E' V' : ℝ} {x y : List Bool} {k : ℕ}
+    (C : OracleTracePair q x y k E V) (hE : E ≤ E') (hV : V ≤ V') :
+    OracleTracePair q x y k E' V' :=
+  { C with
+    first_simulation := fun F hF => (C.first_simulation F hF).trans hE
+    second_simulation := fun F hF => (C.second_simulation F hF).trans hE
+    moment := C.moment.trans hV }
+
+def initialErrorCap (J K m k : ℕ) (q ω w : ℝ) : ℝ :=
+  Real.sqrt (6*(4*(initialMomentRatio K m q w/(1-initialMomentRatio K m q w)))^k)+
+    (2*Real.exp (-((K*m:ℕ):ℝ)*q^4/64)+Real.exp (-(J:ℝ)*ω^4/16))
+
+/-- Both required initial comparisons are genuine approximate complete-trace
+ oracles. The metadata budgets have been eliminated: only explicit finite
+ scalar bounds and literal period geometry remain to be instantiated. -/
+theorem actual_initial_oracles {J K m R : ℕ}
+    (hK : 0 < K) (hm : 128 ≤ m) (hKm : K ≤ m) (hR : 0 < R)
+    (off : Fin m) (hi : 1 ≤ off.val ∧ off.val+1 < m)
+    (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 2)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=K) (fun _ => off)))
+    (a : ℤ) (s k : ℕ) (hk : 1 ≤ k) {q w V W : ℝ}
+    (hq : q∈Set.Ioc 0 (1/2:ℝ))
+    (hseed : 16 ≤ q^3*(m:ℝ)) (hcount : 8 ≤ ((K*m:ℕ):ℝ)*q^2)
+    (ha : (J*initialSideCount q m+J:ℕ) ≤ a)
+    (ha' : a+(2^s:ℕ) ≤ (J*initialSideCount q m+K:ℕ))
+    (hleft : (K:ℝ)/3 ≤ 2*((a-(J*initialSideCount q m:ℕ):ℤ)-J)+1)
+    (hright : (K:ℝ)/3 ≤ 2*K-(2*((a-(J*initialSideCount q m:ℕ):ℤ)+(2:ℝ)^s)+1))
+    (hM : ((2*(2^s):ℕ):ℝ) < (K:ℝ)/(3*(R+1:ℝ)))
+    (hw : w=((6*(2*k:ℕ):ℝ)*(2/q)))
+    (hscale : initialMomentRatio K m q w ≤ 1/2)
+    (hclip : Real.sqrt (6*(4*(initialMomentRatio K m q w/(1-initialMomentRatio K m q w)))^k) ≤ 1/2)
+    (hV : 0 ≤ V) (hW : 0 ≤ W)
+    (hSV : 64*initialSignedCap J K m R s q (initialOverlapLower q) w ≤ V)
+    (hPW : 64*initialPureCap J K m R s q (initialOverlapLower q) w ≤ W) :
+    Nonempty (OracleTracePair q
+      (motifPeriodWord off hi (J*(2*initialSideCount q m+1)+K) s a (-1))
+      (motifPeriodWord off hi (J*(2*initialSideCount q m+1)+K) s a 1) k
+      (initialErrorCap J K m k q (initialOverlapLower q) w) V) ∧
+    Nonempty (OracleTracePair q
+      (pureAtomRun (initialPeriod m off.val) (J*(2*initialSideCount q m+1)+K))
+      (motifPeriodWord off hi (J*(2*initialSideCount q m+1)+K) s a 1) k
+      (initialErrorCap J K m k q (initialOverlapLower q) w) W) := by
+  let t := initialSideCount q m
+  let ω := initialOverlapLower q
+  let c : ℤ := a-(J*t:ℕ)
+  have hω : 0 < ω := initialOverlapLower_pos q
+  have hω1 : ω < 1/8 := initialOverlapLower_lt_eighth hq.1
+  have hqi : q∈Set.Ioo 0 1 := ⟨hq.1,by linarith [hq.2]⟩
+  have hqc : q∈Set.Icc 0 1 := ⟨hq.1.le,hqi.2.le⟩
+  let u := pureAtomRun (initialPeriod m off.val) t
+  let v := pureAtomRun (initialPeriod m off.val) (t+1)
+  let N := max u.length v.length
+  have hu : u.length ≤ N := le_max_left _ _
+  have hv : v.length ≤ N := le_max_right _ _
+  have hov : ω ≤ listTraceOverlap q hqc u v := initial_pure_trace_overlap hq (by omega)
+    hseed off (by linarith)
+  let M : PairedMechanisms (BoundedTrace N) J := tracePairedMechanisms hqc hω
+    (fun _ => u) (fun _ => v) N (fun _ => hu) (fun _ => hv) (fun _ => hov)
+  have hfirst (i : Fin J) : SimulatesTrace q (M.first i) Subtype.val u :=
+    listTraceLaw_simulates q hqc u N hu
+  have hsecond (i : Fin J) : SimulatesTrace q (M.second i) Subtype.val v :=
+    listTraceLaw_simulates q hqc v N hv
+  have htail := initial_metadata_bad_mass M hq hcount (fun _ => rfl)
+    (show ω^2∈Set.Icc (0:ℝ) 1 from ⟨sq_nonneg _,by nlinarith⟩)
+  have hsigned (z : InitialMetadata (BoundedTrace N) J (K*m))
+      (hz : initialMetadataGood q ω z) :
+      64*initialSignedBudget K m z.2.val R s (posteriorAnchorSeries c (fun i => (z.1 i).1))
+        ((z.2.val:ℝ)/(K*m:ℕ)) q w ≤ V := by
+    apply le_trans _ hSV
+    apply mul_le_mul_of_nonneg_left _ (by norm_num)
+    exact initial_signed_budget_cap _ c R s hq hz.1.1 (by have H:=z.2.isLt; omega)
+      (mul_div_cancel₀ _ (by exact_mod_cast (Nat.mul_ne_zero (by omega : K≠0) (by omega : m≠0))))
+      hz.1.2 hz.2
+  have hpure (z : InitialMetadata (BoundedTrace N) J (K*m))
+      (hz : initialMetadataGood q ω z) :
+      64*initialPureBudget K m z.2.val R s (posteriorAnchorSeries c (fun i => (z.1 i).1))
+        ((z.2.val:ℝ)/(K*m:ℕ)) q w ≤ W := by
+    apply le_trans _ hPW
+    apply mul_le_mul_of_nonneg_left _ (by norm_num)
+    exact initial_pure_budget_cap _ c R s hq hz.1.1 (by have H:=z.2.isLt; omega)
+      (mul_div_cancel₀ _ (by exact_mod_cast (Nat.mul_ne_zero (by omega : K≠0) (by omega : m≠0))))
+      hz.1.2 hz.2
+  have hsim (τ : ℝ) := initial_sigma_simulates M t s a τ off hi hsafe hqi
+    Subtype.val hfirst hsecond ha ha'
+  have hp := initial_pure_sigma_simulates (K:=K) M t off hqi Subtype.val hfirst hsecond
+  obtain ⟨S⟩ := initial_sigma_clipped_exists M hK hm hKm hR off hi hoff hsafe c s k hk
+    hleft hright hq hM hw hscale hclip hV hsigned
+  obtain ⟨P⟩ := initial_pure_sigma_clipped_exists M hK hm hKm hR off hi hoff hsafe c s k hk
+    hleft hright hq hw hscale hclip hW hpure
+  constructor
+  · obtain ⟨O⟩ := S.tracePair (decodeInitialObservation Subtype.val) (hsim (-1)) (hsim 1)
+    exact ⟨O.weaken (add_le_add le_rfl htail) le_rfl⟩
+  · obtain ⟨O⟩ := P.tracePair (decodeInitialObservation Subtype.val) hp (hsim 1)
+    exact ⟨O.weaken (add_le_add le_rfl htail) le_rfl⟩
+
+end TraceReconstruction
+
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/TraceReconstruction/InitialOverlap.lean b/lean/OAI/Probability/TraceReconstruction/InitialOverlap.lean
index a2ffa863fa275b9cc8e83b324a527127fc5ef3db..ff1ee5a198893e23960fa5261604fbb0bf2cdb4a 100644
--- a/lean/OAI/Probability/TraceReconstruction/InitialOverlap.lean
+++ b/lean/OAI/Probability/TraceReconstruction/InitialOverlap.lean
@@ -1,1186 +1,1186 @@
-import OAI.Probability.TraceReconstruction.Oracles
-
-namespace OAI
-
-/-!
-# InitialOverlap
-
-Literal periods, count overlap, initial low/high energies, genuine densities, and centered moment bounds.
--/
-
-open scoped ENNReal
-noncomputable section
-
-open Filter
-open scoped Topology
-
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem primitive_one_add_score_nonneg {q : ℝ} (hq : q ∈ Set.Ioo 0 1)
-    (a : PrimitiveMark) : 0 ≤ 1 + primitiveScore q a := by
-  have hq1 : 1 ≤ 1/q := (le_div_iff₀ hq.1).mpr (by simpa using hq.2.le)
-  have hq0 := hq.1
-  fin_cases a <;> norm_num [primitiveScore] <;> linarith [show 2/q=2*(1/q) by ring]
-
-/-- Exact conditional law of the genuine periodic primitive observation. -/
-theorem conditionalPeriodLaw_density {u m l : ℕ} (hl : l ≤ u*m)
-    {q : ℝ} (hq : q ∈ Set.Ioo 0 1) (off : Fin u → Fin m) (x : Fin l → PrimitiveMark) :
-    (conditionalPrimitiveLaw hl q hq (defectTypes (periodSlot off))).mass x =
-      ((primitiveMarkLaw q hq false).product l).mass x *
-        averagedDensity (uniformSubsetLaw l (by simpa using hl))
-          (defectAtRank (Finset.univ.image (periodSlot off))) (primitiveScore q) x := by
-  rw [conditionalPrimitiveLaw_density]
-  congr 2
-  funext S
-  exact defectTypes_atRank (periodSlot off) S
-
-/-- Absolute constant in the conditional second-moment exponent. -/
-def primitivePeriodConstant : ℝ := 36*rankSchurConstant^2
-
-theorem primitivePeriodConstant_nonneg : 0 ≤ primitivePeriodConstant := by
-  unfold primitivePeriodConstant
-  positivity
-
-theorem conditionalPeriod_second_moment {u m l : ℕ} (hl : l ≤ u*m)
-    (hm : 32 ≤ m) (hum : u ≤ m) {q θ : ℝ}
-    (hq : q ∈ Set.Ioc 0 (1/2:ℝ))
-    (hθ : θ ∈ Set.Icc (1-q^2) (1-q^2/4))
-    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
-    (off : Fin u → Fin m)
-    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
-    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2) :
-    ((primitiveMarkLaw q ⟨hq.1, by linarith [hq.2]⟩ false).product l).expect (fun x =>
-      averagedDensity (uniformSubsetLaw l (by simpa using hl))
-        (defectAtRank (Finset.univ.image (periodSlot off))) (primitiveScore q) x ^ 2) ≤
-      2*Real.exp (primitivePeriodConstant/q^4 * ((u:ℝ)/m)) := by
-  have hqi : q ∈ Set.Ioo 0 1 := ⟨hq.1, by linarith [hq.2]⟩
-  let P := primitiveMarkLaw q hqi false
-  have hZ : P.expect (primitiveScore q) = 0 := primitive_score_centered hqi
-  have h := period_density_second_moment P (primitiveScore q) hZ hl hm hum hq hθ hmean hv off hoff hdiff
-  refine h.trans (mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr ?_) (by norm_num))
-  have hv0 : 0 ≤ P.expect (fun a => primitiveScore q a ^ 2) :=
-    Finset.sum_nonneg (fun a _ => mul_nonneg (P.nonneg a) (sq_nonneg _))
-  have hv1 : P.expect (fun a => primitiveScore q a ^ 2) ≤ 2/q :=
-    primitive_score_variance_le hqi
-  have hsq : (P.expect (fun a => primitiveScore q a ^ 2))^2 ≤ (2/q)^2 :=
-    pow_le_pow_left₀ hv0 hv1 2
-  calc
-    _ ≤ (2/q)^2*(3*rankSchurConstant/q)^2*((u:ℝ)/m) :=
-      mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hsq (sq_nonneg _)) (by positivity)
-    _ = _ := by unfold primitivePeriodConstant; field_simp; ring
-
-/-- Association converts the two actual conditional period-laws into common
-mass. The two original block totals may differ; only the retained count agrees. -/
-theorem conditionalPeriod_overlap {u v m l : ℕ}
-    (hl : l ≤ u*m) (hl' : l ≤ v*m) (hm : 32 ≤ m) (hum : u ≤ m) (hvm : v ≤ m)
-    {q θ θ' : ℝ} (hq : q ∈ Set.Ioc 0 (1/2:ℝ))
-    (hθ : θ ∈ Set.Icc (1-q^2) (1-q^2/4))
-    (hθ' : θ' ∈ Set.Icc (1-q^2) (1-q^2/4))
-    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hmean' : ((v*m:ℕ):ℝ)*θ'=l)
-    (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ)) (hv' : 1 ≤ ((v*m:ℕ):ℝ)*θ'*(1-θ'))
-    (off : Fin u → Fin m) (off' : Fin v → Fin m)
-    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
-    (hoff' : ∀ j, |((off' j).val:ℝ)-(m:ℝ)/2| ≤ 3)
-    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2)
-    (hdiff' : ∀ j k, |((off' j).val:ℝ)-(off' k).val| ≤ 2) :
-    Real.exp (-(primitivePeriodConstant/q^4*((u:ℝ)/m+(v:ℝ)/m)))/128 ≤
-      FiniteLaw.overlap
-        (conditionalPrimitiveLaw hl q ⟨hq.1, by linarith [hq.2]⟩ (defectTypes (periodSlot off)))
-        (conditionalPrimitiveLaw hl' q ⟨hq.1, by linarith [hq.2]⟩ (defectTypes (periodSlot off'))) := by
-  have hqi : q ∈ Set.Ioo 0 1 := ⟨hq.1, by linarith [hq.2]⟩
-  let P := primitiveMarkLaw q hqi false
-  let R := uniformSubsetLaw l (show l≤Fintype.card (Fin (u*m)) by simpa using hl)
-  let S := uniformSubsetLaw l (show l≤Fintype.card (Fin (v*m)) by simpa using hl')
-  let f := defectAtRank (l:=l) (Finset.univ.image (periodSlot off))
-  let g := defectAtRank (l:=l) (Finset.univ.image (periodSlot off'))
-  let G := averagedDensity R f (primitiveScore q)
-  let H := averagedDensity S g (primitiveScore q)
-  let A := primitivePeriodConstant/q^4*((u:ℝ)/m)
-  let B := primitivePeriodConstant/q^4*((v:ℝ)/m)
-  have h1 := conditionalPeriod_second_moment hl hm hum hq hθ hmean hv off hoff hdiff
-  have h2 := conditionalPeriod_second_moment hl' hm hvm hq hθ' hmean' hv' off' hoff' hdiff'
-  change (P.product l).expect (fun x => G x^2) ≤ 2*Real.exp A at h1
-  change (P.product l).expect (fun x => H x^2) ≤ 2*Real.exp B at h2
-  have ha := averagedDensity_overlap P R S (primitiveScore q)
-    (primitive_score_centered hqi) (primitive_one_add_score_nonneg hqi) f g
-  change 1/32 ≤ (P.product l).expect (fun x => min (G x) (H x)) *
-    (P.product l).expect (fun x => G x^2) * (P.product l).expect (fun x => H x^2) at ha
-  have hG : ∀x, 0 ≤ G x := averagedDensity_nonneg R f _ (primitive_one_add_score_nonneg hqi)
-  have hH : ∀x, 0 ≤ H x := averagedDensity_nonneg S g _ (primitive_one_add_score_nonneg hqi)
-  have ho : 0 ≤ (P.product l).expect (fun x => min (G x) (H x)) :=
-    Finset.sum_nonneg (fun x _ => mul_nonneg ((P.product l).nonneg _) (le_min (hG x) (hH x)))
-  have hm2 : 0 ≤ (P.product l).expect (fun x => H x^2) :=
-    Finset.sum_nonneg (fun x _ => mul_nonneg ((P.product l).nonneg _) (sq_nonneg _))
-  have hb := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left h1 ho) hm2
-  have hc := mul_le_mul_of_nonneg_left h2 (mul_nonneg ho (by positivity : 0≤2*Real.exp A))
-  have he : (P.product l).expect (fun x => min (G x) (H x)) =
-      FiniteLaw.overlap
-        (conditionalPrimitiveLaw hl q hqi (defectTypes (periodSlot off)))
-        (conditionalPrimitiveLaw hl' q hqi (defectTypes (periodSlot off'))) := by
-    unfold FiniteLaw.expect FiniteLaw.overlap
-    apply Finset.sum_congr rfl
-    intro x _
-    rw [conditionalPeriodLaw_density,conditionalPeriodLaw_density]
-    exact mul_min_of_nonneg _ _ ((P.product l).nonneg x)
-  rw [← he]
-  have hd : Real.exp (-(A+B))/128 ≤ (P.product l).expect (fun x => min (G x) (H x)) := by
-    have hx : (1:ℝ)/32 ≤ (P.product l).expect (fun x => min (G x) (H x)) * (4*Real.exp (A+B)) := by
-      calc
-        _ ≤ _ := ha.trans (hb.trans hc)
-        _ = _ := by rw [Real.exp_add]; ring
-    have hp : 0 < 4*Real.exp (A+B) := by positivity
-    calc
-      _ = ((1:ℝ)/32)/(4*Real.exp (A+B)) := by rw [Real.exp_neg]; ring
-      _ ≤ _ := (div_le_iff₀ hp).2 hx
-  convert hd using 1
-  congr 2
-  dsimp [A,B]
-  ring
-
-end TraceReconstruction
-
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- The common count window is admissible for every genuine period count in
-the base case. All hypotheses are elementary size/location inequalities. -/
-theorem primitive_count_window {q θ : ℝ} {m n l : ℕ}
-    (hq : q ∈ Set.Ioc 0 (1/2:ℝ)) (hm : 32 ≤ m)
-    (hscale : 16 ≤ q^3*(m:ℝ))
-    (hnlo : q*(m:ℝ)^2/2 ≤ (n:ℝ)) (hnhi : (n:ℝ) ≤ (m:ℝ)^2)
-    (hmean : (n:ℝ)*θ=l)
-    (hdev : |(l:ℝ)-(n:ℝ)*(1-q^2/2)| ≤ 2*(m:ℝ)) :
-    θ ∈ Set.Icc (1-q^2) (1-q^2/4) ∧
-    1 ≤ (n:ℝ)*θ*(1-θ) ∧
-    Real.sqrt ((n:ℝ)*θ*(1-θ)) ≤ (m:ℝ) ∧
-    ((l:ℝ)-(n:ℝ)*(1-q^2/2))^2 / ((n:ℝ)*(1-q^2/2)*(q^2/2)) ≤ 32/q^3 := by
-  have hm0 : (0:ℝ) < m := by exact_mod_cast (show 0 < m by omega)
-  have hq0 := hq.1
-  have hq2 : q^2 ≤ 1/4 := by nlinarith [hq.2]
-  have hq2p : 0 < q^2 := sq_pos_of_pos hq0
-  have hq3 : 0 < q^3 := pow_pos hq0 3
-  have hn0 : (0:ℝ) < n := lt_of_lt_of_le (by have hq0 := hq.1; positivity) hnlo
-  have hsize : 8*(m:ℝ) ≤ (n:ℝ)*q^2 := by
-    have ha := mul_le_mul_of_nonneg_right hnlo hq2p.le
-    have hb := mul_le_mul_of_nonneg_right hscale hm0.le
-    nlinarith
-  have hθ : θ ∈ Set.Icc (1-q^2) (1-q^2/4) := by
-    have hl := (abs_le.mp hdev).1
-    have hr := (abs_le.mp hdev).2
-    rw [←hmean] at hl hr
-    constructor
-    · nlinarith
-    · nlinarith
-  have ht0 : 0 < θ := by nlinarith [hθ.1]
-  have ht1 : θ < 1 := by nlinarith [hθ.2]
-  have hvlo : (n:ℝ)*q^2/8 ≤ (n:ℝ)*θ*(1-θ) := by
-    have hlow : (1:ℝ)/2 ≤ θ := by linarith [hθ.1]
-    have hhigh : q^2/4 ≤ 1-θ := by linarith [hθ.2]
-    have hp := mul_le_mul hlow hhigh (by positivity : (0:ℝ)≤q^2/4) ht0.le
-    have hp' := mul_le_mul_of_nonneg_left hp hn0.le
-    nlinarith
-  have hv : 1 ≤ (n:ℝ)*θ*(1-θ) := by
-    have hm32 : (32:ℝ) ≤ m := by exact_mod_cast hm
-    linarith
-  have hvhi : (n:ℝ)*θ*(1-θ) ≤ (m:ℝ)^2 := by
-    have hprod : θ*(1-θ) ≤ 1 := by nlinarith [sq_nonneg θ]
-    have hh := mul_le_mul_of_nonneg_left hprod hn0.le
-    nlinarith
-  have hsqrt : Real.sqrt ((n:ℝ)*θ*(1-θ)) ≤ (m:ℝ) := by
-    exact (Real.sqrt_le_iff).mpr ⟨hm0.le,hvhi⟩
-  have hp0 : 0 < 1-q^2/2 := by linarith
-  have hden0 : 0 < (n:ℝ)*(1-q^2/2)*(q^2/2) := by positivity
-  have hden : q^3*(m:ℝ)^2/8 ≤ (n:ℝ)*(1-q^2/2)*(q^2/2) := by
-    have ha := mul_le_mul_of_nonneg_right hnlo hq2p.le
-    have hb := mul_le_mul_of_nonneg_right (show (1:ℝ)/2≤1-q^2/2 by linarith)
-      (mul_nonneg hn0.le (by positivity : (0:ℝ)≤q^2/2))
-    nlinarith
-  have hdev2 : ((l:ℝ)-(n:ℝ)*(1-q^2/2))^2 ≤ 4*(m:ℝ)^2 := by
-    have h := pow_le_pow_left₀ (abs_nonneg _) hdev 2
-    rw [sq_abs] at h
-    nlinarith
-  have hquot : ((l:ℝ)-(n:ℝ)*(1-q^2/2))^2 /
-      ((n:ℝ)*(1-q^2/2)*(q^2/2)) ≤ 32/q^3 := by
-    apply (div_le_div_iff₀ hden0 hq3).mpr
-    have hh := mul_le_mul_of_nonneg_right hdev2 hq3.le
-    nlinarith
-  exact ⟨hθ,hv,hsqrt,hquot⟩
-
-/-- Genuine binomial count mass in the window, uniformly in the vanishing
-working deletion rate, with an absolute explicit exponential cost. -/
-theorem primitive_count_mass_lower {q : ℝ} {m n l : ℕ}
-    (hq : q ∈ Set.Ioc 0 (1/2:ℝ)) (hm : 32 ≤ m)
-    (hscale : 16 ≤ q^3*(m:ℝ))
-    (hnlo : q*(m:ℝ)^2/2 ≤ (n:ℝ)) (hnhi : (n:ℝ) ≤ (m:ℝ)^2)
-    (hdev : |(l:ℝ)-(n:ℝ)*(1-q^2/2)| ≤ 2*(m:ℝ)) :
-    Real.exp (-32/q^3)/(12*(m:ℝ)) ≤ binomialMass n (1-primitiveErasure q) l := by
-  have hm0 : (0:ℝ) < m := by exact_mod_cast (show 0 < m by omega)
-  have hn0 : (0:ℝ) < n := lt_of_lt_of_le (by have hq0 := hq.1; positivity) hnlo
-  let θ : ℝ := (l:ℝ)/n
-  have hmean : (n:ℝ)*θ=l := by dsimp [θ]; field_simp
-  obtain ⟨ht,hv,hsqrt,hquot⟩ := primitive_count_window hq hm hscale hnlo hnhi hmean hdev
-  have hq2 : q^2 ≤ 1/4 := by nlinarith [hq.1,hq.2]
-  have hq2p : 0 < q^2 := sq_pos_of_pos hq.1
-  have ht' : θ ∈ Set.Ioo 0 1 := ⟨by nlinarith [ht.1], by nlinarith [ht.2]⟩
-  have hp : 1-q^2/2 ∈ Set.Ioo 0 1 := ⟨by linarith, by linarith⟩
-  have hl : l ≤ n := by
-    have h := mul_le_mul_of_nonneg_left ht'.2.le hn0.le
-    rw [hmean,mul_one] at h
-    exact_mod_cast h
-  have hmass := binomialMass_near_mean_lower n l ht' hp hl hmean hv
-  have hden : 0 < 12*Real.sqrt ((n:ℝ)*θ*(1-θ)) := by positivity
-  have hexp : Real.exp (-32/q^3) ≤ Real.exp (-((l:ℝ)-(n:ℝ)*(1-q^2/2))^2 /
-      ((n:ℝ)*(1-q^2/2)*(1-(1-q^2/2)))) := by
-    apply Real.exp_le_exp.mpr
-    have he : 1-(1-q^2/2)=q^2/2 := by ring
-    rw [he]
-    simpa only [neg_div] using neg_le_neg hquot
-  calc
-    _ ≤ Real.exp (-((l:ℝ)-(n:ℝ)*(1-q^2/2))^2 /
-        ((n:ℝ)*(1-q^2/2)*(1-(1-q^2/2)))) / (12*Real.sqrt ((n:ℝ)*θ*(1-θ))) := by
-      exact div_le_div₀ (Real.exp_nonneg _) hexp hden (by nlinarith)
-    _ ≤ _ := by simpa only [primitiveErasure] using hmass
-
-
-/-- The actual base count, with no redefinition of the working channel. -/
-def initialSideCount (q : ℝ) (m : ℕ) : ℕ := ⌊q*(m:ℝ)⌋₊
-
-theorem initialSideCount_bounds {q : ℝ} {m : ℕ}
-    (hq : q ∈ Set.Ioc 0 (1/2:ℝ)) (hm : 32 ≤ m)
-    (hscale : 16 ≤ q^3*(m:ℝ)) :
-    q*(m:ℝ)/2 ≤ (initialSideCount q m:ℝ) ∧
-    (initialSideCount q m:ℝ) ≤ q*(m:ℝ) ∧ initialSideCount q m+1 ≤ m := by
-  have hm0 : (0:ℝ) ≤ m := Nat.cast_nonneg _
-  have hm32 : (32:ℝ) ≤ m := by exact_mod_cast hm
-  have hq0 := hq.1
-  have hq2 : q^2 ≤ 1 := by nlinarith [hq.2]
-  have hqm : 2 ≤ q*(m:ℝ) := by
-    have hh := mul_le_mul_of_nonneg_right hq2 (mul_nonneg hq0.le hm0)
-    nlinarith
-  have hlow := Nat.lt_floor_add_one (q*(m:ℝ))
-  have hhigh := Nat.floor_le (mul_nonneg hq0.le hm0)
-  change q*(m:ℝ) < (initialSideCount q m:ℝ)+1 at hlow
-  change (initialSideCount q m:ℝ) ≤ q*(m:ℝ) at hhigh
-  refine ⟨by linarith,hhigh,?_⟩
-  have hh := mul_le_mul_of_nonneg_right hq.2 hm0
-  have : (initialSideCount q m:ℝ)+1 ≤ (m:ℝ) := by linarith
-  exact_mod_cast this
-
-/-- An injective enumeration of m common integer counts; no asymptotic
-cardinality argument is left implicit. -/
-def commonCountIndex {m : ℕ} (k : ℕ) (p : ℝ) (hp : p ∈ Set.Icc 0 1) (j : Fin m) :
-    Fin ((k+1)*m+1) :=
-  ⟨⌊p*((k*m:ℕ):ℝ)⌋₊+j.val,by
-    have ha : (⌊p*((k*m:ℕ):ℝ)⌋₊:ℝ) ≤ ((k*m:ℕ):ℝ) := by
-      exact (Nat.floor_le (mul_nonneg hp.1 (Nat.cast_nonneg _))).trans
-        (mul_le_of_le_one_left (Nat.cast_nonneg _) hp.2)
-    have ha' : ⌊p*((k*m:ℕ):ℝ)⌋₊ ≤ k*m := by exact_mod_cast ha
-    have hj := j.isLt
-    rw [Nat.add_mul,one_mul]
-    omega⟩
-
-theorem commonCountIndex_injective {m : ℕ} (k : ℕ) (p : ℝ) (hp : p ∈ Set.Icc 0 1) :
-    Function.Injective (commonCountIndex (m:=m) k p hp) := by
-  intro a b h
-  have hv := congrArg Fin.val h
-  simp only [commonCountIndex,Nat.add_left_cancel_iff] at hv
-  exact Fin.ext hv
-
-theorem commonCountIndex_deviation {m : ℕ} (k : ℕ) (p : ℝ) (hp : p ∈ Set.Icc 0 1)
-    (j : Fin m) :
-    |((commonCountIndex k p hp j).val:ℝ)-((k*m:ℕ):ℝ)*p| ≤ 2*(m:ℝ) ∧
-    |((commonCountIndex k p hp j).val:ℝ)-(((k+1)*m:ℕ):ℝ)*p| ≤ 2*(m:ℝ) := by
-  have hm : (1:ℝ) ≤ m := by exact_mod_cast (Nat.succ_le_of_lt (Nat.zero_lt_of_lt j.isLt))
-  have hj : (j.val:ℝ) < m := by exact_mod_cast j.isLt
-  have hj0 : (0:ℝ) ≤ j.val := Nat.cast_nonneg _
-  have ha := Nat.floor_le (mul_nonneg hp.1 (Nat.cast_nonneg (k*m)))
-  have hb := Nat.lt_floor_add_one (p*((k*m:ℕ):ℝ))
-  have hp0 := hp.1
-  have hp1 := hp.2
-  have hmp : 0 ≤ (m:ℝ)*p := mul_nonneg (Nat.cast_nonneg _) hp0
-  have hmp' : (m:ℝ)*p ≤ m := mul_le_of_le_one_right (Nat.cast_nonneg _) hp1
-  change |((⌊p*((k*m:ℕ):ℝ)⌋₊+j.val:ℕ):ℝ)-((k*m:ℕ):ℝ)*p| ≤ _ ∧
-    |((⌊p*((k*m:ℕ):ℝ)⌋₊+j.val:ℕ):ℝ)-(((k+1)*m:ℕ):ℝ)*p| ≤ _
-  have hn : (((k+1)*m:ℕ):ℝ)=((k*m:ℕ):ℝ)+(m:ℝ) := by push_cast; ring
-  rw [hn]
-  push_cast
-  constructor <;> apply abs_le.mpr <;> constructor <;> push_cast at ha hb <;> nlinarith
-
-end TraceReconstruction
-
-
-
-namespace TraceReconstruction
-open scoped Classical
-
-theorem ordinaryRun_replicate (n : ℕ) :
-    (List.replicate n [false,true]).flatten = ordinaryRun n := by
-  induction n with
-  | zero => rfl
-  | succ n ih => simp [List.replicate_succ,ordinaryRun,ih]
-
-/-- Unique primitive parsing identifies the rank model with the literal
-period words in the manuscript, not just a law on unlabeled locations. -/
-theorem primitive_period_word {m : ℕ} (off : Fin m) :
-    (List.ofFn (fun j : Fin m => primitiveBlock (decide (j=off)))).flatten =
-      initialPeriod m off.val := by
-  induction m with
-  | zero => exact Fin.elim0 off
-  | succ m ih =>
-    refine Fin.cases ?_ (fun d => ?_) off
-    · simp [List.ofFn_succ,primitiveBlock,initialPeriod,ordinaryRun,ordinaryRun_replicate]
-    · rw [List.ofFn_succ,List.flatten_cons]
-      have hf : primitiveBlock (decide ((0:Fin (m+1))=d.succ))=[false,true] := by
-        simp [primitiveBlock,Ne.symm (Fin.succ_ne_zero d)]
-      rw [hf]
-      simp only [Fin.succ_inj]
-      rw [ih d]
-      simp [initialPeriod,ordinaryRun,Nat.sub_sub,Nat.add_comm]
-
-/-- Membership in the true periodic defect set is exactly the within-period
-slot test, uniformly for all deterministic period types. -/
-theorem periodSlot_range_iff {u m : ℕ} (off : Fin u → Fin m) (i : Fin u) (j : Fin m) :
-    (⟨i.val*m+j.val,by
-      have hi := Nat.mul_le_mul_right m (Nat.succ_le_of_lt i.isLt)
-      have hj := j.isLt
-      nlinarith⟩ : Fin (u*m)) ∈ Set.range (periodSlot off) ↔ j=off i := by
-  constructor
-  · rintro ⟨a,ha⟩
-    have hval := congrArg Fin.val ha
-    change a.val*m+(off a).val=i.val*m+j.val at hval
-    have hm : 0 < m := Nat.zero_lt_of_lt j.isLt
-    have hdiv := congrArg (fun k : ℕ => k/m) hval
-    simp only [Nat.mul_comm _ m, Nat.mul_add_div hm, Nat.div_eq_of_lt (off a).isLt,
-      Nat.div_eq_of_lt j.isLt,add_zero] at hdiv
-    have hai : a=i := Fin.ext hdiv
-    subst a
-    apply Fin.ext
-    omega
-  · intro h
-    refine ⟨i,?_⟩
-    apply Fin.ext
-    simp only [periodSlot]
-    exact congrArg (fun k:Fin m => i.val*m+k.val) h.symm
-
-/-- The simulator's actual primitive list is the unpadded source gadget. -/
-theorem periodic_primitive_word {u m : ℕ} (off : Fin u → Fin m) :
-    (((List.ofFn (defectTypes (periodSlot off))).map primitiveBlock).flatten) =
-      periodString m u off := by
-  rw [List.map_ofFn,List.ofFn_mul]
-  rw [List.flatten_flatten,List.map_ofFn]
-  unfold periodString
-  congr 1
-  apply congrArg List.ofFn
-  funext i
-  simp only [Function.comp_apply]
-  have he : (fun j:Fin m => primitiveBlock (defectTypes (periodSlot off)
-      ⟨i.val*m+j.val,by
-        have hi := Nat.mul_le_mul_right m (Nat.succ_le_of_lt i.isLt)
-        have hj := j.isLt
-        nlinarith⟩)) = fun j:Fin m => primitiveBlock (decide (j=off i)) := by
-    funext j
-    simp only [defectTypes,periodSlot_range_iff]
-  rw [he]
-  exact primitive_period_word (off i)
-
-theorem periodString_constant {u m : ℕ} (off : Fin m) :
-    periodString m u (fun _ => off) = pureAtomRun (initialPeriod m off.val) u := by
-  simp only [periodString,List.ofFn_const,pureAtomRun]
-
-end TraceReconstruction
-
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem primitive_count_admissible {q : ℝ} {m n l : ℕ}
-    (hq : q ∈ Set.Ioc 0 (1/2:ℝ)) (hm : 32 ≤ m)
-    (hscale : 16 ≤ q^3*(m:ℝ))
-    (hnlo : q*(m:ℝ)^2/2 ≤ (n:ℝ)) (hnhi : (n:ℝ) ≤ (m:ℝ)^2)
-    (hdev : |(l:ℝ)-(n:ℝ)*(1-q^2/2)| ≤ 2*(m:ℝ)) :
-    ∃ θ : ℝ, l ≤ n ∧ θ ∈ Set.Icc (1-q^2) (1-q^2/4) ∧
-      (n:ℝ)*θ=l ∧ 1 ≤ (n:ℝ)*θ*(1-θ) := by
-  have hm0 : (0:ℝ) < m := by exact_mod_cast (show 0 < m by omega)
-  have hq0 := hq.1
-  have hn0 : (0:ℝ) < n := lt_of_lt_of_le (by positivity) hnlo
-  let θ : ℝ := (l:ℝ)/n
-  have hmean : (n:ℝ)*θ=l := by dsimp [θ]; field_simp
-  obtain ⟨ht,hv,_,_⟩ := primitive_count_window hq hm hscale hnlo hnhi hmean hdev
-  have hl : l ≤ n := by
-    have ht1 : θ ≤ 1 := by nlinarith [ht.2,sq_nonneg q]
-    have hh := mul_le_mul_of_nonneg_left ht1 hn0.le
-    rw [hmean,mul_one] at hh
-    exact_mod_cast hh
-  exact ⟨θ,hl,ht,hmean,hv⟩
-
-/-- The common-mass constant depends only on the working deletion rate. -/
-def initialOverlapLower (q : ℝ) : ℝ :=
-  Real.exp (-(32+3*primitivePeriodConstant)/q^3)/1536
-
-theorem initialOverlapLower_pos (q : ℝ) : 0 < initialOverlapLower q := by
-  unfold initialOverlapLower
-  positivity
-
-theorem initialOverlapLower_lt_eighth {q : ℝ} (hq : 0 < q) :
-    initialOverlapLower q < 1/8 := by
-  have hx : -(32+3*primitivePeriodConstant)/q^3 ≤ 0 := by
-    have hc := primitivePeriodConstant_nonneg
-    apply div_nonpos_of_nonpos_of_nonneg <;> nlinarith [pow_pos hq 3]
-  have he : Real.exp (-(32+3*primitivePeriodConstant)/q^3) ≤ 1 := by
-    simpa using Real.exp_le_exp.mpr hx
-  unfold initialOverlapLower
-  linarith
-
-/-- Unconditioned common mass of the two genuine count-and-mark simulators.
-This includes all retained counts, not a conditioned or changed channel. -/
-theorem initial_counted_overlap {q : ℝ} {m : ℕ}
-    (hq : q ∈ Set.Ioc 0 (1/2:ℝ)) (hm : 32 ≤ m)
-    (hscale : 16 ≤ q^3*(m:ℝ)) (off : Fin m)
-    (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 3) :
-    let k := initialSideCount q m
-    initialOverlapLower q ≤ FiniteLaw.overlap
-      (countedPrimitiveLaw (N:=(k+1)*m) (Nat.mul_le_mul_right m (Nat.le_succ k))
-        ⟨hq.1,by linarith [hq.2]⟩ (defectTypes (periodSlot (u:=k) (fun _ => off))))
-      (countedPrimitiveLaw (N:=(k+1)*m) le_rfl
-        ⟨hq.1,by linarith [hq.2]⟩ (defectTypes (periodSlot (u:=k+1) (fun _ => off)))) := by
-  let k := initialSideCount q m
-  let N := (k+1)*m
-  have hq0 := hq.1
-  have hqi : q ∈ Set.Ioo 0 1 := ⟨hq.1,by linarith [hq.2]⟩
-  have hm0 : (0:ℝ) < m := by exact_mod_cast (show 0 < m by omega)
-  have hq2 : q^2 ≤ 1/4 := by nlinarith [hq.2]
-  let p := 1-primitiveErasure q
-  have hp : p ∈ Set.Icc 0 1 := primitiveRetention_mem hqi
-  have hp' : p=1-q^2/2 := rfl
-  obtain ⟨hklo,hkhi,hk1⟩ := initialSideCount_bounds hq hm hscale
-  change q*(m:ℝ)/2 ≤ (k:ℝ) at hklo
-  change (k:ℝ) ≤ q*(m:ℝ) at hkhi
-  change k+1 ≤ m at hk1
-  have hk : k ≤ m := by omega
-  have hnlo : q*(m:ℝ)^2/2 ≤ ((k*m:ℕ):ℝ) := by
-    have hh := mul_le_mul_of_nonneg_right hklo hm0.le
-    push_cast
-    nlinarith
-  have hnhi : ((k*m:ℕ):ℝ) ≤ (m:ℝ)^2 := by
-    exact_mod_cast (show k*m ≤ m^2 by simpa [pow_two] using Nat.mul_le_mul_right m hk)
-  have hNlo : q*(m:ℝ)^2/2 ≤ (N:ℝ) := by
-    exact hnlo.trans (by exact_mod_cast Nat.mul_le_mul_right m (Nat.le_succ k))
-  have hNhi : (N:ℝ) ≤ (m:ℝ)^2 := by
-    exact_mod_cast (show N ≤ m^2 by simpa [N,pow_two] using Nat.mul_le_mul_right m hk1)
-  have hqm : 1 ≤ q*(m:ℝ) := by
-    have hh := mul_le_mul_of_nonneg_right (show q^2≤1 by linarith) (mul_nonneg hq0.le hm0.le)
-    nlinarith
-  have hrate : (k:ℝ)/(m:ℝ)+((k+1:ℕ):ℝ)/(m:ℝ) ≤ 3*q := by
-    rw [←add_div]
-    apply (div_le_iff₀ hm0).mpr
-    push_cast
-    nlinarith
-  have harg : primitivePeriodConstant/q^4*((k:ℝ)/m+((k+1:ℕ):ℝ)/m) ≤
-      3*primitivePeriodConstant/q^3 := by
-    calc
-      _ ≤ primitivePeriodConstant/q^4*(3*q) :=
-        mul_le_mul_of_nonneg_left hrate (div_nonneg primitivePeriodConstant_nonneg (by positivity))
-      _ = _ := by field_simp
-  let C₁ := boundedCountLaw (N:=N) (Nat.mul_le_mul_right m (Nat.le_succ k)) p hp
-  let C₂ := boundedCountLaw (N:=N) le_rfl p hp
-  let K := boundedConditionalPrimitiveLaw hqi (defectTypes (periodSlot (u:=k) (fun _ => off))) (N:=N)
-  let L := boundedConditionalPrimitiveLaw hqi (defectTypes (periodSlot (u:=k+1) (fun _ => off))) (N:=N)
-  let f : Fin (N+1) → ℝ := fun l => min (C₁.mass l) (C₂.mass l) * FiniteLaw.overlap (K l) (L l)
-  let a : ℝ := Real.exp (-32/q^3)/(12*(m:ℝ))
-  let b : ℝ := Real.exp (-(3*primitivePeriodConstant)/q^3)/128
-  have ha0 : 0 ≤ a := by dsimp [a]; positivity
-  have hb0 : 0 ≤ b := by dsimp [b]; positivity
-  have hf0 : ∀l, 0 ≤ f l := fun l =>
-    mul_nonneg (le_min (C₁.nonneg l) (C₂.nonneg l)) (FiniteLaw.overlap_nonneg _ _)
-  let ix : Fin m → Fin (N+1) := commonCountIndex k p hp
-  have hterm : ∀j : Fin m, a*b ≤ f (ix j) := by
-    intro j
-    let l := (ix j).val
-    have hd := commonCountIndex_deviation k p hp j
-    change |(l:ℝ)-((k*m:ℕ):ℝ)*p| ≤ 2*(m:ℝ) ∧ |(l:ℝ)-(N:ℝ)*p| ≤ 2*(m:ℝ) at hd
-    rw [hp'] at hd
-    obtain ⟨θ,hl,hθ,hmean,hv⟩ := primitive_count_admissible hq hm hscale hnlo hnhi hd.1
-    obtain ⟨θ',hl',hθ',hmean',hv'⟩ := primitive_count_admissible hq hm hscale hNlo hNhi hd.2
-    have hm1 := primitive_count_mass_lower hq hm hscale hnlo hnhi hd.1
-    have hm2 := primitive_count_mass_lower hq hm hscale hNlo hNhi hd.2
-    have hmin : a ≤ min (C₁.mass (ix j)) (C₂.mass (ix j)) := le_min hm1 hm2
-    have hc := conditionalPeriod_overlap hl hl' hm hk hk1 hq hθ hθ' hmean hmean' hv hv'
-      (fun _ => off) (fun _ => off) (fun _ => hoff) (fun _ => hoff)
-      (fun _ _ => by simp) (fun _ _ => by simp)
-    have hb : b ≤ FiniteLaw.overlap (K (ix j)) (L (ix j)) := by
-      dsimp only [K,L,boundedConditionalPrimitiveLaw]
-      rw [dite_eq_left hl,dite_eq_left hl']
-      refine le_trans ?_ hc
-      apply div_le_div_of_nonneg_right _ (by norm_num : (0:ℝ)≤128)
-      apply Real.exp_le_exp.mpr
-      simpa only [neg_div] using neg_le_neg harg
-    exact mul_le_mul hmin hb hb0 (le_min (C₁.nonneg _) (C₂.nonneg _))
-  have hsub : (∑j : Fin m, f (ix j)) ≤ ∑l : Fin (N+1), f l := by
-    have hi := commonCountIndex_injective (m:=m) k p hp
-    calc
-      _ = ∑l ∈ Finset.univ.image ix, f l := by
-        rw [Finset.sum_image]
-        exact fun a _ b _ hab => hi hab
-      _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _)
-        (fun l _ _ => hf0 l)
-  have hsum : (m:ℝ)*(a*b) ≤ ∑l : Fin (N+1), f l := by
-    calc
-      _ = ∑_j : Fin m, a*b := by simp
-      _ ≤ ∑j : Fin m, f (ix j) := Finset.sum_le_sum (fun j _ => hterm j)
-      _ ≤ _ := hsub
-  have he : (m:ℝ)*(a*b)=initialOverlapLower q := by
-    dsimp [a,b,initialOverlapLower]
-    rw [div_mul_div_comm,←Real.exp_add]
-    have hx : -32/q^3 + -(3*primitivePeriodConstant)/q^3 = -(32+3*primitivePeriodConstant)/q^3 := by ring
-    rw [hx]
-    field_simp
-    ring
-  rw [he] at hsum
-  exact hsum.trans (FiniteLaw.overlap_sigma_lower C₁ C₂ K L)
-
-/-- Data processing for common exact simulators, stated as a common-mass
-inequality and allowing unequal original lengths. -/
-theorem simulations_overlap_le {α : Type*} [Fintype α] {q : ℝ}
-    (hq : q ∈ Set.Icc 0 1) (P Q : FiniteLaw α) (decode : α → List Bool)
-    (x y : List Bool) (hP : SimulatesTrace q P decode x)
-    (hQ : SimulatesTrace q Q decode y) :
-    P.overlap Q ≤ listTraceOverlap q hq x y := by
-  have h := approximations_trace_tv hq P Q decode x y 0 0
-    (max x.length y.length) (le_max_left _ _) (le_max_right _ _)
-    hP.approximate hQ.approximate
-  simp only [add_zero] at h
-  unfold listTraceOverlap
-  rw [FiniteLaw.overlap_eq_one_sub_tv, FiniteLaw.overlap_eq_one_sub_tv]
-  linarith
-
-/-- The initial hidden-bit overlap for ordinary complete deletion traces of
-consecutive pure runs. This is the unconditioned original channel. -/
-theorem initial_pure_trace_overlap {q : ℝ} {m : ℕ}
-    (hq : q ∈ Set.Ioc 0 (1/2:ℝ)) (hm : 32 ≤ m)
-    (hscale : 16 ≤ q^3*(m:ℝ)) (off : Fin m)
-    (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 3) :
-    let k := initialSideCount q m
-    initialOverlapLower q ≤ listTraceOverlap q ⟨hq.1.le, by linarith [hq.2]⟩
-      (pureAtomRun (initialPeriod m off.val) k)
-      (pureAtomRun (initialPeriod m off.val) (k+1)) := by
-  let k := initialSideCount q m
-  have hqi : q ∈ Set.Ioo 0 1 := ⟨hq.1,by linarith [hq.2]⟩
-  have hca := countedPrimitiveLaw_simulates (N:=(k+1)*m)
-    (Nat.mul_le_mul_right m (Nat.le_succ k)) hqi
-    (defectTypes (periodSlot (u:=k) (fun _ => off)))
-  have hcb := countedPrimitiveLaw_simulates (N:=(k+1)*m) le_rfl hqi
-    (defectTypes (periodSlot (u:=k+1) (fun _ => off)))
-  rw [periodic_primitive_word,periodString_constant] at hca hcb
-  exact (initial_counted_overlap hq hm hscale off hoff).trans
-    (simulations_overlap_le ⟨hq.1.le,by linarith [hq.2]⟩ _ _ _ _ _ hca hcb)
-
-end TraceReconstruction
-
-
-
-/-! From the actual displaced period words to their conditional oracle
-moments.  The coefficient bounds are the rank kernels,
-not assumptions on an abstract pair. -/
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- An interior period slot admits the three literal shifts. -/
-def oneStepOffset {m : ℕ} (off : Fin m)
-    (hi : 1  ≤  off.val ∧ off.val + 1 < m) (a : Fin 3) : Fin m :=
-  ⟨off.val+a.val-1,by have ha := a.isLt; omega⟩
-
-theorem oneStepOffset_add_one {m : ℕ} (off : Fin m)
-    (hi : 1  ≤  off.val ∧ off.val + 1 < m) (a : Fin 3) :
-    (oneStepOffset off hi a).val+1=off.val+a.val := by
-  dsimp [oneStepOffset]
-  omega
-
-theorem oneStepOffset_distance {m : ℕ} (off : Fin m)
-    (hi : 1  ≤  off.val ∧ off.val + 1 < m) (a : Fin 3) :
-    |((oneStepOffset off hi a).val:ℝ)-(off.val:ℝ)|  ≤  1 := by
-  have he := oneStepOffset_add_one off hi a
-  have ha : (a.val:ℝ)  ≤  2 := by exact_mod_cast (show a.val ≤ 2 by omega)
-  have ha0 : (0:ℝ)  ≤  a.val := Nat.cast_nonneg _
-  have he' : ((oneStepOffset off hi a).val:ℝ)+1=(off.val:ℝ)+a.val := by exact_mod_cast he
-  rw [abs_le]
-  constructor <;> linarith
-
-theorem constantPeriod_safe {u m : ℕ} (hm : 3  ≤  m) (off : Fin m)
-    (hi : 1  ≤  off.val ∧ off.val + 1 < m) :
-    SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)) := by
-  constructor
-  · intro i
-    change 1  ≤  i.val*m+off.val ∧ i.val*m+off.val+1<u*m
-    have h := Nat.mul_le_mul_right m (Nat.succ_le_of_lt i.isLt)
-    constructor <;> nlinarith [hi.1,hi.2]
-  · intro i j hij
-    change i.val*m+off.val+3 ≤ j.val*m+off.val
-    have h := Nat.mul_le_mul_right m (Nat.succ_le_of_lt hij)
-    nlinarith
-
-theorem shifted_constantPeriod {u m : ℕ} (off : Fin m)
-    (hi : 1  ≤  off.val ∧ off.val + 1 < m)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
-    (a : Fin u → Fin 3) :
-    shiftedTag (periodSlot (fun _ => off)) hsafe a =
-      periodSlot (fun i => oneStepOffset off hi (a i)) := by
-  funext i
-  apply Fin.ext
-  change i.val*m+off.val+(a i).val-1=i.val*m+(off.val+(a i).val-1)
-  omega
-
-/-- Arbitrary deterministic motif orientations remain genuine one-defect
-periods with the exact geometry used in the high-degree rank bound. -/
-theorem signedPeriod_geometry {u m : ℕ} (off : Fin m)
-    (hi : 1  ≤  off.val ∧ off.val + 1 < m)
-    (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 2)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
-    (s : ℕ) (X : ℤ) (τ : ℝ) :
-    ∃ off' : Fin u → Fin m,
-      signedPeriodDefects off hsafe s X τ = periodSlot off' ∧
-      (∀j, |((off' j).val:ℝ)-(m:ℝ)/2| ≤ 3) ∧
-      (∀j k, |((off' j).val:ℝ)-(off' k).val| ≤ 2) := by
-  let a : Fin u → Fin 3 := fun j =>
-    motifShift (motifIndicator s ((j.val:ℤ)-X)) ((prouhetSeries s).coeff ((j.val:ℤ)-X)) τ
-  refine ⟨fun j => oneStepOffset off hi (a j),shifted_constantPeriod off hi hsafe a,?_,?_⟩
-  · intro j
-    have h := oneStepOffset_distance off hi (a j)
-    have ht := abs_add_le (((oneStepOffset off hi (a j)).val:ℝ)-(off.val:ℝ))
-      ((off.val:ℝ)-(m:ℝ)/2)
-    rw [sub_add_sub_cancel] at ht
-    linarith
-  · intro j k
-    have hj := oneStepOffset_distance off hi (a j)
-    have hk := oneStepOffset_distance off hi (a k)
-    rw [abs_le] at hj hk ⊢
-    constructor <;> linarith
-
-
-/-- Every moment of the actual primitive score has the source uniform
-bound. -/
-theorem primitive_score_evenMoment {q : ℝ} (hq : q∈Set.Ioo 0 1) (k : ℕ) :
-    (primitiveMarkLaw q hq false).expect (fun z => |primitiveScore q z|^(2*k))  ≤ 
-      (2/q)^(2*k) := by
-  calc
-    _  ≤  (primitiveMarkLaw q hq false).expect (fun _ => (2/q)^(2*k)) := by
-      apply Finset.sum_le_sum
-      intro z _
-      exact mul_le_mul_of_nonneg_left
-        (pow_le_pow_left₀ (abs_nonneg _) (primitive_score_bound hq z) _) ((primitiveMarkLaw q hq false).nonneg z)
-    _ = _ := (primitiveMarkLaw q hq false).expect_const _
-
-/-- Hypercontractivity now applies directly to the actual signed initial
-density difference under its ordinary-mark reference law. -/
-theorem initial_moment_of_energy {u m l : ℕ} (hl : l ≤ u*m) (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
-    (π : IntSeries) (s k : ℕ) (hk : 1 ≤ k) {q : ℝ} (hq : q∈Set.Ioo 0 1) :
-    ((primitiveMarkLaw q hq false).product l).evenMomentSq
-      (multilinear (fun σ => mixedMotifCoefficient hl off hsafe π s 1 σ-
-        mixedMotifCoefficient hl off hsafe π s (-1) σ) (primitiveScore q)) k  ≤ 
-      multilinearEnergy (fun σ => mixedMotifCoefficient hl off hsafe π s 1 σ-
-        mixedMotifCoefficient hl off hsafe π s (-1) σ) ((6*(2*k:ℕ):ℝ)*(2/q)) := by
-  apply multilinear_moment_sq_scaled _ _ k hk
-  · exact primitive_score_centered hq
-  · exact div_pos (by norm_num) hq.1
-  · exact primitive_score_evenMoment hq k
-
-
-/-- Jensen's inequality on each homogeneous degree. -/
-theorem degree_square_mixture_le {Ω : Type*} [Fintype Ω] {n : ℕ}
-    (P : FiniteLaw Ω) (A : Ω → (Fin n → Bool) → ℝ) (r : ℕ) :
-    (∑σ : Fin n → Bool, if monomialDegree σ=r then
-      (P.expect (fun ω => A ω σ))^2 else 0)  ≤ 
-    P.expect (fun ω => ∑σ : Fin n → Bool,
-      if monomialDegree σ=r then (A ω σ)^2 else 0) := by
-  unfold FiniteLaw.expect
-  simp_rw [Finset.mul_sum]
-  rw [Finset.sum_comm]
-  apply Finset.sum_le_sum
-  intro σ _
-  by_cases h : monomialDegree σ=r
-  · simp only [ite_eq_left h]
-    have H := weighted_sum_sq_le Finset.univ P.mass (fun ω => A ω σ) (fun ω => P.nonneg ω)
-    simpa only [P.total,one_mul] using H
-  · simp only [ite_eq_right h,mul_zero,Finset.sum_const_zero,le_refl]
-
-theorem degree_square_sub_le {n : ℕ} (A B : (Fin n → Bool) → ℝ) (r : ℕ) :
-    (∑σ : Fin n → Bool, if monomialDegree σ=r then (A σ-B σ)^2 else 0)  ≤ 
-    2*(∑σ : Fin n → Bool, if monomialDegree σ=r then A σ^2 else 0)+
-    2*(∑σ : Fin n → Bool, if monomialDegree σ=r then B σ^2 else 0) := by
-  simp only [Finset.mul_sum,←Finset.sum_add_distrib]
-  apply Finset.sum_le_sum
-  intro σ _
-  split_ifs <;> nlinarith [sq_nonneg (A σ+B σ)]
-
-theorem mixedMotifCoefficient_expect {u m l : ℕ} (hl : l ≤ u*m) (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
-    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
-    (s : ℕ) (τ : ℝ) (σ : Fin l → Bool) :
-    mixedMotifCoefficient hl off hsafe π s τ σ =
-      (seriesProbabilityLaw π hπ hπsum).expect (fun X =>
-        averagedCoefficient (uniformSubsetLaw l (by simpa using hl))
-          (defectAtRank (Finset.univ.image (signedPeriodDefects off hsafe s X.val τ))) σ) := by
-  unfold mixedMotifCoefficient FiniteLaw.expect seriesProbabilityLaw
-  exact (Finset.sum_coe_sort _ _).symm
-
-theorem mixedMotifCoefficient_empty {u m l : ℕ} (hl : l ≤ u*m) (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
-    (π : IntSeries) (hπsum : ∑X∈π.coeff.support,π.coeff X=1) (s : ℕ) (τ : ℝ) :
-    mixedMotifCoefficient hl off hsafe π s τ (fun _ => false)=1 := by
-  simp only [mixedMotifCoefficient,averagedCoefficient_empty,mul_one,hπsum]
-
-/-- Uniform high-degree estimate after the real hidden-anchor mixture. -/
-theorem mixedMotif_degree_le {u m l r : ℕ} (hl : l ≤ u*m) (hr : 0 < r)
-    (hru : r ≤ u) (hm : 32 ≤ m) (hum : u ≤ m) (off : Fin m)
-    (hi : 1 ≤ off.val ∧ off.val+1 < m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 2)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
-    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
-    (s : ℕ) (τ : ℝ) {q θ : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
-    (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
-    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ)) :
-    (∑σ : Fin l → Bool, if monomialDegree σ=r then
-      (mixedMotifCoefficient hl off hsafe π s τ σ)^2 else 0)  ≤ 
-      ((3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/((m:ℝ)*r)))^r := by
-  simp_rw [mixedMotifCoefficient_expect hl off hsafe π hπ hπsum]
-  refine (degree_square_mixture_le _ _ r).trans ?_
-  calc
-    _  ≤  (seriesProbabilityLaw π hπ hπsum).expect (fun _ =>
-        ((3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/((m:ℝ)*r)))^r) := by
-      apply FiniteLaw.expect_mono
-      intro X
-      obtain ⟨d,hd,hnear,hclose⟩ := signedPeriod_geometry off hi hoff hsafe s X.val τ
-      rw [hd]
-      exact period_degree_square_sum hl hr hru hm hum hq hθ hmean hv d hnear hclose
-    _ = _ := FiniteLaw.expect_const _ _
-
-/-- There are never more selected ranks than actual defect periods, even
-in the hidden-anchor mixture. -/
-theorem mixedMotif_degree_vanish {u m l r : ℕ} (hl : l ≤ u*m) (hru : u < r) (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
-    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
-    (s : ℕ) (τ : ℝ) :
-    (∑σ : Fin l → Bool, if monomialDegree σ=r then
-      (mixedMotifCoefficient hl off hsafe π s τ σ)^2 else 0)=0 := by
-  apply le_antisymm
-  · simp_rw [mixedMotifCoefficient_expect hl off hsafe π hπ hπsum]
-    refine (degree_square_mixture_le _ _ r).trans ?_
-    simp_rw [defect_degree_bound hl _ (signedPeriodDefects_strictMono off hsafe s _ τ) hru]
-    rw [FiniteLaw.expect_const]
-  · exact Finset.sum_nonneg (fun σ _ => by split_ifs <;> positivity)
-
-/-- The signed difference has no constant term. -/
-theorem signedMotif_degree_zero {u m l : ℕ} (hl : l ≤ u*m) (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
-    (π : IntSeries) (hπsum : ∑X∈π.coeff.support,π.coeff X=1) (s : ℕ) :
-    (∑σ : Fin l → Bool, if monomialDegree σ=0 then
-      (mixedMotifCoefficient hl off hsafe π s 1 σ-
-        mixedMotifCoefficient hl off hsafe π s (-1) σ)^2 else 0)=0 := by
-  simp_rw [monomialDegree_eq_zero_iff]
-  simp [mixedMotifCoefficient_empty hl off hsafe π hπsum]
-
-/-- Low-degree cancellation is for exactly the coefficients of the
-conditional motif densities, without a rank surrogate at the boundary. -/
-theorem signedMotif_degree_cancellation {u m l r : ℕ}
-    (hu : 0 < u) (hm : 128 ≤ m) (hr : 0 < r) (hl : l ≤ u*m)
-    (off : Fin m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 3)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
-    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
-    (s : ℕ) (hcentral : CentralMotifSupport u s π)
-    {θ : ℝ} (hθ : θ∈Set.Ioo 0 1) (hhalf : 1/2 ≤ θ)
-    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
-    (hM : ((2*(2^s):ℕ):ℝ)<(u:ℝ)/(3*(r+1:ℝ))) :
-    (∑σ : Fin l → Bool, if monomialDegree σ=r then
-      (mixedMotifCoefficient hl off hsafe π s 1 σ-
-        mixedMotifCoefficient hl off hsafe π s (-1) σ)^2 else 0)  ≤ 
-      8*(Fintype.card (Fin r → GapDifference):ℝ)*
-        (∑k : Fin r → GapDifference, if Odd (firstDifferenceCount k) then
-          signedPatternBound u m r s π θ k else 0) +
-      2*((Fintype.card (IncreasingTuple u r):ℝ)*
-        (2*(((r+1:ℕ):ℝ)*θ^r/binomialMass (u*m) θ l * (2*Real.exp (-(m:ℝ)/(1024*u))))))^2 := by
-  refine (sum_degree_coeff_sq_le _).trans ?_
-  simp_rw [mixedMotifCoefficient_rank_difference]
-  exact initial_signed_energy hu hm hr hl off hoff hsafe π hπ hπsum s hcentral
-    hθ hhalf hmean hv hM
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-open FiniteLaw
-
-/-- A finite tail is no larger than the full geometric tail. -/
-theorem finite_geometric_tail_le {a : ℝ} (ha : 0 ≤ a) (ha1 : a < 1) (N R : ℕ) :
-    (∑r ∈ Finset.range N, if R ≤ r then a^r else 0) ≤ a^R/(1-a) := by
-  have ht : HasSum (fun n : ℕ => a^(n+R)) (a^R/(1-a)) := by
-    simpa only [pow_add, mul_comm, div_eq_mul_inv] using
-      (hasSum_geometric_of_lt_one ha ha1).mul_left (a^R)
-  have hsum : (∑r ∈ Finset.range N, if R ≤ r then a^r else 0) =
-      ∑r ∈ (Finset.range N).filter (R ≤ ·), a^r := by
-    rw [Finset.sum_filter]
-  rw [hsum]
-  calc
-    _ = ∑r ∈ ((Finset.range N).filter (R ≤ ·)).image (fun r => r-R), a^(r+R) := by
-      rw [Finset.sum_image]
-      · apply Finset.sum_congr rfl
-        intro r hr
-        rw [Nat.sub_add_cancel (Finset.mem_filter.mp hr).2]
-      · intro r hr s hs h
-        dsimp only at h
-        have hrR := (Finset.mem_filter.mp hr).2
-        have hsR := (Finset.mem_filter.mp hs).2
-        omega
-    _ ≤ _ := by
-      rw [←ht.tsum_eq]
-      exact ht.summable.sum_le_tsum _ (fun _ _ => by positivity)
-
-/-- Splitting the exact homogeneous expansion, not a surrogate polynomial. -/
-theorem multilinearEnergy_low_high {n R : ℕ} (A : (Fin n → Bool) → ℝ)
-    (w a K : ℝ) (hw : 0 ≤ w) (ha : 0 ≤ a) (ha1 : a < 1) (hK : 0 ≤ K)
-    (L : ℕ → ℝ)
-    (hlow : ∀r < R, (∑σ : Fin n → Bool, if monomialDegree σ=r then A σ^2 else 0) ≤ L r)
-    (hhigh : ∀r, R ≤ r → w^(2*r)*(∑σ : Fin n → Bool,
-      if monomialDegree σ=r then A σ^2 else 0) ≤ K*a^r)
-    (hL : ∀r, 0 ≤ L r) :
-    multilinearEnergy A w ≤ (∑r ∈ Finset.range R, w^(2*r)*L r)+K*a^R/(1-a) := by
-  rw [multilinearEnergy_degrees]
-  calc
-    _ ≤ ∑r ∈ Finset.range (n+1),
-        ((if r < R then w^(2*r)*L r else 0)+(if R ≤ r then K*a^r else 0)) := by
-      apply Finset.sum_le_sum
-      intro r _
-      by_cases h : r < R
-      · simp only [ite_eq_left h,ite_eq_right (show ¬R ≤ r by omega),add_zero]
-        exact mul_le_mul_of_nonneg_left (hlow r h) (by positivity)
-      · simp only [ite_eq_right h,ite_eq_left (show R ≤ r by omega),zero_add]
-        exact hhigh r (by omega)
-    _ ≤ _ := by
-      rw [Finset.sum_add_distrib]
-      apply add_le_add
-      · rw [←Finset.sum_filter]
-        exact Finset.sum_le_sum_of_subset_of_nonneg
-          (by intro r hr; exact Finset.mem_range.mpr (Finset.mem_filter.mp hr).2)
-          (fun r _ _ => mul_nonneg (by positivity) (hL r))
-      · have he : (∑r ∈ Finset.range (n+1), if R ≤ r then K*a^r else 0)=
-            K*∑r ∈ Finset.range (n+1), if R ≤ r then a^r else 0 := by
-          rw [Finset.mul_sum]
-          apply Finset.sum_congr rfl
-          intro r _
-          split_ifs <;> ring
-        rw [he,mul_div_assoc]
-        exact mul_le_mul_of_nonneg_left (finite_geometric_tail_le ha ha1 _ _) hK
-
-/-- The signed high-degree bound includes the true anchor mixture. -/
-theorem signedMotif_degree_geometric {u m l r : ℕ} (hl : l ≤ u*m) (hr : 0 < r)
-    (hm : 32 ≤ m) (hum : u ≤ m) (off : Fin m)
-    (hi : 1 ≤ off.val ∧ off.val+1 < m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 2)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
-    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
-    (s : ℕ) {q θ : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
-    (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
-    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ)) (w : ℝ) :
-    w^(2*r)*(∑σ : Fin l → Bool, if monomialDegree σ=r then
-      (mixedMotifCoefficient hl off hsafe π s 1 σ-
-        mixedMotifCoefficient hl off hsafe π s (-1) σ)^2 else 0) ≤
-      4*(w^2*(3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/m))^r := by
-  have hw0 : 0 ≤ w^(2*r) := by rw [pow_mul]; positivity
-  have hp : 0 ≤ 3*rankSchurConstant/q := by positivity [rankSchurConstant_pos,hq.1]
-  have hb := degree_square_sub_le (mixedMotifCoefficient hl off hsafe π s 1)
-    (mixedMotifCoefficient hl off hsafe π s (-1)) r
-  by_cases hru : r ≤ u
-  · have hplus := mixedMotif_degree_le hl hr hru hm hum off hi hoff hsafe π hπ hπsum
-      s 1 hq hθ hmean hv
-    have hminus := mixedMotif_degree_le hl hr hru hm hum off hi hoff hsafe π hπ hπsum
-      s (-1) hq hθ hmean hv
-    have hsqrt : Real.sqrt ((u:ℝ)/((m:ℝ)*r)) ≤ Real.sqrt ((u:ℝ)/m) := by
-      apply Real.sqrt_le_sqrt
-      apply div_le_div_of_nonneg_left (by positivity) (by exact_mod_cast (show 0 < m by omega))
-      have hr1 : (1:ℝ) ≤ r := by exact_mod_cast hr
-      nlinarith
-    calc
-      _ ≤ w^(2*r)*(4*((3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/((m:ℝ)*r)))^r) :=
-        mul_le_mul_of_nonneg_left (by linarith) (by rw [pow_mul]; positivity)
-      _ ≤ w^(2*r)*(4*((3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/m))^r) := by
-        gcongr
-      _ = _ := by rw [pow_mul, mul_pow, mul_pow]; ring
-  · rw [mixedMotif_degree_vanish hl (by omega) off hsafe π hπ hπsum s 1,
-      mixedMotif_degree_vanish hl (by omega) off hsafe π hπ hπsum s (-1)] at hb
-    have hz : (∑σ : Fin l → Bool, if monomialDegree σ=r then
-        (mixedMotifCoefficient hl off hsafe π s 1 σ-
-          mixedMotifCoefficient hl off hsafe π s (-1) σ)^2 else 0)=0 := by
-      apply le_antisymm (by linarith)
-      exact Finset.sum_nonneg (fun σ _ => by split_ifs <;> positivity)
-    rw [hz,mul_zero]
-    positivity
-
-/-- Explicit low-degree majorant from the signed finite-difference estimate. -/
-def signedInitialDegreeBound (u m r s : ℕ) (π : IntSeries) (θ : ℝ) (l : ℕ) : ℝ :=
-  if r=0 then 0 else
-    8*(Fintype.card (Fin r → GapDifference):ℝ)*
-      (∑k : Fin r → GapDifference, if Odd (firstDifferenceCount k) then
-        signedPatternBound u m r s π θ k else 0) +
-    2*((Fintype.card (IncreasingTuple u r):ℝ)*
-      (2*(((r+1:ℕ):ℝ)*θ^r/binomialMass (u*m) θ l * (2*Real.exp (-(m:ℝ)/(1024*u))))))^2
-
-theorem signedInitialDegreeBound_nonneg (u m r s : ℕ) (π : IntSeries) (θ : ℝ) (l : ℕ) :
-    0 ≤ signedInitialDegreeBound u m r s π θ l := by
-  unfold signedInitialDegreeBound
-  split_ifs
-  · exact le_rfl
-  · apply add_nonneg _ (by positivity)
-    apply mul_nonneg (by positivity)
-    apply Finset.sum_nonneg
-    intro k _
-    split_ifs
-    · unfold signedPatternBound
-      split_ifs <;> apply mul_nonneg _ (seriesL2Sq_nonneg _)
-      all_goals dsimp only [periodDifferencePrefactor, lengthDifferenceConstant, periodCentralSumConstant, periodBoundedSumConstant]
-      all_goals positivity
-    · exact le_rfl
-
-/-- A fully finite norm estimate for the actual low and high degree pieces.
-Only numerical admissible-count and geometric conditions remain to be chosen. -/
-theorem signed_initial_full_energy {u m l R : ℕ} (hu : 0 < u) (hm : 128 ≤ m)
-    (hum : u ≤ m) (hR : 0 < R) (hl : l ≤ u*m) (off : Fin m)
-    (hi : 1 ≤ off.val ∧ off.val+1 < m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 2)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
-    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
-    (s : ℕ) (hcentral : CentralMotifSupport u s π)
-    {q θ w : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
-    (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
-    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
-    (hM : ((2*(2^s):ℕ):ℝ) < (u:ℝ)/(3*(R+1:ℝ)))
-    (hw : 0 ≤ w) (ha : w^2*(3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/m) < 1) :
-    multilinearEnergy (fun σ => mixedMotifCoefficient hl off hsafe π s 1 σ-
-        mixedMotifCoefficient hl off hsafe π s (-1) σ) w ≤
-      (∑r ∈ Finset.range R, w^(2*r)*signedInitialDegreeBound u m r s π θ l)+
-        4*(w^2*(3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/m))^R /
-          (1-w^2*(3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/m)) := by
-  have hq2 : q^2 ≤ (1/4:ℝ) := by
-    calc q^2 ≤ (1/2:ℝ)^2 := pow_le_pow_left₀ hq.1.le hq.2 2
-         _ = _ := by norm_num
-  have hθ0 : θ∈Set.Ioo 0 1 := by constructor <;> nlinarith [hq.1,hθ.1,hθ.2]
-  have hhalf : 1/2 ≤ θ := by nlinarith [hθ.1]
-  apply multilinearEnergy_low_high _ w _ 4 hw (by positivity [rankSchurConstant_pos,hq.1]) ha
-    (by norm_num) (signedInitialDegreeBound u m · s π θ l) _ _
-    (fun r => signedInitialDegreeBound_nonneg _ _ _ _ _ _ _)
-  · intro r hr
-    by_cases hr0 : r=0
-    · subst r
-      rw [signedMotif_degree_zero hl off hsafe π hπsum s]
-      exact signedInitialDegreeBound_nonneg _ _ _ _ _ _ _
-    · have hMr : ((2*(2^s):ℕ):ℝ) < (u:ℝ)/(3*(r+1:ℝ)) := by
-        apply hM.trans_le
-        apply div_le_div_of_nonneg_left (by positivity) (by positivity)
-        exact mul_le_mul_of_nonneg_left (by exact_mod_cast (show r+1 ≤ R+1 by omega)) (by norm_num)
-      simpa only [signedInitialDegreeBound,ite_eq_right hr0] using
-        signedMotif_degree_cancellation hu hm (Nat.pos_of_ne_zero hr0) hl off
-          (hoff.trans (by norm_num)) hsafe π hπ hπsum s hcentral hθ0 hhalf hmean hv hMr
-  · intro r hr
-    exact signedMotif_degree_geometric hl (by omega) (by omega) hum off hi hoff hsafe π hπ
-      hπsum s hq hθ hmean hv w
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-open FiniteLaw
-
-theorem primitive_reference_mass_pos {q : ℝ} (hq : q∈Set.Ioo 0 1) (z : PrimitiveMark) :
-    0 < (primitiveMarkLaw q hq false).mass z := by
-  change 0 < primitiveMarkMass q false z
-  apply div_pos _ (primitive_retain_pos hq)
-  fin_cases z <;> simp [ordinaryPrimitiveMass,primitiveErasure] <;> nlinarith [hq.1,hq.2]
-
-theorem primitive_product_mass_pos {q : ℝ} (hq : q∈Set.Ioo 0 1) {l : ℕ}
-    (x : Fin l → PrimitiveMark) : 0 < ((primitiveMarkLaw q hq false).product l).mass x :=
-by
-  change 0 < ∏ i, (primitiveMarkLaw q hq false).mass (x i)
-  exact Finset.prod_pos (fun _ _ => primitive_reference_mass_pos hq _)
-
-/-- The density in the actual base experiment, not an independent model. -/
-def initialMotifDensity {u m l : ℕ} (hl : l ≤ u*m) (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
-    (π : IntSeries) (s : ℕ) (τ q : ℝ) : (Fin l → PrimitiveMark) → ℝ :=
-  multilinear (mixedMotifCoefficient hl off hsafe π s τ) (primitiveScore q)
-
-theorem initialMotifDensity_nonneg {u m l : ℕ} (hl : l ≤ u*m) (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
-    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
-    (s : ℕ) (τ : ℝ) {q : ℝ} (hq : q∈Set.Ioo 0 1) (x : Fin l → PrimitiveMark) :
-    0 ≤ initialMotifDensity hl off hsafe π s τ q x := by
-  have h := (conditionalMotifLaw hl off hsafe hq π hπ hπsum s τ).nonneg x
-  rw [conditionalMotifLaw_density] at h
-  exact nonneg_of_mul_nonneg_right h (primitive_product_mass_pos hq x)
-
-theorem initialMotifDensity_mean {u m l : ℕ} (hl : l ≤ u*m) (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
-    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
-    (s : ℕ) (τ : ℝ) {q : ℝ} (hq : q∈Set.Ioo 0 1) :
-    ((primitiveMarkLaw q hq false).product l).expect (initialMotifDensity hl off hsafe π s τ q)=1 := by
-  have h := (conditionalMotifLaw hl off hsafe hq π hπ hπsum s τ).total
-  simpa only [conditionalMotifLaw_density,FiniteLaw.expect,initialMotifDensity] using h
-
-theorem initialMotifLaw_withDensity {u m l : ℕ} (hl : l ≤ u*m) (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
-    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
-    (s : ℕ) (τ : ℝ) {q : ℝ} (hq : q∈Set.Ioo 0 1) :
-    conditionalMotifLaw hl off hsafe hq π hπ hπsum s τ =
-      ((primitiveMarkLaw q hq false).product l).withDensity
-        (initialMotifDensity hl off hsafe π s τ q)
-        (initialMotifDensity_nonneg hl off hsafe π hπ hπsum s τ hq)
-        (initialMotifDensity_mean hl off hsafe π hπ hπsum s τ hq) := by
-  apply FiniteLaw.ext
-  funext x
-  exact conditionalMotifLaw_density hl off hsafe hq π hπ hπsum s τ x
-
-theorem multilinear_sub {α : Type*} {l : ℕ} (A B : (Fin l → Bool) → ℝ)
-    (Z : α → ℝ) (x : Fin l → α) :
-    multilinear (fun σ => A σ-B σ) Z x = multilinear A Z x-multilinear B Z x := by
-  simp only [multilinear,sub_mul,Finset.sum_sub_distrib]
-
-/-- The constant-term deletion of a polynomial. -/
-def nonconstantCoefficient {l : ℕ} (A : (Fin l → Bool) → ℝ) (σ : Fin l → Bool) : ℝ :=
-  if σ=(fun _ => false) then 0 else A σ
-
-theorem nonconstant_multilinear {α : Type*} {l : ℕ} (A : (Fin l → Bool) → ℝ)
-    (Z : α → ℝ) (x : Fin l → α) :
-    multilinear (nonconstantCoefficient A) Z x=multilinear A Z x-A (fun _ => false) := by
-  have he : ∀σ, nonconstantCoefficient A σ=A σ-(if σ=(fun _ => false) then A σ else 0) := by
-    intro σ
-    by_cases h : σ=(fun _ => false) <;> simp [nonconstantCoefficient,h]
-  rw [show nonconstantCoefficient A = (fun σ => A σ-(if σ=(fun _ => false) then A σ else 0)) from funext he, multilinear_sub]
-  congr 1
-  simp [multilinear,ite_mul]
-
-theorem nonconstant_degree_zero {l : ℕ} (A : (Fin l → Bool) → ℝ) :
-    (∑σ : Fin l → Bool, if monomialDegree σ=0 then (nonconstantCoefficient A σ)^2 else 0)=0 := by
-  simp [monomialDegree_eq_zero_iff,nonconstantCoefficient]
-
-theorem nonconstant_degree_pos {l r : ℕ} (A : (Fin l → Bool) → ℝ) (hr : 0 < r) :
-    (∑σ : Fin l → Bool, if monomialDegree σ=r then (nonconstantCoefficient A σ)^2 else 0)=
-      ∑σ : Fin l → Bool, if monomialDegree σ=r then (A σ)^2 else 0 := by
-  apply Finset.sum_congr rfl
-  intro σ _
-  by_cases h : σ=(fun _ => false)
-  · subst σ
-    simp [monomialDegree,Ne.symm (Nat.ne_of_gt hr)]
-  · simp [nonconstantCoefficient,h]
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-open FiniteLaw
-
-theorem mixedMotif_degree_geometric {u m l r : ℕ} (hl : l ≤ u*m) (hr : 0 < r)
-    (hm : 32 ≤ m) (hum : u ≤ m) (off : Fin m)
-    (hi : 1 ≤ off.val ∧ off.val+1 < m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 2)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
-    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
-    (s : ℕ) (τ : ℝ) {q θ : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
-    (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
-    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ)) (w : ℝ) :
-    w^(2*r)*(∑σ : Fin l → Bool, if monomialDegree σ=r then
-      mixedMotifCoefficient hl off hsafe π s τ σ^2 else 0) ≤
-      (w^2*(3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/m))^r := by
-  have hw : 0 ≤ w^(2*r) := by rw [pow_mul]; positivity
-  have hp : 0 ≤ 3*rankSchurConstant/q := by positivity [rankSchurConstant_pos,hq.1]
-  by_cases hru : r ≤ u
-  · have hb := mixedMotif_degree_le hl hr hru hm hum off hi hoff hsafe π hπ hπsum
-      s τ hq hθ hmean hv
-    have hsqrt : Real.sqrt ((u:ℝ)/((m:ℝ)*r)) ≤ Real.sqrt ((u:ℝ)/m) := by
-      apply Real.sqrt_le_sqrt
-      apply div_le_div_of_nonneg_left (by positivity) (by exact_mod_cast (show 0 < m by omega))
-      have hr1 : (1:ℝ) ≤ r := by exact_mod_cast hr
-      nlinarith
-    calc
-      _ ≤ w^(2*r)*((3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/((m:ℝ)*r)))^r :=
-        mul_le_mul_of_nonneg_left hb hw
-      _ ≤ w^(2*r)*((3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/m))^r := by gcongr
-      _ = _ := by rw [pow_mul,mul_pow,mul_pow]; ring
-  · rw [mixedMotif_degree_vanish hl (by omega) off hsafe π hπ hπsum s τ,mul_zero]
-    positivity
-
-/-- Clipping input: the actual conditional density is close to one. -/
-theorem initial_centered_energy {u m l : ℕ} (hl : l ≤ u*m)
-    (hm : 32 ≤ m) (hum : u ≤ m) (off : Fin m)
-    (hi : 1 ≤ off.val ∧ off.val+1 < m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 2)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
-    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
-    (s : ℕ) (τ : ℝ) {q θ w : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
-    (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
-    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
-    (hw : 0 ≤ w) (ha : w^2*(3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/m) < 1) :
-    multilinearEnergy (nonconstantCoefficient (mixedMotifCoefficient hl off hsafe π s τ)) w ≤
-      (w^2*(3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/m)) /
-        (1-w^2*(3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/m)) := by
-  have H := multilinearEnergy_low_high (R:=1)
-    (nonconstantCoefficient (mixedMotifCoefficient hl off hsafe π s τ)) w _ 1 hw
-    (by positivity [rankSchurConstant_pos,hq.1]) ha (by norm_num) (fun _ => 0)
-    (fun r hr => by
-      have he : r=0 := by omega
-      subst r
-      rw [nonconstant_degree_zero])
-    (fun r hr => by
-      rw [nonconstant_degree_pos _ (by omega)]
-      simpa only [one_mul] using mixedMotif_degree_geometric hl (by omega) hm hum off
-        hi hoff hsafe π hπ hπsum s τ hq hθ hmean hv w)
-    (fun _ => le_rfl)
-  simpa using H
-
-theorem initial_centered_moment {u m l : ℕ} (hl : l ≤ u*m)
-    (hm : 32 ≤ m) (hum : u ≤ m) (off : Fin m)
-    (hi : 1 ≤ off.val ∧ off.val+1 < m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 2)
-    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
-    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
-    (s k : ℕ) (hk : 1 ≤ k) (τ : ℝ) {q θ w : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
-    (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
-    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
-    (hw : w=((6*(2*k:ℕ):ℝ)*(2/q)))
-    (ha : w^2*(3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/m) < 1) :
-    ((primitiveMarkLaw q ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩ false).product l).evenMomentSq
-      (fun x => initialMotifDensity hl off hsafe π s τ q x-1) k ≤
-      (w^2*(3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/m)) /
-        (1-w^2*(3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/m)) := by
-  have hq' : q∈Set.Ioo 0 1 := ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩
-  have he : (fun x => initialMotifDensity hl off hsafe π s τ q x-1) =
-      multilinear (nonconstantCoefficient (mixedMotifCoefficient hl off hsafe π s τ))
-        (primitiveScore q) := by
-    funext x
-    rw [nonconstant_multilinear,mixedMotifCoefficient_empty hl off hsafe π hπsum]
-    rfl
-  rw [he]
-  apply (multilinear_moment_sq_scaled (primitiveMarkLaw q hq' false) (primitiveScore q) k hk (primitive_score_centered hq')
-    (div_pos (by norm_num) hq.1) (primitive_score_evenMoment hq' k) l _).trans
-  rw [←hw]
-  apply initial_centered_energy hl hm hum off hi hoff hsafe π hπ hπsum s τ hq hθ hmean hv _ ha
-  rw [hw]; positivity [hq.1]
-
-end TraceReconstruction
-
-end
-end
-end
-end
-end
-end
-end
-end
-
-end OAI
+import OAI.Probability.TraceReconstruction.Oracles
+
+namespace OAI
+
+/-!
+# InitialOverlap
+
+Literal periods, count overlap, initial low/high energies, genuine densities, and centered moment bounds.
+-/
+
+open scoped ENNReal
+noncomputable section
+
+open Filter
+open scoped Topology
+
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem primitive_one_add_score_nonneg {q : ℝ} (hq : q ∈ Set.Ioo 0 1)
+    (a : PrimitiveMark) : 0 ≤ 1 + primitiveScore q a := by
+  have hq1 : 1 ≤ 1/q := (le_div_iff₀ hq.1).mpr (by simpa using hq.2.le)
+  have hq0 := hq.1
+  fin_cases a <;> norm_num [primitiveScore] <;> linarith [show 2/q=2*(1/q) by ring]
+
+/-- Exact conditional law of the genuine periodic primitive observation. -/
+theorem conditionalPeriodLaw_density {u m l : ℕ} (hl : l ≤ u*m)
+    {q : ℝ} (hq : q ∈ Set.Ioo 0 1) (off : Fin u → Fin m) (x : Fin l → PrimitiveMark) :
+    (conditionalPrimitiveLaw hl q hq (defectTypes (periodSlot off))).mass x =
+      ((primitiveMarkLaw q hq false).product l).mass x *
+        averagedDensity (uniformSubsetLaw l (by simpa using hl))
+          (defectAtRank (Finset.univ.image (periodSlot off))) (primitiveScore q) x := by
+  rw [conditionalPrimitiveLaw_density]
+  congr 2
+  funext S
+  exact defectTypes_atRank (periodSlot off) S
+
+/-- Absolute constant in the conditional second-moment exponent. -/
+def primitivePeriodConstant : ℝ := 36*rankSchurConstant^2
+
+theorem primitivePeriodConstant_nonneg : 0 ≤ primitivePeriodConstant := by
+  unfold primitivePeriodConstant
+  positivity
+
+theorem conditionalPeriod_second_moment {u m l : ℕ} (hl : l ≤ u*m)
+    (hm : 32 ≤ m) (hum : u ≤ m) {q θ : ℝ}
+    (hq : q ∈ Set.Ioc 0 (1/2:ℝ))
+    (hθ : θ ∈ Set.Icc (1-q^2) (1-q^2/4))
+    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
+    (off : Fin u → Fin m)
+    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
+    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2) :
+    ((primitiveMarkLaw q ⟨hq.1, by linarith [hq.2]⟩ false).product l).expect (fun x =>
+      averagedDensity (uniformSubsetLaw l (by simpa using hl))
+        (defectAtRank (Finset.univ.image (periodSlot off))) (primitiveScore q) x ^ 2) ≤
+      2*Real.exp (primitivePeriodConstant/q^4 * ((u:ℝ)/m)) := by
+  have hqi : q ∈ Set.Ioo 0 1 := ⟨hq.1, by linarith [hq.2]⟩
+  let P := primitiveMarkLaw q hqi false
+  have hZ : P.expect (primitiveScore q) = 0 := primitive_score_centered hqi
+  have h := period_density_second_moment P (primitiveScore q) hZ hl hm hum hq hθ hmean hv off hoff hdiff
+  refine h.trans (mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr ?_) (by norm_num))
+  have hv0 : 0 ≤ P.expect (fun a => primitiveScore q a ^ 2) :=
+    Finset.sum_nonneg (fun a _ => mul_nonneg (P.nonneg a) (sq_nonneg _))
+  have hv1 : P.expect (fun a => primitiveScore q a ^ 2) ≤ 2/q :=
+    primitive_score_variance_le hqi
+  have hsq : (P.expect (fun a => primitiveScore q a ^ 2))^2 ≤ (2/q)^2 :=
+    pow_le_pow_left₀ hv0 hv1 2
+  calc
+    _ ≤ (2/q)^2*(3*rankSchurConstant/q)^2*((u:ℝ)/m) :=
+      mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hsq (sq_nonneg _)) (by positivity)
+    _ = _ := by unfold primitivePeriodConstant; field_simp; ring
+
+/-- Association converts the two actual conditional period-laws into common
+mass. The two original block totals may differ; only the retained count agrees. -/
+theorem conditionalPeriod_overlap {u v m l : ℕ}
+    (hl : l ≤ u*m) (hl' : l ≤ v*m) (hm : 32 ≤ m) (hum : u ≤ m) (hvm : v ≤ m)
+    {q θ θ' : ℝ} (hq : q ∈ Set.Ioc 0 (1/2:ℝ))
+    (hθ : θ ∈ Set.Icc (1-q^2) (1-q^2/4))
+    (hθ' : θ' ∈ Set.Icc (1-q^2) (1-q^2/4))
+    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hmean' : ((v*m:ℕ):ℝ)*θ'=l)
+    (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ)) (hv' : 1 ≤ ((v*m:ℕ):ℝ)*θ'*(1-θ'))
+    (off : Fin u → Fin m) (off' : Fin v → Fin m)
+    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
+    (hoff' : ∀ j, |((off' j).val:ℝ)-(m:ℝ)/2| ≤ 3)
+    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2)
+    (hdiff' : ∀ j k, |((off' j).val:ℝ)-(off' k).val| ≤ 2) :
+    Real.exp (-(primitivePeriodConstant/q^4*((u:ℝ)/m+(v:ℝ)/m)))/128 ≤
+      FiniteLaw.overlap
+        (conditionalPrimitiveLaw hl q ⟨hq.1, by linarith [hq.2]⟩ (defectTypes (periodSlot off)))
+        (conditionalPrimitiveLaw hl' q ⟨hq.1, by linarith [hq.2]⟩ (defectTypes (periodSlot off'))) := by
+  have hqi : q ∈ Set.Ioo 0 1 := ⟨hq.1, by linarith [hq.2]⟩
+  let P := primitiveMarkLaw q hqi false
+  let R := uniformSubsetLaw l (show l≤Fintype.card (Fin (u*m)) by simpa using hl)
+  let S := uniformSubsetLaw l (show l≤Fintype.card (Fin (v*m)) by simpa using hl')
+  let f := defectAtRank (l:=l) (Finset.univ.image (periodSlot off))
+  let g := defectAtRank (l:=l) (Finset.univ.image (periodSlot off'))
+  let G := averagedDensity R f (primitiveScore q)
+  let H := averagedDensity S g (primitiveScore q)
+  let A := primitivePeriodConstant/q^4*((u:ℝ)/m)
+  let B := primitivePeriodConstant/q^4*((v:ℝ)/m)
+  have h1 := conditionalPeriod_second_moment hl hm hum hq hθ hmean hv off hoff hdiff
+  have h2 := conditionalPeriod_second_moment hl' hm hvm hq hθ' hmean' hv' off' hoff' hdiff'
+  change (P.product l).expect (fun x => G x^2) ≤ 2*Real.exp A at h1
+  change (P.product l).expect (fun x => H x^2) ≤ 2*Real.exp B at h2
+  have ha := averagedDensity_overlap P R S (primitiveScore q)
+    (primitive_score_centered hqi) (primitive_one_add_score_nonneg hqi) f g
+  change 1/32 ≤ (P.product l).expect (fun x => min (G x) (H x)) *
+    (P.product l).expect (fun x => G x^2) * (P.product l).expect (fun x => H x^2) at ha
+  have hG : ∀x, 0 ≤ G x := averagedDensity_nonneg R f _ (primitive_one_add_score_nonneg hqi)
+  have hH : ∀x, 0 ≤ H x := averagedDensity_nonneg S g _ (primitive_one_add_score_nonneg hqi)
+  have ho : 0 ≤ (P.product l).expect (fun x => min (G x) (H x)) :=
+    Finset.sum_nonneg (fun x _ => mul_nonneg ((P.product l).nonneg _) (le_min (hG x) (hH x)))
+  have hm2 : 0 ≤ (P.product l).expect (fun x => H x^2) :=
+    Finset.sum_nonneg (fun x _ => mul_nonneg ((P.product l).nonneg _) (sq_nonneg _))
+  have hb := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left h1 ho) hm2
+  have hc := mul_le_mul_of_nonneg_left h2 (mul_nonneg ho (by positivity : 0≤2*Real.exp A))
+  have he : (P.product l).expect (fun x => min (G x) (H x)) =
+      FiniteLaw.overlap
+        (conditionalPrimitiveLaw hl q hqi (defectTypes (periodSlot off)))
+        (conditionalPrimitiveLaw hl' q hqi (defectTypes (periodSlot off'))) := by
+    unfold FiniteLaw.expect FiniteLaw.overlap
+    apply Finset.sum_congr rfl
+    intro x _
+    rw [conditionalPeriodLaw_density,conditionalPeriodLaw_density]
+    exact mul_min_of_nonneg _ _ ((P.product l).nonneg x)
+  rw [← he]
+  have hd : Real.exp (-(A+B))/128 ≤ (P.product l).expect (fun x => min (G x) (H x)) := by
+    have hx : (1:ℝ)/32 ≤ (P.product l).expect (fun x => min (G x) (H x)) * (4*Real.exp (A+B)) := by
+      calc
+        _ ≤ _ := ha.trans (hb.trans hc)
+        _ = _ := by rw [Real.exp_add]; ring
+    have hp : 0 < 4*Real.exp (A+B) := by positivity
+    calc
+      _ = ((1:ℝ)/32)/(4*Real.exp (A+B)) := by rw [Real.exp_neg]; ring
+      _ ≤ _ := (div_le_iff₀ hp).2 hx
+  convert hd using 1
+  congr 2
+  dsimp [A,B]
+  ring
+
+end TraceReconstruction
+
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- The common count window is admissible for every genuine period count in
+the base case. All hypotheses are elementary size/location inequalities. -/
+theorem primitive_count_window {q θ : ℝ} {m n l : ℕ}
+    (hq : q ∈ Set.Ioc 0 (1/2:ℝ)) (hm : 32 ≤ m)
+    (hscale : 16 ≤ q^3*(m:ℝ))
+    (hnlo : q*(m:ℝ)^2/2 ≤ (n:ℝ)) (hnhi : (n:ℝ) ≤ (m:ℝ)^2)
+    (hmean : (n:ℝ)*θ=l)
+    (hdev : |(l:ℝ)-(n:ℝ)*(1-q^2/2)| ≤ 2*(m:ℝ)) :
+    θ ∈ Set.Icc (1-q^2) (1-q^2/4) ∧
+    1 ≤ (n:ℝ)*θ*(1-θ) ∧
+    Real.sqrt ((n:ℝ)*θ*(1-θ)) ≤ (m:ℝ) ∧
+    ((l:ℝ)-(n:ℝ)*(1-q^2/2))^2 / ((n:ℝ)*(1-q^2/2)*(q^2/2)) ≤ 32/q^3 := by
+  have hm0 : (0:ℝ) < m := by exact_mod_cast (show 0 < m by omega)
+  have hq0 := hq.1
+  have hq2 : q^2 ≤ 1/4 := by nlinarith [hq.2]
+  have hq2p : 0 < q^2 := sq_pos_of_pos hq0
+  have hq3 : 0 < q^3 := pow_pos hq0 3
+  have hn0 : (0:ℝ) < n := lt_of_lt_of_le (by have hq0 := hq.1; positivity) hnlo
+  have hsize : 8*(m:ℝ) ≤ (n:ℝ)*q^2 := by
+    have ha := mul_le_mul_of_nonneg_right hnlo hq2p.le
+    have hb := mul_le_mul_of_nonneg_right hscale hm0.le
+    nlinarith
+  have hθ : θ ∈ Set.Icc (1-q^2) (1-q^2/4) := by
+    have hl := (abs_le.mp hdev).1
+    have hr := (abs_le.mp hdev).2
+    rw [←hmean] at hl hr
+    constructor
+    · nlinarith
+    · nlinarith
+  have ht0 : 0 < θ := by nlinarith [hθ.1]
+  have ht1 : θ < 1 := by nlinarith [hθ.2]
+  have hvlo : (n:ℝ)*q^2/8 ≤ (n:ℝ)*θ*(1-θ) := by
+    have hlow : (1:ℝ)/2 ≤ θ := by linarith [hθ.1]
+    have hhigh : q^2/4 ≤ 1-θ := by linarith [hθ.2]
+    have hp := mul_le_mul hlow hhigh (by positivity : (0:ℝ)≤q^2/4) ht0.le
+    have hp' := mul_le_mul_of_nonneg_left hp hn0.le
+    nlinarith
+  have hv : 1 ≤ (n:ℝ)*θ*(1-θ) := by
+    have hm32 : (32:ℝ) ≤ m := by exact_mod_cast hm
+    linarith
+  have hvhi : (n:ℝ)*θ*(1-θ) ≤ (m:ℝ)^2 := by
+    have hprod : θ*(1-θ) ≤ 1 := by nlinarith [sq_nonneg θ]
+    have hh := mul_le_mul_of_nonneg_left hprod hn0.le
+    nlinarith
+  have hsqrt : Real.sqrt ((n:ℝ)*θ*(1-θ)) ≤ (m:ℝ) := by
+    exact (Real.sqrt_le_iff).mpr ⟨hm0.le,hvhi⟩
+  have hp0 : 0 < 1-q^2/2 := by linarith
+  have hden0 : 0 < (n:ℝ)*(1-q^2/2)*(q^2/2) := by positivity
+  have hden : q^3*(m:ℝ)^2/8 ≤ (n:ℝ)*(1-q^2/2)*(q^2/2) := by
+    have ha := mul_le_mul_of_nonneg_right hnlo hq2p.le
+    have hb := mul_le_mul_of_nonneg_right (show (1:ℝ)/2≤1-q^2/2 by linarith)
+      (mul_nonneg hn0.le (by positivity : (0:ℝ)≤q^2/2))
+    nlinarith
+  have hdev2 : ((l:ℝ)-(n:ℝ)*(1-q^2/2))^2 ≤ 4*(m:ℝ)^2 := by
+    have h := pow_le_pow_left₀ (abs_nonneg _) hdev 2
+    rw [sq_abs] at h
+    nlinarith
+  have hquot : ((l:ℝ)-(n:ℝ)*(1-q^2/2))^2 /
+      ((n:ℝ)*(1-q^2/2)*(q^2/2)) ≤ 32/q^3 := by
+    apply (div_le_div_iff₀ hden0 hq3).mpr
+    have hh := mul_le_mul_of_nonneg_right hdev2 hq3.le
+    nlinarith
+  exact ⟨hθ,hv,hsqrt,hquot⟩
+
+/-- Genuine binomial count mass in the window, uniformly in the vanishing
+working deletion rate, with an absolute explicit exponential cost. -/
+theorem primitive_count_mass_lower {q : ℝ} {m n l : ℕ}
+    (hq : q ∈ Set.Ioc 0 (1/2:ℝ)) (hm : 32 ≤ m)
+    (hscale : 16 ≤ q^3*(m:ℝ))
+    (hnlo : q*(m:ℝ)^2/2 ≤ (n:ℝ)) (hnhi : (n:ℝ) ≤ (m:ℝ)^2)
+    (hdev : |(l:ℝ)-(n:ℝ)*(1-q^2/2)| ≤ 2*(m:ℝ)) :
+    Real.exp (-32/q^3)/(12*(m:ℝ)) ≤ binomialMass n (1-primitiveErasure q) l := by
+  have hm0 : (0:ℝ) < m := by exact_mod_cast (show 0 < m by omega)
+  have hn0 : (0:ℝ) < n := lt_of_lt_of_le (by have hq0 := hq.1; positivity) hnlo
+  let θ : ℝ := (l:ℝ)/n
+  have hmean : (n:ℝ)*θ=l := by dsimp [θ]; field_simp
+  obtain ⟨ht,hv,hsqrt,hquot⟩ := primitive_count_window hq hm hscale hnlo hnhi hmean hdev
+  have hq2 : q^2 ≤ 1/4 := by nlinarith [hq.1,hq.2]
+  have hq2p : 0 < q^2 := sq_pos_of_pos hq.1
+  have ht' : θ ∈ Set.Ioo 0 1 := ⟨by nlinarith [ht.1], by nlinarith [ht.2]⟩
+  have hp : 1-q^2/2 ∈ Set.Ioo 0 1 := ⟨by linarith, by linarith⟩
+  have hl : l ≤ n := by
+    have h := mul_le_mul_of_nonneg_left ht'.2.le hn0.le
+    rw [hmean,mul_one] at h
+    exact_mod_cast h
+  have hmass := binomialMass_near_mean_lower n l ht' hp hl hmean hv
+  have hden : 0 < 12*Real.sqrt ((n:ℝ)*θ*(1-θ)) := by positivity
+  have hexp : Real.exp (-32/q^3) ≤ Real.exp (-((l:ℝ)-(n:ℝ)*(1-q^2/2))^2 /
+      ((n:ℝ)*(1-q^2/2)*(1-(1-q^2/2)))) := by
+    apply Real.exp_le_exp.mpr
+    have he : 1-(1-q^2/2)=q^2/2 := by ring
+    rw [he]
+    simpa only [neg_div] using neg_le_neg hquot
+  calc
+    _ ≤ Real.exp (-((l:ℝ)-(n:ℝ)*(1-q^2/2))^2 /
+        ((n:ℝ)*(1-q^2/2)*(1-(1-q^2/2)))) / (12*Real.sqrt ((n:ℝ)*θ*(1-θ))) := by
+      exact div_le_div₀ (Real.exp_nonneg _) hexp hden (by nlinarith)
+    _ ≤ _ := by simpa only [primitiveErasure] using hmass
+
+
+/-- The actual base count, with no redefinition of the working channel. -/
+def initialSideCount (q : ℝ) (m : ℕ) : ℕ := ⌊q*(m:ℝ)⌋₊
+
+theorem initialSideCount_bounds {q : ℝ} {m : ℕ}
+    (hq : q ∈ Set.Ioc 0 (1/2:ℝ)) (hm : 32 ≤ m)
+    (hscale : 16 ≤ q^3*(m:ℝ)) :
+    q*(m:ℝ)/2 ≤ (initialSideCount q m:ℝ) ∧
+    (initialSideCount q m:ℝ) ≤ q*(m:ℝ) ∧ initialSideCount q m+1 ≤ m := by
+  have hm0 : (0:ℝ) ≤ m := Nat.cast_nonneg _
+  have hm32 : (32:ℝ) ≤ m := by exact_mod_cast hm
+  have hq0 := hq.1
+  have hq2 : q^2 ≤ 1 := by nlinarith [hq.2]
+  have hqm : 2 ≤ q*(m:ℝ) := by
+    have hh := mul_le_mul_of_nonneg_right hq2 (mul_nonneg hq0.le hm0)
+    nlinarith
+  have hlow := Nat.lt_floor_add_one (q*(m:ℝ))
+  have hhigh := Nat.floor_le (mul_nonneg hq0.le hm0)
+  change q*(m:ℝ) < (initialSideCount q m:ℝ)+1 at hlow
+  change (initialSideCount q m:ℝ) ≤ q*(m:ℝ) at hhigh
+  refine ⟨by linarith,hhigh,?_⟩
+  have hh := mul_le_mul_of_nonneg_right hq.2 hm0
+  have : (initialSideCount q m:ℝ)+1 ≤ (m:ℝ) := by linarith
+  exact_mod_cast this
+
+/-- An injective enumeration of m common integer counts; no asymptotic
+cardinality argument is left implicit. -/
+def commonCountIndex {m : ℕ} (k : ℕ) (p : ℝ) (hp : p ∈ Set.Icc 0 1) (j : Fin m) :
+    Fin ((k+1)*m+1) :=
+  ⟨⌊p*((k*m:ℕ):ℝ)⌋₊+j.val,by
+    have ha : (⌊p*((k*m:ℕ):ℝ)⌋₊:ℝ) ≤ ((k*m:ℕ):ℝ) := by
+      exact (Nat.floor_le (mul_nonneg hp.1 (Nat.cast_nonneg _))).trans
+        (mul_le_of_le_one_left (Nat.cast_nonneg _) hp.2)
+    have ha' : ⌊p*((k*m:ℕ):ℝ)⌋₊ ≤ k*m := by exact_mod_cast ha
+    have hj := j.isLt
+    rw [Nat.add_mul,one_mul]
+    omega⟩
+
+theorem commonCountIndex_injective {m : ℕ} (k : ℕ) (p : ℝ) (hp : p ∈ Set.Icc 0 1) :
+    Function.Injective (commonCountIndex (m:=m) k p hp) := by
+  intro a b h
+  have hv := congrArg Fin.val h
+  simp only [commonCountIndex,Nat.add_left_cancel_iff] at hv
+  exact Fin.ext hv
+
+theorem commonCountIndex_deviation {m : ℕ} (k : ℕ) (p : ℝ) (hp : p ∈ Set.Icc 0 1)
+    (j : Fin m) :
+    |((commonCountIndex k p hp j).val:ℝ)-((k*m:ℕ):ℝ)*p| ≤ 2*(m:ℝ) ∧
+    |((commonCountIndex k p hp j).val:ℝ)-(((k+1)*m:ℕ):ℝ)*p| ≤ 2*(m:ℝ) := by
+  have hm : (1:ℝ) ≤ m := by exact_mod_cast (Nat.succ_le_of_lt (Nat.zero_lt_of_lt j.isLt))
+  have hj : (j.val:ℝ) < m := by exact_mod_cast j.isLt
+  have hj0 : (0:ℝ) ≤ j.val := Nat.cast_nonneg _
+  have ha := Nat.floor_le (mul_nonneg hp.1 (Nat.cast_nonneg (k*m)))
+  have hb := Nat.lt_floor_add_one (p*((k*m:ℕ):ℝ))
+  have hp0 := hp.1
+  have hp1 := hp.2
+  have hmp : 0 ≤ (m:ℝ)*p := mul_nonneg (Nat.cast_nonneg _) hp0
+  have hmp' : (m:ℝ)*p ≤ m := mul_le_of_le_one_right (Nat.cast_nonneg _) hp1
+  change |((⌊p*((k*m:ℕ):ℝ)⌋₊+j.val:ℕ):ℝ)-((k*m:ℕ):ℝ)*p| ≤ _ ∧
+    |((⌊p*((k*m:ℕ):ℝ)⌋₊+j.val:ℕ):ℝ)-(((k+1)*m:ℕ):ℝ)*p| ≤ _
+  have hn : (((k+1)*m:ℕ):ℝ)=((k*m:ℕ):ℝ)+(m:ℝ) := by push_cast; ring
+  rw [hn]
+  push_cast
+  constructor <;> apply abs_le.mpr <;> constructor <;> push_cast at ha hb <;> nlinarith
+
+end TraceReconstruction
+
+
+
+namespace TraceReconstruction
+open scoped Classical
+
+theorem ordinaryRun_replicate (n : ℕ) :
+    (List.replicate n [false,true]).flatten = ordinaryRun n := by
+  induction n with
+  | zero => rfl
+  | succ n ih => simp [List.replicate_succ,ordinaryRun,ih]
+
+/-- Unique primitive parsing identifies the rank model with the literal
+period words in the manuscript, not just a law on unlabeled locations. -/
+theorem primitive_period_word {m : ℕ} (off : Fin m) :
+    (List.ofFn (fun j : Fin m => primitiveBlock (decide (j=off)))).flatten =
+      initialPeriod m off.val := by
+  induction m with
+  | zero => exact Fin.elim0 off
+  | succ m ih =>
+    refine Fin.cases ?_ (fun d => ?_) off
+    · simp [List.ofFn_succ,primitiveBlock,initialPeriod,ordinaryRun,ordinaryRun_replicate]
+    · rw [List.ofFn_succ,List.flatten_cons]
+      have hf : primitiveBlock (decide ((0:Fin (m+1))=d.succ))=[false,true] := by
+        simp [primitiveBlock,Ne.symm (Fin.succ_ne_zero d)]
+      rw [hf]
+      simp only [Fin.succ_inj]
+      rw [ih d]
+      simp [initialPeriod,ordinaryRun,Nat.sub_sub,Nat.add_comm]
+
+/-- Membership in the true periodic defect set is exactly the within-period
+slot test, uniformly for all deterministic period types. -/
+theorem periodSlot_range_iff {u m : ℕ} (off : Fin u → Fin m) (i : Fin u) (j : Fin m) :
+    (⟨i.val*m+j.val,by
+      have hi := Nat.mul_le_mul_right m (Nat.succ_le_of_lt i.isLt)
+      have hj := j.isLt
+      nlinarith⟩ : Fin (u*m)) ∈ Set.range (periodSlot off) ↔ j=off i := by
+  constructor
+  · rintro ⟨a,ha⟩
+    have hval := congrArg Fin.val ha
+    change a.val*m+(off a).val=i.val*m+j.val at hval
+    have hm : 0 < m := Nat.zero_lt_of_lt j.isLt
+    have hdiv := congrArg (fun k : ℕ => k/m) hval
+    simp only [Nat.mul_comm _ m, Nat.mul_add_div hm, Nat.div_eq_of_lt (off a).isLt,
+      Nat.div_eq_of_lt j.isLt,add_zero] at hdiv
+    have hai : a=i := Fin.ext hdiv
+    subst a
+    apply Fin.ext
+    omega
+  · intro h
+    refine ⟨i,?_⟩
+    apply Fin.ext
+    simp only [periodSlot]
+    exact congrArg (fun k:Fin m => i.val*m+k.val) h.symm
+
+/-- The simulator's actual primitive list is the unpadded source gadget. -/
+theorem periodic_primitive_word {u m : ℕ} (off : Fin u → Fin m) :
+    (((List.ofFn (defectTypes (periodSlot off))).map primitiveBlock).flatten) =
+      periodString m u off := by
+  rw [List.map_ofFn,List.ofFn_mul]
+  rw [List.flatten_flatten,List.map_ofFn]
+  unfold periodString
+  congr 1
+  apply congrArg List.ofFn
+  funext i
+  simp only [Function.comp_apply]
+  have he : (fun j:Fin m => primitiveBlock (defectTypes (periodSlot off)
+      ⟨i.val*m+j.val,by
+        have hi := Nat.mul_le_mul_right m (Nat.succ_le_of_lt i.isLt)
+        have hj := j.isLt
+        nlinarith⟩)) = fun j:Fin m => primitiveBlock (decide (j=off i)) := by
+    funext j
+    simp only [defectTypes,periodSlot_range_iff]
+  rw [he]
+  exact primitive_period_word (off i)
+
+theorem periodString_constant {u m : ℕ} (off : Fin m) :
+    periodString m u (fun _ => off) = pureAtomRun (initialPeriod m off.val) u := by
+  simp only [periodString,List.ofFn_const,pureAtomRun]
+
+end TraceReconstruction
+
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem primitive_count_admissible {q : ℝ} {m n l : ℕ}
+    (hq : q ∈ Set.Ioc 0 (1/2:ℝ)) (hm : 32 ≤ m)
+    (hscale : 16 ≤ q^3*(m:ℝ))
+    (hnlo : q*(m:ℝ)^2/2 ≤ (n:ℝ)) (hnhi : (n:ℝ) ≤ (m:ℝ)^2)
+    (hdev : |(l:ℝ)-(n:ℝ)*(1-q^2/2)| ≤ 2*(m:ℝ)) :
+    ∃ θ : ℝ, l ≤ n ∧ θ ∈ Set.Icc (1-q^2) (1-q^2/4) ∧
+      (n:ℝ)*θ=l ∧ 1 ≤ (n:ℝ)*θ*(1-θ) := by
+  have hm0 : (0:ℝ) < m := by exact_mod_cast (show 0 < m by omega)
+  have hq0 := hq.1
+  have hn0 : (0:ℝ) < n := lt_of_lt_of_le (by positivity) hnlo
+  let θ : ℝ := (l:ℝ)/n
+  have hmean : (n:ℝ)*θ=l := by dsimp [θ]; field_simp
+  obtain ⟨ht,hv,_,_⟩ := primitive_count_window hq hm hscale hnlo hnhi hmean hdev
+  have hl : l ≤ n := by
+    have ht1 : θ ≤ 1 := by nlinarith [ht.2,sq_nonneg q]
+    have hh := mul_le_mul_of_nonneg_left ht1 hn0.le
+    rw [hmean,mul_one] at hh
+    exact_mod_cast hh
+  exact ⟨θ,hl,ht,hmean,hv⟩
+
+/-- The common-mass constant depends only on the working deletion rate. -/
+def initialOverlapLower (q : ℝ) : ℝ :=
+  Real.exp (-(32+3*primitivePeriodConstant)/q^3)/1536
+
+theorem initialOverlapLower_pos (q : ℝ) : 0 < initialOverlapLower q := by
+  unfold initialOverlapLower
+  positivity
+
+theorem initialOverlapLower_lt_eighth {q : ℝ} (hq : 0 < q) :
+    initialOverlapLower q < 1/8 := by
+  have hx : -(32+3*primitivePeriodConstant)/q^3 ≤ 0 := by
+    have hc := primitivePeriodConstant_nonneg
+    apply div_nonpos_of_nonpos_of_nonneg <;> nlinarith [pow_pos hq 3]
+  have he : Real.exp (-(32+3*primitivePeriodConstant)/q^3) ≤ 1 := by
+    simpa using Real.exp_le_exp.mpr hx
+  unfold initialOverlapLower
+  linarith
+
+/-- Unconditioned common mass of the two genuine count-and-mark simulators.
+This includes all retained counts, not a conditioned or changed channel. -/
+theorem initial_counted_overlap {q : ℝ} {m : ℕ}
+    (hq : q ∈ Set.Ioc 0 (1/2:ℝ)) (hm : 32 ≤ m)
+    (hscale : 16 ≤ q^3*(m:ℝ)) (off : Fin m)
+    (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 3) :
+    let k := initialSideCount q m
+    initialOverlapLower q ≤ FiniteLaw.overlap
+      (countedPrimitiveLaw (N:=(k+1)*m) (Nat.mul_le_mul_right m (Nat.le_succ k))
+        ⟨hq.1,by linarith [hq.2]⟩ (defectTypes (periodSlot (u:=k) (fun _ => off))))
+      (countedPrimitiveLaw (N:=(k+1)*m) le_rfl
+        ⟨hq.1,by linarith [hq.2]⟩ (defectTypes (periodSlot (u:=k+1) (fun _ => off)))) := by
+  let k := initialSideCount q m
+  let N := (k+1)*m
+  have hq0 := hq.1
+  have hqi : q ∈ Set.Ioo 0 1 := ⟨hq.1,by linarith [hq.2]⟩
+  have hm0 : (0:ℝ) < m := by exact_mod_cast (show 0 < m by omega)
+  have hq2 : q^2 ≤ 1/4 := by nlinarith [hq.2]
+  let p := 1-primitiveErasure q
+  have hp : p ∈ Set.Icc 0 1 := primitiveRetention_mem hqi
+  have hp' : p=1-q^2/2 := rfl
+  obtain ⟨hklo,hkhi,hk1⟩ := initialSideCount_bounds hq hm hscale
+  change q*(m:ℝ)/2 ≤ (k:ℝ) at hklo
+  change (k:ℝ) ≤ q*(m:ℝ) at hkhi
+  change k+1 ≤ m at hk1
+  have hk : k ≤ m := by omega
+  have hnlo : q*(m:ℝ)^2/2 ≤ ((k*m:ℕ):ℝ) := by
+    have hh := mul_le_mul_of_nonneg_right hklo hm0.le
+    push_cast
+    nlinarith
+  have hnhi : ((k*m:ℕ):ℝ) ≤ (m:ℝ)^2 := by
+    exact_mod_cast (show k*m ≤ m^2 by simpa [pow_two] using Nat.mul_le_mul_right m hk)
+  have hNlo : q*(m:ℝ)^2/2 ≤ (N:ℝ) := by
+    exact hnlo.trans (by exact_mod_cast Nat.mul_le_mul_right m (Nat.le_succ k))
+  have hNhi : (N:ℝ) ≤ (m:ℝ)^2 := by
+    exact_mod_cast (show N ≤ m^2 by simpa [N,pow_two] using Nat.mul_le_mul_right m hk1)
+  have hqm : 1 ≤ q*(m:ℝ) := by
+    have hh := mul_le_mul_of_nonneg_right (show q^2≤1 by linarith) (mul_nonneg hq0.le hm0.le)
+    nlinarith
+  have hrate : (k:ℝ)/(m:ℝ)+((k+1:ℕ):ℝ)/(m:ℝ) ≤ 3*q := by
+    rw [←add_div]
+    apply (div_le_iff₀ hm0).mpr
+    push_cast
+    nlinarith
+  have harg : primitivePeriodConstant/q^4*((k:ℝ)/m+((k+1:ℕ):ℝ)/m) ≤
+      3*primitivePeriodConstant/q^3 := by
+    calc
+      _ ≤ primitivePeriodConstant/q^4*(3*q) :=
+        mul_le_mul_of_nonneg_left hrate (div_nonneg primitivePeriodConstant_nonneg (by positivity))
+      _ = _ := by field_simp
+  let C₁ := boundedCountLaw (N:=N) (Nat.mul_le_mul_right m (Nat.le_succ k)) p hp
+  let C₂ := boundedCountLaw (N:=N) le_rfl p hp
+  let K := boundedConditionalPrimitiveLaw hqi (defectTypes (periodSlot (u:=k) (fun _ => off))) (N:=N)
+  let L := boundedConditionalPrimitiveLaw hqi (defectTypes (periodSlot (u:=k+1) (fun _ => off))) (N:=N)
+  let f : Fin (N+1) → ℝ := fun l => min (C₁.mass l) (C₂.mass l) * FiniteLaw.overlap (K l) (L l)
+  let a : ℝ := Real.exp (-32/q^3)/(12*(m:ℝ))
+  let b : ℝ := Real.exp (-(3*primitivePeriodConstant)/q^3)/128
+  have ha0 : 0 ≤ a := by dsimp [a]; positivity
+  have hb0 : 0 ≤ b := by dsimp [b]; positivity
+  have hf0 : ∀l, 0 ≤ f l := fun l =>
+    mul_nonneg (le_min (C₁.nonneg l) (C₂.nonneg l)) (FiniteLaw.overlap_nonneg _ _)
+  let ix : Fin m → Fin (N+1) := commonCountIndex k p hp
+  have hterm : ∀j : Fin m, a*b ≤ f (ix j) := by
+    intro j
+    let l := (ix j).val
+    have hd := commonCountIndex_deviation k p hp j
+    change |(l:ℝ)-((k*m:ℕ):ℝ)*p| ≤ 2*(m:ℝ) ∧ |(l:ℝ)-(N:ℝ)*p| ≤ 2*(m:ℝ) at hd
+    rw [hp'] at hd
+    obtain ⟨θ,hl,hθ,hmean,hv⟩ := primitive_count_admissible hq hm hscale hnlo hnhi hd.1
+    obtain ⟨θ',hl',hθ',hmean',hv'⟩ := primitive_count_admissible hq hm hscale hNlo hNhi hd.2
+    have hm1 := primitive_count_mass_lower hq hm hscale hnlo hnhi hd.1
+    have hm2 := primitive_count_mass_lower hq hm hscale hNlo hNhi hd.2
+    have hmin : a ≤ min (C₁.mass (ix j)) (C₂.mass (ix j)) := le_min hm1 hm2
+    have hc := conditionalPeriod_overlap hl hl' hm hk hk1 hq hθ hθ' hmean hmean' hv hv'
+      (fun _ => off) (fun _ => off) (fun _ => hoff) (fun _ => hoff)
+      (fun _ _ => by simp) (fun _ _ => by simp)
+    have hb : b ≤ FiniteLaw.overlap (K (ix j)) (L (ix j)) := by
+      dsimp only [K,L,boundedConditionalPrimitiveLaw]
+      rw [dite_eq_left hl,dite_eq_left hl']
+      refine le_trans ?_ hc
+      apply div_le_div_of_nonneg_right _ (by norm_num : (0:ℝ)≤128)
+      apply Real.exp_le_exp.mpr
+      simpa only [neg_div] using neg_le_neg harg
+    exact mul_le_mul hmin hb hb0 (le_min (C₁.nonneg _) (C₂.nonneg _))
+  have hsub : (∑j : Fin m, f (ix j)) ≤ ∑l : Fin (N+1), f l := by
+    have hi := commonCountIndex_injective (m:=m) k p hp
+    calc
+      _ = ∑l ∈ Finset.univ.image ix, f l := by
+        rw [Finset.sum_image]
+        exact fun a _ b _ hab => hi hab
+      _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _)
+        (fun l _ _ => hf0 l)
+  have hsum : (m:ℝ)*(a*b) ≤ ∑l : Fin (N+1), f l := by
+    calc
+      _ = ∑_j : Fin m, a*b := by simp
+      _ ≤ ∑j : Fin m, f (ix j) := Finset.sum_le_sum (fun j _ => hterm j)
+      _ ≤ _ := hsub
+  have he : (m:ℝ)*(a*b)=initialOverlapLower q := by
+    dsimp [a,b,initialOverlapLower]
+    rw [div_mul_div_comm,←Real.exp_add]
+    have hx : -32/q^3 + -(3*primitivePeriodConstant)/q^3 = -(32+3*primitivePeriodConstant)/q^3 := by ring
+    rw [hx]
+    field_simp
+    ring
+  rw [he] at hsum
+  exact hsum.trans (FiniteLaw.overlap_sigma_lower C₁ C₂ K L)
+
+/-- Data processing for common exact simulators, stated as a common-mass
+inequality and allowing unequal original lengths. -/
+theorem simulations_overlap_le {α : Type*} [Fintype α] {q : ℝ}
+    (hq : q ∈ Set.Icc 0 1) (P Q : FiniteLaw α) (decode : α → List Bool)
+    (x y : List Bool) (hP : SimulatesTrace q P decode x)
+    (hQ : SimulatesTrace q Q decode y) :
+    P.overlap Q ≤ listTraceOverlap q hq x y := by
+  have h := approximations_trace_tv hq P Q decode x y 0 0
+    (max x.length y.length) (le_max_left _ _) (le_max_right _ _)
+    hP.approximate hQ.approximate
+  simp only [add_zero] at h
+  unfold listTraceOverlap
+  rw [FiniteLaw.overlap_eq_one_sub_tv, FiniteLaw.overlap_eq_one_sub_tv]
+  linarith
+
+/-- The initial hidden-bit overlap for ordinary complete deletion traces of
+consecutive pure runs. This is the unconditioned original channel. -/
+theorem initial_pure_trace_overlap {q : ℝ} {m : ℕ}
+    (hq : q ∈ Set.Ioc 0 (1/2:ℝ)) (hm : 32 ≤ m)
+    (hscale : 16 ≤ q^3*(m:ℝ)) (off : Fin m)
+    (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 3) :
+    let k := initialSideCount q m
+    initialOverlapLower q ≤ listTraceOverlap q ⟨hq.1.le, by linarith [hq.2]⟩
+      (pureAtomRun (initialPeriod m off.val) k)
+      (pureAtomRun (initialPeriod m off.val) (k+1)) := by
+  let k := initialSideCount q m
+  have hqi : q ∈ Set.Ioo 0 1 := ⟨hq.1,by linarith [hq.2]⟩
+  have hca := countedPrimitiveLaw_simulates (N:=(k+1)*m)
+    (Nat.mul_le_mul_right m (Nat.le_succ k)) hqi
+    (defectTypes (periodSlot (u:=k) (fun _ => off)))
+  have hcb := countedPrimitiveLaw_simulates (N:=(k+1)*m) le_rfl hqi
+    (defectTypes (periodSlot (u:=k+1) (fun _ => off)))
+  rw [periodic_primitive_word,periodString_constant] at hca hcb
+  exact (initial_counted_overlap hq hm hscale off hoff).trans
+    (simulations_overlap_le ⟨hq.1.le,by linarith [hq.2]⟩ _ _ _ _ _ hca hcb)
+
+end TraceReconstruction
+
+
+
+/-! From the actual displaced period words to their conditional oracle
+moments.  The coefficient bounds are the rank kernels,
+not assumptions on an abstract pair. -/
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- An interior period slot admits the three literal shifts. -/
+def oneStepOffset {m : ℕ} (off : Fin m)
+    (hi : 1  ≤  off.val ∧ off.val + 1 < m) (a : Fin 3) : Fin m :=
+  ⟨off.val+a.val-1,by have ha := a.isLt; omega⟩
+
+theorem oneStepOffset_add_one {m : ℕ} (off : Fin m)
+    (hi : 1  ≤  off.val ∧ off.val + 1 < m) (a : Fin 3) :
+    (oneStepOffset off hi a).val+1=off.val+a.val := by
+  dsimp [oneStepOffset]
+  omega
+
+theorem oneStepOffset_distance {m : ℕ} (off : Fin m)
+    (hi : 1  ≤  off.val ∧ off.val + 1 < m) (a : Fin 3) :
+    |((oneStepOffset off hi a).val:ℝ)-(off.val:ℝ)|  ≤  1 := by
+  have he := oneStepOffset_add_one off hi a
+  have ha : (a.val:ℝ)  ≤  2 := by exact_mod_cast (show a.val ≤ 2 by omega)
+  have ha0 : (0:ℝ)  ≤  a.val := Nat.cast_nonneg _
+  have he' : ((oneStepOffset off hi a).val:ℝ)+1=(off.val:ℝ)+a.val := by exact_mod_cast he
+  rw [abs_le]
+  constructor <;> linarith
+
+theorem constantPeriod_safe {u m : ℕ} (hm : 3  ≤  m) (off : Fin m)
+    (hi : 1  ≤  off.val ∧ off.val + 1 < m) :
+    SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)) := by
+  constructor
+  · intro i
+    change 1  ≤  i.val*m+off.val ∧ i.val*m+off.val+1<u*m
+    have h := Nat.mul_le_mul_right m (Nat.succ_le_of_lt i.isLt)
+    constructor <;> nlinarith [hi.1,hi.2]
+  · intro i j hij
+    change i.val*m+off.val+3 ≤ j.val*m+off.val
+    have h := Nat.mul_le_mul_right m (Nat.succ_le_of_lt hij)
+    nlinarith
+
+theorem shifted_constantPeriod {u m : ℕ} (off : Fin m)
+    (hi : 1  ≤  off.val ∧ off.val + 1 < m)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
+    (a : Fin u → Fin 3) :
+    shiftedTag (periodSlot (fun _ => off)) hsafe a =
+      periodSlot (fun i => oneStepOffset off hi (a i)) := by
+  funext i
+  apply Fin.ext
+  change i.val*m+off.val+(a i).val-1=i.val*m+(off.val+(a i).val-1)
+  omega
+
+/-- Arbitrary deterministic motif orientations remain genuine one-defect
+periods with the exact geometry used in the high-degree rank bound. -/
+theorem signedPeriod_geometry {u m : ℕ} (off : Fin m)
+    (hi : 1  ≤  off.val ∧ off.val + 1 < m)
+    (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 2)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
+    (s : ℕ) (X : ℤ) (τ : ℝ) :
+    ∃ off' : Fin u → Fin m,
+      signedPeriodDefects off hsafe s X τ = periodSlot off' ∧
+      (∀j, |((off' j).val:ℝ)-(m:ℝ)/2| ≤ 3) ∧
+      (∀j k, |((off' j).val:ℝ)-(off' k).val| ≤ 2) := by
+  let a : Fin u → Fin 3 := fun j =>
+    motifShift (motifIndicator s ((j.val:ℤ)-X)) ((prouhetSeries s).coeff ((j.val:ℤ)-X)) τ
+  refine ⟨fun j => oneStepOffset off hi (a j),shifted_constantPeriod off hi hsafe a,?_,?_⟩
+  · intro j
+    have h := oneStepOffset_distance off hi (a j)
+    have ht := abs_add_le (((oneStepOffset off hi (a j)).val:ℝ)-(off.val:ℝ))
+      ((off.val:ℝ)-(m:ℝ)/2)
+    rw [sub_add_sub_cancel] at ht
+    linarith
+  · intro j k
+    have hj := oneStepOffset_distance off hi (a j)
+    have hk := oneStepOffset_distance off hi (a k)
+    rw [abs_le] at hj hk ⊢
+    constructor <;> linarith
+
+
+/-- Every moment of the actual primitive score has the source uniform
+bound. -/
+theorem primitive_score_evenMoment {q : ℝ} (hq : q∈Set.Ioo 0 1) (k : ℕ) :
+    (primitiveMarkLaw q hq false).expect (fun z => |primitiveScore q z|^(2*k))  ≤ 
+      (2/q)^(2*k) := by
+  calc
+    _  ≤  (primitiveMarkLaw q hq false).expect (fun _ => (2/q)^(2*k)) := by
+      apply Finset.sum_le_sum
+      intro z _
+      exact mul_le_mul_of_nonneg_left
+        (pow_le_pow_left₀ (abs_nonneg _) (primitive_score_bound hq z) _) ((primitiveMarkLaw q hq false).nonneg z)
+    _ = _ := (primitiveMarkLaw q hq false).expect_const _
+
+/-- Hypercontractivity now applies directly to the actual signed initial
+density difference under its ordinary-mark reference law. -/
+theorem initial_moment_of_energy {u m l : ℕ} (hl : l ≤ u*m) (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
+    (π : IntSeries) (s k : ℕ) (hk : 1 ≤ k) {q : ℝ} (hq : q∈Set.Ioo 0 1) :
+    ((primitiveMarkLaw q hq false).product l).evenMomentSq
+      (multilinear (fun σ => mixedMotifCoefficient hl off hsafe π s 1 σ-
+        mixedMotifCoefficient hl off hsafe π s (-1) σ) (primitiveScore q)) k  ≤ 
+      multilinearEnergy (fun σ => mixedMotifCoefficient hl off hsafe π s 1 σ-
+        mixedMotifCoefficient hl off hsafe π s (-1) σ) ((6*(2*k:ℕ):ℝ)*(2/q)) := by
+  apply multilinear_moment_sq_scaled _ _ k hk
+  · exact primitive_score_centered hq
+  · exact div_pos (by norm_num) hq.1
+  · exact primitive_score_evenMoment hq k
+
+
+/-- Jensen's inequality on each homogeneous degree. -/
+theorem degree_square_mixture_le {Ω : Type*} [Fintype Ω] {n : ℕ}
+    (P : FiniteLaw Ω) (A : Ω → (Fin n → Bool) → ℝ) (r : ℕ) :
+    (∑σ : Fin n → Bool, if monomialDegree σ=r then
+      (P.expect (fun ω => A ω σ))^2 else 0)  ≤ 
+    P.expect (fun ω => ∑σ : Fin n → Bool,
+      if monomialDegree σ=r then (A ω σ)^2 else 0) := by
+  unfold FiniteLaw.expect
+  simp_rw [Finset.mul_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_le_sum
+  intro σ _
+  by_cases h : monomialDegree σ=r
+  · simp only [ite_eq_left h]
+    have H := weighted_sum_sq_le Finset.univ P.mass (fun ω => A ω σ) (fun ω => P.nonneg ω)
+    simpa only [P.total,one_mul] using H
+  · simp only [ite_eq_right h,mul_zero,Finset.sum_const_zero,le_refl]
+
+theorem degree_square_sub_le {n : ℕ} (A B : (Fin n → Bool) → ℝ) (r : ℕ) :
+    (∑σ : Fin n → Bool, if monomialDegree σ=r then (A σ-B σ)^2 else 0)  ≤ 
+    2*(∑σ : Fin n → Bool, if monomialDegree σ=r then A σ^2 else 0)+
+    2*(∑σ : Fin n → Bool, if monomialDegree σ=r then B σ^2 else 0) := by
+  simp only [Finset.mul_sum,←Finset.sum_add_distrib]
+  apply Finset.sum_le_sum
+  intro σ _
+  split_ifs <;> nlinarith [sq_nonneg (A σ+B σ)]
+
+theorem mixedMotifCoefficient_expect {u m l : ℕ} (hl : l ≤ u*m) (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
+    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
+    (s : ℕ) (τ : ℝ) (σ : Fin l → Bool) :
+    mixedMotifCoefficient hl off hsafe π s τ σ =
+      (seriesProbabilityLaw π hπ hπsum).expect (fun X =>
+        averagedCoefficient (uniformSubsetLaw l (by simpa using hl))
+          (defectAtRank (Finset.univ.image (signedPeriodDefects off hsafe s X.val τ))) σ) := by
+  unfold mixedMotifCoefficient FiniteLaw.expect seriesProbabilityLaw
+  exact (Finset.sum_coe_sort _ _).symm
+
+theorem mixedMotifCoefficient_empty {u m l : ℕ} (hl : l ≤ u*m) (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
+    (π : IntSeries) (hπsum : ∑X∈π.coeff.support,π.coeff X=1) (s : ℕ) (τ : ℝ) :
+    mixedMotifCoefficient hl off hsafe π s τ (fun _ => false)=1 := by
+  simp only [mixedMotifCoefficient,averagedCoefficient_empty,mul_one,hπsum]
+
+/-- Uniform high-degree estimate after the real hidden-anchor mixture. -/
+theorem mixedMotif_degree_le {u m l r : ℕ} (hl : l ≤ u*m) (hr : 0 < r)
+    (hru : r ≤ u) (hm : 32 ≤ m) (hum : u ≤ m) (off : Fin m)
+    (hi : 1 ≤ off.val ∧ off.val+1 < m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 2)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
+    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
+    (s : ℕ) (τ : ℝ) {q θ : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
+    (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
+    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ)) :
+    (∑σ : Fin l → Bool, if monomialDegree σ=r then
+      (mixedMotifCoefficient hl off hsafe π s τ σ)^2 else 0)  ≤ 
+      ((3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/((m:ℝ)*r)))^r := by
+  simp_rw [mixedMotifCoefficient_expect hl off hsafe π hπ hπsum]
+  refine (degree_square_mixture_le _ _ r).trans ?_
+  calc
+    _  ≤  (seriesProbabilityLaw π hπ hπsum).expect (fun _ =>
+        ((3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/((m:ℝ)*r)))^r) := by
+      apply FiniteLaw.expect_mono
+      intro X
+      obtain ⟨d,hd,hnear,hclose⟩ := signedPeriod_geometry off hi hoff hsafe s X.val τ
+      rw [hd]
+      exact period_degree_square_sum hl hr hru hm hum hq hθ hmean hv d hnear hclose
+    _ = _ := FiniteLaw.expect_const _ _
+
+/-- There are never more selected ranks than actual defect periods, even
+in the hidden-anchor mixture. -/
+theorem mixedMotif_degree_vanish {u m l r : ℕ} (hl : l ≤ u*m) (hru : u < r) (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
+    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
+    (s : ℕ) (τ : ℝ) :
+    (∑σ : Fin l → Bool, if monomialDegree σ=r then
+      (mixedMotifCoefficient hl off hsafe π s τ σ)^2 else 0)=0 := by
+  apply le_antisymm
+  · simp_rw [mixedMotifCoefficient_expect hl off hsafe π hπ hπsum]
+    refine (degree_square_mixture_le _ _ r).trans ?_
+    simp_rw [defect_degree_bound hl _ (signedPeriodDefects_strictMono off hsafe s _ τ) hru]
+    rw [FiniteLaw.expect_const]
+  · exact Finset.sum_nonneg (fun σ _ => by split_ifs <;> positivity)
+
+/-- The signed difference has no constant term. -/
+theorem signedMotif_degree_zero {u m l : ℕ} (hl : l ≤ u*m) (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
+    (π : IntSeries) (hπsum : ∑X∈π.coeff.support,π.coeff X=1) (s : ℕ) :
+    (∑σ : Fin l → Bool, if monomialDegree σ=0 then
+      (mixedMotifCoefficient hl off hsafe π s 1 σ-
+        mixedMotifCoefficient hl off hsafe π s (-1) σ)^2 else 0)=0 := by
+  simp_rw [monomialDegree_eq_zero_iff]
+  simp [mixedMotifCoefficient_empty hl off hsafe π hπsum]
+
+/-- Low-degree cancellation is for exactly the coefficients of the
+conditional motif densities, without a rank surrogate at the boundary. -/
+theorem signedMotif_degree_cancellation {u m l r : ℕ}
+    (hu : 0 < u) (hm : 128 ≤ m) (hr : 0 < r) (hl : l ≤ u*m)
+    (off : Fin m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 3)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
+    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
+    (s : ℕ) (hcentral : CentralMotifSupport u s π)
+    {θ : ℝ} (hθ : θ∈Set.Ioo 0 1) (hhalf : 1/2 ≤ θ)
+    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
+    (hM : ((2*(2^s):ℕ):ℝ)<(u:ℝ)/(3*(r+1:ℝ))) :
+    (∑σ : Fin l → Bool, if monomialDegree σ=r then
+      (mixedMotifCoefficient hl off hsafe π s 1 σ-
+        mixedMotifCoefficient hl off hsafe π s (-1) σ)^2 else 0)  ≤ 
+      8*(Fintype.card (Fin r → GapDifference):ℝ)*
+        (∑k : Fin r → GapDifference, if Odd (firstDifferenceCount k) then
+          signedPatternBound u m r s π θ k else 0) +
+      2*((Fintype.card (IncreasingTuple u r):ℝ)*
+        (2*(((r+1:ℕ):ℝ)*θ^r/binomialMass (u*m) θ l * (2*Real.exp (-(m:ℝ)/(1024*u))))))^2 := by
+  refine (sum_degree_coeff_sq_le _).trans ?_
+  simp_rw [mixedMotifCoefficient_rank_difference]
+  exact initial_signed_energy hu hm hr hl off hoff hsafe π hπ hπsum s hcentral
+    hθ hhalf hmean hv hM
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+open FiniteLaw
+
+/-- A finite tail is no larger than the full geometric tail. -/
+theorem finite_geometric_tail_le {a : ℝ} (ha : 0 ≤ a) (ha1 : a < 1) (N R : ℕ) :
+    (∑r ∈ Finset.range N, if R ≤ r then a^r else 0) ≤ a^R/(1-a) := by
+  have ht : HasSum (fun n : ℕ => a^(n+R)) (a^R/(1-a)) := by
+    simpa only [pow_add, mul_comm, div_eq_mul_inv] using
+      (hasSum_geometric_of_lt_one ha ha1).mul_left (a^R)
+  have hsum : (∑r ∈ Finset.range N, if R ≤ r then a^r else 0) =
+      ∑r ∈ (Finset.range N).filter (R ≤ ·), a^r := by
+    rw [Finset.sum_filter]
+  rw [hsum]
+  calc
+    _ = ∑r ∈ ((Finset.range N).filter (R ≤ ·)).image (fun r => r-R), a^(r+R) := by
+      rw [Finset.sum_image]
+      · apply Finset.sum_congr rfl
+        intro r hr
+        rw [Nat.sub_add_cancel (Finset.mem_filter.mp hr).2]
+      · intro r hr s hs h
+        dsimp only at h
+        have hrR := (Finset.mem_filter.mp hr).2
+        have hsR := (Finset.mem_filter.mp hs).2
+        omega
+    _ ≤ _ := by
+      rw [←ht.tsum_eq]
+      exact ht.summable.sum_le_tsum _ (fun _ _ => by positivity)
+
+/-- Splitting the exact homogeneous expansion, not a surrogate polynomial. -/
+theorem multilinearEnergy_low_high {n R : ℕ} (A : (Fin n → Bool) → ℝ)
+    (w a K : ℝ) (hw : 0 ≤ w) (ha : 0 ≤ a) (ha1 : a < 1) (hK : 0 ≤ K)
+    (L : ℕ → ℝ)
+    (hlow : ∀r < R, (∑σ : Fin n → Bool, if monomialDegree σ=r then A σ^2 else 0) ≤ L r)
+    (hhigh : ∀r, R ≤ r → w^(2*r)*(∑σ : Fin n → Bool,
+      if monomialDegree σ=r then A σ^2 else 0) ≤ K*a^r)
+    (hL : ∀r, 0 ≤ L r) :
+    multilinearEnergy A w ≤ (∑r ∈ Finset.range R, w^(2*r)*L r)+K*a^R/(1-a) := by
+  rw [multilinearEnergy_degrees]
+  calc
+    _ ≤ ∑r ∈ Finset.range (n+1),
+        ((if r < R then w^(2*r)*L r else 0)+(if R ≤ r then K*a^r else 0)) := by
+      apply Finset.sum_le_sum
+      intro r _
+      by_cases h : r < R
+      · simp only [ite_eq_left h,ite_eq_right (show ¬R ≤ r by omega),add_zero]
+        exact mul_le_mul_of_nonneg_left (hlow r h) (by positivity)
+      · simp only [ite_eq_right h,ite_eq_left (show R ≤ r by omega),zero_add]
+        exact hhigh r (by omega)
+    _ ≤ _ := by
+      rw [Finset.sum_add_distrib]
+      apply add_le_add
+      · rw [←Finset.sum_filter]
+        exact Finset.sum_le_sum_of_subset_of_nonneg
+          (by intro r hr; exact Finset.mem_range.mpr (Finset.mem_filter.mp hr).2)
+          (fun r _ _ => mul_nonneg (by positivity) (hL r))
+      · have he : (∑r ∈ Finset.range (n+1), if R ≤ r then K*a^r else 0)=
+            K*∑r ∈ Finset.range (n+1), if R ≤ r then a^r else 0 := by
+          rw [Finset.mul_sum]
+          apply Finset.sum_congr rfl
+          intro r _
+          split_ifs <;> ring
+        rw [he,mul_div_assoc]
+        exact mul_le_mul_of_nonneg_left (finite_geometric_tail_le ha ha1 _ _) hK
+
+/-- The signed high-degree bound includes the true anchor mixture. -/
+theorem signedMotif_degree_geometric {u m l r : ℕ} (hl : l ≤ u*m) (hr : 0 < r)
+    (hm : 32 ≤ m) (hum : u ≤ m) (off : Fin m)
+    (hi : 1 ≤ off.val ∧ off.val+1 < m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 2)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
+    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
+    (s : ℕ) {q θ : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
+    (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
+    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ)) (w : ℝ) :
+    w^(2*r)*(∑σ : Fin l → Bool, if monomialDegree σ=r then
+      (mixedMotifCoefficient hl off hsafe π s 1 σ-
+        mixedMotifCoefficient hl off hsafe π s (-1) σ)^2 else 0) ≤
+      4*(w^2*(3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/m))^r := by
+  have hw0 : 0 ≤ w^(2*r) := by rw [pow_mul]; positivity
+  have hp : 0 ≤ 3*rankSchurConstant/q := by positivity [rankSchurConstant_pos,hq.1]
+  have hb := degree_square_sub_le (mixedMotifCoefficient hl off hsafe π s 1)
+    (mixedMotifCoefficient hl off hsafe π s (-1)) r
+  by_cases hru : r ≤ u
+  · have hplus := mixedMotif_degree_le hl hr hru hm hum off hi hoff hsafe π hπ hπsum
+      s 1 hq hθ hmean hv
+    have hminus := mixedMotif_degree_le hl hr hru hm hum off hi hoff hsafe π hπ hπsum
+      s (-1) hq hθ hmean hv
+    have hsqrt : Real.sqrt ((u:ℝ)/((m:ℝ)*r)) ≤ Real.sqrt ((u:ℝ)/m) := by
+      apply Real.sqrt_le_sqrt
+      apply div_le_div_of_nonneg_left (by positivity) (by exact_mod_cast (show 0 < m by omega))
+      have hr1 : (1:ℝ) ≤ r := by exact_mod_cast hr
+      nlinarith
+    calc
+      _ ≤ w^(2*r)*(4*((3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/((m:ℝ)*r)))^r) :=
+        mul_le_mul_of_nonneg_left (by linarith) (by rw [pow_mul]; positivity)
+      _ ≤ w^(2*r)*(4*((3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/m))^r) := by
+        gcongr
+      _ = _ := by rw [pow_mul, mul_pow, mul_pow]; ring
+  · rw [mixedMotif_degree_vanish hl (by omega) off hsafe π hπ hπsum s 1,
+      mixedMotif_degree_vanish hl (by omega) off hsafe π hπ hπsum s (-1)] at hb
+    have hz : (∑σ : Fin l → Bool, if monomialDegree σ=r then
+        (mixedMotifCoefficient hl off hsafe π s 1 σ-
+          mixedMotifCoefficient hl off hsafe π s (-1) σ)^2 else 0)=0 := by
+      apply le_antisymm (by linarith)
+      exact Finset.sum_nonneg (fun σ _ => by split_ifs <;> positivity)
+    rw [hz,mul_zero]
+    positivity
+
+/-- Explicit low-degree majorant from the signed finite-difference estimate. -/
+def signedInitialDegreeBound (u m r s : ℕ) (π : IntSeries) (θ : ℝ) (l : ℕ) : ℝ :=
+  if r=0 then 0 else
+    8*(Fintype.card (Fin r → GapDifference):ℝ)*
+      (∑k : Fin r → GapDifference, if Odd (firstDifferenceCount k) then
+        signedPatternBound u m r s π θ k else 0) +
+    2*((Fintype.card (IncreasingTuple u r):ℝ)*
+      (2*(((r+1:ℕ):ℝ)*θ^r/binomialMass (u*m) θ l * (2*Real.exp (-(m:ℝ)/(1024*u))))))^2
+
+theorem signedInitialDegreeBound_nonneg (u m r s : ℕ) (π : IntSeries) (θ : ℝ) (l : ℕ) :
+    0 ≤ signedInitialDegreeBound u m r s π θ l := by
+  unfold signedInitialDegreeBound
+  split_ifs
+  · exact le_rfl
+  · apply add_nonneg _ (by positivity)
+    apply mul_nonneg (by positivity)
+    apply Finset.sum_nonneg
+    intro k _
+    split_ifs
+    · unfold signedPatternBound
+      split_ifs <;> apply mul_nonneg _ (seriesL2Sq_nonneg _)
+      all_goals dsimp only [periodDifferencePrefactor, lengthDifferenceConstant, periodCentralSumConstant, periodBoundedSumConstant]
+      all_goals positivity
+    · exact le_rfl
+
+/-- A fully finite norm estimate for the actual low and high degree pieces.
+Only numerical admissible-count and geometric conditions remain to be chosen. -/
+theorem signed_initial_full_energy {u m l R : ℕ} (hu : 0 < u) (hm : 128 ≤ m)
+    (hum : u ≤ m) (hR : 0 < R) (hl : l ≤ u*m) (off : Fin m)
+    (hi : 1 ≤ off.val ∧ off.val+1 < m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 2)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
+    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
+    (s : ℕ) (hcentral : CentralMotifSupport u s π)
+    {q θ w : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
+    (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
+    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
+    (hM : ((2*(2^s):ℕ):ℝ) < (u:ℝ)/(3*(R+1:ℝ)))
+    (hw : 0 ≤ w) (ha : w^2*(3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/m) < 1) :
+    multilinearEnergy (fun σ => mixedMotifCoefficient hl off hsafe π s 1 σ-
+        mixedMotifCoefficient hl off hsafe π s (-1) σ) w ≤
+      (∑r ∈ Finset.range R, w^(2*r)*signedInitialDegreeBound u m r s π θ l)+
+        4*(w^2*(3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/m))^R /
+          (1-w^2*(3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/m)) := by
+  have hq2 : q^2 ≤ (1/4:ℝ) := by
+    calc q^2 ≤ (1/2:ℝ)^2 := pow_le_pow_left₀ hq.1.le hq.2 2
+         _ = _ := by norm_num
+  have hθ0 : θ∈Set.Ioo 0 1 := by constructor <;> nlinarith [hq.1,hθ.1,hθ.2]
+  have hhalf : 1/2 ≤ θ := by nlinarith [hθ.1]
+  apply multilinearEnergy_low_high _ w _ 4 hw (by positivity [rankSchurConstant_pos,hq.1]) ha
+    (by norm_num) (signedInitialDegreeBound u m · s π θ l) _ _
+    (fun r => signedInitialDegreeBound_nonneg _ _ _ _ _ _ _)
+  · intro r hr
+    by_cases hr0 : r=0
+    · subst r
+      rw [signedMotif_degree_zero hl off hsafe π hπsum s]
+      exact signedInitialDegreeBound_nonneg _ _ _ _ _ _ _
+    · have hMr : ((2*(2^s):ℕ):ℝ) < (u:ℝ)/(3*(r+1:ℝ)) := by
+        apply hM.trans_le
+        apply div_le_div_of_nonneg_left (by positivity) (by positivity)
+        exact mul_le_mul_of_nonneg_left (by exact_mod_cast (show r+1 ≤ R+1 by omega)) (by norm_num)
+      simpa only [signedInitialDegreeBound,ite_eq_right hr0] using
+        signedMotif_degree_cancellation hu hm (Nat.pos_of_ne_zero hr0) hl off
+          (hoff.trans (by norm_num)) hsafe π hπ hπsum s hcentral hθ0 hhalf hmean hv hMr
+  · intro r hr
+    exact signedMotif_degree_geometric hl (by omega) (by omega) hum off hi hoff hsafe π hπ
+      hπsum s hq hθ hmean hv w
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+open FiniteLaw
+
+theorem primitive_reference_mass_pos {q : ℝ} (hq : q∈Set.Ioo 0 1) (z : PrimitiveMark) :
+    0 < (primitiveMarkLaw q hq false).mass z := by
+  change 0 < primitiveMarkMass q false z
+  apply div_pos _ (primitive_retain_pos hq)
+  fin_cases z <;> simp [ordinaryPrimitiveMass,primitiveErasure] <;> nlinarith [hq.1,hq.2]
+
+theorem primitive_product_mass_pos {q : ℝ} (hq : q∈Set.Ioo 0 1) {l : ℕ}
+    (x : Fin l → PrimitiveMark) : 0 < ((primitiveMarkLaw q hq false).product l).mass x :=
+by
+  change 0 < ∏ i, (primitiveMarkLaw q hq false).mass (x i)
+  exact Finset.prod_pos (fun _ _ => primitive_reference_mass_pos hq _)
+
+/-- The density in the actual base experiment, not an independent model. -/
+def initialMotifDensity {u m l : ℕ} (hl : l ≤ u*m) (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
+    (π : IntSeries) (s : ℕ) (τ q : ℝ) : (Fin l → PrimitiveMark) → ℝ :=
+  multilinear (mixedMotifCoefficient hl off hsafe π s τ) (primitiveScore q)
+
+theorem initialMotifDensity_nonneg {u m l : ℕ} (hl : l ≤ u*m) (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
+    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
+    (s : ℕ) (τ : ℝ) {q : ℝ} (hq : q∈Set.Ioo 0 1) (x : Fin l → PrimitiveMark) :
+    0 ≤ initialMotifDensity hl off hsafe π s τ q x := by
+  have h := (conditionalMotifLaw hl off hsafe hq π hπ hπsum s τ).nonneg x
+  rw [conditionalMotifLaw_density] at h
+  exact nonneg_of_mul_nonneg_right h (primitive_product_mass_pos hq x)
+
+theorem initialMotifDensity_mean {u m l : ℕ} (hl : l ≤ u*m) (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
+    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
+    (s : ℕ) (τ : ℝ) {q : ℝ} (hq : q∈Set.Ioo 0 1) :
+    ((primitiveMarkLaw q hq false).product l).expect (initialMotifDensity hl off hsafe π s τ q)=1 := by
+  have h := (conditionalMotifLaw hl off hsafe hq π hπ hπsum s τ).total
+  simpa only [conditionalMotifLaw_density,FiniteLaw.expect,initialMotifDensity] using h
+
+theorem initialMotifLaw_withDensity {u m l : ℕ} (hl : l ≤ u*m) (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
+    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
+    (s : ℕ) (τ : ℝ) {q : ℝ} (hq : q∈Set.Ioo 0 1) :
+    conditionalMotifLaw hl off hsafe hq π hπ hπsum s τ =
+      ((primitiveMarkLaw q hq false).product l).withDensity
+        (initialMotifDensity hl off hsafe π s τ q)
+        (initialMotifDensity_nonneg hl off hsafe π hπ hπsum s τ hq)
+        (initialMotifDensity_mean hl off hsafe π hπ hπsum s τ hq) := by
+  apply FiniteLaw.ext
+  funext x
+  exact conditionalMotifLaw_density hl off hsafe hq π hπ hπsum s τ x
+
+theorem multilinear_sub {α : Type*} {l : ℕ} (A B : (Fin l → Bool) → ℝ)
+    (Z : α → ℝ) (x : Fin l → α) :
+    multilinear (fun σ => A σ-B σ) Z x = multilinear A Z x-multilinear B Z x := by
+  simp only [multilinear,sub_mul,Finset.sum_sub_distrib]
+
+/-- The constant-term deletion of a polynomial. -/
+def nonconstantCoefficient {l : ℕ} (A : (Fin l → Bool) → ℝ) (σ : Fin l → Bool) : ℝ :=
+  if σ=(fun _ => false) then 0 else A σ
+
+theorem nonconstant_multilinear {α : Type*} {l : ℕ} (A : (Fin l → Bool) → ℝ)
+    (Z : α → ℝ) (x : Fin l → α) :
+    multilinear (nonconstantCoefficient A) Z x=multilinear A Z x-A (fun _ => false) := by
+  have he : ∀σ, nonconstantCoefficient A σ=A σ-(if σ=(fun _ => false) then A σ else 0) := by
+    intro σ
+    by_cases h : σ=(fun _ => false) <;> simp [nonconstantCoefficient,h]
+  rw [show nonconstantCoefficient A = (fun σ => A σ-(if σ=(fun _ => false) then A σ else 0)) from funext he, multilinear_sub]
+  congr 1
+  simp [multilinear,ite_mul]
+
+theorem nonconstant_degree_zero {l : ℕ} (A : (Fin l → Bool) → ℝ) :
+    (∑σ : Fin l → Bool, if monomialDegree σ=0 then (nonconstantCoefficient A σ)^2 else 0)=0 := by
+  simp [monomialDegree_eq_zero_iff,nonconstantCoefficient]
+
+theorem nonconstant_degree_pos {l r : ℕ} (A : (Fin l → Bool) → ℝ) (hr : 0 < r) :
+    (∑σ : Fin l → Bool, if monomialDegree σ=r then (nonconstantCoefficient A σ)^2 else 0)=
+      ∑σ : Fin l → Bool, if monomialDegree σ=r then (A σ)^2 else 0 := by
+  apply Finset.sum_congr rfl
+  intro σ _
+  by_cases h : σ=(fun _ => false)
+  · subst σ
+    simp [monomialDegree,Ne.symm (Nat.ne_of_gt hr)]
+  · simp [nonconstantCoefficient,h]
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+open FiniteLaw
+
+theorem mixedMotif_degree_geometric {u m l r : ℕ} (hl : l ≤ u*m) (hr : 0 < r)
+    (hm : 32 ≤ m) (hum : u ≤ m) (off : Fin m)
+    (hi : 1 ≤ off.val ∧ off.val+1 < m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 2)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
+    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
+    (s : ℕ) (τ : ℝ) {q θ : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
+    (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
+    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ)) (w : ℝ) :
+    w^(2*r)*(∑σ : Fin l → Bool, if monomialDegree σ=r then
+      mixedMotifCoefficient hl off hsafe π s τ σ^2 else 0) ≤
+      (w^2*(3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/m))^r := by
+  have hw : 0 ≤ w^(2*r) := by rw [pow_mul]; positivity
+  have hp : 0 ≤ 3*rankSchurConstant/q := by positivity [rankSchurConstant_pos,hq.1]
+  by_cases hru : r ≤ u
+  · have hb := mixedMotif_degree_le hl hr hru hm hum off hi hoff hsafe π hπ hπsum
+      s τ hq hθ hmean hv
+    have hsqrt : Real.sqrt ((u:ℝ)/((m:ℝ)*r)) ≤ Real.sqrt ((u:ℝ)/m) := by
+      apply Real.sqrt_le_sqrt
+      apply div_le_div_of_nonneg_left (by positivity) (by exact_mod_cast (show 0 < m by omega))
+      have hr1 : (1:ℝ) ≤ r := by exact_mod_cast hr
+      nlinarith
+    calc
+      _ ≤ w^(2*r)*((3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/((m:ℝ)*r)))^r :=
+        mul_le_mul_of_nonneg_left hb hw
+      _ ≤ w^(2*r)*((3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/m))^r := by gcongr
+      _ = _ := by rw [pow_mul,mul_pow,mul_pow]; ring
+  · rw [mixedMotif_degree_vanish hl (by omega) off hsafe π hπ hπsum s τ,mul_zero]
+    positivity
+
+/-- Clipping input: the actual conditional density is close to one. -/
+theorem initial_centered_energy {u m l : ℕ} (hl : l ≤ u*m)
+    (hm : 32 ≤ m) (hum : u ≤ m) (off : Fin m)
+    (hi : 1 ≤ off.val ∧ off.val+1 < m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 2)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
+    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
+    (s : ℕ) (τ : ℝ) {q θ w : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
+    (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
+    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
+    (hw : 0 ≤ w) (ha : w^2*(3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/m) < 1) :
+    multilinearEnergy (nonconstantCoefficient (mixedMotifCoefficient hl off hsafe π s τ)) w ≤
+      (w^2*(3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/m)) /
+        (1-w^2*(3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/m)) := by
+  have H := multilinearEnergy_low_high (R:=1)
+    (nonconstantCoefficient (mixedMotifCoefficient hl off hsafe π s τ)) w _ 1 hw
+    (by positivity [rankSchurConstant_pos,hq.1]) ha (by norm_num) (fun _ => 0)
+    (fun r hr => by
+      have he : r=0 := by omega
+      subst r
+      rw [nonconstant_degree_zero])
+    (fun r hr => by
+      rw [nonconstant_degree_pos _ (by omega)]
+      simpa only [one_mul] using mixedMotif_degree_geometric hl (by omega) hm hum off
+        hi hoff hsafe π hπ hπsum s τ hq hθ hmean hv w)
+    (fun _ => le_rfl)
+  simpa using H
+
+theorem initial_centered_moment {u m l : ℕ} (hl : l ≤ u*m)
+    (hm : 32 ≤ m) (hum : u ≤ m) (off : Fin m)
+    (hi : 1 ≤ off.val ∧ off.val+1 < m) (hoff : |(off.val:ℝ)-(m:ℝ)/2| ≤ 2)
+    (hsafe : SafeTaggedTuple (periodSlot (u:=u) (fun _ => off)))
+    (π : IntSeries) (hπ : ∀X, 0 ≤ π.coeff X) (hπsum : ∑X∈π.coeff.support,π.coeff X=1)
+    (s k : ℕ) (hk : 1 ≤ k) (τ : ℝ) {q θ w : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
+    (hθ : θ∈Set.Icc (1-q^2) (1-q^2/4))
+    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
+    (hw : w=((6*(2*k:ℕ):ℝ)*(2/q)))
+    (ha : w^2*(3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/m) < 1) :
+    ((primitiveMarkLaw q ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩ false).product l).evenMomentSq
+      (fun x => initialMotifDensity hl off hsafe π s τ q x-1) k ≤
+      (w^2*(3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/m)) /
+        (1-w^2*(3*rankSchurConstant/q)*Real.sqrt ((u:ℝ)/m)) := by
+  have hq' : q∈Set.Ioo 0 1 := ⟨hq.1,lt_of_le_of_lt hq.2 (by norm_num)⟩
+  have he : (fun x => initialMotifDensity hl off hsafe π s τ q x-1) =
+      multilinear (nonconstantCoefficient (mixedMotifCoefficient hl off hsafe π s τ))
+        (primitiveScore q) := by
+    funext x
+    rw [nonconstant_multilinear,mixedMotifCoefficient_empty hl off hsafe π hπsum]
+    rfl
+  rw [he]
+  apply (multilinear_moment_sq_scaled (primitiveMarkLaw q hq' false) (primitiveScore q) k hk (primitive_score_centered hq')
+    (div_pos (by norm_num) hq.1) (primitive_score_evenMoment hq' k) l _).trans
+  rw [←hw]
+  apply initial_centered_energy hl hm hum off hi hoff hsafe π hπ hπsum s τ hq hθ hmean hv _ ha
+  rw [hw]; positivity [hq.1]
+
+end TraceReconstruction
+
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/TraceReconstruction/InitialRates.lean b/lean/OAI/Probability/TraceReconstruction/InitialRates.lean
index 06e7935bfd9736336f2b7abe3c43dcf64df9f28f..a9b55b0cbf27702ac1eabc9e94e51db1729eace9 100644
--- a/lean/OAI/Probability/TraceReconstruction/InitialRates.lean
+++ b/lean/OAI/Probability/TraceReconstruction/InitialRates.lean
@@ -1,1401 +1,1401 @@
-import OAI.Probability.TraceReconstruction.InitialOracle
-import OAI.Probability.TraceReconstruction.Amplification
-
-namespace OAI
-
-/-!
-# InitialRates
-
-Initial geometric parameters, moment depths, error and overlap rates, and production of the base induction pair.
--/
-
-open scoped ENNReal
-noncomputable section
-
-open Filter
-open scoped Topology
-
-
-noncomputable section
-namespace TraceReconstruction
-open Filter
-open scoped Topology Classical
-
-variable {N q ω w : ℕ → ℝ} {J u m : ℕ → ℕ}
-
-/-- Exact power of the sharp initial rank prefactor; all channel losses are
- subpower at the source's smallest scale. -/
-theorem initialPrefactor_rate (hN : Tendsto N atTop atTop)
-    {α β : ℝ} (hu : PowerRate N (fun t=>(u t:ℝ)) α)
-    (hm : PowerRate N (fun t=>(m t:ℝ)) β) (hq : PowerRate N q 0) (r d : ℕ) :
-    PowerRate N (fun t=>initialPrefactorCap (u t) (m t) r d (q t))
-      (α/2-β*((d:ℝ)+(r:ℝ)/2)) := by
-  have hd : PowerRate N (fun t=>initialDifferenceCap d (q t)) 0 := by
-    have H := ((PowerRate.const hN (by positivity : (0:ℝ)<8*((d:ℝ)+1))).div (hq.pow 2)).sqrt.const_mul hN (by norm_num : (0:ℝ)<2)
-    simpa only [initialDifferenceCap,zero_mul,zero_sub,neg_zero,zero_div] using H
-  have hnum := (((hu.mul hm).const_mul hN (by norm_num : (0:ℝ)<144)).mul_const hN
-    (by positivity : (0:ℝ)<(2:ℝ)^(2*r))).mul (hd.pow (2*d+r+2))
-  have hden := ((hm.div_const hN (by norm_num : (0:ℝ)<4)).pow d).mul
-    (((hu.mul hm).div_const hN (by positivity : (0:ℝ)<2*(r+1:ℝ))).sqrt)
-  have H := hnum.div (hden.mul ((hm.div_const hN (by norm_num : (0:ℝ)<4)).sqrt.pow (r+1)))
-  convert H using 1
-  · ext t; simp only [initialPrefactorCap,Nat.cast_mul]
-  · push_cast; ring
-
-theorem periodCentral_rate (hN : Tendsto N atTop atTop) {α : ℝ}
-    (hu : PowerRate N (fun t=>(u t:ℝ)) α) {r : ℕ} (hr : 0<r) :
-    PowerRate N (fun t=>periodCentralSumConstant (u t) r) (α*((r:ℝ)-3)/2) := by
-  have H := ((hu.div_const hN (by positivity : (0:ℝ)<3*(r+1:ℝ))).inv hN).const_mul hN
-    (by positivity : (0:ℝ)<((r+1:ℕ):ℝ)^2)
-  have H' := H.mul (((hu.const_mul hN (by norm_num : (0:ℝ)<2)).sqrt.const_mul hN
-    (by norm_num : (0:ℝ)<2)).pow (r-1))
-  convert H' using 1
-  · ext t; simp only [periodCentralSumConstant,Nat.cast_mul,Nat.cast_ofNat]
-  · rw [Nat.cast_sub (by omega : 1≤r)]; norm_num; ring
-
-theorem periodBounded_rate (hN : Tendsto N atTop atTop) {α : ℝ}
-    (hu : PowerRate N (fun t=>(u t:ℝ)) α) {r : ℕ} (hr : 2≤r) (M : ℕ) :
-    PowerRate N (fun t=>periodBoundedSumConstant (u t) r M) (α*((r:ℝ)-4)/2) := by
-  have H := (((hu.div_const hN (by positivity : (0:ℝ)<3*(r+1:ℝ))).inv hN).const_mul hN
-    (by positivity : (0:ℝ)<((r+1:ℕ):ℝ)^2)).mul_const hN (by positivity : (0:ℝ)<(M:ℝ)+1)
-  have H' := H.mul (((hu.const_mul hN (by norm_num : (0:ℝ)<2)).sqrt.const_mul hN
-    (by norm_num : (0:ℝ)<2)).pow (r-2))
-  convert H' using 1
-  · ext t; simp only [periodBoundedSumConstant,Nat.cast_mul,Nat.cast_ofNat]
-  · rw [Nat.cast_sub hr]; norm_num; ring
-
-theorem hiddenSigned_rate (hN : Tendsto N atTop atTop) {γ : ℝ} (hγ : 0<γ)
-    (hJ : PowerRate N (fun t=>(J t:ℝ)) γ) (hω : PowerRate N ω 0) (s : ℕ) :
-    PowerRate N (fun t=>hiddenSignedL2Bound (J t) s (ω t)) (-γ*((s:ℝ)+1/2)) := by
-  have hden := (hJ.mul (hω.pow 2)).add_const hN (by simpa using hγ) (by norm_num : (0:ℝ)≤2)
-  have hroot := ((PowerRate.const hN (by positivity : (0:ℝ)<4*((s:ℝ)+1))).div hden).sqrt
-  have hp := ((hroot.const_mul hN (by norm_num : (0:ℝ)<2)).pow s).pow 2
-  have H := (hp.const_mul hN (by positivity : (0:ℝ)<((2:ℝ)^(s*s))^2)).mul hroot
-  convert H using 1
-  · rfl
-  · push_cast; ring
-
-theorem hiddenUnsigned_rate (hN : Tendsto N atTop atTop) {γ : ℝ} (hγ : 0<γ)
-    (hJ : PowerRate N (fun t=>(J t:ℝ)) γ) (hω : PowerRate N ω 0) (s : ℕ) :
-    PowerRate N (fun t=>hiddenUnsignedL2Bound (J t) s (ω t)) (-γ/2) := by
-  have hden := (hJ.mul (hω.pow 2)).add_const hN (by simpa using hγ) (by norm_num : (0:ℝ)≤2)
-  have H := (((PowerRate.const hN (by norm_num : (0:ℝ)<4)).div hden).sqrt).const_mul hN
-    (by positivity : (0:ℝ)<((2:ℝ)^s)^2)
-  convert H using 1
-  · rfl
-  · ring
-
-/-- The two eliminated large gaps are quantitatively visible in this exponent. -/
-theorem initial_central_rate (hN : Tendsto N atTop atTop)
-    (hu : PowerRate N (fun t=>(u t:ℝ)) (123/250))
-    (hm : PowerRate N (fun t=>(m t:ℝ)) (62/125)) (hq : PowerRate N q 0)
-    {r : ℕ} (hr : 0<r) (d : ℕ) :
-    PowerRate N (fun t=>initialPrefactorCap (u t) (m t) r d (q t)*periodCentralSumConstant (u t) r)
-      (-123/250-(62/125)*(d:ℝ)-(1/500)*(r:ℝ)) := by
-  convert (initialPrefactor_rate hN hu hm hq r d).mul (periodCentral_rate hN hu hr) using 1; ring
-
-theorem initial_bounded_rate (hN : Tendsto N atTop atTop)
-    (hu : PowerRate N (fun t=>(u t:ℝ)) (123/250))
-    (hm : PowerRate N (fun t=>(m t:ℝ)) (62/125)) (hq : PowerRate N q 0)
-    {r : ℕ} (hr : 2≤r) (d M : ℕ) :
-    PowerRate N (fun t=>initialPrefactorCap (u t) (m t) r d (q t)*periodBoundedSumConstant (u t) r M)
-      (-369/500-(62/125)*(d:ℝ)-(1/500)*(r:ℝ)) := by
-  convert (initialPrefactor_rate hN hu hm hq r d).mul (periodBounded_rate hN hu hr M) using 1; ring
-
-/-- Fixed low degrees already beat the signed and pure source score exponents. -/
-theorem initial_signed_pattern_upper (hN : Tendsto N atTop atTop)
-    (hu : PowerRate N (fun t=>(u t:ℝ)) (123/250))
-    (hm : PowerRate N (fun t=>(m t:ℝ)) (62/125)) (hq : PowerRate N q 0)
-    (hJ : PowerRate N (fun t=>(J t:ℝ)) (1/125)) (hω : PowerRate N ω 0)
-    {r : ℕ} (hr : 0<r) (k : Fin r→GapDifference) (hk : Odd (firstDifferenceCount k)) :
-    PowerUpper N (fun t=>initialSignedPatternCap (J t) (u t) (m t) r 300 (q t) (ω t) k) (-223/100) := by
-  by_cases hlin : ∃a,k=linearPattern a
-  · have H := (initial_central_rate hN hu hm hq hr 1).mul
-      (hiddenSigned_rate hN (by norm_num) hJ hω 300)
-    have hb : -123/250-(62/125)*(1:ℝ)-(1/500)*(r:ℝ)+(-(1/125)*((300:ℝ)+1/2)) ≤ -223/100 := by
-      have hh : (0:ℝ)≤r := Nat.cast_nonneg r
-      linarith
-    simpa only [initialSignedPatternCap,ite_eq_left hlin] using (H.upper hN).mono (by simpa using hb)
-  · have hd : 3≤∑h,(k h).order := odd_nonlinear_order k hk hlin
-    obtain ⟨a,b,hab,_,_⟩ := odd_pattern_pair k hk hlin
-    have hr2 : 2≤r := by have := a.isLt; have := b.isLt; have := hab; omega
-    have H := (initial_bounded_rate hN hu hm hq hr2 (∑h,(k h).order) (2*(2^300))).mul
-      (hiddenUnsigned_rate hN (by norm_num) hJ hω 300)
-    have hb : -369/500-(62/125)*(∑h,(k h).order:ℕ)-(1/500)*(r:ℝ)+(-(1/125)/2) ≤ -223/100 := by
-      have hd' : (3:ℝ)≤(∑h,(k h).order:ℕ) := by exact_mod_cast hd
-      have hr' : (2:ℝ)≤r := by exact_mod_cast hr2
-      linarith
-    simpa only [initialSignedPatternCap,ite_eq_right hlin] using (H.upper hN).mono (by simpa using hb)
-
-theorem initial_pure_pattern_upper (hN : Tendsto N atTop atTop)
-    (hu : PowerRate N (fun t=>(u t:ℝ)) (123/250))
-    (hm : PowerRate N (fun t=>(m t:ℝ)) (62/125)) (hq : PowerRate N q 0)
-    (hJ : PowerRate N (fun t=>(J t:ℝ)) (1/125)) (hω : PowerRate N ω 0)
-    {r : ℕ} (hr : 0<r) (k : Fin r→GapDifference) (hk : k≠unchangedPattern r) :
-    PowerUpper N (fun t=>initialPurePatternCap (J t) (u t) (m t) r 300 (q t) (ω t) k) (-37/25) := by
-  by_cases hlin : ∃a,k=linearPattern a
-  · have H := (initial_central_rate hN hu hm hq hr 1).mul
-      (hiddenSigned_rate hN (by norm_num) hJ hω 300)
-    have hb : -123/250-(62/125)*(1:ℝ)-(1/500)*(r:ℝ)+(-(1/125)*((300:ℝ)+1/2)) ≤ -37/25 := by
-      have hh : (0:ℝ)≤r := Nat.cast_nonneg r
-      linarith
-    simpa only [initialPurePatternCap,ite_eq_left hlin] using (H.upper hN).mono (by simpa using hb)
-  · have hd := (nonconstant_pattern_linear_or_order_two k hk).resolve_left hlin
-    have H := (initial_central_rate hN hu hm hq hr (∑h,(k h).order)).mul
-      (hiddenUnsigned_rate hN (by norm_num) hJ hω 300)
-    have hb : -123/250-(62/125)*(∑h,(k h).order:ℕ)-(1/500)*(r:ℝ)+(-(1/125)/2) ≤ -37/25 := by
-      have hd' : (2:ℝ)≤(∑h,(k h).order:ℕ) := by exact_mod_cast hd
-      have hr' : (0:ℝ)≤r := Nat.cast_nonneg r
-      linarith
-    simpa only [initialPurePatternCap,ite_eq_right hlin] using (H.upper hN).mono (by simpa using hb)
-
-theorem increasingTuple_card_le (u r : ℕ) : Fintype.card (IncreasingTuple u r) ≤ u^r := by
-  simpa only [Fintype.card_fun,Fintype.card_fin] using
-    Fintype.card_subtype_le (fun v : Fin r→Fin u => StrictMono v)
-
-theorem initialRankTail_upper (hN : Tendsto N atTop atTop)
-    (hu : PowerRate N (fun t=>(u t:ℝ)) (123/250))
-    (hm : PowerRate N (fun t=>(m t:ℝ)) (62/125)) (r : ℕ) (a : ℝ) :
-    PowerUpper N (fun t=>initialRankTailCap (u t) (m t) r) a := by
-  have hcard : PowerUpper N (fun t=>(Fintype.card (IncreasingTuple (u t) r):ℝ)) ((123/250)*(r:ℝ)) := by
-    apply (hu.pow r).upper hN |>.of_abs_le
-    exact Eventually.of_forall fun t=>by
-      rw [abs_of_nonneg (Nat.cast_nonneg _),abs_of_nonneg (by positivity)]
-      exact_mod_cast increasingTuple_card_le (u t) r
-  have hpoly := hcard.mul hN (((hu.mul hm).sqrt.const_mul hN
-    (by positivity : (0:ℝ)<48*((r+1:ℕ):ℝ))).upper hN)
-  have hexpr : PowerRate N (fun t=>(m t:ℝ)/(1024*(u t:ℝ))) (1/250) := by
-    convert hm.div (hu.const_mul hN (by norm_num : (0:ℝ)<1024)) using 1; norm_num
-  have hexp := hexpr.exp_neg_upper hN (by norm_num) (a/2-((123/250)*(r:ℝ)+(123/250+62/125)/2))
-  have H := ((PowerRate.const hN (by norm_num : (0:ℝ)<2)).upper hN).mul hN
-    ((hpoly.mul hN hexp).pow hN 2)
-  convert H using 1
-  · ext t
-    simp only [initialRankTailCap,Nat.cast_mul,neg_div]
-    ring
-  · push_cast; ring
-
-theorem initialMomentRatio_rate (hN : Tendsto N atTop atTop)
-    (hu : PowerRate N (fun t=>(u t:ℝ)) (123/250))
-    (hm : PowerRate N (fun t=>(m t:ℝ)) (62/125)) (hq : PowerRate N q 0)
-    (hw : PowerRate N w 0) :
-    PowerRate N (fun t=>initialMomentRatio (u t) (m t) (q t) (w t)) (-1/500) := by
-  have H := ((hw.pow 2).mul ((PowerRate.const hN
-    (by positivity [rankSchurConstant_pos] : (0:ℝ)<3*rankSchurConstant)).div hq)).mul (hu.div hm).sqrt
-  convert H using 1
-  · rfl
-  · norm_num
-
-theorem initial_fixed_high_tail_rate (hN : Tendsto N atTop atTop)
-    (hu : PowerRate N (fun t=>(u t:ℝ)) (123/250))
-    (hm : PowerRate N (fun t=>(m t:ℝ)) (62/125)) (hq : PowerRate N q 0)
-    (hw : PowerRate N w 0) :
-    PowerRate N (fun t=>4*(initialMomentRatio (u t) (m t) (q t) (w t))^3000/
-      (1-initialMomentRatio (u t) (m t) (q t) (w t))) (-6) := by
-  have hρ := initialMomentRatio_rate hN hu hm hq hw
-  have hz := (hρ.upper hN).tendsto_zero hN (by norm_num)
-  have hden : PowerRate N (fun t=>1-initialMomentRatio (u t) (m t) (q t) (w t)) 0 := by
-    apply PowerRate.of_tendsto hN (by norm_num : (0:ℝ)<1)
-    simpa using tendsto_const_nhds.sub hz
-  convert ((hρ.pow 3000).const_mul hN (by norm_num : (0:ℝ)<4)).div hden using 1
-  norm_num
-
-theorem initialSignedDegree_upper (hN : Tendsto N atTop atTop)
-    (hu : PowerRate N (fun t=>(u t:ℝ)) (123/250))
-    (hm : PowerRate N (fun t=>(m t:ℝ)) (62/125)) (hq : PowerRate N q 0)
-    (hJ : PowerRate N (fun t=>(J t:ℝ)) (1/125)) (hω : PowerRate N ω 0) (r : ℕ) :
-    PowerUpper N (fun t=>initialSignedDegreeCap (J t) (u t) (m t) r 300 (q t) (ω t)) (-223/100) := by
-  by_cases hr : r=0
-  · simpa only [initialSignedDegreeCap,ite_eq_left hr] using PowerUpper.zero hN (-223/100)
-  have hs : PowerUpper N (fun t=>∑k : Fin r→GapDifference,
-      if Odd (firstDifferenceCount k) then initialSignedPatternCap (J t) (u t) (m t) r 300 (q t) (ω t) k else 0) (-223/100) := by
-    apply PowerUpper.sum hN Finset.univ
-    intro k _
-    by_cases hk : Odd (firstDifferenceCount k)
-    · simpa only [ite_eq_left hk] using initial_signed_pattern_upper hN hu hm hq hJ hω (by omega) k hk
-    · simpa only [ite_eq_right hk] using PowerUpper.zero hN (-223/100)
-  have hc : (0:ℝ)<8*(Fintype.card (Fin r→GapDifference):ℝ) := by
-    have hp : 0<Fintype.card (Fin r→GapDifference) := Fintype.card_pos_iff.mpr ⟨fun _=>.unchanged⟩
-    positivity
-  have H := (((PowerRate.const hN hc).upper hN).mul hN hs)
-  simp only [zero_add] at H
-  simpa only [initialSignedDegreeCap,ite_eq_right hr] using H.add hN (initialRankTail_upper hN hu hm r (-223/100))
-
-theorem initialPureDegree_upper (hN : Tendsto N atTop atTop)
-    (hu : PowerRate N (fun t=>(u t:ℝ)) (123/250))
-    (hm : PowerRate N (fun t=>(m t:ℝ)) (62/125)) (hq : PowerRate N q 0)
-    (hJ : PowerRate N (fun t=>(J t:ℝ)) (1/125)) (hω : PowerRate N ω 0) (r : ℕ) :
-    PowerUpper N (fun t=>initialPureDegreeCap (J t) (u t) (m t) r 300 (q t) (ω t)) (-37/25) := by
-  by_cases hr : r=0
-  · simpa only [initialPureDegreeCap,ite_eq_left hr] using PowerUpper.zero hN (-37/25)
-  have hs : PowerUpper N (fun t=>∑k : Fin r→GapDifference,
-      if k=unchangedPattern r then 0 else initialPurePatternCap (J t) (u t) (m t) r 300 (q t) (ω t) k) (-37/25) := by
-    apply PowerUpper.sum hN Finset.univ
-    intro k _
-    by_cases hk : k=unchangedPattern r
-    · simpa only [ite_eq_left hk] using PowerUpper.zero hN (-37/25)
-    · simpa only [ite_eq_right hk] using initial_pure_pattern_upper hN hu hm hq hJ hω (by omega) k hk
-  have hc : (0:ℝ)<2*(Fintype.card (Fin r→GapDifference):ℝ) := by
-    have hp : 0<Fintype.card (Fin r→GapDifference) := Fintype.card_pos_iff.mpr ⟨fun _=>.unchanged⟩
-    positivity
-  have H := (((PowerRate.const hN hc).upper hN).mul hN hs)
-  simp only [zero_add] at H
-  simpa only [initialPureDegreeCap,ite_eq_right hr] using H.add hN (initialRankTail_upper hN hu hm r (-37/25))
-
-theorem initialSignedCap_upper (hN : Tendsto N atTop atTop)
-    (hu : PowerRate N (fun t=>(u t:ℝ)) (123/250))
-    (hm : PowerRate N (fun t=>(m t:ℝ)) (62/125)) (hq : PowerRate N q 0)
-    (hJ : PowerRate N (fun t=>(J t:ℝ)) (1/125)) (hω : PowerRate N ω 0) (hw : PowerRate N w 0) :
-    PowerUpper N (fun t=>initialSignedCap (J t) (u t) (m t) 3000 300 (q t) (ω t) (w t)) (-223/100) := by
-  have hs : PowerUpper N (fun t=>∑r∈Finset.range 3000,(w t)^(2*r)*
-      initialSignedDegreeCap (J t) (u t) (m t) r 300 (q t) (ω t)) (-223/100) := by
-    apply PowerUpper.sum hN (Finset.range 3000)
-    intro r _
-    simpa only [zero_mul,zero_add] using ((hw.pow (2*r)).upper hN).mul hN (initialSignedDegree_upper hN hu hm hq hJ hω r)
-  exact hs.add hN ((initial_fixed_high_tail_rate hN hu hm hq hw).upper hN |>.mono (by norm_num))
-
-theorem initialPureCap_upper (hN : Tendsto N atTop atTop)
-    (hu : PowerRate N (fun t=>(u t:ℝ)) (123/250))
-    (hm : PowerRate N (fun t=>(m t:ℝ)) (62/125)) (hq : PowerRate N q 0)
-    (hJ : PowerRate N (fun t=>(J t:ℝ)) (1/125)) (hω : PowerRate N ω 0) (hw : PowerRate N w 0) :
-    PowerUpper N (fun t=>initialPureCap (J t) (u t) (m t) 3000 300 (q t) (ω t) (w t)) (-37/25) := by
-  have hs : PowerUpper N (fun t=>∑r∈Finset.range 3000,(w t)^(2*r)*
-      initialPureDegreeCap (J t) (u t) (m t) r 300 (q t) (ω t)) (-37/25) := by
-    apply PowerUpper.sum hN (Finset.range 3000)
-    intro r _
-    simpa only [zero_mul,zero_add] using ((hw.pow (2*r)).upper hN).mul hN (initialPureDegree_upper hN hu hm hq hJ hω r)
-  exact hs.add hN ((initial_fixed_high_tail_rate hN hu hm hq hw).upper hN |>.mono (by norm_num))
-
-/-- Actual scalar budgets, not assumed coefficient bounds: all report- and
- count-independent caps are strictly below the source clipping allowances. -/
-theorem initial_score_caps_eventually (hN : Tendsto N atTop atTop)
-    (hu : PowerRate N (fun t=>(u t:ℝ)) (123/250))
-    (hm : PowerRate N (fun t=>(m t:ℝ)) (62/125)) (hq : PowerRate N q 0)
-    (hJ : PowerRate N (fun t=>(J t:ℝ)) (1/125)) (hω : PowerRate N ω 0) (hw : PowerRate N w 0) :
-    ∀ᶠt in atTop,
-      64*initialSignedCap (J t) (u t) (m t) 3000 300 (q t) (ω t) (w t) ≤ (N t)^(-11/5:ℝ) ∧
-      64*initialPureCap (J t) (u t) (m t) 3000 300 (q t) (ω t) (w t) ≤ (N t)^(-7/5:ℝ) := by
-  have hc := (PowerRate.const hN (by norm_num : (0:ℝ)<64)).upper hN
-  have hs := hc.mul hN (initialSignedCap_upper hN hu hm hq hJ hω hw)
-  have hp := hc.mul hN (initialPureCap_upper hN hu hm hq hJ hω hw)
-  filter_upwards [hs (-11/5) (by norm_num),hp (-7/5) (by norm_num)] with t hs hp
-  exact ⟨(le_abs_self _).trans hs,(le_abs_self _).trans hp⟩
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open Filter
-open scoped Topology Classical
-
-variable {N f g : ℕ→ℝ} {α β : ℝ}
-
-theorem PowerRate.sub_lower (hN : Tendsto N atTop atTop)
-    (hf : PowerRate N f α) (hg : PowerRate N g β) (hβα : β<α) :
-    PowerRate N (fun t=>f t-g t) α := by
-  have hz := ((hg.div hf).upper hN).tendsto_zero hN (sub_neg.mpr hβα)
-  apply hf.of_ratio hN (by norm_num : (0:ℝ)<1)
-  have H : Tendsto (fun t=>1-g t/f t) atTop (𝓝 1) := by
-    simpa using tendsto_const_nhds.sub hz
-  apply H.congr'
-  filter_upwards [hf.1] with t ht
-  field_simp [ht.ne']
-
-theorem PowerRate.nat_sub_lower (hN : Tendsto N atTop atTop) {f g : ℕ→ℕ}
-    (hf : PowerRate N (fun t=>(f t:ℝ)) α)
-    (hg : PowerRate N (fun t=>(g t:ℝ)) β) (hβα : β<α) :
-    PowerRate N (fun t=>(f t-g t:ℕ)) α := by
-  have H := hf.sub_lower hN hg hβα
-  apply H.congr
-  filter_upwards [H.1] with t ht
-  have hg : g t≤f t := by exact_mod_cast (sub_pos.mp ht).le
-  rw [Nat.cast_sub hg]
-
-/-- Literal integer choices of the initial-pair construction, including the outer floors. -/
-def sourceInitialM (N : ℝ) : ℕ := ⌊N^(62/125:ℝ)⌋₊
-def sourceInitialK (N : ℝ) : ℕ := ⌊N^(123/250:ℝ)⌋₊
-def sourceInitialPeriodSize (N : ℝ) : ℕ := 2*sourceInitialM N-1
-def sourceInitialCapacity (N : ℝ) : ℕ := ⌊N/(sourceInitialPeriodSize N:ℝ)⌋₊
-def sourceInitialJ (q N : ℝ) : ℕ :=
-  ⌊((sourceInitialCapacity N:ℝ)-sourceInitialK N)/
-    (2*(initialSideCount q (sourceInitialM N):ℝ)+1)⌋₊
-def sourceInitialP (q N : ℝ) : ℕ :=
-  sourceInitialJ q N*(2*initialSideCount q (sourceInitialM N)+1)+sourceInitialK N
-/-- One-based manuscript anchors translated once to zero-based list indices. -/
-def sourceInitialAnchor (q N : ℝ) : ℤ := ((sourceInitialP q N/2:ℕ):ℤ)-1
-
-theorem sourceInitialM_rate (hN : Tendsto N atTop atTop) :
-    PowerRate N (fun t=>(sourceInitialM (N t):ℝ)) (62/125) := by
-  have H := ((PowerRate.base hN).rpow (62/125)).nat_floor hN (by norm_num)
-  simpa only [sourceInitialM,one_mul] using H
-
-theorem sourceInitialK_rate (hN : Tendsto N atTop atTop) :
-    PowerRate N (fun t=>(sourceInitialK (N t):ℝ)) (123/250) := by
-  have H := ((PowerRate.base hN).rpow (123/250)).nat_floor hN (by norm_num)
-  simpa only [sourceInitialK,one_mul] using H
-
-theorem sourceInitialPeriodSize_rate (hN : Tendsto N atTop atTop) :
-    PowerRate N (fun t=>(sourceInitialPeriodSize (N t):ℝ)) (62/125) := by
-  have hh : PowerRate N (fun t=>(2*sourceInitialM (N t):ℕ)) (62/125) := by
-    simpa only [Nat.cast_mul,Nat.cast_ofNat] using (sourceInitialM_rate hN).const_mul hN (by norm_num : (0:ℝ)<2)
-  exact hh.nat_sub_lower hN (PowerRate.const hN (by norm_num : (0:ℝ)<(1:ℕ))) (by norm_num)
-
-theorem sourceInitialCapacity_rate (hN : Tendsto N atTop atTop) :
-    PowerRate N (fun t=>(sourceInitialCapacity (N t):ℝ)) (63/125) := by
-  have H := ((PowerRate.base hN).div (sourceInitialPeriodSize_rate hN)).nat_floor hN (by norm_num)
-  convert H using 1
-  · rfl
-  · norm_num
-
-theorem sourceInitialSide_rate (hN : Tendsto N atTop atTop) {q : ℕ→ℝ}
-    (hq : PowerRate N q 0) :
-    PowerRate N (fun t=>(initialSideCount (q t) (sourceInitialM (N t)):ℝ)) (62/125) := by
-  have H := (hq.mul (sourceInitialM_rate hN)).nat_floor hN (by norm_num)
-  simpa only [initialSideCount,zero_add] using H
-
-theorem sourceInitialJ_rate (hN : Tendsto N atTop atTop) {q : ℕ→ℝ}
-    (hq : PowerRate N q 0) :
-    PowerRate N (fun t=>(sourceInitialJ (q t) (N t):ℝ)) (1/125) := by
-  have hn := (sourceInitialCapacity_rate hN).sub_lower hN (sourceInitialK_rate hN) (by norm_num)
-  have hd := ((sourceInitialSide_rate hN hq).const_mul hN (by norm_num : (0:ℝ)<2)).add_const hN
-    (by norm_num) (by norm_num : (0:ℝ)≤1)
-  have H := (hn.div hd).nat_floor hN (by norm_num)
-  convert H using 1
-  · rfl
-  · norm_num
-
-theorem initialOverlap_rate (hN : Tendsto N atTop atTop) {q : ℕ→ℝ}
-    (_hq : ∀t,0<q t)
-    (hΛ : Tendsto (fun t=>(q t)^(-3:ℤ)/Real.log (N t)) atTop (𝓝 0)) :
-    PowerRate N (fun t=>initialOverlapLower (q t)) 0 := by
-  refine ⟨Eventually.of_forall (fun t=>initialOverlapLower_pos _),?_⟩
-  have H := (hΛ.const_mul (-(32+3*primitivePeriodConstant))).sub
-    ((Real.tendsto_log_atTop.comp hN).const_div_atTop (Real.log 1536))
-  simp only [mul_zero,sub_zero] at H
-  apply H.congr'
-  filter_upwards [] with t
-  rw [initialOverlapLower,Real.log_div (Real.exp_pos _).ne' (by norm_num),Real.log_exp]
-  simp only [zpow_neg,zpow_ofNat,Function.comp_apply]
-  ring
-
-theorem deletion_rate_zero (hN : Tendsto N atTop atTop) {q : ℕ→ℝ}
-    (hq : ∀t,q t∈Set.Ioc 0 1)
-    (hΛ : Tendsto (fun t=>(q t)^(-3:ℤ)/Real.log (N t)) atTop (𝓝 0)) :
-    PowerRate N q 0 := by
-  refine ⟨Eventually.of_forall (fun t=>(hq t).1),?_⟩
-  apply squeeze_zero_norm' _ hΛ
-  filter_upwards [hN.eventually_gt_atTop 1] with t hNt
-  have hp := (hq t).1
-  have hle := (hq t).2
-  have hqinv : 1≤(q t)⁻¹ := (one_le_inv₀ hp).mpr hle
-  have hlog : |Real.log (q t)| ≤ (q t)^(-3:ℤ) := by
-    rw [abs_of_nonpos (Real.log_nonpos hp.le hle),←Real.log_inv, zpow_neg,zpow_ofNat]
-    have hh := Real.log_le_sub_one_of_pos (inv_pos.mpr hp)
-    have hcube : (q t)⁻¹ ≤ ((q t)⁻¹)^3 := by nlinarith [sq_nonneg ((q t)⁻¹-1)]
-    rw [←inv_pow]
-    linarith
-  rw [Real.norm_eq_abs,abs_div,abs_of_pos (Real.log_pos hNt)]
-  exact div_le_div_of_nonneg_right hlog (Real.log_pos hNt).le
-
-/-- Mechanical realization of the base scalar score budgets from the actual
- integer choices and the unchanged known channel. -/
-theorem source_initial_score_caps {q w : ℕ→ℝ} (hN : Tendsto N atTop atTop)
-    (hq : ∀t,q t∈Set.Ioc 0 1)
-    (hΛ : Tendsto (fun t=>(q t)^(-3:ℤ)/Real.log (N t)) atTop (𝓝 0))
-    (hw : PowerRate N w 0) :
-    ∀ᶠt in atTop,
-      64*initialSignedCap (sourceInitialJ (q t) (N t)) (sourceInitialK (N t))
-        (sourceInitialM (N t)) 3000 300 (q t) (initialOverlapLower (q t)) (w t) ≤ (N t)^(-11/5:ℝ) ∧
-      64*initialPureCap (sourceInitialJ (q t) (N t)) (sourceInitialK (N t))
-        (sourceInitialM (N t)) 3000 300 (q t) (initialOverlapLower (q t)) (w t) ≤ (N t)^(-7/5:ℝ) := by
-  exact initial_score_caps_eventually hN (sourceInitialK_rate hN) (sourceInitialM_rate hN)
-    (deletion_rate_zero hN hq hΛ) (sourceInitialJ_rate hN (deletion_rate_zero hN hq hΛ))
-    (initialOverlap_rate hN (fun t=>(hq t).1) hΛ) hw
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open Filter
-open scoped Topology Classical
-
-/-- The manuscript's one-based middle defect, translated to a list index. -/
-def sourceInitialOffset (m : ℕ) (hm : 4 ≤ m) : Fin m :=
-  ⟨m/2-1,by omega⟩
-
-theorem sourceInitialOffset_interior (m : ℕ) (hm : 4 ≤ m) :
-    1 ≤ (sourceInitialOffset m hm).val ∧ (sourceInitialOffset m hm).val+1 < m := by
-  simp only [sourceInitialOffset]; omega
-
-theorem sourceInitialOffset_middle (m : ℕ) (hm : 4 ≤ m) :
-    |((sourceInitialOffset m hm).val:ℝ)-(m:ℝ)/2| ≤ 2 := by
-  have h1 : 2*(m/2) ≤ m := Nat.mul_div_le m 2
-  have h2 : m<2*(m/2+1) := by omega
-  have h3 : 1 ≤ m/2 := by omega
-  simp only [sourceInitialOffset,Nat.cast_sub h3,Nat.cast_one]
-  rw [abs_le]
-  have hh1 : (2:ℝ)*(m/2:ℕ) ≤ m := by exact_mod_cast h1
-  have hh2 : (m:ℝ)<2*((m/2:ℕ)+1) := by exact_mod_cast h2
-  constructor <;> linarith
-
-theorem sourceInitialOffset_safe (u m : ℕ) (hm : 4 ≤ m) :
-    SafeTaggedTuple (periodSlot (u:=u) (fun _=>sourceInitialOffset m hm)) := by
-  have hi := sourceInitialOffset_interior m hm
-  exact constant_period_safe _ hi.1 hi.2 (by omega) id strictMono_id
-
-/-- A literal arithmetic margin implies every base window placement condition. -/
-theorem initial_anchor_geometry (J K t s : ℕ) (hK : 12*(J+2^s+1) ≤ K) :
-    let a : ℤ := (((J*(2*t+1)+K)/2:ℕ):ℤ)-1
-    (J*t+J:ℕ) ≤ a ∧ a+(2^s:ℕ) ≤ (J*t+K:ℕ) ∧
-    (K:ℝ)/3 ≤ 2*((a-(J*t:ℕ):ℤ)-J)+1 ∧
-    (K:ℝ)/3 ≤ 2*K-(2*((a-(J*t:ℕ):ℤ)+(2:ℝ)^s)+1) := by
-  have he : (J*(2*t+1)+K)/2=J*t+(J+K)/2 := by
-    rw [show J*(2*t+1)+K=2*(J*t)+(J+K) by ring,Nat.mul_add_div (by norm_num)]
-  have hlo : 2*((J+K)/2) ≤ J+K := Nat.mul_div_le _ 2
-  have hhi : J+K<2*((J+K)/2+1) := by omega
-  have hK1 : 12*(J+1) ≤ K := by nlinarith [Nat.zero_le (2^s)]
-  dsimp only
-  rw [he]
-  have hi1 : J+1 ≤ (J+K)/2 := by omega
-  have hi2 : (J+K)/2+(2^s:ℕ) ≤ K+1 := by omega
-  constructor
-  · apply (le_sub_iff_add_le).mpr
-    exact_mod_cast (show J*t+J+1 ≤ J*t+(J+K)/2 from by omega)
-  constructor
-  · push_cast only [Nat.cast_add]
-    omega
-  have hl : (2:ℝ)*((J+K)/2:ℕ) ≤ J+K := by exact_mod_cast hlo
-  have hh : (J:ℝ)+K<2*(((J+K)/2:ℕ)+1) := by exact_mod_cast hhi
-  have hk : (12:ℝ)*((J:ℝ)+(2:ℝ)^s+1) ≤ K := by exact_mod_cast hK
-  simp only [Nat.cast_add,Int.cast_sub,Int.cast_add,Int.cast_natCast,Int.cast_one,Int.cast_mul,Nat.cast_mul]
-  constructor <;> nlinarith [show (0:ℝ) ≤ (2:ℝ)^s by positivity]
-
-theorem prouhetSeries_coeff_zero (s : ℕ) : (prouhetSeries s).coeff 0=1 := by
-  induction s with
-  | zero => simp [prouhetSeries]
-  | succ s ih =>
-    have hp : (0:ℤ)<(2^s:ℕ) := by positivity
-    rw [prouhetSeries_succ_coeff,ih,(prouhetSeries_coeff_spec s).1 _ (Or.inl (by omega)),sub_zero]
-
-/-- Literal source initial words really are different, not only their formal templates. -/
-theorem motifPeriodWord_ne {m : ℕ} (off : Fin m) (hi : 1 ≤ off.val ∧ off.val+1 < m)
-    (u s : ℕ) (a : ℤ) (ha : 0 ≤ a) (hau : a<u) :
-    motifPeriodWord off hi u s a (-1) ≠ motifPeriodWord off hi u s a 1 := by
-  intro he
-  have H := periodString_injective m u he
-  have hj : a.toNat<u := by omega
-  have h := congrArg (fun f=>((f ⟨a.toNat,hj⟩):Fin m).val) H
-  have ha' : (a.toNat:ℤ)=a := Int.toNat_of_nonneg ha
-  simp only [motifOffset,ha',sub_self,prouhetSeries_coeff_zero] at h
-  have hpos : (0:ℤ)<(2^s:ℕ) := by positivity
-  norm_num [motifIndicator,hpos,motifShift] at h
-
-/-- Floors in the initial construction waste no more than one long slot. -/
-theorem initial_period_count_bounds {N : ℝ} {m K t : ℕ}
-    (hL : 0<2*m-1) (hN : 0 ≤ N) (hK : (K:ℝ) ≤ ⌊N/(2*m-1:ℕ)⌋₊) :
-    let F : ℕ := ⌊N/(2*m-1:ℕ)⌋₊
-    let J : ℕ := ⌊((F:ℝ)-K)/(2*t+1:ℕ)⌋₊
-    let P : ℕ := J*(2*t+1)+K
-    (P:ℝ)*(2*m-1:ℕ) ≤ N ∧ N<((P:ℝ)+(2*t+1:ℕ)+1)*(2*m-1:ℕ) := by
-  dsimp only
-  set F : ℕ := ⌊N/(2*m-1:ℕ)⌋₊
-  set J : ℕ := ⌊((F:ℝ)-K)/(2*t+1:ℕ)⌋₊
-  have hd : (0:ℝ)<(2*t+1:ℕ) := by positivity
-  have hl : (0:ℝ)<(2*m-1:ℕ) := by exact_mod_cast hL
-  have hjlo := Nat.floor_le (div_nonneg (sub_nonneg.mpr hK) hd.le)
-  have hjhi := Nat.lt_floor_add_one (((F:ℝ)-K)/(2*t+1:ℕ))
-  change (J:ℝ) ≤ _ at hjlo
-  change _<(J:ℝ)+1 at hjhi
-  have hflo := Nat.floor_le (div_nonneg hN hl.le)
-  have hfhi := Nat.lt_floor_add_one (N/(2*m-1:ℕ))
-  change (F:ℝ) ≤ _ at hflo
-  change _<(F:ℝ)+1 at hfhi
-  have hp : (J:ℝ)*(2*t+1:ℕ)+K ≤ F := by
-    have H := (le_div_iff₀ hd).mp hjlo
-    linarith
-  have hp' : (F:ℝ)<(J:ℝ)*(2*t+1:ℕ)+K+(2*t+1:ℕ) := by
-    have H := (div_lt_iff₀ hd).mp hjhi
-    nlinarith
-  constructor
-  · push_cast
-    simpa only [Nat.cast_add,Nat.cast_mul,Nat.cast_ofNat,Nat.cast_one] using
-      (mul_le_mul_of_nonneg_right hp hl.le).trans ((le_div_iff₀ hl).mp hflo)
-  · have H := (div_lt_iff₀ hl).mp hfhi
-    push_cast at hp' ⊢
-    nlinarith
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open Filter
-open scoped Topology Classical
-
-/-- Source moment order is twice this integer. -/
-def sourceHalfMoment (B D : ℕ) (n : ℝ) (i : ℕ) : ℕ :=
-  ⌈(10^8:ℝ)*2^(10*B)*(totalScoreExponent B D+B+1)*Real.log n/
-    Real.log (sourceTargetScale B D n i)⌉₊
-
-theorem sourceEpsilon_le_one (B D : ℕ) : sourceEpsilon B D ≤ 1 := by
-  unfold sourceEpsilon
-  apply (div_le_one (by positivity)).mpr
-  have H : (1:ℝ) ≤ 2^(4*B+100) := one_le_pow₀ (by norm_num)
-  nlinarith [show (0:ℝ) ≤ D by positivity]
-
-theorem sourceTargetScale_tendsto (B : ℕ) (ζ : ℝ) (hB : 2 ≤ B) (hζ : ζ∈Set.Ioo 0 1)
-    {n Λ H : ℕ→ℝ} (i : ℕ→ℕ) (hn : Tendsto n atTop atTop) (hH : Tendsto H atTop atTop)
-    (hΛ : ∀t,1 ≤ Λ t) (hrel : ∀t,Real.log (n t)=Λ t*H t) :
-    Tendsto (fun t=>sourceTargetScale B (sourceDepth B ζ (H t)) (n t) (i t)) atTop atTop := by
-  have hlog : Tendsto (fun t=>Real.log (sourceTargetScale B (sourceDepth B ζ (H t)) (n t) (i t))) atTop atTop := by
-    apply tendsto_atTop_mono' atTop _ (((tendsto_rpow_atTop hζ.1).comp hH).atTop_div_const (by norm_num : (0:ℝ)<2))
-    filter_upwards [hH.eventually_ge_atTop 1,hn.eventually_gt_atTop 1] with t ht hnt
-    have hl := sourceTargetScale_log_lower B ζ (H t) (n t) (i t) hB hζ.2.le ht hnt
-    have he : Real.log (n t)/(2*(H t)^(1-ζ))=Λ t*(H t)^ζ/2 := by
-      rw [hrel t,Real.rpow_sub (by linarith : 0<H t),Real.rpow_one]
-      have hp : (H t)^ζ ≠ 0 := (Real.rpow_pos_of_pos (by linarith : 0<H t) _).ne'
-      field_simp
-    rw [he] at hl
-    exact (by nlinarith [hΛ t,Real.rpow_nonneg (by linarith : 0 ≤ H t) ζ] : (H t)^ζ/2 ≤ Λ t*(H t)^ζ/2).trans hl
-  apply tendsto_atTop_mono' atTop _ hlog
-  filter_upwards [hn.eventually_gt_atTop 1] with t ht
-  exact Real.log_le_self (Real.rpow_nonneg (by linarith : 0 ≤ n t) _)
-
-theorem source_uniform_log_absorption (B h : ℕ) (ζ : ℝ) (hB : 2 ≤ B) (hζ : ζ∈Set.Ioo 0 1)
-    {n Λ H : ℕ→ℝ} (i : ℕ→ℕ) (hn : Tendsto n atTop atTop) (hH : Tendsto H atTop atTop)
-    (hΛ : ∀t,1 ≤ Λ t) (hrel : ∀t,Real.log (n t)=Λ t*H t) :
-    Tendsto (fun t=>((sourceDepth B ζ (H t):ℝ)+1)^h*Λ t/
-      Real.log (sourceTargetScale B (sourceDepth B ζ (H t)) (n t) (i t))) atTop (𝓝 0) := by
-  have hh := source_uniform_absorption B h ζ hB hζ i hn hH (fun t=>lt_of_lt_of_le (by norm_num) (hΛ t)) hrel
-  apply squeeze_zero' _ _ hh
-  · filter_upwards [(sourceTargetScale_tendsto B ζ hB hζ i hn hH hΛ hrel).eventually_gt_atTop 1] with t ht
-    exact div_nonneg (mul_nonneg (by positivity) (by linarith [hΛ t])) (Real.log_pos ht).le
-  · filter_upwards [(sourceTargetScale_tendsto B ζ hB hζ i hn hH hΛ hrel).eventually_gt_atTop 1] with t ht
-    apply div_le_div_of_nonneg_left (by positivity [hΛ t])
-      (mul_pos (sourceEpsilon_pos _ _) (Real.log_pos ht))
-    exact mul_le_of_le_one_left (Real.log_pos ht).le (sourceEpsilon_le_one _ _)
-
-theorem totalScoreExponent_bounds (B D : ℕ) (hB : 2 ≤ B) :
-    11/5 ≤ totalScoreExponent B D ∧ totalScoreExponent B D ≤ 3+(D:ℝ)*B := by
-  have hdim (i : ℕ) : (2:ℝ) ≤ (if i<warmupBlocks B then (2:ℕ) else B) ∧
-      (((if i<warmupBlocks B then (2:ℕ) else B):ℕ):ℝ) ≤ B := by
-    split_ifs <;> norm_cast
-  have hlo : 0 ≤ ∑i∈Finset.range D,blockGain (if i<warmupBlocks B then 2 else B) := by
-    apply Finset.sum_nonneg
-    intro i _
-    unfold blockGain
-    linarith [(hdim i).1]
-  have hhi : (∑i∈Finset.range D,blockGain (if i<warmupBlocks B then 2 else B)) ≤ (D:ℝ)*B := by
-    calc
-      _ ≤ ∑_i∈Finset.range D,(B:ℝ) := by
-        apply Finset.sum_le_sum
-        intro i _
-        unfold blockGain
-        linarith [(hdim i).1,(hdim i).2]
-      _ = _ := by simp
-  unfold totalScoreExponent
-  constructor <;> linarith
-
-theorem sourceHalfMoment_bounds (B D i : ℕ) {n : ℝ} (hB : 2 ≤ B) (hn : 1<n) :
-    1 ≤ sourceHalfMoment B D n i ∧
-    (sourceHalfMoment B D n i:ℝ) ≤
-      ((10^8:ℝ)*2^(10*B)*((B:ℝ)+4)*2+1)*((D:ℝ)+1)*2^(B*D) := by
-  have hp := sourceScaleProduct_lower B D i hB
-  have hp0 : 0<sourceScaleProduct B D i := lt_of_lt_of_le (by positivity) hp
-  have he : (10^8:ℝ)*2^(10*B)*(totalScoreExponent B D+B+1)*Real.log n/
-      Real.log (sourceTargetScale B D n i)=
-      (10^8:ℝ)*2^(10*B)*(totalScoreExponent B D+B+1)/sourceScaleProduct B D i := by
-    rw [sourceTargetScale,Real.log_rpow (by linarith : 0<n)]
-    field_simp [(Real.log_pos hn).ne']
-  have hd := totalScoreExponent_bounds B D hB
-  have hA : 0<(10^8:ℝ)*2^(10*B)*(totalScoreExponent B D+B+1) := by positivity [hd.1]
-  unfold sourceHalfMoment
-  rw [he]
-  constructor
-  · exact Nat.one_le_ceil_iff.mpr (div_pos hA hp0)
-  have hc := Nat.ceil_lt_add_one (div_nonneg hA.le hp0.le)
-  have hb := div_le_div_of_nonneg_left hA.le (by positivity) hp
-  have he' : (10^8:ℝ)*2^(10*B)*(totalScoreExponent B D+B+1)/(1/(2*2^(B*D)))=
-      (10^8:ℝ)*2^(10*B)*(totalScoreExponent B D+B+1)*2*2^(B*D) := by field_simp
-  rw [he'] at hb
-  have hdd : totalScoreExponent B D+B+1 ≤ ((B:ℝ)+4)*((D:ℝ)+1) := by nlinarith [hd.2]
-  have hh := mul_le_mul_of_nonneg_right
-    (mul_le_mul_of_nonneg_left hdd (by positivity : (0:ℝ) ≤ 10^8*2^(10*B)))
-    (by positivity : (0:ℝ) ≤ 2*2^(B*D))
-  have hp1 : (1:ℝ) ≤ 2^(B*D) := one_le_pow₀ (by norm_num)
-  nlinarith [show (0:ℝ) ≤ D by positivity]
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open Filter
-open scoped Topology Classical
-
-variable {N f g : ℕ→ℝ} {α β : ℝ}
-
-theorem PowerRate.eventually_lt (hN : Tendsto N atTop atTop)
-    (hf : PowerRate N f α) (hg : PowerRate N g β) (hab : α<β) :
-    ∀ᶠt in atTop, f t<g t := by
-  have hz := ((hf.div hg).upper hN).tendsto_zero hN (sub_neg.mpr hab)
-  filter_upwards [hz.eventually_lt_const (by norm_num : (0:ℝ)<1),hg.1] with t ht hp
-  exact (div_lt_one hp).mp ht
-
-theorem PowerRate.log_base (hN : Tendsto N atTop atTop) :
-    PowerRate N (fun t=>Real.log (N t)) 0 := by
-  refine ⟨(hN.eventually_gt_atTop 1).mono (fun _ ht=>Real.log_pos ht),?_⟩
-  convert
-    (isLittleO_log_rpow_atTop (by norm_num : (0:ℝ)<1)).tendsto_div_nhds_zero.comp
-      (Real.tendsto_log_atTop.comp hN) using 1
-  ext t
-  simp only [Function.comp_apply,Real.rpow_one]
-
-/-- Only arithmetic predicates on the literal source integers. -/
-structure SourceInitialGeometry (q N : ℝ) : Prop where
-  m_large : 128 ≤ sourceInitialM N
-  K_pos : 0 < sourceInitialK N
-  K_le_m : sourceInitialK N ≤ sourceInitialM N
-  seed : 16 ≤ q^3*(sourceInitialM N:ℝ)
-  count : 8 ≤ ((sourceInitialK N*sourceInitialM N:ℕ):ℝ)*q^2
-  anchor_margin : 12*(sourceInitialJ q N+2^300+1) ≤ sourceInitialK N
-  smoothing_margin : ((2*(2^300):ℕ):ℝ) < (sourceInitialK N:ℝ)/(3*(3000+1:ℝ))
-  length_lower : N/2 ≤ (sourceInitialP q N:ℝ)*(sourceInitialPeriodSize N:ℝ)
-  length_upper : (sourceInitialP q N:ℝ)*(sourceInitialPeriodSize N:ℝ) ≤ N
-
-/-- All floor, seed, count, placement and length restrictions are automatic
- in the actual joint regime; no change of the deletion rate at the base scale. -/
-theorem source_initial_geometry {q : ℕ→ℝ} (hN : Tendsto N atTop atTop)
-    (hq : ∀t,q t∈Set.Ioc 0 1)
-    (hΛ : Tendsto (fun t=>(q t)^(-3:ℤ)/Real.log (N t)) atTop (𝓝 0)) :
-    ∀ᶠt in atTop,SourceInitialGeometry (q t) (N t) := by
-  have hm := sourceInitialM_rate hN
-  have hK := sourceInitialK_rate hN
-  have hqrate := deletion_rate_zero hN hq hΛ
-  have hJ := sourceInitialJ_rate hN hqrate
-  have hL := sourceInitialPeriodSize_rate hN
-  have hF := sourceInitialCapacity_rate hN
-  have ht := sourceInitialSide_rate hN hqrate
-  have hm128 := (hm.tendsto_atTop hN (by norm_num)).eventually_ge_atTop 128
-  have hKm := hK.eventually_lt hN hm (by norm_num)
-  have hKF := hK.eventually_lt hN hF (by norm_num)
-  have hseed := ((hqrate.pow 3).mul hm).tendsto_atTop hN (by norm_num)
-  have hcount := ((hK.mul hm).mul (hqrate.pow 2)).tendsto_atTop hN (by norm_num)
-  have hanchor := ((hJ.add_const hN (by norm_num) (by positivity : (0:ℝ)≤2^300+1)).const_mul
-    hN (by norm_num : (0:ℝ)<12)).eventually_lt hN hK (by norm_num)
-  have hsm := (hK.tendsto_atTop hN (by norm_num)).eventually_gt_atTop ((2*(2^300):ℝ)*(3*3001))
-  have hloss : PowerRate N (fun t=>(2*(initialSideCount (q t) (sourceInitialM (N t)):ℝ)+2)*
-      (sourceInitialPeriodSize (N t):ℝ)) (124/125) := by
-    convert ((ht.const_mul hN (by norm_num : (0:ℝ)<2)).add_const hN
-      (by norm_num) (by norm_num : (0:ℝ)≤2)).mul hL using 1
-    norm_num
-  have hloss' := hloss.eventually_lt hN ((PowerRate.base hN).div_const hN (by norm_num : (0:ℝ)<2)) (by norm_num)
-  filter_upwards [hN.eventually_gt_atTop 0,hm128,hK.1,hKm,hKF,
-    hseed.eventually_ge_atTop 16,hcount.eventually_ge_atTop 8,hanchor,hsm,hloss'] with t hNt hm128 hKpos hKm hKF hs hc ha hsm hloss
-  have hmn : 128 ≤ sourceInitialM (N t) := by exact_mod_cast hm128
-  have hKn : 0 < sourceInitialK (N t) := by exact_mod_cast hKpos
-  have hLpos : 0 < sourceInitialPeriodSize (N t) := by unfold sourceInitialPeriodSize; omega
-  have hcnt := initial_period_count_bounds (m:=sourceInitialM (N t))
-    (K:=sourceInitialK (N t)) (t:=initialSideCount (q t) (sourceInitialM (N t)))
-    (N:=N t) hLpos hNt.le hKF.le
-  have hcnt' : (sourceInitialP (q t) (N t):ℝ)*(sourceInitialPeriodSize (N t):ℝ) ≤ N t ∧
-      N t<((sourceInitialP (q t) (N t):ℝ)+(2*initialSideCount (q t) (sourceInitialM (N t))+1:ℕ)+1)*
-        (sourceInitialPeriodSize (N t):ℝ) := by
-    simpa only [sourceInitialP,sourceInitialJ,sourceInitialCapacity,sourceInitialPeriodSize,
-      Nat.cast_add,Nat.cast_mul,Nat.cast_ofNat,Nat.cast_one] using hcnt
-  refine ⟨hmn,hKn,?_,hs,?_,?_,?_,?_,hcnt'.1⟩
-  · exact_mod_cast hKm.le
-  · simpa only [Nat.cast_mul] using hc
-  · apply (Nat.cast_le (α:=ℝ)).mp
-    simp only [Nat.cast_mul,Nat.cast_add,Nat.cast_pow,Nat.cast_ofNat,Nat.cast_one]
-    simpa only [add_assoc] using ha.le
-  · apply (lt_div_iff₀ (by norm_num : (0:ℝ)<3*(3000+1))).mpr
-    simpa only [Nat.cast_mul,Nat.cast_pow,Nat.cast_ofNat,show (3000+1:ℝ)=3001 by norm_num] using hsm
-  · have H := hcnt'.2
-    push_cast at H
-    nlinarith only [H,hloss]
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open Filter
-open scoped Topology Classical
-
-/-- The exact varying even moment order is subpower at EVERY working scale. -/
-theorem sourceHalfMoment_rate (B : ℕ) (ζ : ℝ) (hB : 2 ≤ B) (hζ : ζ∈Set.Ioo 0 1)
-    {n Λ H : ℕ→ℝ} (i : ℕ→ℕ) (hn : Tendsto n atTop atTop) (hH : Tendsto H atTop atTop)
-    (hΛ : ∀t,1 ≤ Λ t) (hrel : ∀t,Real.log (n t)=Λ t*H t) :
-    PowerRate (fun t=>sourceTargetScale B (sourceDepth B ζ (H t)) (n t) (i t))
-      (fun t=>(sourceHalfMoment B (sourceDepth B ζ (H t)) (n t) (i t):ℝ)) 0 := by
-  let C : ℝ := ((10^8:ℝ)*2^(10*B)*((B:ℝ)+4)*2+1)
-  let A : ℝ := C+1+(B:ℝ)*Real.log 2
-  have hC : 0<C := by dsimp [C]; positivity
-  have hA : 0<A := by dsimp [A]; positivity
-  have hscale := sourceTargetScale_tendsto B ζ hB hζ i hn hH hΛ hrel
-  have hlim := (source_uniform_log_absorption B 1 ζ hB hζ i hn hH hΛ hrel).const_mul A
-  simp only [pow_one,mul_zero] at hlim
-  refine ⟨?_,?_⟩
-  · filter_upwards [hn.eventually_gt_atTop 1] with t ht
-    have hk := (sourceHalfMoment_bounds B (sourceDepth B ζ (H t)) (i t) hB ht).1
-    exact_mod_cast (show 0<sourceHalfMoment B (sourceDepth B ζ (H t)) (n t) (i t) by omega)
-  apply squeeze_zero' _ _ hlim
-  · filter_upwards [hn.eventually_gt_atTop 1,hscale.eventually_gt_atTop 1] with t ht hs
-    have hk := (sourceHalfMoment_bounds B (sourceDepth B ζ (H t)) (i t) hB ht).1
-    exact div_nonneg (Real.log_nonneg (by exact_mod_cast hk)) (Real.log_pos hs).le
-  · filter_upwards [hn.eventually_gt_atTop 1,hscale.eventually_gt_atTop 1] with t ht hs
-    let D := sourceDepth B ζ (H t)
-    let k := sourceHalfMoment B D (n t) (i t)
-    have hk := sourceHalfMoment_bounds B D (i t) hB ht
-    have hkpos : 0<(k:ℝ) := by exact_mod_cast (show 0<k by dsimp [k]; omega)
-    have hlog := Real.log_le_log hkpos hk.2
-    change Real.log (k:ℝ) ≤ Real.log (C*((D:ℝ)+1)*2^(B*D)) at hlog
-    rw [Real.log_mul (mul_pos hC (by positivity)).ne' (by positivity),
-      Real.log_mul hC.ne' (by positivity),Real.log_pow] at hlog
-    have hD : Real.log ((D:ℝ)+1) ≤ D := by
-      simpa using Real.log_le_sub_one_of_pos (by positivity : (0:ℝ)<(D:ℝ)+1)
-    have hc := Real.log_le_sub_one_of_pos hC
-    have hb : 0 ≤ (B:ℝ)*Real.log 2 := by positivity
-    have hd : 0 ≤ (D:ℝ) := Nat.cast_nonneg _
-    have hl : Real.log (k:ℝ) ≤ A*((D:ℝ)+1) := by
-      dsimp [A]
-      push_cast at hlog
-      nlinarith
-    have hl' : Real.log (k:ℝ) ≤ A*((D:ℝ)+1)*Λ t :=
-      hl.trans (le_mul_of_one_le_right (by positivity) (hΛ t))
-    have HH := div_le_div_of_nonneg_right hl' (Real.log_pos hs).le
-    simpa only [mul_div_assoc,mul_assoc] using HH
-
-/-- Actual primitive-score weight at the base scale. -/
-def sourceInitialWeight (q : ℝ) (k : ℕ) : ℝ := (6*(2*k:ℕ):ℝ)*(2/q)
-
-theorem sourceInitialWeight_rate {N q : ℕ→ℝ} {k : ℕ→ℕ}
-    (hN : Tendsto N atTop atTop) (hq : PowerRate N q 0)
-    (hk : PowerRate N (fun t=>(k t:ℝ)) 0) :
-    PowerRate N (fun t=>sourceInitialWeight (q t) (k t)) 0 := by
-  have H := (hk.const_mul hN (by norm_num : (0:ℝ)<24)).div hq
-  convert H using 1
-  · ext t
-    simp only [sourceInitialWeight,Nat.cast_mul,Nat.cast_ofNat]
-    ring
-  · norm_num
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open Filter
-open scoped Topology Classical
-
-variable {N q w : ℕ→ℝ} {k : ℕ→ℕ}
-
-def initialClipFactor (N q w : ℝ) : ℝ :=
-  4*(initialMomentRatio (sourceInitialK N) (sourceInitialM N) q w/
-    (1-initialMomentRatio (sourceInitialK N) (sourceInitialM N) q w))
-
-theorem initialClipFactor_rate (hN : Tendsto N atTop atTop)
-    (hq : PowerRate N q 0) (hw : PowerRate N w 0) :
-    PowerRate N (fun t=>initialClipFactor (N t) (q t) (w t)) (-1/500) := by
-  have hρ := initialMomentRatio_rate hN (sourceInitialK_rate hN) (sourceInitialM_rate hN) hq hw
-  have hden := (PowerRate.const hN (by norm_num : (0:ℝ)<1)).sub_lower hN hρ (by norm_num)
-  simpa only [initialClipFactor,sub_zero] using (hρ.div hden).const_mul hN (by norm_num : (0:ℝ)<4)
-
-/-- The source moment is genuinely used in clipping: the exponent is linear
- in that moment, not a fixed negative power of the intermediate length. -/
-theorem initial_clip_exp_bound (hN : Tendsto N atTop atTop)
-    (hq : PowerRate N q 0) (hw : PowerRate N w 0) (hk : ∀ᶠt in atTop,1 ≤ k t) :
-    ∀ᶠt in atTop,
-      Real.sqrt (6*(initialClipFactor (N t) (q t) (w t))^(k t)) ≤
-        Real.exp (-(k t:ℝ)*Real.log (N t)/2000) := by
-  have hfac := initialClipFactor_rate hN hq hw
-  have hu := hfac.upper hN (-3/2000) (by norm_num)
-  filter_upwards [hk,hfac.1,hu,hN.eventually_gt_atTop 1,
-    (Real.tendsto_log_atTop.comp hN).eventually_ge_atTop (2000*Real.log 6)] with t hk hp hu hn hlog
-  rw [abs_of_pos hp] at hu
-  have hh := Real.log_le_log hp hu
-  rw [Real.log_rpow (by linarith : 0<N t)] at hh
-  have hk' : (1:ℝ) ≤ k t := by exact_mod_cast hk
-  have hmul := mul_le_mul_of_nonneg_left hh (by positivity : (0:ℝ) ≤ k t)
-  have hl : 0 ≤ Real.log (N t) := (Real.log_pos hn).le
-  have hg : Real.log 6 ≤ (k t:ℝ)*Real.log (N t)/2000 := by
-    change 2000*Real.log 6 ≤ Real.log (N t) at hlog
-    nlinarith [mul_nonneg (sub_nonneg.mpr hk') hl]
-  have hexp : Real.sqrt (6*initialClipFactor (N t) (q t) (w t)^(k t)) =
-      Real.exp ((Real.log 6+(k t:ℝ)*Real.log (initialClipFactor (N t) (q t) (w t)))/2) := by
-    rw [Real.sqrt_eq_rpow,Real.rpow_def_of_pos (by positivity),Real.log_mul (by norm_num) (pow_pos hp _).ne',Real.log_pow]
-    congr 1
-    ring
-  rw [hexp]
-  exact Real.exp_le_exp.mpr (by nlinarith)
-
-/-- The fixed metadata tails also beat the varying moment times log-scale.
- Their exponents have orders .988 and .008 respectively, while k log N
- has order zero. -/
-theorem initial_metadata_exp_bound (hN : Tendsto N atTop atTop)
-    (hq : PowerRate N q 0) (hω : PowerRate N (fun t=>initialOverlapLower (q t)) 0)
-    (hk : PowerRate N (fun t=>(k t:ℝ)) 0) :
-    ∀ᶠt in atTop,
-      2*Real.exp (-((sourceInitialK (N t)*sourceInitialM (N t):ℕ):ℝ)*(q t)^4/64)+
-        Real.exp (-(sourceInitialJ (q t) (N t):ℝ)*(initialOverlapLower (q t))^4/16) ≤
-        3*Real.exp (-(k t:ℝ)*Real.log (N t)) := by
-  have hlow : PowerRate N (fun t=>(k t:ℝ)*Real.log (N t)) 0 := by
-    simpa using hk.mul (PowerRate.log_base hN)
-  have hfirst := (((sourceInitialK_rate hN).mul (sourceInitialM_rate hN)).mul
-    (hq.pow 4)).div_const hN (by norm_num : (0:ℝ)<64)
-  have hsecond := ((sourceInitialJ_rate hN hq).mul (hω.pow 4)).div_const hN (by norm_num : (0:ℝ)<16)
-  filter_upwards [hlow.eventually_lt hN hfirst (by norm_num),hlow.eventually_lt hN hsecond (by norm_num)] with t h1 h2
-  have he1 := Real.exp_le_exp.mpr (neg_le_neg h1.le)
-  have he2 := Real.exp_le_exp.mpr (neg_le_neg h2.le)
-  push_cast
-  simp only [neg_div,neg_mul] at he1 he2 ⊢
-  linarith
-
-/-- A completely quantitative sufficient moment inequality for the actual
- initial error. This is later discharged by the literal ceiling r_0/2. -/
-theorem initial_error_eventually {n A : ℕ→ℝ} (hN : Tendsto N atTop atTop)
-    (hn : Tendsto n atTop atTop) (hA : ∀ᶠt in atTop,1 ≤ A t)
-    (hq : ∀t,q t∈Set.Ioc 0 1)
-    (hΛ : Tendsto (fun t=>(q t)^(-3:ℤ)/Real.log (N t)) atTop (𝓝 0))
-    (hk : PowerRate N (fun t=>(k t:ℝ)) 0) (hk1 : ∀ᶠt in atTop,1 ≤ k t)
-    (hmoment : ∀ᶠt in atTop, 10^6*A t*Real.log (n t) ≤ (k t:ℝ)*Real.log (N t)) :
-    ∀ᶠt in atTop,
-      initialErrorCap (sourceInitialJ (q t) (N t)) (sourceInitialK (N t))
-        (sourceInitialM (N t)) (k t) (q t) (initialOverlapLower (q t))
-        (sourceInitialWeight (q t) (k t)) ≤ (n t)^(-100*A t) := by
-  have hqrate := deletion_rate_zero hN hq hΛ
-  have hω := initialOverlap_rate hN (fun t=>(hq t).1) hΛ
-  have hw := sourceInitialWeight_rate hN hqrate hk
-  have hclip := initial_clip_exp_bound hN hqrate hw hk1
-  have hmeta := initial_metadata_exp_bound hN hqrate hω hk
-  filter_upwards [hclip,hmeta,hmoment,hA,hn.eventually_gt_atTop 1,
-    (Real.tendsto_log_atTop.comp hn).eventually_ge_atTop (Real.log 4)] with t hc hm hmom hA hn hlog
-  have hAn : 0 ≤ A t*Real.log (n t) := mul_nonneg (by linarith) (Real.log_pos hn).le
-  have he1 : Real.exp (-(k t:ℝ)*Real.log (N t)/2000) ≤ Real.exp (-500*(A t*Real.log (n t))) :=
-    Real.exp_le_exp.mpr (by nlinarith)
-  have he2 : Real.exp (-(k t:ℝ)*Real.log (N t)) ≤ Real.exp (-500*(A t*Real.log (n t))) :=
-    Real.exp_le_exp.mpr (by nlinarith)
-  have herr : initialErrorCap (sourceInitialJ (q t) (N t)) (sourceInitialK (N t))
-      (sourceInitialM (N t)) (k t) (q t) (initialOverlapLower (q t))
-      (sourceInitialWeight (q t) (k t)) ≤ 4*Real.exp (-500*(A t*Real.log (n t))) := by
-    unfold initialErrorCap
-    change Real.sqrt (6*initialClipFactor (N t) (q t) (sourceInitialWeight (q t) (k t))^(k t))+_ ≤ _
-    nlinarith only [hc,hm,he1,he2]
-  apply herr.trans
-  rw [Real.rpow_def_of_pos (by linarith : 0<n t)]
-  calc
-    4*Real.exp (-500*(A t*Real.log (n t))) =
-        Real.exp (Real.log 4-500*(A t*Real.log (n t))) := by
-      rw [Real.exp_sub,Real.exp_log (by norm_num)]
-      rw [show -500*(A t*Real.log (n t)) = -(500*(A t*Real.log (n t))) by ring, Real.exp_neg]
-      rfl
-    _ ≤ Real.exp (Real.log (n t)*(-100*A t)) := by
-      apply Real.exp_le_exp.mpr
-      have hAL : Real.log 4 ≤ A t*Real.log (n t) := hlog.trans (le_mul_of_one_le_left (Real.log_pos hn).le hA)
-      nlinarith
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open Filter
-open scoped Topology Classical
-
-variable {N f g : ℕ→ℝ} {α β : ℝ}
-
-theorem PowerRate.add_lower (hN : Tendsto N atTop atTop)
-    (hf : PowerRate N f α) (hg : PowerRate N g β) (hβα : β<α) :
-    PowerRate N (fun t=>f t+g t) α := by
-  have hz := ((hg.div hf).upper hN).tendsto_zero hN (sub_neg.mpr hβα)
-  apply hf.of_ratio hN (by norm_num : (0:ℝ)<1)
-  have H : Tendsto (fun t=>1+g t/f t) atTop (𝓝 1) := by
-    simpa using tendsto_const_nhds.add hz
-  apply H.congr'
-  filter_upwards [hf.1] with t ht
-  field_simp [ht.ne']
-
-theorem PowerRate.nat_ceil (hN : Tendsto N atTop atTop) (hf : PowerRate N f α)
-    (ha : 0<α) : PowerRate N (fun t=>(⌈f t⌉₊:ℝ)) α := by
-  exact hf.of_ratio hN (by norm_num : (0:ℝ)<1)
-    (tendsto_nat_ceil_div_atTop.comp (hf.tendsto_atTop hN ha))
-
-def sourceOverlapConstant (q : ℝ) : ℝ := (2:ℝ)^40/(q*initialOverlapLower q)
-def sourceInitialCount (q N : ℝ) : ℕ :=
-  ⌈sourceOverlapConstant q*(initialSideCount q (sourceInitialM N):ℝ)*(sourceInitialP q N:ℝ)⌉₊
-
-theorem sourceInitialP_rate {q : ℕ→ℝ} (hN : Tendsto N atTop atTop)
-    (hq : PowerRate N q 0) :
-    PowerRate N (fun t=>(sourceInitialP (q t) (N t):ℝ)) (63/125) := by
-  have hj := sourceInitialJ_rate hN hq
-  have ht := ((sourceInitialSide_rate hN hq).const_mul hN (by norm_num : (0:ℝ)<2)).add_const hN
-    (by norm_num) (by norm_num : (0:ℝ)≤1)
-  have H := (hj.mul ht).add_lower hN (sourceInitialK_rate hN) (by norm_num)
-  convert H using 1
-  · ext t
-    simp only [sourceInitialP,Nat.cast_add,Nat.cast_mul,Nat.cast_ofNat,Nat.cast_one]
-  · norm_num
-
-theorem sourceOverlapConstant_rate {q : ℕ→ℝ} (hN : Tendsto N atTop atTop)
-    (hq : PowerRate N q 0) (hω : PowerRate N (fun t=>initialOverlapLower (q t)) 0) :
-    PowerRate N (fun t=>sourceOverlapConstant (q t)) 0 := by
-  simpa only [sourceOverlapConstant,add_zero,sub_zero] using
-    (PowerRate.const hN (by positivity : (0:ℝ)<(2:ℝ)^40)).div (hq.mul hω)
-
-theorem sourceInitialCount_rate {q : ℕ→ℝ} (hN : Tendsto N atTop atTop)
-    (hq : PowerRate N q 0) (hω : PowerRate N (fun t=>initialOverlapLower (q t)) 0) :
-    PowerRate N (fun t=>(sourceInitialCount (q t) (N t):ℝ)) 1 := by
-  have H := (((sourceOverlapConstant_rate hN hq hω).mul (sourceInitialSide_rate hN hq)).mul
-    (sourceInitialP_rate hN hq)).nat_ceil hN (by norm_num)
-  convert H using 1
-  · rfl
-  · norm_num
-
-theorem sourceOverlapConstant_pos {q : ℝ} (hq : 0<q) : 0<sourceOverlapConstant q := by
-  unfold sourceOverlapConstant
-  positivity [initialOverlapLower_pos q]
-
-theorem sourceOverlapConstant_large {q : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ)) :
-    1073741824 ≤ initialOverlapLower q*sourceOverlapConstant q := by
-  have hω := initialOverlapLower_pos q
-  have he : initialOverlapLower q*sourceOverlapConstant q=(2:ℝ)^40/q := by
-    unfold sourceOverlapConstant
-    field_simp
-  rw [he]
-  apply (le_div_iff₀ hq.1).mpr
-  nlinarith [hq.2]
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open Filter
-open scoped Topology Classical
-
-def sourceInitialWord (q N : ℝ) (τ : ℝ) : List Bool :=
-  if hm : 4 ≤ sourceInitialM N then
-    motifPeriodWord (sourceInitialOffset (sourceInitialM N) hm)
-      (sourceInitialOffset_interior _ hm) (sourceInitialP q N) 300 (sourceInitialAnchor q N) τ
-  else []
-
-def sourceInitialPure (q N : ℝ) : List Bool :=
-  pureAtomRun (initialPeriod (sourceInitialM N) (sourceInitialM N/2-1)) (sourceInitialP q N)
-
-def sourceInitialError (q N : ℝ) (k : ℕ) : ℝ :=
-  initialErrorCap (sourceInitialJ q N) (sourceInitialK N) (sourceInitialM N) k q
-    (initialOverlapLower q) (sourceInitialWeight q k)
-
-theorem sourceInitialWord_length {q N : ℝ} (hm : 4 ≤ sourceInitialM N) (τ : ℝ) :
-    (sourceInitialWord q N τ).length=sourceInitialP q N*sourceInitialPeriodSize N := by
-  rw [sourceInitialWord,dite_eq_left hm,motifPeriodWord_length]
-  rfl
-
-theorem sourceInitialWord_different {q N : ℝ} (H : SourceInitialGeometry q N) :
-    sourceInitialWord q N (-1) ≠ sourceInitialWord q N 1 := by
-  have hm : 4 ≤ sourceInitialM N := (by norm_num : 4 ≤ 128).trans H.m_large
-  have ha := initial_anchor_geometry (sourceInitialJ q N) (sourceInitialK N)
-    (initialSideCount q (sourceInitialM N)) 300 H.anchor_margin
-  have hap : (0:ℤ) ≤ sourceInitialAnchor q N := (Int.natCast_nonneg _).trans ha.1
-  have hau : sourceInitialAnchor q N<(sourceInitialP q N:ℤ) := by
-    have h1 := ha.2.1
-    have he : (sourceInitialJ q N*initialSideCount q (sourceInitialM N)+sourceInitialK N:ℕ) ≤ sourceInitialP q N := by
-      unfold sourceInitialP
-      apply Nat.add_le_add_right
-      apply Nat.mul_le_mul_left
-      omega
-    have hp : (0:ℤ)<(2^300:ℕ) := by positivity
-    have hh : ((sourceInitialJ q N*initialSideCount q (sourceInitialM N)+sourceInitialK N:ℕ):ℤ) ≤ sourceInitialP q N := Int.ofNat_le.mpr he
-    exact (lt_add_of_pos_right (sourceInitialAnchor q N) hp).trans_le (h1.trans hh)
-  simp only [sourceInitialWord,dite_eq_left hm]
-  exact motifPeriodWord_ne _ _ _ _ _ hap hau
-
-/-- The actual initial proof, with no oracle or coefficient-bound hypothesis.
- The only moment restriction is the quantitative source ceiling inequality. -/
-theorem source_initial_oracles_eventually {N q n A : ℕ→ℝ} {k : ℕ→ℕ}
-    (hN : Tendsto N atTop atTop) (hn : Tendsto n atTop atTop)
-    (hA : ∀ᶠt in atTop,1 ≤ A t) (hq : ∀t,q t∈Set.Ioc 0 (1/2:ℝ))
-    (hΛ : Tendsto (fun t=>(q t)^(-3:ℤ)/Real.log (N t)) atTop (𝓝 0))
-    (hk : PowerRate N (fun t=>(k t:ℝ)) 0) (hk1 : ∀ᶠt in atTop,1 ≤ k t)
-    (hmoment : ∀ᶠt in atTop,10^6*A t*Real.log (n t) ≤ (k t:ℝ)*Real.log (N t)) :
-    ∀ᶠt in atTop,
-      SourceInitialGeometry (q t) (N t) ∧
-      sourceInitialError (q t) (N t) (k t) ≤ (n t)^(-100*A t) ∧
-      Nonempty (OracleTracePair (q t) (sourceInitialWord (q t) (N t) (-1))
-        (sourceInitialWord (q t) (N t) 1) (k t) (sourceInitialError (q t) (N t) (k t)) ((N t)^(-11/5:ℝ))) ∧
-      Nonempty (OracleTracePair (q t) (sourceInitialPure (q t) (N t))
-        (sourceInitialWord (q t) (N t) 1) (k t) (sourceInitialError (q t) (N t) (k t)) ((N t)^(-7/5:ℝ))) := by
-  have hq' (t) : q t ∈ Set.Ioc 0 1 := ⟨(hq t).1,by linarith [(hq t).2]⟩
-  have hqr := deletion_rate_zero hN hq' hΛ
-  have hw := sourceInitialWeight_rate hN hqr hk
-  have herr := initial_error_eventually hN hn hA hq' hΛ hk hk1 hmoment
-  have hgeo := source_initial_geometry hN hq' hΛ
-  have hcap := source_initial_score_caps hN hq' hΛ hw
-  have hρ := (initialMomentRatio_rate hN (sourceInitialK_rate hN) (sourceInitialM_rate hN) hqr hw).upper hN |>.tendsto_zero hN (by norm_num)
-  have hsmall := ((tendsto_rpow_neg_atTop (by norm_num : (0:ℝ)<100)).comp hn).eventually_le_const (by norm_num : (0:ℝ)<1/2)
-  filter_upwards [hgeo,hcap,herr,hk1,hA,hn.eventually_gt_atTop 1,hN.eventually_gt_atTop 0,
-    hρ.eventually_le_const (by norm_num : (0:ℝ)<1/2),hsmall] with t hg hcap he hk1 ha hnt hNt hρ hsmall
-  have hm : 4 ≤ sourceInitialM (N t) := (by norm_num : 4 ≤ 128).trans hg.m_large
-  let off := sourceInitialOffset (sourceInitialM (N t)) hm
-  have hi := sourceInitialOffset_interior _ hm
-  have hanchor := initial_anchor_geometry (sourceInitialJ (q t) (N t)) (sourceInitialK (N t))
-    (initialSideCount (q t) (sourceInitialM (N t))) 300 hg.anchor_margin
-  change _ ≤ sourceInitialAnchor (q t) (N t) ∧ sourceInitialAnchor (q t) (N t)+(2^300:ℕ) ≤ _ ∧ _ at hanchor
-  have he' : sourceInitialError (q t) (N t) (k t) ≤ 1/2 := by
-    exact he.trans ((Real.rpow_le_rpow_of_exponent_le hnt.le (by linarith : -100*A t ≤ -100)).trans hsmall)
-  have hc : Real.sqrt (6*(4*(initialMomentRatio (sourceInitialK (N t)) (sourceInitialM (N t)) (q t)
-      (sourceInitialWeight (q t) (k t))/(1-initialMomentRatio (sourceInitialK (N t))
-      (sourceInitialM (N t)) (q t) (sourceInitialWeight (q t) (k t)))))^(k t)) ≤ 1/2 := by
-    have ht : 0 ≤ 2*Real.exp (-((sourceInitialK (N t)*sourceInitialM (N t):ℕ):ℝ)*(q t)^4/64)+
-      Real.exp (-(sourceInitialJ (q t) (N t):ℝ)*(initialOverlapLower (q t))^4/16) := by positivity
-    unfold sourceInitialError initialErrorCap at he'
-    linarith only [he',ht]
-  have HH := actual_initial_oracles hg.K_pos hg.m_large hg.K_le_m (by norm_num : 0<3000)
-    off hi (sourceInitialOffset_middle _ hm) (sourceInitialOffset_safe _ _ hm)
-    (sourceInitialAnchor (q t) (N t)) 300 (k t) hk1 (hq t) hg.seed hg.count
-    hanchor.1 hanchor.2.1 hanchor.2.2.1 hanchor.2.2.2 hg.smoothing_margin
-    (rfl : sourceInitialWeight (q t) (k t)=_) hρ hc
-    (Real.rpow_nonneg hNt.le _) (Real.rpow_nonneg hNt.le _) hcap.1 hcap.2
-  refine ⟨hg,he,?_,?_⟩
-  · simpa only [sourceInitialWord,dite_eq_left hm,sourceInitialP,sourceInitialError,off] using HH.1
-  · simpa only [sourceInitialWord,dite_eq_left hm,sourceInitialP,sourceInitialError,sourceInitialPure,
-      off,sourceInitialOffset] using HH.2
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open Filter
-open scoped Topology Classical
-
-/-- Literal count instantiation; the harmless ceiling is included. -/
-theorem sourceInitialCount_bounds {q N : ℝ} (hq : q ∈ Set.Ioc 0 (1/2:ℝ))
-    (H : SourceInitialGeometry q N) :
-    sourceInitialP q N*sourceInitialPeriodSize N  ≤  sourceInitialCount q N ∧
-    (sourceInitialCount q N:ℝ)  ≤  (sourceOverlapConstant q)^2*
-      ((sourceInitialP q N*sourceInitialPeriodSize N:ℕ):ℝ) := by
-  let C := sourceOverlapConstant q
-  let m := sourceInitialM N
-  let t := initialSideCount q m
-  let P := sourceInitialP q N
-  let ell := sourceInitialPeriodSize N
-  have hC : 0<C := sourceOverlapConstant_pos hq.1
-  have hm : 32 ≤ m := (by norm_num : 32 ≤ 128).trans H.m_large
-  have ht := initialSideCount_bounds hq hm H.seed
-  have hω := initialOverlapLower_lt_eighth hq.1
-  have hω0 := initialOverlapLower_pos q
-  have hCq : 4  ≤  C*q := by
-    have he : C*q=(2:ℝ)^40/initialOverlapLower q := by dsimp [C,sourceOverlapConstant]; field_simp [hq.1.ne']
-    rw [he]
-    exact (le_div_iff₀ hω0).mpr (by nlinarith)
-  have hC2 : 2  ≤  C := by nlinarith [hq.2]
-  have hell : (ell:ℝ)=2*(m:ℝ)-1 := by
-    dsimp [ell,sourceInitialPeriodSize,m]
-    rw [Nat.cast_sub (by omega),Nat.cast_mul,Nat.cast_ofNat,Nat.cast_one]
-  have hell1 : (1:ℝ) ≤ ell := by
-    rw [hell]
-    have hml : (32:ℝ) ≤ m := by exact_mod_cast hm
-    linarith only [hml]
-  have hPm : 1 ≤ P := by dsimp [P,sourceInitialP]; have := H.K_pos; omega
-  have hP : (1:ℝ) ≤ P := by exact_mod_cast hPm
-  have hPt : 0 ≤ (P:ℝ) := Nat.cast_nonneg _
-  have hm0 : (0:ℝ) ≤ m := Nat.cast_nonneg _
-  have hlower : (ell:ℝ) ≤ C*(t:ℝ) := by
-    have hh := mul_le_mul_of_nonneg_left ht.1 hC.le
-    have hh' := mul_le_mul_of_nonneg_right hCq hm0
-    rw [hell]
-    dsimp [t] at *
-    nlinarith only [hh,hh']
-  have hct : 0 ≤ C*(t:ℝ)*(P:ℝ) := by positivity
-  have hceil : C*(t:ℝ)*(P:ℝ) ≤ (sourceInitialCount q N:ℝ) := Nat.le_ceil _
-  have hu : (sourceInitialCount q N:ℝ)<C*(t:ℝ)*(P:ℝ)+1 := Nat.ceil_lt_add_one hct
-  constructor
-  · apply (Nat.cast_le (α:=ℝ)).mp
-    simp only [Nat.cast_mul]
-    simpa only [mul_comm (ell:ℝ) (P:ℝ)] using (mul_le_mul_of_nonneg_right hlower hPt).trans hceil
-  · have htell : (t:ℝ) ≤ ell := by
-      have hh := mul_le_mul_of_nonneg_right hq.2 hm0
-      rw [hell]
-      have hml : (32:ℝ) ≤ m := by exact_mod_cast hm
-      linarith [ht.2.1]
-    have he1 : (1:ℝ) ≤ (P:ℝ)*ell := one_le_mul_of_one_le_of_one_le hP hell1
-    have hu' := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left htell hC.le) hPt
-    have hc : C+1 ≤ C^2 := by nlinarith
-    have hh := mul_le_mul_of_nonneg_right hc (mul_nonneg hPt (zero_le_one.trans hell1))
-    change _  ≤  C^2*((P*ell:ℕ):ℝ)
-    rw [Nat.cast_mul]
-    nlinarith only [hu,hu',he1,hh]
-
-/-- Genuine overlap stability for complete traces of powers. -/
-theorem OracleTracePair.transfer_power_overlap {q E V p : ℝ} {x y : List Bool} {k F : ℕ}
-    (C : OracleTracePair q x y k E V) (hq : q ∈ Set.Icc 0 1) (hk : 1 ≤ k)
-    (hov : p ≤ listTraceOverlap q hq (pureAtomRun y F) (pureAtomRun y (F+1))) :
-    p-(Real.sqrt ((F:ℝ)*V)+2*(F:ℝ)*E)-
-      (Real.sqrt (((F+1:ℕ):ℝ)*V)+2*((F+1:ℕ):ℝ)*E)  ≤ 
-      listTraceOverlap q hq (pureAtomRun x F) (pureAtomRun x (F+1)) := by
-  let a := pureAtomRun y F
-  let b := pureAtomRun y (F+1)
-  let u := pureAtomRun x F
-  let v := pureAtomRun x (F+1)
-  let M := max (max a.length b.length) (max u.length v.length)
-  have ha : a.length ≤ M := (le_max_left _ _).trans (le_max_left _ _)
-  have hb : b.length ≤ M := (le_max_right _ _).trans (le_max_left _ _)
-  have hu : u.length ≤ M := (le_max_left _ _).trans (le_max_right _ _)
-  have hv : v.length ≤ M := (le_max_right _ _).trans (le_max_right _ _)
-  have H₁ := C.power_tv hq hk F M hu ha
-  have H₂ := C.power_tv hq hk (F+1) M hv hb
-  rw [listTraceOverlap_ambient hq _ _ M ha hb,FiniteLaw.overlap_eq_one_sub_tv] at hov
-  rw [listTraceOverlap_ambient hq _ _ M hu hv,FiniteLaw.overlap_eq_one_sub_tv]
-  have HH := (FiniteLaw.tv_triangle (listTraceLaw q hq u M hu) (listTraceLaw q hq a M ha)
-    (listTraceLaw q hq v M hv)).trans
-    (add_le_add le_rfl (FiniteLaw.tv_triangle (listTraceLaw q hq a M ha)
-      (listTraceLaw q hq b M hb) (listTraceLaw q hq v M hv)))
-  rw [FiniteLaw.tv_symm (listTraceLaw q hq b M hb) (listTraceLaw q hq v M hv)] at HH
-  linarith only [HH,H₁,H₂,hov]
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open Filter
-open scoped Topology Classical
-
-theorem powerTransferBudget_mono {V W E : ℝ} {F G : ℕ} (hE : 0 ≤ E)
-    (hV : 0 ≤ V) (hVW : V ≤ W) (hFG : F ≤ G) :
-    Real.sqrt ((F:ℝ)*V)+2*(F:ℝ)*E ≤ Real.sqrt ((G:ℝ)*W)+2*(G:ℝ)*E := by
-  have hfg : (F:ℝ) ≤ G := by exact_mod_cast hFG
-  apply add_le_add
-  · exact Real.sqrt_le_sqrt (mul_le_mul hfg hVW hV (Nat.cast_nonneg _))
-  · nlinarith only [mul_le_mul_of_nonneg_right hfg hE]
-
-theorem source_initial_power_overlap {q N E : ℝ} {k : ℕ}
-    (hq : q ∈ Set.Ioc 0 (1/2:ℝ)) (H : SourceInitialGeometry q N)
-    (hN : 1 ≤ N) (hk : 1 ≤ k)
-    (C : OracleTracePair q (sourceInitialWord q N (-1)) (sourceInitialWord q N 1)
-      k E (N^(-11/5:ℝ)))
-    (U : OracleTracePair q (sourceInitialPure q N) (sourceInitialWord q N 1)
-      k E (N^(-7/5:ℝ)))
-    (hsmall : Real.sqrt (((sourceInitialCount q N+1:ℕ):ℝ)*(N^(-7/5:ℝ)))+
-      2*((sourceInitialCount q N+1:ℕ):ℝ)*E ≤ 1/16) :
-    initialOverlapLower q ≤ listTraceOverlap q ⟨hq.1.le,by linarith [hq.2]⟩
-      (pureAtomRun (sourceInitialWord q N (-1)) (sourceInitialCount q N))
-      (pureAtomRun (sourceInitialWord q N (-1)) (sourceInitialCount q N+1)) := by
-  let F := sourceInitialCount q N
-  have hm : 4 ≤ sourceInitialM N := (by norm_num : 4 ≤ 128).trans H.m_large
-  let off := sourceInitialOffset (sourceInitialM N) hm
-  have ht := initialSideCount_bounds hq ((by norm_num : 32 ≤ 128).trans H.m_large) H.seed
-  have hqm : 2 ≤ q*(sourceInitialM N:ℝ) := by
-    have hh := mul_le_mul_of_nonneg_right (show q^2 ≤ 1 by nlinarith [hq.1,hq.2])
-      (mul_nonneg hq.1.le (Nat.cast_nonneg (sourceInitialM N)))
-    nlinarith [H.seed]
-  have ht1 : 1 ≤ initialSideCount q (sourceInitialM N) := by
-    have hh : (1:ℝ) ≤ initialSideCount q (sourceInitialM N) := by linarith [ht.1]
-    exact_mod_cast hh
-  have hP : 1 ≤ sourceInitialP q N := by unfold sourceInitialP; have := H.K_pos; omega
-  have hω : initialOverlapLower q ∈ Set.Ioc 0 (1/8:ℝ) :=
-    ⟨initialOverlapLower_pos q,(initialOverlapLower_lt_eighth hq.1).le⟩
-  have hqc : q ∈ Set.Icc 0 1 := ⟨hq.1.le,by linarith [hq.2]⟩
-  have hu := U.lifted_power_overlap hqc hk ht1 hP hω (sourceOverlapConstant_large hq)
-    (Nat.le_ceil _) (initial_pure_trace_overlap hq
-      ((by norm_num : 32 ≤ 128).trans H.m_large) H.seed off ((sourceInitialOffset_middle _ hm).trans (by norm_num)))
-  have hv0 := Real.rpow_nonneg (by linarith : 0 ≤ N) (-11/5:ℝ)
-  have hw0 := Real.rpow_nonneg (by linarith : 0 ≤ N) (-7/5:ℝ)
-  have hvw : N^(-11/5:ℝ) ≤ N^(-7/5:ℝ) := Real.rpow_le_rpow_of_exponent_le hN (by norm_num)
-  have hs0 := (powerTransferBudget_mono C.error_nonneg hw0 le_rfl (Nat.le_succ F)).trans hsmall
-  have hs1 := (powerTransferBudget_mono C.error_nonneg hv0 hvw (Nat.le_succ F)).trans hsmall
-  have hs2 := (powerTransferBudget_mono C.error_nonneg hv0 hvw (le_refl (F+1))).trans hsmall
-  have htrans := C.transfer_power_overlap hqc hk hu
-  have hω' := hω.2
-  dsimp only [F,sourceInitialCount] at hs0 hs1 hs2 hsmall ⊢
-  linarith only [htrans,hs0,hs1,hs2,hsmall,hω']
-
-/-- The exact initial overlap error is negligible, including channel-dependent
- count factors. This is a consequence, not a hypothesis of the initial pair. -/
-theorem source_initial_overlap_budget_eventually {q N E : ℕ→ℝ}
-    (hN : Tendsto N atTop atTop) (hq : PowerRate N q 0)
-    (hω : PowerRate N (fun t=>initialOverlapLower (q t)) 0)
-    (hE : ∀ᶠt in atTop,0 ≤ E t ∧ E t ≤ (N t)^(-100:ℝ)) :
-    ∀ᶠt in atTop,Real.sqrt (((sourceInitialCount (q t) (N t)+1:ℕ):ℝ)*((N t)^(-7/5:ℝ)))+
-      2*((sourceInitialCount (q t) (N t)+1:ℕ):ℝ)*E t ≤ 1/16 := by
-  have hF := (sourceInitialCount_rate hN hq hω).add_const hN (by norm_num) (by norm_num : (0:ℝ) ≤ 1)
-  have hw := (PowerRate.base hN).rpow (-7/5:ℝ)
-  have he := (PowerRate.base hN).rpow (-100:ℝ)
-  have hs := ((hF.mul hw).sqrt.upper hN).tendsto_zero hN (by norm_num)
-  have ht := (((hF.mul he).const_mul hN (by norm_num : (0:ℝ)<2)).upper hN).tendsto_zero hN (by norm_num)
-  have hsum := hs.add ht
-  rw [zero_add] at hsum
-  have HH := hsum.eventually_le_const (by norm_num : (0:ℝ)<1/16)
-  filter_upwards [HH,hE] with t ht he
-  simp only [Nat.cast_add,Nat.cast_one]
-  simp only [←mul_assoc] at ht
-  apply le_trans _ ht
-  apply add_le_add le_rfl
-  exact mul_le_mul_of_nonneg_left he.2 (by positivity)
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open Filter
-open scoped Topology Classical
-
-theorem source_initial_level_eventually {N q n A : ℕ→ℝ} {k : ℕ→ℕ}
-    (hN : Tendsto N atTop atTop) (hn : Tendsto n atTop atTop)
-    (hA : ∀ᶠt in atTop,1 ≤ A t) (hq : ∀t,q t ∈ Set.Ioc 0 (1/2:ℝ))
-    (hΛ : Tendsto (fun t=>(q t)^(-3:ℤ)/Real.log (N t)) atTop (𝓝 0))
-    (hk : PowerRate N (fun t=>(k t:ℝ)) 0) (hk1 : ∀ᶠt in atTop,1 ≤ k t)
-    (hmoment : ∀ᶠt in atTop,10^6*A t*Real.log (n t) ≤ (k t:ℝ)*Real.log (N t))
-    (hscale : ∀ᶠt in atTop,N t ≤ n t) :
-    ∀ᶠt in atTop,Nonempty (ConcreteLevelPair (q t) ⟨(hq t).1.le,by linarith [(hq t).2]⟩
-      (N t) ((n t)^(-100*A t)) (11/5) (initialOverlapLower (q t))
-      ((sourceOverlapConstant (q t))^2) 0 (k t)) := by
-  have hor := source_initial_oracles_eventually hN hn hA hq hΛ hk hk1 hmoment
-  have hq' (t) : q t ∈ Set.Ioc 0 1 := ⟨(hq t).1,by linarith [(hq t).2]⟩
-  have hqr := deletion_rate_zero hN hq' hΛ
-  have hωr := initialOverlap_rate hN (fun t=>(hq t).1) hΛ
-  have he : ∀ᶠt in atTop,0 ≤ sourceInitialError (q t) (N t) (k t) ∧
-      sourceInitialError (q t) (N t) (k t) ≤ (N t)^(-100:ℝ) := by
-    filter_upwards [hor,hA,hscale,hN.eventually_gt_atTop 1] with t ht ha hs hNt
-    obtain ⟨C⟩ := ht.2.2.1
-    refine ⟨C.error_nonneg,ht.2.1.trans ?_⟩
-    calc
-      (n t)^(-100*A t) ≤ (n t)^(-100:ℝ) := Real.rpow_le_rpow_of_exponent_le (hNt.le.trans hs) (by linarith)
-      _ ≤ (N t)^(-100:ℝ) := Real.rpow_le_rpow_of_nonpos (by linarith) hs (by norm_num)
-  have hsmall := source_initial_overlap_budget_eventually hN hqr hωr he
-  filter_upwards [hor,hsmall,hk1,hN.eventually_gt_atTop 2,hn.eventually_gt_atTop 0] with t ht hs hk1 hNt hnt
-  obtain ⟨H,herr,⟨C⟩,⟨U⟩⟩ := ht
-  have hm : 4 ≤ sourceInitialM (N t) := (by norm_num : 4 ≤ 128).trans H.m_large
-  have hlen := sourceInitialWord_length (q:=q t) hm (-1)
-  have hlen' := sourceInitialWord_length (q:=q t) hm 1
-  have hh := sourceInitialCount_bounds (hq t) H
-  have hlower : N t/2 ≤ ((sourceInitialWord (q t) (N t) (-1)).length:ℝ) := by
-    rw [hlen,Nat.cast_mul]; exact H.length_lower
-  have hupper : ((sourceInitialWord (q t) (N t) (-1)).length:ℝ) ≤ N t := by
-    rw [hlen,Nat.cast_mul]; exact H.length_upper
-  have hell : (1:ℝ) ≤ (sourceInitialWord (q t) (N t) (-1)).length := by linarith only [hNt,hlower]
-  have hp : 0 < ((sourceInitialWord (q t) (N t) (-1)).length:ℝ) := by linarith only [hell]
-  refine ⟨C.toConcrete ⟨(hq t).1.le,by linarith [(hq t).2]⟩
-    (hlen.trans hlen'.symm) (sourceInitialWord_different H) hlower hupper ?_ (by norm_num)
-    (by simpa only [hlen] using hh.1) (by simpa only [hlen] using hh.2) ?_⟩
-  · simp only [Nat.cast_zero,zero_add,one_mul]
-    apply (div_le_iff₀ hp).mpr
-    exact herr.trans (le_mul_of_one_le_right (Real.rpow_nonneg hnt.le _) hell)
-  · exact source_initial_power_overlap (hq t) H (by linarith) hk1 C U hs
-
-/-- All backwards targets are no larger than n, with the source epsilon. -/
-theorem sourceScaleProduct_le_one (B D i : ℕ) (hB : 2 ≤ B) : sourceScaleProduct B D i ≤ 1 := by
-  apply Finset.prod_le_one₀
-  · intro j _
-    have hdim : sourceDimension B j ≤ B := by unfold sourceDimension; split_ifs <;> omega
-    have he := sourceEpsilon_pos B D
-    have hh := sourceEpsilon_small B D
-    have hp : (1:ℝ) ≤ 2^B := one_le_pow₀ (by norm_num)
-    have hd : 0 ≤ (D:ℝ) := Nat.cast_nonneg _
-    have hu : (2:ℝ)^B*sourceEpsilon B D ≤ 1/2 := by
-      nlinarith only [hh,mul_nonneg hd (mul_nonneg (le_trans zero_le_one hp) he.le)]
-    have hx := scaleA_zero_lower (sourceDimension B j) B (sourceEpsilon B D) hdim he.le
-    apply le_trans _ hx
-    exact mul_nonneg (by positivity) (by nlinarith only [hu,he])
-  · intro j _
-    have he := sourceEpsilon_pos B D
-    have hp : (1:ℝ) ≤ 2^(sourceDimension B j) := one_le_pow₀ (by norm_num)
-    unfold scaleA
-    norm_num only [pow_zero,mul_one]
-    have hh := (div_le_iff₀ (by positivity : (0:ℝ)<2^(sourceDimension B j))).mpr
-      (show 1+sourceEpsilon B D ≤ (1+sourceEpsilon B D)*2^(sourceDimension B j) from
-        le_mul_of_one_le_right (by linarith only [he]) hp)
-    linarith only [hh]
-
-theorem sourceTargetScale_le (B D i : ℕ) (n : ℝ) (hB : 2 ≤ B) (hn : 1 ≤ n) :
-    sourceTargetScale B D n i ≤ n := by
-  simpa only [sourceTargetScale,Real.rpow_one] using Real.rpow_le_rpow_of_exponent_le hn (sourceScaleProduct_le_one B D i hB)
-
-theorem sourceHalfMoment_sufficient (B D i : ℕ) {n : ℝ} (hB : 2 ≤ B) (hn : 1 < n) :
-    10^6*(totalScoreExponent B D+B+1)*Real.log n ≤
-      (sourceHalfMoment B D n i:ℝ)*Real.log (sourceTargetScale B D n i) := by
-  have hlo := sourceScaleProduct_lower B D i hB
-  have hs : 0 < sourceScaleProduct B D i := lt_of_lt_of_le (by positivity) hlo
-  have hl : 0 < Real.log (sourceTargetScale B D n i) := by
-    rw [sourceTargetScale,Real.log_rpow (by linarith : 0<n)]
-    exact mul_pos hs (Real.log_pos hn)
-  have hk := (div_le_iff₀ hl).mp (Nat.le_ceil
-    ((10^8:ℝ)*2^(10*B)*(totalScoreExponent B D+B+1)*Real.log n/
-      Real.log (sourceTargetScale B D n i)))
-  apply le_trans _ hk
-  have hd := (totalScoreExponent_bounds B D hB).1
-  have hA : 0 ≤ (totalScoreExponent B D+B+1)*Real.log n := by positivity [hd,Real.log_pos hn]
-  have hp : (1:ℝ) ≤ 2^(10*B) := one_le_pow₀ (by norm_num)
-  have hc : (10^6:ℝ) ≤ 10^8*2^(10*B) := by linarith only [hp]
-  nlinarith only [mul_le_mul_of_nonneg_right hc hA]
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open Filter
-open scoped Topology Classical
-
-/-- The source base invariant, with literal target scales, moment ceiling,
- error threshold, count and overlap, uniformly in the joint deletion regime. -/
-theorem source_initial_at_smallest_scale (B : ℕ) (ζ : ℝ) (hB : 2 ≤ B)
-    (hζ : ζ∈Set.Ioo 0 1) {n q : ℕ→ℝ} (hn : Tendsto n atTop atTop)
-    (hq : ∀t,q t∈Set.Ioc 0 (1/2:ℝ))
-    (hH : Tendsto (fun t=>(q t)^3*Real.log (n t)) atTop atTop) :
-    ∀ᶠt in atTop,Nonempty (ConcreteLevelPair (q t) ⟨(hq t).1.le,by linarith [(hq t).2]⟩
-      (sourceTargetScale B (sourceDepth B ζ ((q t)^3*Real.log (n t))) (n t) 0)
-      ((n t)^(-100*(totalScoreExponent B (sourceDepth B ζ ((q t)^3*Real.log (n t)))+B+1)))
-      (11/5) (initialOverlapLower (q t)) ((sourceOverlapConstant (q t))^2) 0
-      (sourceHalfMoment B (sourceDepth B ζ ((q t)^3*Real.log (n t))) (n t) 0)) := by
-  let H := fun t=>(q t)^3*Real.log (n t)
-  let Λ := fun t=>(q t)^(-3:ℤ)
-  have hΛ (t) : 1 ≤ Λ t := by
-    dsimp [Λ]
-    rw [zpow_neg,zpow_ofNat,one_le_inv₀ (pow_pos (hq t).1 _)]
-    exact pow_le_one₀ (hq t).1.le (by linarith [(hq t).2])
-  have hrel (t) : Real.log (n t)=Λ t*H t := by
-    dsimp [Λ,H]
-    rw [zpow_neg,zpow_ofNat]
-    field_simp [(hq t).1.ne']
-  let D := fun t=>sourceDepth B ζ (H t)
-  let N := fun t=>sourceTargetScale B (D t) (n t) 0
-  have hN := sourceTargetScale_tendsto B ζ hB hζ (fun _=>0) hn hH hΛ hrel
-  have hl := source_uniform_log_absorption B 0 ζ hB hζ (fun _=>0) hn hH hΛ hrel
-  simp only [pow_zero,one_mul] at hl
-  have hk := sourceHalfMoment_rate B ζ hB hζ (fun _=>0) hn hH hΛ hrel
-  apply source_initial_level_eventually hN hn _ hq hl hk
-  · filter_upwards [hn.eventually_gt_atTop 1] with t ht
-    exact (sourceHalfMoment_bounds B (D t) 0 hB ht).1
-  · filter_upwards [hn.eventually_gt_atTop 1] with t ht
-    exact sourceHalfMoment_sufficient B (D t) 0 hB ht
-  · filter_upwards [hn.eventually_ge_atTop 1] with t ht
-    exact sourceTargetScale_le B (D t) 0 (n t) hB ht
-  · filter_upwards [] with t
-    have hd := (totalScoreExponent_bounds B (D t) hB).1
-    have hbn : (0:ℝ) ≤ B := Nat.cast_nonneg _
-    linarith only [hd,hbn]
-
-end TraceReconstruction
-
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-
-end OAI
+import OAI.Probability.TraceReconstruction.InitialOracle
+import OAI.Probability.TraceReconstruction.Amplification
+
+namespace OAI
+
+/-!
+# InitialRates
+
+Initial geometric parameters, moment depths, error and overlap rates, and production of the base induction pair.
+-/
+
+open scoped ENNReal
+noncomputable section
+
+open Filter
+open scoped Topology
+
+
+noncomputable section
+namespace TraceReconstruction
+open Filter
+open scoped Topology Classical
+
+variable {N q ω w : ℕ → ℝ} {J u m : ℕ → ℕ}
+
+/-- Exact power of the sharp initial rank prefactor; all channel losses are
+ subpower at the source's smallest scale. -/
+theorem initialPrefactor_rate (hN : Tendsto N atTop atTop)
+    {α β : ℝ} (hu : PowerRate N (fun t=>(u t:ℝ)) α)
+    (hm : PowerRate N (fun t=>(m t:ℝ)) β) (hq : PowerRate N q 0) (r d : ℕ) :
+    PowerRate N (fun t=>initialPrefactorCap (u t) (m t) r d (q t))
+      (α/2-β*((d:ℝ)+(r:ℝ)/2)) := by
+  have hd : PowerRate N (fun t=>initialDifferenceCap d (q t)) 0 := by
+    have H := ((PowerRate.const hN (by positivity : (0:ℝ)<8*((d:ℝ)+1))).div (hq.pow 2)).sqrt.const_mul hN (by norm_num : (0:ℝ)<2)
+    simpa only [initialDifferenceCap,zero_mul,zero_sub,neg_zero,zero_div] using H
+  have hnum := (((hu.mul hm).const_mul hN (by norm_num : (0:ℝ)<144)).mul_const hN
+    (by positivity : (0:ℝ)<(2:ℝ)^(2*r))).mul (hd.pow (2*d+r+2))
+  have hden := ((hm.div_const hN (by norm_num : (0:ℝ)<4)).pow d).mul
+    (((hu.mul hm).div_const hN (by positivity : (0:ℝ)<2*(r+1:ℝ))).sqrt)
+  have H := hnum.div (hden.mul ((hm.div_const hN (by norm_num : (0:ℝ)<4)).sqrt.pow (r+1)))
+  convert H using 1
+  · ext t; simp only [initialPrefactorCap,Nat.cast_mul]
+  · push_cast; ring
+
+theorem periodCentral_rate (hN : Tendsto N atTop atTop) {α : ℝ}
+    (hu : PowerRate N (fun t=>(u t:ℝ)) α) {r : ℕ} (hr : 0<r) :
+    PowerRate N (fun t=>periodCentralSumConstant (u t) r) (α*((r:ℝ)-3)/2) := by
+  have H := ((hu.div_const hN (by positivity : (0:ℝ)<3*(r+1:ℝ))).inv hN).const_mul hN
+    (by positivity : (0:ℝ)<((r+1:ℕ):ℝ)^2)
+  have H' := H.mul (((hu.const_mul hN (by norm_num : (0:ℝ)<2)).sqrt.const_mul hN
+    (by norm_num : (0:ℝ)<2)).pow (r-1))
+  convert H' using 1
+  · ext t; simp only [periodCentralSumConstant,Nat.cast_mul,Nat.cast_ofNat]
+  · rw [Nat.cast_sub (by omega : 1≤r)]; norm_num; ring
+
+theorem periodBounded_rate (hN : Tendsto N atTop atTop) {α : ℝ}
+    (hu : PowerRate N (fun t=>(u t:ℝ)) α) {r : ℕ} (hr : 2≤r) (M : ℕ) :
+    PowerRate N (fun t=>periodBoundedSumConstant (u t) r M) (α*((r:ℝ)-4)/2) := by
+  have H := (((hu.div_const hN (by positivity : (0:ℝ)<3*(r+1:ℝ))).inv hN).const_mul hN
+    (by positivity : (0:ℝ)<((r+1:ℕ):ℝ)^2)).mul_const hN (by positivity : (0:ℝ)<(M:ℝ)+1)
+  have H' := H.mul (((hu.const_mul hN (by norm_num : (0:ℝ)<2)).sqrt.const_mul hN
+    (by norm_num : (0:ℝ)<2)).pow (r-2))
+  convert H' using 1
+  · ext t; simp only [periodBoundedSumConstant,Nat.cast_mul,Nat.cast_ofNat]
+  · rw [Nat.cast_sub hr]; norm_num; ring
+
+theorem hiddenSigned_rate (hN : Tendsto N atTop atTop) {γ : ℝ} (hγ : 0<γ)
+    (hJ : PowerRate N (fun t=>(J t:ℝ)) γ) (hω : PowerRate N ω 0) (s : ℕ) :
+    PowerRate N (fun t=>hiddenSignedL2Bound (J t) s (ω t)) (-γ*((s:ℝ)+1/2)) := by
+  have hden := (hJ.mul (hω.pow 2)).add_const hN (by simpa using hγ) (by norm_num : (0:ℝ)≤2)
+  have hroot := ((PowerRate.const hN (by positivity : (0:ℝ)<4*((s:ℝ)+1))).div hden).sqrt
+  have hp := ((hroot.const_mul hN (by norm_num : (0:ℝ)<2)).pow s).pow 2
+  have H := (hp.const_mul hN (by positivity : (0:ℝ)<((2:ℝ)^(s*s))^2)).mul hroot
+  convert H using 1
+  · rfl
+  · push_cast; ring
+
+theorem hiddenUnsigned_rate (hN : Tendsto N atTop atTop) {γ : ℝ} (hγ : 0<γ)
+    (hJ : PowerRate N (fun t=>(J t:ℝ)) γ) (hω : PowerRate N ω 0) (s : ℕ) :
+    PowerRate N (fun t=>hiddenUnsignedL2Bound (J t) s (ω t)) (-γ/2) := by
+  have hden := (hJ.mul (hω.pow 2)).add_const hN (by simpa using hγ) (by norm_num : (0:ℝ)≤2)
+  have H := (((PowerRate.const hN (by norm_num : (0:ℝ)<4)).div hden).sqrt).const_mul hN
+    (by positivity : (0:ℝ)<((2:ℝ)^s)^2)
+  convert H using 1
+  · rfl
+  · ring
+
+/-- The two eliminated large gaps are quantitatively visible in this exponent. -/
+theorem initial_central_rate (hN : Tendsto N atTop atTop)
+    (hu : PowerRate N (fun t=>(u t:ℝ)) (123/250))
+    (hm : PowerRate N (fun t=>(m t:ℝ)) (62/125)) (hq : PowerRate N q 0)
+    {r : ℕ} (hr : 0<r) (d : ℕ) :
+    PowerRate N (fun t=>initialPrefactorCap (u t) (m t) r d (q t)*periodCentralSumConstant (u t) r)
+      (-123/250-(62/125)*(d:ℝ)-(1/500)*(r:ℝ)) := by
+  convert (initialPrefactor_rate hN hu hm hq r d).mul (periodCentral_rate hN hu hr) using 1; ring
+
+theorem initial_bounded_rate (hN : Tendsto N atTop atTop)
+    (hu : PowerRate N (fun t=>(u t:ℝ)) (123/250))
+    (hm : PowerRate N (fun t=>(m t:ℝ)) (62/125)) (hq : PowerRate N q 0)
+    {r : ℕ} (hr : 2≤r) (d M : ℕ) :
+    PowerRate N (fun t=>initialPrefactorCap (u t) (m t) r d (q t)*periodBoundedSumConstant (u t) r M)
+      (-369/500-(62/125)*(d:ℝ)-(1/500)*(r:ℝ)) := by
+  convert (initialPrefactor_rate hN hu hm hq r d).mul (periodBounded_rate hN hu hr M) using 1; ring
+
+/-- Fixed low degrees already beat the signed and pure source score exponents. -/
+theorem initial_signed_pattern_upper (hN : Tendsto N atTop atTop)
+    (hu : PowerRate N (fun t=>(u t:ℝ)) (123/250))
+    (hm : PowerRate N (fun t=>(m t:ℝ)) (62/125)) (hq : PowerRate N q 0)
+    (hJ : PowerRate N (fun t=>(J t:ℝ)) (1/125)) (hω : PowerRate N ω 0)
+    {r : ℕ} (hr : 0<r) (k : Fin r→GapDifference) (hk : Odd (firstDifferenceCount k)) :
+    PowerUpper N (fun t=>initialSignedPatternCap (J t) (u t) (m t) r 300 (q t) (ω t) k) (-223/100) := by
+  by_cases hlin : ∃a,k=linearPattern a
+  · have H := (initial_central_rate hN hu hm hq hr 1).mul
+      (hiddenSigned_rate hN (by norm_num) hJ hω 300)
+    have hb : -123/250-(62/125)*(1:ℝ)-(1/500)*(r:ℝ)+(-(1/125)*((300:ℝ)+1/2)) ≤ -223/100 := by
+      have hh : (0:ℝ)≤r := Nat.cast_nonneg r
+      linarith
+    simpa only [initialSignedPatternCap,ite_eq_left hlin] using (H.upper hN).mono (by simpa using hb)
+  · have hd : 3≤∑h,(k h).order := odd_nonlinear_order k hk hlin
+    obtain ⟨a,b,hab,_,_⟩ := odd_pattern_pair k hk hlin
+    have hr2 : 2≤r := by have := a.isLt; have := b.isLt; have := hab; omega
+    have H := (initial_bounded_rate hN hu hm hq hr2 (∑h,(k h).order) (2*(2^300))).mul
+      (hiddenUnsigned_rate hN (by norm_num) hJ hω 300)
+    have hb : -369/500-(62/125)*(∑h,(k h).order:ℕ)-(1/500)*(r:ℝ)+(-(1/125)/2) ≤ -223/100 := by
+      have hd' : (3:ℝ)≤(∑h,(k h).order:ℕ) := by exact_mod_cast hd
+      have hr' : (2:ℝ)≤r := by exact_mod_cast hr2
+      linarith
+    simpa only [initialSignedPatternCap,ite_eq_right hlin] using (H.upper hN).mono (by simpa using hb)
+
+theorem initial_pure_pattern_upper (hN : Tendsto N atTop atTop)
+    (hu : PowerRate N (fun t=>(u t:ℝ)) (123/250))
+    (hm : PowerRate N (fun t=>(m t:ℝ)) (62/125)) (hq : PowerRate N q 0)
+    (hJ : PowerRate N (fun t=>(J t:ℝ)) (1/125)) (hω : PowerRate N ω 0)
+    {r : ℕ} (hr : 0<r) (k : Fin r→GapDifference) (hk : k≠unchangedPattern r) :
+    PowerUpper N (fun t=>initialPurePatternCap (J t) (u t) (m t) r 300 (q t) (ω t) k) (-37/25) := by
+  by_cases hlin : ∃a,k=linearPattern a
+  · have H := (initial_central_rate hN hu hm hq hr 1).mul
+      (hiddenSigned_rate hN (by norm_num) hJ hω 300)
+    have hb : -123/250-(62/125)*(1:ℝ)-(1/500)*(r:ℝ)+(-(1/125)*((300:ℝ)+1/2)) ≤ -37/25 := by
+      have hh : (0:ℝ)≤r := Nat.cast_nonneg r
+      linarith
+    simpa only [initialPurePatternCap,ite_eq_left hlin] using (H.upper hN).mono (by simpa using hb)
+  · have hd := (nonconstant_pattern_linear_or_order_two k hk).resolve_left hlin
+    have H := (initial_central_rate hN hu hm hq hr (∑h,(k h).order)).mul
+      (hiddenUnsigned_rate hN (by norm_num) hJ hω 300)
+    have hb : -123/250-(62/125)*(∑h,(k h).order:ℕ)-(1/500)*(r:ℝ)+(-(1/125)/2) ≤ -37/25 := by
+      have hd' : (2:ℝ)≤(∑h,(k h).order:ℕ) := by exact_mod_cast hd
+      have hr' : (0:ℝ)≤r := Nat.cast_nonneg r
+      linarith
+    simpa only [initialPurePatternCap,ite_eq_right hlin] using (H.upper hN).mono (by simpa using hb)
+
+theorem increasingTuple_card_le (u r : ℕ) : Fintype.card (IncreasingTuple u r) ≤ u^r := by
+  simpa only [Fintype.card_fun,Fintype.card_fin] using
+    Fintype.card_subtype_le (fun v : Fin r→Fin u => StrictMono v)
+
+theorem initialRankTail_upper (hN : Tendsto N atTop atTop)
+    (hu : PowerRate N (fun t=>(u t:ℝ)) (123/250))
+    (hm : PowerRate N (fun t=>(m t:ℝ)) (62/125)) (r : ℕ) (a : ℝ) :
+    PowerUpper N (fun t=>initialRankTailCap (u t) (m t) r) a := by
+  have hcard : PowerUpper N (fun t=>(Fintype.card (IncreasingTuple (u t) r):ℝ)) ((123/250)*(r:ℝ)) := by
+    apply (hu.pow r).upper hN |>.of_abs_le
+    exact Eventually.of_forall fun t=>by
+      rw [abs_of_nonneg (Nat.cast_nonneg _),abs_of_nonneg (by positivity)]
+      exact_mod_cast increasingTuple_card_le (u t) r
+  have hpoly := hcard.mul hN (((hu.mul hm).sqrt.const_mul hN
+    (by positivity : (0:ℝ)<48*((r+1:ℕ):ℝ))).upper hN)
+  have hexpr : PowerRate N (fun t=>(m t:ℝ)/(1024*(u t:ℝ))) (1/250) := by
+    convert hm.div (hu.const_mul hN (by norm_num : (0:ℝ)<1024)) using 1; norm_num
+  have hexp := hexpr.exp_neg_upper hN (by norm_num) (a/2-((123/250)*(r:ℝ)+(123/250+62/125)/2))
+  have H := ((PowerRate.const hN (by norm_num : (0:ℝ)<2)).upper hN).mul hN
+    ((hpoly.mul hN hexp).pow hN 2)
+  convert H using 1
+  · ext t
+    simp only [initialRankTailCap,Nat.cast_mul,neg_div]
+    ring
+  · push_cast; ring
+
+theorem initialMomentRatio_rate (hN : Tendsto N atTop atTop)
+    (hu : PowerRate N (fun t=>(u t:ℝ)) (123/250))
+    (hm : PowerRate N (fun t=>(m t:ℝ)) (62/125)) (hq : PowerRate N q 0)
+    (hw : PowerRate N w 0) :
+    PowerRate N (fun t=>initialMomentRatio (u t) (m t) (q t) (w t)) (-1/500) := by
+  have H := ((hw.pow 2).mul ((PowerRate.const hN
+    (by positivity [rankSchurConstant_pos] : (0:ℝ)<3*rankSchurConstant)).div hq)).mul (hu.div hm).sqrt
+  convert H using 1
+  · rfl
+  · norm_num
+
+theorem initial_fixed_high_tail_rate (hN : Tendsto N atTop atTop)
+    (hu : PowerRate N (fun t=>(u t:ℝ)) (123/250))
+    (hm : PowerRate N (fun t=>(m t:ℝ)) (62/125)) (hq : PowerRate N q 0)
+    (hw : PowerRate N w 0) :
+    PowerRate N (fun t=>4*(initialMomentRatio (u t) (m t) (q t) (w t))^3000/
+      (1-initialMomentRatio (u t) (m t) (q t) (w t))) (-6) := by
+  have hρ := initialMomentRatio_rate hN hu hm hq hw
+  have hz := (hρ.upper hN).tendsto_zero hN (by norm_num)
+  have hden : PowerRate N (fun t=>1-initialMomentRatio (u t) (m t) (q t) (w t)) 0 := by
+    apply PowerRate.of_tendsto hN (by norm_num : (0:ℝ)<1)
+    simpa using tendsto_const_nhds.sub hz
+  convert ((hρ.pow 3000).const_mul hN (by norm_num : (0:ℝ)<4)).div hden using 1
+  norm_num
+
+theorem initialSignedDegree_upper (hN : Tendsto N atTop atTop)
+    (hu : PowerRate N (fun t=>(u t:ℝ)) (123/250))
+    (hm : PowerRate N (fun t=>(m t:ℝ)) (62/125)) (hq : PowerRate N q 0)
+    (hJ : PowerRate N (fun t=>(J t:ℝ)) (1/125)) (hω : PowerRate N ω 0) (r : ℕ) :
+    PowerUpper N (fun t=>initialSignedDegreeCap (J t) (u t) (m t) r 300 (q t) (ω t)) (-223/100) := by
+  by_cases hr : r=0
+  · simpa only [initialSignedDegreeCap,ite_eq_left hr] using PowerUpper.zero hN (-223/100)
+  have hs : PowerUpper N (fun t=>∑k : Fin r→GapDifference,
+      if Odd (firstDifferenceCount k) then initialSignedPatternCap (J t) (u t) (m t) r 300 (q t) (ω t) k else 0) (-223/100) := by
+    apply PowerUpper.sum hN Finset.univ
+    intro k _
+    by_cases hk : Odd (firstDifferenceCount k)
+    · simpa only [ite_eq_left hk] using initial_signed_pattern_upper hN hu hm hq hJ hω (by omega) k hk
+    · simpa only [ite_eq_right hk] using PowerUpper.zero hN (-223/100)
+  have hc : (0:ℝ)<8*(Fintype.card (Fin r→GapDifference):ℝ) := by
+    have hp : 0<Fintype.card (Fin r→GapDifference) := Fintype.card_pos_iff.mpr ⟨fun _=>.unchanged⟩
+    positivity
+  have H := (((PowerRate.const hN hc).upper hN).mul hN hs)
+  simp only [zero_add] at H
+  simpa only [initialSignedDegreeCap,ite_eq_right hr] using H.add hN (initialRankTail_upper hN hu hm r (-223/100))
+
+theorem initialPureDegree_upper (hN : Tendsto N atTop atTop)
+    (hu : PowerRate N (fun t=>(u t:ℝ)) (123/250))
+    (hm : PowerRate N (fun t=>(m t:ℝ)) (62/125)) (hq : PowerRate N q 0)
+    (hJ : PowerRate N (fun t=>(J t:ℝ)) (1/125)) (hω : PowerRate N ω 0) (r : ℕ) :
+    PowerUpper N (fun t=>initialPureDegreeCap (J t) (u t) (m t) r 300 (q t) (ω t)) (-37/25) := by
+  by_cases hr : r=0
+  · simpa only [initialPureDegreeCap,ite_eq_left hr] using PowerUpper.zero hN (-37/25)
+  have hs : PowerUpper N (fun t=>∑k : Fin r→GapDifference,
+      if k=unchangedPattern r then 0 else initialPurePatternCap (J t) (u t) (m t) r 300 (q t) (ω t) k) (-37/25) := by
+    apply PowerUpper.sum hN Finset.univ
+    intro k _
+    by_cases hk : k=unchangedPattern r
+    · simpa only [ite_eq_left hk] using PowerUpper.zero hN (-37/25)
+    · simpa only [ite_eq_right hk] using initial_pure_pattern_upper hN hu hm hq hJ hω (by omega) k hk
+  have hc : (0:ℝ)<2*(Fintype.card (Fin r→GapDifference):ℝ) := by
+    have hp : 0<Fintype.card (Fin r→GapDifference) := Fintype.card_pos_iff.mpr ⟨fun _=>.unchanged⟩
+    positivity
+  have H := (((PowerRate.const hN hc).upper hN).mul hN hs)
+  simp only [zero_add] at H
+  simpa only [initialPureDegreeCap,ite_eq_right hr] using H.add hN (initialRankTail_upper hN hu hm r (-37/25))
+
+theorem initialSignedCap_upper (hN : Tendsto N atTop atTop)
+    (hu : PowerRate N (fun t=>(u t:ℝ)) (123/250))
+    (hm : PowerRate N (fun t=>(m t:ℝ)) (62/125)) (hq : PowerRate N q 0)
+    (hJ : PowerRate N (fun t=>(J t:ℝ)) (1/125)) (hω : PowerRate N ω 0) (hw : PowerRate N w 0) :
+    PowerUpper N (fun t=>initialSignedCap (J t) (u t) (m t) 3000 300 (q t) (ω t) (w t)) (-223/100) := by
+  have hs : PowerUpper N (fun t=>∑r∈Finset.range 3000,(w t)^(2*r)*
+      initialSignedDegreeCap (J t) (u t) (m t) r 300 (q t) (ω t)) (-223/100) := by
+    apply PowerUpper.sum hN (Finset.range 3000)
+    intro r _
+    simpa only [zero_mul,zero_add] using ((hw.pow (2*r)).upper hN).mul hN (initialSignedDegree_upper hN hu hm hq hJ hω r)
+  exact hs.add hN ((initial_fixed_high_tail_rate hN hu hm hq hw).upper hN |>.mono (by norm_num))
+
+theorem initialPureCap_upper (hN : Tendsto N atTop atTop)
+    (hu : PowerRate N (fun t=>(u t:ℝ)) (123/250))
+    (hm : PowerRate N (fun t=>(m t:ℝ)) (62/125)) (hq : PowerRate N q 0)
+    (hJ : PowerRate N (fun t=>(J t:ℝ)) (1/125)) (hω : PowerRate N ω 0) (hw : PowerRate N w 0) :
+    PowerUpper N (fun t=>initialPureCap (J t) (u t) (m t) 3000 300 (q t) (ω t) (w t)) (-37/25) := by
+  have hs : PowerUpper N (fun t=>∑r∈Finset.range 3000,(w t)^(2*r)*
+      initialPureDegreeCap (J t) (u t) (m t) r 300 (q t) (ω t)) (-37/25) := by
+    apply PowerUpper.sum hN (Finset.range 3000)
+    intro r _
+    simpa only [zero_mul,zero_add] using ((hw.pow (2*r)).upper hN).mul hN (initialPureDegree_upper hN hu hm hq hJ hω r)
+  exact hs.add hN ((initial_fixed_high_tail_rate hN hu hm hq hw).upper hN |>.mono (by norm_num))
+
+/-- Actual scalar budgets, not assumed coefficient bounds: all report- and
+ count-independent caps are strictly below the source clipping allowances. -/
+theorem initial_score_caps_eventually (hN : Tendsto N atTop atTop)
+    (hu : PowerRate N (fun t=>(u t:ℝ)) (123/250))
+    (hm : PowerRate N (fun t=>(m t:ℝ)) (62/125)) (hq : PowerRate N q 0)
+    (hJ : PowerRate N (fun t=>(J t:ℝ)) (1/125)) (hω : PowerRate N ω 0) (hw : PowerRate N w 0) :
+    ∀ᶠt in atTop,
+      64*initialSignedCap (J t) (u t) (m t) 3000 300 (q t) (ω t) (w t) ≤ (N t)^(-11/5:ℝ) ∧
+      64*initialPureCap (J t) (u t) (m t) 3000 300 (q t) (ω t) (w t) ≤ (N t)^(-7/5:ℝ) := by
+  have hc := (PowerRate.const hN (by norm_num : (0:ℝ)<64)).upper hN
+  have hs := hc.mul hN (initialSignedCap_upper hN hu hm hq hJ hω hw)
+  have hp := hc.mul hN (initialPureCap_upper hN hu hm hq hJ hω hw)
+  filter_upwards [hs (-11/5) (by norm_num),hp (-7/5) (by norm_num)] with t hs hp
+  exact ⟨(le_abs_self _).trans hs,(le_abs_self _).trans hp⟩
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open Filter
+open scoped Topology Classical
+
+variable {N f g : ℕ→ℝ} {α β : ℝ}
+
+theorem PowerRate.sub_lower (hN : Tendsto N atTop atTop)
+    (hf : PowerRate N f α) (hg : PowerRate N g β) (hβα : β<α) :
+    PowerRate N (fun t=>f t-g t) α := by
+  have hz := ((hg.div hf).upper hN).tendsto_zero hN (sub_neg.mpr hβα)
+  apply hf.of_ratio hN (by norm_num : (0:ℝ)<1)
+  have H : Tendsto (fun t=>1-g t/f t) atTop (𝓝 1) := by
+    simpa using tendsto_const_nhds.sub hz
+  apply H.congr'
+  filter_upwards [hf.1] with t ht
+  field_simp [ht.ne']
+
+theorem PowerRate.nat_sub_lower (hN : Tendsto N atTop atTop) {f g : ℕ→ℕ}
+    (hf : PowerRate N (fun t=>(f t:ℝ)) α)
+    (hg : PowerRate N (fun t=>(g t:ℝ)) β) (hβα : β<α) :
+    PowerRate N (fun t=>(f t-g t:ℕ)) α := by
+  have H := hf.sub_lower hN hg hβα
+  apply H.congr
+  filter_upwards [H.1] with t ht
+  have hg : g t≤f t := by exact_mod_cast (sub_pos.mp ht).le
+  rw [Nat.cast_sub hg]
+
+/-- Literal integer choices of the initial-pair construction, including the outer floors. -/
+def sourceInitialM (N : ℝ) : ℕ := ⌊N^(62/125:ℝ)⌋₊
+def sourceInitialK (N : ℝ) : ℕ := ⌊N^(123/250:ℝ)⌋₊
+def sourceInitialPeriodSize (N : ℝ) : ℕ := 2*sourceInitialM N-1
+def sourceInitialCapacity (N : ℝ) : ℕ := ⌊N/(sourceInitialPeriodSize N:ℝ)⌋₊
+def sourceInitialJ (q N : ℝ) : ℕ :=
+  ⌊((sourceInitialCapacity N:ℝ)-sourceInitialK N)/
+    (2*(initialSideCount q (sourceInitialM N):ℝ)+1)⌋₊
+def sourceInitialP (q N : ℝ) : ℕ :=
+  sourceInitialJ q N*(2*initialSideCount q (sourceInitialM N)+1)+sourceInitialK N
+/-- One-based manuscript anchors translated once to zero-based list indices. -/
+def sourceInitialAnchor (q N : ℝ) : ℤ := ((sourceInitialP q N/2:ℕ):ℤ)-1
+
+theorem sourceInitialM_rate (hN : Tendsto N atTop atTop) :
+    PowerRate N (fun t=>(sourceInitialM (N t):ℝ)) (62/125) := by
+  have H := ((PowerRate.base hN).rpow (62/125)).nat_floor hN (by norm_num)
+  simpa only [sourceInitialM,one_mul] using H
+
+theorem sourceInitialK_rate (hN : Tendsto N atTop atTop) :
+    PowerRate N (fun t=>(sourceInitialK (N t):ℝ)) (123/250) := by
+  have H := ((PowerRate.base hN).rpow (123/250)).nat_floor hN (by norm_num)
+  simpa only [sourceInitialK,one_mul] using H
+
+theorem sourceInitialPeriodSize_rate (hN : Tendsto N atTop atTop) :
+    PowerRate N (fun t=>(sourceInitialPeriodSize (N t):ℝ)) (62/125) := by
+  have hh : PowerRate N (fun t=>(2*sourceInitialM (N t):ℕ)) (62/125) := by
+    simpa only [Nat.cast_mul,Nat.cast_ofNat] using (sourceInitialM_rate hN).const_mul hN (by norm_num : (0:ℝ)<2)
+  exact hh.nat_sub_lower hN (PowerRate.const hN (by norm_num : (0:ℝ)<(1:ℕ))) (by norm_num)
+
+theorem sourceInitialCapacity_rate (hN : Tendsto N atTop atTop) :
+    PowerRate N (fun t=>(sourceInitialCapacity (N t):ℝ)) (63/125) := by
+  have H := ((PowerRate.base hN).div (sourceInitialPeriodSize_rate hN)).nat_floor hN (by norm_num)
+  convert H using 1
+  · rfl
+  · norm_num
+
+theorem sourceInitialSide_rate (hN : Tendsto N atTop atTop) {q : ℕ→ℝ}
+    (hq : PowerRate N q 0) :
+    PowerRate N (fun t=>(initialSideCount (q t) (sourceInitialM (N t)):ℝ)) (62/125) := by
+  have H := (hq.mul (sourceInitialM_rate hN)).nat_floor hN (by norm_num)
+  simpa only [initialSideCount,zero_add] using H
+
+theorem sourceInitialJ_rate (hN : Tendsto N atTop atTop) {q : ℕ→ℝ}
+    (hq : PowerRate N q 0) :
+    PowerRate N (fun t=>(sourceInitialJ (q t) (N t):ℝ)) (1/125) := by
+  have hn := (sourceInitialCapacity_rate hN).sub_lower hN (sourceInitialK_rate hN) (by norm_num)
+  have hd := ((sourceInitialSide_rate hN hq).const_mul hN (by norm_num : (0:ℝ)<2)).add_const hN
+    (by norm_num) (by norm_num : (0:ℝ)≤1)
+  have H := (hn.div hd).nat_floor hN (by norm_num)
+  convert H using 1
+  · rfl
+  · norm_num
+
+theorem initialOverlap_rate (hN : Tendsto N atTop atTop) {q : ℕ→ℝ}
+    (_hq : ∀t,0<q t)
+    (hΛ : Tendsto (fun t=>(q t)^(-3:ℤ)/Real.log (N t)) atTop (𝓝 0)) :
+    PowerRate N (fun t=>initialOverlapLower (q t)) 0 := by
+  refine ⟨Eventually.of_forall (fun t=>initialOverlapLower_pos _),?_⟩
+  have H := (hΛ.const_mul (-(32+3*primitivePeriodConstant))).sub
+    ((Real.tendsto_log_atTop.comp hN).const_div_atTop (Real.log 1536))
+  simp only [mul_zero,sub_zero] at H
+  apply H.congr'
+  filter_upwards [] with t
+  rw [initialOverlapLower,Real.log_div (Real.exp_pos _).ne' (by norm_num),Real.log_exp]
+  simp only [zpow_neg,zpow_ofNat,Function.comp_apply]
+  ring
+
+theorem deletion_rate_zero (hN : Tendsto N atTop atTop) {q : ℕ→ℝ}
+    (hq : ∀t,q t∈Set.Ioc 0 1)
+    (hΛ : Tendsto (fun t=>(q t)^(-3:ℤ)/Real.log (N t)) atTop (𝓝 0)) :
+    PowerRate N q 0 := by
+  refine ⟨Eventually.of_forall (fun t=>(hq t).1),?_⟩
+  apply squeeze_zero_norm' _ hΛ
+  filter_upwards [hN.eventually_gt_atTop 1] with t hNt
+  have hp := (hq t).1
+  have hle := (hq t).2
+  have hqinv : 1≤(q t)⁻¹ := (one_le_inv₀ hp).mpr hle
+  have hlog : |Real.log (q t)| ≤ (q t)^(-3:ℤ) := by
+    rw [abs_of_nonpos (Real.log_nonpos hp.le hle),←Real.log_inv, zpow_neg,zpow_ofNat]
+    have hh := Real.log_le_sub_one_of_pos (inv_pos.mpr hp)
+    have hcube : (q t)⁻¹ ≤ ((q t)⁻¹)^3 := by nlinarith [sq_nonneg ((q t)⁻¹-1)]
+    rw [←inv_pow]
+    linarith
+  rw [Real.norm_eq_abs,abs_div,abs_of_pos (Real.log_pos hNt)]
+  exact div_le_div_of_nonneg_right hlog (Real.log_pos hNt).le
+
+/-- Mechanical realization of the base scalar score budgets from the actual
+ integer choices and the unchanged known channel. -/
+theorem source_initial_score_caps {q w : ℕ→ℝ} (hN : Tendsto N atTop atTop)
+    (hq : ∀t,q t∈Set.Ioc 0 1)
+    (hΛ : Tendsto (fun t=>(q t)^(-3:ℤ)/Real.log (N t)) atTop (𝓝 0))
+    (hw : PowerRate N w 0) :
+    ∀ᶠt in atTop,
+      64*initialSignedCap (sourceInitialJ (q t) (N t)) (sourceInitialK (N t))
+        (sourceInitialM (N t)) 3000 300 (q t) (initialOverlapLower (q t)) (w t) ≤ (N t)^(-11/5:ℝ) ∧
+      64*initialPureCap (sourceInitialJ (q t) (N t)) (sourceInitialK (N t))
+        (sourceInitialM (N t)) 3000 300 (q t) (initialOverlapLower (q t)) (w t) ≤ (N t)^(-7/5:ℝ) := by
+  exact initial_score_caps_eventually hN (sourceInitialK_rate hN) (sourceInitialM_rate hN)
+    (deletion_rate_zero hN hq hΛ) (sourceInitialJ_rate hN (deletion_rate_zero hN hq hΛ))
+    (initialOverlap_rate hN (fun t=>(hq t).1) hΛ) hw
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open Filter
+open scoped Topology Classical
+
+/-- The manuscript's one-based middle defect, translated to a list index. -/
+def sourceInitialOffset (m : ℕ) (hm : 4 ≤ m) : Fin m :=
+  ⟨m/2-1,by omega⟩
+
+theorem sourceInitialOffset_interior (m : ℕ) (hm : 4 ≤ m) :
+    1 ≤ (sourceInitialOffset m hm).val ∧ (sourceInitialOffset m hm).val+1 < m := by
+  simp only [sourceInitialOffset]; omega
+
+theorem sourceInitialOffset_middle (m : ℕ) (hm : 4 ≤ m) :
+    |((sourceInitialOffset m hm).val:ℝ)-(m:ℝ)/2| ≤ 2 := by
+  have h1 : 2*(m/2) ≤ m := Nat.mul_div_le m 2
+  have h2 : m<2*(m/2+1) := by omega
+  have h3 : 1 ≤ m/2 := by omega
+  simp only [sourceInitialOffset,Nat.cast_sub h3,Nat.cast_one]
+  rw [abs_le]
+  have hh1 : (2:ℝ)*(m/2:ℕ) ≤ m := by exact_mod_cast h1
+  have hh2 : (m:ℝ)<2*((m/2:ℕ)+1) := by exact_mod_cast h2
+  constructor <;> linarith
+
+theorem sourceInitialOffset_safe (u m : ℕ) (hm : 4 ≤ m) :
+    SafeTaggedTuple (periodSlot (u:=u) (fun _=>sourceInitialOffset m hm)) := by
+  have hi := sourceInitialOffset_interior m hm
+  exact constant_period_safe _ hi.1 hi.2 (by omega) id strictMono_id
+
+/-- A literal arithmetic margin implies every base window placement condition. -/
+theorem initial_anchor_geometry (J K t s : ℕ) (hK : 12*(J+2^s+1) ≤ K) :
+    let a : ℤ := (((J*(2*t+1)+K)/2:ℕ):ℤ)-1
+    (J*t+J:ℕ) ≤ a ∧ a+(2^s:ℕ) ≤ (J*t+K:ℕ) ∧
+    (K:ℝ)/3 ≤ 2*((a-(J*t:ℕ):ℤ)-J)+1 ∧
+    (K:ℝ)/3 ≤ 2*K-(2*((a-(J*t:ℕ):ℤ)+(2:ℝ)^s)+1) := by
+  have he : (J*(2*t+1)+K)/2=J*t+(J+K)/2 := by
+    rw [show J*(2*t+1)+K=2*(J*t)+(J+K) by ring,Nat.mul_add_div (by norm_num)]
+  have hlo : 2*((J+K)/2) ≤ J+K := Nat.mul_div_le _ 2
+  have hhi : J+K<2*((J+K)/2+1) := by omega
+  have hK1 : 12*(J+1) ≤ K := by nlinarith [Nat.zero_le (2^s)]
+  dsimp only
+  rw [he]
+  have hi1 : J+1 ≤ (J+K)/2 := by omega
+  have hi2 : (J+K)/2+(2^s:ℕ) ≤ K+1 := by omega
+  constructor
+  · apply (le_sub_iff_add_le).mpr
+    exact_mod_cast (show J*t+J+1 ≤ J*t+(J+K)/2 from by omega)
+  constructor
+  · push_cast only [Nat.cast_add]
+    omega
+  have hl : (2:ℝ)*((J+K)/2:ℕ) ≤ J+K := by exact_mod_cast hlo
+  have hh : (J:ℝ)+K<2*(((J+K)/2:ℕ)+1) := by exact_mod_cast hhi
+  have hk : (12:ℝ)*((J:ℝ)+(2:ℝ)^s+1) ≤ K := by exact_mod_cast hK
+  simp only [Nat.cast_add,Int.cast_sub,Int.cast_add,Int.cast_natCast,Int.cast_one,Int.cast_mul,Nat.cast_mul]
+  constructor <;> nlinarith [show (0:ℝ) ≤ (2:ℝ)^s by positivity]
+
+theorem prouhetSeries_coeff_zero (s : ℕ) : (prouhetSeries s).coeff 0=1 := by
+  induction s with
+  | zero => simp [prouhetSeries]
+  | succ s ih =>
+    have hp : (0:ℤ)<(2^s:ℕ) := by positivity
+    rw [prouhetSeries_succ_coeff,ih,(prouhetSeries_coeff_spec s).1 _ (Or.inl (by omega)),sub_zero]
+
+/-- Literal source initial words really are different, not only their formal templates. -/
+theorem motifPeriodWord_ne {m : ℕ} (off : Fin m) (hi : 1 ≤ off.val ∧ off.val+1 < m)
+    (u s : ℕ) (a : ℤ) (ha : 0 ≤ a) (hau : a<u) :
+    motifPeriodWord off hi u s a (-1) ≠ motifPeriodWord off hi u s a 1 := by
+  intro he
+  have H := periodString_injective m u he
+  have hj : a.toNat<u := by omega
+  have h := congrArg (fun f=>((f ⟨a.toNat,hj⟩):Fin m).val) H
+  have ha' : (a.toNat:ℤ)=a := Int.toNat_of_nonneg ha
+  simp only [motifOffset,ha',sub_self,prouhetSeries_coeff_zero] at h
+  have hpos : (0:ℤ)<(2^s:ℕ) := by positivity
+  norm_num [motifIndicator,hpos,motifShift] at h
+
+/-- Floors in the initial construction waste no more than one long slot. -/
+theorem initial_period_count_bounds {N : ℝ} {m K t : ℕ}
+    (hL : 0<2*m-1) (hN : 0 ≤ N) (hK : (K:ℝ) ≤ ⌊N/(2*m-1:ℕ)⌋₊) :
+    let F : ℕ := ⌊N/(2*m-1:ℕ)⌋₊
+    let J : ℕ := ⌊((F:ℝ)-K)/(2*t+1:ℕ)⌋₊
+    let P : ℕ := J*(2*t+1)+K
+    (P:ℝ)*(2*m-1:ℕ) ≤ N ∧ N<((P:ℝ)+(2*t+1:ℕ)+1)*(2*m-1:ℕ) := by
+  dsimp only
+  set F : ℕ := ⌊N/(2*m-1:ℕ)⌋₊
+  set J : ℕ := ⌊((F:ℝ)-K)/(2*t+1:ℕ)⌋₊
+  have hd : (0:ℝ)<(2*t+1:ℕ) := by positivity
+  have hl : (0:ℝ)<(2*m-1:ℕ) := by exact_mod_cast hL
+  have hjlo := Nat.floor_le (div_nonneg (sub_nonneg.mpr hK) hd.le)
+  have hjhi := Nat.lt_floor_add_one (((F:ℝ)-K)/(2*t+1:ℕ))
+  change (J:ℝ) ≤ _ at hjlo
+  change _<(J:ℝ)+1 at hjhi
+  have hflo := Nat.floor_le (div_nonneg hN hl.le)
+  have hfhi := Nat.lt_floor_add_one (N/(2*m-1:ℕ))
+  change (F:ℝ) ≤ _ at hflo
+  change _<(F:ℝ)+1 at hfhi
+  have hp : (J:ℝ)*(2*t+1:ℕ)+K ≤ F := by
+    have H := (le_div_iff₀ hd).mp hjlo
+    linarith
+  have hp' : (F:ℝ)<(J:ℝ)*(2*t+1:ℕ)+K+(2*t+1:ℕ) := by
+    have H := (div_lt_iff₀ hd).mp hjhi
+    nlinarith
+  constructor
+  · push_cast
+    simpa only [Nat.cast_add,Nat.cast_mul,Nat.cast_ofNat,Nat.cast_one] using
+      (mul_le_mul_of_nonneg_right hp hl.le).trans ((le_div_iff₀ hl).mp hflo)
+  · have H := (div_lt_iff₀ hl).mp hfhi
+    push_cast at hp' ⊢
+    nlinarith
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open Filter
+open scoped Topology Classical
+
+/-- Source moment order is twice this integer. -/
+def sourceHalfMoment (B D : ℕ) (n : ℝ) (i : ℕ) : ℕ :=
+  ⌈(10^8:ℝ)*2^(10*B)*(totalScoreExponent B D+B+1)*Real.log n/
+    Real.log (sourceTargetScale B D n i)⌉₊
+
+theorem sourceEpsilon_le_one (B D : ℕ) : sourceEpsilon B D ≤ 1 := by
+  unfold sourceEpsilon
+  apply (div_le_one (by positivity)).mpr
+  have H : (1:ℝ) ≤ 2^(4*B+100) := one_le_pow₀ (by norm_num)
+  nlinarith [show (0:ℝ) ≤ D by positivity]
+
+theorem sourceTargetScale_tendsto (B : ℕ) (ζ : ℝ) (hB : 2 ≤ B) (hζ : ζ∈Set.Ioo 0 1)
+    {n Λ H : ℕ→ℝ} (i : ℕ→ℕ) (hn : Tendsto n atTop atTop) (hH : Tendsto H atTop atTop)
+    (hΛ : ∀t,1 ≤ Λ t) (hrel : ∀t,Real.log (n t)=Λ t*H t) :
+    Tendsto (fun t=>sourceTargetScale B (sourceDepth B ζ (H t)) (n t) (i t)) atTop atTop := by
+  have hlog : Tendsto (fun t=>Real.log (sourceTargetScale B (sourceDepth B ζ (H t)) (n t) (i t))) atTop atTop := by
+    apply tendsto_atTop_mono' atTop _ (((tendsto_rpow_atTop hζ.1).comp hH).atTop_div_const (by norm_num : (0:ℝ)<2))
+    filter_upwards [hH.eventually_ge_atTop 1,hn.eventually_gt_atTop 1] with t ht hnt
+    have hl := sourceTargetScale_log_lower B ζ (H t) (n t) (i t) hB hζ.2.le ht hnt
+    have he : Real.log (n t)/(2*(H t)^(1-ζ))=Λ t*(H t)^ζ/2 := by
+      rw [hrel t,Real.rpow_sub (by linarith : 0<H t),Real.rpow_one]
+      have hp : (H t)^ζ ≠ 0 := (Real.rpow_pos_of_pos (by linarith : 0<H t) _).ne'
+      field_simp
+    rw [he] at hl
+    exact (by nlinarith [hΛ t,Real.rpow_nonneg (by linarith : 0 ≤ H t) ζ] : (H t)^ζ/2 ≤ Λ t*(H t)^ζ/2).trans hl
+  apply tendsto_atTop_mono' atTop _ hlog
+  filter_upwards [hn.eventually_gt_atTop 1] with t ht
+  exact Real.log_le_self (Real.rpow_nonneg (by linarith : 0 ≤ n t) _)
+
+theorem source_uniform_log_absorption (B h : ℕ) (ζ : ℝ) (hB : 2 ≤ B) (hζ : ζ∈Set.Ioo 0 1)
+    {n Λ H : ℕ→ℝ} (i : ℕ→ℕ) (hn : Tendsto n atTop atTop) (hH : Tendsto H atTop atTop)
+    (hΛ : ∀t,1 ≤ Λ t) (hrel : ∀t,Real.log (n t)=Λ t*H t) :
+    Tendsto (fun t=>((sourceDepth B ζ (H t):ℝ)+1)^h*Λ t/
+      Real.log (sourceTargetScale B (sourceDepth B ζ (H t)) (n t) (i t))) atTop (𝓝 0) := by
+  have hh := source_uniform_absorption B h ζ hB hζ i hn hH (fun t=>lt_of_lt_of_le (by norm_num) (hΛ t)) hrel
+  apply squeeze_zero' _ _ hh
+  · filter_upwards [(sourceTargetScale_tendsto B ζ hB hζ i hn hH hΛ hrel).eventually_gt_atTop 1] with t ht
+    exact div_nonneg (mul_nonneg (by positivity) (by linarith [hΛ t])) (Real.log_pos ht).le
+  · filter_upwards [(sourceTargetScale_tendsto B ζ hB hζ i hn hH hΛ hrel).eventually_gt_atTop 1] with t ht
+    apply div_le_div_of_nonneg_left (by positivity [hΛ t])
+      (mul_pos (sourceEpsilon_pos _ _) (Real.log_pos ht))
+    exact mul_le_of_le_one_left (Real.log_pos ht).le (sourceEpsilon_le_one _ _)
+
+theorem totalScoreExponent_bounds (B D : ℕ) (hB : 2 ≤ B) :
+    11/5 ≤ totalScoreExponent B D ∧ totalScoreExponent B D ≤ 3+(D:ℝ)*B := by
+  have hdim (i : ℕ) : (2:ℝ) ≤ (if i<warmupBlocks B then (2:ℕ) else B) ∧
+      (((if i<warmupBlocks B then (2:ℕ) else B):ℕ):ℝ) ≤ B := by
+    split_ifs <;> norm_cast
+  have hlo : 0 ≤ ∑i∈Finset.range D,blockGain (if i<warmupBlocks B then 2 else B) := by
+    apply Finset.sum_nonneg
+    intro i _
+    unfold blockGain
+    linarith [(hdim i).1]
+  have hhi : (∑i∈Finset.range D,blockGain (if i<warmupBlocks B then 2 else B)) ≤ (D:ℝ)*B := by
+    calc
+      _ ≤ ∑_i∈Finset.range D,(B:ℝ) := by
+        apply Finset.sum_le_sum
+        intro i _
+        unfold blockGain
+        linarith [(hdim i).1,(hdim i).2]
+      _ = _ := by simp
+  unfold totalScoreExponent
+  constructor <;> linarith
+
+theorem sourceHalfMoment_bounds (B D i : ℕ) {n : ℝ} (hB : 2 ≤ B) (hn : 1<n) :
+    1 ≤ sourceHalfMoment B D n i ∧
+    (sourceHalfMoment B D n i:ℝ) ≤
+      ((10^8:ℝ)*2^(10*B)*((B:ℝ)+4)*2+1)*((D:ℝ)+1)*2^(B*D) := by
+  have hp := sourceScaleProduct_lower B D i hB
+  have hp0 : 0<sourceScaleProduct B D i := lt_of_lt_of_le (by positivity) hp
+  have he : (10^8:ℝ)*2^(10*B)*(totalScoreExponent B D+B+1)*Real.log n/
+      Real.log (sourceTargetScale B D n i)=
+      (10^8:ℝ)*2^(10*B)*(totalScoreExponent B D+B+1)/sourceScaleProduct B D i := by
+    rw [sourceTargetScale,Real.log_rpow (by linarith : 0<n)]
+    field_simp [(Real.log_pos hn).ne']
+  have hd := totalScoreExponent_bounds B D hB
+  have hA : 0<(10^8:ℝ)*2^(10*B)*(totalScoreExponent B D+B+1) := by positivity [hd.1]
+  unfold sourceHalfMoment
+  rw [he]
+  constructor
+  · exact Nat.one_le_ceil_iff.mpr (div_pos hA hp0)
+  have hc := Nat.ceil_lt_add_one (div_nonneg hA.le hp0.le)
+  have hb := div_le_div_of_nonneg_left hA.le (by positivity) hp
+  have he' : (10^8:ℝ)*2^(10*B)*(totalScoreExponent B D+B+1)/(1/(2*2^(B*D)))=
+      (10^8:ℝ)*2^(10*B)*(totalScoreExponent B D+B+1)*2*2^(B*D) := by field_simp
+  rw [he'] at hb
+  have hdd : totalScoreExponent B D+B+1 ≤ ((B:ℝ)+4)*((D:ℝ)+1) := by nlinarith [hd.2]
+  have hh := mul_le_mul_of_nonneg_right
+    (mul_le_mul_of_nonneg_left hdd (by positivity : (0:ℝ) ≤ 10^8*2^(10*B)))
+    (by positivity : (0:ℝ) ≤ 2*2^(B*D))
+  have hp1 : (1:ℝ) ≤ 2^(B*D) := one_le_pow₀ (by norm_num)
+  nlinarith [show (0:ℝ) ≤ D by positivity]
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open Filter
+open scoped Topology Classical
+
+variable {N f g : ℕ→ℝ} {α β : ℝ}
+
+theorem PowerRate.eventually_lt (hN : Tendsto N atTop atTop)
+    (hf : PowerRate N f α) (hg : PowerRate N g β) (hab : α<β) :
+    ∀ᶠt in atTop, f t<g t := by
+  have hz := ((hf.div hg).upper hN).tendsto_zero hN (sub_neg.mpr hab)
+  filter_upwards [hz.eventually_lt_const (by norm_num : (0:ℝ)<1),hg.1] with t ht hp
+  exact (div_lt_one hp).mp ht
+
+theorem PowerRate.log_base (hN : Tendsto N atTop atTop) :
+    PowerRate N (fun t=>Real.log (N t)) 0 := by
+  refine ⟨(hN.eventually_gt_atTop 1).mono (fun _ ht=>Real.log_pos ht),?_⟩
+  convert
+    (isLittleO_log_rpow_atTop (by norm_num : (0:ℝ)<1)).tendsto_div_nhds_zero.comp
+      (Real.tendsto_log_atTop.comp hN) using 1
+  ext t
+  simp only [Function.comp_apply,Real.rpow_one]
+
+/-- Only arithmetic predicates on the literal source integers. -/
+structure SourceInitialGeometry (q N : ℝ) : Prop where
+  m_large : 128 ≤ sourceInitialM N
+  K_pos : 0 < sourceInitialK N
+  K_le_m : sourceInitialK N ≤ sourceInitialM N
+  seed : 16 ≤ q^3*(sourceInitialM N:ℝ)
+  count : 8 ≤ ((sourceInitialK N*sourceInitialM N:ℕ):ℝ)*q^2
+  anchor_margin : 12*(sourceInitialJ q N+2^300+1) ≤ sourceInitialK N
+  smoothing_margin : ((2*(2^300):ℕ):ℝ) < (sourceInitialK N:ℝ)/(3*(3000+1:ℝ))
+  length_lower : N/2 ≤ (sourceInitialP q N:ℝ)*(sourceInitialPeriodSize N:ℝ)
+  length_upper : (sourceInitialP q N:ℝ)*(sourceInitialPeriodSize N:ℝ) ≤ N
+
+/-- All floor, seed, count, placement and length restrictions are automatic
+ in the actual joint regime; no change of the deletion rate at the base scale. -/
+theorem source_initial_geometry {q : ℕ→ℝ} (hN : Tendsto N atTop atTop)
+    (hq : ∀t,q t∈Set.Ioc 0 1)
+    (hΛ : Tendsto (fun t=>(q t)^(-3:ℤ)/Real.log (N t)) atTop (𝓝 0)) :
+    ∀ᶠt in atTop,SourceInitialGeometry (q t) (N t) := by
+  have hm := sourceInitialM_rate hN
+  have hK := sourceInitialK_rate hN
+  have hqrate := deletion_rate_zero hN hq hΛ
+  have hJ := sourceInitialJ_rate hN hqrate
+  have hL := sourceInitialPeriodSize_rate hN
+  have hF := sourceInitialCapacity_rate hN
+  have ht := sourceInitialSide_rate hN hqrate
+  have hm128 := (hm.tendsto_atTop hN (by norm_num)).eventually_ge_atTop 128
+  have hKm := hK.eventually_lt hN hm (by norm_num)
+  have hKF := hK.eventually_lt hN hF (by norm_num)
+  have hseed := ((hqrate.pow 3).mul hm).tendsto_atTop hN (by norm_num)
+  have hcount := ((hK.mul hm).mul (hqrate.pow 2)).tendsto_atTop hN (by norm_num)
+  have hanchor := ((hJ.add_const hN (by norm_num) (by positivity : (0:ℝ)≤2^300+1)).const_mul
+    hN (by norm_num : (0:ℝ)<12)).eventually_lt hN hK (by norm_num)
+  have hsm := (hK.tendsto_atTop hN (by norm_num)).eventually_gt_atTop ((2*(2^300):ℝ)*(3*3001))
+  have hloss : PowerRate N (fun t=>(2*(initialSideCount (q t) (sourceInitialM (N t)):ℝ)+2)*
+      (sourceInitialPeriodSize (N t):ℝ)) (124/125) := by
+    convert ((ht.const_mul hN (by norm_num : (0:ℝ)<2)).add_const hN
+      (by norm_num) (by norm_num : (0:ℝ)≤2)).mul hL using 1
+    norm_num
+  have hloss' := hloss.eventually_lt hN ((PowerRate.base hN).div_const hN (by norm_num : (0:ℝ)<2)) (by norm_num)
+  filter_upwards [hN.eventually_gt_atTop 0,hm128,hK.1,hKm,hKF,
+    hseed.eventually_ge_atTop 16,hcount.eventually_ge_atTop 8,hanchor,hsm,hloss'] with t hNt hm128 hKpos hKm hKF hs hc ha hsm hloss
+  have hmn : 128 ≤ sourceInitialM (N t) := by exact_mod_cast hm128
+  have hKn : 0 < sourceInitialK (N t) := by exact_mod_cast hKpos
+  have hLpos : 0 < sourceInitialPeriodSize (N t) := by unfold sourceInitialPeriodSize; omega
+  have hcnt := initial_period_count_bounds (m:=sourceInitialM (N t))
+    (K:=sourceInitialK (N t)) (t:=initialSideCount (q t) (sourceInitialM (N t)))
+    (N:=N t) hLpos hNt.le hKF.le
+  have hcnt' : (sourceInitialP (q t) (N t):ℝ)*(sourceInitialPeriodSize (N t):ℝ) ≤ N t ∧
+      N t<((sourceInitialP (q t) (N t):ℝ)+(2*initialSideCount (q t) (sourceInitialM (N t))+1:ℕ)+1)*
+        (sourceInitialPeriodSize (N t):ℝ) := by
+    simpa only [sourceInitialP,sourceInitialJ,sourceInitialCapacity,sourceInitialPeriodSize,
+      Nat.cast_add,Nat.cast_mul,Nat.cast_ofNat,Nat.cast_one] using hcnt
+  refine ⟨hmn,hKn,?_,hs,?_,?_,?_,?_,hcnt'.1⟩
+  · exact_mod_cast hKm.le
+  · simpa only [Nat.cast_mul] using hc
+  · apply (Nat.cast_le (α:=ℝ)).mp
+    simp only [Nat.cast_mul,Nat.cast_add,Nat.cast_pow,Nat.cast_ofNat,Nat.cast_one]
+    simpa only [add_assoc] using ha.le
+  · apply (lt_div_iff₀ (by norm_num : (0:ℝ)<3*(3000+1))).mpr
+    simpa only [Nat.cast_mul,Nat.cast_pow,Nat.cast_ofNat,show (3000+1:ℝ)=3001 by norm_num] using hsm
+  · have H := hcnt'.2
+    push_cast at H
+    nlinarith only [H,hloss]
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open Filter
+open scoped Topology Classical
+
+/-- The exact varying even moment order is subpower at EVERY working scale. -/
+theorem sourceHalfMoment_rate (B : ℕ) (ζ : ℝ) (hB : 2 ≤ B) (hζ : ζ∈Set.Ioo 0 1)
+    {n Λ H : ℕ→ℝ} (i : ℕ→ℕ) (hn : Tendsto n atTop atTop) (hH : Tendsto H atTop atTop)
+    (hΛ : ∀t,1 ≤ Λ t) (hrel : ∀t,Real.log (n t)=Λ t*H t) :
+    PowerRate (fun t=>sourceTargetScale B (sourceDepth B ζ (H t)) (n t) (i t))
+      (fun t=>(sourceHalfMoment B (sourceDepth B ζ (H t)) (n t) (i t):ℝ)) 0 := by
+  let C : ℝ := ((10^8:ℝ)*2^(10*B)*((B:ℝ)+4)*2+1)
+  let A : ℝ := C+1+(B:ℝ)*Real.log 2
+  have hC : 0<C := by dsimp [C]; positivity
+  have hA : 0<A := by dsimp [A]; positivity
+  have hscale := sourceTargetScale_tendsto B ζ hB hζ i hn hH hΛ hrel
+  have hlim := (source_uniform_log_absorption B 1 ζ hB hζ i hn hH hΛ hrel).const_mul A
+  simp only [pow_one,mul_zero] at hlim
+  refine ⟨?_,?_⟩
+  · filter_upwards [hn.eventually_gt_atTop 1] with t ht
+    have hk := (sourceHalfMoment_bounds B (sourceDepth B ζ (H t)) (i t) hB ht).1
+    exact_mod_cast (show 0<sourceHalfMoment B (sourceDepth B ζ (H t)) (n t) (i t) by omega)
+  apply squeeze_zero' _ _ hlim
+  · filter_upwards [hn.eventually_gt_atTop 1,hscale.eventually_gt_atTop 1] with t ht hs
+    have hk := (sourceHalfMoment_bounds B (sourceDepth B ζ (H t)) (i t) hB ht).1
+    exact div_nonneg (Real.log_nonneg (by exact_mod_cast hk)) (Real.log_pos hs).le
+  · filter_upwards [hn.eventually_gt_atTop 1,hscale.eventually_gt_atTop 1] with t ht hs
+    let D := sourceDepth B ζ (H t)
+    let k := sourceHalfMoment B D (n t) (i t)
+    have hk := sourceHalfMoment_bounds B D (i t) hB ht
+    have hkpos : 0<(k:ℝ) := by exact_mod_cast (show 0<k by dsimp [k]; omega)
+    have hlog := Real.log_le_log hkpos hk.2
+    change Real.log (k:ℝ) ≤ Real.log (C*((D:ℝ)+1)*2^(B*D)) at hlog
+    rw [Real.log_mul (mul_pos hC (by positivity)).ne' (by positivity),
+      Real.log_mul hC.ne' (by positivity),Real.log_pow] at hlog
+    have hD : Real.log ((D:ℝ)+1) ≤ D := by
+      simpa using Real.log_le_sub_one_of_pos (by positivity : (0:ℝ)<(D:ℝ)+1)
+    have hc := Real.log_le_sub_one_of_pos hC
+    have hb : 0 ≤ (B:ℝ)*Real.log 2 := by positivity
+    have hd : 0 ≤ (D:ℝ) := Nat.cast_nonneg _
+    have hl : Real.log (k:ℝ) ≤ A*((D:ℝ)+1) := by
+      dsimp [A]
+      push_cast at hlog
+      nlinarith
+    have hl' : Real.log (k:ℝ) ≤ A*((D:ℝ)+1)*Λ t :=
+      hl.trans (le_mul_of_one_le_right (by positivity) (hΛ t))
+    have HH := div_le_div_of_nonneg_right hl' (Real.log_pos hs).le
+    simpa only [mul_div_assoc,mul_assoc] using HH
+
+/-- Actual primitive-score weight at the base scale. -/
+def sourceInitialWeight (q : ℝ) (k : ℕ) : ℝ := (6*(2*k:ℕ):ℝ)*(2/q)
+
+theorem sourceInitialWeight_rate {N q : ℕ→ℝ} {k : ℕ→ℕ}
+    (hN : Tendsto N atTop atTop) (hq : PowerRate N q 0)
+    (hk : PowerRate N (fun t=>(k t:ℝ)) 0) :
+    PowerRate N (fun t=>sourceInitialWeight (q t) (k t)) 0 := by
+  have H := (hk.const_mul hN (by norm_num : (0:ℝ)<24)).div hq
+  convert H using 1
+  · ext t
+    simp only [sourceInitialWeight,Nat.cast_mul,Nat.cast_ofNat]
+    ring
+  · norm_num
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open Filter
+open scoped Topology Classical
+
+variable {N q w : ℕ→ℝ} {k : ℕ→ℕ}
+
+def initialClipFactor (N q w : ℝ) : ℝ :=
+  4*(initialMomentRatio (sourceInitialK N) (sourceInitialM N) q w/
+    (1-initialMomentRatio (sourceInitialK N) (sourceInitialM N) q w))
+
+theorem initialClipFactor_rate (hN : Tendsto N atTop atTop)
+    (hq : PowerRate N q 0) (hw : PowerRate N w 0) :
+    PowerRate N (fun t=>initialClipFactor (N t) (q t) (w t)) (-1/500) := by
+  have hρ := initialMomentRatio_rate hN (sourceInitialK_rate hN) (sourceInitialM_rate hN) hq hw
+  have hden := (PowerRate.const hN (by norm_num : (0:ℝ)<1)).sub_lower hN hρ (by norm_num)
+  simpa only [initialClipFactor,sub_zero] using (hρ.div hden).const_mul hN (by norm_num : (0:ℝ)<4)
+
+/-- The source moment is genuinely used in clipping: the exponent is linear
+ in that moment, not a fixed negative power of the intermediate length. -/
+theorem initial_clip_exp_bound (hN : Tendsto N atTop atTop)
+    (hq : PowerRate N q 0) (hw : PowerRate N w 0) (hk : ∀ᶠt in atTop,1 ≤ k t) :
+    ∀ᶠt in atTop,
+      Real.sqrt (6*(initialClipFactor (N t) (q t) (w t))^(k t)) ≤
+        Real.exp (-(k t:ℝ)*Real.log (N t)/2000) := by
+  have hfac := initialClipFactor_rate hN hq hw
+  have hu := hfac.upper hN (-3/2000) (by norm_num)
+  filter_upwards [hk,hfac.1,hu,hN.eventually_gt_atTop 1,
+    (Real.tendsto_log_atTop.comp hN).eventually_ge_atTop (2000*Real.log 6)] with t hk hp hu hn hlog
+  rw [abs_of_pos hp] at hu
+  have hh := Real.log_le_log hp hu
+  rw [Real.log_rpow (by linarith : 0<N t)] at hh
+  have hk' : (1:ℝ) ≤ k t := by exact_mod_cast hk
+  have hmul := mul_le_mul_of_nonneg_left hh (by positivity : (0:ℝ) ≤ k t)
+  have hl : 0 ≤ Real.log (N t) := (Real.log_pos hn).le
+  have hg : Real.log 6 ≤ (k t:ℝ)*Real.log (N t)/2000 := by
+    change 2000*Real.log 6 ≤ Real.log (N t) at hlog
+    nlinarith [mul_nonneg (sub_nonneg.mpr hk') hl]
+  have hexp : Real.sqrt (6*initialClipFactor (N t) (q t) (w t)^(k t)) =
+      Real.exp ((Real.log 6+(k t:ℝ)*Real.log (initialClipFactor (N t) (q t) (w t)))/2) := by
+    rw [Real.sqrt_eq_rpow,Real.rpow_def_of_pos (by positivity),Real.log_mul (by norm_num) (pow_pos hp _).ne',Real.log_pow]
+    congr 1
+    ring
+  rw [hexp]
+  exact Real.exp_le_exp.mpr (by nlinarith)
+
+/-- The fixed metadata tails also beat the varying moment times log-scale.
+ Their exponents have orders .988 and .008 respectively, while k log N
+ has order zero. -/
+theorem initial_metadata_exp_bound (hN : Tendsto N atTop atTop)
+    (hq : PowerRate N q 0) (hω : PowerRate N (fun t=>initialOverlapLower (q t)) 0)
+    (hk : PowerRate N (fun t=>(k t:ℝ)) 0) :
+    ∀ᶠt in atTop,
+      2*Real.exp (-((sourceInitialK (N t)*sourceInitialM (N t):ℕ):ℝ)*(q t)^4/64)+
+        Real.exp (-(sourceInitialJ (q t) (N t):ℝ)*(initialOverlapLower (q t))^4/16) ≤
+        3*Real.exp (-(k t:ℝ)*Real.log (N t)) := by
+  have hlow : PowerRate N (fun t=>(k t:ℝ)*Real.log (N t)) 0 := by
+    simpa using hk.mul (PowerRate.log_base hN)
+  have hfirst := (((sourceInitialK_rate hN).mul (sourceInitialM_rate hN)).mul
+    (hq.pow 4)).div_const hN (by norm_num : (0:ℝ)<64)
+  have hsecond := ((sourceInitialJ_rate hN hq).mul (hω.pow 4)).div_const hN (by norm_num : (0:ℝ)<16)
+  filter_upwards [hlow.eventually_lt hN hfirst (by norm_num),hlow.eventually_lt hN hsecond (by norm_num)] with t h1 h2
+  have he1 := Real.exp_le_exp.mpr (neg_le_neg h1.le)
+  have he2 := Real.exp_le_exp.mpr (neg_le_neg h2.le)
+  push_cast
+  simp only [neg_div,neg_mul] at he1 he2 ⊢
+  linarith
+
+/-- A completely quantitative sufficient moment inequality for the actual
+ initial error. This is later discharged by the literal ceiling r_0/2. -/
+theorem initial_error_eventually {n A : ℕ→ℝ} (hN : Tendsto N atTop atTop)
+    (hn : Tendsto n atTop atTop) (hA : ∀ᶠt in atTop,1 ≤ A t)
+    (hq : ∀t,q t∈Set.Ioc 0 1)
+    (hΛ : Tendsto (fun t=>(q t)^(-3:ℤ)/Real.log (N t)) atTop (𝓝 0))
+    (hk : PowerRate N (fun t=>(k t:ℝ)) 0) (hk1 : ∀ᶠt in atTop,1 ≤ k t)
+    (hmoment : ∀ᶠt in atTop, 10^6*A t*Real.log (n t) ≤ (k t:ℝ)*Real.log (N t)) :
+    ∀ᶠt in atTop,
+      initialErrorCap (sourceInitialJ (q t) (N t)) (sourceInitialK (N t))
+        (sourceInitialM (N t)) (k t) (q t) (initialOverlapLower (q t))
+        (sourceInitialWeight (q t) (k t)) ≤ (n t)^(-100*A t) := by
+  have hqrate := deletion_rate_zero hN hq hΛ
+  have hω := initialOverlap_rate hN (fun t=>(hq t).1) hΛ
+  have hw := sourceInitialWeight_rate hN hqrate hk
+  have hclip := initial_clip_exp_bound hN hqrate hw hk1
+  have hmeta := initial_metadata_exp_bound hN hqrate hω hk
+  filter_upwards [hclip,hmeta,hmoment,hA,hn.eventually_gt_atTop 1,
+    (Real.tendsto_log_atTop.comp hn).eventually_ge_atTop (Real.log 4)] with t hc hm hmom hA hn hlog
+  have hAn : 0 ≤ A t*Real.log (n t) := mul_nonneg (by linarith) (Real.log_pos hn).le
+  have he1 : Real.exp (-(k t:ℝ)*Real.log (N t)/2000) ≤ Real.exp (-500*(A t*Real.log (n t))) :=
+    Real.exp_le_exp.mpr (by nlinarith)
+  have he2 : Real.exp (-(k t:ℝ)*Real.log (N t)) ≤ Real.exp (-500*(A t*Real.log (n t))) :=
+    Real.exp_le_exp.mpr (by nlinarith)
+  have herr : initialErrorCap (sourceInitialJ (q t) (N t)) (sourceInitialK (N t))
+      (sourceInitialM (N t)) (k t) (q t) (initialOverlapLower (q t))
+      (sourceInitialWeight (q t) (k t)) ≤ 4*Real.exp (-500*(A t*Real.log (n t))) := by
+    unfold initialErrorCap
+    change Real.sqrt (6*initialClipFactor (N t) (q t) (sourceInitialWeight (q t) (k t))^(k t))+_ ≤ _
+    nlinarith only [hc,hm,he1,he2]
+  apply herr.trans
+  rw [Real.rpow_def_of_pos (by linarith : 0<n t)]
+  calc
+    4*Real.exp (-500*(A t*Real.log (n t))) =
+        Real.exp (Real.log 4-500*(A t*Real.log (n t))) := by
+      rw [Real.exp_sub,Real.exp_log (by norm_num)]
+      rw [show -500*(A t*Real.log (n t)) = -(500*(A t*Real.log (n t))) by ring, Real.exp_neg]
+      rfl
+    _ ≤ Real.exp (Real.log (n t)*(-100*A t)) := by
+      apply Real.exp_le_exp.mpr
+      have hAL : Real.log 4 ≤ A t*Real.log (n t) := hlog.trans (le_mul_of_one_le_left (Real.log_pos hn).le hA)
+      nlinarith
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open Filter
+open scoped Topology Classical
+
+variable {N f g : ℕ→ℝ} {α β : ℝ}
+
+theorem PowerRate.add_lower (hN : Tendsto N atTop atTop)
+    (hf : PowerRate N f α) (hg : PowerRate N g β) (hβα : β<α) :
+    PowerRate N (fun t=>f t+g t) α := by
+  have hz := ((hg.div hf).upper hN).tendsto_zero hN (sub_neg.mpr hβα)
+  apply hf.of_ratio hN (by norm_num : (0:ℝ)<1)
+  have H : Tendsto (fun t=>1+g t/f t) atTop (𝓝 1) := by
+    simpa using tendsto_const_nhds.add hz
+  apply H.congr'
+  filter_upwards [hf.1] with t ht
+  field_simp [ht.ne']
+
+theorem PowerRate.nat_ceil (hN : Tendsto N atTop atTop) (hf : PowerRate N f α)
+    (ha : 0<α) : PowerRate N (fun t=>(⌈f t⌉₊:ℝ)) α := by
+  exact hf.of_ratio hN (by norm_num : (0:ℝ)<1)
+    (tendsto_nat_ceil_div_atTop.comp (hf.tendsto_atTop hN ha))
+
+def sourceOverlapConstant (q : ℝ) : ℝ := (2:ℝ)^40/(q*initialOverlapLower q)
+def sourceInitialCount (q N : ℝ) : ℕ :=
+  ⌈sourceOverlapConstant q*(initialSideCount q (sourceInitialM N):ℝ)*(sourceInitialP q N:ℝ)⌉₊
+
+theorem sourceInitialP_rate {q : ℕ→ℝ} (hN : Tendsto N atTop atTop)
+    (hq : PowerRate N q 0) :
+    PowerRate N (fun t=>(sourceInitialP (q t) (N t):ℝ)) (63/125) := by
+  have hj := sourceInitialJ_rate hN hq
+  have ht := ((sourceInitialSide_rate hN hq).const_mul hN (by norm_num : (0:ℝ)<2)).add_const hN
+    (by norm_num) (by norm_num : (0:ℝ)≤1)
+  have H := (hj.mul ht).add_lower hN (sourceInitialK_rate hN) (by norm_num)
+  convert H using 1
+  · ext t
+    simp only [sourceInitialP,Nat.cast_add,Nat.cast_mul,Nat.cast_ofNat,Nat.cast_one]
+  · norm_num
+
+theorem sourceOverlapConstant_rate {q : ℕ→ℝ} (hN : Tendsto N atTop atTop)
+    (hq : PowerRate N q 0) (hω : PowerRate N (fun t=>initialOverlapLower (q t)) 0) :
+    PowerRate N (fun t=>sourceOverlapConstant (q t)) 0 := by
+  simpa only [sourceOverlapConstant,add_zero,sub_zero] using
+    (PowerRate.const hN (by positivity : (0:ℝ)<(2:ℝ)^40)).div (hq.mul hω)
+
+theorem sourceInitialCount_rate {q : ℕ→ℝ} (hN : Tendsto N atTop atTop)
+    (hq : PowerRate N q 0) (hω : PowerRate N (fun t=>initialOverlapLower (q t)) 0) :
+    PowerRate N (fun t=>(sourceInitialCount (q t) (N t):ℝ)) 1 := by
+  have H := (((sourceOverlapConstant_rate hN hq hω).mul (sourceInitialSide_rate hN hq)).mul
+    (sourceInitialP_rate hN hq)).nat_ceil hN (by norm_num)
+  convert H using 1
+  · rfl
+  · norm_num
+
+theorem sourceOverlapConstant_pos {q : ℝ} (hq : 0<q) : 0<sourceOverlapConstant q := by
+  unfold sourceOverlapConstant
+  positivity [initialOverlapLower_pos q]
+
+theorem sourceOverlapConstant_large {q : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ)) :
+    1073741824 ≤ initialOverlapLower q*sourceOverlapConstant q := by
+  have hω := initialOverlapLower_pos q
+  have he : initialOverlapLower q*sourceOverlapConstant q=(2:ℝ)^40/q := by
+    unfold sourceOverlapConstant
+    field_simp
+  rw [he]
+  apply (le_div_iff₀ hq.1).mpr
+  nlinarith [hq.2]
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open Filter
+open scoped Topology Classical
+
+def sourceInitialWord (q N : ℝ) (τ : ℝ) : List Bool :=
+  if hm : 4 ≤ sourceInitialM N then
+    motifPeriodWord (sourceInitialOffset (sourceInitialM N) hm)
+      (sourceInitialOffset_interior _ hm) (sourceInitialP q N) 300 (sourceInitialAnchor q N) τ
+  else []
+
+def sourceInitialPure (q N : ℝ) : List Bool :=
+  pureAtomRun (initialPeriod (sourceInitialM N) (sourceInitialM N/2-1)) (sourceInitialP q N)
+
+def sourceInitialError (q N : ℝ) (k : ℕ) : ℝ :=
+  initialErrorCap (sourceInitialJ q N) (sourceInitialK N) (sourceInitialM N) k q
+    (initialOverlapLower q) (sourceInitialWeight q k)
+
+theorem sourceInitialWord_length {q N : ℝ} (hm : 4 ≤ sourceInitialM N) (τ : ℝ) :
+    (sourceInitialWord q N τ).length=sourceInitialP q N*sourceInitialPeriodSize N := by
+  rw [sourceInitialWord,dite_eq_left hm,motifPeriodWord_length]
+  rfl
+
+theorem sourceInitialWord_different {q N : ℝ} (H : SourceInitialGeometry q N) :
+    sourceInitialWord q N (-1) ≠ sourceInitialWord q N 1 := by
+  have hm : 4 ≤ sourceInitialM N := (by norm_num : 4 ≤ 128).trans H.m_large
+  have ha := initial_anchor_geometry (sourceInitialJ q N) (sourceInitialK N)
+    (initialSideCount q (sourceInitialM N)) 300 H.anchor_margin
+  have hap : (0:ℤ) ≤ sourceInitialAnchor q N := (Int.natCast_nonneg _).trans ha.1
+  have hau : sourceInitialAnchor q N<(sourceInitialP q N:ℤ) := by
+    have h1 := ha.2.1
+    have he : (sourceInitialJ q N*initialSideCount q (sourceInitialM N)+sourceInitialK N:ℕ) ≤ sourceInitialP q N := by
+      unfold sourceInitialP
+      apply Nat.add_le_add_right
+      apply Nat.mul_le_mul_left
+      omega
+    have hp : (0:ℤ)<(2^300:ℕ) := by positivity
+    have hh : ((sourceInitialJ q N*initialSideCount q (sourceInitialM N)+sourceInitialK N:ℕ):ℤ) ≤ sourceInitialP q N := Int.ofNat_le.mpr he
+    exact (lt_add_of_pos_right (sourceInitialAnchor q N) hp).trans_le (h1.trans hh)
+  simp only [sourceInitialWord,dite_eq_left hm]
+  exact motifPeriodWord_ne _ _ _ _ _ hap hau
+
+/-- The actual initial proof, with no oracle or coefficient-bound hypothesis.
+ The only moment restriction is the quantitative source ceiling inequality. -/
+theorem source_initial_oracles_eventually {N q n A : ℕ→ℝ} {k : ℕ→ℕ}
+    (hN : Tendsto N atTop atTop) (hn : Tendsto n atTop atTop)
+    (hA : ∀ᶠt in atTop,1 ≤ A t) (hq : ∀t,q t∈Set.Ioc 0 (1/2:ℝ))
+    (hΛ : Tendsto (fun t=>(q t)^(-3:ℤ)/Real.log (N t)) atTop (𝓝 0))
+    (hk : PowerRate N (fun t=>(k t:ℝ)) 0) (hk1 : ∀ᶠt in atTop,1 ≤ k t)
+    (hmoment : ∀ᶠt in atTop,10^6*A t*Real.log (n t) ≤ (k t:ℝ)*Real.log (N t)) :
+    ∀ᶠt in atTop,
+      SourceInitialGeometry (q t) (N t) ∧
+      sourceInitialError (q t) (N t) (k t) ≤ (n t)^(-100*A t) ∧
+      Nonempty (OracleTracePair (q t) (sourceInitialWord (q t) (N t) (-1))
+        (sourceInitialWord (q t) (N t) 1) (k t) (sourceInitialError (q t) (N t) (k t)) ((N t)^(-11/5:ℝ))) ∧
+      Nonempty (OracleTracePair (q t) (sourceInitialPure (q t) (N t))
+        (sourceInitialWord (q t) (N t) 1) (k t) (sourceInitialError (q t) (N t) (k t)) ((N t)^(-7/5:ℝ))) := by
+  have hq' (t) : q t ∈ Set.Ioc 0 1 := ⟨(hq t).1,by linarith [(hq t).2]⟩
+  have hqr := deletion_rate_zero hN hq' hΛ
+  have hw := sourceInitialWeight_rate hN hqr hk
+  have herr := initial_error_eventually hN hn hA hq' hΛ hk hk1 hmoment
+  have hgeo := source_initial_geometry hN hq' hΛ
+  have hcap := source_initial_score_caps hN hq' hΛ hw
+  have hρ := (initialMomentRatio_rate hN (sourceInitialK_rate hN) (sourceInitialM_rate hN) hqr hw).upper hN |>.tendsto_zero hN (by norm_num)
+  have hsmall := ((tendsto_rpow_neg_atTop (by norm_num : (0:ℝ)<100)).comp hn).eventually_le_const (by norm_num : (0:ℝ)<1/2)
+  filter_upwards [hgeo,hcap,herr,hk1,hA,hn.eventually_gt_atTop 1,hN.eventually_gt_atTop 0,
+    hρ.eventually_le_const (by norm_num : (0:ℝ)<1/2),hsmall] with t hg hcap he hk1 ha hnt hNt hρ hsmall
+  have hm : 4 ≤ sourceInitialM (N t) := (by norm_num : 4 ≤ 128).trans hg.m_large
+  let off := sourceInitialOffset (sourceInitialM (N t)) hm
+  have hi := sourceInitialOffset_interior _ hm
+  have hanchor := initial_anchor_geometry (sourceInitialJ (q t) (N t)) (sourceInitialK (N t))
+    (initialSideCount (q t) (sourceInitialM (N t))) 300 hg.anchor_margin
+  change _ ≤ sourceInitialAnchor (q t) (N t) ∧ sourceInitialAnchor (q t) (N t)+(2^300:ℕ) ≤ _ ∧ _ at hanchor
+  have he' : sourceInitialError (q t) (N t) (k t) ≤ 1/2 := by
+    exact he.trans ((Real.rpow_le_rpow_of_exponent_le hnt.le (by linarith : -100*A t ≤ -100)).trans hsmall)
+  have hc : Real.sqrt (6*(4*(initialMomentRatio (sourceInitialK (N t)) (sourceInitialM (N t)) (q t)
+      (sourceInitialWeight (q t) (k t))/(1-initialMomentRatio (sourceInitialK (N t))
+      (sourceInitialM (N t)) (q t) (sourceInitialWeight (q t) (k t)))))^(k t)) ≤ 1/2 := by
+    have ht : 0 ≤ 2*Real.exp (-((sourceInitialK (N t)*sourceInitialM (N t):ℕ):ℝ)*(q t)^4/64)+
+      Real.exp (-(sourceInitialJ (q t) (N t):ℝ)*(initialOverlapLower (q t))^4/16) := by positivity
+    unfold sourceInitialError initialErrorCap at he'
+    linarith only [he',ht]
+  have HH := actual_initial_oracles hg.K_pos hg.m_large hg.K_le_m (by norm_num : 0<3000)
+    off hi (sourceInitialOffset_middle _ hm) (sourceInitialOffset_safe _ _ hm)
+    (sourceInitialAnchor (q t) (N t)) 300 (k t) hk1 (hq t) hg.seed hg.count
+    hanchor.1 hanchor.2.1 hanchor.2.2.1 hanchor.2.2.2 hg.smoothing_margin
+    (rfl : sourceInitialWeight (q t) (k t)=_) hρ hc
+    (Real.rpow_nonneg hNt.le _) (Real.rpow_nonneg hNt.le _) hcap.1 hcap.2
+  refine ⟨hg,he,?_,?_⟩
+  · simpa only [sourceInitialWord,dite_eq_left hm,sourceInitialP,sourceInitialError,off] using HH.1
+  · simpa only [sourceInitialWord,dite_eq_left hm,sourceInitialP,sourceInitialError,sourceInitialPure,
+      off,sourceInitialOffset] using HH.2
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open Filter
+open scoped Topology Classical
+
+/-- Literal count instantiation; the harmless ceiling is included. -/
+theorem sourceInitialCount_bounds {q N : ℝ} (hq : q ∈ Set.Ioc 0 (1/2:ℝ))
+    (H : SourceInitialGeometry q N) :
+    sourceInitialP q N*sourceInitialPeriodSize N  ≤  sourceInitialCount q N ∧
+    (sourceInitialCount q N:ℝ)  ≤  (sourceOverlapConstant q)^2*
+      ((sourceInitialP q N*sourceInitialPeriodSize N:ℕ):ℝ) := by
+  let C := sourceOverlapConstant q
+  let m := sourceInitialM N
+  let t := initialSideCount q m
+  let P := sourceInitialP q N
+  let ell := sourceInitialPeriodSize N
+  have hC : 0<C := sourceOverlapConstant_pos hq.1
+  have hm : 32 ≤ m := (by norm_num : 32 ≤ 128).trans H.m_large
+  have ht := initialSideCount_bounds hq hm H.seed
+  have hω := initialOverlapLower_lt_eighth hq.1
+  have hω0 := initialOverlapLower_pos q
+  have hCq : 4  ≤  C*q := by
+    have he : C*q=(2:ℝ)^40/initialOverlapLower q := by dsimp [C,sourceOverlapConstant]; field_simp [hq.1.ne']
+    rw [he]
+    exact (le_div_iff₀ hω0).mpr (by nlinarith)
+  have hC2 : 2  ≤  C := by nlinarith [hq.2]
+  have hell : (ell:ℝ)=2*(m:ℝ)-1 := by
+    dsimp [ell,sourceInitialPeriodSize,m]
+    rw [Nat.cast_sub (by omega),Nat.cast_mul,Nat.cast_ofNat,Nat.cast_one]
+  have hell1 : (1:ℝ) ≤ ell := by
+    rw [hell]
+    have hml : (32:ℝ) ≤ m := by exact_mod_cast hm
+    linarith only [hml]
+  have hPm : 1 ≤ P := by dsimp [P,sourceInitialP]; have := H.K_pos; omega
+  have hP : (1:ℝ) ≤ P := by exact_mod_cast hPm
+  have hPt : 0 ≤ (P:ℝ) := Nat.cast_nonneg _
+  have hm0 : (0:ℝ) ≤ m := Nat.cast_nonneg _
+  have hlower : (ell:ℝ) ≤ C*(t:ℝ) := by
+    have hh := mul_le_mul_of_nonneg_left ht.1 hC.le
+    have hh' := mul_le_mul_of_nonneg_right hCq hm0
+    rw [hell]
+    dsimp [t] at *
+    nlinarith only [hh,hh']
+  have hct : 0 ≤ C*(t:ℝ)*(P:ℝ) := by positivity
+  have hceil : C*(t:ℝ)*(P:ℝ) ≤ (sourceInitialCount q N:ℝ) := Nat.le_ceil _
+  have hu : (sourceInitialCount q N:ℝ)<C*(t:ℝ)*(P:ℝ)+1 := Nat.ceil_lt_add_one hct
+  constructor
+  · apply (Nat.cast_le (α:=ℝ)).mp
+    simp only [Nat.cast_mul]
+    simpa only [mul_comm (ell:ℝ) (P:ℝ)] using (mul_le_mul_of_nonneg_right hlower hPt).trans hceil
+  · have htell : (t:ℝ) ≤ ell := by
+      have hh := mul_le_mul_of_nonneg_right hq.2 hm0
+      rw [hell]
+      have hml : (32:ℝ) ≤ m := by exact_mod_cast hm
+      linarith [ht.2.1]
+    have he1 : (1:ℝ) ≤ (P:ℝ)*ell := one_le_mul_of_one_le_of_one_le hP hell1
+    have hu' := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left htell hC.le) hPt
+    have hc : C+1 ≤ C^2 := by nlinarith
+    have hh := mul_le_mul_of_nonneg_right hc (mul_nonneg hPt (zero_le_one.trans hell1))
+    change _  ≤  C^2*((P*ell:ℕ):ℝ)
+    rw [Nat.cast_mul]
+    nlinarith only [hu,hu',he1,hh]
+
+/-- Genuine overlap stability for complete traces of powers. -/
+theorem OracleTracePair.transfer_power_overlap {q E V p : ℝ} {x y : List Bool} {k F : ℕ}
+    (C : OracleTracePair q x y k E V) (hq : q ∈ Set.Icc 0 1) (hk : 1 ≤ k)
+    (hov : p ≤ listTraceOverlap q hq (pureAtomRun y F) (pureAtomRun y (F+1))) :
+    p-(Real.sqrt ((F:ℝ)*V)+2*(F:ℝ)*E)-
+      (Real.sqrt (((F+1:ℕ):ℝ)*V)+2*((F+1:ℕ):ℝ)*E)  ≤ 
+      listTraceOverlap q hq (pureAtomRun x F) (pureAtomRun x (F+1)) := by
+  let a := pureAtomRun y F
+  let b := pureAtomRun y (F+1)
+  let u := pureAtomRun x F
+  let v := pureAtomRun x (F+1)
+  let M := max (max a.length b.length) (max u.length v.length)
+  have ha : a.length ≤ M := (le_max_left _ _).trans (le_max_left _ _)
+  have hb : b.length ≤ M := (le_max_right _ _).trans (le_max_left _ _)
+  have hu : u.length ≤ M := (le_max_left _ _).trans (le_max_right _ _)
+  have hv : v.length ≤ M := (le_max_right _ _).trans (le_max_right _ _)
+  have H₁ := C.power_tv hq hk F M hu ha
+  have H₂ := C.power_tv hq hk (F+1) M hv hb
+  rw [listTraceOverlap_ambient hq _ _ M ha hb,FiniteLaw.overlap_eq_one_sub_tv] at hov
+  rw [listTraceOverlap_ambient hq _ _ M hu hv,FiniteLaw.overlap_eq_one_sub_tv]
+  have HH := (FiniteLaw.tv_triangle (listTraceLaw q hq u M hu) (listTraceLaw q hq a M ha)
+    (listTraceLaw q hq v M hv)).trans
+    (add_le_add le_rfl (FiniteLaw.tv_triangle (listTraceLaw q hq a M ha)
+      (listTraceLaw q hq b M hb) (listTraceLaw q hq v M hv)))
+  rw [FiniteLaw.tv_symm (listTraceLaw q hq b M hb) (listTraceLaw q hq v M hv)] at HH
+  linarith only [HH,H₁,H₂,hov]
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open Filter
+open scoped Topology Classical
+
+theorem powerTransferBudget_mono {V W E : ℝ} {F G : ℕ} (hE : 0 ≤ E)
+    (hV : 0 ≤ V) (hVW : V ≤ W) (hFG : F ≤ G) :
+    Real.sqrt ((F:ℝ)*V)+2*(F:ℝ)*E ≤ Real.sqrt ((G:ℝ)*W)+2*(G:ℝ)*E := by
+  have hfg : (F:ℝ) ≤ G := by exact_mod_cast hFG
+  apply add_le_add
+  · exact Real.sqrt_le_sqrt (mul_le_mul hfg hVW hV (Nat.cast_nonneg _))
+  · nlinarith only [mul_le_mul_of_nonneg_right hfg hE]
+
+theorem source_initial_power_overlap {q N E : ℝ} {k : ℕ}
+    (hq : q ∈ Set.Ioc 0 (1/2:ℝ)) (H : SourceInitialGeometry q N)
+    (hN : 1 ≤ N) (hk : 1 ≤ k)
+    (C : OracleTracePair q (sourceInitialWord q N (-1)) (sourceInitialWord q N 1)
+      k E (N^(-11/5:ℝ)))
+    (U : OracleTracePair q (sourceInitialPure q N) (sourceInitialWord q N 1)
+      k E (N^(-7/5:ℝ)))
+    (hsmall : Real.sqrt (((sourceInitialCount q N+1:ℕ):ℝ)*(N^(-7/5:ℝ)))+
+      2*((sourceInitialCount q N+1:ℕ):ℝ)*E ≤ 1/16) :
+    initialOverlapLower q ≤ listTraceOverlap q ⟨hq.1.le,by linarith [hq.2]⟩
+      (pureAtomRun (sourceInitialWord q N (-1)) (sourceInitialCount q N))
+      (pureAtomRun (sourceInitialWord q N (-1)) (sourceInitialCount q N+1)) := by
+  let F := sourceInitialCount q N
+  have hm : 4 ≤ sourceInitialM N := (by norm_num : 4 ≤ 128).trans H.m_large
+  let off := sourceInitialOffset (sourceInitialM N) hm
+  have ht := initialSideCount_bounds hq ((by norm_num : 32 ≤ 128).trans H.m_large) H.seed
+  have hqm : 2 ≤ q*(sourceInitialM N:ℝ) := by
+    have hh := mul_le_mul_of_nonneg_right (show q^2 ≤ 1 by nlinarith [hq.1,hq.2])
+      (mul_nonneg hq.1.le (Nat.cast_nonneg (sourceInitialM N)))
+    nlinarith [H.seed]
+  have ht1 : 1 ≤ initialSideCount q (sourceInitialM N) := by
+    have hh : (1:ℝ) ≤ initialSideCount q (sourceInitialM N) := by linarith [ht.1]
+    exact_mod_cast hh
+  have hP : 1 ≤ sourceInitialP q N := by unfold sourceInitialP; have := H.K_pos; omega
+  have hω : initialOverlapLower q ∈ Set.Ioc 0 (1/8:ℝ) :=
+    ⟨initialOverlapLower_pos q,(initialOverlapLower_lt_eighth hq.1).le⟩
+  have hqc : q ∈ Set.Icc 0 1 := ⟨hq.1.le,by linarith [hq.2]⟩
+  have hu := U.lifted_power_overlap hqc hk ht1 hP hω (sourceOverlapConstant_large hq)
+    (Nat.le_ceil _) (initial_pure_trace_overlap hq
+      ((by norm_num : 32 ≤ 128).trans H.m_large) H.seed off ((sourceInitialOffset_middle _ hm).trans (by norm_num)))
+  have hv0 := Real.rpow_nonneg (by linarith : 0 ≤ N) (-11/5:ℝ)
+  have hw0 := Real.rpow_nonneg (by linarith : 0 ≤ N) (-7/5:ℝ)
+  have hvw : N^(-11/5:ℝ) ≤ N^(-7/5:ℝ) := Real.rpow_le_rpow_of_exponent_le hN (by norm_num)
+  have hs0 := (powerTransferBudget_mono C.error_nonneg hw0 le_rfl (Nat.le_succ F)).trans hsmall
+  have hs1 := (powerTransferBudget_mono C.error_nonneg hv0 hvw (Nat.le_succ F)).trans hsmall
+  have hs2 := (powerTransferBudget_mono C.error_nonneg hv0 hvw (le_refl (F+1))).trans hsmall
+  have htrans := C.transfer_power_overlap hqc hk hu
+  have hω' := hω.2
+  dsimp only [F,sourceInitialCount] at hs0 hs1 hs2 hsmall ⊢
+  linarith only [htrans,hs0,hs1,hs2,hsmall,hω']
+
+/-- The exact initial overlap error is negligible, including channel-dependent
+ count factors. This is a consequence, not a hypothesis of the initial pair. -/
+theorem source_initial_overlap_budget_eventually {q N E : ℕ→ℝ}
+    (hN : Tendsto N atTop atTop) (hq : PowerRate N q 0)
+    (hω : PowerRate N (fun t=>initialOverlapLower (q t)) 0)
+    (hE : ∀ᶠt in atTop,0 ≤ E t ∧ E t ≤ (N t)^(-100:ℝ)) :
+    ∀ᶠt in atTop,Real.sqrt (((sourceInitialCount (q t) (N t)+1:ℕ):ℝ)*((N t)^(-7/5:ℝ)))+
+      2*((sourceInitialCount (q t) (N t)+1:ℕ):ℝ)*E t ≤ 1/16 := by
+  have hF := (sourceInitialCount_rate hN hq hω).add_const hN (by norm_num) (by norm_num : (0:ℝ) ≤ 1)
+  have hw := (PowerRate.base hN).rpow (-7/5:ℝ)
+  have he := (PowerRate.base hN).rpow (-100:ℝ)
+  have hs := ((hF.mul hw).sqrt.upper hN).tendsto_zero hN (by norm_num)
+  have ht := (((hF.mul he).const_mul hN (by norm_num : (0:ℝ)<2)).upper hN).tendsto_zero hN (by norm_num)
+  have hsum := hs.add ht
+  rw [zero_add] at hsum
+  have HH := hsum.eventually_le_const (by norm_num : (0:ℝ)<1/16)
+  filter_upwards [HH,hE] with t ht he
+  simp only [Nat.cast_add,Nat.cast_one]
+  simp only [←mul_assoc] at ht
+  apply le_trans _ ht
+  apply add_le_add le_rfl
+  exact mul_le_mul_of_nonneg_left he.2 (by positivity)
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open Filter
+open scoped Topology Classical
+
+theorem source_initial_level_eventually {N q n A : ℕ→ℝ} {k : ℕ→ℕ}
+    (hN : Tendsto N atTop atTop) (hn : Tendsto n atTop atTop)
+    (hA : ∀ᶠt in atTop,1 ≤ A t) (hq : ∀t,q t ∈ Set.Ioc 0 (1/2:ℝ))
+    (hΛ : Tendsto (fun t=>(q t)^(-3:ℤ)/Real.log (N t)) atTop (𝓝 0))
+    (hk : PowerRate N (fun t=>(k t:ℝ)) 0) (hk1 : ∀ᶠt in atTop,1 ≤ k t)
+    (hmoment : ∀ᶠt in atTop,10^6*A t*Real.log (n t) ≤ (k t:ℝ)*Real.log (N t))
+    (hscale : ∀ᶠt in atTop,N t ≤ n t) :
+    ∀ᶠt in atTop,Nonempty (ConcreteLevelPair (q t) ⟨(hq t).1.le,by linarith [(hq t).2]⟩
+      (N t) ((n t)^(-100*A t)) (11/5) (initialOverlapLower (q t))
+      ((sourceOverlapConstant (q t))^2) 0 (k t)) := by
+  have hor := source_initial_oracles_eventually hN hn hA hq hΛ hk hk1 hmoment
+  have hq' (t) : q t ∈ Set.Ioc 0 1 := ⟨(hq t).1,by linarith [(hq t).2]⟩
+  have hqr := deletion_rate_zero hN hq' hΛ
+  have hωr := initialOverlap_rate hN (fun t=>(hq t).1) hΛ
+  have he : ∀ᶠt in atTop,0 ≤ sourceInitialError (q t) (N t) (k t) ∧
+      sourceInitialError (q t) (N t) (k t) ≤ (N t)^(-100:ℝ) := by
+    filter_upwards [hor,hA,hscale,hN.eventually_gt_atTop 1] with t ht ha hs hNt
+    obtain ⟨C⟩ := ht.2.2.1
+    refine ⟨C.error_nonneg,ht.2.1.trans ?_⟩
+    calc
+      (n t)^(-100*A t) ≤ (n t)^(-100:ℝ) := Real.rpow_le_rpow_of_exponent_le (hNt.le.trans hs) (by linarith)
+      _ ≤ (N t)^(-100:ℝ) := Real.rpow_le_rpow_of_nonpos (by linarith) hs (by norm_num)
+  have hsmall := source_initial_overlap_budget_eventually hN hqr hωr he
+  filter_upwards [hor,hsmall,hk1,hN.eventually_gt_atTop 2,hn.eventually_gt_atTop 0] with t ht hs hk1 hNt hnt
+  obtain ⟨H,herr,⟨C⟩,⟨U⟩⟩ := ht
+  have hm : 4 ≤ sourceInitialM (N t) := (by norm_num : 4 ≤ 128).trans H.m_large
+  have hlen := sourceInitialWord_length (q:=q t) hm (-1)
+  have hlen' := sourceInitialWord_length (q:=q t) hm 1
+  have hh := sourceInitialCount_bounds (hq t) H
+  have hlower : N t/2 ≤ ((sourceInitialWord (q t) (N t) (-1)).length:ℝ) := by
+    rw [hlen,Nat.cast_mul]; exact H.length_lower
+  have hupper : ((sourceInitialWord (q t) (N t) (-1)).length:ℝ) ≤ N t := by
+    rw [hlen,Nat.cast_mul]; exact H.length_upper
+  have hell : (1:ℝ) ≤ (sourceInitialWord (q t) (N t) (-1)).length := by linarith only [hNt,hlower]
+  have hp : 0 < ((sourceInitialWord (q t) (N t) (-1)).length:ℝ) := by linarith only [hell]
+  refine ⟨C.toConcrete ⟨(hq t).1.le,by linarith [(hq t).2]⟩
+    (hlen.trans hlen'.symm) (sourceInitialWord_different H) hlower hupper ?_ (by norm_num)
+    (by simpa only [hlen] using hh.1) (by simpa only [hlen] using hh.2) ?_⟩
+  · simp only [Nat.cast_zero,zero_add,one_mul]
+    apply (div_le_iff₀ hp).mpr
+    exact herr.trans (le_mul_of_one_le_right (Real.rpow_nonneg hnt.le _) hell)
+  · exact source_initial_power_overlap (hq t) H (by linarith) hk1 C U hs
+
+/-- All backwards targets are no larger than n, with the source epsilon. -/
+theorem sourceScaleProduct_le_one (B D i : ℕ) (hB : 2 ≤ B) : sourceScaleProduct B D i ≤ 1 := by
+  apply Finset.prod_le_one₀
+  · intro j _
+    have hdim : sourceDimension B j ≤ B := by unfold sourceDimension; split_ifs <;> omega
+    have he := sourceEpsilon_pos B D
+    have hh := sourceEpsilon_small B D
+    have hp : (1:ℝ) ≤ 2^B := one_le_pow₀ (by norm_num)
+    have hd : 0 ≤ (D:ℝ) := Nat.cast_nonneg _
+    have hu : (2:ℝ)^B*sourceEpsilon B D ≤ 1/2 := by
+      nlinarith only [hh,mul_nonneg hd (mul_nonneg (le_trans zero_le_one hp) he.le)]
+    have hx := scaleA_zero_lower (sourceDimension B j) B (sourceEpsilon B D) hdim he.le
+    apply le_trans _ hx
+    exact mul_nonneg (by positivity) (by nlinarith only [hu,he])
+  · intro j _
+    have he := sourceEpsilon_pos B D
+    have hp : (1:ℝ) ≤ 2^(sourceDimension B j) := one_le_pow₀ (by norm_num)
+    unfold scaleA
+    norm_num only [pow_zero,mul_one]
+    have hh := (div_le_iff₀ (by positivity : (0:ℝ)<2^(sourceDimension B j))).mpr
+      (show 1+sourceEpsilon B D ≤ (1+sourceEpsilon B D)*2^(sourceDimension B j) from
+        le_mul_of_one_le_right (by linarith only [he]) hp)
+    linarith only [hh]
+
+theorem sourceTargetScale_le (B D i : ℕ) (n : ℝ) (hB : 2 ≤ B) (hn : 1 ≤ n) :
+    sourceTargetScale B D n i ≤ n := by
+  simpa only [sourceTargetScale,Real.rpow_one] using Real.rpow_le_rpow_of_exponent_le hn (sourceScaleProduct_le_one B D i hB)
+
+theorem sourceHalfMoment_sufficient (B D i : ℕ) {n : ℝ} (hB : 2 ≤ B) (hn : 1 < n) :
+    10^6*(totalScoreExponent B D+B+1)*Real.log n ≤
+      (sourceHalfMoment B D n i:ℝ)*Real.log (sourceTargetScale B D n i) := by
+  have hlo := sourceScaleProduct_lower B D i hB
+  have hs : 0 < sourceScaleProduct B D i := lt_of_lt_of_le (by positivity) hlo
+  have hl : 0 < Real.log (sourceTargetScale B D n i) := by
+    rw [sourceTargetScale,Real.log_rpow (by linarith : 0<n)]
+    exact mul_pos hs (Real.log_pos hn)
+  have hk := (div_le_iff₀ hl).mp (Nat.le_ceil
+    ((10^8:ℝ)*2^(10*B)*(totalScoreExponent B D+B+1)*Real.log n/
+      Real.log (sourceTargetScale B D n i)))
+  apply le_trans _ hk
+  have hd := (totalScoreExponent_bounds B D hB).1
+  have hA : 0 ≤ (totalScoreExponent B D+B+1)*Real.log n := by positivity [hd,Real.log_pos hn]
+  have hp : (1:ℝ) ≤ 2^(10*B) := one_le_pow₀ (by norm_num)
+  have hc : (10^6:ℝ) ≤ 10^8*2^(10*B) := by linarith only [hp]
+  nlinarith only [mul_le_mul_of_nonneg_right hc hA]
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open Filter
+open scoped Topology Classical
+
+/-- The source base invariant, with literal target scales, moment ceiling,
+ error threshold, count and overlap, uniformly in the joint deletion regime. -/
+theorem source_initial_at_smallest_scale (B : ℕ) (ζ : ℝ) (hB : 2 ≤ B)
+    (hζ : ζ∈Set.Ioo 0 1) {n q : ℕ→ℝ} (hn : Tendsto n atTop atTop)
+    (hq : ∀t,q t∈Set.Ioc 0 (1/2:ℝ))
+    (hH : Tendsto (fun t=>(q t)^3*Real.log (n t)) atTop atTop) :
+    ∀ᶠt in atTop,Nonempty (ConcreteLevelPair (q t) ⟨(hq t).1.le,by linarith [(hq t).2]⟩
+      (sourceTargetScale B (sourceDepth B ζ ((q t)^3*Real.log (n t))) (n t) 0)
+      ((n t)^(-100*(totalScoreExponent B (sourceDepth B ζ ((q t)^3*Real.log (n t)))+B+1)))
+      (11/5) (initialOverlapLower (q t)) ((sourceOverlapConstant (q t))^2) 0
+      (sourceHalfMoment B (sourceDepth B ζ ((q t)^3*Real.log (n t))) (n t) 0)) := by
+  let H := fun t=>(q t)^3*Real.log (n t)
+  let Λ := fun t=>(q t)^(-3:ℤ)
+  have hΛ (t) : 1 ≤ Λ t := by
+    dsimp [Λ]
+    rw [zpow_neg,zpow_ofNat,one_le_inv₀ (pow_pos (hq t).1 _)]
+    exact pow_le_one₀ (hq t).1.le (by linarith [(hq t).2])
+  have hrel (t) : Real.log (n t)=Λ t*H t := by
+    dsimp [Λ,H]
+    rw [zpow_neg,zpow_ofNat]
+    field_simp [(hq t).1.ne']
+  let D := fun t=>sourceDepth B ζ (H t)
+  let N := fun t=>sourceTargetScale B (D t) (n t) 0
+  have hN := sourceTargetScale_tendsto B ζ hB hζ (fun _=>0) hn hH hΛ hrel
+  have hl := source_uniform_log_absorption B 0 ζ hB hζ (fun _=>0) hn hH hΛ hrel
+  simp only [pow_zero,one_mul] at hl
+  have hk := sourceHalfMoment_rate B ζ hB hζ (fun _=>0) hn hH hΛ hrel
+  apply source_initial_level_eventually hN hn _ hq hl hk
+  · filter_upwards [hn.eventually_gt_atTop 1] with t ht
+    exact (sourceHalfMoment_bounds B (D t) 0 hB ht).1
+  · filter_upwards [hn.eventually_gt_atTop 1] with t ht
+    exact sourceHalfMoment_sufficient B (D t) 0 hB ht
+  · filter_upwards [hn.eventually_ge_atTop 1] with t ht
+    exact sourceTargetScale_le B (D t) 0 (n t) hB ht
+  · filter_upwards [] with t
+    have hd := (totalScoreExponent_bounds B (D t) hB).1
+    have hbn : (0:ℝ) ≤ B := Nat.cast_nonneg _
+    linarith only [hd,hbn]
+
+end TraceReconstruction
+
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/TraceReconstruction/Kernels.lean b/lean/OAI/Probability/TraceReconstruction/Kernels.lean
index ce904ef39cafd87bce0942b20dc3911f137456a0..cd6d91896749232dd8ff0650db60905059493d3b 100644
--- a/lean/OAI/Probability/TraceReconstruction/Kernels.lean
+++ b/lean/OAI/Probability/TraceReconstruction/Kernels.lean
@@ -1,649 +1,649 @@
-import OAI.Probability.TraceReconstruction.Binomial
-
-namespace OAI
-
-/-!
-# Kernels
-
-Constrained rank-gap sums, correlated anchor kernels, window kernels, lattice Laplace sums, and geometric degree bounds.
--/
-
-open scoped ENNReal
-noncomputable section
-
-noncomputable section
-namespace TraceReconstruction
-
-/-- Eliminate one output gap by its prescribed total before dropping the
-other rank constraints. This is the exact finite summation step used in
-the conditional rank-kernel estimate. -/
-theorem constrained_sum_product_le (r L : ℕ)
-    (f : Fin (r + 1) → Fin (L + 1) → ℝ)
-    (hf : ∀ i k, 0 ≤ f i k) (M : ℝ) (hM : 0 ≤ M)
-    (hbound : ∀ k, f 0 k ≤ M) :
-    (∑ z : Fin (r + 1) → Fin (L + 1),
-      if ∑ j, (z j).val = L then ∏ j, f j (z j) else 0) ≤
-        M * ∏ i : Fin r, ∑ k, f i.succ k := by
-  classical
-  rw [← (Fin.consEquiv (fun _ : Fin (r + 1) => Fin (L + 1))).sum_comp]
-  rw [Fintype.sum_prod_type, Finset.sum_comm]
-  have hpoint (z : Fin r → Fin (L + 1)) :
-      (∑ k : Fin (L + 1),
-        if k.val + ∑ i, (z i).val = L then f 0 k * ∏ i, f i.succ (z i) else 0) ≤
-          M * ∏ i, f i.succ (z i) := by
-    by_cases hz : ∑ i, (z i).val ≤ L
-    · let k₀ : Fin (L + 1) := ⟨L - ∑ i, (z i).val, by omega⟩
-      have heq (k : Fin (L + 1)) : k.val + ∑ i, (z i).val = L ↔ k = k₀ := by
-        change _ ↔ k = ⟨L - ∑ i, (z i).val, _⟩
-        rw [Fin.ext_iff]
-        simp only
-        omega
-      simp_rw [heq]
-      rw [Fintype.sum_ite_eq']
-      exact mul_le_mul_of_nonneg_right (hbound k₀) (Finset.prod_nonneg fun i _ => hf _ _)
-    · have heq (k : Fin (L + 1)) : ¬ (k.val + ∑ i, (z i).val = L) := by omega
-      simp_rw [ite_eq_right (heq _), Finset.sum_const_zero]
-      exact mul_nonneg hM (Finset.prod_nonneg fun i _ => hf _ _)
-  calc
-    _ ≤ ∑ z : Fin r → Fin (L + 1), M * ∏ i, f i.succ (z i) := by
-      apply Finset.sum_le_sum
-      intro z _
-      simpa only [Fin.consEquiv_apply, Fin.sum_univ_succ, Fin.prod_univ_succ,
-        Fin.cons_zero, Fin.cons_succ] using hpoint z
-    _ = _ := by
-      rw [← Finset.mul_sum, ← Fintype.prod_sum (fun (i : Fin r) (k : Fin (L + 1)) => f i.succ k)]
-
-/-- The product of two rank kernels has one scalar denominator factor,
-one uniform gap factor and independent one-dimensional cross-overlaps. -/
-theorem constrained_cross_kernel_le (r L : ℕ)
-    (f g : Fin (r + 1) → Fin (L + 1) → ℝ)
-    (hf : ∀ i k, 0 ≤ f i k) (hg : ∀ i k, 0 ≤ g i k)
-    (M : ℝ) (hM : 0 ≤ M) (hbound : ∀ k, f 0 k * g 0 k ≤ M) :
-    (∑ z : Fin (r + 1) → Fin (L + 1),
-      if ∑ j, (z j).val = L then (∏ j, f j (z j)) * (∏ j, g j (z j)) else 0) ≤
-        M * ∏ i : Fin r, ∑ k, f i.succ k * g i.succ k := by
-  simp_rw [← Finset.prod_mul_distrib]
-  exact constrained_sum_product_le r L (fun i k => f i k * g i k)
-    (fun i k => mul_nonneg (hf i k) (hg i k)) M hM hbound
-
-end TraceReconstruction
-
-namespace TraceReconstruction
-
-/-! ## Correlated sibling-anchor estimate
-
-This is the analytic summation step in the correlated-anchor estimate, subsection
-“Summing the correlated anchor kernels”. The common latent draw is kept
-inside the joint kernel, so independence of the two displaced locations
-is NOT assumed. Conditional independence only factors the two local
-kernels for each fixed common draw. All arrays here have finite support;
-the integer-jitter application uses a finite interval containing that support.
--/
-
-section CorrelatedAnchors
-
-variable {Ω I J : Type*} [Fintype Ω] [Fintype I] [Fintype J]
-
-/-- Mixture of conditional products; `p` is the common displacement law. -/
-def jointKernel (p : Ω → ℝ) (a : Ω → I → ℝ) (b : Ω → J → ℝ)
-    (i : I) (j : J) : ℝ := ∑ w, p w * a w i * b w j
-
-omit [Fintype I] [Fintype J] in
-theorem jointKernel_nonneg
-    (p : Ω → ℝ) (a : Ω → I → ℝ) (b : Ω → J → ℝ)
-    (hp : ∀ w, 0 ≤ p w) (ha : ∀ w i, 0 ≤ a w i)
-    (hb : ∀ w j, 0 ≤ b w j) (i : I) (j : J) :
-    0 ≤ jointKernel p a b i j := by
-  exact Finset.sum_nonneg fun w _ => mul_nonneg (mul_nonneg (hp w) (ha w i)) (hb w j)
-
-theorem sum_jointKernel
-    (p : Ω → ℝ) (a : Ω → I → ℝ) (b : Ω → J → ℝ) (L M : ℝ)
-    (hp : ∑ w, p w = 1)
-    (ha : ∀ w, ∑ i, a w i = L) (hb : ∀ w, ∑ j, b w j = M) :
-    ∑ i, ∑ j, jointKernel p a b i j = L * M := by
-  simp only [jointKernel]
-  have hcomm (i : I) :
-      (∑ j : J, ∑ w : Ω, p w * a w i * b w j) =
-        ∑ w : Ω, ∑ j : J, p w * a w i * b w j := Finset.sum_comm
-  simp_rw [hcomm]
-  simp_rw [← Finset.mul_sum, hb]
-  rw [Finset.sum_comm]
-  simp_rw [← Finset.sum_mul, ← Finset.mul_sum, ha]
-  rw [← Finset.sum_mul, hp, one_mul]
-
-omit [Fintype I] [Fintype J] in
-theorem jointKernel_le
-    (p : Ω → ℝ) (a : Ω → I → ℝ) (b : Ω → J → ℝ) (A B : ℝ)
-    (hp : ∀ w, 0 ≤ p w) (ha : ∀ w i, 0 ≤ a w i)
-    (hB : 0 ≤ B) (hb : ∀ w j, b w j ≤ B)
-    (hA : ∀ i, ∑ w, p w * a w i ≤ A) (i : I) (j : J) :
-    jointKernel p a b i j ≤ A * B := by
-  calc
-    jointKernel p a b i j ≤ ∑ w, (p w * a w i) * B := by
-      exact Finset.sum_le_sum fun w _ =>
-        mul_le_mul_of_nonneg_left (hb w j) (mul_nonneg (hp w) (ha w i))
-    _ = (∑ w, p w * a w i) * B := (Finset.sum_mul _ _ _).symm
-    _ ≤ A * B := mul_le_mul_of_nonneg_right (hA i) hB
-
-/-- The joint squared-sum estimate retaining the common jitter. In the
-    manuscript A=L/(omega*sqrt(T*t)) and B=M/(omega*sqrt(t)),
-    up to the explicit absolute binomial constants. -/
-theorem correlated_anchor_square_sum
-    (p : Ω → ℝ) (a : Ω → I → ℝ) (b : Ω → J → ℝ)
-    (L M A B : ℝ)
-    (hp0 : ∀ w, 0 ≤ p w) (hp1 : ∑ w, p w = 1)
-    (ha0 : ∀ w i, 0 ≤ a w i) (hb0 : ∀ w j, 0 ≤ b w j)
-    (ha1 : ∀ w, ∑ i, a w i = L) (hb1 : ∀ w, ∑ j, b w j = M)
-    (hB : 0 ≤ B) (hb : ∀ w j, b w j ≤ B)
-    (hA : ∀ i, ∑ w, p w * a w i ≤ A) :
-    ∑ i, ∑ j, (jointKernel p a b i j) ^ 2 ≤ (A * B) * (L * M) := by
-  calc
-    ∑ i, ∑ j, (jointKernel p a b i j) ^ 2 ≤
-        ∑ i, ∑ j, (A * B) * jointKernel p a b i j := by
-      apply Finset.sum_le_sum
-      intro i _
-      apply Finset.sum_le_sum
-      intro j _
-      have h0 := jointKernel_nonneg p a b hp0 ha0 hb0 i j
-      have h1 := jointKernel_le p a b A B hp0 ha0 hB hb hA i j
-      nlinarith
-    _ = (A * B) * (∑ i, ∑ j, jointKernel p a b i j) := by
-      simp_rw [← Finset.mul_sum]
-    _ = (A * B) * (L * M) := by rw [sum_jointKernel p a b L M hp1 ha1 hb1]
-
-end CorrelatedAnchors
-
-
-end TraceReconstruction
-
-namespace TraceReconstruction
-
-/-- Truncating output ranks cannot increase a binomial cross-overlap. -/
-theorem binomial_overlap_truncate (g h L : ℕ) {p : ℝ}
-    (hp : p ∈ Set.Icc 0 1) :
-    (∑ k : Fin (L + 1), binomialMass g p k * binomialMass h p k) ≤
-      ∑ k ∈ Finset.range (h + 1), binomialMass g p k * binomialMass h p k := by
-  rw [Fin.sum_univ_eq_sum_range (fun k => binomialMass g p k * binomialMass h p k) (L + 1)]
-  by_cases hL : L ≤ h
-  · apply Finset.sum_le_sum_of_subset_of_nonneg (Finset.range_mono (by omega))
-    intro k _ _
-    exact mul_nonneg (binomialMass_nonneg g k hp) (binomialMass_nonneg h k hp)
-  · apply le_of_eq
-    symm
-    apply Finset.sum_subset (Finset.range_mono (by omega))
-    intro k _ hk
-    have hhk : h < k := by
-      simp only [Finset.mem_range, not_lt] at hk
-      omega
-    simp [binomialMass, Nat.choose_eq_zero_of_lt hhk]
-
-/-- Gaussian cross-overlap with the prefactor attached to either prescribed
-first length, removing the order restriction in the tilting proof. -/
-theorem binomial_gaussian_overlap_general (g h L : ℕ) {p : ℝ}
-    (hp : p ∈ Set.Ico (1 / 2 : ℝ) 1) (hS : 0 < g + h) :
-    (∑ k : Fin (L + 1), binomialMass g p k * binomialMass h p k) ≤
-      Real.sqrt (2 / (((g : ℝ) + 1) * p * (1 - p))) *
-        Real.exp (-((g : ℝ) - h) ^ 2 / (16 * ((g : ℝ) + h))) := by
-  have hp0 : 0 < p := by linarith [hp.1]
-  have hpq : 0 < p * (1 - p) := mul_pos hp0 (sub_pos.mpr hp.2)
-  by_cases hhg : h ≤ g
-  · exact (binomial_overlap_truncate g h L ⟨hp0.le, hp.2.le⟩).trans
-      (binomial_gaussian_overlap g h hp hhg hS)
-  · have hgh : g ≤ h := by omega
-    have ho := (binomial_overlap_truncate h g L ⟨hp0.le, hp.2.le⟩).trans
-      (binomial_gaussian_overlap h g hp hgh (by omega))
-    have hb : Real.sqrt (2 / (((h : ℝ) + 1) * p * (1 - p))) ≤
-        Real.sqrt (2 / (((g : ℝ) + 1) * p * (1 - p))) := by
-      apply Real.sqrt_le_sqrt
-      apply div_le_div_of_nonneg_left (by norm_num)
-        (by nlinarith only [mul_pos (show (0 : ℝ) < (g : ℝ) + 1 by positivity) hpq])
-      have hcast : (g : ℝ) ≤ h := by exact_mod_cast hgh
-      have ht := mul_le_mul_of_nonneg_right (show (g : ℝ) + 1 ≤ (h : ℝ) + 1 by linarith) hpq.le
-      nlinarith only [ht]
-    have heq : -((h : ℝ) - g) ^ 2 / (16 * ((h : ℝ) + g)) =
-        -((g : ℝ) - h) ^ 2 / (16 * ((g : ℝ) + h)) := by ring
-    rw [heq] at ho
-    exact (by simpa only [mul_comm] using ho :
-      (∑ k : Fin (L + 1), binomialMass g p k * binomialMass h p k) ≤ _).trans
-        (mul_le_mul_of_nonneg_right hb (Real.exp_nonneg _))
-
-/-- The exact conditional rank kernel, including the fixed count
-normalization, before any coarse-gap estimates or restrictions. -/
-def rankKernel (r n l : ℕ) (p : ℝ) (g : Fin (r + 1) → ℕ)
-    (z : Fin (r + 1) → Fin (l + 1)) : ℝ :=
-  if r ≤ l ∧ ∑ j, (z j).val = l - r then
-    p ^ r / binomialMass n p l * ∏ j, binomialMass (g j) p (z j) else 0
-
-/-- Largest-gap elimination for the conditional rank kernel. The displayed
-constant is absolute; no asymptotic proposal or Stirling theorem is assumed. -/
-theorem rankKernel_cross_le (r n l : ℕ) {p : ℝ}
-    (hp : p ∈ Set.Ico (1 / 2 : ℝ) 1) (hl : l ≤ n)
-    (hmean : (n : ℝ) * p = l) (hv : 1 ≤ (n : ℝ) * p * (1 - p))
-    (g h : Fin (r + 1) → ℕ) :
-    (∑ z : Fin (r + 1) → Fin (l - r + 1),
-      if ∑ j, (z j).val = l - r then
-        (p ^ r / binomialMass n p l) ^ 2 *
-          (∏ j, binomialMass (g j) p (z j)) *
-          (∏ j, binomialMass (h j) p (z j)) else 0) ≤
-      (4608 * n / ((g 0 : ℝ) + 1)) *
-        ∏ i : Fin r, ∑ k : Fin (l - r + 1),
-          binomialMass (g i.succ) p k * binomialMass (h i.succ) p k := by
-  have hp0 : 0 < p := by linarith [hp.1]
-  have hpq : 0 < p * (1 - p) := mul_pos hp0 (sub_pos.mpr hp.2)
-  have hden : 0 < ((g 0 : ℝ) + 1) * p * (1 - p) := by
-    nlinarith only [mul_pos (show (0 : ℝ) < (g 0 : ℝ) + 1 by positivity) hpq]
-  have hc : (p ^ r / binomialMass n p l) ^ 2 ≤ 144 * ((n : ℝ) * p * (1 - p)) := by
-    rw [div_eq_mul_inv, mul_pow]
-    apply (mul_le_mul_of_nonneg_right
-      (show (p ^ r) ^ 2 ≤ (1 : ℝ) by
-        exact pow_le_one₀ (pow_nonneg hp0.le _) (pow_le_one₀ hp0.le hp.2.le))
-      (sq_nonneg _)).trans
-    simpa only [one_mul] using binomialMass_mean_inverse_sq n l ⟨hp0, hp.2⟩ hl hmean hv
-  have hx := constrained_cross_kernel_le r (l - r)
-    (fun i k => binomialMass (g i) p k) (fun i k => binomialMass (h i) p k)
-    (fun i k => binomialMass_nonneg _ _ ⟨hp0.le, hp.2.le⟩)
-    (fun i k => binomialMass_nonneg _ _ ⟨hp0.le, hp.2.le⟩)
-    (32 / (((g 0 : ℝ) + 1) * p * (1 - p))) (by positivity)
-    (fun k => binomial_product_sup (g 0) (h 0) k hp)
-  have hprod : 0 ≤ ∏ i : Fin r, ∑ k : Fin (l - r + 1),
-      binomialMass (g i.succ) p k * binomialMass (h i.succ) p k := by
-    apply Finset.prod_nonneg
-    intro i _
-    exact Finset.sum_nonneg fun k _ => mul_nonneg
-      (binomialMass_nonneg _ _ ⟨hp0.le, hp.2.le⟩)
-      (binomialMass_nonneg _ _ ⟨hp0.le, hp.2.le⟩)
-  calc
-    _ = (p ^ r / binomialMass n p l) ^ 2 *
-      (∑ z : Fin (r + 1) → Fin (l - r + 1),
-        if ∑ j, (z j).val = l - r then
-          (∏ j, binomialMass (g j) p (z j)) *
-          (∏ j, binomialMass (h j) p (z j)) else 0) := by
-      rw [Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro z _
-      split_ifs <;> ring
-    _ ≤ (p ^ r / binomialMass n p l) ^ 2 *
-      ((32 / (((g 0 : ℝ) + 1) * p * (1 - p))) * _) :=
-      mul_le_mul_of_nonneg_left hx (sq_nonneg _)
-    _ ≤ (144 * ((n : ℝ) * p * (1 - p))) *
-      ((32 / (((g 0 : ℝ) + 1) * p * (1 - p))) * _) :=
-      mul_le_mul_of_nonneg_right hc (mul_nonneg (by positivity) hprod)
-    _ = _ := by
-      have hq : 1 - p ≠ 0 := ne_of_gt (sub_pos.mpr hp.2)
-      field_simp
-      ring
-
-end TraceReconstruction
-
-namespace TraceReconstruction
-theorem fairSeries_total (n : ℕ) : ∑' z : ℤ, (fairSeries n).coeff z = 1 := by
-  have h := fairSeries_L1 n
-  unfold seriesL1 at h
-  simpa only [abs_of_nonneg (fairSeries_nonneg n _)] using h
-
-theorem fairSeries_sum_finite_le_one (n : ℕ) (A : Finset ℤ) (c : ℤ) :
-    ∑ z ∈ A, (fairSeries n).coeff (c-z) ≤ 1 := by
-  have hs : Summable (fun z : ℤ => (fairSeries n).coeff (c-z)) := by
-    exact (Equiv.subLeft c).summable_iff.mpr (series_summable (fairSeries n) id rfl)
-  have hh : (∑' z : ℤ, (fairSeries n).coeff (c-z)) = 1 := by
-    simpa using ((Equiv.subLeft c).tsum_eq (fun z => (fairSeries n).coeff z)).trans
-      (fairSeries_total n)
-  rw [← hh]
-  exact Summable.sum_le_tsum A (fun _ _ => fairSeries_nonneg n _) hs
-
-def windowKernel (n L : ℕ) (z : ℤ) : ℝ :=
-  ∑ a ∈ Finset.range L, (fairSeries n).coeff ((a:ℤ)-z)
-
-theorem windowKernel_nonneg (n L : ℕ) (z : ℤ) : 0 ≤ windowKernel n L z :=
-  Finset.sum_nonneg (fun _ _ => fairSeries_nonneg n _)
-
-theorem windowKernel_sup (n L : ℕ) (z : ℤ) :
-    windowKernel n L z ≤ L*binomialBound n := by
-  calc
-    _ ≤ ∑ _a ∈ Finset.range L, binomialBound n := by
-      apply Finset.sum_le_sum
-      intro a _
-      exact (le_abs_self _).trans (fairSeries_coeff_bound n _)
-    _ = _ := by simp
-
-theorem windowKernel_sum (n L : ℕ) (A : Finset ℤ) (c : ℤ) :
-    ∑ z ∈ A, windowKernel n L (z-c) ≤ L := by
-  unfold windowKernel
-  rw [Finset.sum_comm]
-  calc
-    _ ≤ ∑ _a ∈ Finset.range L, (1:ℝ) := by
-      apply Finset.sum_le_sum
-      intro a _
-      simpa only [show ∀ z : ℤ, (a:ℤ)-(z-c)=(a:ℤ)+c-z from fun z => by omega]
-        using fairSeries_sum_finite_le_one n A ((a:ℤ)+c)
-    _ = _ := by simp
-
-theorem windowKernel_energy (n L : ℕ) (A : Finset ℤ) (c : ℤ) :
-    ∑ z ∈ A, windowKernel n L (z-c)^2 ≤ L^2*binomialBound n := by
-  calc
-    _ ≤ ∑ z ∈ A, windowKernel n L (z-c)*(L*binomialBound n) := by
-      apply Finset.sum_le_sum
-      intro z _
-      simpa only [pow_two] using mul_le_mul_of_nonneg_left
-        (windowKernel_sup n L (z-c)) (windowKernel_nonneg n L (z-c))
-    _ = (∑ z ∈ A, windowKernel n L (z-c))*(L*binomialBound n) := by rw [Finset.sum_mul]
-    _ ≤ L*(L*binomialBound n) := mul_le_mul_of_nonneg_right
-      (windowKernel_sum n L A c) (by positivity [binomialBound_nonneg n])
-    _ = _ := by ring
-
-def jointWindowKernel (g n₁ n₂ L : ℕ) (z₁ z₂ : ℤ) : ℝ :=
-  ∑ v ∈ (fairSeries g).coeff.support,
-    (fairSeries g).coeff v * windowKernel n₁ L (z₁+v) * windowKernel n₂ L (z₂+v)
-
-theorem windowKernel_average (g n L : ℕ) (z : ℤ) :
-    (∑ v ∈ (fairSeries g).coeff.support, (fairSeries g).coeff v * windowKernel n L (z+v)) =
-      windowKernel (g+n) L z := by
-  unfold windowKernel
-  simp_rw [Finset.mul_sum]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro a _
-  rw [fairSeries_add, AddMonoidAlgebra.coeff_mul_apply_left, Finsupp.sum]
-  apply Finset.sum_congr rfl
-  intro v _
-  congr 2
-  omega
-
-theorem jointWindowKernel_nonneg (g n₁ n₂ L : ℕ) (z₁ z₂ : ℤ) :
-    0 ≤ jointWindowKernel g n₁ n₂ L z₁ z₂ := by
-  apply Finset.sum_nonneg
-  intro v _
-  exact mul_nonneg (mul_nonneg (fairSeries_nonneg g v) (windowKernel_nonneg n₁ L _))
-    (windowKernel_nonneg n₂ L _)
-
-theorem jointWindowKernel_sup (g n₁ n₂ L : ℕ) (z₁ z₂ : ℤ) :
-    jointWindowKernel g n₁ n₂ L z₁ z₂ ≤ L^2*binomialBound g*binomialBound n₂ := by
-  have hbound : binomialBound (g+n₁) ≤ binomialBound g := by
-    apply Real.sqrt_le_sqrt
-    apply div_le_div_of_nonneg_left (by norm_num) (by positivity)
-    simp only [Nat.cast_add]; linarith [show (0:ℝ) ≤ n₁ by positivity]
-  calc
-    _ ≤ ∑ v ∈ (fairSeries g).coeff.support,
-        (fairSeries g).coeff v * windowKernel n₁ L (z₁+v) * (L*binomialBound n₂) := by
-      apply Finset.sum_le_sum
-      intro v _
-      exact mul_le_mul_of_nonneg_left (windowKernel_sup n₂ L _) (mul_nonneg
-        (fairSeries_nonneg g v) (windowKernel_nonneg n₁ L _))
-    _ = windowKernel (g+n₁) L z₁ * (L*binomialBound n₂) := by
-      rw [← Finset.sum_mul, windowKernel_average]
-    _ ≤ (L*binomialBound g)*(L*binomialBound n₂) := by
-      apply mul_le_mul_of_nonneg_right _ (mul_nonneg (by positivity) (binomialBound_nonneg n₂))
-      exact (windowKernel_sup (g+n₁) L z₁).trans
-        (mul_le_mul_of_nonneg_left hbound (by positivity))
-    _ = _ := by ring
-
-theorem jointWindowKernel_sum (g n₁ n₂ L : ℕ) (A B : Finset ℤ) (c₁ c₂ : ℤ) :
-    ∑ z₁ ∈ A, ∑ z₂ ∈ B, jointWindowKernel g n₁ n₂ L (z₁-c₁) (z₂-c₂) ≤ L^2 := by
-  unfold jointWindowKernel
-  simp_rw [Finset.sum_comm (s := B)]
-  rw [Finset.sum_comm]
-  calc
-    _ = ∑ v ∈ (fairSeries g).coeff.support, (fairSeries g).coeff v *
-        (∑ z₁ ∈ A, windowKernel n₁ L ((z₁-c₁)+v)) *
-        (∑ z₂ ∈ B, windowKernel n₂ L ((z₂-c₂)+v)) := by
-      apply Finset.sum_congr rfl
-      intro v _
-      simp_rw [Finset.mul_sum, Finset.sum_mul]
-      rw [Finset.sum_comm]
-    _ ≤ ∑ v ∈ (fairSeries g).coeff.support, (fairSeries g).coeff v * (L*L) := by
-      apply Finset.sum_le_sum
-      intro v _
-      have h1 : (∑ z₁ ∈ A, windowKernel n₁ L ((z₁-c₁)+v)) ≤ L := by
-        simpa only [show ∀ z : ℤ, (z-c₁)+v = z-(c₁-v) from fun z => by omega]
-          using windowKernel_sum n₁ L A (c₁-v)
-      have h2 : (∑ z₂ ∈ B, windowKernel n₂ L ((z₂-c₂)+v)) ≤ L := by
-        simpa only [show ∀ z : ℤ, (z-c₂)+v = z-(c₂-v) from fun z => by omega]
-          using windowKernel_sum n₂ L B (c₂-v)
-      rw [mul_assoc]
-      apply mul_le_mul_of_nonneg_left _ (fairSeries_nonneg g v)
-      exact mul_le_mul h1 h2 (Finset.sum_nonneg (fun z _ => windowKernel_nonneg n₂ L _)) (by positivity)
-    _ = L^2 := by
-      rw [← Finset.sum_mul]
-      have ht : (∑ v ∈ (fairSeries g).coeff.support, (fairSeries g).coeff v) = 1 := by
-        have h := fairSeries_L1 g
-        rw [seriesL1_eq_sum] at h
-        simpa only [abs_of_nonneg (fairSeries_nonneg g _)] using h
-      rw [ht]; ring
-
-theorem jointWindowKernel_energy (g n₁ n₂ L : ℕ) (A B : Finset ℤ) (c₁ c₂ : ℤ) :
-    ∑ z₁ ∈ A, ∑ z₂ ∈ B, (jointWindowKernel g n₁ n₂ L (z₁-c₁) (z₂-c₂))^2 ≤
-      L^4*binomialBound g*binomialBound n₂ := by
-  calc
-    _ ≤ ∑ z₁ ∈ A, ∑ z₂ ∈ B, jointWindowKernel g n₁ n₂ L (z₁-c₁) (z₂-c₂) *
-        (L^2*binomialBound g*binomialBound n₂) := by
-      apply Finset.sum_le_sum
-      intro z₁ _
-      apply Finset.sum_le_sum
-      intro z₂ _
-      simpa only [pow_two] using mul_le_mul_of_nonneg_left
-        (jointWindowKernel_sup g n₁ n₂ L _ _) (jointWindowKernel_nonneg g n₁ n₂ L _ _)
-    _ = (∑ z₁ ∈ A, ∑ z₂ ∈ B, jointWindowKernel g n₁ n₂ L (z₁-c₁) (z₂-c₂)) *
-        (L^2*binomialBound g*binomialBound n₂) := by simp_rw [Finset.sum_mul]
-    _ ≤ L^2*(L^2*binomialBound g*binomialBound n₂) := mul_le_mul_of_nonneg_right
-      (jointWindowKernel_sum g n₁ n₂ L A B c₁ c₂)
-      (mul_nonneg (mul_nonneg (by positivity) (binomialBound_nonneg g)) (binomialBound_nonneg n₂))
-    _ = _ := by ring
-end TraceReconstruction
-
-open Filter
-open scoped Topology
-
-
-namespace TraceReconstruction
-
-/-- Elementary discrete integral comparison, including the singular first cell. -/
-theorem sum_inverse_sqrt (N : ℕ) :
-    (∑ k ∈ Finset.range N, (Real.sqrt (k+1:ℝ))⁻¹) ≤ 2*Real.sqrt N := by
-  induction N with
-  | zero => simp
-  | succ N ih =>
-    rw [Finset.sum_range_succ]
-    have hn : 0 ≤ (N:ℝ) := by positivity
-    have hp : 0 < Real.sqrt (N+1:ℝ) := Real.sqrt_pos.2 (by positivity)
-    have hs : Real.sqrt (N:ℝ) ≤ Real.sqrt (N+1:ℝ) := Real.sqrt_le_sqrt (by linarith)
-    have h1 := Real.sq_sqrt hn
-    have h2 := Real.sq_sqrt (show 0 ≤ (N+1:ℝ) by positivity)
-    have hstep : (Real.sqrt (N+1:ℝ))⁻¹ ≤ 2*(Real.sqrt (N+1:ℝ)-Real.sqrt N) := by
-      apply (inv_le_iff_one_le_mul₀ hp).2
-      nlinarith [sq_nonneg (Real.sqrt (N+1:ℝ)-Real.sqrt N)]
-    push_cast
-    linarith
-
-theorem exp_gap_lower {h : ℝ} (hh : 0 < h) (h1 : h ≤ 1) :
-    h/2 ≤ 1-Real.exp (-h) := by
-  have he := Real.add_one_le_exp h
-  have hp : 0 < Real.exp h := Real.exp_pos h
-  have hp' : 0 < 1+h := by linarith
-  have hinv : (Real.exp h)⁻¹ ≤ (1+h)⁻¹ := inv_anti₀ hp' (by simpa [add_comm] using he)
-  rw [Real.exp_neg]
-  have hi : (1+h)⁻¹*(1+h)=1 := inv_mul_cancel₀ (ne_of_gt hp')
-  nlinarith [mul_nonneg hh.le (sub_nonneg.mpr h1)]
-
-theorem sum_exp_decay {h : ℝ} (hh : 0 < h) (h1 : h ≤ 1) (M : ℕ) :
-    (∑ k ∈ Finset.range M, Real.exp (-h*(k:ℝ))) ≤ 2/h := by
-  have he : 0 ≤ Real.exp (-h) := (Real.exp_pos _).le
-  have he1 : Real.exp (-h) < 1 := Real.exp_lt_one_iff.mpr (by linarith)
-  have hf : (∑ k ∈ Finset.range M, (Real.exp (-h))^k) ≤ (1-Real.exp (-h))⁻¹ := by
-    rw [inv_eq_one_div, le_div_iff₀ (by linarith : 0 < 1-Real.exp (-h)), geom_sum_mul_neg]
-    exact sub_le_self _ (pow_nonneg he M)
-  have hi : (1-Real.exp (-h))⁻¹ ≤ 2/h := by
-    rw [inv_le_iff_one_le_mul₀ (by linarith : 0<1-Real.exp (-h))]
-    have hb := exp_gap_lower hh h1
-    have := mul_le_mul_of_nonneg_right hb (show 0 ≤ 2/h by positivity)
-    nlinarith [div_mul_cancel₀ (2:ℝ) hh.ne']
-  calc
-    _ = ∑ k ∈ Finset.range M, (Real.exp (-h))^k := by
-      apply Finset.sum_congr rfl
-      intro k _
-      rw [← Real.exp_nat_mul]
-      congr 1
-      ring
-    _ ≤ _ := hf.trans hi
-
-/-- The uniform lattice Laplace estimate used by the initial-pair construction, with an absolute constant. -/
-theorem coarse_laplace_sum {h : ℝ} (hh : 0 < h) (h1 : h ≤ 1) (M : ℕ) :
-    (∑ k ∈ Finset.range M, (Real.sqrt (k+1:ℝ))⁻¹ *
-      Real.exp (-h*(k+1:ℝ))) ≤ 6/Real.sqrt h := by
-  let N := ⌈1/h⌉₊
-  have hN : 1/h ≤ (N:ℝ) := Nat.le_ceil _
-  have hNup : (N:ℝ) < 1/h+1 := Nat.ceil_lt_add_one (by positivity)
-  have hs : 0 < Real.sqrt h := Real.sqrt_pos.2 hh
-  have hNsqrt : Real.sqrt N ≤ 2/Real.sqrt h := by
-    have hn : (N:ℝ)*h ≤ 2 := by
-      have hx := mul_le_mul_of_nonneg_right hNup.le hh.le
-      have hc : (1/h)*h=1 := div_mul_cancel₀ _ hh.ne'
-      nlinarith
-    have hprod : Real.sqrt N * Real.sqrt h ≤ 2 := by
-      have hsqh := Real.sq_sqrt hh.le
-      have hsqN := Real.sq_sqrt (show 0≤(N:ℝ) by positivity)
-      have hnon : 0≤Real.sqrt N*Real.sqrt h := by positivity
-      nlinarith [sq_nonneg (Real.sqrt N*Real.sqrt h)]
-    exact (le_div_iff₀ hs).2 hprod
-  let low := (Finset.range M).filter (fun k => k<N)
-  let high := (Finset.range M).filter (fun k => ¬k<N)
-  have hlo : (∑ k ∈ low, (Real.sqrt (k+1:ℝ))⁻¹ * Real.exp (-h*(k+1:ℝ))) ≤
-      2*Real.sqrt N := by
-    calc
-      _ ≤ ∑ k ∈ low, (Real.sqrt (k+1:ℝ))⁻¹ := by
-        apply Finset.sum_le_sum
-        intro k hk
-        exact mul_le_of_le_one_right (by positivity) (Real.exp_le_one_iff.mpr (by nlinarith [show (0:ℝ)≤k by positivity]))
-      _ ≤ ∑ k ∈ Finset.range N, (Real.sqrt (k+1:ℝ))⁻¹ := by
-        apply Finset.sum_le_sum_of_subset_of_nonneg
-        · intro k hk
-          exact Finset.mem_range.mpr (Finset.mem_filter.mp hk).2
-        · intros; positivity
-      _ ≤ _ := sum_inverse_sqrt N
-  have hhi : (∑ k ∈ high, (Real.sqrt (k+1:ℝ))⁻¹ * Real.exp (-h*(k+1:ℝ))) ≤
-      Real.sqrt h * (2/h) := by
-    calc
-      _ ≤ ∑ k ∈ high, Real.sqrt h * Real.exp (-h*(k:ℝ)) := by
-        apply Finset.sum_le_sum
-        intro k hk
-        have hkN : N ≤ k := Nat.le_of_not_gt (Finset.mem_filter.mp hk).2
-        have hkR : 1/h ≤ (k+1:ℝ) := hN.trans (by exact_mod_cast (by omega : N≤k+1))
-        have hinv : (Real.sqrt (k+1:ℝ))⁻¹ ≤ Real.sqrt h := by
-          apply (inv_le_iff_one_le_mul₀ (Real.sqrt_pos.2 (by positivity))).2
-          have hmul : 1 ≤ h*(k+1:ℝ) := by simpa [mul_comm] using (div_le_iff₀ hh).mp hkR
-          have hsqh := Real.sq_sqrt hh.le
-          have hsqk := Real.sq_sqrt (show 0≤(k+1:ℝ) by positivity)
-          nlinarith [mul_nonneg (Real.sqrt_nonneg h) (Real.sqrt_nonneg (k+1:ℝ))]
-        apply mul_le_mul hinv _ (Real.exp_pos _).le (Real.sqrt_nonneg _)
-        exact Real.exp_le_exp.mpr (by nlinarith)
-      _ ≤ ∑ k ∈ Finset.range M, Real.sqrt h * Real.exp (-h*(k:ℝ)) := by
-        apply Finset.sum_le_sum_of_subset_of_nonneg (Finset.filter_subset _ _)
-        intros; positivity
-      _ = Real.sqrt h * ∑ k ∈ Finset.range M, Real.exp (-h*(k:ℝ)) := by rw [Finset.mul_sum]
-      _ ≤ _ := mul_le_mul_of_nonneg_left (sum_exp_decay hh h1 M) (Real.sqrt_nonneg h)
-  have hsum : (∑ k ∈ low, (Real.sqrt (k+1:ℝ))⁻¹ * Real.exp (-h*(k+1:ℝ))) +
-      (∑ k ∈ high, (Real.sqrt (k+1:ℝ))⁻¹ * Real.exp (-h*(k+1:ℝ))) =
-      ∑ k ∈ Finset.range M, (Real.sqrt (k+1:ℝ))⁻¹ * Real.exp (-h*(k+1:ℝ)) :=
-    Finset.sum_filter_add_sum_filter_not _ _ _
-  have hcancel : Real.sqrt h*(2/h)=2/Real.sqrt h := by
-    have hsq := Real.sq_sqrt hh.le
-    field_simp
-    nlinarith
-  rw [← hsum]
-  rw [hcancel] at hhi
-  have heq : 2*(2/Real.sqrt h)+2/Real.sqrt h=6/Real.sqrt h := by ring
-  linarith
-
-
-/-- Exponential tilting removes the coarse-gap simplex constraint without a dimension loss. -/
-theorem coarse_simplex_sum (r M : ℕ) {h u : ℝ} (hh : 0<h) (h1 : h≤1) :
-    (∑ v : Fin r → Fin M, if (∑ i, ((v i).val+1:ℝ)) ≤ u then
-      ∏ i, (Real.sqrt ((v i).val+1:ℝ))⁻¹ else 0) ≤
-      Real.exp (h*u) * (6/Real.sqrt h)^r := by
-  have hev (v : Fin r → Fin M) :
-      ∏ i, Real.exp (-h*((v i).val+1:ℝ)) =
-        Real.exp (-h*∑ i, ((v i).val+1:ℝ)) := by
-    rw [← Real.exp_sum, Finset.mul_sum]
-  have hbound :
-      (∑ v : Fin r → Fin M, if (∑ i, ((v i).val+1:ℝ)) ≤ u then
-        ∏ i, (Real.sqrt ((v i).val+1:ℝ))⁻¹ else 0) ≤
-      Real.exp (h*u) * ∑ v : Fin r → Fin M,
-        ∏ i, ((Real.sqrt ((v i).val+1:ℝ))⁻¹ * Real.exp (-h*((v i).val+1:ℝ))) := by
-    rw [Finset.mul_sum]
-    apply Finset.sum_le_sum
-    intro v _
-    rw [Finset.prod_mul_distrib, hev]
-    split_ifs with hv
-    · have he : 1 ≤ Real.exp (h*u) * Real.exp (-h*∑ i, ((v i).val+1:ℝ)) := by
-        rw [← Real.exp_add]
-        apply Real.one_le_exp_iff.mpr
-        nlinarith
-      have hp : 0 ≤ ∏ i, (Real.sqrt ((v i).val+1:ℝ))⁻¹ := by positivity
-      nlinarith [mul_le_mul_of_nonneg_left he hp]
-    · positivity
-  apply hbound.trans
-  apply mul_le_mul_of_nonneg_left _ (Real.exp_pos _).le
-  rw [← Fintype.prod_sum (fun (_i : Fin r) (k : Fin M) =>
-    (Real.sqrt (k.val+1:ℝ))⁻¹ * Real.exp (-h*(k.val+1:ℝ)))]
-  have hc : (6/Real.sqrt h)^r = ∏ _i : Fin r, (6/Real.sqrt h) := by simp [div_pow]
-  rw [hc]
-  apply Finset.prod_le_prod₀
-  · intros; positivity
-  · intro i _
-    rw [Fin.sum_univ_eq_sum_range (fun k : ℕ =>
-      (Real.sqrt (k+1:ℝ))⁻¹ * Real.exp (-h*(k+1:ℝ))) M]
-    exact coarse_laplace_sum hh h1 M
-
-end TraceReconstruction
-
-
-namespace TraceReconstruction
-
-/-- The degree-series estimate in the common-mass argument, with no asymptotic notation. -/
-theorem degree_series_bound (a : ℝ) (N : ℕ) :
-    (∑ r ∈ Finset.range N, a^r / (Real.sqrt (r:ℝ))^r)^2 ≤
-      2 * Real.exp (2*a^2) := by
-  let u (r : ℕ) : ℝ := (Real.sqrt ((1/2:ℝ)^r))
-  let v (r : ℕ) : ℝ := (Real.sqrt ((2:ℝ)^r))*a^r/(Real.sqrt (r:ℝ))^r
-  have huv (r : ℕ) : u r * v r = a^r/(Real.sqrt (r:ℝ))^r := by
-    unfold u v
-    have he : Real.sqrt ((1/2:ℝ)^r) * Real.sqrt ((2:ℝ)^r)=1 := by
-      rw [← Real.sqrt_mul (by positivity), ← mul_pow]
-      norm_num
-    calc
-      _ = (Real.sqrt ((1/2:ℝ)^r) * Real.sqrt ((2:ℝ)^r)) * (a^r/(Real.sqrt (r:ℝ))^r) := by ring
-      _ = _ := by rw [he, one_mul]
-  have hu : (∑ r ∈ Finset.range N, (u r)^2) ≤ 2 := by
-    simp only [u, Real.sq_sqrt (by positivity : 0 ≤ (1/2:ℝ)^_)]
-    have hg := geom_sum_mul_neg (1/2:ℝ) N
-    have hp : 0≤(1/2:ℝ)^N := by positivity
-    linarith
-  have hv : (∑ r ∈ Finset.range N, (v r)^2) ≤ Real.exp (2*a^2) := by
-    apply le_trans _ (Real.sum_le_exp_of_nonneg (by positivity) N)
-    apply Finset.sum_le_sum
-    intro r _
-    have hr : 0 < ((r.factorial):ℝ) := by exact_mod_cast Nat.factorial_pos r
-    have hrpow : 0 < (r:ℝ)^r := by
-      cases r with
-      | zero => norm_num
-      | succ r => positivity
-    have hfac : (r.factorial:ℝ) ≤ (r:ℝ)^r := by exact_mod_cast Nat.factorial_le_pow r
-    have hveq : (v r)^2 = (2*a^2)^r / (r:ℝ)^r := by
-      unfold v
-      rw [div_pow, mul_pow, Real.sq_sqrt (by positivity)]
-      have hd : ((Real.sqrt (r:ℝ))^r)^2=(r:ℝ)^r := by
-        rw [← pow_mul, mul_comm r 2, pow_mul, Real.sq_sqrt (by positivity)]
-      rw [hd]
-      rw [← pow_mul, mul_comm r 2, pow_mul]
-      ring
-    rw [hveq]
-    exact div_le_div_of_nonneg_left (by positivity) hr hfac
-  have hCS := Finset.sum_mul_sq_le_sq_mul_sq (Finset.range N) u v
-  simp only [huv] at hCS
-  exact hCS.trans (mul_le_mul hu hv (by positivity) (by norm_num))
-
-end TraceReconstruction
-
-end
-end
-
-end OAI
+import OAI.Probability.TraceReconstruction.Binomial
+
+namespace OAI
+
+/-!
+# Kernels
+
+Constrained rank-gap sums, correlated anchor kernels, window kernels, lattice Laplace sums, and geometric degree bounds.
+-/
+
+open scoped ENNReal
+noncomputable section
+
+noncomputable section
+namespace TraceReconstruction
+
+/-- Eliminate one output gap by its prescribed total before dropping the
+other rank constraints. This is the exact finite summation step used in
+the conditional rank-kernel estimate. -/
+theorem constrained_sum_product_le (r L : ℕ)
+    (f : Fin (r + 1) → Fin (L + 1) → ℝ)
+    (hf : ∀ i k, 0 ≤ f i k) (M : ℝ) (hM : 0 ≤ M)
+    (hbound : ∀ k, f 0 k ≤ M) :
+    (∑ z : Fin (r + 1) → Fin (L + 1),
+      if ∑ j, (z j).val = L then ∏ j, f j (z j) else 0) ≤
+        M * ∏ i : Fin r, ∑ k, f i.succ k := by
+  classical
+  rw [← (Fin.consEquiv (fun _ : Fin (r + 1) => Fin (L + 1))).sum_comp]
+  rw [Fintype.sum_prod_type, Finset.sum_comm]
+  have hpoint (z : Fin r → Fin (L + 1)) :
+      (∑ k : Fin (L + 1),
+        if k.val + ∑ i, (z i).val = L then f 0 k * ∏ i, f i.succ (z i) else 0) ≤
+          M * ∏ i, f i.succ (z i) := by
+    by_cases hz : ∑ i, (z i).val ≤ L
+    · let k₀ : Fin (L + 1) := ⟨L - ∑ i, (z i).val, by omega⟩
+      have heq (k : Fin (L + 1)) : k.val + ∑ i, (z i).val = L ↔ k = k₀ := by
+        change _ ↔ k = ⟨L - ∑ i, (z i).val, _⟩
+        rw [Fin.ext_iff]
+        simp only
+        omega
+      simp_rw [heq]
+      rw [Fintype.sum_ite_eq']
+      exact mul_le_mul_of_nonneg_right (hbound k₀) (Finset.prod_nonneg fun i _ => hf _ _)
+    · have heq (k : Fin (L + 1)) : ¬ (k.val + ∑ i, (z i).val = L) := by omega
+      simp_rw [ite_eq_right (heq _), Finset.sum_const_zero]
+      exact mul_nonneg hM (Finset.prod_nonneg fun i _ => hf _ _)
+  calc
+    _ ≤ ∑ z : Fin r → Fin (L + 1), M * ∏ i, f i.succ (z i) := by
+      apply Finset.sum_le_sum
+      intro z _
+      simpa only [Fin.consEquiv_apply, Fin.sum_univ_succ, Fin.prod_univ_succ,
+        Fin.cons_zero, Fin.cons_succ] using hpoint z
+    _ = _ := by
+      rw [← Finset.mul_sum, ← Fintype.prod_sum (fun (i : Fin r) (k : Fin (L + 1)) => f i.succ k)]
+
+/-- The product of two rank kernels has one scalar denominator factor,
+one uniform gap factor and independent one-dimensional cross-overlaps. -/
+theorem constrained_cross_kernel_le (r L : ℕ)
+    (f g : Fin (r + 1) → Fin (L + 1) → ℝ)
+    (hf : ∀ i k, 0 ≤ f i k) (hg : ∀ i k, 0 ≤ g i k)
+    (M : ℝ) (hM : 0 ≤ M) (hbound : ∀ k, f 0 k * g 0 k ≤ M) :
+    (∑ z : Fin (r + 1) → Fin (L + 1),
+      if ∑ j, (z j).val = L then (∏ j, f j (z j)) * (∏ j, g j (z j)) else 0) ≤
+        M * ∏ i : Fin r, ∑ k, f i.succ k * g i.succ k := by
+  simp_rw [← Finset.prod_mul_distrib]
+  exact constrained_sum_product_le r L (fun i k => f i k * g i k)
+    (fun i k => mul_nonneg (hf i k) (hg i k)) M hM hbound
+
+end TraceReconstruction
+
+namespace TraceReconstruction
+
+/-! ## Correlated sibling-anchor estimate
+
+This is the analytic summation step in the correlated-anchor estimate, subsection
+“Summing the correlated anchor kernels”. The common latent draw is kept
+inside the joint kernel, so independence of the two displaced locations
+is NOT assumed. Conditional independence only factors the two local
+kernels for each fixed common draw. All arrays here have finite support;
+the integer-jitter application uses a finite interval containing that support.
+-/
+
+section CorrelatedAnchors
+
+variable {Ω I J : Type*} [Fintype Ω] [Fintype I] [Fintype J]
+
+/-- Mixture of conditional products; `p` is the common displacement law. -/
+def jointKernel (p : Ω → ℝ) (a : Ω → I → ℝ) (b : Ω → J → ℝ)
+    (i : I) (j : J) : ℝ := ∑ w, p w * a w i * b w j
+
+omit [Fintype I] [Fintype J] in
+theorem jointKernel_nonneg
+    (p : Ω → ℝ) (a : Ω → I → ℝ) (b : Ω → J → ℝ)
+    (hp : ∀ w, 0 ≤ p w) (ha : ∀ w i, 0 ≤ a w i)
+    (hb : ∀ w j, 0 ≤ b w j) (i : I) (j : J) :
+    0 ≤ jointKernel p a b i j := by
+  exact Finset.sum_nonneg fun w _ => mul_nonneg (mul_nonneg (hp w) (ha w i)) (hb w j)
+
+theorem sum_jointKernel
+    (p : Ω → ℝ) (a : Ω → I → ℝ) (b : Ω → J → ℝ) (L M : ℝ)
+    (hp : ∑ w, p w = 1)
+    (ha : ∀ w, ∑ i, a w i = L) (hb : ∀ w, ∑ j, b w j = M) :
+    ∑ i, ∑ j, jointKernel p a b i j = L * M := by
+  simp only [jointKernel]
+  have hcomm (i : I) :
+      (∑ j : J, ∑ w : Ω, p w * a w i * b w j) =
+        ∑ w : Ω, ∑ j : J, p w * a w i * b w j := Finset.sum_comm
+  simp_rw [hcomm]
+  simp_rw [← Finset.mul_sum, hb]
+  rw [Finset.sum_comm]
+  simp_rw [← Finset.sum_mul, ← Finset.mul_sum, ha]
+  rw [← Finset.sum_mul, hp, one_mul]
+
+omit [Fintype I] [Fintype J] in
+theorem jointKernel_le
+    (p : Ω → ℝ) (a : Ω → I → ℝ) (b : Ω → J → ℝ) (A B : ℝ)
+    (hp : ∀ w, 0 ≤ p w) (ha : ∀ w i, 0 ≤ a w i)
+    (hB : 0 ≤ B) (hb : ∀ w j, b w j ≤ B)
+    (hA : ∀ i, ∑ w, p w * a w i ≤ A) (i : I) (j : J) :
+    jointKernel p a b i j ≤ A * B := by
+  calc
+    jointKernel p a b i j ≤ ∑ w, (p w * a w i) * B := by
+      exact Finset.sum_le_sum fun w _ =>
+        mul_le_mul_of_nonneg_left (hb w j) (mul_nonneg (hp w) (ha w i))
+    _ = (∑ w, p w * a w i) * B := (Finset.sum_mul _ _ _).symm
+    _ ≤ A * B := mul_le_mul_of_nonneg_right (hA i) hB
+
+/-- The joint squared-sum estimate retaining the common jitter. In the
+    manuscript A=L/(omega*sqrt(T*t)) and B=M/(omega*sqrt(t)),
+    up to the explicit absolute binomial constants. -/
+theorem correlated_anchor_square_sum
+    (p : Ω → ℝ) (a : Ω → I → ℝ) (b : Ω → J → ℝ)
+    (L M A B : ℝ)
+    (hp0 : ∀ w, 0 ≤ p w) (hp1 : ∑ w, p w = 1)
+    (ha0 : ∀ w i, 0 ≤ a w i) (hb0 : ∀ w j, 0 ≤ b w j)
+    (ha1 : ∀ w, ∑ i, a w i = L) (hb1 : ∀ w, ∑ j, b w j = M)
+    (hB : 0 ≤ B) (hb : ∀ w j, b w j ≤ B)
+    (hA : ∀ i, ∑ w, p w * a w i ≤ A) :
+    ∑ i, ∑ j, (jointKernel p a b i j) ^ 2 ≤ (A * B) * (L * M) := by
+  calc
+    ∑ i, ∑ j, (jointKernel p a b i j) ^ 2 ≤
+        ∑ i, ∑ j, (A * B) * jointKernel p a b i j := by
+      apply Finset.sum_le_sum
+      intro i _
+      apply Finset.sum_le_sum
+      intro j _
+      have h0 := jointKernel_nonneg p a b hp0 ha0 hb0 i j
+      have h1 := jointKernel_le p a b A B hp0 ha0 hB hb hA i j
+      nlinarith
+    _ = (A * B) * (∑ i, ∑ j, jointKernel p a b i j) := by
+      simp_rw [← Finset.mul_sum]
+    _ = (A * B) * (L * M) := by rw [sum_jointKernel p a b L M hp1 ha1 hb1]
+
+end CorrelatedAnchors
+
+
+end TraceReconstruction
+
+namespace TraceReconstruction
+
+/-- Truncating output ranks cannot increase a binomial cross-overlap. -/
+theorem binomial_overlap_truncate (g h L : ℕ) {p : ℝ}
+    (hp : p ∈ Set.Icc 0 1) :
+    (∑ k : Fin (L + 1), binomialMass g p k * binomialMass h p k) ≤
+      ∑ k ∈ Finset.range (h + 1), binomialMass g p k * binomialMass h p k := by
+  rw [Fin.sum_univ_eq_sum_range (fun k => binomialMass g p k * binomialMass h p k) (L + 1)]
+  by_cases hL : L ≤ h
+  · apply Finset.sum_le_sum_of_subset_of_nonneg (Finset.range_mono (by omega))
+    intro k _ _
+    exact mul_nonneg (binomialMass_nonneg g k hp) (binomialMass_nonneg h k hp)
+  · apply le_of_eq
+    symm
+    apply Finset.sum_subset (Finset.range_mono (by omega))
+    intro k _ hk
+    have hhk : h < k := by
+      simp only [Finset.mem_range, not_lt] at hk
+      omega
+    simp [binomialMass, Nat.choose_eq_zero_of_lt hhk]
+
+/-- Gaussian cross-overlap with the prefactor attached to either prescribed
+first length, removing the order restriction in the tilting proof. -/
+theorem binomial_gaussian_overlap_general (g h L : ℕ) {p : ℝ}
+    (hp : p ∈ Set.Ico (1 / 2 : ℝ) 1) (hS : 0 < g + h) :
+    (∑ k : Fin (L + 1), binomialMass g p k * binomialMass h p k) ≤
+      Real.sqrt (2 / (((g : ℝ) + 1) * p * (1 - p))) *
+        Real.exp (-((g : ℝ) - h) ^ 2 / (16 * ((g : ℝ) + h))) := by
+  have hp0 : 0 < p := by linarith [hp.1]
+  have hpq : 0 < p * (1 - p) := mul_pos hp0 (sub_pos.mpr hp.2)
+  by_cases hhg : h ≤ g
+  · exact (binomial_overlap_truncate g h L ⟨hp0.le, hp.2.le⟩).trans
+      (binomial_gaussian_overlap g h hp hhg hS)
+  · have hgh : g ≤ h := by omega
+    have ho := (binomial_overlap_truncate h g L ⟨hp0.le, hp.2.le⟩).trans
+      (binomial_gaussian_overlap h g hp hgh (by omega))
+    have hb : Real.sqrt (2 / (((h : ℝ) + 1) * p * (1 - p))) ≤
+        Real.sqrt (2 / (((g : ℝ) + 1) * p * (1 - p))) := by
+      apply Real.sqrt_le_sqrt
+      apply div_le_div_of_nonneg_left (by norm_num)
+        (by nlinarith only [mul_pos (show (0 : ℝ) < (g : ℝ) + 1 by positivity) hpq])
+      have hcast : (g : ℝ) ≤ h := by exact_mod_cast hgh
+      have ht := mul_le_mul_of_nonneg_right (show (g : ℝ) + 1 ≤ (h : ℝ) + 1 by linarith) hpq.le
+      nlinarith only [ht]
+    have heq : -((h : ℝ) - g) ^ 2 / (16 * ((h : ℝ) + g)) =
+        -((g : ℝ) - h) ^ 2 / (16 * ((g : ℝ) + h)) := by ring
+    rw [heq] at ho
+    exact (by simpa only [mul_comm] using ho :
+      (∑ k : Fin (L + 1), binomialMass g p k * binomialMass h p k) ≤ _).trans
+        (mul_le_mul_of_nonneg_right hb (Real.exp_nonneg _))
+
+/-- The exact conditional rank kernel, including the fixed count
+normalization, before any coarse-gap estimates or restrictions. -/
+def rankKernel (r n l : ℕ) (p : ℝ) (g : Fin (r + 1) → ℕ)
+    (z : Fin (r + 1) → Fin (l + 1)) : ℝ :=
+  if r ≤ l ∧ ∑ j, (z j).val = l - r then
+    p ^ r / binomialMass n p l * ∏ j, binomialMass (g j) p (z j) else 0
+
+/-- Largest-gap elimination for the conditional rank kernel. The displayed
+constant is absolute; no asymptotic proposal or Stirling theorem is assumed. -/
+theorem rankKernel_cross_le (r n l : ℕ) {p : ℝ}
+    (hp : p ∈ Set.Ico (1 / 2 : ℝ) 1) (hl : l ≤ n)
+    (hmean : (n : ℝ) * p = l) (hv : 1 ≤ (n : ℝ) * p * (1 - p))
+    (g h : Fin (r + 1) → ℕ) :
+    (∑ z : Fin (r + 1) → Fin (l - r + 1),
+      if ∑ j, (z j).val = l - r then
+        (p ^ r / binomialMass n p l) ^ 2 *
+          (∏ j, binomialMass (g j) p (z j)) *
+          (∏ j, binomialMass (h j) p (z j)) else 0) ≤
+      (4608 * n / ((g 0 : ℝ) + 1)) *
+        ∏ i : Fin r, ∑ k : Fin (l - r + 1),
+          binomialMass (g i.succ) p k * binomialMass (h i.succ) p k := by
+  have hp0 : 0 < p := by linarith [hp.1]
+  have hpq : 0 < p * (1 - p) := mul_pos hp0 (sub_pos.mpr hp.2)
+  have hden : 0 < ((g 0 : ℝ) + 1) * p * (1 - p) := by
+    nlinarith only [mul_pos (show (0 : ℝ) < (g 0 : ℝ) + 1 by positivity) hpq]
+  have hc : (p ^ r / binomialMass n p l) ^ 2 ≤ 144 * ((n : ℝ) * p * (1 - p)) := by
+    rw [div_eq_mul_inv, mul_pow]
+    apply (mul_le_mul_of_nonneg_right
+      (show (p ^ r) ^ 2 ≤ (1 : ℝ) by
+        exact pow_le_one₀ (pow_nonneg hp0.le _) (pow_le_one₀ hp0.le hp.2.le))
+      (sq_nonneg _)).trans
+    simpa only [one_mul] using binomialMass_mean_inverse_sq n l ⟨hp0, hp.2⟩ hl hmean hv
+  have hx := constrained_cross_kernel_le r (l - r)
+    (fun i k => binomialMass (g i) p k) (fun i k => binomialMass (h i) p k)
+    (fun i k => binomialMass_nonneg _ _ ⟨hp0.le, hp.2.le⟩)
+    (fun i k => binomialMass_nonneg _ _ ⟨hp0.le, hp.2.le⟩)
+    (32 / (((g 0 : ℝ) + 1) * p * (1 - p))) (by positivity)
+    (fun k => binomial_product_sup (g 0) (h 0) k hp)
+  have hprod : 0 ≤ ∏ i : Fin r, ∑ k : Fin (l - r + 1),
+      binomialMass (g i.succ) p k * binomialMass (h i.succ) p k := by
+    apply Finset.prod_nonneg
+    intro i _
+    exact Finset.sum_nonneg fun k _ => mul_nonneg
+      (binomialMass_nonneg _ _ ⟨hp0.le, hp.2.le⟩)
+      (binomialMass_nonneg _ _ ⟨hp0.le, hp.2.le⟩)
+  calc
+    _ = (p ^ r / binomialMass n p l) ^ 2 *
+      (∑ z : Fin (r + 1) → Fin (l - r + 1),
+        if ∑ j, (z j).val = l - r then
+          (∏ j, binomialMass (g j) p (z j)) *
+          (∏ j, binomialMass (h j) p (z j)) else 0) := by
+      rw [Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro z _
+      split_ifs <;> ring
+    _ ≤ (p ^ r / binomialMass n p l) ^ 2 *
+      ((32 / (((g 0 : ℝ) + 1) * p * (1 - p))) * _) :=
+      mul_le_mul_of_nonneg_left hx (sq_nonneg _)
+    _ ≤ (144 * ((n : ℝ) * p * (1 - p))) *
+      ((32 / (((g 0 : ℝ) + 1) * p * (1 - p))) * _) :=
+      mul_le_mul_of_nonneg_right hc (mul_nonneg (by positivity) hprod)
+    _ = _ := by
+      have hq : 1 - p ≠ 0 := ne_of_gt (sub_pos.mpr hp.2)
+      field_simp
+      ring
+
+end TraceReconstruction
+
+namespace TraceReconstruction
+theorem fairSeries_total (n : ℕ) : ∑' z : ℤ, (fairSeries n).coeff z = 1 := by
+  have h := fairSeries_L1 n
+  unfold seriesL1 at h
+  simpa only [abs_of_nonneg (fairSeries_nonneg n _)] using h
+
+theorem fairSeries_sum_finite_le_one (n : ℕ) (A : Finset ℤ) (c : ℤ) :
+    ∑ z ∈ A, (fairSeries n).coeff (c-z) ≤ 1 := by
+  have hs : Summable (fun z : ℤ => (fairSeries n).coeff (c-z)) := by
+    exact (Equiv.subLeft c).summable_iff.mpr (series_summable (fairSeries n) id rfl)
+  have hh : (∑' z : ℤ, (fairSeries n).coeff (c-z)) = 1 := by
+    simpa using ((Equiv.subLeft c).tsum_eq (fun z => (fairSeries n).coeff z)).trans
+      (fairSeries_total n)
+  rw [← hh]
+  exact Summable.sum_le_tsum A (fun _ _ => fairSeries_nonneg n _) hs
+
+def windowKernel (n L : ℕ) (z : ℤ) : ℝ :=
+  ∑ a ∈ Finset.range L, (fairSeries n).coeff ((a:ℤ)-z)
+
+theorem windowKernel_nonneg (n L : ℕ) (z : ℤ) : 0 ≤ windowKernel n L z :=
+  Finset.sum_nonneg (fun _ _ => fairSeries_nonneg n _)
+
+theorem windowKernel_sup (n L : ℕ) (z : ℤ) :
+    windowKernel n L z ≤ L*binomialBound n := by
+  calc
+    _ ≤ ∑ _a ∈ Finset.range L, binomialBound n := by
+      apply Finset.sum_le_sum
+      intro a _
+      exact (le_abs_self _).trans (fairSeries_coeff_bound n _)
+    _ = _ := by simp
+
+theorem windowKernel_sum (n L : ℕ) (A : Finset ℤ) (c : ℤ) :
+    ∑ z ∈ A, windowKernel n L (z-c) ≤ L := by
+  unfold windowKernel
+  rw [Finset.sum_comm]
+  calc
+    _ ≤ ∑ _a ∈ Finset.range L, (1:ℝ) := by
+      apply Finset.sum_le_sum
+      intro a _
+      simpa only [show ∀ z : ℤ, (a:ℤ)-(z-c)=(a:ℤ)+c-z from fun z => by omega]
+        using fairSeries_sum_finite_le_one n A ((a:ℤ)+c)
+    _ = _ := by simp
+
+theorem windowKernel_energy (n L : ℕ) (A : Finset ℤ) (c : ℤ) :
+    ∑ z ∈ A, windowKernel n L (z-c)^2 ≤ L^2*binomialBound n := by
+  calc
+    _ ≤ ∑ z ∈ A, windowKernel n L (z-c)*(L*binomialBound n) := by
+      apply Finset.sum_le_sum
+      intro z _
+      simpa only [pow_two] using mul_le_mul_of_nonneg_left
+        (windowKernel_sup n L (z-c)) (windowKernel_nonneg n L (z-c))
+    _ = (∑ z ∈ A, windowKernel n L (z-c))*(L*binomialBound n) := by rw [Finset.sum_mul]
+    _ ≤ L*(L*binomialBound n) := mul_le_mul_of_nonneg_right
+      (windowKernel_sum n L A c) (by positivity [binomialBound_nonneg n])
+    _ = _ := by ring
+
+def jointWindowKernel (g n₁ n₂ L : ℕ) (z₁ z₂ : ℤ) : ℝ :=
+  ∑ v ∈ (fairSeries g).coeff.support,
+    (fairSeries g).coeff v * windowKernel n₁ L (z₁+v) * windowKernel n₂ L (z₂+v)
+
+theorem windowKernel_average (g n L : ℕ) (z : ℤ) :
+    (∑ v ∈ (fairSeries g).coeff.support, (fairSeries g).coeff v * windowKernel n L (z+v)) =
+      windowKernel (g+n) L z := by
+  unfold windowKernel
+  simp_rw [Finset.mul_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro a _
+  rw [fairSeries_add, AddMonoidAlgebra.coeff_mul_apply_left, Finsupp.sum]
+  apply Finset.sum_congr rfl
+  intro v _
+  congr 2
+  omega
+
+theorem jointWindowKernel_nonneg (g n₁ n₂ L : ℕ) (z₁ z₂ : ℤ) :
+    0 ≤ jointWindowKernel g n₁ n₂ L z₁ z₂ := by
+  apply Finset.sum_nonneg
+  intro v _
+  exact mul_nonneg (mul_nonneg (fairSeries_nonneg g v) (windowKernel_nonneg n₁ L _))
+    (windowKernel_nonneg n₂ L _)
+
+theorem jointWindowKernel_sup (g n₁ n₂ L : ℕ) (z₁ z₂ : ℤ) :
+    jointWindowKernel g n₁ n₂ L z₁ z₂ ≤ L^2*binomialBound g*binomialBound n₂ := by
+  have hbound : binomialBound (g+n₁) ≤ binomialBound g := by
+    apply Real.sqrt_le_sqrt
+    apply div_le_div_of_nonneg_left (by norm_num) (by positivity)
+    simp only [Nat.cast_add]; linarith [show (0:ℝ) ≤ n₁ by positivity]
+  calc
+    _ ≤ ∑ v ∈ (fairSeries g).coeff.support,
+        (fairSeries g).coeff v * windowKernel n₁ L (z₁+v) * (L*binomialBound n₂) := by
+      apply Finset.sum_le_sum
+      intro v _
+      exact mul_le_mul_of_nonneg_left (windowKernel_sup n₂ L _) (mul_nonneg
+        (fairSeries_nonneg g v) (windowKernel_nonneg n₁ L _))
+    _ = windowKernel (g+n₁) L z₁ * (L*binomialBound n₂) := by
+      rw [← Finset.sum_mul, windowKernel_average]
+    _ ≤ (L*binomialBound g)*(L*binomialBound n₂) := by
+      apply mul_le_mul_of_nonneg_right _ (mul_nonneg (by positivity) (binomialBound_nonneg n₂))
+      exact (windowKernel_sup (g+n₁) L z₁).trans
+        (mul_le_mul_of_nonneg_left hbound (by positivity))
+    _ = _ := by ring
+
+theorem jointWindowKernel_sum (g n₁ n₂ L : ℕ) (A B : Finset ℤ) (c₁ c₂ : ℤ) :
+    ∑ z₁ ∈ A, ∑ z₂ ∈ B, jointWindowKernel g n₁ n₂ L (z₁-c₁) (z₂-c₂) ≤ L^2 := by
+  unfold jointWindowKernel
+  simp_rw [Finset.sum_comm (s := B)]
+  rw [Finset.sum_comm]
+  calc
+    _ = ∑ v ∈ (fairSeries g).coeff.support, (fairSeries g).coeff v *
+        (∑ z₁ ∈ A, windowKernel n₁ L ((z₁-c₁)+v)) *
+        (∑ z₂ ∈ B, windowKernel n₂ L ((z₂-c₂)+v)) := by
+      apply Finset.sum_congr rfl
+      intro v _
+      simp_rw [Finset.mul_sum, Finset.sum_mul]
+      rw [Finset.sum_comm]
+    _ ≤ ∑ v ∈ (fairSeries g).coeff.support, (fairSeries g).coeff v * (L*L) := by
+      apply Finset.sum_le_sum
+      intro v _
+      have h1 : (∑ z₁ ∈ A, windowKernel n₁ L ((z₁-c₁)+v)) ≤ L := by
+        simpa only [show ∀ z : ℤ, (z-c₁)+v = z-(c₁-v) from fun z => by omega]
+          using windowKernel_sum n₁ L A (c₁-v)
+      have h2 : (∑ z₂ ∈ B, windowKernel n₂ L ((z₂-c₂)+v)) ≤ L := by
+        simpa only [show ∀ z : ℤ, (z-c₂)+v = z-(c₂-v) from fun z => by omega]
+          using windowKernel_sum n₂ L B (c₂-v)
+      rw [mul_assoc]
+      apply mul_le_mul_of_nonneg_left _ (fairSeries_nonneg g v)
+      exact mul_le_mul h1 h2 (Finset.sum_nonneg (fun z _ => windowKernel_nonneg n₂ L _)) (by positivity)
+    _ = L^2 := by
+      rw [← Finset.sum_mul]
+      have ht : (∑ v ∈ (fairSeries g).coeff.support, (fairSeries g).coeff v) = 1 := by
+        have h := fairSeries_L1 g
+        rw [seriesL1_eq_sum] at h
+        simpa only [abs_of_nonneg (fairSeries_nonneg g _)] using h
+      rw [ht]; ring
+
+theorem jointWindowKernel_energy (g n₁ n₂ L : ℕ) (A B : Finset ℤ) (c₁ c₂ : ℤ) :
+    ∑ z₁ ∈ A, ∑ z₂ ∈ B, (jointWindowKernel g n₁ n₂ L (z₁-c₁) (z₂-c₂))^2 ≤
+      L^4*binomialBound g*binomialBound n₂ := by
+  calc
+    _ ≤ ∑ z₁ ∈ A, ∑ z₂ ∈ B, jointWindowKernel g n₁ n₂ L (z₁-c₁) (z₂-c₂) *
+        (L^2*binomialBound g*binomialBound n₂) := by
+      apply Finset.sum_le_sum
+      intro z₁ _
+      apply Finset.sum_le_sum
+      intro z₂ _
+      simpa only [pow_two] using mul_le_mul_of_nonneg_left
+        (jointWindowKernel_sup g n₁ n₂ L _ _) (jointWindowKernel_nonneg g n₁ n₂ L _ _)
+    _ = (∑ z₁ ∈ A, ∑ z₂ ∈ B, jointWindowKernel g n₁ n₂ L (z₁-c₁) (z₂-c₂)) *
+        (L^2*binomialBound g*binomialBound n₂) := by simp_rw [Finset.sum_mul]
+    _ ≤ L^2*(L^2*binomialBound g*binomialBound n₂) := mul_le_mul_of_nonneg_right
+      (jointWindowKernel_sum g n₁ n₂ L A B c₁ c₂)
+      (mul_nonneg (mul_nonneg (by positivity) (binomialBound_nonneg g)) (binomialBound_nonneg n₂))
+    _ = _ := by ring
+end TraceReconstruction
+
+open Filter
+open scoped Topology
+
+
+namespace TraceReconstruction
+
+/-- Elementary discrete integral comparison, including the singular first cell. -/
+theorem sum_inverse_sqrt (N : ℕ) :
+    (∑ k ∈ Finset.range N, (Real.sqrt (k+1:ℝ))⁻¹) ≤ 2*Real.sqrt N := by
+  induction N with
+  | zero => simp
+  | succ N ih =>
+    rw [Finset.sum_range_succ]
+    have hn : 0 ≤ (N:ℝ) := by positivity
+    have hp : 0 < Real.sqrt (N+1:ℝ) := Real.sqrt_pos.2 (by positivity)
+    have hs : Real.sqrt (N:ℝ) ≤ Real.sqrt (N+1:ℝ) := Real.sqrt_le_sqrt (by linarith)
+    have h1 := Real.sq_sqrt hn
+    have h2 := Real.sq_sqrt (show 0 ≤ (N+1:ℝ) by positivity)
+    have hstep : (Real.sqrt (N+1:ℝ))⁻¹ ≤ 2*(Real.sqrt (N+1:ℝ)-Real.sqrt N) := by
+      apply (inv_le_iff_one_le_mul₀ hp).2
+      nlinarith [sq_nonneg (Real.sqrt (N+1:ℝ)-Real.sqrt N)]
+    push_cast
+    linarith
+
+theorem exp_gap_lower {h : ℝ} (hh : 0 < h) (h1 : h ≤ 1) :
+    h/2 ≤ 1-Real.exp (-h) := by
+  have he := Real.add_one_le_exp h
+  have hp : 0 < Real.exp h := Real.exp_pos h
+  have hp' : 0 < 1+h := by linarith
+  have hinv : (Real.exp h)⁻¹ ≤ (1+h)⁻¹ := inv_anti₀ hp' (by simpa [add_comm] using he)
+  rw [Real.exp_neg]
+  have hi : (1+h)⁻¹*(1+h)=1 := inv_mul_cancel₀ (ne_of_gt hp')
+  nlinarith [mul_nonneg hh.le (sub_nonneg.mpr h1)]
+
+theorem sum_exp_decay {h : ℝ} (hh : 0 < h) (h1 : h ≤ 1) (M : ℕ) :
+    (∑ k ∈ Finset.range M, Real.exp (-h*(k:ℝ))) ≤ 2/h := by
+  have he : 0 ≤ Real.exp (-h) := (Real.exp_pos _).le
+  have he1 : Real.exp (-h) < 1 := Real.exp_lt_one_iff.mpr (by linarith)
+  have hf : (∑ k ∈ Finset.range M, (Real.exp (-h))^k) ≤ (1-Real.exp (-h))⁻¹ := by
+    rw [inv_eq_one_div, le_div_iff₀ (by linarith : 0 < 1-Real.exp (-h)), geom_sum_mul_neg]
+    exact sub_le_self _ (pow_nonneg he M)
+  have hi : (1-Real.exp (-h))⁻¹ ≤ 2/h := by
+    rw [inv_le_iff_one_le_mul₀ (by linarith : 0<1-Real.exp (-h))]
+    have hb := exp_gap_lower hh h1
+    have := mul_le_mul_of_nonneg_right hb (show 0 ≤ 2/h by positivity)
+    nlinarith [div_mul_cancel₀ (2:ℝ) hh.ne']
+  calc
+    _ = ∑ k ∈ Finset.range M, (Real.exp (-h))^k := by
+      apply Finset.sum_congr rfl
+      intro k _
+      rw [← Real.exp_nat_mul]
+      congr 1
+      ring
+    _ ≤ _ := hf.trans hi
+
+/-- The uniform lattice Laplace estimate used by the initial-pair construction, with an absolute constant. -/
+theorem coarse_laplace_sum {h : ℝ} (hh : 0 < h) (h1 : h ≤ 1) (M : ℕ) :
+    (∑ k ∈ Finset.range M, (Real.sqrt (k+1:ℝ))⁻¹ *
+      Real.exp (-h*(k+1:ℝ))) ≤ 6/Real.sqrt h := by
+  let N := ⌈1/h⌉₊
+  have hN : 1/h ≤ (N:ℝ) := Nat.le_ceil _
+  have hNup : (N:ℝ) < 1/h+1 := Nat.ceil_lt_add_one (by positivity)
+  have hs : 0 < Real.sqrt h := Real.sqrt_pos.2 hh
+  have hNsqrt : Real.sqrt N ≤ 2/Real.sqrt h := by
+    have hn : (N:ℝ)*h ≤ 2 := by
+      have hx := mul_le_mul_of_nonneg_right hNup.le hh.le
+      have hc : (1/h)*h=1 := div_mul_cancel₀ _ hh.ne'
+      nlinarith
+    have hprod : Real.sqrt N * Real.sqrt h ≤ 2 := by
+      have hsqh := Real.sq_sqrt hh.le
+      have hsqN := Real.sq_sqrt (show 0≤(N:ℝ) by positivity)
+      have hnon : 0≤Real.sqrt N*Real.sqrt h := by positivity
+      nlinarith [sq_nonneg (Real.sqrt N*Real.sqrt h)]
+    exact (le_div_iff₀ hs).2 hprod
+  let low := (Finset.range M).filter (fun k => k<N)
+  let high := (Finset.range M).filter (fun k => ¬k<N)
+  have hlo : (∑ k ∈ low, (Real.sqrt (k+1:ℝ))⁻¹ * Real.exp (-h*(k+1:ℝ))) ≤
+      2*Real.sqrt N := by
+    calc
+      _ ≤ ∑ k ∈ low, (Real.sqrt (k+1:ℝ))⁻¹ := by
+        apply Finset.sum_le_sum
+        intro k hk
+        exact mul_le_of_le_one_right (by positivity) (Real.exp_le_one_iff.mpr (by nlinarith [show (0:ℝ)≤k by positivity]))
+      _ ≤ ∑ k ∈ Finset.range N, (Real.sqrt (k+1:ℝ))⁻¹ := by
+        apply Finset.sum_le_sum_of_subset_of_nonneg
+        · intro k hk
+          exact Finset.mem_range.mpr (Finset.mem_filter.mp hk).2
+        · intros; positivity
+      _ ≤ _ := sum_inverse_sqrt N
+  have hhi : (∑ k ∈ high, (Real.sqrt (k+1:ℝ))⁻¹ * Real.exp (-h*(k+1:ℝ))) ≤
+      Real.sqrt h * (2/h) := by
+    calc
+      _ ≤ ∑ k ∈ high, Real.sqrt h * Real.exp (-h*(k:ℝ)) := by
+        apply Finset.sum_le_sum
+        intro k hk
+        have hkN : N ≤ k := Nat.le_of_not_gt (Finset.mem_filter.mp hk).2
+        have hkR : 1/h ≤ (k+1:ℝ) := hN.trans (by exact_mod_cast (by omega : N≤k+1))
+        have hinv : (Real.sqrt (k+1:ℝ))⁻¹ ≤ Real.sqrt h := by
+          apply (inv_le_iff_one_le_mul₀ (Real.sqrt_pos.2 (by positivity))).2
+          have hmul : 1 ≤ h*(k+1:ℝ) := by simpa [mul_comm] using (div_le_iff₀ hh).mp hkR
+          have hsqh := Real.sq_sqrt hh.le
+          have hsqk := Real.sq_sqrt (show 0≤(k+1:ℝ) by positivity)
+          nlinarith [mul_nonneg (Real.sqrt_nonneg h) (Real.sqrt_nonneg (k+1:ℝ))]
+        apply mul_le_mul hinv _ (Real.exp_pos _).le (Real.sqrt_nonneg _)
+        exact Real.exp_le_exp.mpr (by nlinarith)
+      _ ≤ ∑ k ∈ Finset.range M, Real.sqrt h * Real.exp (-h*(k:ℝ)) := by
+        apply Finset.sum_le_sum_of_subset_of_nonneg (Finset.filter_subset _ _)
+        intros; positivity
+      _ = Real.sqrt h * ∑ k ∈ Finset.range M, Real.exp (-h*(k:ℝ)) := by rw [Finset.mul_sum]
+      _ ≤ _ := mul_le_mul_of_nonneg_left (sum_exp_decay hh h1 M) (Real.sqrt_nonneg h)
+  have hsum : (∑ k ∈ low, (Real.sqrt (k+1:ℝ))⁻¹ * Real.exp (-h*(k+1:ℝ))) +
+      (∑ k ∈ high, (Real.sqrt (k+1:ℝ))⁻¹ * Real.exp (-h*(k+1:ℝ))) =
+      ∑ k ∈ Finset.range M, (Real.sqrt (k+1:ℝ))⁻¹ * Real.exp (-h*(k+1:ℝ)) :=
+    Finset.sum_filter_add_sum_filter_not _ _ _
+  have hcancel : Real.sqrt h*(2/h)=2/Real.sqrt h := by
+    have hsq := Real.sq_sqrt hh.le
+    field_simp
+    nlinarith
+  rw [← hsum]
+  rw [hcancel] at hhi
+  have heq : 2*(2/Real.sqrt h)+2/Real.sqrt h=6/Real.sqrt h := by ring
+  linarith
+
+
+/-- Exponential tilting removes the coarse-gap simplex constraint without a dimension loss. -/
+theorem coarse_simplex_sum (r M : ℕ) {h u : ℝ} (hh : 0<h) (h1 : h≤1) :
+    (∑ v : Fin r → Fin M, if (∑ i, ((v i).val+1:ℝ)) ≤ u then
+      ∏ i, (Real.sqrt ((v i).val+1:ℝ))⁻¹ else 0) ≤
+      Real.exp (h*u) * (6/Real.sqrt h)^r := by
+  have hev (v : Fin r → Fin M) :
+      ∏ i, Real.exp (-h*((v i).val+1:ℝ)) =
+        Real.exp (-h*∑ i, ((v i).val+1:ℝ)) := by
+    rw [← Real.exp_sum, Finset.mul_sum]
+  have hbound :
+      (∑ v : Fin r → Fin M, if (∑ i, ((v i).val+1:ℝ)) ≤ u then
+        ∏ i, (Real.sqrt ((v i).val+1:ℝ))⁻¹ else 0) ≤
+      Real.exp (h*u) * ∑ v : Fin r → Fin M,
+        ∏ i, ((Real.sqrt ((v i).val+1:ℝ))⁻¹ * Real.exp (-h*((v i).val+1:ℝ))) := by
+    rw [Finset.mul_sum]
+    apply Finset.sum_le_sum
+    intro v _
+    rw [Finset.prod_mul_distrib, hev]
+    split_ifs with hv
+    · have he : 1 ≤ Real.exp (h*u) * Real.exp (-h*∑ i, ((v i).val+1:ℝ)) := by
+        rw [← Real.exp_add]
+        apply Real.one_le_exp_iff.mpr
+        nlinarith
+      have hp : 0 ≤ ∏ i, (Real.sqrt ((v i).val+1:ℝ))⁻¹ := by positivity
+      nlinarith [mul_le_mul_of_nonneg_left he hp]
+    · positivity
+  apply hbound.trans
+  apply mul_le_mul_of_nonneg_left _ (Real.exp_pos _).le
+  rw [← Fintype.prod_sum (fun (_i : Fin r) (k : Fin M) =>
+    (Real.sqrt (k.val+1:ℝ))⁻¹ * Real.exp (-h*(k.val+1:ℝ)))]
+  have hc : (6/Real.sqrt h)^r = ∏ _i : Fin r, (6/Real.sqrt h) := by simp [div_pow]
+  rw [hc]
+  apply Finset.prod_le_prod₀
+  · intros; positivity
+  · intro i _
+    rw [Fin.sum_univ_eq_sum_range (fun k : ℕ =>
+      (Real.sqrt (k+1:ℝ))⁻¹ * Real.exp (-h*(k+1:ℝ))) M]
+    exact coarse_laplace_sum hh h1 M
+
+end TraceReconstruction
+
+
+namespace TraceReconstruction
+
+/-- The degree-series estimate in the common-mass argument, with no asymptotic notation. -/
+theorem degree_series_bound (a : ℝ) (N : ℕ) :
+    (∑ r ∈ Finset.range N, a^r / (Real.sqrt (r:ℝ))^r)^2 ≤
+      2 * Real.exp (2*a^2) := by
+  let u (r : ℕ) : ℝ := (Real.sqrt ((1/2:ℝ)^r))
+  let v (r : ℕ) : ℝ := (Real.sqrt ((2:ℝ)^r))*a^r/(Real.sqrt (r:ℝ))^r
+  have huv (r : ℕ) : u r * v r = a^r/(Real.sqrt (r:ℝ))^r := by
+    unfold u v
+    have he : Real.sqrt ((1/2:ℝ)^r) * Real.sqrt ((2:ℝ)^r)=1 := by
+      rw [← Real.sqrt_mul (by positivity), ← mul_pow]
+      norm_num
+    calc
+      _ = (Real.sqrt ((1/2:ℝ)^r) * Real.sqrt ((2:ℝ)^r)) * (a^r/(Real.sqrt (r:ℝ))^r) := by ring
+      _ = _ := by rw [he, one_mul]
+  have hu : (∑ r ∈ Finset.range N, (u r)^2) ≤ 2 := by
+    simp only [u, Real.sq_sqrt (by positivity : 0 ≤ (1/2:ℝ)^_)]
+    have hg := geom_sum_mul_neg (1/2:ℝ) N
+    have hp : 0≤(1/2:ℝ)^N := by positivity
+    linarith
+  have hv : (∑ r ∈ Finset.range N, (v r)^2) ≤ Real.exp (2*a^2) := by
+    apply le_trans _ (Real.sum_le_exp_of_nonneg (by positivity) N)
+    apply Finset.sum_le_sum
+    intro r _
+    have hr : 0 < ((r.factorial):ℝ) := by exact_mod_cast Nat.factorial_pos r
+    have hrpow : 0 < (r:ℝ)^r := by
+      cases r with
+      | zero => norm_num
+      | succ r => positivity
+    have hfac : (r.factorial:ℝ) ≤ (r:ℝ)^r := by exact_mod_cast Nat.factorial_le_pow r
+    have hveq : (v r)^2 = (2*a^2)^r / (r:ℝ)^r := by
+      unfold v
+      rw [div_pow, mul_pow, Real.sq_sqrt (by positivity)]
+      have hd : ((Real.sqrt (r:ℝ))^r)^2=(r:ℝ)^r := by
+        rw [← pow_mul, mul_comm r 2, pow_mul, Real.sq_sqrt (by positivity)]
+      rw [hd]
+      rw [← pow_mul, mul_comm r 2, pow_mul]
+      ring
+    rw [hveq]
+    exact div_le_div_of_nonneg_left (by positivity) hr hfac
+  have hCS := Finset.sum_mul_sq_le_sq_mul_sq (Finset.range N) u v
+  simp only [huv] at hCS
+  exact hCS.trans (mul_le_mul hu hv (by positivity) (by norm_num))
+
+end TraceReconstruction
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/TraceReconstruction/Layout.lean b/lean/OAI/Probability/TraceReconstruction/Layout.lean
index 1e18b6dba12df82a37fa7de1bfd62fccccdce9ae..1026e5d69cd3de097079dc6e5e6384a759243f5b 100644
--- a/lean/OAI/Probability/TraceReconstruction/Layout.lean
+++ b/lean/OAI/Probability/TraceReconstruction/Layout.lean
@@ -1,1033 +1,1033 @@
-import OAI.Probability.TraceReconstruction.Shapes
-
-namespace OAI
-
-/-!
-# Layout
-
-Multiwindow and paired layouts, coordinate simulations, and literal source types and trees.
--/
-
-open scoped ENNReal
-noncomputable section
-
-open Filter
-open scoped Topology
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- Literal unmarked concatenation of fixed windows with separately reported
-pure pre/post regions. Counts are in atoms, not surviving trace symbols. -/
-def framedConcat {α : Type*} (e : α) {k : ℕ}
-    (pre post : Fin k → ℕ) (W : Fin k → List α) : List α :=
-  (List.ofFn (fun i => List.replicate (pre i) e ++ W i ++
-    List.replicate (post i) e)).flatten
-
-theorem framedConcat_succ {α : Type*} (e : α) {k : ℕ}
-    (pre post : Fin (k+1) → ℕ) (W : Fin (k+1) → List α) :
-    framedConcat e pre post W = List.replicate (pre 0) e ++ W 0 ++
-      List.replicate (post 0) e ++
-      framedConcat e (fun i => pre i.succ) (fun i => post i.succ) (fun i => W i.succ) := by
-  simp [framedConcat,List.ofFn_succ,List.append_assoc]
-
-/-- Only the first, last, and intervening pure lengths are observable in an
-unmarked word. In particular boundaries may jitter across nominal units. -/
-theorem framedConcat_eq_of_gaps {α : Type*} (e : α) (k : ℕ)
-    (pre post pre' post' : Fin (k+1) → ℕ) (W : Fin (k+1) → List α)
-    (u u' : ℕ) (hfirst : u+pre 0=u'+pre' 0)
-    (hgap : ∀i : Fin k, post i.castSucc+pre i.succ=post' i.castSucc+pre' i.succ)
-    (hlast : post (Fin.last k)=post' (Fin.last k)) :
-    List.replicate u e ++ framedConcat e pre post W =
-      List.replicate u' e ++ framedConcat e pre' post' W := by
-  induction k generalizing u u' with
-  | zero =>
-    have hl : post 0=post' 0 := hlast
-    simp only [framedConcat,List.ofFn_succ,List.ofFn_zero,List.flatten_cons,
-      List.flatten_nil,List.append_nil]
-    calc
-      _ = List.replicate (u+pre 0) e ++ W 0 ++ List.replicate (post 0) e := by
-        simp only [List.replicate_add,List.append_assoc]
-      _ = List.replicate (u'+pre' 0) e ++ W 0 ++ List.replicate (post' 0) e := by rw [hfirst,hl]
-      _ = _ := by simp only [List.replicate_add,List.append_assoc]
-  | succ k ih =>
-    rw [framedConcat_succ e pre post W,framedConcat_succ e pre' post' W]
-    have ht := ih (fun i => pre i.succ) (fun i => post i.succ)
-      (fun i => pre' i.succ) (fun i => post' i.succ) (fun i => W i.succ)
-      (post 0) (post' 0) (hgap 0) (fun i => hgap i.succ) hlast
-    calc
-      _ = List.replicate (u+pre 0) e ++ W 0 ++
-          (List.replicate (post 0) e ++ framedConcat e
-            (fun i => pre i.succ) (fun i => post i.succ) (fun i => W i.succ)) := by
-        simp only [List.replicate_add,List.append_assoc]
-      _ = List.replicate (u'+pre' 0) e ++ W 0 ++
-          (List.replicate (post' 0) e ++ framedConcat e
-            (fun i => pre' i.succ) (fun i => post' i.succ) (fun i => W i.succ)) := by rw [hfirst,ht]
-      _ = _ := by simp only [List.replicate_add,List.append_assoc]
-
-/-- Number of atoms strictly before the indicated padded unit. -/
-def framePrefix {k : ℕ} (pre post : Fin k → ℕ) (len : Fin k → ℕ) (i : ℕ) : ℕ :=
-  ∑j : Fin k, if j.val < i then pre j+len j+post j else 0
-
-theorem framePrefix_zero {k : ℕ} (pre post len : Fin k → ℕ) :
-    framePrefix pre post len 0=0 := by simp [framePrefix]
-
-theorem framePrefix_step {k : ℕ} (pre post len : Fin k → ℕ) (i : Fin k) :
-    framePrefix pre post len (i.val+1)=framePrefix pre post len i.val+pre i+len i+post i := by
-  have he : (fun j : Fin k => if j.val < i.val+1 then pre j+len j+post j else 0)=
-      fun j => (if j.val < i.val then pre j+len j+post j else 0)+
-        (if j=i then pre i+len i+post i else 0) := by
-    funext j
-    by_cases h : j=i
-    · subst j; simp
-    · have hn : j.val ≠ i.val := fun h' => h (Fin.ext h')
-      by_cases hlt : j.val < i.val
-      · simp [hlt,show j.val < i.val+1 by omega,h]
-      · simp [hlt,show ¬j.val < i.val+1 by omega,h]
-  simp only [framePrefix,he,Finset.sum_add_distrib,Finset.sum_ite_eq',Finset.mem_univ,ite_true]
-  omega
-
-/-- Equal actual starting positions and equal total counts are enough:
-there is no requirement that a part coincide with its nominal unit. -/
-theorem framedConcat_eq_of_starts {α : Type*} (e : α) (k : ℕ)
-    (pre post pre' post' : Fin (k+1) → ℕ) (W : Fin (k+1) → List α)
-    (hstart : ∀i : Fin (k+1),
-      framePrefix pre post (fun j => (W j).length) i.val+pre i=
-      framePrefix pre' post' (fun j => (W j).length) i.val+pre' i)
-    (htotal : framePrefix pre post (fun j => (W j).length) (k+1)=
-      framePrefix pre' post' (fun j => (W j).length) (k+1)) :
-    framedConcat e pre post W=framedConcat e pre' post' W := by
-  have H := framedConcat_eq_of_gaps e k pre post pre' post' W 0 0
-  simp only [List.replicate_zero,List.nil_append] at H
-  apply H
-  · simpa only [Fin.val_zero,framePrefix_zero,zero_add] using hstart 0
-  · intro i
-    have h₁ := hstart i.castSucc
-    have h₂ := hstart i.succ
-    have h₃ := framePrefix_step pre post (fun j => (W j).length) i.castSucc
-    have h₄ := framePrefix_step pre' post' (fun j => (W j).length) i.castSucc
-    simp only [Fin.val_castSucc,Fin.val_succ] at *
-    omega
-  · have h₁ := hstart (Fin.last k)
-    have h₂ := framePrefix_step pre post (fun j => (W j).length) (Fin.last k)
-    have h₃ := framePrefix_step pre' post' (fun j => (W j).length) (Fin.last k)
-    simp only [Fin.val_last] at *
-    omega
-
-end TraceReconstruction
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem framePrefix_constant {k : ℕ} (pre post len : Fin k → ℕ) (P i : ℕ)
-    (hc : ∀j,pre j+len j+post j=P) (hi : i ≤ k) :
-    framePrefix pre post len i=i*P := by
-  induction i with
-  | zero => simp [framePrefix_zero]
-  | succ i ih =>
-    have h := framePrefix_step pre post len ⟨i,by omega⟩
-    have hh := hc ⟨i,by omega⟩
-    rw [h,ih (by omega),Nat.add_mul,Nat.one_mul]
-    omega
-
-theorem flatten_framedConcat (w : List Bool) {k : ℕ}
-    (pre post : Fin k → ℕ) (W : Fin k → List (List Bool)) :
-    (framedConcat w pre post W).flatten=
-      (List.ofFn (fun i => pureAtomRun w (pre i) ++ (W i).flatten ++
-        pureAtomRun w (post i))).flatten := by
-  rw [framedConcat,List.flatten_flatten,List.map_ofFn]
-  apply congrArg List.flatten
-  apply congrArg List.ofFn
-  funext i
-  simp [Function.comp_apply,List.flatten_append,pureAtomRun]
-
-/-- The deterministic multiwindow identity used by the nested simulator.
-Every motif is covered by its own window. Predetermined pure-region lengths
-may move the boundaries of nominal units; only their exact cumulative counts
-are required. Windows and chunks all remain literal unmarked binary words. -/
-theorem template_group_windows (k P T a L : ℕ) (sign : Fin L → Bool)
-    (w : Bool → List Bool) (τ : Fin (k+1) → Bool)
-    (t pre post : Fin (k+1) → ℕ)
-    (hleft : ∀u,t u ≤ a) (hfit : ∀u,a+L ≤ t u+T)
-    (hunit : ∀u,t u+T ≤ P)
-    (hstart : ∀u,framePrefix pre post (fun _ => T) u.val+pre u=u.val*P+t u)
-    (htotal : framePrefix pre post (fun _ => T) (k+1)=(k+1)*P) :
-    (List.ofFn (fun u => complementaryWords P a L sign w (τ u))).flatten=
-      (List.ofFn (fun u => pureAtomRun (w false) (pre u) ++
-        complementaryWords T (a-t u) L sign w (τ u) ++
-        pureAtomRun (w false) (post u))).flatten := by
-  let W : Fin (k+1) → List (List Bool) := fun u =>
-    List.ofFn (fun i => w (templateType T (a-t u) L sign (τ u) i))
-  let z : Fin (k+1) → ℕ := fun u => P-(t u+T)
-  have hz : ∀u,t u+T+z u=P := by intro u; dsimp [z]; have hu := hunit u; omega
-  have hlen : ∀u,(W u).length=T := by intro u; simp [W]
-  have hp : ∀i,i ≤ k+1 → framePrefix t z (fun u => (W u).length) i=i*P :=
-    fun i hi => framePrefix_constant t z _ P i (fun u => by rw [hlen]; exact hz u) hi
-  have he := framedConcat_eq_of_starts (w false) k t z pre post W
-    (fun u => by simp only [hlen] at hp ⊢; rw [hp _ (by omega),hstart])
-    (by simp only [hlen] at hp ⊢; rw [hp _ le_rfl,htotal])
-  have he' := congrArg List.flatten he
-  rw [flatten_framedConcat,flatten_framedConcat] at he'
-  calc
-    _ = (List.ofFn (fun u => pureAtomRun (w false) (t u) ++ (W u).flatten ++
-          pureAtomRun (w false) (z u))).flatten := by
-      congr 1
-      apply congrArg List.ofFn
-      funext u
-      have hh := template_word_window (t u) T (z u) a L sign (τ u) w (hleft u) (hfit u)
-      rw [hz] at hh
-      simpa only [complementaryWords,atomString_constant,W,atomString,List.ofFn_const,pureAtomRun] using hh
-    _ = _ := he'
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-variable {K : Type*} [Fintype K]
-
-/-- Counts of literal side chunks, with one deterministic pure filler in
- each region. A bundle can contain any even number of paired chunks. -/
-def pairedPreCount {N : ℕ} (left : K → Fin N) (filler : Fin N → ℕ)
-    (first : K → ℕ) (u : Fin N) : ℕ :=
-  filler u + ∑v, if left v=u then first v else 0
-
-def pairedPostCount {N : ℕ} (right : K → Fin N) (filler : Fin N → ℕ)
-    (reserve first : K → ℕ) (u : Fin N) : ℕ :=
-  filler u + ∑v, if right v=u then 2*reserve v-first v else 0
-
-theorem pairedPreCount_cast {N : ℕ} (left : K → Fin N) (filler : Fin N → ℕ)
-    (reserve first : K → ℕ) (A : ℕ)
-    (hfill : ∀u,filler u+(∑v,if left v=u then reserve v else 0)=A) (u : Fin N) :
-    (pairedPreCount left filler first u : ℤ) = A+
-      ∑v, if left v=u then (first v:ℤ)-reserve v else 0 := by
-  have hf : (filler u:ℤ)+(∑v,if left v=u then (reserve v:ℤ) else 0)=A := by
-    exact_mod_cast hfill u
-  have he : (∑v,if left v=u then (first v:ℤ)-reserve v else 0)=
-      (∑v,if left v=u then (first v:ℤ) else 0)-
-        (∑v,if left v=u then (reserve v:ℤ) else 0) := by
-    rw [←Finset.sum_sub_distrib]
-    apply Finset.sum_congr rfl
-    intro v _
-    split_ifs <;> ring
-  simp only [pairedPreCount,Nat.cast_add,Nat.cast_sum,Nat.cast_ite,Nat.cast_zero]
-  rw [he]
-  omega
-
-theorem pairedPostCount_cast {N : ℕ} (right : K → Fin N) (filler : Fin N → ℕ)
-    (reserve first : K → ℕ) (A : ℕ) (hfirst : ∀v,first v ≤ 2*reserve v)
-    (hfill : ∀u,filler u+(∑v,if right v=u then reserve v else 0)=A) (u : Fin N) :
-    (pairedPostCount right filler reserve first u : ℤ) = A-
-      ∑v, if right v=u then (first v:ℤ)-reserve v else 0 := by
-  have hf : (filler u:ℤ)+(∑v,if right v=u then (reserve v:ℤ) else 0)=A := by
-    exact_mod_cast hfill u
-  have he : (∑v,if right v=u then ((2*reserve v-first v:ℕ):ℤ) else 0)=
-      (∑v,if right v=u then (reserve v:ℤ) else 0)-
-        (∑v,if right v=u then (first v:ℤ)-reserve v else 0) := by
-    rw [←Finset.sum_sub_distrib]
-    apply Finset.sum_congr rfl
-    intro v _
-    rw [Nat.cast_sub (hfirst v)]
-    push_cast
-    split_ifs <;> ring
-  simp only [pairedPostCount,Nat.cast_add,Nat.cast_sum,Nat.cast_ite,Nat.cast_zero]
-  rw [he]
-  omega
-
-/-- Prefix sum of a delta at a prescribed label, with no loss from replacing
- its real finite support by an interval. -/
-theorem fin_prefix_delta {N : ℕ} (a : Fin N) (i : ℕ) (x : ℤ) :
-    (∑u : Fin N, if u.val < i then (if a=u then x else 0) else 0)=
-      if a.val < i then x else 0 := by
-  have he : (fun u : Fin N => if u.val < i then (if a=u then x else 0) else 0)=
-      fun u => if u=a then (if a.val < i then x else 0) else 0 := by
-    funext u
-    by_cases h : u=a
-    · subst u; simp
-    · simp [h,Ne.symm h]
-  rw [he]
-  simp
-
-/-- Exact paired-interval contribution to the position of a window. -/
-theorem pair_activity_identity {N : ℕ} (l r u : Fin N) (hlr : l.val ≤ r.val) (x : ℤ) :
-    (if l.val < u.val then x else 0)-(if r.val < u.val then x else 0)+
-      (if l=u then x else 0) =
-      if l.val ≤ u.val ∧ u.val ≤ r.val then x else 0 := by
-  by_cases hlu : l.val < u.val
-  · by_cases hru : r.val < u.val
-    · simp [hlu,hru,show l≠u by intro h; subst u; omega,show ¬u.val ≤ r.val by omega]
-    · simp [hlu,hru,show l≠u by intro h; subst u; omega,show l.val ≤ u.val by omega,show u.val ≤ r.val by omega]
-  · by_cases hlu' : l=u
-    · subst u; simp [hlr,show ¬r.val < l.val by omega]
-    · have hn : l.val≠u.val := fun h => hlu' (Fin.ext h)
-      simp [hlu,hlu',show ¬r.val < u.val by omega,show ¬l.val ≤ u.val by omega]
-
-/-- Cumulative pure chunk counts have exactly the source's localBits-plus-active-
-globalBits jitter. This identity uses the actual chronological pre/post schedule. -/
-theorem pairedLayout_start {N : ℕ} (left right : K → Fin N)
-    (horder : ∀v,(left v).val ≤ (right v).val)
-    (fpre fpost : Fin N → ℕ) (reserve first : K → ℕ)
-    (hfirst : ∀v,first v ≤ 2*reserve v) (A B T : ℕ)
-    (hpre : ∀u,fpre u+(∑v,if left v=u then reserve v else 0)=A)
-    (hpost : ∀u,fpost u+(∑v,if right v=u then reserve v else 0)=B)
-    (u : Fin N) :
-    ((framePrefix (pairedPreCount left fpre first)
-      (pairedPostCount right fpost reserve first) (fun _ => T) u.val+
-      pairedPreCount left fpre first u : ℕ):ℤ)=
-      u.val*(A+T+B)+A+
-        ∑v, if (left v).val ≤ u.val ∧ u.val ≤ (right v).val then
-          (first v:ℤ)-reserve v else 0 := by
-  let d : K → ℤ := fun v => (first v:ℤ)-reserve v
-  have hpr := pairedPreCount_cast left fpre reserve first A hpre
-  have hpo := pairedPostCount_cast right fpost reserve first B hfirst hpost
-  have hsum : (∑j : Fin N, if j.val < u.val then (1:ℤ) else 0)=(u.val:ℤ) := by
-    have hc := framePrefix_constant (fun _ : Fin N => 0) (fun _ => 0) (fun _ => 1) 1 u.val
-      (by simp) (by omega)
-    simpa [framePrefix] using congrArg (fun n : ℕ => (n:ℤ)) hc
-  simp only [Nat.cast_add,framePrefix,Nat.cast_sum,Nat.cast_ite,Nat.cast_zero,hpr,hpo]
-  have hex : (∑j : Fin N,if j.val < u.val then
-        (A+(∑v,if left v=j then d v else 0))+T+
-        (B-(∑v,if right v=j then d v else 0)) else 0)=
-      (u.val:ℤ)*(A+T+B)+
-        ∑v,((if (left v).val < u.val then d v else 0)-
-          (if (right v).val < u.val then d v else 0)) := by
-    calc
-      _ = (∑j : Fin N,(if j.val < u.val then (1:ℤ) else 0)*(A+T+B))+
-          ∑j : Fin N,∑v,
-            ((if j.val < u.val then (if left v=j then d v else 0) else 0)-
-             (if j.val < u.val then (if right v=j then d v else 0) else 0)) := by
-        rw [←Finset.sum_add_distrib]
-        apply Finset.sum_congr rfl
-        intro j _
-        by_cases hj : j.val < u.val
-        · simp [hj,Finset.sum_sub_distrib]; ring
-        · simp [hj]
-      _ = _ := by
-        rw [←Finset.sum_mul,hsum,Finset.sum_comm]
-        simp_rw [Finset.sum_sub_distrib,fin_prefix_delta]
-  change (∑j : Fin N,if j.val < u.val then
-      (A+(∑v,if left v=j then d v else 0))+T+
-      (B-(∑v,if right v=j then d v else 0)) else 0)+
-      (A+∑v,if left v=u then d v else 0)=_
-  rw [hex]
-  have he : (∑v,((if (left v).val < u.val then d v else 0)-
-      (if (right v).val < u.val then d v else 0)))+
-      (∑v,if left v=u then d v else 0)=
-      ∑v,if (left v).val ≤ u.val ∧ u.val ≤ (right v).val then d v else 0 := by
-    rw [←Finset.sum_add_distrib]
-    exact Finset.sum_congr rfl (fun v _ => pair_activity_identity (left v) (right v) u (horder v) (d v))
-  dsimp only [d] at *
-  linarith only [he]
-
-end TraceReconstruction
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-variable {K : Type*} [Fintype K]
-
-theorem pairedLayout_total {N : ℕ} (left right : K → Fin N)
-    (fpre fpost : Fin N → ℕ) (reserve first : K → ℕ)
-    (hfirst : ∀v,first v ≤ 2*reserve v) (A B T : ℕ)
-    (hpre : ∀u,fpre u+(∑v,if left v=u then reserve v else 0)=A)
-    (hpost : ∀u,fpost u+(∑v,if right v=u then reserve v else 0)=B) :
-    framePrefix (pairedPreCount left fpre first)
-      (pairedPostCount right fpost reserve first) (fun _ => T) N=N*(A+T+B) := by
-  have hpr := pairedPreCount_cast left fpre reserve first A hpre
-  have hpo := pairedPostCount_cast right fpost reserve first B hfirst hpost
-  have he : ∀f : K → Fin N,
-      (∑u : Fin N,∑v,if f v=u then (first v:ℤ)-reserve v else 0)=
-        ∑v,((first v:ℤ)-reserve v) := by
-    intro f
-    rw [Finset.sum_comm]
-    simp
-  apply Nat.cast_injective (R:=ℤ)
-  simp only [framePrefix,Fin.isLt,ite_true,Nat.cast_sum,Nat.cast_add,hpr,hpo,
-    Finset.sum_add_distrib,Finset.sum_sub_distrib,he,Finset.sum_const,Finset.card_univ,
-    Fintype.card_fin,nsmul_eq_mul,Nat.cast_mul]
-  push_cast
-  ring
-
-/-- One localBits bundle at each child, and one globalBits bundle at each vertex in
- the first quarter, paired with the reflected last-quarter post region. -/
-abbrev SourceBundle (N : ℕ) := Fin N ⊕ Fin (N/4)
-
-def sourceBundleLeft (N : ℕ) : SourceBundle N → Fin N
-  | .inl u => u
-  | .inr v => ⟨v.val,by have := v.isLt; omega⟩
-
-def sourceBundleRight (N : ℕ) : SourceBundle N → Fin N
-  | .inl u => u
-  | .inr v => ⟨N-1-v.val,by have := v.isLt; omega⟩
-
-theorem sourceBundle_order (N : ℕ) (v : SourceBundle N) :
-    (sourceBundleLeft N v).val ≤ (sourceBundleRight N v).val := by
-  cases v with
-  | inl u => rfl
-  | inr v => dsimp [sourceBundleLeft,sourceBundleRight]; have := v.isLt; omega
-
-theorem sum_injective_delta_le {α β : Type*} [Fintype α] [DecidableEq β]
-    (f : α → β) (hf : Function.Injective f) (u : β) (R : ℕ) :
-    (∑v,if f v=u then R else 0) ≤ R := by
-  by_cases h : ∃v,f v=u
-  · obtain ⟨v,hv⟩ := h
-    have he : ∀j,f j=u ↔ j=v := fun j => by rw [←hv]; exact hf.eq_iff
-    simp [he]
-  · push Not at h
-    simp [h]
-
-theorem sourceBundle_reserve_pre (N R : ℕ) (u : Fin N) :
-    (∑v : SourceBundle N,if sourceBundleLeft N v=u then R else 0) ≤ 2*R := by
-  rw [Fintype.sum_sum_type]
-  change (∑v : Fin N, if v=u then R else 0)+(∑v : Fin (N/4),
-    if (⟨v.val,by have := v.isLt; omega⟩ : Fin N)=u then R else 0) ≤ 2*R
-  simp only [Finset.sum_ite_eq',Finset.mem_univ,ite_true]
-  have h := sum_injective_delta_le
-    (fun v : Fin (N/4) => (⟨v.val,by have := v.isLt; omega⟩ : Fin N))
-    (fun _ _ he => Fin.ext (congrArg (fun z : Fin N => z.val) he)) u R
-  omega
-
-theorem sourceBundle_reserve_post (N R : ℕ) (u : Fin N) :
-    (∑v : SourceBundle N,if sourceBundleRight N v=u then R else 0) ≤ 2*R := by
-  rw [Fintype.sum_sum_type]
-  change (∑v : Fin N, if v=u then R else 0)+(∑v : Fin (N/4),
-    if (⟨N-1-v.val,by have := v.isLt; omega⟩ : Fin N)=u then R else 0) ≤ 2*R
-  simp only [Finset.sum_ite_eq',Finset.mem_univ,ite_true]
-  have hinj : Function.Injective (fun v : Fin (N/4) =>
-      (⟨N-1-v.val,by have := v.isLt; omega⟩ : Fin N)) := by
-    intro v w he
-    have hv := v.isLt
-    have hw := w.isLt
-    have h := congrArg Fin.val he
-    apply Fin.ext
-    dsimp only at h
-    omega
-  have h := sum_injective_delta_le _ hinj u R
-  omega
-
-def sourceBundleCount (N t h : ℕ) (localBits : Fin N → Fin t → Bool)
-    (globalBits : Fin ((N/4)*t) → Bool) : SourceBundle N → ℕ
-  | .inl u => t*h+∑a,bitIndex (localBits u a)
-  | .inr v => t*h+∑a : Fin t,bitIndex (globalBits (globalTrialIndex N t v a))
-
-theorem sourceBundleCount_le (N t h : ℕ) (ht : Even t)
-    (localBits : Fin N → Fin t → Bool) (globalBits : Fin ((N/4)*t) → Bool)
-    (v : SourceBundle N) : sourceBundleCount N t h localBits globalBits v ≤ 2*(t*h+t/2) := by
-  have hh : 2*(t/2)=t := Nat.mul_div_cancel' ((even_iff_two_dvd).mp ht)
-  have hs (b : Fin t → Bool) : (∑a,bitIndex (b a)) ≤ t := by
-    calc
-      _ ≤ ∑_a : Fin t,1 := Finset.sum_le_sum (fun a _ => by cases b a <;> norm_num [bitIndex])
-      _ = t := by simp
-  cases v with
-  | inl u => dsimp [sourceBundleCount]; have := hs (localBits u); omega
-  | inr v => dsimp [sourceBundleCount]; have := hs (fun a => globalBits (globalTrialIndex N t v a)); omega
-
-/-- The actual scheduled bits give precisely the arithmetic source displacement
- Y=H+V, including its common-globalBits plateau and outer-quarter behavior. -/
-theorem sourceBundle_activity (N t h : ℕ)
-    (localBits : Fin N → Fin t → Bool) (globalBits : Fin ((N/4)*t) → Bool) (u : Fin N) :
-    (∑v : SourceBundle N, if (sourceBundleLeft N v).val ≤ u.val ∧
-        u.val ≤ (sourceBundleRight N v).val then
-      (sourceBundleCount N t h localBits globalBits v:ℤ)-(t*h+t/2:ℕ) else 0)=
-      ((∑a : Fin t,(bitIndex (localBits u a):ℤ))-(t/2:ℕ))+
-        scheduledGlobalShift N t globalBits u := by
-  rw [Fintype.sum_sum_type]
-  have he : (∑v : Fin N, if v.val ≤ u.val ∧ u.val ≤ v.val then
-      (sourceBundleCount N t h localBits globalBits (.inl v):ℤ)-(t*h+t/2:ℕ) else 0)=
-      (∑a : Fin t,(bitIndex (localBits u a):ℤ))-(t/2:ℕ) := by
-    have hc : ∀v : Fin N,(v.val ≤ u.val ∧ u.val ≤ v.val) ↔ v=u := by
-      intro v
-      constructor
-      · intro hv; exact Fin.ext (le_antisymm hv.1 hv.2)
-      · intro hv; subst v; exact ⟨le_rfl,le_rfl⟩
-    simp only [hc,Finset.sum_ite_eq',Finset.mem_univ,ite_true,sourceBundleCount,
-      Nat.cast_add,Nat.cast_mul,Nat.cast_sum]
-    ring
-  change (∑v : Fin N, if v.val ≤ u.val ∧ u.val ≤ v.val then
-      (sourceBundleCount N t h localBits globalBits (.inl v):ℤ)-(t*h+t/2:ℕ) else 0)+_=_
-  rw [he]
-  congr 1
-  unfold scheduledGlobalShift
-  apply Finset.sum_congr rfl
-  intro v _
-  have hv := v.isLt
-  have hu := u.isLt
-  have hc : (sourceBundleLeft N (.inr v)).val ≤ u.val ∧
-      u.val ≤ (sourceBundleRight N (.inr v)).val ↔ v.val ≤ u.val ∧ u.val+v.val < N := by
-    dsimp [sourceBundleLeft,sourceBundleRight]; omega
-  simp only [hc]
-  split_ifs
-  · simp only [sourceBundleCount,Nat.cast_add,Nat.cast_mul,Nat.cast_sum]; ring
-  · rfl
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- One-bundle reserved nominal count is integral because the number of
- paired trials is even. The actual filler reserves both local and global
- bundles when both are present in a region. -/
-def sourceLayoutReserve (t h : ℕ) : ℕ := t*h+t/2
-
-def sourcePreFiller (N A t h : ℕ) (u : Fin N) : ℕ :=
-  A-∑v : SourceBundle N,if sourceBundleLeft N v=u then sourceLayoutReserve t h else 0
-
-def sourcePostFiller (N B t h : ℕ) (u : Fin N) : ℕ :=
-  B-∑v : SourceBundle N,if sourceBundleRight N v=u then sourceLayoutReserve t h else 0
-
-def sourcePreCount (N A t h : ℕ) (loc : Fin N → Fin t → Bool)
-    (glob : Fin ((N/4)*t) → Bool) : Fin N → ℕ :=
-  pairedPreCount (sourceBundleLeft N) (sourcePreFiller N A t h)
-    (sourceBundleCount N t h loc glob)
-
-def sourcePostCount (N B t h : ℕ) (loc : Fin N → Fin t → Bool)
-    (glob : Fin ((N/4)*t) → Bool) : Fin N → ℕ :=
-  pairedPostCount (sourceBundleRight N) (sourcePostFiller N B t h)
-    (fun _ => sourceLayoutReserve t h) (sourceBundleCount N t h loc glob)
-
-def sourceDisplacement (N t : ℕ) (loc : Fin N → Fin t → Bool)
-    (glob : Fin ((N/4)*t) → Bool) (u : Fin N) : ℤ :=
-  (∑a : Fin t,(bitIndex (loc u a):ℤ))-(t/2:ℕ)+scheduledGlobalShift N t glob u
-
-def sourceWindowStart (N A t : ℕ) (loc : Fin N → Fin t → Bool)
-    (glob : Fin ((N/4)*t) → Bool) (u : Fin N) : ℕ :=
-  ((A:ℤ)+sourceDisplacement N t loc glob u).toNat
-
-theorem sourcePreFiller_nominal (N A t h : ℕ)
-    (hA : 2*sourceLayoutReserve t h ≤ A) (u : Fin N) :
-    sourcePreFiller N A t h u+
-      (∑v : SourceBundle N,if sourceBundleLeft N v=u then sourceLayoutReserve t h else 0)=A := by
-  exact Nat.sub_add_cancel ((sourceBundle_reserve_pre N _ u).trans hA)
-
-theorem sourcePostFiller_nominal (N B t h : ℕ)
-    (hB : 2*sourceLayoutReserve t h ≤ B) (u : Fin N) :
-    sourcePostFiller N B t h u+
-      (∑v : SourceBundle N,if sourceBundleRight N v=u then sourceLayoutReserve t h else 0)=B := by
-  exact Nat.sub_add_cancel ((sourceBundle_reserve_post N _ u).trans hB)
-
-theorem sourceLayout_exact_start (N A T B t h : ℕ) (ht : Even t)
-    (hA : 2*sourceLayoutReserve t h ≤ A) (hB : 2*sourceLayoutReserve t h ≤ B)
-    (loc : Fin N → Fin t → Bool) (glob : Fin ((N/4)*t) → Bool) (u : Fin N) :
-    ((framePrefix (sourcePreCount N A t h loc glob) (sourcePostCount N B t h loc glob)
-      (fun _ => T) u.val+sourcePreCount N A t h loc glob u:ℕ):ℤ)=
-      u.val*(A+T+B)+A+sourceDisplacement N t loc glob u := by
-  have H := pairedLayout_start (sourceBundleLeft N) (sourceBundleRight N)
-    (sourceBundle_order N) (sourcePreFiller N A t h) (sourcePostFiller N B t h)
-    (fun _ => sourceLayoutReserve t h) (sourceBundleCount N t h loc glob)
-    (sourceBundleCount_le N t h ht loc glob) A B T
-    (sourcePreFiller_nominal N A t h hA) (sourcePostFiller_nominal N B t h hB) u
-  simp only [sourceLayoutReserve] at H
-  rw [sourceBundle_activity] at H
-  exact H
-
-theorem sourceLayout_exact_total (N A T B t h : ℕ) (ht : Even t)
-    (hA : 2*sourceLayoutReserve t h ≤ A) (hB : 2*sourceLayoutReserve t h ≤ B)
-    (loc : Fin N → Fin t → Bool) (glob : Fin ((N/4)*t) → Bool) :
-    framePrefix (sourcePreCount N A t h loc glob) (sourcePostCount N B t h loc glob)
-      (fun _ => T) N=N*(A+T+B) := by
-  exact pairedLayout_total (sourceBundleLeft N) (sourceBundleRight N)
-    (sourcePreFiller N A t h) (sourcePostFiller N B t h)
-    (fun _ => sourceLayoutReserve t h) (sourceBundleCount N t h loc glob)
-    (sourceBundleCount_le N t h ht loc glob) A B T
-    (sourcePreFiller_nominal N A t h hA) (sourcePostFiller_nominal N B t h hB)
-
-/-- The literal surrogate side chunks, after concatenation, are a correct
- partition of the actual parent word whenever the windows fit. This is
- an equality of unmarked binary strings, not an assumed channel identity. -/
-theorem sourceLayout_literal_partition (k A T B t h a L : ℕ) (ht : Even t)
-    (hA : 2*sourceLayoutReserve t h ≤ A) (hB : 2*sourceLayoutReserve t h ≤ B)
-    (loc : Fin (k+1) → Fin t → Bool) (glob : Fin (((k+1)/4)*t) → Bool)
-    (sign : Fin L → Bool) (w : Bool → List Bool) (τ : Fin (k+1) → Bool)
-    (hzero : ∀u,0 ≤ (A:ℤ)+sourceDisplacement (k+1) t loc glob u)
-    (hleft : ∀u,sourceWindowStart (k+1) A t loc glob u ≤ a)
-    (hfit : ∀u,a+L ≤ sourceWindowStart (k+1) A t loc glob u+T)
-    (hunit : ∀u,sourceWindowStart (k+1) A t loc glob u+T ≤ A+T+B) :
-    (List.ofFn (fun u => complementaryWords (A+T+B) a L sign w (τ u))).flatten=
-      (List.ofFn (fun u => pureAtomRun (w false) (sourcePreCount (k+1) A t h loc glob u) ++
-        complementaryWords T (a-sourceWindowStart (k+1) A t loc glob u) L sign w (τ u) ++
-        pureAtomRun (w false) (sourcePostCount (k+1) B t h loc glob u))).flatten := by
-  apply template_group_windows k (A+T+B) T a L sign w τ
-    (sourceWindowStart (k+1) A t loc glob)
-    (sourcePreCount (k+1) A t h loc glob) (sourcePostCount (k+1) B t h loc glob)
-    hleft hfit hunit
-  · intro u
-    apply Nat.cast_injective (R:=ℤ)
-    rw [sourceLayout_exact_start (k+1) A T B t h ht hA hB]
-    simp only [Nat.cast_add,Nat.cast_mul,sourceWindowStart,Int.toNat_of_nonneg (hzero u)]
-    ring
-  · exact sourceLayout_exact_total (k+1) A T B t h ht hA hB loc glob
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- Independent emissions at all literal slots accumulate their actual
- approximation errors. The decoders may depend on the fixed slot ID. -/
-theorem coordinateProduct_approximates {α : Type*} [Fintype α]
-    {n : ℕ} {q : ℝ} (hq : q∈Set.Icc 0 1) (P : Fin n → FiniteLaw α)
-    (f : Fin n → α → List Bool) (w : Fin n → List Bool) (E : Fin n → ℝ)
-    (h : ∀i,ApproximatesTrace q (P i) (f i) (w i) (E i)) :
-    ApproximatesTrace q (FiniteLaw.coordinateProduct P)
-      (fun x => coordinateWord (fun i => f i (x i))) (coordinateWord w) (∑i,E i) := by
-  induction n with
-  | zero =>
-    intro F hF
-    change |(FiniteLaw.coordinateProduct P).expect (fun _ => F [])-F []|≤∑i : Fin 0,E i
-    simp only [FiniteLaw.expect_const,sub_self,abs_zero,Finset.univ_eq_empty,Finset.sum_empty,le_refl]
-  | succ n ih =>
-    intro F hF
-    have ht := ih (fun i => P i.succ) (fun i => f i.succ)
-      (fun i => w i.succ) (fun i => E i.succ) (fun i => h i.succ)
-    have hh := ApproximatesTrace.append hq (h 0) ht F hF
-    rw [FiniteLaw.expect_prod] at hh
-    rw [FiniteLaw.expect_coordinateProduct_succ,coordinateWord_succ,Fin.sum_univ_succ]
-    have he (a : α) (x : Fin n → α) :
-        coordinateWord (fun i => f i ((Fin.cons a x : Fin (n+1) → α) i))=
-          f 0 a++coordinateWord (fun i => f i.succ (x i)) := by
-      rw [coordinateWord_succ]
-      simp only [Fin.cons_zero,Fin.cons_succ]
-    simp_rw [he]
-    exact hh
-
-namespace FiniteLaw
-variable {ι κ α : Type*} [Fintype ι] [Fintype κ] [Fintype α]
-
-/-- Independent emissions indexed by the actual token IDs of the layout. -/
-def indexedProduct (P : ι → FiniteLaw α) : FiniteLaw (ι → α) where
-  mass x := ∏i,(P i).mass (x i)
-  nonneg x := Finset.prod_nonneg (fun i _ => (P i).nonneg _)
-  total := by
-    rw [←Fintype.prod_sum]
-    simp only [(P _).total,Finset.prod_const_one]
-
-/-- Reorder independent emissions by the literal chronological slot order.
-The order is a bijection, so no output is duplicated or silently discarded. -/
-theorem indexedProduct_expect_equiv (P : ι → FiniteLaw α) (e : κ ≃ ι)
-    (F : (ι → α) → ℝ) :
-    (indexedProduct P).expect F=(indexedProduct (fun j => P (e j))).expect
-      (fun x => F (fun i => x (e.symm i))) := by
-  let E : (κ → α) ≃ (ι → α) :=
-    { toFun := fun x i => x (e.symm i)
-      invFun := fun x j => x (e j)
-      left_inv := fun x => by funext j; simp
-      right_inv := fun x => by funext i; simp }
-  unfold expect indexedProduct
-  rw [←E.sum_comp]
-  apply Finset.sum_congr rfl
-  intro x _
-  change (∏i,(P i).mass (x (e.symm i)))*F (fun i => x (e.symm i))=_
-  congr 1
-  rw [←e.prod_comp]
-  simp only [Equiv.symm_apply_apply]
-
-theorem indexedProduct_map (P : ι → FiniteLaw α) {β : Type*} [Fintype β]
-    (f : ι → α → β) :
-    (indexedProduct P).map (fun x i => f i (x i))=
-      indexedProduct (fun i => (P i).map (f i)) := by
-  apply FiniteLaw.ext
-  funext y
-  rw [map_mass]
-  change (∑x : ι → α,if (fun i => f i (x i))=y then ∏i,(P i).mass (x i) else 0)=_
-  simp only [indexedProduct,map_mass]
-  rw [Fintype.prod_sum]
-  apply Finset.sum_congr rfl
-  intro x _
-  by_cases he : (fun i => f i (x i))=y
-  · rw [ite_eq_left he]
-    exact Finset.prod_congr rfl (fun i _ => (ite_eq_left (congrFun he i)).symm)
-  · rw [ite_eq_right he]
-    symm
-    have hi : ∃i,f i (x i)≠y i := by
-      by_contra h
-      push Not at h
-      exact he (funext h)
-    obtain ⟨i,hi⟩ := hi
-    exact Finset.prod_eq_zero (Finset.mem_univ i) (ite_eq_right hi)
-
-end FiniteLaw
-
-/-- An arbitrarily indexed actual layout has the same unconditional trace
- guarantee after its fixed, public chronological ordering is applied. -/
-theorem indexedProduct_approximates {ι α : Type*} [Fintype ι] [Fintype α]
-    {n : ℕ} {q : ℝ} (hq : q∈Set.Icc 0 1) (e : Fin n ≃ ι)
-    (P : ι → FiniteLaw α) (f : ι → α → List Bool) (w : ι → List Bool) (E : ι → ℝ)
-    (h : ∀i,ApproximatesTrace q (P i) (f i) (w i) (E i)) :
-    ApproximatesTrace q (FiniteLaw.indexedProduct P)
-      (fun x => coordinateWord (fun j => f (e j) (x (e j))))
-      (coordinateWord (fun j => w (e j))) (∑i,E i) := by
-  intro F hF
-  rw [FiniteLaw.indexedProduct_expect_equiv P e]
-  simp only [Equiv.symm_apply_apply]
-  have H := coordinateProduct_approximates hq (fun j => P (e j))
-    (fun j => f (e j)) (fun j => w (e j)) (fun j => E (e j)) (fun j => h (e j)) F hF
-  rw [e.sum_comp] at H
-  simp only [FiniteLaw.expect,FiniteLaw.indexedProduct,FiniteLaw.coordinateProduct] at H ⊢
-  convert H using 1
-  congr 3
-  ext x
-  simp
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- The Boolean representation of the source Prouhet signs; true denotes
- a negative type. No sign convention is changed in passing to literal words. -/
-def boolTypeSign (b : Bool) : ℝ := if b then -1 else 1
-
-def prouhetBool (s : ℕ) (i : Fin (2^s)) : Bool :=
-  decide ((prouhetSeries s).coeff (i.val:ℤ)= -1)
-
-theorem boolTypeSign_injective : Function.Injective boolTypeSign := by
-  intro a b h
-  cases a <;> cases b <;> norm_num [boolTypeSign] at *
-
-theorem boolTypeSign_template (s P a : ℕ) (τ : Bool) (i : Fin P) :
-    boolTypeSign (templateType P a (2^s) (prouhetBool s) τ i)=
-      motifChildSign s (a:ℤ) i.val (boolTypeSign τ) := by
-  by_cases hi : a ≤ i.val ∧ i.val < a+2^s
-  · have hz : 0 ≤ (i.val:ℤ)-a ∧ (i.val:ℤ)-a < (2^s:ℕ) := by omega
-    have he : ((i.val-a:ℕ):ℤ)=(i.val:ℤ)-a := by omega
-    have hc := (abs_eq (by norm_num : (0:ℝ)≤1)).mp
-      ((prouhetSeries_coeff_spec s).2 _ hz.1 hz.2)
-    have hχ : motifIndicator s ((i.val:ℤ)-a)=1 := by simp only [motifIndicator,ite_eq_left hz]
-    rcases hc with hc | hc <;> cases τ <;>
-      norm_num [templateType,hi,prouhetBool,he,boolTypeSign,motifChildSign,hχ,hc]
-  · have hz : ¬(0 ≤ (i.val:ℤ)-a ∧ (i.val:ℤ)-a < (2^s:ℕ)) := by omega
-    have hc : (prouhetSeries s).coeff ((i.val:ℤ)-a)=0 :=
-      (prouhetSeries_coeff_spec s).1 _ (by omega)
-    have hχ : motifIndicator s ((i.val:ℤ)-a)=0 := by simp only [motifIndicator,ite_eq_right hz]
-    simp [templateType,hi,boolTypeSign,motifChildSign,hχ,hc]
-
-theorem sourceWindowStart_cast_of_fit (N P T t : ℕ)
-    (loc : Fin N → Fin t → Bool) (glob : Fin ((N/4)*t) → Bool)
-    (hP : 2*T ≤ P) (hT : 16 ≤ T) (u : Fin N)
-    (hY : |8*sourceDisplacement N t loc glob u| ≤ (T:ℤ)) :
-    (sourceWindowStart N ((P-T)/2) t loc glob u:ℤ)=
-      (((P-T)/2:ℕ):ℤ)+sourceDisplacement N t loc glob u := by
-  apply Int.toNat_of_nonneg
-  have hy := abs_le.mp hY
-  omega
-
-/-- A zero-based centered version of the floor-fit calculation. -/
-theorem sourceWindowStart_fit (N P T t L : ℕ)
-    (loc : Fin N → Fin t → Bool) (glob : Fin ((N/4)*t) → Bool)
-    (hP : 2*T ≤ P) (hT : 16 ≤ T) (heven : Even T) (hL : 8*L ≤ T)
-    (u : Fin N) (hY : |8*sourceDisplacement N t loc glob u| ≤ (T:ℤ)) :
-    sourceWindowStart N ((P-T)/2) t loc glob u+T ≤ P ∧
-    sourceWindowStart N ((P-T)/2) t loc glob u ≤ P/2 ∧
-    P/2+L ≤ sourceWindowStart N ((P-T)/2) t loc glob u+T :=
-  centered_window_floor_fit P T L _ _ hP hT heven hL hY
-    (sourceWindowStart_cast_of_fit N P T t loc glob hP hT u hY)
-
-/-- The zero-based centered translated anchor, including both half-trial corrections. -/
-theorem sourceWindowStart_anchor (N P T t L : ℕ)
-    (loc : Fin N → Fin t → Bool) (glob : Fin ((N/4)*t) → Bool)
-    (hP : 2*T ≤ P) (hT : 16 ≤ T) (heven : Even T) (hL : 8*L ≤ T)
-    (u : Fin N) (hY : |8*sourceDisplacement N t loc glob u| ≤ (T:ℤ)) :
-    ((P/2-sourceWindowStart N ((P-T)/2) t loc glob u:ℕ):ℤ)=
-      (T/2:ℕ)+(t/2:ℕ)-(∑a : Fin t,(bitIndex (loc u a):ℤ))-
-        scheduledGlobalShift N t glob u := by
-  have hfit := sourceWindowStart_fit N P T t L loc glob hP hT heven hL u hY
-  have hc := sourceWindowStart_cast_of_fit N P T t loc glob hP hT u hY
-  have ha := centered_window_floor_anchor P T (by omega) heven
-  have hA : (P-T)/2 ≤ P/2 := Nat.div_le_div_right (Nat.sub_le P T)
-  have ha' : ((P/2:ℕ):ℤ)-((P-T)/2:ℕ)=(T/2:ℕ) := by omega
-  rw [Nat.cast_sub hfit.2.1,hc]
-  dsimp only [sourceDisplacement]
-  linarith only [ha']
-
-/-- The manuscript numbers motif slots from one. Thus its actual zero-based
- onset is P/2-1, not P/2. The strict margins include this endpoint correction. -/
-theorem sourceWindowStart_oneBased_fit (N P T t L : ℕ)
-    (loc : Fin N → Fin t → Bool) (glob : Fin ((N/4)*t) → Bool)
-    (hP : 2*T ≤ P) (hT : 16 ≤ T) (heven : Even T) (hL : 8*L ≤ T)
-    (u : Fin N) (hY : |8*sourceDisplacement N t loc glob u| ≤ (T:ℤ)) :
-    sourceWindowStart N ((P-T)/2) t loc glob u+T ≤ P ∧
-    sourceWindowStart N ((P-T)/2) t loc glob u ≤ P/2-1 ∧
-    P/2-1+L ≤ sourceWindowStart N ((P-T)/2) t loc glob u+T := by
-  have hc := sourceWindowStart_cast_of_fit N P T t loc glob hP hT u hY
-  have ha := centered_window_floor_anchor P T (by omega) heven
-  have hy := abs_le.mp hY
-  omega
-
-/-- The exact source anchor in zero-based coordinates. The minus one is
- forced by the one-based motif-label convention. -/
-theorem sourceWindowStart_oneBased_anchor (N P T t L : ℕ)
-    (loc : Fin N → Fin t → Bool) (glob : Fin ((N/4)*t) → Bool)
-    (hP : 2*T ≤ P) (hT : 16 ≤ T) (heven : Even T) (hL : 8*L ≤ T)
-    (u : Fin N) (hY : |8*sourceDisplacement N t loc glob u| ≤ (T:ℤ)) :
-    ((P/2-1-sourceWindowStart N ((P-T)/2) t loc glob u:ℕ):ℤ)=
-      (T/2:ℕ)-1+(t/2:ℕ)-(∑a : Fin t,(bitIndex (loc u a):ℤ))-
-        scheduledGlobalShift N t glob u := by
-  have hf := sourceWindowStart_oneBased_fit N P T t L loc glob hP hT heven hL u hY
-  have ha := sourceWindowStart_anchor N P T t L loc glob hP hT heven hL u hY
-  omega
-
-/-- The actual recursively substituted source block words. Indices in P
- are shifted down by one, but physical binary strings are unchanged. -/
-def sourceBlockWord (s : ℕ) (P : ℕ → ℕ) (atom : Bool → List Bool) : ℕ → Bool → List Bool
-  | 0 => atom
-  | j+1 => complementaryWords (P j) (P j/2-1) (2^s) (prouhetBool s)
-      (sourceBlockWord s P atom j)
-
-def sourceBlockLength (P : ℕ → ℕ) (S : ℕ) : ℕ → ℕ
-  | 0 => S
-  | j+1 => P j*sourceBlockLength P S j
-
-theorem sourceBlockWord_length (s : ℕ) (P : ℕ → ℕ) (atom : Bool → List Bool) (S : ℕ)
-    (hw : ∀τ,(atom τ).length=S) (j : ℕ) (τ : Bool) :
-    (sourceBlockWord s P atom j τ).length=sourceBlockLength P S j := by
-  induction j generalizing τ with
-  | zero => exact hw τ
-  | succ j ih => exact complementaryWords_length _ _ _ _ _ _ (fun τ => ih τ) τ
-
-theorem sourceBlockWord_injective (s : ℕ) (P : ℕ → ℕ) (atom : Bool → List Bool) (S : ℕ)
-    (hw : ∀τ,(atom τ).length=S) (hinj : Function.Injective atom) (j : ℕ)
-    (hP : ∀k<j,0<P k) : Function.Injective (sourceBlockWord s P atom j) := by
-  induction j with
-  | zero => exact hinj
-  | succ j ih =>
-    apply complementaryWords_injective _ _ _ (sourceBlockLength P S j)
-      _ _ (sourceBlockWord_length s P atom S hw j)
-    · exact ih (fun k hk => hP k (by omega))
-    · positivity
-    · have h := hP j (Nat.lt_succ_self j); omega
-
-end TraceReconstruction
-
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- Total Boolean type, also on failed layouts; no conditioning is hidden
- in the definition. -/
-def integerTemplateType (N s : ℕ) (X : ℤ) (τ : Bool) (u : Fin N) : Bool :=
-  decide (motifChildSign s X u.val (boolTypeSign τ)= -1)
-
-theorem boolTypeSign_integerTemplate (N s : ℕ) (X : ℤ) (τ : Bool) (u : Fin N) :
-    boolTypeSign (integerTemplateType N s X τ u)=motifChildSign s X u.val (boolTypeSign τ) := by
-  have hτ : IsTypeSign (boolTypeSign τ) := by cases τ <;> norm_num [IsTypeSign,boolTypeSign]
-  rcases motifChildSign_isSign s X u.val hτ with h | h <;>
-    (simp only [integerTemplateType,h]; norm_num [boolTypeSign])
-
-theorem integerTemplateType_nat (N s a : ℕ) (τ : Bool) :
-    integerTemplateType N s (a:ℤ) τ=templateType N a (2^s) (prouhetBool s) τ := by
-  funext u
-  apply boolTypeSign_injective
-  rw [boolTypeSign_integerTemplate,boolTypeSign_template]
-
-/-- Global trials at a parent use its children's local-trial count. This
- fixes the index shift between source level labels and zero-based arrays. -/
-def sourceScheduledTrials (T n : ℕ → ℕ) : ℕ → ℕ
-  | 0 => 0
-  | b+1 => (T (b+1)/4)*n b
-
-def sourceScheduledShift (T n : ℕ → ℕ) :
-    (b : ℕ) → (Fin (sourceScheduledTrials T n b) → Bool) → Fin (T b) → ℤ
-  | 0, _, _ => 0
-  | b+1, glob, u => scheduledGlobalShift (T (b+1)) (n b) glob u
-
-def sourceLocalCenter (T n : ℕ → ℕ) (b : ℕ) : ℤ :=
-  (T b/2:ℕ)-1+(n b/2:ℕ)
-
-def sourceNodeAnchor (T n : ℕ → ℕ) (center : ℕ → ℤ) (b : ℕ)
-    (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool (b+1)) (incoming : ℤ) : ℤ :=
-  center b-(∑i,(bitIndex (bits.1 i):ℤ))-incoming
-
-/-- All actual leaf Booleans on the complete deterministic window tree. -/
-def sourceTreeLeafBool (T n : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ) :
-    (b : ℕ) → WindowAssignments T n (sourceScheduledTrials T n) Bool b → ℤ → Bool → WindowLeaf T b → Bool
-  | 0, _, _, τ, _ => τ
-  | b+1, bits, incoming, τ, v =>
-      sourceTreeLeafBool T n s center b (bits.2.2 v.1)
-        (sourceScheduledShift T n b bits.2.1 v.1)
-        (integerTemplateType (T b) s (sourceNodeAnchor T n center b bits incoming) τ v.1) v.2
-
-theorem sourceTreeLeafBool_sign (T n : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (b : ℕ) (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool b)
-    (incoming : ℤ) (τ : Bool) (v : WindowLeaf T b) :
-    boolTypeSign (sourceTreeLeafBool T n s center b bits incoming τ v)=
-      sourceTreeLeafSign T n (sourceScheduledTrials T n) s center
-        (sourceScheduledShift T n) b bits incoming (boolTypeSign τ) v := by
-  induction b generalizing incoming τ with
-  | zero => rfl
-  | succ b ih =>
-    change boolTypeSign (sourceTreeLeafBool T n s center b _ _ _ v.2)=_
-    rw [ih,boolTypeSign_integerTemplate]
-    rfl
-
-def sourceNominalPre (P T : ℕ → ℕ) (b : ℕ) : ℕ := (P b-T b)/2
-def sourceNominalPost (P T : ℕ → ℕ) (b : ℕ) : ℕ := P b-T b-sourceNominalPre P T b
-
-theorem sourceNominal_total (P T : ℕ → ℕ) (b : ℕ) (h : T b ≤ P b) :
-    sourceNominalPre P T b+T b+sourceNominalPost P T b=P b := by
-  dsimp [sourceNominalPre,sourceNominalPost]
-  omega
-
-/-- The actual center before recursively cutting its pure side regions. -/
-def sourceCenterWord (s : ℕ) (P T n : ℕ → ℕ) (center : ℕ → ℤ) (atom : Bool → List Bool) :
-    (b : ℕ) → WindowAssignments T n (sourceScheduledTrials T n) Bool b → ℤ → Bool → List Bool
-  | 0, _, _, τ => atom τ
-  | b+1, bits, incoming, τ => atomString (T b) (sourceBlockWord s P atom b)
-      (integerTemplateType (T b) s (sourceNodeAnchor T n center b bits incoming) τ)
-
-/-- The unconditional literal surrogate partition: local and common-global
- pairs contribute their real selected pure lengths, on every bit assignment. -/
-def sourceTreeWord (s : ℕ) (P T n h : ℕ → ℕ) (center : ℕ → ℤ) (atom : Bool → List Bool) :
-    (b : ℕ) → WindowAssignments T n (sourceScheduledTrials T n) Bool b → ℤ → Bool → List Bool
-  | 0, _, _, τ => atom τ
-  | 1, bits, incoming, τ => sourceCenterWord s P T n center atom 1 bits incoming τ
-  | b+2, bits, incoming, τ =>
-      let loc := fun u => (bits.2.2 u).1
-      let glob := bits.2.1
-      coordinateWord (fun u : Fin (T (b+1)) =>
-        pureAtomRun (sourceBlockWord s P atom b false)
-          (sourcePreCount (T (b+1)) (sourceNominalPre P T b) (n b) (h b) loc glob u) ++
-        sourceTreeWord s P T n h center atom (b+1) (bits.2.2 u)
-          (sourceScheduledShift T n (b+1) glob u)
-          (integerTemplateType (T (b+1)) s (sourceNodeAnchor T n center (b+1) bits incoming) τ u) ++
-        pureAtomRun (sourceBlockWord s P atom b false)
-          (sourcePostCount (T (b+1)) (sourceNominalPost P T b) (n b) (h b) loc glob u))
-
-/-- The actual fit event uses a joint test of all displacements. It does not
- assert independence and will only be charged probabilistically afterwards. -/
-def SourceTreeFit (T n : ℕ → ℕ) :
-    (b : ℕ) → WindowAssignments T n (sourceScheduledTrials T n) Bool b → Prop
-  | 0, _ => True
-  | 1, _ => True
-  | b+2, bits =>
-      (∀u, |8*sourceDisplacement (T (b+1)) (n b) (fun v => (bits.2.2 v).1) bits.2.1 u| ≤ (T b:ℤ)) ∧
-      ∀u, SourceTreeFit T n (b+1) (bits.2.2 u)
-
-/-- Numerical layout conditions from the source. These contain no word or
- statistical conclusions, only the genuine integer margins. -/
-structure SourceLayoutMargins (s : ℕ) (P T n h : ℕ → ℕ) (b : ℕ) : Prop where
-  positive : ∀j<b, 0<T j
-  large : ∀j<b, 16≤T j
-  even_window : ∀j<b, Even (T j)
-  ambient : ∀j<b, 2*T j≤P j
-  motif : ∀j<b, 8*2^s≤T j
-  even_trials : ∀j<b, Even (n j)
-  pre_reserve : ∀j<b, 2*sourceLayoutReserve (n j) (h j)≤ sourceNominalPre P T j
-  post_reserve : ∀j<b, 2*sourceLayoutReserve (n j) (h j)≤ sourceNominalPost P T j
-
-theorem SourceLayoutMargins.restrict {s : ℕ} {P T n h : ℕ → ℕ} {b c : ℕ}
-    (H : SourceLayoutMargins s P T n h b) (hc : c≤b) : SourceLayoutMargins s P T n h c := by
-  constructor <;> intro j hj
-  · exact H.positive j (lt_of_lt_of_le hj hc)
-  · exact H.large j (lt_of_lt_of_le hj hc)
-  · exact H.even_window j (lt_of_lt_of_le hj hc)
-  · exact H.ambient j (lt_of_lt_of_le hj hc)
-  · exact H.motif j (lt_of_lt_of_le hj hc)
-  · exact H.even_trials j (lt_of_lt_of_le hj hc)
-  · exact H.pre_reserve j (lt_of_lt_of_le hj hc)
-  · exact H.post_reserve j (lt_of_lt_of_le hj hc)
-
-/-- Every recursively cut surrogate center is literally the actual center on
- the joint fit event. The source type assignment is retained at all leaves. -/
-theorem sourceTreeWord_eq_center (s : ℕ) (P T n h : ℕ → ℕ) (center : ℕ → ℤ)
-    (atom : Bool → List Bool) (b : ℕ) (H : SourceLayoutMargins s P T n h b)
-    (hc : ∀j, j+1<b → center j=sourceLocalCenter T n j)
-    (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool b)
-    (incoming : ℤ) (τ : Bool) (hfit : SourceTreeFit T n b bits) :
-    sourceTreeWord s P T n h center atom b bits incoming τ=
-      sourceCenterWord s P T n center atom b bits incoming τ := by
-  induction b using Nat.twoStepInduction generalizing incoming τ with
-  | zero => rfl
-  | one => rfl
-  | more b ih₁ ih₂ =>
-    change (Fin (n (b+1)) → Bool) × (Fin ((T (b+1)/4)*n b) → Bool) ×
-      (Fin (T (b+1)) → WindowAssignments T n (sourceScheduledTrials T n) Bool (b+1)) at bits
-    let loc : Fin (T (b+1)) → Fin (n b) → Bool := fun u => (bits.2.2 u).1
-    let glob := bits.2.1
-    let a := P b/2-1
-    let A := sourceNominalPre P T b
-    let B := sourceNominalPost P T b
-    let start := sourceWindowStart (T (b+1)) A (n b) loc glob
-    let θ := integerTemplateType (T (b+1)) s (sourceNodeAnchor T n center (b+1) bits incoming) τ
-    have hb : b<b+2 := by omega
-    have htotal := sourceNominal_total P T b (by have hh := H.ambient b hb; omega)
-    have hf (u : Fin (T (b+1))) := sourceWindowStart_oneBased_fit (T (b+1)) (P b) (T b) (n b)
-      (2^s) loc glob (H.ambient b hb) (H.large b hb) (H.even_window b hb) (H.motif b hb) u (hfit.1 u)
-    have hx (u : Fin (T (b+1))) :
-        sourceNodeAnchor T n center b (bits.2.2 u) (sourceScheduledShift T n (b+1) glob u)=((a-start u:ℕ):ℤ) := by
-      change sourceNodeAnchor T n center b (bits.2.2 u) (sourceScheduledShift T n (b+1) glob u)=
-        ((P b/2-1-sourceWindowStart (T (b+1)) ((P b-T b)/2) (n b) loc glob u:ℕ):ℤ)
-      rw [sourceWindowStart_oneBased_anchor (T (b+1)) (P b) (T b) (n b) (2^s) loc glob
-        (H.ambient b hb) (H.large b hb) (H.even_window b hb) (H.motif b hb) u (hfit.1 u)]
-      simp only [sourceNodeAnchor,hc b (by omega),sourceLocalCenter,sourceScheduledShift,loc]
-    have hchild (u : Fin (T (b+1))) :
-        sourceTreeWord s P T n h center atom (b+1) (bits.2.2 u)
-          (sourceScheduledShift T n (b+1) glob u) (θ u)=
-        complementaryWords (T b) (a-start u) (2^s) (prouhetBool s) (sourceBlockWord s P atom b) (θ u) := by
-      rw [ih₂ (H.restrict (by omega)) (fun j hj => hc j (by omega)) _ _ _ (hfit.2 u)]
-      simp only [sourceCenterWord,hx,integerTemplateType_nat,complementaryWords]
-    have hstartnonneg (u : Fin (T (b+1))) : 0≤(A:ℤ)+sourceDisplacement (T (b+1)) (n b) loc glob u := by
-      have he := sourceWindowStart_cast_of_fit (T (b+1)) (P b) (T b) (n b) loc glob
-        (H.ambient b hb) (H.large b hb) u (hfit.1 u)
-      have hn : (0:ℤ)≤(start u:ℕ) := Int.natCast_nonneg _
-      change (start u:ℤ)=(A:ℤ)+_ at he
-      omega
-    have hp := H.positive (b+1) (by omega)
-    obtain ⟨k,hk⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : T (b+1)≠0)
-    have hpart : (List.ofFn (fun u => complementaryWords (P b) a (2^s) (prouhetBool s)
-          (sourceBlockWord s P atom b) (θ u))).flatten=
-        coordinateWord (fun u => pureAtomRun (sourceBlockWord s P atom b false)
-          (sourcePreCount (T (b+1)) A (n b) (h b) loc glob u) ++
-          complementaryWords (T b) (a-start u) (2^s) (prouhetBool s) (sourceBlockWord s P atom b) (θ u) ++
-          pureAtomRun (sourceBlockWord s P atom b false) (sourcePostCount (T (b+1)) B (n b) (h b) loc glob u)) := by
-      have hh := sourceLayout_literal_partition k A (T b) B (n b) (h b) a (2^s)
-      have hk' : k+1=T (b+1) := hk.symm
-      rw [hk'] at hh
-      rw [show A+T b+B=P b from htotal] at hh
-      exact hh (H.even_trials b hb) (H.pre_reserve b hb) (H.post_reserve b hb) loc glob
-        (prouhetBool s) (sourceBlockWord s P atom b) θ hstartnonneg
-        (fun u => (hf u).2.1) (fun u => (hf u).2.2) (fun u => (hf u).1)
-    change coordinateWord (fun u : Fin (T (b+1)) =>
-      pureAtomRun (sourceBlockWord s P atom b false) (sourcePreCount (T (b+1)) A (n b) (h b) loc glob u) ++
-      sourceTreeWord s P T n h center atom (b+1) (bits.2.2 u)
-        (sourceScheduledShift T n (b+1) glob u) (θ u) ++
-      pureAtomRun (sourceBlockWord s P atom b false) (sourcePostCount (T (b+1)) B (n b) (h b) loc glob u))=
-        atomString (T (b+1)) (sourceBlockWord s P atom (b+1)) θ
-    calc
-      _ = coordinateWord (fun u : Fin (T (b+1)) =>
-        pureAtomRun (sourceBlockWord s P atom b false) (sourcePreCount (T (b+1)) A (n b) (h b) loc glob u) ++
-        complementaryWords (T b) (a-start u) (2^s) (prouhetBool s) (sourceBlockWord s P atom b) (θ u) ++
-        pureAtomRun (sourceBlockWord s P atom b false) (sourcePostCount (T (b+1)) B (n b) (h b) loc glob u)) := by
-          apply congrArg coordinateWord
-          funext u
-          exact congrArg (fun w => pureAtomRun (sourceBlockWord s P atom b false)
-            (sourcePreCount (T (b+1)) A (n b) (h b) loc glob u) ++ w ++
-              pureAtomRun (sourceBlockWord s P atom b false)
-                (sourcePostCount (T (b+1)) B (n b) (h b) loc glob u)) (hchild u)
-      _ = _ := hpart.symm
-
-end TraceReconstruction
-
-end
-end
-end
-end
-end
-end
-end
-end
-end
-
-end OAI
+import OAI.Probability.TraceReconstruction.Shapes
+
+namespace OAI
+
+/-!
+# Layout
+
+Multiwindow and paired layouts, coordinate simulations, and literal source types and trees.
+-/
+
+open scoped ENNReal
+noncomputable section
+
+open Filter
+open scoped Topology
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- Literal unmarked concatenation of fixed windows with separately reported
+pure pre/post regions. Counts are in atoms, not surviving trace symbols. -/
+def framedConcat {α : Type*} (e : α) {k : ℕ}
+    (pre post : Fin k → ℕ) (W : Fin k → List α) : List α :=
+  (List.ofFn (fun i => List.replicate (pre i) e ++ W i ++
+    List.replicate (post i) e)).flatten
+
+theorem framedConcat_succ {α : Type*} (e : α) {k : ℕ}
+    (pre post : Fin (k+1) → ℕ) (W : Fin (k+1) → List α) :
+    framedConcat e pre post W = List.replicate (pre 0) e ++ W 0 ++
+      List.replicate (post 0) e ++
+      framedConcat e (fun i => pre i.succ) (fun i => post i.succ) (fun i => W i.succ) := by
+  simp [framedConcat,List.ofFn_succ,List.append_assoc]
+
+/-- Only the first, last, and intervening pure lengths are observable in an
+unmarked word. In particular boundaries may jitter across nominal units. -/
+theorem framedConcat_eq_of_gaps {α : Type*} (e : α) (k : ℕ)
+    (pre post pre' post' : Fin (k+1) → ℕ) (W : Fin (k+1) → List α)
+    (u u' : ℕ) (hfirst : u+pre 0=u'+pre' 0)
+    (hgap : ∀i : Fin k, post i.castSucc+pre i.succ=post' i.castSucc+pre' i.succ)
+    (hlast : post (Fin.last k)=post' (Fin.last k)) :
+    List.replicate u e ++ framedConcat e pre post W =
+      List.replicate u' e ++ framedConcat e pre' post' W := by
+  induction k generalizing u u' with
+  | zero =>
+    have hl : post 0=post' 0 := hlast
+    simp only [framedConcat,List.ofFn_succ,List.ofFn_zero,List.flatten_cons,
+      List.flatten_nil,List.append_nil]
+    calc
+      _ = List.replicate (u+pre 0) e ++ W 0 ++ List.replicate (post 0) e := by
+        simp only [List.replicate_add,List.append_assoc]
+      _ = List.replicate (u'+pre' 0) e ++ W 0 ++ List.replicate (post' 0) e := by rw [hfirst,hl]
+      _ = _ := by simp only [List.replicate_add,List.append_assoc]
+  | succ k ih =>
+    rw [framedConcat_succ e pre post W,framedConcat_succ e pre' post' W]
+    have ht := ih (fun i => pre i.succ) (fun i => post i.succ)
+      (fun i => pre' i.succ) (fun i => post' i.succ) (fun i => W i.succ)
+      (post 0) (post' 0) (hgap 0) (fun i => hgap i.succ) hlast
+    calc
+      _ = List.replicate (u+pre 0) e ++ W 0 ++
+          (List.replicate (post 0) e ++ framedConcat e
+            (fun i => pre i.succ) (fun i => post i.succ) (fun i => W i.succ)) := by
+        simp only [List.replicate_add,List.append_assoc]
+      _ = List.replicate (u'+pre' 0) e ++ W 0 ++
+          (List.replicate (post' 0) e ++ framedConcat e
+            (fun i => pre' i.succ) (fun i => post' i.succ) (fun i => W i.succ)) := by rw [hfirst,ht]
+      _ = _ := by simp only [List.replicate_add,List.append_assoc]
+
+/-- Number of atoms strictly before the indicated padded unit. -/
+def framePrefix {k : ℕ} (pre post : Fin k → ℕ) (len : Fin k → ℕ) (i : ℕ) : ℕ :=
+  ∑j : Fin k, if j.val < i then pre j+len j+post j else 0
+
+theorem framePrefix_zero {k : ℕ} (pre post len : Fin k → ℕ) :
+    framePrefix pre post len 0=0 := by simp [framePrefix]
+
+theorem framePrefix_step {k : ℕ} (pre post len : Fin k → ℕ) (i : Fin k) :
+    framePrefix pre post len (i.val+1)=framePrefix pre post len i.val+pre i+len i+post i := by
+  have he : (fun j : Fin k => if j.val < i.val+1 then pre j+len j+post j else 0)=
+      fun j => (if j.val < i.val then pre j+len j+post j else 0)+
+        (if j=i then pre i+len i+post i else 0) := by
+    funext j
+    by_cases h : j=i
+    · subst j; simp
+    · have hn : j.val ≠ i.val := fun h' => h (Fin.ext h')
+      by_cases hlt : j.val < i.val
+      · simp [hlt,show j.val < i.val+1 by omega,h]
+      · simp [hlt,show ¬j.val < i.val+1 by omega,h]
+  simp only [framePrefix,he,Finset.sum_add_distrib,Finset.sum_ite_eq',Finset.mem_univ,ite_true]
+  omega
+
+/-- Equal actual starting positions and equal total counts are enough:
+there is no requirement that a part coincide with its nominal unit. -/
+theorem framedConcat_eq_of_starts {α : Type*} (e : α) (k : ℕ)
+    (pre post pre' post' : Fin (k+1) → ℕ) (W : Fin (k+1) → List α)
+    (hstart : ∀i : Fin (k+1),
+      framePrefix pre post (fun j => (W j).length) i.val+pre i=
+      framePrefix pre' post' (fun j => (W j).length) i.val+pre' i)
+    (htotal : framePrefix pre post (fun j => (W j).length) (k+1)=
+      framePrefix pre' post' (fun j => (W j).length) (k+1)) :
+    framedConcat e pre post W=framedConcat e pre' post' W := by
+  have H := framedConcat_eq_of_gaps e k pre post pre' post' W 0 0
+  simp only [List.replicate_zero,List.nil_append] at H
+  apply H
+  · simpa only [Fin.val_zero,framePrefix_zero,zero_add] using hstart 0
+  · intro i
+    have h₁ := hstart i.castSucc
+    have h₂ := hstart i.succ
+    have h₃ := framePrefix_step pre post (fun j => (W j).length) i.castSucc
+    have h₄ := framePrefix_step pre' post' (fun j => (W j).length) i.castSucc
+    simp only [Fin.val_castSucc,Fin.val_succ] at *
+    omega
+  · have h₁ := hstart (Fin.last k)
+    have h₂ := framePrefix_step pre post (fun j => (W j).length) (Fin.last k)
+    have h₃ := framePrefix_step pre' post' (fun j => (W j).length) (Fin.last k)
+    simp only [Fin.val_last] at *
+    omega
+
+end TraceReconstruction
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem framePrefix_constant {k : ℕ} (pre post len : Fin k → ℕ) (P i : ℕ)
+    (hc : ∀j,pre j+len j+post j=P) (hi : i ≤ k) :
+    framePrefix pre post len i=i*P := by
+  induction i with
+  | zero => simp [framePrefix_zero]
+  | succ i ih =>
+    have h := framePrefix_step pre post len ⟨i,by omega⟩
+    have hh := hc ⟨i,by omega⟩
+    rw [h,ih (by omega),Nat.add_mul,Nat.one_mul]
+    omega
+
+theorem flatten_framedConcat (w : List Bool) {k : ℕ}
+    (pre post : Fin k → ℕ) (W : Fin k → List (List Bool)) :
+    (framedConcat w pre post W).flatten=
+      (List.ofFn (fun i => pureAtomRun w (pre i) ++ (W i).flatten ++
+        pureAtomRun w (post i))).flatten := by
+  rw [framedConcat,List.flatten_flatten,List.map_ofFn]
+  apply congrArg List.flatten
+  apply congrArg List.ofFn
+  funext i
+  simp [Function.comp_apply,List.flatten_append,pureAtomRun]
+
+/-- The deterministic multiwindow identity used by the nested simulator.
+Every motif is covered by its own window. Predetermined pure-region lengths
+may move the boundaries of nominal units; only their exact cumulative counts
+are required. Windows and chunks all remain literal unmarked binary words. -/
+theorem template_group_windows (k P T a L : ℕ) (sign : Fin L → Bool)
+    (w : Bool → List Bool) (τ : Fin (k+1) → Bool)
+    (t pre post : Fin (k+1) → ℕ)
+    (hleft : ∀u,t u ≤ a) (hfit : ∀u,a+L ≤ t u+T)
+    (hunit : ∀u,t u+T ≤ P)
+    (hstart : ∀u,framePrefix pre post (fun _ => T) u.val+pre u=u.val*P+t u)
+    (htotal : framePrefix pre post (fun _ => T) (k+1)=(k+1)*P) :
+    (List.ofFn (fun u => complementaryWords P a L sign w (τ u))).flatten=
+      (List.ofFn (fun u => pureAtomRun (w false) (pre u) ++
+        complementaryWords T (a-t u) L sign w (τ u) ++
+        pureAtomRun (w false) (post u))).flatten := by
+  let W : Fin (k+1) → List (List Bool) := fun u =>
+    List.ofFn (fun i => w (templateType T (a-t u) L sign (τ u) i))
+  let z : Fin (k+1) → ℕ := fun u => P-(t u+T)
+  have hz : ∀u,t u+T+z u=P := by intro u; dsimp [z]; have hu := hunit u; omega
+  have hlen : ∀u,(W u).length=T := by intro u; simp [W]
+  have hp : ∀i,i ≤ k+1 → framePrefix t z (fun u => (W u).length) i=i*P :=
+    fun i hi => framePrefix_constant t z _ P i (fun u => by rw [hlen]; exact hz u) hi
+  have he := framedConcat_eq_of_starts (w false) k t z pre post W
+    (fun u => by simp only [hlen] at hp ⊢; rw [hp _ (by omega),hstart])
+    (by simp only [hlen] at hp ⊢; rw [hp _ le_rfl,htotal])
+  have he' := congrArg List.flatten he
+  rw [flatten_framedConcat,flatten_framedConcat] at he'
+  calc
+    _ = (List.ofFn (fun u => pureAtomRun (w false) (t u) ++ (W u).flatten ++
+          pureAtomRun (w false) (z u))).flatten := by
+      congr 1
+      apply congrArg List.ofFn
+      funext u
+      have hh := template_word_window (t u) T (z u) a L sign (τ u) w (hleft u) (hfit u)
+      rw [hz] at hh
+      simpa only [complementaryWords,atomString_constant,W,atomString,List.ofFn_const,pureAtomRun] using hh
+    _ = _ := he'
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+variable {K : Type*} [Fintype K]
+
+/-- Counts of literal side chunks, with one deterministic pure filler in
+ each region. A bundle can contain any even number of paired chunks. -/
+def pairedPreCount {N : ℕ} (left : K → Fin N) (filler : Fin N → ℕ)
+    (first : K → ℕ) (u : Fin N) : ℕ :=
+  filler u + ∑v, if left v=u then first v else 0
+
+def pairedPostCount {N : ℕ} (right : K → Fin N) (filler : Fin N → ℕ)
+    (reserve first : K → ℕ) (u : Fin N) : ℕ :=
+  filler u + ∑v, if right v=u then 2*reserve v-first v else 0
+
+theorem pairedPreCount_cast {N : ℕ} (left : K → Fin N) (filler : Fin N → ℕ)
+    (reserve first : K → ℕ) (A : ℕ)
+    (hfill : ∀u,filler u+(∑v,if left v=u then reserve v else 0)=A) (u : Fin N) :
+    (pairedPreCount left filler first u : ℤ) = A+
+      ∑v, if left v=u then (first v:ℤ)-reserve v else 0 := by
+  have hf : (filler u:ℤ)+(∑v,if left v=u then (reserve v:ℤ) else 0)=A := by
+    exact_mod_cast hfill u
+  have he : (∑v,if left v=u then (first v:ℤ)-reserve v else 0)=
+      (∑v,if left v=u then (first v:ℤ) else 0)-
+        (∑v,if left v=u then (reserve v:ℤ) else 0) := by
+    rw [←Finset.sum_sub_distrib]
+    apply Finset.sum_congr rfl
+    intro v _
+    split_ifs <;> ring
+  simp only [pairedPreCount,Nat.cast_add,Nat.cast_sum,Nat.cast_ite,Nat.cast_zero]
+  rw [he]
+  omega
+
+theorem pairedPostCount_cast {N : ℕ} (right : K → Fin N) (filler : Fin N → ℕ)
+    (reserve first : K → ℕ) (A : ℕ) (hfirst : ∀v,first v ≤ 2*reserve v)
+    (hfill : ∀u,filler u+(∑v,if right v=u then reserve v else 0)=A) (u : Fin N) :
+    (pairedPostCount right filler reserve first u : ℤ) = A-
+      ∑v, if right v=u then (first v:ℤ)-reserve v else 0 := by
+  have hf : (filler u:ℤ)+(∑v,if right v=u then (reserve v:ℤ) else 0)=A := by
+    exact_mod_cast hfill u
+  have he : (∑v,if right v=u then ((2*reserve v-first v:ℕ):ℤ) else 0)=
+      (∑v,if right v=u then (reserve v:ℤ) else 0)-
+        (∑v,if right v=u then (first v:ℤ)-reserve v else 0) := by
+    rw [←Finset.sum_sub_distrib]
+    apply Finset.sum_congr rfl
+    intro v _
+    rw [Nat.cast_sub (hfirst v)]
+    push_cast
+    split_ifs <;> ring
+  simp only [pairedPostCount,Nat.cast_add,Nat.cast_sum,Nat.cast_ite,Nat.cast_zero]
+  rw [he]
+  omega
+
+/-- Prefix sum of a delta at a prescribed label, with no loss from replacing
+ its real finite support by an interval. -/
+theorem fin_prefix_delta {N : ℕ} (a : Fin N) (i : ℕ) (x : ℤ) :
+    (∑u : Fin N, if u.val < i then (if a=u then x else 0) else 0)=
+      if a.val < i then x else 0 := by
+  have he : (fun u : Fin N => if u.val < i then (if a=u then x else 0) else 0)=
+      fun u => if u=a then (if a.val < i then x else 0) else 0 := by
+    funext u
+    by_cases h : u=a
+    · subst u; simp
+    · simp [h,Ne.symm h]
+  rw [he]
+  simp
+
+/-- Exact paired-interval contribution to the position of a window. -/
+theorem pair_activity_identity {N : ℕ} (l r u : Fin N) (hlr : l.val ≤ r.val) (x : ℤ) :
+    (if l.val < u.val then x else 0)-(if r.val < u.val then x else 0)+
+      (if l=u then x else 0) =
+      if l.val ≤ u.val ∧ u.val ≤ r.val then x else 0 := by
+  by_cases hlu : l.val < u.val
+  · by_cases hru : r.val < u.val
+    · simp [hlu,hru,show l≠u by intro h; subst u; omega,show ¬u.val ≤ r.val by omega]
+    · simp [hlu,hru,show l≠u by intro h; subst u; omega,show l.val ≤ u.val by omega,show u.val ≤ r.val by omega]
+  · by_cases hlu' : l=u
+    · subst u; simp [hlr,show ¬r.val < l.val by omega]
+    · have hn : l.val≠u.val := fun h => hlu' (Fin.ext h)
+      simp [hlu,hlu',show ¬r.val < u.val by omega,show ¬l.val ≤ u.val by omega]
+
+/-- Cumulative pure chunk counts have exactly the source's localBits-plus-active-
+globalBits jitter. This identity uses the actual chronological pre/post schedule. -/
+theorem pairedLayout_start {N : ℕ} (left right : K → Fin N)
+    (horder : ∀v,(left v).val ≤ (right v).val)
+    (fpre fpost : Fin N → ℕ) (reserve first : K → ℕ)
+    (hfirst : ∀v,first v ≤ 2*reserve v) (A B T : ℕ)
+    (hpre : ∀u,fpre u+(∑v,if left v=u then reserve v else 0)=A)
+    (hpost : ∀u,fpost u+(∑v,if right v=u then reserve v else 0)=B)
+    (u : Fin N) :
+    ((framePrefix (pairedPreCount left fpre first)
+      (pairedPostCount right fpost reserve first) (fun _ => T) u.val+
+      pairedPreCount left fpre first u : ℕ):ℤ)=
+      u.val*(A+T+B)+A+
+        ∑v, if (left v).val ≤ u.val ∧ u.val ≤ (right v).val then
+          (first v:ℤ)-reserve v else 0 := by
+  let d : K → ℤ := fun v => (first v:ℤ)-reserve v
+  have hpr := pairedPreCount_cast left fpre reserve first A hpre
+  have hpo := pairedPostCount_cast right fpost reserve first B hfirst hpost
+  have hsum : (∑j : Fin N, if j.val < u.val then (1:ℤ) else 0)=(u.val:ℤ) := by
+    have hc := framePrefix_constant (fun _ : Fin N => 0) (fun _ => 0) (fun _ => 1) 1 u.val
+      (by simp) (by omega)
+    simpa [framePrefix] using congrArg (fun n : ℕ => (n:ℤ)) hc
+  simp only [Nat.cast_add,framePrefix,Nat.cast_sum,Nat.cast_ite,Nat.cast_zero,hpr,hpo]
+  have hex : (∑j : Fin N,if j.val < u.val then
+        (A+(∑v,if left v=j then d v else 0))+T+
+        (B-(∑v,if right v=j then d v else 0)) else 0)=
+      (u.val:ℤ)*(A+T+B)+
+        ∑v,((if (left v).val < u.val then d v else 0)-
+          (if (right v).val < u.val then d v else 0)) := by
+    calc
+      _ = (∑j : Fin N,(if j.val < u.val then (1:ℤ) else 0)*(A+T+B))+
+          ∑j : Fin N,∑v,
+            ((if j.val < u.val then (if left v=j then d v else 0) else 0)-
+             (if j.val < u.val then (if right v=j then d v else 0) else 0)) := by
+        rw [←Finset.sum_add_distrib]
+        apply Finset.sum_congr rfl
+        intro j _
+        by_cases hj : j.val < u.val
+        · simp [hj,Finset.sum_sub_distrib]; ring
+        · simp [hj]
+      _ = _ := by
+        rw [←Finset.sum_mul,hsum,Finset.sum_comm]
+        simp_rw [Finset.sum_sub_distrib,fin_prefix_delta]
+  change (∑j : Fin N,if j.val < u.val then
+      (A+(∑v,if left v=j then d v else 0))+T+
+      (B-(∑v,if right v=j then d v else 0)) else 0)+
+      (A+∑v,if left v=u then d v else 0)=_
+  rw [hex]
+  have he : (∑v,((if (left v).val < u.val then d v else 0)-
+      (if (right v).val < u.val then d v else 0)))+
+      (∑v,if left v=u then d v else 0)=
+      ∑v,if (left v).val ≤ u.val ∧ u.val ≤ (right v).val then d v else 0 := by
+    rw [←Finset.sum_add_distrib]
+    exact Finset.sum_congr rfl (fun v _ => pair_activity_identity (left v) (right v) u (horder v) (d v))
+  dsimp only [d] at *
+  linarith only [he]
+
+end TraceReconstruction
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+variable {K : Type*} [Fintype K]
+
+theorem pairedLayout_total {N : ℕ} (left right : K → Fin N)
+    (fpre fpost : Fin N → ℕ) (reserve first : K → ℕ)
+    (hfirst : ∀v,first v ≤ 2*reserve v) (A B T : ℕ)
+    (hpre : ∀u,fpre u+(∑v,if left v=u then reserve v else 0)=A)
+    (hpost : ∀u,fpost u+(∑v,if right v=u then reserve v else 0)=B) :
+    framePrefix (pairedPreCount left fpre first)
+      (pairedPostCount right fpost reserve first) (fun _ => T) N=N*(A+T+B) := by
+  have hpr := pairedPreCount_cast left fpre reserve first A hpre
+  have hpo := pairedPostCount_cast right fpost reserve first B hfirst hpost
+  have he : ∀f : K → Fin N,
+      (∑u : Fin N,∑v,if f v=u then (first v:ℤ)-reserve v else 0)=
+        ∑v,((first v:ℤ)-reserve v) := by
+    intro f
+    rw [Finset.sum_comm]
+    simp
+  apply Nat.cast_injective (R:=ℤ)
+  simp only [framePrefix,Fin.isLt,ite_true,Nat.cast_sum,Nat.cast_add,hpr,hpo,
+    Finset.sum_add_distrib,Finset.sum_sub_distrib,he,Finset.sum_const,Finset.card_univ,
+    Fintype.card_fin,nsmul_eq_mul,Nat.cast_mul]
+  push_cast
+  ring
+
+/-- One localBits bundle at each child, and one globalBits bundle at each vertex in
+ the first quarter, paired with the reflected last-quarter post region. -/
+abbrev SourceBundle (N : ℕ) := Fin N ⊕ Fin (N/4)
+
+def sourceBundleLeft (N : ℕ) : SourceBundle N → Fin N
+  | .inl u => u
+  | .inr v => ⟨v.val,by have := v.isLt; omega⟩
+
+def sourceBundleRight (N : ℕ) : SourceBundle N → Fin N
+  | .inl u => u
+  | .inr v => ⟨N-1-v.val,by have := v.isLt; omega⟩
+
+theorem sourceBundle_order (N : ℕ) (v : SourceBundle N) :
+    (sourceBundleLeft N v).val ≤ (sourceBundleRight N v).val := by
+  cases v with
+  | inl u => rfl
+  | inr v => dsimp [sourceBundleLeft,sourceBundleRight]; have := v.isLt; omega
+
+theorem sum_injective_delta_le {α β : Type*} [Fintype α] [DecidableEq β]
+    (f : α → β) (hf : Function.Injective f) (u : β) (R : ℕ) :
+    (∑v,if f v=u then R else 0) ≤ R := by
+  by_cases h : ∃v,f v=u
+  · obtain ⟨v,hv⟩ := h
+    have he : ∀j,f j=u ↔ j=v := fun j => by rw [←hv]; exact hf.eq_iff
+    simp [he]
+  · push Not at h
+    simp [h]
+
+theorem sourceBundle_reserve_pre (N R : ℕ) (u : Fin N) :
+    (∑v : SourceBundle N,if sourceBundleLeft N v=u then R else 0) ≤ 2*R := by
+  rw [Fintype.sum_sum_type]
+  change (∑v : Fin N, if v=u then R else 0)+(∑v : Fin (N/4),
+    if (⟨v.val,by have := v.isLt; omega⟩ : Fin N)=u then R else 0) ≤ 2*R
+  simp only [Finset.sum_ite_eq',Finset.mem_univ,ite_true]
+  have h := sum_injective_delta_le
+    (fun v : Fin (N/4) => (⟨v.val,by have := v.isLt; omega⟩ : Fin N))
+    (fun _ _ he => Fin.ext (congrArg (fun z : Fin N => z.val) he)) u R
+  omega
+
+theorem sourceBundle_reserve_post (N R : ℕ) (u : Fin N) :
+    (∑v : SourceBundle N,if sourceBundleRight N v=u then R else 0) ≤ 2*R := by
+  rw [Fintype.sum_sum_type]
+  change (∑v : Fin N, if v=u then R else 0)+(∑v : Fin (N/4),
+    if (⟨N-1-v.val,by have := v.isLt; omega⟩ : Fin N)=u then R else 0) ≤ 2*R
+  simp only [Finset.sum_ite_eq',Finset.mem_univ,ite_true]
+  have hinj : Function.Injective (fun v : Fin (N/4) =>
+      (⟨N-1-v.val,by have := v.isLt; omega⟩ : Fin N)) := by
+    intro v w he
+    have hv := v.isLt
+    have hw := w.isLt
+    have h := congrArg Fin.val he
+    apply Fin.ext
+    dsimp only at h
+    omega
+  have h := sum_injective_delta_le _ hinj u R
+  omega
+
+def sourceBundleCount (N t h : ℕ) (localBits : Fin N → Fin t → Bool)
+    (globalBits : Fin ((N/4)*t) → Bool) : SourceBundle N → ℕ
+  | .inl u => t*h+∑a,bitIndex (localBits u a)
+  | .inr v => t*h+∑a : Fin t,bitIndex (globalBits (globalTrialIndex N t v a))
+
+theorem sourceBundleCount_le (N t h : ℕ) (ht : Even t)
+    (localBits : Fin N → Fin t → Bool) (globalBits : Fin ((N/4)*t) → Bool)
+    (v : SourceBundle N) : sourceBundleCount N t h localBits globalBits v ≤ 2*(t*h+t/2) := by
+  have hh : 2*(t/2)=t := Nat.mul_div_cancel' ((even_iff_two_dvd).mp ht)
+  have hs (b : Fin t → Bool) : (∑a,bitIndex (b a)) ≤ t := by
+    calc
+      _ ≤ ∑_a : Fin t,1 := Finset.sum_le_sum (fun a _ => by cases b a <;> norm_num [bitIndex])
+      _ = t := by simp
+  cases v with
+  | inl u => dsimp [sourceBundleCount]; have := hs (localBits u); omega
+  | inr v => dsimp [sourceBundleCount]; have := hs (fun a => globalBits (globalTrialIndex N t v a)); omega
+
+/-- The actual scheduled bits give precisely the arithmetic source displacement
+ Y=H+V, including its common-globalBits plateau and outer-quarter behavior. -/
+theorem sourceBundle_activity (N t h : ℕ)
+    (localBits : Fin N → Fin t → Bool) (globalBits : Fin ((N/4)*t) → Bool) (u : Fin N) :
+    (∑v : SourceBundle N, if (sourceBundleLeft N v).val ≤ u.val ∧
+        u.val ≤ (sourceBundleRight N v).val then
+      (sourceBundleCount N t h localBits globalBits v:ℤ)-(t*h+t/2:ℕ) else 0)=
+      ((∑a : Fin t,(bitIndex (localBits u a):ℤ))-(t/2:ℕ))+
+        scheduledGlobalShift N t globalBits u := by
+  rw [Fintype.sum_sum_type]
+  have he : (∑v : Fin N, if v.val ≤ u.val ∧ u.val ≤ v.val then
+      (sourceBundleCount N t h localBits globalBits (.inl v):ℤ)-(t*h+t/2:ℕ) else 0)=
+      (∑a : Fin t,(bitIndex (localBits u a):ℤ))-(t/2:ℕ) := by
+    have hc : ∀v : Fin N,(v.val ≤ u.val ∧ u.val ≤ v.val) ↔ v=u := by
+      intro v
+      constructor
+      · intro hv; exact Fin.ext (le_antisymm hv.1 hv.2)
+      · intro hv; subst v; exact ⟨le_rfl,le_rfl⟩
+    simp only [hc,Finset.sum_ite_eq',Finset.mem_univ,ite_true,sourceBundleCount,
+      Nat.cast_add,Nat.cast_mul,Nat.cast_sum]
+    ring
+  change (∑v : Fin N, if v.val ≤ u.val ∧ u.val ≤ v.val then
+      (sourceBundleCount N t h localBits globalBits (.inl v):ℤ)-(t*h+t/2:ℕ) else 0)+_=_
+  rw [he]
+  congr 1
+  unfold scheduledGlobalShift
+  apply Finset.sum_congr rfl
+  intro v _
+  have hv := v.isLt
+  have hu := u.isLt
+  have hc : (sourceBundleLeft N (.inr v)).val ≤ u.val ∧
+      u.val ≤ (sourceBundleRight N (.inr v)).val ↔ v.val ≤ u.val ∧ u.val+v.val < N := by
+    dsimp [sourceBundleLeft,sourceBundleRight]; omega
+  simp only [hc]
+  split_ifs
+  · simp only [sourceBundleCount,Nat.cast_add,Nat.cast_mul,Nat.cast_sum]; ring
+  · rfl
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- One-bundle reserved nominal count is integral because the number of
+ paired trials is even. The actual filler reserves both local and global
+ bundles when both are present in a region. -/
+def sourceLayoutReserve (t h : ℕ) : ℕ := t*h+t/2
+
+def sourcePreFiller (N A t h : ℕ) (u : Fin N) : ℕ :=
+  A-∑v : SourceBundle N,if sourceBundleLeft N v=u then sourceLayoutReserve t h else 0
+
+def sourcePostFiller (N B t h : ℕ) (u : Fin N) : ℕ :=
+  B-∑v : SourceBundle N,if sourceBundleRight N v=u then sourceLayoutReserve t h else 0
+
+def sourcePreCount (N A t h : ℕ) (loc : Fin N → Fin t → Bool)
+    (glob : Fin ((N/4)*t) → Bool) : Fin N → ℕ :=
+  pairedPreCount (sourceBundleLeft N) (sourcePreFiller N A t h)
+    (sourceBundleCount N t h loc glob)
+
+def sourcePostCount (N B t h : ℕ) (loc : Fin N → Fin t → Bool)
+    (glob : Fin ((N/4)*t) → Bool) : Fin N → ℕ :=
+  pairedPostCount (sourceBundleRight N) (sourcePostFiller N B t h)
+    (fun _ => sourceLayoutReserve t h) (sourceBundleCount N t h loc glob)
+
+def sourceDisplacement (N t : ℕ) (loc : Fin N → Fin t → Bool)
+    (glob : Fin ((N/4)*t) → Bool) (u : Fin N) : ℤ :=
+  (∑a : Fin t,(bitIndex (loc u a):ℤ))-(t/2:ℕ)+scheduledGlobalShift N t glob u
+
+def sourceWindowStart (N A t : ℕ) (loc : Fin N → Fin t → Bool)
+    (glob : Fin ((N/4)*t) → Bool) (u : Fin N) : ℕ :=
+  ((A:ℤ)+sourceDisplacement N t loc glob u).toNat
+
+theorem sourcePreFiller_nominal (N A t h : ℕ)
+    (hA : 2*sourceLayoutReserve t h ≤ A) (u : Fin N) :
+    sourcePreFiller N A t h u+
+      (∑v : SourceBundle N,if sourceBundleLeft N v=u then sourceLayoutReserve t h else 0)=A := by
+  exact Nat.sub_add_cancel ((sourceBundle_reserve_pre N _ u).trans hA)
+
+theorem sourcePostFiller_nominal (N B t h : ℕ)
+    (hB : 2*sourceLayoutReserve t h ≤ B) (u : Fin N) :
+    sourcePostFiller N B t h u+
+      (∑v : SourceBundle N,if sourceBundleRight N v=u then sourceLayoutReserve t h else 0)=B := by
+  exact Nat.sub_add_cancel ((sourceBundle_reserve_post N _ u).trans hB)
+
+theorem sourceLayout_exact_start (N A T B t h : ℕ) (ht : Even t)
+    (hA : 2*sourceLayoutReserve t h ≤ A) (hB : 2*sourceLayoutReserve t h ≤ B)
+    (loc : Fin N → Fin t → Bool) (glob : Fin ((N/4)*t) → Bool) (u : Fin N) :
+    ((framePrefix (sourcePreCount N A t h loc glob) (sourcePostCount N B t h loc glob)
+      (fun _ => T) u.val+sourcePreCount N A t h loc glob u:ℕ):ℤ)=
+      u.val*(A+T+B)+A+sourceDisplacement N t loc glob u := by
+  have H := pairedLayout_start (sourceBundleLeft N) (sourceBundleRight N)
+    (sourceBundle_order N) (sourcePreFiller N A t h) (sourcePostFiller N B t h)
+    (fun _ => sourceLayoutReserve t h) (sourceBundleCount N t h loc glob)
+    (sourceBundleCount_le N t h ht loc glob) A B T
+    (sourcePreFiller_nominal N A t h hA) (sourcePostFiller_nominal N B t h hB) u
+  simp only [sourceLayoutReserve] at H
+  rw [sourceBundle_activity] at H
+  exact H
+
+theorem sourceLayout_exact_total (N A T B t h : ℕ) (ht : Even t)
+    (hA : 2*sourceLayoutReserve t h ≤ A) (hB : 2*sourceLayoutReserve t h ≤ B)
+    (loc : Fin N → Fin t → Bool) (glob : Fin ((N/4)*t) → Bool) :
+    framePrefix (sourcePreCount N A t h loc glob) (sourcePostCount N B t h loc glob)
+      (fun _ => T) N=N*(A+T+B) := by
+  exact pairedLayout_total (sourceBundleLeft N) (sourceBundleRight N)
+    (sourcePreFiller N A t h) (sourcePostFiller N B t h)
+    (fun _ => sourceLayoutReserve t h) (sourceBundleCount N t h loc glob)
+    (sourceBundleCount_le N t h ht loc glob) A B T
+    (sourcePreFiller_nominal N A t h hA) (sourcePostFiller_nominal N B t h hB)
+
+/-- The literal surrogate side chunks, after concatenation, are a correct
+ partition of the actual parent word whenever the windows fit. This is
+ an equality of unmarked binary strings, not an assumed channel identity. -/
+theorem sourceLayout_literal_partition (k A T B t h a L : ℕ) (ht : Even t)
+    (hA : 2*sourceLayoutReserve t h ≤ A) (hB : 2*sourceLayoutReserve t h ≤ B)
+    (loc : Fin (k+1) → Fin t → Bool) (glob : Fin (((k+1)/4)*t) → Bool)
+    (sign : Fin L → Bool) (w : Bool → List Bool) (τ : Fin (k+1) → Bool)
+    (hzero : ∀u,0 ≤ (A:ℤ)+sourceDisplacement (k+1) t loc glob u)
+    (hleft : ∀u,sourceWindowStart (k+1) A t loc glob u ≤ a)
+    (hfit : ∀u,a+L ≤ sourceWindowStart (k+1) A t loc glob u+T)
+    (hunit : ∀u,sourceWindowStart (k+1) A t loc glob u+T ≤ A+T+B) :
+    (List.ofFn (fun u => complementaryWords (A+T+B) a L sign w (τ u))).flatten=
+      (List.ofFn (fun u => pureAtomRun (w false) (sourcePreCount (k+1) A t h loc glob u) ++
+        complementaryWords T (a-sourceWindowStart (k+1) A t loc glob u) L sign w (τ u) ++
+        pureAtomRun (w false) (sourcePostCount (k+1) B t h loc glob u))).flatten := by
+  apply template_group_windows k (A+T+B) T a L sign w τ
+    (sourceWindowStart (k+1) A t loc glob)
+    (sourcePreCount (k+1) A t h loc glob) (sourcePostCount (k+1) B t h loc glob)
+    hleft hfit hunit
+  · intro u
+    apply Nat.cast_injective (R:=ℤ)
+    rw [sourceLayout_exact_start (k+1) A T B t h ht hA hB]
+    simp only [Nat.cast_add,Nat.cast_mul,sourceWindowStart,Int.toNat_of_nonneg (hzero u)]
+    ring
+  · exact sourceLayout_exact_total (k+1) A T B t h ht hA hB loc glob
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- Independent emissions at all literal slots accumulate their actual
+ approximation errors. The decoders may depend on the fixed slot ID. -/
+theorem coordinateProduct_approximates {α : Type*} [Fintype α]
+    {n : ℕ} {q : ℝ} (hq : q∈Set.Icc 0 1) (P : Fin n → FiniteLaw α)
+    (f : Fin n → α → List Bool) (w : Fin n → List Bool) (E : Fin n → ℝ)
+    (h : ∀i,ApproximatesTrace q (P i) (f i) (w i) (E i)) :
+    ApproximatesTrace q (FiniteLaw.coordinateProduct P)
+      (fun x => coordinateWord (fun i => f i (x i))) (coordinateWord w) (∑i,E i) := by
+  induction n with
+  | zero =>
+    intro F hF
+    change |(FiniteLaw.coordinateProduct P).expect (fun _ => F [])-F []|≤∑i : Fin 0,E i
+    simp only [FiniteLaw.expect_const,sub_self,abs_zero,Finset.univ_eq_empty,Finset.sum_empty,le_refl]
+  | succ n ih =>
+    intro F hF
+    have ht := ih (fun i => P i.succ) (fun i => f i.succ)
+      (fun i => w i.succ) (fun i => E i.succ) (fun i => h i.succ)
+    have hh := ApproximatesTrace.append hq (h 0) ht F hF
+    rw [FiniteLaw.expect_prod] at hh
+    rw [FiniteLaw.expect_coordinateProduct_succ,coordinateWord_succ,Fin.sum_univ_succ]
+    have he (a : α) (x : Fin n → α) :
+        coordinateWord (fun i => f i ((Fin.cons a x : Fin (n+1) → α) i))=
+          f 0 a++coordinateWord (fun i => f i.succ (x i)) := by
+      rw [coordinateWord_succ]
+      simp only [Fin.cons_zero,Fin.cons_succ]
+    simp_rw [he]
+    exact hh
+
+namespace FiniteLaw
+variable {ι κ α : Type*} [Fintype ι] [Fintype κ] [Fintype α]
+
+/-- Independent emissions indexed by the actual token IDs of the layout. -/
+def indexedProduct (P : ι → FiniteLaw α) : FiniteLaw (ι → α) where
+  mass x := ∏i,(P i).mass (x i)
+  nonneg x := Finset.prod_nonneg (fun i _ => (P i).nonneg _)
+  total := by
+    rw [←Fintype.prod_sum]
+    simp only [(P _).total,Finset.prod_const_one]
+
+/-- Reorder independent emissions by the literal chronological slot order.
+The order is a bijection, so no output is duplicated or silently discarded. -/
+theorem indexedProduct_expect_equiv (P : ι → FiniteLaw α) (e : κ ≃ ι)
+    (F : (ι → α) → ℝ) :
+    (indexedProduct P).expect F=(indexedProduct (fun j => P (e j))).expect
+      (fun x => F (fun i => x (e.symm i))) := by
+  let E : (κ → α) ≃ (ι → α) :=
+    { toFun := fun x i => x (e.symm i)
+      invFun := fun x j => x (e j)
+      left_inv := fun x => by funext j; simp
+      right_inv := fun x => by funext i; simp }
+  unfold expect indexedProduct
+  rw [←E.sum_comp]
+  apply Finset.sum_congr rfl
+  intro x _
+  change (∏i,(P i).mass (x (e.symm i)))*F (fun i => x (e.symm i))=_
+  congr 1
+  rw [←e.prod_comp]
+  simp only [Equiv.symm_apply_apply]
+
+theorem indexedProduct_map (P : ι → FiniteLaw α) {β : Type*} [Fintype β]
+    (f : ι → α → β) :
+    (indexedProduct P).map (fun x i => f i (x i))=
+      indexedProduct (fun i => (P i).map (f i)) := by
+  apply FiniteLaw.ext
+  funext y
+  rw [map_mass]
+  change (∑x : ι → α,if (fun i => f i (x i))=y then ∏i,(P i).mass (x i) else 0)=_
+  simp only [indexedProduct,map_mass]
+  rw [Fintype.prod_sum]
+  apply Finset.sum_congr rfl
+  intro x _
+  by_cases he : (fun i => f i (x i))=y
+  · rw [ite_eq_left he]
+    exact Finset.prod_congr rfl (fun i _ => (ite_eq_left (congrFun he i)).symm)
+  · rw [ite_eq_right he]
+    symm
+    have hi : ∃i,f i (x i)≠y i := by
+      by_contra h
+      push Not at h
+      exact he (funext h)
+    obtain ⟨i,hi⟩ := hi
+    exact Finset.prod_eq_zero (Finset.mem_univ i) (ite_eq_right hi)
+
+end FiniteLaw
+
+/-- An arbitrarily indexed actual layout has the same unconditional trace
+ guarantee after its fixed, public chronological ordering is applied. -/
+theorem indexedProduct_approximates {ι α : Type*} [Fintype ι] [Fintype α]
+    {n : ℕ} {q : ℝ} (hq : q∈Set.Icc 0 1) (e : Fin n ≃ ι)
+    (P : ι → FiniteLaw α) (f : ι → α → List Bool) (w : ι → List Bool) (E : ι → ℝ)
+    (h : ∀i,ApproximatesTrace q (P i) (f i) (w i) (E i)) :
+    ApproximatesTrace q (FiniteLaw.indexedProduct P)
+      (fun x => coordinateWord (fun j => f (e j) (x (e j))))
+      (coordinateWord (fun j => w (e j))) (∑i,E i) := by
+  intro F hF
+  rw [FiniteLaw.indexedProduct_expect_equiv P e]
+  simp only [Equiv.symm_apply_apply]
+  have H := coordinateProduct_approximates hq (fun j => P (e j))
+    (fun j => f (e j)) (fun j => w (e j)) (fun j => E (e j)) (fun j => h (e j)) F hF
+  rw [e.sum_comp] at H
+  simp only [FiniteLaw.expect,FiniteLaw.indexedProduct,FiniteLaw.coordinateProduct] at H ⊢
+  convert H using 1
+  congr 3
+  ext x
+  simp
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- The Boolean representation of the source Prouhet signs; true denotes
+ a negative type. No sign convention is changed in passing to literal words. -/
+def boolTypeSign (b : Bool) : ℝ := if b then -1 else 1
+
+def prouhetBool (s : ℕ) (i : Fin (2^s)) : Bool :=
+  decide ((prouhetSeries s).coeff (i.val:ℤ)= -1)
+
+theorem boolTypeSign_injective : Function.Injective boolTypeSign := by
+  intro a b h
+  cases a <;> cases b <;> norm_num [boolTypeSign] at *
+
+theorem boolTypeSign_template (s P a : ℕ) (τ : Bool) (i : Fin P) :
+    boolTypeSign (templateType P a (2^s) (prouhetBool s) τ i)=
+      motifChildSign s (a:ℤ) i.val (boolTypeSign τ) := by
+  by_cases hi : a ≤ i.val ∧ i.val < a+2^s
+  · have hz : 0 ≤ (i.val:ℤ)-a ∧ (i.val:ℤ)-a < (2^s:ℕ) := by omega
+    have he : ((i.val-a:ℕ):ℤ)=(i.val:ℤ)-a := by omega
+    have hc := (abs_eq (by norm_num : (0:ℝ)≤1)).mp
+      ((prouhetSeries_coeff_spec s).2 _ hz.1 hz.2)
+    have hχ : motifIndicator s ((i.val:ℤ)-a)=1 := by simp only [motifIndicator,ite_eq_left hz]
+    rcases hc with hc | hc <;> cases τ <;>
+      norm_num [templateType,hi,prouhetBool,he,boolTypeSign,motifChildSign,hχ,hc]
+  · have hz : ¬(0 ≤ (i.val:ℤ)-a ∧ (i.val:ℤ)-a < (2^s:ℕ)) := by omega
+    have hc : (prouhetSeries s).coeff ((i.val:ℤ)-a)=0 :=
+      (prouhetSeries_coeff_spec s).1 _ (by omega)
+    have hχ : motifIndicator s ((i.val:ℤ)-a)=0 := by simp only [motifIndicator,ite_eq_right hz]
+    simp [templateType,hi,boolTypeSign,motifChildSign,hχ,hc]
+
+theorem sourceWindowStart_cast_of_fit (N P T t : ℕ)
+    (loc : Fin N → Fin t → Bool) (glob : Fin ((N/4)*t) → Bool)
+    (hP : 2*T ≤ P) (hT : 16 ≤ T) (u : Fin N)
+    (hY : |8*sourceDisplacement N t loc glob u| ≤ (T:ℤ)) :
+    (sourceWindowStart N ((P-T)/2) t loc glob u:ℤ)=
+      (((P-T)/2:ℕ):ℤ)+sourceDisplacement N t loc glob u := by
+  apply Int.toNat_of_nonneg
+  have hy := abs_le.mp hY
+  omega
+
+/-- A zero-based centered version of the floor-fit calculation. -/
+theorem sourceWindowStart_fit (N P T t L : ℕ)
+    (loc : Fin N → Fin t → Bool) (glob : Fin ((N/4)*t) → Bool)
+    (hP : 2*T ≤ P) (hT : 16 ≤ T) (heven : Even T) (hL : 8*L ≤ T)
+    (u : Fin N) (hY : |8*sourceDisplacement N t loc glob u| ≤ (T:ℤ)) :
+    sourceWindowStart N ((P-T)/2) t loc glob u+T ≤ P ∧
+    sourceWindowStart N ((P-T)/2) t loc glob u ≤ P/2 ∧
+    P/2+L ≤ sourceWindowStart N ((P-T)/2) t loc glob u+T :=
+  centered_window_floor_fit P T L _ _ hP hT heven hL hY
+    (sourceWindowStart_cast_of_fit N P T t loc glob hP hT u hY)
+
+/-- The zero-based centered translated anchor, including both half-trial corrections. -/
+theorem sourceWindowStart_anchor (N P T t L : ℕ)
+    (loc : Fin N → Fin t → Bool) (glob : Fin ((N/4)*t) → Bool)
+    (hP : 2*T ≤ P) (hT : 16 ≤ T) (heven : Even T) (hL : 8*L ≤ T)
+    (u : Fin N) (hY : |8*sourceDisplacement N t loc glob u| ≤ (T:ℤ)) :
+    ((P/2-sourceWindowStart N ((P-T)/2) t loc glob u:ℕ):ℤ)=
+      (T/2:ℕ)+(t/2:ℕ)-(∑a : Fin t,(bitIndex (loc u a):ℤ))-
+        scheduledGlobalShift N t glob u := by
+  have hfit := sourceWindowStart_fit N P T t L loc glob hP hT heven hL u hY
+  have hc := sourceWindowStart_cast_of_fit N P T t loc glob hP hT u hY
+  have ha := centered_window_floor_anchor P T (by omega) heven
+  have hA : (P-T)/2 ≤ P/2 := Nat.div_le_div_right (Nat.sub_le P T)
+  have ha' : ((P/2:ℕ):ℤ)-((P-T)/2:ℕ)=(T/2:ℕ) := by omega
+  rw [Nat.cast_sub hfit.2.1,hc]
+  dsimp only [sourceDisplacement]
+  linarith only [ha']
+
+/-- The manuscript numbers motif slots from one. Thus its actual zero-based
+ onset is P/2-1, not P/2. The strict margins include this endpoint correction. -/
+theorem sourceWindowStart_oneBased_fit (N P T t L : ℕ)
+    (loc : Fin N → Fin t → Bool) (glob : Fin ((N/4)*t) → Bool)
+    (hP : 2*T ≤ P) (hT : 16 ≤ T) (heven : Even T) (hL : 8*L ≤ T)
+    (u : Fin N) (hY : |8*sourceDisplacement N t loc glob u| ≤ (T:ℤ)) :
+    sourceWindowStart N ((P-T)/2) t loc glob u+T ≤ P ∧
+    sourceWindowStart N ((P-T)/2) t loc glob u ≤ P/2-1 ∧
+    P/2-1+L ≤ sourceWindowStart N ((P-T)/2) t loc glob u+T := by
+  have hc := sourceWindowStart_cast_of_fit N P T t loc glob hP hT u hY
+  have ha := centered_window_floor_anchor P T (by omega) heven
+  have hy := abs_le.mp hY
+  omega
+
+/-- The exact source anchor in zero-based coordinates. The minus one is
+ forced by the one-based motif-label convention. -/
+theorem sourceWindowStart_oneBased_anchor (N P T t L : ℕ)
+    (loc : Fin N → Fin t → Bool) (glob : Fin ((N/4)*t) → Bool)
+    (hP : 2*T ≤ P) (hT : 16 ≤ T) (heven : Even T) (hL : 8*L ≤ T)
+    (u : Fin N) (hY : |8*sourceDisplacement N t loc glob u| ≤ (T:ℤ)) :
+    ((P/2-1-sourceWindowStart N ((P-T)/2) t loc glob u:ℕ):ℤ)=
+      (T/2:ℕ)-1+(t/2:ℕ)-(∑a : Fin t,(bitIndex (loc u a):ℤ))-
+        scheduledGlobalShift N t glob u := by
+  have hf := sourceWindowStart_oneBased_fit N P T t L loc glob hP hT heven hL u hY
+  have ha := sourceWindowStart_anchor N P T t L loc glob hP hT heven hL u hY
+  omega
+
+/-- The actual recursively substituted source block words. Indices in P
+ are shifted down by one, but physical binary strings are unchanged. -/
+def sourceBlockWord (s : ℕ) (P : ℕ → ℕ) (atom : Bool → List Bool) : ℕ → Bool → List Bool
+  | 0 => atom
+  | j+1 => complementaryWords (P j) (P j/2-1) (2^s) (prouhetBool s)
+      (sourceBlockWord s P atom j)
+
+def sourceBlockLength (P : ℕ → ℕ) (S : ℕ) : ℕ → ℕ
+  | 0 => S
+  | j+1 => P j*sourceBlockLength P S j
+
+theorem sourceBlockWord_length (s : ℕ) (P : ℕ → ℕ) (atom : Bool → List Bool) (S : ℕ)
+    (hw : ∀τ,(atom τ).length=S) (j : ℕ) (τ : Bool) :
+    (sourceBlockWord s P atom j τ).length=sourceBlockLength P S j := by
+  induction j generalizing τ with
+  | zero => exact hw τ
+  | succ j ih => exact complementaryWords_length _ _ _ _ _ _ (fun τ => ih τ) τ
+
+theorem sourceBlockWord_injective (s : ℕ) (P : ℕ → ℕ) (atom : Bool → List Bool) (S : ℕ)
+    (hw : ∀τ,(atom τ).length=S) (hinj : Function.Injective atom) (j : ℕ)
+    (hP : ∀k<j,0<P k) : Function.Injective (sourceBlockWord s P atom j) := by
+  induction j with
+  | zero => exact hinj
+  | succ j ih =>
+    apply complementaryWords_injective _ _ _ (sourceBlockLength P S j)
+      _ _ (sourceBlockWord_length s P atom S hw j)
+    · exact ih (fun k hk => hP k (by omega))
+    · positivity
+    · have h := hP j (Nat.lt_succ_self j); omega
+
+end TraceReconstruction
+
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- Total Boolean type, also on failed layouts; no conditioning is hidden
+ in the definition. -/
+def integerTemplateType (N s : ℕ) (X : ℤ) (τ : Bool) (u : Fin N) : Bool :=
+  decide (motifChildSign s X u.val (boolTypeSign τ)= -1)
+
+theorem boolTypeSign_integerTemplate (N s : ℕ) (X : ℤ) (τ : Bool) (u : Fin N) :
+    boolTypeSign (integerTemplateType N s X τ u)=motifChildSign s X u.val (boolTypeSign τ) := by
+  have hτ : IsTypeSign (boolTypeSign τ) := by cases τ <;> norm_num [IsTypeSign,boolTypeSign]
+  rcases motifChildSign_isSign s X u.val hτ with h | h <;>
+    (simp only [integerTemplateType,h]; norm_num [boolTypeSign])
+
+theorem integerTemplateType_nat (N s a : ℕ) (τ : Bool) :
+    integerTemplateType N s (a:ℤ) τ=templateType N a (2^s) (prouhetBool s) τ := by
+  funext u
+  apply boolTypeSign_injective
+  rw [boolTypeSign_integerTemplate,boolTypeSign_template]
+
+/-- Global trials at a parent use its children's local-trial count. This
+ fixes the index shift between source level labels and zero-based arrays. -/
+def sourceScheduledTrials (T n : ℕ → ℕ) : ℕ → ℕ
+  | 0 => 0
+  | b+1 => (T (b+1)/4)*n b
+
+def sourceScheduledShift (T n : ℕ → ℕ) :
+    (b : ℕ) → (Fin (sourceScheduledTrials T n b) → Bool) → Fin (T b) → ℤ
+  | 0, _, _ => 0
+  | b+1, glob, u => scheduledGlobalShift (T (b+1)) (n b) glob u
+
+def sourceLocalCenter (T n : ℕ → ℕ) (b : ℕ) : ℤ :=
+  (T b/2:ℕ)-1+(n b/2:ℕ)
+
+def sourceNodeAnchor (T n : ℕ → ℕ) (center : ℕ → ℤ) (b : ℕ)
+    (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool (b+1)) (incoming : ℤ) : ℤ :=
+  center b-(∑i,(bitIndex (bits.1 i):ℤ))-incoming
+
+/-- All actual leaf Booleans on the complete deterministic window tree. -/
+def sourceTreeLeafBool (T n : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ) :
+    (b : ℕ) → WindowAssignments T n (sourceScheduledTrials T n) Bool b → ℤ → Bool → WindowLeaf T b → Bool
+  | 0, _, _, τ, _ => τ
+  | b+1, bits, incoming, τ, v =>
+      sourceTreeLeafBool T n s center b (bits.2.2 v.1)
+        (sourceScheduledShift T n b bits.2.1 v.1)
+        (integerTemplateType (T b) s (sourceNodeAnchor T n center b bits incoming) τ v.1) v.2
+
+theorem sourceTreeLeafBool_sign (T n : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (b : ℕ) (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool b)
+    (incoming : ℤ) (τ : Bool) (v : WindowLeaf T b) :
+    boolTypeSign (sourceTreeLeafBool T n s center b bits incoming τ v)=
+      sourceTreeLeafSign T n (sourceScheduledTrials T n) s center
+        (sourceScheduledShift T n) b bits incoming (boolTypeSign τ) v := by
+  induction b generalizing incoming τ with
+  | zero => rfl
+  | succ b ih =>
+    change boolTypeSign (sourceTreeLeafBool T n s center b _ _ _ v.2)=_
+    rw [ih,boolTypeSign_integerTemplate]
+    rfl
+
+def sourceNominalPre (P T : ℕ → ℕ) (b : ℕ) : ℕ := (P b-T b)/2
+def sourceNominalPost (P T : ℕ → ℕ) (b : ℕ) : ℕ := P b-T b-sourceNominalPre P T b
+
+theorem sourceNominal_total (P T : ℕ → ℕ) (b : ℕ) (h : T b ≤ P b) :
+    sourceNominalPre P T b+T b+sourceNominalPost P T b=P b := by
+  dsimp [sourceNominalPre,sourceNominalPost]
+  omega
+
+/-- The actual center before recursively cutting its pure side regions. -/
+def sourceCenterWord (s : ℕ) (P T n : ℕ → ℕ) (center : ℕ → ℤ) (atom : Bool → List Bool) :
+    (b : ℕ) → WindowAssignments T n (sourceScheduledTrials T n) Bool b → ℤ → Bool → List Bool
+  | 0, _, _, τ => atom τ
+  | b+1, bits, incoming, τ => atomString (T b) (sourceBlockWord s P atom b)
+      (integerTemplateType (T b) s (sourceNodeAnchor T n center b bits incoming) τ)
+
+/-- The unconditional literal surrogate partition: local and common-global
+ pairs contribute their real selected pure lengths, on every bit assignment. -/
+def sourceTreeWord (s : ℕ) (P T n h : ℕ → ℕ) (center : ℕ → ℤ) (atom : Bool → List Bool) :
+    (b : ℕ) → WindowAssignments T n (sourceScheduledTrials T n) Bool b → ℤ → Bool → List Bool
+  | 0, _, _, τ => atom τ
+  | 1, bits, incoming, τ => sourceCenterWord s P T n center atom 1 bits incoming τ
+  | b+2, bits, incoming, τ =>
+      let loc := fun u => (bits.2.2 u).1
+      let glob := bits.2.1
+      coordinateWord (fun u : Fin (T (b+1)) =>
+        pureAtomRun (sourceBlockWord s P atom b false)
+          (sourcePreCount (T (b+1)) (sourceNominalPre P T b) (n b) (h b) loc glob u) ++
+        sourceTreeWord s P T n h center atom (b+1) (bits.2.2 u)
+          (sourceScheduledShift T n (b+1) glob u)
+          (integerTemplateType (T (b+1)) s (sourceNodeAnchor T n center (b+1) bits incoming) τ u) ++
+        pureAtomRun (sourceBlockWord s P atom b false)
+          (sourcePostCount (T (b+1)) (sourceNominalPost P T b) (n b) (h b) loc glob u))
+
+/-- The actual fit event uses a joint test of all displacements. It does not
+ assert independence and will only be charged probabilistically afterwards. -/
+def SourceTreeFit (T n : ℕ → ℕ) :
+    (b : ℕ) → WindowAssignments T n (sourceScheduledTrials T n) Bool b → Prop
+  | 0, _ => True
+  | 1, _ => True
+  | b+2, bits =>
+      (∀u, |8*sourceDisplacement (T (b+1)) (n b) (fun v => (bits.2.2 v).1) bits.2.1 u| ≤ (T b:ℤ)) ∧
+      ∀u, SourceTreeFit T n (b+1) (bits.2.2 u)
+
+/-- Numerical layout conditions from the source. These contain no word or
+ statistical conclusions, only the genuine integer margins. -/
+structure SourceLayoutMargins (s : ℕ) (P T n h : ℕ → ℕ) (b : ℕ) : Prop where
+  positive : ∀j<b, 0<T j
+  large : ∀j<b, 16≤T j
+  even_window : ∀j<b, Even (T j)
+  ambient : ∀j<b, 2*T j≤P j
+  motif : ∀j<b, 8*2^s≤T j
+  even_trials : ∀j<b, Even (n j)
+  pre_reserve : ∀j<b, 2*sourceLayoutReserve (n j) (h j)≤ sourceNominalPre P T j
+  post_reserve : ∀j<b, 2*sourceLayoutReserve (n j) (h j)≤ sourceNominalPost P T j
+
+theorem SourceLayoutMargins.restrict {s : ℕ} {P T n h : ℕ → ℕ} {b c : ℕ}
+    (H : SourceLayoutMargins s P T n h b) (hc : c≤b) : SourceLayoutMargins s P T n h c := by
+  constructor <;> intro j hj
+  · exact H.positive j (lt_of_lt_of_le hj hc)
+  · exact H.large j (lt_of_lt_of_le hj hc)
+  · exact H.even_window j (lt_of_lt_of_le hj hc)
+  · exact H.ambient j (lt_of_lt_of_le hj hc)
+  · exact H.motif j (lt_of_lt_of_le hj hc)
+  · exact H.even_trials j (lt_of_lt_of_le hj hc)
+  · exact H.pre_reserve j (lt_of_lt_of_le hj hc)
+  · exact H.post_reserve j (lt_of_lt_of_le hj hc)
+
+/-- Every recursively cut surrogate center is literally the actual center on
+ the joint fit event. The source type assignment is retained at all leaves. -/
+theorem sourceTreeWord_eq_center (s : ℕ) (P T n h : ℕ → ℕ) (center : ℕ → ℤ)
+    (atom : Bool → List Bool) (b : ℕ) (H : SourceLayoutMargins s P T n h b)
+    (hc : ∀j, j+1<b → center j=sourceLocalCenter T n j)
+    (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool b)
+    (incoming : ℤ) (τ : Bool) (hfit : SourceTreeFit T n b bits) :
+    sourceTreeWord s P T n h center atom b bits incoming τ=
+      sourceCenterWord s P T n center atom b bits incoming τ := by
+  induction b using Nat.twoStepInduction generalizing incoming τ with
+  | zero => rfl
+  | one => rfl
+  | more b ih₁ ih₂ =>
+    change (Fin (n (b+1)) → Bool) × (Fin ((T (b+1)/4)*n b) → Bool) ×
+      (Fin (T (b+1)) → WindowAssignments T n (sourceScheduledTrials T n) Bool (b+1)) at bits
+    let loc : Fin (T (b+1)) → Fin (n b) → Bool := fun u => (bits.2.2 u).1
+    let glob := bits.2.1
+    let a := P b/2-1
+    let A := sourceNominalPre P T b
+    let B := sourceNominalPost P T b
+    let start := sourceWindowStart (T (b+1)) A (n b) loc glob
+    let θ := integerTemplateType (T (b+1)) s (sourceNodeAnchor T n center (b+1) bits incoming) τ
+    have hb : b<b+2 := by omega
+    have htotal := sourceNominal_total P T b (by have hh := H.ambient b hb; omega)
+    have hf (u : Fin (T (b+1))) := sourceWindowStart_oneBased_fit (T (b+1)) (P b) (T b) (n b)
+      (2^s) loc glob (H.ambient b hb) (H.large b hb) (H.even_window b hb) (H.motif b hb) u (hfit.1 u)
+    have hx (u : Fin (T (b+1))) :
+        sourceNodeAnchor T n center b (bits.2.2 u) (sourceScheduledShift T n (b+1) glob u)=((a-start u:ℕ):ℤ) := by
+      change sourceNodeAnchor T n center b (bits.2.2 u) (sourceScheduledShift T n (b+1) glob u)=
+        ((P b/2-1-sourceWindowStart (T (b+1)) ((P b-T b)/2) (n b) loc glob u:ℕ):ℤ)
+      rw [sourceWindowStart_oneBased_anchor (T (b+1)) (P b) (T b) (n b) (2^s) loc glob
+        (H.ambient b hb) (H.large b hb) (H.even_window b hb) (H.motif b hb) u (hfit.1 u)]
+      simp only [sourceNodeAnchor,hc b (by omega),sourceLocalCenter,sourceScheduledShift,loc]
+    have hchild (u : Fin (T (b+1))) :
+        sourceTreeWord s P T n h center atom (b+1) (bits.2.2 u)
+          (sourceScheduledShift T n (b+1) glob u) (θ u)=
+        complementaryWords (T b) (a-start u) (2^s) (prouhetBool s) (sourceBlockWord s P atom b) (θ u) := by
+      rw [ih₂ (H.restrict (by omega)) (fun j hj => hc j (by omega)) _ _ _ (hfit.2 u)]
+      simp only [sourceCenterWord,hx,integerTemplateType_nat,complementaryWords]
+    have hstartnonneg (u : Fin (T (b+1))) : 0≤(A:ℤ)+sourceDisplacement (T (b+1)) (n b) loc glob u := by
+      have he := sourceWindowStart_cast_of_fit (T (b+1)) (P b) (T b) (n b) loc glob
+        (H.ambient b hb) (H.large b hb) u (hfit.1 u)
+      have hn : (0:ℤ)≤(start u:ℕ) := Int.natCast_nonneg _
+      change (start u:ℤ)=(A:ℤ)+_ at he
+      omega
+    have hp := H.positive (b+1) (by omega)
+    obtain ⟨k,hk⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : T (b+1)≠0)
+    have hpart : (List.ofFn (fun u => complementaryWords (P b) a (2^s) (prouhetBool s)
+          (sourceBlockWord s P atom b) (θ u))).flatten=
+        coordinateWord (fun u => pureAtomRun (sourceBlockWord s P atom b false)
+          (sourcePreCount (T (b+1)) A (n b) (h b) loc glob u) ++
+          complementaryWords (T b) (a-start u) (2^s) (prouhetBool s) (sourceBlockWord s P atom b) (θ u) ++
+          pureAtomRun (sourceBlockWord s P atom b false) (sourcePostCount (T (b+1)) B (n b) (h b) loc glob u)) := by
+      have hh := sourceLayout_literal_partition k A (T b) B (n b) (h b) a (2^s)
+      have hk' : k+1=T (b+1) := hk.symm
+      rw [hk'] at hh
+      rw [show A+T b+B=P b from htotal] at hh
+      exact hh (H.even_trials b hb) (H.pre_reserve b hb) (H.post_reserve b hb) loc glob
+        (prouhetBool s) (sourceBlockWord s P atom b) θ hstartnonneg
+        (fun u => (hf u).2.1) (fun u => (hf u).2.2) (fun u => (hf u).1)
+    change coordinateWord (fun u : Fin (T (b+1)) =>
+      pureAtomRun (sourceBlockWord s P atom b false) (sourcePreCount (T (b+1)) A (n b) (h b) loc glob u) ++
+      sourceTreeWord s P T n h center atom (b+1) (bits.2.2 u)
+        (sourceScheduledShift T n (b+1) glob u) (θ u) ++
+      pureAtomRun (sourceBlockWord s P atom b false) (sourcePostCount (T (b+1)) B (n b) (h b) loc glob u))=
+        atomString (T (b+1)) (sourceBlockWord s P atom (b+1)) θ
+    calc
+      _ = coordinateWord (fun u : Fin (T (b+1)) =>
+        pureAtomRun (sourceBlockWord s P atom b false) (sourcePreCount (T (b+1)) A (n b) (h b) loc glob u) ++
+        complementaryWords (T b) (a-start u) (2^s) (prouhetBool s) (sourceBlockWord s P atom b) (θ u) ++
+        pureAtomRun (sourceBlockWord s P atom b false) (sourcePostCount (T (b+1)) B (n b) (h b) loc glob u)) := by
+          apply congrArg coordinateWord
+          funext u
+          exact congrArg (fun w => pureAtomRun (sourceBlockWord s P atom b false)
+            (sourcePreCount (T (b+1)) A (n b) (h b) loc glob u) ++ w ++
+              pureAtomRun (sourceBlockWord s P atom b false)
+                (sourcePostCount (T (b+1)) B (n b) (h b) loc glob u)) (hchild u)
+      _ = _ := hpart.symm
+
+end TraceReconstruction
+
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/TraceReconstruction/LayoutRates.lean b/lean/OAI/Probability/TraceReconstruction/LayoutRates.lean
index 0337da5bc6a37fed47fe1be0a6674277ed497110..333929b85252786fc6a6198bdce281eb00b7e29e 100644
--- a/lean/OAI/Probability/TraceReconstruction/LayoutRates.lean
+++ b/lean/OAI/Probability/TraceReconstruction/LayoutRates.lean
@@ -1,1182 +1,1182 @@
-import OAI.Probability.TraceReconstruction.InitialRates
-
-namespace OAI
-
-/-!
-# LayoutRates
-
-Integer block schedules, rounding, length and trial margins, sparse gaps, and forced-anchor decay.
--/
-
-open scoped ENNReal
-noncomputable section
-
-open Filter
-open scoped Topology
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- The source integer window size, with zero-based nesting index. -/
-def sourceWindowSize (N : ℝ) (b : ℕ) (ε : ℝ) (j : ℕ) : ℕ :=
-  4*⌊N^(scaleKappa b (j+1) ε)/4⌋₊
-
-/-- At a nonroot node the source trial count is an even integer. -/
-def sourceLocalTrials (P h : ℕ) : ℕ := 2*(P/(100*(2*h+1)))
-
-/-- The root schedule count: the literal floor after subtracting its window. -/
-def sourceRootTrials (N : ℝ) (S T h : ℕ) : ℕ :=
-  (⌊N/(S:ℝ)⌋₊-T)/(2*h+1)
-
-/-- Literal length and overlap-count state of one nested source block.
- `copies` and `trials` refer to the transition into this state. -/
-structure SourceArithmeticStage where
-  length : ℕ
-  count : ℕ
-  copies : ℕ
-  trials : ℕ
-
-def sourceArithmeticStages (C N : ℝ) (b : ℕ) (ε : ℝ) (ell k : ℕ) :
-    ℕ → SourceArithmeticStage
-  | 0 => ⟨ell,k,1,0⟩
-  | j+1 =>
-    let prev := sourceArithmeticStages C N b ε ell k j
-    let T := sourceWindowSize N b ε j
-    let J := sourceRootTrials N prev.length T prev.count
-    let P := if j+1=b then J*(2*prev.count+1)+T
-      else ⌊N^(scaleA b (j+1) ε)/(prev.length:ℝ)⌋₊
-    let m := if j+1=b then J else sourceLocalTrials P prev.count
-    ⟨prev.length*P,⌈C*prev.count*P⌉₊,P,m⟩
-
-def sourceArithmeticP (C N : ℝ) (b : ℕ) (ε : ℝ) (ell k : ℕ) (j : ℕ) : ℕ :=
-  (sourceArithmeticStages C N b ε ell k (j+1)).copies
-
-def sourceArithmeticTrials (C N : ℝ) (b : ℕ) (ε : ℝ) (ell k : ℕ) (j : ℕ) : ℕ :=
-  (sourceArithmeticStages C N b ε ell k (j+1)).trials
-
-def sourceArithmeticLength (C N : ℝ) (b : ℕ) (ε : ℝ) (ell k : ℕ) (j : ℕ) : ℕ :=
-  (sourceArithmeticStages C N b ε ell k j).length
-
-def sourceArithmeticCount (C N : ℝ) (b : ℕ) (ε : ℝ) (ell k : ℕ) (j : ℕ) : ℕ :=
-  (sourceArithmeticStages C N b ε ell k j).count
-
-theorem sourceLocalTrials_even (P h : ℕ) : Even (sourceLocalTrials P h) :=
-  even_two_mul _
-
-theorem sourceLocalTrials_budget (P h : ℕ) :
-    50*sourceLocalTrials P h*(2*h+1) ≤ P := by
-  have H := Nat.mul_div_le P (100*(2*h+1))
-  unfold sourceLocalTrials
-  nlinarith only [H]
-
-theorem sourceLocalTrials_reserves (P T h : ℕ) (hT : 2*T ≤ P) :
-    2*sourceLayoutReserve (sourceLocalTrials P h) h ≤ (P-T)/2 ∧
-    2*sourceLayoutReserve (sourceLocalTrials P h) h ≤ P-T-(P-T)/2 := by
-  have H := sourceLocalTrials_budget P h
-  rw [mul_assoc] at H
-  have he : sourceLocalTrials P h/2=P/(100*(2*h+1)) := by
-    unfold sourceLocalTrials
-    omega
-  have hr : 2*sourceLayoutReserve (sourceLocalTrials P h) h=
-      sourceLocalTrials P h*(2*h+1) := by
-    unfold sourceLayoutReserve
-    rw [he]
-    unfold sourceLocalTrials
-    ring
-  rw [hr]
-  have hm : 2*(sourceLocalTrials P h*(2*h+1)) ≤ P-T := by omega
-  constructor <;> omega
-
-theorem sourceWindowSize_even (N : ℝ) (b : ℕ) (ε : ℝ) (j : ℕ) :
-    Even (sourceWindowSize N b ε j) := by
-  unfold sourceWindowSize
-  exact ⟨2*⌊N^(scaleKappa b (j+1) ε)/4⌋₊,by ring⟩
-
-theorem sourceArithmeticLength_step (C N : ℝ) (b : ℕ) (ε : ℝ) (ell k j : ℕ) :
-    sourceArithmeticLength C N b ε ell k (j+1)=
-      sourceArithmeticLength C N b ε ell k j*sourceArithmeticP C N b ε ell k j := rfl
-
-theorem sourceArithmeticCount_step (C N : ℝ) (b : ℕ) (ε : ℝ) (ell k j : ℕ) :
-    sourceArithmeticCount C N b ε ell k (j+1)=
-      ⌈C*sourceArithmeticCount C N b ε ell k j*sourceArithmeticP C N b ε ell k j⌉₊ := rfl
-
-theorem sourceArithmetic_inner_trials (C N : ℝ) (b : ℕ) (ε : ℝ) (ell k j : ℕ)
-    (hj : j+1≠b) : sourceArithmeticTrials C N b ε ell k j=
-      sourceLocalTrials (sourceArithmeticP C N b ε ell k j) (sourceArithmeticCount C N b ε ell k j) := by
-  simp only [sourceArithmeticTrials,sourceArithmeticP,sourceArithmeticCount,sourceArithmeticStages,ite_eq_right hj]
-
-theorem sourceArithmetic_root_copies (C N : ℝ) (b : ℕ) (ε : ℝ) (ell k : ℕ) (hb : 1 ≤ b) :
-    sourceArithmeticP C N b ε ell k (b-1)=
-      sourceArithmeticTrials C N b ε ell k (b-1)*(2*sourceArithmeticCount C N b ε ell k (b-1)+1)+
-        sourceWindowSize N b ε (b-1) := by
-  simp only [sourceArithmeticP,sourceArithmeticTrials,sourceArithmeticCount,sourceArithmeticStages,
-    Nat.sub_add_cancel hb,ite_true]
-
-theorem sourceArithmetic_count_lift (C N : ℝ) (b : ℕ) (ε : ℝ) (ell k j : ℕ) :
-    C*sourceArithmeticCount C N b ε ell k j*sourceArithmeticP C N b ε ell k j ≤
-      sourceArithmeticCount C N b ε ell k (j+1) := Nat.le_ceil _
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- Rounding the source windows loses at most a factor two. -/
-theorem sourceWindowSize_bounds {N ε : ℝ} {b j : ℕ}
-    (h : 8 ≤ N^(scaleKappa b (j+1) ε)) :
-    N^(scaleKappa b (j+1) ε)/2 ≤ (sourceWindowSize N b ε j:ℝ) ∧
-      (sourceWindowSize N b ε j:ℝ) ≤ N^(scaleKappa b (j+1) ε) := by
-  have hlo := Nat.lt_floor_add_one (N^(scaleKappa b (j+1) ε)/4)
-  have hhi := Nat.floor_le (show 0 ≤ N^(scaleKappa b (j+1) ε)/4 by linarith)
-  unfold sourceWindowSize
-  push_cast
-  constructor <;> linarith only [h,hlo,hhi]
-
-/-- The nonroot even trial count is between one and two times its unrounded
- quotient once the quotient is at least two. -/
-theorem sourceLocalTrials_bounds (P h : ℕ) (hlarge : (200:ℝ)*(2*h+1) ≤ P) :
-    (P:ℝ)/(100*(2*h+1)) ≤ (sourceLocalTrials P h:ℝ) ∧
-      (sourceLocalTrials P h:ℝ) ≤ (P:ℝ)/(50*(2*h+1)) := by
-  have hd : 0 < (100:ℝ)*(2*h+1) := by positivity
-  have hx : (2:ℝ) ≤ (P:ℝ)/(100*(2*h+1)) := (le_div_iff₀ hd).mpr (by linarith only [hlarge])
-  have he : (sourceLocalTrials P h:ℝ)=2*(⌊(P:ℝ)/(100*(2*h+1))⌋₊:ℝ) := by
-    unfold sourceLocalTrials
-    rw [show (100:ℝ)*(2*h+1)=((100*(2*h+1):ℕ):ℝ) by push_cast; ring,Nat.floor_div_eq_div]
-    norm_cast
-  rw [he]
-  have hlo := Nat.lt_floor_add_one ((P:ℝ)/(100*(2*h+1)))
-  have hhi := Nat.floor_le (div_nonneg (Nat.cast_nonneg P) hd.le)
-  constructor
-  · linarith only [hx,hlo]
-  · have heq : (P:ℝ)/(50*(2*h+1))=2*((P:ℝ)/(100*(2*h+1))) := by field_simp; ring
-    rw [heq]
-    linarith only [hhi]
-
-theorem source_floor_length_bounds {X : ℝ} {S : ℕ} (hS : 0 < S) (hX : 2*(S:ℝ) ≤ X) :
-    X/2 ≤ ((S*⌊X/(S:ℝ)⌋₊:ℕ):ℝ) ∧ ((S*⌊X/(S:ℝ)⌋₊:ℕ):ℝ) ≤ X := by
-  have hs : (0:ℝ) < S := by exact_mod_cast hS
-  have hlo := Nat.lt_floor_add_one (X/(S:ℝ))
-  have hhi := Nat.floor_le (div_nonneg (show 0 ≤ X by linarith only [hX,hs]) hs.le)
-  have hh := mul_lt_mul_of_pos_left hlo hs
-  have hh' := mul_le_mul_of_nonneg_left hhi hs.le
-  rw [mul_add,mul_div_cancel₀ _ hs.ne'] at hh
-  rw [mul_div_cancel₀ _ hs.ne'] at hh'
-  push_cast
-  constructor <;> nlinarith only [hX,hh,hh']
-
-/-- The real floor and integer division versions of the root are identical
- whenever its remaining ambient count is nonnegative. -/
-theorem sourceRootTrials_floor {N : ℝ} {S T h : ℕ}
-    (hT : T ≤ ⌊N/(S:ℝ)⌋₊) :
-    sourceRootTrials N S T h=⌊((⌊N/(S:ℝ)⌋₊:ℝ)-T)/(2*h+1)⌋₊ := by
-  unfold sourceRootTrials
-  rw [←Nat.cast_sub hT]
-  rw [show (2:ℝ)*h+1=((2*h+1:ℕ):ℝ) by push_cast; ring,Nat.floor_div_eq_div]
-
-theorem source_root_length_bounds {N : ℝ} {S T h : ℕ} (hS : 0 < S)
-    (hT : T ≤ ⌊N/(S:ℝ)⌋₊) (hN : (2*h+2:ℝ)*S ≤ N/2) :
-    N/2 ≤ (S:ℝ)*(sourceRootTrials N S T h*(2*h+1)+T) ∧
-      (S:ℝ)*(sourceRootTrials N S T h*(2*h+1)+T) ≤ N := by
-  have hs : (0:ℝ)<S := by exact_mod_cast hS
-  have hd : 0 < 2*h+1 := by omega
-  have hi := Nat.div_mul_le_self (⌊N/(S:ℝ)⌋₊-T) (2*h+1)
-  have hj := Nat.lt_mul_div_succ (⌊N/(S:ℝ)⌋₊-T) hd
-  rw [Nat.mul_comm (2*h+1)] at hj
-  have hp : sourceRootTrials N S T h*(2*h+1)+T ≤ ⌊N/(S:ℝ)⌋₊ := by
-    unfold sourceRootTrials
-    omega
-  have hp' : ⌊N/(S:ℝ)⌋₊ < (sourceRootTrials N S T h+1)*(2*h+1)+T := by
-    unfold sourceRootTrials
-    omega
-  have hN0 : 0 ≤ N := by
-    have hh : (0:ℝ) ≤ (2*h+2:ℝ)*S := by positivity
-    linarith only [hN,hh]
-  have hfloor := Nat.floor_le (div_nonneg hN0 hs.le)
-  have hfloor' := Nat.lt_floor_add_one (N/(S:ℝ))
-  have hpr : (sourceRootTrials N S T h:ℝ)*(2*h+1)+T ≤ (⌊N/(S:ℝ)⌋₊:ℝ) := by exact_mod_cast hp
-  have hpr' : (⌊N/(S:ℝ)⌋₊:ℝ) < (sourceRootTrials N S T h+1:ℝ)*(2*h+1)+T := by exact_mod_cast hp'
-  constructor
-  · have hf : N/(S:ℝ) < (sourceRootTrials N S T h:ℝ)*(2*h+1)+T+(2*h+2) := by
-      nlinarith only [hfloor',hpr']
-    have hh := mul_lt_mul_of_pos_left hf hs
-    rw [mul_div_cancel₀ _ hs.ne'] at hh
-    nlinarith only [hh,hN]
-  · have hh := mul_le_mul_of_nonneg_left (hpr.trans hfloor) hs.le
-    simpa only [mul_div_cancel₀ _ hs.ne'] using hh
-
-theorem source_ceil_count_bounds {C F : ℝ} {S h P : ℕ} (hC : 2 ≤ C) (hF : 1 ≤ F)
-    (hS : 1 ≤ S) (hP : 1 ≤ P) (hlo : S ≤ h) (hhi : (h:ℝ) ≤ F*S) :
-    S*P ≤ ⌈C*h*P⌉₊ ∧ (⌈C*h*P⌉₊:ℝ) ≤ C^2*F*(S*P) := by
-  have hs : (1:ℝ) ≤ S := by exact_mod_cast hS
-  have hp : (1:ℝ) ≤ P := by exact_mod_cast hP
-  have hh : (S:ℝ) ≤ h := by exact_mod_cast hlo
-  have hceil := Nat.ceil_lt_add_one (show 0 ≤ C*(h:ℝ)*P by positivity)
-  constructor
-  · have hbound : (S:ℝ)*P ≤ C*h*P := by
-      have hh' := mul_le_mul_of_nonneg_right hh (by positivity : (0:ℝ) ≤ P)
-      have hc := mul_le_mul_of_nonneg_right (show (1:ℝ) ≤ C by linarith) (show 0 ≤ (h:ℝ)*P by positivity)
-      nlinarith only [hh',hc]
-    exact_mod_cast hbound.trans (Nat.le_ceil (C*(h:ℝ)*P))
-  · have hmul := mul_le_mul_of_nonneg_left hhi (show 0 ≤ C*(P:ℝ) by positivity)
-    have hbase : (1:ℝ) ≤ F*(S*P) := one_le_mul_of_one_le_of_one_le hF (one_le_mul_of_one_le_of_one_le hs hp)
-    have hgain : 1 ≤ C*(C-1) := by nlinarith only [hC]
-    have hgap := mul_le_mul hgain hbase (by norm_num) (by linarith only [hgain])
-    nlinarith only [hceil,hmul,hgap]
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem source_scale_target_step {N ε : ℝ} (b j : ℕ) (hN : 0 < N) :
-    N^(scaleA b (j+1) ε)=(N^(scaleA b j ε))^2*N^ε := by
-  rw [←Real.rpow_natCast,←Real.rpow_mul hN.le,←Real.rpow_add hN]
-  congr 1
-  have hh := scaleA_step b j ε
-  norm_num only [Nat.cast_ofNat]
-  linarith only [hh]
-
-theorem sourceWindowSize_le_previous {N ε : ℝ} {b j : ℕ}
-    (hN : 1 ≤ N) (he : 0 ≤ ε) :
-    (sourceWindowSize N b ε j:ℝ) ≤ N^(scaleA b j ε) := by
-  have hf := Nat.floor_le (show 0 ≤ N^(scaleKappa b (j+1) ε)/4 by positivity)
-  have hk : scaleKappa b (j+1) ε ≤ scaleA b j ε := by
-    unfold scaleKappa
-    simp only [Nat.add_sub_cancel]
-    have hh : (1:ℝ) ≤ 2^(j+1) := one_le_pow₀ (by norm_num)
-    nlinarith only [mul_nonneg (sub_nonneg.mpr hh) he]
-  apply le_trans _ (Real.rpow_le_rpow_of_exponent_le hN hk)
-  unfold sourceWindowSize
-  push_cast
-  linarith only [hf]
-
-/-- The root's subtraction and rounding margins follow from the one
- subpower comparison N^epsilon >= 8F. -/
-theorem source_root_margins {N X u F : ℝ} {S h T : ℕ}
-    (hS : 1 ≤ S) (hX : (S:ℝ) ≤ X) (hF : 1 ≤ F) (hu : 8*F ≤ u)
-    (hh : (h:ℝ) ≤ F*S) (hT : (T:ℝ) ≤ X) (hrel : X^2*u=N) :
-    T ≤ ⌊N/(S:ℝ)⌋₊ ∧ (2*(h:ℝ)+2)*S ≤ N/2 := by
-  have hs : (1:ℝ) ≤ S := by exact_mod_cast hS
-  have hx : 1 ≤ X := hs.trans hX
-  have hx0 : 0 ≤ X := by linarith only [hx]
-  have hs0 : (0:ℝ) < S := by linarith only [hs]
-  have hfs : 1 ≤ F*S := by nlinarith only [hF,hs]
-  have hsX : (S:ℝ)^2 ≤ X^2 := sq_le_sq₀ hs0.le hx0 |>.mpr hX
-  have h8 : 8*F*X^2 ≤ N := by nlinarith only [mul_le_mul_of_nonneg_right hu (sq_nonneg X),hrel]
-  constructor
-  · apply Nat.le_floor
-    apply (le_div_iff₀ hs0).mpr
-    have htX := mul_le_mul hT hX hs0.le hx0
-    nlinarith only [htX,h8,mul_le_mul_of_nonneg_right hF (sq_nonneg X)]
-  · have hhS := mul_le_mul_of_nonneg_right hh hs0.le
-    have hS2 := mul_le_mul_of_nonneg_right hfs hs0.le
-    have hFsq := mul_le_mul_of_nonneg_left hsX (by linarith only [hF] : 0 ≤ F)
-    nlinarith only [hhS,hS2,hFsq,h8]
-
-theorem source_arithmetic_inner_copies (C N : ℝ) (b : ℕ) (ε : ℝ) (ell k j : ℕ)
-    (hj : j+1≠b) : sourceArithmeticP C N b ε ell k j=
-      ⌊N^(scaleA b (j+1) ε)/(sourceArithmeticLength C N b ε ell k j:ℝ)⌋₊ := by
-  simp only [sourceArithmeticP,sourceArithmeticLength,sourceArithmeticStages,ite_eq_right hj]
-
-theorem source_arithmetic_root_copies (C N : ℝ) (b : ℕ) (ε : ℝ) (ell k j : ℕ)
-    (hj : j+1=b) : sourceArithmeticP C N b ε ell k j=
-      sourceRootTrials N (sourceArithmeticLength C N b ε ell k j) (sourceWindowSize N b ε j)
-        (sourceArithmeticCount C N b ε ell k j)*
-          (2*sourceArithmeticCount C N b ε ell k j+1)+sourceWindowSize N b ε j := by
-  simp only [sourceArithmeticP,sourceArithmeticLength,sourceArithmeticCount,sourceArithmeticStages,ite_eq_left hj]
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- All nested lengths and overlap counts for the literal floors/ceilings.
- Only the scalar subpower margin is needed for every rounding step. -/
-theorem source_arithmetic_lengths {C N F ε : ℝ} {b ell k : ℕ}
-    (hC : 2 ≤ C) (hN : 1 ≤ N) (hF : 1 ≤ F) (he : 0 ≤ ε)
-    (ha : ∀j ≤ b,0 ≤ scaleA b j ε)
-    (hsmall : 8*C^(2*b)*F ≤ N^ε)
-    (hell : N^(scaleA b 0 ε)/2 ≤ (ell:ℝ)) (hell' : (ell:ℝ) ≤ N^(scaleA b 0 ε))
-    (hk : ell ≤ k) (hk' : (k:ℝ) ≤ F*ell) :
-    ∀j ≤ b,
-      N^(scaleA b j ε)/2 ≤ (sourceArithmeticLength C N b ε ell k j:ℝ) ∧
-      (sourceArithmeticLength C N b ε ell k j:ℝ) ≤ N^(scaleA b j ε) ∧
-      sourceArithmeticLength C N b ε ell k j ≤ sourceArithmeticCount C N b ε ell k j ∧
-      (sourceArithmeticCount C N b ε ell k j:ℝ) ≤
-        C^(2*j)*F*sourceArithmeticLength C N b ε ell k j := by
-  have hN0 : 0 < N := by linarith only [hN]
-  have hC1 : 1 ≤ C := by linarith only [hC]
-  intro j
-  induction j with
-  | zero =>
-    intro _
-    simpa only [sourceArithmeticLength,sourceArithmeticCount,sourceArithmeticStages,
-      Nat.mul_zero,pow_zero,one_mul] using And.intro hell (And.intro hell' (And.intro hk hk'))
-  | succ j ih =>
-    intro hj
-    obtain ⟨hlo,hhi,hcount,hcount'⟩ := ih (by omega)
-    let S := sourceArithmeticLength C N b ε ell k j
-    let H := sourceArithmeticCount C N b ε ell k j
-    let P := sourceArithmeticP C N b ε ell k j
-    let Fj := C^(2*j)*F
-    have hFj : 1 ≤ Fj := one_le_mul_of_one_le_of_one_le (one_le_pow₀ hC1) hF
-    have hS0 : (0:ℝ)<S := lt_of_lt_of_le (by positivity) hlo
-    have hS : 1 ≤ S := by exact_mod_cast (show 0 < S from Nat.cast_pos.mp hS0)
-    have hfactor : Fj ≤ C^(2*b)*F := mul_le_mul_of_nonneg_right
-      (pow_le_pow_right₀ hC1 (by omega)) (by linarith only [hF])
-    have hu : 8*Fj ≤ N^ε := (mul_le_mul_of_nonneg_left hfactor (by norm_num)).trans (by simpa only [mul_assoc] using hsmall)
-    have hx : 1 ≤ N^(scaleA b j ε) := Real.one_le_rpow hN (ha j (by omega))
-    have hlength : N^(scaleA b (j+1) ε)/2 ≤ ((S*P:ℕ):ℝ) ∧
-        ((S*P:ℕ):ℝ) ≤ N^(scaleA b (j+1) ε) := by
-      by_cases hr : j+1=b
-      · have hrel : (N^(scaleA b j ε))^2*N^ε=N := by
-          rw [←source_scale_target_step b j hN0,hr,scaleA_top,Real.rpow_one]
-        have hmargin := source_root_margins hS hhi hFj hu hcount'
-          (sourceWindowSize_le_previous (b:=b) (j:=j) hN he) hrel
-        have hlen := source_root_length_bounds (Nat.pos_of_ne_zero (by omega : S≠0)) hmargin.1 hmargin.2
-        have hp : P=sourceRootTrials N S (sourceWindowSize N b ε j) H*(2*H+1)+sourceWindowSize N b ε j :=
-          source_arithmetic_root_copies C N b ε ell k j hr
-        rw [hp,hr,scaleA_top,Real.rpow_one]
-        simpa only [Nat.cast_mul,Nat.cast_add,Nat.cast_ofNat,Nat.cast_one] using hlen
-      · have hstep := source_scale_target_step (ε:=ε) b j hN0
-        have hX : 2*(S:ℝ) ≤ N^(scaleA b (j+1) ε) := by
-          rw [hstep]
-          have hu' : 2 ≤ N^ε := by linarith only [hu,hFj]
-          have hxx : N^(scaleA b j ε) ≤ (N^(scaleA b j ε))^2 := by nlinarith only [hx]
-          nlinarith only [hhi,hxx,mul_le_mul_of_nonneg_left hu' (sq_nonneg (N^(scaleA b j ε)))]
-        have hp : P=⌊N^(scaleA b (j+1) ε)/(S:ℝ)⌋₊ :=
-          source_arithmetic_inner_copies C N b ε ell k j hr
-        rw [hp]
-        exact source_floor_length_bounds (by omega) hX
-    have hP : 1 ≤ P := by
-      have hp : (0:ℝ)<(S*P:ℕ) := lt_of_lt_of_le (by positivity) hlength.1
-      have hp' : 0<S*P := Nat.cast_pos.mp hp
-      by_contra hp0
-      have he : P=0 := by omega
-      simp only [he,Nat.mul_zero,lt_self_iff_false] at hp'
-    have hceil := source_ceil_count_bounds hC hFj hS hP hcount hcount'
-    rw [sourceArithmeticLength_step,sourceArithmeticCount_step]
-    refine ⟨hlength.1,hlength.2,hceil.1,?_⟩
-    have heq : C^2*Fj=C^(2*(j+1))*F := by
-      dsimp [Fj]
-      rw [show 2*(j+1)=2+2*j by omega,pow_add]
-      ring
-    simpa only [←Nat.cast_mul,heq] using hceil.2
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem source_copies_bounds {X u : ℝ} {S P : ℕ} (hX : 0 < X)
-    (hSlo : X/2 ≤ S) (hShi : (S:ℝ) ≤ X)
-    (hlo : X^2*u/2 ≤ ((S*P:ℕ):ℝ)) (hhi : ((S*P:ℕ):ℝ) ≤ X^2*u) :
-    X*u/2 ≤ (P:ℝ) ∧ (P:ℝ) ≤ 2*X*u := by
-  have hp := (Nat.cast_nonneg P : (0:ℝ) ≤ P)
-  have hlo' := mul_le_mul_of_nonneg_right hShi hp
-  have hhi' := mul_le_mul_of_nonneg_right hSlo hp
-  push_cast at hlo hhi
-  constructor <;> nlinarith only [hX,hlo,hhi,hlo',hhi']
-
-theorem source_inner_trials_scale {X u F : ℝ} {S P h : ℕ}
-    (hX : 1 ≤ X) (hF : 1 ≤ F) (hu : 1200*F ≤ u)
-    (hSlo : X/2 ≤ S) (hShi : (S:ℝ) ≤ X) (hhlo : S ≤ h) (hhhi : (h:ℝ) ≤ F*S)
-    (hPlo : X*u/2 ≤ (P:ℝ)) (hPhi : (P:ℝ) ≤ 2*X*u) :
-    u/(600*F) ≤ (sourceLocalTrials P h:ℝ) ∧ (sourceLocalTrials P h:ℝ) ≤ 2*u := by
-  have hx : 0<X := by linarith only [hX]
-  have hf : 0<F := by linarith only [hF]
-  have hfx : 1 ≤ F*X := one_le_mul_of_one_le_of_one_le hF hX
-  have hhu : (h:ℝ) ≤ F*X := hhhi.trans (mul_le_mul_of_nonneg_left hShi hf.le)
-  have hhl : X ≤ 2*(h:ℝ)+1 := by exact_mod_cast (show X ≤ 2*(h:ℝ)+1 by
-    have hh : (S:ℝ) ≤ h := by exact_mod_cast hhlo
-    linarith only [hh,hSlo])
-  have hhh : 2*(h:ℝ)+1 ≤ 3*F*X := by linarith only [hhu,hfx]
-  have hlarge : (200:ℝ)*(2*h+1) ≤ P := by
-    have hum := mul_le_mul_of_nonneg_left hu hx.le
-    nlinarith only [hhh,hPlo,hum]
-  obtain ⟨hlo,hhi⟩ := sourceLocalTrials_bounds P h hlarge
-  have hd : 0 < (100:ℝ)*(2*h+1) := by positivity
-  constructor
-  · apply le_trans _ hlo
-    apply (div_le_div_iff₀ (by positivity : (0:ℝ)<600*F) hd).mpr
-    have hum := mul_le_mul_of_nonneg_left hhh (by linarith only [hu,hF] : 0 ≤ u)
-    have hpm := mul_le_mul_of_nonneg_left hPlo hf.le
-    nlinarith only [hum,hpm]
-  · apply hhi.trans
-    apply (div_le_iff₀ (by positivity : (0:ℝ)<50*(2*h+1))).mpr
-    have hmm := mul_le_mul_of_nonneg_left hhl (by linarith only [hu,hF] : 0 ≤ u)
-    nlinarith only [hPhi,hmm]
-
-theorem source_root_trials_scale {X u F : ℝ} {S P h J T : ℕ}
-    (hX : 1 ≤ X) (hF : 1 ≤ F) (hu : 1200*F ≤ u)
-    (hSlo : X/2 ≤ S) (hShi : (S:ℝ) ≤ X) (hhlo : S ≤ h) (hhhi : (h:ℝ) ≤ F*S)
-    (hPlo : X*u/2 ≤ (P:ℝ)) (hPhi : (P:ℝ) ≤ 2*X*u)
-    (hT : (T:ℝ) ≤ X) (hP : P=J*(2*h+1)+T) :
-    u/(600*F) ≤ (J:ℝ) ∧ (J:ℝ) ≤ 2*u := by
-  have hx : 0<X := by linarith only [hX]
-  have hf : 0<F := by linarith only [hF]
-  have hfx : 1 ≤ F*X := one_le_mul_of_one_le_of_one_le hF hX
-  have hhu : (h:ℝ) ≤ F*X := hhhi.trans (mul_le_mul_of_nonneg_left hShi hf.le)
-  have hhl : X ≤ 2*(h:ℝ)+1 := by
-    have hh : (S:ℝ) ≤ h := by exact_mod_cast hhlo
-    linarith only [hh,hSlo]
-  have hhh : 2*(h:ℝ)+1 ≤ 3*F*X := by linarith only [hhu,hfx]
-  have hp : (P:ℝ)=(J:ℝ)*(2*h+1)+T := by exact_mod_cast hP
-  have hj : (0:ℝ) ≤ J := Nat.cast_nonneg _
-  have hupper := mul_le_mul_of_nonneg_left hhl hj
-  have hlower := mul_le_mul_of_nonneg_left hhh hj
-  constructor
-  · apply (div_le_iff₀ (by positivity : (0:ℝ)<600*F)).mpr
-    have hu' : 4 ≤ u := by linarith only [hu,hF]
-    nlinarith only [hp,hT,hPlo,hlower,mul_le_mul_of_nonneg_left hu' hx.le,hx]
-  · nlinarith only [hupper,hp,hPhi,(Nat.cast_nonneg T : (0:ℝ) ≤ T),hx]
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open Filter
-open scoped Topology Classical
-
-def sourceDegreeCutoff (B : ℕ) : ℕ := 2^(10*B+40)
-
-def sourceCancellationOrder (B D : ℕ) : ℕ :=
-  ⌈100*(totalScoreExponent B D+(sourceDegreeCutoff B:ℝ)+B+10)/sourceEpsilon B D⌉₊
-
-def sourceChannelLogConstant : ℝ := (2:ℝ)^40+1536+(32+3*primitivePeriodConstant)+1
-
-theorem sourceChannelLogConstant_pos : 0 < sourceChannelLogConstant := by
-  unfold sourceChannelLogConstant primitivePeriodConstant
-  positivity
-
-theorem deletion_inverse_cube_ge_one {q : ℝ} (hq : q∈Set.Ioc 0 1) :
-    1 ≤ q^(-3:ℤ) := by
-  rw [zpow_neg,zpow_ofNat,one_le_inv₀ (pow_pos hq.1 _)]
-  exact pow_le_one₀ hq.1.le hq.2
-
-theorem deletion_inverse_le_cube {q : ℝ} (hq : q∈Set.Ioc 0 1) :
-    q⁻¹ ≤ q^(-3:ℤ) := by
-  rw [zpow_neg,zpow_ofNat]
-  apply inv_anti₀ (pow_pos hq.1 3)
-  calc
-    q^3=q*q^2 := by ring
-    _ ≤ q*1 := mul_le_mul_of_nonneg_left (pow_le_one₀ hq.1.le hq.2) hq.1.le
-    _=q := mul_one _
-
-theorem sourceOverlapConstant_log_bound {q : ℝ} (hq : q∈Set.Ioc 0 1) :
-    Real.log (sourceOverlapConstant q) ≤ sourceChannelLogConstant*q^(-3:ℤ) := by
-  have hΛ := deletion_inverse_cube_ge_one hq
-  have hlogq : -Real.log q ≤ q^(-3:ℤ) := (neg_le_neg (Real.neg_inv_le_log hq.1.le)).trans
-    (by simpa only [neg_neg] using deletion_inverse_le_cube hq)
-  have hω : 0 < initialOverlapLower q := initialOverlapLower_pos q
-  have hlogC : Real.log (sourceOverlapConstant q)=
-      Real.log ((2:ℝ)^40)+Real.log 1536+(32+3*primitivePeriodConstant)*q^(-3:ℤ)-Real.log q := by
-    rw [sourceOverlapConstant,Real.log_div (by positivity) (mul_pos hq.1 hω).ne',
-      Real.log_mul hq.1.ne' hω.ne',initialOverlapLower,
-      Real.log_div (Real.exp_pos _).ne' (by norm_num),Real.log_exp]
-    rw [zpow_neg,zpow_ofNat]
-    ring
-  rw [hlogC]
-  have htwo := Real.log_le_self (by positivity : (0:ℝ) ≤ (2:ℝ)^40)
-  have h1536 := Real.log_le_self (by norm_num : (0:ℝ) ≤ 1536)
-  have hc := mul_le_mul_of_nonneg_left hΛ (show 0 ≤ (2:ℝ)^40+1536 by positivity)
-  unfold sourceChannelLogConstant
-  nlinarith only [hlogq,htwo,h1536,hc]
-
-theorem sourceOverlapConstant_ge_two {q : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ)) :
-    2 ≤ sourceOverlapConstant q := by
-  have hC := sourceOverlapConstant_pos hq.1
-  have hω := initialOverlapLower_lt_eighth hq.1
-  have hlarge := sourceOverlapConstant_large hq
-  nlinarith only [hC,hω,hlarge]
-
-/-- The whole block-count loss, including every nesting and every earlier
- block, is bounded by a single exp(O_B((D+1)Lambda)). -/
-theorem source_count_factor_exp_bound {q : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
-    (B D b i : ℕ) (hb : b ≤ B) (hi : i ≤ D) :
-    8*(sourceOverlapConstant q)^(2*b)*(sourceOverlapConstant q)^(2*B*i+2) ≤
-      Real.exp ((8+sourceChannelLogConstant*(2*B+2))*((D:ℝ)+1)*q^(-3:ℤ)) := by
-  let C := sourceOverlapConstant q
-  have hC : 0<C := sourceOverlapConstant_pos hq.1
-  have hC1 : 1 ≤ C := (by norm_num : (1:ℝ) ≤ 2).trans (sourceOverlapConstant_ge_two hq)
-  have hq' : q∈Set.Ioc 0 1 := ⟨hq.1,by linarith only [hq.2]⟩
-  have hΛ := deletion_inverse_cube_ge_one hq'
-  have hk := sourceChannelLogConstant_pos
-  have hlog := sourceOverlapConstant_log_bound hq'
-  have hm : 2*b+(2*B*i+2) ≤ (2*B+2)*(D+1) := by nlinarith only [hb,hi]
-  have hpow := pow_le_pow_right₀ hC1 hm
-  have hlog2 := mul_le_mul_of_nonneg_left hlog (show 0 ≤ (((2*B+2)*(D+1):ℕ):ℝ) by positivity)
-  have he : C^((2*B+2)*(D+1)) ≤
-      Real.exp (sourceChannelLogConstant*(2*B+2)*((D:ℝ)+1)*q^(-3:ℤ)) := by
-    rw [←Real.log_le_iff_le_exp (pow_pos hC _),Real.log_pow]
-    simpa only [Nat.cast_mul,Nat.cast_add,Nat.cast_ofNat,Nat.cast_one,
-      mul_assoc,mul_left_comm,mul_comm] using hlog2
-  have h8 : (8:ℝ) ≤ Real.exp (8*((D:ℝ)+1)*q^(-3:ℤ)) := by
-    apply le_trans (by norm_num : (8:ℝ) ≤ 1+8)
-    apply le_trans _ (Real.add_one_le_exp _)
-    have hd : (1:ℝ) ≤ (D:ℝ)+1 := by linarith only [Nat.cast_nonneg D (α:=ℝ)]
-    have hh := one_le_mul_of_one_le_of_one_le hd hΛ
-    linarith only [hh]
-  calc
-    8*C^(2*b)*C^(2*B*i+2)=8*C^(2*b+(2*B*i+2)) := by rw [pow_add]; ring
-    _ ≤ 8*C^((2*B+2)*(D+1)) := mul_le_mul_of_nonneg_left hpow (by norm_num)
-    _ ≤ Real.exp (8*((D:ℝ)+1)*q^(-3:ℤ))*
-        Real.exp (sourceChannelLogConstant*(2*B+2)*((D:ℝ)+1)*q^(-3:ℤ)) :=
-      mul_le_mul h8 he (pow_nonneg hC.le _) (Real.exp_pos _).le
-    _ = _ := by rw [←Real.exp_add]; congr 1; ring
-
-theorem source_uniform_count_margin (B : ℕ) (ζ : ℝ) (hB : 2 ≤ B) (hζ : ζ∈Set.Ioo 0 1)
-    {n q : ℕ→ℝ} (hn : Tendsto n atTop atTop) (hq : ∀t,q t∈Set.Ioc 0 (1/2:ℝ))
-    (hH : Tendsto (fun t=>(q t)^3*Real.log (n t)) atTop atTop) :
-    ∀ᶠt in atTop,∀i ≤ sourceDepth B ζ ((q t)^3*Real.log (n t)),∀b ≤ B,
-      8*(sourceOverlapConstant (q t))^(2*b)*(sourceOverlapConstant (q t))^(2*B*i+2) ≤
-        (sourceTargetScale B (sourceDepth B ζ ((q t)^3*Real.log (n t))) (n t) (i+1))^
-          (sourceEpsilon B (sourceDepth B ζ ((q t)^3*Real.log (n t)))) := by
-  let K := 8+sourceChannelLogConstant*(2*B+2)
-  have hK : 0 ≤ K := by dsimp [K]; positivity [sourceChannelLogConstant_pos]
-  have hrel (t) : Real.log (n t)=(q t)^(-3:ℤ)*((q t)^3*Real.log (n t)) := by
-    rw [zpow_neg,zpow_ofNat]
-    field_simp [(hq t).1.ne']
-  have H := source_uniform_exp_absorption B 1 ζ K 1 hB hζ hK (by norm_num) hn hH
-    (fun t=>zpow_pos (hq t).1 _) hrel
-  filter_upwards [H] with t ht i hi b hb
-  have hh := source_count_factor_exp_bound (hq t) B (sourceDepth B ζ ((q t)^3*Real.log (n t))) b i hb hi
-  exact hh.trans (by simpa only [pow_one,one_mul] using ht (i+1))
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- Tiny epsilon estimates are uniform in the varying block depth. -/
-theorem sourceEpsilon_power_small (B D m : ℕ) (hm : m ≤ 4*B+100) :
-    (2:ℝ)^m*sourceEpsilon B D ≤ 1 := by
-  have hp : (2:ℝ)^m ≤ 2^(4*B+100) := pow_le_pow_right₀ (by norm_num) hm
-  have hd : (1:ℝ) ≤ (D:ℝ)+1 := by linarith only [Nat.cast_nonneg D (α:=ℝ)]
-  have hmul := mul_le_mul_of_nonneg_left hd (show 0 ≤ (2:ℝ)^(4*B+100) by positivity)
-  unfold sourceEpsilon
-  rw [mul_one_div]
-  exact (div_le_one (by positivity)).mpr (hp.trans (by simpa only [mul_one] using hmul))
-
-theorem scaleA_mono_index (b j k : ℕ) (ε : ℝ) (he : 0 ≤ ε) (hjk : j ≤ k) :
-    scaleA b j ε ≤ scaleA b k ε := by
-  unfold scaleA
-  exact sub_le_sub_right (div_le_div_of_nonneg_right
-    (mul_le_mul_of_nonneg_left (pow_le_pow_right₀ (by norm_num) hjk) (by linarith only [he]))
-      (by positivity)) _
-
-theorem scaleA_le_geometric (b j : ℕ) (ε : ℝ) (he : 0 ≤ ε) (hj : j ≤ b) :
-    scaleA b j ε ≤ (2:ℝ)^j/2^b := by
-  have hp : (2:ℝ)^j ≤ 2^b := pow_le_pow_right₀ (by norm_num) hj
-  have hh := mul_le_mul_of_nonneg_left hp he
-  unfold scaleA
-  apply (sub_le_iff_le_add).mpr
-  apply (div_le_iff₀ (by positivity)).mpr
-  simp only [add_mul]
-  rw [div_mul_cancel₀ _ (by positivity : (2:ℝ)^b ≠ 0)]
-  nlinarith only [hh]
-
-theorem sourceKappa_three_epsilon (B D b j : ℕ) (hb : b ≤ B) (_hj : 1 ≤ j) (hj' : j ≤ b) :
-    3*sourceEpsilon B D ≤ scaleKappa b j (sourceEpsilon B D) := by
-  let ε := sourceEpsilon B D
-  let X := (2:ℝ)^B
-  have he : 0 < ε := sourceEpsilon_pos _ _
-  have hx : 1 ≤ X := one_le_pow₀ (by norm_num)
-  have hsmall := sourceEpsilon_power_small B D (2*B+4) (by omega)
-  have hsmall' : 16*X^2*ε ≤ 1 := by
-    rw [show 2*B+4=B*2+4 by omega,pow_add,pow_mul] at hsmall
-    norm_num only [show (2:ℝ)^4=16 by norm_num] at hsmall
-    dsimp [X,ε]
-    nlinarith only [hsmall]
-  have hgeom := scaleA_zero_lower b B ε hb he.le
-  have hmono := scaleA_mono_index b 0 (j-1) ε he.le (Nat.zero_le _)
-  have hp : (2:ℝ)^j ≤ X := pow_le_pow_right₀ (by norm_num) (hj'.trans hb)
-  have hprod : (3*X+4)*X*ε ≤ 1 := by
-    have hxx : 0 ≤ X*(X-1) := mul_nonneg (by linarith only [hx]) (by linarith only [hx])
-    have hcmp : (3*X+4)*X ≤ 16*X^2 := by nlinarith only [hxx,sq_nonneg X]
-    exact (mul_le_mul_of_nonneg_right hcmp he.le).trans hsmall'
-  have hq : (3*X+4)*ε ≤ 1/X := (le_div_iff₀ (by linarith only [hx])).mpr (by nlinarith only [hprod])
-  have hzero : 1/X-ε ≤ scaleA b 0 ε := by
-    have hh : (1/X)*(1-(X-1)*ε)=1/X-ε+ε/X := by field_simp; ring
-    dsimp [X] at hh
-    rw [hh] at hgeom
-    exact (le_add_of_nonneg_right (div_nonneg he.le (by positivity))).trans hgeom
-  have hpk := mul_le_mul_of_nonneg_right hp he.le
-  unfold scaleKappa
-  change 3*ε ≤ scaleA b (j-1) ε-3*(2^j-1)*ε
-  nlinarith only [hzero,hmono,hq,hpk,he]
-
-theorem sourceA_nonneg (B D b j : ℕ) (hb : b ≤ B) (hb0 : 1 ≤ b) :
-    0 ≤ scaleA b j (sourceEpsilon B D) := by
-  have h := sourceKappa_three_epsilon B D b 1 hb (by omega) hb0
-  unfold scaleKappa at h
-  norm_num only [Nat.sub_self,pow_one] at h
-  have hm := scaleA_mono_index b 0 j (sourceEpsilon B D) (sourceEpsilon_pos B D).le (Nat.zero_le _)
-  linarith only [h,hm,sourceEpsilon_pos B D]
-
-/-- The interior scale mass excludes the bottom window, as required by the
- sparse role count rather than the much larger total leaf count. -/
-def sourceInteriorExponent (b : ℕ) (ε : ℝ) : ℝ :=
-  ∑j∈Finset.range (b-1),scaleKappa b (j+2) ε
-
-theorem two_power_sum_shift (m : ℕ) :
-    (∑j∈Finset.range m,(2:ℝ)^(j+1))=2^(m+1)-2 := by
-  induction m with
-  | zero => norm_num
-  | succ m ih => rw [Finset.sum_range_succ,ih,pow_succ (2:ℝ) (m+1)]; ring
-
-theorem sourceInteriorExponent_upper (b : ℕ) (ε : ℝ) (hb : 1 ≤ b) (he : 0 ≤ ε) :
-    sourceInteriorExponent b ε ≤ 1-2/(2:ℝ)^b := by
-  have hterm (j : ℕ) (hj : j∈Finset.range (b-1)) :
-      scaleKappa b (j+2) ε ≤ (2:ℝ)^(j+1)/2^b := by
-    have hj' := Finset.mem_range.mp hj
-    have ha := scaleA_le_geometric b (j+1) ε he (by omega)
-    have hp : (1:ℝ) ≤ 2^(j+2) := one_le_pow₀ (by norm_num)
-    unfold scaleKappa
-    simp only [show j+2-1=j+1 by omega]
-    linarith only [ha,mul_nonneg (sub_nonneg.mpr hp) he]
-  apply (Finset.sum_le_sum hterm).trans_eq
-  rw [←Finset.sum_div,two_power_sum_shift,Nat.sub_add_cancel hb]
-  field_simp
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem scaleA_zero_simple_lower (b : ℕ) (ε : ℝ) (he : 0 ≤ ε) :
-    1/(2:ℝ)^b-ε ≤ scaleA b 0 ε := by
-  unfold scaleA
-  norm_num only [pow_zero,mul_one]
-  exact sub_le_sub_right (div_le_div_of_nonneg_right (by linarith only [he]) (by positivity)) _
-
-/-- Quantitative sparse gap, keeping its growth in d. Both the two-dimensional
- warmup and the post-warmup dimension satisfy this common estimate. -/
-theorem source_sparse_gap (B D b : ℕ) (d : ℝ) (hb1 : 1 ≤ b) (_hb : b ≤ B)
-    (hd : 11/5 ≤ d) (hcases : b=2 ∨ (b=B ∧ (2:ℝ)^B+2 ≤ d)) :
-    (d+1)/(128*(2:ℝ)^(2*B)) ≤
-      scaleA b 0 (sourceEpsilon B D)*d-sourceInteriorExponent b (sourceEpsilon B D) := by
-  let X := (2:ℝ)^B
-  let ε := sourceEpsilon B D
-  have hx : 1 ≤ X := one_le_pow₀ (by norm_num)
-  have hx0 : 0 < X := by linarith only [hx]
-  have hd0 : 0 ≤ d := by linarith only [hd]
-  have he : 0 < ε := sourceEpsilon_pos B D
-  have hA := mul_le_mul_of_nonneg_right (scaleA_zero_simple_lower b ε he.le) hd0
-  have hI := sourceInteriorExponent_upper b ε hb1 he.le
-  have hcoeff : (d+1)/(64*X^2) ≤ (d+2)/(2:ℝ)^b-1 := by
-    rcases hcases with htwo | ⟨hmain,hdmain⟩
-    · rw [htwo]
-      have hstep : (d+1)/(64*X^2) ≤ (d+1)/64 := by
-        apply div_le_div_of_nonneg_left (by linarith only [hd]) (by norm_num)
-        nlinarith only [hx,sq_nonneg (X-1)]
-      apply hstep.trans
-      norm_num only [show (2:ℝ)^2=4 by norm_num]
-      linarith only [hd]
-    · rw [hmain]
-      change (d+1)/(64*X^2) ≤ (d+2)/X-1
-      apply (div_le_iff₀ (by positivity : (0:ℝ)<64*X^2)).mpr
-      have heq : ((d+2)/X-1)*(64*X^2)=64*X*(d+2-X) := by field_simp
-      rw [heq]
-      have hh : 0 ≤ (d-X-2)*(64*X-1) := mul_nonneg (by dsimp [X]; linarith only [hdmain])
-        (by linarith only [hx])
-      nlinarith only [hh,hx]
-  have hsmall := sourceEpsilon_power_small B D (2*B+8) (by omega)
-  have hsmall' : 256*X^2*ε ≤ 1 := by
-    rw [show 2*B+8=B*2+8 by omega,pow_add,pow_mul] at hsmall
-    norm_num only [show (2:ℝ)^8=256 by norm_num] at hsmall
-    dsimp [X,ε]
-    nlinarith only [hsmall]
-  have herr : ε*d ≤ (d+1)/(256*X^2) := by
-    apply (le_div_iff₀ (by positivity : (0:ℝ)<256*X^2)).mpr
-    have hh := mul_le_mul_of_nonneg_right hsmall' hd0
-    nlinarith only [hh]
-  have hpow : (2:ℝ)^(2*B)=X^2 := by dsimp [X]; rw [Nat.mul_comm,pow_mul]
-  rw [hpow]
-  change (d+1)/(128*X^2) ≤ scaleA b 0 ε*d-sourceInteriorExponent b ε
-  have hp : 0 < (2:ℝ)^b := by positivity
-  have hc : (1/(2:ℝ)^b-ε)*d-(1-2/(2:ℝ)^b)=((d+2)/(2:ℝ)^b-1)-ε*d := by ring
-  have hbase := sub_le_sub hA hI
-  rw [hc] at hbase
-  have hident : (d+1)/(64*X^2)-(d+1)/(256*X^2)=3*(d+1)/(256*X^2) := by ring
-  have hlo := (sub_le_sub hcoeff herr).trans hbase
-  rw [hident] at hlo
-  have hcompare : (d+1)/(128*X^2) ≤ 3*(d+1)/(256*X^2) := by
-    have hh : (d+1)/128 ≤ 3*(d+1)/256 := by linarith only [hd]
-    simpa only [div_div] using div_le_div_of_nonneg_right hh (sq_nonneg X)
-  exact hcompare.trans hlo
-
-theorem totalScoreExponent_step (B i : ℕ) :
-    totalScoreExponent B (i+1)=totalScoreExponent B i+blockGain (sourceDimension B i) := by
-  unfold totalScoreExponent sourceDimension
-  rw [Finset.sum_range_succ]
-  ring
-
-theorem totalScoreExponent_after_warmup (B i : ℕ) (hB : 2 ≤ B) (hi : warmupBlocks B ≤ i) :
-    (2:ℝ)^B+11/5 ≤ totalScoreExponent B i := by
-  rw [totalScoreExponent_exact B i hi]
-  have hm : (0:ℝ) ≤ ((i:ℝ)-warmupBlocks B)*blockGain B :=
-    mul_nonneg (sub_nonneg.mpr (by exact_mod_cast hi)) (blockGain_pos B hB).le
-  unfold warmupBlocks at *
-  push_cast at *
-  linarith only [hm]
-
-theorem source_level_sparse_gap (B D i : ℕ) (hB : 2 ≤ B) :
-    (totalScoreExponent B i+1)/(128*(2:ℝ)^(2*B)) ≤
-      scaleA (sourceDimension B i) 0 (sourceEpsilon B D)*totalScoreExponent B i-
-        sourceInteriorExponent (sourceDimension B i) (sourceEpsilon B D) := by
-  have hb : 2 ≤ sourceDimension B i ∧ sourceDimension B i ≤ B := by
-    unfold sourceDimension
-    split_ifs <;> omega
-  apply source_sparse_gap B D _ _ (by omega) hb.2 (totalScoreExponent_bounds B i hB).1
-  unfold sourceDimension
-  split_ifs with hi
-  · exact Or.inl rfl
-  · exact Or.inr ⟨rfl,by linarith only [totalScoreExponent_after_warmup B i hB (by omega)]⟩
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- Levels outside the finite source block are irrelevant. Their harmless
- positive extension permits the global-index API of the finite tree lemmas. -/
-def sourceBlockWindows (N : ℝ) (b : ℕ) (ε : ℝ) (j : ℕ) : ℕ :=
-  if j<b then sourceWindowSize N b ε j else 24
-
-structure SourceArithmeticBounds (C N F ε : ℝ) (s b ell k : ℕ) : Prop where
-  lengths : ∀j ≤ b,N^(scaleA b j ε)/2 ≤ (sourceArithmeticLength C N b ε ell k j:ℝ) ∧
-    (sourceArithmeticLength C N b ε ell k j:ℝ) ≤ N^(scaleA b j ε)
-  counts : ∀j ≤ b,sourceArithmeticLength C N b ε ell k j ≤ sourceArithmeticCount C N b ε ell k j ∧
-    (sourceArithmeticCount C N b ε ell k j:ℝ) ≤ C^(2*j)*F*sourceArithmeticLength C N b ε ell k j
-  copies : ∀j<b,N^(scaleA b j ε)*N^ε/2 ≤ (sourceArithmeticP C N b ε ell k j:ℝ) ∧
-    (sourceArithmeticP C N b ε ell k j:ℝ) ≤ 2*N^(scaleA b j ε)*N^ε
-  trials : ∀j<b,N^ε/(600*(C^(2*b)*F)) ≤ (sourceArithmeticTrials C N b ε ell k j:ℝ) ∧
-    (sourceArithmeticTrials C N b ε ell k j:ℝ) ≤ 2*N^ε
-  windows : ∀j<b,24 ≤ sourceBlockWindows N b ε j ∧ 24*2^s ≤ sourceBlockWindows N b ε j ∧
-    2*sourceBlockWindows N b ε j ≤ sourceArithmeticP C N b ε ell k j ∧
-    4*(sourceArithmeticTrials C N b ε ell k j+1) ≤ sourceBlockWindows N b ε j
-
-/-- Exact layout inequalities for the literal recursive integer construction.
- The hypotheses are just two scalar subpower margins and the exact scale
- identities, not any probabilistic producer or new overlap premise. -/
-theorem source_arithmetic_geometry {C N F ε : ℝ} {s b ell k : ℕ}
-    (hC : 2 ≤ C) (hN : 1 ≤ N) (hF : 1 ≤ F) (he : 0 ≤ ε)
-    (ha : ∀j ≤ b,0 ≤ scaleA b j ε) (hκ : ∀j<b,3*ε ≤ scaleKappa b (j+1) ε)
-    (hsmall : 1200*C^(2*b)*F ≤ N^ε) (hmotif : 48*((2^s:ℕ):ℝ) ≤ N^ε)
-    (hell : N^(scaleA b 0 ε)/2 ≤ (ell:ℝ)) (hell' : (ell:ℝ) ≤ N^(scaleA b 0 ε))
-    (hk : ell ≤ k) (hk' : (k:ℝ) ≤ F*ell) :
-    SourceArithmeticBounds C N F ε s b ell k := by
-  let G := C^(2*b)*F
-  let u := N^ε
-  have hC1 : 1 ≤ C := by linarith only [hC]
-  have hN0 : 0 < N := by linarith only [hN]
-  have hG : 1 ≤ G := one_le_mul_of_one_le_of_one_le (one_le_pow₀ hC1) hF
-  have hG0 : 0 < G := by linarith only [hG]
-  have hu : 1200*G ≤ u := by simpa only [G,u,mul_assoc] using hsmall
-  have hu8 : 8 ≤ u := by linarith only [hu,hG]
-  have H := source_arithmetic_lengths hC hN hF he ha
-    (show 8*C^(2*b)*F ≤ N^ε by
-      have hh : 8*G ≤ u := by linarith only [hu,hG]
-      simpa only [G,u,mul_assoc] using hh) hell hell' hk hk'
-  have hcopies (j : ℕ) (hj : j<b) :
-      N^(scaleA b j ε)*u/2 ≤ (sourceArithmeticP C N b ε ell k j:ℝ) ∧
-      (sourceArithmeticP C N b ε ell k j:ℝ) ≤ 2*N^(scaleA b j ε)*u := by
-    have hlen := H j (by omega)
-    have hnext := H (j+1) (by omega)
-    rw [source_scale_target_step b j hN0,sourceArithmeticLength_step] at hnext
-    exact source_copies_bounds (Real.rpow_pos_of_pos hN0 _) hlen.1 hlen.2.1 hnext.1 hnext.2.1
-  have htrials (j : ℕ) (hj : j<b) :
-      u/(600*G) ≤ (sourceArithmeticTrials C N b ε ell k j:ℝ) ∧
-      (sourceArithmeticTrials C N b ε ell k j:ℝ) ≤ 2*u := by
-    have hlen := H j (by omega)
-    have hcp := hcopies j hj
-    have hX : 1 ≤ N^(scaleA b j ε) := Real.one_le_rpow hN (ha j (by omega))
-    have hcount : (sourceArithmeticCount C N b ε ell k j:ℝ) ≤
-        G*sourceArithmeticLength C N b ε ell k j := by
-      apply hlen.2.2.2.trans
-      apply mul_le_mul_of_nonneg_right _ (by positivity)
-      exact mul_le_mul_of_nonneg_right (pow_le_pow_right₀ hC1 (by omega)) (by linarith only [hF])
-    by_cases hjroot : j+1=b
-    · have hp : sourceArithmeticP C N b ε ell k j=
-          sourceArithmeticTrials C N b ε ell k j*(2*sourceArithmeticCount C N b ε ell k j+1)+
-            sourceWindowSize N b ε j := by
-        simp only [sourceArithmeticP,sourceArithmeticTrials,sourceArithmeticCount,sourceArithmeticStages,ite_eq_left hjroot]
-      exact source_root_trials_scale hX hG hu hlen.1 hlen.2.1 hlen.2.2.1 hcount hcp.1 hcp.2
-        (sourceWindowSize_le_previous hN he) hp
-    · rw [sourceArithmetic_inner_trials C N b ε ell k j hjroot]
-      exact source_inner_trials_scale hX hG hu hlen.1 hlen.2.1 hlen.2.2.1 hcount hcp.1 hcp.2
-  refine ⟨fun j hj => ⟨(H j hj).1,(H j hj).2.1⟩,fun j hj => (H j hj).2.2,hcopies,htrials,?_⟩
-  intro j hj
-  rw [sourceBlockWindows,ite_eq_left hj]
-  have hp := hcopies j hj
-  have ht := htrials j hj
-  have hu3 : u^3 ≤ N^(scaleKappa b (j+1) ε) := by
-    dsimp [u]
-    rw [←Real.rpow_natCast,←Real.rpow_mul hN0.le]
-    norm_num only [Nat.cast_ofNat]
-    apply Real.rpow_le_rpow_of_exponent_le hN
-    linarith only [hκ j hj]
-  have hu32 : 64*u ≤ u^3 := by
-    have hu_sq : 64 ≤ u^2 := by nlinarith only [hu8,sq_nonneg (u-8)]
-    nlinarith only [mul_le_mul_of_nonneg_right hu_sq (by linarith only [hu8] : 0 ≤ u)]
-  have hw := sourceWindowSize_bounds (show 8 ≤ N^(scaleKappa b (j+1) ε) by linarith only [hu32,hu8,hu3])
-  have hwbig : u^3/2 ≤ (sourceWindowSize N b ε j:ℝ) := le_trans (by linarith only [hu3] : u^3/2 ≤ N^(scaleKappa b (j+1) ε)/2) hw.1
-  have hwp := sourceWindowSize_le_previous (b:=b) (j:=j) hN he
-  have hX : 0 ≤ N^(scaleA b j ε) := (Real.rpow_pos_of_pos hN0 _).le
-  have hambient : (2:ℝ)*(sourceWindowSize N b ε j:ℝ) ≤ sourceArithmeticP C N b ε ell k j := by
-    have hh := mul_le_mul_of_nonneg_right hu8 hX
-    nlinarith only [hp.1,hwp,hh]
-  have h24 : (24:ℝ) ≤ sourceWindowSize N b ε j := by linarith only [hwbig,hu32,hu8]
-  have hmot : (24:ℝ)*((2^s:ℕ):ℝ) ≤ sourceWindowSize N b ε j := by
-    have hu' : 48*((2^s:ℕ):ℝ) ≤ u := hmotif
-    linarith only [hwbig,hu32,hu']
-  have hroot : (4:ℝ)*((sourceArithmeticTrials C N b ε ell k j:ℝ)+1) ≤ sourceWindowSize N b ε j := by
-    linarith only [hwbig,hu32,hu8,ht.2]
-  exact ⟨by exact_mod_cast h24,by exact_mod_cast hmot,by exact_mod_cast hambient,by exact_mod_cast hroot⟩
-
-theorem SourceArithmeticBounds.layout {C N F ε : ℝ} {s b ell k : ℕ}
-    (H : SourceArithmeticBounds C N F ε s b ell k) :
-    SourceLayoutMargins s (sourceArithmeticP C N b ε ell k) (sourceBlockWindows N b ε)
-      (sourceArithmeticTrials C N b ε ell k) (sourceArithmeticCount C N b ε ell k) (b-1) := by
-  constructor
-  · intro j hj
-    have hh := (H.windows j (by omega)).1
-    omega
-  · intro j hj
-    have hh := (H.windows j (by omega)).1
-    omega
-  · intro j hj
-    rw [sourceBlockWindows,ite_eq_left (by omega)]
-    exact sourceWindowSize_even N b ε j
-  · intro j hj
-    exact (H.windows j (by omega)).2.2.1
-  · intro j hj
-    have hh := (H.windows j (by omega)).2.1
-    omega
-  · intro j hj
-    rw [sourceArithmetic_inner_trials C N b ε ell k j (by omega)]
-    exact sourceLocalTrials_even _ _
-  · intro j hj
-    rw [sourceArithmetic_inner_trials C N b ε ell k j (by omega)]
-    exact (sourceLocalTrials_reserves _ _ _ (H.windows j (by omega)).2.2.1).1
-  · intro j hj
-    rw [sourceArithmetic_inner_trials C N b ε ell k j (by omega)]
-    exact (sourceLocalTrials_reserves _ _ _ (H.windows j (by omega)).2.2.1).2
-
-theorem SourceArithmeticBounds.window_positive {C N F ε : ℝ} {s b ell k : ℕ}
-    (H : SourceArithmeticBounds C N F ε s b ell k) (j : ℕ) :
-    0 < sourceBlockWindows N b ε j := by
-  by_cases hj : j<b
-  · have hh := (H.windows j hj).1
-    omega
-  · simp only [sourceBlockWindows,ite_eq_right hj]
-    norm_num
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- Exact exponent of the full binary forced skeleton, in the same bottom-up
- order as `binaryTreeBudget`. -/
-def binaryTreeExponent (e : ℕ → ℝ) : ℕ → ℝ
-  | 0 => 0
-  | b+1 => e b+2*binaryTreeExponent e b
-
-theorem binaryTreeExponent_sum (e : ℕ → ℝ) (b : ℕ) :
-    binaryTreeExponent e b=∑j∈Finset.range b,((2:ℝ)^b/2^(j+1))*e j := by
-  induction b with
-  | zero => simp [binaryTreeExponent]
-  | succ b ih =>
-    rw [binaryTreeExponent,ih,Finset.sum_range_succ,Finset.mul_sum]
-    have he : (∑j∈Finset.range b,((2:ℝ)^(b+1)/2^(j+1))*e j)=
-        ∑j∈Finset.range b,2*((2^b/2^(j+1))*e j) := by
-      apply Finset.sum_congr rfl
-      intro j _
-      rw [pow_succ]
-      ring
-    rw [he]
-    simp only [div_self (by positivity : (2:ℝ)^(b+1)≠0),one_mul]
-    ring
-
-theorem binaryTreeBudget_mono {C D : ℕ → ℝ} (b : ℕ)
-    (hC : ∀j < b,0 ≤ C j) (hCD : ∀j < b,C j ≤ D j) :
-    binaryTreeBudget C b ≤ binaryTreeBudget D b := by
-  induction b with
-  | zero => rfl
-  | succ b ih =>
-    have hc : 0 ≤ binaryTreeBudget C b := by
-      cases b with
-      | zero => norm_num [binaryTreeBudget]
-      | succ j => exact mul_nonneg (hC j (by omega)) (sq_nonneg _)
-    have hd := ih (fun j hj=>hC j (by omega)) (fun j hj=>hCD j (by omega))
-    rw [binaryTreeBudget,binaryTreeBudget]
-    exact mul_le_mul (hCD b (by omega)) (pow_le_pow_left₀ hc hd 2)
-      (sq_nonneg _) ((hC b (by omega)).trans (hCD b (by omega)))
-
-theorem binaryTreeBudget_scaled (A N : ℝ) (hN : 0 < N) (e : ℕ → ℝ) (b : ℕ) :
-    binaryTreeBudget (fun j => A*N^(-e j)) b=
-      binaryTreeBudget (fun _=>A) b*N^(-binaryTreeExponent e b) := by
-  induction b with
-  | zero => simp [binaryTreeBudget,binaryTreeExponent]
-  | succ b ih =>
-    rw [binaryTreeBudget,ih,binaryTreeBudget,binaryTreeExponent,mul_pow]
-    have hr : N^(-e b)*(N^(-binaryTreeExponent e b))^2=
-        N^(-(e b+2*binaryTreeExponent e b)) := by
-      rw [←Real.rpow_natCast,←Real.rpow_mul hN.le,←Real.rpow_add hN]
-      congr 1
-      norm_num only [Nat.cast_ofNat]
-      ring
-    calc
-      _ = (A*(binaryTreeBudget (fun _=>A) b)^2)*
-        (N^(-e b)*(N^(-binaryTreeExponent e b))^2) := by ring
-      _ = _ := by rw [hr]
-
-theorem binaryTreeBudget_const (A : ℝ) (b : ℕ) :
-    binaryTreeBudget (fun _=>A) b=A^(2^b-1) := by
-  induction b with
-  | zero => simp [binaryTreeBudget]
-  | succ b ih =>
-    rw [binaryTreeBudget,ih,←pow_mul,←pow_succ']
-    congr 1
-    have hp : 1 ≤ 2^b := Nat.one_le_pow b 2 (by omega)
-    rw [pow_succ]
-    omega
-
-theorem source_interior_product (T : ℕ → ℕ) (b : ℕ) :
-    (∏j : Fin (b+1),if j.val=0 then 1 else T j.val)=∏j∈Finset.range b,T (j+1) := by
-  rw [Fin.prod_univ_succ]
-  simp only [Fin.val_zero,ite_true,Fin.val_succ,Nat.add_eq_zero_iff,one_ne_zero,and_false,ite_false,one_mul]
-  exact Fin.prod_univ_eq_prod_range (fun j => T (j+1)) b
-
-/-- Pull all extra selected leaves into the sparse-degree small factor.
- Only the mandatory 2^(b+1) bottom leaves pay the old score exponent. -/
-theorem sourceForcedBudget_degree_factor (T n : ℕ → ℕ) (s b r : ℕ) (ω w : ℝ)
-    (hr : 2^(b+1) ≤ r) :
-    w^(2*r)*sourceForcedBudget T n s b r ω=
-      (OccupiedParityTree.shapeRoleConstant r (b+1):ℝ)*
-      (w^2*((2^s:ℕ):ℝ)*(sourceSparseCount T s b:ℝ))^(r-2^(b+1))*
-      (w^2)^(2^(b+1))*
-      ((2*((2^s:ℕ):ℝ)^3*hiddenBinomialCap (n b) ω)*
-        binaryTreeBudget (fun j => 4*((2^s:ℕ):ℝ)^6*
-          hiddenBinomialCap (sourceScheduledTrials T n (j+1)) ω*hiddenBinomialCap (n j) ω) b) := by
-  unfold sourceForcedBudget interiorLabelBudget sourceSparseCount
-  rw [source_interior_product]
-  push_cast
-  have hwr : w^(2*r)=(w^2)^(r-2^(b+1))*(w^2)^(2^(b+1)) := by
-    rw [←pow_add,Nat.sub_add_cancel hr,pow_mul]
-  rw [hwr]
-  simp only [mul_pow,pow_mul]
-  ring
-
-/-- The remaining extra-degree factor costs no scale power. -/
-theorem sourceForcedBudget_degree_le (T n : ℕ → ℕ) (s b r : ℕ) (ω w : ℝ)
-    (hr : 2^(b+1) ≤ r)
-    (hsmall : w^2*((2^s:ℕ):ℝ)*(sourceSparseCount T s b:ℝ) ≤ 1) :
-    w^(2*r)*sourceForcedBudget T n s b r ω ≤
-      (OccupiedParityTree.shapeRoleConstant r (b+1):ℝ)*(w^2)^(2^(b+1))*
-      ((2*((2^s:ℕ):ℝ)^3*hiddenBinomialCap (n b) ω)*
-        binaryTreeBudget (fun j => 4*((2^s:ℕ):ℝ)^6*
-          hiddenBinomialCap (sourceScheduledTrials T n (j+1)) ω*hiddenBinomialCap (n j) ω) b) := by
-  rw [sourceForcedBudget_degree_factor T n s b r ω w hr]
-  have hfac : (w^2*((2^s:ℕ):ℝ)*(sourceSparseCount T s b:ℝ))^(r-2^(b+1)) ≤ 1 :=
-    pow_le_one₀ (by positivity) hsmall
-  have htree := binaryTreeBudget_nonneg
-    (fun j => 4*((2^s:ℕ):ℝ)^6*hiddenBinomialCap (sourceScheduledTrials T n (j+1)) ω*hiddenBinomialCap (n j) ω)
-    (fun j => mul_nonneg (mul_nonneg (by positivity) (hiddenBinomialCap_nonneg _ _))
-      (hiddenBinomialCap_nonneg _ _)) b
-  have hroot := hiddenBinomialCap_nonneg (n b) ω
-  have hmul := mul_le_mul_of_nonneg_left hfac
-    (show 0 ≤ (OccupiedParityTree.shapeRoleConstant r (b+1):ℝ) by positivity)
-  simp only [mul_one] at hmul
-  exact mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hmul (by positivity)) (by positivity)
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem hiddenBinomialCap_rpow {N e F K ω : ℝ} {m : ℕ}
-    (hN : 0 < N) (hF : 0 < F) (hK : 0 ≤ K)
-    (hm : N^e/F ≤ (m:ℝ)) (hcoef : 4*F ≤ K^2*ω^2) :
-    hiddenBinomialCap m ω ≤ K*N^(-e/2) := by
-  have hpow : (N^(-e/2))^2=N^(-e) := by
-    rw [←Real.rpow_natCast,←Real.rpow_mul hN.le]
-    norm_num only [Nat.cast_ofNat]
-    congr 1
-    ring
-  have hprod : N^(-e)*N^e=1 := by rw [←Real.rpow_add hN,neg_add_cancel,Real.rpow_zero]
-  unfold hiddenBinomialCap
-  apply (Real.sqrt_le_left (mul_nonneg hK (Real.rpow_nonneg hN.le _))).mpr
-  rw [mul_pow,hpow]
-  apply (div_le_iff₀ (by positivity : (0:ℝ)<(m:ℝ)*ω^2+2)).mpr
-  have hh : 4 ≤ K^2*ω^2/F := (le_div_iff₀ hF).mpr hcoef
-  calc
-    4 ≤ K^2*ω^2/F := hh
-    _ = K^2*N^(-e)*(N^e/F*ω^2) := by
-      calc
-        _ = (K^2*ω^2/F)*(N^(-e)*N^e) := by rw [hprod,mul_one]
-        _ = _ := by ring
-    _ ≤ K^2*N^(-e)*((m:ℝ)*ω^2+2) := by
-      apply mul_le_mul_of_nonneg_left _ (by positivity)
-      have hh' := mul_le_mul_of_nonneg_right hm (sq_nonneg ω)
-      linarith only [hh']
-
-/-- The exact exponent in the skeleton budget equals the manuscript's
- forcedDecay, not an arbitrary favorable surrogate. -/
-theorem forcedDecay_binary (b : ℕ) (ε d : ℝ) (hb : 1 ≤ b) :
-    forcedDecay b ε d=ε/2+
-      binaryTreeExponent (fun j=>scaleKappa b (j+2) ε/2+ε) (b-1)+
-      (2:ℝ)^b*scaleA b 0 ε*d := by
-  rw [forcedDecay,binaryTreeExponent_sum]
-  congr 2
-  apply Finset.sum_congr rfl
-  intro j _
-  have he : (2:ℝ)^b/2^(j+2)=2^(b-1)/2^(j+1) := by
-    rw [show b=(b-1)+1 by omega,pow_succ]
-    simp only [Nat.add_sub_cancel]
-    rw [show j+2=(j+1)+1 by omega,pow_succ]
-    field_simp
-  rw [he]
-
-def sourceSkeletonPrefactor (s b : ℕ) (K : ℝ) : ℝ :=
-  (2*((2^s:ℕ):ℝ)^3*K)*binaryTreeBudget (fun _=>4*((2^s:ℕ):ℝ)^6*K^2) b
-
-theorem sourceSkeletonPrefactor_nonneg (s b : ℕ) {K : ℝ} (hK : 0 ≤ K) :
-    0 ≤ sourceSkeletonPrefactor s b K := by
-  have h := binaryTreeBudget_nonneg (fun _=>4*((2^s:ℕ):ℝ)^6*K^2) (fun _=>by positivity) b
-  unfold sourceSkeletonPrefactor
-  positivity
-
-theorem source_forced_core_bound (T n : ℕ → ℕ) (s b : ℕ) (ω N ε K : ℝ)
-    (hN : 0 < N) (hK : 0 ≤ K)
-    (hn : ∀j ≤ b,hiddenBinomialCap (n j) ω ≤ K*N^(-ε/2))
-    (hg : ∀j < b,hiddenBinomialCap (sourceScheduledTrials T n (j+1)) ω ≤
-      K*N^(-(scaleKappa (b+1) (j+2) ε+ε)/2)) :
-    (2*((2^s:ℕ):ℝ)^3*hiddenBinomialCap (n b) ω)*
-      binaryTreeBudget (fun j=>4*((2^s:ℕ):ℝ)^6*
-        hiddenBinomialCap (sourceScheduledTrials T n (j+1)) ω*hiddenBinomialCap (n j) ω) b ≤
-      sourceSkeletonPrefactor s b K*
-        N^(-(ε/2+binaryTreeExponent (fun j=>scaleKappa (b+1) (j+2) ε/2+ε) b)) := by
-  let A := 4*((2^s:ℕ):ℝ)^6*K^2
-  let e := fun j=>scaleKappa (b+1) (j+2) ε/2+ε
-  have hpoints (j : ℕ) (hj : j<b) :
-      4*((2^s:ℕ):ℝ)^6*hiddenBinomialCap (sourceScheduledTrials T n (j+1)) ω*
-          hiddenBinomialCap (n j) ω ≤ A*N^(-e j) := by
-    calc
-      _ ≤ 4*((2^s:ℕ):ℝ)^6*(K*N^(-(scaleKappa (b+1) (j+2) ε+ε)/2))*(K*N^(-ε/2)) :=
-        mul_le_mul (mul_le_mul_of_nonneg_left (hg j hj) (by positivity)) (hn j (by omega))
-          (hiddenBinomialCap_nonneg _ _) (by positivity)
-      _ = _ := by
-        have hr : N^(-(scaleKappa (b+1) (j+2) ε+ε)/2)*N^(-ε/2)=N^(-e j) := by
-          rw [←Real.rpow_add hN]
-          congr 1
-          dsimp [e]
-          ring
-        dsimp [A]
-        calc
-          _ = 4*((2^s:ℕ):ℝ)^6*K^2*
-            (N^(-(scaleKappa (b+1) (j+2) ε+ε)/2)*N^(-ε/2)) := by ring
-          _ = _ := by rw [hr]
-  have htree := binaryTreeBudget_mono b
-    (fun j _=>mul_nonneg (mul_nonneg (by positivity) (hiddenBinomialCap_nonneg _ _))
-      (hiddenBinomialCap_nonneg _ _)) hpoints
-  rw [binaryTreeBudget_scaled A N hN e b] at htree
-  have ht0 := binaryTreeBudget_nonneg (fun j=>4*((2^s:ℕ):ℝ)^6*
-      hiddenBinomialCap (sourceScheduledTrials T n (j+1)) ω*hiddenBinomialCap (n j) ω)
-    (fun j=>mul_nonneg (mul_nonneg (by positivity) (hiddenBinomialCap_nonneg _ _))
-      (hiddenBinomialCap_nonneg _ _)) b
-  calc
-    _ ≤ (2*((2^s:ℕ):ℝ)^3*(K*N^(-ε/2)))*
-        (binaryTreeBudget (fun _=>A) b*N^(-binaryTreeExponent e b)) :=
-      mul_le_mul (mul_le_mul_of_nonneg_left (hn b le_rfl) (by positivity)) htree ht0 (by positivity)
-    _ = _ := by
-      have hr : N^(-ε/2)*N^(-binaryTreeExponent e b)=N^(-(ε/2+binaryTreeExponent e b)) := by
-        rw [←Real.rpow_add hN]
-        congr 1
-        ring
-      dsimp [sourceSkeletonPrefactor]
-      calc
-        _ = (2*((2^s:ℕ):ℝ)^3*K)*binaryTreeBudget (fun _=>A) b*
-          (N^(-ε/2)*N^(-binaryTreeExponent e b)) := by ring
-        _ = _ := by rw [hr]
-
-/-- The sharp forced degree has the full decay gain, after all actual
- source caps and the old-score factor have been multiplied. -/
-theorem source_forced_degree_decay (T n : ℕ → ℕ) (s b r : ℕ) (ω N ε d K W w : ℝ)
-    (hN : 0 < N) (hK : 0 ≤ K) (hW : 0 ≤ W) (hr : 2^(b+1) ≤ r)
-    (hsmall : w^2*((2^s:ℕ):ℝ)*(sourceSparseCount T s b:ℝ) ≤ 1)
-    (hw : w^2 ≤ W*N^(-scaleA (b+1) 0 ε*d))
-    (hn : ∀j ≤ b,hiddenBinomialCap (n j) ω ≤ K*N^(-ε/2))
-    (hg : ∀j < b,hiddenBinomialCap (sourceScheduledTrials T n (j+1)) ω ≤
-      K*N^(-(scaleKappa (b+1) (j+2) ε+ε)/2)) :
-    w^(2*r)*sourceForcedBudget T n s b r ω ≤
-      (OccupiedParityTree.shapeRoleConstant r (b+1):ℝ)*W^(2^(b+1))*
-        sourceSkeletonPrefactor s b K*N^(-forcedDecay (b+1) ε d) := by
-  have hstart := sourceForcedBudget_degree_le T n s b r ω w hr hsmall
-  have hcore := source_forced_core_bound T n s b ω N ε K hN hK hn hg
-  have hpow := pow_le_pow_left₀ (sq_nonneg w) hw (2^(b+1))
-  have hpref := sourceSkeletonPrefactor_nonneg s b hK
-  have hcore0 := sourceForcedBudget_nonneg T n s b r ω
-  have ht := binaryTreeBudget_nonneg (fun j=>4*((2^s:ℕ):ℝ)^6*
-      hiddenBinomialCap (sourceScheduledTrials T n (j+1)) ω*hiddenBinomialCap (n j) ω)
-    (fun j=>mul_nonneg (mul_nonneg (by positivity) (hiddenBinomialCap_nonneg _ _))
-      (hiddenBinomialCap_nonneg _ _)) b
-  have hroot := hiddenBinomialCap_nonneg (n b) ω
-  apply hstart.trans
-  apply le_trans (mul_le_mul
-    (mul_le_mul_of_nonneg_left hpow (by positivity)) hcore (by positivity) (by positivity))
-  apply le_of_eq
-  rw [mul_pow]
-  have hrp : (N^(-scaleA (b+1) 0 ε*d))^(2^(b+1))*
-      N^(-(ε/2+binaryTreeExponent (fun j=>scaleKappa (b+1) (j+2) ε/2+ε) b))=
-      N^(-forcedDecay (b+1) ε d) := by
-    rw [←Real.rpow_natCast,←Real.rpow_mul hN.le,←Real.rpow_add hN,
-      forcedDecay_binary (b+1) ε d (by omega)]
-    simp only [Nat.add_sub_cancel,Nat.cast_pow,Nat.cast_ofNat]
-    congr 1
-    ring
-  calc
-    _ = (OccupiedParityTree.shapeRoleConstant r (b+1):ℝ)*W^(2^(b+1))*sourceSkeletonPrefactor s b K*
-      ((N^(-scaleA (b+1) 0 ε*d))^(2^(b+1))*
-        N^(-(ε/2+binaryTreeExponent (fun j=>scaleKappa (b+1) (j+2) ε/2+ε) b))) := by ring
-    _ = _ := by rw [hrp]
-
-end TraceReconstruction
-
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-
-end OAI
+import OAI.Probability.TraceReconstruction.InitialRates
+
+namespace OAI
+
+/-!
+# LayoutRates
+
+Integer block schedules, rounding, length and trial margins, sparse gaps, and forced-anchor decay.
+-/
+
+open scoped ENNReal
+noncomputable section
+
+open Filter
+open scoped Topology
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- The source integer window size, with zero-based nesting index. -/
+def sourceWindowSize (N : ℝ) (b : ℕ) (ε : ℝ) (j : ℕ) : ℕ :=
+  4*⌊N^(scaleKappa b (j+1) ε)/4⌋₊
+
+/-- At a nonroot node the source trial count is an even integer. -/
+def sourceLocalTrials (P h : ℕ) : ℕ := 2*(P/(100*(2*h+1)))
+
+/-- The root schedule count: the literal floor after subtracting its window. -/
+def sourceRootTrials (N : ℝ) (S T h : ℕ) : ℕ :=
+  (⌊N/(S:ℝ)⌋₊-T)/(2*h+1)
+
+/-- Literal length and overlap-count state of one nested source block.
+ `copies` and `trials` refer to the transition into this state. -/
+structure SourceArithmeticStage where
+  length : ℕ
+  count : ℕ
+  copies : ℕ
+  trials : ℕ
+
+def sourceArithmeticStages (C N : ℝ) (b : ℕ) (ε : ℝ) (ell k : ℕ) :
+    ℕ → SourceArithmeticStage
+  | 0 => ⟨ell,k,1,0⟩
+  | j+1 =>
+    let prev := sourceArithmeticStages C N b ε ell k j
+    let T := sourceWindowSize N b ε j
+    let J := sourceRootTrials N prev.length T prev.count
+    let P := if j+1=b then J*(2*prev.count+1)+T
+      else ⌊N^(scaleA b (j+1) ε)/(prev.length:ℝ)⌋₊
+    let m := if j+1=b then J else sourceLocalTrials P prev.count
+    ⟨prev.length*P,⌈C*prev.count*P⌉₊,P,m⟩
+
+def sourceArithmeticP (C N : ℝ) (b : ℕ) (ε : ℝ) (ell k : ℕ) (j : ℕ) : ℕ :=
+  (sourceArithmeticStages C N b ε ell k (j+1)).copies
+
+def sourceArithmeticTrials (C N : ℝ) (b : ℕ) (ε : ℝ) (ell k : ℕ) (j : ℕ) : ℕ :=
+  (sourceArithmeticStages C N b ε ell k (j+1)).trials
+
+def sourceArithmeticLength (C N : ℝ) (b : ℕ) (ε : ℝ) (ell k : ℕ) (j : ℕ) : ℕ :=
+  (sourceArithmeticStages C N b ε ell k j).length
+
+def sourceArithmeticCount (C N : ℝ) (b : ℕ) (ε : ℝ) (ell k : ℕ) (j : ℕ) : ℕ :=
+  (sourceArithmeticStages C N b ε ell k j).count
+
+theorem sourceLocalTrials_even (P h : ℕ) : Even (sourceLocalTrials P h) :=
+  even_two_mul _
+
+theorem sourceLocalTrials_budget (P h : ℕ) :
+    50*sourceLocalTrials P h*(2*h+1) ≤ P := by
+  have H := Nat.mul_div_le P (100*(2*h+1))
+  unfold sourceLocalTrials
+  nlinarith only [H]
+
+theorem sourceLocalTrials_reserves (P T h : ℕ) (hT : 2*T ≤ P) :
+    2*sourceLayoutReserve (sourceLocalTrials P h) h ≤ (P-T)/2 ∧
+    2*sourceLayoutReserve (sourceLocalTrials P h) h ≤ P-T-(P-T)/2 := by
+  have H := sourceLocalTrials_budget P h
+  rw [mul_assoc] at H
+  have he : sourceLocalTrials P h/2=P/(100*(2*h+1)) := by
+    unfold sourceLocalTrials
+    omega
+  have hr : 2*sourceLayoutReserve (sourceLocalTrials P h) h=
+      sourceLocalTrials P h*(2*h+1) := by
+    unfold sourceLayoutReserve
+    rw [he]
+    unfold sourceLocalTrials
+    ring
+  rw [hr]
+  have hm : 2*(sourceLocalTrials P h*(2*h+1)) ≤ P-T := by omega
+  constructor <;> omega
+
+theorem sourceWindowSize_even (N : ℝ) (b : ℕ) (ε : ℝ) (j : ℕ) :
+    Even (sourceWindowSize N b ε j) := by
+  unfold sourceWindowSize
+  exact ⟨2*⌊N^(scaleKappa b (j+1) ε)/4⌋₊,by ring⟩
+
+theorem sourceArithmeticLength_step (C N : ℝ) (b : ℕ) (ε : ℝ) (ell k j : ℕ) :
+    sourceArithmeticLength C N b ε ell k (j+1)=
+      sourceArithmeticLength C N b ε ell k j*sourceArithmeticP C N b ε ell k j := rfl
+
+theorem sourceArithmeticCount_step (C N : ℝ) (b : ℕ) (ε : ℝ) (ell k j : ℕ) :
+    sourceArithmeticCount C N b ε ell k (j+1)=
+      ⌈C*sourceArithmeticCount C N b ε ell k j*sourceArithmeticP C N b ε ell k j⌉₊ := rfl
+
+theorem sourceArithmetic_inner_trials (C N : ℝ) (b : ℕ) (ε : ℝ) (ell k j : ℕ)
+    (hj : j+1≠b) : sourceArithmeticTrials C N b ε ell k j=
+      sourceLocalTrials (sourceArithmeticP C N b ε ell k j) (sourceArithmeticCount C N b ε ell k j) := by
+  simp only [sourceArithmeticTrials,sourceArithmeticP,sourceArithmeticCount,sourceArithmeticStages,ite_eq_right hj]
+
+theorem sourceArithmetic_root_copies (C N : ℝ) (b : ℕ) (ε : ℝ) (ell k : ℕ) (hb : 1 ≤ b) :
+    sourceArithmeticP C N b ε ell k (b-1)=
+      sourceArithmeticTrials C N b ε ell k (b-1)*(2*sourceArithmeticCount C N b ε ell k (b-1)+1)+
+        sourceWindowSize N b ε (b-1) := by
+  simp only [sourceArithmeticP,sourceArithmeticTrials,sourceArithmeticCount,sourceArithmeticStages,
+    Nat.sub_add_cancel hb,ite_true]
+
+theorem sourceArithmetic_count_lift (C N : ℝ) (b : ℕ) (ε : ℝ) (ell k j : ℕ) :
+    C*sourceArithmeticCount C N b ε ell k j*sourceArithmeticP C N b ε ell k j ≤
+      sourceArithmeticCount C N b ε ell k (j+1) := Nat.le_ceil _
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- Rounding the source windows loses at most a factor two. -/
+theorem sourceWindowSize_bounds {N ε : ℝ} {b j : ℕ}
+    (h : 8 ≤ N^(scaleKappa b (j+1) ε)) :
+    N^(scaleKappa b (j+1) ε)/2 ≤ (sourceWindowSize N b ε j:ℝ) ∧
+      (sourceWindowSize N b ε j:ℝ) ≤ N^(scaleKappa b (j+1) ε) := by
+  have hlo := Nat.lt_floor_add_one (N^(scaleKappa b (j+1) ε)/4)
+  have hhi := Nat.floor_le (show 0 ≤ N^(scaleKappa b (j+1) ε)/4 by linarith)
+  unfold sourceWindowSize
+  push_cast
+  constructor <;> linarith only [h,hlo,hhi]
+
+/-- The nonroot even trial count is between one and two times its unrounded
+ quotient once the quotient is at least two. -/
+theorem sourceLocalTrials_bounds (P h : ℕ) (hlarge : (200:ℝ)*(2*h+1) ≤ P) :
+    (P:ℝ)/(100*(2*h+1)) ≤ (sourceLocalTrials P h:ℝ) ∧
+      (sourceLocalTrials P h:ℝ) ≤ (P:ℝ)/(50*(2*h+1)) := by
+  have hd : 0 < (100:ℝ)*(2*h+1) := by positivity
+  have hx : (2:ℝ) ≤ (P:ℝ)/(100*(2*h+1)) := (le_div_iff₀ hd).mpr (by linarith only [hlarge])
+  have he : (sourceLocalTrials P h:ℝ)=2*(⌊(P:ℝ)/(100*(2*h+1))⌋₊:ℝ) := by
+    unfold sourceLocalTrials
+    rw [show (100:ℝ)*(2*h+1)=((100*(2*h+1):ℕ):ℝ) by push_cast; ring,Nat.floor_div_eq_div]
+    norm_cast
+  rw [he]
+  have hlo := Nat.lt_floor_add_one ((P:ℝ)/(100*(2*h+1)))
+  have hhi := Nat.floor_le (div_nonneg (Nat.cast_nonneg P) hd.le)
+  constructor
+  · linarith only [hx,hlo]
+  · have heq : (P:ℝ)/(50*(2*h+1))=2*((P:ℝ)/(100*(2*h+1))) := by field_simp; ring
+    rw [heq]
+    linarith only [hhi]
+
+theorem source_floor_length_bounds {X : ℝ} {S : ℕ} (hS : 0 < S) (hX : 2*(S:ℝ) ≤ X) :
+    X/2 ≤ ((S*⌊X/(S:ℝ)⌋₊:ℕ):ℝ) ∧ ((S*⌊X/(S:ℝ)⌋₊:ℕ):ℝ) ≤ X := by
+  have hs : (0:ℝ) < S := by exact_mod_cast hS
+  have hlo := Nat.lt_floor_add_one (X/(S:ℝ))
+  have hhi := Nat.floor_le (div_nonneg (show 0 ≤ X by linarith only [hX,hs]) hs.le)
+  have hh := mul_lt_mul_of_pos_left hlo hs
+  have hh' := mul_le_mul_of_nonneg_left hhi hs.le
+  rw [mul_add,mul_div_cancel₀ _ hs.ne'] at hh
+  rw [mul_div_cancel₀ _ hs.ne'] at hh'
+  push_cast
+  constructor <;> nlinarith only [hX,hh,hh']
+
+/-- The real floor and integer division versions of the root are identical
+ whenever its remaining ambient count is nonnegative. -/
+theorem sourceRootTrials_floor {N : ℝ} {S T h : ℕ}
+    (hT : T ≤ ⌊N/(S:ℝ)⌋₊) :
+    sourceRootTrials N S T h=⌊((⌊N/(S:ℝ)⌋₊:ℝ)-T)/(2*h+1)⌋₊ := by
+  unfold sourceRootTrials
+  rw [←Nat.cast_sub hT]
+  rw [show (2:ℝ)*h+1=((2*h+1:ℕ):ℝ) by push_cast; ring,Nat.floor_div_eq_div]
+
+theorem source_root_length_bounds {N : ℝ} {S T h : ℕ} (hS : 0 < S)
+    (hT : T ≤ ⌊N/(S:ℝ)⌋₊) (hN : (2*h+2:ℝ)*S ≤ N/2) :
+    N/2 ≤ (S:ℝ)*(sourceRootTrials N S T h*(2*h+1)+T) ∧
+      (S:ℝ)*(sourceRootTrials N S T h*(2*h+1)+T) ≤ N := by
+  have hs : (0:ℝ)<S := by exact_mod_cast hS
+  have hd : 0 < 2*h+1 := by omega
+  have hi := Nat.div_mul_le_self (⌊N/(S:ℝ)⌋₊-T) (2*h+1)
+  have hj := Nat.lt_mul_div_succ (⌊N/(S:ℝ)⌋₊-T) hd
+  rw [Nat.mul_comm (2*h+1)] at hj
+  have hp : sourceRootTrials N S T h*(2*h+1)+T ≤ ⌊N/(S:ℝ)⌋₊ := by
+    unfold sourceRootTrials
+    omega
+  have hp' : ⌊N/(S:ℝ)⌋₊ < (sourceRootTrials N S T h+1)*(2*h+1)+T := by
+    unfold sourceRootTrials
+    omega
+  have hN0 : 0 ≤ N := by
+    have hh : (0:ℝ) ≤ (2*h+2:ℝ)*S := by positivity
+    linarith only [hN,hh]
+  have hfloor := Nat.floor_le (div_nonneg hN0 hs.le)
+  have hfloor' := Nat.lt_floor_add_one (N/(S:ℝ))
+  have hpr : (sourceRootTrials N S T h:ℝ)*(2*h+1)+T ≤ (⌊N/(S:ℝ)⌋₊:ℝ) := by exact_mod_cast hp
+  have hpr' : (⌊N/(S:ℝ)⌋₊:ℝ) < (sourceRootTrials N S T h+1:ℝ)*(2*h+1)+T := by exact_mod_cast hp'
+  constructor
+  · have hf : N/(S:ℝ) < (sourceRootTrials N S T h:ℝ)*(2*h+1)+T+(2*h+2) := by
+      nlinarith only [hfloor',hpr']
+    have hh := mul_lt_mul_of_pos_left hf hs
+    rw [mul_div_cancel₀ _ hs.ne'] at hh
+    nlinarith only [hh,hN]
+  · have hh := mul_le_mul_of_nonneg_left (hpr.trans hfloor) hs.le
+    simpa only [mul_div_cancel₀ _ hs.ne'] using hh
+
+theorem source_ceil_count_bounds {C F : ℝ} {S h P : ℕ} (hC : 2 ≤ C) (hF : 1 ≤ F)
+    (hS : 1 ≤ S) (hP : 1 ≤ P) (hlo : S ≤ h) (hhi : (h:ℝ) ≤ F*S) :
+    S*P ≤ ⌈C*h*P⌉₊ ∧ (⌈C*h*P⌉₊:ℝ) ≤ C^2*F*(S*P) := by
+  have hs : (1:ℝ) ≤ S := by exact_mod_cast hS
+  have hp : (1:ℝ) ≤ P := by exact_mod_cast hP
+  have hh : (S:ℝ) ≤ h := by exact_mod_cast hlo
+  have hceil := Nat.ceil_lt_add_one (show 0 ≤ C*(h:ℝ)*P by positivity)
+  constructor
+  · have hbound : (S:ℝ)*P ≤ C*h*P := by
+      have hh' := mul_le_mul_of_nonneg_right hh (by positivity : (0:ℝ) ≤ P)
+      have hc := mul_le_mul_of_nonneg_right (show (1:ℝ) ≤ C by linarith) (show 0 ≤ (h:ℝ)*P by positivity)
+      nlinarith only [hh',hc]
+    exact_mod_cast hbound.trans (Nat.le_ceil (C*(h:ℝ)*P))
+  · have hmul := mul_le_mul_of_nonneg_left hhi (show 0 ≤ C*(P:ℝ) by positivity)
+    have hbase : (1:ℝ) ≤ F*(S*P) := one_le_mul_of_one_le_of_one_le hF (one_le_mul_of_one_le_of_one_le hs hp)
+    have hgain : 1 ≤ C*(C-1) := by nlinarith only [hC]
+    have hgap := mul_le_mul hgain hbase (by norm_num) (by linarith only [hgain])
+    nlinarith only [hceil,hmul,hgap]
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem source_scale_target_step {N ε : ℝ} (b j : ℕ) (hN : 0 < N) :
+    N^(scaleA b (j+1) ε)=(N^(scaleA b j ε))^2*N^ε := by
+  rw [←Real.rpow_natCast,←Real.rpow_mul hN.le,←Real.rpow_add hN]
+  congr 1
+  have hh := scaleA_step b j ε
+  norm_num only [Nat.cast_ofNat]
+  linarith only [hh]
+
+theorem sourceWindowSize_le_previous {N ε : ℝ} {b j : ℕ}
+    (hN : 1 ≤ N) (he : 0 ≤ ε) :
+    (sourceWindowSize N b ε j:ℝ) ≤ N^(scaleA b j ε) := by
+  have hf := Nat.floor_le (show 0 ≤ N^(scaleKappa b (j+1) ε)/4 by positivity)
+  have hk : scaleKappa b (j+1) ε ≤ scaleA b j ε := by
+    unfold scaleKappa
+    simp only [Nat.add_sub_cancel]
+    have hh : (1:ℝ) ≤ 2^(j+1) := one_le_pow₀ (by norm_num)
+    nlinarith only [mul_nonneg (sub_nonneg.mpr hh) he]
+  apply le_trans _ (Real.rpow_le_rpow_of_exponent_le hN hk)
+  unfold sourceWindowSize
+  push_cast
+  linarith only [hf]
+
+/-- The root's subtraction and rounding margins follow from the one
+ subpower comparison N^epsilon >= 8F. -/
+theorem source_root_margins {N X u F : ℝ} {S h T : ℕ}
+    (hS : 1 ≤ S) (hX : (S:ℝ) ≤ X) (hF : 1 ≤ F) (hu : 8*F ≤ u)
+    (hh : (h:ℝ) ≤ F*S) (hT : (T:ℝ) ≤ X) (hrel : X^2*u=N) :
+    T ≤ ⌊N/(S:ℝ)⌋₊ ∧ (2*(h:ℝ)+2)*S ≤ N/2 := by
+  have hs : (1:ℝ) ≤ S := by exact_mod_cast hS
+  have hx : 1 ≤ X := hs.trans hX
+  have hx0 : 0 ≤ X := by linarith only [hx]
+  have hs0 : (0:ℝ) < S := by linarith only [hs]
+  have hfs : 1 ≤ F*S := by nlinarith only [hF,hs]
+  have hsX : (S:ℝ)^2 ≤ X^2 := sq_le_sq₀ hs0.le hx0 |>.mpr hX
+  have h8 : 8*F*X^2 ≤ N := by nlinarith only [mul_le_mul_of_nonneg_right hu (sq_nonneg X),hrel]
+  constructor
+  · apply Nat.le_floor
+    apply (le_div_iff₀ hs0).mpr
+    have htX := mul_le_mul hT hX hs0.le hx0
+    nlinarith only [htX,h8,mul_le_mul_of_nonneg_right hF (sq_nonneg X)]
+  · have hhS := mul_le_mul_of_nonneg_right hh hs0.le
+    have hS2 := mul_le_mul_of_nonneg_right hfs hs0.le
+    have hFsq := mul_le_mul_of_nonneg_left hsX (by linarith only [hF] : 0 ≤ F)
+    nlinarith only [hhS,hS2,hFsq,h8]
+
+theorem source_arithmetic_inner_copies (C N : ℝ) (b : ℕ) (ε : ℝ) (ell k j : ℕ)
+    (hj : j+1≠b) : sourceArithmeticP C N b ε ell k j=
+      ⌊N^(scaleA b (j+1) ε)/(sourceArithmeticLength C N b ε ell k j:ℝ)⌋₊ := by
+  simp only [sourceArithmeticP,sourceArithmeticLength,sourceArithmeticStages,ite_eq_right hj]
+
+theorem source_arithmetic_root_copies (C N : ℝ) (b : ℕ) (ε : ℝ) (ell k j : ℕ)
+    (hj : j+1=b) : sourceArithmeticP C N b ε ell k j=
+      sourceRootTrials N (sourceArithmeticLength C N b ε ell k j) (sourceWindowSize N b ε j)
+        (sourceArithmeticCount C N b ε ell k j)*
+          (2*sourceArithmeticCount C N b ε ell k j+1)+sourceWindowSize N b ε j := by
+  simp only [sourceArithmeticP,sourceArithmeticLength,sourceArithmeticCount,sourceArithmeticStages,ite_eq_left hj]
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- All nested lengths and overlap counts for the literal floors/ceilings.
+ Only the scalar subpower margin is needed for every rounding step. -/
+theorem source_arithmetic_lengths {C N F ε : ℝ} {b ell k : ℕ}
+    (hC : 2 ≤ C) (hN : 1 ≤ N) (hF : 1 ≤ F) (he : 0 ≤ ε)
+    (ha : ∀j ≤ b,0 ≤ scaleA b j ε)
+    (hsmall : 8*C^(2*b)*F ≤ N^ε)
+    (hell : N^(scaleA b 0 ε)/2 ≤ (ell:ℝ)) (hell' : (ell:ℝ) ≤ N^(scaleA b 0 ε))
+    (hk : ell ≤ k) (hk' : (k:ℝ) ≤ F*ell) :
+    ∀j ≤ b,
+      N^(scaleA b j ε)/2 ≤ (sourceArithmeticLength C N b ε ell k j:ℝ) ∧
+      (sourceArithmeticLength C N b ε ell k j:ℝ) ≤ N^(scaleA b j ε) ∧
+      sourceArithmeticLength C N b ε ell k j ≤ sourceArithmeticCount C N b ε ell k j ∧
+      (sourceArithmeticCount C N b ε ell k j:ℝ) ≤
+        C^(2*j)*F*sourceArithmeticLength C N b ε ell k j := by
+  have hN0 : 0 < N := by linarith only [hN]
+  have hC1 : 1 ≤ C := by linarith only [hC]
+  intro j
+  induction j with
+  | zero =>
+    intro _
+    simpa only [sourceArithmeticLength,sourceArithmeticCount,sourceArithmeticStages,
+      Nat.mul_zero,pow_zero,one_mul] using And.intro hell (And.intro hell' (And.intro hk hk'))
+  | succ j ih =>
+    intro hj
+    obtain ⟨hlo,hhi,hcount,hcount'⟩ := ih (by omega)
+    let S := sourceArithmeticLength C N b ε ell k j
+    let H := sourceArithmeticCount C N b ε ell k j
+    let P := sourceArithmeticP C N b ε ell k j
+    let Fj := C^(2*j)*F
+    have hFj : 1 ≤ Fj := one_le_mul_of_one_le_of_one_le (one_le_pow₀ hC1) hF
+    have hS0 : (0:ℝ)<S := lt_of_lt_of_le (by positivity) hlo
+    have hS : 1 ≤ S := by exact_mod_cast (show 0 < S from Nat.cast_pos.mp hS0)
+    have hfactor : Fj ≤ C^(2*b)*F := mul_le_mul_of_nonneg_right
+      (pow_le_pow_right₀ hC1 (by omega)) (by linarith only [hF])
+    have hu : 8*Fj ≤ N^ε := (mul_le_mul_of_nonneg_left hfactor (by norm_num)).trans (by simpa only [mul_assoc] using hsmall)
+    have hx : 1 ≤ N^(scaleA b j ε) := Real.one_le_rpow hN (ha j (by omega))
+    have hlength : N^(scaleA b (j+1) ε)/2 ≤ ((S*P:ℕ):ℝ) ∧
+        ((S*P:ℕ):ℝ) ≤ N^(scaleA b (j+1) ε) := by
+      by_cases hr : j+1=b
+      · have hrel : (N^(scaleA b j ε))^2*N^ε=N := by
+          rw [←source_scale_target_step b j hN0,hr,scaleA_top,Real.rpow_one]
+        have hmargin := source_root_margins hS hhi hFj hu hcount'
+          (sourceWindowSize_le_previous (b:=b) (j:=j) hN he) hrel
+        have hlen := source_root_length_bounds (Nat.pos_of_ne_zero (by omega : S≠0)) hmargin.1 hmargin.2
+        have hp : P=sourceRootTrials N S (sourceWindowSize N b ε j) H*(2*H+1)+sourceWindowSize N b ε j :=
+          source_arithmetic_root_copies C N b ε ell k j hr
+        rw [hp,hr,scaleA_top,Real.rpow_one]
+        simpa only [Nat.cast_mul,Nat.cast_add,Nat.cast_ofNat,Nat.cast_one] using hlen
+      · have hstep := source_scale_target_step (ε:=ε) b j hN0
+        have hX : 2*(S:ℝ) ≤ N^(scaleA b (j+1) ε) := by
+          rw [hstep]
+          have hu' : 2 ≤ N^ε := by linarith only [hu,hFj]
+          have hxx : N^(scaleA b j ε) ≤ (N^(scaleA b j ε))^2 := by nlinarith only [hx]
+          nlinarith only [hhi,hxx,mul_le_mul_of_nonneg_left hu' (sq_nonneg (N^(scaleA b j ε)))]
+        have hp : P=⌊N^(scaleA b (j+1) ε)/(S:ℝ)⌋₊ :=
+          source_arithmetic_inner_copies C N b ε ell k j hr
+        rw [hp]
+        exact source_floor_length_bounds (by omega) hX
+    have hP : 1 ≤ P := by
+      have hp : (0:ℝ)<(S*P:ℕ) := lt_of_lt_of_le (by positivity) hlength.1
+      have hp' : 0<S*P := Nat.cast_pos.mp hp
+      by_contra hp0
+      have he : P=0 := by omega
+      simp only [he,Nat.mul_zero,lt_self_iff_false] at hp'
+    have hceil := source_ceil_count_bounds hC hFj hS hP hcount hcount'
+    rw [sourceArithmeticLength_step,sourceArithmeticCount_step]
+    refine ⟨hlength.1,hlength.2,hceil.1,?_⟩
+    have heq : C^2*Fj=C^(2*(j+1))*F := by
+      dsimp [Fj]
+      rw [show 2*(j+1)=2+2*j by omega,pow_add]
+      ring
+    simpa only [←Nat.cast_mul,heq] using hceil.2
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem source_copies_bounds {X u : ℝ} {S P : ℕ} (hX : 0 < X)
+    (hSlo : X/2 ≤ S) (hShi : (S:ℝ) ≤ X)
+    (hlo : X^2*u/2 ≤ ((S*P:ℕ):ℝ)) (hhi : ((S*P:ℕ):ℝ) ≤ X^2*u) :
+    X*u/2 ≤ (P:ℝ) ∧ (P:ℝ) ≤ 2*X*u := by
+  have hp := (Nat.cast_nonneg P : (0:ℝ) ≤ P)
+  have hlo' := mul_le_mul_of_nonneg_right hShi hp
+  have hhi' := mul_le_mul_of_nonneg_right hSlo hp
+  push_cast at hlo hhi
+  constructor <;> nlinarith only [hX,hlo,hhi,hlo',hhi']
+
+theorem source_inner_trials_scale {X u F : ℝ} {S P h : ℕ}
+    (hX : 1 ≤ X) (hF : 1 ≤ F) (hu : 1200*F ≤ u)
+    (hSlo : X/2 ≤ S) (hShi : (S:ℝ) ≤ X) (hhlo : S ≤ h) (hhhi : (h:ℝ) ≤ F*S)
+    (hPlo : X*u/2 ≤ (P:ℝ)) (hPhi : (P:ℝ) ≤ 2*X*u) :
+    u/(600*F) ≤ (sourceLocalTrials P h:ℝ) ∧ (sourceLocalTrials P h:ℝ) ≤ 2*u := by
+  have hx : 0<X := by linarith only [hX]
+  have hf : 0<F := by linarith only [hF]
+  have hfx : 1 ≤ F*X := one_le_mul_of_one_le_of_one_le hF hX
+  have hhu : (h:ℝ) ≤ F*X := hhhi.trans (mul_le_mul_of_nonneg_left hShi hf.le)
+  have hhl : X ≤ 2*(h:ℝ)+1 := by exact_mod_cast (show X ≤ 2*(h:ℝ)+1 by
+    have hh : (S:ℝ) ≤ h := by exact_mod_cast hhlo
+    linarith only [hh,hSlo])
+  have hhh : 2*(h:ℝ)+1 ≤ 3*F*X := by linarith only [hhu,hfx]
+  have hlarge : (200:ℝ)*(2*h+1) ≤ P := by
+    have hum := mul_le_mul_of_nonneg_left hu hx.le
+    nlinarith only [hhh,hPlo,hum]
+  obtain ⟨hlo,hhi⟩ := sourceLocalTrials_bounds P h hlarge
+  have hd : 0 < (100:ℝ)*(2*h+1) := by positivity
+  constructor
+  · apply le_trans _ hlo
+    apply (div_le_div_iff₀ (by positivity : (0:ℝ)<600*F) hd).mpr
+    have hum := mul_le_mul_of_nonneg_left hhh (by linarith only [hu,hF] : 0 ≤ u)
+    have hpm := mul_le_mul_of_nonneg_left hPlo hf.le
+    nlinarith only [hum,hpm]
+  · apply hhi.trans
+    apply (div_le_iff₀ (by positivity : (0:ℝ)<50*(2*h+1))).mpr
+    have hmm := mul_le_mul_of_nonneg_left hhl (by linarith only [hu,hF] : 0 ≤ u)
+    nlinarith only [hPhi,hmm]
+
+theorem source_root_trials_scale {X u F : ℝ} {S P h J T : ℕ}
+    (hX : 1 ≤ X) (hF : 1 ≤ F) (hu : 1200*F ≤ u)
+    (hSlo : X/2 ≤ S) (hShi : (S:ℝ) ≤ X) (hhlo : S ≤ h) (hhhi : (h:ℝ) ≤ F*S)
+    (hPlo : X*u/2 ≤ (P:ℝ)) (hPhi : (P:ℝ) ≤ 2*X*u)
+    (hT : (T:ℝ) ≤ X) (hP : P=J*(2*h+1)+T) :
+    u/(600*F) ≤ (J:ℝ) ∧ (J:ℝ) ≤ 2*u := by
+  have hx : 0<X := by linarith only [hX]
+  have hf : 0<F := by linarith only [hF]
+  have hfx : 1 ≤ F*X := one_le_mul_of_one_le_of_one_le hF hX
+  have hhu : (h:ℝ) ≤ F*X := hhhi.trans (mul_le_mul_of_nonneg_left hShi hf.le)
+  have hhl : X ≤ 2*(h:ℝ)+1 := by
+    have hh : (S:ℝ) ≤ h := by exact_mod_cast hhlo
+    linarith only [hh,hSlo]
+  have hhh : 2*(h:ℝ)+1 ≤ 3*F*X := by linarith only [hhu,hfx]
+  have hp : (P:ℝ)=(J:ℝ)*(2*h+1)+T := by exact_mod_cast hP
+  have hj : (0:ℝ) ≤ J := Nat.cast_nonneg _
+  have hupper := mul_le_mul_of_nonneg_left hhl hj
+  have hlower := mul_le_mul_of_nonneg_left hhh hj
+  constructor
+  · apply (div_le_iff₀ (by positivity : (0:ℝ)<600*F)).mpr
+    have hu' : 4 ≤ u := by linarith only [hu,hF]
+    nlinarith only [hp,hT,hPlo,hlower,mul_le_mul_of_nonneg_left hu' hx.le,hx]
+  · nlinarith only [hupper,hp,hPhi,(Nat.cast_nonneg T : (0:ℝ) ≤ T),hx]
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open Filter
+open scoped Topology Classical
+
+def sourceDegreeCutoff (B : ℕ) : ℕ := 2^(10*B+40)
+
+def sourceCancellationOrder (B D : ℕ) : ℕ :=
+  ⌈100*(totalScoreExponent B D+(sourceDegreeCutoff B:ℝ)+B+10)/sourceEpsilon B D⌉₊
+
+def sourceChannelLogConstant : ℝ := (2:ℝ)^40+1536+(32+3*primitivePeriodConstant)+1
+
+theorem sourceChannelLogConstant_pos : 0 < sourceChannelLogConstant := by
+  unfold sourceChannelLogConstant primitivePeriodConstant
+  positivity
+
+theorem deletion_inverse_cube_ge_one {q : ℝ} (hq : q∈Set.Ioc 0 1) :
+    1 ≤ q^(-3:ℤ) := by
+  rw [zpow_neg,zpow_ofNat,one_le_inv₀ (pow_pos hq.1 _)]
+  exact pow_le_one₀ hq.1.le hq.2
+
+theorem deletion_inverse_le_cube {q : ℝ} (hq : q∈Set.Ioc 0 1) :
+    q⁻¹ ≤ q^(-3:ℤ) := by
+  rw [zpow_neg,zpow_ofNat]
+  apply inv_anti₀ (pow_pos hq.1 3)
+  calc
+    q^3=q*q^2 := by ring
+    _ ≤ q*1 := mul_le_mul_of_nonneg_left (pow_le_one₀ hq.1.le hq.2) hq.1.le
+    _=q := mul_one _
+
+theorem sourceOverlapConstant_log_bound {q : ℝ} (hq : q∈Set.Ioc 0 1) :
+    Real.log (sourceOverlapConstant q) ≤ sourceChannelLogConstant*q^(-3:ℤ) := by
+  have hΛ := deletion_inverse_cube_ge_one hq
+  have hlogq : -Real.log q ≤ q^(-3:ℤ) := (neg_le_neg (Real.neg_inv_le_log hq.1.le)).trans
+    (by simpa only [neg_neg] using deletion_inverse_le_cube hq)
+  have hω : 0 < initialOverlapLower q := initialOverlapLower_pos q
+  have hlogC : Real.log (sourceOverlapConstant q)=
+      Real.log ((2:ℝ)^40)+Real.log 1536+(32+3*primitivePeriodConstant)*q^(-3:ℤ)-Real.log q := by
+    rw [sourceOverlapConstant,Real.log_div (by positivity) (mul_pos hq.1 hω).ne',
+      Real.log_mul hq.1.ne' hω.ne',initialOverlapLower,
+      Real.log_div (Real.exp_pos _).ne' (by norm_num),Real.log_exp]
+    rw [zpow_neg,zpow_ofNat]
+    ring
+  rw [hlogC]
+  have htwo := Real.log_le_self (by positivity : (0:ℝ) ≤ (2:ℝ)^40)
+  have h1536 := Real.log_le_self (by norm_num : (0:ℝ) ≤ 1536)
+  have hc := mul_le_mul_of_nonneg_left hΛ (show 0 ≤ (2:ℝ)^40+1536 by positivity)
+  unfold sourceChannelLogConstant
+  nlinarith only [hlogq,htwo,h1536,hc]
+
+theorem sourceOverlapConstant_ge_two {q : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ)) :
+    2 ≤ sourceOverlapConstant q := by
+  have hC := sourceOverlapConstant_pos hq.1
+  have hω := initialOverlapLower_lt_eighth hq.1
+  have hlarge := sourceOverlapConstant_large hq
+  nlinarith only [hC,hω,hlarge]
+
+/-- The whole block-count loss, including every nesting and every earlier
+ block, is bounded by a single exp(O_B((D+1)Lambda)). -/
+theorem source_count_factor_exp_bound {q : ℝ} (hq : q∈Set.Ioc 0 (1/2:ℝ))
+    (B D b i : ℕ) (hb : b ≤ B) (hi : i ≤ D) :
+    8*(sourceOverlapConstant q)^(2*b)*(sourceOverlapConstant q)^(2*B*i+2) ≤
+      Real.exp ((8+sourceChannelLogConstant*(2*B+2))*((D:ℝ)+1)*q^(-3:ℤ)) := by
+  let C := sourceOverlapConstant q
+  have hC : 0<C := sourceOverlapConstant_pos hq.1
+  have hC1 : 1 ≤ C := (by norm_num : (1:ℝ) ≤ 2).trans (sourceOverlapConstant_ge_two hq)
+  have hq' : q∈Set.Ioc 0 1 := ⟨hq.1,by linarith only [hq.2]⟩
+  have hΛ := deletion_inverse_cube_ge_one hq'
+  have hk := sourceChannelLogConstant_pos
+  have hlog := sourceOverlapConstant_log_bound hq'
+  have hm : 2*b+(2*B*i+2) ≤ (2*B+2)*(D+1) := by nlinarith only [hb,hi]
+  have hpow := pow_le_pow_right₀ hC1 hm
+  have hlog2 := mul_le_mul_of_nonneg_left hlog (show 0 ≤ (((2*B+2)*(D+1):ℕ):ℝ) by positivity)
+  have he : C^((2*B+2)*(D+1)) ≤
+      Real.exp (sourceChannelLogConstant*(2*B+2)*((D:ℝ)+1)*q^(-3:ℤ)) := by
+    rw [←Real.log_le_iff_le_exp (pow_pos hC _),Real.log_pow]
+    simpa only [Nat.cast_mul,Nat.cast_add,Nat.cast_ofNat,Nat.cast_one,
+      mul_assoc,mul_left_comm,mul_comm] using hlog2
+  have h8 : (8:ℝ) ≤ Real.exp (8*((D:ℝ)+1)*q^(-3:ℤ)) := by
+    apply le_trans (by norm_num : (8:ℝ) ≤ 1+8)
+    apply le_trans _ (Real.add_one_le_exp _)
+    have hd : (1:ℝ) ≤ (D:ℝ)+1 := by linarith only [Nat.cast_nonneg D (α:=ℝ)]
+    have hh := one_le_mul_of_one_le_of_one_le hd hΛ
+    linarith only [hh]
+  calc
+    8*C^(2*b)*C^(2*B*i+2)=8*C^(2*b+(2*B*i+2)) := by rw [pow_add]; ring
+    _ ≤ 8*C^((2*B+2)*(D+1)) := mul_le_mul_of_nonneg_left hpow (by norm_num)
+    _ ≤ Real.exp (8*((D:ℝ)+1)*q^(-3:ℤ))*
+        Real.exp (sourceChannelLogConstant*(2*B+2)*((D:ℝ)+1)*q^(-3:ℤ)) :=
+      mul_le_mul h8 he (pow_nonneg hC.le _) (Real.exp_pos _).le
+    _ = _ := by rw [←Real.exp_add]; congr 1; ring
+
+theorem source_uniform_count_margin (B : ℕ) (ζ : ℝ) (hB : 2 ≤ B) (hζ : ζ∈Set.Ioo 0 1)
+    {n q : ℕ→ℝ} (hn : Tendsto n atTop atTop) (hq : ∀t,q t∈Set.Ioc 0 (1/2:ℝ))
+    (hH : Tendsto (fun t=>(q t)^3*Real.log (n t)) atTop atTop) :
+    ∀ᶠt in atTop,∀i ≤ sourceDepth B ζ ((q t)^3*Real.log (n t)),∀b ≤ B,
+      8*(sourceOverlapConstant (q t))^(2*b)*(sourceOverlapConstant (q t))^(2*B*i+2) ≤
+        (sourceTargetScale B (sourceDepth B ζ ((q t)^3*Real.log (n t))) (n t) (i+1))^
+          (sourceEpsilon B (sourceDepth B ζ ((q t)^3*Real.log (n t)))) := by
+  let K := 8+sourceChannelLogConstant*(2*B+2)
+  have hK : 0 ≤ K := by dsimp [K]; positivity [sourceChannelLogConstant_pos]
+  have hrel (t) : Real.log (n t)=(q t)^(-3:ℤ)*((q t)^3*Real.log (n t)) := by
+    rw [zpow_neg,zpow_ofNat]
+    field_simp [(hq t).1.ne']
+  have H := source_uniform_exp_absorption B 1 ζ K 1 hB hζ hK (by norm_num) hn hH
+    (fun t=>zpow_pos (hq t).1 _) hrel
+  filter_upwards [H] with t ht i hi b hb
+  have hh := source_count_factor_exp_bound (hq t) B (sourceDepth B ζ ((q t)^3*Real.log (n t))) b i hb hi
+  exact hh.trans (by simpa only [pow_one,one_mul] using ht (i+1))
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- Tiny epsilon estimates are uniform in the varying block depth. -/
+theorem sourceEpsilon_power_small (B D m : ℕ) (hm : m ≤ 4*B+100) :
+    (2:ℝ)^m*sourceEpsilon B D ≤ 1 := by
+  have hp : (2:ℝ)^m ≤ 2^(4*B+100) := pow_le_pow_right₀ (by norm_num) hm
+  have hd : (1:ℝ) ≤ (D:ℝ)+1 := by linarith only [Nat.cast_nonneg D (α:=ℝ)]
+  have hmul := mul_le_mul_of_nonneg_left hd (show 0 ≤ (2:ℝ)^(4*B+100) by positivity)
+  unfold sourceEpsilon
+  rw [mul_one_div]
+  exact (div_le_one (by positivity)).mpr (hp.trans (by simpa only [mul_one] using hmul))
+
+theorem scaleA_mono_index (b j k : ℕ) (ε : ℝ) (he : 0 ≤ ε) (hjk : j ≤ k) :
+    scaleA b j ε ≤ scaleA b k ε := by
+  unfold scaleA
+  exact sub_le_sub_right (div_le_div_of_nonneg_right
+    (mul_le_mul_of_nonneg_left (pow_le_pow_right₀ (by norm_num) hjk) (by linarith only [he]))
+      (by positivity)) _
+
+theorem scaleA_le_geometric (b j : ℕ) (ε : ℝ) (he : 0 ≤ ε) (hj : j ≤ b) :
+    scaleA b j ε ≤ (2:ℝ)^j/2^b := by
+  have hp : (2:ℝ)^j ≤ 2^b := pow_le_pow_right₀ (by norm_num) hj
+  have hh := mul_le_mul_of_nonneg_left hp he
+  unfold scaleA
+  apply (sub_le_iff_le_add).mpr
+  apply (div_le_iff₀ (by positivity)).mpr
+  simp only [add_mul]
+  rw [div_mul_cancel₀ _ (by positivity : (2:ℝ)^b ≠ 0)]
+  nlinarith only [hh]
+
+theorem sourceKappa_three_epsilon (B D b j : ℕ) (hb : b ≤ B) (_hj : 1 ≤ j) (hj' : j ≤ b) :
+    3*sourceEpsilon B D ≤ scaleKappa b j (sourceEpsilon B D) := by
+  let ε := sourceEpsilon B D
+  let X := (2:ℝ)^B
+  have he : 0 < ε := sourceEpsilon_pos _ _
+  have hx : 1 ≤ X := one_le_pow₀ (by norm_num)
+  have hsmall := sourceEpsilon_power_small B D (2*B+4) (by omega)
+  have hsmall' : 16*X^2*ε ≤ 1 := by
+    rw [show 2*B+4=B*2+4 by omega,pow_add,pow_mul] at hsmall
+    norm_num only [show (2:ℝ)^4=16 by norm_num] at hsmall
+    dsimp [X,ε]
+    nlinarith only [hsmall]
+  have hgeom := scaleA_zero_lower b B ε hb he.le
+  have hmono := scaleA_mono_index b 0 (j-1) ε he.le (Nat.zero_le _)
+  have hp : (2:ℝ)^j ≤ X := pow_le_pow_right₀ (by norm_num) (hj'.trans hb)
+  have hprod : (3*X+4)*X*ε ≤ 1 := by
+    have hxx : 0 ≤ X*(X-1) := mul_nonneg (by linarith only [hx]) (by linarith only [hx])
+    have hcmp : (3*X+4)*X ≤ 16*X^2 := by nlinarith only [hxx,sq_nonneg X]
+    exact (mul_le_mul_of_nonneg_right hcmp he.le).trans hsmall'
+  have hq : (3*X+4)*ε ≤ 1/X := (le_div_iff₀ (by linarith only [hx])).mpr (by nlinarith only [hprod])
+  have hzero : 1/X-ε ≤ scaleA b 0 ε := by
+    have hh : (1/X)*(1-(X-1)*ε)=1/X-ε+ε/X := by field_simp; ring
+    dsimp [X] at hh
+    rw [hh] at hgeom
+    exact (le_add_of_nonneg_right (div_nonneg he.le (by positivity))).trans hgeom
+  have hpk := mul_le_mul_of_nonneg_right hp he.le
+  unfold scaleKappa
+  change 3*ε ≤ scaleA b (j-1) ε-3*(2^j-1)*ε
+  nlinarith only [hzero,hmono,hq,hpk,he]
+
+theorem sourceA_nonneg (B D b j : ℕ) (hb : b ≤ B) (hb0 : 1 ≤ b) :
+    0 ≤ scaleA b j (sourceEpsilon B D) := by
+  have h := sourceKappa_three_epsilon B D b 1 hb (by omega) hb0
+  unfold scaleKappa at h
+  norm_num only [Nat.sub_self,pow_one] at h
+  have hm := scaleA_mono_index b 0 j (sourceEpsilon B D) (sourceEpsilon_pos B D).le (Nat.zero_le _)
+  linarith only [h,hm,sourceEpsilon_pos B D]
+
+/-- The interior scale mass excludes the bottom window, as required by the
+ sparse role count rather than the much larger total leaf count. -/
+def sourceInteriorExponent (b : ℕ) (ε : ℝ) : ℝ :=
+  ∑j∈Finset.range (b-1),scaleKappa b (j+2) ε
+
+theorem two_power_sum_shift (m : ℕ) :
+    (∑j∈Finset.range m,(2:ℝ)^(j+1))=2^(m+1)-2 := by
+  induction m with
+  | zero => norm_num
+  | succ m ih => rw [Finset.sum_range_succ,ih,pow_succ (2:ℝ) (m+1)]; ring
+
+theorem sourceInteriorExponent_upper (b : ℕ) (ε : ℝ) (hb : 1 ≤ b) (he : 0 ≤ ε) :
+    sourceInteriorExponent b ε ≤ 1-2/(2:ℝ)^b := by
+  have hterm (j : ℕ) (hj : j∈Finset.range (b-1)) :
+      scaleKappa b (j+2) ε ≤ (2:ℝ)^(j+1)/2^b := by
+    have hj' := Finset.mem_range.mp hj
+    have ha := scaleA_le_geometric b (j+1) ε he (by omega)
+    have hp : (1:ℝ) ≤ 2^(j+2) := one_le_pow₀ (by norm_num)
+    unfold scaleKappa
+    simp only [show j+2-1=j+1 by omega]
+    linarith only [ha,mul_nonneg (sub_nonneg.mpr hp) he]
+  apply (Finset.sum_le_sum hterm).trans_eq
+  rw [←Finset.sum_div,two_power_sum_shift,Nat.sub_add_cancel hb]
+  field_simp
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem scaleA_zero_simple_lower (b : ℕ) (ε : ℝ) (he : 0 ≤ ε) :
+    1/(2:ℝ)^b-ε ≤ scaleA b 0 ε := by
+  unfold scaleA
+  norm_num only [pow_zero,mul_one]
+  exact sub_le_sub_right (div_le_div_of_nonneg_right (by linarith only [he]) (by positivity)) _
+
+/-- Quantitative sparse gap, keeping its growth in d. Both the two-dimensional
+ warmup and the post-warmup dimension satisfy this common estimate. -/
+theorem source_sparse_gap (B D b : ℕ) (d : ℝ) (hb1 : 1 ≤ b) (_hb : b ≤ B)
+    (hd : 11/5 ≤ d) (hcases : b=2 ∨ (b=B ∧ (2:ℝ)^B+2 ≤ d)) :
+    (d+1)/(128*(2:ℝ)^(2*B)) ≤
+      scaleA b 0 (sourceEpsilon B D)*d-sourceInteriorExponent b (sourceEpsilon B D) := by
+  let X := (2:ℝ)^B
+  let ε := sourceEpsilon B D
+  have hx : 1 ≤ X := one_le_pow₀ (by norm_num)
+  have hx0 : 0 < X := by linarith only [hx]
+  have hd0 : 0 ≤ d := by linarith only [hd]
+  have he : 0 < ε := sourceEpsilon_pos B D
+  have hA := mul_le_mul_of_nonneg_right (scaleA_zero_simple_lower b ε he.le) hd0
+  have hI := sourceInteriorExponent_upper b ε hb1 he.le
+  have hcoeff : (d+1)/(64*X^2) ≤ (d+2)/(2:ℝ)^b-1 := by
+    rcases hcases with htwo | ⟨hmain,hdmain⟩
+    · rw [htwo]
+      have hstep : (d+1)/(64*X^2) ≤ (d+1)/64 := by
+        apply div_le_div_of_nonneg_left (by linarith only [hd]) (by norm_num)
+        nlinarith only [hx,sq_nonneg (X-1)]
+      apply hstep.trans
+      norm_num only [show (2:ℝ)^2=4 by norm_num]
+      linarith only [hd]
+    · rw [hmain]
+      change (d+1)/(64*X^2) ≤ (d+2)/X-1
+      apply (div_le_iff₀ (by positivity : (0:ℝ)<64*X^2)).mpr
+      have heq : ((d+2)/X-1)*(64*X^2)=64*X*(d+2-X) := by field_simp
+      rw [heq]
+      have hh : 0 ≤ (d-X-2)*(64*X-1) := mul_nonneg (by dsimp [X]; linarith only [hdmain])
+        (by linarith only [hx])
+      nlinarith only [hh,hx]
+  have hsmall := sourceEpsilon_power_small B D (2*B+8) (by omega)
+  have hsmall' : 256*X^2*ε ≤ 1 := by
+    rw [show 2*B+8=B*2+8 by omega,pow_add,pow_mul] at hsmall
+    norm_num only [show (2:ℝ)^8=256 by norm_num] at hsmall
+    dsimp [X,ε]
+    nlinarith only [hsmall]
+  have herr : ε*d ≤ (d+1)/(256*X^2) := by
+    apply (le_div_iff₀ (by positivity : (0:ℝ)<256*X^2)).mpr
+    have hh := mul_le_mul_of_nonneg_right hsmall' hd0
+    nlinarith only [hh]
+  have hpow : (2:ℝ)^(2*B)=X^2 := by dsimp [X]; rw [Nat.mul_comm,pow_mul]
+  rw [hpow]
+  change (d+1)/(128*X^2) ≤ scaleA b 0 ε*d-sourceInteriorExponent b ε
+  have hp : 0 < (2:ℝ)^b := by positivity
+  have hc : (1/(2:ℝ)^b-ε)*d-(1-2/(2:ℝ)^b)=((d+2)/(2:ℝ)^b-1)-ε*d := by ring
+  have hbase := sub_le_sub hA hI
+  rw [hc] at hbase
+  have hident : (d+1)/(64*X^2)-(d+1)/(256*X^2)=3*(d+1)/(256*X^2) := by ring
+  have hlo := (sub_le_sub hcoeff herr).trans hbase
+  rw [hident] at hlo
+  have hcompare : (d+1)/(128*X^2) ≤ 3*(d+1)/(256*X^2) := by
+    have hh : (d+1)/128 ≤ 3*(d+1)/256 := by linarith only [hd]
+    simpa only [div_div] using div_le_div_of_nonneg_right hh (sq_nonneg X)
+  exact hcompare.trans hlo
+
+theorem totalScoreExponent_step (B i : ℕ) :
+    totalScoreExponent B (i+1)=totalScoreExponent B i+blockGain (sourceDimension B i) := by
+  unfold totalScoreExponent sourceDimension
+  rw [Finset.sum_range_succ]
+  ring
+
+theorem totalScoreExponent_after_warmup (B i : ℕ) (hB : 2 ≤ B) (hi : warmupBlocks B ≤ i) :
+    (2:ℝ)^B+11/5 ≤ totalScoreExponent B i := by
+  rw [totalScoreExponent_exact B i hi]
+  have hm : (0:ℝ) ≤ ((i:ℝ)-warmupBlocks B)*blockGain B :=
+    mul_nonneg (sub_nonneg.mpr (by exact_mod_cast hi)) (blockGain_pos B hB).le
+  unfold warmupBlocks at *
+  push_cast at *
+  linarith only [hm]
+
+theorem source_level_sparse_gap (B D i : ℕ) (hB : 2 ≤ B) :
+    (totalScoreExponent B i+1)/(128*(2:ℝ)^(2*B)) ≤
+      scaleA (sourceDimension B i) 0 (sourceEpsilon B D)*totalScoreExponent B i-
+        sourceInteriorExponent (sourceDimension B i) (sourceEpsilon B D) := by
+  have hb : 2 ≤ sourceDimension B i ∧ sourceDimension B i ≤ B := by
+    unfold sourceDimension
+    split_ifs <;> omega
+  apply source_sparse_gap B D _ _ (by omega) hb.2 (totalScoreExponent_bounds B i hB).1
+  unfold sourceDimension
+  split_ifs with hi
+  · exact Or.inl rfl
+  · exact Or.inr ⟨rfl,by linarith only [totalScoreExponent_after_warmup B i hB (by omega)]⟩
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- Levels outside the finite source block are irrelevant. Their harmless
+ positive extension permits the global-index API of the finite tree lemmas. -/
+def sourceBlockWindows (N : ℝ) (b : ℕ) (ε : ℝ) (j : ℕ) : ℕ :=
+  if j<b then sourceWindowSize N b ε j else 24
+
+structure SourceArithmeticBounds (C N F ε : ℝ) (s b ell k : ℕ) : Prop where
+  lengths : ∀j ≤ b,N^(scaleA b j ε)/2 ≤ (sourceArithmeticLength C N b ε ell k j:ℝ) ∧
+    (sourceArithmeticLength C N b ε ell k j:ℝ) ≤ N^(scaleA b j ε)
+  counts : ∀j ≤ b,sourceArithmeticLength C N b ε ell k j ≤ sourceArithmeticCount C N b ε ell k j ∧
+    (sourceArithmeticCount C N b ε ell k j:ℝ) ≤ C^(2*j)*F*sourceArithmeticLength C N b ε ell k j
+  copies : ∀j<b,N^(scaleA b j ε)*N^ε/2 ≤ (sourceArithmeticP C N b ε ell k j:ℝ) ∧
+    (sourceArithmeticP C N b ε ell k j:ℝ) ≤ 2*N^(scaleA b j ε)*N^ε
+  trials : ∀j<b,N^ε/(600*(C^(2*b)*F)) ≤ (sourceArithmeticTrials C N b ε ell k j:ℝ) ∧
+    (sourceArithmeticTrials C N b ε ell k j:ℝ) ≤ 2*N^ε
+  windows : ∀j<b,24 ≤ sourceBlockWindows N b ε j ∧ 24*2^s ≤ sourceBlockWindows N b ε j ∧
+    2*sourceBlockWindows N b ε j ≤ sourceArithmeticP C N b ε ell k j ∧
+    4*(sourceArithmeticTrials C N b ε ell k j+1) ≤ sourceBlockWindows N b ε j
+
+/-- Exact layout inequalities for the literal recursive integer construction.
+ The hypotheses are just two scalar subpower margins and the exact scale
+ identities, not any probabilistic producer or new overlap premise. -/
+theorem source_arithmetic_geometry {C N F ε : ℝ} {s b ell k : ℕ}
+    (hC : 2 ≤ C) (hN : 1 ≤ N) (hF : 1 ≤ F) (he : 0 ≤ ε)
+    (ha : ∀j ≤ b,0 ≤ scaleA b j ε) (hκ : ∀j<b,3*ε ≤ scaleKappa b (j+1) ε)
+    (hsmall : 1200*C^(2*b)*F ≤ N^ε) (hmotif : 48*((2^s:ℕ):ℝ) ≤ N^ε)
+    (hell : N^(scaleA b 0 ε)/2 ≤ (ell:ℝ)) (hell' : (ell:ℝ) ≤ N^(scaleA b 0 ε))
+    (hk : ell ≤ k) (hk' : (k:ℝ) ≤ F*ell) :
+    SourceArithmeticBounds C N F ε s b ell k := by
+  let G := C^(2*b)*F
+  let u := N^ε
+  have hC1 : 1 ≤ C := by linarith only [hC]
+  have hN0 : 0 < N := by linarith only [hN]
+  have hG : 1 ≤ G := one_le_mul_of_one_le_of_one_le (one_le_pow₀ hC1) hF
+  have hG0 : 0 < G := by linarith only [hG]
+  have hu : 1200*G ≤ u := by simpa only [G,u,mul_assoc] using hsmall
+  have hu8 : 8 ≤ u := by linarith only [hu,hG]
+  have H := source_arithmetic_lengths hC hN hF he ha
+    (show 8*C^(2*b)*F ≤ N^ε by
+      have hh : 8*G ≤ u := by linarith only [hu,hG]
+      simpa only [G,u,mul_assoc] using hh) hell hell' hk hk'
+  have hcopies (j : ℕ) (hj : j<b) :
+      N^(scaleA b j ε)*u/2 ≤ (sourceArithmeticP C N b ε ell k j:ℝ) ∧
+      (sourceArithmeticP C N b ε ell k j:ℝ) ≤ 2*N^(scaleA b j ε)*u := by
+    have hlen := H j (by omega)
+    have hnext := H (j+1) (by omega)
+    rw [source_scale_target_step b j hN0,sourceArithmeticLength_step] at hnext
+    exact source_copies_bounds (Real.rpow_pos_of_pos hN0 _) hlen.1 hlen.2.1 hnext.1 hnext.2.1
+  have htrials (j : ℕ) (hj : j<b) :
+      u/(600*G) ≤ (sourceArithmeticTrials C N b ε ell k j:ℝ) ∧
+      (sourceArithmeticTrials C N b ε ell k j:ℝ) ≤ 2*u := by
+    have hlen := H j (by omega)
+    have hcp := hcopies j hj
+    have hX : 1 ≤ N^(scaleA b j ε) := Real.one_le_rpow hN (ha j (by omega))
+    have hcount : (sourceArithmeticCount C N b ε ell k j:ℝ) ≤
+        G*sourceArithmeticLength C N b ε ell k j := by
+      apply hlen.2.2.2.trans
+      apply mul_le_mul_of_nonneg_right _ (by positivity)
+      exact mul_le_mul_of_nonneg_right (pow_le_pow_right₀ hC1 (by omega)) (by linarith only [hF])
+    by_cases hjroot : j+1=b
+    · have hp : sourceArithmeticP C N b ε ell k j=
+          sourceArithmeticTrials C N b ε ell k j*(2*sourceArithmeticCount C N b ε ell k j+1)+
+            sourceWindowSize N b ε j := by
+        simp only [sourceArithmeticP,sourceArithmeticTrials,sourceArithmeticCount,sourceArithmeticStages,ite_eq_left hjroot]
+      exact source_root_trials_scale hX hG hu hlen.1 hlen.2.1 hlen.2.2.1 hcount hcp.1 hcp.2
+        (sourceWindowSize_le_previous hN he) hp
+    · rw [sourceArithmetic_inner_trials C N b ε ell k j hjroot]
+      exact source_inner_trials_scale hX hG hu hlen.1 hlen.2.1 hlen.2.2.1 hcount hcp.1 hcp.2
+  refine ⟨fun j hj => ⟨(H j hj).1,(H j hj).2.1⟩,fun j hj => (H j hj).2.2,hcopies,htrials,?_⟩
+  intro j hj
+  rw [sourceBlockWindows,ite_eq_left hj]
+  have hp := hcopies j hj
+  have ht := htrials j hj
+  have hu3 : u^3 ≤ N^(scaleKappa b (j+1) ε) := by
+    dsimp [u]
+    rw [←Real.rpow_natCast,←Real.rpow_mul hN0.le]
+    norm_num only [Nat.cast_ofNat]
+    apply Real.rpow_le_rpow_of_exponent_le hN
+    linarith only [hκ j hj]
+  have hu32 : 64*u ≤ u^3 := by
+    have hu_sq : 64 ≤ u^2 := by nlinarith only [hu8,sq_nonneg (u-8)]
+    nlinarith only [mul_le_mul_of_nonneg_right hu_sq (by linarith only [hu8] : 0 ≤ u)]
+  have hw := sourceWindowSize_bounds (show 8 ≤ N^(scaleKappa b (j+1) ε) by linarith only [hu32,hu8,hu3])
+  have hwbig : u^3/2 ≤ (sourceWindowSize N b ε j:ℝ) := le_trans (by linarith only [hu3] : u^3/2 ≤ N^(scaleKappa b (j+1) ε)/2) hw.1
+  have hwp := sourceWindowSize_le_previous (b:=b) (j:=j) hN he
+  have hX : 0 ≤ N^(scaleA b j ε) := (Real.rpow_pos_of_pos hN0 _).le
+  have hambient : (2:ℝ)*(sourceWindowSize N b ε j:ℝ) ≤ sourceArithmeticP C N b ε ell k j := by
+    have hh := mul_le_mul_of_nonneg_right hu8 hX
+    nlinarith only [hp.1,hwp,hh]
+  have h24 : (24:ℝ) ≤ sourceWindowSize N b ε j := by linarith only [hwbig,hu32,hu8]
+  have hmot : (24:ℝ)*((2^s:ℕ):ℝ) ≤ sourceWindowSize N b ε j := by
+    have hu' : 48*((2^s:ℕ):ℝ) ≤ u := hmotif
+    linarith only [hwbig,hu32,hu']
+  have hroot : (4:ℝ)*((sourceArithmeticTrials C N b ε ell k j:ℝ)+1) ≤ sourceWindowSize N b ε j := by
+    linarith only [hwbig,hu32,hu8,ht.2]
+  exact ⟨by exact_mod_cast h24,by exact_mod_cast hmot,by exact_mod_cast hambient,by exact_mod_cast hroot⟩
+
+theorem SourceArithmeticBounds.layout {C N F ε : ℝ} {s b ell k : ℕ}
+    (H : SourceArithmeticBounds C N F ε s b ell k) :
+    SourceLayoutMargins s (sourceArithmeticP C N b ε ell k) (sourceBlockWindows N b ε)
+      (sourceArithmeticTrials C N b ε ell k) (sourceArithmeticCount C N b ε ell k) (b-1) := by
+  constructor
+  · intro j hj
+    have hh := (H.windows j (by omega)).1
+    omega
+  · intro j hj
+    have hh := (H.windows j (by omega)).1
+    omega
+  · intro j hj
+    rw [sourceBlockWindows,ite_eq_left (by omega)]
+    exact sourceWindowSize_even N b ε j
+  · intro j hj
+    exact (H.windows j (by omega)).2.2.1
+  · intro j hj
+    have hh := (H.windows j (by omega)).2.1
+    omega
+  · intro j hj
+    rw [sourceArithmetic_inner_trials C N b ε ell k j (by omega)]
+    exact sourceLocalTrials_even _ _
+  · intro j hj
+    rw [sourceArithmetic_inner_trials C N b ε ell k j (by omega)]
+    exact (sourceLocalTrials_reserves _ _ _ (H.windows j (by omega)).2.2.1).1
+  · intro j hj
+    rw [sourceArithmetic_inner_trials C N b ε ell k j (by omega)]
+    exact (sourceLocalTrials_reserves _ _ _ (H.windows j (by omega)).2.2.1).2
+
+theorem SourceArithmeticBounds.window_positive {C N F ε : ℝ} {s b ell k : ℕ}
+    (H : SourceArithmeticBounds C N F ε s b ell k) (j : ℕ) :
+    0 < sourceBlockWindows N b ε j := by
+  by_cases hj : j<b
+  · have hh := (H.windows j hj).1
+    omega
+  · simp only [sourceBlockWindows,ite_eq_right hj]
+    norm_num
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- Exact exponent of the full binary forced skeleton, in the same bottom-up
+ order as `binaryTreeBudget`. -/
+def binaryTreeExponent (e : ℕ → ℝ) : ℕ → ℝ
+  | 0 => 0
+  | b+1 => e b+2*binaryTreeExponent e b
+
+theorem binaryTreeExponent_sum (e : ℕ → ℝ) (b : ℕ) :
+    binaryTreeExponent e b=∑j∈Finset.range b,((2:ℝ)^b/2^(j+1))*e j := by
+  induction b with
+  | zero => simp [binaryTreeExponent]
+  | succ b ih =>
+    rw [binaryTreeExponent,ih,Finset.sum_range_succ,Finset.mul_sum]
+    have he : (∑j∈Finset.range b,((2:ℝ)^(b+1)/2^(j+1))*e j)=
+        ∑j∈Finset.range b,2*((2^b/2^(j+1))*e j) := by
+      apply Finset.sum_congr rfl
+      intro j _
+      rw [pow_succ]
+      ring
+    rw [he]
+    simp only [div_self (by positivity : (2:ℝ)^(b+1)≠0),one_mul]
+    ring
+
+theorem binaryTreeBudget_mono {C D : ℕ → ℝ} (b : ℕ)
+    (hC : ∀j < b,0 ≤ C j) (hCD : ∀j < b,C j ≤ D j) :
+    binaryTreeBudget C b ≤ binaryTreeBudget D b := by
+  induction b with
+  | zero => rfl
+  | succ b ih =>
+    have hc : 0 ≤ binaryTreeBudget C b := by
+      cases b with
+      | zero => norm_num [binaryTreeBudget]
+      | succ j => exact mul_nonneg (hC j (by omega)) (sq_nonneg _)
+    have hd := ih (fun j hj=>hC j (by omega)) (fun j hj=>hCD j (by omega))
+    rw [binaryTreeBudget,binaryTreeBudget]
+    exact mul_le_mul (hCD b (by omega)) (pow_le_pow_left₀ hc hd 2)
+      (sq_nonneg _) ((hC b (by omega)).trans (hCD b (by omega)))
+
+theorem binaryTreeBudget_scaled (A N : ℝ) (hN : 0 < N) (e : ℕ → ℝ) (b : ℕ) :
+    binaryTreeBudget (fun j => A*N^(-e j)) b=
+      binaryTreeBudget (fun _=>A) b*N^(-binaryTreeExponent e b) := by
+  induction b with
+  | zero => simp [binaryTreeBudget,binaryTreeExponent]
+  | succ b ih =>
+    rw [binaryTreeBudget,ih,binaryTreeBudget,binaryTreeExponent,mul_pow]
+    have hr : N^(-e b)*(N^(-binaryTreeExponent e b))^2=
+        N^(-(e b+2*binaryTreeExponent e b)) := by
+      rw [←Real.rpow_natCast,←Real.rpow_mul hN.le,←Real.rpow_add hN]
+      congr 1
+      norm_num only [Nat.cast_ofNat]
+      ring
+    calc
+      _ = (A*(binaryTreeBudget (fun _=>A) b)^2)*
+        (N^(-e b)*(N^(-binaryTreeExponent e b))^2) := by ring
+      _ = _ := by rw [hr]
+
+theorem binaryTreeBudget_const (A : ℝ) (b : ℕ) :
+    binaryTreeBudget (fun _=>A) b=A^(2^b-1) := by
+  induction b with
+  | zero => simp [binaryTreeBudget]
+  | succ b ih =>
+    rw [binaryTreeBudget,ih,←pow_mul,←pow_succ']
+    congr 1
+    have hp : 1 ≤ 2^b := Nat.one_le_pow b 2 (by omega)
+    rw [pow_succ]
+    omega
+
+theorem source_interior_product (T : ℕ → ℕ) (b : ℕ) :
+    (∏j : Fin (b+1),if j.val=0 then 1 else T j.val)=∏j∈Finset.range b,T (j+1) := by
+  rw [Fin.prod_univ_succ]
+  simp only [Fin.val_zero,ite_true,Fin.val_succ,Nat.add_eq_zero_iff,one_ne_zero,and_false,ite_false,one_mul]
+  exact Fin.prod_univ_eq_prod_range (fun j => T (j+1)) b
+
+/-- Pull all extra selected leaves into the sparse-degree small factor.
+ Only the mandatory 2^(b+1) bottom leaves pay the old score exponent. -/
+theorem sourceForcedBudget_degree_factor (T n : ℕ → ℕ) (s b r : ℕ) (ω w : ℝ)
+    (hr : 2^(b+1) ≤ r) :
+    w^(2*r)*sourceForcedBudget T n s b r ω=
+      (OccupiedParityTree.shapeRoleConstant r (b+1):ℝ)*
+      (w^2*((2^s:ℕ):ℝ)*(sourceSparseCount T s b:ℝ))^(r-2^(b+1))*
+      (w^2)^(2^(b+1))*
+      ((2*((2^s:ℕ):ℝ)^3*hiddenBinomialCap (n b) ω)*
+        binaryTreeBudget (fun j => 4*((2^s:ℕ):ℝ)^6*
+          hiddenBinomialCap (sourceScheduledTrials T n (j+1)) ω*hiddenBinomialCap (n j) ω) b) := by
+  unfold sourceForcedBudget interiorLabelBudget sourceSparseCount
+  rw [source_interior_product]
+  push_cast
+  have hwr : w^(2*r)=(w^2)^(r-2^(b+1))*(w^2)^(2^(b+1)) := by
+    rw [←pow_add,Nat.sub_add_cancel hr,pow_mul]
+  rw [hwr]
+  simp only [mul_pow,pow_mul]
+  ring
+
+/-- The remaining extra-degree factor costs no scale power. -/
+theorem sourceForcedBudget_degree_le (T n : ℕ → ℕ) (s b r : ℕ) (ω w : ℝ)
+    (hr : 2^(b+1) ≤ r)
+    (hsmall : w^2*((2^s:ℕ):ℝ)*(sourceSparseCount T s b:ℝ) ≤ 1) :
+    w^(2*r)*sourceForcedBudget T n s b r ω ≤
+      (OccupiedParityTree.shapeRoleConstant r (b+1):ℝ)*(w^2)^(2^(b+1))*
+      ((2*((2^s:ℕ):ℝ)^3*hiddenBinomialCap (n b) ω)*
+        binaryTreeBudget (fun j => 4*((2^s:ℕ):ℝ)^6*
+          hiddenBinomialCap (sourceScheduledTrials T n (j+1)) ω*hiddenBinomialCap (n j) ω) b) := by
+  rw [sourceForcedBudget_degree_factor T n s b r ω w hr]
+  have hfac : (w^2*((2^s:ℕ):ℝ)*(sourceSparseCount T s b:ℝ))^(r-2^(b+1)) ≤ 1 :=
+    pow_le_one₀ (by positivity) hsmall
+  have htree := binaryTreeBudget_nonneg
+    (fun j => 4*((2^s:ℕ):ℝ)^6*hiddenBinomialCap (sourceScheduledTrials T n (j+1)) ω*hiddenBinomialCap (n j) ω)
+    (fun j => mul_nonneg (mul_nonneg (by positivity) (hiddenBinomialCap_nonneg _ _))
+      (hiddenBinomialCap_nonneg _ _)) b
+  have hroot := hiddenBinomialCap_nonneg (n b) ω
+  have hmul := mul_le_mul_of_nonneg_left hfac
+    (show 0 ≤ (OccupiedParityTree.shapeRoleConstant r (b+1):ℝ) by positivity)
+  simp only [mul_one] at hmul
+  exact mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hmul (by positivity)) (by positivity)
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem hiddenBinomialCap_rpow {N e F K ω : ℝ} {m : ℕ}
+    (hN : 0 < N) (hF : 0 < F) (hK : 0 ≤ K)
+    (hm : N^e/F ≤ (m:ℝ)) (hcoef : 4*F ≤ K^2*ω^2) :
+    hiddenBinomialCap m ω ≤ K*N^(-e/2) := by
+  have hpow : (N^(-e/2))^2=N^(-e) := by
+    rw [←Real.rpow_natCast,←Real.rpow_mul hN.le]
+    norm_num only [Nat.cast_ofNat]
+    congr 1
+    ring
+  have hprod : N^(-e)*N^e=1 := by rw [←Real.rpow_add hN,neg_add_cancel,Real.rpow_zero]
+  unfold hiddenBinomialCap
+  apply (Real.sqrt_le_left (mul_nonneg hK (Real.rpow_nonneg hN.le _))).mpr
+  rw [mul_pow,hpow]
+  apply (div_le_iff₀ (by positivity : (0:ℝ)<(m:ℝ)*ω^2+2)).mpr
+  have hh : 4 ≤ K^2*ω^2/F := (le_div_iff₀ hF).mpr hcoef
+  calc
+    4 ≤ K^2*ω^2/F := hh
+    _ = K^2*N^(-e)*(N^e/F*ω^2) := by
+      calc
+        _ = (K^2*ω^2/F)*(N^(-e)*N^e) := by rw [hprod,mul_one]
+        _ = _ := by ring
+    _ ≤ K^2*N^(-e)*((m:ℝ)*ω^2+2) := by
+      apply mul_le_mul_of_nonneg_left _ (by positivity)
+      have hh' := mul_le_mul_of_nonneg_right hm (sq_nonneg ω)
+      linarith only [hh']
+
+/-- The exact exponent in the skeleton budget equals the manuscript's
+ forcedDecay, not an arbitrary favorable surrogate. -/
+theorem forcedDecay_binary (b : ℕ) (ε d : ℝ) (hb : 1 ≤ b) :
+    forcedDecay b ε d=ε/2+
+      binaryTreeExponent (fun j=>scaleKappa b (j+2) ε/2+ε) (b-1)+
+      (2:ℝ)^b*scaleA b 0 ε*d := by
+  rw [forcedDecay,binaryTreeExponent_sum]
+  congr 2
+  apply Finset.sum_congr rfl
+  intro j _
+  have he : (2:ℝ)^b/2^(j+2)=2^(b-1)/2^(j+1) := by
+    rw [show b=(b-1)+1 by omega,pow_succ]
+    simp only [Nat.add_sub_cancel]
+    rw [show j+2=(j+1)+1 by omega,pow_succ]
+    field_simp
+  rw [he]
+
+def sourceSkeletonPrefactor (s b : ℕ) (K : ℝ) : ℝ :=
+  (2*((2^s:ℕ):ℝ)^3*K)*binaryTreeBudget (fun _=>4*((2^s:ℕ):ℝ)^6*K^2) b
+
+theorem sourceSkeletonPrefactor_nonneg (s b : ℕ) {K : ℝ} (hK : 0 ≤ K) :
+    0 ≤ sourceSkeletonPrefactor s b K := by
+  have h := binaryTreeBudget_nonneg (fun _=>4*((2^s:ℕ):ℝ)^6*K^2) (fun _=>by positivity) b
+  unfold sourceSkeletonPrefactor
+  positivity
+
+theorem source_forced_core_bound (T n : ℕ → ℕ) (s b : ℕ) (ω N ε K : ℝ)
+    (hN : 0 < N) (hK : 0 ≤ K)
+    (hn : ∀j ≤ b,hiddenBinomialCap (n j) ω ≤ K*N^(-ε/2))
+    (hg : ∀j < b,hiddenBinomialCap (sourceScheduledTrials T n (j+1)) ω ≤
+      K*N^(-(scaleKappa (b+1) (j+2) ε+ε)/2)) :
+    (2*((2^s:ℕ):ℝ)^3*hiddenBinomialCap (n b) ω)*
+      binaryTreeBudget (fun j=>4*((2^s:ℕ):ℝ)^6*
+        hiddenBinomialCap (sourceScheduledTrials T n (j+1)) ω*hiddenBinomialCap (n j) ω) b ≤
+      sourceSkeletonPrefactor s b K*
+        N^(-(ε/2+binaryTreeExponent (fun j=>scaleKappa (b+1) (j+2) ε/2+ε) b)) := by
+  let A := 4*((2^s:ℕ):ℝ)^6*K^2
+  let e := fun j=>scaleKappa (b+1) (j+2) ε/2+ε
+  have hpoints (j : ℕ) (hj : j<b) :
+      4*((2^s:ℕ):ℝ)^6*hiddenBinomialCap (sourceScheduledTrials T n (j+1)) ω*
+          hiddenBinomialCap (n j) ω ≤ A*N^(-e j) := by
+    calc
+      _ ≤ 4*((2^s:ℕ):ℝ)^6*(K*N^(-(scaleKappa (b+1) (j+2) ε+ε)/2))*(K*N^(-ε/2)) :=
+        mul_le_mul (mul_le_mul_of_nonneg_left (hg j hj) (by positivity)) (hn j (by omega))
+          (hiddenBinomialCap_nonneg _ _) (by positivity)
+      _ = _ := by
+        have hr : N^(-(scaleKappa (b+1) (j+2) ε+ε)/2)*N^(-ε/2)=N^(-e j) := by
+          rw [←Real.rpow_add hN]
+          congr 1
+          dsimp [e]
+          ring
+        dsimp [A]
+        calc
+          _ = 4*((2^s:ℕ):ℝ)^6*K^2*
+            (N^(-(scaleKappa (b+1) (j+2) ε+ε)/2)*N^(-ε/2)) := by ring
+          _ = _ := by rw [hr]
+  have htree := binaryTreeBudget_mono b
+    (fun j _=>mul_nonneg (mul_nonneg (by positivity) (hiddenBinomialCap_nonneg _ _))
+      (hiddenBinomialCap_nonneg _ _)) hpoints
+  rw [binaryTreeBudget_scaled A N hN e b] at htree
+  have ht0 := binaryTreeBudget_nonneg (fun j=>4*((2^s:ℕ):ℝ)^6*
+      hiddenBinomialCap (sourceScheduledTrials T n (j+1)) ω*hiddenBinomialCap (n j) ω)
+    (fun j=>mul_nonneg (mul_nonneg (by positivity) (hiddenBinomialCap_nonneg _ _))
+      (hiddenBinomialCap_nonneg _ _)) b
+  calc
+    _ ≤ (2*((2^s:ℕ):ℝ)^3*(K*N^(-ε/2)))*
+        (binaryTreeBudget (fun _=>A) b*N^(-binaryTreeExponent e b)) :=
+      mul_le_mul (mul_le_mul_of_nonneg_left (hn b le_rfl) (by positivity)) htree ht0 (by positivity)
+    _ = _ := by
+      have hr : N^(-ε/2)*N^(-binaryTreeExponent e b)=N^(-(ε/2+binaryTreeExponent e b)) := by
+        rw [←Real.rpow_add hN]
+        congr 1
+        ring
+      dsimp [sourceSkeletonPrefactor]
+      calc
+        _ = (2*((2^s:ℕ):ℝ)^3*K)*binaryTreeBudget (fun _=>A) b*
+          (N^(-ε/2)*N^(-binaryTreeExponent e b)) := by ring
+        _ = _ := by rw [hr]
+
+/-- The sharp forced degree has the full decay gain, after all actual
+ source caps and the old-score factor have been multiplied. -/
+theorem source_forced_degree_decay (T n : ℕ → ℕ) (s b r : ℕ) (ω N ε d K W w : ℝ)
+    (hN : 0 < N) (hK : 0 ≤ K) (hW : 0 ≤ W) (hr : 2^(b+1) ≤ r)
+    (hsmall : w^2*((2^s:ℕ):ℝ)*(sourceSparseCount T s b:ℝ) ≤ 1)
+    (hw : w^2 ≤ W*N^(-scaleA (b+1) 0 ε*d))
+    (hn : ∀j ≤ b,hiddenBinomialCap (n j) ω ≤ K*N^(-ε/2))
+    (hg : ∀j < b,hiddenBinomialCap (sourceScheduledTrials T n (j+1)) ω ≤
+      K*N^(-(scaleKappa (b+1) (j+2) ε+ε)/2)) :
+    w^(2*r)*sourceForcedBudget T n s b r ω ≤
+      (OccupiedParityTree.shapeRoleConstant r (b+1):ℝ)*W^(2^(b+1))*
+        sourceSkeletonPrefactor s b K*N^(-forcedDecay (b+1) ε d) := by
+  have hstart := sourceForcedBudget_degree_le T n s b r ω w hr hsmall
+  have hcore := source_forced_core_bound T n s b ω N ε K hN hK hn hg
+  have hpow := pow_le_pow_left₀ (sq_nonneg w) hw (2^(b+1))
+  have hpref := sourceSkeletonPrefactor_nonneg s b hK
+  have hcore0 := sourceForcedBudget_nonneg T n s b r ω
+  have ht := binaryTreeBudget_nonneg (fun j=>4*((2^s:ℕ):ℝ)^6*
+      hiddenBinomialCap (sourceScheduledTrials T n (j+1)) ω*hiddenBinomialCap (n j) ω)
+    (fun j=>mul_nonneg (mul_nonneg (by positivity) (hiddenBinomialCap_nonneg _ _))
+      (hiddenBinomialCap_nonneg _ _)) b
+  have hroot := hiddenBinomialCap_nonneg (n b) ω
+  apply hstart.trans
+  apply le_trans (mul_le_mul
+    (mul_le_mul_of_nonneg_left hpow (by positivity)) hcore (by positivity) (by positivity))
+  apply le_of_eq
+  rw [mul_pow]
+  have hrp : (N^(-scaleA (b+1) 0 ε*d))^(2^(b+1))*
+      N^(-(ε/2+binaryTreeExponent (fun j=>scaleKappa (b+1) (j+2) ε/2+ε) b))=
+      N^(-forcedDecay (b+1) ε d) := by
+    rw [←Real.rpow_natCast,←Real.rpow_mul hN.le,←Real.rpow_add hN,
+      forcedDecay_binary (b+1) ε d (by omega)]
+    simp only [Nat.add_sub_cancel,Nat.cast_pow,Nat.cast_ofNat]
+    congr 1
+    ring
+  calc
+    _ = (OccupiedParityTree.shapeRoleConstant r (b+1):ℝ)*W^(2^(b+1))*sourceSkeletonPrefactor s b K*
+      ((N^(-scaleA (b+1) 0 ε*d))^(2^(b+1))*
+        N^(-(ε/2+binaryTreeExponent (fun j=>scaleKappa (b+1) (j+2) ε/2+ε) b))) := by ring
+    _ = _ := by rw [hrp]
+
+end TraceReconstruction
+
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/TraceReconstruction/LeafOracle.lean b/lean/OAI/Probability/TraceReconstruction/LeafOracle.lean
index 2b6ef00d1fcd31eefe15e360e3a2c6b84861cedd..578f82b24f9c7faa75cdc7cc869506acd24d2573 100644
--- a/lean/OAI/Probability/TraceReconstruction/LeafOracle.lean
+++ b/lean/OAI/Probability/TraceReconstruction/LeafOracle.lean
@@ -1,1320 +1,1320 @@
-import OAI.Probability.TraceReconstruction.InitialOverlap
-
-namespace OAI
-
-/-!
-# LeafOracle
-
-Recursive leaf oracles, sparse high tails, occupied shapes, posterior smoothing, source bit trees, and singleton cancellation.
--/
-
-open scoped ENNReal
-noncomputable section
-
-open Filter
-open scoped Topology
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-open FiniteLaw
-
-/-- Complete window-tree leaf indices, with counts numbered from the bottom.
-The tree is fixed before any occupied subset or orientation is selected. -/
-def WindowLeaf (T : ℕ → ℕ) : ℕ → Type
-  | 0 => Fin 1
-  | b+1 => Fin (T b) × WindowLeaf T b
-
-noncomputable instance windowLeafFintype (T : ℕ → ℕ) : (b : ℕ) → Fintype (WindowLeaf T b)
-  | 0 => inferInstanceAs (Fintype (Fin 1))
-  | b+1 => by
-    letI := windowLeafFintype T b
-    exact inferInstanceAs (Fintype (Fin (T b) × WindowLeaf T b))
-
-/-- A motif's actual outgoing type, allowing every integer displacement. -/
-def motifChildSign (s : ℕ) (X : ℤ) (h : ℕ) (τ : ℝ) : ℝ :=
-  1-motifIndicator s ((h:ℤ)-X)+(prouhetSeries s).coeff ((h:ℤ)-X)*τ
-
-theorem motifChildSign_isSign (s : ℕ) (X : ℤ) (h : ℕ) {τ : ℝ} (hτ : IsTypeSign τ) :
-    IsTypeSign (motifChildSign s X h τ) :=
-  motif_child_sign (prouhet_motif_value s ((h:ℤ)-X)) hτ
-
-theorem motifChildSign_negative_active (s : ℕ) (X : ℤ) (h : ℕ) (τ : ℝ)
-    (ht : motifChildSign s X h τ= -1) : 0 ≤ (h:ℤ)-X ∧ (h:ℤ)-X<(2^s:ℕ) := by
-  by_contra hn
-  have hz := (prouhetSeries_coeff_spec s).1 ((h:ℤ)-X) (by omega)
-  simp only [motifChildSign,motifIndicator,ite_eq_right hn,hz,zero_mul,sub_zero,add_zero] at ht
-  norm_num at ht
-
-/-- Literal top-down type recursion in the source's unconditioned surrogate.
-`X path` is the root anchor or `w_j-Y_path`. Dependence of different paths
-through the same global jitter is retained by this single full-tree map. -/
-def windowLeafSign (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ) :
-    (b : ℕ) → List ℕ → ℝ → WindowLeaf T b → ℝ
-  | 0, _, τ, _ => τ
-  | b+1, path, τ, v => windowLeafSign T s X b (path++[v.1.val])
-      (motifChildSign s (X path) v.1.val τ) v.2
-
-theorem windowLeafSign_isSign (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
-    (b : ℕ) (path : List ℕ) {τ : ℝ} (hτ : IsTypeSign τ) (v : WindowLeaf T b) :
-    IsTypeSign (windowLeafSign T s X b path τ v) := by
-  induction b generalizing path τ with
-  | zero => exact hτ
-  | succ b ih => exact ih _ (motifChildSign_isSign s _ _ hτ) v.2
-
-def windowLeafType (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
-    (b : ℕ) (path : List ℕ) (τ : ℝ) (v : WindowLeaf T b) : Bool :=
-  decide (windowLeafSign T s X b path τ v= -1)
-
-theorem windowLeafType_weight (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
-    (b : ℕ) (path : List ℕ) {τ : ℝ} (hτ : IsTypeSign τ) (v : WindowLeaf T b) :
-    (if windowLeafType T s X b path τ v then (1:ℝ) else 0)=
-      (1-windowLeafSign T s X b path τ v)/2 := by
-  rcases windowLeafSign_isSign T s X b path hτ v with hp | hm
-  · norm_num [windowLeafType,hp]
-  · norm_num [windowLeafType,hm]
-
-/-- A finite indexing of all leaves, not just the active or fitted leaves. -/
-def windowFinType (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
-    (b : ℕ) (τ : ℝ) (i : Fin (Fintype.card (WindowLeaf T b))) : Bool :=
-  windowLeafType T s X b [] τ ((Fintype.equivFin (WindowLeaf T b)).symm i)
-
-/-- The actual mixed leaf oracle at arbitrary exact metadata. The one latent
-law governs the whole map X; no independent-anchor law is substituted. -/
-def recursiveLeafLaw {Ω α : Type*} [Fintype Ω] [Fintype α]
-    (R : FiniteLaw Ω) (X : Ω → List ℕ → ℤ) (T : ℕ → ℕ) (s b : ℕ)
-    (P Q : FiniteLaw α) (τ : ℝ) :
-    FiniteLaw (Fin (Fintype.card (WindowLeaf T b)) → α) :=
-  R.mixture (fun ω => coordinateProduct (fun i => if windowFinType T s (X ω) b τ i then Q else P))
-
-theorem recursiveLeafLaw_density {Ω α : Type*} [Fintype Ω] [Fintype α]
-    (R : FiniteLaw Ω) (X : Ω → List ℕ → ℤ) (T : ℕ → ℕ) (s b : ℕ)
-    (P Q : FiniteLaw α) (Z : α → ℝ) (hZ : ∀a,Q.mass a=P.mass a*(1+Z a))
-    (τ : ℝ) (x : Fin (Fintype.card (WindowLeaf T b)) → α) :
-    (recursiveLeafLaw R X T s b P Q τ).mass x =
-      (P.product (Fintype.card (WindowLeaf T b))).mass x *
-        multilinear (averagedCoefficient R (fun ω => windowFinType T s (X ω) b τ)) Z x := by
-  rw [recursiveLeafLaw,leaf_oracle_density P Q Z hZ,averagedDensity_expansion]
-  rfl
-
-/-- Pointwise recursive coefficient before averaging over the actual bits. -/
-theorem recursive_selectionCoefficient {Ω : Type*} [Fintype Ω]
-    (X : Ω → List ℕ → ℤ) (T : ℕ → ℕ) (s b : ℕ) (ω : Ω)
-    {τ : ℝ} (hτ : IsTypeSign τ) (σ : Fin (Fintype.card (WindowLeaf T b)) → Bool) :
-    selectionCoefficient (windowFinType T s (X ω) b τ) σ =
-      ∏i, if σ i then
-        (1-windowLeafSign T s (X ω) b [] τ ((Fintype.equivFin (WindowLeaf T b)).symm i))/2 else 1 := by
-  unfold selectionCoefficient
-  apply Finset.prod_congr rfl
-  intro i _
-  cases σ i
-  · rfl
-  · exact windowLeafType_weight T s (X ω) b [] hτ _
-
-theorem motif_negative_count (s N : ℕ) (X : ℤ) (τ : ℝ) :
-    (Finset.univ.filter (fun i : Fin N => motifChildSign s X i.val τ= -1)).card ≤ 2^s := by
-  have H := Finset.card_le_card_of_injOn
-    (s:=Finset.univ.filter (fun i : Fin N => motifChildSign s X i.val τ= -1))
-    (t:=Finset.range (2^s)) (fun i : Fin N => ((i.val:ℤ)-X).toNat) ?_ ?_
-  · simpa using H
-  · intro i hi
-    have h := motifChildSign_negative_active s X i.val τ (Finset.mem_filter.mp hi).2
-    apply Finset.mem_range.mpr
-    dsimp only
-    omega
-  · intro i hi j hj he
-    have h := motifChildSign_negative_active s X i.val τ (Finset.mem_filter.mp hi).2
-    have h' := motifChildSign_negative_active s X j.val τ (Finset.mem_filter.mp hj).2
-    apply Fin.ext
-    dsimp only at he
-    omega
-
-/-- Exact number of minus leaves in a fixed realization, without a fit condition. -/
-def windowMinusCount (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
-    (b : ℕ) (path : List ℕ) (τ : ℝ) : ℕ :=
-  ∑v : WindowLeaf T b, if windowLeafSign T s X b path τ v= -1 then 1 else 0
-
-theorem windowMinusCount_succ (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
-    (b : ℕ) (path : List ℕ) (τ : ℝ) :
-    windowMinusCount T s X (b+1) path τ =
-      ∑i : Fin (T b), windowMinusCount T s X b (path++[i.val])
-        (motifChildSign s (X path) i.val τ) := by
-  unfold windowMinusCount
-  change (∑v : Fin (T b) × WindowLeaf T b, _) = _
-  rw [Fintype.sum_prod_type]
-  rfl
-
-/-- Source the block layout high-degree budget, bottom motif length times all
-upper window counts. It holds for every latent realization, typical or not. -/
-theorem windowMinusCount_bound (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
-    (b : ℕ) (path : List ℕ) (τ : ℝ) :
-    windowMinusCount T s X (b+1) path τ ≤
-      2^s * ∏j∈Finset.range b, T (j+1) := by
-  induction b generalizing path τ with
-  | zero =>
-    rw [windowMinusCount_succ]
-    change (∑i : Fin (T 0), ∑v : Fin 1,
-      if motifChildSign s (X path) i.val τ= -1 then (1:ℕ) else 0) ≤ _
-    simp only [Fin.sum_univ_one,Finset.range_zero,Finset.prod_empty,mul_one]
-    simpa only [Finset.card_filter] using motif_negative_count s (T 0) (X path) τ
-  | succ b ih =>
-    rw [windowMinusCount_succ]
-    calc
-      _ ≤ ∑i : Fin (T (b+1)), 2^s*∏j∈Finset.range b,T (j+1) :=
-        Finset.sum_le_sum (fun _ _ => ih _ _)
-      _ = _ := by simp [Finset.prod_range_succ]; ring
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-open FiniteLaw
-
-def boolSupport {n : ℕ} (θ : Fin n → Bool) : Finset (Fin n) :=
-  Finset.univ.filter (fun i => θ i=true)
-
-theorem boolSupport_injective {n : ℕ} : Function.Injective (boolSupport (n:=n)) := by
-  intro θ φ h
-  funext i
-  have hi : (θ i=true) ↔ (φ i=true) := by
-    simpa [boolSupport] using Finset.ext_iff.mp h i
-  cases hθ : θ i <;> cases hφ : φ i <;> simp_all
-
-theorem selectionCoefficient_eq_support {n : ℕ} (θ σ : Fin n → Bool) :
-    selectionCoefficient θ σ = if boolSupport σ ⊆ boolSupport θ then 1 else 0 := by
-  by_cases h : boolSupport σ ⊆ boolSupport θ
-  · rw [ite_eq_left h]
-    apply Finset.prod_eq_one
-    intro i _
-    cases hs : σ i
-    · simp
-    · have ht : θ i=true := by simpa [boolSupport] using h (by simp [boolSupport,hs])
-      simp [ht]
-  · rw [ite_eq_right h]
-    obtain ⟨i,hi,hni⟩ := Finset.not_subset.mp h
-    have hs : σ i=true := by simpa [boolSupport] using hi
-    have ht : θ i=false := by simpa [boolSupport] using hni
-    apply Finset.prod_eq_zero (Finset.mem_univ i)
-    simp [hs,ht]
-
-/-- Uniform cardinality bound for the actual selected monomials of a sparse
-0/1 coefficient vector; mixing does not add a squared-cardinality loss. -/
-theorem selection_degree_sum_le {n : ℕ} (θ : Fin n → Bool) (r : ℕ) :
-    (∑σ : Fin n → Bool, if monomialDegree σ=r then selectionCoefficient θ σ else 0) ≤
-      ((monomialDegree θ).choose r:ℝ) := by
-  have hc : (Finset.univ.filter (fun σ : Fin n → Bool =>
-      monomialDegree σ=r ∧ boolSupport σ ⊆ boolSupport θ)).card ≤
-        ((boolSupport θ).powersetCard r).card := by
-    apply Finset.card_le_card_of_injOn boolSupport
-    · intro σ hσ
-      rcases (Finset.mem_filter.mp hσ).2 with ⟨hd,hs⟩
-      exact Finset.mem_powersetCard.mpr ⟨hs,hd⟩
-    · exact boolSupport_injective.injOn
-  rw [Finset.card_powersetCard] at hc
-  have he : (∑σ : Fin n → Bool, if monomialDegree σ=r then selectionCoefficient θ σ else 0) =
-      ((Finset.univ.filter (fun σ : Fin n → Bool =>
-        monomialDegree σ=r ∧ boolSupport σ ⊆ boolSupport θ)).card:ℝ) := by
-    simp_rw [selectionCoefficient_eq_support,←ite_and]
-    exact Finset.sum_boole _ _
-  rw [he]
-  exact_mod_cast hc
-
-theorem averaged_sparse_degree {Ω : Type*} [Fintype Ω] {n A : ℕ}
-    (R : FiniteLaw Ω) (θ : Ω → Fin n → Bool)
-    (hA : ∀ω, monomialDegree (θ ω) ≤ A) (r : ℕ) :
-    (∑σ : Fin n → Bool, if monomialDegree σ=r then averagedCoefficient R θ σ^2 else 0) ≤
-      (A.choose r:ℝ) := by
-  calc
-    _ ≤ ∑σ : Fin n → Bool, if monomialDegree σ=r then averagedCoefficient R θ σ else 0 := by
-      apply Finset.sum_le_sum
-      intro σ _
-      split_ifs
-      · have hc := coefficient_nonneg R θ σ
-        have h1 := coefficient_le_one R θ σ
-        nlinarith [mul_nonneg hc (sub_nonneg.mpr h1)]
-      · rfl
-    _ = ∑ω, R.mass ω * ∑σ : Fin n → Bool,
-        if monomialDegree σ=r then selectionCoefficient (θ ω) σ else 0 := by
-      unfold averagedCoefficient FiniteLaw.expect
-      simp_rw [Finset.mul_sum]
-      rw [Finset.sum_comm]
-      apply Finset.sum_congr rfl
-      intro σ _
-      by_cases h : monomialDegree σ=r <;> simp [h]
-    _ ≤ ∑ω, R.mass ω * (A.choose r:ℝ) := by
-      apply Finset.sum_le_sum
-      intro ω _
-      apply mul_le_mul_of_nonneg_left _ (R.nonneg ω)
-      exact (selection_degree_sum_le (θ ω) r).trans (by
-        exact_mod_cast Nat.choose_le_choose r (hA ω))
-    _ = _ := by rw [←Finset.sum_mul,R.total,one_mul]
-
-/-- Genuine sparse complete-tree leaf count in the exact Fin indexing used
-by the independent-product reference law. -/
-theorem windowFinType_degree (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
-    (b : ℕ) (τ : ℝ) :
-    monomialDegree (windowFinType T s X b τ) = windowMinusCount T s X b [] τ := by
-  have h := Equiv.sum_comp (Fintype.equivFin (WindowLeaf T b)).symm
-    (fun v => if windowLeafSign T s X b [] τ v= -1 then (1:ℕ) else 0)
-  simpa only [monomialDegree,Finset.card_filter,windowFinType,windowLeafType,
-    decide_eq_true_eq,windowMinusCount] using h
-
-theorem recursive_degree_bound {Ω : Type*} [Fintype Ω]
-    (R : FiniteLaw Ω) (X : Ω → List ℕ → ℤ) (T : ℕ → ℕ) (s b r : ℕ) (τ : ℝ) :
-    (∑σ, if monomialDegree σ=r then
-      averagedCoefficient R (fun ω => windowFinType T s (X ω) (b+1) τ) σ^2 else 0) ≤
-        ((2^s*∏j∈Finset.range b,T (j+1)).choose r:ℝ) := by
-  apply averaged_sparse_degree
-  intro ω
-  rw [windowFinType_degree]
-  exact windowMinusCount_bound T s (X ω) b [] τ
-
-def degreeTail {n : ℕ} (A : (Fin n → Bool) → ℝ) (k : ℕ) (σ : Fin n → Bool) : ℝ :=
-  if k ≤ monomialDegree σ then A σ else 0
-
-theorem degreeTail_degree {n r k : ℕ} (A : (Fin n → Bool) → ℝ) :
-    (∑σ, if monomialDegree σ=r then degreeTail A k σ^2 else 0) =
-      if k ≤ r then ∑σ, if monomialDegree σ=r then A σ^2 else 0 else 0 := by
-  by_cases h : k ≤ r
-  · rw [ite_eq_left h]
-    apply Finset.sum_congr rfl
-    intro σ _
-    by_cases he : monomialDegree σ=r <;> simp [degreeTail,he,h]
-  · rw [ite_eq_right h]
-    apply Finset.sum_eq_zero
-    intro σ _
-    by_cases he : monomialDegree σ=r <;> simp [degreeTail,he,h]
-
-/-- Geometric high-degree energy of an actual mixed sparse leaf law. -/
-theorem averaged_sparse_tail_energy {Ω : Type*} [Fintype Ω] {n A₀ : ℕ}
-    (R : FiniteLaw Ω) (θ : Ω → Fin n → Bool)
-    (hA : ∀ω, monomialDegree (θ ω) ≤ A₀) (k : ℕ) {w : ℝ}
-    (hw : 0 ≤ w) (ha : w^2*A₀ < 1) :
-    multilinearEnergy (degreeTail (averagedCoefficient R θ) k) w ≤
-      (w^2*A₀)^k/(1-w^2*A₀) := by
-  have H := multilinearEnergy_low_high (R:=k)
-    (degreeTail (averagedCoefficient R θ) k) w (w^2*A₀) 1 hw (by positivity) ha
-    (by norm_num) (fun _ => 0) ?_ ?_ (fun _ => le_rfl)
-  · simpa using H
-  · intro r hr
-    rw [degreeTail_degree,ite_eq_right (by omega)]
-  · intro r hr
-    rw [degreeTail_degree,ite_eq_left hr,one_mul]
-    have hc : (A₀.choose r:ℝ) ≤ (A₀:ℝ)^r := by exact_mod_cast Nat.choose_le_pow A₀ r
-    calc
-      _ ≤ w^(2*r)*(A₀:ℝ)^r := mul_le_mul_of_nonneg_left
-        ((averaged_sparse_degree R θ hA r).trans hc) (by positivity)
-      _ = _ := by rw [pow_mul,mul_pow]
-
-/-- Actual recursive-oracle high tail, including arbitrary correlated latent
-bits and displacements. This supplies the block layout's unconditioned high-degree
-input; low degrees still require the signed occupied-tree cancellation. -/
-theorem recursive_high_moment {Ω α : Type*} [Fintype Ω] [Fintype α]
-    (R : FiniteLaw Ω) (X : Ω → List ℕ → ℤ) (T : ℕ → ℕ) (s b r₀ k : ℕ) (τ : ℝ)
-    (P : FiniteLaw α) (Z : α → ℝ) (hk : 1 ≤ k) (hcenter : P.expect Z=0)
-    {L : ℝ} (hL : 0 < L) (hMoment : P.expect (fun a => |Z a|^(2*k)) ≤ L^(2*k))
-    (ha : ((6*(2*k:ℕ):ℝ)*L)^2 * (2^s*∏j∈Finset.range b,T (j+1):ℕ) < 1) :
-    (P.product (Fintype.card (WindowLeaf T (b+1)))).evenMomentSq
-      (multilinear (degreeTail (averagedCoefficient R
-        (fun ω => windowFinType T s (X ω) (b+1) τ)) r₀) Z) k ≤
-      (((6*(2*k:ℕ):ℝ)*L)^2*(2^s*∏j∈Finset.range b,T (j+1):ℕ))^r₀/
-        (1-((6*(2*k:ℕ):ℝ)*L)^2*(2^s*∏j∈Finset.range b,T (j+1):ℕ)) := by
-  apply (multilinear_moment_sq_scaled P Z k hk hcenter hL hMoment _ _).trans
-  apply averaged_sparse_tail_energy R _ _ r₀ (by positivity) ha
-  intro ω
-  rw [windowFinType_degree]
-  exact windowMinusCount_bound T s (X ω) b [] τ
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-open FiniteLaw
-
-def selectedChildren {α β : Type*} [DecidableEq α] (S : Finset (α × β)) : Finset α :=
-  S.image Prod.fst
-
-def selectedChildLeaves {α β : Type*} [DecidableEq α] [DecidableEq β]
-    (S : Finset (α × β)) (h : α) : Finset β :=
-  (S.filter (fun v => v.1=h)).image Prod.snd
-
-@[to_additive]
-theorem selected_prod_children {α β M : Type*} [DecidableEq α] [DecidableEq β]
-    [CommMonoid M] (S : Finset (α × β)) (f : α × β → M) :
-    (∏h∈selectedChildren S, ∏v∈selectedChildLeaves S h, f (h,v)) = ∏v∈S, f v := by
-  have he (h : α) : (∏v∈selectedChildLeaves S h, f (h,v)) =
-      ∏v∈S.filter (fun v => v.1=h), f v := by
-    unfold selectedChildLeaves
-    rw [Finset.prod_image]
-    · apply Finset.prod_congr rfl
-      intro v hv
-      rw [←(Finset.mem_filter.mp hv).2]
-    · intro v hv w hw h
-      apply Prod.ext
-      · exact (Finset.mem_filter.mp hv).2.trans (Finset.mem_filter.mp hw).2.symm
-      · exact h
-  simp_rw [he]
-  exact Finset.prod_fiberwise_of_maps_to (fun v hv => Finset.mem_image.mpr ⟨v,hv,rfl⟩) f
-
-@[to_additive]
-theorem selected_prod_index {α M : Type*} [DecidableEq α] [CommMonoid M]
-    (S : Finset α) (f : α → M) :
-    (∏i : Fin S.card, f (S.equivFin.symm i).val) = ∏v∈S, f v := by
-  have h := Equiv.prod_comp S.equivFin.symm (fun v : S => f v.val)
-  simpa only [Finset.prod_coe_sort] using h
-
-/-- Occupied shape of an actual subset of a complete window tree. Only
-occupied children are enumerated, so its leaves are exactly the selected
-monomial variables, not the other complete-tree leaves. -/
-def occupiedWindowShape (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ) :
-    (b : ℕ) → (path : List ℕ) → Finset (WindowLeaf T (b+1)) → CoefficientShape b
-  | 0, path, S => .bottom S.card
-      (fun i => motifIndicator s (((S.equivFin.symm i).val.1.val:ℤ)-X path))
-      (fun i => (prouhetSeries s).coeff (((S.equivFin.symm i).val.1.val:ℤ)-X path))
-  | b+1, path, S => .node (selectedChildren S).card
-      (fun i => motifIndicator s ((((selectedChildren S).equivFin.symm i).val.val:ℤ)-X path))
-      (fun i => (prouhetSeries s).coeff ((((selectedChildren S).equivFin.symm i).val.val:ℤ)-X path))
-      (fun i => occupiedWindowShape T s X b
-        (path++[((selectedChildren S).equivFin.symm i).val.val])
-        (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val))
-
-theorem occupiedWindowShape_admissible (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
-    (b : ℕ) (path : List ℕ) (S : Finset (WindowLeaf T (b+1))) :
-    (occupiedWindowShape T s X b path S).Admissible := by
-  induction b generalizing path with
-  | zero => exact fun _ => prouhet_motif_value s _
-  | succ b ih => exact ⟨fun _ => prouhet_motif_value s _, fun _ => ih _ _⟩
-
-/-- The literal coefficient product equals the evaluation of its occupied
-shape; this is before any averaging or parity cancellation. -/
-theorem occupiedWindowShape_eval (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
-    (b : ℕ) (path : List ℕ) (S : Finset (WindowLeaf T (b+1))) (τ : ℝ) :
-    (occupiedWindowShape T s X b path S).eval τ =
-      ∏v∈S, (1-windowLeafSign T s X (b+1) path τ v)/2 := by
-  induction b generalizing path τ with
-  | zero =>
-    change (∏i : Fin S.card, _) = _
-    refine (selected_prod_index S (fun v : Fin (T 0) × Fin 1 =>
-      (motifIndicator s ((v.1.val:ℤ)-X path) -
-        (prouhetSeries s).coeff ((v.1.val:ℤ)-X path)*τ)/2)).trans ?_
-    apply Finset.prod_congr rfl
-    intro v hv
-    simp only [windowLeafSign,motifChildSign]
-    ring
-  | succ b ih =>
-    change (∏i : Fin (selectedChildren S).card, _) = _
-    simp_rw [ih]
-    refine (selected_prod_index (selectedChildren S) (fun h : Fin (T (b+1)) =>
-      ∏v∈selectedChildLeaves S h,
-        (1-windowLeafSign T s X (b+1) (path++[h.val])
-          (motifChildSign s (X path) h.val τ) v)/2)).trans ?_
-    exact selected_prod_children S (fun v =>
-      (1-windowLeafSign T s X (b+1+1) path τ v)/2)
-
-theorem selected_children_card {α β : Type*} [DecidableEq α] [DecidableEq β]
-    (S : Finset (α × β)) :
-    (∑h∈selectedChildren S, (selectedChildLeaves S h).card) = S.card := by
-  simpa only [Finset.sum_const,smul_eq_mul,mul_one] using
-    selected_sum_children S (fun _ => (1:ℕ))
-
-theorem selectedChildLeaves_nonempty {α β : Type*} [DecidableEq α] [DecidableEq β]
-    (S : Finset (α × β)) (h : α) (hh : h∈selectedChildren S) :
-    (selectedChildLeaves S h).Nonempty := by
-  obtain ⟨v,hv,rfl⟩ := Finset.mem_image.mp hh
-  exact ⟨v.2,Finset.mem_image.mpr ⟨v,Finset.mem_filter.mpr ⟨hv,rfl⟩,rfl⟩⟩
-
-theorem occupiedWindowShape_leafCount (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
-    (b : ℕ) (path : List ℕ) (S : Finset (WindowLeaf T (b+1)))
-    (a : (occupiedWindowShape T s X b path S).Choice) :
-    ((occupiedWindowShape T s X b path S).termTree a).leafCount = S.card := by
-  induction b generalizing path with
-  | zero => simp [occupiedWindowShape,CoefficientShape.termTree,OccupiedParityTree.leafCount]
-  | succ b ih =>
-    change (∑i : Fin (selectedChildren S).card, _) = _
-    simp_rw [ih]
-    exact (selected_sum_index (selectedChildren S) (fun h =>
-      (selectedChildLeaves S h).card)).trans (selected_children_card S)
-
-theorem occupiedWindowShape_occupied (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
-    (b : ℕ) (path : List ℕ) (S : Finset (WindowLeaf T (b+1))) (hS : S.Nonempty)
-    (a : (occupiedWindowShape T s X b path S).Choice) :
-    ((occupiedWindowShape T s X b path S).termTree a).occupied := by
-  induction b generalizing path with
-  | zero => exact ⟨hS.card_pos,fun _ => trivial⟩
-  | succ b ih =>
-    refine ⟨(Finset.image_nonempty.mpr hS).card_pos,fun i => ?_⟩
-    exact ih _ _ (selectedChildLeaves_nonempty S _
-      ((selectedChildren S).equivFin.symm i).property) (a.1 i)
-
-/-- Forced-leaf count for a term of an actual selected monomial. -/
-theorem occupiedWindowShape_forced_count (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
-    (b : ℕ) (path : List ℕ) (S : Finset (WindowLeaf T (b+1)))
-    (a : (occupiedWindowShape T s X b path S).Choice)
-    (hw : (occupiedWindowShape T s X b path S).weight a ≠ 0)
-    (ho : ((occupiedWindowShape T s X b path S).termTree a).isOdd)
-    (hn : ((occupiedWindowShape T s X b path S).termTree a).noSingleton) :
-    2^(b+1) ≤ S.card := by
-  simpa only [occupiedWindowShape_leafCount] using
-    (occupiedWindowShape T s X b path S).surviving_leaves a hw ho hn
-
-/-- The role choices depend only on the selected subset, not on any hidden
-orientation. This common index type permits averaging before norm estimates. -/
-def OccupiedWindowChoice (T : ℕ → ℕ) :
-    (b : ℕ) → Finset (WindowLeaf T (b+1)) → Type
-  | 0, S => Fin S.card → Fin 3
-  | b+1, S =>
-      ((i : Fin (selectedChildren S).card) → OccupiedWindowChoice T b
-        (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val)) ×
-      (Fin (selectedChildren S).card → Fin 3)
-
-noncomputable instance occupiedWindowChoiceFintype (T : ℕ → ℕ) :
-    (b : ℕ) → (S : Finset (WindowLeaf T (b+1))) → Fintype (OccupiedWindowChoice T b S)
-  | 0, S => by unfold OccupiedWindowChoice; infer_instance
-  | b+1, S => by
-      letI := fun i : Fin (selectedChildren S).card =>
-        occupiedWindowChoiceFintype T b
-          (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val)
-      unfold OccupiedWindowChoice
-      infer_instance
-
-noncomputable def occupiedWindowChoiceEquiv (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ) :
-    (b : ℕ) → (path : List ℕ) → (S : Finset (WindowLeaf T (b+1))) →
-      OccupiedWindowChoice T b S ≃ (occupiedWindowShape T s X b path S).Choice
-  | 0, _, _ => Equiv.refl _
-  | b+1, path, S => Equiv.prodCongr
-      (Equiv.piCongrRight fun i => occupiedWindowChoiceEquiv T s X b
-        (path++[((selectedChildren S).equivFin.symm i).val.val])
-        (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val))
-      (Equiv.refl _)
-
-/-- The parity tree of a role choice is independent of all displacement bits. -/
-noncomputable def occupiedWindowTerm (T : ℕ → ℕ) :
-    (b : ℕ) → (S : Finset (WindowLeaf T (b+1))) →
-      OccupiedWindowChoice T b S → OccupiedParityTree (b+1)
-  | 0, S, a => .node S.card (fun _ => .leaf) a
-  | b+1, S, a => .node (selectedChildren S).card
-      (fun i => occupiedWindowTerm T b
-        (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val) (a.1 i)) a.2
-
- theorem occupiedWindowTerm_eq (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
-    (b : ℕ) (path : List ℕ) (S : Finset (WindowLeaf T (b+1)))
-    (a : OccupiedWindowChoice T b S) :
-    (occupiedWindowShape T s X b path S).termTree
-      (occupiedWindowChoiceEquiv T s X b path S a) = occupiedWindowTerm T b S a := by
-  induction b generalizing path with
-  | zero => rfl
-  | succ b ih =>
-    change OccupiedParityTree.node _ (fun i => _) _ = OccupiedParityTree.node _ _ _
-    congr 1
-    funext i
-    exact ih _ _ _
-
-noncomputable def occupiedWindowWeight (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
-    (b : ℕ) (path : List ℕ) (S : Finset (WindowLeaf T (b+1)))
-    (a : OccupiedWindowChoice T b S) : ℝ :=
-  (occupiedWindowShape T s X b path S).weight
-    (occupiedWindowChoiceEquiv T s X b path S a)
-
-/-- Exact source parity formula for the actual selected leaves, using a
-single role index independent of the latent map. No fit conditioning. -/
-theorem occupiedWindow_difference (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
-    (b : ℕ) (path : List ℕ) (S : Finset (WindowLeaf T (b+1))) :
-    (∏v∈S, (1-windowLeafSign T s X (b+1) path 1 v)/2) -
-      (∏v∈S, (1-windowLeafSign T s X (b+1) path (-1) v)/2) =
-      2*∑a : OccupiedWindowChoice T b S,
-        if (occupiedWindowTerm T b S a).isOdd then occupiedWindowWeight T s X b path S a else 0 := by
-  rw [←occupiedWindowShape_eval,←occupiedWindowShape_eval,
-    CoefficientShape.difference_expansion _ (occupiedWindowShape_admissible T s X b path S)]
-  congr 1
-  rw [←(occupiedWindowChoiceEquiv T s X b path S).sum_comp]
-  apply Finset.sum_congr rfl
-  intro a _
-  rw [occupiedWindowTerm_eq]
-  rfl
-
-/-- Averaging the complete exact expansion keeps all correlations of the
-common hidden map. The indicator of an odd root is deterministic. -/
-theorem occupiedWindow_averaged_difference {Ω : Type*} [Fintype Ω]
-    (R : FiniteLaw Ω) (T : ℕ → ℕ) (s : ℕ) (X : Ω → List ℕ → ℤ)
-    (b : ℕ) (path : List ℕ) (S : Finset (WindowLeaf T (b+1))) :
-    R.expect (fun ω => ∏v∈S, (1-windowLeafSign T s (X ω) (b+1) path 1 v)/2) -
-      R.expect (fun ω => ∏v∈S, (1-windowLeafSign T s (X ω) (b+1) path (-1) v)/2) =
-      2*∑a : OccupiedWindowChoice T b S,
-        if (occupiedWindowTerm T b S a).isOdd then
-          R.expect (fun ω => occupiedWindowWeight T s (X ω) b path S a) else 0 := by
-  unfold expect
-  rw [←Finset.sum_sub_distrib]
-  simp_rw [←mul_sub,occupiedWindow_difference,mul_left_comm (R.mass _) 2,
-    Finset.mul_sum]
-  rw [Finset.sum_comm]
-  simp_rw [←Finset.mul_sum]
-  congr 1
-  apply Finset.sum_congr rfl
-  intro a _
-  by_cases h : (occupiedWindowTerm T b S a).isOdd <;> simp [h]
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-open FiniteLaw
-
-/-- Arbitrary functions of the actual posterior count have the translated
-fair-binomial law, not only its generating polynomial. -/
-theorem posterior_count_average {ι : Type*} [Fintype ι]
-    (d : ι → Option Bool) (F : ℤ → ℝ) :
-    (∑ b : ι → Bool, (∏i, (flagPosterior (d i)).mass (b i)) *
-      F (∑i, (bitIndex (b i):ℤ))) =
-    ∑k∈Finset.range ((∑i, hiddenTrial (d i))+1),
-      fairBinomial (∑i, hiddenTrial (d i)) k * F ((∑i,(visibleOnes (d i):ℤ))+k) := by
-  have H := congrArg (fun p : IntSeries => Finsupp.linearCombination ℝ F p.coeff)
-    (posterior_count_series d)
-  rw [fairSeries_sum,Finset.mul_sum] at H
-  simp only [AddMonoidAlgebra.single_mul_single,one_mul,
-    AddMonoidAlgebra.coeff_sum,AddMonoidAlgebra.coeff_single,map_sum,
-    Finsupp.linearCombination_single,smul_eq_mul] at H
-  exact H
-
-theorem fairBinomial_reflect (n k : ℕ) (hk : k ≤ n) :
-    fairBinomial n (n-k) = fairBinomial n k := by
-  simp only [fairBinomial,Nat.choose_symm hk]
-
-/-- Reflection of the remaining hidden count. The known visible contribution
-is kept, and no positivity assumption on the metadata mass is needed. -/
-theorem posterior_count_average_reflected {ι : Type*} [Fintype ι]
-    (d : ι → Option Bool) (F : ℤ → ℝ) :
-    (∑ b : ι → Bool, (∏i, (flagPosterior (d i)).mass (b i)) *
-      F (∑i, (bitIndex (b i):ℤ))) =
-    ∑k∈Finset.range ((∑i, hiddenTrial (d i))+1),
-      fairBinomial (∑i, hiddenTrial (d i)) k *
-        F ((∑i,(visibleOnes (d i):ℤ))+(∑i,hiddenTrial (d i))-k) := by
-  rw [posterior_count_average]
-  rw [←Finset.sum_range_reflect]
-  apply Finset.sum_congr rfl
-  intro k hk
-  have hk' : k ≤ ∑i,hiddenTrial (d i) := by simpa only [Finset.mem_range,Nat.lt_succ_iff] using hk
-  simp only [Nat.add_sub_cancel,fairBinomial_reflect _ _ hk',Nat.cast_sub hk']
-  congr 2
-  ring
-
-/-- The actual local signed motif averaged over posterior orientations is
-exactly a translated smoothed Prouhet polynomial. -/
-theorem posterior_signed_motif {ι : Type*} [Fintype ι]
-    (d : ι → Option Bool) (s : ℕ) (z : ℤ) :
-    (∑b : ι → Bool, (∏i,(flagPosterior (d i)).mass (b i)) *
-      (prouhetSeries s).coeff (z+∑i,(bitIndex (b i):ℤ))) =
-      (prouhetSeries s * fairSeries (∑i,hiddenTrial (d i))).coeff
-        (z+(∑i,(visibleOnes (d i):ℤ))+(∑i,hiddenTrial (d i))) := by
-  rw [posterior_count_average_reflected d (fun a => (prouhetSeries s).coeff (z+a)),
-    fairSeries_sum,Finset.mul_sum,
-    AddMonoidAlgebra.coeff_sum,Finsupp.finsetSum_apply]
-  apply Finset.sum_congr rfl
-  intro k hk
-  rw [AddMonoidAlgebra.coeff_mul_single_apply]
-  simp only [mul_comm]
-  congr 2
-  omega
-
-/-- Singleton smoothing for literal hidden orientation bits. Other mechanisms
-can now be conditioned on without changing this bound. -/
-theorem posterior_signed_motif_bound {ι : Type*} [Fintype ι]
-    (d : ι → Option Bool) (s : ℕ) (z : ℤ) :
-    |∑b : ι → Bool, (∏i,(flagPosterior (d i)).mass (b i)) *
-      (prouhetSeries s).coeff (z+∑i,(bitIndex (b i):ℤ))| ≤
-      (2:ℝ)^(s*s)*((2*binomialBound ((∑i,hiddenTrial (d i))/(s+1)))^s *
-        binomialBound ((∑i,hiddenTrial (d i))/(s+1))) := by
-  rw [posterior_signed_motif]
-  exact (signed_binomial_smoothing _ s).1 _
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-open FiniteLaw
-
-/-- The literal unsigned window evaluated at an affine fair-binomial shift. -/
-theorem windowKernel_binomial (n L : ℕ) (z : ℤ) :
-    windowKernel n L z =
-      ∑k∈Finset.range (n+1), fairBinomial n k *
-        (if 0 ≤ z+(k:ℤ) ∧ z+(k:ℤ)<L then (1:ℝ) else 0) := by
-  unfold windowKernel
-  simp_rw [fairSeries_sum,AddMonoidAlgebra.coeff_sum,Finsupp.finsetSum_apply,
-    AddMonoidAlgebra.coeff_single,Finsupp.single_apply]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro k hk
-  rw [←geomSeries_coeff,geomSeries]
-  simp only [AddMonoidAlgebra.single_pow,one_pow,nsmul_eq_mul,mul_one,
-    AddMonoidAlgebra.coeff_sum,Finsupp.finsetSum_apply,
-    AddMonoidAlgebra.coeff_single,Finsupp.single_apply,Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro a ha
-  by_cases h : (a:ℤ)=z+k
-  · have h' : (k:ℤ)=(a:ℤ)-z := by omega
-    simp only [ite_eq_left h,ite_eq_left h',mul_one]
-  · have h' : (k:ℤ)≠(a:ℤ)-z := by omega
-    simp only [ite_eq_right h,ite_eq_right h',mul_zero]
-
-/-- Exact local anchor kernel for the actual posterior bits, with the known
-visible contribution retained as a translation. -/
-theorem posterior_unsigned_motif {ι : Type*} [Fintype ι]
-    (d : ι → Option Bool) (s : ℕ) (z : ℤ) :
-    (∑b : ι → Bool, (∏i,(flagPosterior (d i)).mass (b i)) *
-      motifIndicator s (z+∑i,(bitIndex (b i):ℤ))) =
-      windowKernel (∑i,hiddenTrial (d i)) (2^s) (z+∑i,(visibleOnes (d i):ℤ)) := by
-  rw [posterior_count_average d (fun a => motifIndicator s (z+a)),windowKernel_binomial]
-  apply Finset.sum_congr rfl
-  intro k hk
-  simp only [motifIndicator,add_assoc]
-
-/-- The finite-series functional form of the posterior count identity. -/
-theorem posterior_count_support {ι : Type*} [Fintype ι]
-    (d : ι → Option Bool) (F : ℤ → ℝ) :
-    (∑b : ι → Bool, (∏i,(flagPosterior (d i)).mass (b i))*F (∑i,(bitIndex (b i):ℤ))) =
-    ∑v∈(fairSeries (∑i,hiddenTrial (d i))).coeff.support,
-      (fairSeries (∑i,hiddenTrial (d i))).coeff v * F ((∑i,(visibleOnes (d i):ℤ))+v) := by
-  rw [posterior_count_average]
-  have H := congrArg (fun p : IntSeries =>
-    Finsupp.linearCombination ℝ (fun v => F ((∑i,(visibleOnes (d i):ℤ))+v)) p.coeff)
-    (fairSeries_sum (∑i,hiddenTrial (d i)))
-  simp only [AddMonoidAlgebra.coeff_sum,AddMonoidAlgebra.coeff_single,map_sum,
-    Finsupp.linearCombination_single,smul_eq_mul] at H
-  exact H.symm
-
-/-- The actual two-sibling anchor multiplier. Global bits are shared by the
-two siblings until AFTER averaging; their local bits are independent. -/
-def posteriorJointAnchors {ιg ι₁ ι₂ : Type*} [Fintype ιg] [Fintype ι₁] [Fintype ι₂]
-    (dg : ιg → Option Bool) (d₁ : ι₁ → Option Bool) (d₂ : ι₂ → Option Bool)
-    (s : ℕ) (z₁ z₂ : ℤ) : ℝ :=
-  ∑bg : ιg → Bool, (∏i,(flagPosterior (dg i)).mass (bg i)) *
-    (∑b₁ : ι₁ → Bool, (∏i,(flagPosterior (d₁ i)).mass (b₁ i)) *
-      motifIndicator s (z₁+(∑i,(bitIndex (bg i):ℤ))+(∑i,(bitIndex (b₁ i):ℤ)))) *
-    (∑b₂ : ι₂ → Bool, (∏i,(flagPosterior (d₂ i)).mass (b₂ i)) *
-      motifIndicator s (z₂+(∑i,(bitIndex (bg i):ℤ))+(∑i,(bitIndex (b₂ i):ℤ))))
-
-theorem posteriorJointAnchors_eq {ιg ι₁ ι₂ : Type*} [Fintype ιg] [Fintype ι₁] [Fintype ι₂]
-    (dg : ιg → Option Bool) (d₁ : ι₁ → Option Bool) (d₂ : ι₂ → Option Bool)
-    (s : ℕ) (z₁ z₂ : ℤ) :
-    posteriorJointAnchors dg d₁ d₂ s z₁ z₂ =
-    jointWindowKernel (∑i,hiddenTrial (dg i)) (∑i,hiddenTrial (d₁ i))
-      (∑i,hiddenTrial (d₂ i)) (2^s)
-      (z₁+(∑i,(visibleOnes (dg i):ℤ))+(∑i,(visibleOnes (d₁ i):ℤ)))
-      (z₂+(∑i,(visibleOnes (dg i):ℤ))+(∑i,(visibleOnes (d₂ i):ℤ))) := by
-  unfold posteriorJointAnchors
-  simp_rw [posterior_unsigned_motif,mul_assoc]
-  rw [posterior_count_support dg (fun v =>
-    windowKernel (∑i,hiddenTrial (d₁ i)) (2^s) (z₁+v+∑i,(visibleOnes (d₁ i):ℤ)) *
-    windowKernel (∑i,hiddenTrial (d₂ i)) (2^s) (z₂+v+∑i,(visibleOnes (d₂ i):ℤ)))]
-  unfold jointWindowKernel
-  apply Finset.sum_congr rfl
-  intro v hv
-  have h₁ : z₁+((∑i,(visibleOnes (dg i):ℤ))+v)+(∑i,(visibleOnes (d₁ i):ℤ)) =
-      z₁+(∑i,(visibleOnes (dg i):ℤ))+(∑i,(visibleOnes (d₁ i):ℤ))+v := by ring
-  have h₂ : z₂+((∑i,(visibleOnes (dg i):ℤ))+v)+(∑i,(visibleOnes (d₂ i):ℤ)) =
-      z₂+(∑i,(visibleOnes (dg i):ℤ))+(∑i,(visibleOnes (d₂ i):ℤ))+v := by ring
-  rw [h₁,h₂,mul_assoc]
-
-/-- Quantitative sibling summation now directly bounds genuine posterior
-bits, not an independent-anchor surrogate. -/
-theorem posterior_sibling_anchor_energy {ιg ι₁ ι₂ : Type*}
-    [Fintype ιg] [Fintype ι₁] [Fintype ι₂]
-    (dg : ιg → Option Bool) (d₁ : ι₁ → Option Bool) (d₂ : ι₂ → Option Bool)
-    (s : ℕ) (A B C D : Finset ℤ) (c₁ c₂ : ℤ) :
-    (∑a∈A, ∑b∈B, ∑_c∈closeLabels C (2^s) a, ∑_d∈closeLabels D (2^s) b,
-      (posteriorJointAnchors dg d₁ d₂ s (a-c₁) (b-c₂))^2) ≤
-    4*((2^s:ℕ):ℝ)^6*binomialBound (∑i,hiddenTrial (dg i))*
-      binomialBound (∑i,hiddenTrial (d₂ i)) := by
-  simp_rw [posteriorJointAnchors_eq]
-  have he : ∀a c u v : ℤ, a-c+u+v = a-(c-u-v) := by intros; omega
-  simp_rw [he]
-  exact sibling_anchor_energy (∑i,hiddenTrial (dg i)) (∑i,hiddenTrial (d₁ i))
-    (∑i,hiddenTrial (d₂ i)) (2^s) A B C D
-    (c₁-(∑i,(visibleOnes (dg i):ℤ))-(∑i,(visibleOnes (d₁ i):ℤ)))
-    (c₂-(∑i,(visibleOnes (dg i):ℤ))-(∑i,(visibleOnes (d₂ i):ℤ)))
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-open FiniteLaw
-
-namespace FiniteLaw
-variable {α β : Type*} [Fintype α] [Fintype β]
-
-theorem expect_mul_const (P : FiniteLaw α) (f : α → ℝ) (c : ℝ) :
-    P.expect (fun a => f a*c) = P.expect f*c := by
-  unfold expect
-  simp only [←mul_assoc,Finset.sum_mul]
-
-theorem expect_const_mul (P : FiniteLaw α) (f : α → ℝ) (c : ℝ) :
-    P.expect (fun a => c*f a) = c*P.expect f := by
-  unfold expect
-  simp only [mul_left_comm (P.mass _) c,Finset.mul_sum]
-
-theorem coordinateProduct_expect_product {n : ℕ} (P : Fin n → FiniteLaw α)
-    (f : Fin n → α → ℝ) :
-    (coordinateProduct P).expect (fun x => ∏i,f i (x i)) = ∏i,(P i).expect (f i) := by
-  unfold expect coordinateProduct
-  simp only [←Finset.prod_mul_distrib]
-  exact (Fintype.prod_sum (fun (i : Fin n) (a : α) => (P i).mass a*f i a)).symm
-
-theorem expect_prod_split (P : FiniteLaw α) (Q : FiniteLaw β)
-    (f : α → ℝ) (g : β → ℝ) :
-    (P.prod Q).expect (fun x => f x.1*g x.2) = P.expect f * Q.expect g := by
-  rw [expect_prod]
-  simp_rw [expect_const_mul]
-  exact expect_mul_const P f _
-
-end FiniteLaw
-
-/-- All local and global orientation collections on the complete index tree,
-including unoccupied vertices and paths which do not fit. The type is fixed
-before any selected monomial or role choices. `n` is the local trial count,
-`g` the number of global paired trials, and T the full window counts. -/
-def WindowAssignments (T n g : ℕ → ℕ) (A : Type) : ℕ → Type
-  | 0 => PUnit
-  | b+1 => (Fin (n b) → A) × (Fin (g b) → A) ×
-      (Fin (T b) → WindowAssignments T n g A b)
-
-noncomputable instance windowAssignmentsFintype (T n g : ℕ → ℕ) (A : Type) [Fintype A] :
-    (b : ℕ) → Fintype (WindowAssignments T n g A b)
-  | 0 => inferInstanceAs (Fintype PUnit)
-  | b+1 => by
-      letI := windowAssignmentsFintype T n g A b
-      exact inferInstanceAs (Fintype ((Fin (n b) → A) × (Fin (g b) → A) ×
-        (Fin (T b) → WindowAssignments T n g A b)))
-
-def posteriorArray {n : ℕ} (d : Fin n → Option Bool) : FiniteLaw (Fin n → Bool) :=
-  coordinateProduct (fun i => flagPosterior (d i))
-
-/-- Actual independent posterior orientations at exact non-leaf metadata.
-The common global bits in a group are drawn only once. -/
-def windowPosteriorLaw (T n g : ℕ → ℕ) : (b : ℕ) →
-    WindowAssignments T n g (Option Bool) b → FiniteLaw (WindowAssignments T n g Bool b)
-  | 0, _ => ⟨fun _ => 1,fun _ => by norm_num,by change (∑_ : PUnit,(1:ℝ))=1; simp⟩
-  | b+1, d => (posteriorArray d.1).prod ((posteriorArray d.2.1).prod
-      (coordinateProduct (fun i => windowPosteriorLaw T n g b (d.2.2 i))))
-
-/-- Displacements from the jointly generated bit tree. The ancestor global
-map may vary off its plateau. No independent jitter per child is introduced.
-The fixed center includes the deterministic half-trial corrections. -/
-def sourceTreeAnchors (T n g : ℕ → ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ) :
-    (b : ℕ) → WindowAssignments T n g Bool b → ℤ → List ℕ → ℤ
-  | 0, _, _, _ => 0
-  | b+1, bits, incoming, [] => center b - (∑i,(bitIndex (bits.1 i):ℤ))-incoming
-  | b+1, bits, _, h::tail => if hh : h<T b then
-      sourceTreeAnchors T n g center globalShift b (bits.2.2 ⟨h,hh⟩)
-        (globalShift b bits.2.1 ⟨h,hh⟩) tail else 0
-
-theorem sourceTreeAnchors_root (T n g : ℕ → ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (b : ℕ) (bits : WindowAssignments T n g Bool (b+1)) (incoming : ℤ) :
-    sourceTreeAnchors T n g center globalShift (b+1) bits incoming [] =
-      center b - (∑i,(bitIndex (bits.1 i):ℤ))-incoming := rfl
-
-theorem sourceTreeAnchors_child (T n g : ℕ → ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (b : ℕ) (bits : WindowAssignments T n g Bool (b+1)) (incoming : ℤ)
-    (h : Fin (T b)) (tail : List ℕ) :
-    sourceTreeAnchors T n g center globalShift (b+1) bits incoming (h.val::tail) =
-      sourceTreeAnchors T n g center globalShift b (bits.2.2 h)
-        (globalShift b bits.2.1 h) tail := by
-  simp only [sourceTreeAnchors,dite_eq_left h.isLt]
-
-/-- Moving the origin of a subtree preserves the literal type recursion. -/
-theorem windowLeafSign_prefix (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
-    (b : ℕ) (pre path : List ℕ) (τ : ℝ) (v : WindowLeaf T b) :
-    windowLeafSign T s X b (pre++path) τ v =
-      windowLeafSign T s (fun tail => X (pre++tail)) b path τ v := by
-  induction b generalizing path τ with
-  | zero => rfl
-  | succ b ih =>
-    change windowLeafSign T s X b _ _ v.2 = _
-    rw [List.append_assoc,ih]
-    rfl
-
-/-- Exact recursive leaf signs for the complete bit tree, with one joint
-parent-global shift supplied to each sibling's local mechanism. -/
-def sourceTreeLeafSign (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ) :
-    (b : ℕ) → WindowAssignments T n g Bool b → ℤ → ℝ → WindowLeaf T b → ℝ
-  | 0, _, _, τ, _ => τ
-  | b+1, bits, incoming, τ, v =>
-      sourceTreeLeafSign T n g s center globalShift b (bits.2.2 v.1)
-        (globalShift b bits.2.1 v.1)
-        (motifChildSign s (center b-(∑i,(bitIndex (bits.1 i):ℤ))-incoming) v.1.val τ) v.2
-
-theorem sourceTreeLeafSign_eq (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (b : ℕ) (bits : WindowAssignments T n g Bool b) (incoming : ℤ)
-    (τ : ℝ) (v : WindowLeaf T b) :
-    sourceTreeLeafSign T n g s center globalShift b bits incoming τ v =
-      windowLeafSign T s (sourceTreeAnchors T n g center globalShift b bits incoming) b [] τ v := by
-  induction b generalizing incoming τ with
-  | zero => rfl
-  | succ b ih =>
-    change sourceTreeLeafSign T n g s center globalShift b _ _ _ v.2 =
-      windowLeafSign T s _ b [v.1.val] _ v.2
-    rw [ih]
-    have hp := windowLeafSign_prefix T s
-      (sourceTreeAnchors T n g center globalShift (b+1) bits incoming)
-      b [v.1.val] []
-      (motifChildSign s (center b-(∑i,(bitIndex (bits.1 i):ℤ))-incoming) v.1.val τ) v.2
-    simp only [List.append_nil,List.singleton_append,sourceTreeAnchors_child] at hp
-    exact hp.symm
-
-end TraceReconstruction
-
--- Actual subtree coefficient factorization and singleton cancellation.
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-open FiniteLaw
-
-/-- The factor at the current vertex of a fixed actual occupied role choice.
-It only uses that vertex's displacement, never any descendant mechanism. -/
-def occupiedRootWeight (T : ℕ → ℕ) (s : ℕ) :
-    (b : ℕ) → (S : Finset (WindowLeaf T (b+1))) →
-      OccupiedWindowChoice T b S → ℤ → ℝ
-  | 0, S, a, X => ∏i : Fin S.card,
-      bottomRoleWeight (motifIndicator s (((S.equivFin.symm i).val.1.val:ℤ)-X))
-        ((prouhetSeries s).coeff (((S.equivFin.symm i).val.1.val:ℤ)-X)) (a i)
-  | b+1, S, a, X => ∏i : Fin (selectedChildren S).card,
-      @internalRoleWeight (occupiedWindowTerm T b
-        (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val) (a.1 i)).isOdd
-        (Classical.propDecidable _)
-        (motifIndicator s ((((selectedChildren S).equivFin.symm i).val.val:ℤ)-X))
-        ((prouhetSeries s).coeff ((((selectedChildren S).equivFin.symm i).val.val:ℤ)-X)) (a.2 i)
-
-theorem occupiedWindowWeight_zero (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
-    (path : List ℕ) (S : Finset (WindowLeaf T 1)) (a : OccupiedWindowChoice T 0 S) :
-    occupiedWindowWeight T s X 0 path S a = occupiedRootWeight T s 0 S a (X path) := rfl
-
-theorem occupiedWindowWeight_succ (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
-    (b : ℕ) (path : List ℕ) (S : Finset (WindowLeaf T (b+1+1)))
-    (a : OccupiedWindowChoice T (b+1) S) :
-    occupiedWindowWeight T s X (b+1) path S a =
-      occupiedRootWeight T s (b+1) S a (X path) *
-        ∏i : Fin (selectedChildren S).card,
-          occupiedWindowWeight T s X b
-            (path++[((selectedChildren S).equivFin.symm i).val.val])
-            (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val) (a.1 i) := by
-  unfold occupiedWindowWeight
-  change (∏i, _ * @internalRoleWeight
-    ((occupiedWindowShape T s X b
-      (path++[((selectedChildren S).equivFin.symm i).val.val])
-      (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val)).termTree
-      (occupiedWindowChoiceEquiv T s X b
-        (path++[((selectedChildren S).equivFin.symm i).val.val])
-        (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val) (a.1 i))).isOdd
-        (Classical.propDecidable _) _ _ _) = _
-  simp_rw [occupiedWindowTerm_eq]
-  rw [Finset.prod_mul_distrib,mul_comm]
-  rfl
-
-/-- Moving a subtree's path origin does not change any of its role factors. -/
-theorem occupiedWindowWeight_prefix (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
-    (b : ℕ) (pre path : List ℕ) (S : Finset (WindowLeaf T (b+1)))
-    (a : OccupiedWindowChoice T b S) :
-    occupiedWindowWeight T s X b (pre++path) S a =
-      occupiedWindowWeight T s (fun tail => X (pre++tail)) b path S a := by
-  induction b generalizing path with
-  | zero => rfl
-  | succ b ih =>
-    rw [occupiedWindowWeight_succ,occupiedWindowWeight_succ]
-    congr 1
-    apply Finset.prod_congr rfl
-    intro i hi
-    rw [List.append_assoc,ih]
-
-theorem bottomRoleWeight_abs_le_one (χ σ : ℝ) (h : IsMotifValue χ σ) (a : Fin 3) :
-    |bottomRoleWeight χ σ a| ≤ 1 := by
-  rcases h with ⟨rfl,rfl⟩ | ⟨rfl,rfl|rfl⟩ <;>
-    fin_cases a <;> norm_num [bottomRoleWeight]
-
-theorem internalRoleWeight_abs_le_one (odd : Prop) [Decidable odd]
-    (χ σ : ℝ) (h : IsMotifValue χ σ) (a : Fin 3) :
-    |internalRoleWeight odd χ σ a| ≤ 1 := by
-  rcases h with ⟨rfl,rfl⟩ | ⟨rfl,rfl|rfl⟩ <;>
-    fin_cases a <;> by_cases ho : odd <;> norm_num [internalRoleWeight,roleWeight,ho]
-
-theorem occupiedRootWeight_abs_le_one (T : ℕ → ℕ) (s b : ℕ)
-    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S) (X : ℤ) :
-    |occupiedRootWeight T s b S a X| ≤ 1 := by
-  cases b with
-  | zero =>
-    rw [occupiedRootWeight,Finset.abs_prod]
-    exact Finset.prod_le_one₀ (fun _ _ => abs_nonneg _)
-      (fun i _ => bottomRoleWeight_abs_le_one _ _ (prouhet_motif_value s _) (a i))
-  | succ b =>
-    rw [occupiedRootWeight,Finset.abs_prod]
-    exact Finset.prod_le_one₀ (fun _ _ => abs_nonneg _)
-      (fun i _ => internalRoleWeight_abs_le_one _ _ _ (prouhet_motif_value s _) (a.2 i))
-
-theorem occupiedWindowWeight_abs_le_one (T : ℕ → ℕ) (s : ℕ)
-    (X : List ℕ → ℤ) (b : ℕ) (path : List ℕ) (S : Finset (WindowLeaf T (b+1)))
-    (a : OccupiedWindowChoice T b S) : |occupiedWindowWeight T s X b path S a| ≤ 1 := by
-  induction b generalizing path with
-  | zero => exact occupiedRootWeight_abs_le_one T s 0 S a _
-  | succ b ih =>
-    rw [occupiedWindowWeight_succ,abs_mul,Finset.abs_prod]
-    exact (mul_le_of_le_one_left (by positivity)
-      (occupiedRootWeight_abs_le_one T s (b+1) S a _)).trans
-      (Finset.prod_le_one₀ (fun _ _ => abs_nonneg _) (fun i _ => ih _ _ _))
-
-/-- A singleton signed role factors off its signed motif with a scalar
-independent of all displacements. Bottom zero-role terms really vanish. -/
-theorem bottom_singleton_factor {n : ℕ} (a : Fin n → Fin 3) (i : Fin n)
-    (hi : a i=2) (hrest : ∀j,j≠i → a j=0) :
-    ∃c : ℝ, |c|≤1 ∧ ∀χ σ : Fin n → ℝ,
-      (∏j,bottomRoleWeight (χ j) (σ j) (a j)) = c*σ i := by
-  let c : ℝ := -(∏j∈Finset.univ.erase i,(0:ℝ))/2
-  refine ⟨c,?_,?_⟩
-  · have hp : 0 ≤ ∏j∈Finset.univ.erase i,(0:ℝ) := Finset.prod_nonneg (fun _ _ => le_rfl)
-    have hp' : (∏j∈Finset.univ.erase i,(0:ℝ)) ≤ 1 := Finset.prod_le_one₀
-      (fun _ _ => le_rfl) (fun _ _ => by norm_num)
-    dsimp [c]
-    rw [abs_div,abs_neg,abs_of_nonneg hp]
-    rw [abs_of_pos (by norm_num : (0:ℝ)<2)]
-    linarith
-  · intro χ σ
-    rw [←Finset.prod_erase_mul _ _ (Finset.mem_univ i)]
-    have he : (∏j∈Finset.univ.erase i,bottomRoleWeight (χ j) (σ j) (a j)) =
-        ∏j∈Finset.univ.erase i,(0:ℝ) := by
-      apply Finset.prod_congr rfl
-      intro j hj
-      simp only [hrest j (Finset.mem_erase.mp hj).1,bottomRoleWeight,↓reduceIte]
-    rw [he,hi]
-    simp only [bottomRoleWeight,show (2:Fin 3)≠0 by decide,
-      show (2:Fin 3)≠1 by decide,↓reduceIte]
-    dsimp [c]
-    ring
-
-theorem internal_singleton_factor {n : ℕ} (odd : Fin n → Prop)
-    (a : Fin n → Fin 3) (i : Fin n) (hi : a i=2) (hrest : ∀j,j≠i → a j=0) :
-    ∃c : ℝ, |c|≤1 ∧ ∀χ σ : Fin n → ℝ,
-      (∏j,@internalRoleWeight (odd j) (Classical.propDecidable _) (χ j) (σ j) (a j)) = c*σ i := by
-  refine ⟨if odd i then 1 else 0,?_,?_⟩
-  · split_ifs <;> norm_num
-  · intro χ σ
-    rw [Finset.prod_eq_single i]
-    · by_cases ho : odd i <;> simp [internalRoleWeight,roleWeight,hi,ho]
-    · intro j hj hji
-      by_cases ho : odd j <;> simp [internalRoleWeight,roleWeight,hrest j hji,ho]
-    · simp
-
-namespace FiniteLaw
-
-/-- Unoccupied independent coordinates integrate to one. -/
-theorem coordinateProduct_expect_subtype {α : Type*} [Fintype α] {n : ℕ}
-    (P : Fin n → FiniteLaw α) (S : Finset (Fin n)) (f : S → α → ℝ) :
-    (coordinateProduct P).expect (fun x => ∏i : S, f i (x i.val)) =
-      ∏i : S, (P i.val).expect (f i) := by
-  have scalar : ∀f : S → ℝ,
-      (∏j : Fin n, if hj : j∈S then f ⟨j,hj⟩ else 1) = ∏i : S,f i := by
-    intro f
-    calc
-      _ = ∏j∈S, if hj : j∈S then f ⟨j,hj⟩ else 1 :=
-        (Finset.prod_subset S.subset_univ (fun j _ hj => dite_eq_right hj)).symm
-      _ = ∏i : S, if hj : i.val∈S then f ⟨i.val,hj⟩ else 1 :=
-        Finset.prod_subtype S (fun _ => Iff.rfl) _
-      _ = _ := by simp only [dite_eq_left (Subtype.property _)]
-  let F : Fin n → α → ℝ := fun j a => if hj : j∈S then f ⟨j,hj⟩ a else 1
-  have hp : ∀x : Fin n → α, (∏j,F j (x j)) = ∏i : S,f i (x i.val) := by
-    intro x
-    exact scalar (fun i => f i (x i.val))
-  rw [show (fun x : Fin n → α => ∏i : S,f i (x i.val)) = (fun x => ∏j,F j (x j)) by
-    funext x; exact (hp x).symm]
-  rw [coordinateProduct_expect_product]
-  have he : ∀j, (P j).expect (F j) = if hj : j∈S then (P j).expect (f ⟨j,hj⟩) else 1 := by
-    intro j
-    by_cases hj : j∈S
-    · simp only [F,dite_eq_left hj]
-    · simp only [F,dite_eq_right hj,expect_const]
-  simp_rw [he]
-  exact scalar (fun i => (P i.val).expect (f i))
-
-theorem coordinateProduct_expect_selected {α : Type*} [Fintype α] {n : ℕ}
-    (P : Fin n → FiniteLaw α) (S : Finset (Fin n)) (f : Fin S.card → α → ℝ) :
-    (coordinateProduct P).expect (fun x => ∏i : Fin S.card, f i (x (S.equivFin.symm i).val)) =
-      ∏i : Fin S.card, (P (S.equivFin.symm i).val).expect (f i) := by
-  have hp : ∀x : Fin n → α,
-      (∏i : Fin S.card,f i (x (S.equivFin.symm i).val)) =
-        ∏i : S,f (S.equivFin i) (x i.val) := by
-    intro x
-    simpa only [Equiv.apply_symm_apply] using
-      Equiv.prod_comp S.equivFin.symm (fun i : S => f (S.equivFin i) (x i.val))
-  simp_rw [hp,coordinateProduct_expect_subtype]
-  symm
-  simpa only [Equiv.apply_symm_apply] using
-    Equiv.prod_comp S.equivFin.symm (fun i : S => (P i.val).expect (f (S.equivFin i)))
-
-theorem expect_prod_selected {α β γ : Type*} [Fintype α] [Fintype β] [Fintype γ]
-    {n : ℕ} (P : FiniteLaw α) (Q : FiniteLaw β) (R : Fin n → FiniteLaw γ)
-    (S : Finset (Fin n)) (f : α → ℝ) (h : β → Fin S.card → γ → ℝ) :
-    (P.prod (Q.prod (coordinateProduct R))).expect (fun x =>
-      f x.1 * ∏i : Fin S.card, h x.2.1 i (x.2.2 (S.equivFin.symm i).val)) =
-      P.expect f * Q.expect (fun z => ∏i : Fin S.card,
-        (R (S.equivFin.symm i).val).expect (h z i)) := by
-  simp_rw [expect_prod,expect_const_mul,coordinateProduct_expect_selected]
-  rw [expect_mul_const]
-
-theorem abs_expect_le_const {α : Type*} [Fintype α] (P : FiniteLaw α)
-    (f : α → ℝ) (c : ℝ) (h : ∀a, |f a|≤c) : |P.expect f|≤c :=
-  (P.abs_expect_le f).trans ((P.expect_mono h).trans_eq (P.expect_const c))
-
-end FiniteLaw
-
-
-/-- A fixed real coefficient term evaluated on the actual complete bit tree.
-The selected role choice is fixed independently of these bits. -/
-def sourceRoleTerm (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (b : ℕ) (bits : WindowAssignments T n g Bool (b+1)) (incoming : ℤ)
-    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S) : ℝ :=
-  occupiedWindowWeight T s (sourceTreeAnchors T n g center globalShift (b+1) bits incoming)
-    b [] S a
-
-def sourceRoleAverage (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ)
-    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S) : ℝ :=
-  (windowPosteriorLaw T n g (b+1) d).expect
-    (fun bits => sourceRoleTerm T n g s center globalShift b bits incoming S a)
-
-theorem sourceRoleTerm_zero (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (bits : WindowAssignments T n g Bool 1) (incoming : ℤ)
-    (S : Finset (WindowLeaf T 1)) (a : OccupiedWindowChoice T 0 S) :
-    sourceRoleTerm T n g s center globalShift 0 bits incoming S a =
-      occupiedRootWeight T s 0 S a (center 0-(∑i,(bitIndex (bits.1 i):ℤ))-incoming) := rfl
-
-theorem sourceRoleTerm_succ (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (b : ℕ) (bits : WindowAssignments T n g Bool (b+1+1)) (incoming : ℤ)
-    (S : Finset (WindowLeaf T (b+1+1))) (a : OccupiedWindowChoice T (b+1) S) :
-    sourceRoleTerm T n g s center globalShift (b+1) bits incoming S a =
-      occupiedRootWeight T s (b+1) S a (center (b+1)-(∑i,(bitIndex (bits.1 i):ℤ))-incoming) *
-        ∏i : Fin (selectedChildren S).card,
-          sourceRoleTerm T n g s center globalShift b
-            (bits.2.2 ((selectedChildren S).equivFin.symm i).val)
-            (globalShift (b+1) bits.2.1 ((selectedChildren S).equivFin.symm i).val)
-            (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val) (a.1 i) := by
-  unfold sourceRoleTerm
-  rw [occupiedWindowWeight_succ]
-  congr 1
-  apply Finset.prod_congr rfl
-  intro i hi
-  have hp := occupiedWindowWeight_prefix T s
-    (sourceTreeAnchors T n g center globalShift (b+1+1) bits incoming) b
-    [((selectedChildren S).equivFin.symm i).val.val] []
-    (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val) (a.1 i)
-  simpa only [List.append_nil,List.nil_append,List.singleton_append,sourceTreeAnchors_child] using hp
-
-theorem sourceRoleAverage_zero (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (d : WindowAssignments T n g (Option Bool) 1) (incoming : ℤ)
-    (S : Finset (WindowLeaf T 1)) (a : OccupiedWindowChoice T 0 S) :
-    sourceRoleAverage T n g s center globalShift 0 d incoming S a =
-      (posteriorArray d.1).expect (fun loc =>
-        occupiedRootWeight T s 0 S a (center 0-(∑i,(bitIndex (loc i):ℤ))-incoming)) := by
-  rw [sourceRoleAverage]
-  simp_rw [sourceRoleTerm_zero]
-  rcases d with ⟨dl,dg,dc⟩
-  change ((posteriorArray dl).prod ((posteriorArray dg).prod
-    (coordinateProduct (fun i => windowPosteriorLaw T n g 0 (dc i))))).expect
-      (fun bits => occupiedRootWeight T s 0 S a
-        (center 0-(∑i,(bitIndex (bits.1 i):ℤ))-incoming)) = _
-  rw [expect_prod]
-  simp only [expect_const]
-
-theorem sourceRoleAverage_succ (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1+1)) (incoming : ℤ)
-    (S : Finset (WindowLeaf T (b+1+1))) (a : OccupiedWindowChoice T (b+1) S) :
-    sourceRoleAverage T n g s center globalShift (b+1) d incoming S a =
-      (posteriorArray d.1).expect (fun loc =>
-        occupiedRootWeight T s (b+1) S a (center (b+1)-(∑i,(bitIndex (loc i):ℤ))-incoming)) *
-      (posteriorArray d.2.1).expect (fun glob =>
-        ∏i : Fin (selectedChildren S).card,
-          sourceRoleAverage T n g s center globalShift b
-            (d.2.2 ((selectedChildren S).equivFin.symm i).val)
-            (globalShift (b+1) glob ((selectedChildren S).equivFin.symm i).val)
-            (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val) (a.1 i)) := by
-  rw [sourceRoleAverage]
-  simp_rw [sourceRoleTerm_succ]
-  exact FiniteLaw.expect_prod_selected (posteriorArray d.1) (posteriorArray d.2.1)
-    (fun i => windowPosteriorLaw T n g (b+1) (d.2.2 i)) (selectedChildren S)
-    (fun loc => occupiedRootWeight T s (b+1) S a
-      (center (b+1)-(∑i,(bitIndex (loc i):ℤ))-incoming))
-    (fun glob i bits => sourceRoleTerm T n g s center globalShift b bits
-      (globalShift (b+1) glob ((selectedChildren S).equivFin.symm i).val)
-      (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val) (a.1 i))
-
-theorem sourceRoleAverage_abs_le_one (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ)
-    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S) :
-    |sourceRoleAverage T n g s center globalShift b d incoming S a|≤1 := by
-  apply FiniteLaw.abs_expect_le_const
-  intro bits
-  exact occupiedWindowWeight_abs_le_one T s _ b [] S a
-
-/-- The exact signed smoothing cost read off the visible metadata alone. -/
-def posteriorSmoothingCost {ι : Type*} [Fintype ι] (s : ℕ) (d : ι → Option Bool) : ℝ :=
-  (2:ℝ)^(s*s)*((2*binomialBound ((∑i,hiddenTrial (d i))/(s+1)))^s *
-    binomialBound ((∑i,hiddenTrial (d i))/(s+1)))
-
-/-- A purely numerical typical-metadata condition on every local collection.
-It includes unoccupied vertices and is fixed before a coefficient is chosen. -/
-def MetadataSmoothingCap (T n g : ℕ → ℕ) (s : ℕ) (η : ℝ) :
-    (b : ℕ) → WindowAssignments T n g (Option Bool) b → Prop
-  | 0, _ => True
-  | b+1, d => posteriorSmoothingCost s d.1 ≤ η ∧
-      ∀h, MetadataSmoothingCap T n g s η b (d.2.2 h)
-
-def occupiedRootHasSingleton (T : ℕ → ℕ) : (b : ℕ) →
-    (S : Finset (WindowLeaf T (b+1))) → OccupiedWindowChoice T b S → Prop
-  | 0, _, a => ∃i, a i=2 ∧ ∀j,j≠i → a j=0
-  | _+1, _, a => ∃i, a.2 i=2 ∧ ∀j,j≠i → a.2 j=0
-
-theorem occupiedRoot_singleton_factor (T : ℕ → ℕ) (s b : ℕ)
-    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S)
-    (h : occupiedRootHasSingleton T b S a) :
-    ∃c : ℝ, |c|≤1 ∧ ∃label : ℤ, ∀X,
-      occupiedRootWeight T s b S a X = c*(prouhetSeries s).coeff (label-X) := by
-  cases b with
-  | zero =>
-    obtain ⟨i,hi,hr⟩ := h
-    obtain ⟨c,hc,hfac⟩ := bottom_singleton_factor a i hi hr
-    refine ⟨c,hc,((S.equivFin.symm i).val.1.val:ℤ),fun X => ?_⟩
-    exact hfac _ _
-  | succ b =>
-    obtain ⟨i,hi,hr⟩ := h
-    obtain ⟨c,hc,hfac⟩ := internal_singleton_factor
-      (fun j => (occupiedWindowTerm T b
-        (selectedChildLeaves S ((selectedChildren S).equivFin.symm j).val) (a.1 j)).isOdd)
-      a.2 i hi hr
-    refine ⟨c,hc,(((selectedChildren S).equivFin.symm i).val.val:ℤ),fun X => ?_⟩
-    exact hfac _ _
-
-theorem occupiedRoot_singleton_average (T : ℕ → ℕ) (s b : ℕ)
-    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S)
-    (h : occupiedRootHasSingleton T b S a) {m : ℕ} (d : Fin m → Option Bool)
-    (center incoming : ℤ) :
-    |(posteriorArray d).expect (fun loc => occupiedRootWeight T s b S a
-      (center-(∑i,(bitIndex (loc i):ℤ))-incoming))| ≤ posteriorSmoothingCost s d := by
-  obtain ⟨c,hc,label,hfac⟩ := occupiedRoot_singleton_factor T s b S a h
-  simp_rw [hfac,expect_const_mul,abs_mul]
-  have he : ∀loc : Fin m → Bool,
-      label-(center-(∑i,(bitIndex (loc i):ℤ))-incoming) =
-        (label-center+incoming)+(∑i,(bitIndex (loc i):ℤ)) := by intro loc; omega
-  simp_rw [he]
-  have hb : |(posteriorArray d).expect (fun loc =>
-      (prouhetSeries s).coeff ((label-center+incoming)+(∑i,(bitIndex (loc i):ℤ))))| ≤
-        posteriorSmoothingCost s d := by
-    have hb := posterior_signed_motif_bound d s (label-center+incoming)
-    unfold posteriorArray coordinateProduct expect posteriorSmoothingCost
-    convert hb using 1
-    congr 3
-    exact Subsingleton.elim _ _
-  calc
-    _ ≤ 1*|(posteriorArray d).expect (fun loc =>
-        (prouhetSeries s).coeff ((label-center+incoming)+(∑i,(bitIndex (loc i):ℤ))))| :=
-      mul_le_mul_of_nonneg_right hc (abs_nonneg _)
-    _ ≤ _ := by simpa only [one_mul] using hb
-
-theorem abs_prod_le_coordinate {ι : Type*} [Fintype ι] (f : ι → ℝ) (i : ι)
-    (h : ∀j,j≠i → |f j|≤1) : |∏j,f j|≤|f i| := by
-  rw [Finset.abs_prod,←Finset.prod_erase_mul _ _ (Finset.mem_univ i)]
-  have he : (∏j∈Finset.univ.erase i,|f j|)≤1 :=
-    Finset.prod_le_one₀ (fun _ _ => abs_nonneg _) (fun j hj => h j (Finset.mem_erase.mp hj).1)
-  simpa only [one_mul] using mul_le_mul_of_nonneg_right he (abs_nonneg (f i))
-
-/-- Literal singleton cancellation for the complete posterior bit tree.
-No successful-fit event is conditioned on, and the common parent-global
-jitter is retained until after the child factors have averaged. -/
-theorem sourceRoleAverage_singleton (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ)
-    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S)
-    (η : ℝ) (hcap : MetadataSmoothingCap T n g s η (b+1) d)
-    (hsing : ¬(occupiedWindowTerm T b S a).noSingleton) :
-    |sourceRoleAverage T n g s center globalShift b d incoming S a|≤η := by
-  induction b generalizing incoming with
-  | zero =>
-    have hh : Odd (signedRoleCount a) ∧ activeRoleCount a=1 := by
-      by_contra! h
-      apply hsing
-      exact ⟨fun ho he => h ho he,fun _ => trivial⟩
-    have hr : occupiedRootHasSingleton T 0 S a :=
-      (odd_roles_singleton_or_branch a hh.1).resolve_right (by omega)
-    rw [sourceRoleAverage_zero]
-    exact (occupiedRoot_singleton_average T s 0 S a hr d.1 _ _).trans hcap.1
-  | succ b ih =>
-    rw [sourceRoleAverage_succ,abs_mul]
-    have hroot : |(posteriorArray d.1).expect (fun loc =>
-        occupiedRootWeight T s (b+1) S a
-          (center (b+1)-(∑i,(bitIndex (loc i):ℤ))-incoming))|≤1 :=
-      FiniteLaw.abs_expect_le_const _ _ 1 (fun _ => occupiedRootWeight_abs_le_one T s (b+1) S a _)
-    have hglob : |(posteriorArray d.2.1).expect (fun glob =>
-        ∏i : Fin (selectedChildren S).card,
-          sourceRoleAverage T n g s center globalShift b
-            (d.2.2 ((selectedChildren S).equivFin.symm i).val)
-            (globalShift (b+1) glob ((selectedChildren S).equivFin.symm i).val)
-            (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val) (a.1 i))|≤1 := by
-      apply FiniteLaw.abs_expect_le_const
-      intro glob
-      rw [Finset.abs_prod]
-      exact Finset.prod_le_one₀ (fun _ _ => abs_nonneg _)
-        (fun i _ => sourceRoleAverage_abs_le_one T n g s center globalShift b _ _ _ _)
-    by_cases hr : occupiedRootHasSingleton T (b+1) S a
-    · have hh := (occupiedRoot_singleton_average T s (b+1) S a hr d.1
-        (center (b+1)) incoming).trans hcap.1
-      exact (mul_le_mul_of_nonneg_left hglob (abs_nonneg _)).trans (by simpa using hh)
-    · have hex : ∃i, ¬(occupiedWindowTerm T b
-          (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val) (a.1 i)).noSingleton := by
-        by_contra! h
-        apply hsing
-        refine ⟨?_,h⟩
-        intro ho he
-        have hh := odd_roles_singleton_or_branch a.2 ho
-        exact hr (hh.resolve_right (by omega))
-      obtain ⟨i,hi⟩ := hex
-      have hsmall : |(posteriorArray d.2.1).expect (fun glob =>
-          ∏j : Fin (selectedChildren S).card,
-            sourceRoleAverage T n g s center globalShift b
-              (d.2.2 ((selectedChildren S).equivFin.symm j).val)
-              (globalShift (b+1) glob ((selectedChildren S).equivFin.symm j).val)
-              (selectedChildLeaves S ((selectedChildren S).equivFin.symm j).val) (a.1 j))|≤η := by
-        apply FiniteLaw.abs_expect_le_const
-        intro glob
-        apply (abs_prod_le_coordinate _ i (fun j _ =>
-          sourceRoleAverage_abs_le_one T n g s center globalShift b _ _ _ _)).trans
-        exact ih _ _ _ _ (hcap.2 _) hi
-      exact (mul_le_mul_of_nonneg_right hroot (abs_nonneg _)).trans (by simpa using hsmall)
-
-end TraceReconstruction
-
-end
-end
-end
-end
-end
-end
-end
-end
-
-end OAI
+import OAI.Probability.TraceReconstruction.InitialOverlap
+
+namespace OAI
+
+/-!
+# LeafOracle
+
+Recursive leaf oracles, sparse high tails, occupied shapes, posterior smoothing, source bit trees, and singleton cancellation.
+-/
+
+open scoped ENNReal
+noncomputable section
+
+open Filter
+open scoped Topology
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+open FiniteLaw
+
+/-- Complete window-tree leaf indices, with counts numbered from the bottom.
+The tree is fixed before any occupied subset or orientation is selected. -/
+def WindowLeaf (T : ℕ → ℕ) : ℕ → Type
+  | 0 => Fin 1
+  | b+1 => Fin (T b) × WindowLeaf T b
+
+noncomputable instance windowLeafFintype (T : ℕ → ℕ) : (b : ℕ) → Fintype (WindowLeaf T b)
+  | 0 => inferInstanceAs (Fintype (Fin 1))
+  | b+1 => by
+    letI := windowLeafFintype T b
+    exact inferInstanceAs (Fintype (Fin (T b) × WindowLeaf T b))
+
+/-- A motif's actual outgoing type, allowing every integer displacement. -/
+def motifChildSign (s : ℕ) (X : ℤ) (h : ℕ) (τ : ℝ) : ℝ :=
+  1-motifIndicator s ((h:ℤ)-X)+(prouhetSeries s).coeff ((h:ℤ)-X)*τ
+
+theorem motifChildSign_isSign (s : ℕ) (X : ℤ) (h : ℕ) {τ : ℝ} (hτ : IsTypeSign τ) :
+    IsTypeSign (motifChildSign s X h τ) :=
+  motif_child_sign (prouhet_motif_value s ((h:ℤ)-X)) hτ
+
+theorem motifChildSign_negative_active (s : ℕ) (X : ℤ) (h : ℕ) (τ : ℝ)
+    (ht : motifChildSign s X h τ= -1) : 0 ≤ (h:ℤ)-X ∧ (h:ℤ)-X<(2^s:ℕ) := by
+  by_contra hn
+  have hz := (prouhetSeries_coeff_spec s).1 ((h:ℤ)-X) (by omega)
+  simp only [motifChildSign,motifIndicator,ite_eq_right hn,hz,zero_mul,sub_zero,add_zero] at ht
+  norm_num at ht
+
+/-- Literal top-down type recursion in the source's unconditioned surrogate.
+`X path` is the root anchor or `w_j-Y_path`. Dependence of different paths
+through the same global jitter is retained by this single full-tree map. -/
+def windowLeafSign (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ) :
+    (b : ℕ) → List ℕ → ℝ → WindowLeaf T b → ℝ
+  | 0, _, τ, _ => τ
+  | b+1, path, τ, v => windowLeafSign T s X b (path++[v.1.val])
+      (motifChildSign s (X path) v.1.val τ) v.2
+
+theorem windowLeafSign_isSign (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
+    (b : ℕ) (path : List ℕ) {τ : ℝ} (hτ : IsTypeSign τ) (v : WindowLeaf T b) :
+    IsTypeSign (windowLeafSign T s X b path τ v) := by
+  induction b generalizing path τ with
+  | zero => exact hτ
+  | succ b ih => exact ih _ (motifChildSign_isSign s _ _ hτ) v.2
+
+def windowLeafType (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
+    (b : ℕ) (path : List ℕ) (τ : ℝ) (v : WindowLeaf T b) : Bool :=
+  decide (windowLeafSign T s X b path τ v= -1)
+
+theorem windowLeafType_weight (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
+    (b : ℕ) (path : List ℕ) {τ : ℝ} (hτ : IsTypeSign τ) (v : WindowLeaf T b) :
+    (if windowLeafType T s X b path τ v then (1:ℝ) else 0)=
+      (1-windowLeafSign T s X b path τ v)/2 := by
+  rcases windowLeafSign_isSign T s X b path hτ v with hp | hm
+  · norm_num [windowLeafType,hp]
+  · norm_num [windowLeafType,hm]
+
+/-- A finite indexing of all leaves, not just the active or fitted leaves. -/
+def windowFinType (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
+    (b : ℕ) (τ : ℝ) (i : Fin (Fintype.card (WindowLeaf T b))) : Bool :=
+  windowLeafType T s X b [] τ ((Fintype.equivFin (WindowLeaf T b)).symm i)
+
+/-- The actual mixed leaf oracle at arbitrary exact metadata. The one latent
+law governs the whole map X; no independent-anchor law is substituted. -/
+def recursiveLeafLaw {Ω α : Type*} [Fintype Ω] [Fintype α]
+    (R : FiniteLaw Ω) (X : Ω → List ℕ → ℤ) (T : ℕ → ℕ) (s b : ℕ)
+    (P Q : FiniteLaw α) (τ : ℝ) :
+    FiniteLaw (Fin (Fintype.card (WindowLeaf T b)) → α) :=
+  R.mixture (fun ω => coordinateProduct (fun i => if windowFinType T s (X ω) b τ i then Q else P))
+
+theorem recursiveLeafLaw_density {Ω α : Type*} [Fintype Ω] [Fintype α]
+    (R : FiniteLaw Ω) (X : Ω → List ℕ → ℤ) (T : ℕ → ℕ) (s b : ℕ)
+    (P Q : FiniteLaw α) (Z : α → ℝ) (hZ : ∀a,Q.mass a=P.mass a*(1+Z a))
+    (τ : ℝ) (x : Fin (Fintype.card (WindowLeaf T b)) → α) :
+    (recursiveLeafLaw R X T s b P Q τ).mass x =
+      (P.product (Fintype.card (WindowLeaf T b))).mass x *
+        multilinear (averagedCoefficient R (fun ω => windowFinType T s (X ω) b τ)) Z x := by
+  rw [recursiveLeafLaw,leaf_oracle_density P Q Z hZ,averagedDensity_expansion]
+  rfl
+
+/-- Pointwise recursive coefficient before averaging over the actual bits. -/
+theorem recursive_selectionCoefficient {Ω : Type*} [Fintype Ω]
+    (X : Ω → List ℕ → ℤ) (T : ℕ → ℕ) (s b : ℕ) (ω : Ω)
+    {τ : ℝ} (hτ : IsTypeSign τ) (σ : Fin (Fintype.card (WindowLeaf T b)) → Bool) :
+    selectionCoefficient (windowFinType T s (X ω) b τ) σ =
+      ∏i, if σ i then
+        (1-windowLeafSign T s (X ω) b [] τ ((Fintype.equivFin (WindowLeaf T b)).symm i))/2 else 1 := by
+  unfold selectionCoefficient
+  apply Finset.prod_congr rfl
+  intro i _
+  cases σ i
+  · rfl
+  · exact windowLeafType_weight T s (X ω) b [] hτ _
+
+theorem motif_negative_count (s N : ℕ) (X : ℤ) (τ : ℝ) :
+    (Finset.univ.filter (fun i : Fin N => motifChildSign s X i.val τ= -1)).card ≤ 2^s := by
+  have H := Finset.card_le_card_of_injOn
+    (s:=Finset.univ.filter (fun i : Fin N => motifChildSign s X i.val τ= -1))
+    (t:=Finset.range (2^s)) (fun i : Fin N => ((i.val:ℤ)-X).toNat) ?_ ?_
+  · simpa using H
+  · intro i hi
+    have h := motifChildSign_negative_active s X i.val τ (Finset.mem_filter.mp hi).2
+    apply Finset.mem_range.mpr
+    dsimp only
+    omega
+  · intro i hi j hj he
+    have h := motifChildSign_negative_active s X i.val τ (Finset.mem_filter.mp hi).2
+    have h' := motifChildSign_negative_active s X j.val τ (Finset.mem_filter.mp hj).2
+    apply Fin.ext
+    dsimp only at he
+    omega
+
+/-- Exact number of minus leaves in a fixed realization, without a fit condition. -/
+def windowMinusCount (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
+    (b : ℕ) (path : List ℕ) (τ : ℝ) : ℕ :=
+  ∑v : WindowLeaf T b, if windowLeafSign T s X b path τ v= -1 then 1 else 0
+
+theorem windowMinusCount_succ (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
+    (b : ℕ) (path : List ℕ) (τ : ℝ) :
+    windowMinusCount T s X (b+1) path τ =
+      ∑i : Fin (T b), windowMinusCount T s X b (path++[i.val])
+        (motifChildSign s (X path) i.val τ) := by
+  unfold windowMinusCount
+  change (∑v : Fin (T b) × WindowLeaf T b, _) = _
+  rw [Fintype.sum_prod_type]
+  rfl
+
+/-- Source the block layout high-degree budget, bottom motif length times all
+upper window counts. It holds for every latent realization, typical or not. -/
+theorem windowMinusCount_bound (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
+    (b : ℕ) (path : List ℕ) (τ : ℝ) :
+    windowMinusCount T s X (b+1) path τ ≤
+      2^s * ∏j∈Finset.range b, T (j+1) := by
+  induction b generalizing path τ with
+  | zero =>
+    rw [windowMinusCount_succ]
+    change (∑i : Fin (T 0), ∑v : Fin 1,
+      if motifChildSign s (X path) i.val τ= -1 then (1:ℕ) else 0) ≤ _
+    simp only [Fin.sum_univ_one,Finset.range_zero,Finset.prod_empty,mul_one]
+    simpa only [Finset.card_filter] using motif_negative_count s (T 0) (X path) τ
+  | succ b ih =>
+    rw [windowMinusCount_succ]
+    calc
+      _ ≤ ∑i : Fin (T (b+1)), 2^s*∏j∈Finset.range b,T (j+1) :=
+        Finset.sum_le_sum (fun _ _ => ih _ _)
+      _ = _ := by simp [Finset.prod_range_succ]; ring
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+open FiniteLaw
+
+def boolSupport {n : ℕ} (θ : Fin n → Bool) : Finset (Fin n) :=
+  Finset.univ.filter (fun i => θ i=true)
+
+theorem boolSupport_injective {n : ℕ} : Function.Injective (boolSupport (n:=n)) := by
+  intro θ φ h
+  funext i
+  have hi : (θ i=true) ↔ (φ i=true) := by
+    simpa [boolSupport] using Finset.ext_iff.mp h i
+  cases hθ : θ i <;> cases hφ : φ i <;> simp_all
+
+theorem selectionCoefficient_eq_support {n : ℕ} (θ σ : Fin n → Bool) :
+    selectionCoefficient θ σ = if boolSupport σ ⊆ boolSupport θ then 1 else 0 := by
+  by_cases h : boolSupport σ ⊆ boolSupport θ
+  · rw [ite_eq_left h]
+    apply Finset.prod_eq_one
+    intro i _
+    cases hs : σ i
+    · simp
+    · have ht : θ i=true := by simpa [boolSupport] using h (by simp [boolSupport,hs])
+      simp [ht]
+  · rw [ite_eq_right h]
+    obtain ⟨i,hi,hni⟩ := Finset.not_subset.mp h
+    have hs : σ i=true := by simpa [boolSupport] using hi
+    have ht : θ i=false := by simpa [boolSupport] using hni
+    apply Finset.prod_eq_zero (Finset.mem_univ i)
+    simp [hs,ht]
+
+/-- Uniform cardinality bound for the actual selected monomials of a sparse
+0/1 coefficient vector; mixing does not add a squared-cardinality loss. -/
+theorem selection_degree_sum_le {n : ℕ} (θ : Fin n → Bool) (r : ℕ) :
+    (∑σ : Fin n → Bool, if monomialDegree σ=r then selectionCoefficient θ σ else 0) ≤
+      ((monomialDegree θ).choose r:ℝ) := by
+  have hc : (Finset.univ.filter (fun σ : Fin n → Bool =>
+      monomialDegree σ=r ∧ boolSupport σ ⊆ boolSupport θ)).card ≤
+        ((boolSupport θ).powersetCard r).card := by
+    apply Finset.card_le_card_of_injOn boolSupport
+    · intro σ hσ
+      rcases (Finset.mem_filter.mp hσ).2 with ⟨hd,hs⟩
+      exact Finset.mem_powersetCard.mpr ⟨hs,hd⟩
+    · exact boolSupport_injective.injOn
+  rw [Finset.card_powersetCard] at hc
+  have he : (∑σ : Fin n → Bool, if monomialDegree σ=r then selectionCoefficient θ σ else 0) =
+      ((Finset.univ.filter (fun σ : Fin n → Bool =>
+        monomialDegree σ=r ∧ boolSupport σ ⊆ boolSupport θ)).card:ℝ) := by
+    simp_rw [selectionCoefficient_eq_support,←ite_and]
+    exact Finset.sum_boole _ _
+  rw [he]
+  exact_mod_cast hc
+
+theorem averaged_sparse_degree {Ω : Type*} [Fintype Ω] {n A : ℕ}
+    (R : FiniteLaw Ω) (θ : Ω → Fin n → Bool)
+    (hA : ∀ω, monomialDegree (θ ω) ≤ A) (r : ℕ) :
+    (∑σ : Fin n → Bool, if monomialDegree σ=r then averagedCoefficient R θ σ^2 else 0) ≤
+      (A.choose r:ℝ) := by
+  calc
+    _ ≤ ∑σ : Fin n → Bool, if monomialDegree σ=r then averagedCoefficient R θ σ else 0 := by
+      apply Finset.sum_le_sum
+      intro σ _
+      split_ifs
+      · have hc := coefficient_nonneg R θ σ
+        have h1 := coefficient_le_one R θ σ
+        nlinarith [mul_nonneg hc (sub_nonneg.mpr h1)]
+      · rfl
+    _ = ∑ω, R.mass ω * ∑σ : Fin n → Bool,
+        if monomialDegree σ=r then selectionCoefficient (θ ω) σ else 0 := by
+      unfold averagedCoefficient FiniteLaw.expect
+      simp_rw [Finset.mul_sum]
+      rw [Finset.sum_comm]
+      apply Finset.sum_congr rfl
+      intro σ _
+      by_cases h : monomialDegree σ=r <;> simp [h]
+    _ ≤ ∑ω, R.mass ω * (A.choose r:ℝ) := by
+      apply Finset.sum_le_sum
+      intro ω _
+      apply mul_le_mul_of_nonneg_left _ (R.nonneg ω)
+      exact (selection_degree_sum_le (θ ω) r).trans (by
+        exact_mod_cast Nat.choose_le_choose r (hA ω))
+    _ = _ := by rw [←Finset.sum_mul,R.total,one_mul]
+
+/-- Genuine sparse complete-tree leaf count in the exact Fin indexing used
+by the independent-product reference law. -/
+theorem windowFinType_degree (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
+    (b : ℕ) (τ : ℝ) :
+    monomialDegree (windowFinType T s X b τ) = windowMinusCount T s X b [] τ := by
+  have h := Equiv.sum_comp (Fintype.equivFin (WindowLeaf T b)).symm
+    (fun v => if windowLeafSign T s X b [] τ v= -1 then (1:ℕ) else 0)
+  simpa only [monomialDegree,Finset.card_filter,windowFinType,windowLeafType,
+    decide_eq_true_eq,windowMinusCount] using h
+
+theorem recursive_degree_bound {Ω : Type*} [Fintype Ω]
+    (R : FiniteLaw Ω) (X : Ω → List ℕ → ℤ) (T : ℕ → ℕ) (s b r : ℕ) (τ : ℝ) :
+    (∑σ, if monomialDegree σ=r then
+      averagedCoefficient R (fun ω => windowFinType T s (X ω) (b+1) τ) σ^2 else 0) ≤
+        ((2^s*∏j∈Finset.range b,T (j+1)).choose r:ℝ) := by
+  apply averaged_sparse_degree
+  intro ω
+  rw [windowFinType_degree]
+  exact windowMinusCount_bound T s (X ω) b [] τ
+
+def degreeTail {n : ℕ} (A : (Fin n → Bool) → ℝ) (k : ℕ) (σ : Fin n → Bool) : ℝ :=
+  if k ≤ monomialDegree σ then A σ else 0
+
+theorem degreeTail_degree {n r k : ℕ} (A : (Fin n → Bool) → ℝ) :
+    (∑σ, if monomialDegree σ=r then degreeTail A k σ^2 else 0) =
+      if k ≤ r then ∑σ, if monomialDegree σ=r then A σ^2 else 0 else 0 := by
+  by_cases h : k ≤ r
+  · rw [ite_eq_left h]
+    apply Finset.sum_congr rfl
+    intro σ _
+    by_cases he : monomialDegree σ=r <;> simp [degreeTail,he,h]
+  · rw [ite_eq_right h]
+    apply Finset.sum_eq_zero
+    intro σ _
+    by_cases he : monomialDegree σ=r <;> simp [degreeTail,he,h]
+
+/-- Geometric high-degree energy of an actual mixed sparse leaf law. -/
+theorem averaged_sparse_tail_energy {Ω : Type*} [Fintype Ω] {n A₀ : ℕ}
+    (R : FiniteLaw Ω) (θ : Ω → Fin n → Bool)
+    (hA : ∀ω, monomialDegree (θ ω) ≤ A₀) (k : ℕ) {w : ℝ}
+    (hw : 0 ≤ w) (ha : w^2*A₀ < 1) :
+    multilinearEnergy (degreeTail (averagedCoefficient R θ) k) w ≤
+      (w^2*A₀)^k/(1-w^2*A₀) := by
+  have H := multilinearEnergy_low_high (R:=k)
+    (degreeTail (averagedCoefficient R θ) k) w (w^2*A₀) 1 hw (by positivity) ha
+    (by norm_num) (fun _ => 0) ?_ ?_ (fun _ => le_rfl)
+  · simpa using H
+  · intro r hr
+    rw [degreeTail_degree,ite_eq_right (by omega)]
+  · intro r hr
+    rw [degreeTail_degree,ite_eq_left hr,one_mul]
+    have hc : (A₀.choose r:ℝ) ≤ (A₀:ℝ)^r := by exact_mod_cast Nat.choose_le_pow A₀ r
+    calc
+      _ ≤ w^(2*r)*(A₀:ℝ)^r := mul_le_mul_of_nonneg_left
+        ((averaged_sparse_degree R θ hA r).trans hc) (by positivity)
+      _ = _ := by rw [pow_mul,mul_pow]
+
+/-- Actual recursive-oracle high tail, including arbitrary correlated latent
+bits and displacements. This supplies the block layout's unconditioned high-degree
+input; low degrees still require the signed occupied-tree cancellation. -/
+theorem recursive_high_moment {Ω α : Type*} [Fintype Ω] [Fintype α]
+    (R : FiniteLaw Ω) (X : Ω → List ℕ → ℤ) (T : ℕ → ℕ) (s b r₀ k : ℕ) (τ : ℝ)
+    (P : FiniteLaw α) (Z : α → ℝ) (hk : 1 ≤ k) (hcenter : P.expect Z=0)
+    {L : ℝ} (hL : 0 < L) (hMoment : P.expect (fun a => |Z a|^(2*k)) ≤ L^(2*k))
+    (ha : ((6*(2*k:ℕ):ℝ)*L)^2 * (2^s*∏j∈Finset.range b,T (j+1):ℕ) < 1) :
+    (P.product (Fintype.card (WindowLeaf T (b+1)))).evenMomentSq
+      (multilinear (degreeTail (averagedCoefficient R
+        (fun ω => windowFinType T s (X ω) (b+1) τ)) r₀) Z) k ≤
+      (((6*(2*k:ℕ):ℝ)*L)^2*(2^s*∏j∈Finset.range b,T (j+1):ℕ))^r₀/
+        (1-((6*(2*k:ℕ):ℝ)*L)^2*(2^s*∏j∈Finset.range b,T (j+1):ℕ)) := by
+  apply (multilinear_moment_sq_scaled P Z k hk hcenter hL hMoment _ _).trans
+  apply averaged_sparse_tail_energy R _ _ r₀ (by positivity) ha
+  intro ω
+  rw [windowFinType_degree]
+  exact windowMinusCount_bound T s (X ω) b [] τ
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+open FiniteLaw
+
+def selectedChildren {α β : Type*} [DecidableEq α] (S : Finset (α × β)) : Finset α :=
+  S.image Prod.fst
+
+def selectedChildLeaves {α β : Type*} [DecidableEq α] [DecidableEq β]
+    (S : Finset (α × β)) (h : α) : Finset β :=
+  (S.filter (fun v => v.1=h)).image Prod.snd
+
+@[to_additive]
+theorem selected_prod_children {α β M : Type*} [DecidableEq α] [DecidableEq β]
+    [CommMonoid M] (S : Finset (α × β)) (f : α × β → M) :
+    (∏h∈selectedChildren S, ∏v∈selectedChildLeaves S h, f (h,v)) = ∏v∈S, f v := by
+  have he (h : α) : (∏v∈selectedChildLeaves S h, f (h,v)) =
+      ∏v∈S.filter (fun v => v.1=h), f v := by
+    unfold selectedChildLeaves
+    rw [Finset.prod_image]
+    · apply Finset.prod_congr rfl
+      intro v hv
+      rw [←(Finset.mem_filter.mp hv).2]
+    · intro v hv w hw h
+      apply Prod.ext
+      · exact (Finset.mem_filter.mp hv).2.trans (Finset.mem_filter.mp hw).2.symm
+      · exact h
+  simp_rw [he]
+  exact Finset.prod_fiberwise_of_maps_to (fun v hv => Finset.mem_image.mpr ⟨v,hv,rfl⟩) f
+
+@[to_additive]
+theorem selected_prod_index {α M : Type*} [DecidableEq α] [CommMonoid M]
+    (S : Finset α) (f : α → M) :
+    (∏i : Fin S.card, f (S.equivFin.symm i).val) = ∏v∈S, f v := by
+  have h := Equiv.prod_comp S.equivFin.symm (fun v : S => f v.val)
+  simpa only [Finset.prod_coe_sort] using h
+
+/-- Occupied shape of an actual subset of a complete window tree. Only
+occupied children are enumerated, so its leaves are exactly the selected
+monomial variables, not the other complete-tree leaves. -/
+def occupiedWindowShape (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ) :
+    (b : ℕ) → (path : List ℕ) → Finset (WindowLeaf T (b+1)) → CoefficientShape b
+  | 0, path, S => .bottom S.card
+      (fun i => motifIndicator s (((S.equivFin.symm i).val.1.val:ℤ)-X path))
+      (fun i => (prouhetSeries s).coeff (((S.equivFin.symm i).val.1.val:ℤ)-X path))
+  | b+1, path, S => .node (selectedChildren S).card
+      (fun i => motifIndicator s ((((selectedChildren S).equivFin.symm i).val.val:ℤ)-X path))
+      (fun i => (prouhetSeries s).coeff ((((selectedChildren S).equivFin.symm i).val.val:ℤ)-X path))
+      (fun i => occupiedWindowShape T s X b
+        (path++[((selectedChildren S).equivFin.symm i).val.val])
+        (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val))
+
+theorem occupiedWindowShape_admissible (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
+    (b : ℕ) (path : List ℕ) (S : Finset (WindowLeaf T (b+1))) :
+    (occupiedWindowShape T s X b path S).Admissible := by
+  induction b generalizing path with
+  | zero => exact fun _ => prouhet_motif_value s _
+  | succ b ih => exact ⟨fun _ => prouhet_motif_value s _, fun _ => ih _ _⟩
+
+/-- The literal coefficient product equals the evaluation of its occupied
+shape; this is before any averaging or parity cancellation. -/
+theorem occupiedWindowShape_eval (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
+    (b : ℕ) (path : List ℕ) (S : Finset (WindowLeaf T (b+1))) (τ : ℝ) :
+    (occupiedWindowShape T s X b path S).eval τ =
+      ∏v∈S, (1-windowLeafSign T s X (b+1) path τ v)/2 := by
+  induction b generalizing path τ with
+  | zero =>
+    change (∏i : Fin S.card, _) = _
+    refine (selected_prod_index S (fun v : Fin (T 0) × Fin 1 =>
+      (motifIndicator s ((v.1.val:ℤ)-X path) -
+        (prouhetSeries s).coeff ((v.1.val:ℤ)-X path)*τ)/2)).trans ?_
+    apply Finset.prod_congr rfl
+    intro v hv
+    simp only [windowLeafSign,motifChildSign]
+    ring
+  | succ b ih =>
+    change (∏i : Fin (selectedChildren S).card, _) = _
+    simp_rw [ih]
+    refine (selected_prod_index (selectedChildren S) (fun h : Fin (T (b+1)) =>
+      ∏v∈selectedChildLeaves S h,
+        (1-windowLeafSign T s X (b+1) (path++[h.val])
+          (motifChildSign s (X path) h.val τ) v)/2)).trans ?_
+    exact selected_prod_children S (fun v =>
+      (1-windowLeafSign T s X (b+1+1) path τ v)/2)
+
+theorem selected_children_card {α β : Type*} [DecidableEq α] [DecidableEq β]
+    (S : Finset (α × β)) :
+    (∑h∈selectedChildren S, (selectedChildLeaves S h).card) = S.card := by
+  simpa only [Finset.sum_const,smul_eq_mul,mul_one] using
+    selected_sum_children S (fun _ => (1:ℕ))
+
+theorem selectedChildLeaves_nonempty {α β : Type*} [DecidableEq α] [DecidableEq β]
+    (S : Finset (α × β)) (h : α) (hh : h∈selectedChildren S) :
+    (selectedChildLeaves S h).Nonempty := by
+  obtain ⟨v,hv,rfl⟩ := Finset.mem_image.mp hh
+  exact ⟨v.2,Finset.mem_image.mpr ⟨v,Finset.mem_filter.mpr ⟨hv,rfl⟩,rfl⟩⟩
+
+theorem occupiedWindowShape_leafCount (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
+    (b : ℕ) (path : List ℕ) (S : Finset (WindowLeaf T (b+1)))
+    (a : (occupiedWindowShape T s X b path S).Choice) :
+    ((occupiedWindowShape T s X b path S).termTree a).leafCount = S.card := by
+  induction b generalizing path with
+  | zero => simp [occupiedWindowShape,CoefficientShape.termTree,OccupiedParityTree.leafCount]
+  | succ b ih =>
+    change (∑i : Fin (selectedChildren S).card, _) = _
+    simp_rw [ih]
+    exact (selected_sum_index (selectedChildren S) (fun h =>
+      (selectedChildLeaves S h).card)).trans (selected_children_card S)
+
+theorem occupiedWindowShape_occupied (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
+    (b : ℕ) (path : List ℕ) (S : Finset (WindowLeaf T (b+1))) (hS : S.Nonempty)
+    (a : (occupiedWindowShape T s X b path S).Choice) :
+    ((occupiedWindowShape T s X b path S).termTree a).occupied := by
+  induction b generalizing path with
+  | zero => exact ⟨hS.card_pos,fun _ => trivial⟩
+  | succ b ih =>
+    refine ⟨(Finset.image_nonempty.mpr hS).card_pos,fun i => ?_⟩
+    exact ih _ _ (selectedChildLeaves_nonempty S _
+      ((selectedChildren S).equivFin.symm i).property) (a.1 i)
+
+/-- Forced-leaf count for a term of an actual selected monomial. -/
+theorem occupiedWindowShape_forced_count (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
+    (b : ℕ) (path : List ℕ) (S : Finset (WindowLeaf T (b+1)))
+    (a : (occupiedWindowShape T s X b path S).Choice)
+    (hw : (occupiedWindowShape T s X b path S).weight a ≠ 0)
+    (ho : ((occupiedWindowShape T s X b path S).termTree a).isOdd)
+    (hn : ((occupiedWindowShape T s X b path S).termTree a).noSingleton) :
+    2^(b+1) ≤ S.card := by
+  simpa only [occupiedWindowShape_leafCount] using
+    (occupiedWindowShape T s X b path S).surviving_leaves a hw ho hn
+
+/-- The role choices depend only on the selected subset, not on any hidden
+orientation. This common index type permits averaging before norm estimates. -/
+def OccupiedWindowChoice (T : ℕ → ℕ) :
+    (b : ℕ) → Finset (WindowLeaf T (b+1)) → Type
+  | 0, S => Fin S.card → Fin 3
+  | b+1, S =>
+      ((i : Fin (selectedChildren S).card) → OccupiedWindowChoice T b
+        (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val)) ×
+      (Fin (selectedChildren S).card → Fin 3)
+
+noncomputable instance occupiedWindowChoiceFintype (T : ℕ → ℕ) :
+    (b : ℕ) → (S : Finset (WindowLeaf T (b+1))) → Fintype (OccupiedWindowChoice T b S)
+  | 0, S => by unfold OccupiedWindowChoice; infer_instance
+  | b+1, S => by
+      letI := fun i : Fin (selectedChildren S).card =>
+        occupiedWindowChoiceFintype T b
+          (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val)
+      unfold OccupiedWindowChoice
+      infer_instance
+
+noncomputable def occupiedWindowChoiceEquiv (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ) :
+    (b : ℕ) → (path : List ℕ) → (S : Finset (WindowLeaf T (b+1))) →
+      OccupiedWindowChoice T b S ≃ (occupiedWindowShape T s X b path S).Choice
+  | 0, _, _ => Equiv.refl _
+  | b+1, path, S => Equiv.prodCongr
+      (Equiv.piCongrRight fun i => occupiedWindowChoiceEquiv T s X b
+        (path++[((selectedChildren S).equivFin.symm i).val.val])
+        (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val))
+      (Equiv.refl _)
+
+/-- The parity tree of a role choice is independent of all displacement bits. -/
+noncomputable def occupiedWindowTerm (T : ℕ → ℕ) :
+    (b : ℕ) → (S : Finset (WindowLeaf T (b+1))) →
+      OccupiedWindowChoice T b S → OccupiedParityTree (b+1)
+  | 0, S, a => .node S.card (fun _ => .leaf) a
+  | b+1, S, a => .node (selectedChildren S).card
+      (fun i => occupiedWindowTerm T b
+        (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val) (a.1 i)) a.2
+
+ theorem occupiedWindowTerm_eq (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
+    (b : ℕ) (path : List ℕ) (S : Finset (WindowLeaf T (b+1)))
+    (a : OccupiedWindowChoice T b S) :
+    (occupiedWindowShape T s X b path S).termTree
+      (occupiedWindowChoiceEquiv T s X b path S a) = occupiedWindowTerm T b S a := by
+  induction b generalizing path with
+  | zero => rfl
+  | succ b ih =>
+    change OccupiedParityTree.node _ (fun i => _) _ = OccupiedParityTree.node _ _ _
+    congr 1
+    funext i
+    exact ih _ _ _
+
+noncomputable def occupiedWindowWeight (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
+    (b : ℕ) (path : List ℕ) (S : Finset (WindowLeaf T (b+1)))
+    (a : OccupiedWindowChoice T b S) : ℝ :=
+  (occupiedWindowShape T s X b path S).weight
+    (occupiedWindowChoiceEquiv T s X b path S a)
+
+/-- Exact source parity formula for the actual selected leaves, using a
+single role index independent of the latent map. No fit conditioning. -/
+theorem occupiedWindow_difference (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
+    (b : ℕ) (path : List ℕ) (S : Finset (WindowLeaf T (b+1))) :
+    (∏v∈S, (1-windowLeafSign T s X (b+1) path 1 v)/2) -
+      (∏v∈S, (1-windowLeafSign T s X (b+1) path (-1) v)/2) =
+      2*∑a : OccupiedWindowChoice T b S,
+        if (occupiedWindowTerm T b S a).isOdd then occupiedWindowWeight T s X b path S a else 0 := by
+  rw [←occupiedWindowShape_eval,←occupiedWindowShape_eval,
+    CoefficientShape.difference_expansion _ (occupiedWindowShape_admissible T s X b path S)]
+  congr 1
+  rw [←(occupiedWindowChoiceEquiv T s X b path S).sum_comp]
+  apply Finset.sum_congr rfl
+  intro a _
+  rw [occupiedWindowTerm_eq]
+  rfl
+
+/-- Averaging the complete exact expansion keeps all correlations of the
+common hidden map. The indicator of an odd root is deterministic. -/
+theorem occupiedWindow_averaged_difference {Ω : Type*} [Fintype Ω]
+    (R : FiniteLaw Ω) (T : ℕ → ℕ) (s : ℕ) (X : Ω → List ℕ → ℤ)
+    (b : ℕ) (path : List ℕ) (S : Finset (WindowLeaf T (b+1))) :
+    R.expect (fun ω => ∏v∈S, (1-windowLeafSign T s (X ω) (b+1) path 1 v)/2) -
+      R.expect (fun ω => ∏v∈S, (1-windowLeafSign T s (X ω) (b+1) path (-1) v)/2) =
+      2*∑a : OccupiedWindowChoice T b S,
+        if (occupiedWindowTerm T b S a).isOdd then
+          R.expect (fun ω => occupiedWindowWeight T s (X ω) b path S a) else 0 := by
+  unfold expect
+  rw [←Finset.sum_sub_distrib]
+  simp_rw [←mul_sub,occupiedWindow_difference,mul_left_comm (R.mass _) 2,
+    Finset.mul_sum]
+  rw [Finset.sum_comm]
+  simp_rw [←Finset.mul_sum]
+  congr 1
+  apply Finset.sum_congr rfl
+  intro a _
+  by_cases h : (occupiedWindowTerm T b S a).isOdd <;> simp [h]
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+open FiniteLaw
+
+/-- Arbitrary functions of the actual posterior count have the translated
+fair-binomial law, not only its generating polynomial. -/
+theorem posterior_count_average {ι : Type*} [Fintype ι]
+    (d : ι → Option Bool) (F : ℤ → ℝ) :
+    (∑ b : ι → Bool, (∏i, (flagPosterior (d i)).mass (b i)) *
+      F (∑i, (bitIndex (b i):ℤ))) =
+    ∑k∈Finset.range ((∑i, hiddenTrial (d i))+1),
+      fairBinomial (∑i, hiddenTrial (d i)) k * F ((∑i,(visibleOnes (d i):ℤ))+k) := by
+  have H := congrArg (fun p : IntSeries => Finsupp.linearCombination ℝ F p.coeff)
+    (posterior_count_series d)
+  rw [fairSeries_sum,Finset.mul_sum] at H
+  simp only [AddMonoidAlgebra.single_mul_single,one_mul,
+    AddMonoidAlgebra.coeff_sum,AddMonoidAlgebra.coeff_single,map_sum,
+    Finsupp.linearCombination_single,smul_eq_mul] at H
+  exact H
+
+theorem fairBinomial_reflect (n k : ℕ) (hk : k ≤ n) :
+    fairBinomial n (n-k) = fairBinomial n k := by
+  simp only [fairBinomial,Nat.choose_symm hk]
+
+/-- Reflection of the remaining hidden count. The known visible contribution
+is kept, and no positivity assumption on the metadata mass is needed. -/
+theorem posterior_count_average_reflected {ι : Type*} [Fintype ι]
+    (d : ι → Option Bool) (F : ℤ → ℝ) :
+    (∑ b : ι → Bool, (∏i, (flagPosterior (d i)).mass (b i)) *
+      F (∑i, (bitIndex (b i):ℤ))) =
+    ∑k∈Finset.range ((∑i, hiddenTrial (d i))+1),
+      fairBinomial (∑i, hiddenTrial (d i)) k *
+        F ((∑i,(visibleOnes (d i):ℤ))+(∑i,hiddenTrial (d i))-k) := by
+  rw [posterior_count_average]
+  rw [←Finset.sum_range_reflect]
+  apply Finset.sum_congr rfl
+  intro k hk
+  have hk' : k ≤ ∑i,hiddenTrial (d i) := by simpa only [Finset.mem_range,Nat.lt_succ_iff] using hk
+  simp only [Nat.add_sub_cancel,fairBinomial_reflect _ _ hk',Nat.cast_sub hk']
+  congr 2
+  ring
+
+/-- The actual local signed motif averaged over posterior orientations is
+exactly a translated smoothed Prouhet polynomial. -/
+theorem posterior_signed_motif {ι : Type*} [Fintype ι]
+    (d : ι → Option Bool) (s : ℕ) (z : ℤ) :
+    (∑b : ι → Bool, (∏i,(flagPosterior (d i)).mass (b i)) *
+      (prouhetSeries s).coeff (z+∑i,(bitIndex (b i):ℤ))) =
+      (prouhetSeries s * fairSeries (∑i,hiddenTrial (d i))).coeff
+        (z+(∑i,(visibleOnes (d i):ℤ))+(∑i,hiddenTrial (d i))) := by
+  rw [posterior_count_average_reflected d (fun a => (prouhetSeries s).coeff (z+a)),
+    fairSeries_sum,Finset.mul_sum,
+    AddMonoidAlgebra.coeff_sum,Finsupp.finsetSum_apply]
+  apply Finset.sum_congr rfl
+  intro k hk
+  rw [AddMonoidAlgebra.coeff_mul_single_apply]
+  simp only [mul_comm]
+  congr 2
+  omega
+
+/-- Singleton smoothing for literal hidden orientation bits. Other mechanisms
+can now be conditioned on without changing this bound. -/
+theorem posterior_signed_motif_bound {ι : Type*} [Fintype ι]
+    (d : ι → Option Bool) (s : ℕ) (z : ℤ) :
+    |∑b : ι → Bool, (∏i,(flagPosterior (d i)).mass (b i)) *
+      (prouhetSeries s).coeff (z+∑i,(bitIndex (b i):ℤ))| ≤
+      (2:ℝ)^(s*s)*((2*binomialBound ((∑i,hiddenTrial (d i))/(s+1)))^s *
+        binomialBound ((∑i,hiddenTrial (d i))/(s+1))) := by
+  rw [posterior_signed_motif]
+  exact (signed_binomial_smoothing _ s).1 _
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+open FiniteLaw
+
+/-- The literal unsigned window evaluated at an affine fair-binomial shift. -/
+theorem windowKernel_binomial (n L : ℕ) (z : ℤ) :
+    windowKernel n L z =
+      ∑k∈Finset.range (n+1), fairBinomial n k *
+        (if 0 ≤ z+(k:ℤ) ∧ z+(k:ℤ)<L then (1:ℝ) else 0) := by
+  unfold windowKernel
+  simp_rw [fairSeries_sum,AddMonoidAlgebra.coeff_sum,Finsupp.finsetSum_apply,
+    AddMonoidAlgebra.coeff_single,Finsupp.single_apply]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro k hk
+  rw [←geomSeries_coeff,geomSeries]
+  simp only [AddMonoidAlgebra.single_pow,one_pow,nsmul_eq_mul,mul_one,
+    AddMonoidAlgebra.coeff_sum,Finsupp.finsetSum_apply,
+    AddMonoidAlgebra.coeff_single,Finsupp.single_apply,Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro a ha
+  by_cases h : (a:ℤ)=z+k
+  · have h' : (k:ℤ)=(a:ℤ)-z := by omega
+    simp only [ite_eq_left h,ite_eq_left h',mul_one]
+  · have h' : (k:ℤ)≠(a:ℤ)-z := by omega
+    simp only [ite_eq_right h,ite_eq_right h',mul_zero]
+
+/-- Exact local anchor kernel for the actual posterior bits, with the known
+visible contribution retained as a translation. -/
+theorem posterior_unsigned_motif {ι : Type*} [Fintype ι]
+    (d : ι → Option Bool) (s : ℕ) (z : ℤ) :
+    (∑b : ι → Bool, (∏i,(flagPosterior (d i)).mass (b i)) *
+      motifIndicator s (z+∑i,(bitIndex (b i):ℤ))) =
+      windowKernel (∑i,hiddenTrial (d i)) (2^s) (z+∑i,(visibleOnes (d i):ℤ)) := by
+  rw [posterior_count_average d (fun a => motifIndicator s (z+a)),windowKernel_binomial]
+  apply Finset.sum_congr rfl
+  intro k hk
+  simp only [motifIndicator,add_assoc]
+
+/-- The finite-series functional form of the posterior count identity. -/
+theorem posterior_count_support {ι : Type*} [Fintype ι]
+    (d : ι → Option Bool) (F : ℤ → ℝ) :
+    (∑b : ι → Bool, (∏i,(flagPosterior (d i)).mass (b i))*F (∑i,(bitIndex (b i):ℤ))) =
+    ∑v∈(fairSeries (∑i,hiddenTrial (d i))).coeff.support,
+      (fairSeries (∑i,hiddenTrial (d i))).coeff v * F ((∑i,(visibleOnes (d i):ℤ))+v) := by
+  rw [posterior_count_average]
+  have H := congrArg (fun p : IntSeries =>
+    Finsupp.linearCombination ℝ (fun v => F ((∑i,(visibleOnes (d i):ℤ))+v)) p.coeff)
+    (fairSeries_sum (∑i,hiddenTrial (d i)))
+  simp only [AddMonoidAlgebra.coeff_sum,AddMonoidAlgebra.coeff_single,map_sum,
+    Finsupp.linearCombination_single,smul_eq_mul] at H
+  exact H.symm
+
+/-- The actual two-sibling anchor multiplier. Global bits are shared by the
+two siblings until AFTER averaging; their local bits are independent. -/
+def posteriorJointAnchors {ιg ι₁ ι₂ : Type*} [Fintype ιg] [Fintype ι₁] [Fintype ι₂]
+    (dg : ιg → Option Bool) (d₁ : ι₁ → Option Bool) (d₂ : ι₂ → Option Bool)
+    (s : ℕ) (z₁ z₂ : ℤ) : ℝ :=
+  ∑bg : ιg → Bool, (∏i,(flagPosterior (dg i)).mass (bg i)) *
+    (∑b₁ : ι₁ → Bool, (∏i,(flagPosterior (d₁ i)).mass (b₁ i)) *
+      motifIndicator s (z₁+(∑i,(bitIndex (bg i):ℤ))+(∑i,(bitIndex (b₁ i):ℤ)))) *
+    (∑b₂ : ι₂ → Bool, (∏i,(flagPosterior (d₂ i)).mass (b₂ i)) *
+      motifIndicator s (z₂+(∑i,(bitIndex (bg i):ℤ))+(∑i,(bitIndex (b₂ i):ℤ))))
+
+theorem posteriorJointAnchors_eq {ιg ι₁ ι₂ : Type*} [Fintype ιg] [Fintype ι₁] [Fintype ι₂]
+    (dg : ιg → Option Bool) (d₁ : ι₁ → Option Bool) (d₂ : ι₂ → Option Bool)
+    (s : ℕ) (z₁ z₂ : ℤ) :
+    posteriorJointAnchors dg d₁ d₂ s z₁ z₂ =
+    jointWindowKernel (∑i,hiddenTrial (dg i)) (∑i,hiddenTrial (d₁ i))
+      (∑i,hiddenTrial (d₂ i)) (2^s)
+      (z₁+(∑i,(visibleOnes (dg i):ℤ))+(∑i,(visibleOnes (d₁ i):ℤ)))
+      (z₂+(∑i,(visibleOnes (dg i):ℤ))+(∑i,(visibleOnes (d₂ i):ℤ))) := by
+  unfold posteriorJointAnchors
+  simp_rw [posterior_unsigned_motif,mul_assoc]
+  rw [posterior_count_support dg (fun v =>
+    windowKernel (∑i,hiddenTrial (d₁ i)) (2^s) (z₁+v+∑i,(visibleOnes (d₁ i):ℤ)) *
+    windowKernel (∑i,hiddenTrial (d₂ i)) (2^s) (z₂+v+∑i,(visibleOnes (d₂ i):ℤ)))]
+  unfold jointWindowKernel
+  apply Finset.sum_congr rfl
+  intro v hv
+  have h₁ : z₁+((∑i,(visibleOnes (dg i):ℤ))+v)+(∑i,(visibleOnes (d₁ i):ℤ)) =
+      z₁+(∑i,(visibleOnes (dg i):ℤ))+(∑i,(visibleOnes (d₁ i):ℤ))+v := by ring
+  have h₂ : z₂+((∑i,(visibleOnes (dg i):ℤ))+v)+(∑i,(visibleOnes (d₂ i):ℤ)) =
+      z₂+(∑i,(visibleOnes (dg i):ℤ))+(∑i,(visibleOnes (d₂ i):ℤ))+v := by ring
+  rw [h₁,h₂,mul_assoc]
+
+/-- Quantitative sibling summation now directly bounds genuine posterior
+bits, not an independent-anchor surrogate. -/
+theorem posterior_sibling_anchor_energy {ιg ι₁ ι₂ : Type*}
+    [Fintype ιg] [Fintype ι₁] [Fintype ι₂]
+    (dg : ιg → Option Bool) (d₁ : ι₁ → Option Bool) (d₂ : ι₂ → Option Bool)
+    (s : ℕ) (A B C D : Finset ℤ) (c₁ c₂ : ℤ) :
+    (∑a∈A, ∑b∈B, ∑_c∈closeLabels C (2^s) a, ∑_d∈closeLabels D (2^s) b,
+      (posteriorJointAnchors dg d₁ d₂ s (a-c₁) (b-c₂))^2) ≤
+    4*((2^s:ℕ):ℝ)^6*binomialBound (∑i,hiddenTrial (dg i))*
+      binomialBound (∑i,hiddenTrial (d₂ i)) := by
+  simp_rw [posteriorJointAnchors_eq]
+  have he : ∀a c u v : ℤ, a-c+u+v = a-(c-u-v) := by intros; omega
+  simp_rw [he]
+  exact sibling_anchor_energy (∑i,hiddenTrial (dg i)) (∑i,hiddenTrial (d₁ i))
+    (∑i,hiddenTrial (d₂ i)) (2^s) A B C D
+    (c₁-(∑i,(visibleOnes (dg i):ℤ))-(∑i,(visibleOnes (d₁ i):ℤ)))
+    (c₂-(∑i,(visibleOnes (dg i):ℤ))-(∑i,(visibleOnes (d₂ i):ℤ)))
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+open FiniteLaw
+
+namespace FiniteLaw
+variable {α β : Type*} [Fintype α] [Fintype β]
+
+theorem expect_mul_const (P : FiniteLaw α) (f : α → ℝ) (c : ℝ) :
+    P.expect (fun a => f a*c) = P.expect f*c := by
+  unfold expect
+  simp only [←mul_assoc,Finset.sum_mul]
+
+theorem expect_const_mul (P : FiniteLaw α) (f : α → ℝ) (c : ℝ) :
+    P.expect (fun a => c*f a) = c*P.expect f := by
+  unfold expect
+  simp only [mul_left_comm (P.mass _) c,Finset.mul_sum]
+
+theorem coordinateProduct_expect_product {n : ℕ} (P : Fin n → FiniteLaw α)
+    (f : Fin n → α → ℝ) :
+    (coordinateProduct P).expect (fun x => ∏i,f i (x i)) = ∏i,(P i).expect (f i) := by
+  unfold expect coordinateProduct
+  simp only [←Finset.prod_mul_distrib]
+  exact (Fintype.prod_sum (fun (i : Fin n) (a : α) => (P i).mass a*f i a)).symm
+
+theorem expect_prod_split (P : FiniteLaw α) (Q : FiniteLaw β)
+    (f : α → ℝ) (g : β → ℝ) :
+    (P.prod Q).expect (fun x => f x.1*g x.2) = P.expect f * Q.expect g := by
+  rw [expect_prod]
+  simp_rw [expect_const_mul]
+  exact expect_mul_const P f _
+
+end FiniteLaw
+
+/-- All local and global orientation collections on the complete index tree,
+including unoccupied vertices and paths which do not fit. The type is fixed
+before any selected monomial or role choices. `n` is the local trial count,
+`g` the number of global paired trials, and T the full window counts. -/
+def WindowAssignments (T n g : ℕ → ℕ) (A : Type) : ℕ → Type
+  | 0 => PUnit
+  | b+1 => (Fin (n b) → A) × (Fin (g b) → A) ×
+      (Fin (T b) → WindowAssignments T n g A b)
+
+noncomputable instance windowAssignmentsFintype (T n g : ℕ → ℕ) (A : Type) [Fintype A] :
+    (b : ℕ) → Fintype (WindowAssignments T n g A b)
+  | 0 => inferInstanceAs (Fintype PUnit)
+  | b+1 => by
+      letI := windowAssignmentsFintype T n g A b
+      exact inferInstanceAs (Fintype ((Fin (n b) → A) × (Fin (g b) → A) ×
+        (Fin (T b) → WindowAssignments T n g A b)))
+
+def posteriorArray {n : ℕ} (d : Fin n → Option Bool) : FiniteLaw (Fin n → Bool) :=
+  coordinateProduct (fun i => flagPosterior (d i))
+
+/-- Actual independent posterior orientations at exact non-leaf metadata.
+The common global bits in a group are drawn only once. -/
+def windowPosteriorLaw (T n g : ℕ → ℕ) : (b : ℕ) →
+    WindowAssignments T n g (Option Bool) b → FiniteLaw (WindowAssignments T n g Bool b)
+  | 0, _ => ⟨fun _ => 1,fun _ => by norm_num,by change (∑_ : PUnit,(1:ℝ))=1; simp⟩
+  | b+1, d => (posteriorArray d.1).prod ((posteriorArray d.2.1).prod
+      (coordinateProduct (fun i => windowPosteriorLaw T n g b (d.2.2 i))))
+
+/-- Displacements from the jointly generated bit tree. The ancestor global
+map may vary off its plateau. No independent jitter per child is introduced.
+The fixed center includes the deterministic half-trial corrections. -/
+def sourceTreeAnchors (T n g : ℕ → ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ) :
+    (b : ℕ) → WindowAssignments T n g Bool b → ℤ → List ℕ → ℤ
+  | 0, _, _, _ => 0
+  | b+1, bits, incoming, [] => center b - (∑i,(bitIndex (bits.1 i):ℤ))-incoming
+  | b+1, bits, _, h::tail => if hh : h<T b then
+      sourceTreeAnchors T n g center globalShift b (bits.2.2 ⟨h,hh⟩)
+        (globalShift b bits.2.1 ⟨h,hh⟩) tail else 0
+
+theorem sourceTreeAnchors_root (T n g : ℕ → ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (b : ℕ) (bits : WindowAssignments T n g Bool (b+1)) (incoming : ℤ) :
+    sourceTreeAnchors T n g center globalShift (b+1) bits incoming [] =
+      center b - (∑i,(bitIndex (bits.1 i):ℤ))-incoming := rfl
+
+theorem sourceTreeAnchors_child (T n g : ℕ → ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (b : ℕ) (bits : WindowAssignments T n g Bool (b+1)) (incoming : ℤ)
+    (h : Fin (T b)) (tail : List ℕ) :
+    sourceTreeAnchors T n g center globalShift (b+1) bits incoming (h.val::tail) =
+      sourceTreeAnchors T n g center globalShift b (bits.2.2 h)
+        (globalShift b bits.2.1 h) tail := by
+  simp only [sourceTreeAnchors,dite_eq_left h.isLt]
+
+/-- Moving the origin of a subtree preserves the literal type recursion. -/
+theorem windowLeafSign_prefix (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
+    (b : ℕ) (pre path : List ℕ) (τ : ℝ) (v : WindowLeaf T b) :
+    windowLeafSign T s X b (pre++path) τ v =
+      windowLeafSign T s (fun tail => X (pre++tail)) b path τ v := by
+  induction b generalizing path τ with
+  | zero => rfl
+  | succ b ih =>
+    change windowLeafSign T s X b _ _ v.2 = _
+    rw [List.append_assoc,ih]
+    rfl
+
+/-- Exact recursive leaf signs for the complete bit tree, with one joint
+parent-global shift supplied to each sibling's local mechanism. -/
+def sourceTreeLeafSign (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ) :
+    (b : ℕ) → WindowAssignments T n g Bool b → ℤ → ℝ → WindowLeaf T b → ℝ
+  | 0, _, _, τ, _ => τ
+  | b+1, bits, incoming, τ, v =>
+      sourceTreeLeafSign T n g s center globalShift b (bits.2.2 v.1)
+        (globalShift b bits.2.1 v.1)
+        (motifChildSign s (center b-(∑i,(bitIndex (bits.1 i):ℤ))-incoming) v.1.val τ) v.2
+
+theorem sourceTreeLeafSign_eq (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (b : ℕ) (bits : WindowAssignments T n g Bool b) (incoming : ℤ)
+    (τ : ℝ) (v : WindowLeaf T b) :
+    sourceTreeLeafSign T n g s center globalShift b bits incoming τ v =
+      windowLeafSign T s (sourceTreeAnchors T n g center globalShift b bits incoming) b [] τ v := by
+  induction b generalizing incoming τ with
+  | zero => rfl
+  | succ b ih =>
+    change sourceTreeLeafSign T n g s center globalShift b _ _ _ v.2 =
+      windowLeafSign T s _ b [v.1.val] _ v.2
+    rw [ih]
+    have hp := windowLeafSign_prefix T s
+      (sourceTreeAnchors T n g center globalShift (b+1) bits incoming)
+      b [v.1.val] []
+      (motifChildSign s (center b-(∑i,(bitIndex (bits.1 i):ℤ))-incoming) v.1.val τ) v.2
+    simp only [List.append_nil,List.singleton_append,sourceTreeAnchors_child] at hp
+    exact hp.symm
+
+end TraceReconstruction
+
+-- Actual subtree coefficient factorization and singleton cancellation.
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+open FiniteLaw
+
+/-- The factor at the current vertex of a fixed actual occupied role choice.
+It only uses that vertex's displacement, never any descendant mechanism. -/
+def occupiedRootWeight (T : ℕ → ℕ) (s : ℕ) :
+    (b : ℕ) → (S : Finset (WindowLeaf T (b+1))) →
+      OccupiedWindowChoice T b S → ℤ → ℝ
+  | 0, S, a, X => ∏i : Fin S.card,
+      bottomRoleWeight (motifIndicator s (((S.equivFin.symm i).val.1.val:ℤ)-X))
+        ((prouhetSeries s).coeff (((S.equivFin.symm i).val.1.val:ℤ)-X)) (a i)
+  | b+1, S, a, X => ∏i : Fin (selectedChildren S).card,
+      @internalRoleWeight (occupiedWindowTerm T b
+        (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val) (a.1 i)).isOdd
+        (Classical.propDecidable _)
+        (motifIndicator s ((((selectedChildren S).equivFin.symm i).val.val:ℤ)-X))
+        ((prouhetSeries s).coeff ((((selectedChildren S).equivFin.symm i).val.val:ℤ)-X)) (a.2 i)
+
+theorem occupiedWindowWeight_zero (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
+    (path : List ℕ) (S : Finset (WindowLeaf T 1)) (a : OccupiedWindowChoice T 0 S) :
+    occupiedWindowWeight T s X 0 path S a = occupiedRootWeight T s 0 S a (X path) := rfl
+
+theorem occupiedWindowWeight_succ (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
+    (b : ℕ) (path : List ℕ) (S : Finset (WindowLeaf T (b+1+1)))
+    (a : OccupiedWindowChoice T (b+1) S) :
+    occupiedWindowWeight T s X (b+1) path S a =
+      occupiedRootWeight T s (b+1) S a (X path) *
+        ∏i : Fin (selectedChildren S).card,
+          occupiedWindowWeight T s X b
+            (path++[((selectedChildren S).equivFin.symm i).val.val])
+            (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val) (a.1 i) := by
+  unfold occupiedWindowWeight
+  change (∏i, _ * @internalRoleWeight
+    ((occupiedWindowShape T s X b
+      (path++[((selectedChildren S).equivFin.symm i).val.val])
+      (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val)).termTree
+      (occupiedWindowChoiceEquiv T s X b
+        (path++[((selectedChildren S).equivFin.symm i).val.val])
+        (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val) (a.1 i))).isOdd
+        (Classical.propDecidable _) _ _ _) = _
+  simp_rw [occupiedWindowTerm_eq]
+  rw [Finset.prod_mul_distrib,mul_comm]
+  rfl
+
+/-- Moving a subtree's path origin does not change any of its role factors. -/
+theorem occupiedWindowWeight_prefix (T : ℕ → ℕ) (s : ℕ) (X : List ℕ → ℤ)
+    (b : ℕ) (pre path : List ℕ) (S : Finset (WindowLeaf T (b+1)))
+    (a : OccupiedWindowChoice T b S) :
+    occupiedWindowWeight T s X b (pre++path) S a =
+      occupiedWindowWeight T s (fun tail => X (pre++tail)) b path S a := by
+  induction b generalizing path with
+  | zero => rfl
+  | succ b ih =>
+    rw [occupiedWindowWeight_succ,occupiedWindowWeight_succ]
+    congr 1
+    apply Finset.prod_congr rfl
+    intro i hi
+    rw [List.append_assoc,ih]
+
+theorem bottomRoleWeight_abs_le_one (χ σ : ℝ) (h : IsMotifValue χ σ) (a : Fin 3) :
+    |bottomRoleWeight χ σ a| ≤ 1 := by
+  rcases h with ⟨rfl,rfl⟩ | ⟨rfl,rfl|rfl⟩ <;>
+    fin_cases a <;> norm_num [bottomRoleWeight]
+
+theorem internalRoleWeight_abs_le_one (odd : Prop) [Decidable odd]
+    (χ σ : ℝ) (h : IsMotifValue χ σ) (a : Fin 3) :
+    |internalRoleWeight odd χ σ a| ≤ 1 := by
+  rcases h with ⟨rfl,rfl⟩ | ⟨rfl,rfl|rfl⟩ <;>
+    fin_cases a <;> by_cases ho : odd <;> norm_num [internalRoleWeight,roleWeight,ho]
+
+theorem occupiedRootWeight_abs_le_one (T : ℕ → ℕ) (s b : ℕ)
+    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S) (X : ℤ) :
+    |occupiedRootWeight T s b S a X| ≤ 1 := by
+  cases b with
+  | zero =>
+    rw [occupiedRootWeight,Finset.abs_prod]
+    exact Finset.prod_le_one₀ (fun _ _ => abs_nonneg _)
+      (fun i _ => bottomRoleWeight_abs_le_one _ _ (prouhet_motif_value s _) (a i))
+  | succ b =>
+    rw [occupiedRootWeight,Finset.abs_prod]
+    exact Finset.prod_le_one₀ (fun _ _ => abs_nonneg _)
+      (fun i _ => internalRoleWeight_abs_le_one _ _ _ (prouhet_motif_value s _) (a.2 i))
+
+theorem occupiedWindowWeight_abs_le_one (T : ℕ → ℕ) (s : ℕ)
+    (X : List ℕ → ℤ) (b : ℕ) (path : List ℕ) (S : Finset (WindowLeaf T (b+1)))
+    (a : OccupiedWindowChoice T b S) : |occupiedWindowWeight T s X b path S a| ≤ 1 := by
+  induction b generalizing path with
+  | zero => exact occupiedRootWeight_abs_le_one T s 0 S a _
+  | succ b ih =>
+    rw [occupiedWindowWeight_succ,abs_mul,Finset.abs_prod]
+    exact (mul_le_of_le_one_left (by positivity)
+      (occupiedRootWeight_abs_le_one T s (b+1) S a _)).trans
+      (Finset.prod_le_one₀ (fun _ _ => abs_nonneg _) (fun i _ => ih _ _ _))
+
+/-- A singleton signed role factors off its signed motif with a scalar
+independent of all displacements. Bottom zero-role terms really vanish. -/
+theorem bottom_singleton_factor {n : ℕ} (a : Fin n → Fin 3) (i : Fin n)
+    (hi : a i=2) (hrest : ∀j,j≠i → a j=0) :
+    ∃c : ℝ, |c|≤1 ∧ ∀χ σ : Fin n → ℝ,
+      (∏j,bottomRoleWeight (χ j) (σ j) (a j)) = c*σ i := by
+  let c : ℝ := -(∏j∈Finset.univ.erase i,(0:ℝ))/2
+  refine ⟨c,?_,?_⟩
+  · have hp : 0 ≤ ∏j∈Finset.univ.erase i,(0:ℝ) := Finset.prod_nonneg (fun _ _ => le_rfl)
+    have hp' : (∏j∈Finset.univ.erase i,(0:ℝ)) ≤ 1 := Finset.prod_le_one₀
+      (fun _ _ => le_rfl) (fun _ _ => by norm_num)
+    dsimp [c]
+    rw [abs_div,abs_neg,abs_of_nonneg hp]
+    rw [abs_of_pos (by norm_num : (0:ℝ)<2)]
+    linarith
+  · intro χ σ
+    rw [←Finset.prod_erase_mul _ _ (Finset.mem_univ i)]
+    have he : (∏j∈Finset.univ.erase i,bottomRoleWeight (χ j) (σ j) (a j)) =
+        ∏j∈Finset.univ.erase i,(0:ℝ) := by
+      apply Finset.prod_congr rfl
+      intro j hj
+      simp only [hrest j (Finset.mem_erase.mp hj).1,bottomRoleWeight,↓reduceIte]
+    rw [he,hi]
+    simp only [bottomRoleWeight,show (2:Fin 3)≠0 by decide,
+      show (2:Fin 3)≠1 by decide,↓reduceIte]
+    dsimp [c]
+    ring
+
+theorem internal_singleton_factor {n : ℕ} (odd : Fin n → Prop)
+    (a : Fin n → Fin 3) (i : Fin n) (hi : a i=2) (hrest : ∀j,j≠i → a j=0) :
+    ∃c : ℝ, |c|≤1 ∧ ∀χ σ : Fin n → ℝ,
+      (∏j,@internalRoleWeight (odd j) (Classical.propDecidable _) (χ j) (σ j) (a j)) = c*σ i := by
+  refine ⟨if odd i then 1 else 0,?_,?_⟩
+  · split_ifs <;> norm_num
+  · intro χ σ
+    rw [Finset.prod_eq_single i]
+    · by_cases ho : odd i <;> simp [internalRoleWeight,roleWeight,hi,ho]
+    · intro j hj hji
+      by_cases ho : odd j <;> simp [internalRoleWeight,roleWeight,hrest j hji,ho]
+    · simp
+
+namespace FiniteLaw
+
+/-- Unoccupied independent coordinates integrate to one. -/
+theorem coordinateProduct_expect_subtype {α : Type*} [Fintype α] {n : ℕ}
+    (P : Fin n → FiniteLaw α) (S : Finset (Fin n)) (f : S → α → ℝ) :
+    (coordinateProduct P).expect (fun x => ∏i : S, f i (x i.val)) =
+      ∏i : S, (P i.val).expect (f i) := by
+  have scalar : ∀f : S → ℝ,
+      (∏j : Fin n, if hj : j∈S then f ⟨j,hj⟩ else 1) = ∏i : S,f i := by
+    intro f
+    calc
+      _ = ∏j∈S, if hj : j∈S then f ⟨j,hj⟩ else 1 :=
+        (Finset.prod_subset S.subset_univ (fun j _ hj => dite_eq_right hj)).symm
+      _ = ∏i : S, if hj : i.val∈S then f ⟨i.val,hj⟩ else 1 :=
+        Finset.prod_subtype S (fun _ => Iff.rfl) _
+      _ = _ := by simp only [dite_eq_left (Subtype.property _)]
+  let F : Fin n → α → ℝ := fun j a => if hj : j∈S then f ⟨j,hj⟩ a else 1
+  have hp : ∀x : Fin n → α, (∏j,F j (x j)) = ∏i : S,f i (x i.val) := by
+    intro x
+    exact scalar (fun i => f i (x i.val))
+  rw [show (fun x : Fin n → α => ∏i : S,f i (x i.val)) = (fun x => ∏j,F j (x j)) by
+    funext x; exact (hp x).symm]
+  rw [coordinateProduct_expect_product]
+  have he : ∀j, (P j).expect (F j) = if hj : j∈S then (P j).expect (f ⟨j,hj⟩) else 1 := by
+    intro j
+    by_cases hj : j∈S
+    · simp only [F,dite_eq_left hj]
+    · simp only [F,dite_eq_right hj,expect_const]
+  simp_rw [he]
+  exact scalar (fun i => (P i.val).expect (f i))
+
+theorem coordinateProduct_expect_selected {α : Type*} [Fintype α] {n : ℕ}
+    (P : Fin n → FiniteLaw α) (S : Finset (Fin n)) (f : Fin S.card → α → ℝ) :
+    (coordinateProduct P).expect (fun x => ∏i : Fin S.card, f i (x (S.equivFin.symm i).val)) =
+      ∏i : Fin S.card, (P (S.equivFin.symm i).val).expect (f i) := by
+  have hp : ∀x : Fin n → α,
+      (∏i : Fin S.card,f i (x (S.equivFin.symm i).val)) =
+        ∏i : S,f (S.equivFin i) (x i.val) := by
+    intro x
+    simpa only [Equiv.apply_symm_apply] using
+      Equiv.prod_comp S.equivFin.symm (fun i : S => f (S.equivFin i) (x i.val))
+  simp_rw [hp,coordinateProduct_expect_subtype]
+  symm
+  simpa only [Equiv.apply_symm_apply] using
+    Equiv.prod_comp S.equivFin.symm (fun i : S => (P i.val).expect (f (S.equivFin i)))
+
+theorem expect_prod_selected {α β γ : Type*} [Fintype α] [Fintype β] [Fintype γ]
+    {n : ℕ} (P : FiniteLaw α) (Q : FiniteLaw β) (R : Fin n → FiniteLaw γ)
+    (S : Finset (Fin n)) (f : α → ℝ) (h : β → Fin S.card → γ → ℝ) :
+    (P.prod (Q.prod (coordinateProduct R))).expect (fun x =>
+      f x.1 * ∏i : Fin S.card, h x.2.1 i (x.2.2 (S.equivFin.symm i).val)) =
+      P.expect f * Q.expect (fun z => ∏i : Fin S.card,
+        (R (S.equivFin.symm i).val).expect (h z i)) := by
+  simp_rw [expect_prod,expect_const_mul,coordinateProduct_expect_selected]
+  rw [expect_mul_const]
+
+theorem abs_expect_le_const {α : Type*} [Fintype α] (P : FiniteLaw α)
+    (f : α → ℝ) (c : ℝ) (h : ∀a, |f a|≤c) : |P.expect f|≤c :=
+  (P.abs_expect_le f).trans ((P.expect_mono h).trans_eq (P.expect_const c))
+
+end FiniteLaw
+
+
+/-- A fixed real coefficient term evaluated on the actual complete bit tree.
+The selected role choice is fixed independently of these bits. -/
+def sourceRoleTerm (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (b : ℕ) (bits : WindowAssignments T n g Bool (b+1)) (incoming : ℤ)
+    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S) : ℝ :=
+  occupiedWindowWeight T s (sourceTreeAnchors T n g center globalShift (b+1) bits incoming)
+    b [] S a
+
+def sourceRoleAverage (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ)
+    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S) : ℝ :=
+  (windowPosteriorLaw T n g (b+1) d).expect
+    (fun bits => sourceRoleTerm T n g s center globalShift b bits incoming S a)
+
+theorem sourceRoleTerm_zero (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (bits : WindowAssignments T n g Bool 1) (incoming : ℤ)
+    (S : Finset (WindowLeaf T 1)) (a : OccupiedWindowChoice T 0 S) :
+    sourceRoleTerm T n g s center globalShift 0 bits incoming S a =
+      occupiedRootWeight T s 0 S a (center 0-(∑i,(bitIndex (bits.1 i):ℤ))-incoming) := rfl
+
+theorem sourceRoleTerm_succ (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (b : ℕ) (bits : WindowAssignments T n g Bool (b+1+1)) (incoming : ℤ)
+    (S : Finset (WindowLeaf T (b+1+1))) (a : OccupiedWindowChoice T (b+1) S) :
+    sourceRoleTerm T n g s center globalShift (b+1) bits incoming S a =
+      occupiedRootWeight T s (b+1) S a (center (b+1)-(∑i,(bitIndex (bits.1 i):ℤ))-incoming) *
+        ∏i : Fin (selectedChildren S).card,
+          sourceRoleTerm T n g s center globalShift b
+            (bits.2.2 ((selectedChildren S).equivFin.symm i).val)
+            (globalShift (b+1) bits.2.1 ((selectedChildren S).equivFin.symm i).val)
+            (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val) (a.1 i) := by
+  unfold sourceRoleTerm
+  rw [occupiedWindowWeight_succ]
+  congr 1
+  apply Finset.prod_congr rfl
+  intro i hi
+  have hp := occupiedWindowWeight_prefix T s
+    (sourceTreeAnchors T n g center globalShift (b+1+1) bits incoming) b
+    [((selectedChildren S).equivFin.symm i).val.val] []
+    (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val) (a.1 i)
+  simpa only [List.append_nil,List.nil_append,List.singleton_append,sourceTreeAnchors_child] using hp
+
+theorem sourceRoleAverage_zero (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (d : WindowAssignments T n g (Option Bool) 1) (incoming : ℤ)
+    (S : Finset (WindowLeaf T 1)) (a : OccupiedWindowChoice T 0 S) :
+    sourceRoleAverage T n g s center globalShift 0 d incoming S a =
+      (posteriorArray d.1).expect (fun loc =>
+        occupiedRootWeight T s 0 S a (center 0-(∑i,(bitIndex (loc i):ℤ))-incoming)) := by
+  rw [sourceRoleAverage]
+  simp_rw [sourceRoleTerm_zero]
+  rcases d with ⟨dl,dg,dc⟩
+  change ((posteriorArray dl).prod ((posteriorArray dg).prod
+    (coordinateProduct (fun i => windowPosteriorLaw T n g 0 (dc i))))).expect
+      (fun bits => occupiedRootWeight T s 0 S a
+        (center 0-(∑i,(bitIndex (bits.1 i):ℤ))-incoming)) = _
+  rw [expect_prod]
+  simp only [expect_const]
+
+theorem sourceRoleAverage_succ (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1+1)) (incoming : ℤ)
+    (S : Finset (WindowLeaf T (b+1+1))) (a : OccupiedWindowChoice T (b+1) S) :
+    sourceRoleAverage T n g s center globalShift (b+1) d incoming S a =
+      (posteriorArray d.1).expect (fun loc =>
+        occupiedRootWeight T s (b+1) S a (center (b+1)-(∑i,(bitIndex (loc i):ℤ))-incoming)) *
+      (posteriorArray d.2.1).expect (fun glob =>
+        ∏i : Fin (selectedChildren S).card,
+          sourceRoleAverage T n g s center globalShift b
+            (d.2.2 ((selectedChildren S).equivFin.symm i).val)
+            (globalShift (b+1) glob ((selectedChildren S).equivFin.symm i).val)
+            (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val) (a.1 i)) := by
+  rw [sourceRoleAverage]
+  simp_rw [sourceRoleTerm_succ]
+  exact FiniteLaw.expect_prod_selected (posteriorArray d.1) (posteriorArray d.2.1)
+    (fun i => windowPosteriorLaw T n g (b+1) (d.2.2 i)) (selectedChildren S)
+    (fun loc => occupiedRootWeight T s (b+1) S a
+      (center (b+1)-(∑i,(bitIndex (loc i):ℤ))-incoming))
+    (fun glob i bits => sourceRoleTerm T n g s center globalShift b bits
+      (globalShift (b+1) glob ((selectedChildren S).equivFin.symm i).val)
+      (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val) (a.1 i))
+
+theorem sourceRoleAverage_abs_le_one (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ)
+    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S) :
+    |sourceRoleAverage T n g s center globalShift b d incoming S a|≤1 := by
+  apply FiniteLaw.abs_expect_le_const
+  intro bits
+  exact occupiedWindowWeight_abs_le_one T s _ b [] S a
+
+/-- The exact signed smoothing cost read off the visible metadata alone. -/
+def posteriorSmoothingCost {ι : Type*} [Fintype ι] (s : ℕ) (d : ι → Option Bool) : ℝ :=
+  (2:ℝ)^(s*s)*((2*binomialBound ((∑i,hiddenTrial (d i))/(s+1)))^s *
+    binomialBound ((∑i,hiddenTrial (d i))/(s+1)))
+
+/-- A purely numerical typical-metadata condition on every local collection.
+It includes unoccupied vertices and is fixed before a coefficient is chosen. -/
+def MetadataSmoothingCap (T n g : ℕ → ℕ) (s : ℕ) (η : ℝ) :
+    (b : ℕ) → WindowAssignments T n g (Option Bool) b → Prop
+  | 0, _ => True
+  | b+1, d => posteriorSmoothingCost s d.1 ≤ η ∧
+      ∀h, MetadataSmoothingCap T n g s η b (d.2.2 h)
+
+def occupiedRootHasSingleton (T : ℕ → ℕ) : (b : ℕ) →
+    (S : Finset (WindowLeaf T (b+1))) → OccupiedWindowChoice T b S → Prop
+  | 0, _, a => ∃i, a i=2 ∧ ∀j,j≠i → a j=0
+  | _+1, _, a => ∃i, a.2 i=2 ∧ ∀j,j≠i → a.2 j=0
+
+theorem occupiedRoot_singleton_factor (T : ℕ → ℕ) (s b : ℕ)
+    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S)
+    (h : occupiedRootHasSingleton T b S a) :
+    ∃c : ℝ, |c|≤1 ∧ ∃label : ℤ, ∀X,
+      occupiedRootWeight T s b S a X = c*(prouhetSeries s).coeff (label-X) := by
+  cases b with
+  | zero =>
+    obtain ⟨i,hi,hr⟩ := h
+    obtain ⟨c,hc,hfac⟩ := bottom_singleton_factor a i hi hr
+    refine ⟨c,hc,((S.equivFin.symm i).val.1.val:ℤ),fun X => ?_⟩
+    exact hfac _ _
+  | succ b =>
+    obtain ⟨i,hi,hr⟩ := h
+    obtain ⟨c,hc,hfac⟩ := internal_singleton_factor
+      (fun j => (occupiedWindowTerm T b
+        (selectedChildLeaves S ((selectedChildren S).equivFin.symm j).val) (a.1 j)).isOdd)
+      a.2 i hi hr
+    refine ⟨c,hc,(((selectedChildren S).equivFin.symm i).val.val:ℤ),fun X => ?_⟩
+    exact hfac _ _
+
+theorem occupiedRoot_singleton_average (T : ℕ → ℕ) (s b : ℕ)
+    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S)
+    (h : occupiedRootHasSingleton T b S a) {m : ℕ} (d : Fin m → Option Bool)
+    (center incoming : ℤ) :
+    |(posteriorArray d).expect (fun loc => occupiedRootWeight T s b S a
+      (center-(∑i,(bitIndex (loc i):ℤ))-incoming))| ≤ posteriorSmoothingCost s d := by
+  obtain ⟨c,hc,label,hfac⟩ := occupiedRoot_singleton_factor T s b S a h
+  simp_rw [hfac,expect_const_mul,abs_mul]
+  have he : ∀loc : Fin m → Bool,
+      label-(center-(∑i,(bitIndex (loc i):ℤ))-incoming) =
+        (label-center+incoming)+(∑i,(bitIndex (loc i):ℤ)) := by intro loc; omega
+  simp_rw [he]
+  have hb : |(posteriorArray d).expect (fun loc =>
+      (prouhetSeries s).coeff ((label-center+incoming)+(∑i,(bitIndex (loc i):ℤ))))| ≤
+        posteriorSmoothingCost s d := by
+    have hb := posterior_signed_motif_bound d s (label-center+incoming)
+    unfold posteriorArray coordinateProduct expect posteriorSmoothingCost
+    convert hb using 1
+    congr 3
+    exact Subsingleton.elim _ _
+  calc
+    _ ≤ 1*|(posteriorArray d).expect (fun loc =>
+        (prouhetSeries s).coeff ((label-center+incoming)+(∑i,(bitIndex (loc i):ℤ))))| :=
+      mul_le_mul_of_nonneg_right hc (abs_nonneg _)
+    _ ≤ _ := by simpa only [one_mul] using hb
+
+theorem abs_prod_le_coordinate {ι : Type*} [Fintype ι] (f : ι → ℝ) (i : ι)
+    (h : ∀j,j≠i → |f j|≤1) : |∏j,f j|≤|f i| := by
+  rw [Finset.abs_prod,←Finset.prod_erase_mul _ _ (Finset.mem_univ i)]
+  have he : (∏j∈Finset.univ.erase i,|f j|)≤1 :=
+    Finset.prod_le_one₀ (fun _ _ => abs_nonneg _) (fun j hj => h j (Finset.mem_erase.mp hj).1)
+  simpa only [one_mul] using mul_le_mul_of_nonneg_right he (abs_nonneg (f i))
+
+/-- Literal singleton cancellation for the complete posterior bit tree.
+No successful-fit event is conditioned on, and the common parent-global
+jitter is retained until after the child factors have averaged. -/
+theorem sourceRoleAverage_singleton (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (b : ℕ) (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ)
+    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S)
+    (η : ℝ) (hcap : MetadataSmoothingCap T n g s η (b+1) d)
+    (hsing : ¬(occupiedWindowTerm T b S a).noSingleton) :
+    |sourceRoleAverage T n g s center globalShift b d incoming S a|≤η := by
+  induction b generalizing incoming with
+  | zero =>
+    have hh : Odd (signedRoleCount a) ∧ activeRoleCount a=1 := by
+      by_contra! h
+      apply hsing
+      exact ⟨fun ho he => h ho he,fun _ => trivial⟩
+    have hr : occupiedRootHasSingleton T 0 S a :=
+      (odd_roles_singleton_or_branch a hh.1).resolve_right (by omega)
+    rw [sourceRoleAverage_zero]
+    exact (occupiedRoot_singleton_average T s 0 S a hr d.1 _ _).trans hcap.1
+  | succ b ih =>
+    rw [sourceRoleAverage_succ,abs_mul]
+    have hroot : |(posteriorArray d.1).expect (fun loc =>
+        occupiedRootWeight T s (b+1) S a
+          (center (b+1)-(∑i,(bitIndex (loc i):ℤ))-incoming))|≤1 :=
+      FiniteLaw.abs_expect_le_const _ _ 1 (fun _ => occupiedRootWeight_abs_le_one T s (b+1) S a _)
+    have hglob : |(posteriorArray d.2.1).expect (fun glob =>
+        ∏i : Fin (selectedChildren S).card,
+          sourceRoleAverage T n g s center globalShift b
+            (d.2.2 ((selectedChildren S).equivFin.symm i).val)
+            (globalShift (b+1) glob ((selectedChildren S).equivFin.symm i).val)
+            (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val) (a.1 i))|≤1 := by
+      apply FiniteLaw.abs_expect_le_const
+      intro glob
+      rw [Finset.abs_prod]
+      exact Finset.prod_le_one₀ (fun _ _ => abs_nonneg _)
+        (fun i _ => sourceRoleAverage_abs_le_one T n g s center globalShift b _ _ _ _)
+    by_cases hr : occupiedRootHasSingleton T (b+1) S a
+    · have hh := (occupiedRoot_singleton_average T s (b+1) S a hr d.1
+        (center (b+1)) incoming).trans hcap.1
+      exact (mul_le_mul_of_nonneg_left hglob (abs_nonneg _)).trans (by simpa using hh)
+    · have hex : ∃i, ¬(occupiedWindowTerm T b
+          (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val) (a.1 i)).noSingleton := by
+        by_contra! h
+        apply hsing
+        refine ⟨?_,h⟩
+        intro ho he
+        have hh := odd_roles_singleton_or_branch a.2 ho
+        exact hr (hh.resolve_right (by omega))
+      obtain ⟨i,hi⟩ := hex
+      have hsmall : |(posteriorArray d.2.1).expect (fun glob =>
+          ∏j : Fin (selectedChildren S).card,
+            sourceRoleAverage T n g s center globalShift b
+              (d.2.2 ((selectedChildren S).equivFin.symm j).val)
+              (globalShift (b+1) glob ((selectedChildren S).equivFin.symm j).val)
+              (selectedChildLeaves S ((selectedChildren S).equivFin.symm j).val) (a.1 j))|≤η := by
+        apply FiniteLaw.abs_expect_le_const
+        intro glob
+        apply (abs_prod_le_coordinate _ i (fun j _ =>
+          sourceRoleAverage_abs_le_one T n g s center globalShift b _ _ _ _)).trans
+        exact ih _ _ _ _ (hcap.2 _) hi
+      exact (mul_le_mul_of_nonneg_right hroot (abs_nonneg _)).trans (by simpa using hsmall)
+
+end TraceReconstruction
+
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/TraceReconstruction/Metadata.lean b/lean/OAI/Probability/TraceReconstruction/Metadata.lean
index bf1824c051dd0759ac56095cf6723eba93893888..d97dfcfd9de594612e39823b52bfc478b31ebd50 100644
--- a/lean/OAI/Probability/TraceReconstruction/Metadata.lean
+++ b/lean/OAI/Probability/TraceReconstruction/Metadata.lean
@@ -1,772 +1,772 @@
-import OAI.Probability.TraceReconstruction.Layout
-
-namespace OAI
-
-/-!
-# Metadata
-
-Hidden metadata, fair jitter, heterogeneous emissions, independent window trials, concentration, and fit.
--/
-
-open scoped ENNReal
-noncomputable section
-
-open Filter
-open scoped Topology
-
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-namespace FiniteLaw
-variable {α : Type*} [Fintype α]
-
-theorem pairedReportMass_none (P Q H : FiniteLaw α) (ω : ℝ) (z : α × α) :
-    pairedReportMass P Q H ω (none,z)=pairedHiddenMass H ω z := by
-  simp only [pairedReportMass,Fintype.sum_bool,pairedFlagMass]
-  ring
-
-theorem pairedReport_hidden_expect (P Q H : FiniteLaw α) {ω : ℝ} (hω : 0≤ω)
-    (hP : ∀a,ω*H.mass a≤P.mass a) (hQ : ∀a,ω*H.mass a≤Q.mass a) (F : ℕ → ℝ) :
-    (pairedReport P Q H hω hP hQ).expect (fun z => F (hiddenTrial z.1))=
-      ω^2*F 1+(1-ω^2)*F 0 := by
-  let R := pairedReport P Q H hω hP hQ
-  have hp : R.expect (fun z => if z.1=none then (1:ℝ) else 0)=ω^2 := by
-    simp only [expect,R,pairedReport,Fintype.sum_prod_type,Fintype.sum_option]
-    simp only [reduceCtorEq,ite_false,mul_zero,Finset.sum_const_zero,add_zero]
-    simpa only [pairedReportMass_none,Fintype.sum_prod_type,ite_true,mul_one] using sum_pairedHiddenMass H ω
-  have he (z : Option Bool × (α × α)) : F (hiddenTrial z.1)=
-      F 0+(F 1-F 0)*(if z.1=none then 1 else 0) := by
-    rcases z with ⟨d,z⟩; cases d <;> simp [hiddenTrial]
-  simp_rw [he]
-  have hadd : R.expect (fun z => F 0+(F 1-F 0)*(if z.1=none then 1 else 0))=
-      R.expect (fun _ => F 0)+R.expect (fun z => (F 1-F 0)*(if z.1=none then 1 else 0)) := by
-    simp only [expect,mul_add,Finset.sum_add_distrib]
-  rw [hadd,expect_const,expect_const_mul,hp]
-  ring
-
-/-- Centered hidden-flag MGF, derived from the literal paired experiment. -/
-theorem pairedReport_hidden_mgf (P Q H : FiniteLaw α) {ω t : ℝ} (hω : 0≤ω)
-    (hP : ∀a,ω*H.mass a≤P.mass a) (hQ : ∀a,ω*H.mass a≤Q.mass a)
-    (hp : ω^2∈Set.Icc 0 1) (ht : |t|≤1) :
-    (pairedReport P Q H hω hP hQ).expect (fun z =>
-      Real.exp (t*((hiddenTrial z.1:ℕ)-ω^2)))≤Real.exp (t^2) := by
-  rw [pairedReport_hidden_expect P Q H hω hP hQ (fun k => Real.exp (t*((k:ℝ)-ω^2)))]
-  have H := binomialMass_centered_mgf 1 hp ht
-  norm_num [Finset.sum_range_succ,binomialMass] at H ⊢
-  convert H using 1 ; ring
-
-end FiniteLaw
-
-namespace PairedMechanisms
-variable {Side : Type*} [Fintype Side] {m : ℕ} (M : PairedMechanisms Side m)
-
-def hiddenCount (z : Reports Side m) : ℕ := ∑i,hiddenTrial (z i).1
-
-/-- No independence of displacements is presumed: only the flags of the
- distinct literal paired mechanisms are independent. -/
-theorem hiddenCount_mgf {ω t : ℝ} (hw : ∀i,M.hiddenWeight i=ω)
-    (hp : ω^2∈Set.Icc 0 1) (ht : |t|≤1) :
-    M.reports.expect (fun z => Real.exp (t*((hiddenCount z:ℕ)-(m:ℝ)*ω^2)))≤
-      Real.exp ((m:ℝ)*t^2) := by
-  have he (z : Reports Side m) :
-      Real.exp (t*((hiddenCount z:ℕ)-(m:ℝ)*ω^2))=
-        ∏i : Fin m,Real.exp (t*((hiddenTrial (z i).1:ℕ)-ω^2)) := by
-    rw [←Real.exp_sum]
-    congr 1
-    simp only [mul_sub,Finset.mul_sum,Finset.sum_sub_distrib,Finset.sum_const,Finset.card_univ,Fintype.card_fin,
-      nsmul_eq_mul,hiddenCount,Nat.cast_sum]
-    ring
-  simp_rw [he]
-  rw [reports,FiniteLaw.coordinateProduct_expect_product _
-    (fun (_ : Fin m) (z : Option Bool × (Side × Side)) => Real.exp (t*((hiddenTrial z.1:ℕ)-ω^2))) ]
-  calc
-    _ ≤ ∏i : Fin m,Real.exp (t^2) := by
-      apply Finset.prod_le_prod₀
-      · intro i _
-        exact Finset.sum_nonneg (fun z _ => mul_nonneg
-          ((FiniteLaw.pairedReport (M.first i) (M.second i) (M.common i) (M.weight_nonneg i)
-            (M.common_first i) (M.common_second i)).nonneg z) (Real.exp_nonneg _))
-      · intro i _
-        have h := FiniteLaw.pairedReport_hidden_mgf (M.first i) (M.second i) (M.common i)
-          (M.weight_nonneg i) (M.common_first i) (M.common_second i) (by simpa [hw i] using hp) ht
-        simpa only [hw i] using h
-    _ = _ := by simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fin,Real.exp_nat_mul]
-
-/-- A quantitative unconditional bound for failure of a source hidden-count
- condition. The exponent ω⁴m is sufficient for the uniform scale margins. -/
-theorem hiddenCount_small {ω : ℝ} (hw : ∀i,M.hiddenWeight i=ω) (hp : ω^2∈Set.Icc 0 1) :
-    M.reports.expect (fun z => if (hiddenCount z:ℝ)<(m:ℝ)*ω^2/2 then 1 else 0)≤
-      Real.exp (-(m:ℝ)*ω^4/16) := by
-  have ht : 0≤ω^2/4 := by positivity
-  have ht' : |-(ω^2/4)|≤1 := by rw [abs_neg,abs_of_nonneg ht]; linarith [hp.2]
-  have H := finite_exp_markov Finset.univ M.reports.mass
-    (fun z => -((hiddenCount z:ℕ)-(m:ℝ)*ω^2)) (fun z _ => M.reports.nonneg z)
-    ((m:ℝ)*ω^2/2) (ω^2/4) ht
-  have hm := M.hiddenCount_mgf hw hp ht'
-  have hsub : M.reports.expect (fun z => if (hiddenCount z:ℝ)<(m:ℝ)*ω^2/2 then 1 else 0)≤
-      ∑z∈Finset.univ.filter (fun z => (m:ℝ)*ω^2/2≤-((hiddenCount z:ℕ)-(m:ℝ)*ω^2)),M.reports.mass z := by
-    rw [Finset.sum_filter]
-    apply Finset.sum_le_sum
-    intro z _
-    dsimp only
-    by_cases hz : (hiddenCount z:ℝ)<(m:ℝ)*ω^2/2
-    · rw [ite_eq_left hz,ite_eq_left (by linarith : (m:ℝ)*ω^2/2≤-((hiddenCount z:ℕ)-(m:ℝ)*ω^2)),mul_one]
-    · rw [ite_eq_right hz,mul_zero]; split_ifs <;> positivity [M.reports.nonneg z]
-  calc
-    _ ≤ _ := hsub.trans H
-    _ ≤ Real.exp (-(ω^2/4)*((m:ℝ)*ω^2/2))*Real.exp ((m:ℝ)*(-(ω^2/4))^2) := by
-      apply mul_le_mul_of_nonneg_left _ (Real.exp_nonneg _)
-      simpa only [FiniteLaw.expect,neg_mul,mul_neg,neg_neg] using hm
-    _ = _ := by rw [←Real.exp_add]; congr 1; ring
-
-/-- Exact Bayes averaging under all metadata, including impossible reports. -/
-theorem posterior_average (F : (Fin m → Bool) → ℝ) :
-    M.reports.expect (fun z => (orientationsGiven z).expect F)=
-      (fairOrientations m).expect F := by
-  unfold FiniteLaw.expect
-  simp only [Finset.mul_sum]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro b _
-  have hb (z : Reports Side m) : M.reports.mass z*(orientationsGiven z).mass b=
-      (fairOrientations m).mass b*(M.reportsGiven b).mass z := by
-    change (∏i,FiniteLaw.pairedReportMass (M.first i) (M.second i) (M.common i) (M.hiddenWeight i) (z i))*
-      (∏i,(FiniteLaw.flagPosterior (z i).1).mass (b i))=_
-    rw [←FiniteLaw.pairedFlags_product_bayes]
-    rw [Finset.prod_mul_distrib]
-    rfl
-  simp_rw [←mul_assoc,hb,mul_assoc]
-  rw [←Finset.mul_sum,←Finset.sum_mul,(M.reportsGiven b).total,one_mul]
-
-/-- Markov's inequality for the actual conditional failure probability.
- This does not condition any orientation averages on fitting paths. -/
-theorem posterior_bad_mass {a : ℝ} (_ha : 0<a) (bad : (Fin m → Bool) → Prop)
-    [DecidablePred bad] :
-    a*M.reports.expect (fun z => if a≤(orientationsGiven z).expect (fun b => if bad b then 1 else 0)
-      then 1 else 0) ≤ (fairOrientations m).expect (fun b => if bad b then 1 else 0) := by
-  rw [←M.posterior_average,←FiniteLaw.expect_const_mul]
-  apply FiniteLaw.expect_mono
-  intro z
-  split_ifs with hz
-  · simpa using hz
-  · simp only [mul_zero]
-    exact Finset.sum_nonneg (fun b _ => mul_nonneg ((orientationsGiven z).nonneg b)
-      (by dsimp only; split_ifs <;> norm_num))
-
-end PairedMechanisms
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-namespace FiniteLaw
-variable {ι α : Type*} [Fintype ι] [Fintype α]
-
-theorem indexedProduct_expect_product (P : ι → FiniteLaw α) (F : ι → α → ℝ) :
-    (indexedProduct P).expect (fun x => ∏i,F i (x i))=∏i,(P i).expect (F i) := by
-  simp only [expect,indexedProduct,←Finset.prod_mul_distrib]
-  exact (Fintype.prod_sum (fun i a => (P i).mass a*F i a)).symm
-
-end FiniteLaw
-
-/-- The unconditional orientation distribution on any actual finite trial ID set. -/
-def fairField (ι : Type*) [Fintype ι] : FiniteLaw (ι → Bool) :=
-  FiniteLaw.indexedProduct (fun _ => FiniteLaw.flagPosterior none)
-
-def fairLinearSum {ι : Type*} [Fintype ι] (c : ι → ℝ) (bits : ι → Bool) : ℝ :=
-  ∑i,c i*((bitIndex (bits i):ℝ)-1/2)
-
-/-- Exact MGF calculation for each paired orientation bit. -/
-theorem fairBit_mgf (t : ℝ) :
-    (FiniteLaw.flagPosterior none).expect (fun b => Real.exp (t*((bitIndex b:ℝ)-1/2)))=
-      Real.cosh (t/2) := by
-  simp only [FiniteLaw.expect,Fintype.sum_bool,FiniteLaw.flagPosterior,bitIndex]
-  norm_num only [Bool.false_eq_true,↓reduceIte,Nat.cast_zero,Nat.cast_one,decide_true,decide_false]
-  rw [Real.cosh_eq]
-  ring_nf
-
-/-- Hoeffding's exponential estimate for the actual independently fair bits,
- allowing deterministic zero coefficients for completed/nonactive global pairs. -/
-theorem fairLinearSum_mgf {ι : Type*} [Fintype ι] (c : ι → ℝ) (t : ℝ) :
-    (fairField ι).expect (fun bits => Real.exp (t*fairLinearSum c bits))≤
-      Real.exp ((∑i,(c i)^2)*t^2/8) := by
-  have he (bits : ι → Bool) : Real.exp (t*fairLinearSum c bits)=
-      ∏i,Real.exp ((t*c i)*((bitIndex (bits i):ℝ)-1/2)) := by
-    rw [←Real.exp_sum]
-    congr 1
-    simp only [fairLinearSum,Finset.mul_sum,mul_assoc]
-  simp_rw [he]
-  rw [fairField,FiniteLaw.indexedProduct_expect_product _
-    (fun i b => Real.exp ((t*c i)*((bitIndex b:ℝ)-1/2)))]
-  simp_rw [fairBit_mgf]
-  calc
-    _ ≤ ∏i,Real.exp ((t*c i)^2/8) := by
-      apply Finset.prod_le_prod₀
-      · intro i _; exact (Real.cosh_pos _).le
-      · intro i _
-        convert Real.cosh_le_exp_half_sq (t*c i/2) using 1
-        congr 1
-        ring
-    _ = _ := by
-      rw [←Real.exp_sum]
-      congr 1
-      simp only [Finset.sum_mul,Finset.sum_div]
-      apply Finset.sum_congr rfl
-      intro i _
-      ring
-
-theorem fairLinearSum_mgf_card {ι : Type*} [Fintype ι] (c : ι → ℝ)
-    (hc : ∀i,|c i|≤1) (t : ℝ) :
-    (fairField ι).expect (fun bits => Real.exp (t*fairLinearSum c bits))≤
-      Real.exp ((Fintype.card ι:ℝ)*t^2/8) := by
-  apply (fairLinearSum_mgf c t).trans
-  apply Real.exp_le_exp.mpr
-  have hs : (∑i,(c i)^2)≤(Fintype.card ι:ℝ) := by
-    calc
-      _ ≤ ∑_i : ι,(1:ℝ) := by
-        apply Finset.sum_le_sum
-        intro i _
-        have h:=abs_le.mp (hc i)
-        nlinarith [sq_nonneg (c i)]
-      _ = _ := by simp
-  gcongr
-
-/-- Two-sided finite Hoeffding bound with the source's sharp quadratic scale. -/
-theorem fairLinearSum_tail_variance {ι : Type*} [Fintype ι] (c : ι → ℝ)
-    {V : ℝ} (hV : 0<V) (hc : (∑i,(c i)^2)≤V) {a : ℝ} (ha : 0≤a) :
-    (fairField ι).expect (fun bits => if a≤|fairLinearSum c bits| then 1 else 0)≤
-      2*Real.exp (-2*a^2/V) := by
-  have oneSide (d : ι → ℝ) (hd : (∑i,(d i)^2)≤V) :
-      (fairField ι).expect (fun bits => if a≤fairLinearSum d bits then 1 else 0)≤
-        Real.exp (-2*a^2/V) := by
-    have H := finite_exp_markov Finset.univ (fairField ι).mass (fairLinearSum d)
-      (fun bits _ => (fairField ι).nonneg bits) a (4*a/V) (by positivity)
-    have hm : (fairField ι).expect (fun bits => Real.exp ((4*a/V)*fairLinearSum d bits))≤
-        Real.exp (V*(4*a/V)^2/8) := by
-      apply (fairLinearSum_mgf d (4*a/V)).trans
-      apply Real.exp_le_exp.mpr
-      gcongr
-    have he : (fairField ι).expect (fun bits => if a≤fairLinearSum d bits then 1 else 0)=
-        ∑bits∈Finset.univ.filter (fun bits => a≤fairLinearSum d bits),(fairField ι).mass bits := by
-      simp only [FiniteLaw.expect,Finset.sum_filter,mul_ite,mul_one,mul_zero]
-    rw [he]
-    calc
-      _ ≤ _ := H
-      _ ≤ Real.exp (-(4*a/V)*a)*
-          Real.exp (V*(4*a/V)^2/8) := by
-        exact mul_le_mul_of_nonneg_left hm (Real.exp_nonneg _)
-      _ = _ := by
-        rw [←Real.exp_add]
-        congr 1
-        field_simp [ne_of_gt hV]
-        ring
-  have hneg (bits : ι → Bool) : fairLinearSum (fun i => -c i) bits= -fairLinearSum c bits := by
-    simp only [fairLinearSum,neg_mul,Finset.sum_neg_distrib]
-  have hnegc : (∑i,(-c i)^2)≤V := by simpa only [neg_sq] using hc
-  have Hpos := oneSide c hc
-  have Hneg := oneSide (fun i => -c i) hnegc
-  have hb : (fairField ι).expect (fun bits => if a≤|fairLinearSum c bits| then 1 else 0)≤
-      (fairField ι).expect (fun bits => if a≤fairLinearSum c bits then 1 else 0)+
-      (fairField ι).expect (fun bits => if a≤fairLinearSum (fun i => -c i) bits then 1 else 0) := by
-    have hpoint (bits : ι → Bool) :
-        (if a≤|fairLinearSum c bits| then (1:ℝ) else 0)≤
-        (if a≤fairLinearSum c bits then 1 else 0)+(if a≤fairLinearSum (fun i => -c i) bits then 1 else 0) := by
-      rw [hneg]
-      by_cases h : 0≤fairLinearSum c bits
-      · rw [abs_of_nonneg h]; split_ifs <;> norm_num
-      · rw [abs_of_neg (lt_of_not_ge h)]; split_ifs <;> norm_num
-    unfold FiniteLaw.expect
-    rw [←Finset.sum_add_distrib]
-    apply Finset.sum_le_sum
-    intro bits _
-    simpa only [mul_add] using mul_le_mul_of_nonneg_left (hpoint bits) ((fairField ι).nonneg bits)
-  linarith
-
-/-- Cardinal specialization, without imposing any extra computation limits. -/
-theorem fairLinearSum_tail {ι : Type*} [Fintype ι] (c : ι → ℝ)
-    (hc : ∀i,|c i|≤1) (hn : 0<Fintype.card ι) {a : ℝ} (ha : 0≤a) :
-    (fairField ι).expect (fun bits => if a≤|fairLinearSum c bits| then 1 else 0)≤
-      2*Real.exp (-2*a^2/(Fintype.card ι:ℝ)) := by
-  apply fairLinearSum_tail_variance c (by exact_mod_cast hn) _ ha
-  calc
-    _ ≤ ∑_i : ι,(1:ℝ) := by
-      apply Finset.sum_le_sum
-      intro i _
-      have h:=abs_le.mp (hc i)
-      nlinarith [sq_nonneg (c i)]
-    _ = _ := by simp
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-namespace FiniteLaw
-variable {ι : Type*} [Fintype ι] [DecidableEq ι] {A : ι → Type*} [∀i,Fintype (A i)]
-
-/-- Independent observations at literal slot IDs, allowing different outcome
- types (side traces and old leaf oracles carry genuinely different data). -/
-def dependentProduct (P : ∀i,FiniteLaw (A i)) : FiniteLaw (∀i,A i) where
-  mass x := ∏i,(P i).mass (x i)
-  nonneg x := Finset.prod_nonneg (fun i _ => (P i).nonneg _)
-  total := by rw [←Fintype.prod_sum]; simp only [(P _).total,Finset.prod_const_one]
-
-theorem dependentProduct_expect_product (P : ∀i,FiniteLaw (A i)) (F : ∀i,A i → ℝ) :
-    (dependentProduct P).expect (fun x => ∏i,F i (x i))=∏i,(P i).expect (F i) := by
-  simp only [expect,dependentProduct,←Finset.prod_mul_distrib]
-  exact (Fintype.prod_sum (fun i a => (P i).mass a*F i a)).symm
-
-theorem expect_dependentProduct_succ {n : ℕ} {A : Fin (n+1) → Type*}
-    [∀i,Fintype (A i)] (P : ∀i,FiniteLaw (A i)) (F : (∀i,A i) → ℝ) :
-    (dependentProduct P).expect F=(P 0).expect (fun a =>
-      (dependentProduct (fun i : Fin n => P i.succ)).expect (fun x => F (Fin.cons a x))) := by
-  change (∑x : ∀i,A i,(∏i,(P i).mass (x i))*F x)=_
-  rw [←(Fin.consEquiv A).sum_comp]
-  simp only [Fintype.sum_prod_type,Fin.consEquiv,Equiv.coe_fn_mk,
-    Fin.prod_univ_succ,Fin.cons_zero,Fin.cons_succ,expect,dependentProduct,Finset.mul_sum,mul_assoc]
-
-/-- A genuine dependent Pi-law is invariant under a fixed bijective relabeling
- of slots. No observation is duplicated and no random order is transmitted. -/
-theorem dependentProduct_expect_equiv {κ : Type*} [Fintype κ] [DecidableEq κ] (P : ∀i,FiniteLaw (A i))
-    (e : κ ≃ ι) (F : (∀i,A i) → ℝ) :
-    (dependentProduct P).expect F=(dependentProduct (fun j => P (e j))).expect
-      (fun x => F (fun i => cast (congrArg A (e.apply_symm_apply i)) (x (e.symm i)))) := by
-  let E : (∀j,A (e j)) ≃ (∀i,A i) := Equiv.piCongrLeft A e
-  have he (x : ∀j,A (e j)) (i : ι) : E x i=
-      cast (congrArg A (e.apply_symm_apply i)) (x (e.symm i)) := by
-    exact Equiv.piCongrLeft_apply_eq_cast x i
-  unfold expect dependentProduct
-  rw [←E.sum_comp]
-  apply Finset.sum_congr rfl
-  intro x _
-  change (∏i,(P i).mass (E x i))*F (E x)=
-    (∏j,(P (e j)).mass (x j))*F (fun i => cast (congrArg A (e.apply_symm_apply i)) (x (e.symm i)))
-  apply congrArg₂ (· * ·)
-  · rw [←e.prod_comp]
-    apply Finset.prod_congr rfl
-    intro j _
-    congr 1
-    exact Equiv.piCongrLeft_apply_apply A e x j
-  · congr 1
-    funext i
-    exact he x i
-
-end FiniteLaw
-
-/-- Independent heterogeneous emissions in a literal finite chronological
- order accumulate precisely the sum of their trace approximation costs. -/
-theorem dependentCoordinate_approximates {n : ℕ} {A : Fin n → Type*}
-    [∀i,Fintype (A i)] {q : ℝ} (hq : q∈Set.Icc 0 1)
-    (P : ∀i,FiniteLaw (A i)) (f : ∀i,A i → List Bool)
-    (w : Fin n → List Bool) (E : Fin n → ℝ)
-    (h : ∀i,ApproximatesTrace q (P i) (f i) (w i) (E i)) :
-    ApproximatesTrace q (FiniteLaw.dependentProduct P)
-      (fun x => coordinateWord (fun i => f i (x i))) (coordinateWord w) (∑i,E i) := by
-  induction n with
-  | zero =>
-    intro F hF
-    change |(FiniteLaw.dependentProduct P).expect (fun _ => F [])-F []|≤∑i : Fin 0,E i
-    simp only [FiniteLaw.expect_const,sub_self,abs_zero,Finset.univ_eq_empty,Finset.sum_empty,le_refl]
-  | succ n ih =>
-    intro F hF
-    have ht := ih (fun i => P i.succ) (fun i => f i.succ)
-      (fun i => w i.succ) (fun i => E i.succ) (fun i => h i.succ)
-    have hh := ApproximatesTrace.append hq (h 0) ht F hF
-    rw [FiniteLaw.expect_prod] at hh
-    rw [FiniteLaw.expect_dependentProduct_succ,coordinateWord_succ,Fin.sum_univ_succ]
-    have he (a : A 0) (x : ∀i : Fin n,A i.succ) :
-        coordinateWord (fun i => f i ((Fin.cons a x : ∀i,A i) i))=
-          f 0 a++coordinateWord (fun i => f i.succ (x i)) := by
-      rw [coordinateWord_succ]
-      simp only [Fin.cons_zero,Fin.cons_succ]
-    simp_rw [he]
-    exact hh
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- Actual IDs of every local or global paired trial on the whole window tree. -/
-def WindowTrial (T n g : ℕ → ℕ) : ℕ → Type
-  | 0 => PEmpty
-  | b+1 => Fin (n b) ⊕ (Fin (g b) ⊕ (Fin (T b) × WindowTrial T n g b))
-
-noncomputable instance windowTrialFintype (T n g : ℕ → ℕ) :
-    (b : ℕ) → Fintype (WindowTrial T n g b)
-  | 0 => inferInstanceAs (Fintype PEmpty)
-  | b+1 => by
-    letI := windowTrialFintype T n g b
-    exact inferInstanceAs (Fintype (Fin (n b) ⊕ (Fin (g b) ⊕ (Fin (T b) × WindowTrial T n g b))))
-
-/-- A fixed bijection, independent of every bit, reveal and hypothesis. -/
-def windowAssignmentEquiv (T n g : ℕ → ℕ) (A : Type) :
-    (b : ℕ) → WindowAssignments T n g A b ≃ (WindowTrial T n g b → A)
-  | 0 =>
-    { toFun := fun _ i => PEmpty.elim i
-      invFun := fun _ => PUnit.unit
-      left_inv := fun x => by cases x; rfl
-      right_inv := fun x => by funext i; exact PEmpty.elim i }
-  | b+1 =>
-    { toFun := fun x i => match i with
-        | .inl k => x.1 k
-        | .inr (.inl k) => x.2.1 k
-        | .inr (.inr (u,k)) => windowAssignmentEquiv T n g A b (x.2.2 u) k
-      invFun := fun x => (fun k => x (.inl k),fun k => x (.inr (.inl k)),
-        fun u => (windowAssignmentEquiv T n g A b).symm (fun k => x (.inr (.inr (u,k)))))
-      left_inv := fun x => by
-        rcases x with ⟨loc,glob,kids⟩
-        change (loc,glob,fun u => (windowAssignmentEquiv T n g A b).symm
-          (windowAssignmentEquiv T n g A b (kids u)))=(loc,glob,kids)
-        simp only [Equiv.symm_apply_apply]
-
-      right_inv := fun x => by
-        funext i
-        rcases i with k | k
-        · rfl
-        · rcases k with k | ⟨u,k⟩
-          · rfl
-          · exact congrFun ((windowAssignmentEquiv T n g A b).apply_symm_apply
-              (fun k => x (.inr (.inr (u,k))))) k }
-
-theorem windowPosterior_mass_flat (T n g : ℕ → ℕ) (b : ℕ)
-    (d : WindowAssignments T n g (Option Bool) b) (bits : WindowAssignments T n g Bool b) :
-    (windowPosteriorLaw T n g b d).mass bits=
-      ∏i : WindowTrial T n g b,(FiniteLaw.flagPosterior
-        (windowAssignmentEquiv T n g (Option Bool) b d i)).mass
-          (windowAssignmentEquiv T n g Bool b bits i) := by
-  induction b with
-  | zero =>
-    change (1:ℝ)=∏i : PEmpty,_
-    simp
-  | succ b ih =>
-    change (∏i,(FiniteLaw.flagPosterior (d.1 i)).mass (bits.1 i))*
-      ((∏i,(FiniteLaw.flagPosterior (d.2.1 i)).mass (bits.2.1 i))*
-        ∏u,(windowPosteriorLaw T n g b (d.2.2 u)).mass (bits.2.2 u))=_
-    simp_rw [ih]
-    change _=∏i : Fin (n b) ⊕ (Fin (g b) ⊕ (Fin (T b) × WindowTrial T n g b)),
-      (FiniteLaw.flagPosterior (match i with
-        | .inl k => d.1 k
-        | .inr (.inl k) => d.2.1 k
-        | .inr (.inr (u,k)) => windowAssignmentEquiv T n g (Option Bool) b (d.2.2 u) k)).mass
-      (match i with
-        | .inl k => bits.1 k
-        | .inr (.inl k) => bits.2.1 k
-        | .inr (.inr (u,k)) => windowAssignmentEquiv T n g Bool b (bits.2.2 u) k)
-    simp only [Fintype.prod_sum_type,Fintype.prod_prod_type]
-
-/-- Exact flat posterior distribution, including all paths and zero-mass
- metadata. This is the same independent bit law, not a new assumption. -/
-theorem windowPosterior_expect_flat (T n g : ℕ → ℕ) (b : ℕ)
-    (d : WindowAssignments T n g (Option Bool) b)
-    (F : WindowAssignments T n g Bool b → ℝ) :
-    (windowPosteriorLaw T n g b d).expect F=
-      (FiniteLaw.indexedProduct (fun i : WindowTrial T n g b => FiniteLaw.flagPosterior
-        (windowAssignmentEquiv T n g (Option Bool) b d i))).expect
-          (fun bits => F ((windowAssignmentEquiv T n g Bool b).symm bits)) := by
-  unfold FiniteLaw.expect FiniteLaw.indexedProduct
-  rw [←(windowAssignmentEquiv T n g Bool b).sum_comp]
-  apply Finset.sum_congr rfl
-  intro bits _
-  dsimp only
-  rw [Equiv.symm_apply_apply,windowPosterior_mass_flat]
-
-def noReveals (T n g : ℕ → ℕ) (b : ℕ) : WindowAssignments T n g (Option Bool) b :=
-  (windowAssignmentEquiv T n g (Option Bool) b).symm (fun _ => none)
-
-theorem windowPosterior_fair_flat (T n g : ℕ → ℕ) (b : ℕ)
-    (F : WindowAssignments T n g Bool b → ℝ) :
-    (windowPosteriorLaw T n g b (noReveals T n g b)).expect F=
-      (FiniteLaw.indexedProduct (fun _ : WindowTrial T n g b => FiniteLaw.flagPosterior none)).expect
-        (fun bits => F ((windowAssignmentEquiv T n g Bool b).symm bits)) := by
-  rw [windowPosterior_expect_flat]
-  simp only [noReveals,Equiv.apply_symm_apply]
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- The literal global schedule is indexed by a fixed pair/block bijection. -/
-def globalTrialEquiv (N t : ℕ) : (Fin (N/4) × Fin t) ≃ Fin ((N/4)*t) :=
-  Equiv.ofBijective _ (globalTrialIndex_bijective N t)
-
-def sourceGlobalCoefficient (N t : ℕ) (u : Fin N) (k : Fin ((N/4)*t)) : ℝ :=
-  let v := ((globalTrialEquiv N t).symm k).1
-  if v.val≤u.val ∧ u.val+v.val<N then 1 else 0
-
-def sourceGroupCoefficient (T n : ℕ → ℕ) (b : ℕ) (u : Fin (T (b+1))) :
-    WindowTrial T n (sourceScheduledTrials T n) (b+2) → ℝ
-  | .inl _ => 0
-  | .inr (.inl k) => sourceGlobalCoefficient (T (b+1)) (n b) u k
-  | .inr (.inr (v,.inl _)) => if v=u then 1 else 0
-  | .inr (.inr (_,.inr _)) => 0
-
-theorem even_half_cast (t : ℕ) (ht : Even t) : ((t/2:ℕ):ℝ)=(t:ℝ)/2 := by
-  have h : 2*(t/2)=t := Nat.mul_div_cancel' ((even_iff_two_dvd).mp ht)
-  have hr : (2:ℝ)*(t/2:ℕ)=t := by exact_mod_cast h
-  linarith
-
-theorem scheduledGlobalShift_linear (N t : ℕ) (ht : Even t)
-    (bits : Fin ((N/4)*t) → Bool) (u : Fin N) :
-    (scheduledGlobalShift N t bits u:ℝ)=
-      ∑k,sourceGlobalCoefficient N t u k*((bitIndex (bits k):ℝ)-1/2) := by
-  rw [←(globalTrialEquiv N t).sum_comp]
-  simp only [Fintype.sum_prod_type,sourceGlobalCoefficient,Equiv.symm_apply_apply]
-  unfold scheduledGlobalShift
-  simp only [Int.cast_sum,Int.cast_ite,Int.cast_sub,Int.cast_natCast,Int.cast_zero]
-  apply Finset.sum_congr rfl
-  intro v _
-  split_ifs with hv
-  · simp only [one_mul,Finset.sum_sub_distrib,Finset.sum_const,Finset.card_univ,
-      Fintype.card_fin,nsmul_eq_mul]
-    rw [even_half_cast t ht]
-    simp only [globalTrialEquiv,Equiv.ofBijective_apply]
-    ring
-  · simp only [zero_mul,Finset.sum_const_zero]
-
-theorem sourceDisplacement_linear (T n : ℕ → ℕ) (b : ℕ) (ht : Even (n b))
-    (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool (b+2)) (u : Fin (T (b+1))) :
-    (sourceDisplacement (T (b+1)) (n b) (fun v => (bits.2.2 v).1) bits.2.1 u:ℝ)=
-      fairLinearSum (sourceGroupCoefficient T n b u)
-        (windowAssignmentEquiv T n (sourceScheduledTrials T n) Bool (b+2) bits) := by
-  rcases bits with ⟨root,global,children⟩
-  change Fin ((T (b+1)/4)*n b) → Bool at global
-  change Fin (T (b+1)) → ((Fin (n b) → Bool) ×
-    (Fin (sourceScheduledTrials T n b) → Bool) ×
-    (Fin (T b) → WindowAssignments T n (sourceScheduledTrials T n) Bool b)) at children
-  unfold fairLinearSum
-  change _=∑i : Fin (n (b+1)) ⊕ (Fin ((T (b+1)/4)*n b) ⊕
-      (Fin (T (b+1)) × (Fin (n b) ⊕ (Fin (sourceScheduledTrials T n b) ⊕
-        (Fin (T b) × WindowTrial T n (sourceScheduledTrials T n) b))))),_
-  simp only [Fintype.sum_sum_type,Fintype.sum_prod_type,sourceGroupCoefficient,
-    zero_mul,Finset.sum_const_zero,zero_add,add_zero]
-  change _=(∑k,sourceGlobalCoefficient (T (b+1)) (n b) u k*
-    ((bitIndex (global k):ℝ)-1/2))+
-    (∑v : Fin (T (b+1)),∑a : Fin (n b),
-      (if v=u then (1:ℝ) else 0)*((bitIndex ((children v).1 a):ℝ)-1/2))
-  have hl : (∑v : Fin (T (b+1)),∑a : Fin (n b),
-      (if v=u then (1:ℝ) else 0)*((bitIndex ((children v).1 a):ℝ)-1/2))=
-        (∑a : Fin (n b),(bitIndex ((children u).1 a):ℝ))-(n b:ℝ)/2 := by
-    simp only [ite_mul,one_mul,zero_mul]
-    rw [show (∑v : Fin (T (b+1)),∑a : Fin (n b),
-      if v=u then ((bitIndex ((children v).1 a):ℝ)-1/2) else 0)=
-        ∑v : Fin (T (b+1)),if v=u then
-          (∑a : Fin (n b),((bitIndex ((children v).1 a):ℝ)-1/2)) else 0 from by
-            apply Finset.sum_congr rfl
-            intro v _
-            split_ifs <;> simp]
-    rw [Finset.sum_ite_eq',ite_eq_left (Finset.mem_univ u)]
-    simp only [Finset.sum_sub_distrib,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]
-    ring
-  rw [hl,←scheduledGlobalShift_linear (T (b+1)) (n b) ht global u]
-  simp only [sourceDisplacement,Int.cast_add,Int.cast_sub,Int.cast_sum,Int.cast_natCast,
-    even_half_cast _ ht]
-  ring
-
-theorem sourceGroupCoefficient_variance (T n : ℕ → ℕ) (b : ℕ) (u : Fin (T (b+1))) :
-    (∑i,(sourceGroupCoefficient T n b u i)^2)≤((T (b+1):ℝ)+1)*(n b:ℝ) := by
-  change (∑i : Fin (n (b+1)) ⊕ (Fin ((T (b+1)/4)*n b) ⊕
-      (Fin (T (b+1)) × (Fin (n b) ⊕ (Fin (sourceScheduledTrials T n b) ⊕
-        (Fin (T b) × WindowTrial T n (sourceScheduledTrials T n) b))))),_)≤_
-  simp only [Fintype.sum_sum_type,Fintype.sum_prod_type,sourceGroupCoefficient,
-    zero_pow (by omega : 2≠0),Finset.sum_const_zero,zero_add,add_zero]
-  have hg : (∑k : Fin ((T (b+1)/4)*n b),(sourceGlobalCoefficient (T (b+1)) (n b) u k)^2)≤
-      ((T (b+1)/4)*n b:ℕ) := by
-    calc
-      _ ≤ ∑_k : Fin ((T (b+1)/4)*n b),(1:ℝ) := by
-        apply Finset.sum_le_sum
-        intro k _
-        dsimp only [sourceGlobalCoefficient]
-        split_ifs <;> norm_num
-      _ = _ := by simp
-  have hl : (∑v : Fin (T (b+1)),∑_a : Fin (n b),(if v=u then (1:ℝ) else 0)^2)=(n b:ℝ) := by
-    simp only [ite_pow,one_pow,zero_pow (by omega : 2≠0)]
-    simp
-  rw [hl]
-  have hdiv : ((T (b+1)/4:ℕ):ℝ)≤T (b+1) := by exact_mod_cast Nat.div_le_self (T (b+1)) 4
-  push_cast at hg
-  have hm := mul_le_mul_of_nonneg_right hdiv (Nat.cast_nonneg (n b): (0:ℝ)≤n b)
-  nlinarith only [hg,hm]
-
-/-- Actual per-window unconditional fit tail under the full tree's fair law.
- No independence between different window displacements is asserted. -/
-theorem sourceDisplacement_tail (T n : ℕ → ℕ) (b : ℕ) (ht : Even (n b))
-    (hn : 0<n b) (u : Fin (T (b+1))) {a : ℝ} (ha : 0≤a) :
-    (windowPosteriorLaw T n (sourceScheduledTrials T n) (b+2)
-      (noReveals T n (sourceScheduledTrials T n) (b+2))).expect
-      (fun bits => if a≤|(sourceDisplacement (T (b+1)) (n b)
-        (fun v => (bits.2.2 v).1) bits.2.1 u:ℝ)| then 1 else 0)≤
-      2*Real.exp (-2*a^2/(((T (b+1):ℝ)+1)*(n b:ℝ))) := by
-  rw [windowPosterior_fair_flat]
-  simp_rw [sourceDisplacement_linear T n b ht,Equiv.apply_symm_apply]
-  exact fairLinearSum_tail_variance (sourceGroupCoefficient T n b u)
-    (by positivity) (sourceGroupCoefficient_variance T n b u) ha
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-namespace FiniteLaw
-variable {α ι : Type*} [Fintype α] [Fintype ι]
-
-theorem expect_fintype_sum (P : FiniteLaw α) (f : ι → α → ℝ) :
-    P.expect (fun a => ∑i,f i a)=∑i,P.expect (f i) := by
-  simp only [expect,Finset.mul_sum]
-  exact Finset.sum_comm
-
-theorem expect_add (P : FiniteLaw α) (f g : α → ℝ) :
-    P.expect (fun a => f a+g a)=P.expect f+P.expect g := by
-  simp only [expect,mul_add,Finset.sum_add_distrib]
-
-theorem coordinateProduct_expect_eval {n : ℕ} (P : Fin n → FiniteLaw α)
-    (u : Fin n) (f : α → ℝ) :
-    (coordinateProduct P).expect (fun x => f (x u))=(P u).expect f := by
-  let F : Fin n → α → ℝ := fun j a => if j=u then f a else 1
-  have hp (x : Fin n → α) : (∏j,F j (x j))=f (x u) := by simp [F]
-  rw [show (fun x : Fin n → α => f (x u))=(fun x => ∏j,F j (x j)) by
-    funext x; exact (hp x).symm,coordinateProduct_expect_product]
-  have hj (j : Fin n) : (P j).expect (F j)=if j=u then (P j).expect f else 1 := by
-    by_cases h : j=u <;> simp [F,h,expect_const]
-  simp_rw [hj]
-  simp
-
-end FiniteLaw
-
-theorem noReveals_succ (T n g : ℕ → ℕ) (b : ℕ) :
-    noReveals T n g (b+1)=(fun _ => none,fun _ => none,fun _ => noReveals T n g b) := rfl
-
-theorem windowPosterior_child (T n g : ℕ → ℕ) (b : ℕ)
-    (d : WindowAssignments T n g (Option Bool) (b+1)) (u : Fin (T b))
-    (F : WindowAssignments T n g Bool b → ℝ) :
-    (windowPosteriorLaw T n g (b+1) d).expect (fun bits => F (bits.2.2 u))=
-      (windowPosteriorLaw T n g b (d.2.2 u)).expect F := by
-  rcases d with ⟨loc,glob,kids⟩
-  change ((posteriorArray loc).prod ((posteriorArray glob).prod
-    (FiniteLaw.coordinateProduct (fun i => windowPosteriorLaw T n g b (kids i))))).expect
-      (fun bits => F (bits.2.2 u))=_
-  rw [FiniteLaw.expect_prod]
-  simp_rw [FiniteLaw.expect_prod,FiniteLaw.coordinateProduct_expect_eval,FiniteLaw.expect_const]
-
-theorem failure_forall_le_sum {ι : Type*} [Fintype ι] (p : ι → Prop) [DecidablePred p] [Decidable (∀i,p i)] :
-    (if ∀i,p i then (0:ℝ) else 1)≤∑i,if p i then 0 else 1 := by
-  by_cases hp : ∀i,p i
-  · simp only [ite_eq_left hp]
-    exact Finset.sum_nonneg (fun i _ => by split_ifs <;> norm_num)
-  · simp only [ite_eq_right hp]
-    push Not at hp
-    obtain ⟨i,hi⟩ := hp
-    calc
-      (1:ℝ) = if p i then 0 else 1 := by simp [hi]
-      _ ≤ ∑j,if p j then 0 else 1 := by
-        apply Finset.single_le_sum (f:=fun j => if p j then (0:ℝ) else 1)
-        · intro j _; split_ifs <;> norm_num
-        · exact Finset.mem_univ i
-
-/-- Explicit finite tree union bound. The source's scale estimates will make
- this number exponentially small; no independence of sibling shifts is used. -/
-def sourceFitTail (T n : ℕ → ℕ) : ℕ → ℝ
-  | 0 => 0
-  | 1 => 0
-  | b+2 => (T (b+1):ℝ)*(2*Real.exp (-2*((T b:ℝ)/8)^2/(((T (b+1):ℝ)+1)*(n b:ℝ)))+
-      sourceFitTail T n (b+1))
-
-theorem sourceTreeFit_tail (T n : ℕ → ℕ) (b : ℕ)
-    (hn : ∀j<b,0<n j) (he : ∀j<b,Even (n j)) :
-    (windowPosteriorLaw T n (sourceScheduledTrials T n) b
-      (noReveals T n (sourceScheduledTrials T n) b)).expect
-        (fun bits => if SourceTreeFit T n b bits then 0 else 1) ≤ sourceFitTail T n b := by
-  induction b using Nat.twoStepInduction with
-  | zero =>
-    have hh (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool 0) : SourceTreeFit T n 0 bits := trivial
-    simp only [ite_eq_left (hh _),FiniteLaw.expect_const,sourceFitTail,le_refl]
-  | one =>
-    have hh (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool 1) : SourceTreeFit T n 1 bits := trivial
-    simp only [ite_eq_left (hh _),FiniteLaw.expect_const,sourceFitTail,le_refl]
-  | more b ih0 ih =>
-    let P := windowPosteriorLaw T n (sourceScheduledTrials T n) (b+2)
-      (noReveals T n (sourceScheduledTrials T n) (b+2))
-    let badShift : WindowAssignments T n (sourceScheduledTrials T n) Bool (b+2) → Fin (T (b+1)) → Prop :=
-      fun bits u => (T b:ℝ)/8≤|(sourceDisplacement (T (b+1)) (n b)
-        (fun v => (bits.2.2 v).1) bits.2.1 u:ℝ)|
-    have hp (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool (b+2)) :
-        (if SourceTreeFit T n (b+2) bits then (0:ℝ) else 1)≤
-          (∑u : Fin (T (b+1)),if badShift bits u then 1 else 0)+
-          (∑u : Fin (T (b+1)),if SourceTreeFit T n (b+1) (bits.2.2 u) then 0 else 1) := by
-      let p := fun u : Fin (T (b+1)) => |8*sourceDisplacement (T (b+1)) (n b)
-        (fun v => (bits.2.2 v).1) bits.2.1 u|≤(T b:ℤ)
-      have hh : (if SourceTreeFit T n (b+2) bits then (0:ℝ) else 1)≤
-          (if ∀u,p u then 0 else 1)+(if ∀u,SourceTreeFit T n (b+1) (bits.2.2 u) then 0 else 1) := by
-        by_cases H : SourceTreeFit T n (b+2) bits
-        · rw [ite_eq_left H]
-          split_ifs <;> norm_num
-        · rw [ite_eq_right H]
-          by_cases h1 : ∀u,p u
-          · have h2 : ¬∀u,SourceTreeFit T n (b+1) (bits.2.2 u) := fun h2 => H ⟨h1,h2⟩
-            rw [ite_eq_left h1,ite_eq_right h2]
-            norm_num
-          · rw [ite_eq_right h1]
-            split_ifs <;> norm_num
-      apply hh.trans
-      apply add_le_add
-      · apply (failure_forall_le_sum p).trans
-        apply Finset.sum_le_sum
-        intro u _
-        by_cases hu : p u
-        · simp only [ite_eq_left hu]
-          split_ifs <;> norm_num
-        · have hr : (T b:ℝ)<8*|(sourceDisplacement (T (b+1)) (n b)
-              (fun v => (bits.2.2 v).1) bits.2.1 u:ℝ)| := by
-            have hi : (T b:ℤ) < |8*sourceDisplacement (T (b+1)) (n b)
-              (fun v => (bits.2.2 v).1) bits.2.1 u| := lt_of_not_ge hu
-            have hi' : (T b:ℝ) < |(8*sourceDisplacement (T (b+1)) (n b)
-              (fun v => (bits.2.2 v).1) bits.2.1 u:ℤ)| := by exact_mod_cast hi
-            norm_num only [Int.cast_abs,Int.cast_mul,Int.cast_ofNat,abs_mul,abs_of_pos (by norm_num : (0:ℝ)<8)] at hi'
-            exact hi'
-          have hb : badShift bits u := by dsimp [badShift]; linarith
-          simp [hu,hb]
-      · exact failure_forall_le_sum _
-    calc
-      _ ≤ P.expect (fun bits => (∑u : Fin (T (b+1)),if badShift bits u then 1 else 0)+
-          (∑u : Fin (T (b+1)),if SourceTreeFit T n (b+1) (bits.2.2 u) then 0 else 1)) :=
-            P.expect_mono hp
-      _ = (∑u : Fin (T (b+1)),P.expect (fun bits => if badShift bits u then 1 else 0))+
-          (∑u : Fin (T (b+1)),P.expect (fun bits => if SourceTreeFit T n (b+1) (bits.2.2 u) then 0 else 1)) := by
-            rw [FiniteLaw.expect_add,FiniteLaw.expect_fintype_sum,FiniteLaw.expect_fintype_sum]
-      _ ≤ (∑_u : Fin (T (b+1)),2*Real.exp (-2*((T b:ℝ)/8)^2/(((T (b+1):ℝ)+1)*(n b:ℝ))))+
-          ∑_u : Fin (T (b+1)),sourceFitTail T n (b+1) := by
-            apply add_le_add
-            · apply Finset.sum_le_sum
-              intro u _
-              exact sourceDisplacement_tail T n b (he b (by omega)) (hn b (by omega)) u (by positivity)
-            · apply Finset.sum_le_sum
-              intro u _
-              dsimp only [P]
-              rw [windowPosterior_child T n (sourceScheduledTrials T n) (b+1)
-                (noReveals T n (sourceScheduledTrials T n) (b+2)) u
-                (fun x => if SourceTreeFit T n (b+1) x then 0 else 1)]
-              rw [noReveals_succ]
-              exact ih (fun j hj => hn j (by omega)) (fun j hj => he j (by omega))
-      _ = sourceFitTail T n (b+2) := by simp [sourceFitTail,mul_add]
-
-end TraceReconstruction
-
-end
-end
-end
-end
-end
-end
-end
-
-end OAI
+import OAI.Probability.TraceReconstruction.Layout
+
+namespace OAI
+
+/-!
+# Metadata
+
+Hidden metadata, fair jitter, heterogeneous emissions, independent window trials, concentration, and fit.
+-/
+
+open scoped ENNReal
+noncomputable section
+
+open Filter
+open scoped Topology
+
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+namespace FiniteLaw
+variable {α : Type*} [Fintype α]
+
+theorem pairedReportMass_none (P Q H : FiniteLaw α) (ω : ℝ) (z : α × α) :
+    pairedReportMass P Q H ω (none,z)=pairedHiddenMass H ω z := by
+  simp only [pairedReportMass,Fintype.sum_bool,pairedFlagMass]
+  ring
+
+theorem pairedReport_hidden_expect (P Q H : FiniteLaw α) {ω : ℝ} (hω : 0≤ω)
+    (hP : ∀a,ω*H.mass a≤P.mass a) (hQ : ∀a,ω*H.mass a≤Q.mass a) (F : ℕ → ℝ) :
+    (pairedReport P Q H hω hP hQ).expect (fun z => F (hiddenTrial z.1))=
+      ω^2*F 1+(1-ω^2)*F 0 := by
+  let R := pairedReport P Q H hω hP hQ
+  have hp : R.expect (fun z => if z.1=none then (1:ℝ) else 0)=ω^2 := by
+    simp only [expect,R,pairedReport,Fintype.sum_prod_type,Fintype.sum_option]
+    simp only [reduceCtorEq,ite_false,mul_zero,Finset.sum_const_zero,add_zero]
+    simpa only [pairedReportMass_none,Fintype.sum_prod_type,ite_true,mul_one] using sum_pairedHiddenMass H ω
+  have he (z : Option Bool × (α × α)) : F (hiddenTrial z.1)=
+      F 0+(F 1-F 0)*(if z.1=none then 1 else 0) := by
+    rcases z with ⟨d,z⟩; cases d <;> simp [hiddenTrial]
+  simp_rw [he]
+  have hadd : R.expect (fun z => F 0+(F 1-F 0)*(if z.1=none then 1 else 0))=
+      R.expect (fun _ => F 0)+R.expect (fun z => (F 1-F 0)*(if z.1=none then 1 else 0)) := by
+    simp only [expect,mul_add,Finset.sum_add_distrib]
+  rw [hadd,expect_const,expect_const_mul,hp]
+  ring
+
+/-- Centered hidden-flag MGF, derived from the literal paired experiment. -/
+theorem pairedReport_hidden_mgf (P Q H : FiniteLaw α) {ω t : ℝ} (hω : 0≤ω)
+    (hP : ∀a,ω*H.mass a≤P.mass a) (hQ : ∀a,ω*H.mass a≤Q.mass a)
+    (hp : ω^2∈Set.Icc 0 1) (ht : |t|≤1) :
+    (pairedReport P Q H hω hP hQ).expect (fun z =>
+      Real.exp (t*((hiddenTrial z.1:ℕ)-ω^2)))≤Real.exp (t^2) := by
+  rw [pairedReport_hidden_expect P Q H hω hP hQ (fun k => Real.exp (t*((k:ℝ)-ω^2)))]
+  have H := binomialMass_centered_mgf 1 hp ht
+  norm_num [Finset.sum_range_succ,binomialMass] at H ⊢
+  convert H using 1 ; ring
+
+end FiniteLaw
+
+namespace PairedMechanisms
+variable {Side : Type*} [Fintype Side] {m : ℕ} (M : PairedMechanisms Side m)
+
+def hiddenCount (z : Reports Side m) : ℕ := ∑i,hiddenTrial (z i).1
+
+/-- No independence of displacements is presumed: only the flags of the
+ distinct literal paired mechanisms are independent. -/
+theorem hiddenCount_mgf {ω t : ℝ} (hw : ∀i,M.hiddenWeight i=ω)
+    (hp : ω^2∈Set.Icc 0 1) (ht : |t|≤1) :
+    M.reports.expect (fun z => Real.exp (t*((hiddenCount z:ℕ)-(m:ℝ)*ω^2)))≤
+      Real.exp ((m:ℝ)*t^2) := by
+  have he (z : Reports Side m) :
+      Real.exp (t*((hiddenCount z:ℕ)-(m:ℝ)*ω^2))=
+        ∏i : Fin m,Real.exp (t*((hiddenTrial (z i).1:ℕ)-ω^2)) := by
+    rw [←Real.exp_sum]
+    congr 1
+    simp only [mul_sub,Finset.mul_sum,Finset.sum_sub_distrib,Finset.sum_const,Finset.card_univ,Fintype.card_fin,
+      nsmul_eq_mul,hiddenCount,Nat.cast_sum]
+    ring
+  simp_rw [he]
+  rw [reports,FiniteLaw.coordinateProduct_expect_product _
+    (fun (_ : Fin m) (z : Option Bool × (Side × Side)) => Real.exp (t*((hiddenTrial z.1:ℕ)-ω^2))) ]
+  calc
+    _ ≤ ∏i : Fin m,Real.exp (t^2) := by
+      apply Finset.prod_le_prod₀
+      · intro i _
+        exact Finset.sum_nonneg (fun z _ => mul_nonneg
+          ((FiniteLaw.pairedReport (M.first i) (M.second i) (M.common i) (M.weight_nonneg i)
+            (M.common_first i) (M.common_second i)).nonneg z) (Real.exp_nonneg _))
+      · intro i _
+        have h := FiniteLaw.pairedReport_hidden_mgf (M.first i) (M.second i) (M.common i)
+          (M.weight_nonneg i) (M.common_first i) (M.common_second i) (by simpa [hw i] using hp) ht
+        simpa only [hw i] using h
+    _ = _ := by simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fin,Real.exp_nat_mul]
+
+/-- A quantitative unconditional bound for failure of a source hidden-count
+ condition. The exponent ω⁴m is sufficient for the uniform scale margins. -/
+theorem hiddenCount_small {ω : ℝ} (hw : ∀i,M.hiddenWeight i=ω) (hp : ω^2∈Set.Icc 0 1) :
+    M.reports.expect (fun z => if (hiddenCount z:ℝ)<(m:ℝ)*ω^2/2 then 1 else 0)≤
+      Real.exp (-(m:ℝ)*ω^4/16) := by
+  have ht : 0≤ω^2/4 := by positivity
+  have ht' : |-(ω^2/4)|≤1 := by rw [abs_neg,abs_of_nonneg ht]; linarith [hp.2]
+  have H := finite_exp_markov Finset.univ M.reports.mass
+    (fun z => -((hiddenCount z:ℕ)-(m:ℝ)*ω^2)) (fun z _ => M.reports.nonneg z)
+    ((m:ℝ)*ω^2/2) (ω^2/4) ht
+  have hm := M.hiddenCount_mgf hw hp ht'
+  have hsub : M.reports.expect (fun z => if (hiddenCount z:ℝ)<(m:ℝ)*ω^2/2 then 1 else 0)≤
+      ∑z∈Finset.univ.filter (fun z => (m:ℝ)*ω^2/2≤-((hiddenCount z:ℕ)-(m:ℝ)*ω^2)),M.reports.mass z := by
+    rw [Finset.sum_filter]
+    apply Finset.sum_le_sum
+    intro z _
+    dsimp only
+    by_cases hz : (hiddenCount z:ℝ)<(m:ℝ)*ω^2/2
+    · rw [ite_eq_left hz,ite_eq_left (by linarith : (m:ℝ)*ω^2/2≤-((hiddenCount z:ℕ)-(m:ℝ)*ω^2)),mul_one]
+    · rw [ite_eq_right hz,mul_zero]; split_ifs <;> positivity [M.reports.nonneg z]
+  calc
+    _ ≤ _ := hsub.trans H
+    _ ≤ Real.exp (-(ω^2/4)*((m:ℝ)*ω^2/2))*Real.exp ((m:ℝ)*(-(ω^2/4))^2) := by
+      apply mul_le_mul_of_nonneg_left _ (Real.exp_nonneg _)
+      simpa only [FiniteLaw.expect,neg_mul,mul_neg,neg_neg] using hm
+    _ = _ := by rw [←Real.exp_add]; congr 1; ring
+
+/-- Exact Bayes averaging under all metadata, including impossible reports. -/
+theorem posterior_average (F : (Fin m → Bool) → ℝ) :
+    M.reports.expect (fun z => (orientationsGiven z).expect F)=
+      (fairOrientations m).expect F := by
+  unfold FiniteLaw.expect
+  simp only [Finset.mul_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro b _
+  have hb (z : Reports Side m) : M.reports.mass z*(orientationsGiven z).mass b=
+      (fairOrientations m).mass b*(M.reportsGiven b).mass z := by
+    change (∏i,FiniteLaw.pairedReportMass (M.first i) (M.second i) (M.common i) (M.hiddenWeight i) (z i))*
+      (∏i,(FiniteLaw.flagPosterior (z i).1).mass (b i))=_
+    rw [←FiniteLaw.pairedFlags_product_bayes]
+    rw [Finset.prod_mul_distrib]
+    rfl
+  simp_rw [←mul_assoc,hb,mul_assoc]
+  rw [←Finset.mul_sum,←Finset.sum_mul,(M.reportsGiven b).total,one_mul]
+
+/-- Markov's inequality for the actual conditional failure probability.
+ This does not condition any orientation averages on fitting paths. -/
+theorem posterior_bad_mass {a : ℝ} (_ha : 0<a) (bad : (Fin m → Bool) → Prop)
+    [DecidablePred bad] :
+    a*M.reports.expect (fun z => if a≤(orientationsGiven z).expect (fun b => if bad b then 1 else 0)
+      then 1 else 0) ≤ (fairOrientations m).expect (fun b => if bad b then 1 else 0) := by
+  rw [←M.posterior_average,←FiniteLaw.expect_const_mul]
+  apply FiniteLaw.expect_mono
+  intro z
+  split_ifs with hz
+  · simpa using hz
+  · simp only [mul_zero]
+    exact Finset.sum_nonneg (fun b _ => mul_nonneg ((orientationsGiven z).nonneg b)
+      (by dsimp only; split_ifs <;> norm_num))
+
+end PairedMechanisms
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+namespace FiniteLaw
+variable {ι α : Type*} [Fintype ι] [Fintype α]
+
+theorem indexedProduct_expect_product (P : ι → FiniteLaw α) (F : ι → α → ℝ) :
+    (indexedProduct P).expect (fun x => ∏i,F i (x i))=∏i,(P i).expect (F i) := by
+  simp only [expect,indexedProduct,←Finset.prod_mul_distrib]
+  exact (Fintype.prod_sum (fun i a => (P i).mass a*F i a)).symm
+
+end FiniteLaw
+
+/-- The unconditional orientation distribution on any actual finite trial ID set. -/
+def fairField (ι : Type*) [Fintype ι] : FiniteLaw (ι → Bool) :=
+  FiniteLaw.indexedProduct (fun _ => FiniteLaw.flagPosterior none)
+
+def fairLinearSum {ι : Type*} [Fintype ι] (c : ι → ℝ) (bits : ι → Bool) : ℝ :=
+  ∑i,c i*((bitIndex (bits i):ℝ)-1/2)
+
+/-- Exact MGF calculation for each paired orientation bit. -/
+theorem fairBit_mgf (t : ℝ) :
+    (FiniteLaw.flagPosterior none).expect (fun b => Real.exp (t*((bitIndex b:ℝ)-1/2)))=
+      Real.cosh (t/2) := by
+  simp only [FiniteLaw.expect,Fintype.sum_bool,FiniteLaw.flagPosterior,bitIndex]
+  norm_num only [Bool.false_eq_true,↓reduceIte,Nat.cast_zero,Nat.cast_one,decide_true,decide_false]
+  rw [Real.cosh_eq]
+  ring_nf
+
+/-- Hoeffding's exponential estimate for the actual independently fair bits,
+ allowing deterministic zero coefficients for completed/nonactive global pairs. -/
+theorem fairLinearSum_mgf {ι : Type*} [Fintype ι] (c : ι → ℝ) (t : ℝ) :
+    (fairField ι).expect (fun bits => Real.exp (t*fairLinearSum c bits))≤
+      Real.exp ((∑i,(c i)^2)*t^2/8) := by
+  have he (bits : ι → Bool) : Real.exp (t*fairLinearSum c bits)=
+      ∏i,Real.exp ((t*c i)*((bitIndex (bits i):ℝ)-1/2)) := by
+    rw [←Real.exp_sum]
+    congr 1
+    simp only [fairLinearSum,Finset.mul_sum,mul_assoc]
+  simp_rw [he]
+  rw [fairField,FiniteLaw.indexedProduct_expect_product _
+    (fun i b => Real.exp ((t*c i)*((bitIndex b:ℝ)-1/2)))]
+  simp_rw [fairBit_mgf]
+  calc
+    _ ≤ ∏i,Real.exp ((t*c i)^2/8) := by
+      apply Finset.prod_le_prod₀
+      · intro i _; exact (Real.cosh_pos _).le
+      · intro i _
+        convert Real.cosh_le_exp_half_sq (t*c i/2) using 1
+        congr 1
+        ring
+    _ = _ := by
+      rw [←Real.exp_sum]
+      congr 1
+      simp only [Finset.sum_mul,Finset.sum_div]
+      apply Finset.sum_congr rfl
+      intro i _
+      ring
+
+theorem fairLinearSum_mgf_card {ι : Type*} [Fintype ι] (c : ι → ℝ)
+    (hc : ∀i,|c i|≤1) (t : ℝ) :
+    (fairField ι).expect (fun bits => Real.exp (t*fairLinearSum c bits))≤
+      Real.exp ((Fintype.card ι:ℝ)*t^2/8) := by
+  apply (fairLinearSum_mgf c t).trans
+  apply Real.exp_le_exp.mpr
+  have hs : (∑i,(c i)^2)≤(Fintype.card ι:ℝ) := by
+    calc
+      _ ≤ ∑_i : ι,(1:ℝ) := by
+        apply Finset.sum_le_sum
+        intro i _
+        have h:=abs_le.mp (hc i)
+        nlinarith [sq_nonneg (c i)]
+      _ = _ := by simp
+  gcongr
+
+/-- Two-sided finite Hoeffding bound with the source's sharp quadratic scale. -/
+theorem fairLinearSum_tail_variance {ι : Type*} [Fintype ι] (c : ι → ℝ)
+    {V : ℝ} (hV : 0<V) (hc : (∑i,(c i)^2)≤V) {a : ℝ} (ha : 0≤a) :
+    (fairField ι).expect (fun bits => if a≤|fairLinearSum c bits| then 1 else 0)≤
+      2*Real.exp (-2*a^2/V) := by
+  have oneSide (d : ι → ℝ) (hd : (∑i,(d i)^2)≤V) :
+      (fairField ι).expect (fun bits => if a≤fairLinearSum d bits then 1 else 0)≤
+        Real.exp (-2*a^2/V) := by
+    have H := finite_exp_markov Finset.univ (fairField ι).mass (fairLinearSum d)
+      (fun bits _ => (fairField ι).nonneg bits) a (4*a/V) (by positivity)
+    have hm : (fairField ι).expect (fun bits => Real.exp ((4*a/V)*fairLinearSum d bits))≤
+        Real.exp (V*(4*a/V)^2/8) := by
+      apply (fairLinearSum_mgf d (4*a/V)).trans
+      apply Real.exp_le_exp.mpr
+      gcongr
+    have he : (fairField ι).expect (fun bits => if a≤fairLinearSum d bits then 1 else 0)=
+        ∑bits∈Finset.univ.filter (fun bits => a≤fairLinearSum d bits),(fairField ι).mass bits := by
+      simp only [FiniteLaw.expect,Finset.sum_filter,mul_ite,mul_one,mul_zero]
+    rw [he]
+    calc
+      _ ≤ _ := H
+      _ ≤ Real.exp (-(4*a/V)*a)*
+          Real.exp (V*(4*a/V)^2/8) := by
+        exact mul_le_mul_of_nonneg_left hm (Real.exp_nonneg _)
+      _ = _ := by
+        rw [←Real.exp_add]
+        congr 1
+        field_simp [ne_of_gt hV]
+        ring
+  have hneg (bits : ι → Bool) : fairLinearSum (fun i => -c i) bits= -fairLinearSum c bits := by
+    simp only [fairLinearSum,neg_mul,Finset.sum_neg_distrib]
+  have hnegc : (∑i,(-c i)^2)≤V := by simpa only [neg_sq] using hc
+  have Hpos := oneSide c hc
+  have Hneg := oneSide (fun i => -c i) hnegc
+  have hb : (fairField ι).expect (fun bits => if a≤|fairLinearSum c bits| then 1 else 0)≤
+      (fairField ι).expect (fun bits => if a≤fairLinearSum c bits then 1 else 0)+
+      (fairField ι).expect (fun bits => if a≤fairLinearSum (fun i => -c i) bits then 1 else 0) := by
+    have hpoint (bits : ι → Bool) :
+        (if a≤|fairLinearSum c bits| then (1:ℝ) else 0)≤
+        (if a≤fairLinearSum c bits then 1 else 0)+(if a≤fairLinearSum (fun i => -c i) bits then 1 else 0) := by
+      rw [hneg]
+      by_cases h : 0≤fairLinearSum c bits
+      · rw [abs_of_nonneg h]; split_ifs <;> norm_num
+      · rw [abs_of_neg (lt_of_not_ge h)]; split_ifs <;> norm_num
+    unfold FiniteLaw.expect
+    rw [←Finset.sum_add_distrib]
+    apply Finset.sum_le_sum
+    intro bits _
+    simpa only [mul_add] using mul_le_mul_of_nonneg_left (hpoint bits) ((fairField ι).nonneg bits)
+  linarith
+
+/-- Cardinal specialization, without imposing any extra computation limits. -/
+theorem fairLinearSum_tail {ι : Type*} [Fintype ι] (c : ι → ℝ)
+    (hc : ∀i,|c i|≤1) (hn : 0<Fintype.card ι) {a : ℝ} (ha : 0≤a) :
+    (fairField ι).expect (fun bits => if a≤|fairLinearSum c bits| then 1 else 0)≤
+      2*Real.exp (-2*a^2/(Fintype.card ι:ℝ)) := by
+  apply fairLinearSum_tail_variance c (by exact_mod_cast hn) _ ha
+  calc
+    _ ≤ ∑_i : ι,(1:ℝ) := by
+      apply Finset.sum_le_sum
+      intro i _
+      have h:=abs_le.mp (hc i)
+      nlinarith [sq_nonneg (c i)]
+    _ = _ := by simp
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+namespace FiniteLaw
+variable {ι : Type*} [Fintype ι] [DecidableEq ι] {A : ι → Type*} [∀i,Fintype (A i)]
+
+/-- Independent observations at literal slot IDs, allowing different outcome
+ types (side traces and old leaf oracles carry genuinely different data). -/
+def dependentProduct (P : ∀i,FiniteLaw (A i)) : FiniteLaw (∀i,A i) where
+  mass x := ∏i,(P i).mass (x i)
+  nonneg x := Finset.prod_nonneg (fun i _ => (P i).nonneg _)
+  total := by rw [←Fintype.prod_sum]; simp only [(P _).total,Finset.prod_const_one]
+
+theorem dependentProduct_expect_product (P : ∀i,FiniteLaw (A i)) (F : ∀i,A i → ℝ) :
+    (dependentProduct P).expect (fun x => ∏i,F i (x i))=∏i,(P i).expect (F i) := by
+  simp only [expect,dependentProduct,←Finset.prod_mul_distrib]
+  exact (Fintype.prod_sum (fun i a => (P i).mass a*F i a)).symm
+
+theorem expect_dependentProduct_succ {n : ℕ} {A : Fin (n+1) → Type*}
+    [∀i,Fintype (A i)] (P : ∀i,FiniteLaw (A i)) (F : (∀i,A i) → ℝ) :
+    (dependentProduct P).expect F=(P 0).expect (fun a =>
+      (dependentProduct (fun i : Fin n => P i.succ)).expect (fun x => F (Fin.cons a x))) := by
+  change (∑x : ∀i,A i,(∏i,(P i).mass (x i))*F x)=_
+  rw [←(Fin.consEquiv A).sum_comp]
+  simp only [Fintype.sum_prod_type,Fin.consEquiv,Equiv.coe_fn_mk,
+    Fin.prod_univ_succ,Fin.cons_zero,Fin.cons_succ,expect,dependentProduct,Finset.mul_sum,mul_assoc]
+
+/-- A genuine dependent Pi-law is invariant under a fixed bijective relabeling
+ of slots. No observation is duplicated and no random order is transmitted. -/
+theorem dependentProduct_expect_equiv {κ : Type*} [Fintype κ] [DecidableEq κ] (P : ∀i,FiniteLaw (A i))
+    (e : κ ≃ ι) (F : (∀i,A i) → ℝ) :
+    (dependentProduct P).expect F=(dependentProduct (fun j => P (e j))).expect
+      (fun x => F (fun i => cast (congrArg A (e.apply_symm_apply i)) (x (e.symm i)))) := by
+  let E : (∀j,A (e j)) ≃ (∀i,A i) := Equiv.piCongrLeft A e
+  have he (x : ∀j,A (e j)) (i : ι) : E x i=
+      cast (congrArg A (e.apply_symm_apply i)) (x (e.symm i)) := by
+    exact Equiv.piCongrLeft_apply_eq_cast x i
+  unfold expect dependentProduct
+  rw [←E.sum_comp]
+  apply Finset.sum_congr rfl
+  intro x _
+  change (∏i,(P i).mass (E x i))*F (E x)=
+    (∏j,(P (e j)).mass (x j))*F (fun i => cast (congrArg A (e.apply_symm_apply i)) (x (e.symm i)))
+  apply congrArg₂ (· * ·)
+  · rw [←e.prod_comp]
+    apply Finset.prod_congr rfl
+    intro j _
+    congr 1
+    exact Equiv.piCongrLeft_apply_apply A e x j
+  · congr 1
+    funext i
+    exact he x i
+
+end FiniteLaw
+
+/-- Independent heterogeneous emissions in a literal finite chronological
+ order accumulate precisely the sum of their trace approximation costs. -/
+theorem dependentCoordinate_approximates {n : ℕ} {A : Fin n → Type*}
+    [∀i,Fintype (A i)] {q : ℝ} (hq : q∈Set.Icc 0 1)
+    (P : ∀i,FiniteLaw (A i)) (f : ∀i,A i → List Bool)
+    (w : Fin n → List Bool) (E : Fin n → ℝ)
+    (h : ∀i,ApproximatesTrace q (P i) (f i) (w i) (E i)) :
+    ApproximatesTrace q (FiniteLaw.dependentProduct P)
+      (fun x => coordinateWord (fun i => f i (x i))) (coordinateWord w) (∑i,E i) := by
+  induction n with
+  | zero =>
+    intro F hF
+    change |(FiniteLaw.dependentProduct P).expect (fun _ => F [])-F []|≤∑i : Fin 0,E i
+    simp only [FiniteLaw.expect_const,sub_self,abs_zero,Finset.univ_eq_empty,Finset.sum_empty,le_refl]
+  | succ n ih =>
+    intro F hF
+    have ht := ih (fun i => P i.succ) (fun i => f i.succ)
+      (fun i => w i.succ) (fun i => E i.succ) (fun i => h i.succ)
+    have hh := ApproximatesTrace.append hq (h 0) ht F hF
+    rw [FiniteLaw.expect_prod] at hh
+    rw [FiniteLaw.expect_dependentProduct_succ,coordinateWord_succ,Fin.sum_univ_succ]
+    have he (a : A 0) (x : ∀i : Fin n,A i.succ) :
+        coordinateWord (fun i => f i ((Fin.cons a x : ∀i,A i) i))=
+          f 0 a++coordinateWord (fun i => f i.succ (x i)) := by
+      rw [coordinateWord_succ]
+      simp only [Fin.cons_zero,Fin.cons_succ]
+    simp_rw [he]
+    exact hh
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- Actual IDs of every local or global paired trial on the whole window tree. -/
+def WindowTrial (T n g : ℕ → ℕ) : ℕ → Type
+  | 0 => PEmpty
+  | b+1 => Fin (n b) ⊕ (Fin (g b) ⊕ (Fin (T b) × WindowTrial T n g b))
+
+noncomputable instance windowTrialFintype (T n g : ℕ → ℕ) :
+    (b : ℕ) → Fintype (WindowTrial T n g b)
+  | 0 => inferInstanceAs (Fintype PEmpty)
+  | b+1 => by
+    letI := windowTrialFintype T n g b
+    exact inferInstanceAs (Fintype (Fin (n b) ⊕ (Fin (g b) ⊕ (Fin (T b) × WindowTrial T n g b))))
+
+/-- A fixed bijection, independent of every bit, reveal and hypothesis. -/
+def windowAssignmentEquiv (T n g : ℕ → ℕ) (A : Type) :
+    (b : ℕ) → WindowAssignments T n g A b ≃ (WindowTrial T n g b → A)
+  | 0 =>
+    { toFun := fun _ i => PEmpty.elim i
+      invFun := fun _ => PUnit.unit
+      left_inv := fun x => by cases x; rfl
+      right_inv := fun x => by funext i; exact PEmpty.elim i }
+  | b+1 =>
+    { toFun := fun x i => match i with
+        | .inl k => x.1 k
+        | .inr (.inl k) => x.2.1 k
+        | .inr (.inr (u,k)) => windowAssignmentEquiv T n g A b (x.2.2 u) k
+      invFun := fun x => (fun k => x (.inl k),fun k => x (.inr (.inl k)),
+        fun u => (windowAssignmentEquiv T n g A b).symm (fun k => x (.inr (.inr (u,k)))))
+      left_inv := fun x => by
+        rcases x with ⟨loc,glob,kids⟩
+        change (loc,glob,fun u => (windowAssignmentEquiv T n g A b).symm
+          (windowAssignmentEquiv T n g A b (kids u)))=(loc,glob,kids)
+        simp only [Equiv.symm_apply_apply]
+
+      right_inv := fun x => by
+        funext i
+        rcases i with k | k
+        · rfl
+        · rcases k with k | ⟨u,k⟩
+          · rfl
+          · exact congrFun ((windowAssignmentEquiv T n g A b).apply_symm_apply
+              (fun k => x (.inr (.inr (u,k))))) k }
+
+theorem windowPosterior_mass_flat (T n g : ℕ → ℕ) (b : ℕ)
+    (d : WindowAssignments T n g (Option Bool) b) (bits : WindowAssignments T n g Bool b) :
+    (windowPosteriorLaw T n g b d).mass bits=
+      ∏i : WindowTrial T n g b,(FiniteLaw.flagPosterior
+        (windowAssignmentEquiv T n g (Option Bool) b d i)).mass
+          (windowAssignmentEquiv T n g Bool b bits i) := by
+  induction b with
+  | zero =>
+    change (1:ℝ)=∏i : PEmpty,_
+    simp
+  | succ b ih =>
+    change (∏i,(FiniteLaw.flagPosterior (d.1 i)).mass (bits.1 i))*
+      ((∏i,(FiniteLaw.flagPosterior (d.2.1 i)).mass (bits.2.1 i))*
+        ∏u,(windowPosteriorLaw T n g b (d.2.2 u)).mass (bits.2.2 u))=_
+    simp_rw [ih]
+    change _=∏i : Fin (n b) ⊕ (Fin (g b) ⊕ (Fin (T b) × WindowTrial T n g b)),
+      (FiniteLaw.flagPosterior (match i with
+        | .inl k => d.1 k
+        | .inr (.inl k) => d.2.1 k
+        | .inr (.inr (u,k)) => windowAssignmentEquiv T n g (Option Bool) b (d.2.2 u) k)).mass
+      (match i with
+        | .inl k => bits.1 k
+        | .inr (.inl k) => bits.2.1 k
+        | .inr (.inr (u,k)) => windowAssignmentEquiv T n g Bool b (bits.2.2 u) k)
+    simp only [Fintype.prod_sum_type,Fintype.prod_prod_type]
+
+/-- Exact flat posterior distribution, including all paths and zero-mass
+ metadata. This is the same independent bit law, not a new assumption. -/
+theorem windowPosterior_expect_flat (T n g : ℕ → ℕ) (b : ℕ)
+    (d : WindowAssignments T n g (Option Bool) b)
+    (F : WindowAssignments T n g Bool b → ℝ) :
+    (windowPosteriorLaw T n g b d).expect F=
+      (FiniteLaw.indexedProduct (fun i : WindowTrial T n g b => FiniteLaw.flagPosterior
+        (windowAssignmentEquiv T n g (Option Bool) b d i))).expect
+          (fun bits => F ((windowAssignmentEquiv T n g Bool b).symm bits)) := by
+  unfold FiniteLaw.expect FiniteLaw.indexedProduct
+  rw [←(windowAssignmentEquiv T n g Bool b).sum_comp]
+  apply Finset.sum_congr rfl
+  intro bits _
+  dsimp only
+  rw [Equiv.symm_apply_apply,windowPosterior_mass_flat]
+
+def noReveals (T n g : ℕ → ℕ) (b : ℕ) : WindowAssignments T n g (Option Bool) b :=
+  (windowAssignmentEquiv T n g (Option Bool) b).symm (fun _ => none)
+
+theorem windowPosterior_fair_flat (T n g : ℕ → ℕ) (b : ℕ)
+    (F : WindowAssignments T n g Bool b → ℝ) :
+    (windowPosteriorLaw T n g b (noReveals T n g b)).expect F=
+      (FiniteLaw.indexedProduct (fun _ : WindowTrial T n g b => FiniteLaw.flagPosterior none)).expect
+        (fun bits => F ((windowAssignmentEquiv T n g Bool b).symm bits)) := by
+  rw [windowPosterior_expect_flat]
+  simp only [noReveals,Equiv.apply_symm_apply]
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- The literal global schedule is indexed by a fixed pair/block bijection. -/
+def globalTrialEquiv (N t : ℕ) : (Fin (N/4) × Fin t) ≃ Fin ((N/4)*t) :=
+  Equiv.ofBijective _ (globalTrialIndex_bijective N t)
+
+def sourceGlobalCoefficient (N t : ℕ) (u : Fin N) (k : Fin ((N/4)*t)) : ℝ :=
+  let v := ((globalTrialEquiv N t).symm k).1
+  if v.val≤u.val ∧ u.val+v.val<N then 1 else 0
+
+def sourceGroupCoefficient (T n : ℕ → ℕ) (b : ℕ) (u : Fin (T (b+1))) :
+    WindowTrial T n (sourceScheduledTrials T n) (b+2) → ℝ
+  | .inl _ => 0
+  | .inr (.inl k) => sourceGlobalCoefficient (T (b+1)) (n b) u k
+  | .inr (.inr (v,.inl _)) => if v=u then 1 else 0
+  | .inr (.inr (_,.inr _)) => 0
+
+theorem even_half_cast (t : ℕ) (ht : Even t) : ((t/2:ℕ):ℝ)=(t:ℝ)/2 := by
+  have h : 2*(t/2)=t := Nat.mul_div_cancel' ((even_iff_two_dvd).mp ht)
+  have hr : (2:ℝ)*(t/2:ℕ)=t := by exact_mod_cast h
+  linarith
+
+theorem scheduledGlobalShift_linear (N t : ℕ) (ht : Even t)
+    (bits : Fin ((N/4)*t) → Bool) (u : Fin N) :
+    (scheduledGlobalShift N t bits u:ℝ)=
+      ∑k,sourceGlobalCoefficient N t u k*((bitIndex (bits k):ℝ)-1/2) := by
+  rw [←(globalTrialEquiv N t).sum_comp]
+  simp only [Fintype.sum_prod_type,sourceGlobalCoefficient,Equiv.symm_apply_apply]
+  unfold scheduledGlobalShift
+  simp only [Int.cast_sum,Int.cast_ite,Int.cast_sub,Int.cast_natCast,Int.cast_zero]
+  apply Finset.sum_congr rfl
+  intro v _
+  split_ifs with hv
+  · simp only [one_mul,Finset.sum_sub_distrib,Finset.sum_const,Finset.card_univ,
+      Fintype.card_fin,nsmul_eq_mul]
+    rw [even_half_cast t ht]
+    simp only [globalTrialEquiv,Equiv.ofBijective_apply]
+    ring
+  · simp only [zero_mul,Finset.sum_const_zero]
+
+theorem sourceDisplacement_linear (T n : ℕ → ℕ) (b : ℕ) (ht : Even (n b))
+    (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool (b+2)) (u : Fin (T (b+1))) :
+    (sourceDisplacement (T (b+1)) (n b) (fun v => (bits.2.2 v).1) bits.2.1 u:ℝ)=
+      fairLinearSum (sourceGroupCoefficient T n b u)
+        (windowAssignmentEquiv T n (sourceScheduledTrials T n) Bool (b+2) bits) := by
+  rcases bits with ⟨root,global,children⟩
+  change Fin ((T (b+1)/4)*n b) → Bool at global
+  change Fin (T (b+1)) → ((Fin (n b) → Bool) ×
+    (Fin (sourceScheduledTrials T n b) → Bool) ×
+    (Fin (T b) → WindowAssignments T n (sourceScheduledTrials T n) Bool b)) at children
+  unfold fairLinearSum
+  change _=∑i : Fin (n (b+1)) ⊕ (Fin ((T (b+1)/4)*n b) ⊕
+      (Fin (T (b+1)) × (Fin (n b) ⊕ (Fin (sourceScheduledTrials T n b) ⊕
+        (Fin (T b) × WindowTrial T n (sourceScheduledTrials T n) b))))),_
+  simp only [Fintype.sum_sum_type,Fintype.sum_prod_type,sourceGroupCoefficient,
+    zero_mul,Finset.sum_const_zero,zero_add,add_zero]
+  change _=(∑k,sourceGlobalCoefficient (T (b+1)) (n b) u k*
+    ((bitIndex (global k):ℝ)-1/2))+
+    (∑v : Fin (T (b+1)),∑a : Fin (n b),
+      (if v=u then (1:ℝ) else 0)*((bitIndex ((children v).1 a):ℝ)-1/2))
+  have hl : (∑v : Fin (T (b+1)),∑a : Fin (n b),
+      (if v=u then (1:ℝ) else 0)*((bitIndex ((children v).1 a):ℝ)-1/2))=
+        (∑a : Fin (n b),(bitIndex ((children u).1 a):ℝ))-(n b:ℝ)/2 := by
+    simp only [ite_mul,one_mul,zero_mul]
+    rw [show (∑v : Fin (T (b+1)),∑a : Fin (n b),
+      if v=u then ((bitIndex ((children v).1 a):ℝ)-1/2) else 0)=
+        ∑v : Fin (T (b+1)),if v=u then
+          (∑a : Fin (n b),((bitIndex ((children v).1 a):ℝ)-1/2)) else 0 from by
+            apply Finset.sum_congr rfl
+            intro v _
+            split_ifs <;> simp]
+    rw [Finset.sum_ite_eq',ite_eq_left (Finset.mem_univ u)]
+    simp only [Finset.sum_sub_distrib,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]
+    ring
+  rw [hl,←scheduledGlobalShift_linear (T (b+1)) (n b) ht global u]
+  simp only [sourceDisplacement,Int.cast_add,Int.cast_sub,Int.cast_sum,Int.cast_natCast,
+    even_half_cast _ ht]
+  ring
+
+theorem sourceGroupCoefficient_variance (T n : ℕ → ℕ) (b : ℕ) (u : Fin (T (b+1))) :
+    (∑i,(sourceGroupCoefficient T n b u i)^2)≤((T (b+1):ℝ)+1)*(n b:ℝ) := by
+  change (∑i : Fin (n (b+1)) ⊕ (Fin ((T (b+1)/4)*n b) ⊕
+      (Fin (T (b+1)) × (Fin (n b) ⊕ (Fin (sourceScheduledTrials T n b) ⊕
+        (Fin (T b) × WindowTrial T n (sourceScheduledTrials T n) b))))),_)≤_
+  simp only [Fintype.sum_sum_type,Fintype.sum_prod_type,sourceGroupCoefficient,
+    zero_pow (by omega : 2≠0),Finset.sum_const_zero,zero_add,add_zero]
+  have hg : (∑k : Fin ((T (b+1)/4)*n b),(sourceGlobalCoefficient (T (b+1)) (n b) u k)^2)≤
+      ((T (b+1)/4)*n b:ℕ) := by
+    calc
+      _ ≤ ∑_k : Fin ((T (b+1)/4)*n b),(1:ℝ) := by
+        apply Finset.sum_le_sum
+        intro k _
+        dsimp only [sourceGlobalCoefficient]
+        split_ifs <;> norm_num
+      _ = _ := by simp
+  have hl : (∑v : Fin (T (b+1)),∑_a : Fin (n b),(if v=u then (1:ℝ) else 0)^2)=(n b:ℝ) := by
+    simp only [ite_pow,one_pow,zero_pow (by omega : 2≠0)]
+    simp
+  rw [hl]
+  have hdiv : ((T (b+1)/4:ℕ):ℝ)≤T (b+1) := by exact_mod_cast Nat.div_le_self (T (b+1)) 4
+  push_cast at hg
+  have hm := mul_le_mul_of_nonneg_right hdiv (Nat.cast_nonneg (n b): (0:ℝ)≤n b)
+  nlinarith only [hg,hm]
+
+/-- Actual per-window unconditional fit tail under the full tree's fair law.
+ No independence between different window displacements is asserted. -/
+theorem sourceDisplacement_tail (T n : ℕ → ℕ) (b : ℕ) (ht : Even (n b))
+    (hn : 0<n b) (u : Fin (T (b+1))) {a : ℝ} (ha : 0≤a) :
+    (windowPosteriorLaw T n (sourceScheduledTrials T n) (b+2)
+      (noReveals T n (sourceScheduledTrials T n) (b+2))).expect
+      (fun bits => if a≤|(sourceDisplacement (T (b+1)) (n b)
+        (fun v => (bits.2.2 v).1) bits.2.1 u:ℝ)| then 1 else 0)≤
+      2*Real.exp (-2*a^2/(((T (b+1):ℝ)+1)*(n b:ℝ))) := by
+  rw [windowPosterior_fair_flat]
+  simp_rw [sourceDisplacement_linear T n b ht,Equiv.apply_symm_apply]
+  exact fairLinearSum_tail_variance (sourceGroupCoefficient T n b u)
+    (by positivity) (sourceGroupCoefficient_variance T n b u) ha
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+namespace FiniteLaw
+variable {α ι : Type*} [Fintype α] [Fintype ι]
+
+theorem expect_fintype_sum (P : FiniteLaw α) (f : ι → α → ℝ) :
+    P.expect (fun a => ∑i,f i a)=∑i,P.expect (f i) := by
+  simp only [expect,Finset.mul_sum]
+  exact Finset.sum_comm
+
+theorem expect_add (P : FiniteLaw α) (f g : α → ℝ) :
+    P.expect (fun a => f a+g a)=P.expect f+P.expect g := by
+  simp only [expect,mul_add,Finset.sum_add_distrib]
+
+theorem coordinateProduct_expect_eval {n : ℕ} (P : Fin n → FiniteLaw α)
+    (u : Fin n) (f : α → ℝ) :
+    (coordinateProduct P).expect (fun x => f (x u))=(P u).expect f := by
+  let F : Fin n → α → ℝ := fun j a => if j=u then f a else 1
+  have hp (x : Fin n → α) : (∏j,F j (x j))=f (x u) := by simp [F]
+  rw [show (fun x : Fin n → α => f (x u))=(fun x => ∏j,F j (x j)) by
+    funext x; exact (hp x).symm,coordinateProduct_expect_product]
+  have hj (j : Fin n) : (P j).expect (F j)=if j=u then (P j).expect f else 1 := by
+    by_cases h : j=u <;> simp [F,h,expect_const]
+  simp_rw [hj]
+  simp
+
+end FiniteLaw
+
+theorem noReveals_succ (T n g : ℕ → ℕ) (b : ℕ) :
+    noReveals T n g (b+1)=(fun _ => none,fun _ => none,fun _ => noReveals T n g b) := rfl
+
+theorem windowPosterior_child (T n g : ℕ → ℕ) (b : ℕ)
+    (d : WindowAssignments T n g (Option Bool) (b+1)) (u : Fin (T b))
+    (F : WindowAssignments T n g Bool b → ℝ) :
+    (windowPosteriorLaw T n g (b+1) d).expect (fun bits => F (bits.2.2 u))=
+      (windowPosteriorLaw T n g b (d.2.2 u)).expect F := by
+  rcases d with ⟨loc,glob,kids⟩
+  change ((posteriorArray loc).prod ((posteriorArray glob).prod
+    (FiniteLaw.coordinateProduct (fun i => windowPosteriorLaw T n g b (kids i))))).expect
+      (fun bits => F (bits.2.2 u))=_
+  rw [FiniteLaw.expect_prod]
+  simp_rw [FiniteLaw.expect_prod,FiniteLaw.coordinateProduct_expect_eval,FiniteLaw.expect_const]
+
+theorem failure_forall_le_sum {ι : Type*} [Fintype ι] (p : ι → Prop) [DecidablePred p] [Decidable (∀i,p i)] :
+    (if ∀i,p i then (0:ℝ) else 1)≤∑i,if p i then 0 else 1 := by
+  by_cases hp : ∀i,p i
+  · simp only [ite_eq_left hp]
+    exact Finset.sum_nonneg (fun i _ => by split_ifs <;> norm_num)
+  · simp only [ite_eq_right hp]
+    push Not at hp
+    obtain ⟨i,hi⟩ := hp
+    calc
+      (1:ℝ) = if p i then 0 else 1 := by simp [hi]
+      _ ≤ ∑j,if p j then 0 else 1 := by
+        apply Finset.single_le_sum (f:=fun j => if p j then (0:ℝ) else 1)
+        · intro j _; split_ifs <;> norm_num
+        · exact Finset.mem_univ i
+
+/-- Explicit finite tree union bound. The source's scale estimates will make
+ this number exponentially small; no independence of sibling shifts is used. -/
+def sourceFitTail (T n : ℕ → ℕ) : ℕ → ℝ
+  | 0 => 0
+  | 1 => 0
+  | b+2 => (T (b+1):ℝ)*(2*Real.exp (-2*((T b:ℝ)/8)^2/(((T (b+1):ℝ)+1)*(n b:ℝ)))+
+      sourceFitTail T n (b+1))
+
+theorem sourceTreeFit_tail (T n : ℕ → ℕ) (b : ℕ)
+    (hn : ∀j<b,0<n j) (he : ∀j<b,Even (n j)) :
+    (windowPosteriorLaw T n (sourceScheduledTrials T n) b
+      (noReveals T n (sourceScheduledTrials T n) b)).expect
+        (fun bits => if SourceTreeFit T n b bits then 0 else 1) ≤ sourceFitTail T n b := by
+  induction b using Nat.twoStepInduction with
+  | zero =>
+    have hh (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool 0) : SourceTreeFit T n 0 bits := trivial
+    simp only [ite_eq_left (hh _),FiniteLaw.expect_const,sourceFitTail,le_refl]
+  | one =>
+    have hh (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool 1) : SourceTreeFit T n 1 bits := trivial
+    simp only [ite_eq_left (hh _),FiniteLaw.expect_const,sourceFitTail,le_refl]
+  | more b ih0 ih =>
+    let P := windowPosteriorLaw T n (sourceScheduledTrials T n) (b+2)
+      (noReveals T n (sourceScheduledTrials T n) (b+2))
+    let badShift : WindowAssignments T n (sourceScheduledTrials T n) Bool (b+2) → Fin (T (b+1)) → Prop :=
+      fun bits u => (T b:ℝ)/8≤|(sourceDisplacement (T (b+1)) (n b)
+        (fun v => (bits.2.2 v).1) bits.2.1 u:ℝ)|
+    have hp (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool (b+2)) :
+        (if SourceTreeFit T n (b+2) bits then (0:ℝ) else 1)≤
+          (∑u : Fin (T (b+1)),if badShift bits u then 1 else 0)+
+          (∑u : Fin (T (b+1)),if SourceTreeFit T n (b+1) (bits.2.2 u) then 0 else 1) := by
+      let p := fun u : Fin (T (b+1)) => |8*sourceDisplacement (T (b+1)) (n b)
+        (fun v => (bits.2.2 v).1) bits.2.1 u|≤(T b:ℤ)
+      have hh : (if SourceTreeFit T n (b+2) bits then (0:ℝ) else 1)≤
+          (if ∀u,p u then 0 else 1)+(if ∀u,SourceTreeFit T n (b+1) (bits.2.2 u) then 0 else 1) := by
+        by_cases H : SourceTreeFit T n (b+2) bits
+        · rw [ite_eq_left H]
+          split_ifs <;> norm_num
+        · rw [ite_eq_right H]
+          by_cases h1 : ∀u,p u
+          · have h2 : ¬∀u,SourceTreeFit T n (b+1) (bits.2.2 u) := fun h2 => H ⟨h1,h2⟩
+            rw [ite_eq_left h1,ite_eq_right h2]
+            norm_num
+          · rw [ite_eq_right h1]
+            split_ifs <;> norm_num
+      apply hh.trans
+      apply add_le_add
+      · apply (failure_forall_le_sum p).trans
+        apply Finset.sum_le_sum
+        intro u _
+        by_cases hu : p u
+        · simp only [ite_eq_left hu]
+          split_ifs <;> norm_num
+        · have hr : (T b:ℝ)<8*|(sourceDisplacement (T (b+1)) (n b)
+              (fun v => (bits.2.2 v).1) bits.2.1 u:ℝ)| := by
+            have hi : (T b:ℤ) < |8*sourceDisplacement (T (b+1)) (n b)
+              (fun v => (bits.2.2 v).1) bits.2.1 u| := lt_of_not_ge hu
+            have hi' : (T b:ℝ) < |(8*sourceDisplacement (T (b+1)) (n b)
+              (fun v => (bits.2.2 v).1) bits.2.1 u:ℤ)| := by exact_mod_cast hi
+            norm_num only [Int.cast_abs,Int.cast_mul,Int.cast_ofNat,abs_mul,abs_of_pos (by norm_num : (0:ℝ)<8)] at hi'
+            exact hi'
+          have hb : badShift bits u := by dsimp [badShift]; linarith
+          simp [hu,hb]
+      · exact failure_forall_le_sum _
+    calc
+      _ ≤ P.expect (fun bits => (∑u : Fin (T (b+1)),if badShift bits u then 1 else 0)+
+          (∑u : Fin (T (b+1)),if SourceTreeFit T n (b+1) (bits.2.2 u) then 0 else 1)) :=
+            P.expect_mono hp
+      _ = (∑u : Fin (T (b+1)),P.expect (fun bits => if badShift bits u then 1 else 0))+
+          (∑u : Fin (T (b+1)),P.expect (fun bits => if SourceTreeFit T n (b+1) (bits.2.2 u) then 0 else 1)) := by
+            rw [FiniteLaw.expect_add,FiniteLaw.expect_fintype_sum,FiniteLaw.expect_fintype_sum]
+      _ ≤ (∑_u : Fin (T (b+1)),2*Real.exp (-2*((T b:ℝ)/8)^2/(((T (b+1):ℝ)+1)*(n b:ℝ))))+
+          ∑_u : Fin (T (b+1)),sourceFitTail T n (b+1) := by
+            apply add_le_add
+            · apply Finset.sum_le_sum
+              intro u _
+              exact sourceDisplacement_tail T n b (he b (by omega)) (hn b (by omega)) u (by positivity)
+            · apply Finset.sum_le_sum
+              intro u _
+              dsimp only [P]
+              rw [windowPosterior_child T n (sourceScheduledTrials T n) (b+1)
+                (noReveals T n (sourceScheduledTrials T n) (b+2)) u
+                (fun x => if SourceTreeFit T n (b+1) x then 0 else 1)]
+              rw [noReveals_succ]
+              exact ih (fun j hj => hn j (by omega)) (fun j hj => he j (by omega))
+      _ = sourceFitTail T n (b+2) := by simp [sourceFitTail,mul_add]
+
+end TraceReconstruction
+
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/TraceReconstruction/Model.lean b/lean/OAI/Probability/TraceReconstruction/Model.lean
index b5e91214b2af88dcd12c914b7146fa2a0d4ae392..b662bc6c2de055afe95d8158682b1a43d08cac45 100644
--- a/lean/OAI/Probability/TraceReconstruction/Model.lean
+++ b/lean/OAI/Probability/TraceReconstruction/Model.lean
@@ -1,548 +1,548 @@
-import OAI.Probability.TraceReconstruction.FiniteLaw
-
-namespace OAI
-
-/-!
-# Model
-
-Independent deletion, complete trace observations, randomized estimators, ENNReal fixed-budget complexity, concatenation, and literal input words.
--/
-
-open scoped ENNReal
-noncomputable section
-
-open scoped ENNReal
-noncomputable section
-
-namespace TraceReconstruction
-
-abbrev Word (n : ℕ) := Fin n → Bool
-abbrev Mask (n : ℕ) := Fin n → Bool
-abbrev Trace := List Bool
-
-/-- `true` retains the bit. The observer gets only the resulting list. -/
-def traceOfMask {n : ℕ} (x : Word n) (a : Mask n) : Trace :=
-  ((List.finRange n).filter fun i => a i).map x
-
-/-- The deletion-mask probability, with deletion probability `q`. -/
-def maskWeight {n : ℕ} (q : ℝ) (a : Mask n) : ℝ :=
-  ∏ i, if a i then 1 - q else q
-
-/-- Law of a single complete trace. -/
-def traceMass {n : ℕ} (q : ℝ) (x : Word n) (w : Trace) : ℝ :=
-  ∑ a : Mask n, if traceOfMask x a = w then maskWeight q a else 0
-
-/-- All possible complete outputs from length-n inputs, with lengths included. -/
-def traceSpace (n : ℕ) : Finset Trace :=
-  Finset.univ.biUnion fun k : Fin (n + 1) =>
-    Finset.univ.image fun x : Word k.val => List.ofFn x
-
-/-- 1/2 sum of absolute differences, on the full finite output space. -/
-def traceTV {n : ℕ} (q : ℝ) (x y : Word n) : ℝ :=
-  (1 / 2 : ℝ) * ∑ w ∈ traceSpace n, |traceMass q x w - traceMass q y w|
-
-/-- The estimator may depend arbitrarily on n, q and every complete sample.
-    Its value is its conditional probability of reporting the given input. -/
-def IsEstimator {n m : ℕ} (A : (Fin m → Trace) → Word n → ℝ) : Prop :=
-  (∀ z x, 0 ≤ A z x) ∧ (∀ z, ∑ x : Word n, A z x = 1)
-
-/-- Independent masks generate independent complete traces; no retained
-    positions, boundaries or other latent variables are revealed to A. -/
-def reconstructionSuccess {n m : ℕ} (q : ℝ)
-    (A : (Fin m → Trace) → Word n → ℝ) (x : Word n) : ℝ :=
-  ∑ a : Fin m → Mask n,
-    (∏ j, maskWeight q (a j)) * A (fun j => traceOfMask x (a j)) x
-
-def SufficientBudget (n : ℕ) (q s : ℝ) (m : ℕ) : Prop :=
-  ∃ A : (Fin m → Trace) → Word n → ℝ,
-    IsEstimator A ∧ ∀ x, s ≤ reconstructionSuccess q A x
-
-/-- Natural fixed sample budget; infimum of the empty set is infinity. -/
-noncomputable def sampleComplexity (n : ℕ) (q s : ℝ) : ℝ≥0∞ :=
-  sInf {t | ∃ m : ℕ, SufficientBudget n q s m ∧ t = (m : ℝ≥0∞)}
-
-/-- The real infimum is a minimum for n>0 (a nonempty finite set). -/
-noncomputable def minTraceTV (n : ℕ) (q : ℝ) : ℝ :=
-  sInf {t | ∃ x y : Word n, x ≠ y ∧ t = traceTV q x y}
-
-end TraceReconstruction
-
-noncomputable section
-
-namespace TraceReconstruction
-
-
-
-
-/-! ## Exact finite channel laws -/
-
-theorem traceOfMask_length_le {n : ℕ} (x : Word n) (a : Mask n) :
-    (traceOfMask x a).length ≤ n := by
-  simpa [traceOfMask] using List.length_filter_le (fun i => a i) (List.finRange n)
-
-theorem mem_traceSpace_of_length_le {n : ℕ} (w : Trace) (h : w.length ≤ n) :
-    w ∈ traceSpace n := by
-  apply Finset.mem_biUnion.mpr
-  refine ⟨⟨w.length, Nat.lt_succ_of_le h⟩, Finset.mem_univ _, ?_⟩
-  exact Finset.mem_image.mpr ⟨w.get, Finset.mem_univ _, List.ofFn_get w⟩
-
-theorem length_le_of_mem_traceSpace {n : ℕ} {w : Trace} (h : w ∈ traceSpace n) :
-    w.length ≤ n := by
-  obtain ⟨k, _, hk⟩ := Finset.mem_biUnion.mp h
-  obtain ⟨x, _, rfl⟩ := Finset.mem_image.mp hk
-  simpa using Nat.le_of_lt_succ k.isLt
-
-@[simp] theorem mem_traceSpace {n : ℕ} {w : Trace} :
-    w ∈ traceSpace n ↔ w.length ≤ n :=
-  ⟨length_le_of_mem_traceSpace, mem_traceSpace_of_length_le w⟩
-
-theorem traceOfMask_mem {n : ℕ} (x : Word n) (a : Mask n) :
-    traceOfMask x a ∈ traceSpace n :=
-  mem_traceSpace_of_length_le _ (traceOfMask_length_le x a)
-
-theorem maskWeight_nonneg {n : ℕ} {q : ℝ} (hq : q ∈ Set.Icc 0 1) (a : Mask n) :
-    0 ≤ maskWeight q a := by
-  apply Finset.prod_nonneg
-  intro i _
-  split <;> linarith [hq.1, hq.2]
-
-@[simp] theorem sum_maskWeight {n : ℕ} (q : ℝ) :
-    ∑ a : Mask n, maskWeight q a = 1 := by
-  unfold maskWeight
-  change (∑ a : Fin n → Bool, ∏ i, (fun (_ : Fin n) (b : Bool) =>
-    if b then 1 - q else q) i (a i)) = 1
-  rw [← Fintype.prod_sum (fun (_ : Fin n) (b : Bool) => if b then 1 - q else q)]
-  simp
-
-theorem traceMass_nonneg {n : ℕ} {q : ℝ} (hq : q ∈ Set.Icc 0 1)
-    (x : Word n) (w : Trace) : 0 ≤ traceMass q x w := by
-  apply Finset.sum_nonneg
-  intro a _
-  split
-  · exact maskWeight_nonneg hq a
-  · exact le_rfl
-
-theorem traceMass_eq_zero_of_not_mem {n : ℕ} (q : ℝ) (x : Word n) (w : Trace)
-    (hw : w ∉ traceSpace n) : traceMass q x w = 0 := by
-  apply Finset.sum_eq_zero
-  intro a _
-  have hne : traceOfMask x a ≠ w := by
-    intro h
-    exact hw (h ▸ traceOfMask_mem x a)
-  simp [hne]
-
-@[simp] theorem sum_traceMass {n : ℕ} (q : ℝ) (x : Word n) :
-    ∑ w ∈ traceSpace n, traceMass q x w = 1 := by
-  unfold traceMass
-  rw [Finset.sum_comm]
-  simp [traceOfMask_length_le]
-
-theorem traceTV_nonneg {n : ℕ} (q : ℝ) (x y : Word n) : 0 ≤ traceTV q x y := by
-  unfold traceTV
-  positivity
-
-@[simp] theorem traceTV_self {n : ℕ} (q : ℝ) (x : Word n) : traceTV q x x = 0 := by
-  simp [traceTV]
-
-theorem traceTV_comm {n : ℕ} (q : ℝ) (x y : Word n) : traceTV q x y = traceTV q y x := by
-  simp only [traceTV, abs_sub_comm]
-
-theorem traceTV_le_one {n : ℕ} {q : ℝ} (hq : q ∈ Set.Icc 0 1) (x y : Word n) :
-    traceTV q x y ≤ 1 := by
-  have h : ∑ w ∈ traceSpace n, |traceMass q x w - traceMass q y w| ≤
-      ∑ w ∈ traceSpace n, (traceMass q x w + traceMass q y w) := by
-    apply Finset.sum_le_sum
-    intro w _
-    exact abs_sub_le_iff.mpr
-      ⟨by linarith [traceMass_nonneg hq y w], by linarith [traceMass_nonneg hq x w]⟩
-  rw [Finset.sum_add_distrib, sum_traceMass, sum_traceMass] at h
-  unfold traceTV
-  linarith
-
-
-end TraceReconstruction
-
-namespace TraceReconstruction
-namespace FiniteLaw
-variable {α β γ : Type*} [Fintype α] [Fintype β] [Fintype γ]
-
-@[ext] theorem ext {P Q : FiniteLaw α} (h : P.mass = Q.mass) : P = Q := by
-  cases P with
-  | mk pm pn pt => cases Q with
-    | mk qm qn qt => cases h; rfl
-
-noncomputable def pure (a : α) : FiniteLaw α := by
-  classical
-  exact ⟨(fun b => if a = b then 1 else 0),
-    (fun b => by split_ifs <;> norm_num), by simp⟩
-
-noncomputable def map (P : FiniteLaw α) (f : α → β) : FiniteLaw β :=
-  P.bind (fun a => pure (f a))
-
-theorem map_mass [DecidableEq β] (P : FiniteLaw α) (f : α → β) (b : β) :
-    (P.map f).mass b = ∑ a, if f a = b then P.mass a else 0 := by
-  classical
-  unfold map bind pure
-  simp only
-  apply Finset.sum_congr rfl
-  intro a _
-  split_ifs <;> simp
-
-theorem expectation_map (P : FiniteLaw α) (f : α → β) (F : β → ℝ) :
-    (∑ b, (P.map f).mass b * F b) = ∑ a, P.mass a * F (f a) := by
-  classical
-  simp_rw [map_mass, Finset.sum_mul]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro a _
-  simp
-
-theorem tv_map_le (P Q : FiniteLaw α) (f : α → β) :
-    tv (P.map f) (Q.map f) ≤ tv P Q := tv_bind_le P Q _
-
-end FiniteLaw
-
-abbrev BoundedTrace (n : ℕ) := ↥(traceSpace n)
-
-def maskLaw {n : ℕ} (q : ℝ) (hq : q ∈ Set.Icc 0 1) : FiniteLaw (Mask n) :=
-  ⟨maskWeight q, maskWeight_nonneg hq, sum_maskWeight q⟩
-
-def completeTraceLaw {n : ℕ} (q : ℝ) (hq : q ∈ Set.Icc 0 1) (x : Word n) :
-    FiniteLaw (BoundedTrace n) where
-  mass w := traceMass q x w
-  nonneg w := traceMass_nonneg hq x w
-  total := by rw [Finset.sum_coe_sort]; exact sum_traceMass q x
-
-theorem completeTraceLaw_tv {n : ℕ} (q : ℝ) (hq : q ∈ Set.Icc 0 1) (x y : Word n) :
-    (completeTraceLaw q hq x).tv (completeTraceLaw q hq y) = traceTV q x y := by
-  change (1 / 2 : ℝ) * (∑ w : BoundedTrace n, |traceMass q x w - traceMass q y w|) = _
-  rw [Finset.sum_coe_sort (traceSpace n) (fun w => |traceMass q x w - traceMass q y w|)]
-  rfl
-
-theorem completeTraceLaw_mask {n : ℕ} (q : ℝ) (hq : q ∈ Set.Icc 0 1) (x : Word n) :
-    completeTraceLaw q hq x =
-      (maskLaw q hq).map (fun a : Mask n => ⟨traceOfMask x a, traceOfMask_mem x a⟩) := by
-  apply FiniteLaw.ext
-  funext w
-  rw [FiniteLaw.map_mass]
-  unfold completeTraceLaw traceMass maskLaw
-  simp only
-  apply Finset.sum_congr rfl
-  intro a _
-  rcases w with ⟨w, hw⟩
-  simp only [Subtype.mk.injEq]
-
-theorem trace_expectation {n : ℕ} (q : ℝ) (x : Word n) (F : Trace → ℝ) :
-    (∑ w ∈ traceSpace n, traceMass q x w * F w) =
-      ∑ a : Mask n, maskWeight q a * F (traceOfMask x a) := by
-  classical
-  unfold traceMass
-  simp_rw [Finset.sum_mul]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro a _
-  simp [traceOfMask_length_le]
-
-/-- Concatenation of exact binary inputs, not an encoded surrogate. -/
-def appendWord {n m : ℕ} (x : Word n) (y : Word m) : Word (n + m) := Fin.append x y
-
-theorem trace_filterMap {n : ℕ} (x : Word n) (a : Mask n) :
-    traceOfMask x a = (List.ofFn fun i => if a i then some (x i) else none).filterMap id := by
-  have hlist (l : List (Fin n)) :
-      (l.filter fun i => a i).map x = l.filterMap (fun i => if a i then some (x i) else none) := by
-    induction l with
-    | nil => rfl
-    | cons i l ih => cases hai : a i <;> simp [hai, ih]
-  unfold traceOfMask
-  rw [hlist]
-  have heq : (List.ofFn fun i => if a i then some (x i) else none) =
-      (List.finRange n).map (fun i => if a i then some (x i) else none) := by
-    rw [← List.ofFn_id, List.map_ofFn]
-    rfl
-  rw [heq, List.filterMap_map]
-  rfl
-
-theorem traceOfMask_append {n m : ℕ} (x : Word n) (y : Word m)
-    (a : Mask n) (b : Mask m) :
-    traceOfMask (appendWord x y) (Fin.append a b) = traceOfMask x a ++ traceOfMask y b := by
-  rw [trace_filterMap, List.ofFn_add, List.filterMap_append, trace_filterMap, trace_filterMap]
-  congr 2 <;> apply congrArg List.ofFn <;> funext i
-  · change (if Fin.append a b (Fin.castAdd m i) then some (Fin.append x y (Fin.castAdd m i)) else none) = _
-    simp
-  · simp [appendWord]
-
-theorem maskWeight_append {n m : ℕ} (q : ℝ) (a : Mask n) (b : Mask m) :
-    maskWeight q (Fin.append a b) = maskWeight q a * maskWeight q b := by
-  unfold maskWeight
-  rw [Fin.prod_univ_add]
-  simp
-
-theorem concatenated_trace_expectation {n m : ℕ} (q : ℝ) (x : Word n) (y : Word m)
-    (F : Trace → ℝ) :
-    (∑ a : Mask (n + m), maskWeight q a * F (traceOfMask (appendWord x y) a)) =
-      ∑ a : Mask n, ∑ b : Mask m,
-        maskWeight q a * maskWeight q b * F (traceOfMask x a ++ traceOfMask y b) := by
-  rw [← (Fin.appendEquiv n m).sum_comp]
-  rw [Fintype.sum_prod_type]
-  apply Finset.sum_congr rfl
-  intro a _
-  apply Finset.sum_congr rfl
-  intro b _
-  change maskWeight q (Fin.append a b) * F (traceOfMask (appendWord x y) (Fin.append a b)) = _
-  rw [maskWeight_append, traceOfMask_append]
-
-end TraceReconstruction
-
-namespace TraceReconstruction
-
-/-- The four complete traces possible from the ordinary primitive `01`. -/
-abbrev PrimitiveMark := Fin 4
-
-def primitiveOutput : PrimitiveMark → List Bool := ![[], [false], [true], [false, true]]
-def ordinaryPrimitiveMass (q : ℝ) : PrimitiveMark → ℝ :=
-  ![q ^ 2, q * (1 - q), q * (1 - q), (1 - q) ^ 2]
-def defectPrimitiveMass (q : ℝ) : PrimitiveMark → ℝ := ![q, 0, 1 - q, 0]
-def primitiveErasure (q : ℝ) : ℝ := q ^ 2 / 2
-
-def primitiveMarkMass (q : ℝ) (isDefect : Bool) (z : PrimitiveMark) : ℝ :=
-  ((if isDefect then defectPrimitiveMass q z else ordinaryPrimitiveMass q z) -
-    if z = 0 then primitiveErasure q else 0) / (1 - primitiveErasure q)
-
-theorem primitive_retain_pos {q : ℝ} (hq : q ∈ Set.Ioo 0 1) :
-    0 < 1 - primitiveErasure q := by
-  unfold primitiveErasure
-  nlinarith [hq.1, hq.2]
-
-theorem primitiveMarkMass_nonneg {q : ℝ} (hq : q ∈ Set.Ioo 0 1)
-    (b : Bool) (z : PrimitiveMark) : 0 ≤ primitiveMarkMass q b z := by
-  apply div_nonneg _ (primitive_retain_pos hq).le
-  cases b <;> fin_cases z <;>
-    simp [ordinaryPrimitiveMass, defectPrimitiveMass, primitiveErasure] <;>
-    nlinarith [hq.1, hq.2]
-
-theorem primitiveMarkMass_sum {q : ℝ} (hq : q ∈ Set.Ioo 0 1) (b : Bool) :
-    ∑ z : PrimitiveMark, primitiveMarkMass q b z = 1 := by
-  have hd := ne_of_gt (primitive_retain_pos hq)
-  unfold primitiveMarkMass
-  rw [← Finset.sum_div]
-  apply (div_eq_one_iff_eq hd).mpr
-  cases b <;> simp [ordinaryPrimitiveMass, defectPrimitiveMass, primitiveErasure, Fin.sum_univ_succ]
-  all_goals ring
-
-def primitiveMarkLaw (q : ℝ) (hq : q ∈ Set.Ioo 0 1) (b : Bool) : FiniteLaw PrimitiveMark :=
-  ⟨primitiveMarkMass q b, primitiveMarkMass_nonneg hq b, primitiveMarkMass_sum hq b⟩
-
-/-- Emitting the conditional mark, or silently erasing the block, gives
-exactly the ordinary primitive-output probabilities. Silent erasure and an
-emitted empty mark are distinct latent events. -/
-theorem primitiveMark_simulation {q : ℝ} (hq : q ∈ Set.Ioo 0 1)
-    (b : Bool) (z : PrimitiveMark) :
-    (if z = 0 then primitiveErasure q else 0) +
-      (1 - primitiveErasure q) * primitiveMarkMass q b z =
-        if b then defectPrimitiveMass q z else ordinaryPrimitiveMass q z := by
-  unfold primitiveMarkMass
-  rw [mul_div_cancel₀ _ (ne_of_gt (primitive_retain_pos hq))]
-  ring
-
-def primitiveScore (q : ℝ) : PrimitiveMark → ℝ := ![2 / q - 2, -1, 1 / q - 1, -1]
-
-theorem primitive_score_identity {q : ℝ} (hq : q ∈ Set.Ioo 0 1) (z : PrimitiveMark) :
-    primitiveMarkMass q true z = primitiveMarkMass q false z * (1 + primitiveScore q z) := by
-  have hq0 := ne_of_gt hq.1
-  have hd := ne_of_gt (primitive_retain_pos hq)
-  fin_cases z <;>
-    simp [primitiveMarkMass, ordinaryPrimitiveMass, defectPrimitiveMass, primitiveScore, primitiveErasure] <;>
-    field_simp
-  all_goals ring
-
-theorem primitive_score_centered {q : ℝ} (hq : q ∈ Set.Ioo 0 1) :
-    ∑ z : PrimitiveMark, primitiveMarkMass q false z * primitiveScore q z = 0 := by
-  have hi := primitiveMarkMass_sum hq true
-  simp_rw [primitive_score_identity hq, mul_add, mul_one, Finset.sum_add_distrib,
-    primitiveMarkMass_sum hq false] at hi
-  linarith
-
-theorem primitive_score_bound {q : ℝ} (hq : q ∈ Set.Ioo 0 1) (z : PrimitiveMark) :
-    |primitiveScore q z| ≤ 2 / q := by
-  have hi : 1 ≤ 1 / q := (le_div_iff₀ hq.1).mpr (by linarith [hq.2])
-  simp only [one_div] at hi
-  have he : 2 / q = 2 * (1 / q) := by ring
-  fin_cases z <;> simp [primitiveScore, abs_le, he] <;> (try constructor) <;> linarith
-
-theorem primitive_score_variance_exact {q : ℝ} (hq : q ∈ Set.Ioo 0 1) :
-    (∑ z : PrimitiveMark, primitiveMarkMass q false z * primitiveScore q z ^ 2) =
-      (1 / q - 2 * q + q ^ 2) / (1 - q ^ 2 / 2) := by
-  have hq0 := ne_of_gt hq.1
-  have hd := ne_of_gt (primitive_retain_pos hq)
-  simp only [primitiveMarkMass, primitiveScore, ordinaryPrimitiveMass, primitiveErasure,
-    Bool.false_eq_true, ↓reduceIte]
-  simp [Fin.sum_univ_succ]
-  field_simp
-  ring
-
-theorem primitive_score_variance_le {q : ℝ} (hq : q ∈ Set.Ioo 0 1) :
-    (∑ z : PrimitiveMark, primitiveMarkMass q false z * primitiveScore q z ^ 2) ≤ 2 / q := by
-  rw [primitive_score_variance_exact hq]
-  have hd : 0 < 1 - q ^ 2 / 2 := primitive_retain_pos hq
-  apply (div_le_div_iff₀ hd hq.1).mpr
-  rw [add_mul, sub_mul, one_div_mul_cancel (ne_of_gt hq.1)]
-  have hpow : q ^ 3 ≤ q ^ 2 := by
-    calc
-      _ = q ^ 2 * q := by ring
-      _ ≤ q ^ 2 * 1 := mul_le_mul_of_nonneg_left hq.2.le (sq_nonneg _)
-      _ = _ := mul_one _
-  nlinarith only [hpow, sq_nonneg q]
-
-end TraceReconstruction
-
-namespace TraceReconstruction
-def ordinaryRun : ℕ → List Bool
-  | 0 => []
-  | m + 1 => false :: true :: ordinaryRun m
-
-@[simp] theorem ordinaryRun_length (m : ℕ) : (ordinaryRun m).length = 2 * m := by
-  induction m with
-  | zero => rfl
-  | succ m ih => simp only [ordinaryRun, List.length_cons, ih]; omega
-
-theorem ordinaryRun_add (a b : ℕ) : ordinaryRun (a + b) = ordinaryRun a ++ ordinaryRun b := by
-  induction a with
-  | zero => simp [ordinaryRun]
-  | succ a ih => simp [Nat.succ_add, ordinaryRun, ih]
-
-/-- A period of m primitives with its unique `1` defect at zero-based slot d. -/
-def initialPeriod (m d : ℕ) : List Bool :=
-  ordinaryRun d ++ [true] ++ ordinaryRun (m - 1 - d)
-
-theorem initialPeriod_length {m d : ℕ} (hd : d < m) :
-    (initialPeriod m d).length = 2 * m - 1 := by
-  simp only [initialPeriod, List.length_append, ordinaryRun_length, List.length_singleton]
-  omega
-
-/-- Distinct defect slots differ before either period ends, independently of future text. -/
-theorem initialPeriod_tail_ne {m d e : ℕ} (hde : d < e) (he : e < m)
-    (u v : List Bool) : initialPeriod m d ++ u ≠ initialPeriod m e ++ v := by
-  obtain ⟨a, ha⟩ := Nat.exists_eq_add_of_le (show d + 1 ≤ e by omega)
-  subst e
-  unfold initialPeriod
-  rw [Nat.add_assoc d 1 a, ordinaryRun_add d (1 + a)]
-  simp only [Nat.add_assoc] at *
-  simp only [Nat.one_add, ordinaryRun, List.append_assoc, List.cons_append, List.nil_append]
-  intro h
-  have hh := List.append_cancel_left h
-  have hf := congrArg List.head? hh
-  simp only [List.head?_cons, Option.some.injEq] at hf
-  cases hf
-
-theorem initialPeriod_injective (m : ℕ) :
-    Function.Injective (fun d : Fin m => initialPeriod m d) := by
-  intro d e h
-  by_contra hne
-  have hv : d.val ≠ e.val := fun hv => hne (Fin.ext hv)
-  rcases lt_or_gt_of_ne hv with hlt | hgt
-  · exact initialPeriod_tail_ne hlt e.isLt [] [] (by simpa using h)
-  · exact initialPeriod_tail_ne hgt d.isLt [] [] (by simpa using h.symm)
-
-def periodString (m P : ℕ) (d : Fin P → Fin m) : List Bool :=
-  (List.ofFn (fun i => initialPeriod m (d i))).flatten
-
-theorem periodString_cons (m P : ℕ) (d : Fin (P + 1) → Fin m) :
-    periodString m (P + 1) d = initialPeriod m (d 0) ++
-      periodString m P (fun i => d i.succ) := by
-  simp only [periodString, List.ofFn_succ, List.flatten_cons]
-
-theorem periodString_length (m P : ℕ) (d : Fin P → Fin m) :
-    (periodString m P d).length = P * (2 * m - 1) := by
-  induction P with
-  | zero => simp [periodString]
-  | succ P ih =>
-    rw [periodString_cons, List.length_append, initialPeriod_length (d 0).isLt, ih]
-    rw [Nat.add_mul, one_mul, Nat.add_comm]
-
-theorem periodString_injective (m P : ℕ) : Function.Injective (periodString m P) := by
-  induction P with
-  | zero => intro d e _; exact Subsingleton.elim _ _
-  | succ P ih =>
-    intro d e h
-    rw [periodString_cons, periodString_cons] at h
-    have hz : d 0 = e 0 := by
-      by_contra hne
-      have hv : (d 0).val ≠ (e 0).val := fun hv => hne (Fin.ext hv)
-      rcases lt_or_gt_of_ne hv with hlt | hgt
-      · exact initialPeriod_tail_ne hlt (e 0).isLt _ _ h
-      · exact initialPeriod_tail_ne hgt (d 0).isLt _ _ h.symm
-    rw [hz] at h
-    have ht := ih (List.append_cancel_left h)
-    funext i
-    refine Fin.cases hz (fun j => congrFun ht j) i
-end TraceReconstruction
-
-namespace TraceReconstruction
-/-- Uniformly sized binary atoms concatenated without boundary marks. -/
-def atomString (P : ℕ) (w : Bool → List Bool) (θ : Fin P → Bool) : List Bool :=
-  (List.ofFn (fun i => w (θ i))).flatten
-
-theorem atomString_cons (P : ℕ) (w : Bool → List Bool) (θ : Fin (P + 1) → Bool) :
-    atomString (P + 1) w θ = w (θ 0) ++ atomString P w (fun i => θ i.succ) := by
-  simp only [atomString, List.ofFn_succ, List.flatten_cons]
-
-theorem atomString_length (P S : ℕ) (w : Bool → List Bool) (hw : ∀ b, (w b).length = S)
-    (θ : Fin P → Bool) : (atomString P w θ).length = P * S := by
-  induction P with
-  | zero => simp [atomString]
-  | succ P ih => rw [atomString_cons, List.length_append, hw, ih, Nat.add_mul, one_mul, Nat.add_comm]
-
-theorem atomString_injective (P S : ℕ) (w : Bool → List Bool)
-    (hw : ∀ b, (w b).length = S) (hinj : Function.Injective w) :
-    Function.Injective (atomString P w) := by
-  induction P with
-  | zero => intro θ φ _; exact Subsingleton.elim _ _
-  | succ P ih =>
-    intro θ φ h
-    rw [atomString_cons, atomString_cons] at h
-    have hh := congrArg (List.take S) h
-    have hθ (u : List Bool) (b : Bool) : (w b ++ u).take S = w b := by
-      rw [← hw b, List.take_left]
-    rw [hθ, hθ] at hh
-    have hz := hinj hh
-    rw [hz] at h
-    have ht := ih (List.append_cancel_left h)
-    funext i
-    exact Fin.cases hz (fun j => congrFun ht j) i
-
-/-- Zero-based notation for the source complementary template; `true` is minus.
-The motif itself is fixed independently of the hypothesis. -/
-def templateType (P a L : ℕ) (sign : Fin L → Bool) (τ : Bool) (i : Fin P) : Bool :=
-  if h : a ≤ i.val ∧ i.val < a + L then xor τ (sign ⟨i.val - a, by omega⟩) else false
-
-theorem templateType_at (P a L : ℕ) (sign : Fin L → Bool) (τ : Bool)
-    (hL : 0 < L) (ha : a < P) :
-    templateType P a L sign τ ⟨a, ha⟩ = xor τ (sign ⟨0, hL⟩) := by
-  simp [templateType, hL]
-
-theorem templateType_injective (P a L : ℕ) (sign : Fin L → Bool)
-    (hL : 0 < L) (ha : a < P) :
-    Function.Injective (templateType P a L sign) := by
-  intro τ υ h
-  have he := congrFun h ⟨a, ha⟩
-  rw [templateType_at P a L sign τ hL ha, templateType_at P a L sign υ hL ha] at he
-  cases hs : sign ⟨0, hL⟩ <;> cases τ <;> cases υ <;> simp_all
-
-def complementaryWords (P a L : ℕ) (sign : Fin L → Bool) (w : Bool → List Bool)
-    (τ : Bool) : List Bool := atomString P w (templateType P a L sign τ)
-
-theorem complementaryWords_length (P a L S : ℕ) (sign : Fin L → Bool)
-    (w : Bool → List Bool) (hw : ∀ b, (w b).length = S) (τ : Bool) :
-    (complementaryWords P a L sign w τ).length = P * S := atomString_length P S w hw _
-
-theorem complementaryWords_injective (P a L S : ℕ) (sign : Fin L → Bool)
-    (w : Bool → List Bool) (hw : ∀ b, (w b).length = S) (hinj : Function.Injective w)
-    (hL : 0 < L) (ha : a < P) : Function.Injective (complementaryWords P a L sign w) :=
-  (atomString_injective P S w hw hinj).comp (templateType_injective P a L sign hL ha)
-end TraceReconstruction
-
-end
-end
-end
-
-end OAI
+import OAI.Probability.TraceReconstruction.FiniteLaw
+
+namespace OAI
+
+/-!
+# Model
+
+Independent deletion, complete trace observations, randomized estimators, ENNReal fixed-budget complexity, concatenation, and literal input words.
+-/
+
+open scoped ENNReal
+noncomputable section
+
+open scoped ENNReal
+noncomputable section
+
+namespace TraceReconstruction
+
+abbrev Word (n : ℕ) := Fin n → Bool
+abbrev Mask (n : ℕ) := Fin n → Bool
+abbrev Trace := List Bool
+
+/-- `true` retains the bit. The observer gets only the resulting list. -/
+def traceOfMask {n : ℕ} (x : Word n) (a : Mask n) : Trace :=
+  ((List.finRange n).filter fun i => a i).map x
+
+/-- The deletion-mask probability, with deletion probability `q`. -/
+def maskWeight {n : ℕ} (q : ℝ) (a : Mask n) : ℝ :=
+  ∏ i, if a i then 1 - q else q
+
+/-- Law of a single complete trace. -/
+def traceMass {n : ℕ} (q : ℝ) (x : Word n) (w : Trace) : ℝ :=
+  ∑ a : Mask n, if traceOfMask x a = w then maskWeight q a else 0
+
+/-- All possible complete outputs from length-n inputs, with lengths included. -/
+def traceSpace (n : ℕ) : Finset Trace :=
+  Finset.univ.biUnion fun k : Fin (n + 1) =>
+    Finset.univ.image fun x : Word k.val => List.ofFn x
+
+/-- 1/2 sum of absolute differences, on the full finite output space. -/
+def traceTV {n : ℕ} (q : ℝ) (x y : Word n) : ℝ :=
+  (1 / 2 : ℝ) * ∑ w ∈ traceSpace n, |traceMass q x w - traceMass q y w|
+
+/-- The estimator may depend arbitrarily on n, q and every complete sample.
+    Its value is its conditional probability of reporting the given input. -/
+def IsEstimator {n m : ℕ} (A : (Fin m → Trace) → Word n → ℝ) : Prop :=
+  (∀ z x, 0 ≤ A z x) ∧ (∀ z, ∑ x : Word n, A z x = 1)
+
+/-- Independent masks generate independent complete traces; no retained
+    positions, boundaries or other latent variables are revealed to A. -/
+def reconstructionSuccess {n m : ℕ} (q : ℝ)
+    (A : (Fin m → Trace) → Word n → ℝ) (x : Word n) : ℝ :=
+  ∑ a : Fin m → Mask n,
+    (∏ j, maskWeight q (a j)) * A (fun j => traceOfMask x (a j)) x
+
+def SufficientBudget (n : ℕ) (q s : ℝ) (m : ℕ) : Prop :=
+  ∃ A : (Fin m → Trace) → Word n → ℝ,
+    IsEstimator A ∧ ∀ x, s ≤ reconstructionSuccess q A x
+
+/-- Natural fixed sample budget; infimum of the empty set is infinity. -/
+noncomputable def sampleComplexity (n : ℕ) (q s : ℝ) : ℝ≥0∞ :=
+  sInf {t | ∃ m : ℕ, SufficientBudget n q s m ∧ t = (m : ℝ≥0∞)}
+
+/-- The real infimum is a minimum for n>0 (a nonempty finite set). -/
+noncomputable def minTraceTV (n : ℕ) (q : ℝ) : ℝ :=
+  sInf {t | ∃ x y : Word n, x ≠ y ∧ t = traceTV q x y}
+
+end TraceReconstruction
+
+noncomputable section
+
+namespace TraceReconstruction
+
+
+
+
+/-! ## Exact finite channel laws -/
+
+theorem traceOfMask_length_le {n : ℕ} (x : Word n) (a : Mask n) :
+    (traceOfMask x a).length ≤ n := by
+  simpa [traceOfMask] using List.length_filter_le (fun i => a i) (List.finRange n)
+
+theorem mem_traceSpace_of_length_le {n : ℕ} (w : Trace) (h : w.length ≤ n) :
+    w ∈ traceSpace n := by
+  apply Finset.mem_biUnion.mpr
+  refine ⟨⟨w.length, Nat.lt_succ_of_le h⟩, Finset.mem_univ _, ?_⟩
+  exact Finset.mem_image.mpr ⟨w.get, Finset.mem_univ _, List.ofFn_get w⟩
+
+theorem length_le_of_mem_traceSpace {n : ℕ} {w : Trace} (h : w ∈ traceSpace n) :
+    w.length ≤ n := by
+  obtain ⟨k, _, hk⟩ := Finset.mem_biUnion.mp h
+  obtain ⟨x, _, rfl⟩ := Finset.mem_image.mp hk
+  simpa using Nat.le_of_lt_succ k.isLt
+
+@[simp] theorem mem_traceSpace {n : ℕ} {w : Trace} :
+    w ∈ traceSpace n ↔ w.length ≤ n :=
+  ⟨length_le_of_mem_traceSpace, mem_traceSpace_of_length_le w⟩
+
+theorem traceOfMask_mem {n : ℕ} (x : Word n) (a : Mask n) :
+    traceOfMask x a ∈ traceSpace n :=
+  mem_traceSpace_of_length_le _ (traceOfMask_length_le x a)
+
+theorem maskWeight_nonneg {n : ℕ} {q : ℝ} (hq : q ∈ Set.Icc 0 1) (a : Mask n) :
+    0 ≤ maskWeight q a := by
+  apply Finset.prod_nonneg
+  intro i _
+  split <;> linarith [hq.1, hq.2]
+
+@[simp] theorem sum_maskWeight {n : ℕ} (q : ℝ) :
+    ∑ a : Mask n, maskWeight q a = 1 := by
+  unfold maskWeight
+  change (∑ a : Fin n → Bool, ∏ i, (fun (_ : Fin n) (b : Bool) =>
+    if b then 1 - q else q) i (a i)) = 1
+  rw [← Fintype.prod_sum (fun (_ : Fin n) (b : Bool) => if b then 1 - q else q)]
+  simp
+
+theorem traceMass_nonneg {n : ℕ} {q : ℝ} (hq : q ∈ Set.Icc 0 1)
+    (x : Word n) (w : Trace) : 0 ≤ traceMass q x w := by
+  apply Finset.sum_nonneg
+  intro a _
+  split
+  · exact maskWeight_nonneg hq a
+  · exact le_rfl
+
+theorem traceMass_eq_zero_of_not_mem {n : ℕ} (q : ℝ) (x : Word n) (w : Trace)
+    (hw : w ∉ traceSpace n) : traceMass q x w = 0 := by
+  apply Finset.sum_eq_zero
+  intro a _
+  have hne : traceOfMask x a ≠ w := by
+    intro h
+    exact hw (h ▸ traceOfMask_mem x a)
+  simp [hne]
+
+@[simp] theorem sum_traceMass {n : ℕ} (q : ℝ) (x : Word n) :
+    ∑ w ∈ traceSpace n, traceMass q x w = 1 := by
+  unfold traceMass
+  rw [Finset.sum_comm]
+  simp [traceOfMask_length_le]
+
+theorem traceTV_nonneg {n : ℕ} (q : ℝ) (x y : Word n) : 0 ≤ traceTV q x y := by
+  unfold traceTV
+  positivity
+
+@[simp] theorem traceTV_self {n : ℕ} (q : ℝ) (x : Word n) : traceTV q x x = 0 := by
+  simp [traceTV]
+
+theorem traceTV_comm {n : ℕ} (q : ℝ) (x y : Word n) : traceTV q x y = traceTV q y x := by
+  simp only [traceTV, abs_sub_comm]
+
+theorem traceTV_le_one {n : ℕ} {q : ℝ} (hq : q ∈ Set.Icc 0 1) (x y : Word n) :
+    traceTV q x y ≤ 1 := by
+  have h : ∑ w ∈ traceSpace n, |traceMass q x w - traceMass q y w| ≤
+      ∑ w ∈ traceSpace n, (traceMass q x w + traceMass q y w) := by
+    apply Finset.sum_le_sum
+    intro w _
+    exact abs_sub_le_iff.mpr
+      ⟨by linarith [traceMass_nonneg hq y w], by linarith [traceMass_nonneg hq x w]⟩
+  rw [Finset.sum_add_distrib, sum_traceMass, sum_traceMass] at h
+  unfold traceTV
+  linarith
+
+
+end TraceReconstruction
+
+namespace TraceReconstruction
+namespace FiniteLaw
+variable {α β γ : Type*} [Fintype α] [Fintype β] [Fintype γ]
+
+@[ext] theorem ext {P Q : FiniteLaw α} (h : P.mass = Q.mass) : P = Q := by
+  cases P with
+  | mk pm pn pt => cases Q with
+    | mk qm qn qt => cases h; rfl
+
+noncomputable def pure (a : α) : FiniteLaw α := by
+  classical
+  exact ⟨(fun b => if a = b then 1 else 0),
+    (fun b => by split_ifs <;> norm_num), by simp⟩
+
+noncomputable def map (P : FiniteLaw α) (f : α → β) : FiniteLaw β :=
+  P.bind (fun a => pure (f a))
+
+theorem map_mass [DecidableEq β] (P : FiniteLaw α) (f : α → β) (b : β) :
+    (P.map f).mass b = ∑ a, if f a = b then P.mass a else 0 := by
+  classical
+  unfold map bind pure
+  simp only
+  apply Finset.sum_congr rfl
+  intro a _
+  split_ifs <;> simp
+
+theorem expectation_map (P : FiniteLaw α) (f : α → β) (F : β → ℝ) :
+    (∑ b, (P.map f).mass b * F b) = ∑ a, P.mass a * F (f a) := by
+  classical
+  simp_rw [map_mass, Finset.sum_mul]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro a _
+  simp
+
+theorem tv_map_le (P Q : FiniteLaw α) (f : α → β) :
+    tv (P.map f) (Q.map f) ≤ tv P Q := tv_bind_le P Q _
+
+end FiniteLaw
+
+abbrev BoundedTrace (n : ℕ) := ↥(traceSpace n)
+
+def maskLaw {n : ℕ} (q : ℝ) (hq : q ∈ Set.Icc 0 1) : FiniteLaw (Mask n) :=
+  ⟨maskWeight q, maskWeight_nonneg hq, sum_maskWeight q⟩
+
+def completeTraceLaw {n : ℕ} (q : ℝ) (hq : q ∈ Set.Icc 0 1) (x : Word n) :
+    FiniteLaw (BoundedTrace n) where
+  mass w := traceMass q x w
+  nonneg w := traceMass_nonneg hq x w
+  total := by rw [Finset.sum_coe_sort]; exact sum_traceMass q x
+
+theorem completeTraceLaw_tv {n : ℕ} (q : ℝ) (hq : q ∈ Set.Icc 0 1) (x y : Word n) :
+    (completeTraceLaw q hq x).tv (completeTraceLaw q hq y) = traceTV q x y := by
+  change (1 / 2 : ℝ) * (∑ w : BoundedTrace n, |traceMass q x w - traceMass q y w|) = _
+  rw [Finset.sum_coe_sort (traceSpace n) (fun w => |traceMass q x w - traceMass q y w|)]
+  rfl
+
+theorem completeTraceLaw_mask {n : ℕ} (q : ℝ) (hq : q ∈ Set.Icc 0 1) (x : Word n) :
+    completeTraceLaw q hq x =
+      (maskLaw q hq).map (fun a : Mask n => ⟨traceOfMask x a, traceOfMask_mem x a⟩) := by
+  apply FiniteLaw.ext
+  funext w
+  rw [FiniteLaw.map_mass]
+  unfold completeTraceLaw traceMass maskLaw
+  simp only
+  apply Finset.sum_congr rfl
+  intro a _
+  rcases w with ⟨w, hw⟩
+  simp only [Subtype.mk.injEq]
+
+theorem trace_expectation {n : ℕ} (q : ℝ) (x : Word n) (F : Trace → ℝ) :
+    (∑ w ∈ traceSpace n, traceMass q x w * F w) =
+      ∑ a : Mask n, maskWeight q a * F (traceOfMask x a) := by
+  classical
+  unfold traceMass
+  simp_rw [Finset.sum_mul]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro a _
+  simp [traceOfMask_length_le]
+
+/-- Concatenation of exact binary inputs, not an encoded surrogate. -/
+def appendWord {n m : ℕ} (x : Word n) (y : Word m) : Word (n + m) := Fin.append x y
+
+theorem trace_filterMap {n : ℕ} (x : Word n) (a : Mask n) :
+    traceOfMask x a = (List.ofFn fun i => if a i then some (x i) else none).filterMap id := by
+  have hlist (l : List (Fin n)) :
+      (l.filter fun i => a i).map x = l.filterMap (fun i => if a i then some (x i) else none) := by
+    induction l with
+    | nil => rfl
+    | cons i l ih => cases hai : a i <;> simp [hai, ih]
+  unfold traceOfMask
+  rw [hlist]
+  have heq : (List.ofFn fun i => if a i then some (x i) else none) =
+      (List.finRange n).map (fun i => if a i then some (x i) else none) := by
+    rw [← List.ofFn_id, List.map_ofFn]
+    rfl
+  rw [heq, List.filterMap_map]
+  rfl
+
+theorem traceOfMask_append {n m : ℕ} (x : Word n) (y : Word m)
+    (a : Mask n) (b : Mask m) :
+    traceOfMask (appendWord x y) (Fin.append a b) = traceOfMask x a ++ traceOfMask y b := by
+  rw [trace_filterMap, List.ofFn_add, List.filterMap_append, trace_filterMap, trace_filterMap]
+  congr 2 <;> apply congrArg List.ofFn <;> funext i
+  · change (if Fin.append a b (Fin.castAdd m i) then some (Fin.append x y (Fin.castAdd m i)) else none) = _
+    simp
+  · simp [appendWord]
+
+theorem maskWeight_append {n m : ℕ} (q : ℝ) (a : Mask n) (b : Mask m) :
+    maskWeight q (Fin.append a b) = maskWeight q a * maskWeight q b := by
+  unfold maskWeight
+  rw [Fin.prod_univ_add]
+  simp
+
+theorem concatenated_trace_expectation {n m : ℕ} (q : ℝ) (x : Word n) (y : Word m)
+    (F : Trace → ℝ) :
+    (∑ a : Mask (n + m), maskWeight q a * F (traceOfMask (appendWord x y) a)) =
+      ∑ a : Mask n, ∑ b : Mask m,
+        maskWeight q a * maskWeight q b * F (traceOfMask x a ++ traceOfMask y b) := by
+  rw [← (Fin.appendEquiv n m).sum_comp]
+  rw [Fintype.sum_prod_type]
+  apply Finset.sum_congr rfl
+  intro a _
+  apply Finset.sum_congr rfl
+  intro b _
+  change maskWeight q (Fin.append a b) * F (traceOfMask (appendWord x y) (Fin.append a b)) = _
+  rw [maskWeight_append, traceOfMask_append]
+
+end TraceReconstruction
+
+namespace TraceReconstruction
+
+/-- The four complete traces possible from the ordinary primitive `01`. -/
+abbrev PrimitiveMark := Fin 4
+
+def primitiveOutput : PrimitiveMark → List Bool := ![[], [false], [true], [false, true]]
+def ordinaryPrimitiveMass (q : ℝ) : PrimitiveMark → ℝ :=
+  ![q ^ 2, q * (1 - q), q * (1 - q), (1 - q) ^ 2]
+def defectPrimitiveMass (q : ℝ) : PrimitiveMark → ℝ := ![q, 0, 1 - q, 0]
+def primitiveErasure (q : ℝ) : ℝ := q ^ 2 / 2
+
+def primitiveMarkMass (q : ℝ) (isDefect : Bool) (z : PrimitiveMark) : ℝ :=
+  ((if isDefect then defectPrimitiveMass q z else ordinaryPrimitiveMass q z) -
+    if z = 0 then primitiveErasure q else 0) / (1 - primitiveErasure q)
+
+theorem primitive_retain_pos {q : ℝ} (hq : q ∈ Set.Ioo 0 1) :
+    0 < 1 - primitiveErasure q := by
+  unfold primitiveErasure
+  nlinarith [hq.1, hq.2]
+
+theorem primitiveMarkMass_nonneg {q : ℝ} (hq : q ∈ Set.Ioo 0 1)
+    (b : Bool) (z : PrimitiveMark) : 0 ≤ primitiveMarkMass q b z := by
+  apply div_nonneg _ (primitive_retain_pos hq).le
+  cases b <;> fin_cases z <;>
+    simp [ordinaryPrimitiveMass, defectPrimitiveMass, primitiveErasure] <;>
+    nlinarith [hq.1, hq.2]
+
+theorem primitiveMarkMass_sum {q : ℝ} (hq : q ∈ Set.Ioo 0 1) (b : Bool) :
+    ∑ z : PrimitiveMark, primitiveMarkMass q b z = 1 := by
+  have hd := ne_of_gt (primitive_retain_pos hq)
+  unfold primitiveMarkMass
+  rw [← Finset.sum_div]
+  apply (div_eq_one_iff_eq hd).mpr
+  cases b <;> simp [ordinaryPrimitiveMass, defectPrimitiveMass, primitiveErasure, Fin.sum_univ_succ]
+  all_goals ring
+
+def primitiveMarkLaw (q : ℝ) (hq : q ∈ Set.Ioo 0 1) (b : Bool) : FiniteLaw PrimitiveMark :=
+  ⟨primitiveMarkMass q b, primitiveMarkMass_nonneg hq b, primitiveMarkMass_sum hq b⟩
+
+/-- Emitting the conditional mark, or silently erasing the block, gives
+exactly the ordinary primitive-output probabilities. Silent erasure and an
+emitted empty mark are distinct latent events. -/
+theorem primitiveMark_simulation {q : ℝ} (hq : q ∈ Set.Ioo 0 1)
+    (b : Bool) (z : PrimitiveMark) :
+    (if z = 0 then primitiveErasure q else 0) +
+      (1 - primitiveErasure q) * primitiveMarkMass q b z =
+        if b then defectPrimitiveMass q z else ordinaryPrimitiveMass q z := by
+  unfold primitiveMarkMass
+  rw [mul_div_cancel₀ _ (ne_of_gt (primitive_retain_pos hq))]
+  ring
+
+def primitiveScore (q : ℝ) : PrimitiveMark → ℝ := ![2 / q - 2, -1, 1 / q - 1, -1]
+
+theorem primitive_score_identity {q : ℝ} (hq : q ∈ Set.Ioo 0 1) (z : PrimitiveMark) :
+    primitiveMarkMass q true z = primitiveMarkMass q false z * (1 + primitiveScore q z) := by
+  have hq0 := ne_of_gt hq.1
+  have hd := ne_of_gt (primitive_retain_pos hq)
+  fin_cases z <;>
+    simp [primitiveMarkMass, ordinaryPrimitiveMass, defectPrimitiveMass, primitiveScore, primitiveErasure] <;>
+    field_simp
+  all_goals ring
+
+theorem primitive_score_centered {q : ℝ} (hq : q ∈ Set.Ioo 0 1) :
+    ∑ z : PrimitiveMark, primitiveMarkMass q false z * primitiveScore q z = 0 := by
+  have hi := primitiveMarkMass_sum hq true
+  simp_rw [primitive_score_identity hq, mul_add, mul_one, Finset.sum_add_distrib,
+    primitiveMarkMass_sum hq false] at hi
+  linarith
+
+theorem primitive_score_bound {q : ℝ} (hq : q ∈ Set.Ioo 0 1) (z : PrimitiveMark) :
+    |primitiveScore q z| ≤ 2 / q := by
+  have hi : 1 ≤ 1 / q := (le_div_iff₀ hq.1).mpr (by linarith [hq.2])
+  simp only [one_div] at hi
+  have he : 2 / q = 2 * (1 / q) := by ring
+  fin_cases z <;> simp [primitiveScore, abs_le, he] <;> (try constructor) <;> linarith
+
+theorem primitive_score_variance_exact {q : ℝ} (hq : q ∈ Set.Ioo 0 1) :
+    (∑ z : PrimitiveMark, primitiveMarkMass q false z * primitiveScore q z ^ 2) =
+      (1 / q - 2 * q + q ^ 2) / (1 - q ^ 2 / 2) := by
+  have hq0 := ne_of_gt hq.1
+  have hd := ne_of_gt (primitive_retain_pos hq)
+  simp only [primitiveMarkMass, primitiveScore, ordinaryPrimitiveMass, primitiveErasure,
+    Bool.false_eq_true, ↓reduceIte]
+  simp [Fin.sum_univ_succ]
+  field_simp
+  ring
+
+theorem primitive_score_variance_le {q : ℝ} (hq : q ∈ Set.Ioo 0 1) :
+    (∑ z : PrimitiveMark, primitiveMarkMass q false z * primitiveScore q z ^ 2) ≤ 2 / q := by
+  rw [primitive_score_variance_exact hq]
+  have hd : 0 < 1 - q ^ 2 / 2 := primitive_retain_pos hq
+  apply (div_le_div_iff₀ hd hq.1).mpr
+  rw [add_mul, sub_mul, one_div_mul_cancel (ne_of_gt hq.1)]
+  have hpow : q ^ 3 ≤ q ^ 2 := by
+    calc
+      _ = q ^ 2 * q := by ring
+      _ ≤ q ^ 2 * 1 := mul_le_mul_of_nonneg_left hq.2.le (sq_nonneg _)
+      _ = _ := mul_one _
+  nlinarith only [hpow, sq_nonneg q]
+
+end TraceReconstruction
+
+namespace TraceReconstruction
+def ordinaryRun : ℕ → List Bool
+  | 0 => []
+  | m + 1 => false :: true :: ordinaryRun m
+
+@[simp] theorem ordinaryRun_length (m : ℕ) : (ordinaryRun m).length = 2 * m := by
+  induction m with
+  | zero => rfl
+  | succ m ih => simp only [ordinaryRun, List.length_cons, ih]; omega
+
+theorem ordinaryRun_add (a b : ℕ) : ordinaryRun (a + b) = ordinaryRun a ++ ordinaryRun b := by
+  induction a with
+  | zero => simp [ordinaryRun]
+  | succ a ih => simp [Nat.succ_add, ordinaryRun, ih]
+
+/-- A period of m primitives with its unique `1` defect at zero-based slot d. -/
+def initialPeriod (m d : ℕ) : List Bool :=
+  ordinaryRun d ++ [true] ++ ordinaryRun (m - 1 - d)
+
+theorem initialPeriod_length {m d : ℕ} (hd : d < m) :
+    (initialPeriod m d).length = 2 * m - 1 := by
+  simp only [initialPeriod, List.length_append, ordinaryRun_length, List.length_singleton]
+  omega
+
+/-- Distinct defect slots differ before either period ends, independently of future text. -/
+theorem initialPeriod_tail_ne {m d e : ℕ} (hde : d < e) (he : e < m)
+    (u v : List Bool) : initialPeriod m d ++ u ≠ initialPeriod m e ++ v := by
+  obtain ⟨a, ha⟩ := Nat.exists_eq_add_of_le (show d + 1 ≤ e by omega)
+  subst e
+  unfold initialPeriod
+  rw [Nat.add_assoc d 1 a, ordinaryRun_add d (1 + a)]
+  simp only [Nat.add_assoc] at *
+  simp only [Nat.one_add, ordinaryRun, List.append_assoc, List.cons_append, List.nil_append]
+  intro h
+  have hh := List.append_cancel_left h
+  have hf := congrArg List.head? hh
+  simp only [List.head?_cons, Option.some.injEq] at hf
+  cases hf
+
+theorem initialPeriod_injective (m : ℕ) :
+    Function.Injective (fun d : Fin m => initialPeriod m d) := by
+  intro d e h
+  by_contra hne
+  have hv : d.val ≠ e.val := fun hv => hne (Fin.ext hv)
+  rcases lt_or_gt_of_ne hv with hlt | hgt
+  · exact initialPeriod_tail_ne hlt e.isLt [] [] (by simpa using h)
+  · exact initialPeriod_tail_ne hgt d.isLt [] [] (by simpa using h.symm)
+
+def periodString (m P : ℕ) (d : Fin P → Fin m) : List Bool :=
+  (List.ofFn (fun i => initialPeriod m (d i))).flatten
+
+theorem periodString_cons (m P : ℕ) (d : Fin (P + 1) → Fin m) :
+    periodString m (P + 1) d = initialPeriod m (d 0) ++
+      periodString m P (fun i => d i.succ) := by
+  simp only [periodString, List.ofFn_succ, List.flatten_cons]
+
+theorem periodString_length (m P : ℕ) (d : Fin P → Fin m) :
+    (periodString m P d).length = P * (2 * m - 1) := by
+  induction P with
+  | zero => simp [periodString]
+  | succ P ih =>
+    rw [periodString_cons, List.length_append, initialPeriod_length (d 0).isLt, ih]
+    rw [Nat.add_mul, one_mul, Nat.add_comm]
+
+theorem periodString_injective (m P : ℕ) : Function.Injective (periodString m P) := by
+  induction P with
+  | zero => intro d e _; exact Subsingleton.elim _ _
+  | succ P ih =>
+    intro d e h
+    rw [periodString_cons, periodString_cons] at h
+    have hz : d 0 = e 0 := by
+      by_contra hne
+      have hv : (d 0).val ≠ (e 0).val := fun hv => hne (Fin.ext hv)
+      rcases lt_or_gt_of_ne hv with hlt | hgt
+      · exact initialPeriod_tail_ne hlt (e 0).isLt _ _ h
+      · exact initialPeriod_tail_ne hgt (d 0).isLt _ _ h.symm
+    rw [hz] at h
+    have ht := ih (List.append_cancel_left h)
+    funext i
+    refine Fin.cases hz (fun j => congrFun ht j) i
+end TraceReconstruction
+
+namespace TraceReconstruction
+/-- Uniformly sized binary atoms concatenated without boundary marks. -/
+def atomString (P : ℕ) (w : Bool → List Bool) (θ : Fin P → Bool) : List Bool :=
+  (List.ofFn (fun i => w (θ i))).flatten
+
+theorem atomString_cons (P : ℕ) (w : Bool → List Bool) (θ : Fin (P + 1) → Bool) :
+    atomString (P + 1) w θ = w (θ 0) ++ atomString P w (fun i => θ i.succ) := by
+  simp only [atomString, List.ofFn_succ, List.flatten_cons]
+
+theorem atomString_length (P S : ℕ) (w : Bool → List Bool) (hw : ∀ b, (w b).length = S)
+    (θ : Fin P → Bool) : (atomString P w θ).length = P * S := by
+  induction P with
+  | zero => simp [atomString]
+  | succ P ih => rw [atomString_cons, List.length_append, hw, ih, Nat.add_mul, one_mul, Nat.add_comm]
+
+theorem atomString_injective (P S : ℕ) (w : Bool → List Bool)
+    (hw : ∀ b, (w b).length = S) (hinj : Function.Injective w) :
+    Function.Injective (atomString P w) := by
+  induction P with
+  | zero => intro θ φ _; exact Subsingleton.elim _ _
+  | succ P ih =>
+    intro θ φ h
+    rw [atomString_cons, atomString_cons] at h
+    have hh := congrArg (List.take S) h
+    have hθ (u : List Bool) (b : Bool) : (w b ++ u).take S = w b := by
+      rw [← hw b, List.take_left]
+    rw [hθ, hθ] at hh
+    have hz := hinj hh
+    rw [hz] at h
+    have ht := ih (List.append_cancel_left h)
+    funext i
+    exact Fin.cases hz (fun j => congrFun ht j) i
+
+/-- Zero-based notation for the source complementary template; `true` is minus.
+The motif itself is fixed independently of the hypothesis. -/
+def templateType (P a L : ℕ) (sign : Fin L → Bool) (τ : Bool) (i : Fin P) : Bool :=
+  if h : a ≤ i.val ∧ i.val < a + L then xor τ (sign ⟨i.val - a, by omega⟩) else false
+
+theorem templateType_at (P a L : ℕ) (sign : Fin L → Bool) (τ : Bool)
+    (hL : 0 < L) (ha : a < P) :
+    templateType P a L sign τ ⟨a, ha⟩ = xor τ (sign ⟨0, hL⟩) := by
+  simp [templateType, hL]
+
+theorem templateType_injective (P a L : ℕ) (sign : Fin L → Bool)
+    (hL : 0 < L) (ha : a < P) :
+    Function.Injective (templateType P a L sign) := by
+  intro τ υ h
+  have he := congrFun h ⟨a, ha⟩
+  rw [templateType_at P a L sign τ hL ha, templateType_at P a L sign υ hL ha] at he
+  cases hs : sign ⟨0, hL⟩ <;> cases τ <;> cases υ <;> simp_all
+
+def complementaryWords (P a L : ℕ) (sign : Fin L → Bool) (w : Bool → List Bool)
+    (τ : Bool) : List Bool := atomString P w (templateType P a L sign τ)
+
+theorem complementaryWords_length (P a L S : ℕ) (sign : Fin L → Bool)
+    (w : Bool → List Bool) (hw : ∀ b, (w b).length = S) (τ : Bool) :
+    (complementaryWords P a L sign w τ).length = P * S := atomString_length P S w hw _
+
+theorem complementaryWords_injective (P a L S : ℕ) (sign : Fin L → Bool)
+    (w : Bool → List Bool) (hw : ∀ b, (w b).length = S) (hinj : Function.Injective w)
+    (hL : 0 < L) (ha : a < P) : Function.Injective (complementaryWords P a L sign w) :=
+  (atomString_injective P S w hw hinj).comp (templateType_injective P a L sign hL ha)
+end TraceReconstruction
+
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/TraceReconstruction/Oracles.lean b/lean/OAI/Probability/TraceReconstruction/Oracles.lean
index 80264348b00a626b676fb44fe8ab7db8cb9d895c..43bbd5da98fde82637d496ac7c7e7abb7374e30e 100644
--- a/lean/OAI/Probability/TraceReconstruction/Oracles.lean
+++ b/lean/OAI/Probability/TraceReconstruction/Oracles.lean
@@ -1,1697 +1,1697 @@
-import OAI.Probability.TraceReconstruction.InitialMoments
-
-namespace OAI
-
-/-!
-# Oracles
-
-TV kernels and trace simulation, anchor costs, counted primitives, common-part lifting, concrete oracle pairs, and paired-flag metadata.
--/
-
-open scoped ENNReal
-noncomputable section
-
-open Filter
-open scoped Topology
-
-
-noncomputable section
-namespace TraceReconstruction.FiniteLaw
-open scoped Classical
-variable {α β γ : Type*} [Fintype α] [Fintype β] [Fintype γ]
-
-@[ext] theorem law_ext {P Q : FiniteLaw α} (h : ∀a,P.mass a=Q.mass a) : P=Q := by
-  cases P; cases Q; congr; exact funext h
-
-theorem tv_self (P : FiniteLaw α) : P.tv P=0 := by simp [tv]
-
-theorem tv_le_one (P Q : FiniteLaw α) : P.tv Q≤1 := by
-  unfold tv
-  have h : (∑a,|P.mass a-Q.mass a|)≤2 := by
-    calc
-      _ ≤ ∑a, (P.mass a+Q.mass a) := Finset.sum_le_sum (fun a _ =>
-        (abs_sub (P.mass a) (Q.mass a)).trans_eq (by rw [abs_of_nonneg (P.nonneg a),abs_of_nonneg (Q.nonneg a)]))
-      _ = 2 := by rw [Finset.sum_add_distrib,P.total,Q.total]; norm_num
-  linarith
-
-theorem expect_bind (P : FiniteLaw α) (K : α → FiniteLaw β) (f : β → ℝ) :
-    (P.bind K).expect f=P.expect (fun a => (K a).expect f) := by
-  unfold expect bind
-  simp_rw [Finset.sum_mul,Finset.mul_sum]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro a _
-  apply Finset.sum_congr rfl
-  intro b _
-  ring
-
-theorem bind_assoc (P : FiniteLaw α) (K : α → FiniteLaw β) (L : β → FiniteLaw γ) :
-    (P.bind K).bind L=P.bind (fun a => (K a).bind L) := by
-  apply law_ext
-  intro c
-  simp only [bind,Finset.sum_mul,Finset.mul_sum]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro a _
-  apply Finset.sum_congr rfl
-  intro b _
-  ring
-
-theorem tv_bind_kernel_le (P : FiniteLaw α) (K L : α → FiniteLaw β) :
-    (P.bind K).tv (P.bind L) ≤ P.expect (fun a => (K a).tv (L a)) := by
-  unfold tv bind expect
-  simp only
-  calc
-    _ = (1/2:ℝ)*∑b, |∑a,P.mass a*((K a).mass b-(L a).mass b)| := by
-      congr 1
-      apply Finset.sum_congr rfl
-      intro b _
-      congr 1
-      rw [← Finset.sum_sub_distrib]
-      apply Finset.sum_congr rfl
-      intro a _
-      ring
-    _ ≤ (1/2:ℝ)*∑b,∑a,P.mass a*|(K a).mass b-(L a).mass b| := by
-      apply mul_le_mul_of_nonneg_left _ (by norm_num)
-      apply Finset.sum_le_sum
-      intro b _
-      simpa only [abs_mul,abs_of_nonneg (P.nonneg _)] using
-        Finset.abs_sum_le_sum_abs (fun a => P.mass a*((K a).mass b-(L a).mass b)) Finset.univ
-    _ = _ := by
-      rw [Finset.sum_comm,Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro a _
-      rw [← Finset.mul_sum]
-      ring
-
-/-- An invalid layout is charged only its unconditional probability. -/
-theorem tv_bind_fit_le (P : FiniteLaw α) (K L : α → FiniteLaw β)
-    (good : α → Prop) [DecidablePred good] (E : ℝ)
-    (hfit : ∀a,good a → (K a).tv (L a)≤E) :
-    (P.bind K).tv (P.bind L) ≤ P.expect (fun a => if good a then E else 1) := by
-  apply (tv_bind_kernel_le P K L).trans
-  apply expect_mono
-  intro a
-  split_ifs with h
-  · exact hfit a h
-  · exact tv_le_one _ _
-
-theorem tv_prod_eq_left (P Q : FiniteLaw α) (R : FiniteLaw β) :
-    (P.prod R).tv (Q.prod R)=P.tv Q := by
-  unfold tv prod
-  congr 1
-  rw [Fintype.sum_prod_type]
-  apply Finset.sum_congr rfl
-  intro a _
-  simp_rw [← sub_mul,abs_mul,abs_of_nonneg (R.nonneg _),← Finset.mul_sum,R.total,mul_one]
-
-theorem tv_prod_eq_right (P : FiniteLaw α) (Q R : FiniteLaw β) :
-    (P.prod Q).tv (P.prod R)=Q.tv R := by
-  unfold tv prod
-  congr 1
-  rw [Fintype.sum_prod_type]
-  simp_rw [← mul_sub,abs_mul,abs_of_nonneg (P.nonneg _),← Finset.mul_sum]
-  rw [← Finset.sum_mul,P.total,one_mul]
-
-theorem tv_prod_le (P Q : FiniteLaw α) (R S : FiniteLaw β) :
-    (P.prod R).tv (Q.prod S) ≤ P.tv Q+R.tv S := by
-  have H := tv_triangle (P.prod R) (Q.prod R) (Q.prod S)
-  rwa [tv_prod_eq_left,tv_prod_eq_right] at H
-
-end TraceReconstruction.FiniteLaw
-
-
-noncomputable section
-namespace TraceReconstruction.FiniteLaw
-open scoped Classical
-variable {α : Type*} [Fintype α]
-
-theorem coordinateProduct_succ_tv {n : ℕ} (P Q : Fin (n+1) → FiniteLaw α) :
-    (coordinateProduct P).tv (coordinateProduct Q) =
-      ((P 0).prod (coordinateProduct (fun i => P i.succ))).tv
-        ((Q 0).prod (coordinateProduct (fun i => Q i.succ))) := by
-  change (1/2:ℝ)*(∑x : Fin (n+1) → α, |(∏i,(P i).mass (x i))-(∏i,(Q i).mass (x i))|)=
-    (1/2:ℝ)*∑x : α × (Fin n → α), |(P 0).mass x.1*(∏i,(P i.succ).mass (x.2 i))-(Q 0).mass x.1*(∏i,(Q i.succ).mass (x.2 i))|
-  congr 1
-  rw [← (Fin.consEquiv (fun _ : Fin (n+1) => α)).sum_comp]
-  apply Finset.sum_congr rfl
-  intro x _
-  simp only [Fin.consEquiv,Equiv.coe_fn_mk,Fin.prod_univ_succ,Fin.cons_zero,Fin.cons_succ]
-
-theorem tv_coordinateProduct_le {n : ℕ} (P Q : Fin n → FiniteLaw α) :
-    (coordinateProduct P).tv (coordinateProduct Q) ≤ ∑i, (P i).tv (Q i) := by
-  induction n with
-  | zero => simp [tv,coordinateProduct]
-  | succ n ih =>
-    rw [coordinateProduct_succ_tv,Fin.sum_univ_succ]
-    exact (tv_prod_le _ _ _ _).trans (by linarith [ih (fun i => P i.succ) (fun i => Q i.succ)])
-
-theorem expect_coordinateProduct_succ {n : ℕ} (P : Fin (n+1) → FiniteLaw α)
-    (F : (Fin (n+1) → α) → ℝ) :
-    (coordinateProduct P).expect F =
-      (P 0).expect (fun a => (coordinateProduct (fun i => P i.succ)).expect
-        (fun x => F (Fin.cons a x))) := by
-  change (∑x : Fin (n+1) → α, (∏i,(P i).mass (x i))*F x)=
-    ∑a : α, (P 0).mass a*(∑x : Fin n → α, (∏i,(P i.succ).mass (x i))*F (Fin.cons a x))
-  rw [← (Fin.consEquiv (fun _ : Fin (n+1) => α)).sum_comp,Fintype.sum_prod_type]
-  simp only [Fin.consEquiv,Equiv.coe_fn_mk,Fin.prod_univ_succ,Fin.cons_zero,Fin.cons_succ]
-  simp_rw [Finset.mul_sum,mul_assoc]
-
-end TraceReconstruction.FiniteLaw
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem FiniteLaw.expect_sub {α : Type*} [Fintype α] (P : FiniteLaw α)
-    (F G : α → ℝ) : P.expect (fun a => F a-G a)=P.expect F-P.expect G := by
-  simp only [FiniteLaw.expect,mul_sub,Finset.sum_sub_distrib]
-
-theorem FiniteLaw.abs_expect_le {α : Type*} [Fintype α] (P : FiniteLaw α) (F : α → ℝ) :
-    |P.expect F|≤P.expect (fun a => |F a|) := by
-  simpa only [FiniteLaw.expect,abs_mul,abs_of_nonneg (P.nonneg _)] using
-    Finset.abs_sum_le_sum_abs (fun a => P.mass a*F a) Finset.univ
-
-/-- A finite auxiliary observation, decoded by a fixed function, simulates the
-usual deletion channel on a literal binary word. Test functions range over the
-observable traces only, never latent boundaries. -/
-def SimulatesTrace {α : Type*} [Fintype α] (q : ℝ) (P : FiniteLaw α)
-    (decode : α → List Bool) (w : List Bool) : Prop :=
-  ∀ F : List Bool → ℝ, P.expect (fun a => F (decode a))=listDeletionExpectation q w F
-
-def ApproximatesTrace {α : Type*} [Fintype α] (q : ℝ) (P : FiniteLaw α)
-    (decode : α → List Bool) (w : List Bool) (E : ℝ) : Prop :=
-  ∀ F : List Bool → ℝ, (∀t,F t∈Set.Icc 0 1) →
-    |P.expect (fun a => F (decode a))-listDeletionExpectation q w F|≤E
-
-theorem listDeletionExpectation_const (q c : ℝ) (w : List Bool) :
-    listDeletionExpectation q w (fun _ => c)=c := by
-  induction w with
-  | nil => rfl
-  | cons b w ih => simp only [listDeletionExpectation,ih]; ring
-
-theorem listDeletionExpectation_linear (q : ℝ) (w : List Bool)
-    (a b : ℝ) (F G : List Bool → ℝ) :
-    listDeletionExpectation q w (fun t => a*F t+b*G t)=
-      a*listDeletionExpectation q w F+b*listDeletionExpectation q w G := by
-  induction w generalizing F G with
-  | nil => rfl
-  | cons c w ih => simp only [listDeletionExpectation,ih]; ring
-
-theorem listDeletionExpectation_mono {q : ℝ} (hq : q∈Set.Icc 0 1)
-    (w : List Bool) {F G : List Bool → ℝ} (h : ∀t,F t≤G t) :
-    listDeletionExpectation q w F≤listDeletionExpectation q w G := by
-  induction w generalizing F G with
-  | nil => exact h []
-  | cons b w ih =>
-    exact add_le_add (mul_le_mul_of_nonneg_left (ih h) hq.1)
-      (mul_le_mul_of_nonneg_left (ih (fun t => h (b::t))) (sub_nonneg.mpr hq.2))
-
-theorem listDeletionExpectation_range {q : ℝ} (hq : q∈Set.Icc 0 1)
-    (w : List Bool) (F : List Bool → ℝ) (hF : ∀t,F t∈Set.Icc 0 1) :
-    listDeletionExpectation q w F∈Set.Icc 0 1 := by
-  constructor
-  · simpa only [listDeletionExpectation_const] using
-      listDeletionExpectation_mono hq w (fun t => (hF t).1)
-  · simpa only [listDeletionExpectation_const] using
-      listDeletionExpectation_mono hq w (fun t => (hF t).2)
-
-theorem listDeletionExpectation_sum {α : Type*} [Fintype α] (q : ℝ)
-    (w : List Bool) (F : α → List Bool → ℝ) :
-    listDeletionExpectation q w (fun t => ∑a,F a t)=
-      ∑a,listDeletionExpectation q w (F a) := by
-  induction w generalizing F with
-  | nil => rfl
-  | cons b w ih =>
-    simp only [listDeletionExpectation,ih,Finset.mul_sum,Finset.sum_add_distrib]
-
-theorem FiniteLaw.expect_listDeletionExpectation {α : Type*} [Fintype α]
-    (P : FiniteLaw α) (q : ℝ) (w : List Bool) (F : α → List Bool → ℝ) :
-    P.expect (fun a => listDeletionExpectation q w (F a))=
-      listDeletionExpectation q w (fun t => P.expect (fun a => F a t)) := by
-  unfold FiniteLaw.expect
-  rw [listDeletionExpectation_sum]
-  apply Finset.sum_congr rfl
-  intro a _
-  have H := listDeletionExpectation_linear q w (P.mass a) 0 (F a) (fun _ => 0)
-  simpa only [zero_mul,add_zero] using H.symm
-
-theorem SimulatesTrace.append {α β : Type*} [Fintype α] [Fintype β]
-    {q : ℝ} {P : FiniteLaw α} {Q : FiniteLaw β} {f : α → List Bool} {g : β → List Bool}
-    {x y : List Bool} (hP : SimulatesTrace q P f x) (hQ : SimulatesTrace q Q g y) :
-    SimulatesTrace q (P.prod Q) (fun a => f a.1++g a.2) (x++y) := by
-  intro F
-  rw [FiniteLaw.expect_prod,listDeletionExpectation_append]
-  change P.expect (fun a => Q.expect (fun b => F (f a++g b)))=_
-  unfold SimulatesTrace at hP hQ
-  calc
-    _ = P.expect (fun a => listDeletionExpectation q y (fun t => F (f a++t))) :=
-      congrArg P.expect (funext fun a => hQ (fun t => F (f a++t)))
-    _ = _ := hP (fun t => listDeletionExpectation q y (fun u => F (t++u)))
-
-theorem SimulatesTrace.approximate {α : Type*} [Fintype α]
-    {q : ℝ} {P : FiniteLaw α} {f : α → List Bool} {w : List Bool}
-    (h : SimulatesTrace q P f w) : ApproximatesTrace q P f w 0 := by
-  intro F _
-  rw [h,sub_self,abs_zero]
-
-theorem ApproximatesTrace.append {α β : Type*} [Fintype α] [Fintype β]
-    {q : ℝ} (hq : q∈Set.Icc 0 1) {P : FiniteLaw α} {Q : FiniteLaw β}
-    {f : α → List Bool} {g : β → List Bool} {x y : List Bool} {E D : ℝ}
-    (hP : ApproximatesTrace q P f x E) (hQ : ApproximatesTrace q Q g y D) :
-    ApproximatesTrace q (P.prod Q) (fun a => f a.1++g a.2) (x++y) (E+D) := by
-  intro F hF
-  rw [FiniteLaw.expect_prod,listDeletionExpectation_append]
-  let H : List Bool → ℝ := fun t => listDeletionExpectation q y (fun u => F (t++u))
-  have hH : ∀t,H t∈Set.Icc 0 1 := fun t => listDeletionExpectation_range hq y _ (fun u => hF (t++u))
-  have h1 := hP H hH
-  have h2 : |P.expect (fun a => Q.expect (fun b => F (f a++g b)))-P.expect (fun a => H (f a))|≤D := by
-    rw [← FiniteLaw.expect_sub]
-    calc
-      _ ≤ P.expect (fun a => |Q.expect (fun b => F (f a++g b))-H (f a)|) := FiniteLaw.abs_expect_le _ _
-      _ ≤ P.expect (fun _ => D) := FiniteLaw.expect_mono _ (fun a => hQ _ (fun t => hF (f a++t)))
-      _ = D := FiniteLaw.expect_const _ _
-  exact (abs_sub_le _ _ _).trans (by linarith only [h1,h2])
-
-/-- Any randomized partition into exactly the same literal word gives exactly
-its usual trace after boundaries are forgotten. No fit conditioning occurs. -/
-theorem SimulatesTrace.random_partition {α Ω : Type*} [Fintype α] [Fintype Ω]
-    {q : ℝ} (P : FiniteLaw Ω) (K : Ω → FiniteLaw α) (decode : α → List Bool)
-    (w : List Bool) (h : ∀ω,SimulatesTrace q (K ω) decode w) :
-    SimulatesTrace q (P.bind K) decode w := by
-  intro F
-  rw [FiniteLaw.expect_bind]
-  unfold SimulatesTrace at h
-  simp_rw [h]
-  exact FiniteLaw.expect_const _ _
-
-/-- A failed deterministic layout is charged its unconditional probability;
-the conditional law on all metadata, not a fit-conditioned law, is used. -/
-theorem ApproximatesTrace.random_partition {α Ω : Type*} [Fintype α] [Fintype Ω]
-    {q : ℝ} (hq : q∈Set.Icc 0 1) (P : FiniteLaw Ω) (K : Ω → FiniteLaw α)
-    (decode : α → List Bool) (w : List Bool) (good : Ω → Prop) [DecidablePred good]
-    (E : ℝ) (h : ∀ω,good ω → ApproximatesTrace q (K ω) decode w E) :
-    ApproximatesTrace q (P.bind K) decode w (P.expect (fun ω => if good ω then E else 1)) := by
-  intro F hF
-  rw [FiniteLaw.expect_bind,← FiniteLaw.expect_const P (listDeletionExpectation q w F),← FiniteLaw.expect_sub]
-  apply (FiniteLaw.abs_expect_le _ _).trans
-  apply FiniteLaw.expect_mono
-  intro ω
-  split_ifs with hgood
-  · exact h ω hgood F hF
-  · have hc := listDeletionExpectation_range hq w F hF
-    have hn := FiniteLaw.expect_mono (K ω) (fun a => (hF (decode a)).1)
-    have hu := FiniteLaw.expect_mono (K ω) (fun a => (hF (decode a)).2)
-    rw [FiniteLaw.expect_const] at hn hu
-    exact abs_le.mpr ⟨by linarith [hc.2],by linarith [hc.1]⟩
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- Admissible close child labels, without relaxing any plateau restrictions
-on the anchor. An arbitrary finite label domain is retained. -/
-def closeLabels (A : Finset ℤ) (L : ℕ) (a : ℤ) : Finset ℤ :=
-  A.filter (fun c => |c-a| < (L:ℤ))
-
-theorem closeLabels_card (A : Finset ℤ) (L : ℕ) (a : ℤ) :
-    (closeLabels A L a).card ≤ 2*L := by
-  have hsub : closeLabels A L a ⊆ Finset.Ioo (a-(L:ℤ)) (a+(L:ℤ)) := by
-    intro c hc
-    have h := abs_lt.mp ((Finset.mem_filter.mp hc).2)
-    exact Finset.mem_Ioo.mpr ⟨by omega,by omega⟩
-  have H := Finset.card_le_card hsub
-  rw [Int.card_Ioo] at H
-  omega
-
-/-- Root anchor and close-child energy. This keeps the actual labels finite,
-including any additional source plateau cutoff. -/
-theorem root_anchor_energy (n L : ℕ) (A B : Finset ℤ) (c : ℤ) :
-    (∑ a∈A, ∑ _d∈closeLabels B L a, windowKernel n L (a-c)^2) ≤
-      2*L^3*binomialBound n := by
-  calc
-    _ ≤ ∑a∈A, (2*L:ℝ)*windowKernel n L (a-c)^2 := by
-      apply Finset.sum_le_sum
-      intro a _
-      simp only [Finset.sum_const,nsmul_eq_mul]
-      exact mul_le_mul_of_nonneg_right (by exact_mod_cast closeLabels_card B L a) (sq_nonneg _)
-    _ = (2*L:ℝ)*(∑a∈A,windowKernel n L (a-c)^2) := by rw [Finset.mul_sum]
-    _ ≤ (2*L:ℝ)*(L^2*binomialBound n) :=
-      mul_le_mul_of_nonneg_left (windowKernel_energy n L A c) (by positivity)
-    _ = _ := by ring
-
-/-- The shared global jitter stays in the joint kernel until after squaring.
-The two close labels cost only (2L)^2, uniformly in parent labels. -/
-theorem sibling_anchor_energy (g n₁ n₂ L : ℕ) (A B C D : Finset ℤ) (c₁ c₂ : ℤ) :
-    (∑ a∈A, ∑ b∈B, ∑ _c∈closeLabels C L a, ∑ _d∈closeLabels D L b,
-      jointWindowKernel g n₁ n₂ L (a-c₁) (b-c₂)^2) ≤
-      4*L^6*binomialBound g*binomialBound n₂ := by
-  calc
-    _ ≤ ∑a∈A, ∑b∈B, (2*L:ℝ)^2*jointWindowKernel g n₁ n₂ L (a-c₁) (b-c₂)^2 := by
-      apply Finset.sum_le_sum
-      intro a _
-      apply Finset.sum_le_sum
-      intro b _
-      simp only [Finset.sum_const,nsmul_eq_mul]
-      have hc : ((closeLabels C L a).card:ℝ)≤2*L := by exact_mod_cast closeLabels_card C L a
-      have hd : ((closeLabels D L b).card:ℝ)≤2*L := by exact_mod_cast closeLabels_card D L b
-      have hm := mul_le_mul hc hd (by positivity) (by positivity : (0:ℝ)≤2*L)
-      rw [mul_assoc] at hm
-      nlinarith [mul_le_mul_of_nonneg_right hm (sq_nonneg (jointWindowKernel g n₁ n₂ L (a-c₁) (b-c₂)))]
-    _ = (2*L:ℝ)^2*(∑a∈A,∑b∈B,jointWindowKernel g n₁ n₂ L (a-c₁) (b-c₂)^2) := by
-      simp_rw [Finset.mul_sum]
-    _ ≤ (2*L:ℝ)^2*(L^4*binomialBound g*binomialBound n₂) :=
-      mul_le_mul_of_nonneg_left (jointWindowKernel_energy g n₁ n₂ L A B c₁ c₂) (by positivity)
-    _ = _ := by ring
-
-/-- Exact bottom-up summation of any finite binary anchor model. The state
-contains its actual index path, so group-dependent kernels are allowed. -/
-def binaryTreeSum {α : Type*} [Fintype α] (K : ℕ → α → α × α → ℝ) : ℕ → α → ℝ
-  | 0, _ => 1
-  | n+1, a => ∑b, K n a b*binaryTreeSum K n b.1*binaryTreeSum K n b.2
-
-def binaryTreeBudget (C : ℕ → ℝ) : ℕ → ℝ
-  | 0 => 1
-  | n+1 => C n*(binaryTreeBudget C n)^2
-
-theorem binaryTreeBudget_nonneg (C : ℕ → ℝ) (hC : ∀n,0≤C n) (n : ℕ) :
-    0≤binaryTreeBudget C n := by
-  cases n with
-  | zero => norm_num [binaryTreeBudget]
-  | succ n => exact mul_nonneg (hC n) (sq_nonneg _)
-
-theorem binaryTreeSum_nonneg {α : Type*} [Fintype α]
-    (K : ℕ → α → α × α → ℝ) (hK : ∀n a b,0≤K n a b) (n : ℕ) (a : α) :
-    0≤binaryTreeSum K n a := by
-  induction n generalizing a with
-  | zero => norm_num [binaryTreeSum]
-  | succ n ih => exact Finset.sum_nonneg (fun b _ => mul_nonneg (mul_nonneg (hK n a b) (ih b.1)) (ih b.2))
-
-/-- Uniform local anchor square sums multiply without decoupling the two
-correlated offsets of a sibling group. Domain/plateau indicators can be
-included in K and are never removed by conditioning. -/
-theorem binaryTreeSum_le {α : Type*} [Fintype α]
-    (K : ℕ → α → α × α → ℝ) (hK : ∀n a b,0≤K n a b)
-    (C : ℕ → ℝ) (hC : ∀n,0≤C n) (hlocal : ∀n a, (∑b,K n a b)≤C n)
-    (n : ℕ) (a : α) : binaryTreeSum K n a≤binaryTreeBudget C n := by
-  induction n generalizing a with
-  | zero => rfl
-  | succ n ih =>
-    calc
-      _ ≤ ∑b, K n a b*(binaryTreeBudget C n)^2 := by
-        apply Finset.sum_le_sum
-        intro b _
-        rw [mul_assoc,pow_two]
-        exact mul_le_mul_of_nonneg_left (mul_le_mul (ih b.1) (ih b.2)
-          (binaryTreeSum_nonneg K hK n b.2) (binaryTreeBudget_nonneg C hC n)) (hK n a b)
-      _ = (∑b,K n a b)*(binaryTreeBudget C n)^2 := by rw [Finset.sum_mul]
-      _ ≤ C n*(binaryTreeBudget C n)^2 := mul_le_mul_of_nonneg_right (hlocal n a) (sq_nonneg _)
-      _ = _ := rfl
-
-end TraceReconstruction
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem hypergeomMass_middle_sq (V M m v : ℕ) {p : ℝ}
-    (hp : p∈Set.Icc (1/4:ℝ) (3/4:ℝ)) (hMV : M≤V)
-    (hN : 16≤V+M) (hm : m≤V+M) (hmean : (m:ℝ)=(V+M:ℕ)*p) :
-    hypergeomMass V M m v^2≤4096/((M:ℝ)+1) := by
-  have hp' : p∈Set.Ioo (0:ℝ) 1 := by constructor <;> linarith [hp.1,hp.2]
-  have hpp : (1/8:ℝ)≤p*(1-p) := by nlinarith [sq_nonneg (p-1/2),mul_nonneg (sub_nonneg.mpr hp.1) (sub_nonneg.mpr hp.2)]
-  have hNr : (16:ℝ)≤(V:ℝ)+M := by exact_mod_cast hN
-  have hVr : (M:ℝ)≤V := by exact_mod_cast hMV
-  have hv : 1≤(V+M:ℕ)*p*(1-p) := by
-    push_cast
-    nlinarith [Nat.cast_nonneg (α:=ℝ) V,Nat.cast_nonneg (α:=ℝ) M]
-  have H := hypergeomMass_sq_upper V M m v hp' hm hmean hv
-  apply H.trans
-  have hd : 0<((V:ℝ)+1)*((M:ℝ)+1)*p*(1-p) := by positivity [hp'.1,sub_pos.mpr hp'.2]
-  apply (div_le_iff₀ hd).2
-  have hM : 0<(M:ℝ)+1 := by positivity
-  have he : 4096/((M:ℝ)+1)*(((V:ℝ)+1)*((M:ℝ)+1)*p*(1-p))=
-      4096*((V:ℝ)+1)*(p*(1-p)) := by field_simp
-  rw [he]
-  push_cast
-  nlinarith [mul_le_mul_of_nonneg_left hpp (show 0≤(V:ℝ)+1 by positivity)]
-
-theorem hypergeomMass_middle_le (V M m v : ℕ) {p : ℝ}
-    (hp : p∈Set.Icc (1/4:ℝ) (3/4:ℝ)) (hMV : M≤V)
-    (hN : 16≤V+M) (hm : m≤V+M) (hmean : (m:ℝ)=(V+M:ℕ)*p) :
-    hypergeomMass V M m v≤64/Real.sqrt ((M:ℝ)+1) := by
-  have H := hypergeomMass_middle_sq V M m v hp hMV hN hm hmean
-  have hs : 0<Real.sqrt ((M:ℝ)+1) := Real.sqrt_pos.2 (by positivity)
-  have he : (64/Real.sqrt ((M:ℝ)+1))^2=4096/((M:ℝ)+1) := by
-    rw [div_pow,Real.sq_sqrt (by positivity)]; norm_num
-  rw [←he] at H
-  exact (sq_le_sq₀ (hypergeomMass_nonneg _ _ _ _) (by positivity)).1 H
-
-theorem sum_range_extend_zero {f : ℕ→ℝ} {m n : ℕ} (hmn : m≤n)
-    (hz : ∀k,m≤k→f k=0) : ∑k∈Finset.range n,f k=∑k∈Finset.range m,f k := by
-  symm
-  apply Finset.sum_subset (Finset.range_mono hmn)
-  intro k _ hk
-  exact hz k (by simpa using hk)
-
-theorem hypergeomMass_above_total (V M m v : ℕ) (h : m<v) :
-    hypergeomMass V M m v=0 := by simp only [hypergeomMass,ite_eq_right (by omega : ¬v≤ m)]
-
-def hypergeomLaw (V M m : ℕ) (hm : m≤V+M) : FiniteLaw (Fin (V+M+1)) where
-  mass v := hypergeomMass V M m v
-  nonneg _ := hypergeomMass_nonneg _ _ _ _
-  total := by
-    rw [Fin.sum_univ_eq_sum_range]
-    rw [sum_range_extend_zero (Nat.add_le_add_right hm 1) (fun k hk =>
-      hypergeomMass_above_total V M m k (by omega))]
-    exact sum_hypergeomMass V M m hm
-
-theorem hypergeomMass_complement (V M m v : ℕ) (hv : v≤ m) :
-    hypergeomMass V M m v=hypergeomMass M V m (m-v) := by
-  simp only [hypergeomMass,ite_eq_left hv,ite_eq_left (Nat.sub_le _ _),Nat.sub_sub_self hv]
-  rw [Nat.add_comm M V,mul_comm]
-
-theorem hypergeomLaw_adjacent_tv (V M m : ℕ) (hM : M≤ m) (hV : m<V) :
-    (hypergeomLaw V M (m+1) (by omega)).tv (hypergeomLaw V M m (by omega))=
-      adjacentHypergeomTV V M m := by
-  unfold FiniteLaw.tv
-  simp only [hypergeomLaw]
-  rw [Fin.sum_univ_eq_sum_range (fun v => |hypergeomMass V M (m+1) v-
-    hypergeomMass V M m v|)]
-  rw [sum_range_extend_zero (show m+2≤V+M+1 by omega) (fun k hk => by
-    rw [hypergeomMass_above_total V M (m+1) k (by omega),
-      hypergeomMass_above_total V M m k (by omega),sub_self,abs_zero])]
-  rw [←Finset.sum_range_reflect (fun v => |hypergeomMass V M (m+1) v-
-    hypergeomMass V M m v|) (m+2)]
-  rw [show m+2=(m+1)+1 by omega,Finset.sum_range_succ']
-  have hzero : |hypergeomMass V M (m+1) (m+1)-hypergeomMass V M m (m+1)|=
-      hypergeomMass M V (m+1) 0 := by
-    rw [hypergeomMass_above_total V M m (m+1) (by omega),sub_zero,
-      hypergeomMass_complement V M (m+1) (m+1) le_rfl,Nat.sub_self,
-      abs_of_nonneg (hypergeomMass_nonneg _ _ _ _)]
-  have he (k : ℕ) (hk : k∈Finset.range (m+1)) :
-      |hypergeomMass V M (m+1) ((m+1+1)-1-(k+1))-
-       hypergeomMass V M m ((m+1+1)-1-(k+1))|=
-      |hypergeomMass M V (m+1) (k+1)-hypergeomMass M V m k| := by
-    have hk' : k≤ m := by simpa only [Finset.mem_range,Nat.lt_succ_iff] using hk
-    rw [show (m+1+1)-1-(k+1)=m-k by omega,
-      hypergeomMass_complement V M (m+1) (m-k) (by omega),
-      hypergeomMass_complement V M m (m-k) (by omega)]
-    rw [show m+1-(m-k)=k+1 by omega,show m-(m-k)=k by omega]
-  rw [Finset.sum_congr rfl he]
-  simp only [Nat.add_sub_cancel,Nat.sub_zero,hzero]
-  rw [sum_range_extend_zero (Nat.add_le_add_right hM 1) (fun k hk => by
-    rw [hypergeomMass_high M V (m+1) (k+1) (by omega),
-      hypergeomMass_high M V m k (by omega),sub_self,abs_zero])]
-  unfold adjacentHypergeomTV
-  ring
-
-theorem adjacentHypergeomTV_middle_le (V M m : ℕ) {p : ℝ}
-    (hp : p∈Set.Icc (1/4:ℝ) (3/4:ℝ)) (hMV : M≤V)
-    (hN : 16≤V+M) (hM : M≤ m) (hV : m<V) (hmean : (m:ℝ)=(V+M:ℕ)*p) :
-    adjacentHypergeomTV V M m≤64/Real.sqrt ((M:ℝ)+1) := by
-  apply (adjacentHypergeomTV_le_mass V M m hM hV).trans
-  rw [hypergeomMass_complement M V m _ (by
-    exact (hypergeomHiddenMode_le (V-1) M m (by omega)).trans hM)]
-  exact hypergeomMass_middle_le V M m _ hp hMV hN (by omega) hmean
-
-theorem hypergeomLaw_adjacent_middle (V M m : ℕ) (hN : 16≤V+M)
-    (hM : M≤ m) (hV : m<V) (hlow : V+M≤4*m) (hhigh : 4*m≤3*(V+M)) :
-    (hypergeomLaw V M m (by omega)).tv (hypergeomLaw V M (m+1) (by omega))≤
-      64/Real.sqrt ((M:ℝ)+1) := by
-  have hn : 0<(V+M:ℕ) := by omega
-  have hnr : (0:ℝ)<(V+M:ℕ) := by exact_mod_cast hn
-  have hL : ((V+M:ℕ):ℝ)≤4*(m:ℝ) := by exact_mod_cast hlow
-  have hU : 4*(m:ℝ)≤3*((V+M:ℕ):ℝ) := by exact_mod_cast hhigh
-  rw [FiniteLaw.tv_symm,hypergeomLaw_adjacent_tv V M m hM hV]
-  have hp : (m:ℝ)/(V+M:ℕ)∈Set.Icc (1/4:ℝ) (3/4:ℝ) := by
-    constructor
-    · apply (le_div_iff₀ hnr).2; linarith
-    · apply (div_le_iff₀ hnr).2; linarith
-  exact adjacentHypergeomTV_middle_le V M m hp (by omega) hN hM hV
-    (by field_simp)
-
-theorem FiniteLaw.tv_chain {α : Type*} [Fintype α] (P : ℕ→FiniteLaw α) (n : ℕ) :
-    (P 0).tv (P n)≤∑i∈Finset.range n,(P i).tv (P (i+1)) := by
-  induction n with
-  | zero => simp [FiniteLaw.tv_self]
-  | succ n ih =>
-    rw [Finset.sum_range_succ]
-    linarith [FiniteLaw.tv_triangle (P 0) (P n) (P (n+1))]
-
-theorem hypergeomLaw_steps (V M m P : ℕ) (hN : 16≤V+M)
-    (hM : M≤ m) (hV : m+P<V) (hlow : V+M≤4*m) (hhigh : 4*(m+P)≤3*(V+M)) :
-    (hypergeomLaw V M m (by omega)).tv (hypergeomLaw V M (m+P) (by omega))≤
-      (P:ℝ)*64/Real.sqrt ((M:ℝ)+1) := by
-  let Q : ℕ→FiniteLaw (Fin (V+M+1)) := fun j =>
-    hypergeomLaw V M (m+min j P) (by omega)
-  have H := FiniteLaw.tv_chain Q P
-  have hQ0 : Q 0=hypergeomLaw V M m (by omega) := by simp [Q]
-  have hQP : Q P=hypergeomLaw V M (m+P) (by omega) := by simp [Q]
-  rw [hQ0,hQP] at H
-  apply H.trans
-  calc
-    _≤∑j∈Finset.range P,64/Real.sqrt ((M:ℝ)+1) := by
-      apply Finset.sum_le_sum
-      intro j hj
-      have hj' : j<P := Finset.mem_range.mp hj
-      simp only [Q,min_eq_left (by omega : j≤P),min_eq_left (by omega : j+1≤P)]
-      exact hypergeomLaw_adjacent_middle V M (m+j) hN (by omega) (by omega)
-        (by omega) (by omega)
-    _=_ := by simp [mul_div_assoc]
-
-end TraceReconstruction
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-variable {α β : Type*} [Fintype α] [Fintype β]
-
-def visibleFiberEquiv (m : ℕ) (S : Finset α) (hS : S.card≤ m) :
-    {T : FixedSubset (α⊕β) m // T.val.toLeft=S} ≃ FixedSubset β (m-S.card) where
-  toFun T := ⟨T.val.val.toRight,by
-    have hc := Finset.card_toLeft_add_card_toRight (u:=T.val.val)
-    rw [T.property,T.val.property] at hc
-    omega⟩
-  invFun R := ⟨⟨S.disjSum R.val,by simp only [Finset.card_disjSum,R.property]; omega⟩,by simp⟩
-  left_inv T := by
-    apply Subtype.ext; apply Subtype.ext
-    simpa only [T.property] using (Finset.toLeft_disjSum_toRight (u:=T.val.val))
-  right_inv R := by apply Subtype.ext; exact Finset.toRight_disjSum
-
-def visibleScheduleLaw (m : ℕ) (hm : m≤Fintype.card α+Fintype.card β) : FiniteLaw (Finset α) :=
-  (uniformSubsetLaw (α:=α⊕β) m (by simpa using hm)).map (fun T => T.val.toLeft)
-
-theorem visibleScheduleLaw_mass (m : ℕ) (hm : m≤Fintype.card α+Fintype.card β) (S : Finset α) :
-    (visibleScheduleLaw (α:=α) (β:=β) m hm).mass S=
-      if S.card≤ m then ((Fintype.card β).choose (m-S.card):ℝ)/
-        ((Fintype.card α+Fintype.card β).choose m:ℝ) else 0 := by
-  unfold visibleScheduleLaw
-  rw [FiniteLaw.map_mass]
-  by_cases hS : S.card≤ m
-  · rw [ite_eq_left hS]
-    simp only [uniformSubsetLaw,Fintype.card_sum]
-    rw [←Finset.sum_filter]
-    simp only [Finset.sum_const,nsmul_eq_mul]
-    have he : (Finset.univ.filter (fun T : FixedSubset (α⊕β) m => T.val.toLeft=S)).card=
-        (Fintype.card β).choose (m-S.card) := by
-      rw [←Fintype.card_subtype]
-      exact (Fintype.card_congr (visibleFiberEquiv (β:=β) m S hS)).trans (card_fixedSubset _)
-    rw [he]
-    ring
-  · rw [ite_eq_right hS]
-    apply Finset.sum_eq_zero
-    intro T _
-    have hnot : T.val.toLeft≠S := by
-      intro he
-      apply hS
-      simpa only [he,T.property] using (Finset.card_toLeft_le (u:=T.val))
-    rw [ite_eq_right hnot]
-
-theorem sum_finset_card (f : ℕ→ℝ) :
-    (∑S : Finset α,f S.card)=∑v : Fin (Fintype.card α+1),
-      ((Fintype.card α).choose v.val:ℝ)*f v.val := by
-  let g : Finset α→Fin (Fintype.card α+1) := fun S =>
-    ⟨S.card,by have H:=Finset.card_le_univ S; omega⟩
-  rw [←Fintype.sum_fiberwise g (fun S => f S.card)]
-  apply Finset.sum_congr rfl
-  intro v _
-  have hc : Fintype.card {S : Finset α // g S=v}=(Fintype.card α).choose v.val := by
-    let e : {S : Finset α // g S=v} ≃ FixedSubset α v.val :=
-      { toFun := fun S => ⟨S.val,by have H:=congrArg Fin.val S.property; exact H⟩
-        invFun := fun S => ⟨S.val,by apply Fin.ext; exact S.property⟩
-        left_inv := fun _ => rfl
-        right_inv := fun _ => rfl }
-    exact (Fintype.card_congr e).trans (card_fixedSubset _)
-  calc
-    _=∑_S : {S : Finset α // g S=v},f v.val := by
-      apply Finset.sum_congr rfl
-      intro S _
-      congr 1
-      exact congrArg Fin.val S.property
-    _=_ := by simp [hc]
-
-end TraceReconstruction
-
-namespace TraceReconstruction
-open scoped Classical
-variable {α β : Type*} [Fintype α] [Fintype β]
-
-theorem visibleScheduleLaw_tv (m m' : ℕ)
-    (hm : m≤Fintype.card α+Fintype.card β) (hm' : m'≤Fintype.card α+Fintype.card β) :
-    (visibleScheduleLaw (α:=α) (β:=β) m hm).tv (visibleScheduleLaw m' hm')=
-      (hypergeomLaw (Fintype.card α) (Fintype.card β) m hm).tv
-        (hypergeomLaw (Fintype.card α) (Fintype.card β) m' hm') := by
-  unfold FiniteLaw.tv
-  simp_rw [visibleScheduleLaw_mass]
-  rw [sum_finset_card (α:=α) (fun v =>
-    |(if v≤ m then ((Fintype.card β).choose (m-v):ℝ)/
-      ((Fintype.card α+Fintype.card β).choose m:ℝ) else 0)-
-     (if v≤ m' then ((Fintype.card β).choose (m'-v):ℝ)/
-      ((Fintype.card α+Fintype.card β).choose m':ℝ) else 0)|)]
-  have he (v : ℕ) :
-      ((Fintype.card α).choose v:ℝ)*
-        |(if v≤ m then ((Fintype.card β).choose (m-v):ℝ)/
-          ((Fintype.card α+Fintype.card β).choose m:ℝ) else 0)-
-         (if v≤ m' then ((Fintype.card β).choose (m'-v):ℝ)/
-          ((Fintype.card α+Fintype.card β).choose m':ℝ) else 0)|=
-      |hypergeomMass (Fintype.card α) (Fintype.card β) m v-
-       hypergeomMass (Fintype.card α) (Fintype.card β) m' v| := by
-    rw [←abs_of_nonneg (Nat.cast_nonneg (α:=ℝ) ((Fintype.card α).choose v)),←abs_mul,mul_sub]
-    simp only [mul_ite,mul_zero,hypergeomMass,mul_div_assoc]
-  simp_rw [he]
-  simp only [hypergeomLaw]
-  rw [Fin.sum_univ_eq_sum_range (fun v => |hypergeomMass (Fintype.card α) (Fintype.card β) m v-
-       hypergeomMass (Fintype.card α) (Fintype.card β) m' v|)]
-  rw [Fin.sum_univ_eq_sum_range (n:=Fintype.card α+Fintype.card β+1)
-    (fun v => |hypergeomMass (Fintype.card α) (Fintype.card β) m v-
-       hypergeomMass (Fintype.card α) (Fintype.card β) m' v|)]
-  congr 1
-  symm
-  apply sum_range_extend_zero (by omega)
-  intro k hk
-  rw [hypergeomMass_high _ _ _ _ (by omega),hypergeomMass_high _ _ _ _ (by omega),sub_self,abs_zero]
-
-theorem visibleScheduleLaw_steps (m P : ℕ)
-    (hN : 16≤Fintype.card α+Fintype.card β) (hM : Fintype.card β≤ m)
-    (hV : m+P<Fintype.card α) (hlow : Fintype.card α+Fintype.card β≤4*m)
-    (hhigh : 4*(m+P)≤3*(Fintype.card α+Fintype.card β)) :
-    (visibleScheduleLaw (α:=α) (β:=β) m (by omega)).tv
-      (visibleScheduleLaw (β:=β) (m+P) (by omega))≤
-      (P:ℝ)*64/Real.sqrt ((Fintype.card β:ℝ)+1) := by
-  rw [visibleScheduleLaw_tv]
-  exact hypergeomLaw_steps _ _ m P hN hM hV hlow hhigh
-
-end TraceReconstruction
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-namespace FiniteLaw
-variable {α β γ : Type*} [Fintype α] [Fintype β] [Fintype γ]
-
-theorem ext_expect {P Q : FiniteLaw α} (h : ∀f : α→ℝ,P.expect f=Q.expect f) : P=Q := by
-  apply ext
-  funext a
-  have H:=h (fun b => if b=a then 1 else 0)
-  simpa only [expect,mul_ite,mul_one,mul_zero,Fintype.sum_ite_eq'] using H
-
-theorem expect_map (P : FiniteLaw α) (f : α→β) (G : β→ℝ) :
-    (P.map f).expect G=P.expect (G∘f) := expectation_map P f G
-
-theorem map_map (P : FiniteLaw α) (f : α→β) (g : β→γ) :
-    (P.map f).map g=P.map (g∘f) := by
-  apply ext_expect
-  intro G
-  rw [expect_map,expect_map,expect_map]
-  rfl
-
-theorem map_id (P : FiniteLaw α) : P.map id=P := by
-  apply ext_expect
-  intro G
-  rw [expect_map]
-  rfl
-
-theorem map_equiv_mass (P : FiniteLaw α) (e : α≃β) (b : β) :
-    (P.map e).mass b=P.mass (e.symm b) := by
-  rw [map_mass]
-  simp only [←e.eq_symm_apply,Fintype.sum_ite_eq']
-
-theorem tv_map_equiv (P Q : FiniteLaw α) (e : α≃β) :
-    (P.map e).tv (Q.map e)=P.tv Q := by
-  unfold tv
-  simp_rw [map_equiv_mass]
-  rw [e.symm.sum_comp (fun a => |P.mass a-Q.mass a|)]
-
-end FiniteLaw
-
-variable {α β : Type*} [Fintype α] [Fintype β]
-
-def fixedSubsetEquiv (e : α≃β) (m : ℕ) : FixedSubset α m ≃ FixedSubset β m where
-  toFun S := ⟨S.val.map e.toEmbedding,by simpa only [Finset.card_map] using S.property⟩
-  invFun T := ⟨T.val.map e.symm.toEmbedding,by simpa only [Finset.card_map] using T.property⟩
-  left_inv S := by apply Subtype.ext; exact e.finsetCongr.left_inv S.val
-  right_inv T := by apply Subtype.ext; exact e.finsetCongr.right_inv T.val
-
-theorem uniformSubsetLaw_map (e : α≃β) (m : ℕ) (hm : m≤Fintype.card α) :
-    (uniformSubsetLaw m hm).map (fixedSubsetEquiv e m)=
-      uniformSubsetLaw m (by simpa only [←Fintype.card_congr e] using hm) := by
-  apply FiniteLaw.ext
-  funext T
-  rw [FiniteLaw.map_equiv_mass]
-  simp only [uniformSubsetLaw,Fintype.card_congr e]
-
-end TraceReconstruction
-
-namespace TraceReconstruction
-open scoped Classical
-variable {α : Type*} [Fintype α]
-
-abbrev HiddenIndex (H : Finset α) := {a : α // a∈H}
-abbrev VisibleIndex (H : Finset α) := {a : α // a∉H}
-
-def splitIndexEquiv (H : Finset α) : VisibleIndex H⊕HiddenIndex H ≃ α :=
-  (Equiv.sumComm _ _).trans (Equiv.sumCompl (fun a => a∈H))
-
-theorem card_splitIndex (H : Finset α) :
-    Fintype.card (VisibleIndex H)+Fintype.card (HiddenIndex H)=Fintype.card α := by
-  simpa only [Fintype.card_sum] using Fintype.card_congr (splitIndexEquiv H)
-
-omit [Fintype α] in
-theorem card_hiddenIndex (H : Finset α) : Fintype.card (HiddenIndex H)=H.card := by
-  exact Fintype.card_coe H
-
-def eraseScheduleLaw (H : Finset α) (m : ℕ) (hm : m≤Fintype.card α) : FiniteLaw (Finset α) :=
-  (uniformSubsetLaw m hm).map (fun S => S.val\H)
-
-omit [Fintype α] in
-theorem erase_splitIndex (H : Finset α) (S : Finset (VisibleIndex H⊕HiddenIndex H)) :
-    (S.map (splitIndexEquiv H).toEmbedding)\H=
-      S.toLeft.map ⟨Subtype.val,Subtype.val_injective⟩ := by
-  ext a
-  constructor
-  · intro ha
-    obtain ⟨ha,hn⟩:=Finset.mem_sdiff.mp ha
-    obtain ⟨b,hb,rfl⟩:=Finset.mem_map.mp ha
-    cases b with
-    | inl b =>
-      apply Finset.mem_map.mpr
-      exact ⟨b,by simpa using hb,rfl⟩
-    | inr b => exact False.elim (hn b.property)
-  · intro ha
-    obtain ⟨b,hb,rfl⟩:=Finset.mem_map.mp ha
-    apply Finset.mem_sdiff.mpr
-    constructor
-    · exact Finset.mem_map.mpr ⟨Sum.inl b,by simpa using hb,rfl⟩
-    · exact b.property
-
-end TraceReconstruction
-
-namespace TraceReconstruction
-open scoped Classical
-variable {α : Type*} [Fintype α]
-
-theorem eraseScheduleLaw_eq (H : Finset α) (m : ℕ) (hm : m≤Fintype.card α) :
-    eraseScheduleLaw H m hm=
-      (visibleScheduleLaw (α:=VisibleIndex H) (β:=HiddenIndex H) m
-        (by simpa only [card_splitIndex] using hm)).map
-        (fun S => S.map ⟨Subtype.val,Subtype.val_injective⟩) := by
-  have h:=uniformSubsetLaw_map (splitIndexEquiv H) m
-    (by simpa only [Fintype.card_sum,card_splitIndex] using hm)
-  unfold eraseScheduleLaw visibleScheduleLaw
-  rw [←h,FiniteLaw.map_map,FiniteLaw.map_map]
-  congr 1
-  funext S
-  exact erase_splitIndex H S.val
-
-theorem eraseScheduleLaw_steps (H : Finset α) (m P : ℕ)
-    (hN : 16≤Fintype.card α) (hM : H.card≤ m)
-    (hV : m+P+H.card<Fintype.card α) (hlow : Fintype.card α≤4*m)
-    (hhigh : 4*(m+P)≤3*Fintype.card α) :
-    (eraseScheduleLaw H m (by omega)).tv (eraseScheduleLaw H (m+P) (by omega))≤
-    (P:ℝ)*64/Real.sqrt ((H.card:ℝ)+1) := by
-  rw [eraseScheduleLaw_eq,eraseScheduleLaw_eq]
-  refine (FiniteLaw.tv_map_le _ _ _).trans ?_
-  have hc:=card_splitIndex H
-  have hd:=card_hiddenIndex H
-  have h:=visibleScheduleLaw_steps (α:=VisibleIndex H) (β:=HiddenIndex H) m P
-    (by omega) (by omega) (by omega) (by omega) (by omega)
-  simpa only [hd] using h
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-namespace FiniteLaw
-variable {ι : Type*} [Fintype ι] {β : ι → Type*} [∀i,Fintype (β i)]
-
-def sigma (P : FiniteLaw ι) (K : ∀i, FiniteLaw (β i)) : FiniteLaw (Σi,β i) where
-  mass a:=P.mass a.1*(K a.1).mass a.2
-  nonneg a:=mul_nonneg (P.nonneg _) ((K _).nonneg _)
-  total:=by
-    rw [Fintype.sum_sigma]
-    simp_rw [←Finset.mul_sum,(K _).total,mul_one]
-    exact P.total
-
-theorem expect_sigma (P : FiniteLaw ι) (K : ∀i,FiniteLaw (β i)) (F : (Σi,β i)→ℝ) :
-    (P.sigma K).expect F=P.expect (fun i => (K i).expect (fun b => F ⟨i,b⟩)) := by
-  simp only [expect,sigma,Fintype.sum_sigma,Finset.mul_sum,mul_assoc]
-
-theorem overlap_sigma_lower (P Q : FiniteLaw ι) (K L : ∀i,FiniteLaw (β i)) :
-    (∑i,min (P.mass i) (Q.mass i)*overlap (K i) (L i))≤overlap (P.sigma K) (Q.sigma L) := by
-  simp only [overlap,sigma,Fintype.sum_sigma,Finset.mul_sum]
-  apply Finset.sum_le_sum
-  intro i _
-  apply Finset.sum_le_sum
-  intro b _
-  apply le_min
-  · exact mul_le_mul (min_le_left _ _) (min_le_left _ _) (le_min ((K i).nonneg b) ((L i).nonneg b)) (P.nonneg i)
-  · exact mul_le_mul (min_le_right _ _) (min_le_right _ _) (le_min ((K i).nonneg b) ((L i).nonneg b)) (Q.nonneg i)
-end FiniteLaw
-
-theorem sum_binomialMass_cutoff {n N : ℕ} (hn : n≤N) (p : ℝ) :
-    ∑l : Fin (N+1),binomialMass n p l.val=1 := by
-  rw [Fin.sum_univ_eq_sum_range (fun k => binomialMass n p k) (N+1)]
-  rw [←sum_binomialMass n p]
-  symm
-  apply Finset.sum_subset (Finset.range_mono (by omega))
-  intro k _ hk
-  have hn' : n<k := by simp only [Finset.mem_range] at hk; omega
-  simp [binomialMass,Nat.choose_eq_zero_of_lt hn']
-
-def boundedCountLaw {n N : ℕ} (hn : n≤N) (p : ℝ) (hp : p∈Set.Icc 0 1) : FiniteLaw (Fin (N+1)) where
-  mass l:=binomialMass n p l.val
-  nonneg l:=binomialMass_nonneg n l.val hp
-  total:=sum_binomialMass_cutoff hn p
-
-abbrev BoundedMarks (N : ℕ) := Σl : Fin (N+1),Fin l.val→PrimitiveMark
-
-def boundedConditionalPrimitiveLaw {n N : ℕ} {q : ℝ} (hq : q∈Set.Ioo 0 1)
-    (θ : Fin n→Bool) (l : Fin (N+1)) : FiniteLaw (Fin l.val→PrimitiveMark) :=
-  if h : l.val≤n then conditionalPrimitiveLaw h q hq θ else (primitiveMarkLaw q hq false).product l.val
-
-theorem primitiveRetention_mem {q : ℝ} (hq : q∈Set.Ioo 0 1) :
-    1-primitiveErasure q∈Set.Icc 0 1 := by
-  constructor
-  · exact (primitive_retain_pos hq).le
-  · unfold primitiveErasure; nlinarith [sq_nonneg q]
-
-def countedPrimitiveLaw {n N : ℕ} (hn : n≤N) {q : ℝ} (hq : q∈Set.Ioo 0 1)
-    (θ : Fin n→Bool) : FiniteLaw (BoundedMarks N) :=
-  (boundedCountLaw hn (1-primitiveErasure q) (primitiveRetention_mem hq)).sigma
-    (boundedConditionalPrimitiveLaw hq θ)
-
-def decodeBoundedMarks {N : ℕ} (a : BoundedMarks N) : List Bool := joinPrimitiveMarks (List.ofFn a.2)
-
-theorem countedPrimitiveLaw_expect {n N : ℕ} (hn : n≤N) {q : ℝ} (hq : q∈Set.Ioo 0 1)
-    (θ : Fin n→Bool) (F : List PrimitiveMark→ℝ) :
-    (countedPrimitiveLaw hn hq θ).expect (fun a => F (List.ofFn a.2))=
-      silentMarkExpectation q (List.ofFn θ) F := by
-  rw [countedPrimitiveLaw,FiniteLaw.expect_sigma,silentMarkExpectation_count_mixture q hq θ F]
-  change (∑l : Fin (N+1),binomialMass n (1-primitiveErasure q) l.val*
-    (boundedConditionalPrimitiveLaw hq θ l).expect (fun x => F (List.ofFn x)))=_
-  let f : ℕ→ℝ := fun l => if hl:l≤n then binomialMass n (1-primitiveErasure q) l*
-    (conditionalPrimitiveLaw hl q hq θ).expect (fun x => F (List.ofFn x)) else 0
-  have hleft : (∑l : Fin (N+1),binomialMass n (1-primitiveErasure q) l.val*
-    (boundedConditionalPrimitiveLaw hq θ l).expect (fun x => F (List.ofFn x)))=
-      ∑l : Fin (N+1),f l.val := by
-    apply Finset.sum_congr rfl
-    intro l _
-    by_cases hl:l.val≤n
-    · simp only [boundedConditionalPrimitiveLaw,hl,dite_eq_left,f]
-    · simp [boundedConditionalPrimitiveLaw,hl,f,binomialMass,Nat.choose_eq_zero_of_lt (Nat.lt_of_not_ge hl)]
-  rw [hleft,Fin.sum_univ_eq_sum_range f (N+1)]
-  have he : (∑l∈Finset.range (n+1),f l)=∑l∈Finset.range (N+1),f l := by
-    apply Finset.sum_subset (Finset.range_mono (by omega))
-    intro l _ hl
-    simp only [Finset.mem_range] at hl
-    simp [f,show ¬l≤n by omega]
-  rw [←he,←Fin.sum_univ_eq_sum_range f (n+1)]
-  apply Finset.sum_congr rfl
-  intro l _
-  simp only [f,dite_eq_left (Nat.le_of_lt_succ l.isLt)]
-
-theorem countedPrimitiveLaw_simulates {n N : ℕ} (hn : n≤N) {q : ℝ} (hq : q∈Set.Ioo 0 1)
-    (θ : Fin n→Bool) :
-    SimulatesTrace q (countedPrimitiveLaw hn hq θ) decodeBoundedMarks
-      (((List.ofFn θ).map primitiveBlock).flatten) := by
-  intro F
-  rw [primitive_run_simulation hq]
-  exact countedPrimitiveLaw_expect hn hq θ (fun a => F (joinPrimitiveMarks a))
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-variable {α : Type*} [Fintype α]
-
-theorem subsetLaw_expect_card (p : ℝ) (hp : p∈Set.Icc 0 1) (f : ℕ→ℝ) :
-    (subsetLaw (α:=α) p hp).expect (fun S => f S.card)=
-      ∑k∈Finset.range (Fintype.card α+1),binomialMass (Fintype.card α) p k*f k := by
-  change (∑S : Finset α,p^S.card*(1-p)^(Fintype.card α-S.card)*f S.card)=_
-  rw [sum_finset_card (α:=α) (fun k => p^k*(1-p)^(Fintype.card α-k)*f k),
-    Fin.sum_univ_eq_sum_range (fun k => ((Fintype.card α).choose k:ℝ)*(p^k*(1-p)^(Fintype.card α-k)*f k))]
-  apply Finset.sum_congr rfl
-  intro k _
-  unfold binomialMass
-  ring
-
-theorem subsetLaw_variance (p : ℝ) (hp : p∈Set.Icc 0 1) :
-    (subsetLaw (α:=α) p hp).expect (fun S => ((S.card:ℝ)-Fintype.card α*p)^2)=
-      (Fintype.card α:ℝ)*p*(1-p) := by
-  rw [subsetLaw_expect_card p hp (fun k => ((k:ℝ)-Fintype.card α*p)^2)]
-  exact binomialMass_variance _ p
-
-def hiddenScheduleLaw (p : ℝ) (hp : p∈Set.Icc 0 1) (m : ℕ) (hm : m≤Fintype.card α) :
-    FiniteLaw (Finset α×Finset α) :=
-  (subsetLaw p hp).bind (fun H => (eraseScheduleLaw H m hm).map (fun S => (H,S)))
-
-theorem hiddenScheduleLaw_tv (p : ℝ) (hp : p∈Set.Icc 0 1) (m m' : ℕ)
-    (hm : m≤Fintype.card α) (hm' : m'≤Fintype.card α) :
-    (hiddenScheduleLaw p hp m hm).tv (hiddenScheduleLaw p hp m' hm') ≤
-      (subsetLaw (α:=α) p hp).expect (fun H =>
-        (eraseScheduleLaw H m hm).tv (eraseScheduleLaw H m' hm')) := by
-  unfold hiddenScheduleLaw
-  refine (FiniteLaw.tv_bind_kernel_le _ _ _).trans ?_
-  apply FiniteLaw.expect_mono
-  intro H
-  exact FiniteLaw.tv_map_le _ _ _
-
-theorem good_hiddenSchedule_bound {G P : ℕ} {p : ℝ}
-    (hG : 8≤G) (hP : 4*P≤G) (hp : p∈Set.Ioc 0 (1/8))
-    (H : Finset (Fin (2*G)))
-    (hlo : p*(2*G:ℝ)/2≤H.card) (hhi : (H.card:ℝ)≤2*p*(2*G:ℝ)) :
-    (eraseScheduleLaw H G (by simp only [Fintype.card_fin]; omega)).tv
-      (eraseScheduleLaw H (G+P) (by simp only [Fintype.card_fin]; omega))≤
-      (P:ℝ)*64/Real.sqrt (p*(2*G:ℝ)/2) := by
-  have hgr : (0:ℝ)<G := by exact_mod_cast (show 0<G by omega)
-  have hhr : (H.card:ℝ)≤(G:ℝ)/2 := by nlinarith [mul_le_mul_of_nonneg_right hp.2 hgr.le]
-  have hh : 2*H.card≤G := by exact_mod_cast (show 2*(H.card:ℝ)≤G by linarith)
-  have h:=eraseScheduleLaw_steps H G P (by simp only [Fintype.card_fin]; omega)
-    (by omega) (by simp only [Fintype.card_fin]; omega)
-    (by simp only [Fintype.card_fin]; omega) (by simp only [Fintype.card_fin]; omega)
-  apply h.trans
-  apply div_le_div_of_nonneg_left (by positivity)
-    (Real.sqrt_pos.2 (by positivity [hp.1]))
-    (Real.sqrt_le_sqrt (by linarith))
-
-theorem bad_hiddenSchedule_variance {μ x : ℝ} (hμ : 0<μ)
-    (hbad : ¬(μ/2≤x ∧ x≤2*μ)) :
-    1≤4*(x-μ)^2/μ^2 := by
-  rw [le_div_iff₀ (sq_pos_of_pos hμ)]
-  push Not at hbad
-  by_cases h : μ/2≤x
-  · have hx:=hbad h
-    nlinarith [sq_nonneg (x-2*μ)]
-  · have hx : x<μ/2 := lt_of_not_ge h
-    nlinarith [sq_nonneg (x-μ/2)]
-
-theorem hiddenScheduleLaw_steps {G P : ℕ} {p : ℝ}
-    (hG : 8≤G) (hP : 4*P≤G) (hp : p∈Set.Ioc 0 (1/8)) :
-    (hiddenScheduleLaw (α:=Fin (2*G)) p ⟨hp.1.le,by linarith [hp.2]⟩ G
-      (by simp only [Fintype.card_fin]; omega)).tv
-      (hiddenScheduleLaw p ⟨hp.1.le,by linarith [hp.2]⟩ (G+P)
-        (by simp only [Fintype.card_fin]; omega))≤
-      (P:ℝ)*64/Real.sqrt (p*(2*G:ℝ)/2)+4/(p*(2*G:ℝ)) := by
-  let μ : ℝ := p*(2*G:ℝ)
-  have hgr : (0:ℝ)<G := by exact_mod_cast (show 0<G by omega)
-  have hμ : 0<μ := by dsimp [μ]; positivity [hp.1]
-  have hμ0 : μ≠0 := ne_of_gt hμ
-  let E : ℝ := (P:ℝ)*64/Real.sqrt (μ/2)
-  have hE : 0≤E := by dsimp [E]; positivity
-  apply (hiddenScheduleLaw_tv p ⟨hp.1.le,by linarith [hp.2]⟩ G (G+P) _ _).trans
-  have he := (subsetLaw (α:=Fin (2*G)) p ⟨hp.1.le,by linarith [hp.2]⟩).expect_mono
-    (g:=fun H => E+4*((H.card:ℝ)-μ)^2/μ^2) (fun H => by
-      by_cases hg : μ/2≤(H.card:ℝ) ∧ (H.card:ℝ)≤2*μ
-      · exact (good_hiddenSchedule_bound hG hP hp H hg.1 (by simpa only [μ,mul_assoc] using hg.2)).trans
-          (le_add_of_nonneg_right (by positivity))
-      · exact (FiniteLaw.tv_le_one _ _).trans
-          ((bad_hiddenSchedule_variance hμ hg).trans (by linarith)))
-  apply he.trans
-  have hv:=subsetLaw_variance (α:=Fin (2*G)) p ⟨hp.1.le,by linarith [hp.2]⟩
-  simp only [Fintype.card_fin,Nat.cast_mul,Nat.cast_ofNat] at hv
-  change (∑H,(subsetLaw p _).mass H*(E+4*((H.card:ℝ)-μ)^2/μ^2))≤E+4/μ
-  simp_rw [mul_add, Finset.sum_add_distrib]
-  rw [← Finset.sum_mul,FiniteLaw.total,one_mul]
-  have heq : (∑H : Finset (Fin (2*G)),(subsetLaw p ⟨hp.1.le,by linarith [hp.2]⟩).mass H*(4*((H.card:ℝ)-μ)^2/μ^2))=
-      4/μ^2*(2*(G:ℝ)*p*(1-p)) := by
-    rw [←hv,FiniteLaw.expect,Finset.mul_sum]
-    apply Finset.sum_congr rfl
-    intro H _
-    dsimp [μ]
-    ring
-  rw [heq]
-  have heq' : 4/μ^2*(2*(G:ℝ)*p*(1-p))=4/μ*(1-p) := by
-    have h : 2*(G:ℝ)*p=μ := by dsimp [μ]; ring
-    rw [h]
-    field_simp
-  rw [heq']
-  have h:=mul_le_mul_of_nonneg_left (show 1-p≤1 by linarith [hp.1]) (show 0≤4/μ by positivity)
-  linarith
-
-end TraceReconstruction
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-namespace FiniteLaw
-variable {α β γ : Type*} [Fintype α] [Fintype β] [Fintype γ]
-
-theorem expect_comm (P : FiniteLaw α) (Q : FiniteLaw β) (F : α→β→ℝ) :
-    P.expect (fun a => Q.expect (F a))=Q.expect (fun b => P.expect (fun a => F a b)) := by
-  simp only [expect,Finset.mul_sum]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro b _
-  apply Finset.sum_congr rfl
-  intro a _
-  ring
-
-theorem bind_map (P : FiniteLaw α) (f : α→β) (K : β→FiniteLaw γ) :
-    (P.map f).bind K=P.bind (K∘f) := by
-  apply ext_expect
-  intro F
-  rw [expect_bind,expect_map,expect_bind]
-  rfl
-
-theorem map_bind (P : FiniteLaw α) (K : α→FiniteLaw β) (f : β→γ) :
-    (P.bind K).map f=P.bind (fun a => (K a).map f) := by
-  apply ext_expect
-  intro F
-  simp only [expect_bind,expect_map]
-
-theorem bind_comm (P : FiniteLaw α) (Q : FiniteLaw β) (K : α→β→FiniteLaw γ) :
-    P.bind (fun a => Q.bind (K a))=Q.bind (fun b => P.bind (fun a => K a b)) := by
-  apply ext_expect
-  intro F
-  simp only [expect_bind]
-  exact expect_comm P Q (fun a b => (K a b).expect F)
-
-end FiniteLaw
-variable {α : Type*} [Fintype α]
-
-theorem subsetWeight_mask {n : ℕ} (p : ℝ) (a : Mask n) :
-    subsetWeight p (maskSubset a)=maskWeight (1-p) a := by
-  rw [maskWeight_card]
-  simp only [subsetWeight,Fintype.card_fin,sub_sub_cancel]
-
-theorem coordinateProduct_common_flags {n : ℕ} (p : ℝ) (hp : p∈Set.Icc 0 1)
-    (U : FiniteLaw α) (R P : Fin n→FiniteLaw α)
-    (hP : ∀ i x,(P i).mass x=p*U.mass x+(1-p)*(R i).mass x) :
-    (subsetLaw p hp).bind (fun H : Finset (Fin n) =>
-      FiniteLaw.coordinateProduct (fun i => if i∈H then U else R i))=
-      FiniteLaw.coordinateProduct P := by
-  apply FiniteLaw.ext
-  funext x
-  change (∑H : Finset (Fin n),subsetWeight p H*(∏i,(if i∈H then U else R i).mass (x i)))=
-    ∏i,(P i).mass (x i)
-  rw [←(maskSubsetEquiv n).sum_comp]
-  change (∑a : Mask n,subsetWeight p (maskSubset a)*(∏i,(if i∈maskSubset a then U else R i).mass (x i)))=_
-  simp_rw [subsetWeight_mask,maskWeight,←Finset.prod_mul_distrib]
-  have he (a : Mask n) (i : Fin n) :
-      (if a i then 1-(1-p) else 1-p)*(if i∈maskSubset a then U else R i).mass (x i)=
-      if a i then p*U.mass (x i) else (1-p)*(R i).mass (x i) := by
-    cases ha : a i <;> simp [maskSubset,ha]
-  simp_rw [he]
-  rw [←Fintype.prod_sum (fun i (b : Bool) => if b then p*U.mass (x i) else (1-p)*(R i).mass (x i))]
-  apply Finset.prod_congr rfl
-  intro i _
-  simp only [Fintype.sum_bool,ite_true,Bool.false_eq_true,ite_false]
-  rw [hP]
-
-end TraceReconstruction
-
-namespace TraceReconstruction
-open scoped Classical
-variable {α : Type*} [Fintype α]
-
-def schedulePayload {n : ℕ} (U P Q : FiniteLaw α) (z : Finset (Fin n)×Finset (Fin n)) :
-    FiniteLaw (Fin n→α) :=
-  FiniteLaw.coordinateProduct (fun i => if i∈z.1 then U else if i∈z.2 then Q else P)
-
-def uniformSchedulePayload {n : ℕ} (P Q : FiniteLaw α) (m : ℕ) (hm : m≤n) :
-    FiniteLaw (Fin n→α) :=
-  (uniformSubsetLaw (α:=Fin n) m (by simpa using hm)).bind
-    (fun S => FiniteLaw.coordinateProduct (fun i => if i∈S.val then Q else P))
-
-theorem hiddenSchedule_payload_exact {n : ℕ} (p : ℝ) (hp : p∈Set.Icc 0 1)
-    (U P' Q' P Q : FiniteLaw α)
-    (hP : ∀ x,P.mass x=p*U.mass x+(1-p)*P'.mass x)
-    (hQ : ∀ x,Q.mass x=p*U.mass x+(1-p)*Q'.mass x)
-    (m : ℕ) (hm : m≤n) :
-    (hiddenScheduleLaw (α:=Fin n) p hp m (by simpa using hm)).bind (schedulePayload U P' Q')=
-      uniformSchedulePayload P Q m hm := by
-  unfold hiddenScheduleLaw eraseScheduleLaw
-  rw [FiniteLaw.bind_assoc]
-  simp_rw [FiniteLaw.bind_map]
-  rw [FiniteLaw.bind_comm]
-  unfold uniformSchedulePayload
-  congr 1
-  funext S
-  trans (subsetLaw p hp).bind (fun H : Finset (Fin n) =>
-    FiniteLaw.coordinateProduct (fun i => if i∈H then U else if i∈S.val then Q' else P'))
-  · congr 1
-    funext H
-    dsimp only [Function.comp_def,schedulePayload]
-    congr 1
-    funext i
-    by_cases h : i∈H <;> by_cases h' : i∈S.val <;> simp [h,h']
-  apply coordinateProduct_common_flags
-  intro i x
-  by_cases h : i∈S.val
-  · simpa [h] using hQ x
-  · simpa [h] using hP x
-
-theorem uniformSchedulePayload_steps {G R : ℕ} {p : ℝ}
-    (hG : 8≤G) (hR : 4*R≤G) (hp : p∈Set.Ioc 0 (1/8))
-    (P Q : FiniteLaw α) (hov : p≤FiniteLaw.overlap P Q) :
-    (uniformSchedulePayload (n:=2*G) P Q G (by omega)).tv
-      (uniformSchedulePayload P Q (G+R) (by omega))≤
-      (R:ℝ)*64/Real.sqrt (p*(2*G:ℝ)/2)+4/(p*(2*G:ℝ)) := by
-  have hp1 : p∈Set.Ioo 0 1 := ⟨hp.1,by linarith [hp.2]⟩
-  obtain ⟨U,P',Q',hP,hQ⟩:=FiniteLaw.common_part_decomposition P Q hp1 hov
-  rw [←hiddenSchedule_payload_exact p ⟨hp.1.le,hp1.2.le⟩ U P' Q' P Q hP hQ,
-    ←hiddenSchedule_payload_exact p ⟨hp.1.le,hp1.2.le⟩ U P' Q' P Q hP hQ]
-  exact (FiniteLaw.tv_bind_le _ _ _).trans (hiddenScheduleLaw_steps hG hR hp)
-
-end TraceReconstruction
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-def coordinateWord {n : ℕ} (w : Fin n→List Bool) : List Bool := (List.ofFn w).flatten
-
-theorem coordinateWord_succ {n : ℕ} (w : Fin (n+1)→List Bool) :
-    coordinateWord w=w 0++coordinateWord (fun i => w i.succ) := by
-  simp only [coordinateWord,List.ofFn_succ,List.flatten_cons]
-
-theorem coordinateWord_cons {n : ℕ} (w : List Bool) (v : Fin n→List Bool) :
-    coordinateWord (Fin.cons w v)=w++coordinateWord v := by
-  rw [coordinateWord_succ]
-  simp only [Fin.cons_zero,Fin.cons_succ]
-
-theorem coordinateProduct_simulates {α : Type*} [Fintype α]
-    {n : ℕ} {q : ℝ} (P : Fin n→FiniteLaw α) (f : α→List Bool) (w : Fin n→List Bool)
-    (h : ∀i,SimulatesTrace q (P i) f (w i)) :
-    SimulatesTrace q (FiniteLaw.coordinateProduct P) (fun a => coordinateWord (fun i => f (a i)))
-      (coordinateWord w) := by
-  induction n with
-  | zero =>
-    intro F
-    change (FiniteLaw.coordinateProduct P).expect (fun _ => F [])=F []
-    exact FiniteLaw.expect_const _ _
-  | succ n ih =>
-    intro F
-    rw [FiniteLaw.expect_coordinateProduct_succ,coordinateWord_succ,listDeletionExpectation_append]
-    have hj (a : α) (x : Fin n→α) :
-        coordinateWord (fun i => f ((Fin.cons a x : Fin (n+1)→α) i))=f a++coordinateWord (fun i => f (x i)) := by
-      rw [coordinateWord_succ]
-      simp only [Fin.cons_zero,Fin.cons_succ]
-    simp_rw [hj]
-    have ht:=ih (fun i => P i.succ) (fun i => w i.succ) (fun i => h i.succ)
-    calc
-      _=(P 0).expect (fun a => listDeletionExpectation q (coordinateWord (fun i => w i.succ))
-          (fun t => F (f a++t))) := by
-        congr 1
-        funext a
-        exact ht (fun t => F (f a++t))
-      _=_ := h 0 (fun t => listDeletionExpectation q (coordinateWord (fun i => w i.succ)) (fun u => F (t++u)))
-
-theorem sum_subset_choices {n : ℕ} (S : Finset (Fin n)) (k : ℕ) :
-    (∑i : Fin n,if i∈S then k+1 else k)=n*k+S.card := by
-  have he (i : Fin n) : (if i∈S then k+1 else k)=k+(if i∈S then 1 else 0) := by
-    split_ifs <;> omega
-  simp_rw [he]
-  simp only [Finset.sum_add_distrib,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,
-    Finset.sum_boole,Finset.filter_mem_eq_inter,Finset.univ_inter,Nat.cast_id]
-
-theorem coordinateWord_pureAtomRun {n : ℕ} (w : List Bool) (k : Fin n→ℕ) :
-    coordinateWord (fun i => pureAtomRun w (k i))=pureAtomRun w (∑i,k i) := by
-  change (List.ofFn (pureAtomRun w ∘ k)).flatten=_
-  rw [←List.map_ofFn,pureAtomRun_partition,List.sum_ofFn]
-
-theorem uniform_schedule_word {n m : ℕ} (S : FixedSubset (Fin n) m) (w : List Bool) (k : ℕ) :
-    coordinateWord (fun i => if i∈S.val then pureAtomRun w (k+1) else pureAtomRun w k)=
-      pureAtomRun w (n*k+m) := by
-  have he (i : Fin n) : (if i∈S.val then pureAtomRun w (k+1) else pureAtomRun w k)=
-      pureAtomRun w (if i∈S.val then k+1 else k) := by split_ifs <;> rfl
-  simp_rw [he]
-  rw [coordinateWord_pureAtomRun,sum_subset_choices,S.property]
-
-end TraceReconstruction
-noncomputable section
-namespace TraceReconstruction
-
-theorem lifting_count_scale {F k P : ℕ} {C p : ℝ}
-    (hk : 1≤k) (hP : 1≤P) (hp : p∈Set.Ioc 0 (1/8))
-    (hC : 1073741824≤p*C) (hF : C*k*P≤F) :
-    (1048576:ℝ)*(P:ℝ)^2≤p*((F*P/(2*k+1):ℕ):ℝ) := by
-  let G := F*P/(2*k+1)
-  have hkr : (1:ℝ)≤k := by exact_mod_cast hk
-  have hPr : (1:ℝ)≤P := by exact_mod_cast hP
-  have hGr : (0:ℝ)≤G := Nat.cast_nonneg G
-  have hnat:=Nat.lt_mul_div_succ (F*P) (show 0<2*k+1 by omega)
-  have hupper : (F:ℝ)*P<(2*(k:ℝ)+1)*((G:ℝ)+1) := by exact_mod_cast hnat
-  have hF' := mul_le_mul_of_nonneg_right hF (show (0:ℝ)≤P by positivity)
-  have hupper' := mul_lt_mul_of_pos_left hupper hp.1
-  have hlower:=mul_le_mul_of_nonneg_left hF' hp.1.le
-  have hC' := mul_le_mul_of_nonneg_right hC (show (0:ℝ)≤(k:ℝ)*(P:ℝ)^2 by positivity)
-  have hden : (2*(k:ℝ)+1)*(p*((G:ℝ)+1))≤3*(k:ℝ)*(p*((G:ℝ)+1)) := by
-    apply mul_le_mul_of_nonneg_right (by linarith)
-    positivity [hp.1.le]
-  have hmain : (1073741824:ℝ)*(P:ℝ)^2<3*p*((G:ℝ)+1) := by
-    have h : (k:ℝ)*(1073741824*(P:ℝ)^2)<(k:ℝ)*(3*p*((G:ℝ)+1)) := by
-      nlinarith only [hlower,hupper',hC',hden]
-    exact (mul_lt_mul_iff_of_pos_left (by linarith : (0:ℝ)<k)).mp h
-  change (1048576:ℝ)*(P:ℝ)^2≤p*(G:ℝ)
-  nlinarith [sq_nonneg ((P:ℝ)-1),hp.2]
-
-theorem lifting_scale_valid {G P : ℕ} {p : ℝ}
-    (hP : 1≤P) (hp : p∈Set.Ioc 0 (1/8))
-    (hscale : (1048576:ℝ)*(P:ℝ)^2≤p*(G:ℝ)) : 8≤G ∧ 4*P≤G := by
-  have hPr : (1:ℝ)≤P := by exact_mod_cast hP
-  have hmul:=mul_le_mul_of_nonneg_right hp.2 (Nat.cast_nonneg (α:=ℝ) G)
-  have hG : (8:ℝ)≤G := by nlinarith [sq_nonneg ((P:ℝ)-1)]
-  have h4 : 4*(P:ℝ)≤G := by nlinarith [sq_nonneg ((P:ℝ)-1)]
-  constructor
-  · exact_mod_cast hG
-  · exact_mod_cast h4
-
-theorem lifting_numeric_bound {G P : ℕ} {p : ℝ}
-    (hP : 1≤P) (_hp : p∈Set.Ioc 0 (1/8))
-    (hscale : (1048576:ℝ)*(P:ℝ)^2≤p*(G:ℝ)) :
-    (P:ℝ)*64/Real.sqrt (p*(2*G:ℝ)/2)+4/(p*(2*G:ℝ))≤1/3 := by
-  have hPr : (1:ℝ)≤P := by exact_mod_cast hP
-  have hs : 0<p*(G:ℝ) := by nlinarith [sq_nonneg ((P:ℝ)-1)]
-  have hre : p*(2*G:ℝ)/2=p*G := by ring
-  have hden : (0:ℝ)<p*(2*G:ℝ) := by nlinarith
-  have hsq : 1024*(P:ℝ)≤Real.sqrt (p*(G:ℝ)) := by
-    apply (Real.le_sqrt (by positivity) (le_of_lt hs)).2
-    nlinarith only [hscale]
-  have hspos:=Real.sqrt_pos.2 hs
-  have hfirst : (P:ℝ)*64/Real.sqrt (p*(2*G:ℝ)/2)≤1/16 := by
-    rw [hre,div_le_iff₀ hspos]
-    nlinarith only [hsq]
-  have hsecond : 4/(p*(2*G:ℝ))≤1/16 := by
-    rw [div_le_iff₀ hden]
-    nlinarith [sq_nonneg ((P:ℝ)-1)]
-  linarith
-
-end TraceReconstruction
-universe u
-namespace TraceReconstruction
-open scoped Classical
-variable {α : Type u} [Fintype α]
-
-theorem maskLaw_simulates {q : ℝ} (hq : q∈Set.Icc 0 1) (w : List Bool) :
-    SimulatesTrace q (maskLaw q hq) (traceOfMask w.get) w := by
-  intro F
-  have h:=listDeletionExpectation_ofFn q w.get F
-  rw [List.ofFn_get] at h
-  exact h.symm
-
-theorem uniformSchedulePayload_simulates {n m k : ℕ} {q : ℝ} (hm : m≤n)
-    (P Q : FiniteLaw α) (f : α→List Bool) (w : List Bool)
-    (hP : SimulatesTrace q P f (pureAtomRun w k))
-    (hQ : SimulatesTrace q Q f (pureAtomRun w (k+1))) :
-    SimulatesTrace q (uniformSchedulePayload P Q m hm)
-      (fun a => coordinateWord (fun i => f (a i))) (pureAtomRun w (n*k+m)) := by
-  apply SimulatesTrace.random_partition
-  intro S
-  rw [←uniform_schedule_word S w k]
-  apply coordinateProduct_simulates
-  intro i
-  by_cases h : i∈S.val
-  · simpa only [h,ite_true] using hQ
-  · simpa only [h,ite_false] using hP
-
-/-- Explicit source common-part lifting, using finite auxiliary observations
-that decode exactly to ordinary complete traces of the adjacent pure powers. -/
-theorem adjacent_power_lift {F k R : ℕ} {C p q : ℝ}
-    (hk : 1≤k) (hR : 1≤R) (hp : p∈Set.Ioc 0 (1/8)) (hq : q∈Set.Icc 0 1)
-    (hC : 1073741824≤p*C) (hF : C*k*R≤F)
-    (P Q : FiniteLaw α) (f : α→List Bool) (w : List Bool)
-    (hP : SimulatesTrace q P f (pureAtomRun w k))
-    (hQ : SimulatesTrace q Q f (pureAtomRun w (k+1)))
-    (hov : p≤FiniteLaw.overlap P Q) :
-    ∃ (β : Type u) (_ : Fintype β) (P' Q' : FiniteLaw β) (g : β→List Bool),
-      SimulatesTrace q P' g (pureAtomRun w (F*R)) ∧
-      SimulatesTrace q Q' g (pureAtomRun w (F*R+R)) ∧
-      (2/3:ℝ)≤FiniteLaw.overlap P' Q' := by
-  let G := F*R/(2*k+1)
-  have hs : (1048576:ℝ)*(R:ℝ)^2≤p*(G:ℝ) := lifting_count_scale hk hR hp hC hF
-  obtain ⟨hG,hr⟩:=lifting_scale_valid hR hp hs
-  let A:=uniformSchedulePayload (n:=2*G) P Q G (by omega)
-  let B:=uniformSchedulePayload (n:=2*G) P Q (G+R) (by omega)
-  have htv : A.tv B≤1/3 :=
-    (uniformSchedulePayload_steps hG hr hp P Q hov).trans (lifting_numeric_bound hR hp hs)
-  let tail:=pureAtomRun w (F*R-G*(2*k+1))
-  let S:=maskLaw (n:=tail.length) q hq
-  let g : (Fin (2*G)→α)×Mask tail.length→List Bool :=
-    fun a => coordinateWord (fun i => f (a.1 i))++traceOfMask tail.get a.2
-  refine ⟨(Fin (2*G)→α)×Mask tail.length,inferInstance,A.prod S,B.prod S,g,?_,?_,?_⟩
-  · have ha:=uniformSchedulePayload_simulates (n:=2*G) (by omega : G≤2*G) P Q f w hP hQ
-    have ht:=(ha.append (maskLaw_simulates hq tail))
-    have hh : G*(2*k+1)≤F*R := Nat.div_mul_le_self _ _
-    have he : 2*G*k+G+(F*R-G*(2*k+1))=F*R := by
-      have he0 : G*(2*k+1)=2*G*k+G := by ring
-      omega
-    simpa only [tail,←pureAtomRun_add,he] using ht
-  · have hb:=uniformSchedulePayload_simulates (n:=2*G) (by omega : G+R≤2*G) P Q f w hP hQ
-    have ht:=(hb.append (maskLaw_simulates hq tail))
-    have hh : G*(2*k+1)≤F*R := Nat.div_mul_le_self _ _
-    have he : 2*G*k+(G+R)+(F*R-G*(2*k+1))=F*R+R := by
-      have he0 : G*(2*k+1)=2*G*k+G := by ring
-      omega
-    simpa only [tail,←pureAtomRun_add,he] using ht
-  · rw [FiniteLaw.overlap_eq_one_sub_tv,FiniteLaw.tv_prod_eq_left]
-    linarith
-
-end TraceReconstruction
-
-
-
-/-!
-Concrete objects for the scale inequalities, Definition 3 (the level invariant).
-The observable channel remains the complete deletion trace of a literal word.
--/
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-namespace FiniteLaw
-variable {α : Type*} [Fintype α]
-
-/-- The maximizing test for total variation on a finite sample space. -/
-theorem tv_eq_positive_test (P Q : FiniteLaw α) :
-    P.tv Q = P.expect (fun a => if Q.mass a ≤ P.mass a then 1 else 0) -
-      Q.expect (fun a => if Q.mass a ≤ P.mass a then 1 else 0) := by
-  have hz : (∑ a, (P.mass a - Q.mass a)) = 0 := by
-    rw [Finset.sum_sub_distrib, P.total, Q.total, sub_self]
-  have he (a : α) :
-      |P.mass a-Q.mass a|+(P.mass a-Q.mass a) =
-        2*((P.mass a-Q.mass a)*(if Q.mass a ≤ P.mass a then 1 else 0)) := by
-    split_ifs with h
-    · rw [abs_of_nonneg (sub_nonneg.mpr h)]; ring
-    · rw [abs_of_neg (sub_neg.mpr (lt_of_not_ge h))]; ring
-  have hs := congrArg (fun f : α → ℝ => ∑ a, f a) (funext he)
-  simp only [Finset.sum_add_distrib, hz, add_zero, ← Finset.mul_sum] at hs
-  simp only [tv, expect]
-  rw [← Finset.sum_sub_distrib]
-  simp_rw [← sub_mul]
-  linarith only [hs]
-
-/-- Dual characterization, sufficient for simulators with different latent spaces. -/
-theorem tv_le_of_tests (P Q : FiniteLaw α) {E : ℝ}
-    (h : ∀ f : α → ℝ, (∀ a, f a ∈ Set.Icc 0 1) →
-      P.expect f-Q.expect f ≤ E) : P.tv Q ≤ E := by
-  rw [tv_eq_positive_test]
-  exact h _ (fun a => by split_ifs <;> norm_num)
-
-theorem abs_expect_sub_le_tv (P Q : FiniteLaw α) (f : α → ℝ)
-    (hf : ∀ a, f a ∈ Set.Icc 0 1) : |P.expect f-Q.expect f| ≤ P.tv Q := by
-  apply abs_le.mpr
-  constructor
-  · have h := expectation_diff_le_tv Q P f hf
-    rw [tv_symm Q P] at h
-    change Q.expect f-P.expect f ≤ P.tv Q at h
-    linarith
-  · exact expectation_diff_le_tv P Q f hf
-end FiniteLaw
-
-/-- The genuine complete-trace law embedded in a specified finite trace space.
-The extra elements have mass zero, not a cemetery mass. -/
-def listTraceLaw (q : ℝ) (hq : q ∈ Set.Icc 0 1) (w : List Bool)
-    (M : ℕ) (hw : w.length ≤ M) : FiniteLaw (BoundedTrace M) :=
-  (maskLaw (n:=w.length) q hq).map (fun a =>
-    ⟨traceOfMask w.get a, mem_traceSpace_of_length_le _
-      ((traceOfMask_length_le w.get a).trans hw)⟩)
-
-theorem listTraceLaw_expect (q : ℝ) (hq : q ∈ Set.Icc 0 1) (w : List Bool)
-    (M : ℕ) (hw : w.length ≤ M) (F : List Bool → ℝ) :
-    (listTraceLaw q hq w M hw).expect (fun a => F a.val) =
-      listDeletionExpectation q w F := by
-  rw [listTraceLaw, FiniteLaw.expect_map]
-  have h := listDeletionExpectation_ofFn q w.get F
-  rw [List.ofFn_get] at h
-  exact h.symm
-
-theorem listTraceLaw_simulates (q : ℝ) (hq : q ∈ Set.Icc 0 1) (w : List Bool)
-    (M : ℕ) (hw : w.length ≤ M) :
-    SimulatesTrace q (listTraceLaw q hq w M hw) Subtype.val w :=
-  listTraceLaw_expect q hq w M hw
-
-/-- Ordinary overlap of complete traces, including unequal input lengths. -/
-def listTraceOverlap (q : ℝ) (hq : q ∈ Set.Icc 0 1) (x y : List Bool) : ℝ :=
-  let M := max x.length y.length
-  FiniteLaw.overlap (listTraceLaw q hq x M (le_max_left _ _))
-    (listTraceLaw q hq y M (le_max_right _ _))
-
-/-- Common approximate simulation transfers distinguishability to genuine
-complete traces, without revealing the latent partition. -/
-theorem approximations_trace_tv {α : Type*} [Fintype α]
-    {q : ℝ} (hq : q ∈ Set.Icc 0 1) (P Q : FiniteLaw α) (decode : α → List Bool)
-    (x y : List Bool) (Ex Ey : ℝ) (M : ℕ) (hx : x.length ≤ M) (hy : y.length ≤ M)
-    (hP : ApproximatesTrace q P decode x Ex)
-    (hQ : ApproximatesTrace q Q decode y Ey) :
-    (listTraceLaw q hq x M hx).tv (listTraceLaw q hq y M hy) ≤ P.tv Q+Ex+Ey := by
-  apply FiniteLaw.tv_le_of_tests
-  intro f hf
-  let F : List Bool → ℝ := fun t => if h : t ∈ traceSpace M then f ⟨t,h⟩ else 0
-  have hF : ∀ t, F t ∈ Set.Icc 0 1 := by
-    intro t
-    dsimp [F]
-    split_ifs with h
-    · exact hf ⟨t,h⟩
-    · norm_num
-  have hFx : (listTraceLaw q hq x M hx).expect f = listDeletionExpectation q x F := by
-    rw [← listTraceLaw_expect q hq x M hx F]
-    congr 1
-    funext t
-    simp only [F, dite_eq_left t.property]
-  have hFy : (listTraceLaw q hq y M hy).expect f = listDeletionExpectation q y F := by
-    rw [← listTraceLaw_expect q hq y M hy F]
-    congr 1
-    funext t
-    simp only [F, dite_eq_left t.property]
-  rw [hFx,hFy]
-  have hp := hP F hF
-  have hq' := hQ F hF
-  have ht := FiniteLaw.abs_expect_sub_le_tv P Q (fun a => F (decode a)) (fun a => hF _)
-  linarith [le_abs_self (P.expect (fun a => F (decode a))-Q.expect (fun a => F (decode a))),
-    (abs_le.mp hp).1, (abs_le.mp hp).2, (abs_le.mp hq').1, (abs_le.mp hq').2]
-
-/-- A literal witness of the source level invariant. `halfOrder` is half its
-positive even moment order. `countFactor` is the prescribed upper multiplier
-`C^(2*B*i+O_B(1))`, to be instantiated uniformly in the actual construction.
-Finite outcomes suffice for the source's finite latent experiments, and any
-randomized common decoder can be included as an additional latent draw. -/
-structure ConcreteLevelPair (q : ℝ) (hq : q ∈ Set.Icc 0 1)
-    (N Δ d ω countFactor : ℝ) (i halfOrder : ℕ) where
-  left : List Bool
-  right : List Bool
-  equalLength : left.length = right.length
-  different : left ≠ right
-  length_lower : N/2 ≤ (left.length : ℝ)
-  length_upper : (left.length : ℝ) ≤ N
-  Outcome : Type
-  finiteOutcome : Fintype Outcome
-  firstLaw : @FiniteLaw Outcome finiteOutcome
-  secondLaw : @FiniteLaw Outcome finiteOutcome
-  decode : Outcome → List Bool
-  error : ℝ
-  error_nonneg : 0 ≤ error
-  normalized_error : error/(left.length : ℝ) ≤ ((i:ℝ)+1)*Δ
-  first_simulation : @ApproximatesTrace Outcome finiteOutcome q firstLaw decode left error
-  second_simulation : @ApproximatesTrace Outcome finiteOutcome q secondLaw decode right error
-  score : Outcome → ℝ
-  density : ∀ a, secondLaw.mass a = firstLaw.mass a*(1+score a)
-  score_bound : @FiniteLaw.evenMomentSq Outcome finiteOutcome firstLaw score halfOrder ≤ N ^ (-d)
-  overlapCount : ℕ
-  overlapCount_lower : left.length ≤ overlapCount
-  overlapCount_upper : (overlapCount:ℝ) ≤ countFactor*(left.length:ℝ)
-  power_overlap : ω ≤ listTraceOverlap q hq (pureAtomRun left overlapCount)
-    (pureAtomRun left (overlapCount+1))
-
-end TraceReconstruction
-
-
-
-/-! The actual finite paired-flag experiment.  We separate its unconditional
-orientation-first generation (used for trace simulation) from its exact
-metadata-first Bayes factorization (used for the density estimates).
-Neither construction conditions on successful window layout. -/
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-namespace FiniteLaw
-variable {α β γ : Type*} [Fintype α] [Fintype β] [Fintype γ]
-
-/-- A joint law retaining both the latent variable and its conditional draw. -/
-def joint (P : FiniteLaw α) (K : α → FiniteLaw β) : FiniteLaw (α × β) where
-  mass z := P.mass z.1 * (K z.1).mass z.2
-  nonneg z := mul_nonneg (P.nonneg _) ((K _).nonneg _)
-  total := by
-    rw [Fintype.sum_prod_type]
-    simp_rw [← Finset.mul_sum, (K _).total, mul_one]
-    exact P.total
-
-theorem expect_joint (P : FiniteLaw α) (K : α → FiniteLaw β) (F : α × β → ℝ) :
-    (P.joint K).expect F = P.expect (fun a => (K a).expect (fun b => F (a,b))) := by
-  simp only [expect, joint, Fintype.sum_prod_type, Finset.mul_sum, mul_assoc]
-
-theorem expect_mixture (P : FiniteLaw α) (K : α → FiniteLaw β) (F : β → ℝ) :
-    (P.mixture K).expect F = P.expect (fun a => (K a).expect F) := by
-  simp only [expect, mixture, Finset.sum_mul, Finset.mul_sum, mul_assoc]
-  exact Finset.sum_comm
-
-theorem mixture_eq_bind (P : FiniteLaw α) (K : α → FiniteLaw β) :
-    P.mixture K = P.bind K := rfl
-
-/-- Independent maps commute with the literal finite product law. -/
-theorem coordinateProduct_map {n : ℕ} (P : Fin n → FiniteLaw α)
-    (f : Fin n → α → β) :
-    (coordinateProduct P).map (fun a i => f i (a i)) =
-      coordinateProduct (fun i => (P i).map (f i)) := by
-  apply FiniteLaw.ext
-  funext b
-  rw [map_mass]
-  change (∑ a : Fin n → α, if (fun i => f i (a i)) = b then ∏ i, (P i).mass (a i) else 0) =
-    ∏ i, ((P i).map (f i)).mass (b i)
-  simp_rw [map_mass]
-  rw [Fintype.prod_sum]
-  apply Finset.sum_congr rfl
-  intro a _
-  by_cases hab : (fun i => f i (a i)) = b
-  · rw [ite_eq_left hab]
-    apply Finset.prod_congr rfl
-    intro i _
-    rw [ite_eq_left (congrFun hab i)]
-  · rw [ite_eq_right hab]
-    symm
-    have hi : ∃ i, f i (a i) ≠ b i := by
-      by_contra h
-      push Not at h
-      exact hab (funext h)
-    obtain ⟨i,hi⟩ := hi
-    exact Finset.prod_eq_zero (Finset.mem_univ i) (ite_eq_right hi)
-
-/-- Forgetting the report recovers the two independent side emissions. -/
-theorem pairedFlagLaw_map (P Q H : FiniteLaw α) {ω : ℝ} (hω : 0 ≤ ω)
-    (hP : ∀ a, ω*H.mass a ≤ P.mass a) (hQ : ∀ a, ω*H.mass a ≤ Q.mass a)
-    (b : Bool) :
-    (pairedFlagLaw P Q H hω hP hQ b).map Prod.snd = orientedProd P Q b := by
-  apply FiniteLaw.ext
-  funext z
-  rw [map_mass, Fintype.sum_prod_type]
-  simp only [pairedFlagLaw]
-  simp_rw [Finset.sum_ite_eq']
-  simp only [Finset.mem_univ, ite_true]
-  exact pairedFlag_simulation P Q H ω b z
-
-end FiniteLaw
-
-variable {Side Leaf : Type*} [Fintype Side] [Fintype Leaf] {m n : ℕ}
-
-/-- Data of the paired mechanisms, with common submass proved for each side.
-All words and their actual trace laws will be substituted in these fields. -/
-structure PairedMechanisms (Side : Type*) [Fintype Side] (m : ℕ) where
-  first : Fin m → FiniteLaw Side
-  second : Fin m → FiniteLaw Side
-  common : Fin m → FiniteLaw Side
-  hiddenWeight : Fin m → ℝ
-  weight_nonneg : ∀ i, 0 ≤ hiddenWeight i
-  common_first : ∀ i a, hiddenWeight i*(common i).mass a ≤ (first i).mass a
-  common_second : ∀ i a, hiddenWeight i*(common i).mass a ≤ (second i).mass a
-
-abbrev Reports (Side : Type*) (m : ℕ) := Fin m → Option Bool × (Side × Side)
-
-namespace PairedMechanisms
-variable (M : PairedMechanisms Side m)
-
-def reports : FiniteLaw (Reports Side m) :=
-  FiniteLaw.coordinateProduct (fun i => FiniteLaw.pairedReport (M.first i) (M.second i)
-    (M.common i) (M.weight_nonneg i) (M.common_first i) (M.common_second i))
-
-def reportsGiven (b : Fin m → Bool) : FiniteLaw (Reports Side m) :=
-  FiniteLaw.coordinateProduct (fun i => FiniteLaw.pairedFlagLaw (M.first i) (M.second i)
-    (M.common i) (M.weight_nonneg i) (M.common_first i) (M.common_second i) (b i))
-
-/-- Bit orientations, prior to generating any emissions. -/
-def fairOrientations (m : ℕ) : FiniteLaw (Fin m → Bool) :=
-  (FiniteLaw.flagPosterior none).product m
-
-/-- Exact posterior, with zero-mass metadata treated by the same canonical
-formula. In particular unseen orientations are independent fair bits. -/
-def orientationsGiven (z : Reports Side m) : FiniteLaw (Fin m → Bool) :=
-  FiniteLaw.coordinateProduct (fun i => FiniteLaw.flagPosterior (z i).1)
-
-def selectedLeafLaw (P Q : FiniteLaw Leaf) (θ : Fin n → Bool) :
-    FiniteLaw (Fin n → Leaf) :=
-  FiniteLaw.coordinateProduct (fun i => if θ i then Q else P)
-
-/-- Actual orientation-first, unconditioned augmented observation. -/
-def orientationOracle (P Q : FiniteLaw Leaf) (θ : (Fin m → Bool) → Fin n → Bool) :
-    FiniteLaw (Reports Side m × (Fin n → Leaf)) :=
-  (fairOrientations m).mixture (fun b => (M.reportsGiven b).prod (selectedLeafLaw P Q (θ b)))
-
-/-- Metadata-first factorization of exactly the same law, not a new oracle. -/
-def canonicalOracle (P Q : FiniteLaw Leaf) (θ : (Fin m → Bool) → Fin n → Bool) :
-    FiniteLaw (Reports Side m × (Fin n → Leaf)) :=
-  M.reports.joint (fun z => (orientationsGiven z).mixture (fun b => selectedLeafLaw P Q (θ b)))
-
-theorem oracle_factorization (P Q : FiniteLaw Leaf) (θ : (Fin m → Bool) → Fin n → Bool) :
-    M.orientationOracle P Q θ = M.canonicalOracle P Q θ := by
-  apply FiniteLaw.ext
-  funext zx
-  rcases zx with ⟨z,x⟩
-  change (∑ b, (∏ i : Fin m, (1/2:ℝ)) *
-      ((∏ i, FiniteLaw.pairedFlagMass (M.first i) (M.second i) (M.common i)
-        (M.hiddenWeight i) (b i) (z i)) * (selectedLeafLaw P Q (θ b)).mass x)) =
-    (∏ i, FiniteLaw.pairedReportMass (M.first i) (M.second i) (M.common i)
-      (M.hiddenWeight i) (z i)) *
-      (∑ b, (∏ i, (FiniteLaw.flagPosterior (z i).1).mass (b i)) * (selectedLeafLaw P Q (θ b)).mass x)
-  rw [Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro b _
-  rw [← mul_assoc, ← Finset.prod_mul_distrib,
-    FiniteLaw.pairedFlags_product_bayes, mul_assoc]
-
-/-- The reference is independent old-first-law emissions, even after reporting
-all side traces. This equality is exact for every metadata value. -/
-theorem canonicalOracle_density (P Q : FiniteLaw Leaf) (Z : Leaf → ℝ)
-    (hZ : ∀ a, Q.mass a = P.mass a*(1+Z a))
-    (θ : (Fin m → Bool) → Fin n → Bool) (z : Reports Side m) (x : Fin n → Leaf) :
-    (M.canonicalOracle P Q θ).mass (z,x) =
-      (M.reports.prod (P.product n)).mass (z,x) *
-        averagedDensity (orientationsGiven z) θ Z x := by
-  change M.reports.mass z *
-    ((orientationsGiven z).mixture (fun b => FiniteLaw.coordinateProduct
-      (fun i => if θ b i then Q else P))).mass x = _
-  rw [leaf_oracle_density P Q Z hZ]
-  exact (mul_assoc _ _ _).symm
-
-/-- The full side metadata has a common law for both hypotheses. -/
-theorem canonicalOracle_metadata (P Q : FiniteLaw Leaf)
-    (θ : (Fin m → Bool) → Fin n → Bool) (F : Reports Side m → ℝ) :
-    (M.canonicalOracle P Q θ).expect (fun z => F z.1) = M.reports.expect F := by
-  rw [canonicalOracle, FiniteLaw.expect_joint]
-  simp only [FiniteLaw.expect_const]
-
-/-- Conditional on every orientation, forgetting flags gives independent side
-emissions. This is the unconditional simulation side of the construction. -/
-theorem reportsGiven_forget (b : Fin m → Bool) :
-    (M.reportsGiven b).map (fun z i => (z i).2) =
-      FiniteLaw.coordinateProduct (fun i => FiniteLaw.orientedProd (M.first i) (M.second i) (b i)) := by
-  rw [reportsGiven, FiniteLaw.coordinateProduct_map]
-  congr 1
-  funext i
-  exact FiniteLaw.pairedFlagLaw_map _ _ _ _ _ _ _
-
-end PairedMechanisms
-end TraceReconstruction
-
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-
-end OAI
+import OAI.Probability.TraceReconstruction.InitialMoments
+
+namespace OAI
+
+/-!
+# Oracles
+
+TV kernels and trace simulation, anchor costs, counted primitives, common-part lifting, concrete oracle pairs, and paired-flag metadata.
+-/
+
+open scoped ENNReal
+noncomputable section
+
+open Filter
+open scoped Topology
+
+
+noncomputable section
+namespace TraceReconstruction.FiniteLaw
+open scoped Classical
+variable {α β γ : Type*} [Fintype α] [Fintype β] [Fintype γ]
+
+@[ext] theorem law_ext {P Q : FiniteLaw α} (h : ∀a,P.mass a=Q.mass a) : P=Q := by
+  cases P; cases Q; congr; exact funext h
+
+theorem tv_self (P : FiniteLaw α) : P.tv P=0 := by simp [tv]
+
+theorem tv_le_one (P Q : FiniteLaw α) : P.tv Q≤1 := by
+  unfold tv
+  have h : (∑a,|P.mass a-Q.mass a|)≤2 := by
+    calc
+      _ ≤ ∑a, (P.mass a+Q.mass a) := Finset.sum_le_sum (fun a _ =>
+        (abs_sub (P.mass a) (Q.mass a)).trans_eq (by rw [abs_of_nonneg (P.nonneg a),abs_of_nonneg (Q.nonneg a)]))
+      _ = 2 := by rw [Finset.sum_add_distrib,P.total,Q.total]; norm_num
+  linarith
+
+theorem expect_bind (P : FiniteLaw α) (K : α → FiniteLaw β) (f : β → ℝ) :
+    (P.bind K).expect f=P.expect (fun a => (K a).expect f) := by
+  unfold expect bind
+  simp_rw [Finset.sum_mul,Finset.mul_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro a _
+  apply Finset.sum_congr rfl
+  intro b _
+  ring
+
+theorem bind_assoc (P : FiniteLaw α) (K : α → FiniteLaw β) (L : β → FiniteLaw γ) :
+    (P.bind K).bind L=P.bind (fun a => (K a).bind L) := by
+  apply law_ext
+  intro c
+  simp only [bind,Finset.sum_mul,Finset.mul_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro a _
+  apply Finset.sum_congr rfl
+  intro b _
+  ring
+
+theorem tv_bind_kernel_le (P : FiniteLaw α) (K L : α → FiniteLaw β) :
+    (P.bind K).tv (P.bind L) ≤ P.expect (fun a => (K a).tv (L a)) := by
+  unfold tv bind expect
+  simp only
+  calc
+    _ = (1/2:ℝ)*∑b, |∑a,P.mass a*((K a).mass b-(L a).mass b)| := by
+      congr 1
+      apply Finset.sum_congr rfl
+      intro b _
+      congr 1
+      rw [← Finset.sum_sub_distrib]
+      apply Finset.sum_congr rfl
+      intro a _
+      ring
+    _ ≤ (1/2:ℝ)*∑b,∑a,P.mass a*|(K a).mass b-(L a).mass b| := by
+      apply mul_le_mul_of_nonneg_left _ (by norm_num)
+      apply Finset.sum_le_sum
+      intro b _
+      simpa only [abs_mul,abs_of_nonneg (P.nonneg _)] using
+        Finset.abs_sum_le_sum_abs (fun a => P.mass a*((K a).mass b-(L a).mass b)) Finset.univ
+    _ = _ := by
+      rw [Finset.sum_comm,Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro a _
+      rw [← Finset.mul_sum]
+      ring
+
+/-- An invalid layout is charged only its unconditional probability. -/
+theorem tv_bind_fit_le (P : FiniteLaw α) (K L : α → FiniteLaw β)
+    (good : α → Prop) [DecidablePred good] (E : ℝ)
+    (hfit : ∀a,good a → (K a).tv (L a)≤E) :
+    (P.bind K).tv (P.bind L) ≤ P.expect (fun a => if good a then E else 1) := by
+  apply (tv_bind_kernel_le P K L).trans
+  apply expect_mono
+  intro a
+  split_ifs with h
+  · exact hfit a h
+  · exact tv_le_one _ _
+
+theorem tv_prod_eq_left (P Q : FiniteLaw α) (R : FiniteLaw β) :
+    (P.prod R).tv (Q.prod R)=P.tv Q := by
+  unfold tv prod
+  congr 1
+  rw [Fintype.sum_prod_type]
+  apply Finset.sum_congr rfl
+  intro a _
+  simp_rw [← sub_mul,abs_mul,abs_of_nonneg (R.nonneg _),← Finset.mul_sum,R.total,mul_one]
+
+theorem tv_prod_eq_right (P : FiniteLaw α) (Q R : FiniteLaw β) :
+    (P.prod Q).tv (P.prod R)=Q.tv R := by
+  unfold tv prod
+  congr 1
+  rw [Fintype.sum_prod_type]
+  simp_rw [← mul_sub,abs_mul,abs_of_nonneg (P.nonneg _),← Finset.mul_sum]
+  rw [← Finset.sum_mul,P.total,one_mul]
+
+theorem tv_prod_le (P Q : FiniteLaw α) (R S : FiniteLaw β) :
+    (P.prod R).tv (Q.prod S) ≤ P.tv Q+R.tv S := by
+  have H := tv_triangle (P.prod R) (Q.prod R) (Q.prod S)
+  rwa [tv_prod_eq_left,tv_prod_eq_right] at H
+
+end TraceReconstruction.FiniteLaw
+
+
+noncomputable section
+namespace TraceReconstruction.FiniteLaw
+open scoped Classical
+variable {α : Type*} [Fintype α]
+
+theorem coordinateProduct_succ_tv {n : ℕ} (P Q : Fin (n+1) → FiniteLaw α) :
+    (coordinateProduct P).tv (coordinateProduct Q) =
+      ((P 0).prod (coordinateProduct (fun i => P i.succ))).tv
+        ((Q 0).prod (coordinateProduct (fun i => Q i.succ))) := by
+  change (1/2:ℝ)*(∑x : Fin (n+1) → α, |(∏i,(P i).mass (x i))-(∏i,(Q i).mass (x i))|)=
+    (1/2:ℝ)*∑x : α × (Fin n → α), |(P 0).mass x.1*(∏i,(P i.succ).mass (x.2 i))-(Q 0).mass x.1*(∏i,(Q i.succ).mass (x.2 i))|
+  congr 1
+  rw [← (Fin.consEquiv (fun _ : Fin (n+1) => α)).sum_comp]
+  apply Finset.sum_congr rfl
+  intro x _
+  simp only [Fin.consEquiv,Equiv.coe_fn_mk,Fin.prod_univ_succ,Fin.cons_zero,Fin.cons_succ]
+
+theorem tv_coordinateProduct_le {n : ℕ} (P Q : Fin n → FiniteLaw α) :
+    (coordinateProduct P).tv (coordinateProduct Q) ≤ ∑i, (P i).tv (Q i) := by
+  induction n with
+  | zero => simp [tv,coordinateProduct]
+  | succ n ih =>
+    rw [coordinateProduct_succ_tv,Fin.sum_univ_succ]
+    exact (tv_prod_le _ _ _ _).trans (by linarith [ih (fun i => P i.succ) (fun i => Q i.succ)])
+
+theorem expect_coordinateProduct_succ {n : ℕ} (P : Fin (n+1) → FiniteLaw α)
+    (F : (Fin (n+1) → α) → ℝ) :
+    (coordinateProduct P).expect F =
+      (P 0).expect (fun a => (coordinateProduct (fun i => P i.succ)).expect
+        (fun x => F (Fin.cons a x))) := by
+  change (∑x : Fin (n+1) → α, (∏i,(P i).mass (x i))*F x)=
+    ∑a : α, (P 0).mass a*(∑x : Fin n → α, (∏i,(P i.succ).mass (x i))*F (Fin.cons a x))
+  rw [← (Fin.consEquiv (fun _ : Fin (n+1) => α)).sum_comp,Fintype.sum_prod_type]
+  simp only [Fin.consEquiv,Equiv.coe_fn_mk,Fin.prod_univ_succ,Fin.cons_zero,Fin.cons_succ]
+  simp_rw [Finset.mul_sum,mul_assoc]
+
+end TraceReconstruction.FiniteLaw
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem FiniteLaw.expect_sub {α : Type*} [Fintype α] (P : FiniteLaw α)
+    (F G : α → ℝ) : P.expect (fun a => F a-G a)=P.expect F-P.expect G := by
+  simp only [FiniteLaw.expect,mul_sub,Finset.sum_sub_distrib]
+
+theorem FiniteLaw.abs_expect_le {α : Type*} [Fintype α] (P : FiniteLaw α) (F : α → ℝ) :
+    |P.expect F|≤P.expect (fun a => |F a|) := by
+  simpa only [FiniteLaw.expect,abs_mul,abs_of_nonneg (P.nonneg _)] using
+    Finset.abs_sum_le_sum_abs (fun a => P.mass a*F a) Finset.univ
+
+/-- A finite auxiliary observation, decoded by a fixed function, simulates the
+usual deletion channel on a literal binary word. Test functions range over the
+observable traces only, never latent boundaries. -/
+def SimulatesTrace {α : Type*} [Fintype α] (q : ℝ) (P : FiniteLaw α)
+    (decode : α → List Bool) (w : List Bool) : Prop :=
+  ∀ F : List Bool → ℝ, P.expect (fun a => F (decode a))=listDeletionExpectation q w F
+
+def ApproximatesTrace {α : Type*} [Fintype α] (q : ℝ) (P : FiniteLaw α)
+    (decode : α → List Bool) (w : List Bool) (E : ℝ) : Prop :=
+  ∀ F : List Bool → ℝ, (∀t,F t∈Set.Icc 0 1) →
+    |P.expect (fun a => F (decode a))-listDeletionExpectation q w F|≤E
+
+theorem listDeletionExpectation_const (q c : ℝ) (w : List Bool) :
+    listDeletionExpectation q w (fun _ => c)=c := by
+  induction w with
+  | nil => rfl
+  | cons b w ih => simp only [listDeletionExpectation,ih]; ring
+
+theorem listDeletionExpectation_linear (q : ℝ) (w : List Bool)
+    (a b : ℝ) (F G : List Bool → ℝ) :
+    listDeletionExpectation q w (fun t => a*F t+b*G t)=
+      a*listDeletionExpectation q w F+b*listDeletionExpectation q w G := by
+  induction w generalizing F G with
+  | nil => rfl
+  | cons c w ih => simp only [listDeletionExpectation,ih]; ring
+
+theorem listDeletionExpectation_mono {q : ℝ} (hq : q∈Set.Icc 0 1)
+    (w : List Bool) {F G : List Bool → ℝ} (h : ∀t,F t≤G t) :
+    listDeletionExpectation q w F≤listDeletionExpectation q w G := by
+  induction w generalizing F G with
+  | nil => exact h []
+  | cons b w ih =>
+    exact add_le_add (mul_le_mul_of_nonneg_left (ih h) hq.1)
+      (mul_le_mul_of_nonneg_left (ih (fun t => h (b::t))) (sub_nonneg.mpr hq.2))
+
+theorem listDeletionExpectation_range {q : ℝ} (hq : q∈Set.Icc 0 1)
+    (w : List Bool) (F : List Bool → ℝ) (hF : ∀t,F t∈Set.Icc 0 1) :
+    listDeletionExpectation q w F∈Set.Icc 0 1 := by
+  constructor
+  · simpa only [listDeletionExpectation_const] using
+      listDeletionExpectation_mono hq w (fun t => (hF t).1)
+  · simpa only [listDeletionExpectation_const] using
+      listDeletionExpectation_mono hq w (fun t => (hF t).2)
+
+theorem listDeletionExpectation_sum {α : Type*} [Fintype α] (q : ℝ)
+    (w : List Bool) (F : α → List Bool → ℝ) :
+    listDeletionExpectation q w (fun t => ∑a,F a t)=
+      ∑a,listDeletionExpectation q w (F a) := by
+  induction w generalizing F with
+  | nil => rfl
+  | cons b w ih =>
+    simp only [listDeletionExpectation,ih,Finset.mul_sum,Finset.sum_add_distrib]
+
+theorem FiniteLaw.expect_listDeletionExpectation {α : Type*} [Fintype α]
+    (P : FiniteLaw α) (q : ℝ) (w : List Bool) (F : α → List Bool → ℝ) :
+    P.expect (fun a => listDeletionExpectation q w (F a))=
+      listDeletionExpectation q w (fun t => P.expect (fun a => F a t)) := by
+  unfold FiniteLaw.expect
+  rw [listDeletionExpectation_sum]
+  apply Finset.sum_congr rfl
+  intro a _
+  have H := listDeletionExpectation_linear q w (P.mass a) 0 (F a) (fun _ => 0)
+  simpa only [zero_mul,add_zero] using H.symm
+
+theorem SimulatesTrace.append {α β : Type*} [Fintype α] [Fintype β]
+    {q : ℝ} {P : FiniteLaw α} {Q : FiniteLaw β} {f : α → List Bool} {g : β → List Bool}
+    {x y : List Bool} (hP : SimulatesTrace q P f x) (hQ : SimulatesTrace q Q g y) :
+    SimulatesTrace q (P.prod Q) (fun a => f a.1++g a.2) (x++y) := by
+  intro F
+  rw [FiniteLaw.expect_prod,listDeletionExpectation_append]
+  change P.expect (fun a => Q.expect (fun b => F (f a++g b)))=_
+  unfold SimulatesTrace at hP hQ
+  calc
+    _ = P.expect (fun a => listDeletionExpectation q y (fun t => F (f a++t))) :=
+      congrArg P.expect (funext fun a => hQ (fun t => F (f a++t)))
+    _ = _ := hP (fun t => listDeletionExpectation q y (fun u => F (t++u)))
+
+theorem SimulatesTrace.approximate {α : Type*} [Fintype α]
+    {q : ℝ} {P : FiniteLaw α} {f : α → List Bool} {w : List Bool}
+    (h : SimulatesTrace q P f w) : ApproximatesTrace q P f w 0 := by
+  intro F _
+  rw [h,sub_self,abs_zero]
+
+theorem ApproximatesTrace.append {α β : Type*} [Fintype α] [Fintype β]
+    {q : ℝ} (hq : q∈Set.Icc 0 1) {P : FiniteLaw α} {Q : FiniteLaw β}
+    {f : α → List Bool} {g : β → List Bool} {x y : List Bool} {E D : ℝ}
+    (hP : ApproximatesTrace q P f x E) (hQ : ApproximatesTrace q Q g y D) :
+    ApproximatesTrace q (P.prod Q) (fun a => f a.1++g a.2) (x++y) (E+D) := by
+  intro F hF
+  rw [FiniteLaw.expect_prod,listDeletionExpectation_append]
+  let H : List Bool → ℝ := fun t => listDeletionExpectation q y (fun u => F (t++u))
+  have hH : ∀t,H t∈Set.Icc 0 1 := fun t => listDeletionExpectation_range hq y _ (fun u => hF (t++u))
+  have h1 := hP H hH
+  have h2 : |P.expect (fun a => Q.expect (fun b => F (f a++g b)))-P.expect (fun a => H (f a))|≤D := by
+    rw [← FiniteLaw.expect_sub]
+    calc
+      _ ≤ P.expect (fun a => |Q.expect (fun b => F (f a++g b))-H (f a)|) := FiniteLaw.abs_expect_le _ _
+      _ ≤ P.expect (fun _ => D) := FiniteLaw.expect_mono _ (fun a => hQ _ (fun t => hF (f a++t)))
+      _ = D := FiniteLaw.expect_const _ _
+  exact (abs_sub_le _ _ _).trans (by linarith only [h1,h2])
+
+/-- Any randomized partition into exactly the same literal word gives exactly
+its usual trace after boundaries are forgotten. No fit conditioning occurs. -/
+theorem SimulatesTrace.random_partition {α Ω : Type*} [Fintype α] [Fintype Ω]
+    {q : ℝ} (P : FiniteLaw Ω) (K : Ω → FiniteLaw α) (decode : α → List Bool)
+    (w : List Bool) (h : ∀ω,SimulatesTrace q (K ω) decode w) :
+    SimulatesTrace q (P.bind K) decode w := by
+  intro F
+  rw [FiniteLaw.expect_bind]
+  unfold SimulatesTrace at h
+  simp_rw [h]
+  exact FiniteLaw.expect_const _ _
+
+/-- A failed deterministic layout is charged its unconditional probability;
+the conditional law on all metadata, not a fit-conditioned law, is used. -/
+theorem ApproximatesTrace.random_partition {α Ω : Type*} [Fintype α] [Fintype Ω]
+    {q : ℝ} (hq : q∈Set.Icc 0 1) (P : FiniteLaw Ω) (K : Ω → FiniteLaw α)
+    (decode : α → List Bool) (w : List Bool) (good : Ω → Prop) [DecidablePred good]
+    (E : ℝ) (h : ∀ω,good ω → ApproximatesTrace q (K ω) decode w E) :
+    ApproximatesTrace q (P.bind K) decode w (P.expect (fun ω => if good ω then E else 1)) := by
+  intro F hF
+  rw [FiniteLaw.expect_bind,← FiniteLaw.expect_const P (listDeletionExpectation q w F),← FiniteLaw.expect_sub]
+  apply (FiniteLaw.abs_expect_le _ _).trans
+  apply FiniteLaw.expect_mono
+  intro ω
+  split_ifs with hgood
+  · exact h ω hgood F hF
+  · have hc := listDeletionExpectation_range hq w F hF
+    have hn := FiniteLaw.expect_mono (K ω) (fun a => (hF (decode a)).1)
+    have hu := FiniteLaw.expect_mono (K ω) (fun a => (hF (decode a)).2)
+    rw [FiniteLaw.expect_const] at hn hu
+    exact abs_le.mpr ⟨by linarith [hc.2],by linarith [hc.1]⟩
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- Admissible close child labels, without relaxing any plateau restrictions
+on the anchor. An arbitrary finite label domain is retained. -/
+def closeLabels (A : Finset ℤ) (L : ℕ) (a : ℤ) : Finset ℤ :=
+  A.filter (fun c => |c-a| < (L:ℤ))
+
+theorem closeLabels_card (A : Finset ℤ) (L : ℕ) (a : ℤ) :
+    (closeLabels A L a).card ≤ 2*L := by
+  have hsub : closeLabels A L a ⊆ Finset.Ioo (a-(L:ℤ)) (a+(L:ℤ)) := by
+    intro c hc
+    have h := abs_lt.mp ((Finset.mem_filter.mp hc).2)
+    exact Finset.mem_Ioo.mpr ⟨by omega,by omega⟩
+  have H := Finset.card_le_card hsub
+  rw [Int.card_Ioo] at H
+  omega
+
+/-- Root anchor and close-child energy. This keeps the actual labels finite,
+including any additional source plateau cutoff. -/
+theorem root_anchor_energy (n L : ℕ) (A B : Finset ℤ) (c : ℤ) :
+    (∑ a∈A, ∑ _d∈closeLabels B L a, windowKernel n L (a-c)^2) ≤
+      2*L^3*binomialBound n := by
+  calc
+    _ ≤ ∑a∈A, (2*L:ℝ)*windowKernel n L (a-c)^2 := by
+      apply Finset.sum_le_sum
+      intro a _
+      simp only [Finset.sum_const,nsmul_eq_mul]
+      exact mul_le_mul_of_nonneg_right (by exact_mod_cast closeLabels_card B L a) (sq_nonneg _)
+    _ = (2*L:ℝ)*(∑a∈A,windowKernel n L (a-c)^2) := by rw [Finset.mul_sum]
+    _ ≤ (2*L:ℝ)*(L^2*binomialBound n) :=
+      mul_le_mul_of_nonneg_left (windowKernel_energy n L A c) (by positivity)
+    _ = _ := by ring
+
+/-- The shared global jitter stays in the joint kernel until after squaring.
+The two close labels cost only (2L)^2, uniformly in parent labels. -/
+theorem sibling_anchor_energy (g n₁ n₂ L : ℕ) (A B C D : Finset ℤ) (c₁ c₂ : ℤ) :
+    (∑ a∈A, ∑ b∈B, ∑ _c∈closeLabels C L a, ∑ _d∈closeLabels D L b,
+      jointWindowKernel g n₁ n₂ L (a-c₁) (b-c₂)^2) ≤
+      4*L^6*binomialBound g*binomialBound n₂ := by
+  calc
+    _ ≤ ∑a∈A, ∑b∈B, (2*L:ℝ)^2*jointWindowKernel g n₁ n₂ L (a-c₁) (b-c₂)^2 := by
+      apply Finset.sum_le_sum
+      intro a _
+      apply Finset.sum_le_sum
+      intro b _
+      simp only [Finset.sum_const,nsmul_eq_mul]
+      have hc : ((closeLabels C L a).card:ℝ)≤2*L := by exact_mod_cast closeLabels_card C L a
+      have hd : ((closeLabels D L b).card:ℝ)≤2*L := by exact_mod_cast closeLabels_card D L b
+      have hm := mul_le_mul hc hd (by positivity) (by positivity : (0:ℝ)≤2*L)
+      rw [mul_assoc] at hm
+      nlinarith [mul_le_mul_of_nonneg_right hm (sq_nonneg (jointWindowKernel g n₁ n₂ L (a-c₁) (b-c₂)))]
+    _ = (2*L:ℝ)^2*(∑a∈A,∑b∈B,jointWindowKernel g n₁ n₂ L (a-c₁) (b-c₂)^2) := by
+      simp_rw [Finset.mul_sum]
+    _ ≤ (2*L:ℝ)^2*(L^4*binomialBound g*binomialBound n₂) :=
+      mul_le_mul_of_nonneg_left (jointWindowKernel_energy g n₁ n₂ L A B c₁ c₂) (by positivity)
+    _ = _ := by ring
+
+/-- Exact bottom-up summation of any finite binary anchor model. The state
+contains its actual index path, so group-dependent kernels are allowed. -/
+def binaryTreeSum {α : Type*} [Fintype α] (K : ℕ → α → α × α → ℝ) : ℕ → α → ℝ
+  | 0, _ => 1
+  | n+1, a => ∑b, K n a b*binaryTreeSum K n b.1*binaryTreeSum K n b.2
+
+def binaryTreeBudget (C : ℕ → ℝ) : ℕ → ℝ
+  | 0 => 1
+  | n+1 => C n*(binaryTreeBudget C n)^2
+
+theorem binaryTreeBudget_nonneg (C : ℕ → ℝ) (hC : ∀n,0≤C n) (n : ℕ) :
+    0≤binaryTreeBudget C n := by
+  cases n with
+  | zero => norm_num [binaryTreeBudget]
+  | succ n => exact mul_nonneg (hC n) (sq_nonneg _)
+
+theorem binaryTreeSum_nonneg {α : Type*} [Fintype α]
+    (K : ℕ → α → α × α → ℝ) (hK : ∀n a b,0≤K n a b) (n : ℕ) (a : α) :
+    0≤binaryTreeSum K n a := by
+  induction n generalizing a with
+  | zero => norm_num [binaryTreeSum]
+  | succ n ih => exact Finset.sum_nonneg (fun b _ => mul_nonneg (mul_nonneg (hK n a b) (ih b.1)) (ih b.2))
+
+/-- Uniform local anchor square sums multiply without decoupling the two
+correlated offsets of a sibling group. Domain/plateau indicators can be
+included in K and are never removed by conditioning. -/
+theorem binaryTreeSum_le {α : Type*} [Fintype α]
+    (K : ℕ → α → α × α → ℝ) (hK : ∀n a b,0≤K n a b)
+    (C : ℕ → ℝ) (hC : ∀n,0≤C n) (hlocal : ∀n a, (∑b,K n a b)≤C n)
+    (n : ℕ) (a : α) : binaryTreeSum K n a≤binaryTreeBudget C n := by
+  induction n generalizing a with
+  | zero => rfl
+  | succ n ih =>
+    calc
+      _ ≤ ∑b, K n a b*(binaryTreeBudget C n)^2 := by
+        apply Finset.sum_le_sum
+        intro b _
+        rw [mul_assoc,pow_two]
+        exact mul_le_mul_of_nonneg_left (mul_le_mul (ih b.1) (ih b.2)
+          (binaryTreeSum_nonneg K hK n b.2) (binaryTreeBudget_nonneg C hC n)) (hK n a b)
+      _ = (∑b,K n a b)*(binaryTreeBudget C n)^2 := by rw [Finset.sum_mul]
+      _ ≤ C n*(binaryTreeBudget C n)^2 := mul_le_mul_of_nonneg_right (hlocal n a) (sq_nonneg _)
+      _ = _ := rfl
+
+end TraceReconstruction
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem hypergeomMass_middle_sq (V M m v : ℕ) {p : ℝ}
+    (hp : p∈Set.Icc (1/4:ℝ) (3/4:ℝ)) (hMV : M≤V)
+    (hN : 16≤V+M) (hm : m≤V+M) (hmean : (m:ℝ)=(V+M:ℕ)*p) :
+    hypergeomMass V M m v^2≤4096/((M:ℝ)+1) := by
+  have hp' : p∈Set.Ioo (0:ℝ) 1 := by constructor <;> linarith [hp.1,hp.2]
+  have hpp : (1/8:ℝ)≤p*(1-p) := by nlinarith [sq_nonneg (p-1/2),mul_nonneg (sub_nonneg.mpr hp.1) (sub_nonneg.mpr hp.2)]
+  have hNr : (16:ℝ)≤(V:ℝ)+M := by exact_mod_cast hN
+  have hVr : (M:ℝ)≤V := by exact_mod_cast hMV
+  have hv : 1≤(V+M:ℕ)*p*(1-p) := by
+    push_cast
+    nlinarith [Nat.cast_nonneg (α:=ℝ) V,Nat.cast_nonneg (α:=ℝ) M]
+  have H := hypergeomMass_sq_upper V M m v hp' hm hmean hv
+  apply H.trans
+  have hd : 0<((V:ℝ)+1)*((M:ℝ)+1)*p*(1-p) := by positivity [hp'.1,sub_pos.mpr hp'.2]
+  apply (div_le_iff₀ hd).2
+  have hM : 0<(M:ℝ)+1 := by positivity
+  have he : 4096/((M:ℝ)+1)*(((V:ℝ)+1)*((M:ℝ)+1)*p*(1-p))=
+      4096*((V:ℝ)+1)*(p*(1-p)) := by field_simp
+  rw [he]
+  push_cast
+  nlinarith [mul_le_mul_of_nonneg_left hpp (show 0≤(V:ℝ)+1 by positivity)]
+
+theorem hypergeomMass_middle_le (V M m v : ℕ) {p : ℝ}
+    (hp : p∈Set.Icc (1/4:ℝ) (3/4:ℝ)) (hMV : M≤V)
+    (hN : 16≤V+M) (hm : m≤V+M) (hmean : (m:ℝ)=(V+M:ℕ)*p) :
+    hypergeomMass V M m v≤64/Real.sqrt ((M:ℝ)+1) := by
+  have H := hypergeomMass_middle_sq V M m v hp hMV hN hm hmean
+  have hs : 0<Real.sqrt ((M:ℝ)+1) := Real.sqrt_pos.2 (by positivity)
+  have he : (64/Real.sqrt ((M:ℝ)+1))^2=4096/((M:ℝ)+1) := by
+    rw [div_pow,Real.sq_sqrt (by positivity)]; norm_num
+  rw [←he] at H
+  exact (sq_le_sq₀ (hypergeomMass_nonneg _ _ _ _) (by positivity)).1 H
+
+theorem sum_range_extend_zero {f : ℕ→ℝ} {m n : ℕ} (hmn : m≤n)
+    (hz : ∀k,m≤k→f k=0) : ∑k∈Finset.range n,f k=∑k∈Finset.range m,f k := by
+  symm
+  apply Finset.sum_subset (Finset.range_mono hmn)
+  intro k _ hk
+  exact hz k (by simpa using hk)
+
+theorem hypergeomMass_above_total (V M m v : ℕ) (h : m<v) :
+    hypergeomMass V M m v=0 := by simp only [hypergeomMass,ite_eq_right (by omega : ¬v≤ m)]
+
+def hypergeomLaw (V M m : ℕ) (hm : m≤V+M) : FiniteLaw (Fin (V+M+1)) where
+  mass v := hypergeomMass V M m v
+  nonneg _ := hypergeomMass_nonneg _ _ _ _
+  total := by
+    rw [Fin.sum_univ_eq_sum_range]
+    rw [sum_range_extend_zero (Nat.add_le_add_right hm 1) (fun k hk =>
+      hypergeomMass_above_total V M m k (by omega))]
+    exact sum_hypergeomMass V M m hm
+
+theorem hypergeomMass_complement (V M m v : ℕ) (hv : v≤ m) :
+    hypergeomMass V M m v=hypergeomMass M V m (m-v) := by
+  simp only [hypergeomMass,ite_eq_left hv,ite_eq_left (Nat.sub_le _ _),Nat.sub_sub_self hv]
+  rw [Nat.add_comm M V,mul_comm]
+
+theorem hypergeomLaw_adjacent_tv (V M m : ℕ) (hM : M≤ m) (hV : m<V) :
+    (hypergeomLaw V M (m+1) (by omega)).tv (hypergeomLaw V M m (by omega))=
+      adjacentHypergeomTV V M m := by
+  unfold FiniteLaw.tv
+  simp only [hypergeomLaw]
+  rw [Fin.sum_univ_eq_sum_range (fun v => |hypergeomMass V M (m+1) v-
+    hypergeomMass V M m v|)]
+  rw [sum_range_extend_zero (show m+2≤V+M+1 by omega) (fun k hk => by
+    rw [hypergeomMass_above_total V M (m+1) k (by omega),
+      hypergeomMass_above_total V M m k (by omega),sub_self,abs_zero])]
+  rw [←Finset.sum_range_reflect (fun v => |hypergeomMass V M (m+1) v-
+    hypergeomMass V M m v|) (m+2)]
+  rw [show m+2=(m+1)+1 by omega,Finset.sum_range_succ']
+  have hzero : |hypergeomMass V M (m+1) (m+1)-hypergeomMass V M m (m+1)|=
+      hypergeomMass M V (m+1) 0 := by
+    rw [hypergeomMass_above_total V M m (m+1) (by omega),sub_zero,
+      hypergeomMass_complement V M (m+1) (m+1) le_rfl,Nat.sub_self,
+      abs_of_nonneg (hypergeomMass_nonneg _ _ _ _)]
+  have he (k : ℕ) (hk : k∈Finset.range (m+1)) :
+      |hypergeomMass V M (m+1) ((m+1+1)-1-(k+1))-
+       hypergeomMass V M m ((m+1+1)-1-(k+1))|=
+      |hypergeomMass M V (m+1) (k+1)-hypergeomMass M V m k| := by
+    have hk' : k≤ m := by simpa only [Finset.mem_range,Nat.lt_succ_iff] using hk
+    rw [show (m+1+1)-1-(k+1)=m-k by omega,
+      hypergeomMass_complement V M (m+1) (m-k) (by omega),
+      hypergeomMass_complement V M m (m-k) (by omega)]
+    rw [show m+1-(m-k)=k+1 by omega,show m-(m-k)=k by omega]
+  rw [Finset.sum_congr rfl he]
+  simp only [Nat.add_sub_cancel,Nat.sub_zero,hzero]
+  rw [sum_range_extend_zero (Nat.add_le_add_right hM 1) (fun k hk => by
+    rw [hypergeomMass_high M V (m+1) (k+1) (by omega),
+      hypergeomMass_high M V m k (by omega),sub_self,abs_zero])]
+  unfold adjacentHypergeomTV
+  ring
+
+theorem adjacentHypergeomTV_middle_le (V M m : ℕ) {p : ℝ}
+    (hp : p∈Set.Icc (1/4:ℝ) (3/4:ℝ)) (hMV : M≤V)
+    (hN : 16≤V+M) (hM : M≤ m) (hV : m<V) (hmean : (m:ℝ)=(V+M:ℕ)*p) :
+    adjacentHypergeomTV V M m≤64/Real.sqrt ((M:ℝ)+1) := by
+  apply (adjacentHypergeomTV_le_mass V M m hM hV).trans
+  rw [hypergeomMass_complement M V m _ (by
+    exact (hypergeomHiddenMode_le (V-1) M m (by omega)).trans hM)]
+  exact hypergeomMass_middle_le V M m _ hp hMV hN (by omega) hmean
+
+theorem hypergeomLaw_adjacent_middle (V M m : ℕ) (hN : 16≤V+M)
+    (hM : M≤ m) (hV : m<V) (hlow : V+M≤4*m) (hhigh : 4*m≤3*(V+M)) :
+    (hypergeomLaw V M m (by omega)).tv (hypergeomLaw V M (m+1) (by omega))≤
+      64/Real.sqrt ((M:ℝ)+1) := by
+  have hn : 0<(V+M:ℕ) := by omega
+  have hnr : (0:ℝ)<(V+M:ℕ) := by exact_mod_cast hn
+  have hL : ((V+M:ℕ):ℝ)≤4*(m:ℝ) := by exact_mod_cast hlow
+  have hU : 4*(m:ℝ)≤3*((V+M:ℕ):ℝ) := by exact_mod_cast hhigh
+  rw [FiniteLaw.tv_symm,hypergeomLaw_adjacent_tv V M m hM hV]
+  have hp : (m:ℝ)/(V+M:ℕ)∈Set.Icc (1/4:ℝ) (3/4:ℝ) := by
+    constructor
+    · apply (le_div_iff₀ hnr).2; linarith
+    · apply (div_le_iff₀ hnr).2; linarith
+  exact adjacentHypergeomTV_middle_le V M m hp (by omega) hN hM hV
+    (by field_simp)
+
+theorem FiniteLaw.tv_chain {α : Type*} [Fintype α] (P : ℕ→FiniteLaw α) (n : ℕ) :
+    (P 0).tv (P n)≤∑i∈Finset.range n,(P i).tv (P (i+1)) := by
+  induction n with
+  | zero => simp [FiniteLaw.tv_self]
+  | succ n ih =>
+    rw [Finset.sum_range_succ]
+    linarith [FiniteLaw.tv_triangle (P 0) (P n) (P (n+1))]
+
+theorem hypergeomLaw_steps (V M m P : ℕ) (hN : 16≤V+M)
+    (hM : M≤ m) (hV : m+P<V) (hlow : V+M≤4*m) (hhigh : 4*(m+P)≤3*(V+M)) :
+    (hypergeomLaw V M m (by omega)).tv (hypergeomLaw V M (m+P) (by omega))≤
+      (P:ℝ)*64/Real.sqrt ((M:ℝ)+1) := by
+  let Q : ℕ→FiniteLaw (Fin (V+M+1)) := fun j =>
+    hypergeomLaw V M (m+min j P) (by omega)
+  have H := FiniteLaw.tv_chain Q P
+  have hQ0 : Q 0=hypergeomLaw V M m (by omega) := by simp [Q]
+  have hQP : Q P=hypergeomLaw V M (m+P) (by omega) := by simp [Q]
+  rw [hQ0,hQP] at H
+  apply H.trans
+  calc
+    _≤∑j∈Finset.range P,64/Real.sqrt ((M:ℝ)+1) := by
+      apply Finset.sum_le_sum
+      intro j hj
+      have hj' : j<P := Finset.mem_range.mp hj
+      simp only [Q,min_eq_left (by omega : j≤P),min_eq_left (by omega : j+1≤P)]
+      exact hypergeomLaw_adjacent_middle V M (m+j) hN (by omega) (by omega)
+        (by omega) (by omega)
+    _=_ := by simp [mul_div_assoc]
+
+end TraceReconstruction
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+variable {α β : Type*} [Fintype α] [Fintype β]
+
+def visibleFiberEquiv (m : ℕ) (S : Finset α) (hS : S.card≤ m) :
+    {T : FixedSubset (α⊕β) m // T.val.toLeft=S} ≃ FixedSubset β (m-S.card) where
+  toFun T := ⟨T.val.val.toRight,by
+    have hc := Finset.card_toLeft_add_card_toRight (u:=T.val.val)
+    rw [T.property,T.val.property] at hc
+    omega⟩
+  invFun R := ⟨⟨S.disjSum R.val,by simp only [Finset.card_disjSum,R.property]; omega⟩,by simp⟩
+  left_inv T := by
+    apply Subtype.ext; apply Subtype.ext
+    simpa only [T.property] using (Finset.toLeft_disjSum_toRight (u:=T.val.val))
+  right_inv R := by apply Subtype.ext; exact Finset.toRight_disjSum
+
+def visibleScheduleLaw (m : ℕ) (hm : m≤Fintype.card α+Fintype.card β) : FiniteLaw (Finset α) :=
+  (uniformSubsetLaw (α:=α⊕β) m (by simpa using hm)).map (fun T => T.val.toLeft)
+
+theorem visibleScheduleLaw_mass (m : ℕ) (hm : m≤Fintype.card α+Fintype.card β) (S : Finset α) :
+    (visibleScheduleLaw (α:=α) (β:=β) m hm).mass S=
+      if S.card≤ m then ((Fintype.card β).choose (m-S.card):ℝ)/
+        ((Fintype.card α+Fintype.card β).choose m:ℝ) else 0 := by
+  unfold visibleScheduleLaw
+  rw [FiniteLaw.map_mass]
+  by_cases hS : S.card≤ m
+  · rw [ite_eq_left hS]
+    simp only [uniformSubsetLaw,Fintype.card_sum]
+    rw [←Finset.sum_filter]
+    simp only [Finset.sum_const,nsmul_eq_mul]
+    have he : (Finset.univ.filter (fun T : FixedSubset (α⊕β) m => T.val.toLeft=S)).card=
+        (Fintype.card β).choose (m-S.card) := by
+      rw [←Fintype.card_subtype]
+      exact (Fintype.card_congr (visibleFiberEquiv (β:=β) m S hS)).trans (card_fixedSubset _)
+    rw [he]
+    ring
+  · rw [ite_eq_right hS]
+    apply Finset.sum_eq_zero
+    intro T _
+    have hnot : T.val.toLeft≠S := by
+      intro he
+      apply hS
+      simpa only [he,T.property] using (Finset.card_toLeft_le (u:=T.val))
+    rw [ite_eq_right hnot]
+
+theorem sum_finset_card (f : ℕ→ℝ) :
+    (∑S : Finset α,f S.card)=∑v : Fin (Fintype.card α+1),
+      ((Fintype.card α).choose v.val:ℝ)*f v.val := by
+  let g : Finset α→Fin (Fintype.card α+1) := fun S =>
+    ⟨S.card,by have H:=Finset.card_le_univ S; omega⟩
+  rw [←Fintype.sum_fiberwise g (fun S => f S.card)]
+  apply Finset.sum_congr rfl
+  intro v _
+  have hc : Fintype.card {S : Finset α // g S=v}=(Fintype.card α).choose v.val := by
+    let e : {S : Finset α // g S=v} ≃ FixedSubset α v.val :=
+      { toFun := fun S => ⟨S.val,by have H:=congrArg Fin.val S.property; exact H⟩
+        invFun := fun S => ⟨S.val,by apply Fin.ext; exact S.property⟩
+        left_inv := fun _ => rfl
+        right_inv := fun _ => rfl }
+    exact (Fintype.card_congr e).trans (card_fixedSubset _)
+  calc
+    _=∑_S : {S : Finset α // g S=v},f v.val := by
+      apply Finset.sum_congr rfl
+      intro S _
+      congr 1
+      exact congrArg Fin.val S.property
+    _=_ := by simp [hc]
+
+end TraceReconstruction
+
+namespace TraceReconstruction
+open scoped Classical
+variable {α β : Type*} [Fintype α] [Fintype β]
+
+theorem visibleScheduleLaw_tv (m m' : ℕ)
+    (hm : m≤Fintype.card α+Fintype.card β) (hm' : m'≤Fintype.card α+Fintype.card β) :
+    (visibleScheduleLaw (α:=α) (β:=β) m hm).tv (visibleScheduleLaw m' hm')=
+      (hypergeomLaw (Fintype.card α) (Fintype.card β) m hm).tv
+        (hypergeomLaw (Fintype.card α) (Fintype.card β) m' hm') := by
+  unfold FiniteLaw.tv
+  simp_rw [visibleScheduleLaw_mass]
+  rw [sum_finset_card (α:=α) (fun v =>
+    |(if v≤ m then ((Fintype.card β).choose (m-v):ℝ)/
+      ((Fintype.card α+Fintype.card β).choose m:ℝ) else 0)-
+     (if v≤ m' then ((Fintype.card β).choose (m'-v):ℝ)/
+      ((Fintype.card α+Fintype.card β).choose m':ℝ) else 0)|)]
+  have he (v : ℕ) :
+      ((Fintype.card α).choose v:ℝ)*
+        |(if v≤ m then ((Fintype.card β).choose (m-v):ℝ)/
+          ((Fintype.card α+Fintype.card β).choose m:ℝ) else 0)-
+         (if v≤ m' then ((Fintype.card β).choose (m'-v):ℝ)/
+          ((Fintype.card α+Fintype.card β).choose m':ℝ) else 0)|=
+      |hypergeomMass (Fintype.card α) (Fintype.card β) m v-
+       hypergeomMass (Fintype.card α) (Fintype.card β) m' v| := by
+    rw [←abs_of_nonneg (Nat.cast_nonneg (α:=ℝ) ((Fintype.card α).choose v)),←abs_mul,mul_sub]
+    simp only [mul_ite,mul_zero,hypergeomMass,mul_div_assoc]
+  simp_rw [he]
+  simp only [hypergeomLaw]
+  rw [Fin.sum_univ_eq_sum_range (fun v => |hypergeomMass (Fintype.card α) (Fintype.card β) m v-
+       hypergeomMass (Fintype.card α) (Fintype.card β) m' v|)]
+  rw [Fin.sum_univ_eq_sum_range (n:=Fintype.card α+Fintype.card β+1)
+    (fun v => |hypergeomMass (Fintype.card α) (Fintype.card β) m v-
+       hypergeomMass (Fintype.card α) (Fintype.card β) m' v|)]
+  congr 1
+  symm
+  apply sum_range_extend_zero (by omega)
+  intro k hk
+  rw [hypergeomMass_high _ _ _ _ (by omega),hypergeomMass_high _ _ _ _ (by omega),sub_self,abs_zero]
+
+theorem visibleScheduleLaw_steps (m P : ℕ)
+    (hN : 16≤Fintype.card α+Fintype.card β) (hM : Fintype.card β≤ m)
+    (hV : m+P<Fintype.card α) (hlow : Fintype.card α+Fintype.card β≤4*m)
+    (hhigh : 4*(m+P)≤3*(Fintype.card α+Fintype.card β)) :
+    (visibleScheduleLaw (α:=α) (β:=β) m (by omega)).tv
+      (visibleScheduleLaw (β:=β) (m+P) (by omega))≤
+      (P:ℝ)*64/Real.sqrt ((Fintype.card β:ℝ)+1) := by
+  rw [visibleScheduleLaw_tv]
+  exact hypergeomLaw_steps _ _ m P hN hM hV hlow hhigh
+
+end TraceReconstruction
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+namespace FiniteLaw
+variable {α β γ : Type*} [Fintype α] [Fintype β] [Fintype γ]
+
+theorem ext_expect {P Q : FiniteLaw α} (h : ∀f : α→ℝ,P.expect f=Q.expect f) : P=Q := by
+  apply ext
+  funext a
+  have H:=h (fun b => if b=a then 1 else 0)
+  simpa only [expect,mul_ite,mul_one,mul_zero,Fintype.sum_ite_eq'] using H
+
+theorem expect_map (P : FiniteLaw α) (f : α→β) (G : β→ℝ) :
+    (P.map f).expect G=P.expect (G∘f) := expectation_map P f G
+
+theorem map_map (P : FiniteLaw α) (f : α→β) (g : β→γ) :
+    (P.map f).map g=P.map (g∘f) := by
+  apply ext_expect
+  intro G
+  rw [expect_map,expect_map,expect_map]
+  rfl
+
+theorem map_id (P : FiniteLaw α) : P.map id=P := by
+  apply ext_expect
+  intro G
+  rw [expect_map]
+  rfl
+
+theorem map_equiv_mass (P : FiniteLaw α) (e : α≃β) (b : β) :
+    (P.map e).mass b=P.mass (e.symm b) := by
+  rw [map_mass]
+  simp only [←e.eq_symm_apply,Fintype.sum_ite_eq']
+
+theorem tv_map_equiv (P Q : FiniteLaw α) (e : α≃β) :
+    (P.map e).tv (Q.map e)=P.tv Q := by
+  unfold tv
+  simp_rw [map_equiv_mass]
+  rw [e.symm.sum_comp (fun a => |P.mass a-Q.mass a|)]
+
+end FiniteLaw
+
+variable {α β : Type*} [Fintype α] [Fintype β]
+
+def fixedSubsetEquiv (e : α≃β) (m : ℕ) : FixedSubset α m ≃ FixedSubset β m where
+  toFun S := ⟨S.val.map e.toEmbedding,by simpa only [Finset.card_map] using S.property⟩
+  invFun T := ⟨T.val.map e.symm.toEmbedding,by simpa only [Finset.card_map] using T.property⟩
+  left_inv S := by apply Subtype.ext; exact e.finsetCongr.left_inv S.val
+  right_inv T := by apply Subtype.ext; exact e.finsetCongr.right_inv T.val
+
+theorem uniformSubsetLaw_map (e : α≃β) (m : ℕ) (hm : m≤Fintype.card α) :
+    (uniformSubsetLaw m hm).map (fixedSubsetEquiv e m)=
+      uniformSubsetLaw m (by simpa only [←Fintype.card_congr e] using hm) := by
+  apply FiniteLaw.ext
+  funext T
+  rw [FiniteLaw.map_equiv_mass]
+  simp only [uniformSubsetLaw,Fintype.card_congr e]
+
+end TraceReconstruction
+
+namespace TraceReconstruction
+open scoped Classical
+variable {α : Type*} [Fintype α]
+
+abbrev HiddenIndex (H : Finset α) := {a : α // a∈H}
+abbrev VisibleIndex (H : Finset α) := {a : α // a∉H}
+
+def splitIndexEquiv (H : Finset α) : VisibleIndex H⊕HiddenIndex H ≃ α :=
+  (Equiv.sumComm _ _).trans (Equiv.sumCompl (fun a => a∈H))
+
+theorem card_splitIndex (H : Finset α) :
+    Fintype.card (VisibleIndex H)+Fintype.card (HiddenIndex H)=Fintype.card α := by
+  simpa only [Fintype.card_sum] using Fintype.card_congr (splitIndexEquiv H)
+
+omit [Fintype α] in
+theorem card_hiddenIndex (H : Finset α) : Fintype.card (HiddenIndex H)=H.card := by
+  exact Fintype.card_coe H
+
+def eraseScheduleLaw (H : Finset α) (m : ℕ) (hm : m≤Fintype.card α) : FiniteLaw (Finset α) :=
+  (uniformSubsetLaw m hm).map (fun S => S.val\H)
+
+omit [Fintype α] in
+theorem erase_splitIndex (H : Finset α) (S : Finset (VisibleIndex H⊕HiddenIndex H)) :
+    (S.map (splitIndexEquiv H).toEmbedding)\H=
+      S.toLeft.map ⟨Subtype.val,Subtype.val_injective⟩ := by
+  ext a
+  constructor
+  · intro ha
+    obtain ⟨ha,hn⟩:=Finset.mem_sdiff.mp ha
+    obtain ⟨b,hb,rfl⟩:=Finset.mem_map.mp ha
+    cases b with
+    | inl b =>
+      apply Finset.mem_map.mpr
+      exact ⟨b,by simpa using hb,rfl⟩
+    | inr b => exact False.elim (hn b.property)
+  · intro ha
+    obtain ⟨b,hb,rfl⟩:=Finset.mem_map.mp ha
+    apply Finset.mem_sdiff.mpr
+    constructor
+    · exact Finset.mem_map.mpr ⟨Sum.inl b,by simpa using hb,rfl⟩
+    · exact b.property
+
+end TraceReconstruction
+
+namespace TraceReconstruction
+open scoped Classical
+variable {α : Type*} [Fintype α]
+
+theorem eraseScheduleLaw_eq (H : Finset α) (m : ℕ) (hm : m≤Fintype.card α) :
+    eraseScheduleLaw H m hm=
+      (visibleScheduleLaw (α:=VisibleIndex H) (β:=HiddenIndex H) m
+        (by simpa only [card_splitIndex] using hm)).map
+        (fun S => S.map ⟨Subtype.val,Subtype.val_injective⟩) := by
+  have h:=uniformSubsetLaw_map (splitIndexEquiv H) m
+    (by simpa only [Fintype.card_sum,card_splitIndex] using hm)
+  unfold eraseScheduleLaw visibleScheduleLaw
+  rw [←h,FiniteLaw.map_map,FiniteLaw.map_map]
+  congr 1
+  funext S
+  exact erase_splitIndex H S.val
+
+theorem eraseScheduleLaw_steps (H : Finset α) (m P : ℕ)
+    (hN : 16≤Fintype.card α) (hM : H.card≤ m)
+    (hV : m+P+H.card<Fintype.card α) (hlow : Fintype.card α≤4*m)
+    (hhigh : 4*(m+P)≤3*Fintype.card α) :
+    (eraseScheduleLaw H m (by omega)).tv (eraseScheduleLaw H (m+P) (by omega))≤
+    (P:ℝ)*64/Real.sqrt ((H.card:ℝ)+1) := by
+  rw [eraseScheduleLaw_eq,eraseScheduleLaw_eq]
+  refine (FiniteLaw.tv_map_le _ _ _).trans ?_
+  have hc:=card_splitIndex H
+  have hd:=card_hiddenIndex H
+  have h:=visibleScheduleLaw_steps (α:=VisibleIndex H) (β:=HiddenIndex H) m P
+    (by omega) (by omega) (by omega) (by omega) (by omega)
+  simpa only [hd] using h
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+namespace FiniteLaw
+variable {ι : Type*} [Fintype ι] {β : ι → Type*} [∀i,Fintype (β i)]
+
+def sigma (P : FiniteLaw ι) (K : ∀i, FiniteLaw (β i)) : FiniteLaw (Σi,β i) where
+  mass a:=P.mass a.1*(K a.1).mass a.2
+  nonneg a:=mul_nonneg (P.nonneg _) ((K _).nonneg _)
+  total:=by
+    rw [Fintype.sum_sigma]
+    simp_rw [←Finset.mul_sum,(K _).total,mul_one]
+    exact P.total
+
+theorem expect_sigma (P : FiniteLaw ι) (K : ∀i,FiniteLaw (β i)) (F : (Σi,β i)→ℝ) :
+    (P.sigma K).expect F=P.expect (fun i => (K i).expect (fun b => F ⟨i,b⟩)) := by
+  simp only [expect,sigma,Fintype.sum_sigma,Finset.mul_sum,mul_assoc]
+
+theorem overlap_sigma_lower (P Q : FiniteLaw ι) (K L : ∀i,FiniteLaw (β i)) :
+    (∑i,min (P.mass i) (Q.mass i)*overlap (K i) (L i))≤overlap (P.sigma K) (Q.sigma L) := by
+  simp only [overlap,sigma,Fintype.sum_sigma,Finset.mul_sum]
+  apply Finset.sum_le_sum
+  intro i _
+  apply Finset.sum_le_sum
+  intro b _
+  apply le_min
+  · exact mul_le_mul (min_le_left _ _) (min_le_left _ _) (le_min ((K i).nonneg b) ((L i).nonneg b)) (P.nonneg i)
+  · exact mul_le_mul (min_le_right _ _) (min_le_right _ _) (le_min ((K i).nonneg b) ((L i).nonneg b)) (Q.nonneg i)
+end FiniteLaw
+
+theorem sum_binomialMass_cutoff {n N : ℕ} (hn : n≤N) (p : ℝ) :
+    ∑l : Fin (N+1),binomialMass n p l.val=1 := by
+  rw [Fin.sum_univ_eq_sum_range (fun k => binomialMass n p k) (N+1)]
+  rw [←sum_binomialMass n p]
+  symm
+  apply Finset.sum_subset (Finset.range_mono (by omega))
+  intro k _ hk
+  have hn' : n<k := by simp only [Finset.mem_range] at hk; omega
+  simp [binomialMass,Nat.choose_eq_zero_of_lt hn']
+
+def boundedCountLaw {n N : ℕ} (hn : n≤N) (p : ℝ) (hp : p∈Set.Icc 0 1) : FiniteLaw (Fin (N+1)) where
+  mass l:=binomialMass n p l.val
+  nonneg l:=binomialMass_nonneg n l.val hp
+  total:=sum_binomialMass_cutoff hn p
+
+abbrev BoundedMarks (N : ℕ) := Σl : Fin (N+1),Fin l.val→PrimitiveMark
+
+def boundedConditionalPrimitiveLaw {n N : ℕ} {q : ℝ} (hq : q∈Set.Ioo 0 1)
+    (θ : Fin n→Bool) (l : Fin (N+1)) : FiniteLaw (Fin l.val→PrimitiveMark) :=
+  if h : l.val≤n then conditionalPrimitiveLaw h q hq θ else (primitiveMarkLaw q hq false).product l.val
+
+theorem primitiveRetention_mem {q : ℝ} (hq : q∈Set.Ioo 0 1) :
+    1-primitiveErasure q∈Set.Icc 0 1 := by
+  constructor
+  · exact (primitive_retain_pos hq).le
+  · unfold primitiveErasure; nlinarith [sq_nonneg q]
+
+def countedPrimitiveLaw {n N : ℕ} (hn : n≤N) {q : ℝ} (hq : q∈Set.Ioo 0 1)
+    (θ : Fin n→Bool) : FiniteLaw (BoundedMarks N) :=
+  (boundedCountLaw hn (1-primitiveErasure q) (primitiveRetention_mem hq)).sigma
+    (boundedConditionalPrimitiveLaw hq θ)
+
+def decodeBoundedMarks {N : ℕ} (a : BoundedMarks N) : List Bool := joinPrimitiveMarks (List.ofFn a.2)
+
+theorem countedPrimitiveLaw_expect {n N : ℕ} (hn : n≤N) {q : ℝ} (hq : q∈Set.Ioo 0 1)
+    (θ : Fin n→Bool) (F : List PrimitiveMark→ℝ) :
+    (countedPrimitiveLaw hn hq θ).expect (fun a => F (List.ofFn a.2))=
+      silentMarkExpectation q (List.ofFn θ) F := by
+  rw [countedPrimitiveLaw,FiniteLaw.expect_sigma,silentMarkExpectation_count_mixture q hq θ F]
+  change (∑l : Fin (N+1),binomialMass n (1-primitiveErasure q) l.val*
+    (boundedConditionalPrimitiveLaw hq θ l).expect (fun x => F (List.ofFn x)))=_
+  let f : ℕ→ℝ := fun l => if hl:l≤n then binomialMass n (1-primitiveErasure q) l*
+    (conditionalPrimitiveLaw hl q hq θ).expect (fun x => F (List.ofFn x)) else 0
+  have hleft : (∑l : Fin (N+1),binomialMass n (1-primitiveErasure q) l.val*
+    (boundedConditionalPrimitiveLaw hq θ l).expect (fun x => F (List.ofFn x)))=
+      ∑l : Fin (N+1),f l.val := by
+    apply Finset.sum_congr rfl
+    intro l _
+    by_cases hl:l.val≤n
+    · simp only [boundedConditionalPrimitiveLaw,hl,dite_eq_left,f]
+    · simp [boundedConditionalPrimitiveLaw,hl,f,binomialMass,Nat.choose_eq_zero_of_lt (Nat.lt_of_not_ge hl)]
+  rw [hleft,Fin.sum_univ_eq_sum_range f (N+1)]
+  have he : (∑l∈Finset.range (n+1),f l)=∑l∈Finset.range (N+1),f l := by
+    apply Finset.sum_subset (Finset.range_mono (by omega))
+    intro l _ hl
+    simp only [Finset.mem_range] at hl
+    simp [f,show ¬l≤n by omega]
+  rw [←he,←Fin.sum_univ_eq_sum_range f (n+1)]
+  apply Finset.sum_congr rfl
+  intro l _
+  simp only [f,dite_eq_left (Nat.le_of_lt_succ l.isLt)]
+
+theorem countedPrimitiveLaw_simulates {n N : ℕ} (hn : n≤N) {q : ℝ} (hq : q∈Set.Ioo 0 1)
+    (θ : Fin n→Bool) :
+    SimulatesTrace q (countedPrimitiveLaw hn hq θ) decodeBoundedMarks
+      (((List.ofFn θ).map primitiveBlock).flatten) := by
+  intro F
+  rw [primitive_run_simulation hq]
+  exact countedPrimitiveLaw_expect hn hq θ (fun a => F (joinPrimitiveMarks a))
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+variable {α : Type*} [Fintype α]
+
+theorem subsetLaw_expect_card (p : ℝ) (hp : p∈Set.Icc 0 1) (f : ℕ→ℝ) :
+    (subsetLaw (α:=α) p hp).expect (fun S => f S.card)=
+      ∑k∈Finset.range (Fintype.card α+1),binomialMass (Fintype.card α) p k*f k := by
+  change (∑S : Finset α,p^S.card*(1-p)^(Fintype.card α-S.card)*f S.card)=_
+  rw [sum_finset_card (α:=α) (fun k => p^k*(1-p)^(Fintype.card α-k)*f k),
+    Fin.sum_univ_eq_sum_range (fun k => ((Fintype.card α).choose k:ℝ)*(p^k*(1-p)^(Fintype.card α-k)*f k))]
+  apply Finset.sum_congr rfl
+  intro k _
+  unfold binomialMass
+  ring
+
+theorem subsetLaw_variance (p : ℝ) (hp : p∈Set.Icc 0 1) :
+    (subsetLaw (α:=α) p hp).expect (fun S => ((S.card:ℝ)-Fintype.card α*p)^2)=
+      (Fintype.card α:ℝ)*p*(1-p) := by
+  rw [subsetLaw_expect_card p hp (fun k => ((k:ℝ)-Fintype.card α*p)^2)]
+  exact binomialMass_variance _ p
+
+def hiddenScheduleLaw (p : ℝ) (hp : p∈Set.Icc 0 1) (m : ℕ) (hm : m≤Fintype.card α) :
+    FiniteLaw (Finset α×Finset α) :=
+  (subsetLaw p hp).bind (fun H => (eraseScheduleLaw H m hm).map (fun S => (H,S)))
+
+theorem hiddenScheduleLaw_tv (p : ℝ) (hp : p∈Set.Icc 0 1) (m m' : ℕ)
+    (hm : m≤Fintype.card α) (hm' : m'≤Fintype.card α) :
+    (hiddenScheduleLaw p hp m hm).tv (hiddenScheduleLaw p hp m' hm') ≤
+      (subsetLaw (α:=α) p hp).expect (fun H =>
+        (eraseScheduleLaw H m hm).tv (eraseScheduleLaw H m' hm')) := by
+  unfold hiddenScheduleLaw
+  refine (FiniteLaw.tv_bind_kernel_le _ _ _).trans ?_
+  apply FiniteLaw.expect_mono
+  intro H
+  exact FiniteLaw.tv_map_le _ _ _
+
+theorem good_hiddenSchedule_bound {G P : ℕ} {p : ℝ}
+    (hG : 8≤G) (hP : 4*P≤G) (hp : p∈Set.Ioc 0 (1/8))
+    (H : Finset (Fin (2*G)))
+    (hlo : p*(2*G:ℝ)/2≤H.card) (hhi : (H.card:ℝ)≤2*p*(2*G:ℝ)) :
+    (eraseScheduleLaw H G (by simp only [Fintype.card_fin]; omega)).tv
+      (eraseScheduleLaw H (G+P) (by simp only [Fintype.card_fin]; omega))≤
+      (P:ℝ)*64/Real.sqrt (p*(2*G:ℝ)/2) := by
+  have hgr : (0:ℝ)<G := by exact_mod_cast (show 0<G by omega)
+  have hhr : (H.card:ℝ)≤(G:ℝ)/2 := by nlinarith [mul_le_mul_of_nonneg_right hp.2 hgr.le]
+  have hh : 2*H.card≤G := by exact_mod_cast (show 2*(H.card:ℝ)≤G by linarith)
+  have h:=eraseScheduleLaw_steps H G P (by simp only [Fintype.card_fin]; omega)
+    (by omega) (by simp only [Fintype.card_fin]; omega)
+    (by simp only [Fintype.card_fin]; omega) (by simp only [Fintype.card_fin]; omega)
+  apply h.trans
+  apply div_le_div_of_nonneg_left (by positivity)
+    (Real.sqrt_pos.2 (by positivity [hp.1]))
+    (Real.sqrt_le_sqrt (by linarith))
+
+theorem bad_hiddenSchedule_variance {μ x : ℝ} (hμ : 0<μ)
+    (hbad : ¬(μ/2≤x ∧ x≤2*μ)) :
+    1≤4*(x-μ)^2/μ^2 := by
+  rw [le_div_iff₀ (sq_pos_of_pos hμ)]
+  push Not at hbad
+  by_cases h : μ/2≤x
+  · have hx:=hbad h
+    nlinarith [sq_nonneg (x-2*μ)]
+  · have hx : x<μ/2 := lt_of_not_ge h
+    nlinarith [sq_nonneg (x-μ/2)]
+
+theorem hiddenScheduleLaw_steps {G P : ℕ} {p : ℝ}
+    (hG : 8≤G) (hP : 4*P≤G) (hp : p∈Set.Ioc 0 (1/8)) :
+    (hiddenScheduleLaw (α:=Fin (2*G)) p ⟨hp.1.le,by linarith [hp.2]⟩ G
+      (by simp only [Fintype.card_fin]; omega)).tv
+      (hiddenScheduleLaw p ⟨hp.1.le,by linarith [hp.2]⟩ (G+P)
+        (by simp only [Fintype.card_fin]; omega))≤
+      (P:ℝ)*64/Real.sqrt (p*(2*G:ℝ)/2)+4/(p*(2*G:ℝ)) := by
+  let μ : ℝ := p*(2*G:ℝ)
+  have hgr : (0:ℝ)<G := by exact_mod_cast (show 0<G by omega)
+  have hμ : 0<μ := by dsimp [μ]; positivity [hp.1]
+  have hμ0 : μ≠0 := ne_of_gt hμ
+  let E : ℝ := (P:ℝ)*64/Real.sqrt (μ/2)
+  have hE : 0≤E := by dsimp [E]; positivity
+  apply (hiddenScheduleLaw_tv p ⟨hp.1.le,by linarith [hp.2]⟩ G (G+P) _ _).trans
+  have he := (subsetLaw (α:=Fin (2*G)) p ⟨hp.1.le,by linarith [hp.2]⟩).expect_mono
+    (g:=fun H => E+4*((H.card:ℝ)-μ)^2/μ^2) (fun H => by
+      by_cases hg : μ/2≤(H.card:ℝ) ∧ (H.card:ℝ)≤2*μ
+      · exact (good_hiddenSchedule_bound hG hP hp H hg.1 (by simpa only [μ,mul_assoc] using hg.2)).trans
+          (le_add_of_nonneg_right (by positivity))
+      · exact (FiniteLaw.tv_le_one _ _).trans
+          ((bad_hiddenSchedule_variance hμ hg).trans (by linarith)))
+  apply he.trans
+  have hv:=subsetLaw_variance (α:=Fin (2*G)) p ⟨hp.1.le,by linarith [hp.2]⟩
+  simp only [Fintype.card_fin,Nat.cast_mul,Nat.cast_ofNat] at hv
+  change (∑H,(subsetLaw p _).mass H*(E+4*((H.card:ℝ)-μ)^2/μ^2))≤E+4/μ
+  simp_rw [mul_add, Finset.sum_add_distrib]
+  rw [← Finset.sum_mul,FiniteLaw.total,one_mul]
+  have heq : (∑H : Finset (Fin (2*G)),(subsetLaw p ⟨hp.1.le,by linarith [hp.2]⟩).mass H*(4*((H.card:ℝ)-μ)^2/μ^2))=
+      4/μ^2*(2*(G:ℝ)*p*(1-p)) := by
+    rw [←hv,FiniteLaw.expect,Finset.mul_sum]
+    apply Finset.sum_congr rfl
+    intro H _
+    dsimp [μ]
+    ring
+  rw [heq]
+  have heq' : 4/μ^2*(2*(G:ℝ)*p*(1-p))=4/μ*(1-p) := by
+    have h : 2*(G:ℝ)*p=μ := by dsimp [μ]; ring
+    rw [h]
+    field_simp
+  rw [heq']
+  have h:=mul_le_mul_of_nonneg_left (show 1-p≤1 by linarith [hp.1]) (show 0≤4/μ by positivity)
+  linarith
+
+end TraceReconstruction
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+namespace FiniteLaw
+variable {α β γ : Type*} [Fintype α] [Fintype β] [Fintype γ]
+
+theorem expect_comm (P : FiniteLaw α) (Q : FiniteLaw β) (F : α→β→ℝ) :
+    P.expect (fun a => Q.expect (F a))=Q.expect (fun b => P.expect (fun a => F a b)) := by
+  simp only [expect,Finset.mul_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro b _
+  apply Finset.sum_congr rfl
+  intro a _
+  ring
+
+theorem bind_map (P : FiniteLaw α) (f : α→β) (K : β→FiniteLaw γ) :
+    (P.map f).bind K=P.bind (K∘f) := by
+  apply ext_expect
+  intro F
+  rw [expect_bind,expect_map,expect_bind]
+  rfl
+
+theorem map_bind (P : FiniteLaw α) (K : α→FiniteLaw β) (f : β→γ) :
+    (P.bind K).map f=P.bind (fun a => (K a).map f) := by
+  apply ext_expect
+  intro F
+  simp only [expect_bind,expect_map]
+
+theorem bind_comm (P : FiniteLaw α) (Q : FiniteLaw β) (K : α→β→FiniteLaw γ) :
+    P.bind (fun a => Q.bind (K a))=Q.bind (fun b => P.bind (fun a => K a b)) := by
+  apply ext_expect
+  intro F
+  simp only [expect_bind]
+  exact expect_comm P Q (fun a b => (K a b).expect F)
+
+end FiniteLaw
+variable {α : Type*} [Fintype α]
+
+theorem subsetWeight_mask {n : ℕ} (p : ℝ) (a : Mask n) :
+    subsetWeight p (maskSubset a)=maskWeight (1-p) a := by
+  rw [maskWeight_card]
+  simp only [subsetWeight,Fintype.card_fin,sub_sub_cancel]
+
+theorem coordinateProduct_common_flags {n : ℕ} (p : ℝ) (hp : p∈Set.Icc 0 1)
+    (U : FiniteLaw α) (R P : Fin n→FiniteLaw α)
+    (hP : ∀ i x,(P i).mass x=p*U.mass x+(1-p)*(R i).mass x) :
+    (subsetLaw p hp).bind (fun H : Finset (Fin n) =>
+      FiniteLaw.coordinateProduct (fun i => if i∈H then U else R i))=
+      FiniteLaw.coordinateProduct P := by
+  apply FiniteLaw.ext
+  funext x
+  change (∑H : Finset (Fin n),subsetWeight p H*(∏i,(if i∈H then U else R i).mass (x i)))=
+    ∏i,(P i).mass (x i)
+  rw [←(maskSubsetEquiv n).sum_comp]
+  change (∑a : Mask n,subsetWeight p (maskSubset a)*(∏i,(if i∈maskSubset a then U else R i).mass (x i)))=_
+  simp_rw [subsetWeight_mask,maskWeight,←Finset.prod_mul_distrib]
+  have he (a : Mask n) (i : Fin n) :
+      (if a i then 1-(1-p) else 1-p)*(if i∈maskSubset a then U else R i).mass (x i)=
+      if a i then p*U.mass (x i) else (1-p)*(R i).mass (x i) := by
+    cases ha : a i <;> simp [maskSubset,ha]
+  simp_rw [he]
+  rw [←Fintype.prod_sum (fun i (b : Bool) => if b then p*U.mass (x i) else (1-p)*(R i).mass (x i))]
+  apply Finset.prod_congr rfl
+  intro i _
+  simp only [Fintype.sum_bool,ite_true,Bool.false_eq_true,ite_false]
+  rw [hP]
+
+end TraceReconstruction
+
+namespace TraceReconstruction
+open scoped Classical
+variable {α : Type*} [Fintype α]
+
+def schedulePayload {n : ℕ} (U P Q : FiniteLaw α) (z : Finset (Fin n)×Finset (Fin n)) :
+    FiniteLaw (Fin n→α) :=
+  FiniteLaw.coordinateProduct (fun i => if i∈z.1 then U else if i∈z.2 then Q else P)
+
+def uniformSchedulePayload {n : ℕ} (P Q : FiniteLaw α) (m : ℕ) (hm : m≤n) :
+    FiniteLaw (Fin n→α) :=
+  (uniformSubsetLaw (α:=Fin n) m (by simpa using hm)).bind
+    (fun S => FiniteLaw.coordinateProduct (fun i => if i∈S.val then Q else P))
+
+theorem hiddenSchedule_payload_exact {n : ℕ} (p : ℝ) (hp : p∈Set.Icc 0 1)
+    (U P' Q' P Q : FiniteLaw α)
+    (hP : ∀ x,P.mass x=p*U.mass x+(1-p)*P'.mass x)
+    (hQ : ∀ x,Q.mass x=p*U.mass x+(1-p)*Q'.mass x)
+    (m : ℕ) (hm : m≤n) :
+    (hiddenScheduleLaw (α:=Fin n) p hp m (by simpa using hm)).bind (schedulePayload U P' Q')=
+      uniformSchedulePayload P Q m hm := by
+  unfold hiddenScheduleLaw eraseScheduleLaw
+  rw [FiniteLaw.bind_assoc]
+  simp_rw [FiniteLaw.bind_map]
+  rw [FiniteLaw.bind_comm]
+  unfold uniformSchedulePayload
+  congr 1
+  funext S
+  trans (subsetLaw p hp).bind (fun H : Finset (Fin n) =>
+    FiniteLaw.coordinateProduct (fun i => if i∈H then U else if i∈S.val then Q' else P'))
+  · congr 1
+    funext H
+    dsimp only [Function.comp_def,schedulePayload]
+    congr 1
+    funext i
+    by_cases h : i∈H <;> by_cases h' : i∈S.val <;> simp [h,h']
+  apply coordinateProduct_common_flags
+  intro i x
+  by_cases h : i∈S.val
+  · simpa [h] using hQ x
+  · simpa [h] using hP x
+
+theorem uniformSchedulePayload_steps {G R : ℕ} {p : ℝ}
+    (hG : 8≤G) (hR : 4*R≤G) (hp : p∈Set.Ioc 0 (1/8))
+    (P Q : FiniteLaw α) (hov : p≤FiniteLaw.overlap P Q) :
+    (uniformSchedulePayload (n:=2*G) P Q G (by omega)).tv
+      (uniformSchedulePayload P Q (G+R) (by omega))≤
+      (R:ℝ)*64/Real.sqrt (p*(2*G:ℝ)/2)+4/(p*(2*G:ℝ)) := by
+  have hp1 : p∈Set.Ioo 0 1 := ⟨hp.1,by linarith [hp.2]⟩
+  obtain ⟨U,P',Q',hP,hQ⟩:=FiniteLaw.common_part_decomposition P Q hp1 hov
+  rw [←hiddenSchedule_payload_exact p ⟨hp.1.le,hp1.2.le⟩ U P' Q' P Q hP hQ,
+    ←hiddenSchedule_payload_exact p ⟨hp.1.le,hp1.2.le⟩ U P' Q' P Q hP hQ]
+  exact (FiniteLaw.tv_bind_le _ _ _).trans (hiddenScheduleLaw_steps hG hR hp)
+
+end TraceReconstruction
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+def coordinateWord {n : ℕ} (w : Fin n→List Bool) : List Bool := (List.ofFn w).flatten
+
+theorem coordinateWord_succ {n : ℕ} (w : Fin (n+1)→List Bool) :
+    coordinateWord w=w 0++coordinateWord (fun i => w i.succ) := by
+  simp only [coordinateWord,List.ofFn_succ,List.flatten_cons]
+
+theorem coordinateWord_cons {n : ℕ} (w : List Bool) (v : Fin n→List Bool) :
+    coordinateWord (Fin.cons w v)=w++coordinateWord v := by
+  rw [coordinateWord_succ]
+  simp only [Fin.cons_zero,Fin.cons_succ]
+
+theorem coordinateProduct_simulates {α : Type*} [Fintype α]
+    {n : ℕ} {q : ℝ} (P : Fin n→FiniteLaw α) (f : α→List Bool) (w : Fin n→List Bool)
+    (h : ∀i,SimulatesTrace q (P i) f (w i)) :
+    SimulatesTrace q (FiniteLaw.coordinateProduct P) (fun a => coordinateWord (fun i => f (a i)))
+      (coordinateWord w) := by
+  induction n with
+  | zero =>
+    intro F
+    change (FiniteLaw.coordinateProduct P).expect (fun _ => F [])=F []
+    exact FiniteLaw.expect_const _ _
+  | succ n ih =>
+    intro F
+    rw [FiniteLaw.expect_coordinateProduct_succ,coordinateWord_succ,listDeletionExpectation_append]
+    have hj (a : α) (x : Fin n→α) :
+        coordinateWord (fun i => f ((Fin.cons a x : Fin (n+1)→α) i))=f a++coordinateWord (fun i => f (x i)) := by
+      rw [coordinateWord_succ]
+      simp only [Fin.cons_zero,Fin.cons_succ]
+    simp_rw [hj]
+    have ht:=ih (fun i => P i.succ) (fun i => w i.succ) (fun i => h i.succ)
+    calc
+      _=(P 0).expect (fun a => listDeletionExpectation q (coordinateWord (fun i => w i.succ))
+          (fun t => F (f a++t))) := by
+        congr 1
+        funext a
+        exact ht (fun t => F (f a++t))
+      _=_ := h 0 (fun t => listDeletionExpectation q (coordinateWord (fun i => w i.succ)) (fun u => F (t++u)))
+
+theorem sum_subset_choices {n : ℕ} (S : Finset (Fin n)) (k : ℕ) :
+    (∑i : Fin n,if i∈S then k+1 else k)=n*k+S.card := by
+  have he (i : Fin n) : (if i∈S then k+1 else k)=k+(if i∈S then 1 else 0) := by
+    split_ifs <;> omega
+  simp_rw [he]
+  simp only [Finset.sum_add_distrib,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,
+    Finset.sum_boole,Finset.filter_mem_eq_inter,Finset.univ_inter,Nat.cast_id]
+
+theorem coordinateWord_pureAtomRun {n : ℕ} (w : List Bool) (k : Fin n→ℕ) :
+    coordinateWord (fun i => pureAtomRun w (k i))=pureAtomRun w (∑i,k i) := by
+  change (List.ofFn (pureAtomRun w ∘ k)).flatten=_
+  rw [←List.map_ofFn,pureAtomRun_partition,List.sum_ofFn]
+
+theorem uniform_schedule_word {n m : ℕ} (S : FixedSubset (Fin n) m) (w : List Bool) (k : ℕ) :
+    coordinateWord (fun i => if i∈S.val then pureAtomRun w (k+1) else pureAtomRun w k)=
+      pureAtomRun w (n*k+m) := by
+  have he (i : Fin n) : (if i∈S.val then pureAtomRun w (k+1) else pureAtomRun w k)=
+      pureAtomRun w (if i∈S.val then k+1 else k) := by split_ifs <;> rfl
+  simp_rw [he]
+  rw [coordinateWord_pureAtomRun,sum_subset_choices,S.property]
+
+end TraceReconstruction
+noncomputable section
+namespace TraceReconstruction
+
+theorem lifting_count_scale {F k P : ℕ} {C p : ℝ}
+    (hk : 1≤k) (hP : 1≤P) (hp : p∈Set.Ioc 0 (1/8))
+    (hC : 1073741824≤p*C) (hF : C*k*P≤F) :
+    (1048576:ℝ)*(P:ℝ)^2≤p*((F*P/(2*k+1):ℕ):ℝ) := by
+  let G := F*P/(2*k+1)
+  have hkr : (1:ℝ)≤k := by exact_mod_cast hk
+  have hPr : (1:ℝ)≤P := by exact_mod_cast hP
+  have hGr : (0:ℝ)≤G := Nat.cast_nonneg G
+  have hnat:=Nat.lt_mul_div_succ (F*P) (show 0<2*k+1 by omega)
+  have hupper : (F:ℝ)*P<(2*(k:ℝ)+1)*((G:ℝ)+1) := by exact_mod_cast hnat
+  have hF' := mul_le_mul_of_nonneg_right hF (show (0:ℝ)≤P by positivity)
+  have hupper' := mul_lt_mul_of_pos_left hupper hp.1
+  have hlower:=mul_le_mul_of_nonneg_left hF' hp.1.le
+  have hC' := mul_le_mul_of_nonneg_right hC (show (0:ℝ)≤(k:ℝ)*(P:ℝ)^2 by positivity)
+  have hden : (2*(k:ℝ)+1)*(p*((G:ℝ)+1))≤3*(k:ℝ)*(p*((G:ℝ)+1)) := by
+    apply mul_le_mul_of_nonneg_right (by linarith)
+    positivity [hp.1.le]
+  have hmain : (1073741824:ℝ)*(P:ℝ)^2<3*p*((G:ℝ)+1) := by
+    have h : (k:ℝ)*(1073741824*(P:ℝ)^2)<(k:ℝ)*(3*p*((G:ℝ)+1)) := by
+      nlinarith only [hlower,hupper',hC',hden]
+    exact (mul_lt_mul_iff_of_pos_left (by linarith : (0:ℝ)<k)).mp h
+  change (1048576:ℝ)*(P:ℝ)^2≤p*(G:ℝ)
+  nlinarith [sq_nonneg ((P:ℝ)-1),hp.2]
+
+theorem lifting_scale_valid {G P : ℕ} {p : ℝ}
+    (hP : 1≤P) (hp : p∈Set.Ioc 0 (1/8))
+    (hscale : (1048576:ℝ)*(P:ℝ)^2≤p*(G:ℝ)) : 8≤G ∧ 4*P≤G := by
+  have hPr : (1:ℝ)≤P := by exact_mod_cast hP
+  have hmul:=mul_le_mul_of_nonneg_right hp.2 (Nat.cast_nonneg (α:=ℝ) G)
+  have hG : (8:ℝ)≤G := by nlinarith [sq_nonneg ((P:ℝ)-1)]
+  have h4 : 4*(P:ℝ)≤G := by nlinarith [sq_nonneg ((P:ℝ)-1)]
+  constructor
+  · exact_mod_cast hG
+  · exact_mod_cast h4
+
+theorem lifting_numeric_bound {G P : ℕ} {p : ℝ}
+    (hP : 1≤P) (_hp : p∈Set.Ioc 0 (1/8))
+    (hscale : (1048576:ℝ)*(P:ℝ)^2≤p*(G:ℝ)) :
+    (P:ℝ)*64/Real.sqrt (p*(2*G:ℝ)/2)+4/(p*(2*G:ℝ))≤1/3 := by
+  have hPr : (1:ℝ)≤P := by exact_mod_cast hP
+  have hs : 0<p*(G:ℝ) := by nlinarith [sq_nonneg ((P:ℝ)-1)]
+  have hre : p*(2*G:ℝ)/2=p*G := by ring
+  have hden : (0:ℝ)<p*(2*G:ℝ) := by nlinarith
+  have hsq : 1024*(P:ℝ)≤Real.sqrt (p*(G:ℝ)) := by
+    apply (Real.le_sqrt (by positivity) (le_of_lt hs)).2
+    nlinarith only [hscale]
+  have hspos:=Real.sqrt_pos.2 hs
+  have hfirst : (P:ℝ)*64/Real.sqrt (p*(2*G:ℝ)/2)≤1/16 := by
+    rw [hre,div_le_iff₀ hspos]
+    nlinarith only [hsq]
+  have hsecond : 4/(p*(2*G:ℝ))≤1/16 := by
+    rw [div_le_iff₀ hden]
+    nlinarith [sq_nonneg ((P:ℝ)-1)]
+  linarith
+
+end TraceReconstruction
+universe u
+namespace TraceReconstruction
+open scoped Classical
+variable {α : Type u} [Fintype α]
+
+theorem maskLaw_simulates {q : ℝ} (hq : q∈Set.Icc 0 1) (w : List Bool) :
+    SimulatesTrace q (maskLaw q hq) (traceOfMask w.get) w := by
+  intro F
+  have h:=listDeletionExpectation_ofFn q w.get F
+  rw [List.ofFn_get] at h
+  exact h.symm
+
+theorem uniformSchedulePayload_simulates {n m k : ℕ} {q : ℝ} (hm : m≤n)
+    (P Q : FiniteLaw α) (f : α→List Bool) (w : List Bool)
+    (hP : SimulatesTrace q P f (pureAtomRun w k))
+    (hQ : SimulatesTrace q Q f (pureAtomRun w (k+1))) :
+    SimulatesTrace q (uniformSchedulePayload P Q m hm)
+      (fun a => coordinateWord (fun i => f (a i))) (pureAtomRun w (n*k+m)) := by
+  apply SimulatesTrace.random_partition
+  intro S
+  rw [←uniform_schedule_word S w k]
+  apply coordinateProduct_simulates
+  intro i
+  by_cases h : i∈S.val
+  · simpa only [h,ite_true] using hQ
+  · simpa only [h,ite_false] using hP
+
+/-- Explicit source common-part lifting, using finite auxiliary observations
+that decode exactly to ordinary complete traces of the adjacent pure powers. -/
+theorem adjacent_power_lift {F k R : ℕ} {C p q : ℝ}
+    (hk : 1≤k) (hR : 1≤R) (hp : p∈Set.Ioc 0 (1/8)) (hq : q∈Set.Icc 0 1)
+    (hC : 1073741824≤p*C) (hF : C*k*R≤F)
+    (P Q : FiniteLaw α) (f : α→List Bool) (w : List Bool)
+    (hP : SimulatesTrace q P f (pureAtomRun w k))
+    (hQ : SimulatesTrace q Q f (pureAtomRun w (k+1)))
+    (hov : p≤FiniteLaw.overlap P Q) :
+    ∃ (β : Type u) (_ : Fintype β) (P' Q' : FiniteLaw β) (g : β→List Bool),
+      SimulatesTrace q P' g (pureAtomRun w (F*R)) ∧
+      SimulatesTrace q Q' g (pureAtomRun w (F*R+R)) ∧
+      (2/3:ℝ)≤FiniteLaw.overlap P' Q' := by
+  let G := F*R/(2*k+1)
+  have hs : (1048576:ℝ)*(R:ℝ)^2≤p*(G:ℝ) := lifting_count_scale hk hR hp hC hF
+  obtain ⟨hG,hr⟩:=lifting_scale_valid hR hp hs
+  let A:=uniformSchedulePayload (n:=2*G) P Q G (by omega)
+  let B:=uniformSchedulePayload (n:=2*G) P Q (G+R) (by omega)
+  have htv : A.tv B≤1/3 :=
+    (uniformSchedulePayload_steps hG hr hp P Q hov).trans (lifting_numeric_bound hR hp hs)
+  let tail:=pureAtomRun w (F*R-G*(2*k+1))
+  let S:=maskLaw (n:=tail.length) q hq
+  let g : (Fin (2*G)→α)×Mask tail.length→List Bool :=
+    fun a => coordinateWord (fun i => f (a.1 i))++traceOfMask tail.get a.2
+  refine ⟨(Fin (2*G)→α)×Mask tail.length,inferInstance,A.prod S,B.prod S,g,?_,?_,?_⟩
+  · have ha:=uniformSchedulePayload_simulates (n:=2*G) (by omega : G≤2*G) P Q f w hP hQ
+    have ht:=(ha.append (maskLaw_simulates hq tail))
+    have hh : G*(2*k+1)≤F*R := Nat.div_mul_le_self _ _
+    have he : 2*G*k+G+(F*R-G*(2*k+1))=F*R := by
+      have he0 : G*(2*k+1)=2*G*k+G := by ring
+      omega
+    simpa only [tail,←pureAtomRun_add,he] using ht
+  · have hb:=uniformSchedulePayload_simulates (n:=2*G) (by omega : G+R≤2*G) P Q f w hP hQ
+    have ht:=(hb.append (maskLaw_simulates hq tail))
+    have hh : G*(2*k+1)≤F*R := Nat.div_mul_le_self _ _
+    have he : 2*G*k+(G+R)+(F*R-G*(2*k+1))=F*R+R := by
+      have he0 : G*(2*k+1)=2*G*k+G := by ring
+      omega
+    simpa only [tail,←pureAtomRun_add,he] using ht
+  · rw [FiniteLaw.overlap_eq_one_sub_tv,FiniteLaw.tv_prod_eq_left]
+    linarith
+
+end TraceReconstruction
+
+
+
+/-!
+Concrete objects for the scale inequalities, Definition 3 (the level invariant).
+The observable channel remains the complete deletion trace of a literal word.
+-/
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+namespace FiniteLaw
+variable {α : Type*} [Fintype α]
+
+/-- The maximizing test for total variation on a finite sample space. -/
+theorem tv_eq_positive_test (P Q : FiniteLaw α) :
+    P.tv Q = P.expect (fun a => if Q.mass a ≤ P.mass a then 1 else 0) -
+      Q.expect (fun a => if Q.mass a ≤ P.mass a then 1 else 0) := by
+  have hz : (∑ a, (P.mass a - Q.mass a)) = 0 := by
+    rw [Finset.sum_sub_distrib, P.total, Q.total, sub_self]
+  have he (a : α) :
+      |P.mass a-Q.mass a|+(P.mass a-Q.mass a) =
+        2*((P.mass a-Q.mass a)*(if Q.mass a ≤ P.mass a then 1 else 0)) := by
+    split_ifs with h
+    · rw [abs_of_nonneg (sub_nonneg.mpr h)]; ring
+    · rw [abs_of_neg (sub_neg.mpr (lt_of_not_ge h))]; ring
+  have hs := congrArg (fun f : α → ℝ => ∑ a, f a) (funext he)
+  simp only [Finset.sum_add_distrib, hz, add_zero, ← Finset.mul_sum] at hs
+  simp only [tv, expect]
+  rw [← Finset.sum_sub_distrib]
+  simp_rw [← sub_mul]
+  linarith only [hs]
+
+/-- Dual characterization, sufficient for simulators with different latent spaces. -/
+theorem tv_le_of_tests (P Q : FiniteLaw α) {E : ℝ}
+    (h : ∀ f : α → ℝ, (∀ a, f a ∈ Set.Icc 0 1) →
+      P.expect f-Q.expect f ≤ E) : P.tv Q ≤ E := by
+  rw [tv_eq_positive_test]
+  exact h _ (fun a => by split_ifs <;> norm_num)
+
+theorem abs_expect_sub_le_tv (P Q : FiniteLaw α) (f : α → ℝ)
+    (hf : ∀ a, f a ∈ Set.Icc 0 1) : |P.expect f-Q.expect f| ≤ P.tv Q := by
+  apply abs_le.mpr
+  constructor
+  · have h := expectation_diff_le_tv Q P f hf
+    rw [tv_symm Q P] at h
+    change Q.expect f-P.expect f ≤ P.tv Q at h
+    linarith
+  · exact expectation_diff_le_tv P Q f hf
+end FiniteLaw
+
+/-- The genuine complete-trace law embedded in a specified finite trace space.
+The extra elements have mass zero, not a cemetery mass. -/
+def listTraceLaw (q : ℝ) (hq : q ∈ Set.Icc 0 1) (w : List Bool)
+    (M : ℕ) (hw : w.length ≤ M) : FiniteLaw (BoundedTrace M) :=
+  (maskLaw (n:=w.length) q hq).map (fun a =>
+    ⟨traceOfMask w.get a, mem_traceSpace_of_length_le _
+      ((traceOfMask_length_le w.get a).trans hw)⟩)
+
+theorem listTraceLaw_expect (q : ℝ) (hq : q ∈ Set.Icc 0 1) (w : List Bool)
+    (M : ℕ) (hw : w.length ≤ M) (F : List Bool → ℝ) :
+    (listTraceLaw q hq w M hw).expect (fun a => F a.val) =
+      listDeletionExpectation q w F := by
+  rw [listTraceLaw, FiniteLaw.expect_map]
+  have h := listDeletionExpectation_ofFn q w.get F
+  rw [List.ofFn_get] at h
+  exact h.symm
+
+theorem listTraceLaw_simulates (q : ℝ) (hq : q ∈ Set.Icc 0 1) (w : List Bool)
+    (M : ℕ) (hw : w.length ≤ M) :
+    SimulatesTrace q (listTraceLaw q hq w M hw) Subtype.val w :=
+  listTraceLaw_expect q hq w M hw
+
+/-- Ordinary overlap of complete traces, including unequal input lengths. -/
+def listTraceOverlap (q : ℝ) (hq : q ∈ Set.Icc 0 1) (x y : List Bool) : ℝ :=
+  let M := max x.length y.length
+  FiniteLaw.overlap (listTraceLaw q hq x M (le_max_left _ _))
+    (listTraceLaw q hq y M (le_max_right _ _))
+
+/-- Common approximate simulation transfers distinguishability to genuine
+complete traces, without revealing the latent partition. -/
+theorem approximations_trace_tv {α : Type*} [Fintype α]
+    {q : ℝ} (hq : q ∈ Set.Icc 0 1) (P Q : FiniteLaw α) (decode : α → List Bool)
+    (x y : List Bool) (Ex Ey : ℝ) (M : ℕ) (hx : x.length ≤ M) (hy : y.length ≤ M)
+    (hP : ApproximatesTrace q P decode x Ex)
+    (hQ : ApproximatesTrace q Q decode y Ey) :
+    (listTraceLaw q hq x M hx).tv (listTraceLaw q hq y M hy) ≤ P.tv Q+Ex+Ey := by
+  apply FiniteLaw.tv_le_of_tests
+  intro f hf
+  let F : List Bool → ℝ := fun t => if h : t ∈ traceSpace M then f ⟨t,h⟩ else 0
+  have hF : ∀ t, F t ∈ Set.Icc 0 1 := by
+    intro t
+    dsimp [F]
+    split_ifs with h
+    · exact hf ⟨t,h⟩
+    · norm_num
+  have hFx : (listTraceLaw q hq x M hx).expect f = listDeletionExpectation q x F := by
+    rw [← listTraceLaw_expect q hq x M hx F]
+    congr 1
+    funext t
+    simp only [F, dite_eq_left t.property]
+  have hFy : (listTraceLaw q hq y M hy).expect f = listDeletionExpectation q y F := by
+    rw [← listTraceLaw_expect q hq y M hy F]
+    congr 1
+    funext t
+    simp only [F, dite_eq_left t.property]
+  rw [hFx,hFy]
+  have hp := hP F hF
+  have hq' := hQ F hF
+  have ht := FiniteLaw.abs_expect_sub_le_tv P Q (fun a => F (decode a)) (fun a => hF _)
+  linarith [le_abs_self (P.expect (fun a => F (decode a))-Q.expect (fun a => F (decode a))),
+    (abs_le.mp hp).1, (abs_le.mp hp).2, (abs_le.mp hq').1, (abs_le.mp hq').2]
+
+/-- A literal witness of the source level invariant. `halfOrder` is half its
+positive even moment order. `countFactor` is the prescribed upper multiplier
+`C^(2*B*i+O_B(1))`, to be instantiated uniformly in the actual construction.
+Finite outcomes suffice for the source's finite latent experiments, and any
+randomized common decoder can be included as an additional latent draw. -/
+structure ConcreteLevelPair (q : ℝ) (hq : q ∈ Set.Icc 0 1)
+    (N Δ d ω countFactor : ℝ) (i halfOrder : ℕ) where
+  left : List Bool
+  right : List Bool
+  equalLength : left.length = right.length
+  different : left ≠ right
+  length_lower : N/2 ≤ (left.length : ℝ)
+  length_upper : (left.length : ℝ) ≤ N
+  Outcome : Type
+  finiteOutcome : Fintype Outcome
+  firstLaw : @FiniteLaw Outcome finiteOutcome
+  secondLaw : @FiniteLaw Outcome finiteOutcome
+  decode : Outcome → List Bool
+  error : ℝ
+  error_nonneg : 0 ≤ error
+  normalized_error : error/(left.length : ℝ) ≤ ((i:ℝ)+1)*Δ
+  first_simulation : @ApproximatesTrace Outcome finiteOutcome q firstLaw decode left error
+  second_simulation : @ApproximatesTrace Outcome finiteOutcome q secondLaw decode right error
+  score : Outcome → ℝ
+  density : ∀ a, secondLaw.mass a = firstLaw.mass a*(1+score a)
+  score_bound : @FiniteLaw.evenMomentSq Outcome finiteOutcome firstLaw score halfOrder ≤ N ^ (-d)
+  overlapCount : ℕ
+  overlapCount_lower : left.length ≤ overlapCount
+  overlapCount_upper : (overlapCount:ℝ) ≤ countFactor*(left.length:ℝ)
+  power_overlap : ω ≤ listTraceOverlap q hq (pureAtomRun left overlapCount)
+    (pureAtomRun left (overlapCount+1))
+
+end TraceReconstruction
+
+
+
+/-! The actual finite paired-flag experiment.  We separate its unconditional
+orientation-first generation (used for trace simulation) from its exact
+metadata-first Bayes factorization (used for the density estimates).
+Neither construction conditions on successful window layout. -/
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+namespace FiniteLaw
+variable {α β γ : Type*} [Fintype α] [Fintype β] [Fintype γ]
+
+/-- A joint law retaining both the latent variable and its conditional draw. -/
+def joint (P : FiniteLaw α) (K : α → FiniteLaw β) : FiniteLaw (α × β) where
+  mass z := P.mass z.1 * (K z.1).mass z.2
+  nonneg z := mul_nonneg (P.nonneg _) ((K _).nonneg _)
+  total := by
+    rw [Fintype.sum_prod_type]
+    simp_rw [← Finset.mul_sum, (K _).total, mul_one]
+    exact P.total
+
+theorem expect_joint (P : FiniteLaw α) (K : α → FiniteLaw β) (F : α × β → ℝ) :
+    (P.joint K).expect F = P.expect (fun a => (K a).expect (fun b => F (a,b))) := by
+  simp only [expect, joint, Fintype.sum_prod_type, Finset.mul_sum, mul_assoc]
+
+theorem expect_mixture (P : FiniteLaw α) (K : α → FiniteLaw β) (F : β → ℝ) :
+    (P.mixture K).expect F = P.expect (fun a => (K a).expect F) := by
+  simp only [expect, mixture, Finset.sum_mul, Finset.mul_sum, mul_assoc]
+  exact Finset.sum_comm
+
+theorem mixture_eq_bind (P : FiniteLaw α) (K : α → FiniteLaw β) :
+    P.mixture K = P.bind K := rfl
+
+/-- Independent maps commute with the literal finite product law. -/
+theorem coordinateProduct_map {n : ℕ} (P : Fin n → FiniteLaw α)
+    (f : Fin n → α → β) :
+    (coordinateProduct P).map (fun a i => f i (a i)) =
+      coordinateProduct (fun i => (P i).map (f i)) := by
+  apply FiniteLaw.ext
+  funext b
+  rw [map_mass]
+  change (∑ a : Fin n → α, if (fun i => f i (a i)) = b then ∏ i, (P i).mass (a i) else 0) =
+    ∏ i, ((P i).map (f i)).mass (b i)
+  simp_rw [map_mass]
+  rw [Fintype.prod_sum]
+  apply Finset.sum_congr rfl
+  intro a _
+  by_cases hab : (fun i => f i (a i)) = b
+  · rw [ite_eq_left hab]
+    apply Finset.prod_congr rfl
+    intro i _
+    rw [ite_eq_left (congrFun hab i)]
+  · rw [ite_eq_right hab]
+    symm
+    have hi : ∃ i, f i (a i) ≠ b i := by
+      by_contra h
+      push Not at h
+      exact hab (funext h)
+    obtain ⟨i,hi⟩ := hi
+    exact Finset.prod_eq_zero (Finset.mem_univ i) (ite_eq_right hi)
+
+/-- Forgetting the report recovers the two independent side emissions. -/
+theorem pairedFlagLaw_map (P Q H : FiniteLaw α) {ω : ℝ} (hω : 0 ≤ ω)
+    (hP : ∀ a, ω*H.mass a ≤ P.mass a) (hQ : ∀ a, ω*H.mass a ≤ Q.mass a)
+    (b : Bool) :
+    (pairedFlagLaw P Q H hω hP hQ b).map Prod.snd = orientedProd P Q b := by
+  apply FiniteLaw.ext
+  funext z
+  rw [map_mass, Fintype.sum_prod_type]
+  simp only [pairedFlagLaw]
+  simp_rw [Finset.sum_ite_eq']
+  simp only [Finset.mem_univ, ite_true]
+  exact pairedFlag_simulation P Q H ω b z
+
+end FiniteLaw
+
+variable {Side Leaf : Type*} [Fintype Side] [Fintype Leaf] {m n : ℕ}
+
+/-- Data of the paired mechanisms, with common submass proved for each side.
+All words and their actual trace laws will be substituted in these fields. -/
+structure PairedMechanisms (Side : Type*) [Fintype Side] (m : ℕ) where
+  first : Fin m → FiniteLaw Side
+  second : Fin m → FiniteLaw Side
+  common : Fin m → FiniteLaw Side
+  hiddenWeight : Fin m → ℝ
+  weight_nonneg : ∀ i, 0 ≤ hiddenWeight i
+  common_first : ∀ i a, hiddenWeight i*(common i).mass a ≤ (first i).mass a
+  common_second : ∀ i a, hiddenWeight i*(common i).mass a ≤ (second i).mass a
+
+abbrev Reports (Side : Type*) (m : ℕ) := Fin m → Option Bool × (Side × Side)
+
+namespace PairedMechanisms
+variable (M : PairedMechanisms Side m)
+
+def reports : FiniteLaw (Reports Side m) :=
+  FiniteLaw.coordinateProduct (fun i => FiniteLaw.pairedReport (M.first i) (M.second i)
+    (M.common i) (M.weight_nonneg i) (M.common_first i) (M.common_second i))
+
+def reportsGiven (b : Fin m → Bool) : FiniteLaw (Reports Side m) :=
+  FiniteLaw.coordinateProduct (fun i => FiniteLaw.pairedFlagLaw (M.first i) (M.second i)
+    (M.common i) (M.weight_nonneg i) (M.common_first i) (M.common_second i) (b i))
+
+/-- Bit orientations, prior to generating any emissions. -/
+def fairOrientations (m : ℕ) : FiniteLaw (Fin m → Bool) :=
+  (FiniteLaw.flagPosterior none).product m
+
+/-- Exact posterior, with zero-mass metadata treated by the same canonical
+formula. In particular unseen orientations are independent fair bits. -/
+def orientationsGiven (z : Reports Side m) : FiniteLaw (Fin m → Bool) :=
+  FiniteLaw.coordinateProduct (fun i => FiniteLaw.flagPosterior (z i).1)
+
+def selectedLeafLaw (P Q : FiniteLaw Leaf) (θ : Fin n → Bool) :
+    FiniteLaw (Fin n → Leaf) :=
+  FiniteLaw.coordinateProduct (fun i => if θ i then Q else P)
+
+/-- Actual orientation-first, unconditioned augmented observation. -/
+def orientationOracle (P Q : FiniteLaw Leaf) (θ : (Fin m → Bool) → Fin n → Bool) :
+    FiniteLaw (Reports Side m × (Fin n → Leaf)) :=
+  (fairOrientations m).mixture (fun b => (M.reportsGiven b).prod (selectedLeafLaw P Q (θ b)))
+
+/-- Metadata-first factorization of exactly the same law, not a new oracle. -/
+def canonicalOracle (P Q : FiniteLaw Leaf) (θ : (Fin m → Bool) → Fin n → Bool) :
+    FiniteLaw (Reports Side m × (Fin n → Leaf)) :=
+  M.reports.joint (fun z => (orientationsGiven z).mixture (fun b => selectedLeafLaw P Q (θ b)))
+
+theorem oracle_factorization (P Q : FiniteLaw Leaf) (θ : (Fin m → Bool) → Fin n → Bool) :
+    M.orientationOracle P Q θ = M.canonicalOracle P Q θ := by
+  apply FiniteLaw.ext
+  funext zx
+  rcases zx with ⟨z,x⟩
+  change (∑ b, (∏ i : Fin m, (1/2:ℝ)) *
+      ((∏ i, FiniteLaw.pairedFlagMass (M.first i) (M.second i) (M.common i)
+        (M.hiddenWeight i) (b i) (z i)) * (selectedLeafLaw P Q (θ b)).mass x)) =
+    (∏ i, FiniteLaw.pairedReportMass (M.first i) (M.second i) (M.common i)
+      (M.hiddenWeight i) (z i)) *
+      (∑ b, (∏ i, (FiniteLaw.flagPosterior (z i).1).mass (b i)) * (selectedLeafLaw P Q (θ b)).mass x)
+  rw [Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro b _
+  rw [← mul_assoc, ← Finset.prod_mul_distrib,
+    FiniteLaw.pairedFlags_product_bayes, mul_assoc]
+
+/-- The reference is independent old-first-law emissions, even after reporting
+all side traces. This equality is exact for every metadata value. -/
+theorem canonicalOracle_density (P Q : FiniteLaw Leaf) (Z : Leaf → ℝ)
+    (hZ : ∀ a, Q.mass a = P.mass a*(1+Z a))
+    (θ : (Fin m → Bool) → Fin n → Bool) (z : Reports Side m) (x : Fin n → Leaf) :
+    (M.canonicalOracle P Q θ).mass (z,x) =
+      (M.reports.prod (P.product n)).mass (z,x) *
+        averagedDensity (orientationsGiven z) θ Z x := by
+  change M.reports.mass z *
+    ((orientationsGiven z).mixture (fun b => FiniteLaw.coordinateProduct
+      (fun i => if θ b i then Q else P))).mass x = _
+  rw [leaf_oracle_density P Q Z hZ]
+  exact (mul_assoc _ _ _).symm
+
+/-- The full side metadata has a common law for both hypotheses. -/
+theorem canonicalOracle_metadata (P Q : FiniteLaw Leaf)
+    (θ : (Fin m → Bool) → Fin n → Bool) (F : Reports Side m → ℝ) :
+    (M.canonicalOracle P Q θ).expect (fun z => F z.1) = M.reports.expect F := by
+  rw [canonicalOracle, FiniteLaw.expect_joint]
+  simp only [FiniteLaw.expect_const]
+
+/-- Conditional on every orientation, forgetting flags gives independent side
+emissions. This is the unconditional simulation side of the construction. -/
+theorem reportsGiven_forget (b : Fin m → Bool) :
+    (M.reportsGiven b).map (fun z i => (z i).2) =
+      FiniteLaw.coordinateProduct (fun i => FiniteLaw.orientedProd (M.first i) (M.second i) (b i)) := by
+  rw [reportsGiven, FiniteLaw.coordinateProduct_map]
+  congr 1
+  funext i
+  exact FiniteLaw.pairedFlagLaw_map _ _ _ _ _ _ _
+
+end PairedMechanisms
+end TraceReconstruction
+
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/TraceReconstruction/Parity.lean b/lean/OAI/Probability/TraceReconstruction/Parity.lean
index 058a7cb9c790a67ee0eb75b44de76fee9c947e02..300b49cea1dccbbf1424da422e9a3d3cbcac64a5 100644
--- a/lean/OAI/Probability/TraceReconstruction/Parity.lean
+++ b/lean/OAI/Probability/TraceReconstruction/Parity.lean
@@ -1,504 +1,504 @@
-import Mathlib
-
-namespace OAI
-
-/-!
-# Parity
-
-Parity-tree roles, forced leaves, coefficient shapes, and exact unsigned/signed motif expansions.
--/
-
-open scoped ENNReal
-noncomputable section
-
-namespace TraceReconstruction
-
-/-- Three roles of an active odd child in the manuscript parity expansion:
-constant, unsigned motif, and signed motif. -/
-def roleWeight (χ σ : ℝ) (r : Fin 3) : ℝ :=
-  if r=0 then 1 else if r=1 then -χ else σ
-
-def signedRoleCount {n : ℕ} (r : Fin n → Fin 3) : ℕ :=
-  (Finset.univ.filter fun i => r i=2).card
-
-def activeRoleCount {n : ℕ} (r : Fin n → Fin 3) : ℕ :=
-  (Finset.univ.filter fun i => r i≠0).card
-
-theorem role_product_expansion {n : ℕ} (χ σ : Fin n → ℝ) (τ : ℝ) :
-    (∏ i : Fin n, (1 - χ i + σ i * τ)) =
-      ∑ r : Fin n → Fin 3, (∏ i, roleWeight (χ i) (σ i) (r i)) * τ^signedRoleCount r := by
-  have hp (r : Fin n → Fin 3) : ∏ i, (if r i=2 then τ else (1:ℝ)) = τ^signedRoleCount r := by
-    rw [← Finset.prod_filter]
-    simp only [Finset.prod_const, signedRoleCount]
-  simp_rw [← hp, ← Finset.prod_mul_distrib]
-  rw [← Fintype.prod_sum (fun (i : Fin n) (r : Fin 3) => roleWeight (χ i) (σ i) r * (if r=2 then τ else 1))]
-  apply Finset.prod_congr rfl
-  intro i _
-  simp [Fin.sum_univ_three, roleWeight]
-  ring
-
-theorem odd_role_difference {n : ℕ} (χ σ : Fin n → ℝ) :
-    (∏ i : Fin n, (1 - χ i + σ i)) - (∏ i : Fin n, (1 - χ i - σ i)) =
-      2*∑ r : Fin n → Fin 3,
-        if Odd (signedRoleCount r) then ∏ i, roleWeight (χ i) (σ i) (r i) else 0 := by
-  have hplus := role_product_expansion χ σ 1
-  have hminus := role_product_expansion χ σ (-1)
-  simp only [mul_one, one_pow] at hplus
-  simp only [mul_neg_one, ← sub_eq_add_neg] at hminus
-  rw [hplus,hminus, ← Finset.sum_sub_distrib, Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro r _
-  by_cases hr : Odd (signedRoleCount r)
-  · rw [ite_eq_left hr, hr.neg_one_pow]
-    ring
-  · have he : Even (signedRoleCount r) := Nat.not_odd_iff_even.mp hr
-    rw [ite_eq_right hr, he.neg_one_pow]
-    ring
-
-theorem signedRoleCount_le_active {n : ℕ} (r : Fin n → Fin 3) :
-    signedRoleCount r ≤ activeRoleCount r := by
-  apply Finset.card_le_card
-  intro i hi
-  simp only [Finset.mem_filter, Finset.mem_univ, true_and] at hi ⊢
-  rw [hi]
-  decide
-
-/-- A surviving odd node either has a unique active signed motif (the
-local high-order cancellation case), or at least two active children. -/
-theorem odd_roles_singleton_or_branch {n : ℕ} (r : Fin n → Fin 3)
-    (hodd : Odd (signedRoleCount r)) :
-    (∃ i, r i=2 ∧ ∀ j, j≠i → r j=0) ∨ 2 ≤ activeRoleCount r := by
-  have hpos : 0 < signedRoleCount r := hodd.pos
-  have hle := signedRoleCount_le_active r
-  by_cases h : 2 ≤ activeRoleCount r
-  · exact Or.inr h
-  have ha : activeRoleCount r=1 := by omega
-  obtain ⟨i,hi⟩ := Finset.card_eq_one.mp ha
-  have hmem : ∀ j, r j≠0 ↔ j=i := by
-    intro j
-    have he := Finset.ext_iff.mp hi j
-    simpa only [Finset.mem_filter, Finset.mem_univ, true_and, Finset.mem_singleton] using he
-  have hs : ∃ j, r j=2 := by
-    have hs0 : (Finset.univ.filter fun j => r j=2).Nonempty := Finset.card_pos.mp hpos
-    obtain ⟨j,hj⟩ := hs0
-    exact ⟨j, (Finset.mem_filter.mp hj).2⟩
-  obtain ⟨j,hj⟩ := hs
-  have hij := (hmem j).mp (by rw [hj]; decide)
-  subst j
-  exact Or.inl ⟨i,hj,fun j hji => Classical.byContradiction (fun hn => hji ((hmem j).mp hn))⟩
-
-/-- Finite occupied trees with at least two continuing odd children at
-each positive depth. This records exactly the no-singleton branch of
-the parity argument; the arity and all extra children remain arbitrary. -/
-inductive BranchingTree : ℕ → Type
-  | leaf : BranchingTree 0
-  | node {b : ℕ} (n : ℕ) (children : Fin n → BranchingTree b) (branch : 2 ≤ n) : BranchingTree (b+1)
-
-def BranchingTree.leafCount : {b : ℕ} → BranchingTree b → ℕ
-  | _, .leaf => 1
-  | _, .node _ children _ => ∑ i, (children i).leafCount
-
-def BranchingTree.levelCount : {b : ℕ} → BranchingTree b → ℕ → ℕ
-  | _, .leaf, h => if h=0 then 1 else 0
-  | _, .node _ children _, h => if h=0 then 1 else ∑ i, (children i).levelCount (h-1)
-
-theorem BranchingTree.leafCount_lower {b : ℕ} (t : BranchingTree b) :
-    2^b ≤ t.leafCount := by
-  induction t with
-  | leaf => simp [leafCount]
-  | @node b n children hn ih =>
-    have h := Finset.sum_le_sum (fun i (_ : i ∈ (Finset.univ : Finset (Fin n))) => ih i)
-    simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, smul_eq_mul] at h
-    have hmul := Nat.mul_le_mul_right (2^b) hn
-    simp only [leafCount]
-    rw [pow_succ]
-    omega
-
-
-
-/-- A term in the parity expansion retains the occupied ordered shape,
-all roles, and all extra children, rather than assuming it is binary. -/
-inductive OccupiedParityTree : ℕ → Type
-  | leaf : OccupiedParityTree 0
-  | node {b : ℕ} (n : ℕ) (children : Fin n → OccupiedParityTree b)
-      (roles : Fin n → Fin 3) : OccupiedParityTree (b+1)
-
-def OccupiedParityTree.isOdd : {b : ℕ} → OccupiedParityTree b → Prop
-  | _, .leaf => True
-  | _, .node _ _ roles => Odd (signedRoleCount roles)
-
-def OccupiedParityTree.leafCount : {b : ℕ} → OccupiedParityTree b → ℕ
-  | _, .leaf => 1
-  | _, .node _ children _ => ∑ i, (children i).leafCount
-
-/-- Only odd-role child terms may acquire an active parent multiplier. -/
-def OccupiedParityTree.valid : {b : ℕ} → OccupiedParityTree b → Prop
-  | _, .leaf => True
-  | _, .node _ children roles =>
-      (∀ i, roles i ≠ 0 → (children i).isOdd) ∧ ∀ i, (children i).valid
-
-def OccupiedParityTree.noSingleton : {b : ℕ} → OccupiedParityTree b → Prop
-  | _, .leaf => True
-  | _, .node _ children roles =>
-      (Odd (signedRoleCount roles) → activeRoleCount roles ≠ 1) ∧
-        ∀ i, (children i).noSingleton
-
-/-- The exact forced binary number of selected leaves, with all extra
-occupied vertices present. The hypotheses here are checked properties
-of a parity term, not a statistical or construction theorem. -/
-theorem OccupiedParityTree.forced_leaves {b : ℕ} (t : OccupiedParityTree b)
-    (hv : t.valid) (hn : t.noSingleton) (ho : t.isOdd) : 2^b ≤ t.leafCount := by
-  induction t with
-  | leaf => simp [leafCount]
-  | @node b n children roles ih =>
-    have hod : Odd (signedRoleCount roles) := ho
-    have hpos := hod.pos
-    have hle := signedRoleCount_le_active roles
-    have hne := hn.1 hod
-    have ha : 2 ≤ activeRoleCount roles := by omega
-    have hsum : (∑ i : Fin n, if roles i≠0 then 2^b else 0) ≤
-        ∑ i : Fin n, (children i).leafCount := by
-      apply Finset.sum_le_sum
-      intro i _
-      split_ifs with hi
-      · exact ih i (hv.2 i) (hn.2 i) (hv.1 i hi)
-      · exact Nat.zero_le _
-    have hs : (∑ i : Fin n, if roles i≠0 then 2^b else 0) = activeRoleCount roles * 2^b := by
-      rw [← Finset.sum_filter]
-      simp only [Finset.sum_const, smul_eq_mul, activeRoleCount]
-    rw [hs] at hsum
-    change 2^(b+1) ≤ _
-    rw [pow_succ]
-    exact (show 2^b*2 ≤ activeRoleCount roles*2^b by
-      simpa only [Nat.mul_comm] using Nat.mul_le_mul_right (2^b) ha).trans hsum
-
-end TraceReconstruction
-
-open Filter
-open scoped Topology
-
-
-noncomputable section
-namespace TraceReconstruction
-attribute [local instance] Classical.propDecidable
-
-/-- The literal unsigned/signed motif values at one child index. -/
-def IsMotifValue (χ σ : ℝ) : Prop :=
-  (χ=0 ∧ σ=0) ∨ (χ=1 ∧ (σ=1 ∨ σ= -1))
-
-def IsTypeSign (τ : ℝ) : Prop := τ=1 ∨ τ= -1
-
-theorem motif_child_sign {χ σ τ : ℝ} (hm : IsMotifValue χ σ) (ht : IsTypeSign τ) :
-    IsTypeSign (1-χ+σ*τ) := by
-  rcases hm with ⟨rfl,rfl⟩ | ⟨rfl,hσ⟩
-  · exact Or.inl (by ring)
-  · rcases hσ with rfl | rfl <;> rcases ht with rfl | rfl <;>
-      norm_num [IsTypeSign]
-
-theorem sign_pow_parity {τ : ℝ} (ht : IsTypeSign τ) (n : ℕ) :
-    τ^n = if Odd n then τ else 1 := by
-  rcases ht with rfl | rfl
-  · simp
-  · split_ifs with hn
-    · exact hn.neg_one_pow
-    · exact (Nat.not_odd_iff_even.mp hn).neg_one_pow
-
-/-- Product expansion with an arbitrary three-role scalar per child; the
-only sign variable is the current vertex type. -/
-theorem arbitrary_role_expansion {n : ℕ} (w : Fin n → Fin 3 → ℝ) (τ : ℝ) :
-    (∏ i, ∑ a : Fin 3, w i a * (if a=2 then τ else 1)) =
-      ∑ a : Fin n → Fin 3, (∏ i, w i (a i))*τ^signedRoleCount a := by
-  rw [Finset.prod_univ_sum]
-  simp only [Fintype.piFinset_univ,Finset.prod_mul_distrib]
-  apply Finset.sum_congr rfl
-  intro a _
-  congr 1
-  rw [← Finset.prod_filter]
-  simp [signedRoleCount]
-
-/-- At the last internal level every selected leaf has to be active.
-Role zero has weight zero; unsigned and signed roles are respectively
-χ/2 and -σ/2 in (χ-στ)/2. -/
-def bottomRoleWeight (χ σ : ℝ) (a : Fin 3) : ℝ :=
-  if a=0 then 0 else if a=1 then χ/2 else -σ/2
-
-def internalRoleWeight (odd : Prop) [Decidable odd] (χ σ : ℝ) (a : Fin 3) : ℝ :=
-  if odd then roleWeight χ σ a else if a=0 then 1 else 0
-
-theorem bottomRole_sum (χ σ τ : ℝ) :
-    (∑ a : Fin 3, bottomRoleWeight χ σ a*(if a=2 then τ else 1)) = (χ-σ*τ)/2 := by
-  rw [Fin.sum_univ_three]
-  have h01 : (0:Fin 3) ≠ 1 := by decide
-  have h02 : (0:Fin 3) ≠ 2 := by decide
-  have h12 : (1:Fin 3) ≠ 2 := by decide
-  simp only [bottomRoleWeight,h02,h12,Ne.symm h01,Ne.symm h02,Ne.symm h12,
-    ↓reduceIte,zero_add,mul_one]
-  ring
-
-theorem internalRole_sum (odd : Prop) [Decidable odd] (χ σ τ : ℝ) :
-    (∑ a : Fin 3, internalRoleWeight odd χ σ a*(if a=2 then τ else 1)) =
-      if odd then 1-χ+σ*τ else 1 := by
-  rw [Fin.sum_univ_three]
-  have h01 : (0:Fin 3) ≠ 1 := by decide
-  have h02 : (0:Fin 3) ≠ 2 := by decide
-  have h12 : (1:Fin 3) ≠ 2 := by decide
-  by_cases h : odd <;> simp only [internalRoleWeight,roleWeight,h,h02,h12,
-    Ne.symm h01,Ne.symm h02,Ne.symm h12,↓reduceIte,zero_mul,add_zero,mul_one]
-  ring
-
-/-- An arbitrary occupied ordered subtree, with motif values at its actual
-child labels. Depth zero here means a bottom sibling group, so it already
-has one internal level. No binary-shape assumption is made. -/
-inductive CoefficientShape : ℕ → Type
-  | bottom (n : ℕ) (χ σ : Fin n → ℝ) : CoefficientShape 0
-  | node {b : ℕ} (n : ℕ) (χ σ : Fin n → ℝ)
-      (children : Fin n → CoefficientShape b) : CoefficientShape (b+1)
-
-namespace CoefficientShape
-
-def Admissible : {b : ℕ} → CoefficientShape b → Prop
-  | _, .bottom _ χ σ => ∀ i, IsMotifValue (χ i) (σ i)
-  | _, .node _ χ σ children => (∀ i, IsMotifValue (χ i) (σ i)) ∧ ∀ i, (children i).Admissible
-
-def eval : {b : ℕ} → CoefficientShape b → ℝ → ℝ
-  | _, .bottom _ χ σ, τ => ∏ i, (χ i-σ i*τ)/2
-  | _, .node _ χ σ children, τ => ∏ i, (children i).eval (1-χ i+σ i*τ)
-
-def Choice : {b : ℕ} → CoefficientShape b → Type
-  | _, .bottom n _ _ => Fin n → Fin 3
-  | _, .node n _ _ children => ((i : Fin n) → (children i).Choice) × (Fin n → Fin 3)
-
-noncomputable instance choiceFintype : {b : ℕ} → (t : CoefficientShape b) → Fintype t.Choice
-  | _, .bottom _ _ _ => by unfold Choice; infer_instance
-  | _, .node _ _ _ children => by
-      letI := fun i => choiceFintype (children i)
-      unfold Choice
-      infer_instance
-
-def termTree : {b : ℕ} → (t : CoefficientShape b) → t.Choice → OccupiedParityTree (b+1)
-  | _, .bottom n _ _, a => .node n (fun _ => .leaf) a
-  | _, .node n _ _ children, a => .node n (fun i => (children i).termTree (a.1 i)) a.2
-
-noncomputable def weight : {b : ℕ} → (t : CoefficientShape b) → t.Choice → ℝ
-  | _, .bottom _ χ σ, a => ∏ i, bottomRoleWeight (χ i) (σ i) (a i)
-  | _, .node _ χ σ children, a => ∏ i,
-      (children i).weight (a.1 i) *
-        @internalRoleWeight ((children i).termTree (a.1 i)).isOdd (Classical.propDecidable _)
-          (χ i) (σ i) (a.2 i)
-
-/-- Full finite coefficient expansion, before any orientation average or
-fit-event conditioning. This is the literal algebra behind source
-the correlated-anchor estimate, the parity roles being retained in every term. -/
-theorem expansion {b : ℕ} (t : CoefficientShape b) (ha : t.Admissible)
-    {τ : ℝ} (ht : IsTypeSign τ) :
-    t.eval τ = ∑ a : t.Choice, t.weight a * (if (t.termTree a).isOdd then τ else 1) := by
-  classical
-  induction t generalizing τ with
-  | bottom n χ σ =>
-    have H := arbitrary_role_expansion (fun i => bottomRoleWeight (χ i) (σ i)) τ
-    simp_rw [bottomRole_sum,sign_pow_parity ht] at H
-    trans ∑ a : Fin n → Fin 3, (∏ i, bottomRoleWeight (χ i) (σ i) (a i)) *
-      (if Odd (signedRoleCount a) then τ else 1)
-    · exact H
-    · apply Finset.sum_congr rfl
-      intro a _
-      apply congrArg (fun z => (∏ i, bottomRoleWeight (χ i) (σ i) (a i))*z)
-      change _ = @ite ℝ (Odd (signedRoleCount a)) (Classical.propDecidable _) τ 1
-      split_ifs <;> rfl
-  | @node b n χ σ children ih =>
-    change (∏ i, (children i).eval (1-χ i+σ i*τ)) = _
-    have he (i : Fin n) := ih i (ha.2 i) (motif_child_sign (ha.1 i) ht)
-    simp_rw [he]
-    rw [Finset.prod_univ_sum]
-    simp only [Fintype.piFinset_univ]
-    change (∑ a : (i : Fin n) → (children i).Choice,
-      ∏ i, (children i).weight (a i)*
-        (if ((children i).termTree (a i)).isOdd then 1-χ i+σ i*τ else 1)) =
-      ∑ a : ((i : Fin n) → (children i).Choice) × (Fin n → Fin 3),
-        (∏ i, (children i).weight (a.1 i)*
-          internalRoleWeight ((children i).termTree (a.1 i)).isOdd (χ i) (σ i) (a.2 i))*
-        (@ite ℝ (Odd (signedRoleCount a.2)) (Classical.propDecidable _) τ 1)
-    rw [Fintype.sum_prod_type]
-    apply Finset.sum_congr rfl
-    intro a _
-    have H := arbitrary_role_expansion (fun i =>
-      internalRoleWeight ((children i).termTree (a i)).isOdd (χ i) (σ i)) τ
-    simp_rw [internalRole_sum,sign_pow_parity ht] at H
-    rw [Finset.prod_mul_distrib,H,Finset.mul_sum]
-    apply Finset.sum_congr rfl
-    intro a' _
-    rw [Finset.prod_mul_distrib,mul_assoc]
-    congr 2
-    split_ifs <;> rfl
-
-/-- Nonzero terms automatically satisfy the source active-odd rule. -/
-theorem term_valid {b : ℕ} (t : CoefficientShape b) (a : t.Choice)
-    (h : t.weight a ≠ 0) : (t.termTree a).valid := by
-  classical
-  induction t with
-  | bottom n χ σ => exact ⟨fun _ _ => trivial,fun _ => trivial⟩
-  | @node b n χ σ children ih =>
-    have hp (i : Fin n) : (children i).weight (a.1 i) *
-        internalRoleWeight ((children i).termTree (a.1 i)).isOdd (χ i) (σ i) (a.2 i) ≠ 0 :=
-      (Finset.prod_ne_zero_iff.mp h) i (Finset.mem_univ i)
-    constructor
-    · intro i hai
-      by_contra hn
-      have hz := (mul_ne_zero_iff.mp (hp i)).2
-      simp [internalRoleWeight,hn,hai] at hz
-    · intro i
-      exact ih i (a.1 i) (mul_ne_zero_iff.mp (hp i)).1
-
-/-- The difference of hypotheses consists exactly of odd root terms. -/
-theorem difference_expansion {b : ℕ} (t : CoefficientShape b) (ha : t.Admissible) :
-    t.eval 1-t.eval (-1) = 2*∑ a : t.Choice,
-      if (t.termTree a).isOdd then t.weight a else 0 := by
-  classical
-  rw [expansion t ha (Or.inl rfl),expansion t ha (Or.inr rfl),
-    ← Finset.sum_sub_distrib,Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro a _
-  split_ifs <;> ring
-
-/-- A nonzero surviving term without a singleton odd vertex has the full
-forced binary number of leaves. This follows from the actual expansion,
-not an assumed parity interface. -/
-theorem surviving_leaves {b : ℕ} (t : CoefficientShape b) (a : t.Choice)
-    (hw : t.weight a ≠ 0) (ho : (t.termTree a).isOdd)
-    (hn : (t.termTree a).noSingleton) : 2^(b+1) ≤ (t.termTree a).leafCount :=
-  (t.termTree a).forced_leaves (t.term_valid a hw) hn ho
-
-end CoefficientShape
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-namespace OccupiedParityTree
-
-/-- The actual binary selection sits inside an arbitrary occupied parity
-term, retaining its unselected children rather than assuming binary input. -/
-inductive Forced : {b : ℕ} → OccupiedParityTree b → Type
-  | leaf : Forced .leaf
-  | node {b n : ℕ} {children : Fin n → OccupiedParityTree b} {roles : Fin n → Fin 3}
-      (a c : Fin n) (hne : a≠c) (ha : roles a≠0) (hc : roles c≠0)
-      (left : Forced (children a)) (right : Forced (children c)) :
-      Forced (.node n children roles)
-
-/-- Every surviving nonsingleton term has an embedded binary subtree.
-The selection depends only on the role-marked shape. -/
-theorem forced_nonempty {b : ℕ} (t : OccupiedParityTree b)
-    (hv : t.valid) (hn : t.noSingleton) (ho : t.isOdd) : Nonempty t.Forced := by
-  classical
-  induction t with
-  | leaf => exact ⟨.leaf⟩
-  | @node b n children roles ih =>
-    have hod : Odd (signedRoleCount roles) := ho
-    have ha : 2 ≤ activeRoleCount roles := by
-      have := signedRoleCount_le_active roles
-      have := hod.pos
-      have := hn.1 hod
-      omega
-    obtain ⟨a,ha,c,hc,hne⟩ := Finset.one_lt_card.mp ha
-    have ha' : roles a≠0 := (Finset.mem_filter.mp ha).2
-    have hc' : roles c≠0 := (Finset.mem_filter.mp hc).2
-    obtain ⟨la⟩ := ih a (hv.2 a) (hn.2 a) (hv.1 a ha')
-    obtain ⟨lc⟩ := ih c (hv.2 c) (hn.2 c) (hv.1 c hc')
-    exact ⟨.node a c hne ha' hc' la lc⟩
-
-/-- Occupied shapes have at least one selected leaf below every vertex. -/
-def occupied : {b : ℕ} → OccupiedParityTree b → Prop
-  | _, .leaf => True
-  | _, .node n children _ => 0<n ∧ ∀ i, (children i).occupied
-
-/-- Number of vertices at a given distance below the root. -/
-def levelCount : {b : ℕ} → OccupiedParityTree b → ℕ → ℕ
-  | _, _, 0 => 1
-  | _, .leaf, _+1 => 0
-  | _, .node _ children _, k+1 => ∑ i, (children i).levelCount k
-
-theorem leafCount_pos {b : ℕ} (t : OccupiedParityTree b) (ht : t.occupied) :
-    0<t.leafCount := by
-  induction t with
-  | leaf => exact Nat.zero_lt_one
-  | @node b n children roles ih =>
-    let i : Fin n := ⟨0,ht.1⟩
-    have hp := ih i (ht.2 i)
-    change 0 < ∑ k, (children k).leafCount
-    exact hp.trans_le (Finset.single_le_sum (fun k _ => Nat.zero_le ((children k).leafCount)) (Finset.mem_univ i))
-
-theorem levelCount_le_leaves {b : ℕ} (t : OccupiedParityTree b) (ht : t.occupied)
-    (k : ℕ) : t.levelCount k ≤ t.leafCount := by
-  induction t generalizing k with
-  | leaf => cases k <;> norm_num [levelCount,leafCount]
-  | @node b n children roles ih =>
-    cases k with
-    | zero => exact leafCount_pos (.node n children roles) ht
-    | succ k => exact Finset.sum_le_sum (fun i _ => ih i (ht.2 i) k)
-
-/-- Two chosen coordinates supply an additive gain in a finite sum. -/
-theorem sum_two_improvements {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (x y : ι → ℕ) {a c : ι} (hne : a≠c) (A B : ℕ)
-    (hrest : ∀ i, i≠a → i≠c → x i≤y i)
-    (ha : x a+A≤y a+B) (hc : x c+A≤y c+B) :
-    (∑ i, x i)+2*A ≤ (∑ i, y i)+2*B := by
-  have hc' : c∈Finset.univ.erase a := by simp [hne.symm]
-  have hx1 := Finset.sum_erase_add Finset.univ x (Finset.mem_univ a)
-  have hy1 := Finset.sum_erase_add Finset.univ y (Finset.mem_univ a)
-  have hx2 := Finset.sum_erase_add (Finset.univ.erase a) x hc'
-  have hy2 := Finset.sum_erase_add (Finset.univ.erase a) y hc'
-  have hrem : (∑ i∈(Finset.univ.erase a).erase c, x i) ≤
-      ∑ i∈(Finset.univ.erase a).erase c, y i := by
-    apply Finset.sum_le_sum
-    intro i hi
-    exact hrest i (Finset.mem_erase.mp (Finset.mem_erase.mp hi).2).1
-      (Finset.mem_erase.mp hi).1
-  omega
-
-theorem Forced.leaf_lower {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) :
-    2^b ≤ t.leafCount := by
-  classical
-  induction F with
-  | leaf => simp [leafCount]
-  | @node b n children roles a c hne ha hc left right ihl ihr =>
-    have hh := sum_two_improvements (fun _ : Fin n => 0)
-      (fun i => (children i).leafCount) hne (2^b) 0
-      (fun _ _ _ => Nat.zero_le _) (by simpa using ihl) (by simpa using ihr)
-    simpa only [Finset.sum_const_zero,zero_add,add_zero,zero_mul,pow_succ,Nat.mul_comm,leafCount] using hh
-
-/-- At every positive nonroot level, each unforced occupied vertex costs
- an extra leaf. This is the exact combinatorial inequality used when
- additional (nonbinary) labels are summed in the correlated-anchor estimate. -/
-theorem Forced.level_extra {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced)
-    (ht : t.occupied) (k : ℕ) :
-    t.levelCount k+2^b ≤ t.leafCount+2^k := by
-  classical
-  induction F generalizing k with
-  | leaf => cases k <;> simp [levelCount,leafCount]
-  | @node b n children roles a c hne ha hc left right ihl ihr =>
-    cases k with
-    | zero =>
-      have hl := (Forced.node a c hne ha hc left right).leaf_lower
-      simpa only [levelCount,pow_zero,Nat.add_comm] using Nat.add_le_add_left hl 1
-    | succ k =>
-      have hh := sum_two_improvements
-        (fun i => (children i).levelCount k) (fun i => (children i).leafCount)
-        hne (2^b) (2^k) (fun i _ _ => levelCount_le_leaves _ (ht.2 i) k)
-        (ihl (ht.2 a) k) (ihr (ht.2 c) k)
-      simpa only [levelCount,leafCount,pow_succ,Nat.mul_comm] using hh
-
-theorem Forced.nonforced_vertices_le_extra_leaves {b : ℕ} {t : OccupiedParityTree b}
-    (F : t.Forced) (ht : t.occupied) (k : ℕ) :
-    t.levelCount k-2^k ≤ t.leafCount-2^b := by
-  have := F.level_extra ht k
-  have := F.leaf_lower
-  omega
-
-end OccupiedParityTree
-end TraceReconstruction
-
-end
-end
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-!
+# Parity
+
+Parity-tree roles, forced leaves, coefficient shapes, and exact unsigned/signed motif expansions.
+-/
+
+open scoped ENNReal
+noncomputable section
+
+namespace TraceReconstruction
+
+/-- Three roles of an active odd child in the manuscript parity expansion:
+constant, unsigned motif, and signed motif. -/
+def roleWeight (χ σ : ℝ) (r : Fin 3) : ℝ :=
+  if r=0 then 1 else if r=1 then -χ else σ
+
+def signedRoleCount {n : ℕ} (r : Fin n → Fin 3) : ℕ :=
+  (Finset.univ.filter fun i => r i=2).card
+
+def activeRoleCount {n : ℕ} (r : Fin n → Fin 3) : ℕ :=
+  (Finset.univ.filter fun i => r i≠0).card
+
+theorem role_product_expansion {n : ℕ} (χ σ : Fin n → ℝ) (τ : ℝ) :
+    (∏ i : Fin n, (1 - χ i + σ i * τ)) =
+      ∑ r : Fin n → Fin 3, (∏ i, roleWeight (χ i) (σ i) (r i)) * τ^signedRoleCount r := by
+  have hp (r : Fin n → Fin 3) : ∏ i, (if r i=2 then τ else (1:ℝ)) = τ^signedRoleCount r := by
+    rw [← Finset.prod_filter]
+    simp only [Finset.prod_const, signedRoleCount]
+  simp_rw [← hp, ← Finset.prod_mul_distrib]
+  rw [← Fintype.prod_sum (fun (i : Fin n) (r : Fin 3) => roleWeight (χ i) (σ i) r * (if r=2 then τ else 1))]
+  apply Finset.prod_congr rfl
+  intro i _
+  simp [Fin.sum_univ_three, roleWeight]
+  ring
+
+theorem odd_role_difference {n : ℕ} (χ σ : Fin n → ℝ) :
+    (∏ i : Fin n, (1 - χ i + σ i)) - (∏ i : Fin n, (1 - χ i - σ i)) =
+      2*∑ r : Fin n → Fin 3,
+        if Odd (signedRoleCount r) then ∏ i, roleWeight (χ i) (σ i) (r i) else 0 := by
+  have hplus := role_product_expansion χ σ 1
+  have hminus := role_product_expansion χ σ (-1)
+  simp only [mul_one, one_pow] at hplus
+  simp only [mul_neg_one, ← sub_eq_add_neg] at hminus
+  rw [hplus,hminus, ← Finset.sum_sub_distrib, Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro r _
+  by_cases hr : Odd (signedRoleCount r)
+  · rw [ite_eq_left hr, hr.neg_one_pow]
+    ring
+  · have he : Even (signedRoleCount r) := Nat.not_odd_iff_even.mp hr
+    rw [ite_eq_right hr, he.neg_one_pow]
+    ring
+
+theorem signedRoleCount_le_active {n : ℕ} (r : Fin n → Fin 3) :
+    signedRoleCount r ≤ activeRoleCount r := by
+  apply Finset.card_le_card
+  intro i hi
+  simp only [Finset.mem_filter, Finset.mem_univ, true_and] at hi ⊢
+  rw [hi]
+  decide
+
+/-- A surviving odd node either has a unique active signed motif (the
+local high-order cancellation case), or at least two active children. -/
+theorem odd_roles_singleton_or_branch {n : ℕ} (r : Fin n → Fin 3)
+    (hodd : Odd (signedRoleCount r)) :
+    (∃ i, r i=2 ∧ ∀ j, j≠i → r j=0) ∨ 2 ≤ activeRoleCount r := by
+  have hpos : 0 < signedRoleCount r := hodd.pos
+  have hle := signedRoleCount_le_active r
+  by_cases h : 2 ≤ activeRoleCount r
+  · exact Or.inr h
+  have ha : activeRoleCount r=1 := by omega
+  obtain ⟨i,hi⟩ := Finset.card_eq_one.mp ha
+  have hmem : ∀ j, r j≠0 ↔ j=i := by
+    intro j
+    have he := Finset.ext_iff.mp hi j
+    simpa only [Finset.mem_filter, Finset.mem_univ, true_and, Finset.mem_singleton] using he
+  have hs : ∃ j, r j=2 := by
+    have hs0 : (Finset.univ.filter fun j => r j=2).Nonempty := Finset.card_pos.mp hpos
+    obtain ⟨j,hj⟩ := hs0
+    exact ⟨j, (Finset.mem_filter.mp hj).2⟩
+  obtain ⟨j,hj⟩ := hs
+  have hij := (hmem j).mp (by rw [hj]; decide)
+  subst j
+  exact Or.inl ⟨i,hj,fun j hji => Classical.byContradiction (fun hn => hji ((hmem j).mp hn))⟩
+
+/-- Finite occupied trees with at least two continuing odd children at
+each positive depth. This records exactly the no-singleton branch of
+the parity argument; the arity and all extra children remain arbitrary. -/
+inductive BranchingTree : ℕ → Type
+  | leaf : BranchingTree 0
+  | node {b : ℕ} (n : ℕ) (children : Fin n → BranchingTree b) (branch : 2 ≤ n) : BranchingTree (b+1)
+
+def BranchingTree.leafCount : {b : ℕ} → BranchingTree b → ℕ
+  | _, .leaf => 1
+  | _, .node _ children _ => ∑ i, (children i).leafCount
+
+def BranchingTree.levelCount : {b : ℕ} → BranchingTree b → ℕ → ℕ
+  | _, .leaf, h => if h=0 then 1 else 0
+  | _, .node _ children _, h => if h=0 then 1 else ∑ i, (children i).levelCount (h-1)
+
+theorem BranchingTree.leafCount_lower {b : ℕ} (t : BranchingTree b) :
+    2^b ≤ t.leafCount := by
+  induction t with
+  | leaf => simp [leafCount]
+  | @node b n children hn ih =>
+    have h := Finset.sum_le_sum (fun i (_ : i ∈ (Finset.univ : Finset (Fin n))) => ih i)
+    simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, smul_eq_mul] at h
+    have hmul := Nat.mul_le_mul_right (2^b) hn
+    simp only [leafCount]
+    rw [pow_succ]
+    omega
+
+
+
+/-- A term in the parity expansion retains the occupied ordered shape,
+all roles, and all extra children, rather than assuming it is binary. -/
+inductive OccupiedParityTree : ℕ → Type
+  | leaf : OccupiedParityTree 0
+  | node {b : ℕ} (n : ℕ) (children : Fin n → OccupiedParityTree b)
+      (roles : Fin n → Fin 3) : OccupiedParityTree (b+1)
+
+def OccupiedParityTree.isOdd : {b : ℕ} → OccupiedParityTree b → Prop
+  | _, .leaf => True
+  | _, .node _ _ roles => Odd (signedRoleCount roles)
+
+def OccupiedParityTree.leafCount : {b : ℕ} → OccupiedParityTree b → ℕ
+  | _, .leaf => 1
+  | _, .node _ children _ => ∑ i, (children i).leafCount
+
+/-- Only odd-role child terms may acquire an active parent multiplier. -/
+def OccupiedParityTree.valid : {b : ℕ} → OccupiedParityTree b → Prop
+  | _, .leaf => True
+  | _, .node _ children roles =>
+      (∀ i, roles i ≠ 0 → (children i).isOdd) ∧ ∀ i, (children i).valid
+
+def OccupiedParityTree.noSingleton : {b : ℕ} → OccupiedParityTree b → Prop
+  | _, .leaf => True
+  | _, .node _ children roles =>
+      (Odd (signedRoleCount roles) → activeRoleCount roles ≠ 1) ∧
+        ∀ i, (children i).noSingleton
+
+/-- The exact forced binary number of selected leaves, with all extra
+occupied vertices present. The hypotheses here are checked properties
+of a parity term, not a statistical or construction theorem. -/
+theorem OccupiedParityTree.forced_leaves {b : ℕ} (t : OccupiedParityTree b)
+    (hv : t.valid) (hn : t.noSingleton) (ho : t.isOdd) : 2^b ≤ t.leafCount := by
+  induction t with
+  | leaf => simp [leafCount]
+  | @node b n children roles ih =>
+    have hod : Odd (signedRoleCount roles) := ho
+    have hpos := hod.pos
+    have hle := signedRoleCount_le_active roles
+    have hne := hn.1 hod
+    have ha : 2 ≤ activeRoleCount roles := by omega
+    have hsum : (∑ i : Fin n, if roles i≠0 then 2^b else 0) ≤
+        ∑ i : Fin n, (children i).leafCount := by
+      apply Finset.sum_le_sum
+      intro i _
+      split_ifs with hi
+      · exact ih i (hv.2 i) (hn.2 i) (hv.1 i hi)
+      · exact Nat.zero_le _
+    have hs : (∑ i : Fin n, if roles i≠0 then 2^b else 0) = activeRoleCount roles * 2^b := by
+      rw [← Finset.sum_filter]
+      simp only [Finset.sum_const, smul_eq_mul, activeRoleCount]
+    rw [hs] at hsum
+    change 2^(b+1) ≤ _
+    rw [pow_succ]
+    exact (show 2^b*2 ≤ activeRoleCount roles*2^b by
+      simpa only [Nat.mul_comm] using Nat.mul_le_mul_right (2^b) ha).trans hsum
+
+end TraceReconstruction
+
+open Filter
+open scoped Topology
+
+
+noncomputable section
+namespace TraceReconstruction
+attribute [local instance] Classical.propDecidable
+
+/-- The literal unsigned/signed motif values at one child index. -/
+def IsMotifValue (χ σ : ℝ) : Prop :=
+  (χ=0 ∧ σ=0) ∨ (χ=1 ∧ (σ=1 ∨ σ= -1))
+
+def IsTypeSign (τ : ℝ) : Prop := τ=1 ∨ τ= -1
+
+theorem motif_child_sign {χ σ τ : ℝ} (hm : IsMotifValue χ σ) (ht : IsTypeSign τ) :
+    IsTypeSign (1-χ+σ*τ) := by
+  rcases hm with ⟨rfl,rfl⟩ | ⟨rfl,hσ⟩
+  · exact Or.inl (by ring)
+  · rcases hσ with rfl | rfl <;> rcases ht with rfl | rfl <;>
+      norm_num [IsTypeSign]
+
+theorem sign_pow_parity {τ : ℝ} (ht : IsTypeSign τ) (n : ℕ) :
+    τ^n = if Odd n then τ else 1 := by
+  rcases ht with rfl | rfl
+  · simp
+  · split_ifs with hn
+    · exact hn.neg_one_pow
+    · exact (Nat.not_odd_iff_even.mp hn).neg_one_pow
+
+/-- Product expansion with an arbitrary three-role scalar per child; the
+only sign variable is the current vertex type. -/
+theorem arbitrary_role_expansion {n : ℕ} (w : Fin n → Fin 3 → ℝ) (τ : ℝ) :
+    (∏ i, ∑ a : Fin 3, w i a * (if a=2 then τ else 1)) =
+      ∑ a : Fin n → Fin 3, (∏ i, w i (a i))*τ^signedRoleCount a := by
+  rw [Finset.prod_univ_sum]
+  simp only [Fintype.piFinset_univ,Finset.prod_mul_distrib]
+  apply Finset.sum_congr rfl
+  intro a _
+  congr 1
+  rw [← Finset.prod_filter]
+  simp [signedRoleCount]
+
+/-- At the last internal level every selected leaf has to be active.
+Role zero has weight zero; unsigned and signed roles are respectively
+χ/2 and -σ/2 in (χ-στ)/2. -/
+def bottomRoleWeight (χ σ : ℝ) (a : Fin 3) : ℝ :=
+  if a=0 then 0 else if a=1 then χ/2 else -σ/2
+
+def internalRoleWeight (odd : Prop) [Decidable odd] (χ σ : ℝ) (a : Fin 3) : ℝ :=
+  if odd then roleWeight χ σ a else if a=0 then 1 else 0
+
+theorem bottomRole_sum (χ σ τ : ℝ) :
+    (∑ a : Fin 3, bottomRoleWeight χ σ a*(if a=2 then τ else 1)) = (χ-σ*τ)/2 := by
+  rw [Fin.sum_univ_three]
+  have h01 : (0:Fin 3) ≠ 1 := by decide
+  have h02 : (0:Fin 3) ≠ 2 := by decide
+  have h12 : (1:Fin 3) ≠ 2 := by decide
+  simp only [bottomRoleWeight,h02,h12,Ne.symm h01,Ne.symm h02,Ne.symm h12,
+    ↓reduceIte,zero_add,mul_one]
+  ring
+
+theorem internalRole_sum (odd : Prop) [Decidable odd] (χ σ τ : ℝ) :
+    (∑ a : Fin 3, internalRoleWeight odd χ σ a*(if a=2 then τ else 1)) =
+      if odd then 1-χ+σ*τ else 1 := by
+  rw [Fin.sum_univ_three]
+  have h01 : (0:Fin 3) ≠ 1 := by decide
+  have h02 : (0:Fin 3) ≠ 2 := by decide
+  have h12 : (1:Fin 3) ≠ 2 := by decide
+  by_cases h : odd <;> simp only [internalRoleWeight,roleWeight,h,h02,h12,
+    Ne.symm h01,Ne.symm h02,Ne.symm h12,↓reduceIte,zero_mul,add_zero,mul_one]
+  ring
+
+/-- An arbitrary occupied ordered subtree, with motif values at its actual
+child labels. Depth zero here means a bottom sibling group, so it already
+has one internal level. No binary-shape assumption is made. -/
+inductive CoefficientShape : ℕ → Type
+  | bottom (n : ℕ) (χ σ : Fin n → ℝ) : CoefficientShape 0
+  | node {b : ℕ} (n : ℕ) (χ σ : Fin n → ℝ)
+      (children : Fin n → CoefficientShape b) : CoefficientShape (b+1)
+
+namespace CoefficientShape
+
+def Admissible : {b : ℕ} → CoefficientShape b → Prop
+  | _, .bottom _ χ σ => ∀ i, IsMotifValue (χ i) (σ i)
+  | _, .node _ χ σ children => (∀ i, IsMotifValue (χ i) (σ i)) ∧ ∀ i, (children i).Admissible
+
+def eval : {b : ℕ} → CoefficientShape b → ℝ → ℝ
+  | _, .bottom _ χ σ, τ => ∏ i, (χ i-σ i*τ)/2
+  | _, .node _ χ σ children, τ => ∏ i, (children i).eval (1-χ i+σ i*τ)
+
+def Choice : {b : ℕ} → CoefficientShape b → Type
+  | _, .bottom n _ _ => Fin n → Fin 3
+  | _, .node n _ _ children => ((i : Fin n) → (children i).Choice) × (Fin n → Fin 3)
+
+noncomputable instance choiceFintype : {b : ℕ} → (t : CoefficientShape b) → Fintype t.Choice
+  | _, .bottom _ _ _ => by unfold Choice; infer_instance
+  | _, .node _ _ _ children => by
+      letI := fun i => choiceFintype (children i)
+      unfold Choice
+      infer_instance
+
+def termTree : {b : ℕ} → (t : CoefficientShape b) → t.Choice → OccupiedParityTree (b+1)
+  | _, .bottom n _ _, a => .node n (fun _ => .leaf) a
+  | _, .node n _ _ children, a => .node n (fun i => (children i).termTree (a.1 i)) a.2
+
+noncomputable def weight : {b : ℕ} → (t : CoefficientShape b) → t.Choice → ℝ
+  | _, .bottom _ χ σ, a => ∏ i, bottomRoleWeight (χ i) (σ i) (a i)
+  | _, .node _ χ σ children, a => ∏ i,
+      (children i).weight (a.1 i) *
+        @internalRoleWeight ((children i).termTree (a.1 i)).isOdd (Classical.propDecidable _)
+          (χ i) (σ i) (a.2 i)
+
+/-- Full finite coefficient expansion, before any orientation average or
+fit-event conditioning. This is the literal algebra behind source
+the correlated-anchor estimate, the parity roles being retained in every term. -/
+theorem expansion {b : ℕ} (t : CoefficientShape b) (ha : t.Admissible)
+    {τ : ℝ} (ht : IsTypeSign τ) :
+    t.eval τ = ∑ a : t.Choice, t.weight a * (if (t.termTree a).isOdd then τ else 1) := by
+  classical
+  induction t generalizing τ with
+  | bottom n χ σ =>
+    have H := arbitrary_role_expansion (fun i => bottomRoleWeight (χ i) (σ i)) τ
+    simp_rw [bottomRole_sum,sign_pow_parity ht] at H
+    trans ∑ a : Fin n → Fin 3, (∏ i, bottomRoleWeight (χ i) (σ i) (a i)) *
+      (if Odd (signedRoleCount a) then τ else 1)
+    · exact H
+    · apply Finset.sum_congr rfl
+      intro a _
+      apply congrArg (fun z => (∏ i, bottomRoleWeight (χ i) (σ i) (a i))*z)
+      change _ = @ite ℝ (Odd (signedRoleCount a)) (Classical.propDecidable _) τ 1
+      split_ifs <;> rfl
+  | @node b n χ σ children ih =>
+    change (∏ i, (children i).eval (1-χ i+σ i*τ)) = _
+    have he (i : Fin n) := ih i (ha.2 i) (motif_child_sign (ha.1 i) ht)
+    simp_rw [he]
+    rw [Finset.prod_univ_sum]
+    simp only [Fintype.piFinset_univ]
+    change (∑ a : (i : Fin n) → (children i).Choice,
+      ∏ i, (children i).weight (a i)*
+        (if ((children i).termTree (a i)).isOdd then 1-χ i+σ i*τ else 1)) =
+      ∑ a : ((i : Fin n) → (children i).Choice) × (Fin n → Fin 3),
+        (∏ i, (children i).weight (a.1 i)*
+          internalRoleWeight ((children i).termTree (a.1 i)).isOdd (χ i) (σ i) (a.2 i))*
+        (@ite ℝ (Odd (signedRoleCount a.2)) (Classical.propDecidable _) τ 1)
+    rw [Fintype.sum_prod_type]
+    apply Finset.sum_congr rfl
+    intro a _
+    have H := arbitrary_role_expansion (fun i =>
+      internalRoleWeight ((children i).termTree (a i)).isOdd (χ i) (σ i)) τ
+    simp_rw [internalRole_sum,sign_pow_parity ht] at H
+    rw [Finset.prod_mul_distrib,H,Finset.mul_sum]
+    apply Finset.sum_congr rfl
+    intro a' _
+    rw [Finset.prod_mul_distrib,mul_assoc]
+    congr 2
+    split_ifs <;> rfl
+
+/-- Nonzero terms automatically satisfy the source active-odd rule. -/
+theorem term_valid {b : ℕ} (t : CoefficientShape b) (a : t.Choice)
+    (h : t.weight a ≠ 0) : (t.termTree a).valid := by
+  classical
+  induction t with
+  | bottom n χ σ => exact ⟨fun _ _ => trivial,fun _ => trivial⟩
+  | @node b n χ σ children ih =>
+    have hp (i : Fin n) : (children i).weight (a.1 i) *
+        internalRoleWeight ((children i).termTree (a.1 i)).isOdd (χ i) (σ i) (a.2 i) ≠ 0 :=
+      (Finset.prod_ne_zero_iff.mp h) i (Finset.mem_univ i)
+    constructor
+    · intro i hai
+      by_contra hn
+      have hz := (mul_ne_zero_iff.mp (hp i)).2
+      simp [internalRoleWeight,hn,hai] at hz
+    · intro i
+      exact ih i (a.1 i) (mul_ne_zero_iff.mp (hp i)).1
+
+/-- The difference of hypotheses consists exactly of odd root terms. -/
+theorem difference_expansion {b : ℕ} (t : CoefficientShape b) (ha : t.Admissible) :
+    t.eval 1-t.eval (-1) = 2*∑ a : t.Choice,
+      if (t.termTree a).isOdd then t.weight a else 0 := by
+  classical
+  rw [expansion t ha (Or.inl rfl),expansion t ha (Or.inr rfl),
+    ← Finset.sum_sub_distrib,Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro a _
+  split_ifs <;> ring
+
+/-- A nonzero surviving term without a singleton odd vertex has the full
+forced binary number of leaves. This follows from the actual expansion,
+not an assumed parity interface. -/
+theorem surviving_leaves {b : ℕ} (t : CoefficientShape b) (a : t.Choice)
+    (hw : t.weight a ≠ 0) (ho : (t.termTree a).isOdd)
+    (hn : (t.termTree a).noSingleton) : 2^(b+1) ≤ (t.termTree a).leafCount :=
+  (t.termTree a).forced_leaves (t.term_valid a hw) hn ho
+
+end CoefficientShape
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+namespace OccupiedParityTree
+
+/-- The actual binary selection sits inside an arbitrary occupied parity
+term, retaining its unselected children rather than assuming binary input. -/
+inductive Forced : {b : ℕ} → OccupiedParityTree b → Type
+  | leaf : Forced .leaf
+  | node {b n : ℕ} {children : Fin n → OccupiedParityTree b} {roles : Fin n → Fin 3}
+      (a c : Fin n) (hne : a≠c) (ha : roles a≠0) (hc : roles c≠0)
+      (left : Forced (children a)) (right : Forced (children c)) :
+      Forced (.node n children roles)
+
+/-- Every surviving nonsingleton term has an embedded binary subtree.
+The selection depends only on the role-marked shape. -/
+theorem forced_nonempty {b : ℕ} (t : OccupiedParityTree b)
+    (hv : t.valid) (hn : t.noSingleton) (ho : t.isOdd) : Nonempty t.Forced := by
+  classical
+  induction t with
+  | leaf => exact ⟨.leaf⟩
+  | @node b n children roles ih =>
+    have hod : Odd (signedRoleCount roles) := ho
+    have ha : 2 ≤ activeRoleCount roles := by
+      have := signedRoleCount_le_active roles
+      have := hod.pos
+      have := hn.1 hod
+      omega
+    obtain ⟨a,ha,c,hc,hne⟩ := Finset.one_lt_card.mp ha
+    have ha' : roles a≠0 := (Finset.mem_filter.mp ha).2
+    have hc' : roles c≠0 := (Finset.mem_filter.mp hc).2
+    obtain ⟨la⟩ := ih a (hv.2 a) (hn.2 a) (hv.1 a ha')
+    obtain ⟨lc⟩ := ih c (hv.2 c) (hn.2 c) (hv.1 c hc')
+    exact ⟨.node a c hne ha' hc' la lc⟩
+
+/-- Occupied shapes have at least one selected leaf below every vertex. -/
+def occupied : {b : ℕ} → OccupiedParityTree b → Prop
+  | _, .leaf => True
+  | _, .node n children _ => 0<n ∧ ∀ i, (children i).occupied
+
+/-- Number of vertices at a given distance below the root. -/
+def levelCount : {b : ℕ} → OccupiedParityTree b → ℕ → ℕ
+  | _, _, 0 => 1
+  | _, .leaf, _+1 => 0
+  | _, .node _ children _, k+1 => ∑ i, (children i).levelCount k
+
+theorem leafCount_pos {b : ℕ} (t : OccupiedParityTree b) (ht : t.occupied) :
+    0<t.leafCount := by
+  induction t with
+  | leaf => exact Nat.zero_lt_one
+  | @node b n children roles ih =>
+    let i : Fin n := ⟨0,ht.1⟩
+    have hp := ih i (ht.2 i)
+    change 0 < ∑ k, (children k).leafCount
+    exact hp.trans_le (Finset.single_le_sum (fun k _ => Nat.zero_le ((children k).leafCount)) (Finset.mem_univ i))
+
+theorem levelCount_le_leaves {b : ℕ} (t : OccupiedParityTree b) (ht : t.occupied)
+    (k : ℕ) : t.levelCount k ≤ t.leafCount := by
+  induction t generalizing k with
+  | leaf => cases k <;> norm_num [levelCount,leafCount]
+  | @node b n children roles ih =>
+    cases k with
+    | zero => exact leafCount_pos (.node n children roles) ht
+    | succ k => exact Finset.sum_le_sum (fun i _ => ih i (ht.2 i) k)
+
+/-- Two chosen coordinates supply an additive gain in a finite sum. -/
+theorem sum_two_improvements {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (x y : ι → ℕ) {a c : ι} (hne : a≠c) (A B : ℕ)
+    (hrest : ∀ i, i≠a → i≠c → x i≤y i)
+    (ha : x a+A≤y a+B) (hc : x c+A≤y c+B) :
+    (∑ i, x i)+2*A ≤ (∑ i, y i)+2*B := by
+  have hc' : c∈Finset.univ.erase a := by simp [hne.symm]
+  have hx1 := Finset.sum_erase_add Finset.univ x (Finset.mem_univ a)
+  have hy1 := Finset.sum_erase_add Finset.univ y (Finset.mem_univ a)
+  have hx2 := Finset.sum_erase_add (Finset.univ.erase a) x hc'
+  have hy2 := Finset.sum_erase_add (Finset.univ.erase a) y hc'
+  have hrem : (∑ i∈(Finset.univ.erase a).erase c, x i) ≤
+      ∑ i∈(Finset.univ.erase a).erase c, y i := by
+    apply Finset.sum_le_sum
+    intro i hi
+    exact hrest i (Finset.mem_erase.mp (Finset.mem_erase.mp hi).2).1
+      (Finset.mem_erase.mp hi).1
+  omega
+
+theorem Forced.leaf_lower {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) :
+    2^b ≤ t.leafCount := by
+  classical
+  induction F with
+  | leaf => simp [leafCount]
+  | @node b n children roles a c hne ha hc left right ihl ihr =>
+    have hh := sum_two_improvements (fun _ : Fin n => 0)
+      (fun i => (children i).leafCount) hne (2^b) 0
+      (fun _ _ _ => Nat.zero_le _) (by simpa using ihl) (by simpa using ihr)
+    simpa only [Finset.sum_const_zero,zero_add,add_zero,zero_mul,pow_succ,Nat.mul_comm,leafCount] using hh
+
+/-- At every positive nonroot level, each unforced occupied vertex costs
+ an extra leaf. This is the exact combinatorial inequality used when
+ additional (nonbinary) labels are summed in the correlated-anchor estimate. -/
+theorem Forced.level_extra {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced)
+    (ht : t.occupied) (k : ℕ) :
+    t.levelCount k+2^b ≤ t.leafCount+2^k := by
+  classical
+  induction F generalizing k with
+  | leaf => cases k <;> simp [levelCount,leafCount]
+  | @node b n children roles a c hne ha hc left right ihl ihr =>
+    cases k with
+    | zero =>
+      have hl := (Forced.node a c hne ha hc left right).leaf_lower
+      simpa only [levelCount,pow_zero,Nat.add_comm] using Nat.add_le_add_left hl 1
+    | succ k =>
+      have hh := sum_two_improvements
+        (fun i => (children i).levelCount k) (fun i => (children i).leafCount)
+        hne (2^b) (2^k) (fun i _ _ => levelCount_le_leaves _ (ht.2 i) k)
+        (ihl (ht.2 a) k) (ihr (ht.2 c) k)
+      simpa only [levelCount,leafCount,pow_succ,Nat.mul_comm] using hh
+
+theorem Forced.nonforced_vertices_le_extra_leaves {b : ℕ} {t : OccupiedParityTree b}
+    (F : t.Forced) (ht : t.occupied) (k : ℕ) :
+    t.levelCount k-2^k ≤ t.leafCount-2^b := by
+  have := F.level_extra ht k
+  have := F.leaf_lower
+  omega
+
+end OccupiedParityTree
+end TraceReconstruction
+
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/TraceReconstruction/Patterns.lean b/lean/OAI/Probability/TraceReconstruction/Patterns.lean
index 035ecec5c5edf09cf2fd951180badae9903eecde..95cf0a4dc37cf945734e5d8af638a1703bc1a2bb 100644
--- a/lean/OAI/Probability/TraceReconstruction/Patterns.lean
+++ b/lean/OAI/Probability/TraceReconstruction/Patterns.lean
@@ -1,735 +1,735 @@
-import OAI.Probability.TraceReconstruction.Differences
-import OAI.Probability.TraceReconstruction.InitialDensity
-
-namespace OAI
-
-/-!
-# Patterns
-
-Pattern serialization, motif norms, flag posteriors, signed rank coefficients, translated binomials, and support geometry.
--/
-
-open scoped ENNReal
-noncomputable section
-
-open Filter
-open scoped Topology
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem seriesL2Sq_partial {α : Type*} [Fintype α] (f : IntSeries)
-    (j : α → ℤ) (hj : Function.Injective j) :
-    (∑ a, f.coeff (j a)^2) ≤ seriesL2Sq f := by
-  classical
-  let A := Finset.univ.image j
-  have he : (∑ a, f.coeff (j a)^2)=∑ z∈A, f.coeff z^2 := by
-    rw [Finset.sum_image]
-    exact fun _ _ _ _ h => hj h
-  rw [he]
-  calc
-    _ ≤ ∑ z∈f.coeff.support ∪ A, f.coeff z^2 :=
-      Finset.sum_le_sum_of_subset_of_nonneg Finset.subset_union_right (fun _ _ _ => sq_nonneg _)
-    _ = _ := by
-      rw [seriesL2Sq_eq_sum]
-      symm
-      apply Finset.sum_subset Finset.subset_union_left
-      intro z _ hz
-      simp [Finsupp.notMem_support_iff.mp hz]
-
-/-- The cancellation estimate after disjoint rank clipping, retaining the
- actual fixed-period kernels and all anchor correlations. -/
-theorem period_pattern_norm {u m l r : ℕ}
-    (hm : 32 ≤ m) (hr : 0 < r) (hl : l ≤ u*m)
-    (off : Fin u → Fin m)
-    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
-    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2)
-    (hsafe : ∀ j : IncreasingTuple u r, SafeTaggedTuple (periodSlot off ∘ j.val))
-    (k : Fin r → GapDifference) {θ : ℝ} (hθ : θ ∈ Set.Ioo 0 1)
-    (hhalf : 1/2 ≤ θ) (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
-    (a : Fin r) (A : IncreasingTuple u r → ℝ) (w : Fin u → ℝ)
-    (hA : ∀ j, A j^2 ≤ w (j.val a)^2)
-    (hw : ∀ h, w h ≠ 0 → (u:ℝ)/3 ≤ 2*h.val+1 ∧ (u:ℝ)/3 ≤ 2*u-(2*h.val+1)) :
-    (∑ v, (∑ j, if PeriodRankRegion m θ j v then
-      A j*positionalRankDifference hl (periodSlot off ∘ j.val) (hsafe j) k v else 0)^2) ≤
-      periodDifferencePrefactor u m r (∑ h, (k h).order) θ *
-        periodCentralSumConstant u r * ∑ h, w h^2 := by
-  have hC : 0 ≤ periodDifferencePrefactor u m r (∑ h, (k h).order) θ := by
-    unfold periodDifferencePrefactor lengthDifferenceConstant
-    positivity
-  calc
-    _ ≤ ∑ j, A j^2*∑ v, positionalRankDifference hl (periodSlot off ∘ j.val) (hsafe j) k v^2 :=
-      period_clipped_square_sum (by omega) hhalf _ A
-    _ ≤ ∑ j, A j^2*(periodDifferencePrefactor u m r (∑ h, (k h).order) θ *
-        ∏ i, (Real.sqrt (coarsePeriodGap j.val i))⁻¹) := by
-      apply Finset.sum_le_sum
-      intro j _
-      exact mul_le_mul_of_nonneg_left
-        (period_positionalRankDifference_bound hm hr hl off hoff hdiff j (hsafe j) k hθ hmean hv)
-        (sq_nonneg _)
-    _ = periodDifferencePrefactor u m r (∑ h, (k h).order) θ *
-        ∑ j, A j^2*∏ i, (Real.sqrt (coarsePeriodGap j.val i))⁻¹ := by
-      rw [Finset.mul_sum]; congr 1; funext j; ring
-    _ ≤ _ := by
-      rw [mul_assoc]
-      exact mul_le_mul_of_nonneg_left (weighted_period_gap_sum a A w hA hw) hC
-
-theorem period_pattern_bounded_norm {u m l r M : ℕ}
-    (hm : 32 ≤ m) (hr : 0 < r) (hl : l ≤ u*m)
-    (off : Fin u → Fin m)
-    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
-    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2)
-    (hsafe : ∀ j : IncreasingTuple u r, SafeTaggedTuple (periodSlot off ∘ j.val))
-    (k : Fin r → GapDifference) {θ : ℝ} (hθ : θ ∈ Set.Ioo 0 1)
-    (hhalf : 1/2 ≤ θ) (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
-    (a : Fin r) (c : Fin (r+1)) (A : IncreasingTuple u r → ℝ) (w : Fin u → ℝ)
-    (hA : ∀ j, A j^2 ≤ w (j.val a)^2)
-    (hsupport : ∀ j, A j ≠ 0 → coarsePeriodGap j.val c ≤ M)
-    (hw : ∀ h, w h ≠ 0 → (u:ℝ)/3 ≤ 2*h.val+1 ∧ (u:ℝ)/3 ≤ 2*u-(2*h.val+1))
-    (hM : (M:ℝ) < (u:ℝ)/(3*(r+1:ℝ))) :
-    (∑ v, (∑ j, if PeriodRankRegion m θ j v then
-      A j*positionalRankDifference hl (periodSlot off ∘ j.val) (hsafe j) k v else 0)^2) ≤
-      periodDifferencePrefactor u m r (∑ h, (k h).order) θ *
-        periodBoundedSumConstant u r M * ∑ h, w h^2 := by
-  have hC : 0 ≤ periodDifferencePrefactor u m r (∑ h, (k h).order) θ := by
-    unfold periodDifferencePrefactor lengthDifferenceConstant
-    positivity
-  calc
-    _ ≤ ∑ j, A j^2*∑ v, positionalRankDifference hl (periodSlot off ∘ j.val) (hsafe j) k v^2 :=
-      period_clipped_square_sum (by omega) hhalf _ A
-    _ ≤ ∑ j, A j^2*(periodDifferencePrefactor u m r (∑ h, (k h).order) θ *
-        ∏ i, (Real.sqrt (coarsePeriodGap j.val i))⁻¹) := by
-      apply Finset.sum_le_sum
-      intro j _
-      exact mul_le_mul_of_nonneg_left
-        (period_positionalRankDifference_bound hm hr hl off hoff hdiff j (hsafe j) k hθ hmean hv)
-        (sq_nonneg _)
-    _ = periodDifferencePrefactor u m r (∑ h, (k h).order) θ *
-        ∑ j, A j^2*∏ i, (Real.sqrt (coarsePeriodGap j.val i))⁻¹ := by
-      rw [Finset.mul_sum]; congr 1; funext j; ring
-    _ ≤ _ := by
-      rw [mul_assoc]
-      exact mul_le_mul_of_nonneg_left (weighted_period_bounded_gap_sum a c A w hA hsupport hw hM) hC
-
-/-- Two involved motif positions force a short *fixed* internal coarse gap.
- The chosen gap is independent of the actual labels and of the offset law. -/
-theorem motif_pattern_bounded_gap {u r : ℕ} (π : IntSeries) (s : ℕ)
-    (j : IncreasingTuple u (r+1)) (k : Fin (r+1) → GapDifference)
-    (i : Fin r) (b : Fin (r+1)) (hib : i.castSucc < b)
-    (hi : k i.castSucc ≠ .unchanged) (hb : k b ≠ .unchanged)
-    (hA : motifPatternMultiplier π s (fun a => (j.val a).val) k ≠ 0) :
-    coarsePeriodGap j.val i.castSucc.succ ≤ 2*(2^s) := by
-  have hs := motif_multiplier_bounded_separation π s (fun a => (j.val a).val) k hA i.castSucc b hi hb
-  have hj1 := j.property (show i.castSucc < i.succ by exact Fin.castSucc_lt_succ)
-  have hj2 := j.property.monotone (show i.succ ≤ b by simpa only [Fin.le_iff_val_le_val,Fin.lt_def,
-    Fin.val_succ,Fin.val_castSucc] using (Nat.succ_le_of_lt hib))
-  have h1 : (j.val i.castSucc).val < (j.val i.succ).val := hj1
-  have h2 : (j.val i.succ).val ≤ (j.val b).val := hj2
-  have hh := (abs_lt.mp hs).1
-  rw [coarsePeriodGap_middle]
-  omega
-
-end TraceReconstruction
-
-
-namespace TraceReconstruction
-
-theorem centered_window_floor_fit (P T L t : ℕ) (Y : ℤ)
-    (hP : 2*T ≤ P) (hT : 16 ≤ T) (heven : Even T)
-    (hL : 8*L ≤ T) (hY : |8*Y| ≤ (T:ℤ))
-    (ht : (t:ℤ)=(((P-T)/2:ℕ):ℤ)+Y) :
-    t+T ≤ P ∧ t ≤ P/2 ∧ P/2+L ≤ t+T := by
-  obtain ⟨k,hk⟩ := heven
-  have hy := abs_le.mp hY
-  omega
-
-theorem centered_window_floor_anchor (P T : ℕ) (hTP : T ≤ P) (heven : Even T) :
-    P/2-(P-T)/2=T/2 := by
-  obtain ⟨k,hk⟩ := heven
-  omega
-
-theorem atomString_append (n m : ℕ) (w : Bool → List Bool) (θ : Fin (n+m) → Bool) :
-    atomString (n+m) w θ =
-      atomString n w (fun i => θ (i.castLE (Nat.le_add_right n m))) ++
-      atomString m w (fun i => θ (i.natAdd n)) := by
-  simp only [atomString,List.ofFn_add,List.flatten_append]
-
-theorem atomString_three (n m k : ℕ) (w : Bool → List Bool) (θ : Fin (n+m+k) → Bool) :
-    atomString (n+m+k) w θ =
-      atomString n w (fun i => θ ⟨i.val,by omega⟩) ++
-      atomString m w (fun i => θ ⟨n+i.val,by omega⟩) ++
-      atomString k w (fun i => θ ⟨n+m+i.val,by omega⟩) := by
-  rw [atomString_append,atomString_append]
-  rfl
-
-/-- A full deterministic template equals a pure prefix, the actual translated
- window, and a pure suffix. This equality is in unmarked binary strings. -/
-theorem template_word_window (t T z a L : ℕ) (sign : Fin L → Bool) (τ : Bool)
-    (w : Bool → List Bool) (hleft : t ≤ a) (hright : a+L ≤ t+T) :
-    atomString (t+T+z) w (templateType (t+T+z) a L sign τ) =
-      atomString t w (fun _ => false) ++
-      atomString T w (templateType T (a-t) L sign τ) ++
-      atomString z w (fun _ => false) := by
-  rw [atomString_three]
-  congr 2
-  · congr 1
-    funext i
-    exact templateType_outside_window sign τ hleft hright ⟨i.val,by omega⟩ (Or.inl i.isLt)
-  · congr 1
-    funext i
-    exact templateType_inside_window sign τ (by omega) hleft i
-  · funext i
-    exact templateType_outside_window sign τ hleft hright ⟨t+T+i.val,by omega⟩ (Or.inr (by dsimp; omega))
-
-def pureAtomRun (w : List Bool) (n : ℕ) : List Bool :=
-  (List.replicate n w).flatten
-
-theorem atomString_constant (n : ℕ) (w : Bool → List Bool) (b : Bool) :
-    atomString n w (fun _ => b)=pureAtomRun (w b) n := by
-  simp [atomString,pureAtomRun]
-
-theorem pureAtomRun_add (w : List Bool) (n m : ℕ) :
-    pureAtomRun w (n+m)=pureAtomRun w n ++ pureAtomRun w m := by
-  simp only [pureAtomRun,List.replicate_add,List.flatten_append]
-
-/-- Splitting or regrouping a pure region into reported chunks changes no
- unmarked string. In particular arbitrary fixed chunk IDs are harmless. -/
-theorem pureAtomRun_partition (w : List Bool) (ns : List ℕ) :
-    (ns.map (pureAtomRun w)).flatten=pureAtomRun w ns.sum := by
-  induction ns with
-  | nil => simp [pureAtomRun]
-  | cons n ns ih => simp [ih,pureAtomRun_add]
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction.FiniteLaw
-open scoped Classical
-variable {α : Type*} [Fintype α]
-
-/-- Exact orientation law after a report: only invisible bits remain random. -/
-def flagPosterior (d : Option Bool) : FiniteLaw Bool where
-  mass b := match d with
-    | none => 1/2
-    | some c => if b=c then 1 else 0
-  nonneg := by
-    intro b
-    cases d with
-    | none => norm_num
-    | some c => dsimp; split_ifs <;> norm_num
-  total := by cases d with
-    | none => norm_num [Fintype.sum_bool]
-    | some c => simp
-
-def pairedReportMass (P Q H : FiniteLaw α) (ω : ℝ)
-    (z : Option Bool × (α × α)) : ℝ :=
-  ∑ b : Bool, (1/2 : ℝ)*pairedFlagMass P Q H ω b z
-
-/-- Pointwise Bayes factorization, valid even at zero probability reports. -/
-theorem pairedFlag_bayes (P Q H : FiniteLaw α) (ω : ℝ)
-    (z : Option Bool × (α × α)) (b : Bool) :
-    (1/2 : ℝ)*pairedFlagMass P Q H ω b z =
-      pairedReportMass P Q H ω z * (flagPosterior z.1).mass b := by
-  rcases z with ⟨d,z⟩
-  cases d with
-  | none => simp [pairedReportMass,pairedFlagMass,flagPosterior]; ring
-  | some c => cases b <;> cases c <;>
-      simp [pairedReportMass,pairedFlagMass,flagPosterior]
-
-theorem pairedReportMass_nonneg (P Q H : FiniteLaw α) {ω : ℝ} (hω : 0 ≤ ω)
-    (hP : ∀ a, ω*H.mass a ≤ P.mass a) (hQ : ∀ a, ω*H.mass a ≤ Q.mass a)
-    (z : Option Bool × (α × α)) : 0 ≤ pairedReportMass P Q H ω z := by
-  apply Finset.sum_nonneg
-  intro b _
-  exact mul_nonneg (by norm_num) (pairedFlagMass_nonneg P Q H hω hP hQ b z)
-
-theorem pairedReportMass_total (P Q H : FiniteLaw α) (ω : ℝ) :
-    (∑ z, pairedReportMass P Q H ω z)=1 := by
-  unfold pairedReportMass
-  rw [Finset.sum_comm]
-  simp_rw [← Finset.mul_sum,pairedFlagMass_sum]
-  norm_num [Fintype.sum_bool]
-
-def pairedReport (P Q H : FiniteLaw α) {ω : ℝ} (hω : 0 ≤ ω)
-    (hP : ∀ a, ω*H.mass a ≤ P.mass a) (hQ : ∀ a, ω*H.mass a ≤ Q.mass a) :
-    FiniteLaw (Option Bool × (α × α)) :=
-  ⟨pairedReportMass P Q H ω, pairedReportMass_nonneg P Q H hω hP hQ,
-    pairedReportMass_total P Q H ω⟩
-
-/-- Conditioning solely on exact reports does not correlate mechanisms.
- This is the actual joint-mass factorization, not an independence hypothesis. -/
-theorem pairedFlags_product_bayes {ι : Type*} [Fintype ι]
-    (P Q H : ι → FiniteLaw α) (ω : ι → ℝ)
-    (z : ι → Option Bool × (α × α)) (b : ι → Bool) :
-    (∏ i, (1/2 : ℝ)*pairedFlagMass (P i) (Q i) (H i) (ω i) (b i) (z i)) =
-      (∏ i, pairedReportMass (P i) (Q i) (H i) (ω i) (z i)) *
-        ∏ i, (flagPosterior (z i).1).mass (b i) := by
-  simp_rw [pairedFlag_bayes]
-  exact Finset.prod_mul_distrib
-
-theorem pairedFlags_report_marginal {ι : Type*} [Fintype ι]
-    (P Q H : ι → FiniteLaw α) (ω : ι → ℝ)
-    (z : ι → Option Bool × (α × α)) :
-    (∑ b : ι → Bool, ∏ i, (1/2 : ℝ)*pairedFlagMass (P i) (Q i) (H i) (ω i) (b i) (z i)) =
-      ∏ i, pairedReportMass (P i) (Q i) (H i) (ω i) (z i) := by
-  exact (Fintype.prod_sum (fun i b => (1/2 : ℝ)*pairedFlagMass (P i) (Q i) (H i) (ω i) b (z i))).symm
-
-end TraceReconstruction.FiniteLaw
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-def bitIndex (b : Bool) : ℕ := if b then 1 else 0
-
-def posteriorBitSeries (d : Option Bool) : IntSeries :=
-  ∑ b : Bool, AddMonoidAlgebra.single (bitIndex b : ℤ) ((FiniteLaw.flagPosterior d).mass b)
-
-def visibleOnes (d : Option Bool) : ℕ := if d=some true then 1 else 0
-
-def hiddenTrial (d : Option Bool) : ℕ := if d=none then 1 else 0
-
-theorem posteriorBitSeries_factor (d : Option Bool) :
-    posteriorBitSeries d = AddMonoidAlgebra.single (visibleOnes d : ℤ) 1 * fairSeries (hiddenTrial d) := by
-  cases d with
-  | none => simp [posteriorBitSeries,FiniteLaw.flagPosterior,bitIndex,visibleOnes,hiddenTrial,fairSeries,
-      add_comm]
-  | some b => cases b <;> simp [posteriorBitSeries,FiniteLaw.flagPosterior,bitIndex,visibleOnes,hiddenTrial,
-      fairSeries]
-
-/-- The count distribution of the literal posterior orientation bits is
- a translated fair binomial, including reports of probability zero. -/
-theorem posterior_count_series {ι : Type*} [Fintype ι] (d : ι → Option Bool) :
-    (∑ b : ι → Bool, AddMonoidAlgebra.single (∑ i, (bitIndex (b i):ℤ))
-      (∏ i, (FiniteLaw.flagPosterior (d i)).mass (b i))) =
-      AddMonoidAlgebra.single (∑ i, (visibleOnes (d i):ℤ)) 1 *
-        fairSeries (∑ i, hiddenTrial (d i)) := by
-  have he : (∑ b : ι → Bool, AddMonoidAlgebra.single (∑ i, (bitIndex (b i):ℤ))
-      (∏ i, (FiniteLaw.flagPosterior (d i)).mass (b i))) = ∏ i, posteriorBitSeries (d i) := by
-    rw [show (∏ i, posteriorBitSeries (d i)) =
-      ∏ i, ∑ b : Bool, AddMonoidAlgebra.single (bitIndex b : ℤ) ((FiniteLaw.flagPosterior (d i)).mass b) from rfl]
-    rw [Fintype.prod_sum]
-    apply Finset.sum_congr rfl
-    intro b _
-    rw [AddMonoidAlgebra.prod_single]
-  rw [he]
-  simp_rw [posteriorBitSeries_factor]
-  rw [Finset.prod_mul_distrib,AddMonoidAlgebra.prod_single]
-  simp only [Finset.prod_const_one]
-  congr 1
-  unfold fairSeries
-  exact Finset.prod_pow_eq_pow_sum _ _ _
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem geomSeries_coeff (k : ℕ) (z : ℤ) :
-    (geomSeries k).coeff z = if 0 ≤ z ∧ z < k then 1 else 0 := by
-  unfold geomSeries
-  simp only [AddMonoidAlgebra.single_pow,one_pow,nsmul_eq_mul,mul_one,
-    AddMonoidAlgebra.coeff_sum,Finsupp.finsetSum_apply,AddMonoidAlgebra.coeff_single,Finsupp.single_apply]
-  by_cases hz : 0 ≤ z ∧ z < k
-  · obtain ⟨a,rfl⟩ := Int.eq_ofNat_of_zero_le hz.1
-    rw [ite_eq_left hz,Finset.sum_eq_single a]
-    · simp
-    · intro b _ hb
-      exact ite_eq_right (by exact_mod_cast hb)
-    · intro ha
-      have ha' : a < k := by exact_mod_cast hz.2
-      exact (ha (Finset.mem_range.mpr ha')).elim
-  · rw [ite_eq_right hz]
-    apply Finset.sum_eq_zero
-    intro a ha
-    apply ite_eq_right
-    intro haz
-    subst z
-    exact hz ⟨by positivity,by exact_mod_cast Finset.mem_range.mp ha⟩
-
-theorem motif_unsigned_convolution (π : IntSeries) (s : ℕ) (z : ℤ) :
-    (∑ X ∈ π.coeff.support, π.coeff X*motifIndicator s (z-X)) =
-      (geomSeries (2^s)*π).coeff z := by
-  rw [mul_comm,AddMonoidAlgebra.coeff_mul_apply_left,Finsupp.sum]
-  apply Finset.sum_congr rfl
-  intro X _
-  rw [geomSeries_coeff]
-  simp only [motifIndicator,sub_eq_neg_add]
-
-theorem motifPatternMultiplier_sq_le {r : ℕ} (π : IntSeries)
-    (hπ : ∀ X, 0 ≤ π.coeff X) (s : ℕ) (j : Fin r → ℤ)
-    (k : Fin r → GapDifference) (h : Fin r) (hk : k h ≠ .unchanged) :
-    (motifPatternMultiplier π s j k)^2 ≤ ((geomSeries (2^s)*π).coeff (j h))^2 := by
-  have ha := motifPatternMultiplier_abs_le π hπ s j k h hk
-  rw [motif_unsigned_convolution] at ha
-  have hh := pow_le_pow_left₀ (abs_nonneg (motifPatternMultiplier π s j k)) ha 2
-  simpa only [sq_abs] using hh
-
-theorem unsigned_motif_energy (π : IntSeries) (s : ℕ) :
-    seriesL2Sq (geomSeries (2^s)*π) ≤ ((2:ℝ)^s)^2 * seriesL2Sq π := by
-  refine (seriesL2Sq_mul_le _ _).trans ?_
-  apply mul_le_mul_of_nonneg_right _ (seriesL2Sq_nonneg _)
-  exact pow_le_pow_left₀ (seriesL1_nonneg _) (by simpa using geomSeries_L1_le (2^s)) 2
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-
-/-- Elementary chi-square upper bound on the Bernoulli relative entropy.
- No Stirling estimate is used in the subsequent count overlap. -/
-theorem bernoulli_entropy_upper {t p : ℝ} (ht : t ∈ Set.Ioo 0 1) (hp : p ∈ Set.Ioo 0 1) :
-    t*Real.log (t/p)+(1-t)*Real.log ((1-t)/(1-p)) ≤
-      (t-p)^2/(p*(1-p)) := by
-  have h1 := mul_le_mul_of_nonneg_left (Real.log_le_sub_one_of_pos (div_pos ht.1 hp.1)) ht.1.le
-  have h2 := mul_le_mul_of_nonneg_left
-    (Real.log_le_sub_one_of_pos (div_pos (sub_pos.mpr ht.2) (sub_pos.mpr hp.2)))
-    (sub_nonneg.mpr ht.2.le)
-  have hid : t*(t/p-1)+(1-t)*((1-t)/(1-p)-1)=(t-p)^2/(p*(1-p)) := by
-    field_simp [hp.1.ne', (sub_pos.mpr hp.2).ne']
-    ring
-  linarith
-
-/-- Exact likelihood change from the mean-matched Bernoulli law. -/
-theorem binomialMass_likelihood (n l : ℕ) {t p : ℝ}
-    (ht : t ∈ Set.Ioo 0 1) (hp : p ∈ Set.Ioo 0 1) :
-    binomialMass n p l = binomialMass n t l *
-      Real.exp (-((l:ℝ)*Real.log (t/p)+((n-l:ℕ):ℝ)*Real.log ((1-t)/(1-p)))) := by
-  have h1 : Real.exp (-((l:ℝ)*Real.log (t/p)))=(p/t)^l := by
-    rw [show -((l:ℝ)*Real.log (t/p))=(l:ℝ)*Real.log (p/t) by
-      rw [Real.log_div ht.1.ne' hp.1.ne',Real.log_div hp.1.ne' ht.1.ne']; ring,
-      Real.exp_nat_mul,Real.exp_log (div_pos hp.1 ht.1)]
-  have h2 : Real.exp (-(((n-l:ℕ):ℝ)*Real.log ((1-t)/(1-p))))=((1-p)/(1-t))^(n-l) := by
-    rw [show -(((n-l:ℕ):ℝ)*Real.log ((1-t)/(1-p)))=
-      ((n-l:ℕ):ℝ)*Real.log ((1-p)/(1-t)) by
-      rw [Real.log_div (sub_pos.mpr ht.2).ne' (sub_pos.mpr hp.2).ne',
-        Real.log_div (sub_pos.mpr hp.2).ne' (sub_pos.mpr ht.2).ne']; ring,
-      Real.exp_nat_mul,Real.exp_log (div_pos (sub_pos.mpr hp.2) (sub_pos.mpr ht.2))]
-  rw [neg_add,Real.exp_add,h1,h2]
-  unfold binomialMass
-  rw [div_pow,div_pow]
-  field_simp [ht.1.ne', (sub_pos.mpr ht.2).ne']
-
-/-- Pointwise lower mass at every interior integer, with mean-matched
- variance at least one. It applies uniformly even as deletion vanishes. -/
-theorem binomialMass_near_mean_lower (n l : ℕ) {t p : ℝ}
-    (ht : t ∈ Set.Ioo 0 1) (hp : p ∈ Set.Ioo 0 1)
-    (hl : l ≤ n) (hmean : (n:ℝ)*t=l) (hv : 1 ≤ (n:ℝ)*t*(1-t)) :
-    Real.exp (-((l:ℝ)-(n:ℝ)*p)^2/((n:ℝ)*p*(1-p))) /
-      (12*Real.sqrt ((n:ℝ)*t*(1-t))) ≤ binomialMass n p l := by
-  have hn : (0:ℝ)<n := by nlinarith [mul_pos ht.1 (sub_pos.mpr ht.2)]
-  have hnl : ((n-l:ℕ):ℝ)=(n:ℝ)-(l:ℝ) := by exact Nat.cast_sub hl
-  have he : (l:ℝ)*Real.log (t/p)+((n-l:ℕ):ℝ)*Real.log ((1-t)/(1-p)) ≤
-      ((l:ℝ)-(n:ℝ)*p)^2/((n:ℝ)*p*(1-p)) := by
-    have H := mul_le_mul_of_nonneg_left (bernoulli_entropy_upper ht hp) hn.le
-    calc
-      _ = (n:ℝ)*(t*Real.log (t/p)+(1-t)*Real.log ((1-t)/(1-p))) := by
-        rw [hnl,← hmean]; ring
-      _ ≤ (n:ℝ)*((t-p)^2/(p*(1-p))) := H
-      _ = _ := by
-        rw [← hmean]
-        field_simp [hn.ne', hp.1.ne', (sub_pos.mpr hp.2).ne']
-  rw [binomialMass_likelihood n l ht hp]
-  calc
-    _ = (1/(12*Real.sqrt ((n:ℝ)*t*(1-t)))) *
-      Real.exp (-((l:ℝ)-(n:ℝ)*p)^2/((n:ℝ)*p*(1-p))) := by ring
-    _ ≤ (1/(12*Real.sqrt ((n:ℝ)*t*(1-t)))) *
-      Real.exp (-((l:ℝ)*Real.log (t/p)+((n-l:ℕ):ℝ)*Real.log ((1-t)/(1-p)))) := by
-      apply mul_le_mul_of_nonneg_left _ (by positivity)
-      apply Real.exp_le_exp.mpr
-      simpa only [neg_div] using neg_le_neg he
-    _ ≤ _ := mul_le_mul_of_nonneg_right (binomialMass_mean_lower n l ht hl hmean hv)
-      (Real.exp_nonneg _)
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-def signedPeriodDefects {u m : ℕ} (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
-    (s : ℕ) (X : ℤ) (τ : ℝ) : Fin u → Fin (u*m) :=
-  shiftedTag (periodSlot (fun _ => off)) hsafe
-    (fun j => motifShift (motifIndicator s ((j.val:ℤ)-X)) ((prouhetSeries s).coeff ((j.val:ℤ)-X)) τ)
-
-theorem signedPeriodDefects_strictMono {u m : ℕ} (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
-    (s : ℕ) (X : ℤ) (τ : ℝ) : StrictMono (signedPeriodDefects off hsafe s X τ) :=
-  shiftedTag_strictMono _ _ _
-
-theorem SafeTaggedTuple.comp {n u r : ℕ} {p : Fin u → Fin n}
-    (hp : SafeTaggedTuple p) (j : Fin r → Fin u) (hj : StrictMono j) :
-    SafeTaggedTuple (p ∘ j) :=
-  ⟨fun i => hp.1 (j i),fun i k hik => hp.2 (j i) (j k) (hj hik)⟩
-
-theorem shiftedTag_comp {n u r : ℕ} (p : Fin u → Fin n) (hp : SafeTaggedTuple p)
-    (a : Fin u → Fin 3) (j : Fin r → Fin u) (hj : StrictMono j) :
-    shiftedTag p hp a ∘ j = shiftedTag (p ∘ j) (hp.comp j hj) (a ∘ j) := by
-  funext i
-  apply Fin.ext
-  rfl
-
-/-- Exact degree-r coefficient difference of the actual randomized motif
- in the marked central run, prior to any rank clipping. -/
-def signedRankCoefficient {u m l r : ℕ} (hl : l ≤ u*m) (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
-    (π : IntSeries) (s : ℕ) (v : IncreasingTuple l r) : ℝ :=
-  ∑ X ∈ π.coeff.support, π.coeff X *
-    (rankDefectCoefficient hl (signedPeriodDefects off hsafe s X 1) v -
-     rankDefectCoefficient hl (signedPeriodDefects off hsafe s X (-1)) v)
-
-theorem signedRankCoefficient_expansion {u m l r : ℕ} (hl : l ≤ u*m) (off : Fin m)
-    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
-    (π : IntSeries) (s : ℕ) (v : IncreasingTuple l r) :
-    signedRankCoefficient hl off hsafe π s v =
-      ∑ k : Fin r → GapDifference, if Odd (firstDifferenceCount k) then
-        2*∑ j : IncreasingTuple u r,
-          motifPatternMultiplier π s (fun i => (j.val i).val) k *
-            positionalRankDifference hl (periodSlot (fun _ => off) ∘ j.val)
-              (hsafe.comp j.val j.property) k v else 0 := by
-  unfold signedRankCoefficient
-  simp_rw [rankDefectCoefficient_eq _ _ (signedPeriodDefects_strictMono _ _ _ _ _),
-    ← Finset.sum_sub_distrib,Finset.mul_sum]
-  rw [Finset.sum_comm]
-  have he (j : IncreasingTuple u r) :
-      (∑ X ∈ π.coeff.support, π.coeff X *
-        (subsetRankKernel hl (signedPeriodDefects off hsafe s X 1 ∘ j.val) v.val -
-         subsetRankKernel hl (signedPeriodDefects off hsafe s X (-1) ∘ j.val) v.val)) =
-      ∑ k : Fin r → GapDifference,
-        (if Odd (firstDifferenceCount k) then 2*motifPatternMultiplier π s (fun i => (j.val i).val) k else 0)*
-          positionalRankDifference hl (periodSlot (fun _ => off) ∘ j.val)
-            (hsafe.comp j.val j.property) k v := by
-    have htag (X : ℤ) (τ : ℝ) :
-        signedPeriodDefects off hsafe s X τ ∘ j.val =
-        shiftedTag (periodSlot (fun _ => off) ∘ j.val) (hsafe.comp j.val j.property)
-          (fun i => motifShift (motifIndicator s ((j.val i).val-X))
-            ((prouhetSeries s).coeff ((j.val i).val-X)) τ) := by
-      funext i; apply Fin.ext; rfl
-    simp_rw [htag]
-    simpa only [positionalRankDifference]
-      using mixed_signed_shift_expansion π s (fun i => (j.val i).val)
-        (fun a => subsetRankKernel hl
-          (shiftedTag (periodSlot (fun _ => off) ∘ j.val) (hsafe.comp j.val j.property) a) v.val)
-  simp_rw [he]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro k _
-  split_ifs with hk
-  · apply Finset.sum_congr rfl
-    intro j _; ring
-  · simp only [zero_mul,Finset.sum_const_zero]
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-def translateSeries (c : ℤ) (f : IntSeries) : IntSeries :=
-  AddMonoidAlgebra.single c 1*f
-
-theorem translateSeries_coeff (c : ℤ) (f : IntSeries) (z : ℤ) :
-    (translateSeries c f).coeff z = f.coeff (z-c) := by
-  rw [translateSeries,AddMonoidAlgebra.coeff_single_mul_apply,one_mul]
-  congr 1; omega
-
-theorem translateSeries_L2Sq (c : ℤ) (f : IntSeries) :
-    seriesL2Sq (translateSeries c f)=seriesL2Sq f := by
-  unfold seriesL2Sq
-  simp_rw [translateSeries_coeff]
-  exact (Equiv.subRight c).tsum_eq (fun z => f.coeff z^2)
-
-theorem translateSeries_L1 (c : ℤ) (f : IntSeries) :
-    seriesL1 (translateSeries c f)=seriesL1 f := by
-  unfold seriesL1
-  simp_rw [translateSeries_coeff]
-  exact (Equiv.subRight c).tsum_eq (fun z => |f.coeff z|)
-
-theorem mul_translateSeries (c : ℤ) (f g : IntSeries) :
-    f*translateSeries c g=translateSeries c (f*g) := by
-  unfold translateSeries
-  ring
-
-theorem fairSeries_support (g : ℕ) (z : ℤ) (hz : (fairSeries g).coeff z ≠ 0) :
-    0≤z ∧ z≤g := by
-  have h0 : 0≤z := by
-    by_contra h
-    exact hz (fairSeries_coeff_neg g z (by omega))
-  obtain ⟨k,rfl⟩ := Int.eq_ofNat_of_zero_le h0
-  refine ⟨by positivity,?_⟩
-  by_contra hk
-  rw [fairSeries_coeff_nat,fairBinomial,Nat.choose_eq_zero_of_lt (by omega)] at hz
-  simp at hz
-
-theorem fairSeries_reflect (g : ℕ) (z : ℤ) :
-    (fairSeries g).coeff ((g:ℤ)-z)=(fairSeries g).coeff z := by
-  by_cases hz : 0≤z ∧ z≤g
-  · obtain ⟨k,rfl⟩ := Int.eq_ofNat_of_zero_le hz.1
-    have hk : k≤g := by exact_mod_cast hz.2
-    rw [← Nat.cast_sub hk,fairSeries_coeff_nat,fairSeries_coeff_nat]
-    simp only [fairBinomial,Nat.choose_symm hk]
-  · have h1 : (fairSeries g).coeff z=0 := by
-      by_contra h; exact hz (fairSeries_support g z h)
-    have h2 : (fairSeries g).coeff ((g:ℤ)-z)=0 := by
-      by_contra h
-      have hh := fairSeries_support g ((g:ℤ)-z) h
-      exact hz ⟨by omega,by omega⟩
-    rw [h1,h2]
-
-/-- Reflection in the posterior count is exactly an ordinary translated
- fair binomial; it does not change the smoothing channel. -/
-theorem affine_reflected_binomial (g : ℕ) (c z : ℤ) :
-    (fairSeries g).coeff (c-z)=
-      (translateSeries (c-g) (fairSeries g)).coeff z := by
-  rw [translateSeries_coeff,← fairSeries_reflect g (c-z)]
-  congr 1; omega
-
-theorem translated_binomial_nonneg (g : ℕ) (c z : ℤ) :
-    0≤(translateSeries c (fairSeries g)).coeff z := by
-  rw [translateSeries_coeff]
-  exact fairSeries_nonneg _ _
-
-theorem translated_binomial_support (g : ℕ) (c z : ℤ)
-    (hz : (translateSeries c (fairSeries g)).coeff z ≠ 0) :
-    c≤z ∧ z≤c+g := by
-  rw [translateSeries_coeff] at hz
-  have h := fairSeries_support g (z-c) hz
-  omega
-
-theorem translated_binomial_smoothing (g s : ℕ) (c : ℤ) :
-    seriesL2Sq (prouhetSeries s*translateSeries c (fairSeries g)) ≤
-      ((2:ℝ)^(s*s))^2*(((2*binomialBound (g/(s+1)))^s)^2*binomialBound (g/(s+1))) := by
-  rw [mul_translateSeries,translateSeries_L2Sq]
-  exact (signed_binomial_smoothing g s).2
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem linearPattern_order {r : ℕ} (a : Fin r) :
-    (∑ h, ((linearPattern a) h).order)=1 := by
-  classical
-  have he (h : Fin r) : ((linearPattern a) h).order=if h=a then 1 else 0 := by
-    by_cases hh : h=a <;> simp [linearPattern,hh,GapDifference.order]
-  simp_rw [he]
-  simp
-
-theorem first_second_counts_linear {r : ℕ} (k : Fin r → GapDifference)
-    (h1 : firstDifferenceCount k=1) (h2 : secondDifferenceCount k=0) :
-    ∃ a : Fin r, k=linearPattern a := by
-  classical
-  obtain ⟨a,ha⟩ := Finset.card_eq_one.mp h1
-  refine ⟨a,funext fun b => ?_⟩
-  have hfirst : k b = .first ↔ b=a := by
-    have hh := congrArg (fun A : Finset (Fin r) => b∈A) ha
-    simpa only [Finset.mem_filter,Finset.mem_univ,true_and,Finset.mem_singleton] using (iff_of_eq hh)
-  have hsecond : k b ≠ .second := by
-    have hh := Finset.card_eq_zero.mp h2
-    intro hb
-    have hmem : b∈(Finset.univ.filter fun b => k b=.second) := by simp [hb]
-    rw [hh] at hmem
-    exact Finset.notMem_empty _ hmem
-  by_cases hb : b=a
-  · have hf := hfirst.mpr hb
-    simpa [linearPattern,hb] using hf
-  · have hf := mt hfirst.mp hb
-    cases hk : k b <;> simp_all [linearPattern]
-
-theorem odd_pattern_pair {r : ℕ} (k : Fin r → GapDifference)
-    (hk : Odd (firstDifferenceCount k)) (hnl : ¬∃a, k=linearPattern a) :
-    ∃ a b : Fin r, a<b ∧ k a≠.unchanged ∧ k b≠.unchanged := by
-  classical
-  have hp : 0<firstDifferenceCount k := hk.pos
-  obtain ⟨a,ha⟩ := Finset.card_pos.mp hp
-  have ha' : k a=.first := (Finset.mem_filter.mp ha).2
-  by_contra hh
-  push Not at hh
-  apply hnl
-  refine ⟨a,funext fun b => ?_⟩
-  by_cases hb : b=a
-  · subst b; simp [linearPattern,ha']
-  · have hba : k b=.unchanged := by
-      rcases lt_or_gt_of_ne hb with hba|hab
-      · by_contra hb'
-        exact (show k a≠.unchanged by simp [ha']) (hh b a hba hb')
-      · exact hh a b hab (by simp [ha'])
-    simp [linearPattern,hb,hba]
-
-theorem odd_nonlinear_order {r : ℕ} (k : Fin r → GapDifference)
-    (hk : Odd (firstDifferenceCount k)) (hnl : ¬∃a, k=linearPattern a) :
-    3≤∑ h, (k h).order := by
-  rcases odd_pattern_linear_or_order_three k hk with h|h
-  · exact (hnl (first_second_counts_linear k h.1 h.2)).elim
-  · exact h
-
-/-- An actual convolution witness, retaining the common motif offset. -/
-theorem motif_convolution_support (f π : IntSeries) (L : ℕ)
-    (hf : ∀z, f.coeff z≠0 → 0≤z ∧ z<L) (z : ℤ) (hz : (f*π).coeff z≠0) :
-    ∃X∈π.coeff.support, X≤z ∧ z<X+L := by
-  rw [mul_comm,AddMonoidAlgebra.coeff_mul_apply_left,Finsupp.sum] at hz
-  obtain ⟨X,hX,hv⟩ := Finset.exists_ne_zero_of_sum_ne_zero hz
-  have hh := hf (-X+z) (mul_ne_zero_iff.mp hv).2
-  exact ⟨X,hX,by omega,by omega⟩
-
-/-- A support criterion obtained from the actual side-count range. -/
-def CentralMotifSupport (u s : ℕ) (π : IntSeries) : Prop :=
-  ∀X∈π.coeff.support,
-    (u:ℝ)/3≤2*(X:ℝ)+1 ∧ (u:ℝ)/3≤2*u-(2*((X:ℝ)+(2:ℝ)^s)+1)
-
-theorem motif_convolution_central {u s : ℕ} (π f : IntSeries)
-    (hπ : CentralMotifSupport u s π)
-    (hf : ∀z, f.coeff z≠0 → 0≤z ∧ z<(2^s:ℕ))
-    (h : Fin u) (hh : (f*π).coeff (h.val:ℤ)≠0) :
-    (u:ℝ)/3≤2*h.val+1 ∧ (u:ℝ)/3≤2*u-(2*h.val+1) := by
-  obtain ⟨X,hX,hL,hR⟩ := motif_convolution_support f π (2^s) hf h.val hh
-  have hL' : (X:ℝ)≤h.val := by exact_mod_cast hL
-  have hR' : (h.val:ℝ)<(X:ℝ)+(2:ℝ)^s := by exact_mod_cast hR
-  have hs := hπ X hX
-  constructor <;> linarith
-
-theorem prouhet_convolution_central {u s : ℕ} (π : IntSeries)
-    (hπ : CentralMotifSupport u s π) (h : Fin u)
-    (hh : (prouhetSeries s*π).coeff (h.val:ℤ)≠0) :
-    (u:ℝ)/3≤2*h.val+1 ∧ (u:ℝ)/3≤2*u-(2*h.val+1) := by
-  apply motif_convolution_central π _ hπ _ h hh
-  intro z hz
-  constructor
-  · by_contra h; exact hz ((prouhetSeries_coeff_spec s).1 z (Or.inl (by omega)))
-  · by_contra h; exact hz ((prouhetSeries_coeff_spec s).1 z (Or.inr (by omega)))
-
-theorem unsigned_convolution_central {u s : ℕ} (π : IntSeries)
-    (hπ : CentralMotifSupport u s π) (h : Fin u)
-    (hh : (geomSeries (2^s)*π).coeff (h.val:ℤ)≠0) :
-    (u:ℝ)/3≤2*h.val+1 ∧ (u:ℝ)/3≤2*u-(2*h.val+1) := by
-  apply motif_convolution_central π _ hπ _ h hh
-  intro z hz
-  by_contra hs
-  rw [geomSeries_coeff,ite_eq_right hs] at hz
-  exact hz rfl
-
-end TraceReconstruction
-
-end
-end
-end
-end
-end
-end
-end
-end
-end
-
-end OAI
+import OAI.Probability.TraceReconstruction.Differences
+import OAI.Probability.TraceReconstruction.InitialDensity
+
+namespace OAI
+
+/-!
+# Patterns
+
+Pattern serialization, motif norms, flag posteriors, signed rank coefficients, translated binomials, and support geometry.
+-/
+
+open scoped ENNReal
+noncomputable section
+
+open Filter
+open scoped Topology
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem seriesL2Sq_partial {α : Type*} [Fintype α] (f : IntSeries)
+    (j : α → ℤ) (hj : Function.Injective j) :
+    (∑ a, f.coeff (j a)^2) ≤ seriesL2Sq f := by
+  classical
+  let A := Finset.univ.image j
+  have he : (∑ a, f.coeff (j a)^2)=∑ z∈A, f.coeff z^2 := by
+    rw [Finset.sum_image]
+    exact fun _ _ _ _ h => hj h
+  rw [he]
+  calc
+    _ ≤ ∑ z∈f.coeff.support ∪ A, f.coeff z^2 :=
+      Finset.sum_le_sum_of_subset_of_nonneg Finset.subset_union_right (fun _ _ _ => sq_nonneg _)
+    _ = _ := by
+      rw [seriesL2Sq_eq_sum]
+      symm
+      apply Finset.sum_subset Finset.subset_union_left
+      intro z _ hz
+      simp [Finsupp.notMem_support_iff.mp hz]
+
+/-- The cancellation estimate after disjoint rank clipping, retaining the
+ actual fixed-period kernels and all anchor correlations. -/
+theorem period_pattern_norm {u m l r : ℕ}
+    (hm : 32 ≤ m) (hr : 0 < r) (hl : l ≤ u*m)
+    (off : Fin u → Fin m)
+    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
+    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2)
+    (hsafe : ∀ j : IncreasingTuple u r, SafeTaggedTuple (periodSlot off ∘ j.val))
+    (k : Fin r → GapDifference) {θ : ℝ} (hθ : θ ∈ Set.Ioo 0 1)
+    (hhalf : 1/2 ≤ θ) (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
+    (a : Fin r) (A : IncreasingTuple u r → ℝ) (w : Fin u → ℝ)
+    (hA : ∀ j, A j^2 ≤ w (j.val a)^2)
+    (hw : ∀ h, w h ≠ 0 → (u:ℝ)/3 ≤ 2*h.val+1 ∧ (u:ℝ)/3 ≤ 2*u-(2*h.val+1)) :
+    (∑ v, (∑ j, if PeriodRankRegion m θ j v then
+      A j*positionalRankDifference hl (periodSlot off ∘ j.val) (hsafe j) k v else 0)^2) ≤
+      periodDifferencePrefactor u m r (∑ h, (k h).order) θ *
+        periodCentralSumConstant u r * ∑ h, w h^2 := by
+  have hC : 0 ≤ periodDifferencePrefactor u m r (∑ h, (k h).order) θ := by
+    unfold periodDifferencePrefactor lengthDifferenceConstant
+    positivity
+  calc
+    _ ≤ ∑ j, A j^2*∑ v, positionalRankDifference hl (periodSlot off ∘ j.val) (hsafe j) k v^2 :=
+      period_clipped_square_sum (by omega) hhalf _ A
+    _ ≤ ∑ j, A j^2*(periodDifferencePrefactor u m r (∑ h, (k h).order) θ *
+        ∏ i, (Real.sqrt (coarsePeriodGap j.val i))⁻¹) := by
+      apply Finset.sum_le_sum
+      intro j _
+      exact mul_le_mul_of_nonneg_left
+        (period_positionalRankDifference_bound hm hr hl off hoff hdiff j (hsafe j) k hθ hmean hv)
+        (sq_nonneg _)
+    _ = periodDifferencePrefactor u m r (∑ h, (k h).order) θ *
+        ∑ j, A j^2*∏ i, (Real.sqrt (coarsePeriodGap j.val i))⁻¹ := by
+      rw [Finset.mul_sum]; congr 1; funext j; ring
+    _ ≤ _ := by
+      rw [mul_assoc]
+      exact mul_le_mul_of_nonneg_left (weighted_period_gap_sum a A w hA hw) hC
+
+theorem period_pattern_bounded_norm {u m l r M : ℕ}
+    (hm : 32 ≤ m) (hr : 0 < r) (hl : l ≤ u*m)
+    (off : Fin u → Fin m)
+    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
+    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2)
+    (hsafe : ∀ j : IncreasingTuple u r, SafeTaggedTuple (periodSlot off ∘ j.val))
+    (k : Fin r → GapDifference) {θ : ℝ} (hθ : θ ∈ Set.Ioo 0 1)
+    (hhalf : 1/2 ≤ θ) (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
+    (a : Fin r) (c : Fin (r+1)) (A : IncreasingTuple u r → ℝ) (w : Fin u → ℝ)
+    (hA : ∀ j, A j^2 ≤ w (j.val a)^2)
+    (hsupport : ∀ j, A j ≠ 0 → coarsePeriodGap j.val c ≤ M)
+    (hw : ∀ h, w h ≠ 0 → (u:ℝ)/3 ≤ 2*h.val+1 ∧ (u:ℝ)/3 ≤ 2*u-(2*h.val+1))
+    (hM : (M:ℝ) < (u:ℝ)/(3*(r+1:ℝ))) :
+    (∑ v, (∑ j, if PeriodRankRegion m θ j v then
+      A j*positionalRankDifference hl (periodSlot off ∘ j.val) (hsafe j) k v else 0)^2) ≤
+      periodDifferencePrefactor u m r (∑ h, (k h).order) θ *
+        periodBoundedSumConstant u r M * ∑ h, w h^2 := by
+  have hC : 0 ≤ periodDifferencePrefactor u m r (∑ h, (k h).order) θ := by
+    unfold periodDifferencePrefactor lengthDifferenceConstant
+    positivity
+  calc
+    _ ≤ ∑ j, A j^2*∑ v, positionalRankDifference hl (periodSlot off ∘ j.val) (hsafe j) k v^2 :=
+      period_clipped_square_sum (by omega) hhalf _ A
+    _ ≤ ∑ j, A j^2*(periodDifferencePrefactor u m r (∑ h, (k h).order) θ *
+        ∏ i, (Real.sqrt (coarsePeriodGap j.val i))⁻¹) := by
+      apply Finset.sum_le_sum
+      intro j _
+      exact mul_le_mul_of_nonneg_left
+        (period_positionalRankDifference_bound hm hr hl off hoff hdiff j (hsafe j) k hθ hmean hv)
+        (sq_nonneg _)
+    _ = periodDifferencePrefactor u m r (∑ h, (k h).order) θ *
+        ∑ j, A j^2*∏ i, (Real.sqrt (coarsePeriodGap j.val i))⁻¹ := by
+      rw [Finset.mul_sum]; congr 1; funext j; ring
+    _ ≤ _ := by
+      rw [mul_assoc]
+      exact mul_le_mul_of_nonneg_left (weighted_period_bounded_gap_sum a c A w hA hsupport hw hM) hC
+
+/-- Two involved motif positions force a short *fixed* internal coarse gap.
+ The chosen gap is independent of the actual labels and of the offset law. -/
+theorem motif_pattern_bounded_gap {u r : ℕ} (π : IntSeries) (s : ℕ)
+    (j : IncreasingTuple u (r+1)) (k : Fin (r+1) → GapDifference)
+    (i : Fin r) (b : Fin (r+1)) (hib : i.castSucc < b)
+    (hi : k i.castSucc ≠ .unchanged) (hb : k b ≠ .unchanged)
+    (hA : motifPatternMultiplier π s (fun a => (j.val a).val) k ≠ 0) :
+    coarsePeriodGap j.val i.castSucc.succ ≤ 2*(2^s) := by
+  have hs := motif_multiplier_bounded_separation π s (fun a => (j.val a).val) k hA i.castSucc b hi hb
+  have hj1 := j.property (show i.castSucc < i.succ by exact Fin.castSucc_lt_succ)
+  have hj2 := j.property.monotone (show i.succ ≤ b by simpa only [Fin.le_iff_val_le_val,Fin.lt_def,
+    Fin.val_succ,Fin.val_castSucc] using (Nat.succ_le_of_lt hib))
+  have h1 : (j.val i.castSucc).val < (j.val i.succ).val := hj1
+  have h2 : (j.val i.succ).val ≤ (j.val b).val := hj2
+  have hh := (abs_lt.mp hs).1
+  rw [coarsePeriodGap_middle]
+  omega
+
+end TraceReconstruction
+
+
+namespace TraceReconstruction
+
+theorem centered_window_floor_fit (P T L t : ℕ) (Y : ℤ)
+    (hP : 2*T ≤ P) (hT : 16 ≤ T) (heven : Even T)
+    (hL : 8*L ≤ T) (hY : |8*Y| ≤ (T:ℤ))
+    (ht : (t:ℤ)=(((P-T)/2:ℕ):ℤ)+Y) :
+    t+T ≤ P ∧ t ≤ P/2 ∧ P/2+L ≤ t+T := by
+  obtain ⟨k,hk⟩ := heven
+  have hy := abs_le.mp hY
+  omega
+
+theorem centered_window_floor_anchor (P T : ℕ) (hTP : T ≤ P) (heven : Even T) :
+    P/2-(P-T)/2=T/2 := by
+  obtain ⟨k,hk⟩ := heven
+  omega
+
+theorem atomString_append (n m : ℕ) (w : Bool → List Bool) (θ : Fin (n+m) → Bool) :
+    atomString (n+m) w θ =
+      atomString n w (fun i => θ (i.castLE (Nat.le_add_right n m))) ++
+      atomString m w (fun i => θ (i.natAdd n)) := by
+  simp only [atomString,List.ofFn_add,List.flatten_append]
+
+theorem atomString_three (n m k : ℕ) (w : Bool → List Bool) (θ : Fin (n+m+k) → Bool) :
+    atomString (n+m+k) w θ =
+      atomString n w (fun i => θ ⟨i.val,by omega⟩) ++
+      atomString m w (fun i => θ ⟨n+i.val,by omega⟩) ++
+      atomString k w (fun i => θ ⟨n+m+i.val,by omega⟩) := by
+  rw [atomString_append,atomString_append]
+  rfl
+
+/-- A full deterministic template equals a pure prefix, the actual translated
+ window, and a pure suffix. This equality is in unmarked binary strings. -/
+theorem template_word_window (t T z a L : ℕ) (sign : Fin L → Bool) (τ : Bool)
+    (w : Bool → List Bool) (hleft : t ≤ a) (hright : a+L ≤ t+T) :
+    atomString (t+T+z) w (templateType (t+T+z) a L sign τ) =
+      atomString t w (fun _ => false) ++
+      atomString T w (templateType T (a-t) L sign τ) ++
+      atomString z w (fun _ => false) := by
+  rw [atomString_three]
+  congr 2
+  · congr 1
+    funext i
+    exact templateType_outside_window sign τ hleft hright ⟨i.val,by omega⟩ (Or.inl i.isLt)
+  · congr 1
+    funext i
+    exact templateType_inside_window sign τ (by omega) hleft i
+  · funext i
+    exact templateType_outside_window sign τ hleft hright ⟨t+T+i.val,by omega⟩ (Or.inr (by dsimp; omega))
+
+def pureAtomRun (w : List Bool) (n : ℕ) : List Bool :=
+  (List.replicate n w).flatten
+
+theorem atomString_constant (n : ℕ) (w : Bool → List Bool) (b : Bool) :
+    atomString n w (fun _ => b)=pureAtomRun (w b) n := by
+  simp [atomString,pureAtomRun]
+
+theorem pureAtomRun_add (w : List Bool) (n m : ℕ) :
+    pureAtomRun w (n+m)=pureAtomRun w n ++ pureAtomRun w m := by
+  simp only [pureAtomRun,List.replicate_add,List.flatten_append]
+
+/-- Splitting or regrouping a pure region into reported chunks changes no
+ unmarked string. In particular arbitrary fixed chunk IDs are harmless. -/
+theorem pureAtomRun_partition (w : List Bool) (ns : List ℕ) :
+    (ns.map (pureAtomRun w)).flatten=pureAtomRun w ns.sum := by
+  induction ns with
+  | nil => simp [pureAtomRun]
+  | cons n ns ih => simp [ih,pureAtomRun_add]
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction.FiniteLaw
+open scoped Classical
+variable {α : Type*} [Fintype α]
+
+/-- Exact orientation law after a report: only invisible bits remain random. -/
+def flagPosterior (d : Option Bool) : FiniteLaw Bool where
+  mass b := match d with
+    | none => 1/2
+    | some c => if b=c then 1 else 0
+  nonneg := by
+    intro b
+    cases d with
+    | none => norm_num
+    | some c => dsimp; split_ifs <;> norm_num
+  total := by cases d with
+    | none => norm_num [Fintype.sum_bool]
+    | some c => simp
+
+def pairedReportMass (P Q H : FiniteLaw α) (ω : ℝ)
+    (z : Option Bool × (α × α)) : ℝ :=
+  ∑ b : Bool, (1/2 : ℝ)*pairedFlagMass P Q H ω b z
+
+/-- Pointwise Bayes factorization, valid even at zero probability reports. -/
+theorem pairedFlag_bayes (P Q H : FiniteLaw α) (ω : ℝ)
+    (z : Option Bool × (α × α)) (b : Bool) :
+    (1/2 : ℝ)*pairedFlagMass P Q H ω b z =
+      pairedReportMass P Q H ω z * (flagPosterior z.1).mass b := by
+  rcases z with ⟨d,z⟩
+  cases d with
+  | none => simp [pairedReportMass,pairedFlagMass,flagPosterior]; ring
+  | some c => cases b <;> cases c <;>
+      simp [pairedReportMass,pairedFlagMass,flagPosterior]
+
+theorem pairedReportMass_nonneg (P Q H : FiniteLaw α) {ω : ℝ} (hω : 0 ≤ ω)
+    (hP : ∀ a, ω*H.mass a ≤ P.mass a) (hQ : ∀ a, ω*H.mass a ≤ Q.mass a)
+    (z : Option Bool × (α × α)) : 0 ≤ pairedReportMass P Q H ω z := by
+  apply Finset.sum_nonneg
+  intro b _
+  exact mul_nonneg (by norm_num) (pairedFlagMass_nonneg P Q H hω hP hQ b z)
+
+theorem pairedReportMass_total (P Q H : FiniteLaw α) (ω : ℝ) :
+    (∑ z, pairedReportMass P Q H ω z)=1 := by
+  unfold pairedReportMass
+  rw [Finset.sum_comm]
+  simp_rw [← Finset.mul_sum,pairedFlagMass_sum]
+  norm_num [Fintype.sum_bool]
+
+def pairedReport (P Q H : FiniteLaw α) {ω : ℝ} (hω : 0 ≤ ω)
+    (hP : ∀ a, ω*H.mass a ≤ P.mass a) (hQ : ∀ a, ω*H.mass a ≤ Q.mass a) :
+    FiniteLaw (Option Bool × (α × α)) :=
+  ⟨pairedReportMass P Q H ω, pairedReportMass_nonneg P Q H hω hP hQ,
+    pairedReportMass_total P Q H ω⟩
+
+/-- Conditioning solely on exact reports does not correlate mechanisms.
+ This is the actual joint-mass factorization, not an independence hypothesis. -/
+theorem pairedFlags_product_bayes {ι : Type*} [Fintype ι]
+    (P Q H : ι → FiniteLaw α) (ω : ι → ℝ)
+    (z : ι → Option Bool × (α × α)) (b : ι → Bool) :
+    (∏ i, (1/2 : ℝ)*pairedFlagMass (P i) (Q i) (H i) (ω i) (b i) (z i)) =
+      (∏ i, pairedReportMass (P i) (Q i) (H i) (ω i) (z i)) *
+        ∏ i, (flagPosterior (z i).1).mass (b i) := by
+  simp_rw [pairedFlag_bayes]
+  exact Finset.prod_mul_distrib
+
+theorem pairedFlags_report_marginal {ι : Type*} [Fintype ι]
+    (P Q H : ι → FiniteLaw α) (ω : ι → ℝ)
+    (z : ι → Option Bool × (α × α)) :
+    (∑ b : ι → Bool, ∏ i, (1/2 : ℝ)*pairedFlagMass (P i) (Q i) (H i) (ω i) (b i) (z i)) =
+      ∏ i, pairedReportMass (P i) (Q i) (H i) (ω i) (z i) := by
+  exact (Fintype.prod_sum (fun i b => (1/2 : ℝ)*pairedFlagMass (P i) (Q i) (H i) (ω i) b (z i))).symm
+
+end TraceReconstruction.FiniteLaw
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+def bitIndex (b : Bool) : ℕ := if b then 1 else 0
+
+def posteriorBitSeries (d : Option Bool) : IntSeries :=
+  ∑ b : Bool, AddMonoidAlgebra.single (bitIndex b : ℤ) ((FiniteLaw.flagPosterior d).mass b)
+
+def visibleOnes (d : Option Bool) : ℕ := if d=some true then 1 else 0
+
+def hiddenTrial (d : Option Bool) : ℕ := if d=none then 1 else 0
+
+theorem posteriorBitSeries_factor (d : Option Bool) :
+    posteriorBitSeries d = AddMonoidAlgebra.single (visibleOnes d : ℤ) 1 * fairSeries (hiddenTrial d) := by
+  cases d with
+  | none => simp [posteriorBitSeries,FiniteLaw.flagPosterior,bitIndex,visibleOnes,hiddenTrial,fairSeries,
+      add_comm]
+  | some b => cases b <;> simp [posteriorBitSeries,FiniteLaw.flagPosterior,bitIndex,visibleOnes,hiddenTrial,
+      fairSeries]
+
+/-- The count distribution of the literal posterior orientation bits is
+ a translated fair binomial, including reports of probability zero. -/
+theorem posterior_count_series {ι : Type*} [Fintype ι] (d : ι → Option Bool) :
+    (∑ b : ι → Bool, AddMonoidAlgebra.single (∑ i, (bitIndex (b i):ℤ))
+      (∏ i, (FiniteLaw.flagPosterior (d i)).mass (b i))) =
+      AddMonoidAlgebra.single (∑ i, (visibleOnes (d i):ℤ)) 1 *
+        fairSeries (∑ i, hiddenTrial (d i)) := by
+  have he : (∑ b : ι → Bool, AddMonoidAlgebra.single (∑ i, (bitIndex (b i):ℤ))
+      (∏ i, (FiniteLaw.flagPosterior (d i)).mass (b i))) = ∏ i, posteriorBitSeries (d i) := by
+    rw [show (∏ i, posteriorBitSeries (d i)) =
+      ∏ i, ∑ b : Bool, AddMonoidAlgebra.single (bitIndex b : ℤ) ((FiniteLaw.flagPosterior (d i)).mass b) from rfl]
+    rw [Fintype.prod_sum]
+    apply Finset.sum_congr rfl
+    intro b _
+    rw [AddMonoidAlgebra.prod_single]
+  rw [he]
+  simp_rw [posteriorBitSeries_factor]
+  rw [Finset.prod_mul_distrib,AddMonoidAlgebra.prod_single]
+  simp only [Finset.prod_const_one]
+  congr 1
+  unfold fairSeries
+  exact Finset.prod_pow_eq_pow_sum _ _ _
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem geomSeries_coeff (k : ℕ) (z : ℤ) :
+    (geomSeries k).coeff z = if 0 ≤ z ∧ z < k then 1 else 0 := by
+  unfold geomSeries
+  simp only [AddMonoidAlgebra.single_pow,one_pow,nsmul_eq_mul,mul_one,
+    AddMonoidAlgebra.coeff_sum,Finsupp.finsetSum_apply,AddMonoidAlgebra.coeff_single,Finsupp.single_apply]
+  by_cases hz : 0 ≤ z ∧ z < k
+  · obtain ⟨a,rfl⟩ := Int.eq_ofNat_of_zero_le hz.1
+    rw [ite_eq_left hz,Finset.sum_eq_single a]
+    · simp
+    · intro b _ hb
+      exact ite_eq_right (by exact_mod_cast hb)
+    · intro ha
+      have ha' : a < k := by exact_mod_cast hz.2
+      exact (ha (Finset.mem_range.mpr ha')).elim
+  · rw [ite_eq_right hz]
+    apply Finset.sum_eq_zero
+    intro a ha
+    apply ite_eq_right
+    intro haz
+    subst z
+    exact hz ⟨by positivity,by exact_mod_cast Finset.mem_range.mp ha⟩
+
+theorem motif_unsigned_convolution (π : IntSeries) (s : ℕ) (z : ℤ) :
+    (∑ X ∈ π.coeff.support, π.coeff X*motifIndicator s (z-X)) =
+      (geomSeries (2^s)*π).coeff z := by
+  rw [mul_comm,AddMonoidAlgebra.coeff_mul_apply_left,Finsupp.sum]
+  apply Finset.sum_congr rfl
+  intro X _
+  rw [geomSeries_coeff]
+  simp only [motifIndicator,sub_eq_neg_add]
+
+theorem motifPatternMultiplier_sq_le {r : ℕ} (π : IntSeries)
+    (hπ : ∀ X, 0 ≤ π.coeff X) (s : ℕ) (j : Fin r → ℤ)
+    (k : Fin r → GapDifference) (h : Fin r) (hk : k h ≠ .unchanged) :
+    (motifPatternMultiplier π s j k)^2 ≤ ((geomSeries (2^s)*π).coeff (j h))^2 := by
+  have ha := motifPatternMultiplier_abs_le π hπ s j k h hk
+  rw [motif_unsigned_convolution] at ha
+  have hh := pow_le_pow_left₀ (abs_nonneg (motifPatternMultiplier π s j k)) ha 2
+  simpa only [sq_abs] using hh
+
+theorem unsigned_motif_energy (π : IntSeries) (s : ℕ) :
+    seriesL2Sq (geomSeries (2^s)*π) ≤ ((2:ℝ)^s)^2 * seriesL2Sq π := by
+  refine (seriesL2Sq_mul_le _ _).trans ?_
+  apply mul_le_mul_of_nonneg_right _ (seriesL2Sq_nonneg _)
+  exact pow_le_pow_left₀ (seriesL1_nonneg _) (by simpa using geomSeries_L1_le (2^s)) 2
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+
+/-- Elementary chi-square upper bound on the Bernoulli relative entropy.
+ No Stirling estimate is used in the subsequent count overlap. -/
+theorem bernoulli_entropy_upper {t p : ℝ} (ht : t ∈ Set.Ioo 0 1) (hp : p ∈ Set.Ioo 0 1) :
+    t*Real.log (t/p)+(1-t)*Real.log ((1-t)/(1-p)) ≤
+      (t-p)^2/(p*(1-p)) := by
+  have h1 := mul_le_mul_of_nonneg_left (Real.log_le_sub_one_of_pos (div_pos ht.1 hp.1)) ht.1.le
+  have h2 := mul_le_mul_of_nonneg_left
+    (Real.log_le_sub_one_of_pos (div_pos (sub_pos.mpr ht.2) (sub_pos.mpr hp.2)))
+    (sub_nonneg.mpr ht.2.le)
+  have hid : t*(t/p-1)+(1-t)*((1-t)/(1-p)-1)=(t-p)^2/(p*(1-p)) := by
+    field_simp [hp.1.ne', (sub_pos.mpr hp.2).ne']
+    ring
+  linarith
+
+/-- Exact likelihood change from the mean-matched Bernoulli law. -/
+theorem binomialMass_likelihood (n l : ℕ) {t p : ℝ}
+    (ht : t ∈ Set.Ioo 0 1) (hp : p ∈ Set.Ioo 0 1) :
+    binomialMass n p l = binomialMass n t l *
+      Real.exp (-((l:ℝ)*Real.log (t/p)+((n-l:ℕ):ℝ)*Real.log ((1-t)/(1-p)))) := by
+  have h1 : Real.exp (-((l:ℝ)*Real.log (t/p)))=(p/t)^l := by
+    rw [show -((l:ℝ)*Real.log (t/p))=(l:ℝ)*Real.log (p/t) by
+      rw [Real.log_div ht.1.ne' hp.1.ne',Real.log_div hp.1.ne' ht.1.ne']; ring,
+      Real.exp_nat_mul,Real.exp_log (div_pos hp.1 ht.1)]
+  have h2 : Real.exp (-(((n-l:ℕ):ℝ)*Real.log ((1-t)/(1-p))))=((1-p)/(1-t))^(n-l) := by
+    rw [show -(((n-l:ℕ):ℝ)*Real.log ((1-t)/(1-p)))=
+      ((n-l:ℕ):ℝ)*Real.log ((1-p)/(1-t)) by
+      rw [Real.log_div (sub_pos.mpr ht.2).ne' (sub_pos.mpr hp.2).ne',
+        Real.log_div (sub_pos.mpr hp.2).ne' (sub_pos.mpr ht.2).ne']; ring,
+      Real.exp_nat_mul,Real.exp_log (div_pos (sub_pos.mpr hp.2) (sub_pos.mpr ht.2))]
+  rw [neg_add,Real.exp_add,h1,h2]
+  unfold binomialMass
+  rw [div_pow,div_pow]
+  field_simp [ht.1.ne', (sub_pos.mpr ht.2).ne']
+
+/-- Pointwise lower mass at every interior integer, with mean-matched
+ variance at least one. It applies uniformly even as deletion vanishes. -/
+theorem binomialMass_near_mean_lower (n l : ℕ) {t p : ℝ}
+    (ht : t ∈ Set.Ioo 0 1) (hp : p ∈ Set.Ioo 0 1)
+    (hl : l ≤ n) (hmean : (n:ℝ)*t=l) (hv : 1 ≤ (n:ℝ)*t*(1-t)) :
+    Real.exp (-((l:ℝ)-(n:ℝ)*p)^2/((n:ℝ)*p*(1-p))) /
+      (12*Real.sqrt ((n:ℝ)*t*(1-t))) ≤ binomialMass n p l := by
+  have hn : (0:ℝ)<n := by nlinarith [mul_pos ht.1 (sub_pos.mpr ht.2)]
+  have hnl : ((n-l:ℕ):ℝ)=(n:ℝ)-(l:ℝ) := by exact Nat.cast_sub hl
+  have he : (l:ℝ)*Real.log (t/p)+((n-l:ℕ):ℝ)*Real.log ((1-t)/(1-p)) ≤
+      ((l:ℝ)-(n:ℝ)*p)^2/((n:ℝ)*p*(1-p)) := by
+    have H := mul_le_mul_of_nonneg_left (bernoulli_entropy_upper ht hp) hn.le
+    calc
+      _ = (n:ℝ)*(t*Real.log (t/p)+(1-t)*Real.log ((1-t)/(1-p))) := by
+        rw [hnl,← hmean]; ring
+      _ ≤ (n:ℝ)*((t-p)^2/(p*(1-p))) := H
+      _ = _ := by
+        rw [← hmean]
+        field_simp [hn.ne', hp.1.ne', (sub_pos.mpr hp.2).ne']
+  rw [binomialMass_likelihood n l ht hp]
+  calc
+    _ = (1/(12*Real.sqrt ((n:ℝ)*t*(1-t)))) *
+      Real.exp (-((l:ℝ)-(n:ℝ)*p)^2/((n:ℝ)*p*(1-p))) := by ring
+    _ ≤ (1/(12*Real.sqrt ((n:ℝ)*t*(1-t)))) *
+      Real.exp (-((l:ℝ)*Real.log (t/p)+((n-l:ℕ):ℝ)*Real.log ((1-t)/(1-p)))) := by
+      apply mul_le_mul_of_nonneg_left _ (by positivity)
+      apply Real.exp_le_exp.mpr
+      simpa only [neg_div] using neg_le_neg he
+    _ ≤ _ := mul_le_mul_of_nonneg_right (binomialMass_mean_lower n l ht hl hmean hv)
+      (Real.exp_nonneg _)
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+def signedPeriodDefects {u m : ℕ} (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
+    (s : ℕ) (X : ℤ) (τ : ℝ) : Fin u → Fin (u*m) :=
+  shiftedTag (periodSlot (fun _ => off)) hsafe
+    (fun j => motifShift (motifIndicator s ((j.val:ℤ)-X)) ((prouhetSeries s).coeff ((j.val:ℤ)-X)) τ)
+
+theorem signedPeriodDefects_strictMono {u m : ℕ} (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
+    (s : ℕ) (X : ℤ) (τ : ℝ) : StrictMono (signedPeriodDefects off hsafe s X τ) :=
+  shiftedTag_strictMono _ _ _
+
+theorem SafeTaggedTuple.comp {n u r : ℕ} {p : Fin u → Fin n}
+    (hp : SafeTaggedTuple p) (j : Fin r → Fin u) (hj : StrictMono j) :
+    SafeTaggedTuple (p ∘ j) :=
+  ⟨fun i => hp.1 (j i),fun i k hik => hp.2 (j i) (j k) (hj hik)⟩
+
+theorem shiftedTag_comp {n u r : ℕ} (p : Fin u → Fin n) (hp : SafeTaggedTuple p)
+    (a : Fin u → Fin 3) (j : Fin r → Fin u) (hj : StrictMono j) :
+    shiftedTag p hp a ∘ j = shiftedTag (p ∘ j) (hp.comp j hj) (a ∘ j) := by
+  funext i
+  apply Fin.ext
+  rfl
+
+/-- Exact degree-r coefficient difference of the actual randomized motif
+ in the marked central run, prior to any rank clipping. -/
+def signedRankCoefficient {u m l r : ℕ} (hl : l ≤ u*m) (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
+    (π : IntSeries) (s : ℕ) (v : IncreasingTuple l r) : ℝ :=
+  ∑ X ∈ π.coeff.support, π.coeff X *
+    (rankDefectCoefficient hl (signedPeriodDefects off hsafe s X 1) v -
+     rankDefectCoefficient hl (signedPeriodDefects off hsafe s X (-1)) v)
+
+theorem signedRankCoefficient_expansion {u m l r : ℕ} (hl : l ≤ u*m) (off : Fin m)
+    (hsafe : SafeTaggedTuple (periodSlot (u := u) (fun _ => off)))
+    (π : IntSeries) (s : ℕ) (v : IncreasingTuple l r) :
+    signedRankCoefficient hl off hsafe π s v =
+      ∑ k : Fin r → GapDifference, if Odd (firstDifferenceCount k) then
+        2*∑ j : IncreasingTuple u r,
+          motifPatternMultiplier π s (fun i => (j.val i).val) k *
+            positionalRankDifference hl (periodSlot (fun _ => off) ∘ j.val)
+              (hsafe.comp j.val j.property) k v else 0 := by
+  unfold signedRankCoefficient
+  simp_rw [rankDefectCoefficient_eq _ _ (signedPeriodDefects_strictMono _ _ _ _ _),
+    ← Finset.sum_sub_distrib,Finset.mul_sum]
+  rw [Finset.sum_comm]
+  have he (j : IncreasingTuple u r) :
+      (∑ X ∈ π.coeff.support, π.coeff X *
+        (subsetRankKernel hl (signedPeriodDefects off hsafe s X 1 ∘ j.val) v.val -
+         subsetRankKernel hl (signedPeriodDefects off hsafe s X (-1) ∘ j.val) v.val)) =
+      ∑ k : Fin r → GapDifference,
+        (if Odd (firstDifferenceCount k) then 2*motifPatternMultiplier π s (fun i => (j.val i).val) k else 0)*
+          positionalRankDifference hl (periodSlot (fun _ => off) ∘ j.val)
+            (hsafe.comp j.val j.property) k v := by
+    have htag (X : ℤ) (τ : ℝ) :
+        signedPeriodDefects off hsafe s X τ ∘ j.val =
+        shiftedTag (periodSlot (fun _ => off) ∘ j.val) (hsafe.comp j.val j.property)
+          (fun i => motifShift (motifIndicator s ((j.val i).val-X))
+            ((prouhetSeries s).coeff ((j.val i).val-X)) τ) := by
+      funext i; apply Fin.ext; rfl
+    simp_rw [htag]
+    simpa only [positionalRankDifference]
+      using mixed_signed_shift_expansion π s (fun i => (j.val i).val)
+        (fun a => subsetRankKernel hl
+          (shiftedTag (periodSlot (fun _ => off) ∘ j.val) (hsafe.comp j.val j.property) a) v.val)
+  simp_rw [he]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro k _
+  split_ifs with hk
+  · apply Finset.sum_congr rfl
+    intro j _; ring
+  · simp only [zero_mul,Finset.sum_const_zero]
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+def translateSeries (c : ℤ) (f : IntSeries) : IntSeries :=
+  AddMonoidAlgebra.single c 1*f
+
+theorem translateSeries_coeff (c : ℤ) (f : IntSeries) (z : ℤ) :
+    (translateSeries c f).coeff z = f.coeff (z-c) := by
+  rw [translateSeries,AddMonoidAlgebra.coeff_single_mul_apply,one_mul]
+  congr 1; omega
+
+theorem translateSeries_L2Sq (c : ℤ) (f : IntSeries) :
+    seriesL2Sq (translateSeries c f)=seriesL2Sq f := by
+  unfold seriesL2Sq
+  simp_rw [translateSeries_coeff]
+  exact (Equiv.subRight c).tsum_eq (fun z => f.coeff z^2)
+
+theorem translateSeries_L1 (c : ℤ) (f : IntSeries) :
+    seriesL1 (translateSeries c f)=seriesL1 f := by
+  unfold seriesL1
+  simp_rw [translateSeries_coeff]
+  exact (Equiv.subRight c).tsum_eq (fun z => |f.coeff z|)
+
+theorem mul_translateSeries (c : ℤ) (f g : IntSeries) :
+    f*translateSeries c g=translateSeries c (f*g) := by
+  unfold translateSeries
+  ring
+
+theorem fairSeries_support (g : ℕ) (z : ℤ) (hz : (fairSeries g).coeff z ≠ 0) :
+    0≤z ∧ z≤g := by
+  have h0 : 0≤z := by
+    by_contra h
+    exact hz (fairSeries_coeff_neg g z (by omega))
+  obtain ⟨k,rfl⟩ := Int.eq_ofNat_of_zero_le h0
+  refine ⟨by positivity,?_⟩
+  by_contra hk
+  rw [fairSeries_coeff_nat,fairBinomial,Nat.choose_eq_zero_of_lt (by omega)] at hz
+  simp at hz
+
+theorem fairSeries_reflect (g : ℕ) (z : ℤ) :
+    (fairSeries g).coeff ((g:ℤ)-z)=(fairSeries g).coeff z := by
+  by_cases hz : 0≤z ∧ z≤g
+  · obtain ⟨k,rfl⟩ := Int.eq_ofNat_of_zero_le hz.1
+    have hk : k≤g := by exact_mod_cast hz.2
+    rw [← Nat.cast_sub hk,fairSeries_coeff_nat,fairSeries_coeff_nat]
+    simp only [fairBinomial,Nat.choose_symm hk]
+  · have h1 : (fairSeries g).coeff z=0 := by
+      by_contra h; exact hz (fairSeries_support g z h)
+    have h2 : (fairSeries g).coeff ((g:ℤ)-z)=0 := by
+      by_contra h
+      have hh := fairSeries_support g ((g:ℤ)-z) h
+      exact hz ⟨by omega,by omega⟩
+    rw [h1,h2]
+
+/-- Reflection in the posterior count is exactly an ordinary translated
+ fair binomial; it does not change the smoothing channel. -/
+theorem affine_reflected_binomial (g : ℕ) (c z : ℤ) :
+    (fairSeries g).coeff (c-z)=
+      (translateSeries (c-g) (fairSeries g)).coeff z := by
+  rw [translateSeries_coeff,← fairSeries_reflect g (c-z)]
+  congr 1; omega
+
+theorem translated_binomial_nonneg (g : ℕ) (c z : ℤ) :
+    0≤(translateSeries c (fairSeries g)).coeff z := by
+  rw [translateSeries_coeff]
+  exact fairSeries_nonneg _ _
+
+theorem translated_binomial_support (g : ℕ) (c z : ℤ)
+    (hz : (translateSeries c (fairSeries g)).coeff z ≠ 0) :
+    c≤z ∧ z≤c+g := by
+  rw [translateSeries_coeff] at hz
+  have h := fairSeries_support g (z-c) hz
+  omega
+
+theorem translated_binomial_smoothing (g s : ℕ) (c : ℤ) :
+    seriesL2Sq (prouhetSeries s*translateSeries c (fairSeries g)) ≤
+      ((2:ℝ)^(s*s))^2*(((2*binomialBound (g/(s+1)))^s)^2*binomialBound (g/(s+1))) := by
+  rw [mul_translateSeries,translateSeries_L2Sq]
+  exact (signed_binomial_smoothing g s).2
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem linearPattern_order {r : ℕ} (a : Fin r) :
+    (∑ h, ((linearPattern a) h).order)=1 := by
+  classical
+  have he (h : Fin r) : ((linearPattern a) h).order=if h=a then 1 else 0 := by
+    by_cases hh : h=a <;> simp [linearPattern,hh,GapDifference.order]
+  simp_rw [he]
+  simp
+
+theorem first_second_counts_linear {r : ℕ} (k : Fin r → GapDifference)
+    (h1 : firstDifferenceCount k=1) (h2 : secondDifferenceCount k=0) :
+    ∃ a : Fin r, k=linearPattern a := by
+  classical
+  obtain ⟨a,ha⟩ := Finset.card_eq_one.mp h1
+  refine ⟨a,funext fun b => ?_⟩
+  have hfirst : k b = .first ↔ b=a := by
+    have hh := congrArg (fun A : Finset (Fin r) => b∈A) ha
+    simpa only [Finset.mem_filter,Finset.mem_univ,true_and,Finset.mem_singleton] using (iff_of_eq hh)
+  have hsecond : k b ≠ .second := by
+    have hh := Finset.card_eq_zero.mp h2
+    intro hb
+    have hmem : b∈(Finset.univ.filter fun b => k b=.second) := by simp [hb]
+    rw [hh] at hmem
+    exact Finset.notMem_empty _ hmem
+  by_cases hb : b=a
+  · have hf := hfirst.mpr hb
+    simpa [linearPattern,hb] using hf
+  · have hf := mt hfirst.mp hb
+    cases hk : k b <;> simp_all [linearPattern]
+
+theorem odd_pattern_pair {r : ℕ} (k : Fin r → GapDifference)
+    (hk : Odd (firstDifferenceCount k)) (hnl : ¬∃a, k=linearPattern a) :
+    ∃ a b : Fin r, a<b ∧ k a≠.unchanged ∧ k b≠.unchanged := by
+  classical
+  have hp : 0<firstDifferenceCount k := hk.pos
+  obtain ⟨a,ha⟩ := Finset.card_pos.mp hp
+  have ha' : k a=.first := (Finset.mem_filter.mp ha).2
+  by_contra hh
+  push Not at hh
+  apply hnl
+  refine ⟨a,funext fun b => ?_⟩
+  by_cases hb : b=a
+  · subst b; simp [linearPattern,ha']
+  · have hba : k b=.unchanged := by
+      rcases lt_or_gt_of_ne hb with hba|hab
+      · by_contra hb'
+        exact (show k a≠.unchanged by simp [ha']) (hh b a hba hb')
+      · exact hh a b hab (by simp [ha'])
+    simp [linearPattern,hb,hba]
+
+theorem odd_nonlinear_order {r : ℕ} (k : Fin r → GapDifference)
+    (hk : Odd (firstDifferenceCount k)) (hnl : ¬∃a, k=linearPattern a) :
+    3≤∑ h, (k h).order := by
+  rcases odd_pattern_linear_or_order_three k hk with h|h
+  · exact (hnl (first_second_counts_linear k h.1 h.2)).elim
+  · exact h
+
+/-- An actual convolution witness, retaining the common motif offset. -/
+theorem motif_convolution_support (f π : IntSeries) (L : ℕ)
+    (hf : ∀z, f.coeff z≠0 → 0≤z ∧ z<L) (z : ℤ) (hz : (f*π).coeff z≠0) :
+    ∃X∈π.coeff.support, X≤z ∧ z<X+L := by
+  rw [mul_comm,AddMonoidAlgebra.coeff_mul_apply_left,Finsupp.sum] at hz
+  obtain ⟨X,hX,hv⟩ := Finset.exists_ne_zero_of_sum_ne_zero hz
+  have hh := hf (-X+z) (mul_ne_zero_iff.mp hv).2
+  exact ⟨X,hX,by omega,by omega⟩
+
+/-- A support criterion obtained from the actual side-count range. -/
+def CentralMotifSupport (u s : ℕ) (π : IntSeries) : Prop :=
+  ∀X∈π.coeff.support,
+    (u:ℝ)/3≤2*(X:ℝ)+1 ∧ (u:ℝ)/3≤2*u-(2*((X:ℝ)+(2:ℝ)^s)+1)
+
+theorem motif_convolution_central {u s : ℕ} (π f : IntSeries)
+    (hπ : CentralMotifSupport u s π)
+    (hf : ∀z, f.coeff z≠0 → 0≤z ∧ z<(2^s:ℕ))
+    (h : Fin u) (hh : (f*π).coeff (h.val:ℤ)≠0) :
+    (u:ℝ)/3≤2*h.val+1 ∧ (u:ℝ)/3≤2*u-(2*h.val+1) := by
+  obtain ⟨X,hX,hL,hR⟩ := motif_convolution_support f π (2^s) hf h.val hh
+  have hL' : (X:ℝ)≤h.val := by exact_mod_cast hL
+  have hR' : (h.val:ℝ)<(X:ℝ)+(2:ℝ)^s := by exact_mod_cast hR
+  have hs := hπ X hX
+  constructor <;> linarith
+
+theorem prouhet_convolution_central {u s : ℕ} (π : IntSeries)
+    (hπ : CentralMotifSupport u s π) (h : Fin u)
+    (hh : (prouhetSeries s*π).coeff (h.val:ℤ)≠0) :
+    (u:ℝ)/3≤2*h.val+1 ∧ (u:ℝ)/3≤2*u-(2*h.val+1) := by
+  apply motif_convolution_central π _ hπ _ h hh
+  intro z hz
+  constructor
+  · by_contra h; exact hz ((prouhetSeries_coeff_spec s).1 z (Or.inl (by omega)))
+  · by_contra h; exact hz ((prouhetSeries_coeff_spec s).1 z (Or.inr (by omega)))
+
+theorem unsigned_convolution_central {u s : ℕ} (π : IntSeries)
+    (hπ : CentralMotifSupport u s π) (h : Fin u)
+    (hh : (geomSeries (2^s)*π).coeff (h.val:ℤ)≠0) :
+    (u:ℝ)/3≤2*h.val+1 ∧ (u:ℝ)/3≤2*u-(2*h.val+1) := by
+  apply motif_convolution_central π _ hπ _ h hh
+  intro z hz
+  by_contra hs
+  rw [geomSeries_coeff,ite_eq_right hs] at hz
+  exact hz rfl
+
+end TraceReconstruction
+
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/TraceReconstruction/Propagation.lean b/lean/OAI/Probability/TraceReconstruction/Propagation.lean
index e442a3b5bd7be69a53006f7336e1cf800f99cae2..76775335e646014c55d721c0b14a36c10b52238c 100644
--- a/lean/OAI/Probability/TraceReconstruction/Propagation.lean
+++ b/lean/OAI/Probability/TraceReconstruction/Propagation.lean
@@ -1,1268 +1,1268 @@
-import OAI.Probability.TraceReconstruction.ErrorRates
-
-namespace OAI
-
-/-!
-# Propagation
-
-Moment restriction, scale margins, sharp budget closure, clipping and error propagation, and dense/warmup overlap.
--/
-
-open scoped ENNReal
-noncomputable section
-
-open Filter
-open scoped Topology
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-namespace FiniteLaw
-variable {α : Type*} [Fintype α]
-
-theorem evenMomentSq_mono_order (P : FiniteLaw α) (F : α→ℝ) {j k : ℕ}
-    (hj : 1≤j) (hjk : j≤k) : P.evenMomentSq F j ≤ P.evenMomentSq F k := by
-  have hjp : (0:ℝ)<j := by exact_mod_cast (show 0<j by omega)
-  have hkp : (0:ℝ)<k := by exact_mod_cast (show 0<k by omega)
-  have hp : 1≤(k:ℝ)/j := (one_le_div hjp).mpr (by exact_mod_cast hjk)
-  have he (a : α) : (|F a|^(2*j))^((k:ℝ)/j)=|F a|^(2*k) := by
-    rw [←Real.rpow_natCast,←Real.rpow_mul (abs_nonneg _),←Real.rpow_natCast]
-    congr 1
-    push_cast
-    field_simp
-  have H := Real.rpow_arith_mean_le_arith_mean_rpow Finset.univ P.mass
-    (fun a=>|F a|^(2*j)) (fun a _=>P.nonneg a) P.total (fun _ _=>by positivity) hp
-  simp_rw [he] at H
-  have hr := Real.rpow_le_rpow (Real.rpow_nonneg (P.expect_nonneg _ (fun _=>by positivity)) _) H
-    (show 0≤1/(k:ℝ) by positivity)
-  change (P.expect (fun a=>|F a|^(2*j))^((k:ℝ)/j))^(1/(k:ℝ)) ≤ _ at hr
-  rw [←Real.rpow_mul (P.expect_nonneg _ (fun _=>by positivity))] at hr
-  have he' : (k:ℝ)/j*(1/(k:ℝ))=1/(j:ℝ) := by field_simp
-  rw [he'] at hr
-  exact hr
-
-end FiniteLaw
-
-def OracleTracePair.restrict {q E V : ℝ} {x y : List Bool} {j k : ℕ}
-    (C : OracleTracePair q x y k E V) (hj : 1≤j) (hjk : j≤k) :
-    OracleTracePair q x y j E V := by
-  let := C.finiteOutcome
-  exact {C with moment := (C.first.evenMomentSq_mono_order C.score hj hjk).trans C.moment}
-
-/-- Null-mass outcomes are assigned zero score. This proves, rather than
- assumes, the pointwise positivity needed by the finite simulation API. -/
-def ConcreteLevelPair.oracle {q N Δ d ω F : ℝ} {hq : q∈Set.Icc 0 1} {i k : ℕ}
-    (C : ConcreteLevelPair q hq N Δ d ω F i k) :
-    OracleTracePair q C.left C.right k C.error (N^(-d)) := by
-  let := C.finiteOutcome
-  let Z := fun a=>if C.firstLaw.mass a=0 then 0 else C.score a
-  have hden (a) : C.secondLaw.mass a=C.firstLaw.mass a*(1+Z a) := by
-    dsimp [Z]
-    split_ifs with h
-    · simpa only [h,zero_mul] using C.density a
-    · exact C.density a
-  have hpos (a) : 0≤1+Z a := by
-    by_cases h : C.firstLaw.mass a=0
-    · simp [Z,h]
-    · have hp : 0<C.firstLaw.mass a := lt_of_le_of_ne (C.firstLaw.nonneg a) (Ne.symm h)
-      exact (mul_nonneg_iff_of_pos_left hp).mp (by rw [←hden]; exact C.secondLaw.nonneg a)
-  have hc : C.firstLaw.expect Z=0 := by
-    have hsum := congrArg (fun f : C.Outcome→ℝ=>∑a,f a) (funext hden)
-    simp only [Finset.sum_add_distrib,mul_add,mul_one] at hsum
-    rw [C.firstLaw.total,C.secondLaw.total] at hsum
-    dsimp [FiniteLaw.expect]
-    linarith only [hsum]
-  have hm : C.firstLaw.evenMomentSq Z k=C.firstLaw.evenMomentSq C.score k := by
-    unfold FiniteLaw.evenMomentSq FiniteLaw.expect
-    congr 1
-    apply Finset.sum_congr rfl
-    intro a _
-    dsimp [Z]
-    split_ifs with h
-    · simp only [h,zero_mul]
-    · rfl
-  exact {
-    Outcome := C.Outcome
-    finiteOutcome := C.finiteOutcome
-    first := C.firstLaw
-    second := C.secondLaw
-    decode := C.decode
-    first_simulation := C.first_simulation
-    second_simulation := C.second_simulation
-    score := Z
-    density := hden
-    likelihood_nonneg := hpos
-    centered := hc
-    moment := hm.le.trans C.score_bound
-  }
-
-theorem sourceScaleProduct_step_le (B D i : ℕ) (hB : 2≤B) (hi : i<D) :
-    sourceScaleProduct B D i ≤ sourceScaleProduct B D (i+1) := by
-  have he := sourceEpsilon_pos B D
-  have hp : (1:ℝ)≤2^(sourceDimension B i) := one_le_pow₀ (by norm_num)
-  have ha : scaleA (sourceDimension B i) 0 (sourceEpsilon B D)≤1 := by
-    unfold scaleA
-    norm_num only [pow_zero,mul_one]
-    have hh := (div_le_iff₀ (by positivity : (0:ℝ)<2^(sourceDimension B i))).mpr
-      (le_mul_of_one_le_right (by linarith only [he] : 0≤1+sourceEpsilon B D) hp)
-    linarith only [hh]
-  have hs : 0 ≤ sourceScaleProduct B D (i+1) := (lt_of_lt_of_le (by positivity) (sourceScaleProduct_lower B D (i+1) hB)).le
-  have hh := mul_le_mul_of_nonneg_right ha hs
-  unfold sourceScaleProduct at *
-  rw [Finset.prod_eq_prod_Ico_succ_bot hi]
-  simpa only [one_mul] using hh
-
-theorem sourceHalfMoment_step (B D i : ℕ) {n : ℝ} (hB : 2≤B) (hi : i<D) (hn : 1<n) :
-    sourceHalfMoment B D n (i+1) ≤ sourceHalfMoment B D n i := by
-  have hlog : 0<Real.log (sourceTargetScale B D n i) := by
-    rw [sourceTargetScale,Real.log_rpow (by linarith only [hn] : 0<n)]
-    exact mul_pos (lt_of_lt_of_le (by positivity) (sourceScaleProduct_lower B D i hB)) (Real.log_pos hn)
-  have hlog' : Real.log (sourceTargetScale B D n i) ≤ Real.log (sourceTargetScale B D n (i+1)) := by
-    rw [sourceTargetScale,sourceTargetScale,Real.log_rpow (by linarith only [hn] : 0<n),Real.log_rpow (by linarith only [hn] : 0<n)]
-    exact mul_le_mul_of_nonneg_right (sourceScaleProduct_step_le B D i hB hi) (Real.log_pos hn).le
-  unfold sourceHalfMoment
-  apply Nat.ceil_mono
-  have hd := (totalScoreExponent_bounds B D hB).1
-  have hl := (Real.log_pos hn).le
-  exact div_le_div_of_nonneg_left (by positivity) hlog hlog'
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-def sourceGapFloor (B : ℕ) : ℝ := 1/(128*(2:ℝ)^(2*B))
-
-theorem sourceGapFloor_pos (B : ℕ) : 0<sourceGapFloor B := by unfold sourceGapFloor; positivity
-
-theorem sourceGapFloor_le_one (B : ℕ) : sourceGapFloor B≤1 := by
-  unfold sourceGapFloor
-  apply (div_le_one (by positivity)).mpr
-  have hh : (1:ℝ)≤2^(2*B) := one_le_pow₀ (by norm_num)
-  linarith only [hh]
-
-theorem sourceEpsilon_gap_small (B D : ℕ) : 1000*sourceEpsilon B D ≤ sourceGapFloor B := by
-  have hp : (1000:ℝ)*128*2^(2*B) ≤ 2^(4*B+100) := by
-    have hpow : (2:ℝ)^(2*B+17) ≤ 2^(4*B+100) := pow_le_pow_right₀ (by norm_num) (by omega)
-    rw [pow_add] at hpow
-    norm_num only [show (2:ℝ)^17=131072 by norm_num] at hpow
-    have hpos : 0≤(2:ℝ)^(2*B) := by positivity
-    linarith only [hpow,hpos]
-  have hd : 1≤(D:ℝ)+1 := by linarith [Nat.cast_nonneg D (α:=ℝ)]
-  have hh := mul_le_mul_of_nonneg_left hd (show 0≤(2:ℝ)^(4*B+100) by positivity)
-  unfold sourceEpsilon sourceGapFloor
-  rw [←mul_div_assoc, mul_one]
-  apply (div_le_div_iff₀ (by positivity) (by positivity)).mpr
-  nlinarith only [hp,hh]
-
-theorem sourceHalfMoment_strong (B D i : ℕ) {n : ℝ} (hB : 2≤B) (hn : 1<n) :
-    (10^8:ℝ)*2^(10*B)*(totalScoreExponent B D+B+1)*Real.log n ≤
-      (sourceHalfMoment B D n i:ℝ)*Real.log (sourceTargetScale B D n i) := by
-  have hlog : 0<Real.log (sourceTargetScale B D n i) := by
-    rw [sourceTargetScale,Real.log_rpow (by linarith only [hn] : 0<n)]
-    exact mul_pos (lt_of_lt_of_le (by positivity) (sourceScaleProduct_lower B D i hB)) (Real.log_pos hn)
-  exact (div_le_iff₀ hlog).mp (Nat.le_ceil _)
-
-theorem source_clipping_power_delta (B D i : ℕ) {n gap : ℝ}
-    (hB : 2≤B) (hn : 4≤n) (hg : sourceGapFloor B ≤ gap) :
-    3*(sourceTargetScale B D n i)^(-gap*(sourceHalfMoment B D n i:ℝ)/4) ≤
-      n^(-200*(totalScoreExponent B D+B+1)) := by
-  let N := sourceTargetScale B D n i
-  let A := totalScoreExponent B D+B+1
-  let k := sourceHalfMoment B D n i
-  have hn1 : 1<n := by linarith only [hn]
-  have hn0 : 0<n := by linarith only [hn]
-  have hN : 0<N := Real.rpow_pos_of_pos hn0 _
-  have hA : 1≤A := by dsimp [A]; linarith only [(totalScoreExponent_bounds B D hB).1,Nat.cast_nonneg B (α:=ℝ)]
-  have hlogN : 0≤Real.log N := by
-    dsimp [N]
-    rw [sourceTargetScale,Real.log_rpow hn0]
-    exact mul_nonneg (lt_of_lt_of_le (by positivity) (sourceScaleProduct_lower B D i hB)).le (Real.log_pos hn1).le
-  have hk := sourceHalfMoment_strong B D i hB hn1
-  change (10^8:ℝ)*2^(10*B)*A*Real.log n ≤ (k:ℝ)*Real.log N at hk
-  have hp : (2:ℝ)^(2*B) ≤ 2^(10*B) := pow_le_pow_right₀ (by norm_num) (by omega)
-  have hc : 1000 ≤ sourceGapFloor B/4*((10^8:ℝ)*2^(10*B)) := by
-    unfold sourceGapFloor
-    calc
-      (1000:ℝ) ≤ (10^8*2^(10*B))/(512*2^(2*B)) := by
-        apply (le_div_iff₀ (by positivity : (0:ℝ)<512*2^(2*B))).mpr
-        nlinarith only [hp,show (0:ℝ) ≤ 2^(2*B) by positivity]
-      _ = _ := by field_simp; ring
-  have hh := mul_le_mul_of_nonneg_left hk (show 0 ≤ sourceGapFloor B/4 by positivity [sourceGapFloor_pos B])
-  have hh' := mul_le_mul_of_nonneg_right hc (show 0≤A*Real.log n by positivity [Real.log_pos hn1])
-  have hhg := mul_le_mul_of_nonneg_right hg (show 0≤(k:ℝ)*Real.log N/4 by positivity)
-  have hdec : 1000*A*Real.log n ≤ gap*(k:ℝ)/4*Real.log N := by nlinarith only [hh,hh',hhg]
-  have h3 : Real.log 3 ≤ A*Real.log n := by
-    have hl : Real.log 3 ≤ Real.log n := Real.log_le_log (by norm_num) (by linarith only [hn])
-    exact hl.trans (le_mul_of_one_le_left (Real.log_pos hn1).le hA)
-  rw [show (3:ℝ)=Real.exp (Real.log 3) by rw [Real.exp_log (by norm_num)],Real.rpow_def_of_pos hN,←Real.exp_add,
-    Real.rpow_def_of_pos hn0]
-  exact Real.exp_le_exp.mpr (by nlinarith only [hdec,h3,mul_nonneg (by linarith only [hA] : 0≤A) (Real.log_pos hn1).le])
-
-end TraceReconstruction
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-def sourceWorkSize (B D : ℕ) : ℝ :=
-  totalScoreExponent B D+(sourceDegreeCutoff B:ℝ)+B+10
-
-theorem sourceWorkSize_pos (B D : ℕ) (hB : 2≤ B) : 0< sourceWorkSize B D := by
-  unfold sourceWorkSize
-  have hd := (totalScoreExponent_bounds B D hB).1
-  have hr := Nat.cast_nonneg (sourceDegreeCutoff B) (α:=ℝ)
-  have hb := Nat.cast_nonneg B (α:=ℝ)
-  linarith only [hd,hr,hb]
-
-theorem sourceCancellationOrder_lower (B D : ℕ) :
-    100*sourceWorkSize B D ≤ (sourceCancellationOrder B D:ℝ)*sourceEpsilon B D := by
-  exact (div_le_iff₀ (sourceEpsilon_pos B D)).mp (Nat.le_ceil _)
-
-theorem source_smoothing_absorb {B D : ℕ} {N K ω : ℝ} {m : ℕ}
-    (hB : 2≤ B) (hN : 1≤ N) (hK : 0≤ K)
-    (hm : hiddenBinomialCap m ω ≤ K*N^(-sourceEpsilon B D/2))
-    (hpref : sourceSmoothingPrefactor (sourceCancellationOrder B D) K ≤ N^(sourceEpsilon B D)) :
-    hiddenSmoothingBound m (sourceCancellationOrder B D) ω ≤ N^(-20*sourceWorkSize B D) := by
-  have hN0 : 0< N := by linarith only [hN]
-  apply (hiddenSmoothingBound_scale hN0 hK hm).trans
-  apply (mul_le_mul_of_nonneg_right hpref (by positivity)).trans
-  rw [←Real.rpow_add hN0]
-  apply Real.rpow_le_rpow_of_exponent_le hN
-  have hs := sourceCancellationOrder_lower B D
-  have hw := sourceWorkSize_pos B D hB
-  have he := sourceEpsilon_pos B D
-  have he' : sourceEpsilon B D ≤ 1 := by
-    simpa only [pow_zero,one_mul] using sourceEpsilon_power_small B D 0 (by omega)
-  have hW : 10≤ sourceWorkSize B D := by
-    unfold sourceWorkSize
-    have hd := (totalScoreExponent_bounds B D hB).1
-    have hr := Nat.cast_nonneg (sourceDegreeCutoff B) (α:=ℝ)
-    have hb := Nat.cast_nonneg B (α:=ℝ)
-    linarith only [hd,hr,hb]
-  nlinarith only [hs,he,he',hW]
-
-/-- The actual sparse count controls both extra forced roles and clipping. -/
-theorem sparse_power_absorb {N ε gap W L x : ℝ}
-    (hN : 1≤ N) (he : 0≤ε) (hg : 2*ε≤ gap) (hW : 0≤ W) (hL : 1≤ L)
-    (hx : x≤ W*L*N^(-gap)) (hfac : 8*W*L^2≤ N^ε) :
-    x≤ N^(-gap/2)/8 ∧ x≤1/2 ∧ x*L≤1 := by
-  have hN0 : 0< N := by linarith only [hN]
-  have hL0 : 0≤ L := by linarith only [hL]
-  have hg0 : 0≤ gap := by linarith only [he,hg]
-  have hwl : 8*W*L≤ N^ε := by
-    have hh : L≤ L^2 := by nlinarith only [hL,sq_nonneg (L-1)]
-    exact (mul_le_mul_of_nonneg_left hh (by positivity : 0≤8*W)).trans hfac
-  have hx' : x≤ N^(ε-gap)/8 := by
-    have hh := mul_le_mul_of_nonneg_right hwl (Real.rpow_pos_of_pos hN0 (-gap)).le
-    rw [←Real.rpow_add hN0] at hh
-    have heq : ε+-gap=ε-gap := by ring
-    rw [heq] at hh
-    nlinarith only [hx,hh]
-  have hsmall := Real.rpow_le_rpow_of_exponent_le hN (show ε-gap≤-gap/2 by linarith only [hg])
-  have hxhalf : x≤ N^(-gap/2)/8 := hx'.trans (div_le_div_of_nonneg_right hsmall (by norm_num))
-  have hone : N^(-gap/2)≤1 := Real.rpow_le_one_of_one_le_of_nonpos hN (by linarith only [hg0])
-  refine ⟨hxhalf,by linarith only [hxhalf,hone],?_⟩
-  have hh := mul_le_mul_of_nonneg_right hfac (Real.rpow_pos_of_pos hN0 (-gap)).le
-  rw [←Real.rpow_add hN0] at hh
-  have hh' := mul_le_mul_of_nonneg_right hx hL0
-  have hr := Real.rpow_le_one_of_one_le_of_nonpos hN (show ε+-gap≤0 by linarith only [hg,he])
-  nlinarith only [hh,hh',hr]
-
-theorem source_weight_square {N e d L : ℝ} {k : ℕ} (hN : 0< N)
-    (hL : L=N^(-e*d/2)) :
-    ((6*(2*k:ℕ):ℝ)*L)^2=144*(k:ℝ)^2*N^(-e*d) := by
-  rw [hL,mul_pow]
-  have hr : (N^(-e*d/2))^2=N^(-e*d) := by
-    rw [←Real.rpow_natCast,←Real.rpow_mul hN.le]
-    norm_num only [Nat.cast_ofNat]
-    congr 1
-    ring
-  rw [hr]
-  push_cast
-  ring
-
-theorem source_high_tail_margin (B : ℕ) {d : ℝ} (_hB : 2≤ B) (hd : 0≤ d) :
-    d+blockGain B+1 ≤ ((d+1)*sourceGapFloor B)/2*(sourceDegreeCutoff B:ℝ) := by
-  have hp : (2:ℝ)^(2*B+8)*(B+2) ≤ 2^(10*B+40) := by
-    have hb : (B:ℝ)+2 ≤ (2:ℝ)^(B+2) := by
-      have hh : B+1 < 2^(B+1) := Nat.lt_two_pow_self
-      have hh' : B+2≤2^(B+1) := by omega
-      have hcast : (B:ℝ)+2≤(2:ℝ)^(B+1) := by exact_mod_cast hh'
-      exact hcast.trans (pow_le_pow_right₀ (by norm_num) (by omega))
-    have hh := mul_le_mul_of_nonneg_left hb (show (0:ℝ)≤2^(2*B+8) by positivity)
-    rw [←pow_add] at hh
-    exact hh.trans (pow_le_pow_right₀ (by norm_num) (by omega))
-  have hc : (B:ℝ)+2 ≤ sourceGapFloor B/2*(sourceDegreeCutoff B:ℝ) := by
-    unfold sourceGapFloor sourceDegreeCutoff
-    push_cast
-    calc
-      (B:ℝ)+2 ≤ 2^(10*B+40)/(256*2^(2*B)) := by
-        apply (le_div_iff₀ (by positivity : (0:ℝ)<256*2^(2*B))).mpr
-        rw [pow_add] at hp
-        norm_num only [show (2:ℝ)^8=256 by norm_num] at hp
-        nlinarith only [hp]
-      _ = _ := by field_simp; norm_num
-  have hh := mul_le_mul_of_nonneg_left hc (show 0≤ d+1 by linarith only [hd])
-  unfold blockGain
-  have hb : (0:ℝ)≤ B := Nat.cast_nonneg _
-  nlinarith only [hh,hd,hb,mul_nonneg hd hb]
-
-end TraceReconstruction
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem rpow_coefficient_absorb {N ε e d c : ℝ} (hN : 1≤ N)
-    (hc : 3*c≤ N^ε) (he : ε+e≤ d) : c*N^e ≤ N^d/3 := by
-  have hN0 : 0< N := by linarith only [hN]
-  have hh := mul_le_mul_of_nonneg_right hc (Real.rpow_pos_of_pos hN0 e).le
-  rw [←Real.rpow_add hN0] at hh
-  have hr := Real.rpow_le_rpow_of_exponent_le hN he
-  linarith only [hh,hr]
-
-/-- Closing the three explicit sharp classes, with the source's fixed
- finite degree cutoff and no hidden dependence on N in their constants. -/
-theorem sharp_energy_absorb {N ε d M Fdec gap Cr Cf E : ℝ} {R : ℕ}
-    (hN : 1≤ N) (he : ε≤1) (hM : d+(R:ℝ)+1≤ M) (hd : 0≤ d) (_hR : 0≤(R:ℝ))
-    (hbad : 384*Cr≤ N^ε) (hforce : 192*Cf≤ N^ε) (htail : 1536≤ N^ε)
-    (hfdec : d+ε≤ Fdec) (hgap : d+ε≤ gap*(R:ℝ))
-    (hE : E ≤ Cr*N^R*((N^(-20*M))^2+(N^(-20*M))^2)+Cf*N^(-Fdec)+8*N^(-gap*(R:ℝ))) :
-    64*E≤ N^(-d) := by
-  have hN0 : 0< N := by linarith only [hN]
-  have hpow : N^R*(N^(-20*M))^2=N^((R:ℝ)-40*M) := by
-    rw [←Real.rpow_natCast (N^(-20*M)) 2,←Real.rpow_mul hN0.le,←Real.rpow_natCast N R,←Real.rpow_add hN0]
-    norm_num only [Nat.cast_ofNat]
-    congr 1
-    ring
-  have hb := rpow_coefficient_absorb hN (c:=128*Cr) (by nlinarith only [hbad])
-    (e:=(R:ℝ)-40*M) (d:=-d) (by linarith only [hM,hd,he])
-  have hf := rpow_coefficient_absorb hN (c:=64*Cf) (by nlinarith only [hforce])
-    (e:=-Fdec) (d:=-d) (by linarith only [hfdec])
-  have ht := rpow_coefficient_absorb hN (c:=512) (by norm_num at htail ⊢; exact htail)
-    (e:=-gap*(R:ℝ)) (d:=-d) (by linarith only [hgap])
-  have hh := mul_le_mul_of_nonneg_left hE (by norm_num : (0:ℝ)≤64)
-  have heq : 64*(Cr*N^R*((N^(-20*M))^2+(N^(-20*M))^2)+Cf*N^(-Fdec)+8*N^(-gap*(R:ℝ))) =
-      128*Cr*N^((R:ℝ)-40*M)+64*Cf*N^(-Fdec)+512*N^(-gap*(R:ℝ)) := by
-    calc
-      _ = 128*Cr*(N^R*(N^(-20*M))^2)+64*Cf*N^(-Fdec)+512*N^(-gap*(R:ℝ)) := by ring
-      _ = _ := by rw [hpow]
-  rw [heq] at hh
-  linarith only [hh,hb,hf,ht]
-
-end TraceReconstruction
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem sourceDimension_bounds (B i : ℕ) (hB : 2≤ B) :
-    2≤ sourceDimension B i ∧ sourceDimension B i≤ B := by
-  unfold sourceDimension
-  split_ifs <;> omega
-
-theorem totalScoreExponent_mono (B : ℕ) (hB : 2≤ B) : Monotone (totalScoreExponent B) := by
-  apply monotone_nat_of_le_succ
-  intro i
-  rw [totalScoreExponent_step]
-  exact le_add_of_nonneg_right (blockGain_pos _ (sourceDimension_bounds B i hB).1).le
-
-theorem sourceTargetScale_bounds (B D i : ℕ) {n : ℝ} (hB : 2≤ B) (hn : 1≤ n) :
-    1≤ sourceTargetScale B D n i ∧ sourceTargetScale B D n i≤ n := by
-  have hp : 0≤ sourceScaleProduct B D i :=
-    (show (0:ℝ)≤1/(2*2^(B*D)) by positivity).trans (sourceScaleProduct_lower B D i hB)
-  refine ⟨Real.one_le_rpow hn hp,?_⟩
-  simpa only [sourceTargetScale,Real.rpow_one] using Real.rpow_le_rpow_of_exponent_le hn (sourceScaleProduct_le_one B D i hB)
-
-theorem ConcreteLevelPair.error_le {q N Δ d ω F : ℝ} {hq : q∈Set.Icc 0 1} {i k : ℕ}
-    (C : ConcreteLevelPair q hq N Δ d ω F i k) (hN : 0< N) (hΔ : 0≤Δ) :
-    C.error≤((i:ℝ)+1)*Δ*N := by
-  have hl : 0<(C.left.length:ℝ) := lt_of_lt_of_le (by linarith only [hN]) C.length_lower
-  exact ((div_le_iff₀ hl).mp C.normalized_error).trans
-    (mul_le_mul_of_nonneg_left C.length_upper (by positivity))
-
-theorem source_warmup_score_margin (B D : ℕ) {d : ℝ} (hB : 2≤ B)
-    (hd : 11/5≤ d) (hd' : d≤3+(B:ℝ)*D) :
-    27/50≤ scaleA 2 0 (sourceEpsilon B D)*d ∧ scaleA 2 1 (sourceEpsilon B D)≤1/2 := by
-  have he := sourceEpsilon_pos B D
-  have hloss := sourceEpsilon_loss B D 2 d hB (by linarith only [hd]) hd'
-  norm_num only [show (2:ℝ)^2=4 by norm_num,Nat.cast_ofNat] at hloss
-  have hlo := mul_le_mul_of_nonneg_right (scaleA_zero_simple_lower 2 (sourceEpsilon B D) he.le)
-    (show 0≤ d by linarith only [hd])
-  norm_num only [show (2:ℝ)^2=4 by norm_num] at hlo
-  constructor
-  · nlinarith only [hloss,hlo,hd,he]
-  · have hh := scaleA_le_geometric 2 1 (sourceEpsilon B D) he.le (by omega)
-    norm_num at hh ⊢
-    exact hh
-
-theorem source_main_score_margin (B D : ℕ) {d : ℝ} (hB : 2≤ B)
-    (hd : (2:ℝ)^B+2≤ d) (hd' : d≤3+(B:ℝ)*D) :
-    1+1/(2:ℝ)^B ≤ scaleA B 0 (sourceEpsilon B D)*d := by
-  let X := (2:ℝ)^B
-  let ε := sourceEpsilon B D
-  have hX : 4≤ X := by
-    have hh := pow_le_pow_right₀ (by norm_num : (1:ℝ)≤2) hB
-    norm_num at hh
-    exact hh
-  have hX0 : 0< X := by linarith only [hX]
-  have he : 0<ε := sourceEpsilon_pos _ _
-  have hd0 : 0≤ d := by dsimp [X] at hX; linarith only [hd,hX]
-  have hloss := sourceEpsilon_loss B D B d le_rfl hd0 hd'
-  change ((X-1)*d+3*B*(X/2))*ε<1/100 at hloss
-  have hnonneg : 0≤3*(B:ℝ)*(X/2)*ε := by positivity
-  have hde : 0≤ d*ε := mul_nonneg hd0 he.le
-  have hmul : X*d*ε≤1/50 := by
-    have hxx : 0≤(X-2)*(d*ε) := mul_nonneg (by linarith only [hX]) hde
-    nlinarith only [hloss,hnonneg,hxx]
-  have hlo := mul_le_mul_of_nonneg_right (scaleA_zero_simple_lower B ε he.le) hd0
-  change (1/X-ε)*d≤ scaleA B 0 ε*d at hlo
-  have hgoal : 1+1/X≤(1/X-ε)*d := by
-    apply (mul_le_mul_iff_left₀ hX0).mp
-    field_simp
-    change X+2 ≤ d at hd
-    nlinarith only [hd,hmul]
-  exact hgoal.trans hlo
-
-end TraceReconstruction
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem sourceSparseCount_ge_one (T : ℕ→ℕ) (s b : ℕ) (hT : ∀j<b,1 ≤ T (j+1)) :
-    1 ≤ sourceSparseCount T s b := by
-  unfold sourceSparseCount
-  apply one_le_mul_of_one_le_of_one_le (one_le_pow₀ (by omega))
-  exact Finset.one_le_prod (fun j hj=>hT j (Finset.mem_range.mp hj))
-
-/-- Scalar closure of the exact sparse/forced block budget. All assumptions
- here are explicit elementary coefficient bounds, not oracle conclusions. -/
-theorem source_block_energy_close (T n : ℕ→ℕ) (s b R : ℕ) (ω N ε d dnew K W M gap : ℝ)
-    (hN : 1 ≤ N) (he : 0 ≤ ε) (he1 : ε ≤ 1) (hK : 0 ≤ K) (hW : 0 ≤ W)
-    (hd : 0 ≤ dnew) (hM : dnew+(R:ℝ)+1 ≤ M) (hg : 2*ε ≤ gap)
-    (hf : dnew+ε ≤ forcedDecay (b+1) ε d) (hR : dnew+ε ≤ gap/2*(R:ℝ))
-    (hfac : 8*W*((2^s:ℕ):ℝ)^2 ≤ N^ε)
-    (hbad : 384*sourceBadRoleFactor b R ≤ N^ε)
-    (hforce : 192*(sourceForcedRoleFactor b R*W^(2^(b+1))*sourceSkeletonPrefactor s b K) ≤ N^ε)
-    (htail : 1536 ≤ N^ε)
-    (hleaf : (SourceLeafCount T (b+1):ℝ) ≤ N)
-    (hcount : 1 ≤ sourceSparseCount T s b)
-    (k : ℕ) (L : ℝ)
-    (hw : ((6*(2*k:ℕ):ℝ)*L)^2 ≤ W*N^(-scaleA (b+1) 0 ε*d))
-    (hsparse : ((6*(2*k:ℕ):ℝ)*L)^2*(sourceSparseCount T s b:ℝ) ≤
-      W*((2^s:ℕ):ℝ)*N^(-gap))
-    (hn : ∀j ≤ b,hiddenBinomialCap (n j) ω ≤ K*N^(-ε/2))
-    (hgcap : ∀j<b,hiddenBinomialCap (sourceScheduledTrials T n (j+1)) ω ≤
-      K*N^(-(scaleKappa (b+1) (j+2) ε+ε)/2)) :
-    ((6*(2*k:ℕ):ℝ)*L)^2*(sourceSparseCount T s b:ℝ) ≤ N^(-gap/2)/8 ∧
-    ((6*(2*k:ℕ):ℝ)*L)^2*(sourceSparseCount T s b:ℝ) ≤ 1/2 ∧
-    64*sourceSharpEnergyBudget T n s b R ω (N^(-20*M)) (N^(-20*M)) ((6*(2*k:ℕ):ℝ)*L) ≤ N^(-dnew) := by
-  let w := (6*(2*k:ℕ):ℝ)*L
-  have hmot : 1 ≤ ((2^s:ℕ):ℝ) := by exact_mod_cast (one_le_pow₀ (by omega : 1 ≤ (2:ℕ)) : 1 ≤ (2:ℕ)^s)
-  have hx := sparse_power_absorb hN he hg hW hmot hsparse hfac
-  have hwone : w^2 ≤ 1 := by
-    have hc : (1:ℝ) ≤ sourceSparseCount T s b := by exact_mod_cast hcount
-    have hh := mul_le_mul_of_nonneg_left hc (sq_nonneg w)
-    dsimp [w] at hh
-    nlinarith only [hh,hx.2.1]
-  have hsmall : w^2*((2^s:ℕ):ℝ)*(sourceSparseCount T s b:ℝ) ≤ 1 := by
-    dsimp [w]
-    nlinarith only [hx.2.2]
-  have hr : w^2*(sourceSparseCount T s b:ℝ) ≤ 1*N^(-(gap/2)) := by
-    have hp : 0 ≤ N^(-gap/2) := Real.rpow_nonneg (by linarith only [hN]) _
-    dsimp [w]
-    rw [←neg_div]
-    nlinarith only [hx.1,hp]
-  have hnum := source_sharp_numeric T n s b R ω (N^(-20*M)) (N^(-20*M)) w N ε d K W 1 (gap/2)
-    hN hK hW (by norm_num) hleaf hwone hw hsmall hx.2.1 hr hn hgcap
-  simp only [one_pow,mul_one] at hnum
-  refine ⟨hx.1,hx.2.1,?_⟩
-  exact sharp_energy_absorb hN he1 hM hd (Nat.cast_nonneg _) hbad hforce htail hf hR hnum
-
-end TraceReconstruction
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- Instantiation at the manuscript's literal block dimension, integer
- schedules, source cutoff, moment envelope, and full next score exponent. -/
-theorem source_level_energy (B D i ell h k : ℕ) (C N F ω K : ℝ)
-    (hB : 2 ≤ B) (hi : i<D) (hN : 1 ≤ N) (hC : 1 ≤ C) (hF : 1 ≤ F) (hω : 0<ω)
-    (H : SourceArithmeticBounds C N F (sourceEpsilon B D) (sourceCancellationOrder B D)
-      (sourceDimension B i) ell h)
-    (hk : (k:ℝ) ≤ sourceMomentEnvelope B D)
-    (hK : 200*(C^(2*sourceDimension B i)*F)/ω ≤ K)
-    (hfac : 8*sourceWeightEnvelope B D*((2^(sourceCancellationOrder B D):ℕ):ℝ)^2 ≤ N^(sourceEpsilon B D))
-    (hbad : 384*sourceBadRoleFactor (sourceDimension B i-1) (sourceDegreeCutoff B) ≤ N^(sourceEpsilon B D))
-    (hforce : 192*(sourceForcedRoleFactor (sourceDimension B i-1) (sourceDegreeCutoff B)*
-        (sourceWeightEnvelope B D)^(2^(sourceDimension B i))*
-        sourceSkeletonPrefactor (sourceCancellationOrder B D) (sourceDimension B i-1) K) ≤ N^(sourceEpsilon B D))
-    (htail : 1536 ≤ N^(sourceEpsilon B D))
-    (hsmooth : sourceSmoothingPrefactor (sourceCancellationOrder B D) K ≤ N^(sourceEpsilon B D)) :
-    let b := sourceDimension B i
-    let ε := sourceEpsilon B D
-    let s := sourceCancellationOrder B D
-    let T := sourceBlockWindows N b ε
-    let t := sourceArithmeticTrials C N b ε ell h
-    let L := N^(-scaleA b 0 ε*totalScoreExponent B i/2)
-    (((6*(2*k:ℕ):ℝ)*L)^2*(sourceSparseCount T s (b-1):ℝ) ≤
-        N^(-((totalScoreExponent B i+1)*sourceGapFloor B)/2)/8) ∧
-    (((6*(2*k:ℕ):ℝ)*L)^2*(sourceSparseCount T s (b-1):ℝ) ≤ 1/2) ∧
-    (64*sourceSharpEnergyBudget T t s (b-1) (sourceDegreeCutoff B) ω
-       (N^(-20*sourceWorkSize B D)) (N^(-20*sourceWorkSize B D)) ((6*(2*k:ℕ):ℝ)*L) ≤
-       N^(-totalScoreExponent B (i+1))) ∧
-    (∀j<b,hiddenSmoothingBound (t j) s ω ≤ N^(-20*sourceWorkSize B D)) := by
-  let b := sourceDimension B i
-  let ε := sourceEpsilon B D
-  let s := sourceCancellationOrder B D
-  let d := totalScoreExponent B i
-  let dnew := totalScoreExponent B (i+1)
-  let W := sourceWeightEnvelope B D
-  let gap := (d+1)*sourceGapFloor B
-  let T := sourceBlockWindows N b ε
-  let t := sourceArithmeticTrials C N b ε ell h
-  let L := N^(-scaleA b 0 ε*d/2)
-  have hb := sourceDimension_bounds B i hB
-  have hb1 : b-1+1=b := by dsimp [b]; omega
-  have he := sourceEpsilon_pos B D
-  have he1 : ε ≤ 1/100 := by
-    have hh := sourceEpsilon_gap_small B D
-    have hg := sourceGapFloor_le_one B
-    dsimp [ε]
-    linarith only [hh,hg]
-  have hd : 0 ≤ d := le_trans (by norm_num) (totalScoreExponent_bounds B i hB).1
-  have hdnew : 0 ≤ dnew := le_trans (by norm_num) (totalScoreExponent_bounds B (i+1) hB).1
-  have hdle : d ≤ totalScoreExponent B D := totalScoreExponent_mono B hB (Nat.le_of_lt hi)
-  have hdnewle : dnew ≤ totalScoreExponent B D := totalScoreExponent_mono B hB (by omega)
-  have hdd : d ≤ 3+(B:ℝ)*D := by
-    have hh := (totalScoreExponent_bounds B D hB).2
-    nlinarith only [hdle,hh]
-  have hN0 : 0<N := by linarith only [hN]
-  have hK0 : 0 ≤ K := le_trans (by positivity : 0 ≤ 200*(C^(2*sourceDimension B i)*F)/ω) hK
-  have hW : 0 ≤ W := by dsimp [W,sourceWeightEnvelope]; exact mul_nonneg (by norm_num) (sq_nonneg _)
-  have hM : dnew+(sourceDegreeCutoff B:ℝ)+1 ≤ sourceWorkSize B D := by
-    unfold sourceWorkSize
-    linarith only [hdnewle,Nat.cast_nonneg B (α:=ℝ)]
-  have hg : 2*ε ≤ gap := by
-    have hf := sourceGapFloor_pos B
-    have hh := sourceEpsilon_gap_small B D
-    dsimp [gap,ε]
-    nlinarith only [hh,hd,mul_nonneg hd hf.le,he]
-  have hfdec : dnew+ε ≤ forcedDecay b ε d := by
-    have hh := forcedDecay_margin B D b d (by dsimp [b]; omega) hb.2 hd hdd
-    have heq : dnew=d+blockGain b := totalScoreExponent_step B i
-    rw [heq]
-    unfold blockGain
-    linarith only [hh,he1]
-  have hR : dnew+ε ≤ gap/2*(sourceDegreeCutoff B:ℝ) := by
-    have hh := source_high_tail_margin B hB hd
-    have heq : dnew=d+blockGain b := totalScoreExponent_step B i
-    have hbg : blockGain b ≤ blockGain B := by unfold blockGain; exact (by
-      have hcast : (b:ℝ) ≤ B := by exact_mod_cast hb.2
-      linarith only [hcast])
-    dsimp [gap]
-    rw [heq]
-    linarith only [hh,hbg,he1]
-  have hw : ((6*(2*k:ℕ):ℝ)*L)^2 ≤ W*N^(-scaleA b 0 ε*d) := by
-    rw [source_weight_square hN0 (show L=N^(-scaleA b 0 ε*d/2) from rfl)]
-    have hh := mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (Nat.cast_nonneg k) hk 2) (by norm_num : (0:ℝ)≤144)
-    exact mul_le_mul_of_nonneg_right hh (Real.rpow_pos_of_pos hN0 _).le
-  have hsparse : ((6*(2*k:ℕ):ℝ)*L)^2*(sourceSparseCount T s (b-1):ℝ) ≤ W*((2^s:ℕ):ℝ)*N^(-gap) := by
-    apply (source_sparse_weight_bound N ε d W _ b s hN0 hW hw).trans
-    apply mul_le_mul_of_nonneg_left _ (by positivity)
-    apply Real.rpow_le_rpow_of_exponent_le hN
-    have hh := source_level_sparse_gap B D i hB
-    have heq : (d+1)/(128*(2:ℝ)^(2*B))=gap := by dsimp [gap,sourceGapFloor]; ring
-    rw [heq] at hh
-    linarith only [hh]
-  have hcaps := H.binomial_caps hN0 hC hF hω
-  have hncap (j) (hj : j<b) : hiddenBinomialCap (t j) ω ≤ K*N^(-ε/2) :=
-    (hcaps.1 j hj).trans (mul_le_mul_of_nonneg_right hK (Real.rpow_pos_of_pos hN0 _).le)
-  have hgcap (j) (hj : j<b-1) : hiddenBinomialCap (sourceScheduledTrials T t (j+1)) ω ≤
-      K*N^(-(scaleKappa b (j+2) ε+ε)/2) :=
-    (hcaps.2 j (by omega)).trans (mul_le_mul_of_nonneg_right hK (Real.rpow_pos_of_pos hN0 _).le)
-  have hc : 1 ≤ sourceSparseCount T s (b-1) := sourceSparseCount_ge_one T s (b-1)
-    (fun j _=>Nat.succ_le_of_lt (H.window_positive (j+1)))
-  have HE := source_block_energy_close T t s (b-1) (sourceDegreeCutoff B) ω N ε d dnew K W
-    (sourceWorkSize B D) gap hN he.le (by linarith only [he1]) hK0 hW hdnew hM hg
-    (by rw [hb1]; exact hfdec) hR hfac hbad (by rw [hb1]; exact hforce) htail
-    (by rw [hb1]; exact H.leaf_bound hN0) hc k L (by rw [hb1]; exact hw) hsparse
-    (fun j hj=>hncap j (by omega)) (fun j hj=>by rw [hb1]; exact hgcap j hj)
-  refine ⟨HE.1,HE.2.1,HE.2.2,?_⟩
-  intro j hj
-  exact source_smoothing_absorb hB hN hK0 (hncap j hj) hsmooth
-
-end TraceReconstruction
-noncomputable section
-namespace TraceReconstruction
-open Filter
-open scoped Topology Classical
-
-structure SourceNumericMargins (B D : ℕ) (q N : ℝ) : Prop where
-  geometry : 1200*sourceCapEnvelope B D q ≤ N^(sourceEpsilon B D)
-  motif : 48*(2:ℝ)^(sourceCancellationOrder B D) ≤ N^(sourceEpsilon B D)
-  sparse : 8*sourceWeightEnvelope B D*((2:ℝ)^(sourceCancellationOrder B D))^2 ≤ N^(sourceEpsilon B D)
-  smoothing : sourceSmoothingPrefactor (sourceCancellationOrder B D) (sourceCapEnvelope B D q) ≤ N^(sourceEpsilon B D)
-  bad : ∀b ≤ B,384*sourceBadRoleFactor b (sourceDegreeCutoff B) ≤ N^(sourceEpsilon B D)
-  force : ∀b ≤ B,192*(sourceForcedRoleFactor b (sourceDegreeCutoff B)*(sourceWeightEnvelope B D)^(2^(b+1))*
-    sourceSkeletonPrefactor (sourceCancellationOrder B D) b (sourceCapEnvelope B D q)) ≤ N^(sourceEpsilon B D)
-  tail : 1536 ≤ N^(sourceEpsilon B D)
-  error : 64*(sourceCapEnvelope B D q+1)*((D:ℝ)+1) ≤ N^(sourceEpsilon B D)
-  overlap : 1099511627776*(sourceCapEnvelope B D q+1)*(sourceWeightEnvelope B D+1)^2*
-    ((2:ℝ)^(sourceCancellationOrder B D))^(B+2) ≤ N^(sourceEpsilon B D)
-
-theorem sourceWeightEnvelope_nonneg (B D : ℕ) : 0 ≤ sourceWeightEnvelope B D := by
-  unfold sourceWeightEnvelope
-  exact mul_nonneg (by norm_num) (sq_nonneg _)
-
-theorem sourceForcedRoleFactor_nonneg (b R : ℕ) : 0 ≤ sourceForcedRoleFactor b R := by
-  unfold sourceForcedRoleFactor
-  exact Finset.sum_nonneg (fun _ _=>by positivity)
-
-theorem source_forced_prefactor_expUpper (B b : ℕ) (hB : 2 ≤ B) :
-    SourceExpUpper (fun D q=>192*(sourceForcedRoleFactor b (sourceDegreeCutoff B)*
-      (sourceWeightEnvelope B D)^(2^(b+1))*
-      sourceSkeletonPrefactor (sourceCancellationOrder B D) b (sourceCapEnvelope B D q))) := by
-  have H := (((SourceExpUpper.const (sourceForcedRoleFactor b (sourceDegreeCutoff B))).mul
-    ((sourceWeightEnvelope_expUpper B).pow (fun _ _ _=>sourceWeightEnvelope_nonneg _ _) (2^(b+1)))
-    (fun _ _ _=>sourceForcedRoleFactor_nonneg _ _) (fun _ _ _=>pow_nonneg (sourceWeightEnvelope_nonneg _ _) _)).mul
-      (sourceSkeletonPrefactor_expUpper B b hB)
-      (fun _ _ _=>mul_nonneg (sourceForcedRoleFactor_nonneg _ _) (pow_nonneg (sourceWeightEnvelope_nonneg _ _) _))
-      (fun D q hq=>sourceSkeletonPrefactor_nonneg _ _ (sourceCapEnvelope_pos B D hq.1).le))
-  exact (SourceExpUpper.const 192).mul H (fun _ _ _=>by norm_num)
-    (fun D q hq=>mul_nonneg (mul_nonneg (sourceForcedRoleFactor_nonneg _ _)
-      (pow_nonneg (sourceWeightEnvelope_nonneg _ _) _))
-        (sourceSkeletonPrefactor_nonneg _ _ (sourceCapEnvelope_pos B D hq.1).le))
-
-theorem source_overlap_prefactor_expUpper (B : ℕ) (hB : 2 ≤ B) :
-    SourceExpUpper (fun D q=>1099511627776*(sourceCapEnvelope B D q+1)*(sourceWeightEnvelope B D+1)^2*
-      ((2:ℝ)^(sourceCancellationOrder B D))^(B+2)) := by
-  have hK := (sourceCapEnvelope_expUpper B).add (SourceExpUpper.const 1)
-  have hW := (sourceWeightEnvelope_expUpper B).add (SourceExpUpper.const 1)
-  have hL := sourceMotif_expUpper B hB
-  exact (((SourceExpUpper.const 1099511627776).mul hK (fun _ _ _=>by norm_num)
-    (fun D q hq=>by linarith only [sourceCapEnvelope_pos B D hq.1])).mul
-      (hW.pow (fun D _ _=>by linarith only [sourceWeightEnvelope_nonneg B D]) 2)
-      (fun D q hq=>by positivity [sourceCapEnvelope_pos B D hq.1]) (fun _ _ _=>sq_nonneg _)).mul
-        (hL.pow (fun _ _ _=>by positivity) (B+2))
-        (fun D q hq=>by positivity [sourceCapEnvelope_pos B D hq.1]) (fun _ _ _=>by positivity)
-
-theorem source_numeric_margins_eventually (B : ℕ) (ζ : ℝ) (hB : 2 ≤ B) (hζ : ζ∈Set.Ioo 0 1)
-    {n q : ℕ→ℝ} (hn : Tendsto n atTop atTop) (hq : ∀t,q t∈Set.Ioc 0 (1/2:ℝ))
-    (hH : Tendsto (fun t=>(q t)^3*Real.log (n t)) atTop atTop) :
-    ∀ᶠt in atTop,∀i,SourceNumericMargins B (sourceDepth B ζ ((q t)^3*Real.log (n t))) (q t)
-      (sourceTargetScale B (sourceDepth B ζ ((q t)^3*Real.log (n t))) (n t) i) := by
-  have hu (f : ℕ→ℝ→ℝ) (hf : SourceExpUpper f) :=
-    SourceExpUpper.uniform f hf B ζ 1 hB hζ (by norm_num) hn hq hH
-  have hgeom := hu _ ((SourceExpUpper.const 1200).mul (sourceCapEnvelope_expUpper B)
-    (fun _ _ _=>by norm_num) (fun D q hq=>(sourceCapEnvelope_pos B D hq.1).le))
-  have hmot := hu _ ((SourceExpUpper.const 48).mul (sourceMotif_expUpper B hB)
-    (fun _ _ _=>by norm_num) (fun _ _ _=>by positivity))
-  have hsparse := hu _ (((SourceExpUpper.const 8).mul (sourceWeightEnvelope_expUpper B)
-    (fun _ _ _=>by norm_num) (fun _ _ _=>sourceWeightEnvelope_nonneg _ _)).mul
-      ((sourceMotif_expUpper B hB).pow (fun _ _ _=>by positivity) 2)
-      (fun _ _ _=>mul_nonneg (by norm_num) (sourceWeightEnvelope_nonneg _ _)) (fun _ _ _=>sq_nonneg _))
-  have hsmooth := hu _ (sourceSmoothingPrefactor_expUpper B hB)
-  have hbad : ∀ᶠt in atTop,∀b∈Finset.range (B+1),∀i,
-      384*sourceBadRoleFactor b (sourceDegreeCutoff B) ≤
-        (sourceTargetScale B (sourceDepth B ζ ((q t)^3*Real.log (n t))) (n t) i)^
-          (sourceEpsilon B (sourceDepth B ζ ((q t)^3*Real.log (n t)))) := by
-    apply (Filter.eventually_all_finset _).mpr
-    intro b _
-    simpa only [one_mul] using hu _ (SourceExpUpper.const (384*sourceBadRoleFactor b (sourceDegreeCutoff B)))
-  have hforce : ∀ᶠt in atTop,∀b∈Finset.range (B+1),∀i,
-      192*(sourceForcedRoleFactor b (sourceDegreeCutoff B)*
-        (sourceWeightEnvelope B (sourceDepth B ζ ((q t)^3*Real.log (n t))))^(2^(b+1))*
-        sourceSkeletonPrefactor (sourceCancellationOrder B (sourceDepth B ζ ((q t)^3*Real.log (n t))))
-          b (sourceCapEnvelope B (sourceDepth B ζ ((q t)^3*Real.log (n t))) (q t))) ≤
-        (sourceTargetScale B (sourceDepth B ζ ((q t)^3*Real.log (n t))) (n t) i)^
-          (sourceEpsilon B (sourceDepth B ζ ((q t)^3*Real.log (n t)))) := by
-    apply (Filter.eventually_all_finset _).mpr
-    intro b _
-    simpa only [one_mul] using hu _ (source_forced_prefactor_expUpper B b hB)
-  have htail := hu _ (SourceExpUpper.const 1536)
-  have herr := hu _ (((SourceExpUpper.const 64).mul
-    ((sourceCapEnvelope_expUpper B).add (SourceExpUpper.const 1))
-    (fun _ _ _=>by norm_num) (fun D q hq=>by positivity [sourceCapEnvelope_pos B D hq.1])).mul
-    SourcePolyUpper.depth.expUpper (fun D q hq=>by positivity [sourceCapEnvelope_pos B D hq.1])
-      (fun _ _ _=>by positivity))
-  have hov := hu _ (source_overlap_prefactor_expUpper B hB)
-  filter_upwards [hgeom,hmot,hsparse,hsmooth,hbad,hforce,htail,herr,hov] with t hg hm hs hsm hb hf ht he ho
-  intro i
-  refine ⟨?_,?_,?_,?_,fun b h=>hb b (Finset.mem_range.mpr (by omega)) i,
-    fun b h=>hf b (Finset.mem_range.mpr (by omega)) i,?_,?_,?_⟩ <;>
-    first | simpa only [one_mul] using hg i | simpa only [one_mul] using hm i |
-      simpa only [one_mul] using hs i | simpa only [one_mul] using hsm i |
-      simpa only [one_mul] using ht i | simpa only [one_mul] using he i | simpa only [one_mul] using ho i
-
-end TraceReconstruction
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem source_count_envelope (B D i b : ℕ) {q : ℝ}
-    (hq : q∈Set.Ioc 0 (1/2:ℝ)) (hi : i≤ D) (hb : b≤ B) :
-    (sourceOverlapConstant q)^(2*b)*(sourceOverlapConstant q)^(2*B*i+2) ≤
-      (sourceOverlapConstant q)^(2*B*(D+1)+2) := by
-  rw [←pow_add]
-  apply pow_le_pow_right₀ ((by norm_num : (1:ℝ)≤ 2).trans (sourceOverlapConstant_ge_two hq))
-  have hh := Nat.mul_le_mul_left (2*B) hi
-  nlinarith only [hh,hb]
-
-theorem source_cap_envelope (B D i b : ℕ) {q : ℝ}
-    (hq : q∈Set.Ioc 0 (1/2:ℝ)) (hi : i≤ D) (hb : b≤ B) :
-    200*((sourceOverlapConstant q)^(2*b)*(sourceOverlapConstant q)^(2*B*i+2))/initialOverlapLower q ≤
-      sourceCapEnvelope B D q := by
-  dsimp [sourceCapEnvelope]
-  rw [div_eq_mul_inv]
-  exact mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left (source_count_envelope B D i b hq hi hb)
-    (by norm_num)) (inv_pos.mpr (initialOverlapLower_pos q)).le
-
-theorem source_local_envelope (B D i b : ℕ) {q : ℝ}
-    (hq : q∈Set.Ioc 0 (1/2:ℝ)) (hi : i≤ D) (hb : b≤ B) :
-    (sourceOverlapConstant q)^(2*b)*(sourceOverlapConstant q)^(2*B*i+2) ≤
-      sourceCapEnvelope B D q := by
-  have hω := initialOverlapLower_lt_eighth hq.1
-  have hω0 := initialOverlapLower_pos q
-  have hG : 0 ≤ (sourceOverlapConstant q)^(2*b)*(sourceOverlapConstant q)^(2*B*i+2) := by
-    positivity [sourceOverlapConstant_pos hq.1]
-  apply le_trans _ (source_cap_envelope B D i b hq hi hb)
-  apply (le_div_iff₀ hω0).mpr
-  nlinarith only [hω,hG,mul_nonneg hG (show 0 ≤ 200-initialOverlapLower q by linarith only [hω])]
-
-theorem SourceArithmeticBounds.count_bound {C N F ε : ℝ} {s b ell k j : ℕ}
-    (H : SourceArithmeticBounds C N F ε s b ell k) (hC : 1≤ C) (hF : 0≤ F) (hN : 1≤ N)
-    (hj : j≤ b) (ha : 0≤ scaleA b j ε) :
-    ((sourceArithmeticCount C N b ε ell k j+1:ℕ):ℝ) ≤
-      (C^(2*b)*F+1)*N^(scaleA b j ε) := by
-  have hc := (H.counts j hj).2
-  have hc' : (sourceArithmeticCount C N b ε ell k j:ℝ) ≤
-      (C^(2*b)*F)*N^(scaleA b j ε) := by
-    apply hc.trans
-    exact mul_le_mul (mul_le_mul_of_nonneg_right (pow_le_pow_right₀ hC (by omega)) hF)
-      (H.lengths j hj).2 (by positivity) (by positivity)
-  have ho := Real.one_le_rpow hN ha
-  push_cast
-  nlinarith only [hc',ho]
-
-theorem sourceErrorUnit_small {B D : ℕ} {n : ℝ} (hB : 2≤ B) (hn : 4≤ n) :
-    4*n^(-200*(totalScoreExponent B D+B+1)) ≤ n^(-100*(totalScoreExponent B D+B+1)) ∧
-    4*n^(-200*(totalScoreExponent B D+B+1)) ≤ 1/2 := by
-  let A := totalScoreExponent B D+B+1
-  have hA : 1≤ A := by dsimp [A]; linarith only [(totalScoreExponent_bounds B D hB).1,Nat.cast_nonneg B (α:=ℝ)]
-  have hn0 : 0<n := by linarith only [hn]
-  have hn1 : 1≤ n := by linarith only [hn]
-  have hpow : 4≤ n^(100*A) := hn.trans (by
-    simpa only [Real.rpow_one] using Real.rpow_le_rpow_of_exponent_le hn1 (show 1≤ 100*A by linarith only [hA]))
-  have hh := mul_le_mul_of_nonneg_right hpow (Real.rpow_pos_of_pos hn0 (-200*A)).le
-  rw [←Real.rpow_add hn0] at hh
-  have heq : 100*A+(-200*A)=-100*A := by ring
-  rw [heq] at hh
-  have hsmall : n^(-100*A) ≤ 1/2 := by
-    have hh' := Real.rpow_le_rpow_of_exponent_le hn1 (show -100*A≤ (-1:ℝ) by linarith only [hA])
-    rw [Real.rpow_neg_one] at hh'
-    exact hh'.trans (by apply (inv_le_comm₀ hn0 (by norm_num)).mpr; norm_num; linarith only [hn])
-  exact ⟨hh,hh.trans hsmall⟩
-
-end TraceReconstruction
-noncomputable section
-namespace TraceReconstruction
-open Filter
-open scoped Topology Classical
-
-def sourceTailDenominator (B D : ℕ) (q : ℝ) : ℝ :=
-  512+9600*sourceCapEnvelope B D q*(initialOverlapLower q)⁻¹^4
-
-def sourceTailExponent (B D : ℕ) : ℝ :=
-  300*(totalScoreExponent B D+B+1)+40*sourceWorkSize B D+5
-
-theorem sourceTailDenominator_pos (B D : ℕ) {q : ℝ} (hq : 0<q) :
-    0<sourceTailDenominator B D q := by
-  unfold sourceTailDenominator
-  positivity [sourceCapEnvelope_pos B D hq]
-
-theorem sourceTailDenominator_expUpper (B : ℕ) : SourceExpUpper (sourceTailDenominator B) := by
-  have H := ((SourceExpUpper.const 9600).mul (sourceCapEnvelope_expUpper B)
-    (fun _ _ _=>by norm_num) (fun D q hq=>(sourceCapEnvelope_pos B D hq.1).le)).mul
-      (initialOverlapLower_inverse_expUpper.pow (fun _ q _=>(inv_pos.mpr (initialOverlapLower_pos q)).le) 4)
-      (fun D q hq=>by positivity [sourceCapEnvelope_pos B D hq.1]) (fun _ _ _=>by positivity)
-  exact (SourceExpUpper.const 512).add H
-
-theorem sourceWorkSize_poly (B : ℕ) (hB : 2 ≤ B) : SourcePolyUpper (sourceWorkSize B) := by
-  exact (((totalScoreExponent_poly B hB).add (SourcePolyUpper.const (sourceDegreeCutoff B:ℝ))).add
-    (SourcePolyUpper.const (B:ℝ))).add (SourcePolyUpper.const 10)
-
-theorem sourceTailExponent_expUpper (B : ℕ) (hB : 2 ≤ B) :
-    SourceExpUpper (fun D _=>sourceTailExponent B D) := by
-  have ha := ((totalScoreExponent_poly B hB).add (SourcePolyUpper.const (B:ℝ))).add (SourcePolyUpper.const 1)
-  have hu := (SourcePolyUpper.const 300).mul ha (fun _=>by norm_num)
-    (fun D=>by linarith only [(totalScoreExponent_bounds B D hB).1,Nat.cast_nonneg B (α:=ℝ)])
-  have hw := (SourcePolyUpper.const 40).mul (sourceWorkSize_poly B hB) (fun _=>by norm_num)
-    (fun D=>(sourceWorkSize_pos B D hB).le)
-  exact ((hu.add hw).add (SourcePolyUpper.const 5)).expUpper
-
-theorem source_tails_eventually (B : ℕ) (ζ : ℝ) (hB : 2 ≤ B) (hζ : ζ∈Set.Ioo 0 1)
-    {n q : ℕ→ℝ} (hn : Tendsto n atTop atTop) (hq : ∀t,q t∈Set.Ioc 0 (1/2:ℝ))
-    (hH : Tendsto (fun t=>(q t)^3*Real.log (n t)) atTop atTop) :
-    ∀ᶠt in atTop,∀i,2*(B:ℝ)*Real.exp
-      (-(sourceTargetScale B (sourceDepth B ζ ((q t)^3*Real.log (n t))) (n t) i)^
-        (sourceEpsilon B (sourceDepth B ζ ((q t)^3*Real.log (n t))))/
-        sourceTailDenominator B (sourceDepth B ζ ((q t)^3*Real.log (n t))) (q t)) ≤
-      (n t)^(-sourceTailExponent B (sourceDepth B ζ ((q t)^3*Real.log (n t)))) := by
-  apply (sourceTailDenominator_expUpper B).uniform_tail (SourceExpUpper.const (2*(B:ℝ)))
-    (sourceTailExponent_expUpper B hB) (fun D q hq=>sourceTailDenominator_pos B D hq.1)
-    (fun _ _ _=>by positivity) (fun D _ _=>?_) B ζ hB hζ hn hq hH
-  unfold sourceTailExponent
-  have ha : 0 ≤ totalScoreExponent B D+B+1 := by
-    linarith only [(totalScoreExponent_bounds B D hB).1,Nat.cast_nonneg B (α:=ℝ)]
-  have hw := (sourceWorkSize_pos B D hB).le
-  linarith only [ha,hw]
-
-/-- Turning the common uniform exponential bound into the two exact
- scheduled-error terms, including the division by the clipping threshold. -/
-theorem source_exponential_error_budget {B D b s ell k : ℕ} {C N F ω f n : ℝ}
-    (H : SourceArithmeticBounds C N F (sourceEpsilon B D) s b ell k)
-    (hB : 2 ≤ B) (hb : b ≤ B) (hC : 0<C) (hF : 0<F) (hω : 0<ω)
-    (hn : 1 ≤ n) (hN : 1 ≤ N) (hNn : N ≤ n)
-    (hf512 : 512 ≤ f) (hfmeta : 9600*(C^(2*b)*F)/ω^4 ≤ f)
-    (htail : 2*(B:ℝ)*Real.exp (-(N^(sourceEpsilon B D)/f)) ≤ n^(-sourceTailExponent B D)) :
-    sourceEffectiveHiddenTail (sourceBlockWindows N b (sourceEpsilon B D))
-      (sourceArithmeticTrials C N b (sourceEpsilon B D) ell k) b ω ≤ n^(-200*(totalScoreExponent B D+B+1)) ∧
-    sourceFitTail (sourceBlockWindows N b (sourceEpsilon B D))
-      (sourceArithmeticTrials C N b (sourceEpsilon B D) ell k) b /
-      N^(-20*sourceWorkSize B D) ≤ n^(-200*(totalScoreExponent B D+B+1)) ∧
-    sourceFitTail (sourceBlockWindows N b (sourceEpsilon B D))
-      (sourceArithmeticTrials C N b (sourceEpsilon B D) ell k) b ≤ n^(-200*(totalScoreExponent B D+B+1)) := by
-  let A := totalScoreExponent B D+B+1
-  let M := sourceWorkSize B D
-  let ε := sourceEpsilon B D
-  have hN0 : 0<N := by linarith only [hN]
-  have hn0 : 0<n := by linarith only [hn]
-  have hf0 : 0<f := by linarith only [hf512]
-  have hA : 0 ≤ A := by dsimp [A]; linarith only [(totalScoreExponent_bounds B D hB).1,Nat.cast_nonneg B (α:=ℝ)]
-  have hM : 0 ≤ M := (sourceWorkSize_pos B D hB).le
-  have hexpfit : Real.exp (-(N^ε/512)) ≤ Real.exp (-(N^ε/f)) := by
-    apply Real.exp_le_exp.mpr
-    exact neg_le_neg (div_le_div_of_nonneg_left (Real.rpow_pos_of_pos hN0 ε).le (by norm_num) hf512)
-  have hexpmeta : Real.exp (-(N^ε/(600*(C^(2*b)*F)))*ω^4/16) ≤ Real.exp (-(N^ε/f)) := by
-    have hden : 0<9600*(C^(2*b)*F)/ω^4 := by positivity
-    have hh := div_le_div_of_nonneg_left (Real.rpow_pos_of_pos hN0 ε).le hden hfmeta
-    have heq : N^ε/(9600*(C^(2*b)*F)/ω^4)=N^ε/(600*(C^(2*b)*F))*ω^4/16 := by field_simp; ring
-    rw [heq] at hh
-    apply Real.exp_le_exp.mpr
-    nlinarith only [hh]
-  have hcommon : 2*(b:ℝ)*N*Real.exp (-(N^ε/f)) ≤ n^(1-sourceTailExponent B D) := by
-    have hcast : (b:ℝ) ≤ B := by exact_mod_cast hb
-    have hh := mul_le_mul_of_nonneg_left htail hn0.le
-    have hcoeff : 2*(b:ℝ)*N ≤ n*(2*(B:ℝ)) := by
-      have hh' := mul_le_mul hcast hNn (by linarith only [hN]) (Nat.cast_nonneg B (α:=ℝ))
-      nlinarith only [hh']
-    have hp := mul_le_mul_of_nonneg_right hcoeff (Real.exp_pos (-(N^ε/f))).le
-    have heq : n*(n^(-sourceTailExponent B D))=n^(1-sourceTailExponent B D) := by
-      calc
-        _ = n^(1:ℝ)*n^(-sourceTailExponent B D) := by rw [Real.rpow_one]
-        _ = _ := by rw [←Real.rpow_add hn0]; rfl
-    nlinarith only [hh,hp,heq]
-  have hfit := H.fit_tail hC hF hN (sourceEpsilon_pos B D).le
-    (fun v hv=>sourceKappa_three_epsilon B D b (v+1) hb (by omega) (by omega))
-  have hm := H.metadata_tail (ω:=ω) hN0
-  have hf' := hfit.trans ((mul_le_mul_of_nonneg_left hexpfit (by positivity : 0 ≤ 2*(b:ℝ)*N)).trans hcommon)
-  have hm' := hm.trans ((mul_le_mul_of_nonneg_left hexpmeta (by positivity : 0 ≤ 2*(b:ℝ)*N)).trans hcommon)
-  have hpow : n^(1-sourceTailExponent B D) ≤ n^(-200*A) := by
-    apply Real.rpow_le_rpow_of_exponent_le hn
-    dsimp [sourceTailExponent]
-    dsimp [A,M] at *
-    linarith only [hA,hM]
-  refine ⟨hm'.trans hpow,?_,hf'.trans hpow⟩
-  rw [div_eq_mul_inv,←Real.rpow_neg hN0.le]
-  simp only [neg_mul,neg_neg]
-  have hr : N^(20*M) ≤ n^(20*M) := Real.rpow_le_rpow hN0.le hNn (by positivity)
-  apply (mul_le_mul hf' hr (Real.rpow_pos_of_pos hN0 _).le (Real.rpow_pos_of_pos hn0 _).le).trans
-  rw [←Real.rpow_add hn0]
-  apply Real.rpow_le_rpow_of_exponent_le hn
-  dsimp [sourceTailExponent]
-  dsimp [A,M] at *
-  linarith only [hA,hM]
-
-end TraceReconstruction
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem rpow_constant_absorb {N ε e c M : ℝ} (hN : 1 ≤ N) (hM : 0<M)
-    (hc : M*c ≤ N^ε) (he : ε+e ≤ 0) : c*N^e ≤ 1/M := by
-  have hN0 : 0<N := by linarith only [hN]
-  have hh := mul_le_mul_of_nonneg_right hc (Real.rpow_pos_of_pos hN0 e).le
-  rw [←Real.rpow_add hN0] at hh
-  have hr := Real.rpow_le_rpow_of_exponent_le hN he
-  rw [Real.rpow_zero] at hr
-  apply (le_div_iff₀ hM).mpr
-  nlinarith only [hh,hr]
-
-theorem overlap_sqrt_budget {v e : ℝ} (hv : v ≤ 1/4096) (he : e ≤ 1/8) :
-    Real.sqrt v+e ≤ 1/4 := by
-  have hh : Real.sqrt v ≤ 1/64 := (Real.sqrt_le_left (by norm_num)).mpr (by norm_num at hv ⊢; exact hv)
-  linarith only [hh,he]
-
-/-- Generic dense-template estimate, used for the first warmup step and
- every step of a post-warmup block. -/
-theorem source_dense_variance_budget {N ε e a K L V h : ℝ} {j : ℕ}
-    (hN : 1 ≤ N) (_hK : 0 ≤ K) (hL : 0 ≤ L) (hh : 0 ≤ h)
-    (hcount : h ≤ (K+1)*N^a) (hV : V ≤ N^(-e))
-    (hc : 4096*(K+1)*L^j ≤ N^ε) (he : ε+a ≤ e) :
-    h*(L^j*V) ≤ 1/4096 := by
-  have hN0 : 0<N := by linarith only [hN]
-  have hv := mul_le_mul_of_nonneg_left hV (show 0 ≤ h*L^j by positivity)
-  have ht := mul_le_mul_of_nonneg_right hcount (show 0 ≤ L^j*N^(-e) by positivity)
-  have heq : ((K+1)*N^a)*(L^j*N^(-e))=((K+1)*L^j)*N^(a-e) := by
-    rw [sub_eq_add_neg,Real.rpow_add hN0]
-    ring
-  rw [heq] at ht
-  have hb := rpow_constant_absorb hN (M:=4096) (by norm_num)
-    (c:=(K+1)*L^j) (by nlinarith only [hc]) (e:=a-e) (by linarith only [he])
-  nlinarith only [hv,ht,hb]
-
-/-- Exact scalar one-window refinement. Its quadratic sparse term is what
- makes the second warmup overlap possible at old score exponent 11/5. -/
-theorem source_warmup_variance_budget {N ε e M K W L w T h E : ℝ}
-    (hN : 1 ≤ N) (hK : 0 ≤ K) (hW : 0 ≤ W) (hL : 0 ≤ L)
-    (hT : 0 ≤ T) (hh : 0 ≤ h) (hcount : h ≤ (K+1)*N)
-    (hwindow : T ≤ N) (hw : w^2 ≤ W*N^(-e))
-    (henergy : E ≤ 64*(72*T*w^2*(N^(-20*M))^2+8*(w^2*L)^2))
-    (hbad : 8192*(4608*(K+1)*L*W) ≤ N^ε)
-    (hquad : 8192*(512*(K+1)*W^2*L^3) ≤ N^ε)
-    (hebad : ε+2-e-40*M ≤ 0) (hequad : ε+1-2*e ≤ 0) :
-    h*(L*E) ≤ 1/4096 := by
-  have hN0 : 0<N := by linarith only [hN]
-  have ha : 0 ≤ N^(-20*M) := (Real.rpow_pos_of_pos hN0 _).le
-  have hw0 : 0 ≤ W*N^(-e) := by positivity
-  have hb0 : 0 ≤ 64*(72*T*w^2*(N^(-20*M))^2+8*(w^2*L)^2) := by positivity
-  have H := (mul_le_mul_of_nonneg_left henergy (show 0 ≤ h*L by positivity)).trans
-    (mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hcount hL) hb0)
-  have ht := mul_le_mul hwindow hw (sq_nonneg _) hN0.le
-  have hs := pow_le_pow_left₀ (mul_nonneg (sq_nonneg w) hL) (mul_le_mul_of_nonneg_right hw hL) 2
-  have hbd := mul_le_mul_of_nonneg_left ht (sq_nonneg (N^(-20*M)))
-  have hm := mul_le_mul_of_nonneg_left (show 64*(72*T*w^2*(N^(-20*M))^2+8*(w^2*L)^2) ≤
-      64*(72*(N*(W*N^(-e)))*(N^(-20*M))^2+8*((W*N^(-e))*L)^2) by nlinarith only [hbd,hs])
-    (show 0 ≤ ((K+1)*N)*L by positivity)
-  have hp1 : N*N^(-e)*(N^(-20*M))^2=N^(1-e-40*M) := by
-    rw [←Real.rpow_natCast (N^(-20*M)) 2,←Real.rpow_mul hN0.le]
-    norm_num only [Nat.cast_ofNat]
-    nth_rw 1 [←Real.rpow_one N]
-    rw [←Real.rpow_add hN0,←Real.rpow_add hN0]
-    congr 1
-    ring
-  have hp2 : (N^(-e))^2=N^(-2*e) := by
-    rw [←Real.rpow_natCast,←Real.rpow_mul hN0.le]
-    norm_num only [Nat.cast_ofNat]
-    congr 1
-    ring
-  have hpow1 : N^((1:ℝ))*N^(1-e-40*M)=N^(2-e-40*M) := by rw [←Real.rpow_add hN0]; congr 1; ring
-  have hpow2 : N^((1:ℝ))*N^(-2*e)=N^(1-2*e) := by rw [←Real.rpow_add hN0]; congr 1; ring
-  rw [Real.rpow_one] at hpow1 hpow2
-  have heq : (((K+1)*N)*L)*(64*(72*(N*(W*N^(-e)))*(N^(-20*M))^2+8*((W*N^(-e))*L)^2))=
-      (4608*(K+1)*L*W)*N^(2-e-40*M)+(512*(K+1)*W^2*L^3)*N^(1-2*e) := by
-    calc
-      _ = (4608*(K+1)*L*W)*N*(N*N^(-e)*(N^(-20*M))^2)+
-        (512*(K+1)*W^2*L^3)*N*(N^(-e))^2 := by ring
-      _ = _ := by rw [hp1,hp2,mul_assoc _ N _,hpow1,mul_assoc _ N _,hpow2]
-  rw [heq] at hm
-  have hb := rpow_constant_absorb hN (M:=8192) (by norm_num) hbad (e:=2-e-40*M) (by linarith only [hebad])
-  have hq := rpow_constant_absorb hN (M:=8192) (by norm_num) hquad (e:=1-2*e) (by linarith only [hequad])
-  nlinarith only [H,hm,hb,hq]
-
-theorem sourceBlockLength_factor (P : ℕ→ℕ) (ell j : ℕ) :
-    sourceBlockLength P ell j=ell*sourceBlockLength P 1 j := by
-  induction j with
-  | zero => simp only [sourceBlockLength,mul_one]
-  | succ j ih => rw [sourceBlockLength,sourceBlockLength,ih]; ring
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem source_delta_power_budget {B D : ℕ} {n N ε : ℝ} (hB : 2 ≤ B)
-    (hn : 1 ≤ n) (hN : 1 ≤ N) (hNn : N ≤ n) (he : 0 ≤ ε) (he1 : ε ≤ 1) :
-    n^(-100*(totalScoreExponent B D+B+1))*N^(2+ε) ≤ 1 := by
-  have hn0 : 0<n := by linarith only [hn]
-  have hN0 : 0<N := by linarith only [hN]
-  have hA : 1 ≤ totalScoreExponent B D+B+1 := by
-    linarith only [(totalScoreExponent_bounds B D hB).1,Nat.cast_nonneg B (α:=ℝ)]
-  apply (mul_le_mul_of_nonneg_left (Real.rpow_le_rpow hN0.le hNn (by linarith only [he]))
-    (Real.rpow_pos_of_pos hn0 (-100*(totalScoreExponent B D+B+1))).le).trans
-  rw [←Real.rpow_add hn0]
-  simpa only [Real.rpow_zero] using Real.rpow_le_rpow_of_exponent_le hn
-    (show -100*(totalScoreExponent B D+B+1)+(2+ε) ≤ 0 by linarith only [hA,he1])
-
-/-- Uniform repeated old simulation error; this is the error term in every
- dense nested-overlap estimate. -/
-theorem source_repeated_error_budget {n N ε K E t h : ℝ} {B D : ℕ}
-    (hB : 2 ≤ B) (hn : 1 ≤ n) (hN : 1 ≤ N) (hNn : N ≤ n)
-    (he : 0 ≤ ε) (he1 : ε ≤ 1) (hh : 0 ≤ h) (_hK : 0 ≤ K)
-    (hcount : h ≤ (K+1)*N)
-    (hcost : t*E ≤ ((D:ℝ)+1)*n^(-100*(totalScoreExponent B D+B+1))*N)
-    (hmargin : 64*(K+1)*((D:ℝ)+1) ≤ N^ε) : 2*h*(t*E) ≤ 1/32 := by
-  let Δ := n^(-100*(totalScoreExponent B D+B+1))
-  have hn0 : 0<n := by linarith only [hn]
-  have hΔ : 0 ≤ Δ := (Real.rpow_pos_of_pos hn0 _).le
-  have hc := mul_le_mul_of_nonneg_left hcost (show 0 ≤ 2*h by positivity)
-  have ht := mul_le_mul_of_nonneg_right hcount (show 0 ≤ 2*((D:ℝ)+1)*Δ*N by positivity)
-  have hm := mul_le_mul_of_nonneg_right hmargin (show 0 ≤ Δ*N^2 by positivity)
-  have hp := source_delta_power_budget (D:=D) hB hn hN hNn he he1
-  have hl : N^ε*N^2=N^(2+ε) := by
-    rw [←Real.rpow_natCast,←Real.rpow_add (by linarith only [hN] : 0<N)]
-    norm_num only [Nat.cast_ofNat]
-    congr 1
-    ring
-  have hm' : 64*(K+1)*((D:ℝ)+1)*Δ*N^2 ≤ Δ*N^(2+ε) := by
-    calc
-      _ = (64*(K+1)*((D:ℝ)+1))*(Δ*N^2) := by ring
-      _ ≤ N^ε*(Δ*N^2) := hm
-      _ = _ := by rw [show N^ε*(Δ*N^2)=Δ*(N^ε*N^2) by ring,hl]
-  change Δ*N^(2+ε) ≤ 1 at hp
-  nlinarith only [hc,ht,hm',hp]
-
-theorem source_old_error_cost {C N F ε E Δ : ℝ} {s b ell k j i D : ℕ}
-    (H : SourceArithmeticBounds C N F ε s b ell k) (hN : 1 ≤ N)
-    (hE : E ≤ ((i:ℝ)+1)*Δ*ell) (hΔ : 0 ≤ Δ) (hi : i ≤ D) (hj : j ≤ b)
-    (ha : scaleA b j ε ≤ 1) :
-    (sourceBlockLength (sourceArithmeticP C N b ε ell k) 1 j:ℝ)*E ≤
-      ((D:ℝ)+1)*Δ*N := by
-  let t := sourceBlockLength (sourceArithmeticP C N b ε ell k) 1 j
-  have hh := mul_le_mul_of_nonneg_left hE (Nat.cast_nonneg t (α:=ℝ))
-  have hl : (ell:ℝ)*t ≤ N := by
-    have hl' := (H.lengths j hj).2
-    rw [sourceArithmeticLength_eq_block,sourceBlockLength_factor] at hl'
-    push_cast at hl'
-    exact hl'.trans (by simpa only [Real.rpow_one] using Real.rpow_le_rpow_of_exponent_le hN ha)
-  have hi' : (i:ℝ) ≤ D := by exact_mod_cast hi
-  have hh' := mul_le_mul_of_nonneg_left hl (show 0 ≤ ((D:ℝ)+1)*Δ by positivity)
-  have hcoef := mul_le_mul_of_nonneg_right hi' (show 0 ≤ Δ*(ell:ℝ)*t by positivity)
-  dsimp [t] at *
-  nlinarith only [hh,hh',hcoef]
-
-/-- The normalized error invariant propagates through actual leaf products,
- with one fresh delta paid by clipping and fitting. -/
-theorem normalized_block_error {E Δ clip fit : ℝ} {ell newell leaves i : ℕ}
-    (hell : 1 ≤ newell) (hE : E ≤ ((i:ℝ)+1)*Δ*ell) (hΔ : 0 ≤ Δ)
-    (hleaf : leaves*ell ≤ newell) (hnew : clip+fit ≤ Δ) :
-    (clip+(leaves:ℝ)*E+fit)/(newell:ℝ) ≤ ((i:ℝ)+2)*Δ := by
-  have hl0 : (0:ℝ)<newell := by exact_mod_cast (show 0<newell by omega)
-  apply (div_le_iff₀ hl0).mpr
-  have hleaf' : (leaves:ℝ)*ell ≤ newell := by exact_mod_cast hleaf
-  have h1 := mul_le_mul_of_nonneg_left hE (Nat.cast_nonneg leaves (α:=ℝ))
-  have h2 := mul_le_mul_of_nonneg_left hleaf' (show 0 ≤ ((i:ℝ)+1)*Δ by positivity)
-  have h3 : Δ ≤ Δ*newell := le_mul_of_one_le_right hΔ (by exact_mod_cast hell)
-  nlinarith only [h1,h2,h3,hnew]
-
-end TraceReconstruction
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- All three genuine clipping losses, and fitting, share the same global
- error unit. The reference power is the actual half-moment at this scale. -/
-theorem source_level_clipping (B D j s b : ℕ) (T t : ℕ→ℕ) (n ω a gap L : ℝ)
-    (hB : 2 ≤ B) (hn : 4 ≤ n) (hg : sourceGapFloor B ≤ gap)
-    (hmeta : sourceEffectiveHiddenTail T t (b+1) ω ≤ n^(-200*(totalScoreExponent B D+B+1)))
-    (hfit : sourceFitTail T t (b+1)/a ≤ n^(-200*(totalScoreExponent B D+B+1)))
-    (hfit' : sourceFitTail T t (b+1) ≤ n^(-200*(totalScoreExponent B D+B+1)))
-    (hhalf : ((6*(2*sourceHalfMoment B D n j:ℕ):ℝ)*L)^2*sourceSparseCount T s b ≤ 1/2)
-    (hscale : ((6*(2*sourceHalfMoment B D n j:ℕ):ℝ)*L)^2*sourceSparseCount T s b ≤
-      (sourceTargetScale B D n j)^(-gap/2)/8) :
-    sourceClippingCost T t s b (sourceHalfMoment B D n j) ω a L ≤ 1/2 ∧
-    sourceClippingCost T t s b (sourceHalfMoment B D n j) ω a L+sourceFitTail T t (b+1) ≤
-      n^(-100*(totalScoreExponent B D+B+1)) := by
-  have hN : 0<sourceTargetScale B D n j := Real.rpow_pos_of_pos (by linarith only [hn]) _
-  have hc := (source_clipping_moment_power T s b (sourceHalfMoment B D n j)
-    (sourceTargetScale B D n j) gap L hN hhalf hscale).trans (source_clipping_power_delta B D j hB hn hg)
-  have hu := sourceErrorUnit_small (D:=D) hB hn
-  have hunit : 0 ≤ n^(-200*(totalScoreExponent B D+B+1)) := (Real.rpow_pos_of_pos (by linarith only [hn]) _).le
-  unfold sourceClippingCost
-  constructor <;> linarith only [hmeta,hfit,hfit',hc,hu.1,hu.2,hunit]
-
-/-- At one window the root metadata can still fail; only the fitting loss
- vanishes. We charge the root metadata rather than discard it. -/
-theorem source_one_window_clipping (B D j s : ℕ) (T t : ℕ) (n ω a gap L : ℝ)
-    (hB : 2 ≤ B) (hn : 4 ≤ n) (hg : sourceGapFloor B ≤ gap)
-    (hmeta : sourceEffectiveHiddenTail (fun _=>T) (fun _=>t) 1 ω ≤ n^(-200*(totalScoreExponent B D+B+1)))
-    (hhalf : ((6*(2*sourceHalfMoment B D n j:ℕ):ℝ)*L)^2*(2:ℝ)^s ≤ 1/2)
-    (hscale : ((6*(2*sourceHalfMoment B D n j:ℕ):ℝ)*L)^2*(2:ℝ)^s ≤
-      (sourceTargetScale B D n j)^(-gap/2)/8) :
-    sourceClippingCost (fun _=>T) (fun _=>t) s 0 (sourceHalfMoment B D n j) ω a L ≤ 1/2 ∧
-    sourceClippingCost (fun _=>T) (fun _=>t) s 0 (sourceHalfMoment B D n j) ω a L ≤
-      n^(-100*(totalScoreExponent B D+B+1)) := by
-  have hu : 0 ≤ n^(-200*(totalScoreExponent B D+B+1)) :=
-    (Real.rpow_pos_of_pos (by linarith only [hn]) _).le
-  simpa only [sourceFitTail,add_zero] using source_level_clipping B D j s 0
-    (fun _=>T) (fun _=>t) n ω a gap L hB hn hg hmeta
-    (by simpa only [sourceFitTail,zero_div] using hu) (by simpa only [sourceFitTail] using hu)
-    (by simpa only [sourceSparseCount,Finset.range_zero,Finset.prod_empty,Nat.mul_one,Nat.cast_pow,Nat.cast_ofNat] using hhalf)
-    (by simpa only [sourceSparseCount,Finset.range_zero,Finset.prod_empty,Nat.mul_one,Nat.cast_pow,Nat.cast_ofNat] using hscale)
-
-end TraceReconstruction
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem source_overlap_prefactors {K W L X : ℝ} {B j : ℕ}
-    (hK : 0 ≤ K) (hW : 0 ≤ W) (hL : 1 ≤ L) (hB : 2 ≤ B) (hj : j ≤ B+2)
-    (hX : 1099511627776*(K+1)*(W+1)^2*L^(B+2) ≤ X) :
-    4096*(K+1)*L^j ≤ X ∧
-    8192*(4608*(K+1)*L*W) ≤ X ∧
-    8192*(512*(K+1)*W^2*L^3) ≤ X := by
-  have hK1 : 0 ≤ K+1 := by linarith only [hK]
-  have hw1 : 1 ≤ (W+1)^2 := by nlinarith only [hW,sq_nonneg W]
-  have hww : W ≤ (W+1)^2 := by nlinarith only [hW,sq_nonneg W]
-  have hw2 : W^2 ≤ (W+1)^2 := by nlinarith only [hW]
-  have hL0 : 0 ≤ L := by linarith only [hL]
-  have hjpow := pow_le_pow_right₀ hL hj
-  have hlpow : L ≤ L^(B+2) := by simpa only [pow_one] using pow_le_pow_right₀ hL (show 1 ≤ B+2 by omega)
-  have h3pow := pow_le_pow_right₀ hL (show 3 ≤ B+2 by omega)
-  have hb0 : 0 ≤ (K+1)*(W+1)^2*L^(B+2) := by positivity
-  have hfirst : (K+1)*L^j ≤ (K+1)*(W+1)^2*L^(B+2) := by
-    have hp := mul_le_mul_of_nonneg_left hjpow hK1
-    have hp' := mul_le_mul_of_nonneg_right hw1 (show 0 ≤ (K+1)*L^(B+2) by positivity)
-    nlinarith only [hp,hp']
-  have hsecond : (K+1)*L*W ≤ (K+1)*(W+1)^2*L^(B+2) := by
-    have hp := mul_le_mul_of_nonneg_left (mul_le_mul hlpow hww hW (by positivity)) hK1
-    nlinarith only [hp]
-  have hthird : (K+1)*W^2*L^3 ≤ (K+1)*(W+1)^2*L^(B+2) := by
-    have hp := mul_le_mul_of_nonneg_left (mul_le_mul hw2 h3pow (by positivity) (by positivity)) hK1
-    nlinarith only [hp]
-  constructor
-  · nlinarith only [hfirst,hX,hb0]
-  constructor <;> nlinarith only [hsecond,hthird,hX,hb0]
-
-/-- Complete dense-overlap numerical criterion for the literal schedules,
- including the error carried by the old exact simulator. -/
-theorem source_literal_dense_overlap {C N F ε E Δ V K W L n e : ℝ}
-    {s b ell k j i D B : ℕ} (H : SourceArithmeticBounds C N F ε s b ell k)
-    (hB : 2 ≤ B) (hb : b ≤ B) (hC : 1 ≤ C) (hF : 0 ≤ F)
-    (hn : 1 ≤ n) (hN : 1 ≤ N) (hNn : N ≤ n) (he : 0 ≤ ε) (he1 : ε ≤ 1)
-    (hK : 0 ≤ K) (hW : 0 ≤ W) (hL : 1 ≤ L) (hLdef : L=(2:ℝ)^s)
-    (hG : C^(2*b)*F ≤ K) (hi : i ≤ D) (hj : j ≤ b)
-    (ha : 0 ≤ scaleA b j ε) (ha1 : scaleA b j ε ≤ 1)
-    (hE : E ≤ ((i:ℝ)+1)*Δ*ell) (hΔ : Δ=n^(-100*(totalScoreExponent B D+B+1)))
-    (hV : V ≤ N^(-e)) (hgap : ε+scaleA b j ε ≤ e)
-    (hmargin : 64*(K+1)*((D:ℝ)+1) ≤ N^ε)
-    (hoverlap : 1099511627776*(K+1)*(W+1)^2*L^(B+2) ≤ N^ε) :
-    Real.sqrt (((sourceArithmeticCount C N b ε ell k j+1:ℕ):ℝ)*
-      (((2^s:ℕ):ℝ)^j*V))+
-      2*((sourceArithmeticCount C N b ε ell k j+1:ℕ):ℝ)*
-      ((sourceBlockLength (sourceArithmeticP C N b ε ell k) 1 j:ℝ)*E) ≤ 1/4 := by
-  let h := ((sourceArithmeticCount C N b ε ell k j+1:ℕ):ℝ)
-  have hn0 : 0<n := by linarith only [hn]
-  have hN0 : 0<N := by linarith only [hN]
-  have hh0 : 0 ≤ h := Nat.cast_nonneg _
-  have hc : h ≤ (K+1)*N^(scaleA b j ε) := by
-    exact (H.count_bound hC hF hN hj ha).trans
-      (mul_le_mul_of_nonneg_right (add_le_add hG le_rfl) (Real.rpow_pos_of_pos hN0 _).le)
-  have hc' : h ≤ (K+1)*N := hc.trans (mul_le_mul_of_nonneg_left
-    (by simpa only [Real.rpow_one] using Real.rpow_le_rpow_of_exponent_le hN ha1)
-    (by linarith only [hK]))
-  have hpref := (source_overlap_prefactors hK hW hL hB (show j ≤ B+2 by omega) hoverlap).1
-  have hv := source_dense_variance_budget hN hK (show 0 ≤ L by linarith only [hL]) hh0 hc hV hpref hgap
-  have hcost := source_old_error_cost H hN hE (by rw [hΔ]; exact (Real.rpow_pos_of_pos hn0 _).le) hi hj ha1
-  rw [hΔ] at hcost
-  have herr := source_repeated_error_budget hB hn hN hNn he he1 hh0 hK hc' hcost hmargin
-  have hres := overlap_sqrt_budget hv (show 2*h*((sourceBlockLength (sourceArithmeticP C N b ε ell k) 1 j:ℝ)*E) ≤ 1/8 by linarith only [herr])
-  simpa only [hLdef,Nat.cast_pow,Nat.cast_ofNat] using hres
-
-end TraceReconstruction
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem SourceArithmeticBounds.length_positive {C N F ε : ℝ} {s b ell k j : ℕ}
-    (H : SourceArithmeticBounds C N F ε s b ell k) (hN : 0<N) (hj : j ≤ b) :
-    1 ≤ sourceArithmeticLength C N b ε ell k j := by
-  have hp : (0:ℝ)<sourceArithmeticLength C N b ε ell k j :=
-    (show 0<N^(scaleA b j ε)/2 by positivity).trans_le (H.lengths j hj).1
-  have hh : 0<sourceArithmeticLength C N b ε ell k j := by exact_mod_cast hp
-  omega
-
-theorem SourceArithmeticBounds.copies_le_target {C N F ε : ℝ} {s b ell k j : ℕ}
-    (H : SourceArithmeticBounds C N F ε s b ell k) (hN : 1 ≤ N) (hj : j<b)
-    (ha : scaleA b (j+1) ε ≤ 1) : (sourceArithmeticP C N b ε ell k j:ℝ) ≤ N := by
-  have hL := H.length_positive (by linarith only [hN]) (j:=j) (by omega)
-  have hp : sourceArithmeticP C N b ε ell k j ≤ sourceArithmeticLength C N b ε ell k (j+1) := by
-    rw [sourceArithmeticLength_step]
-    simpa only [Nat.mul_comm] using Nat.le_mul_of_pos_right (sourceArithmeticP C N b ε ell k j) hL
-  exact (show (sourceArithmeticP C N b ε ell k j:ℝ) ≤ sourceArithmeticLength C N b ε ell k (j+1) by exact_mod_cast hp).trans
-    ((H.lengths (j+1) (by omega)).2.trans (by
-      simpa only [Real.rpow_one] using Real.rpow_le_rpow_of_exponent_le hN ha))
-
-/-- Both contributions to the error in the second warmup overlap are paid
- by the common delta margin. The old term uses T₀ ≤ P₀, not an independent
- full-block overlap hypothesis. -/
-theorem source_warmup_error_budget {C N F ε E Δ clip K n : ℝ} {s ell k i D B : ℕ}
-    (H : SourceArithmeticBounds C N F ε s 2 ell k)
-    (hB : 2 ≤ B) (hC : 1 ≤ C) (hF : 0 ≤ F) (hn : 1 ≤ n) (hN : 1 ≤ N) (hNn : N ≤ n)
-    (he : 0 ≤ ε) (he1 : ε ≤ 1) (hK : 0 ≤ K) (hG : C^4*F ≤ K) (hi : i ≤ D)
-    (hE : E ≤ ((i:ℝ)+1)*Δ*ell) (hE0 : 0 ≤ E) (hΔ : Δ=n^(-100*(totalScoreExponent B D+B+1)))
-    (hclip : clip ≤ Δ) (hmargin : 64*(K+1)*((D:ℝ)+1) ≤ N^ε) :
-    2*((sourceArithmeticCount C N 2 ε ell k 2+1:ℕ):ℝ)*
-      ((sourceArithmeticP C N 2 ε ell k 1:ℝ)*
-        (clip+(sourceBlockWindows N 2 ε 0:ℝ)*E)) ≤ 1/8 := by
-  let P := sourceArithmeticP C N 2 ε ell k
-  let T := sourceBlockWindows N 2 ε
-  let h := ((sourceArithmeticCount C N 2 ε ell k 2+1:ℕ):ℝ)
-  have hn0 : 0<n := by linarith only [hn]
-  have hΔ0 : 0 ≤ Δ := by rw [hΔ]; exact (Real.rpow_pos_of_pos hn0 _).le
-  have hh0 : 0 ≤ h := Nat.cast_nonneg _
-  have hc : h ≤ (K+1)*N := by
-    have hh := H.count_bound hC hF hN (j:=2) le_rfl (by rw [scaleA_top]; norm_num)
-    simpa only [scaleA_top,Real.rpow_one] using hh.trans
-      (mul_le_mul_of_nonneg_right (by simpa only [show (2:ℕ)*2=4 by rfl] using add_le_add hG (show (1:ℝ) ≤ 1 from le_rfl)) (Real.rpow_pos_of_pos (by linarith only [hN]) _).le)
-  have hcost := source_old_error_cost H hN hE hΔ0 hi (j:=2) le_rfl (by rw [scaleA_top])
-  rw [hΔ] at hcost
-  have herr := source_repeated_error_budget hB hn hN hNn he he1 hh0 hK hc hcost hmargin
-  have hp1 : (P 1:ℝ) ≤ N := H.copies_le_target hN (by omega) (by rw [show (1:ℕ)+1=2 by rfl,scaleA_top])
-  have hclipcost : (P 1:ℝ)*clip ≤ ((D:ℝ)+1)*Δ*N := by
-    have h1 := mul_le_mul_of_nonneg_left hclip (Nat.cast_nonneg (P 1) (α:=ℝ))
-    have h2 := mul_le_mul_of_nonneg_right hp1 hΔ0
-    have h3 := le_mul_of_one_le_left (show 0 ≤ Δ*N by positivity) (show (1:ℝ) ≤ D+1 by linarith only [Nat.cast_nonneg D (α:=ℝ)])
-    nlinarith only [h1,h2,h3]
-  rw [hΔ] at hclipcost
-  have hcliperr := source_repeated_error_budget hB hn hN hNn he he1 hh0 hK hc hclipcost hmargin
-  have hT : (T 0:ℝ) ≤ P 0 := by
-    exact_mod_cast (show T 0 ≤ P 0 from (by have h := (H.windows 0 (by omega)).2.2.1; change 2*T 0 ≤ P 0 at h; omega))
-  have hold := mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_right hT hE0)
-    (show 0 ≤ 2*h*(P 1:ℝ) by positivity)
-  have hfactor : (sourceBlockLength P 1 2:ℝ)=(P 1:ℝ)*(P 0:ℝ) := by
-    simp only [sourceBlockLength,Nat.cast_mul,mul_one]
-  change 2*h*((sourceBlockLength P 1 2:ℝ)*E) ≤ 1/32 at herr
-  rw [hfactor] at herr
-  nlinarith only [herr,hcliperr,hold]
-
-end TraceReconstruction
-
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-
-end OAI
+import OAI.Probability.TraceReconstruction.ErrorRates
+
+namespace OAI
+
+/-!
+# Propagation
+
+Moment restriction, scale margins, sharp budget closure, clipping and error propagation, and dense/warmup overlap.
+-/
+
+open scoped ENNReal
+noncomputable section
+
+open Filter
+open scoped Topology
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+namespace FiniteLaw
+variable {α : Type*} [Fintype α]
+
+theorem evenMomentSq_mono_order (P : FiniteLaw α) (F : α→ℝ) {j k : ℕ}
+    (hj : 1≤j) (hjk : j≤k) : P.evenMomentSq F j ≤ P.evenMomentSq F k := by
+  have hjp : (0:ℝ)<j := by exact_mod_cast (show 0<j by omega)
+  have hkp : (0:ℝ)<k := by exact_mod_cast (show 0<k by omega)
+  have hp : 1≤(k:ℝ)/j := (one_le_div hjp).mpr (by exact_mod_cast hjk)
+  have he (a : α) : (|F a|^(2*j))^((k:ℝ)/j)=|F a|^(2*k) := by
+    rw [←Real.rpow_natCast,←Real.rpow_mul (abs_nonneg _),←Real.rpow_natCast]
+    congr 1
+    push_cast
+    field_simp
+  have H := Real.rpow_arith_mean_le_arith_mean_rpow Finset.univ P.mass
+    (fun a=>|F a|^(2*j)) (fun a _=>P.nonneg a) P.total (fun _ _=>by positivity) hp
+  simp_rw [he] at H
+  have hr := Real.rpow_le_rpow (Real.rpow_nonneg (P.expect_nonneg _ (fun _=>by positivity)) _) H
+    (show 0≤1/(k:ℝ) by positivity)
+  change (P.expect (fun a=>|F a|^(2*j))^((k:ℝ)/j))^(1/(k:ℝ)) ≤ _ at hr
+  rw [←Real.rpow_mul (P.expect_nonneg _ (fun _=>by positivity))] at hr
+  have he' : (k:ℝ)/j*(1/(k:ℝ))=1/(j:ℝ) := by field_simp
+  rw [he'] at hr
+  exact hr
+
+end FiniteLaw
+
+def OracleTracePair.restrict {q E V : ℝ} {x y : List Bool} {j k : ℕ}
+    (C : OracleTracePair q x y k E V) (hj : 1≤j) (hjk : j≤k) :
+    OracleTracePair q x y j E V := by
+  let := C.finiteOutcome
+  exact {C with moment := (C.first.evenMomentSq_mono_order C.score hj hjk).trans C.moment}
+
+/-- Null-mass outcomes are assigned zero score. This proves, rather than
+ assumes, the pointwise positivity needed by the finite simulation API. -/
+def ConcreteLevelPair.oracle {q N Δ d ω F : ℝ} {hq : q∈Set.Icc 0 1} {i k : ℕ}
+    (C : ConcreteLevelPair q hq N Δ d ω F i k) :
+    OracleTracePair q C.left C.right k C.error (N^(-d)) := by
+  let := C.finiteOutcome
+  let Z := fun a=>if C.firstLaw.mass a=0 then 0 else C.score a
+  have hden (a) : C.secondLaw.mass a=C.firstLaw.mass a*(1+Z a) := by
+    dsimp [Z]
+    split_ifs with h
+    · simpa only [h,zero_mul] using C.density a
+    · exact C.density a
+  have hpos (a) : 0≤1+Z a := by
+    by_cases h : C.firstLaw.mass a=0
+    · simp [Z,h]
+    · have hp : 0<C.firstLaw.mass a := lt_of_le_of_ne (C.firstLaw.nonneg a) (Ne.symm h)
+      exact (mul_nonneg_iff_of_pos_left hp).mp (by rw [←hden]; exact C.secondLaw.nonneg a)
+  have hc : C.firstLaw.expect Z=0 := by
+    have hsum := congrArg (fun f : C.Outcome→ℝ=>∑a,f a) (funext hden)
+    simp only [Finset.sum_add_distrib,mul_add,mul_one] at hsum
+    rw [C.firstLaw.total,C.secondLaw.total] at hsum
+    dsimp [FiniteLaw.expect]
+    linarith only [hsum]
+  have hm : C.firstLaw.evenMomentSq Z k=C.firstLaw.evenMomentSq C.score k := by
+    unfold FiniteLaw.evenMomentSq FiniteLaw.expect
+    congr 1
+    apply Finset.sum_congr rfl
+    intro a _
+    dsimp [Z]
+    split_ifs with h
+    · simp only [h,zero_mul]
+    · rfl
+  exact {
+    Outcome := C.Outcome
+    finiteOutcome := C.finiteOutcome
+    first := C.firstLaw
+    second := C.secondLaw
+    decode := C.decode
+    first_simulation := C.first_simulation
+    second_simulation := C.second_simulation
+    score := Z
+    density := hden
+    likelihood_nonneg := hpos
+    centered := hc
+    moment := hm.le.trans C.score_bound
+  }
+
+theorem sourceScaleProduct_step_le (B D i : ℕ) (hB : 2≤B) (hi : i<D) :
+    sourceScaleProduct B D i ≤ sourceScaleProduct B D (i+1) := by
+  have he := sourceEpsilon_pos B D
+  have hp : (1:ℝ)≤2^(sourceDimension B i) := one_le_pow₀ (by norm_num)
+  have ha : scaleA (sourceDimension B i) 0 (sourceEpsilon B D)≤1 := by
+    unfold scaleA
+    norm_num only [pow_zero,mul_one]
+    have hh := (div_le_iff₀ (by positivity : (0:ℝ)<2^(sourceDimension B i))).mpr
+      (le_mul_of_one_le_right (by linarith only [he] : 0≤1+sourceEpsilon B D) hp)
+    linarith only [hh]
+  have hs : 0 ≤ sourceScaleProduct B D (i+1) := (lt_of_lt_of_le (by positivity) (sourceScaleProduct_lower B D (i+1) hB)).le
+  have hh := mul_le_mul_of_nonneg_right ha hs
+  unfold sourceScaleProduct at *
+  rw [Finset.prod_eq_prod_Ico_succ_bot hi]
+  simpa only [one_mul] using hh
+
+theorem sourceHalfMoment_step (B D i : ℕ) {n : ℝ} (hB : 2≤B) (hi : i<D) (hn : 1<n) :
+    sourceHalfMoment B D n (i+1) ≤ sourceHalfMoment B D n i := by
+  have hlog : 0<Real.log (sourceTargetScale B D n i) := by
+    rw [sourceTargetScale,Real.log_rpow (by linarith only [hn] : 0<n)]
+    exact mul_pos (lt_of_lt_of_le (by positivity) (sourceScaleProduct_lower B D i hB)) (Real.log_pos hn)
+  have hlog' : Real.log (sourceTargetScale B D n i) ≤ Real.log (sourceTargetScale B D n (i+1)) := by
+    rw [sourceTargetScale,sourceTargetScale,Real.log_rpow (by linarith only [hn] : 0<n),Real.log_rpow (by linarith only [hn] : 0<n)]
+    exact mul_le_mul_of_nonneg_right (sourceScaleProduct_step_le B D i hB hi) (Real.log_pos hn).le
+  unfold sourceHalfMoment
+  apply Nat.ceil_mono
+  have hd := (totalScoreExponent_bounds B D hB).1
+  have hl := (Real.log_pos hn).le
+  exact div_le_div_of_nonneg_left (by positivity) hlog hlog'
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+def sourceGapFloor (B : ℕ) : ℝ := 1/(128*(2:ℝ)^(2*B))
+
+theorem sourceGapFloor_pos (B : ℕ) : 0<sourceGapFloor B := by unfold sourceGapFloor; positivity
+
+theorem sourceGapFloor_le_one (B : ℕ) : sourceGapFloor B≤1 := by
+  unfold sourceGapFloor
+  apply (div_le_one (by positivity)).mpr
+  have hh : (1:ℝ)≤2^(2*B) := one_le_pow₀ (by norm_num)
+  linarith only [hh]
+
+theorem sourceEpsilon_gap_small (B D : ℕ) : 1000*sourceEpsilon B D ≤ sourceGapFloor B := by
+  have hp : (1000:ℝ)*128*2^(2*B) ≤ 2^(4*B+100) := by
+    have hpow : (2:ℝ)^(2*B+17) ≤ 2^(4*B+100) := pow_le_pow_right₀ (by norm_num) (by omega)
+    rw [pow_add] at hpow
+    norm_num only [show (2:ℝ)^17=131072 by norm_num] at hpow
+    have hpos : 0≤(2:ℝ)^(2*B) := by positivity
+    linarith only [hpow,hpos]
+  have hd : 1≤(D:ℝ)+1 := by linarith [Nat.cast_nonneg D (α:=ℝ)]
+  have hh := mul_le_mul_of_nonneg_left hd (show 0≤(2:ℝ)^(4*B+100) by positivity)
+  unfold sourceEpsilon sourceGapFloor
+  rw [←mul_div_assoc, mul_one]
+  apply (div_le_div_iff₀ (by positivity) (by positivity)).mpr
+  nlinarith only [hp,hh]
+
+theorem sourceHalfMoment_strong (B D i : ℕ) {n : ℝ} (hB : 2≤B) (hn : 1<n) :
+    (10^8:ℝ)*2^(10*B)*(totalScoreExponent B D+B+1)*Real.log n ≤
+      (sourceHalfMoment B D n i:ℝ)*Real.log (sourceTargetScale B D n i) := by
+  have hlog : 0<Real.log (sourceTargetScale B D n i) := by
+    rw [sourceTargetScale,Real.log_rpow (by linarith only [hn] : 0<n)]
+    exact mul_pos (lt_of_lt_of_le (by positivity) (sourceScaleProduct_lower B D i hB)) (Real.log_pos hn)
+  exact (div_le_iff₀ hlog).mp (Nat.le_ceil _)
+
+theorem source_clipping_power_delta (B D i : ℕ) {n gap : ℝ}
+    (hB : 2≤B) (hn : 4≤n) (hg : sourceGapFloor B ≤ gap) :
+    3*(sourceTargetScale B D n i)^(-gap*(sourceHalfMoment B D n i:ℝ)/4) ≤
+      n^(-200*(totalScoreExponent B D+B+1)) := by
+  let N := sourceTargetScale B D n i
+  let A := totalScoreExponent B D+B+1
+  let k := sourceHalfMoment B D n i
+  have hn1 : 1<n := by linarith only [hn]
+  have hn0 : 0<n := by linarith only [hn]
+  have hN : 0<N := Real.rpow_pos_of_pos hn0 _
+  have hA : 1≤A := by dsimp [A]; linarith only [(totalScoreExponent_bounds B D hB).1,Nat.cast_nonneg B (α:=ℝ)]
+  have hlogN : 0≤Real.log N := by
+    dsimp [N]
+    rw [sourceTargetScale,Real.log_rpow hn0]
+    exact mul_nonneg (lt_of_lt_of_le (by positivity) (sourceScaleProduct_lower B D i hB)).le (Real.log_pos hn1).le
+  have hk := sourceHalfMoment_strong B D i hB hn1
+  change (10^8:ℝ)*2^(10*B)*A*Real.log n ≤ (k:ℝ)*Real.log N at hk
+  have hp : (2:ℝ)^(2*B) ≤ 2^(10*B) := pow_le_pow_right₀ (by norm_num) (by omega)
+  have hc : 1000 ≤ sourceGapFloor B/4*((10^8:ℝ)*2^(10*B)) := by
+    unfold sourceGapFloor
+    calc
+      (1000:ℝ) ≤ (10^8*2^(10*B))/(512*2^(2*B)) := by
+        apply (le_div_iff₀ (by positivity : (0:ℝ)<512*2^(2*B))).mpr
+        nlinarith only [hp,show (0:ℝ) ≤ 2^(2*B) by positivity]
+      _ = _ := by field_simp; ring
+  have hh := mul_le_mul_of_nonneg_left hk (show 0 ≤ sourceGapFloor B/4 by positivity [sourceGapFloor_pos B])
+  have hh' := mul_le_mul_of_nonneg_right hc (show 0≤A*Real.log n by positivity [Real.log_pos hn1])
+  have hhg := mul_le_mul_of_nonneg_right hg (show 0≤(k:ℝ)*Real.log N/4 by positivity)
+  have hdec : 1000*A*Real.log n ≤ gap*(k:ℝ)/4*Real.log N := by nlinarith only [hh,hh',hhg]
+  have h3 : Real.log 3 ≤ A*Real.log n := by
+    have hl : Real.log 3 ≤ Real.log n := Real.log_le_log (by norm_num) (by linarith only [hn])
+    exact hl.trans (le_mul_of_one_le_left (Real.log_pos hn1).le hA)
+  rw [show (3:ℝ)=Real.exp (Real.log 3) by rw [Real.exp_log (by norm_num)],Real.rpow_def_of_pos hN,←Real.exp_add,
+    Real.rpow_def_of_pos hn0]
+  exact Real.exp_le_exp.mpr (by nlinarith only [hdec,h3,mul_nonneg (by linarith only [hA] : 0≤A) (Real.log_pos hn1).le])
+
+end TraceReconstruction
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+def sourceWorkSize (B D : ℕ) : ℝ :=
+  totalScoreExponent B D+(sourceDegreeCutoff B:ℝ)+B+10
+
+theorem sourceWorkSize_pos (B D : ℕ) (hB : 2≤ B) : 0< sourceWorkSize B D := by
+  unfold sourceWorkSize
+  have hd := (totalScoreExponent_bounds B D hB).1
+  have hr := Nat.cast_nonneg (sourceDegreeCutoff B) (α:=ℝ)
+  have hb := Nat.cast_nonneg B (α:=ℝ)
+  linarith only [hd,hr,hb]
+
+theorem sourceCancellationOrder_lower (B D : ℕ) :
+    100*sourceWorkSize B D ≤ (sourceCancellationOrder B D:ℝ)*sourceEpsilon B D := by
+  exact (div_le_iff₀ (sourceEpsilon_pos B D)).mp (Nat.le_ceil _)
+
+theorem source_smoothing_absorb {B D : ℕ} {N K ω : ℝ} {m : ℕ}
+    (hB : 2≤ B) (hN : 1≤ N) (hK : 0≤ K)
+    (hm : hiddenBinomialCap m ω ≤ K*N^(-sourceEpsilon B D/2))
+    (hpref : sourceSmoothingPrefactor (sourceCancellationOrder B D) K ≤ N^(sourceEpsilon B D)) :
+    hiddenSmoothingBound m (sourceCancellationOrder B D) ω ≤ N^(-20*sourceWorkSize B D) := by
+  have hN0 : 0< N := by linarith only [hN]
+  apply (hiddenSmoothingBound_scale hN0 hK hm).trans
+  apply (mul_le_mul_of_nonneg_right hpref (by positivity)).trans
+  rw [←Real.rpow_add hN0]
+  apply Real.rpow_le_rpow_of_exponent_le hN
+  have hs := sourceCancellationOrder_lower B D
+  have hw := sourceWorkSize_pos B D hB
+  have he := sourceEpsilon_pos B D
+  have he' : sourceEpsilon B D ≤ 1 := by
+    simpa only [pow_zero,one_mul] using sourceEpsilon_power_small B D 0 (by omega)
+  have hW : 10≤ sourceWorkSize B D := by
+    unfold sourceWorkSize
+    have hd := (totalScoreExponent_bounds B D hB).1
+    have hr := Nat.cast_nonneg (sourceDegreeCutoff B) (α:=ℝ)
+    have hb := Nat.cast_nonneg B (α:=ℝ)
+    linarith only [hd,hr,hb]
+  nlinarith only [hs,he,he',hW]
+
+/-- The actual sparse count controls both extra forced roles and clipping. -/
+theorem sparse_power_absorb {N ε gap W L x : ℝ}
+    (hN : 1≤ N) (he : 0≤ε) (hg : 2*ε≤ gap) (hW : 0≤ W) (hL : 1≤ L)
+    (hx : x≤ W*L*N^(-gap)) (hfac : 8*W*L^2≤ N^ε) :
+    x≤ N^(-gap/2)/8 ∧ x≤1/2 ∧ x*L≤1 := by
+  have hN0 : 0< N := by linarith only [hN]
+  have hL0 : 0≤ L := by linarith only [hL]
+  have hg0 : 0≤ gap := by linarith only [he,hg]
+  have hwl : 8*W*L≤ N^ε := by
+    have hh : L≤ L^2 := by nlinarith only [hL,sq_nonneg (L-1)]
+    exact (mul_le_mul_of_nonneg_left hh (by positivity : 0≤8*W)).trans hfac
+  have hx' : x≤ N^(ε-gap)/8 := by
+    have hh := mul_le_mul_of_nonneg_right hwl (Real.rpow_pos_of_pos hN0 (-gap)).le
+    rw [←Real.rpow_add hN0] at hh
+    have heq : ε+-gap=ε-gap := by ring
+    rw [heq] at hh
+    nlinarith only [hx,hh]
+  have hsmall := Real.rpow_le_rpow_of_exponent_le hN (show ε-gap≤-gap/2 by linarith only [hg])
+  have hxhalf : x≤ N^(-gap/2)/8 := hx'.trans (div_le_div_of_nonneg_right hsmall (by norm_num))
+  have hone : N^(-gap/2)≤1 := Real.rpow_le_one_of_one_le_of_nonpos hN (by linarith only [hg0])
+  refine ⟨hxhalf,by linarith only [hxhalf,hone],?_⟩
+  have hh := mul_le_mul_of_nonneg_right hfac (Real.rpow_pos_of_pos hN0 (-gap)).le
+  rw [←Real.rpow_add hN0] at hh
+  have hh' := mul_le_mul_of_nonneg_right hx hL0
+  have hr := Real.rpow_le_one_of_one_le_of_nonpos hN (show ε+-gap≤0 by linarith only [hg,he])
+  nlinarith only [hh,hh',hr]
+
+theorem source_weight_square {N e d L : ℝ} {k : ℕ} (hN : 0< N)
+    (hL : L=N^(-e*d/2)) :
+    ((6*(2*k:ℕ):ℝ)*L)^2=144*(k:ℝ)^2*N^(-e*d) := by
+  rw [hL,mul_pow]
+  have hr : (N^(-e*d/2))^2=N^(-e*d) := by
+    rw [←Real.rpow_natCast,←Real.rpow_mul hN.le]
+    norm_num only [Nat.cast_ofNat]
+    congr 1
+    ring
+  rw [hr]
+  push_cast
+  ring
+
+theorem source_high_tail_margin (B : ℕ) {d : ℝ} (_hB : 2≤ B) (hd : 0≤ d) :
+    d+blockGain B+1 ≤ ((d+1)*sourceGapFloor B)/2*(sourceDegreeCutoff B:ℝ) := by
+  have hp : (2:ℝ)^(2*B+8)*(B+2) ≤ 2^(10*B+40) := by
+    have hb : (B:ℝ)+2 ≤ (2:ℝ)^(B+2) := by
+      have hh : B+1 < 2^(B+1) := Nat.lt_two_pow_self
+      have hh' : B+2≤2^(B+1) := by omega
+      have hcast : (B:ℝ)+2≤(2:ℝ)^(B+1) := by exact_mod_cast hh'
+      exact hcast.trans (pow_le_pow_right₀ (by norm_num) (by omega))
+    have hh := mul_le_mul_of_nonneg_left hb (show (0:ℝ)≤2^(2*B+8) by positivity)
+    rw [←pow_add] at hh
+    exact hh.trans (pow_le_pow_right₀ (by norm_num) (by omega))
+  have hc : (B:ℝ)+2 ≤ sourceGapFloor B/2*(sourceDegreeCutoff B:ℝ) := by
+    unfold sourceGapFloor sourceDegreeCutoff
+    push_cast
+    calc
+      (B:ℝ)+2 ≤ 2^(10*B+40)/(256*2^(2*B)) := by
+        apply (le_div_iff₀ (by positivity : (0:ℝ)<256*2^(2*B))).mpr
+        rw [pow_add] at hp
+        norm_num only [show (2:ℝ)^8=256 by norm_num] at hp
+        nlinarith only [hp]
+      _ = _ := by field_simp; norm_num
+  have hh := mul_le_mul_of_nonneg_left hc (show 0≤ d+1 by linarith only [hd])
+  unfold blockGain
+  have hb : (0:ℝ)≤ B := Nat.cast_nonneg _
+  nlinarith only [hh,hd,hb,mul_nonneg hd hb]
+
+end TraceReconstruction
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem rpow_coefficient_absorb {N ε e d c : ℝ} (hN : 1≤ N)
+    (hc : 3*c≤ N^ε) (he : ε+e≤ d) : c*N^e ≤ N^d/3 := by
+  have hN0 : 0< N := by linarith only [hN]
+  have hh := mul_le_mul_of_nonneg_right hc (Real.rpow_pos_of_pos hN0 e).le
+  rw [←Real.rpow_add hN0] at hh
+  have hr := Real.rpow_le_rpow_of_exponent_le hN he
+  linarith only [hh,hr]
+
+/-- Closing the three explicit sharp classes, with the source's fixed
+ finite degree cutoff and no hidden dependence on N in their constants. -/
+theorem sharp_energy_absorb {N ε d M Fdec gap Cr Cf E : ℝ} {R : ℕ}
+    (hN : 1≤ N) (he : ε≤1) (hM : d+(R:ℝ)+1≤ M) (hd : 0≤ d) (_hR : 0≤(R:ℝ))
+    (hbad : 384*Cr≤ N^ε) (hforce : 192*Cf≤ N^ε) (htail : 1536≤ N^ε)
+    (hfdec : d+ε≤ Fdec) (hgap : d+ε≤ gap*(R:ℝ))
+    (hE : E ≤ Cr*N^R*((N^(-20*M))^2+(N^(-20*M))^2)+Cf*N^(-Fdec)+8*N^(-gap*(R:ℝ))) :
+    64*E≤ N^(-d) := by
+  have hN0 : 0< N := by linarith only [hN]
+  have hpow : N^R*(N^(-20*M))^2=N^((R:ℝ)-40*M) := by
+    rw [←Real.rpow_natCast (N^(-20*M)) 2,←Real.rpow_mul hN0.le,←Real.rpow_natCast N R,←Real.rpow_add hN0]
+    norm_num only [Nat.cast_ofNat]
+    congr 1
+    ring
+  have hb := rpow_coefficient_absorb hN (c:=128*Cr) (by nlinarith only [hbad])
+    (e:=(R:ℝ)-40*M) (d:=-d) (by linarith only [hM,hd,he])
+  have hf := rpow_coefficient_absorb hN (c:=64*Cf) (by nlinarith only [hforce])
+    (e:=-Fdec) (d:=-d) (by linarith only [hfdec])
+  have ht := rpow_coefficient_absorb hN (c:=512) (by norm_num at htail ⊢; exact htail)
+    (e:=-gap*(R:ℝ)) (d:=-d) (by linarith only [hgap])
+  have hh := mul_le_mul_of_nonneg_left hE (by norm_num : (0:ℝ)≤64)
+  have heq : 64*(Cr*N^R*((N^(-20*M))^2+(N^(-20*M))^2)+Cf*N^(-Fdec)+8*N^(-gap*(R:ℝ))) =
+      128*Cr*N^((R:ℝ)-40*M)+64*Cf*N^(-Fdec)+512*N^(-gap*(R:ℝ)) := by
+    calc
+      _ = 128*Cr*(N^R*(N^(-20*M))^2)+64*Cf*N^(-Fdec)+512*N^(-gap*(R:ℝ)) := by ring
+      _ = _ := by rw [hpow]
+  rw [heq] at hh
+  linarith only [hh,hb,hf,ht]
+
+end TraceReconstruction
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem sourceDimension_bounds (B i : ℕ) (hB : 2≤ B) :
+    2≤ sourceDimension B i ∧ sourceDimension B i≤ B := by
+  unfold sourceDimension
+  split_ifs <;> omega
+
+theorem totalScoreExponent_mono (B : ℕ) (hB : 2≤ B) : Monotone (totalScoreExponent B) := by
+  apply monotone_nat_of_le_succ
+  intro i
+  rw [totalScoreExponent_step]
+  exact le_add_of_nonneg_right (blockGain_pos _ (sourceDimension_bounds B i hB).1).le
+
+theorem sourceTargetScale_bounds (B D i : ℕ) {n : ℝ} (hB : 2≤ B) (hn : 1≤ n) :
+    1≤ sourceTargetScale B D n i ∧ sourceTargetScale B D n i≤ n := by
+  have hp : 0≤ sourceScaleProduct B D i :=
+    (show (0:ℝ)≤1/(2*2^(B*D)) by positivity).trans (sourceScaleProduct_lower B D i hB)
+  refine ⟨Real.one_le_rpow hn hp,?_⟩
+  simpa only [sourceTargetScale,Real.rpow_one] using Real.rpow_le_rpow_of_exponent_le hn (sourceScaleProduct_le_one B D i hB)
+
+theorem ConcreteLevelPair.error_le {q N Δ d ω F : ℝ} {hq : q∈Set.Icc 0 1} {i k : ℕ}
+    (C : ConcreteLevelPair q hq N Δ d ω F i k) (hN : 0< N) (hΔ : 0≤Δ) :
+    C.error≤((i:ℝ)+1)*Δ*N := by
+  have hl : 0<(C.left.length:ℝ) := lt_of_lt_of_le (by linarith only [hN]) C.length_lower
+  exact ((div_le_iff₀ hl).mp C.normalized_error).trans
+    (mul_le_mul_of_nonneg_left C.length_upper (by positivity))
+
+theorem source_warmup_score_margin (B D : ℕ) {d : ℝ} (hB : 2≤ B)
+    (hd : 11/5≤ d) (hd' : d≤3+(B:ℝ)*D) :
+    27/50≤ scaleA 2 0 (sourceEpsilon B D)*d ∧ scaleA 2 1 (sourceEpsilon B D)≤1/2 := by
+  have he := sourceEpsilon_pos B D
+  have hloss := sourceEpsilon_loss B D 2 d hB (by linarith only [hd]) hd'
+  norm_num only [show (2:ℝ)^2=4 by norm_num,Nat.cast_ofNat] at hloss
+  have hlo := mul_le_mul_of_nonneg_right (scaleA_zero_simple_lower 2 (sourceEpsilon B D) he.le)
+    (show 0≤ d by linarith only [hd])
+  norm_num only [show (2:ℝ)^2=4 by norm_num] at hlo
+  constructor
+  · nlinarith only [hloss,hlo,hd,he]
+  · have hh := scaleA_le_geometric 2 1 (sourceEpsilon B D) he.le (by omega)
+    norm_num at hh ⊢
+    exact hh
+
+theorem source_main_score_margin (B D : ℕ) {d : ℝ} (hB : 2≤ B)
+    (hd : (2:ℝ)^B+2≤ d) (hd' : d≤3+(B:ℝ)*D) :
+    1+1/(2:ℝ)^B ≤ scaleA B 0 (sourceEpsilon B D)*d := by
+  let X := (2:ℝ)^B
+  let ε := sourceEpsilon B D
+  have hX : 4≤ X := by
+    have hh := pow_le_pow_right₀ (by norm_num : (1:ℝ)≤2) hB
+    norm_num at hh
+    exact hh
+  have hX0 : 0< X := by linarith only [hX]
+  have he : 0<ε := sourceEpsilon_pos _ _
+  have hd0 : 0≤ d := by dsimp [X] at hX; linarith only [hd,hX]
+  have hloss := sourceEpsilon_loss B D B d le_rfl hd0 hd'
+  change ((X-1)*d+3*B*(X/2))*ε<1/100 at hloss
+  have hnonneg : 0≤3*(B:ℝ)*(X/2)*ε := by positivity
+  have hde : 0≤ d*ε := mul_nonneg hd0 he.le
+  have hmul : X*d*ε≤1/50 := by
+    have hxx : 0≤(X-2)*(d*ε) := mul_nonneg (by linarith only [hX]) hde
+    nlinarith only [hloss,hnonneg,hxx]
+  have hlo := mul_le_mul_of_nonneg_right (scaleA_zero_simple_lower B ε he.le) hd0
+  change (1/X-ε)*d≤ scaleA B 0 ε*d at hlo
+  have hgoal : 1+1/X≤(1/X-ε)*d := by
+    apply (mul_le_mul_iff_left₀ hX0).mp
+    field_simp
+    change X+2 ≤ d at hd
+    nlinarith only [hd,hmul]
+  exact hgoal.trans hlo
+
+end TraceReconstruction
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem sourceSparseCount_ge_one (T : ℕ→ℕ) (s b : ℕ) (hT : ∀j<b,1 ≤ T (j+1)) :
+    1 ≤ sourceSparseCount T s b := by
+  unfold sourceSparseCount
+  apply one_le_mul_of_one_le_of_one_le (one_le_pow₀ (by omega))
+  exact Finset.one_le_prod (fun j hj=>hT j (Finset.mem_range.mp hj))
+
+/-- Scalar closure of the exact sparse/forced block budget. All assumptions
+ here are explicit elementary coefficient bounds, not oracle conclusions. -/
+theorem source_block_energy_close (T n : ℕ→ℕ) (s b R : ℕ) (ω N ε d dnew K W M gap : ℝ)
+    (hN : 1 ≤ N) (he : 0 ≤ ε) (he1 : ε ≤ 1) (hK : 0 ≤ K) (hW : 0 ≤ W)
+    (hd : 0 ≤ dnew) (hM : dnew+(R:ℝ)+1 ≤ M) (hg : 2*ε ≤ gap)
+    (hf : dnew+ε ≤ forcedDecay (b+1) ε d) (hR : dnew+ε ≤ gap/2*(R:ℝ))
+    (hfac : 8*W*((2^s:ℕ):ℝ)^2 ≤ N^ε)
+    (hbad : 384*sourceBadRoleFactor b R ≤ N^ε)
+    (hforce : 192*(sourceForcedRoleFactor b R*W^(2^(b+1))*sourceSkeletonPrefactor s b K) ≤ N^ε)
+    (htail : 1536 ≤ N^ε)
+    (hleaf : (SourceLeafCount T (b+1):ℝ) ≤ N)
+    (hcount : 1 ≤ sourceSparseCount T s b)
+    (k : ℕ) (L : ℝ)
+    (hw : ((6*(2*k:ℕ):ℝ)*L)^2 ≤ W*N^(-scaleA (b+1) 0 ε*d))
+    (hsparse : ((6*(2*k:ℕ):ℝ)*L)^2*(sourceSparseCount T s b:ℝ) ≤
+      W*((2^s:ℕ):ℝ)*N^(-gap))
+    (hn : ∀j ≤ b,hiddenBinomialCap (n j) ω ≤ K*N^(-ε/2))
+    (hgcap : ∀j<b,hiddenBinomialCap (sourceScheduledTrials T n (j+1)) ω ≤
+      K*N^(-(scaleKappa (b+1) (j+2) ε+ε)/2)) :
+    ((6*(2*k:ℕ):ℝ)*L)^2*(sourceSparseCount T s b:ℝ) ≤ N^(-gap/2)/8 ∧
+    ((6*(2*k:ℕ):ℝ)*L)^2*(sourceSparseCount T s b:ℝ) ≤ 1/2 ∧
+    64*sourceSharpEnergyBudget T n s b R ω (N^(-20*M)) (N^(-20*M)) ((6*(2*k:ℕ):ℝ)*L) ≤ N^(-dnew) := by
+  let w := (6*(2*k:ℕ):ℝ)*L
+  have hmot : 1 ≤ ((2^s:ℕ):ℝ) := by exact_mod_cast (one_le_pow₀ (by omega : 1 ≤ (2:ℕ)) : 1 ≤ (2:ℕ)^s)
+  have hx := sparse_power_absorb hN he hg hW hmot hsparse hfac
+  have hwone : w^2 ≤ 1 := by
+    have hc : (1:ℝ) ≤ sourceSparseCount T s b := by exact_mod_cast hcount
+    have hh := mul_le_mul_of_nonneg_left hc (sq_nonneg w)
+    dsimp [w] at hh
+    nlinarith only [hh,hx.2.1]
+  have hsmall : w^2*((2^s:ℕ):ℝ)*(sourceSparseCount T s b:ℝ) ≤ 1 := by
+    dsimp [w]
+    nlinarith only [hx.2.2]
+  have hr : w^2*(sourceSparseCount T s b:ℝ) ≤ 1*N^(-(gap/2)) := by
+    have hp : 0 ≤ N^(-gap/2) := Real.rpow_nonneg (by linarith only [hN]) _
+    dsimp [w]
+    rw [←neg_div]
+    nlinarith only [hx.1,hp]
+  have hnum := source_sharp_numeric T n s b R ω (N^(-20*M)) (N^(-20*M)) w N ε d K W 1 (gap/2)
+    hN hK hW (by norm_num) hleaf hwone hw hsmall hx.2.1 hr hn hgcap
+  simp only [one_pow,mul_one] at hnum
+  refine ⟨hx.1,hx.2.1,?_⟩
+  exact sharp_energy_absorb hN he1 hM hd (Nat.cast_nonneg _) hbad hforce htail hf hR hnum
+
+end TraceReconstruction
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- Instantiation at the manuscript's literal block dimension, integer
+ schedules, source cutoff, moment envelope, and full next score exponent. -/
+theorem source_level_energy (B D i ell h k : ℕ) (C N F ω K : ℝ)
+    (hB : 2 ≤ B) (hi : i<D) (hN : 1 ≤ N) (hC : 1 ≤ C) (hF : 1 ≤ F) (hω : 0<ω)
+    (H : SourceArithmeticBounds C N F (sourceEpsilon B D) (sourceCancellationOrder B D)
+      (sourceDimension B i) ell h)
+    (hk : (k:ℝ) ≤ sourceMomentEnvelope B D)
+    (hK : 200*(C^(2*sourceDimension B i)*F)/ω ≤ K)
+    (hfac : 8*sourceWeightEnvelope B D*((2^(sourceCancellationOrder B D):ℕ):ℝ)^2 ≤ N^(sourceEpsilon B D))
+    (hbad : 384*sourceBadRoleFactor (sourceDimension B i-1) (sourceDegreeCutoff B) ≤ N^(sourceEpsilon B D))
+    (hforce : 192*(sourceForcedRoleFactor (sourceDimension B i-1) (sourceDegreeCutoff B)*
+        (sourceWeightEnvelope B D)^(2^(sourceDimension B i))*
+        sourceSkeletonPrefactor (sourceCancellationOrder B D) (sourceDimension B i-1) K) ≤ N^(sourceEpsilon B D))
+    (htail : 1536 ≤ N^(sourceEpsilon B D))
+    (hsmooth : sourceSmoothingPrefactor (sourceCancellationOrder B D) K ≤ N^(sourceEpsilon B D)) :
+    let b := sourceDimension B i
+    let ε := sourceEpsilon B D
+    let s := sourceCancellationOrder B D
+    let T := sourceBlockWindows N b ε
+    let t := sourceArithmeticTrials C N b ε ell h
+    let L := N^(-scaleA b 0 ε*totalScoreExponent B i/2)
+    (((6*(2*k:ℕ):ℝ)*L)^2*(sourceSparseCount T s (b-1):ℝ) ≤
+        N^(-((totalScoreExponent B i+1)*sourceGapFloor B)/2)/8) ∧
+    (((6*(2*k:ℕ):ℝ)*L)^2*(sourceSparseCount T s (b-1):ℝ) ≤ 1/2) ∧
+    (64*sourceSharpEnergyBudget T t s (b-1) (sourceDegreeCutoff B) ω
+       (N^(-20*sourceWorkSize B D)) (N^(-20*sourceWorkSize B D)) ((6*(2*k:ℕ):ℝ)*L) ≤
+       N^(-totalScoreExponent B (i+1))) ∧
+    (∀j<b,hiddenSmoothingBound (t j) s ω ≤ N^(-20*sourceWorkSize B D)) := by
+  let b := sourceDimension B i
+  let ε := sourceEpsilon B D
+  let s := sourceCancellationOrder B D
+  let d := totalScoreExponent B i
+  let dnew := totalScoreExponent B (i+1)
+  let W := sourceWeightEnvelope B D
+  let gap := (d+1)*sourceGapFloor B
+  let T := sourceBlockWindows N b ε
+  let t := sourceArithmeticTrials C N b ε ell h
+  let L := N^(-scaleA b 0 ε*d/2)
+  have hb := sourceDimension_bounds B i hB
+  have hb1 : b-1+1=b := by dsimp [b]; omega
+  have he := sourceEpsilon_pos B D
+  have he1 : ε ≤ 1/100 := by
+    have hh := sourceEpsilon_gap_small B D
+    have hg := sourceGapFloor_le_one B
+    dsimp [ε]
+    linarith only [hh,hg]
+  have hd : 0 ≤ d := le_trans (by norm_num) (totalScoreExponent_bounds B i hB).1
+  have hdnew : 0 ≤ dnew := le_trans (by norm_num) (totalScoreExponent_bounds B (i+1) hB).1
+  have hdle : d ≤ totalScoreExponent B D := totalScoreExponent_mono B hB (Nat.le_of_lt hi)
+  have hdnewle : dnew ≤ totalScoreExponent B D := totalScoreExponent_mono B hB (by omega)
+  have hdd : d ≤ 3+(B:ℝ)*D := by
+    have hh := (totalScoreExponent_bounds B D hB).2
+    nlinarith only [hdle,hh]
+  have hN0 : 0<N := by linarith only [hN]
+  have hK0 : 0 ≤ K := le_trans (by positivity : 0 ≤ 200*(C^(2*sourceDimension B i)*F)/ω) hK
+  have hW : 0 ≤ W := by dsimp [W,sourceWeightEnvelope]; exact mul_nonneg (by norm_num) (sq_nonneg _)
+  have hM : dnew+(sourceDegreeCutoff B:ℝ)+1 ≤ sourceWorkSize B D := by
+    unfold sourceWorkSize
+    linarith only [hdnewle,Nat.cast_nonneg B (α:=ℝ)]
+  have hg : 2*ε ≤ gap := by
+    have hf := sourceGapFloor_pos B
+    have hh := sourceEpsilon_gap_small B D
+    dsimp [gap,ε]
+    nlinarith only [hh,hd,mul_nonneg hd hf.le,he]
+  have hfdec : dnew+ε ≤ forcedDecay b ε d := by
+    have hh := forcedDecay_margin B D b d (by dsimp [b]; omega) hb.2 hd hdd
+    have heq : dnew=d+blockGain b := totalScoreExponent_step B i
+    rw [heq]
+    unfold blockGain
+    linarith only [hh,he1]
+  have hR : dnew+ε ≤ gap/2*(sourceDegreeCutoff B:ℝ) := by
+    have hh := source_high_tail_margin B hB hd
+    have heq : dnew=d+blockGain b := totalScoreExponent_step B i
+    have hbg : blockGain b ≤ blockGain B := by unfold blockGain; exact (by
+      have hcast : (b:ℝ) ≤ B := by exact_mod_cast hb.2
+      linarith only [hcast])
+    dsimp [gap]
+    rw [heq]
+    linarith only [hh,hbg,he1]
+  have hw : ((6*(2*k:ℕ):ℝ)*L)^2 ≤ W*N^(-scaleA b 0 ε*d) := by
+    rw [source_weight_square hN0 (show L=N^(-scaleA b 0 ε*d/2) from rfl)]
+    have hh := mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (Nat.cast_nonneg k) hk 2) (by norm_num : (0:ℝ)≤144)
+    exact mul_le_mul_of_nonneg_right hh (Real.rpow_pos_of_pos hN0 _).le
+  have hsparse : ((6*(2*k:ℕ):ℝ)*L)^2*(sourceSparseCount T s (b-1):ℝ) ≤ W*((2^s:ℕ):ℝ)*N^(-gap) := by
+    apply (source_sparse_weight_bound N ε d W _ b s hN0 hW hw).trans
+    apply mul_le_mul_of_nonneg_left _ (by positivity)
+    apply Real.rpow_le_rpow_of_exponent_le hN
+    have hh := source_level_sparse_gap B D i hB
+    have heq : (d+1)/(128*(2:ℝ)^(2*B))=gap := by dsimp [gap,sourceGapFloor]; ring
+    rw [heq] at hh
+    linarith only [hh]
+  have hcaps := H.binomial_caps hN0 hC hF hω
+  have hncap (j) (hj : j<b) : hiddenBinomialCap (t j) ω ≤ K*N^(-ε/2) :=
+    (hcaps.1 j hj).trans (mul_le_mul_of_nonneg_right hK (Real.rpow_pos_of_pos hN0 _).le)
+  have hgcap (j) (hj : j<b-1) : hiddenBinomialCap (sourceScheduledTrials T t (j+1)) ω ≤
+      K*N^(-(scaleKappa b (j+2) ε+ε)/2) :=
+    (hcaps.2 j (by omega)).trans (mul_le_mul_of_nonneg_right hK (Real.rpow_pos_of_pos hN0 _).le)
+  have hc : 1 ≤ sourceSparseCount T s (b-1) := sourceSparseCount_ge_one T s (b-1)
+    (fun j _=>Nat.succ_le_of_lt (H.window_positive (j+1)))
+  have HE := source_block_energy_close T t s (b-1) (sourceDegreeCutoff B) ω N ε d dnew K W
+    (sourceWorkSize B D) gap hN he.le (by linarith only [he1]) hK0 hW hdnew hM hg
+    (by rw [hb1]; exact hfdec) hR hfac hbad (by rw [hb1]; exact hforce) htail
+    (by rw [hb1]; exact H.leaf_bound hN0) hc k L (by rw [hb1]; exact hw) hsparse
+    (fun j hj=>hncap j (by omega)) (fun j hj=>by rw [hb1]; exact hgcap j hj)
+  refine ⟨HE.1,HE.2.1,HE.2.2,?_⟩
+  intro j hj
+  exact source_smoothing_absorb hB hN hK0 (hncap j hj) hsmooth
+
+end TraceReconstruction
+noncomputable section
+namespace TraceReconstruction
+open Filter
+open scoped Topology Classical
+
+structure SourceNumericMargins (B D : ℕ) (q N : ℝ) : Prop where
+  geometry : 1200*sourceCapEnvelope B D q ≤ N^(sourceEpsilon B D)
+  motif : 48*(2:ℝ)^(sourceCancellationOrder B D) ≤ N^(sourceEpsilon B D)
+  sparse : 8*sourceWeightEnvelope B D*((2:ℝ)^(sourceCancellationOrder B D))^2 ≤ N^(sourceEpsilon B D)
+  smoothing : sourceSmoothingPrefactor (sourceCancellationOrder B D) (sourceCapEnvelope B D q) ≤ N^(sourceEpsilon B D)
+  bad : ∀b ≤ B,384*sourceBadRoleFactor b (sourceDegreeCutoff B) ≤ N^(sourceEpsilon B D)
+  force : ∀b ≤ B,192*(sourceForcedRoleFactor b (sourceDegreeCutoff B)*(sourceWeightEnvelope B D)^(2^(b+1))*
+    sourceSkeletonPrefactor (sourceCancellationOrder B D) b (sourceCapEnvelope B D q)) ≤ N^(sourceEpsilon B D)
+  tail : 1536 ≤ N^(sourceEpsilon B D)
+  error : 64*(sourceCapEnvelope B D q+1)*((D:ℝ)+1) ≤ N^(sourceEpsilon B D)
+  overlap : 1099511627776*(sourceCapEnvelope B D q+1)*(sourceWeightEnvelope B D+1)^2*
+    ((2:ℝ)^(sourceCancellationOrder B D))^(B+2) ≤ N^(sourceEpsilon B D)
+
+theorem sourceWeightEnvelope_nonneg (B D : ℕ) : 0 ≤ sourceWeightEnvelope B D := by
+  unfold sourceWeightEnvelope
+  exact mul_nonneg (by norm_num) (sq_nonneg _)
+
+theorem sourceForcedRoleFactor_nonneg (b R : ℕ) : 0 ≤ sourceForcedRoleFactor b R := by
+  unfold sourceForcedRoleFactor
+  exact Finset.sum_nonneg (fun _ _=>by positivity)
+
+theorem source_forced_prefactor_expUpper (B b : ℕ) (hB : 2 ≤ B) :
+    SourceExpUpper (fun D q=>192*(sourceForcedRoleFactor b (sourceDegreeCutoff B)*
+      (sourceWeightEnvelope B D)^(2^(b+1))*
+      sourceSkeletonPrefactor (sourceCancellationOrder B D) b (sourceCapEnvelope B D q))) := by
+  have H := (((SourceExpUpper.const (sourceForcedRoleFactor b (sourceDegreeCutoff B))).mul
+    ((sourceWeightEnvelope_expUpper B).pow (fun _ _ _=>sourceWeightEnvelope_nonneg _ _) (2^(b+1)))
+    (fun _ _ _=>sourceForcedRoleFactor_nonneg _ _) (fun _ _ _=>pow_nonneg (sourceWeightEnvelope_nonneg _ _) _)).mul
+      (sourceSkeletonPrefactor_expUpper B b hB)
+      (fun _ _ _=>mul_nonneg (sourceForcedRoleFactor_nonneg _ _) (pow_nonneg (sourceWeightEnvelope_nonneg _ _) _))
+      (fun D q hq=>sourceSkeletonPrefactor_nonneg _ _ (sourceCapEnvelope_pos B D hq.1).le))
+  exact (SourceExpUpper.const 192).mul H (fun _ _ _=>by norm_num)
+    (fun D q hq=>mul_nonneg (mul_nonneg (sourceForcedRoleFactor_nonneg _ _)
+      (pow_nonneg (sourceWeightEnvelope_nonneg _ _) _))
+        (sourceSkeletonPrefactor_nonneg _ _ (sourceCapEnvelope_pos B D hq.1).le))
+
+theorem source_overlap_prefactor_expUpper (B : ℕ) (hB : 2 ≤ B) :
+    SourceExpUpper (fun D q=>1099511627776*(sourceCapEnvelope B D q+1)*(sourceWeightEnvelope B D+1)^2*
+      ((2:ℝ)^(sourceCancellationOrder B D))^(B+2)) := by
+  have hK := (sourceCapEnvelope_expUpper B).add (SourceExpUpper.const 1)
+  have hW := (sourceWeightEnvelope_expUpper B).add (SourceExpUpper.const 1)
+  have hL := sourceMotif_expUpper B hB
+  exact (((SourceExpUpper.const 1099511627776).mul hK (fun _ _ _=>by norm_num)
+    (fun D q hq=>by linarith only [sourceCapEnvelope_pos B D hq.1])).mul
+      (hW.pow (fun D _ _=>by linarith only [sourceWeightEnvelope_nonneg B D]) 2)
+      (fun D q hq=>by positivity [sourceCapEnvelope_pos B D hq.1]) (fun _ _ _=>sq_nonneg _)).mul
+        (hL.pow (fun _ _ _=>by positivity) (B+2))
+        (fun D q hq=>by positivity [sourceCapEnvelope_pos B D hq.1]) (fun _ _ _=>by positivity)
+
+theorem source_numeric_margins_eventually (B : ℕ) (ζ : ℝ) (hB : 2 ≤ B) (hζ : ζ∈Set.Ioo 0 1)
+    {n q : ℕ→ℝ} (hn : Tendsto n atTop atTop) (hq : ∀t,q t∈Set.Ioc 0 (1/2:ℝ))
+    (hH : Tendsto (fun t=>(q t)^3*Real.log (n t)) atTop atTop) :
+    ∀ᶠt in atTop,∀i,SourceNumericMargins B (sourceDepth B ζ ((q t)^3*Real.log (n t))) (q t)
+      (sourceTargetScale B (sourceDepth B ζ ((q t)^3*Real.log (n t))) (n t) i) := by
+  have hu (f : ℕ→ℝ→ℝ) (hf : SourceExpUpper f) :=
+    SourceExpUpper.uniform f hf B ζ 1 hB hζ (by norm_num) hn hq hH
+  have hgeom := hu _ ((SourceExpUpper.const 1200).mul (sourceCapEnvelope_expUpper B)
+    (fun _ _ _=>by norm_num) (fun D q hq=>(sourceCapEnvelope_pos B D hq.1).le))
+  have hmot := hu _ ((SourceExpUpper.const 48).mul (sourceMotif_expUpper B hB)
+    (fun _ _ _=>by norm_num) (fun _ _ _=>by positivity))
+  have hsparse := hu _ (((SourceExpUpper.const 8).mul (sourceWeightEnvelope_expUpper B)
+    (fun _ _ _=>by norm_num) (fun _ _ _=>sourceWeightEnvelope_nonneg _ _)).mul
+      ((sourceMotif_expUpper B hB).pow (fun _ _ _=>by positivity) 2)
+      (fun _ _ _=>mul_nonneg (by norm_num) (sourceWeightEnvelope_nonneg _ _)) (fun _ _ _=>sq_nonneg _))
+  have hsmooth := hu _ (sourceSmoothingPrefactor_expUpper B hB)
+  have hbad : ∀ᶠt in atTop,∀b∈Finset.range (B+1),∀i,
+      384*sourceBadRoleFactor b (sourceDegreeCutoff B) ≤
+        (sourceTargetScale B (sourceDepth B ζ ((q t)^3*Real.log (n t))) (n t) i)^
+          (sourceEpsilon B (sourceDepth B ζ ((q t)^3*Real.log (n t)))) := by
+    apply (Filter.eventually_all_finset _).mpr
+    intro b _
+    simpa only [one_mul] using hu _ (SourceExpUpper.const (384*sourceBadRoleFactor b (sourceDegreeCutoff B)))
+  have hforce : ∀ᶠt in atTop,∀b∈Finset.range (B+1),∀i,
+      192*(sourceForcedRoleFactor b (sourceDegreeCutoff B)*
+        (sourceWeightEnvelope B (sourceDepth B ζ ((q t)^3*Real.log (n t))))^(2^(b+1))*
+        sourceSkeletonPrefactor (sourceCancellationOrder B (sourceDepth B ζ ((q t)^3*Real.log (n t))))
+          b (sourceCapEnvelope B (sourceDepth B ζ ((q t)^3*Real.log (n t))) (q t))) ≤
+        (sourceTargetScale B (sourceDepth B ζ ((q t)^3*Real.log (n t))) (n t) i)^
+          (sourceEpsilon B (sourceDepth B ζ ((q t)^3*Real.log (n t)))) := by
+    apply (Filter.eventually_all_finset _).mpr
+    intro b _
+    simpa only [one_mul] using hu _ (source_forced_prefactor_expUpper B b hB)
+  have htail := hu _ (SourceExpUpper.const 1536)
+  have herr := hu _ (((SourceExpUpper.const 64).mul
+    ((sourceCapEnvelope_expUpper B).add (SourceExpUpper.const 1))
+    (fun _ _ _=>by norm_num) (fun D q hq=>by positivity [sourceCapEnvelope_pos B D hq.1])).mul
+    SourcePolyUpper.depth.expUpper (fun D q hq=>by positivity [sourceCapEnvelope_pos B D hq.1])
+      (fun _ _ _=>by positivity))
+  have hov := hu _ (source_overlap_prefactor_expUpper B hB)
+  filter_upwards [hgeom,hmot,hsparse,hsmooth,hbad,hforce,htail,herr,hov] with t hg hm hs hsm hb hf ht he ho
+  intro i
+  refine ⟨?_,?_,?_,?_,fun b h=>hb b (Finset.mem_range.mpr (by omega)) i,
+    fun b h=>hf b (Finset.mem_range.mpr (by omega)) i,?_,?_,?_⟩ <;>
+    first | simpa only [one_mul] using hg i | simpa only [one_mul] using hm i |
+      simpa only [one_mul] using hs i | simpa only [one_mul] using hsm i |
+      simpa only [one_mul] using ht i | simpa only [one_mul] using he i | simpa only [one_mul] using ho i
+
+end TraceReconstruction
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem source_count_envelope (B D i b : ℕ) {q : ℝ}
+    (hq : q∈Set.Ioc 0 (1/2:ℝ)) (hi : i≤ D) (hb : b≤ B) :
+    (sourceOverlapConstant q)^(2*b)*(sourceOverlapConstant q)^(2*B*i+2) ≤
+      (sourceOverlapConstant q)^(2*B*(D+1)+2) := by
+  rw [←pow_add]
+  apply pow_le_pow_right₀ ((by norm_num : (1:ℝ)≤ 2).trans (sourceOverlapConstant_ge_two hq))
+  have hh := Nat.mul_le_mul_left (2*B) hi
+  nlinarith only [hh,hb]
+
+theorem source_cap_envelope (B D i b : ℕ) {q : ℝ}
+    (hq : q∈Set.Ioc 0 (1/2:ℝ)) (hi : i≤ D) (hb : b≤ B) :
+    200*((sourceOverlapConstant q)^(2*b)*(sourceOverlapConstant q)^(2*B*i+2))/initialOverlapLower q ≤
+      sourceCapEnvelope B D q := by
+  dsimp [sourceCapEnvelope]
+  rw [div_eq_mul_inv]
+  exact mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left (source_count_envelope B D i b hq hi hb)
+    (by norm_num)) (inv_pos.mpr (initialOverlapLower_pos q)).le
+
+theorem source_local_envelope (B D i b : ℕ) {q : ℝ}
+    (hq : q∈Set.Ioc 0 (1/2:ℝ)) (hi : i≤ D) (hb : b≤ B) :
+    (sourceOverlapConstant q)^(2*b)*(sourceOverlapConstant q)^(2*B*i+2) ≤
+      sourceCapEnvelope B D q := by
+  have hω := initialOverlapLower_lt_eighth hq.1
+  have hω0 := initialOverlapLower_pos q
+  have hG : 0 ≤ (sourceOverlapConstant q)^(2*b)*(sourceOverlapConstant q)^(2*B*i+2) := by
+    positivity [sourceOverlapConstant_pos hq.1]
+  apply le_trans _ (source_cap_envelope B D i b hq hi hb)
+  apply (le_div_iff₀ hω0).mpr
+  nlinarith only [hω,hG,mul_nonneg hG (show 0 ≤ 200-initialOverlapLower q by linarith only [hω])]
+
+theorem SourceArithmeticBounds.count_bound {C N F ε : ℝ} {s b ell k j : ℕ}
+    (H : SourceArithmeticBounds C N F ε s b ell k) (hC : 1≤ C) (hF : 0≤ F) (hN : 1≤ N)
+    (hj : j≤ b) (ha : 0≤ scaleA b j ε) :
+    ((sourceArithmeticCount C N b ε ell k j+1:ℕ):ℝ) ≤
+      (C^(2*b)*F+1)*N^(scaleA b j ε) := by
+  have hc := (H.counts j hj).2
+  have hc' : (sourceArithmeticCount C N b ε ell k j:ℝ) ≤
+      (C^(2*b)*F)*N^(scaleA b j ε) := by
+    apply hc.trans
+    exact mul_le_mul (mul_le_mul_of_nonneg_right (pow_le_pow_right₀ hC (by omega)) hF)
+      (H.lengths j hj).2 (by positivity) (by positivity)
+  have ho := Real.one_le_rpow hN ha
+  push_cast
+  nlinarith only [hc',ho]
+
+theorem sourceErrorUnit_small {B D : ℕ} {n : ℝ} (hB : 2≤ B) (hn : 4≤ n) :
+    4*n^(-200*(totalScoreExponent B D+B+1)) ≤ n^(-100*(totalScoreExponent B D+B+1)) ∧
+    4*n^(-200*(totalScoreExponent B D+B+1)) ≤ 1/2 := by
+  let A := totalScoreExponent B D+B+1
+  have hA : 1≤ A := by dsimp [A]; linarith only [(totalScoreExponent_bounds B D hB).1,Nat.cast_nonneg B (α:=ℝ)]
+  have hn0 : 0<n := by linarith only [hn]
+  have hn1 : 1≤ n := by linarith only [hn]
+  have hpow : 4≤ n^(100*A) := hn.trans (by
+    simpa only [Real.rpow_one] using Real.rpow_le_rpow_of_exponent_le hn1 (show 1≤ 100*A by linarith only [hA]))
+  have hh := mul_le_mul_of_nonneg_right hpow (Real.rpow_pos_of_pos hn0 (-200*A)).le
+  rw [←Real.rpow_add hn0] at hh
+  have heq : 100*A+(-200*A)=-100*A := by ring
+  rw [heq] at hh
+  have hsmall : n^(-100*A) ≤ 1/2 := by
+    have hh' := Real.rpow_le_rpow_of_exponent_le hn1 (show -100*A≤ (-1:ℝ) by linarith only [hA])
+    rw [Real.rpow_neg_one] at hh'
+    exact hh'.trans (by apply (inv_le_comm₀ hn0 (by norm_num)).mpr; norm_num; linarith only [hn])
+  exact ⟨hh,hh.trans hsmall⟩
+
+end TraceReconstruction
+noncomputable section
+namespace TraceReconstruction
+open Filter
+open scoped Topology Classical
+
+def sourceTailDenominator (B D : ℕ) (q : ℝ) : ℝ :=
+  512+9600*sourceCapEnvelope B D q*(initialOverlapLower q)⁻¹^4
+
+def sourceTailExponent (B D : ℕ) : ℝ :=
+  300*(totalScoreExponent B D+B+1)+40*sourceWorkSize B D+5
+
+theorem sourceTailDenominator_pos (B D : ℕ) {q : ℝ} (hq : 0<q) :
+    0<sourceTailDenominator B D q := by
+  unfold sourceTailDenominator
+  positivity [sourceCapEnvelope_pos B D hq]
+
+theorem sourceTailDenominator_expUpper (B : ℕ) : SourceExpUpper (sourceTailDenominator B) := by
+  have H := ((SourceExpUpper.const 9600).mul (sourceCapEnvelope_expUpper B)
+    (fun _ _ _=>by norm_num) (fun D q hq=>(sourceCapEnvelope_pos B D hq.1).le)).mul
+      (initialOverlapLower_inverse_expUpper.pow (fun _ q _=>(inv_pos.mpr (initialOverlapLower_pos q)).le) 4)
+      (fun D q hq=>by positivity [sourceCapEnvelope_pos B D hq.1]) (fun _ _ _=>by positivity)
+  exact (SourceExpUpper.const 512).add H
+
+theorem sourceWorkSize_poly (B : ℕ) (hB : 2 ≤ B) : SourcePolyUpper (sourceWorkSize B) := by
+  exact (((totalScoreExponent_poly B hB).add (SourcePolyUpper.const (sourceDegreeCutoff B:ℝ))).add
+    (SourcePolyUpper.const (B:ℝ))).add (SourcePolyUpper.const 10)
+
+theorem sourceTailExponent_expUpper (B : ℕ) (hB : 2 ≤ B) :
+    SourceExpUpper (fun D _=>sourceTailExponent B D) := by
+  have ha := ((totalScoreExponent_poly B hB).add (SourcePolyUpper.const (B:ℝ))).add (SourcePolyUpper.const 1)
+  have hu := (SourcePolyUpper.const 300).mul ha (fun _=>by norm_num)
+    (fun D=>by linarith only [(totalScoreExponent_bounds B D hB).1,Nat.cast_nonneg B (α:=ℝ)])
+  have hw := (SourcePolyUpper.const 40).mul (sourceWorkSize_poly B hB) (fun _=>by norm_num)
+    (fun D=>(sourceWorkSize_pos B D hB).le)
+  exact ((hu.add hw).add (SourcePolyUpper.const 5)).expUpper
+
+theorem source_tails_eventually (B : ℕ) (ζ : ℝ) (hB : 2 ≤ B) (hζ : ζ∈Set.Ioo 0 1)
+    {n q : ℕ→ℝ} (hn : Tendsto n atTop atTop) (hq : ∀t,q t∈Set.Ioc 0 (1/2:ℝ))
+    (hH : Tendsto (fun t=>(q t)^3*Real.log (n t)) atTop atTop) :
+    ∀ᶠt in atTop,∀i,2*(B:ℝ)*Real.exp
+      (-(sourceTargetScale B (sourceDepth B ζ ((q t)^3*Real.log (n t))) (n t) i)^
+        (sourceEpsilon B (sourceDepth B ζ ((q t)^3*Real.log (n t))))/
+        sourceTailDenominator B (sourceDepth B ζ ((q t)^3*Real.log (n t))) (q t)) ≤
+      (n t)^(-sourceTailExponent B (sourceDepth B ζ ((q t)^3*Real.log (n t)))) := by
+  apply (sourceTailDenominator_expUpper B).uniform_tail (SourceExpUpper.const (2*(B:ℝ)))
+    (sourceTailExponent_expUpper B hB) (fun D q hq=>sourceTailDenominator_pos B D hq.1)
+    (fun _ _ _=>by positivity) (fun D _ _=>?_) B ζ hB hζ hn hq hH
+  unfold sourceTailExponent
+  have ha : 0 ≤ totalScoreExponent B D+B+1 := by
+    linarith only [(totalScoreExponent_bounds B D hB).1,Nat.cast_nonneg B (α:=ℝ)]
+  have hw := (sourceWorkSize_pos B D hB).le
+  linarith only [ha,hw]
+
+/-- Turning the common uniform exponential bound into the two exact
+ scheduled-error terms, including the division by the clipping threshold. -/
+theorem source_exponential_error_budget {B D b s ell k : ℕ} {C N F ω f n : ℝ}
+    (H : SourceArithmeticBounds C N F (sourceEpsilon B D) s b ell k)
+    (hB : 2 ≤ B) (hb : b ≤ B) (hC : 0<C) (hF : 0<F) (hω : 0<ω)
+    (hn : 1 ≤ n) (hN : 1 ≤ N) (hNn : N ≤ n)
+    (hf512 : 512 ≤ f) (hfmeta : 9600*(C^(2*b)*F)/ω^4 ≤ f)
+    (htail : 2*(B:ℝ)*Real.exp (-(N^(sourceEpsilon B D)/f)) ≤ n^(-sourceTailExponent B D)) :
+    sourceEffectiveHiddenTail (sourceBlockWindows N b (sourceEpsilon B D))
+      (sourceArithmeticTrials C N b (sourceEpsilon B D) ell k) b ω ≤ n^(-200*(totalScoreExponent B D+B+1)) ∧
+    sourceFitTail (sourceBlockWindows N b (sourceEpsilon B D))
+      (sourceArithmeticTrials C N b (sourceEpsilon B D) ell k) b /
+      N^(-20*sourceWorkSize B D) ≤ n^(-200*(totalScoreExponent B D+B+1)) ∧
+    sourceFitTail (sourceBlockWindows N b (sourceEpsilon B D))
+      (sourceArithmeticTrials C N b (sourceEpsilon B D) ell k) b ≤ n^(-200*(totalScoreExponent B D+B+1)) := by
+  let A := totalScoreExponent B D+B+1
+  let M := sourceWorkSize B D
+  let ε := sourceEpsilon B D
+  have hN0 : 0<N := by linarith only [hN]
+  have hn0 : 0<n := by linarith only [hn]
+  have hf0 : 0<f := by linarith only [hf512]
+  have hA : 0 ≤ A := by dsimp [A]; linarith only [(totalScoreExponent_bounds B D hB).1,Nat.cast_nonneg B (α:=ℝ)]
+  have hM : 0 ≤ M := (sourceWorkSize_pos B D hB).le
+  have hexpfit : Real.exp (-(N^ε/512)) ≤ Real.exp (-(N^ε/f)) := by
+    apply Real.exp_le_exp.mpr
+    exact neg_le_neg (div_le_div_of_nonneg_left (Real.rpow_pos_of_pos hN0 ε).le (by norm_num) hf512)
+  have hexpmeta : Real.exp (-(N^ε/(600*(C^(2*b)*F)))*ω^4/16) ≤ Real.exp (-(N^ε/f)) := by
+    have hden : 0<9600*(C^(2*b)*F)/ω^4 := by positivity
+    have hh := div_le_div_of_nonneg_left (Real.rpow_pos_of_pos hN0 ε).le hden hfmeta
+    have heq : N^ε/(9600*(C^(2*b)*F)/ω^4)=N^ε/(600*(C^(2*b)*F))*ω^4/16 := by field_simp; ring
+    rw [heq] at hh
+    apply Real.exp_le_exp.mpr
+    nlinarith only [hh]
+  have hcommon : 2*(b:ℝ)*N*Real.exp (-(N^ε/f)) ≤ n^(1-sourceTailExponent B D) := by
+    have hcast : (b:ℝ) ≤ B := by exact_mod_cast hb
+    have hh := mul_le_mul_of_nonneg_left htail hn0.le
+    have hcoeff : 2*(b:ℝ)*N ≤ n*(2*(B:ℝ)) := by
+      have hh' := mul_le_mul hcast hNn (by linarith only [hN]) (Nat.cast_nonneg B (α:=ℝ))
+      nlinarith only [hh']
+    have hp := mul_le_mul_of_nonneg_right hcoeff (Real.exp_pos (-(N^ε/f))).le
+    have heq : n*(n^(-sourceTailExponent B D))=n^(1-sourceTailExponent B D) := by
+      calc
+        _ = n^(1:ℝ)*n^(-sourceTailExponent B D) := by rw [Real.rpow_one]
+        _ = _ := by rw [←Real.rpow_add hn0]; rfl
+    nlinarith only [hh,hp,heq]
+  have hfit := H.fit_tail hC hF hN (sourceEpsilon_pos B D).le
+    (fun v hv=>sourceKappa_three_epsilon B D b (v+1) hb (by omega) (by omega))
+  have hm := H.metadata_tail (ω:=ω) hN0
+  have hf' := hfit.trans ((mul_le_mul_of_nonneg_left hexpfit (by positivity : 0 ≤ 2*(b:ℝ)*N)).trans hcommon)
+  have hm' := hm.trans ((mul_le_mul_of_nonneg_left hexpmeta (by positivity : 0 ≤ 2*(b:ℝ)*N)).trans hcommon)
+  have hpow : n^(1-sourceTailExponent B D) ≤ n^(-200*A) := by
+    apply Real.rpow_le_rpow_of_exponent_le hn
+    dsimp [sourceTailExponent]
+    dsimp [A,M] at *
+    linarith only [hA,hM]
+  refine ⟨hm'.trans hpow,?_,hf'.trans hpow⟩
+  rw [div_eq_mul_inv,←Real.rpow_neg hN0.le]
+  simp only [neg_mul,neg_neg]
+  have hr : N^(20*M) ≤ n^(20*M) := Real.rpow_le_rpow hN0.le hNn (by positivity)
+  apply (mul_le_mul hf' hr (Real.rpow_pos_of_pos hN0 _).le (Real.rpow_pos_of_pos hn0 _).le).trans
+  rw [←Real.rpow_add hn0]
+  apply Real.rpow_le_rpow_of_exponent_le hn
+  dsimp [sourceTailExponent]
+  dsimp [A,M] at *
+  linarith only [hA,hM]
+
+end TraceReconstruction
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem rpow_constant_absorb {N ε e c M : ℝ} (hN : 1 ≤ N) (hM : 0<M)
+    (hc : M*c ≤ N^ε) (he : ε+e ≤ 0) : c*N^e ≤ 1/M := by
+  have hN0 : 0<N := by linarith only [hN]
+  have hh := mul_le_mul_of_nonneg_right hc (Real.rpow_pos_of_pos hN0 e).le
+  rw [←Real.rpow_add hN0] at hh
+  have hr := Real.rpow_le_rpow_of_exponent_le hN he
+  rw [Real.rpow_zero] at hr
+  apply (le_div_iff₀ hM).mpr
+  nlinarith only [hh,hr]
+
+theorem overlap_sqrt_budget {v e : ℝ} (hv : v ≤ 1/4096) (he : e ≤ 1/8) :
+    Real.sqrt v+e ≤ 1/4 := by
+  have hh : Real.sqrt v ≤ 1/64 := (Real.sqrt_le_left (by norm_num)).mpr (by norm_num at hv ⊢; exact hv)
+  linarith only [hh,he]
+
+/-- Generic dense-template estimate, used for the first warmup step and
+ every step of a post-warmup block. -/
+theorem source_dense_variance_budget {N ε e a K L V h : ℝ} {j : ℕ}
+    (hN : 1 ≤ N) (_hK : 0 ≤ K) (hL : 0 ≤ L) (hh : 0 ≤ h)
+    (hcount : h ≤ (K+1)*N^a) (hV : V ≤ N^(-e))
+    (hc : 4096*(K+1)*L^j ≤ N^ε) (he : ε+a ≤ e) :
+    h*(L^j*V) ≤ 1/4096 := by
+  have hN0 : 0<N := by linarith only [hN]
+  have hv := mul_le_mul_of_nonneg_left hV (show 0 ≤ h*L^j by positivity)
+  have ht := mul_le_mul_of_nonneg_right hcount (show 0 ≤ L^j*N^(-e) by positivity)
+  have heq : ((K+1)*N^a)*(L^j*N^(-e))=((K+1)*L^j)*N^(a-e) := by
+    rw [sub_eq_add_neg,Real.rpow_add hN0]
+    ring
+  rw [heq] at ht
+  have hb := rpow_constant_absorb hN (M:=4096) (by norm_num)
+    (c:=(K+1)*L^j) (by nlinarith only [hc]) (e:=a-e) (by linarith only [he])
+  nlinarith only [hv,ht,hb]
+
+/-- Exact scalar one-window refinement. Its quadratic sparse term is what
+ makes the second warmup overlap possible at old score exponent 11/5. -/
+theorem source_warmup_variance_budget {N ε e M K W L w T h E : ℝ}
+    (hN : 1 ≤ N) (hK : 0 ≤ K) (hW : 0 ≤ W) (hL : 0 ≤ L)
+    (hT : 0 ≤ T) (hh : 0 ≤ h) (hcount : h ≤ (K+1)*N)
+    (hwindow : T ≤ N) (hw : w^2 ≤ W*N^(-e))
+    (henergy : E ≤ 64*(72*T*w^2*(N^(-20*M))^2+8*(w^2*L)^2))
+    (hbad : 8192*(4608*(K+1)*L*W) ≤ N^ε)
+    (hquad : 8192*(512*(K+1)*W^2*L^3) ≤ N^ε)
+    (hebad : ε+2-e-40*M ≤ 0) (hequad : ε+1-2*e ≤ 0) :
+    h*(L*E) ≤ 1/4096 := by
+  have hN0 : 0<N := by linarith only [hN]
+  have ha : 0 ≤ N^(-20*M) := (Real.rpow_pos_of_pos hN0 _).le
+  have hw0 : 0 ≤ W*N^(-e) := by positivity
+  have hb0 : 0 ≤ 64*(72*T*w^2*(N^(-20*M))^2+8*(w^2*L)^2) := by positivity
+  have H := (mul_le_mul_of_nonneg_left henergy (show 0 ≤ h*L by positivity)).trans
+    (mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hcount hL) hb0)
+  have ht := mul_le_mul hwindow hw (sq_nonneg _) hN0.le
+  have hs := pow_le_pow_left₀ (mul_nonneg (sq_nonneg w) hL) (mul_le_mul_of_nonneg_right hw hL) 2
+  have hbd := mul_le_mul_of_nonneg_left ht (sq_nonneg (N^(-20*M)))
+  have hm := mul_le_mul_of_nonneg_left (show 64*(72*T*w^2*(N^(-20*M))^2+8*(w^2*L)^2) ≤
+      64*(72*(N*(W*N^(-e)))*(N^(-20*M))^2+8*((W*N^(-e))*L)^2) by nlinarith only [hbd,hs])
+    (show 0 ≤ ((K+1)*N)*L by positivity)
+  have hp1 : N*N^(-e)*(N^(-20*M))^2=N^(1-e-40*M) := by
+    rw [←Real.rpow_natCast (N^(-20*M)) 2,←Real.rpow_mul hN0.le]
+    norm_num only [Nat.cast_ofNat]
+    nth_rw 1 [←Real.rpow_one N]
+    rw [←Real.rpow_add hN0,←Real.rpow_add hN0]
+    congr 1
+    ring
+  have hp2 : (N^(-e))^2=N^(-2*e) := by
+    rw [←Real.rpow_natCast,←Real.rpow_mul hN0.le]
+    norm_num only [Nat.cast_ofNat]
+    congr 1
+    ring
+  have hpow1 : N^((1:ℝ))*N^(1-e-40*M)=N^(2-e-40*M) := by rw [←Real.rpow_add hN0]; congr 1; ring
+  have hpow2 : N^((1:ℝ))*N^(-2*e)=N^(1-2*e) := by rw [←Real.rpow_add hN0]; congr 1; ring
+  rw [Real.rpow_one] at hpow1 hpow2
+  have heq : (((K+1)*N)*L)*(64*(72*(N*(W*N^(-e)))*(N^(-20*M))^2+8*((W*N^(-e))*L)^2))=
+      (4608*(K+1)*L*W)*N^(2-e-40*M)+(512*(K+1)*W^2*L^3)*N^(1-2*e) := by
+    calc
+      _ = (4608*(K+1)*L*W)*N*(N*N^(-e)*(N^(-20*M))^2)+
+        (512*(K+1)*W^2*L^3)*N*(N^(-e))^2 := by ring
+      _ = _ := by rw [hp1,hp2,mul_assoc _ N _,hpow1,mul_assoc _ N _,hpow2]
+  rw [heq] at hm
+  have hb := rpow_constant_absorb hN (M:=8192) (by norm_num) hbad (e:=2-e-40*M) (by linarith only [hebad])
+  have hq := rpow_constant_absorb hN (M:=8192) (by norm_num) hquad (e:=1-2*e) (by linarith only [hequad])
+  nlinarith only [H,hm,hb,hq]
+
+theorem sourceBlockLength_factor (P : ℕ→ℕ) (ell j : ℕ) :
+    sourceBlockLength P ell j=ell*sourceBlockLength P 1 j := by
+  induction j with
+  | zero => simp only [sourceBlockLength,mul_one]
+  | succ j ih => rw [sourceBlockLength,sourceBlockLength,ih]; ring
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem source_delta_power_budget {B D : ℕ} {n N ε : ℝ} (hB : 2 ≤ B)
+    (hn : 1 ≤ n) (hN : 1 ≤ N) (hNn : N ≤ n) (he : 0 ≤ ε) (he1 : ε ≤ 1) :
+    n^(-100*(totalScoreExponent B D+B+1))*N^(2+ε) ≤ 1 := by
+  have hn0 : 0<n := by linarith only [hn]
+  have hN0 : 0<N := by linarith only [hN]
+  have hA : 1 ≤ totalScoreExponent B D+B+1 := by
+    linarith only [(totalScoreExponent_bounds B D hB).1,Nat.cast_nonneg B (α:=ℝ)]
+  apply (mul_le_mul_of_nonneg_left (Real.rpow_le_rpow hN0.le hNn (by linarith only [he]))
+    (Real.rpow_pos_of_pos hn0 (-100*(totalScoreExponent B D+B+1))).le).trans
+  rw [←Real.rpow_add hn0]
+  simpa only [Real.rpow_zero] using Real.rpow_le_rpow_of_exponent_le hn
+    (show -100*(totalScoreExponent B D+B+1)+(2+ε) ≤ 0 by linarith only [hA,he1])
+
+/-- Uniform repeated old simulation error; this is the error term in every
+ dense nested-overlap estimate. -/
+theorem source_repeated_error_budget {n N ε K E t h : ℝ} {B D : ℕ}
+    (hB : 2 ≤ B) (hn : 1 ≤ n) (hN : 1 ≤ N) (hNn : N ≤ n)
+    (he : 0 ≤ ε) (he1 : ε ≤ 1) (hh : 0 ≤ h) (_hK : 0 ≤ K)
+    (hcount : h ≤ (K+1)*N)
+    (hcost : t*E ≤ ((D:ℝ)+1)*n^(-100*(totalScoreExponent B D+B+1))*N)
+    (hmargin : 64*(K+1)*((D:ℝ)+1) ≤ N^ε) : 2*h*(t*E) ≤ 1/32 := by
+  let Δ := n^(-100*(totalScoreExponent B D+B+1))
+  have hn0 : 0<n := by linarith only [hn]
+  have hΔ : 0 ≤ Δ := (Real.rpow_pos_of_pos hn0 _).le
+  have hc := mul_le_mul_of_nonneg_left hcost (show 0 ≤ 2*h by positivity)
+  have ht := mul_le_mul_of_nonneg_right hcount (show 0 ≤ 2*((D:ℝ)+1)*Δ*N by positivity)
+  have hm := mul_le_mul_of_nonneg_right hmargin (show 0 ≤ Δ*N^2 by positivity)
+  have hp := source_delta_power_budget (D:=D) hB hn hN hNn he he1
+  have hl : N^ε*N^2=N^(2+ε) := by
+    rw [←Real.rpow_natCast,←Real.rpow_add (by linarith only [hN] : 0<N)]
+    norm_num only [Nat.cast_ofNat]
+    congr 1
+    ring
+  have hm' : 64*(K+1)*((D:ℝ)+1)*Δ*N^2 ≤ Δ*N^(2+ε) := by
+    calc
+      _ = (64*(K+1)*((D:ℝ)+1))*(Δ*N^2) := by ring
+      _ ≤ N^ε*(Δ*N^2) := hm
+      _ = _ := by rw [show N^ε*(Δ*N^2)=Δ*(N^ε*N^2) by ring,hl]
+  change Δ*N^(2+ε) ≤ 1 at hp
+  nlinarith only [hc,ht,hm',hp]
+
+theorem source_old_error_cost {C N F ε E Δ : ℝ} {s b ell k j i D : ℕ}
+    (H : SourceArithmeticBounds C N F ε s b ell k) (hN : 1 ≤ N)
+    (hE : E ≤ ((i:ℝ)+1)*Δ*ell) (hΔ : 0 ≤ Δ) (hi : i ≤ D) (hj : j ≤ b)
+    (ha : scaleA b j ε ≤ 1) :
+    (sourceBlockLength (sourceArithmeticP C N b ε ell k) 1 j:ℝ)*E ≤
+      ((D:ℝ)+1)*Δ*N := by
+  let t := sourceBlockLength (sourceArithmeticP C N b ε ell k) 1 j
+  have hh := mul_le_mul_of_nonneg_left hE (Nat.cast_nonneg t (α:=ℝ))
+  have hl : (ell:ℝ)*t ≤ N := by
+    have hl' := (H.lengths j hj).2
+    rw [sourceArithmeticLength_eq_block,sourceBlockLength_factor] at hl'
+    push_cast at hl'
+    exact hl'.trans (by simpa only [Real.rpow_one] using Real.rpow_le_rpow_of_exponent_le hN ha)
+  have hi' : (i:ℝ) ≤ D := by exact_mod_cast hi
+  have hh' := mul_le_mul_of_nonneg_left hl (show 0 ≤ ((D:ℝ)+1)*Δ by positivity)
+  have hcoef := mul_le_mul_of_nonneg_right hi' (show 0 ≤ Δ*(ell:ℝ)*t by positivity)
+  dsimp [t] at *
+  nlinarith only [hh,hh',hcoef]
+
+/-- The normalized error invariant propagates through actual leaf products,
+ with one fresh delta paid by clipping and fitting. -/
+theorem normalized_block_error {E Δ clip fit : ℝ} {ell newell leaves i : ℕ}
+    (hell : 1 ≤ newell) (hE : E ≤ ((i:ℝ)+1)*Δ*ell) (hΔ : 0 ≤ Δ)
+    (hleaf : leaves*ell ≤ newell) (hnew : clip+fit ≤ Δ) :
+    (clip+(leaves:ℝ)*E+fit)/(newell:ℝ) ≤ ((i:ℝ)+2)*Δ := by
+  have hl0 : (0:ℝ)<newell := by exact_mod_cast (show 0<newell by omega)
+  apply (div_le_iff₀ hl0).mpr
+  have hleaf' : (leaves:ℝ)*ell ≤ newell := by exact_mod_cast hleaf
+  have h1 := mul_le_mul_of_nonneg_left hE (Nat.cast_nonneg leaves (α:=ℝ))
+  have h2 := mul_le_mul_of_nonneg_left hleaf' (show 0 ≤ ((i:ℝ)+1)*Δ by positivity)
+  have h3 : Δ ≤ Δ*newell := le_mul_of_one_le_right hΔ (by exact_mod_cast hell)
+  nlinarith only [h1,h2,h3,hnew]
+
+end TraceReconstruction
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- All three genuine clipping losses, and fitting, share the same global
+ error unit. The reference power is the actual half-moment at this scale. -/
+theorem source_level_clipping (B D j s b : ℕ) (T t : ℕ→ℕ) (n ω a gap L : ℝ)
+    (hB : 2 ≤ B) (hn : 4 ≤ n) (hg : sourceGapFloor B ≤ gap)
+    (hmeta : sourceEffectiveHiddenTail T t (b+1) ω ≤ n^(-200*(totalScoreExponent B D+B+1)))
+    (hfit : sourceFitTail T t (b+1)/a ≤ n^(-200*(totalScoreExponent B D+B+1)))
+    (hfit' : sourceFitTail T t (b+1) ≤ n^(-200*(totalScoreExponent B D+B+1)))
+    (hhalf : ((6*(2*sourceHalfMoment B D n j:ℕ):ℝ)*L)^2*sourceSparseCount T s b ≤ 1/2)
+    (hscale : ((6*(2*sourceHalfMoment B D n j:ℕ):ℝ)*L)^2*sourceSparseCount T s b ≤
+      (sourceTargetScale B D n j)^(-gap/2)/8) :
+    sourceClippingCost T t s b (sourceHalfMoment B D n j) ω a L ≤ 1/2 ∧
+    sourceClippingCost T t s b (sourceHalfMoment B D n j) ω a L+sourceFitTail T t (b+1) ≤
+      n^(-100*(totalScoreExponent B D+B+1)) := by
+  have hN : 0<sourceTargetScale B D n j := Real.rpow_pos_of_pos (by linarith only [hn]) _
+  have hc := (source_clipping_moment_power T s b (sourceHalfMoment B D n j)
+    (sourceTargetScale B D n j) gap L hN hhalf hscale).trans (source_clipping_power_delta B D j hB hn hg)
+  have hu := sourceErrorUnit_small (D:=D) hB hn
+  have hunit : 0 ≤ n^(-200*(totalScoreExponent B D+B+1)) := (Real.rpow_pos_of_pos (by linarith only [hn]) _).le
+  unfold sourceClippingCost
+  constructor <;> linarith only [hmeta,hfit,hfit',hc,hu.1,hu.2,hunit]
+
+/-- At one window the root metadata can still fail; only the fitting loss
+ vanishes. We charge the root metadata rather than discard it. -/
+theorem source_one_window_clipping (B D j s : ℕ) (T t : ℕ) (n ω a gap L : ℝ)
+    (hB : 2 ≤ B) (hn : 4 ≤ n) (hg : sourceGapFloor B ≤ gap)
+    (hmeta : sourceEffectiveHiddenTail (fun _=>T) (fun _=>t) 1 ω ≤ n^(-200*(totalScoreExponent B D+B+1)))
+    (hhalf : ((6*(2*sourceHalfMoment B D n j:ℕ):ℝ)*L)^2*(2:ℝ)^s ≤ 1/2)
+    (hscale : ((6*(2*sourceHalfMoment B D n j:ℕ):ℝ)*L)^2*(2:ℝ)^s ≤
+      (sourceTargetScale B D n j)^(-gap/2)/8) :
+    sourceClippingCost (fun _=>T) (fun _=>t) s 0 (sourceHalfMoment B D n j) ω a L ≤ 1/2 ∧
+    sourceClippingCost (fun _=>T) (fun _=>t) s 0 (sourceHalfMoment B D n j) ω a L ≤
+      n^(-100*(totalScoreExponent B D+B+1)) := by
+  have hu : 0 ≤ n^(-200*(totalScoreExponent B D+B+1)) :=
+    (Real.rpow_pos_of_pos (by linarith only [hn]) _).le
+  simpa only [sourceFitTail,add_zero] using source_level_clipping B D j s 0
+    (fun _=>T) (fun _=>t) n ω a gap L hB hn hg hmeta
+    (by simpa only [sourceFitTail,zero_div] using hu) (by simpa only [sourceFitTail] using hu)
+    (by simpa only [sourceSparseCount,Finset.range_zero,Finset.prod_empty,Nat.mul_one,Nat.cast_pow,Nat.cast_ofNat] using hhalf)
+    (by simpa only [sourceSparseCount,Finset.range_zero,Finset.prod_empty,Nat.mul_one,Nat.cast_pow,Nat.cast_ofNat] using hscale)
+
+end TraceReconstruction
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem source_overlap_prefactors {K W L X : ℝ} {B j : ℕ}
+    (hK : 0 ≤ K) (hW : 0 ≤ W) (hL : 1 ≤ L) (hB : 2 ≤ B) (hj : j ≤ B+2)
+    (hX : 1099511627776*(K+1)*(W+1)^2*L^(B+2) ≤ X) :
+    4096*(K+1)*L^j ≤ X ∧
+    8192*(4608*(K+1)*L*W) ≤ X ∧
+    8192*(512*(K+1)*W^2*L^3) ≤ X := by
+  have hK1 : 0 ≤ K+1 := by linarith only [hK]
+  have hw1 : 1 ≤ (W+1)^2 := by nlinarith only [hW,sq_nonneg W]
+  have hww : W ≤ (W+1)^2 := by nlinarith only [hW,sq_nonneg W]
+  have hw2 : W^2 ≤ (W+1)^2 := by nlinarith only [hW]
+  have hL0 : 0 ≤ L := by linarith only [hL]
+  have hjpow := pow_le_pow_right₀ hL hj
+  have hlpow : L ≤ L^(B+2) := by simpa only [pow_one] using pow_le_pow_right₀ hL (show 1 ≤ B+2 by omega)
+  have h3pow := pow_le_pow_right₀ hL (show 3 ≤ B+2 by omega)
+  have hb0 : 0 ≤ (K+1)*(W+1)^2*L^(B+2) := by positivity
+  have hfirst : (K+1)*L^j ≤ (K+1)*(W+1)^2*L^(B+2) := by
+    have hp := mul_le_mul_of_nonneg_left hjpow hK1
+    have hp' := mul_le_mul_of_nonneg_right hw1 (show 0 ≤ (K+1)*L^(B+2) by positivity)
+    nlinarith only [hp,hp']
+  have hsecond : (K+1)*L*W ≤ (K+1)*(W+1)^2*L^(B+2) := by
+    have hp := mul_le_mul_of_nonneg_left (mul_le_mul hlpow hww hW (by positivity)) hK1
+    nlinarith only [hp]
+  have hthird : (K+1)*W^2*L^3 ≤ (K+1)*(W+1)^2*L^(B+2) := by
+    have hp := mul_le_mul_of_nonneg_left (mul_le_mul hw2 h3pow (by positivity) (by positivity)) hK1
+    nlinarith only [hp]
+  constructor
+  · nlinarith only [hfirst,hX,hb0]
+  constructor <;> nlinarith only [hsecond,hthird,hX,hb0]
+
+/-- Complete dense-overlap numerical criterion for the literal schedules,
+ including the error carried by the old exact simulator. -/
+theorem source_literal_dense_overlap {C N F ε E Δ V K W L n e : ℝ}
+    {s b ell k j i D B : ℕ} (H : SourceArithmeticBounds C N F ε s b ell k)
+    (hB : 2 ≤ B) (hb : b ≤ B) (hC : 1 ≤ C) (hF : 0 ≤ F)
+    (hn : 1 ≤ n) (hN : 1 ≤ N) (hNn : N ≤ n) (he : 0 ≤ ε) (he1 : ε ≤ 1)
+    (hK : 0 ≤ K) (hW : 0 ≤ W) (hL : 1 ≤ L) (hLdef : L=(2:ℝ)^s)
+    (hG : C^(2*b)*F ≤ K) (hi : i ≤ D) (hj : j ≤ b)
+    (ha : 0 ≤ scaleA b j ε) (ha1 : scaleA b j ε ≤ 1)
+    (hE : E ≤ ((i:ℝ)+1)*Δ*ell) (hΔ : Δ=n^(-100*(totalScoreExponent B D+B+1)))
+    (hV : V ≤ N^(-e)) (hgap : ε+scaleA b j ε ≤ e)
+    (hmargin : 64*(K+1)*((D:ℝ)+1) ≤ N^ε)
+    (hoverlap : 1099511627776*(K+1)*(W+1)^2*L^(B+2) ≤ N^ε) :
+    Real.sqrt (((sourceArithmeticCount C N b ε ell k j+1:ℕ):ℝ)*
+      (((2^s:ℕ):ℝ)^j*V))+
+      2*((sourceArithmeticCount C N b ε ell k j+1:ℕ):ℝ)*
+      ((sourceBlockLength (sourceArithmeticP C N b ε ell k) 1 j:ℝ)*E) ≤ 1/4 := by
+  let h := ((sourceArithmeticCount C N b ε ell k j+1:ℕ):ℝ)
+  have hn0 : 0<n := by linarith only [hn]
+  have hN0 : 0<N := by linarith only [hN]
+  have hh0 : 0 ≤ h := Nat.cast_nonneg _
+  have hc : h ≤ (K+1)*N^(scaleA b j ε) := by
+    exact (H.count_bound hC hF hN hj ha).trans
+      (mul_le_mul_of_nonneg_right (add_le_add hG le_rfl) (Real.rpow_pos_of_pos hN0 _).le)
+  have hc' : h ≤ (K+1)*N := hc.trans (mul_le_mul_of_nonneg_left
+    (by simpa only [Real.rpow_one] using Real.rpow_le_rpow_of_exponent_le hN ha1)
+    (by linarith only [hK]))
+  have hpref := (source_overlap_prefactors hK hW hL hB (show j ≤ B+2 by omega) hoverlap).1
+  have hv := source_dense_variance_budget hN hK (show 0 ≤ L by linarith only [hL]) hh0 hc hV hpref hgap
+  have hcost := source_old_error_cost H hN hE (by rw [hΔ]; exact (Real.rpow_pos_of_pos hn0 _).le) hi hj ha1
+  rw [hΔ] at hcost
+  have herr := source_repeated_error_budget hB hn hN hNn he he1 hh0 hK hc' hcost hmargin
+  have hres := overlap_sqrt_budget hv (show 2*h*((sourceBlockLength (sourceArithmeticP C N b ε ell k) 1 j:ℝ)*E) ≤ 1/8 by linarith only [herr])
+  simpa only [hLdef,Nat.cast_pow,Nat.cast_ofNat] using hres
+
+end TraceReconstruction
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem SourceArithmeticBounds.length_positive {C N F ε : ℝ} {s b ell k j : ℕ}
+    (H : SourceArithmeticBounds C N F ε s b ell k) (hN : 0<N) (hj : j ≤ b) :
+    1 ≤ sourceArithmeticLength C N b ε ell k j := by
+  have hp : (0:ℝ)<sourceArithmeticLength C N b ε ell k j :=
+    (show 0<N^(scaleA b j ε)/2 by positivity).trans_le (H.lengths j hj).1
+  have hh : 0<sourceArithmeticLength C N b ε ell k j := by exact_mod_cast hp
+  omega
+
+theorem SourceArithmeticBounds.copies_le_target {C N F ε : ℝ} {s b ell k j : ℕ}
+    (H : SourceArithmeticBounds C N F ε s b ell k) (hN : 1 ≤ N) (hj : j<b)
+    (ha : scaleA b (j+1) ε ≤ 1) : (sourceArithmeticP C N b ε ell k j:ℝ) ≤ N := by
+  have hL := H.length_positive (by linarith only [hN]) (j:=j) (by omega)
+  have hp : sourceArithmeticP C N b ε ell k j ≤ sourceArithmeticLength C N b ε ell k (j+1) := by
+    rw [sourceArithmeticLength_step]
+    simpa only [Nat.mul_comm] using Nat.le_mul_of_pos_right (sourceArithmeticP C N b ε ell k j) hL
+  exact (show (sourceArithmeticP C N b ε ell k j:ℝ) ≤ sourceArithmeticLength C N b ε ell k (j+1) by exact_mod_cast hp).trans
+    ((H.lengths (j+1) (by omega)).2.trans (by
+      simpa only [Real.rpow_one] using Real.rpow_le_rpow_of_exponent_le hN ha))
+
+/-- Both contributions to the error in the second warmup overlap are paid
+ by the common delta margin. The old term uses T₀ ≤ P₀, not an independent
+ full-block overlap hypothesis. -/
+theorem source_warmup_error_budget {C N F ε E Δ clip K n : ℝ} {s ell k i D B : ℕ}
+    (H : SourceArithmeticBounds C N F ε s 2 ell k)
+    (hB : 2 ≤ B) (hC : 1 ≤ C) (hF : 0 ≤ F) (hn : 1 ≤ n) (hN : 1 ≤ N) (hNn : N ≤ n)
+    (he : 0 ≤ ε) (he1 : ε ≤ 1) (hK : 0 ≤ K) (hG : C^4*F ≤ K) (hi : i ≤ D)
+    (hE : E ≤ ((i:ℝ)+1)*Δ*ell) (hE0 : 0 ≤ E) (hΔ : Δ=n^(-100*(totalScoreExponent B D+B+1)))
+    (hclip : clip ≤ Δ) (hmargin : 64*(K+1)*((D:ℝ)+1) ≤ N^ε) :
+    2*((sourceArithmeticCount C N 2 ε ell k 2+1:ℕ):ℝ)*
+      ((sourceArithmeticP C N 2 ε ell k 1:ℝ)*
+        (clip+(sourceBlockWindows N 2 ε 0:ℝ)*E)) ≤ 1/8 := by
+  let P := sourceArithmeticP C N 2 ε ell k
+  let T := sourceBlockWindows N 2 ε
+  let h := ((sourceArithmeticCount C N 2 ε ell k 2+1:ℕ):ℝ)
+  have hn0 : 0<n := by linarith only [hn]
+  have hΔ0 : 0 ≤ Δ := by rw [hΔ]; exact (Real.rpow_pos_of_pos hn0 _).le
+  have hh0 : 0 ≤ h := Nat.cast_nonneg _
+  have hc : h ≤ (K+1)*N := by
+    have hh := H.count_bound hC hF hN (j:=2) le_rfl (by rw [scaleA_top]; norm_num)
+    simpa only [scaleA_top,Real.rpow_one] using hh.trans
+      (mul_le_mul_of_nonneg_right (by simpa only [show (2:ℕ)*2=4 by rfl] using add_le_add hG (show (1:ℝ) ≤ 1 from le_rfl)) (Real.rpow_pos_of_pos (by linarith only [hN]) _).le)
+  have hcost := source_old_error_cost H hN hE hΔ0 hi (j:=2) le_rfl (by rw [scaleA_top])
+  rw [hΔ] at hcost
+  have herr := source_repeated_error_budget hB hn hN hNn he he1 hh0 hK hc hcost hmargin
+  have hp1 : (P 1:ℝ) ≤ N := H.copies_le_target hN (by omega) (by rw [show (1:ℕ)+1=2 by rfl,scaleA_top])
+  have hclipcost : (P 1:ℝ)*clip ≤ ((D:ℝ)+1)*Δ*N := by
+    have h1 := mul_le_mul_of_nonneg_left hclip (Nat.cast_nonneg (P 1) (α:=ℝ))
+    have h2 := mul_le_mul_of_nonneg_right hp1 hΔ0
+    have h3 := le_mul_of_one_le_left (show 0 ≤ Δ*N by positivity) (show (1:ℝ) ≤ D+1 by linarith only [Nat.cast_nonneg D (α:=ℝ)])
+    nlinarith only [h1,h2,h3]
+  rw [hΔ] at hclipcost
+  have hcliperr := source_repeated_error_budget hB hn hN hNn he he1 hh0 hK hc hclipcost hmargin
+  have hT : (T 0:ℝ) ≤ P 0 := by
+    exact_mod_cast (show T 0 ≤ P 0 from (by have h := (H.windows 0 (by omega)).2.2.1; change 2*T 0 ≤ P 0 at h; omega))
+  have hold := mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_right hT hE0)
+    (show 0 ≤ 2*h*(P 1:ℝ) by positivity)
+  have hfactor : (sourceBlockLength P 1 2:ℝ)=(P 1:ℝ)*(P 0:ℝ) := by
+    simp only [sourceBlockLength,Nat.cast_mul,mul_one]
+  change 2*h*((sourceBlockLength P 1 2:ℝ)*E) ≤ 1/32 at herr
+  rw [hfactor] at herr
+  nlinarith only [herr,hcliperr,hold]
+
+end TraceReconstruction
+
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/TraceReconstruction/Ranks.lean b/lean/OAI/Probability/TraceReconstruction/Ranks.lean
index 1e20fbd4e932950c92a992b6b495fe6e6d8f39b9..1a6a98c3d27679a17682ff83967dbc147d8a548a 100644
--- a/lean/OAI/Probability/TraceReconstruction/Ranks.lean
+++ b/lean/OAI/Probability/TraceReconstruction/Ranks.lean
@@ -1,1831 +1,1831 @@
-import OAI.Probability.TraceReconstruction.FiniteLaw
-import OAI.Probability.TraceReconstruction.Kernels
-import OAI.Probability.TraceReconstruction.Scales
-
-namespace OAI
-
-/-!
-# Ranks
-
-Retained subsets, conditional rank experiments, tagged gaps, period geometry, Gaussian Schur estimates, and uniform square sums.
--/
-
-open scoped ENNReal
-noncomputable section
-
-open Filter
-open scoped Topology
-
-
-namespace TraceReconstruction
-
-variable {α : Type*} [Fintype α]
-
-def subsetWeight (p : ℝ) (S : Finset α) : ℝ :=
-  p^S.card * (1-p)^(Fintype.card α-S.card)
-
-theorem sum_subsetWeight (p : ℝ) : (∑ S : Finset α, subsetWeight p S)=1 := by
-  classical
-  have h := Finset.sum_pow_mul_eq_add_pow p (1-p) (Finset.univ : Finset α)
-  simpa [subsetWeight] using h
-
-theorem sum_fixedCard_subsetWeight (p : ℝ) (l : ℕ) :
-    (∑ S : Finset α, if S.card=l then subsetWeight p S else 0) =
-      binomialMass (Fintype.card α) p l := by
-  classical
-  rw [← Finset.sum_filter]
-  have hset : (Finset.univ.filter (fun S : Finset α => S.card=l)) =
-      (Finset.univ : Finset α).powersetCard l := by ext; simp
-  rw [hset]
-  have he : ∀ S ∈ (Finset.univ : Finset α).powersetCard l,
-      subsetWeight p S=p^l*(1-p)^(Fintype.card α-l) := by
-    classical
-    intro S hS
-    simp only [subsetWeight, (Finset.mem_powersetCard.mp hS).2]
-  rw [Finset.sum_congr rfl he]
-  simp only [Finset.sum_const, Finset.card_powersetCard, Finset.card_univ, nsmul_eq_mul,
-    binomialMass]
-  ring
-
-def subsetLaw (p : ℝ) (hp : p ∈ Set.Icc 0 1) : FiniteLaw (Finset α) where
-  mass := subsetWeight p
-  nonneg S := by unfold subsetWeight; positivity [hp.1, sub_nonneg.mpr hp.2]
-  total := sum_subsetWeight p
-
-abbrev FixedSubset (α : Type*) [Fintype α] (l : ℕ) := {S : Finset α // S.card=l}
-
-theorem card_fixedSubset (l : ℕ) :
-    Fintype.card (FixedSubset α l)=(Fintype.card α).choose l := by
-  classical
-  rw [Fintype.card_subtype]
-  have hset : (Finset.univ.filter (fun S : Finset α => S.card=l)) =
-      (Finset.univ : Finset α).powersetCard l := by ext; simp
-  rw [hset, Finset.card_powersetCard, Finset.card_univ]
-
-def uniformSubsetLaw (l : ℕ) (hl : l ≤ Fintype.card α) : FiniteLaw (FixedSubset α l) where
-  mass _ := 1 / ((Fintype.card α).choose l : ℝ)
-  nonneg _ := by positivity
-  total := by
-    classical
-    simp only [Finset.sum_const, Finset.card_univ, card_fixedSubset, nsmul_eq_mul]
-    exact mul_one_div_cancel (by exact_mod_cast (Nat.choose_pos hl).ne')
-
-/-- Conditional Bernoulli sampling, at every interior parameter, is exactly uniform. -/
-theorem conditional_subset_uniform {p : ℝ} (hp : p ∈ Set.Ioo 0 1)
-    {l : ℕ} (hl : l ≤ Fintype.card α) (S : FixedSubset α l) :
-    subsetWeight p S.val / binomialMass (Fintype.card α) p l =
-      (uniformSubsetLaw l hl).mass S := by
-  classical
-  have hp0 : p≠0 := hp.1.ne'
-  have hq0 : 1-p≠0 := by linarith [hp.2]
-  have hc : (((Fintype.card α).choose l):ℝ)≠0 := by exact_mod_cast (Nat.choose_pos hl).ne'
-  simp only [subsetWeight, S.property, binomialMass, uniformSubsetLaw]
-  field_simp
-
-/-- Slot at an observed return rank, with the original total order. -/
-def slotAtRank {n l : ℕ} (S : FixedSubset (Fin n) l) (v : Fin l) : Fin n :=
-  (S.val.orderIsoOfFin S.property v).val
-
-theorem slotAtRank_mem {n l : ℕ} (S : FixedSubset (Fin n) l) (v : Fin l) :
-    slotAtRank S v ∈ S.val := (S.val.orderIsoOfFin S.property v).property
-
-theorem slotAtRank_strictMono {n l : ℕ} (S : FixedSubset (Fin n) l) :
-    StrictMono (slotAtRank S) := by
-  classical
-  intro a b hab
-  exact (S.val.orderIsoOfFin S.property).strictMono hab
-
-def defectAtRank {n l : ℕ} (D : Finset (Fin n)) (S : FixedSubset (Fin n) l)
-    (v : Fin l) : Bool := decide (slotAtRank S v ∈ D)
-
-def taggedRankEvent {n l r : ℕ} (p : Fin r → Fin n) (v : Fin r → Fin l)
-    (S : FixedSubset (Fin n) l) : Prop := ∀ i, slotAtRank S (v i)=p i
-
-/-- Unhypothesized rank kernel: defined from the actual uniformly retained subset. -/
-def subsetRankKernel {n l r : ℕ} (hl : l≤n) (p : Fin r → Fin n) (v : Fin r → Fin l) : ℝ := by
-  classical
-  exact ∑ S : FixedSubset (Fin n) l, if taggedRankEvent p v S then
-    (uniformSubsetLaw l (by simpa using hl)).mass S else 0
-
-theorem subsetRankKernel_nonneg {n l r : ℕ} (hl : l≤n)
-    (p : Fin r → Fin n) (v : Fin r → Fin l) : 0 ≤ subsetRankKernel hl p v := by
-  classical
-  unfold subsetRankKernel
-  apply Finset.sum_nonneg
-  intro S _
-  split_ifs
-  · exact (uniformSubsetLaw l (by simpa using hl)).nonneg S
-  · rfl
-
-theorem subsetRankKernel_le_one {n l r : ℕ} (hl : l≤n)
-    (p : Fin r → Fin n) (v : Fin r → Fin l) : subsetRankKernel hl p v ≤ 1 := by
-  classical
-  unfold subsetRankKernel
-  rw [← (uniformSubsetLaw (α := Fin n) l (by simpa using hl)).total]
-  apply Finset.sum_le_sum
-  intro S _
-  split_ifs
-  · rfl
-  · exact (uniformSubsetLaw l (by simpa using hl)).nonneg S
-
-
-/-- Canonical slot universe split at the tagged positions. Actual order is only
-needed to identify each component with its before/between/after interval. -/
-abbrev GapUniverse (r d : ℕ) (g : Fin d → ℕ) := Fin r ⊕ ((i : Fin d) × Fin (g i))
-
-def gapDecode {r d : ℕ} {g : Fin d → ℕ} (S : Finset (GapUniverse r d g))
-    (i : Fin d) : Finset (Fin (g i)) :=
-  Finset.univ.filter fun a => Sum.inr ⟨i,a⟩ ∈ S
-
-def gapEncode {r d : ℕ} {g : Fin d → ℕ} (A : (i : Fin d) → Finset (Fin (g i))) :
-    Finset (GapUniverse r d g) := by
-  classical
-  exact Finset.univ.filter fun x => match x with
-    | Sum.inl _ => True
-    | Sum.inr a => a.2 ∈ A a.1
-
-@[simp] theorem mem_gapEncode_left {r d : ℕ} {g : Fin d → ℕ}
-    (A : (i : Fin d) → Finset (Fin (g i))) (a : Fin r) :
-    Sum.inl a ∈ gapEncode A := by simp [gapEncode]
-
-@[simp] theorem mem_gapEncode_right {r d : ℕ} {g : Fin d → ℕ}
-    (A : (i : Fin d) → Finset (Fin (g i))) (i : Fin d) (a : Fin (g i)) :
-    Sum.inr ⟨i,a⟩ ∈ gapEncode (r := r) A ↔ a ∈ A i := by simp [gapEncode]
-
-@[simp] theorem gapDecode_encode {r d : ℕ} {g : Fin d → ℕ}
-    (A : (i : Fin d) → Finset (Fin (g i))) (i : Fin d) :
-    gapDecode (gapEncode (r := r) A) i=A i := by ext; simp [gapDecode]
-
-theorem gapEncode_decode {r d : ℕ} {g : Fin d → ℕ}
-    (S : Finset (GapUniverse r d g)) (hS : ∀ a, Sum.inl a∈S) :
-    gapEncode (gapDecode S)=S := by
-  ext x
-  cases x with
-  | inl a => simp [hS]
-  | inr a => rcases a with ⟨i,a⟩; simp [gapDecode]
-
-def gapEvent {r d : ℕ} {g : Fin d → ℕ} (z : Fin d → ℕ)
-    (S : Finset (GapUniverse r d g)) : Prop :=
-  (∀ a, Sum.inl a∈S) ∧ ∀ i, (gapDecode S i).card=z i
-
-instance gapEvent_decidable {r d : ℕ} {g : Fin d → ℕ} (z : Fin d → ℕ)
-    (S : Finset (GapUniverse r d g)) : Decidable (gapEvent z S) := by
-  unfold gapEvent
-  infer_instance
-
-def gapEventEquiv {r d : ℕ} {g z : Fin d → ℕ} :
-    {S : Finset (GapUniverse r d g) // gapEvent z S} ≃
-      ((i : Fin d) → FixedSubset (Fin (g i)) (z i)) where
-  toFun S i := ⟨gapDecode S.val i, S.property.2 i⟩
-  invFun A := ⟨gapEncode (fun i => (A i).val), by
-    constructor
-    · intro a; simp
-    · intro i; simp only [gapDecode_encode, (A i).property]⟩
-  left_inv S := by
-    apply Subtype.ext
-    exact gapEncode_decode S.val S.property.1
-  right_inv A := by
-    funext i
-    apply Subtype.ext
-    change gapDecode (gapEncode (fun i => (A i).val)) i = (A i).val
-    exact gapDecode_encode _ _
-
-theorem card_gapEvent {r d : ℕ} (g z : Fin d → ℕ) :
-    Fintype.card {S : Finset (GapUniverse r d g) // gapEvent z S} =
-      ∏ i, (g i).choose (z i) := by
-  classical
-  rw [Fintype.card_congr gapEventEquiv, Fintype.card_pi]
-  simp only [card_fixedSubset, Fintype.card_fin]
-
-theorem card_gapEncode {r d : ℕ} {g : Fin d → ℕ}
-    (A : (i : Fin d) → Finset (Fin (g i))) :
-    (gapEncode (r := r) A).card = r + ∑ i, (A i).card := by
-  classical
-  rw [Finset.card_eq_sum_ones]
-  simp only [gapEncode, Finset.sum_filter]
-  rw [Fintype.sum_sum_type, Fintype.sum_sigma]
-  simp
-  apply Finset.sum_congr rfl
-  intro i _
-  calc
-    _ = ∑ x : Fin (g i), if x ∈ A i then (1:ℕ) else 0 := by
-      apply Finset.sum_congr rfl
-      intro x _
-      split_ifs <;> rfl
-    _ = _ := (Finset.card_eq_sum_ite (Finset.subset_univ (A i))).symm
-
-theorem card_of_gapEvent {r d : ℕ} {g z : Fin d → ℕ}
-    (S : Finset (GapUniverse r d g)) (hS : gapEvent z S) :
-    S.card = r+∑ i, z i := by
-  rw [← gapEncode_decode S hS.1, card_gapEncode]
-  simp only [hS.2]
-
-
-/-- The exact unconditioned tagged-gap probability; no asymptotic estimate is used. -/
-theorem gapEvent_bernoulli_probability {r d : ℕ} (g z : Fin d → ℕ)
-    (hz : ∀ i, z i≤g i) (p : ℝ) :
-    (∑ S : Finset (GapUniverse r d g), if gapEvent z S then subsetWeight p S else 0) =
-      p^r * ∏ i, binomialMass (g i) p (z i) := by
-  classical
-  have hcount : (∑ S : Finset (GapUniverse r d g),
-      if gapEvent z S then (1:ℝ) else 0) = ∏ i, ((g i).choose (z i):ℝ) := by
-    rw [Finset.sum_boole, ← Fintype.card_subtype, card_gapEvent, Nat.cast_prod]
-  have hnum : (∑ S : Finset (GapUniverse r d g), if gapEvent z S then subsetWeight p S else 0) =
-      (∏ i, ((g i).choose (z i):ℝ)) *
-        (p^(r+∑ i, z i)*(1-p)^(r+∑ i, g i-(r+∑ i, z i))) := by
-    rw [← hcount, Finset.sum_mul]
-    apply Finset.sum_congr rfl
-    intro S _
-    split_ifs with hS
-    · simp only [one_mul, subsetWeight, card_of_gapEvent S hS,
-        Fintype.card_sum, Fintype.card_fin, Fintype.card_sigma]
-    · exact (zero_mul _).symm
-  have hsub : (∑ i, g i)-(∑ i, z i)=∑ i, (g i-z i) := by
-    exact (Finset.sum_tsub_distrib Finset.univ (fun i _ => hz i)).symm
-  rw [hnum, Nat.add_sub_add_left, hsub, pow_add]
-  simp only [binomialMass, Finset.prod_mul_distrib, Finset.prod_pow_eq_pow_sum]
-  ring
-
-/-- Conditional Bernoulli representation of the source rank kernel in gap coordinates. -/
-theorem gapEvent_conditional_formula {r d : ℕ} (g z : Fin d → ℕ)
-    (hz : ∀ i, z i≤g i) (p : ℝ) :
-    (∑ S : Finset (GapUniverse r d g), if gapEvent z S then subsetWeight p S else 0) /
-      binomialMass (r+∑ i, g i) p (r+∑ i, z i) =
-    p^r / binomialMass (r+∑ i, g i) p (r+∑ i, z i) *
-      ∏ i, binomialMass (g i) p (z i) := by
-  rw [gapEvent_bernoulli_probability g z hz]
-  ring
-
-end TraceReconstruction
-
-
-namespace TraceReconstruction
-
-theorem slotAtRank_lower_card {n l : ℕ} (S : FixedSubset (Fin n) l) (v : Fin l) :
-    (S.val.filter fun j => j < slotAtRank S v).card = v.val := by
-  classical
-  let e := S.val.orderEmbOfFin S.property
-  have hslot (i:Fin l) : e i = slotAtRank S i := rfl
-  have hmap : (Finset.Iio v).map e.toEmbedding = S.val.filter (fun j => j<e v) := by
-    ext j
-    simp only [Finset.mem_map, Finset.mem_Iio, Finset.mem_filter, RelEmbedding.coe_toEmbedding]
-    constructor
-    · rintro ⟨i, hi, rfl⟩
-      exact ⟨S.val.orderEmbOfFin_mem S.property i, e.strictMono hi⟩
-    · rintro ⟨hj,hv⟩
-      obtain ⟨i,hi⟩ := (S.val.orderIsoOfFin S.property).surjective ⟨j,hj⟩
-      have hei : e i=j := congrArg Subtype.val hi
-      refine ⟨i, ?_, hei⟩
-      exact e.lt_iff_lt.mp (hei.symm ▸ hv)
-  rw [← hslot, ← hmap, Finset.card_map, Fin.card_Iio]
-
-theorem slotAtRank_eq_iff {n l : ℕ} (S : FixedSubset (Fin n) l) (v : Fin l) (p : Fin n) :
-    slotAtRank S v=p ↔ p∈S.val ∧ (S.val.filter fun j => j<p).card=v.val := by
-  classical
-  constructor
-  · intro h
-    rw [← h]
-    exact ⟨slotAtRank_mem S v, slotAtRank_lower_card S v⟩
-  · rintro ⟨hp,hcard⟩
-    obtain ⟨i,hi⟩ := (S.val.orderIsoOfFin S.property).surjective ⟨p,hp⟩
-    have he : slotAtRank S i=p := congrArg Subtype.val hi
-    have hcv := slotAtRank_lower_card S i
-    rw [he,hcard] at hcv
-    have hiv : i=v := Fin.ext hcv.symm
-    simpa [hiv] using he
-
-/-- Bounds for the open gaps of an increasing tuple; slots and ranks are zero-based. -/
-def tupleLower {n r : ℕ} (p : Fin r → Fin n) : Fin (r+1) → ℕ :=
-  Fin.cons 0 (fun i => (p i).val + 1)
-
-def tupleUpper {n r : ℕ} (p : Fin r → Fin n) : Fin (r+1) → ℕ :=
-  Fin.snoc (fun i => (p i).val) n
-
-def tupleGap {n r : ℕ} (p : Fin r → Fin n) (j : Fin (r+1)) : ℕ :=
-  tupleUpper p j - tupleLower p j
-
-@[simp] theorem tupleLower_zero {n r : ℕ} (p : Fin r → Fin n) : tupleLower p 0 = 0 :=
-  Fin.cons_zero _ _
-@[simp] theorem tupleLower_succ {n r : ℕ} (p : Fin r → Fin n) (i : Fin r) :
-    tupleLower p i.succ = (p i).val + 1 := Fin.cons_succ _ _ _
-@[simp] theorem tupleUpper_castSucc {n r : ℕ} (p : Fin r → Fin n) (i : Fin r) :
-    tupleUpper p i.castSucc = (p i).val := Fin.snoc_castSucc _ _ _
-@[simp] theorem tupleUpper_last {n r : ℕ} (p : Fin r → Fin n) :
-    tupleUpper p (Fin.last r) = n := Fin.snoc_last _ _
-
-theorem tupleLower_le_tag {n r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p)
-    (i : Fin r) (j : Fin (r+1)) (hji : j ≤ i.castSucc) :
-    tupleLower p j ≤ (p i).val := by
-  induction j using Fin.cases with
-  | zero => simp
-  | succ a =>
-    simp only [tupleLower_succ]
-    have hai : a < i := by
-      change a.val < i.val
-      have hh : a.val + 1 ≤ i.val := hji
-      omega
-    exact hp hai
-
-theorem tag_lt_tupleLower {n r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p)
-    (i : Fin r) (j : Fin (r+1)) (hij : i.castSucc < j) :
-    (p i).val < tupleLower p j := by
-  induction j using Fin.cases with
-  | zero => exact False.elim (by simp at hij)
-  | succ a =>
-    simp only [tupleLower_succ]
-    have hia : i ≤ a := by
-      change i.val ≤ a.val
-      have hh : i.val < a.val + 1 := hij
-      omega
-    exact Nat.lt_succ_of_le (hp.monotone hia)
-
-theorem tupleUpper_le_total {n r : ℕ} (p : Fin r → Fin n) (j : Fin (r+1)) :
-    tupleUpper p j ≤ n := by
-  induction j using Fin.lastCases with
-  | last => simp
-  | cast i => simpa only [tupleUpper_castSucc] using (p i).isLt.le
-
-theorem tupleLower_le_total {n r : ℕ} (p : Fin r → Fin n) (j : Fin (r+1)) :
-    tupleLower p j ≤ n := by
-  induction j using Fin.cases with
-  | zero => simp
-  | succ i => simpa only [tupleLower_succ] using Nat.succ_le_of_lt (p i).isLt
-
-theorem tupleLower_le_upper {n r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p)
-    (j : Fin (r+1)) : tupleLower p j ≤ tupleUpper p j := by
-  induction j using Fin.lastCases with
-  | last => simpa using tupleLower_le_total p (Fin.last r)
-  | cast i => simpa using tupleLower_le_tag p hp i i.castSucc le_rfl
-
-theorem tupleGap_total {n r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p) :
-    r + ∑ j, tupleGap p j = n := by
-  have hrs : r ≤ n := by simpa using Fintype.card_le_of_injective p hp.injective
-  have ha : (∑ j, tupleLower p j) = (∑ i, (p i).val) + r := by
-    rw [Fin.sum_univ_succ]
-    simp [Finset.sum_add_distrib]
-  have hb : (∑ j, tupleUpper p j) = (∑ i, (p i).val) + n := by
-    rw [Fin.sum_univ_castSucc]
-    simp
-  simp only [tupleGap]
-  rw [Finset.sum_tsub_distrib _ (fun j _ => tupleLower_le_upper p hp j), ha, hb]
-  omega
-
-/-- Maps abstract gap coordinates to their actual source slots. -/
-def tupleGapMap {n r : ℕ} (p : Fin r → Fin n) :
-    GapUniverse r (r+1) (tupleGap p) → Fin n
-  | .inl i => p i
-  | .inr ⟨j,a⟩ => ⟨tupleLower p j + a.val, by
-      have hx := a.isLt
-      have hu := tupleUpper_le_total p j
-      simp only [tupleGap] at hx
-      omega⟩
-
-theorem tupleGapMap_gap_lt_tag {n r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p)
-    (i : Fin r) (j : Fin (r+1)) (a : Fin (tupleGap p j)) :
-    tupleGapMap p (.inr ⟨j,a⟩) < p i ↔ j ≤ i.castSucc := by
-  constructor
-  · intro h
-    by_contra hn
-    have hx := tag_lt_tupleLower p hp i j (lt_of_not_ge hn)
-    change tupleLower p j + a.val < (p i).val at h
-    omega
-  · intro h
-    have hji : j.val < r := lt_of_le_of_lt (show j.val ≤ i.val from h) i.isLt
-    let k : Fin r := ⟨j.val, hji⟩
-    have hjk : j = k.castSucc := Fin.ext rfl
-    have ht : tupleUpper p j ≤ (p i).val := by
-      rw [hjk, tupleUpper_castSucc]
-      exact hp.monotone (show k ≤ i from h)
-    have hx := a.isLt
-    change tupleLower p j + a.val < (p i).val
-    unfold tupleGap at hx
-    omega
-
-theorem tupleGapMap_tag_lt_gap {n r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p)
-    (i : Fin r) (j : Fin (r+1)) (a : Fin (tupleGap p j)) :
-    p i < tupleGapMap p (.inr ⟨j,a⟩) ↔ i.castSucc < j := by
-  constructor
-  · intro h
-    by_contra hn
-    have hx := (tupleGapMap_gap_lt_tag p hp i j a).2 (le_of_not_gt hn)
-    exact (lt_asymm h hx)
-  · intro h
-    have hx := tag_lt_tupleLower p hp i j h
-    change (p i).val < tupleLower p j + a.val
-    omega
-
-theorem tupleGapMap_injective {n r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p) :
-    Function.Injective (tupleGapMap p) := by
-  rintro (i | ⟨j,a⟩) (i' | ⟨j',a'⟩) h
-  · exact congrArg Sum.inl (hp.injective h)
-  · have hn : p i ≠ tupleGapMap p (.inr ⟨j',a'⟩) := by
-      rcases le_or_gt j' i.castSucc with hj | hj
-      · exact ne_of_gt ((tupleGapMap_gap_lt_tag p hp i j' a').2 hj)
-      · exact ne_of_lt ((tupleGapMap_tag_lt_gap p hp i j' a').2 hj)
-    exact False.elim (hn h)
-  · have hn : p i' ≠ tupleGapMap p (.inr ⟨j,a⟩) := by
-      rcases le_or_gt j i'.castSucc with hj | hj
-      · exact ne_of_gt ((tupleGapMap_gap_lt_tag p hp i' j a).2 hj)
-      · exact ne_of_lt ((tupleGapMap_tag_lt_gap p hp i' j a).2 hj)
-    exact False.elim (hn h.symm)
-  · have hj : j=j' := by
-      by_contra hn
-      rcases lt_or_gt_of_ne hn with hj | hj
-      · let i : Fin r := ⟨j.val, lt_of_lt_of_le hj (Nat.le_of_lt_succ j'.isLt)⟩
-        have hji : j=i.castSucc := Fin.ext rfl
-        have h1 := (tupleGapMap_gap_lt_tag p hp i j a).2 (by rw [hji])
-        have h2 := (tupleGapMap_tag_lt_gap p hp i j' a').2 (by rwa [← hji])
-        rw [h] at h1
-        exact (lt_asymm h1 h2)
-      · let i : Fin r := ⟨j'.val, lt_of_lt_of_le hj (Nat.le_of_lt_succ j.isLt)⟩
-        have hji : j'=i.castSucc := Fin.ext rfl
-        have h1 := (tupleGapMap_gap_lt_tag p hp i j' a').2 (by rw [hji])
-        have h2 := (tupleGapMap_tag_lt_gap p hp i j a).2 (by rwa [← hji])
-        rw [h] at h2
-        exact (lt_asymm h1 h2)
-    subst j'
-    have ha : a=a' := by
-      apply Fin.ext
-      have hh := congrArg Fin.val h
-      change tupleLower p j + a.val = tupleLower p j + a'.val at hh
-      omega
-    subst a'
-    rfl
-
-def tupleGapEquiv {n r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p) :
-    GapUniverse r (r+1) (tupleGap p) ≃ Fin n :=
-  Equiv.ofBijective (tupleGapMap p) ((Fintype.bijective_iff_injective_and_card _).2
-    ⟨tupleGapMap_injective p hp, by
-      simp only [Fintype.card_sum, Fintype.card_sigma, Fintype.card_fin]
-      exact tupleGap_total p hp⟩)
-
-/-- The count below a tagged slot is its tag index plus the retained gap counts. -/
-theorem gapEncoded_lower_card {n r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p)
-    (A : (j : Fin (r+1)) → Finset (Fin (tupleGap p j))) (i : Fin r) :
-    (((gapEncode A).map (tupleGapEquiv p hp).toEmbedding).filter
-      fun x => x < p i).card = i.val + ∑ j ∈ Finset.Iic i.castSucc, (A j).card := by
-  classical
-  have hmap : (((gapEncode A).map (tupleGapEquiv p hp).toEmbedding).filter
-      fun x => x < p i) =
-      ((gapEncode A).filter fun x => tupleGapMap p x < p i).map
-        (tupleGapEquiv p hp).toEmbedding := by
-    rw [Finset.filter_map]
-    rfl
-  rw [hmap, Finset.card_map, Finset.card_eq_sum_ones]
-  simp only [Finset.sum_filter, gapEncode]
-  rw [Fintype.sum_sum_type, Fintype.sum_sigma]
-  simp only
-  simp only [↓reduceIte]
-  have hleft : (∑ a : Fin r, if tupleGapMap p (.inl a) < p i then (1:ℕ) else 0) = i.val := by
-    simp only [tupleGapMap, hp.lt_iff_lt]
-    have hx : (Finset.univ.filter fun a : Fin r => a < i) = Finset.Iio i := by ext; simp
-    rw [← Finset.sum_filter, hx, Finset.sum_const, Fin.card_Iio, smul_eq_mul, mul_one]
-  rw [hleft]
-  congr 1
-  simp_rw [tupleGapMap_gap_lt_tag p hp]
-  have hr (j : Fin (r+1)) :
-      (∑ a : Fin (tupleGap p j), if a ∈ A j then
-        (if j ≤ i.castSucc then (1:ℕ) else 0) else 0) =
-      if j ≤ i.castSucc then (A j).card else 0 := by
-    by_cases hj : j ≤ i.castSucc
-    · simp only [hj, ite_true]
-      exact (Finset.card_eq_sum_ite (Finset.subset_univ (A j))).symm
-    · simp [hj]
-  rw [Finset.sum_congr rfl (fun j _ => hr j), ← Finset.sum_filter]
-  congr 1
-  ext j
-  simp
-
-@[simp] theorem gapEncode_univ {r d : ℕ} {g : Fin d → ℕ} :
-    gapEncode (r := r) (fun i => (Finset.univ : Finset (Fin (g i)))) = Finset.univ := by
-  ext x
-  cases x with
-  | inl i => simp
-  | inr p => rcases p with ⟨j,a⟩; simp
-
-theorem tupleGap_prefix {n r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p)
-    (i : Fin r) : i.val + ∑ j ∈ Finset.Iic i.castSucc, tupleGap p j = (p i).val := by
-  have h := gapEncoded_lower_card p hp (fun _ => Finset.univ) i
-  have hset : (Finset.univ.filter fun x : Fin n => x < p i) = Finset.Iio (p i) := by
-    ext; simp
-  simpa only [gapEncode_univ, Finset.map_univ_equiv,
-    hset, Fin.card_Iio, Finset.card_univ, Fintype.card_fin] using h.symm
-
-/-- Initial-segment sums determine a family of natural numbers. -/
-theorem fin_prefix_sum_injective {d : ℕ} (f g : Fin d → ℕ)
-    (h : ∀ i, (∑ j ∈ Finset.Iic i, f j) = ∑ j ∈ Finset.Iic i, g j) : f=g := by
-  funext i
-  induction i using WellFoundedLT.induction with
-  | ind i ih =>
-    have hset : Finset.Iic i = insert i (Finset.Iio i) := by ext; simp
-    have hprev : (∑ j ∈ Finset.Iio i, f j) = ∑ j ∈ Finset.Iio i, g j := by
-      apply Finset.sum_congr rfl
-      intro j hj
-      exact ih j (Finset.mem_Iio.mp hj)
-    have hh := h i
-    rw [hset, Finset.sum_insert (by simp), Finset.sum_insert (by simp), hprev] at hh
-    omega
-
-def pullGapSet {n r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p)
-    (S : Finset (Fin n)) : Finset (GapUniverse r (r+1) (tupleGap p)) :=
-  S.map (tupleGapEquiv p hp).symm.toEmbedding
-
-@[simp] theorem mem_pullGapSet {n r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p)
-    (S : Finset (Fin n)) (x : GapUniverse r (r+1) (tupleGap p)) :
-    x ∈ pullGapSet p hp S ↔ tupleGapMap p x ∈ S := by
-  simp [pullGapSet, tupleGapEquiv]
-
-@[simp] theorem map_pullGapSet {n r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p)
-    (S : Finset (Fin n)) :
-    (pullGapSet p hp S).map (tupleGapEquiv p hp).toEmbedding = S := by
-  ext x
-  simp only [pullGapSet, Finset.mem_map_equiv, Equiv.symm_symm, Equiv.apply_symm_apply]
-
-@[simp] theorem pullGapSet_map {n r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p)
-    (T : Finset (GapUniverse r (r+1) (tupleGap p))) :
-    pullGapSet p hp (T.map (tupleGapEquiv p hp).toEmbedding) = T := by
-  ext x
-  simp only [pullGapSet, Finset.mem_map_equiv, Equiv.symm_symm, Equiv.symm_apply_apply]
-
-theorem pullGapSet_lower_card {n r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p)
-    (S : Finset (Fin n)) (hS : ∀ i, p i ∈ S) (i : Fin r) :
-    (S.filter fun x => x < p i).card =
-      i.val + ∑ j ∈ Finset.Iic i.castSucc, (gapDecode (pullGapSet p hp S) j).card := by
-  have hT : ∀ a, Sum.inl a ∈ pullGapSet p hp S := by
-    intro a
-    simpa only [mem_pullGapSet, tupleGapMap] using hS a
-  have h := gapEncoded_lower_card p hp (gapDecode (pullGapSet p hp S)) i
-  rwa [gapEncode_decode _ hT, map_pullGapSet] at h
-
-theorem pullGapSet_total_card {n r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p)
-    (S : Finset (Fin n)) (hS : ∀ i, p i ∈ S) :
-    S.card = r + ∑ j, (gapDecode (pullGapSet p hp S) j).card := by
-  have hT : ∀ a, Sum.inl a ∈ pullGapSet p hp S := by
-    intro a
-    simpa only [mem_pullGapSet, tupleGapMap] using hS a
-  have h := card_gapEncode (r := r) (gapDecode (pullGapSet p hp S))
-  rw [gapEncode_decode _ hT] at h
-  simpa only [pullGapSet, Finset.card_map] using h
-
-theorem taggedRankEvent_iff_gapEvent {n l r : ℕ} (p : Fin r → Fin n)
-    (hp : StrictMono p) (v : Fin r → Fin l) (hv : StrictMono v)
-    (S : FixedSubset (Fin n) l) :
-    taggedRankEvent p v S ↔ gapEvent (tupleGap v) (pullGapSet p hp S.val) := by
-  constructor
-  · intro h
-    have hS : ∀ i, p i ∈ S.val := fun i => (slotAtRank_eq_iff S (v i) (p i)).mp (h i) |>.1
-    have hbefore : ∀ i, (S.val.filter fun x => x < p i).card = (v i).val :=
-      fun i => (slotAtRank_eq_iff S (v i) (p i)).mp (h i) |>.2
-    refine ⟨?_, ?_⟩
-    · intro i
-      simpa only [mem_pullGapSet, tupleGapMap] using hS i
-    · have heq : (fun j => (gapDecode (pullGapSet p hp S.val) j).card) = tupleGap v := by
-        apply fin_prefix_sum_injective
-        intro j
-        induction j using Fin.lastCases with
-        | last =>
-          have htot := pullGapSet_total_card p hp S.val hS
-          have hvtot := tupleGap_total v hv
-          have hset : Finset.Iic (Fin.last r) = Finset.univ := by ext; simp [Fin.le_last]
-          rw [hset]
-          rw [S.property] at htot
-          omega
-        | cast i =>
-          have hpre := pullGapSet_lower_card p hp S.val hS i
-          rw [hbefore i] at hpre
-          have hvpre := tupleGap_prefix v hv i
-          omega
-      intro j
-      exact congrFun heq j
-  · intro h
-    have hS : ∀ i, p i ∈ S.val := by
-      intro i
-      simpa only [mem_pullGapSet, tupleGapMap] using h.1 i
-    intro i
-    apply (slotAtRank_eq_iff S (v i) (p i)).2
-    refine ⟨hS i, ?_⟩
-    rw [pullGapSet_lower_card p hp S.val hS i]
-    simp_rw [h.2]
-    exact tupleGap_prefix v hv i
-
-instance taggedRankEvent_decidable {n l r : ℕ} (p : Fin r → Fin n)
-    (v : Fin r → Fin l) (S : FixedSubset (Fin n) l) : Decidable (taggedRankEvent p v S) := by
-  unfold taggedRankEvent
-  infer_instance
-
-/-- The actual event of seeing specified source slots at specified output ranks
-is in bijection with independent choices of the retained slots in each gap. -/
-def rankGapEventEquiv {n l r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p)
-    (v : Fin r → Fin l) (hv : StrictMono v) :
-    {S : FixedSubset (Fin n) l // taggedRankEvent p v S} ≃
-      {T : Finset (GapUniverse r (r+1) (tupleGap p)) // gapEvent (tupleGap v) T} where
-  toFun S := ⟨pullGapSet p hp S.val.val, (taggedRankEvent_iff_gapEvent p hp v hv S.val).mp S.property⟩
-  invFun T :=
-    let S : FixedSubset (Fin n) l := ⟨T.val.map (tupleGapEquiv p hp).toEmbedding, by
-      rw [Finset.card_map, card_of_gapEvent T.val T.property, tupleGap_total v hv]⟩
-    ⟨S, (taggedRankEvent_iff_gapEvent p hp v hv S).mpr (by
-      change gapEvent (tupleGap v) (pullGapSet p hp (T.val.map (tupleGapEquiv p hp).toEmbedding))
-      rw [pullGapSet_map]
-      exact T.property)⟩
-  left_inv S := by
-    apply Subtype.ext
-    apply Subtype.ext
-    exact map_pullGapSet p hp S.val.val
-  right_inv T := by
-    apply Subtype.ext
-    exact pullGapSet_map p hp T.val
-
-theorem card_taggedRankEvent {n l r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p)
-    (v : Fin r → Fin l) (hv : StrictMono v) :
-    Fintype.card {S : FixedSubset (Fin n) l // taggedRankEvent p v S} =
-      ∏ j, (tupleGap p j).choose (tupleGap v j) := by
-  classical
-  rw [Fintype.card_congr (rankGapEventEquiv p hp v hv), card_gapEvent]
-
-/-- Exact counting formula, including impossible rank/gap profiles. -/
-theorem subsetRankKernel_choose_formula {n l r : ℕ} (hl : l ≤ n)
-    (p : Fin r → Fin n) (hp : StrictMono p) (v : Fin r → Fin l) (hv : StrictMono v) :
-    subsetRankKernel hl p v =
-      (∏ j, ((tupleGap p j).choose (tupleGap v j) : ℝ)) / (n.choose l : ℝ) := by
-  classical
-  unfold subsetRankKernel
-  simp only [uniformSubsetLaw, Fintype.card_fin]
-  rw [← Finset.sum_filter, Finset.sum_const, nsmul_eq_mul,
-    ← Fintype.card_subtype, card_taggedRankEvent p hp v hv, Nat.cast_prod]
-  ring
-
-theorem gapEvent_conditional_choose {r d : ℕ} (g z : Fin d → ℕ)
-    (p : ℝ) (hp : p ∈ Set.Ioo 0 1) :
-    (∑ S : Finset (GapUniverse r d g), if gapEvent z S then subsetWeight p S else 0) /
-      binomialMass (r+∑ i, g i) p (r+∑ i, z i) =
-    (∏ i, ((g i).choose (z i):ℝ)) /
-      ((r+∑ i, g i).choose (r+∑ i, z i) : ℝ) := by
-  classical
-  have hcount : (∑ S : Finset (GapUniverse r d g),
-      if gapEvent z S then (1:ℝ) else 0) = ∏ i, ((g i).choose (z i):ℝ) := by
-    rw [Finset.sum_boole, ← Fintype.card_subtype, card_gapEvent, Nat.cast_prod]
-  have hnum : (∑ S : Finset (GapUniverse r d g), if gapEvent z S then subsetWeight p S else 0) =
-      (∏ i, ((g i).choose (z i):ℝ)) *
-        (p^(r+∑ i, z i)*(1-p)^(r+∑ i, g i-(r+∑ i, z i))) := by
-    rw [← hcount, Finset.sum_mul]
-    apply Finset.sum_congr rfl
-    intro S _
-    split_ifs with hS
-    · simp only [one_mul, subsetWeight, card_of_gapEvent S hS,
-        Fintype.card_sum, Fintype.card_fin, Fintype.card_sigma]
-    · exact (zero_mul _).symm
-  rw [hnum, binomialMass]
-  have hp0 : p ≠ 0 := hp.1.ne'
-  have hq0 : 1-p ≠ 0 := by linarith [hp.2]
-  rw [mul_assoc, mul_div_mul_right _ _ (mul_ne_zero (pow_ne_zero _ hp0) (pow_ne_zero _ hq0))]
-
-/-- Conditional product formula for the kernel defined from actual retained
-subsets, not an assumed auxiliary kernel. Source the initial-pair construction 50–62. -/
-theorem subsetRankKernel_binomial_formula {n l r : ℕ} (hl : l ≤ n)
-    (p : Fin r → Fin n) (hp : StrictMono p) (v : Fin r → Fin l) (hv : StrictMono v)
-    (θ : ℝ) (hθ : θ ∈ Set.Ioo 0 1) :
-    subsetRankKernel hl p v = θ^r / binomialMass n θ l *
-      ∏ j, binomialMass (tupleGap p j) θ (tupleGap v j) := by
-  classical
-  by_cases hz : ∀ j, tupleGap v j ≤ tupleGap p j
-  · have h1 := gapEvent_conditional_choose (r := r) (tupleGap p) (tupleGap v) θ hθ
-    have h2 := gapEvent_conditional_formula (r := r) (tupleGap p) (tupleGap v) hz θ
-    rw [tupleGap_total p hp, tupleGap_total v hv] at h1 h2
-    rw [subsetRankKernel_choose_formula hl p hp v hv, ← h1, h2]
-  · push Not at hz
-    obtain ⟨j,hj⟩ := hz
-    have hc : (tupleGap p j).choose (tupleGap v j) = 0 := Nat.choose_eq_zero_of_lt hj
-    have hz1 : (∏ j, ((tupleGap p j).choose (tupleGap v j) : ℝ))=0 :=
-      Finset.prod_eq_zero (Finset.mem_univ j) (by simp [hc])
-    have hz2 : (∏ j, binomialMass (tupleGap p j) θ (tupleGap v j))=0 :=
-      Finset.prod_eq_zero (Finset.mem_univ j) (by simp [binomialMass, hc])
-    rw [subsetRankKernel_choose_formula hl p hp v hv, hz1, hz2]
-    simp
-
-end TraceReconstruction
-
-namespace TraceReconstruction
-
-/-- Twice the coarse period gaps (endpoints are half-integral before doubling). -/
-def coarsePeriodGap {u r : ℕ} (j : Fin r → Fin u) (i : Fin (r+1)) : ℕ :=
-  2 * tupleGap j i + if i = 0 ∨ i = Fin.last r then 1 else 2
-
-theorem coarsePeriodGap_pos {u r : ℕ} (j : Fin r → Fin u) (i : Fin (r+1)) :
-    1 ≤ coarsePeriodGap j i := by
-  unfold coarsePeriodGap
-  split_ifs <;> omega
-
-theorem coarsePeriodGap_sum {u r : ℕ} (hr : 0 < r) (j : Fin r → Fin u)
-    (hj : StrictMono j) : ∑ i, coarsePeriodGap j i = 2*u := by
-  classical
-  have hc : (∑ i : Fin (r+1), if i = 0 ∨ i = Fin.last r then 1 else 2) = 2*r := by
-    obtain ⟨d,rfl⟩ := Nat.exists_eq_succ_of_ne_zero (Nat.ne_of_gt hr)
-    rw [Fin.sum_univ_succ]
-    simp only [true_or, ↓reduceIte, Fin.succ_ne_zero, false_or]
-    rw [Fin.sum_univ_castSucc]
-    simp only [Fin.succ_last, ↓reduceIte]
-    have hx (i : Fin d) : i.castSucc.succ ≠ Fin.last (d+1) := by
-      intro h
-      have hv := congrArg Fin.val h
-      simp only [Fin.val_succ, Fin.val_castSucc, Fin.val_last] at hv
-      omega
-    simp only [hx, ↓reduceIte, Finset.sum_const, Finset.card_univ, Fintype.card_fin,
-      smul_eq_mul]
-    omega
-  simp only [coarsePeriodGap, Finset.sum_add_distrib, ← Finset.mul_sum]
-  have ht := tupleGap_total j hj
-  omega
-
-theorem coarsePeriodGap_le {u r : ℕ} (hr : 0 < r) (j : Fin r → Fin u)
-    (hj : StrictMono j) (i : Fin (r+1)) : coarsePeriodGap j i ≤ 2*u := by
-  rw [← coarsePeriodGap_sum hr j hj]
-  exact Finset.single_le_sum (fun _ _ => Nat.zero_le _) (Finset.mem_univ i)
-
-theorem tupleGap_injective {n r : ℕ} {p q : Fin r → Fin n}
-    (hp : StrictMono p) (hq : StrictMono q) (h : tupleGap p = tupleGap q) : p=q := by
-  funext i
-  apply Fin.ext
-  have hp' := tupleGap_prefix p hp i
-  have hq' := tupleGap_prefix q hq i
-  rw [h] at hp'
-  omega
-
-theorem coarsePeriodGap_injective {u r : ℕ} {j k : Fin r → Fin u}
-    (hj : StrictMono j) (hk : StrictMono k) (h : coarsePeriodGap j = coarsePeriodGap k) :
-    j=k := by
-  apply tupleGap_injective hj hk
-  funext i
-  have hi := congrFun h i
-  unfold coarsePeriodGap at hi
-  omega
-
-/-- Actual defect slot in a period, with zero-based period and slot indices. -/
-def periodSlot {u m : ℕ} (off : Fin u → Fin m) (j : Fin u) : Fin (u*m) :=
-  ⟨j.val*m + (off j).val, by
-    have h := (off j).isLt
-    have h' := Nat.mul_le_mul_right m (Nat.succ_le_of_lt j.isLt)
-    nlinarith⟩
-
-theorem periodSlot_strictMono {u m : ℕ} (off : Fin u → Fin m) :
-    StrictMono (periodSlot off) := by
-  intro j k hjk
-  change j.val*m + (off j).val < k.val*m + (off k).val
-  have hjk' : j.val+1 ≤ k.val := hjk
-  have ho := (off j).isLt
-  have hm := Nat.mul_le_mul_right m hjk'
-  nlinarith
-
-theorem tupleGap_zero {n r : ℕ} (p : Fin (r+1) → Fin n) :
-    tupleGap p 0 = (p 0).val := by
-  have he : (0 : Fin (r+2)) = (0 : Fin (r+1)).castSucc := rfl
-  unfold tupleGap
-  rw [tupleLower_zero, Nat.sub_zero, he, tupleUpper_castSucc]
-
-theorem tupleGap_last {n r : ℕ} (p : Fin (r+1) → Fin n) :
-    tupleGap p (Fin.last (r+1)) = n - ((p (Fin.last r)).val+1) := by
-  unfold tupleGap
-  rw [tupleUpper_last, ← Fin.succ_last, tupleLower_succ]
-
-theorem tupleGap_middle {n r : ℕ} (p : Fin (r+1) → Fin n) (i : Fin r) :
-    tupleGap p i.castSucc.succ = (p i.succ).val - ((p i.castSucc).val+1) := by
-  unfold tupleGap
-  rw [tupleLower_succ, Fin.succ_castSucc, tupleUpper_castSucc]
-
-theorem coarsePeriodGap_zero {u r : ℕ} (j : Fin (r+1) → Fin u) :
-    coarsePeriodGap j 0 = 2*(j 0).val+1 := by
-  simp only [coarsePeriodGap, tupleGap_zero, true_or, ↓reduceIte]
-
-theorem coarsePeriodGap_last {u r : ℕ} (j : Fin (r+1) → Fin u) :
-    coarsePeriodGap j (Fin.last (r+1)) = 2*(u-((j (Fin.last r)).val+1))+1 := by
-  simp only [coarsePeriodGap, tupleGap_last, or_true, ↓reduceIte]
-
-theorem coarsePeriodGap_middle {u r : ℕ} (j : Fin (r+1) → Fin u) (i : Fin r) :
-    coarsePeriodGap j i.castSucc.succ = 2*((j i.succ).val-((j i.castSucc).val+1))+2 := by
-  have hn : i.castSucc.succ ≠ Fin.last (r+1) := by
-    intro h
-    have hv := congrArg Fin.val h
-    simp only [Fin.val_succ, Fin.val_castSucc, Fin.val_last] at hv
-    omega
-  simp only [coarsePeriodGap, tupleGap_middle, Fin.succ_ne_zero, hn, or_self, ↓reduceIte]
-
-/-- Actual bounded defect displacements give the O(1) gap errors of the initial-pair construction. -/
-theorem period_gap_rounding {u m r : ℕ} (hr : 0 < r)
-    (off : Fin u → Fin m)
-    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
-    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2)
-    (j : Fin r → Fin u) (hj : StrictMono j) (i : Fin (r+1)) :
-    |(tupleGap (periodSlot off ∘ j) i:ℝ) - (m:ℝ)*coarsePeriodGap j i/2| ≤ 4 := by
-  obtain ⟨d,rfl⟩ := Nat.exists_eq_succ_of_ne_zero (Nat.ne_of_gt hr)
-  have hp := (periodSlot_strictMono off).comp hj
-  induction i using Fin.cases with
-  | zero =>
-    rw [tupleGap_zero, coarsePeriodGap_zero]
-    simp only [Function.comp_apply, periodSlot, Nat.cast_add, Nat.cast_mul, Nat.cast_ofNat,
-      Nat.cast_one]
-    have offset_bound := abs_le.mp (hoff (j 0))
-    apply abs_le.mpr
-    constructor <;> linarith
-  | succ i =>
-    induction i using Fin.lastCases with
-    | last =>
-      rw [Fin.succ_last, tupleGap_last, coarsePeriodGap_last]
-      have hcut : (j (Fin.last d)).val+1 ≤ u := Nat.succ_le_of_lt (j _).isLt
-      have hslot : ((periodSlot off ∘ j) (Fin.last d)).val+1 ≤ u*m :=
-        Nat.succ_le_of_lt (((periodSlot off ∘ j) (Fin.last d))).isLt
-      simp only [Nat.cast_add, Nat.cast_mul, Nat.cast_ofNat]
-      rw [Nat.cast_sub hslot, Nat.cast_sub hcut]
-      simp only [Function.comp_apply, periodSlot, Nat.cast_add, Nat.cast_mul, Nat.cast_one]
-      have h := abs_le.mp (hoff (j (Fin.last d)))
-      apply abs_le.mpr
-      constructor <;> linarith
-    | cast i =>
-      rw [tupleGap_middle, coarsePeriodGap_middle]
-      have hcut : (j i.castSucc).val+1 ≤ (j i.succ).val := hj (Fin.castSucc_lt_succ (i := i))
-      have hslot : ((periodSlot off ∘ j) i.castSucc).val+1 ≤
-          ((periodSlot off ∘ j) i.succ).val := hp (Fin.castSucc_lt_succ (i := i))
-      simp only [Nat.cast_add, Nat.cast_mul, Nat.cast_ofNat]
-      rw [Nat.cast_sub hslot, Nat.cast_sub hcut]
-      simp only [Function.comp_apply, periodSlot, Nat.cast_add, Nat.cast_mul, Nat.cast_one]
-      have h := abs_le.mp (hdiff (j i.succ) (j i.castSucc))
-      apply abs_le.mpr
-      constructor <;> linarith
-
-end TraceReconstruction
-
-namespace TraceReconstruction
-
-/-- An increasing tuple enumerates a monomial's distinct observed ranks. -/
-abbrev IncreasingTuple (n r : ℕ) := {v : Fin r → Fin n // StrictMono v}
-
-noncomputable instance increasingTupleFintype (n r : ℕ) : Fintype (IncreasingTuple n r) :=
-  Fintype.ofFinite _
-
-theorem tupleGap_le_remainder {n r : ℕ} (v : IncreasingTuple n r) (i : Fin (r+1)) :
-    tupleGap v.val i ≤ n-r := by
-  have ht := tupleGap_total v.val v.property
-  have hle : tupleGap v.val i ≤ ∑ j, tupleGap v.val j :=
-    Finset.single_le_sum (fun _ _ => Nat.zero_le _) (Finset.mem_univ i)
-  omega
-
-def rankGapProfile {n r : ℕ} (σ : Equiv.Perm (Fin (r+1))) (v : IncreasingTuple n r) :
-    Fin (r+1) → Fin (n-r+1) :=
-  fun i => ⟨tupleGap v.val (σ i), Nat.lt_succ_of_le (tupleGap_le_remainder v (σ i))⟩
-
-theorem rankGapProfile_sum {n r : ℕ} (σ : Equiv.Perm (Fin (r+1)))
-    (v : IncreasingTuple n r) : ∑ i, (rankGapProfile σ v i).val = n-r := by
-  change (∑ i, tupleGap v.val (σ i)) = _
-  rw [Equiv.sum_comp σ]
-  have ht := tupleGap_total v.val v.property
-  omega
-
-theorem rankGapProfile_injective {n r : ℕ} (σ : Equiv.Perm (Fin (r+1))) :
-    Function.Injective (rankGapProfile (n := n) σ) := by
-  intro v w h
-  apply Subtype.ext
-  funext i
-  apply Fin.ext
-  have he : tupleGap v.val = tupleGap w.val := by
-    funext j
-    have hh := congrArg Fin.val (congrFun h (σ.symm j))
-    simpa only [rankGapProfile, Equiv.apply_symm_apply] using hh
-  have hv := tupleGap_prefix v.val v.property i
-  have hw := tupleGap_prefix w.val w.property i
-  rw [he] at hv
-  omega
-
-/-- Summing nonnegative kernels over actual profiles may drop any remaining constraints. -/
-theorem sum_injective_profile_le {α β : Type*} [Fintype α] [Fintype β]
-    (profile : α → β) (hinj : Function.Injective profile)
-    (P : β → Prop) [DecidablePred P] (hp : ∀ a, P (profile a))
-    (f : β → ℝ) (hf : ∀ b, 0 ≤ f b) :
-    (∑ a, f (profile a)) ≤ ∑ b, if P b then f b else 0 := by
-  classical
-  let S := Finset.univ.image profile
-  have hs : ∀ b ∈ S, P b := by
-    intro b hb
-    obtain ⟨a,_,rfl⟩ := Finset.mem_image.mp hb
-    exact hp a
-  calc
-    _ = ∑ b ∈ S, f b := by
-      symm
-      exact Finset.sum_image (fun _ _ _ _ h => hinj h)
-    _ = ∑ b ∈ S, if P b then f b else 0 := by
-      apply Finset.sum_congr rfl
-      intro b hb
-      rw [ite_eq_left (hs b hb)]
-    _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _) (by
-      intro b _ _
-      split_ifs
-      · exact hf b
-      · rfl)
-
-/-- The actual conditional rank kernel, summed over distinct observed ranks,
-reduces to the constrained finite product estimate with any eliminated gap. -/
-theorem subsetRankKernel_cross_le {n l r : ℕ} (hl : l ≤ n)
-    (p q : Fin r → Fin n) (hp : StrictMono p) (hq : StrictMono q)
-    (θ : ℝ) (hθ : θ ∈ Set.Ioo 0 1) (e : Fin (r+1))
-    (M : ℝ) (hM : 0 ≤ M)
-    (hbound : ∀ k : Fin (l-r+1),
-      binomialMass (tupleGap p e) θ k * binomialMass (tupleGap q e) θ k ≤ M) :
-    (∑ v : IncreasingTuple l r,
-      subsetRankKernel hl p v.val * subsetRankKernel hl q v.val) ≤
-      (θ^r / binomialMass n θ l)^2 * M *
-        ∏ i : Fin r, ∑ k : Fin (l-r+1),
-          binomialMass (tupleGap p (Equiv.swap 0 e i.succ)) θ k *
-          binomialMass (tupleGap q (Equiv.swap 0 e i.succ)) θ k := by
-  classical
-  let σ : Equiv.Perm (Fin (r+1)) := Equiv.swap 0 e
-  let f : Fin (r+1) → Fin (l-r+1) → ℝ := fun i k => binomialMass (tupleGap p (σ i)) θ k
-  let g : Fin (r+1) → Fin (l-r+1) → ℝ := fun i k => binomialMass (tupleGap q (σ i)) θ k
-  have hf : ∀ i k, 0 ≤ f i k := by
-    intro i k
-    dsimp [f,binomialMass]
-    positivity [hθ.1.le, sub_nonneg.mpr hθ.2.le]
-  have hg : ∀ i k, 0 ≤ g i k := by
-    intro i k
-    dsimp [g,binomialMass]
-    positivity [hθ.1.le, sub_nonneg.mpr hθ.2.le]
-  let F (z : Fin (r+1) → Fin (l-r+1)) := (∏ i, f i (z i)) * (∏ i, g i (z i))
-  have hF : ∀ z, 0 ≤ F z := by
-    intro z
-    exact mul_nonneg (Finset.prod_nonneg fun i _ => hf i _) (Finset.prod_nonneg fun i _ => hg i _)
-  have hpoint (v : IncreasingTuple l r) :
-      subsetRankKernel hl p v.val * subsetRankKernel hl q v.val =
-        (θ^r / binomialMass n θ l)^2 * F (rankGapProfile σ v) := by
-    rw [subsetRankKernel_binomial_formula hl p hp v.val v.property θ hθ,
-      subsetRankKernel_binomial_formula hl q hq v.val v.property θ hθ]
-    dsimp only [F,f,g,rankGapProfile]
-    rw [Equiv.prod_comp σ (fun j => binomialMass (tupleGap p j) θ (tupleGap v.val j)),
-      Equiv.prod_comp σ (fun j => binomialMass (tupleGap q j) θ (tupleGap v.val j))]
-    ring
-  simp_rw [hpoint]
-  rw [← Finset.mul_sum]
-  calc
-    _ ≤ (θ^r / binomialMass n θ l)^2 *
-        ∑ z : Fin (r+1) → Fin (l-r+1), if ∑ i, (z i).val = l-r then F z else 0 := by
-      apply mul_le_mul_of_nonneg_left _ (sq_nonneg _)
-      exact sum_injective_profile_le (rankGapProfile σ) (rankGapProfile_injective σ)
-        (fun z => ∑ i, (z i).val = l-r) (rankGapProfile_sum σ) F hF
-    _ ≤ (θ^r / binomialMass n θ l)^2 * (M * ∏ i : Fin r, ∑ k, f i.succ k * g i.succ k) := by
-      apply mul_le_mul_of_nonneg_left _ (sq_nonneg _)
-      exact constrained_cross_kernel_le r (l-r) f g hf hg M hM (by
-        intro k
-        simpa only [f,g,σ,Equiv.swap_apply_left] using hbound k)
-    _ = _ := by dsimp [f,g,σ]; ring
-
-end TraceReconstruction
-
-namespace TraceReconstruction
-
-theorem finite_geometric_distance (S : Finset ℤ) (a : ℤ) {ρ : ℝ}
-    (hρ : ρ∈Set.Ico 0 1) :
-    (∑ z ∈ S, ρ^(z-a).natAbs) ≤ 2/(1-ρ) := by
-  have hs : Summable (fun n:ℕ => ρ^n) := summable_geometric_of_lt_one hρ.1 hρ.2
-  have hnat : Summable (fun n:ℕ => ρ^((n:ℤ).natAbs)) := by simpa using hs
-  have hneg : Summable (fun n:ℕ => ρ^((-(n:ℤ)).natAbs)) := by simpa using hs
-  have hz : Summable (fun z:ℤ => ρ^z.natAbs) := hnat.of_nat_of_neg hneg
-  have ht : (∑' z:ℤ, ρ^z.natAbs)=2/(1-ρ)-1 := by
-    rw [Summable.tsum_of_nat_of_neg (f := fun z : ℤ => ρ^z.natAbs) hnat hneg]
-    simp only [Int.natAbs_natCast, Int.natAbs_neg, Int.natAbs_zero, pow_zero]
-    rw [tsum_geometric_of_lt_one hρ.1 hρ.2]
-    ring
-  have hshift : Summable (fun z:ℤ => ρ^(z-a).natAbs) :=
-    hz.comp_injective (sub_left_injective)
-  have hts : (∑' z:ℤ, ρ^(z-a).natAbs)=(∑' z:ℤ, ρ^z.natAbs) :=
-    (Equiv.subRight a).tsum_eq (fun z:ℤ => ρ^z.natAbs)
-  have hbound := hshift.sum_le_tsum S (fun z _ => pow_nonneg hρ.1 _)
-  rw [hts,ht] at hbound
-  linarith
-
-theorem finite_gaussian_distance (S : Finset ℤ) (a : ℤ) {c : ℝ}
-    (hc : 0<c) :
-    (∑ z ∈ S, Real.exp (-c*((z:ℝ)-a)^2)) ≤ 2/(1-Real.exp (-c)) := by
-  apply le_trans _ (finite_geometric_distance S a ⟨(Real.exp_pos _).le, by
-    rw [Real.exp_lt_one_iff]; linarith⟩)
-  apply Finset.sum_le_sum
-  intro z _
-  have hcast (w : ℤ) : (w.natAbs:ℝ)=|(w:ℝ)| := by
-    rw [← Int.cast_abs, ← Int.natCast_natAbs, Int.cast_natCast]
-  have he : Real.exp (-c)^(z-a).natAbs = Real.exp (-c*|(z:ℝ)-a|) := by
-    rw [← Real.exp_nat_mul]
-    congr 1
-    rw [mul_comm, hcast, Int.cast_sub]
-  rw [he]
-  apply Real.exp_le_exp.mpr
-  have hd : |(z:ℝ)-a| ≤ ((z:ℝ)-a)^2 := by
-    have hnat : (z-a).natAbs ≤ (z-a).natAbs^2 := by
-      nlinarith only [Nat.zero_le (z-a).natAbs, (Nat.mul_self_le_mul_self_iff.mpr (show 0≤(z-a).natAbs by omega))]
-    have hr : ((z-a).natAbs:ℝ) ≤ ((z-a).natAbs:ℝ)^2 := by exact_mod_cast hnat
-    simpa only [hcast, Int.cast_sub, sq_abs] using hr
-  nlinarith [mul_le_mul_of_nonneg_left hd hc.le]
-
-end TraceReconstruction
-
-namespace TraceReconstruction
-
-theorem fin_geometric_distance (N : ℕ) (a : ℤ) {ρ : ℝ} (hρ : ρ∈Set.Ico 0 1) :
-    (∑ z : Fin N, ρ^((z.val:ℤ)-a).natAbs) ≤ 2/(1-ρ) := by
-  let e : ℕ ↪ ℤ := ⟨Int.ofNat, Nat.cast_injective⟩
-  have h := finite_geometric_distance ((Finset.range N).map e) a hρ
-  rw [Finset.sum_map] at h
-  rw [Fin.sum_univ_eq_sum_range (fun z:ℕ => ρ^((z:ℤ)-a).natAbs) N]
-  exact h
-
-theorem coarse_laplace_sum_with_zero {h : ℝ} (hh : 0<h) (h1 : h≤1) (M : ℕ) :
-    (∑ k : Fin (M+1), (Real.sqrt (k.val:ℝ))⁻¹ * Real.exp (-h*(k.val:ℝ))) ≤
-      6/Real.sqrt h := by
-  rw [Fin.sum_univ_succ]
-  simp only [Fin.val_zero, Nat.cast_zero, Real.sqrt_zero, inv_zero, zero_mul, zero_add]
-  have hc := coarse_laplace_sum hh h1 M
-  rw [← Fin.sum_univ_eq_sum_range (fun k : ℕ =>
-    (Real.sqrt (k+1:ℝ))⁻¹ * Real.exp (-h*(k+1:ℝ))) M] at hc
-  simpa only [Fin.val_succ, Nat.cast_add, Nat.cast_one] using hc
-
-/-- The total constraint eliminates one coarse gap with no cost in its size.
-The zero-extension of x^(-1/2) avoids any singular integral convention. -/
-theorem coarse_constrained_inverse_sum (r U : ℕ) {h : ℝ} (hh : 0<h) (h1 : h≤1) :
-    (∑ γ : Fin (r+1) → Fin (U+1),
-      if ∑ i, (γ i).val=U then ∏ i : Fin r, (Real.sqrt ((γ i.succ).val:ℝ))⁻¹ else 0) ≤
-      Real.exp (h*U) * (6/Real.sqrt h)^r := by
-  classical
-  let f : Fin (r+1) → Fin (U+1) → ℝ := Fin.cases
-    (fun k => Real.exp (-h*(k.val:ℝ)))
-    (fun _ k => (Real.sqrt (k.val:ℝ))⁻¹ * Real.exp (-h*(k.val:ℝ)))
-  have hf : ∀ i k, 0≤f i k := by
-    intro i k
-    refine Fin.cases ?_ (fun _ => ?_) i <;> dsimp [f] <;> positivity
-  have hf0 : ∀ k, f 0 k≤1 := by
-    intro k
-    change Real.exp (-h*(k.val:ℝ))≤1
-    apply Real.exp_le_one_iff.mpr
-    nlinarith [show (0:ℝ)≤k.val by positivity]
-  have hid (γ : Fin (r+1) → Fin (U+1)) (hγ : ∑ i, (γ i).val=U) :
-      ∏ i : Fin r, (Real.sqrt ((γ i.succ).val:ℝ))⁻¹ =
-        Real.exp (h*U) * ∏ i, f i (γ i) := by
-    have hγr : ((γ 0).val:ℝ) + ∑ i:Fin r, ((γ i.succ).val:ℝ)=(U:ℝ) := by
-      exact_mod_cast (show (γ 0).val+∑ i:Fin r, (γ i.succ).val=U by
-        simpa only [Fin.sum_univ_succ] using hγ)
-    simp only [Fin.prod_univ_succ, f, Fin.cases_zero, Fin.cases_succ,
-      Finset.prod_mul_distrib, ← Real.exp_sum]
-    have he : Real.exp (h*U) * Real.exp (-h*((γ 0).val:ℝ)) *
-        Real.exp (∑ i:Fin r, -h*((γ i.succ).val:ℝ))=1 := by
-      rw [← Real.exp_add, ← Real.exp_add, ← Finset.mul_sum]
-      rw [show h*(U:ℝ)+ -h*((γ 0).val:ℝ)+ -h*∑ i:Fin r, ((γ i.succ).val:ℝ)=0 by
-        rw [← hγr]
-        ring]
-      exact Real.exp_zero
-    calc
-      _ = (Real.exp (h*U) * Real.exp (-h*((γ 0).val:ℝ)) *
-        Real.exp (∑ i:Fin r, -h*((γ i.succ).val:ℝ))) *
-          (∏ i:Fin r, (Real.sqrt ((γ i.succ).val:ℝ))⁻¹) := by rw [he, one_mul]
-      _ = _ := by ring
-  have hrew : (∑ γ : Fin (r+1) → Fin (U+1),
-      if ∑ i, (γ i).val=U then ∏ i : Fin r, (Real.sqrt ((γ i.succ).val:ℝ))⁻¹ else 0) =
-      Real.exp (h*U) * ∑ γ : Fin (r+1) → Fin (U+1),
-        if ∑ i, (γ i).val=U then ∏ i, f i (γ i) else 0 := by
-    rw [Finset.mul_sum]
-    apply Finset.sum_congr rfl
-    intro γ _
-    split_ifs with hγ
-    · exact hid γ hγ
-    · simp
-  rw [hrew]
-  apply mul_le_mul_of_nonneg_left _ (Real.exp_nonneg _)
-  apply (constrained_sum_product_le r U f hf 1 (by norm_num) hf0).trans
-  simp only [one_mul]
-  calc
-    _ ≤ ∏ _i : Fin r, (6/Real.sqrt h) := by
-      apply Finset.prod_le_prod₀
-      · intro i _; exact Finset.sum_nonneg (fun k _ => hf i.succ k)
-      · intro i _; exact coarse_laplace_sum_with_zero hh h1 U
-    _ = _ := by simp [div_pow]
-
-/-- The squared norm of an aggregate equals the sum of the cross-kernels. -/
-theorem sum_sq_sum_kernel {α β : Type*} [Fintype β] (S : Finset α) (K : α → β → ℝ) :
-    (∑ z, (∑ a ∈ S, K a z)^2) =
-      ∑ a ∈ S, ∑ b ∈ S, ∑ z, K a z*K b z := by
-  simp_rw [pow_two, Finset.sum_mul, Finset.mul_sum]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro a _
-  exact Finset.sum_comm
-
-/-- Schur summation over all pairs of coarse-gap profiles, retaining the
-largest-gap total constraint before dropping other restrictions. -/
-theorem coarse_kernel_square_sum {β : Type*} [Fintype β] (r U : ℕ)
-    (S : Finset (Fin (r+1) → Fin (U+1)))
-    (K : (Fin (r+1) → Fin (U+1)) → β → ℝ) {M ρ h : ℝ}
-    (hM : 0≤M) (hρ : ρ∈Set.Ico 0 1) (hh : 0<h) (h1 : h≤1)
-    (hS : ∀ γ∈S, ∑ i, (γ i).val=U)
-    (hcross : ∀ γ∈S, ∀ γ'∈S, (∑ z, K γ z*K γ' z) ≤
-      M * ∏ i:Fin r, ((Real.sqrt ((γ i.succ).val:ℝ))⁻¹ *
-        ρ^(((γ' i.succ).val:ℤ)-(γ i.succ).val).natAbs)) :
-    (∑ z, (∑ γ∈S, K γ z)^2) ≤
-      M * (2/(1-ρ))^r * Real.exp (h*U) * (6/Real.sqrt h)^r := by
-  classical
-  let A (γ : Fin (r+1) → Fin (U+1)) :=
-    ∏ i:Fin r, (Real.sqrt ((γ i.succ).val:ℝ))⁻¹
-  have hA (γ) : 0≤A γ := by dsimp [A]; positivity
-  have hinner (γ : Fin (r+1) → Fin (U+1)) :
-      (∑ γ'∈S, ∏ i:Fin r, ρ^(((γ' i.succ).val:ℤ)-(γ i.succ).val).natAbs) ≤
-        (2/(1-ρ))^r := by
-    let f : Fin (r+1) → Fin (U+1) → ℝ := Fin.cases (fun _ => 1)
-      (fun i k => ρ^((k.val:ℤ)-(γ i.succ).val).natAbs)
-    have hf : ∀ i k, 0≤f i k := by
-      intro i k
-      refine Fin.cases ?_ (fun _ => ?_) i
-      · norm_num [f]
-      · exact pow_nonneg hρ.1 _
-    have hsub : (∑ γ'∈S, ∏ i:Fin r, ρ^(((γ' i.succ).val:ℤ)-(γ i.succ).val).natAbs) ≤
-        ∑ γ' : Fin (r+1) → Fin (U+1), if ∑ i, (γ' i).val=U then ∏ i, f i (γ' i) else 0 := by
-      calc
-        _ = ∑ γ'∈S, if ∑ i, (γ' i).val=U then ∏ i, f i (γ' i) else 0 := by
-          apply Finset.sum_congr rfl
-          intro γ' hγ'
-          simp only [hS γ' hγ', ↓reduceIte, Fin.prod_univ_succ, f, Fin.cases_zero,
-            Fin.cases_succ, one_mul]
-        _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _) (by
-          intro γ' _ _
-          split_ifs
-          · exact Finset.prod_nonneg (fun i _ => hf i _)
-          · rfl)
-    apply hsub.trans
-    apply (constrained_sum_product_le r U f hf 1 (by norm_num) (by intro k; rfl)).trans
-    simp only [one_mul]
-    calc
-      _ ≤ ∏ _i:Fin r, (2/(1-ρ)) := by
-        apply Finset.prod_le_prod₀
-        · intro i _; exact Finset.sum_nonneg (fun k _ => hf i.succ k)
-        · intro i _; exact fin_geometric_distance (U+1) ((γ i.succ).val:ℤ) hρ
-      _ = _ := by simp [div_pow]
-  have houter : (∑ γ∈S, A γ) ≤ Real.exp (h*U)*(6/Real.sqrt h)^r := by
-    apply le_trans _ (coarse_constrained_inverse_sum r U hh h1)
-    calc
-      _ = ∑ γ∈S, if ∑ i, (γ i).val=U then A γ else 0 := by
-        apply Finset.sum_congr rfl
-        intro γ hγ
-        rw [ite_eq_left (hS γ hγ)]
-      _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _) (by
-        intro γ _ _
-        split_ifs
-        · exact hA γ
-        · rfl)
-  rw [sum_sq_sum_kernel]
-  calc
-    _ ≤ ∑ γ∈S, ∑ γ'∈S, M * ∏ i:Fin r, ((Real.sqrt ((γ i.succ).val:ℝ))⁻¹ *
-        ρ^(((γ' i.succ).val:ℤ)-(γ i.succ).val).natAbs) := by
-      exact Finset.sum_le_sum (fun γ hγ => Finset.sum_le_sum (fun γ' hγ' => hcross γ hγ γ' hγ'))
-    _ = ∑ γ∈S, M*A γ * ∑ γ'∈S, ∏ i:Fin r, ρ^(((γ' i.succ).val:ℤ)-(γ i.succ).val).natAbs := by
-      simp only [Finset.prod_mul_distrib, A, Finset.mul_sum, mul_assoc]
-    _ ≤ ∑ γ∈S, M*A γ*(2/(1-ρ))^r := by
-      exact Finset.sum_le_sum (fun γ _ => mul_le_mul_of_nonneg_left (hinner γ) (mul_nonneg hM (hA γ)))
-    _ = M*(2/(1-ρ))^r * ∑ γ∈S, A γ := by
-      rw [Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intros
-      ring
-    _ ≤ M*(2/(1-ρ))^r * (Real.exp (h*U)*(6/Real.sqrt h)^r) := by
-      apply mul_le_mul_of_nonneg_left houter
-      exact mul_nonneg hM (pow_nonneg (div_nonneg (by norm_num) (by linarith [hρ.2])) _)
-    _ = _ := by ring
-
-theorem coarse_kernel_rectangular_sum {β : Type*} [Fintype β] (r U : ℕ)
-    (S T : Finset (Fin (r+1) → Fin (U+1)))
-    (K G : (Fin (r+1) → Fin (U+1)) → β → ℝ) {M ρ h : ℝ}
-    (hM : 0≤M) (hρ : ρ∈Set.Ico 0 1) (hh : 0<h) (h1 : h≤1)
-    (hS : ∀ γ∈S, ∑ i, (γ i).val=U)
-    (hT : ∀ γ∈T, ∑ i, (γ i).val=U)
-    (hcross : ∀ γ∈S, ∀ γ'∈T, (∑ z, K γ z*G γ' z) ≤
-      M * ∏ i:Fin r, ((Real.sqrt ((γ i.succ).val:ℝ))⁻¹ *
-        ρ^(((γ' i.succ).val:ℤ)-(γ i.succ).val).natAbs)) :
-    (∑ γ∈S, ∑ γ'∈T, ∑ z, K γ z*G γ' z) ≤
-      M * (2/(1-ρ))^r * Real.exp (h*U) * (6/Real.sqrt h)^r := by
-  classical
-  let A (γ : Fin (r+1) → Fin (U+1)) :=
-    ∏ i:Fin r, (Real.sqrt ((γ i.succ).val:ℝ))⁻¹
-  have hA (γ) : 0≤A γ := by dsimp [A]; positivity
-  have hinner (γ : Fin (r+1) → Fin (U+1)) :
-      (∑ γ'∈T, ∏ i:Fin r, ρ^(((γ' i.succ).val:ℤ)-(γ i.succ).val).natAbs) ≤
-        (2/(1-ρ))^r := by
-    let f : Fin (r+1) → Fin (U+1) → ℝ := Fin.cases (fun _ => 1)
-      (fun i k => ρ^((k.val:ℤ)-(γ i.succ).val).natAbs)
-    have hf : ∀ i k, 0≤f i k := by
-      intro i k
-      refine Fin.cases ?_ (fun _ => ?_) i
-      · norm_num [f]
-      · exact pow_nonneg hρ.1 _
-    have hsub : (∑ γ'∈T, ∏ i:Fin r, ρ^(((γ' i.succ).val:ℤ)-(γ i.succ).val).natAbs) ≤
-        ∑ γ' : Fin (r+1) → Fin (U+1), if ∑ i, (γ' i).val=U then ∏ i, f i (γ' i) else 0 := by
-      calc
-        _ = ∑ γ'∈T, if ∑ i, (γ' i).val=U then ∏ i, f i (γ' i) else 0 := by
-          apply Finset.sum_congr rfl
-          intro γ' hγ'
-          simp only [hT γ' hγ', ↓reduceIte, Fin.prod_univ_succ, f, Fin.cases_zero,
-            Fin.cases_succ, one_mul]
-        _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _) (by
-          intro γ' _ _
-          split_ifs
-          · exact Finset.prod_nonneg (fun i _ => hf i _)
-          · rfl)
-    apply hsub.trans
-    apply (constrained_sum_product_le r U f hf 1 (by norm_num) (by intro k; rfl)).trans
-    simp only [one_mul]
-    calc
-      _ ≤ ∏ _i:Fin r, (2/(1-ρ)) := by
-        apply Finset.prod_le_prod₀
-        · intro i _; exact Finset.sum_nonneg (fun k _ => hf i.succ k)
-        · intro i _; exact fin_geometric_distance (U+1) ((γ i.succ).val:ℤ) hρ
-      _ = _ := by simp [div_pow]
-  have houter : (∑ γ∈S, A γ) ≤ Real.exp (h*U)*(6/Real.sqrt h)^r := by
-    apply le_trans _ (coarse_constrained_inverse_sum r U hh h1)
-    calc
-      _ = ∑ γ∈S, if ∑ i, (γ i).val=U then A γ else 0 := by
-        apply Finset.sum_congr rfl
-        intro γ hγ
-        rw [ite_eq_left (hS γ hγ)]
-      _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _) (by
-        intro γ _ _
-        split_ifs
-        · exact hA γ
-        · rfl)
-  calc
-    _ ≤ ∑ γ∈S, ∑ γ'∈T, M * ∏ i:Fin r, ((Real.sqrt ((γ i.succ).val:ℝ))⁻¹ *
-        ρ^(((γ' i.succ).val:ℤ)-(γ i.succ).val).natAbs) := by
-      exact Finset.sum_le_sum (fun γ hγ => Finset.sum_le_sum (fun γ' hγ' => hcross γ hγ γ' hγ'))
-    _ = ∑ γ∈S, M*A γ * ∑ γ'∈T, ∏ i:Fin r, ρ^(((γ' i.succ).val:ℤ)-(γ i.succ).val).natAbs := by
-      simp only [Finset.prod_mul_distrib, A, Finset.mul_sum, mul_assoc]
-    _ ≤ ∑ γ∈S, M*A γ*(2/(1-ρ))^r := by
-      exact Finset.sum_le_sum (fun γ _ => mul_le_mul_of_nonneg_left (hinner γ) (mul_nonneg hM (hA γ)))
-    _ = M*(2/(1-ρ))^r * ∑ γ∈S, A γ := by
-      rw [Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intros
-      ring
-    _ ≤ M*(2/(1-ρ))^r * (Real.exp (h*U)*(6/Real.sqrt h)^r) := by
-      apply mul_le_mul_of_nonneg_left houter
-      exact mul_nonneg hM (pow_nonneg (div_nonneg (by norm_num) (by linarith [hρ.2])) _)
-    _ = _ := by ring
-
-
-end TraceReconstruction
-
-namespace TraceReconstruction
-
-/-- Geometry of actual period gaps: the half-integer coarse gaps are
-represented by positive integers a,b, with bounded slot-rounding errors. -/
-theorem coarse_gap_geometry (m U a b g k : ℕ) (hm : 32 ≤ m) (hUm : U  ≤  2*m)
-    (ha : 1 ≤ a) (hb : 1 ≤ b) (haU : a ≤ U) (hbU : b ≤ U)
-    (hg : |(g:ℝ)-(m:ℝ)*a/2| ≤ 4) (hk : |(k:ℝ)-(m:ℝ)*b/2| ≤ 4) :
-    (m:ℝ)*a/4  ≤  (g:ℝ)+1 ∧ 0<(g:ℝ)+k ∧
-      ((a:ℝ)-b)^2/1024  ≤  ((g:ℝ)-k)^2/(16*((g:ℝ)+k)) := by
-  have hmR : (32:ℝ) ≤ m := by exact_mod_cast hm
-  have hUmR : (U:ℝ)  ≤  2*m := by exact_mod_cast hUm
-  have haR : (1:ℝ) ≤ a := by exact_mod_cast ha
-  have hbR : (1:ℝ) ≤ b := by exact_mod_cast hb
-  have haUR : (a:ℝ) ≤ U := by exact_mod_cast haU
-  have hbUR : (b:ℝ) ≤ U := by exact_mod_cast hbU
-  have hga := abs_le.mp hg
-  have hkb := abs_le.mp hk
-  have hma : (32:ℝ) ≤ (m:ℝ)*a := by nlinarith
-  have hmb : (32:ℝ) ≤ (m:ℝ)*b := by nlinarith
-  have hlo : (m:ℝ)*a/4 ≤ (g:ℝ)+1 := by nlinarith only [hga.1,hma]
-  have hsum0 : 0<(g:ℝ)+k := by nlinarith only [hga.1,hkb.1,hma,hmb]
-  refine ⟨hlo,hsum0,?_⟩
-  by_cases hab : a=b
-  · simp only [hab, sub_self, zero_pow (by decide : 2 ≠ 0), zero_div]
-    exact div_nonneg (sq_nonneg _) (by positivity)
-  have hd1 : 1 ≤ |(a:ℝ)-b| := by
-    rcases lt_or_gt_of_ne hab with hab | hab
-    · rw [abs_of_neg (sub_neg.mpr (by exact_mod_cast hab : (a:ℝ) < b))]
-      have hx : (a:ℝ)+1 ≤ b := by exact_mod_cast hab
-      linarith
-    · rw [abs_of_pos (sub_pos.mpr (by exact_mod_cast hab : (b:ℝ) < a))]
-      have hx : (b:ℝ)+1 ≤ a := by exact_mod_cast hab
-      linarith
-  have herr : |((g:ℝ)-k)-(m:ℝ)/2*((a:ℝ)-b)| ≤ 8 := by
-    calc
-      _ = |((g:ℝ)-(m:ℝ)*a/2)-((k:ℝ)-(m:ℝ)*b/2)| := by congr 1; ring
-      _  ≤  |(g:ℝ)-(m:ℝ)*a/2|+|(k:ℝ)-(m:ℝ)*b/2| := abs_sub _ _
-      _  ≤  _ := by linarith
-  have hd : (m:ℝ)/4*|(a:ℝ)-b|  ≤  |(g:ℝ)-k| := by
-    have ht := abs_sub_abs_le_abs_sub ((m:ℝ)/2*((a:ℝ)-b)) ((g:ℝ)-k)
-    rw [abs_sub_comm ((m:ℝ)/2*((a:ℝ)-b)) ((g:ℝ)-k)] at ht
-    rw [abs_mul, abs_of_nonneg (by positivity : 0 ≤ (m:ℝ)/2)] at ht
-    have hmabs : (32:ℝ) ≤ (m:ℝ)*|(a:ℝ)-b| := by nlinarith only [hmR,hd1]
-    nlinarith only [ht,herr,hmabs]
-  have hsq : (m:ℝ)^2*((a:ℝ)-b)^2/16 ≤ ((g:ℝ)-k)^2 := by
-    have hx := mul_self_le_mul_self (by positivity : 0 ≤ (m:ℝ)/4*|(a:ℝ)-b|) hd
-    nlinarith only [hx, sq_abs ((a:ℝ)-b), sq_abs ((g:ℝ)-k)]
-  have hsum : (g:ℝ)+k  ≤  4*(m:ℝ)^2 := by
-    have hu1 : (1:ℝ) ≤ U := by linarith
-    have hmu : (32:ℝ) ≤ (m:ℝ)*U := by nlinarith only [hmR,hu1]
-    have hmaU := mul_le_mul_of_nonneg_left haUR (show (0:ℝ) ≤ m by positivity)
-    have hmbU := mul_le_mul_of_nonneg_left hbUR (show (0:ℝ) ≤ m by positivity)
-    have hUmM := mul_le_mul_of_nonneg_left hUmR (show (0:ℝ) ≤ m by positivity)
-    nlinarith only [hga.2,hkb.2,hmaU,hmbU,hmu,hUmM]
-  apply (le_div_iff₀ (by positivity : 0<16*((g:ℝ)+k))).mpr
-  have hx := mul_le_mul_of_nonneg_right hsum (sq_nonneg ((a:ℝ)-b))
-  nlinarith only [hx,hsq]
-
-/-- From integer Gaussian decay to a summable geometric profile. -/
-theorem exp_integer_square_le (a b : ℕ) {c : ℝ} (hc : 0 ≤ c) :
-    Real.exp (-c * ((a:ℝ)-b)^2) ≤
-      (Real.exp (-c)) ^ ((a:ℤ)-b).natAbs := by
-  rw [← Real.exp_nat_mul]
-  apply Real.exp_le_exp.mpr
-  have hd : (((a:ℤ)-b).natAbs : ℝ) = |(a:ℝ)-b| := by
-    rw [← Int.cast_natCast, Int.natCast_natAbs, Int.cast_abs, Int.cast_sub]
-    simp only [Int.cast_natCast]
-  have hsq : (((a:ℤ)-b).natAbs:ℝ) ≤ (((a:ℤ)-b).natAbs:ℝ)^2 := by
-    exact_mod_cast (show ((a:ℤ)-b).natAbs ≤ ((a:ℤ)-b).natAbs^2 by
-      simpa only [pow_two] using Nat.le_mul_self ((a:ℤ)-b).natAbs)
-  rw [hd, sq_abs] at hsq
-  have hmul := mul_le_mul_of_nonneg_left hsq hc
-  rw [hd]
-  nlinarith only [hmul]
-
-theorem coarse_binomial_overlap (m U a b g k L : ℕ)
-    (hm : 32 ≤ m) (hUm : U ≤ 2*m)
-    (ha : 1 ≤ a) (hb : 1 ≤ b) (haU : a ≤ U) (hbU : b ≤ U)
-    (hg : |(g:ℝ)-(m:ℝ)*a/2| ≤ 4) (hk : |(k:ℝ)-(m:ℝ)*b/2| ≤ 4)
-    {θ : ℝ} (hθ : θ ∈ Set.Ico (1/2:ℝ) 1) :
-    (∑ z : Fin (L+1), binomialMass g θ z * binomialMass k θ z) ≤
-      Real.sqrt (8 / ((m:ℝ)*θ*(1-θ))) * (Real.sqrt (a:ℝ))⁻¹ *
-        (Real.exp (-(1/1024:ℝ))) ^ ((a:ℤ)-b).natAbs := by
-  have geo := coarse_gap_geometry m U a b g k hm hUm ha hb haU hbU hg hk
-  have hsum : 0 < g+k := by exact_mod_cast geo.2.1
-  have hθ0 : 0 < θ := by linarith [hθ.1]
-  have hθ1 : 0 < 1-θ := sub_pos.mpr hθ.2
-  have varpos : 0 < θ*(1-θ) := mul_pos hθ0 hθ1
-  have hm0 : (0:ℝ) < m := by exact_mod_cast (show 0 < m by omega)
-  have ha0 : (0:ℝ) < a := by exact_mod_cast ha
-  have hpref : Real.sqrt (2 / (((g:ℝ)+1)*θ*(1-θ))) ≤
-      Real.sqrt (8 / ((m:ℝ)*θ*(1-θ))) * (Real.sqrt (a:ℝ))⁻¹ := by
-    rw [← Real.sqrt_inv, ← Real.sqrt_mul (by positivity : 0 ≤ 8/((m:ℝ)*θ*(1-θ)))]
-    apply Real.sqrt_le_sqrt
-    have hdg : 0 < ((g:ℝ)+1)*θ*(1-θ) := by nlinarith
-    have hdm : 0 < (m:ℝ)*θ*(1-θ) := by nlinarith
-    apply (div_le_iff₀ hdg).mpr
-    have hdiv : 0 < (m:ℝ)*θ*(1-θ)*(a:ℝ) := by positivity
-    have heq : (8 / ((m:ℝ)*θ*(1-θ)) * (a:ℝ)⁻¹) * (((g:ℝ)+1)*θ*(1-θ)) =
-        (8*((g:ℝ)+1)) / ((m:ℝ)*a) := by
-      field_simp [ne_of_gt hθ0, ne_of_gt hθ1, ne_of_gt hm0, ne_of_gt ha0]
-    rw [heq]
-    apply (le_div_iff₀ (mul_pos hm0 ha0)).mpr
-    linarith [geo.1]
-  have hexp : Real.exp (-((g:ℝ)-k)^2/(16*((g:ℝ)+k))) ≤
-      (Real.exp (-(1/1024:ℝ))) ^ ((a:ℤ)-b).natAbs := by
-    apply le_trans _ (exp_integer_square_le a b (by norm_num : (0:ℝ)≤1/1024))
-    apply Real.exp_le_exp.mpr
-    convert neg_le_neg geo.2.2 using 2 <;> first | rfl | ring
-  exact (binomial_gaussian_overlap_general g k L hθ hsum).trans
-    (mul_le_mul hpref hexp (Real.exp_nonneg _) (by positivity))
-
-end TraceReconstruction
-
-
-namespace TraceReconstruction
-
-theorem int_natAbs_sub_comm (a b : ℤ) : (a-b).natAbs = (b-a).natAbs := by
-  rw [← Int.natAbs_neg (a-b),neg_sub]
-
-theorem subsetRankKernel_cross_normalized {n l r : ℕ} (hl : l  ≤  n)
-    (p q : Fin r → Fin n) (hp : StrictMono p) (hq : StrictMono q)
-    {θ : ℝ} (hθ : θ  ∈  Set.Ico (1/2:ℝ) 1)
-    (hmean : (n:ℝ)*θ=l) (hv : 1 ≤ (n:ℝ)*θ*(1-θ)) (e : Fin (r+1)) :
-    (∑ v : IncreasingTuple l r,
-      subsetRankKernel hl p v.val * subsetRankKernel hl q v.val)  ≤ 
-      (4608*n/((tupleGap p e:ℝ)+1))*
-        ∏ i : Fin r, ∑ k : Fin (l-r+1),
-          binomialMass (tupleGap p (Equiv.swap 0 e i.succ)) θ k *
-          binomialMass (tupleGap q (Equiv.swap 0 e i.succ)) θ k := by
-  have hθ0 : 0 < θ := by linarith [hθ.1]
-  have hqpos : 0 < 1-θ := sub_pos.mpr hθ.2
-  have hc : (θ^r/binomialMass n θ l)^2  ≤  144*((n:ℝ)*θ*(1-θ)) := by
-    rw [div_eq_mul_inv,mul_pow]
-    apply (mul_le_mul_of_nonneg_right
-      (show (θ^r)^2  ≤  (1:ℝ) by
-        exact pow_le_one₀ (pow_nonneg hθ0.le _) (pow_le_one₀ hθ0.le hθ.2.le))
-      (sq_nonneg _)).trans
-    simpa only [one_mul] using binomialMass_mean_inverse_sq n l ⟨hθ0,hθ.2⟩ hl hmean hv
-  have hprod : 0  ≤  ∏ i : Fin r, ∑ k : Fin (l-r+1),
-      binomialMass (tupleGap p (Equiv.swap 0 e i.succ)) θ k *
-      binomialMass (tupleGap q (Equiv.swap 0 e i.succ)) θ k := by
-    apply Finset.prod_nonneg
-    intro i _
-    exact Finset.sum_nonneg fun k _ => mul_nonneg
-      (binomialMass_nonneg _ _ ⟨hθ0.le,hθ.2.le⟩)
-      (binomialMass_nonneg _ _ ⟨hθ0.le,hθ.2.le⟩)
-  apply (subsetRankKernel_cross_le hl p q hp hq θ ⟨hθ0,hθ.2⟩ e
-    (32/(((tupleGap p e:ℝ)+1)*θ*(1-θ))) (by positivity)
-    (fun k => binomial_product_sup _ _ k hθ)).trans
-  calc
-    _  ≤  (144*((n:ℝ)*θ*(1-θ)))*(32/(((tupleGap p e:ℝ)+1)*θ*(1-θ)))*_ := by
-      exact mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hc (by positivity)) hprod
-    _ = _ := by
-      have hq0 : 1-θ ≠ 0 := ne_of_gt (sub_pos.mpr hθ.2)
-      field_simp
-      ring
-
-def periodProfile {u r : ℕ} (hr : 0 < r) (σ : Equiv.Perm (Fin (r+1)))
-    (j : IncreasingTuple u r) : Fin (r+1) → Fin (2*u+1) :=
-  fun i => ⟨coarsePeriodGap j.val (σ i),
-    Nat.lt_succ_of_le (coarsePeriodGap_le hr j.val j.property (σ i))⟩
-
-theorem periodProfile_sum {u r : ℕ} (hr : 0 < r) (σ : Equiv.Perm (Fin (r+1)))
-    (j : IncreasingTuple u r) : ∑ i, (periodProfile hr σ j i).val = 2*u := by
-  change (∑ i, coarsePeriodGap j.val (σ i)) = _
-  rw [Equiv.sum_comp σ]
-  exact coarsePeriodGap_sum hr j.val j.property
-
-theorem periodProfile_injective {u r : ℕ} (hr : 0 < r) (σ : Equiv.Perm (Fin (r+1))) :
-    Function.Injective (periodProfile (u:=u) hr σ) := by
-  intro j k h
-  apply Subtype.ext
-  apply coarsePeriodGap_injective j.property k.property
-  funext i
-  have he := congrArg Fin.val (congrFun h (σ.symm i))
-  simpa only [periodProfile,Equiv.apply_symm_apply] using he
-
-theorem period_kernel_cross {u m l r : ℕ} (hl : l ≤ u*m) (hr : 0 < r)
-    (hm : 32 ≤ m) (hum : u ≤ m) {θ : ℝ} (hθ : θ ∈ Set.Ico (1/2:ℝ) 1)
-    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
-    (off : Fin u → Fin m)
-    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
-    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2)
-    (j k : IncreasingTuple u r) (e : Fin (r+1))
-    (he : 2*u  ≤  (r+1)*coarsePeriodGap j.val e) :
-    (∑ v : IncreasingTuple l r,
-      subsetRankKernel hl (periodSlot off ∘ j.val) v.val *
-      subsetRankKernel hl (periodSlot off ∘ k.val) v.val)  ≤ 
-      (9216*(r+1:ℕ) : ℝ)* (Real.sqrt (8/((m:ℝ)*θ*(1-θ))))^r *
-        ∏ i : Fin r, (Real.sqrt ((periodProfile hr (Equiv.swap 0 e) j i.succ).val:ℝ))⁻¹ *
-          (Real.exp (-(1/1024:ℝ)))^(((periodProfile hr (Equiv.swap 0 e) k i.succ).val:ℤ)-
-            (periodProfile hr (Equiv.swap 0 e) j i.succ).val).natAbs := by
-  classical
-  have hθ0 : 0 < θ := by linarith [hθ.1]
-  have hm0 : (0:ℝ) < m := by exact_mod_cast (show 0 < m by omega)
-  have hUm : 2*u ≤ 2*m := Nat.mul_le_mul_left 2 hum
-  have hg (v : IncreasingTuple u r) (i : Fin (r+1)) :=
-    period_gap_rounding hr off hoff hdiff v.val v.property i
-  have hgeo := coarse_gap_geometry m (2*u) (coarsePeriodGap j.val e)
-    (coarsePeriodGap k.val e) (tupleGap (periodSlot off ∘ j.val) e)
-    (tupleGap (periodSlot off ∘ k.val) e) hm hUm
-    (coarsePeriodGap_pos j.val e) (coarsePeriodGap_pos k.val e)
-    (coarsePeriodGap_le hr j.val j.property e) (coarsePeriodGap_le hr k.val k.property e)
-    (hg j e) (hg k e)
-  have hlarge : (4608 * (u*m:ℕ) : ℝ)/((tupleGap (periodSlot off ∘ j.val) e:ℝ)+1)  ≤ 
-      9216*(r+1:ℕ) := by
-    have heR : 2*(u:ℝ)  ≤  (r+1:ℕ)* (coarsePeriodGap j.val e:ℝ) := by exact_mod_cast he
-    apply (div_le_iff₀ (by positivity : 0 < (tupleGap (periodSlot off ∘ j.val) e:ℝ)+1)).mpr
-    have hx := mul_le_mul_of_nonneg_right hgeo.1
-      (show (0:ℝ) ≤ (r+1:ℕ) by positivity)
-    have hy := mul_le_mul_of_nonneg_left heR hm0.le
-    simp only [Nat.cast_mul]
-    nlinarith only [hx,hy]
-  have hover (i : Fin r) :
-      (∑ z : Fin (l-r+1),
-        binomialMass (tupleGap (periodSlot off ∘ j.val) (Equiv.swap 0 e i.succ)) θ z *
-        binomialMass (tupleGap (periodSlot off ∘ k.val) (Equiv.swap 0 e i.succ)) θ z)  ≤ 
-      Real.sqrt (8/((m:ℝ)*θ*(1-θ))) *
-        ((Real.sqrt ((periodProfile hr (Equiv.swap 0 e) j i.succ).val:ℝ))⁻¹ *
-          (Real.exp (-(1/1024:ℝ)))^(((periodProfile hr (Equiv.swap 0 e) k i.succ).val:ℤ)-
-            (periodProfile hr (Equiv.swap 0 e) j i.succ).val).natAbs) := by
-    have ho := coarse_binomial_overlap m (2*u) _ _ _ _ (l-r) hm hUm
-      (coarsePeriodGap_pos j.val _) (coarsePeriodGap_pos k.val _)
-      (coarsePeriodGap_le hr j.val j.property _) (coarsePeriodGap_le hr k.val k.property _)
-      (hg j (Equiv.swap 0 e i.succ)) (hg k (Equiv.swap 0 e i.succ)) hθ
-    rw [int_natAbs_sub_comm] at ho
-    simpa only [periodProfile,mul_assoc] using ho
-  have hn (i : Fin r) : 0 ≤  ∑ z : Fin (l-r+1),
-      binomialMass (tupleGap (periodSlot off ∘ j.val) (Equiv.swap 0 e i.succ)) θ z *
-      binomialMass (tupleGap (periodSlot off ∘ k.val) (Equiv.swap 0 e i.succ)) θ z := by
-    exact Finset.sum_nonneg fun z _ => mul_nonneg
-      (binomialMass_nonneg _ _ ⟨hθ0.le,hθ.2.le⟩) (binomialMass_nonneg _ _ ⟨hθ0.le,hθ.2.le⟩)
-  apply (subsetRankKernel_cross_normalized hl _ _
-    ((periodSlot_strictMono off).comp j.property) ((periodSlot_strictMono off).comp k.property)
-    hθ hmean hv e).trans
-  calc
-    _  ≤  (9216*(r+1:ℕ) : ℝ)*
-        ∏ i : Fin r, (Real.sqrt (8/((m:ℝ)*θ*(1-θ))) *
-        ((Real.sqrt ((periodProfile hr (Equiv.swap 0 e) j i.succ).val:ℝ))⁻¹ *
-          (Real.exp (-(1/1024:ℝ)))^(((periodProfile hr (Equiv.swap 0 e) k i.succ).val:ℤ)-
-            (periodProfile hr (Equiv.swap 0 e) j i.succ).val).natAbs)) := by
-      apply mul_le_mul hlarge (Finset.prod_le_prod₀ (fun i _ => hn i) (fun i _ => hover i))
-        (Finset.prod_nonneg (fun i _ => hn i)) (by positivity)
-    _ = _ := by rw [Finset.prod_mul_distrib]; simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fin]; ring
-
-end TraceReconstruction
-
-
-namespace TraceReconstruction
-
-theorem coarse_profile_geometric_sum {α : Type*} [Fintype α] (r U : ℕ)
-    (profile : α → Fin (r+1) → Fin (U+1)) (hinj : Function.Injective profile)
-    (htotal : ∀ a, ∑ i, (profile a i).val=U)
-    (γ : Fin (r+1) → Fin (U+1)) {ρ : ℝ} (hρ : ρ∈Set.Ico 0 1) :
-    (∑ a, ∏ i : Fin r, ρ^(((profile a i.succ).val:ℤ)-(γ i.succ).val).natAbs) ≤
-      (2/(1-ρ))^r := by
-  classical
-  let f : Fin (r+1) → Fin (U+1) → ℝ := Fin.cases (fun _ => 1)
-    (fun i k => ρ^((k.val:ℤ)-(γ i.succ).val).natAbs)
-  have hf : ∀ i k, 0≤f i k := by
-    intro i k
-    refine Fin.cases ?_ (fun _ => ?_) i
-    · norm_num [f]
-    · exact pow_nonneg hρ.1 _
-  have hid (δ : Fin (r+1) → Fin (U+1)) :
-      (∏ i : Fin r, ρ^(((δ i.succ).val:ℤ)-(γ i.succ).val).natAbs) = ∏ i, f i (δ i) := by
-    simp only [Fin.prod_univ_succ,f,Fin.cases_zero,Fin.cases_succ,one_mul]
-  simp_rw [hid]
-  apply (sum_injective_profile_le profile hinj (fun δ => ∑ i, (δ i).val=U)
-    htotal (fun δ => ∏ i, f i (δ i)) (fun δ => Finset.prod_nonneg (fun i _ => hf i _))).trans
-  apply (constrained_sum_product_le r U f hf 1 (by norm_num) (by intro k; rfl)).trans
-  simp only [one_mul]
-  calc
-    _ ≤ ∏ _i : Fin r, (2/(1-ρ)) := by
-      apply Finset.prod_le_prod₀
-      · exact fun i _ => Finset.sum_nonneg (fun k _ => hf i.succ k)
-      · exact fun i _ => fin_geometric_distance (U+1) ((γ i.succ).val:ℤ) hρ
-    _ = _ := by simp [div_pow]
-
-theorem coarse_profile_inverse_sum {α : Type*} [Fintype α] (r U : ℕ)
-    (profile : α → Fin (r+1) → Fin (U+1)) (hinj : Function.Injective profile)
-    (htotal : ∀ a, ∑ i, (profile a i).val=U)
-    {h : ℝ} (hh : 0<h) (h1 : h≤1) :
-    (∑ a, ∏ i : Fin r, (Real.sqrt ((profile a i.succ).val:ℝ))⁻¹) ≤
-      Real.exp (h*U)*(6/Real.sqrt h)^r := by
-  classical
-  apply le_trans _ (coarse_constrained_inverse_sum r U hh h1)
-  exact sum_injective_profile_le profile hinj (fun δ => ∑ i, (δ i).val=U)
-    htotal (fun δ => ∏ i : Fin r, (Real.sqrt ((δ i.succ).val:ℝ))⁻¹) (fun δ => by positivity)
-
-/-- Coarse Schur bound for actual injective profiles, with no extension of
-kernels outside their geometric domain. -/
-theorem coarse_profile_bilinear_sum {α β : Type*} [Fintype α] [Fintype β]
-    (r U : ℕ) (p : α → Fin (r+1) → Fin (U+1)) (q : β → Fin (r+1) → Fin (U+1))
-    (hp : Function.Injective p) (hq : Function.Injective q)
-    (hpt : ∀ a, ∑ i, (p a i).val=U) (hqt : ∀ b, ∑ i, (q b i).val=U)
-    (K : α → β → ℝ) {M ρ h : ℝ}
-    (hM : 0≤M) (hρ : ρ∈Set.Ico 0 1) (hh : 0<h) (h1 : h≤1)
-    (hcross : ∀ a b, K a b ≤ M * ∏ i : Fin r,
-      (Real.sqrt ((p a i.succ).val:ℝ))⁻¹ *
-        ρ^(((q b i.succ).val:ℤ)-(p a i.succ).val).natAbs) :
-    (∑ a, ∑ b, K a b) ≤ M*(2/(1-ρ))^r * Real.exp (h*U)*(6/Real.sqrt h)^r := by
-  classical
-  let A (a : α) := ∏ i : Fin r, (Real.sqrt ((p a i.succ).val:ℝ))⁻¹
-  have hA (a) : 0≤A a := by dsimp [A]; positivity
-  have hinner (a) := coarse_profile_geometric_sum r U q hq hqt (p a) hρ
-  have houter := coarse_profile_inverse_sum r U p hp hpt hh h1
-  calc
-    _ ≤ ∑ a, ∑ b, M * ∏ i : Fin r,
-        (Real.sqrt ((p a i.succ).val:ℝ))⁻¹ *
-          ρ^(((q b i.succ).val:ℤ)-(p a i.succ).val).natAbs :=
-      Finset.sum_le_sum (fun a _ => Finset.sum_le_sum (fun b _ => hcross a b))
-    _ = ∑ a, M*A a * ∑ b, ∏ i : Fin r, ρ^(((q b i.succ).val:ℤ)-(p a i.succ).val).natAbs := by
-      simp only [Finset.prod_mul_distrib,A,Finset.mul_sum,mul_assoc]
-    _ ≤ ∑ a, M*A a*(2/(1-ρ))^r :=
-      Finset.sum_le_sum (fun a _ => mul_le_mul_of_nonneg_left (hinner a) (mul_nonneg hM (hA a)))
-    _ = M*(2/(1-ρ))^r * ∑ a, A a := by
-      rw [Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intros
-      ring
-    _ ≤ M*(2/(1-ρ))^r * (Real.exp (h*U)*(6/Real.sqrt h)^r) := by
-      apply mul_le_mul_of_nonneg_left houter
-      exact mul_nonneg hM (pow_nonneg (div_nonneg (by norm_num) (by linarith [hρ.2])) _)
-    _ = _ := by ring
-
-end TraceReconstruction
-
-namespace TraceReconstruction
-
-def LargePeriodTuple (u r : ℕ) (e : Fin (r+1)) :=
-  {j : IncreasingTuple u r // 2*u ≤ (r+1)*coarsePeriodGap j.val e}
-
-noncomputable instance largePeriodTupleFintype (u r : ℕ) (e : Fin (r+1)) :
-    Fintype (LargePeriodTuple u r e) := by
-  unfold LargePeriodTuple
-  exact Fintype.ofFinite _
-
-theorem exists_large_period_gap {u r : ℕ} (hr : 0 < r) (j : IncreasingTuple u r) :
-    ∃ e : Fin (r+1), 2*u ≤ (r+1)*coarsePeriodGap j.val e := by
-  obtain ⟨e,_,he⟩ := Finset.exists_max_image Finset.univ (coarsePeriodGap j.val)
-    Finset.univ_nonempty
-  refine ⟨e,?_⟩
-  calc
-    _ = ∑ i, coarsePeriodGap j.val i := (coarsePeriodGap_sum hr j.val j.property).symm
-    _ ≤ ∑ _i : Fin (r+1), coarsePeriodGap j.val e := Finset.sum_le_sum (fun i hi => he i hi)
-    _ = _ := by simp
-
-/-- Unnormalized but explicit source rank-kernel square-sum estimate. The
-only remaining factors are elementary scale simplifications. -/
-theorem period_rank_square_sum_raw {u m l r : ℕ} (hl : l ≤ u*m) (hr : 0 < r)
-    (hm : 32 ≤ m) (hum : u ≤ m) {θ h : ℝ} (hθ : θ ∈ Set.Ico (1/2:ℝ) 1)
-    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
-    (hh : 0 < h) (h1 : h ≤ 1)
-    (off : Fin u → Fin m)
-    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
-    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2) :
-    (∑ v : IncreasingTuple l r,
-      (∑ j : IncreasingTuple u r, subsetRankKernel hl (periodSlot off ∘ j.val) v.val)^2) ≤
-      9216*(r+1:ℕ)^2 * (Real.sqrt (8/((m:ℝ)*θ*(1-θ))))^r *
-        (2/(1-Real.exp (-(1/1024:ℝ))))^r * Real.exp (h*(2*u:ℕ)) * (6/Real.sqrt h)^r := by
-  classical
-  let K (j : IncreasingTuple u r) (v : IncreasingTuple l r) :=
-    subsetRankKernel hl (periodSlot off ∘ j.val) v.val
-  let C (j k : IncreasingTuple u r) := ∑ v : IncreasingTuple l r, K j v*K k v
-  have hC (j k) : 0 ≤ C j k := by
-    exact Finset.sum_nonneg (fun v _ => mul_nonneg
-      (subsetRankKernel_nonneg hl _ _) (subsetRankKernel_nonneg hl _ _))
-  have hρ : Real.exp (-(1/1024:ℝ)) ∈ Set.Ico 0 1 := by
-    refine ⟨(Real.exp_pos _).le,?_⟩
-    rw [Real.exp_lt_one_iff]
-    norm_num
-  have hschur (e : Fin (r+1)) :
-      (∑ j : LargePeriodTuple u r e, ∑ k : IncreasingTuple u r, C j.val k) ≤
-        ((9216*(r+1:ℕ):ℝ)*(Real.sqrt (8/((m:ℝ)*θ*(1-θ))))^r) *
-          (2/(1-Real.exp (-(1/1024:ℝ))))^r * Real.exp (h*(2*u:ℕ)) * (6/Real.sqrt h)^r := by
-    apply coarse_profile_bilinear_sum r (2*u)
-      (fun j : LargePeriodTuple u r e => periodProfile hr (Equiv.swap 0 e) j.val)
-      (periodProfile hr (Equiv.swap 0 e))
-      ((periodProfile_injective hr (Equiv.swap 0 e)).comp Subtype.val_injective)
-      (periodProfile_injective hr (Equiv.swap 0 e))
-      (fun j => periodProfile_sum hr _ j.val) (periodProfile_sum hr _)
-      (fun j k => C j.val k) (by positivity) hρ hh h1
-    intro j k
-    exact period_kernel_cross hl hr hm hum hθ hmean hv off hoff hdiff j.val k e j.property
-  have hcover (j : IncreasingTuple u r) :
-      (∑ k : IncreasingTuple u r, C j k) ≤ ∑ e : Fin (r+1),
-        if 2*u ≤ (r+1)*coarsePeriodGap j.val e then ∑ k : IncreasingTuple u r, C j k else 0 := by
-    obtain ⟨e,he⟩ := exists_large_period_gap hr j
-    have hn : ∀ e' : Fin (r+1), 0 ≤
-        if 2*u ≤ (r+1)*coarsePeriodGap j.val e' then ∑ k : IncreasingTuple u r, C j k else 0 := by
-      intro e'
-      split_ifs
-      · exact Finset.sum_nonneg (fun k _ => hC j k)
-      · rfl
-    simpa only [ite_eq_left he] using Finset.single_le_sum (fun e' _ => hn e') (Finset.mem_univ e)
-  have heq (e : Fin (r+1)) :
-      (∑ j : IncreasingTuple u r,
-        if 2*u ≤ (r+1)*coarsePeriodGap j.val e then ∑ k : IncreasingTuple u r, C j k else 0) =
-      ∑ j : LargePeriodTuple u r e, ∑ k : IncreasingTuple u r, C j.val k := by
-    rw [← Finset.sum_filter]
-    exact Finset.sum_subtype _ (by intro j; simp) _
-  calc
-    _ = ∑ j : IncreasingTuple u r, ∑ k : IncreasingTuple u r, C j k := by
-      exact sum_sq_sum_kernel Finset.univ K
-    _ ≤ ∑ j : IncreasingTuple u r, ∑ e : Fin (r+1),
-        if 2*u ≤ (r+1)*coarsePeriodGap j.val e then ∑ k : IncreasingTuple u r, C j k else 0 :=
-      Finset.sum_le_sum (fun j _ => hcover j)
-    _ = ∑ e : Fin (r+1), ∑ j : LargePeriodTuple u r e,
-        ∑ k : IncreasingTuple u r, C j.val k := by
-      rw [Finset.sum_comm]
-      exact Finset.sum_congr rfl (fun e _ => heq e)
-    _ ≤ ∑ _e : Fin (r+1),
-        ((9216*(r+1:ℕ):ℝ)*(Real.sqrt (8/((m:ℝ)*θ*(1-θ))))^r) *
-          (2/(1-Real.exp (-(1/1024:ℝ))))^r * Real.exp (h*(2*u:ℕ)) * (6/Real.sqrt h)^r :=
-      Finset.sum_le_sum (fun e _ => hschur e)
-    _ = _ := by simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]; ring
-
-end TraceReconstruction
-
-namespace TraceReconstruction
-
-theorem rank_polynomial_factor {r : ℕ} (hr : 0 < r) :
-    (9216:ℝ)*(r+1:ℕ)^2 ≤ (36864:ℝ)^r := by
-  have h1 : (9216:ℝ) ≤ (9216:ℝ)^r := le_self_pow₀ (by norm_num) (Nat.ne_of_gt hr)
-  have h2 : ((r+1:ℕ):ℝ) ≤ (2:ℝ)^r := by exact_mod_cast nat_add_one_le_two_pow r
-  have h3 := pow_le_pow_left₀ (by positivity : (0:ℝ) ≤ (r+1:ℕ)) h2 2
-  have h4 : ((2:ℝ)^r)^2 = (4:ℝ)^r := by
-    rw [← pow_mul,Nat.mul_comm r 2,pow_mul]
-    norm_num
-  calc
-    _ ≤ (9216:ℝ)^r*((2:ℝ)^r)^2 := mul_le_mul h1 h3 (by positivity) (by positivity)
-    _ = _ := by rw [h4,← mul_pow]; norm_num
-
-noncomputable def rankSchurConstant : ℝ :=
-  884736 * (2/(1-Real.exp (-(1/1024:ℝ)))) * Real.exp 1
-
-theorem rankSchurConstant_pos : 0 < rankSchurConstant := by
-  have hρ : Real.exp (-(1/1024:ℝ)) < 1 := by rw [Real.exp_lt_one_iff]; norm_num
-  unfold rankSchurConstant
-  positivity
-
-/-- Rank-kernel square sum, with an explicit absolute constant, uniformly
-in the defect types and admissible conditional count. -/
-theorem period_rank_square_sum {u m l r : ℕ} (hl : l ≤ u*m) (hr : 0 < r)
-    (hru : r ≤ u) (hm : 32 ≤ m) (hum : u ≤ m) {θ : ℝ} (hθ : θ ∈ Set.Ico (1/2:ℝ) 1)
-    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
-    (off : Fin u → Fin m)
-    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
-    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2) :
-    (∑ v : IncreasingTuple l r,
-      (∑ j : IncreasingTuple u r, subsetRankKernel hl (periodSlot off ∘ j.val) v.val)^2) ≤
-      (rankSchurConstant * Real.sqrt ((u:ℝ)/((m:ℝ)*r*θ*(1-θ))))^r := by
-  have hr0 : (0:ℝ) < r := by exact_mod_cast hr
-  have hu0 : (0:ℝ) < u := by exact_mod_cast (show 0 < u by omega)
-  have hm0 : (0:ℝ) < m := by exact_mod_cast (show 0 < m by omega)
-  have hθ0 : 0 < θ := by linarith [hθ.1]
-  have hq0 : 0 < 1-θ := sub_pos.mpr hθ.2
-  let h : ℝ := (r:ℝ)/(2*u)
-  have hh : 0 < h := div_pos hr0 (by positivity)
-  have h1 : h ≤ 1 := by
-    apply (div_le_one (by positivity : (0:ℝ) < 2*u)).mpr
-    have hruR : (r:ℝ) ≤ u := by exact_mod_cast hru
-    linarith
-  have hx : h*(2*u:ℕ) = r := by
-    dsimp [h]
-    simp only [Nat.cast_mul,Nat.cast_ofNat]
-    field_simp
-  have he : Real.exp (h*(2*u:ℕ)) = (Real.exp 1)^r := by
-    rw [hx,← Real.exp_nat_mul]
-    simp
-  have hρ : 0 ≤ 2/(1-Real.exp (-(1/1024:ℝ))) := by
-    have hn : Real.exp (-(1/1024:ℝ)) < 1 := by rw [Real.exp_lt_one_iff]; norm_num
-    positivity
-  have hp : 0 ≤ (u:ℝ)/((m:ℝ)*r*θ*(1-θ)) := by positivity
-  have hs : Real.sqrt (8/((m:ℝ)*θ*(1-θ))) * (6/Real.sqrt h) ≤
-      24 * Real.sqrt ((u:ℝ)/((m:ℝ)*r*θ*(1-θ))) := by
-    have hsq : (Real.sqrt (8/((m:ℝ)*θ*(1-θ))) * (6/Real.sqrt h))^2 =
-        576 * ((u:ℝ)/((m:ℝ)*r*θ*(1-θ))) := by
-      rw [mul_pow,div_pow,Real.sq_sqrt (by positivity : (0:ℝ) ≤ 8/((m:ℝ)*θ*(1-θ))),
-        Real.sq_sqrt hh.le]
-      dsimp [h]
-      field_simp
-      ring
-    nlinarith [Real.sq_sqrt hp, Real.sqrt_nonneg ((u:ℝ)/((m:ℝ)*r*θ*(1-θ)))]
-  apply (period_rank_square_sum_raw hl hr hm hum hθ hmean hv hh h1 off hoff hdiff).trans
-  rw [he]
-  calc
-    _ ≤ (36864:ℝ)^r * (Real.sqrt (8/((m:ℝ)*θ*(1-θ))))^r *
-        (2/(1-Real.exp (-(1/1024:ℝ))))^r * (Real.exp 1)^r * (6/Real.sqrt h)^r := by
-      gcongr
-      exact rank_polynomial_factor hr
-    _ = ((36864:ℝ)*(2/(1-Real.exp (-(1/1024:ℝ))))*Real.exp 1 *
-        (Real.sqrt (8/((m:ℝ)*θ*(1-θ)))*(6/Real.sqrt h)))^r := by
-      simp only [mul_pow]
-      ring
-    _ ≤ ((36864:ℝ)*(2/(1-Real.exp (-(1/1024:ℝ))))*Real.exp 1 *
-        (24 * Real.sqrt ((u:ℝ)/((m:ℝ)*r*θ*(1-θ)))))^r := by
-      apply pow_le_pow_left₀ (by positivity)
-      exact mul_le_mul_of_nonneg_left hs (by positivity)
-    _ = _ := by congr 1; unfold rankSchurConstant; ring
-
-end TraceReconstruction
-
-namespace TraceReconstruction
-
-/-- The source's channel-uniform rank-kernel estimate (base-kernel),
-with the absolute constant made explicit. -/
-theorem period_rank_square_sum_delta {u m l r : ℕ} (hl : l ≤ u*m) (hr : 0 < r)
-    (hru : r ≤ u) (hm : 32 ≤ m) (hum : u ≤ m) {δ θ : ℝ}
-    (hδ : δ ∈ Set.Ioc 0 (1/2:ℝ))
-    (hθ : θ ∈ Set.Icc (1-δ^2) (1-δ^2/4))
-    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
-    (off : Fin u → Fin m)
-    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
-    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2) :
-    (∑ v : IncreasingTuple l r,
-      (∑ j : IncreasingTuple u r, subsetRankKernel hl (periodSlot off ∘ j.val) v.val)^2) ≤
-      ((3*rankSchurConstant/δ) * Real.sqrt ((u:ℝ)/((m:ℝ)*r)))^r := by
-  have hd0 : 0 < δ := hδ.1
-  have hd2 : 0 < δ^2 := sq_pos_of_pos hδ.1
-  have hd2b : δ^2 ≤ 1/4 := by nlinarith [hδ.1,hδ.2]
-  have ht : θ ∈ Set.Ico (1/2:ℝ) 1 := by
-    constructor <;> nlinarith [hθ.1,hθ.2]
-  have ht0 : 0 < θ := by linarith [ht.1]
-  have hq0 : 0 < 1-θ := sub_pos.mpr ht.2
-  have hr0 : (0:ℝ) < r := by exact_mod_cast hr
-  have hm0 : (0:ℝ) < m := by exact_mod_cast (show 0 < m by omega)
-  have hu0 : (0:ℝ) ≤ u := Nat.cast_nonneg u
-  have hvar : δ^2/8 ≤ θ*(1-θ) := by
-    have hq : δ^2/4 ≤ 1-θ := by linarith [hθ.2]
-    nlinarith [mul_le_mul ht.1 hq (by positivity) ht0.le]
-  have hdiv : (u:ℝ)/((m:ℝ)*r*θ*(1-θ)) ≤
-      (3/δ)^2*((u:ℝ)/((m:ℝ)*r)) := by
-    have hden : (0:ℝ) < (m:ℝ)*r*θ*(1-θ) := by positivity
-    apply (div_le_iff₀ hden).mpr
-    have he : (3/δ)^2 * ((u:ℝ)/((m:ℝ)*r)) * ((m:ℝ)*r*θ*(1-θ)) =
-        ((u:ℝ)*9*θ*(1-θ))/δ^2 := by field_simp; ring
-    rw [he]
-    apply (le_div_iff₀ hd2).mpr
-    nlinarith [mul_le_mul_of_nonneg_left hvar (by positivity : (0:ℝ) ≤ 9*(u:ℝ))]
-  have hs : Real.sqrt ((u:ℝ)/((m:ℝ)*r*θ*(1-θ))) ≤
-      (3/δ)*Real.sqrt ((u:ℝ)/((m:ℝ)*r)) := by
-    calc
-      _ ≤ Real.sqrt ((3/δ)^2*((u:ℝ)/((m:ℝ)*r))) := Real.sqrt_le_sqrt hdiv
-      _ = _ := by
-        rw [Real.sqrt_mul (sq_nonneg _),Real.sqrt_sq (by positivity)]
-  apply (period_rank_square_sum hl hr hru hm hum ht hmean hv off hoff hdiff).trans
-  apply pow_le_pow_left₀ (by positivity [rankSchurConstant_pos])
-  calc
-    _ ≤ rankSchurConstant*((3/δ)*Real.sqrt ((u:ℝ)/((m:ℝ)*r))) :=
-      mul_le_mul_of_nonneg_left hs rankSchurConstant_pos.le
-    _ = _ := by ring
-
-end TraceReconstruction
-
-end
-
-end OAI
+import OAI.Probability.TraceReconstruction.FiniteLaw
+import OAI.Probability.TraceReconstruction.Kernels
+import OAI.Probability.TraceReconstruction.Scales
+
+namespace OAI
+
+/-!
+# Ranks
+
+Retained subsets, conditional rank experiments, tagged gaps, period geometry, Gaussian Schur estimates, and uniform square sums.
+-/
+
+open scoped ENNReal
+noncomputable section
+
+open Filter
+open scoped Topology
+
+
+namespace TraceReconstruction
+
+variable {α : Type*} [Fintype α]
+
+def subsetWeight (p : ℝ) (S : Finset α) : ℝ :=
+  p^S.card * (1-p)^(Fintype.card α-S.card)
+
+theorem sum_subsetWeight (p : ℝ) : (∑ S : Finset α, subsetWeight p S)=1 := by
+  classical
+  have h := Finset.sum_pow_mul_eq_add_pow p (1-p) (Finset.univ : Finset α)
+  simpa [subsetWeight] using h
+
+theorem sum_fixedCard_subsetWeight (p : ℝ) (l : ℕ) :
+    (∑ S : Finset α, if S.card=l then subsetWeight p S else 0) =
+      binomialMass (Fintype.card α) p l := by
+  classical
+  rw [← Finset.sum_filter]
+  have hset : (Finset.univ.filter (fun S : Finset α => S.card=l)) =
+      (Finset.univ : Finset α).powersetCard l := by ext; simp
+  rw [hset]
+  have he : ∀ S ∈ (Finset.univ : Finset α).powersetCard l,
+      subsetWeight p S=p^l*(1-p)^(Fintype.card α-l) := by
+    classical
+    intro S hS
+    simp only [subsetWeight, (Finset.mem_powersetCard.mp hS).2]
+  rw [Finset.sum_congr rfl he]
+  simp only [Finset.sum_const, Finset.card_powersetCard, Finset.card_univ, nsmul_eq_mul,
+    binomialMass]
+  ring
+
+def subsetLaw (p : ℝ) (hp : p ∈ Set.Icc 0 1) : FiniteLaw (Finset α) where
+  mass := subsetWeight p
+  nonneg S := by unfold subsetWeight; positivity [hp.1, sub_nonneg.mpr hp.2]
+  total := sum_subsetWeight p
+
+abbrev FixedSubset (α : Type*) [Fintype α] (l : ℕ) := {S : Finset α // S.card=l}
+
+theorem card_fixedSubset (l : ℕ) :
+    Fintype.card (FixedSubset α l)=(Fintype.card α).choose l := by
+  classical
+  rw [Fintype.card_subtype]
+  have hset : (Finset.univ.filter (fun S : Finset α => S.card=l)) =
+      (Finset.univ : Finset α).powersetCard l := by ext; simp
+  rw [hset, Finset.card_powersetCard, Finset.card_univ]
+
+def uniformSubsetLaw (l : ℕ) (hl : l ≤ Fintype.card α) : FiniteLaw (FixedSubset α l) where
+  mass _ := 1 / ((Fintype.card α).choose l : ℝ)
+  nonneg _ := by positivity
+  total := by
+    classical
+    simp only [Finset.sum_const, Finset.card_univ, card_fixedSubset, nsmul_eq_mul]
+    exact mul_one_div_cancel (by exact_mod_cast (Nat.choose_pos hl).ne')
+
+/-- Conditional Bernoulli sampling, at every interior parameter, is exactly uniform. -/
+theorem conditional_subset_uniform {p : ℝ} (hp : p ∈ Set.Ioo 0 1)
+    {l : ℕ} (hl : l ≤ Fintype.card α) (S : FixedSubset α l) :
+    subsetWeight p S.val / binomialMass (Fintype.card α) p l =
+      (uniformSubsetLaw l hl).mass S := by
+  classical
+  have hp0 : p≠0 := hp.1.ne'
+  have hq0 : 1-p≠0 := by linarith [hp.2]
+  have hc : (((Fintype.card α).choose l):ℝ)≠0 := by exact_mod_cast (Nat.choose_pos hl).ne'
+  simp only [subsetWeight, S.property, binomialMass, uniformSubsetLaw]
+  field_simp
+
+/-- Slot at an observed return rank, with the original total order. -/
+def slotAtRank {n l : ℕ} (S : FixedSubset (Fin n) l) (v : Fin l) : Fin n :=
+  (S.val.orderIsoOfFin S.property v).val
+
+theorem slotAtRank_mem {n l : ℕ} (S : FixedSubset (Fin n) l) (v : Fin l) :
+    slotAtRank S v ∈ S.val := (S.val.orderIsoOfFin S.property v).property
+
+theorem slotAtRank_strictMono {n l : ℕ} (S : FixedSubset (Fin n) l) :
+    StrictMono (slotAtRank S) := by
+  classical
+  intro a b hab
+  exact (S.val.orderIsoOfFin S.property).strictMono hab
+
+def defectAtRank {n l : ℕ} (D : Finset (Fin n)) (S : FixedSubset (Fin n) l)
+    (v : Fin l) : Bool := decide (slotAtRank S v ∈ D)
+
+def taggedRankEvent {n l r : ℕ} (p : Fin r → Fin n) (v : Fin r → Fin l)
+    (S : FixedSubset (Fin n) l) : Prop := ∀ i, slotAtRank S (v i)=p i
+
+/-- Unhypothesized rank kernel: defined from the actual uniformly retained subset. -/
+def subsetRankKernel {n l r : ℕ} (hl : l≤n) (p : Fin r → Fin n) (v : Fin r → Fin l) : ℝ := by
+  classical
+  exact ∑ S : FixedSubset (Fin n) l, if taggedRankEvent p v S then
+    (uniformSubsetLaw l (by simpa using hl)).mass S else 0
+
+theorem subsetRankKernel_nonneg {n l r : ℕ} (hl : l≤n)
+    (p : Fin r → Fin n) (v : Fin r → Fin l) : 0 ≤ subsetRankKernel hl p v := by
+  classical
+  unfold subsetRankKernel
+  apply Finset.sum_nonneg
+  intro S _
+  split_ifs
+  · exact (uniformSubsetLaw l (by simpa using hl)).nonneg S
+  · rfl
+
+theorem subsetRankKernel_le_one {n l r : ℕ} (hl : l≤n)
+    (p : Fin r → Fin n) (v : Fin r → Fin l) : subsetRankKernel hl p v ≤ 1 := by
+  classical
+  unfold subsetRankKernel
+  rw [← (uniformSubsetLaw (α := Fin n) l (by simpa using hl)).total]
+  apply Finset.sum_le_sum
+  intro S _
+  split_ifs
+  · rfl
+  · exact (uniformSubsetLaw l (by simpa using hl)).nonneg S
+
+
+/-- Canonical slot universe split at the tagged positions. Actual order is only
+needed to identify each component with its before/between/after interval. -/
+abbrev GapUniverse (r d : ℕ) (g : Fin d → ℕ) := Fin r ⊕ ((i : Fin d) × Fin (g i))
+
+def gapDecode {r d : ℕ} {g : Fin d → ℕ} (S : Finset (GapUniverse r d g))
+    (i : Fin d) : Finset (Fin (g i)) :=
+  Finset.univ.filter fun a => Sum.inr ⟨i,a⟩ ∈ S
+
+def gapEncode {r d : ℕ} {g : Fin d → ℕ} (A : (i : Fin d) → Finset (Fin (g i))) :
+    Finset (GapUniverse r d g) := by
+  classical
+  exact Finset.univ.filter fun x => match x with
+    | Sum.inl _ => True
+    | Sum.inr a => a.2 ∈ A a.1
+
+@[simp] theorem mem_gapEncode_left {r d : ℕ} {g : Fin d → ℕ}
+    (A : (i : Fin d) → Finset (Fin (g i))) (a : Fin r) :
+    Sum.inl a ∈ gapEncode A := by simp [gapEncode]
+
+@[simp] theorem mem_gapEncode_right {r d : ℕ} {g : Fin d → ℕ}
+    (A : (i : Fin d) → Finset (Fin (g i))) (i : Fin d) (a : Fin (g i)) :
+    Sum.inr ⟨i,a⟩ ∈ gapEncode (r := r) A ↔ a ∈ A i := by simp [gapEncode]
+
+@[simp] theorem gapDecode_encode {r d : ℕ} {g : Fin d → ℕ}
+    (A : (i : Fin d) → Finset (Fin (g i))) (i : Fin d) :
+    gapDecode (gapEncode (r := r) A) i=A i := by ext; simp [gapDecode]
+
+theorem gapEncode_decode {r d : ℕ} {g : Fin d → ℕ}
+    (S : Finset (GapUniverse r d g)) (hS : ∀ a, Sum.inl a∈S) :
+    gapEncode (gapDecode S)=S := by
+  ext x
+  cases x with
+  | inl a => simp [hS]
+  | inr a => rcases a with ⟨i,a⟩; simp [gapDecode]
+
+def gapEvent {r d : ℕ} {g : Fin d → ℕ} (z : Fin d → ℕ)
+    (S : Finset (GapUniverse r d g)) : Prop :=
+  (∀ a, Sum.inl a∈S) ∧ ∀ i, (gapDecode S i).card=z i
+
+instance gapEvent_decidable {r d : ℕ} {g : Fin d → ℕ} (z : Fin d → ℕ)
+    (S : Finset (GapUniverse r d g)) : Decidable (gapEvent z S) := by
+  unfold gapEvent
+  infer_instance
+
+def gapEventEquiv {r d : ℕ} {g z : Fin d → ℕ} :
+    {S : Finset (GapUniverse r d g) // gapEvent z S} ≃
+      ((i : Fin d) → FixedSubset (Fin (g i)) (z i)) where
+  toFun S i := ⟨gapDecode S.val i, S.property.2 i⟩
+  invFun A := ⟨gapEncode (fun i => (A i).val), by
+    constructor
+    · intro a; simp
+    · intro i; simp only [gapDecode_encode, (A i).property]⟩
+  left_inv S := by
+    apply Subtype.ext
+    exact gapEncode_decode S.val S.property.1
+  right_inv A := by
+    funext i
+    apply Subtype.ext
+    change gapDecode (gapEncode (fun i => (A i).val)) i = (A i).val
+    exact gapDecode_encode _ _
+
+theorem card_gapEvent {r d : ℕ} (g z : Fin d → ℕ) :
+    Fintype.card {S : Finset (GapUniverse r d g) // gapEvent z S} =
+      ∏ i, (g i).choose (z i) := by
+  classical
+  rw [Fintype.card_congr gapEventEquiv, Fintype.card_pi]
+  simp only [card_fixedSubset, Fintype.card_fin]
+
+theorem card_gapEncode {r d : ℕ} {g : Fin d → ℕ}
+    (A : (i : Fin d) → Finset (Fin (g i))) :
+    (gapEncode (r := r) A).card = r + ∑ i, (A i).card := by
+  classical
+  rw [Finset.card_eq_sum_ones]
+  simp only [gapEncode, Finset.sum_filter]
+  rw [Fintype.sum_sum_type, Fintype.sum_sigma]
+  simp
+  apply Finset.sum_congr rfl
+  intro i _
+  calc
+    _ = ∑ x : Fin (g i), if x ∈ A i then (1:ℕ) else 0 := by
+      apply Finset.sum_congr rfl
+      intro x _
+      split_ifs <;> rfl
+    _ = _ := (Finset.card_eq_sum_ite (Finset.subset_univ (A i))).symm
+
+theorem card_of_gapEvent {r d : ℕ} {g z : Fin d → ℕ}
+    (S : Finset (GapUniverse r d g)) (hS : gapEvent z S) :
+    S.card = r+∑ i, z i := by
+  rw [← gapEncode_decode S hS.1, card_gapEncode]
+  simp only [hS.2]
+
+
+/-- The exact unconditioned tagged-gap probability; no asymptotic estimate is used. -/
+theorem gapEvent_bernoulli_probability {r d : ℕ} (g z : Fin d → ℕ)
+    (hz : ∀ i, z i≤g i) (p : ℝ) :
+    (∑ S : Finset (GapUniverse r d g), if gapEvent z S then subsetWeight p S else 0) =
+      p^r * ∏ i, binomialMass (g i) p (z i) := by
+  classical
+  have hcount : (∑ S : Finset (GapUniverse r d g),
+      if gapEvent z S then (1:ℝ) else 0) = ∏ i, ((g i).choose (z i):ℝ) := by
+    rw [Finset.sum_boole, ← Fintype.card_subtype, card_gapEvent, Nat.cast_prod]
+  have hnum : (∑ S : Finset (GapUniverse r d g), if gapEvent z S then subsetWeight p S else 0) =
+      (∏ i, ((g i).choose (z i):ℝ)) *
+        (p^(r+∑ i, z i)*(1-p)^(r+∑ i, g i-(r+∑ i, z i))) := by
+    rw [← hcount, Finset.sum_mul]
+    apply Finset.sum_congr rfl
+    intro S _
+    split_ifs with hS
+    · simp only [one_mul, subsetWeight, card_of_gapEvent S hS,
+        Fintype.card_sum, Fintype.card_fin, Fintype.card_sigma]
+    · exact (zero_mul _).symm
+  have hsub : (∑ i, g i)-(∑ i, z i)=∑ i, (g i-z i) := by
+    exact (Finset.sum_tsub_distrib Finset.univ (fun i _ => hz i)).symm
+  rw [hnum, Nat.add_sub_add_left, hsub, pow_add]
+  simp only [binomialMass, Finset.prod_mul_distrib, Finset.prod_pow_eq_pow_sum]
+  ring
+
+/-- Conditional Bernoulli representation of the source rank kernel in gap coordinates. -/
+theorem gapEvent_conditional_formula {r d : ℕ} (g z : Fin d → ℕ)
+    (hz : ∀ i, z i≤g i) (p : ℝ) :
+    (∑ S : Finset (GapUniverse r d g), if gapEvent z S then subsetWeight p S else 0) /
+      binomialMass (r+∑ i, g i) p (r+∑ i, z i) =
+    p^r / binomialMass (r+∑ i, g i) p (r+∑ i, z i) *
+      ∏ i, binomialMass (g i) p (z i) := by
+  rw [gapEvent_bernoulli_probability g z hz]
+  ring
+
+end TraceReconstruction
+
+
+namespace TraceReconstruction
+
+theorem slotAtRank_lower_card {n l : ℕ} (S : FixedSubset (Fin n) l) (v : Fin l) :
+    (S.val.filter fun j => j < slotAtRank S v).card = v.val := by
+  classical
+  let e := S.val.orderEmbOfFin S.property
+  have hslot (i:Fin l) : e i = slotAtRank S i := rfl
+  have hmap : (Finset.Iio v).map e.toEmbedding = S.val.filter (fun j => j<e v) := by
+    ext j
+    simp only [Finset.mem_map, Finset.mem_Iio, Finset.mem_filter, RelEmbedding.coe_toEmbedding]
+    constructor
+    · rintro ⟨i, hi, rfl⟩
+      exact ⟨S.val.orderEmbOfFin_mem S.property i, e.strictMono hi⟩
+    · rintro ⟨hj,hv⟩
+      obtain ⟨i,hi⟩ := (S.val.orderIsoOfFin S.property).surjective ⟨j,hj⟩
+      have hei : e i=j := congrArg Subtype.val hi
+      refine ⟨i, ?_, hei⟩
+      exact e.lt_iff_lt.mp (hei.symm ▸ hv)
+  rw [← hslot, ← hmap, Finset.card_map, Fin.card_Iio]
+
+theorem slotAtRank_eq_iff {n l : ℕ} (S : FixedSubset (Fin n) l) (v : Fin l) (p : Fin n) :
+    slotAtRank S v=p ↔ p∈S.val ∧ (S.val.filter fun j => j<p).card=v.val := by
+  classical
+  constructor
+  · intro h
+    rw [← h]
+    exact ⟨slotAtRank_mem S v, slotAtRank_lower_card S v⟩
+  · rintro ⟨hp,hcard⟩
+    obtain ⟨i,hi⟩ := (S.val.orderIsoOfFin S.property).surjective ⟨p,hp⟩
+    have he : slotAtRank S i=p := congrArg Subtype.val hi
+    have hcv := slotAtRank_lower_card S i
+    rw [he,hcard] at hcv
+    have hiv : i=v := Fin.ext hcv.symm
+    simpa [hiv] using he
+
+/-- Bounds for the open gaps of an increasing tuple; slots and ranks are zero-based. -/
+def tupleLower {n r : ℕ} (p : Fin r → Fin n) : Fin (r+1) → ℕ :=
+  Fin.cons 0 (fun i => (p i).val + 1)
+
+def tupleUpper {n r : ℕ} (p : Fin r → Fin n) : Fin (r+1) → ℕ :=
+  Fin.snoc (fun i => (p i).val) n
+
+def tupleGap {n r : ℕ} (p : Fin r → Fin n) (j : Fin (r+1)) : ℕ :=
+  tupleUpper p j - tupleLower p j
+
+@[simp] theorem tupleLower_zero {n r : ℕ} (p : Fin r → Fin n) : tupleLower p 0 = 0 :=
+  Fin.cons_zero _ _
+@[simp] theorem tupleLower_succ {n r : ℕ} (p : Fin r → Fin n) (i : Fin r) :
+    tupleLower p i.succ = (p i).val + 1 := Fin.cons_succ _ _ _
+@[simp] theorem tupleUpper_castSucc {n r : ℕ} (p : Fin r → Fin n) (i : Fin r) :
+    tupleUpper p i.castSucc = (p i).val := Fin.snoc_castSucc _ _ _
+@[simp] theorem tupleUpper_last {n r : ℕ} (p : Fin r → Fin n) :
+    tupleUpper p (Fin.last r) = n := Fin.snoc_last _ _
+
+theorem tupleLower_le_tag {n r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p)
+    (i : Fin r) (j : Fin (r+1)) (hji : j ≤ i.castSucc) :
+    tupleLower p j ≤ (p i).val := by
+  induction j using Fin.cases with
+  | zero => simp
+  | succ a =>
+    simp only [tupleLower_succ]
+    have hai : a < i := by
+      change a.val < i.val
+      have hh : a.val + 1 ≤ i.val := hji
+      omega
+    exact hp hai
+
+theorem tag_lt_tupleLower {n r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p)
+    (i : Fin r) (j : Fin (r+1)) (hij : i.castSucc < j) :
+    (p i).val < tupleLower p j := by
+  induction j using Fin.cases with
+  | zero => exact False.elim (by simp at hij)
+  | succ a =>
+    simp only [tupleLower_succ]
+    have hia : i ≤ a := by
+      change i.val ≤ a.val
+      have hh : i.val < a.val + 1 := hij
+      omega
+    exact Nat.lt_succ_of_le (hp.monotone hia)
+
+theorem tupleUpper_le_total {n r : ℕ} (p : Fin r → Fin n) (j : Fin (r+1)) :
+    tupleUpper p j ≤ n := by
+  induction j using Fin.lastCases with
+  | last => simp
+  | cast i => simpa only [tupleUpper_castSucc] using (p i).isLt.le
+
+theorem tupleLower_le_total {n r : ℕ} (p : Fin r → Fin n) (j : Fin (r+1)) :
+    tupleLower p j ≤ n := by
+  induction j using Fin.cases with
+  | zero => simp
+  | succ i => simpa only [tupleLower_succ] using Nat.succ_le_of_lt (p i).isLt
+
+theorem tupleLower_le_upper {n r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p)
+    (j : Fin (r+1)) : tupleLower p j ≤ tupleUpper p j := by
+  induction j using Fin.lastCases with
+  | last => simpa using tupleLower_le_total p (Fin.last r)
+  | cast i => simpa using tupleLower_le_tag p hp i i.castSucc le_rfl
+
+theorem tupleGap_total {n r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p) :
+    r + ∑ j, tupleGap p j = n := by
+  have hrs : r ≤ n := by simpa using Fintype.card_le_of_injective p hp.injective
+  have ha : (∑ j, tupleLower p j) = (∑ i, (p i).val) + r := by
+    rw [Fin.sum_univ_succ]
+    simp [Finset.sum_add_distrib]
+  have hb : (∑ j, tupleUpper p j) = (∑ i, (p i).val) + n := by
+    rw [Fin.sum_univ_castSucc]
+    simp
+  simp only [tupleGap]
+  rw [Finset.sum_tsub_distrib _ (fun j _ => tupleLower_le_upper p hp j), ha, hb]
+  omega
+
+/-- Maps abstract gap coordinates to their actual source slots. -/
+def tupleGapMap {n r : ℕ} (p : Fin r → Fin n) :
+    GapUniverse r (r+1) (tupleGap p) → Fin n
+  | .inl i => p i
+  | .inr ⟨j,a⟩ => ⟨tupleLower p j + a.val, by
+      have hx := a.isLt
+      have hu := tupleUpper_le_total p j
+      simp only [tupleGap] at hx
+      omega⟩
+
+theorem tupleGapMap_gap_lt_tag {n r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p)
+    (i : Fin r) (j : Fin (r+1)) (a : Fin (tupleGap p j)) :
+    tupleGapMap p (.inr ⟨j,a⟩) < p i ↔ j ≤ i.castSucc := by
+  constructor
+  · intro h
+    by_contra hn
+    have hx := tag_lt_tupleLower p hp i j (lt_of_not_ge hn)
+    change tupleLower p j + a.val < (p i).val at h
+    omega
+  · intro h
+    have hji : j.val < r := lt_of_le_of_lt (show j.val ≤ i.val from h) i.isLt
+    let k : Fin r := ⟨j.val, hji⟩
+    have hjk : j = k.castSucc := Fin.ext rfl
+    have ht : tupleUpper p j ≤ (p i).val := by
+      rw [hjk, tupleUpper_castSucc]
+      exact hp.monotone (show k ≤ i from h)
+    have hx := a.isLt
+    change tupleLower p j + a.val < (p i).val
+    unfold tupleGap at hx
+    omega
+
+theorem tupleGapMap_tag_lt_gap {n r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p)
+    (i : Fin r) (j : Fin (r+1)) (a : Fin (tupleGap p j)) :
+    p i < tupleGapMap p (.inr ⟨j,a⟩) ↔ i.castSucc < j := by
+  constructor
+  · intro h
+    by_contra hn
+    have hx := (tupleGapMap_gap_lt_tag p hp i j a).2 (le_of_not_gt hn)
+    exact (lt_asymm h hx)
+  · intro h
+    have hx := tag_lt_tupleLower p hp i j h
+    change (p i).val < tupleLower p j + a.val
+    omega
+
+theorem tupleGapMap_injective {n r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p) :
+    Function.Injective (tupleGapMap p) := by
+  rintro (i | ⟨j,a⟩) (i' | ⟨j',a'⟩) h
+  · exact congrArg Sum.inl (hp.injective h)
+  · have hn : p i ≠ tupleGapMap p (.inr ⟨j',a'⟩) := by
+      rcases le_or_gt j' i.castSucc with hj | hj
+      · exact ne_of_gt ((tupleGapMap_gap_lt_tag p hp i j' a').2 hj)
+      · exact ne_of_lt ((tupleGapMap_tag_lt_gap p hp i j' a').2 hj)
+    exact False.elim (hn h)
+  · have hn : p i' ≠ tupleGapMap p (.inr ⟨j,a⟩) := by
+      rcases le_or_gt j i'.castSucc with hj | hj
+      · exact ne_of_gt ((tupleGapMap_gap_lt_tag p hp i' j a).2 hj)
+      · exact ne_of_lt ((tupleGapMap_tag_lt_gap p hp i' j a).2 hj)
+    exact False.elim (hn h.symm)
+  · have hj : j=j' := by
+      by_contra hn
+      rcases lt_or_gt_of_ne hn with hj | hj
+      · let i : Fin r := ⟨j.val, lt_of_lt_of_le hj (Nat.le_of_lt_succ j'.isLt)⟩
+        have hji : j=i.castSucc := Fin.ext rfl
+        have h1 := (tupleGapMap_gap_lt_tag p hp i j a).2 (by rw [hji])
+        have h2 := (tupleGapMap_tag_lt_gap p hp i j' a').2 (by rwa [← hji])
+        rw [h] at h1
+        exact (lt_asymm h1 h2)
+      · let i : Fin r := ⟨j'.val, lt_of_lt_of_le hj (Nat.le_of_lt_succ j.isLt)⟩
+        have hji : j'=i.castSucc := Fin.ext rfl
+        have h1 := (tupleGapMap_gap_lt_tag p hp i j' a').2 (by rw [hji])
+        have h2 := (tupleGapMap_tag_lt_gap p hp i j a).2 (by rwa [← hji])
+        rw [h] at h2
+        exact (lt_asymm h1 h2)
+    subst j'
+    have ha : a=a' := by
+      apply Fin.ext
+      have hh := congrArg Fin.val h
+      change tupleLower p j + a.val = tupleLower p j + a'.val at hh
+      omega
+    subst a'
+    rfl
+
+def tupleGapEquiv {n r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p) :
+    GapUniverse r (r+1) (tupleGap p) ≃ Fin n :=
+  Equiv.ofBijective (tupleGapMap p) ((Fintype.bijective_iff_injective_and_card _).2
+    ⟨tupleGapMap_injective p hp, by
+      simp only [Fintype.card_sum, Fintype.card_sigma, Fintype.card_fin]
+      exact tupleGap_total p hp⟩)
+
+/-- The count below a tagged slot is its tag index plus the retained gap counts. -/
+theorem gapEncoded_lower_card {n r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p)
+    (A : (j : Fin (r+1)) → Finset (Fin (tupleGap p j))) (i : Fin r) :
+    (((gapEncode A).map (tupleGapEquiv p hp).toEmbedding).filter
+      fun x => x < p i).card = i.val + ∑ j ∈ Finset.Iic i.castSucc, (A j).card := by
+  classical
+  have hmap : (((gapEncode A).map (tupleGapEquiv p hp).toEmbedding).filter
+      fun x => x < p i) =
+      ((gapEncode A).filter fun x => tupleGapMap p x < p i).map
+        (tupleGapEquiv p hp).toEmbedding := by
+    rw [Finset.filter_map]
+    rfl
+  rw [hmap, Finset.card_map, Finset.card_eq_sum_ones]
+  simp only [Finset.sum_filter, gapEncode]
+  rw [Fintype.sum_sum_type, Fintype.sum_sigma]
+  simp only
+  simp only [↓reduceIte]
+  have hleft : (∑ a : Fin r, if tupleGapMap p (.inl a) < p i then (1:ℕ) else 0) = i.val := by
+    simp only [tupleGapMap, hp.lt_iff_lt]
+    have hx : (Finset.univ.filter fun a : Fin r => a < i) = Finset.Iio i := by ext; simp
+    rw [← Finset.sum_filter, hx, Finset.sum_const, Fin.card_Iio, smul_eq_mul, mul_one]
+  rw [hleft]
+  congr 1
+  simp_rw [tupleGapMap_gap_lt_tag p hp]
+  have hr (j : Fin (r+1)) :
+      (∑ a : Fin (tupleGap p j), if a ∈ A j then
+        (if j ≤ i.castSucc then (1:ℕ) else 0) else 0) =
+      if j ≤ i.castSucc then (A j).card else 0 := by
+    by_cases hj : j ≤ i.castSucc
+    · simp only [hj, ite_true]
+      exact (Finset.card_eq_sum_ite (Finset.subset_univ (A j))).symm
+    · simp [hj]
+  rw [Finset.sum_congr rfl (fun j _ => hr j), ← Finset.sum_filter]
+  congr 1
+  ext j
+  simp
+
+@[simp] theorem gapEncode_univ {r d : ℕ} {g : Fin d → ℕ} :
+    gapEncode (r := r) (fun i => (Finset.univ : Finset (Fin (g i)))) = Finset.univ := by
+  ext x
+  cases x with
+  | inl i => simp
+  | inr p => rcases p with ⟨j,a⟩; simp
+
+theorem tupleGap_prefix {n r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p)
+    (i : Fin r) : i.val + ∑ j ∈ Finset.Iic i.castSucc, tupleGap p j = (p i).val := by
+  have h := gapEncoded_lower_card p hp (fun _ => Finset.univ) i
+  have hset : (Finset.univ.filter fun x : Fin n => x < p i) = Finset.Iio (p i) := by
+    ext; simp
+  simpa only [gapEncode_univ, Finset.map_univ_equiv,
+    hset, Fin.card_Iio, Finset.card_univ, Fintype.card_fin] using h.symm
+
+/-- Initial-segment sums determine a family of natural numbers. -/
+theorem fin_prefix_sum_injective {d : ℕ} (f g : Fin d → ℕ)
+    (h : ∀ i, (∑ j ∈ Finset.Iic i, f j) = ∑ j ∈ Finset.Iic i, g j) : f=g := by
+  funext i
+  induction i using WellFoundedLT.induction with
+  | ind i ih =>
+    have hset : Finset.Iic i = insert i (Finset.Iio i) := by ext; simp
+    have hprev : (∑ j ∈ Finset.Iio i, f j) = ∑ j ∈ Finset.Iio i, g j := by
+      apply Finset.sum_congr rfl
+      intro j hj
+      exact ih j (Finset.mem_Iio.mp hj)
+    have hh := h i
+    rw [hset, Finset.sum_insert (by simp), Finset.sum_insert (by simp), hprev] at hh
+    omega
+
+def pullGapSet {n r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p)
+    (S : Finset (Fin n)) : Finset (GapUniverse r (r+1) (tupleGap p)) :=
+  S.map (tupleGapEquiv p hp).symm.toEmbedding
+
+@[simp] theorem mem_pullGapSet {n r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p)
+    (S : Finset (Fin n)) (x : GapUniverse r (r+1) (tupleGap p)) :
+    x ∈ pullGapSet p hp S ↔ tupleGapMap p x ∈ S := by
+  simp [pullGapSet, tupleGapEquiv]
+
+@[simp] theorem map_pullGapSet {n r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p)
+    (S : Finset (Fin n)) :
+    (pullGapSet p hp S).map (tupleGapEquiv p hp).toEmbedding = S := by
+  ext x
+  simp only [pullGapSet, Finset.mem_map_equiv, Equiv.symm_symm, Equiv.apply_symm_apply]
+
+@[simp] theorem pullGapSet_map {n r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p)
+    (T : Finset (GapUniverse r (r+1) (tupleGap p))) :
+    pullGapSet p hp (T.map (tupleGapEquiv p hp).toEmbedding) = T := by
+  ext x
+  simp only [pullGapSet, Finset.mem_map_equiv, Equiv.symm_symm, Equiv.symm_apply_apply]
+
+theorem pullGapSet_lower_card {n r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p)
+    (S : Finset (Fin n)) (hS : ∀ i, p i ∈ S) (i : Fin r) :
+    (S.filter fun x => x < p i).card =
+      i.val + ∑ j ∈ Finset.Iic i.castSucc, (gapDecode (pullGapSet p hp S) j).card := by
+  have hT : ∀ a, Sum.inl a ∈ pullGapSet p hp S := by
+    intro a
+    simpa only [mem_pullGapSet, tupleGapMap] using hS a
+  have h := gapEncoded_lower_card p hp (gapDecode (pullGapSet p hp S)) i
+  rwa [gapEncode_decode _ hT, map_pullGapSet] at h
+
+theorem pullGapSet_total_card {n r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p)
+    (S : Finset (Fin n)) (hS : ∀ i, p i ∈ S) :
+    S.card = r + ∑ j, (gapDecode (pullGapSet p hp S) j).card := by
+  have hT : ∀ a, Sum.inl a ∈ pullGapSet p hp S := by
+    intro a
+    simpa only [mem_pullGapSet, tupleGapMap] using hS a
+  have h := card_gapEncode (r := r) (gapDecode (pullGapSet p hp S))
+  rw [gapEncode_decode _ hT] at h
+  simpa only [pullGapSet, Finset.card_map] using h
+
+theorem taggedRankEvent_iff_gapEvent {n l r : ℕ} (p : Fin r → Fin n)
+    (hp : StrictMono p) (v : Fin r → Fin l) (hv : StrictMono v)
+    (S : FixedSubset (Fin n) l) :
+    taggedRankEvent p v S ↔ gapEvent (tupleGap v) (pullGapSet p hp S.val) := by
+  constructor
+  · intro h
+    have hS : ∀ i, p i ∈ S.val := fun i => (slotAtRank_eq_iff S (v i) (p i)).mp (h i) |>.1
+    have hbefore : ∀ i, (S.val.filter fun x => x < p i).card = (v i).val :=
+      fun i => (slotAtRank_eq_iff S (v i) (p i)).mp (h i) |>.2
+    refine ⟨?_, ?_⟩
+    · intro i
+      simpa only [mem_pullGapSet, tupleGapMap] using hS i
+    · have heq : (fun j => (gapDecode (pullGapSet p hp S.val) j).card) = tupleGap v := by
+        apply fin_prefix_sum_injective
+        intro j
+        induction j using Fin.lastCases with
+        | last =>
+          have htot := pullGapSet_total_card p hp S.val hS
+          have hvtot := tupleGap_total v hv
+          have hset : Finset.Iic (Fin.last r) = Finset.univ := by ext; simp [Fin.le_last]
+          rw [hset]
+          rw [S.property] at htot
+          omega
+        | cast i =>
+          have hpre := pullGapSet_lower_card p hp S.val hS i
+          rw [hbefore i] at hpre
+          have hvpre := tupleGap_prefix v hv i
+          omega
+      intro j
+      exact congrFun heq j
+  · intro h
+    have hS : ∀ i, p i ∈ S.val := by
+      intro i
+      simpa only [mem_pullGapSet, tupleGapMap] using h.1 i
+    intro i
+    apply (slotAtRank_eq_iff S (v i) (p i)).2
+    refine ⟨hS i, ?_⟩
+    rw [pullGapSet_lower_card p hp S.val hS i]
+    simp_rw [h.2]
+    exact tupleGap_prefix v hv i
+
+instance taggedRankEvent_decidable {n l r : ℕ} (p : Fin r → Fin n)
+    (v : Fin r → Fin l) (S : FixedSubset (Fin n) l) : Decidable (taggedRankEvent p v S) := by
+  unfold taggedRankEvent
+  infer_instance
+
+/-- The actual event of seeing specified source slots at specified output ranks
+is in bijection with independent choices of the retained slots in each gap. -/
+def rankGapEventEquiv {n l r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p)
+    (v : Fin r → Fin l) (hv : StrictMono v) :
+    {S : FixedSubset (Fin n) l // taggedRankEvent p v S} ≃
+      {T : Finset (GapUniverse r (r+1) (tupleGap p)) // gapEvent (tupleGap v) T} where
+  toFun S := ⟨pullGapSet p hp S.val.val, (taggedRankEvent_iff_gapEvent p hp v hv S.val).mp S.property⟩
+  invFun T :=
+    let S : FixedSubset (Fin n) l := ⟨T.val.map (tupleGapEquiv p hp).toEmbedding, by
+      rw [Finset.card_map, card_of_gapEvent T.val T.property, tupleGap_total v hv]⟩
+    ⟨S, (taggedRankEvent_iff_gapEvent p hp v hv S).mpr (by
+      change gapEvent (tupleGap v) (pullGapSet p hp (T.val.map (tupleGapEquiv p hp).toEmbedding))
+      rw [pullGapSet_map]
+      exact T.property)⟩
+  left_inv S := by
+    apply Subtype.ext
+    apply Subtype.ext
+    exact map_pullGapSet p hp S.val.val
+  right_inv T := by
+    apply Subtype.ext
+    exact pullGapSet_map p hp T.val
+
+theorem card_taggedRankEvent {n l r : ℕ} (p : Fin r → Fin n) (hp : StrictMono p)
+    (v : Fin r → Fin l) (hv : StrictMono v) :
+    Fintype.card {S : FixedSubset (Fin n) l // taggedRankEvent p v S} =
+      ∏ j, (tupleGap p j).choose (tupleGap v j) := by
+  classical
+  rw [Fintype.card_congr (rankGapEventEquiv p hp v hv), card_gapEvent]
+
+/-- Exact counting formula, including impossible rank/gap profiles. -/
+theorem subsetRankKernel_choose_formula {n l r : ℕ} (hl : l ≤ n)
+    (p : Fin r → Fin n) (hp : StrictMono p) (v : Fin r → Fin l) (hv : StrictMono v) :
+    subsetRankKernel hl p v =
+      (∏ j, ((tupleGap p j).choose (tupleGap v j) : ℝ)) / (n.choose l : ℝ) := by
+  classical
+  unfold subsetRankKernel
+  simp only [uniformSubsetLaw, Fintype.card_fin]
+  rw [← Finset.sum_filter, Finset.sum_const, nsmul_eq_mul,
+    ← Fintype.card_subtype, card_taggedRankEvent p hp v hv, Nat.cast_prod]
+  ring
+
+theorem gapEvent_conditional_choose {r d : ℕ} (g z : Fin d → ℕ)
+    (p : ℝ) (hp : p ∈ Set.Ioo 0 1) :
+    (∑ S : Finset (GapUniverse r d g), if gapEvent z S then subsetWeight p S else 0) /
+      binomialMass (r+∑ i, g i) p (r+∑ i, z i) =
+    (∏ i, ((g i).choose (z i):ℝ)) /
+      ((r+∑ i, g i).choose (r+∑ i, z i) : ℝ) := by
+  classical
+  have hcount : (∑ S : Finset (GapUniverse r d g),
+      if gapEvent z S then (1:ℝ) else 0) = ∏ i, ((g i).choose (z i):ℝ) := by
+    rw [Finset.sum_boole, ← Fintype.card_subtype, card_gapEvent, Nat.cast_prod]
+  have hnum : (∑ S : Finset (GapUniverse r d g), if gapEvent z S then subsetWeight p S else 0) =
+      (∏ i, ((g i).choose (z i):ℝ)) *
+        (p^(r+∑ i, z i)*(1-p)^(r+∑ i, g i-(r+∑ i, z i))) := by
+    rw [← hcount, Finset.sum_mul]
+    apply Finset.sum_congr rfl
+    intro S _
+    split_ifs with hS
+    · simp only [one_mul, subsetWeight, card_of_gapEvent S hS,
+        Fintype.card_sum, Fintype.card_fin, Fintype.card_sigma]
+    · exact (zero_mul _).symm
+  rw [hnum, binomialMass]
+  have hp0 : p ≠ 0 := hp.1.ne'
+  have hq0 : 1-p ≠ 0 := by linarith [hp.2]
+  rw [mul_assoc, mul_div_mul_right _ _ (mul_ne_zero (pow_ne_zero _ hp0) (pow_ne_zero _ hq0))]
+
+/-- Conditional product formula for the kernel defined from actual retained
+subsets, not an assumed auxiliary kernel. Source the initial-pair construction 50–62. -/
+theorem subsetRankKernel_binomial_formula {n l r : ℕ} (hl : l ≤ n)
+    (p : Fin r → Fin n) (hp : StrictMono p) (v : Fin r → Fin l) (hv : StrictMono v)
+    (θ : ℝ) (hθ : θ ∈ Set.Ioo 0 1) :
+    subsetRankKernel hl p v = θ^r / binomialMass n θ l *
+      ∏ j, binomialMass (tupleGap p j) θ (tupleGap v j) := by
+  classical
+  by_cases hz : ∀ j, tupleGap v j ≤ tupleGap p j
+  · have h1 := gapEvent_conditional_choose (r := r) (tupleGap p) (tupleGap v) θ hθ
+    have h2 := gapEvent_conditional_formula (r := r) (tupleGap p) (tupleGap v) hz θ
+    rw [tupleGap_total p hp, tupleGap_total v hv] at h1 h2
+    rw [subsetRankKernel_choose_formula hl p hp v hv, ← h1, h2]
+  · push Not at hz
+    obtain ⟨j,hj⟩ := hz
+    have hc : (tupleGap p j).choose (tupleGap v j) = 0 := Nat.choose_eq_zero_of_lt hj
+    have hz1 : (∏ j, ((tupleGap p j).choose (tupleGap v j) : ℝ))=0 :=
+      Finset.prod_eq_zero (Finset.mem_univ j) (by simp [hc])
+    have hz2 : (∏ j, binomialMass (tupleGap p j) θ (tupleGap v j))=0 :=
+      Finset.prod_eq_zero (Finset.mem_univ j) (by simp [binomialMass, hc])
+    rw [subsetRankKernel_choose_formula hl p hp v hv, hz1, hz2]
+    simp
+
+end TraceReconstruction
+
+namespace TraceReconstruction
+
+/-- Twice the coarse period gaps (endpoints are half-integral before doubling). -/
+def coarsePeriodGap {u r : ℕ} (j : Fin r → Fin u) (i : Fin (r+1)) : ℕ :=
+  2 * tupleGap j i + if i = 0 ∨ i = Fin.last r then 1 else 2
+
+theorem coarsePeriodGap_pos {u r : ℕ} (j : Fin r → Fin u) (i : Fin (r+1)) :
+    1 ≤ coarsePeriodGap j i := by
+  unfold coarsePeriodGap
+  split_ifs <;> omega
+
+theorem coarsePeriodGap_sum {u r : ℕ} (hr : 0 < r) (j : Fin r → Fin u)
+    (hj : StrictMono j) : ∑ i, coarsePeriodGap j i = 2*u := by
+  classical
+  have hc : (∑ i : Fin (r+1), if i = 0 ∨ i = Fin.last r then 1 else 2) = 2*r := by
+    obtain ⟨d,rfl⟩ := Nat.exists_eq_succ_of_ne_zero (Nat.ne_of_gt hr)
+    rw [Fin.sum_univ_succ]
+    simp only [true_or, ↓reduceIte, Fin.succ_ne_zero, false_or]
+    rw [Fin.sum_univ_castSucc]
+    simp only [Fin.succ_last, ↓reduceIte]
+    have hx (i : Fin d) : i.castSucc.succ ≠ Fin.last (d+1) := by
+      intro h
+      have hv := congrArg Fin.val h
+      simp only [Fin.val_succ, Fin.val_castSucc, Fin.val_last] at hv
+      omega
+    simp only [hx, ↓reduceIte, Finset.sum_const, Finset.card_univ, Fintype.card_fin,
+      smul_eq_mul]
+    omega
+  simp only [coarsePeriodGap, Finset.sum_add_distrib, ← Finset.mul_sum]
+  have ht := tupleGap_total j hj
+  omega
+
+theorem coarsePeriodGap_le {u r : ℕ} (hr : 0 < r) (j : Fin r → Fin u)
+    (hj : StrictMono j) (i : Fin (r+1)) : coarsePeriodGap j i ≤ 2*u := by
+  rw [← coarsePeriodGap_sum hr j hj]
+  exact Finset.single_le_sum (fun _ _ => Nat.zero_le _) (Finset.mem_univ i)
+
+theorem tupleGap_injective {n r : ℕ} {p q : Fin r → Fin n}
+    (hp : StrictMono p) (hq : StrictMono q) (h : tupleGap p = tupleGap q) : p=q := by
+  funext i
+  apply Fin.ext
+  have hp' := tupleGap_prefix p hp i
+  have hq' := tupleGap_prefix q hq i
+  rw [h] at hp'
+  omega
+
+theorem coarsePeriodGap_injective {u r : ℕ} {j k : Fin r → Fin u}
+    (hj : StrictMono j) (hk : StrictMono k) (h : coarsePeriodGap j = coarsePeriodGap k) :
+    j=k := by
+  apply tupleGap_injective hj hk
+  funext i
+  have hi := congrFun h i
+  unfold coarsePeriodGap at hi
+  omega
+
+/-- Actual defect slot in a period, with zero-based period and slot indices. -/
+def periodSlot {u m : ℕ} (off : Fin u → Fin m) (j : Fin u) : Fin (u*m) :=
+  ⟨j.val*m + (off j).val, by
+    have h := (off j).isLt
+    have h' := Nat.mul_le_mul_right m (Nat.succ_le_of_lt j.isLt)
+    nlinarith⟩
+
+theorem periodSlot_strictMono {u m : ℕ} (off : Fin u → Fin m) :
+    StrictMono (periodSlot off) := by
+  intro j k hjk
+  change j.val*m + (off j).val < k.val*m + (off k).val
+  have hjk' : j.val+1 ≤ k.val := hjk
+  have ho := (off j).isLt
+  have hm := Nat.mul_le_mul_right m hjk'
+  nlinarith
+
+theorem tupleGap_zero {n r : ℕ} (p : Fin (r+1) → Fin n) :
+    tupleGap p 0 = (p 0).val := by
+  have he : (0 : Fin (r+2)) = (0 : Fin (r+1)).castSucc := rfl
+  unfold tupleGap
+  rw [tupleLower_zero, Nat.sub_zero, he, tupleUpper_castSucc]
+
+theorem tupleGap_last {n r : ℕ} (p : Fin (r+1) → Fin n) :
+    tupleGap p (Fin.last (r+1)) = n - ((p (Fin.last r)).val+1) := by
+  unfold tupleGap
+  rw [tupleUpper_last, ← Fin.succ_last, tupleLower_succ]
+
+theorem tupleGap_middle {n r : ℕ} (p : Fin (r+1) → Fin n) (i : Fin r) :
+    tupleGap p i.castSucc.succ = (p i.succ).val - ((p i.castSucc).val+1) := by
+  unfold tupleGap
+  rw [tupleLower_succ, Fin.succ_castSucc, tupleUpper_castSucc]
+
+theorem coarsePeriodGap_zero {u r : ℕ} (j : Fin (r+1) → Fin u) :
+    coarsePeriodGap j 0 = 2*(j 0).val+1 := by
+  simp only [coarsePeriodGap, tupleGap_zero, true_or, ↓reduceIte]
+
+theorem coarsePeriodGap_last {u r : ℕ} (j : Fin (r+1) → Fin u) :
+    coarsePeriodGap j (Fin.last (r+1)) = 2*(u-((j (Fin.last r)).val+1))+1 := by
+  simp only [coarsePeriodGap, tupleGap_last, or_true, ↓reduceIte]
+
+theorem coarsePeriodGap_middle {u r : ℕ} (j : Fin (r+1) → Fin u) (i : Fin r) :
+    coarsePeriodGap j i.castSucc.succ = 2*((j i.succ).val-((j i.castSucc).val+1))+2 := by
+  have hn : i.castSucc.succ ≠ Fin.last (r+1) := by
+    intro h
+    have hv := congrArg Fin.val h
+    simp only [Fin.val_succ, Fin.val_castSucc, Fin.val_last] at hv
+    omega
+  simp only [coarsePeriodGap, tupleGap_middle, Fin.succ_ne_zero, hn, or_self, ↓reduceIte]
+
+/-- Actual bounded defect displacements give the O(1) gap errors of the initial-pair construction. -/
+theorem period_gap_rounding {u m r : ℕ} (hr : 0 < r)
+    (off : Fin u → Fin m)
+    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
+    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2)
+    (j : Fin r → Fin u) (hj : StrictMono j) (i : Fin (r+1)) :
+    |(tupleGap (periodSlot off ∘ j) i:ℝ) - (m:ℝ)*coarsePeriodGap j i/2| ≤ 4 := by
+  obtain ⟨d,rfl⟩ := Nat.exists_eq_succ_of_ne_zero (Nat.ne_of_gt hr)
+  have hp := (periodSlot_strictMono off).comp hj
+  induction i using Fin.cases with
+  | zero =>
+    rw [tupleGap_zero, coarsePeriodGap_zero]
+    simp only [Function.comp_apply, periodSlot, Nat.cast_add, Nat.cast_mul, Nat.cast_ofNat,
+      Nat.cast_one]
+    have offset_bound := abs_le.mp (hoff (j 0))
+    apply abs_le.mpr
+    constructor <;> linarith
+  | succ i =>
+    induction i using Fin.lastCases with
+    | last =>
+      rw [Fin.succ_last, tupleGap_last, coarsePeriodGap_last]
+      have hcut : (j (Fin.last d)).val+1 ≤ u := Nat.succ_le_of_lt (j _).isLt
+      have hslot : ((periodSlot off ∘ j) (Fin.last d)).val+1 ≤ u*m :=
+        Nat.succ_le_of_lt (((periodSlot off ∘ j) (Fin.last d))).isLt
+      simp only [Nat.cast_add, Nat.cast_mul, Nat.cast_ofNat]
+      rw [Nat.cast_sub hslot, Nat.cast_sub hcut]
+      simp only [Function.comp_apply, periodSlot, Nat.cast_add, Nat.cast_mul, Nat.cast_one]
+      have h := abs_le.mp (hoff (j (Fin.last d)))
+      apply abs_le.mpr
+      constructor <;> linarith
+    | cast i =>
+      rw [tupleGap_middle, coarsePeriodGap_middle]
+      have hcut : (j i.castSucc).val+1 ≤ (j i.succ).val := hj (Fin.castSucc_lt_succ (i := i))
+      have hslot : ((periodSlot off ∘ j) i.castSucc).val+1 ≤
+          ((periodSlot off ∘ j) i.succ).val := hp (Fin.castSucc_lt_succ (i := i))
+      simp only [Nat.cast_add, Nat.cast_mul, Nat.cast_ofNat]
+      rw [Nat.cast_sub hslot, Nat.cast_sub hcut]
+      simp only [Function.comp_apply, periodSlot, Nat.cast_add, Nat.cast_mul, Nat.cast_one]
+      have h := abs_le.mp (hdiff (j i.succ) (j i.castSucc))
+      apply abs_le.mpr
+      constructor <;> linarith
+
+end TraceReconstruction
+
+namespace TraceReconstruction
+
+/-- An increasing tuple enumerates a monomial's distinct observed ranks. -/
+abbrev IncreasingTuple (n r : ℕ) := {v : Fin r → Fin n // StrictMono v}
+
+noncomputable instance increasingTupleFintype (n r : ℕ) : Fintype (IncreasingTuple n r) :=
+  Fintype.ofFinite _
+
+theorem tupleGap_le_remainder {n r : ℕ} (v : IncreasingTuple n r) (i : Fin (r+1)) :
+    tupleGap v.val i ≤ n-r := by
+  have ht := tupleGap_total v.val v.property
+  have hle : tupleGap v.val i ≤ ∑ j, tupleGap v.val j :=
+    Finset.single_le_sum (fun _ _ => Nat.zero_le _) (Finset.mem_univ i)
+  omega
+
+def rankGapProfile {n r : ℕ} (σ : Equiv.Perm (Fin (r+1))) (v : IncreasingTuple n r) :
+    Fin (r+1) → Fin (n-r+1) :=
+  fun i => ⟨tupleGap v.val (σ i), Nat.lt_succ_of_le (tupleGap_le_remainder v (σ i))⟩
+
+theorem rankGapProfile_sum {n r : ℕ} (σ : Equiv.Perm (Fin (r+1)))
+    (v : IncreasingTuple n r) : ∑ i, (rankGapProfile σ v i).val = n-r := by
+  change (∑ i, tupleGap v.val (σ i)) = _
+  rw [Equiv.sum_comp σ]
+  have ht := tupleGap_total v.val v.property
+  omega
+
+theorem rankGapProfile_injective {n r : ℕ} (σ : Equiv.Perm (Fin (r+1))) :
+    Function.Injective (rankGapProfile (n := n) σ) := by
+  intro v w h
+  apply Subtype.ext
+  funext i
+  apply Fin.ext
+  have he : tupleGap v.val = tupleGap w.val := by
+    funext j
+    have hh := congrArg Fin.val (congrFun h (σ.symm j))
+    simpa only [rankGapProfile, Equiv.apply_symm_apply] using hh
+  have hv := tupleGap_prefix v.val v.property i
+  have hw := tupleGap_prefix w.val w.property i
+  rw [he] at hv
+  omega
+
+/-- Summing nonnegative kernels over actual profiles may drop any remaining constraints. -/
+theorem sum_injective_profile_le {α β : Type*} [Fintype α] [Fintype β]
+    (profile : α → β) (hinj : Function.Injective profile)
+    (P : β → Prop) [DecidablePred P] (hp : ∀ a, P (profile a))
+    (f : β → ℝ) (hf : ∀ b, 0 ≤ f b) :
+    (∑ a, f (profile a)) ≤ ∑ b, if P b then f b else 0 := by
+  classical
+  let S := Finset.univ.image profile
+  have hs : ∀ b ∈ S, P b := by
+    intro b hb
+    obtain ⟨a,_,rfl⟩ := Finset.mem_image.mp hb
+    exact hp a
+  calc
+    _ = ∑ b ∈ S, f b := by
+      symm
+      exact Finset.sum_image (fun _ _ _ _ h => hinj h)
+    _ = ∑ b ∈ S, if P b then f b else 0 := by
+      apply Finset.sum_congr rfl
+      intro b hb
+      rw [ite_eq_left (hs b hb)]
+    _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _) (by
+      intro b _ _
+      split_ifs
+      · exact hf b
+      · rfl)
+
+/-- The actual conditional rank kernel, summed over distinct observed ranks,
+reduces to the constrained finite product estimate with any eliminated gap. -/
+theorem subsetRankKernel_cross_le {n l r : ℕ} (hl : l ≤ n)
+    (p q : Fin r → Fin n) (hp : StrictMono p) (hq : StrictMono q)
+    (θ : ℝ) (hθ : θ ∈ Set.Ioo 0 1) (e : Fin (r+1))
+    (M : ℝ) (hM : 0 ≤ M)
+    (hbound : ∀ k : Fin (l-r+1),
+      binomialMass (tupleGap p e) θ k * binomialMass (tupleGap q e) θ k ≤ M) :
+    (∑ v : IncreasingTuple l r,
+      subsetRankKernel hl p v.val * subsetRankKernel hl q v.val) ≤
+      (θ^r / binomialMass n θ l)^2 * M *
+        ∏ i : Fin r, ∑ k : Fin (l-r+1),
+          binomialMass (tupleGap p (Equiv.swap 0 e i.succ)) θ k *
+          binomialMass (tupleGap q (Equiv.swap 0 e i.succ)) θ k := by
+  classical
+  let σ : Equiv.Perm (Fin (r+1)) := Equiv.swap 0 e
+  let f : Fin (r+1) → Fin (l-r+1) → ℝ := fun i k => binomialMass (tupleGap p (σ i)) θ k
+  let g : Fin (r+1) → Fin (l-r+1) → ℝ := fun i k => binomialMass (tupleGap q (σ i)) θ k
+  have hf : ∀ i k, 0 ≤ f i k := by
+    intro i k
+    dsimp [f,binomialMass]
+    positivity [hθ.1.le, sub_nonneg.mpr hθ.2.le]
+  have hg : ∀ i k, 0 ≤ g i k := by
+    intro i k
+    dsimp [g,binomialMass]
+    positivity [hθ.1.le, sub_nonneg.mpr hθ.2.le]
+  let F (z : Fin (r+1) → Fin (l-r+1)) := (∏ i, f i (z i)) * (∏ i, g i (z i))
+  have hF : ∀ z, 0 ≤ F z := by
+    intro z
+    exact mul_nonneg (Finset.prod_nonneg fun i _ => hf i _) (Finset.prod_nonneg fun i _ => hg i _)
+  have hpoint (v : IncreasingTuple l r) :
+      subsetRankKernel hl p v.val * subsetRankKernel hl q v.val =
+        (θ^r / binomialMass n θ l)^2 * F (rankGapProfile σ v) := by
+    rw [subsetRankKernel_binomial_formula hl p hp v.val v.property θ hθ,
+      subsetRankKernel_binomial_formula hl q hq v.val v.property θ hθ]
+    dsimp only [F,f,g,rankGapProfile]
+    rw [Equiv.prod_comp σ (fun j => binomialMass (tupleGap p j) θ (tupleGap v.val j)),
+      Equiv.prod_comp σ (fun j => binomialMass (tupleGap q j) θ (tupleGap v.val j))]
+    ring
+  simp_rw [hpoint]
+  rw [← Finset.mul_sum]
+  calc
+    _ ≤ (θ^r / binomialMass n θ l)^2 *
+        ∑ z : Fin (r+1) → Fin (l-r+1), if ∑ i, (z i).val = l-r then F z else 0 := by
+      apply mul_le_mul_of_nonneg_left _ (sq_nonneg _)
+      exact sum_injective_profile_le (rankGapProfile σ) (rankGapProfile_injective σ)
+        (fun z => ∑ i, (z i).val = l-r) (rankGapProfile_sum σ) F hF
+    _ ≤ (θ^r / binomialMass n θ l)^2 * (M * ∏ i : Fin r, ∑ k, f i.succ k * g i.succ k) := by
+      apply mul_le_mul_of_nonneg_left _ (sq_nonneg _)
+      exact constrained_cross_kernel_le r (l-r) f g hf hg M hM (by
+        intro k
+        simpa only [f,g,σ,Equiv.swap_apply_left] using hbound k)
+    _ = _ := by dsimp [f,g,σ]; ring
+
+end TraceReconstruction
+
+namespace TraceReconstruction
+
+theorem finite_geometric_distance (S : Finset ℤ) (a : ℤ) {ρ : ℝ}
+    (hρ : ρ∈Set.Ico 0 1) :
+    (∑ z ∈ S, ρ^(z-a).natAbs) ≤ 2/(1-ρ) := by
+  have hs : Summable (fun n:ℕ => ρ^n) := summable_geometric_of_lt_one hρ.1 hρ.2
+  have hnat : Summable (fun n:ℕ => ρ^((n:ℤ).natAbs)) := by simpa using hs
+  have hneg : Summable (fun n:ℕ => ρ^((-(n:ℤ)).natAbs)) := by simpa using hs
+  have hz : Summable (fun z:ℤ => ρ^z.natAbs) := hnat.of_nat_of_neg hneg
+  have ht : (∑' z:ℤ, ρ^z.natAbs)=2/(1-ρ)-1 := by
+    rw [Summable.tsum_of_nat_of_neg (f := fun z : ℤ => ρ^z.natAbs) hnat hneg]
+    simp only [Int.natAbs_natCast, Int.natAbs_neg, Int.natAbs_zero, pow_zero]
+    rw [tsum_geometric_of_lt_one hρ.1 hρ.2]
+    ring
+  have hshift : Summable (fun z:ℤ => ρ^(z-a).natAbs) :=
+    hz.comp_injective (sub_left_injective)
+  have hts : (∑' z:ℤ, ρ^(z-a).natAbs)=(∑' z:ℤ, ρ^z.natAbs) :=
+    (Equiv.subRight a).tsum_eq (fun z:ℤ => ρ^z.natAbs)
+  have hbound := hshift.sum_le_tsum S (fun z _ => pow_nonneg hρ.1 _)
+  rw [hts,ht] at hbound
+  linarith
+
+theorem finite_gaussian_distance (S : Finset ℤ) (a : ℤ) {c : ℝ}
+    (hc : 0<c) :
+    (∑ z ∈ S, Real.exp (-c*((z:ℝ)-a)^2)) ≤ 2/(1-Real.exp (-c)) := by
+  apply le_trans _ (finite_geometric_distance S a ⟨(Real.exp_pos _).le, by
+    rw [Real.exp_lt_one_iff]; linarith⟩)
+  apply Finset.sum_le_sum
+  intro z _
+  have hcast (w : ℤ) : (w.natAbs:ℝ)=|(w:ℝ)| := by
+    rw [← Int.cast_abs, ← Int.natCast_natAbs, Int.cast_natCast]
+  have he : Real.exp (-c)^(z-a).natAbs = Real.exp (-c*|(z:ℝ)-a|) := by
+    rw [← Real.exp_nat_mul]
+    congr 1
+    rw [mul_comm, hcast, Int.cast_sub]
+  rw [he]
+  apply Real.exp_le_exp.mpr
+  have hd : |(z:ℝ)-a| ≤ ((z:ℝ)-a)^2 := by
+    have hnat : (z-a).natAbs ≤ (z-a).natAbs^2 := by
+      nlinarith only [Nat.zero_le (z-a).natAbs, (Nat.mul_self_le_mul_self_iff.mpr (show 0≤(z-a).natAbs by omega))]
+    have hr : ((z-a).natAbs:ℝ) ≤ ((z-a).natAbs:ℝ)^2 := by exact_mod_cast hnat
+    simpa only [hcast, Int.cast_sub, sq_abs] using hr
+  nlinarith [mul_le_mul_of_nonneg_left hd hc.le]
+
+end TraceReconstruction
+
+namespace TraceReconstruction
+
+theorem fin_geometric_distance (N : ℕ) (a : ℤ) {ρ : ℝ} (hρ : ρ∈Set.Ico 0 1) :
+    (∑ z : Fin N, ρ^((z.val:ℤ)-a).natAbs) ≤ 2/(1-ρ) := by
+  let e : ℕ ↪ ℤ := ⟨Int.ofNat, Nat.cast_injective⟩
+  have h := finite_geometric_distance ((Finset.range N).map e) a hρ
+  rw [Finset.sum_map] at h
+  rw [Fin.sum_univ_eq_sum_range (fun z:ℕ => ρ^((z:ℤ)-a).natAbs) N]
+  exact h
+
+theorem coarse_laplace_sum_with_zero {h : ℝ} (hh : 0<h) (h1 : h≤1) (M : ℕ) :
+    (∑ k : Fin (M+1), (Real.sqrt (k.val:ℝ))⁻¹ * Real.exp (-h*(k.val:ℝ))) ≤
+      6/Real.sqrt h := by
+  rw [Fin.sum_univ_succ]
+  simp only [Fin.val_zero, Nat.cast_zero, Real.sqrt_zero, inv_zero, zero_mul, zero_add]
+  have hc := coarse_laplace_sum hh h1 M
+  rw [← Fin.sum_univ_eq_sum_range (fun k : ℕ =>
+    (Real.sqrt (k+1:ℝ))⁻¹ * Real.exp (-h*(k+1:ℝ))) M] at hc
+  simpa only [Fin.val_succ, Nat.cast_add, Nat.cast_one] using hc
+
+/-- The total constraint eliminates one coarse gap with no cost in its size.
+The zero-extension of x^(-1/2) avoids any singular integral convention. -/
+theorem coarse_constrained_inverse_sum (r U : ℕ) {h : ℝ} (hh : 0<h) (h1 : h≤1) :
+    (∑ γ : Fin (r+1) → Fin (U+1),
+      if ∑ i, (γ i).val=U then ∏ i : Fin r, (Real.sqrt ((γ i.succ).val:ℝ))⁻¹ else 0) ≤
+      Real.exp (h*U) * (6/Real.sqrt h)^r := by
+  classical
+  let f : Fin (r+1) → Fin (U+1) → ℝ := Fin.cases
+    (fun k => Real.exp (-h*(k.val:ℝ)))
+    (fun _ k => (Real.sqrt (k.val:ℝ))⁻¹ * Real.exp (-h*(k.val:ℝ)))
+  have hf : ∀ i k, 0≤f i k := by
+    intro i k
+    refine Fin.cases ?_ (fun _ => ?_) i <;> dsimp [f] <;> positivity
+  have hf0 : ∀ k, f 0 k≤1 := by
+    intro k
+    change Real.exp (-h*(k.val:ℝ))≤1
+    apply Real.exp_le_one_iff.mpr
+    nlinarith [show (0:ℝ)≤k.val by positivity]
+  have hid (γ : Fin (r+1) → Fin (U+1)) (hγ : ∑ i, (γ i).val=U) :
+      ∏ i : Fin r, (Real.sqrt ((γ i.succ).val:ℝ))⁻¹ =
+        Real.exp (h*U) * ∏ i, f i (γ i) := by
+    have hγr : ((γ 0).val:ℝ) + ∑ i:Fin r, ((γ i.succ).val:ℝ)=(U:ℝ) := by
+      exact_mod_cast (show (γ 0).val+∑ i:Fin r, (γ i.succ).val=U by
+        simpa only [Fin.sum_univ_succ] using hγ)
+    simp only [Fin.prod_univ_succ, f, Fin.cases_zero, Fin.cases_succ,
+      Finset.prod_mul_distrib, ← Real.exp_sum]
+    have he : Real.exp (h*U) * Real.exp (-h*((γ 0).val:ℝ)) *
+        Real.exp (∑ i:Fin r, -h*((γ i.succ).val:ℝ))=1 := by
+      rw [← Real.exp_add, ← Real.exp_add, ← Finset.mul_sum]
+      rw [show h*(U:ℝ)+ -h*((γ 0).val:ℝ)+ -h*∑ i:Fin r, ((γ i.succ).val:ℝ)=0 by
+        rw [← hγr]
+        ring]
+      exact Real.exp_zero
+    calc
+      _ = (Real.exp (h*U) * Real.exp (-h*((γ 0).val:ℝ)) *
+        Real.exp (∑ i:Fin r, -h*((γ i.succ).val:ℝ))) *
+          (∏ i:Fin r, (Real.sqrt ((γ i.succ).val:ℝ))⁻¹) := by rw [he, one_mul]
+      _ = _ := by ring
+  have hrew : (∑ γ : Fin (r+1) → Fin (U+1),
+      if ∑ i, (γ i).val=U then ∏ i : Fin r, (Real.sqrt ((γ i.succ).val:ℝ))⁻¹ else 0) =
+      Real.exp (h*U) * ∑ γ : Fin (r+1) → Fin (U+1),
+        if ∑ i, (γ i).val=U then ∏ i, f i (γ i) else 0 := by
+    rw [Finset.mul_sum]
+    apply Finset.sum_congr rfl
+    intro γ _
+    split_ifs with hγ
+    · exact hid γ hγ
+    · simp
+  rw [hrew]
+  apply mul_le_mul_of_nonneg_left _ (Real.exp_nonneg _)
+  apply (constrained_sum_product_le r U f hf 1 (by norm_num) hf0).trans
+  simp only [one_mul]
+  calc
+    _ ≤ ∏ _i : Fin r, (6/Real.sqrt h) := by
+      apply Finset.prod_le_prod₀
+      · intro i _; exact Finset.sum_nonneg (fun k _ => hf i.succ k)
+      · intro i _; exact coarse_laplace_sum_with_zero hh h1 U
+    _ = _ := by simp [div_pow]
+
+/-- The squared norm of an aggregate equals the sum of the cross-kernels. -/
+theorem sum_sq_sum_kernel {α β : Type*} [Fintype β] (S : Finset α) (K : α → β → ℝ) :
+    (∑ z, (∑ a ∈ S, K a z)^2) =
+      ∑ a ∈ S, ∑ b ∈ S, ∑ z, K a z*K b z := by
+  simp_rw [pow_two, Finset.sum_mul, Finset.mul_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro a _
+  exact Finset.sum_comm
+
+/-- Schur summation over all pairs of coarse-gap profiles, retaining the
+largest-gap total constraint before dropping other restrictions. -/
+theorem coarse_kernel_square_sum {β : Type*} [Fintype β] (r U : ℕ)
+    (S : Finset (Fin (r+1) → Fin (U+1)))
+    (K : (Fin (r+1) → Fin (U+1)) → β → ℝ) {M ρ h : ℝ}
+    (hM : 0≤M) (hρ : ρ∈Set.Ico 0 1) (hh : 0<h) (h1 : h≤1)
+    (hS : ∀ γ∈S, ∑ i, (γ i).val=U)
+    (hcross : ∀ γ∈S, ∀ γ'∈S, (∑ z, K γ z*K γ' z) ≤
+      M * ∏ i:Fin r, ((Real.sqrt ((γ i.succ).val:ℝ))⁻¹ *
+        ρ^(((γ' i.succ).val:ℤ)-(γ i.succ).val).natAbs)) :
+    (∑ z, (∑ γ∈S, K γ z)^2) ≤
+      M * (2/(1-ρ))^r * Real.exp (h*U) * (6/Real.sqrt h)^r := by
+  classical
+  let A (γ : Fin (r+1) → Fin (U+1)) :=
+    ∏ i:Fin r, (Real.sqrt ((γ i.succ).val:ℝ))⁻¹
+  have hA (γ) : 0≤A γ := by dsimp [A]; positivity
+  have hinner (γ : Fin (r+1) → Fin (U+1)) :
+      (∑ γ'∈S, ∏ i:Fin r, ρ^(((γ' i.succ).val:ℤ)-(γ i.succ).val).natAbs) ≤
+        (2/(1-ρ))^r := by
+    let f : Fin (r+1) → Fin (U+1) → ℝ := Fin.cases (fun _ => 1)
+      (fun i k => ρ^((k.val:ℤ)-(γ i.succ).val).natAbs)
+    have hf : ∀ i k, 0≤f i k := by
+      intro i k
+      refine Fin.cases ?_ (fun _ => ?_) i
+      · norm_num [f]
+      · exact pow_nonneg hρ.1 _
+    have hsub : (∑ γ'∈S, ∏ i:Fin r, ρ^(((γ' i.succ).val:ℤ)-(γ i.succ).val).natAbs) ≤
+        ∑ γ' : Fin (r+1) → Fin (U+1), if ∑ i, (γ' i).val=U then ∏ i, f i (γ' i) else 0 := by
+      calc
+        _ = ∑ γ'∈S, if ∑ i, (γ' i).val=U then ∏ i, f i (γ' i) else 0 := by
+          apply Finset.sum_congr rfl
+          intro γ' hγ'
+          simp only [hS γ' hγ', ↓reduceIte, Fin.prod_univ_succ, f, Fin.cases_zero,
+            Fin.cases_succ, one_mul]
+        _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _) (by
+          intro γ' _ _
+          split_ifs
+          · exact Finset.prod_nonneg (fun i _ => hf i _)
+          · rfl)
+    apply hsub.trans
+    apply (constrained_sum_product_le r U f hf 1 (by norm_num) (by intro k; rfl)).trans
+    simp only [one_mul]
+    calc
+      _ ≤ ∏ _i:Fin r, (2/(1-ρ)) := by
+        apply Finset.prod_le_prod₀
+        · intro i _; exact Finset.sum_nonneg (fun k _ => hf i.succ k)
+        · intro i _; exact fin_geometric_distance (U+1) ((γ i.succ).val:ℤ) hρ
+      _ = _ := by simp [div_pow]
+  have houter : (∑ γ∈S, A γ) ≤ Real.exp (h*U)*(6/Real.sqrt h)^r := by
+    apply le_trans _ (coarse_constrained_inverse_sum r U hh h1)
+    calc
+      _ = ∑ γ∈S, if ∑ i, (γ i).val=U then A γ else 0 := by
+        apply Finset.sum_congr rfl
+        intro γ hγ
+        rw [ite_eq_left (hS γ hγ)]
+      _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _) (by
+        intro γ _ _
+        split_ifs
+        · exact hA γ
+        · rfl)
+  rw [sum_sq_sum_kernel]
+  calc
+    _ ≤ ∑ γ∈S, ∑ γ'∈S, M * ∏ i:Fin r, ((Real.sqrt ((γ i.succ).val:ℝ))⁻¹ *
+        ρ^(((γ' i.succ).val:ℤ)-(γ i.succ).val).natAbs) := by
+      exact Finset.sum_le_sum (fun γ hγ => Finset.sum_le_sum (fun γ' hγ' => hcross γ hγ γ' hγ'))
+    _ = ∑ γ∈S, M*A γ * ∑ γ'∈S, ∏ i:Fin r, ρ^(((γ' i.succ).val:ℤ)-(γ i.succ).val).natAbs := by
+      simp only [Finset.prod_mul_distrib, A, Finset.mul_sum, mul_assoc]
+    _ ≤ ∑ γ∈S, M*A γ*(2/(1-ρ))^r := by
+      exact Finset.sum_le_sum (fun γ _ => mul_le_mul_of_nonneg_left (hinner γ) (mul_nonneg hM (hA γ)))
+    _ = M*(2/(1-ρ))^r * ∑ γ∈S, A γ := by
+      rw [Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intros
+      ring
+    _ ≤ M*(2/(1-ρ))^r * (Real.exp (h*U)*(6/Real.sqrt h)^r) := by
+      apply mul_le_mul_of_nonneg_left houter
+      exact mul_nonneg hM (pow_nonneg (div_nonneg (by norm_num) (by linarith [hρ.2])) _)
+    _ = _ := by ring
+
+theorem coarse_kernel_rectangular_sum {β : Type*} [Fintype β] (r U : ℕ)
+    (S T : Finset (Fin (r+1) → Fin (U+1)))
+    (K G : (Fin (r+1) → Fin (U+1)) → β → ℝ) {M ρ h : ℝ}
+    (hM : 0≤M) (hρ : ρ∈Set.Ico 0 1) (hh : 0<h) (h1 : h≤1)
+    (hS : ∀ γ∈S, ∑ i, (γ i).val=U)
+    (hT : ∀ γ∈T, ∑ i, (γ i).val=U)
+    (hcross : ∀ γ∈S, ∀ γ'∈T, (∑ z, K γ z*G γ' z) ≤
+      M * ∏ i:Fin r, ((Real.sqrt ((γ i.succ).val:ℝ))⁻¹ *
+        ρ^(((γ' i.succ).val:ℤ)-(γ i.succ).val).natAbs)) :
+    (∑ γ∈S, ∑ γ'∈T, ∑ z, K γ z*G γ' z) ≤
+      M * (2/(1-ρ))^r * Real.exp (h*U) * (6/Real.sqrt h)^r := by
+  classical
+  let A (γ : Fin (r+1) → Fin (U+1)) :=
+    ∏ i:Fin r, (Real.sqrt ((γ i.succ).val:ℝ))⁻¹
+  have hA (γ) : 0≤A γ := by dsimp [A]; positivity
+  have hinner (γ : Fin (r+1) → Fin (U+1)) :
+      (∑ γ'∈T, ∏ i:Fin r, ρ^(((γ' i.succ).val:ℤ)-(γ i.succ).val).natAbs) ≤
+        (2/(1-ρ))^r := by
+    let f : Fin (r+1) → Fin (U+1) → ℝ := Fin.cases (fun _ => 1)
+      (fun i k => ρ^((k.val:ℤ)-(γ i.succ).val).natAbs)
+    have hf : ∀ i k, 0≤f i k := by
+      intro i k
+      refine Fin.cases ?_ (fun _ => ?_) i
+      · norm_num [f]
+      · exact pow_nonneg hρ.1 _
+    have hsub : (∑ γ'∈T, ∏ i:Fin r, ρ^(((γ' i.succ).val:ℤ)-(γ i.succ).val).natAbs) ≤
+        ∑ γ' : Fin (r+1) → Fin (U+1), if ∑ i, (γ' i).val=U then ∏ i, f i (γ' i) else 0 := by
+      calc
+        _ = ∑ γ'∈T, if ∑ i, (γ' i).val=U then ∏ i, f i (γ' i) else 0 := by
+          apply Finset.sum_congr rfl
+          intro γ' hγ'
+          simp only [hT γ' hγ', ↓reduceIte, Fin.prod_univ_succ, f, Fin.cases_zero,
+            Fin.cases_succ, one_mul]
+        _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _) (by
+          intro γ' _ _
+          split_ifs
+          · exact Finset.prod_nonneg (fun i _ => hf i _)
+          · rfl)
+    apply hsub.trans
+    apply (constrained_sum_product_le r U f hf 1 (by norm_num) (by intro k; rfl)).trans
+    simp only [one_mul]
+    calc
+      _ ≤ ∏ _i:Fin r, (2/(1-ρ)) := by
+        apply Finset.prod_le_prod₀
+        · intro i _; exact Finset.sum_nonneg (fun k _ => hf i.succ k)
+        · intro i _; exact fin_geometric_distance (U+1) ((γ i.succ).val:ℤ) hρ
+      _ = _ := by simp [div_pow]
+  have houter : (∑ γ∈S, A γ) ≤ Real.exp (h*U)*(6/Real.sqrt h)^r := by
+    apply le_trans _ (coarse_constrained_inverse_sum r U hh h1)
+    calc
+      _ = ∑ γ∈S, if ∑ i, (γ i).val=U then A γ else 0 := by
+        apply Finset.sum_congr rfl
+        intro γ hγ
+        rw [ite_eq_left (hS γ hγ)]
+      _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _) (by
+        intro γ _ _
+        split_ifs
+        · exact hA γ
+        · rfl)
+  calc
+    _ ≤ ∑ γ∈S, ∑ γ'∈T, M * ∏ i:Fin r, ((Real.sqrt ((γ i.succ).val:ℝ))⁻¹ *
+        ρ^(((γ' i.succ).val:ℤ)-(γ i.succ).val).natAbs) := by
+      exact Finset.sum_le_sum (fun γ hγ => Finset.sum_le_sum (fun γ' hγ' => hcross γ hγ γ' hγ'))
+    _ = ∑ γ∈S, M*A γ * ∑ γ'∈T, ∏ i:Fin r, ρ^(((γ' i.succ).val:ℤ)-(γ i.succ).val).natAbs := by
+      simp only [Finset.prod_mul_distrib, A, Finset.mul_sum, mul_assoc]
+    _ ≤ ∑ γ∈S, M*A γ*(2/(1-ρ))^r := by
+      exact Finset.sum_le_sum (fun γ _ => mul_le_mul_of_nonneg_left (hinner γ) (mul_nonneg hM (hA γ)))
+    _ = M*(2/(1-ρ))^r * ∑ γ∈S, A γ := by
+      rw [Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intros
+      ring
+    _ ≤ M*(2/(1-ρ))^r * (Real.exp (h*U)*(6/Real.sqrt h)^r) := by
+      apply mul_le_mul_of_nonneg_left houter
+      exact mul_nonneg hM (pow_nonneg (div_nonneg (by norm_num) (by linarith [hρ.2])) _)
+    _ = _ := by ring
+
+
+end TraceReconstruction
+
+namespace TraceReconstruction
+
+/-- Geometry of actual period gaps: the half-integer coarse gaps are
+represented by positive integers a,b, with bounded slot-rounding errors. -/
+theorem coarse_gap_geometry (m U a b g k : ℕ) (hm : 32 ≤ m) (hUm : U  ≤  2*m)
+    (ha : 1 ≤ a) (hb : 1 ≤ b) (haU : a ≤ U) (hbU : b ≤ U)
+    (hg : |(g:ℝ)-(m:ℝ)*a/2| ≤ 4) (hk : |(k:ℝ)-(m:ℝ)*b/2| ≤ 4) :
+    (m:ℝ)*a/4  ≤  (g:ℝ)+1 ∧ 0<(g:ℝ)+k ∧
+      ((a:ℝ)-b)^2/1024  ≤  ((g:ℝ)-k)^2/(16*((g:ℝ)+k)) := by
+  have hmR : (32:ℝ) ≤ m := by exact_mod_cast hm
+  have hUmR : (U:ℝ)  ≤  2*m := by exact_mod_cast hUm
+  have haR : (1:ℝ) ≤ a := by exact_mod_cast ha
+  have hbR : (1:ℝ) ≤ b := by exact_mod_cast hb
+  have haUR : (a:ℝ) ≤ U := by exact_mod_cast haU
+  have hbUR : (b:ℝ) ≤ U := by exact_mod_cast hbU
+  have hga := abs_le.mp hg
+  have hkb := abs_le.mp hk
+  have hma : (32:ℝ) ≤ (m:ℝ)*a := by nlinarith
+  have hmb : (32:ℝ) ≤ (m:ℝ)*b := by nlinarith
+  have hlo : (m:ℝ)*a/4 ≤ (g:ℝ)+1 := by nlinarith only [hga.1,hma]
+  have hsum0 : 0<(g:ℝ)+k := by nlinarith only [hga.1,hkb.1,hma,hmb]
+  refine ⟨hlo,hsum0,?_⟩
+  by_cases hab : a=b
+  · simp only [hab, sub_self, zero_pow (by decide : 2 ≠ 0), zero_div]
+    exact div_nonneg (sq_nonneg _) (by positivity)
+  have hd1 : 1 ≤ |(a:ℝ)-b| := by
+    rcases lt_or_gt_of_ne hab with hab | hab
+    · rw [abs_of_neg (sub_neg.mpr (by exact_mod_cast hab : (a:ℝ) < b))]
+      have hx : (a:ℝ)+1 ≤ b := by exact_mod_cast hab
+      linarith
+    · rw [abs_of_pos (sub_pos.mpr (by exact_mod_cast hab : (b:ℝ) < a))]
+      have hx : (b:ℝ)+1 ≤ a := by exact_mod_cast hab
+      linarith
+  have herr : |((g:ℝ)-k)-(m:ℝ)/2*((a:ℝ)-b)| ≤ 8 := by
+    calc
+      _ = |((g:ℝ)-(m:ℝ)*a/2)-((k:ℝ)-(m:ℝ)*b/2)| := by congr 1; ring
+      _  ≤  |(g:ℝ)-(m:ℝ)*a/2|+|(k:ℝ)-(m:ℝ)*b/2| := abs_sub _ _
+      _  ≤  _ := by linarith
+  have hd : (m:ℝ)/4*|(a:ℝ)-b|  ≤  |(g:ℝ)-k| := by
+    have ht := abs_sub_abs_le_abs_sub ((m:ℝ)/2*((a:ℝ)-b)) ((g:ℝ)-k)
+    rw [abs_sub_comm ((m:ℝ)/2*((a:ℝ)-b)) ((g:ℝ)-k)] at ht
+    rw [abs_mul, abs_of_nonneg (by positivity : 0 ≤ (m:ℝ)/2)] at ht
+    have hmabs : (32:ℝ) ≤ (m:ℝ)*|(a:ℝ)-b| := by nlinarith only [hmR,hd1]
+    nlinarith only [ht,herr,hmabs]
+  have hsq : (m:ℝ)^2*((a:ℝ)-b)^2/16 ≤ ((g:ℝ)-k)^2 := by
+    have hx := mul_self_le_mul_self (by positivity : 0 ≤ (m:ℝ)/4*|(a:ℝ)-b|) hd
+    nlinarith only [hx, sq_abs ((a:ℝ)-b), sq_abs ((g:ℝ)-k)]
+  have hsum : (g:ℝ)+k  ≤  4*(m:ℝ)^2 := by
+    have hu1 : (1:ℝ) ≤ U := by linarith
+    have hmu : (32:ℝ) ≤ (m:ℝ)*U := by nlinarith only [hmR,hu1]
+    have hmaU := mul_le_mul_of_nonneg_left haUR (show (0:ℝ) ≤ m by positivity)
+    have hmbU := mul_le_mul_of_nonneg_left hbUR (show (0:ℝ) ≤ m by positivity)
+    have hUmM := mul_le_mul_of_nonneg_left hUmR (show (0:ℝ) ≤ m by positivity)
+    nlinarith only [hga.2,hkb.2,hmaU,hmbU,hmu,hUmM]
+  apply (le_div_iff₀ (by positivity : 0<16*((g:ℝ)+k))).mpr
+  have hx := mul_le_mul_of_nonneg_right hsum (sq_nonneg ((a:ℝ)-b))
+  nlinarith only [hx,hsq]
+
+/-- From integer Gaussian decay to a summable geometric profile. -/
+theorem exp_integer_square_le (a b : ℕ) {c : ℝ} (hc : 0 ≤ c) :
+    Real.exp (-c * ((a:ℝ)-b)^2) ≤
+      (Real.exp (-c)) ^ ((a:ℤ)-b).natAbs := by
+  rw [← Real.exp_nat_mul]
+  apply Real.exp_le_exp.mpr
+  have hd : (((a:ℤ)-b).natAbs : ℝ) = |(a:ℝ)-b| := by
+    rw [← Int.cast_natCast, Int.natCast_natAbs, Int.cast_abs, Int.cast_sub]
+    simp only [Int.cast_natCast]
+  have hsq : (((a:ℤ)-b).natAbs:ℝ) ≤ (((a:ℤ)-b).natAbs:ℝ)^2 := by
+    exact_mod_cast (show ((a:ℤ)-b).natAbs ≤ ((a:ℤ)-b).natAbs^2 by
+      simpa only [pow_two] using Nat.le_mul_self ((a:ℤ)-b).natAbs)
+  rw [hd, sq_abs] at hsq
+  have hmul := mul_le_mul_of_nonneg_left hsq hc
+  rw [hd]
+  nlinarith only [hmul]
+
+theorem coarse_binomial_overlap (m U a b g k L : ℕ)
+    (hm : 32 ≤ m) (hUm : U ≤ 2*m)
+    (ha : 1 ≤ a) (hb : 1 ≤ b) (haU : a ≤ U) (hbU : b ≤ U)
+    (hg : |(g:ℝ)-(m:ℝ)*a/2| ≤ 4) (hk : |(k:ℝ)-(m:ℝ)*b/2| ≤ 4)
+    {θ : ℝ} (hθ : θ ∈ Set.Ico (1/2:ℝ) 1) :
+    (∑ z : Fin (L+1), binomialMass g θ z * binomialMass k θ z) ≤
+      Real.sqrt (8 / ((m:ℝ)*θ*(1-θ))) * (Real.sqrt (a:ℝ))⁻¹ *
+        (Real.exp (-(1/1024:ℝ))) ^ ((a:ℤ)-b).natAbs := by
+  have geo := coarse_gap_geometry m U a b g k hm hUm ha hb haU hbU hg hk
+  have hsum : 0 < g+k := by exact_mod_cast geo.2.1
+  have hθ0 : 0 < θ := by linarith [hθ.1]
+  have hθ1 : 0 < 1-θ := sub_pos.mpr hθ.2
+  have varpos : 0 < θ*(1-θ) := mul_pos hθ0 hθ1
+  have hm0 : (0:ℝ) < m := by exact_mod_cast (show 0 < m by omega)
+  have ha0 : (0:ℝ) < a := by exact_mod_cast ha
+  have hpref : Real.sqrt (2 / (((g:ℝ)+1)*θ*(1-θ))) ≤
+      Real.sqrt (8 / ((m:ℝ)*θ*(1-θ))) * (Real.sqrt (a:ℝ))⁻¹ := by
+    rw [← Real.sqrt_inv, ← Real.sqrt_mul (by positivity : 0 ≤ 8/((m:ℝ)*θ*(1-θ)))]
+    apply Real.sqrt_le_sqrt
+    have hdg : 0 < ((g:ℝ)+1)*θ*(1-θ) := by nlinarith
+    have hdm : 0 < (m:ℝ)*θ*(1-θ) := by nlinarith
+    apply (div_le_iff₀ hdg).mpr
+    have hdiv : 0 < (m:ℝ)*θ*(1-θ)*(a:ℝ) := by positivity
+    have heq : (8 / ((m:ℝ)*θ*(1-θ)) * (a:ℝ)⁻¹) * (((g:ℝ)+1)*θ*(1-θ)) =
+        (8*((g:ℝ)+1)) / ((m:ℝ)*a) := by
+      field_simp [ne_of_gt hθ0, ne_of_gt hθ1, ne_of_gt hm0, ne_of_gt ha0]
+    rw [heq]
+    apply (le_div_iff₀ (mul_pos hm0 ha0)).mpr
+    linarith [geo.1]
+  have hexp : Real.exp (-((g:ℝ)-k)^2/(16*((g:ℝ)+k))) ≤
+      (Real.exp (-(1/1024:ℝ))) ^ ((a:ℤ)-b).natAbs := by
+    apply le_trans _ (exp_integer_square_le a b (by norm_num : (0:ℝ)≤1/1024))
+    apply Real.exp_le_exp.mpr
+    convert neg_le_neg geo.2.2 using 2 <;> first | rfl | ring
+  exact (binomial_gaussian_overlap_general g k L hθ hsum).trans
+    (mul_le_mul hpref hexp (Real.exp_nonneg _) (by positivity))
+
+end TraceReconstruction
+
+
+namespace TraceReconstruction
+
+theorem int_natAbs_sub_comm (a b : ℤ) : (a-b).natAbs = (b-a).natAbs := by
+  rw [← Int.natAbs_neg (a-b),neg_sub]
+
+theorem subsetRankKernel_cross_normalized {n l r : ℕ} (hl : l  ≤  n)
+    (p q : Fin r → Fin n) (hp : StrictMono p) (hq : StrictMono q)
+    {θ : ℝ} (hθ : θ  ∈  Set.Ico (1/2:ℝ) 1)
+    (hmean : (n:ℝ)*θ=l) (hv : 1 ≤ (n:ℝ)*θ*(1-θ)) (e : Fin (r+1)) :
+    (∑ v : IncreasingTuple l r,
+      subsetRankKernel hl p v.val * subsetRankKernel hl q v.val)  ≤ 
+      (4608*n/((tupleGap p e:ℝ)+1))*
+        ∏ i : Fin r, ∑ k : Fin (l-r+1),
+          binomialMass (tupleGap p (Equiv.swap 0 e i.succ)) θ k *
+          binomialMass (tupleGap q (Equiv.swap 0 e i.succ)) θ k := by
+  have hθ0 : 0 < θ := by linarith [hθ.1]
+  have hqpos : 0 < 1-θ := sub_pos.mpr hθ.2
+  have hc : (θ^r/binomialMass n θ l)^2  ≤  144*((n:ℝ)*θ*(1-θ)) := by
+    rw [div_eq_mul_inv,mul_pow]
+    apply (mul_le_mul_of_nonneg_right
+      (show (θ^r)^2  ≤  (1:ℝ) by
+        exact pow_le_one₀ (pow_nonneg hθ0.le _) (pow_le_one₀ hθ0.le hθ.2.le))
+      (sq_nonneg _)).trans
+    simpa only [one_mul] using binomialMass_mean_inverse_sq n l ⟨hθ0,hθ.2⟩ hl hmean hv
+  have hprod : 0  ≤  ∏ i : Fin r, ∑ k : Fin (l-r+1),
+      binomialMass (tupleGap p (Equiv.swap 0 e i.succ)) θ k *
+      binomialMass (tupleGap q (Equiv.swap 0 e i.succ)) θ k := by
+    apply Finset.prod_nonneg
+    intro i _
+    exact Finset.sum_nonneg fun k _ => mul_nonneg
+      (binomialMass_nonneg _ _ ⟨hθ0.le,hθ.2.le⟩)
+      (binomialMass_nonneg _ _ ⟨hθ0.le,hθ.2.le⟩)
+  apply (subsetRankKernel_cross_le hl p q hp hq θ ⟨hθ0,hθ.2⟩ e
+    (32/(((tupleGap p e:ℝ)+1)*θ*(1-θ))) (by positivity)
+    (fun k => binomial_product_sup _ _ k hθ)).trans
+  calc
+    _  ≤  (144*((n:ℝ)*θ*(1-θ)))*(32/(((tupleGap p e:ℝ)+1)*θ*(1-θ)))*_ := by
+      exact mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hc (by positivity)) hprod
+    _ = _ := by
+      have hq0 : 1-θ ≠ 0 := ne_of_gt (sub_pos.mpr hθ.2)
+      field_simp
+      ring
+
+def periodProfile {u r : ℕ} (hr : 0 < r) (σ : Equiv.Perm (Fin (r+1)))
+    (j : IncreasingTuple u r) : Fin (r+1) → Fin (2*u+1) :=
+  fun i => ⟨coarsePeriodGap j.val (σ i),
+    Nat.lt_succ_of_le (coarsePeriodGap_le hr j.val j.property (σ i))⟩
+
+theorem periodProfile_sum {u r : ℕ} (hr : 0 < r) (σ : Equiv.Perm (Fin (r+1)))
+    (j : IncreasingTuple u r) : ∑ i, (periodProfile hr σ j i).val = 2*u := by
+  change (∑ i, coarsePeriodGap j.val (σ i)) = _
+  rw [Equiv.sum_comp σ]
+  exact coarsePeriodGap_sum hr j.val j.property
+
+theorem periodProfile_injective {u r : ℕ} (hr : 0 < r) (σ : Equiv.Perm (Fin (r+1))) :
+    Function.Injective (periodProfile (u:=u) hr σ) := by
+  intro j k h
+  apply Subtype.ext
+  apply coarsePeriodGap_injective j.property k.property
+  funext i
+  have he := congrArg Fin.val (congrFun h (σ.symm i))
+  simpa only [periodProfile,Equiv.apply_symm_apply] using he
+
+theorem period_kernel_cross {u m l r : ℕ} (hl : l ≤ u*m) (hr : 0 < r)
+    (hm : 32 ≤ m) (hum : u ≤ m) {θ : ℝ} (hθ : θ ∈ Set.Ico (1/2:ℝ) 1)
+    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
+    (off : Fin u → Fin m)
+    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
+    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2)
+    (j k : IncreasingTuple u r) (e : Fin (r+1))
+    (he : 2*u  ≤  (r+1)*coarsePeriodGap j.val e) :
+    (∑ v : IncreasingTuple l r,
+      subsetRankKernel hl (periodSlot off ∘ j.val) v.val *
+      subsetRankKernel hl (periodSlot off ∘ k.val) v.val)  ≤ 
+      (9216*(r+1:ℕ) : ℝ)* (Real.sqrt (8/((m:ℝ)*θ*(1-θ))))^r *
+        ∏ i : Fin r, (Real.sqrt ((periodProfile hr (Equiv.swap 0 e) j i.succ).val:ℝ))⁻¹ *
+          (Real.exp (-(1/1024:ℝ)))^(((periodProfile hr (Equiv.swap 0 e) k i.succ).val:ℤ)-
+            (periodProfile hr (Equiv.swap 0 e) j i.succ).val).natAbs := by
+  classical
+  have hθ0 : 0 < θ := by linarith [hθ.1]
+  have hm0 : (0:ℝ) < m := by exact_mod_cast (show 0 < m by omega)
+  have hUm : 2*u ≤ 2*m := Nat.mul_le_mul_left 2 hum
+  have hg (v : IncreasingTuple u r) (i : Fin (r+1)) :=
+    period_gap_rounding hr off hoff hdiff v.val v.property i
+  have hgeo := coarse_gap_geometry m (2*u) (coarsePeriodGap j.val e)
+    (coarsePeriodGap k.val e) (tupleGap (periodSlot off ∘ j.val) e)
+    (tupleGap (periodSlot off ∘ k.val) e) hm hUm
+    (coarsePeriodGap_pos j.val e) (coarsePeriodGap_pos k.val e)
+    (coarsePeriodGap_le hr j.val j.property e) (coarsePeriodGap_le hr k.val k.property e)
+    (hg j e) (hg k e)
+  have hlarge : (4608 * (u*m:ℕ) : ℝ)/((tupleGap (periodSlot off ∘ j.val) e:ℝ)+1)  ≤ 
+      9216*(r+1:ℕ) := by
+    have heR : 2*(u:ℝ)  ≤  (r+1:ℕ)* (coarsePeriodGap j.val e:ℝ) := by exact_mod_cast he
+    apply (div_le_iff₀ (by positivity : 0 < (tupleGap (periodSlot off ∘ j.val) e:ℝ)+1)).mpr
+    have hx := mul_le_mul_of_nonneg_right hgeo.1
+      (show (0:ℝ) ≤ (r+1:ℕ) by positivity)
+    have hy := mul_le_mul_of_nonneg_left heR hm0.le
+    simp only [Nat.cast_mul]
+    nlinarith only [hx,hy]
+  have hover (i : Fin r) :
+      (∑ z : Fin (l-r+1),
+        binomialMass (tupleGap (periodSlot off ∘ j.val) (Equiv.swap 0 e i.succ)) θ z *
+        binomialMass (tupleGap (periodSlot off ∘ k.val) (Equiv.swap 0 e i.succ)) θ z)  ≤ 
+      Real.sqrt (8/((m:ℝ)*θ*(1-θ))) *
+        ((Real.sqrt ((periodProfile hr (Equiv.swap 0 e) j i.succ).val:ℝ))⁻¹ *
+          (Real.exp (-(1/1024:ℝ)))^(((periodProfile hr (Equiv.swap 0 e) k i.succ).val:ℤ)-
+            (periodProfile hr (Equiv.swap 0 e) j i.succ).val).natAbs) := by
+    have ho := coarse_binomial_overlap m (2*u) _ _ _ _ (l-r) hm hUm
+      (coarsePeriodGap_pos j.val _) (coarsePeriodGap_pos k.val _)
+      (coarsePeriodGap_le hr j.val j.property _) (coarsePeriodGap_le hr k.val k.property _)
+      (hg j (Equiv.swap 0 e i.succ)) (hg k (Equiv.swap 0 e i.succ)) hθ
+    rw [int_natAbs_sub_comm] at ho
+    simpa only [periodProfile,mul_assoc] using ho
+  have hn (i : Fin r) : 0 ≤  ∑ z : Fin (l-r+1),
+      binomialMass (tupleGap (periodSlot off ∘ j.val) (Equiv.swap 0 e i.succ)) θ z *
+      binomialMass (tupleGap (periodSlot off ∘ k.val) (Equiv.swap 0 e i.succ)) θ z := by
+    exact Finset.sum_nonneg fun z _ => mul_nonneg
+      (binomialMass_nonneg _ _ ⟨hθ0.le,hθ.2.le⟩) (binomialMass_nonneg _ _ ⟨hθ0.le,hθ.2.le⟩)
+  apply (subsetRankKernel_cross_normalized hl _ _
+    ((periodSlot_strictMono off).comp j.property) ((periodSlot_strictMono off).comp k.property)
+    hθ hmean hv e).trans
+  calc
+    _  ≤  (9216*(r+1:ℕ) : ℝ)*
+        ∏ i : Fin r, (Real.sqrt (8/((m:ℝ)*θ*(1-θ))) *
+        ((Real.sqrt ((periodProfile hr (Equiv.swap 0 e) j i.succ).val:ℝ))⁻¹ *
+          (Real.exp (-(1/1024:ℝ)))^(((periodProfile hr (Equiv.swap 0 e) k i.succ).val:ℤ)-
+            (periodProfile hr (Equiv.swap 0 e) j i.succ).val).natAbs)) := by
+      apply mul_le_mul hlarge (Finset.prod_le_prod₀ (fun i _ => hn i) (fun i _ => hover i))
+        (Finset.prod_nonneg (fun i _ => hn i)) (by positivity)
+    _ = _ := by rw [Finset.prod_mul_distrib]; simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fin]; ring
+
+end TraceReconstruction
+
+
+namespace TraceReconstruction
+
+theorem coarse_profile_geometric_sum {α : Type*} [Fintype α] (r U : ℕ)
+    (profile : α → Fin (r+1) → Fin (U+1)) (hinj : Function.Injective profile)
+    (htotal : ∀ a, ∑ i, (profile a i).val=U)
+    (γ : Fin (r+1) → Fin (U+1)) {ρ : ℝ} (hρ : ρ∈Set.Ico 0 1) :
+    (∑ a, ∏ i : Fin r, ρ^(((profile a i.succ).val:ℤ)-(γ i.succ).val).natAbs) ≤
+      (2/(1-ρ))^r := by
+  classical
+  let f : Fin (r+1) → Fin (U+1) → ℝ := Fin.cases (fun _ => 1)
+    (fun i k => ρ^((k.val:ℤ)-(γ i.succ).val).natAbs)
+  have hf : ∀ i k, 0≤f i k := by
+    intro i k
+    refine Fin.cases ?_ (fun _ => ?_) i
+    · norm_num [f]
+    · exact pow_nonneg hρ.1 _
+  have hid (δ : Fin (r+1) → Fin (U+1)) :
+      (∏ i : Fin r, ρ^(((δ i.succ).val:ℤ)-(γ i.succ).val).natAbs) = ∏ i, f i (δ i) := by
+    simp only [Fin.prod_univ_succ,f,Fin.cases_zero,Fin.cases_succ,one_mul]
+  simp_rw [hid]
+  apply (sum_injective_profile_le profile hinj (fun δ => ∑ i, (δ i).val=U)
+    htotal (fun δ => ∏ i, f i (δ i)) (fun δ => Finset.prod_nonneg (fun i _ => hf i _))).trans
+  apply (constrained_sum_product_le r U f hf 1 (by norm_num) (by intro k; rfl)).trans
+  simp only [one_mul]
+  calc
+    _ ≤ ∏ _i : Fin r, (2/(1-ρ)) := by
+      apply Finset.prod_le_prod₀
+      · exact fun i _ => Finset.sum_nonneg (fun k _ => hf i.succ k)
+      · exact fun i _ => fin_geometric_distance (U+1) ((γ i.succ).val:ℤ) hρ
+    _ = _ := by simp [div_pow]
+
+theorem coarse_profile_inverse_sum {α : Type*} [Fintype α] (r U : ℕ)
+    (profile : α → Fin (r+1) → Fin (U+1)) (hinj : Function.Injective profile)
+    (htotal : ∀ a, ∑ i, (profile a i).val=U)
+    {h : ℝ} (hh : 0<h) (h1 : h≤1) :
+    (∑ a, ∏ i : Fin r, (Real.sqrt ((profile a i.succ).val:ℝ))⁻¹) ≤
+      Real.exp (h*U)*(6/Real.sqrt h)^r := by
+  classical
+  apply le_trans _ (coarse_constrained_inverse_sum r U hh h1)
+  exact sum_injective_profile_le profile hinj (fun δ => ∑ i, (δ i).val=U)
+    htotal (fun δ => ∏ i : Fin r, (Real.sqrt ((δ i.succ).val:ℝ))⁻¹) (fun δ => by positivity)
+
+/-- Coarse Schur bound for actual injective profiles, with no extension of
+kernels outside their geometric domain. -/
+theorem coarse_profile_bilinear_sum {α β : Type*} [Fintype α] [Fintype β]
+    (r U : ℕ) (p : α → Fin (r+1) → Fin (U+1)) (q : β → Fin (r+1) → Fin (U+1))
+    (hp : Function.Injective p) (hq : Function.Injective q)
+    (hpt : ∀ a, ∑ i, (p a i).val=U) (hqt : ∀ b, ∑ i, (q b i).val=U)
+    (K : α → β → ℝ) {M ρ h : ℝ}
+    (hM : 0≤M) (hρ : ρ∈Set.Ico 0 1) (hh : 0<h) (h1 : h≤1)
+    (hcross : ∀ a b, K a b ≤ M * ∏ i : Fin r,
+      (Real.sqrt ((p a i.succ).val:ℝ))⁻¹ *
+        ρ^(((q b i.succ).val:ℤ)-(p a i.succ).val).natAbs) :
+    (∑ a, ∑ b, K a b) ≤ M*(2/(1-ρ))^r * Real.exp (h*U)*(6/Real.sqrt h)^r := by
+  classical
+  let A (a : α) := ∏ i : Fin r, (Real.sqrt ((p a i.succ).val:ℝ))⁻¹
+  have hA (a) : 0≤A a := by dsimp [A]; positivity
+  have hinner (a) := coarse_profile_geometric_sum r U q hq hqt (p a) hρ
+  have houter := coarse_profile_inverse_sum r U p hp hpt hh h1
+  calc
+    _ ≤ ∑ a, ∑ b, M * ∏ i : Fin r,
+        (Real.sqrt ((p a i.succ).val:ℝ))⁻¹ *
+          ρ^(((q b i.succ).val:ℤ)-(p a i.succ).val).natAbs :=
+      Finset.sum_le_sum (fun a _ => Finset.sum_le_sum (fun b _ => hcross a b))
+    _ = ∑ a, M*A a * ∑ b, ∏ i : Fin r, ρ^(((q b i.succ).val:ℤ)-(p a i.succ).val).natAbs := by
+      simp only [Finset.prod_mul_distrib,A,Finset.mul_sum,mul_assoc]
+    _ ≤ ∑ a, M*A a*(2/(1-ρ))^r :=
+      Finset.sum_le_sum (fun a _ => mul_le_mul_of_nonneg_left (hinner a) (mul_nonneg hM (hA a)))
+    _ = M*(2/(1-ρ))^r * ∑ a, A a := by
+      rw [Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intros
+      ring
+    _ ≤ M*(2/(1-ρ))^r * (Real.exp (h*U)*(6/Real.sqrt h)^r) := by
+      apply mul_le_mul_of_nonneg_left houter
+      exact mul_nonneg hM (pow_nonneg (div_nonneg (by norm_num) (by linarith [hρ.2])) _)
+    _ = _ := by ring
+
+end TraceReconstruction
+
+namespace TraceReconstruction
+
+def LargePeriodTuple (u r : ℕ) (e : Fin (r+1)) :=
+  {j : IncreasingTuple u r // 2*u ≤ (r+1)*coarsePeriodGap j.val e}
+
+noncomputable instance largePeriodTupleFintype (u r : ℕ) (e : Fin (r+1)) :
+    Fintype (LargePeriodTuple u r e) := by
+  unfold LargePeriodTuple
+  exact Fintype.ofFinite _
+
+theorem exists_large_period_gap {u r : ℕ} (hr : 0 < r) (j : IncreasingTuple u r) :
+    ∃ e : Fin (r+1), 2*u ≤ (r+1)*coarsePeriodGap j.val e := by
+  obtain ⟨e,_,he⟩ := Finset.exists_max_image Finset.univ (coarsePeriodGap j.val)
+    Finset.univ_nonempty
+  refine ⟨e,?_⟩
+  calc
+    _ = ∑ i, coarsePeriodGap j.val i := (coarsePeriodGap_sum hr j.val j.property).symm
+    _ ≤ ∑ _i : Fin (r+1), coarsePeriodGap j.val e := Finset.sum_le_sum (fun i hi => he i hi)
+    _ = _ := by simp
+
+/-- Unnormalized but explicit source rank-kernel square-sum estimate. The
+only remaining factors are elementary scale simplifications. -/
+theorem period_rank_square_sum_raw {u m l r : ℕ} (hl : l ≤ u*m) (hr : 0 < r)
+    (hm : 32 ≤ m) (hum : u ≤ m) {θ h : ℝ} (hθ : θ ∈ Set.Ico (1/2:ℝ) 1)
+    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
+    (hh : 0 < h) (h1 : h ≤ 1)
+    (off : Fin u → Fin m)
+    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
+    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2) :
+    (∑ v : IncreasingTuple l r,
+      (∑ j : IncreasingTuple u r, subsetRankKernel hl (periodSlot off ∘ j.val) v.val)^2) ≤
+      9216*(r+1:ℕ)^2 * (Real.sqrt (8/((m:ℝ)*θ*(1-θ))))^r *
+        (2/(1-Real.exp (-(1/1024:ℝ))))^r * Real.exp (h*(2*u:ℕ)) * (6/Real.sqrt h)^r := by
+  classical
+  let K (j : IncreasingTuple u r) (v : IncreasingTuple l r) :=
+    subsetRankKernel hl (periodSlot off ∘ j.val) v.val
+  let C (j k : IncreasingTuple u r) := ∑ v : IncreasingTuple l r, K j v*K k v
+  have hC (j k) : 0 ≤ C j k := by
+    exact Finset.sum_nonneg (fun v _ => mul_nonneg
+      (subsetRankKernel_nonneg hl _ _) (subsetRankKernel_nonneg hl _ _))
+  have hρ : Real.exp (-(1/1024:ℝ)) ∈ Set.Ico 0 1 := by
+    refine ⟨(Real.exp_pos _).le,?_⟩
+    rw [Real.exp_lt_one_iff]
+    norm_num
+  have hschur (e : Fin (r+1)) :
+      (∑ j : LargePeriodTuple u r e, ∑ k : IncreasingTuple u r, C j.val k) ≤
+        ((9216*(r+1:ℕ):ℝ)*(Real.sqrt (8/((m:ℝ)*θ*(1-θ))))^r) *
+          (2/(1-Real.exp (-(1/1024:ℝ))))^r * Real.exp (h*(2*u:ℕ)) * (6/Real.sqrt h)^r := by
+    apply coarse_profile_bilinear_sum r (2*u)
+      (fun j : LargePeriodTuple u r e => periodProfile hr (Equiv.swap 0 e) j.val)
+      (periodProfile hr (Equiv.swap 0 e))
+      ((periodProfile_injective hr (Equiv.swap 0 e)).comp Subtype.val_injective)
+      (periodProfile_injective hr (Equiv.swap 0 e))
+      (fun j => periodProfile_sum hr _ j.val) (periodProfile_sum hr _)
+      (fun j k => C j.val k) (by positivity) hρ hh h1
+    intro j k
+    exact period_kernel_cross hl hr hm hum hθ hmean hv off hoff hdiff j.val k e j.property
+  have hcover (j : IncreasingTuple u r) :
+      (∑ k : IncreasingTuple u r, C j k) ≤ ∑ e : Fin (r+1),
+        if 2*u ≤ (r+1)*coarsePeriodGap j.val e then ∑ k : IncreasingTuple u r, C j k else 0 := by
+    obtain ⟨e,he⟩ := exists_large_period_gap hr j
+    have hn : ∀ e' : Fin (r+1), 0 ≤
+        if 2*u ≤ (r+1)*coarsePeriodGap j.val e' then ∑ k : IncreasingTuple u r, C j k else 0 := by
+      intro e'
+      split_ifs
+      · exact Finset.sum_nonneg (fun k _ => hC j k)
+      · rfl
+    simpa only [ite_eq_left he] using Finset.single_le_sum (fun e' _ => hn e') (Finset.mem_univ e)
+  have heq (e : Fin (r+1)) :
+      (∑ j : IncreasingTuple u r,
+        if 2*u ≤ (r+1)*coarsePeriodGap j.val e then ∑ k : IncreasingTuple u r, C j k else 0) =
+      ∑ j : LargePeriodTuple u r e, ∑ k : IncreasingTuple u r, C j.val k := by
+    rw [← Finset.sum_filter]
+    exact Finset.sum_subtype _ (by intro j; simp) _
+  calc
+    _ = ∑ j : IncreasingTuple u r, ∑ k : IncreasingTuple u r, C j k := by
+      exact sum_sq_sum_kernel Finset.univ K
+    _ ≤ ∑ j : IncreasingTuple u r, ∑ e : Fin (r+1),
+        if 2*u ≤ (r+1)*coarsePeriodGap j.val e then ∑ k : IncreasingTuple u r, C j k else 0 :=
+      Finset.sum_le_sum (fun j _ => hcover j)
+    _ = ∑ e : Fin (r+1), ∑ j : LargePeriodTuple u r e,
+        ∑ k : IncreasingTuple u r, C j.val k := by
+      rw [Finset.sum_comm]
+      exact Finset.sum_congr rfl (fun e _ => heq e)
+    _ ≤ ∑ _e : Fin (r+1),
+        ((9216*(r+1:ℕ):ℝ)*(Real.sqrt (8/((m:ℝ)*θ*(1-θ))))^r) *
+          (2/(1-Real.exp (-(1/1024:ℝ))))^r * Real.exp (h*(2*u:ℕ)) * (6/Real.sqrt h)^r :=
+      Finset.sum_le_sum (fun e _ => hschur e)
+    _ = _ := by simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]; ring
+
+end TraceReconstruction
+
+namespace TraceReconstruction
+
+theorem rank_polynomial_factor {r : ℕ} (hr : 0 < r) :
+    (9216:ℝ)*(r+1:ℕ)^2 ≤ (36864:ℝ)^r := by
+  have h1 : (9216:ℝ) ≤ (9216:ℝ)^r := le_self_pow₀ (by norm_num) (Nat.ne_of_gt hr)
+  have h2 : ((r+1:ℕ):ℝ) ≤ (2:ℝ)^r := by exact_mod_cast nat_add_one_le_two_pow r
+  have h3 := pow_le_pow_left₀ (by positivity : (0:ℝ) ≤ (r+1:ℕ)) h2 2
+  have h4 : ((2:ℝ)^r)^2 = (4:ℝ)^r := by
+    rw [← pow_mul,Nat.mul_comm r 2,pow_mul]
+    norm_num
+  calc
+    _ ≤ (9216:ℝ)^r*((2:ℝ)^r)^2 := mul_le_mul h1 h3 (by positivity) (by positivity)
+    _ = _ := by rw [h4,← mul_pow]; norm_num
+
+noncomputable def rankSchurConstant : ℝ :=
+  884736 * (2/(1-Real.exp (-(1/1024:ℝ)))) * Real.exp 1
+
+theorem rankSchurConstant_pos : 0 < rankSchurConstant := by
+  have hρ : Real.exp (-(1/1024:ℝ)) < 1 := by rw [Real.exp_lt_one_iff]; norm_num
+  unfold rankSchurConstant
+  positivity
+
+/-- Rank-kernel square sum, with an explicit absolute constant, uniformly
+in the defect types and admissible conditional count. -/
+theorem period_rank_square_sum {u m l r : ℕ} (hl : l ≤ u*m) (hr : 0 < r)
+    (hru : r ≤ u) (hm : 32 ≤ m) (hum : u ≤ m) {θ : ℝ} (hθ : θ ∈ Set.Ico (1/2:ℝ) 1)
+    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
+    (off : Fin u → Fin m)
+    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
+    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2) :
+    (∑ v : IncreasingTuple l r,
+      (∑ j : IncreasingTuple u r, subsetRankKernel hl (periodSlot off ∘ j.val) v.val)^2) ≤
+      (rankSchurConstant * Real.sqrt ((u:ℝ)/((m:ℝ)*r*θ*(1-θ))))^r := by
+  have hr0 : (0:ℝ) < r := by exact_mod_cast hr
+  have hu0 : (0:ℝ) < u := by exact_mod_cast (show 0 < u by omega)
+  have hm0 : (0:ℝ) < m := by exact_mod_cast (show 0 < m by omega)
+  have hθ0 : 0 < θ := by linarith [hθ.1]
+  have hq0 : 0 < 1-θ := sub_pos.mpr hθ.2
+  let h : ℝ := (r:ℝ)/(2*u)
+  have hh : 0 < h := div_pos hr0 (by positivity)
+  have h1 : h ≤ 1 := by
+    apply (div_le_one (by positivity : (0:ℝ) < 2*u)).mpr
+    have hruR : (r:ℝ) ≤ u := by exact_mod_cast hru
+    linarith
+  have hx : h*(2*u:ℕ) = r := by
+    dsimp [h]
+    simp only [Nat.cast_mul,Nat.cast_ofNat]
+    field_simp
+  have he : Real.exp (h*(2*u:ℕ)) = (Real.exp 1)^r := by
+    rw [hx,← Real.exp_nat_mul]
+    simp
+  have hρ : 0 ≤ 2/(1-Real.exp (-(1/1024:ℝ))) := by
+    have hn : Real.exp (-(1/1024:ℝ)) < 1 := by rw [Real.exp_lt_one_iff]; norm_num
+    positivity
+  have hp : 0 ≤ (u:ℝ)/((m:ℝ)*r*θ*(1-θ)) := by positivity
+  have hs : Real.sqrt (8/((m:ℝ)*θ*(1-θ))) * (6/Real.sqrt h) ≤
+      24 * Real.sqrt ((u:ℝ)/((m:ℝ)*r*θ*(1-θ))) := by
+    have hsq : (Real.sqrt (8/((m:ℝ)*θ*(1-θ))) * (6/Real.sqrt h))^2 =
+        576 * ((u:ℝ)/((m:ℝ)*r*θ*(1-θ))) := by
+      rw [mul_pow,div_pow,Real.sq_sqrt (by positivity : (0:ℝ) ≤ 8/((m:ℝ)*θ*(1-θ))),
+        Real.sq_sqrt hh.le]
+      dsimp [h]
+      field_simp
+      ring
+    nlinarith [Real.sq_sqrt hp, Real.sqrt_nonneg ((u:ℝ)/((m:ℝ)*r*θ*(1-θ)))]
+  apply (period_rank_square_sum_raw hl hr hm hum hθ hmean hv hh h1 off hoff hdiff).trans
+  rw [he]
+  calc
+    _ ≤ (36864:ℝ)^r * (Real.sqrt (8/((m:ℝ)*θ*(1-θ))))^r *
+        (2/(1-Real.exp (-(1/1024:ℝ))))^r * (Real.exp 1)^r * (6/Real.sqrt h)^r := by
+      gcongr
+      exact rank_polynomial_factor hr
+    _ = ((36864:ℝ)*(2/(1-Real.exp (-(1/1024:ℝ))))*Real.exp 1 *
+        (Real.sqrt (8/((m:ℝ)*θ*(1-θ)))*(6/Real.sqrt h)))^r := by
+      simp only [mul_pow]
+      ring
+    _ ≤ ((36864:ℝ)*(2/(1-Real.exp (-(1/1024:ℝ))))*Real.exp 1 *
+        (24 * Real.sqrt ((u:ℝ)/((m:ℝ)*r*θ*(1-θ)))))^r := by
+      apply pow_le_pow_left₀ (by positivity)
+      exact mul_le_mul_of_nonneg_left hs (by positivity)
+    _ = _ := by congr 1; unfold rankSchurConstant; ring
+
+end TraceReconstruction
+
+namespace TraceReconstruction
+
+/-- The source's channel-uniform rank-kernel estimate (base-kernel),
+with the absolute constant made explicit. -/
+theorem period_rank_square_sum_delta {u m l r : ℕ} (hl : l ≤ u*m) (hr : 0 < r)
+    (hru : r ≤ u) (hm : 32 ≤ m) (hum : u ≤ m) {δ θ : ℝ}
+    (hδ : δ ∈ Set.Ioc 0 (1/2:ℝ))
+    (hθ : θ ∈ Set.Icc (1-δ^2) (1-δ^2/4))
+    (hmean : ((u*m:ℕ):ℝ)*θ=l) (hv : 1 ≤ ((u*m:ℕ):ℝ)*θ*(1-θ))
+    (off : Fin u → Fin m)
+    (hoff : ∀ j, |((off j).val:ℝ)-(m:ℝ)/2| ≤ 3)
+    (hdiff : ∀ j k, |((off j).val:ℝ)-(off k).val| ≤ 2) :
+    (∑ v : IncreasingTuple l r,
+      (∑ j : IncreasingTuple u r, subsetRankKernel hl (periodSlot off ∘ j.val) v.val)^2) ≤
+      ((3*rankSchurConstant/δ) * Real.sqrt ((u:ℝ)/((m:ℝ)*r)))^r := by
+  have hd0 : 0 < δ := hδ.1
+  have hd2 : 0 < δ^2 := sq_pos_of_pos hδ.1
+  have hd2b : δ^2 ≤ 1/4 := by nlinarith [hδ.1,hδ.2]
+  have ht : θ ∈ Set.Ico (1/2:ℝ) 1 := by
+    constructor <;> nlinarith [hθ.1,hθ.2]
+  have ht0 : 0 < θ := by linarith [ht.1]
+  have hq0 : 0 < 1-θ := sub_pos.mpr ht.2
+  have hr0 : (0:ℝ) < r := by exact_mod_cast hr
+  have hm0 : (0:ℝ) < m := by exact_mod_cast (show 0 < m by omega)
+  have hu0 : (0:ℝ) ≤ u := Nat.cast_nonneg u
+  have hvar : δ^2/8 ≤ θ*(1-θ) := by
+    have hq : δ^2/4 ≤ 1-θ := by linarith [hθ.2]
+    nlinarith [mul_le_mul ht.1 hq (by positivity) ht0.le]
+  have hdiv : (u:ℝ)/((m:ℝ)*r*θ*(1-θ)) ≤
+      (3/δ)^2*((u:ℝ)/((m:ℝ)*r)) := by
+    have hden : (0:ℝ) < (m:ℝ)*r*θ*(1-θ) := by positivity
+    apply (div_le_iff₀ hden).mpr
+    have he : (3/δ)^2 * ((u:ℝ)/((m:ℝ)*r)) * ((m:ℝ)*r*θ*(1-θ)) =
+        ((u:ℝ)*9*θ*(1-θ))/δ^2 := by field_simp; ring
+    rw [he]
+    apply (le_div_iff₀ hd2).mpr
+    nlinarith [mul_le_mul_of_nonneg_left hvar (by positivity : (0:ℝ) ≤ 9*(u:ℝ))]
+  have hs : Real.sqrt ((u:ℝ)/((m:ℝ)*r*θ*(1-θ))) ≤
+      (3/δ)*Real.sqrt ((u:ℝ)/((m:ℝ)*r)) := by
+    calc
+      _ ≤ Real.sqrt ((3/δ)^2*((u:ℝ)/((m:ℝ)*r))) := Real.sqrt_le_sqrt hdiv
+      _ = _ := by
+        rw [Real.sqrt_mul (sq_nonneg _),Real.sqrt_sq (by positivity)]
+  apply (period_rank_square_sum hl hr hru hm hum ht hmean hv off hoff hdiff).trans
+  apply pow_le_pow_left₀ (by positivity [rankSchurConstant_pos])
+  calc
+    _ ≤ rankSchurConstant*((3/δ)*Real.sqrt ((u:ℝ)/((m:ℝ)*r))) :=
+      mul_le_mul_of_nonneg_left hs rankSchurConstant_pos.le
+    _ = _ := by ring
+
+end TraceReconstruction
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/TraceReconstruction/Restoration.lean b/lean/OAI/Probability/TraceReconstruction/Restoration.lean
index 936499cc964e68dc66a29ef4dcc16d94ad2d0184..7b80a78c81e10d0007989a166cab6bfc39f4cfcd 100644
--- a/lean/OAI/Probability/TraceReconstruction/Restoration.lean
+++ b/lean/OAI/Probability/TraceReconstruction/Restoration.lean
@@ -1,217 +1,217 @@
-import OAI.Probability.TraceReconstruction.Amplification
-import OAI.Probability.TraceReconstruction.Propagation
-
-namespace OAI
-
-/-!
-# Restoration
-
-Restoration to honest complete traces, degradation to the requested channel, and exact-length padding.
--/
-
-open scoped ENNReal
-noncomputable section
-
-open Filter
-open scoped Topology
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-namespace FiniteLaw
-variable {α β : Type*} [Fintype α] [Fintype β]
-
-theorem approximate_hellinger_transfer (P Q : FiniteLaw α) (K : α→FiniteLaw β)
-    (R S : FiniteLaw β) {E V : ℝ} (hH : P.hellingerSq Q ≤ V)
-    (hR : R.tv (P.bind K) ≤ E) (hS : S.tv (Q.bind K) ≤ E) (m : ℕ) :
-    (R.product m).tv (S.product m) ≤ Real.sqrt ((m:ℝ)*V)+2*m*E := by
-  have hmid := (tv_le_sqrt_hellingerSq ((P.bind K).product m) ((Q.bind K).product m)).trans
-    (Real.sqrt_le_sqrt ((hellingerSq_product_le _ _ m).trans
-      (mul_le_mul_of_nonneg_left ((hellingerSq_bind_le P Q K).trans hH) (Nat.cast_nonneg m))))
-  have hr := (tv_product_le R (P.bind K) m).trans (mul_le_mul_of_nonneg_left hR (Nat.cast_nonneg m))
-  have hs := (tv_product_le S (Q.bind K) m).trans (mul_le_mul_of_nonneg_left hS (Nat.cast_nonneg m))
-  rw [tv_symm (S.product m)] at hs
-  linarith only [hr,hs,hmid,tv_triangle (R.product m) ((P.bind K).product m) (S.product m),
-    tv_triangle ((P.bind K).product m) ((Q.bind K).product m) (S.product m)]
-
-end FiniteLaw
-
-/-- A total common finite projection. The fallback is never used by a
- genuine trace of a length-at-most-n input. -/
-def boundedTraceProjection (n : ℕ) (w : List Bool) : BoundedTrace n :=
-  if h : w∈traceSpace n then ⟨w,h⟩ else ⟨[],mem_traceSpace_of_length_le [] (by simp)⟩
-
-theorem boundedTraceProjection_val {n : ℕ} {w : List Bool} (hw : w.length ≤ n) :
-    (boundedTraceProjection n w).val=w := by
-  simp only [boundedTraceProjection,dite_eq_left (mem_traceSpace_of_length_le w hw)]
-
-theorem listTraceLaw_projection (q : ℝ) (hq : q∈Set.Icc 0 1) (w : List Bool)
-    (n : ℕ) (hw : w.length≤n) (f : BoundedTrace n→ℝ) :
-    (listTraceLaw q hq w n hw).expect f=
-      listDeletionExpectation q w (fun z=>f (boundedTraceProjection n z)) := by
-  rw [←listTraceLaw_expect q hq w n hw]
-  congr 1
-  funext t
-  congr 1
-  apply Subtype.ext
-  exact (boundedTraceProjection_val (length_le_of_mem_traceSpace t.property)).symm
-
-theorem ApproximatesTrace.projected_tv {α : Type*} [Fintype α] {q E : ℝ}
-    {P : FiniteLaw α} {decode : α→List Bool} {w : List Bool}
-    (H : ApproximatesTrace q P decode w E) (hq : q∈Set.Icc 0 1)
-    (n : ℕ) (hw : w.length≤n) :
-    (listTraceLaw q hq w n hw).tv (P.map (fun a=>boundedTraceProjection n (decode a))) ≤ E := by
-  apply FiniteLaw.tv_le_of_tests
-  intro f hf
-  rw [listTraceLaw_projection,FiniteLaw.expect_map]
-  simp only [Function.comp_def]
-  have hh := H (fun z=>f (boundedTraceProjection n z)) (fun z=>hf _)
-  linarith only [(abs_le.mp hh).1]
-
-/-- Restoration for m independent complete observations, not a single
- concatenated trace. -/
-theorem HellingerTracePair.sample_tv {q E V : ℝ} {x y : List Bool}
-    (C : HellingerTracePair q x y E V) (hq : q∈Set.Icc 0 1)
-    (m n : ℕ) (hx : x.length≤n) (hy : y.length≤n) :
-    ((listTraceLaw q hq x n hx).product m).tv ((listTraceLaw q hq y n hy).product m) ≤
-      Real.sqrt ((m:ℝ)*V)+2*m*E := by
-  let := C.finiteOutcome
-  exact FiniteLaw.approximate_hellinger_transfer C.first C.second
-    (fun a=>FiniteLaw.pure (boundedTraceProjection n (C.decode a))) _ _ C.bound
-    (C.first_simulation.projected_tv hq n hx) (C.second_simulation.projected_tv hq n hy) m
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- Independent deletion composes on the actual complete output list. -/
-theorem listDeletionExpectation_comp (r t : ℝ) (w : List Bool) (F : List Bool→ℝ) :
-    listDeletionExpectation r w (fun z=>listDeletionExpectation t z F)=
-      listDeletionExpectation (r+(1-r)*t) w F := by
-  induction w generalizing F with
-  | nil => rfl
-  | cons b w ih =>
-    simp only [listDeletionExpectation]
-    rw [listDeletionExpectation_linear,ih,ih]
-    ring
-
-def extraDeletion (r q : ℝ) : ℝ := (q-r)/(1-r)
-
-theorem extraDeletion_bounds {r q : ℝ} (hr : r<1) (hrq : r≤q) (hq : q≤1) :
-    extraDeletion r q∈Set.Icc 0 1 := by
-  constructor
-  · exact div_nonneg (sub_nonneg.mpr hrq) (sub_pos.mpr hr).le
-  · apply (div_le_iff₀ (sub_pos.mpr hr)).mpr
-    linarith only [hq]
-
-theorem extraDeletion_comp {r q : ℝ} (hr : r<1) : r+(1-r)*extraDeletion r q=q := by
-  unfold extraDeletion
-  field_simp [(sub_pos.mpr hr).ne']
-  ring
-
-theorem listTraceLaw_degrade {r q : ℝ} (hr : r∈Set.Ico 0 1) (hq : q∈Set.Icc 0 1)
-    (hrq : r≤q) (w : List Bool) (n : ℕ) (hw : w.length≤n) :
-    listTraceLaw q hq w n hw=(listTraceLaw r ⟨hr.1,hr.2.le⟩ w n hw).bind
-      (fun z=>listTraceLaw (extraDeletion r q) (extraDeletion_bounds hr.2 hrq hq.2)
-        z.val n (length_le_of_mem_traceSpace z.property)) := by
-  apply FiniteLaw.ext_expect
-  intro f
-  rw [FiniteLaw.expect_bind]
-  simp_rw [listTraceLaw_projection]
-  rw [listTraceLaw_expect r ⟨hr.1,hr.2.le⟩ w n hw
-    (fun z=>listDeletionExpectation (extraDeletion r q) z (fun u=>f (boundedTraceProjection n u))),
-    listDeletionExpectation_comp,extraDeletion_comp hr.2]
-
-/-- The requested higher-deletion channel preserves the exact m-sample
- restoration estimate, through one common per-sample Markov kernel. -/
-theorem HellingerTracePair.degraded_sample_tv {r q E V : ℝ} {x y : List Bool}
-    (C : HellingerTracePair r x y E V) (hr : r∈Set.Ico 0 1) (hq : q∈Set.Icc 0 1)
-    (hrq : r≤q) (m n : ℕ) (hx : x.length≤n) (hy : y.length≤n) :
-    ((listTraceLaw q hq x n hx).product m).tv ((listTraceLaw q hq y n hy).product m) ≤
-      Real.sqrt ((m:ℝ)*V)+2*m*E := by
-  let := C.finiteOutcome
-  let K : BoundedTrace n→FiniteLaw (BoundedTrace n) := fun z=>
-    listTraceLaw (extraDeletion r q) (extraDeletion_bounds hr.2 hrq hq.2)
-      z.val n (length_le_of_mem_traceSpace z.property)
-  let f := fun z=>boundedTraceProjection n (C.decode z)
-  have hP := C.first_simulation.projected_tv ⟨hr.1,hr.2.le⟩ n hx
-  have hQ := C.second_simulation.projected_tv ⟨hr.1,hr.2.le⟩ n hy
-  have hp := (FiniteLaw.tv_bind_le (listTraceLaw r ⟨hr.1,hr.2.le⟩ x n hx) (C.first.map f) K).trans hP
-  have hq' := (FiniteLaw.tv_bind_le (listTraceLaw r ⟨hr.1,hr.2.le⟩ y n hy) (C.second.map f) K).trans hQ
-  rw [←listTraceLaw_degrade hr hq hrq x n hx,FiniteLaw.bind_map] at hp
-  rw [←listTraceLaw_degrade hr hq hrq y n hy,FiniteLaw.bind_map] at hq'
-  exact FiniteLaw.approximate_hellinger_transfer C.first C.second (fun z=>K (f z)) _ _ C.bound hp hq' m
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-def wordOfLength {n : ℕ} (w : List Bool) (h : w.length=n) : Word n := h ▸ w.get
-
-theorem ofFn_wordOfLength {n : ℕ} (w : List Bool) (h : w.length=n) :
-    List.ofFn (wordOfLength w h)=w := by
-  subst n
-  exact List.ofFn_get w
-
-theorem listTraceLaw_ofFn {n : ℕ} (q : ℝ) (hq : q∈Set.Icc 0 1) (x : Word n) :
-    listTraceLaw q hq (List.ofFn x) n (by simp)=completeTraceLaw q hq x := by
-  apply FiniteLaw.ext_expect
-  intro f
-  rw [listTraceLaw_projection,listDeletionExpectation_ofFn,completeTraceLaw_mask,FiniteLaw.expect_map]
-  unfold FiniteLaw.expect maskLaw
-  apply Finset.sum_congr rfl
-  intro a _
-  congr 2
-  apply Subtype.ext
-  exact boundedTraceProjection_val (traceOfMask_length_le x a)
-
-theorem listTraceLaw_wordOfLength {n : ℕ} (q : ℝ) (hq : q∈Set.Icc 0 1)
-    (w : List Bool) (hw : w.length=n) :
-    listTraceLaw q hq w n hw.le=completeTraceLaw q hq (wordOfLength w hw) := by
-  have hh := listTraceLaw_ofFn q hq (wordOfLength w hw)
-  simpa only [ofFn_wordOfLength] using hh
-
-/-- Literal exact-length padding and degradation of a level witness. -/
-theorem ConcreteLevelPair.restored_words {r q N Δ d ω F : ℝ} {hr : r∈Set.Icc 0 1}
-    {i k : ℕ} (U : ConcreteLevelPair r hr N Δ d ω F i k)
-    (hr1 : r<1) (hq : q∈Set.Icc 0 1) (hrq : r≤q) (hk : 1≤k)
-    (n : ℕ) (hN : N≤n) :
-    ∃x y : Word n,x≠y ∧ ∀m : ℕ,
-      ((completeTraceLaw q hq x).product m).tv ((completeTraceLaw q hq y).product m) ≤
-        Real.sqrt ((m:ℝ)*N^(-d))+2*m*U.error := by
-  have hlen : U.left.length≤n := by exact_mod_cast U.length_upper.trans hN
-  let tail := List.replicate (n-U.left.length) false
-  let x := U.left++tail
-  let y := U.right++tail
-  have hx : x.length=n := by simp only [x,tail,List.length_append,List.length_replicate]; omega
-  have hy : y.length=n := by simp only [y,tail,List.length_append,List.length_replicate,←U.equalLength]; omega
-  have hxy : x≠y := fun h=>U.different (List.append_cancel_right h)
-  let C := (U.oracle.pad hr [] tail).hellinger hk
-  refine ⟨wordOfLength x hx,wordOfLength y hy,?_,?_⟩
-  · intro hh
-    have hh' := congrArg List.ofFn hh
-    rw [ofFn_wordOfLength,ofFn_wordOfLength] at hh'
-    exact hxy hh'
-  · intro m
-    have HH := C.degraded_sample_tv ⟨hr.1,hr1⟩ hq hrq m n
-      (by simpa only [List.nil_append] using hx.le) (by simpa only [List.nil_append] using hy.le)
-    change ((listTraceLaw q hq x n hx.le).product m).tv ((listTraceLaw q hq y n hy.le).product m) ≤ _ at HH
-    rw [listTraceLaw_wordOfLength q hq x hx,listTraceLaw_wordOfLength q hq y hy] at HH
-    exact HH
-
-end TraceReconstruction
-
-end
-end
-end
-end
-
-end OAI
+import OAI.Probability.TraceReconstruction.Amplification
+import OAI.Probability.TraceReconstruction.Propagation
+
+namespace OAI
+
+/-!
+# Restoration
+
+Restoration to honest complete traces, degradation to the requested channel, and exact-length padding.
+-/
+
+open scoped ENNReal
+noncomputable section
+
+open Filter
+open scoped Topology
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+namespace FiniteLaw
+variable {α β : Type*} [Fintype α] [Fintype β]
+
+theorem approximate_hellinger_transfer (P Q : FiniteLaw α) (K : α→FiniteLaw β)
+    (R S : FiniteLaw β) {E V : ℝ} (hH : P.hellingerSq Q ≤ V)
+    (hR : R.tv (P.bind K) ≤ E) (hS : S.tv (Q.bind K) ≤ E) (m : ℕ) :
+    (R.product m).tv (S.product m) ≤ Real.sqrt ((m:ℝ)*V)+2*m*E := by
+  have hmid := (tv_le_sqrt_hellingerSq ((P.bind K).product m) ((Q.bind K).product m)).trans
+    (Real.sqrt_le_sqrt ((hellingerSq_product_le _ _ m).trans
+      (mul_le_mul_of_nonneg_left ((hellingerSq_bind_le P Q K).trans hH) (Nat.cast_nonneg m))))
+  have hr := (tv_product_le R (P.bind K) m).trans (mul_le_mul_of_nonneg_left hR (Nat.cast_nonneg m))
+  have hs := (tv_product_le S (Q.bind K) m).trans (mul_le_mul_of_nonneg_left hS (Nat.cast_nonneg m))
+  rw [tv_symm (S.product m)] at hs
+  linarith only [hr,hs,hmid,tv_triangle (R.product m) ((P.bind K).product m) (S.product m),
+    tv_triangle ((P.bind K).product m) ((Q.bind K).product m) (S.product m)]
+
+end FiniteLaw
+
+/-- A total common finite projection. The fallback is never used by a
+ genuine trace of a length-at-most-n input. -/
+def boundedTraceProjection (n : ℕ) (w : List Bool) : BoundedTrace n :=
+  if h : w∈traceSpace n then ⟨w,h⟩ else ⟨[],mem_traceSpace_of_length_le [] (by simp)⟩
+
+theorem boundedTraceProjection_val {n : ℕ} {w : List Bool} (hw : w.length ≤ n) :
+    (boundedTraceProjection n w).val=w := by
+  simp only [boundedTraceProjection,dite_eq_left (mem_traceSpace_of_length_le w hw)]
+
+theorem listTraceLaw_projection (q : ℝ) (hq : q∈Set.Icc 0 1) (w : List Bool)
+    (n : ℕ) (hw : w.length≤n) (f : BoundedTrace n→ℝ) :
+    (listTraceLaw q hq w n hw).expect f=
+      listDeletionExpectation q w (fun z=>f (boundedTraceProjection n z)) := by
+  rw [←listTraceLaw_expect q hq w n hw]
+  congr 1
+  funext t
+  congr 1
+  apply Subtype.ext
+  exact (boundedTraceProjection_val (length_le_of_mem_traceSpace t.property)).symm
+
+theorem ApproximatesTrace.projected_tv {α : Type*} [Fintype α] {q E : ℝ}
+    {P : FiniteLaw α} {decode : α→List Bool} {w : List Bool}
+    (H : ApproximatesTrace q P decode w E) (hq : q∈Set.Icc 0 1)
+    (n : ℕ) (hw : w.length≤n) :
+    (listTraceLaw q hq w n hw).tv (P.map (fun a=>boundedTraceProjection n (decode a))) ≤ E := by
+  apply FiniteLaw.tv_le_of_tests
+  intro f hf
+  rw [listTraceLaw_projection,FiniteLaw.expect_map]
+  simp only [Function.comp_def]
+  have hh := H (fun z=>f (boundedTraceProjection n z)) (fun z=>hf _)
+  linarith only [(abs_le.mp hh).1]
+
+/-- Restoration for m independent complete observations, not a single
+ concatenated trace. -/
+theorem HellingerTracePair.sample_tv {q E V : ℝ} {x y : List Bool}
+    (C : HellingerTracePair q x y E V) (hq : q∈Set.Icc 0 1)
+    (m n : ℕ) (hx : x.length≤n) (hy : y.length≤n) :
+    ((listTraceLaw q hq x n hx).product m).tv ((listTraceLaw q hq y n hy).product m) ≤
+      Real.sqrt ((m:ℝ)*V)+2*m*E := by
+  let := C.finiteOutcome
+  exact FiniteLaw.approximate_hellinger_transfer C.first C.second
+    (fun a=>FiniteLaw.pure (boundedTraceProjection n (C.decode a))) _ _ C.bound
+    (C.first_simulation.projected_tv hq n hx) (C.second_simulation.projected_tv hq n hy) m
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- Independent deletion composes on the actual complete output list. -/
+theorem listDeletionExpectation_comp (r t : ℝ) (w : List Bool) (F : List Bool→ℝ) :
+    listDeletionExpectation r w (fun z=>listDeletionExpectation t z F)=
+      listDeletionExpectation (r+(1-r)*t) w F := by
+  induction w generalizing F with
+  | nil => rfl
+  | cons b w ih =>
+    simp only [listDeletionExpectation]
+    rw [listDeletionExpectation_linear,ih,ih]
+    ring
+
+def extraDeletion (r q : ℝ) : ℝ := (q-r)/(1-r)
+
+theorem extraDeletion_bounds {r q : ℝ} (hr : r<1) (hrq : r≤q) (hq : q≤1) :
+    extraDeletion r q∈Set.Icc 0 1 := by
+  constructor
+  · exact div_nonneg (sub_nonneg.mpr hrq) (sub_pos.mpr hr).le
+  · apply (div_le_iff₀ (sub_pos.mpr hr)).mpr
+    linarith only [hq]
+
+theorem extraDeletion_comp {r q : ℝ} (hr : r<1) : r+(1-r)*extraDeletion r q=q := by
+  unfold extraDeletion
+  field_simp [(sub_pos.mpr hr).ne']
+  ring
+
+theorem listTraceLaw_degrade {r q : ℝ} (hr : r∈Set.Ico 0 1) (hq : q∈Set.Icc 0 1)
+    (hrq : r≤q) (w : List Bool) (n : ℕ) (hw : w.length≤n) :
+    listTraceLaw q hq w n hw=(listTraceLaw r ⟨hr.1,hr.2.le⟩ w n hw).bind
+      (fun z=>listTraceLaw (extraDeletion r q) (extraDeletion_bounds hr.2 hrq hq.2)
+        z.val n (length_le_of_mem_traceSpace z.property)) := by
+  apply FiniteLaw.ext_expect
+  intro f
+  rw [FiniteLaw.expect_bind]
+  simp_rw [listTraceLaw_projection]
+  rw [listTraceLaw_expect r ⟨hr.1,hr.2.le⟩ w n hw
+    (fun z=>listDeletionExpectation (extraDeletion r q) z (fun u=>f (boundedTraceProjection n u))),
+    listDeletionExpectation_comp,extraDeletion_comp hr.2]
+
+/-- The requested higher-deletion channel preserves the exact m-sample
+ restoration estimate, through one common per-sample Markov kernel. -/
+theorem HellingerTracePair.degraded_sample_tv {r q E V : ℝ} {x y : List Bool}
+    (C : HellingerTracePair r x y E V) (hr : r∈Set.Ico 0 1) (hq : q∈Set.Icc 0 1)
+    (hrq : r≤q) (m n : ℕ) (hx : x.length≤n) (hy : y.length≤n) :
+    ((listTraceLaw q hq x n hx).product m).tv ((listTraceLaw q hq y n hy).product m) ≤
+      Real.sqrt ((m:ℝ)*V)+2*m*E := by
+  let := C.finiteOutcome
+  let K : BoundedTrace n→FiniteLaw (BoundedTrace n) := fun z=>
+    listTraceLaw (extraDeletion r q) (extraDeletion_bounds hr.2 hrq hq.2)
+      z.val n (length_le_of_mem_traceSpace z.property)
+  let f := fun z=>boundedTraceProjection n (C.decode z)
+  have hP := C.first_simulation.projected_tv ⟨hr.1,hr.2.le⟩ n hx
+  have hQ := C.second_simulation.projected_tv ⟨hr.1,hr.2.le⟩ n hy
+  have hp := (FiniteLaw.tv_bind_le (listTraceLaw r ⟨hr.1,hr.2.le⟩ x n hx) (C.first.map f) K).trans hP
+  have hq' := (FiniteLaw.tv_bind_le (listTraceLaw r ⟨hr.1,hr.2.le⟩ y n hy) (C.second.map f) K).trans hQ
+  rw [←listTraceLaw_degrade hr hq hrq x n hx,FiniteLaw.bind_map] at hp
+  rw [←listTraceLaw_degrade hr hq hrq y n hy,FiniteLaw.bind_map] at hq'
+  exact FiniteLaw.approximate_hellinger_transfer C.first C.second (fun z=>K (f z)) _ _ C.bound hp hq' m
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+def wordOfLength {n : ℕ} (w : List Bool) (h : w.length=n) : Word n := h ▸ w.get
+
+theorem ofFn_wordOfLength {n : ℕ} (w : List Bool) (h : w.length=n) :
+    List.ofFn (wordOfLength w h)=w := by
+  subst n
+  exact List.ofFn_get w
+
+theorem listTraceLaw_ofFn {n : ℕ} (q : ℝ) (hq : q∈Set.Icc 0 1) (x : Word n) :
+    listTraceLaw q hq (List.ofFn x) n (by simp)=completeTraceLaw q hq x := by
+  apply FiniteLaw.ext_expect
+  intro f
+  rw [listTraceLaw_projection,listDeletionExpectation_ofFn,completeTraceLaw_mask,FiniteLaw.expect_map]
+  unfold FiniteLaw.expect maskLaw
+  apply Finset.sum_congr rfl
+  intro a _
+  congr 2
+  apply Subtype.ext
+  exact boundedTraceProjection_val (traceOfMask_length_le x a)
+
+theorem listTraceLaw_wordOfLength {n : ℕ} (q : ℝ) (hq : q∈Set.Icc 0 1)
+    (w : List Bool) (hw : w.length=n) :
+    listTraceLaw q hq w n hw.le=completeTraceLaw q hq (wordOfLength w hw) := by
+  have hh := listTraceLaw_ofFn q hq (wordOfLength w hw)
+  simpa only [ofFn_wordOfLength] using hh
+
+/-- Literal exact-length padding and degradation of a level witness. -/
+theorem ConcreteLevelPair.restored_words {r q N Δ d ω F : ℝ} {hr : r∈Set.Icc 0 1}
+    {i k : ℕ} (U : ConcreteLevelPair r hr N Δ d ω F i k)
+    (hr1 : r<1) (hq : q∈Set.Icc 0 1) (hrq : r≤q) (hk : 1≤k)
+    (n : ℕ) (hN : N≤n) :
+    ∃x y : Word n,x≠y ∧ ∀m : ℕ,
+      ((completeTraceLaw q hq x).product m).tv ((completeTraceLaw q hq y).product m) ≤
+        Real.sqrt ((m:ℝ)*N^(-d))+2*m*U.error := by
+  have hlen : U.left.length≤n := by exact_mod_cast U.length_upper.trans hN
+  let tail := List.replicate (n-U.left.length) false
+  let x := U.left++tail
+  let y := U.right++tail
+  have hx : x.length=n := by simp only [x,tail,List.length_append,List.length_replicate]; omega
+  have hy : y.length=n := by simp only [y,tail,List.length_append,List.length_replicate,←U.equalLength]; omega
+  have hxy : x≠y := fun h=>U.different (List.append_cancel_right h)
+  let C := (U.oracle.pad hr [] tail).hellinger hk
+  refine ⟨wordOfLength x hx,wordOfLength y hy,?_,?_⟩
+  · intro hh
+    have hh' := congrArg List.ofFn hh
+    rw [ofFn_wordOfLength,ofFn_wordOfLength] at hh'
+    exact hxy hh'
+  · intro m
+    have HH := C.degraded_sample_tv ⟨hr.1,hr1⟩ hq hrq m n
+      (by simpa only [List.nil_append] using hx.le) (by simpa only [List.nil_append] using hy.le)
+    change ((listTraceLaw q hq x n hx.le).product m).tv ((listTraceLaw q hq y n hy.le).product m) ≤ _ at HH
+    rw [listTraceLaw_wordOfLength q hq x hx,listTraceLaw_wordOfLength q hq y hy] at HH
+    exact HH
+
+end TraceReconstruction
+
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/TraceReconstruction/Scales.lean b/lean/OAI/Probability/TraceReconstruction/Scales.lean
index e8e1aaa7b77489ba951287cb3cd8444e07e09b9f..86c9ae03ff5e160bdbc440d4af4cb7feaa0ab0a1 100644
--- a/lean/OAI/Probability/TraceReconstruction/Scales.lean
+++ b/lean/OAI/Probability/TraceReconstruction/Scales.lean
@@ -1,740 +1,740 @@
-import Mathlib
-
-namespace OAI
-
-/-!
-# Scales
-
-Exact block exponents, deletion reduction, induction depth, backward target scales, and uniform power absorption.
--/
-
-open scoped ENNReal
-noncomputable section
-
-namespace TraceReconstruction
-/-- The exact real scale exponents in the scale inequalities; natural indices are zero-based. -/
-def scaleA (b j : ℕ) (ε : ℝ) : ℝ := (1+ε)* (2:ℝ)^j / (2:ℝ)^b - ε
-
-def scaleKappa (b j : ℕ) (ε : ℝ) : ℝ :=
-  scaleA b (j-1) ε - 3*((2:ℝ)^j-1)*ε
-
-def sourceEpsilon (B D : ℕ) : ℝ := 1 / ((2:ℝ)^(4*B+100) * (D+1))
-
-theorem scaleA_top (b : ℕ) (ε : ℝ) : scaleA b b ε = 1 := by
-  unfold scaleA
-  field_simp
-  ring
-
-theorem scaleA_step (b j : ℕ) (ε : ℝ) : scaleA b (j+1) ε - 2*scaleA b j ε = ε := by
-  unfold scaleA
-  rw [pow_succ]
-  ring
-
-theorem scaleKappa_step (b j : ℕ) (ε : ℝ) (hj : 1 ≤ j) :
-    2*scaleKappa b j ε - scaleKappa b (j+1) ε - ε = ε := by
-  unfold scaleKappa scaleA
-  simp only [Nat.add_sub_cancel]
-  rw [show j = (j-1)+1 from by omega, pow_succ]
-  simp only [Nat.add_sub_cancel, pow_succ]
-  ring
-
-def forcedDecay (b : ℕ) (ε d : ℝ) : ℝ :=
-  ε/2 + ∑ k ∈ Finset.range (b-1),
-    ((2:ℝ)^b / (2:ℝ)^(k+2))*(scaleKappa b (k+2) ε / 2 + ε) +
-  (2:ℝ)^b * scaleA b 0 ε * d
-
-theorem forcedDecay_summand (b k : ℕ) (ε : ℝ) :
-    ((2:ℝ)^b / (2:ℝ)^(k+2))*(scaleKappa b (k+2) ε / 2 + ε) =
-      (1+ε)/4 - 3*((2:ℝ)^b/2)*ε + ((2:ℝ)^b/(2:ℝ)^(k+1))*ε := by
-  unfold scaleKappa scaleA
-  simp only [show k+2-1=k+1 by omega, pow_succ]
-  field_simp
-  ring
-
-theorem forcedDecay_lower (b : ℕ) (ε d : ℝ) (hb : 1 ≤ b) (he : 0 ≤ ε) :
-    d + ((b:ℝ)-1)/4 - (((2:ℝ)^b-1)*d + 3*b*((2:ℝ)^b/2))*ε ≤
-      forcedDecay b ε d := by
-  have hterm (k : ℕ) : 1/4 - 3*((2:ℝ)^b/2)*ε ≤
-      ((2:ℝ)^b / (2:ℝ)^(k+2))*(scaleKappa b (k+2) ε / 2 + ε) := by
-    rw [forcedDecay_summand]
-    have hp : 0 ≤ ((2:ℝ)^b / (2:ℝ)^(k+1))*ε := by positivity
-    linarith
-  have hsum := Finset.sum_le_sum (fun k (_ : k ∈ Finset.range (b-1)) => hterm k)
-  simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul] at hsum
-  rw [Nat.cast_sub hb, Nat.cast_one] at hsum
-  have ha : (2:ℝ)^b * scaleA b 0 ε * d = d - ((2:ℝ)^b-1)*d*ε := by
-    unfold scaleA
-    simp only [pow_zero, mul_one]
-    field_simp
-    ring
-  unfold forcedDecay
-  rw [ha]
-  have hp : 0 ≤ (2:ℝ)^b*ε := by positivity
-  nlinarith
-
-theorem nat_add_one_le_two_pow (n : ℕ) : n+1 ≤ 2^n := by
-  induction n with
-  | zero => norm_num
-  | succ n ih => rw [pow_succ]; omega
-
-theorem sourceEpsilon_pos (B D : ℕ) : 0 < sourceEpsilon B D := by
-  unfold sourceEpsilon
-  positivity
-
-theorem sourceEpsilon_loss (B D b : ℕ) (d : ℝ) (hb : b ≤ B)
-    (hd0 : 0 ≤ d) (hd : d ≤ 3 + (B:ℝ)*D) :
-    (((2:ℝ)^b-1)*d + 3*b*((2:ℝ)^b/2))*sourceEpsilon B D < 1/100 := by
-  have hpow : (2:ℝ)^b ≤ (2:ℝ)^B := pow_le_pow_right₀ (by norm_num) hb
-  have hB : (B:ℝ)+1 ≤ (2:ℝ)^B := by exact_mod_cast nat_add_one_le_two_pow B
-  have hB0 : (0:ℝ) ≤ B := by positivity
-  have hD0 : (0:ℝ) ≤ D := by positivity
-  have hh : (((2:ℝ)^b-1)*d + 3*b*((2:ℝ)^b/2)) ≤
-      4*((2:ℝ)^B)^2*((D:ℝ)+1) := by
-    have hb' : (b:ℝ) ≤ B := by exact_mod_cast hb
-    have h1 := mul_le_mul_of_nonneg_right (show (2:ℝ)^b-1 ≤ (2:ℝ)^B by linarith) hd0
-    have h2 := mul_le_mul_of_nonneg_left hd (show 0 ≤ (2:ℝ)^B by positivity)
-    have h3 := mul_le_mul hb' hpow (by positivity : (0:ℝ) ≤ (2:ℝ)^b) hB0
-    have h4 := mul_le_mul_of_nonneg_left hB hD0
-    have h5 := mul_le_mul_of_nonneg_left h4 (show 0 ≤ (2:ℝ)^B by positivity)
-    have h6 := mul_le_mul_of_nonneg_left hB (show 0 ≤ (2:ℝ)^B by positivity)
-    nlinarith [sq_nonneg ((2:ℝ)^B)]
-  apply lt_of_le_of_lt (mul_le_mul_of_nonneg_right hh (sourceEpsilon_pos B D).le)
-  unfold sourceEpsilon
-  have hpos : 0 < (2:ℝ)^(4*B+100) := by positivity
-  have hp2 : (2:ℝ)^(2*B) = ((2:ℝ)^B)^2 := by rw [mul_comm 2 B, pow_mul]
-  have hp4 : (2:ℝ)^(4*B+100) = ((2:ℝ)^B)^2 * (2:ℝ)^(2*B+100) := by
-    rw [show 4*B+100=2*B+(2*B+100) by omega, pow_add, hp2]
-  have hh2 : (400:ℝ) < (2:ℝ)^(2*B+100) := by
-    have h := pow_le_pow_right₀ (by norm_num : (1:ℝ) ≤ 2) (by omega : 9 ≤ 2*B+100)
-    norm_num at h ⊢
-    linarith
-  rw [mul_one_div]
-  apply (div_lt_iff₀ (by positivity)).mpr
-  rw [hp4]
-  have := mul_lt_mul_of_pos_left hh2 (show 0 < ((2:ℝ)^B)^2*((D:ℝ)+1) by positivity)
-  nlinarith
-
-theorem forcedDecay_margin (B D b : ℕ) (d : ℝ) (hb1 : 1 ≤ b) (hb : b ≤ B)
-    (hd0 : 0 ≤ d) (hd : d ≤ 3 + (B:ℝ)*D) :
-    d + ((b:ℝ)-1)/4 - 1/20 + 3/100 < forcedDecay b (sourceEpsilon B D) d := by
-  have h := forcedDecay_lower b (sourceEpsilon B D) d hb1 (sourceEpsilon_pos B D).le
-  have hg := sourceEpsilon_loss B D b d hb hd0 hd
-  linarith
-end TraceReconstruction
-
-open Filter
-open scoped Topology
-
-
-open Filter
-open scoped Topology
-namespace TraceReconstruction
-
-def workingDeletion (q : ℝ) : ℝ := min q (1/2)
-def blockGain (b : ℕ) : ℝ := ((b:ℝ)-1)/4-1/20
-def warmupBlocks (B : ℕ) : ℕ := 5*2^B
-def sourceDepth (B : ℕ) (ζ H : ℝ) : ℕ := ⌊(1-ζ)*Real.log H/((B:ℝ)*Real.log 2)⌋₊
-def totalScoreExponent (B D : ℕ) : ℝ :=
-  11/5+∑ i ∈ Finset.range D, blockGain (if i<warmupBlocks B then 2 else B)
-def finalSlope (B : ℕ) (ζ : ℝ) : ℝ := (1-ζ)*blockGain B/((B:ℝ)*Real.log 2)
-
-theorem workingDeletion_bounds {q : ℝ} (hq : q ∈ Set.Ioo 0 1) :
-    workingDeletion q ∈ Set.Ioo 0 1 ∧ q/2 ≤ workingDeletion q ∧ workingDeletion q ≤ q := by
-  unfold workingDeletion
-  constructor
-  · constructor
-    · exact lt_min hq.1 (by norm_num)
-    · exact (min_le_left _ _).trans_lt hq.2
-  constructor
-  · exact le_min (by linarith [hq.1]) (by linarith [hq.2])
-  · exact min_le_left _ _
-
-theorem workingScale_bounds {q L : ℝ} (hq : q ∈ Set.Ioo 0 1) (hL : 0 ≤ L) :
-    q^3*L/8 ≤ (workingDeletion q)^3*L ∧ (workingDeletion q)^3*L ≤ q^3*L := by
-  have hw := workingDeletion_bounds hq
-  have hl := pow_le_pow_left₀ (div_nonneg hq.1.le (by norm_num)) hw.2.1 3
-  have hu := pow_le_pow_left₀ hw.1.1.le hw.2.2 3
-  constructor
-  · have := mul_le_mul_of_nonneg_right hl hL
-    nlinarith only [this]
-  · exact mul_le_mul_of_nonneg_right hu hL
-
-theorem blockGain_two : blockGain 2 = 1/5 := by norm_num [blockGain]
-theorem blockGain_pos (B : ℕ) (hB : 2 ≤ B) : 0 < blockGain B := by
-  have hB' : (2:ℝ) ≤ B := by exact_mod_cast hB
-  unfold blockGain
-  linarith
-
-theorem totalScoreExponent_exact (B D : ℕ) (hD : warmupBlocks B ≤ D) :
-    totalScoreExponent B D = 11/5 + (warmupBlocks B:ℝ)/5 +
-      ((D:ℝ)-warmupBlocks B)*blockGain B := by
-  unfold totalScoreExponent
-  rw [← Nat.add_sub_of_le hD, Finset.sum_range_add]
-  have hfirst : (∑ i ∈ Finset.range (warmupBlocks B),
-      blockGain (if i<warmupBlocks B then 2 else B)) = (warmupBlocks B:ℝ)/5 := by
-    simp only [Finset.sum_congr rfl (fun i hi => congrArg blockGain (ite_eq_left (Finset.mem_range.mp hi))), blockGain_two]
-    simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul]
-    ring
-  rw [hfirst]
-  have hlast : (∑ i ∈ Finset.range (D-warmupBlocks B),
-      blockGain (if warmupBlocks B+i<warmupBlocks B then 2 else B)) =
-      ((D:ℝ)-warmupBlocks B)*blockGain B := by
-    simp only [show ∀ i, ¬warmupBlocks B+i<warmupBlocks B from fun i => by omega, ite_false,
-      Finset.sum_const, Finset.card_range, nsmul_eq_mul, Nat.cast_sub hD]
-  rw [hlast]
-  simp only [Nat.add_sub_of_le hD]
-  ring
-
-theorem sourceDepth_lower (B : ℕ) (ζ H : ℝ) :
-    (1-ζ)*Real.log H/((B:ℝ)*Real.log 2)-1 < (sourceDepth B ζ H:ℝ) := by
-  have := Nat.lt_floor_add_one ((1-ζ)*Real.log H/((B:ℝ)*Real.log 2))
-  change _ < (sourceDepth B ζ H:ℝ)+1 at this
-  linarith
-
-theorem totalScoreExponent_slope_lower (B : ℕ) (ζ H : ℝ) (hB : 2 ≤ B)
-    (hD : warmupBlocks B ≤ sourceDepth B ζ H) :
-    finalSlope B ζ * Real.log H -
-      (blockGain B + (warmupBlocks B:ℝ)*blockGain B) <
-      totalScoreExponent B (sourceDepth B ζ H) := by
-  rw [totalScoreExponent_exact B _ hD]
-  have hh := mul_lt_mul_of_pos_right (sourceDepth_lower B ζ H) (blockGain_pos B hB)
-  unfold finalSlope
-  have he : ((1-ζ)*Real.log H/((B:ℝ)*Real.log 2))*blockGain B =
-      ((1-ζ)*blockGain B/((B:ℝ)*Real.log 2))*Real.log H := by ring
-  rw [sub_mul, he] at hh
-  have hw : (0:ℝ) ≤ warmupBlocks B := by positivity
-  linarith
-
-theorem sourceDepth_tendsto (B : ℕ) (ζ : ℝ) (hB : 2 ≤ B) (hζ : ζ < 1)
-    {H : ℕ → ℝ} (hH : Tendsto H atTop atTop) :
-    Tendsto (fun k => sourceDepth B ζ (H k)) atTop atTop := by
-  have hden : 0 < (B:ℝ)*Real.log 2 := by
-    apply mul_pos
-    · exact_mod_cast (show 0 < B by omega)
-    · exact Real.log_pos (by norm_num)
-  have ht := (Real.tendsto_log_atTop.comp hH).const_mul_atTop (sub_pos.mpr hζ)
-  have ht' := ht.atTop_div_const hden
-  exact tendsto_nat_floor_atTop.comp ht'
-
-/-- A fixed multiplicative constant is absorbed by any strict exponent
-slope margin, with two independently diverging instance parameters. -/
-theorem eventually_exponent_margin {N H d : ℕ → ℝ} {α c K a : ℝ}
-    (hN : Tendsto N atTop atTop) (hH : Tendsto H atTop atTop)
-    (hα : c < α) (ha : 0 < a)
-    (hd : ∀ᶠ k in atTop, α*Real.log (H k)-K ≤ d k) :
-    ∀ᶠ k in atTop, (N k)^(c*Real.log (H k)) ≤ a*(N k)^(d k) := by
-  have hlog := (Real.tendsto_log_atTop.comp hH).const_mul_atTop (sub_pos.mpr hα)
-  filter_upwards [hd, hlog.eventually_ge_atTop (K+1), hN.eventually_ge_atTop (max 1 (1/a))] with k hd hlog hN
-  have hN1 : 1 ≤ N k := (le_max_left _ _).trans hN
-  have hNa : 1/a ≤ N k := (le_max_right _ _).trans hN
-  simp only [Function.comp_apply] at hlog
-  have hex : c*Real.log (H k)+1 ≤ d k := by nlinarith only [hd,hlog]
-  have hp := Real.rpow_le_rpow_of_exponent_le hN1 hex
-  rw [Real.rpow_add (by linarith : 0 < N k), Real.rpow_one] at hp
-  have hapos : 1 ≤ a*N k := (div_le_iff₀ ha).mp hNa |>.trans_eq (mul_comm _ _)
-  calc
-    (N k)^(c*Real.log (H k)) ≤ (a*N k)*(N k)^(c*Real.log (H k)) := by
-      exact le_mul_of_one_le_left (Real.rpow_nonneg (by linarith) _) hapos
-    _ ≤ a*(N k)^(d k) := by
-      have := mul_le_mul_of_nonneg_left hp ha.le
-      nlinarith only [this]
-
-end TraceReconstruction
-
-open Filter
-open scoped Topology
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-open Filter
-
-/-- Literal source block dimensions, with zero-based block number. -/
-def sourceDimension (B i : ℕ) : ℕ := if i < warmupBlocks B then 2 else B
-
-def sourceScaleProduct (B D i : ℕ) : ℝ :=
-  ∏j∈Finset.Ico i D,scaleA (sourceDimension B j) 0 (sourceEpsilon B D)
-
-/-- The backwards target scales as an exact product of source exponents. -/
-def sourceTargetScale (B D : ℕ) (n : ℝ) (i : ℕ) : ℝ := n ^ sourceScaleProduct B D i
-
-theorem sourceTargetScale_top (B D : ℕ) (n : ℝ) : sourceTargetScale B D n D=n := by
-  simp [sourceTargetScale,sourceScaleProduct]
-
-theorem sourceTargetScale_step (B D i : ℕ) (n : ℝ) (hn : 0 ≤ n) (hi : i < D) :
-    sourceTargetScale B D n i =
-      (sourceTargetScale B D n (i+1))^(scaleA (sourceDimension B i) 0 (sourceEpsilon B D)) := by
-  unfold sourceTargetScale sourceScaleProduct
-  rw [←Real.rpow_mul hn,Finset.prod_eq_prod_Ico_succ_bot hi]
-  rw [mul_comm]
-
-theorem sourceEpsilon_small (B D : ℕ) :
-    ((D:ℝ)+1)*(2:ℝ)^B*sourceEpsilon B D ≤ 1/2 := by
-  have hp : 2*(2:ℝ)^B ≤ (2:ℝ)^(4*B+100) := by
-    rw [←pow_succ']
-    exact pow_le_pow_right₀ (by norm_num) (by omega)
-  unfold sourceEpsilon
-  have hD : 0 < (D:ℝ)+1 := by positivity
-  have hpow : 0 < (2:ℝ)^(4*B+100) := by positivity
-  apply (le_div_iff₀ (by norm_num : (0:ℝ)<2)).mpr
-  have he : ((D:ℝ)+1)*2^B*(1/(2^(4*B+100)*((D:ℝ)+1)))*2 =
-      (2*2^B)/2^(4*B+100) := by field_simp
-  rw [he]
-  exact (div_le_one hpow).mpr hp
-
-theorem scaleA_zero_lower (b B : ℕ) (ε : ℝ) (hb : b ≤ B) (he : 0 ≤ ε) :
-    (1/((2:ℝ)^B))*(1-((2:ℝ)^B-1)*ε) ≤ scaleA b 0 ε := by
-  have hp : (2:ℝ)^b ≤ 2^B := pow_le_pow_right₀ (by norm_num) hb
-  have hpos : 0 < (2:ℝ)^b := by positivity
-  have hposB : 0 < (2:ℝ)^B := by positivity
-  have hd := div_le_div_of_nonneg_left (show 0 ≤ 1+ε by linarith) hpos hp
-  have heq : (1/((2:ℝ)^B))*(1-((2:ℝ)^B-1)*ε) = (1+ε)/2^B-ε := by
-    field_simp; ring
-  rw [heq]
-  unfold scaleA
-  norm_num only [pow_zero,mul_one]
-  linarith only [hd]
-
-theorem sourceScaleProduct_lower (B D i : ℕ) (hB : 2 ≤ B) :
-    1/(2*((2:ℝ)^(B*D))) ≤ sourceScaleProduct B D i := by
-  let ε := sourceEpsilon B D
-  let u := ((2:ℝ)^B-1)*ε
-  let a := (1/(2:ℝ)^B)*(1-u)
-  have hε : 0 < ε := sourceEpsilon_pos _ _
-  have hp : (1:ℝ) ≤ 2^B := one_le_pow₀ (by norm_num)
-  have hu : 0 ≤ u := mul_nonneg (sub_nonneg.mpr hp) hε.le
-  have hsmall := sourceEpsilon_small B D
-  have hu' : ((D:ℝ)+1)*u ≤ 1/2 := by
-    dsimp [u]
-    dsimp [ε] at *
-    nlinarith only [mul_nonneg (show 0 ≤ (D:ℝ)+1 by positivity) hε.le,hsmall]
-  have hu1 : u ≤ 1/2 := by nlinarith only [hu',mul_nonneg (Nat.cast_nonneg D) hu]
-  have ha : 0 ≤ a := mul_nonneg (by positivity) (by linarith only [hu1])
-  have ha1 : a ≤ 1 := by
-    dsimp [a]
-    have h1 : 1/(2:ℝ)^B ≤ 1 := (div_le_one (by positivity)).mpr hp
-    exact (mul_le_mul_of_nonneg_left (by linarith only [hu] : 1-u ≤ 1) (by positivity)).trans (by simpa using h1)
-  have hb : 1/2 ≤ (1-u)^D := by
-    have H := one_add_mul_le_pow (a := -u) (by linarith only [hu1]) D
-    have hd : (D:ℝ)*u ≤ 1/2 := by nlinarith only [hu',hu]
-    simpa only [sub_eq_add_neg] using (show (1:ℝ)/2 ≤ (1+-u)^D by nlinarith only [H,hd])
-  have hc : (Finset.Ico i D).card ≤ D := by simp only [Nat.card_Ico]; omega
-  calc
-    _ = (1/(2:ℝ)^B)^D*(1/2) := by rw [div_pow,one_pow,←pow_mul]; field_simp
-    _ ≤ (1/(2:ℝ)^B)^D*(1-u)^D := mul_le_mul_of_nonneg_left hb (by positivity)
-    _ = a^D := (mul_pow _ _ _).symm
-    _ ≤ a^(Finset.Ico i D).card := pow_le_pow_of_le_one ha ha1 hc
-    _ = ∏j∈Finset.Ico i D,a := by simp
-    _ ≤ sourceScaleProduct B D i := by
-      apply Finset.prod_le_prod₀ (fun _ _ => ha)
-      intro j _
-      apply scaleA_zero_lower _ _ _ _ hε.le
-      unfold sourceDimension
-      split_ifs <;> omega
-
-theorem sourceDepth_upper (B : ℕ) (ζ H : ℝ) (_hB : 2 ≤ B)
-    (hζ : ζ ≤ 1) (hH : 1 ≤ H) :
-    (sourceDepth B ζ H:ℝ) ≤ (1-ζ)*Real.log H/((B:ℝ)*Real.log 2) := by
-  apply Nat.floor_le
-  exact div_nonneg (mul_nonneg (sub_nonneg.mpr hζ) (Real.log_nonneg hH))
-    (mul_nonneg (Nat.cast_nonneg _) (Real.log_nonneg (by norm_num)))
-
-theorem sourceDepth_power (B : ℕ) (ζ H : ℝ) (hB : 2 ≤ B)
-    (hζ : ζ ≤ 1) (hH : 1 ≤ H) :
-    (2:ℝ)^(B*sourceDepth B ζ H) ≤ H^(1-ζ) := by
-  have hden : 0 < (B:ℝ)*Real.log 2 :=
-    mul_pos (by exact_mod_cast (show 0 < B by omega)) (Real.log_pos (by norm_num))
-  have hd := (le_div_iff₀ hden).mp (sourceDepth_upper B ζ H hB hζ hH)
-  apply (Real.log_le_log_iff (by positivity) (Real.rpow_pos_of_pos (by linarith) _)).mp
-  rw [Real.log_pow,Real.log_rpow (by linarith : 0 < H)]
-  push_cast
-  nlinarith only [hd]
-
-/-- Source smallest-scale lower bound, with an absolute explicit factor. -/
-theorem sourceTargetScale_log_lower (B : ℕ) (ζ H n : ℝ) (i : ℕ) (hB : 2 ≤ B)
-    (hζ : ζ ≤ 1) (hH : 1 ≤ H) (hn : 1 < n) :
-    Real.log n/(2*H^(1-ζ)) ≤ Real.log (sourceTargetScale B (sourceDepth B ζ H) n i) := by
-  have hp := sourceDepth_power B ζ H hB hζ hH
-  have hs := sourceScaleProduct_lower B (sourceDepth B ζ H) i hB
-  have hfrac : 1/(2*H^(1-ζ)) ≤ 1/(2*(2:ℝ)^(B*sourceDepth B ζ H)) :=
-    div_le_div_of_nonneg_left (by norm_num) (by positivity) (by linarith only [hp])
-  have hh := mul_le_mul_of_nonneg_right (hfrac.trans hs) (Real.log_pos hn).le
-  unfold sourceTargetScale
-  rw [Real.log_rpow (by linarith)]
-  simpa only [div_eq_mul_inv,one_mul,mul_comm] using hh
-
-/-- Polynomial depth losses are negligible compared with the H-zeta gain.
- This statement has no bounded-deletion-rate premise. -/
-theorem sourceDepth_polynomial_absorption (B h : ℕ) (ζ : ℝ) (hB : 2 ≤ B)
-    (hζ : ζ∈Set.Ioo 0 1) {H : ℕ → ℝ} (hH : Tendsto H atTop atTop) :
-    Tendsto (fun k => ((sourceDepth B ζ (H k):ℝ)+1)^h/(H k)^ζ) atTop (nhds 0) := by
-  let A := (1-ζ)/((B:ℝ)*Real.log 2)+1
-  have hd : 0 < (B:ℝ)*Real.log 2 :=
-    mul_pos (by exact_mod_cast (show 0 < B by omega)) (Real.log_pos (by norm_num))
-  have hA : 0 < A := by dsimp [A]; positivity [sub_pos.mpr hζ.2]
-  have hl := (isLittleO_log_rpow_rpow_atTop (h:ℝ) hζ.1).tendsto_div_nhds_zero.comp hH
-  simp only [Real.rpow_natCast] at hl
-  have hx := hl.const_mul (A^h)
-  simp only [mul_zero] at hx
-  apply squeeze_zero' _ _ hx
-  · filter_upwards [hH.eventually_gt_atTop 0] with k hk
-    exact div_nonneg (by positivity) (Real.rpow_pos_of_pos hk _).le
-  · filter_upwards [hH.eventually_ge_atTop 1,
-      (Real.tendsto_log_atTop.comp hH).eventually_ge_atTop 1] with k hk hklog
-    have hD := sourceDepth_upper B ζ (H k) hB hζ.2.le hk
-    have hDa : (sourceDepth B ζ (H k):ℝ)+1 ≤ A*Real.log (H k) := by
-      dsimp [A]
-      change 1 ≤ Real.log (H k) at hklog
-      calc
-        _ ≤ (1-ζ)*Real.log (H k)/((B:ℝ)*Real.log 2)+Real.log (H k) := add_le_add hD hklog
-        _ = _ := by ring
-    have hh := pow_le_pow_left₀ (by positivity : 0 ≤ (sourceDepth B ζ (H k):ℝ)+1) hDa h
-    rw [mul_pow] at hh
-    exact (div_le_div_of_nonneg_right hh (by positivity)).trans_eq (by dsimp only [Function.comp_def]; ring)
-
-/-- Uniform quantitative scale margin.  This is the source's Lambda*H^zeta
- bound, before any little-o notation or parameter choices. -/
-theorem sourceTargetScale_epsilon_log_lower (B : ℕ) (ζ H n Λ : ℝ) (i : ℕ)
-    (hB : 2 ≤ B) (hζ : ζ ≤ 1) (hH : 1 ≤ H) (hn : 1 < n)
-    (hΛ : 0 < Λ) (hrel : Real.log n=Λ*H) :
-    Λ*H^ζ/(2*(2:ℝ)^(4*B+100)*((sourceDepth B ζ H:ℝ)+1)) ≤
-      sourceEpsilon B (sourceDepth B ζ H)*
-        Real.log (sourceTargetScale B (sourceDepth B ζ H) n i) := by
-  have h := mul_le_mul_of_nonneg_left (sourceTargetScale_log_lower B ζ H n i hB hζ hH hn)
-    (sourceEpsilon_pos B (sourceDepth B ζ H)).le
-  apply le_trans _ h
-  rw [hrel,Real.rpow_sub (by linarith : 0 < H), Real.rpow_one]
-  unfold sourceEpsilon
-  have hH0 : H≠0 := by linarith
-  have hHz : H^ζ≠0 := (Real.rpow_pos_of_pos (by linarith : 0 < H) _).ne'
-  have hd : (sourceDepth B ζ H:ℝ)+1≠0 := by positivity
-  have hp : (2:ℝ)^(4*B+100)≠0 := by positivity
-  apply le_of_eq
-  field_simp
-
-theorem source_absorption_ratio_bound (B h : ℕ) (ζ H n Λ : ℝ) (i : ℕ)
-    (hB : 2 ≤ B) (hζ : ζ ≤ 1) (hH : 1 ≤ H) (hn : 1 < n)
-    (hΛ : 0 < Λ) (hrel : Real.log n=Λ*H) :
-    ((sourceDepth B ζ H:ℝ)+1)^h*Λ /
-      (sourceEpsilon B (sourceDepth B ζ H)*
-        Real.log (sourceTargetScale B (sourceDepth B ζ H) n i)) ≤
-      (2*(2:ℝ)^(4*B+100))*(((sourceDepth B ζ H:ℝ)+1)^(h+1)/H^ζ) := by
-  have hHz : 0 < H^ζ := Real.rpow_pos_of_pos (by linarith) _
-  have hlow := sourceTargetScale_epsilon_log_lower B ζ H n Λ i hB hζ hH hn hΛ hrel
-  have hh := div_le_div_of_nonneg_left
-    (by positivity : 0 ≤ ((sourceDepth B ζ H:ℝ)+1)^h*Λ)
-    (by positivity : 0 < Λ*H^ζ/(2*(2:ℝ)^(4*B+100)*((sourceDepth B ζ H:ℝ)+1))) hlow
-  apply hh.trans_eq
-  rw [pow_succ]
-  field_simp; ring
-
-/-- Joint-regime uniform absorption at the genuine backwards scales,
- with an arbitrary index choice and with the channel unchanged across scales. -/
-theorem source_uniform_absorption (B h : ℕ) (ζ : ℝ) (hB : 2 ≤ B) (hζ : ζ∈Set.Ioo 0 1)
-    {n Λ H : ℕ → ℝ} (i : ℕ → ℕ) (hn : Tendsto n atTop atTop) (hH : Tendsto H atTop atTop)
-    (hΛ : ∀k,0 < Λ k) (hrel : ∀k,Real.log (n k)=Λ k*H k) :
-    Tendsto (fun k => ((sourceDepth B ζ (H k):ℝ)+1)^h*Λ k /
-      (sourceEpsilon B (sourceDepth B ζ (H k))*
-        Real.log (sourceTargetScale B (sourceDepth B ζ (H k)) (n k) (i k)))) atTop (nhds 0) := by
-  have hb := (sourceDepth_polynomial_absorption B (h+1) ζ hB hζ hH).const_mul
-    (2*(2:ℝ)^(4*B+100))
-  simp only [mul_zero] at hb
-  apply squeeze_zero' _ _ hb
-  · filter_upwards [hH.eventually_ge_atTop 1,hn.eventually_gt_atTop 1] with k hk hnk
-    have hh := sourceTargetScale_epsilon_log_lower B ζ (H k) (n k) (Λ k) (i k)
-      hB hζ.2.le hk hnk (hΛ k) (hrel k)
-    have hΛk := hΛ k
-    have hHk : 0 < H k := by linarith
-    exact div_nonneg (by positivity) ((by positivity :
-      0 ≤ Λ k*(H k)^ζ/(2*(2:ℝ)^(4*B+100)*((sourceDepth B ζ (H k):ℝ)+1))).trans hh)
-  · filter_upwards [hH.eventually_ge_atTop 1,hn.eventually_gt_atTop 1] with k hk hnk
-    exact source_absorption_ratio_bound B h ζ (H k) (n k) (Λ k) (i k)
-      hB hζ.2.le hk hnk (hΛ k) (hrel k)
-
-theorem source_uniform_exp_absorption (B h : ℕ) (ζ C δ : ℝ) (hB : 2 ≤ B)
-    (hζ : ζ∈Set.Ioo 0 1) (hC : 0 ≤ C) (hδ : 0 < δ)
-    {n Λ H : ℕ → ℝ} (hn : Tendsto n atTop atTop) (hH : Tendsto H atTop atTop)
-    (hΛ : ∀k,0 < Λ k) (hrel : ∀k,Real.log (n k)=Λ k*H k) :
-    ∀ᶠk in atTop,∀i,
-      Real.exp (C*((sourceDepth B ζ (H k):ℝ)+1)^h*Λ k) ≤
-        (sourceTargetScale B (sourceDepth B ζ (H k)) (n k) i)^
-          (δ*sourceEpsilon B (sourceDepth B ζ (H k))) := by
-  have hb := (sourceDepth_polynomial_absorption B (h+1) ζ hB hζ hH).const_mul
-    (C*(2*(2:ℝ)^(4*B+100)))
-  simp only [mul_zero] at hb
-  filter_upwards [hH.eventually_ge_atTop 1,hn.eventually_gt_atTop 1,
-    hb.eventually_lt_const hδ] with k hk hnk hbound
-  intro i
-  have hratio := source_absorption_ratio_bound B h ζ (H k) (n k) (Λ k) i
-    hB hζ.2.le hk hnk (hΛ k) (hrel k)
-  have hlower := sourceTargetScale_epsilon_log_lower B ζ (H k) (n k) (Λ k) i
-    hB hζ.2.le hk hnk (hΛ k) (hrel k)
-  have hΛk := hΛ k
-  have hHk : 0 < H k := by linarith
-  have hden : 0 < sourceEpsilon B (sourceDepth B ζ (H k))*
-      Real.log (sourceTargetScale B (sourceDepth B ζ (H k)) (n k) i) :=
-    lt_of_lt_of_le (by positivity) hlower
-  have hMul := mul_le_mul_of_nonneg_left hratio hC
-  have H' : C*(((sourceDepth B ζ (H k):ℝ)+1)^h*Λ k)/
-      (sourceEpsilon B (sourceDepth B ζ (H k))*
-        Real.log (sourceTargetScale B (sourceDepth B ζ (H k)) (n k) i)) ≤ δ := by
-    calc
-      _ = C*(((sourceDepth B ζ (H k):ℝ)+1)^h*Λ k /
-        (sourceEpsilon B (sourceDepth B ζ (H k))*
-          Real.log (sourceTargetScale B (sourceDepth B ζ (H k)) (n k) i))) := by ring
-      _ ≤ _ := hMul.trans (by simpa only [mul_assoc] using hbound.le)
-  have HH := (div_le_iff₀ hden).mp H'
-  have hN : 0 < sourceTargetScale B (sourceDepth B ζ (H k)) (n k) i :=
-    Real.rpow_pos_of_pos (by linarith) _
-  rw [Real.rpow_def_of_pos hN]
-  exact Real.exp_le_exp.mpr (by nlinarith only [HH])
-
-/-- Direct instantiation with the manuscript's known deletion rate: Lambda
- is q^{-3}, H=q^3 log n, never recomputed at an intermediate scale. -/
-theorem source_deletion_uniform_absorption (B h : ℕ) (ζ : ℝ) (hB : 2 ≤ B)
-    (hζ : ζ∈Set.Ioo 0 1) {n q : ℕ → ℝ} (i : ℕ → ℕ)
-    (hq : ∀k,0 < q k) (hn : Tendsto n atTop atTop)
-    (hH : Tendsto (fun k => (q k)^3*Real.log (n k)) atTop atTop) :
-    Tendsto (fun k =>
-      ((sourceDepth B ζ ((q k)^3*Real.log (n k)):ℝ)+1)^h*(q k)^(-3:ℤ) /
-      (sourceEpsilon B (sourceDepth B ζ ((q k)^3*Real.log (n k)))*
-        Real.log (sourceTargetScale B (sourceDepth B ζ ((q k)^3*Real.log (n k)))
-          (n k) (i k)))) atTop (nhds 0) := by
-  apply source_uniform_absorption B h ζ hB hζ i hn hH
-  · intro k
-    exact zpow_pos (hq k) _
-  · intro k
-    simp only [zpow_neg,zpow_ofNat]
-    field_simp [(hq k).ne']
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open Filter
-open scoped Topology
-
-/-- Logarithmic order along a genuine diverging positive scale. -/
-def PowerRate (N f : ℕ → ℝ) (a : ℝ) : Prop :=
-  (∀ᶠ t in Filter.atTop, 0 < f t) ∧ Tendsto (fun t => Real.log (f t) / Real.log (N t)) Filter.atTop (𝓝 a)
-
-/-- A robust upper order, with arbitrary strictly positive exponent slack. -/
-def PowerUpper (N f : ℕ → ℝ) (a : ℝ) : Prop :=
-  ∀ b, a < b → ∀ᶠ t in Filter.atTop, |f t| ≤ (N t)^b
-
-variable {N f g : ℕ → ℝ} {a b : ℝ}
-
-theorem PowerRate.congr (h : PowerRate N f a) (he : f =ᶠ[Filter.atTop] g) : PowerRate N g a := by
-  constructor
-  · filter_upwards [h.1,he] with t ht he; simpa [←he] using ht
-  · exact h.2.congr' (he.mono fun t ht => by simp [ht])
-
-theorem PowerRate.const (hN : Tendsto N Filter.atTop Filter.atTop) {c : ℝ} (hc : 0<c) :
-    PowerRate N (fun _ => c) 0 := by
-  exact ⟨Eventually.of_forall (fun _ => hc), (Real.tendsto_log_atTop.comp hN).const_div_atTop _⟩
-
-theorem PowerRate.base (hN : Tendsto N Filter.atTop Filter.atTop) : PowerRate N N 1 := by
-  have hn := hN.eventually_gt_atTop 1
-  refine ⟨hn.mono fun _ h => by linarith, ?_⟩
-  apply tendsto_congr' _ |>.mpr tendsto_const_nhds
-  filter_upwards [hn] with t ht
-  exact div_self (ne_of_gt (Real.log_pos ht))
-
-theorem PowerRate.mul (hf : PowerRate N f a) (hg : PowerRate N g b) :
-    PowerRate N (fun t => f t*g t) (a+b) := by
-  refine ⟨by filter_upwards [hf.1,hg.1] with t h ht; positivity, ?_⟩
-  apply (hf.2.add hg.2).congr'
-  filter_upwards [hf.1,hg.1] with t h ht
-  rw [Real.log_mul h.ne' ht.ne',add_div]
-
-theorem PowerRate.div (hf : PowerRate N f a) (hg : PowerRate N g b) :
-    PowerRate N (fun t => f t/g t) (a-b) := by
-  refine ⟨by filter_upwards [hf.1,hg.1] with t h ht; positivity, ?_⟩
-  apply (hf.2.sub hg.2).congr'
-  filter_upwards [hf.1,hg.1] with t h ht
-  rw [Real.log_div h.ne' ht.ne',sub_div]
-
-theorem PowerRate.rpow (hf : PowerRate N f a) (c : ℝ) :
-    PowerRate N (fun t => (f t)^c) (a*c) := by
-  refine ⟨by filter_upwards [hf.1] with t h; positivity, ?_⟩
-  apply (hf.2.mul_const c).congr'
-  filter_upwards [hf.1] with t h
-  rw [Real.log_rpow h]
-  ring
-
-theorem PowerRate.pow (hf : PowerRate N f a) (c : ℕ) :
-    PowerRate N (fun t => (f t)^c) (a*c) := by
-  simpa only [Real.rpow_natCast] using hf.rpow (c:ℝ)
-
-theorem PowerRate.sqrt (hf : PowerRate N f a) :
-    PowerRate N (fun t => Real.sqrt (f t)) (a/2) := by
-  simpa only [Real.sqrt_eq_rpow,div_eq_mul_inv, one_mul] using hf.rpow (1/2:ℝ)
-
-theorem PowerRate.inv (hN : Tendsto N Filter.atTop Filter.atTop) (hf : PowerRate N f a) :
-    PowerRate N (fun t=>(f t)⁻¹) (-a) := by
-  simpa only [one_div,zero_sub] using (PowerRate.const hN (by norm_num : (0:ℝ)<1)).div hf
-
-theorem PowerRate.const_mul (hN : Tendsto N Filter.atTop Filter.atTop) (hf : PowerRate N f a)
-    {c : ℝ} (hc : 0<c) : PowerRate N (fun t=>c*f t) a := by
-  simpa only [zero_add] using (PowerRate.const hN hc).mul hf
-
-theorem PowerRate.mul_const (hN : Tendsto N Filter.atTop Filter.atTop) (hf : PowerRate N f a)
-    {c : ℝ} (hc : 0<c) : PowerRate N (fun t=>f t*c) a := by
-  simpa only [add_zero] using hf.mul (PowerRate.const hN hc)
-
-theorem PowerRate.div_const (hN : Tendsto N Filter.atTop Filter.atTop) (hf : PowerRate N f a)
-    {c : ℝ} (hc : 0<c) : PowerRate N (fun t=>f t/c) a := by
-  simpa only [sub_zero] using hf.div (PowerRate.const hN hc)
-
-theorem PowerRate.upper (hN : Tendsto N Filter.atTop Filter.atTop) (hf : PowerRate N f a) :
-    PowerUpper N f a := by
-  intro b hb
-  filter_upwards [hN.eventually_gt_atTop 1,hf.1,hf.2.eventually_lt_const hb] with t hn hp hab
-  rw [abs_of_pos hp,←Real.log_le_log_iff hp (Real.rpow_pos_of_pos (by linarith) b),Real.log_rpow (by linarith : 0<N t)]
-  exact (div_lt_iff₀ (Real.log_pos hn)).mp hab |>.le
-
-theorem PowerRate.lower (hN : Tendsto N Filter.atTop Filter.atTop) (hf : PowerRate N f a) {c : ℝ}
-    (hc : c<a) : ∀ᶠ t in Filter.atTop, (N t)^c ≤ f t := by
-  filter_upwards [hN.eventually_gt_atTop 1,hf.1,hf.2.eventually_const_lt hc] with t hn hp hac
-  rw [←Real.log_le_log_iff (Real.rpow_pos_of_pos (by linarith) c) hp, Real.log_rpow (by linarith : 0<N t)]
-  exact (lt_div_iff₀ (Real.log_pos hn)).mp hac |>.le
-
-theorem PowerRate.tendsto_atTop (hN : Tendsto N Filter.atTop Filter.atTop) (hf : PowerRate N f a) (ha : 0<a) :
-    Tendsto f Filter.atTop Filter.atTop := by
-  exact tendsto_atTop_mono' Filter.atTop (hf.lower hN (show a/2<a by linarith))
-    ((tendsto_rpow_atTop (show 0<a/2 by linarith)).comp hN)
-
-theorem PowerUpper.mono (hf : PowerUpper N f a) (hab : a≤b) : PowerUpper N f b :=
-  fun c hc => hf c (lt_of_le_of_lt hab hc)
-
-theorem PowerUpper.congr (hf : PowerUpper N f a) (he : f =ᶠ[Filter.atTop] g) : PowerUpper N g a := by
-  intro b hb
-  filter_upwards [hf b hb,he] with t h he
-  simpa [←he] using h
-
-theorem PowerUpper.of_abs_le (hg : PowerUpper N g a) (h : ∀ᶠ t in Filter.atTop, |f t|≤|g t|) :
-    PowerUpper N f a := by
-  intro b hb
-  filter_upwards [hg b hb,h] with t ht hh
-  exact hh.trans ht
-
-theorem PowerUpper.zero (hN : Tendsto N Filter.atTop Filter.atTop) (a : ℝ) : PowerUpper N (fun _=>0) a := by
-  intro b hb
-  filter_upwards [hN.eventually_gt_atTop 0] with t ht
-  simpa using (Real.rpow_pos_of_pos ht b).le
-
-theorem PowerUpper.mul (hN : Tendsto N Filter.atTop Filter.atTop) (hf : PowerUpper N f a)
-    (hg : PowerUpper N g b) : PowerUpper N (fun t=>f t*g t) (a+b) := by
-  intro c hc
-  let δ := (c-(a+b))/2
-  have hd : 0<δ := by dsimp [δ]; linarith
-  filter_upwards [hN.eventually_gt_atTop 0,hf (a+δ) (by linarith),hg (b+δ) (by linarith)] with t hn ht hs
-  calc |f t*g t|=|f t| * |g t| := abs_mul _ _
-       _ ≤ (N t)^(a+δ)*(N t)^(b+δ) := mul_le_mul ht hs (abs_nonneg _) (by positivity)
-       _ = (N t)^c := by rw [←Real.rpow_add hn]; congr 1; dsimp [δ]; ring
-
-theorem PowerUpper.pow (hN : Tendsto N Filter.atTop Filter.atTop) (hf : PowerUpper N f a)
-    (r : ℕ) : PowerUpper N (fun t=>(f t)^r) (a*r) := by
-  induction r with
-  | zero => simpa using (PowerRate.const hN (by norm_num : (0:ℝ)<1)).upper hN
-  | succ r ih =>
-    have H := ih.mul hN hf
-    convert H using 1
-    · ext t; exact pow_succ _ _
-    · push_cast; ring
-
-theorem PowerUpper.add (hN : Tendsto N Filter.atTop Filter.atTop) (hf : PowerUpper N f a)
-    (hg : PowerUpper N g a) : PowerUpper N (fun t=>f t+g t) a := by
-  intro c hc
-  let d := (a+c)/2
-  have had : a<d := by dsimp [d]; linarith
-  have hdc : d<c := by dsimp [d]; linarith
-  have hpow := ((tendsto_rpow_atTop (sub_pos.mpr hdc)).comp hN).eventually_ge_atTop 2
-  filter_upwards [hN.eventually_gt_atTop 0,hf d had,hg d had,hpow] with t hn hf hg hp
-  calc |f t+g t| ≤ |f t|+|g t| := abs_add_le _ _
-       _ ≤ 2*(N t)^d := by linarith
-       _ ≤ (N t)^(c-d)*(N t)^d := mul_le_mul_of_nonneg_right hp (by positivity)
-       _ = (N t)^c := by rw [←Real.rpow_add hn]; congr 1; ring
-
-theorem PowerUpper.sum {ι : Type*} (hN : Tendsto N Filter.atTop Filter.atTop) (S : Finset ι)
-    (f : ι→ℕ→ℝ) (h : ∀i∈S, PowerUpper N (f i) a) :
-    PowerUpper N (fun t=>∑i∈S,f i t) a := by
-  classical
-  induction S using Finset.induction_on with
-  | empty => simpa using PowerUpper.zero hN a
-  | @insert i S hi ih =>
-    have hh := (h i (Finset.mem_insert_self _ _)).add hN
-      (ih (fun j hj=>h j (Finset.mem_insert_of_mem hj)))
-    simpa only [Finset.sum_insert hi] using hh
-
-theorem PowerRate.add_const (hN : Tendsto N Filter.atTop Filter.atTop) (hf : PowerRate N f a)
-    (ha : 0<a) {c : ℝ} (hc : 0≤c) : PowerRate N (fun t=>f t+c) a := by
-  have ht := hf.tendsto_atTop hN ha
-  have hratio : Tendsto (fun t=>1+c/f t) Filter.atTop (𝓝 1) := by
-    simpa using (tendsto_const_nhds.add (ht.const_div_atTop c))
-  have hl : Tendsto (fun t=>Real.log (1+c/f t)/Real.log (N t)) Filter.atTop (𝓝 0) :=
-    (hratio.log one_ne_zero).div_atTop (Real.tendsto_log_atTop.comp hN)
-  refine ⟨by filter_upwards [hf.1] with t h; positivity, ?_⟩
-  have H := hf.2.add hl
-  simp only [add_zero] at H
-  apply H.congr'
-  filter_upwards [hf.1] with t hp
-  have hh : 0<1+c/f t := by positivity
-  rw [←add_div,←Real.log_mul hp.ne' hh.ne']
-  congr 2
-  field_simp
-
-theorem PowerRate.of_tendsto (hN : Tendsto N Filter.atTop Filter.atTop)
-    {c : ℝ} (hc : 0<c) (hf : Tendsto f Filter.atTop (𝓝 c)) : PowerRate N f 0 := by
-  exact ⟨hf.eventually_const_lt hc,(hf.log hc.ne').div_atTop (Real.tendsto_log_atTop.comp hN)⟩
-
-theorem PowerUpper.tendsto_zero (hN : Tendsto N Filter.atTop Filter.atTop)
-    (hf : PowerUpper N f a) (ha : a<0) : Tendsto f Filter.atTop (𝓝 0) := by
-  have H := (tendsto_rpow_neg_atTop (show 0< -(a/2) by linarith)).comp hN
-  simp only [neg_neg] at H
-  exact squeeze_zero_norm' (by simpa only [Real.norm_eq_abs,Function.comp_apply] using hf (a/2) (by linarith)) H
-
-theorem PowerRate.of_ratio (hN : Tendsto N Filter.atTop Filter.atTop) (hf : PowerRate N f a)
-    {c : ℝ} (hc : 0<c) (hr : Tendsto (fun t=>g t/f t) Filter.atTop (𝓝 c)) : PowerRate N g a := by
-  have hp : ∀ᶠ t in Filter.atTop, 0<g t := by
-    filter_upwards [hf.1,hr.eventually_const_lt hc] with t ht hgt
-    exact (div_pos_iff.mp hgt).resolve_right (by intro h; linarith [h.2]) |>.1
-  refine ⟨hp,?_⟩
-  have hh := (hr.log hc.ne').div_atTop (Real.tendsto_log_atTop.comp hN)
-  have H := hf.2.add hh
-  simp only [add_zero] at H
-  apply H.congr'
-  filter_upwards [hf.1,hp] with t hf hg
-  simp only [Function.comp_apply]
-  rw [Real.log_div hg.ne' hf.ne']
-  ring
-
-theorem PowerRate.nat_floor (hN : Tendsto N Filter.atTop Filter.atTop) (hf : PowerRate N f a)
-    (ha : 0<a) : PowerRate N (fun t=>(⌊f t⌋₊:ℝ)) a := by
-  exact hf.of_ratio hN (by norm_num : (0:ℝ)<1)
-    (tendsto_nat_floor_div_atTop.comp (hf.tendsto_atTop hN ha))
-
-theorem PowerRate.exp_neg_upper (hN : Tendsto N Filter.atTop Filter.atTop)
-    (hf : PowerRate N f a) (ha : 0<a) (c : ℝ) :
-    PowerUpper N (fun t=>Real.exp (-f t)) c := by
-  intro b hb
-  have hh := ((isLittleO_log_rpow_atTop (show 0<a/2 by linarith)).tendsto_div_nhds_zero).comp hN
-  have hsmall := (hh.const_mul (-b)).eventually_lt_const (show (-b)*0<1 by simp)
-  filter_upwards [hN.eventually_gt_atTop 0,hf.lower hN (show a/2<a by linarith),hsmall] with t hn hl hs
-  rw [abs_of_pos (Real.exp_pos _),Real.rpow_def_of_pos hn]
-  apply Real.exp_le_exp.mpr
-  have hpos := Real.rpow_pos_of_pos hn (a/2)
-  have hle : -b*Real.log (N t) ≤ (N t)^(a/2) := by
-    have H := (div_lt_iff₀ hpos).mp (show (-b*Real.log (N t))/(N t)^(a/2)<1 by simpa only [mul_div_assoc,Function.comp_apply] using hs)
-    linarith
-  linarith
-
-end TraceReconstruction
-
-end
-end
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-!
+# Scales
+
+Exact block exponents, deletion reduction, induction depth, backward target scales, and uniform power absorption.
+-/
+
+open scoped ENNReal
+noncomputable section
+
+namespace TraceReconstruction
+/-- The exact real scale exponents in the scale inequalities; natural indices are zero-based. -/
+def scaleA (b j : ℕ) (ε : ℝ) : ℝ := (1+ε)* (2:ℝ)^j / (2:ℝ)^b - ε
+
+def scaleKappa (b j : ℕ) (ε : ℝ) : ℝ :=
+  scaleA b (j-1) ε - 3*((2:ℝ)^j-1)*ε
+
+def sourceEpsilon (B D : ℕ) : ℝ := 1 / ((2:ℝ)^(4*B+100) * (D+1))
+
+theorem scaleA_top (b : ℕ) (ε : ℝ) : scaleA b b ε = 1 := by
+  unfold scaleA
+  field_simp
+  ring
+
+theorem scaleA_step (b j : ℕ) (ε : ℝ) : scaleA b (j+1) ε - 2*scaleA b j ε = ε := by
+  unfold scaleA
+  rw [pow_succ]
+  ring
+
+theorem scaleKappa_step (b j : ℕ) (ε : ℝ) (hj : 1 ≤ j) :
+    2*scaleKappa b j ε - scaleKappa b (j+1) ε - ε = ε := by
+  unfold scaleKappa scaleA
+  simp only [Nat.add_sub_cancel]
+  rw [show j = (j-1)+1 from by omega, pow_succ]
+  simp only [Nat.add_sub_cancel, pow_succ]
+  ring
+
+def forcedDecay (b : ℕ) (ε d : ℝ) : ℝ :=
+  ε/2 + ∑ k ∈ Finset.range (b-1),
+    ((2:ℝ)^b / (2:ℝ)^(k+2))*(scaleKappa b (k+2) ε / 2 + ε) +
+  (2:ℝ)^b * scaleA b 0 ε * d
+
+theorem forcedDecay_summand (b k : ℕ) (ε : ℝ) :
+    ((2:ℝ)^b / (2:ℝ)^(k+2))*(scaleKappa b (k+2) ε / 2 + ε) =
+      (1+ε)/4 - 3*((2:ℝ)^b/2)*ε + ((2:ℝ)^b/(2:ℝ)^(k+1))*ε := by
+  unfold scaleKappa scaleA
+  simp only [show k+2-1=k+1 by omega, pow_succ]
+  field_simp
+  ring
+
+theorem forcedDecay_lower (b : ℕ) (ε d : ℝ) (hb : 1 ≤ b) (he : 0 ≤ ε) :
+    d + ((b:ℝ)-1)/4 - (((2:ℝ)^b-1)*d + 3*b*((2:ℝ)^b/2))*ε ≤
+      forcedDecay b ε d := by
+  have hterm (k : ℕ) : 1/4 - 3*((2:ℝ)^b/2)*ε ≤
+      ((2:ℝ)^b / (2:ℝ)^(k+2))*(scaleKappa b (k+2) ε / 2 + ε) := by
+    rw [forcedDecay_summand]
+    have hp : 0 ≤ ((2:ℝ)^b / (2:ℝ)^(k+1))*ε := by positivity
+    linarith
+  have hsum := Finset.sum_le_sum (fun k (_ : k ∈ Finset.range (b-1)) => hterm k)
+  simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul] at hsum
+  rw [Nat.cast_sub hb, Nat.cast_one] at hsum
+  have ha : (2:ℝ)^b * scaleA b 0 ε * d = d - ((2:ℝ)^b-1)*d*ε := by
+    unfold scaleA
+    simp only [pow_zero, mul_one]
+    field_simp
+    ring
+  unfold forcedDecay
+  rw [ha]
+  have hp : 0 ≤ (2:ℝ)^b*ε := by positivity
+  nlinarith
+
+theorem nat_add_one_le_two_pow (n : ℕ) : n+1 ≤ 2^n := by
+  induction n with
+  | zero => norm_num
+  | succ n ih => rw [pow_succ]; omega
+
+theorem sourceEpsilon_pos (B D : ℕ) : 0 < sourceEpsilon B D := by
+  unfold sourceEpsilon
+  positivity
+
+theorem sourceEpsilon_loss (B D b : ℕ) (d : ℝ) (hb : b ≤ B)
+    (hd0 : 0 ≤ d) (hd : d ≤ 3 + (B:ℝ)*D) :
+    (((2:ℝ)^b-1)*d + 3*b*((2:ℝ)^b/2))*sourceEpsilon B D < 1/100 := by
+  have hpow : (2:ℝ)^b ≤ (2:ℝ)^B := pow_le_pow_right₀ (by norm_num) hb
+  have hB : (B:ℝ)+1 ≤ (2:ℝ)^B := by exact_mod_cast nat_add_one_le_two_pow B
+  have hB0 : (0:ℝ) ≤ B := by positivity
+  have hD0 : (0:ℝ) ≤ D := by positivity
+  have hh : (((2:ℝ)^b-1)*d + 3*b*((2:ℝ)^b/2)) ≤
+      4*((2:ℝ)^B)^2*((D:ℝ)+1) := by
+    have hb' : (b:ℝ) ≤ B := by exact_mod_cast hb
+    have h1 := mul_le_mul_of_nonneg_right (show (2:ℝ)^b-1 ≤ (2:ℝ)^B by linarith) hd0
+    have h2 := mul_le_mul_of_nonneg_left hd (show 0 ≤ (2:ℝ)^B by positivity)
+    have h3 := mul_le_mul hb' hpow (by positivity : (0:ℝ) ≤ (2:ℝ)^b) hB0
+    have h4 := mul_le_mul_of_nonneg_left hB hD0
+    have h5 := mul_le_mul_of_nonneg_left h4 (show 0 ≤ (2:ℝ)^B by positivity)
+    have h6 := mul_le_mul_of_nonneg_left hB (show 0 ≤ (2:ℝ)^B by positivity)
+    nlinarith [sq_nonneg ((2:ℝ)^B)]
+  apply lt_of_le_of_lt (mul_le_mul_of_nonneg_right hh (sourceEpsilon_pos B D).le)
+  unfold sourceEpsilon
+  have hpos : 0 < (2:ℝ)^(4*B+100) := by positivity
+  have hp2 : (2:ℝ)^(2*B) = ((2:ℝ)^B)^2 := by rw [mul_comm 2 B, pow_mul]
+  have hp4 : (2:ℝ)^(4*B+100) = ((2:ℝ)^B)^2 * (2:ℝ)^(2*B+100) := by
+    rw [show 4*B+100=2*B+(2*B+100) by omega, pow_add, hp2]
+  have hh2 : (400:ℝ) < (2:ℝ)^(2*B+100) := by
+    have h := pow_le_pow_right₀ (by norm_num : (1:ℝ) ≤ 2) (by omega : 9 ≤ 2*B+100)
+    norm_num at h ⊢
+    linarith
+  rw [mul_one_div]
+  apply (div_lt_iff₀ (by positivity)).mpr
+  rw [hp4]
+  have := mul_lt_mul_of_pos_left hh2 (show 0 < ((2:ℝ)^B)^2*((D:ℝ)+1) by positivity)
+  nlinarith
+
+theorem forcedDecay_margin (B D b : ℕ) (d : ℝ) (hb1 : 1 ≤ b) (hb : b ≤ B)
+    (hd0 : 0 ≤ d) (hd : d ≤ 3 + (B:ℝ)*D) :
+    d + ((b:ℝ)-1)/4 - 1/20 + 3/100 < forcedDecay b (sourceEpsilon B D) d := by
+  have h := forcedDecay_lower b (sourceEpsilon B D) d hb1 (sourceEpsilon_pos B D).le
+  have hg := sourceEpsilon_loss B D b d hb hd0 hd
+  linarith
+end TraceReconstruction
+
+open Filter
+open scoped Topology
+
+
+open Filter
+open scoped Topology
+namespace TraceReconstruction
+
+def workingDeletion (q : ℝ) : ℝ := min q (1/2)
+def blockGain (b : ℕ) : ℝ := ((b:ℝ)-1)/4-1/20
+def warmupBlocks (B : ℕ) : ℕ := 5*2^B
+def sourceDepth (B : ℕ) (ζ H : ℝ) : ℕ := ⌊(1-ζ)*Real.log H/((B:ℝ)*Real.log 2)⌋₊
+def totalScoreExponent (B D : ℕ) : ℝ :=
+  11/5+∑ i ∈ Finset.range D, blockGain (if i<warmupBlocks B then 2 else B)
+def finalSlope (B : ℕ) (ζ : ℝ) : ℝ := (1-ζ)*blockGain B/((B:ℝ)*Real.log 2)
+
+theorem workingDeletion_bounds {q : ℝ} (hq : q ∈ Set.Ioo 0 1) :
+    workingDeletion q ∈ Set.Ioo 0 1 ∧ q/2 ≤ workingDeletion q ∧ workingDeletion q ≤ q := by
+  unfold workingDeletion
+  constructor
+  · constructor
+    · exact lt_min hq.1 (by norm_num)
+    · exact (min_le_left _ _).trans_lt hq.2
+  constructor
+  · exact le_min (by linarith [hq.1]) (by linarith [hq.2])
+  · exact min_le_left _ _
+
+theorem workingScale_bounds {q L : ℝ} (hq : q ∈ Set.Ioo 0 1) (hL : 0 ≤ L) :
+    q^3*L/8 ≤ (workingDeletion q)^3*L ∧ (workingDeletion q)^3*L ≤ q^3*L := by
+  have hw := workingDeletion_bounds hq
+  have hl := pow_le_pow_left₀ (div_nonneg hq.1.le (by norm_num)) hw.2.1 3
+  have hu := pow_le_pow_left₀ hw.1.1.le hw.2.2 3
+  constructor
+  · have := mul_le_mul_of_nonneg_right hl hL
+    nlinarith only [this]
+  · exact mul_le_mul_of_nonneg_right hu hL
+
+theorem blockGain_two : blockGain 2 = 1/5 := by norm_num [blockGain]
+theorem blockGain_pos (B : ℕ) (hB : 2 ≤ B) : 0 < blockGain B := by
+  have hB' : (2:ℝ) ≤ B := by exact_mod_cast hB
+  unfold blockGain
+  linarith
+
+theorem totalScoreExponent_exact (B D : ℕ) (hD : warmupBlocks B ≤ D) :
+    totalScoreExponent B D = 11/5 + (warmupBlocks B:ℝ)/5 +
+      ((D:ℝ)-warmupBlocks B)*blockGain B := by
+  unfold totalScoreExponent
+  rw [← Nat.add_sub_of_le hD, Finset.sum_range_add]
+  have hfirst : (∑ i ∈ Finset.range (warmupBlocks B),
+      blockGain (if i<warmupBlocks B then 2 else B)) = (warmupBlocks B:ℝ)/5 := by
+    simp only [Finset.sum_congr rfl (fun i hi => congrArg blockGain (ite_eq_left (Finset.mem_range.mp hi))), blockGain_two]
+    simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul]
+    ring
+  rw [hfirst]
+  have hlast : (∑ i ∈ Finset.range (D-warmupBlocks B),
+      blockGain (if warmupBlocks B+i<warmupBlocks B then 2 else B)) =
+      ((D:ℝ)-warmupBlocks B)*blockGain B := by
+    simp only [show ∀ i, ¬warmupBlocks B+i<warmupBlocks B from fun i => by omega, ite_false,
+      Finset.sum_const, Finset.card_range, nsmul_eq_mul, Nat.cast_sub hD]
+  rw [hlast]
+  simp only [Nat.add_sub_of_le hD]
+  ring
+
+theorem sourceDepth_lower (B : ℕ) (ζ H : ℝ) :
+    (1-ζ)*Real.log H/((B:ℝ)*Real.log 2)-1 < (sourceDepth B ζ H:ℝ) := by
+  have := Nat.lt_floor_add_one ((1-ζ)*Real.log H/((B:ℝ)*Real.log 2))
+  change _ < (sourceDepth B ζ H:ℝ)+1 at this
+  linarith
+
+theorem totalScoreExponent_slope_lower (B : ℕ) (ζ H : ℝ) (hB : 2 ≤ B)
+    (hD : warmupBlocks B ≤ sourceDepth B ζ H) :
+    finalSlope B ζ * Real.log H -
+      (blockGain B + (warmupBlocks B:ℝ)*blockGain B) <
+      totalScoreExponent B (sourceDepth B ζ H) := by
+  rw [totalScoreExponent_exact B _ hD]
+  have hh := mul_lt_mul_of_pos_right (sourceDepth_lower B ζ H) (blockGain_pos B hB)
+  unfold finalSlope
+  have he : ((1-ζ)*Real.log H/((B:ℝ)*Real.log 2))*blockGain B =
+      ((1-ζ)*blockGain B/((B:ℝ)*Real.log 2))*Real.log H := by ring
+  rw [sub_mul, he] at hh
+  have hw : (0:ℝ) ≤ warmupBlocks B := by positivity
+  linarith
+
+theorem sourceDepth_tendsto (B : ℕ) (ζ : ℝ) (hB : 2 ≤ B) (hζ : ζ < 1)
+    {H : ℕ → ℝ} (hH : Tendsto H atTop atTop) :
+    Tendsto (fun k => sourceDepth B ζ (H k)) atTop atTop := by
+  have hden : 0 < (B:ℝ)*Real.log 2 := by
+    apply mul_pos
+    · exact_mod_cast (show 0 < B by omega)
+    · exact Real.log_pos (by norm_num)
+  have ht := (Real.tendsto_log_atTop.comp hH).const_mul_atTop (sub_pos.mpr hζ)
+  have ht' := ht.atTop_div_const hden
+  exact tendsto_nat_floor_atTop.comp ht'
+
+/-- A fixed multiplicative constant is absorbed by any strict exponent
+slope margin, with two independently diverging instance parameters. -/
+theorem eventually_exponent_margin {N H d : ℕ → ℝ} {α c K a : ℝ}
+    (hN : Tendsto N atTop atTop) (hH : Tendsto H atTop atTop)
+    (hα : c < α) (ha : 0 < a)
+    (hd : ∀ᶠ k in atTop, α*Real.log (H k)-K ≤ d k) :
+    ∀ᶠ k in atTop, (N k)^(c*Real.log (H k)) ≤ a*(N k)^(d k) := by
+  have hlog := (Real.tendsto_log_atTop.comp hH).const_mul_atTop (sub_pos.mpr hα)
+  filter_upwards [hd, hlog.eventually_ge_atTop (K+1), hN.eventually_ge_atTop (max 1 (1/a))] with k hd hlog hN
+  have hN1 : 1 ≤ N k := (le_max_left _ _).trans hN
+  have hNa : 1/a ≤ N k := (le_max_right _ _).trans hN
+  simp only [Function.comp_apply] at hlog
+  have hex : c*Real.log (H k)+1 ≤ d k := by nlinarith only [hd,hlog]
+  have hp := Real.rpow_le_rpow_of_exponent_le hN1 hex
+  rw [Real.rpow_add (by linarith : 0 < N k), Real.rpow_one] at hp
+  have hapos : 1 ≤ a*N k := (div_le_iff₀ ha).mp hNa |>.trans_eq (mul_comm _ _)
+  calc
+    (N k)^(c*Real.log (H k)) ≤ (a*N k)*(N k)^(c*Real.log (H k)) := by
+      exact le_mul_of_one_le_left (Real.rpow_nonneg (by linarith) _) hapos
+    _ ≤ a*(N k)^(d k) := by
+      have := mul_le_mul_of_nonneg_left hp ha.le
+      nlinarith only [this]
+
+end TraceReconstruction
+
+open Filter
+open scoped Topology
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+open Filter
+
+/-- Literal source block dimensions, with zero-based block number. -/
+def sourceDimension (B i : ℕ) : ℕ := if i < warmupBlocks B then 2 else B
+
+def sourceScaleProduct (B D i : ℕ) : ℝ :=
+  ∏j∈Finset.Ico i D,scaleA (sourceDimension B j) 0 (sourceEpsilon B D)
+
+/-- The backwards target scales as an exact product of source exponents. -/
+def sourceTargetScale (B D : ℕ) (n : ℝ) (i : ℕ) : ℝ := n ^ sourceScaleProduct B D i
+
+theorem sourceTargetScale_top (B D : ℕ) (n : ℝ) : sourceTargetScale B D n D=n := by
+  simp [sourceTargetScale,sourceScaleProduct]
+
+theorem sourceTargetScale_step (B D i : ℕ) (n : ℝ) (hn : 0 ≤ n) (hi : i < D) :
+    sourceTargetScale B D n i =
+      (sourceTargetScale B D n (i+1))^(scaleA (sourceDimension B i) 0 (sourceEpsilon B D)) := by
+  unfold sourceTargetScale sourceScaleProduct
+  rw [←Real.rpow_mul hn,Finset.prod_eq_prod_Ico_succ_bot hi]
+  rw [mul_comm]
+
+theorem sourceEpsilon_small (B D : ℕ) :
+    ((D:ℝ)+1)*(2:ℝ)^B*sourceEpsilon B D ≤ 1/2 := by
+  have hp : 2*(2:ℝ)^B ≤ (2:ℝ)^(4*B+100) := by
+    rw [←pow_succ']
+    exact pow_le_pow_right₀ (by norm_num) (by omega)
+  unfold sourceEpsilon
+  have hD : 0 < (D:ℝ)+1 := by positivity
+  have hpow : 0 < (2:ℝ)^(4*B+100) := by positivity
+  apply (le_div_iff₀ (by norm_num : (0:ℝ)<2)).mpr
+  have he : ((D:ℝ)+1)*2^B*(1/(2^(4*B+100)*((D:ℝ)+1)))*2 =
+      (2*2^B)/2^(4*B+100) := by field_simp
+  rw [he]
+  exact (div_le_one hpow).mpr hp
+
+theorem scaleA_zero_lower (b B : ℕ) (ε : ℝ) (hb : b ≤ B) (he : 0 ≤ ε) :
+    (1/((2:ℝ)^B))*(1-((2:ℝ)^B-1)*ε) ≤ scaleA b 0 ε := by
+  have hp : (2:ℝ)^b ≤ 2^B := pow_le_pow_right₀ (by norm_num) hb
+  have hpos : 0 < (2:ℝ)^b := by positivity
+  have hposB : 0 < (2:ℝ)^B := by positivity
+  have hd := div_le_div_of_nonneg_left (show 0 ≤ 1+ε by linarith) hpos hp
+  have heq : (1/((2:ℝ)^B))*(1-((2:ℝ)^B-1)*ε) = (1+ε)/2^B-ε := by
+    field_simp; ring
+  rw [heq]
+  unfold scaleA
+  norm_num only [pow_zero,mul_one]
+  linarith only [hd]
+
+theorem sourceScaleProduct_lower (B D i : ℕ) (hB : 2 ≤ B) :
+    1/(2*((2:ℝ)^(B*D))) ≤ sourceScaleProduct B D i := by
+  let ε := sourceEpsilon B D
+  let u := ((2:ℝ)^B-1)*ε
+  let a := (1/(2:ℝ)^B)*(1-u)
+  have hε : 0 < ε := sourceEpsilon_pos _ _
+  have hp : (1:ℝ) ≤ 2^B := one_le_pow₀ (by norm_num)
+  have hu : 0 ≤ u := mul_nonneg (sub_nonneg.mpr hp) hε.le
+  have hsmall := sourceEpsilon_small B D
+  have hu' : ((D:ℝ)+1)*u ≤ 1/2 := by
+    dsimp [u]
+    dsimp [ε] at *
+    nlinarith only [mul_nonneg (show 0 ≤ (D:ℝ)+1 by positivity) hε.le,hsmall]
+  have hu1 : u ≤ 1/2 := by nlinarith only [hu',mul_nonneg (Nat.cast_nonneg D) hu]
+  have ha : 0 ≤ a := mul_nonneg (by positivity) (by linarith only [hu1])
+  have ha1 : a ≤ 1 := by
+    dsimp [a]
+    have h1 : 1/(2:ℝ)^B ≤ 1 := (div_le_one (by positivity)).mpr hp
+    exact (mul_le_mul_of_nonneg_left (by linarith only [hu] : 1-u ≤ 1) (by positivity)).trans (by simpa using h1)
+  have hb : 1/2 ≤ (1-u)^D := by
+    have H := one_add_mul_le_pow (a := -u) (by linarith only [hu1]) D
+    have hd : (D:ℝ)*u ≤ 1/2 := by nlinarith only [hu',hu]
+    simpa only [sub_eq_add_neg] using (show (1:ℝ)/2 ≤ (1+-u)^D by nlinarith only [H,hd])
+  have hc : (Finset.Ico i D).card ≤ D := by simp only [Nat.card_Ico]; omega
+  calc
+    _ = (1/(2:ℝ)^B)^D*(1/2) := by rw [div_pow,one_pow,←pow_mul]; field_simp
+    _ ≤ (1/(2:ℝ)^B)^D*(1-u)^D := mul_le_mul_of_nonneg_left hb (by positivity)
+    _ = a^D := (mul_pow _ _ _).symm
+    _ ≤ a^(Finset.Ico i D).card := pow_le_pow_of_le_one ha ha1 hc
+    _ = ∏j∈Finset.Ico i D,a := by simp
+    _ ≤ sourceScaleProduct B D i := by
+      apply Finset.prod_le_prod₀ (fun _ _ => ha)
+      intro j _
+      apply scaleA_zero_lower _ _ _ _ hε.le
+      unfold sourceDimension
+      split_ifs <;> omega
+
+theorem sourceDepth_upper (B : ℕ) (ζ H : ℝ) (_hB : 2 ≤ B)
+    (hζ : ζ ≤ 1) (hH : 1 ≤ H) :
+    (sourceDepth B ζ H:ℝ) ≤ (1-ζ)*Real.log H/((B:ℝ)*Real.log 2) := by
+  apply Nat.floor_le
+  exact div_nonneg (mul_nonneg (sub_nonneg.mpr hζ) (Real.log_nonneg hH))
+    (mul_nonneg (Nat.cast_nonneg _) (Real.log_nonneg (by norm_num)))
+
+theorem sourceDepth_power (B : ℕ) (ζ H : ℝ) (hB : 2 ≤ B)
+    (hζ : ζ ≤ 1) (hH : 1 ≤ H) :
+    (2:ℝ)^(B*sourceDepth B ζ H) ≤ H^(1-ζ) := by
+  have hden : 0 < (B:ℝ)*Real.log 2 :=
+    mul_pos (by exact_mod_cast (show 0 < B by omega)) (Real.log_pos (by norm_num))
+  have hd := (le_div_iff₀ hden).mp (sourceDepth_upper B ζ H hB hζ hH)
+  apply (Real.log_le_log_iff (by positivity) (Real.rpow_pos_of_pos (by linarith) _)).mp
+  rw [Real.log_pow,Real.log_rpow (by linarith : 0 < H)]
+  push_cast
+  nlinarith only [hd]
+
+/-- Source smallest-scale lower bound, with an absolute explicit factor. -/
+theorem sourceTargetScale_log_lower (B : ℕ) (ζ H n : ℝ) (i : ℕ) (hB : 2 ≤ B)
+    (hζ : ζ ≤ 1) (hH : 1 ≤ H) (hn : 1 < n) :
+    Real.log n/(2*H^(1-ζ)) ≤ Real.log (sourceTargetScale B (sourceDepth B ζ H) n i) := by
+  have hp := sourceDepth_power B ζ H hB hζ hH
+  have hs := sourceScaleProduct_lower B (sourceDepth B ζ H) i hB
+  have hfrac : 1/(2*H^(1-ζ)) ≤ 1/(2*(2:ℝ)^(B*sourceDepth B ζ H)) :=
+    div_le_div_of_nonneg_left (by norm_num) (by positivity) (by linarith only [hp])
+  have hh := mul_le_mul_of_nonneg_right (hfrac.trans hs) (Real.log_pos hn).le
+  unfold sourceTargetScale
+  rw [Real.log_rpow (by linarith)]
+  simpa only [div_eq_mul_inv,one_mul,mul_comm] using hh
+
+/-- Polynomial depth losses are negligible compared with the H-zeta gain.
+ This statement has no bounded-deletion-rate premise. -/
+theorem sourceDepth_polynomial_absorption (B h : ℕ) (ζ : ℝ) (hB : 2 ≤ B)
+    (hζ : ζ∈Set.Ioo 0 1) {H : ℕ → ℝ} (hH : Tendsto H atTop atTop) :
+    Tendsto (fun k => ((sourceDepth B ζ (H k):ℝ)+1)^h/(H k)^ζ) atTop (nhds 0) := by
+  let A := (1-ζ)/((B:ℝ)*Real.log 2)+1
+  have hd : 0 < (B:ℝ)*Real.log 2 :=
+    mul_pos (by exact_mod_cast (show 0 < B by omega)) (Real.log_pos (by norm_num))
+  have hA : 0 < A := by dsimp [A]; positivity [sub_pos.mpr hζ.2]
+  have hl := (isLittleO_log_rpow_rpow_atTop (h:ℝ) hζ.1).tendsto_div_nhds_zero.comp hH
+  simp only [Real.rpow_natCast] at hl
+  have hx := hl.const_mul (A^h)
+  simp only [mul_zero] at hx
+  apply squeeze_zero' _ _ hx
+  · filter_upwards [hH.eventually_gt_atTop 0] with k hk
+    exact div_nonneg (by positivity) (Real.rpow_pos_of_pos hk _).le
+  · filter_upwards [hH.eventually_ge_atTop 1,
+      (Real.tendsto_log_atTop.comp hH).eventually_ge_atTop 1] with k hk hklog
+    have hD := sourceDepth_upper B ζ (H k) hB hζ.2.le hk
+    have hDa : (sourceDepth B ζ (H k):ℝ)+1 ≤ A*Real.log (H k) := by
+      dsimp [A]
+      change 1 ≤ Real.log (H k) at hklog
+      calc
+        _ ≤ (1-ζ)*Real.log (H k)/((B:ℝ)*Real.log 2)+Real.log (H k) := add_le_add hD hklog
+        _ = _ := by ring
+    have hh := pow_le_pow_left₀ (by positivity : 0 ≤ (sourceDepth B ζ (H k):ℝ)+1) hDa h
+    rw [mul_pow] at hh
+    exact (div_le_div_of_nonneg_right hh (by positivity)).trans_eq (by dsimp only [Function.comp_def]; ring)
+
+/-- Uniform quantitative scale margin.  This is the source's Lambda*H^zeta
+ bound, before any little-o notation or parameter choices. -/
+theorem sourceTargetScale_epsilon_log_lower (B : ℕ) (ζ H n Λ : ℝ) (i : ℕ)
+    (hB : 2 ≤ B) (hζ : ζ ≤ 1) (hH : 1 ≤ H) (hn : 1 < n)
+    (hΛ : 0 < Λ) (hrel : Real.log n=Λ*H) :
+    Λ*H^ζ/(2*(2:ℝ)^(4*B+100)*((sourceDepth B ζ H:ℝ)+1)) ≤
+      sourceEpsilon B (sourceDepth B ζ H)*
+        Real.log (sourceTargetScale B (sourceDepth B ζ H) n i) := by
+  have h := mul_le_mul_of_nonneg_left (sourceTargetScale_log_lower B ζ H n i hB hζ hH hn)
+    (sourceEpsilon_pos B (sourceDepth B ζ H)).le
+  apply le_trans _ h
+  rw [hrel,Real.rpow_sub (by linarith : 0 < H), Real.rpow_one]
+  unfold sourceEpsilon
+  have hH0 : H≠0 := by linarith
+  have hHz : H^ζ≠0 := (Real.rpow_pos_of_pos (by linarith : 0 < H) _).ne'
+  have hd : (sourceDepth B ζ H:ℝ)+1≠0 := by positivity
+  have hp : (2:ℝ)^(4*B+100)≠0 := by positivity
+  apply le_of_eq
+  field_simp
+
+theorem source_absorption_ratio_bound (B h : ℕ) (ζ H n Λ : ℝ) (i : ℕ)
+    (hB : 2 ≤ B) (hζ : ζ ≤ 1) (hH : 1 ≤ H) (hn : 1 < n)
+    (hΛ : 0 < Λ) (hrel : Real.log n=Λ*H) :
+    ((sourceDepth B ζ H:ℝ)+1)^h*Λ /
+      (sourceEpsilon B (sourceDepth B ζ H)*
+        Real.log (sourceTargetScale B (sourceDepth B ζ H) n i)) ≤
+      (2*(2:ℝ)^(4*B+100))*(((sourceDepth B ζ H:ℝ)+1)^(h+1)/H^ζ) := by
+  have hHz : 0 < H^ζ := Real.rpow_pos_of_pos (by linarith) _
+  have hlow := sourceTargetScale_epsilon_log_lower B ζ H n Λ i hB hζ hH hn hΛ hrel
+  have hh := div_le_div_of_nonneg_left
+    (by positivity : 0 ≤ ((sourceDepth B ζ H:ℝ)+1)^h*Λ)
+    (by positivity : 0 < Λ*H^ζ/(2*(2:ℝ)^(4*B+100)*((sourceDepth B ζ H:ℝ)+1))) hlow
+  apply hh.trans_eq
+  rw [pow_succ]
+  field_simp; ring
+
+/-- Joint-regime uniform absorption at the genuine backwards scales,
+ with an arbitrary index choice and with the channel unchanged across scales. -/
+theorem source_uniform_absorption (B h : ℕ) (ζ : ℝ) (hB : 2 ≤ B) (hζ : ζ∈Set.Ioo 0 1)
+    {n Λ H : ℕ → ℝ} (i : ℕ → ℕ) (hn : Tendsto n atTop atTop) (hH : Tendsto H atTop atTop)
+    (hΛ : ∀k,0 < Λ k) (hrel : ∀k,Real.log (n k)=Λ k*H k) :
+    Tendsto (fun k => ((sourceDepth B ζ (H k):ℝ)+1)^h*Λ k /
+      (sourceEpsilon B (sourceDepth B ζ (H k))*
+        Real.log (sourceTargetScale B (sourceDepth B ζ (H k)) (n k) (i k)))) atTop (nhds 0) := by
+  have hb := (sourceDepth_polynomial_absorption B (h+1) ζ hB hζ hH).const_mul
+    (2*(2:ℝ)^(4*B+100))
+  simp only [mul_zero] at hb
+  apply squeeze_zero' _ _ hb
+  · filter_upwards [hH.eventually_ge_atTop 1,hn.eventually_gt_atTop 1] with k hk hnk
+    have hh := sourceTargetScale_epsilon_log_lower B ζ (H k) (n k) (Λ k) (i k)
+      hB hζ.2.le hk hnk (hΛ k) (hrel k)
+    have hΛk := hΛ k
+    have hHk : 0 < H k := by linarith
+    exact div_nonneg (by positivity) ((by positivity :
+      0 ≤ Λ k*(H k)^ζ/(2*(2:ℝ)^(4*B+100)*((sourceDepth B ζ (H k):ℝ)+1))).trans hh)
+  · filter_upwards [hH.eventually_ge_atTop 1,hn.eventually_gt_atTop 1] with k hk hnk
+    exact source_absorption_ratio_bound B h ζ (H k) (n k) (Λ k) (i k)
+      hB hζ.2.le hk hnk (hΛ k) (hrel k)
+
+theorem source_uniform_exp_absorption (B h : ℕ) (ζ C δ : ℝ) (hB : 2 ≤ B)
+    (hζ : ζ∈Set.Ioo 0 1) (hC : 0 ≤ C) (hδ : 0 < δ)
+    {n Λ H : ℕ → ℝ} (hn : Tendsto n atTop atTop) (hH : Tendsto H atTop atTop)
+    (hΛ : ∀k,0 < Λ k) (hrel : ∀k,Real.log (n k)=Λ k*H k) :
+    ∀ᶠk in atTop,∀i,
+      Real.exp (C*((sourceDepth B ζ (H k):ℝ)+1)^h*Λ k) ≤
+        (sourceTargetScale B (sourceDepth B ζ (H k)) (n k) i)^
+          (δ*sourceEpsilon B (sourceDepth B ζ (H k))) := by
+  have hb := (sourceDepth_polynomial_absorption B (h+1) ζ hB hζ hH).const_mul
+    (C*(2*(2:ℝ)^(4*B+100)))
+  simp only [mul_zero] at hb
+  filter_upwards [hH.eventually_ge_atTop 1,hn.eventually_gt_atTop 1,
+    hb.eventually_lt_const hδ] with k hk hnk hbound
+  intro i
+  have hratio := source_absorption_ratio_bound B h ζ (H k) (n k) (Λ k) i
+    hB hζ.2.le hk hnk (hΛ k) (hrel k)
+  have hlower := sourceTargetScale_epsilon_log_lower B ζ (H k) (n k) (Λ k) i
+    hB hζ.2.le hk hnk (hΛ k) (hrel k)
+  have hΛk := hΛ k
+  have hHk : 0 < H k := by linarith
+  have hden : 0 < sourceEpsilon B (sourceDepth B ζ (H k))*
+      Real.log (sourceTargetScale B (sourceDepth B ζ (H k)) (n k) i) :=
+    lt_of_lt_of_le (by positivity) hlower
+  have hMul := mul_le_mul_of_nonneg_left hratio hC
+  have H' : C*(((sourceDepth B ζ (H k):ℝ)+1)^h*Λ k)/
+      (sourceEpsilon B (sourceDepth B ζ (H k))*
+        Real.log (sourceTargetScale B (sourceDepth B ζ (H k)) (n k) i)) ≤ δ := by
+    calc
+      _ = C*(((sourceDepth B ζ (H k):ℝ)+1)^h*Λ k /
+        (sourceEpsilon B (sourceDepth B ζ (H k))*
+          Real.log (sourceTargetScale B (sourceDepth B ζ (H k)) (n k) i))) := by ring
+      _ ≤ _ := hMul.trans (by simpa only [mul_assoc] using hbound.le)
+  have HH := (div_le_iff₀ hden).mp H'
+  have hN : 0 < sourceTargetScale B (sourceDepth B ζ (H k)) (n k) i :=
+    Real.rpow_pos_of_pos (by linarith) _
+  rw [Real.rpow_def_of_pos hN]
+  exact Real.exp_le_exp.mpr (by nlinarith only [HH])
+
+/-- Direct instantiation with the manuscript's known deletion rate: Lambda
+ is q^{-3}, H=q^3 log n, never recomputed at an intermediate scale. -/
+theorem source_deletion_uniform_absorption (B h : ℕ) (ζ : ℝ) (hB : 2 ≤ B)
+    (hζ : ζ∈Set.Ioo 0 1) {n q : ℕ → ℝ} (i : ℕ → ℕ)
+    (hq : ∀k,0 < q k) (hn : Tendsto n atTop atTop)
+    (hH : Tendsto (fun k => (q k)^3*Real.log (n k)) atTop atTop) :
+    Tendsto (fun k =>
+      ((sourceDepth B ζ ((q k)^3*Real.log (n k)):ℝ)+1)^h*(q k)^(-3:ℤ) /
+      (sourceEpsilon B (sourceDepth B ζ ((q k)^3*Real.log (n k)))*
+        Real.log (sourceTargetScale B (sourceDepth B ζ ((q k)^3*Real.log (n k)))
+          (n k) (i k)))) atTop (nhds 0) := by
+  apply source_uniform_absorption B h ζ hB hζ i hn hH
+  · intro k
+    exact zpow_pos (hq k) _
+  · intro k
+    simp only [zpow_neg,zpow_ofNat]
+    field_simp [(hq k).ne']
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open Filter
+open scoped Topology
+
+/-- Logarithmic order along a genuine diverging positive scale. -/
+def PowerRate (N f : ℕ → ℝ) (a : ℝ) : Prop :=
+  (∀ᶠ t in Filter.atTop, 0 < f t) ∧ Tendsto (fun t => Real.log (f t) / Real.log (N t)) Filter.atTop (𝓝 a)
+
+/-- A robust upper order, with arbitrary strictly positive exponent slack. -/
+def PowerUpper (N f : ℕ → ℝ) (a : ℝ) : Prop :=
+  ∀ b, a < b → ∀ᶠ t in Filter.atTop, |f t| ≤ (N t)^b
+
+variable {N f g : ℕ → ℝ} {a b : ℝ}
+
+theorem PowerRate.congr (h : PowerRate N f a) (he : f =ᶠ[Filter.atTop] g) : PowerRate N g a := by
+  constructor
+  · filter_upwards [h.1,he] with t ht he; simpa [←he] using ht
+  · exact h.2.congr' (he.mono fun t ht => by simp [ht])
+
+theorem PowerRate.const (hN : Tendsto N Filter.atTop Filter.atTop) {c : ℝ} (hc : 0<c) :
+    PowerRate N (fun _ => c) 0 := by
+  exact ⟨Eventually.of_forall (fun _ => hc), (Real.tendsto_log_atTop.comp hN).const_div_atTop _⟩
+
+theorem PowerRate.base (hN : Tendsto N Filter.atTop Filter.atTop) : PowerRate N N 1 := by
+  have hn := hN.eventually_gt_atTop 1
+  refine ⟨hn.mono fun _ h => by linarith, ?_⟩
+  apply tendsto_congr' _ |>.mpr tendsto_const_nhds
+  filter_upwards [hn] with t ht
+  exact div_self (ne_of_gt (Real.log_pos ht))
+
+theorem PowerRate.mul (hf : PowerRate N f a) (hg : PowerRate N g b) :
+    PowerRate N (fun t => f t*g t) (a+b) := by
+  refine ⟨by filter_upwards [hf.1,hg.1] with t h ht; positivity, ?_⟩
+  apply (hf.2.add hg.2).congr'
+  filter_upwards [hf.1,hg.1] with t h ht
+  rw [Real.log_mul h.ne' ht.ne',add_div]
+
+theorem PowerRate.div (hf : PowerRate N f a) (hg : PowerRate N g b) :
+    PowerRate N (fun t => f t/g t) (a-b) := by
+  refine ⟨by filter_upwards [hf.1,hg.1] with t h ht; positivity, ?_⟩
+  apply (hf.2.sub hg.2).congr'
+  filter_upwards [hf.1,hg.1] with t h ht
+  rw [Real.log_div h.ne' ht.ne',sub_div]
+
+theorem PowerRate.rpow (hf : PowerRate N f a) (c : ℝ) :
+    PowerRate N (fun t => (f t)^c) (a*c) := by
+  refine ⟨by filter_upwards [hf.1] with t h; positivity, ?_⟩
+  apply (hf.2.mul_const c).congr'
+  filter_upwards [hf.1] with t h
+  rw [Real.log_rpow h]
+  ring
+
+theorem PowerRate.pow (hf : PowerRate N f a) (c : ℕ) :
+    PowerRate N (fun t => (f t)^c) (a*c) := by
+  simpa only [Real.rpow_natCast] using hf.rpow (c:ℝ)
+
+theorem PowerRate.sqrt (hf : PowerRate N f a) :
+    PowerRate N (fun t => Real.sqrt (f t)) (a/2) := by
+  simpa only [Real.sqrt_eq_rpow,div_eq_mul_inv, one_mul] using hf.rpow (1/2:ℝ)
+
+theorem PowerRate.inv (hN : Tendsto N Filter.atTop Filter.atTop) (hf : PowerRate N f a) :
+    PowerRate N (fun t=>(f t)⁻¹) (-a) := by
+  simpa only [one_div,zero_sub] using (PowerRate.const hN (by norm_num : (0:ℝ)<1)).div hf
+
+theorem PowerRate.const_mul (hN : Tendsto N Filter.atTop Filter.atTop) (hf : PowerRate N f a)
+    {c : ℝ} (hc : 0<c) : PowerRate N (fun t=>c*f t) a := by
+  simpa only [zero_add] using (PowerRate.const hN hc).mul hf
+
+theorem PowerRate.mul_const (hN : Tendsto N Filter.atTop Filter.atTop) (hf : PowerRate N f a)
+    {c : ℝ} (hc : 0<c) : PowerRate N (fun t=>f t*c) a := by
+  simpa only [add_zero] using hf.mul (PowerRate.const hN hc)
+
+theorem PowerRate.div_const (hN : Tendsto N Filter.atTop Filter.atTop) (hf : PowerRate N f a)
+    {c : ℝ} (hc : 0<c) : PowerRate N (fun t=>f t/c) a := by
+  simpa only [sub_zero] using hf.div (PowerRate.const hN hc)
+
+theorem PowerRate.upper (hN : Tendsto N Filter.atTop Filter.atTop) (hf : PowerRate N f a) :
+    PowerUpper N f a := by
+  intro b hb
+  filter_upwards [hN.eventually_gt_atTop 1,hf.1,hf.2.eventually_lt_const hb] with t hn hp hab
+  rw [abs_of_pos hp,←Real.log_le_log_iff hp (Real.rpow_pos_of_pos (by linarith) b),Real.log_rpow (by linarith : 0<N t)]
+  exact (div_lt_iff₀ (Real.log_pos hn)).mp hab |>.le
+
+theorem PowerRate.lower (hN : Tendsto N Filter.atTop Filter.atTop) (hf : PowerRate N f a) {c : ℝ}
+    (hc : c<a) : ∀ᶠ t in Filter.atTop, (N t)^c ≤ f t := by
+  filter_upwards [hN.eventually_gt_atTop 1,hf.1,hf.2.eventually_const_lt hc] with t hn hp hac
+  rw [←Real.log_le_log_iff (Real.rpow_pos_of_pos (by linarith) c) hp, Real.log_rpow (by linarith : 0<N t)]
+  exact (lt_div_iff₀ (Real.log_pos hn)).mp hac |>.le
+
+theorem PowerRate.tendsto_atTop (hN : Tendsto N Filter.atTop Filter.atTop) (hf : PowerRate N f a) (ha : 0<a) :
+    Tendsto f Filter.atTop Filter.atTop := by
+  exact tendsto_atTop_mono' Filter.atTop (hf.lower hN (show a/2<a by linarith))
+    ((tendsto_rpow_atTop (show 0<a/2 by linarith)).comp hN)
+
+theorem PowerUpper.mono (hf : PowerUpper N f a) (hab : a≤b) : PowerUpper N f b :=
+  fun c hc => hf c (lt_of_le_of_lt hab hc)
+
+theorem PowerUpper.congr (hf : PowerUpper N f a) (he : f =ᶠ[Filter.atTop] g) : PowerUpper N g a := by
+  intro b hb
+  filter_upwards [hf b hb,he] with t h he
+  simpa [←he] using h
+
+theorem PowerUpper.of_abs_le (hg : PowerUpper N g a) (h : ∀ᶠ t in Filter.atTop, |f t|≤|g t|) :
+    PowerUpper N f a := by
+  intro b hb
+  filter_upwards [hg b hb,h] with t ht hh
+  exact hh.trans ht
+
+theorem PowerUpper.zero (hN : Tendsto N Filter.atTop Filter.atTop) (a : ℝ) : PowerUpper N (fun _=>0) a := by
+  intro b hb
+  filter_upwards [hN.eventually_gt_atTop 0] with t ht
+  simpa using (Real.rpow_pos_of_pos ht b).le
+
+theorem PowerUpper.mul (hN : Tendsto N Filter.atTop Filter.atTop) (hf : PowerUpper N f a)
+    (hg : PowerUpper N g b) : PowerUpper N (fun t=>f t*g t) (a+b) := by
+  intro c hc
+  let δ := (c-(a+b))/2
+  have hd : 0<δ := by dsimp [δ]; linarith
+  filter_upwards [hN.eventually_gt_atTop 0,hf (a+δ) (by linarith),hg (b+δ) (by linarith)] with t hn ht hs
+  calc |f t*g t|=|f t| * |g t| := abs_mul _ _
+       _ ≤ (N t)^(a+δ)*(N t)^(b+δ) := mul_le_mul ht hs (abs_nonneg _) (by positivity)
+       _ = (N t)^c := by rw [←Real.rpow_add hn]; congr 1; dsimp [δ]; ring
+
+theorem PowerUpper.pow (hN : Tendsto N Filter.atTop Filter.atTop) (hf : PowerUpper N f a)
+    (r : ℕ) : PowerUpper N (fun t=>(f t)^r) (a*r) := by
+  induction r with
+  | zero => simpa using (PowerRate.const hN (by norm_num : (0:ℝ)<1)).upper hN
+  | succ r ih =>
+    have H := ih.mul hN hf
+    convert H using 1
+    · ext t; exact pow_succ _ _
+    · push_cast; ring
+
+theorem PowerUpper.add (hN : Tendsto N Filter.atTop Filter.atTop) (hf : PowerUpper N f a)
+    (hg : PowerUpper N g a) : PowerUpper N (fun t=>f t+g t) a := by
+  intro c hc
+  let d := (a+c)/2
+  have had : a<d := by dsimp [d]; linarith
+  have hdc : d<c := by dsimp [d]; linarith
+  have hpow := ((tendsto_rpow_atTop (sub_pos.mpr hdc)).comp hN).eventually_ge_atTop 2
+  filter_upwards [hN.eventually_gt_atTop 0,hf d had,hg d had,hpow] with t hn hf hg hp
+  calc |f t+g t| ≤ |f t|+|g t| := abs_add_le _ _
+       _ ≤ 2*(N t)^d := by linarith
+       _ ≤ (N t)^(c-d)*(N t)^d := mul_le_mul_of_nonneg_right hp (by positivity)
+       _ = (N t)^c := by rw [←Real.rpow_add hn]; congr 1; ring
+
+theorem PowerUpper.sum {ι : Type*} (hN : Tendsto N Filter.atTop Filter.atTop) (S : Finset ι)
+    (f : ι→ℕ→ℝ) (h : ∀i∈S, PowerUpper N (f i) a) :
+    PowerUpper N (fun t=>∑i∈S,f i t) a := by
+  classical
+  induction S using Finset.induction_on with
+  | empty => simpa using PowerUpper.zero hN a
+  | @insert i S hi ih =>
+    have hh := (h i (Finset.mem_insert_self _ _)).add hN
+      (ih (fun j hj=>h j (Finset.mem_insert_of_mem hj)))
+    simpa only [Finset.sum_insert hi] using hh
+
+theorem PowerRate.add_const (hN : Tendsto N Filter.atTop Filter.atTop) (hf : PowerRate N f a)
+    (ha : 0<a) {c : ℝ} (hc : 0≤c) : PowerRate N (fun t=>f t+c) a := by
+  have ht := hf.tendsto_atTop hN ha
+  have hratio : Tendsto (fun t=>1+c/f t) Filter.atTop (𝓝 1) := by
+    simpa using (tendsto_const_nhds.add (ht.const_div_atTop c))
+  have hl : Tendsto (fun t=>Real.log (1+c/f t)/Real.log (N t)) Filter.atTop (𝓝 0) :=
+    (hratio.log one_ne_zero).div_atTop (Real.tendsto_log_atTop.comp hN)
+  refine ⟨by filter_upwards [hf.1] with t h; positivity, ?_⟩
+  have H := hf.2.add hl
+  simp only [add_zero] at H
+  apply H.congr'
+  filter_upwards [hf.1] with t hp
+  have hh : 0<1+c/f t := by positivity
+  rw [←add_div,←Real.log_mul hp.ne' hh.ne']
+  congr 2
+  field_simp
+
+theorem PowerRate.of_tendsto (hN : Tendsto N Filter.atTop Filter.atTop)
+    {c : ℝ} (hc : 0<c) (hf : Tendsto f Filter.atTop (𝓝 c)) : PowerRate N f 0 := by
+  exact ⟨hf.eventually_const_lt hc,(hf.log hc.ne').div_atTop (Real.tendsto_log_atTop.comp hN)⟩
+
+theorem PowerUpper.tendsto_zero (hN : Tendsto N Filter.atTop Filter.atTop)
+    (hf : PowerUpper N f a) (ha : a<0) : Tendsto f Filter.atTop (𝓝 0) := by
+  have H := (tendsto_rpow_neg_atTop (show 0< -(a/2) by linarith)).comp hN
+  simp only [neg_neg] at H
+  exact squeeze_zero_norm' (by simpa only [Real.norm_eq_abs,Function.comp_apply] using hf (a/2) (by linarith)) H
+
+theorem PowerRate.of_ratio (hN : Tendsto N Filter.atTop Filter.atTop) (hf : PowerRate N f a)
+    {c : ℝ} (hc : 0<c) (hr : Tendsto (fun t=>g t/f t) Filter.atTop (𝓝 c)) : PowerRate N g a := by
+  have hp : ∀ᶠ t in Filter.atTop, 0<g t := by
+    filter_upwards [hf.1,hr.eventually_const_lt hc] with t ht hgt
+    exact (div_pos_iff.mp hgt).resolve_right (by intro h; linarith [h.2]) |>.1
+  refine ⟨hp,?_⟩
+  have hh := (hr.log hc.ne').div_atTop (Real.tendsto_log_atTop.comp hN)
+  have H := hf.2.add hh
+  simp only [add_zero] at H
+  apply H.congr'
+  filter_upwards [hf.1,hp] with t hf hg
+  simp only [Function.comp_apply]
+  rw [Real.log_div hg.ne' hf.ne']
+  ring
+
+theorem PowerRate.nat_floor (hN : Tendsto N Filter.atTop Filter.atTop) (hf : PowerRate N f a)
+    (ha : 0<a) : PowerRate N (fun t=>(⌊f t⌋₊:ℝ)) a := by
+  exact hf.of_ratio hN (by norm_num : (0:ℝ)<1)
+    (tendsto_nat_floor_div_atTop.comp (hf.tendsto_atTop hN ha))
+
+theorem PowerRate.exp_neg_upper (hN : Tendsto N Filter.atTop Filter.atTop)
+    (hf : PowerRate N f a) (ha : 0<a) (c : ℝ) :
+    PowerUpper N (fun t=>Real.exp (-f t)) c := by
+  intro b hb
+  have hh := ((isLittleO_log_rpow_atTop (show 0<a/2 by linarith)).tendsto_div_nhds_zero).comp hN
+  have hsmall := (hh.const_mul (-b)).eventually_lt_const (show (-b)*0<1 by simp)
+  filter_upwards [hN.eventually_gt_atTop 0,hf.lower hN (show a/2<a by linarith),hsmall] with t hn hl hs
+  rw [abs_of_pos (Real.exp_pos _),Real.rpow_def_of_pos hn]
+  apply Real.exp_le_exp.mpr
+  have hpos := Real.rpow_pos_of_pos hn (a/2)
+  have hle : -b*Real.log (N t) ≤ (N t)^(a/2) := by
+    have H := (div_lt_iff₀ hpos).mp (show (-b*Real.log (N t))/(N t)^(a/2)<1 by simpa only [mul_div_assoc,Function.comp_apply] using hs)
+    linarith
+  linarith
+
+end TraceReconstruction
+
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/TraceReconstruction/Shapes.lean b/lean/OAI/Probability/TraceReconstruction/Shapes.lean
index 7aaeb0161457e5962ee66bf4dfcd0955202c1ab9..e887035fcb83288396403fc993ef70563f3f1386 100644
--- a/lean/OAI/Probability/TraceReconstruction/Shapes.lean
+++ b/lean/OAI/Probability/TraceReconstruction/Shapes.lean
@@ -1,942 +1,942 @@
-import OAI.Probability.TraceReconstruction.Anchors
-
-namespace OAI
-
-/-!
-# Shapes
-
-Global schedules, edge labels, forced/extra coordinate splitting, finite shape codes, and occupied-label counts.
--/
-
-open scoped ENNReal
-noncomputable section
-
-open Filter
-open scoped Topology
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-open FiniteLaw
-
-/-- Source central third, in the zero-based child label convention. The
-shift of the source one-based indices is recorded in center parameters. -/
-def centralThird (N : ℕ) : Finset (Fin N) :=
-  Finset.univ.filter (fun u => N ≤ 3*u.val ∧ 3*u.val ≤ 2*N)
-
-/-- One actual global-pair orientation, ordered by its pre-region block and
-its local identifier there. -/
-def globalTrialIndex (N t : ℕ) (v : Fin (N/4)) (a : Fin t) : Fin ((N/4)*t) :=
-  ⟨v.val*t+a.val, by nlinarith [v.isLt,a.isLt]⟩
-
-/-- Literal net global displacement immediately before a child window:
-a pre block at v+1 remains active through its matched post region N-v.
-The paired displacement has even t trials in each affected region. -/
-def scheduledGlobalShift (N t : ℕ) (bits : Fin ((N/4)*t) → Bool) (u : Fin N) : ℤ :=
-  ∑v : Fin (N/4), if v.val ≤ u.val ∧ u.val+v.val<N then
-    (∑a : Fin t, (bitIndex (bits (globalTrialIndex N t v a)):ℤ))-(t/2:ℕ) else 0
-
-theorem centralThird_active (N : ℕ) (u : Fin N) (hu : u∈centralThird N)
-    (v : Fin (N/4)) : v.val≤u.val ∧ u.val+v.val<N := by
-  have huu := (Finset.mem_filter.mp hu).2
-  have hv := v.isLt
-  have hd := Nat.mod_add_div N 4
-  have hm := Nat.mod_lt N (by omega : 0<4)
-  omega
-
-theorem globalTrialIndex_bijective (N t : ℕ) :
-    Function.Bijective (fun x : Fin (N/4) × Fin t => globalTrialIndex N t x.1 x.2) := by
-  constructor
-  · intro x y he
-    have he' := congrArg Fin.val he
-    change x.1.val*t+x.2.val = y.1.val*t+y.2.val at he'
-    have hx := x.2.isLt
-    have hy := y.2.isLt
-    have hv : x.1.val=y.1.val := by
-      by_contra h
-      rcases lt_or_gt_of_ne h with h | h <;> nlinarith
-    have ha : x.2.val=y.2.val := by rw [hv] at he'; omega
-    exact Prod.ext (Fin.ext hv) (Fin.ext ha)
-  · intro i
-    have ht : 0<t := by
-      by_contra h
-      have hz : t=0 := by omega
-      have := i.isLt
-      simp only [hz,Nat.mul_zero] at this
-      omega
-    refine ⟨(⟨i.val/t, (Nat.div_lt_iff_lt_mul ht).mpr i.isLt⟩,
-      ⟨i.val%t,Nat.mod_lt _ ht⟩),?_⟩
-    apply Fin.ext
-    change i.val/t*t+i.val%t=i.val
-    simpa only [Nat.mul_comm] using Nat.div_add_mod i.val t
-
-theorem globalTrial_sum (N t : ℕ) (f : Fin ((N/4)*t) → ℤ) :
-    (∑v : Fin (N/4),∑a : Fin t,f (globalTrialIndex N t v a)) = ∑i,f i := by
-  let e := Equiv.ofBijective _ (globalTrialIndex_bijective N t)
-  have h := e.sum_comp f
-  simpa only [e,Fintype.sum_prod_type,Equiv.ofBijective_apply] using h
-
-theorem scheduledGlobalShift_plateau (N t : ℕ) (ht : Even t)
-    (bits : Fin ((N/4)*t) → Bool) (u : Fin N) (hu : u∈centralThird N) :
-    scheduledGlobalShift N t bits u = (∑i,(bitIndex (bits i):ℤ))-(((N/4)*t)/2:ℕ) := by
-  have hd : 2 ∣ t := (even_iff_two_dvd).mp ht
-  have hh := Nat.mul_div_assoc (N/4) hd
-  calc
-    _ = ∑v : Fin (N/4),
-        ((∑a : Fin t, (bitIndex (bits (globalTrialIndex N t v a)):ℤ))-(t/2:ℕ)) := by
-      apply Finset.sum_congr rfl
-      intro v hv
-      exact ite_eq_left (centralThird_active N u hu v)
-    _ = (∑i,(bitIndex (bits i):ℤ))-(N/4:ℕ)*(t/2:ℕ) := by
-      rw [Finset.sum_sub_distrib]
-      congr 1
-      · exact globalTrial_sum N t (fun i => (bitIndex (bits i):ℤ))
-      · simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]
-    _ = _ := by congr 1; exact_mod_cast hh.symm
-
-
-/-- The actual complete-tree global schedule has the exact plateau identity
-needed by the already proved correlated anchor energy theorem. -/
-def sourceGlobalTrials (T t : ℕ → ℕ) (b : ℕ) : ℕ := (T b/4)*t b
-
-def sourceGlobalShift (T t : ℕ → ℕ) (b : ℕ)
-    (bits : Fin (sourceGlobalTrials T t b) → Bool) (u : Fin (T b)) : ℤ :=
-  scheduledGlobalShift (T b) (t b) bits u
-
-theorem sourceGlobalShift_hasPlateau (T t : ℕ → ℕ) (ht : ∀b,Even (t b)) :
-    HasGlobalPlateau T (sourceGlobalTrials T t) (fun b => centralThird (T b))
-      (fun b => (sourceGlobalTrials T t b/2:ℕ)) (sourceGlobalShift T t) := by
-  intro b bits u hu
-  exact scheduledGlobalShift_plateau (T b) (t b) (ht b) bits u hu
-end TraceReconstruction
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-namespace OccupiedParityTree
-
-/-- One actual child-label coordinate of an arbitrary ordered occupied
-shape. Unlike a complete r-ary padding, this contains no unused edges. -/
-def Edge : {b : ℕ} → OccupiedParityTree b → Type
-  | _, .leaf => PEmpty
-  | _, .node n children _ => Fin n ⊕ ((i : Fin n) × Edge (children i))
-
-noncomputable instance edgeFintype : {b : ℕ} → (t : OccupiedParityTree b) → Fintype t.Edge
-  | _, .leaf => inferInstanceAs (Fintype PEmpty)
-  | _, .node n children roles => by
-      letI := fun i => edgeFintype (children i)
-      exact inferInstanceAs (Fintype (Fin n ⊕ ((i : Fin n) × Edge (children i))))
-
-def Edge.height : {b : ℕ} → {t : OccupiedParityTree b} → t.Edge → ℕ
-  | _, .leaf, e => e.elim
-  | _, @OccupiedParityTree.node b _n _children _roles, e =>
-      match e with
-      | Sum.inl _ => b
-      | Sum.inr z => Edge.height z.2
-
-/-- All literal label assignments to this particular ordered shape. -/
-def Labels (T : ℕ → ℕ) {b : ℕ} (t : OccupiedParityTree b) :=
-  (e : t.Edge) → Fin (T e.height)
-
-noncomputable instance labelsFintype (T : ℕ → ℕ) {b : ℕ} (t : OccupiedParityTree b) :
-    Fintype (t.Labels T) := inferInstanceAs (Fintype ((e : t.Edge) → Fin (T e.height)))
-
-def Labels.child (T : ℕ → ℕ) {b n : ℕ} {children : Fin n → OccupiedParityTree b}
-    {roles : Fin n → Fin 3} (x : (OccupiedParityTree.node n children roles).Labels T)
-    (i : Fin n) : (children i).Labels T := fun e => x (Sum.inr ⟨i,e⟩)
-
-/-- Projection to the actual labels of a fixed forced subtree. The shape
-selection is independent of these labels and of all random orientations. -/
-def Forced.labels (T : ℕ → ℕ) : {b : ℕ} → {t : OccupiedParityTree b} →
-    t.Forced → t.Labels T → BinaryWindowLabels T b
-  | _, _, .leaf, _ => PUnit.unit
-  | _, _, .node a c _ _ _ left right, x =>
-      (x (Sum.inl a),x (Sum.inl c),left.labels T (x.child T a),right.labels T (x.child T c))
-
-/-- Edge addresses in a complete binary tree, independent of all actual
-numeric labels. False and true are its two children. -/
-def BinaryEdge : ℕ → Type
-  | 0 => PEmpty
-  | b+1 => Bool ⊕ (Bool × BinaryEdge b)
-
-noncomputable instance binaryEdgeFintype : (b : ℕ) → Fintype (BinaryEdge b)
-  | 0 => inferInstanceAs (Fintype PEmpty)
-  | b+1 => by
-      letI := binaryEdgeFintype b
-      exact inferInstanceAs (Fintype (Bool ⊕ (Bool × BinaryEdge b)))
-
-def BinaryEdge.height : {b : ℕ} → BinaryEdge b → ℕ
-  | 0, e => e.elim
-  | b+1, Sum.inl _ => b
-  | _b+1, Sum.inr e => BinaryEdge.height e.2
-
-/-- The forced binary tree is embedded in the actual occupied edge set. -/
-def Forced.edge : {b : ℕ} → {t : OccupiedParityTree b} → t.Forced → BinaryEdge b → t.Edge
-  | _, _, .leaf, e => e.elim
-  | _, _, .node a _ _ _ _ _ _, Sum.inl false => Sum.inl a
-  | _, _, .node _ c _ _ _ _ _, Sum.inl true => Sum.inl c
-  | _, _, .node a _ _ _ _ left _, Sum.inr (false,e) => Sum.inr ⟨a,left.edge e⟩
-  | _, _, .node _ c _ _ _ _ right, Sum.inr (true,e) => Sum.inr ⟨c,right.edge e⟩
-
-theorem Forced.edge_injective {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) :
-    Function.Injective F.edge := by
-  induction F with
-  | leaf => intro i; exact i.elim
-  | @node b n children roles a c hne ha hc left right ihl ihr =>
-    intro i j he
-    cases i with
-    | inl i =>
-      cases j with
-      | inl j =>
-        cases i <;> cases j
-        · rfl
-        · exact (hne (Sum.inl.inj he)).elim
-        · exact (hne (Sum.inl.inj he).symm).elim
-        · rfl
-      | inr j => cases i <;> rcases j with ⟨j,e⟩ <;> cases j <;> cases he
-    | inr i =>
-      cases j with
-      | inl j => rcases i with ⟨i,e⟩; cases i <;> cases j <;> cases he
-      | inr j =>
-        rcases i with ⟨i,e⟩
-        rcases j with ⟨j,f⟩
-        cases i <;> cases j
-        · have hh := Sum.inr.inj he
-          have hh' : left.edge e=left.edge f := by injection hh
-          exact congrArg (fun e => Sum.inr (false,e)) (ihl hh')
-        · have hh := congrArg Sigma.fst (Sum.inr.inj he)
-          exact (hne hh).elim
-        · have hh := congrArg Sigma.fst (Sum.inr.inj he)
-          exact (hne hh.symm).elim
-        · have hh := Sum.inr.inj he
-          have hh' : right.edge e=right.edge f := by injection hh
-          exact congrArg (fun e => Sum.inr (true,e)) (ihr hh')
-
-theorem Forced.edge_height {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced)
-    (e : BinaryEdge b) : (F.edge e).height=e.height := by
-  induction F with
-  | leaf => exact e.elim
-  | @node b n children roles a c hne ha hc left right ihl ihr =>
-    cases e with
-    | inl i => cases i <;> rfl
-    | inr z => rcases z with ⟨i,e⟩; cases i; exact ihl e; exact ihr e
-
-/-- The finite number of edge coordinates at a given child height. -/
-def edgeCount {b : ℕ} (t : OccupiedParityTree b) (h : ℕ) : ℕ :=
-  Fintype.card {e : t.Edge // e.height=h}
-
-theorem edgeCount_leaf (h : ℕ) : edgeCount OccupiedParityTree.leaf h=0 := by
-  unfold edgeCount
-  exact Fintype.card_eq_zero_iff.mpr ⟨fun e => e.val.elim⟩
-
-theorem edgeCount_node {b n : ℕ} (children : Fin n → OccupiedParityTree b)
-    (roles : Fin n → Fin 3) (h : ℕ) :
-    edgeCount (.node n children roles) h = (if b=h then n else 0)+∑i,edgeCount (children i) h := by
-  unfold edgeCount
-  simp only [Fintype.card_subtype, Finset.card_filter]
-  change (∑e : Fin n ⊕ ((i : Fin n) × (children i).Edge),if Edge.height (t := .node n children roles) e=h then 1 else 0)=_
-  rw [Fintype.sum_sum_type]
-  simp only [Edge.height,Fintype.sum_sigma]
-
-  split_ifs <;> simp_all <;> simp only [Finset.card_filter] <;> rfl
-
-theorem Edge.height_lt {b : ℕ} {t : OccupiedParityTree b} (e : t.Edge) : e.height<b := by
-  induction t with
-  | leaf => exact e.elim
-  | @node b n children roles ih =>
-    cases e with
-    | inl i => exact Nat.lt_succ_self b
-    | inr z => exact Nat.lt_succ_of_lt (ih z.1 z.2)
-
-theorem edgeCount_outside {b : ℕ} (t : OccupiedParityTree b) (h : ℕ) (hh : b≤h) :
-    t.edgeCount h=0 := by
-  apply Fintype.card_eq_zero_iff.mpr
-  refine ⟨fun e => ?_⟩
-  have hc := e.val.height_lt
-  rw [e.property] at hc
-  omega
-
-theorem edgeCount_level {b : ℕ} (t : OccupiedParityTree b) (h : ℕ) (hh : h<b) :
-    t.edgeCount h=t.levelCount (b-h) := by
-  induction t with
-  | leaf => omega
-  | @node b n children roles ih =>
-    rw [edgeCount_node]
-    by_cases he : h=b
-    · subst h
-      have hz (i : Fin n) := edgeCount_outside (children i) b le_rfl
-      simp only [hz,Finset.sum_const_zero,add_zero,Nat.add_sub_cancel_left,
-        levelCount,Finset.sum_const,Finset.card_univ,Fintype.card_fin,smul_eq_mul,mul_one]
-      rfl
-    · have hh' : h<b := by omega
-      have hd : b+1-h=(b-h)+1 := by omega
-      rw [ite_eq_right (Ne.symm he),zero_add,hd]
-      exact Finset.sum_congr rfl (fun i _ => ih i hh')
-
-def BinaryEdgeLabels (T : ℕ → ℕ) (b : ℕ) := (e : BinaryEdge b) → Fin (T e.height)
-
-noncomputable instance binaryEdgeLabelsFintype (T : ℕ → ℕ) (b : ℕ) :
-    Fintype (BinaryEdgeLabels T b) := inferInstanceAs (Fintype ((e : BinaryEdge b) → Fin (T e.height)))
-
-/-- Exact coordinates of the recursively defined forced label tuple. -/
-def binaryLabelsEquiv (T : ℕ → ℕ) : (b : ℕ) → BinaryEdgeLabels T b ≃ BinaryWindowLabels T b
-  | 0 =>
-    { toFun := fun _ => PUnit.unit
-      invFun := fun _ e => e.elim
-      left_inv := fun _ => by funext e; exact e.elim
-      right_inv := fun y => by cases y; rfl }
-  | b+1 =>
-    { toFun := fun x => (x (Sum.inl false),x (Sum.inl true),
-        binaryLabelsEquiv T b (fun e => x (Sum.inr (false,e))),
-        binaryLabelsEquiv T b (fun e => x (Sum.inr (true,e))))
-      invFun := fun x e => match e with
-        | Sum.inl false => x.1
-        | Sum.inl true => x.2.1
-        | Sum.inr (false,e) => (binaryLabelsEquiv T b).symm x.2.2.1 e
-        | Sum.inr (true,e) => (binaryLabelsEquiv T b).symm x.2.2.2 e
-      left_inv := fun x => by
-        funext e
-        cases e with
-        | inl j => cases j <;> rfl
-        | inr z =>
-          rcases z with ⟨j,e⟩
-          cases j <;> exact congrFun ((binaryLabelsEquiv T b).symm_apply_apply _) e
-      right_inv := fun x => by
-        rcases x with ⟨a,c,l,r⟩
-        simp only [Equiv.apply_symm_apply]
-        rfl }
-
-/-- Restriction to the real forced coordinates, with only equality transport
-of the child-height index. -/
-def Forced.restrict (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced)
-    (x : t.Labels T) : BinaryEdgeLabels T b := fun e =>
-  Fin.cast (congrArg T (F.edge_height e)) (x (F.edge e))
-
-theorem Forced.restrict_labels (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced)
-    (x : t.Labels T) : binaryLabelsEquiv T b (F.restrict T x)=F.labels T x := by
-  induction F with
-  | leaf => rfl
-  | @node b n children roles a c hne ha hc left right ihl ihr =>
-    change (x (Sum.inl a),x (Sum.inl c),_,_) = (x (Sum.inl a),x (Sum.inl c),_,_)
-    refine Prod.ext rfl (Prod.ext rfl (Prod.ext ?_ ?_))
-    · exact ihl (x.child T a)
-    · exact ihr (x.child T c)
-
-end OccupiedParityTree
-end TraceReconstruction
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-namespace OccupiedParityTree
-
-def Forced.ExtraEdge {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) :=
-  {e : t.Edge // e∉Set.range F.edge}
-
-noncomputable instance forcedExtraEdgeFintype {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) :
-    Fintype F.ExtraEdge := inferInstanceAs (Fintype {e : t.Edge // e∉Set.range F.edge})
-
-def Forced.ExtraLabels (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) :=
-  (e : F.ExtraEdge) → Fin (T e.val.height)
-
-noncomputable instance forcedExtraLabelsFintype (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b}
-    (F : t.Forced) : Fintype (F.ExtraLabels T) :=
-  inferInstanceAs (Fintype ((e : F.ExtraEdge) → Fin (T e.val.height)))
-
-/-- Exact bijective partition of all actual edge coordinates into the
-forced binary coordinates and the genuine extra coordinates. -/
-noncomputable def Forced.labelsEquiv (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) :
-    t.Labels T ≃ BinaryWindowLabels T b × F.ExtraLabels T := by
-  classical
-  letI : DecidablePred (Set.range F.edge) := fun _ => Classical.propDecidable _
-  exact (Equiv.piEquivPiSubtypeProd (Set.range F.edge) (fun e : t.Edge => Fin (T e.height))).trans
-    (Equiv.prodCongr
-      (((Equiv.piCongrLeft (fun e : Set.range F.edge => Fin (T e.val.height))
-        (Equiv.ofInjective F.edge F.edge_injective)).symm.trans
-          (Equiv.piCongrRight fun e => finCongr (congrArg T (F.edge_height e)))).trans
-            (binaryLabelsEquiv T b))
-      (Equiv.refl _))
-
-theorem Forced.labelsEquiv_fst (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced)
-    (x : t.Labels T) : (F.labelsEquiv T x).1=F.labels T x := by
-  exact F.restrict_labels T x
-
-theorem Forced.labelsEquiv_snd (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced)
-    (x : t.Labels T) : (F.labelsEquiv T x).2=(fun e : F.ExtraEdge => x e.val) := rfl
-
-/-- Literal finite Fubini over forced labels and all remaining labels. -/
-theorem Forced.sum_labels (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced)
-    (f : BinaryWindowLabels T b → F.ExtraLabels T → ℝ) :
-    (∑x : t.Labels T,f (F.labels T x) (fun e => x e.val)) =
-      ∑p : BinaryWindowLabels T b, ∑z : F.ExtraLabels T,f p z := by
-  have h := (F.labelsEquiv T).sum_comp (fun x => f x.1 x.2)
-  simpa only [Forced.labelsEquiv_fst,Forced.labelsEquiv_snd,Fintype.sum_prod_type] using h
-
-def Forced.ExtraLeafEdge {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) :=
-  {e : F.ExtraEdge // e.val.height=0}
-
-def Forced.ExtraInteriorEdge {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) :=
-  {e : F.ExtraEdge // e.val.height≠0}
-
-noncomputable instance extraLeafEdgeFintype {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) :
-    Fintype F.ExtraLeafEdge := inferInstanceAs (Fintype {e : F.ExtraEdge // e.val.height=0})
-
-noncomputable instance extraInteriorEdgeFintype {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) :
-    Fintype F.ExtraInteriorEdge := inferInstanceAs (Fintype {e : F.ExtraEdge // e.val.height≠0})
-
-def Forced.InteriorLabels (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) :=
-  (e : F.ExtraInteriorEdge) → Fin (T e.val.val.height)
-
-noncomputable instance extraInteriorLabelsFintype (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b}
-    (F : t.Forced) : Fintype (F.InteriorLabels T) :=
-  inferInstanceAs (Fintype ((e : F.ExtraInteriorEdge) → Fin (T e.val.val.height)))
-
-/-- Extra bottom labels are separated from all nonleaf labels without
-introducing any T₁ factor. -/
-noncomputable def Forced.extraLabelsEquiv (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b}
-    (F : t.Forced) : F.ExtraLabels T ≃ (F.ExtraLeafEdge → Fin (T 0)) × F.InteriorLabels T :=
-  (Equiv.piEquivPiSubtypeProd (fun e : F.ExtraEdge => e.val.height=0)
-    (fun e : F.ExtraEdge => Fin (T e.val.height))).trans
-    (Equiv.prodCongr (Equiv.piCongrRight fun e => finCongr (congrArg T e.property)) (Equiv.refl _))
-
-end OccupiedParityTree
-end TraceReconstruction
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-namespace OccupiedParityTree
-
-/-- The parent address of an edge in the full deterministic numerical index
-tree. In particular it does not contain the edge's own child label. -/
-def Edge.parentPath (T : ℕ → ℕ) : {b : ℕ} → {t : OccupiedParityTree b} →
-    t.Labels T → t.Edge → List ℕ
-  | _, .leaf, _, e => e.elim
-  | _, .node _n _children _roles, x, e => match e with
-    | Sum.inl _ => []
-    | Sum.inr z => (x (Sum.inl z.1)).val :: Edge.parentPath T (x.child T z.1) z.2
-
-/-- Every ancestor label of an extra selected leaf is an interior label. -/
-theorem Edge.parentPath_congr (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b}
-    (x y : t.Labels T) (h : ∀e : t.Edge, e.height≠0 → (x e).val=(y e).val)
-    (e : t.Edge) : e.parentPath T x=e.parentPath T y := by
-  induction t with
-  | leaf => exact e.elim
-  | @node b n children roles ih =>
-    cases e with
-    | inl i => rfl
-    | inr z =>
-      have hb : b≠0 := by have := z.2.height_lt; omega
-      change (x (Sum.inl z.1)).val :: _ = (y (Sum.inl z.1)).val :: _
-      congr 1
-      · exact h (Sum.inl z.1) hb
-      · exact ih z.1 (x.child T z.1) (y.child T z.1)
-          (fun e he => h (Sum.inr ⟨z.1,e⟩) he) z.2
-
-theorem Forced.labels_value_eq (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b}
-    (F : t.Forced) (x y : t.Labels T) (h : F.labels T x=F.labels T y) (j : BinaryEdge b) :
-    (x (F.edge j)).val=(y (F.edge j)).val := by
-  have hh := (binaryLabelsEquiv T b).injective
-    ((F.restrict_labels T x).trans (h.trans (F.restrict_labels T y).symm))
-  exact congrArg (fun z : BinaryEdgeLabels T b => (z j).val) hh
-
-/-- The combined exact split into forced, extra-leaf and extra-interior labels. -/
-noncomputable def Forced.fullLabelsEquiv (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b}
-    (F : t.Forced) : t.Labels T ≃
-      BinaryWindowLabels T b × (F.ExtraLeafEdge → Fin (T 0)) × F.InteriorLabels T :=
-  (F.labelsEquiv T).trans (Equiv.prodCongr (Equiv.refl _) (F.extraLabelsEquiv T))
-
-theorem Forced.fullLabelsEquiv_fst (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b}
-    (F : t.Forced) (x : t.Labels T) : (F.fullLabelsEquiv T x).1=F.labels T x :=
-  F.labelsEquiv_fst T x
-
-theorem Forced.fullLabelsEquiv_leaf_value (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b}
-    (F : t.Forced) (x : t.Labels T) (e : F.ExtraLeafEdge) :
-    ((F.fullLabelsEquiv T x).2.1 e).val=(x e.val.val).val := rfl
-
-theorem Forced.fullLabelsEquiv_interior_value (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b}
-    (F : t.Forced) (x : t.Labels T) (e : F.ExtraInteriorEdge) :
-    ((F.fullLabelsEquiv T x).2.2 e).val=(x e.val.val).val := rfl
-
-/-- Changing only extra leaf labels changes no ancestor address, even when
-all the complete-tree mechanisms share their source global bits. -/
-theorem Forced.parentPath_extraLeaf_invariant (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b}
-    (F : t.Forced) (p : BinaryWindowLabels T b) (h k : F.ExtraLeafEdge → Fin (T 0))
-    (z : F.InteriorLabels T) (e : t.Edge) :
-    e.parentPath T ((F.fullLabelsEquiv T).symm (p,h,z)) =
-      e.parentPath T ((F.fullLabelsEquiv T).symm (p,k,z)) := by
-  let x := (F.fullLabelsEquiv T).symm (p,h,z)
-  let y := (F.fullLabelsEquiv T).symm (p,k,z)
-  have hx : F.fullLabelsEquiv T x=(p,h,z) := Equiv.apply_symm_apply _ _
-  have hy : F.fullLabelsEquiv T y=(p,k,z) := Equiv.apply_symm_apply _ _
-  have hp : F.labels T x=F.labels T y := by
-    rw [←F.fullLabelsEquiv_fst,←F.fullLabelsEquiv_fst,hx,hy]
-  apply Edge.parentPath_congr T x y
-  intro i hi
-  by_cases hr : i∈Set.range F.edge
-  · obtain ⟨j,rfl⟩ := hr
-    exact F.labels_value_eq T x y hp j
-  · let j : F.ExtraInteriorEdge := ⟨⟨i,hr⟩,hi⟩
-    have hvx := congrArg (fun a => (a.2.2 j).val) hx
-    have hvy := congrArg (fun a => (a.2.2 j).val) hy
-    have hv := hvx.trans hvy.symm
-    simpa only [Forced.fullLabelsEquiv_interior_value] using hv
-
-end OccupiedParityTree
-end TraceReconstruction
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-namespace OccupiedParityTree
-
-/-- A finite ordered-role shape code with at most r children at every
-vertex. There are no numerical child labels in this type. -/
-def FiniteRoleShape (r : ℕ) : ℕ → Type
-  | 0 => PUnit
-  | b+1 => (n : Fin (r+1)) × (Fin n.val → FiniteRoleShape r b) × (Fin n.val → Fin 3)
-
-noncomputable instance finiteRoleShapeFintype (r : ℕ) : (b : ℕ) → Fintype (FiniteRoleShape r b)
-  | 0 => inferInstanceAs (Fintype PUnit)
-  | b+1 => by
-    letI := finiteRoleShapeFintype r b
-    exact inferInstanceAs (Fintype ((n : Fin (r+1)) ×
-      (Fin n.val → FiniteRoleShape r b) × (Fin n.val → Fin 3)))
-
-def FiniteRoleShape.toTree (r : ℕ) : (b : ℕ) → FiniteRoleShape r b → OccupiedParityTree b
-  | 0, _ => .leaf
-  | b+1, x => .node x.1.val (fun i => (x.2.1 i).toTree r b) x.2.2
-
-theorem occupied_arity_le_leaves {b n : ℕ} (children : Fin n → OccupiedParityTree b)
-    (roles : Fin n → Fin 3) (ht : (OccupiedParityTree.node n children roles).occupied) :
-    n≤(OccupiedParityTree.node n children roles).leafCount := by
-  have h (i : Fin n) : 1≤(children i).leafCount := leafCount_pos _ (ht.2 i)
-  simpa only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,smul_eq_mul,mul_one,leafCount]
-    using Finset.sum_le_sum (s:=Finset.univ) (fun i _ => h i)
-
-theorem child_le_leaves {b n : ℕ} (children : Fin n → OccupiedParityTree b)
-    (roles : Fin n → Fin 3) (i : Fin n) :
-    (children i).leafCount≤(OccupiedParityTree.node n children roles).leafCount :=
-  Finset.single_le_sum (fun j _ => Nat.zero_le ((children j).leafCount)) (Finset.mem_univ i)
-
-/-- Every occupied shape with r selected leaves really has a finite code
-whose domain depends only on r and the depth, not on any window size. -/
-theorem exists_finiteRoleShape {b : ℕ} (t : OccupiedParityTree b) (r : ℕ)
-    (ht : t.occupied) (hr : t.leafCount≤r) :
-    ∃x : FiniteRoleShape r b,x.toTree r b=t := by
-  induction t with
-  | leaf => exact ⟨PUnit.unit,rfl⟩
-  | @node b n children roles ih =>
-    have hn : n<r+1 := by have := occupied_arity_le_leaves children roles ht; omega
-    have hx (i : Fin n) : ∃x : FiniteRoleShape r b,x.toTree r b=children i :=
-      ih i (ht.2 i) ((child_le_leaves children roles i).trans hr)
-    choose x hx using hx
-    refine ⟨⟨⟨n,hn⟩,x,roles⟩,?_⟩
-    change OccupiedParityTree.node n (fun i => (x i).toTree r b) roles = _
-    congr 1
-    exact funext hx
-
-/-- The manuscript's finite shape/role constant, now a literal natural
-number independent of the actual child-label sizes and all channel data. -/
-def shapeRoleConstant (r b : ℕ) : ℕ := Fintype.card (FiniteRoleShape r b)
-
-theorem finite_bounded_occupied_shapes (b r : ℕ) :
-    Finite {t : OccupiedParityTree b // t.occupied ∧ t.leafCount≤r} := by
-  let f : {t : OccupiedParityTree b // t.occupied ∧ t.leafCount≤r} → FiniteRoleShape r b :=
-    fun t => Classical.choose (exists_finiteRoleShape t.val r t.property.1 t.property.2)
-  apply Finite.of_injective f
-  intro x y he
-  apply Subtype.ext
-  have hx := Classical.choose_spec (exists_finiteRoleShape x.val r x.property.1 x.property.2)
-  have hy := Classical.choose_spec (exists_finiteRoleShape y.val r y.property.1 y.property.2)
-  dsimp only [f] at he
-  exact hx.symm.trans ((congrArg (FiniteRoleShape.toTree r b) he).trans hy)
-end OccupiedParityTree
-end TraceReconstruction
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-open FiniteLaw
-
-/-- Extra-leaf index set version of the whole-bit-law sum. -/
-def indexedExtraAverage {Ω E : Type*} [Fintype Ω] [Fintype E] (P : FiniteLaw Ω)
-    (s N : ℕ) (f : Ω → ℝ) (X : Ω → E → ℤ) (h : E → Fin N) : ℝ :=
-  P.expect (fun ω => f ω*∏j,motifIndicator s (((h j).val:ℤ)-X ω j))
-
-theorem indexedExtraAverage_nonneg {Ω E : Type*} [Fintype Ω] [Fintype E] (P : FiniteLaw Ω)
-    (s N : ℕ) (f : Ω → ℝ) (hf : ∀ω,0≤f ω) (X : Ω → E → ℤ)
-    (h : E → Fin N) : 0 ≤ indexedExtraAverage P s N f X h := by
-  apply Finset.sum_nonneg
-  intro ω hω
-  apply mul_nonneg (P.nonneg ω)
-  exact mul_nonneg (hf ω) (Finset.prod_nonneg (fun j _ => motifIndicator_nonneg _ _))
-
-theorem indexedExtraAverage_sum {Ω E : Type*} [Fintype Ω] [Fintype E] (P : FiniteLaw Ω)
-    (s N : ℕ) (f : Ω → ℝ) (hf : ∀ω,0≤f ω) (X : Ω → E → ℤ) :
-    (∑h : E → Fin N,indexedExtraAverage P s N f X h) ≤
-      ((2^s:ℕ):ℝ)^(Fintype.card E)*P.expect f := by
-  unfold indexedExtraAverage expect
-  rw [Finset.sum_comm]
-  simp_rw [←mul_assoc,←Finset.mul_sum]
-  calc
-    _ ≤ ∑ω,P.mass ω*f ω*(((2^s:ℕ):ℝ)^(Fintype.card E)) := by
-      apply Finset.sum_le_sum
-      intro ω hω
-      apply mul_le_mul_of_nonneg_left _ (mul_nonneg (P.nonneg _) (hf ω))
-      have he := Fintype.prod_sum (fun (j : E) (h : Fin N) => motifIndicator s ((h.val:ℤ)-X ω j))
-      rw [←he]
-      calc
-        _ ≤ ∏_j : E,((2^s:ℕ):ℝ) := by
-          apply Finset.prod_le_prod₀
-          · intro j hj
-            exact Finset.sum_nonneg (fun h hh => motifIndicator_nonneg _ _)
-          · intro j hj
-            exact motifIndicator_finite_sum s N (X ω j)
-        _ = _ := by simp
-    _ = _ := by rw [←Finset.sum_mul]; ring
-
-theorem indexedExtraAverage_energy {Ω E : Type*} [Fintype Ω] [Fintype E] (P : FiniteLaw Ω)
-    (s N : ℕ) (f : Ω → ℝ) (hf : ∀ω,0≤f ω) (X : Ω → E → ℤ) :
-    (∑h : E → Fin N,indexedExtraAverage P s N f X h ^ 2) ≤
-      ((2^s:ℕ):ℝ)^(2*Fintype.card E)*(P.expect f)^2 := by
-  have hn (h : E → Fin N) := indexedExtraAverage_nonneg P s N f hf X h
-  calc
-    _ ≤ (∑h : E → Fin N,indexedExtraAverage P s N f X h)^2 :=
-      Finset.sum_sq_le_sq_sum_of_nonneg (fun h _ => hn h)
-    _ ≤ (((2^s:ℕ):ℝ)^(Fintype.card E)*P.expect f)^2 := by
-      apply pow_le_pow_left₀ (Finset.sum_nonneg (fun h _ => hn h))
-      exact indexedExtraAverage_sum P s N f hf X
-    _ = _ := by rw [mul_pow,←pow_mul]; congr 2; omega
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-open OccupiedParityTree
-
-/-- Node-wise assembly of genuine edge labels. -/
-def OccupiedParityTree.Labels.node (T : ℕ → ℕ) {b k : ℕ}
-    {children : Fin k → OccupiedParityTree b} {roles : Fin k → Fin 3}
-    (h : Fin k → Fin (T b)) (z : (i : Fin k) → (children i).Labels T) :
-    (OccupiedParityTree.node k children roles).Labels T :=
-  fun e => match e with
-  | Sum.inl i => h i
-  | Sum.inr q => z q.1 q.2
-
-/-- The real numeric labels of the selected occupied configuration. -/
-def occupiedTermLabels (T : ℕ → ℕ) : (b : ℕ) →
-    (S : Finset (WindowLeaf T (b+1))) → (a : OccupiedWindowChoice T b S) →
-      (occupiedWindowTerm T b S a).Labels T
-  | 0, S, _ => Labels.node T (fun i => (S.equivFin.symm i).val.1) (fun _ e => e.elim)
-  | b+1, S, a => Labels.node T (fun i => ((selectedChildren S).equivFin.symm i).val)
-      (fun i => occupiedTermLabels T b
-        (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val) (a.1 i))
-
-theorem occupiedTermLabels_forced (T : ℕ → ℕ) (b : ℕ)
-    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S)
-    (F : (occupiedWindowTerm T b S a).Forced) :
-    F.labels T (occupiedTermLabels T b S a)=forcedWindowLabels T b S a F := by
-  induction b with
-  | zero =>
-    cases F with
-    | node i j hne hi hj left right => cases left; cases right; rfl
-  | succ b ih =>
-    rcases a with ⟨u,r⟩
-    cases F with
-    | node i j hne hi hj left right =>
-      change
-        (((selectedChildren S).equivFin.symm i).val,
-         ((selectedChildren S).equivFin.symm j).val,
-         left.labels T (occupiedTermLabels T b (selectedChildLeaves S
-           ((selectedChildren S).equivFin.symm i).val) (u i)),
-         right.labels T (occupiedTermLabels T b (selectedChildLeaves S
-           ((selectedChildren S).equivFin.symm j).val) (u j))) =
-        (((selectedChildren S).equivFin.symm i).val,
-         ((selectedChildren S).equivFin.symm j).val,
-         forcedWindowLabels T b (selectedChildLeaves S
-           ((selectedChildren S).equivFin.symm i).val) (u i) left,
-         forcedWindowLabels T b (selectedChildLeaves S
-           ((selectedChildren S).equivFin.symm j).val) (u j) right)
-      exact congrArg (fun z => (((selectedChildren S).equivFin.symm i).val,
-        ((selectedChildren S).equivFin.symm j).val,z)) (Prod.ext (ih _ _ left) (ih _ _ right))
-
-namespace OccupiedParityTree
-/-- Reading the set of leaf addresses from a shape and numeric edge labels.
-Repeated or unordered coordinates are allowed; the true encoding is a
-particular subset of this unrestricted label domain. -/
-def labelledLeaves (T : ℕ → ℕ) : {b : ℕ} → (t : OccupiedParityTree b) →
-    t.Labels T → Finset (WindowLeaf T b)
-  | 0, .leaf, _ => {(⟨0,by decide⟩ : Fin 1)}
-  | b+1, .node n children roles, x => Finset.univ.biUnion fun i : Fin n =>
-      ((children i).labelledLeaves T (x.child T i)).image (fun v => (x (Sum.inl i),v))
-end OccupiedParityTree
-
-theorem mem_selectedChildLeaves {α β : Type*} [DecidableEq α] [DecidableEq β]
-    (S : Finset (α×β)) (h : α) (v : β) : v∈selectedChildLeaves S h ↔ (h,v)∈S := by
-  constructor
-  · intro hm
-    obtain ⟨w,hw,hv⟩ := Finset.mem_image.mp hm
-    have he : w=(h,v) := Prod.ext (Finset.mem_filter.mp hw).2 hv
-    exact he ▸ (Finset.mem_filter.mp hw).1
-  · intro hm
-    exact Finset.mem_image.mpr ⟨(h,v),Finset.mem_filter.mpr ⟨hm,rfl⟩,rfl⟩
-
-theorem occupiedTermLabels_leaves (T : ℕ → ℕ) (b : ℕ)
-    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S) :
-    (occupiedWindowTerm T b S a).labelledLeaves T (occupiedTermLabels T b S a)=S := by
-  induction b with
-  | zero =>
-    ext v
-    constructor
-    · intro hv
-      obtain ⟨i,hi,hv⟩ := Finset.mem_biUnion.mp hv
-      obtain ⟨w,hw,hvw⟩ := Finset.mem_image.mp hv
-      have he : (S.equivFin.symm i).val=v := by
-        apply Prod.ext
-        · change (occupiedTermLabels T 0 S a (Sum.inl i)) = v.1
-          have hfst := congrArg (Prod.fst : Fin (T 0) × Fin 1 → Fin (T 0)) hvw
-          exact hfst
-        · exact Subsingleton.elim (α := Fin 1) _ _
-      exact he ▸ (S.equivFin.symm i).property
-    · intro hv
-      let i := S.equivFin ⟨v,hv⟩
-      apply Finset.mem_biUnion.mpr
-      refine ⟨i,Finset.mem_univ _,?_⟩
-      apply Finset.mem_image.mpr
-      refine ⟨(⟨0,by decide⟩ : Fin 1),Finset.mem_singleton_self _,?_⟩
-      change ((S.equivFin.symm i).val.1,(⟨0,by decide⟩ : Fin 1))=v
-      have he : (S.equivFin.symm i).val=v := congrArg Subtype.val (S.equivFin.symm_apply_apply _)
-      apply Prod.ext
-      · change (S.equivFin.symm i).val.1=v.1
-        exact congrArg (Prod.fst : Fin (T 0) × Fin 1 → Fin (T 0)) he
-      · exact Subsingleton.elim (α := Fin 1) _ _
-  | succ b ih =>
-    rcases a with ⟨u,r⟩
-    simp only [occupiedWindowTerm, labelledLeaves, occupiedTermLabels, Labels.node]
-    have hc (i : Fin (selectedChildren S).card) :
-        (Labels.node T (roles := r) (fun i => ((selectedChildren S).equivFin.symm i).val)
-          (fun i => occupiedTermLabels T b (selectedChildLeaves S
-            ((selectedChildren S).equivFin.symm i).val) (u i))).child T i =
-        occupiedTermLabels T b (selectedChildLeaves S
-            ((selectedChildren S).equivFin.symm i).val) (u i) := by rfl
-    simp_rw [hc,ih]
-    ext v
-    constructor
-    · intro hv
-      obtain ⟨i,hi,hv⟩ := Finset.mem_biUnion.mp hv
-      obtain ⟨w,hw,rfl⟩ := Finset.mem_image.mp hv
-      exact (mem_selectedChildLeaves _ _ _).mp hw
-    · intro hv
-      have hc : v.1∈selectedChildren S := Finset.mem_image.mpr ⟨v,hv,rfl⟩
-      let i := (selectedChildren S).equivFin ⟨v.1,hc⟩
-      have he : ((selectedChildren S).equivFin.symm i).val=v.1 :=
-        congrArg Subtype.val ((selectedChildren S).equivFin.symm_apply_apply _)
-      apply Finset.mem_biUnion.mpr
-      refine ⟨i,Finset.mem_univ _,?_⟩
-      apply Finset.mem_image.mpr
-      refine ⟨v.2,?_,?_⟩
-      · rw [he]
-        exact (mem_selectedChildLeaves _ _ _).mpr hv
-      · exact Prod.ext he rfl
-
-/-- The shape remembers all the role choices once the actual selected set
-is fixed. -/
-theorem occupiedWindowTerm_injective (T : ℕ → ℕ) (b : ℕ)
-    (S : Finset (WindowLeaf T (b+1))) : Function.Injective (occupiedWindowTerm T b S) := by
-  induction b with
-  | zero =>
-    intro a c h
-    exact eq_of_heq (OccupiedParityTree.node.inj h).2.2
-  | succ b ih =>
-    intro a c h
-    apply Prod.ext
-    · funext i
-      exact ih _ ((congrFun (eq_of_heq (OccupiedParityTree.node.inj h).2.1) i))
-    · exact eq_of_heq (OccupiedParityTree.node.inj h).2.2
-
-/-- Unrestricted shape/label space really covers the coefficient index
-without multiplicity. -/
-def occupiedTermEncoding (T : ℕ → ℕ) (b : ℕ)
-    (w : (S : Finset (WindowLeaf T (b+1))) × OccupiedWindowChoice T b S) :
-    (t : OccupiedParityTree (b+1)) × t.Labels T :=
-  ⟨occupiedWindowTerm T b w.1 w.2,occupiedTermLabels T b w.1 w.2⟩
-
-theorem occupiedTermEncoding_injective (T : ℕ → ℕ) (b : ℕ) :
-    Function.Injective (occupiedTermEncoding T b) := by
-  intro w v he
-  have hS := congrArg (fun z : (t : OccupiedParityTree (b+1)) × t.Labels T =>
-    z.1.labelledLeaves T z.2) he
-  change (occupiedWindowTerm T b w.1 w.2).labelledLeaves T (occupiedTermLabels T b w.1 w.2)=
-    (occupiedWindowTerm T b v.1 v.2).labelledLeaves T (occupiedTermLabels T b v.1 v.2) at hS
-  simp only [occupiedTermLabels_leaves] at hS
-  cases w with
-  | mk S a =>
-    cases v with
-    | mk V c =>
-      dsimp only at hS
-      subst V
-      have ht := congrArg Sigma.fst he
-      have ha := occupiedWindowTerm_injective T b S ht
-      change a=c at ha
-      subst c
-      rfl
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-namespace OccupiedParityTree
-
-noncomputable def Forced.edgeEquiv {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) :
-    BinaryEdge b ⊕ F.ExtraEdge ≃ t.Edge := by
-  classical
-  letI : DecidablePred (Set.range F.edge) := fun _ => Classical.propDecidable _
-  exact (Equiv.sumCongr (Equiv.ofInjective F.edge F.edge_injective) (Equiv.refl _)).trans
-    (Equiv.sumCompl (Set.range F.edge))
-
-def binaryEdgeCount (b h : ℕ) : ℕ := Fintype.card {e : BinaryEdge b // e.height=h}
-
-theorem BinaryEdge.height_lt {b : ℕ} (e : BinaryEdge b) : e.height<b := by
-  induction b with
-  | zero => exact e.elim
-  | succ b ih =>
-    cases e with
-    | inl i => exact Nat.lt_succ_self _
-    | inr z => exact Nat.lt_succ_of_lt (ih z.2)
-
-theorem binaryEdgeCount_outside (b h : ℕ) (hh : b≤h) : binaryEdgeCount b h=0 := by
-  apply Fintype.card_eq_zero_iff.mpr
-  refine ⟨fun e => ?_⟩
-  have hc := e.val.height_lt
-  rw [e.property] at hc
-  omega
-
-theorem binaryEdgeCount_succ (b h : ℕ) :
-    binaryEdgeCount (b+1) h = (if b=h then 2 else 0)+2*binaryEdgeCount b h := by
-  unfold binaryEdgeCount
-  simp only [Fintype.card_subtype,Finset.card_filter]
-  change (∑e : Bool ⊕ (Bool × BinaryEdge b), if BinaryEdge.height (b:=b+1) e=h then 1 else 0)=_
-  rw [Fintype.sum_sum_type,Fintype.sum_prod_type]
-  by_cases hb : b=h <;> simp [BinaryEdge.height,hb,two_mul] <;> simp only [Finset.card_filter] <;> rfl
-
-theorem binaryEdgeCount_eq (b h : ℕ) (hh : h<b) : binaryEdgeCount b h=2^(b-h) := by
-  induction b with
-  | zero => omega
-  | succ b ih =>
-    rw [binaryEdgeCount_succ]
-    by_cases he : b=h
-    · subst h
-      rw [ite_eq_left rfl,binaryEdgeCount_outside b b le_rfl]
-      simp
-    · rw [ite_eq_right he,zero_add,ih (by omega)]
-      have hsub : b+1-h=(b-h)+1 := by omega
-      rw [hsub,pow_succ,Nat.mul_comm]
-
-def Forced.extraCount {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) (h : ℕ) : ℕ :=
-  Fintype.card {e : F.ExtraEdge // e.val.height=h}
-
-theorem Forced.count_partition {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) (h : ℕ) :
-    F.extraCount h+binaryEdgeCount b h=t.edgeCount h := by
-  have he := F.edgeEquiv.sum_comp (fun e : t.Edge => if e.height=h then (1:ℕ) else 0)
-  rw [Fintype.sum_sum_type] at he
-  change (∑e : BinaryEdge b, if (F.edge e).height=h then 1 else 0)+
-    (∑e : F.ExtraEdge, if e.val.height=h then 1 else 0) =
-      (∑e : t.Edge, if e.height=h then 1 else 0) at he
-  simp_rw [F.edge_height] at he
-  unfold Forced.extraCount binaryEdgeCount edgeCount
-  simp only [Fintype.card_subtype,Finset.card_filter]
-  omega
-
-theorem levelCount_bottom {b : ℕ} (t : OccupiedParityTree b) : t.levelCount b=t.leafCount := by
-  induction t with
-  | leaf => rfl
-  | node n children roles ih => exact Finset.sum_congr rfl (fun i _ => ih i)
-
-/-- The extra bottom coordinates are precisely r-2^b, including the empty
-height-zero case. -/
-theorem Forced.card_extraLeafEdge {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) :
-    Fintype.card F.ExtraLeafEdge=t.leafCount-2^b := by
-  cases b with
-  | zero =>
-    cases t
-    change F.extraCount 0 = _
-    have h := F.count_partition 0
-    rw [edgeCount_leaf] at h
-    simp only [leafCount,pow_zero,Nat.sub_self]
-    omega
-  | succ b =>
-    change F.extraCount 0 = _
-    have h := F.count_partition 0
-    rw [binaryEdgeCount_eq (b+1) 0 (by omega),edgeCount_level t 0 (by omega)] at h
-    simp only [Nat.sub_zero,levelCount_bottom] at h
-    omega
-
-/-- Exact source nonforced vertex bound, now for genuine extra edge-label
-coordinates rather than an abstract level-count surrogate. -/
-theorem Forced.extraCount_le {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced)
-    (ht : t.occupied) (h : ℕ) : F.extraCount h≤t.leafCount-2^b := by
-  by_cases hh : h<b
-  · have hp := F.count_partition h
-    rw [binaryEdgeCount_eq b h hh,edgeCount_level t h hh] at hp
-    have Hb := F.nonforced_vertices_le_extra_leaves ht (b-h)
-    omega
-  · have hp := F.count_partition h
-    rw [edgeCount_outside t h (by omega)] at hp
-    omega
-
-/-- Extra nonleaf numeric labels grouped by their actual child height. -/
-def Forced.interiorHeight {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced)
-    (e : F.ExtraInteriorEdge) : Fin b := ⟨e.val.val.height,e.val.val.height_lt⟩
-
-theorem Forced.interiorFiber_le {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced)
-    (ht : t.occupied) (h : Fin b) :
-    Fintype.card {e : F.ExtraInteriorEdge // F.interiorHeight e=h} ≤ t.leafCount-2^b := by
-  apply le_trans _ (F.extraCount_le ht h.val)
-  let f : {e : F.ExtraInteriorEdge // F.interiorHeight e=h} →
-      {e : F.ExtraEdge // e.val.height=h.val} := fun e =>
-    ⟨e.val.val,congrArg Fin.val e.property⟩
-  apply Fintype.card_le_of_injective f
-  intro u v he
-  have he' : u.val.val=v.val.val := congrArg
-    (fun q : {e : F.ExtraEdge // e.val.height=h.val} => q.val) he
-  exact Subtype.ext (Subtype.ext he')
-
-theorem Forced.card_interiorLabels_le (T : ℕ → ℕ) (hT : ∀j,0<T j)
-    {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) (ht : t.occupied) :
-    Fintype.card (F.InteriorLabels T) ≤
-      (∏j : Fin b,if j.val=0 then 1 else T j.val)^(t.leafCount-2^b) := by
-  change Fintype.card ((e : F.ExtraInteriorEdge) → Fin (T e.val.val.height)) ≤ _
-  rw [Fintype.card_pi]
-  simp only [Fintype.card_fin]
-  have hp := Fintype.prod_fiberwise' F.interiorHeight (fun j : Fin b => T j.val)
-  change (∏j : Fin b, ∏_e : {e : F.ExtraInteriorEdge // F.interiorHeight e=j}, T j.val) =
-    ∏e : F.ExtraInteriorEdge,T e.val.val.height at hp
-  rw [←hp]
-  simp only [Finset.prod_const,Finset.card_univ]
-  rw [←Finset.prod_pow]
-  apply Finset.prod_le_prod
-  intro j hj
-  by_cases hz : j.val=0
-  · have he : Fintype.card {e : F.ExtraInteriorEdge // F.interiorHeight e=j}=0 := by
-      apply Fintype.card_eq_zero_iff.mpr
-      refine ⟨fun e => ?_⟩
-      exact e.val.property ((congrArg Fin.val e.property).trans hz)
-    rw [he,ite_eq_left hz,pow_zero,one_pow]
-  · rw [ite_eq_right hz]
-    exact Nat.pow_le_pow_right (hT j.val) (F.interiorFiber_le ht j)
-
-end OccupiedParityTree
-end TraceReconstruction
-
-end
-end
-end
-end
-end
-end
-end
-end
-end
-
-end OAI
+import OAI.Probability.TraceReconstruction.Anchors
+
+namespace OAI
+
+/-!
+# Shapes
+
+Global schedules, edge labels, forced/extra coordinate splitting, finite shape codes, and occupied-label counts.
+-/
+
+open scoped ENNReal
+noncomputable section
+
+open Filter
+open scoped Topology
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+open FiniteLaw
+
+/-- Source central third, in the zero-based child label convention. The
+shift of the source one-based indices is recorded in center parameters. -/
+def centralThird (N : ℕ) : Finset (Fin N) :=
+  Finset.univ.filter (fun u => N ≤ 3*u.val ∧ 3*u.val ≤ 2*N)
+
+/-- One actual global-pair orientation, ordered by its pre-region block and
+its local identifier there. -/
+def globalTrialIndex (N t : ℕ) (v : Fin (N/4)) (a : Fin t) : Fin ((N/4)*t) :=
+  ⟨v.val*t+a.val, by nlinarith [v.isLt,a.isLt]⟩
+
+/-- Literal net global displacement immediately before a child window:
+a pre block at v+1 remains active through its matched post region N-v.
+The paired displacement has even t trials in each affected region. -/
+def scheduledGlobalShift (N t : ℕ) (bits : Fin ((N/4)*t) → Bool) (u : Fin N) : ℤ :=
+  ∑v : Fin (N/4), if v.val ≤ u.val ∧ u.val+v.val<N then
+    (∑a : Fin t, (bitIndex (bits (globalTrialIndex N t v a)):ℤ))-(t/2:ℕ) else 0
+
+theorem centralThird_active (N : ℕ) (u : Fin N) (hu : u∈centralThird N)
+    (v : Fin (N/4)) : v.val≤u.val ∧ u.val+v.val<N := by
+  have huu := (Finset.mem_filter.mp hu).2
+  have hv := v.isLt
+  have hd := Nat.mod_add_div N 4
+  have hm := Nat.mod_lt N (by omega : 0<4)
+  omega
+
+theorem globalTrialIndex_bijective (N t : ℕ) :
+    Function.Bijective (fun x : Fin (N/4) × Fin t => globalTrialIndex N t x.1 x.2) := by
+  constructor
+  · intro x y he
+    have he' := congrArg Fin.val he
+    change x.1.val*t+x.2.val = y.1.val*t+y.2.val at he'
+    have hx := x.2.isLt
+    have hy := y.2.isLt
+    have hv : x.1.val=y.1.val := by
+      by_contra h
+      rcases lt_or_gt_of_ne h with h | h <;> nlinarith
+    have ha : x.2.val=y.2.val := by rw [hv] at he'; omega
+    exact Prod.ext (Fin.ext hv) (Fin.ext ha)
+  · intro i
+    have ht : 0<t := by
+      by_contra h
+      have hz : t=0 := by omega
+      have := i.isLt
+      simp only [hz,Nat.mul_zero] at this
+      omega
+    refine ⟨(⟨i.val/t, (Nat.div_lt_iff_lt_mul ht).mpr i.isLt⟩,
+      ⟨i.val%t,Nat.mod_lt _ ht⟩),?_⟩
+    apply Fin.ext
+    change i.val/t*t+i.val%t=i.val
+    simpa only [Nat.mul_comm] using Nat.div_add_mod i.val t
+
+theorem globalTrial_sum (N t : ℕ) (f : Fin ((N/4)*t) → ℤ) :
+    (∑v : Fin (N/4),∑a : Fin t,f (globalTrialIndex N t v a)) = ∑i,f i := by
+  let e := Equiv.ofBijective _ (globalTrialIndex_bijective N t)
+  have h := e.sum_comp f
+  simpa only [e,Fintype.sum_prod_type,Equiv.ofBijective_apply] using h
+
+theorem scheduledGlobalShift_plateau (N t : ℕ) (ht : Even t)
+    (bits : Fin ((N/4)*t) → Bool) (u : Fin N) (hu : u∈centralThird N) :
+    scheduledGlobalShift N t bits u = (∑i,(bitIndex (bits i):ℤ))-(((N/4)*t)/2:ℕ) := by
+  have hd : 2 ∣ t := (even_iff_two_dvd).mp ht
+  have hh := Nat.mul_div_assoc (N/4) hd
+  calc
+    _ = ∑v : Fin (N/4),
+        ((∑a : Fin t, (bitIndex (bits (globalTrialIndex N t v a)):ℤ))-(t/2:ℕ)) := by
+      apply Finset.sum_congr rfl
+      intro v hv
+      exact ite_eq_left (centralThird_active N u hu v)
+    _ = (∑i,(bitIndex (bits i):ℤ))-(N/4:ℕ)*(t/2:ℕ) := by
+      rw [Finset.sum_sub_distrib]
+      congr 1
+      · exact globalTrial_sum N t (fun i => (bitIndex (bits i):ℤ))
+      · simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]
+    _ = _ := by congr 1; exact_mod_cast hh.symm
+
+
+/-- The actual complete-tree global schedule has the exact plateau identity
+needed by the already proved correlated anchor energy theorem. -/
+def sourceGlobalTrials (T t : ℕ → ℕ) (b : ℕ) : ℕ := (T b/4)*t b
+
+def sourceGlobalShift (T t : ℕ → ℕ) (b : ℕ)
+    (bits : Fin (sourceGlobalTrials T t b) → Bool) (u : Fin (T b)) : ℤ :=
+  scheduledGlobalShift (T b) (t b) bits u
+
+theorem sourceGlobalShift_hasPlateau (T t : ℕ → ℕ) (ht : ∀b,Even (t b)) :
+    HasGlobalPlateau T (sourceGlobalTrials T t) (fun b => centralThird (T b))
+      (fun b => (sourceGlobalTrials T t b/2:ℕ)) (sourceGlobalShift T t) := by
+  intro b bits u hu
+  exact scheduledGlobalShift_plateau (T b) (t b) (ht b) bits u hu
+end TraceReconstruction
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+namespace OccupiedParityTree
+
+/-- One actual child-label coordinate of an arbitrary ordered occupied
+shape. Unlike a complete r-ary padding, this contains no unused edges. -/
+def Edge : {b : ℕ} → OccupiedParityTree b → Type
+  | _, .leaf => PEmpty
+  | _, .node n children _ => Fin n ⊕ ((i : Fin n) × Edge (children i))
+
+noncomputable instance edgeFintype : {b : ℕ} → (t : OccupiedParityTree b) → Fintype t.Edge
+  | _, .leaf => inferInstanceAs (Fintype PEmpty)
+  | _, .node n children roles => by
+      letI := fun i => edgeFintype (children i)
+      exact inferInstanceAs (Fintype (Fin n ⊕ ((i : Fin n) × Edge (children i))))
+
+def Edge.height : {b : ℕ} → {t : OccupiedParityTree b} → t.Edge → ℕ
+  | _, .leaf, e => e.elim
+  | _, @OccupiedParityTree.node b _n _children _roles, e =>
+      match e with
+      | Sum.inl _ => b
+      | Sum.inr z => Edge.height z.2
+
+/-- All literal label assignments to this particular ordered shape. -/
+def Labels (T : ℕ → ℕ) {b : ℕ} (t : OccupiedParityTree b) :=
+  (e : t.Edge) → Fin (T e.height)
+
+noncomputable instance labelsFintype (T : ℕ → ℕ) {b : ℕ} (t : OccupiedParityTree b) :
+    Fintype (t.Labels T) := inferInstanceAs (Fintype ((e : t.Edge) → Fin (T e.height)))
+
+def Labels.child (T : ℕ → ℕ) {b n : ℕ} {children : Fin n → OccupiedParityTree b}
+    {roles : Fin n → Fin 3} (x : (OccupiedParityTree.node n children roles).Labels T)
+    (i : Fin n) : (children i).Labels T := fun e => x (Sum.inr ⟨i,e⟩)
+
+/-- Projection to the actual labels of a fixed forced subtree. The shape
+selection is independent of these labels and of all random orientations. -/
+def Forced.labels (T : ℕ → ℕ) : {b : ℕ} → {t : OccupiedParityTree b} →
+    t.Forced → t.Labels T → BinaryWindowLabels T b
+  | _, _, .leaf, _ => PUnit.unit
+  | _, _, .node a c _ _ _ left right, x =>
+      (x (Sum.inl a),x (Sum.inl c),left.labels T (x.child T a),right.labels T (x.child T c))
+
+/-- Edge addresses in a complete binary tree, independent of all actual
+numeric labels. False and true are its two children. -/
+def BinaryEdge : ℕ → Type
+  | 0 => PEmpty
+  | b+1 => Bool ⊕ (Bool × BinaryEdge b)
+
+noncomputable instance binaryEdgeFintype : (b : ℕ) → Fintype (BinaryEdge b)
+  | 0 => inferInstanceAs (Fintype PEmpty)
+  | b+1 => by
+      letI := binaryEdgeFintype b
+      exact inferInstanceAs (Fintype (Bool ⊕ (Bool × BinaryEdge b)))
+
+def BinaryEdge.height : {b : ℕ} → BinaryEdge b → ℕ
+  | 0, e => e.elim
+  | b+1, Sum.inl _ => b
+  | _b+1, Sum.inr e => BinaryEdge.height e.2
+
+/-- The forced binary tree is embedded in the actual occupied edge set. -/
+def Forced.edge : {b : ℕ} → {t : OccupiedParityTree b} → t.Forced → BinaryEdge b → t.Edge
+  | _, _, .leaf, e => e.elim
+  | _, _, .node a _ _ _ _ _ _, Sum.inl false => Sum.inl a
+  | _, _, .node _ c _ _ _ _ _, Sum.inl true => Sum.inl c
+  | _, _, .node a _ _ _ _ left _, Sum.inr (false,e) => Sum.inr ⟨a,left.edge e⟩
+  | _, _, .node _ c _ _ _ _ right, Sum.inr (true,e) => Sum.inr ⟨c,right.edge e⟩
+
+theorem Forced.edge_injective {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) :
+    Function.Injective F.edge := by
+  induction F with
+  | leaf => intro i; exact i.elim
+  | @node b n children roles a c hne ha hc left right ihl ihr =>
+    intro i j he
+    cases i with
+    | inl i =>
+      cases j with
+      | inl j =>
+        cases i <;> cases j
+        · rfl
+        · exact (hne (Sum.inl.inj he)).elim
+        · exact (hne (Sum.inl.inj he).symm).elim
+        · rfl
+      | inr j => cases i <;> rcases j with ⟨j,e⟩ <;> cases j <;> cases he
+    | inr i =>
+      cases j with
+      | inl j => rcases i with ⟨i,e⟩; cases i <;> cases j <;> cases he
+      | inr j =>
+        rcases i with ⟨i,e⟩
+        rcases j with ⟨j,f⟩
+        cases i <;> cases j
+        · have hh := Sum.inr.inj he
+          have hh' : left.edge e=left.edge f := by injection hh
+          exact congrArg (fun e => Sum.inr (false,e)) (ihl hh')
+        · have hh := congrArg Sigma.fst (Sum.inr.inj he)
+          exact (hne hh).elim
+        · have hh := congrArg Sigma.fst (Sum.inr.inj he)
+          exact (hne hh.symm).elim
+        · have hh := Sum.inr.inj he
+          have hh' : right.edge e=right.edge f := by injection hh
+          exact congrArg (fun e => Sum.inr (true,e)) (ihr hh')
+
+theorem Forced.edge_height {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced)
+    (e : BinaryEdge b) : (F.edge e).height=e.height := by
+  induction F with
+  | leaf => exact e.elim
+  | @node b n children roles a c hne ha hc left right ihl ihr =>
+    cases e with
+    | inl i => cases i <;> rfl
+    | inr z => rcases z with ⟨i,e⟩; cases i; exact ihl e; exact ihr e
+
+/-- The finite number of edge coordinates at a given child height. -/
+def edgeCount {b : ℕ} (t : OccupiedParityTree b) (h : ℕ) : ℕ :=
+  Fintype.card {e : t.Edge // e.height=h}
+
+theorem edgeCount_leaf (h : ℕ) : edgeCount OccupiedParityTree.leaf h=0 := by
+  unfold edgeCount
+  exact Fintype.card_eq_zero_iff.mpr ⟨fun e => e.val.elim⟩
+
+theorem edgeCount_node {b n : ℕ} (children : Fin n → OccupiedParityTree b)
+    (roles : Fin n → Fin 3) (h : ℕ) :
+    edgeCount (.node n children roles) h = (if b=h then n else 0)+∑i,edgeCount (children i) h := by
+  unfold edgeCount
+  simp only [Fintype.card_subtype, Finset.card_filter]
+  change (∑e : Fin n ⊕ ((i : Fin n) × (children i).Edge),if Edge.height (t := .node n children roles) e=h then 1 else 0)=_
+  rw [Fintype.sum_sum_type]
+  simp only [Edge.height,Fintype.sum_sigma]
+
+  split_ifs <;> simp_all <;> simp only [Finset.card_filter] <;> rfl
+
+theorem Edge.height_lt {b : ℕ} {t : OccupiedParityTree b} (e : t.Edge) : e.height<b := by
+  induction t with
+  | leaf => exact e.elim
+  | @node b n children roles ih =>
+    cases e with
+    | inl i => exact Nat.lt_succ_self b
+    | inr z => exact Nat.lt_succ_of_lt (ih z.1 z.2)
+
+theorem edgeCount_outside {b : ℕ} (t : OccupiedParityTree b) (h : ℕ) (hh : b≤h) :
+    t.edgeCount h=0 := by
+  apply Fintype.card_eq_zero_iff.mpr
+  refine ⟨fun e => ?_⟩
+  have hc := e.val.height_lt
+  rw [e.property] at hc
+  omega
+
+theorem edgeCount_level {b : ℕ} (t : OccupiedParityTree b) (h : ℕ) (hh : h<b) :
+    t.edgeCount h=t.levelCount (b-h) := by
+  induction t with
+  | leaf => omega
+  | @node b n children roles ih =>
+    rw [edgeCount_node]
+    by_cases he : h=b
+    · subst h
+      have hz (i : Fin n) := edgeCount_outside (children i) b le_rfl
+      simp only [hz,Finset.sum_const_zero,add_zero,Nat.add_sub_cancel_left,
+        levelCount,Finset.sum_const,Finset.card_univ,Fintype.card_fin,smul_eq_mul,mul_one]
+      rfl
+    · have hh' : h<b := by omega
+      have hd : b+1-h=(b-h)+1 := by omega
+      rw [ite_eq_right (Ne.symm he),zero_add,hd]
+      exact Finset.sum_congr rfl (fun i _ => ih i hh')
+
+def BinaryEdgeLabels (T : ℕ → ℕ) (b : ℕ) := (e : BinaryEdge b) → Fin (T e.height)
+
+noncomputable instance binaryEdgeLabelsFintype (T : ℕ → ℕ) (b : ℕ) :
+    Fintype (BinaryEdgeLabels T b) := inferInstanceAs (Fintype ((e : BinaryEdge b) → Fin (T e.height)))
+
+/-- Exact coordinates of the recursively defined forced label tuple. -/
+def binaryLabelsEquiv (T : ℕ → ℕ) : (b : ℕ) → BinaryEdgeLabels T b ≃ BinaryWindowLabels T b
+  | 0 =>
+    { toFun := fun _ => PUnit.unit
+      invFun := fun _ e => e.elim
+      left_inv := fun _ => by funext e; exact e.elim
+      right_inv := fun y => by cases y; rfl }
+  | b+1 =>
+    { toFun := fun x => (x (Sum.inl false),x (Sum.inl true),
+        binaryLabelsEquiv T b (fun e => x (Sum.inr (false,e))),
+        binaryLabelsEquiv T b (fun e => x (Sum.inr (true,e))))
+      invFun := fun x e => match e with
+        | Sum.inl false => x.1
+        | Sum.inl true => x.2.1
+        | Sum.inr (false,e) => (binaryLabelsEquiv T b).symm x.2.2.1 e
+        | Sum.inr (true,e) => (binaryLabelsEquiv T b).symm x.2.2.2 e
+      left_inv := fun x => by
+        funext e
+        cases e with
+        | inl j => cases j <;> rfl
+        | inr z =>
+          rcases z with ⟨j,e⟩
+          cases j <;> exact congrFun ((binaryLabelsEquiv T b).symm_apply_apply _) e
+      right_inv := fun x => by
+        rcases x with ⟨a,c,l,r⟩
+        simp only [Equiv.apply_symm_apply]
+        rfl }
+
+/-- Restriction to the real forced coordinates, with only equality transport
+of the child-height index. -/
+def Forced.restrict (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced)
+    (x : t.Labels T) : BinaryEdgeLabels T b := fun e =>
+  Fin.cast (congrArg T (F.edge_height e)) (x (F.edge e))
+
+theorem Forced.restrict_labels (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced)
+    (x : t.Labels T) : binaryLabelsEquiv T b (F.restrict T x)=F.labels T x := by
+  induction F with
+  | leaf => rfl
+  | @node b n children roles a c hne ha hc left right ihl ihr =>
+    change (x (Sum.inl a),x (Sum.inl c),_,_) = (x (Sum.inl a),x (Sum.inl c),_,_)
+    refine Prod.ext rfl (Prod.ext rfl (Prod.ext ?_ ?_))
+    · exact ihl (x.child T a)
+    · exact ihr (x.child T c)
+
+end OccupiedParityTree
+end TraceReconstruction
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+namespace OccupiedParityTree
+
+def Forced.ExtraEdge {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) :=
+  {e : t.Edge // e∉Set.range F.edge}
+
+noncomputable instance forcedExtraEdgeFintype {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) :
+    Fintype F.ExtraEdge := inferInstanceAs (Fintype {e : t.Edge // e∉Set.range F.edge})
+
+def Forced.ExtraLabels (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) :=
+  (e : F.ExtraEdge) → Fin (T e.val.height)
+
+noncomputable instance forcedExtraLabelsFintype (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b}
+    (F : t.Forced) : Fintype (F.ExtraLabels T) :=
+  inferInstanceAs (Fintype ((e : F.ExtraEdge) → Fin (T e.val.height)))
+
+/-- Exact bijective partition of all actual edge coordinates into the
+forced binary coordinates and the genuine extra coordinates. -/
+noncomputable def Forced.labelsEquiv (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) :
+    t.Labels T ≃ BinaryWindowLabels T b × F.ExtraLabels T := by
+  classical
+  letI : DecidablePred (Set.range F.edge) := fun _ => Classical.propDecidable _
+  exact (Equiv.piEquivPiSubtypeProd (Set.range F.edge) (fun e : t.Edge => Fin (T e.height))).trans
+    (Equiv.prodCongr
+      (((Equiv.piCongrLeft (fun e : Set.range F.edge => Fin (T e.val.height))
+        (Equiv.ofInjective F.edge F.edge_injective)).symm.trans
+          (Equiv.piCongrRight fun e => finCongr (congrArg T (F.edge_height e)))).trans
+            (binaryLabelsEquiv T b))
+      (Equiv.refl _))
+
+theorem Forced.labelsEquiv_fst (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced)
+    (x : t.Labels T) : (F.labelsEquiv T x).1=F.labels T x := by
+  exact F.restrict_labels T x
+
+theorem Forced.labelsEquiv_snd (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced)
+    (x : t.Labels T) : (F.labelsEquiv T x).2=(fun e : F.ExtraEdge => x e.val) := rfl
+
+/-- Literal finite Fubini over forced labels and all remaining labels. -/
+theorem Forced.sum_labels (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced)
+    (f : BinaryWindowLabels T b → F.ExtraLabels T → ℝ) :
+    (∑x : t.Labels T,f (F.labels T x) (fun e => x e.val)) =
+      ∑p : BinaryWindowLabels T b, ∑z : F.ExtraLabels T,f p z := by
+  have h := (F.labelsEquiv T).sum_comp (fun x => f x.1 x.2)
+  simpa only [Forced.labelsEquiv_fst,Forced.labelsEquiv_snd,Fintype.sum_prod_type] using h
+
+def Forced.ExtraLeafEdge {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) :=
+  {e : F.ExtraEdge // e.val.height=0}
+
+def Forced.ExtraInteriorEdge {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) :=
+  {e : F.ExtraEdge // e.val.height≠0}
+
+noncomputable instance extraLeafEdgeFintype {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) :
+    Fintype F.ExtraLeafEdge := inferInstanceAs (Fintype {e : F.ExtraEdge // e.val.height=0})
+
+noncomputable instance extraInteriorEdgeFintype {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) :
+    Fintype F.ExtraInteriorEdge := inferInstanceAs (Fintype {e : F.ExtraEdge // e.val.height≠0})
+
+def Forced.InteriorLabels (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) :=
+  (e : F.ExtraInteriorEdge) → Fin (T e.val.val.height)
+
+noncomputable instance extraInteriorLabelsFintype (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b}
+    (F : t.Forced) : Fintype (F.InteriorLabels T) :=
+  inferInstanceAs (Fintype ((e : F.ExtraInteriorEdge) → Fin (T e.val.val.height)))
+
+/-- Extra bottom labels are separated from all nonleaf labels without
+introducing any T₁ factor. -/
+noncomputable def Forced.extraLabelsEquiv (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b}
+    (F : t.Forced) : F.ExtraLabels T ≃ (F.ExtraLeafEdge → Fin (T 0)) × F.InteriorLabels T :=
+  (Equiv.piEquivPiSubtypeProd (fun e : F.ExtraEdge => e.val.height=0)
+    (fun e : F.ExtraEdge => Fin (T e.val.height))).trans
+    (Equiv.prodCongr (Equiv.piCongrRight fun e => finCongr (congrArg T e.property)) (Equiv.refl _))
+
+end OccupiedParityTree
+end TraceReconstruction
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+namespace OccupiedParityTree
+
+/-- The parent address of an edge in the full deterministic numerical index
+tree. In particular it does not contain the edge's own child label. -/
+def Edge.parentPath (T : ℕ → ℕ) : {b : ℕ} → {t : OccupiedParityTree b} →
+    t.Labels T → t.Edge → List ℕ
+  | _, .leaf, _, e => e.elim
+  | _, .node _n _children _roles, x, e => match e with
+    | Sum.inl _ => []
+    | Sum.inr z => (x (Sum.inl z.1)).val :: Edge.parentPath T (x.child T z.1) z.2
+
+/-- Every ancestor label of an extra selected leaf is an interior label. -/
+theorem Edge.parentPath_congr (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b}
+    (x y : t.Labels T) (h : ∀e : t.Edge, e.height≠0 → (x e).val=(y e).val)
+    (e : t.Edge) : e.parentPath T x=e.parentPath T y := by
+  induction t with
+  | leaf => exact e.elim
+  | @node b n children roles ih =>
+    cases e with
+    | inl i => rfl
+    | inr z =>
+      have hb : b≠0 := by have := z.2.height_lt; omega
+      change (x (Sum.inl z.1)).val :: _ = (y (Sum.inl z.1)).val :: _
+      congr 1
+      · exact h (Sum.inl z.1) hb
+      · exact ih z.1 (x.child T z.1) (y.child T z.1)
+          (fun e he => h (Sum.inr ⟨z.1,e⟩) he) z.2
+
+theorem Forced.labels_value_eq (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b}
+    (F : t.Forced) (x y : t.Labels T) (h : F.labels T x=F.labels T y) (j : BinaryEdge b) :
+    (x (F.edge j)).val=(y (F.edge j)).val := by
+  have hh := (binaryLabelsEquiv T b).injective
+    ((F.restrict_labels T x).trans (h.trans (F.restrict_labels T y).symm))
+  exact congrArg (fun z : BinaryEdgeLabels T b => (z j).val) hh
+
+/-- The combined exact split into forced, extra-leaf and extra-interior labels. -/
+noncomputable def Forced.fullLabelsEquiv (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b}
+    (F : t.Forced) : t.Labels T ≃
+      BinaryWindowLabels T b × (F.ExtraLeafEdge → Fin (T 0)) × F.InteriorLabels T :=
+  (F.labelsEquiv T).trans (Equiv.prodCongr (Equiv.refl _) (F.extraLabelsEquiv T))
+
+theorem Forced.fullLabelsEquiv_fst (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b}
+    (F : t.Forced) (x : t.Labels T) : (F.fullLabelsEquiv T x).1=F.labels T x :=
+  F.labelsEquiv_fst T x
+
+theorem Forced.fullLabelsEquiv_leaf_value (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b}
+    (F : t.Forced) (x : t.Labels T) (e : F.ExtraLeafEdge) :
+    ((F.fullLabelsEquiv T x).2.1 e).val=(x e.val.val).val := rfl
+
+theorem Forced.fullLabelsEquiv_interior_value (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b}
+    (F : t.Forced) (x : t.Labels T) (e : F.ExtraInteriorEdge) :
+    ((F.fullLabelsEquiv T x).2.2 e).val=(x e.val.val).val := rfl
+
+/-- Changing only extra leaf labels changes no ancestor address, even when
+all the complete-tree mechanisms share their source global bits. -/
+theorem Forced.parentPath_extraLeaf_invariant (T : ℕ → ℕ) {b : ℕ} {t : OccupiedParityTree b}
+    (F : t.Forced) (p : BinaryWindowLabels T b) (h k : F.ExtraLeafEdge → Fin (T 0))
+    (z : F.InteriorLabels T) (e : t.Edge) :
+    e.parentPath T ((F.fullLabelsEquiv T).symm (p,h,z)) =
+      e.parentPath T ((F.fullLabelsEquiv T).symm (p,k,z)) := by
+  let x := (F.fullLabelsEquiv T).symm (p,h,z)
+  let y := (F.fullLabelsEquiv T).symm (p,k,z)
+  have hx : F.fullLabelsEquiv T x=(p,h,z) := Equiv.apply_symm_apply _ _
+  have hy : F.fullLabelsEquiv T y=(p,k,z) := Equiv.apply_symm_apply _ _
+  have hp : F.labels T x=F.labels T y := by
+    rw [←F.fullLabelsEquiv_fst,←F.fullLabelsEquiv_fst,hx,hy]
+  apply Edge.parentPath_congr T x y
+  intro i hi
+  by_cases hr : i∈Set.range F.edge
+  · obtain ⟨j,rfl⟩ := hr
+    exact F.labels_value_eq T x y hp j
+  · let j : F.ExtraInteriorEdge := ⟨⟨i,hr⟩,hi⟩
+    have hvx := congrArg (fun a => (a.2.2 j).val) hx
+    have hvy := congrArg (fun a => (a.2.2 j).val) hy
+    have hv := hvx.trans hvy.symm
+    simpa only [Forced.fullLabelsEquiv_interior_value] using hv
+
+end OccupiedParityTree
+end TraceReconstruction
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+namespace OccupiedParityTree
+
+/-- A finite ordered-role shape code with at most r children at every
+vertex. There are no numerical child labels in this type. -/
+def FiniteRoleShape (r : ℕ) : ℕ → Type
+  | 0 => PUnit
+  | b+1 => (n : Fin (r+1)) × (Fin n.val → FiniteRoleShape r b) × (Fin n.val → Fin 3)
+
+noncomputable instance finiteRoleShapeFintype (r : ℕ) : (b : ℕ) → Fintype (FiniteRoleShape r b)
+  | 0 => inferInstanceAs (Fintype PUnit)
+  | b+1 => by
+    letI := finiteRoleShapeFintype r b
+    exact inferInstanceAs (Fintype ((n : Fin (r+1)) ×
+      (Fin n.val → FiniteRoleShape r b) × (Fin n.val → Fin 3)))
+
+def FiniteRoleShape.toTree (r : ℕ) : (b : ℕ) → FiniteRoleShape r b → OccupiedParityTree b
+  | 0, _ => .leaf
+  | b+1, x => .node x.1.val (fun i => (x.2.1 i).toTree r b) x.2.2
+
+theorem occupied_arity_le_leaves {b n : ℕ} (children : Fin n → OccupiedParityTree b)
+    (roles : Fin n → Fin 3) (ht : (OccupiedParityTree.node n children roles).occupied) :
+    n≤(OccupiedParityTree.node n children roles).leafCount := by
+  have h (i : Fin n) : 1≤(children i).leafCount := leafCount_pos _ (ht.2 i)
+  simpa only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,smul_eq_mul,mul_one,leafCount]
+    using Finset.sum_le_sum (s:=Finset.univ) (fun i _ => h i)
+
+theorem child_le_leaves {b n : ℕ} (children : Fin n → OccupiedParityTree b)
+    (roles : Fin n → Fin 3) (i : Fin n) :
+    (children i).leafCount≤(OccupiedParityTree.node n children roles).leafCount :=
+  Finset.single_le_sum (fun j _ => Nat.zero_le ((children j).leafCount)) (Finset.mem_univ i)
+
+/-- Every occupied shape with r selected leaves really has a finite code
+whose domain depends only on r and the depth, not on any window size. -/
+theorem exists_finiteRoleShape {b : ℕ} (t : OccupiedParityTree b) (r : ℕ)
+    (ht : t.occupied) (hr : t.leafCount≤r) :
+    ∃x : FiniteRoleShape r b,x.toTree r b=t := by
+  induction t with
+  | leaf => exact ⟨PUnit.unit,rfl⟩
+  | @node b n children roles ih =>
+    have hn : n<r+1 := by have := occupied_arity_le_leaves children roles ht; omega
+    have hx (i : Fin n) : ∃x : FiniteRoleShape r b,x.toTree r b=children i :=
+      ih i (ht.2 i) ((child_le_leaves children roles i).trans hr)
+    choose x hx using hx
+    refine ⟨⟨⟨n,hn⟩,x,roles⟩,?_⟩
+    change OccupiedParityTree.node n (fun i => (x i).toTree r b) roles = _
+    congr 1
+    exact funext hx
+
+/-- The manuscript's finite shape/role constant, now a literal natural
+number independent of the actual child-label sizes and all channel data. -/
+def shapeRoleConstant (r b : ℕ) : ℕ := Fintype.card (FiniteRoleShape r b)
+
+theorem finite_bounded_occupied_shapes (b r : ℕ) :
+    Finite {t : OccupiedParityTree b // t.occupied ∧ t.leafCount≤r} := by
+  let f : {t : OccupiedParityTree b // t.occupied ∧ t.leafCount≤r} → FiniteRoleShape r b :=
+    fun t => Classical.choose (exists_finiteRoleShape t.val r t.property.1 t.property.2)
+  apply Finite.of_injective f
+  intro x y he
+  apply Subtype.ext
+  have hx := Classical.choose_spec (exists_finiteRoleShape x.val r x.property.1 x.property.2)
+  have hy := Classical.choose_spec (exists_finiteRoleShape y.val r y.property.1 y.property.2)
+  dsimp only [f] at he
+  exact hx.symm.trans ((congrArg (FiniteRoleShape.toTree r b) he).trans hy)
+end OccupiedParityTree
+end TraceReconstruction
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+open FiniteLaw
+
+/-- Extra-leaf index set version of the whole-bit-law sum. -/
+def indexedExtraAverage {Ω E : Type*} [Fintype Ω] [Fintype E] (P : FiniteLaw Ω)
+    (s N : ℕ) (f : Ω → ℝ) (X : Ω → E → ℤ) (h : E → Fin N) : ℝ :=
+  P.expect (fun ω => f ω*∏j,motifIndicator s (((h j).val:ℤ)-X ω j))
+
+theorem indexedExtraAverage_nonneg {Ω E : Type*} [Fintype Ω] [Fintype E] (P : FiniteLaw Ω)
+    (s N : ℕ) (f : Ω → ℝ) (hf : ∀ω,0≤f ω) (X : Ω → E → ℤ)
+    (h : E → Fin N) : 0 ≤ indexedExtraAverage P s N f X h := by
+  apply Finset.sum_nonneg
+  intro ω hω
+  apply mul_nonneg (P.nonneg ω)
+  exact mul_nonneg (hf ω) (Finset.prod_nonneg (fun j _ => motifIndicator_nonneg _ _))
+
+theorem indexedExtraAverage_sum {Ω E : Type*} [Fintype Ω] [Fintype E] (P : FiniteLaw Ω)
+    (s N : ℕ) (f : Ω → ℝ) (hf : ∀ω,0≤f ω) (X : Ω → E → ℤ) :
+    (∑h : E → Fin N,indexedExtraAverage P s N f X h) ≤
+      ((2^s:ℕ):ℝ)^(Fintype.card E)*P.expect f := by
+  unfold indexedExtraAverage expect
+  rw [Finset.sum_comm]
+  simp_rw [←mul_assoc,←Finset.mul_sum]
+  calc
+    _ ≤ ∑ω,P.mass ω*f ω*(((2^s:ℕ):ℝ)^(Fintype.card E)) := by
+      apply Finset.sum_le_sum
+      intro ω hω
+      apply mul_le_mul_of_nonneg_left _ (mul_nonneg (P.nonneg _) (hf ω))
+      have he := Fintype.prod_sum (fun (j : E) (h : Fin N) => motifIndicator s ((h.val:ℤ)-X ω j))
+      rw [←he]
+      calc
+        _ ≤ ∏_j : E,((2^s:ℕ):ℝ) := by
+          apply Finset.prod_le_prod₀
+          · intro j hj
+            exact Finset.sum_nonneg (fun h hh => motifIndicator_nonneg _ _)
+          · intro j hj
+            exact motifIndicator_finite_sum s N (X ω j)
+        _ = _ := by simp
+    _ = _ := by rw [←Finset.sum_mul]; ring
+
+theorem indexedExtraAverage_energy {Ω E : Type*} [Fintype Ω] [Fintype E] (P : FiniteLaw Ω)
+    (s N : ℕ) (f : Ω → ℝ) (hf : ∀ω,0≤f ω) (X : Ω → E → ℤ) :
+    (∑h : E → Fin N,indexedExtraAverage P s N f X h ^ 2) ≤
+      ((2^s:ℕ):ℝ)^(2*Fintype.card E)*(P.expect f)^2 := by
+  have hn (h : E → Fin N) := indexedExtraAverage_nonneg P s N f hf X h
+  calc
+    _ ≤ (∑h : E → Fin N,indexedExtraAverage P s N f X h)^2 :=
+      Finset.sum_sq_le_sq_sum_of_nonneg (fun h _ => hn h)
+    _ ≤ (((2^s:ℕ):ℝ)^(Fintype.card E)*P.expect f)^2 := by
+      apply pow_le_pow_left₀ (Finset.sum_nonneg (fun h _ => hn h))
+      exact indexedExtraAverage_sum P s N f hf X
+    _ = _ := by rw [mul_pow,←pow_mul]; congr 2; omega
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+open OccupiedParityTree
+
+/-- Node-wise assembly of genuine edge labels. -/
+def OccupiedParityTree.Labels.node (T : ℕ → ℕ) {b k : ℕ}
+    {children : Fin k → OccupiedParityTree b} {roles : Fin k → Fin 3}
+    (h : Fin k → Fin (T b)) (z : (i : Fin k) → (children i).Labels T) :
+    (OccupiedParityTree.node k children roles).Labels T :=
+  fun e => match e with
+  | Sum.inl i => h i
+  | Sum.inr q => z q.1 q.2
+
+/-- The real numeric labels of the selected occupied configuration. -/
+def occupiedTermLabels (T : ℕ → ℕ) : (b : ℕ) →
+    (S : Finset (WindowLeaf T (b+1))) → (a : OccupiedWindowChoice T b S) →
+      (occupiedWindowTerm T b S a).Labels T
+  | 0, S, _ => Labels.node T (fun i => (S.equivFin.symm i).val.1) (fun _ e => e.elim)
+  | b+1, S, a => Labels.node T (fun i => ((selectedChildren S).equivFin.symm i).val)
+      (fun i => occupiedTermLabels T b
+        (selectedChildLeaves S ((selectedChildren S).equivFin.symm i).val) (a.1 i))
+
+theorem occupiedTermLabels_forced (T : ℕ → ℕ) (b : ℕ)
+    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S)
+    (F : (occupiedWindowTerm T b S a).Forced) :
+    F.labels T (occupiedTermLabels T b S a)=forcedWindowLabels T b S a F := by
+  induction b with
+  | zero =>
+    cases F with
+    | node i j hne hi hj left right => cases left; cases right; rfl
+  | succ b ih =>
+    rcases a with ⟨u,r⟩
+    cases F with
+    | node i j hne hi hj left right =>
+      change
+        (((selectedChildren S).equivFin.symm i).val,
+         ((selectedChildren S).equivFin.symm j).val,
+         left.labels T (occupiedTermLabels T b (selectedChildLeaves S
+           ((selectedChildren S).equivFin.symm i).val) (u i)),
+         right.labels T (occupiedTermLabels T b (selectedChildLeaves S
+           ((selectedChildren S).equivFin.symm j).val) (u j))) =
+        (((selectedChildren S).equivFin.symm i).val,
+         ((selectedChildren S).equivFin.symm j).val,
+         forcedWindowLabels T b (selectedChildLeaves S
+           ((selectedChildren S).equivFin.symm i).val) (u i) left,
+         forcedWindowLabels T b (selectedChildLeaves S
+           ((selectedChildren S).equivFin.symm j).val) (u j) right)
+      exact congrArg (fun z => (((selectedChildren S).equivFin.symm i).val,
+        ((selectedChildren S).equivFin.symm j).val,z)) (Prod.ext (ih _ _ left) (ih _ _ right))
+
+namespace OccupiedParityTree
+/-- Reading the set of leaf addresses from a shape and numeric edge labels.
+Repeated or unordered coordinates are allowed; the true encoding is a
+particular subset of this unrestricted label domain. -/
+def labelledLeaves (T : ℕ → ℕ) : {b : ℕ} → (t : OccupiedParityTree b) →
+    t.Labels T → Finset (WindowLeaf T b)
+  | 0, .leaf, _ => {(⟨0,by decide⟩ : Fin 1)}
+  | b+1, .node n children roles, x => Finset.univ.biUnion fun i : Fin n =>
+      ((children i).labelledLeaves T (x.child T i)).image (fun v => (x (Sum.inl i),v))
+end OccupiedParityTree
+
+theorem mem_selectedChildLeaves {α β : Type*} [DecidableEq α] [DecidableEq β]
+    (S : Finset (α×β)) (h : α) (v : β) : v∈selectedChildLeaves S h ↔ (h,v)∈S := by
+  constructor
+  · intro hm
+    obtain ⟨w,hw,hv⟩ := Finset.mem_image.mp hm
+    have he : w=(h,v) := Prod.ext (Finset.mem_filter.mp hw).2 hv
+    exact he ▸ (Finset.mem_filter.mp hw).1
+  · intro hm
+    exact Finset.mem_image.mpr ⟨(h,v),Finset.mem_filter.mpr ⟨hm,rfl⟩,rfl⟩
+
+theorem occupiedTermLabels_leaves (T : ℕ → ℕ) (b : ℕ)
+    (S : Finset (WindowLeaf T (b+1))) (a : OccupiedWindowChoice T b S) :
+    (occupiedWindowTerm T b S a).labelledLeaves T (occupiedTermLabels T b S a)=S := by
+  induction b with
+  | zero =>
+    ext v
+    constructor
+    · intro hv
+      obtain ⟨i,hi,hv⟩ := Finset.mem_biUnion.mp hv
+      obtain ⟨w,hw,hvw⟩ := Finset.mem_image.mp hv
+      have he : (S.equivFin.symm i).val=v := by
+        apply Prod.ext
+        · change (occupiedTermLabels T 0 S a (Sum.inl i)) = v.1
+          have hfst := congrArg (Prod.fst : Fin (T 0) × Fin 1 → Fin (T 0)) hvw
+          exact hfst
+        · exact Subsingleton.elim (α := Fin 1) _ _
+      exact he ▸ (S.equivFin.symm i).property
+    · intro hv
+      let i := S.equivFin ⟨v,hv⟩
+      apply Finset.mem_biUnion.mpr
+      refine ⟨i,Finset.mem_univ _,?_⟩
+      apply Finset.mem_image.mpr
+      refine ⟨(⟨0,by decide⟩ : Fin 1),Finset.mem_singleton_self _,?_⟩
+      change ((S.equivFin.symm i).val.1,(⟨0,by decide⟩ : Fin 1))=v
+      have he : (S.equivFin.symm i).val=v := congrArg Subtype.val (S.equivFin.symm_apply_apply _)
+      apply Prod.ext
+      · change (S.equivFin.symm i).val.1=v.1
+        exact congrArg (Prod.fst : Fin (T 0) × Fin 1 → Fin (T 0)) he
+      · exact Subsingleton.elim (α := Fin 1) _ _
+  | succ b ih =>
+    rcases a with ⟨u,r⟩
+    simp only [occupiedWindowTerm, labelledLeaves, occupiedTermLabels, Labels.node]
+    have hc (i : Fin (selectedChildren S).card) :
+        (Labels.node T (roles := r) (fun i => ((selectedChildren S).equivFin.symm i).val)
+          (fun i => occupiedTermLabels T b (selectedChildLeaves S
+            ((selectedChildren S).equivFin.symm i).val) (u i))).child T i =
+        occupiedTermLabels T b (selectedChildLeaves S
+            ((selectedChildren S).equivFin.symm i).val) (u i) := by rfl
+    simp_rw [hc,ih]
+    ext v
+    constructor
+    · intro hv
+      obtain ⟨i,hi,hv⟩ := Finset.mem_biUnion.mp hv
+      obtain ⟨w,hw,rfl⟩ := Finset.mem_image.mp hv
+      exact (mem_selectedChildLeaves _ _ _).mp hw
+    · intro hv
+      have hc : v.1∈selectedChildren S := Finset.mem_image.mpr ⟨v,hv,rfl⟩
+      let i := (selectedChildren S).equivFin ⟨v.1,hc⟩
+      have he : ((selectedChildren S).equivFin.symm i).val=v.1 :=
+        congrArg Subtype.val ((selectedChildren S).equivFin.symm_apply_apply _)
+      apply Finset.mem_biUnion.mpr
+      refine ⟨i,Finset.mem_univ _,?_⟩
+      apply Finset.mem_image.mpr
+      refine ⟨v.2,?_,?_⟩
+      · rw [he]
+        exact (mem_selectedChildLeaves _ _ _).mpr hv
+      · exact Prod.ext he rfl
+
+/-- The shape remembers all the role choices once the actual selected set
+is fixed. -/
+theorem occupiedWindowTerm_injective (T : ℕ → ℕ) (b : ℕ)
+    (S : Finset (WindowLeaf T (b+1))) : Function.Injective (occupiedWindowTerm T b S) := by
+  induction b with
+  | zero =>
+    intro a c h
+    exact eq_of_heq (OccupiedParityTree.node.inj h).2.2
+  | succ b ih =>
+    intro a c h
+    apply Prod.ext
+    · funext i
+      exact ih _ ((congrFun (eq_of_heq (OccupiedParityTree.node.inj h).2.1) i))
+    · exact eq_of_heq (OccupiedParityTree.node.inj h).2.2
+
+/-- Unrestricted shape/label space really covers the coefficient index
+without multiplicity. -/
+def occupiedTermEncoding (T : ℕ → ℕ) (b : ℕ)
+    (w : (S : Finset (WindowLeaf T (b+1))) × OccupiedWindowChoice T b S) :
+    (t : OccupiedParityTree (b+1)) × t.Labels T :=
+  ⟨occupiedWindowTerm T b w.1 w.2,occupiedTermLabels T b w.1 w.2⟩
+
+theorem occupiedTermEncoding_injective (T : ℕ → ℕ) (b : ℕ) :
+    Function.Injective (occupiedTermEncoding T b) := by
+  intro w v he
+  have hS := congrArg (fun z : (t : OccupiedParityTree (b+1)) × t.Labels T =>
+    z.1.labelledLeaves T z.2) he
+  change (occupiedWindowTerm T b w.1 w.2).labelledLeaves T (occupiedTermLabels T b w.1 w.2)=
+    (occupiedWindowTerm T b v.1 v.2).labelledLeaves T (occupiedTermLabels T b v.1 v.2) at hS
+  simp only [occupiedTermLabels_leaves] at hS
+  cases w with
+  | mk S a =>
+    cases v with
+    | mk V c =>
+      dsimp only at hS
+      subst V
+      have ht := congrArg Sigma.fst he
+      have ha := occupiedWindowTerm_injective T b S ht
+      change a=c at ha
+      subst c
+      rfl
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+namespace OccupiedParityTree
+
+noncomputable def Forced.edgeEquiv {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) :
+    BinaryEdge b ⊕ F.ExtraEdge ≃ t.Edge := by
+  classical
+  letI : DecidablePred (Set.range F.edge) := fun _ => Classical.propDecidable _
+  exact (Equiv.sumCongr (Equiv.ofInjective F.edge F.edge_injective) (Equiv.refl _)).trans
+    (Equiv.sumCompl (Set.range F.edge))
+
+def binaryEdgeCount (b h : ℕ) : ℕ := Fintype.card {e : BinaryEdge b // e.height=h}
+
+theorem BinaryEdge.height_lt {b : ℕ} (e : BinaryEdge b) : e.height<b := by
+  induction b with
+  | zero => exact e.elim
+  | succ b ih =>
+    cases e with
+    | inl i => exact Nat.lt_succ_self _
+    | inr z => exact Nat.lt_succ_of_lt (ih z.2)
+
+theorem binaryEdgeCount_outside (b h : ℕ) (hh : b≤h) : binaryEdgeCount b h=0 := by
+  apply Fintype.card_eq_zero_iff.mpr
+  refine ⟨fun e => ?_⟩
+  have hc := e.val.height_lt
+  rw [e.property] at hc
+  omega
+
+theorem binaryEdgeCount_succ (b h : ℕ) :
+    binaryEdgeCount (b+1) h = (if b=h then 2 else 0)+2*binaryEdgeCount b h := by
+  unfold binaryEdgeCount
+  simp only [Fintype.card_subtype,Finset.card_filter]
+  change (∑e : Bool ⊕ (Bool × BinaryEdge b), if BinaryEdge.height (b:=b+1) e=h then 1 else 0)=_
+  rw [Fintype.sum_sum_type,Fintype.sum_prod_type]
+  by_cases hb : b=h <;> simp [BinaryEdge.height,hb,two_mul] <;> simp only [Finset.card_filter] <;> rfl
+
+theorem binaryEdgeCount_eq (b h : ℕ) (hh : h<b) : binaryEdgeCount b h=2^(b-h) := by
+  induction b with
+  | zero => omega
+  | succ b ih =>
+    rw [binaryEdgeCount_succ]
+    by_cases he : b=h
+    · subst h
+      rw [ite_eq_left rfl,binaryEdgeCount_outside b b le_rfl]
+      simp
+    · rw [ite_eq_right he,zero_add,ih (by omega)]
+      have hsub : b+1-h=(b-h)+1 := by omega
+      rw [hsub,pow_succ,Nat.mul_comm]
+
+def Forced.extraCount {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) (h : ℕ) : ℕ :=
+  Fintype.card {e : F.ExtraEdge // e.val.height=h}
+
+theorem Forced.count_partition {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) (h : ℕ) :
+    F.extraCount h+binaryEdgeCount b h=t.edgeCount h := by
+  have he := F.edgeEquiv.sum_comp (fun e : t.Edge => if e.height=h then (1:ℕ) else 0)
+  rw [Fintype.sum_sum_type] at he
+  change (∑e : BinaryEdge b, if (F.edge e).height=h then 1 else 0)+
+    (∑e : F.ExtraEdge, if e.val.height=h then 1 else 0) =
+      (∑e : t.Edge, if e.height=h then 1 else 0) at he
+  simp_rw [F.edge_height] at he
+  unfold Forced.extraCount binaryEdgeCount edgeCount
+  simp only [Fintype.card_subtype,Finset.card_filter]
+  omega
+
+theorem levelCount_bottom {b : ℕ} (t : OccupiedParityTree b) : t.levelCount b=t.leafCount := by
+  induction t with
+  | leaf => rfl
+  | node n children roles ih => exact Finset.sum_congr rfl (fun i _ => ih i)
+
+/-- The extra bottom coordinates are precisely r-2^b, including the empty
+height-zero case. -/
+theorem Forced.card_extraLeafEdge {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) :
+    Fintype.card F.ExtraLeafEdge=t.leafCount-2^b := by
+  cases b with
+  | zero =>
+    cases t
+    change F.extraCount 0 = _
+    have h := F.count_partition 0
+    rw [edgeCount_leaf] at h
+    simp only [leafCount,pow_zero,Nat.sub_self]
+    omega
+  | succ b =>
+    change F.extraCount 0 = _
+    have h := F.count_partition 0
+    rw [binaryEdgeCount_eq (b+1) 0 (by omega),edgeCount_level t 0 (by omega)] at h
+    simp only [Nat.sub_zero,levelCount_bottom] at h
+    omega
+
+/-- Exact source nonforced vertex bound, now for genuine extra edge-label
+coordinates rather than an abstract level-count surrogate. -/
+theorem Forced.extraCount_le {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced)
+    (ht : t.occupied) (h : ℕ) : F.extraCount h≤t.leafCount-2^b := by
+  by_cases hh : h<b
+  · have hp := F.count_partition h
+    rw [binaryEdgeCount_eq b h hh,edgeCount_level t h hh] at hp
+    have Hb := F.nonforced_vertices_le_extra_leaves ht (b-h)
+    omega
+  · have hp := F.count_partition h
+    rw [edgeCount_outside t h (by omega)] at hp
+    omega
+
+/-- Extra nonleaf numeric labels grouped by their actual child height. -/
+def Forced.interiorHeight {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced)
+    (e : F.ExtraInteriorEdge) : Fin b := ⟨e.val.val.height,e.val.val.height_lt⟩
+
+theorem Forced.interiorFiber_le {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced)
+    (ht : t.occupied) (h : Fin b) :
+    Fintype.card {e : F.ExtraInteriorEdge // F.interiorHeight e=h} ≤ t.leafCount-2^b := by
+  apply le_trans _ (F.extraCount_le ht h.val)
+  let f : {e : F.ExtraInteriorEdge // F.interiorHeight e=h} →
+      {e : F.ExtraEdge // e.val.height=h.val} := fun e =>
+    ⟨e.val.val,congrArg Fin.val e.property⟩
+  apply Fintype.card_le_of_injective f
+  intro u v he
+  have he' : u.val.val=v.val.val := congrArg
+    (fun q : {e : F.ExtraEdge // e.val.height=h.val} => q.val) he
+  exact Subtype.ext (Subtype.ext he')
+
+theorem Forced.card_interiorLabels_le (T : ℕ → ℕ) (hT : ∀j,0<T j)
+    {b : ℕ} {t : OccupiedParityTree b} (F : t.Forced) (ht : t.occupied) :
+    Fintype.card (F.InteriorLabels T) ≤
+      (∏j : Fin b,if j.val=0 then 1 else T j.val)^(t.leafCount-2^b) := by
+  change Fintype.card ((e : F.ExtraInteriorEdge) → Fin (T e.val.val.height)) ≤ _
+  rw [Fintype.card_pi]
+  simp only [Fintype.card_fin]
+  have hp := Fintype.prod_fiberwise' F.interiorHeight (fun j : Fin b => T j.val)
+  change (∏j : Fin b, ∏_e : {e : F.ExtraInteriorEdge // F.interiorHeight e=j}, T j.val) =
+    ∏e : F.ExtraInteriorEdge,T e.val.val.height at hp
+  rw [←hp]
+  simp only [Finset.prod_const,Finset.card_univ]
+  rw [←Finset.prod_pow]
+  apply Finset.prod_le_prod
+  intro j hj
+  by_cases hz : j.val=0
+  · have he : Fintype.card {e : F.ExtraInteriorEdge // F.interiorHeight e=j}=0 := by
+      apply Fintype.card_eq_zero_iff.mpr
+      refine ⟨fun e => ?_⟩
+      exact e.val.property ((congrArg Fin.val e.property).trans hz)
+    rw [he,ite_eq_left hz,pow_zero,one_pow]
+  · rw [ite_eq_right hz]
+    exact Nat.pow_le_pow_right (hT j.val) (F.interiorFiber_le ht j)
+
+end OccupiedParityTree
+end TraceReconstruction
+
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/TraceReconstruction/Simulation.lean b/lean/OAI/Probability/TraceReconstruction/Simulation.lean
index 698ff1ad5f48539859bf40946d74323083563fdc..bfd3bdeb940d7e3e2576540977f404c21919c136 100644
--- a/lean/OAI/Probability/TraceReconstruction/Simulation.lean
+++ b/lean/OAI/Probability/TraceReconstruction/Simulation.lean
@@ -1,1313 +1,1313 @@
-import OAI.Probability.TraceReconstruction.Energy
-import OAI.Probability.TraceReconstruction.Metadata
-
-namespace OAI
-
-/-!
-# Simulation
-
-Canonical and filled simulations, central fit, root words, chronology, slots, finite codes, and complete-trace decoding.
--/
-
-open scoped ENNReal
-noncomputable section
-
-open Filter
-open scoped Topology
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- Heterogeneous emissions retain their actual outcome type while moving to
- the one fixed public chronological order. -/
-theorem dependentProduct_approximates {ι : Type*} [Fintype ι] [DecidableEq ι]
-    {A : ι → Type*} [∀i,Fintype (A i)] {k : ℕ} {q : ℝ} (hq : q∈Set.Icc 0 1)
-    (e : Fin k ≃ ι) (P : ∀i,FiniteLaw (A i)) (f : ∀i,A i → List Bool)
-    (w : ι → List Bool) (E : ι → ℝ)
-    (h : ∀i,ApproximatesTrace q (P i) (f i) (w i) (E i)) :
-    ApproximatesTrace q (FiniteLaw.dependentProduct P)
-      (fun x => coordinateWord (fun j => f (e j) (x (e j))))
-      (coordinateWord (fun j => w (e j))) (∑i,E i) := by
-  intro F hF
-  let G := Equiv.piCongrLeft A e
-  have he (x : ∀j,A (e j)) (j : Fin k) :
-      cast (congrArg A (e.apply_symm_apply (e j))) (x (e.symm (e j)))=x j := by
-    rw [←Equiv.piCongrLeft_apply_eq_cast]
-    exact Equiv.piCongrLeft_apply_apply A e x j
-  rw [FiniteLaw.dependentProduct_expect_equiv P e]
-  simp_rw [he]
-  have H := dependentCoordinate_approximates hq (fun j => P (e j))
-    (fun j => f (e j)) (fun j => w (e j)) (fun j => E (e j)) (fun j => h (e j)) F hF
-  rwa [e.sum_comp] at H
-
-/-- Each paired mechanism owns two distinct side slots; fillers and old leaf
- oracles have their own IDs. This type is independent of every orientation. -/
-abbrev EmissionSlot (m k n : ℕ) := (Fin m × Bool) ⊕ (Fin k ⊕ Fin n)
-
-def EmissionOutcome (Side Leaf : Type) {m k n : ℕ} : EmissionSlot m k n → Type
-  | .inl _ => Side
-  | .inr (.inl _) => Side
-  | .inr (.inr _) => Leaf
-
-instance emissionOutcomeFintype {Side Leaf : Type} [Fintype Side] [Fintype Leaf]
-    {m k n : ℕ} (i : EmissionSlot m k n) : Fintype (EmissionOutcome Side Leaf i) := by
-  rcases i with a | a
-  · exact inferInstanceAs (Fintype Side)
-  · rcases a with a | a
-    · exact inferInstanceAs (Fintype Side)
-    · exact inferInstanceAs (Fintype Leaf)
-
-/-- Actual arrays and slot observations are bijective, not a lossy interface. -/
-def emissionAssignmentEquiv (Side Leaf : Type) (m k n : ℕ) :
-    ((Fin m → Side × Side) × (Fin k → Side) × (Fin n → Leaf)) ≃
-      (∀i : EmissionSlot m k n,EmissionOutcome Side Leaf i) where
-  toFun x i := match i with
-    | .inl (j,false) => (x.1 j).1
-    | .inl (j,true) => (x.1 j).2
-    | .inr (.inl j) => x.2.1 j
-    | .inr (.inr j) => x.2.2 j
-  invFun x := (fun j => (x (.inl (j,false)),x (.inl (j,true))),
-    fun j => x (.inr (.inl j)),fun j => x (.inr (.inr j)))
-  left_inv x := by
-    rcases x with ⟨z,f,l⟩
-    change (fun j => ((z j).1,(z j).2),f,l)=(z,f,l)
-    simp only [Prod.mk.eta]
-  right_inv x := by
-    funext i
-    rcases i with ⟨j,b⟩ | i
-    · cases b <;> rfl
-    · cases i <;> rfl
-
-namespace PairedMechanisms
-variable {Side Leaf : Type} [Fintype Side] [Fintype Leaf] {m k n : ℕ}
-variable (M : PairedMechanisms Side m)
-
-def slotLaw (R : Fin k → FiniteLaw Side) (P Q : FiniteLaw Leaf)
-    (bits : Fin m → Bool) (θ : Fin n → Bool) :
-    (i : EmissionSlot m k n) → FiniteLaw (EmissionOutcome Side Leaf i)
-  | .inl (j,false) => if bits j then M.second j else M.first j
-  | .inl (j,true) => if bits j then M.first j else M.second j
-  | .inr (.inl j) => R j
-  | .inr (.inr j) => if θ j then Q else P
-
-theorem slotLaw_mass (R : Fin k → FiniteLaw Side) (P Q : FiniteLaw Leaf)
-    (bits : Fin m → Bool) (θ : Fin n → Bool)
-    (x : (Fin m → Side × Side) × (Fin k → Side) × (Fin n → Leaf)) :
-    (FiniteLaw.dependentProduct (M.slotLaw R P Q bits θ)).mass
-      (emissionAssignmentEquiv Side Leaf m k n x)=
-    ((FiniteLaw.coordinateProduct (fun j => FiniteLaw.orientedProd (M.first j) (M.second j) (bits j))).prod
-      ((FiniteLaw.coordinateProduct R).prod (selectedLeafLaw P Q θ))).mass x := by
-  change (∏i : EmissionSlot m k n,_)=_
-  rw [Fintype.prod_sum_type,Fintype.prod_sum_type,Fintype.prod_prod_type]
-  simp only [Fintype.prod_bool,slotLaw,emissionAssignmentEquiv,Equiv.coe_fn_mk]
-  change (∏j : Fin m,_) * ((∏j : Fin k,_) * ∏j : Fin n,_)=
-    (∏j : Fin m,_) * ((∏j : Fin k,_) * ∏j : Fin n,_)
-  congr 1
-  apply Finset.prod_congr rfl
-  intro j _
-  cases hb : bits j <;> simp [FiniteLaw.orientedProd,hb,FiniteLaw.prod,mul_comm]
-
-theorem slotLaw_expect (R : Fin k → FiniteLaw Side) (P Q : FiniteLaw Leaf)
-    (bits : Fin m → Bool) (θ : Fin n → Bool)
-    (F : (∀i : EmissionSlot m k n,EmissionOutcome Side Leaf i) → ℝ) :
-    ((FiniteLaw.coordinateProduct (fun j => FiniteLaw.orientedProd (M.first j) (M.second j) (bits j))).prod
-      ((FiniteLaw.coordinateProduct R).prod (selectedLeafLaw P Q θ))).expect
-        (fun x => F (emissionAssignmentEquiv Side Leaf m k n x))=
-      (FiniteLaw.dependentProduct (M.slotLaw R P Q bits θ)).expect F := by
-  unfold FiniteLaw.expect
-  rw [←(emissionAssignmentEquiv Side Leaf m k n).sum_comp]
-  simp_rw [slotLaw_mass]
-
-/-- Forgetting only the visible flags recovers the genuine independent side,
- filler and leaf laws at every orientation, before any fit restriction. -/
-theorem reportsGiven_slot_expect (R : Fin k → FiniteLaw Side) (P Q : FiniteLaw Leaf)
-    (bits : Fin m → Bool) (θ : Fin n → Bool)
-    (F : (∀i : EmissionSlot m k n,EmissionOutcome Side Leaf i) → ℝ) :
-    ((M.reportsGiven bits).prod ((FiniteLaw.coordinateProduct R).prod (selectedLeafLaw P Q θ))).expect
-      (fun x => F (emissionAssignmentEquiv Side Leaf m k n (fun j => (x.1 j).2,x.2)))=
-      (FiniteLaw.dependentProduct (M.slotLaw R P Q bits θ)).expect F := by
-  rw [FiniteLaw.expect_prod]
-  change (M.reportsGiven bits).expect ((fun z : Fin m → Side × Side =>
-    ((FiniteLaw.coordinateProduct R).prod (selectedLeafLaw P Q θ)).expect
-      (fun b => F (emissionAssignmentEquiv Side Leaf m k n (z,b)))) ∘
-        (fun z j => (z j).2))=_
-  rw [←FiniteLaw.expect_map]
-  rw [reportsGiven_forget]
-  have H := slotLaw_expect M R P Q bits θ F
-  rw [FiniteLaw.expect_prod] at H
-  exact H
-
-/-- The common, bit-blind projection simply forgets flags and concatenates in
- the publicly fixed ID order. -/
-def slotDecode (f : Side → List Bool) (g : Leaf → List Bool) :
-    (i : EmissionSlot m k n) → EmissionOutcome Side Leaf i → List Bool
-  | .inl _,x => f x
-  | .inr (.inl _),x => f x
-  | .inr (.inr _),x => g x
-
-def slotWord (U V : Fin m → List Bool) (W : Fin k → List Bool)
-    (atom : Bool → List Bool) (bits : Fin m → Bool) (θ : Fin n → Bool) :
-    EmissionSlot m k n → List Bool
-  | .inl (j,false) => if bits j then V j else U j
-  | .inl (j,true) => if bits j then U j else V j
-  | .inr (.inl j) => W j
-  | .inr (.inr j) => atom (θ j)
-
-def slotCost (E : ℝ) : EmissionSlot m k n → ℝ
-  | .inl _ => 0
-  | .inr (.inl _) => 0
-  | .inr (.inr _) => E
-
-theorem slotCost_sum (E : ℝ) : (∑i : EmissionSlot m k n,slotCost E i)=(n:ℝ)*E := by
-  simp only [Fintype.sum_sum_type,slotCost,Finset.sum_const_zero,
-    zero_add,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]
-
-theorem slotLaw_simulation {q E : ℝ} (R : Fin k → FiniteLaw Side) (P Q : FiniteLaw Leaf)
-    (f : Side → List Bool) (g : Leaf → List Bool) (U V : Fin m → List Bool)
-    (W : Fin k → List Bool) (atom : Bool → List Bool)
-    (hU : ∀j,SimulatesTrace q (M.first j) f (U j))
-    (hV : ∀j,SimulatesTrace q (M.second j) f (V j))
-    (hW : ∀j,SimulatesTrace q (R j) f (W j))
-    (hP : ApproximatesTrace q P g (atom false) E) (hQ : ApproximatesTrace q Q g (atom true) E)
-    (bits : Fin m → Bool) (θ : Fin n → Bool) (i : EmissionSlot m k n) :
-    ApproximatesTrace q (M.slotLaw R P Q bits θ i) (slotDecode f g i)
-      (slotWord U V W atom bits θ i) (slotCost E i) := by
-  rcases i with ⟨j,b⟩ | i
-  · cases b <;> cases hb : bits j <;>
-      simp only [slotLaw,slotWord,slotCost,hb,Bool.false_eq_true,ite_false,ite_true]
-    · exact (hU j).approximate
-    · exact (hV j).approximate
-    · exact (hV j).approximate
-    · exact (hU j).approximate
-  · rcases i with j | j
-    · exact (hW j).approximate
-    · cases hb : θ j <;> simp only [slotLaw,slotWord,slotCost,hb,Bool.false_eq_true]
-      · exact hP
-      · exact hQ
-
-/-- Exact conditional-on-orientation simulation boundary. The orientation is
- not part of the common projection, and no fit conditioning enters its law. -/
-theorem reportsGiven_simulation {q E : ℝ} (hq : q∈Set.Icc 0 1)
-    {L : ℕ} (e : Fin L ≃ EmissionSlot m k n)
-    (R : Fin k → FiniteLaw Side) (P Q : FiniteLaw Leaf)
-    (f : Side → List Bool) (g : Leaf → List Bool) (U V : Fin m → List Bool)
-    (W : Fin k → List Bool) (atom : Bool → List Bool)
-    (hU : ∀j,SimulatesTrace q (M.first j) f (U j))
-    (hV : ∀j,SimulatesTrace q (M.second j) f (V j))
-    (hW : ∀j,SimulatesTrace q (R j) f (W j))
-    (hP : ApproximatesTrace q P g (atom false) E) (hQ : ApproximatesTrace q Q g (atom true) E)
-    (bits : Fin m → Bool) (θ : Fin n → Bool) :
-    ApproximatesTrace q
-      ((M.reportsGiven bits).prod ((FiniteLaw.coordinateProduct R).prod (selectedLeafLaw P Q θ)))
-      (fun x => coordinateWord (fun j => slotDecode f g (e j)
-        (emissionAssignmentEquiv Side Leaf m k n (fun i => (x.1 i).2,x.2) (e j))))
-      (coordinateWord (fun j => slotWord U V W atom bits θ (e j))) ((n:ℝ)*E) := by
-  intro F hF
-  dsimp only
-  rw [M.reportsGiven_slot_expect R P Q bits θ
-    (fun z => F (coordinateWord (fun j => slotDecode f g (e j) (z (e j)))))]
-  have H := dependentProduct_approximates hq e (M.slotLaw R P Q bits θ) (slotDecode f g)
-    (slotWord U V W atom bits θ) (slotCost E)
-    (M.slotLaw_simulation R P Q f g U V W atom hU hV hW hP hQ bits θ) F hF
-  rwa [slotCost_sum] at H
-
-end PairedMechanisms
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-namespace PairedMechanisms
-variable {Side Leaf : Type} [Fintype Side] [Fintype Leaf] {m k n : ℕ}
-variable (M : PairedMechanisms Side m)
-
-/-- The actual metadata-first oracle with independent genuine fillers has
- exactly the orientation-first product experiment, in the displayed order. -/
-theorem canonical_filled_expect (R : Fin k → FiniteLaw Side) (P Q : FiniteLaw Leaf)
-    (θ : (Fin m → Bool) → Fin n → Bool)
-    (F : Reports Side m × (Fin k → Side) × (Fin n → Leaf) → ℝ) :
-    ((M.canonicalOracle P Q θ).prod (FiniteLaw.coordinateProduct R)).expect
-      (fun x => F (x.1.1,x.2,x.1.2))=
-    (fairOrientations m).expect (fun bits =>
-      ((M.reportsGiven bits).prod ((FiniteLaw.coordinateProduct R).prod
-        (selectedLeafLaw P Q (θ bits)))).expect F) := by
-  rw [←M.oracle_factorization,orientationOracle,FiniteLaw.expect_prod,FiniteLaw.expect_mixture]
-  simp_rw [FiniteLaw.expect_prod]
-  congr 1
-  funext bits
-  congr 1
-  funext z
-  exact FiniteLaw.expect_comm _ _ (fun leaf fill => F (z,fill,leaf))
-
-/-- Unconditional common simulation of the actual full canonical oracle.
- The only layout charge is the unconditional fair-orientation failure mass.
- No orientation or side metadata is conditioned on successful fitting. -/
-theorem canonical_filled_simulation {q E : ℝ} (hq : q∈Set.Icc 0 1) (hE : 0≤E)
-    {L : ℕ} (e : Fin L ≃ EmissionSlot m k n)
-    (R : Fin k → FiniteLaw Side) (P Q : FiniteLaw Leaf)
-    (f : Side → List Bool) (g : Leaf → List Bool) (U V : Fin m → List Bool)
-    (W : Fin k → List Bool) (atom : Bool → List Bool)
-    (hU : ∀j,SimulatesTrace q (M.first j) f (U j))
-    (hV : ∀j,SimulatesTrace q (M.second j) f (V j))
-    (hW : ∀j,SimulatesTrace q (R j) f (W j))
-    (hP : ApproximatesTrace q P g (atom false) E) (hQ : ApproximatesTrace q Q g (atom true) E)
-    (θ : (Fin m → Bool) → Fin n → Bool) (w : List Bool)
-    (good : (Fin m → Bool) → Prop) [DecidablePred good]
-    (hword : ∀bits,good bits → coordinateWord
-      (fun j => slotWord U V W atom bits (θ bits) (e j))=w) :
-    ApproximatesTrace q ((M.canonicalOracle P Q θ).prod (FiniteLaw.coordinateProduct R))
-      (fun x => coordinateWord (fun j => slotDecode f g (e j)
-        (emissionAssignmentEquiv Side Leaf m k n (fun i => (x.1.1 i).2,x.2,x.1.2) (e j))))
-      w ((n:ℝ)*E+(fairOrientations m).expect (fun bits => if good bits then 0 else 1)) := by
-  intro F hF
-  dsimp only
-  rw [M.canonical_filled_expect R P Q θ (fun x => F (coordinateWord (fun j => slotDecode f g (e j)
-    (emissionAssignmentEquiv Side Leaf m k n (fun i => (x.1 i).2,x.2) (e j)))))]
-  rw [←FiniteLaw.expect_const (fairOrientations m) (listDeletionExpectation q w F),←FiniteLaw.expect_sub]
-  apply (FiniteLaw.abs_expect_le _ _).trans
-  calc
-    _ ≤ (fairOrientations m).expect (fun bits => (n:ℝ)*E+(if good bits then 0 else 1)) := by
-      apply FiniteLaw.expect_mono
-      intro bits
-      by_cases hb : good bits
-      · simp only [ite_eq_left hb,add_zero]
-        have H := M.reportsGiven_simulation hq e R P Q f g U V W atom hU hV hW hP hQ bits (θ bits) F hF
-        rwa [hword bits hb] at H
-      · simp only [ite_eq_right hb]
-        have hc := listDeletionExpectation_range hq w F hF
-        have hlo := FiniteLaw.expect_mono
-          ((M.reportsGiven bits).prod ((FiniteLaw.coordinateProduct R).prod (selectedLeafLaw P Q (θ bits))))
-          (fun x => (hF (coordinateWord (fun j => slotDecode f g (e j)
-            (emissionAssignmentEquiv Side Leaf m k n (fun i => (x.1 i).2,x.2) (e j))))).1)
-        have hhi := FiniteLaw.expect_mono
-          ((M.reportsGiven bits).prod ((FiniteLaw.coordinateProduct R).prod (selectedLeafLaw P Q (θ bits))))
-          (fun x => (hF (coordinateWord (fun j => slotDecode f g (e j)
-            (emissionAssignmentEquiv Side Leaf m k n (fun i => (x.1 i).2,x.2) (e j))))).2)
-        rw [FiniteLaw.expect_const] at hlo hhi
-        have hne : 0≤(n:ℝ)*E := mul_nonneg (Nat.cast_nonneg _) hE
-        apply abs_sub_le_iff.mpr
-        constructor <;> linarith [hc.1,hc.2]
-    _ = _ := by
-      unfold FiniteLaw.expect
-      simp only [mul_add,Finset.sum_add_distrib,←Finset.sum_mul,(fairOrientations m).total,one_mul]
-
-end PairedMechanisms
-end TraceReconstruction
-
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- The one-based source offset still puts the complete motif in the central
- third under the true fit threshold. The stronger numerical margin is an
- eventual consequence of L=o(T), not an additional stochastic event. -/
-theorem source_motif_central (T t s : ℕ) (bits : Fin t → Bool) (v : ℤ)
-    (hT : 24 ≤ T) (he : Even T) (hL : 24*2^s ≤ T)
-    (hfit : |8*((∑i,(bitIndex (bits i):ℤ))-(t/2:ℕ)+v)| ≤ (T:ℤ))
-    (u : Fin T)
-    (hu : motifIndicator s ((u.val:ℤ)-((T/2:ℕ)-1+(t/2:ℕ))+
-      (∑i,(bitIndex (bits i):ℤ))+v)≠0) : u∈centralThird T := by
-  have hsupport : 0≤(u.val:ℤ)-((T/2:ℕ)-1+(t/2:ℕ))+
-      (∑i,(bitIndex (bits i):ℤ))+v ∧
-      (u.val:ℤ)-((T/2:ℕ)-1+(t/2:ℕ))+(∑i,(bitIndex (bits i):ℤ))+v < (2^s:ℕ) := by
-    by_contra h
-    apply hu
-    exact ite_eq_right h
-  have hh := abs_le.mp hfit
-  obtain ⟨k,hk⟩ := he
-  have hhalf : 2*(T/2)=T := by omega
-  apply Finset.mem_filter.mpr
-  refine ⟨Finset.mem_univ _,?_⟩
-  omega
-
-/-- Fit controls every internal source motif, retaining all orientations in
- the law. The incoming displacement of the root is supplied separately. -/
-theorem sourceTreeFit_central (T n : ℕ → ℕ) (s b : ℕ)
-    (center : ℕ → ℤ) (hc : ∀j≤b,center j=sourceLocalCenter T n j)
-    (hT : ∀j≤b,24≤T j) (he : ∀j≤b,Even (T j)) (hL : ∀j≤b,24*2^s≤T j)
-    (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool (b+1)) (incoming : ℤ)
-    (hroot : |8*((∑i,(bitIndex (bits.1 i):ℤ))-(n b/2:ℕ)+incoming)|≤(T b:ℤ))
-    (hfit : SourceTreeFit T n (b+1) bits) :
-    SourceCentral T n (sourceScheduledTrials T n) s center (sourceScheduledShift T n)
-      (fun j => centralThird (T j)) (b+1) bits incoming := by
-  induction b generalizing incoming with
-  | zero =>
-    constructor
-    · intro u hu
-      rw [hc 0 (by omega)] at hu
-      exact source_motif_central (T 0) (n 0) s bits.1 incoming
-        (hT 0 (by omega)) (he 0 (by omega)) (hL 0 (by omega)) hroot u hu
-    · intro u; trivial
-  | succ b ih =>
-    constructor
-    · intro u hu
-      rw [hc (b+1) (by omega)] at hu
-      exact source_motif_central (T (b+1)) (n (b+1)) s bits.1 incoming
-        (hT (b+1) (by omega)) (he (b+1) (by omega)) (hL (b+1) (by omega)) hroot u hu
-    · intro u
-      apply ih (fun j hj => hc j (by omega)) (fun j hj => hT j (by omega))
-        (fun j hj => he j (by omega)) (fun j hj => hL j (by omega)) _ _ _ (hfit.2 u)
-      exact hfit.1 u
-
-/-- The actual root side count has no parity restriction. Its entire range
- lies inside the fit displacement interval when J=o(T). -/
-theorem source_root_fit (T J : ℕ) (bits : Fin J → Bool) (h : 4*(J+1)≤T) :
-    |8*((∑i,(bitIndex (bits i):ℤ))-(J/2:ℕ))|≤(T:ℤ) := by
-  have hlo : (0:ℤ)≤∑i,(bitIndex (bits i):ℤ) := Finset.sum_nonneg (fun _ _ => Int.natCast_nonneg _)
-  have hhi : (∑i,(bitIndex (bits i):ℤ))≤J := by
-    calc
-      _ ≤ ∑_i : Fin J,(1:ℤ) := by
-        apply Finset.sum_le_sum
-        intro i _
-        cases bits i <;> norm_num [bitIndex]
-      _ = J := by simp
-  apply abs_le.mpr
-  omega
-
-theorem sourceNoncentralMass_le_fit (T n : ℕ → ℕ) (s b : ℕ)
-    (center : ℕ → ℤ) (hc : ∀j≤b,center j=sourceLocalCenter T n j)
-    (hT : ∀j≤b,24≤T j) (he : ∀j≤b,Even (T j)) (hL : ∀j≤b,24*2^s≤T j)
-    (hroot : 4*(n b+1)≤T b)
-    (d : WindowAssignments T n (sourceScheduledTrials T n) (Option Bool) (b+1)) :
-    sourceNoncentralMass T n (sourceScheduledTrials T n) s center (sourceScheduledShift T n)
-      (fun j => centralThird (T j)) (b+1) d 0 ≤
-    (windowPosteriorLaw T n (sourceScheduledTrials T n) (b+1) d).expect
-      (fun bits => if SourceTreeFit T n (b+1) bits then 0 else 1) := by
-  apply FiniteLaw.expect_mono
-  intro bits
-  by_cases hf : SourceTreeFit T n (b+1) bits
-  · have hh := sourceTreeFit_central T n s b center hc hT he hL bits 0
-      (by simpa only [add_zero] using source_root_fit (T b) (n b) bits.1 hroot) hf
-    simp only [ite_eq_left hf,ite_eq_left hh,le_refl]
-  · simp only [ite_eq_right hf]
-    split_ifs <;> norm_num
-
-end TraceReconstruction
-
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- Only interior trials are paired in even bundles. The top J may be odd. -/
-theorem sourceTreeWord_eq_center_interior (s : ℕ) (P T n h : ℕ → ℕ) (center : ℕ → ℤ)
-    (atom : Bool → List Bool) (b : ℕ) (H : SourceLayoutMargins s P T n h (b-1))
-    (htop : ∀j<b,0<T j)
-    (hc : ∀j, j+1<b → center j=sourceLocalCenter T n j)
-    (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool b)
-    (incoming : ℤ) (τ : Bool) (hfit : SourceTreeFit T n b bits) :
-    sourceTreeWord s P T n h center atom b bits incoming τ=
-      sourceCenterWord s P T n center atom b bits incoming τ := by
-  induction b using Nat.twoStepInduction generalizing incoming τ with
-  | zero => rfl
-  | one => rfl
-  | more b ih₁ ih₂ =>
-    change (Fin (n (b+1)) → Bool) × (Fin ((T (b+1)/4)*n b) → Bool) ×
-      (Fin (T (b+1)) → WindowAssignments T n (sourceScheduledTrials T n) Bool (b+1)) at bits
-    let loc : Fin (T (b+1)) → Fin (n b) → Bool := fun u => (bits.2.2 u).1
-    let glob := bits.2.1
-    let a := P b/2-1
-    let A := sourceNominalPre P T b
-    let B := sourceNominalPost P T b
-    let start := sourceWindowStart (T (b+1)) A (n b) loc glob
-    let θ := integerTemplateType (T (b+1)) s (sourceNodeAnchor T n center (b+1) bits incoming) τ
-    have hb : b<b+2-1 := by omega
-    have htotal := sourceNominal_total P T b (by have hh := H.ambient b hb; omega)
-    have hf (u : Fin (T (b+1))) := sourceWindowStart_oneBased_fit (T (b+1)) (P b) (T b) (n b)
-      (2^s) loc glob (H.ambient b hb) (H.large b hb) (H.even_window b hb) (H.motif b hb) u (hfit.1 u)
-    have hx (u : Fin (T (b+1))) :
-        sourceNodeAnchor T n center b (bits.2.2 u) (sourceScheduledShift T n (b+1) glob u)=((a-start u:ℕ):ℤ) := by
-      change sourceNodeAnchor T n center b (bits.2.2 u) (sourceScheduledShift T n (b+1) glob u)=
-        ((P b/2-1-sourceWindowStart (T (b+1)) ((P b-T b)/2) (n b) loc glob u:ℕ):ℤ)
-      rw [sourceWindowStart_oneBased_anchor (T (b+1)) (P b) (T b) (n b) (2^s) loc glob
-        (H.ambient b hb) (H.large b hb) (H.even_window b hb) (H.motif b hb) u (hfit.1 u)]
-      simp only [sourceNodeAnchor,hc b (by omega),sourceLocalCenter,sourceScheduledShift,loc]
-    have hchild (u : Fin (T (b+1))) :
-        sourceTreeWord s P T n h center atom (b+1) (bits.2.2 u)
-          (sourceScheduledShift T n (b+1) glob u) (θ u)=
-        complementaryWords (T b) (a-start u) (2^s) (prouhetBool s) (sourceBlockWord s P atom b) (θ u) := by
-      rw [ih₂ (H.restrict (by omega)) (fun j hj => htop j (by omega)) (fun j hj => hc j (by omega)) _ _ _ (hfit.2 u)]
-      simp only [sourceCenterWord,hx,integerTemplateType_nat,complementaryWords]
-    have hstartnonneg (u : Fin (T (b+1))) : 0≤(A:ℤ)+sourceDisplacement (T (b+1)) (n b) loc glob u := by
-      have he := sourceWindowStart_cast_of_fit (T (b+1)) (P b) (T b) (n b) loc glob
-        (H.ambient b hb) (H.large b hb) u (hfit.1 u)
-      have hn : (0:ℤ)≤(start u:ℕ) := Int.natCast_nonneg _
-      change (start u:ℤ)=(A:ℤ)+_ at he
-      omega
-    have hp := htop (b+1) (by omega)
-    obtain ⟨k,hk⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : T (b+1)≠0)
-    have hpart : (List.ofFn (fun u => complementaryWords (P b) a (2^s) (prouhetBool s)
-          (sourceBlockWord s P atom b) (θ u))).flatten=
-        coordinateWord (fun u => pureAtomRun (sourceBlockWord s P atom b false)
-          (sourcePreCount (T (b+1)) A (n b) (h b) loc glob u) ++
-          complementaryWords (T b) (a-start u) (2^s) (prouhetBool s) (sourceBlockWord s P atom b) (θ u) ++
-          pureAtomRun (sourceBlockWord s P atom b false) (sourcePostCount (T (b+1)) B (n b) (h b) loc glob u)) := by
-      have hh := sourceLayout_literal_partition k A (T b) B (n b) (h b) a (2^s)
-      have hk' : k+1=T (b+1) := hk.symm
-      rw [hk'] at hh
-      rw [show A+T b+B=P b from htotal] at hh
-      exact hh (H.even_trials b hb) (H.pre_reserve b hb) (H.post_reserve b hb) loc glob
-        (prouhetBool s) (sourceBlockWord s P atom b) θ hstartnonneg
-        (fun u => (hf u).2.1) (fun u => (hf u).2.2) (fun u => (hf u).1)
-    change coordinateWord (fun u : Fin (T (b+1)) =>
-      pureAtomRun (sourceBlockWord s P atom b false) (sourcePreCount (T (b+1)) A (n b) (h b) loc glob u) ++
-      sourceTreeWord s P T n h center atom (b+1) (bits.2.2 u)
-        (sourceScheduledShift T n (b+1) glob u) (θ u) ++
-      pureAtomRun (sourceBlockWord s P atom b false) (sourcePostCount (T (b+1)) B (n b) (h b) loc glob u))=
-        atomString (T (b+1)) (sourceBlockWord s P atom (b+1)) θ
-    calc
-      _ = coordinateWord (fun u : Fin (T (b+1)) =>
-        pureAtomRun (sourceBlockWord s P atom b false) (sourcePreCount (T (b+1)) A (n b) (h b) loc glob u) ++
-        complementaryWords (T b) (a-start u) (2^s) (prouhetBool s) (sourceBlockWord s P atom b) (θ u) ++
-        pureAtomRun (sourceBlockWord s P atom b false) (sourcePostCount (T (b+1)) B (n b) (h b) loc glob u)) := by
-          apply congrArg coordinateWord
-          funext u
-          exact congrArg (fun w => pureAtomRun (sourceBlockWord s P atom b false)
-            (sourcePreCount (T (b+1)) A (n b) (h b) loc glob u) ++ w ++
-              pureAtomRun (sourceBlockWord s P atom b false)
-                (sourcePostCount (T (b+1)) B (n b) (h b) loc glob u)) (hchild u)
-      _ = _ := hpart.symm
-
-
-
-/-- Full literal unconditioned root partition, including both members of
- every outer paired side. Its top trial count is the source's J, even or odd. -/
-def sourceFullWord (s : ℕ) (P T n h : ℕ → ℕ) (atom : Bool → List Bool) (b : ℕ)
-    (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool (b+1)) (τ : Bool) : List Bool :=
-  pureAtomRun (sourceBlockWord s P atom b false) (n b*h b+∑i,bitIndex (bits.1 i)) ++
-  sourceTreeWord s P T n h (sourceLocalCenter T n) atom (b+1) bits 0 τ ++
-  pureAtomRun (sourceBlockWord s P atom b false) (n b*(h b+1)-∑i,bitIndex (bits.1 i))
-
-theorem bitIndex_sum_le {J : ℕ} (bits : Fin J → Bool) : (∑i,bitIndex (bits i))≤J := by
-  calc
-    _ ≤ ∑_i : Fin J,1 := by
-      apply Finset.sum_le_sum
-      intro i _
-      cases bits i <;> norm_num [bitIndex]
-    _ = J := by simp
-
-/-- The source root's floor anchor and paired side counts have their literal
- unmarked word interpretation, without requiring J to be even. -/
-theorem sourceFullWord_eq_block (s : ℕ) (P T n h : ℕ → ℕ) (atom : Bool → List Bool)
-    (b : ℕ) (H : SourceLayoutMargins s P T n h b) (hpos : 0<T b)
-    (hroot : P b=n b*(2*h b+1)+T b) (he : Even (T b))
-    (hJ : 4*(n b+1)≤T b) (hL : 8*2^s≤T b)
-    (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool (b+1))
-    (τ : Bool) (hfit : SourceTreeFit T n (b+1) bits) :
-    sourceFullWord s P T n h atom b bits τ=sourceBlockWord s P atom (b+1) τ := by
-  let E : ℕ := ∑i,bitIndex (bits.1 i)
-  let t := n b*h b+E
-  let z := n b*(h b+1)-E
-  let a := P b/2-1
-  have hE : E≤n b := bitIndex_sum_le bits.1
-  have hnat : n b≤n b*(h b+1) := by nlinarith
-  have htotal : t+T b+z=P b := by
-    have hprod : n b*(h b+1)=n b*h b+n b := by ring
-    have hprod2 : n b*(2*h b+1)=2*(n b*h b)+n b := by ring
-    dsimp [t,z]
-    omega
-  obtain ⟨k,hk⟩ := he
-  have hhalf : 2*(T b/2)=T b := by omega
-  have hP : P b=2*(n b*h b)+n b+T b := by rw [hroot]; ring
-  have hPa : P b/2=n b*h b+T b/2+n b/2 := by omega
-  have ht : t≤a := by dsimp [t,a]; omega
-  have hz : a+2^s≤t+T b := by dsimp [t,a]; omega
-  have hanchor : sourceNodeAnchor T n (sourceLocalCenter T n) b bits 0=((a-t:ℕ):ℤ) := by
-    have hEc : (E:ℤ)=∑i,(bitIndex (bits.1 i):ℤ) := by simp [E]
-    dsimp only [sourceNodeAnchor,sourceLocalCenter]
-    have ha : a-t=T b/2-1+n b/2-E := by dsimp [a,t]; omega
-    have hTe : 1≤T b/2 := by omega
-    have hEe : E≤T b/2-1+n b/2 := by omega
-    rw [ha,Nat.cast_sub hEe,Nat.cast_add,Nat.cast_sub hTe,hEc]
-    push_cast
-    ring
-  have ho (j : ℕ) (hj : j<b+1) : 0<T j := by
-    by_cases heq : j=b
-    · simpa only [heq] using hpos
-    · exact H.positive j (by omega)
-  have hcenter := sourceTreeWord_eq_center_interior s P T n h (sourceLocalCenter T n) atom
-    (b+1) (by simpa using H) ho (fun _ _ => rfl) bits 0 τ hfit
-  have hpart := template_word_window t (T b) z a (2^s) (prouhetBool s) τ
-    (sourceBlockWord s P atom b) ht hz
-  rw [htotal] at hpart
-  rw [atomString_constant,atomString_constant] at hpart
-  unfold sourceFullWord
-  rw [hcenter]
-  change pureAtomRun (sourceBlockWord s P atom b false) t ++
-    atomString (T b) (sourceBlockWord s P atom b)
-      (integerTemplateType (T b) s (sourceNodeAnchor T n (sourceLocalCenter T n) b bits 0) τ) ++
-    pureAtomRun (sourceBlockWord s P atom b false) z = _
-  rw [hanchor,integerTemplateType_nat]
-  exact hpart.symm
-
-end TraceReconstruction
-
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- Pure filler IDs. At bottom leaf windows the two fillers have zero length;
- they are deterministic empty outputs, not extra information. -/
-def WindowFiller (T : ℕ → ℕ) : ℕ → Type
-  | 0 => PEmpty
-  | b+1 => (Fin (T b) × Bool) ⊕ (Fin (T b) × WindowFiller T b)
-
-noncomputable instance windowFillerFintype (T : ℕ → ℕ) : (b : ℕ) → Fintype (WindowFiller T b)
-  | 0 => inferInstanceAs (Fintype PEmpty)
-  | b+1 => by
-    letI := windowFillerFintype T b
-    exact inferInstanceAs (Fintype ((Fin (T b) × Bool) ⊕ (Fin (T b) × WindowFiller T b)))
-
-abbrev TreeEmission (T n g : ℕ → ℕ) (b : ℕ) :=
-  (WindowTrial T n g b × Bool) ⊕ (WindowFiller T b ⊕ WindowLeaf T b)
-
-def TreeEmission.up (T n g : ℕ → ℕ) (b : ℕ) (u : Fin (T b)) :
-    TreeEmission T n g b → TreeEmission T n g (b+1)
-  | .inl (i,p) => .inl (.inr (.inr (u,i)),p)
-  | .inr (.inl f) => .inr (.inl (.inr (u,f)))
-  | .inr (.inr v) => .inr (.inr (u,v))
-
-def treeLocalSlots (T n g : ℕ → ℕ) (b : ℕ) (p : Bool) : List (TreeEmission T n g (b+1)) :=
-  List.ofFn (fun k : Fin (n b) => Sum.inl (Sum.inl k,p))
-
-def treeGlobalSlots (T n g : ℕ → ℕ) (b : ℕ) (p : Bool)
-    (pos : Fin (g b) → Fin (T b)) (u : Fin (T b)) : List (TreeEmission T n g (b+1)) :=
-  ((List.ofFn id : List (Fin (g b))).filter (fun k => pos k=u)).map
-    (fun k => Sum.inl (Sum.inr (Sum.inl k),p))
-
-/-- Public chronological IDs. Each child carries its own local paired sides;
- the group's global sides and deterministic fillers surround that child.
- The order is fixed independently of all bits and both hypotheses. -/
-def treeChronology (T n g : ℕ → ℕ)
-    (pre post : (b : ℕ) → Fin (g b) → Fin (T b)) :
-    (b : ℕ) → List (TreeEmission T n g b)
-  | 0 => [.inr (.inr (show WindowLeaf T 0 from (0 : Fin 1)))]
-  | b+1 => treeLocalSlots T n g b false ++
-      (List.ofFn (fun u : Fin (T b) =>
-        treeGlobalSlots T n g b false (pre b) u ++
-        [.inr (.inl (.inl (u,false)))] ++
-        (treeChronology T n g pre post b).map (TreeEmission.up T n g b u) ++
-        treeGlobalSlots T n g b true (post b) u ++
-        [.inr (.inl (.inl (u,true)))])).flatten ++
-      treeLocalSlots T n g b true
-
-theorem sum_fiber_list_length {α ι : Type*} [Fintype ι] [DecidableEq ι] (pos : α → ι) (l : List α) :
-    (∑i,(l.filter (fun a => pos a=i)).length)=l.length := by
-  induction l with
-  | nil => simp
-  | cons a l ih =>
-    have hh (i : ι) : ((a::l).filter (fun a => pos a=i)).length=
-        (if pos a=i then 1 else 0)+(l.filter (fun a => pos a=i)).length := by
-      by_cases h : pos a=i <;> simp [h,Nat.add_comm]
-    simp_rw [hh]
-    rw [Finset.sum_add_distrib,ih]
-    simp [Nat.add_comm]
-
-theorem treeChronology_length (T n g : ℕ → ℕ)
-    (pre post : (b : ℕ) → Fin (g b) → Fin (T b)) (b : ℕ) :
-    (treeChronology T n g pre post b).length=Fintype.card (TreeEmission T n g b) := by
-  induction b with
-  | zero =>
-    change 1=Fintype.card ((PEmpty × Bool) ⊕ (PEmpty ⊕ Fin 1))
-    simp
-  | succ b ih =>
-    have hp := sum_fiber_list_length (pre b) (List.ofFn id : List (Fin (g b)))
-    have hq := sum_fiber_list_length (post b) (List.ofFn id : List (Fin (g b)))
-    simp only [List.length_ofFn] at hp hq
-    change _=Fintype.card (((Fin (n b) ⊕ (Fin (g b) ⊕ (Fin (T b) × WindowTrial T n g b))) × Bool) ⊕
-      (((Fin (T b) × Bool) ⊕ (Fin (T b) × WindowFiller T b)) ⊕ (Fin (T b) × WindowLeaf T b)))
-    simp only [treeChronology,treeLocalSlots,treeGlobalSlots,List.length_append,
-      List.length_flatten,List.map_ofFn,Function.comp_apply,List.length_ofFn,List.length_map,List.length_cons,List.length_nil,
-      Finset.sum_add_distrib,List.sum_ofFn,ih,Fintype.card_sum,Fintype.card_prod,Fintype.card_fin,Fintype.card_bool]
-    rw [hp,hq]
-    simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,smul_eq_mul]
-    ring
-
-theorem treeChronology_mem (T n g : ℕ → ℕ)
-    (pre post : (b : ℕ) → Fin (g b) → Fin (T b)) (b : ℕ)
-    (x : TreeEmission T n g b) : x∈treeChronology T n g pre post b := by
-  induction b with
-  | zero =>
-    rcases x with ⟨i,p⟩ | f | v
-    · exact PEmpty.elim i
-    · exact PEmpty.elim f
-    · change Fin 1 at v
-      have hv : v=0 := Subsingleton.elim _ _
-      subst v; exact List.mem_singleton.mpr rfl
-  | succ b ih =>
-    have hc (u : Fin (T b)) (x : TreeEmission T n g (b+1))
-        (hx : x∈treeGlobalSlots T n g b false (pre b) u ++ [.inr (.inl (.inl (u,false)))] ++
-          (treeChronology T n g pre post b).map (TreeEmission.up T n g b u) ++
-          treeGlobalSlots T n g b true (post b) u ++ [.inr (.inl (.inl (u,true)))]) :
-        x∈treeChronology T n g pre post (b+1) := by
-      apply List.mem_append_left
-      apply List.mem_append_right
-      apply List.mem_flatten.mpr
-      exact ⟨_,List.mem_ofFn.mpr ⟨u,rfl⟩,hx⟩
-    rcases x with ⟨k,p⟩ | f | v
-    · rcases k with k | k | ⟨u,k⟩
-      · cases p <;>
-          simp [treeChronology,treeLocalSlots,List.mem_ofFn]
-      · cases p
-        · apply hc (pre b k)
-          simp only [List.mem_append,treeGlobalSlots,List.mem_map,List.mem_filter,List.mem_ofFn]
-          exact Or.inl (Or.inl (Or.inl (Or.inl ⟨k,⟨⟨k,rfl⟩,by simp⟩,rfl⟩)))
-        · apply hc (post b k)
-          simp only [List.mem_append,treeGlobalSlots,List.mem_map,List.mem_filter,List.mem_ofFn]
-          exact Or.inl (Or.inr ⟨k,⟨⟨k,rfl⟩,by simp⟩,rfl⟩)
-      · apply hc u
-        have hm : TreeEmission.up T n g b u (.inl (k,p))∈
-            (treeChronology T n g pre post b).map (TreeEmission.up T n g b u) := List.mem_map.mpr ⟨_,ih _,rfl⟩
-        exact List.mem_append_left _ (List.mem_append_left _ (List.mem_append_right _ hm))
-    · rcases f with ⟨u,p⟩ | ⟨u,f⟩
-      · cases p <;> apply hc u <;> simp
-      · apply hc u
-        have hm : TreeEmission.up T n g b u (.inr (.inl f))∈
-            (treeChronology T n g pre post b).map (TreeEmission.up T n g b u) := List.mem_map.mpr ⟨_,ih _,rfl⟩
-        exact List.mem_append_left _ (List.mem_append_left _ (List.mem_append_right _ hm))
-    · rcases v with ⟨u,v⟩
-      apply hc u
-      have hm : TreeEmission.up T n g b u (.inr (.inr v))∈
-          (treeChronology T n g pre post b).map (TreeEmission.up T n g b u) := List.mem_map.mpr ⟨_,ih _,rfl⟩
-      exact List.mem_append_left _ (List.mem_append_left _ (List.mem_append_right _ hm))
-
-theorem treeChronology_nodup (T n g : ℕ → ℕ)
-    (pre post : (b : ℕ) → Fin (g b) → Fin (T b)) (b : ℕ) :
-    (treeChronology T n g pre post b).Nodup := by
-  have hs : (treeChronology T n g pre post b).toFinset=Finset.univ := by
-    ext x
-    simp only [List.mem_toFinset,Finset.mem_univ,iff_true]
-    exact treeChronology_mem T n g pre post b x
-  have hh : (treeChronology T n g pre post b).toFinset.card=(treeChronology T n g pre post b).length := by
-    rw [hs,Finset.card_univ,treeChronology_length]
-  exact Multiset.coe_nodup.mp ((Multiset.toFinset_card_eq_card_iff_nodup
-    (m := (treeChronology T n g pre post b : Multiset (TreeEmission T n g b)))).mp hh)
-
-
-/-- A genuine complete chronological bijection, not an assumed layout. -/
-def treeChronologyEquiv (T n g : ℕ → ℕ)
-    (pre post : (b : ℕ) → Fin (g b) → Fin (T b)) (b : ℕ) :
-    Fin (treeChronology T n g pre post b).length ≃ TreeEmission T n g b :=
-  List.Nodup.getEquivOfForallMemList _ (treeChronology_nodup T n g pre post b)
-    (treeChronology_mem T n g pre post b)
-
-end TraceReconstruction
-
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-def pairedSideCount (h : ℕ) (bit side : Bool) : ℕ := h+bitIndex (Bool.xor bit side)
-
-theorem pairedSideCount_false (h : ℕ) (bit : Bool) :
-    pairedSideCount h bit false=h+bitIndex bit := by cases bit <;> rfl
-
-theorem pairedSideCount_true (h : ℕ) (bit : Bool) :
-    pairedSideCount h bit true=h+1-bitIndex bit := by cases bit <;> simp [pairedSideCount,bitIndex]
-
-theorem pairedSideCount_sum_pre {t h : ℕ} (bits : Fin t → Bool) :
-    (∑i,pairedSideCount h (bits i) false)=t*h+∑i,bitIndex (bits i) := by
-  simp only [pairedSideCount_false,Finset.sum_add_distrib,Finset.sum_const,
-    Finset.card_univ,Fintype.card_fin,smul_eq_mul]
-
-theorem pairedSideCount_sum_total {t h : ℕ} (bits : Fin t → Bool) :
-    (∑i,pairedSideCount h (bits i) true)+(∑i,bitIndex (bits i))=t*(h+1) := by
-  rw [←Finset.sum_add_distrib]
-  calc
-    _ = ∑_i : Fin t,(h+1) := by
-      apply Finset.sum_congr rfl
-      intro i _
-      cases bits i <;> simp [pairedSideCount,bitIndex]
-    _ = _ := by simp
-
-theorem pairedSideCount_sum_post {t h : ℕ} (bits : Fin t → Bool) :
-    (∑i,pairedSideCount h (bits i) true)=t*(h+1)-∑i,bitIndex (bits i) := by
-  have := pairedSideCount_sum_total (h:=h) bits
-  omega
-
-/-- Global trial positions are ID functions, not positions inferred from a
- realized output or orientation. -/
-def sourceGlobalPre (N t : ℕ) (k : Fin ((N/4)*t)) : Fin N :=
-  ⟨((globalTrialEquiv N t).symm k).1.val,by
-    have := ((globalTrialEquiv N t).symm k).1.isLt; omega⟩
-
-def sourceGlobalPost (N t : ℕ) (k : Fin ((N/4)*t)) : Fin N :=
-  ⟨N-1-((globalTrialEquiv N t).symm k).1.val,by
-    have := ((globalTrialEquiv N t).symm k).1.isLt; omega⟩
-
-theorem sum_globalSides (N t h : ℕ) (bits : Fin ((N/4)*t) → Bool)
-    (side : Bool) (pos : Fin (N/4) → Fin N) (u : Fin N) :
-    (∑k : Fin ((N/4)*t), if pos ((globalTrialEquiv N t).symm k).1=u then
-      pairedSideCount h (bits k) side else 0)=
-    ∑v : Fin (N/4),if pos v=u then
-      (∑a : Fin t,pairedSideCount h (bits (globalTrialIndex N t v a)) side) else 0 := by
-  rw [←(globalTrialEquiv N t).sum_comp,Fintype.sum_prod_type]
-  apply Finset.sum_congr rfl
-  intro v _
-  simp only [Equiv.symm_apply_apply]
-  split_ifs with hv
-  · rfl
-  · simp only [Finset.sum_const_zero]
-
-theorem sourcePreCount_slots (N A t h : ℕ) (loc : Fin N → Fin t → Bool)
-    (glob : Fin ((N/4)*t) → Bool) (u : Fin N) :
-    sourcePreCount N A t h loc glob u=
-      (∑k : Fin ((N/4)*t),if sourceGlobalPre N t k=u then pairedSideCount h (glob k) false else 0)+
-      sourcePreFiller N A t h u+(∑i,pairedSideCount h (loc u i) false) := by
-  rw [pairedSideCount_sum_pre]
-  change sourcePreFiller N A t h u+(∑v : SourceBundle N,
-    if sourceBundleLeft N v=u then sourceBundleCount N t h loc glob v else 0)=_
-  rw [Fintype.sum_sum_type]
-  have hl : (∑v : Fin N,if sourceBundleLeft N (.inl v)=u then
-      sourceBundleCount N t h loc glob (.inl v) else 0)=t*h+∑i,bitIndex (loc u i) := by
-    change (∑v : Fin N,if v=u then (t*h+(∑i,bitIndex (loc v i))) else 0)=_
-    simp
-  rw [hl]
-  have hg := sum_globalSides N t h glob false
-    (fun v => ⟨v.val,by have := v.isLt; omega⟩) u
-  simp only [pairedSideCount_sum_pre] at hg
-  change (∑k,if sourceGlobalPre N t k=u then pairedSideCount h (glob k) false else 0)=_ at hg
-  rw [hg]
-  change _+(_+_) = _+_+_
-  have he : (∑v : Fin (N/4),if sourceBundleLeft N (.inr v)=u then
-      sourceBundleCount N t h loc glob (.inr v) else 0)=
-      ∑v : Fin (N/4),if (⟨v.val,by have := v.isLt; omega⟩ : Fin N)=u then
-        (t*h+∑i,bitIndex (glob (globalTrialIndex N t v i))) else 0 := by
-    apply Finset.sum_congr rfl
-    intro v _
-    rfl
-  rw [he]
-  omega
-
-theorem sourcePostCount_slots (N B t h : ℕ) (ht : Even t) (loc : Fin N → Fin t → Bool)
-    (glob : Fin ((N/4)*t) → Bool) (u : Fin N) :
-    sourcePostCount N B t h loc glob u=
-      (∑i,pairedSideCount h (loc u i) true)+
-      (∑k : Fin ((N/4)*t),if sourceGlobalPost N t k=u then pairedSideCount h (glob k) true else 0)+
-      sourcePostFiller N B t h u := by
-  have hh : 2*(t/2)=t := Nat.mul_div_cancel' ((even_iff_two_dvd).mp ht)
-  have hc (bits : Fin t → Bool) :
-      2*sourceLayoutReserve t h-(t*h+∑i,bitIndex (bits i))=
-        ∑i,pairedSideCount h (bits i) true := by
-    rw [pairedSideCount_sum_post]
-    dsimp only [sourceLayoutReserve]
-    have hs := bitIndex_sum_le bits
-    have he : 2*(t*h+t/2)=t*h+t*(h+1) := by
-      rw [Nat.mul_add,Nat.mul_add,Nat.mul_one,hh]
-      omega
-    rw [he]
-    omega
-  change sourcePostFiller N B t h u+(∑v : SourceBundle N,
-    if sourceBundleRight N v=u then 2*sourceLayoutReserve t h-sourceBundleCount N t h loc glob v else 0)=_
-  rw [Fintype.sum_sum_type]
-  have hl : (∑v : Fin N,if sourceBundleRight N (.inl v)=u then
-      2*sourceLayoutReserve t h-sourceBundleCount N t h loc glob (.inl v) else 0)=
-      ∑i,pairedSideCount h (loc u i) true := by
-    simp only [sourceBundleRight,sourceBundleCount,Finset.sum_ite_eq',Finset.mem_univ,ite_true,hc]
-  rw [hl]
-  have hg := sum_globalSides N t h glob true
-    (fun v => ⟨N-1-v.val,by have := v.isLt; omega⟩) u
-  change (∑k,if sourceGlobalPost N t k=u then pairedSideCount h (glob k) true else 0)=_ at hg
-  rw [hg]
-  simp only [sourceBundleRight,sourceBundleCount,hc]
-  omega
-
-/-- Flattening selected equal-atom chunks sums exactly their genuine lengths. -/
-theorem filter_chunks_flatten {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (l : List ι) (p : ι → Bool) (c : ι → ℕ) (w : List Bool) :
-    ((l.filter p).map (fun i => pureAtomRun w (c i))).flatten=
-      pureAtomRun w ((l.map (fun i => if p i then c i else 0)).sum) := by
-  calc
-    _ = pureAtomRun w (((l.filter p).map c).sum) := by
-      simpa only [List.map_map,Function.comp_def] using pureAtomRun_partition w ((l.filter p).map c)
-    _ = _ := by
-      congr 1
-      induction l with
-      | nil => rfl
-      | cons i l ih => cases hp : p i <;> simp [hp,ih]
-
-end TraceReconstruction
-
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-def sourceScheduledPre (T n : ℕ → ℕ) :
-    (b : ℕ) → Fin (sourceScheduledTrials T n b) → Fin (T b)
-  | 0,k => Fin.elim0 k
-  | b+1,k => sourceGlobalPre (T (b+1)) (n b) k
-
-def sourceScheduledPost (T n : ℕ → ℕ) :
-    (b : ℕ) → Fin (sourceScheduledTrials T n b) → Fin (T b)
-  | 0,k => Fin.elim0 k
-  | b+1,k => sourceGlobalPost (T (b+1)) (n b) k
-
-/-- Two actual deterministic pure binary side words for each paired trial ID. -/
-def sourceTrialWord (s : ℕ) (P T n h : ℕ → ℕ) (atom : Bool → List Bool) :
-    (b : ℕ) → WindowTrial T n (sourceScheduledTrials T n) b → Bool → List Bool
-  | 0,i,_ => PEmpty.elim i
-  | b+1,.inl _,c => pureAtomRun (sourceBlockWord s P atom b false) (h b+bitIndex c)
-  | 1,.inr (.inl k),_ => Fin.elim0 k
-  | b+2,.inr (.inl _),c => pureAtomRun (sourceBlockWord s P atom b false) (h b+bitIndex c)
-  | b+1,.inr (.inr (_,i)),c => sourceTrialWord s P T n h atom b i c
-
-/-- Fixed pure fillers, including the deterministic empty bottom fillers. -/
-def sourceFillerWord (s : ℕ) (P T n h : ℕ → ℕ) (atom : Bool → List Bool) :
-    (b : ℕ) → WindowFiller T b → List Bool
-  | 0,i => PEmpty.elim i
-  | b+1,.inl (u,p) => match b,u with
-    | 0,_ => []
-    | b+1,u => pureAtomRun (sourceBlockWord s P atom b false)
-        (if p then sourcePostFiller (T (b+1)) (sourceNominalPost P T b) (n b) (h b) u
-          else sourcePreFiller (T (b+1)) (sourceNominalPre P T b) (n b) (h b) u)
-  | b+1,.inr (_,i) => sourceFillerWord s P T n h atom b i
-
-def sourceSlotWord (s : ℕ) (P T n h : ℕ → ℕ) (center : ℕ → ℤ) (atom : Bool → List Bool)
-    (b : ℕ) (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool b)
-    (incoming : ℤ) (τ : Bool) : TreeEmission T n (sourceScheduledTrials T n) b → List Bool
-  | .inl (i,side) => sourceTrialWord s P T n h atom b i
-      (Bool.xor (windowAssignmentEquiv T n (sourceScheduledTrials T n) Bool b bits i) side)
-  | .inr (.inl i) => sourceFillerWord s P T n h atom b i
-  | .inr (.inr v) => atom (sourceTreeLeafBool T n s center b bits incoming τ v)
-
-def sourceChronologicalWord (s : ℕ) (P T n h : ℕ → ℕ) (center : ℕ → ℤ) (atom : Bool → List Bool)
-    (b : ℕ) (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool b)
-    (incoming : ℤ) (τ : Bool) : List Bool :=
-  ((treeChronology T n (sourceScheduledTrials T n) (sourceScheduledPre T n) (sourceScheduledPost T n) b).map
-    (sourceSlotWord s P T n h center atom b bits incoming τ)).flatten
-
-theorem sourceSlotWord_up (s : ℕ) (P T n h : ℕ → ℕ) (center : ℕ → ℤ) (atom : Bool → List Bool)
-    (b : ℕ) (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool (b+1))
-    (incoming : ℤ) (τ : Bool) (u : Fin (T b)) (x : TreeEmission T n (sourceScheduledTrials T n) b) :
-    sourceSlotWord s P T n h center atom (b+1) bits incoming τ (TreeEmission.up T n (sourceScheduledTrials T n) b u x)=
-      sourceSlotWord s P T n h center atom b (bits.2.2 u) (sourceScheduledShift T n b bits.2.1 u)
-        (integerTemplateType (T b) s (sourceNodeAnchor T n center b bits incoming) τ u) x := by
-  rcases x with ⟨i,p⟩ | f | v <;> rfl
-
-theorem treeChronology_eval (T n g : ℕ → ℕ)
-    (pre post : (b : ℕ) → Fin (g b) → Fin (T b)) (b : ℕ)
-    (f : TreeEmission T n g (b+1) → List Bool) :
-    ((treeChronology T n g pre post (b+1)).map f).flatten=
-    ((treeLocalSlots T n g b false).map f).flatten ++
-      coordinateWord (fun u => ((treeGlobalSlots T n g b false (pre b) u).map f).flatten ++
-        f (.inr (.inl (.inl (u,false)))) ++
-        ((treeChronology T n g pre post b).map (fun x => f (TreeEmission.up T n g b u x))).flatten ++
-        ((treeGlobalSlots T n g b true (post b) u).map f).flatten ++
-        f (.inr (.inl (.inl (u,true))))) ++
-      ((treeLocalSlots T n g b true).map f).flatten := by
-  simp only [treeChronology,List.map_append,List.flatten_append,List.map_flatten,
-    List.map_ofFn,coordinateWord]
-  congr 2
-  rw [List.flatten_flatten,List.map_ofFn]
-  apply congrArg List.flatten
-  apply congrArg List.ofFn
-  funext u
-  simp only [Function.comp_apply,List.map_append,List.flatten_append,List.map_cons,List.map_nil,
-    List.flatten_cons,List.flatten_nil,List.append_nil,List.map_map]
-  rfl
-
-theorem sourceLocalSlots_word (s : ℕ) (P T n h : ℕ → ℕ) (center : ℕ → ℤ) (atom : Bool → List Bool)
-    (b : ℕ) (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool (b+1))
-    (incoming : ℤ) (τ side : Bool) :
-    ((treeLocalSlots T n (sourceScheduledTrials T n) b side).map
-      (sourceSlotWord s P T n h center atom (b+1) bits incoming τ)).flatten=
-      pureAtomRun (sourceBlockWord s P atom b false) (∑i,pairedSideCount (h b) (bits.1 i) side) := by
-  rcases bits with ⟨loc,glob,kids⟩
-  simp only [treeLocalSlots,List.map_ofFn]
-  change (List.ofFn (fun i : Fin (n b) => pureAtomRun (sourceBlockWord s P atom b false)
-    (pairedSideCount (h b) (loc i) side))).flatten=_
-  exact coordinateWord_pureAtomRun _ _
-
-theorem sourceGlobalSlots_word (s : ℕ) (P T n h : ℕ → ℕ) (center : ℕ → ℤ) (atom : Bool → List Bool)
-    (b : ℕ) (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool (b+2))
-    (incoming : ℤ) (τ side : Bool) (pos : Fin ((T (b+1)/4)*n b) → Fin (T (b+1))) (u : Fin (T (b+1))) :
-    ((treeGlobalSlots T n (sourceScheduledTrials T n) (b+1) side pos u).map
-      (sourceSlotWord s P T n h center atom (b+2) bits incoming τ)).flatten=
-      pureAtomRun (sourceBlockWord s P atom b false)
-        (∑k,if pos k=u then pairedSideCount (h b) (bits.2.1 k) side else 0) := by
-  rcases bits with ⟨loc,glob,kids⟩
-  change Fin ((T (b+1)/4)*n b) → Bool at glob
-  simp only [treeGlobalSlots,List.map_map]
-  change ((List.filter (fun k => decide (pos k=u)) (List.ofFn id)).map
-    (fun i => pureAtomRun (sourceBlockWord s P atom b false) (pairedSideCount (h b) (glob i) side))).flatten=_
-  rw [filter_chunks_flatten,List.map_ofFn,List.sum_ofFn]
-  simp only [Function.comp_apply,id_eq,decide_eq_true_eq]
-
-end TraceReconstruction
-
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-def sourceFramedWord (s : ℕ) (P T n h : ℕ → ℕ) (center : ℕ → ℤ) (atom : Bool → List Bool) :
-    (b : ℕ) → WindowAssignments T n (sourceScheduledTrials T n) Bool b → ℤ → Bool → List Bool
-  | 0,_,_,τ => atom τ
-  | b+1,bits,incoming,τ =>
-      pureAtomRun (sourceBlockWord s P atom b false) (∑i,pairedSideCount (h b) (bits.1 i) false) ++
-      sourceTreeWord s P T n h center atom (b+1) bits incoming τ ++
-      pureAtomRun (sourceBlockWord s P atom b false) (∑i,pairedSideCount (h b) (bits.1 i) true)
-
-theorem sourceChronologicalWord_eq_framed (s : ℕ) (P T n h : ℕ → ℕ) (center : ℕ → ℤ)
-    (atom : Bool → List Bool) (b : ℕ) (hev : ∀j,j+1<b → Even (n j))
-    (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool b) (incoming : ℤ) (τ : Bool) :
-    sourceChronologicalWord s P T n h center atom b bits incoming τ=
-      sourceFramedWord s P T n h center atom b bits incoming τ := by
-  induction b generalizing incoming τ with
-  | zero => simp [sourceChronologicalWord,treeChronology,sourceSlotWord,sourceTreeLeafBool,sourceFramedWord]
-  | succ b ih =>
-    rcases bits with ⟨loc,glob,kids⟩
-    unfold sourceChronologicalWord
-    rw [treeChronology_eval]
-    erw [sourceLocalSlots_word,sourceLocalSlots_word]
-    change _=pureAtomRun (sourceBlockWord s P atom b false) (∑i,pairedSideCount (h b) (loc i) false) ++
-      sourceTreeWord s P T n h center atom (b+1) (loc,glob,kids) incoming τ ++
-      pureAtomRun (sourceBlockWord s P atom b false) (∑i,pairedSideCount (h b) (loc i) true)
-    apply congrArg (fun w => pureAtomRun (sourceBlockWord s P atom b false)
-      (∑i,pairedSideCount (h b) (loc i) false) ++ w ++
-      pureAtomRun (sourceBlockWord s P atom b false) (∑i,pairedSideCount (h b) (loc i) true))
-    have child (u : Fin (T b)) :
-        ((treeChronology T n (sourceScheduledTrials T n) (sourceScheduledPre T n) (sourceScheduledPost T n) b).map
-          (fun x => sourceSlotWord s P T n h center atom (b+1) (loc,glob,kids) incoming τ
-            (TreeEmission.up T n (sourceScheduledTrials T n) b u x))).flatten=
-        sourceFramedWord s P T n h center atom b (kids u) (sourceScheduledShift T n b glob u)
-          (integerTemplateType (T b) s (sourceNodeAnchor T n center b (loc,glob,kids) incoming) τ u) := by
-      have he : (fun x : TreeEmission T n (sourceScheduledTrials T n) b =>
-          sourceSlotWord s P T n h center atom (b+1) (loc,glob,kids) incoming τ
-            (TreeEmission.up T n (sourceScheduledTrials T n) b u x)) =
-          sourceSlotWord s P T n h center atom b (kids u) (sourceScheduledShift T n b glob u)
-            (integerTemplateType (T b) s (sourceNodeAnchor T n center b (loc,glob,kids) incoming) τ u) := by
-        funext x
-        exact sourceSlotWord_up s P T n h center atom b (loc,glob,kids) incoming τ u x
-      erw [he]
-      exact ih (fun j hj => hev j (by omega)) (kids u) _ _
-    cases b with
-    | zero =>
-      change coordinateWord _=atomString (T 0) atom _
-      apply congrArg coordinateWord
-      funext u
-      rw [child]
-      simp only [treeGlobalSlots,sourceScheduledTrials,List.ofFn_zero,List.filter_nil,List.map_nil,
-        List.flatten_nil,List.nil_append,List.append_nil,sourceSlotWord,sourceFillerWord,sourceFramedWord]
-    | succ b =>
-      change coordinateWord _=coordinateWord _
-      apply congrArg coordinateWord
-      funext u
-      erw [sourceGlobalSlots_word,sourceGlobalSlots_word,child]
-      change _=pureAtomRun (sourceBlockWord s P atom b false)
-          (sourcePreCount (T (b+1)) (sourceNominalPre P T b) (n b) (h b) (fun v => (kids v).1) glob u) ++
-        sourceTreeWord s P T n h center atom (b+1) (kids u) (sourceScheduledShift T n (b+1) glob u)
-          (integerTemplateType (T (b+1)) s (sourceNodeAnchor T n center (b+1) (loc,glob,kids) incoming) τ u) ++
-        pureAtomRun (sourceBlockWord s P atom b false)
-          (sourcePostCount (T (b+1)) (sourceNominalPost P T b) (n b) (h b) (fun v => (kids v).1) glob u)
-      erw [sourcePreCount_slots,sourcePostCount_slots _ _ _ _ (hev b (by omega))]
-      change pureAtomRun _ _ ++ pureAtomRun _ _ ++
-        (pureAtomRun _ _ ++ sourceTreeWord s P T n h center atom (b+1) (kids u)
-          (sourceScheduledShift T n (b+1) glob u)
-          (integerTemplateType (T (b+1)) s (sourceNodeAnchor T n center (b+1) (loc,glob,kids) incoming) τ u) ++
-          pureAtomRun _ _) ++ pureAtomRun _ _ ++ pureAtomRun _ _ = _
-      simp only [sourceScheduledPre,sourceScheduledPost,pureAtomRun_add,List.append_assoc,
-        Bool.false_eq_true,ite_false,ite_true]
-      rfl
-
-/-- Chronological side/filler/leaf slots produce the SAME full source word
- on the joint fit event. No orientation is removed or conditioned away. -/
-theorem sourceChronologicalWord_eq_block (s : ℕ) (P T n h : ℕ → ℕ) (atom : Bool → List Bool)
-    (b : ℕ) (H : SourceLayoutMargins s P T n h b)
-    (hP : P b=n b*(2*h b+1)+T b) (hT : Even (T b))
-    (hlarge : 4*(n b+1)≤T b) (hmotif : 8*2^s≤T b)
-    (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool (b+1)) (τ : Bool)
-    (hfit : SourceTreeFit T n (b+1) bits) :
-    sourceChronologicalWord s P T n h (sourceLocalCenter T n) atom (b+1) bits 0 τ=
-      sourceBlockWord s P atom (b+1) τ := by
-  rw [sourceChronologicalWord_eq_framed s P T n h (sourceLocalCenter T n) atom (b+1)
-    (fun j hj => H.even_trials j (by omega))]
-  have hf := sourceFullWord_eq_block s P T n h atom b H (by omega) hP hT hlarge hmotif bits τ hfit
-  change sourceFullWord s P T n h atom b bits τ=_ at hf
-  simpa only [sourceFramedWord,sourceFullWord,pairedSideCount_sum_pre,pairedSideCount_sum_post] using hf
-
-end TraceReconstruction
-
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-def finEnumeration (α : Type) [Fintype α] : Fin (Fintype.card α) ≃ α := (Fintype.equivFin α).symm
-
-abbrev SourceTrialCount (T n : ℕ → ℕ) (b : ℕ) := Fintype.card (WindowTrial T n (sourceScheduledTrials T n) b)
-abbrev SourceFillerCount (T : ℕ → ℕ) (b : ℕ) := Fintype.card (WindowFiller T b)
-abbrev SourceLeafCount (T : ℕ → ℕ) (b : ℕ) := Fintype.card (WindowLeaf T b)
-
-def sourceEmissionEnum (T n : ℕ → ℕ) (b : ℕ) :
-    EmissionSlot (SourceTrialCount T n b) (SourceFillerCount T b) (SourceLeafCount T b) ≃
-      TreeEmission T n (sourceScheduledTrials T n) b :=
-  Equiv.sumCongr (Equiv.prodCongr (finEnumeration _) (Equiv.refl Bool))
-    (Equiv.sumCongr (finEnumeration _) (finEnumeration _))
-
-def sourceAssignmentCode (T n : ℕ → ℕ) (A : Type) (b : ℕ) :
-    WindowAssignments T n (sourceScheduledTrials T n) A b ≃ (Fin (SourceTrialCount T n b) → A) where
-  toFun bits j := windowAssignmentEquiv T n (sourceScheduledTrials T n) A b bits (finEnumeration _ j)
-  invFun bits := (windowAssignmentEquiv T n (sourceScheduledTrials T n) A b).symm
-    (fun i => bits ((finEnumeration _).symm i))
-  left_inv bits := by simp only [Equiv.apply_symm_apply,Equiv.symm_apply_apply]
-  right_inv bits := by
-    funext j
-    simp only [Equiv.apply_symm_apply,Equiv.symm_apply_apply]
-
-theorem sourceAssignmentCode_eval (T n : ℕ → ℕ) (A : Type) (b : ℕ)
-    (bits : Fin (SourceTrialCount T n b) → A) (j : Fin (SourceTrialCount T n b)) :
-    windowAssignmentEquiv T n (sourceScheduledTrials T n) A b
-      ((sourceAssignmentCode T n A b).symm bits) (finEnumeration _ j)=bits j := by
-  change (sourceAssignmentCode T n A b ((sourceAssignmentCode T n A b).symm bits)) j=bits j
-  rw [Equiv.apply_symm_apply]
-
-def sourceChronologyCode (T n : ℕ → ℕ) (b : ℕ) :
-    Fin (treeChronology T n (sourceScheduledTrials T n) (sourceScheduledPre T n) (sourceScheduledPost T n) b).length ≃
-      EmissionSlot (SourceTrialCount T n b) (SourceFillerCount T b) (SourceLeafCount T b) :=
-  (treeChronologyEquiv T n (sourceScheduledTrials T n) (sourceScheduledPre T n) (sourceScheduledPost T n) b).trans
-    (sourceEmissionEnum T n b).symm
-
-def sourceFlatLeafType (T n : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ) (b : ℕ) (incoming : ℤ) (τ : Bool)
-    (bits : Fin (SourceTrialCount T n b) → Bool) (j : Fin (SourceLeafCount T b)) : Bool :=
-  sourceTreeLeafBool T n s center b ((sourceAssignmentCode T n Bool b).symm bits) incoming τ (finEnumeration _ j)
-
-def sourceFlatTrialWord (s : ℕ) (P T n h : ℕ → ℕ) (atom : Bool → List Bool) (b : ℕ)
-    (c : Bool) (j : Fin (SourceTrialCount T n b)) : List Bool :=
-  sourceTrialWord s P T n h atom b (finEnumeration _ j) c
-
-def sourceFlatFillerWord (s : ℕ) (P T n h : ℕ → ℕ) (atom : Bool → List Bool) (b : ℕ)
-    (j : Fin (SourceFillerCount T b)) : List Bool :=
-  sourceFillerWord s P T n h atom b (finEnumeration _ j)
-
-theorem sourceSlotWord_code (s : ℕ) (P T n h : ℕ → ℕ) (center : ℕ → ℤ) (atom : Bool → List Bool)
-    (b : ℕ) (bits : Fin (SourceTrialCount T n b) → Bool) (incoming : ℤ) (τ : Bool)
-    (x : EmissionSlot (SourceTrialCount T n b) (SourceFillerCount T b) (SourceLeafCount T b)) :
-    sourceSlotWord s P T n h center atom b ((sourceAssignmentCode T n Bool b).symm bits) incoming τ
-        (sourceEmissionEnum T n b x)=
-      PairedMechanisms.slotWord (sourceFlatTrialWord s P T n h atom b false)
-        (sourceFlatTrialWord s P T n h atom b true) (sourceFlatFillerWord s P T n h atom b) atom bits
-        (sourceFlatLeafType T n s center b incoming τ bits) x := by
-  rcases x with ⟨j,p⟩ | f | v
-  · cases p <;>
-      change sourceTrialWord s P T n h atom b (finEnumeration _ j)
-        (Bool.xor (windowAssignmentEquiv T n (sourceScheduledTrials T n) Bool b
-          ((sourceAssignmentCode T n Bool b).symm bits) (finEnumeration _ j)) _) = _ <;>
-      rw [sourceAssignmentCode_eval] <;> cases he : bits j <;> simp [PairedMechanisms.slotWord,he,sourceFlatTrialWord]
-  · rfl
-  · rfl
-
-theorem treeChronologyEquiv_word (T n g : ℕ → ℕ)
-    (pre post : (b : ℕ) → Fin (g b) → Fin (T b)) (b : ℕ)
-    (f : TreeEmission T n g b → List Bool) :
-    coordinateWord (fun j => f (treeChronologyEquiv T n g pre post b j))=
-      ((treeChronology T n g pre post b).map f).flatten := by
-  unfold coordinateWord treeChronologyEquiv
-  change (List.ofFn (fun j => f ((treeChronology T n g pre post b).get j))).flatten=_
-  have hh : List.map f (List.ofFn (treeChronology T n g pre post b).get) =
-      List.ofFn (f ∘ (treeChronology T n g pre post b).get) := List.map_ofFn
-  rw [List.ofFn_get] at hh
-  exact congrArg List.flatten hh.symm
-
-theorem source_slotCode_word (s : ℕ) (P T n h : ℕ → ℕ) (center : ℕ → ℤ) (atom : Bool → List Bool)
-    (b : ℕ) (bits : Fin (SourceTrialCount T n b) → Bool) (incoming : ℤ) (τ : Bool) :
-    coordinateWord (fun j => PairedMechanisms.slotWord (sourceFlatTrialWord s P T n h atom b false)
-      (sourceFlatTrialWord s P T n h atom b true) (sourceFlatFillerWord s P T n h atom b) atom bits
-      (sourceFlatLeafType T n s center b incoming τ bits) (sourceChronologyCode T n b j))=
-    sourceChronologicalWord s P T n h center atom b ((sourceAssignmentCode T n Bool b).symm bits) incoming τ := by
-  simp_rw [←sourceSlotWord_code]
-  simp only [sourceChronologyCode,Equiv.trans_apply,Equiv.apply_symm_apply]
-  exact treeChronologyEquiv_word T n (sourceScheduledTrials T n) (sourceScheduledPre T n)
-    (sourceScheduledPost T n) b _
-
-end TraceReconstruction
-
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- Reindex the genuine independent orientations, retaining the whole source tree. -/
-theorem source_fair_code_expect (T n : ℕ → ℕ) (b : ℕ)
-    (F : WindowAssignments T n (sourceScheduledTrials T n) Bool b → ℝ) :
-    (PairedMechanisms.fairOrientations (SourceTrialCount T n b)).expect
-      (fun bits => F ((sourceAssignmentCode T n Bool b).symm bits)) =
-    (windowPosteriorLaw T n (sourceScheduledTrials T n) b
-      (noReveals T n (sourceScheduledTrials T n) b)).expect F := by
-  rw [windowPosterior_fair_flat,FiniteLaw.indexedProduct_expect_equiv _ (finEnumeration _)]
-  simp only [PairedMechanisms.fairOrientations,FiniteLaw.expect,FiniteLaw.product,
-    FiniteLaw.indexedProduct,sourceAssignmentCode,Equiv.coe_fn_symm_mk]
-  apply Finset.sum_congr (by ext; simp)
-  intro bits _
-  rfl
-
-/-- The fit event uses only interior even bundles. In particular no parity
- assumption is made on the top root count J. -/
-theorem sourceTreeFit_tail_interior (T n : ℕ → ℕ) (b : ℕ)
-    (hn : ∀j,j+1<b → 0<n j) (he : ∀j,j+1<b → Even (n j)) :
-    (windowPosteriorLaw T n (sourceScheduledTrials T n) b
-      (noReveals T n (sourceScheduledTrials T n) b)).expect
-        (fun bits => if SourceTreeFit T n b bits then 0 else 1) ≤ sourceFitTail T n b := by
-  induction b using Nat.twoStepInduction with
-  | zero =>
-    have hh (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool 0) : SourceTreeFit T n 0 bits := trivial
-    simp only [ite_eq_left (hh _),FiniteLaw.expect_const,sourceFitTail,le_refl]
-  | one =>
-    have hh (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool 1) : SourceTreeFit T n 1 bits := trivial
-    simp only [ite_eq_left (hh _),FiniteLaw.expect_const,sourceFitTail,le_refl]
-  | more b ih0 ih =>
-    let P := windowPosteriorLaw T n (sourceScheduledTrials T n) (b+2)
-      (noReveals T n (sourceScheduledTrials T n) (b+2))
-    let badShift : WindowAssignments T n (sourceScheduledTrials T n) Bool (b+2) → Fin (T (b+1)) → Prop :=
-      fun bits u => (T b:ℝ)/8≤|(sourceDisplacement (T (b+1)) (n b)
-        (fun v => (bits.2.2 v).1) bits.2.1 u:ℝ)|
-    have hp (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool (b+2)) :
-        (if SourceTreeFit T n (b+2) bits then (0:ℝ) else 1)≤
-          (∑u : Fin (T (b+1)),if badShift bits u then 1 else 0)+
-          (∑u : Fin (T (b+1)),if SourceTreeFit T n (b+1) (bits.2.2 u) then 0 else 1) := by
-      let p := fun u : Fin (T (b+1)) => |8*sourceDisplacement (T (b+1)) (n b)
-        (fun v => (bits.2.2 v).1) bits.2.1 u|≤(T b:ℤ)
-      have hh : (if SourceTreeFit T n (b+2) bits then (0:ℝ) else 1)≤
-          (if ∀u,p u then 0 else 1)+(if ∀u,SourceTreeFit T n (b+1) (bits.2.2 u) then 0 else 1) := by
-        by_cases H : SourceTreeFit T n (b+2) bits
-        · rw [ite_eq_left H]
-          split_ifs <;> norm_num
-        · rw [ite_eq_right H]
-          by_cases h1 : ∀u,p u
-          · have h2 : ¬∀u,SourceTreeFit T n (b+1) (bits.2.2 u) := fun h2 => H ⟨h1,h2⟩
-            rw [ite_eq_left h1,ite_eq_right h2]
-            norm_num
-          · rw [ite_eq_right h1]
-            split_ifs <;> norm_num
-      apply hh.trans
-      apply add_le_add
-      · apply (failure_forall_le_sum p).trans
-        apply Finset.sum_le_sum
-        intro u _
-        by_cases hu : p u
-        · simp only [ite_eq_left hu]
-          split_ifs <;> norm_num
-        · have hr : (T b:ℝ)<8*|(sourceDisplacement (T (b+1)) (n b)
-              (fun v => (bits.2.2 v).1) bits.2.1 u:ℝ)| := by
-            have hi : (T b:ℤ) < |8*sourceDisplacement (T (b+1)) (n b)
-              (fun v => (bits.2.2 v).1) bits.2.1 u| := lt_of_not_ge hu
-            have hi' : (T b:ℝ) < |(8*sourceDisplacement (T (b+1)) (n b)
-              (fun v => (bits.2.2 v).1) bits.2.1 u:ℤ)| := by exact_mod_cast hi
-            norm_num only [Int.cast_abs,Int.cast_mul,Int.cast_ofNat,abs_mul,abs_of_pos (by norm_num : (0:ℝ)<8)] at hi'
-            exact hi'
-          have hb : badShift bits u := by dsimp [badShift]; linarith
-          simp [hu,hb]
-      · exact failure_forall_le_sum _
-    calc
-      _ ≤ P.expect (fun bits => (∑u : Fin (T (b+1)),if badShift bits u then 1 else 0)+
-          (∑u : Fin (T (b+1)),if SourceTreeFit T n (b+1) (bits.2.2 u) then 0 else 1)) :=
-            P.expect_mono hp
-      _ = (∑u : Fin (T (b+1)),P.expect (fun bits => if badShift bits u then 1 else 0))+
-          (∑u : Fin (T (b+1)),P.expect (fun bits => if SourceTreeFit T n (b+1) (bits.2.2 u) then 0 else 1)) := by
-            rw [FiniteLaw.expect_add,FiniteLaw.expect_fintype_sum,FiniteLaw.expect_fintype_sum]
-      _ ≤ (∑_u : Fin (T (b+1)),2*Real.exp (-2*((T b:ℝ)/8)^2/(((T (b+1):ℝ)+1)*(n b:ℝ))))+
-          ∑_u : Fin (T (b+1)),sourceFitTail T n (b+1) := by
-            apply add_le_add
-            · apply Finset.sum_le_sum
-              intro u _
-              exact sourceDisplacement_tail T n b (he b (by omega)) (hn b (by omega)) u (by positivity)
-            · apply Finset.sum_le_sum
-              intro u _
-              dsimp only [P]
-              rw [windowPosterior_child T n (sourceScheduledTrials T n) (b+1)
-                (noReveals T n (sourceScheduledTrials T n) (b+2)) u
-                (fun x => if SourceTreeFit T n (b+1) x then 0 else 1)]
-              rw [noReveals_succ]
-              exact ih (fun j hj => hn j (by omega)) (fun j hj => he j (by omega))
-      _ = sourceFitTail T n (b+2) := by simp [sourceFitTail,mul_add]
-
-
-/-- Honest source joint-fit failure after the actual finite coding. -/
-theorem source_flat_fit_tail (T n : ℕ → ℕ) (b : ℕ)
-    (hn : ∀j,j+1<b → 0<n j) (he : ∀j,j+1<b → Even (n j)) :
-    (PairedMechanisms.fairOrientations (SourceTrialCount T n b)).expect
-      (fun bits => if SourceTreeFit T n b ((sourceAssignmentCode T n Bool b).symm bits) then 0 else 1)
-      ≤ sourceFitTail T n b := by
-  rw [source_fair_code_expect T n b (fun bits => if SourceTreeFit T n b bits then 0 else 1)]
-  exact sourceTreeFit_tail_interior T n b hn he
-
-/-- A fixed public decoder. It does not know the hidden orientations or the
- input hypothesis, and it drops rather than reveals all pairing flags. -/
-def sourceOracleDecode {Side Leaf : Type} (T n : ℕ → ℕ) (b : ℕ)
-    (f : Side → List Bool) (g : Leaf → List Bool) :
-    ((Reports Side (SourceTrialCount T n b) × (Fin (SourceLeafCount T b) → Leaf)) ×
-      (Fin (SourceFillerCount T b) → Side)) → List Bool := fun x =>
-  coordinateWord (fun j => PairedMechanisms.slotDecode f g (sourceChronologyCode T n b j)
-    (emissionAssignmentEquiv Side Leaf (SourceTrialCount T n b) (SourceFillerCount T b)
-      (SourceLeafCount T b) (fun i => (x.1.1 i).2,x.2,x.1.2) (sourceChronologyCode T n b j)))
-
-/-- The complete canonical oracle now simulates the LITERAL source block.
- All IDs, words, leaf types and chronology instantiate the actual construction.
- The only failed-layout cost is its original unconditional fair-bit mass. -/
-theorem source_canonical_simulation {Side Leaf : Type} [Fintype Side] [Fintype Leaf]
-    {q E : ℝ} (hq : q∈Set.Icc 0 1) (hE : 0≤E)
-    (s : ℕ) (P T n h : ℕ → ℕ) (atom : Bool → List Bool) (b : ℕ)
-    (H : SourceLayoutMargins s P T n h b)
-    (hP : P b=n b*(2*h b+1)+T b) (hT : Even (T b))
-    (hlarge : 4*(n b+1)≤T b) (hmotif : 8*2^s≤T b)
-    (hn : ∀j<b,0<n j)
-    (M : PairedMechanisms Side (SourceTrialCount T n (b+1)))
-    (R : Fin (SourceFillerCount T (b+1)) → FiniteLaw Side) (A B : FiniteLaw Leaf)
-    (f : Side → List Bool) (g : Leaf → List Bool)
-    (hU : ∀j,SimulatesTrace q (M.first j) f (sourceFlatTrialWord s P T n h atom (b+1) false j))
-    (hV : ∀j,SimulatesTrace q (M.second j) f (sourceFlatTrialWord s P T n h atom (b+1) true j))
-    (hW : ∀j,SimulatesTrace q (R j) f (sourceFlatFillerWord s P T n h atom (b+1) j))
-    (hA : ApproximatesTrace q A g (atom false) E) (hB : ApproximatesTrace q B g (atom true) E)
-    (τ : Bool) :
-    ApproximatesTrace q
-      ((M.canonicalOracle A B (sourceFlatLeafType T n s (sourceLocalCenter T n) (b+1) 0 τ)).prod
-        (FiniteLaw.coordinateProduct R))
-      (sourceOracleDecode T n (b+1) f g) (sourceBlockWord s P atom (b+1) τ)
-      ((SourceLeafCount T (b+1):ℝ)*E+sourceFitTail T n (b+1)) := by
-  let good := fun bits : Fin (SourceTrialCount T n (b+1)) → Bool =>
-    SourceTreeFit T n (b+1) ((sourceAssignmentCode T n Bool (b+1)).symm bits)
-  have hw (bits : Fin (SourceTrialCount T n (b+1)) → Bool) (hg : good bits) :=
-    sourceChronologicalWord_eq_block s P T n h atom b H hP hT hlarge hmotif
-      ((sourceAssignmentCode T n Bool (b+1)).symm bits) τ hg
-  have hs := M.canonical_filled_simulation hq hE (sourceChronologyCode T n (b+1)) R A B f g
-    (sourceFlatTrialWord s P T n h atom (b+1) false)
-    (sourceFlatTrialWord s P T n h atom (b+1) true)
-    (sourceFlatFillerWord s P T n h atom (b+1)) atom hU hV hW hA hB
-    (sourceFlatLeafType T n s (sourceLocalCenter T n) (b+1) 0 τ)
-    (sourceBlockWord s P atom (b+1) τ) good (fun bits hg => by
-      rw [source_slotCode_word]
-      exact hw bits hg)
-  intro F hF
-  exact (hs F hF).trans (add_le_add le_rfl (source_flat_fit_tail T n (b+1) (fun j hj => hn j (by omega))
-    (fun j hj => H.even_trials j (by omega))))
-
-end TraceReconstruction
-
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-
-end OAI
+import OAI.Probability.TraceReconstruction.Energy
+import OAI.Probability.TraceReconstruction.Metadata
+
+namespace OAI
+
+/-!
+# Simulation
+
+Canonical and filled simulations, central fit, root words, chronology, slots, finite codes, and complete-trace decoding.
+-/
+
+open scoped ENNReal
+noncomputable section
+
+open Filter
+open scoped Topology
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- Heterogeneous emissions retain their actual outcome type while moving to
+ the one fixed public chronological order. -/
+theorem dependentProduct_approximates {ι : Type*} [Fintype ι] [DecidableEq ι]
+    {A : ι → Type*} [∀i,Fintype (A i)] {k : ℕ} {q : ℝ} (hq : q∈Set.Icc 0 1)
+    (e : Fin k ≃ ι) (P : ∀i,FiniteLaw (A i)) (f : ∀i,A i → List Bool)
+    (w : ι → List Bool) (E : ι → ℝ)
+    (h : ∀i,ApproximatesTrace q (P i) (f i) (w i) (E i)) :
+    ApproximatesTrace q (FiniteLaw.dependentProduct P)
+      (fun x => coordinateWord (fun j => f (e j) (x (e j))))
+      (coordinateWord (fun j => w (e j))) (∑i,E i) := by
+  intro F hF
+  let G := Equiv.piCongrLeft A e
+  have he (x : ∀j,A (e j)) (j : Fin k) :
+      cast (congrArg A (e.apply_symm_apply (e j))) (x (e.symm (e j)))=x j := by
+    rw [←Equiv.piCongrLeft_apply_eq_cast]
+    exact Equiv.piCongrLeft_apply_apply A e x j
+  rw [FiniteLaw.dependentProduct_expect_equiv P e]
+  simp_rw [he]
+  have H := dependentCoordinate_approximates hq (fun j => P (e j))
+    (fun j => f (e j)) (fun j => w (e j)) (fun j => E (e j)) (fun j => h (e j)) F hF
+  rwa [e.sum_comp] at H
+
+/-- Each paired mechanism owns two distinct side slots; fillers and old leaf
+ oracles have their own IDs. This type is independent of every orientation. -/
+abbrev EmissionSlot (m k n : ℕ) := (Fin m × Bool) ⊕ (Fin k ⊕ Fin n)
+
+def EmissionOutcome (Side Leaf : Type) {m k n : ℕ} : EmissionSlot m k n → Type
+  | .inl _ => Side
+  | .inr (.inl _) => Side
+  | .inr (.inr _) => Leaf
+
+instance emissionOutcomeFintype {Side Leaf : Type} [Fintype Side] [Fintype Leaf]
+    {m k n : ℕ} (i : EmissionSlot m k n) : Fintype (EmissionOutcome Side Leaf i) := by
+  rcases i with a | a
+  · exact inferInstanceAs (Fintype Side)
+  · rcases a with a | a
+    · exact inferInstanceAs (Fintype Side)
+    · exact inferInstanceAs (Fintype Leaf)
+
+/-- Actual arrays and slot observations are bijective, not a lossy interface. -/
+def emissionAssignmentEquiv (Side Leaf : Type) (m k n : ℕ) :
+    ((Fin m → Side × Side) × (Fin k → Side) × (Fin n → Leaf)) ≃
+      (∀i : EmissionSlot m k n,EmissionOutcome Side Leaf i) where
+  toFun x i := match i with
+    | .inl (j,false) => (x.1 j).1
+    | .inl (j,true) => (x.1 j).2
+    | .inr (.inl j) => x.2.1 j
+    | .inr (.inr j) => x.2.2 j
+  invFun x := (fun j => (x (.inl (j,false)),x (.inl (j,true))),
+    fun j => x (.inr (.inl j)),fun j => x (.inr (.inr j)))
+  left_inv x := by
+    rcases x with ⟨z,f,l⟩
+    change (fun j => ((z j).1,(z j).2),f,l)=(z,f,l)
+    simp only [Prod.mk.eta]
+  right_inv x := by
+    funext i
+    rcases i with ⟨j,b⟩ | i
+    · cases b <;> rfl
+    · cases i <;> rfl
+
+namespace PairedMechanisms
+variable {Side Leaf : Type} [Fintype Side] [Fintype Leaf] {m k n : ℕ}
+variable (M : PairedMechanisms Side m)
+
+def slotLaw (R : Fin k → FiniteLaw Side) (P Q : FiniteLaw Leaf)
+    (bits : Fin m → Bool) (θ : Fin n → Bool) :
+    (i : EmissionSlot m k n) → FiniteLaw (EmissionOutcome Side Leaf i)
+  | .inl (j,false) => if bits j then M.second j else M.first j
+  | .inl (j,true) => if bits j then M.first j else M.second j
+  | .inr (.inl j) => R j
+  | .inr (.inr j) => if θ j then Q else P
+
+theorem slotLaw_mass (R : Fin k → FiniteLaw Side) (P Q : FiniteLaw Leaf)
+    (bits : Fin m → Bool) (θ : Fin n → Bool)
+    (x : (Fin m → Side × Side) × (Fin k → Side) × (Fin n → Leaf)) :
+    (FiniteLaw.dependentProduct (M.slotLaw R P Q bits θ)).mass
+      (emissionAssignmentEquiv Side Leaf m k n x)=
+    ((FiniteLaw.coordinateProduct (fun j => FiniteLaw.orientedProd (M.first j) (M.second j) (bits j))).prod
+      ((FiniteLaw.coordinateProduct R).prod (selectedLeafLaw P Q θ))).mass x := by
+  change (∏i : EmissionSlot m k n,_)=_
+  rw [Fintype.prod_sum_type,Fintype.prod_sum_type,Fintype.prod_prod_type]
+  simp only [Fintype.prod_bool,slotLaw,emissionAssignmentEquiv,Equiv.coe_fn_mk]
+  change (∏j : Fin m,_) * ((∏j : Fin k,_) * ∏j : Fin n,_)=
+    (∏j : Fin m,_) * ((∏j : Fin k,_) * ∏j : Fin n,_)
+  congr 1
+  apply Finset.prod_congr rfl
+  intro j _
+  cases hb : bits j <;> simp [FiniteLaw.orientedProd,hb,FiniteLaw.prod,mul_comm]
+
+theorem slotLaw_expect (R : Fin k → FiniteLaw Side) (P Q : FiniteLaw Leaf)
+    (bits : Fin m → Bool) (θ : Fin n → Bool)
+    (F : (∀i : EmissionSlot m k n,EmissionOutcome Side Leaf i) → ℝ) :
+    ((FiniteLaw.coordinateProduct (fun j => FiniteLaw.orientedProd (M.first j) (M.second j) (bits j))).prod
+      ((FiniteLaw.coordinateProduct R).prod (selectedLeafLaw P Q θ))).expect
+        (fun x => F (emissionAssignmentEquiv Side Leaf m k n x))=
+      (FiniteLaw.dependentProduct (M.slotLaw R P Q bits θ)).expect F := by
+  unfold FiniteLaw.expect
+  rw [←(emissionAssignmentEquiv Side Leaf m k n).sum_comp]
+  simp_rw [slotLaw_mass]
+
+/-- Forgetting only the visible flags recovers the genuine independent side,
+ filler and leaf laws at every orientation, before any fit restriction. -/
+theorem reportsGiven_slot_expect (R : Fin k → FiniteLaw Side) (P Q : FiniteLaw Leaf)
+    (bits : Fin m → Bool) (θ : Fin n → Bool)
+    (F : (∀i : EmissionSlot m k n,EmissionOutcome Side Leaf i) → ℝ) :
+    ((M.reportsGiven bits).prod ((FiniteLaw.coordinateProduct R).prod (selectedLeafLaw P Q θ))).expect
+      (fun x => F (emissionAssignmentEquiv Side Leaf m k n (fun j => (x.1 j).2,x.2)))=
+      (FiniteLaw.dependentProduct (M.slotLaw R P Q bits θ)).expect F := by
+  rw [FiniteLaw.expect_prod]
+  change (M.reportsGiven bits).expect ((fun z : Fin m → Side × Side =>
+    ((FiniteLaw.coordinateProduct R).prod (selectedLeafLaw P Q θ)).expect
+      (fun b => F (emissionAssignmentEquiv Side Leaf m k n (z,b)))) ∘
+        (fun z j => (z j).2))=_
+  rw [←FiniteLaw.expect_map]
+  rw [reportsGiven_forget]
+  have H := slotLaw_expect M R P Q bits θ F
+  rw [FiniteLaw.expect_prod] at H
+  exact H
+
+/-- The common, bit-blind projection simply forgets flags and concatenates in
+ the publicly fixed ID order. -/
+def slotDecode (f : Side → List Bool) (g : Leaf → List Bool) :
+    (i : EmissionSlot m k n) → EmissionOutcome Side Leaf i → List Bool
+  | .inl _,x => f x
+  | .inr (.inl _),x => f x
+  | .inr (.inr _),x => g x
+
+def slotWord (U V : Fin m → List Bool) (W : Fin k → List Bool)
+    (atom : Bool → List Bool) (bits : Fin m → Bool) (θ : Fin n → Bool) :
+    EmissionSlot m k n → List Bool
+  | .inl (j,false) => if bits j then V j else U j
+  | .inl (j,true) => if bits j then U j else V j
+  | .inr (.inl j) => W j
+  | .inr (.inr j) => atom (θ j)
+
+def slotCost (E : ℝ) : EmissionSlot m k n → ℝ
+  | .inl _ => 0
+  | .inr (.inl _) => 0
+  | .inr (.inr _) => E
+
+theorem slotCost_sum (E : ℝ) : (∑i : EmissionSlot m k n,slotCost E i)=(n:ℝ)*E := by
+  simp only [Fintype.sum_sum_type,slotCost,Finset.sum_const_zero,
+    zero_add,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]
+
+theorem slotLaw_simulation {q E : ℝ} (R : Fin k → FiniteLaw Side) (P Q : FiniteLaw Leaf)
+    (f : Side → List Bool) (g : Leaf → List Bool) (U V : Fin m → List Bool)
+    (W : Fin k → List Bool) (atom : Bool → List Bool)
+    (hU : ∀j,SimulatesTrace q (M.first j) f (U j))
+    (hV : ∀j,SimulatesTrace q (M.second j) f (V j))
+    (hW : ∀j,SimulatesTrace q (R j) f (W j))
+    (hP : ApproximatesTrace q P g (atom false) E) (hQ : ApproximatesTrace q Q g (atom true) E)
+    (bits : Fin m → Bool) (θ : Fin n → Bool) (i : EmissionSlot m k n) :
+    ApproximatesTrace q (M.slotLaw R P Q bits θ i) (slotDecode f g i)
+      (slotWord U V W atom bits θ i) (slotCost E i) := by
+  rcases i with ⟨j,b⟩ | i
+  · cases b <;> cases hb : bits j <;>
+      simp only [slotLaw,slotWord,slotCost,hb,Bool.false_eq_true,ite_false,ite_true]
+    · exact (hU j).approximate
+    · exact (hV j).approximate
+    · exact (hV j).approximate
+    · exact (hU j).approximate
+  · rcases i with j | j
+    · exact (hW j).approximate
+    · cases hb : θ j <;> simp only [slotLaw,slotWord,slotCost,hb,Bool.false_eq_true]
+      · exact hP
+      · exact hQ
+
+/-- Exact conditional-on-orientation simulation boundary. The orientation is
+ not part of the common projection, and no fit conditioning enters its law. -/
+theorem reportsGiven_simulation {q E : ℝ} (hq : q∈Set.Icc 0 1)
+    {L : ℕ} (e : Fin L ≃ EmissionSlot m k n)
+    (R : Fin k → FiniteLaw Side) (P Q : FiniteLaw Leaf)
+    (f : Side → List Bool) (g : Leaf → List Bool) (U V : Fin m → List Bool)
+    (W : Fin k → List Bool) (atom : Bool → List Bool)
+    (hU : ∀j,SimulatesTrace q (M.first j) f (U j))
+    (hV : ∀j,SimulatesTrace q (M.second j) f (V j))
+    (hW : ∀j,SimulatesTrace q (R j) f (W j))
+    (hP : ApproximatesTrace q P g (atom false) E) (hQ : ApproximatesTrace q Q g (atom true) E)
+    (bits : Fin m → Bool) (θ : Fin n → Bool) :
+    ApproximatesTrace q
+      ((M.reportsGiven bits).prod ((FiniteLaw.coordinateProduct R).prod (selectedLeafLaw P Q θ)))
+      (fun x => coordinateWord (fun j => slotDecode f g (e j)
+        (emissionAssignmentEquiv Side Leaf m k n (fun i => (x.1 i).2,x.2) (e j))))
+      (coordinateWord (fun j => slotWord U V W atom bits θ (e j))) ((n:ℝ)*E) := by
+  intro F hF
+  dsimp only
+  rw [M.reportsGiven_slot_expect R P Q bits θ
+    (fun z => F (coordinateWord (fun j => slotDecode f g (e j) (z (e j)))))]
+  have H := dependentProduct_approximates hq e (M.slotLaw R P Q bits θ) (slotDecode f g)
+    (slotWord U V W atom bits θ) (slotCost E)
+    (M.slotLaw_simulation R P Q f g U V W atom hU hV hW hP hQ bits θ) F hF
+  rwa [slotCost_sum] at H
+
+end PairedMechanisms
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+namespace PairedMechanisms
+variable {Side Leaf : Type} [Fintype Side] [Fintype Leaf] {m k n : ℕ}
+variable (M : PairedMechanisms Side m)
+
+/-- The actual metadata-first oracle with independent genuine fillers has
+ exactly the orientation-first product experiment, in the displayed order. -/
+theorem canonical_filled_expect (R : Fin k → FiniteLaw Side) (P Q : FiniteLaw Leaf)
+    (θ : (Fin m → Bool) → Fin n → Bool)
+    (F : Reports Side m × (Fin k → Side) × (Fin n → Leaf) → ℝ) :
+    ((M.canonicalOracle P Q θ).prod (FiniteLaw.coordinateProduct R)).expect
+      (fun x => F (x.1.1,x.2,x.1.2))=
+    (fairOrientations m).expect (fun bits =>
+      ((M.reportsGiven bits).prod ((FiniteLaw.coordinateProduct R).prod
+        (selectedLeafLaw P Q (θ bits)))).expect F) := by
+  rw [←M.oracle_factorization,orientationOracle,FiniteLaw.expect_prod,FiniteLaw.expect_mixture]
+  simp_rw [FiniteLaw.expect_prod]
+  congr 1
+  funext bits
+  congr 1
+  funext z
+  exact FiniteLaw.expect_comm _ _ (fun leaf fill => F (z,fill,leaf))
+
+/-- Unconditional common simulation of the actual full canonical oracle.
+ The only layout charge is the unconditional fair-orientation failure mass.
+ No orientation or side metadata is conditioned on successful fitting. -/
+theorem canonical_filled_simulation {q E : ℝ} (hq : q∈Set.Icc 0 1) (hE : 0≤E)
+    {L : ℕ} (e : Fin L ≃ EmissionSlot m k n)
+    (R : Fin k → FiniteLaw Side) (P Q : FiniteLaw Leaf)
+    (f : Side → List Bool) (g : Leaf → List Bool) (U V : Fin m → List Bool)
+    (W : Fin k → List Bool) (atom : Bool → List Bool)
+    (hU : ∀j,SimulatesTrace q (M.first j) f (U j))
+    (hV : ∀j,SimulatesTrace q (M.second j) f (V j))
+    (hW : ∀j,SimulatesTrace q (R j) f (W j))
+    (hP : ApproximatesTrace q P g (atom false) E) (hQ : ApproximatesTrace q Q g (atom true) E)
+    (θ : (Fin m → Bool) → Fin n → Bool) (w : List Bool)
+    (good : (Fin m → Bool) → Prop) [DecidablePred good]
+    (hword : ∀bits,good bits → coordinateWord
+      (fun j => slotWord U V W atom bits (θ bits) (e j))=w) :
+    ApproximatesTrace q ((M.canonicalOracle P Q θ).prod (FiniteLaw.coordinateProduct R))
+      (fun x => coordinateWord (fun j => slotDecode f g (e j)
+        (emissionAssignmentEquiv Side Leaf m k n (fun i => (x.1.1 i).2,x.2,x.1.2) (e j))))
+      w ((n:ℝ)*E+(fairOrientations m).expect (fun bits => if good bits then 0 else 1)) := by
+  intro F hF
+  dsimp only
+  rw [M.canonical_filled_expect R P Q θ (fun x => F (coordinateWord (fun j => slotDecode f g (e j)
+    (emissionAssignmentEquiv Side Leaf m k n (fun i => (x.1 i).2,x.2) (e j)))))]
+  rw [←FiniteLaw.expect_const (fairOrientations m) (listDeletionExpectation q w F),←FiniteLaw.expect_sub]
+  apply (FiniteLaw.abs_expect_le _ _).trans
+  calc
+    _ ≤ (fairOrientations m).expect (fun bits => (n:ℝ)*E+(if good bits then 0 else 1)) := by
+      apply FiniteLaw.expect_mono
+      intro bits
+      by_cases hb : good bits
+      · simp only [ite_eq_left hb,add_zero]
+        have H := M.reportsGiven_simulation hq e R P Q f g U V W atom hU hV hW hP hQ bits (θ bits) F hF
+        rwa [hword bits hb] at H
+      · simp only [ite_eq_right hb]
+        have hc := listDeletionExpectation_range hq w F hF
+        have hlo := FiniteLaw.expect_mono
+          ((M.reportsGiven bits).prod ((FiniteLaw.coordinateProduct R).prod (selectedLeafLaw P Q (θ bits))))
+          (fun x => (hF (coordinateWord (fun j => slotDecode f g (e j)
+            (emissionAssignmentEquiv Side Leaf m k n (fun i => (x.1 i).2,x.2) (e j))))).1)
+        have hhi := FiniteLaw.expect_mono
+          ((M.reportsGiven bits).prod ((FiniteLaw.coordinateProduct R).prod (selectedLeafLaw P Q (θ bits))))
+          (fun x => (hF (coordinateWord (fun j => slotDecode f g (e j)
+            (emissionAssignmentEquiv Side Leaf m k n (fun i => (x.1 i).2,x.2) (e j))))).2)
+        rw [FiniteLaw.expect_const] at hlo hhi
+        have hne : 0≤(n:ℝ)*E := mul_nonneg (Nat.cast_nonneg _) hE
+        apply abs_sub_le_iff.mpr
+        constructor <;> linarith [hc.1,hc.2]
+    _ = _ := by
+      unfold FiniteLaw.expect
+      simp only [mul_add,Finset.sum_add_distrib,←Finset.sum_mul,(fairOrientations m).total,one_mul]
+
+end PairedMechanisms
+end TraceReconstruction
+
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- The one-based source offset still puts the complete motif in the central
+ third under the true fit threshold. The stronger numerical margin is an
+ eventual consequence of L=o(T), not an additional stochastic event. -/
+theorem source_motif_central (T t s : ℕ) (bits : Fin t → Bool) (v : ℤ)
+    (hT : 24 ≤ T) (he : Even T) (hL : 24*2^s ≤ T)
+    (hfit : |8*((∑i,(bitIndex (bits i):ℤ))-(t/2:ℕ)+v)| ≤ (T:ℤ))
+    (u : Fin T)
+    (hu : motifIndicator s ((u.val:ℤ)-((T/2:ℕ)-1+(t/2:ℕ))+
+      (∑i,(bitIndex (bits i):ℤ))+v)≠0) : u∈centralThird T := by
+  have hsupport : 0≤(u.val:ℤ)-((T/2:ℕ)-1+(t/2:ℕ))+
+      (∑i,(bitIndex (bits i):ℤ))+v ∧
+      (u.val:ℤ)-((T/2:ℕ)-1+(t/2:ℕ))+(∑i,(bitIndex (bits i):ℤ))+v < (2^s:ℕ) := by
+    by_contra h
+    apply hu
+    exact ite_eq_right h
+  have hh := abs_le.mp hfit
+  obtain ⟨k,hk⟩ := he
+  have hhalf : 2*(T/2)=T := by omega
+  apply Finset.mem_filter.mpr
+  refine ⟨Finset.mem_univ _,?_⟩
+  omega
+
+/-- Fit controls every internal source motif, retaining all orientations in
+ the law. The incoming displacement of the root is supplied separately. -/
+theorem sourceTreeFit_central (T n : ℕ → ℕ) (s b : ℕ)
+    (center : ℕ → ℤ) (hc : ∀j≤b,center j=sourceLocalCenter T n j)
+    (hT : ∀j≤b,24≤T j) (he : ∀j≤b,Even (T j)) (hL : ∀j≤b,24*2^s≤T j)
+    (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool (b+1)) (incoming : ℤ)
+    (hroot : |8*((∑i,(bitIndex (bits.1 i):ℤ))-(n b/2:ℕ)+incoming)|≤(T b:ℤ))
+    (hfit : SourceTreeFit T n (b+1) bits) :
+    SourceCentral T n (sourceScheduledTrials T n) s center (sourceScheduledShift T n)
+      (fun j => centralThird (T j)) (b+1) bits incoming := by
+  induction b generalizing incoming with
+  | zero =>
+    constructor
+    · intro u hu
+      rw [hc 0 (by omega)] at hu
+      exact source_motif_central (T 0) (n 0) s bits.1 incoming
+        (hT 0 (by omega)) (he 0 (by omega)) (hL 0 (by omega)) hroot u hu
+    · intro u; trivial
+  | succ b ih =>
+    constructor
+    · intro u hu
+      rw [hc (b+1) (by omega)] at hu
+      exact source_motif_central (T (b+1)) (n (b+1)) s bits.1 incoming
+        (hT (b+1) (by omega)) (he (b+1) (by omega)) (hL (b+1) (by omega)) hroot u hu
+    · intro u
+      apply ih (fun j hj => hc j (by omega)) (fun j hj => hT j (by omega))
+        (fun j hj => he j (by omega)) (fun j hj => hL j (by omega)) _ _ _ (hfit.2 u)
+      exact hfit.1 u
+
+/-- The actual root side count has no parity restriction. Its entire range
+ lies inside the fit displacement interval when J=o(T). -/
+theorem source_root_fit (T J : ℕ) (bits : Fin J → Bool) (h : 4*(J+1)≤T) :
+    |8*((∑i,(bitIndex (bits i):ℤ))-(J/2:ℕ))|≤(T:ℤ) := by
+  have hlo : (0:ℤ)≤∑i,(bitIndex (bits i):ℤ) := Finset.sum_nonneg (fun _ _ => Int.natCast_nonneg _)
+  have hhi : (∑i,(bitIndex (bits i):ℤ))≤J := by
+    calc
+      _ ≤ ∑_i : Fin J,(1:ℤ) := by
+        apply Finset.sum_le_sum
+        intro i _
+        cases bits i <;> norm_num [bitIndex]
+      _ = J := by simp
+  apply abs_le.mpr
+  omega
+
+theorem sourceNoncentralMass_le_fit (T n : ℕ → ℕ) (s b : ℕ)
+    (center : ℕ → ℤ) (hc : ∀j≤b,center j=sourceLocalCenter T n j)
+    (hT : ∀j≤b,24≤T j) (he : ∀j≤b,Even (T j)) (hL : ∀j≤b,24*2^s≤T j)
+    (hroot : 4*(n b+1)≤T b)
+    (d : WindowAssignments T n (sourceScheduledTrials T n) (Option Bool) (b+1)) :
+    sourceNoncentralMass T n (sourceScheduledTrials T n) s center (sourceScheduledShift T n)
+      (fun j => centralThird (T j)) (b+1) d 0 ≤
+    (windowPosteriorLaw T n (sourceScheduledTrials T n) (b+1) d).expect
+      (fun bits => if SourceTreeFit T n (b+1) bits then 0 else 1) := by
+  apply FiniteLaw.expect_mono
+  intro bits
+  by_cases hf : SourceTreeFit T n (b+1) bits
+  · have hh := sourceTreeFit_central T n s b center hc hT he hL bits 0
+      (by simpa only [add_zero] using source_root_fit (T b) (n b) bits.1 hroot) hf
+    simp only [ite_eq_left hf,ite_eq_left hh,le_refl]
+  · simp only [ite_eq_right hf]
+    split_ifs <;> norm_num
+
+end TraceReconstruction
+
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- Only interior trials are paired in even bundles. The top J may be odd. -/
+theorem sourceTreeWord_eq_center_interior (s : ℕ) (P T n h : ℕ → ℕ) (center : ℕ → ℤ)
+    (atom : Bool → List Bool) (b : ℕ) (H : SourceLayoutMargins s P T n h (b-1))
+    (htop : ∀j<b,0<T j)
+    (hc : ∀j, j+1<b → center j=sourceLocalCenter T n j)
+    (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool b)
+    (incoming : ℤ) (τ : Bool) (hfit : SourceTreeFit T n b bits) :
+    sourceTreeWord s P T n h center atom b bits incoming τ=
+      sourceCenterWord s P T n center atom b bits incoming τ := by
+  induction b using Nat.twoStepInduction generalizing incoming τ with
+  | zero => rfl
+  | one => rfl
+  | more b ih₁ ih₂ =>
+    change (Fin (n (b+1)) → Bool) × (Fin ((T (b+1)/4)*n b) → Bool) ×
+      (Fin (T (b+1)) → WindowAssignments T n (sourceScheduledTrials T n) Bool (b+1)) at bits
+    let loc : Fin (T (b+1)) → Fin (n b) → Bool := fun u => (bits.2.2 u).1
+    let glob := bits.2.1
+    let a := P b/2-1
+    let A := sourceNominalPre P T b
+    let B := sourceNominalPost P T b
+    let start := sourceWindowStart (T (b+1)) A (n b) loc glob
+    let θ := integerTemplateType (T (b+1)) s (sourceNodeAnchor T n center (b+1) bits incoming) τ
+    have hb : b<b+2-1 := by omega
+    have htotal := sourceNominal_total P T b (by have hh := H.ambient b hb; omega)
+    have hf (u : Fin (T (b+1))) := sourceWindowStart_oneBased_fit (T (b+1)) (P b) (T b) (n b)
+      (2^s) loc glob (H.ambient b hb) (H.large b hb) (H.even_window b hb) (H.motif b hb) u (hfit.1 u)
+    have hx (u : Fin (T (b+1))) :
+        sourceNodeAnchor T n center b (bits.2.2 u) (sourceScheduledShift T n (b+1) glob u)=((a-start u:ℕ):ℤ) := by
+      change sourceNodeAnchor T n center b (bits.2.2 u) (sourceScheduledShift T n (b+1) glob u)=
+        ((P b/2-1-sourceWindowStart (T (b+1)) ((P b-T b)/2) (n b) loc glob u:ℕ):ℤ)
+      rw [sourceWindowStart_oneBased_anchor (T (b+1)) (P b) (T b) (n b) (2^s) loc glob
+        (H.ambient b hb) (H.large b hb) (H.even_window b hb) (H.motif b hb) u (hfit.1 u)]
+      simp only [sourceNodeAnchor,hc b (by omega),sourceLocalCenter,sourceScheduledShift,loc]
+    have hchild (u : Fin (T (b+1))) :
+        sourceTreeWord s P T n h center atom (b+1) (bits.2.2 u)
+          (sourceScheduledShift T n (b+1) glob u) (θ u)=
+        complementaryWords (T b) (a-start u) (2^s) (prouhetBool s) (sourceBlockWord s P atom b) (θ u) := by
+      rw [ih₂ (H.restrict (by omega)) (fun j hj => htop j (by omega)) (fun j hj => hc j (by omega)) _ _ _ (hfit.2 u)]
+      simp only [sourceCenterWord,hx,integerTemplateType_nat,complementaryWords]
+    have hstartnonneg (u : Fin (T (b+1))) : 0≤(A:ℤ)+sourceDisplacement (T (b+1)) (n b) loc glob u := by
+      have he := sourceWindowStart_cast_of_fit (T (b+1)) (P b) (T b) (n b) loc glob
+        (H.ambient b hb) (H.large b hb) u (hfit.1 u)
+      have hn : (0:ℤ)≤(start u:ℕ) := Int.natCast_nonneg _
+      change (start u:ℤ)=(A:ℤ)+_ at he
+      omega
+    have hp := htop (b+1) (by omega)
+    obtain ⟨k,hk⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : T (b+1)≠0)
+    have hpart : (List.ofFn (fun u => complementaryWords (P b) a (2^s) (prouhetBool s)
+          (sourceBlockWord s P atom b) (θ u))).flatten=
+        coordinateWord (fun u => pureAtomRun (sourceBlockWord s P atom b false)
+          (sourcePreCount (T (b+1)) A (n b) (h b) loc glob u) ++
+          complementaryWords (T b) (a-start u) (2^s) (prouhetBool s) (sourceBlockWord s P atom b) (θ u) ++
+          pureAtomRun (sourceBlockWord s P atom b false) (sourcePostCount (T (b+1)) B (n b) (h b) loc glob u)) := by
+      have hh := sourceLayout_literal_partition k A (T b) B (n b) (h b) a (2^s)
+      have hk' : k+1=T (b+1) := hk.symm
+      rw [hk'] at hh
+      rw [show A+T b+B=P b from htotal] at hh
+      exact hh (H.even_trials b hb) (H.pre_reserve b hb) (H.post_reserve b hb) loc glob
+        (prouhetBool s) (sourceBlockWord s P atom b) θ hstartnonneg
+        (fun u => (hf u).2.1) (fun u => (hf u).2.2) (fun u => (hf u).1)
+    change coordinateWord (fun u : Fin (T (b+1)) =>
+      pureAtomRun (sourceBlockWord s P atom b false) (sourcePreCount (T (b+1)) A (n b) (h b) loc glob u) ++
+      sourceTreeWord s P T n h center atom (b+1) (bits.2.2 u)
+        (sourceScheduledShift T n (b+1) glob u) (θ u) ++
+      pureAtomRun (sourceBlockWord s P atom b false) (sourcePostCount (T (b+1)) B (n b) (h b) loc glob u))=
+        atomString (T (b+1)) (sourceBlockWord s P atom (b+1)) θ
+    calc
+      _ = coordinateWord (fun u : Fin (T (b+1)) =>
+        pureAtomRun (sourceBlockWord s P atom b false) (sourcePreCount (T (b+1)) A (n b) (h b) loc glob u) ++
+        complementaryWords (T b) (a-start u) (2^s) (prouhetBool s) (sourceBlockWord s P atom b) (θ u) ++
+        pureAtomRun (sourceBlockWord s P atom b false) (sourcePostCount (T (b+1)) B (n b) (h b) loc glob u)) := by
+          apply congrArg coordinateWord
+          funext u
+          exact congrArg (fun w => pureAtomRun (sourceBlockWord s P atom b false)
+            (sourcePreCount (T (b+1)) A (n b) (h b) loc glob u) ++ w ++
+              pureAtomRun (sourceBlockWord s P atom b false)
+                (sourcePostCount (T (b+1)) B (n b) (h b) loc glob u)) (hchild u)
+      _ = _ := hpart.symm
+
+
+
+/-- Full literal unconditioned root partition, including both members of
+ every outer paired side. Its top trial count is the source's J, even or odd. -/
+def sourceFullWord (s : ℕ) (P T n h : ℕ → ℕ) (atom : Bool → List Bool) (b : ℕ)
+    (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool (b+1)) (τ : Bool) : List Bool :=
+  pureAtomRun (sourceBlockWord s P atom b false) (n b*h b+∑i,bitIndex (bits.1 i)) ++
+  sourceTreeWord s P T n h (sourceLocalCenter T n) atom (b+1) bits 0 τ ++
+  pureAtomRun (sourceBlockWord s P atom b false) (n b*(h b+1)-∑i,bitIndex (bits.1 i))
+
+theorem bitIndex_sum_le {J : ℕ} (bits : Fin J → Bool) : (∑i,bitIndex (bits i))≤J := by
+  calc
+    _ ≤ ∑_i : Fin J,1 := by
+      apply Finset.sum_le_sum
+      intro i _
+      cases bits i <;> norm_num [bitIndex]
+    _ = J := by simp
+
+/-- The source root's floor anchor and paired side counts have their literal
+ unmarked word interpretation, without requiring J to be even. -/
+theorem sourceFullWord_eq_block (s : ℕ) (P T n h : ℕ → ℕ) (atom : Bool → List Bool)
+    (b : ℕ) (H : SourceLayoutMargins s P T n h b) (hpos : 0<T b)
+    (hroot : P b=n b*(2*h b+1)+T b) (he : Even (T b))
+    (hJ : 4*(n b+1)≤T b) (hL : 8*2^s≤T b)
+    (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool (b+1))
+    (τ : Bool) (hfit : SourceTreeFit T n (b+1) bits) :
+    sourceFullWord s P T n h atom b bits τ=sourceBlockWord s P atom (b+1) τ := by
+  let E : ℕ := ∑i,bitIndex (bits.1 i)
+  let t := n b*h b+E
+  let z := n b*(h b+1)-E
+  let a := P b/2-1
+  have hE : E≤n b := bitIndex_sum_le bits.1
+  have hnat : n b≤n b*(h b+1) := by nlinarith
+  have htotal : t+T b+z=P b := by
+    have hprod : n b*(h b+1)=n b*h b+n b := by ring
+    have hprod2 : n b*(2*h b+1)=2*(n b*h b)+n b := by ring
+    dsimp [t,z]
+    omega
+  obtain ⟨k,hk⟩ := he
+  have hhalf : 2*(T b/2)=T b := by omega
+  have hP : P b=2*(n b*h b)+n b+T b := by rw [hroot]; ring
+  have hPa : P b/2=n b*h b+T b/2+n b/2 := by omega
+  have ht : t≤a := by dsimp [t,a]; omega
+  have hz : a+2^s≤t+T b := by dsimp [t,a]; omega
+  have hanchor : sourceNodeAnchor T n (sourceLocalCenter T n) b bits 0=((a-t:ℕ):ℤ) := by
+    have hEc : (E:ℤ)=∑i,(bitIndex (bits.1 i):ℤ) := by simp [E]
+    dsimp only [sourceNodeAnchor,sourceLocalCenter]
+    have ha : a-t=T b/2-1+n b/2-E := by dsimp [a,t]; omega
+    have hTe : 1≤T b/2 := by omega
+    have hEe : E≤T b/2-1+n b/2 := by omega
+    rw [ha,Nat.cast_sub hEe,Nat.cast_add,Nat.cast_sub hTe,hEc]
+    push_cast
+    ring
+  have ho (j : ℕ) (hj : j<b+1) : 0<T j := by
+    by_cases heq : j=b
+    · simpa only [heq] using hpos
+    · exact H.positive j (by omega)
+  have hcenter := sourceTreeWord_eq_center_interior s P T n h (sourceLocalCenter T n) atom
+    (b+1) (by simpa using H) ho (fun _ _ => rfl) bits 0 τ hfit
+  have hpart := template_word_window t (T b) z a (2^s) (prouhetBool s) τ
+    (sourceBlockWord s P atom b) ht hz
+  rw [htotal] at hpart
+  rw [atomString_constant,atomString_constant] at hpart
+  unfold sourceFullWord
+  rw [hcenter]
+  change pureAtomRun (sourceBlockWord s P atom b false) t ++
+    atomString (T b) (sourceBlockWord s P atom b)
+      (integerTemplateType (T b) s (sourceNodeAnchor T n (sourceLocalCenter T n) b bits 0) τ) ++
+    pureAtomRun (sourceBlockWord s P atom b false) z = _
+  rw [hanchor,integerTemplateType_nat]
+  exact hpart.symm
+
+end TraceReconstruction
+
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- Pure filler IDs. At bottom leaf windows the two fillers have zero length;
+ they are deterministic empty outputs, not extra information. -/
+def WindowFiller (T : ℕ → ℕ) : ℕ → Type
+  | 0 => PEmpty
+  | b+1 => (Fin (T b) × Bool) ⊕ (Fin (T b) × WindowFiller T b)
+
+noncomputable instance windowFillerFintype (T : ℕ → ℕ) : (b : ℕ) → Fintype (WindowFiller T b)
+  | 0 => inferInstanceAs (Fintype PEmpty)
+  | b+1 => by
+    letI := windowFillerFintype T b
+    exact inferInstanceAs (Fintype ((Fin (T b) × Bool) ⊕ (Fin (T b) × WindowFiller T b)))
+
+abbrev TreeEmission (T n g : ℕ → ℕ) (b : ℕ) :=
+  (WindowTrial T n g b × Bool) ⊕ (WindowFiller T b ⊕ WindowLeaf T b)
+
+def TreeEmission.up (T n g : ℕ → ℕ) (b : ℕ) (u : Fin (T b)) :
+    TreeEmission T n g b → TreeEmission T n g (b+1)
+  | .inl (i,p) => .inl (.inr (.inr (u,i)),p)
+  | .inr (.inl f) => .inr (.inl (.inr (u,f)))
+  | .inr (.inr v) => .inr (.inr (u,v))
+
+def treeLocalSlots (T n g : ℕ → ℕ) (b : ℕ) (p : Bool) : List (TreeEmission T n g (b+1)) :=
+  List.ofFn (fun k : Fin (n b) => Sum.inl (Sum.inl k,p))
+
+def treeGlobalSlots (T n g : ℕ → ℕ) (b : ℕ) (p : Bool)
+    (pos : Fin (g b) → Fin (T b)) (u : Fin (T b)) : List (TreeEmission T n g (b+1)) :=
+  ((List.ofFn id : List (Fin (g b))).filter (fun k => pos k=u)).map
+    (fun k => Sum.inl (Sum.inr (Sum.inl k),p))
+
+/-- Public chronological IDs. Each child carries its own local paired sides;
+ the group's global sides and deterministic fillers surround that child.
+ The order is fixed independently of all bits and both hypotheses. -/
+def treeChronology (T n g : ℕ → ℕ)
+    (pre post : (b : ℕ) → Fin (g b) → Fin (T b)) :
+    (b : ℕ) → List (TreeEmission T n g b)
+  | 0 => [.inr (.inr (show WindowLeaf T 0 from (0 : Fin 1)))]
+  | b+1 => treeLocalSlots T n g b false ++
+      (List.ofFn (fun u : Fin (T b) =>
+        treeGlobalSlots T n g b false (pre b) u ++
+        [.inr (.inl (.inl (u,false)))] ++
+        (treeChronology T n g pre post b).map (TreeEmission.up T n g b u) ++
+        treeGlobalSlots T n g b true (post b) u ++
+        [.inr (.inl (.inl (u,true)))])).flatten ++
+      treeLocalSlots T n g b true
+
+theorem sum_fiber_list_length {α ι : Type*} [Fintype ι] [DecidableEq ι] (pos : α → ι) (l : List α) :
+    (∑i,(l.filter (fun a => pos a=i)).length)=l.length := by
+  induction l with
+  | nil => simp
+  | cons a l ih =>
+    have hh (i : ι) : ((a::l).filter (fun a => pos a=i)).length=
+        (if pos a=i then 1 else 0)+(l.filter (fun a => pos a=i)).length := by
+      by_cases h : pos a=i <;> simp [h,Nat.add_comm]
+    simp_rw [hh]
+    rw [Finset.sum_add_distrib,ih]
+    simp [Nat.add_comm]
+
+theorem treeChronology_length (T n g : ℕ → ℕ)
+    (pre post : (b : ℕ) → Fin (g b) → Fin (T b)) (b : ℕ) :
+    (treeChronology T n g pre post b).length=Fintype.card (TreeEmission T n g b) := by
+  induction b with
+  | zero =>
+    change 1=Fintype.card ((PEmpty × Bool) ⊕ (PEmpty ⊕ Fin 1))
+    simp
+  | succ b ih =>
+    have hp := sum_fiber_list_length (pre b) (List.ofFn id : List (Fin (g b)))
+    have hq := sum_fiber_list_length (post b) (List.ofFn id : List (Fin (g b)))
+    simp only [List.length_ofFn] at hp hq
+    change _=Fintype.card (((Fin (n b) ⊕ (Fin (g b) ⊕ (Fin (T b) × WindowTrial T n g b))) × Bool) ⊕
+      (((Fin (T b) × Bool) ⊕ (Fin (T b) × WindowFiller T b)) ⊕ (Fin (T b) × WindowLeaf T b)))
+    simp only [treeChronology,treeLocalSlots,treeGlobalSlots,List.length_append,
+      List.length_flatten,List.map_ofFn,Function.comp_apply,List.length_ofFn,List.length_map,List.length_cons,List.length_nil,
+      Finset.sum_add_distrib,List.sum_ofFn,ih,Fintype.card_sum,Fintype.card_prod,Fintype.card_fin,Fintype.card_bool]
+    rw [hp,hq]
+    simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,smul_eq_mul]
+    ring
+
+theorem treeChronology_mem (T n g : ℕ → ℕ)
+    (pre post : (b : ℕ) → Fin (g b) → Fin (T b)) (b : ℕ)
+    (x : TreeEmission T n g b) : x∈treeChronology T n g pre post b := by
+  induction b with
+  | zero =>
+    rcases x with ⟨i,p⟩ | f | v
+    · exact PEmpty.elim i
+    · exact PEmpty.elim f
+    · change Fin 1 at v
+      have hv : v=0 := Subsingleton.elim _ _
+      subst v; exact List.mem_singleton.mpr rfl
+  | succ b ih =>
+    have hc (u : Fin (T b)) (x : TreeEmission T n g (b+1))
+        (hx : x∈treeGlobalSlots T n g b false (pre b) u ++ [.inr (.inl (.inl (u,false)))] ++
+          (treeChronology T n g pre post b).map (TreeEmission.up T n g b u) ++
+          treeGlobalSlots T n g b true (post b) u ++ [.inr (.inl (.inl (u,true)))]) :
+        x∈treeChronology T n g pre post (b+1) := by
+      apply List.mem_append_left
+      apply List.mem_append_right
+      apply List.mem_flatten.mpr
+      exact ⟨_,List.mem_ofFn.mpr ⟨u,rfl⟩,hx⟩
+    rcases x with ⟨k,p⟩ | f | v
+    · rcases k with k | k | ⟨u,k⟩
+      · cases p <;>
+          simp [treeChronology,treeLocalSlots,List.mem_ofFn]
+      · cases p
+        · apply hc (pre b k)
+          simp only [List.mem_append,treeGlobalSlots,List.mem_map,List.mem_filter,List.mem_ofFn]
+          exact Or.inl (Or.inl (Or.inl (Or.inl ⟨k,⟨⟨k,rfl⟩,by simp⟩,rfl⟩)))
+        · apply hc (post b k)
+          simp only [List.mem_append,treeGlobalSlots,List.mem_map,List.mem_filter,List.mem_ofFn]
+          exact Or.inl (Or.inr ⟨k,⟨⟨k,rfl⟩,by simp⟩,rfl⟩)
+      · apply hc u
+        have hm : TreeEmission.up T n g b u (.inl (k,p))∈
+            (treeChronology T n g pre post b).map (TreeEmission.up T n g b u) := List.mem_map.mpr ⟨_,ih _,rfl⟩
+        exact List.mem_append_left _ (List.mem_append_left _ (List.mem_append_right _ hm))
+    · rcases f with ⟨u,p⟩ | ⟨u,f⟩
+      · cases p <;> apply hc u <;> simp
+      · apply hc u
+        have hm : TreeEmission.up T n g b u (.inr (.inl f))∈
+            (treeChronology T n g pre post b).map (TreeEmission.up T n g b u) := List.mem_map.mpr ⟨_,ih _,rfl⟩
+        exact List.mem_append_left _ (List.mem_append_left _ (List.mem_append_right _ hm))
+    · rcases v with ⟨u,v⟩
+      apply hc u
+      have hm : TreeEmission.up T n g b u (.inr (.inr v))∈
+          (treeChronology T n g pre post b).map (TreeEmission.up T n g b u) := List.mem_map.mpr ⟨_,ih _,rfl⟩
+      exact List.mem_append_left _ (List.mem_append_left _ (List.mem_append_right _ hm))
+
+theorem treeChronology_nodup (T n g : ℕ → ℕ)
+    (pre post : (b : ℕ) → Fin (g b) → Fin (T b)) (b : ℕ) :
+    (treeChronology T n g pre post b).Nodup := by
+  have hs : (treeChronology T n g pre post b).toFinset=Finset.univ := by
+    ext x
+    simp only [List.mem_toFinset,Finset.mem_univ,iff_true]
+    exact treeChronology_mem T n g pre post b x
+  have hh : (treeChronology T n g pre post b).toFinset.card=(treeChronology T n g pre post b).length := by
+    rw [hs,Finset.card_univ,treeChronology_length]
+  exact Multiset.coe_nodup.mp ((Multiset.toFinset_card_eq_card_iff_nodup
+    (m := (treeChronology T n g pre post b : Multiset (TreeEmission T n g b)))).mp hh)
+
+
+/-- A genuine complete chronological bijection, not an assumed layout. -/
+def treeChronologyEquiv (T n g : ℕ → ℕ)
+    (pre post : (b : ℕ) → Fin (g b) → Fin (T b)) (b : ℕ) :
+    Fin (treeChronology T n g pre post b).length ≃ TreeEmission T n g b :=
+  List.Nodup.getEquivOfForallMemList _ (treeChronology_nodup T n g pre post b)
+    (treeChronology_mem T n g pre post b)
+
+end TraceReconstruction
+
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+def pairedSideCount (h : ℕ) (bit side : Bool) : ℕ := h+bitIndex (Bool.xor bit side)
+
+theorem pairedSideCount_false (h : ℕ) (bit : Bool) :
+    pairedSideCount h bit false=h+bitIndex bit := by cases bit <;> rfl
+
+theorem pairedSideCount_true (h : ℕ) (bit : Bool) :
+    pairedSideCount h bit true=h+1-bitIndex bit := by cases bit <;> simp [pairedSideCount,bitIndex]
+
+theorem pairedSideCount_sum_pre {t h : ℕ} (bits : Fin t → Bool) :
+    (∑i,pairedSideCount h (bits i) false)=t*h+∑i,bitIndex (bits i) := by
+  simp only [pairedSideCount_false,Finset.sum_add_distrib,Finset.sum_const,
+    Finset.card_univ,Fintype.card_fin,smul_eq_mul]
+
+theorem pairedSideCount_sum_total {t h : ℕ} (bits : Fin t → Bool) :
+    (∑i,pairedSideCount h (bits i) true)+(∑i,bitIndex (bits i))=t*(h+1) := by
+  rw [←Finset.sum_add_distrib]
+  calc
+    _ = ∑_i : Fin t,(h+1) := by
+      apply Finset.sum_congr rfl
+      intro i _
+      cases bits i <;> simp [pairedSideCount,bitIndex]
+    _ = _ := by simp
+
+theorem pairedSideCount_sum_post {t h : ℕ} (bits : Fin t → Bool) :
+    (∑i,pairedSideCount h (bits i) true)=t*(h+1)-∑i,bitIndex (bits i) := by
+  have := pairedSideCount_sum_total (h:=h) bits
+  omega
+
+/-- Global trial positions are ID functions, not positions inferred from a
+ realized output or orientation. -/
+def sourceGlobalPre (N t : ℕ) (k : Fin ((N/4)*t)) : Fin N :=
+  ⟨((globalTrialEquiv N t).symm k).1.val,by
+    have := ((globalTrialEquiv N t).symm k).1.isLt; omega⟩
+
+def sourceGlobalPost (N t : ℕ) (k : Fin ((N/4)*t)) : Fin N :=
+  ⟨N-1-((globalTrialEquiv N t).symm k).1.val,by
+    have := ((globalTrialEquiv N t).symm k).1.isLt; omega⟩
+
+theorem sum_globalSides (N t h : ℕ) (bits : Fin ((N/4)*t) → Bool)
+    (side : Bool) (pos : Fin (N/4) → Fin N) (u : Fin N) :
+    (∑k : Fin ((N/4)*t), if pos ((globalTrialEquiv N t).symm k).1=u then
+      pairedSideCount h (bits k) side else 0)=
+    ∑v : Fin (N/4),if pos v=u then
+      (∑a : Fin t,pairedSideCount h (bits (globalTrialIndex N t v a)) side) else 0 := by
+  rw [←(globalTrialEquiv N t).sum_comp,Fintype.sum_prod_type]
+  apply Finset.sum_congr rfl
+  intro v _
+  simp only [Equiv.symm_apply_apply]
+  split_ifs with hv
+  · rfl
+  · simp only [Finset.sum_const_zero]
+
+theorem sourcePreCount_slots (N A t h : ℕ) (loc : Fin N → Fin t → Bool)
+    (glob : Fin ((N/4)*t) → Bool) (u : Fin N) :
+    sourcePreCount N A t h loc glob u=
+      (∑k : Fin ((N/4)*t),if sourceGlobalPre N t k=u then pairedSideCount h (glob k) false else 0)+
+      sourcePreFiller N A t h u+(∑i,pairedSideCount h (loc u i) false) := by
+  rw [pairedSideCount_sum_pre]
+  change sourcePreFiller N A t h u+(∑v : SourceBundle N,
+    if sourceBundleLeft N v=u then sourceBundleCount N t h loc glob v else 0)=_
+  rw [Fintype.sum_sum_type]
+  have hl : (∑v : Fin N,if sourceBundleLeft N (.inl v)=u then
+      sourceBundleCount N t h loc glob (.inl v) else 0)=t*h+∑i,bitIndex (loc u i) := by
+    change (∑v : Fin N,if v=u then (t*h+(∑i,bitIndex (loc v i))) else 0)=_
+    simp
+  rw [hl]
+  have hg := sum_globalSides N t h glob false
+    (fun v => ⟨v.val,by have := v.isLt; omega⟩) u
+  simp only [pairedSideCount_sum_pre] at hg
+  change (∑k,if sourceGlobalPre N t k=u then pairedSideCount h (glob k) false else 0)=_ at hg
+  rw [hg]
+  change _+(_+_) = _+_+_
+  have he : (∑v : Fin (N/4),if sourceBundleLeft N (.inr v)=u then
+      sourceBundleCount N t h loc glob (.inr v) else 0)=
+      ∑v : Fin (N/4),if (⟨v.val,by have := v.isLt; omega⟩ : Fin N)=u then
+        (t*h+∑i,bitIndex (glob (globalTrialIndex N t v i))) else 0 := by
+    apply Finset.sum_congr rfl
+    intro v _
+    rfl
+  rw [he]
+  omega
+
+theorem sourcePostCount_slots (N B t h : ℕ) (ht : Even t) (loc : Fin N → Fin t → Bool)
+    (glob : Fin ((N/4)*t) → Bool) (u : Fin N) :
+    sourcePostCount N B t h loc glob u=
+      (∑i,pairedSideCount h (loc u i) true)+
+      (∑k : Fin ((N/4)*t),if sourceGlobalPost N t k=u then pairedSideCount h (glob k) true else 0)+
+      sourcePostFiller N B t h u := by
+  have hh : 2*(t/2)=t := Nat.mul_div_cancel' ((even_iff_two_dvd).mp ht)
+  have hc (bits : Fin t → Bool) :
+      2*sourceLayoutReserve t h-(t*h+∑i,bitIndex (bits i))=
+        ∑i,pairedSideCount h (bits i) true := by
+    rw [pairedSideCount_sum_post]
+    dsimp only [sourceLayoutReserve]
+    have hs := bitIndex_sum_le bits
+    have he : 2*(t*h+t/2)=t*h+t*(h+1) := by
+      rw [Nat.mul_add,Nat.mul_add,Nat.mul_one,hh]
+      omega
+    rw [he]
+    omega
+  change sourcePostFiller N B t h u+(∑v : SourceBundle N,
+    if sourceBundleRight N v=u then 2*sourceLayoutReserve t h-sourceBundleCount N t h loc glob v else 0)=_
+  rw [Fintype.sum_sum_type]
+  have hl : (∑v : Fin N,if sourceBundleRight N (.inl v)=u then
+      2*sourceLayoutReserve t h-sourceBundleCount N t h loc glob (.inl v) else 0)=
+      ∑i,pairedSideCount h (loc u i) true := by
+    simp only [sourceBundleRight,sourceBundleCount,Finset.sum_ite_eq',Finset.mem_univ,ite_true,hc]
+  rw [hl]
+  have hg := sum_globalSides N t h glob true
+    (fun v => ⟨N-1-v.val,by have := v.isLt; omega⟩) u
+  change (∑k,if sourceGlobalPost N t k=u then pairedSideCount h (glob k) true else 0)=_ at hg
+  rw [hg]
+  simp only [sourceBundleRight,sourceBundleCount,hc]
+  omega
+
+/-- Flattening selected equal-atom chunks sums exactly their genuine lengths. -/
+theorem filter_chunks_flatten {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (l : List ι) (p : ι → Bool) (c : ι → ℕ) (w : List Bool) :
+    ((l.filter p).map (fun i => pureAtomRun w (c i))).flatten=
+      pureAtomRun w ((l.map (fun i => if p i then c i else 0)).sum) := by
+  calc
+    _ = pureAtomRun w (((l.filter p).map c).sum) := by
+      simpa only [List.map_map,Function.comp_def] using pureAtomRun_partition w ((l.filter p).map c)
+    _ = _ := by
+      congr 1
+      induction l with
+      | nil => rfl
+      | cons i l ih => cases hp : p i <;> simp [hp,ih]
+
+end TraceReconstruction
+
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+def sourceScheduledPre (T n : ℕ → ℕ) :
+    (b : ℕ) → Fin (sourceScheduledTrials T n b) → Fin (T b)
+  | 0,k => Fin.elim0 k
+  | b+1,k => sourceGlobalPre (T (b+1)) (n b) k
+
+def sourceScheduledPost (T n : ℕ → ℕ) :
+    (b : ℕ) → Fin (sourceScheduledTrials T n b) → Fin (T b)
+  | 0,k => Fin.elim0 k
+  | b+1,k => sourceGlobalPost (T (b+1)) (n b) k
+
+/-- Two actual deterministic pure binary side words for each paired trial ID. -/
+def sourceTrialWord (s : ℕ) (P T n h : ℕ → ℕ) (atom : Bool → List Bool) :
+    (b : ℕ) → WindowTrial T n (sourceScheduledTrials T n) b → Bool → List Bool
+  | 0,i,_ => PEmpty.elim i
+  | b+1,.inl _,c => pureAtomRun (sourceBlockWord s P atom b false) (h b+bitIndex c)
+  | 1,.inr (.inl k),_ => Fin.elim0 k
+  | b+2,.inr (.inl _),c => pureAtomRun (sourceBlockWord s P atom b false) (h b+bitIndex c)
+  | b+1,.inr (.inr (_,i)),c => sourceTrialWord s P T n h atom b i c
+
+/-- Fixed pure fillers, including the deterministic empty bottom fillers. -/
+def sourceFillerWord (s : ℕ) (P T n h : ℕ → ℕ) (atom : Bool → List Bool) :
+    (b : ℕ) → WindowFiller T b → List Bool
+  | 0,i => PEmpty.elim i
+  | b+1,.inl (u,p) => match b,u with
+    | 0,_ => []
+    | b+1,u => pureAtomRun (sourceBlockWord s P atom b false)
+        (if p then sourcePostFiller (T (b+1)) (sourceNominalPost P T b) (n b) (h b) u
+          else sourcePreFiller (T (b+1)) (sourceNominalPre P T b) (n b) (h b) u)
+  | b+1,.inr (_,i) => sourceFillerWord s P T n h atom b i
+
+def sourceSlotWord (s : ℕ) (P T n h : ℕ → ℕ) (center : ℕ → ℤ) (atom : Bool → List Bool)
+    (b : ℕ) (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool b)
+    (incoming : ℤ) (τ : Bool) : TreeEmission T n (sourceScheduledTrials T n) b → List Bool
+  | .inl (i,side) => sourceTrialWord s P T n h atom b i
+      (Bool.xor (windowAssignmentEquiv T n (sourceScheduledTrials T n) Bool b bits i) side)
+  | .inr (.inl i) => sourceFillerWord s P T n h atom b i
+  | .inr (.inr v) => atom (sourceTreeLeafBool T n s center b bits incoming τ v)
+
+def sourceChronologicalWord (s : ℕ) (P T n h : ℕ → ℕ) (center : ℕ → ℤ) (atom : Bool → List Bool)
+    (b : ℕ) (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool b)
+    (incoming : ℤ) (τ : Bool) : List Bool :=
+  ((treeChronology T n (sourceScheduledTrials T n) (sourceScheduledPre T n) (sourceScheduledPost T n) b).map
+    (sourceSlotWord s P T n h center atom b bits incoming τ)).flatten
+
+theorem sourceSlotWord_up (s : ℕ) (P T n h : ℕ → ℕ) (center : ℕ → ℤ) (atom : Bool → List Bool)
+    (b : ℕ) (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool (b+1))
+    (incoming : ℤ) (τ : Bool) (u : Fin (T b)) (x : TreeEmission T n (sourceScheduledTrials T n) b) :
+    sourceSlotWord s P T n h center atom (b+1) bits incoming τ (TreeEmission.up T n (sourceScheduledTrials T n) b u x)=
+      sourceSlotWord s P T n h center atom b (bits.2.2 u) (sourceScheduledShift T n b bits.2.1 u)
+        (integerTemplateType (T b) s (sourceNodeAnchor T n center b bits incoming) τ u) x := by
+  rcases x with ⟨i,p⟩ | f | v <;> rfl
+
+theorem treeChronology_eval (T n g : ℕ → ℕ)
+    (pre post : (b : ℕ) → Fin (g b) → Fin (T b)) (b : ℕ)
+    (f : TreeEmission T n g (b+1) → List Bool) :
+    ((treeChronology T n g pre post (b+1)).map f).flatten=
+    ((treeLocalSlots T n g b false).map f).flatten ++
+      coordinateWord (fun u => ((treeGlobalSlots T n g b false (pre b) u).map f).flatten ++
+        f (.inr (.inl (.inl (u,false)))) ++
+        ((treeChronology T n g pre post b).map (fun x => f (TreeEmission.up T n g b u x))).flatten ++
+        ((treeGlobalSlots T n g b true (post b) u).map f).flatten ++
+        f (.inr (.inl (.inl (u,true))))) ++
+      ((treeLocalSlots T n g b true).map f).flatten := by
+  simp only [treeChronology,List.map_append,List.flatten_append,List.map_flatten,
+    List.map_ofFn,coordinateWord]
+  congr 2
+  rw [List.flatten_flatten,List.map_ofFn]
+  apply congrArg List.flatten
+  apply congrArg List.ofFn
+  funext u
+  simp only [Function.comp_apply,List.map_append,List.flatten_append,List.map_cons,List.map_nil,
+    List.flatten_cons,List.flatten_nil,List.append_nil,List.map_map]
+  rfl
+
+theorem sourceLocalSlots_word (s : ℕ) (P T n h : ℕ → ℕ) (center : ℕ → ℤ) (atom : Bool → List Bool)
+    (b : ℕ) (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool (b+1))
+    (incoming : ℤ) (τ side : Bool) :
+    ((treeLocalSlots T n (sourceScheduledTrials T n) b side).map
+      (sourceSlotWord s P T n h center atom (b+1) bits incoming τ)).flatten=
+      pureAtomRun (sourceBlockWord s P atom b false) (∑i,pairedSideCount (h b) (bits.1 i) side) := by
+  rcases bits with ⟨loc,glob,kids⟩
+  simp only [treeLocalSlots,List.map_ofFn]
+  change (List.ofFn (fun i : Fin (n b) => pureAtomRun (sourceBlockWord s P atom b false)
+    (pairedSideCount (h b) (loc i) side))).flatten=_
+  exact coordinateWord_pureAtomRun _ _
+
+theorem sourceGlobalSlots_word (s : ℕ) (P T n h : ℕ → ℕ) (center : ℕ → ℤ) (atom : Bool → List Bool)
+    (b : ℕ) (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool (b+2))
+    (incoming : ℤ) (τ side : Bool) (pos : Fin ((T (b+1)/4)*n b) → Fin (T (b+1))) (u : Fin (T (b+1))) :
+    ((treeGlobalSlots T n (sourceScheduledTrials T n) (b+1) side pos u).map
+      (sourceSlotWord s P T n h center atom (b+2) bits incoming τ)).flatten=
+      pureAtomRun (sourceBlockWord s P atom b false)
+        (∑k,if pos k=u then pairedSideCount (h b) (bits.2.1 k) side else 0) := by
+  rcases bits with ⟨loc,glob,kids⟩
+  change Fin ((T (b+1)/4)*n b) → Bool at glob
+  simp only [treeGlobalSlots,List.map_map]
+  change ((List.filter (fun k => decide (pos k=u)) (List.ofFn id)).map
+    (fun i => pureAtomRun (sourceBlockWord s P atom b false) (pairedSideCount (h b) (glob i) side))).flatten=_
+  rw [filter_chunks_flatten,List.map_ofFn,List.sum_ofFn]
+  simp only [Function.comp_apply,id_eq,decide_eq_true_eq]
+
+end TraceReconstruction
+
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+def sourceFramedWord (s : ℕ) (P T n h : ℕ → ℕ) (center : ℕ → ℤ) (atom : Bool → List Bool) :
+    (b : ℕ) → WindowAssignments T n (sourceScheduledTrials T n) Bool b → ℤ → Bool → List Bool
+  | 0,_,_,τ => atom τ
+  | b+1,bits,incoming,τ =>
+      pureAtomRun (sourceBlockWord s P atom b false) (∑i,pairedSideCount (h b) (bits.1 i) false) ++
+      sourceTreeWord s P T n h center atom (b+1) bits incoming τ ++
+      pureAtomRun (sourceBlockWord s P atom b false) (∑i,pairedSideCount (h b) (bits.1 i) true)
+
+theorem sourceChronologicalWord_eq_framed (s : ℕ) (P T n h : ℕ → ℕ) (center : ℕ → ℤ)
+    (atom : Bool → List Bool) (b : ℕ) (hev : ∀j,j+1<b → Even (n j))
+    (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool b) (incoming : ℤ) (τ : Bool) :
+    sourceChronologicalWord s P T n h center atom b bits incoming τ=
+      sourceFramedWord s P T n h center atom b bits incoming τ := by
+  induction b generalizing incoming τ with
+  | zero => simp [sourceChronologicalWord,treeChronology,sourceSlotWord,sourceTreeLeafBool,sourceFramedWord]
+  | succ b ih =>
+    rcases bits with ⟨loc,glob,kids⟩
+    unfold sourceChronologicalWord
+    rw [treeChronology_eval]
+    erw [sourceLocalSlots_word,sourceLocalSlots_word]
+    change _=pureAtomRun (sourceBlockWord s P atom b false) (∑i,pairedSideCount (h b) (loc i) false) ++
+      sourceTreeWord s P T n h center atom (b+1) (loc,glob,kids) incoming τ ++
+      pureAtomRun (sourceBlockWord s P atom b false) (∑i,pairedSideCount (h b) (loc i) true)
+    apply congrArg (fun w => pureAtomRun (sourceBlockWord s P atom b false)
+      (∑i,pairedSideCount (h b) (loc i) false) ++ w ++
+      pureAtomRun (sourceBlockWord s P atom b false) (∑i,pairedSideCount (h b) (loc i) true))
+    have child (u : Fin (T b)) :
+        ((treeChronology T n (sourceScheduledTrials T n) (sourceScheduledPre T n) (sourceScheduledPost T n) b).map
+          (fun x => sourceSlotWord s P T n h center atom (b+1) (loc,glob,kids) incoming τ
+            (TreeEmission.up T n (sourceScheduledTrials T n) b u x))).flatten=
+        sourceFramedWord s P T n h center atom b (kids u) (sourceScheduledShift T n b glob u)
+          (integerTemplateType (T b) s (sourceNodeAnchor T n center b (loc,glob,kids) incoming) τ u) := by
+      have he : (fun x : TreeEmission T n (sourceScheduledTrials T n) b =>
+          sourceSlotWord s P T n h center atom (b+1) (loc,glob,kids) incoming τ
+            (TreeEmission.up T n (sourceScheduledTrials T n) b u x)) =
+          sourceSlotWord s P T n h center atom b (kids u) (sourceScheduledShift T n b glob u)
+            (integerTemplateType (T b) s (sourceNodeAnchor T n center b (loc,glob,kids) incoming) τ u) := by
+        funext x
+        exact sourceSlotWord_up s P T n h center atom b (loc,glob,kids) incoming τ u x
+      erw [he]
+      exact ih (fun j hj => hev j (by omega)) (kids u) _ _
+    cases b with
+    | zero =>
+      change coordinateWord _=atomString (T 0) atom _
+      apply congrArg coordinateWord
+      funext u
+      rw [child]
+      simp only [treeGlobalSlots,sourceScheduledTrials,List.ofFn_zero,List.filter_nil,List.map_nil,
+        List.flatten_nil,List.nil_append,List.append_nil,sourceSlotWord,sourceFillerWord,sourceFramedWord]
+    | succ b =>
+      change coordinateWord _=coordinateWord _
+      apply congrArg coordinateWord
+      funext u
+      erw [sourceGlobalSlots_word,sourceGlobalSlots_word,child]
+      change _=pureAtomRun (sourceBlockWord s P atom b false)
+          (sourcePreCount (T (b+1)) (sourceNominalPre P T b) (n b) (h b) (fun v => (kids v).1) glob u) ++
+        sourceTreeWord s P T n h center atom (b+1) (kids u) (sourceScheduledShift T n (b+1) glob u)
+          (integerTemplateType (T (b+1)) s (sourceNodeAnchor T n center (b+1) (loc,glob,kids) incoming) τ u) ++
+        pureAtomRun (sourceBlockWord s P atom b false)
+          (sourcePostCount (T (b+1)) (sourceNominalPost P T b) (n b) (h b) (fun v => (kids v).1) glob u)
+      erw [sourcePreCount_slots,sourcePostCount_slots _ _ _ _ (hev b (by omega))]
+      change pureAtomRun _ _ ++ pureAtomRun _ _ ++
+        (pureAtomRun _ _ ++ sourceTreeWord s P T n h center atom (b+1) (kids u)
+          (sourceScheduledShift T n (b+1) glob u)
+          (integerTemplateType (T (b+1)) s (sourceNodeAnchor T n center (b+1) (loc,glob,kids) incoming) τ u) ++
+          pureAtomRun _ _) ++ pureAtomRun _ _ ++ pureAtomRun _ _ = _
+      simp only [sourceScheduledPre,sourceScheduledPost,pureAtomRun_add,List.append_assoc,
+        Bool.false_eq_true,ite_false,ite_true]
+      rfl
+
+/-- Chronological side/filler/leaf slots produce the SAME full source word
+ on the joint fit event. No orientation is removed or conditioned away. -/
+theorem sourceChronologicalWord_eq_block (s : ℕ) (P T n h : ℕ → ℕ) (atom : Bool → List Bool)
+    (b : ℕ) (H : SourceLayoutMargins s P T n h b)
+    (hP : P b=n b*(2*h b+1)+T b) (hT : Even (T b))
+    (hlarge : 4*(n b+1)≤T b) (hmotif : 8*2^s≤T b)
+    (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool (b+1)) (τ : Bool)
+    (hfit : SourceTreeFit T n (b+1) bits) :
+    sourceChronologicalWord s P T n h (sourceLocalCenter T n) atom (b+1) bits 0 τ=
+      sourceBlockWord s P atom (b+1) τ := by
+  rw [sourceChronologicalWord_eq_framed s P T n h (sourceLocalCenter T n) atom (b+1)
+    (fun j hj => H.even_trials j (by omega))]
+  have hf := sourceFullWord_eq_block s P T n h atom b H (by omega) hP hT hlarge hmotif bits τ hfit
+  change sourceFullWord s P T n h atom b bits τ=_ at hf
+  simpa only [sourceFramedWord,sourceFullWord,pairedSideCount_sum_pre,pairedSideCount_sum_post] using hf
+
+end TraceReconstruction
+
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+def finEnumeration (α : Type) [Fintype α] : Fin (Fintype.card α) ≃ α := (Fintype.equivFin α).symm
+
+abbrev SourceTrialCount (T n : ℕ → ℕ) (b : ℕ) := Fintype.card (WindowTrial T n (sourceScheduledTrials T n) b)
+abbrev SourceFillerCount (T : ℕ → ℕ) (b : ℕ) := Fintype.card (WindowFiller T b)
+abbrev SourceLeafCount (T : ℕ → ℕ) (b : ℕ) := Fintype.card (WindowLeaf T b)
+
+def sourceEmissionEnum (T n : ℕ → ℕ) (b : ℕ) :
+    EmissionSlot (SourceTrialCount T n b) (SourceFillerCount T b) (SourceLeafCount T b) ≃
+      TreeEmission T n (sourceScheduledTrials T n) b :=
+  Equiv.sumCongr (Equiv.prodCongr (finEnumeration _) (Equiv.refl Bool))
+    (Equiv.sumCongr (finEnumeration _) (finEnumeration _))
+
+def sourceAssignmentCode (T n : ℕ → ℕ) (A : Type) (b : ℕ) :
+    WindowAssignments T n (sourceScheduledTrials T n) A b ≃ (Fin (SourceTrialCount T n b) → A) where
+  toFun bits j := windowAssignmentEquiv T n (sourceScheduledTrials T n) A b bits (finEnumeration _ j)
+  invFun bits := (windowAssignmentEquiv T n (sourceScheduledTrials T n) A b).symm
+    (fun i => bits ((finEnumeration _).symm i))
+  left_inv bits := by simp only [Equiv.apply_symm_apply,Equiv.symm_apply_apply]
+  right_inv bits := by
+    funext j
+    simp only [Equiv.apply_symm_apply,Equiv.symm_apply_apply]
+
+theorem sourceAssignmentCode_eval (T n : ℕ → ℕ) (A : Type) (b : ℕ)
+    (bits : Fin (SourceTrialCount T n b) → A) (j : Fin (SourceTrialCount T n b)) :
+    windowAssignmentEquiv T n (sourceScheduledTrials T n) A b
+      ((sourceAssignmentCode T n A b).symm bits) (finEnumeration _ j)=bits j := by
+  change (sourceAssignmentCode T n A b ((sourceAssignmentCode T n A b).symm bits)) j=bits j
+  rw [Equiv.apply_symm_apply]
+
+def sourceChronologyCode (T n : ℕ → ℕ) (b : ℕ) :
+    Fin (treeChronology T n (sourceScheduledTrials T n) (sourceScheduledPre T n) (sourceScheduledPost T n) b).length ≃
+      EmissionSlot (SourceTrialCount T n b) (SourceFillerCount T b) (SourceLeafCount T b) :=
+  (treeChronologyEquiv T n (sourceScheduledTrials T n) (sourceScheduledPre T n) (sourceScheduledPost T n) b).trans
+    (sourceEmissionEnum T n b).symm
+
+def sourceFlatLeafType (T n : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ) (b : ℕ) (incoming : ℤ) (τ : Bool)
+    (bits : Fin (SourceTrialCount T n b) → Bool) (j : Fin (SourceLeafCount T b)) : Bool :=
+  sourceTreeLeafBool T n s center b ((sourceAssignmentCode T n Bool b).symm bits) incoming τ (finEnumeration _ j)
+
+def sourceFlatTrialWord (s : ℕ) (P T n h : ℕ → ℕ) (atom : Bool → List Bool) (b : ℕ)
+    (c : Bool) (j : Fin (SourceTrialCount T n b)) : List Bool :=
+  sourceTrialWord s P T n h atom b (finEnumeration _ j) c
+
+def sourceFlatFillerWord (s : ℕ) (P T n h : ℕ → ℕ) (atom : Bool → List Bool) (b : ℕ)
+    (j : Fin (SourceFillerCount T b)) : List Bool :=
+  sourceFillerWord s P T n h atom b (finEnumeration _ j)
+
+theorem sourceSlotWord_code (s : ℕ) (P T n h : ℕ → ℕ) (center : ℕ → ℤ) (atom : Bool → List Bool)
+    (b : ℕ) (bits : Fin (SourceTrialCount T n b) → Bool) (incoming : ℤ) (τ : Bool)
+    (x : EmissionSlot (SourceTrialCount T n b) (SourceFillerCount T b) (SourceLeafCount T b)) :
+    sourceSlotWord s P T n h center atom b ((sourceAssignmentCode T n Bool b).symm bits) incoming τ
+        (sourceEmissionEnum T n b x)=
+      PairedMechanisms.slotWord (sourceFlatTrialWord s P T n h atom b false)
+        (sourceFlatTrialWord s P T n h atom b true) (sourceFlatFillerWord s P T n h atom b) atom bits
+        (sourceFlatLeafType T n s center b incoming τ bits) x := by
+  rcases x with ⟨j,p⟩ | f | v
+  · cases p <;>
+      change sourceTrialWord s P T n h atom b (finEnumeration _ j)
+        (Bool.xor (windowAssignmentEquiv T n (sourceScheduledTrials T n) Bool b
+          ((sourceAssignmentCode T n Bool b).symm bits) (finEnumeration _ j)) _) = _ <;>
+      rw [sourceAssignmentCode_eval] <;> cases he : bits j <;> simp [PairedMechanisms.slotWord,he,sourceFlatTrialWord]
+  · rfl
+  · rfl
+
+theorem treeChronologyEquiv_word (T n g : ℕ → ℕ)
+    (pre post : (b : ℕ) → Fin (g b) → Fin (T b)) (b : ℕ)
+    (f : TreeEmission T n g b → List Bool) :
+    coordinateWord (fun j => f (treeChronologyEquiv T n g pre post b j))=
+      ((treeChronology T n g pre post b).map f).flatten := by
+  unfold coordinateWord treeChronologyEquiv
+  change (List.ofFn (fun j => f ((treeChronology T n g pre post b).get j))).flatten=_
+  have hh : List.map f (List.ofFn (treeChronology T n g pre post b).get) =
+      List.ofFn (f ∘ (treeChronology T n g pre post b).get) := List.map_ofFn
+  rw [List.ofFn_get] at hh
+  exact congrArg List.flatten hh.symm
+
+theorem source_slotCode_word (s : ℕ) (P T n h : ℕ → ℕ) (center : ℕ → ℤ) (atom : Bool → List Bool)
+    (b : ℕ) (bits : Fin (SourceTrialCount T n b) → Bool) (incoming : ℤ) (τ : Bool) :
+    coordinateWord (fun j => PairedMechanisms.slotWord (sourceFlatTrialWord s P T n h atom b false)
+      (sourceFlatTrialWord s P T n h atom b true) (sourceFlatFillerWord s P T n h atom b) atom bits
+      (sourceFlatLeafType T n s center b incoming τ bits) (sourceChronologyCode T n b j))=
+    sourceChronologicalWord s P T n h center atom b ((sourceAssignmentCode T n Bool b).symm bits) incoming τ := by
+  simp_rw [←sourceSlotWord_code]
+  simp only [sourceChronologyCode,Equiv.trans_apply,Equiv.apply_symm_apply]
+  exact treeChronologyEquiv_word T n (sourceScheduledTrials T n) (sourceScheduledPre T n)
+    (sourceScheduledPost T n) b _
+
+end TraceReconstruction
+
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- Reindex the genuine independent orientations, retaining the whole source tree. -/
+theorem source_fair_code_expect (T n : ℕ → ℕ) (b : ℕ)
+    (F : WindowAssignments T n (sourceScheduledTrials T n) Bool b → ℝ) :
+    (PairedMechanisms.fairOrientations (SourceTrialCount T n b)).expect
+      (fun bits => F ((sourceAssignmentCode T n Bool b).symm bits)) =
+    (windowPosteriorLaw T n (sourceScheduledTrials T n) b
+      (noReveals T n (sourceScheduledTrials T n) b)).expect F := by
+  rw [windowPosterior_fair_flat,FiniteLaw.indexedProduct_expect_equiv _ (finEnumeration _)]
+  simp only [PairedMechanisms.fairOrientations,FiniteLaw.expect,FiniteLaw.product,
+    FiniteLaw.indexedProduct,sourceAssignmentCode,Equiv.coe_fn_symm_mk]
+  apply Finset.sum_congr (by ext; simp)
+  intro bits _
+  rfl
+
+/-- The fit event uses only interior even bundles. In particular no parity
+ assumption is made on the top root count J. -/
+theorem sourceTreeFit_tail_interior (T n : ℕ → ℕ) (b : ℕ)
+    (hn : ∀j,j+1<b → 0<n j) (he : ∀j,j+1<b → Even (n j)) :
+    (windowPosteriorLaw T n (sourceScheduledTrials T n) b
+      (noReveals T n (sourceScheduledTrials T n) b)).expect
+        (fun bits => if SourceTreeFit T n b bits then 0 else 1) ≤ sourceFitTail T n b := by
+  induction b using Nat.twoStepInduction with
+  | zero =>
+    have hh (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool 0) : SourceTreeFit T n 0 bits := trivial
+    simp only [ite_eq_left (hh _),FiniteLaw.expect_const,sourceFitTail,le_refl]
+  | one =>
+    have hh (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool 1) : SourceTreeFit T n 1 bits := trivial
+    simp only [ite_eq_left (hh _),FiniteLaw.expect_const,sourceFitTail,le_refl]
+  | more b ih0 ih =>
+    let P := windowPosteriorLaw T n (sourceScheduledTrials T n) (b+2)
+      (noReveals T n (sourceScheduledTrials T n) (b+2))
+    let badShift : WindowAssignments T n (sourceScheduledTrials T n) Bool (b+2) → Fin (T (b+1)) → Prop :=
+      fun bits u => (T b:ℝ)/8≤|(sourceDisplacement (T (b+1)) (n b)
+        (fun v => (bits.2.2 v).1) bits.2.1 u:ℝ)|
+    have hp (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool (b+2)) :
+        (if SourceTreeFit T n (b+2) bits then (0:ℝ) else 1)≤
+          (∑u : Fin (T (b+1)),if badShift bits u then 1 else 0)+
+          (∑u : Fin (T (b+1)),if SourceTreeFit T n (b+1) (bits.2.2 u) then 0 else 1) := by
+      let p := fun u : Fin (T (b+1)) => |8*sourceDisplacement (T (b+1)) (n b)
+        (fun v => (bits.2.2 v).1) bits.2.1 u|≤(T b:ℤ)
+      have hh : (if SourceTreeFit T n (b+2) bits then (0:ℝ) else 1)≤
+          (if ∀u,p u then 0 else 1)+(if ∀u,SourceTreeFit T n (b+1) (bits.2.2 u) then 0 else 1) := by
+        by_cases H : SourceTreeFit T n (b+2) bits
+        · rw [ite_eq_left H]
+          split_ifs <;> norm_num
+        · rw [ite_eq_right H]
+          by_cases h1 : ∀u,p u
+          · have h2 : ¬∀u,SourceTreeFit T n (b+1) (bits.2.2 u) := fun h2 => H ⟨h1,h2⟩
+            rw [ite_eq_left h1,ite_eq_right h2]
+            norm_num
+          · rw [ite_eq_right h1]
+            split_ifs <;> norm_num
+      apply hh.trans
+      apply add_le_add
+      · apply (failure_forall_le_sum p).trans
+        apply Finset.sum_le_sum
+        intro u _
+        by_cases hu : p u
+        · simp only [ite_eq_left hu]
+          split_ifs <;> norm_num
+        · have hr : (T b:ℝ)<8*|(sourceDisplacement (T (b+1)) (n b)
+              (fun v => (bits.2.2 v).1) bits.2.1 u:ℝ)| := by
+            have hi : (T b:ℤ) < |8*sourceDisplacement (T (b+1)) (n b)
+              (fun v => (bits.2.2 v).1) bits.2.1 u| := lt_of_not_ge hu
+            have hi' : (T b:ℝ) < |(8*sourceDisplacement (T (b+1)) (n b)
+              (fun v => (bits.2.2 v).1) bits.2.1 u:ℤ)| := by exact_mod_cast hi
+            norm_num only [Int.cast_abs,Int.cast_mul,Int.cast_ofNat,abs_mul,abs_of_pos (by norm_num : (0:ℝ)<8)] at hi'
+            exact hi'
+          have hb : badShift bits u := by dsimp [badShift]; linarith
+          simp [hu,hb]
+      · exact failure_forall_le_sum _
+    calc
+      _ ≤ P.expect (fun bits => (∑u : Fin (T (b+1)),if badShift bits u then 1 else 0)+
+          (∑u : Fin (T (b+1)),if SourceTreeFit T n (b+1) (bits.2.2 u) then 0 else 1)) :=
+            P.expect_mono hp
+      _ = (∑u : Fin (T (b+1)),P.expect (fun bits => if badShift bits u then 1 else 0))+
+          (∑u : Fin (T (b+1)),P.expect (fun bits => if SourceTreeFit T n (b+1) (bits.2.2 u) then 0 else 1)) := by
+            rw [FiniteLaw.expect_add,FiniteLaw.expect_fintype_sum,FiniteLaw.expect_fintype_sum]
+      _ ≤ (∑_u : Fin (T (b+1)),2*Real.exp (-2*((T b:ℝ)/8)^2/(((T (b+1):ℝ)+1)*(n b:ℝ))))+
+          ∑_u : Fin (T (b+1)),sourceFitTail T n (b+1) := by
+            apply add_le_add
+            · apply Finset.sum_le_sum
+              intro u _
+              exact sourceDisplacement_tail T n b (he b (by omega)) (hn b (by omega)) u (by positivity)
+            · apply Finset.sum_le_sum
+              intro u _
+              dsimp only [P]
+              rw [windowPosterior_child T n (sourceScheduledTrials T n) (b+1)
+                (noReveals T n (sourceScheduledTrials T n) (b+2)) u
+                (fun x => if SourceTreeFit T n (b+1) x then 0 else 1)]
+              rw [noReveals_succ]
+              exact ih (fun j hj => hn j (by omega)) (fun j hj => he j (by omega))
+      _ = sourceFitTail T n (b+2) := by simp [sourceFitTail,mul_add]
+
+
+/-- Honest source joint-fit failure after the actual finite coding. -/
+theorem source_flat_fit_tail (T n : ℕ → ℕ) (b : ℕ)
+    (hn : ∀j,j+1<b → 0<n j) (he : ∀j,j+1<b → Even (n j)) :
+    (PairedMechanisms.fairOrientations (SourceTrialCount T n b)).expect
+      (fun bits => if SourceTreeFit T n b ((sourceAssignmentCode T n Bool b).symm bits) then 0 else 1)
+      ≤ sourceFitTail T n b := by
+  rw [source_fair_code_expect T n b (fun bits => if SourceTreeFit T n b bits then 0 else 1)]
+  exact sourceTreeFit_tail_interior T n b hn he
+
+/-- A fixed public decoder. It does not know the hidden orientations or the
+ input hypothesis, and it drops rather than reveals all pairing flags. -/
+def sourceOracleDecode {Side Leaf : Type} (T n : ℕ → ℕ) (b : ℕ)
+    (f : Side → List Bool) (g : Leaf → List Bool) :
+    ((Reports Side (SourceTrialCount T n b) × (Fin (SourceLeafCount T b) → Leaf)) ×
+      (Fin (SourceFillerCount T b) → Side)) → List Bool := fun x =>
+  coordinateWord (fun j => PairedMechanisms.slotDecode f g (sourceChronologyCode T n b j)
+    (emissionAssignmentEquiv Side Leaf (SourceTrialCount T n b) (SourceFillerCount T b)
+      (SourceLeafCount T b) (fun i => (x.1.1 i).2,x.2,x.1.2) (sourceChronologyCode T n b j)))
+
+/-- The complete canonical oracle now simulates the LITERAL source block.
+ All IDs, words, leaf types and chronology instantiate the actual construction.
+ The only failed-layout cost is its original unconditional fair-bit mass. -/
+theorem source_canonical_simulation {Side Leaf : Type} [Fintype Side] [Fintype Leaf]
+    {q E : ℝ} (hq : q∈Set.Icc 0 1) (hE : 0≤E)
+    (s : ℕ) (P T n h : ℕ → ℕ) (atom : Bool → List Bool) (b : ℕ)
+    (H : SourceLayoutMargins s P T n h b)
+    (hP : P b=n b*(2*h b+1)+T b) (hT : Even (T b))
+    (hlarge : 4*(n b+1)≤T b) (hmotif : 8*2^s≤T b)
+    (hn : ∀j<b,0<n j)
+    (M : PairedMechanisms Side (SourceTrialCount T n (b+1)))
+    (R : Fin (SourceFillerCount T (b+1)) → FiniteLaw Side) (A B : FiniteLaw Leaf)
+    (f : Side → List Bool) (g : Leaf → List Bool)
+    (hU : ∀j,SimulatesTrace q (M.first j) f (sourceFlatTrialWord s P T n h atom (b+1) false j))
+    (hV : ∀j,SimulatesTrace q (M.second j) f (sourceFlatTrialWord s P T n h atom (b+1) true j))
+    (hW : ∀j,SimulatesTrace q (R j) f (sourceFlatFillerWord s P T n h atom (b+1) j))
+    (hA : ApproximatesTrace q A g (atom false) E) (hB : ApproximatesTrace q B g (atom true) E)
+    (τ : Bool) :
+    ApproximatesTrace q
+      ((M.canonicalOracle A B (sourceFlatLeafType T n s (sourceLocalCenter T n) (b+1) 0 τ)).prod
+        (FiniteLaw.coordinateProduct R))
+      (sourceOracleDecode T n (b+1) f g) (sourceBlockWord s P atom (b+1) τ)
+      ((SourceLeafCount T (b+1):ℝ)*E+sourceFitTail T n (b+1)) := by
+  let good := fun bits : Fin (SourceTrialCount T n (b+1)) → Bool =>
+    SourceTreeFit T n (b+1) ((sourceAssignmentCode T n Bool (b+1)).symm bits)
+  have hw (bits : Fin (SourceTrialCount T n (b+1)) → Bool) (hg : good bits) :=
+    sourceChronologicalWord_eq_block s P T n h atom b H hP hT hlarge hmotif
+      ((sourceAssignmentCode T n Bool (b+1)).symm bits) τ hg
+  have hs := M.canonical_filled_simulation hq hE (sourceChronologyCode T n (b+1)) R A B f g
+    (sourceFlatTrialWord s P T n h atom (b+1) false)
+    (sourceFlatTrialWord s P T n h atom (b+1) true)
+    (sourceFlatFillerWord s P T n h atom (b+1)) atom hU hV hW hA hB
+    (sourceFlatLeafType T n s (sourceLocalCenter T n) (b+1) 0 τ)
+    (sourceBlockWord s P atom (b+1) τ) good (fun bits hg => by
+      rw [source_slotCode_word]
+      exact hw bits hg)
+  intro F hF
+  exact (hs F hF).trans (add_le_add le_rfl (source_flat_fit_tail T n (b+1) (fun j hj => hn j (by omega))
+    (fun j hj => H.even_trials j (by omega))))
+
+end TraceReconstruction
+
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/TraceReconstruction/SourceOracle.lean b/lean/OAI/Probability/TraceReconstruction/SourceOracle.lean
index 8a47bec4e0fcb0e5240c8047231614e626ca9367..adbd0bb7f52562fe02ff2d5a81814b12b23a2edd 100644
--- a/lean/OAI/Probability/TraceReconstruction/SourceOracle.lean
+++ b/lean/OAI/Probability/TraceReconstruction/SourceOracle.lean
@@ -1,1852 +1,1852 @@
-import OAI.Probability.TraceReconstruction.Simulation
-
-namespace OAI
-
-/-!
-# SourceOracle
-
-Actual source mechanisms and posteriors, typical metadata, sparse degrees, common-event clipping, approximate oracles, and mixed Hellinger bounds.
--/
-
-open scoped ENNReal
-noncomputable section
-
-open Filter
-open scoped Topology
-
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- Ambient finite trace-space choice does not lose any of the genuine
-complete-trace common mass. The embedding adds only zero-mass points. -/
-theorem listTraceOverlap_le_ambient {q : ℝ} (hq : q∈Set.Icc 0 1)
-    (x y : List Bool) (M : ℕ) (hx : x.length≤M) (hy : y.length≤M) :
-    listTraceOverlap q hq x y≤
-      (listTraceLaw q hq x M hx).overlap (listTraceLaw q hq y M hy) := by
-  let M₀ := max x.length y.length
-  have H := approximations_trace_tv hq
-    (listTraceLaw q hq x M₀ (le_max_left _ _))
-    (listTraceLaw q hq y M₀ (le_max_right _ _)) Subtype.val x y 0 0 M hx hy
-    (listTraceLaw_simulates q hq x M₀ _).approximate
-    (listTraceLaw_simulates q hq y M₀ _).approximate
-  simp only [add_zero] at H
-  unfold listTraceOverlap
-  rw [FiniteLaw.overlap_eq_one_sub_tv,FiniteLaw.overlap_eq_one_sub_tv]
-  linarith
-
-theorem listTraceOverlap_ambient {q : ℝ} (hq : q∈Set.Icc 0 1)
-    (x y : List Bool) (M : ℕ) (hx : x.length≤M) (hy : y.length≤M) :
-    listTraceOverlap q hq x y=
-      (listTraceLaw q hq x M hx).overlap (listTraceLaw q hq y M hy) := by
-  apply le_antisymm (listTraceOverlap_le_ambient hq x y M hx hy)
-  exact simulations_overlap_le hq _ _ Subtype.val x y
-    (listTraceLaw_simulates q hq x M hx) (listTraceLaw_simulates q hq y M hy)
-
-namespace FiniteLaw
-variable {α : Type*} [Fintype α]
-theorem commonPart_le_right (P Q : FiniteLaw α) (h : 0<P.overlap Q)
-    {ω : ℝ} (hω : ω≤P.overlap Q) (a : α) :
-    ω*(commonPart P Q h).mass a≤Q.mass a := by
-  change ω*(min (P.mass a) (Q.mass a)/P.overlap Q)≤_
-  calc
-    _ ≤ P.overlap Q*(min (P.mass a) (Q.mass a)/P.overlap Q) :=
-      mul_le_mul_of_nonneg_right hω (div_nonneg (le_min (P.nonneg a) (Q.nonneg a)) h.le)
-    _ = min (P.mass a) (Q.mass a) := by field_simp
-    _ ≤ _ := min_le_right _ _
-end FiniteLaw
-
-/-- The paired experiment uses the actual unseparated complete deletion
-traces, and the actual normalized minimum of their mass functions. -/
-def tracePairedMechanisms {m : ℕ} {q ω : ℝ} (hq : q∈Set.Icc 0 1) (hω : 0<ω)
-    (U V : Fin m → List Bool) (M : ℕ) (hU : ∀j,(U j).length≤M) (hV : ∀j,(V j).length≤M)
-    (hov : ∀j,ω≤listTraceOverlap q hq (U j) (V j)) : PairedMechanisms (BoundedTrace M) m where
-  first j := listTraceLaw q hq (U j) M (hU j)
-  second j := listTraceLaw q hq (V j) M (hV j)
-  common j := FiniteLaw.commonPart _ _ (hω.trans_le ((hov j).trans (listTraceOverlap_le_ambient hq _ _ M _ _)))
-  hiddenWeight _ := ω
-  weight_nonneg _ := hω.le
-  common_first j := FiniteLaw.commonPart_le_left _ _ _
-    ((hov j).trans (listTraceOverlap_le_ambient hq _ _ M _ _))
-  common_second j := FiniteLaw.commonPart_le_right _ _ _
-    ((hov j).trans (listTraceOverlap_le_ambient hq _ _ M _ _))
-
-/-- Every literal side pair draws its overlap from the correct height of the
-source recursion, including reflected global pairs. -/
-theorem sourceTrialWord_overlap {q ω : ℝ} (hq : q∈Set.Icc 0 1)
-    (s : ℕ) (P T n h : ℕ → ℕ) (atom : Bool → List Bool) (b : ℕ)
-    (hov : ∀j<b,ω≤listTraceOverlap q hq
-      (pureAtomRun (sourceBlockWord s P atom j false) (h j))
-      (pureAtomRun (sourceBlockWord s P atom j false) (h j+1)))
-    (i : WindowTrial T n (sourceScheduledTrials T n) b) :
-    ω≤listTraceOverlap q hq (sourceTrialWord s P T n h atom b i false)
-      (sourceTrialWord s P T n h atom b i true) := by
-  induction b with
-  | zero => exact PEmpty.elim i
-  | succ b ih =>
-    rcases i with i | i | ⟨u,i⟩
-    · simpa [sourceTrialWord,bitIndex] using hov b (by omega)
-    · cases b with
-      | zero => exact Fin.elim0 i
-      | succ b => simpa [sourceTrialWord,bitIndex] using hov b (by omega)
-    · exact ih (fun j hj => hov j (by omega)) i
-
-/-- A common finite side-trace space exists without changing any word. -/
-theorem finite_word_bound {ι : Type*} [Fintype ι] (w : ι → List Bool) :
-    ∃M,∀i,(w i).length≤M := by
-  refine ⟨∑i,(w i).length, fun i => ?_⟩
-  exact Finset.single_le_sum (f:=fun i => (w i).length) (fun _ _ => Nat.zero_le _) (Finset.mem_univ i)
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- The actual report flags, in the literal nested trial coordinates. -/
-def sourceReportFlags {Side : Type*} (T n : ℕ → ℕ) (b : ℕ)
-    (z : Reports Side (SourceTrialCount T n b)) :
-    WindowAssignments T n (sourceScheduledTrials T n) (Option Bool) b :=
-  (sourceAssignmentCode T n (Option Bool) b).symm (fun j => (z j).1)
-
-/-- Exact change of coordinates for the posterior of all hidden bits, not
- just separate local marginals. -/
-theorem source_posterior_code_expect {Side : Type*} (T n : ℕ → ℕ) (b : ℕ)
-    (z : Reports Side (SourceTrialCount T n b))
-    (F : WindowAssignments T n (sourceScheduledTrials T n) Bool b → ℝ) :
-    (PairedMechanisms.orientationsGiven z).expect
-      (fun bits => F ((sourceAssignmentCode T n Bool b).symm bits)) =
-    (windowPosteriorLaw T n (sourceScheduledTrials T n) b (sourceReportFlags T n b z)).expect F := by
-  rw [windowPosterior_expect_flat,FiniteLaw.indexedProduct_expect_equiv _ (finEnumeration _)]
-  simp only [sourceReportFlags,sourceAssignmentCode_eval]
-  simp only [PairedMechanisms.orientationsGiven,FiniteLaw.expect,FiniteLaw.coordinateProduct,
-    FiniteLaw.indexedProduct,sourceAssignmentCode,Equiv.coe_fn_symm_mk]
-  apply Finset.sum_congr (by ext; simp)
-  intro bits _
-  rfl
-
-/-- A monomial is precisely its set of actual leaf addresses. -/
-def sourceSubsetCode (T : ℕ → ℕ) (b : ℕ) :
-    (Fin (SourceLeafCount T b) → Bool) ≃ Finset (WindowLeaf T b) :=
-  (maskSubsetEquiv _).trans (Equiv.finsetCongr (finEnumeration _))
-
-theorem sourceSubsetCode_card (T : ℕ → ℕ) (b : ℕ)
-    (σ : Fin (SourceLeafCount T b) → Bool) :
-    (sourceSubsetCode T b σ).card=monomialDegree σ := by
-  simp only [sourceSubsetCode,Equiv.trans_apply,Equiv.finsetCongr_apply,Finset.card_map]
-  rfl
-
-theorem selectionCoefficient_sign {k : ℕ} (θ σ : Fin k → Bool) :
-    selectionCoefficient θ σ=∏j∈maskSubset σ,(1-boolTypeSign (θ j))/2 := by
-  unfold selectionCoefficient maskSubset
-  rw [Finset.prod_filter]
-  apply Finset.prod_congr rfl
-  intro j _
-  cases σ j <;> cases θ j <;> norm_num [boolTypeSign]
-
-theorem source_leaf_selection (T n : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (b : ℕ) (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool b)
-    (incoming : ℤ) (τ : Bool) (σ : Fin (SourceLeafCount T b) → Bool) :
-    selectionCoefficient
-      (fun j => sourceTreeLeafBool T n s center b bits incoming τ (finEnumeration _ j)) σ =
-    ∏v∈sourceSubsetCode T b σ,
-      (1-sourceTreeLeafSign T n (sourceScheduledTrials T n) s center
-        (sourceScheduledShift T n) b bits incoming (boolTypeSign τ) v)/2 := by
-  rw [selectionCoefficient_sign]
-  simp only [sourceSubsetCode,Equiv.trans_apply,Equiv.finsetCongr_apply,Finset.prod_map]
-  apply Finset.prod_congr rfl
-  intro j _
-  rw [sourceTreeLeafBool_sign]
-  rfl
-
-/-- The real canonical-oracle coefficient difference is the exact source
- polynomial coefficient whose forced-tree energy has been proved. -/
-theorem source_canonical_coefficient {Side : Type*} (T n : ℕ → ℕ) (s : ℕ)
-    (center : ℕ → ℤ) (b : ℕ) (incoming : ℤ)
-    (z : Reports Side (SourceTrialCount T n (b+1)))
-    (σ : Fin (SourceLeafCount T (b+1)) → Bool) :
-    averagedCoefficient (PairedMechanisms.orientationsGiven z)
-      (sourceFlatLeafType T n s center (b+1) incoming false) σ -
-    averagedCoefficient (PairedMechanisms.orientationsGiven z)
-      (sourceFlatLeafType T n s center (b+1) incoming true) σ =
-    sourceCoefficient T n (sourceScheduledTrials T n) s center (sourceScheduledShift T n)
-      b (sourceReportFlags T n (b+1) z) incoming (sourceSubsetCode T (b+1) σ) := by
-  unfold averagedCoefficient sourceFlatLeafType
-  rw [source_posterior_code_expect T n (b+1) z
-    (fun bits => selectionCoefficient
-      (fun j => sourceTreeLeafBool T n s center (b+1) bits incoming false (finEnumeration _ j)) σ),
-    source_posterior_code_expect T n (b+1) z
-    (fun bits => selectionCoefficient
-      (fun j => sourceTreeLeafBool T n s center (b+1) bits incoming true (finEnumeration _ j)) σ)]
-  simp_rw [source_leaf_selection]
-  rfl
-
-/-- Reindexing preserves the entire degree-r square sum, including zero
- and degrees larger than the number of real leaves. -/
-theorem source_canonical_degree_sum {Side : Type*} (T n : ℕ → ℕ) (s : ℕ)
-    (center : ℕ → ℤ) (b r : ℕ) (incoming : ℤ)
-    (z : Reports Side (SourceTrialCount T n (b+1))) :
-    (∑σ : Fin (SourceLeafCount T (b+1)) → Bool, if monomialDegree σ=r then
-      (averagedCoefficient (PairedMechanisms.orientationsGiven z)
-        (sourceFlatLeafType T n s center (b+1) incoming false) σ -
-       averagedCoefficient (PairedMechanisms.orientationsGiven z)
-        (sourceFlatLeafType T n s center (b+1) incoming true) σ)^2 else 0) =
-    ∑S : DegreeLeafSubset T b r,
-      (sourceCoefficient T n (sourceScheduledTrials T n) s center (sourceScheduledShift T n)
-        b (sourceReportFlags T n (b+1) z) incoming S.val)^2 := by
-  simp_rw [source_canonical_coefficient,←sourceSubsetCode_card]
-  rw [(sourceSubsetCode T (b+1)).sum_comp
-    (fun S => if S.card=r then (sourceCoefficient T n (sourceScheduledTrials T n) s center
-      (sourceScheduledShift T n) b (sourceReportFlags T n (b+1) z) incoming S)^2 else 0)]
-  let f : DegreeLeafSubset T b r → ℝ := fun S =>
-    (sourceCoefficient T n (sourceScheduledTrials T n) s center (sourceScheduledShift T n)
-      b (sourceReportFlags T n (b+1) z) incoming S.val)^2
-  simpa only [f, dite_eq_ite] using
-    sum_subtype_zero_extend (fun S : Finset (WindowLeaf T (b+1)) => S.card=r) f
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-namespace PairedMechanisms
-variable {Side : Type*} [Fintype Side] {m : ℕ} (M : PairedMechanisms Side m)
-
-def hiddenSubset (S : Finset (Fin m)) (z : Reports Side m) : ℕ := ∑i : S,hiddenTrial (z i.val).1
-
-theorem hiddenSubset_mgf (S : Finset (Fin m)) {ω t : ℝ} (hw : ∀i,M.hiddenWeight i=ω)
-    (hp : ω^2∈Set.Icc 0 1) (ht : |t|≤1) :
-    M.reports.expect (fun z => Real.exp (t*((hiddenSubset S z:ℕ)-(S.card:ℝ)*ω^2)))≤
-      Real.exp ((S.card:ℝ)*t^2) := by
-  have he (z : Reports Side m) :
-      Real.exp (t*((hiddenSubset S z:ℕ)-(S.card:ℝ)*ω^2))=
-        ∏i : S,Real.exp (t*((hiddenTrial (z i.val).1:ℕ)-ω^2)) := by
-    rw [←Real.exp_sum]
-    congr 1
-    simp only [mul_sub,Finset.mul_sum,Finset.sum_sub_distrib,Finset.sum_const,Finset.card_univ,Fintype.card_coe,
-      nsmul_eq_mul,hiddenSubset,Nat.cast_sum]
-    ring
-  simp_rw [he]
-  rw [reports,FiniteLaw.coordinateProduct_expect_subtype _ S
-    (fun (_ : S) (z : Option Bool × (Side × Side)) => Real.exp (t*((hiddenTrial z.1:ℕ)-ω^2)))]
-  calc
-    _ ≤ ∏i : S,Real.exp (t^2) := by
-      apply Finset.prod_le_prod₀
-      · intro i _
-        exact Finset.sum_nonneg (fun z _ => mul_nonneg
-          ((FiniteLaw.pairedReport (M.first i.val) (M.second i.val) (M.common i.val) (M.weight_nonneg i.val)
-            (M.common_first i.val) (M.common_second i.val)).nonneg z) (Real.exp_nonneg _))
-      · intro i _
-        have h := FiniteLaw.pairedReport_hidden_mgf (M.first i.val) (M.second i.val) (M.common i.val)
-          (M.weight_nonneg i.val) (M.common_first i.val) (M.common_second i.val) (by simpa [hw i.val] using hp) ht
-        simpa only [hw i.val] using h
-    _ = _ := by simp only [Finset.prod_const,Finset.card_univ,Fintype.card_coe,Real.exp_nat_mul]
-
-/-- A quantitative unconditional bound for failure of a source hidden-count
- condition. The exponent ω⁴m is sufficient for the uniform scale margins. -/
-theorem hiddenSubset_small (S : Finset (Fin m)) {ω : ℝ} (hw : ∀i,M.hiddenWeight i=ω) (hp : ω^2∈Set.Icc 0 1) :
-    M.reports.expect (fun z => if (hiddenSubset S z:ℝ)<(S.card:ℝ)*ω^2/2 then 1 else 0)≤
-      Real.exp (-(S.card:ℝ)*ω^4/16) := by
-  have ht : 0≤ω^2/4 := by positivity
-  have ht' : |-(ω^2/4)|≤1 := by rw [abs_neg,abs_of_nonneg ht]; linarith [hp.2]
-  have H := finite_exp_markov Finset.univ M.reports.mass
-    (fun z => -((hiddenSubset S z:ℕ)-(S.card:ℝ)*ω^2)) (fun z _ => M.reports.nonneg z)
-    ((S.card:ℝ)*ω^2/2) (ω^2/4) ht
-  have hm := M.hiddenSubset_mgf S hw hp ht'
-  have hsub : M.reports.expect (fun z => if (hiddenSubset S z:ℝ)<(S.card:ℝ)*ω^2/2 then 1 else 0)≤
-      ∑z∈Finset.univ.filter (fun z => (S.card:ℝ)*ω^2/2≤-((hiddenSubset S z:ℕ)-(S.card:ℝ)*ω^2)),M.reports.mass z := by
-    rw [Finset.sum_filter]
-    apply Finset.sum_le_sum
-    intro z _
-    dsimp only
-    by_cases hz : (hiddenSubset S z:ℝ)<(S.card:ℝ)*ω^2/2
-    · rw [ite_eq_left hz,ite_eq_left (by linarith : (S.card:ℝ)*ω^2/2≤-((hiddenSubset S z:ℕ)-(S.card:ℝ)*ω^2)),mul_one]
-    · rw [ite_eq_right hz,mul_zero]; split_ifs <;> positivity [M.reports.nonneg z]
-  calc
-    _ ≤ _ := hsub.trans H
-    _ ≤ Real.exp (-(ω^2/4)*((S.card:ℝ)*ω^2/2))*Real.exp ((S.card:ℝ)*(-(ω^2/4))^2) := by
-      apply mul_le_mul_of_nonneg_left _ (Real.exp_nonneg _)
-      simpa only [FiniteLaw.expect,neg_mul,mul_neg,neg_neg] using hm
-    _ = _ := by rw [←Real.exp_add]; congr 1; ring
-
-
-end PairedMechanisms
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- IDs for precisely one local and one global collection at every genuine
- nonleaf of the complete source window tree. -/
-def WindowCollection (T : ℕ → ℕ) : ℕ → Type
-  | 0 => PEmpty
-  | b+1 => Bool ⊕ (Fin (T b) × WindowCollection T b)
-
-noncomputable instance windowCollectionFintype (T : ℕ → ℕ) :
-    (b : ℕ) → Fintype (WindowCollection T b)
-  | 0 => inferInstanceAs (Fintype PEmpty)
-  | b+1 => by
-    letI := windowCollectionFintype T b
-    exact inferInstanceAs (Fintype (Bool ⊕ (Fin (T b) × WindowCollection T b)))
-
-def collectionCount (T n g : ℕ → ℕ) : (b : ℕ) → WindowCollection T b → ℕ
-  | b+1, .inl false => n b
-  | b+1, .inl true => g b
-  | b+1, .inr (_,v) => collectionCount T n g b v
-
-def collectionTrial (T n g : ℕ → ℕ) :
-    (b : ℕ) → (v : WindowCollection T b) → Fin (collectionCount T n g b v) → WindowTrial T n g b
-  | _+1, .inl false, j => .inl j
-  | _+1, .inl true, j => .inr (.inl j)
-  | b+1, .inr (u,v), j => .inr (.inr (u,collectionTrial T n g b v j))
-
-theorem collectionTrial_injective (T n g : ℕ → ℕ) (b : ℕ) (v : WindowCollection T b) :
-    Function.Injective (collectionTrial T n g b v) := by
-  induction b with
-  | zero => exact PEmpty.elim v
-  | succ b ih =>
-    rcases v with v | ⟨u,v⟩
-    · cases v
-      · intro i j h
-        exact Sum.inl.inj h
-      · intro i j h
-        exact Sum.inl.inj (Sum.inr.inj h)
-    · intro i j h
-      apply ih v
-      exact congrArg Prod.snd (Sum.inr.inj (Sum.inr.inj h))
-
-def collectionFlags (T n g : ℕ → ℕ) {A : Type} :
-    (b : ℕ) → WindowAssignments T n g A b → (v : WindowCollection T b) →
-      Fin (collectionCount T n g b v) → A
-  | _+1, d, .inl false => d.1
-  | _+1, d, .inl true => d.2.1
-  | b+1, d, .inr (u,v) => collectionFlags T n g b (d.2.2 u) v
-
-theorem collectionFlags_trial (T n g : ℕ → ℕ) {A : Type} (b : ℕ)
-    (d : WindowAssignments T n g A b) (v : WindowCollection T b)
-    (j : Fin (collectionCount T n g b v)) :
-    collectionFlags T n g b d v j=
-      windowAssignmentEquiv T n g A b d (collectionTrial T n g b v j) := by
-  induction b with
-  | zero => exact PEmpty.elim v
-  | succ b ih =>
-    rcases v with v | ⟨u,v⟩
-    · cases v <;> rfl
-    · exact ih (d.2.2 u) v j
-
-/-- This is the source hidden-count predicate, before any conditioning on fit. -/
-def HiddenCollectionConditions (T n g : ℕ → ℕ) (ω : ℝ) :
-    (b : ℕ) → WindowAssignments T n g (Option Bool) b → Prop
-  | 0, _ => True
-  | b+1, d =>
-      (n b:ℝ)*ω^2/2≤∑i,(hiddenTrial (d.1 i):ℝ) ∧
-      (g b:ℝ)*ω^2/2≤∑i,(hiddenTrial (d.2.1 i):ℝ) ∧
-      ∀u,HiddenCollectionConditions T n g ω b (d.2.2 u)
-
-theorem hiddenCollectionConditions_iff (T n g : ℕ → ℕ) (ω : ℝ) (b : ℕ)
-    (d : WindowAssignments T n g (Option Bool) b) :
-    HiddenCollectionConditions T n g ω b d ↔
-      ∀v : WindowCollection T b,(collectionCount T n g b v:ℝ)*ω^2/2≤
-        ∑i,(hiddenTrial (collectionFlags T n g b d v i):ℝ) := by
-  induction b with
-  | zero => simp [HiddenCollectionConditions,WindowCollection]
-  | succ b ih =>
-    constructor
-    · intro h v
-      rcases v with v | ⟨u,v⟩
-      · cases v
-        · exact h.1
-        · exact h.2.1
-      · exact (ih (d.2.2 u)).mp (h.2.2 u) v
-    · intro h
-      refine ⟨h (.inl false),h (.inl true),fun u => (ih (d.2.2 u)).mpr ?_⟩
-      intro v
-      exact h (.inr (u,v))
-
-def sourceCollectionIndex (T n : ℕ → ℕ) (b : ℕ) (v : WindowCollection T b) :
-    Fin (collectionCount T n (sourceScheduledTrials T n) b v) → Fin (SourceTrialCount T n b) :=
-  fun j => (finEnumeration _).symm (collectionTrial T n (sourceScheduledTrials T n) b v j)
-
-theorem sourceCollectionIndex_injective (T n : ℕ → ℕ) (b : ℕ) (v : WindowCollection T b) :
-    Function.Injective (sourceCollectionIndex T n b v) :=
-  (finEnumeration _).symm.injective.comp (collectionTrial_injective T n _ b v)
-
-def sourceCollectionSet (T n : ℕ → ℕ) (b : ℕ) (v : WindowCollection T b) :
-    Finset (Fin (SourceTrialCount T n b)) := Finset.univ.image (sourceCollectionIndex T n b v)
-
-theorem sourceCollectionSet_card (T n : ℕ → ℕ) (b : ℕ) (v : WindowCollection T b) :
-    (sourceCollectionSet T n b v).card=collectionCount T n (sourceScheduledTrials T n) b v := by
-  rw [sourceCollectionSet,Finset.card_image_of_injective _ (sourceCollectionIndex_injective T n b v)]
-  simp only [Finset.card_univ,Fintype.card_fin]
-
-theorem sourceCollection_flags {Side : Type*} (T n : ℕ → ℕ) (b : ℕ)
-    (z : Reports Side (SourceTrialCount T n b)) (v : WindowCollection T b)
-    (j : Fin (collectionCount T n (sourceScheduledTrials T n) b v)) :
-    collectionFlags T n (sourceScheduledTrials T n) b (sourceReportFlags T n b z) v j=
-      (z (sourceCollectionIndex T n b v j)).1 := by
-  rw [collectionFlags_trial]
-  have h := sourceAssignmentCode_eval T n (Option Bool) b (fun j => (z j).1)
-    (sourceCollectionIndex T n b v j)
-  simpa only [sourceCollectionIndex,Equiv.apply_symm_apply,sourceReportFlags] using h
-
-theorem sourceCollection_hidden {Side : Type*} (T n : ℕ → ℕ) (b : ℕ)
-    (z : Reports Side (SourceTrialCount T n b)) (v : WindowCollection T b) :
-    PairedMechanisms.hiddenSubset (sourceCollectionSet T n b v) z=
-      ∑i,hiddenTrial (collectionFlags T n (sourceScheduledTrials T n) b (sourceReportFlags T n b z) v i) := by
-  unfold PairedMechanisms.hiddenSubset
-  rw [←Finset.sum_subtype (sourceCollectionSet T n b v) (fun _ => Iff.rfl)
-    (fun j => hiddenTrial (z j).1)]
-  rw [sourceCollectionSet,Finset.sum_image]
-  · simp_rw [sourceCollection_flags]
-  · exact (sourceCollectionIndex_injective T n b v).injOn
-
-/-- Exact finite-union metadata cost; it contains no postulated typical
- event and no independence assertion between different collections. -/
-def sourceHiddenTail (T n : ℕ → ℕ) (b : ℕ) (ω : ℝ) : ℝ :=
-  ∑v : WindowCollection T b,Real.exp (-(collectionCount T n (sourceScheduledTrials T n) b v:ℝ)*ω^4/16)
-
-theorem source_hidden_conditions_tail {Side : Type*} [Fintype Side]
-    (T n : ℕ → ℕ) (b : ℕ) (M : PairedMechanisms Side (SourceTrialCount T n b))
-    (ω : ℝ) (hw : ∀i,M.hiddenWeight i=ω) (hp : ω^2∈Set.Icc 0 1) :
-    M.reports.expect (fun z => if HiddenCollectionConditions T n (sourceScheduledTrials T n) ω b
-      (sourceReportFlags T n b z) then 0 else 1) ≤ sourceHiddenTail T n b ω := by
-  calc
-    _ ≤ M.reports.expect (fun z => ∑v : WindowCollection T b,
-      if (collectionCount T n (sourceScheduledTrials T n) b v:ℝ)*ω^2/2≤
-        ∑i,(hiddenTrial (collectionFlags T n (sourceScheduledTrials T n) b (sourceReportFlags T n b z) v i):ℝ)
-      then 0 else 1) := by
-        apply FiniteLaw.expect_mono
-        intro z
-        simp only [hiddenCollectionConditions_iff]
-        exact failure_forall_le_sum _
-    _ = ∑v : WindowCollection T b,M.reports.expect (fun z =>
-      if (PairedMechanisms.hiddenSubset (sourceCollectionSet T n b v) z:ℝ)<
-        ((sourceCollectionSet T n b v).card:ℝ)*ω^2/2 then 1 else 0) := by
-        rw [FiniteLaw.expect_fintype_sum]
-        apply Finset.sum_congr rfl
-        intro v _
-        apply congrArg (FiniteLaw.expect M.reports)
-        funext z
-        rw [sourceCollectionSet_card,sourceCollection_hidden,Nat.cast_sum]
-        by_cases h : (collectionCount T n (sourceScheduledTrials T n) b v:ℝ)*ω^2/2≤
-          ∑i,(hiddenTrial (collectionFlags T n (sourceScheduledTrials T n) b (sourceReportFlags T n b z) v i):ℝ)
-        · simp [h,not_lt.mpr h]
-        · simp [h,lt_of_not_ge h]
-    _ ≤ _ := by
-      apply Finset.sum_le_sum
-      intro v _
-      simpa only [sourceCollectionSet_card] using M.hiddenSubset_small (sourceCollectionSet T n b v) hw hp
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- Numerical cap forced by at least half the mean number of hidden pairs. -/
-def hiddenBinomialCap (m : ℕ) (ω : ℝ) : ℝ := Real.sqrt (4/((m:ℝ)*ω^2+2))
-
-theorem hiddenBinomialCap_nonneg (m : ℕ) (ω : ℝ) : 0≤hiddenBinomialCap m ω :=
-  Real.sqrt_nonneg _
-
-theorem hidden_binomial_bound (m G : ℕ) (ω : ℝ)
-    (h : (m:ℝ)*ω^2/2≤G) : binomialBound G≤hiddenBinomialCap m ω := by
-  apply Real.sqrt_le_sqrt
-  apply (div_le_div_iff₀ (by positivity : (0:ℝ)<G+1)
-    (by positivity : (0:ℝ)<(m:ℝ)*ω^2+2)).mpr
-  nlinarith
-
-def hiddenSmoothingBound (m s : ℕ) (ω : ℝ) : ℝ :=
-  (2:ℝ)^(s*s)*((2*Real.sqrt (4*((s:ℝ)+1)/((m:ℝ)*ω^2+2)))^s *
-    Real.sqrt (4*((s:ℝ)+1)/((m:ℝ)*ω^2+2)))
-
-theorem hiddenSmoothingBound_nonneg (m s : ℕ) (ω : ℝ) :
-    0≤hiddenSmoothingBound m s ω := by unfold hiddenSmoothingBound; positivity
-
-theorem hidden_smoothing_bound {m : ℕ} (d : Fin m → Option Bool) (s : ℕ) (ω : ℝ)
-    (h : (m:ℝ)*ω^2/2≤∑i,(hiddenTrial (d i):ℝ)) :
-    posteriorSmoothingCost s d≤hiddenSmoothingBound m s ω := by
-  have hn : (m:ℝ)*ω^2/2≤((∑i,hiddenTrial (d i):ℕ):ℝ) := by simpa only [Nat.cast_sum] using h
-  have hb : binomialBound ((∑i,hiddenTrial (d i))/(s+1))≤
-      Real.sqrt (4*((s:ℝ)+1)/((m:ℝ)*ω^2+2)) := by
-    apply (binomialBound_quotient_le _ s).trans
-    apply Real.sqrt_le_sqrt
-    apply (div_le_div_iff₀ (by positivity : (0:ℝ)<((∑i,hiddenTrial (d i):ℕ):ℝ)+1)
-      (by positivity : (0:ℝ)<(m:ℝ)*ω^2+2)).mpr
-    have hh := mul_le_mul_of_nonneg_left hn (by positivity : (0:ℝ)≤4*((s:ℝ)+1))
-    nlinarith
-  apply mul_le_mul_of_nonneg_left _ (by positivity : (0:ℝ)≤2^(s*s))
-  have hnon : 0≤binomialBound ((∑i,hiddenTrial (d i))/(s+1)) := Real.sqrt_nonneg _
-  exact mul_le_mul
-    (pow_le_pow_left₀ (mul_nonneg (by norm_num) hnon) (mul_le_mul_of_nonneg_left hb (by norm_num)) s)
-    hb hnon (by positivity)
-
-/-- Hidden-count typicality implies every local/global anchor cap, for the
- actual source metadata and without a coefficient-dependent event. -/
-theorem hiddenConditions_binomialCaps (T n g : ℕ → ℕ) (ω : ℝ) (b : ℕ)
-    (d : WindowAssignments T n g (Option Bool) b)
-    (h : HiddenCollectionConditions T n g ω b d) :
-    MetadataBinomialCaps T n g (fun j => hiddenBinomialCap (n j) ω)
-      (fun j => hiddenBinomialCap (g j) ω) b d := by
-  induction b with
-  | zero => trivial
-  | succ b ih =>
-    refine ⟨hidden_binomial_bound (n b) _ ω ?_,hidden_binomial_bound (g b) _ ω ?_,fun u => ih _ (h.2.2 u)⟩
-    · simpa only [Nat.cast_sum] using h.1
-    · simpa only [Nat.cast_sum] using h.2.1
-
-theorem hiddenConditions_smoothingCap (T n g : ℕ → ℕ) (ω η : ℝ) (s b : ℕ)
-    (hη : ∀j<b,hiddenSmoothingBound (n j) s ω≤η)
-    (d : WindowAssignments T n g (Option Bool) b)
-    (h : HiddenCollectionConditions T n g ω b d) :
-    MetadataSmoothingCap T n g s η b d := by
-  induction b with
-  | zero => trivial
-  | succ b ih =>
-    exact ⟨(hidden_smoothing_bound d.1 s ω h.1).trans (hη b (by omega)),
-      fun u => ih (fun j hj => hη j (by omega)) _ (h.2.2 u)⟩
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- The actual noncentral posterior mass has the unconditional honest
- fit-error bound. Impossible reports cause no problem in Bayes averaging. -/
-theorem source_noncentral_average {Side : Type*} [Fintype Side]
-    (T n : ℕ → ℕ) (s b : ℕ) (M : PairedMechanisms Side (SourceTrialCount T n (b+1)))
-    (hn : ∀j,j<b → 0<n j) (hne : ∀j,j<b → Even (n j))
-    (hT : ∀j≤b,24≤T j) (he : ∀j≤b,Even (T j)) (hL : ∀j≤b,24*2^s≤T j)
-    (hroot : 4*(n b+1)≤T b) :
-    M.reports.expect (fun z => sourceNoncentralMass T n (sourceScheduledTrials T n) s
-      (sourceLocalCenter T n) (sourceScheduledShift T n) (fun j => centralThird (T j))
-      (b+1) (sourceReportFlags T n (b+1) z) 0) ≤ sourceFitTail T n (b+1) := by
-  calc
-    _ ≤ M.reports.expect (fun z =>
-        (windowPosteriorLaw T n (sourceScheduledTrials T n) (b+1) (sourceReportFlags T n (b+1) z)).expect
-          (fun bits => if SourceTreeFit T n (b+1) bits then 0 else 1)) := by
-      apply FiniteLaw.expect_mono
-      intro z
-      exact sourceNoncentralMass_le_fit T n s b (sourceLocalCenter T n) (fun _ _ => rfl)
-        hT he hL hroot _
-    _ = M.reports.expect (fun z => (PairedMechanisms.orientationsGiven z).expect
-        (fun bits => if SourceTreeFit T n (b+1) ((sourceAssignmentCode T n Bool (b+1)).symm bits) then 0 else 1)) := by
-      apply congrArg (FiniteLaw.expect M.reports)
-      funext z
-      exact (source_posterior_code_expect T n (b+1) z
-        (fun bits => if SourceTreeFit T n (b+1) bits then 0 else 1)).symm
-    _ = _ := M.posterior_average _
-    _ ≤ _ := source_flat_fit_tail T n (b+1) (fun j hj => hn j (by omega)) (fun j hj => hne j (by omega))
-
-/-- Markov for a genuine finite law, stated for the exact failure indicator. -/
-theorem FiniteLaw.positive_markov {α : Type*} [Fintype α] (P : FiniteLaw α)
-    (f : α → ℝ) (hf : ∀x,0≤f x) {a : ℝ} (ha : 0<a) :
-    P.expect (fun x => if a≤f x then 1 else 0) ≤ P.expect f/a := by
-  apply (le_div_iff₀ ha).mpr
-  rw [mul_comm,←FiniteLaw.expect_const_mul]
-  apply FiniteLaw.expect_mono
-  intro x
-  split_ifs with hx
-  · simpa only [mul_one] using hx
-  · simpa only [mul_zero] using hf x
-
-/-- One metadata event, fixed before choosing a coefficient, controls
- local and global smoothing and all internal central-third tests. -/
-def SourceTypical {Side : Type*} (T n : ℕ → ℕ) (s b : ℕ) (ω a : ℝ)
-    (z : Reports Side (SourceTrialCount T n (b+1))) : Prop :=
-  HiddenCollectionConditions T n (sourceScheduledTrials T n) ω (b+1)
-    (sourceReportFlags T n (b+1) z) ∧
-  sourceNoncentralMass T n (sourceScheduledTrials T n) s (sourceLocalCenter T n)
-    (sourceScheduledShift T n) (fun j => centralThird (T j)) (b+1)
-    (sourceReportFlags T n (b+1) z) 0<a
-
-theorem source_typical_failure {Side : Type*} [Fintype Side]
-    (T n : ℕ → ℕ) (s b : ℕ) (M : PairedMechanisms Side (SourceTrialCount T n (b+1)))
-    (ω a : ℝ) (ha : 0<a) (hw : ∀i,M.hiddenWeight i=ω) (hp : ω^2∈Set.Icc 0 1)
-    (hn : ∀j,j<b → 0<n j) (hne : ∀j,j<b → Even (n j))
-    (hT : ∀j≤b,24≤T j) (he : ∀j≤b,Even (T j)) (hL : ∀j≤b,24*2^s≤T j)
-    (hroot : 4*(n b+1)≤T b) :
-    M.reports.expect (fun z => if SourceTypical T n s b ω a z then 0 else 1) ≤
-      sourceHiddenTail T n (b+1) ω+sourceFitTail T n (b+1)/a := by
-  let f := fun z : Reports Side (SourceTrialCount T n (b+1)) =>
-    sourceNoncentralMass T n (sourceScheduledTrials T n) s (sourceLocalCenter T n)
-      (sourceScheduledShift T n) (fun j => centralThird (T j)) (b+1)
-      (sourceReportFlags T n (b+1) z) 0
-  calc
-    _ ≤ M.reports.expect (fun z =>
-        (if HiddenCollectionConditions T n (sourceScheduledTrials T n) ω (b+1)
-          (sourceReportFlags T n (b+1) z) then 0 else 1)+(if a≤f z then 1 else 0)) := by
-      apply FiniteLaw.expect_mono
-      intro z
-      by_cases H : SourceTypical T n s b ω a z
-      · rw [ite_eq_left H]
-        split_ifs <;> norm_num
-      · rw [ite_eq_right H]
-        by_cases h : HiddenCollectionConditions T n (sourceScheduledTrials T n) ω (b+1)
-          (sourceReportFlags T n (b+1) z)
-        · have hf : ¬f z<a := fun hf => H ⟨h,hf⟩
-          rw [ite_eq_left h,ite_eq_left (le_of_not_gt hf)]
-          norm_num
-        · rw [ite_eq_right h]
-          split_ifs <;> norm_num
-    _ ≤ _ := by
-      rw [FiniteLaw.expect_add]
-      apply add_le_add (source_hidden_conditions_tail T n (b+1) M ω hw hp)
-      apply (M.reports.positive_markov f (fun z => sourceNoncentralMass_nonneg T n _ s _ _ _ _ _ _) ha).trans
-      exact div_le_div_of_nonneg_right
-        (source_noncentral_average T n s b M hn hne hT he hL hroot) ha.le
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- Exact common plateau with integer half-count centering. Unlike the
- simplified g/2 formula, this identity needs no parity assumption. -/
-theorem scheduledGlobalShift_plateau_exact (N t : ℕ)
-    (bits : Fin ((N/4)*t) → Bool) (u : Fin N) (hu : u∈centralThird N) :
-    scheduledGlobalShift N t bits u = (∑i,(bitIndex (bits i):ℤ))-(N/4:ℕ)*(t/2:ℕ) := by
-  calc
-    _ = ∑v : Fin (N/4),
-        ((∑a : Fin t, (bitIndex (bits (globalTrialIndex N t v a)):ℤ))-(t/2:ℕ)) := by
-      apply Finset.sum_congr rfl
-      intro v _
-      exact ite_eq_left (centralThird_active N u hu v)
-    _ = _ := by
-      rw [Finset.sum_sub_distrib]
-      congr 1
-      · exact globalTrial_sum N t (fun i => (bitIndex (bits i):ℤ))
-      · simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]
-
-def sourcePlateauCenter (T n : ℕ → ℕ) : ℕ → ℤ
-  | 0 => 0
-  | b+1 => (T (b+1)/4:ℕ)*(n b/2:ℕ)
-
-theorem sourceScheduledShift_hasPlateau (T n : ℕ → ℕ) :
-    HasGlobalPlateau T (sourceScheduledTrials T n) (fun j => centralThird (T j))
-      (sourcePlateauCenter T n) (sourceScheduledShift T n) := by
-  intro b bits u hu
-  cases b with
-  | zero =>
-      change (0:ℤ)=(∑i : Fin 0,(bitIndex (bits i):ℤ))-0
-      simp
-  | succ b => exact scheduledGlobalShift_plateau_exact (T (b+1)) (n b) bits u hu
-
-def sourceForcedBudget (T n : ℕ → ℕ) (s b r : ℕ) (ω : ℝ) : ℝ :=
-  (OccupiedParityTree.shapeRoleConstant r (b+1):ℝ)*interiorLabelBudget T b r *
-    ((2^s:ℕ):ℝ)^(2*(r-2^(b+1))) *
-    ((2*((2^s:ℕ):ℝ)^3*hiddenBinomialCap (n b) ω)*
-      binaryTreeBudget (fun j => 4*((2^s:ℕ):ℝ)^6*
-        hiddenBinomialCap (sourceScheduledTrials T n (j+1)) ω*hiddenBinomialCap (n j) ω) b)
-
-theorem sourceForcedBudget_nonneg (T n : ℕ → ℕ) (s b r : ℕ) (ω : ℝ) :
-    0 ≤ sourceForcedBudget T n s b r ω := by
-  have hB := fun j => hiddenBinomialCap_nonneg (n j) ω
-  have hC := fun j => hiddenBinomialCap_nonneg (sourceScheduledTrials T n j) ω
-  have ht := binaryTreeBudget_nonneg (fun j => 4*((2^s:ℕ):ℝ)^6*
-      hiddenBinomialCap (sourceScheduledTrials T n (j+1)) ω*hiddenBinomialCap (n j) ω)
-    (fun j => mul_nonneg (mul_nonneg (by positivity) (hC (j+1))) (hB j)) b
-  have hb := hB b
-  unfold sourceForcedBudget interiorLabelBudget
-  positivity
-
-/-- Full unabsorbed bound, now with no placeholder anchor-energy hypothesis. -/
-def sourceDegreeBudget (T n : ℕ → ℕ) (s b r : ℕ) (ω η a : ℝ) : ℝ :=
-  4*(3^((b+1)*r):ℕ)*
-    (((SourceLeafCount T (b+1)).choose r:ℕ)*(3^((b+1)*r):ℕ)*(η^2+a^2)+
-      sourceForcedBudget T n s b r ω)
-
-theorem sourceDegreeBudget_nonneg (T n : ℕ → ℕ) (s b r : ℕ) (ω η a : ℝ) :
-    0 ≤ sourceDegreeBudget T n s b r ω η a := by
-  have h := sourceForcedBudget_nonneg T n s b r ω
-  unfold sourceDegreeBudget
-  positivity
-
-theorem source_canonical_degree_bound {Side : Type*} (T n : ℕ → ℕ) (hT : ∀j,0<T j)
-    (s b r : ℕ) (ω η a : ℝ) (ha : 0≤a)
-    (hη : ∀j<b+1,hiddenSmoothingBound (n j) s ω≤η)
-    (z : Reports Side (SourceTrialCount T n (b+1))) (hz : SourceTypical T n s b ω a z) :
-    (∑σ : Fin (SourceLeafCount T (b+1)) → Bool, if monomialDegree σ=r then
-      (averagedCoefficient (PairedMechanisms.orientationsGiven z)
-        (sourceFlatLeafType T n s (sourceLocalCenter T n) (b+1) 0 false) σ -
-       averagedCoefficient (PairedMechanisms.orientationsGiven z)
-        (sourceFlatLeafType T n s (sourceLocalCenter T n) (b+1) 0 true) σ)^2 else 0) ≤
-      sourceDegreeBudget T n s b r ω η a := by
-  rw [source_canonical_degree_sum]
-  apply (sourceCoefficient_degree_energy T n _ s _ _ (fun j => centralThird (T j)) b r
-    (sourceReportFlags T n (b+1) z) 0 η
-    (hiddenConditions_smoothingCap T n _ ω η s (b+1) hη _ hz.1)).trans
-  apply mul_le_mul_of_nonneg_left _ (by positivity)
-  apply add_le_add
-  · apply mul_le_mul_of_nonneg_left _ (by positivity)
-    apply add_le_add_right
-    exact (sq_le_sq₀ (sourceNoncentralMass_nonneg T n _ s _ _ _ _ _ _) ha).mpr hz.2.le
-  · exact degreeForcedRoleEnergy_bound T n _ hT s (sourceLocalCenter T n) _ _ _ (sourceScheduledShift_hasPlateau T n)
-      (fun j => hiddenBinomialCap (n j) ω) (fun j => hiddenBinomialCap (sourceScheduledTrials T n j) ω)
-      (fun j => hiddenBinomialCap_nonneg _ _) (fun j => hiddenBinomialCap_nonneg _ _)
-      b r _ (hiddenConditions_binomialCaps T n _ ω (b+1) _ hz.1) 0
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- Pointwise equality of the literal source types and the signed tree types.
- This holds for every assignment, including overflowing windows. -/
-theorem sourceFlatLeafType_window (T n : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (b : ℕ) (incoming : ℤ) (τ : Bool) (bits : Fin (SourceTrialCount T n b) → Bool) :
-    sourceFlatLeafType T n s center b incoming τ bits =
-      windowFinType T s (sourceTreeAnchors T n (sourceScheduledTrials T n) center
-        (sourceScheduledShift T n) b ((sourceAssignmentCode T n Bool b).symm bits) incoming)
-        b (boolTypeSign τ) := by
-  funext j
-  unfold sourceFlatLeafType windowFinType windowLeafType
-  rw [←sourceTreeLeafSign_eq,←sourceTreeLeafBool_sign]
-  change sourceTreeLeafBool T n s center b _ incoming τ (finEnumeration _ j)=
-    decide (boolTypeSign (sourceTreeLeafBool T n s center b _ incoming τ (finEnumeration _ j))= -1)
-  cases sourceTreeLeafBool T n s center b _ incoming τ (finEnumeration _ j) <;> norm_num [boolTypeSign]
-
-def sourceSparseCount (T : ℕ → ℕ) (s b : ℕ) : ℕ :=
-  2^s*∏j∈Finset.range b,T (j+1)
-
-theorem sourceFlatLeafType_sparse (T n : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (b : ℕ) (incoming : ℤ) (τ : Bool) (bits : Fin (SourceTrialCount T n (b+1)) → Bool) :
-    monomialDegree (sourceFlatLeafType T n s center (b+1) incoming τ bits) ≤ sourceSparseCount T s b := by
-  rw [sourceFlatLeafType_window,windowFinType_degree]
-  exact windowMinusCount_bound T s _ b [] _
-
-def sourceOracleCoefficient {Side : Type*} (T n : ℕ → ℕ) (s b : ℕ)
-    (z : Reports Side (SourceTrialCount T n (b+1))) (τ : Bool) :
-    (Fin (SourceLeafCount T (b+1)) → Bool) → ℝ :=
-  averagedCoefficient (PairedMechanisms.orientationsGiven z)
-    (sourceFlatLeafType T n s (sourceLocalCenter T n) (b+1) 0 τ)
-
-theorem sourceOracleCoefficient_empty {Side : Type*} (T n : ℕ → ℕ) (s b : ℕ)
-    (z : Reports Side (SourceTrialCount T n (b+1))) (τ : Bool) :
-    sourceOracleCoefficient T n s b z τ (fun _ => false)=1 := averagedCoefficient_empty _ _
-
-theorem sourceOracleCoefficient_degree {Side : Type*} (T n : ℕ → ℕ) (s b r : ℕ)
-    (z : Reports Side (SourceTrialCount T n (b+1))) (τ : Bool) :
-    (∑σ, if monomialDegree σ=r then (sourceOracleCoefficient T n s b z τ σ)^2 else 0) ≤
-      ((sourceSparseCount T s b).choose r:ℝ) :=
-  averaged_sparse_degree _ _ (sourceFlatLeafType_sparse T n s _ b 0 τ) r
-
-def sourceSignedCoefficient {Side : Type*} (T n : ℕ → ℕ) (s b : ℕ)
-    (z : Reports Side (SourceTrialCount T n (b+1))) (σ : Fin (SourceLeafCount T (b+1)) → Bool) : ℝ :=
-  sourceOracleCoefficient T n s b z false σ-sourceOracleCoefficient T n s b z true σ
-
-theorem sourceSignedCoefficient_zero {Side : Type*} (T n : ℕ → ℕ) (s b : ℕ)
-    (z : Reports Side (SourceTrialCount T n (b+1))) :
-    (∑σ, if monomialDegree σ=0 then sourceSignedCoefficient T n s b z σ^2 else 0)=0 := by
-  simp [monomialDegree_eq_zero_iff,sourceSignedCoefficient,sourceOracleCoefficient_empty]
-
-theorem sourceSignedCoefficient_degree_geometric {Side : Type*} (T n : ℕ → ℕ) (s b r : ℕ)
-    (z : Reports Side (SourceTrialCount T n (b+1))) (w : ℝ) :
-    w^(2*r)*(∑σ, if monomialDegree σ=r then sourceSignedCoefficient T n s b z σ^2 else 0) ≤
-      4*(w^2*sourceSparseCount T s b)^r := by
-  have h0 : 0 ≤ w^(2*r) := by rw [pow_mul]; positivity
-  have hp := sourceOracleCoefficient_degree T n s b r z false
-  have hm := sourceOracleCoefficient_degree T n s b r z true
-  have hc : ((sourceSparseCount T s b).choose r:ℝ) ≤ (sourceSparseCount T s b:ℝ)^r := by
-    exact_mod_cast Nat.choose_le_pow (sourceSparseCount T s b) r
-  have hd := degree_square_sub_le (sourceOracleCoefficient T n s b z false)
-    (sourceOracleCoefficient T n s b z true) r
-  calc
-    _ ≤ w^(2*r)*(4*(sourceSparseCount T s b:ℝ)^r) :=
-      mul_le_mul_of_nonneg_left (by dsimp only [sourceSignedCoefficient]; linarith) h0
-    _ = _ := by rw [pow_mul,mul_pow]; ring
-
-def sourceEnergyBudget (T n : ℕ → ℕ) (s b R : ℕ) (ω η a w : ℝ) : ℝ :=
-  (∑r∈Finset.range R, w^(2*r)*(if r=0 then 0 else sourceDegreeBudget T n s b r ω η a))+
-    4*(w^2*sourceSparseCount T s b)^R/(1-w^2*sourceSparseCount T s b)
-
-/-- Complete signed energy, with finite low-degree forced-tree bound and
- the geometric high tail. No coefficient estimates remain as hypotheses. -/
-theorem source_canonical_energy {Side : Type*} (T n : ℕ → ℕ) (hT : ∀j,0<T j)
-    (s b R : ℕ) (ω η a w : ℝ) (ha : 0 ≤ a) (hw : 0 ≤ w)
-    (hsmall : w^2*sourceSparseCount T s b<1)
-    (hη : ∀j<b+1,hiddenSmoothingBound (n j) s ω ≤ η)
-    (z : Reports Side (SourceTrialCount T n (b+1))) (hz : SourceTypical T n s b ω a z) :
-    multilinearEnergy (sourceSignedCoefficient T n s b z) w ≤ sourceEnergyBudget T n s b R ω η a w := by
-  apply multilinearEnergy_low_high _ w (w^2*sourceSparseCount T s b) 4 hw (by positivity) hsmall
-    (by norm_num) (fun r => if r=0 then 0 else sourceDegreeBudget T n s b r ω η a)
-  · intro r _
-    by_cases hr : r=0
-    · subst r
-      rw [sourceSignedCoefficient_zero,ite_eq_left rfl]
-    · rw [ite_eq_right hr]
-      exact source_canonical_degree_bound T n hT s b r ω η a ha hη z hz
-  · intro r _
-    exact sourceSignedCoefficient_degree_geometric T n s b r z w
-  · intro r
-    split_ifs
-    · rfl
-    · exact sourceDegreeBudget_nonneg T n s b r ω η a
-
-theorem nonconstant_eq_degreeTail {l : ℕ} (A : (Fin l → Bool) → ℝ) :
-    nonconstantCoefficient A=degreeTail A 1 := by
-  funext σ
-  by_cases h : σ=(fun _ => false)
-  · subst σ; simp [nonconstantCoefficient,degreeTail,monomialDegree]
-  · have hd : 1 ≤ monomialDegree σ := by
-      have := (monomialDegree_eq_zero_iff σ).not.mpr h
-      omega
-    simp [nonconstantCoefficient,degreeTail,h,hd]
-
-/-- The individual density is close to the pure reference without any
- metadata typicality assumption. -/
-theorem source_canonical_centered_energy {Side : Type*} (T n : ℕ → ℕ) (s b : ℕ)
-    (z : Reports Side (SourceTrialCount T n (b+1))) (τ : Bool) {w : ℝ} (hw : 0 ≤ w)
-    (hsmall : w^2*sourceSparseCount T s b<1) :
-    multilinearEnergy (nonconstantCoefficient (sourceOracleCoefficient T n s b z τ)) w ≤
-      (w^2*sourceSparseCount T s b)/(1-w^2*sourceSparseCount T s b) := by
-  rw [nonconstant_eq_degreeTail]
-  simpa only [pow_one,sourceOracleCoefficient] using averaged_sparse_tail_energy _ _
-    (sourceFlatLeafType_sparse T n s _ b 0 τ) 1 hw hsmall
-
-end TraceReconstruction
-
-noncomputable section
-namespace TraceReconstruction.FiniteLaw
-open scoped Classical
-variable {α : Type*} [Fintype α]
-
-theorem abs_expect_pow_le (P : FiniteLaw α) (F : α → ℝ) (t : ℕ) :
-    |P.expect F|^t ≤ P.expect (fun a => |F a|^t) := by
-  have habs : |P.expect F| ≤ P.expect (fun a => |F a|) := by
-    calc
-      _ ≤ ∑a, |P.mass a*F a| := Finset.abs_sum_le_sum_abs _ _
-      _ = _ := by simp only [abs_mul,abs_of_nonneg (P.nonneg _)]; rfl
-  apply (pow_le_pow_left₀ (abs_nonneg _) habs t).trans
-  simpa only [smul_eq_mul,expect] using
-    (convexOn_pow (𝕜:=ℝ) t).map_sum_le (t:=Finset.univ)
-      (w:=P.mass) (p:=fun a => |F a|) (fun a _ => P.nonneg a) P.total (fun a _ => (show |F a| ∈ Set.Ici (0:ℝ) from abs_nonneg (F a)))
-
-theorem density_score_centered (P Q : FiniteLaw α) (Z : α → ℝ)
-    (hZ : ∀a,Q.mass a=P.mass a*(1+Z a)) : P.expect Z=0 := by
-  have h := Q.total
-  simp_rw [hZ,mul_add,mul_one,Finset.sum_add_distrib,P.total] at h
-  change _+P.expect Z=1 at h
-  linarith
-
-/-- Common-event mass difference is controlled by the restricted density
- difference at every moment, by finite Jensen rather than a pointwise bound. -/
-theorem clipping_mass_difference (R : FiniteLaw α) (f g : α → ℝ)
-    (hf : ∀a,0 ≤ f a) (hg : ∀a,0 ≤ g a) (hfm : R.expect f=1) (hgm : R.expect g=1)
-    (A : Finset α) (t : ℕ) :
-    |(R.withDensity f hf hfm).eventMass A-(R.withDensity g hg hgm).eventMass A|^t ≤
-      R.expect (fun a => |if a∈A then g a-f a else 0|^t) := by
-  have he : (R.withDensity f hf hfm).eventMass A-(R.withDensity g hg hgm).eventMass A=
-      -(R.expect (fun a => if a∈A then g a-f a else 0)) := by
-    simp only [expect,mul_ite,mul_zero,←Finset.sum_filter,
-      Finset.filter_mem_eq_inter,Finset.univ_inter,eventMass,withDensity,mul_sub,
-      Finset.sum_sub_distrib]
-    ring
-  rw [he,abs_neg]
-  exact R.abs_expect_pow_le _ t
-
-theorem clipped_raw_moment_restricted (R : FiniteLaw α) (f g : α → ℝ)
-    (hf : ∀a,0 ≤ f a) (hfm : R.expect f=1)
-    (A : Finset α) (hAf : ∀a∈A,1/2 ≤ f a)
-    (hP : 0<(R.withDensity f hf hfm).eventMass A) (k : ℕ) :
-    (condition (R.withDensity f hf hfm) A hP).expect (fun a => |g a/f a-1|^(k+1)) ≤
-      2^k/(R.withDensity f hf hfm).eventMass A *
-        R.expect (fun a => |if a∈A then g a-f a else 0|^(k+1)) := by
-  let g' : α → ℝ := fun a => if a∈A then g a else f a
-  have H := clipped_raw_moment R f g' hf hfm A hAf hP k
-  have he : (condition (R.withDensity f hf hfm) A hP).expect (fun a => |g a/f a-1|^(k+1))=
-      (condition (R.withDensity f hf hfm) A hP).expect (fun a => |g' a/f a-1|^(k+1)) := by
-    apply Finset.sum_congr rfl
-    intro a _
-    by_cases ha : a∈A <;> simp [condition,g',ha]
-  rw [he]
-  convert H using 1
-  congr 2
-  funext a
-  by_cases ha : a∈A <;> simp [g',ha]
-
-/-- Explicit normalized clipping, at half of any positive even moment. This
- constructs the actual conditioned score; no bounded-score hypothesis. -/
-theorem clipped_score_moment (R : FiniteLaw α) (f g : α → ℝ)
-    (hf : ∀a,0 ≤ f a) (hg : ∀a,0 ≤ g a) (hfm : R.expect f=1) (hgm : R.expect g=1)
-    (A : Finset α) (hAf : ∀a∈A,1/2 ≤ f a)
-    (hp : 1/2 ≤ (R.withDensity f hf hfm).eventMass A)
-    (hq : 1/2 ≤ (R.withDensity g hg hgm).eventMass A)
-    (k : ℕ) (hk : 1 ≤ k) (V : ℝ) (hV : 0 ≤ V)
-    (hM : R.expect (fun a => |if a∈A then g a-f a else 0|^(2*k)) ≤ V^k) :
-    let p := (R.withDensity f hf hfm).eventMass A
-    let q := (R.withDensity g hg hgm).eventMass A
-    let hP : 0 < p := lt_of_lt_of_le (by norm_num) hp
-    (condition (R.withDensity f hf hfm) A hP).evenMomentSq (fun a => p/q*(g a/f a)-1) k ≤64*V := by
-  dsimp only
-  let p := (R.withDensity f hf hfm).eventMass A
-  let q := (R.withDensity g hg hgm).eventMass A
-  have hp0 : 0<p := lt_of_lt_of_le (by norm_num) hp
-  have hq0 : 0<q := lt_of_lt_of_le (by norm_num) hq
-  have hp1 : p ≤ 1 := eventMass_le_one _ _
-  have hratio : p/q ≤ 2 := (div_le_iff₀ hq0).mpr (by linarith)
-  have hden : (1:ℝ)/p ≤ 2 := (div_le_iff₀ hp0).mpr (by linarith)
-  have ht : 2*k-1+1=2*k := by omega
-  have hraw := clipped_raw_moment_restricted R f g hf hfm A hAf hp0 (2*k-1)
-  rw [ht] at hraw
-  have hraw' : (condition (R.withDensity f hf hfm) A hp0).expect (fun a => |g a/f a-1|^(2*k)) ≤
-      2^(2*k)*V^k := by
-    apply hraw.trans
-    calc
-      _ ≤ (2^(2*k-1)/p)*V^k := mul_le_mul_of_nonneg_left hM (by positivity)
-      _ ≤ (2^(2*k-1)*2)*V^k := by
-        apply mul_le_mul_of_nonneg_right _ (pow_nonneg hV _)
-        simpa only [mul_one_div] using mul_le_mul_of_nonneg_left hden (by positivity : 0 ≤ (2:ℝ)^(2*k-1))
-      _ = _ := by rw [←pow_succ,ht]
-  have hnorm : |p/q-1|^(2*k) ≤ 2^(2*k)*V^k := by
-    rw [div_sub_one hq0.ne',abs_div,abs_of_pos hq0,div_pow]
-    apply (div_le_iff₀ (pow_pos hq0 _)).mpr
-    have hd := (clipping_mass_difference R f g hf hg hfm hgm A (2*k)).trans hM
-    have hqpow : 1 ≤ (2*q)^(2*k) := one_le_pow₀ (by linarith)
-    calc
-      _ ≤ V^k := hd
-      _ ≤ (2*q)^(2*k)*V^k := by nlinarith [pow_nonneg hV k]
-      _ = _ := by rw [mul_pow]; ring
-  have hs := score_rescaling_moment (condition (R.withDensity f hf hfm) A hp0)
-    (fun a => g a/f a) (p/q) (div_nonneg hp0.le hq0.le) (2*k-1)
-  rw [ht] at hs
-  have hpower : (p/q)^(2*k) ≤ (2:ℝ)^(2*k) :=
-    pow_le_pow_left₀ (div_nonneg hp0.le hq0.le) hratio _
-  have hs' : (condition (R.withDensity f hf hfm) A hp0).expect (fun a => |p/q*(g a/f a)-1|^(2*k)) ≤
-      (64*V)^k := by
-    apply hs.trans
-    calc
-      _ ≤ 2^(2*k-1)*(2^(2*k)*(2^(2*k)*V^k)+2^(2*k)*V^k) := by
-        apply mul_le_mul_of_nonneg_left _ (by positivity)
-        exact add_le_add (mul_le_mul hpower hraw' (by
-          apply Finset.sum_nonneg; intro a _; positivity [(condition (R.withDensity f hf hfm) A hp0).nonneg a]) (by positivity)) hnorm
-      _ ≤ 2^(2*k-1)*(2^(2*k)*(2^(2*k)*V^k)+2^(2*k)*(2^(2*k)*V^k)) := by
-        apply mul_le_mul_of_nonneg_left _ (by positivity)
-        apply add_le_add_right
-        have htwo : (1:ℝ) ≤ 2^(2*k) := one_le_pow₀ (by norm_num)
-        simpa only [one_mul] using mul_le_mul_of_nonneg_right htwo
-          (by positivity : 0 ≤ (2:ℝ)^(2*k)*V^k)
-      _ = (64*V)^k := by
-        have hm : 2^(2*k-1)*2= (2:ℝ)^(2*k) := by rw [←pow_succ,ht]
-        calc
-          _ = (2^(2*k-1)*2)*(2:ℝ)^(2*k)*(2:ℝ)^(2*k)*V^k := by ring
-          _ = _ := by rw [hm,←pow_add,←pow_add,mul_pow]; congr 1; rw [show 2*k+2*k+2*k=6*k by omega,pow_mul]; norm_num
-  unfold evenMomentSq
-  calc
-    _ ≤ ((64*V)^k)^(1/(k:ℝ)) := Real.rpow_le_rpow
-      (by apply Finset.sum_nonneg; intro a _; positivity [(condition (R.withDensity f hf hfm) A hp0).nonneg a]) hs' (by positivity)
-    _ = _ := by rw [←Real.rpow_natCast,one_div,Real.rpow_rpow_inv (by positivity) (by positivity)]
-
-end TraceReconstruction.FiniteLaw
-
-noncomputable section
-namespace TraceReconstruction.FiniteLaw
-open scoped Classical
-variable {α : Type*} [Fintype α]
-
-theorem eventMass_indicator (P : FiniteLaw α) (A : Finset α) :
-    P.eventMass A=P.expect (fun a => if a∈A then 1 else 0) := by
-  simp only [expect,mul_ite,mul_one,mul_zero,←Finset.sum_filter,
-    Finset.filter_mem_eq_inter,Finset.univ_inter,eventMass]
-
-theorem eventMass_compl (P : FiniteLaw α) (A : Finset α) :
-    P.eventMass Aᶜ=1-P.eventMass A := by
-  rw [eventMass_indicator,eventMass_indicator]
-  have he : P.expect (fun a => (if a∈Aᶜ then (1:ℝ) else 0)+(if a∈A then 1 else 0))=1 := by
-    convert P.expect_const 1 using 1
-    congr 1; funext a
-    by_cases ha : a∈A <;> simp [ha]
-  rw [expect_add] at he
-  linarith
-
-theorem density_second_moment_le_six (R : FiniteLaw α) (f : α → ℝ) (k : ℕ) (hk : 1 ≤ k)
-    (hM : R.expect (fun a => |f a-1|^(2*k)) ≤ 1) :
-    R.expect (fun a => f a^2) ≤ 6 := by
-  have h := R.expect_lower_moment_le_two (fun a => f a-1) (2*k) hM 2 (by omega)
-  calc
-    _ ≤ R.expect (fun a => 2*(f a-1)^2+2) := R.expect_mono (fun a => by nlinarith [sq_nonneg (f a-2)])
-    _ = 2*R.expect (fun a => (f a-1)^2)+2 := by rw [expect_add,expect_const_mul,expect_const]
-    _ ≤ _ := by linarith [le_abs_self (R.expect (fun a => (f a-1)^2))]
-
-/-- Cauchy--Schwarz loss on the actual low-reference-density event, for
- either density. The same clipping event is used for both hypotheses. -/
-theorem density_low_event_mass (R : FiniteLaw α) (f g : α → ℝ)
-    (hg : ∀a,0 ≤ g a) (hgm : R.expect g=1) (k : ℕ) (hk : 1 ≤ k) (U : ℝ)
-    (hU : 0 ≤ U) (hU1 : U ≤ 1)
-    (hfM : R.expect (fun a => |f a-1|^(2*k)) ≤ U^k)
-    (hgM : R.expect (fun a => |g a-1|^(2*k)) ≤ U^k) :
-    (R.withDensity g hg hgm).eventMass (Finset.univ.filter (fun a => f a<1/2)) ≤
-      Real.sqrt (6*(4*U)^k) := by
-  let A := Finset.univ.filter (fun a => f a<1/2)
-  have hR : R.eventMass A ≤ (4*U)^k := by
-    calc
-      _ ≤ 2^(2*k)*R.expect (fun a => |f a-1|^(2*k)) :=
-        R.low_density_reference_mass f (2*k) A (fun a ha => (Finset.mem_filter.mp ha).2)
-      _ ≤ 2^(2*k)*U^k := mul_le_mul_of_nonneg_left hfM (by positivity)
-      _ = _ := by rw [pow_mul,mul_pow]; norm_num
-  have hsecond := R.density_second_moment_le_six g k hk (hgM.trans (pow_le_one₀ hU hU1))
-  have hsq : ((R.withDensity g hg hgm).eventMass A)^2 ≤ 6*(4*U)^k := by
-    apply (R.density_bad_mass_sq g A).trans
-    calc
-      _ ≤ R.eventMass A*6 := mul_le_mul_of_nonneg_left hsecond (R.eventMass_nonneg A)
-      _ ≤ _ := by nlinarith
-  exact (Real.le_sqrt ((R.withDensity g hg hgm).eventMass_nonneg A) (by positivity)).mpr hsq
-
-def commonClipEvent (B : Finset α) (f : α → ℝ) : Finset α :=
-  B.filter (fun a => 1/2 ≤ f a)
-
-theorem commonClipEvent_loss (P : FiniteLaw α) (B : Finset α) (f : α → ℝ) :
-    1-P.eventMass (commonClipEvent B f) ≤
-      P.eventMass Bᶜ+P.eventMass (Finset.univ.filter (fun a => f a<1/2)) := by
-  rw [←eventMass_compl,eventMass_indicator,eventMass_indicator,eventMass_indicator,←expect_add]
-  apply P.expect_mono
-  intro a
-  simp only [commonClipEvent,Finset.mem_compl,Finset.mem_filter,Finset.mem_univ,true_and]
-  split_ifs <;> simp_all
-
-
-/-- Quantitative common clipping on any genuine good event. Both bad-event
- costs below are measured under the actual normalized laws, not reference mass. -/
-theorem commonClipEvent_mass (R : FiniteLaw α) (f g : α → ℝ)
-    (hf : ∀a,0 ≤ f a) (hg : ∀a,0 ≤ g a) (hfm : R.expect f=1) (hgm : R.expect g=1)
-    (B : Finset α) (e : ℝ) (hfB : (R.withDensity f hf hfm).eventMass Bᶜ ≤ e)
-    (hgB : (R.withDensity g hg hgm).eventMass Bᶜ ≤ e)
-    (k : ℕ) (hk : 1 ≤ k) (U : ℝ) (hU : 0 ≤ U) (hU1 : U ≤ 1)
-    (hfM : R.expect (fun a => |f a-1|^(2*k)) ≤ U^k)
-    (hgM : R.expect (fun a => |g a-1|^(2*k)) ≤ U^k) :
-    1-(R.withDensity f hf hfm).eventMass (commonClipEvent B f) ≤ e+Real.sqrt (6*(4*U)^k) ∧
-    1-(R.withDensity g hg hgm).eventMass (commonClipEvent B f) ≤ e+Real.sqrt (6*(4*U)^k) := by
-  constructor
-  · exact (commonClipEvent_loss _ B f).trans (add_le_add hfB
-      (density_low_event_mass R f f hf hfm k hk U hU hU1 hfM hfM))
-  · exact (commonClipEvent_loss _ B f).trans (add_le_add hgB
-      (density_low_event_mass R f g hg hgm k hk U hU hU1 hfM hgM))
-
-end TraceReconstruction.FiniteLaw
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- A pair produced by genuine common-event normalized conditioning. The
- outcome space and public decoder remain unchanged. -/
-structure ClippedOracle {α : Type*} [Fintype α] (P Q : FiniteLaw α) (k : ℕ) (e V : ℝ) where
-  first : FiniteLaw α
-  second : FiniteLaw α
-  score : α → ℝ
-  density : ∀a,second.mass a=first.mass a*(1+score a)
-  likelihood_nonneg : ∀a,0 ≤ 1+score a
-  centered : first.expect score=0
-  moment : first.evenMomentSq score k ≤ V
-  first_tv : first.tv P ≤ e
-  second_tv : second.tv Q ≤ e
-
-namespace FiniteLaw
-variable {α : Type*} [Fintype α]
-
-theorem exists_clippedOracle (R : FiniteLaw α) (f g : α → ℝ)
-    (hf : ∀a,0 ≤ f a) (hg : ∀a,0 ≤ g a) (hfm : R.expect f=1) (hgm : R.expect g=1)
-    (B : Finset α) (e : ℝ) (hfB : (R.withDensity f hf hfm).eventMass Bᶜ ≤ e)
-    (hgB : (R.withDensity g hg hgm).eventMass Bᶜ ≤ e)
-    (k : ℕ) (hk : 1 ≤ k) (U : ℝ) (hU : 0 ≤ U) (hU1 : U ≤ 1)
-    (hfM : R.expect (fun a => |f a-1|^(2*k)) ≤ U^k)
-    (hgM : R.expect (fun a => |g a-1|^(2*k)) ≤ U^k)
-    (V : ℝ) (hV : 0 ≤ V)
-    (hM : R.expect (fun a => |if a∈B then g a-f a else 0|^(2*k)) ≤ V^k)
-    (hsmall : e+Real.sqrt (6*(4*U)^k) ≤ 1/2) :
-    Nonempty (ClippedOracle (R.withDensity f hf hfm) (R.withDensity g hg hgm)
-      k (e+Real.sqrt (6*(4*U)^k)) (64*V)) := by
-  let A := commonClipEvent B f
-  let P := R.withDensity f hf hfm
-  let Q := R.withDensity g hg hgm
-  have hA (a : α) (ha : a∈A) : 1/2 ≤ f a := (Finset.mem_filter.mp ha).2
-  have hloss := commonClipEvent_mass R f g hf hg hfm hgm B e hfB hgB k hk U hU hU1 hfM hgM
-  have hp : 1/2 ≤ P.eventMass A := by linarith [hloss.1]
-  have hq : 1/2 ≤ Q.eventMass A := by linarith [hloss.2]
-  have hp0 : 0<P.eventMass A := lt_of_lt_of_le (by norm_num) hp
-  have hq0 : 0<Q.eventMass A := lt_of_lt_of_le (by norm_num) hq
-  let Z : α → ℝ := fun a => P.eventMass A/Q.eventMass A*(g a/f a)-1
-  have hden (a : α) : (Q.condition A hq0).mass a=(P.condition A hp0).mass a*(1+Z a) := by
-    dsimp only [Z]
-    rw [show 1+(P.eventMass A/Q.eventMass A*(g a/f a)-1)=P.eventMass A/Q.eventMass A*(g a/f a) by ring]
-    exact conditioned_density_identity R f g hf hg hfm hgm A
-      (fun a ha => lt_of_lt_of_le (by norm_num) (hA a ha)) hp0 hq0 a
-  have hAM : R.expect (fun a => |if a∈A then g a-f a else 0|^(2*k)) ≤ V^k := by
-    apply le_trans (R.expect_mono _) hM
-    intro a
-    by_cases ha : a∈A
-    · have hb := (Finset.mem_filter.mp ha).1
-      simp only [ite_eq_left ha,ite_eq_left hb,le_refl]
-    · rw [ite_eq_right ha]
-      simp only [abs_zero,zero_pow (by omega : 2*k≠0)]
-      exact pow_nonneg (abs_nonneg _) _
-  exact ⟨{
-    first := P.condition A hp0
-    second := Q.condition A hq0
-    score := Z
-    density := hden
-    likelihood_nonneg := fun a => by
-      dsimp only [Z]
-      have hh := mul_nonneg (div_nonneg hp0.le hq0.le) (div_nonneg (hg a) (hf a))
-      linarith
-    centered := density_score_centered _ _ Z hden
-    moment := clipped_score_moment R f g hf hg hfm hgm A hA hp hq k hk V hV hAM
-    first_tv := by rw [tv_symm,tv_condition]; exact hloss.1
-    second_tv := by rw [tv_symm,tv_condition]; exact hloss.2
-  }⟩
-
-end FiniteLaw
-end TraceReconstruction
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- Exact finite-union metadata cost; it contains no postulated typical
- event and no independence assertion between different collections. -/
-def sourceEffectiveHiddenTail (T n : ℕ → ℕ) (b : ℕ) (ω : ℝ) : ℝ :=
-  ∑v : WindowCollection T b,if collectionCount T n (sourceScheduledTrials T n) b v=0 then 0 else Real.exp (-(collectionCount T n (sourceScheduledTrials T n) b v:ℝ)*ω^4/16)
-
-theorem source_hidden_conditions_effective_tail {Side : Type*} [Fintype Side]
-    (T n : ℕ → ℕ) (b : ℕ) (M : PairedMechanisms Side (SourceTrialCount T n b))
-    (ω : ℝ) (hw : ∀i,M.hiddenWeight i=ω) (hp : ω^2∈Set.Icc 0 1) :
-    M.reports.expect (fun z => if HiddenCollectionConditions T n (sourceScheduledTrials T n) ω b
-      (sourceReportFlags T n b z) then 0 else 1) ≤ sourceEffectiveHiddenTail T n b ω := by
-  calc
-    _ ≤ M.reports.expect (fun z => ∑v : WindowCollection T b,
-      if (collectionCount T n (sourceScheduledTrials T n) b v:ℝ)*ω^2/2≤
-        ∑i,(hiddenTrial (collectionFlags T n (sourceScheduledTrials T n) b (sourceReportFlags T n b z) v i):ℝ)
-      then 0 else 1) := by
-        apply FiniteLaw.expect_mono
-        intro z
-        simp only [hiddenCollectionConditions_iff]
-        exact failure_forall_le_sum _
-    _ = ∑v : WindowCollection T b,M.reports.expect (fun z =>
-      if (PairedMechanisms.hiddenSubset (sourceCollectionSet T n b v) z:ℝ)<
-        ((sourceCollectionSet T n b v).card:ℝ)*ω^2/2 then 1 else 0) := by
-        rw [FiniteLaw.expect_fintype_sum]
-        apply Finset.sum_congr rfl
-        intro v _
-        apply congrArg (FiniteLaw.expect M.reports)
-        funext z
-        rw [sourceCollectionSet_card,sourceCollection_hidden,Nat.cast_sum]
-        by_cases h : (collectionCount T n (sourceScheduledTrials T n) b v:ℝ)*ω^2/2≤
-          ∑i,(hiddenTrial (collectionFlags T n (sourceScheduledTrials T n) b (sourceReportFlags T n b z) v i):ℝ)
-        · simp [h,not_lt.mpr h]
-        · simp [h,lt_of_not_ge h]
-    _ ≤ _ := by
-      apply Finset.sum_le_sum
-      intro v _
-      by_cases hc : collectionCount T n (sourceScheduledTrials T n) b v=0
-      · rw [ite_eq_left hc]
-        have hz : M.reports.expect (fun z =>
-            if (PairedMechanisms.hiddenSubset (sourceCollectionSet T n b v) z:ℝ)<
-              ((sourceCollectionSet T n b v).card:ℝ)*ω^2/2 then 1 else 0)=0 := by
-          have h (z : Reports Side (SourceTrialCount T n b)) :
-              ¬(PairedMechanisms.hiddenSubset (sourceCollectionSet T n b v) z:ℝ)<
-                ((sourceCollectionSet T n b v).card:ℝ)*ω^2/2 := by
-            rw [sourceCollectionSet_card,hc]
-            simp only [Nat.cast_zero,zero_mul,zero_div]
-            exact not_lt.mpr (Nat.cast_nonneg _)
-          simp_rw [ite_eq_right (h _)]
-          exact FiniteLaw.expect_const _ 0
-        exact hz.le
-      · rw [ite_eq_right hc]
-        simpa only [sourceCollectionSet_card] using M.hiddenSubset_small (sourceCollectionSet T n b v) hw hp
-
-theorem source_typical_effective_failure {Side : Type*} [Fintype Side]
-    (T n : ℕ → ℕ) (s b : ℕ) (M : PairedMechanisms Side (SourceTrialCount T n (b+1)))
-    (ω a : ℝ) (ha : 0<a) (hw : ∀i,M.hiddenWeight i=ω) (hp : ω^2∈Set.Icc 0 1)
-    (hn : ∀j,j<b → 0<n j) (hne : ∀j,j<b → Even (n j))
-    (hT : ∀j≤b,24≤T j) (he : ∀j≤b,Even (T j)) (hL : ∀j≤b,24*2^s≤T j)
-    (hroot : 4*(n b+1)≤T b) :
-    M.reports.expect (fun z => if SourceTypical T n s b ω a z then 0 else 1) ≤
-      sourceEffectiveHiddenTail T n (b+1) ω+sourceFitTail T n (b+1)/a := by
-  let f := fun z : Reports Side (SourceTrialCount T n (b+1)) =>
-    sourceNoncentralMass T n (sourceScheduledTrials T n) s (sourceLocalCenter T n)
-      (sourceScheduledShift T n) (fun j => centralThird (T j)) (b+1)
-      (sourceReportFlags T n (b+1) z) 0
-  calc
-    _ ≤ M.reports.expect (fun z =>
-        (if HiddenCollectionConditions T n (sourceScheduledTrials T n) ω (b+1)
-          (sourceReportFlags T n (b+1) z) then 0 else 1)+(if a≤f z then 1 else 0)) := by
-      apply FiniteLaw.expect_mono
-      intro z
-      by_cases H : SourceTypical T n s b ω a z
-      · rw [ite_eq_left H]
-        split_ifs <;> norm_num
-      · rw [ite_eq_right H]
-        by_cases h : HiddenCollectionConditions T n (sourceScheduledTrials T n) ω (b+1)
-          (sourceReportFlags T n (b+1) z)
-        · have hf : ¬f z<a := fun hf => H ⟨h,hf⟩
-          rw [ite_eq_left h,ite_eq_left (le_of_not_gt hf)]
-          norm_num
-        · rw [ite_eq_right h]
-          split_ifs <;> norm_num
-    _ ≤ _ := by
-      rw [FiniteLaw.expect_add]
-      apply add_le_add (source_hidden_conditions_effective_tail T n (b+1) M ω hw hp)
-      apply (M.reports.positive_markov f (fun z => sourceNoncentralMass_nonneg T n _ s _ _ _ _ _ _) ha).trans
-      exact div_le_div_of_nonneg_right
-        (source_noncentral_average T n s b M hn hne hT he hL hroot) ha.le
-
-
-end TraceReconstruction
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- Literal density of the source's unconditioned finite canonical oracle. -/
-def sourceOracleDensity {Side Leaf : Type*} (T n : ℕ → ℕ) (s b : ℕ)
-    (Z : Leaf → ℝ) (τ : Bool)
-    (zx : Reports Side (SourceTrialCount T n (b+1)) × (Fin (SourceLeafCount T (b+1)) → Leaf)) : ℝ :=
-  averagedDensity (PairedMechanisms.orientationsGiven zx.1)
-    (sourceFlatLeafType T n s (sourceLocalCenter T n) (b+1) 0 τ) Z zx.2
-
-theorem sourceOracleDensity_expansion {Side Leaf : Type*} (T n : ℕ → ℕ) (s b : ℕ)
-    (Z : Leaf → ℝ) (τ : Bool)
-    (z : Reports Side (SourceTrialCount T n (b+1))) (x : Fin (SourceLeafCount T (b+1)) → Leaf) :
-    sourceOracleDensity T n s b Z τ (z,x)=multilinear (sourceOracleCoefficient T n s b z τ) Z x :=
-  averagedDensity_expansion _ _ _ _
-
-theorem sourceOracleDensity_nonneg {Side Leaf : Type*} (T n : ℕ → ℕ) (s b : ℕ)
-    (Z : Leaf → ℝ) (hZ : ∀a,0 ≤ 1+Z a) (τ : Bool)
-    (zx : Reports Side (SourceTrialCount T n (b+1)) × (Fin (SourceLeafCount T (b+1)) → Leaf)) :
-    0 ≤ sourceOracleDensity T n s b Z τ zx := averagedDensity_nonneg _ _ _ hZ _
-
-theorem sourceOracleDensity_mean {Side Leaf : Type*} [Fintype Leaf] (T n : ℕ → ℕ) (s b : ℕ)
-    (P : FiniteLaw Leaf) (Z : Leaf → ℝ) (hZ : P.expect Z=0) (τ : Bool)
-    (z : Reports Side (SourceTrialCount T n (b+1))) :
-    (P.product (SourceLeafCount T (b+1))).expect (fun x => sourceOracleDensity T n s b Z τ (z,x))=1 :=
-  averagedDensity_mean P (PairedMechanisms.orientationsGiven z) Z hZ
-    (sourceFlatLeafType T n s (sourceLocalCenter T n) (b+1) 0 τ)
-
-theorem source_canonical_centered_moment {Side Leaf : Type*} [Fintype Leaf]
-    (T n : ℕ → ℕ) (s b k : ℕ) (P : FiniteLaw Leaf) (Z : Leaf → ℝ)
-    (hk : 1 ≤ k) (hcenter : P.expect Z=0) {L : ℝ} (hL : 0<L)
-    (hMoment : P.expect (fun a => |Z a|^(2*k)) ≤ L^(2*k))
-    (hsmall : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b<1)
-    (z : Reports Side (SourceTrialCount T n (b+1))) (τ : Bool) :
-    (P.product (SourceLeafCount T (b+1))).expect
-      (fun x => |sourceOracleDensity T n s b Z τ (z,x)-1|^(2*k)) ≤
-        ((((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b)/
-          (1-((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b))^k := by
-  have he (x : Fin (SourceLeafCount T (b+1)) → Leaf) :
-      sourceOracleDensity T n s b Z τ (z,x)-1=
-        multilinear (nonconstantCoefficient (sourceOracleCoefficient T n s b z τ)) Z x := by
-    rw [nonconstant_multilinear,sourceOracleCoefficient_empty,sourceOracleDensity_expansion]
-  simp_rw [he]
-  apply (multilinear_moment_scaled P Z k hk hcenter hL hMoment _ _).trans
-  exact pow_le_pow_left₀ (multilinearEnergy_nonneg _ _)
-    (source_canonical_centered_energy T n s b z τ (by positivity) hsmall) k
-
-theorem source_canonical_signed_moment {Side Leaf : Type*} [Fintype Leaf]
-    (T n : ℕ → ℕ) (hT : ∀j,0<T j) (s b R k : ℕ) (ω η a : ℝ) (ha : 0 ≤ a)
-    (hη : ∀j<b+1,hiddenSmoothingBound (n j) s ω ≤ η)
-    (P : FiniteLaw Leaf) (Z : Leaf → ℝ) (hk : 1 ≤ k) (hcenter : P.expect Z=0)
-    {L : ℝ} (hL : 0<L) (hMoment : P.expect (fun a => |Z a|^(2*k)) ≤ L^(2*k))
-    (hsmall : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b<1)
-    (z : Reports Side (SourceTrialCount T n (b+1))) (hz : SourceTypical T n s b ω a z) :
-    (P.product (SourceLeafCount T (b+1))).expect
-      (fun x => |sourceOracleDensity T n s b Z true (z,x)-sourceOracleDensity T n s b Z false (z,x)|^(2*k)) ≤
-        (sourceEnergyBudget T n s b R ω η a ((6*(2*k:ℕ):ℝ)*L))^k := by
-  have he (x : Fin (SourceLeafCount T (b+1)) → Leaf) :
-      |sourceOracleDensity T n s b Z true (z,x)-sourceOracleDensity T n s b Z false (z,x)|=
-        |multilinear (sourceSignedCoefficient T n s b z) Z x| := by
-    unfold sourceSignedCoefficient
-    rw [multilinear_sub,←sourceOracleDensity_expansion,←sourceOracleDensity_expansion,abs_sub_comm]
-  simp_rw [he]
-  apply (multilinear_moment_scaled P Z k hk hcenter hL hMoment _ _).trans
-  exact pow_le_pow_left₀ (multilinearEnergy_nonneg _ _)
-    (source_canonical_energy T n hT s b R ω η a _ ha (by positivity) hsmall hη z hz) k
-
-end TraceReconstruction
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-theorem degreeForcedRoleEnergy_zero_of_lt (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
-    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
-    (A : (b : ℕ) → Finset (Fin (T b))) (b r : ℕ) (hr : r<2^(b+1))
-    (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ) :
-    degreeForcedRoleEnergy T n g s center globalShift A b r d incoming=0 := by
-  have hi : IsEmpty (DegreeForcedRole T b r) := ⟨fun w => by
-    obtain ⟨F⟩ := w.property.2
-    have hh := F.leaf_lower
-    rw [occupiedWindowTerm_leafCount,w.property.1] at hh
-    omega⟩
-  let := hi
-  exact Finset.sum_eq_zero (fun w _ => isEmptyElim w)
-
-/-- Below the genuine forced binary leaf threshold there is NO forced
- contribution; retaining even its coarse bound would lose the main exponent. -/
-def sourceSharpDegreeBudget (T n : ℕ → ℕ) (s b r : ℕ) (ω η a : ℝ) : ℝ :=
-  4*(3^((b+1)*r):ℕ)*
-    (((SourceLeafCount T (b+1)).choose r:ℕ)*(3^((b+1)*r):ℕ)*(η^2+a^2)+
-      if 2^(b+1)≤r then sourceForcedBudget T n s b r ω else 0)
-
-theorem sourceSharpDegreeBudget_nonneg (T n : ℕ → ℕ) (s b r : ℕ) (ω η a : ℝ) :
-    0 ≤ sourceSharpDegreeBudget T n s b r ω η a := by
-  have h := sourceForcedBudget_nonneg T n s b r ω
-  unfold sourceSharpDegreeBudget
-  split_ifs <;> positivity
-
-theorem source_canonical_sharp_degree_bound {Side : Type*} (T n : ℕ → ℕ) (hT : ∀j,0<T j)
-    (s b r : ℕ) (ω η a : ℝ) (ha : 0≤a)
-    (hη : ∀j<b+1,hiddenSmoothingBound (n j) s ω≤η)
-    (z : Reports Side (SourceTrialCount T n (b+1))) (hz : SourceTypical T n s b ω a z) :
-    (∑σ : Fin (SourceLeafCount T (b+1)) → Bool, if monomialDegree σ=r then
-      (averagedCoefficient (PairedMechanisms.orientationsGiven z)
-        (sourceFlatLeafType T n s (sourceLocalCenter T n) (b+1) 0 false) σ -
-       averagedCoefficient (PairedMechanisms.orientationsGiven z)
-        (sourceFlatLeafType T n s (sourceLocalCenter T n) (b+1) 0 true) σ)^2 else 0) ≤
-      sourceSharpDegreeBudget T n s b r ω η a := by
-  rw [source_canonical_degree_sum]
-  apply (sourceCoefficient_degree_energy T n _ s _ _ (fun j => centralThird (T j)) b r
-    (sourceReportFlags T n (b+1) z) 0 η
-    (hiddenConditions_smoothingCap T n _ ω η s (b+1) hη _ hz.1)).trans
-  apply mul_le_mul_of_nonneg_left _ (by positivity)
-  apply add_le_add
-  · apply mul_le_mul_of_nonneg_left _ (by positivity)
-    apply add_le_add_right
-    exact (sq_le_sq₀ (sourceNoncentralMass_nonneg T n _ s _ _ _ _ _ _) ha).mpr hz.2.le
-  · by_cases hr : 2^(b+1)≤r
-    · rw [ite_eq_left hr]
-      exact degreeForcedRoleEnergy_bound T n _ hT s (sourceLocalCenter T n) _ _ _ (sourceScheduledShift_hasPlateau T n)
-        (fun j => hiddenBinomialCap (n j) ω) (fun j => hiddenBinomialCap (sourceScheduledTrials T n j) ω)
-        (fun j => hiddenBinomialCap_nonneg _ _) (fun j => hiddenBinomialCap_nonneg _ _)
-        b r _ (hiddenConditions_binomialCaps T n _ ω (b+1) _ hz.1) 0
-    · rw [ite_eq_right hr,degreeForcedRoleEnergy_zero_of_lt T n _ s _ _ _ b r (lt_of_not_ge hr)]
-def sourceSharpEnergyBudget (T n : ℕ → ℕ) (s b R : ℕ) (ω η a w : ℝ) : ℝ :=
-  (∑r∈Finset.range R, w^(2*r)*(if r=0 then 0 else sourceSharpDegreeBudget T n s b r ω η a))+
-    4*(w^2*sourceSparseCount T s b)^R/(1-w^2*sourceSparseCount T s b)
-
-/-- Complete signed energy, with finite low-degree forced-tree bound and
- the geometric high tail. No coefficient estimates remain as hypotheses. -/
-theorem source_canonical_sharp_energy {Side : Type*} (T n : ℕ → ℕ) (hT : ∀j,0<T j)
-    (s b R : ℕ) (ω η a w : ℝ) (ha : 0 ≤ a) (hw : 0 ≤ w)
-    (hsmall : w^2*sourceSparseCount T s b<1)
-    (hη : ∀j<b+1,hiddenSmoothingBound (n j) s ω ≤ η)
-    (z : Reports Side (SourceTrialCount T n (b+1))) (hz : SourceTypical T n s b ω a z) :
-    multilinearEnergy (sourceSignedCoefficient T n s b z) w ≤ sourceSharpEnergyBudget T n s b R ω η a w := by
-  apply multilinearEnergy_low_high _ w (w^2*sourceSparseCount T s b) 4 hw (by positivity) hsmall
-    (by norm_num) (fun r => if r=0 then 0 else sourceSharpDegreeBudget T n s b r ω η a)
-  · intro r _
-    by_cases hr : r=0
-    · subst r
-      rw [sourceSignedCoefficient_zero,ite_eq_left rfl]
-    · rw [ite_eq_right hr]
-      exact source_canonical_sharp_degree_bound T n hT s b r ω η a ha hη z hz
-  · intro r _
-    exact sourceSignedCoefficient_degree_geometric T n s b r z w
-  · intro r
-    split_ifs
-    · rfl
-    · exact sourceSharpDegreeBudget_nonneg T n s b r ω η a
-
-theorem sourceSharpEnergyBudget_nonneg (T n : ℕ → ℕ) (s b R : ℕ) (ω η a w : ℝ)
-    (hsmall : w^2*sourceSparseCount T s b<1) :
-    0 ≤ sourceSharpEnergyBudget T n s b R ω η a w := by
-  apply add_nonneg
-  · apply Finset.sum_nonneg
-    intro r _
-    apply mul_nonneg (pow_nonneg (sq_nonneg w) r |>.trans_eq (pow_mul w 2 r).symm)
-    split_ifs
-    · rfl
-    · exact sourceSharpDegreeBudget_nonneg T n s b r ω η a
-  · exact div_nonneg (by positivity) (sub_nonneg.mpr hsmall.le)
-
-end TraceReconstruction
-
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-local instance (priority := 10000) {α β : Type*} : DecidableEq (α × β) := Classical.decEq _
-
-theorem source_canonical_sharp_signed_moment {Side Leaf : Type*} [Fintype Leaf]
-    (T n : ℕ → ℕ) (hT : ∀j,0<T j) (s b R k : ℕ) (ω η a : ℝ) (ha : 0 ≤ a)
-    (hη : ∀j<b+1,hiddenSmoothingBound (n j) s ω ≤ η)
-    (P : FiniteLaw Leaf) (Z : Leaf → ℝ) (hk : 1 ≤ k) (hcenter : P.expect Z=0)
-    {L : ℝ} (hL : 0<L) (hMoment : P.expect (fun a => |Z a|^(2*k)) ≤ L^(2*k))
-    (hsmall : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b<1)
-    (z : Reports Side (SourceTrialCount T n (b+1))) (hz : SourceTypical T n s b ω a z) :
-    (P.product (SourceLeafCount T (b+1))).expect
-      (fun x => |sourceOracleDensity T n s b Z true (z,x)-sourceOracleDensity T n s b Z false (z,x)|^(2*k)) ≤
-        (sourceSharpEnergyBudget T n s b R ω η a ((6*(2*k:ℕ):ℝ)*L))^k := by
-  have he (x : Fin (SourceLeafCount T (b+1)) → Leaf) :
-      |sourceOracleDensity T n s b Z true (z,x)-sourceOracleDensity T n s b Z false (z,x)|=
-        |multilinear (sourceSignedCoefficient T n s b z) Z x| := by
-    unfold sourceSignedCoefficient
-    rw [multilinear_sub,←sourceOracleDensity_expansion,←sourceOracleDensity_expansion,abs_sub_comm]
-  simp_rw [he]
-  apply (multilinear_moment_scaled P Z k hk hcenter hL hMoment _ _).trans
-  exact pow_le_pow_left₀ (multilinearEnergy_nonneg _ _)
-    (source_canonical_sharp_energy T n hT s b R ω η a _ ha (by positivity) hsmall hη z hz) k
-
-
-def sourceReferenceLaw {Side Leaf : Type*} [Fintype Side] [Fintype Leaf]
-    (T n : ℕ → ℕ) (b : ℕ) (M : PairedMechanisms Side (SourceTrialCount T n (b+1)))
-    (P : FiniteLaw Leaf) :
-    FiniteLaw (Reports Side (SourceTrialCount T n (b+1)) × (Fin (SourceLeafCount T (b+1)) → Leaf)) :=
-  M.reports.prod (P.product (SourceLeafCount T (b+1)))
-
-theorem sourceOracleDensity_total {Side Leaf : Type*} [Fintype Side] [Fintype Leaf]
-    (T n : ℕ → ℕ) (s b : ℕ) (M : PairedMechanisms Side (SourceTrialCount T n (b+1)))
-    (P : FiniteLaw Leaf) (Z : Leaf → ℝ) (hZ : P.expect Z=0) (τ : Bool) :
-    (sourceReferenceLaw T n b M P).expect (sourceOracleDensity T n s b Z τ)=1 := by
-  rw [sourceReferenceLaw,FiniteLaw.expect_prod]
-  simp_rw [sourceOracleDensity_mean T n s b P Z hZ τ]
-  exact M.reports.expect_const 1
-
-theorem sourceOracle_withDensity {Side Leaf : Type*} [Fintype Side] [Fintype Leaf]
-    (T n : ℕ → ℕ) (s b : ℕ) (M : PairedMechanisms Side (SourceTrialCount T n (b+1)))
-    (P Q : FiniteLaw Leaf) (Z : Leaf → ℝ) (hden : ∀a,Q.mass a=P.mass a*(1+Z a))
-    (hZ : ∀a,0 ≤ 1+Z a) (hc : P.expect Z=0) (τ : Bool) :
-    M.canonicalOracle P Q (sourceFlatLeafType T n s (sourceLocalCenter T n) (b+1) 0 τ)=
-      (sourceReferenceLaw T n b M P).withDensity (sourceOracleDensity T n s b Z τ)
-        (sourceOracleDensity_nonneg T n s b Z hZ τ)
-        (sourceOracleDensity_total T n s b M P Z hc τ) := by
-  apply FiniteLaw.ext
-  funext zx
-  exact M.canonicalOracle_density P Q Z hden _ zx.1 zx.2
-
-def sourceTypicalSet {Side Leaf : Type*} [Fintype Side] [Fintype Leaf]
-    (T n : ℕ → ℕ) (s b : ℕ) (ω a : ℝ) :
-    Finset (Reports Side (SourceTrialCount T n (b+1)) × (Fin (SourceLeafCount T (b+1)) → Leaf)) :=
-  Finset.univ.filter (fun x => SourceTypical T n s b ω a x.1)
-
-theorem sourceTypicalSet_mass {Side Leaf : Type*} [Fintype Side] [Fintype Leaf]
-    (T n : ℕ → ℕ) (s b : ℕ) (M : PairedMechanisms Side (SourceTrialCount T n (b+1)))
-    (P Q : FiniteLaw Leaf) (ω a : ℝ) (τ : Bool) :
-    (M.canonicalOracle P Q (sourceFlatLeafType T n s (sourceLocalCenter T n) (b+1) 0 τ)).eventMass
-      (sourceTypicalSet T n s b ω a)ᶜ=
-      M.reports.expect (fun z => if SourceTypical T n s b ω a z then 0 else 1) := by
-  rw [FiniteLaw.eventMass_indicator]
-  calc
-    _ = (M.canonicalOracle P Q (sourceFlatLeafType T n s (sourceLocalCenter T n) (b+1) 0 τ)).expect
-        (fun zx => if SourceTypical T n s b ω a zx.1 then 0 else 1) := by
-      apply congrArg (FiniteLaw.expect _)
-      funext zx
-      simp [sourceTypicalSet]
-    _ = _ := M.canonicalOracle_metadata P Q
-      (sourceFlatLeafType T n s (sourceLocalCenter T n) (b+1) 0 τ)
-      (fun z => if SourceTypical T n s b ω a z then 0 else 1)
-
-def sourceCenteredBudget (T : ℕ → ℕ) (s b k : ℕ) (L : ℝ) : ℝ :=
-  (((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b)/
-    (1-((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b)
-
-def sourceClippingCost (T n : ℕ → ℕ) (s b k : ℕ) (ω a L : ℝ) : ℝ :=
-  sourceEffectiveHiddenTail T n (b+1) ω+sourceFitTail T n (b+1)/a+
-    Real.sqrt (6*(4*sourceCenteredBudget T s b k L)^k)
-
-theorem sourceCenteredBudget_nonneg (T : ℕ → ℕ) (s b k : ℕ) (L : ℝ)
-    (hsmall : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b<1) :
-    0 ≤ sourceCenteredBudget T s b k L :=
-  div_nonneg (by positivity) (sub_nonneg.mpr hsmall.le)
-
-theorem sourceCenteredBudget_le_one (T : ℕ → ℕ) (s b k : ℕ) (L : ℝ)
-    (hscale : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b≤1/2) :
-    sourceCenteredBudget T s b k L ≤ 1 := by
-  have hsmall : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b<1 := by linarith
-  apply (div_le_iff₀ (sub_pos.mpr hsmall)).mpr
-  linarith
-
-theorem source_reference_centered_moment {Side Leaf : Type*} [Fintype Side] [Fintype Leaf]
-    (T n : ℕ → ℕ) (s b k : ℕ) (M : PairedMechanisms Side (SourceTrialCount T n (b+1)))
-    (P : FiniteLaw Leaf) (Z : Leaf → ℝ) (hk : 1 ≤ k) (hc : P.expect Z=0)
-    {L : ℝ} (hL : 0<L) (hMoment : P.expect (fun a => |Z a|^(2*k)) ≤ L^(2*k))
-    (hsmall : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b<1) (τ : Bool) :
-    (sourceReferenceLaw T n b M P).expect
-      (fun zx => |sourceOracleDensity T n s b Z τ zx-1|^(2*k)) ≤ (sourceCenteredBudget T s b k L)^k := by
-  rw [sourceReferenceLaw,FiniteLaw.expect_prod]
-  calc
-    _ ≤ M.reports.expect (fun _ => (sourceCenteredBudget T s b k L)^k) :=
-      M.reports.expect_mono (fun z => source_canonical_centered_moment T n s b k P Z hk hc hL hMoment hsmall z τ)
-    _ = _ := M.reports.expect_const _
-
-theorem source_reference_signed_moment {Side Leaf : Type*} [Fintype Side] [Fintype Leaf]
-    (T n : ℕ → ℕ) (hTpos : ∀j,0<T j) (s b R k : ℕ)
-    (M : PairedMechanisms Side (SourceTrialCount T n (b+1))) (P : FiniteLaw Leaf)
-    (Z : Leaf → ℝ) (ω a η L : ℝ) (ha : 0≤a) (hL : 0<L) (hk : 1 ≤ k) (hc : P.expect Z=0)
-    (hMoment : P.expect (fun a => |Z a|^(2*k)) ≤ L^(2*k))
-    (hsmall : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b<1)
-    (hη : ∀j<b+1,hiddenSmoothingBound (n j) s ω ≤ η) :
-    (sourceReferenceLaw T n b M P).expect (fun zx =>
-      |if zx∈sourceTypicalSet T n s b ω a then sourceOracleDensity T n s b Z true zx-
-        sourceOracleDensity T n s b Z false zx else 0|^(2*k)) ≤
-      (sourceSharpEnergyBudget T n s b R ω η a ((6*(2*k:ℕ):ℝ)*L))^k := by
-  rw [sourceReferenceLaw,FiniteLaw.expect_prod]
-  have hV := sourceSharpEnergyBudget_nonneg T n s b R ω η a ((6*(2*k:ℕ):ℝ)*L) hsmall
-  calc
-    _ ≤ M.reports.expect (fun _ => (sourceSharpEnergyBudget T n s b R ω η a ((6*(2*k:ℕ):ℝ)*L))^k) := by
-      apply M.reports.expect_mono
-      intro z
-      by_cases hz : SourceTypical T n s b ω a z
-      · simp only [sourceTypicalSet,Finset.mem_filter,Finset.mem_univ,true_and,hz,ite_true]
-        exact source_canonical_sharp_signed_moment T n hTpos s b R k ω η a ha hη P Z hk hc hL hMoment hsmall z hz
-      · simp only [sourceTypicalSet,Finset.mem_filter,Finset.mem_univ,true_and,hz,ite_false,
-          abs_zero,zero_pow (by omega : 2*k≠0),FiniteLaw.expect_const]
-        exact pow_nonneg hV k
-    _ = _ := M.reports.expect_const _
-
-theorem source_reference_clipped_exists {Side Leaf : Type*} [Fintype Side] [Fintype Leaf]
-    (T n : ℕ → ℕ) (hTpos : ∀j,0<T j) (s b R k : ℕ)
-    (M : PairedMechanisms Side (SourceTrialCount T n (b+1))) (P : FiniteLaw Leaf)
-    (Z : Leaf → ℝ) (hZ : ∀a,0 ≤ 1+Z a) (hc : P.expect Z=0)
-    (ω a η L : ℝ) (ha : 0≤a) (hL : 0<L) (hk : 1≤k)
-    (hMoment : P.expect (fun a => |Z a|^(2*k)) ≤ L^(2*k))
-    (hscale : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b ≤ 1/2)
-    (hη : ∀j<b+1,hiddenSmoothingBound (n j) s ω ≤ η)
-    (hB : ∀τ,((sourceReferenceLaw T n b M P).withDensity (sourceOracleDensity T n s b Z τ)
-      (sourceOracleDensity_nonneg T n s b Z hZ τ) (sourceOracleDensity_total T n s b M P Z hc τ)).eventMass
-        (sourceTypicalSet T n s b ω a)ᶜ ≤ sourceEffectiveHiddenTail T n (b+1) ω+sourceFitTail T n (b+1)/a)
-    (herror : sourceClippingCost T n s b k ω a L ≤ 1/2) :
-    Nonempty (ClippedOracle
-      ((sourceReferenceLaw T n b M P).withDensity (sourceOracleDensity T n s b Z false)
-        (sourceOracleDensity_nonneg T n s b Z hZ false) (sourceOracleDensity_total T n s b M P Z hc false))
-      ((sourceReferenceLaw T n b M P).withDensity (sourceOracleDensity T n s b Z true)
-        (sourceOracleDensity_nonneg T n s b Z hZ true) (sourceOracleDensity_total T n s b M P Z hc true))
-      k (sourceClippingCost T n s b k ω a L)
-      (64*sourceSharpEnergyBudget T n s b R ω η a ((6*(2*k:ℕ):ℝ)*L))) := by
-  have hsmall : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b<1 := by linarith
-  exact FiniteLaw.exists_clippedOracle (sourceReferenceLaw T n b M P)
-    (sourceOracleDensity T n s b Z false) (sourceOracleDensity T n s b Z true)
-    (sourceOracleDensity_nonneg T n s b Z hZ false) (sourceOracleDensity_nonneg T n s b Z hZ true)
-    (sourceOracleDensity_total T n s b M P Z hc false) (sourceOracleDensity_total T n s b M P Z hc true)
-    (sourceTypicalSet T n s b ω a) _ (hB false) (hB true) k hk (sourceCenteredBudget T s b k L)
-    (sourceCenteredBudget_nonneg T s b k L hsmall) (sourceCenteredBudget_le_one T s b k L hscale)
-    (source_reference_centered_moment T n s b k M P Z hk hc hL hMoment hsmall false)
-    (source_reference_centered_moment T n s b k M P Z hk hc hL hMoment hsmall true)
-    _ (sourceSharpEnergyBudget_nonneg T n s b R ω η a ((6*(2*k:ℕ):ℝ)*L) hsmall)
-    (source_reference_signed_moment T n hTpos s b R k M P Z ω a η L ha hL hk hc hMoment hsmall hη) herror
-
-/-- Complete finite quantitative clipping of the actual recursive source
- oracle. All analytic budgets here are explicit arithmetic in the literal
- branching/trial counts; no coefficient or oracle-existence premise remains. -/
-theorem source_clippedOracle_exists {Side Leaf : Type*} [Fintype Side] [Fintype Leaf]
-    (T n : ℕ → ℕ) (hTpos : ∀j,0<T j) (s b R k : ℕ)
-    (M : PairedMechanisms Side (SourceTrialCount T n (b+1))) (P Q : FiniteLaw Leaf)
-    (Z : Leaf → ℝ) (hden : ∀a,Q.mass a=P.mass a*(1+Z a)) (hZ : ∀a,0 ≤ 1+Z a)
-    (ω a η L : ℝ) (ha : 0<a) (hL : 0<L)
-    (hw : ∀i,M.hiddenWeight i=ω) (hp : ω^2∈Set.Icc 0 1)
-    (hn : ∀j,j<b → 0<n j) (hne : ∀j,j<b → Even (n j))
-    (hT : ∀j≤b,24≤T j) (he : ∀j≤b,Even (T j)) (hMotif : ∀j≤b,24*2^s≤T j)
-    (hroot : 4*(n b+1)≤T b) (hk : 1 ≤ k)
-    (hMoment : P.expect (fun a => |Z a|^(2*k)) ≤ L^(2*k))
-    (hscale : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b ≤ 1/2)
-    (hη : ∀j<b+1,hiddenSmoothingBound (n j) s ω ≤ η)
-    (herror : sourceClippingCost T n s b k ω a L ≤ 1/2) :
-    Nonempty (ClippedOracle
-      (M.canonicalOracle P Q (sourceFlatLeafType T n s (sourceLocalCenter T n) (b+1) 0 false))
-      (M.canonicalOracle P Q (sourceFlatLeafType T n s (sourceLocalCenter T n) (b+1) 0 true))
-      k (sourceClippingCost T n s b k ω a L)
-      (64*sourceSharpEnergyBudget T n s b R ω η a ((6*(2*k:ℕ):ℝ)*L))) := by
-  have hc := FiniteLaw.density_score_centered P Q Z hden
-  have hB (τ : Bool) : ((sourceReferenceLaw T n b M P).withDensity (sourceOracleDensity T n s b Z τ)
-      (sourceOracleDensity_nonneg T n s b Z hZ τ) (sourceOracleDensity_total T n s b M P Z hc τ)).eventMass
-        (sourceTypicalSet T n s b ω a)ᶜ ≤ sourceEffectiveHiddenTail T n (b+1) ω+sourceFitTail T n (b+1)/a := by
-    rw [←sourceOracle_withDensity T n s b M P Q Z hden hZ hc τ,sourceTypicalSet_mass]
-    exact source_typical_effective_failure T n s b M ω a ha hw hp hn hne hT he hMotif hroot
-  rw [sourceOracle_withDensity T n s b M P Q Z hden hZ hc false,
-    sourceOracle_withDensity T n s b M P Q Z hden hZ hc true]
-  exact source_reference_clipped_exists T n hTpos s b R k M P Z hZ hc ω a η L ha.le hL hk hMoment hscale hη hB herror
-
-end TraceReconstruction
-
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-namespace ClippedOracle
-variable {α β : Type*} [Fintype α] [Fintype β]
-    {P Q : FiniteLaw α} {k : ℕ} {e V : ℝ}
-
-def prod (C : ClippedOracle P Q k e V) (R : FiniteLaw β) :
-    ClippedOracle (P.prod R) (Q.prod R) k e V where
-  first := C.first.prod R
-  second := C.second.prod R
-  score := fun x => C.score x.1
-  density x := by
-    change C.second.mass x.1*R.mass x.2=(C.first.mass x.1*R.mass x.2)*(1+C.score x.1)
-    rw [C.density]; ring
-  likelihood_nonneg x := C.likelihood_nonneg x.1
-  centered := by
-    rw [FiniteLaw.expect_prod]
-    simp_rw [FiniteLaw.expect_const]
-    exact C.centered
-  moment := by
-    unfold FiniteLaw.evenMomentSq
-    rw [FiniteLaw.expect_prod]
-    simp_rw [FiniteLaw.expect_const]
-    exact C.moment
-  first_tv := by rw [FiniteLaw.tv_prod_eq_left]; exact C.first_tv
-  second_tv := by rw [FiniteLaw.tv_prod_eq_left]; exact C.second_tv
-end ClippedOracle
-
-theorem ApproximatesTrace.of_tv {α : Type*} [Fintype α]
-    {q E e : ℝ} {P Q : FiniteLaw α} {f : α → List Bool} {w : List Bool}
-    (hP : ApproximatesTrace q P f w E) (htv : Q.tv P ≤ e) :
-    ApproximatesTrace q Q f w (e+E) := by
-  intro F hF
-  exact (abs_sub_le (Q.expect (fun a => F (f a))) (P.expect (fun a => F (f a)))
-    (listDeletionExpectation q w F)).trans (add_le_add
-      ((FiniteLaw.abs_expect_sub_le_tv Q P _ (fun a => hF _)).trans htv) (hP F hF))
-
-/-- A finite genuine approximate oracle pair with a common complete-trace
- decoder. It does not include an overlap or any unproved scale conclusion. -/
-structure OracleTracePair (q : ℝ) (x y : List Bool) (k : ℕ) (E V : ℝ) where
-  Outcome : Type
-  finiteOutcome : Fintype Outcome
-  first : @FiniteLaw Outcome finiteOutcome
-  second : @FiniteLaw Outcome finiteOutcome
-  decode : Outcome → List Bool
-  first_simulation : @ApproximatesTrace Outcome finiteOutcome q first decode x E
-  second_simulation : @ApproximatesTrace Outcome finiteOutcome q second decode y E
-  score : Outcome → ℝ
-  density : ∀a,second.mass a=first.mass a*(1+score a)
-  likelihood_nonneg : ∀a,0≤1+score a
-  centered : first.expect score=0
-  moment : @FiniteLaw.evenMomentSq Outcome finiteOutcome first score k ≤ V
-
-/-- The clipping and independent filler metadata preserve the source's fixed
- public trace decoder. -/
-theorem source_clipped_simulation {Side Leaf : Type} [Fintype Side] [Fintype Leaf]
-    {q E e V : ℝ} (hq : q∈Set.Icc 0 1) (hE : 0≤E)
-    (s : ℕ) (P T n h : ℕ → ℕ) (atom : Bool → List Bool) (b k : ℕ)
-    (H : SourceLayoutMargins s P T n h b)
-    (hP : P b=n b*(2*h b+1)+T b) (hT : Even (T b))
-    (hlarge : 4*(n b+1)≤T b) (hmotif : 8*2^s≤T b)
-    (hn : ∀j<b,0<n j)
-    (M : PairedMechanisms Side (SourceTrialCount T n (b+1)))
-    (R : Fin (SourceFillerCount T (b+1)) → FiniteLaw Side) (A B : FiniteLaw Leaf)
-    (f : Side → List Bool) (g : Leaf → List Bool)
-    (hU : ∀j,SimulatesTrace q (M.first j) f (sourceFlatTrialWord s P T n h atom (b+1) false j))
-    (hV : ∀j,SimulatesTrace q (M.second j) f (sourceFlatTrialWord s P T n h atom (b+1) true j))
-    (hW : ∀j,SimulatesTrace q (R j) f (sourceFlatFillerWord s P T n h atom (b+1) j))
-    (hA : ApproximatesTrace q A g (atom false) E) (hB : ApproximatesTrace q B g (atom true) E)
-    (C : ClippedOracle
-      (M.canonicalOracle A B (sourceFlatLeafType T n s (sourceLocalCenter T n) (b+1) 0 false))
-      (M.canonicalOracle A B (sourceFlatLeafType T n s (sourceLocalCenter T n) (b+1) 0 true)) k e V) :
-    Nonempty (OracleTracePair q (sourceBlockWord s P atom (b+1) false)
-      (sourceBlockWord s P atom (b+1) true) k
-      (e+(SourceLeafCount T (b+1):ℝ)*E+sourceFitTail T n (b+1)) V) := by
-  let D := C.prod (FiniteLaw.coordinateProduct R)
-  refine ⟨{
-    Outcome := (Reports Side (SourceTrialCount T n (b+1)) × (Fin (SourceLeafCount T (b+1)) → Leaf)) ×
-      (Fin (SourceFillerCount T (b+1)) → Side)
-    finiteOutcome := inferInstance
-    first := D.first
-    second := D.second
-    decode := sourceOracleDecode T n (b+1) f g
-    score := D.score
-    density := D.density
-    likelihood_nonneg := D.likelihood_nonneg
-    centered := D.centered
-    moment := D.moment
-    first_simulation := ?_
-    second_simulation := ?_
-  }⟩
-  · simpa only [add_assoc] using
-      (source_canonical_simulation hq hE s P T n h atom b H hP hT hlarge hmotif hn M R A B f g hU hV hW hA hB false).of_tv D.first_tv
-  · simpa only [add_assoc] using
-      (source_canonical_simulation hq hE s P T n h atom b H hP hT hlarge hmotif hn M R A B f g hU hV hW hA hB true).of_tv D.second_tv
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction.FiniteLaw
-open scoped Classical
-variable {α : Type*} [Fintype α]
-
-theorem expect_nonneg (P : FiniteLaw α) (F : α → ℝ) (hF : ∀a,0≤F a) : 0≤P.expect F :=
-  Finset.sum_nonneg (fun a _ => mul_nonneg (P.nonneg a) (hF a))
-
-theorem evenMomentSq_nonneg (P : FiniteLaw α) (F : α → ℝ) (k : ℕ) :
-    0≤P.evenMomentSq F k := Real.rpow_nonneg (P.expect_nonneg _ (fun _ => by positivity)) _
-
-theorem evenMomentSq_pow (P : FiniteLaw α) (F : α → ℝ) (k : ℕ) (hk : 1≤k) :
-    (P.evenMomentSq F k)^k=P.expect (fun a => |F a|^(2*k)) := by
-  unfold evenMomentSq
-  rw [one_div,Real.rpow_inv_natCast_pow (P.expect_nonneg _ (fun _ => by positivity)) (by omega)]
-
-theorem expect_even_pow_le_of_moment (P : FiniteLaw α) (F : α → ℝ) (k : ℕ) (hk : 1≤k)
-    {V : ℝ} (h : P.evenMomentSq F k≤V) :
-    P.expect (fun a => |F a|^(2*k))≤V^k := by
-  rw [←P.evenMomentSq_pow F k hk]
-  exact pow_le_pow_left₀ (P.evenMomentSq_nonneg F k) h k
-
-theorem secondMoment_le_evenMomentSq (P : FiniteLaw α) (F : α → ℝ) (k : ℕ) (hk : 1≤k) :
-    P.expect (fun a => F a^2)≤P.evenMomentSq F k := by
-  have hn : 0≤P.expect (fun a => F a^2) := P.expect_nonneg _ (fun _ => sq_nonneg _)
-  have hj := P.abs_expect_pow_le (fun a => F a^2) k
-  simp only [abs_of_nonneg hn,abs_pow,←pow_mul] at hj
-  have hr := Real.rpow_le_rpow (pow_nonneg hn k) hj (by positivity : 0≤(k:ℝ)⁻¹)
-  rw [Real.pow_rpow_inv_natCast hn (by omega)] at hr
-  simpa only [evenMomentSq,one_div] using hr
-
-theorem affinity_coordinateProduct {n : ℕ} (P Q : Fin n → FiniteLaw α) :
-    (coordinateProduct P).affinity (coordinateProduct Q)=∏i,(P i).affinity (Q i) := by
-  unfold affinity coordinateProduct
-  simp only
-  have he (a : Fin n → α) :
-      Real.sqrt (∏i,(P i).mass (a i))*Real.sqrt (∏i,(Q i).mass (a i))=
-        ∏i,Real.sqrt ((P i).mass (a i))*Real.sqrt ((Q i).mass (a i)) := by
-    rw [Real.sqrt_prod _ (fun i _ => (P i).nonneg (a i)),
-      Real.sqrt_prod _ (fun i _ => (Q i).nonneg (a i)),Finset.prod_mul_distrib]
-  simp_rw [he]
-  exact (Fintype.prod_sum (fun (i : Fin n) a => Real.sqrt ((P i).mass a)*Real.sqrt ((Q i).mass a))).symm
-
-theorem one_sub_prod_le_sum {ι : Type*} (S : Finset ι) (f : ι → ℝ)
-    (h0 : ∀i∈S,0≤f i) (h1 : ∀i∈S,f i≤1) :
-    1-∏i∈S,f i≤∑i∈S,(1-f i) := by
-  induction S using Finset.induction_on with
-  | empty => simp
-  | @insert a S ha ih =>
-    rw [Finset.prod_insert ha,Finset.sum_insert ha]
-    have H := ih (fun i hi => h0 i (Finset.mem_insert_of_mem hi))
-      (fun i hi => h1 i (Finset.mem_insert_of_mem hi))
-    have hs : 0≤∑i∈S,(1-f i) := Finset.sum_nonneg (fun i hi => sub_nonneg.mpr (h1 i (Finset.mem_insert_of_mem hi)))
-    have hm := mul_le_mul_of_nonneg_left H (h0 a (Finset.mem_insert_self a S))
-    have hm' := mul_le_mul_of_nonneg_right (h1 a (Finset.mem_insert_self a S)) hs
-    nlinarith
-
-theorem hellingerSq_coordinateProduct_le {n : ℕ} (P Q : Fin n → FiniteLaw α) :
-    (coordinateProduct P).hellingerSq (coordinateProduct Q)≤∑i,(P i).hellingerSq (Q i) := by
-  simp_rw [hellingerSq_eq]
-  rw [affinity_coordinateProduct]
-  have h := one_sub_prod_le_sum Finset.univ (fun i => (P i).affinity (Q i))
-    (fun i _ => affinity_nonneg _ _) (fun i _ => affinity_le_one _ _)
-  have he : (∑i,(2-2*(P i).affinity (Q i)))=2*∑i,(1-(P i).affinity (Q i)) := by
-    rw [Finset.mul_sum]; apply Finset.sum_congr rfl; intro i _; ring
-  rw [he]
-  linarith
-
-@[simp] theorem hellingerSq_self (P : FiniteLaw α) : P.hellingerSq P=0 := by
-  simp [hellingerSq]
-
-theorem hellingerSq_symm (P Q : FiniteLaw α) : P.hellingerSq Q=Q.hellingerSq P := by
-  apply Finset.sum_congr rfl; intro a _; ring
-
-/-- Only differing coordinates cost squared Hellinger distance. This estimate
- does not pay for pure filler coordinates, nor assume independence of traces
- after their common concatenating decoder. -/
-theorem tv_mixedProduct_le {n : ℕ} (P Q : FiniteLaw α) (Z : α → ℝ)
-    (hden : ∀a,Q.mass a=P.mass a*(1+Z a)) (k : ℕ) (hk : 1≤k)
-    {V : ℝ} (hM : P.evenMomentSq Z k≤V) (τ σ : Fin n → Bool) :
-    (coordinateProduct (fun i => if τ i then Q else P)).tv
-      (coordinateProduct (fun i => if σ i then Q else P))≤
-        Real.sqrt ((∑i,if τ i=σ i then (0:ℝ) else 1)*V) := by
-  have hH : P.hellingerSq Q≤V := (P.hellingerSq_le_score Q Z hden).trans
-    ((P.secondMoment_le_evenMomentSq Z k hk).trans hM)
-  apply (tv_le_sqrt_hellingerSq _ _).trans
-  apply Real.sqrt_le_sqrt
-  apply (hellingerSq_coordinateProduct_le _ _).trans
-  rw [Finset.sum_mul]
-  apply Finset.sum_le_sum
-  intro i _
-  cases τ i <;> cases σ i <;> simp only [Bool.false_eq_true,Bool.true_eq_false,ite_false,ite_true,
-    zero_mul,one_mul,hellingerSq_self,le_refl]
-  · exact hH
-  · rw [hellingerSq_symm]; exact hH
-
-end TraceReconstruction.FiniteLaw
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-/-- Literal recursive source oracle with actual complete-trace side laws.
- All remaining hypotheses here are explicit parameter inequalities, the old
- leaf oracle, and its genuine repeated-word overlap invariant. -/
-theorem actual_source_oracle {Leaf : Type} [Fintype Leaf]
-    {q E : ℝ} (hq : q∈Set.Icc 0 1) (hE : 0≤E)
-    (s : ℕ) (P T n h : ℕ → ℕ) (atom : Bool → List Bool) (b R k : ℕ)
-    (H : SourceLayoutMargins s P T n h b)
-    (hTpos : ∀j,0<T j) (hP : P b=n b*(2*h b+1)+T b)
-    (hn : ∀j<b,0<n j) (hT : ∀j≤b,24≤T j) (he : ∀j≤b,Even (T j))
-    (hMotif : ∀j≤b,24*2^s≤T j) (hroot : 4*(n b+1)≤T b)
-    (A B : FiniteLaw Leaf) (g : Leaf → List Bool) (Z : Leaf → ℝ)
-    (hden : ∀a,B.mass a=A.mass a*(1+Z a)) (hZ : ∀a,0≤1+Z a)
-    (hA : ApproximatesTrace q A g (atom false) E) (hB : ApproximatesTrace q B g (atom true) E)
-    (ω a η L : ℝ) (ha : 0<a) (hL : 0<L) (hω : ω∈Set.Ioo 0 1)
-    (hov : ∀j<b+1,ω≤listTraceOverlap q hq
-      (pureAtomRun (sourceBlockWord s P atom j false) (h j))
-      (pureAtomRun (sourceBlockWord s P atom j false) (h j+1)))
-    (hk : 1≤k) (hMoment : A.expect (fun a => |Z a|^(2*k))≤L^(2*k))
-    (hscale : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b≤1/2)
-    (hη : ∀j<b+1,hiddenSmoothingBound (n j) s ω≤η)
-    (herror : sourceClippingCost T n s b k ω a L≤1/2) :
-    Nonempty (OracleTracePair q (sourceBlockWord s P atom (b+1) false)
-      (sourceBlockWord s P atom (b+1) true) k
-      (sourceClippingCost T n s b k ω a L+(SourceLeafCount T (b+1):ℝ)*E+sourceFitTail T n (b+1))
-      (64*sourceSharpEnergyBudget T n s b R ω η a ((6*(2*k:ℕ):ℝ)*L))) := by
-  let U := sourceFlatTrialWord s P T n h atom (b+1) false
-  let V := sourceFlatTrialWord s P T n h atom (b+1) true
-  let W := sourceFlatFillerWord s P T n h atom (b+1)
-  obtain ⟨mU,hmU⟩ := finite_word_bound U
-  obtain ⟨mV,hmV⟩ := finite_word_bound V
-  obtain ⟨mW,hmW⟩ := finite_word_bound W
-  let m := max mU (max mV mW)
-  have hU (j) : (U j).length ≤ m := (hmU j).trans (le_max_left _ _)
-  have hV (j) : (V j).length ≤ m := (hmV j).trans ((le_max_left _ _).trans (le_max_right _ _))
-  have hW (j) : (W j).length ≤ m := (hmW j).trans ((le_max_right _ _).trans (le_max_right _ _))
-  have hov' (j) : ω≤listTraceOverlap q hq (U j) (V j) :=
-    sourceTrialWord_overlap hq s P T n h atom (b+1) hov (finEnumeration _ j)
-  let M := tracePairedMechanisms hq hω.1 U V m hU hV hov'
-  let F : Fin (SourceFillerCount T (b+1)) → FiniteLaw (BoundedTrace m) :=
-    fun j => listTraceLaw q hq (W j) m (hW j)
-  have hp : ω^2∈Set.Icc 0 1 := ⟨sq_nonneg ω,by nlinarith [hω.1,hω.2]⟩
-  obtain ⟨C⟩ := source_clippedOracle_exists T n hTpos s b R k M A B Z hden hZ ω a η L ha hL
-    (fun _ => rfl) hp hn H.even_trials hT he hMotif hroot hk hMoment hscale hη herror
-  apply source_clipped_simulation hq hE s P T n h atom b k H hP (he b le_rfl) hroot
-    (by nlinarith [hMotif b le_rfl]) hn M F A B Subtype.val g
-    (fun j => listTraceLaw_simulates q hq (U j) m (hU j))
-    (fun j => listTraceLaw_simulates q hq (V j) m (hV j))
-    (fun j => listTraceLaw_simulates q hq (W j) m (hW j)) hA hB C
-
-end TraceReconstruction
-
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-
-end OAI
+import OAI.Probability.TraceReconstruction.Simulation
+
+namespace OAI
+
+/-!
+# SourceOracle
+
+Actual source mechanisms and posteriors, typical metadata, sparse degrees, common-event clipping, approximate oracles, and mixed Hellinger bounds.
+-/
+
+open scoped ENNReal
+noncomputable section
+
+open Filter
+open scoped Topology
+
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- Ambient finite trace-space choice does not lose any of the genuine
+complete-trace common mass. The embedding adds only zero-mass points. -/
+theorem listTraceOverlap_le_ambient {q : ℝ} (hq : q∈Set.Icc 0 1)
+    (x y : List Bool) (M : ℕ) (hx : x.length≤M) (hy : y.length≤M) :
+    listTraceOverlap q hq x y≤
+      (listTraceLaw q hq x M hx).overlap (listTraceLaw q hq y M hy) := by
+  let M₀ := max x.length y.length
+  have H := approximations_trace_tv hq
+    (listTraceLaw q hq x M₀ (le_max_left _ _))
+    (listTraceLaw q hq y M₀ (le_max_right _ _)) Subtype.val x y 0 0 M hx hy
+    (listTraceLaw_simulates q hq x M₀ _).approximate
+    (listTraceLaw_simulates q hq y M₀ _).approximate
+  simp only [add_zero] at H
+  unfold listTraceOverlap
+  rw [FiniteLaw.overlap_eq_one_sub_tv,FiniteLaw.overlap_eq_one_sub_tv]
+  linarith
+
+theorem listTraceOverlap_ambient {q : ℝ} (hq : q∈Set.Icc 0 1)
+    (x y : List Bool) (M : ℕ) (hx : x.length≤M) (hy : y.length≤M) :
+    listTraceOverlap q hq x y=
+      (listTraceLaw q hq x M hx).overlap (listTraceLaw q hq y M hy) := by
+  apply le_antisymm (listTraceOverlap_le_ambient hq x y M hx hy)
+  exact simulations_overlap_le hq _ _ Subtype.val x y
+    (listTraceLaw_simulates q hq x M hx) (listTraceLaw_simulates q hq y M hy)
+
+namespace FiniteLaw
+variable {α : Type*} [Fintype α]
+theorem commonPart_le_right (P Q : FiniteLaw α) (h : 0<P.overlap Q)
+    {ω : ℝ} (hω : ω≤P.overlap Q) (a : α) :
+    ω*(commonPart P Q h).mass a≤Q.mass a := by
+  change ω*(min (P.mass a) (Q.mass a)/P.overlap Q)≤_
+  calc
+    _ ≤ P.overlap Q*(min (P.mass a) (Q.mass a)/P.overlap Q) :=
+      mul_le_mul_of_nonneg_right hω (div_nonneg (le_min (P.nonneg a) (Q.nonneg a)) h.le)
+    _ = min (P.mass a) (Q.mass a) := by field_simp
+    _ ≤ _ := min_le_right _ _
+end FiniteLaw
+
+/-- The paired experiment uses the actual unseparated complete deletion
+traces, and the actual normalized minimum of their mass functions. -/
+def tracePairedMechanisms {m : ℕ} {q ω : ℝ} (hq : q∈Set.Icc 0 1) (hω : 0<ω)
+    (U V : Fin m → List Bool) (M : ℕ) (hU : ∀j,(U j).length≤M) (hV : ∀j,(V j).length≤M)
+    (hov : ∀j,ω≤listTraceOverlap q hq (U j) (V j)) : PairedMechanisms (BoundedTrace M) m where
+  first j := listTraceLaw q hq (U j) M (hU j)
+  second j := listTraceLaw q hq (V j) M (hV j)
+  common j := FiniteLaw.commonPart _ _ (hω.trans_le ((hov j).trans (listTraceOverlap_le_ambient hq _ _ M _ _)))
+  hiddenWeight _ := ω
+  weight_nonneg _ := hω.le
+  common_first j := FiniteLaw.commonPart_le_left _ _ _
+    ((hov j).trans (listTraceOverlap_le_ambient hq _ _ M _ _))
+  common_second j := FiniteLaw.commonPart_le_right _ _ _
+    ((hov j).trans (listTraceOverlap_le_ambient hq _ _ M _ _))
+
+/-- Every literal side pair draws its overlap from the correct height of the
+source recursion, including reflected global pairs. -/
+theorem sourceTrialWord_overlap {q ω : ℝ} (hq : q∈Set.Icc 0 1)
+    (s : ℕ) (P T n h : ℕ → ℕ) (atom : Bool → List Bool) (b : ℕ)
+    (hov : ∀j<b,ω≤listTraceOverlap q hq
+      (pureAtomRun (sourceBlockWord s P atom j false) (h j))
+      (pureAtomRun (sourceBlockWord s P atom j false) (h j+1)))
+    (i : WindowTrial T n (sourceScheduledTrials T n) b) :
+    ω≤listTraceOverlap q hq (sourceTrialWord s P T n h atom b i false)
+      (sourceTrialWord s P T n h atom b i true) := by
+  induction b with
+  | zero => exact PEmpty.elim i
+  | succ b ih =>
+    rcases i with i | i | ⟨u,i⟩
+    · simpa [sourceTrialWord,bitIndex] using hov b (by omega)
+    · cases b with
+      | zero => exact Fin.elim0 i
+      | succ b => simpa [sourceTrialWord,bitIndex] using hov b (by omega)
+    · exact ih (fun j hj => hov j (by omega)) i
+
+/-- A common finite side-trace space exists without changing any word. -/
+theorem finite_word_bound {ι : Type*} [Fintype ι] (w : ι → List Bool) :
+    ∃M,∀i,(w i).length≤M := by
+  refine ⟨∑i,(w i).length, fun i => ?_⟩
+  exact Finset.single_le_sum (f:=fun i => (w i).length) (fun _ _ => Nat.zero_le _) (Finset.mem_univ i)
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- The actual report flags, in the literal nested trial coordinates. -/
+def sourceReportFlags {Side : Type*} (T n : ℕ → ℕ) (b : ℕ)
+    (z : Reports Side (SourceTrialCount T n b)) :
+    WindowAssignments T n (sourceScheduledTrials T n) (Option Bool) b :=
+  (sourceAssignmentCode T n (Option Bool) b).symm (fun j => (z j).1)
+
+/-- Exact change of coordinates for the posterior of all hidden bits, not
+ just separate local marginals. -/
+theorem source_posterior_code_expect {Side : Type*} (T n : ℕ → ℕ) (b : ℕ)
+    (z : Reports Side (SourceTrialCount T n b))
+    (F : WindowAssignments T n (sourceScheduledTrials T n) Bool b → ℝ) :
+    (PairedMechanisms.orientationsGiven z).expect
+      (fun bits => F ((sourceAssignmentCode T n Bool b).symm bits)) =
+    (windowPosteriorLaw T n (sourceScheduledTrials T n) b (sourceReportFlags T n b z)).expect F := by
+  rw [windowPosterior_expect_flat,FiniteLaw.indexedProduct_expect_equiv _ (finEnumeration _)]
+  simp only [sourceReportFlags,sourceAssignmentCode_eval]
+  simp only [PairedMechanisms.orientationsGiven,FiniteLaw.expect,FiniteLaw.coordinateProduct,
+    FiniteLaw.indexedProduct,sourceAssignmentCode,Equiv.coe_fn_symm_mk]
+  apply Finset.sum_congr (by ext; simp)
+  intro bits _
+  rfl
+
+/-- A monomial is precisely its set of actual leaf addresses. -/
+def sourceSubsetCode (T : ℕ → ℕ) (b : ℕ) :
+    (Fin (SourceLeafCount T b) → Bool) ≃ Finset (WindowLeaf T b) :=
+  (maskSubsetEquiv _).trans (Equiv.finsetCongr (finEnumeration _))
+
+theorem sourceSubsetCode_card (T : ℕ → ℕ) (b : ℕ)
+    (σ : Fin (SourceLeafCount T b) → Bool) :
+    (sourceSubsetCode T b σ).card=monomialDegree σ := by
+  simp only [sourceSubsetCode,Equiv.trans_apply,Equiv.finsetCongr_apply,Finset.card_map]
+  rfl
+
+theorem selectionCoefficient_sign {k : ℕ} (θ σ : Fin k → Bool) :
+    selectionCoefficient θ σ=∏j∈maskSubset σ,(1-boolTypeSign (θ j))/2 := by
+  unfold selectionCoefficient maskSubset
+  rw [Finset.prod_filter]
+  apply Finset.prod_congr rfl
+  intro j _
+  cases σ j <;> cases θ j <;> norm_num [boolTypeSign]
+
+theorem source_leaf_selection (T n : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (b : ℕ) (bits : WindowAssignments T n (sourceScheduledTrials T n) Bool b)
+    (incoming : ℤ) (τ : Bool) (σ : Fin (SourceLeafCount T b) → Bool) :
+    selectionCoefficient
+      (fun j => sourceTreeLeafBool T n s center b bits incoming τ (finEnumeration _ j)) σ =
+    ∏v∈sourceSubsetCode T b σ,
+      (1-sourceTreeLeafSign T n (sourceScheduledTrials T n) s center
+        (sourceScheduledShift T n) b bits incoming (boolTypeSign τ) v)/2 := by
+  rw [selectionCoefficient_sign]
+  simp only [sourceSubsetCode,Equiv.trans_apply,Equiv.finsetCongr_apply,Finset.prod_map]
+  apply Finset.prod_congr rfl
+  intro j _
+  rw [sourceTreeLeafBool_sign]
+  rfl
+
+/-- The real canonical-oracle coefficient difference is the exact source
+ polynomial coefficient whose forced-tree energy has been proved. -/
+theorem source_canonical_coefficient {Side : Type*} (T n : ℕ → ℕ) (s : ℕ)
+    (center : ℕ → ℤ) (b : ℕ) (incoming : ℤ)
+    (z : Reports Side (SourceTrialCount T n (b+1)))
+    (σ : Fin (SourceLeafCount T (b+1)) → Bool) :
+    averagedCoefficient (PairedMechanisms.orientationsGiven z)
+      (sourceFlatLeafType T n s center (b+1) incoming false) σ -
+    averagedCoefficient (PairedMechanisms.orientationsGiven z)
+      (sourceFlatLeafType T n s center (b+1) incoming true) σ =
+    sourceCoefficient T n (sourceScheduledTrials T n) s center (sourceScheduledShift T n)
+      b (sourceReportFlags T n (b+1) z) incoming (sourceSubsetCode T (b+1) σ) := by
+  unfold averagedCoefficient sourceFlatLeafType
+  rw [source_posterior_code_expect T n (b+1) z
+    (fun bits => selectionCoefficient
+      (fun j => sourceTreeLeafBool T n s center (b+1) bits incoming false (finEnumeration _ j)) σ),
+    source_posterior_code_expect T n (b+1) z
+    (fun bits => selectionCoefficient
+      (fun j => sourceTreeLeafBool T n s center (b+1) bits incoming true (finEnumeration _ j)) σ)]
+  simp_rw [source_leaf_selection]
+  rfl
+
+/-- Reindexing preserves the entire degree-r square sum, including zero
+ and degrees larger than the number of real leaves. -/
+theorem source_canonical_degree_sum {Side : Type*} (T n : ℕ → ℕ) (s : ℕ)
+    (center : ℕ → ℤ) (b r : ℕ) (incoming : ℤ)
+    (z : Reports Side (SourceTrialCount T n (b+1))) :
+    (∑σ : Fin (SourceLeafCount T (b+1)) → Bool, if monomialDegree σ=r then
+      (averagedCoefficient (PairedMechanisms.orientationsGiven z)
+        (sourceFlatLeafType T n s center (b+1) incoming false) σ -
+       averagedCoefficient (PairedMechanisms.orientationsGiven z)
+        (sourceFlatLeafType T n s center (b+1) incoming true) σ)^2 else 0) =
+    ∑S : DegreeLeafSubset T b r,
+      (sourceCoefficient T n (sourceScheduledTrials T n) s center (sourceScheduledShift T n)
+        b (sourceReportFlags T n (b+1) z) incoming S.val)^2 := by
+  simp_rw [source_canonical_coefficient,←sourceSubsetCode_card]
+  rw [(sourceSubsetCode T (b+1)).sum_comp
+    (fun S => if S.card=r then (sourceCoefficient T n (sourceScheduledTrials T n) s center
+      (sourceScheduledShift T n) b (sourceReportFlags T n (b+1) z) incoming S)^2 else 0)]
+  let f : DegreeLeafSubset T b r → ℝ := fun S =>
+    (sourceCoefficient T n (sourceScheduledTrials T n) s center (sourceScheduledShift T n)
+      b (sourceReportFlags T n (b+1) z) incoming S.val)^2
+  simpa only [f, dite_eq_ite] using
+    sum_subtype_zero_extend (fun S : Finset (WindowLeaf T (b+1)) => S.card=r) f
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+namespace PairedMechanisms
+variable {Side : Type*} [Fintype Side] {m : ℕ} (M : PairedMechanisms Side m)
+
+def hiddenSubset (S : Finset (Fin m)) (z : Reports Side m) : ℕ := ∑i : S,hiddenTrial (z i.val).1
+
+theorem hiddenSubset_mgf (S : Finset (Fin m)) {ω t : ℝ} (hw : ∀i,M.hiddenWeight i=ω)
+    (hp : ω^2∈Set.Icc 0 1) (ht : |t|≤1) :
+    M.reports.expect (fun z => Real.exp (t*((hiddenSubset S z:ℕ)-(S.card:ℝ)*ω^2)))≤
+      Real.exp ((S.card:ℝ)*t^2) := by
+  have he (z : Reports Side m) :
+      Real.exp (t*((hiddenSubset S z:ℕ)-(S.card:ℝ)*ω^2))=
+        ∏i : S,Real.exp (t*((hiddenTrial (z i.val).1:ℕ)-ω^2)) := by
+    rw [←Real.exp_sum]
+    congr 1
+    simp only [mul_sub,Finset.mul_sum,Finset.sum_sub_distrib,Finset.sum_const,Finset.card_univ,Fintype.card_coe,
+      nsmul_eq_mul,hiddenSubset,Nat.cast_sum]
+    ring
+  simp_rw [he]
+  rw [reports,FiniteLaw.coordinateProduct_expect_subtype _ S
+    (fun (_ : S) (z : Option Bool × (Side × Side)) => Real.exp (t*((hiddenTrial z.1:ℕ)-ω^2)))]
+  calc
+    _ ≤ ∏i : S,Real.exp (t^2) := by
+      apply Finset.prod_le_prod₀
+      · intro i _
+        exact Finset.sum_nonneg (fun z _ => mul_nonneg
+          ((FiniteLaw.pairedReport (M.first i.val) (M.second i.val) (M.common i.val) (M.weight_nonneg i.val)
+            (M.common_first i.val) (M.common_second i.val)).nonneg z) (Real.exp_nonneg _))
+      · intro i _
+        have h := FiniteLaw.pairedReport_hidden_mgf (M.first i.val) (M.second i.val) (M.common i.val)
+          (M.weight_nonneg i.val) (M.common_first i.val) (M.common_second i.val) (by simpa [hw i.val] using hp) ht
+        simpa only [hw i.val] using h
+    _ = _ := by simp only [Finset.prod_const,Finset.card_univ,Fintype.card_coe,Real.exp_nat_mul]
+
+/-- A quantitative unconditional bound for failure of a source hidden-count
+ condition. The exponent ω⁴m is sufficient for the uniform scale margins. -/
+theorem hiddenSubset_small (S : Finset (Fin m)) {ω : ℝ} (hw : ∀i,M.hiddenWeight i=ω) (hp : ω^2∈Set.Icc 0 1) :
+    M.reports.expect (fun z => if (hiddenSubset S z:ℝ)<(S.card:ℝ)*ω^2/2 then 1 else 0)≤
+      Real.exp (-(S.card:ℝ)*ω^4/16) := by
+  have ht : 0≤ω^2/4 := by positivity
+  have ht' : |-(ω^2/4)|≤1 := by rw [abs_neg,abs_of_nonneg ht]; linarith [hp.2]
+  have H := finite_exp_markov Finset.univ M.reports.mass
+    (fun z => -((hiddenSubset S z:ℕ)-(S.card:ℝ)*ω^2)) (fun z _ => M.reports.nonneg z)
+    ((S.card:ℝ)*ω^2/2) (ω^2/4) ht
+  have hm := M.hiddenSubset_mgf S hw hp ht'
+  have hsub : M.reports.expect (fun z => if (hiddenSubset S z:ℝ)<(S.card:ℝ)*ω^2/2 then 1 else 0)≤
+      ∑z∈Finset.univ.filter (fun z => (S.card:ℝ)*ω^2/2≤-((hiddenSubset S z:ℕ)-(S.card:ℝ)*ω^2)),M.reports.mass z := by
+    rw [Finset.sum_filter]
+    apply Finset.sum_le_sum
+    intro z _
+    dsimp only
+    by_cases hz : (hiddenSubset S z:ℝ)<(S.card:ℝ)*ω^2/2
+    · rw [ite_eq_left hz,ite_eq_left (by linarith : (S.card:ℝ)*ω^2/2≤-((hiddenSubset S z:ℕ)-(S.card:ℝ)*ω^2)),mul_one]
+    · rw [ite_eq_right hz,mul_zero]; split_ifs <;> positivity [M.reports.nonneg z]
+  calc
+    _ ≤ _ := hsub.trans H
+    _ ≤ Real.exp (-(ω^2/4)*((S.card:ℝ)*ω^2/2))*Real.exp ((S.card:ℝ)*(-(ω^2/4))^2) := by
+      apply mul_le_mul_of_nonneg_left _ (Real.exp_nonneg _)
+      simpa only [FiniteLaw.expect,neg_mul,mul_neg,neg_neg] using hm
+    _ = _ := by rw [←Real.exp_add]; congr 1; ring
+
+
+end PairedMechanisms
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- IDs for precisely one local and one global collection at every genuine
+ nonleaf of the complete source window tree. -/
+def WindowCollection (T : ℕ → ℕ) : ℕ → Type
+  | 0 => PEmpty
+  | b+1 => Bool ⊕ (Fin (T b) × WindowCollection T b)
+
+noncomputable instance windowCollectionFintype (T : ℕ → ℕ) :
+    (b : ℕ) → Fintype (WindowCollection T b)
+  | 0 => inferInstanceAs (Fintype PEmpty)
+  | b+1 => by
+    letI := windowCollectionFintype T b
+    exact inferInstanceAs (Fintype (Bool ⊕ (Fin (T b) × WindowCollection T b)))
+
+def collectionCount (T n g : ℕ → ℕ) : (b : ℕ) → WindowCollection T b → ℕ
+  | b+1, .inl false => n b
+  | b+1, .inl true => g b
+  | b+1, .inr (_,v) => collectionCount T n g b v
+
+def collectionTrial (T n g : ℕ → ℕ) :
+    (b : ℕ) → (v : WindowCollection T b) → Fin (collectionCount T n g b v) → WindowTrial T n g b
+  | _+1, .inl false, j => .inl j
+  | _+1, .inl true, j => .inr (.inl j)
+  | b+1, .inr (u,v), j => .inr (.inr (u,collectionTrial T n g b v j))
+
+theorem collectionTrial_injective (T n g : ℕ → ℕ) (b : ℕ) (v : WindowCollection T b) :
+    Function.Injective (collectionTrial T n g b v) := by
+  induction b with
+  | zero => exact PEmpty.elim v
+  | succ b ih =>
+    rcases v with v | ⟨u,v⟩
+    · cases v
+      · intro i j h
+        exact Sum.inl.inj h
+      · intro i j h
+        exact Sum.inl.inj (Sum.inr.inj h)
+    · intro i j h
+      apply ih v
+      exact congrArg Prod.snd (Sum.inr.inj (Sum.inr.inj h))
+
+def collectionFlags (T n g : ℕ → ℕ) {A : Type} :
+    (b : ℕ) → WindowAssignments T n g A b → (v : WindowCollection T b) →
+      Fin (collectionCount T n g b v) → A
+  | _+1, d, .inl false => d.1
+  | _+1, d, .inl true => d.2.1
+  | b+1, d, .inr (u,v) => collectionFlags T n g b (d.2.2 u) v
+
+theorem collectionFlags_trial (T n g : ℕ → ℕ) {A : Type} (b : ℕ)
+    (d : WindowAssignments T n g A b) (v : WindowCollection T b)
+    (j : Fin (collectionCount T n g b v)) :
+    collectionFlags T n g b d v j=
+      windowAssignmentEquiv T n g A b d (collectionTrial T n g b v j) := by
+  induction b with
+  | zero => exact PEmpty.elim v
+  | succ b ih =>
+    rcases v with v | ⟨u,v⟩
+    · cases v <;> rfl
+    · exact ih (d.2.2 u) v j
+
+/-- This is the source hidden-count predicate, before any conditioning on fit. -/
+def HiddenCollectionConditions (T n g : ℕ → ℕ) (ω : ℝ) :
+    (b : ℕ) → WindowAssignments T n g (Option Bool) b → Prop
+  | 0, _ => True
+  | b+1, d =>
+      (n b:ℝ)*ω^2/2≤∑i,(hiddenTrial (d.1 i):ℝ) ∧
+      (g b:ℝ)*ω^2/2≤∑i,(hiddenTrial (d.2.1 i):ℝ) ∧
+      ∀u,HiddenCollectionConditions T n g ω b (d.2.2 u)
+
+theorem hiddenCollectionConditions_iff (T n g : ℕ → ℕ) (ω : ℝ) (b : ℕ)
+    (d : WindowAssignments T n g (Option Bool) b) :
+    HiddenCollectionConditions T n g ω b d ↔
+      ∀v : WindowCollection T b,(collectionCount T n g b v:ℝ)*ω^2/2≤
+        ∑i,(hiddenTrial (collectionFlags T n g b d v i):ℝ) := by
+  induction b with
+  | zero => simp [HiddenCollectionConditions,WindowCollection]
+  | succ b ih =>
+    constructor
+    · intro h v
+      rcases v with v | ⟨u,v⟩
+      · cases v
+        · exact h.1
+        · exact h.2.1
+      · exact (ih (d.2.2 u)).mp (h.2.2 u) v
+    · intro h
+      refine ⟨h (.inl false),h (.inl true),fun u => (ih (d.2.2 u)).mpr ?_⟩
+      intro v
+      exact h (.inr (u,v))
+
+def sourceCollectionIndex (T n : ℕ → ℕ) (b : ℕ) (v : WindowCollection T b) :
+    Fin (collectionCount T n (sourceScheduledTrials T n) b v) → Fin (SourceTrialCount T n b) :=
+  fun j => (finEnumeration _).symm (collectionTrial T n (sourceScheduledTrials T n) b v j)
+
+theorem sourceCollectionIndex_injective (T n : ℕ → ℕ) (b : ℕ) (v : WindowCollection T b) :
+    Function.Injective (sourceCollectionIndex T n b v) :=
+  (finEnumeration _).symm.injective.comp (collectionTrial_injective T n _ b v)
+
+def sourceCollectionSet (T n : ℕ → ℕ) (b : ℕ) (v : WindowCollection T b) :
+    Finset (Fin (SourceTrialCount T n b)) := Finset.univ.image (sourceCollectionIndex T n b v)
+
+theorem sourceCollectionSet_card (T n : ℕ → ℕ) (b : ℕ) (v : WindowCollection T b) :
+    (sourceCollectionSet T n b v).card=collectionCount T n (sourceScheduledTrials T n) b v := by
+  rw [sourceCollectionSet,Finset.card_image_of_injective _ (sourceCollectionIndex_injective T n b v)]
+  simp only [Finset.card_univ,Fintype.card_fin]
+
+theorem sourceCollection_flags {Side : Type*} (T n : ℕ → ℕ) (b : ℕ)
+    (z : Reports Side (SourceTrialCount T n b)) (v : WindowCollection T b)
+    (j : Fin (collectionCount T n (sourceScheduledTrials T n) b v)) :
+    collectionFlags T n (sourceScheduledTrials T n) b (sourceReportFlags T n b z) v j=
+      (z (sourceCollectionIndex T n b v j)).1 := by
+  rw [collectionFlags_trial]
+  have h := sourceAssignmentCode_eval T n (Option Bool) b (fun j => (z j).1)
+    (sourceCollectionIndex T n b v j)
+  simpa only [sourceCollectionIndex,Equiv.apply_symm_apply,sourceReportFlags] using h
+
+theorem sourceCollection_hidden {Side : Type*} (T n : ℕ → ℕ) (b : ℕ)
+    (z : Reports Side (SourceTrialCount T n b)) (v : WindowCollection T b) :
+    PairedMechanisms.hiddenSubset (sourceCollectionSet T n b v) z=
+      ∑i,hiddenTrial (collectionFlags T n (sourceScheduledTrials T n) b (sourceReportFlags T n b z) v i) := by
+  unfold PairedMechanisms.hiddenSubset
+  rw [←Finset.sum_subtype (sourceCollectionSet T n b v) (fun _ => Iff.rfl)
+    (fun j => hiddenTrial (z j).1)]
+  rw [sourceCollectionSet,Finset.sum_image]
+  · simp_rw [sourceCollection_flags]
+  · exact (sourceCollectionIndex_injective T n b v).injOn
+
+/-- Exact finite-union metadata cost; it contains no postulated typical
+ event and no independence assertion between different collections. -/
+def sourceHiddenTail (T n : ℕ → ℕ) (b : ℕ) (ω : ℝ) : ℝ :=
+  ∑v : WindowCollection T b,Real.exp (-(collectionCount T n (sourceScheduledTrials T n) b v:ℝ)*ω^4/16)
+
+theorem source_hidden_conditions_tail {Side : Type*} [Fintype Side]
+    (T n : ℕ → ℕ) (b : ℕ) (M : PairedMechanisms Side (SourceTrialCount T n b))
+    (ω : ℝ) (hw : ∀i,M.hiddenWeight i=ω) (hp : ω^2∈Set.Icc 0 1) :
+    M.reports.expect (fun z => if HiddenCollectionConditions T n (sourceScheduledTrials T n) ω b
+      (sourceReportFlags T n b z) then 0 else 1) ≤ sourceHiddenTail T n b ω := by
+  calc
+    _ ≤ M.reports.expect (fun z => ∑v : WindowCollection T b,
+      if (collectionCount T n (sourceScheduledTrials T n) b v:ℝ)*ω^2/2≤
+        ∑i,(hiddenTrial (collectionFlags T n (sourceScheduledTrials T n) b (sourceReportFlags T n b z) v i):ℝ)
+      then 0 else 1) := by
+        apply FiniteLaw.expect_mono
+        intro z
+        simp only [hiddenCollectionConditions_iff]
+        exact failure_forall_le_sum _
+    _ = ∑v : WindowCollection T b,M.reports.expect (fun z =>
+      if (PairedMechanisms.hiddenSubset (sourceCollectionSet T n b v) z:ℝ)<
+        ((sourceCollectionSet T n b v).card:ℝ)*ω^2/2 then 1 else 0) := by
+        rw [FiniteLaw.expect_fintype_sum]
+        apply Finset.sum_congr rfl
+        intro v _
+        apply congrArg (FiniteLaw.expect M.reports)
+        funext z
+        rw [sourceCollectionSet_card,sourceCollection_hidden,Nat.cast_sum]
+        by_cases h : (collectionCount T n (sourceScheduledTrials T n) b v:ℝ)*ω^2/2≤
+          ∑i,(hiddenTrial (collectionFlags T n (sourceScheduledTrials T n) b (sourceReportFlags T n b z) v i):ℝ)
+        · simp [h,not_lt.mpr h]
+        · simp [h,lt_of_not_ge h]
+    _ ≤ _ := by
+      apply Finset.sum_le_sum
+      intro v _
+      simpa only [sourceCollectionSet_card] using M.hiddenSubset_small (sourceCollectionSet T n b v) hw hp
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- Numerical cap forced by at least half the mean number of hidden pairs. -/
+def hiddenBinomialCap (m : ℕ) (ω : ℝ) : ℝ := Real.sqrt (4/((m:ℝ)*ω^2+2))
+
+theorem hiddenBinomialCap_nonneg (m : ℕ) (ω : ℝ) : 0≤hiddenBinomialCap m ω :=
+  Real.sqrt_nonneg _
+
+theorem hidden_binomial_bound (m G : ℕ) (ω : ℝ)
+    (h : (m:ℝ)*ω^2/2≤G) : binomialBound G≤hiddenBinomialCap m ω := by
+  apply Real.sqrt_le_sqrt
+  apply (div_le_div_iff₀ (by positivity : (0:ℝ)<G+1)
+    (by positivity : (0:ℝ)<(m:ℝ)*ω^2+2)).mpr
+  nlinarith
+
+def hiddenSmoothingBound (m s : ℕ) (ω : ℝ) : ℝ :=
+  (2:ℝ)^(s*s)*((2*Real.sqrt (4*((s:ℝ)+1)/((m:ℝ)*ω^2+2)))^s *
+    Real.sqrt (4*((s:ℝ)+1)/((m:ℝ)*ω^2+2)))
+
+theorem hiddenSmoothingBound_nonneg (m s : ℕ) (ω : ℝ) :
+    0≤hiddenSmoothingBound m s ω := by unfold hiddenSmoothingBound; positivity
+
+theorem hidden_smoothing_bound {m : ℕ} (d : Fin m → Option Bool) (s : ℕ) (ω : ℝ)
+    (h : (m:ℝ)*ω^2/2≤∑i,(hiddenTrial (d i):ℝ)) :
+    posteriorSmoothingCost s d≤hiddenSmoothingBound m s ω := by
+  have hn : (m:ℝ)*ω^2/2≤((∑i,hiddenTrial (d i):ℕ):ℝ) := by simpa only [Nat.cast_sum] using h
+  have hb : binomialBound ((∑i,hiddenTrial (d i))/(s+1))≤
+      Real.sqrt (4*((s:ℝ)+1)/((m:ℝ)*ω^2+2)) := by
+    apply (binomialBound_quotient_le _ s).trans
+    apply Real.sqrt_le_sqrt
+    apply (div_le_div_iff₀ (by positivity : (0:ℝ)<((∑i,hiddenTrial (d i):ℕ):ℝ)+1)
+      (by positivity : (0:ℝ)<(m:ℝ)*ω^2+2)).mpr
+    have hh := mul_le_mul_of_nonneg_left hn (by positivity : (0:ℝ)≤4*((s:ℝ)+1))
+    nlinarith
+  apply mul_le_mul_of_nonneg_left _ (by positivity : (0:ℝ)≤2^(s*s))
+  have hnon : 0≤binomialBound ((∑i,hiddenTrial (d i))/(s+1)) := Real.sqrt_nonneg _
+  exact mul_le_mul
+    (pow_le_pow_left₀ (mul_nonneg (by norm_num) hnon) (mul_le_mul_of_nonneg_left hb (by norm_num)) s)
+    hb hnon (by positivity)
+
+/-- Hidden-count typicality implies every local/global anchor cap, for the
+ actual source metadata and without a coefficient-dependent event. -/
+theorem hiddenConditions_binomialCaps (T n g : ℕ → ℕ) (ω : ℝ) (b : ℕ)
+    (d : WindowAssignments T n g (Option Bool) b)
+    (h : HiddenCollectionConditions T n g ω b d) :
+    MetadataBinomialCaps T n g (fun j => hiddenBinomialCap (n j) ω)
+      (fun j => hiddenBinomialCap (g j) ω) b d := by
+  induction b with
+  | zero => trivial
+  | succ b ih =>
+    refine ⟨hidden_binomial_bound (n b) _ ω ?_,hidden_binomial_bound (g b) _ ω ?_,fun u => ih _ (h.2.2 u)⟩
+    · simpa only [Nat.cast_sum] using h.1
+    · simpa only [Nat.cast_sum] using h.2.1
+
+theorem hiddenConditions_smoothingCap (T n g : ℕ → ℕ) (ω η : ℝ) (s b : ℕ)
+    (hη : ∀j<b,hiddenSmoothingBound (n j) s ω≤η)
+    (d : WindowAssignments T n g (Option Bool) b)
+    (h : HiddenCollectionConditions T n g ω b d) :
+    MetadataSmoothingCap T n g s η b d := by
+  induction b with
+  | zero => trivial
+  | succ b ih =>
+    exact ⟨(hidden_smoothing_bound d.1 s ω h.1).trans (hη b (by omega)),
+      fun u => ih (fun j hj => hη j (by omega)) _ (h.2.2 u)⟩
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- The actual noncentral posterior mass has the unconditional honest
+ fit-error bound. Impossible reports cause no problem in Bayes averaging. -/
+theorem source_noncentral_average {Side : Type*} [Fintype Side]
+    (T n : ℕ → ℕ) (s b : ℕ) (M : PairedMechanisms Side (SourceTrialCount T n (b+1)))
+    (hn : ∀j,j<b → 0<n j) (hne : ∀j,j<b → Even (n j))
+    (hT : ∀j≤b,24≤T j) (he : ∀j≤b,Even (T j)) (hL : ∀j≤b,24*2^s≤T j)
+    (hroot : 4*(n b+1)≤T b) :
+    M.reports.expect (fun z => sourceNoncentralMass T n (sourceScheduledTrials T n) s
+      (sourceLocalCenter T n) (sourceScheduledShift T n) (fun j => centralThird (T j))
+      (b+1) (sourceReportFlags T n (b+1) z) 0) ≤ sourceFitTail T n (b+1) := by
+  calc
+    _ ≤ M.reports.expect (fun z =>
+        (windowPosteriorLaw T n (sourceScheduledTrials T n) (b+1) (sourceReportFlags T n (b+1) z)).expect
+          (fun bits => if SourceTreeFit T n (b+1) bits then 0 else 1)) := by
+      apply FiniteLaw.expect_mono
+      intro z
+      exact sourceNoncentralMass_le_fit T n s b (sourceLocalCenter T n) (fun _ _ => rfl)
+        hT he hL hroot _
+    _ = M.reports.expect (fun z => (PairedMechanisms.orientationsGiven z).expect
+        (fun bits => if SourceTreeFit T n (b+1) ((sourceAssignmentCode T n Bool (b+1)).symm bits) then 0 else 1)) := by
+      apply congrArg (FiniteLaw.expect M.reports)
+      funext z
+      exact (source_posterior_code_expect T n (b+1) z
+        (fun bits => if SourceTreeFit T n (b+1) bits then 0 else 1)).symm
+    _ = _ := M.posterior_average _
+    _ ≤ _ := source_flat_fit_tail T n (b+1) (fun j hj => hn j (by omega)) (fun j hj => hne j (by omega))
+
+/-- Markov for a genuine finite law, stated for the exact failure indicator. -/
+theorem FiniteLaw.positive_markov {α : Type*} [Fintype α] (P : FiniteLaw α)
+    (f : α → ℝ) (hf : ∀x,0≤f x) {a : ℝ} (ha : 0<a) :
+    P.expect (fun x => if a≤f x then 1 else 0) ≤ P.expect f/a := by
+  apply (le_div_iff₀ ha).mpr
+  rw [mul_comm,←FiniteLaw.expect_const_mul]
+  apply FiniteLaw.expect_mono
+  intro x
+  split_ifs with hx
+  · simpa only [mul_one] using hx
+  · simpa only [mul_zero] using hf x
+
+/-- One metadata event, fixed before choosing a coefficient, controls
+ local and global smoothing and all internal central-third tests. -/
+def SourceTypical {Side : Type*} (T n : ℕ → ℕ) (s b : ℕ) (ω a : ℝ)
+    (z : Reports Side (SourceTrialCount T n (b+1))) : Prop :=
+  HiddenCollectionConditions T n (sourceScheduledTrials T n) ω (b+1)
+    (sourceReportFlags T n (b+1) z) ∧
+  sourceNoncentralMass T n (sourceScheduledTrials T n) s (sourceLocalCenter T n)
+    (sourceScheduledShift T n) (fun j => centralThird (T j)) (b+1)
+    (sourceReportFlags T n (b+1) z) 0<a
+
+theorem source_typical_failure {Side : Type*} [Fintype Side]
+    (T n : ℕ → ℕ) (s b : ℕ) (M : PairedMechanisms Side (SourceTrialCount T n (b+1)))
+    (ω a : ℝ) (ha : 0<a) (hw : ∀i,M.hiddenWeight i=ω) (hp : ω^2∈Set.Icc 0 1)
+    (hn : ∀j,j<b → 0<n j) (hne : ∀j,j<b → Even (n j))
+    (hT : ∀j≤b,24≤T j) (he : ∀j≤b,Even (T j)) (hL : ∀j≤b,24*2^s≤T j)
+    (hroot : 4*(n b+1)≤T b) :
+    M.reports.expect (fun z => if SourceTypical T n s b ω a z then 0 else 1) ≤
+      sourceHiddenTail T n (b+1) ω+sourceFitTail T n (b+1)/a := by
+  let f := fun z : Reports Side (SourceTrialCount T n (b+1)) =>
+    sourceNoncentralMass T n (sourceScheduledTrials T n) s (sourceLocalCenter T n)
+      (sourceScheduledShift T n) (fun j => centralThird (T j)) (b+1)
+      (sourceReportFlags T n (b+1) z) 0
+  calc
+    _ ≤ M.reports.expect (fun z =>
+        (if HiddenCollectionConditions T n (sourceScheduledTrials T n) ω (b+1)
+          (sourceReportFlags T n (b+1) z) then 0 else 1)+(if a≤f z then 1 else 0)) := by
+      apply FiniteLaw.expect_mono
+      intro z
+      by_cases H : SourceTypical T n s b ω a z
+      · rw [ite_eq_left H]
+        split_ifs <;> norm_num
+      · rw [ite_eq_right H]
+        by_cases h : HiddenCollectionConditions T n (sourceScheduledTrials T n) ω (b+1)
+          (sourceReportFlags T n (b+1) z)
+        · have hf : ¬f z<a := fun hf => H ⟨h,hf⟩
+          rw [ite_eq_left h,ite_eq_left (le_of_not_gt hf)]
+          norm_num
+        · rw [ite_eq_right h]
+          split_ifs <;> norm_num
+    _ ≤ _ := by
+      rw [FiniteLaw.expect_add]
+      apply add_le_add (source_hidden_conditions_tail T n (b+1) M ω hw hp)
+      apply (M.reports.positive_markov f (fun z => sourceNoncentralMass_nonneg T n _ s _ _ _ _ _ _) ha).trans
+      exact div_le_div_of_nonneg_right
+        (source_noncentral_average T n s b M hn hne hT he hL hroot) ha.le
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- Exact common plateau with integer half-count centering. Unlike the
+ simplified g/2 formula, this identity needs no parity assumption. -/
+theorem scheduledGlobalShift_plateau_exact (N t : ℕ)
+    (bits : Fin ((N/4)*t) → Bool) (u : Fin N) (hu : u∈centralThird N) :
+    scheduledGlobalShift N t bits u = (∑i,(bitIndex (bits i):ℤ))-(N/4:ℕ)*(t/2:ℕ) := by
+  calc
+    _ = ∑v : Fin (N/4),
+        ((∑a : Fin t, (bitIndex (bits (globalTrialIndex N t v a)):ℤ))-(t/2:ℕ)) := by
+      apply Finset.sum_congr rfl
+      intro v _
+      exact ite_eq_left (centralThird_active N u hu v)
+    _ = _ := by
+      rw [Finset.sum_sub_distrib]
+      congr 1
+      · exact globalTrial_sum N t (fun i => (bitIndex (bits i):ℤ))
+      · simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]
+
+def sourcePlateauCenter (T n : ℕ → ℕ) : ℕ → ℤ
+  | 0 => 0
+  | b+1 => (T (b+1)/4:ℕ)*(n b/2:ℕ)
+
+theorem sourceScheduledShift_hasPlateau (T n : ℕ → ℕ) :
+    HasGlobalPlateau T (sourceScheduledTrials T n) (fun j => centralThird (T j))
+      (sourcePlateauCenter T n) (sourceScheduledShift T n) := by
+  intro b bits u hu
+  cases b with
+  | zero =>
+      change (0:ℤ)=(∑i : Fin 0,(bitIndex (bits i):ℤ))-0
+      simp
+  | succ b => exact scheduledGlobalShift_plateau_exact (T (b+1)) (n b) bits u hu
+
+def sourceForcedBudget (T n : ℕ → ℕ) (s b r : ℕ) (ω : ℝ) : ℝ :=
+  (OccupiedParityTree.shapeRoleConstant r (b+1):ℝ)*interiorLabelBudget T b r *
+    ((2^s:ℕ):ℝ)^(2*(r-2^(b+1))) *
+    ((2*((2^s:ℕ):ℝ)^3*hiddenBinomialCap (n b) ω)*
+      binaryTreeBudget (fun j => 4*((2^s:ℕ):ℝ)^6*
+        hiddenBinomialCap (sourceScheduledTrials T n (j+1)) ω*hiddenBinomialCap (n j) ω) b)
+
+theorem sourceForcedBudget_nonneg (T n : ℕ → ℕ) (s b r : ℕ) (ω : ℝ) :
+    0 ≤ sourceForcedBudget T n s b r ω := by
+  have hB := fun j => hiddenBinomialCap_nonneg (n j) ω
+  have hC := fun j => hiddenBinomialCap_nonneg (sourceScheduledTrials T n j) ω
+  have ht := binaryTreeBudget_nonneg (fun j => 4*((2^s:ℕ):ℝ)^6*
+      hiddenBinomialCap (sourceScheduledTrials T n (j+1)) ω*hiddenBinomialCap (n j) ω)
+    (fun j => mul_nonneg (mul_nonneg (by positivity) (hC (j+1))) (hB j)) b
+  have hb := hB b
+  unfold sourceForcedBudget interiorLabelBudget
+  positivity
+
+/-- Full unabsorbed bound, now with no placeholder anchor-energy hypothesis. -/
+def sourceDegreeBudget (T n : ℕ → ℕ) (s b r : ℕ) (ω η a : ℝ) : ℝ :=
+  4*(3^((b+1)*r):ℕ)*
+    (((SourceLeafCount T (b+1)).choose r:ℕ)*(3^((b+1)*r):ℕ)*(η^2+a^2)+
+      sourceForcedBudget T n s b r ω)
+
+theorem sourceDegreeBudget_nonneg (T n : ℕ → ℕ) (s b r : ℕ) (ω η a : ℝ) :
+    0 ≤ sourceDegreeBudget T n s b r ω η a := by
+  have h := sourceForcedBudget_nonneg T n s b r ω
+  unfold sourceDegreeBudget
+  positivity
+
+theorem source_canonical_degree_bound {Side : Type*} (T n : ℕ → ℕ) (hT : ∀j,0<T j)
+    (s b r : ℕ) (ω η a : ℝ) (ha : 0≤a)
+    (hη : ∀j<b+1,hiddenSmoothingBound (n j) s ω≤η)
+    (z : Reports Side (SourceTrialCount T n (b+1))) (hz : SourceTypical T n s b ω a z) :
+    (∑σ : Fin (SourceLeafCount T (b+1)) → Bool, if monomialDegree σ=r then
+      (averagedCoefficient (PairedMechanisms.orientationsGiven z)
+        (sourceFlatLeafType T n s (sourceLocalCenter T n) (b+1) 0 false) σ -
+       averagedCoefficient (PairedMechanisms.orientationsGiven z)
+        (sourceFlatLeafType T n s (sourceLocalCenter T n) (b+1) 0 true) σ)^2 else 0) ≤
+      sourceDegreeBudget T n s b r ω η a := by
+  rw [source_canonical_degree_sum]
+  apply (sourceCoefficient_degree_energy T n _ s _ _ (fun j => centralThird (T j)) b r
+    (sourceReportFlags T n (b+1) z) 0 η
+    (hiddenConditions_smoothingCap T n _ ω η s (b+1) hη _ hz.1)).trans
+  apply mul_le_mul_of_nonneg_left _ (by positivity)
+  apply add_le_add
+  · apply mul_le_mul_of_nonneg_left _ (by positivity)
+    apply add_le_add_right
+    exact (sq_le_sq₀ (sourceNoncentralMass_nonneg T n _ s _ _ _ _ _ _) ha).mpr hz.2.le
+  · exact degreeForcedRoleEnergy_bound T n _ hT s (sourceLocalCenter T n) _ _ _ (sourceScheduledShift_hasPlateau T n)
+      (fun j => hiddenBinomialCap (n j) ω) (fun j => hiddenBinomialCap (sourceScheduledTrials T n j) ω)
+      (fun j => hiddenBinomialCap_nonneg _ _) (fun j => hiddenBinomialCap_nonneg _ _)
+      b r _ (hiddenConditions_binomialCaps T n _ ω (b+1) _ hz.1) 0
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- Pointwise equality of the literal source types and the signed tree types.
+ This holds for every assignment, including overflowing windows. -/
+theorem sourceFlatLeafType_window (T n : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (b : ℕ) (incoming : ℤ) (τ : Bool) (bits : Fin (SourceTrialCount T n b) → Bool) :
+    sourceFlatLeafType T n s center b incoming τ bits =
+      windowFinType T s (sourceTreeAnchors T n (sourceScheduledTrials T n) center
+        (sourceScheduledShift T n) b ((sourceAssignmentCode T n Bool b).symm bits) incoming)
+        b (boolTypeSign τ) := by
+  funext j
+  unfold sourceFlatLeafType windowFinType windowLeafType
+  rw [←sourceTreeLeafSign_eq,←sourceTreeLeafBool_sign]
+  change sourceTreeLeafBool T n s center b _ incoming τ (finEnumeration _ j)=
+    decide (boolTypeSign (sourceTreeLeafBool T n s center b _ incoming τ (finEnumeration _ j))= -1)
+  cases sourceTreeLeafBool T n s center b _ incoming τ (finEnumeration _ j) <;> norm_num [boolTypeSign]
+
+def sourceSparseCount (T : ℕ → ℕ) (s b : ℕ) : ℕ :=
+  2^s*∏j∈Finset.range b,T (j+1)
+
+theorem sourceFlatLeafType_sparse (T n : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (b : ℕ) (incoming : ℤ) (τ : Bool) (bits : Fin (SourceTrialCount T n (b+1)) → Bool) :
+    monomialDegree (sourceFlatLeafType T n s center (b+1) incoming τ bits) ≤ sourceSparseCount T s b := by
+  rw [sourceFlatLeafType_window,windowFinType_degree]
+  exact windowMinusCount_bound T s _ b [] _
+
+def sourceOracleCoefficient {Side : Type*} (T n : ℕ → ℕ) (s b : ℕ)
+    (z : Reports Side (SourceTrialCount T n (b+1))) (τ : Bool) :
+    (Fin (SourceLeafCount T (b+1)) → Bool) → ℝ :=
+  averagedCoefficient (PairedMechanisms.orientationsGiven z)
+    (sourceFlatLeafType T n s (sourceLocalCenter T n) (b+1) 0 τ)
+
+theorem sourceOracleCoefficient_empty {Side : Type*} (T n : ℕ → ℕ) (s b : ℕ)
+    (z : Reports Side (SourceTrialCount T n (b+1))) (τ : Bool) :
+    sourceOracleCoefficient T n s b z τ (fun _ => false)=1 := averagedCoefficient_empty _ _
+
+theorem sourceOracleCoefficient_degree {Side : Type*} (T n : ℕ → ℕ) (s b r : ℕ)
+    (z : Reports Side (SourceTrialCount T n (b+1))) (τ : Bool) :
+    (∑σ, if monomialDegree σ=r then (sourceOracleCoefficient T n s b z τ σ)^2 else 0) ≤
+      ((sourceSparseCount T s b).choose r:ℝ) :=
+  averaged_sparse_degree _ _ (sourceFlatLeafType_sparse T n s _ b 0 τ) r
+
+def sourceSignedCoefficient {Side : Type*} (T n : ℕ → ℕ) (s b : ℕ)
+    (z : Reports Side (SourceTrialCount T n (b+1))) (σ : Fin (SourceLeafCount T (b+1)) → Bool) : ℝ :=
+  sourceOracleCoefficient T n s b z false σ-sourceOracleCoefficient T n s b z true σ
+
+theorem sourceSignedCoefficient_zero {Side : Type*} (T n : ℕ → ℕ) (s b : ℕ)
+    (z : Reports Side (SourceTrialCount T n (b+1))) :
+    (∑σ, if monomialDegree σ=0 then sourceSignedCoefficient T n s b z σ^2 else 0)=0 := by
+  simp [monomialDegree_eq_zero_iff,sourceSignedCoefficient,sourceOracleCoefficient_empty]
+
+theorem sourceSignedCoefficient_degree_geometric {Side : Type*} (T n : ℕ → ℕ) (s b r : ℕ)
+    (z : Reports Side (SourceTrialCount T n (b+1))) (w : ℝ) :
+    w^(2*r)*(∑σ, if monomialDegree σ=r then sourceSignedCoefficient T n s b z σ^2 else 0) ≤
+      4*(w^2*sourceSparseCount T s b)^r := by
+  have h0 : 0 ≤ w^(2*r) := by rw [pow_mul]; positivity
+  have hp := sourceOracleCoefficient_degree T n s b r z false
+  have hm := sourceOracleCoefficient_degree T n s b r z true
+  have hc : ((sourceSparseCount T s b).choose r:ℝ) ≤ (sourceSparseCount T s b:ℝ)^r := by
+    exact_mod_cast Nat.choose_le_pow (sourceSparseCount T s b) r
+  have hd := degree_square_sub_le (sourceOracleCoefficient T n s b z false)
+    (sourceOracleCoefficient T n s b z true) r
+  calc
+    _ ≤ w^(2*r)*(4*(sourceSparseCount T s b:ℝ)^r) :=
+      mul_le_mul_of_nonneg_left (by dsimp only [sourceSignedCoefficient]; linarith) h0
+    _ = _ := by rw [pow_mul,mul_pow]; ring
+
+def sourceEnergyBudget (T n : ℕ → ℕ) (s b R : ℕ) (ω η a w : ℝ) : ℝ :=
+  (∑r∈Finset.range R, w^(2*r)*(if r=0 then 0 else sourceDegreeBudget T n s b r ω η a))+
+    4*(w^2*sourceSparseCount T s b)^R/(1-w^2*sourceSparseCount T s b)
+
+/-- Complete signed energy, with finite low-degree forced-tree bound and
+ the geometric high tail. No coefficient estimates remain as hypotheses. -/
+theorem source_canonical_energy {Side : Type*} (T n : ℕ → ℕ) (hT : ∀j,0<T j)
+    (s b R : ℕ) (ω η a w : ℝ) (ha : 0 ≤ a) (hw : 0 ≤ w)
+    (hsmall : w^2*sourceSparseCount T s b<1)
+    (hη : ∀j<b+1,hiddenSmoothingBound (n j) s ω ≤ η)
+    (z : Reports Side (SourceTrialCount T n (b+1))) (hz : SourceTypical T n s b ω a z) :
+    multilinearEnergy (sourceSignedCoefficient T n s b z) w ≤ sourceEnergyBudget T n s b R ω η a w := by
+  apply multilinearEnergy_low_high _ w (w^2*sourceSparseCount T s b) 4 hw (by positivity) hsmall
+    (by norm_num) (fun r => if r=0 then 0 else sourceDegreeBudget T n s b r ω η a)
+  · intro r _
+    by_cases hr : r=0
+    · subst r
+      rw [sourceSignedCoefficient_zero,ite_eq_left rfl]
+    · rw [ite_eq_right hr]
+      exact source_canonical_degree_bound T n hT s b r ω η a ha hη z hz
+  · intro r _
+    exact sourceSignedCoefficient_degree_geometric T n s b r z w
+  · intro r
+    split_ifs
+    · rfl
+    · exact sourceDegreeBudget_nonneg T n s b r ω η a
+
+theorem nonconstant_eq_degreeTail {l : ℕ} (A : (Fin l → Bool) → ℝ) :
+    nonconstantCoefficient A=degreeTail A 1 := by
+  funext σ
+  by_cases h : σ=(fun _ => false)
+  · subst σ; simp [nonconstantCoefficient,degreeTail,monomialDegree]
+  · have hd : 1 ≤ monomialDegree σ := by
+      have := (monomialDegree_eq_zero_iff σ).not.mpr h
+      omega
+    simp [nonconstantCoefficient,degreeTail,h,hd]
+
+/-- The individual density is close to the pure reference without any
+ metadata typicality assumption. -/
+theorem source_canonical_centered_energy {Side : Type*} (T n : ℕ → ℕ) (s b : ℕ)
+    (z : Reports Side (SourceTrialCount T n (b+1))) (τ : Bool) {w : ℝ} (hw : 0 ≤ w)
+    (hsmall : w^2*sourceSparseCount T s b<1) :
+    multilinearEnergy (nonconstantCoefficient (sourceOracleCoefficient T n s b z τ)) w ≤
+      (w^2*sourceSparseCount T s b)/(1-w^2*sourceSparseCount T s b) := by
+  rw [nonconstant_eq_degreeTail]
+  simpa only [pow_one,sourceOracleCoefficient] using averaged_sparse_tail_energy _ _
+    (sourceFlatLeafType_sparse T n s _ b 0 τ) 1 hw hsmall
+
+end TraceReconstruction
+
+noncomputable section
+namespace TraceReconstruction.FiniteLaw
+open scoped Classical
+variable {α : Type*} [Fintype α]
+
+theorem abs_expect_pow_le (P : FiniteLaw α) (F : α → ℝ) (t : ℕ) :
+    |P.expect F|^t ≤ P.expect (fun a => |F a|^t) := by
+  have habs : |P.expect F| ≤ P.expect (fun a => |F a|) := by
+    calc
+      _ ≤ ∑a, |P.mass a*F a| := Finset.abs_sum_le_sum_abs _ _
+      _ = _ := by simp only [abs_mul,abs_of_nonneg (P.nonneg _)]; rfl
+  apply (pow_le_pow_left₀ (abs_nonneg _) habs t).trans
+  simpa only [smul_eq_mul,expect] using
+    (convexOn_pow (𝕜:=ℝ) t).map_sum_le (t:=Finset.univ)
+      (w:=P.mass) (p:=fun a => |F a|) (fun a _ => P.nonneg a) P.total (fun a _ => (show |F a| ∈ Set.Ici (0:ℝ) from abs_nonneg (F a)))
+
+theorem density_score_centered (P Q : FiniteLaw α) (Z : α → ℝ)
+    (hZ : ∀a,Q.mass a=P.mass a*(1+Z a)) : P.expect Z=0 := by
+  have h := Q.total
+  simp_rw [hZ,mul_add,mul_one,Finset.sum_add_distrib,P.total] at h
+  change _+P.expect Z=1 at h
+  linarith
+
+/-- Common-event mass difference is controlled by the restricted density
+ difference at every moment, by finite Jensen rather than a pointwise bound. -/
+theorem clipping_mass_difference (R : FiniteLaw α) (f g : α → ℝ)
+    (hf : ∀a,0 ≤ f a) (hg : ∀a,0 ≤ g a) (hfm : R.expect f=1) (hgm : R.expect g=1)
+    (A : Finset α) (t : ℕ) :
+    |(R.withDensity f hf hfm).eventMass A-(R.withDensity g hg hgm).eventMass A|^t ≤
+      R.expect (fun a => |if a∈A then g a-f a else 0|^t) := by
+  have he : (R.withDensity f hf hfm).eventMass A-(R.withDensity g hg hgm).eventMass A=
+      -(R.expect (fun a => if a∈A then g a-f a else 0)) := by
+    simp only [expect,mul_ite,mul_zero,←Finset.sum_filter,
+      Finset.filter_mem_eq_inter,Finset.univ_inter,eventMass,withDensity,mul_sub,
+      Finset.sum_sub_distrib]
+    ring
+  rw [he,abs_neg]
+  exact R.abs_expect_pow_le _ t
+
+theorem clipped_raw_moment_restricted (R : FiniteLaw α) (f g : α → ℝ)
+    (hf : ∀a,0 ≤ f a) (hfm : R.expect f=1)
+    (A : Finset α) (hAf : ∀a∈A,1/2 ≤ f a)
+    (hP : 0<(R.withDensity f hf hfm).eventMass A) (k : ℕ) :
+    (condition (R.withDensity f hf hfm) A hP).expect (fun a => |g a/f a-1|^(k+1)) ≤
+      2^k/(R.withDensity f hf hfm).eventMass A *
+        R.expect (fun a => |if a∈A then g a-f a else 0|^(k+1)) := by
+  let g' : α → ℝ := fun a => if a∈A then g a else f a
+  have H := clipped_raw_moment R f g' hf hfm A hAf hP k
+  have he : (condition (R.withDensity f hf hfm) A hP).expect (fun a => |g a/f a-1|^(k+1))=
+      (condition (R.withDensity f hf hfm) A hP).expect (fun a => |g' a/f a-1|^(k+1)) := by
+    apply Finset.sum_congr rfl
+    intro a _
+    by_cases ha : a∈A <;> simp [condition,g',ha]
+  rw [he]
+  convert H using 1
+  congr 2
+  funext a
+  by_cases ha : a∈A <;> simp [g',ha]
+
+/-- Explicit normalized clipping, at half of any positive even moment. This
+ constructs the actual conditioned score; no bounded-score hypothesis. -/
+theorem clipped_score_moment (R : FiniteLaw α) (f g : α → ℝ)
+    (hf : ∀a,0 ≤ f a) (hg : ∀a,0 ≤ g a) (hfm : R.expect f=1) (hgm : R.expect g=1)
+    (A : Finset α) (hAf : ∀a∈A,1/2 ≤ f a)
+    (hp : 1/2 ≤ (R.withDensity f hf hfm).eventMass A)
+    (hq : 1/2 ≤ (R.withDensity g hg hgm).eventMass A)
+    (k : ℕ) (hk : 1 ≤ k) (V : ℝ) (hV : 0 ≤ V)
+    (hM : R.expect (fun a => |if a∈A then g a-f a else 0|^(2*k)) ≤ V^k) :
+    let p := (R.withDensity f hf hfm).eventMass A
+    let q := (R.withDensity g hg hgm).eventMass A
+    let hP : 0 < p := lt_of_lt_of_le (by norm_num) hp
+    (condition (R.withDensity f hf hfm) A hP).evenMomentSq (fun a => p/q*(g a/f a)-1) k ≤64*V := by
+  dsimp only
+  let p := (R.withDensity f hf hfm).eventMass A
+  let q := (R.withDensity g hg hgm).eventMass A
+  have hp0 : 0<p := lt_of_lt_of_le (by norm_num) hp
+  have hq0 : 0<q := lt_of_lt_of_le (by norm_num) hq
+  have hp1 : p ≤ 1 := eventMass_le_one _ _
+  have hratio : p/q ≤ 2 := (div_le_iff₀ hq0).mpr (by linarith)
+  have hden : (1:ℝ)/p ≤ 2 := (div_le_iff₀ hp0).mpr (by linarith)
+  have ht : 2*k-1+1=2*k := by omega
+  have hraw := clipped_raw_moment_restricted R f g hf hfm A hAf hp0 (2*k-1)
+  rw [ht] at hraw
+  have hraw' : (condition (R.withDensity f hf hfm) A hp0).expect (fun a => |g a/f a-1|^(2*k)) ≤
+      2^(2*k)*V^k := by
+    apply hraw.trans
+    calc
+      _ ≤ (2^(2*k-1)/p)*V^k := mul_le_mul_of_nonneg_left hM (by positivity)
+      _ ≤ (2^(2*k-1)*2)*V^k := by
+        apply mul_le_mul_of_nonneg_right _ (pow_nonneg hV _)
+        simpa only [mul_one_div] using mul_le_mul_of_nonneg_left hden (by positivity : 0 ≤ (2:ℝ)^(2*k-1))
+      _ = _ := by rw [←pow_succ,ht]
+  have hnorm : |p/q-1|^(2*k) ≤ 2^(2*k)*V^k := by
+    rw [div_sub_one hq0.ne',abs_div,abs_of_pos hq0,div_pow]
+    apply (div_le_iff₀ (pow_pos hq0 _)).mpr
+    have hd := (clipping_mass_difference R f g hf hg hfm hgm A (2*k)).trans hM
+    have hqpow : 1 ≤ (2*q)^(2*k) := one_le_pow₀ (by linarith)
+    calc
+      _ ≤ V^k := hd
+      _ ≤ (2*q)^(2*k)*V^k := by nlinarith [pow_nonneg hV k]
+      _ = _ := by rw [mul_pow]; ring
+  have hs := score_rescaling_moment (condition (R.withDensity f hf hfm) A hp0)
+    (fun a => g a/f a) (p/q) (div_nonneg hp0.le hq0.le) (2*k-1)
+  rw [ht] at hs
+  have hpower : (p/q)^(2*k) ≤ (2:ℝ)^(2*k) :=
+    pow_le_pow_left₀ (div_nonneg hp0.le hq0.le) hratio _
+  have hs' : (condition (R.withDensity f hf hfm) A hp0).expect (fun a => |p/q*(g a/f a)-1|^(2*k)) ≤
+      (64*V)^k := by
+    apply hs.trans
+    calc
+      _ ≤ 2^(2*k-1)*(2^(2*k)*(2^(2*k)*V^k)+2^(2*k)*V^k) := by
+        apply mul_le_mul_of_nonneg_left _ (by positivity)
+        exact add_le_add (mul_le_mul hpower hraw' (by
+          apply Finset.sum_nonneg; intro a _; positivity [(condition (R.withDensity f hf hfm) A hp0).nonneg a]) (by positivity)) hnorm
+      _ ≤ 2^(2*k-1)*(2^(2*k)*(2^(2*k)*V^k)+2^(2*k)*(2^(2*k)*V^k)) := by
+        apply mul_le_mul_of_nonneg_left _ (by positivity)
+        apply add_le_add_right
+        have htwo : (1:ℝ) ≤ 2^(2*k) := one_le_pow₀ (by norm_num)
+        simpa only [one_mul] using mul_le_mul_of_nonneg_right htwo
+          (by positivity : 0 ≤ (2:ℝ)^(2*k)*V^k)
+      _ = (64*V)^k := by
+        have hm : 2^(2*k-1)*2= (2:ℝ)^(2*k) := by rw [←pow_succ,ht]
+        calc
+          _ = (2^(2*k-1)*2)*(2:ℝ)^(2*k)*(2:ℝ)^(2*k)*V^k := by ring
+          _ = _ := by rw [hm,←pow_add,←pow_add,mul_pow]; congr 1; rw [show 2*k+2*k+2*k=6*k by omega,pow_mul]; norm_num
+  unfold evenMomentSq
+  calc
+    _ ≤ ((64*V)^k)^(1/(k:ℝ)) := Real.rpow_le_rpow
+      (by apply Finset.sum_nonneg; intro a _; positivity [(condition (R.withDensity f hf hfm) A hp0).nonneg a]) hs' (by positivity)
+    _ = _ := by rw [←Real.rpow_natCast,one_div,Real.rpow_rpow_inv (by positivity) (by positivity)]
+
+end TraceReconstruction.FiniteLaw
+
+noncomputable section
+namespace TraceReconstruction.FiniteLaw
+open scoped Classical
+variable {α : Type*} [Fintype α]
+
+theorem eventMass_indicator (P : FiniteLaw α) (A : Finset α) :
+    P.eventMass A=P.expect (fun a => if a∈A then 1 else 0) := by
+  simp only [expect,mul_ite,mul_one,mul_zero,←Finset.sum_filter,
+    Finset.filter_mem_eq_inter,Finset.univ_inter,eventMass]
+
+theorem eventMass_compl (P : FiniteLaw α) (A : Finset α) :
+    P.eventMass Aᶜ=1-P.eventMass A := by
+  rw [eventMass_indicator,eventMass_indicator]
+  have he : P.expect (fun a => (if a∈Aᶜ then (1:ℝ) else 0)+(if a∈A then 1 else 0))=1 := by
+    convert P.expect_const 1 using 1
+    congr 1; funext a
+    by_cases ha : a∈A <;> simp [ha]
+  rw [expect_add] at he
+  linarith
+
+theorem density_second_moment_le_six (R : FiniteLaw α) (f : α → ℝ) (k : ℕ) (hk : 1 ≤ k)
+    (hM : R.expect (fun a => |f a-1|^(2*k)) ≤ 1) :
+    R.expect (fun a => f a^2) ≤ 6 := by
+  have h := R.expect_lower_moment_le_two (fun a => f a-1) (2*k) hM 2 (by omega)
+  calc
+    _ ≤ R.expect (fun a => 2*(f a-1)^2+2) := R.expect_mono (fun a => by nlinarith [sq_nonneg (f a-2)])
+    _ = 2*R.expect (fun a => (f a-1)^2)+2 := by rw [expect_add,expect_const_mul,expect_const]
+    _ ≤ _ := by linarith [le_abs_self (R.expect (fun a => (f a-1)^2))]
+
+/-- Cauchy--Schwarz loss on the actual low-reference-density event, for
+ either density. The same clipping event is used for both hypotheses. -/
+theorem density_low_event_mass (R : FiniteLaw α) (f g : α → ℝ)
+    (hg : ∀a,0 ≤ g a) (hgm : R.expect g=1) (k : ℕ) (hk : 1 ≤ k) (U : ℝ)
+    (hU : 0 ≤ U) (hU1 : U ≤ 1)
+    (hfM : R.expect (fun a => |f a-1|^(2*k)) ≤ U^k)
+    (hgM : R.expect (fun a => |g a-1|^(2*k)) ≤ U^k) :
+    (R.withDensity g hg hgm).eventMass (Finset.univ.filter (fun a => f a<1/2)) ≤
+      Real.sqrt (6*(4*U)^k) := by
+  let A := Finset.univ.filter (fun a => f a<1/2)
+  have hR : R.eventMass A ≤ (4*U)^k := by
+    calc
+      _ ≤ 2^(2*k)*R.expect (fun a => |f a-1|^(2*k)) :=
+        R.low_density_reference_mass f (2*k) A (fun a ha => (Finset.mem_filter.mp ha).2)
+      _ ≤ 2^(2*k)*U^k := mul_le_mul_of_nonneg_left hfM (by positivity)
+      _ = _ := by rw [pow_mul,mul_pow]; norm_num
+  have hsecond := R.density_second_moment_le_six g k hk (hgM.trans (pow_le_one₀ hU hU1))
+  have hsq : ((R.withDensity g hg hgm).eventMass A)^2 ≤ 6*(4*U)^k := by
+    apply (R.density_bad_mass_sq g A).trans
+    calc
+      _ ≤ R.eventMass A*6 := mul_le_mul_of_nonneg_left hsecond (R.eventMass_nonneg A)
+      _ ≤ _ := by nlinarith
+  exact (Real.le_sqrt ((R.withDensity g hg hgm).eventMass_nonneg A) (by positivity)).mpr hsq
+
+def commonClipEvent (B : Finset α) (f : α → ℝ) : Finset α :=
+  B.filter (fun a => 1/2 ≤ f a)
+
+theorem commonClipEvent_loss (P : FiniteLaw α) (B : Finset α) (f : α → ℝ) :
+    1-P.eventMass (commonClipEvent B f) ≤
+      P.eventMass Bᶜ+P.eventMass (Finset.univ.filter (fun a => f a<1/2)) := by
+  rw [←eventMass_compl,eventMass_indicator,eventMass_indicator,eventMass_indicator,←expect_add]
+  apply P.expect_mono
+  intro a
+  simp only [commonClipEvent,Finset.mem_compl,Finset.mem_filter,Finset.mem_univ,true_and]
+  split_ifs <;> simp_all
+
+
+/-- Quantitative common clipping on any genuine good event. Both bad-event
+ costs below are measured under the actual normalized laws, not reference mass. -/
+theorem commonClipEvent_mass (R : FiniteLaw α) (f g : α → ℝ)
+    (hf : ∀a,0 ≤ f a) (hg : ∀a,0 ≤ g a) (hfm : R.expect f=1) (hgm : R.expect g=1)
+    (B : Finset α) (e : ℝ) (hfB : (R.withDensity f hf hfm).eventMass Bᶜ ≤ e)
+    (hgB : (R.withDensity g hg hgm).eventMass Bᶜ ≤ e)
+    (k : ℕ) (hk : 1 ≤ k) (U : ℝ) (hU : 0 ≤ U) (hU1 : U ≤ 1)
+    (hfM : R.expect (fun a => |f a-1|^(2*k)) ≤ U^k)
+    (hgM : R.expect (fun a => |g a-1|^(2*k)) ≤ U^k) :
+    1-(R.withDensity f hf hfm).eventMass (commonClipEvent B f) ≤ e+Real.sqrt (6*(4*U)^k) ∧
+    1-(R.withDensity g hg hgm).eventMass (commonClipEvent B f) ≤ e+Real.sqrt (6*(4*U)^k) := by
+  constructor
+  · exact (commonClipEvent_loss _ B f).trans (add_le_add hfB
+      (density_low_event_mass R f f hf hfm k hk U hU hU1 hfM hfM))
+  · exact (commonClipEvent_loss _ B f).trans (add_le_add hgB
+      (density_low_event_mass R f g hg hgm k hk U hU hU1 hfM hgM))
+
+end TraceReconstruction.FiniteLaw
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- A pair produced by genuine common-event normalized conditioning. The
+ outcome space and public decoder remain unchanged. -/
+structure ClippedOracle {α : Type*} [Fintype α] (P Q : FiniteLaw α) (k : ℕ) (e V : ℝ) where
+  first : FiniteLaw α
+  second : FiniteLaw α
+  score : α → ℝ
+  density : ∀a,second.mass a=first.mass a*(1+score a)
+  likelihood_nonneg : ∀a,0 ≤ 1+score a
+  centered : first.expect score=0
+  moment : first.evenMomentSq score k ≤ V
+  first_tv : first.tv P ≤ e
+  second_tv : second.tv Q ≤ e
+
+namespace FiniteLaw
+variable {α : Type*} [Fintype α]
+
+theorem exists_clippedOracle (R : FiniteLaw α) (f g : α → ℝ)
+    (hf : ∀a,0 ≤ f a) (hg : ∀a,0 ≤ g a) (hfm : R.expect f=1) (hgm : R.expect g=1)
+    (B : Finset α) (e : ℝ) (hfB : (R.withDensity f hf hfm).eventMass Bᶜ ≤ e)
+    (hgB : (R.withDensity g hg hgm).eventMass Bᶜ ≤ e)
+    (k : ℕ) (hk : 1 ≤ k) (U : ℝ) (hU : 0 ≤ U) (hU1 : U ≤ 1)
+    (hfM : R.expect (fun a => |f a-1|^(2*k)) ≤ U^k)
+    (hgM : R.expect (fun a => |g a-1|^(2*k)) ≤ U^k)
+    (V : ℝ) (hV : 0 ≤ V)
+    (hM : R.expect (fun a => |if a∈B then g a-f a else 0|^(2*k)) ≤ V^k)
+    (hsmall : e+Real.sqrt (6*(4*U)^k) ≤ 1/2) :
+    Nonempty (ClippedOracle (R.withDensity f hf hfm) (R.withDensity g hg hgm)
+      k (e+Real.sqrt (6*(4*U)^k)) (64*V)) := by
+  let A := commonClipEvent B f
+  let P := R.withDensity f hf hfm
+  let Q := R.withDensity g hg hgm
+  have hA (a : α) (ha : a∈A) : 1/2 ≤ f a := (Finset.mem_filter.mp ha).2
+  have hloss := commonClipEvent_mass R f g hf hg hfm hgm B e hfB hgB k hk U hU hU1 hfM hgM
+  have hp : 1/2 ≤ P.eventMass A := by linarith [hloss.1]
+  have hq : 1/2 ≤ Q.eventMass A := by linarith [hloss.2]
+  have hp0 : 0<P.eventMass A := lt_of_lt_of_le (by norm_num) hp
+  have hq0 : 0<Q.eventMass A := lt_of_lt_of_le (by norm_num) hq
+  let Z : α → ℝ := fun a => P.eventMass A/Q.eventMass A*(g a/f a)-1
+  have hden (a : α) : (Q.condition A hq0).mass a=(P.condition A hp0).mass a*(1+Z a) := by
+    dsimp only [Z]
+    rw [show 1+(P.eventMass A/Q.eventMass A*(g a/f a)-1)=P.eventMass A/Q.eventMass A*(g a/f a) by ring]
+    exact conditioned_density_identity R f g hf hg hfm hgm A
+      (fun a ha => lt_of_lt_of_le (by norm_num) (hA a ha)) hp0 hq0 a
+  have hAM : R.expect (fun a => |if a∈A then g a-f a else 0|^(2*k)) ≤ V^k := by
+    apply le_trans (R.expect_mono _) hM
+    intro a
+    by_cases ha : a∈A
+    · have hb := (Finset.mem_filter.mp ha).1
+      simp only [ite_eq_left ha,ite_eq_left hb,le_refl]
+    · rw [ite_eq_right ha]
+      simp only [abs_zero,zero_pow (by omega : 2*k≠0)]
+      exact pow_nonneg (abs_nonneg _) _
+  exact ⟨{
+    first := P.condition A hp0
+    second := Q.condition A hq0
+    score := Z
+    density := hden
+    likelihood_nonneg := fun a => by
+      dsimp only [Z]
+      have hh := mul_nonneg (div_nonneg hp0.le hq0.le) (div_nonneg (hg a) (hf a))
+      linarith
+    centered := density_score_centered _ _ Z hden
+    moment := clipped_score_moment R f g hf hg hfm hgm A hA hp hq k hk V hV hAM
+    first_tv := by rw [tv_symm,tv_condition]; exact hloss.1
+    second_tv := by rw [tv_symm,tv_condition]; exact hloss.2
+  }⟩
+
+end FiniteLaw
+end TraceReconstruction
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- Exact finite-union metadata cost; it contains no postulated typical
+ event and no independence assertion between different collections. -/
+def sourceEffectiveHiddenTail (T n : ℕ → ℕ) (b : ℕ) (ω : ℝ) : ℝ :=
+  ∑v : WindowCollection T b,if collectionCount T n (sourceScheduledTrials T n) b v=0 then 0 else Real.exp (-(collectionCount T n (sourceScheduledTrials T n) b v:ℝ)*ω^4/16)
+
+theorem source_hidden_conditions_effective_tail {Side : Type*} [Fintype Side]
+    (T n : ℕ → ℕ) (b : ℕ) (M : PairedMechanisms Side (SourceTrialCount T n b))
+    (ω : ℝ) (hw : ∀i,M.hiddenWeight i=ω) (hp : ω^2∈Set.Icc 0 1) :
+    M.reports.expect (fun z => if HiddenCollectionConditions T n (sourceScheduledTrials T n) ω b
+      (sourceReportFlags T n b z) then 0 else 1) ≤ sourceEffectiveHiddenTail T n b ω := by
+  calc
+    _ ≤ M.reports.expect (fun z => ∑v : WindowCollection T b,
+      if (collectionCount T n (sourceScheduledTrials T n) b v:ℝ)*ω^2/2≤
+        ∑i,(hiddenTrial (collectionFlags T n (sourceScheduledTrials T n) b (sourceReportFlags T n b z) v i):ℝ)
+      then 0 else 1) := by
+        apply FiniteLaw.expect_mono
+        intro z
+        simp only [hiddenCollectionConditions_iff]
+        exact failure_forall_le_sum _
+    _ = ∑v : WindowCollection T b,M.reports.expect (fun z =>
+      if (PairedMechanisms.hiddenSubset (sourceCollectionSet T n b v) z:ℝ)<
+        ((sourceCollectionSet T n b v).card:ℝ)*ω^2/2 then 1 else 0) := by
+        rw [FiniteLaw.expect_fintype_sum]
+        apply Finset.sum_congr rfl
+        intro v _
+        apply congrArg (FiniteLaw.expect M.reports)
+        funext z
+        rw [sourceCollectionSet_card,sourceCollection_hidden,Nat.cast_sum]
+        by_cases h : (collectionCount T n (sourceScheduledTrials T n) b v:ℝ)*ω^2/2≤
+          ∑i,(hiddenTrial (collectionFlags T n (sourceScheduledTrials T n) b (sourceReportFlags T n b z) v i):ℝ)
+        · simp [h,not_lt.mpr h]
+        · simp [h,lt_of_not_ge h]
+    _ ≤ _ := by
+      apply Finset.sum_le_sum
+      intro v _
+      by_cases hc : collectionCount T n (sourceScheduledTrials T n) b v=0
+      · rw [ite_eq_left hc]
+        have hz : M.reports.expect (fun z =>
+            if (PairedMechanisms.hiddenSubset (sourceCollectionSet T n b v) z:ℝ)<
+              ((sourceCollectionSet T n b v).card:ℝ)*ω^2/2 then 1 else 0)=0 := by
+          have h (z : Reports Side (SourceTrialCount T n b)) :
+              ¬(PairedMechanisms.hiddenSubset (sourceCollectionSet T n b v) z:ℝ)<
+                ((sourceCollectionSet T n b v).card:ℝ)*ω^2/2 := by
+            rw [sourceCollectionSet_card,hc]
+            simp only [Nat.cast_zero,zero_mul,zero_div]
+            exact not_lt.mpr (Nat.cast_nonneg _)
+          simp_rw [ite_eq_right (h _)]
+          exact FiniteLaw.expect_const _ 0
+        exact hz.le
+      · rw [ite_eq_right hc]
+        simpa only [sourceCollectionSet_card] using M.hiddenSubset_small (sourceCollectionSet T n b v) hw hp
+
+theorem source_typical_effective_failure {Side : Type*} [Fintype Side]
+    (T n : ℕ → ℕ) (s b : ℕ) (M : PairedMechanisms Side (SourceTrialCount T n (b+1)))
+    (ω a : ℝ) (ha : 0<a) (hw : ∀i,M.hiddenWeight i=ω) (hp : ω^2∈Set.Icc 0 1)
+    (hn : ∀j,j<b → 0<n j) (hne : ∀j,j<b → Even (n j))
+    (hT : ∀j≤b,24≤T j) (he : ∀j≤b,Even (T j)) (hL : ∀j≤b,24*2^s≤T j)
+    (hroot : 4*(n b+1)≤T b) :
+    M.reports.expect (fun z => if SourceTypical T n s b ω a z then 0 else 1) ≤
+      sourceEffectiveHiddenTail T n (b+1) ω+sourceFitTail T n (b+1)/a := by
+  let f := fun z : Reports Side (SourceTrialCount T n (b+1)) =>
+    sourceNoncentralMass T n (sourceScheduledTrials T n) s (sourceLocalCenter T n)
+      (sourceScheduledShift T n) (fun j => centralThird (T j)) (b+1)
+      (sourceReportFlags T n (b+1) z) 0
+  calc
+    _ ≤ M.reports.expect (fun z =>
+        (if HiddenCollectionConditions T n (sourceScheduledTrials T n) ω (b+1)
+          (sourceReportFlags T n (b+1) z) then 0 else 1)+(if a≤f z then 1 else 0)) := by
+      apply FiniteLaw.expect_mono
+      intro z
+      by_cases H : SourceTypical T n s b ω a z
+      · rw [ite_eq_left H]
+        split_ifs <;> norm_num
+      · rw [ite_eq_right H]
+        by_cases h : HiddenCollectionConditions T n (sourceScheduledTrials T n) ω (b+1)
+          (sourceReportFlags T n (b+1) z)
+        · have hf : ¬f z<a := fun hf => H ⟨h,hf⟩
+          rw [ite_eq_left h,ite_eq_left (le_of_not_gt hf)]
+          norm_num
+        · rw [ite_eq_right h]
+          split_ifs <;> norm_num
+    _ ≤ _ := by
+      rw [FiniteLaw.expect_add]
+      apply add_le_add (source_hidden_conditions_effective_tail T n (b+1) M ω hw hp)
+      apply (M.reports.positive_markov f (fun z => sourceNoncentralMass_nonneg T n _ s _ _ _ _ _ _) ha).trans
+      exact div_le_div_of_nonneg_right
+        (source_noncentral_average T n s b M hn hne hT he hL hroot) ha.le
+
+
+end TraceReconstruction
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- Literal density of the source's unconditioned finite canonical oracle. -/
+def sourceOracleDensity {Side Leaf : Type*} (T n : ℕ → ℕ) (s b : ℕ)
+    (Z : Leaf → ℝ) (τ : Bool)
+    (zx : Reports Side (SourceTrialCount T n (b+1)) × (Fin (SourceLeafCount T (b+1)) → Leaf)) : ℝ :=
+  averagedDensity (PairedMechanisms.orientationsGiven zx.1)
+    (sourceFlatLeafType T n s (sourceLocalCenter T n) (b+1) 0 τ) Z zx.2
+
+theorem sourceOracleDensity_expansion {Side Leaf : Type*} (T n : ℕ → ℕ) (s b : ℕ)
+    (Z : Leaf → ℝ) (τ : Bool)
+    (z : Reports Side (SourceTrialCount T n (b+1))) (x : Fin (SourceLeafCount T (b+1)) → Leaf) :
+    sourceOracleDensity T n s b Z τ (z,x)=multilinear (sourceOracleCoefficient T n s b z τ) Z x :=
+  averagedDensity_expansion _ _ _ _
+
+theorem sourceOracleDensity_nonneg {Side Leaf : Type*} (T n : ℕ → ℕ) (s b : ℕ)
+    (Z : Leaf → ℝ) (hZ : ∀a,0 ≤ 1+Z a) (τ : Bool)
+    (zx : Reports Side (SourceTrialCount T n (b+1)) × (Fin (SourceLeafCount T (b+1)) → Leaf)) :
+    0 ≤ sourceOracleDensity T n s b Z τ zx := averagedDensity_nonneg _ _ _ hZ _
+
+theorem sourceOracleDensity_mean {Side Leaf : Type*} [Fintype Leaf] (T n : ℕ → ℕ) (s b : ℕ)
+    (P : FiniteLaw Leaf) (Z : Leaf → ℝ) (hZ : P.expect Z=0) (τ : Bool)
+    (z : Reports Side (SourceTrialCount T n (b+1))) :
+    (P.product (SourceLeafCount T (b+1))).expect (fun x => sourceOracleDensity T n s b Z τ (z,x))=1 :=
+  averagedDensity_mean P (PairedMechanisms.orientationsGiven z) Z hZ
+    (sourceFlatLeafType T n s (sourceLocalCenter T n) (b+1) 0 τ)
+
+theorem source_canonical_centered_moment {Side Leaf : Type*} [Fintype Leaf]
+    (T n : ℕ → ℕ) (s b k : ℕ) (P : FiniteLaw Leaf) (Z : Leaf → ℝ)
+    (hk : 1 ≤ k) (hcenter : P.expect Z=0) {L : ℝ} (hL : 0<L)
+    (hMoment : P.expect (fun a => |Z a|^(2*k)) ≤ L^(2*k))
+    (hsmall : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b<1)
+    (z : Reports Side (SourceTrialCount T n (b+1))) (τ : Bool) :
+    (P.product (SourceLeafCount T (b+1))).expect
+      (fun x => |sourceOracleDensity T n s b Z τ (z,x)-1|^(2*k)) ≤
+        ((((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b)/
+          (1-((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b))^k := by
+  have he (x : Fin (SourceLeafCount T (b+1)) → Leaf) :
+      sourceOracleDensity T n s b Z τ (z,x)-1=
+        multilinear (nonconstantCoefficient (sourceOracleCoefficient T n s b z τ)) Z x := by
+    rw [nonconstant_multilinear,sourceOracleCoefficient_empty,sourceOracleDensity_expansion]
+  simp_rw [he]
+  apply (multilinear_moment_scaled P Z k hk hcenter hL hMoment _ _).trans
+  exact pow_le_pow_left₀ (multilinearEnergy_nonneg _ _)
+    (source_canonical_centered_energy T n s b z τ (by positivity) hsmall) k
+
+theorem source_canonical_signed_moment {Side Leaf : Type*} [Fintype Leaf]
+    (T n : ℕ → ℕ) (hT : ∀j,0<T j) (s b R k : ℕ) (ω η a : ℝ) (ha : 0 ≤ a)
+    (hη : ∀j<b+1,hiddenSmoothingBound (n j) s ω ≤ η)
+    (P : FiniteLaw Leaf) (Z : Leaf → ℝ) (hk : 1 ≤ k) (hcenter : P.expect Z=0)
+    {L : ℝ} (hL : 0<L) (hMoment : P.expect (fun a => |Z a|^(2*k)) ≤ L^(2*k))
+    (hsmall : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b<1)
+    (z : Reports Side (SourceTrialCount T n (b+1))) (hz : SourceTypical T n s b ω a z) :
+    (P.product (SourceLeafCount T (b+1))).expect
+      (fun x => |sourceOracleDensity T n s b Z true (z,x)-sourceOracleDensity T n s b Z false (z,x)|^(2*k)) ≤
+        (sourceEnergyBudget T n s b R ω η a ((6*(2*k:ℕ):ℝ)*L))^k := by
+  have he (x : Fin (SourceLeafCount T (b+1)) → Leaf) :
+      |sourceOracleDensity T n s b Z true (z,x)-sourceOracleDensity T n s b Z false (z,x)|=
+        |multilinear (sourceSignedCoefficient T n s b z) Z x| := by
+    unfold sourceSignedCoefficient
+    rw [multilinear_sub,←sourceOracleDensity_expansion,←sourceOracleDensity_expansion,abs_sub_comm]
+  simp_rw [he]
+  apply (multilinear_moment_scaled P Z k hk hcenter hL hMoment _ _).trans
+  exact pow_le_pow_left₀ (multilinearEnergy_nonneg _ _)
+    (source_canonical_energy T n hT s b R ω η a _ ha (by positivity) hsmall hη z hz) k
+
+end TraceReconstruction
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+theorem degreeForcedRoleEnergy_zero_of_lt (T n g : ℕ → ℕ) (s : ℕ) (center : ℕ → ℤ)
+    (globalShift : (b : ℕ) → (Fin (g b) → Bool) → Fin (T b) → ℤ)
+    (A : (b : ℕ) → Finset (Fin (T b))) (b r : ℕ) (hr : r<2^(b+1))
+    (d : WindowAssignments T n g (Option Bool) (b+1)) (incoming : ℤ) :
+    degreeForcedRoleEnergy T n g s center globalShift A b r d incoming=0 := by
+  have hi : IsEmpty (DegreeForcedRole T b r) := ⟨fun w => by
+    obtain ⟨F⟩ := w.property.2
+    have hh := F.leaf_lower
+    rw [occupiedWindowTerm_leafCount,w.property.1] at hh
+    omega⟩
+  let := hi
+  exact Finset.sum_eq_zero (fun w _ => isEmptyElim w)
+
+/-- Below the genuine forced binary leaf threshold there is NO forced
+ contribution; retaining even its coarse bound would lose the main exponent. -/
+def sourceSharpDegreeBudget (T n : ℕ → ℕ) (s b r : ℕ) (ω η a : ℝ) : ℝ :=
+  4*(3^((b+1)*r):ℕ)*
+    (((SourceLeafCount T (b+1)).choose r:ℕ)*(3^((b+1)*r):ℕ)*(η^2+a^2)+
+      if 2^(b+1)≤r then sourceForcedBudget T n s b r ω else 0)
+
+theorem sourceSharpDegreeBudget_nonneg (T n : ℕ → ℕ) (s b r : ℕ) (ω η a : ℝ) :
+    0 ≤ sourceSharpDegreeBudget T n s b r ω η a := by
+  have h := sourceForcedBudget_nonneg T n s b r ω
+  unfold sourceSharpDegreeBudget
+  split_ifs <;> positivity
+
+theorem source_canonical_sharp_degree_bound {Side : Type*} (T n : ℕ → ℕ) (hT : ∀j,0<T j)
+    (s b r : ℕ) (ω η a : ℝ) (ha : 0≤a)
+    (hη : ∀j<b+1,hiddenSmoothingBound (n j) s ω≤η)
+    (z : Reports Side (SourceTrialCount T n (b+1))) (hz : SourceTypical T n s b ω a z) :
+    (∑σ : Fin (SourceLeafCount T (b+1)) → Bool, if monomialDegree σ=r then
+      (averagedCoefficient (PairedMechanisms.orientationsGiven z)
+        (sourceFlatLeafType T n s (sourceLocalCenter T n) (b+1) 0 false) σ -
+       averagedCoefficient (PairedMechanisms.orientationsGiven z)
+        (sourceFlatLeafType T n s (sourceLocalCenter T n) (b+1) 0 true) σ)^2 else 0) ≤
+      sourceSharpDegreeBudget T n s b r ω η a := by
+  rw [source_canonical_degree_sum]
+  apply (sourceCoefficient_degree_energy T n _ s _ _ (fun j => centralThird (T j)) b r
+    (sourceReportFlags T n (b+1) z) 0 η
+    (hiddenConditions_smoothingCap T n _ ω η s (b+1) hη _ hz.1)).trans
+  apply mul_le_mul_of_nonneg_left _ (by positivity)
+  apply add_le_add
+  · apply mul_le_mul_of_nonneg_left _ (by positivity)
+    apply add_le_add_right
+    exact (sq_le_sq₀ (sourceNoncentralMass_nonneg T n _ s _ _ _ _ _ _) ha).mpr hz.2.le
+  · by_cases hr : 2^(b+1)≤r
+    · rw [ite_eq_left hr]
+      exact degreeForcedRoleEnergy_bound T n _ hT s (sourceLocalCenter T n) _ _ _ (sourceScheduledShift_hasPlateau T n)
+        (fun j => hiddenBinomialCap (n j) ω) (fun j => hiddenBinomialCap (sourceScheduledTrials T n j) ω)
+        (fun j => hiddenBinomialCap_nonneg _ _) (fun j => hiddenBinomialCap_nonneg _ _)
+        b r _ (hiddenConditions_binomialCaps T n _ ω (b+1) _ hz.1) 0
+    · rw [ite_eq_right hr,degreeForcedRoleEnergy_zero_of_lt T n _ s _ _ _ b r (lt_of_not_ge hr)]
+def sourceSharpEnergyBudget (T n : ℕ → ℕ) (s b R : ℕ) (ω η a w : ℝ) : ℝ :=
+  (∑r∈Finset.range R, w^(2*r)*(if r=0 then 0 else sourceSharpDegreeBudget T n s b r ω η a))+
+    4*(w^2*sourceSparseCount T s b)^R/(1-w^2*sourceSparseCount T s b)
+
+/-- Complete signed energy, with finite low-degree forced-tree bound and
+ the geometric high tail. No coefficient estimates remain as hypotheses. -/
+theorem source_canonical_sharp_energy {Side : Type*} (T n : ℕ → ℕ) (hT : ∀j,0<T j)
+    (s b R : ℕ) (ω η a w : ℝ) (ha : 0 ≤ a) (hw : 0 ≤ w)
+    (hsmall : w^2*sourceSparseCount T s b<1)
+    (hη : ∀j<b+1,hiddenSmoothingBound (n j) s ω ≤ η)
+    (z : Reports Side (SourceTrialCount T n (b+1))) (hz : SourceTypical T n s b ω a z) :
+    multilinearEnergy (sourceSignedCoefficient T n s b z) w ≤ sourceSharpEnergyBudget T n s b R ω η a w := by
+  apply multilinearEnergy_low_high _ w (w^2*sourceSparseCount T s b) 4 hw (by positivity) hsmall
+    (by norm_num) (fun r => if r=0 then 0 else sourceSharpDegreeBudget T n s b r ω η a)
+  · intro r _
+    by_cases hr : r=0
+    · subst r
+      rw [sourceSignedCoefficient_zero,ite_eq_left rfl]
+    · rw [ite_eq_right hr]
+      exact source_canonical_sharp_degree_bound T n hT s b r ω η a ha hη z hz
+  · intro r _
+    exact sourceSignedCoefficient_degree_geometric T n s b r z w
+  · intro r
+    split_ifs
+    · rfl
+    · exact sourceSharpDegreeBudget_nonneg T n s b r ω η a
+
+theorem sourceSharpEnergyBudget_nonneg (T n : ℕ → ℕ) (s b R : ℕ) (ω η a w : ℝ)
+    (hsmall : w^2*sourceSparseCount T s b<1) :
+    0 ≤ sourceSharpEnergyBudget T n s b R ω η a w := by
+  apply add_nonneg
+  · apply Finset.sum_nonneg
+    intro r _
+    apply mul_nonneg (pow_nonneg (sq_nonneg w) r |>.trans_eq (pow_mul w 2 r).symm)
+    split_ifs
+    · rfl
+    · exact sourceSharpDegreeBudget_nonneg T n s b r ω η a
+  · exact div_nonneg (by positivity) (sub_nonneg.mpr hsmall.le)
+
+end TraceReconstruction
+
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+local instance (priority := 10000) {α β : Type*} : DecidableEq (α × β) := Classical.decEq _
+
+theorem source_canonical_sharp_signed_moment {Side Leaf : Type*} [Fintype Leaf]
+    (T n : ℕ → ℕ) (hT : ∀j,0<T j) (s b R k : ℕ) (ω η a : ℝ) (ha : 0 ≤ a)
+    (hη : ∀j<b+1,hiddenSmoothingBound (n j) s ω ≤ η)
+    (P : FiniteLaw Leaf) (Z : Leaf → ℝ) (hk : 1 ≤ k) (hcenter : P.expect Z=0)
+    {L : ℝ} (hL : 0<L) (hMoment : P.expect (fun a => |Z a|^(2*k)) ≤ L^(2*k))
+    (hsmall : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b<1)
+    (z : Reports Side (SourceTrialCount T n (b+1))) (hz : SourceTypical T n s b ω a z) :
+    (P.product (SourceLeafCount T (b+1))).expect
+      (fun x => |sourceOracleDensity T n s b Z true (z,x)-sourceOracleDensity T n s b Z false (z,x)|^(2*k)) ≤
+        (sourceSharpEnergyBudget T n s b R ω η a ((6*(2*k:ℕ):ℝ)*L))^k := by
+  have he (x : Fin (SourceLeafCount T (b+1)) → Leaf) :
+      |sourceOracleDensity T n s b Z true (z,x)-sourceOracleDensity T n s b Z false (z,x)|=
+        |multilinear (sourceSignedCoefficient T n s b z) Z x| := by
+    unfold sourceSignedCoefficient
+    rw [multilinear_sub,←sourceOracleDensity_expansion,←sourceOracleDensity_expansion,abs_sub_comm]
+  simp_rw [he]
+  apply (multilinear_moment_scaled P Z k hk hcenter hL hMoment _ _).trans
+  exact pow_le_pow_left₀ (multilinearEnergy_nonneg _ _)
+    (source_canonical_sharp_energy T n hT s b R ω η a _ ha (by positivity) hsmall hη z hz) k
+
+
+def sourceReferenceLaw {Side Leaf : Type*} [Fintype Side] [Fintype Leaf]
+    (T n : ℕ → ℕ) (b : ℕ) (M : PairedMechanisms Side (SourceTrialCount T n (b+1)))
+    (P : FiniteLaw Leaf) :
+    FiniteLaw (Reports Side (SourceTrialCount T n (b+1)) × (Fin (SourceLeafCount T (b+1)) → Leaf)) :=
+  M.reports.prod (P.product (SourceLeafCount T (b+1)))
+
+theorem sourceOracleDensity_total {Side Leaf : Type*} [Fintype Side] [Fintype Leaf]
+    (T n : ℕ → ℕ) (s b : ℕ) (M : PairedMechanisms Side (SourceTrialCount T n (b+1)))
+    (P : FiniteLaw Leaf) (Z : Leaf → ℝ) (hZ : P.expect Z=0) (τ : Bool) :
+    (sourceReferenceLaw T n b M P).expect (sourceOracleDensity T n s b Z τ)=1 := by
+  rw [sourceReferenceLaw,FiniteLaw.expect_prod]
+  simp_rw [sourceOracleDensity_mean T n s b P Z hZ τ]
+  exact M.reports.expect_const 1
+
+theorem sourceOracle_withDensity {Side Leaf : Type*} [Fintype Side] [Fintype Leaf]
+    (T n : ℕ → ℕ) (s b : ℕ) (M : PairedMechanisms Side (SourceTrialCount T n (b+1)))
+    (P Q : FiniteLaw Leaf) (Z : Leaf → ℝ) (hden : ∀a,Q.mass a=P.mass a*(1+Z a))
+    (hZ : ∀a,0 ≤ 1+Z a) (hc : P.expect Z=0) (τ : Bool) :
+    M.canonicalOracle P Q (sourceFlatLeafType T n s (sourceLocalCenter T n) (b+1) 0 τ)=
+      (sourceReferenceLaw T n b M P).withDensity (sourceOracleDensity T n s b Z τ)
+        (sourceOracleDensity_nonneg T n s b Z hZ τ)
+        (sourceOracleDensity_total T n s b M P Z hc τ) := by
+  apply FiniteLaw.ext
+  funext zx
+  exact M.canonicalOracle_density P Q Z hden _ zx.1 zx.2
+
+def sourceTypicalSet {Side Leaf : Type*} [Fintype Side] [Fintype Leaf]
+    (T n : ℕ → ℕ) (s b : ℕ) (ω a : ℝ) :
+    Finset (Reports Side (SourceTrialCount T n (b+1)) × (Fin (SourceLeafCount T (b+1)) → Leaf)) :=
+  Finset.univ.filter (fun x => SourceTypical T n s b ω a x.1)
+
+theorem sourceTypicalSet_mass {Side Leaf : Type*} [Fintype Side] [Fintype Leaf]
+    (T n : ℕ → ℕ) (s b : ℕ) (M : PairedMechanisms Side (SourceTrialCount T n (b+1)))
+    (P Q : FiniteLaw Leaf) (ω a : ℝ) (τ : Bool) :
+    (M.canonicalOracle P Q (sourceFlatLeafType T n s (sourceLocalCenter T n) (b+1) 0 τ)).eventMass
+      (sourceTypicalSet T n s b ω a)ᶜ=
+      M.reports.expect (fun z => if SourceTypical T n s b ω a z then 0 else 1) := by
+  rw [FiniteLaw.eventMass_indicator]
+  calc
+    _ = (M.canonicalOracle P Q (sourceFlatLeafType T n s (sourceLocalCenter T n) (b+1) 0 τ)).expect
+        (fun zx => if SourceTypical T n s b ω a zx.1 then 0 else 1) := by
+      apply congrArg (FiniteLaw.expect _)
+      funext zx
+      simp [sourceTypicalSet]
+    _ = _ := M.canonicalOracle_metadata P Q
+      (sourceFlatLeafType T n s (sourceLocalCenter T n) (b+1) 0 τ)
+      (fun z => if SourceTypical T n s b ω a z then 0 else 1)
+
+def sourceCenteredBudget (T : ℕ → ℕ) (s b k : ℕ) (L : ℝ) : ℝ :=
+  (((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b)/
+    (1-((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b)
+
+def sourceClippingCost (T n : ℕ → ℕ) (s b k : ℕ) (ω a L : ℝ) : ℝ :=
+  sourceEffectiveHiddenTail T n (b+1) ω+sourceFitTail T n (b+1)/a+
+    Real.sqrt (6*(4*sourceCenteredBudget T s b k L)^k)
+
+theorem sourceCenteredBudget_nonneg (T : ℕ → ℕ) (s b k : ℕ) (L : ℝ)
+    (hsmall : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b<1) :
+    0 ≤ sourceCenteredBudget T s b k L :=
+  div_nonneg (by positivity) (sub_nonneg.mpr hsmall.le)
+
+theorem sourceCenteredBudget_le_one (T : ℕ → ℕ) (s b k : ℕ) (L : ℝ)
+    (hscale : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b≤1/2) :
+    sourceCenteredBudget T s b k L ≤ 1 := by
+  have hsmall : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b<1 := by linarith
+  apply (div_le_iff₀ (sub_pos.mpr hsmall)).mpr
+  linarith
+
+theorem source_reference_centered_moment {Side Leaf : Type*} [Fintype Side] [Fintype Leaf]
+    (T n : ℕ → ℕ) (s b k : ℕ) (M : PairedMechanisms Side (SourceTrialCount T n (b+1)))
+    (P : FiniteLaw Leaf) (Z : Leaf → ℝ) (hk : 1 ≤ k) (hc : P.expect Z=0)
+    {L : ℝ} (hL : 0<L) (hMoment : P.expect (fun a => |Z a|^(2*k)) ≤ L^(2*k))
+    (hsmall : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b<1) (τ : Bool) :
+    (sourceReferenceLaw T n b M P).expect
+      (fun zx => |sourceOracleDensity T n s b Z τ zx-1|^(2*k)) ≤ (sourceCenteredBudget T s b k L)^k := by
+  rw [sourceReferenceLaw,FiniteLaw.expect_prod]
+  calc
+    _ ≤ M.reports.expect (fun _ => (sourceCenteredBudget T s b k L)^k) :=
+      M.reports.expect_mono (fun z => source_canonical_centered_moment T n s b k P Z hk hc hL hMoment hsmall z τ)
+    _ = _ := M.reports.expect_const _
+
+theorem source_reference_signed_moment {Side Leaf : Type*} [Fintype Side] [Fintype Leaf]
+    (T n : ℕ → ℕ) (hTpos : ∀j,0<T j) (s b R k : ℕ)
+    (M : PairedMechanisms Side (SourceTrialCount T n (b+1))) (P : FiniteLaw Leaf)
+    (Z : Leaf → ℝ) (ω a η L : ℝ) (ha : 0≤a) (hL : 0<L) (hk : 1 ≤ k) (hc : P.expect Z=0)
+    (hMoment : P.expect (fun a => |Z a|^(2*k)) ≤ L^(2*k))
+    (hsmall : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b<1)
+    (hη : ∀j<b+1,hiddenSmoothingBound (n j) s ω ≤ η) :
+    (sourceReferenceLaw T n b M P).expect (fun zx =>
+      |if zx∈sourceTypicalSet T n s b ω a then sourceOracleDensity T n s b Z true zx-
+        sourceOracleDensity T n s b Z false zx else 0|^(2*k)) ≤
+      (sourceSharpEnergyBudget T n s b R ω η a ((6*(2*k:ℕ):ℝ)*L))^k := by
+  rw [sourceReferenceLaw,FiniteLaw.expect_prod]
+  have hV := sourceSharpEnergyBudget_nonneg T n s b R ω η a ((6*(2*k:ℕ):ℝ)*L) hsmall
+  calc
+    _ ≤ M.reports.expect (fun _ => (sourceSharpEnergyBudget T n s b R ω η a ((6*(2*k:ℕ):ℝ)*L))^k) := by
+      apply M.reports.expect_mono
+      intro z
+      by_cases hz : SourceTypical T n s b ω a z
+      · simp only [sourceTypicalSet,Finset.mem_filter,Finset.mem_univ,true_and,hz,ite_true]
+        exact source_canonical_sharp_signed_moment T n hTpos s b R k ω η a ha hη P Z hk hc hL hMoment hsmall z hz
+      · simp only [sourceTypicalSet,Finset.mem_filter,Finset.mem_univ,true_and,hz,ite_false,
+          abs_zero,zero_pow (by omega : 2*k≠0),FiniteLaw.expect_const]
+        exact pow_nonneg hV k
+    _ = _ := M.reports.expect_const _
+
+theorem source_reference_clipped_exists {Side Leaf : Type*} [Fintype Side] [Fintype Leaf]
+    (T n : ℕ → ℕ) (hTpos : ∀j,0<T j) (s b R k : ℕ)
+    (M : PairedMechanisms Side (SourceTrialCount T n (b+1))) (P : FiniteLaw Leaf)
+    (Z : Leaf → ℝ) (hZ : ∀a,0 ≤ 1+Z a) (hc : P.expect Z=0)
+    (ω a η L : ℝ) (ha : 0≤a) (hL : 0<L) (hk : 1≤k)
+    (hMoment : P.expect (fun a => |Z a|^(2*k)) ≤ L^(2*k))
+    (hscale : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b ≤ 1/2)
+    (hη : ∀j<b+1,hiddenSmoothingBound (n j) s ω ≤ η)
+    (hB : ∀τ,((sourceReferenceLaw T n b M P).withDensity (sourceOracleDensity T n s b Z τ)
+      (sourceOracleDensity_nonneg T n s b Z hZ τ) (sourceOracleDensity_total T n s b M P Z hc τ)).eventMass
+        (sourceTypicalSet T n s b ω a)ᶜ ≤ sourceEffectiveHiddenTail T n (b+1) ω+sourceFitTail T n (b+1)/a)
+    (herror : sourceClippingCost T n s b k ω a L ≤ 1/2) :
+    Nonempty (ClippedOracle
+      ((sourceReferenceLaw T n b M P).withDensity (sourceOracleDensity T n s b Z false)
+        (sourceOracleDensity_nonneg T n s b Z hZ false) (sourceOracleDensity_total T n s b M P Z hc false))
+      ((sourceReferenceLaw T n b M P).withDensity (sourceOracleDensity T n s b Z true)
+        (sourceOracleDensity_nonneg T n s b Z hZ true) (sourceOracleDensity_total T n s b M P Z hc true))
+      k (sourceClippingCost T n s b k ω a L)
+      (64*sourceSharpEnergyBudget T n s b R ω η a ((6*(2*k:ℕ):ℝ)*L))) := by
+  have hsmall : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b<1 := by linarith
+  exact FiniteLaw.exists_clippedOracle (sourceReferenceLaw T n b M P)
+    (sourceOracleDensity T n s b Z false) (sourceOracleDensity T n s b Z true)
+    (sourceOracleDensity_nonneg T n s b Z hZ false) (sourceOracleDensity_nonneg T n s b Z hZ true)
+    (sourceOracleDensity_total T n s b M P Z hc false) (sourceOracleDensity_total T n s b M P Z hc true)
+    (sourceTypicalSet T n s b ω a) _ (hB false) (hB true) k hk (sourceCenteredBudget T s b k L)
+    (sourceCenteredBudget_nonneg T s b k L hsmall) (sourceCenteredBudget_le_one T s b k L hscale)
+    (source_reference_centered_moment T n s b k M P Z hk hc hL hMoment hsmall false)
+    (source_reference_centered_moment T n s b k M P Z hk hc hL hMoment hsmall true)
+    _ (sourceSharpEnergyBudget_nonneg T n s b R ω η a ((6*(2*k:ℕ):ℝ)*L) hsmall)
+    (source_reference_signed_moment T n hTpos s b R k M P Z ω a η L ha hL hk hc hMoment hsmall hη) herror
+
+/-- Complete finite quantitative clipping of the actual recursive source
+ oracle. All analytic budgets here are explicit arithmetic in the literal
+ branching/trial counts; no coefficient or oracle-existence premise remains. -/
+theorem source_clippedOracle_exists {Side Leaf : Type*} [Fintype Side] [Fintype Leaf]
+    (T n : ℕ → ℕ) (hTpos : ∀j,0<T j) (s b R k : ℕ)
+    (M : PairedMechanisms Side (SourceTrialCount T n (b+1))) (P Q : FiniteLaw Leaf)
+    (Z : Leaf → ℝ) (hden : ∀a,Q.mass a=P.mass a*(1+Z a)) (hZ : ∀a,0 ≤ 1+Z a)
+    (ω a η L : ℝ) (ha : 0<a) (hL : 0<L)
+    (hw : ∀i,M.hiddenWeight i=ω) (hp : ω^2∈Set.Icc 0 1)
+    (hn : ∀j,j<b → 0<n j) (hne : ∀j,j<b → Even (n j))
+    (hT : ∀j≤b,24≤T j) (he : ∀j≤b,Even (T j)) (hMotif : ∀j≤b,24*2^s≤T j)
+    (hroot : 4*(n b+1)≤T b) (hk : 1 ≤ k)
+    (hMoment : P.expect (fun a => |Z a|^(2*k)) ≤ L^(2*k))
+    (hscale : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b ≤ 1/2)
+    (hη : ∀j<b+1,hiddenSmoothingBound (n j) s ω ≤ η)
+    (herror : sourceClippingCost T n s b k ω a L ≤ 1/2) :
+    Nonempty (ClippedOracle
+      (M.canonicalOracle P Q (sourceFlatLeafType T n s (sourceLocalCenter T n) (b+1) 0 false))
+      (M.canonicalOracle P Q (sourceFlatLeafType T n s (sourceLocalCenter T n) (b+1) 0 true))
+      k (sourceClippingCost T n s b k ω a L)
+      (64*sourceSharpEnergyBudget T n s b R ω η a ((6*(2*k:ℕ):ℝ)*L))) := by
+  have hc := FiniteLaw.density_score_centered P Q Z hden
+  have hB (τ : Bool) : ((sourceReferenceLaw T n b M P).withDensity (sourceOracleDensity T n s b Z τ)
+      (sourceOracleDensity_nonneg T n s b Z hZ τ) (sourceOracleDensity_total T n s b M P Z hc τ)).eventMass
+        (sourceTypicalSet T n s b ω a)ᶜ ≤ sourceEffectiveHiddenTail T n (b+1) ω+sourceFitTail T n (b+1)/a := by
+    rw [←sourceOracle_withDensity T n s b M P Q Z hden hZ hc τ,sourceTypicalSet_mass]
+    exact source_typical_effective_failure T n s b M ω a ha hw hp hn hne hT he hMotif hroot
+  rw [sourceOracle_withDensity T n s b M P Q Z hden hZ hc false,
+    sourceOracle_withDensity T n s b M P Q Z hden hZ hc true]
+  exact source_reference_clipped_exists T n hTpos s b R k M P Z hZ hc ω a η L ha.le hL hk hMoment hscale hη hB herror
+
+end TraceReconstruction
+
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+namespace ClippedOracle
+variable {α β : Type*} [Fintype α] [Fintype β]
+    {P Q : FiniteLaw α} {k : ℕ} {e V : ℝ}
+
+def prod (C : ClippedOracle P Q k e V) (R : FiniteLaw β) :
+    ClippedOracle (P.prod R) (Q.prod R) k e V where
+  first := C.first.prod R
+  second := C.second.prod R
+  score := fun x => C.score x.1
+  density x := by
+    change C.second.mass x.1*R.mass x.2=(C.first.mass x.1*R.mass x.2)*(1+C.score x.1)
+    rw [C.density]; ring
+  likelihood_nonneg x := C.likelihood_nonneg x.1
+  centered := by
+    rw [FiniteLaw.expect_prod]
+    simp_rw [FiniteLaw.expect_const]
+    exact C.centered
+  moment := by
+    unfold FiniteLaw.evenMomentSq
+    rw [FiniteLaw.expect_prod]
+    simp_rw [FiniteLaw.expect_const]
+    exact C.moment
+  first_tv := by rw [FiniteLaw.tv_prod_eq_left]; exact C.first_tv
+  second_tv := by rw [FiniteLaw.tv_prod_eq_left]; exact C.second_tv
+end ClippedOracle
+
+theorem ApproximatesTrace.of_tv {α : Type*} [Fintype α]
+    {q E e : ℝ} {P Q : FiniteLaw α} {f : α → List Bool} {w : List Bool}
+    (hP : ApproximatesTrace q P f w E) (htv : Q.tv P ≤ e) :
+    ApproximatesTrace q Q f w (e+E) := by
+  intro F hF
+  exact (abs_sub_le (Q.expect (fun a => F (f a))) (P.expect (fun a => F (f a)))
+    (listDeletionExpectation q w F)).trans (add_le_add
+      ((FiniteLaw.abs_expect_sub_le_tv Q P _ (fun a => hF _)).trans htv) (hP F hF))
+
+/-- A finite genuine approximate oracle pair with a common complete-trace
+ decoder. It does not include an overlap or any unproved scale conclusion. -/
+structure OracleTracePair (q : ℝ) (x y : List Bool) (k : ℕ) (E V : ℝ) where
+  Outcome : Type
+  finiteOutcome : Fintype Outcome
+  first : @FiniteLaw Outcome finiteOutcome
+  second : @FiniteLaw Outcome finiteOutcome
+  decode : Outcome → List Bool
+  first_simulation : @ApproximatesTrace Outcome finiteOutcome q first decode x E
+  second_simulation : @ApproximatesTrace Outcome finiteOutcome q second decode y E
+  score : Outcome → ℝ
+  density : ∀a,second.mass a=first.mass a*(1+score a)
+  likelihood_nonneg : ∀a,0≤1+score a
+  centered : first.expect score=0
+  moment : @FiniteLaw.evenMomentSq Outcome finiteOutcome first score k ≤ V
+
+/-- The clipping and independent filler metadata preserve the source's fixed
+ public trace decoder. -/
+theorem source_clipped_simulation {Side Leaf : Type} [Fintype Side] [Fintype Leaf]
+    {q E e V : ℝ} (hq : q∈Set.Icc 0 1) (hE : 0≤E)
+    (s : ℕ) (P T n h : ℕ → ℕ) (atom : Bool → List Bool) (b k : ℕ)
+    (H : SourceLayoutMargins s P T n h b)
+    (hP : P b=n b*(2*h b+1)+T b) (hT : Even (T b))
+    (hlarge : 4*(n b+1)≤T b) (hmotif : 8*2^s≤T b)
+    (hn : ∀j<b,0<n j)
+    (M : PairedMechanisms Side (SourceTrialCount T n (b+1)))
+    (R : Fin (SourceFillerCount T (b+1)) → FiniteLaw Side) (A B : FiniteLaw Leaf)
+    (f : Side → List Bool) (g : Leaf → List Bool)
+    (hU : ∀j,SimulatesTrace q (M.first j) f (sourceFlatTrialWord s P T n h atom (b+1) false j))
+    (hV : ∀j,SimulatesTrace q (M.second j) f (sourceFlatTrialWord s P T n h atom (b+1) true j))
+    (hW : ∀j,SimulatesTrace q (R j) f (sourceFlatFillerWord s P T n h atom (b+1) j))
+    (hA : ApproximatesTrace q A g (atom false) E) (hB : ApproximatesTrace q B g (atom true) E)
+    (C : ClippedOracle
+      (M.canonicalOracle A B (sourceFlatLeafType T n s (sourceLocalCenter T n) (b+1) 0 false))
+      (M.canonicalOracle A B (sourceFlatLeafType T n s (sourceLocalCenter T n) (b+1) 0 true)) k e V) :
+    Nonempty (OracleTracePair q (sourceBlockWord s P atom (b+1) false)
+      (sourceBlockWord s P atom (b+1) true) k
+      (e+(SourceLeafCount T (b+1):ℝ)*E+sourceFitTail T n (b+1)) V) := by
+  let D := C.prod (FiniteLaw.coordinateProduct R)
+  refine ⟨{
+    Outcome := (Reports Side (SourceTrialCount T n (b+1)) × (Fin (SourceLeafCount T (b+1)) → Leaf)) ×
+      (Fin (SourceFillerCount T (b+1)) → Side)
+    finiteOutcome := inferInstance
+    first := D.first
+    second := D.second
+    decode := sourceOracleDecode T n (b+1) f g
+    score := D.score
+    density := D.density
+    likelihood_nonneg := D.likelihood_nonneg
+    centered := D.centered
+    moment := D.moment
+    first_simulation := ?_
+    second_simulation := ?_
+  }⟩
+  · simpa only [add_assoc] using
+      (source_canonical_simulation hq hE s P T n h atom b H hP hT hlarge hmotif hn M R A B f g hU hV hW hA hB false).of_tv D.first_tv
+  · simpa only [add_assoc] using
+      (source_canonical_simulation hq hE s P T n h atom b H hP hT hlarge hmotif hn M R A B f g hU hV hW hA hB true).of_tv D.second_tv
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction.FiniteLaw
+open scoped Classical
+variable {α : Type*} [Fintype α]
+
+theorem expect_nonneg (P : FiniteLaw α) (F : α → ℝ) (hF : ∀a,0≤F a) : 0≤P.expect F :=
+  Finset.sum_nonneg (fun a _ => mul_nonneg (P.nonneg a) (hF a))
+
+theorem evenMomentSq_nonneg (P : FiniteLaw α) (F : α → ℝ) (k : ℕ) :
+    0≤P.evenMomentSq F k := Real.rpow_nonneg (P.expect_nonneg _ (fun _ => by positivity)) _
+
+theorem evenMomentSq_pow (P : FiniteLaw α) (F : α → ℝ) (k : ℕ) (hk : 1≤k) :
+    (P.evenMomentSq F k)^k=P.expect (fun a => |F a|^(2*k)) := by
+  unfold evenMomentSq
+  rw [one_div,Real.rpow_inv_natCast_pow (P.expect_nonneg _ (fun _ => by positivity)) (by omega)]
+
+theorem expect_even_pow_le_of_moment (P : FiniteLaw α) (F : α → ℝ) (k : ℕ) (hk : 1≤k)
+    {V : ℝ} (h : P.evenMomentSq F k≤V) :
+    P.expect (fun a => |F a|^(2*k))≤V^k := by
+  rw [←P.evenMomentSq_pow F k hk]
+  exact pow_le_pow_left₀ (P.evenMomentSq_nonneg F k) h k
+
+theorem secondMoment_le_evenMomentSq (P : FiniteLaw α) (F : α → ℝ) (k : ℕ) (hk : 1≤k) :
+    P.expect (fun a => F a^2)≤P.evenMomentSq F k := by
+  have hn : 0≤P.expect (fun a => F a^2) := P.expect_nonneg _ (fun _ => sq_nonneg _)
+  have hj := P.abs_expect_pow_le (fun a => F a^2) k
+  simp only [abs_of_nonneg hn,abs_pow,←pow_mul] at hj
+  have hr := Real.rpow_le_rpow (pow_nonneg hn k) hj (by positivity : 0≤(k:ℝ)⁻¹)
+  rw [Real.pow_rpow_inv_natCast hn (by omega)] at hr
+  simpa only [evenMomentSq,one_div] using hr
+
+theorem affinity_coordinateProduct {n : ℕ} (P Q : Fin n → FiniteLaw α) :
+    (coordinateProduct P).affinity (coordinateProduct Q)=∏i,(P i).affinity (Q i) := by
+  unfold affinity coordinateProduct
+  simp only
+  have he (a : Fin n → α) :
+      Real.sqrt (∏i,(P i).mass (a i))*Real.sqrt (∏i,(Q i).mass (a i))=
+        ∏i,Real.sqrt ((P i).mass (a i))*Real.sqrt ((Q i).mass (a i)) := by
+    rw [Real.sqrt_prod _ (fun i _ => (P i).nonneg (a i)),
+      Real.sqrt_prod _ (fun i _ => (Q i).nonneg (a i)),Finset.prod_mul_distrib]
+  simp_rw [he]
+  exact (Fintype.prod_sum (fun (i : Fin n) a => Real.sqrt ((P i).mass a)*Real.sqrt ((Q i).mass a))).symm
+
+theorem one_sub_prod_le_sum {ι : Type*} (S : Finset ι) (f : ι → ℝ)
+    (h0 : ∀i∈S,0≤f i) (h1 : ∀i∈S,f i≤1) :
+    1-∏i∈S,f i≤∑i∈S,(1-f i) := by
+  induction S using Finset.induction_on with
+  | empty => simp
+  | @insert a S ha ih =>
+    rw [Finset.prod_insert ha,Finset.sum_insert ha]
+    have H := ih (fun i hi => h0 i (Finset.mem_insert_of_mem hi))
+      (fun i hi => h1 i (Finset.mem_insert_of_mem hi))
+    have hs : 0≤∑i∈S,(1-f i) := Finset.sum_nonneg (fun i hi => sub_nonneg.mpr (h1 i (Finset.mem_insert_of_mem hi)))
+    have hm := mul_le_mul_of_nonneg_left H (h0 a (Finset.mem_insert_self a S))
+    have hm' := mul_le_mul_of_nonneg_right (h1 a (Finset.mem_insert_self a S)) hs
+    nlinarith
+
+theorem hellingerSq_coordinateProduct_le {n : ℕ} (P Q : Fin n → FiniteLaw α) :
+    (coordinateProduct P).hellingerSq (coordinateProduct Q)≤∑i,(P i).hellingerSq (Q i) := by
+  simp_rw [hellingerSq_eq]
+  rw [affinity_coordinateProduct]
+  have h := one_sub_prod_le_sum Finset.univ (fun i => (P i).affinity (Q i))
+    (fun i _ => affinity_nonneg _ _) (fun i _ => affinity_le_one _ _)
+  have he : (∑i,(2-2*(P i).affinity (Q i)))=2*∑i,(1-(P i).affinity (Q i)) := by
+    rw [Finset.mul_sum]; apply Finset.sum_congr rfl; intro i _; ring
+  rw [he]
+  linarith
+
+@[simp] theorem hellingerSq_self (P : FiniteLaw α) : P.hellingerSq P=0 := by
+  simp [hellingerSq]
+
+theorem hellingerSq_symm (P Q : FiniteLaw α) : P.hellingerSq Q=Q.hellingerSq P := by
+  apply Finset.sum_congr rfl; intro a _; ring
+
+/-- Only differing coordinates cost squared Hellinger distance. This estimate
+ does not pay for pure filler coordinates, nor assume independence of traces
+ after their common concatenating decoder. -/
+theorem tv_mixedProduct_le {n : ℕ} (P Q : FiniteLaw α) (Z : α → ℝ)
+    (hden : ∀a,Q.mass a=P.mass a*(1+Z a)) (k : ℕ) (hk : 1≤k)
+    {V : ℝ} (hM : P.evenMomentSq Z k≤V) (τ σ : Fin n → Bool) :
+    (coordinateProduct (fun i => if τ i then Q else P)).tv
+      (coordinateProduct (fun i => if σ i then Q else P))≤
+        Real.sqrt ((∑i,if τ i=σ i then (0:ℝ) else 1)*V) := by
+  have hH : P.hellingerSq Q≤V := (P.hellingerSq_le_score Q Z hden).trans
+    ((P.secondMoment_le_evenMomentSq Z k hk).trans hM)
+  apply (tv_le_sqrt_hellingerSq _ _).trans
+  apply Real.sqrt_le_sqrt
+  apply (hellingerSq_coordinateProduct_le _ _).trans
+  rw [Finset.sum_mul]
+  apply Finset.sum_le_sum
+  intro i _
+  cases τ i <;> cases σ i <;> simp only [Bool.false_eq_true,Bool.true_eq_false,ite_false,ite_true,
+    zero_mul,one_mul,hellingerSq_self,le_refl]
+  · exact hH
+  · rw [hellingerSq_symm]; exact hH
+
+end TraceReconstruction.FiniteLaw
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+/-- Literal recursive source oracle with actual complete-trace side laws.
+ All remaining hypotheses here are explicit parameter inequalities, the old
+ leaf oracle, and its genuine repeated-word overlap invariant. -/
+theorem actual_source_oracle {Leaf : Type} [Fintype Leaf]
+    {q E : ℝ} (hq : q∈Set.Icc 0 1) (hE : 0≤E)
+    (s : ℕ) (P T n h : ℕ → ℕ) (atom : Bool → List Bool) (b R k : ℕ)
+    (H : SourceLayoutMargins s P T n h b)
+    (hTpos : ∀j,0<T j) (hP : P b=n b*(2*h b+1)+T b)
+    (hn : ∀j<b,0<n j) (hT : ∀j≤b,24≤T j) (he : ∀j≤b,Even (T j))
+    (hMotif : ∀j≤b,24*2^s≤T j) (hroot : 4*(n b+1)≤T b)
+    (A B : FiniteLaw Leaf) (g : Leaf → List Bool) (Z : Leaf → ℝ)
+    (hden : ∀a,B.mass a=A.mass a*(1+Z a)) (hZ : ∀a,0≤1+Z a)
+    (hA : ApproximatesTrace q A g (atom false) E) (hB : ApproximatesTrace q B g (atom true) E)
+    (ω a η L : ℝ) (ha : 0<a) (hL : 0<L) (hω : ω∈Set.Ioo 0 1)
+    (hov : ∀j<b+1,ω≤listTraceOverlap q hq
+      (pureAtomRun (sourceBlockWord s P atom j false) (h j))
+      (pureAtomRun (sourceBlockWord s P atom j false) (h j+1)))
+    (hk : 1≤k) (hMoment : A.expect (fun a => |Z a|^(2*k))≤L^(2*k))
+    (hscale : ((6*(2*k:ℕ):ℝ)*L)^2*sourceSparseCount T s b≤1/2)
+    (hη : ∀j<b+1,hiddenSmoothingBound (n j) s ω≤η)
+    (herror : sourceClippingCost T n s b k ω a L≤1/2) :
+    Nonempty (OracleTracePair q (sourceBlockWord s P atom (b+1) false)
+      (sourceBlockWord s P atom (b+1) true) k
+      (sourceClippingCost T n s b k ω a L+(SourceLeafCount T (b+1):ℝ)*E+sourceFitTail T n (b+1))
+      (64*sourceSharpEnergyBudget T n s b R ω η a ((6*(2*k:ℕ):ℝ)*L))) := by
+  let U := sourceFlatTrialWord s P T n h atom (b+1) false
+  let V := sourceFlatTrialWord s P T n h atom (b+1) true
+  let W := sourceFlatFillerWord s P T n h atom (b+1)
+  obtain ⟨mU,hmU⟩ := finite_word_bound U
+  obtain ⟨mV,hmV⟩ := finite_word_bound V
+  obtain ⟨mW,hmW⟩ := finite_word_bound W
+  let m := max mU (max mV mW)
+  have hU (j) : (U j).length ≤ m := (hmU j).trans (le_max_left _ _)
+  have hV (j) : (V j).length ≤ m := (hmV j).trans ((le_max_left _ _).trans (le_max_right _ _))
+  have hW (j) : (W j).length ≤ m := (hmW j).trans ((le_max_right _ _).trans (le_max_right _ _))
+  have hov' (j) : ω≤listTraceOverlap q hq (U j) (V j) :=
+    sourceTrialWord_overlap hq s P T n h atom (b+1) hov (finEnumeration _ j)
+  let M := tracePairedMechanisms hq hω.1 U V m hU hV hov'
+  let F : Fin (SourceFillerCount T (b+1)) → FiniteLaw (BoundedTrace m) :=
+    fun j => listTraceLaw q hq (W j) m (hW j)
+  have hp : ω^2∈Set.Icc 0 1 := ⟨sq_nonneg ω,by nlinarith [hω.1,hω.2]⟩
+  obtain ⟨C⟩ := source_clippedOracle_exists T n hTpos s b R k M A B Z hden hZ ω a η L ha hL
+    (fun _ => rfl) hp hn H.even_trials hT he hMotif hroot hk hMoment hscale hη herror
+  apply source_clipped_simulation hq hE s P T n h atom b k H hP (he b le_rfl) hroot
+    (by nlinarith [hMotif b le_rfl]) hn M F A B Subtype.val g
+    (fun j => listTraceLaw_simulates q hq (U j) m (hU j))
+    (fun j => listTraceLaw_simulates q hq (V j) m (hV j))
+    (fun j => listTraceLaw_simulates q hq (W j) m (hW j)) hA hB C
+
+end TraceReconstruction
+
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/TraceReconstruction/Testing.lean b/lean/OAI/Probability/TraceReconstruction/Testing.lean
index dbe74046ebc1299ad7ecc90f1f9d3c4bd7962d4b..5166ccc4f48105ca59d8855a92ba21a176d160cb 100644
--- a/lean/OAI/Probability/TraceReconstruction/Testing.lean
+++ b/lean/OAI/Probability/TraceReconstruction/Testing.lean
@@ -1,221 +1,221 @@
-import OAI.Probability.TraceReconstruction.Restoration
-
-namespace OAI
-
-/-!
-# Testing
-
-Multi-gadget family restoration and common-reference many-hypothesis statistics for every positive success level.
--/
-
-open scoped ENNReal
-noncomputable section
-
-open Filter
-open scoped Topology
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-namespace FiniteLaw
-variable {α β : Type*} [Fintype α] [Fintype β]
-
-theorem hellingerSq_prod_same_right (P Q : FiniteLaw α) (R : FiniteLaw β) :
-    (P.prod R).hellingerSq (Q.prod R)=P.hellingerSq Q := by
-  unfold hellingerSq prod
-  rw [Fintype.sum_prod_type]
-  have he (a : α) (b : β) :
-      (Real.sqrt (P.mass a*R.mass b)-Real.sqrt (Q.mass a*R.mass b))^2=
-        (Real.sqrt (P.mass a)-Real.sqrt (Q.mass a))^2*R.mass b := by
-    rw [Real.sqrt_mul (P.nonneg a),Real.sqrt_mul (Q.nonneg a),←sub_mul,mul_pow,
-      Real.sq_sqrt (R.nonneg b)]
-  simp_rw [he,←Finset.mul_sum,R.total,mul_one]
-
-end FiniteLaw
-
-def HellingerTracePair.suffix {q E V : ℝ} {x y : List Bool}
-    (C : HellingerTracePair q x y E V) (hq : q∈Set.Icc 0 1) (w : List Bool) :
-    HellingerTracePair q (x++w) (y++w) E V := by
-  let := C.finiteOutcome
-  let R := listTraceLaw q hq w w.length le_rfl
-  exact {
-    Outcome := C.Outcome × BoundedTrace w.length
-    finiteOutcome := inferInstance
-    first := C.first.prod R
-    second := C.second.prod R
-    decode := fun z=>C.decode z.1++z.2.val
-    first_simulation := by
-      simpa only [add_zero] using C.first_simulation.append hq
-        (listTraceLaw_simulates q hq w w.length le_rfl).approximate
-    second_simulation := by
-      simpa only [add_zero] using C.second_simulation.append hq
-        (listTraceLaw_simulates q hq w w.length le_rfl).approximate
-    bound := by rw [FiniteLaw.hellingerSq_prod_same_right]; exact C.bound }
-
-theorem coordinateWord_length_constant {g S : ℕ} (w : Fin g→List Bool)
-    (hw : ∀j,(w j).length=S) : (coordinateWord w).length=g*S := by
-  induction g with
-  | zero => simp [coordinateWord]
-  | succ g ih =>
-    rw [coordinateWord_succ,List.length_append,hw,ih (fun j=>w j.succ) (fun j=>hw j.succ)]
-    simp [Nat.add_mul,Nat.add_comm]
-
-theorem coordinateWord_injective_equal {g S : ℕ} (atom : Bool→List Bool)
-    (hw : ∀τ,(atom τ).length=S) (ha : Function.Injective atom) :
-    Function.Injective (fun θ : Fin g→Bool=>coordinateWord (fun j=>atom (θ j))) := by
-  induction g with
-  | zero => intro θ σ _; exact Subsingleton.elim _ _
-  | succ g ih =>
-    intro θ σ h
-    change coordinateWord (fun j=>atom (θ j))=coordinateWord (fun j=>atom (σ j)) at h
-    rw [coordinateWord_succ,coordinateWord_succ] at h
-    have h0 := List.append_inj_left' h (by rw [coordinateWord_length_constant _ (fun j=>hw (θ j.succ)),coordinateWord_length_constant _ (fun j=>hw (σ j.succ))])
-    have h1 := List.append_inj_right' h (by rw [coordinateWord_length_constant _ (fun j=>hw (θ j.succ)),coordinateWord_length_constant _ (fun j=>hw (σ j.succ))])
-    have ht := ih h1
-    funext j
-    exact Fin.cases (ha h0) (fun k=>congrFun ht k) j
-
-/-- A fixed family of all binary gadget choices, at exact requested length.
- Its common-zero member is uniformly close in m-sample TV to every member.
- This suffices for every fixed positive worst-case success probability. -/
-theorem ConcreteLevelPair.restored_family {r q N Δ d ω F : ℝ} {hr : r∈Set.Icc 0 1}
-    {i k : ℕ} (U : ConcreteLevelPair r hr N Δ d ω F i k)
-    (hr1 : r<1) (hq : q∈Set.Icc 0 1) (hrq : r≤q) (hk : 1≤k)
-    (g n : ℕ) (hN : (g:ℝ)*N≤n) :
-    ∃W : (Fin g→Bool)→Word n,Function.Injective W ∧ ∀θ (m : ℕ),
-      ((completeTraceLaw q hq (W θ)).product m).tv
-        ((completeTraceLaw q hq (W (fun _=>false))).product m) ≤
-          Real.sqrt ((m:ℝ)*g*N^(-d))+2*m*g*U.error := by
-  have hlen : g*U.left.length≤n := by
-    exact_mod_cast (mul_le_mul_of_nonneg_left U.length_upper (Nat.cast_nonneg g)).trans hN
-  let atom := fun (τ : Bool)=>if τ then U.right else U.left
-  have hw (τ : Bool) : (atom τ).length=U.left.length := by cases τ; rfl; exact U.equalLength.symm
-  have ha : Function.Injective atom := by
-    intro τ σ hh
-    cases τ <;> cases σ
-    · rfl
-    · exact False.elim (U.different hh)
-    · exact False.elim (U.different hh.symm)
-    · rfl
-  let tail := List.replicate (n-g*U.left.length) false
-  let w := fun θ : Fin g→Bool=>coordinateWord (fun j=>atom (θ j))++tail
-  have hl (θ : Fin g→Bool) : (w θ).length=n := by
-    simp only [w,tail,List.length_append,List.length_replicate,coordinateWord_length_constant _ (fun j=>hw (θ j))]
-    omega
-  let W := fun θ=>wordOfLength (w θ) (hl θ)
-  refine ⟨W,?_,?_⟩
-  · intro θ σ hh
-    have hh' := congrArg List.ofFn hh
-    simp only [W,ofFn_wordOfLength] at hh'
-    exact coordinateWord_injective_equal atom hw ha (List.append_cancel_right hh')
-  · intro θ m
-    let C := ((U.oracle.hellinger hk).mix hr θ (fun _ : Fin g=>false)).suffix hr tail
-    have hV : (∑j : Fin g,if θ j=false then (0:ℝ) else 1) ≤ g := by
-      calc
-        _ ≤ ∑_j : Fin g,(1:ℝ) := Finset.sum_le_sum (fun j _=>by split_ifs <;> norm_num)
-        _ = g := by simp
-    have HH := C.degraded_sample_tv ⟨hr.1,hr1⟩ hq hrq m n (hl θ).le (hl (fun _=>false)).le
-    change ((listTraceLaw q hq (w θ) n _).product m).tv ((listTraceLaw q hq (w (fun _=>false)) n _).product m) ≤ _ at HH
-    rw [listTraceLaw_wordOfLength q hq _ (hl θ),listTraceLaw_wordOfLength q hq _ (hl (fun _=>false))] at HH
-    apply HH.trans
-    have hN0 : 0≤N := (Nat.cast_nonneg U.left.length).trans U.length_upper
-    have hmul : (m:ℝ)*((∑j : Fin g,if θ j=false then (0:ℝ) else 1)*N^(-d)) ≤ m*g*N^(-d) := by
-      simpa only [mul_assoc] using mul_le_mul_of_nonneg_left
-        (mul_le_mul_of_nonneg_right hV (Real.rpow_nonneg hN0 (-d))) (Nat.cast_nonneg m)
-    have hs := Real.sqrt_le_sqrt hmul
-    simpa only [mul_assoc] using add_le_add hs (le_refl (2*(m:ℝ)*(g*U.error)))
-
-end TraceReconstruction
-
-
-noncomputable section
-namespace TraceReconstruction
-open scoped Classical
-
-namespace FiniteLaw
-variable {α β : Type*} [Fintype α] [Fintype β]
-
-theorem product_map_expect (P : FiniteLaw α) (f : α→β) (m : ℕ)
-    (F : (Fin m→β)→ℝ) :
-    ((P.map f).product m).expect F=(P.product m).expect (fun z=>F (f∘z)) := by
-  induction m with
-  | zero =>
-    simp only [expect,product,Fintype.prod_empty,one_mul,Fintype.sum_unique]
-    congr 1
-    exact Subsingleton.elim _ _
-  | succ m ih =>
-    rw [product_succ_expect,expect_map,product_succ_expect]
-    apply congrArg P.expect
-    funext a
-    change ((P.map f).product m).expect (fun b=>F (Fin.cons (f a) b))=_
-    rw [ih]
-    congr 1
-    funext z
-    congr 1
-    funext j
-    exact Fin.cases rfl (fun _=>rfl) j
-
-end FiniteLaw
-
-theorem reconstructionSuccess_expect {n m : ℕ} (q : ℝ) (hq : q∈Set.Icc 0 1)
-    (A : (Fin m→Trace)→Word n→ℝ) (x : Word n) :
-    reconstructionSuccess q A x=((completeTraceLaw q hq x).product m).expect
-      (fun z=>A (fun j=>(z j).val) x) := by
-  rw [completeTraceLaw_mask,FiniteLaw.product_map_expect]
-  rfl
-
-theorem IsEstimator.le_one {n m : ℕ} {A : (Fin m→Trace)→Word n→ℝ}
-    (hA : IsEstimator A) (z : Fin m→Trace) (x : Word n) : A z x≤1 := by
-  rw [←hA.2 z]
-  exact Finset.single_le_sum (fun y _=>hA.1 z y) (Finset.mem_univ x)
-
-/-- Many-hypothesis testing using a common reference law. The decoder remains
- unrestricted, randomized and sees only complete traces. -/
-theorem family_testing_bound {n m g : ℕ} {q s ε : ℝ} (hq : q∈Set.Icc 0 1)
-    (W : (Fin g→Bool)→Word n) (hW : Function.Injective W)
-    (htv : ∀θ,((completeTraceLaw q hq (W θ)).product m).tv
-      ((completeTraceLaw q hq (W (fun _=>false))).product m)≤ε)
-    (hm : SufficientBudget n q s m) : s≤1/(2:ℝ)^g+ε := by
-  obtain ⟨A,hA,hs⟩ := hm
-  let P := (completeTraceLaw q hq (W (fun _=>false))).product m
-  have hθ (θ : Fin g→Bool) : s≤P.expect (fun z=>A (fun j=>(z j).val) (W θ))+ε := by
-    have hb := FiniteLaw.expectation_diff_le_tv
-      ((completeTraceLaw q hq (W θ)).product m) P
-      (fun z=>A (fun j=>(z j).val) (W θ))
-      (fun z=>⟨hA.1 _ _,hA.le_one _ _⟩)
-    have hh := hs (W θ)
-    rw [reconstructionSuccess_expect q hq] at hh
-    change ((completeTraceLaw q hq (W θ)).product m).expect (fun z=>A (fun j=>(z j).val) (W θ)) - P.expect _ ≤ _ at hb
-    linarith [htv θ]
-  have hsum : (∑θ : Fin g→Bool,P.expect (fun z=>A (fun j=>(z j).val) (W θ)))≤1 := by
-    unfold FiniteLaw.expect
-    rw [Finset.sum_comm]
-    calc
-      _ ≤ ∑z,P.mass z*1 := by
-        apply Finset.sum_le_sum
-        intro z _
-        rw [←Finset.mul_sum]
-        apply mul_le_mul_of_nonneg_left _ (P.nonneg z)
-        calc
-          _ = ∑x ∈ Finset.univ.image W,A (fun j=>(z j).val) x := by
-            rw [Finset.sum_image (fun a _ b _ hab=>hW hab)]
-          _ ≤ ∑x : Word n,A (fun j=>(z j).val) x := Finset.sum_le_sum_of_subset_of_nonneg
-            (Finset.subset_univ _) (fun x _ _=>hA.1 _ _)
-          _ = 1 := hA.2 _
-      _ = 1 := by simp [P.total]
-  have hh := Finset.sum_le_sum (s:=Finset.univ) (fun θ _=>hθ θ)
-  simp only [Finset.sum_add_distrib,Finset.sum_const,Finset.card_univ,
-    Fintype.card_fun,Fintype.card_bool,Fintype.card_fin,nsmul_eq_mul,Nat.cast_pow,Nat.cast_ofNat] at hh
-  have hp : 0<(2:ℝ)^g := by positivity
-  have he : (2:ℝ)^g*(1/(2:ℝ)^g)=1 := by field_simp
-  nlinarith only [hh,hsum,he,hp]
-
-end TraceReconstruction
-
-end
-end
-end
-
-end OAI
+import OAI.Probability.TraceReconstruction.Restoration
+
+namespace OAI
+
+/-!
+# Testing
+
+Multi-gadget family restoration and common-reference many-hypothesis statistics for every positive success level.
+-/
+
+open scoped ENNReal
+noncomputable section
+
+open Filter
+open scoped Topology
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+namespace FiniteLaw
+variable {α β : Type*} [Fintype α] [Fintype β]
+
+theorem hellingerSq_prod_same_right (P Q : FiniteLaw α) (R : FiniteLaw β) :
+    (P.prod R).hellingerSq (Q.prod R)=P.hellingerSq Q := by
+  unfold hellingerSq prod
+  rw [Fintype.sum_prod_type]
+  have he (a : α) (b : β) :
+      (Real.sqrt (P.mass a*R.mass b)-Real.sqrt (Q.mass a*R.mass b))^2=
+        (Real.sqrt (P.mass a)-Real.sqrt (Q.mass a))^2*R.mass b := by
+    rw [Real.sqrt_mul (P.nonneg a),Real.sqrt_mul (Q.nonneg a),←sub_mul,mul_pow,
+      Real.sq_sqrt (R.nonneg b)]
+  simp_rw [he,←Finset.mul_sum,R.total,mul_one]
+
+end FiniteLaw
+
+def HellingerTracePair.suffix {q E V : ℝ} {x y : List Bool}
+    (C : HellingerTracePair q x y E V) (hq : q∈Set.Icc 0 1) (w : List Bool) :
+    HellingerTracePair q (x++w) (y++w) E V := by
+  let := C.finiteOutcome
+  let R := listTraceLaw q hq w w.length le_rfl
+  exact {
+    Outcome := C.Outcome × BoundedTrace w.length
+    finiteOutcome := inferInstance
+    first := C.first.prod R
+    second := C.second.prod R
+    decode := fun z=>C.decode z.1++z.2.val
+    first_simulation := by
+      simpa only [add_zero] using C.first_simulation.append hq
+        (listTraceLaw_simulates q hq w w.length le_rfl).approximate
+    second_simulation := by
+      simpa only [add_zero] using C.second_simulation.append hq
+        (listTraceLaw_simulates q hq w w.length le_rfl).approximate
+    bound := by rw [FiniteLaw.hellingerSq_prod_same_right]; exact C.bound }
+
+theorem coordinateWord_length_constant {g S : ℕ} (w : Fin g→List Bool)
+    (hw : ∀j,(w j).length=S) : (coordinateWord w).length=g*S := by
+  induction g with
+  | zero => simp [coordinateWord]
+  | succ g ih =>
+    rw [coordinateWord_succ,List.length_append,hw,ih (fun j=>w j.succ) (fun j=>hw j.succ)]
+    simp [Nat.add_mul,Nat.add_comm]
+
+theorem coordinateWord_injective_equal {g S : ℕ} (atom : Bool→List Bool)
+    (hw : ∀τ,(atom τ).length=S) (ha : Function.Injective atom) :
+    Function.Injective (fun θ : Fin g→Bool=>coordinateWord (fun j=>atom (θ j))) := by
+  induction g with
+  | zero => intro θ σ _; exact Subsingleton.elim _ _
+  | succ g ih =>
+    intro θ σ h
+    change coordinateWord (fun j=>atom (θ j))=coordinateWord (fun j=>atom (σ j)) at h
+    rw [coordinateWord_succ,coordinateWord_succ] at h
+    have h0 := List.append_inj_left' h (by rw [coordinateWord_length_constant _ (fun j=>hw (θ j.succ)),coordinateWord_length_constant _ (fun j=>hw (σ j.succ))])
+    have h1 := List.append_inj_right' h (by rw [coordinateWord_length_constant _ (fun j=>hw (θ j.succ)),coordinateWord_length_constant _ (fun j=>hw (σ j.succ))])
+    have ht := ih h1
+    funext j
+    exact Fin.cases (ha h0) (fun k=>congrFun ht k) j
+
+/-- A fixed family of all binary gadget choices, at exact requested length.
+ Its common-zero member is uniformly close in m-sample TV to every member.
+ This suffices for every fixed positive worst-case success probability. -/
+theorem ConcreteLevelPair.restored_family {r q N Δ d ω F : ℝ} {hr : r∈Set.Icc 0 1}
+    {i k : ℕ} (U : ConcreteLevelPair r hr N Δ d ω F i k)
+    (hr1 : r<1) (hq : q∈Set.Icc 0 1) (hrq : r≤q) (hk : 1≤k)
+    (g n : ℕ) (hN : (g:ℝ)*N≤n) :
+    ∃W : (Fin g→Bool)→Word n,Function.Injective W ∧ ∀θ (m : ℕ),
+      ((completeTraceLaw q hq (W θ)).product m).tv
+        ((completeTraceLaw q hq (W (fun _=>false))).product m) ≤
+          Real.sqrt ((m:ℝ)*g*N^(-d))+2*m*g*U.error := by
+  have hlen : g*U.left.length≤n := by
+    exact_mod_cast (mul_le_mul_of_nonneg_left U.length_upper (Nat.cast_nonneg g)).trans hN
+  let atom := fun (τ : Bool)=>if τ then U.right else U.left
+  have hw (τ : Bool) : (atom τ).length=U.left.length := by cases τ; rfl; exact U.equalLength.symm
+  have ha : Function.Injective atom := by
+    intro τ σ hh
+    cases τ <;> cases σ
+    · rfl
+    · exact False.elim (U.different hh)
+    · exact False.elim (U.different hh.symm)
+    · rfl
+  let tail := List.replicate (n-g*U.left.length) false
+  let w := fun θ : Fin g→Bool=>coordinateWord (fun j=>atom (θ j))++tail
+  have hl (θ : Fin g→Bool) : (w θ).length=n := by
+    simp only [w,tail,List.length_append,List.length_replicate,coordinateWord_length_constant _ (fun j=>hw (θ j))]
+    omega
+  let W := fun θ=>wordOfLength (w θ) (hl θ)
+  refine ⟨W,?_,?_⟩
+  · intro θ σ hh
+    have hh' := congrArg List.ofFn hh
+    simp only [W,ofFn_wordOfLength] at hh'
+    exact coordinateWord_injective_equal atom hw ha (List.append_cancel_right hh')
+  · intro θ m
+    let C := ((U.oracle.hellinger hk).mix hr θ (fun _ : Fin g=>false)).suffix hr tail
+    have hV : (∑j : Fin g,if θ j=false then (0:ℝ) else 1) ≤ g := by
+      calc
+        _ ≤ ∑_j : Fin g,(1:ℝ) := Finset.sum_le_sum (fun j _=>by split_ifs <;> norm_num)
+        _ = g := by simp
+    have HH := C.degraded_sample_tv ⟨hr.1,hr1⟩ hq hrq m n (hl θ).le (hl (fun _=>false)).le
+    change ((listTraceLaw q hq (w θ) n _).product m).tv ((listTraceLaw q hq (w (fun _=>false)) n _).product m) ≤ _ at HH
+    rw [listTraceLaw_wordOfLength q hq _ (hl θ),listTraceLaw_wordOfLength q hq _ (hl (fun _=>false))] at HH
+    apply HH.trans
+    have hN0 : 0≤N := (Nat.cast_nonneg U.left.length).trans U.length_upper
+    have hmul : (m:ℝ)*((∑j : Fin g,if θ j=false then (0:ℝ) else 1)*N^(-d)) ≤ m*g*N^(-d) := by
+      simpa only [mul_assoc] using mul_le_mul_of_nonneg_left
+        (mul_le_mul_of_nonneg_right hV (Real.rpow_nonneg hN0 (-d))) (Nat.cast_nonneg m)
+    have hs := Real.sqrt_le_sqrt hmul
+    simpa only [mul_assoc] using add_le_add hs (le_refl (2*(m:ℝ)*(g*U.error)))
+
+end TraceReconstruction
+
+
+noncomputable section
+namespace TraceReconstruction
+open scoped Classical
+
+namespace FiniteLaw
+variable {α β : Type*} [Fintype α] [Fintype β]
+
+theorem product_map_expect (P : FiniteLaw α) (f : α→β) (m : ℕ)
+    (F : (Fin m→β)→ℝ) :
+    ((P.map f).product m).expect F=(P.product m).expect (fun z=>F (f∘z)) := by
+  induction m with
+  | zero =>
+    simp only [expect,product,Fintype.prod_empty,one_mul,Fintype.sum_unique]
+    congr 1
+    exact Subsingleton.elim _ _
+  | succ m ih =>
+    rw [product_succ_expect,expect_map,product_succ_expect]
+    apply congrArg P.expect
+    funext a
+    change ((P.map f).product m).expect (fun b=>F (Fin.cons (f a) b))=_
+    rw [ih]
+    congr 1
+    funext z
+    congr 1
+    funext j
+    exact Fin.cases rfl (fun _=>rfl) j
+
+end FiniteLaw
+
+theorem reconstructionSuccess_expect {n m : ℕ} (q : ℝ) (hq : q∈Set.Icc 0 1)
+    (A : (Fin m→Trace)→Word n→ℝ) (x : Word n) :
+    reconstructionSuccess q A x=((completeTraceLaw q hq x).product m).expect
+      (fun z=>A (fun j=>(z j).val) x) := by
+  rw [completeTraceLaw_mask,FiniteLaw.product_map_expect]
+  rfl
+
+theorem IsEstimator.le_one {n m : ℕ} {A : (Fin m→Trace)→Word n→ℝ}
+    (hA : IsEstimator A) (z : Fin m→Trace) (x : Word n) : A z x≤1 := by
+  rw [←hA.2 z]
+  exact Finset.single_le_sum (fun y _=>hA.1 z y) (Finset.mem_univ x)
+
+/-- Many-hypothesis testing using a common reference law. The decoder remains
+ unrestricted, randomized and sees only complete traces. -/
+theorem family_testing_bound {n m g : ℕ} {q s ε : ℝ} (hq : q∈Set.Icc 0 1)
+    (W : (Fin g→Bool)→Word n) (hW : Function.Injective W)
+    (htv : ∀θ,((completeTraceLaw q hq (W θ)).product m).tv
+      ((completeTraceLaw q hq (W (fun _=>false))).product m)≤ε)
+    (hm : SufficientBudget n q s m) : s≤1/(2:ℝ)^g+ε := by
+  obtain ⟨A,hA,hs⟩ := hm
+  let P := (completeTraceLaw q hq (W (fun _=>false))).product m
+  have hθ (θ : Fin g→Bool) : s≤P.expect (fun z=>A (fun j=>(z j).val) (W θ))+ε := by
+    have hb := FiniteLaw.expectation_diff_le_tv
+      ((completeTraceLaw q hq (W θ)).product m) P
+      (fun z=>A (fun j=>(z j).val) (W θ))
+      (fun z=>⟨hA.1 _ _,hA.le_one _ _⟩)
+    have hh := hs (W θ)
+    rw [reconstructionSuccess_expect q hq] at hh
+    change ((completeTraceLaw q hq (W θ)).product m).expect (fun z=>A (fun j=>(z j).val) (W θ)) - P.expect _ ≤ _ at hb
+    linarith [htv θ]
+  have hsum : (∑θ : Fin g→Bool,P.expect (fun z=>A (fun j=>(z j).val) (W θ)))≤1 := by
+    unfold FiniteLaw.expect
+    rw [Finset.sum_comm]
+    calc
+      _ ≤ ∑z,P.mass z*1 := by
+        apply Finset.sum_le_sum
+        intro z _
+        rw [←Finset.mul_sum]
+        apply mul_le_mul_of_nonneg_left _ (P.nonneg z)
+        calc
+          _ = ∑x ∈ Finset.univ.image W,A (fun j=>(z j).val) x := by
+            rw [Finset.sum_image (fun a _ b _ hab=>hW hab)]
+          _ ≤ ∑x : Word n,A (fun j=>(z j).val) x := Finset.sum_le_sum_of_subset_of_nonneg
+            (Finset.subset_univ _) (fun x _ _=>hA.1 _ _)
+          _ = 1 := hA.2 _
+      _ = 1 := by simp [P.total]
+  have hh := Finset.sum_le_sum (s:=Finset.univ) (fun θ _=>hθ θ)
+  simp only [Finset.sum_add_distrib,Finset.sum_const,Finset.card_univ,
+    Fintype.card_fun,Fintype.card_bool,Fintype.card_fin,nsmul_eq_mul,Nat.cast_pow,Nat.cast_ofNat] at hh
+  have hp : 0<(2:ℝ)^g := by positivity
+  have he : (2:ℝ)^g*(1/(2:ℝ)^g)=1 := by field_simp
+  nlinarith only [hh,hsum,he,hp]
+
+end TraceReconstruction
+
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Cayley/Folner.lean b/lean/OAI/Probability/UnimodularPercolation/Cayley/Folner.lean
index 90467876e261471505e9c15308e2c446e00f78d6..4879db9019a96ce72d2e9c28cffb0d5811f64daf 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Cayley/Folner.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Cayley/Folner.lean
@@ -1,133 +1,133 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Main
-import OAI.Probability.UnimodularPercolation.Cayley.Geometry
-
-namespace OAI
-
-/-! The group Følner criterion and nonamenable Cayley graphs. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u
-section GroupGeometry
-variable {Λ : Type u} [Group Λ]
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def rightTranslate (A : Finset Λ) (g : Λ) : Finset Λ :=
-  A.image (fun x => x * g)
-
-@[simp] lemma card_rightTranslate (A : Finset Λ) (g : Λ) :
-    (rightTranslate A g).card = A.card :=
-  Finset.card_image_of_injective _ (Equiv.mulRight g).injective
-
-@[simp] lemma rightTranslate_one (A : Finset Λ) : rightTranslate A 1 = A := by
-  simp [rightTranslate]
-
-lemma rightTranslate_mul (A : Finset Λ) (g h : Λ) :
-    rightTranslate (rightTranslate A g) h = rightTranslate A (g * h) := by
-  simp only [rightTranslate, Finset.image_image, Function.comp_def, mul_assoc]
-
-lemma rightTranslate_sdiff (A B : Finset Λ) (g : Λ) :
-    rightTranslate (A \ B) g = rightTranslate A g \ rightTranslate B g :=
-  Finset.image_sdiff _ _ (Equiv.mulRight g).injective
-
-noncomputable def rightDefect (A : Finset Λ) (g : Λ) : ℕ :=
-  (rightTranslate A g \ A).card
-
-omit [Group Λ] in
-lemma card_sdiff_triangle (A B C : Finset Λ) :
-    (A \ C).card ≤ (A \ B).card + (B \ C).card := by
-  apply (Finset.card_le_card (show A \ C ⊆ (A \ B) ∪ (B \ C) from ?_)).trans
-    (Finset.card_union_le _ _)
-  intro x hx
-  simp only [Finset.mem_sdiff, Finset.mem_union] at hx ⊢
-  by_cases hb : x ∈ B <;> tauto
-
-lemma rightDefect_mul_le (A : Finset Λ) (g h : Λ) :
-    rightDefect A (g * h) ≤ rightDefect A g + rightDefect A h := by
-  calc
-    rightDefect A (g * h) ≤
-        (rightTranslate A (g * h) \ rightTranslate A h).card + rightDefect A h :=
-      card_sdiff_triangle _ _ _
-    _ = rightDefect A g + rightDefect A h := by
-      rw [← rightTranslate_mul, ← rightTranslate_sdiff, card_rightTranslate]
-      rfl
-
-@[simp] lemma rightDefect_inv (A : Finset Λ) (g : Λ) :
-    rightDefect A g⁻¹ = rightDefect A g := by
-  calc
-    rightDefect A g⁻¹ = (rightTranslate (rightTranslate A g⁻¹ \ A) g).card :=
-      (card_rightTranslate _ _).symm
-    _ = (A \ rightTranslate A g).card := by
-      rw [rightTranslate_sdiff, rightTranslate_mul, inv_mul_cancel, rightTranslate_one]
-    _ = rightDefect A g := Finset.card_sdiff_comm (card_rightTranslate A g).symm
-
-lemma rightDefect_le_boundary (S : Finset Λ)
-    (hgen : Subgroup.closure (S : Set Λ) = ⊤) (g : Λ) :
-    ∃ n : ℕ, ∀ A : Finset Λ,
-      rightDefect A g ≤ n * (vertexBoundary (cayleyGraph S) A).ncard := by
-  have hg : g ∈ Subgroup.closure (S : Set Λ) := by rw [hgen]; trivial
-  induction hg using Subgroup.closure_induction with
-  | mem g hg =>
-    refine ⟨1, ?_⟩
-    intro A
-    rw [one_mul]
-    change (rightTranslate A g \ A).card ≤ _
-    rw [← Set.ncard_coe_finset]
-    apply Set.ncard_le_ncard ?_ (finite_vertexBoundary _ (cayley_locallyFinite S) A)
-    intro y hy
-    obtain ⟨hy, hnot⟩ := Finset.mem_sdiff.mp hy
-    obtain ⟨x, hx, rfl⟩ := Finset.mem_image.mp hy
-    exact ⟨hnot, x, hx, cayley_adj_of_mem S x g hg⟩
-  | one =>
-    exact ⟨0, fun A => by simp [rightDefect]⟩
-  | mul g h hg hh ig ih =>
-    obtain ⟨n, hn⟩ := ig
-    obtain ⟨m, hm⟩ := ih
-    refine ⟨n + m, fun A => ?_⟩
-    exact (rightDefect_mul_le A g h).trans (by simpa [add_mul] using Nat.add_le_add (hn A) (hm A))
-  | inv g hg ig =>
-    obtain ⟨n, hn⟩ := ig
-    exact ⟨n, fun A => by simpa only [rightDefect_inv] using hn A⟩
-
-lemma cayley_hV_pos (S : Finset Λ) (hgen : Subgroup.closure (S : Set Λ) = ⊤)
-    (hna : ¬ AmenableGroup Λ) : 0 < hV (cayleyGraph S) := by
-  by_contra hh
-  apply hna
-  intro K ε hε
-  choose n hn using rightDefect_le_boundary S hgen
-  let N : ℕ := (∑ g ∈ K, n g) + 1
-  have hN : (0 : ℝ) < N := by dsimp [N]; positivity
-  have hng (g : Λ) (hg : g ∈ K) : n g ≤ N :=
-    (Finset.single_le_sum (fun _ _ => Nat.zero_le _) hg).trans (Nat.le_add_right _ _)
-  let B : Set ℝ := {r | ∃ A : Finset Λ, A.Nonempty ∧
-    r = (vertexBoundary (cayleyGraph S) A).ncard / (A.card : ℝ)}
-  have hb : BddBelow B := by
-    refine ⟨0, ?_⟩
-    rintro r ⟨A, hA, rfl⟩
-    positivity
-  have hB : B.Nonempty :=
-    ⟨_, {1}, Finset.singleton_nonempty 1, rfl⟩
-  have hlt : sInf B < ε / N :=
-    (le_of_not_gt hh).trans_lt (div_pos hε hN)
-  obtain ⟨r, ⟨A, hA, rfl⟩, hr⟩ := (csInf_lt_iff hb hB).mp hlt
-  have hAc : (0 : ℝ) < A.card := Nat.cast_pos.mpr hA.card_pos
-  have hbound : (N : ℝ) * (vertexBoundary (cayleyGraph S) A).ncard < ε * A.card := by
-    have hh := (div_lt_iff₀ hAc).mp hr
-    have hh' := mul_lt_mul_of_pos_left hh hN
-    have heq : (N : ℝ) * (ε / N * A.card) = ε * A.card := by
-      field_simp
-    rwa [heq] at hh'
-  refine ⟨A, hA, ?_⟩
-  intro g hg
-  have hd : (rightDefect A g : ℝ) ≤ (N : ℝ) *
-      (vertexBoundary (cayleyGraph S) A).ncard := by
-    exact_mod_cast (hn g A).trans (Nat.mul_le_mul_right _ (hng g hg))
-  exact hd.trans_lt hbound
-
-end GroupGeometry
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Main
+import OAI.Probability.UnimodularPercolation.Cayley.Geometry
+
+namespace OAI
+
+/-! The group Følner criterion and nonamenable Cayley graphs. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u
+section GroupGeometry
+variable {Λ : Type u} [Group Λ]
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def rightTranslate (A : Finset Λ) (g : Λ) : Finset Λ :=
+  A.image (fun x => x * g)
+
+@[simp] lemma card_rightTranslate (A : Finset Λ) (g : Λ) :
+    (rightTranslate A g).card = A.card :=
+  Finset.card_image_of_injective _ (Equiv.mulRight g).injective
+
+@[simp] lemma rightTranslate_one (A : Finset Λ) : rightTranslate A 1 = A := by
+  simp [rightTranslate]
+
+lemma rightTranslate_mul (A : Finset Λ) (g h : Λ) :
+    rightTranslate (rightTranslate A g) h = rightTranslate A (g * h) := by
+  simp only [rightTranslate, Finset.image_image, Function.comp_def, mul_assoc]
+
+lemma rightTranslate_sdiff (A B : Finset Λ) (g : Λ) :
+    rightTranslate (A \ B) g = rightTranslate A g \ rightTranslate B g :=
+  Finset.image_sdiff _ _ (Equiv.mulRight g).injective
+
+noncomputable def rightDefect (A : Finset Λ) (g : Λ) : ℕ :=
+  (rightTranslate A g \ A).card
+
+omit [Group Λ] in
+lemma card_sdiff_triangle (A B C : Finset Λ) :
+    (A \ C).card ≤ (A \ B).card + (B \ C).card := by
+  apply (Finset.card_le_card (show A \ C ⊆ (A \ B) ∪ (B \ C) from ?_)).trans
+    (Finset.card_union_le _ _)
+  intro x hx
+  simp only [Finset.mem_sdiff, Finset.mem_union] at hx ⊢
+  by_cases hb : x ∈ B <;> tauto
+
+lemma rightDefect_mul_le (A : Finset Λ) (g h : Λ) :
+    rightDefect A (g * h) ≤ rightDefect A g + rightDefect A h := by
+  calc
+    rightDefect A (g * h) ≤
+        (rightTranslate A (g * h) \ rightTranslate A h).card + rightDefect A h :=
+      card_sdiff_triangle _ _ _
+    _ = rightDefect A g + rightDefect A h := by
+      rw [← rightTranslate_mul, ← rightTranslate_sdiff, card_rightTranslate]
+      rfl
+
+@[simp] lemma rightDefect_inv (A : Finset Λ) (g : Λ) :
+    rightDefect A g⁻¹ = rightDefect A g := by
+  calc
+    rightDefect A g⁻¹ = (rightTranslate (rightTranslate A g⁻¹ \ A) g).card :=
+      (card_rightTranslate _ _).symm
+    _ = (A \ rightTranslate A g).card := by
+      rw [rightTranslate_sdiff, rightTranslate_mul, inv_mul_cancel, rightTranslate_one]
+    _ = rightDefect A g := Finset.card_sdiff_comm (card_rightTranslate A g).symm
+
+lemma rightDefect_le_boundary (S : Finset Λ)
+    (hgen : Subgroup.closure (S : Set Λ) = ⊤) (g : Λ) :
+    ∃ n : ℕ, ∀ A : Finset Λ,
+      rightDefect A g ≤ n * (vertexBoundary (cayleyGraph S) A).ncard := by
+  have hg : g ∈ Subgroup.closure (S : Set Λ) := by rw [hgen]; trivial
+  induction hg using Subgroup.closure_induction with
+  | mem g hg =>
+    refine ⟨1, ?_⟩
+    intro A
+    rw [one_mul]
+    change (rightTranslate A g \ A).card ≤ _
+    rw [← Set.ncard_coe_finset]
+    apply Set.ncard_le_ncard ?_ (finite_vertexBoundary _ (cayley_locallyFinite S) A)
+    intro y hy
+    obtain ⟨hy, hnot⟩ := Finset.mem_sdiff.mp hy
+    obtain ⟨x, hx, rfl⟩ := Finset.mem_image.mp hy
+    exact ⟨hnot, x, hx, cayley_adj_of_mem S x g hg⟩
+  | one =>
+    exact ⟨0, fun A => by simp [rightDefect]⟩
+  | mul g h hg hh ig ih =>
+    obtain ⟨n, hn⟩ := ig
+    obtain ⟨m, hm⟩ := ih
+    refine ⟨n + m, fun A => ?_⟩
+    exact (rightDefect_mul_le A g h).trans (by simpa [add_mul] using Nat.add_le_add (hn A) (hm A))
+  | inv g hg ig =>
+    obtain ⟨n, hn⟩ := ig
+    exact ⟨n, fun A => by simpa only [rightDefect_inv] using hn A⟩
+
+lemma cayley_hV_pos (S : Finset Λ) (hgen : Subgroup.closure (S : Set Λ) = ⊤)
+    (hna : ¬ AmenableGroup Λ) : 0 < hV (cayleyGraph S) := by
+  by_contra hh
+  apply hna
+  intro K ε hε
+  choose n hn using rightDefect_le_boundary S hgen
+  let N : ℕ := (∑ g ∈ K, n g) + 1
+  have hN : (0 : ℝ) < N := by dsimp [N]; positivity
+  have hng (g : Λ) (hg : g ∈ K) : n g ≤ N :=
+    (Finset.single_le_sum (fun _ _ => Nat.zero_le _) hg).trans (Nat.le_add_right _ _)
+  let B : Set ℝ := {r | ∃ A : Finset Λ, A.Nonempty ∧
+    r = (vertexBoundary (cayleyGraph S) A).ncard / (A.card : ℝ)}
+  have hb : BddBelow B := by
+    refine ⟨0, ?_⟩
+    rintro r ⟨A, hA, rfl⟩
+    positivity
+  have hB : B.Nonempty :=
+    ⟨_, {1}, Finset.singleton_nonempty 1, rfl⟩
+  have hlt : sInf B < ε / N :=
+    (le_of_not_gt hh).trans_lt (div_pos hε hN)
+  obtain ⟨r, ⟨A, hA, rfl⟩, hr⟩ := (csInf_lt_iff hb hB).mp hlt
+  have hAc : (0 : ℝ) < A.card := Nat.cast_pos.mpr hA.card_pos
+  have hbound : (N : ℝ) * (vertexBoundary (cayleyGraph S) A).ncard < ε * A.card := by
+    have hh := (div_lt_iff₀ hAc).mp hr
+    have hh' := mul_lt_mul_of_pos_left hh hN
+    have heq : (N : ℝ) * (ε / N * A.card) = ε * A.card := by
+      field_simp
+    rwa [heq] at hh'
+  refine ⟨A, hA, ?_⟩
+  intro g hg
+  have hd : (rightDefect A g : ℝ) ≤ (N : ℝ) *
+      (vertexBoundary (cayleyGraph S) A).ncard := by
+    exact_mod_cast (hn g A).trans (Nat.mul_le_mul_right _ (hng g hg))
+  exact hd.trans_lt hbound
+
+end GroupGeometry
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Cayley/Geometry.lean b/lean/OAI/Probability/UnimodularPercolation/Cayley/Geometry.lean
index dad403e96aaef27ac9f5a0de541fa6c0a306e93f..34bc8b091776c9339a99d9caadcd5ed2babf3e23 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Cayley/Geometry.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Cayley/Geometry.lean
@@ -1,156 +1,156 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Main
-
-namespace OAI
-
-/-! Cayley graphs, translation symmetries and root transport. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-
-noncomputable def AmenableGroup (Λ : Type u) [Group Λ] : Prop := by
-  classical
-  exact ∀ K : Finset Λ, ∀ ε : ℝ, 0 < ε → ∃ A : Finset Λ, A.Nonempty ∧
-    ∀ g ∈ K, (((A.image (fun x => x * g)) \ A).card : ℝ) < ε * A.card
-
-def CayleyEdge {Λ : Type u} [Group Λ] (S : Finset Λ) :=
-  {e : Sym2 Λ // ∃ x : Λ, ∃ s ∈ S, e = s(x, x * s)}
-
-def cayleyGraph {Λ : Type u} [Group Λ] (S : Finset Λ) :
-    BondGraph Λ (CayleyEdge S) := Subtype.val
-
-section GroupGeometry
-variable {Λ : Type u} [Group Λ]
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-lemma amenableGroup_of_finite [Finite Λ] : AmenableGroup Λ := by
-  classical
-  let := Fintype.ofFinite Λ
-  intro K ε hε
-  refine ⟨Finset.univ, Finset.univ_nonempty, ?_⟩
-  intro g hg
-  have he : (Finset.univ.image (fun x : Λ => x * g)) \ Finset.univ = ∅ :=
-    Finset.sdiff_eq_empty_iff_subset.mpr (Finset.subset_univ _)
-  rw [he, Finset.card_empty, Nat.cast_zero, Finset.card_univ]
-  exact mul_pos hε (Nat.cast_pos.mpr (Fintype.card_pos))
-
-lemma infinite_of_nonamenable (h : ¬ AmenableGroup Λ) : Infinite Λ := by
-  by_contra hi
-  let : Finite Λ := not_infinite_iff_finite.mp hi
-  exact h amenableGroup_of_finite
-
-lemma cayleyGraph_injective (S : Finset Λ) : Function.Injective (cayleyGraph S) :=
-  Subtype.val_injective
-
-lemma cayley_adj_of_mem (S : Finset Λ) (x s : Λ) (hs : s ∈ S) :
-    Adj (cayleyGraph S) x (x * s) :=
-  ⟨⟨s(x, x * s), x, s, hs, rfl⟩, rfl⟩
-
-lemma cayley_adj_iff (S : Finset Λ) (hS : ∀ s ∈ S, s⁻¹ ∈ S) (x y : Λ) :
-    Adj (cayleyGraph S) x y ↔ ∃ s ∈ S, x * s = y := by
-  constructor
-  · rintro ⟨⟨e, z, s, hs, rfl⟩, he⟩
-    change s(z, z * s) = s(x, y) at he
-    rcases Sym2.eq_iff.mp he with ⟨hz, hy⟩ | ⟨hz, hx⟩
-    ·
-      exact ⟨s, hs, hz ▸ hy⟩
-    · refine ⟨s⁻¹, hS s hs, ?_⟩
-      rw [← hx, mul_inv_cancel_right, hz]
-  · rintro ⟨s, hs, rfl⟩
-    exact cayley_adj_of_mem S x s hs
-
-lemma cayley_loopless (S : Finset Λ) (hS : ∀ s ∈ S, s⁻¹ ∈ S) (h1 : 1 ∉ S)
-    (x : Λ) : ¬ Adj (cayleyGraph S) x x := by
-  rintro hx
-  obtain ⟨s, hs, he⟩ := (cayley_adj_iff S hS x x).mp hx
-  have : s = 1 := mul_left_cancel (by simpa using he : x * s = x * 1)
-  exact h1 (this ▸ hs)
-
-lemma cayley_locallyFinite (S : Finset Λ) : LocallyFinite (cayleyGraph S) := by
-  intro x
-  let F : Set (Sym2 Λ) :=
-    ((fun s => s(x, x * s)) '' (S : Set Λ)) ∪
-    ((fun s => s(x * s⁻¹, x)) '' (S : Set Λ))
-  have hF : F.Finite :=
-    (S.finite_toSet.image _).union (S.finite_toSet.image _)
-  apply (hF.preimage (cayleyGraph_injective S).injOn).subset
-  rintro ⟨e, z, s, hs, rfl⟩ hx
-  change x ∈ s(z, z * s) at hx
-  rcases Sym2.mem_iff.mp hx with hx | hx
-  · left
-    exact ⟨s, hs, by simp [hx, cayleyGraph]⟩
-  · right
-    refine ⟨s, hs, ?_⟩
-    simp [hx, cayleyGraph]
-
-noncomputable def cayleyEdgeTranslation (S : Finset Λ) (a : Λ) : CayleyEdge S ≃ CayleyEdge S where
-  toFun e := ⟨e.val.map (fun x => a * x), by
-    obtain ⟨e, x, s, hs, rfl⟩ := e
-    exact ⟨a * x, s, hs, by simp [mul_assoc]⟩⟩
-  invFun e := ⟨e.val.map (fun x => a⁻¹ * x), by
-    obtain ⟨e, x, s, hs, rfl⟩ := e
-    exact ⟨a⁻¹ * x, s, hs, by simp [mul_assoc]⟩⟩
-  left_inv e := by
-    apply Subtype.ext
-    simp [Sym2.map_map]
-  right_inv e := by
-    apply Subtype.ext
-    simp [Sym2.map_map]
-
-lemma cayley_translation_isAut (S : Finset Λ) (a : Λ) :
-    IsAut (cayleyGraph S) (Equiv.mulLeft a) (cayleyEdgeTranslation S a) := by
-  intro e
-  rfl
-
-lemma cayley_translation_mem (S : Finset Λ) (a : Λ) :
-    Equiv.mulLeft a ∈ vertexAut (cayleyGraph S) :=
-  ⟨cayleyEdgeTranslation S a, cayley_translation_isAut S a⟩
-
-lemma cayley_quasiTransitive (S : Finset Λ) : QuasiTransitive (cayleyGraph S) := by
-  refine ⟨{1}, ?_⟩
-  intro x
-  exact ⟨1, Finset.mem_singleton_self _, Equiv.mulLeft x, cayleyEdgeTranslation S x,
-    cayley_translation_isAut S x, mul_one x⟩
-
-lemma cayley_connected (S : Finset Λ) (hgen : Subgroup.closure (S : Set Λ) = ⊤) :
-    Connected (cayleyGraph S) := by
-  have paths (g : Λ) : ∀ x : Λ,
-      Relation.ReflTransGen (Adj (cayleyGraph S)) x (x * g) := by
-    have hg : g ∈ Subgroup.closure (S : Set Λ) := by rw [hgen]; trivial
-    induction hg using Subgroup.closure_induction with
-    | mem g hg => exact fun x => .single (cayley_adj_of_mem S x g hg)
-    | one => intro x; simpa using (Relation.ReflTransGen.refl (a := x))
-    | mul a b ha hb ia ib =>
-      intro x
-      simpa [mul_assoc] using (ia x).trans (ib (x * a))
-    | inv a ha ia =>
-      intro x
-      have h := ia (x * a⁻¹)
-      rw [inv_mul_cancel_right] at h
-      let : Std.Symm (Adj (cayleyGraph S)) := ⟨fun _ _ => adj_symm (cayleyGraph S)⟩
-      exact symm h
-  intro x y
-  simpa using paths (x⁻¹ * y) x
-
-lemma cayley_rootTransport (S : Finset Λ) :
-    RootTransport (fun _ : Unit => (1 : Λ)) (fun _ => (1 : ℝ))
-      (vertexAut (cayleyGraph S)) := by
-  intro m hm hr hc
-  simp only [Finset.univ_unique, Finset.sum_singleton, Complex.ofReal_one, one_mul]
-  calc
-    (∑' y, m 1 y) = ∑' y, m y⁻¹ 1 := by
-      apply tsum_congr
-      intro y
-      simpa using (hm (Equiv.mulLeft y⁻¹) (cayley_translation_mem S y⁻¹) 1 y).symm
-    _ = ∑' x, m x 1 := (Equiv.inv Λ).tsum_eq (fun x => m x 1)
-
-end GroupGeometry
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Main
+
+namespace OAI
+
+/-! Cayley graphs, translation symmetries and root transport. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+
+noncomputable def AmenableGroup (Λ : Type u) [Group Λ] : Prop := by
+  classical
+  exact ∀ K : Finset Λ, ∀ ε : ℝ, 0 < ε → ∃ A : Finset Λ, A.Nonempty ∧
+    ∀ g ∈ K, (((A.image (fun x => x * g)) \ A).card : ℝ) < ε * A.card
+
+def CayleyEdge {Λ : Type u} [Group Λ] (S : Finset Λ) :=
+  {e : Sym2 Λ // ∃ x : Λ, ∃ s ∈ S, e = s(x, x * s)}
+
+def cayleyGraph {Λ : Type u} [Group Λ] (S : Finset Λ) :
+    BondGraph Λ (CayleyEdge S) := Subtype.val
+
+section GroupGeometry
+variable {Λ : Type u} [Group Λ]
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+lemma amenableGroup_of_finite [Finite Λ] : AmenableGroup Λ := by
+  classical
+  let := Fintype.ofFinite Λ
+  intro K ε hε
+  refine ⟨Finset.univ, Finset.univ_nonempty, ?_⟩
+  intro g hg
+  have he : (Finset.univ.image (fun x : Λ => x * g)) \ Finset.univ = ∅ :=
+    Finset.sdiff_eq_empty_iff_subset.mpr (Finset.subset_univ _)
+  rw [he, Finset.card_empty, Nat.cast_zero, Finset.card_univ]
+  exact mul_pos hε (Nat.cast_pos.mpr (Fintype.card_pos))
+
+lemma infinite_of_nonamenable (h : ¬ AmenableGroup Λ) : Infinite Λ := by
+  by_contra hi
+  let : Finite Λ := not_infinite_iff_finite.mp hi
+  exact h amenableGroup_of_finite
+
+lemma cayleyGraph_injective (S : Finset Λ) : Function.Injective (cayleyGraph S) :=
+  Subtype.val_injective
+
+lemma cayley_adj_of_mem (S : Finset Λ) (x s : Λ) (hs : s ∈ S) :
+    Adj (cayleyGraph S) x (x * s) :=
+  ⟨⟨s(x, x * s), x, s, hs, rfl⟩, rfl⟩
+
+lemma cayley_adj_iff (S : Finset Λ) (hS : ∀ s ∈ S, s⁻¹ ∈ S) (x y : Λ) :
+    Adj (cayleyGraph S) x y ↔ ∃ s ∈ S, x * s = y := by
+  constructor
+  · rintro ⟨⟨e, z, s, hs, rfl⟩, he⟩
+    change s(z, z * s) = s(x, y) at he
+    rcases Sym2.eq_iff.mp he with ⟨hz, hy⟩ | ⟨hz, hx⟩
+    ·
+      exact ⟨s, hs, hz ▸ hy⟩
+    · refine ⟨s⁻¹, hS s hs, ?_⟩
+      rw [← hx, mul_inv_cancel_right, hz]
+  · rintro ⟨s, hs, rfl⟩
+    exact cayley_adj_of_mem S x s hs
+
+lemma cayley_loopless (S : Finset Λ) (hS : ∀ s ∈ S, s⁻¹ ∈ S) (h1 : 1 ∉ S)
+    (x : Λ) : ¬ Adj (cayleyGraph S) x x := by
+  rintro hx
+  obtain ⟨s, hs, he⟩ := (cayley_adj_iff S hS x x).mp hx
+  have : s = 1 := mul_left_cancel (by simpa using he : x * s = x * 1)
+  exact h1 (this ▸ hs)
+
+lemma cayley_locallyFinite (S : Finset Λ) : LocallyFinite (cayleyGraph S) := by
+  intro x
+  let F : Set (Sym2 Λ) :=
+    ((fun s => s(x, x * s)) '' (S : Set Λ)) ∪
+    ((fun s => s(x * s⁻¹, x)) '' (S : Set Λ))
+  have hF : F.Finite :=
+    (S.finite_toSet.image _).union (S.finite_toSet.image _)
+  apply (hF.preimage (cayleyGraph_injective S).injOn).subset
+  rintro ⟨e, z, s, hs, rfl⟩ hx
+  change x ∈ s(z, z * s) at hx
+  rcases Sym2.mem_iff.mp hx with hx | hx
+  · left
+    exact ⟨s, hs, by simp [hx, cayleyGraph]⟩
+  · right
+    refine ⟨s, hs, ?_⟩
+    simp [hx, cayleyGraph]
+
+noncomputable def cayleyEdgeTranslation (S : Finset Λ) (a : Λ) : CayleyEdge S ≃ CayleyEdge S where
+  toFun e := ⟨e.val.map (fun x => a * x), by
+    obtain ⟨e, x, s, hs, rfl⟩ := e
+    exact ⟨a * x, s, hs, by simp [mul_assoc]⟩⟩
+  invFun e := ⟨e.val.map (fun x => a⁻¹ * x), by
+    obtain ⟨e, x, s, hs, rfl⟩ := e
+    exact ⟨a⁻¹ * x, s, hs, by simp [mul_assoc]⟩⟩
+  left_inv e := by
+    apply Subtype.ext
+    simp [Sym2.map_map]
+  right_inv e := by
+    apply Subtype.ext
+    simp [Sym2.map_map]
+
+lemma cayley_translation_isAut (S : Finset Λ) (a : Λ) :
+    IsAut (cayleyGraph S) (Equiv.mulLeft a) (cayleyEdgeTranslation S a) := by
+  intro e
+  rfl
+
+lemma cayley_translation_mem (S : Finset Λ) (a : Λ) :
+    Equiv.mulLeft a ∈ vertexAut (cayleyGraph S) :=
+  ⟨cayleyEdgeTranslation S a, cayley_translation_isAut S a⟩
+
+lemma cayley_quasiTransitive (S : Finset Λ) : QuasiTransitive (cayleyGraph S) := by
+  refine ⟨{1}, ?_⟩
+  intro x
+  exact ⟨1, Finset.mem_singleton_self _, Equiv.mulLeft x, cayleyEdgeTranslation S x,
+    cayley_translation_isAut S x, mul_one x⟩
+
+lemma cayley_connected (S : Finset Λ) (hgen : Subgroup.closure (S : Set Λ) = ⊤) :
+    Connected (cayleyGraph S) := by
+  have paths (g : Λ) : ∀ x : Λ,
+      Relation.ReflTransGen (Adj (cayleyGraph S)) x (x * g) := by
+    have hg : g ∈ Subgroup.closure (S : Set Λ) := by rw [hgen]; trivial
+    induction hg using Subgroup.closure_induction with
+    | mem g hg => exact fun x => .single (cayley_adj_of_mem S x g hg)
+    | one => intro x; simpa using (Relation.ReflTransGen.refl (a := x))
+    | mul a b ha hb ia ib =>
+      intro x
+      simpa [mul_assoc] using (ia x).trans (ib (x * a))
+    | inv a ha ia =>
+      intro x
+      have h := ia (x * a⁻¹)
+      rw [inv_mul_cancel_right] at h
+      let : Std.Symm (Adj (cayleyGraph S)) := ⟨fun _ _ => adj_symm (cayleyGraph S)⟩
+      exact symm h
+  intro x y
+  simpa using paths (x⁻¹ * y) x
+
+lemma cayley_rootTransport (S : Finset Λ) :
+    RootTransport (fun _ : Unit => (1 : Λ)) (fun _ => (1 : ℝ))
+      (vertexAut (cayleyGraph S)) := by
+  intro m hm hr hc
+  simp only [Finset.univ_unique, Finset.sum_singleton, Complex.ofReal_one, one_mul]
+  calc
+    (∑' y, m 1 y) = ∑' y, m y⁻¹ 1 := by
+      apply tsum_congr
+      intro y
+      simpa using (hm (Equiv.mulLeft y⁻¹) (cayley_translation_mem S y⁻¹) 1 y).symm
+    _ = ∑' x, m x 1 := (Equiv.inv Λ).tsum_eq (fun x => m x 1)
+
+end GroupGeometry
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Cayley/Nonuniqueness.lean b/lean/OAI/Probability/UnimodularPercolation/Cayley/Nonuniqueness.lean
index 1a0c7317e7946f93ffc3a5b58d0a237ed7ec4a98..99c371f779011b7cb479618654a59faa23703416 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Cayley/Nonuniqueness.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Cayley/Nonuniqueness.lean
@@ -1,64 +1,64 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Criticality.FullFiniteness
-import OAI.Probability.UnimodularPercolation.Cayley.Folner
-import OAI.Probability.UnimodularPercolation.Criticality.Phase
-
-namespace OAI
-
-/-! Nonuniqueness of percolation for arbitrary finite symmetric generating sets. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-namespace Percolation
-universe u
-
-theorem arbitrary_cayley {Λ : Type u} [Group Λ] [Group.FG Λ]
-    (hna : ¬ AmenableGroup Λ) (S : Finset Λ)
-    (hsym : ∀ s ∈ S, s⁻¹ ∈ S) (hgen : Subgroup.closure (S : Set Λ) = ⊤)
-    (h1 : 1 ∉ S) :
-    OperatorBounded (cayleyGraph S) (pc (cayleyGraph S)) ∧
-    pc (cayleyGraph S) < ptwo (cayleyGraph S) ∧
-    ptwo (cayleyGraph S) ≤ pu (cayleyGraph S) ∧
-    (∃ p₁ p₂ : unitInterval, pc (cayleyGraph S) < p₁ ∧ p₁ < p₂ ∧ p₂ < 1 ∧
-      ∀ᵐ labels ∂(labelLaw : Measure (CayleyEdge S → unitInterval)),
-        ∀ p ∈ Set.Icc p₁ p₂, (infiniteClusters (cayleyGraph S) (openEdges labels p)).Infinite) ∧
-    (∀ p : unitInterval, pc (cayleyGraph S) < p → p < pu (cayleyGraph S) →
-      ∀ᵐ ω ∂law p, (infiniteClusters (cayleyGraph S) ω).Infinite) := by
-  classical
-  let : Infinite Λ := infinite_of_nonamenable hna
-  let G := cayleyGraph S
-  have hconn : Connected G := cayley_connected S hgen
-  have hloc : LocallyFinite G := cayley_locallyFinite S
-  have hqt : QuasiTransitive G := cayley_quasiTransitive S
-  have hh : 0 < hV G := cayley_hV_pos S hgen hna
-  have hsimple : Function.Injective G := cayleyGraph_injective S
-  have hloop : ∀ x, ¬ Adj G x x := cayley_loopless S hsym h1
-  let : Countable Λ := countable_vertices hloc hconn
-  let : Countable (CayleyEdge S) := countable_edges hloc
-  let root : Unit → Λ := fun _ => 1
-  let weight : Unit → ℝ := fun _ => 1
-  have hw : ∀ i, 0 < weight i := fun _ => zero_lt_one
-  have hw1 : ∑ i, weight i = 1 := by simp [weight]
-  have htr : RootTransport root weight (vertexAut G) := cayley_rootTransport S
-  have hcov : ∀ x, ∃ i, ∃ a ∈ vertexAut G, a (root i) = x := by
-    intro x
-    exact ⟨(), Equiv.mulLeft x, cayley_translation_mem S x, mul_one x⟩
-  have hfinite := critical_finite_ae_of_transport G hconn hloc hqt hh root weight htr hw hw1 hcov
-  have hb : OperatorBounded G (pc G) := by
-    apply critical_bounded_of_transport_finiteness G hsimple hconn hloc hqt hh
-      root weight htr hcov hw hw1
-    intro i
-    filter_upwards [hfinite] with ω hω
-    exact hω (root i)
-  have hgap := strict_gap_of_critical_bounded G hloc hqt
-    (pc_lt_one_of_nonamenable G hconn hloc hqt hh) hb
-  have hm := mainConclusion_of_strict_gap G hconn hloc hqt hgap
-  exact ⟨hb, hgap, hm.2.2.2.1, hm.2.2.2.2,
-    fun p hc hu => infinitely_many_of_pc_lt_lt_pu G hconn hloc hqt p hc hu⟩
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Criticality.FullFiniteness
+import OAI.Probability.UnimodularPercolation.Cayley.Folner
+import OAI.Probability.UnimodularPercolation.Criticality.Phase
+
+namespace OAI
+
+/-! Nonuniqueness of percolation for arbitrary finite symmetric generating sets. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+namespace Percolation
+universe u
+
+theorem arbitrary_cayley {Λ : Type u} [Group Λ] [Group.FG Λ]
+    (hna : ¬ AmenableGroup Λ) (S : Finset Λ)
+    (hsym : ∀ s ∈ S, s⁻¹ ∈ S) (hgen : Subgroup.closure (S : Set Λ) = ⊤)
+    (h1 : 1 ∉ S) :
+    OperatorBounded (cayleyGraph S) (pc (cayleyGraph S)) ∧
+    pc (cayleyGraph S) < ptwo (cayleyGraph S) ∧
+    ptwo (cayleyGraph S) ≤ pu (cayleyGraph S) ∧
+    (∃ p₁ p₂ : unitInterval, pc (cayleyGraph S) < p₁ ∧ p₁ < p₂ ∧ p₂ < 1 ∧
+      ∀ᵐ labels ∂(labelLaw : Measure (CayleyEdge S → unitInterval)),
+        ∀ p ∈ Set.Icc p₁ p₂, (infiniteClusters (cayleyGraph S) (openEdges labels p)).Infinite) ∧
+    (∀ p : unitInterval, pc (cayleyGraph S) < p → p < pu (cayleyGraph S) →
+      ∀ᵐ ω ∂law p, (infiniteClusters (cayleyGraph S) ω).Infinite) := by
+  classical
+  let : Infinite Λ := infinite_of_nonamenable hna
+  let G := cayleyGraph S
+  have hconn : Connected G := cayley_connected S hgen
+  have hloc : LocallyFinite G := cayley_locallyFinite S
+  have hqt : QuasiTransitive G := cayley_quasiTransitive S
+  have hh : 0 < hV G := cayley_hV_pos S hgen hna
+  have hsimple : Function.Injective G := cayleyGraph_injective S
+  have hloop : ∀ x, ¬ Adj G x x := cayley_loopless S hsym h1
+  let : Countable Λ := countable_vertices hloc hconn
+  let : Countable (CayleyEdge S) := countable_edges hloc
+  let root : Unit → Λ := fun _ => 1
+  let weight : Unit → ℝ := fun _ => 1
+  have hw : ∀ i, 0 < weight i := fun _ => zero_lt_one
+  have hw1 : ∑ i, weight i = 1 := by simp [weight]
+  have htr : RootTransport root weight (vertexAut G) := cayley_rootTransport S
+  have hcov : ∀ x, ∃ i, ∃ a ∈ vertexAut G, a (root i) = x := by
+    intro x
+    exact ⟨(), Equiv.mulLeft x, cayley_translation_mem S x, mul_one x⟩
+  have hfinite := critical_finite_ae_of_transport G hconn hloc hqt hh root weight htr hw hw1 hcov
+  have hb : OperatorBounded G (pc G) := by
+    apply critical_bounded_of_transport_finiteness G hsimple hconn hloc hqt hh
+      root weight htr hcov hw hw1
+    intro i
+    filter_upwards [hfinite] with ω hω
+    exact hω (root i)
+  have hgap := strict_gap_of_critical_bounded G hloc hqt
+    (pc_lt_one_of_nonamenable G hconn hloc hqt hh) hb
+  have hm := mainConclusion_of_strict_gap G hconn hloc hqt hgap
+  exact ⟨hb, hgap, hm.2.2.2.1, hm.2.2.2.2,
+    fun p hc hu => infinitely_many_of_pc_lt_lt_pu G hconn hloc hqt p hc hu⟩
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/CayleyMain.lean b/lean/OAI/Probability/UnimodularPercolation/CayleyMain.lean
index 137748b2f13d6a4cd12291a9af5bbe91a6f5b124..34cfe7ea4cb1088aee1a90c9b89f5bcb20663512 100644
--- a/lean/OAI/Probability/UnimodularPercolation/CayleyMain.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/CayleyMain.lean
@@ -1 +1 @@
-import OAI.Probability.UnimodularPercolation.Cayley.Nonuniqueness
+import OAI.Probability.UnimodularPercolation.Cayley.Nonuniqueness
diff --git a/lean/OAI/Probability/UnimodularPercolation/Clusters/ExteriorMoments.lean b/lean/OAI/Probability/UnimodularPercolation/Clusters/ExteriorMoments.lean
index 271902e59796ca035beb1ba29abb8aec40693766..cf1f2007e7ca4344ca4eefd5e11b813f4bc87ca2 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Clusters/ExteriorMoments.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Clusters/ExteriorMoments.lean
@@ -1,1207 +1,1207 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Clusters.KernelMoments
-
-namespace OAI
-
-/-! Exterior Moments for bond percolation. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section BinDiagonalLoss
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def binDiagonalKernel (G : BondGraph V E) (p : unitInterval)
-    (o : V) (n : ℕ) (K : V → V → ℝ) (x y : V) : ℝ :=
-  ∫ ω, binDiagonalMass G o n K ω x y ∂law p
-
-theorem binDiagonalKernel_covariant (G : BondGraph V E) (hloc : LocallyFinite G)
-    (p : unitInterval) (o : V) (n : ℕ) (K : V → V → ℝ)
-    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b)
-    (hK : ∀ x y, K (a x) (a y) = K x y) (x y : V) :
-    binDiagonalKernel G p o n K (a x) (a y) = binDiagonalKernel G p o n K x y := by
-  have hm : Measurable (fun ω => binDiagonalMass G o n K ω x y) :=
-    measurable_finite_local _ _ (fun ω => binDiagonalMass_local G hloc o n K ω x y)
-  have h := integral_map (μ := law p) (measurable_set_preimage b).aemeasurable hm.aestronglyMeasurable
-  rw [law_map_preimage_equiv p b] at h
-  simpa only [binDiagonalKernel,binDiagonalMass_covariant G o n K hab hK] using h.symm
-
-theorem binDiagonalKernel_row_integral (G : BondGraph V E) (hloc : LocallyFinite G)
-    (p : unitInterval) (o : V) (n : ℕ) (K : V → V → ℝ) (x : V) :
-    (∑' y, binDiagonalKernel G p o n K x y) =
-      ∫ ω, binSizeMass G n ω x ^ 2 * ∑ z ∈ binOrbitClass G o n ω x, K x z ∂law p := by
-  rw [←show (∫ ω, ∑' y, binDiagonalMass G o n K ω x y ∂law p) =
-      ∑' y, binDiagonalKernel G p o n K x y from
-    integral_tsum_of_finite_support (law p) (finite_reachBall hloc x (2*n)).toFinset _
-      (fun y => binDiagonalMass_integrable G hloc p o n K x y)
-      (fun ω y hy => binDiagonalMass_outside G o n K ω x y (by simpa only [Set.Finite.mem_toFinset] using hy))]
-  simp only [binDiagonalMass_row]
-
-theorem binDiagonalKernel_column_integral (G : BondGraph V E) (hloc : LocallyFinite G)
-    (p : unitInterval) (o : V) (n : ℕ) (K : V → V → ℝ) (x : V) :
-    (∑' y, binDiagonalKernel G p o n K y x) =
-      ∫ ω, ∑' y, binDiagonalMass G o n K ω y x ∂law p := by
-  symm
-  exact integral_tsum_of_finite_support (law p) (finite_reachBall hloc x (2*n)).toFinset _
-    (fun y => binDiagonalMass_integrable G hloc p o n K y x)
-    (fun ω y hy => binDiagonalMass_outside_column G o n K ω y x (by simpa only [Set.Finite.mem_toFinset] using hy))
-
-theorem integral_binDiagonalLoss (G : BondGraph V E) (hloc : LocallyFinite G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (htr : RootTransport root weight Γ) (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b)
-    (hw : ∀ i, 0 ≤ weight i) (i₀ : ι) (hw₀ : 0 < weight i₀)
-    (p : unitInterval) (n : ℕ) (K : V → V → ℝ)
-    (hcov : ∀ a b, IsAut G a b → ∀ x y, K (a x) (a y) = K x y) :
-    (∫ ω, binSizeMass G n ω (root i₀)^2 *
-      ∑ z ∈ binOrbitClass G (root i₀) n ω (root i₀), K (root i₀) z ∂law p) =
-    ∫ ω, binSizeMass G n ω (root i₀)^2 / (binOrbitClass G (root i₀) n ω (root i₀)).card *
-      ∑ y ∈ binOrbitClass G (root i₀) n ω (root i₀),
-        ∑ z ∈ binOrbitClass G (root i₀) n ω (root i₀), K y z ∂law p := by
-  have hr (x : V) : Summable (binDiagonalKernel G p (root i₀) n K x) := by
-    apply summable_of_ne_finset_zero (s := (finite_reachBall hloc x (2*n)).toFinset)
-    intro y hy
-    simp only [binDiagonalKernel,binDiagonalMass_outside G (root i₀) n K _ x y
-      (by simpa only [Set.Finite.mem_toFinset] using hy),integral_zero]
-  have hc (y : V) : Summable (fun x => binDiagonalKernel G p (root i₀) n K x y) := by
-    apply summable_of_ne_finset_zero (s := (finite_reachBall hloc y (2*n)).toFinset)
-    intro x hx
-    simp only [binDiagonalKernel,binDiagonalMass_outside_column G (root i₀) n K _ x y
-      (by simpa only [Set.Finite.mem_toFinset] using hx),integral_zero]
-  have ht := orbit_kernel_column_eq_row G root weight Γ htr hΓ hw i₀ hw₀
-    (binDiagonalKernel G p (root i₀) n K)
-    (fun a b hab x y => binDiagonalKernel_covariant G hloc p (root i₀) n K hab (hcov a b hab) x y)
-    (fun x y hx => by simp only [binDiagonalKernel,binDiagonalMass_row_zero G _ n K _ x y hx,integral_zero])
-    (fun x y hy => by simp only [binDiagonalKernel,binDiagonalMass_column_zero G _ n K _ x y hy,integral_zero])
-    hr hc (root i₀) ⟨Equiv.refl _,Equiv.refl _,fun _ => by simp,rfl⟩
-  rw [binDiagonalKernel_row_integral G hloc,binDiagonalKernel_column_integral G hloc] at ht
-  simpa only [binDiagonalMass_column] using ht.symm
-
- 
-
-theorem integral_binDiagonalLoss_le (G : BondGraph V E) (hloc : LocallyFinite G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (htr : RootTransport root weight Γ) (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b)
-    (hw : ∀ i, 0 ≤ weight i) (i₀ : ι) (hw₀ : 0 < weight i₀)
-    (p : unitInterval) (n : ℕ) (K : V → V → ℝ)
-    (hcov : ∀ a b, IsAut G a b → ∀ x y, K (a x) (a y) = K x y)
-    (A : Hilbert V →L[ℂ] Hilbert V)
-    (hA : ∀ x y, A (lp.single 2 y (1:ℂ)) x = (K x y : ℂ)) :
-    (∫ ω, binSizeMass G n ω (root i₀)^2 *
-      ∑ z ∈ binOrbitClass G (root i₀) n ω (root i₀), K (root i₀) z ∂law p) ≤
-      ‖A‖ * ∫ ω, binSizeMass G n ω (root i₀)^2 ∂law p := by
-  rw [integral_binDiagonalLoss G hloc root weight Γ htr hΓ hw i₀ hw₀ p n K hcov]
-  have hi : Integrable (fun ω => ∑' y, binDiagonalMass G (root i₀) n K ω y (root i₀)) (law p) := by
-    have he (ω : Set E) : (∑' y, binDiagonalMass G (root i₀) n K ω y (root i₀)) =
-        ∑ y ∈ (finite_reachBall hloc (root i₀) (2*n)).toFinset, binDiagonalMass G (root i₀) n K ω y (root i₀) := by
-      apply tsum_eq_sum
-      intro y hy
-      exact binDiagonalMass_outside_column G (root i₀) n K ω y (root i₀)
-        (by simpa only [Set.Finite.mem_toFinset] using hy)
-    simp only [he]
-    exact integrable_finsetSum _ (fun y _ => binDiagonalMass_integrable G hloc p (root i₀) n K y (root i₀))
-  have hs : Integrable (fun ω => binSizeMass G n ω (root i₀)^2) (law p) :=
-    integrable_finite_local _ p _ (fun ω => congrArg (fun t : ℝ => t^2) (binSizeMass_local G hloc n ω (root i₀)))
-  rw [←integral_const_mul]
-  exact integral_mono (by simpa only [binDiagonalMass_column] using hi) (hs.const_mul ‖A‖)
-    (fun ω => by simpa only [binDiagonalMass_column] using binDiagonalMass_column_le G (root i₀) n K A hA ω (root i₀))
-
-end BinDiagonalLoss
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section DyadicDiagonalStrata
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def fullOrbitClass (G : BondGraph V E) (o : V) (ω : Set E) (x : V) : Finset V :=
-  if h : (cluster G ω x).Finite then h.toFinset.filter (fun y => y ∈ vertexOrbit G o) else ∅
-
-noncomputable def diagonalLossMass (G : BondGraph V E) (o : V) (K : V → V → ℝ)
-    (ω : Set E) (x : V) : ℝ :=
-  ((cluster G ω x).ncard:ℝ)^2 * ∑ z ∈ fullOrbitClass G o ω x, K x z
-
-theorem binSizeMass_zero (G : BondGraph V E) (n : ℕ) (ω : Set E) (x : V)
-    (hw : ¬ ClusterWindow G x n (2*n) ω) : binSizeMass G n ω x = 0 := by
-  simp only [binSizeMass,binMass_zero_of_not_window G n ω x _ hw,tsum_zero]
-
-theorem binOrbitClass_eq_full (G : BondGraph V E) (o : V) (n : ℕ) (ω : Set E) (x : V)
-    (hx : x ∈ vertexOrbit G o) (hw : ClusterWindow G x n (2*n) ω) :
-    binOrbitClass G o n ω x = fullOrbitClass G o ω x := by
-  simp only [binOrbitClass,ite_eq_left hx,binClusterSlice,dite_eq_left hw,fullOrbitClass,dite_eq_left hw.1]
-
-theorem dyadic_window_iff (G : BondGraph V E) (ω : Set E) (x : V) (j : ℕ)
-    (hz : (cluster G ω x).ncard ≠ 0) :
-    ClusterWindow G x (2^j) (2*(2^j)) ω ↔ Nat.log 2 (cluster G ω x).ncard = j := by
-  constructor
-  · intro h
-    apply Nat.log_eq_of_pow_le_of_lt_pow h.2.1
-    simpa only [pow_succ,Nat.mul_comm] using h.2.2
-  · intro h
-    refine ⟨Set.finite_of_ncard_ne_zero hz,?_,?_⟩
-    · rw [←h]
-      exact Nat.pow_log_le_self 2 hz
-    · rw [←h,show 2*2^(Nat.log 2 (cluster G ω x).ncard) = 2^(Nat.log 2 (cluster G ω x).ncard+1) by simp only [pow_succ,Nat.mul_comm]]
-      exact Nat.lt_pow_succ_log_self (by norm_num) _
-
-theorem dyadic_binSizeMass_sq (G : BondGraph V E) (ω : Set E) (x : V) (j : ℕ) :
-    binSizeMass G (2^j) ω x ^ 2 =
-      if Nat.log 2 (cluster G ω x).ncard = j then ((cluster G ω x).ncard:ℝ)^2 else 0 := by
-  by_cases hz : (cluster G ω x).ncard = 0
-  · have hw : ¬ClusterWindow G x (2^j) (2*(2^j)) ω := by
-      intro h
-      have hp : 0 < 2^j := pow_pos (by omega) _
-      have hl := h.2.1
-      omega
-    simp only [binSizeMass_zero G (2^j) ω x hw,hz,Nat.cast_zero,zero_pow (by omega : 2 ≠ 0),ite_self]
-  · by_cases hw : ClusterWindow G x (2^j) (2*(2^j)) ω
-    · rw [binSizeMass_eq G (2^j) ω x hw,ite_eq_left ((dyadic_window_iff G ω x j hz).mp hw)]
-    · rw [binSizeMass_zero G (2^j) ω x hw,ite_eq_right (fun h => hw ((dyadic_window_iff G ω x j hz).mpr h))]
-      norm_num
-
-theorem dyadic_binDiagonalMass (G : BondGraph V E) (o : V) (K : V → V → ℝ)
-    (ω : Set E) (x : V) (hx : x ∈ vertexOrbit G o) (j : ℕ) :
-    binSizeMass G (2^j) ω x ^ 2 * ∑ z ∈ binOrbitClass G o (2^j) ω x, K x z =
-      if Nat.log 2 (cluster G ω x).ncard = j then diagonalLossMass G o K ω x else 0 := by
-  by_cases hw : ClusterWindow G x (2^j) (2*(2^j)) ω
-  · have hz : (cluster G ω x).ncard ≠ 0 := by
-      have hp : 0 < 2^j := pow_pos (by omega) _
-      have hl := hw.2.1
-      omega
-    rw [ite_eq_left ((dyadic_window_iff G ω x j hz).mp hw),binOrbitClass_eq_full G o (2^j) ω x hx hw,
-      binSizeMass_eq G (2^j) ω x hw]
-    rfl
-  · rw [binSizeMass_zero G (2^j) ω x hw]
-    simp only [zero_pow (by omega : 2 ≠ 0),zero_mul]
-    by_cases hz : (cluster G ω x).ncard = 0
-    · simp only [diagonalLossMass,hz,Nat.cast_zero,zero_pow (by omega : 2 ≠ 0),zero_mul,ite_self]
-    · rw [ite_eq_right (fun h => hw ((dyadic_window_iff G ω x j hz).mpr h))]
-
-theorem diagonalLossMass_nonneg (G : BondGraph V E) (o : V) (K : V → V → ℝ)
-    (hK : ∀ x y, 0 ≤ K x y) (ω : Set E) (x : V) : 0 ≤ diagonalLossMass G o K ω x :=
-  mul_nonneg (sq_nonneg _) (Finset.sum_nonneg (fun z _ => hK x z))
-
-theorem diagonalLossMass_le (G : BondGraph V E) (o : V) (K : V → V → ℝ)
-    (L : ℝ) (hrow : ∀ x (S : Finset V), (∑ z ∈ S, K x z) ≤ L) (ω : Set E) (x : V) :
-    diagonalLossMass G o K ω x ≤ L*((cluster G ω x).ncard:ℝ)^2 := by
-  exact (mul_le_mul_of_nonneg_left (hrow x _) (sq_nonneg _)).trans_eq (mul_comm _ _)
-
-end DyadicDiagonalStrata
-end Percolation
-
- 
-namespace Percolation
-open MeasureTheory Filter
-open scoped Topology
-section DiscreteStratumIntegral
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem integral_le_of_discrete_strata {Ω : Type*} [MeasurableSpace Ω]
-    (μ : Measure Ω) (J : Ω → ℕ) (f g : Ω → ℝ) (C L : ℝ)
-    (hf0 : ∀ ω, 0 ≤ f ω) (hg0 : ∀ ω, 0 ≤ g ω)
-    (hfg : ∀ ω, f ω ≤ L*g ω) (hgi : Integrable g μ)
-    (hfi : ∀ j, Integrable (fun ω => if J ω = j then f ω else 0) μ)
-    (hgi' : ∀ j, Integrable (fun ω => if J ω = j then g ω else 0) μ)
-    (h : ∀ j, (∫ ω, (if J ω = j then f ω else 0) ∂μ) ≤
-      C*(∫ ω, (if J ω = j then g ω else 0) ∂μ)) :
-    (∫ ω, f ω ∂μ) ≤ C*(∫ ω, g ω ∂μ) := by
-  let F : ℕ → Ω → ℝ := fun N ω => ∑ j ∈ Finset.range N, if J ω = j then f ω else 0
-  let H : ℕ → Ω → ℝ := fun N ω => ∑ j ∈ Finset.range N, if J ω = j then g ω else 0
-  have he (q : Ω → ℝ) (N : ℕ) (ω : Ω) :
-      (∑ j ∈ Finset.range N, if J ω = j then q ω else 0) = if J ω < N then q ω else 0 := by
-    simp only [Finset.sum_ite_eq,Finset.mem_range]
-  have hFi (N : ℕ) : Integrable (F N) μ := integrable_finsetSum _ (fun j _ => hfi j)
-  have hHi (N : ℕ) : Integrable (H N) μ := integrable_finsetSum _ (fun j _ => hgi' j)
-  have hFbody (N : ℕ) (ω : Ω) : ‖F N ω‖ ≤ L*g ω := by
-    dsimp [F]
-    rw [he]
-    split_ifs
-    · simpa only [Real.norm_eq_abs,abs_of_nonneg (hf0 ω)] using hfg ω
-    · simpa only [norm_zero,abs_zero] using (hf0 ω).trans (hfg ω)
-  have hHbody (N : ℕ) (ω : Ω) : ‖H N ω‖ ≤ g ω := by
-    dsimp [H]
-    rw [he]
-    split_ifs
-    · simp only [abs_of_nonneg (hg0 ω),le_refl]
-    · simpa only [norm_zero,abs_zero] using hg0 ω
-  have ht (q : Ω → ℝ) (ω : Ω) : Tendsto
-      (fun N => ∑ j ∈ Finset.range N, if J ω = j then q ω else 0) atTop (𝓝 (q ω)) := by
-    have he' : (fun N => ∑ j ∈ Finset.range N, if J ω = j then q ω else 0) =ᶠ[atTop] (fun _ => q ω) := by
-      filter_upwards [eventually_ge_atTop (J ω+1)] with N hN
-      rw [he,ite_eq_left (by omega)]
-    exact tendsto_const_nhds.congr' he'.symm
-  have htF := tendsto_integral_of_dominated_convergence (fun ω => L*g ω)
-    (fun N => (hFi N).aestronglyMeasurable) (hgi.const_mul L)
-    (fun N => Eventually.of_forall (hFbody N)) (Eventually.of_forall (ht f))
-  have htH := tendsto_integral_of_dominated_convergence g
-    (fun N => (hHi N).aestronglyMeasurable) hgi
-    (fun N => Eventually.of_forall (hHbody N)) (Eventually.of_forall (ht g))
-  apply le_of_tendsto_of_tendsto htF (tendsto_const_nhds.mul htH)
-  exact Eventually.of_forall (fun N => by
-    change (∫ ω, F N ω ∂μ) ≤ C*(∫ ω, H N ω ∂μ)
-    simp only [F,H,integral_finsetSum _ (fun j _ => hfi j),
-      integral_finsetSum _ (fun j _ => hgi' j),Finset.mul_sum]
-    exact Finset.sum_le_sum (fun j _ => h j))
-
-end DiscreteStratumIntegral
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section DiagonalLoss
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem integral_diagonalLoss_le (G : BondGraph V E) (hloc : LocallyFinite G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (htr : RootTransport root weight Γ) (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b)
-    (hw : ∀ i, 0 ≤ weight i) (i₀ : ι) (hw₀ : 0 < weight i₀)
-    (p : unitInterval) (K : V → V → ℝ) (hK : ∀ x y, 0 ≤ K x y)
-    (hcov : ∀ a b, IsAut G a b → ∀ x y, K (a x) (a y) = K x y)
-    (L : ℝ) (hrow : ∀ x (S : Finset V), (∑ z ∈ S, K x z) ≤ L)
-    (A : Hilbert V →L[ℂ] Hilbert V)
-    (hA : ∀ x y, A (lp.single 2 y (1:ℂ)) x = (K x y : ℂ))
-    (hs : Integrable (fun ω => ((cluster G ω (root i₀)).ncard:ℝ)^2) (law p)) :
-    (∫ ω, diagonalLossMass G (root i₀) K ω (root i₀) ∂law p) ≤
-      ‖A‖*(∫ ω, ((cluster G ω (root i₀)).ncard:ℝ)^2 ∂law p) := by
-  have hx : root i₀ ∈ vertexOrbit G (root i₀) := ⟨Equiv.refl _,Equiv.refl _,fun _ => by simp,rfl⟩
-  have hs' (j : ℕ) : Integrable (fun ω => binSizeMass G (2^j) ω (root i₀)^2) (law p) :=
-    integrable_finite_local _ p _ (fun ω => congrArg (fun t : ℝ => t^2)
-      (binSizeMass_local G hloc (2^j) ω (root i₀)))
-  have hf' (j : ℕ) : Integrable (fun ω => binSizeMass G (2^j) ω (root i₀)^2 *
-      ∑ z ∈ binOrbitClass G (root i₀) (2^j) ω (root i₀), K (root i₀) z) (law p) := by
-    apply integrable_finite_local _ p _
-    intro ω
-    rw [binOrbitClass_local G hloc (root i₀) (2^j) ω (root i₀),binSizeMass_local G hloc (2^j) ω (root i₀)]
-  apply integral_le_of_discrete_strata (law p) (fun ω => Nat.log 2 (cluster G ω (root i₀)).ncard)
-    (fun ω => diagonalLossMass G (root i₀) K ω (root i₀))
-    (fun ω => ((cluster G ω (root i₀)).ncard:ℝ)^2) ‖A‖ L
-    (fun ω => diagonalLossMass_nonneg G (root i₀) K hK ω (root i₀))
-    (fun _ => sq_nonneg _) (fun ω => diagonalLossMass_le G (root i₀) K L hrow ω (root i₀)) hs
-  · intro j
-    simpa only [dyadic_binDiagonalMass G (root i₀) K _ (root i₀) hx] using hf' j
-  · intro j
-    simpa only [dyadic_binSizeMass_sq] using hs' j
-  · intro j
-    have h := integral_binDiagonalLoss_le G hloc root weight Γ htr hΓ hw i₀ hw₀ p (2^j) K hcov A hA
-    simp only [dyadic_binDiagonalMass G (root i₀) K _ (root i₀) hx] at h
-    simpa only [dyadic_binSizeMass_sq] using h
-
- 
-
-theorem subcritical_diagonalLoss_le [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (htr : RootTransport root weight Γ) (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b)
-    (hw : ∀ i, 0 ≤ weight i) (i₀ : ι) (hw₀ : 0 < weight i₀)
-    (p : unitInterval) (hp : p < pc G) :
-    ∃ C : ℝ, 0 ≤ C ∧ ∀ (K : V → V → ℝ), (∀ x y, 0 ≤ K x y) →
-      (∀ a b, IsAut G a b → ∀ x y, K (a x) (a y) = K x y) →
-      ∀ (L : ℝ), (∀ x (S : Finset V), (∑ z ∈ S, K x z) ≤ L) →
-      ∀ (A : Hilbert V →L[ℂ] Hilbert V),
-      (∀ x y, A (lp.single 2 y (1:ℂ)) x = (K x y : ℂ)) →
-      (∫ ω, diagonalLossMass G (root i₀) K ω (root i₀) ∂law p) ≤ ‖A‖*C^3 := by
-  obtain ⟨C,hC,hs⟩ := subcritical_second_moment G hloc hqt p hp
-  refine ⟨C,hC,fun K hK hcov L hrow A hA => ?_⟩
-  exact (integral_diagonalLoss_le G hloc root weight Γ htr hΓ hw i₀ hw₀ p K hK hcov L hrow A hA
-    (hs (root i₀)).1).trans (mul_le_mul_of_nonneg_left (hs (root i₀)).2 (norm_nonneg _))
-
-end DiagonalLoss
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-open scoped ENNReal
-section RootSecondMoment
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem second_moment_le_rootSusceptibility_cube [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (hw : ∀ i, 0 ≤ weight i) (hcov : ∀ x, ∃ i, x ∈ vertexOrbit G (root i))
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (p : unitInterval) (hp : p < pc G) (x : V) :
-    (∫ ω, ((cluster G ω x).ncard:ℝ)^2 ∂law p) ≤
-      (c*rootSusceptibility G root weight p)^3 := by
-  let C := c*rootSusceptibility G root weight p
-  have hC : 0 ≤ C := mul_nonneg hc (rootSusceptibility_nonneg G root weight hw p)
-  have hrow (z : V) : (∑' y, law p {ω | Conn G ω z y}) ≤ ENNReal.ofReal C := by
-    rw [←ennreal_susceptibility G hloc hqt p hp]
-    exact ENNReal.ofReal_le_ofReal (susceptibility_le_rootSusceptibility G root weight hw hcov c hc hmin p z)
-  have hm : Measurable (fun ω : Set E => ((cluster G ω x).ncard:ℝ)^2) :=
-    (measurable_of_countable (fun n : ℕ => (n:ℝ)^2)).comp (measurable_cluster_ncard G hloc x)
-  have hn : 0 ≤ᵐ[law p] (fun ω => ((cluster G ω x).ncard:ℝ)^2) :=
-    Filter.Eventually.of_forall (fun _ => sq_nonneg _)
-  have he : (fun ω => ENNReal.ofReal (((cluster G ω x).ncard:ℝ)^2)) =ᵐ[law p]
-      (fun ω => clusterMass G ω x ^ 2) := by
-    filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
-    rw [clusterMass_eq_ncard G ω x hω,ENNReal.ofReal_pow (Nat.cast_nonneg _)]
-    simp only [ENNReal.ofReal_natCast]
-  have hbound : (∫⁻ ω, ENNReal.ofReal (((cluster G ω x).ncard:ℝ)^2) ∂law p) ≤ ENNReal.ofReal (C^3) := by
-    rw [lintegral_congr_ae he,ENNReal.ofReal_pow hC]
-    exact cluster_second_moment_bound G p _ hrow x
-  rw [integral_eq_lintegral_of_nonneg_ae hn hm.aestronglyMeasurable]
-  exact (ENNReal.toReal_mono ENNReal.ofReal_ne_top hbound).trans_eq (ENNReal.toReal_ofReal (pow_nonneg hC _))
-
-theorem diagonalLoss_le_rootSusceptibility_cube [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (htr : RootTransport root weight Γ) (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b)
-    (hw : ∀ i, 0 ≤ weight i) (i₀ : ι) (hw₀ : 0 < weight i₀)
-    (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i))
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (p : unitInterval) (hp : p < pc G) (K : V → V → ℝ) (hK : ∀ x y, 0 ≤ K x y)
-    (hcov : ∀ a b, IsAut G a b → ∀ x y, K (a x) (a y) = K x y)
-    (L : ℝ) (hrow : ∀ x (S : Finset V), (∑ z ∈ S, K x z) ≤ L)
-    (A : Hilbert V →L[ℂ] Hilbert V)
-    (hA : ∀ x y, A (lp.single 2 y (1:ℂ)) x = (K x y : ℂ)) :
-    (∫ ω, diagonalLossMass G (root i₀) K ω (root i₀) ∂law p) ≤
-      ‖A‖*(c*rootSusceptibility G root weight p)^3 := by
-  obtain ⟨_,_,hs⟩ := subcritical_second_moment G hloc hqt p hp
-  exact (integral_diagonalLoss_le G hloc root weight Γ htr hΓ hw i₀ hw₀ p K hK hcov L hrow A hA
-    (hs (root i₀)).1).trans (mul_le_mul_of_nonneg_left
-      (second_moment_le_rootSusceptibility_cube G hloc hqt root weight hw hcover c hc hmin p hp (root i₀))
-      (norm_nonneg _))
-
-end RootSecondMoment
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section DiagonalSeries
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def diagonalLossTerm (G : BondGraph V E) (o : V) (K : V → V → ℝ)
-    (x y : V) (ω : Set E) : ℝ :=
-  if y ∈ vertexOrbit G o ∧ Conn G ω x y then K x y*((cluster G ω x).ncard:ℝ)^2 else 0
-
-theorem diagonalLossTerm_nonneg (G : BondGraph V E) (o : V) (K : V → V → ℝ)
-    (hK : ∀ x y, 0 ≤ K x y) (x y : V) (ω : Set E) :
-    0 ≤ diagonalLossTerm G o K x y ω := by
-  unfold diagonalLossTerm
-  split_ifs
-  · exact mul_nonneg (hK x y) (sq_nonneg _)
-  · exact le_rfl
-
-theorem diagonalLossTerm_le (G : BondGraph V E) (o : V) (K : V → V → ℝ)
-    (hK : ∀ x y, 0 ≤ K x y) (x y : V) (ω : Set E) :
-    diagonalLossTerm G o K x y ω ≤ K x y*((cluster G ω x).ncard:ℝ)^2 := by
-  unfold diagonalLossTerm
-  split_ifs
-  · exact le_rfl
-  · exact mul_nonneg (hK x y) (sq_nonneg _)
-
-theorem diagonalLossMass_eq_tsum (G : BondGraph V E) (o : V) (K : V → V → ℝ)
-    (x : V) (ω : Set E) :
-    diagonalLossMass G o K ω x = ∑' y, diagonalLossTerm G o K x y ω := by
-  by_cases hf : (cluster G ω x).Finite
-  · let S := fullOrbitClass G o ω x
-    have hs (y : V) : y ∈ S ↔ y ∈ vertexOrbit G o ∧ Conn G ω x y := by
-      dsimp only [S,fullOrbitClass]
-      rw [dite_eq_left hf]
-      simp only [Finset.mem_filter,Set.Finite.mem_toFinset,cluster,Set.mem_ofPred_eq]
-      exact and_comm
-    rw [tsum_eq_sum (s := S) (fun y hy => by
-      simp only [diagonalLossTerm,ite_eq_right (fun h => hy ((hs y).mpr h))])]
-    rw [diagonalLossMass,Finset.mul_sum]
-    apply Finset.sum_congr rfl
-    intro y hy
-    rw [diagonalLossTerm,ite_eq_left ((hs y).mp hy),mul_comm]
-  · have hn := Set.Infinite.ncard hf
-    simp only [diagonalLossMass,diagonalLossTerm,hn,Nat.cast_zero,zero_pow (by omega : 2 ≠ 0),
-      mul_zero,zero_mul,ite_self,tsum_zero]
-
-theorem integrable_diagonalLossTerm [Countable V] [Countable E] (G : BondGraph V E) (o : V) (K : V → V → ℝ)
-    (x y : V) (p : unitInterval)
-    (hs : Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^2) (law p)) :
-    Integrable (diagonalLossTerm G o K x y) (law p) := by
-  by_cases hy : y ∈ vertexOrbit G o
-  · convert (hs.indicator (measurableSet_conn G x y)).const_mul (K x y) using 1
-    ext ω
-    simp only [diagonalLossTerm,hy,true_and,Set.indicator,Set.mem_ofPred_eq]
-    split_ifs <;> simp only [mul_zero]
-  · have he : diagonalLossTerm G o K x y = (fun _ : Set E => (0:ℝ)) := by
-      funext ω
-      simp only [diagonalLossTerm,hy,false_and,ite_false]
-    rw [he]
-    exact integrable_const 0
-
- 
-
-theorem integral_diagonalLoss_eq_series [Countable V] [Countable E]
-    (G : BondGraph V E) (o : V) (K : V → V → ℝ) (hK : ∀ x y, 0 ≤ K x y)
-    (L : ℝ) (hrow : ∀ x (S : Finset V), (∑ z ∈ S, K x z) ≤ L)
-    (x : V) (p : unitInterval)
-    (hs : Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^2) (law p)) :
-    (∫ ω, diagonalLossMass G o K ω x ∂law p) =
-      ∑' y, if y ∈ vertexOrbit G o then
-        K x y*(∫ ω, (if Conn G ω x y then ((cluster G ω x).ncard:ℝ)^2 else 0) ∂law p) else 0 := by
-  have hi (y : V) := integrable_diagonalLossTerm G o K x y p hs
-  have hzero (y : V) (ω : Set E) := diagonalLossTerm_nonneg G o K hK x y ω
-  have hn : Summable (fun y => ∫ ω, ‖diagonalLossTerm G o K x y ω‖ ∂law p) := by
-    simp only [Real.norm_of_nonneg (hzero _ _)]
-    apply summable_of_sum_le (fun y => integral_nonneg (hzero y)) (c := L*(∫ ω, ((cluster G ω x).ncard:ℝ)^2 ∂law p))
-    intro S
-    rw [←integral_finsetSum _ (fun y _ => hi y),←integral_const_mul]
-    apply integral_mono (integrable_finsetSum _ (fun y _ => hi y)) (hs.const_mul L)
-    intro ω
-    calc
-      _ ≤ ∑ y ∈ S, K x y*((cluster G ω x).ncard:ℝ)^2 :=
-        Finset.sum_le_sum (fun y _ => diagonalLossTerm_le G o K hK x y ω)
-      _ = (∑ y ∈ S, K x y)*((cluster G ω x).ncard:ℝ)^2 := (Finset.sum_mul ..).symm
-      _ ≤ _ := mul_le_mul_of_nonneg_right (hrow x S) (sq_nonneg _)
-  rw [show (fun ω => diagonalLossMass G o K ω x) = (fun ω => ∑' y, diagonalLossTerm G o K x y ω) from
-    funext (diagonalLossMass_eq_tsum G o K x)]
-  rw [←integral_tsum_of_summable_integral_norm hi hn]
-  apply tsum_congr
-  intro y
-  by_cases hy : y ∈ vertexOrbit G o
-  · simp only [diagonalLossTerm,hy,true_and,ite_true]
-    rw [←integral_const_mul]
-    apply integral_congr_ae
-    exact Filter.Eventually.of_forall (fun ω => by dsimp only; split_ifs <;> simp only [mul_zero])
-  · simp only [diagonalLossTerm,hy,false_and,ite_false,integral_zero]
-
-end DiagonalSeries
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section RootSusceptibilityLower
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem exists_rootSusceptibility_lower [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
-    (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i))
-    (p₀ : ℝ) (hp₀ : 0 < p₀) :
-    ∃ a : ℝ, 0 < a ∧ ∀ (p : unitInterval), p₀ ≤ (p:ℝ) → p < pc G → ∀ x,
-      a*rootSusceptibility G root weight p ≤ susceptibility G p x := by
-  have hex (ij : ι × ι) : ∃ l : List E, Along G (root ij.1) l (root ij.2) := by
-    obtain ⟨l,hl,_,_⟩ := conn_exists_bond_trail G (conn_univ_of_connected G hconn (root ij.1) (root ij.2))
-    exact ⟨l,hl⟩
-  choose path hpath using hex
-  let N := ∑ ij : ι × ι, (path ij).length
-  have hlen (i j : ι) : (path (i,j)).length ≤ N :=
-    Finset.single_le_sum (f := fun ij : ι × ι => (path ij).length) (fun _ _ => Nat.zero_le _) (Finset.mem_univ (i,j))
-  refine ⟨p₀^N,pow_pos hp₀ N,fun p hp hpcrit x => ?_⟩
-  obtain ⟨j,a,b,hab,hax⟩ := hcover x
-  have hfin (z : V) : Summable (fun y => (law p).real {ω | Conn G (ω ∩ Set.univ) z y}) := by
-    simp only [Set.inter_univ]
-    exact summable_twoPoint_subcritical G hloc hqt p hpcrit z
-  have heq (z : V) : restrictedSusceptibility G Set.univ p z = susceptibility G p z := by
-    simp only [restrictedSusceptibility,susceptibility,Set.inter_univ,twoPoint,Measure.real]
-  have hb (i : ι) : p₀^N*susceptibility G p (root i) ≤ susceptibility G p (root j) := by
-    have h := restrictedSusceptibility_path_lower G Set.univ p (hpath (j,i)) (fun _ _ => Set.mem_univ _) hfin
-    rw [heq,heq] at h
-    have hpow : p₀^N ≤ (p:ℝ)^(path (j,i)).length :=
-      (pow_le_pow_left₀ hp₀.le hp N).trans (pow_le_pow_of_le_one p.2.1 p.2.2 (hlen j i))
-    exact (mul_le_mul_of_nonneg_right hpow (susceptibility_nonneg G p _)).trans h
-  rw [←hax,susceptibility_aut G p hab]
-  calc
-    _ = ∑ i, weight i*(p₀^N*susceptibility G p (root i)) := by
-      simp only [rootSusceptibility,Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro i _
-      ring
-    _ ≤ ∑ i, weight i*susceptibility G p (root j) :=
-      Finset.sum_le_sum (fun i _ => mul_le_mul_of_nonneg_left (hb i) (hw i))
-    _ = susceptibility G p (root j) := by rw [←Finset.sum_mul,hw1,one_mul]
-
-end RootSusceptibilityLower
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-open scoped ENNReal
-section ClusterAssociation
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem association_finiteWitness_ennreal [Countable E] (p : unitInterval)
-    (A B : Set (Set E)) (hA : IsUpperSet A) (hB : IsUpperSet B)
-    (hfA : FiniteWitness A) (hfB : FiniteWitness B) :
-    law p A * law p B ≤ law p (A ∩ B) := by
-  apply (ENNReal.toReal_le_toReal
-    (ENNReal.mul_ne_top (measure_ne_top _ _) (measure_ne_top _ _)) (measure_ne_top _ _)).mp
-  rw [ENNReal.toReal_mul]
-  exact association_finiteWitness p A B hA hB hfA hfB
-
-theorem clusterMass_mul (G : BondGraph V E) (ω : Set E) (x y : V) :
-    clusterMass G ω x * clusterMass G ω y =
-      ∑' a, ∑' b, if Conn G ω x a ∧ Conn G ω y b then (1:ℝ≥0∞) else 0 := by
-  rw [clusterMass,clusterMass,←ENNReal.tsum_mul_right]
-  apply tsum_congr
-  intro a
-  rw [←ENNReal.tsum_mul_left]
-  apply tsum_congr
-  intro b
-  split_ifs <;> simp_all
-
-theorem lintegral_clusterMass_mul [Countable V] [Countable E]
-    (G : BondGraph V E) (p : unitInterval) (x y : V) :
-    (∫⁻ ω, clusterMass G ω x * clusterMass G ω y ∂law p) =
-      ∑' a, ∑' b, law p {ω | Conn G ω x a ∧ Conn G ω y b} := by
-  have hm (a b : V) : Measurable (fun ω : Set E => if Conn G ω x a ∧ Conn G ω y b then (1:ℝ≥0∞) else 0) :=
-    Measurable.ite ((measurableSet_conn G x a).inter (measurableSet_conn G y b)) measurable_const measurable_const
-  simp_rw [clusterMass_mul]
-  rw [lintegral_tsum (fun a => (Measurable.tsum (fun b => hm a b)).aemeasurable)]
-  apply tsum_congr
-  intro a
-  rw [lintegral_tsum (fun b => (hm a b).aemeasurable)]
-  apply tsum_congr
-  intro b
-  have hs : MeasurableSet {ω : Set E | Conn G ω x a ∧ Conn G ω y b} :=
-    (measurableSet_conn G x a).inter (measurableSet_conn G y b)
-  simpa only [Set.indicator,Set.mem_ofPred_eq,Pi.one_apply] using (lintegral_indicator_one (μ := law p) hs)
-
-theorem clusterMass_association [Countable V] [Countable E]
-    (G : BondGraph V E) (p : unitInterval) (x y : V) :
-    (∑' a, law p {ω | Conn G ω x a})*(∑' b, law p {ω | Conn G ω y b}) ≤
-      ∫⁻ ω, clusterMass G ω x * clusterMass G ω y ∂law p := by
-  rw [lintegral_clusterMass_mul,←ENNReal.tsum_mul_right]
-  apply ENNReal.tsum_le_tsum
-  intro a
-  rw [←ENNReal.tsum_mul_left]
-  apply ENNReal.tsum_le_tsum
-  intro b
-  exact association_finiteWitness_ennreal p _ _
-    (fun _ _ hω hc => conn_mono G hω hc) (fun _ _ hω hc => conn_mono G hω hc)
-    (finiteWitness_conn G x a) (finiteWitness_conn G y b)
-
- 
-
-theorem subcritical_cluster_association [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (x y : V) :
-    Integrable (fun ω => ((cluster G ω x).ncard:ℝ)*((cluster G ω y).ncard:ℝ)) (law p) ∧
-    susceptibility G p x * susceptibility G p y ≤
-      ∫ ω, ((cluster G ω x).ncard:ℝ)*((cluster G ω y).ncard:ℝ) ∂law p := by
-  have hm (z : V) : Measurable (fun ω : Set E => ((cluster G ω z).ncard:ℝ)) :=
-    (measurable_of_countable (fun n : ℕ => (n:ℝ))).comp (measurable_cluster_ncard G hloc z)
-  have hn (ω : Set E) : 0 ≤ ((cluster G ω x).ncard:ℝ)*((cluster G ω y).ncard:ℝ) :=
-    mul_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _)
-  obtain ⟨_,_,hsecond⟩ := subcritical_second_moment G hloc hqt p hp
-  have hi : Integrable (fun ω => ((cluster G ω x).ncard:ℝ)*((cluster G ω y).ncard:ℝ)) (law p) := by
-    apply ((hsecond x).1.add (hsecond y).1).mono' ((hm x).mul (hm y)).aestronglyMeasurable
-    exact Filter.Eventually.of_forall (fun ω => by
-      dsimp only [Pi.mul_apply,Pi.add_apply]
-      rw [Real.norm_of_nonneg (hn ω)]
-      nlinarith [sq_nonneg (((cluster G ω x).ncard:ℝ)-((cluster G ω y).ncard:ℝ))])
-  refine ⟨hi,?_⟩
-  have he : (fun ω => ENNReal.ofReal (((cluster G ω x).ncard:ℝ)*((cluster G ω y).ncard:ℝ))) =ᵐ[law p]
-      (fun ω => clusterMass G ω x * clusterMass G ω y) := by
-    filter_upwards [subcritical_cluster_finite_ae G p hp x,subcritical_cluster_finite_ae G p hp y] with ω hx hy
-    rw [clusterMass_eq_ncard G ω x hx,clusterMass_eq_ncard G ω y hy,ENNReal.ofReal_mul (Nat.cast_nonneg _)]
-    simp only [ENNReal.ofReal_natCast]
-  have hb : ENNReal.ofReal (susceptibility G p x * susceptibility G p y) ≤
-      ∫⁻ ω, ENNReal.ofReal (((cluster G ω x).ncard:ℝ)*((cluster G ω y).ncard:ℝ)) ∂law p := by
-    rw [lintegral_congr_ae he,ENNReal.ofReal_mul (susceptibility_nonneg G p x),
-      ennreal_susceptibility G hloc hqt p hp,ennreal_susceptibility G hloc hqt p hp]
-    exact clusterMass_association G p x y
-  have htop : (∫⁻ ω, ENNReal.ofReal (((cluster G ω x).ncard:ℝ)*((cluster G ω y).ncard:ℝ)) ∂law p) ≠ ⊤ :=
-    ne_of_lt ((hasFiniteIntegral_iff_ofReal (Filter.Eventually.of_forall hn)).mp hi.hasFiniteIntegral)
-  rw [integral_eq_lintegral_of_nonneg_ae (Filter.Eventually.of_forall hn) hi.aestronglyMeasurable]
-  simpa only [ENNReal.toReal_ofReal (mul_nonneg (susceptibility_nonneg G p x) (susceptibility_nonneg G p y))] using
-    ENNReal.toReal_mono htop hb
-
-end ClusterAssociation
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section FiniteLocalIndependence
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem integral_mul_finite_local (I R : Finset E) (hdis : Disjoint I R)
-    (p : unitInterval) (f g : Set E → ℝ)
-    (hf : ∀ ω, f ω = f (ω ∩ (I:Set E)))
-    (hg : ∀ ω, g ω = g (ω ∩ (R:Set E))) :
-    (∫ ω, f ω*g ω ∂law p) = (∫ ω, f ω ∂law p)*(∫ ω, g ω ∂law p) := by
-  have hf' := finite_local_of_subset I (I∪R) Finset.subset_union_left f hf
-  have hg' := finite_local_of_subset R (I∪R) Finset.subset_union_right g hg
-  rw [integral_eq_bernoulliPolynomial (I∪R) p _ (fun ω => by rw [hf' ω,hg' ω]),
-    integral_eq_bernoulliPolynomial I p f hf,integral_eq_bernoulliPolynomial R p g hg,
-    bernoulliPolynomial_union I R hdis]
-  have hi (a b : Finset E) (ha : a ⊆ I) (hb : b ⊆ R) :
-      ((a ∪ b : Finset E):Set E) ∩ (I:Set E) = a := by
-    ext e
-    simp only [Finset.coe_union,Set.mem_inter_iff,Set.mem_union,Finset.mem_coe]
-    constructor
-    · rintro ⟨he,hI⟩
-      rcases he with he | he
-      · exact he
-      · exact ((Finset.disjoint_left.mp hdis) hI (hb he)).elim
-    · intro he
-      exact ⟨Or.inl he,ha he⟩
-  have hr (a b : Finset E) (ha : a ⊆ I) (hb : b ⊆ R) :
-      ((a ∪ b : Finset E):Set E) ∩ (R:Set E) = b := by
-    ext e
-    simp only [Finset.coe_union,Set.mem_inter_iff,Set.mem_union,Finset.mem_coe]
-    constructor
-    · rintro ⟨he,hR⟩
-      rcases he with he | he
-      · exact ((Finset.disjoint_left.mp hdis) (ha he) hR).elim
-      · exact he
-    · intro he
-      exact ⟨Or.inr he,hb he⟩
-  trans bernoulliPolynomial I (fun a => f a*bernoulliPolynomial R (fun b => g b) p) p
-  · apply bernoulliPolynomial_congr
-    intro a ha
-    rw [←bernoulliPolynomial_const_mul]
-    apply bernoulliPolynomial_congr
-    intro b hb
-    rw [hf ((a∪b : Finset E):Set E),hg ((a∪b : Finset E):Set E),hi a b ha hb,hr a b ha hb]
-  · exact bernoulliPolynomial_mul_const I _ _ p
-
-theorem complement_incidentSupport (G : BondGraph V E) (hloc : LocallyFinite G) (W : Finset V) :
-    ((incidentSupport G hloc W:Set E))ᶜ = internalEdges G (W:Set V)ᶜ := by
-  ext e
-  simp only [Set.mem_compl_iff,Finset.mem_coe,mem_incidentSupport,internalEdges,Set.mem_ofPred_eq]
-  push Not
-  constructor
-  · intro h x hx hxW
-    exact h x hxW hx
-  · intro h x hxW hx
-    exact h x hx hxW
-
- 
-
-theorem cluster_restrict_incident_eq_iff (G : BondGraph V E) (hloc : LocallyFinite G)
-    (W : Finset V) (Q : Finset E) (hQ : incidentSupport G hloc W ⊆ Q)
-    (ω : Set E) (x : V) :
-    cluster G (ω ∩ (Q:Set E)) x = (W:Set V) ↔ cluster G ω x = W := by
-  have ho : ω \ (Q:Set E) ⊆ internalEdges G (W:Set V)ᶜ := by
-    rw [←complement_incidentSupport G hloc W]
-    exact fun _ h => fun he => h.2 (hQ he)
-  have h := cluster_union_external_eq_iff G (ω ∩ (Q:Set E)) (ω \ (Q:Set E)) x W ho
-  simpa only [Set.inter_union_sdiff] using h.symm
-
-theorem measurableSet_cluster_eq [Countable V] [Countable E]
-    (G : BondGraph V E) (x : V) (W : Set V) : MeasurableSet {ω : Set E | cluster G ω x = W} := by
-  have he : {ω : Set E | cluster G ω x = W} = ⋂ y, {ω : Set E | Conn G ω x y ↔ y ∈ W} := by
-    ext ω
-    simp only [Set.mem_ofPred_eq,Set.mem_iInter,Set.ext_iff,cluster]
-  rw [he]
-  apply MeasurableSet.iInter
-  intro y
-  by_cases hy : y ∈ W
-  · simpa only [hy,iff_true] using measurableSet_conn G x y
-  · simpa only [hy,iff_false,Set.compl_ofPred] using (measurableSet_conn G x y).compl
-
- 
-
-theorem cluster_cell_finite_exterior_independence [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (W : Finset V) (x : V)
-    (R : Finset E) (hR : (R:Set E) ⊆ internalEdges G (W:Set V)ᶜ)
-    (p : unitInterval) (f : Set E → ℝ) (hf : ∀ ω, f ω = f (ω ∩ (R:Set E))) :
-    (∫ ω, (if cluster G ω x = (W:Set V) then (1:ℝ) else 0)*f ω ∂law p) =
-      (law p).real {ω | cluster G ω x = (W:Set V)} * (∫ ω, f ω ∂law p) := by
-  have hdis : Disjoint (incidentSupport G hloc W) R := by
-    rw [←complement_incidentSupport G hloc W] at hR
-    exact Finset.disjoint_left.mpr (fun e he h => hR h he)
-  have hc (ω : Set E) :
-      (if cluster G ω x = (W:Set V) then (1:ℝ) else 0) =
-        (if cluster G (ω ∩ (incidentSupport G hloc W:Set E)) x = (W:Set V) then 1 else 0) := by
-    simp only [cluster_restrict_incident_eq_iff G hloc W _ (by rfl)]
-  rw [integral_mul_finite_local _ R hdis p _ f hc hf]
-  congr 1
-  have hm : MeasurableSet {ω : Set E | cluster G ω x = (W:Set V)} := measurableSet_cluster_eq G x W
-  simpa only [Set.indicator,Set.mem_ofPred_eq,Pi.one_apply] using integral_indicator_one hm
-
-end FiniteLocalIndependence
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-open scoped Topology
-section ExteriorConnectionIndependence
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem measurableSet_restrictedConn [Countable E] (G : BondGraph V E)
-    (S : Set E) (x y : V) : MeasurableSet {ω : Set E | Conn G (ω ∩ S) x y} :=
-  measurableSet_finiteWitness _
-    (fun _ _ hω hc => conn_mono G (Set.inter_subset_inter_left _ hω) hc)
-    (finiteWitness_inter _ (finiteWitness_conn G x y) S)
-
-theorem integral_event_indicator (p : unitInterval) (A : Set (Set E)) (hA : MeasurableSet A) :
-    (∫ ω, (if ω ∈ A then (1:ℝ) else 0) ∂law p) = (law p).real A := by
-  simpa only [Set.indicator,Pi.one_apply] using (integral_indicator_one (μ := law p) hA)
-
- 
-
-theorem cluster_cell_exterior_connection_independent [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (W : Finset V)
-    (x y z : V) (p : unitInterval) :
-    (law p).real {ω | cluster G ω x = (W:Set V) ∧ Conn G (ω ∩ internalEdges G (W:Set V)ᶜ) y z} =
-      (law p).real {ω | cluster G ω x = (W:Set V)} *
-        (law p).real {ω | Conn G (ω ∩ internalEdges G (W:Set V)ᶜ) y z} := by
-  let := Encodable.ofCountable E
-  let S := internalEdges G (W:Set V)ᶜ
-  let R := restrictedBondExhaustion S ∅
-  have hmono : Monotone R := restrictedBondExhaustion_mono S ∅
-  have hsub (n : ℕ) : (R n:Set E) ⊆ S := restrictedBondExhaustion_subset S ∅ (by simp) n
-  have hcover (e : E) (he : e ∈ S) : ∃ n, e ∈ R n := restrictedBondExhaustion_cover S ∅ e he
-  let A : Set (Set E) := {ω | cluster G ω x = (W:Set V)}
-  let B (n : ℕ) : Set (Set E) := {ω | Conn G (ω ∩ (R n:Set E)) y z}
-  let Btop : Set (Set E) := {ω | Conn G (ω ∩ S) y z}
-  have hB : Monotone B := fun n m h ω hc =>
-    conn_mono G (Set.inter_subset_inter_right _ (show (R n:Set E) ⊆ (R m:Set E) from hmono h)) hc
-  have hu : (⋃ n, B n) = Btop := by
-    ext ω
-    simp only [B,Btop,Set.mem_iUnion,Set.mem_ofPred_eq]
-    exact ⟨fun ⟨n,hn⟩ => conn_mono G (Set.inter_subset_inter_right _ (hsub n)) hn,
-      conn_finite_exhaustion G S R hmono hcover⟩
-  have huA : (⋃ n, A ∩ B n) = A ∩ Btop := by rw [←Set.inter_iUnion,hu]
-  have htB := tendsto_measure_iUnion_atTop (μ := law p) hB
-  have htA := tendsto_measure_iUnion_atTop (μ := law p)
-    (show Monotone (fun n => A ∩ B n) from fun _ _ h => Set.inter_subset_inter_right _ (hB h))
-  rw [hu] at htB
-  rw [huA] at htA
-  have htB' := (ENNReal.tendsto_toReal (measure_ne_top (law p) Btop)).comp htB
-  have htA' := (ENNReal.tendsto_toReal (measure_ne_top (law p) (A∩Btop))).comp htA
-  have he (n : ℕ) : (law p).real (A ∩ B n) = (law p).real A * (law p).real (B n) := by
-    have h := cluster_cell_finite_exterior_independence G hloc W x (R n) (hsub n) p
-      (fun ω => if Conn G (ω ∩ (R n:Set E)) y z then (1:ℝ) else 0)
-      (fun ω => by rw [Set.inter_assoc,Set.inter_self])
-    have hf : (∫ ω, (if cluster G ω x = (W:Set V) then (1:ℝ) else 0)*
-        (if Conn G (ω ∩ (R n:Set E)) y z then (1:ℝ) else 0) ∂law p) = (law p).real (A ∩ B n) := by
-      rw [←integral_event_indicator p (A∩B n)
-        ((measurableSet_cluster_eq G x W).inter (measurableSet_restrictedConn G _ y z))]
-      apply integral_congr_ae
-      exact Filter.Eventually.of_forall (fun ω => by
-        dsimp only [A,B,Set.mem_inter_iff,Set.mem_ofPred_eq]
-        split_ifs <;> simp_all)
-    rw [hf] at h
-    rw [show (∫ ω, (if Conn G (ω ∩ (R n:Set E)) y z then (1:ℝ) else 0) ∂law p) =
-      (law p).real (B n) from integral_event_indicator p (B n) (measurableSet_restrictedConn G _ y z)] at h
-    exact h
-  exact tendsto_nhds_unique htA' ((tendsto_const_nhds.mul htB').congr (fun n => (he n).symm))
-
-end ExteriorConnectionIndependence
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section ExposedClusterGeometry
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem conn_exterior_of_cluster_eq (G : BondGraph V E) (ω : Set E)
-    (x y z : V) (W : Set V) (hW : cluster G ω x = W) (hy : y ∉ W) :
-    Conn G (ω ∩ internalEdges G Wᶜ) y z ↔ Conn G ω y z := by
-  refine ⟨conn_mono G Set.inter_subset_left,fun h => ?_⟩
-  have hout (a : V) (ha : Conn G ω y a) : a ∉ W := by
-    intro haW
-    have hxa : Conn G ω x a := by change a ∈ cluster G ω x; rwa [hW]
-    exact hy (hW ▸ (hxa.trans (conn_symm G _ ha) : y ∈ cluster G ω x))
-  induction h with
-  | refl => exact conn_refl G _ y
-  | @tail a b hya hab ih =>
-    obtain ⟨e,he,heq⟩ := hab
-    exact ih.tail ⟨e,⟨he,(mem_internalEdges_of_ends G Wᶜ e a b heq).mpr
-      ⟨hout a hya,hout b (hya.tail ⟨e,he,heq⟩)⟩⟩,heq⟩
-
-theorem cluster_size_eq_tsum_real (G : BondGraph V E) (ω : Set E) (y : V)
-    (hf : (cluster G ω y).Finite) :
-    ((cluster G ω y).ncard:ℝ) = ∑' z, if Conn G ω y z then (1:ℝ) else 0 := by
-  rw [tsum_eq_sum (s := hf.toFinset) (fun z hz => by
-    exact ite_eq_right (fun hc => hz (hf.mem_toFinset.mpr hc)))]
-  rw [show (∑ z ∈ hf.toFinset, if Conn G ω y z then (1:ℝ) else 0) =
-      ∑ _z ∈ hf.toFinset, (1:ℝ) from Finset.sum_congr rfl (fun z hz => ite_eq_left (hf.mem_toFinset.mp hz))]
-  simp only [Finset.sum_const,nsmul_eq_mul,mul_one,←Set.ncard_eq_toFinset_card _ hf]
-
- 
-
-theorem integral_cluster_size_event [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (y : V) (A : Set (Set E)) (hA : MeasurableSet A) :
-    (∫ ω, (if ω ∈ A then ((cluster G ω y).ncard:ℝ) else 0) ∂law p) =
-      ∑' z, (law p).real {ω | ω ∈ A ∧ Conn G ω y z} := by
-  let f (z : V) (ω : Set E) : ℝ := if ω ∈ A ∧ Conn G ω y z then 1 else 0
-  have hm (z : V) : MeasurableSet {ω : Set E | ω ∈ A ∧ Conn G ω y z} :=
-    hA.inter (measurableSet_conn G y z)
-  have hf (z : V) : f z = {ω : Set E | ω ∈ A ∧ Conn G ω y z}.indicator (fun _ => (1:ℝ)) := by
-    funext ω
-    simp only [f,Set.indicator,Set.mem_ofPred_eq]
-  have hi (z : V) : Integrable (f z) (law p) := by
-    rw [hf]
-    exact (integrable_const (1:ℝ)).indicator (hm z)
-  have hn (z : V) (ω : Set E) : 0 ≤ f z ω := by unfold f; split_ifs <;> norm_num
-  have he (z : V) : (∫ ω, f z ω ∂law p) = (law p).real {ω | ω ∈ A ∧ Conn G ω y z} := by
-    rw [hf]
-    exact integral_indicator_one (hm z)
-  have hint : Summable (fun z => ∫ ω, ‖f z ω‖ ∂law p) := by
-    simp_rw [Real.norm_of_nonneg (hn _ _),he]
-    apply (summable_twoPoint_subcritical G hloc hqt p hp y).of_nonneg_of_le
-    · exact fun _ => ENNReal.toReal_nonneg
-    · intro z
-      exact ENNReal.toReal_mono (measure_ne_top _ _) (measure_mono (fun ω h => h.2))
-  calc
-    _ = ∫ ω, ∑' z, f z ω ∂law p := by
-      apply integral_congr_ae
-      filter_upwards [subcritical_cluster_finite_ae G p hp y] with ω hω
-      by_cases hωA : ω ∈ A
-      · simp only [hωA,ite_true,f,true_and]
-        exact cluster_size_eq_tsum_real G ω y hω
-      · simp only [hωA,ite_false,f,false_and,tsum_zero]
-    _ = ∑' z, ∫ ω, f z ω ∂law p := (integral_tsum_of_summable_integral_norm hi hint).symm
-    _ = _ := tsum_congr he
-
-end ExposedClusterGeometry
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section ExteriorCellSusceptibility
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem cluster_cell_exterior_susceptibility [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (W : Finset V) (x y : V) :
-    (∫ ω, (if cluster G ω x = (W:Set V) ∧ ¬Conn G ω x y
-      then ((cluster G ω y).ncard:ℝ) else 0) ∂law p) =
-      (law p).real {ω | cluster G ω x = (W:Set V)} *
-        inducedSusceptibility G (W:Set V)ᶜ p y := by
-  by_cases hy : y ∈ (W:Set V)
-  · rw [inducedSusceptibility,ite_eq_right (fun h => h hy),mul_zero]
-    calc
-      _ = ∫ _ω : Set E, (0:ℝ) ∂law p := by
-        apply integral_congr_ae
-        exact Filter.Eventually.of_forall (fun ω => by
-          dsimp only
-          split_ifs with h
-          · have hc : Conn G ω x y := by change y ∈ cluster G ω x; rwa [h.1]
-            exact (h.2 hc).elim
-          · rfl)
-      _ = 0 := integral_zero _ _
-  · simp only [inducedSusceptibility,Set.mem_compl_iff,hy,not_false_eq_true,ite_true]
-    calc
-      _ = ∫ ω, (if cluster G ω x = (W:Set V) then ((cluster G ω y).ncard:ℝ) else 0) ∂law p := by
-        apply integral_congr_ae
-        exact Filter.Eventually.of_forall (fun ω => by
-          dsimp only
-          by_cases hc : cluster G ω x = (W:Set V)
-          · have hnc : ¬Conn G ω x y := by intro h; exact hy (hc ▸ h)
-            simp [hc,hnc]
-          · simp only [hc,false_and,ite_false])
-      _ = ∑' z, (law p).real {ω | cluster G ω x = (W:Set V) ∧ Conn G ω y z} :=
-        integral_cluster_size_event G hloc hqt p hp y _ (measurableSet_cluster_eq G x W)
-      _ = ∑' z, (law p).real {ω | cluster G ω x = (W:Set V)} *
-          (law p).real {ω | Conn G (ω ∩ internalEdges G (W:Set V)ᶜ) y z} := by
-        apply tsum_congr
-        intro z
-        rw [show {ω : Set E | cluster G ω x = (W:Set V) ∧ Conn G ω y z} =
-            {ω | cluster G ω x = (W:Set V) ∧ Conn G (ω ∩ internalEdges G (W:Set V)ᶜ) y z} from by
-          ext ω
-          exact ⟨fun h => ⟨h.1,(conn_exterior_of_cluster_eq G ω x y z W h.1 hy).mpr h.2⟩,
-            fun h => ⟨h.1,conn_mono G Set.inter_subset_left h.2⟩⟩]
-        exact cluster_cell_exterior_connection_independent G hloc W x y z p
-      _ = _ := tsum_mul_left
-
-theorem restrictedSusceptibility_le_susceptibility [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (S : Set E) (y : V) :
-    restrictedSusceptibility G S p y ≤ susceptibility G p y := by
-  apply Summable.tsum_le_tsum _ (summable_restrictedTwoPoint G hloc hqt p hp S y)
-    (summable_twoPoint_subcritical G hloc hqt p hp y)
-  exact fun z => ENNReal.toReal_mono (measure_ne_top _ _) (measure_mono (fun ω hc => conn_mono G Set.inter_subset_left hc))
-
-theorem inducedSusceptibility_le_susceptibility [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (D : Set V) (y : V) :
-    inducedSusceptibility G D p y ≤ susceptibility G p y := by
-  unfold inducedSusceptibility
-  split_ifs
-  · exact restrictedSusceptibility_le_susceptibility G hloc hqt p hp _ y
-  · exact susceptibility_nonneg G p y
-
-end ExteriorCellSusceptibility
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section ClusterPartition
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem tsum_cluster_cells (G : BondGraph V E) (ω : Set E) (x : V)
-    (hf : (cluster G ω x).Finite) (a : ℝ) :
-    (∑' W : Finset V, if cluster G ω x = (W:Set V) then a else 0) = a := by
-  let W := hf.toFinset
-  have hW : cluster G ω x = (W:Set V) := hf.coe_toFinset.symm
-  rw [tsum_eq_single W (fun U hU => ite_eq_right (fun h => hU (Finset.coe_injective (h.symm.trans hW)))),ite_eq_left hW]
-
- 
-
-theorem integral_cluster_finite_partition [Countable V] [Countable E]
-    (G : BondGraph V E) (p : unitInterval) (hp : p < pc G) (x : V)
-    (f : Set E → ℝ) (hf : Integrable f (law p)) :
-    (∫ ω, f ω ∂law p) = ∑' W : Finset V,
-      ∫ ω, (if cluster G ω x = (W:Set V) then f ω else 0) ∂law p := by
-  let F (W : Finset V) (ω : Set E) := if cluster G ω x = (W:Set V) then f ω else 0
-  have hi (W : Finset V) : Integrable (F W) (law p) := by
-    have he : F W = {ω : Set E | cluster G ω x = (W:Set V)}.indicator f := by
-      funext ω
-      simp only [F,Set.indicator,Set.mem_ofPred_eq]
-      split_ifs <;> rfl
-    rw [he]
-    exact hf.indicator (measurableSet_cluster_eq G x W)
-  have hb (S : Finset (Finset V)) (ω : Set E) : (∑ W ∈ S, ‖F W ω‖) ≤ ‖f ω‖ := by
-    by_cases hex : ∃ W ∈ S, cluster G ω x = (W:Set V)
-    · obtain ⟨W,hWS,hW⟩ := hex
-      rw [Finset.sum_eq_single W]
-      · simp only [F,hW,ite_true,le_refl]
-      · intro U _ hU
-        have hu : cluster G ω x ≠ (U:Set V) := fun h => hU (Finset.coe_injective (h.symm.trans hW))
-        simp only [F,ite_eq_right hu,norm_zero]
-      · exact fun h => (h hWS).elim
-    · have he (W : Finset V) (hW : W ∈ S) : cluster G ω x ≠ (W:Set V) := fun h => hex ⟨W,hW,h⟩
-      calc
-        _ = 0 := Finset.sum_eq_zero (fun W hW => by simp only [F,ite_eq_right (he W hW),norm_zero])
-        _ ≤ _ := norm_nonneg _
-  have hn : Summable (fun W => ∫ ω, ‖F W ω‖ ∂law p) := by
-    apply summable_of_sum_le (fun _ => integral_nonneg (fun _ => norm_nonneg _))
-      (c := ∫ ω, ‖f ω‖ ∂law p)
-    intro S
-    rw [←integral_finsetSum _ (fun W _ => (hi W).norm)]
-    exact integral_mono (integrable_finsetSum _ (fun W _ => (hi W).norm)) hf.norm (hb S)
-  calc
-    _ = ∫ ω, ∑' W : Finset V, F W ω ∂law p := by
-      apply integral_congr_ae
-      filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
-      exact (tsum_cluster_cells G ω x hω (f ω)).symm
-    _ = _ := (integral_tsum_of_summable_integral_norm hi hn).symm
-
-end ClusterPartition
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section FreshExteriorMoment
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def freshExteriorMoment (G : BondGraph V E) (p : unitInterval)
-    (x y : V) (ω : Set E) : ℝ :=
-  ((cluster G ω x).ncard:ℝ) * inducedSusceptibility G (cluster G ω x)ᶜ p y
-
-theorem freshExteriorMoment_eq_tsum (G : BondGraph V E) (p : unitInterval)
-    (x y : V) (ω : Set E) :
-    freshExteriorMoment G p x y ω = ∑' W : Finset V, if cluster G ω x = (W:Set V)
-      then (W.card:ℝ)*inducedSusceptibility G (W:Set V)ᶜ p y else 0 := by
-  by_cases hf : (cluster G ω x).Finite
-  · let W := hf.toFinset
-    have hW : cluster G ω x = (W:Set V) := hf.coe_toFinset.symm
-    rw [tsum_eq_single W (fun U hU => ite_eq_right (fun h => hU (Finset.coe_injective (h.symm.trans hW)))),ite_eq_left hW]
-    simp only [freshExteriorMoment,hW,Set.ncard_coe_finset]
-  · have he (W : Finset V) : cluster G ω x ≠ (W:Set V) := fun h => hf (h ▸ W.finite_toSet)
-    simp only [freshExteriorMoment,Set.Infinite.ncard hf,Nat.cast_zero,zero_mul,ite_eq_right (he _),tsum_zero]
-
-theorem measurable_freshExteriorMoment [Countable V] [Countable E]
-    (G : BondGraph V E) (p : unitInterval) (x y : V) : Measurable (freshExteriorMoment G p x y) := by
-  simp_rw [show freshExteriorMoment G p x y = (fun ω => ∑' W : Finset V, if cluster G ω x = (W:Set V)
-      then (W.card:ℝ)*inducedSusceptibility G (W:Set V)ᶜ p y else 0) from funext (freshExteriorMoment_eq_tsum G p x y)]
-  apply Measurable.tsum
-  intro W
-  exact Measurable.ite (measurableSet_cluster_eq G x W) measurable_const measurable_const
-
-theorem freshExteriorMoment_nonneg (G : BondGraph V E) (p : unitInterval)
-    (x y : V) (ω : Set E) : 0 ≤ freshExteriorMoment G p x y ω :=
-  mul_nonneg (Nat.cast_nonneg _) (inducedSusceptibility_nonneg G _ p y)
-
-theorem integrable_freshExteriorMoment [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (x y : V) :
-    Integrable (freshExteriorMoment G p x y) (law p) := by
-  apply ((susceptibility_eq_integral G hloc hqt p hp x).1.mul_const (susceptibility G p y)).mono'
-    (measurable_freshExteriorMoment G p x y).aestronglyMeasurable
-  exact Filter.Eventually.of_forall (fun ω => by
-    rw [Real.norm_of_nonneg (freshExteriorMoment_nonneg G p x y ω)]
-    exact mul_le_mul_of_nonneg_left (inducedSusceptibility_le_susceptibility G hloc hqt p hp _ y) (Nat.cast_nonneg _))
-
-theorem integrable_distinct_cluster_product [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (x y : V) :
-    Integrable (fun ω => if ¬Conn G ω x y
-      then ((cluster G ω x).ncard:ℝ)*((cluster G ω y).ncard:ℝ) else 0) (law p) := by
-  have he : (fun ω : Set E => if ¬Conn G ω x y
-      then ((cluster G ω x).ncard:ℝ)*((cluster G ω y).ncard:ℝ) else 0) =
-      {ω : Set E | Conn G ω x y}ᶜ.indicator
-        (fun ω => ((cluster G ω x).ncard:ℝ)*((cluster G ω y).ncard:ℝ)) := by
-    funext ω
-    simp only [Set.indicator,Set.mem_compl_iff,Set.mem_ofPred_eq]
-  rw [he]
-  exact (subcritical_cluster_association G hloc hqt p hp x y).1.indicator
-    (measurableSet_conn G x y).compl
-
- 
-
-theorem integral_freshExteriorMoment [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (x y : V) :
-    (∫ ω, freshExteriorMoment G p x y ω ∂law p) =
-      ∫ ω, (if ¬Conn G ω x y then ((cluster G ω x).ncard:ℝ)*((cluster G ω y).ncard:ℝ) else 0) ∂law p := by
-  rw [integral_cluster_finite_partition G p hp x _ (integrable_freshExteriorMoment G hloc hqt p hp x y),
-    integral_cluster_finite_partition G p hp x _ (integrable_distinct_cluster_product G hloc hqt p hp x y)]
-  apply tsum_congr
-  intro W
-  have hl : (∫ ω, (if cluster G ω x = (W:Set V) then freshExteriorMoment G p x y ω else 0) ∂law p) =
-      (W.card:ℝ)*inducedSusceptibility G (W:Set V)ᶜ p y * (law p).real {ω | cluster G ω x = (W:Set V)} := by
-    rw [←integral_event_indicator p _ (measurableSet_cluster_eq G x W),←integral_const_mul]
-    apply integral_congr_ae
-    exact Filter.Eventually.of_forall (fun ω => by
-      dsimp only [Set.mem_ofPred_eq]
-      by_cases hc : cluster G ω x = (W:Set V)
-      · simp only [hc,ite_true,freshExteriorMoment,Set.ncard_coe_finset,mul_one]
-      · simp only [hc,ite_false,mul_zero])
-  have hr : (∫ ω, (if cluster G ω x = (W:Set V) then
-      (if ¬Conn G ω x y then ((cluster G ω x).ncard:ℝ)*((cluster G ω y).ncard:ℝ) else 0) else 0) ∂law p) =
-      (W.card:ℝ)*((law p).real {ω | cluster G ω x = (W:Set V)} * inducedSusceptibility G (W:Set V)ᶜ p y) := by
-    rw [←cluster_cell_exterior_susceptibility G hloc hqt p hp W x y,←integral_const_mul]
-    apply integral_congr_ae
-    exact Filter.Eventually.of_forall (fun ω => by
-      dsimp only
-      by_cases hc : cluster G ω x = (W:Set V)
-      · simp only [hc,true_and,ite_true,Set.ncard_coe_finset]
-        split_ifs <;> simp only [mul_zero]
-      · simp only [hc,false_and,ite_false,mul_zero])
-  rw [hl,hr]
-  ring
-
-end FreshExteriorMoment
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section DistinctClusterMoments
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def distinctClusterMoment (G : BondGraph V E) (p : unitInterval) (x y : V) : ℝ :=
-  ∫ ω, (if ¬Conn G ω x y then ((cluster G ω x).ncard:ℝ)*((cluster G ω y).ncard:ℝ) else 0) ∂law p
-
-theorem distinctClusterMoment_nonneg (G : BondGraph V E) (p : unitInterval) (x y : V) :
-    0 ≤ distinctClusterMoment G p x y := by
-  apply integral_nonneg
-  intro ω
-  dsimp only [Pi.zero_apply]
-  split_ifs <;> positivity
-
-theorem cluster_ncard_covariant (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (ω : Set E) (x : V) :
-    (cluster G (b ⁻¹' ω) x).ncard = (cluster G ω (a x)).ncard := by
-  rw [←cluster_aut G hab,Set.ncard_image_of_injective _ a.injective]
-
-theorem distinctClusterMoment_covariant [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (p : unitInterval)
-    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (x y : V) :
-    distinctClusterMoment G p (a x) (a y) = distinctClusterMoment G p x y := by
-  have hs (z : V) : Measurable (fun ω : Set E => ((cluster G ω z).ncard:ℝ)) :=
-    (measurable_of_countable (fun n : ℕ => (n:ℝ))).comp (measurable_cluster_ncard G hloc z)
-  have hm : Measurable (fun ω : Set E => if ¬Conn G ω x y
-      then ((cluster G ω x).ncard:ℝ)*((cluster G ω y).ncard:ℝ) else 0) :=
-    Measurable.ite (measurableSet_conn G x y).compl ((hs x).mul (hs y)) measurable_const
-  have h := integral_map (μ := law p) (measurable_set_preimage b).aemeasurable hm.aestronglyMeasurable
-  rw [law_map_preimage_equiv p b] at h
-  simpa only [distinctClusterMoment,conn_aut_iff G hab,cluster_ncard_covariant G hab] using h.symm
-
-noncomputable def neighborClusterMoment (G : BondGraph V E) (hloc : LocallyFinite G)
-    (p : unitInterval) (x : V) : ℝ :=
-  ∑ y ∈ neighbors G hloc x, distinctClusterMoment G p x y
-
-theorem neighborClusterMoment_nonneg (G : BondGraph V E) (hloc : LocallyFinite G)
-    (p : unitInterval) (x : V) : 0 ≤ neighborClusterMoment G hloc p x :=
-  Finset.sum_nonneg (fun y _ => distinctClusterMoment_nonneg G p x y)
-
-theorem neighbors_covariant (G : BondGraph V E) (hloc : LocallyFinite G)
-    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (x : V) :
-    neighbors G hloc (a x) = (neighbors G hloc x).map a.toEmbedding := by
-  ext y
-  simp only [mem_neighbors,Finset.mem_map,Equiv.coe_toEmbedding]
-  constructor
-  · intro h
-    refine ⟨a.symm y,?_,by simp⟩
-    simpa only [a.symm_apply_apply] using adj_aut G (isAut_symm G hab) h
-  · rintro ⟨z,hz,rfl⟩
-    exact adj_aut G hab hz
-
-theorem neighborClusterMoment_covariant [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (p : unitInterval)
-    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (x : V) :
-    neighborClusterMoment G hloc p (a x) = neighborClusterMoment G hloc p x := by
-  simp only [neighborClusterMoment,neighbors_covariant G hloc hab,Finset.sum_map,Equiv.coe_toEmbedding,
-    distinctClusterMoment_covariant G hloc p hab]
-
-noncomputable def rootNeighborMoment (G : BondGraph V E) (hloc : LocallyFinite G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (p : unitInterval) : ℝ :=
-  ∑ i, weight i * neighborClusterMoment G hloc p (root i)
-
-theorem neighborMoment_le_rootMoment [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (hw : ∀ i, 0 ≤ weight i) (hcov : ∀ x, ∃ i, x ∈ vertexOrbit G (root i))
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i) (p : unitInterval) (x : V) :
-    neighborClusterMoment G hloc p x ≤ c*rootNeighborMoment G hloc root weight p := by
-  obtain ⟨i,a,b,hab,hax⟩ := hcov x
-  rw [←hax,neighborClusterMoment_covariant G hloc p hab]
-  calc
-    _ ≤ (c*weight i)*neighborClusterMoment G hloc p (root i) := by
-      simpa only [one_mul] using mul_le_mul_of_nonneg_right (hmin i) (neighborClusterMoment_nonneg G hloc p _)
-    _ = c*(weight i*neighborClusterMoment G hloc p (root i)) := mul_assoc _ _ _
-    _ ≤ _ := mul_le_mul_of_nonneg_left
-      (Finset.single_le_sum (fun j _ => mul_nonneg (hw j) (neighborClusterMoment_nonneg G hloc p _)) (Finset.mem_univ i)) hc
-
-end DistinctClusterMoments
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Clusters.KernelMoments
+
+namespace OAI
+
+/-! Exterior Moments for bond percolation. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section BinDiagonalLoss
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def binDiagonalKernel (G : BondGraph V E) (p : unitInterval)
+    (o : V) (n : ℕ) (K : V → V → ℝ) (x y : V) : ℝ :=
+  ∫ ω, binDiagonalMass G o n K ω x y ∂law p
+
+theorem binDiagonalKernel_covariant (G : BondGraph V E) (hloc : LocallyFinite G)
+    (p : unitInterval) (o : V) (n : ℕ) (K : V → V → ℝ)
+    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b)
+    (hK : ∀ x y, K (a x) (a y) = K x y) (x y : V) :
+    binDiagonalKernel G p o n K (a x) (a y) = binDiagonalKernel G p o n K x y := by
+  have hm : Measurable (fun ω => binDiagonalMass G o n K ω x y) :=
+    measurable_finite_local _ _ (fun ω => binDiagonalMass_local G hloc o n K ω x y)
+  have h := integral_map (μ := law p) (measurable_set_preimage b).aemeasurable hm.aestronglyMeasurable
+  rw [law_map_preimage_equiv p b] at h
+  simpa only [binDiagonalKernel,binDiagonalMass_covariant G o n K hab hK] using h.symm
+
+theorem binDiagonalKernel_row_integral (G : BondGraph V E) (hloc : LocallyFinite G)
+    (p : unitInterval) (o : V) (n : ℕ) (K : V → V → ℝ) (x : V) :
+    (∑' y, binDiagonalKernel G p o n K x y) =
+      ∫ ω, binSizeMass G n ω x ^ 2 * ∑ z ∈ binOrbitClass G o n ω x, K x z ∂law p := by
+  rw [←show (∫ ω, ∑' y, binDiagonalMass G o n K ω x y ∂law p) =
+      ∑' y, binDiagonalKernel G p o n K x y from
+    integral_tsum_of_finite_support (law p) (finite_reachBall hloc x (2*n)).toFinset _
+      (fun y => binDiagonalMass_integrable G hloc p o n K x y)
+      (fun ω y hy => binDiagonalMass_outside G o n K ω x y (by simpa only [Set.Finite.mem_toFinset] using hy))]
+  simp only [binDiagonalMass_row]
+
+theorem binDiagonalKernel_column_integral (G : BondGraph V E) (hloc : LocallyFinite G)
+    (p : unitInterval) (o : V) (n : ℕ) (K : V → V → ℝ) (x : V) :
+    (∑' y, binDiagonalKernel G p o n K y x) =
+      ∫ ω, ∑' y, binDiagonalMass G o n K ω y x ∂law p := by
+  symm
+  exact integral_tsum_of_finite_support (law p) (finite_reachBall hloc x (2*n)).toFinset _
+    (fun y => binDiagonalMass_integrable G hloc p o n K y x)
+    (fun ω y hy => binDiagonalMass_outside_column G o n K ω y x (by simpa only [Set.Finite.mem_toFinset] using hy))
+
+theorem integral_binDiagonalLoss (G : BondGraph V E) (hloc : LocallyFinite G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (htr : RootTransport root weight Γ) (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b)
+    (hw : ∀ i, 0 ≤ weight i) (i₀ : ι) (hw₀ : 0 < weight i₀)
+    (p : unitInterval) (n : ℕ) (K : V → V → ℝ)
+    (hcov : ∀ a b, IsAut G a b → ∀ x y, K (a x) (a y) = K x y) :
+    (∫ ω, binSizeMass G n ω (root i₀)^2 *
+      ∑ z ∈ binOrbitClass G (root i₀) n ω (root i₀), K (root i₀) z ∂law p) =
+    ∫ ω, binSizeMass G n ω (root i₀)^2 / (binOrbitClass G (root i₀) n ω (root i₀)).card *
+      ∑ y ∈ binOrbitClass G (root i₀) n ω (root i₀),
+        ∑ z ∈ binOrbitClass G (root i₀) n ω (root i₀), K y z ∂law p := by
+  have hr (x : V) : Summable (binDiagonalKernel G p (root i₀) n K x) := by
+    apply summable_of_ne_finset_zero (s := (finite_reachBall hloc x (2*n)).toFinset)
+    intro y hy
+    simp only [binDiagonalKernel,binDiagonalMass_outside G (root i₀) n K _ x y
+      (by simpa only [Set.Finite.mem_toFinset] using hy),integral_zero]
+  have hc (y : V) : Summable (fun x => binDiagonalKernel G p (root i₀) n K x y) := by
+    apply summable_of_ne_finset_zero (s := (finite_reachBall hloc y (2*n)).toFinset)
+    intro x hx
+    simp only [binDiagonalKernel,binDiagonalMass_outside_column G (root i₀) n K _ x y
+      (by simpa only [Set.Finite.mem_toFinset] using hx),integral_zero]
+  have ht := orbit_kernel_column_eq_row G root weight Γ htr hΓ hw i₀ hw₀
+    (binDiagonalKernel G p (root i₀) n K)
+    (fun a b hab x y => binDiagonalKernel_covariant G hloc p (root i₀) n K hab (hcov a b hab) x y)
+    (fun x y hx => by simp only [binDiagonalKernel,binDiagonalMass_row_zero G _ n K _ x y hx,integral_zero])
+    (fun x y hy => by simp only [binDiagonalKernel,binDiagonalMass_column_zero G _ n K _ x y hy,integral_zero])
+    hr hc (root i₀) ⟨Equiv.refl _,Equiv.refl _,fun _ => by simp,rfl⟩
+  rw [binDiagonalKernel_row_integral G hloc,binDiagonalKernel_column_integral G hloc] at ht
+  simpa only [binDiagonalMass_column] using ht.symm
+
+ 
+
+theorem integral_binDiagonalLoss_le (G : BondGraph V E) (hloc : LocallyFinite G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (htr : RootTransport root weight Γ) (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b)
+    (hw : ∀ i, 0 ≤ weight i) (i₀ : ι) (hw₀ : 0 < weight i₀)
+    (p : unitInterval) (n : ℕ) (K : V → V → ℝ)
+    (hcov : ∀ a b, IsAut G a b → ∀ x y, K (a x) (a y) = K x y)
+    (A : Hilbert V →L[ℂ] Hilbert V)
+    (hA : ∀ x y, A (lp.single 2 y (1:ℂ)) x = (K x y : ℂ)) :
+    (∫ ω, binSizeMass G n ω (root i₀)^2 *
+      ∑ z ∈ binOrbitClass G (root i₀) n ω (root i₀), K (root i₀) z ∂law p) ≤
+      ‖A‖ * ∫ ω, binSizeMass G n ω (root i₀)^2 ∂law p := by
+  rw [integral_binDiagonalLoss G hloc root weight Γ htr hΓ hw i₀ hw₀ p n K hcov]
+  have hi : Integrable (fun ω => ∑' y, binDiagonalMass G (root i₀) n K ω y (root i₀)) (law p) := by
+    have he (ω : Set E) : (∑' y, binDiagonalMass G (root i₀) n K ω y (root i₀)) =
+        ∑ y ∈ (finite_reachBall hloc (root i₀) (2*n)).toFinset, binDiagonalMass G (root i₀) n K ω y (root i₀) := by
+      apply tsum_eq_sum
+      intro y hy
+      exact binDiagonalMass_outside_column G (root i₀) n K ω y (root i₀)
+        (by simpa only [Set.Finite.mem_toFinset] using hy)
+    simp only [he]
+    exact integrable_finsetSum _ (fun y _ => binDiagonalMass_integrable G hloc p (root i₀) n K y (root i₀))
+  have hs : Integrable (fun ω => binSizeMass G n ω (root i₀)^2) (law p) :=
+    integrable_finite_local _ p _ (fun ω => congrArg (fun t : ℝ => t^2) (binSizeMass_local G hloc n ω (root i₀)))
+  rw [←integral_const_mul]
+  exact integral_mono (by simpa only [binDiagonalMass_column] using hi) (hs.const_mul ‖A‖)
+    (fun ω => by simpa only [binDiagonalMass_column] using binDiagonalMass_column_le G (root i₀) n K A hA ω (root i₀))
+
+end BinDiagonalLoss
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section DyadicDiagonalStrata
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def fullOrbitClass (G : BondGraph V E) (o : V) (ω : Set E) (x : V) : Finset V :=
+  if h : (cluster G ω x).Finite then h.toFinset.filter (fun y => y ∈ vertexOrbit G o) else ∅
+
+noncomputable def diagonalLossMass (G : BondGraph V E) (o : V) (K : V → V → ℝ)
+    (ω : Set E) (x : V) : ℝ :=
+  ((cluster G ω x).ncard:ℝ)^2 * ∑ z ∈ fullOrbitClass G o ω x, K x z
+
+theorem binSizeMass_zero (G : BondGraph V E) (n : ℕ) (ω : Set E) (x : V)
+    (hw : ¬ ClusterWindow G x n (2*n) ω) : binSizeMass G n ω x = 0 := by
+  simp only [binSizeMass,binMass_zero_of_not_window G n ω x _ hw,tsum_zero]
+
+theorem binOrbitClass_eq_full (G : BondGraph V E) (o : V) (n : ℕ) (ω : Set E) (x : V)
+    (hx : x ∈ vertexOrbit G o) (hw : ClusterWindow G x n (2*n) ω) :
+    binOrbitClass G o n ω x = fullOrbitClass G o ω x := by
+  simp only [binOrbitClass,ite_eq_left hx,binClusterSlice,dite_eq_left hw,fullOrbitClass,dite_eq_left hw.1]
+
+theorem dyadic_window_iff (G : BondGraph V E) (ω : Set E) (x : V) (j : ℕ)
+    (hz : (cluster G ω x).ncard ≠ 0) :
+    ClusterWindow G x (2^j) (2*(2^j)) ω ↔ Nat.log 2 (cluster G ω x).ncard = j := by
+  constructor
+  · intro h
+    apply Nat.log_eq_of_pow_le_of_lt_pow h.2.1
+    simpa only [pow_succ,Nat.mul_comm] using h.2.2
+  · intro h
+    refine ⟨Set.finite_of_ncard_ne_zero hz,?_,?_⟩
+    · rw [←h]
+      exact Nat.pow_log_le_self 2 hz
+    · rw [←h,show 2*2^(Nat.log 2 (cluster G ω x).ncard) = 2^(Nat.log 2 (cluster G ω x).ncard+1) by simp only [pow_succ,Nat.mul_comm]]
+      exact Nat.lt_pow_succ_log_self (by norm_num) _
+
+theorem dyadic_binSizeMass_sq (G : BondGraph V E) (ω : Set E) (x : V) (j : ℕ) :
+    binSizeMass G (2^j) ω x ^ 2 =
+      if Nat.log 2 (cluster G ω x).ncard = j then ((cluster G ω x).ncard:ℝ)^2 else 0 := by
+  by_cases hz : (cluster G ω x).ncard = 0
+  · have hw : ¬ClusterWindow G x (2^j) (2*(2^j)) ω := by
+      intro h
+      have hp : 0 < 2^j := pow_pos (by omega) _
+      have hl := h.2.1
+      omega
+    simp only [binSizeMass_zero G (2^j) ω x hw,hz,Nat.cast_zero,zero_pow (by omega : 2 ≠ 0),ite_self]
+  · by_cases hw : ClusterWindow G x (2^j) (2*(2^j)) ω
+    · rw [binSizeMass_eq G (2^j) ω x hw,ite_eq_left ((dyadic_window_iff G ω x j hz).mp hw)]
+    · rw [binSizeMass_zero G (2^j) ω x hw,ite_eq_right (fun h => hw ((dyadic_window_iff G ω x j hz).mpr h))]
+      norm_num
+
+theorem dyadic_binDiagonalMass (G : BondGraph V E) (o : V) (K : V → V → ℝ)
+    (ω : Set E) (x : V) (hx : x ∈ vertexOrbit G o) (j : ℕ) :
+    binSizeMass G (2^j) ω x ^ 2 * ∑ z ∈ binOrbitClass G o (2^j) ω x, K x z =
+      if Nat.log 2 (cluster G ω x).ncard = j then diagonalLossMass G o K ω x else 0 := by
+  by_cases hw : ClusterWindow G x (2^j) (2*(2^j)) ω
+  · have hz : (cluster G ω x).ncard ≠ 0 := by
+      have hp : 0 < 2^j := pow_pos (by omega) _
+      have hl := hw.2.1
+      omega
+    rw [ite_eq_left ((dyadic_window_iff G ω x j hz).mp hw),binOrbitClass_eq_full G o (2^j) ω x hx hw,
+      binSizeMass_eq G (2^j) ω x hw]
+    rfl
+  · rw [binSizeMass_zero G (2^j) ω x hw]
+    simp only [zero_pow (by omega : 2 ≠ 0),zero_mul]
+    by_cases hz : (cluster G ω x).ncard = 0
+    · simp only [diagonalLossMass,hz,Nat.cast_zero,zero_pow (by omega : 2 ≠ 0),zero_mul,ite_self]
+    · rw [ite_eq_right (fun h => hw ((dyadic_window_iff G ω x j hz).mpr h))]
+
+theorem diagonalLossMass_nonneg (G : BondGraph V E) (o : V) (K : V → V → ℝ)
+    (hK : ∀ x y, 0 ≤ K x y) (ω : Set E) (x : V) : 0 ≤ diagonalLossMass G o K ω x :=
+  mul_nonneg (sq_nonneg _) (Finset.sum_nonneg (fun z _ => hK x z))
+
+theorem diagonalLossMass_le (G : BondGraph V E) (o : V) (K : V → V → ℝ)
+    (L : ℝ) (hrow : ∀ x (S : Finset V), (∑ z ∈ S, K x z) ≤ L) (ω : Set E) (x : V) :
+    diagonalLossMass G o K ω x ≤ L*((cluster G ω x).ncard:ℝ)^2 := by
+  exact (mul_le_mul_of_nonneg_left (hrow x _) (sq_nonneg _)).trans_eq (mul_comm _ _)
+
+end DyadicDiagonalStrata
+end Percolation
+
+ 
+namespace Percolation
+open MeasureTheory Filter
+open scoped Topology
+section DiscreteStratumIntegral
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem integral_le_of_discrete_strata {Ω : Type*} [MeasurableSpace Ω]
+    (μ : Measure Ω) (J : Ω → ℕ) (f g : Ω → ℝ) (C L : ℝ)
+    (hf0 : ∀ ω, 0 ≤ f ω) (hg0 : ∀ ω, 0 ≤ g ω)
+    (hfg : ∀ ω, f ω ≤ L*g ω) (hgi : Integrable g μ)
+    (hfi : ∀ j, Integrable (fun ω => if J ω = j then f ω else 0) μ)
+    (hgi' : ∀ j, Integrable (fun ω => if J ω = j then g ω else 0) μ)
+    (h : ∀ j, (∫ ω, (if J ω = j then f ω else 0) ∂μ) ≤
+      C*(∫ ω, (if J ω = j then g ω else 0) ∂μ)) :
+    (∫ ω, f ω ∂μ) ≤ C*(∫ ω, g ω ∂μ) := by
+  let F : ℕ → Ω → ℝ := fun N ω => ∑ j ∈ Finset.range N, if J ω = j then f ω else 0
+  let H : ℕ → Ω → ℝ := fun N ω => ∑ j ∈ Finset.range N, if J ω = j then g ω else 0
+  have he (q : Ω → ℝ) (N : ℕ) (ω : Ω) :
+      (∑ j ∈ Finset.range N, if J ω = j then q ω else 0) = if J ω < N then q ω else 0 := by
+    simp only [Finset.sum_ite_eq,Finset.mem_range]
+  have hFi (N : ℕ) : Integrable (F N) μ := integrable_finsetSum _ (fun j _ => hfi j)
+  have hHi (N : ℕ) : Integrable (H N) μ := integrable_finsetSum _ (fun j _ => hgi' j)
+  have hFbody (N : ℕ) (ω : Ω) : ‖F N ω‖ ≤ L*g ω := by
+    dsimp [F]
+    rw [he]
+    split_ifs
+    · simpa only [Real.norm_eq_abs,abs_of_nonneg (hf0 ω)] using hfg ω
+    · simpa only [norm_zero,abs_zero] using (hf0 ω).trans (hfg ω)
+  have hHbody (N : ℕ) (ω : Ω) : ‖H N ω‖ ≤ g ω := by
+    dsimp [H]
+    rw [he]
+    split_ifs
+    · simp only [abs_of_nonneg (hg0 ω),le_refl]
+    · simpa only [norm_zero,abs_zero] using hg0 ω
+  have ht (q : Ω → ℝ) (ω : Ω) : Tendsto
+      (fun N => ∑ j ∈ Finset.range N, if J ω = j then q ω else 0) atTop (𝓝 (q ω)) := by
+    have he' : (fun N => ∑ j ∈ Finset.range N, if J ω = j then q ω else 0) =ᶠ[atTop] (fun _ => q ω) := by
+      filter_upwards [eventually_ge_atTop (J ω+1)] with N hN
+      rw [he,ite_eq_left (by omega)]
+    exact tendsto_const_nhds.congr' he'.symm
+  have htF := tendsto_integral_of_dominated_convergence (fun ω => L*g ω)
+    (fun N => (hFi N).aestronglyMeasurable) (hgi.const_mul L)
+    (fun N => Eventually.of_forall (hFbody N)) (Eventually.of_forall (ht f))
+  have htH := tendsto_integral_of_dominated_convergence g
+    (fun N => (hHi N).aestronglyMeasurable) hgi
+    (fun N => Eventually.of_forall (hHbody N)) (Eventually.of_forall (ht g))
+  apply le_of_tendsto_of_tendsto htF (tendsto_const_nhds.mul htH)
+  exact Eventually.of_forall (fun N => by
+    change (∫ ω, F N ω ∂μ) ≤ C*(∫ ω, H N ω ∂μ)
+    simp only [F,H,integral_finsetSum _ (fun j _ => hfi j),
+      integral_finsetSum _ (fun j _ => hgi' j),Finset.mul_sum]
+    exact Finset.sum_le_sum (fun j _ => h j))
+
+end DiscreteStratumIntegral
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section DiagonalLoss
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem integral_diagonalLoss_le (G : BondGraph V E) (hloc : LocallyFinite G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (htr : RootTransport root weight Γ) (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b)
+    (hw : ∀ i, 0 ≤ weight i) (i₀ : ι) (hw₀ : 0 < weight i₀)
+    (p : unitInterval) (K : V → V → ℝ) (hK : ∀ x y, 0 ≤ K x y)
+    (hcov : ∀ a b, IsAut G a b → ∀ x y, K (a x) (a y) = K x y)
+    (L : ℝ) (hrow : ∀ x (S : Finset V), (∑ z ∈ S, K x z) ≤ L)
+    (A : Hilbert V →L[ℂ] Hilbert V)
+    (hA : ∀ x y, A (lp.single 2 y (1:ℂ)) x = (K x y : ℂ))
+    (hs : Integrable (fun ω => ((cluster G ω (root i₀)).ncard:ℝ)^2) (law p)) :
+    (∫ ω, diagonalLossMass G (root i₀) K ω (root i₀) ∂law p) ≤
+      ‖A‖*(∫ ω, ((cluster G ω (root i₀)).ncard:ℝ)^2 ∂law p) := by
+  have hx : root i₀ ∈ vertexOrbit G (root i₀) := ⟨Equiv.refl _,Equiv.refl _,fun _ => by simp,rfl⟩
+  have hs' (j : ℕ) : Integrable (fun ω => binSizeMass G (2^j) ω (root i₀)^2) (law p) :=
+    integrable_finite_local _ p _ (fun ω => congrArg (fun t : ℝ => t^2)
+      (binSizeMass_local G hloc (2^j) ω (root i₀)))
+  have hf' (j : ℕ) : Integrable (fun ω => binSizeMass G (2^j) ω (root i₀)^2 *
+      ∑ z ∈ binOrbitClass G (root i₀) (2^j) ω (root i₀), K (root i₀) z) (law p) := by
+    apply integrable_finite_local _ p _
+    intro ω
+    rw [binOrbitClass_local G hloc (root i₀) (2^j) ω (root i₀),binSizeMass_local G hloc (2^j) ω (root i₀)]
+  apply integral_le_of_discrete_strata (law p) (fun ω => Nat.log 2 (cluster G ω (root i₀)).ncard)
+    (fun ω => diagonalLossMass G (root i₀) K ω (root i₀))
+    (fun ω => ((cluster G ω (root i₀)).ncard:ℝ)^2) ‖A‖ L
+    (fun ω => diagonalLossMass_nonneg G (root i₀) K hK ω (root i₀))
+    (fun _ => sq_nonneg _) (fun ω => diagonalLossMass_le G (root i₀) K L hrow ω (root i₀)) hs
+  · intro j
+    simpa only [dyadic_binDiagonalMass G (root i₀) K _ (root i₀) hx] using hf' j
+  · intro j
+    simpa only [dyadic_binSizeMass_sq] using hs' j
+  · intro j
+    have h := integral_binDiagonalLoss_le G hloc root weight Γ htr hΓ hw i₀ hw₀ p (2^j) K hcov A hA
+    simp only [dyadic_binDiagonalMass G (root i₀) K _ (root i₀) hx] at h
+    simpa only [dyadic_binSizeMass_sq] using h
+
+ 
+
+theorem subcritical_diagonalLoss_le [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (htr : RootTransport root weight Γ) (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b)
+    (hw : ∀ i, 0 ≤ weight i) (i₀ : ι) (hw₀ : 0 < weight i₀)
+    (p : unitInterval) (hp : p < pc G) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ (K : V → V → ℝ), (∀ x y, 0 ≤ K x y) →
+      (∀ a b, IsAut G a b → ∀ x y, K (a x) (a y) = K x y) →
+      ∀ (L : ℝ), (∀ x (S : Finset V), (∑ z ∈ S, K x z) ≤ L) →
+      ∀ (A : Hilbert V →L[ℂ] Hilbert V),
+      (∀ x y, A (lp.single 2 y (1:ℂ)) x = (K x y : ℂ)) →
+      (∫ ω, diagonalLossMass G (root i₀) K ω (root i₀) ∂law p) ≤ ‖A‖*C^3 := by
+  obtain ⟨C,hC,hs⟩ := subcritical_second_moment G hloc hqt p hp
+  refine ⟨C,hC,fun K hK hcov L hrow A hA => ?_⟩
+  exact (integral_diagonalLoss_le G hloc root weight Γ htr hΓ hw i₀ hw₀ p K hK hcov L hrow A hA
+    (hs (root i₀)).1).trans (mul_le_mul_of_nonneg_left (hs (root i₀)).2 (norm_nonneg _))
+
+end DiagonalLoss
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+open scoped ENNReal
+section RootSecondMoment
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem second_moment_le_rootSusceptibility_cube [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (hw : ∀ i, 0 ≤ weight i) (hcov : ∀ x, ∃ i, x ∈ vertexOrbit G (root i))
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (p : unitInterval) (hp : p < pc G) (x : V) :
+    (∫ ω, ((cluster G ω x).ncard:ℝ)^2 ∂law p) ≤
+      (c*rootSusceptibility G root weight p)^3 := by
+  let C := c*rootSusceptibility G root weight p
+  have hC : 0 ≤ C := mul_nonneg hc (rootSusceptibility_nonneg G root weight hw p)
+  have hrow (z : V) : (∑' y, law p {ω | Conn G ω z y}) ≤ ENNReal.ofReal C := by
+    rw [←ennreal_susceptibility G hloc hqt p hp]
+    exact ENNReal.ofReal_le_ofReal (susceptibility_le_rootSusceptibility G root weight hw hcov c hc hmin p z)
+  have hm : Measurable (fun ω : Set E => ((cluster G ω x).ncard:ℝ)^2) :=
+    (measurable_of_countable (fun n : ℕ => (n:ℝ)^2)).comp (measurable_cluster_ncard G hloc x)
+  have hn : 0 ≤ᵐ[law p] (fun ω => ((cluster G ω x).ncard:ℝ)^2) :=
+    Filter.Eventually.of_forall (fun _ => sq_nonneg _)
+  have he : (fun ω => ENNReal.ofReal (((cluster G ω x).ncard:ℝ)^2)) =ᵐ[law p]
+      (fun ω => clusterMass G ω x ^ 2) := by
+    filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
+    rw [clusterMass_eq_ncard G ω x hω,ENNReal.ofReal_pow (Nat.cast_nonneg _)]
+    simp only [ENNReal.ofReal_natCast]
+  have hbound : (∫⁻ ω, ENNReal.ofReal (((cluster G ω x).ncard:ℝ)^2) ∂law p) ≤ ENNReal.ofReal (C^3) := by
+    rw [lintegral_congr_ae he,ENNReal.ofReal_pow hC]
+    exact cluster_second_moment_bound G p _ hrow x
+  rw [integral_eq_lintegral_of_nonneg_ae hn hm.aestronglyMeasurable]
+  exact (ENNReal.toReal_mono ENNReal.ofReal_ne_top hbound).trans_eq (ENNReal.toReal_ofReal (pow_nonneg hC _))
+
+theorem diagonalLoss_le_rootSusceptibility_cube [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (htr : RootTransport root weight Γ) (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b)
+    (hw : ∀ i, 0 ≤ weight i) (i₀ : ι) (hw₀ : 0 < weight i₀)
+    (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i))
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (p : unitInterval) (hp : p < pc G) (K : V → V → ℝ) (hK : ∀ x y, 0 ≤ K x y)
+    (hcov : ∀ a b, IsAut G a b → ∀ x y, K (a x) (a y) = K x y)
+    (L : ℝ) (hrow : ∀ x (S : Finset V), (∑ z ∈ S, K x z) ≤ L)
+    (A : Hilbert V →L[ℂ] Hilbert V)
+    (hA : ∀ x y, A (lp.single 2 y (1:ℂ)) x = (K x y : ℂ)) :
+    (∫ ω, diagonalLossMass G (root i₀) K ω (root i₀) ∂law p) ≤
+      ‖A‖*(c*rootSusceptibility G root weight p)^3 := by
+  obtain ⟨_,_,hs⟩ := subcritical_second_moment G hloc hqt p hp
+  exact (integral_diagonalLoss_le G hloc root weight Γ htr hΓ hw i₀ hw₀ p K hK hcov L hrow A hA
+    (hs (root i₀)).1).trans (mul_le_mul_of_nonneg_left
+      (second_moment_le_rootSusceptibility_cube G hloc hqt root weight hw hcover c hc hmin p hp (root i₀))
+      (norm_nonneg _))
+
+end RootSecondMoment
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section DiagonalSeries
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def diagonalLossTerm (G : BondGraph V E) (o : V) (K : V → V → ℝ)
+    (x y : V) (ω : Set E) : ℝ :=
+  if y ∈ vertexOrbit G o ∧ Conn G ω x y then K x y*((cluster G ω x).ncard:ℝ)^2 else 0
+
+theorem diagonalLossTerm_nonneg (G : BondGraph V E) (o : V) (K : V → V → ℝ)
+    (hK : ∀ x y, 0 ≤ K x y) (x y : V) (ω : Set E) :
+    0 ≤ diagonalLossTerm G o K x y ω := by
+  unfold diagonalLossTerm
+  split_ifs
+  · exact mul_nonneg (hK x y) (sq_nonneg _)
+  · exact le_rfl
+
+theorem diagonalLossTerm_le (G : BondGraph V E) (o : V) (K : V → V → ℝ)
+    (hK : ∀ x y, 0 ≤ K x y) (x y : V) (ω : Set E) :
+    diagonalLossTerm G o K x y ω ≤ K x y*((cluster G ω x).ncard:ℝ)^2 := by
+  unfold diagonalLossTerm
+  split_ifs
+  · exact le_rfl
+  · exact mul_nonneg (hK x y) (sq_nonneg _)
+
+theorem diagonalLossMass_eq_tsum (G : BondGraph V E) (o : V) (K : V → V → ℝ)
+    (x : V) (ω : Set E) :
+    diagonalLossMass G o K ω x = ∑' y, diagonalLossTerm G o K x y ω := by
+  by_cases hf : (cluster G ω x).Finite
+  · let S := fullOrbitClass G o ω x
+    have hs (y : V) : y ∈ S ↔ y ∈ vertexOrbit G o ∧ Conn G ω x y := by
+      dsimp only [S,fullOrbitClass]
+      rw [dite_eq_left hf]
+      simp only [Finset.mem_filter,Set.Finite.mem_toFinset,cluster,Set.mem_ofPred_eq]
+      exact and_comm
+    rw [tsum_eq_sum (s := S) (fun y hy => by
+      simp only [diagonalLossTerm,ite_eq_right (fun h => hy ((hs y).mpr h))])]
+    rw [diagonalLossMass,Finset.mul_sum]
+    apply Finset.sum_congr rfl
+    intro y hy
+    rw [diagonalLossTerm,ite_eq_left ((hs y).mp hy),mul_comm]
+  · have hn := Set.Infinite.ncard hf
+    simp only [diagonalLossMass,diagonalLossTerm,hn,Nat.cast_zero,zero_pow (by omega : 2 ≠ 0),
+      mul_zero,zero_mul,ite_self,tsum_zero]
+
+theorem integrable_diagonalLossTerm [Countable V] [Countable E] (G : BondGraph V E) (o : V) (K : V → V → ℝ)
+    (x y : V) (p : unitInterval)
+    (hs : Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^2) (law p)) :
+    Integrable (diagonalLossTerm G o K x y) (law p) := by
+  by_cases hy : y ∈ vertexOrbit G o
+  · convert (hs.indicator (measurableSet_conn G x y)).const_mul (K x y) using 1
+    ext ω
+    simp only [diagonalLossTerm,hy,true_and,Set.indicator,Set.mem_ofPred_eq]
+    split_ifs <;> simp only [mul_zero]
+  · have he : diagonalLossTerm G o K x y = (fun _ : Set E => (0:ℝ)) := by
+      funext ω
+      simp only [diagonalLossTerm,hy,false_and,ite_false]
+    rw [he]
+    exact integrable_const 0
+
+ 
+
+theorem integral_diagonalLoss_eq_series [Countable V] [Countable E]
+    (G : BondGraph V E) (o : V) (K : V → V → ℝ) (hK : ∀ x y, 0 ≤ K x y)
+    (L : ℝ) (hrow : ∀ x (S : Finset V), (∑ z ∈ S, K x z) ≤ L)
+    (x : V) (p : unitInterval)
+    (hs : Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^2) (law p)) :
+    (∫ ω, diagonalLossMass G o K ω x ∂law p) =
+      ∑' y, if y ∈ vertexOrbit G o then
+        K x y*(∫ ω, (if Conn G ω x y then ((cluster G ω x).ncard:ℝ)^2 else 0) ∂law p) else 0 := by
+  have hi (y : V) := integrable_diagonalLossTerm G o K x y p hs
+  have hzero (y : V) (ω : Set E) := diagonalLossTerm_nonneg G o K hK x y ω
+  have hn : Summable (fun y => ∫ ω, ‖diagonalLossTerm G o K x y ω‖ ∂law p) := by
+    simp only [Real.norm_of_nonneg (hzero _ _)]
+    apply summable_of_sum_le (fun y => integral_nonneg (hzero y)) (c := L*(∫ ω, ((cluster G ω x).ncard:ℝ)^2 ∂law p))
+    intro S
+    rw [←integral_finsetSum _ (fun y _ => hi y),←integral_const_mul]
+    apply integral_mono (integrable_finsetSum _ (fun y _ => hi y)) (hs.const_mul L)
+    intro ω
+    calc
+      _ ≤ ∑ y ∈ S, K x y*((cluster G ω x).ncard:ℝ)^2 :=
+        Finset.sum_le_sum (fun y _ => diagonalLossTerm_le G o K hK x y ω)
+      _ = (∑ y ∈ S, K x y)*((cluster G ω x).ncard:ℝ)^2 := (Finset.sum_mul ..).symm
+      _ ≤ _ := mul_le_mul_of_nonneg_right (hrow x S) (sq_nonneg _)
+  rw [show (fun ω => diagonalLossMass G o K ω x) = (fun ω => ∑' y, diagonalLossTerm G o K x y ω) from
+    funext (diagonalLossMass_eq_tsum G o K x)]
+  rw [←integral_tsum_of_summable_integral_norm hi hn]
+  apply tsum_congr
+  intro y
+  by_cases hy : y ∈ vertexOrbit G o
+  · simp only [diagonalLossTerm,hy,true_and,ite_true]
+    rw [←integral_const_mul]
+    apply integral_congr_ae
+    exact Filter.Eventually.of_forall (fun ω => by dsimp only; split_ifs <;> simp only [mul_zero])
+  · simp only [diagonalLossTerm,hy,false_and,ite_false,integral_zero]
+
+end DiagonalSeries
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section RootSusceptibilityLower
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem exists_rootSusceptibility_lower [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
+    (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i))
+    (p₀ : ℝ) (hp₀ : 0 < p₀) :
+    ∃ a : ℝ, 0 < a ∧ ∀ (p : unitInterval), p₀ ≤ (p:ℝ) → p < pc G → ∀ x,
+      a*rootSusceptibility G root weight p ≤ susceptibility G p x := by
+  have hex (ij : ι × ι) : ∃ l : List E, Along G (root ij.1) l (root ij.2) := by
+    obtain ⟨l,hl,_,_⟩ := conn_exists_bond_trail G (conn_univ_of_connected G hconn (root ij.1) (root ij.2))
+    exact ⟨l,hl⟩
+  choose path hpath using hex
+  let N := ∑ ij : ι × ι, (path ij).length
+  have hlen (i j : ι) : (path (i,j)).length ≤ N :=
+    Finset.single_le_sum (f := fun ij : ι × ι => (path ij).length) (fun _ _ => Nat.zero_le _) (Finset.mem_univ (i,j))
+  refine ⟨p₀^N,pow_pos hp₀ N,fun p hp hpcrit x => ?_⟩
+  obtain ⟨j,a,b,hab,hax⟩ := hcover x
+  have hfin (z : V) : Summable (fun y => (law p).real {ω | Conn G (ω ∩ Set.univ) z y}) := by
+    simp only [Set.inter_univ]
+    exact summable_twoPoint_subcritical G hloc hqt p hpcrit z
+  have heq (z : V) : restrictedSusceptibility G Set.univ p z = susceptibility G p z := by
+    simp only [restrictedSusceptibility,susceptibility,Set.inter_univ,twoPoint,Measure.real]
+  have hb (i : ι) : p₀^N*susceptibility G p (root i) ≤ susceptibility G p (root j) := by
+    have h := restrictedSusceptibility_path_lower G Set.univ p (hpath (j,i)) (fun _ _ => Set.mem_univ _) hfin
+    rw [heq,heq] at h
+    have hpow : p₀^N ≤ (p:ℝ)^(path (j,i)).length :=
+      (pow_le_pow_left₀ hp₀.le hp N).trans (pow_le_pow_of_le_one p.2.1 p.2.2 (hlen j i))
+    exact (mul_le_mul_of_nonneg_right hpow (susceptibility_nonneg G p _)).trans h
+  rw [←hax,susceptibility_aut G p hab]
+  calc
+    _ = ∑ i, weight i*(p₀^N*susceptibility G p (root i)) := by
+      simp only [rootSusceptibility,Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro i _
+      ring
+    _ ≤ ∑ i, weight i*susceptibility G p (root j) :=
+      Finset.sum_le_sum (fun i _ => mul_le_mul_of_nonneg_left (hb i) (hw i))
+    _ = susceptibility G p (root j) := by rw [←Finset.sum_mul,hw1,one_mul]
+
+end RootSusceptibilityLower
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+open scoped ENNReal
+section ClusterAssociation
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem association_finiteWitness_ennreal [Countable E] (p : unitInterval)
+    (A B : Set (Set E)) (hA : IsUpperSet A) (hB : IsUpperSet B)
+    (hfA : FiniteWitness A) (hfB : FiniteWitness B) :
+    law p A * law p B ≤ law p (A ∩ B) := by
+  apply (ENNReal.toReal_le_toReal
+    (ENNReal.mul_ne_top (measure_ne_top _ _) (measure_ne_top _ _)) (measure_ne_top _ _)).mp
+  rw [ENNReal.toReal_mul]
+  exact association_finiteWitness p A B hA hB hfA hfB
+
+theorem clusterMass_mul (G : BondGraph V E) (ω : Set E) (x y : V) :
+    clusterMass G ω x * clusterMass G ω y =
+      ∑' a, ∑' b, if Conn G ω x a ∧ Conn G ω y b then (1:ℝ≥0∞) else 0 := by
+  rw [clusterMass,clusterMass,←ENNReal.tsum_mul_right]
+  apply tsum_congr
+  intro a
+  rw [←ENNReal.tsum_mul_left]
+  apply tsum_congr
+  intro b
+  split_ifs <;> simp_all
+
+theorem lintegral_clusterMass_mul [Countable V] [Countable E]
+    (G : BondGraph V E) (p : unitInterval) (x y : V) :
+    (∫⁻ ω, clusterMass G ω x * clusterMass G ω y ∂law p) =
+      ∑' a, ∑' b, law p {ω | Conn G ω x a ∧ Conn G ω y b} := by
+  have hm (a b : V) : Measurable (fun ω : Set E => if Conn G ω x a ∧ Conn G ω y b then (1:ℝ≥0∞) else 0) :=
+    Measurable.ite ((measurableSet_conn G x a).inter (measurableSet_conn G y b)) measurable_const measurable_const
+  simp_rw [clusterMass_mul]
+  rw [lintegral_tsum (fun a => (Measurable.tsum (fun b => hm a b)).aemeasurable)]
+  apply tsum_congr
+  intro a
+  rw [lintegral_tsum (fun b => (hm a b).aemeasurable)]
+  apply tsum_congr
+  intro b
+  have hs : MeasurableSet {ω : Set E | Conn G ω x a ∧ Conn G ω y b} :=
+    (measurableSet_conn G x a).inter (measurableSet_conn G y b)
+  simpa only [Set.indicator,Set.mem_ofPred_eq,Pi.one_apply] using (lintegral_indicator_one (μ := law p) hs)
+
+theorem clusterMass_association [Countable V] [Countable E]
+    (G : BondGraph V E) (p : unitInterval) (x y : V) :
+    (∑' a, law p {ω | Conn G ω x a})*(∑' b, law p {ω | Conn G ω y b}) ≤
+      ∫⁻ ω, clusterMass G ω x * clusterMass G ω y ∂law p := by
+  rw [lintegral_clusterMass_mul,←ENNReal.tsum_mul_right]
+  apply ENNReal.tsum_le_tsum
+  intro a
+  rw [←ENNReal.tsum_mul_left]
+  apply ENNReal.tsum_le_tsum
+  intro b
+  exact association_finiteWitness_ennreal p _ _
+    (fun _ _ hω hc => conn_mono G hω hc) (fun _ _ hω hc => conn_mono G hω hc)
+    (finiteWitness_conn G x a) (finiteWitness_conn G y b)
+
+ 
+
+theorem subcritical_cluster_association [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (x y : V) :
+    Integrable (fun ω => ((cluster G ω x).ncard:ℝ)*((cluster G ω y).ncard:ℝ)) (law p) ∧
+    susceptibility G p x * susceptibility G p y ≤
+      ∫ ω, ((cluster G ω x).ncard:ℝ)*((cluster G ω y).ncard:ℝ) ∂law p := by
+  have hm (z : V) : Measurable (fun ω : Set E => ((cluster G ω z).ncard:ℝ)) :=
+    (measurable_of_countable (fun n : ℕ => (n:ℝ))).comp (measurable_cluster_ncard G hloc z)
+  have hn (ω : Set E) : 0 ≤ ((cluster G ω x).ncard:ℝ)*((cluster G ω y).ncard:ℝ) :=
+    mul_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _)
+  obtain ⟨_,_,hsecond⟩ := subcritical_second_moment G hloc hqt p hp
+  have hi : Integrable (fun ω => ((cluster G ω x).ncard:ℝ)*((cluster G ω y).ncard:ℝ)) (law p) := by
+    apply ((hsecond x).1.add (hsecond y).1).mono' ((hm x).mul (hm y)).aestronglyMeasurable
+    exact Filter.Eventually.of_forall (fun ω => by
+      dsimp only [Pi.mul_apply,Pi.add_apply]
+      rw [Real.norm_of_nonneg (hn ω)]
+      nlinarith [sq_nonneg (((cluster G ω x).ncard:ℝ)-((cluster G ω y).ncard:ℝ))])
+  refine ⟨hi,?_⟩
+  have he : (fun ω => ENNReal.ofReal (((cluster G ω x).ncard:ℝ)*((cluster G ω y).ncard:ℝ))) =ᵐ[law p]
+      (fun ω => clusterMass G ω x * clusterMass G ω y) := by
+    filter_upwards [subcritical_cluster_finite_ae G p hp x,subcritical_cluster_finite_ae G p hp y] with ω hx hy
+    rw [clusterMass_eq_ncard G ω x hx,clusterMass_eq_ncard G ω y hy,ENNReal.ofReal_mul (Nat.cast_nonneg _)]
+    simp only [ENNReal.ofReal_natCast]
+  have hb : ENNReal.ofReal (susceptibility G p x * susceptibility G p y) ≤
+      ∫⁻ ω, ENNReal.ofReal (((cluster G ω x).ncard:ℝ)*((cluster G ω y).ncard:ℝ)) ∂law p := by
+    rw [lintegral_congr_ae he,ENNReal.ofReal_mul (susceptibility_nonneg G p x),
+      ennreal_susceptibility G hloc hqt p hp,ennreal_susceptibility G hloc hqt p hp]
+    exact clusterMass_association G p x y
+  have htop : (∫⁻ ω, ENNReal.ofReal (((cluster G ω x).ncard:ℝ)*((cluster G ω y).ncard:ℝ)) ∂law p) ≠ ⊤ :=
+    ne_of_lt ((hasFiniteIntegral_iff_ofReal (Filter.Eventually.of_forall hn)).mp hi.hasFiniteIntegral)
+  rw [integral_eq_lintegral_of_nonneg_ae (Filter.Eventually.of_forall hn) hi.aestronglyMeasurable]
+  simpa only [ENNReal.toReal_ofReal (mul_nonneg (susceptibility_nonneg G p x) (susceptibility_nonneg G p y))] using
+    ENNReal.toReal_mono htop hb
+
+end ClusterAssociation
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section FiniteLocalIndependence
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem integral_mul_finite_local (I R : Finset E) (hdis : Disjoint I R)
+    (p : unitInterval) (f g : Set E → ℝ)
+    (hf : ∀ ω, f ω = f (ω ∩ (I:Set E)))
+    (hg : ∀ ω, g ω = g (ω ∩ (R:Set E))) :
+    (∫ ω, f ω*g ω ∂law p) = (∫ ω, f ω ∂law p)*(∫ ω, g ω ∂law p) := by
+  have hf' := finite_local_of_subset I (I∪R) Finset.subset_union_left f hf
+  have hg' := finite_local_of_subset R (I∪R) Finset.subset_union_right g hg
+  rw [integral_eq_bernoulliPolynomial (I∪R) p _ (fun ω => by rw [hf' ω,hg' ω]),
+    integral_eq_bernoulliPolynomial I p f hf,integral_eq_bernoulliPolynomial R p g hg,
+    bernoulliPolynomial_union I R hdis]
+  have hi (a b : Finset E) (ha : a ⊆ I) (hb : b ⊆ R) :
+      ((a ∪ b : Finset E):Set E) ∩ (I:Set E) = a := by
+    ext e
+    simp only [Finset.coe_union,Set.mem_inter_iff,Set.mem_union,Finset.mem_coe]
+    constructor
+    · rintro ⟨he,hI⟩
+      rcases he with he | he
+      · exact he
+      · exact ((Finset.disjoint_left.mp hdis) hI (hb he)).elim
+    · intro he
+      exact ⟨Or.inl he,ha he⟩
+  have hr (a b : Finset E) (ha : a ⊆ I) (hb : b ⊆ R) :
+      ((a ∪ b : Finset E):Set E) ∩ (R:Set E) = b := by
+    ext e
+    simp only [Finset.coe_union,Set.mem_inter_iff,Set.mem_union,Finset.mem_coe]
+    constructor
+    · rintro ⟨he,hR⟩
+      rcases he with he | he
+      · exact ((Finset.disjoint_left.mp hdis) (ha he) hR).elim
+      · exact he
+    · intro he
+      exact ⟨Or.inr he,hb he⟩
+  trans bernoulliPolynomial I (fun a => f a*bernoulliPolynomial R (fun b => g b) p) p
+  · apply bernoulliPolynomial_congr
+    intro a ha
+    rw [←bernoulliPolynomial_const_mul]
+    apply bernoulliPolynomial_congr
+    intro b hb
+    rw [hf ((a∪b : Finset E):Set E),hg ((a∪b : Finset E):Set E),hi a b ha hb,hr a b ha hb]
+  · exact bernoulliPolynomial_mul_const I _ _ p
+
+theorem complement_incidentSupport (G : BondGraph V E) (hloc : LocallyFinite G) (W : Finset V) :
+    ((incidentSupport G hloc W:Set E))ᶜ = internalEdges G (W:Set V)ᶜ := by
+  ext e
+  simp only [Set.mem_compl_iff,Finset.mem_coe,mem_incidentSupport,internalEdges,Set.mem_ofPred_eq]
+  push Not
+  constructor
+  · intro h x hx hxW
+    exact h x hxW hx
+  · intro h x hxW hx
+    exact h x hx hxW
+
+ 
+
+theorem cluster_restrict_incident_eq_iff (G : BondGraph V E) (hloc : LocallyFinite G)
+    (W : Finset V) (Q : Finset E) (hQ : incidentSupport G hloc W ⊆ Q)
+    (ω : Set E) (x : V) :
+    cluster G (ω ∩ (Q:Set E)) x = (W:Set V) ↔ cluster G ω x = W := by
+  have ho : ω \ (Q:Set E) ⊆ internalEdges G (W:Set V)ᶜ := by
+    rw [←complement_incidentSupport G hloc W]
+    exact fun _ h => fun he => h.2 (hQ he)
+  have h := cluster_union_external_eq_iff G (ω ∩ (Q:Set E)) (ω \ (Q:Set E)) x W ho
+  simpa only [Set.inter_union_sdiff] using h.symm
+
+theorem measurableSet_cluster_eq [Countable V] [Countable E]
+    (G : BondGraph V E) (x : V) (W : Set V) : MeasurableSet {ω : Set E | cluster G ω x = W} := by
+  have he : {ω : Set E | cluster G ω x = W} = ⋂ y, {ω : Set E | Conn G ω x y ↔ y ∈ W} := by
+    ext ω
+    simp only [Set.mem_ofPred_eq,Set.mem_iInter,Set.ext_iff,cluster]
+  rw [he]
+  apply MeasurableSet.iInter
+  intro y
+  by_cases hy : y ∈ W
+  · simpa only [hy,iff_true] using measurableSet_conn G x y
+  · simpa only [hy,iff_false,Set.compl_ofPred] using (measurableSet_conn G x y).compl
+
+ 
+
+theorem cluster_cell_finite_exterior_independence [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (W : Finset V) (x : V)
+    (R : Finset E) (hR : (R:Set E) ⊆ internalEdges G (W:Set V)ᶜ)
+    (p : unitInterval) (f : Set E → ℝ) (hf : ∀ ω, f ω = f (ω ∩ (R:Set E))) :
+    (∫ ω, (if cluster G ω x = (W:Set V) then (1:ℝ) else 0)*f ω ∂law p) =
+      (law p).real {ω | cluster G ω x = (W:Set V)} * (∫ ω, f ω ∂law p) := by
+  have hdis : Disjoint (incidentSupport G hloc W) R := by
+    rw [←complement_incidentSupport G hloc W] at hR
+    exact Finset.disjoint_left.mpr (fun e he h => hR h he)
+  have hc (ω : Set E) :
+      (if cluster G ω x = (W:Set V) then (1:ℝ) else 0) =
+        (if cluster G (ω ∩ (incidentSupport G hloc W:Set E)) x = (W:Set V) then 1 else 0) := by
+    simp only [cluster_restrict_incident_eq_iff G hloc W _ (by rfl)]
+  rw [integral_mul_finite_local _ R hdis p _ f hc hf]
+  congr 1
+  have hm : MeasurableSet {ω : Set E | cluster G ω x = (W:Set V)} := measurableSet_cluster_eq G x W
+  simpa only [Set.indicator,Set.mem_ofPred_eq,Pi.one_apply] using integral_indicator_one hm
+
+end FiniteLocalIndependence
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+open scoped Topology
+section ExteriorConnectionIndependence
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem measurableSet_restrictedConn [Countable E] (G : BondGraph V E)
+    (S : Set E) (x y : V) : MeasurableSet {ω : Set E | Conn G (ω ∩ S) x y} :=
+  measurableSet_finiteWitness _
+    (fun _ _ hω hc => conn_mono G (Set.inter_subset_inter_left _ hω) hc)
+    (finiteWitness_inter _ (finiteWitness_conn G x y) S)
+
+theorem integral_event_indicator (p : unitInterval) (A : Set (Set E)) (hA : MeasurableSet A) :
+    (∫ ω, (if ω ∈ A then (1:ℝ) else 0) ∂law p) = (law p).real A := by
+  simpa only [Set.indicator,Pi.one_apply] using (integral_indicator_one (μ := law p) hA)
+
+ 
+
+theorem cluster_cell_exterior_connection_independent [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (W : Finset V)
+    (x y z : V) (p : unitInterval) :
+    (law p).real {ω | cluster G ω x = (W:Set V) ∧ Conn G (ω ∩ internalEdges G (W:Set V)ᶜ) y z} =
+      (law p).real {ω | cluster G ω x = (W:Set V)} *
+        (law p).real {ω | Conn G (ω ∩ internalEdges G (W:Set V)ᶜ) y z} := by
+  let := Encodable.ofCountable E
+  let S := internalEdges G (W:Set V)ᶜ
+  let R := restrictedBondExhaustion S ∅
+  have hmono : Monotone R := restrictedBondExhaustion_mono S ∅
+  have hsub (n : ℕ) : (R n:Set E) ⊆ S := restrictedBondExhaustion_subset S ∅ (by simp) n
+  have hcover (e : E) (he : e ∈ S) : ∃ n, e ∈ R n := restrictedBondExhaustion_cover S ∅ e he
+  let A : Set (Set E) := {ω | cluster G ω x = (W:Set V)}
+  let B (n : ℕ) : Set (Set E) := {ω | Conn G (ω ∩ (R n:Set E)) y z}
+  let Btop : Set (Set E) := {ω | Conn G (ω ∩ S) y z}
+  have hB : Monotone B := fun n m h ω hc =>
+    conn_mono G (Set.inter_subset_inter_right _ (show (R n:Set E) ⊆ (R m:Set E) from hmono h)) hc
+  have hu : (⋃ n, B n) = Btop := by
+    ext ω
+    simp only [B,Btop,Set.mem_iUnion,Set.mem_ofPred_eq]
+    exact ⟨fun ⟨n,hn⟩ => conn_mono G (Set.inter_subset_inter_right _ (hsub n)) hn,
+      conn_finite_exhaustion G S R hmono hcover⟩
+  have huA : (⋃ n, A ∩ B n) = A ∩ Btop := by rw [←Set.inter_iUnion,hu]
+  have htB := tendsto_measure_iUnion_atTop (μ := law p) hB
+  have htA := tendsto_measure_iUnion_atTop (μ := law p)
+    (show Monotone (fun n => A ∩ B n) from fun _ _ h => Set.inter_subset_inter_right _ (hB h))
+  rw [hu] at htB
+  rw [huA] at htA
+  have htB' := (ENNReal.tendsto_toReal (measure_ne_top (law p) Btop)).comp htB
+  have htA' := (ENNReal.tendsto_toReal (measure_ne_top (law p) (A∩Btop))).comp htA
+  have he (n : ℕ) : (law p).real (A ∩ B n) = (law p).real A * (law p).real (B n) := by
+    have h := cluster_cell_finite_exterior_independence G hloc W x (R n) (hsub n) p
+      (fun ω => if Conn G (ω ∩ (R n:Set E)) y z then (1:ℝ) else 0)
+      (fun ω => by rw [Set.inter_assoc,Set.inter_self])
+    have hf : (∫ ω, (if cluster G ω x = (W:Set V) then (1:ℝ) else 0)*
+        (if Conn G (ω ∩ (R n:Set E)) y z then (1:ℝ) else 0) ∂law p) = (law p).real (A ∩ B n) := by
+      rw [←integral_event_indicator p (A∩B n)
+        ((measurableSet_cluster_eq G x W).inter (measurableSet_restrictedConn G _ y z))]
+      apply integral_congr_ae
+      exact Filter.Eventually.of_forall (fun ω => by
+        dsimp only [A,B,Set.mem_inter_iff,Set.mem_ofPred_eq]
+        split_ifs <;> simp_all)
+    rw [hf] at h
+    rw [show (∫ ω, (if Conn G (ω ∩ (R n:Set E)) y z then (1:ℝ) else 0) ∂law p) =
+      (law p).real (B n) from integral_event_indicator p (B n) (measurableSet_restrictedConn G _ y z)] at h
+    exact h
+  exact tendsto_nhds_unique htA' ((tendsto_const_nhds.mul htB').congr (fun n => (he n).symm))
+
+end ExteriorConnectionIndependence
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section ExposedClusterGeometry
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem conn_exterior_of_cluster_eq (G : BondGraph V E) (ω : Set E)
+    (x y z : V) (W : Set V) (hW : cluster G ω x = W) (hy : y ∉ W) :
+    Conn G (ω ∩ internalEdges G Wᶜ) y z ↔ Conn G ω y z := by
+  refine ⟨conn_mono G Set.inter_subset_left,fun h => ?_⟩
+  have hout (a : V) (ha : Conn G ω y a) : a ∉ W := by
+    intro haW
+    have hxa : Conn G ω x a := by change a ∈ cluster G ω x; rwa [hW]
+    exact hy (hW ▸ (hxa.trans (conn_symm G _ ha) : y ∈ cluster G ω x))
+  induction h with
+  | refl => exact conn_refl G _ y
+  | @tail a b hya hab ih =>
+    obtain ⟨e,he,heq⟩ := hab
+    exact ih.tail ⟨e,⟨he,(mem_internalEdges_of_ends G Wᶜ e a b heq).mpr
+      ⟨hout a hya,hout b (hya.tail ⟨e,he,heq⟩)⟩⟩,heq⟩
+
+theorem cluster_size_eq_tsum_real (G : BondGraph V E) (ω : Set E) (y : V)
+    (hf : (cluster G ω y).Finite) :
+    ((cluster G ω y).ncard:ℝ) = ∑' z, if Conn G ω y z then (1:ℝ) else 0 := by
+  rw [tsum_eq_sum (s := hf.toFinset) (fun z hz => by
+    exact ite_eq_right (fun hc => hz (hf.mem_toFinset.mpr hc)))]
+  rw [show (∑ z ∈ hf.toFinset, if Conn G ω y z then (1:ℝ) else 0) =
+      ∑ _z ∈ hf.toFinset, (1:ℝ) from Finset.sum_congr rfl (fun z hz => ite_eq_left (hf.mem_toFinset.mp hz))]
+  simp only [Finset.sum_const,nsmul_eq_mul,mul_one,←Set.ncard_eq_toFinset_card _ hf]
+
+ 
+
+theorem integral_cluster_size_event [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (y : V) (A : Set (Set E)) (hA : MeasurableSet A) :
+    (∫ ω, (if ω ∈ A then ((cluster G ω y).ncard:ℝ) else 0) ∂law p) =
+      ∑' z, (law p).real {ω | ω ∈ A ∧ Conn G ω y z} := by
+  let f (z : V) (ω : Set E) : ℝ := if ω ∈ A ∧ Conn G ω y z then 1 else 0
+  have hm (z : V) : MeasurableSet {ω : Set E | ω ∈ A ∧ Conn G ω y z} :=
+    hA.inter (measurableSet_conn G y z)
+  have hf (z : V) : f z = {ω : Set E | ω ∈ A ∧ Conn G ω y z}.indicator (fun _ => (1:ℝ)) := by
+    funext ω
+    simp only [f,Set.indicator,Set.mem_ofPred_eq]
+  have hi (z : V) : Integrable (f z) (law p) := by
+    rw [hf]
+    exact (integrable_const (1:ℝ)).indicator (hm z)
+  have hn (z : V) (ω : Set E) : 0 ≤ f z ω := by unfold f; split_ifs <;> norm_num
+  have he (z : V) : (∫ ω, f z ω ∂law p) = (law p).real {ω | ω ∈ A ∧ Conn G ω y z} := by
+    rw [hf]
+    exact integral_indicator_one (hm z)
+  have hint : Summable (fun z => ∫ ω, ‖f z ω‖ ∂law p) := by
+    simp_rw [Real.norm_of_nonneg (hn _ _),he]
+    apply (summable_twoPoint_subcritical G hloc hqt p hp y).of_nonneg_of_le
+    · exact fun _ => ENNReal.toReal_nonneg
+    · intro z
+      exact ENNReal.toReal_mono (measure_ne_top _ _) (measure_mono (fun ω h => h.2))
+  calc
+    _ = ∫ ω, ∑' z, f z ω ∂law p := by
+      apply integral_congr_ae
+      filter_upwards [subcritical_cluster_finite_ae G p hp y] with ω hω
+      by_cases hωA : ω ∈ A
+      · simp only [hωA,ite_true,f,true_and]
+        exact cluster_size_eq_tsum_real G ω y hω
+      · simp only [hωA,ite_false,f,false_and,tsum_zero]
+    _ = ∑' z, ∫ ω, f z ω ∂law p := (integral_tsum_of_summable_integral_norm hi hint).symm
+    _ = _ := tsum_congr he
+
+end ExposedClusterGeometry
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section ExteriorCellSusceptibility
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem cluster_cell_exterior_susceptibility [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (W : Finset V) (x y : V) :
+    (∫ ω, (if cluster G ω x = (W:Set V) ∧ ¬Conn G ω x y
+      then ((cluster G ω y).ncard:ℝ) else 0) ∂law p) =
+      (law p).real {ω | cluster G ω x = (W:Set V)} *
+        inducedSusceptibility G (W:Set V)ᶜ p y := by
+  by_cases hy : y ∈ (W:Set V)
+  · rw [inducedSusceptibility,ite_eq_right (fun h => h hy),mul_zero]
+    calc
+      _ = ∫ _ω : Set E, (0:ℝ) ∂law p := by
+        apply integral_congr_ae
+        exact Filter.Eventually.of_forall (fun ω => by
+          dsimp only
+          split_ifs with h
+          · have hc : Conn G ω x y := by change y ∈ cluster G ω x; rwa [h.1]
+            exact (h.2 hc).elim
+          · rfl)
+      _ = 0 := integral_zero _ _
+  · simp only [inducedSusceptibility,Set.mem_compl_iff,hy,not_false_eq_true,ite_true]
+    calc
+      _ = ∫ ω, (if cluster G ω x = (W:Set V) then ((cluster G ω y).ncard:ℝ) else 0) ∂law p := by
+        apply integral_congr_ae
+        exact Filter.Eventually.of_forall (fun ω => by
+          dsimp only
+          by_cases hc : cluster G ω x = (W:Set V)
+          · have hnc : ¬Conn G ω x y := by intro h; exact hy (hc ▸ h)
+            simp [hc,hnc]
+          · simp only [hc,false_and,ite_false])
+      _ = ∑' z, (law p).real {ω | cluster G ω x = (W:Set V) ∧ Conn G ω y z} :=
+        integral_cluster_size_event G hloc hqt p hp y _ (measurableSet_cluster_eq G x W)
+      _ = ∑' z, (law p).real {ω | cluster G ω x = (W:Set V)} *
+          (law p).real {ω | Conn G (ω ∩ internalEdges G (W:Set V)ᶜ) y z} := by
+        apply tsum_congr
+        intro z
+        rw [show {ω : Set E | cluster G ω x = (W:Set V) ∧ Conn G ω y z} =
+            {ω | cluster G ω x = (W:Set V) ∧ Conn G (ω ∩ internalEdges G (W:Set V)ᶜ) y z} from by
+          ext ω
+          exact ⟨fun h => ⟨h.1,(conn_exterior_of_cluster_eq G ω x y z W h.1 hy).mpr h.2⟩,
+            fun h => ⟨h.1,conn_mono G Set.inter_subset_left h.2⟩⟩]
+        exact cluster_cell_exterior_connection_independent G hloc W x y z p
+      _ = _ := tsum_mul_left
+
+theorem restrictedSusceptibility_le_susceptibility [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (S : Set E) (y : V) :
+    restrictedSusceptibility G S p y ≤ susceptibility G p y := by
+  apply Summable.tsum_le_tsum _ (summable_restrictedTwoPoint G hloc hqt p hp S y)
+    (summable_twoPoint_subcritical G hloc hqt p hp y)
+  exact fun z => ENNReal.toReal_mono (measure_ne_top _ _) (measure_mono (fun ω hc => conn_mono G Set.inter_subset_left hc))
+
+theorem inducedSusceptibility_le_susceptibility [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (D : Set V) (y : V) :
+    inducedSusceptibility G D p y ≤ susceptibility G p y := by
+  unfold inducedSusceptibility
+  split_ifs
+  · exact restrictedSusceptibility_le_susceptibility G hloc hqt p hp _ y
+  · exact susceptibility_nonneg G p y
+
+end ExteriorCellSusceptibility
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section ClusterPartition
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem tsum_cluster_cells (G : BondGraph V E) (ω : Set E) (x : V)
+    (hf : (cluster G ω x).Finite) (a : ℝ) :
+    (∑' W : Finset V, if cluster G ω x = (W:Set V) then a else 0) = a := by
+  let W := hf.toFinset
+  have hW : cluster G ω x = (W:Set V) := hf.coe_toFinset.symm
+  rw [tsum_eq_single W (fun U hU => ite_eq_right (fun h => hU (Finset.coe_injective (h.symm.trans hW)))),ite_eq_left hW]
+
+ 
+
+theorem integral_cluster_finite_partition [Countable V] [Countable E]
+    (G : BondGraph V E) (p : unitInterval) (hp : p < pc G) (x : V)
+    (f : Set E → ℝ) (hf : Integrable f (law p)) :
+    (∫ ω, f ω ∂law p) = ∑' W : Finset V,
+      ∫ ω, (if cluster G ω x = (W:Set V) then f ω else 0) ∂law p := by
+  let F (W : Finset V) (ω : Set E) := if cluster G ω x = (W:Set V) then f ω else 0
+  have hi (W : Finset V) : Integrable (F W) (law p) := by
+    have he : F W = {ω : Set E | cluster G ω x = (W:Set V)}.indicator f := by
+      funext ω
+      simp only [F,Set.indicator,Set.mem_ofPred_eq]
+      split_ifs <;> rfl
+    rw [he]
+    exact hf.indicator (measurableSet_cluster_eq G x W)
+  have hb (S : Finset (Finset V)) (ω : Set E) : (∑ W ∈ S, ‖F W ω‖) ≤ ‖f ω‖ := by
+    by_cases hex : ∃ W ∈ S, cluster G ω x = (W:Set V)
+    · obtain ⟨W,hWS,hW⟩ := hex
+      rw [Finset.sum_eq_single W]
+      · simp only [F,hW,ite_true,le_refl]
+      · intro U _ hU
+        have hu : cluster G ω x ≠ (U:Set V) := fun h => hU (Finset.coe_injective (h.symm.trans hW))
+        simp only [F,ite_eq_right hu,norm_zero]
+      · exact fun h => (h hWS).elim
+    · have he (W : Finset V) (hW : W ∈ S) : cluster G ω x ≠ (W:Set V) := fun h => hex ⟨W,hW,h⟩
+      calc
+        _ = 0 := Finset.sum_eq_zero (fun W hW => by simp only [F,ite_eq_right (he W hW),norm_zero])
+        _ ≤ _ := norm_nonneg _
+  have hn : Summable (fun W => ∫ ω, ‖F W ω‖ ∂law p) := by
+    apply summable_of_sum_le (fun _ => integral_nonneg (fun _ => norm_nonneg _))
+      (c := ∫ ω, ‖f ω‖ ∂law p)
+    intro S
+    rw [←integral_finsetSum _ (fun W _ => (hi W).norm)]
+    exact integral_mono (integrable_finsetSum _ (fun W _ => (hi W).norm)) hf.norm (hb S)
+  calc
+    _ = ∫ ω, ∑' W : Finset V, F W ω ∂law p := by
+      apply integral_congr_ae
+      filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
+      exact (tsum_cluster_cells G ω x hω (f ω)).symm
+    _ = _ := (integral_tsum_of_summable_integral_norm hi hn).symm
+
+end ClusterPartition
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section FreshExteriorMoment
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def freshExteriorMoment (G : BondGraph V E) (p : unitInterval)
+    (x y : V) (ω : Set E) : ℝ :=
+  ((cluster G ω x).ncard:ℝ) * inducedSusceptibility G (cluster G ω x)ᶜ p y
+
+theorem freshExteriorMoment_eq_tsum (G : BondGraph V E) (p : unitInterval)
+    (x y : V) (ω : Set E) :
+    freshExteriorMoment G p x y ω = ∑' W : Finset V, if cluster G ω x = (W:Set V)
+      then (W.card:ℝ)*inducedSusceptibility G (W:Set V)ᶜ p y else 0 := by
+  by_cases hf : (cluster G ω x).Finite
+  · let W := hf.toFinset
+    have hW : cluster G ω x = (W:Set V) := hf.coe_toFinset.symm
+    rw [tsum_eq_single W (fun U hU => ite_eq_right (fun h => hU (Finset.coe_injective (h.symm.trans hW)))),ite_eq_left hW]
+    simp only [freshExteriorMoment,hW,Set.ncard_coe_finset]
+  · have he (W : Finset V) : cluster G ω x ≠ (W:Set V) := fun h => hf (h ▸ W.finite_toSet)
+    simp only [freshExteriorMoment,Set.Infinite.ncard hf,Nat.cast_zero,zero_mul,ite_eq_right (he _),tsum_zero]
+
+theorem measurable_freshExteriorMoment [Countable V] [Countable E]
+    (G : BondGraph V E) (p : unitInterval) (x y : V) : Measurable (freshExteriorMoment G p x y) := by
+  simp_rw [show freshExteriorMoment G p x y = (fun ω => ∑' W : Finset V, if cluster G ω x = (W:Set V)
+      then (W.card:ℝ)*inducedSusceptibility G (W:Set V)ᶜ p y else 0) from funext (freshExteriorMoment_eq_tsum G p x y)]
+  apply Measurable.tsum
+  intro W
+  exact Measurable.ite (measurableSet_cluster_eq G x W) measurable_const measurable_const
+
+theorem freshExteriorMoment_nonneg (G : BondGraph V E) (p : unitInterval)
+    (x y : V) (ω : Set E) : 0 ≤ freshExteriorMoment G p x y ω :=
+  mul_nonneg (Nat.cast_nonneg _) (inducedSusceptibility_nonneg G _ p y)
+
+theorem integrable_freshExteriorMoment [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (x y : V) :
+    Integrable (freshExteriorMoment G p x y) (law p) := by
+  apply ((susceptibility_eq_integral G hloc hqt p hp x).1.mul_const (susceptibility G p y)).mono'
+    (measurable_freshExteriorMoment G p x y).aestronglyMeasurable
+  exact Filter.Eventually.of_forall (fun ω => by
+    rw [Real.norm_of_nonneg (freshExteriorMoment_nonneg G p x y ω)]
+    exact mul_le_mul_of_nonneg_left (inducedSusceptibility_le_susceptibility G hloc hqt p hp _ y) (Nat.cast_nonneg _))
+
+theorem integrable_distinct_cluster_product [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (x y : V) :
+    Integrable (fun ω => if ¬Conn G ω x y
+      then ((cluster G ω x).ncard:ℝ)*((cluster G ω y).ncard:ℝ) else 0) (law p) := by
+  have he : (fun ω : Set E => if ¬Conn G ω x y
+      then ((cluster G ω x).ncard:ℝ)*((cluster G ω y).ncard:ℝ) else 0) =
+      {ω : Set E | Conn G ω x y}ᶜ.indicator
+        (fun ω => ((cluster G ω x).ncard:ℝ)*((cluster G ω y).ncard:ℝ)) := by
+    funext ω
+    simp only [Set.indicator,Set.mem_compl_iff,Set.mem_ofPred_eq]
+  rw [he]
+  exact (subcritical_cluster_association G hloc hqt p hp x y).1.indicator
+    (measurableSet_conn G x y).compl
+
+ 
+
+theorem integral_freshExteriorMoment [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (x y : V) :
+    (∫ ω, freshExteriorMoment G p x y ω ∂law p) =
+      ∫ ω, (if ¬Conn G ω x y then ((cluster G ω x).ncard:ℝ)*((cluster G ω y).ncard:ℝ) else 0) ∂law p := by
+  rw [integral_cluster_finite_partition G p hp x _ (integrable_freshExteriorMoment G hloc hqt p hp x y),
+    integral_cluster_finite_partition G p hp x _ (integrable_distinct_cluster_product G hloc hqt p hp x y)]
+  apply tsum_congr
+  intro W
+  have hl : (∫ ω, (if cluster G ω x = (W:Set V) then freshExteriorMoment G p x y ω else 0) ∂law p) =
+      (W.card:ℝ)*inducedSusceptibility G (W:Set V)ᶜ p y * (law p).real {ω | cluster G ω x = (W:Set V)} := by
+    rw [←integral_event_indicator p _ (measurableSet_cluster_eq G x W),←integral_const_mul]
+    apply integral_congr_ae
+    exact Filter.Eventually.of_forall (fun ω => by
+      dsimp only [Set.mem_ofPred_eq]
+      by_cases hc : cluster G ω x = (W:Set V)
+      · simp only [hc,ite_true,freshExteriorMoment,Set.ncard_coe_finset,mul_one]
+      · simp only [hc,ite_false,mul_zero])
+  have hr : (∫ ω, (if cluster G ω x = (W:Set V) then
+      (if ¬Conn G ω x y then ((cluster G ω x).ncard:ℝ)*((cluster G ω y).ncard:ℝ) else 0) else 0) ∂law p) =
+      (W.card:ℝ)*((law p).real {ω | cluster G ω x = (W:Set V)} * inducedSusceptibility G (W:Set V)ᶜ p y) := by
+    rw [←cluster_cell_exterior_susceptibility G hloc hqt p hp W x y,←integral_const_mul]
+    apply integral_congr_ae
+    exact Filter.Eventually.of_forall (fun ω => by
+      dsimp only
+      by_cases hc : cluster G ω x = (W:Set V)
+      · simp only [hc,true_and,ite_true,Set.ncard_coe_finset]
+        split_ifs <;> simp only [mul_zero]
+      · simp only [hc,false_and,ite_false,mul_zero])
+  rw [hl,hr]
+  ring
+
+end FreshExteriorMoment
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section DistinctClusterMoments
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def distinctClusterMoment (G : BondGraph V E) (p : unitInterval) (x y : V) : ℝ :=
+  ∫ ω, (if ¬Conn G ω x y then ((cluster G ω x).ncard:ℝ)*((cluster G ω y).ncard:ℝ) else 0) ∂law p
+
+theorem distinctClusterMoment_nonneg (G : BondGraph V E) (p : unitInterval) (x y : V) :
+    0 ≤ distinctClusterMoment G p x y := by
+  apply integral_nonneg
+  intro ω
+  dsimp only [Pi.zero_apply]
+  split_ifs <;> positivity
+
+theorem cluster_ncard_covariant (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (ω : Set E) (x : V) :
+    (cluster G (b ⁻¹' ω) x).ncard = (cluster G ω (a x)).ncard := by
+  rw [←cluster_aut G hab,Set.ncard_image_of_injective _ a.injective]
+
+theorem distinctClusterMoment_covariant [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (p : unitInterval)
+    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (x y : V) :
+    distinctClusterMoment G p (a x) (a y) = distinctClusterMoment G p x y := by
+  have hs (z : V) : Measurable (fun ω : Set E => ((cluster G ω z).ncard:ℝ)) :=
+    (measurable_of_countable (fun n : ℕ => (n:ℝ))).comp (measurable_cluster_ncard G hloc z)
+  have hm : Measurable (fun ω : Set E => if ¬Conn G ω x y
+      then ((cluster G ω x).ncard:ℝ)*((cluster G ω y).ncard:ℝ) else 0) :=
+    Measurable.ite (measurableSet_conn G x y).compl ((hs x).mul (hs y)) measurable_const
+  have h := integral_map (μ := law p) (measurable_set_preimage b).aemeasurable hm.aestronglyMeasurable
+  rw [law_map_preimage_equiv p b] at h
+  simpa only [distinctClusterMoment,conn_aut_iff G hab,cluster_ncard_covariant G hab] using h.symm
+
+noncomputable def neighborClusterMoment (G : BondGraph V E) (hloc : LocallyFinite G)
+    (p : unitInterval) (x : V) : ℝ :=
+  ∑ y ∈ neighbors G hloc x, distinctClusterMoment G p x y
+
+theorem neighborClusterMoment_nonneg (G : BondGraph V E) (hloc : LocallyFinite G)
+    (p : unitInterval) (x : V) : 0 ≤ neighborClusterMoment G hloc p x :=
+  Finset.sum_nonneg (fun y _ => distinctClusterMoment_nonneg G p x y)
+
+theorem neighbors_covariant (G : BondGraph V E) (hloc : LocallyFinite G)
+    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (x : V) :
+    neighbors G hloc (a x) = (neighbors G hloc x).map a.toEmbedding := by
+  ext y
+  simp only [mem_neighbors,Finset.mem_map,Equiv.coe_toEmbedding]
+  constructor
+  · intro h
+    refine ⟨a.symm y,?_,by simp⟩
+    simpa only [a.symm_apply_apply] using adj_aut G (isAut_symm G hab) h
+  · rintro ⟨z,hz,rfl⟩
+    exact adj_aut G hab hz
+
+theorem neighborClusterMoment_covariant [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (p : unitInterval)
+    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (x : V) :
+    neighborClusterMoment G hloc p (a x) = neighborClusterMoment G hloc p x := by
+  simp only [neighborClusterMoment,neighbors_covariant G hloc hab,Finset.sum_map,Equiv.coe_toEmbedding,
+    distinctClusterMoment_covariant G hloc p hab]
+
+noncomputable def rootNeighborMoment (G : BondGraph V E) (hloc : LocallyFinite G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (p : unitInterval) : ℝ :=
+  ∑ i, weight i * neighborClusterMoment G hloc p (root i)
+
+theorem neighborMoment_le_rootMoment [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (hw : ∀ i, 0 ≤ weight i) (hcov : ∀ x, ∃ i, x ∈ vertexOrbit G (root i))
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i) (p : unitInterval) (x : V) :
+    neighborClusterMoment G hloc p x ≤ c*rootNeighborMoment G hloc root weight p := by
+  obtain ⟨i,a,b,hab,hax⟩ := hcov x
+  rw [←hax,neighborClusterMoment_covariant G hloc p hab]
+  calc
+    _ ≤ (c*weight i)*neighborClusterMoment G hloc p (root i) := by
+      simpa only [one_mul] using mul_le_mul_of_nonneg_right (hmin i) (neighborClusterMoment_nonneg G hloc p _)
+    _ = c*(weight i*neighborClusterMoment G hloc p (root i)) := mul_assoc _ _ _
+    _ ≤ _ := mul_le_mul_of_nonneg_left
+      (Finset.single_le_sum (fun j _ => mul_nonneg (hw j) (neighborClusterMoment_nonneg G hloc p _)) (Finset.mem_univ i)) hc
+
+end DistinctClusterMoments
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Clusters/KernelMoments.lean b/lean/OAI/Probability/UnimodularPercolation/Clusters/KernelMoments.lean
index 53dd6817c280c4af2e701f8ca532d7db754a06e6..5fddfed454aca4033e8f9fa799d19ee5a22298c0 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Clusters/KernelMoments.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Clusters/KernelMoments.lean
@@ -1,1240 +1,1240 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Clusters.Resampling
-
-namespace OAI
-
-/-! Kernel Moments for bond percolation. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section DeletedDomainPotential
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem inducedSusceptibility_nonneg (G : BondGraph V E) (D : Set V)
-    (p : unitInterval) (x : V) : 0 ≤ inducedSusceptibility G D p x := by
-  unfold inducedSusceptibility
-  split_ifs
-  · exact restrictedSusceptibility_nonneg G _ p x
-  · rfl
-
-noncomputable def deletedBoundaryPotential (G : BondGraph V E) (hloc : LocallyFinite G)
-    (D : Set V) (p : unitInterval) (x : V) (M : ℕ) : ℝ :=
-  ∑ a ∈ ((finite_reachBall hloc x M).toFinset.filter (· ∉ D)),
-    ∑ b ∈ neighbors G hloc a, inducedSusceptibility G D p b
-
-theorem deletedBoundaryPotential_nonneg (G : BondGraph V E) (hloc : LocallyFinite G)
-    (D : Set V) (p : unitInterval) (x : V) (M : ℕ) :
-    0 ≤ deletedBoundaryPotential G hloc D p x M := by
-  exact Finset.sum_nonneg (fun a _ => Finset.sum_nonneg (fun b _ => inducedSusceptibility_nonneg G D p b))
-
-theorem le_deletedBoundaryPotential (G : BondGraph V E) (hloc : LocallyFinite G)
-    (D : Set V) (p : unitInterval) (x : V) (M : ℕ) (a b : V)
-    (ha : a ∈ reachBall G x M) (haD : a ∉ D) (hab : Adj G a b) :
-    inducedSusceptibility G D p b ≤ deletedBoundaryPotential G hloc D p x M := by
-  apply le_trans (Finset.single_le_sum (fun z _ => inducedSusceptibility_nonneg G D p z)
-    ((mem_neighbors G hloc a b).mpr hab))
-  apply Finset.single_le_sum (fun z _ => Finset.sum_nonneg (fun y _ => inducedSusceptibility_nonneg G D p y))
-  exact Finset.mem_filter.mpr ⟨(finite_reachBall hloc x M).mem_toFinset.mpr ha,haD⟩
-
-noncomputable def deletedLocalPotential (G : BondGraph V E) (D : Set V) (p : unitInterval)
-    (ω : Set E) (x : V) (M : ℕ) : ℝ :=
-  if x ∈ D then intrinsicPotentialMax G (ω ∩ internalEdges G D) x M (inducedSusceptibility G D p) else 0
-
-theorem deletedLocalPotential_nonneg (G : BondGraph V E) (hloc : LocallyFinite G)
-    (D : Set V) (p : unitInterval) (ω : Set E) (x : V) (M : ℕ) :
-    0 ≤ deletedLocalPotential G D p ω x M := by
-  unfold deletedLocalPotential
-  split_ifs
-  · rw [intrinsicPotentialMax_eq G hloc]
-    exact inducedSusceptibility_nonneg G D p _
-  · rfl
-
- 
-
-theorem short_path_deleted_potential_bound [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (D : Set V)
-    (p : unitInterval) (hp : 0 < (p:ℝ)) (ω : Set E) (x y : V) (M : ℕ)
-    (l : List E) (hl : Along G x l y) (hlen : l.length ≤ M) (ho : ∀ e ∈ l, e ∈ ω)
-    (hfin : ∀ a, Summable (fun z => (law p).real {η | Conn G (η ∩ internalEdges G D) a z})) :
-    inducedSusceptibility G D p y ≤ deletedLocalPotential G D p ω x M +
-      (p:ℝ)^(-(M:ℤ))*deletedBoundaryPotential G hloc D p x M := by
-  have hpM : 0 ≤ (p:ℝ)^(-(M:ℤ)) := zpow_nonneg p.2.1 _
-  have hn := deletedLocalPotential_nonneg G hloc D p ω x M
-  have hb := deletedBoundaryPotential_nonneg G hloc D p x M
-  by_cases hy : y ∈ D
-  · rcases hl.last_entrance D hy with ⟨hx,hall⟩ | ⟨a,b,e,l₁,l₂,heq,h₁,he,ha,hbD,h₂,hs⟩
-    · have hreach : y ∈ intrinsicBall G (ω ∩ internalEdges G D) x M :=
-        ⟨l,hl,hlen,fun e he => ⟨ho e he,hall e he⟩⟩
-      have hh := le_intrinsicPotentialMax G hloc _ x M (inducedSusceptibility G D p) y hreach
-      change _ ≤ (if x ∈ D then _ else 0) + _
-      rw [ite_eq_left hx]
-      exact hh.trans (le_add_of_nonneg_right (mul_nonneg hpM hb))
-    · have hlen₂ : l₂.length ≤ M := by subst l; simp only [List.length_append,List.length_cons] at hlen; omega
-      have hpath := restrictedSusceptibility_short_path_bound G (internalEdges G D) p hp h₂ hs M hlen₂ hfin
-      have hχ : inducedSusceptibility G D p y ≤ (p:ℝ)^(-(M:ℤ))*inducedSusceptibility G D p b := by
-        simpa only [inducedSusceptibility,ite_eq_left hy,ite_eq_left hbD] using hpath
-      have haBall : a ∈ reachBall G x M := reachBall_mono G x
-        (show l₁.length ≤ M by subst l; simp only [List.length_append,List.length_cons] at hlen; omega) h₁.mem_reachBall
-      exact hχ.trans ((mul_le_mul_of_nonneg_left
-        (le_deletedBoundaryPotential G hloc D p x M a b haBall ha ⟨e,he⟩) hpM).trans
-          (le_add_of_nonneg_left hn))
-  · rw [inducedSusceptibility,ite_eq_right hy]
-    exact add_nonneg hn (mul_nonneg hpM hb)
-
-end DeletedDomainPotential
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section LocalKernelAction
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
-
-theorem summable_mass_action (q : LocalEndpointRule G hloc O M L)
-    (ω : Set E) (x : V) (f : V → ℝ) : Summable (fun y => q.mass ω x y * f y) := by
-  apply summable_of_ne_finset_zero (s := (finite_reachBall hloc x M).toFinset)
-  intro y hy
-  rw [q.outside_ball_zero ω x y (by simpa only [Set.Finite.mem_toFinset] using hy),zero_mul]
-
-theorem summable_kernel_action (q : LocalEndpointRule G hloc O M L)
-    (p : unitInterval) (x : V) (f : V → ℝ) : Summable (fun y => q.kernel p x y * f y) := by
-  apply summable_of_ne_finset_zero (s := (finite_reachBall hloc x M).toFinset)
-  intro y hy
-  rw [q.kernel_outside_ball_zero p x y (by simpa only [Set.Finite.mem_toFinset] using hy),zero_mul]
-
-theorem integrable_mass_action (q : LocalEndpointRule G hloc O M L)
-    (p : unitInterval) (x : V) (f : V → ℝ) :
-    Integrable (fun ω => ∑' y, q.mass ω x y * f y) (law p) := by
-  apply integrable_finite_local (incidentSupport G hloc (finite_reachBall hloc x q.radius).toFinset) p
-  intro ω
-  exact tsum_congr (fun y => congrArg (· * f y) (q.locality ω x y))
-
-theorem kernel_action_integral (q : LocalEndpointRule G hloc O M L)
-    (p : unitInterval) (x : V) (f : V → ℝ) :
-    (∑' y, q.kernel p x y * f y) = ∫ ω, (∑' y, q.mass ω x y * f y) ∂law p := by
-  let B := (finite_reachBall hloc x M).toFinset
-  have h₁ : (∑' y, q.kernel p x y*f y) = ∑ y ∈ B, q.kernel p x y*f y :=
-    tsum_eq_sum fun y hy => by
-      rw [q.kernel_outside_ball_zero p x y (by simpa only [B,Set.Finite.mem_toFinset] using hy),zero_mul]
-  have h₂ (ω : Set E) : (∑' y, q.mass ω x y*f y) = ∑ y ∈ B, q.mass ω x y*f y :=
-    tsum_eq_sum fun y hy => by
-      rw [q.outside_ball_zero ω x y (by simpa only [B,Set.Finite.mem_toFinset] using hy),zero_mul]
-  rw [h₁]
-  simp_rw [h₂]
-  rw [integral_finsetSum B (fun y _ => (q.integrable_mass p x y).mul_const (f y))]
-  apply Finset.sum_congr rfl
-  intro y _
-  exact (integral_mul_const (f y) (fun ω => q.mass ω x y)).symm
-
-theorem mass_action_le (q : LocalEndpointRule G hloc O M L)
-    (ω : Set E) (x : V) (f : V → ℝ) (C : ℝ) (hC : 0 ≤ C)
-    (hf : ∀ y, q.mass ω x y ≠ 0 → f y ≤ C) :
-    (∑' y, q.mass ω x y * f y) ≤ C := by
-  calc
-    _ ≤ ∑' y, q.mass ω x y * C := by
-      apply (q.summable_mass_action ω x f).tsum_le_tsum _ ((q.summable_mass ω x).mul_right C)
-      intro y
-      by_cases h : q.mass ω x y = 0
-      · simp only [h,zero_mul,le_refl]
-      · exact mul_le_mul_of_nonneg_left (hf y h) (q.nonneg ω x y)
-    _ = (∑' y, q.mass ω x y)*C := tsum_mul_right
-    _ ≤ C := by simpa only [one_mul] using mul_le_mul_of_nonneg_right (q.row_le_one ω x) hC
-
-end LocalEndpointRule
-end LocalKernelAction
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section LocalMapDeletedPotential
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem integrable_deletedLocalPotential (G : BondGraph V E) (hloc : LocallyFinite G)
-    (D : Set V) (p : unitInterval) (x : V) (M : ℕ) :
-    Integrable (fun ω => deletedLocalPotential G D p ω x M) (law p) := by
-  by_cases hx : x ∈ D
-  · simp only [deletedLocalPotential,ite_eq_left hx]
-    exact integrable_finite_local _ p _ (fun ω =>
-      intrinsicPotentialMax_domain_local G hloc (internalEdges G D) ω x M (inducedSusceptibility G D p))
-  · simp only [deletedLocalPotential,ite_eq_right hx]
-    exact integrable_const 0
-
-theorem integral_deletedLocalPotential_le
-    [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (D : Set V) (p : unitInterval) (hp : 0 < (p:ℝ)) (hpc : p < pc G) (x : V) (M : ℕ) :
-    (∫ ω, deletedLocalPotential G D p ω x M ∂law p) ≤
-      (1+(1-(p:ℝ))/p*M)*inducedSusceptibility G D p x := by
-  by_cases hx : x ∈ D
-  · simp only [deletedLocalPotential,ite_eq_left hx]
-    exact maximal_induced_susceptibility_bound G hloc hqt D x hx M p hp hpc
-  · simp only [deletedLocalPotential,inducedSusceptibility,ite_eq_right hx,integral_zero,mul_zero,le_refl]
-
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
-
- 
-
-theorem kernel_deleted_potential_bound
-    [Countable V] [Countable E] [Nonempty V]
-    (q : LocalEndpointRule G hloc O M L) (hqt : QuasiTransitive G)
-    (D : Set V) (p : unitInterval) (hp : 0 < (p:ℝ)) (hpc : p < pc G) (x : V) :
-    (∑' y, q.kernel p x y * inducedSusceptibility G D p y) ≤
-      (1+(1-(p:ℝ))/p*M)*inducedSusceptibility G D p x +
-        (p:ℝ)^(-(M:ℤ))*deletedBoundaryPotential G hloc D p x M := by
-  let f := inducedSusceptibility G D p
-  let C := (p:ℝ)^(-(M:ℤ))*deletedBoundaryPotential G hloc D p x M
-  have hC : 0 ≤ C := mul_nonneg (zpow_nonneg p.2.1 _) (deletedBoundaryPotential_nonneg G hloc D p x M)
-  have hpoint (ω : Set E) : (∑' y, q.mass ω x y * f y) ≤ deletedLocalPotential G D p ω x M + C := by
-    apply q.mass_action_le ω x f _ (add_nonneg (deletedLocalPotential_nonneg G hloc D p ω x M) hC)
-    intro y hy
-    obtain ⟨_,_,hxy,hfin,hcap,_⟩ := q.support ω x y hy
-    obtain ⟨l,hl,hlen,ho⟩ := conn_short_of_cluster_cap G ω x y hxy hfin M hcap
-    exact short_path_deleted_potential_bound G hloc D p hp ω x y M l hl hlen ho
-      (summable_restrictedTwoPoint G hloc hqt p hpc (internalEdges G D))
-  rw [q.kernel_action_integral]
-  have hi := integrable_deletedLocalPotential G hloc D p x M
-  calc
-    _ ≤ ∫ ω, deletedLocalPotential G D p ω x M + C ∂law p :=
-      integral_mono (q.integrable_mass_action p x f) (hi.add (integrable_const C)) hpoint
-    _ = (∫ ω, deletedLocalPotential G D p ω x M ∂law p) + C := by
-      rw [integral_add hi (integrable_const C)]; simp
-    _ ≤ _ := by
-      have hh := integral_deletedLocalPotential_le G hloc hqt D p hp hpc x M
-      dsimp [C]
-      linarith
-
-end LocalEndpointRule
-end LocalMapDeletedPotential
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section OrbitKernelTransport
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem orbit_kernel_column_eq_row
-    (G : BondGraph V E) {ι : Type*} [Fintype ι]
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (htr : RootTransport root weight Γ)
-    (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b) (hw : ∀ i, 0 ≤ weight i)
-    (i₀ : ι) (hw₀ : 0 < weight i₀) (K : V → V → ℝ)
-    (hcov : ∀ a b, IsAut G a b → ∀ x y, K (a x) (a y) = K x y)
-    (hout : ∀ x y, x ∉ vertexOrbit G (root i₀) → K x y = 0)
-    (hin : ∀ x y, y ∉ vertexOrbit G (root i₀) → K x y = 0)
-    (hr : ∀ x, Summable (K x)) (hc : ∀ y, Summable (fun x => K x y))
-    (y : V) (hy : y ∈ vertexOrbit G (root i₀)) :
-    (∑' x, K x y) = ∑' z, K (root i₀) z := by
-  let O := vertexOrbit G (root i₀)
-  let w : ι → ℝ := fun i => if root i ∈ O then weight i else 0
-  have hrow (x : V) (hx : x ∈ O) : (∑' z, K x z) = ∑' z, K (root i₀) z := by
-    obtain ⟨a,b,ha,rfl⟩ := hx
-    rw [← a.tsum_eq (fun z => K (a (root i₀)) z)]
-    exact tsum_congr (hcov a b ha (root i₀))
-  have hcol (x : V) (hx : x ∈ O) : (∑' z, K z x) = ∑' z, K z (root i₀) := by
-    obtain ⟨a,b,ha,rfl⟩ := hx
-    rw [← a.tsum_eq (fun z => K z (a (root i₀)))]
-    exact tsum_congr (fun z => hcov a b ha z (root i₀))
-  have ho : root i₀ ∈ O := by
-    refine ⟨Equiv.refl V,Equiv.refl E,?_,rfl⟩
-    intro e; simp
-  have hw' (i : ι) : 0 ≤ w i := by
-    dsimp [w]; split_ifs
-    · exact hw i
-    · rfl
-  have hpos : 0 < ∑ i, w i := by
-    have hi := Finset.single_le_sum (fun i _ => hw' i) (Finset.mem_univ i₀)
-    have he : w i₀ = weight i₀ := ite_eq_left ho
-    rw [he] at hi
-    exact hw₀.trans_le hi
-  have ht := htr.real root weight Γ K
-    (by intro a ha x z; obtain ⟨b,hb⟩ := hΓ a ha; exact hcov a b hb x z) hr hc
-  have h₁ (i : ι) : weight i * (∑' z, K (root i) z) = w i * (∑' z, K (root i₀) z) := by
-    by_cases hi : root i ∈ O
-    · rw [hrow (root i) hi]; simp only [w,ite_eq_left hi]
-    · simp only [w,ite_eq_right hi,zero_mul]
-      have he : (∑' z, K (root i) z) = 0 := by simp only [hout _ _ hi,tsum_zero]
-      rw [he,mul_zero]
-  have h₂ (i : ι) : weight i * (∑' z, K z (root i)) = w i * (∑' z, K z (root i₀)) := by
-    by_cases hi : root i ∈ O
-    · rw [hcol (root i) hi]; simp only [w,ite_eq_left hi]
-    · simp only [w,ite_eq_right hi,zero_mul]
-      have he : (∑' z, K z (root i)) = 0 := by simp only [hin _ _ hi,tsum_zero]
-      rw [he,mul_zero]
-  simp_rw [h₁,h₂] at ht
-  rw [← Finset.sum_mul,← Finset.sum_mul] at ht
-  rw [hcol y hy]
-  exact (mul_left_cancel₀ (ne_of_gt hpos) ht).symm
-
- 
-
-theorem operator_norm_le_absolute_sums (K : V → V → ℝ) (D : ℝ) (hD : 0 ≤ D)
-    (hr : ∀ x (S : Finset V), (∑ y ∈ S, |K x y|) ≤ D)
-    (hc : ∀ y (S : Finset V), (∑ x ∈ S, |K x y|) ≤ D)
-    (A : Hilbert V →L[ℂ] Hilbert V)
-    (hA : ∀ x y, A (lp.single 2 y (1:ℂ)) x = (K x y : ℂ)) : ‖A‖ ≤ D := by
-  obtain ⟨B,hBn,hB⟩ := exists_operator_of_row_column_bound (fun x y => |K x y|) D hD
-    (fun _ _ => abs_nonneg _) hr hc
-  obtain ⟨A',hAn,hA'⟩ := exists_operator_of_matrix_domination
-    (fun x y => |K x y|) K (fun _ _ => abs_nonneg _) (fun _ _ => le_rfl) B hB
-  have he := kernel_extension_unique hA hA'
-  rw [he]
-  exact hAn.trans hBn
-
-end OrbitKernelTransport
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section EndpointDifference
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
-variable (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
-
-noncomputable def kernelDifference (p r : unitInterval) (x y : V) : ℝ :=
-  |q.normalizedKernel o p x y - q.normalizedKernel o r x y|
-noncomputable def differenceRow (p r : unitInterval) : ℝ := ∑' y, q.kernelDifference o p r o y
-
-theorem kernelDifference_nonneg (p r : unitInterval) (x y : V) :
-    0 ≤ q.kernelDifference o p r x y := abs_nonneg _
-
-theorem kernelDifference_covariant (p r : unitInterval)
-    (a : V ≃ V) (b : E ≃ E) (ha : IsAut G a b) (x y : V) :
-    q.kernelDifference o p r (a x) (a y) = q.kernelDifference o p r x y := by
-  simp only [kernelDifference,normalizedKernel,q.kernel_covariant p a b ha,q.kernel_covariant r a b ha]
-
-theorem kernelDifference_outside (p r : unitInterval) (x y : V) (hy : y ∉ reachBall G x M) :
-    q.kernelDifference o p r x y = 0 := by
-  simp only [kernelDifference,normalizedKernel,q.kernel_outside_ball_zero p x y hy,
-    q.kernel_outside_ball_zero r x y hy,mul_zero,sub_self,abs_zero]
-
-theorem kernelDifference_reverse_outside (p r : unitInterval) (x y : V) (hx : x ∉ reachBall G y M) :
-    q.kernelDifference o p r x y = 0 := by
-  simp only [kernelDifference,normalizedKernel,q.kernel_outside_reverse_ball_zero p x y hx,
-    q.kernel_outside_reverse_ball_zero r x y hx,mul_zero,sub_self,abs_zero]
-
-theorem kernelDifference_start (p r : unitInterval) (x y : V) (hx : x ∉ vertexOrbit G o) :
-    q.kernelDifference o p r x y = 0 := by
-  simp only [kernelDifference,normalizedKernel,q.kernel_zero_of_start_not_mem p x y hx,
-    q.kernel_zero_of_start_not_mem r x y hx,mul_zero,sub_self,abs_zero]
-
-theorem kernelDifference_end (p r : unitInterval) (x y : V) (hy : y ∉ vertexOrbit G o) :
-    q.kernelDifference o p r x y = 0 := by
-  simp only [kernelDifference,normalizedKernel,q.kernel_zero_of_end_not_mem p x y hy,
-    q.kernel_zero_of_end_not_mem r x y hy,mul_zero,sub_self,abs_zero]
-
-theorem summable_kernelDifference (p r : unitInterval) (x : V) :
-    Summable (q.kernelDifference o p r x) := by
-  apply summable_of_ne_finset_zero (s := (finite_reachBall hloc x M).toFinset)
-  intro y hy
-  exact q.kernelDifference_outside o p r x y (by simpa only [Set.Finite.mem_toFinset] using hy)
-
-theorem summable_kernelDifference_column (p r : unitInterval) (y : V) :
-    Summable (fun x => q.kernelDifference o p r x y) := by
-  apply summable_of_ne_finset_zero (s := (finite_reachBall hloc y M).toFinset)
-  intro x hx
-  exact q.kernelDifference_reverse_outside o p r x y (by simpa only [Set.Finite.mem_toFinset] using hx)
-
-theorem differenceRow_nonneg (p r : unitInterval) : 0 ≤ q.differenceRow o p r :=
-  tsum_nonneg (q.kernelDifference_nonneg o p r o)
-
-theorem differenceRow_finite (p r : unitInterval) :
-    q.differenceRow o p r = ∑ y ∈ (finite_reachBall hloc o M).toFinset, q.kernelDifference o p r o y := by
-  apply tsum_eq_sum
-  intro y hy
-  exact q.kernelDifference_outside o p r o y (by simpa only [Set.Finite.mem_toFinset] using hy)
-
-theorem kernelDifference_row_le (p r : unitInterval) (x : V) :
-    (∑' y, q.kernelDifference o p r x y) ≤ q.differenceRow o p r := by
-  by_cases hx : x ∈ vertexOrbit G o
-  · obtain ⟨a,b,ha,rfl⟩ := hx
-    rw [← a.tsum_eq (fun y => q.kernelDifference o p r (a o) y)]
-    exact le_of_eq (tsum_congr (q.kernelDifference_covariant o p r a b ha o))
-  · simpa only [q.kernelDifference_start o p r x _ hx,tsum_zero] using q.differenceRow_nonneg o p r
-
- 
-
-theorem norm_sub_le_differenceRow {ι : Type*} [Fintype ι]
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (htr : RootTransport root weight Γ)
-    (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b) (hw : ∀ i, 0 ≤ weight i)
-    (i₀ : ι) (hw₀ : 0 < weight i₀) (ho : o = root i₀)
-    (p r : unitInterval) (A B : Hilbert V →L[ℂ] Hilbert V)
-    (hA : ∀ x y, A (lp.single 2 y (1:ℂ)) x = (q.normalizedKernel o p x y : ℂ))
-    (hB : ∀ x y, B (lp.single 2 y (1:ℂ)) x = (q.normalizedKernel o r x y : ℂ)) :
-    ‖A-B‖ ≤ q.differenceRow o p r := by
-  subst o
-  have hc (y : V) : (∑' x, q.kernelDifference (root i₀) p r x y) ≤ q.differenceRow (root i₀) p r := by
-    by_cases hy : y ∈ vertexOrbit G (root i₀)
-    · exact (orbit_kernel_column_eq_row G root weight Γ htr hΓ hw i₀ hw₀
-        (q.kernelDifference (root i₀) p r) (q.kernelDifference_covariant (root i₀) p r)
-        (q.kernelDifference_start (root i₀) p r) (q.kernelDifference_end (root i₀) p r)
-        (q.summable_kernelDifference (root i₀) p r) (q.summable_kernelDifference_column (root i₀) p r) y hy).le
-    · simpa only [q.kernelDifference_end (root i₀) p r _ y hy,tsum_zero] using
-        q.differenceRow_nonneg (root i₀) p r
-  apply operator_norm_le_absolute_sums
-    (fun x y => q.normalizedKernel (root i₀) p x y-q.normalizedKernel (root i₀) r x y)
-    _ (q.differenceRow_nonneg (root i₀) p r)
-  · intro x S
-    exact ((q.summable_kernelDifference (root i₀) p r x).sum_le_tsum S
-      (fun y _ => q.kernelDifference_nonneg (root i₀) p r x y)).trans (q.kernelDifference_row_le (root i₀) p r x)
-  · intro y S
-    exact ((q.summable_kernelDifference_column (root i₀) p r y).sum_le_tsum S
-      (fun x _ => q.kernelDifference_nonneg (root i₀) p r x y)).trans (hc y)
-  · intro x y
-    change A (lp.single 2 y (1:ℂ)) x - B (lp.single 2 y (1:ℂ)) x = _
-    rw [hA,hB,Complex.ofReal_sub]
-
-end LocalEndpointRule
-end EndpointDifference
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open Filter
-section EndpointOperatorContinuity
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
-
- 
-
-theorem exists_normalized_operator_all {ι : Type*} [Fintype ι]
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (htr : RootTransport root weight Γ)
-    (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b) (hw : ∀ i, 0 ≤ weight i)
-    (i₀ : ι) (hw₀ : 0 < weight i₀)
-    (q : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) M L) (p : unitInterval) :
-    ∃ A : Hilbert V →L[ℂ] Hilbert V, ‖A‖ ≤ 1 ∧
-      ∀ x y, A (lp.single 2 y (1:ℂ)) x = (q.normalizedKernel (root i₀) p x y : ℂ) := by
-  by_cases hu : q.successProbability p (root i₀) = 0
-  · refine ⟨0,by simp,?_⟩
-    intro x y
-    simp only [normalizedKernel,hu,inv_zero,zero_mul,Complex.ofReal_zero,
-      zero_apply,lp.coeFn_zero,Pi.zero_apply]
-  · obtain ⟨A,_,hA,hK⟩ := q.exists_normalized_operator root weight Γ htr hΓ hw i₀ hw₀ p
-      (lt_of_le_of_ne (q.successProbability_nonneg p (root i₀)) (Ne.symm hu))
-    exact ⟨A,hA,hK⟩
-
-theorem continuousAt_differenceRow (o : V)
-    (q : LocalEndpointRule G hloc (vertexOrbit G o) M L) (p₀ : unitInterval)
-    (hu : q.successProbability p₀ o ≠ 0) :
-    ContinuousAt (fun p => q.differenceRow o p p₀) p₀ := by
-  simp_rw [q.differenceRow_finite o]
-  apply tendsto_finsetSum
-  intro y _
-  exact ((((q.continuous_successProbability o).continuousAt.inv₀ hu).mul
-    (q.continuous_kernel o y).continuousAt).sub continuousAt_const).abs
-
- 
-
-theorem exists_continuous_normalized_operators {ι : Type*} [Fintype ι]
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (htr : RootTransport root weight Γ)
-    (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b) (hw : ∀ i, 0 ≤ weight i)
-    (i₀ : ι) (hw₀ : 0 < weight i₀)
-    (q : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) M L) :
-    ∃ A : unitInterval → (Hilbert V →L[ℂ] Hilbert V),
-      (∀ p, ‖A p‖ ≤ 1) ∧
-      (∀ p x y, A p (lp.single 2 y (1:ℂ)) x = (q.normalizedKernel (root i₀) p x y : ℂ)) ∧
-      ∀ p, q.successProbability p (root i₀) ≠ 0 → ContinuousAt A p := by
-  choose A hn hK using q.exists_normalized_operator_all root weight Γ htr hΓ hw i₀ hw₀
-  refine ⟨A,hn,hK,?_⟩
-  intro p hu
-  apply tendsto_iff_norm_sub_tendsto_zero.mpr
-  apply squeeze_zero (fun r => norm_nonneg (A r-A p))
-    (fun r => q.norm_sub_le_differenceRow (root i₀) root weight Γ htr hΓ hw i₀ hw₀ rfl r p (A r) (A p) (hK r) (hK p))
-  have h := (q.continuousAt_differenceRow (root i₀) p hu).tendsto
-  have hz : q.differenceRow (root i₀) p p = 0 := by
-    simp only [differenceRow,kernelDifference,sub_self,abs_zero,tsum_zero]
-  rwa [hz] at h
-
-end LocalEndpointRule
-end EndpointOperatorContinuity
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section ThreeArmGeometry
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem Along.first_entrance {G : BondGraph V E} {x y : V} {l : List E}
-    (hl : Along G x l y) (D : Set V) (hy : y ∈ D) :
-    ∃ z ∈ D, ∃ k : List E, Along G x k z ∧ k ⊆ l ∧ ∀ e ∈ k, e ∉ internalEdges G D := by
-  induction hl with
-  | nil => exact ⟨_,hy,[],.nil _,by simp,by simp⟩
-  | @cons x z y l e he ht ih =>
-    by_cases hx : x ∈ D
-    · exact ⟨x,hx,[],.nil _,by simp,by simp⟩
-    · obtain ⟨w,hw,k,hk,hsub,hout⟩ := ih hy
-      refine ⟨w,hw,e::k,.cons e he hk,?_,?_⟩
-      · intro d hd
-        rcases List.mem_cons.mp hd with rfl | hd
-        · exact List.mem_cons_self
-        · exact List.mem_cons_of_mem _ (hsub hd)
-      · intro d hd
-        rcases List.mem_cons.mp hd with rfl | hd
-        · intro h
-          exact hx ((mem_internalEdges_of_ends G D d x z he).mp h).1
-        · exact hout d hd
-
- 
-
-theorem conn_three_disjoint_arms (G : BondGraph V E) (ω : Set E) (x y z : V)
-    (hxy : Conn G ω x y) (hxz : Conn G ω x z) :
-    ∃ w, ω ∈ DisjointOccurrence {η | Conn G η x w}
-      (DisjointOccurrence {η | Conn G η w y} {η | Conn G η w z}) := by
-  obtain ⟨p⟩ := (conn_iff_reachable G ω x y).mp hxy
-  let t := p.toPath
-  let D : Set V := {a | a ∈ t.val.support}
-  obtain ⟨l,hl,_,ho⟩ := conn_exists_bond_trail G (conn_symm G ω hxz)
-  obtain ⟨w,hw,k,hk,hsub,hout⟩ := hl.first_entrance D t.val.start_mem_support
-  obtain ⟨l₁,h₁,he₁,ho₁⟩ := walk_lift_bonds G ω (t.val.takeUntil w hw)
-  obtain ⟨l₂,h₂,he₂,ho₂⟩ := walk_lift_bonds G ω (t.val.dropUntil w hw)
-  have h₁in (e : E) (he : e ∈ l₁) : e ∈ internalEdges G D := by
-    have hh : G e ∈ (t.val.takeUntil w hw).edges := he₁ ▸ List.mem_map_of_mem he
-    have hh' := t.val.edges_takeUntil_subset_edges hw hh
-    intro a ha
-    exact SimpleGraph.Walk.mem_support_of_mem_edges hh' ha
-  have h₂in (e : E) (he : e ∈ l₂) : e ∈ internalEdges G D := by
-    have hh : G e ∈ (t.val.dropUntil w hw).edges := he₂ ▸ List.mem_map_of_mem he
-    have hh' := t.val.edges_dropUntil_subset_edges hw hh
-    intro a ha
-    exact SimpleGraph.Walk.mem_support_of_mem_edges hh' ha
-  have h12 : Disjoint l₁.toFinset l₂.toFinset := by
-    apply Finset.disjoint_left.mpr
-    intro e he₁' he₂'
-    have hp := t.property.isTrail.disjoint_edges_takeUntil_dropUntil hw
-    exact List.disjoint_left.mp hp (he₁ ▸ List.mem_map_of_mem (List.mem_toFinset.mp he₁'))
-      (he₂ ▸ List.mem_map_of_mem (List.mem_toFinset.mp he₂'))
-  have h13 : Disjoint l₁.toFinset k.toFinset := by
-    apply Finset.disjoint_left.mpr
-    intro e he hk'
-    exact hout e (List.mem_toFinset.mp hk') (h₁in e (List.mem_toFinset.mp he))
-  have h23 : Disjoint l₂.toFinset k.toFinset := by
-    apply Finset.disjoint_left.mpr
-    intro e he hk'
-    exact hout e (List.mem_toFinset.mp hk') (h₂in e (List.mem_toFinset.mp he))
-  have hkopen (e : E) (he : e ∈ k) : e ∈ ω := ho e (hsub he)
-  refine ⟨w,l₁.toFinset,l₂.toFinset ∪ k.toFinset,?_,?_,Finset.disjoint_union_right.mpr ⟨h12,h13⟩,?_,?_⟩
-  · intro e he; exact ho₁ e (List.mem_toFinset.mp he)
-  · intro e he
-    rcases Finset.mem_union.mp he with he | he
-    · exact ho₂ e (List.mem_toFinset.mp he)
-    · exact hkopen e (List.mem_toFinset.mp he)
-  · exact h₁.conn (fun e he => List.mem_toFinset.mpr he)
-  · refine ⟨l₂.toFinset,k.toFinset,?_,?_,h23,?_,?_⟩
-    · exact_mod_cast Finset.subset_union_left
-    · exact_mod_cast Finset.subset_union_right
-    · exact h₂.conn (fun e he => List.mem_toFinset.mpr he)
-    · exact conn_symm G _ (hk.conn (fun e he => List.mem_toFinset.mpr he))
-
-end ThreeArmGeometry
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-open scoped ENNReal
-section ThreeArmProbability
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem bk_ennreal [Countable E] (p : unitInterval) (A B : Set (Set E))
-    (hA : IsUpperSet A) (hB : IsUpperSet B) (hfA : FiniteWitness A) (hfB : FiniteWitness B) :
-    law p (DisjointOccurrence A B) ≤ law p A * law p B := by
-  apply (ENNReal.toReal_le_toReal (measure_ne_top _ _)
-    (ENNReal.mul_ne_top (measure_ne_top _ _) (measure_ne_top _ _))).mp
-  rw [ENNReal.toReal_mul]
-  exact bk p A B hA hB hfA hfB
-
- 
-
-theorem three_arm_bound [Countable V] [Countable E]
-    (G : BondGraph V E) (p : unitInterval) (x y z : V) :
-    law p {ω | Conn G ω x y ∧ Conn G ω x z} ≤
-      ∑' w, law p {ω | Conn G ω x w} * law p {ω | Conn G ω w y} * law p {ω | Conn G ω w z} := by
-  let C := fun a b : V => {ω : Set E | Conn G ω a b}
-  have hup (a b : V) : IsUpperSet (C a b) := fun _ _ h hconn => conn_mono G h hconn
-  have hf (a b : V) : FiniteWitness (C a b) := finiteWitness_conn G a b
-  have hb (w : V) : law p (DisjointOccurrence (C x w) (DisjointOccurrence (C w y) (C w z))) ≤
-      law p (C x w) * law p (C w y) * law p (C w z) := by
-    calc
-      _ ≤ law p (C x w) * law p (DisjointOccurrence (C w y) (C w z)) :=
-        bk_ennreal p _ _ (hup x w) (disjointOccurrence_upper _ _) (hf x w) (finiteWitness_disjointOccurrence _ _)
-      _ ≤ law p (C x w) * (law p (C w y)*law p (C w z)) :=
-        mul_le_mul_right (bk_ennreal p _ _ (hup w y) (hup w z) (hf w y) (hf w z)) _
-      _ = _ := (mul_assoc _ _ _).symm
-  calc
-    _ ≤ law p (⋃ w, DisjointOccurrence (C x w) (DisjointOccurrence (C w y) (C w z))) := by
-      apply measure_mono
-      intro ω h
-      obtain ⟨w,hw⟩ := conn_three_disjoint_arms G ω x y z h.1 h.2
-      exact Set.mem_iUnion.mpr ⟨w,hw⟩
-    _ ≤ ∑' w, law p (DisjointOccurrence (C x w) (DisjointOccurrence (C w y) (C w z))) := measure_iUnion_le _
-    _ ≤ _ := ENNReal.tsum_le_tsum hb
-
- 
-
-theorem three_arm_double_sum_bound [Countable V] [Countable E]
-    (G : BondGraph V E) (p : unitInterval) (C : ℝ≥0∞)
-    (hC : ∀ x, (∑' y, law p {ω | Conn G ω x y}) ≤ C) (x : V) :
-    (∑' y, ∑' z, law p {ω | Conn G ω x y ∧ Conn G ω x z}) ≤ C^3 := by
-  calc
-    _ ≤ ∑' y, ∑' z, ∑' w, law p {ω | Conn G ω x w} * law p {ω | Conn G ω w y} * law p {ω | Conn G ω w z} :=
-      ENNReal.tsum_le_tsum (fun y => ENNReal.tsum_le_tsum (fun z => three_arm_bound G p x y z))
-    _ = ∑' w, law p {ω | Conn G ω x w} * (∑' y, law p {ω | Conn G ω w y}) *
-        (∑' z, law p {ω | Conn G ω w z}) := by
-      calc
-        _ = ∑' y, ∑' w, ∑' z, law p {ω | Conn G ω x w} * law p {ω | Conn G ω w y} * law p {ω | Conn G ω w z} := by
-          apply tsum_congr
-          intro y
-          exact ENNReal.tsum_comm
-        _ = ∑' w, ∑' y, ∑' z, law p {ω | Conn G ω x w} * law p {ω | Conn G ω w y} * law p {ω | Conn G ω w z} := ENNReal.tsum_comm
-        _ = _ := by simp_rw [ENNReal.tsum_mul_left,ENNReal.tsum_mul_right,ENNReal.tsum_mul_left]
-    _ ≤ ∑' w, law p {ω | Conn G ω x w} * C * C :=
-      ENNReal.tsum_le_tsum (fun w => mul_le_mul' (mul_le_mul_right (hC w) _) (hC w))
-    _ = (∑' w, law p {ω | Conn G ω x w}) * C * C := by
-      rw [ENNReal.tsum_mul_right,ENNReal.tsum_mul_right]
-    _ ≤ C*C*C := mul_le_mul_left (mul_le_mul_left (hC x) C) C
-    _ = C^3 := by ring
-
-end ThreeArmProbability
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-open scoped ENNReal
-section ClusterMassMoments
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-noncomputable def clusterMass (G : BondGraph V E) (ω : Set E) (x : V) : ℝ≥0∞ :=
-  ∑' y, if Conn G ω x y then 1 else 0
-
-theorem measurable_clusterMass [Countable V] [Countable E]
-    (G : BondGraph V E) (x : V) : Measurable (fun ω => clusterMass G ω x) := by
-  apply Measurable.tsum
-  intro y
-  exact Measurable.ite (measurableSet_conn G x y) measurable_const measurable_const
-
-theorem clusterMass_eq_ncard (G : BondGraph V E) (ω : Set E) (x : V)
-    (hf : (cluster G ω x).Finite) : clusterMass G ω x = (cluster G ω x).ncard := by
-  change (∑' y, if y ∈ cluster G ω x then (1:ℝ≥0∞) else 0) = _
-  rw [tsum_eq_sum (s := hf.toFinset) (fun y hy => by
-    simp only [Set.Finite.mem_toFinset] at hy
-    simp only [ite_eq_right hy])]
-  have he : (∑ y ∈ hf.toFinset, if y ∈ cluster G ω x then (1:ℝ≥0∞) else 0) = ∑ _y ∈ hf.toFinset, (1:ℝ≥0∞) := by
-    apply Finset.sum_congr rfl
-    intro y hy
-    exact ite_eq_left (hf.mem_toFinset.mp hy)
-  rw [he]
-  simp only [Finset.sum_const,nsmul_eq_mul,mul_one,←Set.ncard_eq_toFinset_card _ hf]
-
-theorem lintegral_clusterMass [Countable V] [Countable E]
-    (G : BondGraph V E) (p : unitInterval) (x : V) :
-    (∫⁻ ω, clusterMass G ω x ∂law p) = ∑' y, law p {ω | Conn G ω x y} := by
-  unfold clusterMass
-  rw [lintegral_tsum (fun y => (Measurable.ite (measurableSet_conn G x y) measurable_const measurable_const).aemeasurable)]
-  apply tsum_congr
-  intro y
-  simpa only [Set.indicator,Set.mem_ofPred_eq,Pi.one_apply] using (lintegral_indicator_one (μ := law p) (measurableSet_conn G x y))
-
-theorem clusterMass_sq (G : BondGraph V E) (ω : Set E) (x : V) :
-    clusterMass G ω x ^ 2 = ∑' y, ∑' z, if Conn G ω x y ∧ Conn G ω x z then (1:ℝ≥0∞) else 0 := by
-  rw [pow_two,clusterMass]
-  rw [←ENNReal.tsum_mul_right]
-  apply tsum_congr
-  intro y
-  rw [←ENNReal.tsum_mul_left]
-  apply tsum_congr
-  intro z
-  split_ifs <;> simp_all
-
-theorem lintegral_clusterMass_sq [Countable V] [Countable E]
-    (G : BondGraph V E) (p : unitInterval) (x : V) :
-    (∫⁻ ω, clusterMass G ω x ^ 2 ∂law p) =
-      ∑' y, ∑' z, law p {ω | Conn G ω x y ∧ Conn G ω x z} := by
-  have hm (y z : V) : Measurable (fun ω : Set E => if Conn G ω x y ∧ Conn G ω x z then (1:ℝ≥0∞) else 0) :=
-    Measurable.ite ((measurableSet_conn G x y).inter (measurableSet_conn G x z)) measurable_const measurable_const
-  simp_rw [clusterMass_sq]
-  rw [lintegral_tsum (fun y => (Measurable.tsum (fun z => hm y z)).aemeasurable)]
-  apply tsum_congr
-  intro y
-  rw [lintegral_tsum (fun z => (hm y z).aemeasurable)]
-  apply tsum_congr
-  intro z
-  have hs : MeasurableSet {ω : Set E | Conn G ω x y ∧ Conn G ω x z} :=
-    (measurableSet_conn G x y).inter (measurableSet_conn G x z)
-  simpa only [Set.indicator,Set.mem_ofPred_eq,Pi.one_apply] using (lintegral_indicator_one (μ := law p) hs)
-
- 
-theorem cluster_second_moment_bound [Countable V] [Countable E]
-    (G : BondGraph V E) (p : unitInterval) (C : ℝ≥0∞)
-    (hC : ∀ x, (∑' y, law p {ω | Conn G ω x y}) ≤ C) (x : V) :
-    (∫⁻ ω, clusterMass G ω x ^ 2 ∂law p) ≤ C^3 := by
-  rw [lintegral_clusterMass_sq]
-  exact three_arm_double_sum_bound G p C hC x
-
-end ClusterMassMoments
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-open scoped ENNReal
-section SubcriticalMoments
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem subcritical_cluster_finite_ae (G : BondGraph V E)
-    (p : unitInterval) (hp : p < pc G) (x : V) :
-    ∀ᵐ ω ∂law p, (cluster G ω x).Finite := by
-  have hz : (law p).real {ω | (cluster G ω x).Infinite} = 0 := by
-    apply le_antisymm _ measureReal_nonneg
-    by_contra! h
-    exact (not_le_of_gt hp) (pc_le_of_infinite_cluster_probability_pos G x p h)
-  apply ae_iff.mpr
-  change law p {ω | (cluster G ω x).Infinite} = 0
-  exact ((ENNReal.toReal_eq_zero_iff _).mp hz).resolve_right (measure_ne_top _ _)
-
-theorem subcritical_ennreal_row_bound [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) :
-    ∃ C : ℝ, 0 ≤ C ∧ ∀ x, (∑' y, law p {ω | Conn G ω x y}) ≤ ENNReal.ofReal C := by
-  obtain ⟨C,hC,hrow⟩ := subcritical_row_bound G hloc hqt p hp
-  refine ⟨C,hC,fun x => ?_⟩
-  have hs := summable_of_sum_le (twoPoint_nonneg G p x) (hrow x)
-  have he : (∑' y, law p {ω | Conn G ω x y}) = ENNReal.ofReal (∑' y, twoPoint G p x y) := by
-    rw [ENNReal.ofReal_tsum_of_nonneg (twoPoint_nonneg G p x) hs]
-    apply tsum_congr
-    intro y
-    exact (ENNReal.ofReal_toReal (measure_ne_top _ _)).symm
-  rw [he]
-  exact ENNReal.ofReal_le_ofReal (hs.tsum_le_of_sum_le (hrow x))
-
- 
-
-theorem subcritical_second_moment [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) :
-    ∃ C : ℝ, 0 ≤ C ∧ ∀ x,
-      Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^2) (law p) ∧
-      (∫ ω, ((cluster G ω x).ncard:ℝ)^2 ∂law p) ≤ C^3 := by
-  obtain ⟨C,hC,hrow⟩ := subcritical_ennreal_row_bound G hloc hqt p hp
-  refine ⟨C,hC,fun x => ?_⟩
-  have hmeas : Measurable (fun ω : Set E => ((cluster G ω x).ncard:ℝ)^2) :=
-    (measurable_of_countable (fun n : ℕ => (n:ℝ)^2)).comp (measurable_cluster_ncard G hloc x)
-  have hn : 0 ≤ᵐ[law p] (fun ω => ((cluster G ω x).ncard:ℝ)^2) :=
-    Filter.Eventually.of_forall (fun _ => sq_nonneg _)
-  have he : (fun ω => ENNReal.ofReal (((cluster G ω x).ncard:ℝ)^2)) =ᵐ[law p]
-      (fun ω => clusterMass G ω x ^ 2) := by
-    filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
-    rw [clusterMass_eq_ncard G ω x hω,ENNReal.ofReal_pow (Nat.cast_nonneg _)]
-    simp only [ENNReal.ofReal_natCast]
-  have hbound : (∫⁻ ω, ENNReal.ofReal (((cluster G ω x).ncard:ℝ)^2) ∂law p) ≤ ENNReal.ofReal (C^3) := by
-    rw [lintegral_congr_ae he,ENNReal.ofReal_pow hC]
-    exact cluster_second_moment_bound G p _ hrow x
-  have hi : Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^2) (law p) :=
-    ⟨hmeas.aestronglyMeasurable,(hasFiniteIntegral_iff_ofReal hn).mpr
-      (hbound.trans_lt ENNReal.ofReal_lt_top)⟩
-  refine ⟨hi,?_⟩
-  rw [integral_eq_lintegral_of_nonneg_ae hn hmeas.aestronglyMeasurable]
-  exact (ENNReal.toReal_mono ENNReal.ofReal_ne_top hbound).trans_eq (ENNReal.toReal_ofReal (pow_nonneg hC _))
-
-end SubcriticalMoments
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-open scoped ENNReal
-section SusceptibilityTypes
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def susceptibility (G : BondGraph V E) (p : unitInterval) (x : V) : ℝ :=
-  ∑' y, twoPoint G p x y
-
-noncomputable def rootSusceptibility (G : BondGraph V E) {ι : Type*} [Fintype ι]
-    (root : ι → V) (weight : ι → ℝ) (p : unitInterval) : ℝ :=
-  ∑ i, weight i * susceptibility G p (root i)
-
-theorem susceptibility_nonneg (G : BondGraph V E) (p : unitInterval) (x : V) :
-    0 ≤ susceptibility G p x := tsum_nonneg (twoPoint_nonneg G p x)
-
-theorem summable_twoPoint_subcritical [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (x : V) : Summable (twoPoint G p x) := by
-  obtain ⟨_,_,hrow⟩ := subcritical_row_bound G hloc hqt p hp
-  exact summable_of_sum_le (twoPoint_nonneg G p x) (hrow x)
-
-theorem ennreal_susceptibility [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (x : V) :
-    ENNReal.ofReal (susceptibility G p x) = ∑' y, law p {ω | Conn G ω x y} := by
-  rw [susceptibility,ENNReal.ofReal_tsum_of_nonneg (twoPoint_nonneg G p x)
-    (summable_twoPoint_subcritical G hloc hqt p hp x)]
-  apply tsum_congr
-  intro y
-  exact ENNReal.ofReal_toReal (measure_ne_top _ _)
-
-theorem susceptibility_aut [Countable V] [Countable E] (G : BondGraph V E)
-    (p : unitInterval) {a : V ≃ V} {b : E ≃ E} (h : IsAut G a b) (x : V) :
-    susceptibility G p (a x) = susceptibility G p x := by
-  unfold susceptibility
-  rw [←a.tsum_eq]
-  simp only [twoPoint_aut G p h]
-
-theorem susceptibility_one_le [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (x : V) : 1 ≤ susceptibility G p x := by
-  have hd : twoPoint G p x x = 1 := by
-    have he : {ω : Set E | Conn G ω x x} = Set.univ := Set.eq_univ_of_forall (fun _ => conn_refl G _ x)
-    simp only [twoPoint,he,measure_univ,ENNReal.toReal_one]
-  rw [←hd]
-  exact (summable_twoPoint_subcritical G hloc hqt p hp x).le_tsum x (fun _ _ => twoPoint_nonneg G p _ _)
-
-theorem rootSusceptibility_nonneg (G : BondGraph V E) {ι : Type*} [Fintype ι]
-    (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 ≤ weight i) (p : unitInterval) :
-    0 ≤ rootSusceptibility G root weight p :=
-  Finset.sum_nonneg (fun i _ => mul_nonneg (hw i) (susceptibility_nonneg G p _))
-
-theorem susceptibility_le_rootSusceptibility [Countable V] [Countable E]
-    (G : BondGraph V E) {ι : Type*} [Fintype ι]
-    (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 ≤ weight i)
-    (hcov : ∀ x, ∃ i, x ∈ vertexOrbit G (root i))
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i) (p : unitInterval) (x : V) :
-    susceptibility G p x ≤ c*rootSusceptibility G root weight p := by
-  obtain ⟨i,a,b,hab,hax⟩ := hcov x
-  rw [←hax,susceptibility_aut G p hab]
-  calc
-    _ ≤ (c*weight i)*susceptibility G p (root i) := by
-      simpa only [one_mul] using mul_le_mul_of_nonneg_right (hmin i) (susceptibility_nonneg G p _)
-    _ = c*(weight i*susceptibility G p (root i)) := mul_assoc _ _ _
-    _ ≤ _ := mul_le_mul_of_nonneg_left
-      (Finset.single_le_sum (fun j _ => mul_nonneg (hw j) (susceptibility_nonneg G p _)) (Finset.mem_univ i)) hc
-
- 
-theorem susceptibility_eq_integral [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (x : V) :
-    Integrable (fun ω => ((cluster G ω x).ncard:ℝ)) (law p) ∧
-      susceptibility G p x = ∫ ω, ((cluster G ω x).ncard:ℝ) ∂law p := by
-  have hm : Measurable (fun ω : Set E => ((cluster G ω x).ncard:ℝ)) :=
-    (measurable_of_countable (fun n : ℕ => (n:ℝ))).comp (measurable_cluster_ncard G hloc x)
-  have hn : 0 ≤ᵐ[law p] (fun ω => ((cluster G ω x).ncard:ℝ)) := Filter.Eventually.of_forall (fun _ => Nat.cast_nonneg _)
-  have he : (fun ω => ENNReal.ofReal ((cluster G ω x).ncard:ℝ)) =ᵐ[law p] (fun ω => clusterMass G ω x) := by
-    filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
-    rw [clusterMass_eq_ncard G ω x hω,ENNReal.ofReal_natCast]
-  have heq : (∫⁻ ω, ENNReal.ofReal ((cluster G ω x).ncard:ℝ) ∂law p) = ENNReal.ofReal (susceptibility G p x) := by
-    rw [lintegral_congr_ae he,lintegral_clusterMass,ennreal_susceptibility G hloc hqt p hp]
-  refine ⟨⟨hm.aestronglyMeasurable,(hasFiniteIntegral_iff_ofReal hn).mpr (by rw [heq]; exact ENNReal.ofReal_lt_top)⟩,?_⟩
-  rw [integral_eq_lintegral_of_nonneg_ae hn hm.aestronglyMeasurable,heq,ENNReal.toReal_ofReal (susceptibility_nonneg G p x)]
-
-end SusceptibilityTypes
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section FiniteKernelNorm
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem kernel_sum_le_norm_card (K : V → V → ℝ)
-    (A : Hilbert V →L[ℂ] Hilbert V)
-    (hA : ∀ x y, A (lp.single 2 y (1:ℂ)) x = (K x y : ℂ)) (F : Finset V) :
-    ∑ x ∈ F, ∑ y ∈ F, K x y ≤ ‖A‖*F.card := by
-  have he : inner ℂ (deltaSum F) (A (deltaSum F)) =
-      ((∑ x ∈ F, ∑ y ∈ F, K x y : ℝ) : ℂ) := by
-    rw [inner_deltaSum]
-    simp only [deltaSum,map_sum,lp.coeFn_sum,Finset.sum_apply,Complex.ofReal_sum]
-    apply Finset.sum_congr rfl
-    intro x _
-    apply Finset.sum_congr rfl
-    intro y _
-    exact hA x y
-  have h := (Complex.re_le_norm (inner ℂ (deltaSum F) (A (deltaSum F)))).trans
-    ((norm_inner_le_norm _ _).trans
-      (mul_le_mul_of_nonneg_left (A.le_opNorm _) (norm_nonneg _)))
-  rw [he,Complex.ofReal_re] at h
-  calc
-    _ ≤ ‖deltaSum F‖*(‖A‖*‖deltaSum F‖) := h
-    _ = ‖A‖*‖deltaSum F‖^2 := by ring
-    _ = _ := by rw [norm_deltaSum_sq]
-
- 
-theorem kernel_average_le_norm (K : V → V → ℝ)
-    (A : Hilbert V →L[ℂ] Hilbert V)
-    (hA : ∀ x y, A (lp.single 2 y (1:ℂ)) x = (K x y : ℂ))
-    (F : Finset V) (hF : F.Nonempty) :
-    (∑ x ∈ F, ∑ y ∈ F, K x y)/(F.card:ℝ) ≤ ‖A‖ := by
-  exact (div_le_iff₀ (Nat.cast_pos.mpr hF.card_pos)).mpr (kernel_sum_le_norm_card K A hA F)
-
-end FiniteKernelNorm
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section BinClusterSlice
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def binClusterSlice (G : BondGraph V E) (o : V) (n : ℕ)
-    (ω : Set E) (x : V) : Finset V :=
-  if h : ClusterWindow G x n (2*n) ω then h.1.toFinset.filter (fun y => y ∈ vertexOrbit G o) else ∅
-
-theorem mem_binClusterSlice (G : BondGraph V E) (o : V) (n : ℕ) (ω : Set E) (x y : V) :
-    y ∈ binClusterSlice G o n ω x ↔ ClusterWindow G x n (2*n) ω ∧ Conn G ω x y ∧ y ∈ vertexOrbit G o := by
-  unfold binClusterSlice
-  split_ifs with h
-  · simp only [Finset.mem_filter,Set.Finite.mem_toFinset,h,true_and,cluster,Set.mem_ofPred_eq]
-  · simp only [Finset.notMem_empty,h,false_and]
-
-theorem binClusterSlice_eq_of_conn (G : BondGraph V E) (o : V) (n : ℕ) (ω : Set E)
-    {x y : V} (h : Conn G ω x y) : binClusterSlice G o n ω x = binClusterSlice G o n ω y := by
-  have hc := cluster_eq_of_conn G ω h
-  have hw : ClusterWindow G x n (2*n) ω ↔ ClusterWindow G y n (2*n) ω := by
-    unfold ClusterWindow
-    rw [hc]
-  ext z
-  simp only [mem_binClusterSlice,hw]
-  have hz : Conn G ω x z ↔ Conn G ω y z := by change z ∈ cluster G ω x ↔ z ∈ cluster G ω y; rw [hc]
-  rw [hz]
-
-theorem binClusterSlice_nonempty (G : BondGraph V E) (o : V) (n : ℕ) (ω : Set E) (x : V)
-    (hw : ClusterWindow G x n (2*n) ω) (hx : x ∈ vertexOrbit G o) :
-    (binClusterSlice G o n ω x).Nonempty :=
-  ⟨x,(mem_binClusterSlice G o n ω x x).mpr ⟨hw,conn_refl _ _ _,hx⟩⟩
-
-theorem binOrbitRowMass_eq_slice (G : BondGraph V E) (o : V) (n : ℕ) (ω : Set E) (x : V)
-    (hx : x ∈ vertexOrbit G o) :
-    binOrbitRowMass G n ω x = (binClusterSlice G o n ω x).card := by
-  by_cases hw : ClusterWindow G x n (2*n) ω
-  · rw [binOrbitRowMass_eq G n ω x hw,vertexOrbit_eq_of_mem G o x hx]
-    simp only [binClusterSlice,dite_eq_left hw]
-    rw [←Finset.sum_filter]
-    simp
-  · rw [binOrbitRowMass_zero G n ω x hw]
-    simp only [binClusterSlice,dite_eq_right hw,Finset.card_empty,Nat.cast_zero]
-
-theorem binSizeMass_eq_of_conn (G : BondGraph V E) (n : ℕ) (ω : Set E)
-    {x y : V} (hc : Conn G ω x y) : binSizeMass G n ω x = binSizeMass G n ω y := by
-  have he := cluster_eq_of_conn G ω hc
-  have hw : ClusterWindow G x n (2*n) ω ↔ ClusterWindow G y n (2*n) ω := by
-    unfold ClusterWindow
-    rw [he]
-  by_cases hx : ClusterWindow G x n (2*n) ω
-  · rw [binSizeMass_eq G n ω x hx,binSizeMass_eq G n ω y (hw.mp hx),he]
-  · have hy := fun h => hx (hw.mpr h)
-    simp only [binSizeMass,binMass_zero_of_not_window G n ω x _ hx,
-      binMass_zero_of_not_window G n ω y _ hy,tsum_zero]
-
-theorem mem_binClusterSlice_iff_mass (G : BondGraph V E) (o : V) (n : ℕ)
-    (ω : Set E) (x y : V) :
-    y ∈ binClusterSlice G o n ω x ↔
-      binConnectionMass G n ω x y = 1 ∧ y ∈ vertexOrbit G o := by
-  rw [mem_binClusterSlice]
-  have he : binConnectionMass G n ω x y =
-      if ClusterWindow G x n (2*n) ω ∧ Conn G ω x y then 1 else 0 := by
-    unfold binConnectionMass ClusterWindow
-    congr 1
-    apply propext; tauto
-  rw [he]
-  by_cases h : ClusterWindow G x n (2*n) ω ∧ Conn G ω x y
-  · rw [ite_eq_left h]
-    simp only [h.1,h.2,true_and]
-  · rw [ite_eq_right h]
-    simp only [zero_ne_one,false_and]
-    tauto
-
-theorem binClusterSlice_local (G : BondGraph V E) (hloc : LocallyFinite G)
-    (o : V) (n : ℕ) (ω : Set E) (x : V) :
-    let Q := incidentSupport G hloc (finite_reachBall hloc x (2*n)).toFinset
-    binClusterSlice G o n ω x = binClusterSlice G o n (ω ∩ (Q : Set E)) x := by
-  intro Q
-  ext y
-  rw [mem_binClusterSlice_iff_mass,mem_binClusterSlice_iff_mass,
-    binConnectionMass_local G hloc n ω x y]
-
-end BinClusterSlice
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section FiniteClassTransport
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-noncomputable def finiteClassTransport (S : V → Finset V) (t : V → ℝ)
-    (K : V → V → ℝ) (x y : V) : ℝ :=
-  if y ∈ S x then t x / (S x).card * ∑ z ∈ S x, K x z else 0
-
-theorem finiteClassTransport_nonneg (S : V → Finset V) (t : V → ℝ)
-    (K : V → V → ℝ) (ht : ∀ x, 0 ≤ t x) (hK : ∀ x y, 0 ≤ K x y) (x y : V) :
-    0 ≤ finiteClassTransport S t K x y := by
-  unfold finiteClassTransport
-  split_ifs
-  · exact mul_nonneg (div_nonneg (ht x) (Nat.cast_nonneg _)) (Finset.sum_nonneg (fun z _ => hK x z))
-  · rfl
-
-theorem finiteClassTransport_row (S : V → Finset V) (t : V → ℝ)
-    (K : V → V → ℝ) (x : V) :
-    (∑' y, finiteClassTransport S t K x y) = t x * ∑ z ∈ S x, K x z := by
-  rw [tsum_eq_sum (s := S x) (fun y hy => (show finiteClassTransport S t K x y = 0 from ite_eq_right hy))]
-  have he : (∑ y ∈ S x, finiteClassTransport S t K x y) =
-      (S x).card * (t x / (S x).card * ∑ z ∈ S x, K x z) := by
-    simp only [finiteClassTransport]
-    rw [Finset.sum_ite_mem]
-    simp only [Finset.inter_self,Finset.sum_const,nsmul_eq_mul]
-  rw [he]
-  by_cases h : (S x).Nonempty
-  · have hc : ((S x).card:ℝ) ≠ 0 := Nat.cast_ne_zero.mpr h.card_pos.ne'
-    field_simp
-  · have hs := Finset.not_nonempty_iff_eq_empty.mp h
-    simp only [hs,Finset.card_empty,Nat.cast_zero,Finset.sum_empty,mul_zero]
-
-theorem finiteClassTransport_column (S : V → Finset V) (t : V → ℝ)
-    (K : V → V → ℝ)
-    (hsym : ∀ x y, y ∈ S x ↔ x ∈ S y)
-    (heq : ∀ x y, y ∈ S x → S y = S x)
-    (ht : ∀ x y, y ∈ S x → t y = t x) (x : V) :
-    (∑' y, finiteClassTransport S t K y x) =
-      t x / (S x).card * ∑ y ∈ S x, ∑ z ∈ S x, K y z := by
-  rw [tsum_eq_sum (s := S x) (fun y hy => (show finiteClassTransport S t K y x = 0 from
-    ite_eq_right (fun h => hy ((hsym x y).mpr h))))]
-  rw [Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro y hy
-  rw [finiteClassTransport,ite_eq_left ((hsym x y).mp hy),heq x y hy,ht x y hy]
-
- 
-theorem finiteClassTransport_column_le (S : V → Finset V) (t : V → ℝ)
-    (K : V → V → ℝ) (ht0 : ∀ x, 0 ≤ t x)
-    (hsym : ∀ x y, y ∈ S x ↔ x ∈ S y)
-    (heq : ∀ x y, y ∈ S x → S y = S x)
-    (ht : ∀ x y, y ∈ S x → t y = t x)
-    (A : Hilbert V →L[ℂ] Hilbert V)
-    (hA : ∀ x y, A (lp.single 2 y (1:ℂ)) x = (K x y : ℂ)) (x : V) :
-    (∑' y, finiteClassTransport S t K y x) ≤ ‖A‖*t x := by
-  rw [finiteClassTransport_column S t K hsym heq ht]
-  by_cases h : (S x).Nonempty
-  · have hh := mul_le_mul_of_nonneg_left (kernel_average_le_norm K A hA (S x) h) (ht0 x)
-    simpa only [div_mul_eq_mul_div,mul_div_assoc,mul_comm (t x) ‖A‖] using hh
-  · have hs := Finset.not_nonempty_iff_eq_empty.mp h
-    simp only [hs,Finset.card_empty,Nat.cast_zero,div_zero,Finset.sum_empty,mul_zero]
-    exact mul_nonneg (norm_nonneg _) (ht0 x)
-
-end FiniteClassTransport
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section BinDiagonalClass
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def binOrbitClass (G : BondGraph V E) (o : V) (n : ℕ)
-    (ω : Set E) (x : V) : Finset V :=
-  if x ∈ vertexOrbit G o then binClusterSlice G o n ω x else ∅
-
-theorem mem_binOrbitClass (G : BondGraph V E) (o : V) (n : ℕ) (ω : Set E) (x y : V) :
-    y ∈ binOrbitClass G o n ω x ↔ x ∈ vertexOrbit G o ∧
-      ClusterWindow G x n (2*n) ω ∧ Conn G ω x y ∧ y ∈ vertexOrbit G o := by
-  unfold binOrbitClass
-  split_ifs with h
-  · simp only [mem_binClusterSlice,h,true_and]
-  · simp only [Finset.notMem_empty,h,false_and]
-
-theorem binOrbitClass_symm (G : BondGraph V E) (o : V) (n : ℕ) (ω : Set E) (x y : V) :
-    y ∈ binOrbitClass G o n ω x ↔ x ∈ binOrbitClass G o n ω y := by
-  have h (x y : V) (hy : y ∈ binOrbitClass G o n ω x) : x ∈ binOrbitClass G o n ω y := by
-    obtain ⟨hx,hw,hc,hy⟩ := (mem_binOrbitClass G o n ω x y).mp hy
-    have hwy : ClusterWindow G y n (2*n) ω := by
-      unfold ClusterWindow at hw ⊢
-      rw [←cluster_eq_of_conn G ω hc]
-      exact hw
-    exact (mem_binOrbitClass G o n ω y x).mpr ⟨hy,hwy,conn_symm G ω hc,hx⟩
-  exact ⟨h x y,h y x⟩
-
-theorem binOrbitClass_eq (G : BondGraph V E) (o : V) (n : ℕ) (ω : Set E) (x y : V)
-    (hy : y ∈ binOrbitClass G o n ω x) : binOrbitClass G o n ω y = binOrbitClass G o n ω x := by
-  obtain ⟨hx,_,hc,hy⟩ := (mem_binOrbitClass G o n ω x y).mp hy
-  simp only [binOrbitClass,ite_eq_left hx,ite_eq_left hy]
-  exact (binClusterSlice_eq_of_conn G o n ω hc).symm
-
-theorem binOrbitClass_mass_eq (G : BondGraph V E) (o : V) (n : ℕ) (ω : Set E) (x y : V)
-    (hy : y ∈ binOrbitClass G o n ω x) : binSizeMass G n ω y = binSizeMass G n ω x :=
-  (binSizeMass_eq_of_conn G n ω ((mem_binOrbitClass G o n ω x y).mp hy).2.2.1).symm
-
-theorem binOrbitClass_local (G : BondGraph V E) (hloc : LocallyFinite G)
-    (o : V) (n : ℕ) (ω : Set E) (x : V) :
-    let Q := incidentSupport G hloc (finite_reachBall hloc x (2*n)).toFinset
-    binOrbitClass G o n ω x = binOrbitClass G o n (ω ∩ (Q : Set E)) x := by
-  intro Q
-  unfold binOrbitClass
-  rw [binClusterSlice_local G hloc o n ω x]
-
-theorem binOrbitClass_mem_covariant (G : BondGraph V E) (o : V)
-    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b)
-    (n : ℕ) (ω : Set E) (x y : V) :
-    y ∈ binOrbitClass G o n (b ⁻¹' ω) x ↔ a y ∈ binOrbitClass G o n ω (a x) := by
-  unfold binOrbitClass
-  simp only [mem_vertexOrbit_aut_iff G o hab]
-  split_ifs
-  · simp only [mem_binClusterSlice_iff_mass,binConnectionMass_covariant G hab,
-      mem_vertexOrbit_aut_iff G o hab]
-  · simp only [Finset.notMem_empty]
-
-theorem binOrbitClass_covariant (G : BondGraph V E) (o : V)
-    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b)
-    (n : ℕ) (ω : Set E) (x : V) :
-    (binOrbitClass G o n (b ⁻¹' ω) x).image a = binOrbitClass G o n ω (a x) := by
-  ext y
-  constructor
-  · rintro hy
-    obtain ⟨z,hz,rfl⟩ := Finset.mem_image.mp hy
-    exact (binOrbitClass_mem_covariant G o hab n ω x z).mp hz
-  · intro hy
-    exact Finset.mem_image.mpr ⟨a.symm y,(binOrbitClass_mem_covariant G o hab n ω x (a.symm y)).mpr
-      (by simpa only [Equiv.apply_symm_apply] using hy),a.apply_symm_apply y⟩
-
-theorem binOrbitClass_reach (G : BondGraph V E) (o : V) (n : ℕ) (ω : Set E) (x y : V)
-    (hy : y ∈ binOrbitClass G o n ω x) : y ∈ reachBall G x (2*n) := by
-  obtain ⟨_,hw,hc,_⟩ := (mem_binOrbitClass G o n ω x y).mp hy
-  exact finite_cluster_subset_reachBall G ω x hw.1 (2*n) hw.2.2.le hc
-
-end BinDiagonalClass
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section BinDiagonalTransport
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def binDiagonalMass (G : BondGraph V E) (o : V) (n : ℕ)
-    (K : V → V → ℝ) (ω : Set E) (x y : V) : ℝ :=
-  finiteClassTransport (binOrbitClass G o n ω) (fun x => binSizeMass G n ω x ^ 2) K x y
-
-theorem binDiagonalMass_nonneg (G : BondGraph V E) (o : V) (n : ℕ)
-    (K : V → V → ℝ) (hK : ∀ x y, 0 ≤ K x y) (ω : Set E) (x y : V) :
-    0 ≤ binDiagonalMass G o n K ω x y :=
-  finiteClassTransport_nonneg _ _ K (fun _ => sq_nonneg _) hK x y
-
-theorem binDiagonalMass_local (G : BondGraph V E) (hloc : LocallyFinite G)
-    (o : V) (n : ℕ) (K : V → V → ℝ) (ω : Set E) (x y : V) :
-    let Q := incidentSupport G hloc (finite_reachBall hloc x (2*n)).toFinset
-    binDiagonalMass G o n K ω x y = binDiagonalMass G o n K (ω ∩ (Q : Set E)) x y := by
-  intro Q
-  simp only [binDiagonalMass,finiteClassTransport]
-  rw [binOrbitClass_local G hloc o n ω x,binSizeMass_local G hloc n ω x]
-
-theorem binDiagonalMass_integrable (G : BondGraph V E) (hloc : LocallyFinite G)
-    (p : unitInterval) (o : V) (n : ℕ) (K : V → V → ℝ) (x y : V) :
-    Integrable (fun ω => binDiagonalMass G o n K ω x y) (law p) :=
-  integrable_finite_local _ p _ (fun ω => binDiagonalMass_local G hloc o n K ω x y)
-
-theorem binDiagonalMass_outside (G : BondGraph V E) (o : V) (n : ℕ)
-    (K : V → V → ℝ) (ω : Set E) (x y : V) (hy : y ∉ reachBall G x (2*n)) :
-    binDiagonalMass G o n K ω x y = 0 :=
-  ite_eq_right (fun h => hy (binOrbitClass_reach G o n ω x y h))
-
-theorem binDiagonalMass_outside_column (G : BondGraph V E) (o : V) (n : ℕ)
-    (K : V → V → ℝ) (ω : Set E) (x y : V) (hx : x ∉ reachBall G y (2*n)) :
-    binDiagonalMass G o n K ω x y = 0 :=
-  ite_eq_right (fun h => hx (binOrbitClass_reach G o n ω y x ((binOrbitClass_symm G o n ω x y).mp h)))
-
-theorem binDiagonalMass_row_zero (G : BondGraph V E) (o : V) (n : ℕ)
-    (K : V → V → ℝ) (ω : Set E) (x y : V) (hx : x ∉ vertexOrbit G o) :
-    binDiagonalMass G o n K ω x y = 0 :=
-  ite_eq_right (fun h => hx ((mem_binOrbitClass G o n ω x y).mp h).1)
-
-theorem binDiagonalMass_column_zero (G : BondGraph V E) (o : V) (n : ℕ)
-    (K : V → V → ℝ) (ω : Set E) (x y : V) (hy : y ∉ vertexOrbit G o) :
-    binDiagonalMass G o n K ω x y = 0 :=
-  ite_eq_right (fun h => hy ((mem_binOrbitClass G o n ω x y).mp h).2.2.2)
-
-theorem binDiagonalMass_covariant (G : BondGraph V E) (o : V) (n : ℕ)
-    (K : V → V → ℝ) {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b)
-    (hK : ∀ x y, K (a x) (a y) = K x y) (ω : Set E) (x y : V) :
-    binDiagonalMass G o n K (b ⁻¹' ω) x y = binDiagonalMass G o n K ω (a x) (a y) := by
-  have hs := binOrbitClass_covariant G o hab n ω x
-  have hc : (binOrbitClass G o n (b ⁻¹' ω) x).card = (binOrbitClass G o n ω (a x)).card := by
-    rw [←hs,Finset.card_image_of_injective _ a.injective]
-  have hsum : (∑ z ∈ binOrbitClass G o n (b ⁻¹' ω) x, K x z) =
-      ∑ z ∈ binOrbitClass G o n ω (a x), K (a x) z := by
-    rw [←hs,Finset.sum_image (fun _ _ _ _ h => a.injective h)]
-    simp only [hK]
-  simp only [binDiagonalMass,finiteClassTransport,binOrbitClass_mem_covariant G o hab,
-    binSizeMass_covariant G n ω hab,hc,hsum]
-
-theorem binDiagonalMass_row (G : BondGraph V E) (o : V) (n : ℕ)
-    (K : V → V → ℝ) (ω : Set E) (x : V) :
-    (∑' y, binDiagonalMass G o n K ω x y) =
-      binSizeMass G n ω x ^ 2 * ∑ z ∈ binOrbitClass G o n ω x, K x z :=
-  finiteClassTransport_row _ _ K x
-
-theorem binDiagonalMass_column (G : BondGraph V E) (o : V) (n : ℕ)
-    (K : V → V → ℝ) (ω : Set E) (x : V) :
-    (∑' y, binDiagonalMass G o n K ω y x) =
-      binSizeMass G n ω x ^ 2 / (binOrbitClass G o n ω x).card *
-        ∑ y ∈ binOrbitClass G o n ω x, ∑ z ∈ binOrbitClass G o n ω x, K y z := by
-  apply finiteClassTransport_column _ _ K (binOrbitClass_symm G o n ω) (binOrbitClass_eq G o n ω)
-  intro x y hy
-  rw [binOrbitClass_mass_eq G o n ω x y hy]
-
-theorem binDiagonalMass_column_le (G : BondGraph V E) (o : V) (n : ℕ)
-    (K : V → V → ℝ) (A : Hilbert V →L[ℂ] Hilbert V)
-    (hA : ∀ x y, A (lp.single 2 y (1:ℂ)) x = (K x y : ℂ)) (ω : Set E) (x : V) :
-    (∑' y, binDiagonalMass G o n K ω y x) ≤ ‖A‖ * binSizeMass G n ω x ^ 2 := by
-  apply finiteClassTransport_column_le _ _ K (fun _ => sq_nonneg _)
-    (binOrbitClass_symm G o n ω) (binOrbitClass_eq G o n ω) _ A hA
-  intro x y hy
-  rw [binOrbitClass_mass_eq G o n ω x y hy]
-
-end BinDiagonalTransport
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Clusters.Resampling
+
+namespace OAI
+
+/-! Kernel Moments for bond percolation. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section DeletedDomainPotential
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem inducedSusceptibility_nonneg (G : BondGraph V E) (D : Set V)
+    (p : unitInterval) (x : V) : 0 ≤ inducedSusceptibility G D p x := by
+  unfold inducedSusceptibility
+  split_ifs
+  · exact restrictedSusceptibility_nonneg G _ p x
+  · rfl
+
+noncomputable def deletedBoundaryPotential (G : BondGraph V E) (hloc : LocallyFinite G)
+    (D : Set V) (p : unitInterval) (x : V) (M : ℕ) : ℝ :=
+  ∑ a ∈ ((finite_reachBall hloc x M).toFinset.filter (· ∉ D)),
+    ∑ b ∈ neighbors G hloc a, inducedSusceptibility G D p b
+
+theorem deletedBoundaryPotential_nonneg (G : BondGraph V E) (hloc : LocallyFinite G)
+    (D : Set V) (p : unitInterval) (x : V) (M : ℕ) :
+    0 ≤ deletedBoundaryPotential G hloc D p x M := by
+  exact Finset.sum_nonneg (fun a _ => Finset.sum_nonneg (fun b _ => inducedSusceptibility_nonneg G D p b))
+
+theorem le_deletedBoundaryPotential (G : BondGraph V E) (hloc : LocallyFinite G)
+    (D : Set V) (p : unitInterval) (x : V) (M : ℕ) (a b : V)
+    (ha : a ∈ reachBall G x M) (haD : a ∉ D) (hab : Adj G a b) :
+    inducedSusceptibility G D p b ≤ deletedBoundaryPotential G hloc D p x M := by
+  apply le_trans (Finset.single_le_sum (fun z _ => inducedSusceptibility_nonneg G D p z)
+    ((mem_neighbors G hloc a b).mpr hab))
+  apply Finset.single_le_sum (fun z _ => Finset.sum_nonneg (fun y _ => inducedSusceptibility_nonneg G D p y))
+  exact Finset.mem_filter.mpr ⟨(finite_reachBall hloc x M).mem_toFinset.mpr ha,haD⟩
+
+noncomputable def deletedLocalPotential (G : BondGraph V E) (D : Set V) (p : unitInterval)
+    (ω : Set E) (x : V) (M : ℕ) : ℝ :=
+  if x ∈ D then intrinsicPotentialMax G (ω ∩ internalEdges G D) x M (inducedSusceptibility G D p) else 0
+
+theorem deletedLocalPotential_nonneg (G : BondGraph V E) (hloc : LocallyFinite G)
+    (D : Set V) (p : unitInterval) (ω : Set E) (x : V) (M : ℕ) :
+    0 ≤ deletedLocalPotential G D p ω x M := by
+  unfold deletedLocalPotential
+  split_ifs
+  · rw [intrinsicPotentialMax_eq G hloc]
+    exact inducedSusceptibility_nonneg G D p _
+  · rfl
+
+ 
+
+theorem short_path_deleted_potential_bound [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (D : Set V)
+    (p : unitInterval) (hp : 0 < (p:ℝ)) (ω : Set E) (x y : V) (M : ℕ)
+    (l : List E) (hl : Along G x l y) (hlen : l.length ≤ M) (ho : ∀ e ∈ l, e ∈ ω)
+    (hfin : ∀ a, Summable (fun z => (law p).real {η | Conn G (η ∩ internalEdges G D) a z})) :
+    inducedSusceptibility G D p y ≤ deletedLocalPotential G D p ω x M +
+      (p:ℝ)^(-(M:ℤ))*deletedBoundaryPotential G hloc D p x M := by
+  have hpM : 0 ≤ (p:ℝ)^(-(M:ℤ)) := zpow_nonneg p.2.1 _
+  have hn := deletedLocalPotential_nonneg G hloc D p ω x M
+  have hb := deletedBoundaryPotential_nonneg G hloc D p x M
+  by_cases hy : y ∈ D
+  · rcases hl.last_entrance D hy with ⟨hx,hall⟩ | ⟨a,b,e,l₁,l₂,heq,h₁,he,ha,hbD,h₂,hs⟩
+    · have hreach : y ∈ intrinsicBall G (ω ∩ internalEdges G D) x M :=
+        ⟨l,hl,hlen,fun e he => ⟨ho e he,hall e he⟩⟩
+      have hh := le_intrinsicPotentialMax G hloc _ x M (inducedSusceptibility G D p) y hreach
+      change _ ≤ (if x ∈ D then _ else 0) + _
+      rw [ite_eq_left hx]
+      exact hh.trans (le_add_of_nonneg_right (mul_nonneg hpM hb))
+    · have hlen₂ : l₂.length ≤ M := by subst l; simp only [List.length_append,List.length_cons] at hlen; omega
+      have hpath := restrictedSusceptibility_short_path_bound G (internalEdges G D) p hp h₂ hs M hlen₂ hfin
+      have hχ : inducedSusceptibility G D p y ≤ (p:ℝ)^(-(M:ℤ))*inducedSusceptibility G D p b := by
+        simpa only [inducedSusceptibility,ite_eq_left hy,ite_eq_left hbD] using hpath
+      have haBall : a ∈ reachBall G x M := reachBall_mono G x
+        (show l₁.length ≤ M by subst l; simp only [List.length_append,List.length_cons] at hlen; omega) h₁.mem_reachBall
+      exact hχ.trans ((mul_le_mul_of_nonneg_left
+        (le_deletedBoundaryPotential G hloc D p x M a b haBall ha ⟨e,he⟩) hpM).trans
+          (le_add_of_nonneg_left hn))
+  · rw [inducedSusceptibility,ite_eq_right hy]
+    exact add_nonneg hn (mul_nonneg hpM hb)
+
+end DeletedDomainPotential
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section LocalKernelAction
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
+
+theorem summable_mass_action (q : LocalEndpointRule G hloc O M L)
+    (ω : Set E) (x : V) (f : V → ℝ) : Summable (fun y => q.mass ω x y * f y) := by
+  apply summable_of_ne_finset_zero (s := (finite_reachBall hloc x M).toFinset)
+  intro y hy
+  rw [q.outside_ball_zero ω x y (by simpa only [Set.Finite.mem_toFinset] using hy),zero_mul]
+
+theorem summable_kernel_action (q : LocalEndpointRule G hloc O M L)
+    (p : unitInterval) (x : V) (f : V → ℝ) : Summable (fun y => q.kernel p x y * f y) := by
+  apply summable_of_ne_finset_zero (s := (finite_reachBall hloc x M).toFinset)
+  intro y hy
+  rw [q.kernel_outside_ball_zero p x y (by simpa only [Set.Finite.mem_toFinset] using hy),zero_mul]
+
+theorem integrable_mass_action (q : LocalEndpointRule G hloc O M L)
+    (p : unitInterval) (x : V) (f : V → ℝ) :
+    Integrable (fun ω => ∑' y, q.mass ω x y * f y) (law p) := by
+  apply integrable_finite_local (incidentSupport G hloc (finite_reachBall hloc x q.radius).toFinset) p
+  intro ω
+  exact tsum_congr (fun y => congrArg (· * f y) (q.locality ω x y))
+
+theorem kernel_action_integral (q : LocalEndpointRule G hloc O M L)
+    (p : unitInterval) (x : V) (f : V → ℝ) :
+    (∑' y, q.kernel p x y * f y) = ∫ ω, (∑' y, q.mass ω x y * f y) ∂law p := by
+  let B := (finite_reachBall hloc x M).toFinset
+  have h₁ : (∑' y, q.kernel p x y*f y) = ∑ y ∈ B, q.kernel p x y*f y :=
+    tsum_eq_sum fun y hy => by
+      rw [q.kernel_outside_ball_zero p x y (by simpa only [B,Set.Finite.mem_toFinset] using hy),zero_mul]
+  have h₂ (ω : Set E) : (∑' y, q.mass ω x y*f y) = ∑ y ∈ B, q.mass ω x y*f y :=
+    tsum_eq_sum fun y hy => by
+      rw [q.outside_ball_zero ω x y (by simpa only [B,Set.Finite.mem_toFinset] using hy),zero_mul]
+  rw [h₁]
+  simp_rw [h₂]
+  rw [integral_finsetSum B (fun y _ => (q.integrable_mass p x y).mul_const (f y))]
+  apply Finset.sum_congr rfl
+  intro y _
+  exact (integral_mul_const (f y) (fun ω => q.mass ω x y)).symm
+
+theorem mass_action_le (q : LocalEndpointRule G hloc O M L)
+    (ω : Set E) (x : V) (f : V → ℝ) (C : ℝ) (hC : 0 ≤ C)
+    (hf : ∀ y, q.mass ω x y ≠ 0 → f y ≤ C) :
+    (∑' y, q.mass ω x y * f y) ≤ C := by
+  calc
+    _ ≤ ∑' y, q.mass ω x y * C := by
+      apply (q.summable_mass_action ω x f).tsum_le_tsum _ ((q.summable_mass ω x).mul_right C)
+      intro y
+      by_cases h : q.mass ω x y = 0
+      · simp only [h,zero_mul,le_refl]
+      · exact mul_le_mul_of_nonneg_left (hf y h) (q.nonneg ω x y)
+    _ = (∑' y, q.mass ω x y)*C := tsum_mul_right
+    _ ≤ C := by simpa only [one_mul] using mul_le_mul_of_nonneg_right (q.row_le_one ω x) hC
+
+end LocalEndpointRule
+end LocalKernelAction
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section LocalMapDeletedPotential
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem integrable_deletedLocalPotential (G : BondGraph V E) (hloc : LocallyFinite G)
+    (D : Set V) (p : unitInterval) (x : V) (M : ℕ) :
+    Integrable (fun ω => deletedLocalPotential G D p ω x M) (law p) := by
+  by_cases hx : x ∈ D
+  · simp only [deletedLocalPotential,ite_eq_left hx]
+    exact integrable_finite_local _ p _ (fun ω =>
+      intrinsicPotentialMax_domain_local G hloc (internalEdges G D) ω x M (inducedSusceptibility G D p))
+  · simp only [deletedLocalPotential,ite_eq_right hx]
+    exact integrable_const 0
+
+theorem integral_deletedLocalPotential_le
+    [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (D : Set V) (p : unitInterval) (hp : 0 < (p:ℝ)) (hpc : p < pc G) (x : V) (M : ℕ) :
+    (∫ ω, deletedLocalPotential G D p ω x M ∂law p) ≤
+      (1+(1-(p:ℝ))/p*M)*inducedSusceptibility G D p x := by
+  by_cases hx : x ∈ D
+  · simp only [deletedLocalPotential,ite_eq_left hx]
+    exact maximal_induced_susceptibility_bound G hloc hqt D x hx M p hp hpc
+  · simp only [deletedLocalPotential,inducedSusceptibility,ite_eq_right hx,integral_zero,mul_zero,le_refl]
+
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
+
+ 
+
+theorem kernel_deleted_potential_bound
+    [Countable V] [Countable E] [Nonempty V]
+    (q : LocalEndpointRule G hloc O M L) (hqt : QuasiTransitive G)
+    (D : Set V) (p : unitInterval) (hp : 0 < (p:ℝ)) (hpc : p < pc G) (x : V) :
+    (∑' y, q.kernel p x y * inducedSusceptibility G D p y) ≤
+      (1+(1-(p:ℝ))/p*M)*inducedSusceptibility G D p x +
+        (p:ℝ)^(-(M:ℤ))*deletedBoundaryPotential G hloc D p x M := by
+  let f := inducedSusceptibility G D p
+  let C := (p:ℝ)^(-(M:ℤ))*deletedBoundaryPotential G hloc D p x M
+  have hC : 0 ≤ C := mul_nonneg (zpow_nonneg p.2.1 _) (deletedBoundaryPotential_nonneg G hloc D p x M)
+  have hpoint (ω : Set E) : (∑' y, q.mass ω x y * f y) ≤ deletedLocalPotential G D p ω x M + C := by
+    apply q.mass_action_le ω x f _ (add_nonneg (deletedLocalPotential_nonneg G hloc D p ω x M) hC)
+    intro y hy
+    obtain ⟨_,_,hxy,hfin,hcap,_⟩ := q.support ω x y hy
+    obtain ⟨l,hl,hlen,ho⟩ := conn_short_of_cluster_cap G ω x y hxy hfin M hcap
+    exact short_path_deleted_potential_bound G hloc D p hp ω x y M l hl hlen ho
+      (summable_restrictedTwoPoint G hloc hqt p hpc (internalEdges G D))
+  rw [q.kernel_action_integral]
+  have hi := integrable_deletedLocalPotential G hloc D p x M
+  calc
+    _ ≤ ∫ ω, deletedLocalPotential G D p ω x M + C ∂law p :=
+      integral_mono (q.integrable_mass_action p x f) (hi.add (integrable_const C)) hpoint
+    _ = (∫ ω, deletedLocalPotential G D p ω x M ∂law p) + C := by
+      rw [integral_add hi (integrable_const C)]; simp
+    _ ≤ _ := by
+      have hh := integral_deletedLocalPotential_le G hloc hqt D p hp hpc x M
+      dsimp [C]
+      linarith
+
+end LocalEndpointRule
+end LocalMapDeletedPotential
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section OrbitKernelTransport
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem orbit_kernel_column_eq_row
+    (G : BondGraph V E) {ι : Type*} [Fintype ι]
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (htr : RootTransport root weight Γ)
+    (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b) (hw : ∀ i, 0 ≤ weight i)
+    (i₀ : ι) (hw₀ : 0 < weight i₀) (K : V → V → ℝ)
+    (hcov : ∀ a b, IsAut G a b → ∀ x y, K (a x) (a y) = K x y)
+    (hout : ∀ x y, x ∉ vertexOrbit G (root i₀) → K x y = 0)
+    (hin : ∀ x y, y ∉ vertexOrbit G (root i₀) → K x y = 0)
+    (hr : ∀ x, Summable (K x)) (hc : ∀ y, Summable (fun x => K x y))
+    (y : V) (hy : y ∈ vertexOrbit G (root i₀)) :
+    (∑' x, K x y) = ∑' z, K (root i₀) z := by
+  let O := vertexOrbit G (root i₀)
+  let w : ι → ℝ := fun i => if root i ∈ O then weight i else 0
+  have hrow (x : V) (hx : x ∈ O) : (∑' z, K x z) = ∑' z, K (root i₀) z := by
+    obtain ⟨a,b,ha,rfl⟩ := hx
+    rw [← a.tsum_eq (fun z => K (a (root i₀)) z)]
+    exact tsum_congr (hcov a b ha (root i₀))
+  have hcol (x : V) (hx : x ∈ O) : (∑' z, K z x) = ∑' z, K z (root i₀) := by
+    obtain ⟨a,b,ha,rfl⟩ := hx
+    rw [← a.tsum_eq (fun z => K z (a (root i₀)))]
+    exact tsum_congr (fun z => hcov a b ha z (root i₀))
+  have ho : root i₀ ∈ O := by
+    refine ⟨Equiv.refl V,Equiv.refl E,?_,rfl⟩
+    intro e; simp
+  have hw' (i : ι) : 0 ≤ w i := by
+    dsimp [w]; split_ifs
+    · exact hw i
+    · rfl
+  have hpos : 0 < ∑ i, w i := by
+    have hi := Finset.single_le_sum (fun i _ => hw' i) (Finset.mem_univ i₀)
+    have he : w i₀ = weight i₀ := ite_eq_left ho
+    rw [he] at hi
+    exact hw₀.trans_le hi
+  have ht := htr.real root weight Γ K
+    (by intro a ha x z; obtain ⟨b,hb⟩ := hΓ a ha; exact hcov a b hb x z) hr hc
+  have h₁ (i : ι) : weight i * (∑' z, K (root i) z) = w i * (∑' z, K (root i₀) z) := by
+    by_cases hi : root i ∈ O
+    · rw [hrow (root i) hi]; simp only [w,ite_eq_left hi]
+    · simp only [w,ite_eq_right hi,zero_mul]
+      have he : (∑' z, K (root i) z) = 0 := by simp only [hout _ _ hi,tsum_zero]
+      rw [he,mul_zero]
+  have h₂ (i : ι) : weight i * (∑' z, K z (root i)) = w i * (∑' z, K z (root i₀)) := by
+    by_cases hi : root i ∈ O
+    · rw [hcol (root i) hi]; simp only [w,ite_eq_left hi]
+    · simp only [w,ite_eq_right hi,zero_mul]
+      have he : (∑' z, K z (root i)) = 0 := by simp only [hin _ _ hi,tsum_zero]
+      rw [he,mul_zero]
+  simp_rw [h₁,h₂] at ht
+  rw [← Finset.sum_mul,← Finset.sum_mul] at ht
+  rw [hcol y hy]
+  exact (mul_left_cancel₀ (ne_of_gt hpos) ht).symm
+
+ 
+
+theorem operator_norm_le_absolute_sums (K : V → V → ℝ) (D : ℝ) (hD : 0 ≤ D)
+    (hr : ∀ x (S : Finset V), (∑ y ∈ S, |K x y|) ≤ D)
+    (hc : ∀ y (S : Finset V), (∑ x ∈ S, |K x y|) ≤ D)
+    (A : Hilbert V →L[ℂ] Hilbert V)
+    (hA : ∀ x y, A (lp.single 2 y (1:ℂ)) x = (K x y : ℂ)) : ‖A‖ ≤ D := by
+  obtain ⟨B,hBn,hB⟩ := exists_operator_of_row_column_bound (fun x y => |K x y|) D hD
+    (fun _ _ => abs_nonneg _) hr hc
+  obtain ⟨A',hAn,hA'⟩ := exists_operator_of_matrix_domination
+    (fun x y => |K x y|) K (fun _ _ => abs_nonneg _) (fun _ _ => le_rfl) B hB
+  have he := kernel_extension_unique hA hA'
+  rw [he]
+  exact hAn.trans hBn
+
+end OrbitKernelTransport
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section EndpointDifference
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
+variable (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
+
+noncomputable def kernelDifference (p r : unitInterval) (x y : V) : ℝ :=
+  |q.normalizedKernel o p x y - q.normalizedKernel o r x y|
+noncomputable def differenceRow (p r : unitInterval) : ℝ := ∑' y, q.kernelDifference o p r o y
+
+theorem kernelDifference_nonneg (p r : unitInterval) (x y : V) :
+    0 ≤ q.kernelDifference o p r x y := abs_nonneg _
+
+theorem kernelDifference_covariant (p r : unitInterval)
+    (a : V ≃ V) (b : E ≃ E) (ha : IsAut G a b) (x y : V) :
+    q.kernelDifference o p r (a x) (a y) = q.kernelDifference o p r x y := by
+  simp only [kernelDifference,normalizedKernel,q.kernel_covariant p a b ha,q.kernel_covariant r a b ha]
+
+theorem kernelDifference_outside (p r : unitInterval) (x y : V) (hy : y ∉ reachBall G x M) :
+    q.kernelDifference o p r x y = 0 := by
+  simp only [kernelDifference,normalizedKernel,q.kernel_outside_ball_zero p x y hy,
+    q.kernel_outside_ball_zero r x y hy,mul_zero,sub_self,abs_zero]
+
+theorem kernelDifference_reverse_outside (p r : unitInterval) (x y : V) (hx : x ∉ reachBall G y M) :
+    q.kernelDifference o p r x y = 0 := by
+  simp only [kernelDifference,normalizedKernel,q.kernel_outside_reverse_ball_zero p x y hx,
+    q.kernel_outside_reverse_ball_zero r x y hx,mul_zero,sub_self,abs_zero]
+
+theorem kernelDifference_start (p r : unitInterval) (x y : V) (hx : x ∉ vertexOrbit G o) :
+    q.kernelDifference o p r x y = 0 := by
+  simp only [kernelDifference,normalizedKernel,q.kernel_zero_of_start_not_mem p x y hx,
+    q.kernel_zero_of_start_not_mem r x y hx,mul_zero,sub_self,abs_zero]
+
+theorem kernelDifference_end (p r : unitInterval) (x y : V) (hy : y ∉ vertexOrbit G o) :
+    q.kernelDifference o p r x y = 0 := by
+  simp only [kernelDifference,normalizedKernel,q.kernel_zero_of_end_not_mem p x y hy,
+    q.kernel_zero_of_end_not_mem r x y hy,mul_zero,sub_self,abs_zero]
+
+theorem summable_kernelDifference (p r : unitInterval) (x : V) :
+    Summable (q.kernelDifference o p r x) := by
+  apply summable_of_ne_finset_zero (s := (finite_reachBall hloc x M).toFinset)
+  intro y hy
+  exact q.kernelDifference_outside o p r x y (by simpa only [Set.Finite.mem_toFinset] using hy)
+
+theorem summable_kernelDifference_column (p r : unitInterval) (y : V) :
+    Summable (fun x => q.kernelDifference o p r x y) := by
+  apply summable_of_ne_finset_zero (s := (finite_reachBall hloc y M).toFinset)
+  intro x hx
+  exact q.kernelDifference_reverse_outside o p r x y (by simpa only [Set.Finite.mem_toFinset] using hx)
+
+theorem differenceRow_nonneg (p r : unitInterval) : 0 ≤ q.differenceRow o p r :=
+  tsum_nonneg (q.kernelDifference_nonneg o p r o)
+
+theorem differenceRow_finite (p r : unitInterval) :
+    q.differenceRow o p r = ∑ y ∈ (finite_reachBall hloc o M).toFinset, q.kernelDifference o p r o y := by
+  apply tsum_eq_sum
+  intro y hy
+  exact q.kernelDifference_outside o p r o y (by simpa only [Set.Finite.mem_toFinset] using hy)
+
+theorem kernelDifference_row_le (p r : unitInterval) (x : V) :
+    (∑' y, q.kernelDifference o p r x y) ≤ q.differenceRow o p r := by
+  by_cases hx : x ∈ vertexOrbit G o
+  · obtain ⟨a,b,ha,rfl⟩ := hx
+    rw [← a.tsum_eq (fun y => q.kernelDifference o p r (a o) y)]
+    exact le_of_eq (tsum_congr (q.kernelDifference_covariant o p r a b ha o))
+  · simpa only [q.kernelDifference_start o p r x _ hx,tsum_zero] using q.differenceRow_nonneg o p r
+
+ 
+
+theorem norm_sub_le_differenceRow {ι : Type*} [Fintype ι]
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (htr : RootTransport root weight Γ)
+    (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b) (hw : ∀ i, 0 ≤ weight i)
+    (i₀ : ι) (hw₀ : 0 < weight i₀) (ho : o = root i₀)
+    (p r : unitInterval) (A B : Hilbert V →L[ℂ] Hilbert V)
+    (hA : ∀ x y, A (lp.single 2 y (1:ℂ)) x = (q.normalizedKernel o p x y : ℂ))
+    (hB : ∀ x y, B (lp.single 2 y (1:ℂ)) x = (q.normalizedKernel o r x y : ℂ)) :
+    ‖A-B‖ ≤ q.differenceRow o p r := by
+  subst o
+  have hc (y : V) : (∑' x, q.kernelDifference (root i₀) p r x y) ≤ q.differenceRow (root i₀) p r := by
+    by_cases hy : y ∈ vertexOrbit G (root i₀)
+    · exact (orbit_kernel_column_eq_row G root weight Γ htr hΓ hw i₀ hw₀
+        (q.kernelDifference (root i₀) p r) (q.kernelDifference_covariant (root i₀) p r)
+        (q.kernelDifference_start (root i₀) p r) (q.kernelDifference_end (root i₀) p r)
+        (q.summable_kernelDifference (root i₀) p r) (q.summable_kernelDifference_column (root i₀) p r) y hy).le
+    · simpa only [q.kernelDifference_end (root i₀) p r _ y hy,tsum_zero] using
+        q.differenceRow_nonneg (root i₀) p r
+  apply operator_norm_le_absolute_sums
+    (fun x y => q.normalizedKernel (root i₀) p x y-q.normalizedKernel (root i₀) r x y)
+    _ (q.differenceRow_nonneg (root i₀) p r)
+  · intro x S
+    exact ((q.summable_kernelDifference (root i₀) p r x).sum_le_tsum S
+      (fun y _ => q.kernelDifference_nonneg (root i₀) p r x y)).trans (q.kernelDifference_row_le (root i₀) p r x)
+  · intro y S
+    exact ((q.summable_kernelDifference_column (root i₀) p r y).sum_le_tsum S
+      (fun x _ => q.kernelDifference_nonneg (root i₀) p r x y)).trans (hc y)
+  · intro x y
+    change A (lp.single 2 y (1:ℂ)) x - B (lp.single 2 y (1:ℂ)) x = _
+    rw [hA,hB,Complex.ofReal_sub]
+
+end LocalEndpointRule
+end EndpointDifference
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open Filter
+section EndpointOperatorContinuity
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
+
+ 
+
+theorem exists_normalized_operator_all {ι : Type*} [Fintype ι]
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (htr : RootTransport root weight Γ)
+    (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b) (hw : ∀ i, 0 ≤ weight i)
+    (i₀ : ι) (hw₀ : 0 < weight i₀)
+    (q : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) M L) (p : unitInterval) :
+    ∃ A : Hilbert V →L[ℂ] Hilbert V, ‖A‖ ≤ 1 ∧
+      ∀ x y, A (lp.single 2 y (1:ℂ)) x = (q.normalizedKernel (root i₀) p x y : ℂ) := by
+  by_cases hu : q.successProbability p (root i₀) = 0
+  · refine ⟨0,by simp,?_⟩
+    intro x y
+    simp only [normalizedKernel,hu,inv_zero,zero_mul,Complex.ofReal_zero,
+      zero_apply,lp.coeFn_zero,Pi.zero_apply]
+  · obtain ⟨A,_,hA,hK⟩ := q.exists_normalized_operator root weight Γ htr hΓ hw i₀ hw₀ p
+      (lt_of_le_of_ne (q.successProbability_nonneg p (root i₀)) (Ne.symm hu))
+    exact ⟨A,hA,hK⟩
+
+theorem continuousAt_differenceRow (o : V)
+    (q : LocalEndpointRule G hloc (vertexOrbit G o) M L) (p₀ : unitInterval)
+    (hu : q.successProbability p₀ o ≠ 0) :
+    ContinuousAt (fun p => q.differenceRow o p p₀) p₀ := by
+  simp_rw [q.differenceRow_finite o]
+  apply tendsto_finsetSum
+  intro y _
+  exact ((((q.continuous_successProbability o).continuousAt.inv₀ hu).mul
+    (q.continuous_kernel o y).continuousAt).sub continuousAt_const).abs
+
+ 
+
+theorem exists_continuous_normalized_operators {ι : Type*} [Fintype ι]
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (htr : RootTransport root weight Γ)
+    (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b) (hw : ∀ i, 0 ≤ weight i)
+    (i₀ : ι) (hw₀ : 0 < weight i₀)
+    (q : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) M L) :
+    ∃ A : unitInterval → (Hilbert V →L[ℂ] Hilbert V),
+      (∀ p, ‖A p‖ ≤ 1) ∧
+      (∀ p x y, A p (lp.single 2 y (1:ℂ)) x = (q.normalizedKernel (root i₀) p x y : ℂ)) ∧
+      ∀ p, q.successProbability p (root i₀) ≠ 0 → ContinuousAt A p := by
+  choose A hn hK using q.exists_normalized_operator_all root weight Γ htr hΓ hw i₀ hw₀
+  refine ⟨A,hn,hK,?_⟩
+  intro p hu
+  apply tendsto_iff_norm_sub_tendsto_zero.mpr
+  apply squeeze_zero (fun r => norm_nonneg (A r-A p))
+    (fun r => q.norm_sub_le_differenceRow (root i₀) root weight Γ htr hΓ hw i₀ hw₀ rfl r p (A r) (A p) (hK r) (hK p))
+  have h := (q.continuousAt_differenceRow (root i₀) p hu).tendsto
+  have hz : q.differenceRow (root i₀) p p = 0 := by
+    simp only [differenceRow,kernelDifference,sub_self,abs_zero,tsum_zero]
+  rwa [hz] at h
+
+end LocalEndpointRule
+end EndpointOperatorContinuity
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section ThreeArmGeometry
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem Along.first_entrance {G : BondGraph V E} {x y : V} {l : List E}
+    (hl : Along G x l y) (D : Set V) (hy : y ∈ D) :
+    ∃ z ∈ D, ∃ k : List E, Along G x k z ∧ k ⊆ l ∧ ∀ e ∈ k, e ∉ internalEdges G D := by
+  induction hl with
+  | nil => exact ⟨_,hy,[],.nil _,by simp,by simp⟩
+  | @cons x z y l e he ht ih =>
+    by_cases hx : x ∈ D
+    · exact ⟨x,hx,[],.nil _,by simp,by simp⟩
+    · obtain ⟨w,hw,k,hk,hsub,hout⟩ := ih hy
+      refine ⟨w,hw,e::k,.cons e he hk,?_,?_⟩
+      · intro d hd
+        rcases List.mem_cons.mp hd with rfl | hd
+        · exact List.mem_cons_self
+        · exact List.mem_cons_of_mem _ (hsub hd)
+      · intro d hd
+        rcases List.mem_cons.mp hd with rfl | hd
+        · intro h
+          exact hx ((mem_internalEdges_of_ends G D d x z he).mp h).1
+        · exact hout d hd
+
+ 
+
+theorem conn_three_disjoint_arms (G : BondGraph V E) (ω : Set E) (x y z : V)
+    (hxy : Conn G ω x y) (hxz : Conn G ω x z) :
+    ∃ w, ω ∈ DisjointOccurrence {η | Conn G η x w}
+      (DisjointOccurrence {η | Conn G η w y} {η | Conn G η w z}) := by
+  obtain ⟨p⟩ := (conn_iff_reachable G ω x y).mp hxy
+  let t := p.toPath
+  let D : Set V := {a | a ∈ t.val.support}
+  obtain ⟨l,hl,_,ho⟩ := conn_exists_bond_trail G (conn_symm G ω hxz)
+  obtain ⟨w,hw,k,hk,hsub,hout⟩ := hl.first_entrance D t.val.start_mem_support
+  obtain ⟨l₁,h₁,he₁,ho₁⟩ := walk_lift_bonds G ω (t.val.takeUntil w hw)
+  obtain ⟨l₂,h₂,he₂,ho₂⟩ := walk_lift_bonds G ω (t.val.dropUntil w hw)
+  have h₁in (e : E) (he : e ∈ l₁) : e ∈ internalEdges G D := by
+    have hh : G e ∈ (t.val.takeUntil w hw).edges := he₁ ▸ List.mem_map_of_mem he
+    have hh' := t.val.edges_takeUntil_subset_edges hw hh
+    intro a ha
+    exact SimpleGraph.Walk.mem_support_of_mem_edges hh' ha
+  have h₂in (e : E) (he : e ∈ l₂) : e ∈ internalEdges G D := by
+    have hh : G e ∈ (t.val.dropUntil w hw).edges := he₂ ▸ List.mem_map_of_mem he
+    have hh' := t.val.edges_dropUntil_subset_edges hw hh
+    intro a ha
+    exact SimpleGraph.Walk.mem_support_of_mem_edges hh' ha
+  have h12 : Disjoint l₁.toFinset l₂.toFinset := by
+    apply Finset.disjoint_left.mpr
+    intro e he₁' he₂'
+    have hp := t.property.isTrail.disjoint_edges_takeUntil_dropUntil hw
+    exact List.disjoint_left.mp hp (he₁ ▸ List.mem_map_of_mem (List.mem_toFinset.mp he₁'))
+      (he₂ ▸ List.mem_map_of_mem (List.mem_toFinset.mp he₂'))
+  have h13 : Disjoint l₁.toFinset k.toFinset := by
+    apply Finset.disjoint_left.mpr
+    intro e he hk'
+    exact hout e (List.mem_toFinset.mp hk') (h₁in e (List.mem_toFinset.mp he))
+  have h23 : Disjoint l₂.toFinset k.toFinset := by
+    apply Finset.disjoint_left.mpr
+    intro e he hk'
+    exact hout e (List.mem_toFinset.mp hk') (h₂in e (List.mem_toFinset.mp he))
+  have hkopen (e : E) (he : e ∈ k) : e ∈ ω := ho e (hsub he)
+  refine ⟨w,l₁.toFinset,l₂.toFinset ∪ k.toFinset,?_,?_,Finset.disjoint_union_right.mpr ⟨h12,h13⟩,?_,?_⟩
+  · intro e he; exact ho₁ e (List.mem_toFinset.mp he)
+  · intro e he
+    rcases Finset.mem_union.mp he with he | he
+    · exact ho₂ e (List.mem_toFinset.mp he)
+    · exact hkopen e (List.mem_toFinset.mp he)
+  · exact h₁.conn (fun e he => List.mem_toFinset.mpr he)
+  · refine ⟨l₂.toFinset,k.toFinset,?_,?_,h23,?_,?_⟩
+    · exact_mod_cast Finset.subset_union_left
+    · exact_mod_cast Finset.subset_union_right
+    · exact h₂.conn (fun e he => List.mem_toFinset.mpr he)
+    · exact conn_symm G _ (hk.conn (fun e he => List.mem_toFinset.mpr he))
+
+end ThreeArmGeometry
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+open scoped ENNReal
+section ThreeArmProbability
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem bk_ennreal [Countable E] (p : unitInterval) (A B : Set (Set E))
+    (hA : IsUpperSet A) (hB : IsUpperSet B) (hfA : FiniteWitness A) (hfB : FiniteWitness B) :
+    law p (DisjointOccurrence A B) ≤ law p A * law p B := by
+  apply (ENNReal.toReal_le_toReal (measure_ne_top _ _)
+    (ENNReal.mul_ne_top (measure_ne_top _ _) (measure_ne_top _ _))).mp
+  rw [ENNReal.toReal_mul]
+  exact bk p A B hA hB hfA hfB
+
+ 
+
+theorem three_arm_bound [Countable V] [Countable E]
+    (G : BondGraph V E) (p : unitInterval) (x y z : V) :
+    law p {ω | Conn G ω x y ∧ Conn G ω x z} ≤
+      ∑' w, law p {ω | Conn G ω x w} * law p {ω | Conn G ω w y} * law p {ω | Conn G ω w z} := by
+  let C := fun a b : V => {ω : Set E | Conn G ω a b}
+  have hup (a b : V) : IsUpperSet (C a b) := fun _ _ h hconn => conn_mono G h hconn
+  have hf (a b : V) : FiniteWitness (C a b) := finiteWitness_conn G a b
+  have hb (w : V) : law p (DisjointOccurrence (C x w) (DisjointOccurrence (C w y) (C w z))) ≤
+      law p (C x w) * law p (C w y) * law p (C w z) := by
+    calc
+      _ ≤ law p (C x w) * law p (DisjointOccurrence (C w y) (C w z)) :=
+        bk_ennreal p _ _ (hup x w) (disjointOccurrence_upper _ _) (hf x w) (finiteWitness_disjointOccurrence _ _)
+      _ ≤ law p (C x w) * (law p (C w y)*law p (C w z)) :=
+        mul_le_mul_right (bk_ennreal p _ _ (hup w y) (hup w z) (hf w y) (hf w z)) _
+      _ = _ := (mul_assoc _ _ _).symm
+  calc
+    _ ≤ law p (⋃ w, DisjointOccurrence (C x w) (DisjointOccurrence (C w y) (C w z))) := by
+      apply measure_mono
+      intro ω h
+      obtain ⟨w,hw⟩ := conn_three_disjoint_arms G ω x y z h.1 h.2
+      exact Set.mem_iUnion.mpr ⟨w,hw⟩
+    _ ≤ ∑' w, law p (DisjointOccurrence (C x w) (DisjointOccurrence (C w y) (C w z))) := measure_iUnion_le _
+    _ ≤ _ := ENNReal.tsum_le_tsum hb
+
+ 
+
+theorem three_arm_double_sum_bound [Countable V] [Countable E]
+    (G : BondGraph V E) (p : unitInterval) (C : ℝ≥0∞)
+    (hC : ∀ x, (∑' y, law p {ω | Conn G ω x y}) ≤ C) (x : V) :
+    (∑' y, ∑' z, law p {ω | Conn G ω x y ∧ Conn G ω x z}) ≤ C^3 := by
+  calc
+    _ ≤ ∑' y, ∑' z, ∑' w, law p {ω | Conn G ω x w} * law p {ω | Conn G ω w y} * law p {ω | Conn G ω w z} :=
+      ENNReal.tsum_le_tsum (fun y => ENNReal.tsum_le_tsum (fun z => three_arm_bound G p x y z))
+    _ = ∑' w, law p {ω | Conn G ω x w} * (∑' y, law p {ω | Conn G ω w y}) *
+        (∑' z, law p {ω | Conn G ω w z}) := by
+      calc
+        _ = ∑' y, ∑' w, ∑' z, law p {ω | Conn G ω x w} * law p {ω | Conn G ω w y} * law p {ω | Conn G ω w z} := by
+          apply tsum_congr
+          intro y
+          exact ENNReal.tsum_comm
+        _ = ∑' w, ∑' y, ∑' z, law p {ω | Conn G ω x w} * law p {ω | Conn G ω w y} * law p {ω | Conn G ω w z} := ENNReal.tsum_comm
+        _ = _ := by simp_rw [ENNReal.tsum_mul_left,ENNReal.tsum_mul_right,ENNReal.tsum_mul_left]
+    _ ≤ ∑' w, law p {ω | Conn G ω x w} * C * C :=
+      ENNReal.tsum_le_tsum (fun w => mul_le_mul' (mul_le_mul_right (hC w) _) (hC w))
+    _ = (∑' w, law p {ω | Conn G ω x w}) * C * C := by
+      rw [ENNReal.tsum_mul_right,ENNReal.tsum_mul_right]
+    _ ≤ C*C*C := mul_le_mul_left (mul_le_mul_left (hC x) C) C
+    _ = C^3 := by ring
+
+end ThreeArmProbability
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+open scoped ENNReal
+section ClusterMassMoments
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+noncomputable def clusterMass (G : BondGraph V E) (ω : Set E) (x : V) : ℝ≥0∞ :=
+  ∑' y, if Conn G ω x y then 1 else 0
+
+theorem measurable_clusterMass [Countable V] [Countable E]
+    (G : BondGraph V E) (x : V) : Measurable (fun ω => clusterMass G ω x) := by
+  apply Measurable.tsum
+  intro y
+  exact Measurable.ite (measurableSet_conn G x y) measurable_const measurable_const
+
+theorem clusterMass_eq_ncard (G : BondGraph V E) (ω : Set E) (x : V)
+    (hf : (cluster G ω x).Finite) : clusterMass G ω x = (cluster G ω x).ncard := by
+  change (∑' y, if y ∈ cluster G ω x then (1:ℝ≥0∞) else 0) = _
+  rw [tsum_eq_sum (s := hf.toFinset) (fun y hy => by
+    simp only [Set.Finite.mem_toFinset] at hy
+    simp only [ite_eq_right hy])]
+  have he : (∑ y ∈ hf.toFinset, if y ∈ cluster G ω x then (1:ℝ≥0∞) else 0) = ∑ _y ∈ hf.toFinset, (1:ℝ≥0∞) := by
+    apply Finset.sum_congr rfl
+    intro y hy
+    exact ite_eq_left (hf.mem_toFinset.mp hy)
+  rw [he]
+  simp only [Finset.sum_const,nsmul_eq_mul,mul_one,←Set.ncard_eq_toFinset_card _ hf]
+
+theorem lintegral_clusterMass [Countable V] [Countable E]
+    (G : BondGraph V E) (p : unitInterval) (x : V) :
+    (∫⁻ ω, clusterMass G ω x ∂law p) = ∑' y, law p {ω | Conn G ω x y} := by
+  unfold clusterMass
+  rw [lintegral_tsum (fun y => (Measurable.ite (measurableSet_conn G x y) measurable_const measurable_const).aemeasurable)]
+  apply tsum_congr
+  intro y
+  simpa only [Set.indicator,Set.mem_ofPred_eq,Pi.one_apply] using (lintegral_indicator_one (μ := law p) (measurableSet_conn G x y))
+
+theorem clusterMass_sq (G : BondGraph V E) (ω : Set E) (x : V) :
+    clusterMass G ω x ^ 2 = ∑' y, ∑' z, if Conn G ω x y ∧ Conn G ω x z then (1:ℝ≥0∞) else 0 := by
+  rw [pow_two,clusterMass]
+  rw [←ENNReal.tsum_mul_right]
+  apply tsum_congr
+  intro y
+  rw [←ENNReal.tsum_mul_left]
+  apply tsum_congr
+  intro z
+  split_ifs <;> simp_all
+
+theorem lintegral_clusterMass_sq [Countable V] [Countable E]
+    (G : BondGraph V E) (p : unitInterval) (x : V) :
+    (∫⁻ ω, clusterMass G ω x ^ 2 ∂law p) =
+      ∑' y, ∑' z, law p {ω | Conn G ω x y ∧ Conn G ω x z} := by
+  have hm (y z : V) : Measurable (fun ω : Set E => if Conn G ω x y ∧ Conn G ω x z then (1:ℝ≥0∞) else 0) :=
+    Measurable.ite ((measurableSet_conn G x y).inter (measurableSet_conn G x z)) measurable_const measurable_const
+  simp_rw [clusterMass_sq]
+  rw [lintegral_tsum (fun y => (Measurable.tsum (fun z => hm y z)).aemeasurable)]
+  apply tsum_congr
+  intro y
+  rw [lintegral_tsum (fun z => (hm y z).aemeasurable)]
+  apply tsum_congr
+  intro z
+  have hs : MeasurableSet {ω : Set E | Conn G ω x y ∧ Conn G ω x z} :=
+    (measurableSet_conn G x y).inter (measurableSet_conn G x z)
+  simpa only [Set.indicator,Set.mem_ofPred_eq,Pi.one_apply] using (lintegral_indicator_one (μ := law p) hs)
+
+ 
+theorem cluster_second_moment_bound [Countable V] [Countable E]
+    (G : BondGraph V E) (p : unitInterval) (C : ℝ≥0∞)
+    (hC : ∀ x, (∑' y, law p {ω | Conn G ω x y}) ≤ C) (x : V) :
+    (∫⁻ ω, clusterMass G ω x ^ 2 ∂law p) ≤ C^3 := by
+  rw [lintegral_clusterMass_sq]
+  exact three_arm_double_sum_bound G p C hC x
+
+end ClusterMassMoments
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+open scoped ENNReal
+section SubcriticalMoments
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem subcritical_cluster_finite_ae (G : BondGraph V E)
+    (p : unitInterval) (hp : p < pc G) (x : V) :
+    ∀ᵐ ω ∂law p, (cluster G ω x).Finite := by
+  have hz : (law p).real {ω | (cluster G ω x).Infinite} = 0 := by
+    apply le_antisymm _ measureReal_nonneg
+    by_contra! h
+    exact (not_le_of_gt hp) (pc_le_of_infinite_cluster_probability_pos G x p h)
+  apply ae_iff.mpr
+  change law p {ω | (cluster G ω x).Infinite} = 0
+  exact ((ENNReal.toReal_eq_zero_iff _).mp hz).resolve_right (measure_ne_top _ _)
+
+theorem subcritical_ennreal_row_bound [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ x, (∑' y, law p {ω | Conn G ω x y}) ≤ ENNReal.ofReal C := by
+  obtain ⟨C,hC,hrow⟩ := subcritical_row_bound G hloc hqt p hp
+  refine ⟨C,hC,fun x => ?_⟩
+  have hs := summable_of_sum_le (twoPoint_nonneg G p x) (hrow x)
+  have he : (∑' y, law p {ω | Conn G ω x y}) = ENNReal.ofReal (∑' y, twoPoint G p x y) := by
+    rw [ENNReal.ofReal_tsum_of_nonneg (twoPoint_nonneg G p x) hs]
+    apply tsum_congr
+    intro y
+    exact (ENNReal.ofReal_toReal (measure_ne_top _ _)).symm
+  rw [he]
+  exact ENNReal.ofReal_le_ofReal (hs.tsum_le_of_sum_le (hrow x))
+
+ 
+
+theorem subcritical_second_moment [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ x,
+      Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^2) (law p) ∧
+      (∫ ω, ((cluster G ω x).ncard:ℝ)^2 ∂law p) ≤ C^3 := by
+  obtain ⟨C,hC,hrow⟩ := subcritical_ennreal_row_bound G hloc hqt p hp
+  refine ⟨C,hC,fun x => ?_⟩
+  have hmeas : Measurable (fun ω : Set E => ((cluster G ω x).ncard:ℝ)^2) :=
+    (measurable_of_countable (fun n : ℕ => (n:ℝ)^2)).comp (measurable_cluster_ncard G hloc x)
+  have hn : 0 ≤ᵐ[law p] (fun ω => ((cluster G ω x).ncard:ℝ)^2) :=
+    Filter.Eventually.of_forall (fun _ => sq_nonneg _)
+  have he : (fun ω => ENNReal.ofReal (((cluster G ω x).ncard:ℝ)^2)) =ᵐ[law p]
+      (fun ω => clusterMass G ω x ^ 2) := by
+    filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
+    rw [clusterMass_eq_ncard G ω x hω,ENNReal.ofReal_pow (Nat.cast_nonneg _)]
+    simp only [ENNReal.ofReal_natCast]
+  have hbound : (∫⁻ ω, ENNReal.ofReal (((cluster G ω x).ncard:ℝ)^2) ∂law p) ≤ ENNReal.ofReal (C^3) := by
+    rw [lintegral_congr_ae he,ENNReal.ofReal_pow hC]
+    exact cluster_second_moment_bound G p _ hrow x
+  have hi : Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^2) (law p) :=
+    ⟨hmeas.aestronglyMeasurable,(hasFiniteIntegral_iff_ofReal hn).mpr
+      (hbound.trans_lt ENNReal.ofReal_lt_top)⟩
+  refine ⟨hi,?_⟩
+  rw [integral_eq_lintegral_of_nonneg_ae hn hmeas.aestronglyMeasurable]
+  exact (ENNReal.toReal_mono ENNReal.ofReal_ne_top hbound).trans_eq (ENNReal.toReal_ofReal (pow_nonneg hC _))
+
+end SubcriticalMoments
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+open scoped ENNReal
+section SusceptibilityTypes
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def susceptibility (G : BondGraph V E) (p : unitInterval) (x : V) : ℝ :=
+  ∑' y, twoPoint G p x y
+
+noncomputable def rootSusceptibility (G : BondGraph V E) {ι : Type*} [Fintype ι]
+    (root : ι → V) (weight : ι → ℝ) (p : unitInterval) : ℝ :=
+  ∑ i, weight i * susceptibility G p (root i)
+
+theorem susceptibility_nonneg (G : BondGraph V E) (p : unitInterval) (x : V) :
+    0 ≤ susceptibility G p x := tsum_nonneg (twoPoint_nonneg G p x)
+
+theorem summable_twoPoint_subcritical [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (x : V) : Summable (twoPoint G p x) := by
+  obtain ⟨_,_,hrow⟩ := subcritical_row_bound G hloc hqt p hp
+  exact summable_of_sum_le (twoPoint_nonneg G p x) (hrow x)
+
+theorem ennreal_susceptibility [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (x : V) :
+    ENNReal.ofReal (susceptibility G p x) = ∑' y, law p {ω | Conn G ω x y} := by
+  rw [susceptibility,ENNReal.ofReal_tsum_of_nonneg (twoPoint_nonneg G p x)
+    (summable_twoPoint_subcritical G hloc hqt p hp x)]
+  apply tsum_congr
+  intro y
+  exact ENNReal.ofReal_toReal (measure_ne_top _ _)
+
+theorem susceptibility_aut [Countable V] [Countable E] (G : BondGraph V E)
+    (p : unitInterval) {a : V ≃ V} {b : E ≃ E} (h : IsAut G a b) (x : V) :
+    susceptibility G p (a x) = susceptibility G p x := by
+  unfold susceptibility
+  rw [←a.tsum_eq]
+  simp only [twoPoint_aut G p h]
+
+theorem susceptibility_one_le [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (x : V) : 1 ≤ susceptibility G p x := by
+  have hd : twoPoint G p x x = 1 := by
+    have he : {ω : Set E | Conn G ω x x} = Set.univ := Set.eq_univ_of_forall (fun _ => conn_refl G _ x)
+    simp only [twoPoint,he,measure_univ,ENNReal.toReal_one]
+  rw [←hd]
+  exact (summable_twoPoint_subcritical G hloc hqt p hp x).le_tsum x (fun _ _ => twoPoint_nonneg G p _ _)
+
+theorem rootSusceptibility_nonneg (G : BondGraph V E) {ι : Type*} [Fintype ι]
+    (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 ≤ weight i) (p : unitInterval) :
+    0 ≤ rootSusceptibility G root weight p :=
+  Finset.sum_nonneg (fun i _ => mul_nonneg (hw i) (susceptibility_nonneg G p _))
+
+theorem susceptibility_le_rootSusceptibility [Countable V] [Countable E]
+    (G : BondGraph V E) {ι : Type*} [Fintype ι]
+    (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 ≤ weight i)
+    (hcov : ∀ x, ∃ i, x ∈ vertexOrbit G (root i))
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i) (p : unitInterval) (x : V) :
+    susceptibility G p x ≤ c*rootSusceptibility G root weight p := by
+  obtain ⟨i,a,b,hab,hax⟩ := hcov x
+  rw [←hax,susceptibility_aut G p hab]
+  calc
+    _ ≤ (c*weight i)*susceptibility G p (root i) := by
+      simpa only [one_mul] using mul_le_mul_of_nonneg_right (hmin i) (susceptibility_nonneg G p _)
+    _ = c*(weight i*susceptibility G p (root i)) := mul_assoc _ _ _
+    _ ≤ _ := mul_le_mul_of_nonneg_left
+      (Finset.single_le_sum (fun j _ => mul_nonneg (hw j) (susceptibility_nonneg G p _)) (Finset.mem_univ i)) hc
+
+ 
+theorem susceptibility_eq_integral [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (x : V) :
+    Integrable (fun ω => ((cluster G ω x).ncard:ℝ)) (law p) ∧
+      susceptibility G p x = ∫ ω, ((cluster G ω x).ncard:ℝ) ∂law p := by
+  have hm : Measurable (fun ω : Set E => ((cluster G ω x).ncard:ℝ)) :=
+    (measurable_of_countable (fun n : ℕ => (n:ℝ))).comp (measurable_cluster_ncard G hloc x)
+  have hn : 0 ≤ᵐ[law p] (fun ω => ((cluster G ω x).ncard:ℝ)) := Filter.Eventually.of_forall (fun _ => Nat.cast_nonneg _)
+  have he : (fun ω => ENNReal.ofReal ((cluster G ω x).ncard:ℝ)) =ᵐ[law p] (fun ω => clusterMass G ω x) := by
+    filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
+    rw [clusterMass_eq_ncard G ω x hω,ENNReal.ofReal_natCast]
+  have heq : (∫⁻ ω, ENNReal.ofReal ((cluster G ω x).ncard:ℝ) ∂law p) = ENNReal.ofReal (susceptibility G p x) := by
+    rw [lintegral_congr_ae he,lintegral_clusterMass,ennreal_susceptibility G hloc hqt p hp]
+  refine ⟨⟨hm.aestronglyMeasurable,(hasFiniteIntegral_iff_ofReal hn).mpr (by rw [heq]; exact ENNReal.ofReal_lt_top)⟩,?_⟩
+  rw [integral_eq_lintegral_of_nonneg_ae hn hm.aestronglyMeasurable,heq,ENNReal.toReal_ofReal (susceptibility_nonneg G p x)]
+
+end SusceptibilityTypes
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section FiniteKernelNorm
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem kernel_sum_le_norm_card (K : V → V → ℝ)
+    (A : Hilbert V →L[ℂ] Hilbert V)
+    (hA : ∀ x y, A (lp.single 2 y (1:ℂ)) x = (K x y : ℂ)) (F : Finset V) :
+    ∑ x ∈ F, ∑ y ∈ F, K x y ≤ ‖A‖*F.card := by
+  have he : inner ℂ (deltaSum F) (A (deltaSum F)) =
+      ((∑ x ∈ F, ∑ y ∈ F, K x y : ℝ) : ℂ) := by
+    rw [inner_deltaSum]
+    simp only [deltaSum,map_sum,lp.coeFn_sum,Finset.sum_apply,Complex.ofReal_sum]
+    apply Finset.sum_congr rfl
+    intro x _
+    apply Finset.sum_congr rfl
+    intro y _
+    exact hA x y
+  have h := (Complex.re_le_norm (inner ℂ (deltaSum F) (A (deltaSum F)))).trans
+    ((norm_inner_le_norm _ _).trans
+      (mul_le_mul_of_nonneg_left (A.le_opNorm _) (norm_nonneg _)))
+  rw [he,Complex.ofReal_re] at h
+  calc
+    _ ≤ ‖deltaSum F‖*(‖A‖*‖deltaSum F‖) := h
+    _ = ‖A‖*‖deltaSum F‖^2 := by ring
+    _ = _ := by rw [norm_deltaSum_sq]
+
+ 
+theorem kernel_average_le_norm (K : V → V → ℝ)
+    (A : Hilbert V →L[ℂ] Hilbert V)
+    (hA : ∀ x y, A (lp.single 2 y (1:ℂ)) x = (K x y : ℂ))
+    (F : Finset V) (hF : F.Nonempty) :
+    (∑ x ∈ F, ∑ y ∈ F, K x y)/(F.card:ℝ) ≤ ‖A‖ := by
+  exact (div_le_iff₀ (Nat.cast_pos.mpr hF.card_pos)).mpr (kernel_sum_le_norm_card K A hA F)
+
+end FiniteKernelNorm
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section BinClusterSlice
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def binClusterSlice (G : BondGraph V E) (o : V) (n : ℕ)
+    (ω : Set E) (x : V) : Finset V :=
+  if h : ClusterWindow G x n (2*n) ω then h.1.toFinset.filter (fun y => y ∈ vertexOrbit G o) else ∅
+
+theorem mem_binClusterSlice (G : BondGraph V E) (o : V) (n : ℕ) (ω : Set E) (x y : V) :
+    y ∈ binClusterSlice G o n ω x ↔ ClusterWindow G x n (2*n) ω ∧ Conn G ω x y ∧ y ∈ vertexOrbit G o := by
+  unfold binClusterSlice
+  split_ifs with h
+  · simp only [Finset.mem_filter,Set.Finite.mem_toFinset,h,true_and,cluster,Set.mem_ofPred_eq]
+  · simp only [Finset.notMem_empty,h,false_and]
+
+theorem binClusterSlice_eq_of_conn (G : BondGraph V E) (o : V) (n : ℕ) (ω : Set E)
+    {x y : V} (h : Conn G ω x y) : binClusterSlice G o n ω x = binClusterSlice G o n ω y := by
+  have hc := cluster_eq_of_conn G ω h
+  have hw : ClusterWindow G x n (2*n) ω ↔ ClusterWindow G y n (2*n) ω := by
+    unfold ClusterWindow
+    rw [hc]
+  ext z
+  simp only [mem_binClusterSlice,hw]
+  have hz : Conn G ω x z ↔ Conn G ω y z := by change z ∈ cluster G ω x ↔ z ∈ cluster G ω y; rw [hc]
+  rw [hz]
+
+theorem binClusterSlice_nonempty (G : BondGraph V E) (o : V) (n : ℕ) (ω : Set E) (x : V)
+    (hw : ClusterWindow G x n (2*n) ω) (hx : x ∈ vertexOrbit G o) :
+    (binClusterSlice G o n ω x).Nonempty :=
+  ⟨x,(mem_binClusterSlice G o n ω x x).mpr ⟨hw,conn_refl _ _ _,hx⟩⟩
+
+theorem binOrbitRowMass_eq_slice (G : BondGraph V E) (o : V) (n : ℕ) (ω : Set E) (x : V)
+    (hx : x ∈ vertexOrbit G o) :
+    binOrbitRowMass G n ω x = (binClusterSlice G o n ω x).card := by
+  by_cases hw : ClusterWindow G x n (2*n) ω
+  · rw [binOrbitRowMass_eq G n ω x hw,vertexOrbit_eq_of_mem G o x hx]
+    simp only [binClusterSlice,dite_eq_left hw]
+    rw [←Finset.sum_filter]
+    simp
+  · rw [binOrbitRowMass_zero G n ω x hw]
+    simp only [binClusterSlice,dite_eq_right hw,Finset.card_empty,Nat.cast_zero]
+
+theorem binSizeMass_eq_of_conn (G : BondGraph V E) (n : ℕ) (ω : Set E)
+    {x y : V} (hc : Conn G ω x y) : binSizeMass G n ω x = binSizeMass G n ω y := by
+  have he := cluster_eq_of_conn G ω hc
+  have hw : ClusterWindow G x n (2*n) ω ↔ ClusterWindow G y n (2*n) ω := by
+    unfold ClusterWindow
+    rw [he]
+  by_cases hx : ClusterWindow G x n (2*n) ω
+  · rw [binSizeMass_eq G n ω x hx,binSizeMass_eq G n ω y (hw.mp hx),he]
+  · have hy := fun h => hx (hw.mpr h)
+    simp only [binSizeMass,binMass_zero_of_not_window G n ω x _ hx,
+      binMass_zero_of_not_window G n ω y _ hy,tsum_zero]
+
+theorem mem_binClusterSlice_iff_mass (G : BondGraph V E) (o : V) (n : ℕ)
+    (ω : Set E) (x y : V) :
+    y ∈ binClusterSlice G o n ω x ↔
+      binConnectionMass G n ω x y = 1 ∧ y ∈ vertexOrbit G o := by
+  rw [mem_binClusterSlice]
+  have he : binConnectionMass G n ω x y =
+      if ClusterWindow G x n (2*n) ω ∧ Conn G ω x y then 1 else 0 := by
+    unfold binConnectionMass ClusterWindow
+    congr 1
+    apply propext; tauto
+  rw [he]
+  by_cases h : ClusterWindow G x n (2*n) ω ∧ Conn G ω x y
+  · rw [ite_eq_left h]
+    simp only [h.1,h.2,true_and]
+  · rw [ite_eq_right h]
+    simp only [zero_ne_one,false_and]
+    tauto
+
+theorem binClusterSlice_local (G : BondGraph V E) (hloc : LocallyFinite G)
+    (o : V) (n : ℕ) (ω : Set E) (x : V) :
+    let Q := incidentSupport G hloc (finite_reachBall hloc x (2*n)).toFinset
+    binClusterSlice G o n ω x = binClusterSlice G o n (ω ∩ (Q : Set E)) x := by
+  intro Q
+  ext y
+  rw [mem_binClusterSlice_iff_mass,mem_binClusterSlice_iff_mass,
+    binConnectionMass_local G hloc n ω x y]
+
+end BinClusterSlice
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section FiniteClassTransport
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+noncomputable def finiteClassTransport (S : V → Finset V) (t : V → ℝ)
+    (K : V → V → ℝ) (x y : V) : ℝ :=
+  if y ∈ S x then t x / (S x).card * ∑ z ∈ S x, K x z else 0
+
+theorem finiteClassTransport_nonneg (S : V → Finset V) (t : V → ℝ)
+    (K : V → V → ℝ) (ht : ∀ x, 0 ≤ t x) (hK : ∀ x y, 0 ≤ K x y) (x y : V) :
+    0 ≤ finiteClassTransport S t K x y := by
+  unfold finiteClassTransport
+  split_ifs
+  · exact mul_nonneg (div_nonneg (ht x) (Nat.cast_nonneg _)) (Finset.sum_nonneg (fun z _ => hK x z))
+  · rfl
+
+theorem finiteClassTransport_row (S : V → Finset V) (t : V → ℝ)
+    (K : V → V → ℝ) (x : V) :
+    (∑' y, finiteClassTransport S t K x y) = t x * ∑ z ∈ S x, K x z := by
+  rw [tsum_eq_sum (s := S x) (fun y hy => (show finiteClassTransport S t K x y = 0 from ite_eq_right hy))]
+  have he : (∑ y ∈ S x, finiteClassTransport S t K x y) =
+      (S x).card * (t x / (S x).card * ∑ z ∈ S x, K x z) := by
+    simp only [finiteClassTransport]
+    rw [Finset.sum_ite_mem]
+    simp only [Finset.inter_self,Finset.sum_const,nsmul_eq_mul]
+  rw [he]
+  by_cases h : (S x).Nonempty
+  · have hc : ((S x).card:ℝ) ≠ 0 := Nat.cast_ne_zero.mpr h.card_pos.ne'
+    field_simp
+  · have hs := Finset.not_nonempty_iff_eq_empty.mp h
+    simp only [hs,Finset.card_empty,Nat.cast_zero,Finset.sum_empty,mul_zero]
+
+theorem finiteClassTransport_column (S : V → Finset V) (t : V → ℝ)
+    (K : V → V → ℝ)
+    (hsym : ∀ x y, y ∈ S x ↔ x ∈ S y)
+    (heq : ∀ x y, y ∈ S x → S y = S x)
+    (ht : ∀ x y, y ∈ S x → t y = t x) (x : V) :
+    (∑' y, finiteClassTransport S t K y x) =
+      t x / (S x).card * ∑ y ∈ S x, ∑ z ∈ S x, K y z := by
+  rw [tsum_eq_sum (s := S x) (fun y hy => (show finiteClassTransport S t K y x = 0 from
+    ite_eq_right (fun h => hy ((hsym x y).mpr h))))]
+  rw [Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro y hy
+  rw [finiteClassTransport,ite_eq_left ((hsym x y).mp hy),heq x y hy,ht x y hy]
+
+ 
+theorem finiteClassTransport_column_le (S : V → Finset V) (t : V → ℝ)
+    (K : V → V → ℝ) (ht0 : ∀ x, 0 ≤ t x)
+    (hsym : ∀ x y, y ∈ S x ↔ x ∈ S y)
+    (heq : ∀ x y, y ∈ S x → S y = S x)
+    (ht : ∀ x y, y ∈ S x → t y = t x)
+    (A : Hilbert V →L[ℂ] Hilbert V)
+    (hA : ∀ x y, A (lp.single 2 y (1:ℂ)) x = (K x y : ℂ)) (x : V) :
+    (∑' y, finiteClassTransport S t K y x) ≤ ‖A‖*t x := by
+  rw [finiteClassTransport_column S t K hsym heq ht]
+  by_cases h : (S x).Nonempty
+  · have hh := mul_le_mul_of_nonneg_left (kernel_average_le_norm K A hA (S x) h) (ht0 x)
+    simpa only [div_mul_eq_mul_div,mul_div_assoc,mul_comm (t x) ‖A‖] using hh
+  · have hs := Finset.not_nonempty_iff_eq_empty.mp h
+    simp only [hs,Finset.card_empty,Nat.cast_zero,div_zero,Finset.sum_empty,mul_zero]
+    exact mul_nonneg (norm_nonneg _) (ht0 x)
+
+end FiniteClassTransport
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section BinDiagonalClass
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def binOrbitClass (G : BondGraph V E) (o : V) (n : ℕ)
+    (ω : Set E) (x : V) : Finset V :=
+  if x ∈ vertexOrbit G o then binClusterSlice G o n ω x else ∅
+
+theorem mem_binOrbitClass (G : BondGraph V E) (o : V) (n : ℕ) (ω : Set E) (x y : V) :
+    y ∈ binOrbitClass G o n ω x ↔ x ∈ vertexOrbit G o ∧
+      ClusterWindow G x n (2*n) ω ∧ Conn G ω x y ∧ y ∈ vertexOrbit G o := by
+  unfold binOrbitClass
+  split_ifs with h
+  · simp only [mem_binClusterSlice,h,true_and]
+  · simp only [Finset.notMem_empty,h,false_and]
+
+theorem binOrbitClass_symm (G : BondGraph V E) (o : V) (n : ℕ) (ω : Set E) (x y : V) :
+    y ∈ binOrbitClass G o n ω x ↔ x ∈ binOrbitClass G o n ω y := by
+  have h (x y : V) (hy : y ∈ binOrbitClass G o n ω x) : x ∈ binOrbitClass G o n ω y := by
+    obtain ⟨hx,hw,hc,hy⟩ := (mem_binOrbitClass G o n ω x y).mp hy
+    have hwy : ClusterWindow G y n (2*n) ω := by
+      unfold ClusterWindow at hw ⊢
+      rw [←cluster_eq_of_conn G ω hc]
+      exact hw
+    exact (mem_binOrbitClass G o n ω y x).mpr ⟨hy,hwy,conn_symm G ω hc,hx⟩
+  exact ⟨h x y,h y x⟩
+
+theorem binOrbitClass_eq (G : BondGraph V E) (o : V) (n : ℕ) (ω : Set E) (x y : V)
+    (hy : y ∈ binOrbitClass G o n ω x) : binOrbitClass G o n ω y = binOrbitClass G o n ω x := by
+  obtain ⟨hx,_,hc,hy⟩ := (mem_binOrbitClass G o n ω x y).mp hy
+  simp only [binOrbitClass,ite_eq_left hx,ite_eq_left hy]
+  exact (binClusterSlice_eq_of_conn G o n ω hc).symm
+
+theorem binOrbitClass_mass_eq (G : BondGraph V E) (o : V) (n : ℕ) (ω : Set E) (x y : V)
+    (hy : y ∈ binOrbitClass G o n ω x) : binSizeMass G n ω y = binSizeMass G n ω x :=
+  (binSizeMass_eq_of_conn G n ω ((mem_binOrbitClass G o n ω x y).mp hy).2.2.1).symm
+
+theorem binOrbitClass_local (G : BondGraph V E) (hloc : LocallyFinite G)
+    (o : V) (n : ℕ) (ω : Set E) (x : V) :
+    let Q := incidentSupport G hloc (finite_reachBall hloc x (2*n)).toFinset
+    binOrbitClass G o n ω x = binOrbitClass G o n (ω ∩ (Q : Set E)) x := by
+  intro Q
+  unfold binOrbitClass
+  rw [binClusterSlice_local G hloc o n ω x]
+
+theorem binOrbitClass_mem_covariant (G : BondGraph V E) (o : V)
+    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b)
+    (n : ℕ) (ω : Set E) (x y : V) :
+    y ∈ binOrbitClass G o n (b ⁻¹' ω) x ↔ a y ∈ binOrbitClass G o n ω (a x) := by
+  unfold binOrbitClass
+  simp only [mem_vertexOrbit_aut_iff G o hab]
+  split_ifs
+  · simp only [mem_binClusterSlice_iff_mass,binConnectionMass_covariant G hab,
+      mem_vertexOrbit_aut_iff G o hab]
+  · simp only [Finset.notMem_empty]
+
+theorem binOrbitClass_covariant (G : BondGraph V E) (o : V)
+    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b)
+    (n : ℕ) (ω : Set E) (x : V) :
+    (binOrbitClass G o n (b ⁻¹' ω) x).image a = binOrbitClass G o n ω (a x) := by
+  ext y
+  constructor
+  · rintro hy
+    obtain ⟨z,hz,rfl⟩ := Finset.mem_image.mp hy
+    exact (binOrbitClass_mem_covariant G o hab n ω x z).mp hz
+  · intro hy
+    exact Finset.mem_image.mpr ⟨a.symm y,(binOrbitClass_mem_covariant G o hab n ω x (a.symm y)).mpr
+      (by simpa only [Equiv.apply_symm_apply] using hy),a.apply_symm_apply y⟩
+
+theorem binOrbitClass_reach (G : BondGraph V E) (o : V) (n : ℕ) (ω : Set E) (x y : V)
+    (hy : y ∈ binOrbitClass G o n ω x) : y ∈ reachBall G x (2*n) := by
+  obtain ⟨_,hw,hc,_⟩ := (mem_binOrbitClass G o n ω x y).mp hy
+  exact finite_cluster_subset_reachBall G ω x hw.1 (2*n) hw.2.2.le hc
+
+end BinDiagonalClass
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section BinDiagonalTransport
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def binDiagonalMass (G : BondGraph V E) (o : V) (n : ℕ)
+    (K : V → V → ℝ) (ω : Set E) (x y : V) : ℝ :=
+  finiteClassTransport (binOrbitClass G o n ω) (fun x => binSizeMass G n ω x ^ 2) K x y
+
+theorem binDiagonalMass_nonneg (G : BondGraph V E) (o : V) (n : ℕ)
+    (K : V → V → ℝ) (hK : ∀ x y, 0 ≤ K x y) (ω : Set E) (x y : V) :
+    0 ≤ binDiagonalMass G o n K ω x y :=
+  finiteClassTransport_nonneg _ _ K (fun _ => sq_nonneg _) hK x y
+
+theorem binDiagonalMass_local (G : BondGraph V E) (hloc : LocallyFinite G)
+    (o : V) (n : ℕ) (K : V → V → ℝ) (ω : Set E) (x y : V) :
+    let Q := incidentSupport G hloc (finite_reachBall hloc x (2*n)).toFinset
+    binDiagonalMass G o n K ω x y = binDiagonalMass G o n K (ω ∩ (Q : Set E)) x y := by
+  intro Q
+  simp only [binDiagonalMass,finiteClassTransport]
+  rw [binOrbitClass_local G hloc o n ω x,binSizeMass_local G hloc n ω x]
+
+theorem binDiagonalMass_integrable (G : BondGraph V E) (hloc : LocallyFinite G)
+    (p : unitInterval) (o : V) (n : ℕ) (K : V → V → ℝ) (x y : V) :
+    Integrable (fun ω => binDiagonalMass G o n K ω x y) (law p) :=
+  integrable_finite_local _ p _ (fun ω => binDiagonalMass_local G hloc o n K ω x y)
+
+theorem binDiagonalMass_outside (G : BondGraph V E) (o : V) (n : ℕ)
+    (K : V → V → ℝ) (ω : Set E) (x y : V) (hy : y ∉ reachBall G x (2*n)) :
+    binDiagonalMass G o n K ω x y = 0 :=
+  ite_eq_right (fun h => hy (binOrbitClass_reach G o n ω x y h))
+
+theorem binDiagonalMass_outside_column (G : BondGraph V E) (o : V) (n : ℕ)
+    (K : V → V → ℝ) (ω : Set E) (x y : V) (hx : x ∉ reachBall G y (2*n)) :
+    binDiagonalMass G o n K ω x y = 0 :=
+  ite_eq_right (fun h => hx (binOrbitClass_reach G o n ω y x ((binOrbitClass_symm G o n ω x y).mp h)))
+
+theorem binDiagonalMass_row_zero (G : BondGraph V E) (o : V) (n : ℕ)
+    (K : V → V → ℝ) (ω : Set E) (x y : V) (hx : x ∉ vertexOrbit G o) :
+    binDiagonalMass G o n K ω x y = 0 :=
+  ite_eq_right (fun h => hx ((mem_binOrbitClass G o n ω x y).mp h).1)
+
+theorem binDiagonalMass_column_zero (G : BondGraph V E) (o : V) (n : ℕ)
+    (K : V → V → ℝ) (ω : Set E) (x y : V) (hy : y ∉ vertexOrbit G o) :
+    binDiagonalMass G o n K ω x y = 0 :=
+  ite_eq_right (fun h => hy ((mem_binOrbitClass G o n ω x y).mp h).2.2.2)
+
+theorem binDiagonalMass_covariant (G : BondGraph V E) (o : V) (n : ℕ)
+    (K : V → V → ℝ) {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b)
+    (hK : ∀ x y, K (a x) (a y) = K x y) (ω : Set E) (x y : V) :
+    binDiagonalMass G o n K (b ⁻¹' ω) x y = binDiagonalMass G o n K ω (a x) (a y) := by
+  have hs := binOrbitClass_covariant G o hab n ω x
+  have hc : (binOrbitClass G o n (b ⁻¹' ω) x).card = (binOrbitClass G o n ω (a x)).card := by
+    rw [←hs,Finset.card_image_of_injective _ a.injective]
+  have hsum : (∑ z ∈ binOrbitClass G o n (b ⁻¹' ω) x, K x z) =
+      ∑ z ∈ binOrbitClass G o n ω (a x), K (a x) z := by
+    rw [←hs,Finset.sum_image (fun _ _ _ _ h => a.injective h)]
+    simp only [hK]
+  simp only [binDiagonalMass,finiteClassTransport,binOrbitClass_mem_covariant G o hab,
+    binSizeMass_covariant G n ω hab,hc,hsum]
+
+theorem binDiagonalMass_row (G : BondGraph V E) (o : V) (n : ℕ)
+    (K : V → V → ℝ) (ω : Set E) (x : V) :
+    (∑' y, binDiagonalMass G o n K ω x y) =
+      binSizeMass G n ω x ^ 2 * ∑ z ∈ binOrbitClass G o n ω x, K x z :=
+  finiteClassTransport_row _ _ K x
+
+theorem binDiagonalMass_column (G : BondGraph V E) (o : V) (n : ℕ)
+    (K : V → V → ℝ) (ω : Set E) (x : V) :
+    (∑' y, binDiagonalMass G o n K ω y x) =
+      binSizeMass G n ω x ^ 2 / (binOrbitClass G o n ω x).card *
+        ∑ y ∈ binOrbitClass G o n ω x, ∑ z ∈ binOrbitClass G o n ω x, K y z := by
+  apply finiteClassTransport_column _ _ K (binOrbitClass_symm G o n ω) (binOrbitClass_eq G o n ω)
+  intro x y hy
+  rw [binOrbitClass_mass_eq G o n ω x y hy]
+
+theorem binDiagonalMass_column_le (G : BondGraph V E) (o : V) (n : ℕ)
+    (K : V → V → ℝ) (A : Hilbert V →L[ℂ] Hilbert V)
+    (hA : ∀ x y, A (lp.single 2 y (1:ℂ)) x = (K x y : ℂ)) (ω : Set E) (x : V) :
+    (∑' y, binDiagonalMass G o n K ω y x) ≤ ‖A‖ * binSizeMass G n ω x ^ 2 := by
+  apply finiteClassTransport_column_le _ _ K (fun _ => sq_nonneg _)
+    (binOrbitClass_symm G o n ω) (binOrbitClass_eq G o n ω) _ A hA
+  intro x y hy
+  rw [binOrbitClass_mass_eq G o n ω x y hy]
+
+end BinDiagonalTransport
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Clusters/LargeContacts.lean b/lean/OAI/Probability/UnimodularPercolation/Clusters/LargeContacts.lean
index 381b42d123b21b577beb61cc293879955f01be6e..fd6d4dd5e5a11464c0faf68f0e994990cba78bd4 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Clusters/LargeContacts.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Clusters/LargeContacts.lean
@@ -1,1229 +1,1229 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Exploration.TailBounds
-
-namespace OAI
-
-/-! Large Contacts for bond percolation. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section UnorderedContactConcentration
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem cluster_contact_tail (G : BondGraph V E) (Q : Finset E)
-    (d : ℕ) (hd : ∀ z, (Q.filter fun e => z ∈ G e).card ≤ d)
-    (x v : V) (p : unitInterval) (m : ℕ) (hdm : 0 < d*m) (a : ℝ) (ha : 0 < a) :
-    bernoulliPolynomial Q (fun ω => if ¬Conn G (ω:Set E) x v ∧
-      min (finiteRootComponent G ω x).card (finiteRootComponent G ω v).card ≤ m ∧
-      2*a ≤ (p:ℝ)*(contactBonds G Q (finiteRootComponent G ω x)
-        (finiteRootComponent G ω v)).card then 1 else 0) p ≤
-      8*Real.exp (-2*a^2/(d*m:ℕ)) := by
-  let f (x v : V) (ω : Finset E) : ℝ :=
-    if ¬Conn G (ω:Set E) x v ∧ (incidentBonds G Q (finiteRootComponent G ω v)).card ≤ d*m ∧
-      2*a ≤ (p:ℝ)*(contactBonds G Q (finiteRootComponent G ω x)
-        (finiteRootComponent G ω v)).card then 1 else 0
-  calc
-    _ ≤ bernoulliPolynomial Q (fun ω => f x v ω+f v x ω) p := by
-      apply bernoulliPolynomial_mono _ _ _ p p.2
-      intro ω _
-      by_cases h : ¬Conn G (ω:Set E) x v ∧
-          min (finiteRootComponent G ω x).card (finiteRootComponent G ω v).card ≤ m ∧
-          2*a ≤ (p:ℝ)*(contactBonds G Q (finiteRootComponent G ω x)
-            (finiteRootComponent G ω v)).card
-      · simp only [ite_eq_left h]
-        rcases le_total (finiteRootComponent G ω v).card (finiteRootComponent G ω x).card with hc | hc
-        · have hsmall : (finiteRootComponent G ω v).card ≤ m := by simpa only [min_eq_right hc] using h.2.1
-          have hh : ¬Conn G (ω:Set E) x v ∧ (incidentBonds G Q (finiteRootComponent G ω v)).card ≤ d*m ∧
-              2*a ≤ (p:ℝ)*(contactBonds G Q (finiteRootComponent G ω x) (finiteRootComponent G ω v)).card :=
-            ⟨h.1,(incidentBonds_card_le G Q _ d hd).trans (Nat.mul_le_mul_left d hsmall),h.2.2⟩
-          simp only [f,ite_eq_left hh]; split_ifs <;> norm_num
-        · have hsmall : (finiteRootComponent G ω x).card ≤ m := by simpa only [min_eq_left hc] using h.2.1
-          have hh : ¬Conn G (ω:Set E) v x ∧ (incidentBonds G Q (finiteRootComponent G ω x)).card ≤ d*m ∧
-              2*a ≤ (p:ℝ)*(contactBonds G Q (finiteRootComponent G ω v) (finiteRootComponent G ω x)).card := by
-            refine ⟨fun hc' => h.1 (conn_symm G _ hc'),
-              (incidentBonds_card_le G Q _ d hd).trans (Nat.mul_le_mul_left d hsmall),?_⟩
-            rw [contactBonds_comm G Q (finiteRootComponent G ω v)]
-            exact h.2.2
-          simp only [f,ite_eq_left hh]; split_ifs <;> norm_num
-      · simp only [ite_eq_right h,f]; split_ifs <;> norm_num
-    _ = bernoulliPolynomial Q (f x v) p+bernoulliPolynomial Q (f v x) p :=
-      bernoulliPolynomial_add ..
-    _ ≤ 4*Real.exp (-2*a^2/(d*m:ℕ))+4*Real.exp (-2*a^2/(d*m:ℕ)) :=
-      add_le_add (ordered_cluster_contact_tail G Q x v p (d*m) hdm a ha)
-        (ordered_cluster_contact_tail G Q v x p (d*m) hdm a ha)
-    _ = _ := by ring
-
-end UnorderedContactConcentration
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section TwoLargeFiniteClusters
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem cluster_exterior_weighted_bound (G : BondGraph V E) (Q : Finset E)
-    (x : V) (p : unitInterval) (F : Finset V → ℝ) (hF : ∀ W, 0 ≤ F W)
-    (f : Finset V → Finset E → ℝ) (K : ℝ)
-    (hf : ∀ W : Finset V, bernoulliPolynomial (exteriorBonds G Q (W:Set V)) (f W) p ≤ K) :
-    bernoulliPolynomial Q (fun ω => F (finiteRootComponent G ω x)*f (finiteRootComponent G ω x)
-      (exteriorBonds G ω (finiteRootComponent G ω x:Set V))) p ≤
-      K*bernoulliPolynomial Q (fun ω => F (finiteRootComponent G ω x)) p := by
-  have hh := cluster_exterior_expectation_bound G Q x p
-    (fun W η => F W*f W η-K*F W) 0 (fun W => by
-      rw [bernoulliPolynomial_sub,bernoulliPolynomial_const_mul,bernoulliPolynomial_const]
-      exact sub_nonpos.mpr (by simpa only [mul_comm K] using mul_le_mul_of_nonneg_left (hf W) (hF W)))
-  rw [bernoulliPolynomial_sub,bernoulliPolynomial_const_mul] at hh
-  exact sub_nonpos.mp hh
-
-theorem finiteClusterTail_domain_mono (G : BondGraph V E) {Q R : Finset E} (hRQ : R ⊆ Q)
-    (x : V) (n : ℕ) (p : unitInterval) : finiteClusterTail G R x n p ≤ finiteClusterTail G Q x n p := by
-  unfold finiteClusterTail
-  rw [←bernoulliPolynomial_restrict R Q hRQ]
-  apply bernoulliPolynomial_mono _ _ _ p p.2
-  intro ω _
-  exact finiteClusterTailIndicator_mono G x n Finset.inter_subset_left
-
- 
-
-theorem two_large_clusters_tail (G : BondGraph V E) (Q : Finset E)
-    (x v : V) (n m : ℕ) (p : unitInterval) :
-    bernoulliPolynomial Q (fun ω => if ¬Conn G (ω:Set E) x v ∧
-      n ≤ (finiteRootComponent G ω x).card ∧ m ≤ (finiteRootComponent G ω v).card
-        then 1 else 0) p ≤ finiteClusterTail G Q v m p*finiteClusterTail G Q x n p := by
-  let F (W : Finset V) : ℝ := if n ≤ W.card then 1 else 0
-  let f (W : Finset V) (η : Finset E) : ℝ := if v ∈ W then 0 else finiteClusterTailIndicator G v m η
-  have hF : ∀ W, 0 ≤ F W := fun W => by dsimp [F]; split_ifs <;> norm_num
-  have hf (W : Finset V) : bernoulliPolynomial (exteriorBonds G Q (W:Set V)) (f W) p ≤
-      finiteClusterTail G Q v m p := by
-    by_cases hv : v ∈ W
-    · simp only [f,ite_eq_left hv,bernoulliPolynomial_zero]
-      exact finiteClusterTail_nonneg G Q v m p
-    · simp only [f,ite_eq_right hv]
-      exact finiteClusterTail_domain_mono G (Finset.filter_subset _ _) v m p
-  have hp := cluster_exterior_weighted_bound G Q x p F hF f (finiteClusterTail G Q v m p) hf
-  have heq : bernoulliPolynomial Q (fun ω => F (finiteRootComponent G ω x)*f (finiteRootComponent G ω x)
-      (exteriorBonds G ω (finiteRootComponent G ω x:Set V))) p =
-      bernoulliPolynomial Q (fun ω => if ¬Conn G (ω:Set E) x v ∧
-      n ≤ (finiteRootComponent G ω x).card ∧ m ≤ (finiteRootComponent G ω v).card then 1 else 0) p := by
-    apply bernoulliPolynomial_congr
-    intro ω _
-    have hv : v ∈ finiteRootComponent G ω x ↔ Conn G (ω:Set E) x v := by
-      change v ∈ (finiteRootComponent G ω x:Set V) ↔ _
-      rw [coe_finiteRootComponent]
-      rfl
-    by_cases hc : Conn G (ω:Set E) x v
-    · simp only [F,f,ite_eq_left (hv.mpr hc),mul_zero,not_true_eq_false,hc,false_and,ite_false]
-    · simp only [F,f,ite_eq_right (fun h => hc (hv.mp h)),finiteClusterTailIndicator,
-        finiteRootComponent_exterior G ω x v hc]
-      split_ifs <;> norm_num <;> grind
-  rw [heq] at hp
-  exact hp
-
-end TwoLargeFiniteClusters
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section EndpointClusterTails
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem finiteClusterTail_of_endpoint_moment [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (q : unitInterval) (α C : ℝ)
-    (hα : 0 ≤ α) (hα1 : α ≤ 1)
-    (hfin : ∀ y, ∀ᵐ ω ∂law q, (cluster G ω y).Finite)
-    (hI : ∀ y, Integrable (fun ω => ((cluster G ω y).ncard:ℝ)^α) (law q))
-    (hC : ∀ y, (∫ ω, ((cluster G ω y).ncard:ℝ)^α ∂law q) ≤ C)
-    (Q : Finset E) (x : V) (p : unitInterval) (hpq : p ≤ q) (n : ℕ) (hn : 1 ≤ n) :
-    finiteClusterTail G Q x n p ≤ 2*C*(n:ℝ)^(-α) := by
-  have hn0 : (0:ℝ) < n := by exact_mod_cast (by omega : 0 < n)
-  let h : unitInterval := ⟨1/(n:ℝ),⟨by positivity,by apply (div_le_one hn0).mpr; exact_mod_cast hn⟩⟩
-  have hh : 0 < (h:ℝ) := by dsimp [h]; positivity
-  have hhpow : (h:ℝ)^α = (n:ℝ)^(-α) := by
-    dsimp [h]
-    rw [one_div,Real.inv_rpow hn0.le,Real.rpow_neg hn0.le]
-  have hhn : 1 ≤ (h:ℝ)*n := by dsimp [h]; rw [one_div,inv_mul_cancel₀ hn0.ne']
-  have hm := cluster_power_mono_integral G hloc x p q hpq α hα (hfin x) (hI x)
-  have hb := markedConnection_le_cluster_power G Q x h p α hα hα1
-    (cluster_finite_ae_mono G x p q hpq (hfin x)) hm.1
-  have hM : markedConnectionPolynomial G Q x h p ≤ C*(n:ℝ)^(-α) := by
-    apply hb.trans
-    rw [hhpow]
-    simpa only [mul_comm C] using
-      mul_le_mul_of_nonneg_left (hm.2.trans (hC x)) (Real.rpow_nonneg (Nat.cast_nonneg n) (-α))
-  have hb' := (finiteClusterTail_le_markedConnection G Q x n p h hhn).trans
-    (mul_le_mul_of_nonneg_left hM (by norm_num : (0:ℝ) ≤ 2))
-  simpa only [mul_assoc] using hb'
-
-theorem two_large_clusters_of_endpoint_moment [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (q : unitInterval) (α C : ℝ)
-    (hα : 0 ≤ α) (hα1 : α ≤ 1) (hC0 : 0 ≤ C)
-    (hfin : ∀ y, ∀ᵐ ω ∂law q, (cluster G ω y).Finite)
-    (hI : ∀ y, Integrable (fun ω => ((cluster G ω y).ncard:ℝ)^α) (law q))
-    (hC : ∀ y, (∫ ω, ((cluster G ω y).ncard:ℝ)^α ∂law q) ≤ C)
-    (Q : Finset E) (x v : V) (p : unitInterval) (hpq : p ≤ q) (n : ℕ) (hn : 1 ≤ n) :
-    bernoulliPolynomial Q (fun ω => if ¬Conn G (ω:Set E) x v ∧
-      n ≤ (finiteRootComponent G ω x).card ∧ n ≤ (finiteRootComponent G ω v).card
-        then 1 else 0) p ≤ 4*C^2*(n:ℝ)^(-2*α) := by
-  have ht (y : V) := finiteClusterTail_of_endpoint_moment G hloc q α C hα hα1 hfin hI hC Q y p hpq n hn
-  have hc : 0 ≤ 2*C*(n:ℝ)^(-α) := by positivity
-  calc
-    _ ≤ finiteClusterTail G Q v n p*finiteClusterTail G Q x n p := two_large_clusters_tail ..
-    _ ≤ (2*C*(n:ℝ)^(-α))*(2*C*(n:ℝ)^(-α)) :=
-      mul_le_mul (ht v) (ht x) (finiteClusterTail_nonneg G Q x n p) hc
-    _ = 4*C^2*(n:ℝ)^(-2*α) := by
-      have hn0 : (0:ℝ) < n := by exact_mod_cast (by omega : 0 < n)
-      rw [show (2*C*(n:ℝ)^(-α))*(2*C*(n:ℝ)^(-α)) =
-        4*C^2*((n:ℝ)^(-α)*(n:ℝ)^(-α)) by ring,←Real.rpow_add hn0]
-      congr 2
-      ring
-
-end EndpointClusterTails
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section ContactBinFirstMoment
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem contactBonds_card_le (G : BondGraph V E) (Q : Finset E)
-    (d : ℕ) (hd : ∀ z, (Q.filter fun e => z ∈ G e).card ≤ d) (A B : Finset V) :
-    (contactBonds G Q A B).card ≤ d*min A.card B.card := by
-  have hh (A B : Finset V) : (contactBonds G Q A B).card ≤ d*A.card := by
-    apply (Finset.card_le_card (t := incidentBonds G Q A) _).trans (incidentBonds_card_le G Q A d hd)
-    intro e he
-    obtain ⟨he,a,ha,b,_,hab⟩ := Finset.mem_filter.mp he
-    exact (mem_incidentBonds G Q A e).mpr ⟨he,a,ha,by rw [hab]; simp⟩
-  rcases le_total A.card B.card with h | h
-  · rw [min_eq_left h]; exact hh A B
-  · rw [min_eq_right h,contactBonds_comm G Q A B]; exact hh B A
-
-noncomputable def clusterContactBin (G : BondGraph V E) (Q : Finset E)
-    (x v : V) (n m : ℕ) (ω : Finset E) : ℝ :=
-  if ¬Conn G (ω:Set E) x v ∧
-    n ≤ min (finiteRootComponent G ω x).card (finiteRootComponent G ω v).card ∧
-    min (finiteRootComponent G ω x).card (finiteRootComponent G ω v).card ≤ m
-  then (contactBonds G Q (finiteRootComponent G ω x) (finiteRootComponent G ω v)).card else 0
-
- 
-
-theorem clusterContactBin_bound (G : BondGraph V E) (Q : Finset E)
-    (d : ℕ) (hd : ∀ z, (Q.filter fun e => z ∈ G e).card ≤ d)
-    (x v : V) (p : unitInterval) (hp : 0 < (p:ℝ)) (n m : ℕ) (hdm : 0 < d*m)
-    (a : ℝ) (ha : 0 < a) :
-    bernoulliPolynomial Q (clusterContactBin G Q x v n m) p ≤
-      (2*a/(p:ℝ))*(finiteClusterTail G Q v n p*finiteClusterTail G Q x n p)+
-      (d*m:ℕ)*(8*Real.exp (-2*a^2/(d*m:ℕ))) := by
-  let D (ω : Finset E) : Prop := ¬Conn G (ω:Set E) x v ∧
-    n ≤ (finiteRootComponent G ω x).card ∧ n ≤ (finiteRootComponent G ω v).card
-  let T (ω : Finset E) : Prop := ¬Conn G (ω:Set E) x v ∧
-    min (finiteRootComponent G ω x).card (finiteRootComponent G ω v).card ≤ m ∧
-    2*a ≤ (p:ℝ)*(contactBonds G Q (finiteRootComponent G ω x) (finiteRootComponent G ω v)).card
-  calc
-    _ ≤ bernoulliPolynomial Q (fun ω => (2*a/(p:ℝ))*(if D ω then 1 else 0)+
-        (d*m:ℕ)*(if T ω then 1 else 0)) p := by
-      apply bernoulliPolynomial_mono _ _ _ p p.2
-      intro ω _
-      unfold clusterContactBin
-      by_cases h : ¬Conn G (ω:Set E) x v ∧
-          n ≤ min (finiteRootComponent G ω x).card (finiteRootComponent G ω v).card ∧
-          min (finiteRootComponent G ω x).card (finiteRootComponent G ω v).card ≤ m
-      · have hD : D ω := ⟨h.1,(le_min_iff.mp h.2.1).1,(le_min_iff.mp h.2.1).2⟩
-        simp only [ite_eq_left h,ite_eq_left hD,mul_one]
-        by_cases ht : T ω
-        · rw [ite_eq_left ht,mul_one]
-          have hk : (contactBonds G Q (finiteRootComponent G ω x) (finiteRootComponent G ω v)).card ≤ d*m :=
-            (contactBonds_card_le G Q d hd _ _).trans (Nat.mul_le_mul_left d h.2.2)
-          have hk' : ((contactBonds G Q (finiteRootComponent G ω x) (finiteRootComponent G ω v)).card:ℝ) ≤ (d*m:ℕ) := by exact_mod_cast hk
-          have hapos : 0 ≤ 2*a/(p:ℝ) := by positivity
-          linarith
-        · rw [ite_eq_right ht,mul_zero,add_zero]
-          apply (le_div_iff₀ hp).mpr
-          have hn : ¬ 2*a ≤ (p:ℝ)*(contactBonds G Q (finiteRootComponent G ω x) (finiteRootComponent G ω v)).card :=
-            fun hh => ht ⟨h.1,h.2.2,hh⟩
-          simpa only [mul_comm] using (not_le.mp hn).le
-      · simp only [ite_eq_right h]
-        split_ifs <;> positivity
-    _ = (2*a/(p:ℝ))*bernoulliPolynomial Q (fun ω => if D ω then 1 else 0) p+
-        (d*m:ℕ)*bernoulliPolynomial Q (fun ω => if T ω then 1 else 0) p := by
-      rw [bernoulliPolynomial_add,bernoulliPolynomial_const_mul,bernoulliPolynomial_const_mul]
-    _ ≤ _ := add_le_add
-      (mul_le_mul_of_nonneg_left (two_large_clusters_tail G Q x v n n p) (by positivity))
-      (mul_le_mul_of_nonneg_left (cluster_contact_tail G Q d hd x v p m hdm a ha) (Nat.cast_nonneg _))
-
-end ContactBinFirstMoment
-end Percolation
-
- 
-namespace Percolation
-section StretchedContactDecay
-
-theorem exists_stretched_exp_volume_bound (d ε r : ℝ) (hd : 0 < d) (hε : 0 < ε) (hr : r ≤ 1) :
-    ∃ C : ℝ, 0 < C ∧ ∀ z : ℝ, 0 < z → z*Real.exp (-(z^ε)/d) ≤ C*z^r := by
-  let s := (1-r)/ε
-  have hs : 0 ≤ s := by dsimp [s]; positivity
-  obtain ⟨K,hK,hKb⟩ := exists_exp_neg_le_rpow s hs
-  refine ⟨K*d^s,by positivity,?_⟩
-  intro z hz
-  have he := mul_le_mul_of_nonneg_left (hKb (z^ε/d) (by positivity)) hz.le
-  have hex : ε*(-s) = r-1 := by dsimp [s]; field_simp; ring
-  have hid : (z^ε/d)^(-s) = z^(r-1)*d^s := by
-    rw [Real.div_rpow (Real.rpow_nonneg hz.le ε) hd.le,←Real.rpow_mul hz.le,hex,
-      Real.rpow_neg hd.le,div_inv_eq_mul]
-  have hzpow : z*z^(r-1) = z^r := by
-    conv_lhs => lhs; rw [←Real.rpow_one z]
-    rw [←Real.rpow_add hz]
-    congr 1
-    ring
-  calc
-    _ ≤ z*(K*(z^ε/d)^(-s)) := by simpa only [neg_div] using he
-    _ = (K*d^s)*(z*z^(r-1)) := by rw [hid]; ring
-    _ = _ := by rw [hzpow]
-
-theorem contact_tail_exponent (d z ε : ℝ) (hd : 0 < d) (hz : 0 < z) :
-    -2*(z^((1:ℝ)/2+ε))^2/(d*(2*z)) = -(z^(2*ε))/d := by
-  have hpow : (z^((1:ℝ)/2+ε))^2 = z*z^(2*ε) := by
-    rw [←Real.rpow_natCast,←Real.rpow_mul hz.le]
-    norm_num only [Nat.cast_ofNat]
-    rw [show ((1:ℝ)/2+ε)*2 = 1+2*ε by ring,Real.rpow_add hz,Real.rpow_one]
-  rw [hpow]
-  field_simp
-
-end StretchedContactDecay
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section UniformContactBins
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem clusterContactBin_power_bound [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (q : unitInterval)
-    (α ε C p₀ : ℝ) (hα : 0 ≤ α) (hα1 : α ≤ 1) (hε : 0 < ε)
-    (hr : (1:ℝ)/2+ε-2*α < 0) (hC0 : 0 < C) (hp₀ : 0 < p₀)
-    (hfin : ∀ y, ∀ᵐ ω ∂law q, (cluster G ω y).Finite)
-    (hI : ∀ y, Integrable (fun ω => ((cluster G ω y).ncard:ℝ)^α) (law q))
-    (hC : ∀ y, (∫ ω, ((cluster G ω y).ncard:ℝ)^α ∂law q) ≤ C)
-    (d : ℕ) (hd : 0 < d) :
-    ∃ A : ℝ, 0 < A ∧ ∀ (Q : Finset E), (∀ z, (Q.filter fun e => z ∈ G e).card ≤ d) →
-      ∀ (x v : V) (p : unitInterval), p₀ ≤ (p:ℝ) → p ≤ q → ∀ n : ℕ, 1 ≤ n →
-      bernoulliPolynomial Q (clusterContactBin G Q x v n (2*n)) p ≤
-        A*(n:ℝ)^((1:ℝ)/2+ε-2*α) := by
-  let r : ℝ := (1:ℝ)/2+ε-2*α
-  have hd' : (0:ℝ) < d := Nat.cast_pos.mpr hd
-  obtain ⟨K,hK,hKb⟩ := exists_stretched_exp_volume_bound (d:ℝ) (2*ε) r hd' (by positivity) (by dsimp [r]; linarith)
-  refine ⟨8*C^2/p₀+16*(d:ℝ)*K,by positivity,?_⟩
-  intro Q hdeg x v p hpp hpq n hn
-  have hn' : 0 < n := by omega
-  have hnr : (0:ℝ) < n := Nat.cast_pos.mpr hn'
-  have hp : 0 < (p:ℝ) := hp₀.trans_le hpp
-  let a : ℝ := (n:ℝ)^((1:ℝ)/2+ε)
-  have ha : 0 < a := Real.rpow_pos_of_pos hnr _
-  have hb := clusterContactBin_bound G Q d hdeg x v p hp n (2*n) (by positivity) a ha
-  have ht (y : V) := finiteClusterTail_of_endpoint_moment G hloc q α C hα hα1 hfin hI hC Q y p hpq n hn
-  have hprod : finiteClusterTail G Q v n p*finiteClusterTail G Q x n p ≤ 4*C^2*(n:ℝ)^(-2*α) := by
-    calc
-      _ ≤ (2*C*(n:ℝ)^(-α))*(2*C*(n:ℝ)^(-α)) :=
-        mul_le_mul (ht v) (ht x) (finiteClusterTail_nonneg G Q x n p) (by positivity)
-      _ = _ := by
-        rw [show (2*C*(n:ℝ)^(-α))*(2*C*(n:ℝ)^(-α)) = 4*C^2*((n:ℝ)^(-α)*(n:ℝ)^(-α)) by ring,
-          ←Real.rpow_add hnr]
-        congr 2
-        ring
-  have hfirst : (2*a/(p:ℝ))*(finiteClusterTail G Q v n p*finiteClusterTail G Q x n p) ≤
-      (8*C^2/p₀)*(n:ℝ)^r := by
-    calc
-      _ ≤ (2*a/p₀)*(4*C^2*(n:ℝ)^(-2*α)) :=
-        mul_le_mul (div_le_div_of_nonneg_left (by positivity) hp₀ hpp) hprod
-          (mul_nonneg (finiteClusterTail_nonneg G Q v n p) (finiteClusterTail_nonneg G Q x n p)) (by positivity)
-      _ = (8*C^2/p₀)*((n:ℝ)^((1:ℝ)/2+ε)*(n:ℝ)^(-2*α)) := by dsimp [a]; ring
-      _ = _ := by rw [←Real.rpow_add hnr]; congr 2; dsimp [r]; ring
-  have hex : -2*a^2/(d*(2*n):ℕ) = -((n:ℝ)^(2*ε))/(d:ℝ) := by
-    simp only [Nat.cast_mul,Nat.cast_ofNat]
-    exact contact_tail_exponent d n ε hd' hnr
-  have hsecond : (d*(2*n):ℕ)*(8*Real.exp (-2*a^2/(d*(2*n):ℕ))) ≤
-      (16*(d:ℝ)*K)*(n:ℝ)^r := by
-    rw [hex]
-    calc
-      _ = (16*(d:ℝ))*((n:ℝ)*Real.exp (-((n:ℝ)^(2*ε))/(d:ℝ))) := by
-        push_cast
-        ring
-      _ ≤ (16*(d:ℝ))*(K*(n:ℝ)^r) := mul_le_mul_of_nonneg_left (hKb n hnr) (by positivity)
-      _ = _ := by ring
-  calc
-    _ ≤ _ := hb
-    _ ≤ (8*C^2/p₀)*(n:ℝ)^r+(16*(d:ℝ)*K)*(n:ℝ)^r := add_le_add hfirst hsecond
-    _ = _ := by dsimp [r]; ring
-
-end UniformContactBins
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section InfiniteContacts
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-def closedContactSet (G : BondGraph V E) (A B : Set V) : Set E :=
-  {e | ∃ a ∈ A, ∃ b ∈ B, G e = s(a,b)}
-
-noncomputable def clusterContactCount (G : BondGraph V E) (ω : Set E) (x v : V) : ℕ :=
-  (closedContactSet G (cluster G ω x) (cluster G ω v)).ncard
-
-noncomputable def fullContactBin (G : BondGraph V E) (x v : V) (n m : ℕ) (ω : Set E) : ℝ :=
-  if ¬Conn G ω x v ∧ n ≤ min (cluster G ω x).ncard (cluster G ω v).ncard ∧
-    min (cluster G ω x).ncard (cluster G ω v).ncard ≤ m
-  then clusterContactCount G ω x v else 0
-
-theorem contactSet_subset_incident (G : BondGraph V E) (hloc : LocallyFinite G)
-    (A : Finset V) (B : Set V) : closedContactSet G (A:Set V) B ⊆ (incidentSupport G hloc A:Set E) := by
-  rintro e ⟨a,ha,b,_,hab⟩
-  exact (mem_incidentSupport G hloc A e).mpr ⟨a,ha,by rw [hab]; simp⟩
-
-theorem coe_contactBonds_eq (G : BondGraph V E) (hloc : LocallyFinite G)
-    (Q : Finset E) (A B : Finset V) (hQ : incidentSupport G hloc A ⊆ Q) :
-    (contactBonds G Q A B : Set E) = closedContactSet G A B := by
-  ext e
-  constructor
-  · exact fun h => (Finset.mem_filter.mp h).2
-  · intro h
-    exact Finset.mem_filter.mpr ⟨hQ (contactSet_subset_incident G hloc A B h),h⟩
-
-theorem contactSet_ncard_le (G : BondGraph V E) (hloc : LocallyFinite G)
-    (d : ℕ) (hd : ∀ z, {e | z ∈ G e}.ncard ≤ d)
-    (A B : Set V) (hA : A.Finite) (hB : B.Finite) :
-    (closedContactSet G A B).ncard ≤ d*min A.ncard B.ncard := by
-  let Q := incidentSupport G hloc (hA.toFinset ∪ hB.toFinset)
-  have hdeg (z : V) : (Q.filter fun e => z ∈ G e).card ≤ d := by
-    calc
-      _ = ((Q.filter fun e => z ∈ G e):Set E).ncard := (Set.ncard_coe_finset _).symm
-      _ ≤ {e | z ∈ G e}.ncard := Set.ncard_le_ncard (fun e he => (Finset.mem_filter.mp he).2) (hloc z)
-      _ ≤ d := hd z
-  have hQ : incidentSupport G hloc hA.toFinset ⊆ Q :=
-    incidentSupport_mono G hloc Finset.subset_union_left
-  have he := coe_contactBonds_eq G hloc Q hA.toFinset hB.toFinset hQ
-  simp only [Set.Finite.coe_toFinset] at he
-  rw [←he,Set.ncard_coe_finset]
-  simpa only [←Set.ncard_eq_toFinset_card] using contactBonds_card_le G Q d hdeg hA.toFinset hB.toFinset
-
-theorem fullContactBin_nonneg (G : BondGraph V E) (x v : V) (n m : ℕ) (ω : Set E) :
-    0 ≤ fullContactBin G x v n m ω := by unfold fullContactBin; split_ifs <;> positivity
-
-theorem fullContactBin_le (G : BondGraph V E) (hloc : LocallyFinite G)
-    (d : ℕ) (hd : ∀ z, {e | z ∈ G e}.ncard ≤ d) (x v : V) (n m : ℕ) (hn : 0 < n) (ω : Set E) :
-    fullContactBin G x v n m ω ≤ (d*m:ℕ) := by
-  unfold fullContactBin
-  split_ifs with h
-  · have hx : (cluster G ω x).Finite := Set.finite_of_ncard_ne_zero (by omega)
-    have hv : (cluster G ω v).Finite := Set.finite_of_ncard_ne_zero (by omega)
-    exact_mod_cast (contactSet_ncard_le G hloc d hd _ _ hx hv).trans (Nat.mul_le_mul_left d h.2.2)
-  · positivity
-
-end InfiniteContacts
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section ContactMeasurability
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem measurable_clusterContactCount [Countable V] [Countable E]
-    (G : BondGraph V E) (x v : V) : Measurable (fun ω : Set E => clusterContactCount G ω x v) := by
-  apply measurable_ncard.comp
-  apply measurable_setOfPred.comp
-  apply Measurable.of_eval
-  intro e
-  apply measurableSet_setOfPred.mp
-  change MeasurableSet {ω : Set E | ∃ a, Conn G ω x a ∧ ∃ b, Conn G ω v b ∧ G e = s(a,b)}
-  simp only [Set.ofPred_exists]
-  apply MeasurableSet.iUnion
-  intro a
-  have he : {ω : Set E | Conn G ω x a ∧ ∃ b, Conn G ω v b ∧ G e = s(a,b)} =
-      {ω | Conn G ω x a} ∩ ⋃ b, {ω | Conn G ω v b ∧ G e = s(a,b)} := by ext ω; simp
-  rw [he]
-  apply (measurableSet_conn G x a).inter
-  apply MeasurableSet.iUnion
-  intro b
-  by_cases h : G e = s(a,b)
-  · simpa only [h,and_true] using measurableSet_conn G v b
-  · simp only [h,and_false,Set.ofPred_false]; exact MeasurableSet.empty
-
-theorem measurable_fullContactBin [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (x v : V) (n m : ℕ) :
-    Measurable (fullContactBin G x v n m) := by
-  have hs : Measurable (fun ω : Set E => min (cluster G ω x).ncard (cluster G ω v).ncard) :=
-    (measurable_cluster_ncard G hloc x).min (measurable_cluster_ncard G hloc v)
-  exact ((measurable_of_countable (fun k : ℕ => (k:ℝ))).comp (measurable_clusterContactCount G x v)).ite
-    (((measurableSet_conn G x v).compl).inter ((measurableSet_le measurable_const hs).inter
-      (measurableSet_le hs measurable_const))) measurable_const
-
-theorem coe_domainBits (Q : Finset E) (ω : Set E) :
-    (domainBits Q ω:Set E) = ω ∩ (Q:Set E) := by ext e; simp only [mem_domainBits,Set.mem_inter_iff,Finset.mem_coe,and_comm]
-
-theorem domainBits_local (Q : Finset E) (ω : Set E) :
-    domainBits Q (ω ∩ (Q:Set E)) = domainBits Q ω := by
-  ext e
-  simp only [mem_domainBits,Set.mem_inter_iff,Finset.mem_coe]
-  tauto
-
-theorem domainBits_of_subset (Q ω : Finset E) (hω : ω ⊆ Q) : domainBits Q (ω:Set E) = ω := by
-  ext e
-  simp only [mem_domainBits,Finset.mem_coe]
-  exact ⟨fun h => h.2,fun h => ⟨hω h,h⟩⟩
-
-theorem integral_domainBits (Q : Finset E) (p : unitInterval) (f : Finset E → ℝ) :
-    (∫ ω, f (domainBits Q ω) ∂law p) = bernoulliPolynomial Q f p := by
-  rw [integral_eq_bernoulliPolynomial Q p _ (fun ω => by rw [domainBits_local])]
-  apply bernoulliPolynomial_congr
-  intro ω hω
-  rw [domainBits_of_subset Q ω hω]
-
-end ContactMeasurability
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section InfiniteContactLimit
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem fullContactBin_eq_finite (G : BondGraph V E) (hloc : LocallyFinite G)
-    (ω : Set E) (x v : V) (hx : (cluster G ω x).Finite) (hv : (cluster G ω v).Finite)
-    (Q : Finset E) (hQx : incidentSupport G hloc hx.toFinset ⊆ Q)
-    (hQv : incidentSupport G hloc hv.toFinset ⊆ Q) (n m : ℕ) :
-    clusterContactBin G Q x v n m (domainBits Q ω) = fullContactBin G x v n m ω := by
-  have hcluster (y : V) (hy : (cluster G ω y).Finite) (hQ : incidentSupport G hloc hy.toFinset ⊆ Q) :
-      cluster G (domainBits Q ω:Set E) y = cluster G ω y := by
-    rw [coe_domainBits]
-    have hh := (cluster_restrict_incident_eq_iff G hloc hy.toFinset Q hQ ω y).mpr
-      (Set.Finite.coe_toFinset hy).symm
-    simpa only [Set.Finite.coe_toFinset] using hh
-  have hcx := hcluster x hx hQx
-  have hcv := hcluster v hv hQv
-  have hFx : finiteRootComponent G (domainBits Q ω) x = hx.toFinset := by
-    apply Finset.coe_injective
-    rw [coe_finiteRootComponent,hcx,Set.Finite.coe_toFinset]
-  have hFv : finiteRootComponent G (domainBits Q ω) v = hv.toFinset := by
-    apply Finset.coe_injective
-    rw [coe_finiteRootComponent,hcv,Set.Finite.coe_toFinset]
-  have hconn : Conn G (domainBits Q ω:Set E) x v ↔ Conn G ω x v := by
-    change v ∈ cluster G (domainBits Q ω:Set E) x ↔ v ∈ cluster G ω x
-    rw [hcx]
-  have hc : (contactBonds G Q hx.toFinset hv.toFinset).card = clusterContactCount G ω x v := by
-    rw [←Set.ncard_coe_finset,coe_contactBonds_eq G hloc Q _ _ hQx]
-    simp only [Set.Finite.coe_toFinset,clusterContactCount]
-  simp only [clusterContactBin,fullContactBin,hconn,hFx,hFv,←Set.ncard_eq_toFinset_card,hc]
-
-theorem clusterContactBin_norm_le (G : BondGraph V E) (Q : Finset E)
-    (d : ℕ) (hd : ∀ z, (Q.filter fun e => z ∈ G e).card ≤ d)
-    (x v : V) (n m : ℕ) (ω : Finset E) : ‖clusterContactBin G Q x v n m ω‖ ≤ (d*m:ℕ) := by
-  unfold clusterContactBin
-  split_ifs with h
-  · rw [Real.norm_natCast]
-    exact_mod_cast (contactBonds_card_le G Q d hd _ _).trans (Nat.mul_le_mul_left d h.2.2)
-  · simp only [norm_zero]; positivity
-
-theorem fullContactBin_integral_le_of_finite [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (d : ℕ)
-    (hd : ∀ z, {e | z ∈ G e}.ncard ≤ d) (x v : V) (p : unitInterval)
-    (hx : ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
-    (hv : ∀ᵐ ω ∂law p, (cluster G ω v).Finite) (n m : ℕ) (B : ℝ)
-    (hB : ∀ Q : Finset E, bernoulliPolynomial Q (clusterContactBin G Q x v n m) p ≤ B) :
-    (∫ ω, fullContactBin G x v n m ω ∂law p) ≤ B := by
-  let : Encodable E := Encodable.ofCountable E
-  let Q := vertexExhaustion (V := E)
-  let f := fun j ω => clusterContactBin G (Q j) x v n m (domainBits (Q j) ω)
-  have hfl (j : ℕ) (ω : Set E) : f j ω = f j (ω ∩ (Q j:Set E)) := by
-    dsimp [f]; rw [domainBits_local]
-  have hfi (j : ℕ) : Integrable (f j) (law p) := integrable_finite_local (Q j) p (f j) (hfl j)
-  have hdeg (j : ℕ) (z : V) : ((Q j).filter fun e => z ∈ G e).card ≤ d := by
-    calc
-      _ = (((Q j).filter fun e => z ∈ G e):Set E).ncard := (Set.ncard_coe_finset _).symm
-      _ ≤ {e | z ∈ G e}.ncard := Set.ncard_le_ncard (fun e he => (Finset.mem_filter.mp he).2) (hloc z)
-      _ ≤ d := hd z
-  have ht : ∀ᵐ ω ∂law p, Tendsto (fun j => f j ω) atTop (nhds (fullContactBin G x v n m ω)) := by
-    filter_upwards [hx,hv] with ω hωx hωv
-    obtain ⟨jx,hjx⟩ := subset_vertexExhaustion (incidentSupport G hloc hωx.toFinset)
-    obtain ⟨jv,hjv⟩ := subset_vertexExhaustion (incidentSupport G hloc hωv.toFinset)
-    have he : (fun j => f j ω) =ᶠ[atTop] (fun _ => fullContactBin G x v n m ω) := by
-      filter_upwards [eventually_ge_atTop (max jx jv)] with j hj
-      exact fullContactBin_eq_finite G hloc ω x v hωx hωv (Q j)
-        (hjx.trans (vertexExhaustion_mono (le_max_left jx jv |>.trans hj)))
-        (hjv.trans (vertexExhaustion_mono (le_max_right jx jv |>.trans hj))) n m
-    exact tendsto_const_nhds.congr' he.symm
-  have hdom := tendsto_integral_of_dominated_convergence (fun _ : Set E => (d*m:ℕ))
-    (fun j => (hfi j).aestronglyMeasurable) (integrable_const _)
-    (fun j => Eventually.of_forall (fun ω => clusterContactBin_norm_le G (Q j) d (hdeg j) x v n m _)) ht
-  apply le_of_tendsto hdom
-  exact Eventually.of_forall (fun j => by rw [show (∫ ω, f j ω ∂law p) =
-    bernoulliPolynomial (Q j) (clusterContactBin G (Q j) x v n m) p from by dsimp [f]; exact integral_domainBits (Q j) p (clusterContactBin G (Q j) x v n m)]; exact hB (Q j))
-
-end InfiniteContactLimit
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section FullContactPower
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem fullContactBin_power_bound [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (q : unitInterval)
-    (α ε C p₀ : ℝ) (hα : 0 ≤ α) (hα1 : α ≤ 1) (hε : 0 < ε)
-    (hr : (1:ℝ)/2+ε-2*α < 0) (hC0 : 0 < C) (hp₀ : 0 < p₀)
-    (hfin : ∀ y, ∀ᵐ ω ∂law q, (cluster G ω y).Finite)
-    (hI : ∀ y, Integrable (fun ω => ((cluster G ω y).ncard:ℝ)^α) (law q))
-    (hC : ∀ y, (∫ ω, ((cluster G ω y).ncard:ℝ)^α ∂law q) ≤ C)
-    (d : ℕ) (hd : 0 < d) (hdeg : ∀ z, {e | z ∈ G e}.ncard ≤ d) :
-    ∃ A : ℝ, 0 < A ∧ ∀ (x v : V) (p : unitInterval), p₀ ≤ (p:ℝ) → p ≤ q →
-      ∀ n : ℕ, 1 ≤ n → (∫ ω, fullContactBin G x v n (2*n) ω ∂law p) ≤
-        A*(n:ℝ)^((1:ℝ)/2+ε-2*α) := by
-  obtain ⟨A,hA,hAb⟩ := clusterContactBin_power_bound G hloc q α ε C p₀ hα hα1 hε hr hC0 hp₀ hfin hI hC d hd
-  refine ⟨A,hA,?_⟩
-  intro x v p hpp hpq n hn
-  apply fullContactBin_integral_le_of_finite G hloc d hdeg x v p
-    (cluster_finite_ae_mono G x p q hpq (hfin x)) (cluster_finite_ae_mono G v p q hpq (hfin v))
-  intro Q
-  apply hAb Q _ x v p hpp hpq n hn
-  intro z
-  calc
-    _ = ((Q.filter fun e => z ∈ G e):Set E).ncard := (Set.ncard_coe_finset _).symm
-    _ ≤ {e | z ∈ G e}.ncard := Set.ncard_le_ncard (fun e he => (Finset.mem_filter.mp he).2) (hloc z)
-    _ ≤ d := hdeg z
-
-theorem integrable_fullContactBin [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (d : ℕ)
-    (hd : ∀ z, {e | z ∈ G e}.ncard ≤ d) (x v : V) (n m : ℕ) (hn : 0 < n) (p : unitInterval) :
-    Integrable (fullContactBin G x v n m) (law p) := by
-  apply Integrable.mono' (integrable_const ((d*m:ℕ):ℝ)) (measurable_fullContactBin G hloc x v n m).aestronglyMeasurable
-  exact Filter.Eventually.of_forall (fun ω => by
-    rw [Real.norm_eq_abs,abs_of_nonneg (fullContactBin_nonneg G x v n m ω)]
-    exact fullContactBin_le G hloc d hd x v n m hn ω)
-
-end FullContactPower
-end Percolation
-
- 
-namespace Percolation
-open MeasureTheory
-section NonnegativeDyadicIntegration
-
- 
-
-theorem integral_le_tsum_majorants {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)
-    (f : Ω → ℝ) (F : ℕ → Ω → ℝ) (hfm : AEStronglyMeasurable f μ)
-    (hf0 : ∀ ω, 0 ≤ f ω) (hF0 : ∀ j ω, 0 ≤ F j ω) (hFi : ∀ j, Integrable (F j) μ)
-    (hs : Summable (fun j => ∫ ω, F j ω ∂μ))
-    (hb : ∀ ω, ENNReal.ofReal (f ω) ≤ ∑' j, ENNReal.ofReal (F j ω)) :
-    Integrable f μ ∧ (∫ ω, f ω ∂μ) ≤ ∑' j, ∫ ω, F j ω ∂μ := by
-  have he : (∫⁻ ω, ∑' j, ENNReal.ofReal (F j ω) ∂μ) =
-      ENNReal.ofReal (∑' j, ∫ ω, F j ω ∂μ) := by
-    rw [lintegral_tsum (fun j => (hFi j).aestronglyMeasurable.aemeasurable.ennreal_ofReal)]
-    simp_rw [←ofReal_integral_eq_lintegral_ofReal (hFi _) (Filter.Eventually.of_forall (hF0 _))]
-    exact (ENNReal.ofReal_tsum_of_nonneg (fun j => integral_nonneg (hF0 j)) hs).symm
-  have hle := (lintegral_mono hb).trans_eq he
-  have hi : Integrable f μ := ⟨hfm,(hasFiniteIntegral_iff_ofReal (Filter.Eventually.of_forall hf0)).mpr
-    (hle.trans_lt ENNReal.ofReal_lt_top)⟩
-  refine ⟨hi,?_⟩
-  rw [←ofReal_integral_eq_lintegral_ofReal hi (Filter.Eventually.of_forall hf0)] at hle
-  exact (ENNReal.ofReal_le_ofReal_iff (tsum_nonneg (fun j => integral_nonneg (hF0 j)))).mp hle
-
-theorem dyadic_rpow_identity (L : ℕ) (hL : 0 < L) (r : ℝ) (j : ℕ) :
-    ((2^j*L:ℕ):ℝ)^r = (L:ℝ)^r*((2:ℝ)^r)^j := by
-  rw [Nat.cast_mul,Nat.cast_pow,Nat.cast_ofNat,Real.mul_rpow (by positivity) (by positivity),
-    ←Real.rpow_natCast,←Real.rpow_mul (by norm_num),mul_comm (j:ℝ) r,Real.rpow_mul (by norm_num),Real.rpow_natCast,mul_comm]
-
-theorem dyadic_rpow_summable (L : ℕ) (hL : 0 < L) (r : ℝ) (hr : r < 0) :
-    Summable (fun j : ℕ => ((2^j*L:ℕ):ℝ)^r) := by
-  simp only [dyadic_rpow_identity L hL r]
-  exact (summable_geometric_of_lt_one (Real.rpow_nonneg (by norm_num) _)
-    (Real.rpow_lt_one_of_one_lt_of_neg (by norm_num) hr)).mul_left _
-
-theorem dyadic_rpow_sum (L : ℕ) (hL : 0 < L) (r : ℝ) (hr : r < 0) :
-    (∑' j : ℕ, ((2^j*L:ℕ):ℝ)^r) = (L:ℝ)^r*(1-(2:ℝ)^r)⁻¹ := by
-  simp only [dyadic_rpow_identity L hL r,tsum_mul_left]
-  rw [tsum_geometric_of_lt_one (Real.rpow_nonneg (by norm_num) _) (Real.rpow_lt_one_of_one_lt_of_neg (by norm_num) hr)]
-
-end NonnegativeDyadicIntegration
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section FullLargeContacts
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def fullLargeContact (G : BondGraph V E) (x v : V) (L : ℕ) (ω : Set E) : ℝ :=
-  if ¬Conn G ω x v ∧ L ≤ min (cluster G ω x).ncard (cluster G ω v).ncard
-  then clusterContactCount G ω x v else 0
-
-theorem fullLargeContact_nonneg (G : BondGraph V E) (x v : V) (L : ℕ) (ω : Set E) :
-    0 ≤ fullLargeContact G x v L ω := by unfold fullLargeContact; split_ifs <;> positivity
-
-theorem measurable_fullLargeContact [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (x v : V) (L : ℕ) :
-    Measurable (fullLargeContact G x v L) := by
-  have hs : Measurable (fun ω : Set E => min (cluster G ω x).ncard (cluster G ω v).ncard) :=
-    (measurable_cluster_ncard G hloc x).min (measurable_cluster_ncard G hloc v)
-  exact ((measurable_of_countable (fun k : ℕ => (k:ℝ))).comp (measurable_clusterContactCount G x v)).ite
-    (((measurableSet_conn G x v).compl).inter (measurableSet_le measurable_const hs)) measurable_const
-
-theorem fullLargeContact_le_dyadic (G : BondGraph V E) (x v : V) (L : ℕ) (hL : 0 < L) (ω : Set E) :
-    ENNReal.ofReal (fullLargeContact G x v L ω) ≤
-      ∑' j : ℕ, ENNReal.ofReal (fullContactBin G x v (2^j*L) (2*(2^j*L)) ω) := by
-  by_cases h : ¬Conn G ω x v ∧ L ≤ min (cluster G ω x).ncard (cluster G ω v).ncard
-  · let k := min (cluster G ω x).ncard (cluster G ω v).ncard
-    let j := Nat.log 2 (k/L)
-    have hq : k/L ≠ 0 := by
-      have : 1 ≤ k/L := (Nat.le_div_iff_mul_le hL).mpr (by rw [one_mul]; exact h.2)
-      omega
-    have hl := (Nat.le_div_iff_mul_le hL).mp (Nat.pow_log_le_self 2 hq)
-    have hu := (Nat.div_lt_iff_lt_mul hL).mp (Nat.lt_pow_succ_log_self (by norm_num : 1 < (2:ℕ)) (k/L))
-    have hbin : fullContactBin G x v (2^j*L) (2*(2^j*L)) ω = fullLargeContact G x v L ω := by
-      rw [fullLargeContact,ite_eq_left h,fullContactBin,ite_eq_left]
-      refine ⟨h.1,hl,?_⟩
-      calc
-        _ = k := rfl
-        _ ≤ 2^(Nat.log 2 (k/L)+1)*L := hu.le
-        _ = _ := by dsimp [j]; rw [pow_succ]; ring
-    exact (le_of_eq (congrArg ENNReal.ofReal hbin.symm)).trans
-      (ENNReal.le_tsum (f := fun scale =>
-        ENNReal.ofReal (fullContactBin G x v (2^scale*L) (2*(2^scale*L)) ω)) j)
-  · simp only [fullLargeContact,ite_eq_right h,ENNReal.ofReal_zero]; exact bot_le
-
- 
-
-theorem fullLargeContact_bound_of_bins [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (d : ℕ)
-    (hd : ∀ z, {e | z ∈ G e}.ncard ≤ d) (x v : V) (p : unitInterval)
-    (A r : ℝ) (_hA : 0 ≤ A) (hr : r < 0)
-    (hB : ∀ n : ℕ, 1 ≤ n → (∫ ω, fullContactBin G x v n (2*n) ω ∂law p) ≤ A*(n:ℝ)^r)
-    (L : ℕ) (hL : 0 < L) :
-    Integrable (fullLargeContact G x v L) (law p) ∧
-      (∫ ω, fullLargeContact G x v L ω ∂law p) ≤ A*(1-(2:ℝ)^r)⁻¹*(L:ℝ)^r := by
-  let F := fun j => fullContactBin G x v (2^j*L) (2*(2^j*L))
-  have hbp (j : ℕ) : 0 < 2^j*L := Nat.mul_pos (pow_pos (by norm_num) _) hL
-  have hFi (j : ℕ) : Integrable (F j) (law p) := integrable_fullContactBin G hloc d hd x v _ _ (hbp j) p
-  have hsumB := (dyadic_rpow_summable L hL r hr).mul_left A
-  have hsum : Summable (fun j => ∫ ω, F j ω ∂law p) :=
-    hsumB.of_nonneg_of_le (fun j => integral_nonneg (fullContactBin_nonneg G x v _ _))
-      (fun j => hB _ (hbp j))
-  obtain ⟨hi,hb⟩ := integral_le_tsum_majorants (law p) (fullLargeContact G x v L) F
-    (measurable_fullLargeContact G hloc x v L).aestronglyMeasurable
-    (fullLargeContact_nonneg G x v L) (fun j => fullContactBin_nonneg G x v _ _) hFi hsum
-    (fullLargeContact_le_dyadic G x v L hL)
-  refine ⟨hi,hb.trans ?_⟩
-  calc
-    _ ≤ ∑' j : ℕ, A*((2^j*L:ℕ):ℝ)^r := hsum.tsum_le_tsum (fun j => hB _ (hbp j)) hsumB
-    _ = _ := by rw [tsum_mul_left,dyadic_rpow_sum L hL r hr]; ring
-
-end FullLargeContacts
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section UniformLargeContacts
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem incident_ncard_le_degree (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) :
-    {e | x ∈ G e}.ncard ≤ degree G x := by
-  have he : ((hloc x).toFinset.filter fun e => x ∈ G e) = (hloc x).toFinset := by
-    ext e
-    simp only [Finset.mem_filter,Set.Finite.mem_toFinset,Set.mem_ofPred_eq,and_self]
-  simpa only [he,←Set.ncard_eq_toFinset_card] using incident_filter_card_le_degree G hloc (hloc x).toFinset x
-
-theorem fullLargeContact_power_bound [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G) (q : unitInterval)
-    (α ε C p₀ : ℝ) (hα : 0 ≤ α) (hα1 : α ≤ 1) (hε : 0 < ε)
-    (hr : (1:ℝ)/2+ε-2*α < 0) (hC0 : 0 < C) (hp₀ : 0 < p₀)
-    (hfin : ∀ y, ∀ᵐ ω ∂law q, (cluster G ω y).Finite)
-    (hI : ∀ y, Integrable (fun ω => ((cluster G ω y).ncard:ℝ)^α) (law q))
-    (hC : ∀ y, (∫ ω, ((cluster G ω y).ncard:ℝ)^α ∂law q) ≤ C) :
-    ∃ A : ℝ, 0 < A ∧ ∀ (x v : V) (p : unitInterval), p₀ ≤ (p:ℝ) → p ≤ q →
-      ∀ L : ℕ, 1 ≤ L → Integrable (fullLargeContact G x v L) (law p) ∧
-        (∫ ω, fullLargeContact G x v L ω ∂law p) ≤ A*(L:ℝ)^((1:ℝ)/2+ε-2*α) := by
-  obtain ⟨d,hd,hdeg⟩ := exists_degree_bound G hqt
-  have hdn (x : V) := (incident_ncard_le_degree G hloc x).trans (hdeg x)
-  obtain ⟨A,hA,hAb⟩ := fullContactBin_power_bound G hloc q α ε C p₀ hα hα1 hε hr hC0 hp₀ hfin hI hC d hd hdn
-  refine ⟨A*(1-(2:ℝ)^((1:ℝ)/2+ε-2*α))⁻¹,?_ ,?_⟩
-  · exact mul_pos hA (inv_pos.mpr (sub_pos.mpr (Real.rpow_lt_one_of_one_lt_of_neg (by norm_num) hr)))
-  · intro x v p hpp hpq L hL
-    exact fullLargeContact_bound_of_bins G hloc d hdn x v p A _ hA.le hr (hAb x v p hpp hpq) L hL
-
- 
-
-theorem critical_large_contact_bound
-    [Countable V] [Countable E] [Nonempty V] [Nonempty E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hconn : Connected G)
-    (hqt : QuasiTransitive G) (hh : 0 < hV G)
-    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
-    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
-    (hp₀ : 0 < (pc G:ℝ)) (hp₁ : (pc G:ℝ) < 1)
-    (hfin : ∀ i, ∀ᵐ ω ∂law (pc G), (cluster G ω (root i)).Finite)
-    (α ε : ℝ) (hα : 0 < α) (hα1 : α < 1/2) (hε : 0 < ε)
-    (hr : (1:ℝ)/2+ε-2*α < 0) :
-    ∃ A : ℝ, 0 < A ∧ ∀ (x v : V) (p : unitInterval), (pc G:ℝ)/2 ≤ (p:ℝ) → p ≤ pc G →
-      ∀ L : ℕ, 1 ≤ L → Integrable (fullLargeContact G x v L) (law p) ∧
-        (∫ ω, fullLargeContact G x v L ω ∂law p) ≤ A*(L:ℝ)^((1:ℝ)/2+ε-2*α) := by
-  have hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i) := by
-    intro x
-    obtain ⟨i,a,ha,hax⟩ := hcov x
-    obtain ⟨b,hab⟩ := hΓ ha
-    exact ⟨i,a,b,hab,hax⟩
-  have hfinite : ∀ x, ∀ᵐ ω ∂law (pc G), (cluster G ω x).Finite := by
-    intro x
-    obtain ⟨i,a,b,hab,rfl⟩ := hcover x
-    exact cluster_finite_ae_covariant G (pc G) hab (root i) (hfin i)
-  obtain ⟨C,hC,hb⟩ := uniform_cluster_power_of_root G hloc (pc G) root hcover α
-    (critical_root_power_integrable G hloc hconn hqt hh root weight Γ hΓ htr hcov hw hw1 hp₀ hp₁ hfin α hα hα1)
-  exact fullLargeContact_power_bound G hloc hqt (pc G) α ε C ((pc G:ℝ)/2) hα.le (by linarith)
-    hε hr hC (by positivity) hfinite (fun x => (hb x).1) (fun x => (hb x).2)
-
-end UniformLargeContacts
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-open scoped ENNReal
-section ExtendedTransport
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem enn_sum_iSup_nat {α : Type*} (s : Finset α) (f : ℕ → α → ℝ≥0∞)
-    (hm : ∀ a, Monotone (fun n => f n a)) :
-    (∑ a ∈ s, ⨆ n, f n a) = ⨆ n, ∑ a ∈ s, f n a := by
-  induction s using Finset.induction_on with
-  | empty => simp
-  | @insert a s ha ih =>
-    simp only [Finset.sum_insert ha,ih]
-    exact ENNReal.iSup_add_iSup_of_monotone (hm a) (fun n m h =>
-      Finset.sum_le_sum fun a _ => hm a h)
-
-theorem enn_tsum_iSup_nat {α : Type*} (f : ℕ → α → ℝ≥0∞)
-    (hm : ∀ a, Monotone (fun n => f n a)) :
-    (∑' a, ⨆ n, f n a) = ⨆ n, ∑' a, f n a := by
-  simp_rw [ENNReal.tsum_eq_iSup_sum,enn_sum_iSup_nat _ f hm]
-  exact iSup_comm
-
- 
-
-theorem RootTransport.ennreal (G : BondGraph V E) (hconn : Connected G)
-    (hloc : LocallyFinite G) {ι : Type*} [Fintype ι]
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hw : ∀ i, 0 ≤ weight i) (m : V → V → ℝ≥0∞)
-    (hi : ∀ a ∈ Γ, ∀ x y, m (a x) (a y) = m x y) :
-    (∑ i, ENNReal.ofReal (weight i) * ∑' y, m (root i) y) =
-      (∑ i, ENNReal.ofReal (weight i) * ∑' x, m x (root i)) := by
-  let cut (n : ℕ) (x y : V) : ℝ≥0∞ :=
-    if transportCut G n x y then min (m x y) n else 0
-  have hc0 (n : ℕ) (x y : V) : cut n x y ≠ ⊤ := by
-    dsimp [cut]
-    split_ifs
-    · exact ne_top_of_le_ne_top (ENNReal.natCast_ne_top n) (min_le_right _ _)
-    · exact ENNReal.zero_ne_top
-  have hcm (x y : V) : Monotone (fun n => cut n x y) := by
-    intro n k hnk
-    dsimp [cut]
-    split_ifs with hn hk
-    · exact min_le_min_left _ (by exact_mod_cast hnk)
-    · exact False.elim (hk (transportCut_mono G hnk hn))
-    · exact zero_le
-    · exact le_refl _
-  have hcs (x y : V) : (⨆ n, cut n x y) = m x y := by
-    apply le_antisymm
-    · refine iSup_le fun n => ?_
-      dsimp [cut]
-      split_ifs
-      · exact min_le_left _ _
-      · exact zero_le
-    · obtain ⟨N,hN⟩ := exists_transportCut G hconn y {x}
-      have hxy := hN x (Finset.mem_singleton_self x)
-      have he : (⨆ k : ℕ, min (m x y) (k:ℝ≥0∞)) = m x y := by
-        rw [←inf_iSup_eq,ENNReal.iSup_natCast,inf_top_eq]
-      rw [←he]
-      refine iSup_le fun k => le_iSup_of_le (max N k) ?_
-      dsimp [cut]
-      rw [ite_eq_left (transportCut_mono G (le_max_left N k) hxy)]
-      exact min_le_min_left _ (by exact_mod_cast le_max_right N k)
-  have hrow (n : ℕ) (x : V) : Summable (fun y => (cut n x y).toReal) := by
-    apply summable_of_ne_finset_zero (s := (finite_reachBall hloc x n).toFinset)
-    intro y hy
-    have hh : ¬transportCut G n x y := fun h => hy ((Set.Finite.mem_toFinset _).mpr h.1)
-    simp only [cut,ite_eq_right hh,ENNReal.toReal_zero]
-  have hcol (n : ℕ) (y : V) : Summable (fun x => (cut n x y).toReal) := by
-    apply summable_of_ne_finset_zero (s := (finite_reachBall hloc y n).toFinset)
-    intro x hx
-    have hh : ¬transportCut G n x y := fun h => hx ((Set.Finite.mem_toFinset _).mpr h.2)
-    simp only [cut,ite_eq_right hh,ENNReal.toReal_zero]
-  have he (n : ℕ) :
-      (∑ i, ENNReal.ofReal (weight i) * ∑' y, cut n (root i) y) =
-        (∑ i, ENNReal.ofReal (weight i) * ∑' x, cut n x (root i)) := by
-    have ht := htr.real root weight Γ (fun x y => (cut n x y).toReal)
-      (fun a ha x y => by
-        obtain ⟨b,hb⟩ := hΓ ha
-        simp only [cut,transportCut_aut G hb,hi a ha]) (hrow n) (hcol n)
-    apply_fun ENNReal.ofReal at ht
-    rw [ENNReal.ofReal_sum_of_nonneg (fun i _ => mul_nonneg (hw i)
-      (tsum_nonneg fun _ => ENNReal.toReal_nonneg)),
-      ENNReal.ofReal_sum_of_nonneg (fun i _ => mul_nonneg (hw i)
-      (tsum_nonneg fun _ => ENNReal.toReal_nonneg))] at ht
-    simp_rw [ENNReal.ofReal_mul (hw _),
-      ENNReal.ofReal_tsum_of_nonneg (fun _ => ENNReal.toReal_nonneg) (hrow n _),
-      ENNReal.ofReal_tsum_of_nonneg (fun _ => ENNReal.toReal_nonneg) (hcol n _),
-      ENNReal.ofReal_toReal (hc0 n _ _)] at ht
-    exact ht
-  have hlim (f : ι → V → ℝ≥0∞) (g : ℕ → ι → V → ℝ≥0∞)
-      (hmono : ∀ i y, Monotone (fun n => g n i y)) (hs : ∀ i y, (⨆ n, g n i y) = f i y) :
-      (∑ i, ENNReal.ofReal (weight i) * ∑' y, f i y) =
-        ⨆ n, ∑ i, ENNReal.ofReal (weight i) * ∑' y, g n i y := by
-    simp_rw [←hs,enn_tsum_iSup_nat _ (hmono _),ENNReal.mul_iSup]
-    exact enn_sum_iSup_nat _ _ (fun i n k h => mul_le_mul_right (ENNReal.tsum_le_tsum
-      (fun y => hmono i y h)) _)
-  rw [hlim (fun i y => m (root i) y) (fun n i y => cut n (root i) y)
-    (fun i y => hcm (root i) y) (fun i y => hcs (root i) y),
-    hlim (fun i x => m x (root i)) (fun n i x => cut n x (root i))
-    (fun i x => hcm x (root i)) (fun i x => hcs x (root i))]
-  exact iSup_congr he
-
-end ExtendedTransport
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-open scoped ENNReal
-section ClusterAveraging
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def clusterAveragingKernel (G : BondGraph V E) (ω : Set E) (x y : V) : ℝ≥0∞ :=
-  if Conn G ω x y then ((cluster G ω x).ncard:ℝ≥0∞)⁻¹ else 0
-
-theorem measurable_clusterAveragingKernel [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (x y : V) :
-    Measurable (fun ω => clusterAveragingKernel G ω x y) := by
-  exact (((measurable_of_countable (fun n : ℕ => (n:ℝ≥0∞))).comp
-    (measurable_cluster_ncard G hloc x)).inv).ite (measurableSet_conn G x y) measurable_const
-
-theorem clusterAveragingKernel_symm (G : BondGraph V E) (ω : Set E) (x y : V) :
-    clusterAveragingKernel G ω x y = clusterAveragingKernel G ω y x := by
-  by_cases h : Conn G ω x y
-  · simp only [clusterAveragingKernel,ite_eq_left h,ite_eq_left (conn_symm G ω h),cluster_eq_of_conn G ω h]
-  · have h' : ¬Conn G ω y x := fun hh => h (conn_symm G ω hh)
-    simp only [clusterAveragingKernel,ite_eq_right h,ite_eq_right h']
-
-theorem cluster_ncard_pos (G : BondGraph V E) (ω : Set E) (x : V)
-    (hf : (cluster G ω x).Finite) : 0 < (cluster G ω x).ncard :=
-  (Set.ncard_pos hf).mpr ⟨x,mem_cluster_self G ω x⟩
-
-theorem clusterAveragingKernel_row (G : BondGraph V E) (ω : Set E) (x : V)
-    (hf : (cluster G ω x).Finite) : (∑' y, clusterAveragingKernel G ω x y) = 1 := by
-  have he : (fun y => clusterAveragingKernel G ω x y) =
-      (fun y => (if Conn G ω x y then (1:ℝ≥0∞) else 0)*((cluster G ω x).ncard:ℝ≥0∞)⁻¹) := by
-    funext y
-    unfold clusterAveragingKernel
-    split_ifs <;> simp
-  rw [he,ENNReal.tsum_mul_right]
-  change clusterMass G ω x * ((cluster G ω x).ncard:ℝ≥0∞)⁻¹ = 1
-  rw [clusterMass_eq_ncard G ω x hf]
-  exact ENNReal.mul_inv_cancel (by exact_mod_cast (cluster_ncard_pos G ω x hf).ne') (ENNReal.natCast_ne_top _)
-
-theorem clusterAveragingKernel_column (G : BondGraph V E) (ω : Set E) (y : V)
-    (hf : (cluster G ω y).Finite) : (∑' x, clusterAveragingKernel G ω x y) = 1 := by
-  simp_rw [clusterAveragingKernel_symm G ω]
-  exact clusterAveragingKernel_row G ω y hf
-
-theorem clusterAveragingKernel_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (ω : Set E) (x y : V) :
-    clusterAveragingKernel G (b ⁻¹' ω) x y = clusterAveragingKernel G ω (a x) (a y) := by
-  simp only [clusterAveragingKernel,conn_aut_iff G hab,cluster_ncard_covariant G hab]
-
- 
-
-theorem root_cluster_average [Countable V] [Countable E]
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hw : ∀ i, 0 ≤ weight i) (p : unitInterval)
-    (hfin : ∀ x, ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
-    (f : Set E → V → ℝ≥0∞) (hf : ∀ x, Measurable (fun ω => f ω x))
-    (hi : ∀ a ∈ Γ, ∀ b, IsAut G a b → ∀ ω x, f (b ⁻¹' ω) x = f ω (a x)) :
-    (∑ i, ENNReal.ofReal (weight i) * ∫⁻ ω, ∑' y,
-      clusterAveragingKernel G ω (root i) y * f ω y ∂law p) =
-    (∑ i, ENNReal.ofReal (weight i) * ∫⁻ ω, f ω (root i) ∂law p) := by
-  let m (x y : V) : ℝ≥0∞ := ∫⁻ ω, clusterAveragingKernel G ω x y * f ω y ∂law p
-  have hm (x y : V) : Measurable (fun ω => clusterAveragingKernel G ω x y * f ω y) :=
-    (measurable_clusterAveragingKernel G hloc x y).mul (hf y)
-  have hinv : ∀ a ∈ Γ, ∀ x y, m (a x) (a y) = m x y := by
-    intro a ha x y
-    obtain ⟨b,hb⟩ := hΓ ha
-    have ht := lintegral_map (μ := law p) (hm x y) (measurable_set_preimage b)
-    rw [law_map_preimage_equiv] at ht
-    dsimp only [m]
-    rw [ht]
-    apply lintegral_congr
-    intro ω
-    rw [clusterAveragingKernel_aut G hb,hi a ha b hb]
-  have ht := htr.ennreal G hconn hloc root weight Γ hΓ hw m hinv
-  have hr (x : V) : (∑' y, m x y) = ∫⁻ ω, ∑' y, clusterAveragingKernel G ω x y*f ω y ∂law p := by
-    exact (lintegral_tsum (fun y => (hm x y).aemeasurable)).symm
-  have hc (y : V) : (∑' x, m x y) = ∫⁻ ω, f ω y ∂law p := by
-    rw [show (∑' x, m x y) = ∫⁻ ω, ∑' x, clusterAveragingKernel G ω x y*f ω y ∂law p from
-      (lintegral_tsum (fun x => (hm x y).aemeasurable)).symm]
-    apply lintegral_congr_ae
-    filter_upwards [hfin y] with ω hω
-    rw [ENNReal.tsum_mul_right,clusterAveragingKernel_column G ω y hω,one_mul]
-  simpa only [hr,hc] using ht
-
-end ClusterAveraging
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-open scoped ENNReal
-section PositiveClusterPairs
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def clusterPairKernel (G : BondGraph V E) (ω : Set E)
-    (J : V → V → ℝ≥0∞) (x y : V) : ℝ≥0∞ :=
-  ∑' a, ∑' b, if Conn G ω x a ∧ Conn G ω y b then J a b else 0
-
- 
-
-noncomputable def clusterPairDensity (G : BondGraph V E) (ω : Set E)
-    (F : V → V → ℝ≥0∞) (x : V) : ℝ≥0∞ :=
-  ((cluster G ω x).ncard:ℝ≥0∞)⁻¹ *
-    ∑' y, ((cluster G ω y).ncard:ℝ≥0∞)⁻¹ * (if ¬Conn G ω x y then F x y else 0)
-
-noncomputable def clusterPairFunctional (G : BondGraph V E) (p : unitInterval)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (F : Set E → V → V → ℝ≥0∞) : ℝ≥0∞ :=
-  ∑ i, ENNReal.ofReal (weight i) * ∫⁻ ω, clusterPairDensity G ω (F ω) (root i) ∂law p
-
-theorem cluster_weighted_resum (G : BondGraph V E) (ω : Set E)
-    (hfin : ∀ x, (cluster G ω x).Finite) (f : V → V → ℝ≥0∞)
-    (hc : ∀ x y, Conn G ω x y → f x y = f y y) :
-    (∑' x, ((cluster G ω x).ncard:ℝ≥0∞)⁻¹ *
-      ∑' y, if Conn G ω x y then f x y else 0) = ∑' y, f y y := by
-  have he (x y : V) : ((cluster G ω x).ncard:ℝ≥0∞)⁻¹ *
-      (if Conn G ω x y then f x y else 0) = clusterAveragingKernel G ω x y*f y y := by
-    by_cases h : Conn G ω x y
-    · simp only [ite_eq_left h,clusterAveragingKernel,hc x y h]
-    · simp only [ite_eq_right h,clusterAveragingKernel,mul_zero,zero_mul]
-  simp_rw [←ENNReal.tsum_mul_left,he]
-  rw [ENNReal.tsum_comm]
-  apply tsum_congr
-  intro y
-  rw [ENNReal.tsum_mul_right,clusterAveragingKernel_column G ω y (hfin y),one_mul]
-
-theorem clusterPairDensity_reroot_pointwise (G : BondGraph V E) (ω : Set E)
-    (hfin : ∀ x, (cluster G ω x).Finite) (J F : V → V → ℝ≥0∞)
-    (hF : ∀ x y a b, Conn G ω x a → Conn G ω y b → F x y = F a b) (x : V) :
-    clusterPairDensity G ω (fun a b => clusterPairKernel G ω J a b * F a b) x =
-      ∑' a, clusterAveragingKernel G ω x a *
-        ∑' b, J a b * (if ¬Conn G ω a b then F a b else 0) := by
-  let K (a b : V) : ℝ≥0∞ := if ¬Conn G ω a b then F a b else 0
-  have hK (a b c d : V) (hac : Conn G ω a c) (hbd : Conn G ω b d) : K a b = K c d := by
-    have he : Conn G ω a b ↔ Conn G ω c d := by
-      constructor
-      · intro h; exact conn_trans G ω (conn_symm G ω hac) (conn_trans G ω h hbd)
-      · intro h; exact conn_trans G ω hac (conn_trans G ω h (conn_symm G ω hbd))
-    simp only [K,he,hF a b c d hac hbd]
-  have hterm (y : V) : ((cluster G ω y).ncard:ℝ≥0∞)⁻¹ *
-      (if ¬Conn G ω x y then clusterPairKernel G ω J x y*F x y else 0) =
-      ∑' a, if Conn G ω x a then ((cluster G ω y).ncard:ℝ≥0∞)⁻¹ *
-        ∑' b, if Conn G ω y b then J a b*K x y else 0 else 0 := by
-    have hcond : (if ¬Conn G ω x y then clusterPairKernel G ω J x y*F x y else 0) =
-        clusterPairKernel G ω J x y*K x y := by dsimp [K]; split_ifs <;> simp
-    rw [hcond,clusterPairKernel,←ENNReal.tsum_mul_right,←ENNReal.tsum_mul_left]
-    apply tsum_congr
-    intro a
-    by_cases hxa : Conn G ω x a
-    · rw [ite_eq_left hxa,←ENNReal.tsum_mul_right]
-      congr 1
-      apply tsum_congr
-      intro b
-      by_cases hyb : Conn G ω y b <;> simp only [hxa,hyb,and_self,and_false,
-        ite_true,ite_false,zero_mul]
-    · simp only [hxa,false_and,ite_false,tsum_zero,zero_mul,mul_zero]
-  unfold clusterPairDensity
-  simp_rw [hterm]
-  rw [ENNReal.tsum_comm,←ENNReal.tsum_mul_left]
-  apply tsum_congr
-  intro a
-  by_cases hxa : Conn G ω x a
-  · simp only [ite_eq_left hxa,clusterAveragingKernel]
-    congr 1
-    rw [cluster_weighted_resum G ω hfin (fun y b => J a b*K x y)
-      (fun y b hyb => congrArg (fun t => J a b*t) (hK x y x b (conn_refl G ω x) hyb))]
-    apply tsum_congr
-    intro b
-    rw [hK x b a b hxa (conn_refl G ω b)]
-  · simp only [ite_eq_right hxa,tsum_zero,mul_zero,clusterAveragingKernel,zero_mul]
-
-end PositiveClusterPairs
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-open scoped ENNReal
-section ClusterPairRerooting
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem clusterPairFunctional_reroot [Countable V] [Countable E]
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hw : ∀ i, 0 ≤ weight i) (p : unitInterval)
-    (hfin : ∀ x, ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
-    (J : V → V → ℝ≥0∞) (hJ : ∀ a ∈ Γ, ∀ x y, J (a x) (a y) = J x y)
-    (F : Set E → V → V → ℝ≥0∞) (hm : ∀ x y, Measurable (fun ω => F ω x y))
-    (hF : ∀ ω x y a b, Conn G ω x a → Conn G ω y b → F ω x y = F ω a b)
-    (hi : ∀ a ∈ Γ, ∀ b, IsAut G a b → ∀ ω x y, F (b ⁻¹' ω) x y = F ω (a x) (a y)) :
-    clusterPairFunctional G p root weight (fun ω x y => clusterPairKernel G ω J x y * F ω x y) =
-      ∑ i, ENNReal.ofReal (weight i) * ∫⁻ ω, ∑' y,
-        J (root i) y * (if ¬Conn G ω (root i) y then F ω (root i) y else 0) ∂law p := by
-  let f (ω : Set E) (x : V) : ℝ≥0∞ :=
-    ∑' y, J x y * (if ¬Conn G ω x y then F ω x y else 0)
-  have hf (x : V) : Measurable (fun ω => f ω x) := by
-    apply Measurable.tsum
-    intro y
-    exact measurable_const.mul ((hm x y).ite (measurableSet_conn G x y).compl measurable_const)
-  have hfe : ∀ a ∈ Γ, ∀ b, IsAut G a b → ∀ ω x, f (b ⁻¹' ω) x = f ω (a x) := by
-    intro a ha b hab ω x
-    dsimp [f]
-    rw [←a.tsum_eq (fun y => J (a x) y * (if ¬Conn G ω (a x) y then F ω (a x) y else 0))]
-    apply tsum_congr
-    intro y
-    rw [hJ a ha,conn_aut_iff G hab,hi a ha b hab]
-  have ht := root_cluster_average G hconn hloc root weight Γ hΓ htr hw p hfin f hf hfe
-  rw [clusterPairFunctional]
-  calc
-    _ = ∑ i, ENNReal.ofReal (weight i) * ∫⁻ ω, ∑' y,
-        clusterAveragingKernel G ω (root i) y*f ω y ∂law p := by
-      apply Finset.sum_congr rfl
-      intro i _
-      congr 1
-      apply lintegral_congr_ae
-      filter_upwards [ae_all_iff.mpr hfin] with ω hω
-      exact clusterPairDensity_reroot_pointwise G ω hω J (F ω) (hF ω) (root i)
-    _ = _ := ht
-
-end ClusterPairRerooting
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-open scoped ENNReal
-section PairReversal
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def clusterPairMass (G : BondGraph V E) (ω : Set E)
-    (F : V → V → ℝ≥0∞) (x y : V) : ℝ≥0∞ :=
-  ((cluster G ω x).ncard:ℝ≥0∞)⁻¹ * ((cluster G ω y).ncard:ℝ≥0∞)⁻¹ *
-    (if ¬Conn G ω x y then F x y else 0)
-
-theorem clusterPairDensity_eq_tsum (G : BondGraph V E) (ω : Set E)
-    (F : V → V → ℝ≥0∞) (x : V) :
-    clusterPairDensity G ω F x = ∑' y, clusterPairMass G ω F x y := by
-  simp only [clusterPairDensity,clusterPairMass,←ENNReal.tsum_mul_left,mul_assoc]
-
-theorem clusterPairMass_swap (G : BondGraph V E) (ω : Set E)
-    (F : V → V → ℝ≥0∞) (x y : V) :
-    clusterPairMass G ω F x y = clusterPairMass G ω (fun a b => F b a) y x := by
-  have he : Conn G ω x y ↔ Conn G ω y x := ⟨conn_symm G ω,conn_symm G ω⟩
-  simp only [clusterPairMass,he,mul_comm]
-
-theorem measurable_clusterPairMass [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G)
-    (F : Set E → V → V → ℝ≥0∞) (hf : ∀ x y, Measurable (fun ω => F ω x y)) (x y : V) :
-    Measurable (fun ω => clusterPairMass G ω (F ω) x y) := by
-  have hs (x : V) : Measurable (fun ω => ((cluster G ω x).ncard:ℝ≥0∞)⁻¹) :=
-    (((measurable_of_countable (fun n : ℕ => (n:ℝ≥0∞))).comp (measurable_cluster_ncard G hloc x))).inv
-  exact ((hs x).mul (hs y)).mul ((hf x y).ite (measurableSet_conn G x y).compl measurable_const)
-
-theorem clusterPairFunctional_swap [Countable V] [Countable E]
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hw : ∀ i, 0 ≤ weight i) (p : unitInterval)
-    (F : Set E → V → V → ℝ≥0∞) (hf : ∀ x y, Measurable (fun ω => F ω x y))
-    (hi : ∀ a ∈ Γ, ∀ b, IsAut G a b → ∀ ω x y, F (b ⁻¹' ω) x y = F ω (a x) (a y)) :
-    clusterPairFunctional G p root weight F =
-      clusterPairFunctional G p root weight (fun ω x y => F ω y x) := by
-  let m (x y : V) : ℝ≥0∞ := ∫⁻ ω, clusterPairMass G ω (F ω) x y ∂law p
-  have hm := measurable_clusterPairMass G hloc F hf
-  have hinv : ∀ a ∈ Γ, ∀ x y, m (a x) (a y) = m x y := by
-    intro a ha x y
-    obtain ⟨b,hb⟩ := hΓ ha
-    have ht := lintegral_map (μ := law p) (hm x y) (measurable_set_preimage b)
-    rw [law_map_preimage_equiv] at ht
-    dsimp only [m]
-    rw [ht]
-    apply lintegral_congr
-    intro ω
-    simp only [clusterPairMass,cluster_ncard_covariant G hb,conn_aut_iff G hb,hi a ha b hb]
-  have ht := htr.ennreal G hconn hloc root weight Γ hΓ hw m hinv
-  have hr (x : V) : (∑' y, m x y) = ∫⁻ ω, clusterPairDensity G ω (F ω) x ∂law p := by
-    simp_rw [clusterPairDensity_eq_tsum]
-    exact (lintegral_tsum (fun y => (hm x y).aemeasurable)).symm
-  have hc (y : V) : (∑' x, m x y) =
-      ∫⁻ ω, clusterPairDensity G ω (fun a b => F ω b a) y ∂law p := by
-    simp_rw [clusterPairDensity_eq_tsum,←clusterPairMass_swap G _ (F _)]
-    exact (lintegral_tsum (fun x => (hm x y).aemeasurable)).symm
-  simpa only [hr,hc,clusterPairFunctional] using ht
-
-end PairReversal
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Exploration.TailBounds
+
+namespace OAI
+
+/-! Large Contacts for bond percolation. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section UnorderedContactConcentration
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem cluster_contact_tail (G : BondGraph V E) (Q : Finset E)
+    (d : ℕ) (hd : ∀ z, (Q.filter fun e => z ∈ G e).card ≤ d)
+    (x v : V) (p : unitInterval) (m : ℕ) (hdm : 0 < d*m) (a : ℝ) (ha : 0 < a) :
+    bernoulliPolynomial Q (fun ω => if ¬Conn G (ω:Set E) x v ∧
+      min (finiteRootComponent G ω x).card (finiteRootComponent G ω v).card ≤ m ∧
+      2*a ≤ (p:ℝ)*(contactBonds G Q (finiteRootComponent G ω x)
+        (finiteRootComponent G ω v)).card then 1 else 0) p ≤
+      8*Real.exp (-2*a^2/(d*m:ℕ)) := by
+  let f (x v : V) (ω : Finset E) : ℝ :=
+    if ¬Conn G (ω:Set E) x v ∧ (incidentBonds G Q (finiteRootComponent G ω v)).card ≤ d*m ∧
+      2*a ≤ (p:ℝ)*(contactBonds G Q (finiteRootComponent G ω x)
+        (finiteRootComponent G ω v)).card then 1 else 0
+  calc
+    _ ≤ bernoulliPolynomial Q (fun ω => f x v ω+f v x ω) p := by
+      apply bernoulliPolynomial_mono _ _ _ p p.2
+      intro ω _
+      by_cases h : ¬Conn G (ω:Set E) x v ∧
+          min (finiteRootComponent G ω x).card (finiteRootComponent G ω v).card ≤ m ∧
+          2*a ≤ (p:ℝ)*(contactBonds G Q (finiteRootComponent G ω x)
+            (finiteRootComponent G ω v)).card
+      · simp only [ite_eq_left h]
+        rcases le_total (finiteRootComponent G ω v).card (finiteRootComponent G ω x).card with hc | hc
+        · have hsmall : (finiteRootComponent G ω v).card ≤ m := by simpa only [min_eq_right hc] using h.2.1
+          have hh : ¬Conn G (ω:Set E) x v ∧ (incidentBonds G Q (finiteRootComponent G ω v)).card ≤ d*m ∧
+              2*a ≤ (p:ℝ)*(contactBonds G Q (finiteRootComponent G ω x) (finiteRootComponent G ω v)).card :=
+            ⟨h.1,(incidentBonds_card_le G Q _ d hd).trans (Nat.mul_le_mul_left d hsmall),h.2.2⟩
+          simp only [f,ite_eq_left hh]; split_ifs <;> norm_num
+        · have hsmall : (finiteRootComponent G ω x).card ≤ m := by simpa only [min_eq_left hc] using h.2.1
+          have hh : ¬Conn G (ω:Set E) v x ∧ (incidentBonds G Q (finiteRootComponent G ω x)).card ≤ d*m ∧
+              2*a ≤ (p:ℝ)*(contactBonds G Q (finiteRootComponent G ω v) (finiteRootComponent G ω x)).card := by
+            refine ⟨fun hc' => h.1 (conn_symm G _ hc'),
+              (incidentBonds_card_le G Q _ d hd).trans (Nat.mul_le_mul_left d hsmall),?_⟩
+            rw [contactBonds_comm G Q (finiteRootComponent G ω v)]
+            exact h.2.2
+          simp only [f,ite_eq_left hh]; split_ifs <;> norm_num
+      · simp only [ite_eq_right h,f]; split_ifs <;> norm_num
+    _ = bernoulliPolynomial Q (f x v) p+bernoulliPolynomial Q (f v x) p :=
+      bernoulliPolynomial_add ..
+    _ ≤ 4*Real.exp (-2*a^2/(d*m:ℕ))+4*Real.exp (-2*a^2/(d*m:ℕ)) :=
+      add_le_add (ordered_cluster_contact_tail G Q x v p (d*m) hdm a ha)
+        (ordered_cluster_contact_tail G Q v x p (d*m) hdm a ha)
+    _ = _ := by ring
+
+end UnorderedContactConcentration
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section TwoLargeFiniteClusters
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem cluster_exterior_weighted_bound (G : BondGraph V E) (Q : Finset E)
+    (x : V) (p : unitInterval) (F : Finset V → ℝ) (hF : ∀ W, 0 ≤ F W)
+    (f : Finset V → Finset E → ℝ) (K : ℝ)
+    (hf : ∀ W : Finset V, bernoulliPolynomial (exteriorBonds G Q (W:Set V)) (f W) p ≤ K) :
+    bernoulliPolynomial Q (fun ω => F (finiteRootComponent G ω x)*f (finiteRootComponent G ω x)
+      (exteriorBonds G ω (finiteRootComponent G ω x:Set V))) p ≤
+      K*bernoulliPolynomial Q (fun ω => F (finiteRootComponent G ω x)) p := by
+  have hh := cluster_exterior_expectation_bound G Q x p
+    (fun W η => F W*f W η-K*F W) 0 (fun W => by
+      rw [bernoulliPolynomial_sub,bernoulliPolynomial_const_mul,bernoulliPolynomial_const]
+      exact sub_nonpos.mpr (by simpa only [mul_comm K] using mul_le_mul_of_nonneg_left (hf W) (hF W)))
+  rw [bernoulliPolynomial_sub,bernoulliPolynomial_const_mul] at hh
+  exact sub_nonpos.mp hh
+
+theorem finiteClusterTail_domain_mono (G : BondGraph V E) {Q R : Finset E} (hRQ : R ⊆ Q)
+    (x : V) (n : ℕ) (p : unitInterval) : finiteClusterTail G R x n p ≤ finiteClusterTail G Q x n p := by
+  unfold finiteClusterTail
+  rw [←bernoulliPolynomial_restrict R Q hRQ]
+  apply bernoulliPolynomial_mono _ _ _ p p.2
+  intro ω _
+  exact finiteClusterTailIndicator_mono G x n Finset.inter_subset_left
+
+ 
+
+theorem two_large_clusters_tail (G : BondGraph V E) (Q : Finset E)
+    (x v : V) (n m : ℕ) (p : unitInterval) :
+    bernoulliPolynomial Q (fun ω => if ¬Conn G (ω:Set E) x v ∧
+      n ≤ (finiteRootComponent G ω x).card ∧ m ≤ (finiteRootComponent G ω v).card
+        then 1 else 0) p ≤ finiteClusterTail G Q v m p*finiteClusterTail G Q x n p := by
+  let F (W : Finset V) : ℝ := if n ≤ W.card then 1 else 0
+  let f (W : Finset V) (η : Finset E) : ℝ := if v ∈ W then 0 else finiteClusterTailIndicator G v m η
+  have hF : ∀ W, 0 ≤ F W := fun W => by dsimp [F]; split_ifs <;> norm_num
+  have hf (W : Finset V) : bernoulliPolynomial (exteriorBonds G Q (W:Set V)) (f W) p ≤
+      finiteClusterTail G Q v m p := by
+    by_cases hv : v ∈ W
+    · simp only [f,ite_eq_left hv,bernoulliPolynomial_zero]
+      exact finiteClusterTail_nonneg G Q v m p
+    · simp only [f,ite_eq_right hv]
+      exact finiteClusterTail_domain_mono G (Finset.filter_subset _ _) v m p
+  have hp := cluster_exterior_weighted_bound G Q x p F hF f (finiteClusterTail G Q v m p) hf
+  have heq : bernoulliPolynomial Q (fun ω => F (finiteRootComponent G ω x)*f (finiteRootComponent G ω x)
+      (exteriorBonds G ω (finiteRootComponent G ω x:Set V))) p =
+      bernoulliPolynomial Q (fun ω => if ¬Conn G (ω:Set E) x v ∧
+      n ≤ (finiteRootComponent G ω x).card ∧ m ≤ (finiteRootComponent G ω v).card then 1 else 0) p := by
+    apply bernoulliPolynomial_congr
+    intro ω _
+    have hv : v ∈ finiteRootComponent G ω x ↔ Conn G (ω:Set E) x v := by
+      change v ∈ (finiteRootComponent G ω x:Set V) ↔ _
+      rw [coe_finiteRootComponent]
+      rfl
+    by_cases hc : Conn G (ω:Set E) x v
+    · simp only [F,f,ite_eq_left (hv.mpr hc),mul_zero,not_true_eq_false,hc,false_and,ite_false]
+    · simp only [F,f,ite_eq_right (fun h => hc (hv.mp h)),finiteClusterTailIndicator,
+        finiteRootComponent_exterior G ω x v hc]
+      split_ifs <;> norm_num <;> grind
+  rw [heq] at hp
+  exact hp
+
+end TwoLargeFiniteClusters
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section EndpointClusterTails
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem finiteClusterTail_of_endpoint_moment [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (q : unitInterval) (α C : ℝ)
+    (hα : 0 ≤ α) (hα1 : α ≤ 1)
+    (hfin : ∀ y, ∀ᵐ ω ∂law q, (cluster G ω y).Finite)
+    (hI : ∀ y, Integrable (fun ω => ((cluster G ω y).ncard:ℝ)^α) (law q))
+    (hC : ∀ y, (∫ ω, ((cluster G ω y).ncard:ℝ)^α ∂law q) ≤ C)
+    (Q : Finset E) (x : V) (p : unitInterval) (hpq : p ≤ q) (n : ℕ) (hn : 1 ≤ n) :
+    finiteClusterTail G Q x n p ≤ 2*C*(n:ℝ)^(-α) := by
+  have hn0 : (0:ℝ) < n := by exact_mod_cast (by omega : 0 < n)
+  let h : unitInterval := ⟨1/(n:ℝ),⟨by positivity,by apply (div_le_one hn0).mpr; exact_mod_cast hn⟩⟩
+  have hh : 0 < (h:ℝ) := by dsimp [h]; positivity
+  have hhpow : (h:ℝ)^α = (n:ℝ)^(-α) := by
+    dsimp [h]
+    rw [one_div,Real.inv_rpow hn0.le,Real.rpow_neg hn0.le]
+  have hhn : 1 ≤ (h:ℝ)*n := by dsimp [h]; rw [one_div,inv_mul_cancel₀ hn0.ne']
+  have hm := cluster_power_mono_integral G hloc x p q hpq α hα (hfin x) (hI x)
+  have hb := markedConnection_le_cluster_power G Q x h p α hα hα1
+    (cluster_finite_ae_mono G x p q hpq (hfin x)) hm.1
+  have hM : markedConnectionPolynomial G Q x h p ≤ C*(n:ℝ)^(-α) := by
+    apply hb.trans
+    rw [hhpow]
+    simpa only [mul_comm C] using
+      mul_le_mul_of_nonneg_left (hm.2.trans (hC x)) (Real.rpow_nonneg (Nat.cast_nonneg n) (-α))
+  have hb' := (finiteClusterTail_le_markedConnection G Q x n p h hhn).trans
+    (mul_le_mul_of_nonneg_left hM (by norm_num : (0:ℝ) ≤ 2))
+  simpa only [mul_assoc] using hb'
+
+theorem two_large_clusters_of_endpoint_moment [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (q : unitInterval) (α C : ℝ)
+    (hα : 0 ≤ α) (hα1 : α ≤ 1) (hC0 : 0 ≤ C)
+    (hfin : ∀ y, ∀ᵐ ω ∂law q, (cluster G ω y).Finite)
+    (hI : ∀ y, Integrable (fun ω => ((cluster G ω y).ncard:ℝ)^α) (law q))
+    (hC : ∀ y, (∫ ω, ((cluster G ω y).ncard:ℝ)^α ∂law q) ≤ C)
+    (Q : Finset E) (x v : V) (p : unitInterval) (hpq : p ≤ q) (n : ℕ) (hn : 1 ≤ n) :
+    bernoulliPolynomial Q (fun ω => if ¬Conn G (ω:Set E) x v ∧
+      n ≤ (finiteRootComponent G ω x).card ∧ n ≤ (finiteRootComponent G ω v).card
+        then 1 else 0) p ≤ 4*C^2*(n:ℝ)^(-2*α) := by
+  have ht (y : V) := finiteClusterTail_of_endpoint_moment G hloc q α C hα hα1 hfin hI hC Q y p hpq n hn
+  have hc : 0 ≤ 2*C*(n:ℝ)^(-α) := by positivity
+  calc
+    _ ≤ finiteClusterTail G Q v n p*finiteClusterTail G Q x n p := two_large_clusters_tail ..
+    _ ≤ (2*C*(n:ℝ)^(-α))*(2*C*(n:ℝ)^(-α)) :=
+      mul_le_mul (ht v) (ht x) (finiteClusterTail_nonneg G Q x n p) hc
+    _ = 4*C^2*(n:ℝ)^(-2*α) := by
+      have hn0 : (0:ℝ) < n := by exact_mod_cast (by omega : 0 < n)
+      rw [show (2*C*(n:ℝ)^(-α))*(2*C*(n:ℝ)^(-α)) =
+        4*C^2*((n:ℝ)^(-α)*(n:ℝ)^(-α)) by ring,←Real.rpow_add hn0]
+      congr 2
+      ring
+
+end EndpointClusterTails
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section ContactBinFirstMoment
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem contactBonds_card_le (G : BondGraph V E) (Q : Finset E)
+    (d : ℕ) (hd : ∀ z, (Q.filter fun e => z ∈ G e).card ≤ d) (A B : Finset V) :
+    (contactBonds G Q A B).card ≤ d*min A.card B.card := by
+  have hh (A B : Finset V) : (contactBonds G Q A B).card ≤ d*A.card := by
+    apply (Finset.card_le_card (t := incidentBonds G Q A) _).trans (incidentBonds_card_le G Q A d hd)
+    intro e he
+    obtain ⟨he,a,ha,b,_,hab⟩ := Finset.mem_filter.mp he
+    exact (mem_incidentBonds G Q A e).mpr ⟨he,a,ha,by rw [hab]; simp⟩
+  rcases le_total A.card B.card with h | h
+  · rw [min_eq_left h]; exact hh A B
+  · rw [min_eq_right h,contactBonds_comm G Q A B]; exact hh B A
+
+noncomputable def clusterContactBin (G : BondGraph V E) (Q : Finset E)
+    (x v : V) (n m : ℕ) (ω : Finset E) : ℝ :=
+  if ¬Conn G (ω:Set E) x v ∧
+    n ≤ min (finiteRootComponent G ω x).card (finiteRootComponent G ω v).card ∧
+    min (finiteRootComponent G ω x).card (finiteRootComponent G ω v).card ≤ m
+  then (contactBonds G Q (finiteRootComponent G ω x) (finiteRootComponent G ω v)).card else 0
+
+ 
+
+theorem clusterContactBin_bound (G : BondGraph V E) (Q : Finset E)
+    (d : ℕ) (hd : ∀ z, (Q.filter fun e => z ∈ G e).card ≤ d)
+    (x v : V) (p : unitInterval) (hp : 0 < (p:ℝ)) (n m : ℕ) (hdm : 0 < d*m)
+    (a : ℝ) (ha : 0 < a) :
+    bernoulliPolynomial Q (clusterContactBin G Q x v n m) p ≤
+      (2*a/(p:ℝ))*(finiteClusterTail G Q v n p*finiteClusterTail G Q x n p)+
+      (d*m:ℕ)*(8*Real.exp (-2*a^2/(d*m:ℕ))) := by
+  let D (ω : Finset E) : Prop := ¬Conn G (ω:Set E) x v ∧
+    n ≤ (finiteRootComponent G ω x).card ∧ n ≤ (finiteRootComponent G ω v).card
+  let T (ω : Finset E) : Prop := ¬Conn G (ω:Set E) x v ∧
+    min (finiteRootComponent G ω x).card (finiteRootComponent G ω v).card ≤ m ∧
+    2*a ≤ (p:ℝ)*(contactBonds G Q (finiteRootComponent G ω x) (finiteRootComponent G ω v)).card
+  calc
+    _ ≤ bernoulliPolynomial Q (fun ω => (2*a/(p:ℝ))*(if D ω then 1 else 0)+
+        (d*m:ℕ)*(if T ω then 1 else 0)) p := by
+      apply bernoulliPolynomial_mono _ _ _ p p.2
+      intro ω _
+      unfold clusterContactBin
+      by_cases h : ¬Conn G (ω:Set E) x v ∧
+          n ≤ min (finiteRootComponent G ω x).card (finiteRootComponent G ω v).card ∧
+          min (finiteRootComponent G ω x).card (finiteRootComponent G ω v).card ≤ m
+      · have hD : D ω := ⟨h.1,(le_min_iff.mp h.2.1).1,(le_min_iff.mp h.2.1).2⟩
+        simp only [ite_eq_left h,ite_eq_left hD,mul_one]
+        by_cases ht : T ω
+        · rw [ite_eq_left ht,mul_one]
+          have hk : (contactBonds G Q (finiteRootComponent G ω x) (finiteRootComponent G ω v)).card ≤ d*m :=
+            (contactBonds_card_le G Q d hd _ _).trans (Nat.mul_le_mul_left d h.2.2)
+          have hk' : ((contactBonds G Q (finiteRootComponent G ω x) (finiteRootComponent G ω v)).card:ℝ) ≤ (d*m:ℕ) := by exact_mod_cast hk
+          have hapos : 0 ≤ 2*a/(p:ℝ) := by positivity
+          linarith
+        · rw [ite_eq_right ht,mul_zero,add_zero]
+          apply (le_div_iff₀ hp).mpr
+          have hn : ¬ 2*a ≤ (p:ℝ)*(contactBonds G Q (finiteRootComponent G ω x) (finiteRootComponent G ω v)).card :=
+            fun hh => ht ⟨h.1,h.2.2,hh⟩
+          simpa only [mul_comm] using (not_le.mp hn).le
+      · simp only [ite_eq_right h]
+        split_ifs <;> positivity
+    _ = (2*a/(p:ℝ))*bernoulliPolynomial Q (fun ω => if D ω then 1 else 0) p+
+        (d*m:ℕ)*bernoulliPolynomial Q (fun ω => if T ω then 1 else 0) p := by
+      rw [bernoulliPolynomial_add,bernoulliPolynomial_const_mul,bernoulliPolynomial_const_mul]
+    _ ≤ _ := add_le_add
+      (mul_le_mul_of_nonneg_left (two_large_clusters_tail G Q x v n n p) (by positivity))
+      (mul_le_mul_of_nonneg_left (cluster_contact_tail G Q d hd x v p m hdm a ha) (Nat.cast_nonneg _))
+
+end ContactBinFirstMoment
+end Percolation
+
+ 
+namespace Percolation
+section StretchedContactDecay
+
+theorem exists_stretched_exp_volume_bound (d ε r : ℝ) (hd : 0 < d) (hε : 0 < ε) (hr : r ≤ 1) :
+    ∃ C : ℝ, 0 < C ∧ ∀ z : ℝ, 0 < z → z*Real.exp (-(z^ε)/d) ≤ C*z^r := by
+  let s := (1-r)/ε
+  have hs : 0 ≤ s := by dsimp [s]; positivity
+  obtain ⟨K,hK,hKb⟩ := exists_exp_neg_le_rpow s hs
+  refine ⟨K*d^s,by positivity,?_⟩
+  intro z hz
+  have he := mul_le_mul_of_nonneg_left (hKb (z^ε/d) (by positivity)) hz.le
+  have hex : ε*(-s) = r-1 := by dsimp [s]; field_simp; ring
+  have hid : (z^ε/d)^(-s) = z^(r-1)*d^s := by
+    rw [Real.div_rpow (Real.rpow_nonneg hz.le ε) hd.le,←Real.rpow_mul hz.le,hex,
+      Real.rpow_neg hd.le,div_inv_eq_mul]
+  have hzpow : z*z^(r-1) = z^r := by
+    conv_lhs => lhs; rw [←Real.rpow_one z]
+    rw [←Real.rpow_add hz]
+    congr 1
+    ring
+  calc
+    _ ≤ z*(K*(z^ε/d)^(-s)) := by simpa only [neg_div] using he
+    _ = (K*d^s)*(z*z^(r-1)) := by rw [hid]; ring
+    _ = _ := by rw [hzpow]
+
+theorem contact_tail_exponent (d z ε : ℝ) (hd : 0 < d) (hz : 0 < z) :
+    -2*(z^((1:ℝ)/2+ε))^2/(d*(2*z)) = -(z^(2*ε))/d := by
+  have hpow : (z^((1:ℝ)/2+ε))^2 = z*z^(2*ε) := by
+    rw [←Real.rpow_natCast,←Real.rpow_mul hz.le]
+    norm_num only [Nat.cast_ofNat]
+    rw [show ((1:ℝ)/2+ε)*2 = 1+2*ε by ring,Real.rpow_add hz,Real.rpow_one]
+  rw [hpow]
+  field_simp
+
+end StretchedContactDecay
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section UniformContactBins
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem clusterContactBin_power_bound [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (q : unitInterval)
+    (α ε C p₀ : ℝ) (hα : 0 ≤ α) (hα1 : α ≤ 1) (hε : 0 < ε)
+    (hr : (1:ℝ)/2+ε-2*α < 0) (hC0 : 0 < C) (hp₀ : 0 < p₀)
+    (hfin : ∀ y, ∀ᵐ ω ∂law q, (cluster G ω y).Finite)
+    (hI : ∀ y, Integrable (fun ω => ((cluster G ω y).ncard:ℝ)^α) (law q))
+    (hC : ∀ y, (∫ ω, ((cluster G ω y).ncard:ℝ)^α ∂law q) ≤ C)
+    (d : ℕ) (hd : 0 < d) :
+    ∃ A : ℝ, 0 < A ∧ ∀ (Q : Finset E), (∀ z, (Q.filter fun e => z ∈ G e).card ≤ d) →
+      ∀ (x v : V) (p : unitInterval), p₀ ≤ (p:ℝ) → p ≤ q → ∀ n : ℕ, 1 ≤ n →
+      bernoulliPolynomial Q (clusterContactBin G Q x v n (2*n)) p ≤
+        A*(n:ℝ)^((1:ℝ)/2+ε-2*α) := by
+  let r : ℝ := (1:ℝ)/2+ε-2*α
+  have hd' : (0:ℝ) < d := Nat.cast_pos.mpr hd
+  obtain ⟨K,hK,hKb⟩ := exists_stretched_exp_volume_bound (d:ℝ) (2*ε) r hd' (by positivity) (by dsimp [r]; linarith)
+  refine ⟨8*C^2/p₀+16*(d:ℝ)*K,by positivity,?_⟩
+  intro Q hdeg x v p hpp hpq n hn
+  have hn' : 0 < n := by omega
+  have hnr : (0:ℝ) < n := Nat.cast_pos.mpr hn'
+  have hp : 0 < (p:ℝ) := hp₀.trans_le hpp
+  let a : ℝ := (n:ℝ)^((1:ℝ)/2+ε)
+  have ha : 0 < a := Real.rpow_pos_of_pos hnr _
+  have hb := clusterContactBin_bound G Q d hdeg x v p hp n (2*n) (by positivity) a ha
+  have ht (y : V) := finiteClusterTail_of_endpoint_moment G hloc q α C hα hα1 hfin hI hC Q y p hpq n hn
+  have hprod : finiteClusterTail G Q v n p*finiteClusterTail G Q x n p ≤ 4*C^2*(n:ℝ)^(-2*α) := by
+    calc
+      _ ≤ (2*C*(n:ℝ)^(-α))*(2*C*(n:ℝ)^(-α)) :=
+        mul_le_mul (ht v) (ht x) (finiteClusterTail_nonneg G Q x n p) (by positivity)
+      _ = _ := by
+        rw [show (2*C*(n:ℝ)^(-α))*(2*C*(n:ℝ)^(-α)) = 4*C^2*((n:ℝ)^(-α)*(n:ℝ)^(-α)) by ring,
+          ←Real.rpow_add hnr]
+        congr 2
+        ring
+  have hfirst : (2*a/(p:ℝ))*(finiteClusterTail G Q v n p*finiteClusterTail G Q x n p) ≤
+      (8*C^2/p₀)*(n:ℝ)^r := by
+    calc
+      _ ≤ (2*a/p₀)*(4*C^2*(n:ℝ)^(-2*α)) :=
+        mul_le_mul (div_le_div_of_nonneg_left (by positivity) hp₀ hpp) hprod
+          (mul_nonneg (finiteClusterTail_nonneg G Q v n p) (finiteClusterTail_nonneg G Q x n p)) (by positivity)
+      _ = (8*C^2/p₀)*((n:ℝ)^((1:ℝ)/2+ε)*(n:ℝ)^(-2*α)) := by dsimp [a]; ring
+      _ = _ := by rw [←Real.rpow_add hnr]; congr 2; dsimp [r]; ring
+  have hex : -2*a^2/(d*(2*n):ℕ) = -((n:ℝ)^(2*ε))/(d:ℝ) := by
+    simp only [Nat.cast_mul,Nat.cast_ofNat]
+    exact contact_tail_exponent d n ε hd' hnr
+  have hsecond : (d*(2*n):ℕ)*(8*Real.exp (-2*a^2/(d*(2*n):ℕ))) ≤
+      (16*(d:ℝ)*K)*(n:ℝ)^r := by
+    rw [hex]
+    calc
+      _ = (16*(d:ℝ))*((n:ℝ)*Real.exp (-((n:ℝ)^(2*ε))/(d:ℝ))) := by
+        push_cast
+        ring
+      _ ≤ (16*(d:ℝ))*(K*(n:ℝ)^r) := mul_le_mul_of_nonneg_left (hKb n hnr) (by positivity)
+      _ = _ := by ring
+  calc
+    _ ≤ _ := hb
+    _ ≤ (8*C^2/p₀)*(n:ℝ)^r+(16*(d:ℝ)*K)*(n:ℝ)^r := add_le_add hfirst hsecond
+    _ = _ := by dsimp [r]; ring
+
+end UniformContactBins
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section InfiniteContacts
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+def closedContactSet (G : BondGraph V E) (A B : Set V) : Set E :=
+  {e | ∃ a ∈ A, ∃ b ∈ B, G e = s(a,b)}
+
+noncomputable def clusterContactCount (G : BondGraph V E) (ω : Set E) (x v : V) : ℕ :=
+  (closedContactSet G (cluster G ω x) (cluster G ω v)).ncard
+
+noncomputable def fullContactBin (G : BondGraph V E) (x v : V) (n m : ℕ) (ω : Set E) : ℝ :=
+  if ¬Conn G ω x v ∧ n ≤ min (cluster G ω x).ncard (cluster G ω v).ncard ∧
+    min (cluster G ω x).ncard (cluster G ω v).ncard ≤ m
+  then clusterContactCount G ω x v else 0
+
+theorem contactSet_subset_incident (G : BondGraph V E) (hloc : LocallyFinite G)
+    (A : Finset V) (B : Set V) : closedContactSet G (A:Set V) B ⊆ (incidentSupport G hloc A:Set E) := by
+  rintro e ⟨a,ha,b,_,hab⟩
+  exact (mem_incidentSupport G hloc A e).mpr ⟨a,ha,by rw [hab]; simp⟩
+
+theorem coe_contactBonds_eq (G : BondGraph V E) (hloc : LocallyFinite G)
+    (Q : Finset E) (A B : Finset V) (hQ : incidentSupport G hloc A ⊆ Q) :
+    (contactBonds G Q A B : Set E) = closedContactSet G A B := by
+  ext e
+  constructor
+  · exact fun h => (Finset.mem_filter.mp h).2
+  · intro h
+    exact Finset.mem_filter.mpr ⟨hQ (contactSet_subset_incident G hloc A B h),h⟩
+
+theorem contactSet_ncard_le (G : BondGraph V E) (hloc : LocallyFinite G)
+    (d : ℕ) (hd : ∀ z, {e | z ∈ G e}.ncard ≤ d)
+    (A B : Set V) (hA : A.Finite) (hB : B.Finite) :
+    (closedContactSet G A B).ncard ≤ d*min A.ncard B.ncard := by
+  let Q := incidentSupport G hloc (hA.toFinset ∪ hB.toFinset)
+  have hdeg (z : V) : (Q.filter fun e => z ∈ G e).card ≤ d := by
+    calc
+      _ = ((Q.filter fun e => z ∈ G e):Set E).ncard := (Set.ncard_coe_finset _).symm
+      _ ≤ {e | z ∈ G e}.ncard := Set.ncard_le_ncard (fun e he => (Finset.mem_filter.mp he).2) (hloc z)
+      _ ≤ d := hd z
+  have hQ : incidentSupport G hloc hA.toFinset ⊆ Q :=
+    incidentSupport_mono G hloc Finset.subset_union_left
+  have he := coe_contactBonds_eq G hloc Q hA.toFinset hB.toFinset hQ
+  simp only [Set.Finite.coe_toFinset] at he
+  rw [←he,Set.ncard_coe_finset]
+  simpa only [←Set.ncard_eq_toFinset_card] using contactBonds_card_le G Q d hdeg hA.toFinset hB.toFinset
+
+theorem fullContactBin_nonneg (G : BondGraph V E) (x v : V) (n m : ℕ) (ω : Set E) :
+    0 ≤ fullContactBin G x v n m ω := by unfold fullContactBin; split_ifs <;> positivity
+
+theorem fullContactBin_le (G : BondGraph V E) (hloc : LocallyFinite G)
+    (d : ℕ) (hd : ∀ z, {e | z ∈ G e}.ncard ≤ d) (x v : V) (n m : ℕ) (hn : 0 < n) (ω : Set E) :
+    fullContactBin G x v n m ω ≤ (d*m:ℕ) := by
+  unfold fullContactBin
+  split_ifs with h
+  · have hx : (cluster G ω x).Finite := Set.finite_of_ncard_ne_zero (by omega)
+    have hv : (cluster G ω v).Finite := Set.finite_of_ncard_ne_zero (by omega)
+    exact_mod_cast (contactSet_ncard_le G hloc d hd _ _ hx hv).trans (Nat.mul_le_mul_left d h.2.2)
+  · positivity
+
+end InfiniteContacts
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section ContactMeasurability
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem measurable_clusterContactCount [Countable V] [Countable E]
+    (G : BondGraph V E) (x v : V) : Measurable (fun ω : Set E => clusterContactCount G ω x v) := by
+  apply measurable_ncard.comp
+  apply measurable_setOfPred.comp
+  apply Measurable.of_eval
+  intro e
+  apply measurableSet_setOfPred.mp
+  change MeasurableSet {ω : Set E | ∃ a, Conn G ω x a ∧ ∃ b, Conn G ω v b ∧ G e = s(a,b)}
+  simp only [Set.ofPred_exists]
+  apply MeasurableSet.iUnion
+  intro a
+  have he : {ω : Set E | Conn G ω x a ∧ ∃ b, Conn G ω v b ∧ G e = s(a,b)} =
+      {ω | Conn G ω x a} ∩ ⋃ b, {ω | Conn G ω v b ∧ G e = s(a,b)} := by ext ω; simp
+  rw [he]
+  apply (measurableSet_conn G x a).inter
+  apply MeasurableSet.iUnion
+  intro b
+  by_cases h : G e = s(a,b)
+  · simpa only [h,and_true] using measurableSet_conn G v b
+  · simp only [h,and_false,Set.ofPred_false]; exact MeasurableSet.empty
+
+theorem measurable_fullContactBin [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (x v : V) (n m : ℕ) :
+    Measurable (fullContactBin G x v n m) := by
+  have hs : Measurable (fun ω : Set E => min (cluster G ω x).ncard (cluster G ω v).ncard) :=
+    (measurable_cluster_ncard G hloc x).min (measurable_cluster_ncard G hloc v)
+  exact ((measurable_of_countable (fun k : ℕ => (k:ℝ))).comp (measurable_clusterContactCount G x v)).ite
+    (((measurableSet_conn G x v).compl).inter ((measurableSet_le measurable_const hs).inter
+      (measurableSet_le hs measurable_const))) measurable_const
+
+theorem coe_domainBits (Q : Finset E) (ω : Set E) :
+    (domainBits Q ω:Set E) = ω ∩ (Q:Set E) := by ext e; simp only [mem_domainBits,Set.mem_inter_iff,Finset.mem_coe,and_comm]
+
+theorem domainBits_local (Q : Finset E) (ω : Set E) :
+    domainBits Q (ω ∩ (Q:Set E)) = domainBits Q ω := by
+  ext e
+  simp only [mem_domainBits,Set.mem_inter_iff,Finset.mem_coe]
+  tauto
+
+theorem domainBits_of_subset (Q ω : Finset E) (hω : ω ⊆ Q) : domainBits Q (ω:Set E) = ω := by
+  ext e
+  simp only [mem_domainBits,Finset.mem_coe]
+  exact ⟨fun h => h.2,fun h => ⟨hω h,h⟩⟩
+
+theorem integral_domainBits (Q : Finset E) (p : unitInterval) (f : Finset E → ℝ) :
+    (∫ ω, f (domainBits Q ω) ∂law p) = bernoulliPolynomial Q f p := by
+  rw [integral_eq_bernoulliPolynomial Q p _ (fun ω => by rw [domainBits_local])]
+  apply bernoulliPolynomial_congr
+  intro ω hω
+  rw [domainBits_of_subset Q ω hω]
+
+end ContactMeasurability
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section InfiniteContactLimit
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem fullContactBin_eq_finite (G : BondGraph V E) (hloc : LocallyFinite G)
+    (ω : Set E) (x v : V) (hx : (cluster G ω x).Finite) (hv : (cluster G ω v).Finite)
+    (Q : Finset E) (hQx : incidentSupport G hloc hx.toFinset ⊆ Q)
+    (hQv : incidentSupport G hloc hv.toFinset ⊆ Q) (n m : ℕ) :
+    clusterContactBin G Q x v n m (domainBits Q ω) = fullContactBin G x v n m ω := by
+  have hcluster (y : V) (hy : (cluster G ω y).Finite) (hQ : incidentSupport G hloc hy.toFinset ⊆ Q) :
+      cluster G (domainBits Q ω:Set E) y = cluster G ω y := by
+    rw [coe_domainBits]
+    have hh := (cluster_restrict_incident_eq_iff G hloc hy.toFinset Q hQ ω y).mpr
+      (Set.Finite.coe_toFinset hy).symm
+    simpa only [Set.Finite.coe_toFinset] using hh
+  have hcx := hcluster x hx hQx
+  have hcv := hcluster v hv hQv
+  have hFx : finiteRootComponent G (domainBits Q ω) x = hx.toFinset := by
+    apply Finset.coe_injective
+    rw [coe_finiteRootComponent,hcx,Set.Finite.coe_toFinset]
+  have hFv : finiteRootComponent G (domainBits Q ω) v = hv.toFinset := by
+    apply Finset.coe_injective
+    rw [coe_finiteRootComponent,hcv,Set.Finite.coe_toFinset]
+  have hconn : Conn G (domainBits Q ω:Set E) x v ↔ Conn G ω x v := by
+    change v ∈ cluster G (domainBits Q ω:Set E) x ↔ v ∈ cluster G ω x
+    rw [hcx]
+  have hc : (contactBonds G Q hx.toFinset hv.toFinset).card = clusterContactCount G ω x v := by
+    rw [←Set.ncard_coe_finset,coe_contactBonds_eq G hloc Q _ _ hQx]
+    simp only [Set.Finite.coe_toFinset,clusterContactCount]
+  simp only [clusterContactBin,fullContactBin,hconn,hFx,hFv,←Set.ncard_eq_toFinset_card,hc]
+
+theorem clusterContactBin_norm_le (G : BondGraph V E) (Q : Finset E)
+    (d : ℕ) (hd : ∀ z, (Q.filter fun e => z ∈ G e).card ≤ d)
+    (x v : V) (n m : ℕ) (ω : Finset E) : ‖clusterContactBin G Q x v n m ω‖ ≤ (d*m:ℕ) := by
+  unfold clusterContactBin
+  split_ifs with h
+  · rw [Real.norm_natCast]
+    exact_mod_cast (contactBonds_card_le G Q d hd _ _).trans (Nat.mul_le_mul_left d h.2.2)
+  · simp only [norm_zero]; positivity
+
+theorem fullContactBin_integral_le_of_finite [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (d : ℕ)
+    (hd : ∀ z, {e | z ∈ G e}.ncard ≤ d) (x v : V) (p : unitInterval)
+    (hx : ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
+    (hv : ∀ᵐ ω ∂law p, (cluster G ω v).Finite) (n m : ℕ) (B : ℝ)
+    (hB : ∀ Q : Finset E, bernoulliPolynomial Q (clusterContactBin G Q x v n m) p ≤ B) :
+    (∫ ω, fullContactBin G x v n m ω ∂law p) ≤ B := by
+  let : Encodable E := Encodable.ofCountable E
+  let Q := vertexExhaustion (V := E)
+  let f := fun j ω => clusterContactBin G (Q j) x v n m (domainBits (Q j) ω)
+  have hfl (j : ℕ) (ω : Set E) : f j ω = f j (ω ∩ (Q j:Set E)) := by
+    dsimp [f]; rw [domainBits_local]
+  have hfi (j : ℕ) : Integrable (f j) (law p) := integrable_finite_local (Q j) p (f j) (hfl j)
+  have hdeg (j : ℕ) (z : V) : ((Q j).filter fun e => z ∈ G e).card ≤ d := by
+    calc
+      _ = (((Q j).filter fun e => z ∈ G e):Set E).ncard := (Set.ncard_coe_finset _).symm
+      _ ≤ {e | z ∈ G e}.ncard := Set.ncard_le_ncard (fun e he => (Finset.mem_filter.mp he).2) (hloc z)
+      _ ≤ d := hd z
+  have ht : ∀ᵐ ω ∂law p, Tendsto (fun j => f j ω) atTop (nhds (fullContactBin G x v n m ω)) := by
+    filter_upwards [hx,hv] with ω hωx hωv
+    obtain ⟨jx,hjx⟩ := subset_vertexExhaustion (incidentSupport G hloc hωx.toFinset)
+    obtain ⟨jv,hjv⟩ := subset_vertexExhaustion (incidentSupport G hloc hωv.toFinset)
+    have he : (fun j => f j ω) =ᶠ[atTop] (fun _ => fullContactBin G x v n m ω) := by
+      filter_upwards [eventually_ge_atTop (max jx jv)] with j hj
+      exact fullContactBin_eq_finite G hloc ω x v hωx hωv (Q j)
+        (hjx.trans (vertexExhaustion_mono (le_max_left jx jv |>.trans hj)))
+        (hjv.trans (vertexExhaustion_mono (le_max_right jx jv |>.trans hj))) n m
+    exact tendsto_const_nhds.congr' he.symm
+  have hdom := tendsto_integral_of_dominated_convergence (fun _ : Set E => (d*m:ℕ))
+    (fun j => (hfi j).aestronglyMeasurable) (integrable_const _)
+    (fun j => Eventually.of_forall (fun ω => clusterContactBin_norm_le G (Q j) d (hdeg j) x v n m _)) ht
+  apply le_of_tendsto hdom
+  exact Eventually.of_forall (fun j => by rw [show (∫ ω, f j ω ∂law p) =
+    bernoulliPolynomial (Q j) (clusterContactBin G (Q j) x v n m) p from by dsimp [f]; exact integral_domainBits (Q j) p (clusterContactBin G (Q j) x v n m)]; exact hB (Q j))
+
+end InfiniteContactLimit
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section FullContactPower
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem fullContactBin_power_bound [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (q : unitInterval)
+    (α ε C p₀ : ℝ) (hα : 0 ≤ α) (hα1 : α ≤ 1) (hε : 0 < ε)
+    (hr : (1:ℝ)/2+ε-2*α < 0) (hC0 : 0 < C) (hp₀ : 0 < p₀)
+    (hfin : ∀ y, ∀ᵐ ω ∂law q, (cluster G ω y).Finite)
+    (hI : ∀ y, Integrable (fun ω => ((cluster G ω y).ncard:ℝ)^α) (law q))
+    (hC : ∀ y, (∫ ω, ((cluster G ω y).ncard:ℝ)^α ∂law q) ≤ C)
+    (d : ℕ) (hd : 0 < d) (hdeg : ∀ z, {e | z ∈ G e}.ncard ≤ d) :
+    ∃ A : ℝ, 0 < A ∧ ∀ (x v : V) (p : unitInterval), p₀ ≤ (p:ℝ) → p ≤ q →
+      ∀ n : ℕ, 1 ≤ n → (∫ ω, fullContactBin G x v n (2*n) ω ∂law p) ≤
+        A*(n:ℝ)^((1:ℝ)/2+ε-2*α) := by
+  obtain ⟨A,hA,hAb⟩ := clusterContactBin_power_bound G hloc q α ε C p₀ hα hα1 hε hr hC0 hp₀ hfin hI hC d hd
+  refine ⟨A,hA,?_⟩
+  intro x v p hpp hpq n hn
+  apply fullContactBin_integral_le_of_finite G hloc d hdeg x v p
+    (cluster_finite_ae_mono G x p q hpq (hfin x)) (cluster_finite_ae_mono G v p q hpq (hfin v))
+  intro Q
+  apply hAb Q _ x v p hpp hpq n hn
+  intro z
+  calc
+    _ = ((Q.filter fun e => z ∈ G e):Set E).ncard := (Set.ncard_coe_finset _).symm
+    _ ≤ {e | z ∈ G e}.ncard := Set.ncard_le_ncard (fun e he => (Finset.mem_filter.mp he).2) (hloc z)
+    _ ≤ d := hdeg z
+
+theorem integrable_fullContactBin [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (d : ℕ)
+    (hd : ∀ z, {e | z ∈ G e}.ncard ≤ d) (x v : V) (n m : ℕ) (hn : 0 < n) (p : unitInterval) :
+    Integrable (fullContactBin G x v n m) (law p) := by
+  apply Integrable.mono' (integrable_const ((d*m:ℕ):ℝ)) (measurable_fullContactBin G hloc x v n m).aestronglyMeasurable
+  exact Filter.Eventually.of_forall (fun ω => by
+    rw [Real.norm_eq_abs,abs_of_nonneg (fullContactBin_nonneg G x v n m ω)]
+    exact fullContactBin_le G hloc d hd x v n m hn ω)
+
+end FullContactPower
+end Percolation
+
+ 
+namespace Percolation
+open MeasureTheory
+section NonnegativeDyadicIntegration
+
+ 
+
+theorem integral_le_tsum_majorants {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)
+    (f : Ω → ℝ) (F : ℕ → Ω → ℝ) (hfm : AEStronglyMeasurable f μ)
+    (hf0 : ∀ ω, 0 ≤ f ω) (hF0 : ∀ j ω, 0 ≤ F j ω) (hFi : ∀ j, Integrable (F j) μ)
+    (hs : Summable (fun j => ∫ ω, F j ω ∂μ))
+    (hb : ∀ ω, ENNReal.ofReal (f ω) ≤ ∑' j, ENNReal.ofReal (F j ω)) :
+    Integrable f μ ∧ (∫ ω, f ω ∂μ) ≤ ∑' j, ∫ ω, F j ω ∂μ := by
+  have he : (∫⁻ ω, ∑' j, ENNReal.ofReal (F j ω) ∂μ) =
+      ENNReal.ofReal (∑' j, ∫ ω, F j ω ∂μ) := by
+    rw [lintegral_tsum (fun j => (hFi j).aestronglyMeasurable.aemeasurable.ennreal_ofReal)]
+    simp_rw [←ofReal_integral_eq_lintegral_ofReal (hFi _) (Filter.Eventually.of_forall (hF0 _))]
+    exact (ENNReal.ofReal_tsum_of_nonneg (fun j => integral_nonneg (hF0 j)) hs).symm
+  have hle := (lintegral_mono hb).trans_eq he
+  have hi : Integrable f μ := ⟨hfm,(hasFiniteIntegral_iff_ofReal (Filter.Eventually.of_forall hf0)).mpr
+    (hle.trans_lt ENNReal.ofReal_lt_top)⟩
+  refine ⟨hi,?_⟩
+  rw [←ofReal_integral_eq_lintegral_ofReal hi (Filter.Eventually.of_forall hf0)] at hle
+  exact (ENNReal.ofReal_le_ofReal_iff (tsum_nonneg (fun j => integral_nonneg (hF0 j)))).mp hle
+
+theorem dyadic_rpow_identity (L : ℕ) (hL : 0 < L) (r : ℝ) (j : ℕ) :
+    ((2^j*L:ℕ):ℝ)^r = (L:ℝ)^r*((2:ℝ)^r)^j := by
+  rw [Nat.cast_mul,Nat.cast_pow,Nat.cast_ofNat,Real.mul_rpow (by positivity) (by positivity),
+    ←Real.rpow_natCast,←Real.rpow_mul (by norm_num),mul_comm (j:ℝ) r,Real.rpow_mul (by norm_num),Real.rpow_natCast,mul_comm]
+
+theorem dyadic_rpow_summable (L : ℕ) (hL : 0 < L) (r : ℝ) (hr : r < 0) :
+    Summable (fun j : ℕ => ((2^j*L:ℕ):ℝ)^r) := by
+  simp only [dyadic_rpow_identity L hL r]
+  exact (summable_geometric_of_lt_one (Real.rpow_nonneg (by norm_num) _)
+    (Real.rpow_lt_one_of_one_lt_of_neg (by norm_num) hr)).mul_left _
+
+theorem dyadic_rpow_sum (L : ℕ) (hL : 0 < L) (r : ℝ) (hr : r < 0) :
+    (∑' j : ℕ, ((2^j*L:ℕ):ℝ)^r) = (L:ℝ)^r*(1-(2:ℝ)^r)⁻¹ := by
+  simp only [dyadic_rpow_identity L hL r,tsum_mul_left]
+  rw [tsum_geometric_of_lt_one (Real.rpow_nonneg (by norm_num) _) (Real.rpow_lt_one_of_one_lt_of_neg (by norm_num) hr)]
+
+end NonnegativeDyadicIntegration
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section FullLargeContacts
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def fullLargeContact (G : BondGraph V E) (x v : V) (L : ℕ) (ω : Set E) : ℝ :=
+  if ¬Conn G ω x v ∧ L ≤ min (cluster G ω x).ncard (cluster G ω v).ncard
+  then clusterContactCount G ω x v else 0
+
+theorem fullLargeContact_nonneg (G : BondGraph V E) (x v : V) (L : ℕ) (ω : Set E) :
+    0 ≤ fullLargeContact G x v L ω := by unfold fullLargeContact; split_ifs <;> positivity
+
+theorem measurable_fullLargeContact [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (x v : V) (L : ℕ) :
+    Measurable (fullLargeContact G x v L) := by
+  have hs : Measurable (fun ω : Set E => min (cluster G ω x).ncard (cluster G ω v).ncard) :=
+    (measurable_cluster_ncard G hloc x).min (measurable_cluster_ncard G hloc v)
+  exact ((measurable_of_countable (fun k : ℕ => (k:ℝ))).comp (measurable_clusterContactCount G x v)).ite
+    (((measurableSet_conn G x v).compl).inter (measurableSet_le measurable_const hs)) measurable_const
+
+theorem fullLargeContact_le_dyadic (G : BondGraph V E) (x v : V) (L : ℕ) (hL : 0 < L) (ω : Set E) :
+    ENNReal.ofReal (fullLargeContact G x v L ω) ≤
+      ∑' j : ℕ, ENNReal.ofReal (fullContactBin G x v (2^j*L) (2*(2^j*L)) ω) := by
+  by_cases h : ¬Conn G ω x v ∧ L ≤ min (cluster G ω x).ncard (cluster G ω v).ncard
+  · let k := min (cluster G ω x).ncard (cluster G ω v).ncard
+    let j := Nat.log 2 (k/L)
+    have hq : k/L ≠ 0 := by
+      have : 1 ≤ k/L := (Nat.le_div_iff_mul_le hL).mpr (by rw [one_mul]; exact h.2)
+      omega
+    have hl := (Nat.le_div_iff_mul_le hL).mp (Nat.pow_log_le_self 2 hq)
+    have hu := (Nat.div_lt_iff_lt_mul hL).mp (Nat.lt_pow_succ_log_self (by norm_num : 1 < (2:ℕ)) (k/L))
+    have hbin : fullContactBin G x v (2^j*L) (2*(2^j*L)) ω = fullLargeContact G x v L ω := by
+      rw [fullLargeContact,ite_eq_left h,fullContactBin,ite_eq_left]
+      refine ⟨h.1,hl,?_⟩
+      calc
+        _ = k := rfl
+        _ ≤ 2^(Nat.log 2 (k/L)+1)*L := hu.le
+        _ = _ := by dsimp [j]; rw [pow_succ]; ring
+    exact (le_of_eq (congrArg ENNReal.ofReal hbin.symm)).trans
+      (ENNReal.le_tsum (f := fun scale =>
+        ENNReal.ofReal (fullContactBin G x v (2^scale*L) (2*(2^scale*L)) ω)) j)
+  · simp only [fullLargeContact,ite_eq_right h,ENNReal.ofReal_zero]; exact bot_le
+
+ 
+
+theorem fullLargeContact_bound_of_bins [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (d : ℕ)
+    (hd : ∀ z, {e | z ∈ G e}.ncard ≤ d) (x v : V) (p : unitInterval)
+    (A r : ℝ) (_hA : 0 ≤ A) (hr : r < 0)
+    (hB : ∀ n : ℕ, 1 ≤ n → (∫ ω, fullContactBin G x v n (2*n) ω ∂law p) ≤ A*(n:ℝ)^r)
+    (L : ℕ) (hL : 0 < L) :
+    Integrable (fullLargeContact G x v L) (law p) ∧
+      (∫ ω, fullLargeContact G x v L ω ∂law p) ≤ A*(1-(2:ℝ)^r)⁻¹*(L:ℝ)^r := by
+  let F := fun j => fullContactBin G x v (2^j*L) (2*(2^j*L))
+  have hbp (j : ℕ) : 0 < 2^j*L := Nat.mul_pos (pow_pos (by norm_num) _) hL
+  have hFi (j : ℕ) : Integrable (F j) (law p) := integrable_fullContactBin G hloc d hd x v _ _ (hbp j) p
+  have hsumB := (dyadic_rpow_summable L hL r hr).mul_left A
+  have hsum : Summable (fun j => ∫ ω, F j ω ∂law p) :=
+    hsumB.of_nonneg_of_le (fun j => integral_nonneg (fullContactBin_nonneg G x v _ _))
+      (fun j => hB _ (hbp j))
+  obtain ⟨hi,hb⟩ := integral_le_tsum_majorants (law p) (fullLargeContact G x v L) F
+    (measurable_fullLargeContact G hloc x v L).aestronglyMeasurable
+    (fullLargeContact_nonneg G x v L) (fun j => fullContactBin_nonneg G x v _ _) hFi hsum
+    (fullLargeContact_le_dyadic G x v L hL)
+  refine ⟨hi,hb.trans ?_⟩
+  calc
+    _ ≤ ∑' j : ℕ, A*((2^j*L:ℕ):ℝ)^r := hsum.tsum_le_tsum (fun j => hB _ (hbp j)) hsumB
+    _ = _ := by rw [tsum_mul_left,dyadic_rpow_sum L hL r hr]; ring
+
+end FullLargeContacts
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section UniformLargeContacts
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem incident_ncard_le_degree (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) :
+    {e | x ∈ G e}.ncard ≤ degree G x := by
+  have he : ((hloc x).toFinset.filter fun e => x ∈ G e) = (hloc x).toFinset := by
+    ext e
+    simp only [Finset.mem_filter,Set.Finite.mem_toFinset,Set.mem_ofPred_eq,and_self]
+  simpa only [he,←Set.ncard_eq_toFinset_card] using incident_filter_card_le_degree G hloc (hloc x).toFinset x
+
+theorem fullLargeContact_power_bound [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G) (q : unitInterval)
+    (α ε C p₀ : ℝ) (hα : 0 ≤ α) (hα1 : α ≤ 1) (hε : 0 < ε)
+    (hr : (1:ℝ)/2+ε-2*α < 0) (hC0 : 0 < C) (hp₀ : 0 < p₀)
+    (hfin : ∀ y, ∀ᵐ ω ∂law q, (cluster G ω y).Finite)
+    (hI : ∀ y, Integrable (fun ω => ((cluster G ω y).ncard:ℝ)^α) (law q))
+    (hC : ∀ y, (∫ ω, ((cluster G ω y).ncard:ℝ)^α ∂law q) ≤ C) :
+    ∃ A : ℝ, 0 < A ∧ ∀ (x v : V) (p : unitInterval), p₀ ≤ (p:ℝ) → p ≤ q →
+      ∀ L : ℕ, 1 ≤ L → Integrable (fullLargeContact G x v L) (law p) ∧
+        (∫ ω, fullLargeContact G x v L ω ∂law p) ≤ A*(L:ℝ)^((1:ℝ)/2+ε-2*α) := by
+  obtain ⟨d,hd,hdeg⟩ := exists_degree_bound G hqt
+  have hdn (x : V) := (incident_ncard_le_degree G hloc x).trans (hdeg x)
+  obtain ⟨A,hA,hAb⟩ := fullContactBin_power_bound G hloc q α ε C p₀ hα hα1 hε hr hC0 hp₀ hfin hI hC d hd hdn
+  refine ⟨A*(1-(2:ℝ)^((1:ℝ)/2+ε-2*α))⁻¹,?_ ,?_⟩
+  · exact mul_pos hA (inv_pos.mpr (sub_pos.mpr (Real.rpow_lt_one_of_one_lt_of_neg (by norm_num) hr)))
+  · intro x v p hpp hpq L hL
+    exact fullLargeContact_bound_of_bins G hloc d hdn x v p A _ hA.le hr (hAb x v p hpp hpq) L hL
+
+ 
+
+theorem critical_large_contact_bound
+    [Countable V] [Countable E] [Nonempty V] [Nonempty E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hconn : Connected G)
+    (hqt : QuasiTransitive G) (hh : 0 < hV G)
+    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
+    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
+    (hp₀ : 0 < (pc G:ℝ)) (hp₁ : (pc G:ℝ) < 1)
+    (hfin : ∀ i, ∀ᵐ ω ∂law (pc G), (cluster G ω (root i)).Finite)
+    (α ε : ℝ) (hα : 0 < α) (hα1 : α < 1/2) (hε : 0 < ε)
+    (hr : (1:ℝ)/2+ε-2*α < 0) :
+    ∃ A : ℝ, 0 < A ∧ ∀ (x v : V) (p : unitInterval), (pc G:ℝ)/2 ≤ (p:ℝ) → p ≤ pc G →
+      ∀ L : ℕ, 1 ≤ L → Integrable (fullLargeContact G x v L) (law p) ∧
+        (∫ ω, fullLargeContact G x v L ω ∂law p) ≤ A*(L:ℝ)^((1:ℝ)/2+ε-2*α) := by
+  have hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i) := by
+    intro x
+    obtain ⟨i,a,ha,hax⟩ := hcov x
+    obtain ⟨b,hab⟩ := hΓ ha
+    exact ⟨i,a,b,hab,hax⟩
+  have hfinite : ∀ x, ∀ᵐ ω ∂law (pc G), (cluster G ω x).Finite := by
+    intro x
+    obtain ⟨i,a,b,hab,rfl⟩ := hcover x
+    exact cluster_finite_ae_covariant G (pc G) hab (root i) (hfin i)
+  obtain ⟨C,hC,hb⟩ := uniform_cluster_power_of_root G hloc (pc G) root hcover α
+    (critical_root_power_integrable G hloc hconn hqt hh root weight Γ hΓ htr hcov hw hw1 hp₀ hp₁ hfin α hα hα1)
+  exact fullLargeContact_power_bound G hloc hqt (pc G) α ε C ((pc G:ℝ)/2) hα.le (by linarith)
+    hε hr hC (by positivity) hfinite (fun x => (hb x).1) (fun x => (hb x).2)
+
+end UniformLargeContacts
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+open scoped ENNReal
+section ExtendedTransport
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem enn_sum_iSup_nat {α : Type*} (s : Finset α) (f : ℕ → α → ℝ≥0∞)
+    (hm : ∀ a, Monotone (fun n => f n a)) :
+    (∑ a ∈ s, ⨆ n, f n a) = ⨆ n, ∑ a ∈ s, f n a := by
+  induction s using Finset.induction_on with
+  | empty => simp
+  | @insert a s ha ih =>
+    simp only [Finset.sum_insert ha,ih]
+    exact ENNReal.iSup_add_iSup_of_monotone (hm a) (fun n m h =>
+      Finset.sum_le_sum fun a _ => hm a h)
+
+theorem enn_tsum_iSup_nat {α : Type*} (f : ℕ → α → ℝ≥0∞)
+    (hm : ∀ a, Monotone (fun n => f n a)) :
+    (∑' a, ⨆ n, f n a) = ⨆ n, ∑' a, f n a := by
+  simp_rw [ENNReal.tsum_eq_iSup_sum,enn_sum_iSup_nat _ f hm]
+  exact iSup_comm
+
+ 
+
+theorem RootTransport.ennreal (G : BondGraph V E) (hconn : Connected G)
+    (hloc : LocallyFinite G) {ι : Type*} [Fintype ι]
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hw : ∀ i, 0 ≤ weight i) (m : V → V → ℝ≥0∞)
+    (hi : ∀ a ∈ Γ, ∀ x y, m (a x) (a y) = m x y) :
+    (∑ i, ENNReal.ofReal (weight i) * ∑' y, m (root i) y) =
+      (∑ i, ENNReal.ofReal (weight i) * ∑' x, m x (root i)) := by
+  let cut (n : ℕ) (x y : V) : ℝ≥0∞ :=
+    if transportCut G n x y then min (m x y) n else 0
+  have hc0 (n : ℕ) (x y : V) : cut n x y ≠ ⊤ := by
+    dsimp [cut]
+    split_ifs
+    · exact ne_top_of_le_ne_top (ENNReal.natCast_ne_top n) (min_le_right _ _)
+    · exact ENNReal.zero_ne_top
+  have hcm (x y : V) : Monotone (fun n => cut n x y) := by
+    intro n k hnk
+    dsimp [cut]
+    split_ifs with hn hk
+    · exact min_le_min_left _ (by exact_mod_cast hnk)
+    · exact False.elim (hk (transportCut_mono G hnk hn))
+    · exact zero_le
+    · exact le_refl _
+  have hcs (x y : V) : (⨆ n, cut n x y) = m x y := by
+    apply le_antisymm
+    · refine iSup_le fun n => ?_
+      dsimp [cut]
+      split_ifs
+      · exact min_le_left _ _
+      · exact zero_le
+    · obtain ⟨N,hN⟩ := exists_transportCut G hconn y {x}
+      have hxy := hN x (Finset.mem_singleton_self x)
+      have he : (⨆ k : ℕ, min (m x y) (k:ℝ≥0∞)) = m x y := by
+        rw [←inf_iSup_eq,ENNReal.iSup_natCast,inf_top_eq]
+      rw [←he]
+      refine iSup_le fun k => le_iSup_of_le (max N k) ?_
+      dsimp [cut]
+      rw [ite_eq_left (transportCut_mono G (le_max_left N k) hxy)]
+      exact min_le_min_left _ (by exact_mod_cast le_max_right N k)
+  have hrow (n : ℕ) (x : V) : Summable (fun y => (cut n x y).toReal) := by
+    apply summable_of_ne_finset_zero (s := (finite_reachBall hloc x n).toFinset)
+    intro y hy
+    have hh : ¬transportCut G n x y := fun h => hy ((Set.Finite.mem_toFinset _).mpr h.1)
+    simp only [cut,ite_eq_right hh,ENNReal.toReal_zero]
+  have hcol (n : ℕ) (y : V) : Summable (fun x => (cut n x y).toReal) := by
+    apply summable_of_ne_finset_zero (s := (finite_reachBall hloc y n).toFinset)
+    intro x hx
+    have hh : ¬transportCut G n x y := fun h => hx ((Set.Finite.mem_toFinset _).mpr h.2)
+    simp only [cut,ite_eq_right hh,ENNReal.toReal_zero]
+  have he (n : ℕ) :
+      (∑ i, ENNReal.ofReal (weight i) * ∑' y, cut n (root i) y) =
+        (∑ i, ENNReal.ofReal (weight i) * ∑' x, cut n x (root i)) := by
+    have ht := htr.real root weight Γ (fun x y => (cut n x y).toReal)
+      (fun a ha x y => by
+        obtain ⟨b,hb⟩ := hΓ ha
+        simp only [cut,transportCut_aut G hb,hi a ha]) (hrow n) (hcol n)
+    apply_fun ENNReal.ofReal at ht
+    rw [ENNReal.ofReal_sum_of_nonneg (fun i _ => mul_nonneg (hw i)
+      (tsum_nonneg fun _ => ENNReal.toReal_nonneg)),
+      ENNReal.ofReal_sum_of_nonneg (fun i _ => mul_nonneg (hw i)
+      (tsum_nonneg fun _ => ENNReal.toReal_nonneg))] at ht
+    simp_rw [ENNReal.ofReal_mul (hw _),
+      ENNReal.ofReal_tsum_of_nonneg (fun _ => ENNReal.toReal_nonneg) (hrow n _),
+      ENNReal.ofReal_tsum_of_nonneg (fun _ => ENNReal.toReal_nonneg) (hcol n _),
+      ENNReal.ofReal_toReal (hc0 n _ _)] at ht
+    exact ht
+  have hlim (f : ι → V → ℝ≥0∞) (g : ℕ → ι → V → ℝ≥0∞)
+      (hmono : ∀ i y, Monotone (fun n => g n i y)) (hs : ∀ i y, (⨆ n, g n i y) = f i y) :
+      (∑ i, ENNReal.ofReal (weight i) * ∑' y, f i y) =
+        ⨆ n, ∑ i, ENNReal.ofReal (weight i) * ∑' y, g n i y := by
+    simp_rw [←hs,enn_tsum_iSup_nat _ (hmono _),ENNReal.mul_iSup]
+    exact enn_sum_iSup_nat _ _ (fun i n k h => mul_le_mul_right (ENNReal.tsum_le_tsum
+      (fun y => hmono i y h)) _)
+  rw [hlim (fun i y => m (root i) y) (fun n i y => cut n (root i) y)
+    (fun i y => hcm (root i) y) (fun i y => hcs (root i) y),
+    hlim (fun i x => m x (root i)) (fun n i x => cut n x (root i))
+    (fun i x => hcm x (root i)) (fun i x => hcs x (root i))]
+  exact iSup_congr he
+
+end ExtendedTransport
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+open scoped ENNReal
+section ClusterAveraging
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def clusterAveragingKernel (G : BondGraph V E) (ω : Set E) (x y : V) : ℝ≥0∞ :=
+  if Conn G ω x y then ((cluster G ω x).ncard:ℝ≥0∞)⁻¹ else 0
+
+theorem measurable_clusterAveragingKernel [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (x y : V) :
+    Measurable (fun ω => clusterAveragingKernel G ω x y) := by
+  exact (((measurable_of_countable (fun n : ℕ => (n:ℝ≥0∞))).comp
+    (measurable_cluster_ncard G hloc x)).inv).ite (measurableSet_conn G x y) measurable_const
+
+theorem clusterAveragingKernel_symm (G : BondGraph V E) (ω : Set E) (x y : V) :
+    clusterAveragingKernel G ω x y = clusterAveragingKernel G ω y x := by
+  by_cases h : Conn G ω x y
+  · simp only [clusterAveragingKernel,ite_eq_left h,ite_eq_left (conn_symm G ω h),cluster_eq_of_conn G ω h]
+  · have h' : ¬Conn G ω y x := fun hh => h (conn_symm G ω hh)
+    simp only [clusterAveragingKernel,ite_eq_right h,ite_eq_right h']
+
+theorem cluster_ncard_pos (G : BondGraph V E) (ω : Set E) (x : V)
+    (hf : (cluster G ω x).Finite) : 0 < (cluster G ω x).ncard :=
+  (Set.ncard_pos hf).mpr ⟨x,mem_cluster_self G ω x⟩
+
+theorem clusterAveragingKernel_row (G : BondGraph V E) (ω : Set E) (x : V)
+    (hf : (cluster G ω x).Finite) : (∑' y, clusterAveragingKernel G ω x y) = 1 := by
+  have he : (fun y => clusterAveragingKernel G ω x y) =
+      (fun y => (if Conn G ω x y then (1:ℝ≥0∞) else 0)*((cluster G ω x).ncard:ℝ≥0∞)⁻¹) := by
+    funext y
+    unfold clusterAveragingKernel
+    split_ifs <;> simp
+  rw [he,ENNReal.tsum_mul_right]
+  change clusterMass G ω x * ((cluster G ω x).ncard:ℝ≥0∞)⁻¹ = 1
+  rw [clusterMass_eq_ncard G ω x hf]
+  exact ENNReal.mul_inv_cancel (by exact_mod_cast (cluster_ncard_pos G ω x hf).ne') (ENNReal.natCast_ne_top _)
+
+theorem clusterAveragingKernel_column (G : BondGraph V E) (ω : Set E) (y : V)
+    (hf : (cluster G ω y).Finite) : (∑' x, clusterAveragingKernel G ω x y) = 1 := by
+  simp_rw [clusterAveragingKernel_symm G ω]
+  exact clusterAveragingKernel_row G ω y hf
+
+theorem clusterAveragingKernel_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (ω : Set E) (x y : V) :
+    clusterAveragingKernel G (b ⁻¹' ω) x y = clusterAveragingKernel G ω (a x) (a y) := by
+  simp only [clusterAveragingKernel,conn_aut_iff G hab,cluster_ncard_covariant G hab]
+
+ 
+
+theorem root_cluster_average [Countable V] [Countable E]
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hw : ∀ i, 0 ≤ weight i) (p : unitInterval)
+    (hfin : ∀ x, ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
+    (f : Set E → V → ℝ≥0∞) (hf : ∀ x, Measurable (fun ω => f ω x))
+    (hi : ∀ a ∈ Γ, ∀ b, IsAut G a b → ∀ ω x, f (b ⁻¹' ω) x = f ω (a x)) :
+    (∑ i, ENNReal.ofReal (weight i) * ∫⁻ ω, ∑' y,
+      clusterAveragingKernel G ω (root i) y * f ω y ∂law p) =
+    (∑ i, ENNReal.ofReal (weight i) * ∫⁻ ω, f ω (root i) ∂law p) := by
+  let m (x y : V) : ℝ≥0∞ := ∫⁻ ω, clusterAveragingKernel G ω x y * f ω y ∂law p
+  have hm (x y : V) : Measurable (fun ω => clusterAveragingKernel G ω x y * f ω y) :=
+    (measurable_clusterAveragingKernel G hloc x y).mul (hf y)
+  have hinv : ∀ a ∈ Γ, ∀ x y, m (a x) (a y) = m x y := by
+    intro a ha x y
+    obtain ⟨b,hb⟩ := hΓ ha
+    have ht := lintegral_map (μ := law p) (hm x y) (measurable_set_preimage b)
+    rw [law_map_preimage_equiv] at ht
+    dsimp only [m]
+    rw [ht]
+    apply lintegral_congr
+    intro ω
+    rw [clusterAveragingKernel_aut G hb,hi a ha b hb]
+  have ht := htr.ennreal G hconn hloc root weight Γ hΓ hw m hinv
+  have hr (x : V) : (∑' y, m x y) = ∫⁻ ω, ∑' y, clusterAveragingKernel G ω x y*f ω y ∂law p := by
+    exact (lintegral_tsum (fun y => (hm x y).aemeasurable)).symm
+  have hc (y : V) : (∑' x, m x y) = ∫⁻ ω, f ω y ∂law p := by
+    rw [show (∑' x, m x y) = ∫⁻ ω, ∑' x, clusterAveragingKernel G ω x y*f ω y ∂law p from
+      (lintegral_tsum (fun x => (hm x y).aemeasurable)).symm]
+    apply lintegral_congr_ae
+    filter_upwards [hfin y] with ω hω
+    rw [ENNReal.tsum_mul_right,clusterAveragingKernel_column G ω y hω,one_mul]
+  simpa only [hr,hc] using ht
+
+end ClusterAveraging
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+open scoped ENNReal
+section PositiveClusterPairs
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def clusterPairKernel (G : BondGraph V E) (ω : Set E)
+    (J : V → V → ℝ≥0∞) (x y : V) : ℝ≥0∞ :=
+  ∑' a, ∑' b, if Conn G ω x a ∧ Conn G ω y b then J a b else 0
+
+ 
+
+noncomputable def clusterPairDensity (G : BondGraph V E) (ω : Set E)
+    (F : V → V → ℝ≥0∞) (x : V) : ℝ≥0∞ :=
+  ((cluster G ω x).ncard:ℝ≥0∞)⁻¹ *
+    ∑' y, ((cluster G ω y).ncard:ℝ≥0∞)⁻¹ * (if ¬Conn G ω x y then F x y else 0)
+
+noncomputable def clusterPairFunctional (G : BondGraph V E) (p : unitInterval)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (F : Set E → V → V → ℝ≥0∞) : ℝ≥0∞ :=
+  ∑ i, ENNReal.ofReal (weight i) * ∫⁻ ω, clusterPairDensity G ω (F ω) (root i) ∂law p
+
+theorem cluster_weighted_resum (G : BondGraph V E) (ω : Set E)
+    (hfin : ∀ x, (cluster G ω x).Finite) (f : V → V → ℝ≥0∞)
+    (hc : ∀ x y, Conn G ω x y → f x y = f y y) :
+    (∑' x, ((cluster G ω x).ncard:ℝ≥0∞)⁻¹ *
+      ∑' y, if Conn G ω x y then f x y else 0) = ∑' y, f y y := by
+  have he (x y : V) : ((cluster G ω x).ncard:ℝ≥0∞)⁻¹ *
+      (if Conn G ω x y then f x y else 0) = clusterAveragingKernel G ω x y*f y y := by
+    by_cases h : Conn G ω x y
+    · simp only [ite_eq_left h,clusterAveragingKernel,hc x y h]
+    · simp only [ite_eq_right h,clusterAveragingKernel,mul_zero,zero_mul]
+  simp_rw [←ENNReal.tsum_mul_left,he]
+  rw [ENNReal.tsum_comm]
+  apply tsum_congr
+  intro y
+  rw [ENNReal.tsum_mul_right,clusterAveragingKernel_column G ω y (hfin y),one_mul]
+
+theorem clusterPairDensity_reroot_pointwise (G : BondGraph V E) (ω : Set E)
+    (hfin : ∀ x, (cluster G ω x).Finite) (J F : V → V → ℝ≥0∞)
+    (hF : ∀ x y a b, Conn G ω x a → Conn G ω y b → F x y = F a b) (x : V) :
+    clusterPairDensity G ω (fun a b => clusterPairKernel G ω J a b * F a b) x =
+      ∑' a, clusterAveragingKernel G ω x a *
+        ∑' b, J a b * (if ¬Conn G ω a b then F a b else 0) := by
+  let K (a b : V) : ℝ≥0∞ := if ¬Conn G ω a b then F a b else 0
+  have hK (a b c d : V) (hac : Conn G ω a c) (hbd : Conn G ω b d) : K a b = K c d := by
+    have he : Conn G ω a b ↔ Conn G ω c d := by
+      constructor
+      · intro h; exact conn_trans G ω (conn_symm G ω hac) (conn_trans G ω h hbd)
+      · intro h; exact conn_trans G ω hac (conn_trans G ω h (conn_symm G ω hbd))
+    simp only [K,he,hF a b c d hac hbd]
+  have hterm (y : V) : ((cluster G ω y).ncard:ℝ≥0∞)⁻¹ *
+      (if ¬Conn G ω x y then clusterPairKernel G ω J x y*F x y else 0) =
+      ∑' a, if Conn G ω x a then ((cluster G ω y).ncard:ℝ≥0∞)⁻¹ *
+        ∑' b, if Conn G ω y b then J a b*K x y else 0 else 0 := by
+    have hcond : (if ¬Conn G ω x y then clusterPairKernel G ω J x y*F x y else 0) =
+        clusterPairKernel G ω J x y*K x y := by dsimp [K]; split_ifs <;> simp
+    rw [hcond,clusterPairKernel,←ENNReal.tsum_mul_right,←ENNReal.tsum_mul_left]
+    apply tsum_congr
+    intro a
+    by_cases hxa : Conn G ω x a
+    · rw [ite_eq_left hxa,←ENNReal.tsum_mul_right]
+      congr 1
+      apply tsum_congr
+      intro b
+      by_cases hyb : Conn G ω y b <;> simp only [hxa,hyb,and_self,and_false,
+        ite_true,ite_false,zero_mul]
+    · simp only [hxa,false_and,ite_false,tsum_zero,zero_mul,mul_zero]
+  unfold clusterPairDensity
+  simp_rw [hterm]
+  rw [ENNReal.tsum_comm,←ENNReal.tsum_mul_left]
+  apply tsum_congr
+  intro a
+  by_cases hxa : Conn G ω x a
+  · simp only [ite_eq_left hxa,clusterAveragingKernel]
+    congr 1
+    rw [cluster_weighted_resum G ω hfin (fun y b => J a b*K x y)
+      (fun y b hyb => congrArg (fun t => J a b*t) (hK x y x b (conn_refl G ω x) hyb))]
+    apply tsum_congr
+    intro b
+    rw [hK x b a b hxa (conn_refl G ω b)]
+  · simp only [ite_eq_right hxa,tsum_zero,mul_zero,clusterAveragingKernel,zero_mul]
+
+end PositiveClusterPairs
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+open scoped ENNReal
+section ClusterPairRerooting
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem clusterPairFunctional_reroot [Countable V] [Countable E]
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hw : ∀ i, 0 ≤ weight i) (p : unitInterval)
+    (hfin : ∀ x, ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
+    (J : V → V → ℝ≥0∞) (hJ : ∀ a ∈ Γ, ∀ x y, J (a x) (a y) = J x y)
+    (F : Set E → V → V → ℝ≥0∞) (hm : ∀ x y, Measurable (fun ω => F ω x y))
+    (hF : ∀ ω x y a b, Conn G ω x a → Conn G ω y b → F ω x y = F ω a b)
+    (hi : ∀ a ∈ Γ, ∀ b, IsAut G a b → ∀ ω x y, F (b ⁻¹' ω) x y = F ω (a x) (a y)) :
+    clusterPairFunctional G p root weight (fun ω x y => clusterPairKernel G ω J x y * F ω x y) =
+      ∑ i, ENNReal.ofReal (weight i) * ∫⁻ ω, ∑' y,
+        J (root i) y * (if ¬Conn G ω (root i) y then F ω (root i) y else 0) ∂law p := by
+  let f (ω : Set E) (x : V) : ℝ≥0∞ :=
+    ∑' y, J x y * (if ¬Conn G ω x y then F ω x y else 0)
+  have hf (x : V) : Measurable (fun ω => f ω x) := by
+    apply Measurable.tsum
+    intro y
+    exact measurable_const.mul ((hm x y).ite (measurableSet_conn G x y).compl measurable_const)
+  have hfe : ∀ a ∈ Γ, ∀ b, IsAut G a b → ∀ ω x, f (b ⁻¹' ω) x = f ω (a x) := by
+    intro a ha b hab ω x
+    dsimp [f]
+    rw [←a.tsum_eq (fun y => J (a x) y * (if ¬Conn G ω (a x) y then F ω (a x) y else 0))]
+    apply tsum_congr
+    intro y
+    rw [hJ a ha,conn_aut_iff G hab,hi a ha b hab]
+  have ht := root_cluster_average G hconn hloc root weight Γ hΓ htr hw p hfin f hf hfe
+  rw [clusterPairFunctional]
+  calc
+    _ = ∑ i, ENNReal.ofReal (weight i) * ∫⁻ ω, ∑' y,
+        clusterAveragingKernel G ω (root i) y*f ω y ∂law p := by
+      apply Finset.sum_congr rfl
+      intro i _
+      congr 1
+      apply lintegral_congr_ae
+      filter_upwards [ae_all_iff.mpr hfin] with ω hω
+      exact clusterPairDensity_reroot_pointwise G ω hω J (F ω) (hF ω) (root i)
+    _ = _ := ht
+
+end ClusterPairRerooting
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+open scoped ENNReal
+section PairReversal
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def clusterPairMass (G : BondGraph V E) (ω : Set E)
+    (F : V → V → ℝ≥0∞) (x y : V) : ℝ≥0∞ :=
+  ((cluster G ω x).ncard:ℝ≥0∞)⁻¹ * ((cluster G ω y).ncard:ℝ≥0∞)⁻¹ *
+    (if ¬Conn G ω x y then F x y else 0)
+
+theorem clusterPairDensity_eq_tsum (G : BondGraph V E) (ω : Set E)
+    (F : V → V → ℝ≥0∞) (x : V) :
+    clusterPairDensity G ω F x = ∑' y, clusterPairMass G ω F x y := by
+  simp only [clusterPairDensity,clusterPairMass,←ENNReal.tsum_mul_left,mul_assoc]
+
+theorem clusterPairMass_swap (G : BondGraph V E) (ω : Set E)
+    (F : V → V → ℝ≥0∞) (x y : V) :
+    clusterPairMass G ω F x y = clusterPairMass G ω (fun a b => F b a) y x := by
+  have he : Conn G ω x y ↔ Conn G ω y x := ⟨conn_symm G ω,conn_symm G ω⟩
+  simp only [clusterPairMass,he,mul_comm]
+
+theorem measurable_clusterPairMass [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G)
+    (F : Set E → V → V → ℝ≥0∞) (hf : ∀ x y, Measurable (fun ω => F ω x y)) (x y : V) :
+    Measurable (fun ω => clusterPairMass G ω (F ω) x y) := by
+  have hs (x : V) : Measurable (fun ω => ((cluster G ω x).ncard:ℝ≥0∞)⁻¹) :=
+    (((measurable_of_countable (fun n : ℕ => (n:ℝ≥0∞))).comp (measurable_cluster_ncard G hloc x))).inv
+  exact ((hs x).mul (hs y)).mul ((hf x y).ite (measurableSet_conn G x y).compl measurable_const)
+
+theorem clusterPairFunctional_swap [Countable V] [Countable E]
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hw : ∀ i, 0 ≤ weight i) (p : unitInterval)
+    (F : Set E → V → V → ℝ≥0∞) (hf : ∀ x y, Measurable (fun ω => F ω x y))
+    (hi : ∀ a ∈ Γ, ∀ b, IsAut G a b → ∀ ω x y, F (b ⁻¹' ω) x y = F ω (a x) (a y)) :
+    clusterPairFunctional G p root weight F =
+      clusterPairFunctional G p root weight (fun ω x y => F ω y x) := by
+  let m (x y : V) : ℝ≥0∞ := ∫⁻ ω, clusterPairMass G ω (F ω) x y ∂law p
+  have hm := measurable_clusterPairMass G hloc F hf
+  have hinv : ∀ a ∈ Γ, ∀ x y, m (a x) (a y) = m x y := by
+    intro a ha x y
+    obtain ⟨b,hb⟩ := hΓ ha
+    have ht := lintegral_map (μ := law p) (hm x y) (measurable_set_preimage b)
+    rw [law_map_preimage_equiv] at ht
+    dsimp only [m]
+    rw [ht]
+    apply lintegral_congr
+    intro ω
+    simp only [clusterPairMass,cluster_ncard_covariant G hb,conn_aut_iff G hb,hi a ha b hb]
+  have ht := htr.ennreal G hconn hloc root weight Γ hΓ hw m hinv
+  have hr (x : V) : (∑' y, m x y) = ∫⁻ ω, clusterPairDensity G ω (F ω) x ∂law p := by
+    simp_rw [clusterPairDensity_eq_tsum]
+    exact (lintegral_tsum (fun y => (hm x y).aemeasurable)).symm
+  have hc (y : V) : (∑' x, m x y) =
+      ∫⁻ ω, clusterPairDensity G ω (fun a b => F ω b a) y ∂law p := by
+    simp_rw [clusterPairDensity_eq_tsum,←clusterPairMass_swap G _ (F _)]
+    exact (lintegral_tsum (fun x => (hm x y).aemeasurable)).symm
+  simpa only [hr,hc,clusterPairFunctional] using ht
+
+end PairReversal
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Clusters/Resampling.lean b/lean/OAI/Probability/UnimodularPercolation/Clusters/Resampling.lean
index cb4e21f0e335a69195c85b3900fe22f9bd88cb96..22ededeb09b11a79bb7c7e7d6e138bb668aa0bd9 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Clusters/Resampling.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Clusters/Resampling.lean
@@ -1,1227 +1,1227 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Kernel.ScaleBudget
-
-namespace OAI
-
-/-! Resampling for bond percolation. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open Filter
-section ContractingIteration
-
- 
-
-theorem exists_efficient_of_contracting_successor {S : Type*}
-    (g H : S → ℝ) (C : ℝ) (hC : 0 ≤ C) (hC1 : C < 1)
-    (hstep : ∀ x, ∃ y, g y ≤ C*g x ∧ 2*H x ≤ H y)
-    (x₀ : S) (hH₀ : 0 < H x₀) (η R : ℝ) (hη : 0 < η) :
-    ∃ x, g x < η ∧ R < H x := by
-  classical
-  choose f hf using hstep
-  let x : ℕ → S := fun n => f^[n] x₀
-  have hx (n : ℕ) : x (n+1) = f (x n) := Function.iterate_succ_apply' f n x₀
-  have hbounds (n : ℕ) : g (x n) ≤ C^n*g x₀ ∧ (2:ℝ)^n*H x₀ ≤ H (x n) := by
-    induction n with
-    | zero => simp only [x,Function.iterate_zero,id_eq,pow_zero,one_mul,le_refl,and_self]
-    | succ n ih =>
-      rw [hx]
-      obtain ⟨hg,hH⟩ := hf (x n)
-      constructor
-      · calc
-          _ ≤ C*g (x n) := hg
-          _ ≤ C*(C^n*g x₀) := mul_le_mul_of_nonneg_left ih.1 hC
-          _ = _ := by rw [pow_succ]; ring
-      · calc
-          _ = 2*((2:ℝ)^n*H x₀) := by rw [pow_succ]; ring
-          _ ≤ 2*H (x n) := mul_le_mul_of_nonneg_left ih.2 (by norm_num)
-          _ ≤ _ := hH
-  have hg : Tendsto (fun n : ℕ => C^n*g x₀) atTop (nhds 0) := by
-    simpa only [zero_mul] using (tendsto_pow_atTop_nhds_zero_of_lt_one hC hC1).mul_const (g x₀)
-  have hH : Tendsto (fun n : ℕ => (2:ℝ)^n*H x₀) atTop atTop :=
-    (tendsto_pow_atTop_atTop_of_one_lt (by norm_num : (1:ℝ) < 2)).atTop_mul_const hH₀
-  obtain ⟨n,hg',hH'⟩ := ((tendsto_order.mp hg).2 η hη |>.and
-    (hH.eventually (eventually_gt_atTop R))).exists
-  exact ⟨x n,(hbounds n).1.trans_lt hg',hH'.trans_le (hbounds n).2⟩
-
-end ContractingIteration
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section EfficientEndpointProfiles
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem efficient_endpoint_profiles_of_divergent_moment
-    [Countable V] [Countable E] [Nonempty V] [Nonempty E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hconn : Connected G)
-    (hqt : QuasiTransitive G) (hh : 0 < hV G)
-    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
-    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
-    (p : unitInterval) (hp : 0 < (p:ℝ)) (hp1 : (p:ℝ) < 1)
-    (hfin : ∀ i, ∀ᵐ ω ∂law p, (cluster G ω (root i)).Finite)
-    (α β ε : ℝ) (hα : 0 < α) (hαβ : α < β)
-    (hε : 0 < ε) (hε1 : ε < 1/2) (hsmall : β+(1/2+ε) < 1)
-    (hdiv : ∃ i, ¬ Integrable (fun ω => ((cluster G ω (root i)).ncard:ℝ)^α) (law p)) :
-    ∃ i, ∀ η : ℝ, 0 < η → ∀ R : ℝ,
-      ∃ q : EndpointProfile G hloc p (root i) (1/2+ε),
-        2 ≤ q.cap ∧ 1 < q.H ∧ R < q.H ∧ q.U+q.m < η*q.H := by
-  have hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i) := by
-    intro x
-    obtain ⟨i,e,he,hx⟩ := hcov x
-    obtain ⟨b,hb⟩ := hΓ he
-    exact ⟨i,e,b,hb,hx⟩
-  obtain ⟨i,hscales⟩ := orbit_scales_of_divergent_moment G hloc root weight Γ hΓ htr hw hw1 hcover p α β hα hαβ hdiv
-  obtain ⟨ξ,hξ,hξ1,hC,hC1⟩ := exists_amplification_contraction β (1/2+ε) (by linarith) (by linarith) hsmall
-  have hsξ : 0 < 1/2+ε+ξ := by linarith
-  obtain ⟨M₀,hM₀⟩ := eventually_atTop.1 (eventually_inverseCodeMass_budget (1/2+ε) ξ (by linarith) hξ)
-  obtain ⟨q₀,hq₀,hH₀⟩ := starting_endpoint_profile G hloc hqt hh p hp hp1 root weight Γ hΓ htr
-    (fun i => (hw i).le) hw1 i (hw i) (hfin i) ε hε hε1 M₀
-  obtain ⟨d,_,hdeg⟩ := exists_degree_bound G hqt
-  let c : ℝ := ∑ j, 1/weight j
-  have hc : 0 ≤ c := Finset.sum_nonneg (fun j _ => (one_div_pos.mpr (hw j)).le)
-  have hmin (j : ι) : 1 ≤ c*weight j := by
-    apply (div_le_iff₀ (hw j)).mp
-    exact Finset.single_le_sum (fun l _ => (one_div_pos.mpr (hw l)).le) (Finset.mem_univ j)
-  let S := {q : EndpointProfile G hloc p (root i) (1/2+ε) // M₀ ≤ q.cap ∧ 1 < q.H}
-  let g : S → ℝ := fun q => q.val.cost ξ/q.val.H
-  let H : S → ℝ := fun q => q.val.H
-  have hstep : ∀ q : S, ∃ q' : S, g q' ≤ amplificationContraction β (1/2+ε) ξ*g q ∧ 2*H q ≤ H q' := by
-    intro q
-    have hcap := hM₀ q.val.cap q.property.1
-    have hH : 0 < q.val.H := lt_trans zero_lt_one q.property.2
-    obtain ⟨r,k,hr,hk,hU,hm,hH',hH''⟩ := q.val.amplify_at_scales hconn hqt root weight Γ hΓ htr hcov hw hw1
-      c hc hmin i p hp1 d hdeg ε ξ β hε hε1 hξ hξ1 hscales hcap.1 hcap.2.1 hcap.2.2 hH M₀
-    have hrH : 1 < r.H := by linarith [q.property.2]
-    refine ⟨⟨r,hr,hrH⟩,?_,hH''⟩
-    exact q.val.cost_contraction r β ξ k hcap.1 hsξ hH hξ1 hk hC hU hm hH'
-  refine ⟨i,?_⟩
-  intro η hη R
-  let D : ℝ := 1+1/(1/2+ε+ξ)
-  have hD : 0 < D := by dsimp [D]; positivity
-  obtain ⟨q,hg,hR⟩ := exists_efficient_of_contracting_successor g H (amplificationContraction β (1/2+ε) ξ)
-    hC hC1 hstep ⟨q₀,hq₀,hH₀⟩ (lt_trans zero_lt_one hH₀) (η/D) R (div_pos hη hD)
-  have hcap := (hM₀ q.val.cap q.property.1).1
-  refine ⟨q.val,hcap,q.property.2,hR,?_⟩
-  have hH : 0 < q.val.H := lt_trans zero_lt_one q.property.2
-  have hcost := q.val.total_cost_le hcap ξ hsξ
-  have hratio : (q.val.U+q.val.m)/q.val.H ≤ D*g q := by
-    have h := div_le_div_of_nonneg_right hcost hH.le
-    simpa only [D,g,mul_div_assoc] using h
-  apply (div_lt_iff₀ hH).mp
-  apply hratio.trans_lt
-  have hg' := (lt_div_iff₀ hD).mp hg
-  simpa only [mul_comm] using hg'
-
-end EfficientEndpointProfiles
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section ShortBoundaryPivotal
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-def ShortClosedBoundary (G : BondGraph V E) (ω : Set E) (x z : V) (M : ℕ) (e : E) : Prop :=
-  e ∉ ω ∧ ¬Conn G ω x z ∧ ∃ a b : V, ∃ l : List E,
-    G e = s(a,b) ∧ Along G x l a ∧ (∀ f ∈ l, f ∈ ω) ∧ l.length < M ∧ Conn G ω b z
-
-theorem bridgeIndex_le_short_trail (G : BondGraph V E) (ω : Set E) (x z : V)
-    (hC : (cluster G ω x).Finite) (e : E) (he : OpenMarkedPivotal G ω {z} x e)
-    (a b : V) (hab : G e = s(a,b)) (l : List E) (hl : Along G x l a)
-    (ho : ∀ f ∈ l, f ∈ ω) : bridgeIndex G ω x z e ≤ l.length+1 := by
-  unfold bridgeIndex
-  have hsub : (separatingBonds G ω x z).filter (fun f => pivotalRank G ω x f ≤ pivotalRank G ω x e) ⊆
-      (e::l).toFinset := by
-    intro f hf
-    obtain ⟨hf,hfe⟩ := Finset.mem_filter.mp hf
-    have hfp := (mem_separatingBonds G ω x z f).mp hf
-    by_cases h : f = e
-    · subst f; simp
-    by_contra hnot
-    have hn : f ∉ l := fun hfl => hnot (List.mem_toFinset.mpr (List.mem_cons_of_mem _ hfl))
-    have hlt : pivotalRank G ω x f < pivotalRank G ω x e := by
-      apply lt_of_le_of_ne hfe
-      intro heq
-      exact h (pivotalRank_injective_of_cluster_finite G ω {z} x hC hfp he heq)
-    have hout := (pivotalRank_lt_iff_of_cluster_finite G ω {z} x hC hfp he).mp hlt
-    have hb : ¬ Conn G (ω \ {f}) x b := ((mem_internalEdges_of_ends G _ e a b hab).mp hout).2
-    apply hb
-    have hxa : Conn G (ω \ {f}) x a := hl.conn (fun g hg => ⟨ho g hg,fun hgf => hn ((Set.mem_singleton_iff.mp hgf) ▸ hg)⟩)
-    exact hxa.tail ⟨e,⟨he.1,fun hEq => h hEq.symm⟩,hab⟩
-  exact (Finset.card_le_card hsub).trans (by simpa only [List.length_cons] using List.toFinset_card_le (e::l))
-
-theorem ShortClosedBoundary.open_pivotal (G : BondGraph V E) (ω : Set E) (x z : V)
-    (M : ℕ) (e : E) (h : ShortClosedBoundary G ω x z M e) :
-    OpenMarkedPivotal G (insert e ω) {z} x e := by
-  obtain ⟨he,hxz,a,b,l,hab,hl,ho,_,hbz⟩ := h
-  refine ⟨Set.mem_insert e ω,⟨z,rfl,?_⟩,?_⟩
-  · have hxa := hl.conn ho
-    exact (conn_mono G (Set.subset_insert e ω) hxa).trans
-      ((Relation.ReflTransGen.single ⟨e,Set.mem_insert e ω,hab⟩).trans
-        (conn_mono G (Set.subset_insert e ω) hbz))
-  · rintro ⟨y,hy,hxy⟩
-    have hy' : y = z := Set.mem_singleton_iff.mp hy
-    subst y
-    apply hxz
-    apply conn_mono G _ hxy
-    intro f hf
-    rcases hf.1 with hfe | hfω
-    · exact (hf.2 (Set.mem_singleton_iff.mpr hfe)).elim
-    · exact hfω
-
-theorem ShortClosedBoundary.bridgeIndex_le (G : BondGraph V E) (ω : Set E) (x z : V)
-    (hω : ω.Finite) (M : ℕ) (e : E) (h : ShortClosedBoundary G ω x z M e) :
-    bridgeIndex G (insert e ω) x z e ≤ M := by
-  have hpiv := ShortClosedBoundary.open_pivotal G ω x z M e h
-  obtain ⟨_,_,a,b,l,hab,hl,ho,hlen,_⟩ := h
-  have hc := cluster_finite_of_finite_edges G (hω.insert e) x
-  have hb := bridgeIndex_le_short_trail G (insert e ω) x z hc e hpiv
-    a b hab l hl (fun f hf => Set.mem_insert_of_mem e (ho f hf))
-  exact hb.trans (by omega)
-
-end ShortBoundaryPivotal
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section ShortBoundaryCounting
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def firstOpenBridgeCount (G : BondGraph V E) (Q : Finset E)
-    (x z : V) (M : ℕ) (ω : Finset E) : ℕ :=
-  (Q.filter (fun e => OpenMarkedPivotal G ω {z} x e ∧ bridgeIndex G ω x z e ≤ M)).card
-
-noncomputable def shortClosedBoundaryCount (G : BondGraph V E) (Q : Finset E)
-    (x z : V) (M : ℕ) (ω : Finset E) : ℕ :=
-  (Q.filter (ShortClosedBoundary G ω x z M)).card
-
-theorem firstOpenBridgeCount_le (G : BondGraph V E) (Q : Finset E)
-    (x z : V) (M : ℕ) (ω : Finset E) :
-    firstOpenBridgeCount G Q x z M ω ≤ if Conn G ω x z then M else 0 := by
-  by_cases hc : Conn G ω x z
-  · rw [ite_eq_left hc]
-    unfold firstOpenBridgeCount
-    let S := Q.filter (fun e => OpenMarkedPivotal G ω {z} x e ∧ bridgeIndex G ω x z e ≤ M)
-    have hm : Set.MapsTo (bridgeIndex G ω x z) (S : Set E) (Finset.Icc 1 M) := by
-      intro e he
-      have he' := (Finset.mem_filter.mp he).2
-      exact Finset.mem_Icc.mpr ⟨bridgeIndex_pos G ω x z he'.1,he'.2⟩
-    have hi : Set.InjOn (bridgeIndex G ω x z) (S : Set E) := by
-      intro e he f hf h
-      exact bridgeIndex_injective G ω x z (cluster_finite_of_finite_edges G ω.finite_toSet x)
-        (Finset.mem_filter.mp he).2.1 (Finset.mem_filter.mp hf).2.1 h
-    simpa only [Nat.card_Icc,Nat.add_sub_cancel] using
-      (Finset.card_le_card_of_injOn (bridgeIndex G ω x z) hm hi)
-  · rw [ite_eq_right hc]
-    unfold firstOpenBridgeCount
-    have he : Q.filter (fun e => OpenMarkedPivotal G ω {z} x e ∧ bridgeIndex G ω x z e ≤ M) = ∅ := by
-      apply Finset.eq_empty_iff_forall_notMem.mpr
-      intro e he
-      obtain ⟨y,hy,hxy⟩ := (Finset.mem_filter.mp he).2.1.2.1
-      exact hc ((Set.mem_singleton_iff.mp hy) ▸ hxy)
-    rw [he,Finset.card_empty]
-
-theorem bernoulliPolynomial_bit_balance (Q : Finset E) (e : E) (he : e ∈ Q)
-    (f : Finset E → ℝ) (p : ℝ) :
-    p*bernoulliPolynomial Q (fun ω => if e ∉ ω then f ω else 0) p =
-      (1-p)*bernoulliPolynomial Q (fun ω => if e ∈ ω then f (ω.erase e) else 0) p := by
-  conv_lhs => rw [←Finset.insert_erase he]
-  conv_rhs => rw [←Finset.insert_erase he]
-  rw [bernoulliPolynomial_insert _ e (Finset.notMem_erase e Q),
-    bernoulliPolynomial_insert _ e (Finset.notMem_erase e Q)]
-  have hc : bernoulliPolynomial (Q.erase e) (fun ω => if e ∉ ω then f ω else 0) p =
-      bernoulliPolynomial (Q.erase e) f p := by
-    apply bernoulliPolynomial_congr
-    intro ω hω
-    simp only [show e ∉ ω from fun h => Finset.notMem_erase e Q (hω h),not_false_eq_true,ite_true]
-  have hco : bernoulliPolynomial (Q.erase e) (fun ω => if e ∉ insert e ω then f (insert e ω) else 0) p = 0 := by
-    simp only [Finset.mem_insert_self,not_true_eq_false,ite_false,bernoulliPolynomial_zero]
-  have ho : bernoulliPolynomial (Q.erase e) (fun ω => if e ∈ ω then f (ω.erase e) else 0) p = 0 := by
-    calc
-      _ = bernoulliPolynomial (Q.erase e) (fun _ => 0) p := by
-        apply bernoulliPolynomial_congr
-        intro ω hω
-        simp only [show e ∉ ω from fun h => Finset.notMem_erase e Q (hω h),ite_false]
-      _ = 0 := bernoulliPolynomial_zero _ _
-  have hoo : bernoulliPolynomial (Q.erase e)
-      (fun ω => if e ∈ insert e ω then f ((insert e ω).erase e) else 0) p =
-      bernoulliPolynomial (Q.erase e) f p := by
-    apply bernoulliPolynomial_congr
-    intro ω hω
-    simp only [Finset.mem_insert_self,ite_true,Finset.erase_insert
-      (show e ∉ ω from fun h => Finset.notMem_erase e Q (hω h))]
-  rw [hc,hco,ho,hoo]
-  ring
-
-end ShortBoundaryCounting
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section ShortBoundaryExpectation
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem bernoulliPolynomial_closed_count_balance (Q : Finset E)
-    (P : Finset E → E → Prop) (hP : ∀ ω e, P ω e → e ∉ ω) (p : ℝ) :
-    p*bernoulliPolynomial Q (fun ω => ((Q.filter (P ω)).card:ℝ)) p =
-      (1-p)*bernoulliPolynomial Q
-        (fun ω => ((Q.filter (fun e => e ∈ ω ∧ P (ω.erase e) e)).card:ℝ)) p := by
-  have hbal (e : E) (he : e ∈ Q) :
-      p*bernoulliPolynomial Q (fun ω => if P ω e then 1 else 0) p =
-        (1-p)*bernoulliPolynomial Q (fun ω => if e ∈ ω ∧ P (ω.erase e) e then 1 else 0) p := by
-    have h := bernoulliPolynomial_bit_balance Q e he (fun ω => if P ω e then 1 else 0) p
-    have hL : bernoulliPolynomial Q (fun ω => if e ∉ ω then (if P ω e then 1 else 0) else 0) p =
-        bernoulliPolynomial Q (fun ω => if P ω e then 1 else 0) p := by
-      apply bernoulliPolynomial_congr
-      intro ω _
-      by_cases hp : P ω e
-      · simp only [hp,hP ω e hp,not_false_eq_true,ite_true]
-      · simp only [hp,ite_false,ite_self]
-    have hR : bernoulliPolynomial Q (fun ω => if e ∈ ω then (if P (ω.erase e) e then 1 else 0) else 0) p =
-        bernoulliPolynomial Q (fun ω => if e ∈ ω ∧ P (ω.erase e) e then 1 else 0) p := by
-      apply bernoulliPolynomial_congr
-      intro ω _
-      by_cases hm : e ∈ ω <;> simp only [hm,true_and,false_and,ite_true,ite_false]
-    rw [hL,hR] at h
-    exact h
-  simp only [Finset.card_filter,Nat.cast_sum,Nat.cast_ite,Nat.cast_one,Nat.cast_zero,
-    bernoulliPolynomial_sum,Finset.mul_sum]
-  exact Finset.sum_congr rfl (fun e he => hbal e he)
-
-theorem shortClosedBoundary_erase_first (G : BondGraph V E) (ω : Finset E)
-    (x z : V) (M : ℕ) (e : E) (he : e ∈ ω)
-    (h : ShortClosedBoundary G (ω.erase e) x z M e) :
-    OpenMarkedPivotal G ω {z} x e ∧ bridgeIndex G ω x z e ≤ M := by
-  have h₁ := ShortClosedBoundary.open_pivotal G (ω.erase e) x z M e h
-  have h₂ := ShortClosedBoundary.bridgeIndex_le G (ω.erase e) x z (ω.erase e).finite_toSet M e h
-  rw [←Finset.coe_insert,Finset.insert_erase he] at h₁ h₂
-  exact ⟨h₁,h₂⟩
-
-theorem shortRecoveredBoundaryCount_le (G : BondGraph V E) (Q ω : Finset E)
-    (x z : V) (M : ℕ) :
-    (Q.filter (fun e => e ∈ ω ∧ ShortClosedBoundary G (ω.erase e) x z M e)).card ≤
-      if Conn G ω x z then M else 0 := by
-  apply le_trans _ (firstOpenBridgeCount_le G Q x z M ω)
-  apply Finset.card_le_card
-  intro e he
-  obtain ⟨heQ,heω,hb⟩ := Finset.mem_filter.mp he
-  exact Finset.mem_filter.mpr ⟨heQ,shortClosedBoundary_erase_first G ω x z M e heω hb⟩
-
- 
-
-theorem finite_short_boundary_expectation (G : BondGraph V E) (Q : Finset E)
-    (x z : V) (M : ℕ) (p : ℝ) (hp : 0 < p) (hp1 : p ≤ 1) :
-    bernoulliPolynomial Q (fun ω => (shortClosedBoundaryCount G Q x z M ω:ℝ)) p ≤
-      ((1-p)/p)*M*bernoulliPolynomial Q (fun ω => if Conn G ω x z then 1 else 0) p := by
-  have hbal := bernoulliPolynomial_closed_count_balance Q
-    (fun ω e => ShortClosedBoundary G ω x z M e) (fun ω e h => h.1) p
-  have hmono := bernoulliPolynomial_mono Q
-    (fun ω => ((Q.filter (fun e => e ∈ ω ∧ ShortClosedBoundary G (ω.erase e) x z M e)).card:ℝ))
-    (fun ω => (M:ℝ)*(if Conn G ω x z then 1 else 0)) p ⟨hp.le,hp1⟩ (by
-      intro ω _
-      have h := shortRecoveredBoundaryCount_le G Q ω x z M
-      by_cases hc : Conn G ω x z
-      · simp only [hc,ite_true,mul_one] at *
-        exact_mod_cast h
-      · simp only [hc,ite_false,mul_zero] at *
-        exact_mod_cast h)
-  rw [bernoulliPolynomial_const_mul] at hmono
-  have hbound := mul_le_mul_of_nonneg_left hmono (sub_nonneg.mpr hp1)
-  rw [←hbal] at hbound
-  change p*bernoulliPolynomial Q (fun ω => (shortClosedBoundaryCount G Q x z M ω:ℝ)) p ≤ _ at hbound
-  have hdiv := (le_div_iff₀ hp).mpr (show
-      bernoulliPolynomial Q (fun ω => (shortClosedBoundaryCount G Q x z M ω:ℝ)) p*p ≤ _ by
-        simpa only [mul_comm p] using hbound)
-  simpa only [div_eq_mul_inv,mul_assoc,mul_left_comm,mul_comm] using hdiv
-
-end ShortBoundaryExpectation
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section ClusterResampling
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem bernoulliPolynomial_commute {F : Type*} (Q : Finset E) (R : Finset F)
-    (f : Finset E → Finset F → ℝ) (p q : ℝ) :
-    bernoulliPolynomial Q (fun a => bernoulliPolynomial R (f a) q) p =
-      bernoulliPolynomial R (fun b => bernoulliPolynomial Q (fun a => f a b) p) q := by
-  simp only [bernoulliPolynomial,Finset.sum_mul]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro b _
-  apply Finset.sum_congr rfl
-  intro a _
-  ring
-
-theorem exteriorBonds_union_blocks (G : BondGraph V E) (Q : Finset E)
-    (W : Set V) (a b : Finset E) (ha : a ⊆ Q \ exteriorBonds G Q W)
-    (hb : b ⊆ exteriorBonds G Q W) : exteriorBonds G (a ∪ b) W = b := by
-  ext e
-  simp only [exteriorBonds,Finset.mem_filter,Finset.mem_union]
-  constructor
-  · rintro ⟨he,heW⟩
-    rcases he with he | he
-    · obtain ⟨heQ,heR⟩ := Finset.mem_sdiff.mp (ha he)
-      exact (heR (Finset.mem_filter.mpr ⟨heQ,heW⟩)).elim
-    · exact he
-  · intro he
-    exact ⟨Or.inr he,(Finset.mem_filter.mp (hb he)).2⟩
-
- 
-theorem cluster_cell_exterior_factor (G : BondGraph V E) (Q : Finset E)
-    (x : V) (W : Set V) (f : Finset E → ℝ) (p : ℝ) :
-    bernoulliPolynomial Q (fun ω =>
-      (if cluster G ω x = W then (1:ℝ) else 0)*f (exteriorBonds G ω W)) p =
-      bernoulliPolynomial (Q \ exteriorBonds G Q W)
-        (fun a => if cluster G a x = W then (1:ℝ) else 0) p *
-      bernoulliPolynomial (exteriorBonds G Q W) f p := by
-  let R := exteriorBonds G Q W
-  let A := Q \ R
-  have hQ : A ∪ R = Q := Finset.sdiff_union_of_subset (Finset.filter_subset _ _)
-  conv_lhs => rw [←hQ,bernoulliPolynomial_union A R Finset.sdiff_disjoint]
-  trans bernoulliPolynomial A (fun a =>
-    (if cluster G a x = W then (1:ℝ) else 0)*bernoulliPolynomial R f p) p
-  · apply bernoulliPolynomial_congr
-    intro a ha
-    rw [←bernoulliPolynomial_const_mul]
-    apply bernoulliPolynomial_congr
-    intro b hb
-    have hc := cluster_union_external_eq_iff G (a:Set E) b x W
-      (fun e he => (Finset.mem_filter.mp (hb he)).2)
-    rw [exteriorBonds_union_blocks G Q W a b ha hb]
-    simp only [Finset.coe_union,hc]
-  · exact bernoulliPolynomial_mul_const A _ _ p
-
-theorem cluster_cell_incident_probability (G : BondGraph V E) (Q : Finset E)
-    (x : V) (W : Set V) (p : ℝ) :
-    bernoulliPolynomial Q (fun ω => if cluster G ω x = W then (1:ℝ) else 0) p =
-      bernoulliPolynomial (Q \ exteriorBonds G Q W)
-        (fun a => if cluster G a x = W then (1:ℝ) else 0) p := by
-  have h := cluster_cell_exterior_factor G Q x W (fun _ => 1) p
-  simpa only [mul_one,bernoulliPolynomial_const] using h
-
-noncomputable def resampleClusterAt (G : BondGraph V E) (Q : Finset E)
-    (W : Set V) (ω η : Finset E) : Finset E :=
-  exteriorBonds G ω W ∪ (η ∩ (Q \ exteriorBonds G Q W))
-
- 
-
-theorem cluster_resample_cell_independent (G : BondGraph V E) (Q : Finset E)
-    (x : V) (W : Set V) (f : Finset E → ℝ) (p : ℝ) :
-    bernoulliPolynomial Q (fun ω => bernoulliPolynomial Q (fun η =>
-      (if cluster G ω x = W then (1:ℝ) else 0)*f (resampleClusterAt G Q W ω η)) p) p =
-      bernoulliPolynomial Q (fun ω => if cluster G ω x = W then (1:ℝ) else 0) p *
-      bernoulliPolynomial Q f p := by
-  let R := exteriorBonds G Q W
-  let A := Q \ R
-  let c := bernoulliPolynomial A (fun a => if cluster G a x = W then (1:ℝ) else 0) p
-  rw [bernoulliPolynomial_commute]
-  trans bernoulliPolynomial Q (fun η => c * bernoulliPolynomial R
-    (fun b => f (b ∪ (η ∩ A))) p) p
-  · apply bernoulliPolynomial_congr
-    intro η _
-    exact cluster_cell_exterior_factor G Q x W (fun b => f (b ∪ (η ∩ A))) p
-  · rw [bernoulliPolynomial_const_mul,bernoulliPolynomial_restrict A Q Finset.sdiff_subset
-      (fun a => bernoulliPolynomial R (fun b => f (b ∪ a)) p) p,
-      bernoulliPolynomial_commute]
-    have hQ : R ∪ A = Q := Finset.union_sdiff_of_subset (Finset.filter_subset _ _)
-    rw [←bernoulliPolynomial_union R A Finset.disjoint_sdiff,hQ,
-      cluster_cell_incident_probability G Q x W p]
-
-end ClusterResampling
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section ResampledBoundary
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem resampleClusterAt_subset (G : BondGraph V E) (Q : Finset E)
-    (W : Set V) (ω η : Finset E) (hω : ω ⊆ Q) :
-    resampleClusterAt G Q W ω η ⊆ Q := by
-  intro e he
-  rcases Finset.mem_union.mp he with he | he
-  · exact hω (Finset.mem_filter.mp he).1
-  · exact (Finset.mem_sdiff.mp (Finset.mem_inter.mp he).2).1
-
-theorem resampleClusterAt_external (G : BondGraph V E) (Q : Finset E)
-    (W : Set V) (ω η : Finset E) {e : E} (he : e ∈ resampleClusterAt G Q W ω η)
-    (heW : e ∈ internalEdges G Wᶜ) : e ∈ ω := by
-  rcases Finset.mem_union.mp he with he | he
-  · exact (Finset.mem_filter.mp he).1
-  · obtain ⟨_,heA⟩ := Finset.mem_inter.mp he
-    obtain ⟨heQ,heR⟩ := Finset.mem_sdiff.mp heA
-    exact (heR (Finset.mem_filter.mpr ⟨heQ,heW⟩)).elim
-
- 
-
-theorem resampled_path_boundary (G : BondGraph V E) (Q ω η : Finset E)
-    (hω : ω ⊆ Q) (x z y : V) (M : ℕ)
-    (hy : Conn G ω z y) (hx : ¬Conn G ω x z)
-    (l : List E) (hl : Along G x l y) (hlen : l.length ≤ M)
-    (ho : ∀ e ∈ l, e ∈ resampleClusterAt G Q (cluster G ω z) ω η) :
-    ∃ e ∈ Q, ShortClosedBoundary G ω x z M e := by
-  let W := cluster G ω z
-  have hxW : x ∉ W := fun h => hx (conn_symm G ω h)
-  rcases hl.first_outside (internalEdges G Wᶜ) with hall | ⟨a,b,e,l₁,l₂,heq,h₁,hab,_,hfirst,hen⟩
-  · have hc : Conn G ((Set.univ:Set E) ∩ internalEdges G Wᶜ) x y :=
-      hl.conn (fun e he => ⟨Set.mem_univ e,hall e he⟩)
-    exact ((conn_internal_endpoint G Wᶜ hxW hc) hy).elim
-  · subst l
-    have heopen := ho e (List.mem_append_right _ (by simp))
-    have heQ := resampleClusterAt_subset G Q W ω η hω heopen
-    have hprefix : ∀ f ∈ l₁, f ∈ ω := by
-      intro f hf
-      exact resampleClusterAt_external G Q W ω η
-        (ho f (List.mem_append_left _ hf)) (hfirst f hf)
-    have hxa : Conn G ((ω:Set E) ∩ internalEdges G Wᶜ) x a :=
-      h₁.conn (fun f hf => ⟨hprefix f hf,hfirst f hf⟩)
-    have haW : a ∉ W := conn_internal_endpoint G Wᶜ hxW hxa
-    have hbW : b ∈ W := by
-      by_contra hbW
-      exact hen ((mem_internalEdges_of_ends G Wᶜ e a b hab).mpr ⟨haW,hbW⟩)
-    have heclosed : e ∉ ω := by
-      intro heω
-      apply haW
-      exact (show Conn G ω z b from hbW).tail ⟨e,heω,hab.trans (Sym2.eq_swap ..)⟩
-    refine ⟨e,heQ,heclosed,hx,a,b,l₁,hab,h₁,hprefix,?_,conn_symm G ω hbW⟩
-    simp only [List.length_append,List.length_cons] at hlen
-    omega
-
-noncomputable def resampleRootCluster (G : BondGraph V E) (Q : Finset E)
-    (z : V) (ω η : Finset E) : Finset E :=
-  resampleClusterAt G Q (cluster G ω z) ω η
-
- 
-theorem resampled_hit_le_boundaryCount (G : BondGraph V E) (Q ω η : Finset E)
-    (hω : ω ⊆ Q) (x z y : V) (M : ℕ)
-    (hpath : ∃ l : List E, Along G x l y ∧ l.length ≤ M ∧
-      ∀ e ∈ l, e ∈ resampleRootCluster G Q z ω η) :
-    (if Conn G ω z y then (1:ℝ) else 0) ≤
-      (if Conn G ω x z then (1:ℝ) else 0) + shortClosedBoundaryCount G Q x z M ω := by
-  by_cases hy : Conn G ω z y
-  · rw [ite_eq_left hy]
-    by_cases hx : Conn G ω x z
-    · rw [ite_eq_left hx]
-      exact le_add_of_nonneg_right (Nat.cast_nonneg _)
-    · rw [ite_eq_right hx,zero_add]
-      obtain ⟨l,hl,hlen,ho⟩ := hpath
-      obtain ⟨e,heQ,he⟩ := resampled_path_boundary G Q ω η hω x z y M hy hx l hl hlen ho
-      have hpos : 0 < shortClosedBoundaryCount G Q x z M ω :=
-        Finset.card_pos.mpr ⟨e,Finset.mem_filter.mpr ⟨heQ,he⟩⟩
-      exact_mod_cast hpos
-  · rw [ite_eq_right hy]
-    positivity
-
-end ResampledBoundary
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section RootResamplingLaw
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem rootCluster_resampling_law (G : BondGraph V E) (Q : Finset E)
-    (x : V) (f : Finset V → Finset E → ℝ) (p : ℝ) :
-    bernoulliPolynomial Q (fun ω => bernoulliPolynomial Q
-      (fun η => f (finiteRootComponent G ω x) (resampleRootCluster G Q x ω η)) p) p =
-      bernoulliPolynomial Q (fun ω => bernoulliPolynomial Q
-        (f (finiteRootComponent G ω x)) p) p := by
-  let F := insert x (searchVertices G Q)
-  let c := fun (W : Finset V) (ω : Finset E) => if cluster G ω x = (W:Set V) then (1:ℝ) else 0
-  have hpart (ω η : Finset E) (hω : ω ⊆ Q) :
-      f (finiteRootComponent G ω x) (resampleRootCluster G Q x ω η) =
-        ∑ W ∈ F.powerset, c W ω * f W (resampleClusterAt G Q W ω η) := by
-    have h := sum_component_partition G Q F x ω hω (by rfl)
-      (fun W => f W (resampleClusterAt G Q W ω η))
-    simpa only [c,ite_mul,one_mul,zero_mul,coe_finiteRootComponent,
-      resampleRootCluster] using h.symm
-  trans ∑ W ∈ F.powerset,
-    bernoulliPolynomial Q (c W) p * bernoulliPolynomial Q (f W) p
-  · trans bernoulliPolynomial Q (fun ω => bernoulliPolynomial Q (fun η =>
-      ∑ W ∈ F.powerset, c W ω * f W (resampleClusterAt G Q W ω η)) p) p
-    · apply bernoulliPolynomial_congr
-      intro ω hω
-      apply bernoulliPolynomial_congr
-      intro η _
-      exact hpart ω η hω
-    · simp only [bernoulliPolynomial_sum]
-      apply Finset.sum_congr rfl
-      intro W _
-      exact cluster_resample_cell_independent G Q x W (f W) p
-  · trans bernoulliPolynomial Q (fun ω => ∑ W ∈ F.powerset,
-      c W ω * bernoulliPolynomial Q (f W) p) p
-    · rw [bernoulliPolynomial_sum]
-      apply Finset.sum_congr rfl
-      intro W _
-      exact (bernoulliPolynomial_mul_const Q (c W) _ p).symm
-    · apply bernoulliPolynomial_congr
-      intro ω hω
-      have h := sum_component_partition G Q F x ω hω (by rfl)
-        (fun W => bernoulliPolynomial Q (f W) p)
-      simpa only [c,ite_mul,one_mul,zero_mul] using h
-
-end RootResamplingLaw
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section FiniteMaximalSusceptibility
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def finiteSusceptibility (G : BondGraph V E) (Q : Finset E)
-    (F : Finset V) (p : ℝ) (x : V) : ℝ :=
-  ∑ z ∈ F, bernoulliPolynomial Q (fun ω => if Conn G ω x z then (1:ℝ) else 0) p
-
-theorem finiteSusceptibility_nonneg (G : BondGraph V E) (Q : Finset E)
-    (F : Finset V) (p : ℝ) (x : V) (hp : p ∈ Set.Icc (0:ℝ) 1) :
-    0 ≤ finiteSusceptibility G Q F p x := by
-  apply Finset.sum_nonneg
-  intro z _
-  exact bernoulliPolynomial_nonneg Q _ p hp (fun _ _ => by positivity)
-
-theorem rootCluster_resampled_selector (G : BondGraph V E) (Q : Finset E)
-    (w : V) (Z : Finset E → V) (p : ℝ) :
-    bernoulliPolynomial Q (fun ω => bernoulliPolynomial Q (fun η =>
-      if Conn G ω w (Z (resampleRootCluster G Q w ω η)) then (1:ℝ) else 0) p) p =
-      bernoulliPolynomial Q (fun η => bernoulliPolynomial Q (fun ω =>
-        if Conn G ω w (Z η) then (1:ℝ) else 0) p) p := by
-  have hm (ω : Finset E) (y : V) :
-      y ∈ finiteRootComponent G ω w ↔ Conn G ω w y := by
-    change y ∈ (finiteRootComponent G ω w : Set V) ↔ _
-    rw [coe_finiteRootComponent]
-    rfl
-  have h := rootCluster_resampling_law G Q w (fun W η => if Z η ∈ W then (1:ℝ) else 0) p
-  simp only [hm] at h
-  exact h.trans (bernoulliPolynomial_commute Q Q _ p p)
-
-theorem finite_selector_connection_bound (G : BondGraph V E) (Q : Finset E)
-    (x w : V) (M : ℕ) (Z : Finset E → V)
-    (hZ : ∀ η ⊆ Q, ∃ l : List E, Along G x l (Z η) ∧ l.length ≤ M ∧ ∀ e ∈ l, e ∈ η)
-    (p : ℝ) (hp : 0 < p) (hp1 : p ≤ 1) :
-    bernoulliPolynomial Q (fun η => bernoulliPolynomial Q (fun ω =>
-      if Conn G ω w (Z η) then (1:ℝ) else 0) p) p ≤
-      (1+(1-p)/p*M)*bernoulliPolynomial Q (fun ω => if Conn G ω x w then (1:ℝ) else 0) p := by
-  rw [←rootCluster_resampled_selector G Q w Z p]
-  have hmono := bernoulliPolynomial_mono Q
-    (fun ω => bernoulliPolynomial Q (fun η =>
-      if Conn G ω w (Z (resampleRootCluster G Q w ω η)) then (1:ℝ) else 0) p)
-    (fun ω => (if Conn G ω x w then (1:ℝ) else 0)+shortClosedBoundaryCount G Q x w M ω)
-    p ⟨hp.le,hp1⟩ (by
-      intro ω hω
-      calc
-        _ ≤ bernoulliPolynomial Q (fun _ =>
-          (if Conn G ω x w then (1:ℝ) else 0)+shortClosedBoundaryCount G Q x w M ω) p := by
-            apply bernoulliPolynomial_mono Q _ _ p ⟨hp.le,hp1⟩
-            intro η _
-            exact resampled_hit_le_boundaryCount G Q ω η hω x w _ M
-              (hZ _ (resampleClusterAt_subset G Q _ ω η hω))
-        _ = _ := bernoulliPolynomial_const Q _ p)
-  rw [bernoulliPolynomial_add] at hmono
-  have hb := finite_short_boundary_expectation G Q x w M p hp hp1
-  calc
-    _ ≤ _ := hmono
-    _ ≤ bernoulliPolynomial Q (fun ω => if Conn G ω x w then (1:ℝ) else 0) p +
-      ((1-p)/p)*M*bernoulliPolynomial Q (fun ω => if Conn G ω x w then (1:ℝ) else 0) p :=
-        add_le_add le_rfl hb
-    _ = _ := by ring
-
- 
-
-theorem finite_selected_susceptibility_bound (G : BondGraph V E) (Q : Finset E)
-    (F : Finset V) (x : V) (M : ℕ) (Z : Finset E → V)
-    (hZ : ∀ η ⊆ Q, ∃ l : List E, Along G x l (Z η) ∧ l.length ≤ M ∧ ∀ e ∈ l, e ∈ η)
-    (p : ℝ) (hp : 0 < p) (hp1 : p ≤ 1) :
-    bernoulliPolynomial Q (fun η => finiteSusceptibility G Q F p (Z η)) p ≤
-      (1+(1-p)/p*M)*finiteSusceptibility G Q F p x := by
-  unfold finiteSusceptibility
-  rw [bernoulliPolynomial_sum,Finset.mul_sum]
-  apply Finset.sum_le_sum
-  intro w _
-  have h := finite_selector_connection_bound G Q x w M Z hZ p hp hp1
-  calc
-    _ = bernoulliPolynomial Q (fun η => bernoulliPolynomial Q (fun ω =>
-        if Conn G ω w (Z η) then (1:ℝ) else 0) p) p := by
-      apply bernoulliPolynomial_congr
-      intro η _
-      apply bernoulliPolynomial_congr
-      intro ω _
-      apply if_congr _ rfl rfl
-      exact ⟨conn_symm G ω,conn_symm G ω⟩
-    _ ≤ _ := h
-
-end FiniteMaximalSusceptibility
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter Topology
-section RestrictedConnectionLimit
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem tendsto_bernoulliPolynomial {ι : Type*} {l : Filter ι}
-    (Q : Finset E) (f : ι → Finset E → ℝ) (g : Finset E → ℝ) (p : ℝ)
-    (h : ∀ a ⊆ Q, Tendsto (fun i => f i a) l (𝓝 (g a))) :
-    Tendsto (fun i => bernoulliPolynomial Q (f i) p) l
-      (𝓝 (bernoulliPolynomial Q g p)) := by
-  unfold bernoulliPolynomial
-  exact tendsto_finsetSum _ (fun a ha =>
-    ((h a (Finset.mem_powerset.mp ha)).mul_const _).mul_const _)
-
-theorem conn_finite_exhaustion (G : BondGraph V E) (S : Set E)
-    (Q : ℕ → Finset E) (hQ : Monotone Q) (hcover : ∀ e ∈ S, ∃ n, e ∈ Q n)
-    {ω : Set E} {x y : V} (h : Conn G (ω ∩ S) x y) :
-    ∃ n, Conn G (ω ∩ (Q n : Set E)) x y := by
-  induction h with
-  | refl => exact ⟨0,conn_refl _ _ _⟩
-  | @tail a b hxa hab ih =>
-    obtain ⟨n,hn⟩ := ih
-    obtain ⟨e,he,hab⟩ := hab
-    obtain ⟨m,hm⟩ := hcover e he.2
-    refine ⟨max n m,?_⟩
-    exact (conn_mono G (Set.inter_subset_inter_right _
-      (show (Q n:Set E) ⊆ (Q (max n m):Set E) from hQ (le_max_left n m))) hn).tail
-      ⟨e,⟨he.1,hQ (le_max_right n m) hm⟩,hab⟩
-
-theorem finite_connection_polynomial_tendsto (G : BondGraph V E) (S : Set E)
-    (Q : ℕ → Finset E) (hQ : Monotone Q) (hsub : ∀ n, (Q n:Set E) ⊆ S)
-    (hcover : ∀ e ∈ S, ∃ n, e ∈ Q n) (p : unitInterval) (x y : V) :
-    Tendsto (fun n => bernoulliPolynomial (Q n) (fun ω => if Conn G ω x y then (1:ℝ) else 0) p)
-      atTop (𝓝 ((law p).real {ω | Conn G (ω ∩ S) x y})) := by
-  have he (n : ℕ) : bernoulliPolynomial (Q n) (fun ω => if Conn G ω x y then (1:ℝ) else 0) p =
-      (law p).real {ω | Conn G (ω ∩ (Q n:Set E)) x y} :=
-    finiteEventPolynomial_restrict (Q n) {ω | Conn G ω x y} p
-  simp_rw [he]
-  have hm : Monotone (fun n => {ω : Set E | Conn G (ω ∩ (Q n:Set E)) x y}) := by
-    intro m n h ω hc
-    exact conn_mono G (Set.inter_subset_inter_right _ (show (Q m:Set E) ⊆ (Q n:Set E) from hQ h)) hc
-  have hu : (⋃ n, {ω : Set E | Conn G (ω ∩ (Q n:Set E)) x y}) =
-      {ω : Set E | Conn G (ω ∩ S) x y} := by
-    ext ω
-    simp only [Set.mem_iUnion,Set.mem_ofPred_eq]
-    exact ⟨fun ⟨n,hn⟩ => conn_mono G (Set.inter_subset_inter_right _ (hsub n)) hn,
-      conn_finite_exhaustion G S Q hQ hcover⟩
-  have ht := tendsto_measure_iUnion_atTop (μ := law p) hm
-  rw [hu] at ht
-  exact (ENNReal.tendsto_toReal (measure_ne_top _ _)).comp ht
-
-noncomputable def restrictedBondExhaustion [Encodable E] (S : Set E) (K : Finset E)
-    (n : ℕ) : Finset E := K ∪ (vertexExhaustion (V := E) n).filter (· ∈ S)
-
-theorem restrictedBondExhaustion_mono [Encodable E] (S : Set E) (K : Finset E) :
-    Monotone (restrictedBondExhaustion S K) := by
-  intro m n h
-  exact Finset.union_subset_union (by rfl) (Finset.filter_subset_filter _ (vertexExhaustion_mono h))
-
-theorem restrictedBondExhaustion_cover [Encodable E] (S : Set E) (K : Finset E)
-    (e : E) (he : e ∈ S) : ∃ n, e ∈ restrictedBondExhaustion S K n := by
-  obtain ⟨n,hn⟩ := subset_vertexExhaustion ({e}:Finset E)
-  exact ⟨n,Finset.mem_union_right _ (Finset.mem_filter.mpr ⟨hn (by simp),he⟩)⟩
-
-theorem restrictedBondExhaustion_subset [Encodable E] (S : Set E) (K : Finset E)
-    (hK : (K:Set E) ⊆ S) (n : ℕ) : (restrictedBondExhaustion S K n:Set E) ⊆ S := by
-  intro e he
-  rcases Finset.mem_union.mp he with he | he
-  · exact hK he
-  · exact (Finset.mem_filter.mp he).2
-
-end RestrictedConnectionLimit
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter Topology
-section RestrictedSelectorBound
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem restricted_selector_connection_bound [Countable E] (G : BondGraph V E)
-    (S : Set E) (K : Finset E) (hK : (K:Set E) ⊆ S)
-    (x w : V) (M : ℕ) (Z : Finset E → V)
-    (hZ : ∀ η ⊆ K, ∃ l : List E, Along G x l (Z η) ∧ l.length ≤ M ∧ ∀ e ∈ l, e ∈ η)
-    (p : unitInterval) (hp : 0 < (p:ℝ)) :
-    bernoulliPolynomial K (fun η => (law p).real {ω | Conn G (ω ∩ S) w (Z η)}) p ≤
-      (1+(1-(p:ℝ))/p*M)*(law p).real {ω | Conn G (ω ∩ S) x w} := by
-  let := Encodable.ofCountable E
-  let Q := restrictedBondExhaustion S K
-  have hlim (a b : V) := finite_connection_polynomial_tendsto G S Q
-    (restrictedBondExhaustion_mono S K) (restrictedBondExhaustion_subset S K hK)
-    (restrictedBondExhaustion_cover S K) p a b
-  have hleft := tendsto_bernoulliPolynomial K
-    (fun n η => bernoulliPolynomial (Q n) (fun ω => if Conn G ω w (Z η) then (1:ℝ) else 0) p)
-    (fun η => (law p).real {ω | Conn G (ω ∩ S) w (Z η)}) p (fun η _ => hlim w (Z η))
-  have hright := (hlim x w).const_mul (1+(1-(p:ℝ))/p*M)
-  apply le_of_tendsto_of_tendsto hleft hright
-  apply Eventually.of_forall
-  intro n
-  have hZQ : ∀ η ⊆ Q n, ∃ l : List E, Along G x l (Z (η ∩ K)) ∧
-      l.length ≤ M ∧ ∀ e ∈ l, e ∈ η := by
-    intro η _
-    obtain ⟨l,hl,hlen,ho⟩ := hZ (η ∩ K) Finset.inter_subset_right
-    exact ⟨l,hl,hlen,fun e he => (Finset.mem_inter.mp (ho e he)).1⟩
-  have h := finite_selector_connection_bound G (Q n) x w M (fun η => Z (η ∩ K)) hZQ p hp p.2.2
-  rw [bernoulliPolynomial_restrict K (Q n) Finset.subset_union_left
-    (fun η => bernoulliPolynomial (Q n) (fun ω => if Conn G ω w (Z η) then (1:ℝ) else 0) p) p] at h
-  exact h
-
-noncomputable def restrictedSusceptibility (G : BondGraph V E) (S : Set E)
-    (p : unitInterval) (x : V) : ℝ :=
-  ∑' y, (law p).real {ω | Conn G (ω ∩ S) x y}
-
-theorem restrictedTwoPoint_le (G : BondGraph V E) (S : Set E)
-    (p : unitInterval) (x y : V) :
-    (law p).real {ω | Conn G (ω ∩ S) x y} ≤ twoPoint G p x y := by
-  exact measureReal_mono (fun _ h => conn_mono G Set.inter_subset_left h)
-
-theorem summable_restrictedTwoPoint [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (S : Set E) (x : V) :
-    Summable (fun y => (law p).real {ω | Conn G (ω ∩ S) x y}) := by
-  obtain ⟨C,_,hC⟩ := subcritical_row_bound G hloc hqt p hp
-  apply summable_of_sum_le (fun _ => measureReal_nonneg)
-  intro F
-  exact (Finset.sum_le_sum (fun y _ => restrictedTwoPoint_le G S p x y)).trans (hC x F)
-
-end RestrictedSelectorBound
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter Topology
-section SelectorSusceptibility
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem summable_bernoulliPolynomial {A : Type*} (K : Finset E)
-    (f : Finset E → A → ℝ) (p : ℝ) (hf : ∀ η ⊆ K, Summable (f η)) :
-    Summable (fun y => bernoulliPolynomial K (fun η => f η y) p) := by
-  unfold bernoulliPolynomial
-  apply summable_sum
-  intro η hη
-  exact ((hf η (Finset.mem_powerset.mp hη)).mul_right _).mul_right _
-
-theorem bernoulliPolynomial_tsum {A : Type*} (K : Finset E)
-    (f : Finset E → A → ℝ) (p : ℝ) (hf : ∀ η ⊆ K, Summable (f η)) :
-    bernoulliPolynomial K (fun η => ∑' y, f η y) p =
-      ∑' y, bernoulliPolynomial K (fun η => f η y) p := by
-  unfold bernoulliPolynomial
-  rw [Summable.tsum_finsetSum (fun η hη =>
-    ((hf η (Finset.mem_powerset.mp hη)).mul_right _).mul_right _)]
-  apply Finset.sum_congr rfl
-  intro η _
-  rw [tsum_mul_right,tsum_mul_right]
-
-theorem restrictedTwoPoint_symm (G : BondGraph V E) (S : Set E)
-    (p : unitInterval) (x y : V) :
-    (law p).real {ω | Conn G (ω ∩ S) x y} =
-      (law p).real {ω | Conn G (ω ∩ S) y x} := by
-  congr 1
-  ext ω
-  exact ⟨conn_symm G _,conn_symm G _⟩
-
- 
-
-theorem restricted_selected_susceptibility_bound [Countable E] (G : BondGraph V E)
-    (S : Set E) (K : Finset E) (hK : (K:Set E) ⊆ S)
-    (x : V) (M : ℕ) (Z : Finset E → V)
-    (hZ : ∀ η ⊆ K, ∃ l : List E, Along G x l (Z η) ∧ l.length ≤ M ∧ ∀ e ∈ l, e ∈ η)
-    (p : unitInterval) (hp : 0 < (p:ℝ))
-    (hfin : ∀ y, Summable (fun z => (law p).real {ω | Conn G (ω ∩ S) y z})) :
-    bernoulliPolynomial K (fun η => restrictedSusceptibility G S p (Z η)) p ≤
-      (1+(1-(p:ℝ))/p*M)*restrictedSusceptibility G S p x := by
-  unfold restrictedSusceptibility
-  rw [bernoulliPolynomial_tsum K _ p (fun η _ => hfin (Z η)),←tsum_mul_left]
-  apply Summable.tsum_le_tsum _ (summable_bernoulliPolynomial K _ p (fun η _ => hfin (Z η)))
-    ((hfin x).mul_left _)
-  intro w
-  have h := restricted_selector_connection_bound G S K hK x w M Z hZ p hp
-  simpa only [restrictedTwoPoint_symm G S p (Z _) w] using h
-
- 
-
-theorem subcritical_selected_susceptibility_bound
-    [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (S : Set E) (K : Finset E) (hK : (K:Set E) ⊆ S)
-    (x : V) (M : ℕ) (Z : Finset E → V)
-    (hZ : ∀ η ⊆ K, ∃ l : List E, Along G x l (Z η) ∧ l.length ≤ M ∧ ∀ e ∈ l, e ∈ η)
-    (p : unitInterval) (hp : 0 < (p:ℝ)) (hpc : p < pc G) :
-    bernoulliPolynomial K (fun η => restrictedSusceptibility G S p (Z η)) p ≤
-      (1+(1-(p:ℝ))/p*M)*restrictedSusceptibility G S p x :=
-  restricted_selected_susceptibility_bound G S K hK x M Z hZ p hp
-    (summable_restrictedTwoPoint G hloc hqt p hpc S)
-
-end SelectorSusceptibility
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section IntrinsicBall
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-def intrinsicBall (G : BondGraph V E) (ω : Set E) (x : V) (M : ℕ) : Set V :=
-  {y | ∃ l : List E, Along G x l y ∧ l.length ≤ M ∧ ∀ e ∈ l, e ∈ ω}
-
-theorem mem_intrinsicBall_self (G : BondGraph V E) (ω : Set E) (x : V) (M : ℕ) :
-    x ∈ intrinsicBall G ω x M := ⟨[],.nil x,Nat.zero_le _,by simp⟩
-
-theorem intrinsicBall_subset_reachBall (G : BondGraph V E) (ω : Set E) (x : V) (M : ℕ) :
-    intrinsicBall G ω x M ⊆ reachBall G x M := by
-  rintro y ⟨l,hl,hlen,_⟩
-  exact reachBall_mono G x hlen hl.mem_reachBall
-
-theorem intrinsicBall_mono (G : BondGraph V E) {ω η : Set E} (h : ω ⊆ η) (x : V) (M : ℕ) :
-    intrinsicBall G ω x M ⊆ intrinsicBall G η x M := by
-  rintro y ⟨l,hl,hlen,ho⟩
-  exact ⟨l,hl,hlen,fun e he => h (ho e he)⟩
-
-theorem Along.bonds_in_incidentBall {G : BondGraph V E} (hloc : LocallyFinite G)
-    {x y : V} {l : List E} (h : Along G x l y) (M : ℕ) (hlen : l.length ≤ M) :
-    ∀ e ∈ l, e ∈ incidentSupport G hloc (finite_reachBall hloc x M).toFinset := by
-  rcases h.first_outside (incidentSupport G hloc (finite_reachBall hloc x M).toFinset : Set E)
-    with h | ⟨a,b,e,l₁,l₂,heq,h₁,he,_,_,hnot⟩
-  · exact h
-  · exfalso
-    apply hnot
-    apply (mem_incidentSupport G hloc _ e).mpr
-    refine ⟨a,?_,by simp [he]⟩
-    apply (finite_reachBall hloc x M).mem_toFinset.mpr
-    apply reachBall_mono G x (show l₁.length ≤ M by subst l; simp only [List.length_append,List.length_cons] at hlen; omega)
-    exact h₁.mem_reachBall
-
-theorem intrinsicBall_restrict (G : BondGraph V E) (hloc : LocallyFinite G)
-    (ω : Set E) (x : V) (M : ℕ) :
-    intrinsicBall G (ω ∩ (incidentSupport G hloc (finite_reachBall hloc x M).toFinset : Set E)) x M =
-      intrinsicBall G ω x M := by
-  apply Set.Subset.antisymm (intrinsicBall_mono G Set.inter_subset_left x M)
-  rintro y ⟨l,hl,hlen,ho⟩
-  exact ⟨l,hl,hlen,fun e he => ⟨ho e he,hl.bonds_in_incidentBall hloc M hlen e he⟩⟩
-
-theorem integral_eq_bernoulliPolynomial (Q : Finset E) (p : unitInterval) (f : Set E → ℝ)
-    (hf : ∀ ω, f ω = f (ω ∩ (Q:Set E))) :
-    (∫ ω, f ω ∂law p) = bernoulliPolynomial Q (fun η => f η) p := by
-  rw [integral_finite_local Q p f hf]
-  simp only [finiteFieldWeight,finiteFieldSet]
-  rw [Finset.sum_coe_sort Q.powerset (fun η => bernoulliWeight Q η p * f η)]
-  unfold bernoulliPolynomial
-  apply Finset.sum_congr rfl
-  intro η _
-  unfold bernoulliWeight
-  ring
-
-end IntrinsicBall
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section IntrinsicMaximum
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem intrinsicBall_exists_max (G : BondGraph V E) (hloc : LocallyFinite G)
-    (ω : Set E) (x : V) (M : ℕ) (f : V → ℝ) :
-    ∃ z ∈ intrinsicBall G ω x M, ∀ y ∈ intrinsicBall G ω x M, f y ≤ f z :=
-  Set.exists_max_image _ f ((finite_reachBall hloc x M).subset
-    (intrinsicBall_subset_reachBall G ω x M)) ⟨x,mem_intrinsicBall_self G ω x M⟩
-
-noncomputable def intrinsicMaximizer (G : BondGraph V E) (hloc : LocallyFinite G)
-    (ω : Set E) (x : V) (M : ℕ) (f : V → ℝ) : V :=
-  Classical.choose (intrinsicBall_exists_max G hloc ω x M f)
-
-theorem intrinsicMaximizer_spec (G : BondGraph V E) (hloc : LocallyFinite G)
-    (ω : Set E) (x : V) (M : ℕ) (f : V → ℝ) :
-    intrinsicMaximizer G hloc ω x M f ∈ intrinsicBall G ω x M ∧
-      ∀ y ∈ intrinsicBall G ω x M, f y ≤ f (intrinsicMaximizer G hloc ω x M f) :=
-  Classical.choose_spec (intrinsicBall_exists_max G hloc ω x M f)
-
-noncomputable def intrinsicPotentialMax (G : BondGraph V E) (ω : Set E)
-    (x : V) (M : ℕ) (f : V → ℝ) : ℝ := sSup (f '' intrinsicBall G ω x M)
-
-theorem intrinsicPotentialMax_eq (G : BondGraph V E) (hloc : LocallyFinite G)
-    (ω : Set E) (x : V) (M : ℕ) (f : V → ℝ) :
-    intrinsicPotentialMax G ω x M f = f (intrinsicMaximizer G hloc ω x M f) := by
-  have h := intrinsicMaximizer_spec G hloc ω x M f
-  apply IsGreatest.csSup_eq
-  exact ⟨⟨_,h.1,rfl⟩,fun _ ⟨y,hy,he⟩ => he ▸ h.2 y hy⟩
-
-theorem le_intrinsicPotentialMax (G : BondGraph V E) (hloc : LocallyFinite G)
-    (ω : Set E) (x : V) (M : ℕ) (f : V → ℝ) (y : V) (hy : y ∈ intrinsicBall G ω x M) :
-    f y ≤ intrinsicPotentialMax G ω x M f := by
-  rw [intrinsicPotentialMax_eq G hloc]
-  exact (intrinsicMaximizer_spec G hloc ω x M f).2 y hy
-
-noncomputable def intrinsicBondSupport (G : BondGraph V E) (hloc : LocallyFinite G)
-    (S : Set E) (x : V) (M : ℕ) : Finset E :=
-  (incidentSupport G hloc (finite_reachBall hloc x M).toFinset).filter (· ∈ S)
-
-theorem intrinsicBondSupport_coe (G : BondGraph V E) (hloc : LocallyFinite G)
-    (S : Set E) (x : V) (M : ℕ) :
-    (intrinsicBondSupport G hloc S x M : Set E) =
-      (incidentSupport G hloc (finite_reachBall hloc x M).toFinset : Set E) ∩ S := by
-  ext e
-  simp [intrinsicBondSupport]
-
-theorem intrinsicBall_domain_restrict (G : BondGraph V E) (hloc : LocallyFinite G)
-    (S ω : Set E) (x : V) (M : ℕ) :
-    intrinsicBall G (ω ∩ (intrinsicBondSupport G hloc S x M : Set E)) x M =
-      intrinsicBall G (ω ∩ S) x M := by
-  rw [intrinsicBondSupport_coe]
-  have he : ω ∩ ((incidentSupport G hloc (finite_reachBall hloc x M).toFinset : Set E) ∩ S) =
-      (ω ∩ S) ∩ (incidentSupport G hloc (finite_reachBall hloc x M).toFinset : Set E) := by
-    ext e; simp [and_left_comm,and_comm,and_assoc]
-  rw [he,intrinsicBall_restrict G hloc]
-
-theorem intrinsicPotentialMax_domain_local (G : BondGraph V E) (hloc : LocallyFinite G)
-    (S ω : Set E) (x : V) (M : ℕ) (f : V → ℝ) :
-    intrinsicPotentialMax G (ω ∩ S) x M f =
-      intrinsicPotentialMax G ((ω ∩ (intrinsicBondSupport G hloc S x M : Set E)) ∩ S) x M f := by
-  have hsub : (intrinsicBondSupport G hloc S x M : Set E) ⊆ S := by
-    rw [intrinsicBondSupport_coe]; exact Set.inter_subset_right
-  rw [Set.inter_assoc,Set.inter_eq_left.mpr hsub]
-  unfold intrinsicPotentialMax
-  rw [intrinsicBall_domain_restrict G hloc]
-
-end IntrinsicMaximum
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section MaximalSusceptibility
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem maximal_restricted_susceptibility_bound
-    [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (S : Set E) (x : V) (M : ℕ) (p : unitInterval)
-    (hp : 0 < (p:ℝ)) (hpc : p < pc G) :
-    (∫ ω, intrinsicPotentialMax G (ω ∩ S) x M (restrictedSusceptibility G S p) ∂law p) ≤
-      (1+(1-(p:ℝ))/p*M)*restrictedSusceptibility G S p x := by
-  let K := intrinsicBondSupport G hloc S x M
-  let f := restrictedSusceptibility G S p
-  let Z : Finset E → V := fun η => intrinsicMaximizer G hloc η x M f
-  have hK : (K:Set E) ⊆ S := by
-    rw [intrinsicBondSupport_coe]; exact Set.inter_subset_right
-  rw [integral_eq_bernoulliPolynomial K p _ (fun ω =>
-    intrinsicPotentialMax_domain_local G hloc S ω x M f)]
-  have he : bernoulliPolynomial K (fun η => intrinsicPotentialMax G ((η:Set E) ∩ S) x M f) p =
-      bernoulliPolynomial K (fun η => f (Z η)) p := by
-    apply bernoulliPolynomial_congr K _ p
-    intro η hη
-    rw [Set.inter_eq_left.mpr (fun e he => hK (hη he))]
-    exact intrinsicPotentialMax_eq G hloc η x M f
-  rw [he]
-  apply subcritical_selected_susceptibility_bound G hloc hqt S K hK x M Z _ p hp hpc
-  intro η _
-  exact (intrinsicMaximizer_spec G hloc η x M f).1
-
- 
-
-noncomputable def inducedSusceptibility (G : BondGraph V E) (D : Set V)
-    (p : unitInterval) (x : V) : ℝ :=
-  if x ∈ D then restrictedSusceptibility G (internalEdges G D) p x else 0
-
-theorem intrinsicBall_internal_subset (G : BondGraph V E) (D : Set V)
-    (ω : Set E) (x : V) (hx : x ∈ D) (M : ℕ) :
-    intrinsicBall G (ω ∩ internalEdges G D) x M ⊆ D := by
-  rintro y ⟨l,hl,_,ho⟩
-  exact conn_internal_endpoint G D hx (hl.conn ho)
-
-theorem inducedSusceptibility_max_eq (G : BondGraph V E) (D : Set V)
-    (ω : Set E) (p : unitInterval) (x : V) (hx : x ∈ D) (M : ℕ) :
-    intrinsicPotentialMax G (ω ∩ internalEdges G D) x M (inducedSusceptibility G D p) =
-      intrinsicPotentialMax G (ω ∩ internalEdges G D) x M (restrictedSusceptibility G (internalEdges G D) p) := by
-  unfold intrinsicPotentialMax
-  congr 1
-  apply Set.image_congr
-  intro y hy
-  exact ite_eq_left (intrinsicBall_internal_subset G D ω x hx M hy)
-
- 
-
-theorem maximal_induced_susceptibility_bound
-    [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (D : Set V) (x : V) (hx : x ∈ D) (M : ℕ) (p : unitInterval)
-    (hp : 0 < (p:ℝ)) (hpc : p < pc G) :
-    (∫ ω, intrinsicPotentialMax G (ω ∩ internalEdges G D) x M (inducedSusceptibility G D p) ∂law p) ≤
-      (1+(1-(p:ℝ))/p*M)*inducedSusceptibility G D p x := by
-  simp_rw [inducedSusceptibility_max_eq G D _ p x hx M]
-  rw [inducedSusceptibility,ite_eq_left hx]
-  exact maximal_restricted_susceptibility_bound G hloc hqt (internalEdges G D) x M p hp hpc
-
-end MaximalSusceptibility
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section RestrictedPathPotential
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem restrictedTwoPoint_adj_lower [Countable E] (G : BondGraph V E)
-    (S : Set E) (p : unitInterval) (x y z : V) (e : E)
-    (he : G e = s(x,y)) (heS : e ∈ S) :
-    (p:ℝ)*(law p).real {ω | Conn G (ω ∩ S) y z} ≤
-      (law p).real {ω | Conn G (ω ∩ S) x z} := by
-  have h := association_finiteWitness p {ω : Set E | e ∈ ω} {ω | Conn G (ω ∩ S) y z}
-    (fun _ _ hω h => hω h)
-    (fun _ _ hω h => conn_mono G (Set.inter_subset_inter_left _ hω) h)
-    (finiteWitness_edge e) (finiteWitness_inter _ (finiteWitness_conn G y z) S)
-  have hp : (law p).real {ω : Set E | e ∈ ω} = (p:ℝ) := by
-    simpa using finite_open_probability p {e}
-  rw [hp] at h
-  exact h.trans (measureReal_mono (fun ω hω =>
-    conn_trans G _ (Relation.ReflTransGen.single ⟨e,⟨hω.1,heS⟩,he⟩) hω.2))
-
-theorem restrictedTwoPoint_path_lower [Countable E] (G : BondGraph V E)
-    (S : Set E) (p : unitInterval) {x y : V} {l : List E}
-    (hl : Along G x l y) (hS : ∀ e ∈ l, e ∈ S) (z : V) :
-    (p:ℝ)^l.length*(law p).real {ω | Conn G (ω ∩ S) y z} ≤
-      (law p).real {ω | Conn G (ω ∩ S) x z} := by
-  induction hl with
-  | nil => simp
-  | @cons x a y l e he ht ih =>
-    have hh := mul_le_mul_of_nonneg_left (ih (fun d hd => hS d (List.mem_cons_of_mem _ hd))) p.2.1
-    have ha := restrictedTwoPoint_adj_lower G S p x a z e he (hS e (by simp))
-    simpa only [List.length_cons,pow_succ',mul_assoc] using hh.trans ha
-
-theorem restrictedSusceptibility_nonneg (G : BondGraph V E) (S : Set E)
-    (p : unitInterval) (x : V) : 0 ≤ restrictedSusceptibility G S p x :=
-  tsum_nonneg (fun _ => measureReal_nonneg)
-
-theorem restrictedSusceptibility_path_lower [Countable E] (G : BondGraph V E)
-    (S : Set E) (p : unitInterval) {x y : V} {l : List E}
-    (hl : Along G x l y) (hS : ∀ e ∈ l, e ∈ S)
-    (hfin : ∀ a, Summable (fun z => (law p).real {ω | Conn G (ω ∩ S) a z})) :
-    (p:ℝ)^l.length*restrictedSusceptibility G S p y ≤ restrictedSusceptibility G S p x := by
-  unfold restrictedSusceptibility
-  rw [←tsum_mul_left]
-  exact ((hfin y).mul_left _).tsum_le_tsum (restrictedTwoPoint_path_lower G S p hl hS) (hfin x)
-
- 
-
-theorem restrictedSusceptibility_short_path_bound [Countable E] (G : BondGraph V E)
-    (S : Set E) (p : unitInterval) (hp : 0 < (p:ℝ)) {x y : V} {l : List E}
-    (hl : Along G x l y) (hS : ∀ e ∈ l, e ∈ S) (M : ℕ) (hlen : l.length ≤ M)
-    (hfin : ∀ a, Summable (fun z => (law p).real {ω | Conn G (ω ∩ S) a z})) :
-    restrictedSusceptibility G S p y ≤ (p:ℝ)^(-(M:ℤ))*restrictedSusceptibility G S p x := by
-  have hh := restrictedSusceptibility_path_lower G S p hl hS hfin
-  have hpow : (p:ℝ)^M ≤ (p:ℝ)^l.length :=
-    pow_le_pow_of_le_one p.2.1 p.2.2 hlen
-  have hbound := (mul_le_mul_of_nonneg_right hpow (restrictedSusceptibility_nonneg G S p y)).trans hh
-  rw [zpow_neg,zpow_natCast,←div_eq_inv_mul]
-  exact (le_div_iff₀ (pow_pos hp M)).mpr (by simpa only [mul_comm] using hbound)
-
-end RestrictedPathPotential
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section ShortPathsDeletedDomain
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem conn_short_of_cluster_cap (G : BondGraph V E) (ω : Set E) (x y : V)
-    (hxy : Conn G ω x y) (hfin : (cluster G ω x).Finite)
-    (M : ℕ) (hM : (cluster G ω x).ncard ≤ M) :
-    ∃ l : List E, Along G x l y ∧ l.length ≤ M ∧ ∀ e ∈ l, e ∈ ω := by
-  obtain ⟨w⟩ := (conn_iff_reachable G ω x y).mp hxy
-  let q := w.toPath
-  have hs : q.val.support.toFinset ⊆ hfin.toFinset := by
-    intro z hz
-    have hz' : z ∈ q.val.support := List.mem_toFinset.mp hz
-    exact hfin.mem_toFinset.mpr ((conn_iff_reachable G ω x z).mpr ⟨q.val.takeUntil z hz'⟩)
-  have hc : q.val.length+1 ≤ (cluster G ω x).ncard := by
-    have hc := Finset.card_le_card hs
-    rwa [List.toFinset_card_of_nodup q.property.support_nodup,
-      SimpleGraph.Walk.length_support,←Set.ncard_eq_toFinset_card _ hfin] at hc
-  obtain ⟨l,hl,he,ho⟩ := walk_lift_bonds G ω q.val
-  have hlen : l.length = q.val.length := by
-    have hh := congrArg List.length he
-    simpa only [List.length_map,SimpleGraph.Walk.length_edges] using hh
-  exact ⟨l,hl,by omega,ho⟩
-
- 
-
-theorem Along.last_entrance {G : BondGraph V E} {x y : V} {l : List E}
-    (hl : Along G x l y) (D : Set V) (hy : y ∈ D) :
-    (x ∈ D ∧ ∀ e ∈ l, e ∈ internalEdges G D) ∨
-      ∃ a b e l₁ l₂, l = l₁ ++ e::l₂ ∧ Along G x l₁ a ∧ G e = s(a,b) ∧
-        a ∉ D ∧ b ∈ D ∧ Along G b l₂ y ∧ ∀ d ∈ l₂, d ∈ internalEdges G D := by
-  induction hl with
-  | nil => exact Or.inl ⟨hy,by simp⟩
-  | @cons x z y l e he ht ih =>
-    rcases ih hy with ⟨hz,hall⟩ | ⟨a,b,d,l₁,l₂,heq,h₁,hd,ha,hb,h₂,hs⟩
-    · by_cases hx : x ∈ D
-      · left
-        refine ⟨hx,?_⟩
-        intro d hd
-        rcases List.mem_cons.mp hd with rfl | hd
-        · exact (mem_internalEdges_of_ends G D d x z he).mpr ⟨hx,hz⟩
-        · exact hall d hd
-      · right
-        exact ⟨x,z,e,[],l,rfl,.nil x,he,hx,hz,ht,hall⟩
-    · right
-      exact ⟨a,b,d,e::l₁,l₂,by simp [heq],.cons e he h₁,hd,ha,hb,h₂,hs⟩
-
-end ShortPathsDeletedDomain
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Kernel.ScaleBudget
+
+namespace OAI
+
+/-! Resampling for bond percolation. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open Filter
+section ContractingIteration
+
+ 
+
+theorem exists_efficient_of_contracting_successor {S : Type*}
+    (g H : S → ℝ) (C : ℝ) (hC : 0 ≤ C) (hC1 : C < 1)
+    (hstep : ∀ x, ∃ y, g y ≤ C*g x ∧ 2*H x ≤ H y)
+    (x₀ : S) (hH₀ : 0 < H x₀) (η R : ℝ) (hη : 0 < η) :
+    ∃ x, g x < η ∧ R < H x := by
+  classical
+  choose f hf using hstep
+  let x : ℕ → S := fun n => f^[n] x₀
+  have hx (n : ℕ) : x (n+1) = f (x n) := Function.iterate_succ_apply' f n x₀
+  have hbounds (n : ℕ) : g (x n) ≤ C^n*g x₀ ∧ (2:ℝ)^n*H x₀ ≤ H (x n) := by
+    induction n with
+    | zero => simp only [x,Function.iterate_zero,id_eq,pow_zero,one_mul,le_refl,and_self]
+    | succ n ih =>
+      rw [hx]
+      obtain ⟨hg,hH⟩ := hf (x n)
+      constructor
+      · calc
+          _ ≤ C*g (x n) := hg
+          _ ≤ C*(C^n*g x₀) := mul_le_mul_of_nonneg_left ih.1 hC
+          _ = _ := by rw [pow_succ]; ring
+      · calc
+          _ = 2*((2:ℝ)^n*H x₀) := by rw [pow_succ]; ring
+          _ ≤ 2*H (x n) := mul_le_mul_of_nonneg_left ih.2 (by norm_num)
+          _ ≤ _ := hH
+  have hg : Tendsto (fun n : ℕ => C^n*g x₀) atTop (nhds 0) := by
+    simpa only [zero_mul] using (tendsto_pow_atTop_nhds_zero_of_lt_one hC hC1).mul_const (g x₀)
+  have hH : Tendsto (fun n : ℕ => (2:ℝ)^n*H x₀) atTop atTop :=
+    (tendsto_pow_atTop_atTop_of_one_lt (by norm_num : (1:ℝ) < 2)).atTop_mul_const hH₀
+  obtain ⟨n,hg',hH'⟩ := ((tendsto_order.mp hg).2 η hη |>.and
+    (hH.eventually (eventually_gt_atTop R))).exists
+  exact ⟨x n,(hbounds n).1.trans_lt hg',hH'.trans_le (hbounds n).2⟩
+
+end ContractingIteration
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section EfficientEndpointProfiles
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem efficient_endpoint_profiles_of_divergent_moment
+    [Countable V] [Countable E] [Nonempty V] [Nonempty E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hconn : Connected G)
+    (hqt : QuasiTransitive G) (hh : 0 < hV G)
+    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
+    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
+    (p : unitInterval) (hp : 0 < (p:ℝ)) (hp1 : (p:ℝ) < 1)
+    (hfin : ∀ i, ∀ᵐ ω ∂law p, (cluster G ω (root i)).Finite)
+    (α β ε : ℝ) (hα : 0 < α) (hαβ : α < β)
+    (hε : 0 < ε) (hε1 : ε < 1/2) (hsmall : β+(1/2+ε) < 1)
+    (hdiv : ∃ i, ¬ Integrable (fun ω => ((cluster G ω (root i)).ncard:ℝ)^α) (law p)) :
+    ∃ i, ∀ η : ℝ, 0 < η → ∀ R : ℝ,
+      ∃ q : EndpointProfile G hloc p (root i) (1/2+ε),
+        2 ≤ q.cap ∧ 1 < q.H ∧ R < q.H ∧ q.U+q.m < η*q.H := by
+  have hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i) := by
+    intro x
+    obtain ⟨i,e,he,hx⟩ := hcov x
+    obtain ⟨b,hb⟩ := hΓ he
+    exact ⟨i,e,b,hb,hx⟩
+  obtain ⟨i,hscales⟩ := orbit_scales_of_divergent_moment G hloc root weight Γ hΓ htr hw hw1 hcover p α β hα hαβ hdiv
+  obtain ⟨ξ,hξ,hξ1,hC,hC1⟩ := exists_amplification_contraction β (1/2+ε) (by linarith) (by linarith) hsmall
+  have hsξ : 0 < 1/2+ε+ξ := by linarith
+  obtain ⟨M₀,hM₀⟩ := eventually_atTop.1 (eventually_inverseCodeMass_budget (1/2+ε) ξ (by linarith) hξ)
+  obtain ⟨q₀,hq₀,hH₀⟩ := starting_endpoint_profile G hloc hqt hh p hp hp1 root weight Γ hΓ htr
+    (fun i => (hw i).le) hw1 i (hw i) (hfin i) ε hε hε1 M₀
+  obtain ⟨d,_,hdeg⟩ := exists_degree_bound G hqt
+  let c : ℝ := ∑ j, 1/weight j
+  have hc : 0 ≤ c := Finset.sum_nonneg (fun j _ => (one_div_pos.mpr (hw j)).le)
+  have hmin (j : ι) : 1 ≤ c*weight j := by
+    apply (div_le_iff₀ (hw j)).mp
+    exact Finset.single_le_sum (fun l _ => (one_div_pos.mpr (hw l)).le) (Finset.mem_univ j)
+  let S := {q : EndpointProfile G hloc p (root i) (1/2+ε) // M₀ ≤ q.cap ∧ 1 < q.H}
+  let g : S → ℝ := fun q => q.val.cost ξ/q.val.H
+  let H : S → ℝ := fun q => q.val.H
+  have hstep : ∀ q : S, ∃ q' : S, g q' ≤ amplificationContraction β (1/2+ε) ξ*g q ∧ 2*H q ≤ H q' := by
+    intro q
+    have hcap := hM₀ q.val.cap q.property.1
+    have hH : 0 < q.val.H := lt_trans zero_lt_one q.property.2
+    obtain ⟨r,k,hr,hk,hU,hm,hH',hH''⟩ := q.val.amplify_at_scales hconn hqt root weight Γ hΓ htr hcov hw hw1
+      c hc hmin i p hp1 d hdeg ε ξ β hε hε1 hξ hξ1 hscales hcap.1 hcap.2.1 hcap.2.2 hH M₀
+    have hrH : 1 < r.H := by linarith [q.property.2]
+    refine ⟨⟨r,hr,hrH⟩,?_,hH''⟩
+    exact q.val.cost_contraction r β ξ k hcap.1 hsξ hH hξ1 hk hC hU hm hH'
+  refine ⟨i,?_⟩
+  intro η hη R
+  let D : ℝ := 1+1/(1/2+ε+ξ)
+  have hD : 0 < D := by dsimp [D]; positivity
+  obtain ⟨q,hg,hR⟩ := exists_efficient_of_contracting_successor g H (amplificationContraction β (1/2+ε) ξ)
+    hC hC1 hstep ⟨q₀,hq₀,hH₀⟩ (lt_trans zero_lt_one hH₀) (η/D) R (div_pos hη hD)
+  have hcap := (hM₀ q.val.cap q.property.1).1
+  refine ⟨q.val,hcap,q.property.2,hR,?_⟩
+  have hH : 0 < q.val.H := lt_trans zero_lt_one q.property.2
+  have hcost := q.val.total_cost_le hcap ξ hsξ
+  have hratio : (q.val.U+q.val.m)/q.val.H ≤ D*g q := by
+    have h := div_le_div_of_nonneg_right hcost hH.le
+    simpa only [D,g,mul_div_assoc] using h
+  apply (div_lt_iff₀ hH).mp
+  apply hratio.trans_lt
+  have hg' := (lt_div_iff₀ hD).mp hg
+  simpa only [mul_comm] using hg'
+
+end EfficientEndpointProfiles
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section ShortBoundaryPivotal
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+def ShortClosedBoundary (G : BondGraph V E) (ω : Set E) (x z : V) (M : ℕ) (e : E) : Prop :=
+  e ∉ ω ∧ ¬Conn G ω x z ∧ ∃ a b : V, ∃ l : List E,
+    G e = s(a,b) ∧ Along G x l a ∧ (∀ f ∈ l, f ∈ ω) ∧ l.length < M ∧ Conn G ω b z
+
+theorem bridgeIndex_le_short_trail (G : BondGraph V E) (ω : Set E) (x z : V)
+    (hC : (cluster G ω x).Finite) (e : E) (he : OpenMarkedPivotal G ω {z} x e)
+    (a b : V) (hab : G e = s(a,b)) (l : List E) (hl : Along G x l a)
+    (ho : ∀ f ∈ l, f ∈ ω) : bridgeIndex G ω x z e ≤ l.length+1 := by
+  unfold bridgeIndex
+  have hsub : (separatingBonds G ω x z).filter (fun f => pivotalRank G ω x f ≤ pivotalRank G ω x e) ⊆
+      (e::l).toFinset := by
+    intro f hf
+    obtain ⟨hf,hfe⟩ := Finset.mem_filter.mp hf
+    have hfp := (mem_separatingBonds G ω x z f).mp hf
+    by_cases h : f = e
+    · subst f; simp
+    by_contra hnot
+    have hn : f ∉ l := fun hfl => hnot (List.mem_toFinset.mpr (List.mem_cons_of_mem _ hfl))
+    have hlt : pivotalRank G ω x f < pivotalRank G ω x e := by
+      apply lt_of_le_of_ne hfe
+      intro heq
+      exact h (pivotalRank_injective_of_cluster_finite G ω {z} x hC hfp he heq)
+    have hout := (pivotalRank_lt_iff_of_cluster_finite G ω {z} x hC hfp he).mp hlt
+    have hb : ¬ Conn G (ω \ {f}) x b := ((mem_internalEdges_of_ends G _ e a b hab).mp hout).2
+    apply hb
+    have hxa : Conn G (ω \ {f}) x a := hl.conn (fun g hg => ⟨ho g hg,fun hgf => hn ((Set.mem_singleton_iff.mp hgf) ▸ hg)⟩)
+    exact hxa.tail ⟨e,⟨he.1,fun hEq => h hEq.symm⟩,hab⟩
+  exact (Finset.card_le_card hsub).trans (by simpa only [List.length_cons] using List.toFinset_card_le (e::l))
+
+theorem ShortClosedBoundary.open_pivotal (G : BondGraph V E) (ω : Set E) (x z : V)
+    (M : ℕ) (e : E) (h : ShortClosedBoundary G ω x z M e) :
+    OpenMarkedPivotal G (insert e ω) {z} x e := by
+  obtain ⟨he,hxz,a,b,l,hab,hl,ho,_,hbz⟩ := h
+  refine ⟨Set.mem_insert e ω,⟨z,rfl,?_⟩,?_⟩
+  · have hxa := hl.conn ho
+    exact (conn_mono G (Set.subset_insert e ω) hxa).trans
+      ((Relation.ReflTransGen.single ⟨e,Set.mem_insert e ω,hab⟩).trans
+        (conn_mono G (Set.subset_insert e ω) hbz))
+  · rintro ⟨y,hy,hxy⟩
+    have hy' : y = z := Set.mem_singleton_iff.mp hy
+    subst y
+    apply hxz
+    apply conn_mono G _ hxy
+    intro f hf
+    rcases hf.1 with hfe | hfω
+    · exact (hf.2 (Set.mem_singleton_iff.mpr hfe)).elim
+    · exact hfω
+
+theorem ShortClosedBoundary.bridgeIndex_le (G : BondGraph V E) (ω : Set E) (x z : V)
+    (hω : ω.Finite) (M : ℕ) (e : E) (h : ShortClosedBoundary G ω x z M e) :
+    bridgeIndex G (insert e ω) x z e ≤ M := by
+  have hpiv := ShortClosedBoundary.open_pivotal G ω x z M e h
+  obtain ⟨_,_,a,b,l,hab,hl,ho,hlen,_⟩ := h
+  have hc := cluster_finite_of_finite_edges G (hω.insert e) x
+  have hb := bridgeIndex_le_short_trail G (insert e ω) x z hc e hpiv
+    a b hab l hl (fun f hf => Set.mem_insert_of_mem e (ho f hf))
+  exact hb.trans (by omega)
+
+end ShortBoundaryPivotal
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section ShortBoundaryCounting
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def firstOpenBridgeCount (G : BondGraph V E) (Q : Finset E)
+    (x z : V) (M : ℕ) (ω : Finset E) : ℕ :=
+  (Q.filter (fun e => OpenMarkedPivotal G ω {z} x e ∧ bridgeIndex G ω x z e ≤ M)).card
+
+noncomputable def shortClosedBoundaryCount (G : BondGraph V E) (Q : Finset E)
+    (x z : V) (M : ℕ) (ω : Finset E) : ℕ :=
+  (Q.filter (ShortClosedBoundary G ω x z M)).card
+
+theorem firstOpenBridgeCount_le (G : BondGraph V E) (Q : Finset E)
+    (x z : V) (M : ℕ) (ω : Finset E) :
+    firstOpenBridgeCount G Q x z M ω ≤ if Conn G ω x z then M else 0 := by
+  by_cases hc : Conn G ω x z
+  · rw [ite_eq_left hc]
+    unfold firstOpenBridgeCount
+    let S := Q.filter (fun e => OpenMarkedPivotal G ω {z} x e ∧ bridgeIndex G ω x z e ≤ M)
+    have hm : Set.MapsTo (bridgeIndex G ω x z) (S : Set E) (Finset.Icc 1 M) := by
+      intro e he
+      have he' := (Finset.mem_filter.mp he).2
+      exact Finset.mem_Icc.mpr ⟨bridgeIndex_pos G ω x z he'.1,he'.2⟩
+    have hi : Set.InjOn (bridgeIndex G ω x z) (S : Set E) := by
+      intro e he f hf h
+      exact bridgeIndex_injective G ω x z (cluster_finite_of_finite_edges G ω.finite_toSet x)
+        (Finset.mem_filter.mp he).2.1 (Finset.mem_filter.mp hf).2.1 h
+    simpa only [Nat.card_Icc,Nat.add_sub_cancel] using
+      (Finset.card_le_card_of_injOn (bridgeIndex G ω x z) hm hi)
+  · rw [ite_eq_right hc]
+    unfold firstOpenBridgeCount
+    have he : Q.filter (fun e => OpenMarkedPivotal G ω {z} x e ∧ bridgeIndex G ω x z e ≤ M) = ∅ := by
+      apply Finset.eq_empty_iff_forall_notMem.mpr
+      intro e he
+      obtain ⟨y,hy,hxy⟩ := (Finset.mem_filter.mp he).2.1.2.1
+      exact hc ((Set.mem_singleton_iff.mp hy) ▸ hxy)
+    rw [he,Finset.card_empty]
+
+theorem bernoulliPolynomial_bit_balance (Q : Finset E) (e : E) (he : e ∈ Q)
+    (f : Finset E → ℝ) (p : ℝ) :
+    p*bernoulliPolynomial Q (fun ω => if e ∉ ω then f ω else 0) p =
+      (1-p)*bernoulliPolynomial Q (fun ω => if e ∈ ω then f (ω.erase e) else 0) p := by
+  conv_lhs => rw [←Finset.insert_erase he]
+  conv_rhs => rw [←Finset.insert_erase he]
+  rw [bernoulliPolynomial_insert _ e (Finset.notMem_erase e Q),
+    bernoulliPolynomial_insert _ e (Finset.notMem_erase e Q)]
+  have hc : bernoulliPolynomial (Q.erase e) (fun ω => if e ∉ ω then f ω else 0) p =
+      bernoulliPolynomial (Q.erase e) f p := by
+    apply bernoulliPolynomial_congr
+    intro ω hω
+    simp only [show e ∉ ω from fun h => Finset.notMem_erase e Q (hω h),not_false_eq_true,ite_true]
+  have hco : bernoulliPolynomial (Q.erase e) (fun ω => if e ∉ insert e ω then f (insert e ω) else 0) p = 0 := by
+    simp only [Finset.mem_insert_self,not_true_eq_false,ite_false,bernoulliPolynomial_zero]
+  have ho : bernoulliPolynomial (Q.erase e) (fun ω => if e ∈ ω then f (ω.erase e) else 0) p = 0 := by
+    calc
+      _ = bernoulliPolynomial (Q.erase e) (fun _ => 0) p := by
+        apply bernoulliPolynomial_congr
+        intro ω hω
+        simp only [show e ∉ ω from fun h => Finset.notMem_erase e Q (hω h),ite_false]
+      _ = 0 := bernoulliPolynomial_zero _ _
+  have hoo : bernoulliPolynomial (Q.erase e)
+      (fun ω => if e ∈ insert e ω then f ((insert e ω).erase e) else 0) p =
+      bernoulliPolynomial (Q.erase e) f p := by
+    apply bernoulliPolynomial_congr
+    intro ω hω
+    simp only [Finset.mem_insert_self,ite_true,Finset.erase_insert
+      (show e ∉ ω from fun h => Finset.notMem_erase e Q (hω h))]
+  rw [hc,hco,ho,hoo]
+  ring
+
+end ShortBoundaryCounting
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section ShortBoundaryExpectation
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem bernoulliPolynomial_closed_count_balance (Q : Finset E)
+    (P : Finset E → E → Prop) (hP : ∀ ω e, P ω e → e ∉ ω) (p : ℝ) :
+    p*bernoulliPolynomial Q (fun ω => ((Q.filter (P ω)).card:ℝ)) p =
+      (1-p)*bernoulliPolynomial Q
+        (fun ω => ((Q.filter (fun e => e ∈ ω ∧ P (ω.erase e) e)).card:ℝ)) p := by
+  have hbal (e : E) (he : e ∈ Q) :
+      p*bernoulliPolynomial Q (fun ω => if P ω e then 1 else 0) p =
+        (1-p)*bernoulliPolynomial Q (fun ω => if e ∈ ω ∧ P (ω.erase e) e then 1 else 0) p := by
+    have h := bernoulliPolynomial_bit_balance Q e he (fun ω => if P ω e then 1 else 0) p
+    have hL : bernoulliPolynomial Q (fun ω => if e ∉ ω then (if P ω e then 1 else 0) else 0) p =
+        bernoulliPolynomial Q (fun ω => if P ω e then 1 else 0) p := by
+      apply bernoulliPolynomial_congr
+      intro ω _
+      by_cases hp : P ω e
+      · simp only [hp,hP ω e hp,not_false_eq_true,ite_true]
+      · simp only [hp,ite_false,ite_self]
+    have hR : bernoulliPolynomial Q (fun ω => if e ∈ ω then (if P (ω.erase e) e then 1 else 0) else 0) p =
+        bernoulliPolynomial Q (fun ω => if e ∈ ω ∧ P (ω.erase e) e then 1 else 0) p := by
+      apply bernoulliPolynomial_congr
+      intro ω _
+      by_cases hm : e ∈ ω <;> simp only [hm,true_and,false_and,ite_true,ite_false]
+    rw [hL,hR] at h
+    exact h
+  simp only [Finset.card_filter,Nat.cast_sum,Nat.cast_ite,Nat.cast_one,Nat.cast_zero,
+    bernoulliPolynomial_sum,Finset.mul_sum]
+  exact Finset.sum_congr rfl (fun e he => hbal e he)
+
+theorem shortClosedBoundary_erase_first (G : BondGraph V E) (ω : Finset E)
+    (x z : V) (M : ℕ) (e : E) (he : e ∈ ω)
+    (h : ShortClosedBoundary G (ω.erase e) x z M e) :
+    OpenMarkedPivotal G ω {z} x e ∧ bridgeIndex G ω x z e ≤ M := by
+  have h₁ := ShortClosedBoundary.open_pivotal G (ω.erase e) x z M e h
+  have h₂ := ShortClosedBoundary.bridgeIndex_le G (ω.erase e) x z (ω.erase e).finite_toSet M e h
+  rw [←Finset.coe_insert,Finset.insert_erase he] at h₁ h₂
+  exact ⟨h₁,h₂⟩
+
+theorem shortRecoveredBoundaryCount_le (G : BondGraph V E) (Q ω : Finset E)
+    (x z : V) (M : ℕ) :
+    (Q.filter (fun e => e ∈ ω ∧ ShortClosedBoundary G (ω.erase e) x z M e)).card ≤
+      if Conn G ω x z then M else 0 := by
+  apply le_trans _ (firstOpenBridgeCount_le G Q x z M ω)
+  apply Finset.card_le_card
+  intro e he
+  obtain ⟨heQ,heω,hb⟩ := Finset.mem_filter.mp he
+  exact Finset.mem_filter.mpr ⟨heQ,shortClosedBoundary_erase_first G ω x z M e heω hb⟩
+
+ 
+
+theorem finite_short_boundary_expectation (G : BondGraph V E) (Q : Finset E)
+    (x z : V) (M : ℕ) (p : ℝ) (hp : 0 < p) (hp1 : p ≤ 1) :
+    bernoulliPolynomial Q (fun ω => (shortClosedBoundaryCount G Q x z M ω:ℝ)) p ≤
+      ((1-p)/p)*M*bernoulliPolynomial Q (fun ω => if Conn G ω x z then 1 else 0) p := by
+  have hbal := bernoulliPolynomial_closed_count_balance Q
+    (fun ω e => ShortClosedBoundary G ω x z M e) (fun ω e h => h.1) p
+  have hmono := bernoulliPolynomial_mono Q
+    (fun ω => ((Q.filter (fun e => e ∈ ω ∧ ShortClosedBoundary G (ω.erase e) x z M e)).card:ℝ))
+    (fun ω => (M:ℝ)*(if Conn G ω x z then 1 else 0)) p ⟨hp.le,hp1⟩ (by
+      intro ω _
+      have h := shortRecoveredBoundaryCount_le G Q ω x z M
+      by_cases hc : Conn G ω x z
+      · simp only [hc,ite_true,mul_one] at *
+        exact_mod_cast h
+      · simp only [hc,ite_false,mul_zero] at *
+        exact_mod_cast h)
+  rw [bernoulliPolynomial_const_mul] at hmono
+  have hbound := mul_le_mul_of_nonneg_left hmono (sub_nonneg.mpr hp1)
+  rw [←hbal] at hbound
+  change p*bernoulliPolynomial Q (fun ω => (shortClosedBoundaryCount G Q x z M ω:ℝ)) p ≤ _ at hbound
+  have hdiv := (le_div_iff₀ hp).mpr (show
+      bernoulliPolynomial Q (fun ω => (shortClosedBoundaryCount G Q x z M ω:ℝ)) p*p ≤ _ by
+        simpa only [mul_comm p] using hbound)
+  simpa only [div_eq_mul_inv,mul_assoc,mul_left_comm,mul_comm] using hdiv
+
+end ShortBoundaryExpectation
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section ClusterResampling
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem bernoulliPolynomial_commute {F : Type*} (Q : Finset E) (R : Finset F)
+    (f : Finset E → Finset F → ℝ) (p q : ℝ) :
+    bernoulliPolynomial Q (fun a => bernoulliPolynomial R (f a) q) p =
+      bernoulliPolynomial R (fun b => bernoulliPolynomial Q (fun a => f a b) p) q := by
+  simp only [bernoulliPolynomial,Finset.sum_mul]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro b _
+  apply Finset.sum_congr rfl
+  intro a _
+  ring
+
+theorem exteriorBonds_union_blocks (G : BondGraph V E) (Q : Finset E)
+    (W : Set V) (a b : Finset E) (ha : a ⊆ Q \ exteriorBonds G Q W)
+    (hb : b ⊆ exteriorBonds G Q W) : exteriorBonds G (a ∪ b) W = b := by
+  ext e
+  simp only [exteriorBonds,Finset.mem_filter,Finset.mem_union]
+  constructor
+  · rintro ⟨he,heW⟩
+    rcases he with he | he
+    · obtain ⟨heQ,heR⟩ := Finset.mem_sdiff.mp (ha he)
+      exact (heR (Finset.mem_filter.mpr ⟨heQ,heW⟩)).elim
+    · exact he
+  · intro he
+    exact ⟨Or.inr he,(Finset.mem_filter.mp (hb he)).2⟩
+
+ 
+theorem cluster_cell_exterior_factor (G : BondGraph V E) (Q : Finset E)
+    (x : V) (W : Set V) (f : Finset E → ℝ) (p : ℝ) :
+    bernoulliPolynomial Q (fun ω =>
+      (if cluster G ω x = W then (1:ℝ) else 0)*f (exteriorBonds G ω W)) p =
+      bernoulliPolynomial (Q \ exteriorBonds G Q W)
+        (fun a => if cluster G a x = W then (1:ℝ) else 0) p *
+      bernoulliPolynomial (exteriorBonds G Q W) f p := by
+  let R := exteriorBonds G Q W
+  let A := Q \ R
+  have hQ : A ∪ R = Q := Finset.sdiff_union_of_subset (Finset.filter_subset _ _)
+  conv_lhs => rw [←hQ,bernoulliPolynomial_union A R Finset.sdiff_disjoint]
+  trans bernoulliPolynomial A (fun a =>
+    (if cluster G a x = W then (1:ℝ) else 0)*bernoulliPolynomial R f p) p
+  · apply bernoulliPolynomial_congr
+    intro a ha
+    rw [←bernoulliPolynomial_const_mul]
+    apply bernoulliPolynomial_congr
+    intro b hb
+    have hc := cluster_union_external_eq_iff G (a:Set E) b x W
+      (fun e he => (Finset.mem_filter.mp (hb he)).2)
+    rw [exteriorBonds_union_blocks G Q W a b ha hb]
+    simp only [Finset.coe_union,hc]
+  · exact bernoulliPolynomial_mul_const A _ _ p
+
+theorem cluster_cell_incident_probability (G : BondGraph V E) (Q : Finset E)
+    (x : V) (W : Set V) (p : ℝ) :
+    bernoulliPolynomial Q (fun ω => if cluster G ω x = W then (1:ℝ) else 0) p =
+      bernoulliPolynomial (Q \ exteriorBonds G Q W)
+        (fun a => if cluster G a x = W then (1:ℝ) else 0) p := by
+  have h := cluster_cell_exterior_factor G Q x W (fun _ => 1) p
+  simpa only [mul_one,bernoulliPolynomial_const] using h
+
+noncomputable def resampleClusterAt (G : BondGraph V E) (Q : Finset E)
+    (W : Set V) (ω η : Finset E) : Finset E :=
+  exteriorBonds G ω W ∪ (η ∩ (Q \ exteriorBonds G Q W))
+
+ 
+
+theorem cluster_resample_cell_independent (G : BondGraph V E) (Q : Finset E)
+    (x : V) (W : Set V) (f : Finset E → ℝ) (p : ℝ) :
+    bernoulliPolynomial Q (fun ω => bernoulliPolynomial Q (fun η =>
+      (if cluster G ω x = W then (1:ℝ) else 0)*f (resampleClusterAt G Q W ω η)) p) p =
+      bernoulliPolynomial Q (fun ω => if cluster G ω x = W then (1:ℝ) else 0) p *
+      bernoulliPolynomial Q f p := by
+  let R := exteriorBonds G Q W
+  let A := Q \ R
+  let c := bernoulliPolynomial A (fun a => if cluster G a x = W then (1:ℝ) else 0) p
+  rw [bernoulliPolynomial_commute]
+  trans bernoulliPolynomial Q (fun η => c * bernoulliPolynomial R
+    (fun b => f (b ∪ (η ∩ A))) p) p
+  · apply bernoulliPolynomial_congr
+    intro η _
+    exact cluster_cell_exterior_factor G Q x W (fun b => f (b ∪ (η ∩ A))) p
+  · rw [bernoulliPolynomial_const_mul,bernoulliPolynomial_restrict A Q Finset.sdiff_subset
+      (fun a => bernoulliPolynomial R (fun b => f (b ∪ a)) p) p,
+      bernoulliPolynomial_commute]
+    have hQ : R ∪ A = Q := Finset.union_sdiff_of_subset (Finset.filter_subset _ _)
+    rw [←bernoulliPolynomial_union R A Finset.disjoint_sdiff,hQ,
+      cluster_cell_incident_probability G Q x W p]
+
+end ClusterResampling
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section ResampledBoundary
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem resampleClusterAt_subset (G : BondGraph V E) (Q : Finset E)
+    (W : Set V) (ω η : Finset E) (hω : ω ⊆ Q) :
+    resampleClusterAt G Q W ω η ⊆ Q := by
+  intro e he
+  rcases Finset.mem_union.mp he with he | he
+  · exact hω (Finset.mem_filter.mp he).1
+  · exact (Finset.mem_sdiff.mp (Finset.mem_inter.mp he).2).1
+
+theorem resampleClusterAt_external (G : BondGraph V E) (Q : Finset E)
+    (W : Set V) (ω η : Finset E) {e : E} (he : e ∈ resampleClusterAt G Q W ω η)
+    (heW : e ∈ internalEdges G Wᶜ) : e ∈ ω := by
+  rcases Finset.mem_union.mp he with he | he
+  · exact (Finset.mem_filter.mp he).1
+  · obtain ⟨_,heA⟩ := Finset.mem_inter.mp he
+    obtain ⟨heQ,heR⟩ := Finset.mem_sdiff.mp heA
+    exact (heR (Finset.mem_filter.mpr ⟨heQ,heW⟩)).elim
+
+ 
+
+theorem resampled_path_boundary (G : BondGraph V E) (Q ω η : Finset E)
+    (hω : ω ⊆ Q) (x z y : V) (M : ℕ)
+    (hy : Conn G ω z y) (hx : ¬Conn G ω x z)
+    (l : List E) (hl : Along G x l y) (hlen : l.length ≤ M)
+    (ho : ∀ e ∈ l, e ∈ resampleClusterAt G Q (cluster G ω z) ω η) :
+    ∃ e ∈ Q, ShortClosedBoundary G ω x z M e := by
+  let W := cluster G ω z
+  have hxW : x ∉ W := fun h => hx (conn_symm G ω h)
+  rcases hl.first_outside (internalEdges G Wᶜ) with hall | ⟨a,b,e,l₁,l₂,heq,h₁,hab,_,hfirst,hen⟩
+  · have hc : Conn G ((Set.univ:Set E) ∩ internalEdges G Wᶜ) x y :=
+      hl.conn (fun e he => ⟨Set.mem_univ e,hall e he⟩)
+    exact ((conn_internal_endpoint G Wᶜ hxW hc) hy).elim
+  · subst l
+    have heopen := ho e (List.mem_append_right _ (by simp))
+    have heQ := resampleClusterAt_subset G Q W ω η hω heopen
+    have hprefix : ∀ f ∈ l₁, f ∈ ω := by
+      intro f hf
+      exact resampleClusterAt_external G Q W ω η
+        (ho f (List.mem_append_left _ hf)) (hfirst f hf)
+    have hxa : Conn G ((ω:Set E) ∩ internalEdges G Wᶜ) x a :=
+      h₁.conn (fun f hf => ⟨hprefix f hf,hfirst f hf⟩)
+    have haW : a ∉ W := conn_internal_endpoint G Wᶜ hxW hxa
+    have hbW : b ∈ W := by
+      by_contra hbW
+      exact hen ((mem_internalEdges_of_ends G Wᶜ e a b hab).mpr ⟨haW,hbW⟩)
+    have heclosed : e ∉ ω := by
+      intro heω
+      apply haW
+      exact (show Conn G ω z b from hbW).tail ⟨e,heω,hab.trans (Sym2.eq_swap ..)⟩
+    refine ⟨e,heQ,heclosed,hx,a,b,l₁,hab,h₁,hprefix,?_,conn_symm G ω hbW⟩
+    simp only [List.length_append,List.length_cons] at hlen
+    omega
+
+noncomputable def resampleRootCluster (G : BondGraph V E) (Q : Finset E)
+    (z : V) (ω η : Finset E) : Finset E :=
+  resampleClusterAt G Q (cluster G ω z) ω η
+
+ 
+theorem resampled_hit_le_boundaryCount (G : BondGraph V E) (Q ω η : Finset E)
+    (hω : ω ⊆ Q) (x z y : V) (M : ℕ)
+    (hpath : ∃ l : List E, Along G x l y ∧ l.length ≤ M ∧
+      ∀ e ∈ l, e ∈ resampleRootCluster G Q z ω η) :
+    (if Conn G ω z y then (1:ℝ) else 0) ≤
+      (if Conn G ω x z then (1:ℝ) else 0) + shortClosedBoundaryCount G Q x z M ω := by
+  by_cases hy : Conn G ω z y
+  · rw [ite_eq_left hy]
+    by_cases hx : Conn G ω x z
+    · rw [ite_eq_left hx]
+      exact le_add_of_nonneg_right (Nat.cast_nonneg _)
+    · rw [ite_eq_right hx,zero_add]
+      obtain ⟨l,hl,hlen,ho⟩ := hpath
+      obtain ⟨e,heQ,he⟩ := resampled_path_boundary G Q ω η hω x z y M hy hx l hl hlen ho
+      have hpos : 0 < shortClosedBoundaryCount G Q x z M ω :=
+        Finset.card_pos.mpr ⟨e,Finset.mem_filter.mpr ⟨heQ,he⟩⟩
+      exact_mod_cast hpos
+  · rw [ite_eq_right hy]
+    positivity
+
+end ResampledBoundary
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section RootResamplingLaw
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem rootCluster_resampling_law (G : BondGraph V E) (Q : Finset E)
+    (x : V) (f : Finset V → Finset E → ℝ) (p : ℝ) :
+    bernoulliPolynomial Q (fun ω => bernoulliPolynomial Q
+      (fun η => f (finiteRootComponent G ω x) (resampleRootCluster G Q x ω η)) p) p =
+      bernoulliPolynomial Q (fun ω => bernoulliPolynomial Q
+        (f (finiteRootComponent G ω x)) p) p := by
+  let F := insert x (searchVertices G Q)
+  let c := fun (W : Finset V) (ω : Finset E) => if cluster G ω x = (W:Set V) then (1:ℝ) else 0
+  have hpart (ω η : Finset E) (hω : ω ⊆ Q) :
+      f (finiteRootComponent G ω x) (resampleRootCluster G Q x ω η) =
+        ∑ W ∈ F.powerset, c W ω * f W (resampleClusterAt G Q W ω η) := by
+    have h := sum_component_partition G Q F x ω hω (by rfl)
+      (fun W => f W (resampleClusterAt G Q W ω η))
+    simpa only [c,ite_mul,one_mul,zero_mul,coe_finiteRootComponent,
+      resampleRootCluster] using h.symm
+  trans ∑ W ∈ F.powerset,
+    bernoulliPolynomial Q (c W) p * bernoulliPolynomial Q (f W) p
+  · trans bernoulliPolynomial Q (fun ω => bernoulliPolynomial Q (fun η =>
+      ∑ W ∈ F.powerset, c W ω * f W (resampleClusterAt G Q W ω η)) p) p
+    · apply bernoulliPolynomial_congr
+      intro ω hω
+      apply bernoulliPolynomial_congr
+      intro η _
+      exact hpart ω η hω
+    · simp only [bernoulliPolynomial_sum]
+      apply Finset.sum_congr rfl
+      intro W _
+      exact cluster_resample_cell_independent G Q x W (f W) p
+  · trans bernoulliPolynomial Q (fun ω => ∑ W ∈ F.powerset,
+      c W ω * bernoulliPolynomial Q (f W) p) p
+    · rw [bernoulliPolynomial_sum]
+      apply Finset.sum_congr rfl
+      intro W _
+      exact (bernoulliPolynomial_mul_const Q (c W) _ p).symm
+    · apply bernoulliPolynomial_congr
+      intro ω hω
+      have h := sum_component_partition G Q F x ω hω (by rfl)
+        (fun W => bernoulliPolynomial Q (f W) p)
+      simpa only [c,ite_mul,one_mul,zero_mul] using h
+
+end RootResamplingLaw
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section FiniteMaximalSusceptibility
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def finiteSusceptibility (G : BondGraph V E) (Q : Finset E)
+    (F : Finset V) (p : ℝ) (x : V) : ℝ :=
+  ∑ z ∈ F, bernoulliPolynomial Q (fun ω => if Conn G ω x z then (1:ℝ) else 0) p
+
+theorem finiteSusceptibility_nonneg (G : BondGraph V E) (Q : Finset E)
+    (F : Finset V) (p : ℝ) (x : V) (hp : p ∈ Set.Icc (0:ℝ) 1) :
+    0 ≤ finiteSusceptibility G Q F p x := by
+  apply Finset.sum_nonneg
+  intro z _
+  exact bernoulliPolynomial_nonneg Q _ p hp (fun _ _ => by positivity)
+
+theorem rootCluster_resampled_selector (G : BondGraph V E) (Q : Finset E)
+    (w : V) (Z : Finset E → V) (p : ℝ) :
+    bernoulliPolynomial Q (fun ω => bernoulliPolynomial Q (fun η =>
+      if Conn G ω w (Z (resampleRootCluster G Q w ω η)) then (1:ℝ) else 0) p) p =
+      bernoulliPolynomial Q (fun η => bernoulliPolynomial Q (fun ω =>
+        if Conn G ω w (Z η) then (1:ℝ) else 0) p) p := by
+  have hm (ω : Finset E) (y : V) :
+      y ∈ finiteRootComponent G ω w ↔ Conn G ω w y := by
+    change y ∈ (finiteRootComponent G ω w : Set V) ↔ _
+    rw [coe_finiteRootComponent]
+    rfl
+  have h := rootCluster_resampling_law G Q w (fun W η => if Z η ∈ W then (1:ℝ) else 0) p
+  simp only [hm] at h
+  exact h.trans (bernoulliPolynomial_commute Q Q _ p p)
+
+theorem finite_selector_connection_bound (G : BondGraph V E) (Q : Finset E)
+    (x w : V) (M : ℕ) (Z : Finset E → V)
+    (hZ : ∀ η ⊆ Q, ∃ l : List E, Along G x l (Z η) ∧ l.length ≤ M ∧ ∀ e ∈ l, e ∈ η)
+    (p : ℝ) (hp : 0 < p) (hp1 : p ≤ 1) :
+    bernoulliPolynomial Q (fun η => bernoulliPolynomial Q (fun ω =>
+      if Conn G ω w (Z η) then (1:ℝ) else 0) p) p ≤
+      (1+(1-p)/p*M)*bernoulliPolynomial Q (fun ω => if Conn G ω x w then (1:ℝ) else 0) p := by
+  rw [←rootCluster_resampled_selector G Q w Z p]
+  have hmono := bernoulliPolynomial_mono Q
+    (fun ω => bernoulliPolynomial Q (fun η =>
+      if Conn G ω w (Z (resampleRootCluster G Q w ω η)) then (1:ℝ) else 0) p)
+    (fun ω => (if Conn G ω x w then (1:ℝ) else 0)+shortClosedBoundaryCount G Q x w M ω)
+    p ⟨hp.le,hp1⟩ (by
+      intro ω hω
+      calc
+        _ ≤ bernoulliPolynomial Q (fun _ =>
+          (if Conn G ω x w then (1:ℝ) else 0)+shortClosedBoundaryCount G Q x w M ω) p := by
+            apply bernoulliPolynomial_mono Q _ _ p ⟨hp.le,hp1⟩
+            intro η _
+            exact resampled_hit_le_boundaryCount G Q ω η hω x w _ M
+              (hZ _ (resampleClusterAt_subset G Q _ ω η hω))
+        _ = _ := bernoulliPolynomial_const Q _ p)
+  rw [bernoulliPolynomial_add] at hmono
+  have hb := finite_short_boundary_expectation G Q x w M p hp hp1
+  calc
+    _ ≤ _ := hmono
+    _ ≤ bernoulliPolynomial Q (fun ω => if Conn G ω x w then (1:ℝ) else 0) p +
+      ((1-p)/p)*M*bernoulliPolynomial Q (fun ω => if Conn G ω x w then (1:ℝ) else 0) p :=
+        add_le_add le_rfl hb
+    _ = _ := by ring
+
+ 
+
+theorem finite_selected_susceptibility_bound (G : BondGraph V E) (Q : Finset E)
+    (F : Finset V) (x : V) (M : ℕ) (Z : Finset E → V)
+    (hZ : ∀ η ⊆ Q, ∃ l : List E, Along G x l (Z η) ∧ l.length ≤ M ∧ ∀ e ∈ l, e ∈ η)
+    (p : ℝ) (hp : 0 < p) (hp1 : p ≤ 1) :
+    bernoulliPolynomial Q (fun η => finiteSusceptibility G Q F p (Z η)) p ≤
+      (1+(1-p)/p*M)*finiteSusceptibility G Q F p x := by
+  unfold finiteSusceptibility
+  rw [bernoulliPolynomial_sum,Finset.mul_sum]
+  apply Finset.sum_le_sum
+  intro w _
+  have h := finite_selector_connection_bound G Q x w M Z hZ p hp hp1
+  calc
+    _ = bernoulliPolynomial Q (fun η => bernoulliPolynomial Q (fun ω =>
+        if Conn G ω w (Z η) then (1:ℝ) else 0) p) p := by
+      apply bernoulliPolynomial_congr
+      intro η _
+      apply bernoulliPolynomial_congr
+      intro ω _
+      apply if_congr _ rfl rfl
+      exact ⟨conn_symm G ω,conn_symm G ω⟩
+    _ ≤ _ := h
+
+end FiniteMaximalSusceptibility
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter Topology
+section RestrictedConnectionLimit
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem tendsto_bernoulliPolynomial {ι : Type*} {l : Filter ι}
+    (Q : Finset E) (f : ι → Finset E → ℝ) (g : Finset E → ℝ) (p : ℝ)
+    (h : ∀ a ⊆ Q, Tendsto (fun i => f i a) l (𝓝 (g a))) :
+    Tendsto (fun i => bernoulliPolynomial Q (f i) p) l
+      (𝓝 (bernoulliPolynomial Q g p)) := by
+  unfold bernoulliPolynomial
+  exact tendsto_finsetSum _ (fun a ha =>
+    ((h a (Finset.mem_powerset.mp ha)).mul_const _).mul_const _)
+
+theorem conn_finite_exhaustion (G : BondGraph V E) (S : Set E)
+    (Q : ℕ → Finset E) (hQ : Monotone Q) (hcover : ∀ e ∈ S, ∃ n, e ∈ Q n)
+    {ω : Set E} {x y : V} (h : Conn G (ω ∩ S) x y) :
+    ∃ n, Conn G (ω ∩ (Q n : Set E)) x y := by
+  induction h with
+  | refl => exact ⟨0,conn_refl _ _ _⟩
+  | @tail a b hxa hab ih =>
+    obtain ⟨n,hn⟩ := ih
+    obtain ⟨e,he,hab⟩ := hab
+    obtain ⟨m,hm⟩ := hcover e he.2
+    refine ⟨max n m,?_⟩
+    exact (conn_mono G (Set.inter_subset_inter_right _
+      (show (Q n:Set E) ⊆ (Q (max n m):Set E) from hQ (le_max_left n m))) hn).tail
+      ⟨e,⟨he.1,hQ (le_max_right n m) hm⟩,hab⟩
+
+theorem finite_connection_polynomial_tendsto (G : BondGraph V E) (S : Set E)
+    (Q : ℕ → Finset E) (hQ : Monotone Q) (hsub : ∀ n, (Q n:Set E) ⊆ S)
+    (hcover : ∀ e ∈ S, ∃ n, e ∈ Q n) (p : unitInterval) (x y : V) :
+    Tendsto (fun n => bernoulliPolynomial (Q n) (fun ω => if Conn G ω x y then (1:ℝ) else 0) p)
+      atTop (𝓝 ((law p).real {ω | Conn G (ω ∩ S) x y})) := by
+  have he (n : ℕ) : bernoulliPolynomial (Q n) (fun ω => if Conn G ω x y then (1:ℝ) else 0) p =
+      (law p).real {ω | Conn G (ω ∩ (Q n:Set E)) x y} :=
+    finiteEventPolynomial_restrict (Q n) {ω | Conn G ω x y} p
+  simp_rw [he]
+  have hm : Monotone (fun n => {ω : Set E | Conn G (ω ∩ (Q n:Set E)) x y}) := by
+    intro m n h ω hc
+    exact conn_mono G (Set.inter_subset_inter_right _ (show (Q m:Set E) ⊆ (Q n:Set E) from hQ h)) hc
+  have hu : (⋃ n, {ω : Set E | Conn G (ω ∩ (Q n:Set E)) x y}) =
+      {ω : Set E | Conn G (ω ∩ S) x y} := by
+    ext ω
+    simp only [Set.mem_iUnion,Set.mem_ofPred_eq]
+    exact ⟨fun ⟨n,hn⟩ => conn_mono G (Set.inter_subset_inter_right _ (hsub n)) hn,
+      conn_finite_exhaustion G S Q hQ hcover⟩
+  have ht := tendsto_measure_iUnion_atTop (μ := law p) hm
+  rw [hu] at ht
+  exact (ENNReal.tendsto_toReal (measure_ne_top _ _)).comp ht
+
+noncomputable def restrictedBondExhaustion [Encodable E] (S : Set E) (K : Finset E)
+    (n : ℕ) : Finset E := K ∪ (vertexExhaustion (V := E) n).filter (· ∈ S)
+
+theorem restrictedBondExhaustion_mono [Encodable E] (S : Set E) (K : Finset E) :
+    Monotone (restrictedBondExhaustion S K) := by
+  intro m n h
+  exact Finset.union_subset_union (by rfl) (Finset.filter_subset_filter _ (vertexExhaustion_mono h))
+
+theorem restrictedBondExhaustion_cover [Encodable E] (S : Set E) (K : Finset E)
+    (e : E) (he : e ∈ S) : ∃ n, e ∈ restrictedBondExhaustion S K n := by
+  obtain ⟨n,hn⟩ := subset_vertexExhaustion ({e}:Finset E)
+  exact ⟨n,Finset.mem_union_right _ (Finset.mem_filter.mpr ⟨hn (by simp),he⟩)⟩
+
+theorem restrictedBondExhaustion_subset [Encodable E] (S : Set E) (K : Finset E)
+    (hK : (K:Set E) ⊆ S) (n : ℕ) : (restrictedBondExhaustion S K n:Set E) ⊆ S := by
+  intro e he
+  rcases Finset.mem_union.mp he with he | he
+  · exact hK he
+  · exact (Finset.mem_filter.mp he).2
+
+end RestrictedConnectionLimit
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter Topology
+section RestrictedSelectorBound
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem restricted_selector_connection_bound [Countable E] (G : BondGraph V E)
+    (S : Set E) (K : Finset E) (hK : (K:Set E) ⊆ S)
+    (x w : V) (M : ℕ) (Z : Finset E → V)
+    (hZ : ∀ η ⊆ K, ∃ l : List E, Along G x l (Z η) ∧ l.length ≤ M ∧ ∀ e ∈ l, e ∈ η)
+    (p : unitInterval) (hp : 0 < (p:ℝ)) :
+    bernoulliPolynomial K (fun η => (law p).real {ω | Conn G (ω ∩ S) w (Z η)}) p ≤
+      (1+(1-(p:ℝ))/p*M)*(law p).real {ω | Conn G (ω ∩ S) x w} := by
+  let := Encodable.ofCountable E
+  let Q := restrictedBondExhaustion S K
+  have hlim (a b : V) := finite_connection_polynomial_tendsto G S Q
+    (restrictedBondExhaustion_mono S K) (restrictedBondExhaustion_subset S K hK)
+    (restrictedBondExhaustion_cover S K) p a b
+  have hleft := tendsto_bernoulliPolynomial K
+    (fun n η => bernoulliPolynomial (Q n) (fun ω => if Conn G ω w (Z η) then (1:ℝ) else 0) p)
+    (fun η => (law p).real {ω | Conn G (ω ∩ S) w (Z η)}) p (fun η _ => hlim w (Z η))
+  have hright := (hlim x w).const_mul (1+(1-(p:ℝ))/p*M)
+  apply le_of_tendsto_of_tendsto hleft hright
+  apply Eventually.of_forall
+  intro n
+  have hZQ : ∀ η ⊆ Q n, ∃ l : List E, Along G x l (Z (η ∩ K)) ∧
+      l.length ≤ M ∧ ∀ e ∈ l, e ∈ η := by
+    intro η _
+    obtain ⟨l,hl,hlen,ho⟩ := hZ (η ∩ K) Finset.inter_subset_right
+    exact ⟨l,hl,hlen,fun e he => (Finset.mem_inter.mp (ho e he)).1⟩
+  have h := finite_selector_connection_bound G (Q n) x w M (fun η => Z (η ∩ K)) hZQ p hp p.2.2
+  rw [bernoulliPolynomial_restrict K (Q n) Finset.subset_union_left
+    (fun η => bernoulliPolynomial (Q n) (fun ω => if Conn G ω w (Z η) then (1:ℝ) else 0) p) p] at h
+  exact h
+
+noncomputable def restrictedSusceptibility (G : BondGraph V E) (S : Set E)
+    (p : unitInterval) (x : V) : ℝ :=
+  ∑' y, (law p).real {ω | Conn G (ω ∩ S) x y}
+
+theorem restrictedTwoPoint_le (G : BondGraph V E) (S : Set E)
+    (p : unitInterval) (x y : V) :
+    (law p).real {ω | Conn G (ω ∩ S) x y} ≤ twoPoint G p x y := by
+  exact measureReal_mono (fun _ h => conn_mono G Set.inter_subset_left h)
+
+theorem summable_restrictedTwoPoint [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (S : Set E) (x : V) :
+    Summable (fun y => (law p).real {ω | Conn G (ω ∩ S) x y}) := by
+  obtain ⟨C,_,hC⟩ := subcritical_row_bound G hloc hqt p hp
+  apply summable_of_sum_le (fun _ => measureReal_nonneg)
+  intro F
+  exact (Finset.sum_le_sum (fun y _ => restrictedTwoPoint_le G S p x y)).trans (hC x F)
+
+end RestrictedSelectorBound
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter Topology
+section SelectorSusceptibility
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem summable_bernoulliPolynomial {A : Type*} (K : Finset E)
+    (f : Finset E → A → ℝ) (p : ℝ) (hf : ∀ η ⊆ K, Summable (f η)) :
+    Summable (fun y => bernoulliPolynomial K (fun η => f η y) p) := by
+  unfold bernoulliPolynomial
+  apply summable_sum
+  intro η hη
+  exact ((hf η (Finset.mem_powerset.mp hη)).mul_right _).mul_right _
+
+theorem bernoulliPolynomial_tsum {A : Type*} (K : Finset E)
+    (f : Finset E → A → ℝ) (p : ℝ) (hf : ∀ η ⊆ K, Summable (f η)) :
+    bernoulliPolynomial K (fun η => ∑' y, f η y) p =
+      ∑' y, bernoulliPolynomial K (fun η => f η y) p := by
+  unfold bernoulliPolynomial
+  rw [Summable.tsum_finsetSum (fun η hη =>
+    ((hf η (Finset.mem_powerset.mp hη)).mul_right _).mul_right _)]
+  apply Finset.sum_congr rfl
+  intro η _
+  rw [tsum_mul_right,tsum_mul_right]
+
+theorem restrictedTwoPoint_symm (G : BondGraph V E) (S : Set E)
+    (p : unitInterval) (x y : V) :
+    (law p).real {ω | Conn G (ω ∩ S) x y} =
+      (law p).real {ω | Conn G (ω ∩ S) y x} := by
+  congr 1
+  ext ω
+  exact ⟨conn_symm G _,conn_symm G _⟩
+
+ 
+
+theorem restricted_selected_susceptibility_bound [Countable E] (G : BondGraph V E)
+    (S : Set E) (K : Finset E) (hK : (K:Set E) ⊆ S)
+    (x : V) (M : ℕ) (Z : Finset E → V)
+    (hZ : ∀ η ⊆ K, ∃ l : List E, Along G x l (Z η) ∧ l.length ≤ M ∧ ∀ e ∈ l, e ∈ η)
+    (p : unitInterval) (hp : 0 < (p:ℝ))
+    (hfin : ∀ y, Summable (fun z => (law p).real {ω | Conn G (ω ∩ S) y z})) :
+    bernoulliPolynomial K (fun η => restrictedSusceptibility G S p (Z η)) p ≤
+      (1+(1-(p:ℝ))/p*M)*restrictedSusceptibility G S p x := by
+  unfold restrictedSusceptibility
+  rw [bernoulliPolynomial_tsum K _ p (fun η _ => hfin (Z η)),←tsum_mul_left]
+  apply Summable.tsum_le_tsum _ (summable_bernoulliPolynomial K _ p (fun η _ => hfin (Z η)))
+    ((hfin x).mul_left _)
+  intro w
+  have h := restricted_selector_connection_bound G S K hK x w M Z hZ p hp
+  simpa only [restrictedTwoPoint_symm G S p (Z _) w] using h
+
+ 
+
+theorem subcritical_selected_susceptibility_bound
+    [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (S : Set E) (K : Finset E) (hK : (K:Set E) ⊆ S)
+    (x : V) (M : ℕ) (Z : Finset E → V)
+    (hZ : ∀ η ⊆ K, ∃ l : List E, Along G x l (Z η) ∧ l.length ≤ M ∧ ∀ e ∈ l, e ∈ η)
+    (p : unitInterval) (hp : 0 < (p:ℝ)) (hpc : p < pc G) :
+    bernoulliPolynomial K (fun η => restrictedSusceptibility G S p (Z η)) p ≤
+      (1+(1-(p:ℝ))/p*M)*restrictedSusceptibility G S p x :=
+  restricted_selected_susceptibility_bound G S K hK x M Z hZ p hp
+    (summable_restrictedTwoPoint G hloc hqt p hpc S)
+
+end SelectorSusceptibility
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section IntrinsicBall
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+def intrinsicBall (G : BondGraph V E) (ω : Set E) (x : V) (M : ℕ) : Set V :=
+  {y | ∃ l : List E, Along G x l y ∧ l.length ≤ M ∧ ∀ e ∈ l, e ∈ ω}
+
+theorem mem_intrinsicBall_self (G : BondGraph V E) (ω : Set E) (x : V) (M : ℕ) :
+    x ∈ intrinsicBall G ω x M := ⟨[],.nil x,Nat.zero_le _,by simp⟩
+
+theorem intrinsicBall_subset_reachBall (G : BondGraph V E) (ω : Set E) (x : V) (M : ℕ) :
+    intrinsicBall G ω x M ⊆ reachBall G x M := by
+  rintro y ⟨l,hl,hlen,_⟩
+  exact reachBall_mono G x hlen hl.mem_reachBall
+
+theorem intrinsicBall_mono (G : BondGraph V E) {ω η : Set E} (h : ω ⊆ η) (x : V) (M : ℕ) :
+    intrinsicBall G ω x M ⊆ intrinsicBall G η x M := by
+  rintro y ⟨l,hl,hlen,ho⟩
+  exact ⟨l,hl,hlen,fun e he => h (ho e he)⟩
+
+theorem Along.bonds_in_incidentBall {G : BondGraph V E} (hloc : LocallyFinite G)
+    {x y : V} {l : List E} (h : Along G x l y) (M : ℕ) (hlen : l.length ≤ M) :
+    ∀ e ∈ l, e ∈ incidentSupport G hloc (finite_reachBall hloc x M).toFinset := by
+  rcases h.first_outside (incidentSupport G hloc (finite_reachBall hloc x M).toFinset : Set E)
+    with h | ⟨a,b,e,l₁,l₂,heq,h₁,he,_,_,hnot⟩
+  · exact h
+  · exfalso
+    apply hnot
+    apply (mem_incidentSupport G hloc _ e).mpr
+    refine ⟨a,?_,by simp [he]⟩
+    apply (finite_reachBall hloc x M).mem_toFinset.mpr
+    apply reachBall_mono G x (show l₁.length ≤ M by subst l; simp only [List.length_append,List.length_cons] at hlen; omega)
+    exact h₁.mem_reachBall
+
+theorem intrinsicBall_restrict (G : BondGraph V E) (hloc : LocallyFinite G)
+    (ω : Set E) (x : V) (M : ℕ) :
+    intrinsicBall G (ω ∩ (incidentSupport G hloc (finite_reachBall hloc x M).toFinset : Set E)) x M =
+      intrinsicBall G ω x M := by
+  apply Set.Subset.antisymm (intrinsicBall_mono G Set.inter_subset_left x M)
+  rintro y ⟨l,hl,hlen,ho⟩
+  exact ⟨l,hl,hlen,fun e he => ⟨ho e he,hl.bonds_in_incidentBall hloc M hlen e he⟩⟩
+
+theorem integral_eq_bernoulliPolynomial (Q : Finset E) (p : unitInterval) (f : Set E → ℝ)
+    (hf : ∀ ω, f ω = f (ω ∩ (Q:Set E))) :
+    (∫ ω, f ω ∂law p) = bernoulliPolynomial Q (fun η => f η) p := by
+  rw [integral_finite_local Q p f hf]
+  simp only [finiteFieldWeight,finiteFieldSet]
+  rw [Finset.sum_coe_sort Q.powerset (fun η => bernoulliWeight Q η p * f η)]
+  unfold bernoulliPolynomial
+  apply Finset.sum_congr rfl
+  intro η _
+  unfold bernoulliWeight
+  ring
+
+end IntrinsicBall
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section IntrinsicMaximum
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem intrinsicBall_exists_max (G : BondGraph V E) (hloc : LocallyFinite G)
+    (ω : Set E) (x : V) (M : ℕ) (f : V → ℝ) :
+    ∃ z ∈ intrinsicBall G ω x M, ∀ y ∈ intrinsicBall G ω x M, f y ≤ f z :=
+  Set.exists_max_image _ f ((finite_reachBall hloc x M).subset
+    (intrinsicBall_subset_reachBall G ω x M)) ⟨x,mem_intrinsicBall_self G ω x M⟩
+
+noncomputable def intrinsicMaximizer (G : BondGraph V E) (hloc : LocallyFinite G)
+    (ω : Set E) (x : V) (M : ℕ) (f : V → ℝ) : V :=
+  Classical.choose (intrinsicBall_exists_max G hloc ω x M f)
+
+theorem intrinsicMaximizer_spec (G : BondGraph V E) (hloc : LocallyFinite G)
+    (ω : Set E) (x : V) (M : ℕ) (f : V → ℝ) :
+    intrinsicMaximizer G hloc ω x M f ∈ intrinsicBall G ω x M ∧
+      ∀ y ∈ intrinsicBall G ω x M, f y ≤ f (intrinsicMaximizer G hloc ω x M f) :=
+  Classical.choose_spec (intrinsicBall_exists_max G hloc ω x M f)
+
+noncomputable def intrinsicPotentialMax (G : BondGraph V E) (ω : Set E)
+    (x : V) (M : ℕ) (f : V → ℝ) : ℝ := sSup (f '' intrinsicBall G ω x M)
+
+theorem intrinsicPotentialMax_eq (G : BondGraph V E) (hloc : LocallyFinite G)
+    (ω : Set E) (x : V) (M : ℕ) (f : V → ℝ) :
+    intrinsicPotentialMax G ω x M f = f (intrinsicMaximizer G hloc ω x M f) := by
+  have h := intrinsicMaximizer_spec G hloc ω x M f
+  apply IsGreatest.csSup_eq
+  exact ⟨⟨_,h.1,rfl⟩,fun _ ⟨y,hy,he⟩ => he ▸ h.2 y hy⟩
+
+theorem le_intrinsicPotentialMax (G : BondGraph V E) (hloc : LocallyFinite G)
+    (ω : Set E) (x : V) (M : ℕ) (f : V → ℝ) (y : V) (hy : y ∈ intrinsicBall G ω x M) :
+    f y ≤ intrinsicPotentialMax G ω x M f := by
+  rw [intrinsicPotentialMax_eq G hloc]
+  exact (intrinsicMaximizer_spec G hloc ω x M f).2 y hy
+
+noncomputable def intrinsicBondSupport (G : BondGraph V E) (hloc : LocallyFinite G)
+    (S : Set E) (x : V) (M : ℕ) : Finset E :=
+  (incidentSupport G hloc (finite_reachBall hloc x M).toFinset).filter (· ∈ S)
+
+theorem intrinsicBondSupport_coe (G : BondGraph V E) (hloc : LocallyFinite G)
+    (S : Set E) (x : V) (M : ℕ) :
+    (intrinsicBondSupport G hloc S x M : Set E) =
+      (incidentSupport G hloc (finite_reachBall hloc x M).toFinset : Set E) ∩ S := by
+  ext e
+  simp [intrinsicBondSupport]
+
+theorem intrinsicBall_domain_restrict (G : BondGraph V E) (hloc : LocallyFinite G)
+    (S ω : Set E) (x : V) (M : ℕ) :
+    intrinsicBall G (ω ∩ (intrinsicBondSupport G hloc S x M : Set E)) x M =
+      intrinsicBall G (ω ∩ S) x M := by
+  rw [intrinsicBondSupport_coe]
+  have he : ω ∩ ((incidentSupport G hloc (finite_reachBall hloc x M).toFinset : Set E) ∩ S) =
+      (ω ∩ S) ∩ (incidentSupport G hloc (finite_reachBall hloc x M).toFinset : Set E) := by
+    ext e; simp [and_left_comm,and_comm,and_assoc]
+  rw [he,intrinsicBall_restrict G hloc]
+
+theorem intrinsicPotentialMax_domain_local (G : BondGraph V E) (hloc : LocallyFinite G)
+    (S ω : Set E) (x : V) (M : ℕ) (f : V → ℝ) :
+    intrinsicPotentialMax G (ω ∩ S) x M f =
+      intrinsicPotentialMax G ((ω ∩ (intrinsicBondSupport G hloc S x M : Set E)) ∩ S) x M f := by
+  have hsub : (intrinsicBondSupport G hloc S x M : Set E) ⊆ S := by
+    rw [intrinsicBondSupport_coe]; exact Set.inter_subset_right
+  rw [Set.inter_assoc,Set.inter_eq_left.mpr hsub]
+  unfold intrinsicPotentialMax
+  rw [intrinsicBall_domain_restrict G hloc]
+
+end IntrinsicMaximum
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section MaximalSusceptibility
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem maximal_restricted_susceptibility_bound
+    [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (S : Set E) (x : V) (M : ℕ) (p : unitInterval)
+    (hp : 0 < (p:ℝ)) (hpc : p < pc G) :
+    (∫ ω, intrinsicPotentialMax G (ω ∩ S) x M (restrictedSusceptibility G S p) ∂law p) ≤
+      (1+(1-(p:ℝ))/p*M)*restrictedSusceptibility G S p x := by
+  let K := intrinsicBondSupport G hloc S x M
+  let f := restrictedSusceptibility G S p
+  let Z : Finset E → V := fun η => intrinsicMaximizer G hloc η x M f
+  have hK : (K:Set E) ⊆ S := by
+    rw [intrinsicBondSupport_coe]; exact Set.inter_subset_right
+  rw [integral_eq_bernoulliPolynomial K p _ (fun ω =>
+    intrinsicPotentialMax_domain_local G hloc S ω x M f)]
+  have he : bernoulliPolynomial K (fun η => intrinsicPotentialMax G ((η:Set E) ∩ S) x M f) p =
+      bernoulliPolynomial K (fun η => f (Z η)) p := by
+    apply bernoulliPolynomial_congr K _ p
+    intro η hη
+    rw [Set.inter_eq_left.mpr (fun e he => hK (hη he))]
+    exact intrinsicPotentialMax_eq G hloc η x M f
+  rw [he]
+  apply subcritical_selected_susceptibility_bound G hloc hqt S K hK x M Z _ p hp hpc
+  intro η _
+  exact (intrinsicMaximizer_spec G hloc η x M f).1
+
+ 
+
+noncomputable def inducedSusceptibility (G : BondGraph V E) (D : Set V)
+    (p : unitInterval) (x : V) : ℝ :=
+  if x ∈ D then restrictedSusceptibility G (internalEdges G D) p x else 0
+
+theorem intrinsicBall_internal_subset (G : BondGraph V E) (D : Set V)
+    (ω : Set E) (x : V) (hx : x ∈ D) (M : ℕ) :
+    intrinsicBall G (ω ∩ internalEdges G D) x M ⊆ D := by
+  rintro y ⟨l,hl,_,ho⟩
+  exact conn_internal_endpoint G D hx (hl.conn ho)
+
+theorem inducedSusceptibility_max_eq (G : BondGraph V E) (D : Set V)
+    (ω : Set E) (p : unitInterval) (x : V) (hx : x ∈ D) (M : ℕ) :
+    intrinsicPotentialMax G (ω ∩ internalEdges G D) x M (inducedSusceptibility G D p) =
+      intrinsicPotentialMax G (ω ∩ internalEdges G D) x M (restrictedSusceptibility G (internalEdges G D) p) := by
+  unfold intrinsicPotentialMax
+  congr 1
+  apply Set.image_congr
+  intro y hy
+  exact ite_eq_left (intrinsicBall_internal_subset G D ω x hx M hy)
+
+ 
+
+theorem maximal_induced_susceptibility_bound
+    [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (D : Set V) (x : V) (hx : x ∈ D) (M : ℕ) (p : unitInterval)
+    (hp : 0 < (p:ℝ)) (hpc : p < pc G) :
+    (∫ ω, intrinsicPotentialMax G (ω ∩ internalEdges G D) x M (inducedSusceptibility G D p) ∂law p) ≤
+      (1+(1-(p:ℝ))/p*M)*inducedSusceptibility G D p x := by
+  simp_rw [inducedSusceptibility_max_eq G D _ p x hx M]
+  rw [inducedSusceptibility,ite_eq_left hx]
+  exact maximal_restricted_susceptibility_bound G hloc hqt (internalEdges G D) x M p hp hpc
+
+end MaximalSusceptibility
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section RestrictedPathPotential
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem restrictedTwoPoint_adj_lower [Countable E] (G : BondGraph V E)
+    (S : Set E) (p : unitInterval) (x y z : V) (e : E)
+    (he : G e = s(x,y)) (heS : e ∈ S) :
+    (p:ℝ)*(law p).real {ω | Conn G (ω ∩ S) y z} ≤
+      (law p).real {ω | Conn G (ω ∩ S) x z} := by
+  have h := association_finiteWitness p {ω : Set E | e ∈ ω} {ω | Conn G (ω ∩ S) y z}
+    (fun _ _ hω h => hω h)
+    (fun _ _ hω h => conn_mono G (Set.inter_subset_inter_left _ hω) h)
+    (finiteWitness_edge e) (finiteWitness_inter _ (finiteWitness_conn G y z) S)
+  have hp : (law p).real {ω : Set E | e ∈ ω} = (p:ℝ) := by
+    simpa using finite_open_probability p {e}
+  rw [hp] at h
+  exact h.trans (measureReal_mono (fun ω hω =>
+    conn_trans G _ (Relation.ReflTransGen.single ⟨e,⟨hω.1,heS⟩,he⟩) hω.2))
+
+theorem restrictedTwoPoint_path_lower [Countable E] (G : BondGraph V E)
+    (S : Set E) (p : unitInterval) {x y : V} {l : List E}
+    (hl : Along G x l y) (hS : ∀ e ∈ l, e ∈ S) (z : V) :
+    (p:ℝ)^l.length*(law p).real {ω | Conn G (ω ∩ S) y z} ≤
+      (law p).real {ω | Conn G (ω ∩ S) x z} := by
+  induction hl with
+  | nil => simp
+  | @cons x a y l e he ht ih =>
+    have hh := mul_le_mul_of_nonneg_left (ih (fun d hd => hS d (List.mem_cons_of_mem _ hd))) p.2.1
+    have ha := restrictedTwoPoint_adj_lower G S p x a z e he (hS e (by simp))
+    simpa only [List.length_cons,pow_succ',mul_assoc] using hh.trans ha
+
+theorem restrictedSusceptibility_nonneg (G : BondGraph V E) (S : Set E)
+    (p : unitInterval) (x : V) : 0 ≤ restrictedSusceptibility G S p x :=
+  tsum_nonneg (fun _ => measureReal_nonneg)
+
+theorem restrictedSusceptibility_path_lower [Countable E] (G : BondGraph V E)
+    (S : Set E) (p : unitInterval) {x y : V} {l : List E}
+    (hl : Along G x l y) (hS : ∀ e ∈ l, e ∈ S)
+    (hfin : ∀ a, Summable (fun z => (law p).real {ω | Conn G (ω ∩ S) a z})) :
+    (p:ℝ)^l.length*restrictedSusceptibility G S p y ≤ restrictedSusceptibility G S p x := by
+  unfold restrictedSusceptibility
+  rw [←tsum_mul_left]
+  exact ((hfin y).mul_left _).tsum_le_tsum (restrictedTwoPoint_path_lower G S p hl hS) (hfin x)
+
+ 
+
+theorem restrictedSusceptibility_short_path_bound [Countable E] (G : BondGraph V E)
+    (S : Set E) (p : unitInterval) (hp : 0 < (p:ℝ)) {x y : V} {l : List E}
+    (hl : Along G x l y) (hS : ∀ e ∈ l, e ∈ S) (M : ℕ) (hlen : l.length ≤ M)
+    (hfin : ∀ a, Summable (fun z => (law p).real {ω | Conn G (ω ∩ S) a z})) :
+    restrictedSusceptibility G S p y ≤ (p:ℝ)^(-(M:ℤ))*restrictedSusceptibility G S p x := by
+  have hh := restrictedSusceptibility_path_lower G S p hl hS hfin
+  have hpow : (p:ℝ)^M ≤ (p:ℝ)^l.length :=
+    pow_le_pow_of_le_one p.2.1 p.2.2 hlen
+  have hbound := (mul_le_mul_of_nonneg_right hpow (restrictedSusceptibility_nonneg G S p y)).trans hh
+  rw [zpow_neg,zpow_natCast,←div_eq_inv_mul]
+  exact (le_div_iff₀ (pow_pos hp M)).mpr (by simpa only [mul_comm] using hbound)
+
+end RestrictedPathPotential
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section ShortPathsDeletedDomain
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem conn_short_of_cluster_cap (G : BondGraph V E) (ω : Set E) (x y : V)
+    (hxy : Conn G ω x y) (hfin : (cluster G ω x).Finite)
+    (M : ℕ) (hM : (cluster G ω x).ncard ≤ M) :
+    ∃ l : List E, Along G x l y ∧ l.length ≤ M ∧ ∀ e ∈ l, e ∈ ω := by
+  obtain ⟨w⟩ := (conn_iff_reachable G ω x y).mp hxy
+  let q := w.toPath
+  have hs : q.val.support.toFinset ⊆ hfin.toFinset := by
+    intro z hz
+    have hz' : z ∈ q.val.support := List.mem_toFinset.mp hz
+    exact hfin.mem_toFinset.mpr ((conn_iff_reachable G ω x z).mpr ⟨q.val.takeUntil z hz'⟩)
+  have hc : q.val.length+1 ≤ (cluster G ω x).ncard := by
+    have hc := Finset.card_le_card hs
+    rwa [List.toFinset_card_of_nodup q.property.support_nodup,
+      SimpleGraph.Walk.length_support,←Set.ncard_eq_toFinset_card _ hfin] at hc
+  obtain ⟨l,hl,he,ho⟩ := walk_lift_bonds G ω q.val
+  have hlen : l.length = q.val.length := by
+    have hh := congrArg List.length he
+    simpa only [List.length_map,SimpleGraph.Walk.length_edges] using hh
+  exact ⟨l,hl,by omega,ho⟩
+
+ 
+
+theorem Along.last_entrance {G : BondGraph V E} {x y : V} {l : List E}
+    (hl : Along G x l y) (D : Set V) (hy : y ∈ D) :
+    (x ∈ D ∧ ∀ e ∈ l, e ∈ internalEdges G D) ∨
+      ∃ a b e l₁ l₂, l = l₁ ++ e::l₂ ∧ Along G x l₁ a ∧ G e = s(a,b) ∧
+        a ∉ D ∧ b ∈ D ∧ Along G b l₂ y ∧ ∀ d ∈ l₂, d ∈ internalEdges G D := by
+  induction hl with
+  | nil => exact Or.inl ⟨hy,by simp⟩
+  | @cons x z y l e he ht ih =>
+    rcases ih hy with ⟨hz,hall⟩ | ⟨a,b,d,l₁,l₂,heq,h₁,hd,ha,hb,h₂,hs⟩
+    · by_cases hx : x ∈ D
+      · left
+        refine ⟨hx,?_⟩
+        intro d hd
+        rcases List.mem_cons.mp hd with rfl | hd
+        · exact (mem_internalEdges_of_ends G D d x z he).mpr ⟨hx,hz⟩
+        · exact hall d hd
+      · right
+        exact ⟨x,z,e,[],l,rfl,.nil x,he,hx,hz,ht,hall⟩
+    · right
+      exact ⟨a,b,d,e::l₁,l₂,by simp [heq],.cons e he h₁,hd,ha,hb,h₂,hs⟩
+
+end ShortPathsDeletedDomain
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Clusters/Uniqueness.lean b/lean/OAI/Probability/UnimodularPercolation/Clusters/Uniqueness.lean
index 457f6bbaefb2f68ec33a29b86f643b1972b408b3..3be618f5a70de961ca03f504567a9edc8b3966e3 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Clusters/Uniqueness.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Clusters/Uniqueness.lean
@@ -1,599 +1,599 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.ProductMeasure.Association
-
-namespace OAI
-
-/-! Cluster events, invariant covers and uniqueness thresholds. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-section InfiniteOrbits
-attribute [local instance] Classical.propDecidable
-
-def vertexOrbit (G : BondGraph V E) (x : V) : Set V :=
-  {y | ∃ a : V ≃ V, ∃ b : E ≃ E, IsAut G a b ∧ a x = y}
-
-theorem adj_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (h : IsAut G a b) {x y : V} (hxy : Adj G x y) : Adj G (a x) (a y) := by
-  obtain ⟨e, he⟩ := hxy
-  exact ⟨b e, by rw [h e, he, Sym2.map_mk]⟩
-
-theorem mem_reachBall_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (h : IsAut G a b) {x y : V} {n : ℕ} (hy : y ∈ reachBall G x n) :
-    a y ∈ reachBall G (a x) n := by
-  induction n generalizing y with
-  | zero => simpa only [reachBall, Set.mem_singleton_iff, Equiv.apply_eq_iff_eq] using hy
-  | succ n ih =>
-    rcases hy with hy | hy
-    · exact Or.inl (ih hy)
-    · change a y ∈ reachBall G (a x) n ∪ ⋃ z ∈ reachBall G (a x) n, {w | Adj G z w}
-      apply Or.inr
-      simp only [Set.mem_iUnion, Set.mem_ofPred_eq] at hy ⊢
-      obtain ⟨z, hz, hzy⟩ := hy
-      exact ⟨a z, ih hz, adj_aut G h hzy⟩
-
- 
-
-theorem vertexOrbit_infinite [Infinite V] (G : BondGraph V E)
-    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G) (x : V) :
-    (vertexOrbit G x).Infinite := by
-  intro hfin
-  obtain ⟨R, hR⟩ := hqt
-  choose n hn using (fun y => exists_mem_reachBall (hconn x y))
-  have hS : (⋃ o ∈ (R : Set V), ⋃ z ∈ vertexOrbit G x, reachBall G z (n o)).Finite :=
-    R.finite_toSet.biUnion fun o _ => hfin.biUnion fun z _ => finite_reachBall hloc z (n o)
-  have he : Set.univ ⊆ ⋃ o ∈ (R : Set V), ⋃ z ∈ vertexOrbit G x, reachBall G z (n o) := by
-    intro y _
-    obtain ⟨o, ho, a, b, hab, hay⟩ := hR y
-    simp only [Set.mem_iUnion]
-    exact ⟨o, ho, a x, ⟨a, b, hab, rfl⟩, hay ▸ mem_reachBall_aut G hab (hn o)⟩
-  exact Set.infinite_univ (hS.subset he)
-
-end InfiniteOrbits
-
-section FinitePairBound
-attribute [local instance] Classical.propDecidable
-open scoped ComplexConjugate
-
-noncomputable def deltaSum (F : Finset V) : Hilbert V := ∑ x ∈ F, lp.single 2 x (1 : ℂ)
-
-@[simp] theorem deltaSum_apply (F : Finset V) (x : V) :
-    deltaSum F x = if x ∈ F then 1 else 0 := by
-  simp only [deltaSum, lp.coeFn_sum, Finset.sum_apply, lp.single_apply]
-  simp
-
-@[simp] theorem inner_deltaSum (F : Finset V) (f : Hilbert V) :
-    inner ℂ (deltaSum F) f = ∑ x ∈ F, f x := by
-  simp [deltaSum, lp.inner_single_left]
-
-theorem norm_deltaSum_sq (F : Finset V) : ‖deltaSum F‖ ^ 2 = (F.card : ℝ) := by
-  rw [← inner_self_eq_norm_sq (𝕜 := ℂ), inner_deltaSum]
-  simp only [map_sum, deltaSum_apply]
-  trans ∑ _x ∈ F, (1 : ℝ)
-  · apply Finset.sum_congr rfl
-    intro x hx
-    simp [hx]
-  · simp
-
- 
-theorem sum_twoPoint_le_norm_card (G : BondGraph V E) (p : unitInterval)
-    {A : Hilbert V →L[ℂ] Hilbert V} (hA : HasKernel G p A) (F : Finset V) :
-    ∑ x ∈ F, ∑ y ∈ F, twoPoint G p x y ≤ ‖A‖ * F.card := by
-  have he : inner ℂ (deltaSum F) (A (deltaSum F)) =
-      ((∑ x ∈ F, ∑ y ∈ F, twoPoint G p x y : ℝ) : ℂ) := by
-    rw [inner_deltaSum]
-    simp only [deltaSum, map_sum]
-    simp only [lp.coeFn_sum, Finset.sum_apply, Complex.ofReal_sum]
-    apply Finset.sum_congr rfl
-    intro x _
-    apply Finset.sum_congr rfl
-    intro y _
-    exact hA x y
-  have h := (Complex.re_le_norm (inner ℂ (deltaSum F) (A (deltaSum F)))).trans
-    ((norm_inner_le_norm _ _).trans
-      (mul_le_mul_of_nonneg_left (A.le_opNorm _) (norm_nonneg _)))
-  rw [he, Complex.ofReal_re] at h
-  calc
-    _ ≤ ‖deltaSum F‖ * (‖A‖ * ‖deltaSum F‖) := h
-    _ = ‖A‖ * ‖deltaSum F‖ ^ 2 := by ring
-    _ = _ := by rw [norm_deltaSum_sq]
-
-end FinitePairBound
-
-section PairCounting
-attribute [local instance] Classical.propDecidable
-
- 
-theorem card_sq_le_image_card_mul_pairs {W : Type*} (S : Finset V) (c : V → W) :
-    (S.card : ℝ) ^ 2 ≤ ((S.image c).card : ℝ) *
-      ∑ x ∈ S, ∑ y ∈ S, if c x = c y then (1 : ℝ) else 0 := by
-  let t := S.image c
-  let f := fun w => ((S.filter (fun x => c x = w)).card : ℝ)
-  have hsum : ∑ w ∈ t, f w = (S.card : ℝ) := by
-    have h := Finset.sum_fiberwise_of_maps_to
-      (fun x (hx : x ∈ S) => Finset.mem_image_of_mem c hx) (fun _ => (1 : ℝ))
-    simpa only [Finset.sum_const, nsmul_eq_mul, mul_one, t, f] using h
-  have hsq : ∑ w ∈ t, f w ^ 2 =
-      ∑ x ∈ S, ∑ y ∈ S, if c x = c y then (1 : ℝ) else 0 := by
-    calc
-      ∑ w ∈ t, f w ^ 2 = ∑ w ∈ t, ∑ x ∈ S with c x = w, f (c x) := by
-        apply Finset.sum_congr rfl
-        intro w hw
-        rw [show ∑ x ∈ S with c x = w, f (c x) =
-            ∑ _x ∈ S with c _x = w, f w from
-          Finset.sum_congr rfl (fun x hx => by rw [(Finset.mem_filter.mp hx).2])]
-        simp only [Finset.sum_const, nsmul_eq_mul, f]
-        ring
-      _ = ∑ x ∈ S, f (c x) := Finset.sum_fiberwise_of_maps_to
-        (fun x hx => Finset.mem_image_of_mem c hx) _
-      _ = _ := by
-        apply Finset.sum_congr rfl
-        intro x hx
-        rw [← Finset.sum_filter]
-        simp only [Finset.sum_const, nsmul_eq_mul, mul_one, f]
-        congr 2
-        ext y
-        simp [eq_comm]
-  have h := Finset.sum_mul_sq_le_sq_mul_sq t (fun _ => (1 : ℝ)) f
-  simpa only [one_mul, one_pow, Finset.sum_const, nsmul_eq_mul, mul_one, hsum, hsq, t] using h
-
-end PairCounting
-
-section PairIntegral
-attribute [local instance] Classical.propDecidable
-variable {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]
-
- 
-theorem square_sum_probability_le (F : Finset V) (S : V → Set Ω)
-    (C : V → V → Set Ω) (hS : ∀ x, MeasurableSet (S x))
-    (hC : ∀ x y, MeasurableSet (C x y)) (n : ℝ)
-    (hpoint : ∀ ω, (∑ x ∈ F, (S x).indicator (fun _ => (1 : ℝ)) ω) ^ 2 ≤
-      n * ∑ x ∈ F, ∑ y ∈ F, (C x y).indicator (fun _ => (1 : ℝ)) ω) :
-    (∑ x ∈ F, μ.real (S x)) ^ 2 ≤ n * ∑ x ∈ F, ∑ y ∈ F, μ.real (C x y) := by
-  let X := fun ω => ∑ x ∈ F, (S x).indicator (fun _ => (1 : ℝ)) ω
-  have hX : MemLp X 2 μ := memLp_finsetSum F
-    (fun x _ => (memLp_const (1 : ℝ)).indicator (hS x))
-  have hCi (x y) : Integrable ((C x y).indicator (fun _ => (1 : ℝ))) μ :=
-    (integrable_const (1 : ℝ)).indicator (hC x y)
-  have hY : Integrable (fun ω => ∑ x ∈ F, ∑ y ∈ F,
-      (C x y).indicator (fun _ => (1 : ℝ)) ω) μ :=
-    integrable_finsetSum F (fun x _ => integrable_finsetSum F (fun y _ => hCi x y))
-  have hv := variance_nonneg X μ
-  rw [variance_eq_sub hX] at hv
-  have hi := integral_mono hX.integrable_sq (hY.const_mul n) hpoint
-  have hx : (∫ ω, X ω ∂μ) = ∑ x ∈ F, μ.real (S x) := by
-    dsimp only [X]
-    rw [integral_finsetSum F (fun x _ => (integrable_const (1 : ℝ)).indicator (hS x))]
-    simp only [integral_indicator_const _ (hS _), smul_eq_mul, mul_one]
-  have hy : (∫ ω, n * ∑ x ∈ F, ∑ y ∈ F,
-      (C x y).indicator (fun _ => (1 : ℝ)) ω ∂μ) =
-      n * ∑ x ∈ F, ∑ y ∈ F, μ.real (C x y) := by
-    rw [integral_const_mul, integral_finsetSum F
-      (fun x _ => integrable_finsetSum F (fun y _ => hCi x y))]
-    congr 1
-    apply Finset.sum_congr rfl
-    intro x _
-    rw [integral_finsetSum F (fun y _ => hCi x y)]
-    simp only [integral_indicator_const _ (hC x _), smul_eq_mul, mul_one]
-  rw [hx] at hv
-  rw [hy] at hi
-  exact (sub_nonneg.mp hv).trans hi
-
-end PairIntegral
-
-section CoupledCovers
-attribute [local instance] Classical.propDecidable
-variable [Countable V] [Countable E]
-
-omit [Countable V] [Countable E] in
-theorem cluster_eq_iff_conn (G : BondGraph V E) (ω : Set E) (x y : V) :
-    cluster G ω x = cluster G ω y ↔ Conn G ω x y := by
-  constructor
-  · intro h
-    have hh := mem_cluster_self G ω y
-    rwa [← h] at hh
-  · exact cluster_eq_of_conn G ω
-
- 
-def TouchedCover (G : BondGraph V E) (p q : unitInterval) (n : ℕ)
-    (labels : E → unitInterval) : Prop :=
-  ∃ f : Fin n → V, ∀ x, (cluster G (openEdges labels p) x).Infinite →
-    ∃ i, Conn G (openEdges labels q) x (f i)
-
-theorem measurableSet_touchedCover (G : BondGraph V E) (p q : unitInterval) (n : ℕ) :
-    MeasurableSet {labels | TouchedCover G p q n labels} := by
-  have he : {labels | TouchedCover G p q n labels} =
-      ⋃ f : Fin n → V, ⋂ x : V, ({labels | (cluster G (openEdges labels p) x).Infinite}ᶜ ∪
-        ⋃ i : Fin n, {labels | Conn G (openEdges labels q) x (f i)}) := by
-    ext labels
-    simp only [TouchedCover, Set.mem_ofPred_eq, Set.mem_iUnion, Set.mem_iInter,
-      Set.mem_union, Set.mem_compl_iff]
-    simp only [← imp_iff_not_or]
-  rw [he]
-  apply MeasurableSet.iUnion
-  intro f
-  apply MeasurableSet.iInter
-  intro x
-  exact ((measurableSet_infinite_cluster G x).preimage (measurable_openEdges p)).compl.union
-    (MeasurableSet.iUnion fun i => (measurableSet_conn G x (f i)).preimage (measurable_openEdges q))
-
-omit [Countable V] [Countable E] in
- 
-theorem survivor_count_sq_le (G : BondGraph V E) (p q : unitInterval) (n : ℕ)
-    (F : Finset V) (labels : E → unitInterval) (h : TouchedCover G p q n labels) :
-    ((F.filter fun x => (cluster G (openEdges labels p) x).Infinite).card : ℝ) ^ 2 ≤
-      n * ∑ x ∈ F, ∑ y ∈ F, if Conn G (openEdges labels q) x y then (1 : ℝ) else 0 := by
-  obtain ⟨f, hf⟩ := h
-  let S := F.filter fun x => (cluster G (openEdges labels p) x).Infinite
-  let c := cluster G (openEdges labels q)
-  have hsub : S.image c ⊆ Finset.univ.image (fun i : Fin n => c (f i)) := by
-    intro C hC
-    obtain ⟨x, hx, rfl⟩ := Finset.mem_image.mp hC
-    obtain ⟨i, hi⟩ := hf x (Finset.mem_filter.mp hx).2
-    exact Finset.mem_image.mpr ⟨i, Finset.mem_univ _, (cluster_eq_of_conn G _ hi).symm⟩
-  have hcard : (S.image c).card ≤ n :=
-    (Finset.card_le_card hsub).trans (by simpa using (Finset.card_image_le (s := (Finset.univ : Finset (Fin n)))
-      (f := fun i : Fin n => c (f i))))
-  have hpair := card_sq_le_image_card_mul_pairs S c
-  have hpairs : ∑ x ∈ S, ∑ y ∈ S, (if c x = c y then (1 : ℝ) else 0) ≤
-      ∑ x ∈ F, ∑ y ∈ F, if Conn G (openEdges labels q) x y then (1 : ℝ) else 0 := by
-    simp only [c, cluster_eq_iff_conn] 
-    calc
-      _ ≤ ∑ x ∈ S, ∑ y ∈ F, if Conn G (openEdges labels q) x y then (1 : ℝ) else 0 := by
-        apply Finset.sum_le_sum
-        intro x _
-        exact Finset.sum_le_sum_of_subset_of_nonneg (Finset.filter_subset _ _)
-          (fun _ _ _ => by split_ifs <;> positivity)
-      _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.filter_subset _ _)
-        (fun _ _ _ => Finset.sum_nonneg fun _ _ => by split_ifs <;> positivity)
-  exact hpair.trans ((mul_le_mul_of_nonneg_left hpairs (Nat.cast_nonneg _)).trans
-    (mul_le_mul_of_nonneg_right (by exact_mod_cast hcard)
-      (Finset.sum_nonneg fun _ _ => Finset.sum_nonneg fun _ _ => by split_ifs <;> positivity)))
-
-end CoupledCovers
-
-section CoverInvariance
-attribute [local instance] Classical.propDecidable
-
- 
-theorem labelLaw_map_precomp (b : E ≃ E) :
-    (labelLaw : Measure (E → unitInterval)).map (fun labels => labels ∘ b) = labelLaw := by
-  have he : (fun labels : E → unitInterval => labels ∘ b) =
-      MeasurableEquiv.piCongrLeft (fun _ : E => unitInterval) b.symm := by
-    funext labels e
-    have h := MeasurableEquiv.piCongrLeft_apply_apply (β := fun _ : E => unitInterval)
-      b.symm labels (b e)
-    simpa only [Equiv.symm_apply_apply, Function.comp_apply] using h.symm
-  unfold labelLaw
-  rw [he]
-  exact Measure.infinitePi_map_piCongrLeft (fun _ : E => (volume : Measure unitInterval)) b.symm
-
-theorem infinite_cluster_precomp (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (h : IsAut G a b) (labels : E → unitInterval) (p : unitInterval) (x : V) :
-    (cluster G (openEdges (labels ∘ b) p) x).Infinite ↔
-      (cluster G (openEdges labels p) (a x)).Infinite := by
-  change (cluster G (b ⁻¹' openEdges labels p) x).Infinite ↔ _
-  rw [← cluster_aut G h]
-  exact (Set.infinite_image_iff (fun _ _ _ _ hab => a.injective hab)).symm
-
-variable [Countable V] [Countable E]
-
-omit [Countable V] [Countable E] in
-theorem touchedCover_precomp (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (h : IsAut G a b) (labels : E → unitInterval) (p q : unitInterval) (n : ℕ) :
-    TouchedCover G p q n (labels ∘ b) ↔ TouchedCover G p q n labels := by
-  constructor
-  · rintro ⟨f, hf⟩
-    refine ⟨fun i => a (f i), fun x hx => ?_⟩
-    have hs : (cluster G (openEdges (labels ∘ b) p) (a.symm x)).Infinite :=
-      (infinite_cluster_precomp G h labels p (a.symm x)).mpr (by simpa using hx)
-    obtain ⟨i, hi⟩ := hf _ hs
-    exact ⟨i, by simpa using (conn_aut_iff G h (openEdges labels q) (a.symm x) (f i)).mp hi⟩
-  · rintro ⟨f, hf⟩
-    refine ⟨fun i => a.symm (f i), fun x hx => ?_⟩
-    obtain ⟨i, hi⟩ := hf (a x) ((infinite_cluster_precomp G h labels p x).mp hx)
-    exact ⟨i, (conn_aut_iff G h (openEdges labels q) x (a.symm (f i))).mpr (by simpa using hi)⟩
-
-theorem cover_survival_aut (G : BondGraph V E) (p q : unitInterval) (n : ℕ)
-    {a : V ≃ V} {b : E ≃ E} (h : IsAut G a b) (x : V) :
-    (labelLaw : Measure (E → unitInterval)).real
-      {labels | TouchedCover G p q n labels ∧ (cluster G (openEdges labels p) (a x)).Infinite} =
-    labelLaw.real {labels | TouchedCover G p q n labels ∧ (cluster G (openEdges labels p) x).Infinite} := by
-  have hm : Measurable (fun labels : E → unitInterval => labels ∘ b) :=
-    Measurable.of_eval (fun e => measurable_pi_apply (b e))
-  have hs : MeasurableSet {labels | TouchedCover G p q n labels ∧
-      (cluster G (openEdges labels p) x).Infinite} :=
-    (measurableSet_touchedCover G p q n).inter
-      ((measurableSet_infinite_cluster G x).preimage (measurable_openEdges p))
-  have he : {labels | TouchedCover G p q n labels ∧
-      (cluster G (openEdges labels p) (a x)).Infinite} =
-      (fun labels : E → unitInterval => labels ∘ b) ⁻¹'
-        {labels | TouchedCover G p q n labels ∧ (cluster G (openEdges labels p) x).Infinite} := by
-    ext labels
-    simp only [Set.mem_ofPred_eq, Set.mem_preimage, touchedCover_precomp G h,
-      infinite_cluster_precomp G h]
-  simp only [Measure.real]
-  rw [he, ← Measure.map_apply hm hs, labelLaw_map_precomp]
-
-end CoverInvariance
-
-section CoverObstruction
-attribute [local instance] Classical.propDecidable
-variable [Countable V] [Countable E]
-
-theorem cover_survival_square_sum_le (G : BondGraph V E) (p q : unitInterval) (n : ℕ)
-    {A : Hilbert V →L[ℂ] Hilbert V} (hA : HasKernel G q A) (F : Finset V) :
-    (∑ x ∈ F, (labelLaw : Measure (E → unitInterval)).real {labels |
-      TouchedCover G p q n labels ∧ (cluster G (openEdges labels p) x).Infinite}) ^ 2 ≤
-      n * ‖A‖ * F.card := by
-  let S := fun x => {labels | TouchedCover G p q n labels ∧
-      (cluster G (openEdges labels p) x).Infinite}
-  let C := fun x y => {labels | Conn G (openEdges labels q) x y}
-  have hS (x) : MeasurableSet (S x) :=
-    (measurableSet_touchedCover G p q n).inter
-      ((measurableSet_infinite_cluster G x).preimage (measurable_openEdges p))
-  have hC (x y) : MeasurableSet (C x y) :=
-    (measurableSet_conn G x y).preimage (measurable_openEdges q)
-  have hpoint (labels : E → unitInterval) :
-      (∑ x ∈ F, (S x).indicator (fun _ => (1 : ℝ)) labels) ^ 2 ≤
-      (n : ℝ) * ∑ x ∈ F, ∑ y ∈ F, (C x y).indicator (fun _ => (1 : ℝ)) labels := by
-    by_cases hc : TouchedCover G p q n labels
-    · have he : ∑ x ∈ F, (S x).indicator (fun _ => (1 : ℝ)) labels =
-          ((F.filter fun x => (cluster G (openEdges labels p) x).Infinite).card : ℝ) := by
-        simp only [S, Set.indicator, Set.mem_ofPred_eq, hc, true_and]
-        rw [← Finset.sum_filter]
-        simp
-      rw [he]
-      simpa only [C, Set.indicator, Set.mem_ofPred_eq] using
-        survivor_count_sq_le G p q n F labels hc
-    · simp only [S, Set.indicator, Set.mem_ofPred_eq, hc, false_and, ite_false,
-        Finset.sum_const_zero, zero_pow (by decide : 2 ≠ 0)]
-      exact mul_nonneg (Nat.cast_nonneg _) (Finset.sum_nonneg fun _ _ =>
-        Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _)
-  have h := square_sum_probability_le (labelLaw : Measure (E → unitInterval)) F S C hS hC
-    (n : ℝ) hpoint
-  have he (x y) : labelLaw.real (C x y) = twoPoint G q x y :=
-    (twoPoint_eq_labelLaw G q x y).symm
-  simp only [he] at h
-  exact h.trans (by simpa only [mul_assoc] using
-    mul_le_mul_of_nonneg_left (sum_twoPoint_le_norm_card G q hA F) (Nat.cast_nonneg n))
-
- 
-
-theorem measure_cover_survival_eq_zero [Infinite V] (G : BondGraph V E)
-    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p q : unitInterval) (hq : OperatorBounded G q) (n : ℕ) (x : V) :
-    (labelLaw : Measure (E → unitInterval)) {labels | TouchedCover G p q n labels ∧
-      (cluster G (openEdges labels p) x).Infinite} = 0 := by
-  apply (measureReal_eq_zero_iff).mp
-  by_contra hnz
-  let θ := (labelLaw : Measure (E → unitInterval)).real {labels | TouchedCover G p q n labels ∧
-      (cluster G (openEdges labels p) x).Infinite}
-  have hθ : 0 < θ := lt_of_le_of_ne measureReal_nonneg (Ne.symm hnz)
-  obtain ⟨A, hA⟩ := hq
-  obtain ⟨m, hm⟩ := exists_nat_gt ((n : ℝ) * ‖A‖ / θ ^ 2 + 1)
-  obtain ⟨F, hF, hcard⟩ := (vertexOrbit_infinite G hconn hloc hqt x).exists_subset_card_eq m
-  have he (y : V) (hy : y ∈ F) : (labelLaw : Measure (E → unitInterval)).real
-      {labels | TouchedCover G p q n labels ∧ (cluster G (openEdges labels p) y).Infinite} = θ := by
-    obtain ⟨a, b, hab, hay⟩ := hF hy
-    rw [← hay]
-    exact cover_survival_aut G p q n hab x
-  have hh := cover_survival_square_sum_le G p q n hA F
-  rw [Finset.sum_congr rfl he] at hh
-  simp only [Finset.sum_const, nsmul_eq_mul, hcard] at hh
-  have hθsq : 0 < θ ^ 2 := sq_pos_of_pos hθ
-  have hm0 : (0 : ℝ) < m := lt_trans (by positivity : (0 : ℝ) < (n : ℝ) * ‖A‖ / θ ^ 2 + 1) hm
-  have hlt : (n : ℝ) * ‖A‖ < (m : ℝ) * θ ^ 2 := by
-    apply (div_lt_iff₀ hθsq).mp
-    linarith
-  nlinarith
-
-end CoverObstruction
-
-section FiniteChanges
-attribute [local instance] Classical.propDecidable
-
- 
-theorem cluster_union_subset (G : BondGraph V E) (ω : Set E) (F : Set E) (x : V) :
-    cluster G (ω ∪ F) x ⊆ cluster G ω x ∪ ⋃ e ∈ F, ⋃ z ∈ {z | z ∈ G e}, cluster G ω z := by
-  intro y hy
-  induction hy with
-  | refl => exact Or.inl (mem_cluster_self G ω x)
-  | @tail y z hy hed ih =>
-    obtain ⟨e, he, heq⟩ := hed
-    rcases he with he | he
-    · have hyz : Conn G ω y z := (conn_refl G ω y).tail ⟨e, he, heq⟩
-      rcases ih with ih | ih
-      · exact Or.inl (conn_trans G ω ih hyz)
-      · right
-        simp only [Set.mem_iUnion, Set.mem_ofPred_eq] at ih ⊢
-        obtain ⟨e', he', w, hw, hwy⟩ := ih
-        exact ⟨e', he', w, hw, conn_trans G ω hwy hyz⟩
-    · right
-      simp only [Set.mem_iUnion, Set.mem_ofPred_eq]
-      exact ⟨e, he, z, heq ▸ by simp, conn_refl G ω z⟩
-
-theorem exists_infinite_cluster_mono (G : BondGraph V E) {ω ω' : Set E} (h : ω ⊆ ω') :
-    (∃ x, (cluster G ω x).Infinite) → ∃ x, (cluster G ω' x).Infinite := by
-  rintro ⟨x, hx⟩
-  exact ⟨x, hx.mono (fun _ hy => conn_mono G h hy)⟩
-
- 
-theorem exists_infinite_cluster_sdiff (G : BondGraph V E) (ω : Set E)
-    {F : Set E} (hF : F.Finite) :
-    (∃ x, (cluster G (ω \ F) x).Infinite) ↔ ∃ x, (cluster G ω x).Infinite := by
-  constructor
-  · exact exists_infinite_cluster_mono G Set.sdiff_subset
-  · contrapose!
-    intro h x
-    have hs : (cluster G (ω \ F) x ∪
-        ⋃ e ∈ F, ⋃ z ∈ {z | z ∈ G e}, cluster G (ω \ F) z).Finite := by
-      apply (h x).union
-      apply hF.biUnion
-      intro e _
-      have he : {z | z ∈ G e}.Finite := by
-        exact (G e).toFinset.finite_toSet.subset (fun z hz => by simpa using hz)
-      exact he.biUnion (fun z _ => h z)
-    apply hs.subset
-    apply Set.Subset.trans (fun y hy => conn_mono G (Set.subset_sdiff_union _ _) hy)
-    exact cluster_union_subset G (ω \ F) F x
-
-end FiniteChanges
-
-section PercolationZeroOne
-attribute [local instance] Classical.propDecidable
-variable [Countable V] [Countable E]
-
-omit [Countable V] [Countable E] in
-theorem infiniteClusters_nonempty_iff (G : BondGraph V E) (ω : Set E) :
-    (infiniteClusters G ω).Nonempty ↔ ∃ x, (cluster G ω x).Infinite := by
-  constructor
-  · rintro ⟨C, hC, x, rfl⟩
-    exact ⟨x, hC⟩
-  · rintro ⟨x, hx⟩
-    exact ⟨cluster G ω x, hx, x, rfl⟩
-
- theorem measurableSet_exists_infinite_cluster (G : BondGraph V E) :
-    MeasurableSet {ω : Set E | ∃ x, (cluster G ω x).Infinite} := by
-  simpa only [Set.ofPred_exists] using MeasurableSet.iUnion (fun x => measurableSet_infinite_cluster G x)
-
- theorem infinite_cluster_zero_one (G : BondGraph V E) (p : unitInterval) :
-    (labelLaw : Measure (E → unitInterval))
-      {labels | ∃ x, (cluster G (openEdges labels p) x).Infinite} = 0 ∨
-    (labelLaw : Measure (E → unitInterval))
-      {labels | ∃ x, (cluster G (openEdges labels p) x).Infinite} = 1 := by
-  let σ : E → MeasurableSpace (E → unitInterval) := fun e =>
-    MeasurableSpace.comap (fun labels : E → unitInterval => labels e) inferInstance
-  have hle (e) : σ e ≤ (inferInstance : MeasurableSpace (E → unitInterval)) := (measurable_pi_apply e).comap_le
-  have hi : iIndep σ (labelLaw : Measure (E → unitInterval)) :=
-    iIndepFun_infinitePi (X := fun (_ : E) (x : unitInterval) => x) (fun _ => measurable_id)
-  have htail : MeasurableSet[Filter.limsup σ Filter.cofinite]
-      {labels | ∃ x, (cluster G (openEdges labels p) x).Infinite} := by
-    rw [Filter.limsup_eq_iInf_iSup]
-    apply MeasurableSpace.measurableSet_iInf.mpr
-    intro s
-    apply MeasurableSpace.measurableSet_iInf.mpr
-    intro hs
-    have hfin : sᶜ.Finite := Filter.mem_cofinite.mp hs
-    let m := ⨆ e ∈ s, σ e
-    have hm : @Measurable (E → unitInterval) (Set E) m inferInstance
-        (fun labels => openEdges labels p ∩ s) := by
-      apply measurable_setOfPred.comp
-      apply Measurable.of_eval
-      intro e
-      change @Measurable (E → unitInterval) Prop m inferInstance (fun labels => labels e ≤ p ∧ e ∈ s)
-      by_cases he : e ∈ s
-      · have hev : @Measurable (E → unitInterval) unitInterval m inferInstance
-            (fun labels => labels e) :=
-          Measurable.of_comap_le (le_iSup_of_le e (le_iSup_of_le he le_rfl))
-        simpa only [he, and_true, Function.comp_def]
-          using (measurable_threshold p).comp hev
-      · simpa only [he, and_false]
-          using (measurable_const : @Measurable (E → unitInterval) Prop m inferInstance
-            (fun _ => False))
-    have heq : {labels : E → unitInterval | ∃ x, (cluster G (openEdges labels p) x).Infinite} =
-        {labels | ∃ x, (cluster G (openEdges labels p ∩ s) x).Infinite} := by
-      ext labels
-      simpa only [Set.sdiff_compl, Set.mem_ofPred_eq] using
-        (exists_infinite_cluster_sdiff G (openEdges labels p) hfin).symm
-    rw [heq]
-    exact (measurableSet_exists_infinite_cluster G).preimage hm
-  exact measure_zero_or_one_of_measurableSet_limsup
-    (s := σ) (f := Filter.cofinite) (p := Set.Finite)
-    (ns := fun F : Finset E => (F : Set E)) hle hi
-    (fun t ht => by simpa using ht)
-    (fun F H => ⟨F ∪ H, Finset.coe_subset.mpr Finset.subset_union_left,
-      Finset.coe_subset.mpr Finset.subset_union_right⟩)
-    (fun F => F.finite_toSet) (fun e => ⟨{e}, by simp⟩) htail
-
-end PercolationZeroOne
-
-section CoupledInterval
-attribute [local instance] Classical.propDecidable
-variable [Countable V] [Countable E]
-
- theorem exists_infinite_cluster_probability_mono (G : BondGraph V E) {p q : unitInterval}
-    (hpq : p ≤ q) :
-    law p {ω | (infiniteClusters G ω).Nonempty} ≤
-      law q {ω | (infiniteClusters G ω).Nonempty} := by
-  have hm : MeasurableSet {ω : Set E | (infiniteClusters G ω).Nonempty} := by
-    simpa only [infiniteClusters_nonempty_iff] using measurableSet_exists_infinite_cluster G
-  rw [← map_openEdges p, ← map_openEdges q,
-    Measure.map_apply (measurable_openEdges p) hm,
-    Measure.map_apply (measurable_openEdges q) hm]
-  apply measure_mono
-  intro labels h
-  simp only [Set.mem_preimage, Set.mem_ofPred_eq, infiniteClusters_nonempty_iff] at h ⊢
-  exact exists_infinite_cluster_mono G (openEdges_mono labels hpq) h
-
- theorem ae_exists_infinite_cluster_of_pc_lt (G : BondGraph V E) (p : unitInterval)
-    (hp : pc G < p) :
-    ∀ᵐ labels ∂(labelLaw : Measure (E → unitInterval)),
-      ∃ x, (cluster G (openEdges labels p) x).Infinite := by
-  obtain ⟨q, hq, hqp⟩ := sInf_lt_iff.mp hp
-  have hpos : 0 < law p {ω | (infiniteClusters G ω).Nonempty} :=
-    lt_of_lt_of_le hq (exists_infinite_cluster_probability_mono G hqp.le)
-  have hm : MeasurableSet {ω : Set E | (infiniteClusters G ω).Nonempty} := by
-    simpa only [infiniteClusters_nonempty_iff] using measurableSet_exists_infinite_cluster G
-  rw [← map_openEdges p, Measure.map_apply (measurable_openEdges p) hm] at hpos
-  simp only [Set.preimage_ofPred_eq, infiniteClusters_nonempty_iff] at hpos
-  have hpone := (infinite_cluster_zero_one G p).resolve_left (ne_of_gt hpos)
-  exact (mem_ae_iff_prob_eq_one
-    ((measurableSet_exists_infinite_cluster G).preimage (measurable_openEdges p))).mpr hpone
-
-omit [Countable V] [Countable E] in
- theorem touchedCover_of_finite_infiniteClusters (G : BondGraph V E)
-    (labels : E → unitInterval) {p r q : unitInterval} (hpr : p ≤ r) (hrq : r ≤ q)
-    (hfinite : (infiniteClusters G (openEdges labels r)).Finite) :
-    ∃ n, TouchedCover G p q n labels := by
-  let C := infiniteClusters G (openEdges labels r)
-  let : Fintype C := hfinite.fintype
-  let f : C → V := fun c => Classical.choose c.property.2
-  have hf (c : C) : c.val = cluster G (openEdges labels r) (f c) :=
-    Classical.choose_spec c.property.2
-  refine ⟨Fintype.card C, fun i => f ((Fintype.equivFin C).symm i), ?_⟩
-  intro x hx
-  have hx' : (cluster G (openEdges labels r) x).Infinite :=
-    hx.mono (fun _ hy => conn_mono G (openEdges_mono labels hpr) hy)
-  let c : C := ⟨cluster G (openEdges labels r) x, hx', x, rfl⟩
-  refine ⟨Fintype.equivFin C c, ?_⟩
-  simp only [Equiv.symm_apply_apply]
-  apply conn_mono G (openEdges_mono labels hrq)
-  change f c ∈ cluster G (openEdges labels r) x
-  change f c ∈ c.val
-  rw [hf c]
-  exact mem_cluster_self G _ _
-
- 
-
- theorem coupled_interval [Infinite V] (G : BondGraph V E) (hconn : Connected G)
-    (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p q : unitInterval) (hp : pc G < p) (hq : OperatorBounded G q) :
-    ∀ᵐ labels ∂(labelLaw : Measure (E → unitInterval)),
-      ∀ r ∈ Set.Icc p q, (infiniteClusters G (openEdges labels r)).Infinite := by
-  have hall : ∀ᵐ labels ∂(labelLaw : Measure (E → unitInterval)), ∀ n x,
-      ¬(TouchedCover G p q n labels ∧ (cluster G (openEdges labels p) x).Infinite) := by
-    rw [ae_all_iff]
-    intro n
-    rw [ae_all_iff]
-    intro x
-    apply ae_iff.mpr
-    simpa only [not_not] using measure_cover_survival_eq_zero G hconn hloc hqt p q hq n x
-  filter_upwards [hall, ae_exists_infinite_cluster_of_pc_lt G p hp] with labels hlabels hex
-  intro r hr
-  by_contra hfinite
-  obtain ⟨n, hn⟩ := touchedCover_of_finite_infiniteClusters G labels hr.1 hr.2
-    (Set.not_infinite.mp hfinite)
-  obtain ⟨x, hx⟩ := hex
-  exact hlabels n x ⟨hn, hx⟩
-
-end CoupledInterval
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.ProductMeasure.Association
+
+namespace OAI
+
+/-! Cluster events, invariant covers and uniqueness thresholds. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+section InfiniteOrbits
+attribute [local instance] Classical.propDecidable
+
+def vertexOrbit (G : BondGraph V E) (x : V) : Set V :=
+  {y | ∃ a : V ≃ V, ∃ b : E ≃ E, IsAut G a b ∧ a x = y}
+
+theorem adj_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (h : IsAut G a b) {x y : V} (hxy : Adj G x y) : Adj G (a x) (a y) := by
+  obtain ⟨e, he⟩ := hxy
+  exact ⟨b e, by rw [h e, he, Sym2.map_mk]⟩
+
+theorem mem_reachBall_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (h : IsAut G a b) {x y : V} {n : ℕ} (hy : y ∈ reachBall G x n) :
+    a y ∈ reachBall G (a x) n := by
+  induction n generalizing y with
+  | zero => simpa only [reachBall, Set.mem_singleton_iff, Equiv.apply_eq_iff_eq] using hy
+  | succ n ih =>
+    rcases hy with hy | hy
+    · exact Or.inl (ih hy)
+    · change a y ∈ reachBall G (a x) n ∪ ⋃ z ∈ reachBall G (a x) n, {w | Adj G z w}
+      apply Or.inr
+      simp only [Set.mem_iUnion, Set.mem_ofPred_eq] at hy ⊢
+      obtain ⟨z, hz, hzy⟩ := hy
+      exact ⟨a z, ih hz, adj_aut G h hzy⟩
+
+ 
+
+theorem vertexOrbit_infinite [Infinite V] (G : BondGraph V E)
+    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G) (x : V) :
+    (vertexOrbit G x).Infinite := by
+  intro hfin
+  obtain ⟨R, hR⟩ := hqt
+  choose n hn using (fun y => exists_mem_reachBall (hconn x y))
+  have hS : (⋃ o ∈ (R : Set V), ⋃ z ∈ vertexOrbit G x, reachBall G z (n o)).Finite :=
+    R.finite_toSet.biUnion fun o _ => hfin.biUnion fun z _ => finite_reachBall hloc z (n o)
+  have he : Set.univ ⊆ ⋃ o ∈ (R : Set V), ⋃ z ∈ vertexOrbit G x, reachBall G z (n o) := by
+    intro y _
+    obtain ⟨o, ho, a, b, hab, hay⟩ := hR y
+    simp only [Set.mem_iUnion]
+    exact ⟨o, ho, a x, ⟨a, b, hab, rfl⟩, hay ▸ mem_reachBall_aut G hab (hn o)⟩
+  exact Set.infinite_univ (hS.subset he)
+
+end InfiniteOrbits
+
+section FinitePairBound
+attribute [local instance] Classical.propDecidable
+open scoped ComplexConjugate
+
+noncomputable def deltaSum (F : Finset V) : Hilbert V := ∑ x ∈ F, lp.single 2 x (1 : ℂ)
+
+@[simp] theorem deltaSum_apply (F : Finset V) (x : V) :
+    deltaSum F x = if x ∈ F then 1 else 0 := by
+  simp only [deltaSum, lp.coeFn_sum, Finset.sum_apply, lp.single_apply]
+  simp
+
+@[simp] theorem inner_deltaSum (F : Finset V) (f : Hilbert V) :
+    inner ℂ (deltaSum F) f = ∑ x ∈ F, f x := by
+  simp [deltaSum, lp.inner_single_left]
+
+theorem norm_deltaSum_sq (F : Finset V) : ‖deltaSum F‖ ^ 2 = (F.card : ℝ) := by
+  rw [← inner_self_eq_norm_sq (𝕜 := ℂ), inner_deltaSum]
+  simp only [map_sum, deltaSum_apply]
+  trans ∑ _x ∈ F, (1 : ℝ)
+  · apply Finset.sum_congr rfl
+    intro x hx
+    simp [hx]
+  · simp
+
+ 
+theorem sum_twoPoint_le_norm_card (G : BondGraph V E) (p : unitInterval)
+    {A : Hilbert V →L[ℂ] Hilbert V} (hA : HasKernel G p A) (F : Finset V) :
+    ∑ x ∈ F, ∑ y ∈ F, twoPoint G p x y ≤ ‖A‖ * F.card := by
+  have he : inner ℂ (deltaSum F) (A (deltaSum F)) =
+      ((∑ x ∈ F, ∑ y ∈ F, twoPoint G p x y : ℝ) : ℂ) := by
+    rw [inner_deltaSum]
+    simp only [deltaSum, map_sum]
+    simp only [lp.coeFn_sum, Finset.sum_apply, Complex.ofReal_sum]
+    apply Finset.sum_congr rfl
+    intro x _
+    apply Finset.sum_congr rfl
+    intro y _
+    exact hA x y
+  have h := (Complex.re_le_norm (inner ℂ (deltaSum F) (A (deltaSum F)))).trans
+    ((norm_inner_le_norm _ _).trans
+      (mul_le_mul_of_nonneg_left (A.le_opNorm _) (norm_nonneg _)))
+  rw [he, Complex.ofReal_re] at h
+  calc
+    _ ≤ ‖deltaSum F‖ * (‖A‖ * ‖deltaSum F‖) := h
+    _ = ‖A‖ * ‖deltaSum F‖ ^ 2 := by ring
+    _ = _ := by rw [norm_deltaSum_sq]
+
+end FinitePairBound
+
+section PairCounting
+attribute [local instance] Classical.propDecidable
+
+ 
+theorem card_sq_le_image_card_mul_pairs {W : Type*} (S : Finset V) (c : V → W) :
+    (S.card : ℝ) ^ 2 ≤ ((S.image c).card : ℝ) *
+      ∑ x ∈ S, ∑ y ∈ S, if c x = c y then (1 : ℝ) else 0 := by
+  let t := S.image c
+  let f := fun w => ((S.filter (fun x => c x = w)).card : ℝ)
+  have hsum : ∑ w ∈ t, f w = (S.card : ℝ) := by
+    have h := Finset.sum_fiberwise_of_maps_to
+      (fun x (hx : x ∈ S) => Finset.mem_image_of_mem c hx) (fun _ => (1 : ℝ))
+    simpa only [Finset.sum_const, nsmul_eq_mul, mul_one, t, f] using h
+  have hsq : ∑ w ∈ t, f w ^ 2 =
+      ∑ x ∈ S, ∑ y ∈ S, if c x = c y then (1 : ℝ) else 0 := by
+    calc
+      ∑ w ∈ t, f w ^ 2 = ∑ w ∈ t, ∑ x ∈ S with c x = w, f (c x) := by
+        apply Finset.sum_congr rfl
+        intro w hw
+        rw [show ∑ x ∈ S with c x = w, f (c x) =
+            ∑ _x ∈ S with c _x = w, f w from
+          Finset.sum_congr rfl (fun x hx => by rw [(Finset.mem_filter.mp hx).2])]
+        simp only [Finset.sum_const, nsmul_eq_mul, f]
+        ring
+      _ = ∑ x ∈ S, f (c x) := Finset.sum_fiberwise_of_maps_to
+        (fun x hx => Finset.mem_image_of_mem c hx) _
+      _ = _ := by
+        apply Finset.sum_congr rfl
+        intro x hx
+        rw [← Finset.sum_filter]
+        simp only [Finset.sum_const, nsmul_eq_mul, mul_one, f]
+        congr 2
+        ext y
+        simp [eq_comm]
+  have h := Finset.sum_mul_sq_le_sq_mul_sq t (fun _ => (1 : ℝ)) f
+  simpa only [one_mul, one_pow, Finset.sum_const, nsmul_eq_mul, mul_one, hsum, hsq, t] using h
+
+end PairCounting
+
+section PairIntegral
+attribute [local instance] Classical.propDecidable
+variable {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]
+
+ 
+theorem square_sum_probability_le (F : Finset V) (S : V → Set Ω)
+    (C : V → V → Set Ω) (hS : ∀ x, MeasurableSet (S x))
+    (hC : ∀ x y, MeasurableSet (C x y)) (n : ℝ)
+    (hpoint : ∀ ω, (∑ x ∈ F, (S x).indicator (fun _ => (1 : ℝ)) ω) ^ 2 ≤
+      n * ∑ x ∈ F, ∑ y ∈ F, (C x y).indicator (fun _ => (1 : ℝ)) ω) :
+    (∑ x ∈ F, μ.real (S x)) ^ 2 ≤ n * ∑ x ∈ F, ∑ y ∈ F, μ.real (C x y) := by
+  let X := fun ω => ∑ x ∈ F, (S x).indicator (fun _ => (1 : ℝ)) ω
+  have hX : MemLp X 2 μ := memLp_finsetSum F
+    (fun x _ => (memLp_const (1 : ℝ)).indicator (hS x))
+  have hCi (x y) : Integrable ((C x y).indicator (fun _ => (1 : ℝ))) μ :=
+    (integrable_const (1 : ℝ)).indicator (hC x y)
+  have hY : Integrable (fun ω => ∑ x ∈ F, ∑ y ∈ F,
+      (C x y).indicator (fun _ => (1 : ℝ)) ω) μ :=
+    integrable_finsetSum F (fun x _ => integrable_finsetSum F (fun y _ => hCi x y))
+  have hv := variance_nonneg X μ
+  rw [variance_eq_sub hX] at hv
+  have hi := integral_mono hX.integrable_sq (hY.const_mul n) hpoint
+  have hx : (∫ ω, X ω ∂μ) = ∑ x ∈ F, μ.real (S x) := by
+    dsimp only [X]
+    rw [integral_finsetSum F (fun x _ => (integrable_const (1 : ℝ)).indicator (hS x))]
+    simp only [integral_indicator_const _ (hS _), smul_eq_mul, mul_one]
+  have hy : (∫ ω, n * ∑ x ∈ F, ∑ y ∈ F,
+      (C x y).indicator (fun _ => (1 : ℝ)) ω ∂μ) =
+      n * ∑ x ∈ F, ∑ y ∈ F, μ.real (C x y) := by
+    rw [integral_const_mul, integral_finsetSum F
+      (fun x _ => integrable_finsetSum F (fun y _ => hCi x y))]
+    congr 1
+    apply Finset.sum_congr rfl
+    intro x _
+    rw [integral_finsetSum F (fun y _ => hCi x y)]
+    simp only [integral_indicator_const _ (hC x _), smul_eq_mul, mul_one]
+  rw [hx] at hv
+  rw [hy] at hi
+  exact (sub_nonneg.mp hv).trans hi
+
+end PairIntegral
+
+section CoupledCovers
+attribute [local instance] Classical.propDecidable
+variable [Countable V] [Countable E]
+
+omit [Countable V] [Countable E] in
+theorem cluster_eq_iff_conn (G : BondGraph V E) (ω : Set E) (x y : V) :
+    cluster G ω x = cluster G ω y ↔ Conn G ω x y := by
+  constructor
+  · intro h
+    have hh := mem_cluster_self G ω y
+    rwa [← h] at hh
+  · exact cluster_eq_of_conn G ω
+
+ 
+def TouchedCover (G : BondGraph V E) (p q : unitInterval) (n : ℕ)
+    (labels : E → unitInterval) : Prop :=
+  ∃ f : Fin n → V, ∀ x, (cluster G (openEdges labels p) x).Infinite →
+    ∃ i, Conn G (openEdges labels q) x (f i)
+
+theorem measurableSet_touchedCover (G : BondGraph V E) (p q : unitInterval) (n : ℕ) :
+    MeasurableSet {labels | TouchedCover G p q n labels} := by
+  have he : {labels | TouchedCover G p q n labels} =
+      ⋃ f : Fin n → V, ⋂ x : V, ({labels | (cluster G (openEdges labels p) x).Infinite}ᶜ ∪
+        ⋃ i : Fin n, {labels | Conn G (openEdges labels q) x (f i)}) := by
+    ext labels
+    simp only [TouchedCover, Set.mem_ofPred_eq, Set.mem_iUnion, Set.mem_iInter,
+      Set.mem_union, Set.mem_compl_iff]
+    simp only [← imp_iff_not_or]
+  rw [he]
+  apply MeasurableSet.iUnion
+  intro f
+  apply MeasurableSet.iInter
+  intro x
+  exact ((measurableSet_infinite_cluster G x).preimage (measurable_openEdges p)).compl.union
+    (MeasurableSet.iUnion fun i => (measurableSet_conn G x (f i)).preimage (measurable_openEdges q))
+
+omit [Countable V] [Countable E] in
+ 
+theorem survivor_count_sq_le (G : BondGraph V E) (p q : unitInterval) (n : ℕ)
+    (F : Finset V) (labels : E → unitInterval) (h : TouchedCover G p q n labels) :
+    ((F.filter fun x => (cluster G (openEdges labels p) x).Infinite).card : ℝ) ^ 2 ≤
+      n * ∑ x ∈ F, ∑ y ∈ F, if Conn G (openEdges labels q) x y then (1 : ℝ) else 0 := by
+  obtain ⟨f, hf⟩ := h
+  let S := F.filter fun x => (cluster G (openEdges labels p) x).Infinite
+  let c := cluster G (openEdges labels q)
+  have hsub : S.image c ⊆ Finset.univ.image (fun i : Fin n => c (f i)) := by
+    intro C hC
+    obtain ⟨x, hx, rfl⟩ := Finset.mem_image.mp hC
+    obtain ⟨i, hi⟩ := hf x (Finset.mem_filter.mp hx).2
+    exact Finset.mem_image.mpr ⟨i, Finset.mem_univ _, (cluster_eq_of_conn G _ hi).symm⟩
+  have hcard : (S.image c).card ≤ n :=
+    (Finset.card_le_card hsub).trans (by simpa using (Finset.card_image_le (s := (Finset.univ : Finset (Fin n)))
+      (f := fun i : Fin n => c (f i))))
+  have hpair := card_sq_le_image_card_mul_pairs S c
+  have hpairs : ∑ x ∈ S, ∑ y ∈ S, (if c x = c y then (1 : ℝ) else 0) ≤
+      ∑ x ∈ F, ∑ y ∈ F, if Conn G (openEdges labels q) x y then (1 : ℝ) else 0 := by
+    simp only [c, cluster_eq_iff_conn] 
+    calc
+      _ ≤ ∑ x ∈ S, ∑ y ∈ F, if Conn G (openEdges labels q) x y then (1 : ℝ) else 0 := by
+        apply Finset.sum_le_sum
+        intro x _
+        exact Finset.sum_le_sum_of_subset_of_nonneg (Finset.filter_subset _ _)
+          (fun _ _ _ => by split_ifs <;> positivity)
+      _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.filter_subset _ _)
+        (fun _ _ _ => Finset.sum_nonneg fun _ _ => by split_ifs <;> positivity)
+  exact hpair.trans ((mul_le_mul_of_nonneg_left hpairs (Nat.cast_nonneg _)).trans
+    (mul_le_mul_of_nonneg_right (by exact_mod_cast hcard)
+      (Finset.sum_nonneg fun _ _ => Finset.sum_nonneg fun _ _ => by split_ifs <;> positivity)))
+
+end CoupledCovers
+
+section CoverInvariance
+attribute [local instance] Classical.propDecidable
+
+ 
+theorem labelLaw_map_precomp (b : E ≃ E) :
+    (labelLaw : Measure (E → unitInterval)).map (fun labels => labels ∘ b) = labelLaw := by
+  have he : (fun labels : E → unitInterval => labels ∘ b) =
+      MeasurableEquiv.piCongrLeft (fun _ : E => unitInterval) b.symm := by
+    funext labels e
+    have h := MeasurableEquiv.piCongrLeft_apply_apply (β := fun _ : E => unitInterval)
+      b.symm labels (b e)
+    simpa only [Equiv.symm_apply_apply, Function.comp_apply] using h.symm
+  unfold labelLaw
+  rw [he]
+  exact Measure.infinitePi_map_piCongrLeft (fun _ : E => (volume : Measure unitInterval)) b.symm
+
+theorem infinite_cluster_precomp (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (h : IsAut G a b) (labels : E → unitInterval) (p : unitInterval) (x : V) :
+    (cluster G (openEdges (labels ∘ b) p) x).Infinite ↔
+      (cluster G (openEdges labels p) (a x)).Infinite := by
+  change (cluster G (b ⁻¹' openEdges labels p) x).Infinite ↔ _
+  rw [← cluster_aut G h]
+  exact (Set.infinite_image_iff (fun _ _ _ _ hab => a.injective hab)).symm
+
+variable [Countable V] [Countable E]
+
+omit [Countable V] [Countable E] in
+theorem touchedCover_precomp (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (h : IsAut G a b) (labels : E → unitInterval) (p q : unitInterval) (n : ℕ) :
+    TouchedCover G p q n (labels ∘ b) ↔ TouchedCover G p q n labels := by
+  constructor
+  · rintro ⟨f, hf⟩
+    refine ⟨fun i => a (f i), fun x hx => ?_⟩
+    have hs : (cluster G (openEdges (labels ∘ b) p) (a.symm x)).Infinite :=
+      (infinite_cluster_precomp G h labels p (a.symm x)).mpr (by simpa using hx)
+    obtain ⟨i, hi⟩ := hf _ hs
+    exact ⟨i, by simpa using (conn_aut_iff G h (openEdges labels q) (a.symm x) (f i)).mp hi⟩
+  · rintro ⟨f, hf⟩
+    refine ⟨fun i => a.symm (f i), fun x hx => ?_⟩
+    obtain ⟨i, hi⟩ := hf (a x) ((infinite_cluster_precomp G h labels p x).mp hx)
+    exact ⟨i, (conn_aut_iff G h (openEdges labels q) x (a.symm (f i))).mpr (by simpa using hi)⟩
+
+theorem cover_survival_aut (G : BondGraph V E) (p q : unitInterval) (n : ℕ)
+    {a : V ≃ V} {b : E ≃ E} (h : IsAut G a b) (x : V) :
+    (labelLaw : Measure (E → unitInterval)).real
+      {labels | TouchedCover G p q n labels ∧ (cluster G (openEdges labels p) (a x)).Infinite} =
+    labelLaw.real {labels | TouchedCover G p q n labels ∧ (cluster G (openEdges labels p) x).Infinite} := by
+  have hm : Measurable (fun labels : E → unitInterval => labels ∘ b) :=
+    Measurable.of_eval (fun e => measurable_pi_apply (b e))
+  have hs : MeasurableSet {labels | TouchedCover G p q n labels ∧
+      (cluster G (openEdges labels p) x).Infinite} :=
+    (measurableSet_touchedCover G p q n).inter
+      ((measurableSet_infinite_cluster G x).preimage (measurable_openEdges p))
+  have he : {labels | TouchedCover G p q n labels ∧
+      (cluster G (openEdges labels p) (a x)).Infinite} =
+      (fun labels : E → unitInterval => labels ∘ b) ⁻¹'
+        {labels | TouchedCover G p q n labels ∧ (cluster G (openEdges labels p) x).Infinite} := by
+    ext labels
+    simp only [Set.mem_ofPred_eq, Set.mem_preimage, touchedCover_precomp G h,
+      infinite_cluster_precomp G h]
+  simp only [Measure.real]
+  rw [he, ← Measure.map_apply hm hs, labelLaw_map_precomp]
+
+end CoverInvariance
+
+section CoverObstruction
+attribute [local instance] Classical.propDecidable
+variable [Countable V] [Countable E]
+
+theorem cover_survival_square_sum_le (G : BondGraph V E) (p q : unitInterval) (n : ℕ)
+    {A : Hilbert V →L[ℂ] Hilbert V} (hA : HasKernel G q A) (F : Finset V) :
+    (∑ x ∈ F, (labelLaw : Measure (E → unitInterval)).real {labels |
+      TouchedCover G p q n labels ∧ (cluster G (openEdges labels p) x).Infinite}) ^ 2 ≤
+      n * ‖A‖ * F.card := by
+  let S := fun x => {labels | TouchedCover G p q n labels ∧
+      (cluster G (openEdges labels p) x).Infinite}
+  let C := fun x y => {labels | Conn G (openEdges labels q) x y}
+  have hS (x) : MeasurableSet (S x) :=
+    (measurableSet_touchedCover G p q n).inter
+      ((measurableSet_infinite_cluster G x).preimage (measurable_openEdges p))
+  have hC (x y) : MeasurableSet (C x y) :=
+    (measurableSet_conn G x y).preimage (measurable_openEdges q)
+  have hpoint (labels : E → unitInterval) :
+      (∑ x ∈ F, (S x).indicator (fun _ => (1 : ℝ)) labels) ^ 2 ≤
+      (n : ℝ) * ∑ x ∈ F, ∑ y ∈ F, (C x y).indicator (fun _ => (1 : ℝ)) labels := by
+    by_cases hc : TouchedCover G p q n labels
+    · have he : ∑ x ∈ F, (S x).indicator (fun _ => (1 : ℝ)) labels =
+          ((F.filter fun x => (cluster G (openEdges labels p) x).Infinite).card : ℝ) := by
+        simp only [S, Set.indicator, Set.mem_ofPred_eq, hc, true_and]
+        rw [← Finset.sum_filter]
+        simp
+      rw [he]
+      simpa only [C, Set.indicator, Set.mem_ofPred_eq] using
+        survivor_count_sq_le G p q n F labels hc
+    · simp only [S, Set.indicator, Set.mem_ofPred_eq, hc, false_and, ite_false,
+        Finset.sum_const_zero, zero_pow (by decide : 2 ≠ 0)]
+      exact mul_nonneg (Nat.cast_nonneg _) (Finset.sum_nonneg fun _ _ =>
+        Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _)
+  have h := square_sum_probability_le (labelLaw : Measure (E → unitInterval)) F S C hS hC
+    (n : ℝ) hpoint
+  have he (x y) : labelLaw.real (C x y) = twoPoint G q x y :=
+    (twoPoint_eq_labelLaw G q x y).symm
+  simp only [he] at h
+  exact h.trans (by simpa only [mul_assoc] using
+    mul_le_mul_of_nonneg_left (sum_twoPoint_le_norm_card G q hA F) (Nat.cast_nonneg n))
+
+ 
+
+theorem measure_cover_survival_eq_zero [Infinite V] (G : BondGraph V E)
+    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p q : unitInterval) (hq : OperatorBounded G q) (n : ℕ) (x : V) :
+    (labelLaw : Measure (E → unitInterval)) {labels | TouchedCover G p q n labels ∧
+      (cluster G (openEdges labels p) x).Infinite} = 0 := by
+  apply (measureReal_eq_zero_iff).mp
+  by_contra hnz
+  let θ := (labelLaw : Measure (E → unitInterval)).real {labels | TouchedCover G p q n labels ∧
+      (cluster G (openEdges labels p) x).Infinite}
+  have hθ : 0 < θ := lt_of_le_of_ne measureReal_nonneg (Ne.symm hnz)
+  obtain ⟨A, hA⟩ := hq
+  obtain ⟨m, hm⟩ := exists_nat_gt ((n : ℝ) * ‖A‖ / θ ^ 2 + 1)
+  obtain ⟨F, hF, hcard⟩ := (vertexOrbit_infinite G hconn hloc hqt x).exists_subset_card_eq m
+  have he (y : V) (hy : y ∈ F) : (labelLaw : Measure (E → unitInterval)).real
+      {labels | TouchedCover G p q n labels ∧ (cluster G (openEdges labels p) y).Infinite} = θ := by
+    obtain ⟨a, b, hab, hay⟩ := hF hy
+    rw [← hay]
+    exact cover_survival_aut G p q n hab x
+  have hh := cover_survival_square_sum_le G p q n hA F
+  rw [Finset.sum_congr rfl he] at hh
+  simp only [Finset.sum_const, nsmul_eq_mul, hcard] at hh
+  have hθsq : 0 < θ ^ 2 := sq_pos_of_pos hθ
+  have hm0 : (0 : ℝ) < m := lt_trans (by positivity : (0 : ℝ) < (n : ℝ) * ‖A‖ / θ ^ 2 + 1) hm
+  have hlt : (n : ℝ) * ‖A‖ < (m : ℝ) * θ ^ 2 := by
+    apply (div_lt_iff₀ hθsq).mp
+    linarith
+  nlinarith
+
+end CoverObstruction
+
+section FiniteChanges
+attribute [local instance] Classical.propDecidable
+
+ 
+theorem cluster_union_subset (G : BondGraph V E) (ω : Set E) (F : Set E) (x : V) :
+    cluster G (ω ∪ F) x ⊆ cluster G ω x ∪ ⋃ e ∈ F, ⋃ z ∈ {z | z ∈ G e}, cluster G ω z := by
+  intro y hy
+  induction hy with
+  | refl => exact Or.inl (mem_cluster_self G ω x)
+  | @tail y z hy hed ih =>
+    obtain ⟨e, he, heq⟩ := hed
+    rcases he with he | he
+    · have hyz : Conn G ω y z := (conn_refl G ω y).tail ⟨e, he, heq⟩
+      rcases ih with ih | ih
+      · exact Or.inl (conn_trans G ω ih hyz)
+      · right
+        simp only [Set.mem_iUnion, Set.mem_ofPred_eq] at ih ⊢
+        obtain ⟨e', he', w, hw, hwy⟩ := ih
+        exact ⟨e', he', w, hw, conn_trans G ω hwy hyz⟩
+    · right
+      simp only [Set.mem_iUnion, Set.mem_ofPred_eq]
+      exact ⟨e, he, z, heq ▸ by simp, conn_refl G ω z⟩
+
+theorem exists_infinite_cluster_mono (G : BondGraph V E) {ω ω' : Set E} (h : ω ⊆ ω') :
+    (∃ x, (cluster G ω x).Infinite) → ∃ x, (cluster G ω' x).Infinite := by
+  rintro ⟨x, hx⟩
+  exact ⟨x, hx.mono (fun _ hy => conn_mono G h hy)⟩
+
+ 
+theorem exists_infinite_cluster_sdiff (G : BondGraph V E) (ω : Set E)
+    {F : Set E} (hF : F.Finite) :
+    (∃ x, (cluster G (ω \ F) x).Infinite) ↔ ∃ x, (cluster G ω x).Infinite := by
+  constructor
+  · exact exists_infinite_cluster_mono G Set.sdiff_subset
+  · contrapose!
+    intro h x
+    have hs : (cluster G (ω \ F) x ∪
+        ⋃ e ∈ F, ⋃ z ∈ {z | z ∈ G e}, cluster G (ω \ F) z).Finite := by
+      apply (h x).union
+      apply hF.biUnion
+      intro e _
+      have he : {z | z ∈ G e}.Finite := by
+        exact (G e).toFinset.finite_toSet.subset (fun z hz => by simpa using hz)
+      exact he.biUnion (fun z _ => h z)
+    apply hs.subset
+    apply Set.Subset.trans (fun y hy => conn_mono G (Set.subset_sdiff_union _ _) hy)
+    exact cluster_union_subset G (ω \ F) F x
+
+end FiniteChanges
+
+section PercolationZeroOne
+attribute [local instance] Classical.propDecidable
+variable [Countable V] [Countable E]
+
+omit [Countable V] [Countable E] in
+theorem infiniteClusters_nonempty_iff (G : BondGraph V E) (ω : Set E) :
+    (infiniteClusters G ω).Nonempty ↔ ∃ x, (cluster G ω x).Infinite := by
+  constructor
+  · rintro ⟨C, hC, x, rfl⟩
+    exact ⟨x, hC⟩
+  · rintro ⟨x, hx⟩
+    exact ⟨cluster G ω x, hx, x, rfl⟩
+
+ theorem measurableSet_exists_infinite_cluster (G : BondGraph V E) :
+    MeasurableSet {ω : Set E | ∃ x, (cluster G ω x).Infinite} := by
+  simpa only [Set.ofPred_exists] using MeasurableSet.iUnion (fun x => measurableSet_infinite_cluster G x)
+
+ theorem infinite_cluster_zero_one (G : BondGraph V E) (p : unitInterval) :
+    (labelLaw : Measure (E → unitInterval))
+      {labels | ∃ x, (cluster G (openEdges labels p) x).Infinite} = 0 ∨
+    (labelLaw : Measure (E → unitInterval))
+      {labels | ∃ x, (cluster G (openEdges labels p) x).Infinite} = 1 := by
+  let σ : E → MeasurableSpace (E → unitInterval) := fun e =>
+    MeasurableSpace.comap (fun labels : E → unitInterval => labels e) inferInstance
+  have hle (e) : σ e ≤ (inferInstance : MeasurableSpace (E → unitInterval)) := (measurable_pi_apply e).comap_le
+  have hi : iIndep σ (labelLaw : Measure (E → unitInterval)) :=
+    iIndepFun_infinitePi (X := fun (_ : E) (x : unitInterval) => x) (fun _ => measurable_id)
+  have htail : MeasurableSet[Filter.limsup σ Filter.cofinite]
+      {labels | ∃ x, (cluster G (openEdges labels p) x).Infinite} := by
+    rw [Filter.limsup_eq_iInf_iSup]
+    apply MeasurableSpace.measurableSet_iInf.mpr
+    intro s
+    apply MeasurableSpace.measurableSet_iInf.mpr
+    intro hs
+    have hfin : sᶜ.Finite := Filter.mem_cofinite.mp hs
+    let m := ⨆ e ∈ s, σ e
+    have hm : @Measurable (E → unitInterval) (Set E) m inferInstance
+        (fun labels => openEdges labels p ∩ s) := by
+      apply measurable_setOfPred.comp
+      apply Measurable.of_eval
+      intro e
+      change @Measurable (E → unitInterval) Prop m inferInstance (fun labels => labels e ≤ p ∧ e ∈ s)
+      by_cases he : e ∈ s
+      · have hev : @Measurable (E → unitInterval) unitInterval m inferInstance
+            (fun labels => labels e) :=
+          Measurable.of_comap_le (le_iSup_of_le e (le_iSup_of_le he le_rfl))
+        simpa only [he, and_true, Function.comp_def]
+          using (measurable_threshold p).comp hev
+      · simpa only [he, and_false]
+          using (measurable_const : @Measurable (E → unitInterval) Prop m inferInstance
+            (fun _ => False))
+    have heq : {labels : E → unitInterval | ∃ x, (cluster G (openEdges labels p) x).Infinite} =
+        {labels | ∃ x, (cluster G (openEdges labels p ∩ s) x).Infinite} := by
+      ext labels
+      simpa only [Set.sdiff_compl, Set.mem_ofPred_eq] using
+        (exists_infinite_cluster_sdiff G (openEdges labels p) hfin).symm
+    rw [heq]
+    exact (measurableSet_exists_infinite_cluster G).preimage hm
+  exact measure_zero_or_one_of_measurableSet_limsup
+    (s := σ) (f := Filter.cofinite) (p := Set.Finite)
+    (ns := fun F : Finset E => (F : Set E)) hle hi
+    (fun t ht => by simpa using ht)
+    (fun F H => ⟨F ∪ H, Finset.coe_subset.mpr Finset.subset_union_left,
+      Finset.coe_subset.mpr Finset.subset_union_right⟩)
+    (fun F => F.finite_toSet) (fun e => ⟨{e}, by simp⟩) htail
+
+end PercolationZeroOne
+
+section CoupledInterval
+attribute [local instance] Classical.propDecidable
+variable [Countable V] [Countable E]
+
+ theorem exists_infinite_cluster_probability_mono (G : BondGraph V E) {p q : unitInterval}
+    (hpq : p ≤ q) :
+    law p {ω | (infiniteClusters G ω).Nonempty} ≤
+      law q {ω | (infiniteClusters G ω).Nonempty} := by
+  have hm : MeasurableSet {ω : Set E | (infiniteClusters G ω).Nonempty} := by
+    simpa only [infiniteClusters_nonempty_iff] using measurableSet_exists_infinite_cluster G
+  rw [← map_openEdges p, ← map_openEdges q,
+    Measure.map_apply (measurable_openEdges p) hm,
+    Measure.map_apply (measurable_openEdges q) hm]
+  apply measure_mono
+  intro labels h
+  simp only [Set.mem_preimage, Set.mem_ofPred_eq, infiniteClusters_nonempty_iff] at h ⊢
+  exact exists_infinite_cluster_mono G (openEdges_mono labels hpq) h
+
+ theorem ae_exists_infinite_cluster_of_pc_lt (G : BondGraph V E) (p : unitInterval)
+    (hp : pc G < p) :
+    ∀ᵐ labels ∂(labelLaw : Measure (E → unitInterval)),
+      ∃ x, (cluster G (openEdges labels p) x).Infinite := by
+  obtain ⟨q, hq, hqp⟩ := sInf_lt_iff.mp hp
+  have hpos : 0 < law p {ω | (infiniteClusters G ω).Nonempty} :=
+    lt_of_lt_of_le hq (exists_infinite_cluster_probability_mono G hqp.le)
+  have hm : MeasurableSet {ω : Set E | (infiniteClusters G ω).Nonempty} := by
+    simpa only [infiniteClusters_nonempty_iff] using measurableSet_exists_infinite_cluster G
+  rw [← map_openEdges p, Measure.map_apply (measurable_openEdges p) hm] at hpos
+  simp only [Set.preimage_ofPred_eq, infiniteClusters_nonempty_iff] at hpos
+  have hpone := (infinite_cluster_zero_one G p).resolve_left (ne_of_gt hpos)
+  exact (mem_ae_iff_prob_eq_one
+    ((measurableSet_exists_infinite_cluster G).preimage (measurable_openEdges p))).mpr hpone
+
+omit [Countable V] [Countable E] in
+ theorem touchedCover_of_finite_infiniteClusters (G : BondGraph V E)
+    (labels : E → unitInterval) {p r q : unitInterval} (hpr : p ≤ r) (hrq : r ≤ q)
+    (hfinite : (infiniteClusters G (openEdges labels r)).Finite) :
+    ∃ n, TouchedCover G p q n labels := by
+  let C := infiniteClusters G (openEdges labels r)
+  let : Fintype C := hfinite.fintype
+  let f : C → V := fun c => Classical.choose c.property.2
+  have hf (c : C) : c.val = cluster G (openEdges labels r) (f c) :=
+    Classical.choose_spec c.property.2
+  refine ⟨Fintype.card C, fun i => f ((Fintype.equivFin C).symm i), ?_⟩
+  intro x hx
+  have hx' : (cluster G (openEdges labels r) x).Infinite :=
+    hx.mono (fun _ hy => conn_mono G (openEdges_mono labels hpr) hy)
+  let c : C := ⟨cluster G (openEdges labels r) x, hx', x, rfl⟩
+  refine ⟨Fintype.equivFin C c, ?_⟩
+  simp only [Equiv.symm_apply_apply]
+  apply conn_mono G (openEdges_mono labels hrq)
+  change f c ∈ cluster G (openEdges labels r) x
+  change f c ∈ c.val
+  rw [hf c]
+  exact mem_cluster_self G _ _
+
+ 
+
+ theorem coupled_interval [Infinite V] (G : BondGraph V E) (hconn : Connected G)
+    (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p q : unitInterval) (hp : pc G < p) (hq : OperatorBounded G q) :
+    ∀ᵐ labels ∂(labelLaw : Measure (E → unitInterval)),
+      ∀ r ∈ Set.Icc p q, (infiniteClusters G (openEdges labels r)).Infinite := by
+  have hall : ∀ᵐ labels ∂(labelLaw : Measure (E → unitInterval)), ∀ n x,
+      ¬(TouchedCover G p q n labels ∧ (cluster G (openEdges labels p) x).Infinite) := by
+    rw [ae_all_iff]
+    intro n
+    rw [ae_all_iff]
+    intro x
+    apply ae_iff.mpr
+    simpa only [not_not] using measure_cover_survival_eq_zero G hconn hloc hqt p q hq n x
+  filter_upwards [hall, ae_exists_infinite_cluster_of_pc_lt G p hp] with labels hlabels hex
+  intro r hr
+  by_contra hfinite
+  obtain ⟨n, hn⟩ := touchedCover_of_finite_infiniteClusters G labels hr.1 hr.2
+    (Set.not_infinite.mp hfinite)
+  obtain ⟨x, hx⟩ := hex
+  exact hlabels n x ⟨hn, hx⟩
+
+end CoupledInterval
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Criticality/ClusterCount.lean b/lean/OAI/Probability/UnimodularPercolation/Criticality/ClusterCount.lean
index 62e8bcb2be82526f62b9c49336868708b9c3ce28..ace38eca82a6b05945c19cd027473e9fbeacf713 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Criticality/ClusterCount.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Criticality/ClusterCount.lean
@@ -1,147 +1,147 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Main
-import OAI.Probability.UnimodularPercolation.Criticality.Connectivity
-import OAI.Probability.UnimodularPercolation.Criticality.Ergodicity
-import OAI.Probability.UnimodularPercolation.Criticality.FiniteChanges
-
-namespace OAI
-
-/-! Exclusion of positive finite critical cluster counts. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal
-universe u v
-variable {V : Type u} {E : Type v}
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem law_invariant_zero_one [Infinite V] (G : BondGraph V E)
-    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (A : Set (Set E)) (hA : MeasurableSet A)
-    (hinv : ∀ a : V ≃ V, ∀ b : E ≃ E, IsAut G a b →
-      (fun ω : Set E => b ⁻¹' ω) ⁻¹' A = A) : law p A = 0 ∨ law p A = 1 := by
-  have hm : law p A = labelLaw ((fun l => openEdges l p) ⁻¹' A) := by
-    rw [←map_openEdges p,Measure.map_apply (measurable_openEdges p) hA]
-  rw [hm]
-  apply label_invariant_zero_one G hconn hloc hqt _ (hA.preimage (measurable_openEdges p))
-  intro a b hab
-  ext l
-  have h := Set.ext_iff.mp (hinv a b hab) (openEdges l p)
-  exact h
-
-theorem unique_infinite_cluster_aut_iff (G : BondGraph V E)
-    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (ω : Set E) :
-    (∃! C, C ∈ infiniteClusters G (b ⁻¹' ω)) ↔ ∃! C, C ∈ infiniteClusters G ω := by
-  have hInf (x : V) : (cluster G (b ⁻¹' ω) x).Infinite ↔ (cluster G ω (a x)).Infinite := by
-    rw [←cluster_aut G hab ω x]
-    exact (Set.infinite_image_iff a.injective.injOn).symm
-  rw [unique_infinite_cluster_iff,unique_infinite_cluster_iff]
-  constructor
-  · rintro ⟨x,hx,h⟩
-    refine ⟨a x,(hInf x).mp hx,?_⟩
-    intro y hy
-    have hh : (cluster G (b ⁻¹' ω) (a.symm y)).Infinite := (hInf _).mpr (by simpa using hy)
-    simpa only [Equiv.apply_symm_apply] using (conn_aut_iff G hab ω x (a.symm y)).mp (h _ hh)
-  · rintro ⟨x,hx,h⟩
-    refine ⟨a.symm x,(hInf _).mpr (by simpa using hx),?_⟩
-    intro y hy
-    apply (conn_aut_iff G hab ω (a.symm x) y).mpr
-    simpa only [Equiv.apply_symm_apply] using h (a y) ((hInf y).mp hy)
-
-theorem critical_unique_probability_zero [Infinite V] (G : BondGraph V E)
-    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (hh : 0 < hV G) : law (pc G) {ω | ∃! C, C ∈ infiniteClusters G ω} = 0 := by
-  let : Countable V := countable_vertices hloc hconn
-  let : Countable E := countable_edges hloc
-  refine (law_invariant_zero_one G hconn hloc hqt (pc G) _
-    (measurableSet_unique_infinite_cluster G) ?_).resolve_right
-    (critical_not_ae_unique G hconn hloc hqt hh)
-  intro a b hab
-  ext ω
-  exact unique_infinite_cluster_aut_iff G hab ω
-
-def InfiniteCover (G : BondGraph V E) (R : Finset V) (ω : Set E) : Prop :=
-  ∀ y, (cluster G ω y).Infinite → ∃ z ∈ R, Conn G ω y z
-
-theorem measurableSet_infiniteCover [Countable V] [Countable E]
-    (G : BondGraph V E) (R : Finset V) : MeasurableSet {ω | InfiniteCover G R ω} := by
-  have he : {ω | InfiniteCover G R ω} =
-      ⋂ y, ({ω | (cluster G ω y).Infinite}ᶜ ∪ ⋃ z ∈ R, {ω | Conn G ω y z}) := by
-    ext ω
-    simp only [InfiniteCover,Set.mem_iInter,Set.mem_union,Set.mem_compl_iff,
-      Set.mem_iUnion,Set.mem_ofPred_eq,imp_iff_not_or]
-    simp only [exists_prop]
-  rw [he]
-  exact MeasurableSet.iInter fun y => (measurableSet_infinite_cluster G y).compl.union
-    (R.measurableSet_biUnion fun z _ => measurableSet_conn G y z)
-
-theorem unique_pos_of_cover_pos [Countable V] [Countable E]
-    (G : BondGraph V E) (hconn : Connected G) (p : unitInterval) (hp : 0 < p)
-    (hp1 : p < 1) (x : V) (R : Finset V)
-    (hpos : 0 < law p {ω | (cluster G ω x).Infinite ∧ InfiniteCover G R ω}) :
-    0 < law p {ω | ∃! C, C ∈ infiniteClusters G ω} := by
-  obtain ⟨F,hF⟩ := finite_connecting_edges G hconn x R
-  apply finiteOverwrite_pos p hp hp1 F F (Finset.Subset.refl _)
-    {ω | (cluster G ω x).Infinite ∧ InfiniteCover G R ω} _
-    (measurableSet_unique_infinite_cluster G) hpos
-  rintro ω ⟨hx,hcover⟩
-  have how : finiteOverwrite F F ω = ω ∪ (F:Set E) := by
-    ext e
-    simp only [finiteOverwrite,Set.mem_union,Set.mem_inter_iff,Set.mem_compl_iff]
-    tauto
-  rw [how]
-  apply finite_opening_unique G ω F x hx
-  intro y hy
-  obtain ⟨z,hz,hyz⟩ := hcover y hy
-  exact conn_trans G _ (conn_mono G Set.subset_union_right (hF z hz))
-    (conn_mono G Set.subset_union_left (conn_symm G _ hyz))
-
-theorem critical_infiniteCover_survival_zero [Infinite V] (G : BondGraph V E)
-    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (hh : 0 < hV G) (x : V) (R : Finset V) :
-    law (pc G) {ω | (cluster G ω x).Infinite ∧ InfiniteCover G R ω} = 0 := by
-  let : Countable V := countable_vertices hloc hconn
-  let : Countable E := countable_edges hloc
-  by_contra h
-  have hp := unique_pos_of_cover_pos G hconn (pc G) (pc_pos G hloc hqt)
-    (pc_lt_one_of_nonamenable G hconn hloc hqt hh) x R (pos_iff_ne_zero.mpr h)
-  rw [critical_unique_probability_zero G hconn hloc hqt hh] at hp
-  exact lt_irrefl _ hp
-
-theorem critical_finite_count_empty_ae [Infinite V] (G : BondGraph V E)
-    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (hh : 0 < hV G) :
-    ∀ᵐ ω ∂law (pc G), (infiniteClusters G ω).Finite → infiniteClusters G ω = ∅ := by
-  let : Countable V := countable_vertices hloc hconn
-  let : Countable E := countable_edges hloc
-  have ha (x : V) (R : Finset V) :
-      ∀ᵐ ω ∂law (pc G), ¬((cluster G ω x).Infinite ∧ InfiniteCover G R ω) := by
-    apply ae_iff.mpr
-    simpa only [not_not] using critical_infiniteCover_survival_zero G hconn hloc hqt hh x R
-  have hall : ∀ᵐ ω ∂law (pc G), ∀ x : V, ∀ R : Finset V,
-      ¬((cluster G ω x).Infinite ∧ InfiniteCover G R ω) := by
-    exact ae_all_iff.mpr fun x => ae_all_iff.mpr (ha x)
-  filter_upwards [hall] with ω hω
-  intro hfin
-  by_contra hn
-  obtain ⟨C,hC⟩ := Set.nonempty_iff_ne_empty.mpr hn
-  obtain ⟨hx,x,rfl⟩ := hC
-  let f : Set V → V := fun C => if hc : C ∈ infiniteClusters G ω then
-    Classical.choose hc.2 else x
-  have hf (C : Set V) (hC : C ∈ infiniteClusters G ω) : C = cluster G ω (f C) := by
-    simp only [f,dite_eq_left hC]
-    exact Classical.choose_spec hC.2
-  let R : Finset V := hfin.toFinset.image f
-  apply hω x R ⟨hx,?_⟩
-  intro y hy
-  have hc : cluster G ω y ∈ infiniteClusters G ω := ⟨hy,y,rfl⟩
-  refine ⟨f (cluster G ω y),Finset.mem_image.mpr ⟨_,hfin.mem_toFinset.mpr hc,rfl⟩,?_⟩
-  have he := hf _ hc
-  exact (Set.ext_iff.mp he _).mpr (mem_cluster_self G ω (f (cluster G ω y)))
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Main
+import OAI.Probability.UnimodularPercolation.Criticality.Connectivity
+import OAI.Probability.UnimodularPercolation.Criticality.Ergodicity
+import OAI.Probability.UnimodularPercolation.Criticality.FiniteChanges
+
+namespace OAI
+
+/-! Exclusion of positive finite critical cluster counts. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal
+universe u v
+variable {V : Type u} {E : Type v}
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem law_invariant_zero_one [Infinite V] (G : BondGraph V E)
+    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (A : Set (Set E)) (hA : MeasurableSet A)
+    (hinv : ∀ a : V ≃ V, ∀ b : E ≃ E, IsAut G a b →
+      (fun ω : Set E => b ⁻¹' ω) ⁻¹' A = A) : law p A = 0 ∨ law p A = 1 := by
+  have hm : law p A = labelLaw ((fun l => openEdges l p) ⁻¹' A) := by
+    rw [←map_openEdges p,Measure.map_apply (measurable_openEdges p) hA]
+  rw [hm]
+  apply label_invariant_zero_one G hconn hloc hqt _ (hA.preimage (measurable_openEdges p))
+  intro a b hab
+  ext l
+  have h := Set.ext_iff.mp (hinv a b hab) (openEdges l p)
+  exact h
+
+theorem unique_infinite_cluster_aut_iff (G : BondGraph V E)
+    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (ω : Set E) :
+    (∃! C, C ∈ infiniteClusters G (b ⁻¹' ω)) ↔ ∃! C, C ∈ infiniteClusters G ω := by
+  have hInf (x : V) : (cluster G (b ⁻¹' ω) x).Infinite ↔ (cluster G ω (a x)).Infinite := by
+    rw [←cluster_aut G hab ω x]
+    exact (Set.infinite_image_iff a.injective.injOn).symm
+  rw [unique_infinite_cluster_iff,unique_infinite_cluster_iff]
+  constructor
+  · rintro ⟨x,hx,h⟩
+    refine ⟨a x,(hInf x).mp hx,?_⟩
+    intro y hy
+    have hh : (cluster G (b ⁻¹' ω) (a.symm y)).Infinite := (hInf _).mpr (by simpa using hy)
+    simpa only [Equiv.apply_symm_apply] using (conn_aut_iff G hab ω x (a.symm y)).mp (h _ hh)
+  · rintro ⟨x,hx,h⟩
+    refine ⟨a.symm x,(hInf _).mpr (by simpa using hx),?_⟩
+    intro y hy
+    apply (conn_aut_iff G hab ω (a.symm x) y).mpr
+    simpa only [Equiv.apply_symm_apply] using h (a y) ((hInf y).mp hy)
+
+theorem critical_unique_probability_zero [Infinite V] (G : BondGraph V E)
+    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (hh : 0 < hV G) : law (pc G) {ω | ∃! C, C ∈ infiniteClusters G ω} = 0 := by
+  let : Countable V := countable_vertices hloc hconn
+  let : Countable E := countable_edges hloc
+  refine (law_invariant_zero_one G hconn hloc hqt (pc G) _
+    (measurableSet_unique_infinite_cluster G) ?_).resolve_right
+    (critical_not_ae_unique G hconn hloc hqt hh)
+  intro a b hab
+  ext ω
+  exact unique_infinite_cluster_aut_iff G hab ω
+
+def InfiniteCover (G : BondGraph V E) (R : Finset V) (ω : Set E) : Prop :=
+  ∀ y, (cluster G ω y).Infinite → ∃ z ∈ R, Conn G ω y z
+
+theorem measurableSet_infiniteCover [Countable V] [Countable E]
+    (G : BondGraph V E) (R : Finset V) : MeasurableSet {ω | InfiniteCover G R ω} := by
+  have he : {ω | InfiniteCover G R ω} =
+      ⋂ y, ({ω | (cluster G ω y).Infinite}ᶜ ∪ ⋃ z ∈ R, {ω | Conn G ω y z}) := by
+    ext ω
+    simp only [InfiniteCover,Set.mem_iInter,Set.mem_union,Set.mem_compl_iff,
+      Set.mem_iUnion,Set.mem_ofPred_eq,imp_iff_not_or]
+    simp only [exists_prop]
+  rw [he]
+  exact MeasurableSet.iInter fun y => (measurableSet_infinite_cluster G y).compl.union
+    (R.measurableSet_biUnion fun z _ => measurableSet_conn G y z)
+
+theorem unique_pos_of_cover_pos [Countable V] [Countable E]
+    (G : BondGraph V E) (hconn : Connected G) (p : unitInterval) (hp : 0 < p)
+    (hp1 : p < 1) (x : V) (R : Finset V)
+    (hpos : 0 < law p {ω | (cluster G ω x).Infinite ∧ InfiniteCover G R ω}) :
+    0 < law p {ω | ∃! C, C ∈ infiniteClusters G ω} := by
+  obtain ⟨F,hF⟩ := finite_connecting_edges G hconn x R
+  apply finiteOverwrite_pos p hp hp1 F F (Finset.Subset.refl _)
+    {ω | (cluster G ω x).Infinite ∧ InfiniteCover G R ω} _
+    (measurableSet_unique_infinite_cluster G) hpos
+  rintro ω ⟨hx,hcover⟩
+  have how : finiteOverwrite F F ω = ω ∪ (F:Set E) := by
+    ext e
+    simp only [finiteOverwrite,Set.mem_union,Set.mem_inter_iff,Set.mem_compl_iff]
+    tauto
+  rw [how]
+  apply finite_opening_unique G ω F x hx
+  intro y hy
+  obtain ⟨z,hz,hyz⟩ := hcover y hy
+  exact conn_trans G _ (conn_mono G Set.subset_union_right (hF z hz))
+    (conn_mono G Set.subset_union_left (conn_symm G _ hyz))
+
+theorem critical_infiniteCover_survival_zero [Infinite V] (G : BondGraph V E)
+    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (hh : 0 < hV G) (x : V) (R : Finset V) :
+    law (pc G) {ω | (cluster G ω x).Infinite ∧ InfiniteCover G R ω} = 0 := by
+  let : Countable V := countable_vertices hloc hconn
+  let : Countable E := countable_edges hloc
+  by_contra h
+  have hp := unique_pos_of_cover_pos G hconn (pc G) (pc_pos G hloc hqt)
+    (pc_lt_one_of_nonamenable G hconn hloc hqt hh) x R (pos_iff_ne_zero.mpr h)
+  rw [critical_unique_probability_zero G hconn hloc hqt hh] at hp
+  exact lt_irrefl _ hp
+
+theorem critical_finite_count_empty_ae [Infinite V] (G : BondGraph V E)
+    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (hh : 0 < hV G) :
+    ∀ᵐ ω ∂law (pc G), (infiniteClusters G ω).Finite → infiniteClusters G ω = ∅ := by
+  let : Countable V := countable_vertices hloc hconn
+  let : Countable E := countable_edges hloc
+  have ha (x : V) (R : Finset V) :
+      ∀ᵐ ω ∂law (pc G), ¬((cluster G ω x).Infinite ∧ InfiniteCover G R ω) := by
+    apply ae_iff.mpr
+    simpa only [not_not] using critical_infiniteCover_survival_zero G hconn hloc hqt hh x R
+  have hall : ∀ᵐ ω ∂law (pc G), ∀ x : V, ∀ R : Finset V,
+      ¬((cluster G ω x).Infinite ∧ InfiniteCover G R ω) := by
+    exact ae_all_iff.mpr fun x => ae_all_iff.mpr (ha x)
+  filter_upwards [hall] with ω hω
+  intro hfin
+  by_contra hn
+  obtain ⟨C,hC⟩ := Set.nonempty_iff_ne_empty.mpr hn
+  obtain ⟨hx,x,rfl⟩ := hC
+  let f : Set V → V := fun C => if hc : C ∈ infiniteClusters G ω then
+    Classical.choose hc.2 else x
+  have hf (C : Set V) (hC : C ∈ infiniteClusters G ω) : C = cluster G ω (f C) := by
+    simp only [f,dite_eq_left hC]
+    exact Classical.choose_spec hC.2
+  let R : Finset V := hfin.toFinset.image f
+  apply hω x R ⟨hx,?_⟩
+  intro y hy
+  have hc : cluster G ω y ∈ infiniteClusters G ω := ⟨hy,y,rfl⟩
+  refine ⟨f (cluster G ω y),Finset.mem_image.mpr ⟨_,hfin.mem_toFinset.mpr hc,rfl⟩,?_⟩
+  have he := hf _ hc
+  exact (Set.ext_iff.mp he _).mpr (mem_cluster_self G ω (f (cluster G ω y)))
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Criticality/Connectivity.lean b/lean/OAI/Probability/UnimodularPercolation/Criticality/Connectivity.lean
index 1a7ec958391826b65dccda83750d0b559a722b93..e10744b4a3dda81c2627da4cd17d028f6280c32f 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Criticality/Connectivity.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Criticality/Connectivity.lean
@@ -1,194 +1,194 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Main
-
-namespace OAI
-
-/-! Critical connection probabilities and nonamenable growth. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-open MeasureTheory Filter
-open scoped Topology
-universe u v
-variable {V : Type u} {E : Type v}
-
-theorem connectivity_twoPoint_mul_le [Countable E] (G : BondGraph V E) (p : unitInterval)
-    (x y z : V) : twoPoint G p x y * twoPoint G p y z ≤ twoPoint G p x z := by
-  have h := association_finiteWitness p {ω | Conn G ω x y} {ω | Conn G ω y z}
-    (fun _ _ hst hxy => conn_mono G hst hxy)
-    (fun _ _ hst hyz => conn_mono G hst hyz)
-    (finiteWitness_conn G x y) (finiteWitness_conn G y z)
-  refine h.trans (measureReal_mono ?_)
-  intro ω hω
-  exact conn_trans G ω hω.1 hω.2
-
-theorem reachBall_split (G : BondGraph V E) (x z : V) (n m : ℕ)
-    (hz : z ∈ reachBall G x (n+m)) :
-    ∃ y ∈ reachBall G x n, z ∈ reachBall G y m := by
-  induction m generalizing z with
-  | zero =>
-      exact ⟨z,by simpa using hz,by simp [reachBall]⟩
-  | succ m ih =>
-      change z ∈ reachBall G x (n+m) ∪ _ at hz
-      rcases hz with hz | hz
-      · obtain ⟨y,hy,hyz⟩ := ih z hz
-        exact ⟨y,hy,Or.inl hyz⟩
-      · simp only [Set.mem_iUnion,Set.mem_ofPred_eq] at hz
-        obtain ⟨z',hz',hzz⟩ := hz
-        obtain ⟨y,hy,hyz⟩ := ih z' hz'
-        refine ⟨y,hy,Or.inr ?_⟩
-        simp only [Set.mem_iUnion,Set.mem_ofPred_eq]
-        exact ⟨z',hyz,hzz⟩
-
-theorem twoPoint_block_lower [Countable E] (G : BondGraph V E) (p : unitInterval)
-    (n : ℕ) (c : ℝ) (hc : 0 ≤ c)
-    (h : ∀ x y, y ∈ reachBall G x n → c ≤ twoPoint G p x y)
-    (m : ℕ) (x y : V) (hy : y ∈ reachBall G x (n*m)) :
-    c^m ≤ twoPoint G p x y := by
-  induction m generalizing x y with
-  | zero =>
-      simp only [mul_zero,reachBall,Set.mem_singleton_iff] at hy
-      subst y
-      simp [twoPoint_self]
-  | succ m ih =>
-      rw [Nat.mul_succ] at hy
-      obtain ⟨z,hz,hzy⟩ := reachBall_split G x y (n*m) n hy
-      calc
-        c^(m+1) = c^m*c := pow_succ _ _
-        _ ≤ twoPoint G p x z * twoPoint G p z y :=
-          mul_le_mul (ih x z hz) (h z y hzy) hc (twoPoint_nonneg G p x z)
-        _ ≤ twoPoint G p x y := connectivity_twoPoint_mul_le G p x z y
-
-theorem isoperimetric_ball_growth (G : BondGraph V E) (hloc : LocallyFinite G)
-    (hh : 0 < hV G) (x : V) (n : ℕ) :
-    (1+hV G)^n ≤ ((reachBall G x n).ncard : ℝ) := by
-  classical
-  induction n with
-  | zero => simp [reachBall]
-  | succ n ih =>
-      let B := (finite_reachBall hloc x n).toFinset
-      have heq : reachBall G x (n+1) = reachBall G x n ∪ vertexBoundary G B := by
-        ext y
-        simp only [reachBall,Set.mem_union,Set.mem_iUnion,Set.mem_ofPred_eq,vertexBoundary]
-        constructor
-        · rintro (hy | ⟨z,hz,hzy⟩)
-          · exact Or.inl hy
-          · by_cases hy : y ∈ reachBall G x n
-            · exact Or.inl hy
-            · exact Or.inr ⟨by simpa [B] using hy,z,by simpa [B] using hz,hzy⟩
-        · rintro (hy | ⟨_,z,hz,hzy⟩)
-          · exact Or.inl hy
-          · exact Or.inr ⟨z,by simpa [B] using hz,hzy⟩
-      have hd : Disjoint (reachBall G x n) (vertexBoundary G B) := by
-        apply Set.disjoint_left.mpr
-        intro y hy hb
-        exact hb.1 (by simpa [B] using hy)
-      have hb := hV_mul_card_le G B
-      have hcard : B.card = (reachBall G x n).ncard :=
-        (Set.ncard_eq_toFinset_card _ _).symm
-      rw [hcard] at hb
-      rw [heq,Set.ncard_union_eq hd (finite_reachBall hloc x n)
-        (finite_vertexBoundary G hloc B),Nat.cast_add,pow_succ]
-      have hh0 : 0 ≤ 1+hV G := by linarith
-      nlinarith [mul_le_mul_of_nonneg_right ih hh0]
-
-theorem subcritical_block_bound [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (hh : 0 < hV G) (p : unitInterval) (hp : p < pc G)
-    (n : ℕ) (c : ℝ) (hc : 0 ≤ c)
-    (h : ∀ x y, y ∈ reachBall G x n → c ≤ twoPoint G p x y) :
-    (1+hV G)^n*c ≤ 1 := by
-  classical
-  by_contra hbad
-  have hgt : 1 < (1+hV G)^n*c := lt_of_not_ge hbad
-  let x : V := Classical.arbitrary V
-  have hbound (m : ℕ) : ((1+hV G)^n*c)^m ≤ susceptibility G p x := by
-    let B := (finite_reachBall hloc x (n*m)).toFinset
-    calc
-      ((1+hV G)^n*c)^m = (1+hV G)^(n*m)*c^m := by rw [mul_pow,pow_mul]
-      _ ≤ (B.card:ℝ)*c^m := by
-        simpa only [B, ← Set.ncard_eq_toFinset_card] using mul_le_mul_of_nonneg_right
-          (isoperimetric_ball_growth G hloc hh x (n*m)) (pow_nonneg hc m)
-      _ = ∑ y ∈ B, c^m := by simp
-      _ ≤ ∑ y ∈ B, twoPoint G p x y := by
-        apply Finset.sum_le_sum
-        intro y hy
-        exact twoPoint_block_lower G p n c hc h m x y (by simpa [B] using hy)
-      _ ≤ susceptibility G p x :=
-        (summable_twoPoint_subcritical G hloc hqt p hp x).sum_le_tsum B
-          (fun y _ => twoPoint_nonneg G p x y)
-  obtain ⟨m,hm⟩ := pow_unbounded_of_one_lt (susceptibility G p x) hgt
-  exact (not_lt_of_ge (hbound m)) hm
-
-theorem uniform_ball_lower_below [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (q : unitInterval) (hq : 0 < q) (n : ℕ) (a c : ℝ) (hac : a < c)
-    (h : ∀ x y, c ≤ twoPoint G q x y) :
-    ∃ p : unitInterval, p < q ∧ ∀ x y, y ∈ reachBall G x n → a < twoPoint G p x y := by
-  classical
-  obtain ⟨R,hR⟩ := hqt
-  obtain ⟨ps,hps,hpsq,ht⟩ := exists_seq_strictMono_tendsto' hq
-  have hev : ∀ᶠ k in atTop, ∀ o ∈ R, ∀ y ∈ reachBall G o n,
-      a < twoPoint G (ps k) o y := by
-    apply (eventually_all_finset R).mpr
-    intro o _
-    apply (eventually_all_finite (finite_reachBall hloc o n)).mpr
-    intro y _
-    exact (twoPoint_tendsto_of_monotone G ps hps.monotone q
-      (fun k => (hpsq k).2.le) ht o y).eventually (eventually_gt_nhds (hac.trans_le (h o y)))
-  obtain ⟨k,hk⟩ := hev.exists
-  refine ⟨ps k,(hpsq k).2,?_⟩
-  intro x y hy
-  obtain ⟨o,ho,g,b,hg,hgx⟩ := hR x
-  have hy' : g.symm y ∈ reachBall G o n := by
-    apply (mem_reachBall_aut_iff G hg o (g.symm y) n).mp
-    simpa only [g.apply_symm_apply,hgx] using hy
-  have hb := hk o ho (g.symm y) hy'
-  have he := twoPoint_aut G (ps k) hg o (g.symm y)
-  rw [g.apply_symm_apply,hgx] at he
-  rwa [he]
-
-theorem critical_twoPoint_no_uniform_positive [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (hh : 0 < hV G) (c : ℝ) (hc : 0 < c) :
-    ¬ (∀ x y, c ≤ twoPoint G (pc G) x y) := by
-  intro h
-  have ha : 0 < c/2 := by positivity
-  obtain ⟨n,hn⟩ := pow_unbounded_of_one_lt (1/(c/2)) (show 1 < 1+hV G by linarith)
-  obtain ⟨p,hp,hball⟩ := uniform_ball_lower_below G hloc hqt (pc G)
-    (pc_pos G hloc hqt) n (c/2) c (by linarith) h
-  have hb := subcritical_block_bound G hloc hqt hh p hp n (c/2) ha.le
-    (fun x y hy => (hball x y hy).le)
-  exact (not_lt_of_ge hb) ((div_lt_iff₀ ha).mp hn)
-
-theorem critical_not_ae_unique [Infinite V] (G : BondGraph V E)
-    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (hh : 0 < hV G) :
-    law (pc G) {ω | ∃! C, C ∈ infiniteClusters G ω} ≠ 1 := by
-  classical
-  let : Countable V := countable_vertices hloc hconn
-  let : Countable E := countable_edges hloc
-  intro hu
-  have hpos : 0 < law (pc G) {ω | (infiniteClusters G ω).Nonempty} := by
-    exact lt_of_lt_of_le (show 0 < law (pc G) {ω | ∃! C, C ∈ infiniteClusters G ω} by
-      rw [hu]; exact zero_lt_one)
-      (measure_mono fun ω hω => ⟨hω.choose,hω.choose_spec.1⟩)
-  obtain ⟨θ,hθ,hroot⟩ := uniform_survival_pos G hconn hqt (pc G) (pc_pos G hloc hqt) hpos
-  have hue : ∀ᵐ ω ∂law (pc G), ∃! C, C ∈ infiniteClusters G ω :=
-    (mem_ae_iff_prob_eq_one (measurableSet_unique_infinite_cluster G)).mpr hu
-  apply critical_twoPoint_no_uniform_positive G hloc hqt hh (θ*θ) (mul_pos hθ hθ)
-  intro x y
-  refine (mul_le_mul (hroot x) (hroot y) hθ.le measureReal_nonneg).trans
-    ((association_infinite_cluster G (pc G) x y).trans ?_)
-  apply ENNReal.toReal_mono (measure_ne_top _ _)
-  apply measure_mono_ae
-  filter_upwards [hue] with ω hω
-  intro hi
-  obtain ⟨r,_,hr⟩ := (unique_infinite_cluster_iff G ω).mp hω
-  exact conn_trans G ω (conn_symm G ω (hr x hi.1)) (hr y hi.2)
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Main
+
+namespace OAI
+
+/-! Critical connection probabilities and nonamenable growth. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+open MeasureTheory Filter
+open scoped Topology
+universe u v
+variable {V : Type u} {E : Type v}
+
+theorem connectivity_twoPoint_mul_le [Countable E] (G : BondGraph V E) (p : unitInterval)
+    (x y z : V) : twoPoint G p x y * twoPoint G p y z ≤ twoPoint G p x z := by
+  have h := association_finiteWitness p {ω | Conn G ω x y} {ω | Conn G ω y z}
+    (fun _ _ hst hxy => conn_mono G hst hxy)
+    (fun _ _ hst hyz => conn_mono G hst hyz)
+    (finiteWitness_conn G x y) (finiteWitness_conn G y z)
+  refine h.trans (measureReal_mono ?_)
+  intro ω hω
+  exact conn_trans G ω hω.1 hω.2
+
+theorem reachBall_split (G : BondGraph V E) (x z : V) (n m : ℕ)
+    (hz : z ∈ reachBall G x (n+m)) :
+    ∃ y ∈ reachBall G x n, z ∈ reachBall G y m := by
+  induction m generalizing z with
+  | zero =>
+      exact ⟨z,by simpa using hz,by simp [reachBall]⟩
+  | succ m ih =>
+      change z ∈ reachBall G x (n+m) ∪ _ at hz
+      rcases hz with hz | hz
+      · obtain ⟨y,hy,hyz⟩ := ih z hz
+        exact ⟨y,hy,Or.inl hyz⟩
+      · simp only [Set.mem_iUnion,Set.mem_ofPred_eq] at hz
+        obtain ⟨z',hz',hzz⟩ := hz
+        obtain ⟨y,hy,hyz⟩ := ih z' hz'
+        refine ⟨y,hy,Or.inr ?_⟩
+        simp only [Set.mem_iUnion,Set.mem_ofPred_eq]
+        exact ⟨z',hyz,hzz⟩
+
+theorem twoPoint_block_lower [Countable E] (G : BondGraph V E) (p : unitInterval)
+    (n : ℕ) (c : ℝ) (hc : 0 ≤ c)
+    (h : ∀ x y, y ∈ reachBall G x n → c ≤ twoPoint G p x y)
+    (m : ℕ) (x y : V) (hy : y ∈ reachBall G x (n*m)) :
+    c^m ≤ twoPoint G p x y := by
+  induction m generalizing x y with
+  | zero =>
+      simp only [mul_zero,reachBall,Set.mem_singleton_iff] at hy
+      subst y
+      simp [twoPoint_self]
+  | succ m ih =>
+      rw [Nat.mul_succ] at hy
+      obtain ⟨z,hz,hzy⟩ := reachBall_split G x y (n*m) n hy
+      calc
+        c^(m+1) = c^m*c := pow_succ _ _
+        _ ≤ twoPoint G p x z * twoPoint G p z y :=
+          mul_le_mul (ih x z hz) (h z y hzy) hc (twoPoint_nonneg G p x z)
+        _ ≤ twoPoint G p x y := connectivity_twoPoint_mul_le G p x z y
+
+theorem isoperimetric_ball_growth (G : BondGraph V E) (hloc : LocallyFinite G)
+    (hh : 0 < hV G) (x : V) (n : ℕ) :
+    (1+hV G)^n ≤ ((reachBall G x n).ncard : ℝ) := by
+  classical
+  induction n with
+  | zero => simp [reachBall]
+  | succ n ih =>
+      let B := (finite_reachBall hloc x n).toFinset
+      have heq : reachBall G x (n+1) = reachBall G x n ∪ vertexBoundary G B := by
+        ext y
+        simp only [reachBall,Set.mem_union,Set.mem_iUnion,Set.mem_ofPred_eq,vertexBoundary]
+        constructor
+        · rintro (hy | ⟨z,hz,hzy⟩)
+          · exact Or.inl hy
+          · by_cases hy : y ∈ reachBall G x n
+            · exact Or.inl hy
+            · exact Or.inr ⟨by simpa [B] using hy,z,by simpa [B] using hz,hzy⟩
+        · rintro (hy | ⟨_,z,hz,hzy⟩)
+          · exact Or.inl hy
+          · exact Or.inr ⟨z,by simpa [B] using hz,hzy⟩
+      have hd : Disjoint (reachBall G x n) (vertexBoundary G B) := by
+        apply Set.disjoint_left.mpr
+        intro y hy hb
+        exact hb.1 (by simpa [B] using hy)
+      have hb := hV_mul_card_le G B
+      have hcard : B.card = (reachBall G x n).ncard :=
+        (Set.ncard_eq_toFinset_card _ _).symm
+      rw [hcard] at hb
+      rw [heq,Set.ncard_union_eq hd (finite_reachBall hloc x n)
+        (finite_vertexBoundary G hloc B),Nat.cast_add,pow_succ]
+      have hh0 : 0 ≤ 1+hV G := by linarith
+      nlinarith [mul_le_mul_of_nonneg_right ih hh0]
+
+theorem subcritical_block_bound [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (hh : 0 < hV G) (p : unitInterval) (hp : p < pc G)
+    (n : ℕ) (c : ℝ) (hc : 0 ≤ c)
+    (h : ∀ x y, y ∈ reachBall G x n → c ≤ twoPoint G p x y) :
+    (1+hV G)^n*c ≤ 1 := by
+  classical
+  by_contra hbad
+  have hgt : 1 < (1+hV G)^n*c := lt_of_not_ge hbad
+  let x : V := Classical.arbitrary V
+  have hbound (m : ℕ) : ((1+hV G)^n*c)^m ≤ susceptibility G p x := by
+    let B := (finite_reachBall hloc x (n*m)).toFinset
+    calc
+      ((1+hV G)^n*c)^m = (1+hV G)^(n*m)*c^m := by rw [mul_pow,pow_mul]
+      _ ≤ (B.card:ℝ)*c^m := by
+        simpa only [B, ← Set.ncard_eq_toFinset_card] using mul_le_mul_of_nonneg_right
+          (isoperimetric_ball_growth G hloc hh x (n*m)) (pow_nonneg hc m)
+      _ = ∑ y ∈ B, c^m := by simp
+      _ ≤ ∑ y ∈ B, twoPoint G p x y := by
+        apply Finset.sum_le_sum
+        intro y hy
+        exact twoPoint_block_lower G p n c hc h m x y (by simpa [B] using hy)
+      _ ≤ susceptibility G p x :=
+        (summable_twoPoint_subcritical G hloc hqt p hp x).sum_le_tsum B
+          (fun y _ => twoPoint_nonneg G p x y)
+  obtain ⟨m,hm⟩ := pow_unbounded_of_one_lt (susceptibility G p x) hgt
+  exact (not_lt_of_ge (hbound m)) hm
+
+theorem uniform_ball_lower_below [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (q : unitInterval) (hq : 0 < q) (n : ℕ) (a c : ℝ) (hac : a < c)
+    (h : ∀ x y, c ≤ twoPoint G q x y) :
+    ∃ p : unitInterval, p < q ∧ ∀ x y, y ∈ reachBall G x n → a < twoPoint G p x y := by
+  classical
+  obtain ⟨R,hR⟩ := hqt
+  obtain ⟨ps,hps,hpsq,ht⟩ := exists_seq_strictMono_tendsto' hq
+  have hev : ∀ᶠ k in atTop, ∀ o ∈ R, ∀ y ∈ reachBall G o n,
+      a < twoPoint G (ps k) o y := by
+    apply (eventually_all_finset R).mpr
+    intro o _
+    apply (eventually_all_finite (finite_reachBall hloc o n)).mpr
+    intro y _
+    exact (twoPoint_tendsto_of_monotone G ps hps.monotone q
+      (fun k => (hpsq k).2.le) ht o y).eventually (eventually_gt_nhds (hac.trans_le (h o y)))
+  obtain ⟨k,hk⟩ := hev.exists
+  refine ⟨ps k,(hpsq k).2,?_⟩
+  intro x y hy
+  obtain ⟨o,ho,g,b,hg,hgx⟩ := hR x
+  have hy' : g.symm y ∈ reachBall G o n := by
+    apply (mem_reachBall_aut_iff G hg o (g.symm y) n).mp
+    simpa only [g.apply_symm_apply,hgx] using hy
+  have hb := hk o ho (g.symm y) hy'
+  have he := twoPoint_aut G (ps k) hg o (g.symm y)
+  rw [g.apply_symm_apply,hgx] at he
+  rwa [he]
+
+theorem critical_twoPoint_no_uniform_positive [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (hh : 0 < hV G) (c : ℝ) (hc : 0 < c) :
+    ¬ (∀ x y, c ≤ twoPoint G (pc G) x y) := by
+  intro h
+  have ha : 0 < c/2 := by positivity
+  obtain ⟨n,hn⟩ := pow_unbounded_of_one_lt (1/(c/2)) (show 1 < 1+hV G by linarith)
+  obtain ⟨p,hp,hball⟩ := uniform_ball_lower_below G hloc hqt (pc G)
+    (pc_pos G hloc hqt) n (c/2) c (by linarith) h
+  have hb := subcritical_block_bound G hloc hqt hh p hp n (c/2) ha.le
+    (fun x y hy => (hball x y hy).le)
+  exact (not_lt_of_ge hb) ((div_lt_iff₀ ha).mp hn)
+
+theorem critical_not_ae_unique [Infinite V] (G : BondGraph V E)
+    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (hh : 0 < hV G) :
+    law (pc G) {ω | ∃! C, C ∈ infiniteClusters G ω} ≠ 1 := by
+  classical
+  let : Countable V := countable_vertices hloc hconn
+  let : Countable E := countable_edges hloc
+  intro hu
+  have hpos : 0 < law (pc G) {ω | (infiniteClusters G ω).Nonempty} := by
+    exact lt_of_lt_of_le (show 0 < law (pc G) {ω | ∃! C, C ∈ infiniteClusters G ω} by
+      rw [hu]; exact zero_lt_one)
+      (measure_mono fun ω hω => ⟨hω.choose,hω.choose_spec.1⟩)
+  obtain ⟨θ,hθ,hroot⟩ := uniform_survival_pos G hconn hqt (pc G) (pc_pos G hloc hqt) hpos
+  have hue : ∀ᵐ ω ∂law (pc G), ∃! C, C ∈ infiniteClusters G ω :=
+    (mem_ae_iff_prob_eq_one (measurableSet_unique_infinite_cluster G)).mpr hu
+  apply critical_twoPoint_no_uniform_positive G hloc hqt hh (θ*θ) (mul_pos hθ hθ)
+  intro x y
+  refine (mul_le_mul (hroot x) (hroot y) hθ.le measureReal_nonneg).trans
+    ((association_infinite_cluster G (pc G) x y).trans ?_)
+  apply ENNReal.toReal_mono (measure_ne_top _ _)
+  apply measure_mono_ae
+  filter_upwards [hue] with ω hω
+  intro hi
+  obtain ⟨r,_,hr⟩ := (unique_infinite_cluster_iff G ω).mp hω
+  exact conn_trans G ω (conn_symm G ω (hr x hi.1)) (hr y hi.2)
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Criticality/Ergodicity.lean b/lean/OAI/Probability/UnimodularPercolation/Criticality/Ergodicity.lean
index 2a0209dfc3c486093d3006b209fa2c513c024068..833e56c3b568211c5bf24fb94a4541d82bf33484 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Criticality/Ergodicity.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Criticality/Ergodicity.lean
@@ -1,146 +1,146 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Main
-
-namespace OAI
-
-/-! Zero-one laws for invariant Bernoulli events. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal symmDiff
-universe u v
-variable {V : Type u} {E : Type v}
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem exists_aut_disjoint_edges [Infinite V] (G : BondGraph V E)
-    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (F : Finset E) : ∃ a : V ≃ V, ∃ b : E ≃ E, IsAut G a b ∧ Disjoint F (F.image b) := by
-  classical
-  let B : Finset V := F.biUnion (fun e => (G e).toFinset)
-  let x : V := Classical.arbitrary V
-  obtain ⟨n,hn⟩ := exists_transportCut G hconn x B
-  have hfin : (⋃ z ∈ (B : Set V), reachBall G z n).Finite :=
-    B.finite_toSet.biUnion (fun z _ => finite_reachBall hloc z n)
-  obtain ⟨y,⟨a,b,hab,hax⟩,hy⟩ :=
-    (vertexOrbit_infinite G hconn hloc hqt x).exists_notMem_finite hfin
-  refine ⟨a,b,hab,Finset.disjoint_left.mpr ?_⟩
-  intro e he heb
-  obtain ⟨e',he',rfl⟩ := Finset.mem_image.mp heb
-  have hm : ∃ z, z ∈ G e' := Sym2.inductionOn (G e') (fun z w => ⟨z,by simp⟩)
-  obtain ⟨z,hz⟩ := hm
-  have hzB : z ∈ B := Finset.mem_biUnion.mpr ⟨e',he',by simpa using hz⟩
-  have hazB : a z ∈ B := by
-    apply Finset.mem_biUnion.mpr
-    refine ⟨b e',he,?_⟩
-    rw [Sym2.mem_toFinset,hab]
-    exact Sym2.mem_map.mpr ⟨z,hz,rfl⟩
-  apply hy
-  simp only [Set.mem_iUnion]
-  refine ⟨a z,hazB,?_⟩
-  rw [←hax]
-  exact mem_reachBall_aut G hab (hn z hzB).1
-
-theorem label_cylinder_independent (F : Finset E) (b : E ≃ E)
-    (hd : Disjoint F (F.image b)) (S : Set (F → unitInterval)) (hS : MeasurableSet S) :
-    (labelLaw : Measure (E → unitInterval))
-      (cylinder F S ∩ (fun l : E → unitInterval => l ∘ b) ⁻¹' cylinder F S) =
-      labelLaw (cylinder F S) * labelLaw (cylinder F S) := by
-  classical
-  have hi : iIndepFun (fun e (l : E → unitInterval) => l e) labelLaw :=
-    iIndepFun_infinitePi (X := fun (_ : E) (x : unitInterval) => x) (fun _ => measurable_id)
-  have h := hi.indepFun_finset F (F.image b) hd (fun e => measurable_pi_apply e)
-  let r : ((F.image b) → unitInterval) → (F → unitInterval) :=
-    fun l e => l ⟨b e,Finset.mem_image.mpr ⟨e,e.property,rfl⟩⟩
-  have hr : Measurable r := Measurable.of_eval (fun _ => measurable_pi_apply _)
-  have h' := h.comp measurable_id hr
-  have he := h'.meas_inter (s := cylinder F S)
-    (t := (fun l : E → unitInterval => l ∘ b) ⁻¹' cylinder F S)
-    ⟨S,hS,rfl⟩ ⟨S,hS,rfl⟩
-  have hm : Measurable (fun l : E → unitInterval => l ∘ b) :=
-    Measurable.of_eval fun e => measurable_pi_apply (b e)
-  rw [←Measure.map_apply hm hS.cylinder,labelLaw_map_precomp] at he
-  exact he
-
-theorem label_invariant_zero_one [Infinite V] (G : BondGraph V E)
-    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (A : Set (E → unitInterval)) (hA : MeasurableSet A)
-    (hinv : ∀ a : V ≃ V, ∀ b : E ≃ E, IsAut G a b →
-      (fun l : E → unitInterval => l ∘ b) ⁻¹' A = A) :
-    labelLaw A = 0 ∨ labelLaw A = 1 := by
-  classical
-  let μ : Measure (E → unitInterval) := labelLaw
-  have hd := Measure.MeasureDense.of_generateFrom_isSetAlgebra_finite μ
-    (isSetAlgebra_measurableCylinders (α := fun _ : E => unitInterval))
-    generateFrom_measurableCylinders.symm
-  have heq : μ.real A ≤ (μ.real A)^2 := by
-    apply le_of_forall_pos_le_add
-    intro ε hε
-    obtain ⟨B,hB,hAB⟩ := hd.approx A hA (measure_ne_top _ _) (ε/4) (by positivity)
-    obtain ⟨F,S,hS,rfl⟩ := (mem_measurableCylinders B).mp hB
-    let B : Set (E → unitInterval) := cylinder F S
-    have hBm : MeasurableSet B := hS.cylinder
-    have hABr : μ.real (A ∆ B) < ε/4 := by
-      exact (ENNReal.toReal_lt_toReal (measure_ne_top _ _) ENNReal.ofReal_ne_top).mpr hAB |>.trans_le
-        (by rw [ENNReal.toReal_ofReal (by positivity)])
-    obtain ⟨a,b,hab,habF⟩ := exists_aut_disjoint_edges G hconn hloc hqt F
-    let φ : (E → unitInterval) → (E → unitInterval) := fun l => l ∘ b
-    have hφ : Measurable φ := Measurable.of_eval fun e => measurable_pi_apply (b e)
-    have hφμ : μ.map φ = μ := labelLaw_map_precomp b
-    have hφA : φ ⁻¹' A = A := hinv a b hab
-    have hφAB : μ.real (A ∆ (φ ⁻¹' B)) = μ.real (A ∆ B) := by
-      conv_lhs => rw [←hφA,←Set.preimage_symmDiff]
-      change (μ (φ ⁻¹' (A ∆ B))).toReal = _
-      rw [←Measure.map_apply hφ (hA.symmDiff hBm),hφμ]
-      rfl
-    have hC : MeasurableSet (B ∩ φ ⁻¹' B) := hBm.inter (hBm.preimage hφ)
-    have hsub : A ∆ (B ∩ φ ⁻¹' B) ⊆ (A ∆ B) ∪ (A ∆ (φ ⁻¹' B)) := by
-      intro l hl
-      simp only [Set.mem_symmDiff,Set.mem_inter_iff,Set.mem_union] at hl ⊢
-      tauto
-    have hAC : μ.real (A ∆ (B ∩ φ ⁻¹' B)) < ε/2 := by
-      calc
-        _ ≤ μ.real ((A ∆ B) ∪ (A ∆ (φ ⁻¹' B))) := measureReal_mono hsub
-        _ ≤ μ.real (A ∆ B) + μ.real (A ∆ (φ ⁻¹' B)) := measureReal_union_le _ _
-        _ < ε/2 := by rw [hφAB]; linarith
-    have hprod : μ.real (B ∩ φ ⁻¹' B) = (μ.real B)^2 := by
-      change (labelLaw (B ∩ φ ⁻¹' B)).toReal = _
-      rw [label_cylinder_independent F b habF S hS,ENNReal.toReal_mul,pow_two]
-      rfl
-    have h1 := abs_measureReal_sub_le_measureReal_symmDiff (μ := μ) hA.nullMeasurableSet hC.nullMeasurableSet
-    have h2 := abs_measureReal_sub_le_measureReal_symmDiff (μ := μ) hA.nullMeasurableSet hBm.nullMeasurableSet
-    have h1' : μ.real A - (μ.real B)^2 < ε/2 := by
-      rw [hprod] at h1
-      exact lt_of_le_of_lt (le_trans (le_abs_self (μ.real A - (μ.real B)^2)) h1) hAC
-    have h2' : μ.real B - μ.real A < ε/4 := by
-      have hh := (neg_le_abs (μ.real A - μ.real B)).trans h2 |>.trans_lt hABr
-      linarith
-    have hA0 : 0 ≤ μ.real A := measureReal_nonneg
-    have hB0 : 0 ≤ μ.real B := measureReal_nonneg
-    have hA1 : μ.real A ≤ 1 := measureReal_le_one
-    have hB1 : μ.real B ≤ 1 := measureReal_le_one
-    have hsq : (μ.real B)^2 ≤ (μ.real A)^2 + ε/2 := by
-      by_cases hle : μ.real B ≤ μ.real A
-      · nlinarith [sq_nonneg (μ.real A-μ.real B)]
-      · have hp := mul_le_mul (le_of_lt h2') (show μ.real B+μ.real A ≤ 2 by linarith)
-          (show 0 ≤ μ.real B+μ.real A by positivity) (show 0 ≤ ε/4 by positivity)
-        nlinarith
-    linarith
-  have hA0 : 0 ≤ μ.real A := measureReal_nonneg
-  have hA1 : μ.real A ≤ 1 := measureReal_le_one
-  have he : μ.real A = 0 ∨ μ.real A = 1 := by
-    have : μ.real A * (μ.real A-1) = 0 := by nlinarith
-    rcases mul_eq_zero.mp this with h | h
-    · exact Or.inl h
-    · exact Or.inr (by linarith)
-  rcases he with h | h
-  · left
-    exact (ENNReal.toReal_eq_zero_iff _).mp h |>.resolve_right (measure_ne_top _ _)
-  · right
-    exact (ENNReal.toReal_eq_one_iff _).mp h
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Main
+
+namespace OAI
+
+/-! Zero-one laws for invariant Bernoulli events. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal symmDiff
+universe u v
+variable {V : Type u} {E : Type v}
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem exists_aut_disjoint_edges [Infinite V] (G : BondGraph V E)
+    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (F : Finset E) : ∃ a : V ≃ V, ∃ b : E ≃ E, IsAut G a b ∧ Disjoint F (F.image b) := by
+  classical
+  let B : Finset V := F.biUnion (fun e => (G e).toFinset)
+  let x : V := Classical.arbitrary V
+  obtain ⟨n,hn⟩ := exists_transportCut G hconn x B
+  have hfin : (⋃ z ∈ (B : Set V), reachBall G z n).Finite :=
+    B.finite_toSet.biUnion (fun z _ => finite_reachBall hloc z n)
+  obtain ⟨y,⟨a,b,hab,hax⟩,hy⟩ :=
+    (vertexOrbit_infinite G hconn hloc hqt x).exists_notMem_finite hfin
+  refine ⟨a,b,hab,Finset.disjoint_left.mpr ?_⟩
+  intro e he heb
+  obtain ⟨e',he',rfl⟩ := Finset.mem_image.mp heb
+  have hm : ∃ z, z ∈ G e' := Sym2.inductionOn (G e') (fun z w => ⟨z,by simp⟩)
+  obtain ⟨z,hz⟩ := hm
+  have hzB : z ∈ B := Finset.mem_biUnion.mpr ⟨e',he',by simpa using hz⟩
+  have hazB : a z ∈ B := by
+    apply Finset.mem_biUnion.mpr
+    refine ⟨b e',he,?_⟩
+    rw [Sym2.mem_toFinset,hab]
+    exact Sym2.mem_map.mpr ⟨z,hz,rfl⟩
+  apply hy
+  simp only [Set.mem_iUnion]
+  refine ⟨a z,hazB,?_⟩
+  rw [←hax]
+  exact mem_reachBall_aut G hab (hn z hzB).1
+
+theorem label_cylinder_independent (F : Finset E) (b : E ≃ E)
+    (hd : Disjoint F (F.image b)) (S : Set (F → unitInterval)) (hS : MeasurableSet S) :
+    (labelLaw : Measure (E → unitInterval))
+      (cylinder F S ∩ (fun l : E → unitInterval => l ∘ b) ⁻¹' cylinder F S) =
+      labelLaw (cylinder F S) * labelLaw (cylinder F S) := by
+  classical
+  have hi : iIndepFun (fun e (l : E → unitInterval) => l e) labelLaw :=
+    iIndepFun_infinitePi (X := fun (_ : E) (x : unitInterval) => x) (fun _ => measurable_id)
+  have h := hi.indepFun_finset F (F.image b) hd (fun e => measurable_pi_apply e)
+  let r : ((F.image b) → unitInterval) → (F → unitInterval) :=
+    fun l e => l ⟨b e,Finset.mem_image.mpr ⟨e,e.property,rfl⟩⟩
+  have hr : Measurable r := Measurable.of_eval (fun _ => measurable_pi_apply _)
+  have h' := h.comp measurable_id hr
+  have he := h'.meas_inter (s := cylinder F S)
+    (t := (fun l : E → unitInterval => l ∘ b) ⁻¹' cylinder F S)
+    ⟨S,hS,rfl⟩ ⟨S,hS,rfl⟩
+  have hm : Measurable (fun l : E → unitInterval => l ∘ b) :=
+    Measurable.of_eval fun e => measurable_pi_apply (b e)
+  rw [←Measure.map_apply hm hS.cylinder,labelLaw_map_precomp] at he
+  exact he
+
+theorem label_invariant_zero_one [Infinite V] (G : BondGraph V E)
+    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (A : Set (E → unitInterval)) (hA : MeasurableSet A)
+    (hinv : ∀ a : V ≃ V, ∀ b : E ≃ E, IsAut G a b →
+      (fun l : E → unitInterval => l ∘ b) ⁻¹' A = A) :
+    labelLaw A = 0 ∨ labelLaw A = 1 := by
+  classical
+  let μ : Measure (E → unitInterval) := labelLaw
+  have hd := Measure.MeasureDense.of_generateFrom_isSetAlgebra_finite μ
+    (isSetAlgebra_measurableCylinders (α := fun _ : E => unitInterval))
+    generateFrom_measurableCylinders.symm
+  have heq : μ.real A ≤ (μ.real A)^2 := by
+    apply le_of_forall_pos_le_add
+    intro ε hε
+    obtain ⟨B,hB,hAB⟩ := hd.approx A hA (measure_ne_top _ _) (ε/4) (by positivity)
+    obtain ⟨F,S,hS,rfl⟩ := (mem_measurableCylinders B).mp hB
+    let B : Set (E → unitInterval) := cylinder F S
+    have hBm : MeasurableSet B := hS.cylinder
+    have hABr : μ.real (A ∆ B) < ε/4 := by
+      exact (ENNReal.toReal_lt_toReal (measure_ne_top _ _) ENNReal.ofReal_ne_top).mpr hAB |>.trans_le
+        (by rw [ENNReal.toReal_ofReal (by positivity)])
+    obtain ⟨a,b,hab,habF⟩ := exists_aut_disjoint_edges G hconn hloc hqt F
+    let φ : (E → unitInterval) → (E → unitInterval) := fun l => l ∘ b
+    have hφ : Measurable φ := Measurable.of_eval fun e => measurable_pi_apply (b e)
+    have hφμ : μ.map φ = μ := labelLaw_map_precomp b
+    have hφA : φ ⁻¹' A = A := hinv a b hab
+    have hφAB : μ.real (A ∆ (φ ⁻¹' B)) = μ.real (A ∆ B) := by
+      conv_lhs => rw [←hφA,←Set.preimage_symmDiff]
+      change (μ (φ ⁻¹' (A ∆ B))).toReal = _
+      rw [←Measure.map_apply hφ (hA.symmDiff hBm),hφμ]
+      rfl
+    have hC : MeasurableSet (B ∩ φ ⁻¹' B) := hBm.inter (hBm.preimage hφ)
+    have hsub : A ∆ (B ∩ φ ⁻¹' B) ⊆ (A ∆ B) ∪ (A ∆ (φ ⁻¹' B)) := by
+      intro l hl
+      simp only [Set.mem_symmDiff,Set.mem_inter_iff,Set.mem_union] at hl ⊢
+      tauto
+    have hAC : μ.real (A ∆ (B ∩ φ ⁻¹' B)) < ε/2 := by
+      calc
+        _ ≤ μ.real ((A ∆ B) ∪ (A ∆ (φ ⁻¹' B))) := measureReal_mono hsub
+        _ ≤ μ.real (A ∆ B) + μ.real (A ∆ (φ ⁻¹' B)) := measureReal_union_le _ _
+        _ < ε/2 := by rw [hφAB]; linarith
+    have hprod : μ.real (B ∩ φ ⁻¹' B) = (μ.real B)^2 := by
+      change (labelLaw (B ∩ φ ⁻¹' B)).toReal = _
+      rw [label_cylinder_independent F b habF S hS,ENNReal.toReal_mul,pow_two]
+      rfl
+    have h1 := abs_measureReal_sub_le_measureReal_symmDiff (μ := μ) hA.nullMeasurableSet hC.nullMeasurableSet
+    have h2 := abs_measureReal_sub_le_measureReal_symmDiff (μ := μ) hA.nullMeasurableSet hBm.nullMeasurableSet
+    have h1' : μ.real A - (μ.real B)^2 < ε/2 := by
+      rw [hprod] at h1
+      exact lt_of_le_of_lt (le_trans (le_abs_self (μ.real A - (μ.real B)^2)) h1) hAC
+    have h2' : μ.real B - μ.real A < ε/4 := by
+      have hh := (neg_le_abs (μ.real A - μ.real B)).trans h2 |>.trans_lt hABr
+      linarith
+    have hA0 : 0 ≤ μ.real A := measureReal_nonneg
+    have hB0 : 0 ≤ μ.real B := measureReal_nonneg
+    have hA1 : μ.real A ≤ 1 := measureReal_le_one
+    have hB1 : μ.real B ≤ 1 := measureReal_le_one
+    have hsq : (μ.real B)^2 ≤ (μ.real A)^2 + ε/2 := by
+      by_cases hle : μ.real B ≤ μ.real A
+      · nlinarith [sq_nonneg (μ.real A-μ.real B)]
+      · have hp := mul_le_mul (le_of_lt h2') (show μ.real B+μ.real A ≤ 2 by linarith)
+          (show 0 ≤ μ.real B+μ.real A by positivity) (show 0 ≤ ε/4 by positivity)
+        nlinarith
+    linarith
+  have hA0 : 0 ≤ μ.real A := measureReal_nonneg
+  have hA1 : μ.real A ≤ 1 := measureReal_le_one
+  have he : μ.real A = 0 ∨ μ.real A = 1 := by
+    have : μ.real A * (μ.real A-1) = 0 := by nlinarith
+    rcases mul_eq_zero.mp this with h | h
+    · exact Or.inl h
+    · exact Or.inr (by linarith)
+  rcases he with h | h
+  · left
+    exact (ENNReal.toReal_eq_zero_iff _).mp h |>.resolve_right (measure_ne_top _ _)
+  · right
+    exact (ENNReal.toReal_eq_one_iff _).mp h
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Criticality/FiniteChanges.lean b/lean/OAI/Probability/UnimodularPercolation/Criticality/FiniteChanges.lean
index 5839868c4685d20d0df1a72b5da0572aa7646d60..0ea2cbc7a8be2f858439fba33a294eddf7e72075 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Criticality/FiniteChanges.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Criticality/FiniteChanges.lean
@@ -1,139 +1,139 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Main
-
-namespace OAI
-
-/-! Finite changes of bond configurations and their positive probabilities. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal
-universe u v
-variable {V : Type u} {E : Type v}
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-def finiteOverwrite (F P : Finset E) (ω : Set E) : Set E :=
-  (ω ∩ (F:Set E)ᶜ) ∪ (P:Set E)
-
-theorem measurable_finiteOverwrite (F P : Finset E) : Measurable (finiteOverwrite F P) :=
-  (measurable_union_const (P:Set E)).comp (measurable_inter_const (F:Set E)ᶜ)
-
-theorem finite_pattern_pos [Countable E] (p : unitInterval) (hp : 0 < p) (hp1 : p < 1)
-    (F P : Finset E) (hPF : P ⊆ F) :
-    0 < (law p : Measure (Set E)) {ω | ω ∩ (F:Set E) = (P:Set E)} := by
-  let R : Set (F:Set E) := Subtype.val ⁻¹' (P:Set E)
-  have he : {ω : Set E | ω ∩ (F:Set E) = (P:Set E)} =
-      (fun ω : Set E => (Subtype.val ⁻¹' ω : Set (F:Set E))) ⁻¹' {R} := by
-    ext ω
-    simp only [Set.mem_ofPred_eq,Set.mem_preimage,Set.mem_singleton_iff]
-    constructor
-    · intro h
-      ext e
-      change e.val ∈ ω ↔ e.val ∈ (P:Set E)
-      have := Set.ext_iff.mp h e.val
-      simpa only [Set.mem_inter_iff,e.property,and_true] using this
-    · intro h
-      ext e
-      simp only [Set.mem_inter_iff,Finset.mem_coe]
-      constructor
-      · rintro ⟨he,hF⟩
-        exact (Set.ext_iff.mp h ⟨e,hF⟩).mp he
-      · intro he
-        have hF := hPF he
-        exact ⟨(Set.ext_iff.mp h ⟨e,hF⟩).mpr he,hF⟩
-  rw [he,←Measure.map_apply (measurable_set_restrict _) (measurableSet_singleton _),law_map_restrict]
-  have hh : 0 < (law p : Measure (Set (F:Set E))).real {R} := by
-    rw [law,setBernoulli_real_singleton p (Set.subset_univ _) Set.finite_univ]
-    exact mul_pos (pow_pos (show (0:ℝ) < p from hp) _)
-      (pow_pos (sub_pos.mpr (show (p:ℝ) < 1 from hp1)) _)
-  exact (ENNReal.toReal_pos_iff.mp hh).1
-
-theorem finiteOverwrite_pos [Countable E] (p : unitInterval) (hp : 0 < p) (hp1 : p < 1)
-    (F P : Finset E) (hPF : P ⊆ F) (A B : Set (Set E))
-    (hB : MeasurableSet B) (hA : 0 < (law p : Measure (Set E)) A)
-    (hAB : ∀ ω ∈ A, finiteOverwrite F P ω ∈ B) : 0 < (law p : Measure (Set E)) B := by
-  let D := (finiteOverwrite F P) ⁻¹' B
-  let C : Set (Set E) := {ω | ω ∩ (F:Set E) = (P:Set E)}
-  have hD : MeasurableSet D := hB.preimage (measurable_finiteOverwrite F P)
-  have hDfix : (fun ω : Set E => ω ∩ (F:Set E)ᶜ) ⁻¹' D = D := by
-    ext ω
-    simp only [D,finiteOverwrite,Set.mem_preimage,Set.inter_assoc,Set.inter_self]
-  have hind := restriction_events_independent p (F:Set E) (F:Set E)ᶜ disjoint_compl_right
-    {(P:Set E)} D (measurableSet_singleton _) hD
-  rw [hDfix] at hind
-  change (law p : Measure (Set E)) (C ∩ D) = law p C * law p D at hind
-  have hposC : 0 < (law p : Measure (Set E)) C := finite_pattern_pos p hp hp1 F P hPF
-  have hposD : 0 < (law p : Measure (Set E)) D := hA.trans_le (measure_mono hAB)
-  have hsub : C ∩ D ⊆ B := by
-    rintro ω ⟨hC,hD⟩
-    have he : finiteOverwrite F P ω = ω := by
-      change ω ∩ (F:Set E) = (P:Set E) at hC
-      rw [finiteOverwrite,←hC]
-      ext e
-      simp only [Set.mem_union,Set.mem_inter_iff,Set.mem_compl_iff]
-      tauto
-    simpa only [D,Set.mem_preimage,he] using hD
-  apply lt_of_lt_of_le _ (measure_mono hsub)
-  rw [hind]
-  exact ENNReal.mul_pos hposC.ne' hposD.ne'
-
-theorem infinite_cluster_union_contains (G : BondGraph V E) (ω : Set E)
-    (F : Set E) (hF : F.Finite) (x : V) (hx : (cluster G (ω ∪ F) x).Infinite) :
-    ∃ z, Conn G (ω ∪ F) x z ∧ (cluster G ω z).Infinite := by
-  by_contra hn
-  have hfin (z : V) : (cluster G ω z ∩ cluster G (ω ∪ F) x).Finite := by
-    by_cases hz : Conn G (ω ∪ F) x z
-    · have hf : (cluster G ω z).Finite := by
-        by_contra hh
-        exact hn ⟨z,hz,hh⟩
-      exact hf.inter_of_left _
-    · have he : cluster G ω z ∩ cluster G (ω ∪ F) x = ∅ := by
-        apply Set.eq_empty_iff_forall_notMem.mpr
-        rintro y ⟨hzy,hxy⟩
-        apply hz
-        exact conn_trans G _ hxy (conn_symm G _ (conn_mono G Set.subset_union_left hzy))
-      rw [he]
-      exact Set.finite_empty
-  have hupper : (cluster G ω x ∩ cluster G (ω ∪ F) x ∪
-      ⋃ e ∈ F, ⋃ z ∈ {z | z ∈ G e}, cluster G ω z ∩ cluster G (ω ∪ F) x).Finite := by
-    apply (hfin x).union
-    apply hF.biUnion
-    intro e _
-    have he : {z | z ∈ G e}.Finite :=
-      (G e).toFinset.finite_toSet.subset (fun z hz => by simpa using hz)
-    exact he.biUnion (fun z _ => hfin z)
-  apply hx (hupper.subset ?_)
-  intro y hy
-  rcases cluster_union_subset G ω F x hy with hy' | hy'
-  · exact Or.inl ⟨hy',hy⟩
-  · right
-    simp only [Set.mem_iUnion,Set.mem_ofPred_eq] at hy' ⊢
-    obtain ⟨e,he,z,hz,hzy⟩ := hy'
-    exact ⟨e,he,z,hz,hzy,hy⟩
-
-theorem finite_opening_unique (G : BondGraph V E) (ω : Set E) (F : Finset E) (x : V)
-    (hx : (cluster G ω x).Infinite)
-    (hjoin : ∀ y, (cluster G ω y).Infinite → Conn G (ω ∪ (F:Set E)) x y) :
-    ∃! C, C ∈ infiniteClusters G (ω ∪ (F:Set E)) := by
-  rw [unique_infinite_cluster_iff]
-  refine ⟨x,hx.mono (fun _ h => conn_mono G Set.subset_union_left h),?_⟩
-  intro y hy
-  obtain ⟨z,hyz,hz⟩ := infinite_cluster_union_contains G ω (F:Set E) F.finite_toSet y hy
-  exact conn_trans G _ (hjoin z hz) (conn_symm G _ hyz)
-
-theorem finite_connecting_edges (G : BondGraph V E) (hconn : Connected G)
-    (x : V) (R : Finset V) : ∃ F : Finset E, ∀ y ∈ R, Conn G (F:Set E) x y := by
-  have hall (y : V) : ∃ F : Finset E, Conn G (F:Set E) x y := by
-    obtain ⟨s,_,hs⟩ := finiteWitness_conn G x y Set.univ (conn_univ_of_connected G hconn x y)
-    exact ⟨s,hs⟩
-  choose F hF using hall
-  refine ⟨R.biUnion F,?_⟩
-  intro y hy
-  exact conn_mono G (fun e he => Finset.mem_biUnion.mpr ⟨y,hy,he⟩) (hF y)
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Main
+
+namespace OAI
+
+/-! Finite changes of bond configurations and their positive probabilities. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal
+universe u v
+variable {V : Type u} {E : Type v}
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+def finiteOverwrite (F P : Finset E) (ω : Set E) : Set E :=
+  (ω ∩ (F:Set E)ᶜ) ∪ (P:Set E)
+
+theorem measurable_finiteOverwrite (F P : Finset E) : Measurable (finiteOverwrite F P) :=
+  (measurable_union_const (P:Set E)).comp (measurable_inter_const (F:Set E)ᶜ)
+
+theorem finite_pattern_pos [Countable E] (p : unitInterval) (hp : 0 < p) (hp1 : p < 1)
+    (F P : Finset E) (hPF : P ⊆ F) :
+    0 < (law p : Measure (Set E)) {ω | ω ∩ (F:Set E) = (P:Set E)} := by
+  let R : Set (F:Set E) := Subtype.val ⁻¹' (P:Set E)
+  have he : {ω : Set E | ω ∩ (F:Set E) = (P:Set E)} =
+      (fun ω : Set E => (Subtype.val ⁻¹' ω : Set (F:Set E))) ⁻¹' {R} := by
+    ext ω
+    simp only [Set.mem_ofPred_eq,Set.mem_preimage,Set.mem_singleton_iff]
+    constructor
+    · intro h
+      ext e
+      change e.val ∈ ω ↔ e.val ∈ (P:Set E)
+      have := Set.ext_iff.mp h e.val
+      simpa only [Set.mem_inter_iff,e.property,and_true] using this
+    · intro h
+      ext e
+      simp only [Set.mem_inter_iff,Finset.mem_coe]
+      constructor
+      · rintro ⟨he,hF⟩
+        exact (Set.ext_iff.mp h ⟨e,hF⟩).mp he
+      · intro he
+        have hF := hPF he
+        exact ⟨(Set.ext_iff.mp h ⟨e,hF⟩).mpr he,hF⟩
+  rw [he,←Measure.map_apply (measurable_set_restrict _) (measurableSet_singleton _),law_map_restrict]
+  have hh : 0 < (law p : Measure (Set (F:Set E))).real {R} := by
+    rw [law,setBernoulli_real_singleton p (Set.subset_univ _) Set.finite_univ]
+    exact mul_pos (pow_pos (show (0:ℝ) < p from hp) _)
+      (pow_pos (sub_pos.mpr (show (p:ℝ) < 1 from hp1)) _)
+  exact (ENNReal.toReal_pos_iff.mp hh).1
+
+theorem finiteOverwrite_pos [Countable E] (p : unitInterval) (hp : 0 < p) (hp1 : p < 1)
+    (F P : Finset E) (hPF : P ⊆ F) (A B : Set (Set E))
+    (hB : MeasurableSet B) (hA : 0 < (law p : Measure (Set E)) A)
+    (hAB : ∀ ω ∈ A, finiteOverwrite F P ω ∈ B) : 0 < (law p : Measure (Set E)) B := by
+  let D := (finiteOverwrite F P) ⁻¹' B
+  let C : Set (Set E) := {ω | ω ∩ (F:Set E) = (P:Set E)}
+  have hD : MeasurableSet D := hB.preimage (measurable_finiteOverwrite F P)
+  have hDfix : (fun ω : Set E => ω ∩ (F:Set E)ᶜ) ⁻¹' D = D := by
+    ext ω
+    simp only [D,finiteOverwrite,Set.mem_preimage,Set.inter_assoc,Set.inter_self]
+  have hind := restriction_events_independent p (F:Set E) (F:Set E)ᶜ disjoint_compl_right
+    {(P:Set E)} D (measurableSet_singleton _) hD
+  rw [hDfix] at hind
+  change (law p : Measure (Set E)) (C ∩ D) = law p C * law p D at hind
+  have hposC : 0 < (law p : Measure (Set E)) C := finite_pattern_pos p hp hp1 F P hPF
+  have hposD : 0 < (law p : Measure (Set E)) D := hA.trans_le (measure_mono hAB)
+  have hsub : C ∩ D ⊆ B := by
+    rintro ω ⟨hC,hD⟩
+    have he : finiteOverwrite F P ω = ω := by
+      change ω ∩ (F:Set E) = (P:Set E) at hC
+      rw [finiteOverwrite,←hC]
+      ext e
+      simp only [Set.mem_union,Set.mem_inter_iff,Set.mem_compl_iff]
+      tauto
+    simpa only [D,Set.mem_preimage,he] using hD
+  apply lt_of_lt_of_le _ (measure_mono hsub)
+  rw [hind]
+  exact ENNReal.mul_pos hposC.ne' hposD.ne'
+
+theorem infinite_cluster_union_contains (G : BondGraph V E) (ω : Set E)
+    (F : Set E) (hF : F.Finite) (x : V) (hx : (cluster G (ω ∪ F) x).Infinite) :
+    ∃ z, Conn G (ω ∪ F) x z ∧ (cluster G ω z).Infinite := by
+  by_contra hn
+  have hfin (z : V) : (cluster G ω z ∩ cluster G (ω ∪ F) x).Finite := by
+    by_cases hz : Conn G (ω ∪ F) x z
+    · have hf : (cluster G ω z).Finite := by
+        by_contra hh
+        exact hn ⟨z,hz,hh⟩
+      exact hf.inter_of_left _
+    · have he : cluster G ω z ∩ cluster G (ω ∪ F) x = ∅ := by
+        apply Set.eq_empty_iff_forall_notMem.mpr
+        rintro y ⟨hzy,hxy⟩
+        apply hz
+        exact conn_trans G _ hxy (conn_symm G _ (conn_mono G Set.subset_union_left hzy))
+      rw [he]
+      exact Set.finite_empty
+  have hupper : (cluster G ω x ∩ cluster G (ω ∪ F) x ∪
+      ⋃ e ∈ F, ⋃ z ∈ {z | z ∈ G e}, cluster G ω z ∩ cluster G (ω ∪ F) x).Finite := by
+    apply (hfin x).union
+    apply hF.biUnion
+    intro e _
+    have he : {z | z ∈ G e}.Finite :=
+      (G e).toFinset.finite_toSet.subset (fun z hz => by simpa using hz)
+    exact he.biUnion (fun z _ => hfin z)
+  apply hx (hupper.subset ?_)
+  intro y hy
+  rcases cluster_union_subset G ω F x hy with hy' | hy'
+  · exact Or.inl ⟨hy',hy⟩
+  · right
+    simp only [Set.mem_iUnion,Set.mem_ofPred_eq] at hy' ⊢
+    obtain ⟨e,he,z,hz,hzy⟩ := hy'
+    exact ⟨e,he,z,hz,hzy,hy⟩
+
+theorem finite_opening_unique (G : BondGraph V E) (ω : Set E) (F : Finset E) (x : V)
+    (hx : (cluster G ω x).Infinite)
+    (hjoin : ∀ y, (cluster G ω y).Infinite → Conn G (ω ∪ (F:Set E)) x y) :
+    ∃! C, C ∈ infiniteClusters G (ω ∪ (F:Set E)) := by
+  rw [unique_infinite_cluster_iff]
+  refine ⟨x,hx.mono (fun _ h => conn_mono G Set.subset_union_left h),?_⟩
+  intro y hy
+  obtain ⟨z,hyz,hz⟩ := infinite_cluster_union_contains G ω (F:Set E) F.finite_toSet y hy
+  exact conn_trans G _ (hjoin z hz) (conn_symm G _ hyz)
+
+theorem finite_connecting_edges (G : BondGraph V E) (hconn : Connected G)
+    (x : V) (R : Finset V) : ∃ F : Finset E, ∀ y ∈ R, Conn G (F:Set E) x y := by
+  have hall (y : V) : ∃ F : Finset E, Conn G (F:Set E) x y := by
+    obtain ⟨s,_,hs⟩ := finiteWitness_conn G x y Set.univ (conn_univ_of_connected G hconn x y)
+    exact ⟨s,hs⟩
+  choose F hF using hall
+  refine ⟨R.biUnion F,?_⟩
+  intro y hy
+  exact conn_mono G (fun e he => Finset.mem_biUnion.mpr ⟨y,hy,he⟩) (hF y)
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Criticality/Finiteness.lean b/lean/OAI/Probability/UnimodularPercolation/Criticality/Finiteness.lean
index 1eca0fcd1f649b7f1e8809e01b4ccab059211658..cc9e444eb03293d92f4ea8edf4359b877ec18db0 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Criticality/Finiteness.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Criticality/Finiteness.lean
@@ -1,466 +1,466 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Deformation.Corridors
-import OAI.Probability.UnimodularPercolation.TwoCluster.Bounds
-
-namespace OAI
-
-/-! Critical two-cluster estimates, finite opening and almost-sure finite clusters. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-open MeasureTheory ProbabilityTheory Filter Topology
-open scoped ENNReal
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-section RootData
-variable [Countable V] [Countable E] [Nonempty V]
-variable (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
-  (hqt : QuasiTransitive G)
-  {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-  (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-  (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
-  (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
-  (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
-  (d : ℕ) (hd : ∀ y, {e | y ∈ G e}.ncard ≤ d)
-
-include hconn hloc hqt hΓ htr hcov hw hw1 hc hmin hd
-
- 
-
-omit [Nonempty V] in
-theorem critical_twoLarge_probability_bound
-    (n : ℕ) (hn : 0 < n) (e : E) (a b : V) (he : G e = s(a,b)) :
-    (pc G:ℝ) * ((n:ℝ) * (law (pc G)).real (twoLargeEvent G a b n)) ≤
-      20*c*Real.sqrt (d*n:ℕ) := by
-  obtain ⟨p,hp,hpc,hlim⟩ := exists_seq_strictMono_tendsto' (pc_pos G hloc hqt)
-  have ht := (continuous_subtype_val.tendsto (pc G) |>.comp hlim).mul
-    ((tendsto_const_nhds (x := (n:ℝ))).mul (twoLarge_probability_tendsto G p hp.monotone (pc G)
-      (fun j => (hpc j).2.le) hlim a b n))
-  apply le_of_tendsto ht
-  apply Eventually.of_forall
-  intro j
-  have hb := twoLarge_probability_bound G hconn hloc root weight Γ hΓ htr hcov hw hw1
-    c hc hmin d hd n hn (p j) (hpc j).1
-      (subcritical_cluster_finite_ae G (p j) (hpc j).2) e a b he
-  change (p j:ℝ) * ((n:ℝ) * (law (p j)).real (twoLargeEvent G a b n)) ≤ _
-  rw [mul_comm (p j:ℝ)]
-  exact (le_div_iff₀ (hpc j).1).mp hb
-
- 
-
-omit [Nonempty V] in
-theorem critical_adjacent_infinite_connected (e : E) (a b : V) (he : G e = s(a,b)) :
-    ∀ᵐ ω ∂law (pc G), (cluster G ω a).Infinite → (cluster G ω b).Infinite → Conn G ω a b := by
-  let A : Set (Set E) := {ω | ¬Conn G ω a b ∧
-    (cluster G ω a).Infinite ∧ (cluster G ω b).Infinite}
-  have hA : MeasurableSet A := (measurableSet_conn G a b).compl.inter
-    ((measurableSet_infinite_cluster G a).inter (measurableSet_infinite_cluster G b))
-  have hp : 0 < (pc G:ℝ) := pc_pos G hloc hqt
-  have hzero : (law (pc G)).real A = 0 := by
-    by_contra hne
-    have hθ : 0 < (law (pc G)).real A := lt_of_le_of_ne measureReal_nonneg (Ne.symm hne)
-    have hb (n : ℕ) (hn : 0 < n) :
-        (pc G:ℝ)*((n:ℝ)*(law (pc G)).real A) ≤ 20*c*Real.sqrt (d*n:ℕ) := by
-      have hs : A ⊆ twoLargeEvent G a b n := by
-        intro ω h
-        refine ⟨h.1,?_,?_⟩
-        · have hx : ω ∈ {ω | (cluster G ω a).Infinite} := h.2.1
-          rw [infinite_cluster_eq_iInter G a] at hx
-          exact Set.mem_iInter.mp hx n
-        · have hx : ω ∈ {ω | (cluster G ω b).Infinite} := h.2.2
-          rw [infinite_cluster_eq_iInter G b] at hx
-          exact Set.mem_iInter.mp hx n
-      exact (mul_le_mul_of_nonneg_left
-        (mul_le_mul_of_nonneg_left (measureReal_mono hs) (Nat.cast_nonneg n)) hp.le).trans
-          (critical_twoLarge_probability_bound G hconn hloc hqt root weight Γ hΓ htr hcov
-            hw hw1 c hc hmin d hd n hn e a b he)
-    let B := 20*c*Real.sqrt (d:ℝ)
-    obtain ⟨m,hm⟩ := exists_nat_gt (B / ((pc G:ℝ)*(law (pc G)).real A) + 1)
-    have hmpos : 0 < (m:ℝ) := lt_trans (by dsimp [B]; positivity) hm
-    have hmn : 0 < m := by exact_mod_cast hmpos
-    have hb' := hb (m^2) (pow_pos hmn _)
-    have hsqrt : Real.sqrt (d*(m^2):ℕ) = Real.sqrt (d:ℝ)*(m:ℝ) := by
-      rw [Nat.cast_mul,Nat.cast_pow,Real.sqrt_mul (Nat.cast_nonneg d),Real.sqrt_sq (Nat.cast_nonneg m)]
-    rw [hsqrt,Nat.cast_pow] at hb'
-    have hcancel : (pc G:ℝ)*(law (pc G)).real A*(m:ℝ) ≤ B := by
-      apply (mul_le_mul_iff_left₀ hmpos).mp
-      change ((pc G:ℝ)*(law (pc G)).real A*(m:ℝ))*(m:ℝ) ≤ B*(m:ℝ)
-      dsimp only [B]
-      nlinarith only [hb']
-    have hlt : B < (pc G:ℝ)*(law (pc G)).real A*(m:ℝ) := by
-      have hdiv : B / ((pc G:ℝ)*(law (pc G)).real A) < (m:ℝ) := by linarith
-      have hh := (div_lt_iff₀ (mul_pos hp hθ)).mp hdiv
-      nlinarith
-    exact (not_lt_of_ge hcancel) hlt
-  have hn : law (pc G) A = 0 := (measureReal_eq_zero_iff).mp hzero
-  have hae : ∀ᵐ ω ∂law (pc G), ω ∉ A := by
-    rw [ae_iff]; simpa using hn
-  filter_upwards [hae] with ω hω ha hb
-  by_contra hc
-  exact hω ⟨hc,ha,hb⟩
-
-end RootData
-end Percolation
-
- 
-
-open MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem measure_union_finset_null [Countable E] (p : unitInterval) (hp : 0 < (p:ℝ))
-    (A : Set (Set E)) (hA : MeasurableSet A) (hzero : law p A = 0) (F : Finset E) :
-    law p {ω | ω ∪ (F:Set E) ∈ A} = 0 := by
-  let B : Set (Set E) := {ω | ω ∪ (F:Set E) ∈ A}
-  let C : Set (Set E) := {ω | (F:Set E) ⊆ ω}
-  have hB : MeasurableSet B := hA.preimage
-    (measurable_set_union_pair.comp (measurable_id.prodMk measurable_const))
-  have hC : MeasurableSet C := by
-    change MeasurableSet {ω | ∀ e ∈ F, e ∈ ω}
-    simp only [Set.ofPred_forall]
-    exact MeasurableSet.iInter fun e => MeasurableSet.iInter fun _ => measurableSet_mem e
-  have heB : (fun ω : Set E => ω ∩ (F:Set E)ᶜ) ⁻¹' B = B := by
-    ext ω
-    have he : (ω ∩ (F:Set E)ᶜ) ∪ (F:Set E) = ω ∪ (F:Set E) := by
-      ext t
-      simp only [Set.mem_union,Set.mem_inter_iff,Set.mem_compl_iff]
-      tauto
-    simp only [Set.mem_preimage,B,Set.mem_ofPred_eq,he]
-  have heC : (fun ω : Set E => ω ∩ (F:Set E)) ⁻¹' C = C := by
-    ext ω
-    simp [C]
-  have hi := restriction_events_independent p (F:Set E)ᶜ (F:Set E)
-    (Set.disjoint_left.mpr (fun _ hx hy => hx hy)) B C hB hC
-  rw [heB,heC] at hi
-  have hBC : law p (B ∩ C) = 0 := measure_mono_null (by
-    intro ω h
-    have he : ω ∪ (F:Set E) = ω := Set.union_eq_left.mpr h.2
-    simpa only [B,Set.mem_ofPred_eq,he] using h.1) hzero
-  have hCp : 0 < (law p).real C := by
-    rw [show C = {ω | (F:Set E) ⊆ ω} from rfl,finite_open_probability]
-    exact pow_pos hp _
-  exact (mul_eq_zero.mp (hi.symm.trans hBC)).resolve_right
-    (fun h => by simp [Measure.real,h] at hCp)
-
-theorem ae_union_finset [Countable E] (p : unitInterval) (hp : 0 < (p:ℝ))
-    (A : Set (Set E)) (hA : MeasurableSet A) (hae : ∀ᵐ ω ∂law p, ω ∈ A) (F : Finset E) :
-    ∀ᵐ ω ∂law p, ω ∪ (F:Set E) ∈ A := by
-  rw [ae_iff] at hae ⊢
-  exact measure_union_finset_null p hp Aᶜ hA.compl hae F
-
- 
-
-theorem atMostOne_of_finite_openings (G : BondGraph V E) (hconn : Connected G) (ω : Set E)
-    (h : ∀ F : Finset E, ∀ a b, Adj G a b →
-      (cluster G (ω ∪ (F:Set E)) a).Infinite →
-      (cluster G (ω ∪ (F:Set E)) b).Infinite → Conn G (ω ∪ (F:Set E)) a b) :
-    ∀ x y, (cluster G ω x).Infinite → (cluster G ω y).Infinite → Conn G ω x y := by
-  intro x y hx hy
-  by_contra hxy
-  have hF : ∀ F : Finset E, ¬Conn G (ω ∪ (F:Set E)) x y := by
-    intro F
-    induction F using Finset.induction_on with
-    | empty => simpa only [Finset.coe_empty,Set.union_empty] using hxy
-    | @insert e F he ih =>
-      have heq : ω ∪ ((insert e F : Finset E):Set E) = insert e (ω ∪ (F:Set E)) := by
-        ext t; simp only [Finset.coe_insert,Set.mem_union,Set.mem_insert_iff]; tauto
-      rw [heq]
-      obtain ⟨⟨a,b⟩,hab'⟩ := Sym2.mk_surjective (G e)
-      have hab : G e = s(a,b) := hab'.symm
-      rw [conn_insert_iff G _ e a b x y hab]
-      rintro (hc | ⟨hxa,hby⟩ | ⟨hxb,hay⟩)
-      · exact ih hc
-      · have ha : (cluster G (ω ∪ (F:Set E)) a).Infinite := by
-          rw [←cluster_eq_of_conn G _ hxa]
-          exact hx.mono (fun z hz => conn_mono G Set.subset_union_left hz)
-        have hb : (cluster G (ω ∪ (F:Set E)) b).Infinite := by
-          rw [cluster_eq_of_conn G _ hby]
-          exact hy.mono (fun z hz => conn_mono G Set.subset_union_left hz)
-        exact ih (conn_trans G _ hxa (conn_trans G _ (h F a b ⟨e,hab⟩ ha hb) hby))
-      · have hb : (cluster G (ω ∪ (F:Set E)) b).Infinite := by
-          rw [←cluster_eq_of_conn G _ hxb]
-          exact hx.mono (fun z hz => conn_mono G Set.subset_union_left hz)
-        have ha : (cluster G (ω ∪ (F:Set E)) a).Infinite := by
-          rw [cluster_eq_of_conn G _ hay]
-          exact hy.mono (fun z hz => conn_mono G Set.subset_union_left hz)
-        exact ih (conn_trans G _ hxb (conn_trans G _
-          (conn_symm G _ (h F a b ⟨e,hab⟩ ha hb)) hay))
-  obtain ⟨F,_,hFc⟩ := finiteWitness_conn G x y Set.univ (conn_univ_of_connected G hconn x y)
-  exact hF F (conn_mono G Set.subset_union_right hFc)
-
-end Percolation
-
- 
-
- 
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter Topology
-open scoped ENNReal
-
- 
-theorem twoPoint_mul_le [Countable E] (G : BondGraph V E) (p : unitInterval)
-    (x y z : V) : twoPoint G p x y * twoPoint G p y z ≤ twoPoint G p x z := by
-  have h := association_finiteWitness p {ω : Set E | Conn G ω x y}
-    {ω : Set E | Conn G ω y z}
-    (fun _ _ hω h => conn_mono G hω h) (fun _ _ hω h => conn_mono G hω h)
-    (finiteWitness_conn G x y) (finiteWitness_conn G y z)
-  exact h.trans (measureReal_mono (fun ω hω => conn_trans G ω hω.1 hω.2))
-
- 
-theorem reachBall_add_split (G : BondGraph V E) (x : V) (n m : ℕ) {y : V}
-    (hy : y ∈ reachBall G x (n+m)) :
-    ∃ z ∈ reachBall G x n, y ∈ reachBall G z m := by
-  induction m generalizing y with
-  | zero => exact ⟨y,hy,Set.mem_singleton y⟩
-  | succ m ih =>
-    rcases hy with hy | hy
-    · obtain ⟨z,hz,hyz⟩ := ih hy
-      exact ⟨z,hz,Or.inl hyz⟩
-    · simp only [Set.mem_iUnion,Set.mem_ofPred_eq] at hy
-      obtain ⟨w,hw,hwy⟩ := hy
-      obtain ⟨z,hz,hwz⟩ := ih hw
-      refine ⟨z,hz,Or.inr ?_⟩
-      exact Set.mem_iUnion.mpr ⟨w,Set.mem_iUnion.mpr ⟨hwz,hwy⟩⟩
-
- 
-theorem twoPoint_lower_on_multiples [Countable E] (G : BondGraph V E)
-    (p : unitInterval) (n : ℕ) (c : ℝ) (hc : 0 ≤ c)
-    (hball : ∀ x y, y ∈ reachBall G x n → c ≤ twoPoint G p x y)
-    (k : ℕ) (x y : V) (hy : y ∈ reachBall G x (n*k)) :
-    c^k ≤ twoPoint G p x y := by
-  induction k generalizing x y with
-  | zero =>
-    have he : y = x := by simpa [reachBall] using hy
-    subst y
-    simp [twoPoint_self]
-  | succ k ih =>
-    rw [Nat.mul_succ] at hy
-    obtain ⟨z,hz,hyz⟩ := reachBall_add_split G x (n*k) n hy
-    calc
-      c^(k+1) = c^k*c := pow_succ _ _
-      _ ≤ twoPoint G p x z * twoPoint G p z y :=
-        mul_le_mul (ih x z hz) (hball z y hyz) hc (twoPoint_nonneg G p x z)
-      _ ≤ twoPoint G p x y := twoPoint_mul_le G p x z y
-
- 
-theorem reachBall_exponential_growth (G : BondGraph V E) (hloc : LocallyFinite G)
-    (hh : 0 ≤ hV G) (x : V) (n : ℕ) :
-    (1+hV G)^n ≤ ((reachBall G x n).ncard : ℝ) := by
-  classical
-  induction n with
-  | zero => simp [reachBall]
-  | succ n ih =>
-    let A := (finite_reachBall hloc x n).toFinset
-    have he : reachBall G x (n+1) = (A : Set V) ∪ vertexBoundary G A := by
-      ext y
-      simp only [reachBall,Set.mem_union,Finset.mem_coe,
-        vertexBoundary,Set.mem_ofPred_eq,Set.mem_iUnion]
-      simp only [A,Set.Finite.mem_toFinset]
-      constructor
-      · rintro (hy | ⟨z,hz,hzy⟩)
-        · exact Or.inl hy
-        · by_cases hy : y ∈ reachBall G x n
-          · exact Or.inl hy
-          · exact Or.inr ⟨hy,z,hz,hzy⟩
-      · rintro (hy | ⟨_,z,hz,hzy⟩)
-        · exact Or.inl hy
-        · exact Or.inr ⟨z,hz,hzy⟩
-    have hd : Disjoint (A : Set V) (vertexBoundary G A) := by
-      exact Set.disjoint_left.mpr (fun y hy hb => hb.1 hy)
-    have hc : (reachBall G x (n+1)).ncard = A.card + (vertexBoundary G A).ncard := by
-      rw [he,Set.ncard_union_eq hd A.finite_toSet (finite_vertexBoundary G hloc A),Set.ncard_coe_finset]
-    have hcard : A.card = (reachBall G x n).ncard :=
-      (Set.ncard_eq_toFinset_card _ (finite_reachBall hloc x n)).symm
-    have hs := hV_mul_card_le G A
-    rw [hcard] at hs
-    rw [hc,hcard,Nat.cast_add,pow_succ]
-    nlinarith [mul_le_mul_of_nonneg_right ih (show 0 ≤ 1+hV G by linarith)]
-
- 
-
-theorem exists_subcritical_ball_lower [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (c : ℝ) (hc : 0 < c) (hcrit : ∀ x y, c ≤ twoPoint G (pc G) x y)
-    (n : ℕ) : ∃ p : unitInterval, p < pc G ∧
-      ∀ x y, y ∈ reachBall G x n → c/2 ≤ twoPoint G p x y := by
-  classical
-  obtain ⟨p,hp,hpc,hlim⟩ := exists_seq_strictMono_tendsto' (pc_pos G hloc hqt)
-  obtain ⟨s,hs⟩ := hqt
-  have hev : ∀ᶠ k in atTop, ∀ x ∈ s, ∀ y ∈ reachBall G x n,
-      c/2 ≤ twoPoint G (p k) x y := by
-    apply (eventually_all_finset s).mpr
-    intro x hx
-    apply (finite_reachBall hloc x n).eventually_all.mpr
-    intro y hy
-    have ht := twoPoint_tendsto_of_monotone G p hp.monotone (pc G)
-      (fun k => (hpc k).2.le) hlim x y
-    have hlt : c/2 < twoPoint G (pc G) x y := lt_of_lt_of_le (by linarith) (hcrit x y)
-    exact (ht.eventually (eventually_gt_nhds hlt)).mono (fun k hk => hk.le)
-  obtain ⟨k,hk⟩ := hev.exists
-  refine ⟨p k,(hpc k).2,?_⟩
-  intro x y hy
-  obtain ⟨o,ho,a,b,ha,hax⟩ := hs x
-  have hy' : a.symm y ∈ reachBall G o n := by
-    have hi := mem_reachBall_aut G (isAut_symm G ha) hy
-    simpa [←hax] using hi
-  have he := twoPoint_aut G (p k) ha o (a.symm y)
-  rw [hax,a.apply_symm_apply] at he
-  rw [he]
-  exact hk o ho (a.symm y) hy'
-
- 
-
-theorem no_uniform_positive_twoPoint_critical [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (hh : 0 < hV G) :
-    ¬ ∃ c : ℝ, 0 < c ∧ ∀ x y, c ≤ twoPoint G (pc G) x y := by
-  rintro ⟨c,hc,hcrit⟩
-  obtain ⟨n,hn⟩ := pow_unbounded_of_one_lt (2/c) (show 1 < 1+hV G by linarith)
-  have hn' : 1 < (c/2)*(1+hV G)^n := by
-    have := (div_lt_iff₀ hc).mp hn
-    nlinarith
-  obtain ⟨p,hp,hball⟩ := exists_subcritical_ball_lower G hloc hqt c hc hcrit n
-  obtain ⟨C,hC,hrow⟩ := subcritical_row_bound G hloc hqt p hp
-  obtain ⟨k,hk⟩ := pow_unbounded_of_one_lt C hn'
-  let x : V := Classical.arbitrary V
-  let A := (finite_reachBall hloc x (n*k)).toFinset
-  have hcard : (reachBall G x (n*k)).ncard = A.card := Set.ncard_eq_toFinset_card _ _
-  have hsum : (A.card:ℝ)*(c/2)^k ≤ ∑ y ∈ A, twoPoint G p x y := by
-    calc
-      _ = ∑ y ∈ A, (c/2)^k := by simp
-      _ ≤ _ := Finset.sum_le_sum fun y hy =>
-        twoPoint_lower_on_multiples G p n (c/2) (by linarith) hball k x y
-          (by simpa [A] using hy)
-  have hg := reachBall_exponential_growth G hloc hh.le x (n*k)
-  rw [hcard] at hg
-  have hpow : ((c/2)*(1+hV G)^n)^k ≤ (A.card:ℝ)*(c/2)^k := by
-    calc
-      _ = (1+hV G)^(n*k)*(c/2)^k := by rw [mul_pow,pow_mul]; ring
-      _ ≤ _ := mul_le_mul_of_nonneg_right hg (pow_nonneg (half_pos hc).le k)
-  exact (not_lt_of_ge (hpow.trans (hsum.trans (hrow x A)))) hk
-
- 
-
-theorem not_unique_infinite_cluster_critical [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
-    (hqt : QuasiTransitive G) (hh : 0 < hV G) :
-    law (pc G) {ω | ∃! C, C ∈ infiniteClusters G ω} ≠ 1 := by
-  intro hu
-  have hp := pc_pos G hloc hqt
-  have hpos : 0 < law (pc G) {ω | (infiniteClusters G ω).Nonempty} :=
-    lt_of_lt_of_le (show 0 < law (pc G) {ω | ∃! C, C ∈ infiniteClusters G ω} by
-      rw [hu]; exact zero_lt_one)
-      (measure_mono fun ω hω => ⟨hω.choose,hω.choose_spec.1⟩)
-  obtain ⟨θ,hθ,hroot⟩ := uniform_survival_pos G hconn hqt (pc G) hp hpos
-  have hue : ∀ᵐ ω ∂law (pc G), ∃! C, C ∈ infiniteClusters G ω :=
-    (mem_ae_iff_prob_eq_one (measurableSet_unique_infinite_cluster G)).mpr hu
-  apply no_uniform_positive_twoPoint_critical G hloc hqt hh
-  refine ⟨θ*θ,mul_pos hθ hθ,fun x y => ?_⟩
-  refine (mul_le_mul (hroot x) (hroot y) hθ.le measureReal_nonneg).trans
-    ((association_infinite_cluster G (pc G) x y).trans ?_)
-  apply ENNReal.toReal_mono (measure_ne_top _ _)
-  apply measure_mono_ae
-  filter_upwards [hue] with ω hω
-  intro hi
-  obtain ⟨r,_,hconnr⟩ := (unique_infinite_cluster_iff G ω).mp hω
-  exact conn_trans G ω (conn_symm G ω (hconnr x hi.1)) (hconnr y hi.2)
-
-end Percolation
-
- 
-
-open MeasureTheory ProbabilityTheory Filter Topology
-open scoped ENNReal
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem critical_finite_of_transport [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
-    (hqt : QuasiTransitive G) (hh : 0 < hV G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
-    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1) :
-    ∀ x, ∀ᵐ ω ∂law (pc G), (cluster G ω x).Finite := by
-  let c : ℝ := ∑ i, 1/weight i
-  have hc : 0 ≤ c := Finset.sum_nonneg (fun i _ => (one_div_pos.mpr (hw i)).le)
-  have hmin (i : ι) : 1 ≤ c*weight i := by
-    apply (div_le_iff₀ (hw i)).mp
-    exact Finset.single_le_sum (fun j _ => (one_div_pos.mpr (hw j)).le) (Finset.mem_univ i)
-  obtain ⟨d,_,hd⟩ := exists_degree_bound G hqt
-  have hdn (x : V) := (incident_ncard_le_degree G hloc x).trans (hd x)
-  let A : Set (Set E) := {ω | ∀ a b, Adj G a b →
-    (cluster G ω a).Infinite → (cluster G ω b).Infinite → Conn G ω a b}
-  have hA : MeasurableSet A := by
-    simp only [A,Set.ofPred_forall]
-    apply MeasurableSet.iInter
-    intro a
-    apply MeasurableSet.iInter
-    intro b
-    apply MeasurableSet.iInter
-    intro _
-    simp only [imp_iff_not_or,Set.ofPred_or]
-    exact (measurableSet_infinite_cluster G a).compl.union
-      ((measurableSet_infinite_cluster G b).compl.union (measurableSet_conn G a b))
-  have hae : ∀ᵐ ω ∂law (pc G), ω ∈ A := by
-    change ∀ᵐ ω ∂law (pc G), ∀ a b, Adj G a b →
-      (cluster G ω a).Infinite → (cluster G ω b).Infinite → Conn G ω a b
-    rw [ae_all_iff]
-    intro a
-    rw [ae_all_iff]
-    intro b
-    rw [ae_all_iff]
-    rintro ⟨e,he⟩
-    exact critical_adjacent_infinite_connected G hconn hloc hqt root weight Γ hΓ htr hcov
-      (fun i => (hw i).le) hw1 c hc hmin d hdn e a b he
-  have haeF : ∀ᵐ ω ∂law (pc G), ∀ F : Finset E, ω ∪ (F:Set E) ∈ A := by
-    rw [ae_all_iff]
-    intro F
-    exact ae_union_finset (pc G) (pc_pos G hloc hqt) A hA hae F
-  have hat : ∀ᵐ ω ∂law (pc G), ∀ x y,
-      (cluster G ω x).Infinite → (cluster G ω y).Infinite → Conn G ω x y := by
-    filter_upwards [haeF] with ω hω
-    exact atMostOne_of_finite_openings G hconn ω hω
-  have hex := infinite_cluster_zero_one G (pc G)
-  have he : (labelLaw : Measure (E → unitInterval))
-      {labels | ∃ x, (cluster G (openEdges labels (pc G)) x).Infinite} =
-      law (pc G) {ω | ∃ x, (cluster G ω x).Infinite} := by
-    rw [←map_openEdges (pc G),Measure.map_apply (measurable_openEdges (pc G))
-      (measurableSet_exists_infinite_cluster G)]
-    rfl
-  rw [he] at hex
-  have hz : law (pc G) {ω | ∃ x, (cluster G ω x).Infinite} = 0 := by
-    rcases hex with hzero | hone
-    · exact hzero
-    · exfalso
-      apply not_unique_infinite_cluster_critical G hconn hloc hqt hh
-      apply (mem_ae_iff_prob_eq_one (measurableSet_unique_infinite_cluster G)).mp
-      have haex : ∀ᵐ ω ∂law (pc G), ∃ x, (cluster G ω x).Infinite :=
-        (mem_ae_iff_prob_eq_one (measurableSet_exists_infinite_cluster G)).mpr hone
-      filter_upwards [hat,haex] with ω hω hex
-      obtain ⟨x,hx⟩ := hex
-      exact (unique_infinite_cluster_iff G ω).mpr ⟨x,hx,fun y hy => hω x y hx hy⟩
-  intro x
-  rw [ae_iff]
-  apply measure_mono_null (t := {ω | ∃ y, (cluster G ω y).Infinite}) ?_ hz
-  intro ω hω
-  exact ⟨x,hω⟩
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Deformation.Corridors
+import OAI.Probability.UnimodularPercolation.TwoCluster.Bounds
+
+namespace OAI
+
+/-! Critical two-cluster estimates, finite opening and almost-sure finite clusters. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+open MeasureTheory ProbabilityTheory Filter Topology
+open scoped ENNReal
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+section RootData
+variable [Countable V] [Countable E] [Nonempty V]
+variable (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
+  (hqt : QuasiTransitive G)
+  {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+  (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+  (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
+  (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
+  (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
+  (d : ℕ) (hd : ∀ y, {e | y ∈ G e}.ncard ≤ d)
+
+include hconn hloc hqt hΓ htr hcov hw hw1 hc hmin hd
+
+ 
+
+omit [Nonempty V] in
+theorem critical_twoLarge_probability_bound
+    (n : ℕ) (hn : 0 < n) (e : E) (a b : V) (he : G e = s(a,b)) :
+    (pc G:ℝ) * ((n:ℝ) * (law (pc G)).real (twoLargeEvent G a b n)) ≤
+      20*c*Real.sqrt (d*n:ℕ) := by
+  obtain ⟨p,hp,hpc,hlim⟩ := exists_seq_strictMono_tendsto' (pc_pos G hloc hqt)
+  have ht := (continuous_subtype_val.tendsto (pc G) |>.comp hlim).mul
+    ((tendsto_const_nhds (x := (n:ℝ))).mul (twoLarge_probability_tendsto G p hp.monotone (pc G)
+      (fun j => (hpc j).2.le) hlim a b n))
+  apply le_of_tendsto ht
+  apply Eventually.of_forall
+  intro j
+  have hb := twoLarge_probability_bound G hconn hloc root weight Γ hΓ htr hcov hw hw1
+    c hc hmin d hd n hn (p j) (hpc j).1
+      (subcritical_cluster_finite_ae G (p j) (hpc j).2) e a b he
+  change (p j:ℝ) * ((n:ℝ) * (law (p j)).real (twoLargeEvent G a b n)) ≤ _
+  rw [mul_comm (p j:ℝ)]
+  exact (le_div_iff₀ (hpc j).1).mp hb
+
+ 
+
+omit [Nonempty V] in
+theorem critical_adjacent_infinite_connected (e : E) (a b : V) (he : G e = s(a,b)) :
+    ∀ᵐ ω ∂law (pc G), (cluster G ω a).Infinite → (cluster G ω b).Infinite → Conn G ω a b := by
+  let A : Set (Set E) := {ω | ¬Conn G ω a b ∧
+    (cluster G ω a).Infinite ∧ (cluster G ω b).Infinite}
+  have hA : MeasurableSet A := (measurableSet_conn G a b).compl.inter
+    ((measurableSet_infinite_cluster G a).inter (measurableSet_infinite_cluster G b))
+  have hp : 0 < (pc G:ℝ) := pc_pos G hloc hqt
+  have hzero : (law (pc G)).real A = 0 := by
+    by_contra hne
+    have hθ : 0 < (law (pc G)).real A := lt_of_le_of_ne measureReal_nonneg (Ne.symm hne)
+    have hb (n : ℕ) (hn : 0 < n) :
+        (pc G:ℝ)*((n:ℝ)*(law (pc G)).real A) ≤ 20*c*Real.sqrt (d*n:ℕ) := by
+      have hs : A ⊆ twoLargeEvent G a b n := by
+        intro ω h
+        refine ⟨h.1,?_,?_⟩
+        · have hx : ω ∈ {ω | (cluster G ω a).Infinite} := h.2.1
+          rw [infinite_cluster_eq_iInter G a] at hx
+          exact Set.mem_iInter.mp hx n
+        · have hx : ω ∈ {ω | (cluster G ω b).Infinite} := h.2.2
+          rw [infinite_cluster_eq_iInter G b] at hx
+          exact Set.mem_iInter.mp hx n
+      exact (mul_le_mul_of_nonneg_left
+        (mul_le_mul_of_nonneg_left (measureReal_mono hs) (Nat.cast_nonneg n)) hp.le).trans
+          (critical_twoLarge_probability_bound G hconn hloc hqt root weight Γ hΓ htr hcov
+            hw hw1 c hc hmin d hd n hn e a b he)
+    let B := 20*c*Real.sqrt (d:ℝ)
+    obtain ⟨m,hm⟩ := exists_nat_gt (B / ((pc G:ℝ)*(law (pc G)).real A) + 1)
+    have hmpos : 0 < (m:ℝ) := lt_trans (by dsimp [B]; positivity) hm
+    have hmn : 0 < m := by exact_mod_cast hmpos
+    have hb' := hb (m^2) (pow_pos hmn _)
+    have hsqrt : Real.sqrt (d*(m^2):ℕ) = Real.sqrt (d:ℝ)*(m:ℝ) := by
+      rw [Nat.cast_mul,Nat.cast_pow,Real.sqrt_mul (Nat.cast_nonneg d),Real.sqrt_sq (Nat.cast_nonneg m)]
+    rw [hsqrt,Nat.cast_pow] at hb'
+    have hcancel : (pc G:ℝ)*(law (pc G)).real A*(m:ℝ) ≤ B := by
+      apply (mul_le_mul_iff_left₀ hmpos).mp
+      change ((pc G:ℝ)*(law (pc G)).real A*(m:ℝ))*(m:ℝ) ≤ B*(m:ℝ)
+      dsimp only [B]
+      nlinarith only [hb']
+    have hlt : B < (pc G:ℝ)*(law (pc G)).real A*(m:ℝ) := by
+      have hdiv : B / ((pc G:ℝ)*(law (pc G)).real A) < (m:ℝ) := by linarith
+      have hh := (div_lt_iff₀ (mul_pos hp hθ)).mp hdiv
+      nlinarith
+    exact (not_lt_of_ge hcancel) hlt
+  have hn : law (pc G) A = 0 := (measureReal_eq_zero_iff).mp hzero
+  have hae : ∀ᵐ ω ∂law (pc G), ω ∉ A := by
+    rw [ae_iff]; simpa using hn
+  filter_upwards [hae] with ω hω ha hb
+  by_contra hc
+  exact hω ⟨hc,ha,hb⟩
+
+end RootData
+end Percolation
+
+ 
+
+open MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem measure_union_finset_null [Countable E] (p : unitInterval) (hp : 0 < (p:ℝ))
+    (A : Set (Set E)) (hA : MeasurableSet A) (hzero : law p A = 0) (F : Finset E) :
+    law p {ω | ω ∪ (F:Set E) ∈ A} = 0 := by
+  let B : Set (Set E) := {ω | ω ∪ (F:Set E) ∈ A}
+  let C : Set (Set E) := {ω | (F:Set E) ⊆ ω}
+  have hB : MeasurableSet B := hA.preimage
+    (measurable_set_union_pair.comp (measurable_id.prodMk measurable_const))
+  have hC : MeasurableSet C := by
+    change MeasurableSet {ω | ∀ e ∈ F, e ∈ ω}
+    simp only [Set.ofPred_forall]
+    exact MeasurableSet.iInter fun e => MeasurableSet.iInter fun _ => measurableSet_mem e
+  have heB : (fun ω : Set E => ω ∩ (F:Set E)ᶜ) ⁻¹' B = B := by
+    ext ω
+    have he : (ω ∩ (F:Set E)ᶜ) ∪ (F:Set E) = ω ∪ (F:Set E) := by
+      ext t
+      simp only [Set.mem_union,Set.mem_inter_iff,Set.mem_compl_iff]
+      tauto
+    simp only [Set.mem_preimage,B,Set.mem_ofPred_eq,he]
+  have heC : (fun ω : Set E => ω ∩ (F:Set E)) ⁻¹' C = C := by
+    ext ω
+    simp [C]
+  have hi := restriction_events_independent p (F:Set E)ᶜ (F:Set E)
+    (Set.disjoint_left.mpr (fun _ hx hy => hx hy)) B C hB hC
+  rw [heB,heC] at hi
+  have hBC : law p (B ∩ C) = 0 := measure_mono_null (by
+    intro ω h
+    have he : ω ∪ (F:Set E) = ω := Set.union_eq_left.mpr h.2
+    simpa only [B,Set.mem_ofPred_eq,he] using h.1) hzero
+  have hCp : 0 < (law p).real C := by
+    rw [show C = {ω | (F:Set E) ⊆ ω} from rfl,finite_open_probability]
+    exact pow_pos hp _
+  exact (mul_eq_zero.mp (hi.symm.trans hBC)).resolve_right
+    (fun h => by simp [Measure.real,h] at hCp)
+
+theorem ae_union_finset [Countable E] (p : unitInterval) (hp : 0 < (p:ℝ))
+    (A : Set (Set E)) (hA : MeasurableSet A) (hae : ∀ᵐ ω ∂law p, ω ∈ A) (F : Finset E) :
+    ∀ᵐ ω ∂law p, ω ∪ (F:Set E) ∈ A := by
+  rw [ae_iff] at hae ⊢
+  exact measure_union_finset_null p hp Aᶜ hA.compl hae F
+
+ 
+
+theorem atMostOne_of_finite_openings (G : BondGraph V E) (hconn : Connected G) (ω : Set E)
+    (h : ∀ F : Finset E, ∀ a b, Adj G a b →
+      (cluster G (ω ∪ (F:Set E)) a).Infinite →
+      (cluster G (ω ∪ (F:Set E)) b).Infinite → Conn G (ω ∪ (F:Set E)) a b) :
+    ∀ x y, (cluster G ω x).Infinite → (cluster G ω y).Infinite → Conn G ω x y := by
+  intro x y hx hy
+  by_contra hxy
+  have hF : ∀ F : Finset E, ¬Conn G (ω ∪ (F:Set E)) x y := by
+    intro F
+    induction F using Finset.induction_on with
+    | empty => simpa only [Finset.coe_empty,Set.union_empty] using hxy
+    | @insert e F he ih =>
+      have heq : ω ∪ ((insert e F : Finset E):Set E) = insert e (ω ∪ (F:Set E)) := by
+        ext t; simp only [Finset.coe_insert,Set.mem_union,Set.mem_insert_iff]; tauto
+      rw [heq]
+      obtain ⟨⟨a,b⟩,hab'⟩ := Sym2.mk_surjective (G e)
+      have hab : G e = s(a,b) := hab'.symm
+      rw [conn_insert_iff G _ e a b x y hab]
+      rintro (hc | ⟨hxa,hby⟩ | ⟨hxb,hay⟩)
+      · exact ih hc
+      · have ha : (cluster G (ω ∪ (F:Set E)) a).Infinite := by
+          rw [←cluster_eq_of_conn G _ hxa]
+          exact hx.mono (fun z hz => conn_mono G Set.subset_union_left hz)
+        have hb : (cluster G (ω ∪ (F:Set E)) b).Infinite := by
+          rw [cluster_eq_of_conn G _ hby]
+          exact hy.mono (fun z hz => conn_mono G Set.subset_union_left hz)
+        exact ih (conn_trans G _ hxa (conn_trans G _ (h F a b ⟨e,hab⟩ ha hb) hby))
+      · have hb : (cluster G (ω ∪ (F:Set E)) b).Infinite := by
+          rw [←cluster_eq_of_conn G _ hxb]
+          exact hx.mono (fun z hz => conn_mono G Set.subset_union_left hz)
+        have ha : (cluster G (ω ∪ (F:Set E)) a).Infinite := by
+          rw [cluster_eq_of_conn G _ hay]
+          exact hy.mono (fun z hz => conn_mono G Set.subset_union_left hz)
+        exact ih (conn_trans G _ hxb (conn_trans G _
+          (conn_symm G _ (h F a b ⟨e,hab⟩ ha hb)) hay))
+  obtain ⟨F,_,hFc⟩ := finiteWitness_conn G x y Set.univ (conn_univ_of_connected G hconn x y)
+  exact hF F (conn_mono G Set.subset_union_right hFc)
+
+end Percolation
+
+ 
+
+ 
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter Topology
+open scoped ENNReal
+
+ 
+theorem twoPoint_mul_le [Countable E] (G : BondGraph V E) (p : unitInterval)
+    (x y z : V) : twoPoint G p x y * twoPoint G p y z ≤ twoPoint G p x z := by
+  have h := association_finiteWitness p {ω : Set E | Conn G ω x y}
+    {ω : Set E | Conn G ω y z}
+    (fun _ _ hω h => conn_mono G hω h) (fun _ _ hω h => conn_mono G hω h)
+    (finiteWitness_conn G x y) (finiteWitness_conn G y z)
+  exact h.trans (measureReal_mono (fun ω hω => conn_trans G ω hω.1 hω.2))
+
+ 
+theorem reachBall_add_split (G : BondGraph V E) (x : V) (n m : ℕ) {y : V}
+    (hy : y ∈ reachBall G x (n+m)) :
+    ∃ z ∈ reachBall G x n, y ∈ reachBall G z m := by
+  induction m generalizing y with
+  | zero => exact ⟨y,hy,Set.mem_singleton y⟩
+  | succ m ih =>
+    rcases hy with hy | hy
+    · obtain ⟨z,hz,hyz⟩ := ih hy
+      exact ⟨z,hz,Or.inl hyz⟩
+    · simp only [Set.mem_iUnion,Set.mem_ofPred_eq] at hy
+      obtain ⟨w,hw,hwy⟩ := hy
+      obtain ⟨z,hz,hwz⟩ := ih hw
+      refine ⟨z,hz,Or.inr ?_⟩
+      exact Set.mem_iUnion.mpr ⟨w,Set.mem_iUnion.mpr ⟨hwz,hwy⟩⟩
+
+ 
+theorem twoPoint_lower_on_multiples [Countable E] (G : BondGraph V E)
+    (p : unitInterval) (n : ℕ) (c : ℝ) (hc : 0 ≤ c)
+    (hball : ∀ x y, y ∈ reachBall G x n → c ≤ twoPoint G p x y)
+    (k : ℕ) (x y : V) (hy : y ∈ reachBall G x (n*k)) :
+    c^k ≤ twoPoint G p x y := by
+  induction k generalizing x y with
+  | zero =>
+    have he : y = x := by simpa [reachBall] using hy
+    subst y
+    simp [twoPoint_self]
+  | succ k ih =>
+    rw [Nat.mul_succ] at hy
+    obtain ⟨z,hz,hyz⟩ := reachBall_add_split G x (n*k) n hy
+    calc
+      c^(k+1) = c^k*c := pow_succ _ _
+      _ ≤ twoPoint G p x z * twoPoint G p z y :=
+        mul_le_mul (ih x z hz) (hball z y hyz) hc (twoPoint_nonneg G p x z)
+      _ ≤ twoPoint G p x y := twoPoint_mul_le G p x z y
+
+ 
+theorem reachBall_exponential_growth (G : BondGraph V E) (hloc : LocallyFinite G)
+    (hh : 0 ≤ hV G) (x : V) (n : ℕ) :
+    (1+hV G)^n ≤ ((reachBall G x n).ncard : ℝ) := by
+  classical
+  induction n with
+  | zero => simp [reachBall]
+  | succ n ih =>
+    let A := (finite_reachBall hloc x n).toFinset
+    have he : reachBall G x (n+1) = (A : Set V) ∪ vertexBoundary G A := by
+      ext y
+      simp only [reachBall,Set.mem_union,Finset.mem_coe,
+        vertexBoundary,Set.mem_ofPred_eq,Set.mem_iUnion]
+      simp only [A,Set.Finite.mem_toFinset]
+      constructor
+      · rintro (hy | ⟨z,hz,hzy⟩)
+        · exact Or.inl hy
+        · by_cases hy : y ∈ reachBall G x n
+          · exact Or.inl hy
+          · exact Or.inr ⟨hy,z,hz,hzy⟩
+      · rintro (hy | ⟨_,z,hz,hzy⟩)
+        · exact Or.inl hy
+        · exact Or.inr ⟨z,hz,hzy⟩
+    have hd : Disjoint (A : Set V) (vertexBoundary G A) := by
+      exact Set.disjoint_left.mpr (fun y hy hb => hb.1 hy)
+    have hc : (reachBall G x (n+1)).ncard = A.card + (vertexBoundary G A).ncard := by
+      rw [he,Set.ncard_union_eq hd A.finite_toSet (finite_vertexBoundary G hloc A),Set.ncard_coe_finset]
+    have hcard : A.card = (reachBall G x n).ncard :=
+      (Set.ncard_eq_toFinset_card _ (finite_reachBall hloc x n)).symm
+    have hs := hV_mul_card_le G A
+    rw [hcard] at hs
+    rw [hc,hcard,Nat.cast_add,pow_succ]
+    nlinarith [mul_le_mul_of_nonneg_right ih (show 0 ≤ 1+hV G by linarith)]
+
+ 
+
+theorem exists_subcritical_ball_lower [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (c : ℝ) (hc : 0 < c) (hcrit : ∀ x y, c ≤ twoPoint G (pc G) x y)
+    (n : ℕ) : ∃ p : unitInterval, p < pc G ∧
+      ∀ x y, y ∈ reachBall G x n → c/2 ≤ twoPoint G p x y := by
+  classical
+  obtain ⟨p,hp,hpc,hlim⟩ := exists_seq_strictMono_tendsto' (pc_pos G hloc hqt)
+  obtain ⟨s,hs⟩ := hqt
+  have hev : ∀ᶠ k in atTop, ∀ x ∈ s, ∀ y ∈ reachBall G x n,
+      c/2 ≤ twoPoint G (p k) x y := by
+    apply (eventually_all_finset s).mpr
+    intro x hx
+    apply (finite_reachBall hloc x n).eventually_all.mpr
+    intro y hy
+    have ht := twoPoint_tendsto_of_monotone G p hp.monotone (pc G)
+      (fun k => (hpc k).2.le) hlim x y
+    have hlt : c/2 < twoPoint G (pc G) x y := lt_of_lt_of_le (by linarith) (hcrit x y)
+    exact (ht.eventually (eventually_gt_nhds hlt)).mono (fun k hk => hk.le)
+  obtain ⟨k,hk⟩ := hev.exists
+  refine ⟨p k,(hpc k).2,?_⟩
+  intro x y hy
+  obtain ⟨o,ho,a,b,ha,hax⟩ := hs x
+  have hy' : a.symm y ∈ reachBall G o n := by
+    have hi := mem_reachBall_aut G (isAut_symm G ha) hy
+    simpa [←hax] using hi
+  have he := twoPoint_aut G (p k) ha o (a.symm y)
+  rw [hax,a.apply_symm_apply] at he
+  rw [he]
+  exact hk o ho (a.symm y) hy'
+
+ 
+
+theorem no_uniform_positive_twoPoint_critical [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (hh : 0 < hV G) :
+    ¬ ∃ c : ℝ, 0 < c ∧ ∀ x y, c ≤ twoPoint G (pc G) x y := by
+  rintro ⟨c,hc,hcrit⟩
+  obtain ⟨n,hn⟩ := pow_unbounded_of_one_lt (2/c) (show 1 < 1+hV G by linarith)
+  have hn' : 1 < (c/2)*(1+hV G)^n := by
+    have := (div_lt_iff₀ hc).mp hn
+    nlinarith
+  obtain ⟨p,hp,hball⟩ := exists_subcritical_ball_lower G hloc hqt c hc hcrit n
+  obtain ⟨C,hC,hrow⟩ := subcritical_row_bound G hloc hqt p hp
+  obtain ⟨k,hk⟩ := pow_unbounded_of_one_lt C hn'
+  let x : V := Classical.arbitrary V
+  let A := (finite_reachBall hloc x (n*k)).toFinset
+  have hcard : (reachBall G x (n*k)).ncard = A.card := Set.ncard_eq_toFinset_card _ _
+  have hsum : (A.card:ℝ)*(c/2)^k ≤ ∑ y ∈ A, twoPoint G p x y := by
+    calc
+      _ = ∑ y ∈ A, (c/2)^k := by simp
+      _ ≤ _ := Finset.sum_le_sum fun y hy =>
+        twoPoint_lower_on_multiples G p n (c/2) (by linarith) hball k x y
+          (by simpa [A] using hy)
+  have hg := reachBall_exponential_growth G hloc hh.le x (n*k)
+  rw [hcard] at hg
+  have hpow : ((c/2)*(1+hV G)^n)^k ≤ (A.card:ℝ)*(c/2)^k := by
+    calc
+      _ = (1+hV G)^(n*k)*(c/2)^k := by rw [mul_pow,pow_mul]; ring
+      _ ≤ _ := mul_le_mul_of_nonneg_right hg (pow_nonneg (half_pos hc).le k)
+  exact (not_lt_of_ge (hpow.trans (hsum.trans (hrow x A)))) hk
+
+ 
+
+theorem not_unique_infinite_cluster_critical [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
+    (hqt : QuasiTransitive G) (hh : 0 < hV G) :
+    law (pc G) {ω | ∃! C, C ∈ infiniteClusters G ω} ≠ 1 := by
+  intro hu
+  have hp := pc_pos G hloc hqt
+  have hpos : 0 < law (pc G) {ω | (infiniteClusters G ω).Nonempty} :=
+    lt_of_lt_of_le (show 0 < law (pc G) {ω | ∃! C, C ∈ infiniteClusters G ω} by
+      rw [hu]; exact zero_lt_one)
+      (measure_mono fun ω hω => ⟨hω.choose,hω.choose_spec.1⟩)
+  obtain ⟨θ,hθ,hroot⟩ := uniform_survival_pos G hconn hqt (pc G) hp hpos
+  have hue : ∀ᵐ ω ∂law (pc G), ∃! C, C ∈ infiniteClusters G ω :=
+    (mem_ae_iff_prob_eq_one (measurableSet_unique_infinite_cluster G)).mpr hu
+  apply no_uniform_positive_twoPoint_critical G hloc hqt hh
+  refine ⟨θ*θ,mul_pos hθ hθ,fun x y => ?_⟩
+  refine (mul_le_mul (hroot x) (hroot y) hθ.le measureReal_nonneg).trans
+    ((association_infinite_cluster G (pc G) x y).trans ?_)
+  apply ENNReal.toReal_mono (measure_ne_top _ _)
+  apply measure_mono_ae
+  filter_upwards [hue] with ω hω
+  intro hi
+  obtain ⟨r,_,hconnr⟩ := (unique_infinite_cluster_iff G ω).mp hω
+  exact conn_trans G ω (conn_symm G ω (hconnr x hi.1)) (hconnr y hi.2)
+
+end Percolation
+
+ 
+
+open MeasureTheory ProbabilityTheory Filter Topology
+open scoped ENNReal
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem critical_finite_of_transport [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
+    (hqt : QuasiTransitive G) (hh : 0 < hV G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
+    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1) :
+    ∀ x, ∀ᵐ ω ∂law (pc G), (cluster G ω x).Finite := by
+  let c : ℝ := ∑ i, 1/weight i
+  have hc : 0 ≤ c := Finset.sum_nonneg (fun i _ => (one_div_pos.mpr (hw i)).le)
+  have hmin (i : ι) : 1 ≤ c*weight i := by
+    apply (div_le_iff₀ (hw i)).mp
+    exact Finset.single_le_sum (fun j _ => (one_div_pos.mpr (hw j)).le) (Finset.mem_univ i)
+  obtain ⟨d,_,hd⟩ := exists_degree_bound G hqt
+  have hdn (x : V) := (incident_ncard_le_degree G hloc x).trans (hd x)
+  let A : Set (Set E) := {ω | ∀ a b, Adj G a b →
+    (cluster G ω a).Infinite → (cluster G ω b).Infinite → Conn G ω a b}
+  have hA : MeasurableSet A := by
+    simp only [A,Set.ofPred_forall]
+    apply MeasurableSet.iInter
+    intro a
+    apply MeasurableSet.iInter
+    intro b
+    apply MeasurableSet.iInter
+    intro _
+    simp only [imp_iff_not_or,Set.ofPred_or]
+    exact (measurableSet_infinite_cluster G a).compl.union
+      ((measurableSet_infinite_cluster G b).compl.union (measurableSet_conn G a b))
+  have hae : ∀ᵐ ω ∂law (pc G), ω ∈ A := by
+    change ∀ᵐ ω ∂law (pc G), ∀ a b, Adj G a b →
+      (cluster G ω a).Infinite → (cluster G ω b).Infinite → Conn G ω a b
+    rw [ae_all_iff]
+    intro a
+    rw [ae_all_iff]
+    intro b
+    rw [ae_all_iff]
+    rintro ⟨e,he⟩
+    exact critical_adjacent_infinite_connected G hconn hloc hqt root weight Γ hΓ htr hcov
+      (fun i => (hw i).le) hw1 c hc hmin d hdn e a b he
+  have haeF : ∀ᵐ ω ∂law (pc G), ∀ F : Finset E, ω ∪ (F:Set E) ∈ A := by
+    rw [ae_all_iff]
+    intro F
+    exact ae_union_finset (pc G) (pc_pos G hloc hqt) A hA hae F
+  have hat : ∀ᵐ ω ∂law (pc G), ∀ x y,
+      (cluster G ω x).Infinite → (cluster G ω y).Infinite → Conn G ω x y := by
+    filter_upwards [haeF] with ω hω
+    exact atMostOne_of_finite_openings G hconn ω hω
+  have hex := infinite_cluster_zero_one G (pc G)
+  have he : (labelLaw : Measure (E → unitInterval))
+      {labels | ∃ x, (cluster G (openEdges labels (pc G)) x).Infinite} =
+      law (pc G) {ω | ∃ x, (cluster G ω x).Infinite} := by
+    rw [←map_openEdges (pc G),Measure.map_apply (measurable_openEdges (pc G))
+      (measurableSet_exists_infinite_cluster G)]
+    rfl
+  rw [he] at hex
+  have hz : law (pc G) {ω | ∃ x, (cluster G ω x).Infinite} = 0 := by
+    rcases hex with hzero | hone
+    · exact hzero
+    · exfalso
+      apply not_unique_infinite_cluster_critical G hconn hloc hqt hh
+      apply (mem_ae_iff_prob_eq_one (measurableSet_unique_infinite_cluster G)).mp
+      have haex : ∀ᵐ ω ∂law (pc G), ∃ x, (cluster G ω x).Infinite :=
+        (mem_ae_iff_prob_eq_one (measurableSet_exists_infinite_cluster G)).mpr hone
+      filter_upwards [hat,haex] with ω hω hex
+      obtain ⟨x,hx⟩ := hex
+      exact (unique_infinite_cluster_iff G ω).mpr ⟨x,hx,fun y hy => hω x y hx hy⟩
+  intro x
+  rw [ae_iff]
+  apply measure_mono_null (t := {ω | ∃ y, (cluster G ω y).Infinite}) ?_ hz
+  intro ω hω
+  exact ⟨x,hω⟩
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Criticality/FullFiniteness.lean b/lean/OAI/Probability/UnimodularPercolation/Criticality/FullFiniteness.lean
index 5306cba0ddc12d3e9d7b617d33162fb079922d07..afb266c8732f18df90dcbcf4076bdefc97fcd6d6 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Criticality/FullFiniteness.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Criticality/FullFiniteness.lean
@@ -1,80 +1,80 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Main
-import OAI.Probability.UnimodularPercolation.Criticality.ClusterCount
-import OAI.Probability.UnimodularPercolation.Encounter.Surgery
-import OAI.Probability.UnimodularPercolation.Encounter.ZeroIntensity
-
-namespace OAI
-
-/-! Almost-sure critical cluster finiteness from the full root transport. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-open MeasureTheory
-universe u v w
-variable {V : Type u} {E : Type v} {ι : Type w}
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem finiteOverwrite_preimage_zero [Countable E] (p : unitInterval)
-    (hp : 0 < p) (hp1 : p < 1) (F P : Finset E) (hPF : P ⊆ F)
-    (B : Set (Set E)) (hB : MeasurableSet B) (hz : law p B = 0) :
-    law p ((finiteOverwrite F P) ⁻¹' B) = 0 := by
-  by_contra hn
-  have hpos := finiteOverwrite_pos p hp hp1 F P hPF
-    ((finiteOverwrite F P) ⁻¹' B) B hB (pos_iff_ne_zero.mpr hn) (fun _ h => h)
-  rw [hz] at hpos
-  exact lt_irrefl _ hpos
-
-theorem critical_finite_count_ae_of_transport [Infinite V] [Fintype ι]
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
-    (hqt : QuasiTransitive G) (hh : 0 < hV G)
-    (root : ι → V) (weight : ι → ℝ)
-    (htr : RootTransport root weight (vertexAut G)) (hw : ∀ i, 0 < weight i)
-    (hw1 : ∑ i, weight i = 1)
-    (hcov : ∀ x, ∃ i, ∃ a ∈ vertexAut G, a (root i) = x) :
-    ∀ᵐ ω ∂law (pc G), (infiniteClusters G ω).Finite := by
-  let : Countable V := countable_vertices hloc hconn
-  let : Countable E := countable_edges hloc
-  have hp := pc_pos G hloc hqt
-  have hp1 := pc_lt_one_of_nonamenable G hconn hloc hqt hh
-  have he (F P : Finset E) (x : V) :
-      ∀ᵐ ω ∂law (pc G), P ⊆ F → ¬ IsEncounter G (finiteOverwrite F P ω) x := by
-    by_cases hPF : P ⊆ F
-    · have hz := finiteOverwrite_preimage_zero (pc G) hp hp1 F P hPF
-        {ω | IsEncounter G ω x} (measurableSet_isEncounter G x)
-        (critical_encounter_zero G hconn hloc hp root weight htr hw hw1 hcov x)
-      apply ae_iff.mpr
-      simpa only [hPF,true_implies,not_not,Set.preimage_ofPred_eq] using hz
-    · filter_upwards [] with ω
-      exact fun h => False.elim (hPF h)
-  have hall : ∀ᵐ ω ∂law (pc G), ∀ F P : Finset E, ∀ x : V,
-      P ⊆ F → ¬ IsEncounter G (finiteOverwrite F P ω) x :=
-    ae_all_iff.mpr fun F => ae_all_iff.mpr fun P => ae_all_iff.mpr (he F P)
-  filter_upwards [hall] with ω hω
-  by_contra hinf
-  obtain ⟨z,hz,hsep⟩ := three_of_infinite_count G ω hinf
-  obtain ⟨F,P,hPF,x,hx⟩ := encounter_surgery_of_three G hconn hloc ω z hz hsep
-  exact hω F P x hPF hx
-
-theorem critical_finite_ae_of_transport [Infinite V] [Fintype ι]
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
-    (hqt : QuasiTransitive G) (hh : 0 < hV G)
-    (root : ι → V) (weight : ι → ℝ)
-    (htr : RootTransport root weight (vertexAut G)) (hw : ∀ i, 0 < weight i)
-    (hw1 : ∑ i, weight i = 1)
-    (hcov : ∀ x, ∃ i, ∃ a ∈ vertexAut G, a (root i) = x) :
-    ∀ᵐ ω ∂law (pc G), ∀ x, (cluster G ω x).Finite := by
-  filter_upwards [critical_finite_count_ae_of_transport G hconn hloc hqt hh
-    root weight htr hw hw1 hcov,critical_finite_count_empty_ae G hconn hloc hqt hh] with ω hf he
-  have hempty := he hf
-  intro x
-  by_contra hx
-  have hc : cluster G ω x ∈ infiniteClusters G ω := ⟨hx,x,rfl⟩
-  rw [hempty] at hc
-  exact hc
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Main
+import OAI.Probability.UnimodularPercolation.Criticality.ClusterCount
+import OAI.Probability.UnimodularPercolation.Encounter.Surgery
+import OAI.Probability.UnimodularPercolation.Encounter.ZeroIntensity
+
+namespace OAI
+
+/-! Almost-sure critical cluster finiteness from the full root transport. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+open MeasureTheory
+universe u v w
+variable {V : Type u} {E : Type v} {ι : Type w}
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem finiteOverwrite_preimage_zero [Countable E] (p : unitInterval)
+    (hp : 0 < p) (hp1 : p < 1) (F P : Finset E) (hPF : P ⊆ F)
+    (B : Set (Set E)) (hB : MeasurableSet B) (hz : law p B = 0) :
+    law p ((finiteOverwrite F P) ⁻¹' B) = 0 := by
+  by_contra hn
+  have hpos := finiteOverwrite_pos p hp hp1 F P hPF
+    ((finiteOverwrite F P) ⁻¹' B) B hB (pos_iff_ne_zero.mpr hn) (fun _ h => h)
+  rw [hz] at hpos
+  exact lt_irrefl _ hpos
+
+theorem critical_finite_count_ae_of_transport [Infinite V] [Fintype ι]
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
+    (hqt : QuasiTransitive G) (hh : 0 < hV G)
+    (root : ι → V) (weight : ι → ℝ)
+    (htr : RootTransport root weight (vertexAut G)) (hw : ∀ i, 0 < weight i)
+    (hw1 : ∑ i, weight i = 1)
+    (hcov : ∀ x, ∃ i, ∃ a ∈ vertexAut G, a (root i) = x) :
+    ∀ᵐ ω ∂law (pc G), (infiniteClusters G ω).Finite := by
+  let : Countable V := countable_vertices hloc hconn
+  let : Countable E := countable_edges hloc
+  have hp := pc_pos G hloc hqt
+  have hp1 := pc_lt_one_of_nonamenable G hconn hloc hqt hh
+  have he (F P : Finset E) (x : V) :
+      ∀ᵐ ω ∂law (pc G), P ⊆ F → ¬ IsEncounter G (finiteOverwrite F P ω) x := by
+    by_cases hPF : P ⊆ F
+    · have hz := finiteOverwrite_preimage_zero (pc G) hp hp1 F P hPF
+        {ω | IsEncounter G ω x} (measurableSet_isEncounter G x)
+        (critical_encounter_zero G hconn hloc hp root weight htr hw hw1 hcov x)
+      apply ae_iff.mpr
+      simpa only [hPF,true_implies,not_not,Set.preimage_ofPred_eq] using hz
+    · filter_upwards [] with ω
+      exact fun h => False.elim (hPF h)
+  have hall : ∀ᵐ ω ∂law (pc G), ∀ F P : Finset E, ∀ x : V,
+      P ⊆ F → ¬ IsEncounter G (finiteOverwrite F P ω) x :=
+    ae_all_iff.mpr fun F => ae_all_iff.mpr fun P => ae_all_iff.mpr (he F P)
+  filter_upwards [hall] with ω hω
+  by_contra hinf
+  obtain ⟨z,hz,hsep⟩ := three_of_infinite_count G ω hinf
+  obtain ⟨F,P,hPF,x,hx⟩ := encounter_surgery_of_three G hconn hloc ω z hz hsep
+  exact hω F P x hPF hx
+
+theorem critical_finite_ae_of_transport [Infinite V] [Fintype ι]
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
+    (hqt : QuasiTransitive G) (hh : 0 < hV G)
+    (root : ι → V) (weight : ι → ℝ)
+    (htr : RootTransport root weight (vertexAut G)) (hw : ∀ i, 0 < weight i)
+    (hw1 : ∑ i, weight i = 1)
+    (hcov : ∀ x, ∃ i, ∃ a ∈ vertexAut G, a (root i) = x) :
+    ∀ᵐ ω ∂law (pc G), ∀ x, (cluster G ω x).Finite := by
+  filter_upwards [critical_finite_count_ae_of_transport G hconn hloc hqt hh
+    root weight htr hw hw1 hcov,critical_finite_count_empty_ae G hconn hloc hqt hh] with ω hf he
+  have hempty := he hf
+  intro x
+  by_contra hx
+  have hc : cluster G ω x ∈ infiniteClusters G ω := ⟨hx,x,rfl⟩
+  rw [hempty] at hc
+  exact hc
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Criticality/FullOperator.lean b/lean/OAI/Probability/UnimodularPercolation/Criticality/FullOperator.lean
index 2975a04307d0ef1751cdcdc2b972b9b02ce8ee04..db1dac7436ea315b8b411aa7841990a19072dd52 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Criticality/FullOperator.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Criticality/FullOperator.lean
@@ -1,59 +1,59 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Main
-import OAI.Probability.UnimodularPercolation.Criticality.FullFiniteness
-import OAI.Probability.UnimodularPercolation.Transport.InvariantRoot
-
-namespace OAI
-
-/-! Critical operator boundedness for the full unimodular automorphism group. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-open MeasureTheory
-universe u v
-variable {V : Type u} {E : Type v}
-variable [TopologicalSpace V] [DiscreteTopology V]
-
-theorem critical_bounded_of_graphUnimodular [Infinite V] (G : BondGraph V E)
-    (hinj : Function.Injective G) (hconn : Connected G) (hloc : LocallyFinite G)
-    (hqt : QuasiTransitive G) (hh : 0 < hV G) (x₀ : V)
-    (hu : GraphUnimodular G hinj hloc hconn x₀) : OperatorBounded G (pc G) := by
-  obtain ⟨n,root,weight,_,hw,hw1,htr,hcov⟩ :=
-    exists_rootTransport_of_unimodular G hinj hloc hconn hqt x₀ hu
-  have hf := critical_finite_ae_of_transport G hconn hloc hqt hh
-    root weight htr hw hw1 hcov
-  apply critical_bounded_of_transport_finiteness G hinj hconn hloc hqt hh
-    root weight htr hcov hw hw1
-  intro i
-  filter_upwards [hf] with ω hω
-  exact hω (root i)
-
-end Percolation
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-noncomputable def FullAutUnimodular (G : BondGraph V E)
-    (hinj : Function.Injective G) (hloc : LocallyFinite G) (hconn : Connected G)
-    (x₀ : V) : Prop := by
-  letI : TopologicalSpace V := ⊥
-  letI : DiscreteTopology V := ⟨rfl⟩
-  exact GraphUnimodular G hinj hloc hconn x₀
-
-theorem critical_operator_fullAut [Infinite V] (G : BondGraph V E)
-    (hinj : Function.Injective G) (_hloop : ∀ x, ¬Adj G x x)
-    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (hh : 0 < hV G) (x₀ : V)
-    (hu : FullAutUnimodular G hinj hloc hconn x₀) :
-    operatorNorm G (pc G) < ⊤ := by
-  let : TopologicalSpace V := ⊥
-  let : DiscreteTopology V := ⟨rfl⟩
-  apply (operatorNorm_lt_top_iff G (pc G)).mpr
-  exact critical_bounded_of_graphUnimodular G hinj hconn hloc hqt hh x₀ hu
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Main
+import OAI.Probability.UnimodularPercolation.Criticality.FullFiniteness
+import OAI.Probability.UnimodularPercolation.Transport.InvariantRoot
+
+namespace OAI
+
+/-! Critical operator boundedness for the full unimodular automorphism group. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+open MeasureTheory
+universe u v
+variable {V : Type u} {E : Type v}
+variable [TopologicalSpace V] [DiscreteTopology V]
+
+theorem critical_bounded_of_graphUnimodular [Infinite V] (G : BondGraph V E)
+    (hinj : Function.Injective G) (hconn : Connected G) (hloc : LocallyFinite G)
+    (hqt : QuasiTransitive G) (hh : 0 < hV G) (x₀ : V)
+    (hu : GraphUnimodular G hinj hloc hconn x₀) : OperatorBounded G (pc G) := by
+  obtain ⟨n,root,weight,_,hw,hw1,htr,hcov⟩ :=
+    exists_rootTransport_of_unimodular G hinj hloc hconn hqt x₀ hu
+  have hf := critical_finite_ae_of_transport G hconn hloc hqt hh
+    root weight htr hw hw1 hcov
+  apply critical_bounded_of_transport_finiteness G hinj hconn hloc hqt hh
+    root weight htr hcov hw hw1
+  intro i
+  filter_upwards [hf] with ω hω
+  exact hω (root i)
+
+end Percolation
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+noncomputable def FullAutUnimodular (G : BondGraph V E)
+    (hinj : Function.Injective G) (hloc : LocallyFinite G) (hconn : Connected G)
+    (x₀ : V) : Prop := by
+  letI : TopologicalSpace V := ⊥
+  letI : DiscreteTopology V := ⟨rfl⟩
+  exact GraphUnimodular G hinj hloc hconn x₀
+
+theorem critical_operator_fullAut [Infinite V] (G : BondGraph V E)
+    (hinj : Function.Injective G) (_hloop : ∀ x, ¬Adj G x x)
+    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (hh : 0 < hV G) (x₀ : V)
+    (hu : FullAutUnimodular G hinj hloc hconn x₀) :
+    operatorNorm G (pc G) < ⊤ := by
+  let : TopologicalSpace V := ⊥
+  let : DiscreteTopology V := ⟨rfl⟩
+  apply (operatorNorm_lt_top_iff G (pc G)).mpr
+  exact critical_bounded_of_graphUnimodular G hinj hconn hloc hqt hh x₀ hu
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Criticality/Operator.lean b/lean/OAI/Probability/UnimodularPercolation/Criticality/Operator.lean
index 7050da574eb4dd7ab9a53c8d7572b46ade2889aa..f31d5ea3d15f271bcfc8639f196ef5d6de727886 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Criticality/Operator.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Criticality/Operator.lean
@@ -1,274 +1,274 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Criticality.Finiteness
-import OAI.Probability.UnimodularPercolation.Criticality.ScaleLimits
-import OAI.Probability.UnimodularPercolation.Deformation.FiniteScale
-import OAI.Probability.UnimodularPercolation.Graph.Automorphisms
-
-namespace OAI
-
-/-! Critical boundedness of the percolation operator under unimodularity and simple edge indexing. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-open MeasureTheory Filter
-open scoped Topology
-
- 
-noncomputable def criticalApproach (q : unitInterval) (n : ℕ) : unitInterval :=
-  ⟨(q:ℝ)-(q:ℝ)/2*Real.exp (-(n:ℝ)),by
-    have he : Real.exp (-(n:ℝ)) ≤ 1 := Real.exp_le_one_iff.mpr (neg_nonpos.mpr (Nat.cast_nonneg n))
-    have hq := q.2
-    have hp := Real.exp_pos (-(n:ℝ))
-    constructor <;> nlinarith only [he,hq.1,hq.2,hp]⟩
-
-theorem criticalApproach_half (q : unitInterval) (n : ℕ) :
-    (q:ℝ)/2 ≤ (criticalApproach q n : ℝ) := by
-  have he : Real.exp (-(n:ℝ)) ≤ 1 := Real.exp_le_one_iff.mpr (neg_nonpos.mpr (Nat.cast_nonneg n))
-  change _ ≤ (q:ℝ)-(q:ℝ)/2*Real.exp (-(n:ℝ))
-  nlinarith only [he,q.2.1]
-
-theorem criticalApproach_lt (q : unitInterval) (hq : 0 < (q:ℝ)) (n : ℕ) :
-    criticalApproach q n < q := by
-  change (q:ℝ)-(q:ℝ)/2*Real.exp (-(n:ℝ)) < q
-  exact sub_lt_self _ (mul_pos (half_pos hq) (Real.exp_pos _))
-
-theorem criticalApproach_gap (q : unitInterval) (n : ℕ) :
-    (q:ℝ)-(criticalApproach q n:ℝ) = (q:ℝ)/2*Real.exp (-(n:ℝ)) := by
-  change (q:ℝ)-((q:ℝ)-(q:ℝ)/2*Real.exp (-(n:ℝ))) = _
-  ring
-
- 
-
-theorem log_sublinear_of_geometric_gap (f : ℕ → ℝ) (p₀ : ℝ) (hp₀ : 0 < p₀)
-    (hf : ∀ n, 1 ≤ f n)
-    (hb : ∀ η : ℝ, 0 < η → ∃ K : ℝ, 0 < K ∧
-      ∀ n : ℕ, f n ≤ K*(p₀*Real.exp (-(n:ℝ)))^(-η)) :
-    Tendsto (fun n => Real.log (f n)/(n:ℝ)) atTop (𝓝 0) := by
-  apply log_sublinear_of_subexp f hf
-  intro η hη
-  obtain ⟨K,hK,hbK⟩ := hb η hη
-  refine ⟨K*p₀^(-η),mul_pos hK (Real.rpow_pos_of_pos hp₀ _),fun n => ?_⟩
-  have he : (p₀*Real.exp (-(n:ℝ)))^(-η) = p₀^(-η)*Real.exp (η*(n:ℝ)) := by
-    rw [Real.mul_rpow hp₀.le (Real.exp_pos _).le,←Real.exp_mul]
-    congr 2
-    ring
-  simpa only [he,mul_assoc] using hbK n
-
-universe u v
-variable {V : Type u} {E : Type v}
-section AnalyticCritical
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Countable V] [Countable E] [Infinite V] [Nonempty E]
-variable {ι : Type*} [Fintype ι] [Nonempty ι]
-
- 
-
-theorem critical_bounded_of_fixed_data
-    (G : BondGraph V E) (hG : Function.Injective G)
-    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G) (hh : 0 < hV G)
-    (a : ℝ) (ha : 0 < a) (root : ι → V) (weight : ι → ℝ)
-    (htr : RootTransport root weight (vertexAut G))
-    (hcov : ∀ x, ∃ r, ∃ g ∈ vertexAut G, g (root r) = x)
-    (hw : ∀ r, 0 < weight r) (hw1 : ∑ r, weight r = 1)
-    (c : ℝ) (hc : 0 < c) (hmin : ∀ r, 1 ≤ c*weight r)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re) (hρ : ‖P‖ < 1)
-    (d : ℕ) (hd : ∀ z, degree G z ≤ d)
-    (hpcrit : 0 < (pc G:ℝ)) (hpcrit₁ : (pc G:ℝ) < 1)
-    (hfin : ∀ i, ∀ᵐ ω ∂law (pc G), (cluster G ω (root i)).Finite) :
-    OperatorBounded G (pc G) := by
-  by_contra hu
-  obtain ⟨C,hC,J,hJ,C₁,hC₁,k₀,hk₀,ε,M,hε,_,hbad⟩ :=
-    exists_finite_scale_obstruction G hG hconn hloc hqt a ha root weight htr hcov hw hw1 c hc hmin
-      P hpos hsym hP hdiag hρ d hd hpcrit hpcrit₁
-  let p := criticalApproach (pc G)
-  have hpnear (n) : (pc G:ℝ)/2 ≤ (p n:ℝ) := criticalApproach_half (pc G) n
-  have hp (n) : p n < pc G := criticalApproach_lt (pc G) hpcrit n
-  have hpzero (n) : 0 < (p n:ℝ) := (half_pos hpcrit).trans_le (hpnear n)
-  let A := fun n => Classical.choose (operatorBounded_of_lt_pc G hloc hqt (p n) (hp n))
-  have hA (n) : HasKernel G (p n) (A n) := Classical.choose_spec (operatorBounded_of_lt_pc G hloc hqt (p n) (hp n))
-  let m := fun n => ‖A n‖
-  let R := fun n => maxSusceptibility G (p n)/m n
-  let Q := fun n => rootTriangle G root weight (p n:ℝ)
-  have hR (n) : 1 ≤ R n := (one_le_div (zero_lt_one.trans_le ((hA n).norm_one_le G (p n) (A n)))).mpr
-    ((hA n).norm_le_maxSusceptibility G hloc hqt (p n) (hp n) (A n))
-  have hQ (n) : 1 ≤ Q n := rootTriangle_one_le G hloc hqt root weight (fun i => (hw i).le) hw1 (p n) (hp n)
-  have hlogR : Tendsto (fun n => Real.log (R n)/(n:ℝ)) atTop (𝓝 0) := by
-    apply log_sublinear_of_geometric_gap R ((pc G:ℝ)/2) (half_pos hpcrit) hR
-    intro η hη
-    obtain ⟨K,hK,hRK⟩ := critical_normRatio_subpower G hloc hconn hqt hh root weight (vertexAut G)
-      (Set.Subset.refl _) htr hcov hw hw1 hpcrit hpcrit₁ hfin hu η hη
-    refine ⟨K,hK,fun n => ?_⟩
-    simpa only [p,criticalApproach_gap] using (hRK (p n) (hpnear n) (hp n) (A n) (hA n)).2
-  have hlogQ : Tendsto (fun n => Real.log (Q n)/(n:ℝ)) atTop (𝓝 0) := by
-    apply log_sublinear_of_geometric_gap Q ((pc G:ℝ)/2) (half_pos hpcrit) hQ
-    intro η hη
-    obtain ⟨K,hK,hQK⟩ := critical_rootTriangle_subpower G hloc hconn hqt hh root weight (vertexAut G)
-      (Set.Subset.refl _) htr hcov hw hw1 hpcrit hpcrit₁ hfin η hη
-    refine ⟨K,hK,fun n => ?_⟩
-    simpa only [p,criticalApproach_gap] using hQK (p n) (hpnear n) (hp n)
-  obtain ⟨cm,hcm,hm⟩ := critical_norm_inverse_lower G hloc hqt hpcrit₁ hu
-  have hmlower (n : ℕ) : (cm/((pc G:ℝ)/2))*Real.exp (n:ℝ) ≤ m n := by
-    have hh := hm (p n) (hpzero n) (hp n) (A n) (hA n)
-    rw [show (pc G:ℝ)-(p n:ℝ) = (pc G:ℝ)/2*Real.exp (-(n:ℝ)) from criticalApproach_gap (pc G) n,
-      Real.exp_neg,div_mul_eq_div_div,div_inv_eq_mul] at hh
-    exact hh
-  have hρpos : 0 < ‖P‖ := by
-    let x : V := Classical.arbitrary V
-    have hh := (hdiag x).trans ((Complex.re_le_norm _).trans (matrixCoeff_norm_le P x x))
-    linarith only [hh]
-  obtain ⟨n,k,B,hk,hB,hsmall,htime,herror,hlarge⟩ := exists_corridor_scales
-    ‖P‖ ((pc G:ℝ)/2) C C₁ ε M (cm/((pc G:ℝ)/2)) J k₀ hρpos hρ (half_pos hpcrit) hC hC₁ hε
-    (div_pos hcm (half_pos hpcrit)) hJ m R Q hR hQ hmlower hlogR hlogQ
-  exact hbad (p n) (hpnear n) (hp n) (A n) (hA n) k hk B hB hsmall htime herror hlarge
-
-end AnalyticCritical
-
- 
-
-theorem nonempty_bonds_of_connected [Infinite V] (G : BondGraph V E) (hconn : Connected G) : Nonempty E := by
-  classical
-  by_contra he
-  let : IsEmpty E := not_nonempty_iff.mp he
-  obtain ⟨x,y,hxy⟩ := exists_pair_ne V
-  have hh := conn_univ_of_connected G hconn x y
-  have heq : (Set.univ : Set E) = ∅ := Set.eq_empty_of_isEmpty _
-  rw [heq,conn_empty_iff] at hh
-  exact hxy hh
-
- 
-
-theorem critical_bounded_of_transport_finiteness [Infinite V]
-    (G : BondGraph V E) (hG : Function.Injective G)
-    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G) (hh : 0 < hV G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (htr : RootTransport root weight (vertexAut G))
-    (hcov : ∀ x, ∃ r, ∃ g ∈ vertexAut G, g (root r) = x)
-    (hw : ∀ r, 0 < weight r) (hw1 : ∑ r, weight r = 1)
-    (hfin : ∀ i, ∀ᵐ ω ∂law (pc G), (cluster G ω (root i)).Finite) :
-    OperatorBounded G (pc G) := by
-  classical
-  let : Countable V := countable_vertices hloc hconn
-  let : Countable E := countable_edges hloc
-  let : Nonempty E := nonempty_bonds_of_connected G hconn
-  let : Nonempty ι := ⟨(hcov (Classical.arbitrary V)).choose⟩
-  let c := ∑ i, (weight i)⁻¹
-  have hc : 0 < c := Finset.sum_pos (fun i _ => inv_pos.mpr (hw i)) Finset.univ_nonempty
-  have hmin (i) : 1 ≤ c*weight i := by
-    have hi : (weight i)⁻¹ ≤ c := Finset.single_le_sum
-      (fun j _ => (inv_pos.mpr (hw j)).le) (Finset.mem_univ i)
-    simpa only [inv_mul_cancel₀ (hw i).ne'] using mul_le_mul_of_nonneg_right hi (hw i).le
-  obtain ⟨d,P,hdpos,hd,hρ,hPs,hP⟩ := exists_strict_walk_operator G hloc hqt hh
-  let a := (1:ℝ)/(2*d)
-  have hdr : (0:ℝ) < d := Nat.cast_pos.mpr hdpos
-  have ha : 0 < a := by dsimp [a]; positivity
-  have had : a*d = 1/2 := by dsimp [a]; field_simp
-  have hPe : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ) := hP
-  have hPpos : PositiveRealMatrix P := by
-    intro x y
-    rw [hPe]
-    refine ⟨rfl,?_⟩
-    exact walkKernel_nonneg G hloc d hd a ha.le (by rw [had]; norm_num) x y
-  have hdiag (x) : 1/2 ≤ (matrixCoeff P x x).re := by
-    rw [hPe,Complex.ofReal_re]
-    exact walkKernel_lazy G hloc d hd a ha.le had.le x
-  exact critical_bounded_of_fixed_data G hG hconn hloc hqt hh a ha root weight htr hcov hw hw1 c hc hmin
-    P hPpos hPs.isSelfAdjoint hPe hdiag hρ d hd (pc_pos G hloc hqt)
-    (pc_lt_one_of_nonamenable G hconn hloc hqt hh) hfin
-
-end Percolation
-
- 
-
- 
-
-namespace Percolation
-open MeasureTheory
-universe u v
-variable {V : Type u} {E : Type v}
-
- 
-
-noncomputable def AutUnimodular [Nonempty V] (G : BondGraph V E)
-    (hG : Function.Injective G) (hconn : Connected G) (hloc : LocallyFinite G) : Prop :=
-  letI : TopologicalSpace V := ⊥
-  letI : DiscreteTopology V := ⟨rfl⟩
-  (graphHaar G hG hconn hloc).IsMulRightInvariant
-
- 
-
-theorem critical_bounded_of_unimodular_finiteness [Infinite V]
-    (G : BondGraph V E) (hG : Function.Injective G)
-    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (hh : 0 < hV G) (huni : AutUnimodular G hG hconn hloc)
-    (hfin : ∀ x, ∀ᵐ ω ∂law (pc G), (cluster G ω x).Finite) :
-    OperatorBounded G (pc G) := by
-  classical
-  let : TopologicalSpace V := ⊥
-  let : DiscreteTopology V := ⟨rfl⟩
-  let : Countable V := countable_vertices hloc hconn
-  let : MeasurableSpace V := borel V
-  let : BorelSpace V := ⟨rfl⟩
-  obtain ⟨ι,hι,hn,root,weight,htr,hcov,hw,hw1⟩ :=
-    exists_rootData_of_unimodular G hG hconn hloc hqt huni
-  let : Fintype ι := hι
-  exact critical_bounded_of_transport_finiteness G hG hconn hloc hqt hh
-    root weight htr hcov hw hw1 (fun i => hfin (root i))
-
- 
-
-theorem critical_finite_of_unimodular [Infinite V]
-    (G : BondGraph V E) (hG : Function.Injective G)
-    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (hh : 0 < hV G) (huni : AutUnimodular G hG hconn hloc) :
-    ∀ x, ∀ᵐ ω ∂law (pc G), (cluster G ω x).Finite := by
-  classical
-  let : TopologicalSpace V := ⊥
-  let : DiscreteTopology V := ⟨rfl⟩
-  let : Countable V := countable_vertices hloc hconn
-  let : Countable E := countable_edges hloc
-  let : MeasurableSpace V := borel V
-  let : BorelSpace V := ⟨rfl⟩
-  obtain ⟨ι,hι,hn,root,weight,htr,hcov,hw,hw1⟩ :=
-    exists_rootData_of_unimodular G hG hconn hloc hqt huni
-  let : Fintype ι := hι
-  exact critical_finite_of_transport G hconn hloc hqt hh root weight
-    (vertexAut G) (Set.Subset.refl _) htr hcov hw hw1
-
- 
-
-theorem critical_bounded_of_unimodular [Infinite V]
-    (G : BondGraph V E) (hG : Function.Injective G)
-    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (hh : 0 < hV G) (huni : AutUnimodular G hG hconn hloc) :
-    OperatorBounded G (pc G) := by
-  exact critical_bounded_of_unimodular_finiteness G hG hconn hloc hqt hh huni
-    (critical_finite_of_unimodular G hG hconn hloc hqt hh huni)
-
-end Percolation
-
- 
-
-namespace Percolation
-open MeasureTheory
-universe u v
-variable {V : Type u} {E : Type v}
-
- 
-
-theorem critical_operator [Infinite V]
-    (G : BondGraph V E) (hG : Function.Injective G)
-    (_hloop : ∀ e x, G e ≠ s(x,x))
-    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (hh : 0 < hV G) (huni : AutUnimodular G hG hconn hloc) :
-    operatorNorm G (pc G) < ⊤
- := by
-  exact (operatorNorm_lt_top_iff G (pc G)).mpr
-    (critical_bounded_of_unimodular G hG hconn hloc hqt hh huni)
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Criticality.Finiteness
+import OAI.Probability.UnimodularPercolation.Criticality.ScaleLimits
+import OAI.Probability.UnimodularPercolation.Deformation.FiniteScale
+import OAI.Probability.UnimodularPercolation.Graph.Automorphisms
+
+namespace OAI
+
+/-! Critical boundedness of the percolation operator under unimodularity and simple edge indexing. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+open MeasureTheory Filter
+open scoped Topology
+
+ 
+noncomputable def criticalApproach (q : unitInterval) (n : ℕ) : unitInterval :=
+  ⟨(q:ℝ)-(q:ℝ)/2*Real.exp (-(n:ℝ)),by
+    have he : Real.exp (-(n:ℝ)) ≤ 1 := Real.exp_le_one_iff.mpr (neg_nonpos.mpr (Nat.cast_nonneg n))
+    have hq := q.2
+    have hp := Real.exp_pos (-(n:ℝ))
+    constructor <;> nlinarith only [he,hq.1,hq.2,hp]⟩
+
+theorem criticalApproach_half (q : unitInterval) (n : ℕ) :
+    (q:ℝ)/2 ≤ (criticalApproach q n : ℝ) := by
+  have he : Real.exp (-(n:ℝ)) ≤ 1 := Real.exp_le_one_iff.mpr (neg_nonpos.mpr (Nat.cast_nonneg n))
+  change _ ≤ (q:ℝ)-(q:ℝ)/2*Real.exp (-(n:ℝ))
+  nlinarith only [he,q.2.1]
+
+theorem criticalApproach_lt (q : unitInterval) (hq : 0 < (q:ℝ)) (n : ℕ) :
+    criticalApproach q n < q := by
+  change (q:ℝ)-(q:ℝ)/2*Real.exp (-(n:ℝ)) < q
+  exact sub_lt_self _ (mul_pos (half_pos hq) (Real.exp_pos _))
+
+theorem criticalApproach_gap (q : unitInterval) (n : ℕ) :
+    (q:ℝ)-(criticalApproach q n:ℝ) = (q:ℝ)/2*Real.exp (-(n:ℝ)) := by
+  change (q:ℝ)-((q:ℝ)-(q:ℝ)/2*Real.exp (-(n:ℝ))) = _
+  ring
+
+ 
+
+theorem log_sublinear_of_geometric_gap (f : ℕ → ℝ) (p₀ : ℝ) (hp₀ : 0 < p₀)
+    (hf : ∀ n, 1 ≤ f n)
+    (hb : ∀ η : ℝ, 0 < η → ∃ K : ℝ, 0 < K ∧
+      ∀ n : ℕ, f n ≤ K*(p₀*Real.exp (-(n:ℝ)))^(-η)) :
+    Tendsto (fun n => Real.log (f n)/(n:ℝ)) atTop (𝓝 0) := by
+  apply log_sublinear_of_subexp f hf
+  intro η hη
+  obtain ⟨K,hK,hbK⟩ := hb η hη
+  refine ⟨K*p₀^(-η),mul_pos hK (Real.rpow_pos_of_pos hp₀ _),fun n => ?_⟩
+  have he : (p₀*Real.exp (-(n:ℝ)))^(-η) = p₀^(-η)*Real.exp (η*(n:ℝ)) := by
+    rw [Real.mul_rpow hp₀.le (Real.exp_pos _).le,←Real.exp_mul]
+    congr 2
+    ring
+  simpa only [he,mul_assoc] using hbK n
+
+universe u v
+variable {V : Type u} {E : Type v}
+section AnalyticCritical
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Countable V] [Countable E] [Infinite V] [Nonempty E]
+variable {ι : Type*} [Fintype ι] [Nonempty ι]
+
+ 
+
+theorem critical_bounded_of_fixed_data
+    (G : BondGraph V E) (hG : Function.Injective G)
+    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G) (hh : 0 < hV G)
+    (a : ℝ) (ha : 0 < a) (root : ι → V) (weight : ι → ℝ)
+    (htr : RootTransport root weight (vertexAut G))
+    (hcov : ∀ x, ∃ r, ∃ g ∈ vertexAut G, g (root r) = x)
+    (hw : ∀ r, 0 < weight r) (hw1 : ∑ r, weight r = 1)
+    (c : ℝ) (hc : 0 < c) (hmin : ∀ r, 1 ≤ c*weight r)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re) (hρ : ‖P‖ < 1)
+    (d : ℕ) (hd : ∀ z, degree G z ≤ d)
+    (hpcrit : 0 < (pc G:ℝ)) (hpcrit₁ : (pc G:ℝ) < 1)
+    (hfin : ∀ i, ∀ᵐ ω ∂law (pc G), (cluster G ω (root i)).Finite) :
+    OperatorBounded G (pc G) := by
+  by_contra hu
+  obtain ⟨C,hC,J,hJ,C₁,hC₁,k₀,hk₀,ε,M,hε,_,hbad⟩ :=
+    exists_finite_scale_obstruction G hG hconn hloc hqt a ha root weight htr hcov hw hw1 c hc hmin
+      P hpos hsym hP hdiag hρ d hd hpcrit hpcrit₁
+  let p := criticalApproach (pc G)
+  have hpnear (n) : (pc G:ℝ)/2 ≤ (p n:ℝ) := criticalApproach_half (pc G) n
+  have hp (n) : p n < pc G := criticalApproach_lt (pc G) hpcrit n
+  have hpzero (n) : 0 < (p n:ℝ) := (half_pos hpcrit).trans_le (hpnear n)
+  let A := fun n => Classical.choose (operatorBounded_of_lt_pc G hloc hqt (p n) (hp n))
+  have hA (n) : HasKernel G (p n) (A n) := Classical.choose_spec (operatorBounded_of_lt_pc G hloc hqt (p n) (hp n))
+  let m := fun n => ‖A n‖
+  let R := fun n => maxSusceptibility G (p n)/m n
+  let Q := fun n => rootTriangle G root weight (p n:ℝ)
+  have hR (n) : 1 ≤ R n := (one_le_div (zero_lt_one.trans_le ((hA n).norm_one_le G (p n) (A n)))).mpr
+    ((hA n).norm_le_maxSusceptibility G hloc hqt (p n) (hp n) (A n))
+  have hQ (n) : 1 ≤ Q n := rootTriangle_one_le G hloc hqt root weight (fun i => (hw i).le) hw1 (p n) (hp n)
+  have hlogR : Tendsto (fun n => Real.log (R n)/(n:ℝ)) atTop (𝓝 0) := by
+    apply log_sublinear_of_geometric_gap R ((pc G:ℝ)/2) (half_pos hpcrit) hR
+    intro η hη
+    obtain ⟨K,hK,hRK⟩ := critical_normRatio_subpower G hloc hconn hqt hh root weight (vertexAut G)
+      (Set.Subset.refl _) htr hcov hw hw1 hpcrit hpcrit₁ hfin hu η hη
+    refine ⟨K,hK,fun n => ?_⟩
+    simpa only [p,criticalApproach_gap] using (hRK (p n) (hpnear n) (hp n) (A n) (hA n)).2
+  have hlogQ : Tendsto (fun n => Real.log (Q n)/(n:ℝ)) atTop (𝓝 0) := by
+    apply log_sublinear_of_geometric_gap Q ((pc G:ℝ)/2) (half_pos hpcrit) hQ
+    intro η hη
+    obtain ⟨K,hK,hQK⟩ := critical_rootTriangle_subpower G hloc hconn hqt hh root weight (vertexAut G)
+      (Set.Subset.refl _) htr hcov hw hw1 hpcrit hpcrit₁ hfin η hη
+    refine ⟨K,hK,fun n => ?_⟩
+    simpa only [p,criticalApproach_gap] using hQK (p n) (hpnear n) (hp n)
+  obtain ⟨cm,hcm,hm⟩ := critical_norm_inverse_lower G hloc hqt hpcrit₁ hu
+  have hmlower (n : ℕ) : (cm/((pc G:ℝ)/2))*Real.exp (n:ℝ) ≤ m n := by
+    have hh := hm (p n) (hpzero n) (hp n) (A n) (hA n)
+    rw [show (pc G:ℝ)-(p n:ℝ) = (pc G:ℝ)/2*Real.exp (-(n:ℝ)) from criticalApproach_gap (pc G) n,
+      Real.exp_neg,div_mul_eq_div_div,div_inv_eq_mul] at hh
+    exact hh
+  have hρpos : 0 < ‖P‖ := by
+    let x : V := Classical.arbitrary V
+    have hh := (hdiag x).trans ((Complex.re_le_norm _).trans (matrixCoeff_norm_le P x x))
+    linarith only [hh]
+  obtain ⟨n,k,B,hk,hB,hsmall,htime,herror,hlarge⟩ := exists_corridor_scales
+    ‖P‖ ((pc G:ℝ)/2) C C₁ ε M (cm/((pc G:ℝ)/2)) J k₀ hρpos hρ (half_pos hpcrit) hC hC₁ hε
+    (div_pos hcm (half_pos hpcrit)) hJ m R Q hR hQ hmlower hlogR hlogQ
+  exact hbad (p n) (hpnear n) (hp n) (A n) (hA n) k hk B hB hsmall htime herror hlarge
+
+end AnalyticCritical
+
+ 
+
+theorem nonempty_bonds_of_connected [Infinite V] (G : BondGraph V E) (hconn : Connected G) : Nonempty E := by
+  classical
+  by_contra he
+  let : IsEmpty E := not_nonempty_iff.mp he
+  obtain ⟨x,y,hxy⟩ := exists_pair_ne V
+  have hh := conn_univ_of_connected G hconn x y
+  have heq : (Set.univ : Set E) = ∅ := Set.eq_empty_of_isEmpty _
+  rw [heq,conn_empty_iff] at hh
+  exact hxy hh
+
+ 
+
+theorem critical_bounded_of_transport_finiteness [Infinite V]
+    (G : BondGraph V E) (hG : Function.Injective G)
+    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G) (hh : 0 < hV G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (htr : RootTransport root weight (vertexAut G))
+    (hcov : ∀ x, ∃ r, ∃ g ∈ vertexAut G, g (root r) = x)
+    (hw : ∀ r, 0 < weight r) (hw1 : ∑ r, weight r = 1)
+    (hfin : ∀ i, ∀ᵐ ω ∂law (pc G), (cluster G ω (root i)).Finite) :
+    OperatorBounded G (pc G) := by
+  classical
+  let : Countable V := countable_vertices hloc hconn
+  let : Countable E := countable_edges hloc
+  let : Nonempty E := nonempty_bonds_of_connected G hconn
+  let : Nonempty ι := ⟨(hcov (Classical.arbitrary V)).choose⟩
+  let c := ∑ i, (weight i)⁻¹
+  have hc : 0 < c := Finset.sum_pos (fun i _ => inv_pos.mpr (hw i)) Finset.univ_nonempty
+  have hmin (i) : 1 ≤ c*weight i := by
+    have hi : (weight i)⁻¹ ≤ c := Finset.single_le_sum
+      (fun j _ => (inv_pos.mpr (hw j)).le) (Finset.mem_univ i)
+    simpa only [inv_mul_cancel₀ (hw i).ne'] using mul_le_mul_of_nonneg_right hi (hw i).le
+  obtain ⟨d,P,hdpos,hd,hρ,hPs,hP⟩ := exists_strict_walk_operator G hloc hqt hh
+  let a := (1:ℝ)/(2*d)
+  have hdr : (0:ℝ) < d := Nat.cast_pos.mpr hdpos
+  have ha : 0 < a := by dsimp [a]; positivity
+  have had : a*d = 1/2 := by dsimp [a]; field_simp
+  have hPe : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ) := hP
+  have hPpos : PositiveRealMatrix P := by
+    intro x y
+    rw [hPe]
+    refine ⟨rfl,?_⟩
+    exact walkKernel_nonneg G hloc d hd a ha.le (by rw [had]; norm_num) x y
+  have hdiag (x) : 1/2 ≤ (matrixCoeff P x x).re := by
+    rw [hPe,Complex.ofReal_re]
+    exact walkKernel_lazy G hloc d hd a ha.le had.le x
+  exact critical_bounded_of_fixed_data G hG hconn hloc hqt hh a ha root weight htr hcov hw hw1 c hc hmin
+    P hPpos hPs.isSelfAdjoint hPe hdiag hρ d hd (pc_pos G hloc hqt)
+    (pc_lt_one_of_nonamenable G hconn hloc hqt hh) hfin
+
+end Percolation
+
+ 
+
+ 
+
+namespace Percolation
+open MeasureTheory
+universe u v
+variable {V : Type u} {E : Type v}
+
+ 
+
+noncomputable def AutUnimodular [Nonempty V] (G : BondGraph V E)
+    (hG : Function.Injective G) (hconn : Connected G) (hloc : LocallyFinite G) : Prop :=
+  letI : TopologicalSpace V := ⊥
+  letI : DiscreteTopology V := ⟨rfl⟩
+  (graphHaar G hG hconn hloc).IsMulRightInvariant
+
+ 
+
+theorem critical_bounded_of_unimodular_finiteness [Infinite V]
+    (G : BondGraph V E) (hG : Function.Injective G)
+    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (hh : 0 < hV G) (huni : AutUnimodular G hG hconn hloc)
+    (hfin : ∀ x, ∀ᵐ ω ∂law (pc G), (cluster G ω x).Finite) :
+    OperatorBounded G (pc G) := by
+  classical
+  let : TopologicalSpace V := ⊥
+  let : DiscreteTopology V := ⟨rfl⟩
+  let : Countable V := countable_vertices hloc hconn
+  let : MeasurableSpace V := borel V
+  let : BorelSpace V := ⟨rfl⟩
+  obtain ⟨ι,hι,hn,root,weight,htr,hcov,hw,hw1⟩ :=
+    exists_rootData_of_unimodular G hG hconn hloc hqt huni
+  let : Fintype ι := hι
+  exact critical_bounded_of_transport_finiteness G hG hconn hloc hqt hh
+    root weight htr hcov hw hw1 (fun i => hfin (root i))
+
+ 
+
+theorem critical_finite_of_unimodular [Infinite V]
+    (G : BondGraph V E) (hG : Function.Injective G)
+    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (hh : 0 < hV G) (huni : AutUnimodular G hG hconn hloc) :
+    ∀ x, ∀ᵐ ω ∂law (pc G), (cluster G ω x).Finite := by
+  classical
+  let : TopologicalSpace V := ⊥
+  let : DiscreteTopology V := ⟨rfl⟩
+  let : Countable V := countable_vertices hloc hconn
+  let : Countable E := countable_edges hloc
+  let : MeasurableSpace V := borel V
+  let : BorelSpace V := ⟨rfl⟩
+  obtain ⟨ι,hι,hn,root,weight,htr,hcov,hw,hw1⟩ :=
+    exists_rootData_of_unimodular G hG hconn hloc hqt huni
+  let : Fintype ι := hι
+  exact critical_finite_of_transport G hconn hloc hqt hh root weight
+    (vertexAut G) (Set.Subset.refl _) htr hcov hw hw1
+
+ 
+
+theorem critical_bounded_of_unimodular [Infinite V]
+    (G : BondGraph V E) (hG : Function.Injective G)
+    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (hh : 0 < hV G) (huni : AutUnimodular G hG hconn hloc) :
+    OperatorBounded G (pc G) := by
+  exact critical_bounded_of_unimodular_finiteness G hG hconn hloc hqt hh huni
+    (critical_finite_of_unimodular G hG hconn hloc hqt hh huni)
+
+end Percolation
+
+ 
+
+namespace Percolation
+open MeasureTheory
+universe u v
+variable {V : Type u} {E : Type v}
+
+ 
+
+theorem critical_operator [Infinite V]
+    (G : BondGraph V E) (hG : Function.Injective G)
+    (_hloop : ∀ e x, G e ≠ s(x,x))
+    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (hh : 0 < hV G) (huni : AutUnimodular G hG hconn hloc) :
+    operatorNorm G (pc G) < ⊤
+ := by
+  exact (operatorNorm_lt_top_iff G (pc G)).mpr
+    (critical_bounded_of_unimodular G hG hconn hloc hqt hh huni)
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Criticality/Phase.lean b/lean/OAI/Probability/UnimodularPercolation/Criticality/Phase.lean
index cb7db30acdec530b0157e5ae3a2809dad1c87104..00e146c6696c296463df9a98929c036b43ea5fad 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Criticality/Phase.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Criticality/Phase.lean
@@ -1,111 +1,111 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Criticality.ClusterCount
-
-namespace OAI
-
-/-! Strict operator-threshold gaps and infinite-cluster phase classification. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem strict_gap_of_critical_bounded [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (hp1 : pc G < 1) (hb : OperatorBounded G (pc G)) : pc G < ptwo G := by
-  obtain ⟨D, hDpos, hD⟩ := exists_degree_bound G hqt
-  obtain ⟨A, hA⟩ := hb
-  let c : ℝ := (4 * D) * ‖A‖
-  have hc : 0 ≤ c := by dsimp [c]; positivity
-  let ε : ℝ := (1 - (pc G : ℝ)) / (2 * (c + 1))
-  have hε : 0 < ε := div_pos (sub_pos.mpr hp1) (by positivity)
-  have heq : ε * (2 * (c + 1)) = 1 - (pc G : ℝ) :=
-    div_mul_cancel₀ _ (ne_of_gt (by positivity : (0 : ℝ) < 2 * (c + 1)))
-  have hr1 : (pc G : ℝ) + ε < 1 := by nlinarith
-  let r : unitInterval := ⟨(pc G : ℝ) + ε, add_nonneg (pc G).2.1 hε.le, hr1.le⟩
-  have hpr : pc G < r := by change (pc G : ℝ) < (pc G : ℝ) + ε; linarith
-  have hrb : OperatorBounded G r := by
-    apply operatorBounded_of_quantitative_gap G hloc D hD (pc G) r
-      (pc_pos G hloc hqt) hpr hr1 A hA
-    change (((pc G : ℝ) + ε) - (pc G : ℝ)) * (4 * D) * ‖A‖ < 1 - ((pc G : ℝ) + ε)
-    have he : (((pc G : ℝ) + ε) - (pc G : ℝ)) * (4 * D) * ‖A‖ = ε * c := by
-      dsimp [c]
-      ring
-    rw [he]
-    nlinarith
-  exact hpr.trans_le (le_sSup hrb)
-
-theorem unique_probability_zero_of_lt_pu [Infinite V]
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
-    (hqt : QuasiTransitive G) (p : unitInterval) (hp : p < pu G) :
-    law p {ω | ∃! C, C ∈ infiniteClusters G ω} = 0 := by
-  let : Countable V := countable_vertices hloc hconn
-  let : Countable E := countable_edges hloc
-  refine (law_invariant_zero_one G hconn hloc hqt p _
-    (measurableSet_unique_infinite_cluster G) ?_).resolve_right ?_
-  · intro a b hab
-    ext ω
-    exact unique_infinite_cluster_aut_iff G hab ω
-  · intro he
-    exact (not_le_of_gt hp) (sInf_le he)
-
-theorem finite_count_empty_ae_of_lt_pu [Infinite V]
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
-    (hqt : QuasiTransitive G) (p : unitInterval) (hp0 : 0 < p) (hp : p < pu G) :
-    ∀ᵐ ω ∂law p, (infiniteClusters G ω).Finite → infiniteClusters G ω = ∅ := by
-  let : Countable V := countable_vertices hloc hconn
-  let : Countable E := countable_edges hloc
-  have hp1 : p < 1 := hp.trans_le (pu G).2.2
-  have ha (x : V) (R : Finset V) :
-      ∀ᵐ ω ∂law p, ¬((cluster G ω x).Infinite ∧ InfiniteCover G R ω) := by
-    have hz : law p {ω | (cluster G ω x).Infinite ∧ InfiniteCover G R ω} = 0 := by
-      by_contra h
-      have ht := unique_pos_of_cover_pos G hconn p hp0 hp1 x R (pos_iff_ne_zero.mpr h)
-      rw [unique_probability_zero_of_lt_pu G hconn hloc hqt p hp] at ht
-      exact lt_irrefl _ ht
-    apply ae_iff.mpr
-    simpa only [not_not] using hz
-  have hall : ∀ᵐ ω ∂law p, ∀ x : V, ∀ R : Finset V,
-      ¬((cluster G ω x).Infinite ∧ InfiniteCover G R ω) :=
-    ae_all_iff.mpr fun x => ae_all_iff.mpr (ha x)
-  filter_upwards [hall] with ω hω
-  intro hfin
-  by_contra hn
-  obtain ⟨C, hC⟩ := Set.nonempty_iff_ne_empty.mpr hn
-  obtain ⟨hx, x, rfl⟩ := hC
-  let f : Set V → V := fun C => if hc : C ∈ infiniteClusters G ω then
-    Classical.choose hc.2 else x
-  have hf (C : Set V) (hC : C ∈ infiniteClusters G ω) : C = cluster G ω (f C) := by
-    simp only [f, dite_eq_left hC]
-    exact Classical.choose_spec hC.2
-  let R : Finset V := hfin.toFinset.image f
-  apply hω x R ⟨hx, ?_⟩
-  intro y hy
-  have hc : cluster G ω y ∈ infiniteClusters G ω := ⟨hy, y, rfl⟩
-  refine ⟨f (cluster G ω y), Finset.mem_image.mpr ⟨_, hfin.mem_toFinset.mpr hc, rfl⟩, ?_⟩
-  have he := hf _ hc
-  exact (Set.ext_iff.mp he _).mpr (mem_cluster_self G ω (f (cluster G ω y)))
-
-theorem infinitely_many_of_pc_lt_lt_pu [Infinite V]
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
-    (hqt : QuasiTransitive G) (p : unitInterval) (hc : pc G < p) (hu : p < pu G) :
-    ∀ᵐ ω ∂law p, (infiniteClusters G ω).Infinite := by
-  let : Countable V := countable_vertices hloc hconn
-  let : Countable E := countable_edges hloc
-  have he : ∀ᵐ ω ∂law p, (infiniteClusters G ω).Nonempty := by
-    rw [← map_openEdges p]
-    apply (ae_map_iff (measurable_openEdges p).aemeasurable ?_).mpr
-    · simpa only [infiniteClusters_nonempty_iff] using ae_exists_infinite_cluster_of_pc_lt G p hc
-    · simpa only [infiniteClusters_nonempty_iff] using measurableSet_exists_infinite_cluster G
-  filter_upwards [he, finite_count_empty_ae_of_lt_pu G hconn hloc hqt p
-    ((pc_pos G hloc hqt).trans hc) hu] with ω hne hf
-  intro hfin
-  exact hne.ne_empty (hf hfin)
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Criticality.ClusterCount
+
+namespace OAI
+
+/-! Strict operator-threshold gaps and infinite-cluster phase classification. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem strict_gap_of_critical_bounded [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (hp1 : pc G < 1) (hb : OperatorBounded G (pc G)) : pc G < ptwo G := by
+  obtain ⟨D, hDpos, hD⟩ := exists_degree_bound G hqt
+  obtain ⟨A, hA⟩ := hb
+  let c : ℝ := (4 * D) * ‖A‖
+  have hc : 0 ≤ c := by dsimp [c]; positivity
+  let ε : ℝ := (1 - (pc G : ℝ)) / (2 * (c + 1))
+  have hε : 0 < ε := div_pos (sub_pos.mpr hp1) (by positivity)
+  have heq : ε * (2 * (c + 1)) = 1 - (pc G : ℝ) :=
+    div_mul_cancel₀ _ (ne_of_gt (by positivity : (0 : ℝ) < 2 * (c + 1)))
+  have hr1 : (pc G : ℝ) + ε < 1 := by nlinarith
+  let r : unitInterval := ⟨(pc G : ℝ) + ε, add_nonneg (pc G).2.1 hε.le, hr1.le⟩
+  have hpr : pc G < r := by change (pc G : ℝ) < (pc G : ℝ) + ε; linarith
+  have hrb : OperatorBounded G r := by
+    apply operatorBounded_of_quantitative_gap G hloc D hD (pc G) r
+      (pc_pos G hloc hqt) hpr hr1 A hA
+    change (((pc G : ℝ) + ε) - (pc G : ℝ)) * (4 * D) * ‖A‖ < 1 - ((pc G : ℝ) + ε)
+    have he : (((pc G : ℝ) + ε) - (pc G : ℝ)) * (4 * D) * ‖A‖ = ε * c := by
+      dsimp [c]
+      ring
+    rw [he]
+    nlinarith
+  exact hpr.trans_le (le_sSup hrb)
+
+theorem unique_probability_zero_of_lt_pu [Infinite V]
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
+    (hqt : QuasiTransitive G) (p : unitInterval) (hp : p < pu G) :
+    law p {ω | ∃! C, C ∈ infiniteClusters G ω} = 0 := by
+  let : Countable V := countable_vertices hloc hconn
+  let : Countable E := countable_edges hloc
+  refine (law_invariant_zero_one G hconn hloc hqt p _
+    (measurableSet_unique_infinite_cluster G) ?_).resolve_right ?_
+  · intro a b hab
+    ext ω
+    exact unique_infinite_cluster_aut_iff G hab ω
+  · intro he
+    exact (not_le_of_gt hp) (sInf_le he)
+
+theorem finite_count_empty_ae_of_lt_pu [Infinite V]
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
+    (hqt : QuasiTransitive G) (p : unitInterval) (hp0 : 0 < p) (hp : p < pu G) :
+    ∀ᵐ ω ∂law p, (infiniteClusters G ω).Finite → infiniteClusters G ω = ∅ := by
+  let : Countable V := countable_vertices hloc hconn
+  let : Countable E := countable_edges hloc
+  have hp1 : p < 1 := hp.trans_le (pu G).2.2
+  have ha (x : V) (R : Finset V) :
+      ∀ᵐ ω ∂law p, ¬((cluster G ω x).Infinite ∧ InfiniteCover G R ω) := by
+    have hz : law p {ω | (cluster G ω x).Infinite ∧ InfiniteCover G R ω} = 0 := by
+      by_contra h
+      have ht := unique_pos_of_cover_pos G hconn p hp0 hp1 x R (pos_iff_ne_zero.mpr h)
+      rw [unique_probability_zero_of_lt_pu G hconn hloc hqt p hp] at ht
+      exact lt_irrefl _ ht
+    apply ae_iff.mpr
+    simpa only [not_not] using hz
+  have hall : ∀ᵐ ω ∂law p, ∀ x : V, ∀ R : Finset V,
+      ¬((cluster G ω x).Infinite ∧ InfiniteCover G R ω) :=
+    ae_all_iff.mpr fun x => ae_all_iff.mpr (ha x)
+  filter_upwards [hall] with ω hω
+  intro hfin
+  by_contra hn
+  obtain ⟨C, hC⟩ := Set.nonempty_iff_ne_empty.mpr hn
+  obtain ⟨hx, x, rfl⟩ := hC
+  let f : Set V → V := fun C => if hc : C ∈ infiniteClusters G ω then
+    Classical.choose hc.2 else x
+  have hf (C : Set V) (hC : C ∈ infiniteClusters G ω) : C = cluster G ω (f C) := by
+    simp only [f, dite_eq_left hC]
+    exact Classical.choose_spec hC.2
+  let R : Finset V := hfin.toFinset.image f
+  apply hω x R ⟨hx, ?_⟩
+  intro y hy
+  have hc : cluster G ω y ∈ infiniteClusters G ω := ⟨hy, y, rfl⟩
+  refine ⟨f (cluster G ω y), Finset.mem_image.mpr ⟨_, hfin.mem_toFinset.mpr hc, rfl⟩, ?_⟩
+  have he := hf _ hc
+  exact (Set.ext_iff.mp he _).mpr (mem_cluster_self G ω (f (cluster G ω y)))
+
+theorem infinitely_many_of_pc_lt_lt_pu [Infinite V]
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
+    (hqt : QuasiTransitive G) (p : unitInterval) (hc : pc G < p) (hu : p < pu G) :
+    ∀ᵐ ω ∂law p, (infiniteClusters G ω).Infinite := by
+  let : Countable V := countable_vertices hloc hconn
+  let : Countable E := countable_edges hloc
+  have he : ∀ᵐ ω ∂law p, (infiniteClusters G ω).Nonempty := by
+    rw [← map_openEdges p]
+    apply (ae_map_iff (measurable_openEdges p).aemeasurable ?_).mpr
+    · simpa only [infiniteClusters_nonempty_iff] using ae_exists_infinite_cluster_of_pc_lt G p hc
+    · simpa only [infiniteClusters_nonempty_iff] using measurableSet_exists_infinite_cluster G
+  filter_upwards [he, finite_count_empty_ae_of_lt_pu G hconn hloc hqt p
+    ((pc_pos G hloc hqt).trans hc) hu] with ω hne hf
+  intro hfin
+  exact hne.ne_empty (hf hfin)
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Criticality/ScaleLimits.lean b/lean/OAI/Probability/UnimodularPercolation/Criticality/ScaleLimits.lean
index 73f24142dd104fd87e3b93085ec4b20c21daa53d..bd5f5cce549a35dbf1f18202b581fbe0d02c8939 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Criticality/ScaleLimits.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Criticality/ScaleLimits.lean
@@ -1,206 +1,206 @@
-import Mathlib
-
-namespace OAI
-
-/-! Analytic scale limits and quantitative parameter estimates. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-open Filter
-open scoped Topology
-
- 
-
-theorem log_sublinear_of_subexp (f : ℕ → ℝ) (hf : ∀ n, 1 ≤ f n)
-    (hb : ∀ η : ℝ, 0 < η → ∃ K : ℝ, 0 < K ∧ ∀ n, f n ≤ K*Real.exp (η*(n:ℝ))) :
-    Tendsto (fun n => Real.log (f n)/(n:ℝ)) atTop (𝓝 0) := by
-  apply Metric.tendsto_nhds.mpr
-  intro ε hε
-  obtain ⟨K,hK,hbK⟩ := hb (ε/2) (half_pos hε)
-  have hlim := tendsto_const_div_atTop_nhds_zero_nat (Real.log K)
-  filter_upwards [hlim.eventually (gt_mem_nhds (half_pos hε)), eventually_ge_atTop (1:ℕ)] with n hn hn1
-  have hnpos : (0:ℝ) < n := Nat.cast_pos.mpr (by omega)
-  have hlog := Real.log_le_log (zero_lt_one.trans_le (hf n)) (hbK n)
-  rw [Real.log_mul hK.ne' (Real.exp_pos _).ne',Real.log_exp] at hlog
-  have hdiv := (div_le_div_of_nonneg_right hlog hnpos.le)
-  rw [add_div,mul_div_cancel_right₀ _ hnpos.ne'] at hdiv
-  rw [Real.dist_eq,sub_zero,abs_of_nonneg (div_nonneg (Real.log_nonneg (hf n)) hnpos.le)]
-  linarith only [hdiv,hn]
-
- 
-theorem tendsto_atTop_of_div_nat_pos (f : ℕ → ℝ) (L : ℝ) (hL : 0 < L)
-    (hf : Tendsto (fun n => f n/(n:ℝ)) atTop (𝓝 L)) : Tendsto f atTop atTop := by
-  have ht := (tendsto_natCast_atTop_atTop (R := ℝ)).atTop_mul_pos hL hf
-  apply ht.congr'
-  filter_upwards [eventually_ge_atTop (1:ℕ)] with n hn
-  exact mul_div_cancel₀ (f n) (by exact_mod_cast (show n ≠ 0 by omega))
-
- 
-theorem tendsto_exp_of_div_nat_neg (f : ℕ → ℝ) (L : ℝ) (hL : L < 0)
-    (hf : Tendsto (fun n => f n/(n:ℝ)) atTop (𝓝 L)) :
-    Tendsto (fun n => Real.exp (f n)) atTop (𝓝 0) := by
-  have hn : Tendsto (fun n => -f n/(n:ℝ)) atTop (𝓝 (-L)) := by
-    simpa only [neg_div] using hf.neg
-  have ht := tendsto_atTop_of_div_nat_pos (fun n => -f n) (-L) (neg_pos.mpr hL) hn
-  simpa only [Function.comp_def,neg_neg] using Real.tendsto_exp_neg_atTop_nhds_zero.comp ht
-
- 
-
-theorem corridor_scale_limits (r C₁ : ℝ) (J : ℕ) (hr : 0 < r) (hC₁ : 0 < C₁) (hJ : 0 < J)
-    (R : ℕ → ℝ) (hR : ∀ n, 1 ≤ R n)
-    (hlogR : Tendsto (fun n => Real.log (R n)/(n:ℝ)) atTop (𝓝 0)) :
-    let k := fun n : ℕ => ⌈r*(n:ℝ)⌉₊
-    let j := fun n : ℕ => J+⌈C₁*Real.log (((k n:ℝ)+1)*R n)⌉₊
-    Tendsto k atTop atTop ∧
-    Tendsto (fun n => (k n:ℝ)/(n:ℝ)) atTop (𝓝 r) ∧
-    Tendsto (fun n => (j n:ℝ)/(n:ℝ)) atTop (𝓝 0) ∧
-    Tendsto (fun n => Real.log (j n:ℝ)/(n:ℝ)) atTop (𝓝 0) := by
-  dsimp only
-  let k := fun n : ℕ => ⌈r*(n:ℝ)⌉₊
-  let j := fun n : ℕ => J+⌈C₁*Real.log (((k n:ℝ)+1)*R n)⌉₊
-  have hkt : Tendsto k atTop atTop :=
-    tendsto_nat_ceil_atTop.comp ((tendsto_natCast_atTop_atTop (R := ℝ)).const_mul_atTop hr)
-  have hk : Tendsto (fun n => (k n:ℝ)/(n:ℝ)) atTop (𝓝 r) :=
-    (tendsto_nat_ceil_mul_div_atTop hr.le).comp tendsto_natCast_atTop_atTop
-  have hkplus : Tendsto (fun n => ((k n:ℝ)+1)/(n:ℝ)) atTop (𝓝 r) := by
-    simpa only [add_div,add_zero] using hk.add (tendsto_const_div_atTop_nhds_zero_nat (1:ℝ))
-  have hkTop : Tendsto (fun n => (k n:ℝ)+1) atTop atTop :=
-    tendsto_atTop_mono (fun n => le_add_of_nonneg_right (show (0:ℝ) ≤ 1 by norm_num))
-      (tendsto_natCast_atTop_atTop.comp hkt)
-  have hlogk : Tendsto (fun n => Real.log ((k n:ℝ)+1)/(n:ℝ)) atTop (𝓝 0) := by
-    have hl : Tendsto (fun n => Real.log ((k n:ℝ)+1)/((k n:ℝ)+1)) atTop (𝓝 0) := by
-      simpa only [Function.comp_def,pow_one,one_mul,add_zero] using
-        (Real.tendsto_pow_log_div_mul_add_atTop 1 0 1 (by norm_num)).comp hkTop
-    have hh := hl.mul hkplus
-    rw [zero_mul] at hh
-    apply hh.congr
-    intro n
-    have hn : (k n:ℝ)+1 ≠ 0 := by positivity
-    field_simp
-  have hupper : Tendsto (fun n : ℕ => ((J:ℝ)+1)/(n:ℝ) + C₁*(Real.log ((k n:ℝ)+1)/(n:ℝ) + Real.log (R n)/(n:ℝ))) atTop (𝓝 0) := by
-    simpa only [add_zero,mul_zero] using
-      (tendsto_const_div_atTop_nhds_zero_nat ((J:ℝ)+1)).add ((hlogk.add hlogR).const_mul C₁)
-  have hjlim : Tendsto (fun n => (j n:ℝ)/(n:ℝ)) atTop (𝓝 0) := by
-    apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hupper
-    · exact Eventually.of_forall (fun n => div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _))
-    filter_upwards [eventually_ge_atTop (1:ℕ)] with n hn
-    have hnpos : (0:ℝ) < n := Nat.cast_pos.mpr (by omega)
-    have hprod : 1 ≤ ((k n:ℝ)+1)*R n := by nlinarith only [hR n,(Nat.cast_nonneg (k n) : (0:ℝ) ≤ _)]
-    have harg : 0 ≤ C₁*Real.log (((k n:ℝ)+1)*R n) := mul_nonneg hC₁.le (Real.log_nonneg hprod)
-    have hceil := (Nat.ceil_lt_add_one harg).le
-    have he : Real.log (((k n:ℝ)+1)*R n) = Real.log ((k n:ℝ)+1)+Real.log (R n) :=
-      Real.log_mul (by positivity) (zero_lt_one.trans_le (hR n)).ne'
-    dsimp only [j]
-    push_cast
-    rw [←add_div,←mul_div_assoc,←add_div,←he]
-    exact (div_le_div_of_nonneg_right (by linarith only [hceil]) hnpos.le)
-  refine ⟨hkt,hk,hjlim,?_⟩
-  apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hjlim
-  · exact Eventually.of_forall (fun n => div_nonneg (Real.log_nonneg (Nat.one_le_cast.mpr (show 1 ≤ j n by dsimp [j]; omega))) (Nat.cast_nonneg n))
-  · exact Eventually.of_forall (fun n => div_le_div_of_nonneg_right (Real.log_le_self (Nat.cast_nonneg (j n))) (Nat.cast_nonneg n))
-
- 
-
-theorem exists_corridor_scales_of_slopes
-    (ρ p₀ C C₁ ε M cm r z : ℝ) (J k₀ : ℕ)
-    (hρ : 0 < ρ) (hp₀ : 0 < p₀) (hC₁ : 0 < C₁) (hε : 0 < ε) (hcm : 0 < cm)
-    (hJ : 0 < J) (hr : 0 < r) (hz : 0 < z)
-    (hzρ : z+r*Real.log ρ < 0) (hzC : z+C*r < 1)
-    (m R Q : ℕ → ℝ) (hR : ∀ n, 1 ≤ R n) (hQ : ∀ n, 1 ≤ Q n)
-    (hm : ∀ n : ℕ, cm*Real.exp (n:ℝ) ≤ m n)
-    (hlogR : Tendsto (fun n => Real.log (R n)/(n:ℝ)) atTop (𝓝 0))
-    (hlogQ : Tendsto (fun n => Real.log (Q n)/(n:ℝ)) atTop (𝓝 0)) :
-    ∃ (n k : ℕ) (B : ℝ), k₀ ≤ k ∧ 0 < B ∧
-      let j := J+⌈C₁*Real.log (((k:ℝ)+1)*R n)⌉₊
-      B*ρ^k ≤ 1/4 ∧ B*(j:ℝ)*Real.exp (C*(k:ℝ)) ≤ m n ∧
-      (R n)^2*Q n*ρ^k ≤ ε*p₀^(2*j) ∧ M < B*p₀^(2*j) := by
-  let k := fun n : ℕ => ⌈r*(n:ℝ)⌉₊
-  let j := fun n : ℕ => J+⌈C₁*Real.log (((k n:ℝ)+1)*R n)⌉₊
-  let B := fun n : ℕ => Real.exp (z*(n:ℝ))
-  obtain ⟨hkt,hk,hj,hlogj⟩ := corridor_scale_limits r C₁ J hr hC₁ hJ R hR hlogR
-  change Tendsto k atTop atTop at hkt
-  change Tendsto (fun n => (k n:ℝ)/(n:ℝ)) atTop (𝓝 r) at hk
-  change Tendsto (fun n => (j n:ℝ)/(n:ℝ)) atTop (𝓝 0) at hj
-  change Tendsto (fun n => Real.log (j n:ℝ)/(n:ℝ)) atTop (𝓝 0) at hlogj
-  have hn : Tendsto (fun n : ℕ => (n:ℝ)/(n:ℝ)) atTop (𝓝 1) := by
-    simpa only [add_zero] using tendsto_natCast_div_add_atTop (0:ℝ)
-  have hβ (n) : Real.exp ((j n:ℝ)*(2*Real.log p₀)) = p₀^(2*j n) := by
-    rw [show (j n:ℝ)*(2*Real.log p₀) = ((2*j n:ℕ):ℝ)*Real.log p₀ by push_cast; ring,
-      Real.exp_nat_mul,Real.exp_log hp₀]
-  have hjpos (n) : (0:ℝ) < j n := Nat.cast_pos.mpr (by dsimp [j]; omega)
-  have hlim₁ : Tendsto (fun n : ℕ => (z*(n:ℝ)+(k n:ℝ)*Real.log ρ)/(n:ℝ)) atTop (𝓝 (z+r*Real.log ρ)) := by
-    simpa only [mul_one,div_mul_eq_mul_div,←mul_div_assoc,←add_div] using
-      (hn.const_mul z).add (hk.mul_const (Real.log ρ))
-  have hdec₁ := tendsto_exp_of_div_nat_neg _ _ hzρ hlim₁
-  have hlim₂ : Tendsto (fun n : ℕ =>
-      (2*Real.log (R n)+Real.log (Q n)+(k n:ℝ)*Real.log ρ-(j n:ℝ)*(2*Real.log p₀))/(n:ℝ))
-      atTop (𝓝 (r*Real.log ρ)) := by
-    simpa only [mul_zero,zero_add,zero_mul,sub_zero,div_mul_eq_mul_div,←mul_div_assoc,←add_div,←sub_div] using
-      (((hlogR.const_mul 2).add hlogQ).add (hk.mul_const (Real.log ρ))).sub (hj.mul_const (2*Real.log p₀))
-  have hsl₂ : r*Real.log ρ < 0 := by linarith only [hz,hzρ]
-  have hdec₂ := tendsto_exp_of_div_nat_neg _ _ hsl₂ hlim₂
-  have hlim₃ : Tendsto (fun n : ℕ =>
-      (z*(n:ℝ)+Real.log (j n:ℝ)+C*(k n:ℝ)-(n:ℝ))/(n:ℝ)) atTop (𝓝 (z+C*r-1)) := by
-    simpa only [mul_one,add_zero,←mul_div_assoc,←add_div,←sub_div] using
-      (((hn.const_mul z).add hlogj).add (hk.const_mul C)).sub hn
-  have hdec₃ := tendsto_exp_of_div_nat_neg _ _ (sub_neg.mpr hzC) hlim₃
-  have hlim₄ : Tendsto (fun n : ℕ =>
-      (z*(n:ℝ)+(j n:ℝ)*(2*Real.log p₀))/(n:ℝ)) atTop (𝓝 z) := by
-    simpa only [mul_one,zero_mul,add_zero,div_mul_eq_mul_div,←mul_div_assoc,←add_div] using
-      (hn.const_mul z).add (hj.mul_const (2*Real.log p₀))
-  have hgrow := Real.tendsto_exp_atTop.comp (tendsto_atTop_of_div_nat_pos _ _ hz hlim₄)
-  have hfinal : ∀ᶠ n : ℕ in atTop,
-      k₀ ≤ k n ∧ B n*ρ^(k n) ≤ 1/4 ∧
-      B n*(j n:ℝ)*Real.exp (C*(k n:ℝ)) ≤ m n ∧
-      (R n)^2*Q n*ρ^(k n) ≤ ε*p₀^(2*j n) ∧ M < B n*p₀^(2*j n) := by
-    filter_upwards [hkt.eventually_ge_atTop k₀,
-      hdec₁.eventually_lt_const (show (0:ℝ) < 1/4 by norm_num),
-      hdec₂.eventually_lt_const hε,hdec₃.eventually_lt_const hcm,
-      hgrow.eventually_gt_atTop M] with n hk₀ h₁ h₂ h₃ h₄
-    have heρ : Real.exp ((k n:ℝ)*Real.log ρ) = ρ^(k n) := by rw [Real.exp_nat_mul,Real.exp_log hρ]
-    have heR : Real.exp (2*Real.log (R n)) = (R n)^2 := by
-      simpa only [Nat.cast_ofNat,Real.exp_log (zero_lt_one.trans_le (hR n))] using Real.exp_nat_mul (Real.log (R n)) 2
-    simp only [Real.exp_add,heρ] at h₁
-    simp only [Real.exp_sub,Real.exp_add,heR,Real.exp_log (zero_lt_one.trans_le (hQ n)),heρ,hβ n] at h₂
-    simp only [Real.exp_sub,Real.exp_add,Real.exp_log (hjpos n)] at h₃
-    change M < Real.exp (z*(n:ℝ)+(j n:ℝ)*(2*Real.log p₀)) at h₄
-    rw [Real.exp_add,hβ n] at h₄
-    refine ⟨hk₀,h₁.le,?_,(div_le_iff₀ (pow_pos hp₀ _)).mp h₂.le,h₄⟩
-    exact ((div_le_iff₀ (Real.exp_pos _)).mp h₃.le).trans (hm n)
-  obtain ⟨n,hn⟩ := hfinal.exists
-  exact ⟨n,k n,B n,hn.1,Real.exp_pos _,hn.2⟩
-
- 
-
-theorem exists_corridor_scales
-    (ρ p₀ C C₁ ε M cm : ℝ) (J k₀ : ℕ)
-    (hρ : 0 < ρ) (hρ₁ : ρ < 1) (hp₀ : 0 < p₀) (hC : 0 < C)
-    (hC₁ : 0 < C₁) (hε : 0 < ε) (hcm : 0 < cm) (hJ : 0 < J)
-    (m R Q : ℕ → ℝ) (hR : ∀ n, 1 ≤ R n) (hQ : ∀ n, 1 ≤ Q n)
-    (hm : ∀ n : ℕ, cm*Real.exp (n:ℝ) ≤ m n)
-    (hlogR : Tendsto (fun n => Real.log (R n)/(n:ℝ)) atTop (𝓝 0))
-    (hlogQ : Tendsto (fun n => Real.log (Q n)/(n:ℝ)) atTop (𝓝 0)) :
-    ∃ (n k : ℕ) (B : ℝ), k₀ ≤ k ∧ 0 < B ∧
-      let j := J+⌈C₁*Real.log (((k:ℝ)+1)*R n)⌉₊
-      B*ρ^k ≤ 1/4 ∧ B*(j:ℝ)*Real.exp (C*(k:ℝ)) ≤ m n ∧
-      (R n)^2*Q n*ρ^k ≤ ε*p₀^(2*j) ∧ M < B*p₀^(2*j) := by
-  let r := 1/(4*(C+1))
-  let z := min (1/4) (-(r*Real.log ρ)/2)
-  have hr : 0 < r := by dsimp [r]; positivity
-  have hlog : Real.log ρ < 0 := by simpa only [Real.log_one] using Real.log_lt_log hρ hρ₁
-  have hz : 0 < z := lt_min (by norm_num) (half_pos (neg_pos.mpr (mul_neg_of_pos_of_neg hr hlog)))
-  have hzr : z+r*Real.log ρ < 0 := by
-    have hh : z ≤ -(r*Real.log ρ)/2 := min_le_right _ _
-    linarith only [hh,mul_neg_of_pos_of_neg hr hlog]
-  have hzc : z+C*r < 1 := by
-    have hz' : z ≤ 1/4 := min_le_left _ _
-    have hr' : r*(4*(C+1)) = 1 := div_mul_cancel₀ 1 (by positivity)
-    nlinarith only [hz',hr',hr]
-  exact exists_corridor_scales_of_slopes ρ p₀ C C₁ ε M cm r z J k₀
-    hρ hp₀ hC₁ hε hcm hJ hr hz hzr hzc m R Q hR hQ hm hlogR hlogQ
-
-end Percolation
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-! Analytic scale limits and quantitative parameter estimates. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+open Filter
+open scoped Topology
+
+ 
+
+theorem log_sublinear_of_subexp (f : ℕ → ℝ) (hf : ∀ n, 1 ≤ f n)
+    (hb : ∀ η : ℝ, 0 < η → ∃ K : ℝ, 0 < K ∧ ∀ n, f n ≤ K*Real.exp (η*(n:ℝ))) :
+    Tendsto (fun n => Real.log (f n)/(n:ℝ)) atTop (𝓝 0) := by
+  apply Metric.tendsto_nhds.mpr
+  intro ε hε
+  obtain ⟨K,hK,hbK⟩ := hb (ε/2) (half_pos hε)
+  have hlim := tendsto_const_div_atTop_nhds_zero_nat (Real.log K)
+  filter_upwards [hlim.eventually (gt_mem_nhds (half_pos hε)), eventually_ge_atTop (1:ℕ)] with n hn hn1
+  have hnpos : (0:ℝ) < n := Nat.cast_pos.mpr (by omega)
+  have hlog := Real.log_le_log (zero_lt_one.trans_le (hf n)) (hbK n)
+  rw [Real.log_mul hK.ne' (Real.exp_pos _).ne',Real.log_exp] at hlog
+  have hdiv := (div_le_div_of_nonneg_right hlog hnpos.le)
+  rw [add_div,mul_div_cancel_right₀ _ hnpos.ne'] at hdiv
+  rw [Real.dist_eq,sub_zero,abs_of_nonneg (div_nonneg (Real.log_nonneg (hf n)) hnpos.le)]
+  linarith only [hdiv,hn]
+
+ 
+theorem tendsto_atTop_of_div_nat_pos (f : ℕ → ℝ) (L : ℝ) (hL : 0 < L)
+    (hf : Tendsto (fun n => f n/(n:ℝ)) atTop (𝓝 L)) : Tendsto f atTop atTop := by
+  have ht := (tendsto_natCast_atTop_atTop (R := ℝ)).atTop_mul_pos hL hf
+  apply ht.congr'
+  filter_upwards [eventually_ge_atTop (1:ℕ)] with n hn
+  exact mul_div_cancel₀ (f n) (by exact_mod_cast (show n ≠ 0 by omega))
+
+ 
+theorem tendsto_exp_of_div_nat_neg (f : ℕ → ℝ) (L : ℝ) (hL : L < 0)
+    (hf : Tendsto (fun n => f n/(n:ℝ)) atTop (𝓝 L)) :
+    Tendsto (fun n => Real.exp (f n)) atTop (𝓝 0) := by
+  have hn : Tendsto (fun n => -f n/(n:ℝ)) atTop (𝓝 (-L)) := by
+    simpa only [neg_div] using hf.neg
+  have ht := tendsto_atTop_of_div_nat_pos (fun n => -f n) (-L) (neg_pos.mpr hL) hn
+  simpa only [Function.comp_def,neg_neg] using Real.tendsto_exp_neg_atTop_nhds_zero.comp ht
+
+ 
+
+theorem corridor_scale_limits (r C₁ : ℝ) (J : ℕ) (hr : 0 < r) (hC₁ : 0 < C₁) (hJ : 0 < J)
+    (R : ℕ → ℝ) (hR : ∀ n, 1 ≤ R n)
+    (hlogR : Tendsto (fun n => Real.log (R n)/(n:ℝ)) atTop (𝓝 0)) :
+    let k := fun n : ℕ => ⌈r*(n:ℝ)⌉₊
+    let j := fun n : ℕ => J+⌈C₁*Real.log (((k n:ℝ)+1)*R n)⌉₊
+    Tendsto k atTop atTop ∧
+    Tendsto (fun n => (k n:ℝ)/(n:ℝ)) atTop (𝓝 r) ∧
+    Tendsto (fun n => (j n:ℝ)/(n:ℝ)) atTop (𝓝 0) ∧
+    Tendsto (fun n => Real.log (j n:ℝ)/(n:ℝ)) atTop (𝓝 0) := by
+  dsimp only
+  let k := fun n : ℕ => ⌈r*(n:ℝ)⌉₊
+  let j := fun n : ℕ => J+⌈C₁*Real.log (((k n:ℝ)+1)*R n)⌉₊
+  have hkt : Tendsto k atTop atTop :=
+    tendsto_nat_ceil_atTop.comp ((tendsto_natCast_atTop_atTop (R := ℝ)).const_mul_atTop hr)
+  have hk : Tendsto (fun n => (k n:ℝ)/(n:ℝ)) atTop (𝓝 r) :=
+    (tendsto_nat_ceil_mul_div_atTop hr.le).comp tendsto_natCast_atTop_atTop
+  have hkplus : Tendsto (fun n => ((k n:ℝ)+1)/(n:ℝ)) atTop (𝓝 r) := by
+    simpa only [add_div,add_zero] using hk.add (tendsto_const_div_atTop_nhds_zero_nat (1:ℝ))
+  have hkTop : Tendsto (fun n => (k n:ℝ)+1) atTop atTop :=
+    tendsto_atTop_mono (fun n => le_add_of_nonneg_right (show (0:ℝ) ≤ 1 by norm_num))
+      (tendsto_natCast_atTop_atTop.comp hkt)
+  have hlogk : Tendsto (fun n => Real.log ((k n:ℝ)+1)/(n:ℝ)) atTop (𝓝 0) := by
+    have hl : Tendsto (fun n => Real.log ((k n:ℝ)+1)/((k n:ℝ)+1)) atTop (𝓝 0) := by
+      simpa only [Function.comp_def,pow_one,one_mul,add_zero] using
+        (Real.tendsto_pow_log_div_mul_add_atTop 1 0 1 (by norm_num)).comp hkTop
+    have hh := hl.mul hkplus
+    rw [zero_mul] at hh
+    apply hh.congr
+    intro n
+    have hn : (k n:ℝ)+1 ≠ 0 := by positivity
+    field_simp
+  have hupper : Tendsto (fun n : ℕ => ((J:ℝ)+1)/(n:ℝ) + C₁*(Real.log ((k n:ℝ)+1)/(n:ℝ) + Real.log (R n)/(n:ℝ))) atTop (𝓝 0) := by
+    simpa only [add_zero,mul_zero] using
+      (tendsto_const_div_atTop_nhds_zero_nat ((J:ℝ)+1)).add ((hlogk.add hlogR).const_mul C₁)
+  have hjlim : Tendsto (fun n => (j n:ℝ)/(n:ℝ)) atTop (𝓝 0) := by
+    apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hupper
+    · exact Eventually.of_forall (fun n => div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _))
+    filter_upwards [eventually_ge_atTop (1:ℕ)] with n hn
+    have hnpos : (0:ℝ) < n := Nat.cast_pos.mpr (by omega)
+    have hprod : 1 ≤ ((k n:ℝ)+1)*R n := by nlinarith only [hR n,(Nat.cast_nonneg (k n) : (0:ℝ) ≤ _)]
+    have harg : 0 ≤ C₁*Real.log (((k n:ℝ)+1)*R n) := mul_nonneg hC₁.le (Real.log_nonneg hprod)
+    have hceil := (Nat.ceil_lt_add_one harg).le
+    have he : Real.log (((k n:ℝ)+1)*R n) = Real.log ((k n:ℝ)+1)+Real.log (R n) :=
+      Real.log_mul (by positivity) (zero_lt_one.trans_le (hR n)).ne'
+    dsimp only [j]
+    push_cast
+    rw [←add_div,←mul_div_assoc,←add_div,←he]
+    exact (div_le_div_of_nonneg_right (by linarith only [hceil]) hnpos.le)
+  refine ⟨hkt,hk,hjlim,?_⟩
+  apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hjlim
+  · exact Eventually.of_forall (fun n => div_nonneg (Real.log_nonneg (Nat.one_le_cast.mpr (show 1 ≤ j n by dsimp [j]; omega))) (Nat.cast_nonneg n))
+  · exact Eventually.of_forall (fun n => div_le_div_of_nonneg_right (Real.log_le_self (Nat.cast_nonneg (j n))) (Nat.cast_nonneg n))
+
+ 
+
+theorem exists_corridor_scales_of_slopes
+    (ρ p₀ C C₁ ε M cm r z : ℝ) (J k₀ : ℕ)
+    (hρ : 0 < ρ) (hp₀ : 0 < p₀) (hC₁ : 0 < C₁) (hε : 0 < ε) (hcm : 0 < cm)
+    (hJ : 0 < J) (hr : 0 < r) (hz : 0 < z)
+    (hzρ : z+r*Real.log ρ < 0) (hzC : z+C*r < 1)
+    (m R Q : ℕ → ℝ) (hR : ∀ n, 1 ≤ R n) (hQ : ∀ n, 1 ≤ Q n)
+    (hm : ∀ n : ℕ, cm*Real.exp (n:ℝ) ≤ m n)
+    (hlogR : Tendsto (fun n => Real.log (R n)/(n:ℝ)) atTop (𝓝 0))
+    (hlogQ : Tendsto (fun n => Real.log (Q n)/(n:ℝ)) atTop (𝓝 0)) :
+    ∃ (n k : ℕ) (B : ℝ), k₀ ≤ k ∧ 0 < B ∧
+      let j := J+⌈C₁*Real.log (((k:ℝ)+1)*R n)⌉₊
+      B*ρ^k ≤ 1/4 ∧ B*(j:ℝ)*Real.exp (C*(k:ℝ)) ≤ m n ∧
+      (R n)^2*Q n*ρ^k ≤ ε*p₀^(2*j) ∧ M < B*p₀^(2*j) := by
+  let k := fun n : ℕ => ⌈r*(n:ℝ)⌉₊
+  let j := fun n : ℕ => J+⌈C₁*Real.log (((k n:ℝ)+1)*R n)⌉₊
+  let B := fun n : ℕ => Real.exp (z*(n:ℝ))
+  obtain ⟨hkt,hk,hj,hlogj⟩ := corridor_scale_limits r C₁ J hr hC₁ hJ R hR hlogR
+  change Tendsto k atTop atTop at hkt
+  change Tendsto (fun n => (k n:ℝ)/(n:ℝ)) atTop (𝓝 r) at hk
+  change Tendsto (fun n => (j n:ℝ)/(n:ℝ)) atTop (𝓝 0) at hj
+  change Tendsto (fun n => Real.log (j n:ℝ)/(n:ℝ)) atTop (𝓝 0) at hlogj
+  have hn : Tendsto (fun n : ℕ => (n:ℝ)/(n:ℝ)) atTop (𝓝 1) := by
+    simpa only [add_zero] using tendsto_natCast_div_add_atTop (0:ℝ)
+  have hβ (n) : Real.exp ((j n:ℝ)*(2*Real.log p₀)) = p₀^(2*j n) := by
+    rw [show (j n:ℝ)*(2*Real.log p₀) = ((2*j n:ℕ):ℝ)*Real.log p₀ by push_cast; ring,
+      Real.exp_nat_mul,Real.exp_log hp₀]
+  have hjpos (n) : (0:ℝ) < j n := Nat.cast_pos.mpr (by dsimp [j]; omega)
+  have hlim₁ : Tendsto (fun n : ℕ => (z*(n:ℝ)+(k n:ℝ)*Real.log ρ)/(n:ℝ)) atTop (𝓝 (z+r*Real.log ρ)) := by
+    simpa only [mul_one,div_mul_eq_mul_div,←mul_div_assoc,←add_div] using
+      (hn.const_mul z).add (hk.mul_const (Real.log ρ))
+  have hdec₁ := tendsto_exp_of_div_nat_neg _ _ hzρ hlim₁
+  have hlim₂ : Tendsto (fun n : ℕ =>
+      (2*Real.log (R n)+Real.log (Q n)+(k n:ℝ)*Real.log ρ-(j n:ℝ)*(2*Real.log p₀))/(n:ℝ))
+      atTop (𝓝 (r*Real.log ρ)) := by
+    simpa only [mul_zero,zero_add,zero_mul,sub_zero,div_mul_eq_mul_div,←mul_div_assoc,←add_div,←sub_div] using
+      (((hlogR.const_mul 2).add hlogQ).add (hk.mul_const (Real.log ρ))).sub (hj.mul_const (2*Real.log p₀))
+  have hsl₂ : r*Real.log ρ < 0 := by linarith only [hz,hzρ]
+  have hdec₂ := tendsto_exp_of_div_nat_neg _ _ hsl₂ hlim₂
+  have hlim₃ : Tendsto (fun n : ℕ =>
+      (z*(n:ℝ)+Real.log (j n:ℝ)+C*(k n:ℝ)-(n:ℝ))/(n:ℝ)) atTop (𝓝 (z+C*r-1)) := by
+    simpa only [mul_one,add_zero,←mul_div_assoc,←add_div,←sub_div] using
+      (((hn.const_mul z).add hlogj).add (hk.const_mul C)).sub hn
+  have hdec₃ := tendsto_exp_of_div_nat_neg _ _ (sub_neg.mpr hzC) hlim₃
+  have hlim₄ : Tendsto (fun n : ℕ =>
+      (z*(n:ℝ)+(j n:ℝ)*(2*Real.log p₀))/(n:ℝ)) atTop (𝓝 z) := by
+    simpa only [mul_one,zero_mul,add_zero,div_mul_eq_mul_div,←mul_div_assoc,←add_div] using
+      (hn.const_mul z).add (hj.mul_const (2*Real.log p₀))
+  have hgrow := Real.tendsto_exp_atTop.comp (tendsto_atTop_of_div_nat_pos _ _ hz hlim₄)
+  have hfinal : ∀ᶠ n : ℕ in atTop,
+      k₀ ≤ k n ∧ B n*ρ^(k n) ≤ 1/4 ∧
+      B n*(j n:ℝ)*Real.exp (C*(k n:ℝ)) ≤ m n ∧
+      (R n)^2*Q n*ρ^(k n) ≤ ε*p₀^(2*j n) ∧ M < B n*p₀^(2*j n) := by
+    filter_upwards [hkt.eventually_ge_atTop k₀,
+      hdec₁.eventually_lt_const (show (0:ℝ) < 1/4 by norm_num),
+      hdec₂.eventually_lt_const hε,hdec₃.eventually_lt_const hcm,
+      hgrow.eventually_gt_atTop M] with n hk₀ h₁ h₂ h₃ h₄
+    have heρ : Real.exp ((k n:ℝ)*Real.log ρ) = ρ^(k n) := by rw [Real.exp_nat_mul,Real.exp_log hρ]
+    have heR : Real.exp (2*Real.log (R n)) = (R n)^2 := by
+      simpa only [Nat.cast_ofNat,Real.exp_log (zero_lt_one.trans_le (hR n))] using Real.exp_nat_mul (Real.log (R n)) 2
+    simp only [Real.exp_add,heρ] at h₁
+    simp only [Real.exp_sub,Real.exp_add,heR,Real.exp_log (zero_lt_one.trans_le (hQ n)),heρ,hβ n] at h₂
+    simp only [Real.exp_sub,Real.exp_add,Real.exp_log (hjpos n)] at h₃
+    change M < Real.exp (z*(n:ℝ)+(j n:ℝ)*(2*Real.log p₀)) at h₄
+    rw [Real.exp_add,hβ n] at h₄
+    refine ⟨hk₀,h₁.le,?_,(div_le_iff₀ (pow_pos hp₀ _)).mp h₂.le,h₄⟩
+    exact ((div_le_iff₀ (Real.exp_pos _)).mp h₃.le).trans (hm n)
+  obtain ⟨n,hn⟩ := hfinal.exists
+  exact ⟨n,k n,B n,hn.1,Real.exp_pos _,hn.2⟩
+
+ 
+
+theorem exists_corridor_scales
+    (ρ p₀ C C₁ ε M cm : ℝ) (J k₀ : ℕ)
+    (hρ : 0 < ρ) (hρ₁ : ρ < 1) (hp₀ : 0 < p₀) (hC : 0 < C)
+    (hC₁ : 0 < C₁) (hε : 0 < ε) (hcm : 0 < cm) (hJ : 0 < J)
+    (m R Q : ℕ → ℝ) (hR : ∀ n, 1 ≤ R n) (hQ : ∀ n, 1 ≤ Q n)
+    (hm : ∀ n : ℕ, cm*Real.exp (n:ℝ) ≤ m n)
+    (hlogR : Tendsto (fun n => Real.log (R n)/(n:ℝ)) atTop (𝓝 0))
+    (hlogQ : Tendsto (fun n => Real.log (Q n)/(n:ℝ)) atTop (𝓝 0)) :
+    ∃ (n k : ℕ) (B : ℝ), k₀ ≤ k ∧ 0 < B ∧
+      let j := J+⌈C₁*Real.log (((k:ℝ)+1)*R n)⌉₊
+      B*ρ^k ≤ 1/4 ∧ B*(j:ℝ)*Real.exp (C*(k:ℝ)) ≤ m n ∧
+      (R n)^2*Q n*ρ^k ≤ ε*p₀^(2*j) ∧ M < B*p₀^(2*j) := by
+  let r := 1/(4*(C+1))
+  let z := min (1/4) (-(r*Real.log ρ)/2)
+  have hr : 0 < r := by dsimp [r]; positivity
+  have hlog : Real.log ρ < 0 := by simpa only [Real.log_one] using Real.log_lt_log hρ hρ₁
+  have hz : 0 < z := lt_min (by norm_num) (half_pos (neg_pos.mpr (mul_neg_of_pos_of_neg hr hlog)))
+  have hzr : z+r*Real.log ρ < 0 := by
+    have hh : z ≤ -(r*Real.log ρ)/2 := min_le_right _ _
+    linarith only [hh,mul_neg_of_pos_of_neg hr hlog]
+  have hzc : z+C*r < 1 := by
+    have hz' : z ≤ 1/4 := min_le_left _ _
+    have hr' : r*(4*(C+1)) = 1 := div_mul_cancel₀ 1 (by positivity)
+    nlinarith only [hz',hr',hr]
+  exact exists_corridor_scales_of_slopes ρ p₀ C C₁ ε M cm r z J k₀
+    hρ hp₀ hC₁ hε hcm hJ hr hz hzr hzc m R Q hR hQ hm hlogR hlogQ
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Criticality/Sharpness.lean b/lean/OAI/Probability/UnimodularPercolation/Criticality/Sharpness.lean
index d2f6607d126df3a004ca2a7178b50e9c191a01ee..8485bd1992d673afa6619c60aef98def9f0a0345 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Criticality/Sharpness.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Criticality/Sharpness.lean
@@ -1,1542 +1,1542 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Kernel.Sprinkling
-
-namespace OAI
-
-/-! Finite Russo formulae, exterior exposure and subcritical sharpness. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-section FiniteRusso
-attribute [local instance] Classical.propDecidable
-attribute [local instance] Classical.decEq
-
- 
-noncomputable def bernoulliPolynomial (s : Finset E) (f : Finset E → ℝ) (p : ℝ) : ℝ :=
-  ∑ t ∈ s.powerset, f t * p ^ t.card * (1-p) ^ (s.card-t.card)
-
-@[simp] theorem bernoulliPolynomial_empty (f : Finset E → ℝ) (p : ℝ) :
-    bernoulliPolynomial ∅ f p = f ∅ := by simp [bernoulliPolynomial]
-
-theorem bernoulliPolynomial_congr (s : Finset E) {f g : Finset E → ℝ}
-    (h : ∀ t ⊆ s, f t = g t) (p : ℝ) :
-    bernoulliPolynomial s f p = bernoulliPolynomial s g p := by
-  apply Finset.sum_congr rfl
-  intro t ht
-  rw [h t (Finset.mem_powerset.mp ht)]
-
-theorem bernoulliPolynomial_sub (s : Finset E) (f g : Finset E → ℝ) (p : ℝ) :
-    bernoulliPolynomial s (fun t => f t-g t) p =
-      bernoulliPolynomial s f p - bernoulliPolynomial s g p := by
-  simp only [bernoulliPolynomial, sub_mul, Finset.sum_sub_distrib]
-
-theorem bernoulliPolynomial_insert (s : Finset E) (e : E) (he : e ∉ s)
-    (f : Finset E → ℝ) (p : ℝ) :
-    bernoulliPolynomial (insert e s) f p =
-      (1-p)*bernoulliPolynomial s f p + p*bernoulliPolynomial s (fun t => f (insert e t)) p := by
-  have hd : Disjoint s.powerset (s.powerset.image (insert e)) := by
-    apply Finset.disjoint_left.mpr
-    intro t ht himg
-    obtain ⟨r, hr, rfl⟩ := Finset.mem_image.mp himg
-    exact he ((Finset.mem_powerset.mp ht) (by simp))
-  have hinj : Set.InjOn (insert e) (↑s.powerset : Set (Finset E)) := by
-    intro t ht r hr h
-    have ht' : e ∉ t := fun h => he (Finset.mem_powerset.mp ht h)
-    have hr' : e ∉ r := fun h => he (Finset.mem_powerset.mp hr h)
-    have hh := congrArg (fun t : Finset E => t.erase e) h
-    simpa [ht', hr'] using hh
-  simp only [bernoulliPolynomial, Finset.powerset_insert, Finset.sum_union hd,
-    Finset.sum_image hinj, Finset.mul_sum, Finset.card_insert_of_notMem he]
-  congr 1
-  · apply Finset.sum_congr rfl
-    intro t ht
-    have hcard := Finset.card_le_card (Finset.mem_powerset.mp ht)
-    have hexp : s.card+1-t.card = (s.card-t.card)+1 := by omega
-    rw [hexp, pow_succ]
-    ring
-  · apply Finset.sum_congr rfl
-    intro t ht
-    have het : e ∉ t := fun h => he (Finset.mem_powerset.mp ht h)
-    simp only [Finset.card_insert_of_notMem het, Nat.add_sub_add_right, pow_succ]
-    ring
-
- 
-theorem hasDerivAt_bernoulliPolynomial (s : Finset E) (f : Finset E → ℝ) (p : ℝ) :
-    HasDerivAt (bernoulliPolynomial s f)
-      (∑ e ∈ s, bernoulliPolynomial (s.erase e) (fun t => f (insert e t)-f t) p) p := by
-  induction s using Finset.induction_on generalizing f with
-  | empty =>
-    convert hasDerivAt_const p (f ∅) using 1 <;> try rfl
-    · funext q
-      exact bernoulliPolynomial_empty f q
-  | @insert e s he ih =>
-    have hd := (((hasDerivAt_const p 1).sub (hasDerivAt_id p)).mul (ih f)).add
-      ((hasDerivAt_id p).mul (ih (fun t => f (insert e t))))
-    have hf : bernoulliPolynomial (insert e s) f =
-        fun q => (1-q)*bernoulliPolynomial s f q +
-          q*bernoulliPolynomial s (fun t => f (insert e t)) q :=
-      funext fun q => bernoulliPolynomial_insert s e he f q
-    rw [hf]
-    convert hd using 1 <;> try rfl
-    have hs : ∑ a ∈ s, bernoulliPolynomial ((insert e s).erase a)
-          (fun t => f (insert a t)-f t) p =
-        (1-p)*(∑ a ∈ s, bernoulliPolynomial (s.erase a) (fun t => f (insert a t)-f t) p) +
-        p*(∑ a ∈ s, bernoulliPolynomial (s.erase a)
-          (fun t => f (insert e (insert a t))-f (insert e t)) p) := by
-      simp only [Finset.mul_sum, ← Finset.sum_add_distrib]
-      apply Finset.sum_congr rfl
-      intro a ha
-      have hae : e ≠ a := (ne_of_mem_of_not_mem ha he).symm
-      rw [Finset.erase_insert_of_ne hae, bernoulliPolynomial_insert _ e
-        (fun hh => he (Finset.mem_of_mem_erase hh))]
-      congr 2
-      apply bernoulliPolynomial_congr
-      intro t _
-      rw [Finset.insert_comm e a]
-    simp only [Finset.sum_insert he, Finset.erase_insert he, zero_sub, neg_one_mul,
-      one_mul, Pi.sub_apply, id_eq]
-    rw [hs, bernoulliPolynomial_sub]
-    ring
-
-@[simp] theorem bernoulliPolynomial_zero (s : Finset E) (p : ℝ) :
-    bernoulliPolynomial s (fun _ => 0) p = 0 := by simp [bernoulliPolynomial]
-
-theorem bernoulliPolynomial_const (s : Finset E) (c p : ℝ) :
-    bernoulliPolynomial s (fun _ => c) p = c := by
-  induction s using Finset.induction_on with
-  | empty => simp
-  | @insert e s he ih => rw [bernoulliPolynomial_insert s e he]; simp only [ih]; ring
-
-theorem bernoulliPolynomial_nonneg (s : Finset E) (f : Finset E → ℝ) (p : ℝ)
-    (hp : p ∈ Set.Icc (0:ℝ) 1) (hf : ∀ t ⊆ s, 0 ≤ f t) :
-    0 ≤ bernoulliPolynomial s f p := by
-  apply Finset.sum_nonneg
-  intro t ht
-  exact mul_nonneg (mul_nonneg (hf t (Finset.mem_powerset.mp ht))
-    (pow_nonneg hp.1 _)) (pow_nonneg (sub_nonneg.mpr hp.2) _)
-
-theorem bernoulliPolynomial_mono (s : Finset E) (f g : Finset E → ℝ) (p : ℝ)
-    (hp : p ∈ Set.Icc (0:ℝ) 1) (hfg : ∀ t ⊆ s, f t ≤ g t) :
-    bernoulliPolynomial s f p ≤ bernoulliPolynomial s g p := by
-  apply Finset.sum_le_sum
-  intro t ht
-  exact mul_le_mul_of_nonneg_right
-    (mul_le_mul_of_nonneg_right (hfg t (Finset.mem_powerset.mp ht)) (pow_nonneg hp.1 _))
-    (pow_nonneg (sub_nonneg.mpr hp.2) _)
-
- 
-theorem bernoulliPolynomial_union (s t : Finset E) (hdis : Disjoint s t)
-    (f : Finset E → ℝ) (p : ℝ) :
-    bernoulliPolynomial (s ∪ t) f p =
-      bernoulliPolynomial s (fun a => bernoulliPolynomial t (fun b => f (a ∪ b)) p) p := by
-  induction s using Finset.induction_on generalizing f with
-  | empty => simp
-  | @insert e s he ih =>
-    have hed : e ∉ t := (Finset.disjoint_insert_left.mp hdis).1
-    have hst : Disjoint s t := (Finset.disjoint_insert_left.mp hdis).2
-    rw [Finset.insert_union, bernoulliPolynomial_insert _ e (by simp [he, hed]),
-      bernoulliPolynomial_insert s e he, ih hst, ih hst]
-    congr 2
-    apply bernoulliPolynomial_congr
-    intro a _
-    apply bernoulliPolynomial_congr
-    intro b _
-    rw [Finset.insert_union]
-
- 
-
-theorem bernoulliPolynomial_closed_difference (s : Finset E) (e : E) (he : e ∈ s)
-    (f : Finset E → ℝ) (p : ℝ) :
-    bernoulliPolynomial s (fun t => if e ∈ t then 0 else f (insert e t)-f t) p =
-      (1-p)*bernoulliPolynomial (s.erase e) (fun t => f (insert e t)-f t) p := by
-  have hes : e ∉ s.erase e := Finset.notMem_erase e s
-  conv_lhs => rw [← Finset.insert_erase he]
-  rw [bernoulliPolynomial_insert _ e hes]
-  have hc : bernoulliPolynomial (s.erase e)
-      (fun t => if e ∈ t then 0 else f (insert e t)-f t) p =
-      bernoulliPolynomial (s.erase e) (fun t => f (insert e t)-f t) p := by
-    apply bernoulliPolynomial_congr
-    intro t ht
-    simp [show e ∉ t from fun h => hes (ht h)]
-  rw [hc]
-  simp
-
-theorem closed_difference_russo (s : Finset E) (f : Finset E → ℝ) (p : ℝ) :
-    (1-p)*deriv (bernoulliPolynomial s f) p =
-      ∑ e ∈ s, bernoulliPolynomial s
-        (fun t => if e ∈ t then 0 else f (insert e t)-f t) p := by
-  rw [(hasDerivAt_bernoulliPolynomial s f p).deriv, Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro e he
-  exact (bernoulliPolynomial_closed_difference s e he f p).symm
-
- 
-noncomputable def finiteEventPolynomial (s : Finset E) (A : Set (Set E)) (p : ℝ) : ℝ :=
-  bernoulliPolynomial s (fun t => if (t : Set E) ∈ A then 1 else 0) p
-
-theorem finiteEventPolynomial_eq (s : Finset E) (A : Set (Set E)) (p : unitInterval) :
-    finiteEventPolynomial s A (p : ℝ) = finiteProbability p s A := by
-  unfold finiteEventPolynomial bernoulliPolynomial finiteProbability
-  apply Finset.sum_congr rfl
-  intro t _
-  by_cases ht : (t : Set E) ∈ A <;> simp [ht]
-
- 
-def closedPivotal (A : Set (Set E)) (e : E) : Set (Set E) :=
-  {ω | e ∉ ω ∧ ω ∉ A ∧ insert e ω ∈ A}
-
-theorem closedPivotal_difference (A : Set (Set E)) (hA : IsUpperSet A) (e : E)
-    (t : Finset E) :
-    (if (t : Set E) ∈ closedPivotal A e then (1:ℝ) else 0) =
-    (if e ∈ t then 0 else
-      (if ((insert e t : Finset E) : Set E) ∈ A then (1:ℝ) else 0) -
-      (if (t : Set E) ∈ A then 1 else 0)) := by
-  by_cases he : e ∈ t
-  · simp [closedPivotal, he]
-  · by_cases ha : (t : Set E) ∈ A
-    · have hi : insert e (t : Set E) ∈ A := hA (Set.subset_insert e _) ha
-      simp [closedPivotal, he, ha, hi]
-    · by_cases hi : insert e (t : Set E) ∈ A <;> simp [closedPivotal, he, ha, hi]
-
- 
-theorem finite_russo_closed (s : Finset E) (A : Set (Set E)) (hA : IsUpperSet A)
-    (p : unitInterval) :
-    (1-(p:ℝ))*deriv (finiteEventPolynomial s A) (p:ℝ) =
-      ∑ e ∈ s, finiteProbability p s (closedPivotal A e) := by
-  rw [show finiteEventPolynomial s A =
-    bernoulliPolynomial s (fun t => if (t : Set E) ∈ A then 1 else 0) from rfl,
-    closed_difference_russo]
-  apply Finset.sum_congr rfl
-  intro e _
-  rw [← finiteEventPolynomial_eq]
-  apply bernoulliPolynomial_congr
-  intro t _
-  exact (closedPivotal_difference A hA e t).symm
-
- 
-
-theorem finite_russo_closed_law [Fintype E] (A : Set (Set E)) (hA : IsUpperSet A)
-    (p : unitInterval) :
-    (1-(p:ℝ))*deriv (finiteEventPolynomial Finset.univ A) (p:ℝ) =
-      ∑ e : E, (law p).real (closedPivotal A e) := by
-  simpa only [finiteProbability_univ] using finite_russo_closed Finset.univ A hA p
-
-end FiniteRusso
-
-section ExteriorExposure
-attribute [local instance] Classical.propDecidable
-attribute [local instance] Classical.decEq
-
- 
-theorem conn_insert_iff (G : BondGraph V E) (ω : Set E) (e : E) (a b x y : V)
-    (he : G e = s(a,b)) :
-    Conn G (insert e ω) x y ↔ Conn G ω x y ∨
-      (Conn G ω x a ∧ Conn G ω b y) ∨ (Conn G ω x b ∧ Conn G ω a y) := by
-  constructor
-  · intro hh
-    induction hh with
-    | refl => exact Or.inl (conn_refl G ω x)
-    | @tail u v huv huv' ih =>
-      obtain ⟨f,hf,hfuv⟩ := huv'
-      rcases Set.mem_insert_iff.mp hf with rfl | hf
-      · have hp : s(a,b) = s(u,v) := he.symm.trans hfuv
-        rcases Sym2.eq_iff.mp hp with ⟨rfl,rfl⟩ | ⟨rfl,rfl⟩
-        · rcases ih with h | ⟨hxa,hba⟩ | ⟨hxb,haa⟩
-          · exact Or.inr (Or.inl ⟨h,conn_refl G ω b⟩)
-          · exact Or.inr (Or.inl ⟨hxa,conn_refl G ω b⟩)
-          · exact Or.inl hxb
-        · rcases ih with h | ⟨hxa,hbb⟩ | ⟨hxb,hab⟩
-          · exact Or.inr (Or.inr ⟨h,conn_refl G ω a⟩)
-          · exact Or.inl hxa
-          · exact Or.inr (Or.inr ⟨hxb,conn_refl G ω a⟩)
-      · have huvω : Conn G ω u v := .single ⟨f,hf,hfuv⟩
-        rcases ih with h | ⟨hxa,hbu⟩ | ⟨hxb,hau⟩
-        · exact Or.inl (h.trans huvω)
-        · exact Or.inr (Or.inl ⟨hxa,hbu.trans huvω⟩)
-        · exact Or.inr (Or.inr ⟨hxb,hau.trans huvω⟩)
-  · intro hh
-    have hm : ω ⊆ insert e ω := Set.subset_insert e ω
-    have hab : Conn G (insert e ω) a b := .single ⟨e,Set.mem_insert e ω,he⟩
-    rcases hh with h | ⟨hxa,hby⟩ | ⟨hxb,hay⟩
-    · exact conn_mono G hm h
-    · exact (conn_mono G hm hxa).trans (hab.trans (conn_mono G hm hby))
-    · exact (conn_mono G hm hxb).trans ((conn_symm G _ hab).trans (conn_mono G hm hay))
-
- 
-def internalEdges (G : BondGraph V E) (S : Set V) : Set E :=
-  {e | ∀ x ∈ G e, x ∈ S}
-
-@[simp] theorem mem_internalEdges_of_ends (G : BondGraph V E) (S : Set V)
-    (e : E) (a b : V) (he : G e = s(a,b)) :
-    e ∈ internalEdges G S ↔ a ∈ S ∧ b ∈ S := by
-  simp [internalEdges, he]
-
- 
-def unexposed (G : BondGraph V E) (Λ : Set V) (ω : Set E) : Set V :=
-  {x | ∀ y, Conn G ω x y → y ∈ Λ}
-
-theorem unexposed_subset (G : BondGraph V E) (Λ : Set V) (ω : Set E) :
-    unexposed G Λ ω ⊆ Λ := fun x hx => hx x (conn_refl G ω x)
-
-theorem mem_unexposed_of_conn (G : BondGraph V E) (Λ : Set V) (ω : Set E)
-    {x y : V} (hx : x ∈ unexposed G Λ ω) (hxy : Conn G ω x y) :
-    y ∈ unexposed G Λ ω := by
-  intro z hyz
-  exact hx z (hxy.trans hyz)
-
- 
-theorem conn_restrict_unexposed (G : BondGraph V E) (Λ : Set V) (ω : Set E)
-    {x y : V} (hx : x ∈ unexposed G Λ ω) (hxy : Conn G ω x y) :
-    Conn G (ω ∩ internalEdges G (unexposed G Λ ω)) x y := by
-  induction hxy with
-  | refl => exact conn_refl _ _ _
-  | @tail y z hxy hyz ih =>
-    obtain ⟨e,he,hends⟩ := hyz
-    have hy : y ∈ unexposed G Λ ω := mem_unexposed_of_conn G Λ ω hx hxy
-    have hz : z ∈ unexposed G Λ ω := mem_unexposed_of_conn G Λ ω hy (.single ⟨e,he,hends⟩)
-    exact ih.tail ⟨e,⟨he,(mem_internalEdges_of_ends G _ e y z hends).mpr ⟨hy,hz⟩⟩,hends⟩
-
- 
-theorem unexposed_boundary_closed (G : BondGraph V E) (Λ : Set V) (ω : Set E)
-    {e : E} {a b : V} (he : G e = s(a,b))
-    (ha : a ∈ unexposed G Λ ω) (hb : b ∉ unexposed G Λ ω) : e ∉ ω := by
-  intro ho
-  exact hb (mem_unexposed_of_conn G Λ ω ha (.single ⟨e,ho,he⟩))
-
- 
-
-theorem unexposed_insert_crossing (G : BondGraph V E) (Λ : Set V) (ω : Set E)
-    {e : E} {a b x : V} (he : G e = s(a,b))
-    (hx : x ∈ unexposed G Λ ω) (_ha : a ∈ unexposed G Λ ω)
-    (hb : b ∉ unexposed G Λ ω) :
-    x ∉ unexposed G Λ (insert e ω) ↔
-      Conn G (ω ∩ internalEdges G (unexposed G Λ ω)) x a := by
-  classical
-  constructor
-  · intro hn
-    obtain ⟨y,hxy,hy⟩ := by simpa [unexposed, not_forall, Classical.not_imp] using hn
-    rcases (conn_insert_iff G ω e a b x y he).mp hxy with h | h | h
-    · exact (hy (hx y h)).elim
-    · exact conn_restrict_unexposed G Λ ω hx h.1
-    · exact (hb (mem_unexposed_of_conn G Λ ω hx h.1)).elim
-  · intro hxa hn
-    have hxa' : Conn G ω x a := conn_mono G Set.inter_subset_left hxa
-    apply hb
-    intro y hby
-    exact hn y ((conn_insert_iff G ω e a b x y he).mpr (Or.inr (Or.inl ⟨hxa',hby⟩)))
-
- 
-theorem not_mem_unexposed_of_conn (G : BondGraph V E) (Λ : Set V) (ω : Set E)
-    {x y : V} (hx : x ∉ unexposed G Λ ω) (hxy : Conn G ω x y) :
-    y ∉ unexposed G Λ ω := by
-  intro hy
-  exact hx (mem_unexposed_of_conn G Λ ω hy (conn_symm G ω hxy))
-
- 
-theorem conn_sdiff_internal_unexposed (G : BondGraph V E) (Λ : Set V) (ω : Set E)
-    {x y : V} (hx : x ∉ unexposed G Λ ω) (hxy : Conn G ω x y) :
-    Conn G (ω \ internalEdges G (unexposed G Λ ω)) x y := by
-  induction hxy with
-  | refl => exact conn_refl _ _ _
-  | @tail y z hxy hyz ih =>
-    obtain ⟨e,he,hends⟩ := hyz
-    have hy : y ∉ unexposed G Λ ω := not_mem_unexposed_of_conn G Λ ω hx hxy
-    exact ih.tail ⟨e,⟨he,fun hi => hy ((mem_internalEdges_of_ends G _ e y z hends).mp hi).1⟩,hends⟩
-
- 
-theorem unexposed_sdiff_internal (G : BondGraph V E) (Λ : Set V) (ω : Set E) :
-    unexposed G Λ (ω \ internalEdges G (unexposed G Λ ω)) = unexposed G Λ ω := by
-  ext x
-  constructor
-  · intro hx
-    by_contra hn
-    apply hn
-    intro y hxy
-    exact hx y (conn_sdiff_internal_unexposed G Λ ω hn hxy)
-  · intro hx y hxy
-    exact hx y (conn_mono G Set.sdiff_subset hxy)
-
- 
-theorem unexposed_union_internal (G : BondGraph V E) (Λ : Set V) (ω η : Set E)
-    (hη : η ⊆ internalEdges G (unexposed G Λ ω)) :
-    unexposed G Λ (ω ∪ η) = unexposed G Λ ω := by
-  ext x
-  constructor
-  · intro hx y hxy
-    exact hx y (conn_mono G Set.subset_union_left hxy)
-  · intro hx y hxy
-    suffices hy : y ∈ unexposed G Λ ω from unexposed_subset G Λ ω hy
-    induction hxy with
-    | refl => exact hx
-    | @tail y z hxy hyz ih =>
-      obtain ⟨e,hωe,he⟩ := hyz
-      rcases hωe with ho | hηe
-      · exact mem_unexposed_of_conn G Λ ω ih (.single ⟨e,ho,he⟩)
-      · exact ((mem_internalEdges_of_ends G _ e y z he).mp (hη hηe)).2
-
- 
-theorem unexposed_union_eq_iff (G : BondGraph V E) (Λ S : Set V) (a b : Set E)
-    (ha : a ⊆ internalEdges G S) (hb : Disjoint b (internalEdges G S)) :
-    unexposed G Λ (a ∪ b) = S ↔ unexposed G Λ b = S := by
-  constructor
-  · intro h
-    have he : (a ∪ b) \ internalEdges G S = b := by
-      ext e
-      constructor
-      · rintro ⟨he,hn⟩
-        exact he.resolve_left (fun he => hn (ha he))
-      · intro he
-        exact ⟨Or.inr he,fun hi => Set.disjoint_left.mp hb he hi⟩
-    simpa only [h, he] using unexposed_sdiff_internal G Λ (a ∪ b)
-  · intro h
-    have he := unexposed_union_internal G Λ b a (h ▸ ha)
-    simpa only [Set.union_comm b a,h] using he
-
- 
-def escapeEvent (G : BondGraph V E) (Λ : Set V) (x : V) : Set (Set E) :=
-  {ω | x ∉ unexposed G Λ ω}
-
-theorem escapeEvent_isUpperSet (G : BondGraph V E) (Λ : Set V) (x : V) :
-    IsUpperSet (escapeEvent G Λ x) := by
-  intro ω η hωη hx hη
-  apply hx
-  intro y hxy
-  exact hη y (conn_mono G hωη hxy)
-
- 
-theorem escape_closed_pivotal_iff (G : BondGraph V E) (Λ : Set V)
-    (ω : Set E) (x : V) (e : E) :
-    (e ∉ ω ∧ ω ∉ escapeEvent G Λ x ∧ insert e ω ∈ escapeEvent G Λ x) ↔
-      x ∈ unexposed G Λ ω ∧ ∃ a b,
-        G e = s(a,b) ∧ a ∈ unexposed G Λ ω ∧ b ∉ unexposed G Λ ω ∧
-        Conn G (ω ∩ internalEdges G (unexposed G Λ ω)) x a := by
-  classical
-  constructor
-  · rintro ⟨_,hx,hn⟩
-    have hx' : x ∈ unexposed G Λ ω := by simpa [escapeEvent] using hx
-    refine ⟨hx',?_⟩
-    obtain ⟨⟨a,b⟩,he'⟩ := Sym2.mk_surjective (G e)
-    have he : G e = s(a,b) := he'.symm
-    obtain ⟨y,hxy,hy⟩ := by simpa [escapeEvent,unexposed,not_forall,Classical.not_imp] using hn
-    rcases (conn_insert_iff G ω e a b x y he).mp hxy with h | h | h
-    · exact (hy (hx' y h)).elim
-    · exact ⟨a,b,he,mem_unexposed_of_conn G Λ ω hx' h.1,
-        fun hb => hy (hb y h.2),conn_restrict_unexposed G Λ ω hx' h.1⟩
-    · exact ⟨b,a,he.trans (Sym2.eq_swap ..),mem_unexposed_of_conn G Λ ω hx' h.1,
-        fun ha => hy (ha y h.2),conn_restrict_unexposed G Λ ω hx' h.1⟩
-  · rintro ⟨hx,a,b,he,ha,hb,hxa⟩
-    exact ⟨unexposed_boundary_closed G Λ ω he ha hb,by simpa [escapeEvent] using hx,
-      (unexposed_insert_crossing G Λ ω he hx ha hb).mpr hxa⟩
-
-attribute [local instance] Classical.decEq
-
- theorem bernoulliPolynomial_add (s : Finset E) (f g : Finset E → ℝ) (p : ℝ) :
-    bernoulliPolynomial s (fun t => f t+g t) p =
-      bernoulliPolynomial s f p + bernoulliPolynomial s g p := by
-  simp only [bernoulliPolynomial, add_mul, Finset.sum_add_distrib]
-
- theorem bernoulliPolynomial_mul_const (s : Finset E) (f : Finset E → ℝ) (c p : ℝ) :
-    bernoulliPolynomial s (fun t => f t*c) p = bernoulliPolynomial s f p*c := by
-  simp only [bernoulliPolynomial, Finset.sum_mul]
-  apply Finset.sum_congr rfl
-  intro t _
-  ring
-
- theorem bernoulliPolynomial_const_mul (s : Finset E) (f : Finset E → ℝ) (c p : ℝ) :
-    bernoulliPolynomial s (fun t => c*f t) p = c*bernoulliPolynomial s f p := by
-  simpa only [mul_comm c] using bernoulliPolynomial_mul_const s f c p
-
- theorem bernoulliPolynomial_sum {ι : Type*} (s : Finset E) (I : Finset ι)
-    (f : ι → Finset E → ℝ) (p : ℝ) :
-    bernoulliPolynomial s (fun t => ∑ i ∈ I, f i t) p =
-      ∑ i ∈ I, bernoulliPolynomial s (f i) p := by
-  classical
-  simp only [bernoulliPolynomial, Finset.sum_mul]
-  exact Finset.sum_comm
-
- 
-theorem bernoulliPolynomial_blocks (s t : Finset E) (hdis : Disjoint s t)
-    (f g : Finset E → ℝ) (p : ℝ) :
-    bernoulliPolynomial (s ∪ t) (fun a => f (a ∩ s)*g (a ∩ t)) p =
-      bernoulliPolynomial s f p * bernoulliPolynomial t g p := by
-  rw [bernoulliPolynomial_union s t hdis]
-  trans bernoulliPolynomial s (fun a => f a*bernoulliPolynomial t g p) p
-  · apply bernoulliPolynomial_congr
-    intro a ha
-    rw [← bernoulliPolynomial_const_mul]
-    apply bernoulliPolynomial_congr
-    intro b hb
-    have has : a ∩ s = a := Finset.inter_eq_left.mpr ha
-    have hbt : b ∩ t = b := Finset.inter_eq_left.mpr hb
-    have hbs : b ∩ s = ∅ := Finset.disjoint_iff_inter_eq_empty.mp
-      (hdis.symm.mono_left hb)
-    have hat : a ∩ t = ∅ := Finset.disjoint_iff_inter_eq_empty.mp
-      (hdis.mono_left ha)
-    simp [Finset.union_inter_distrib_right,has,hbt,hbs,hat]
-  · exact bernoulliPolynomial_mul_const s f _ p
-
- 
-
-theorem bernoulliPolynomial_exposure (G : BondGraph V E) (Λ S : Set V)
-    (s t : Finset E) (hs : (s : Set E) ⊆ internalEdges G S)
-    (ht : Disjoint (t : Set E) (internalEdges G S)) (f : Finset E → ℝ) (p : ℝ) :
-    bernoulliPolynomial (s ∪ t)
-      (fun a => (if unexposed G Λ (a : Set E) = S then (1:ℝ) else 0) * f (a ∩ s)) p =
-    bernoulliPolynomial t (fun b => if unexposed G Λ (b : Set E) = S then (1:ℝ) else 0) p *
-      bernoulliPolynomial s f p := by
-  have hdis : Disjoint s t := by
-    apply Finset.disjoint_left.mpr
-    intro e he hf
-    exact Set.disjoint_left.mp ht hf (hs he)
-  rw [bernoulliPolynomial_union s t hdis]
-  trans bernoulliPolynomial s (fun a =>
-    bernoulliPolynomial t (fun b => if unexposed G Λ (b : Set E) = S then (1:ℝ) else 0) p * f a) p
-  · apply bernoulliPolynomial_congr
-    intro a ha
-    rw [← bernoulliPolynomial_mul_const]
-    apply bernoulliPolynomial_congr
-    intro b hb
-    have hab : unexposed G Λ ((a ∪ b : Finset E) : Set E) = S ↔
-        unexposed G Λ (b : Set E) = S := by
-      rw [Finset.coe_union]
-      exact unexposed_union_eq_iff G Λ S _ _ (fun _ he => hs (ha he))
-        (ht.mono_left (by exact_mod_cast hb))
-    have hbs : b ∩ s = ∅ :=
-      Finset.disjoint_iff_inter_eq_empty.mp (hdis.symm.mono_left hb)
-    rw [Finset.union_inter_distrib_right,Finset.inter_eq_left.mpr ha,hbs,Finset.union_empty]
-    simp only [hab]
-  · exact bernoulliPolynomial_const_mul s f _ p
-
- 
-noncomputable def unexposedRegion (G : BondGraph V E) (Λ : Finset V) (ω : Set E) : Finset V :=
-  ((Set.toFinite (Λ : Set V)).subset (unexposed_subset G (Λ : Set V) ω)).toFinset
-
-@[simp] theorem coe_unexposedRegion (G : BondGraph V E) (Λ : Finset V) (ω : Set E) :
-    (unexposedRegion G Λ ω : Set V) = unexposed G (Λ : Set V) ω := by
-  simp [unexposedRegion]
-
-theorem unexposedRegion_subset (G : BondGraph V E) (Λ : Finset V) (ω : Set E) :
-    unexposedRegion G Λ ω ⊆ Λ := by
-  intro x hx
-  apply unexposed_subset G (Λ : Set V) ω
-  simpa only [← coe_unexposedRegion,Finset.mem_coe] using hx
-
- 
-theorem bernoulliPolynomial_partition_region (G : BondGraph V E) (Λ : Finset V)
-    (F : Finset E) (f : Finset E → ℝ) (p : ℝ) :
-    bernoulliPolynomial F f p = ∑ S ∈ Λ.powerset,
-      bernoulliPolynomial F
-        (fun t => if unexposed G (Λ : Set V) (t : Set E) = (S : Set V) then f t else 0) p := by
-  rw [← bernoulliPolynomial_sum]
-  apply bernoulliPolynomial_congr
-  intro t _
-  symm
-  rw [Finset.sum_eq_single (unexposedRegion G Λ (t : Set E))]
-  · simp
-  · intro S _ hne
-    have hh : unexposed G (Λ : Set V) (t : Set E) ≠ (S : Set V) := by
-      intro he
-      apply hne
-      apply Finset.coe_injective
-      exact he.symm.trans (coe_unexposedRegion G Λ (t : Set E)).symm
-    simp [hh]
-  · intro hn
-    exact (hn (Finset.mem_powerset.mpr (unexposedRegion_subset G Λ (t : Set E)))).elim
-
- 
-noncomputable def internalSupport (G : BondGraph V E) (S : Set V) (F : Finset E) : Finset E :=
-  F.filter (fun e => e ∈ internalEdges G S)
-
- theorem internalSupport_subset (G : BondGraph V E) (S : Set V) (F : Finset E) :
-    internalSupport G S F ⊆ F := Finset.filter_subset _ _
-
- theorem internalSupport_union_sdiff (G : BondGraph V E) (S : Set V) (F : Finset E) :
-    internalSupport G S F ∪ (F \ internalSupport G S F) = F :=
-  Finset.union_sdiff_of_subset (internalSupport_subset G S F)
-
- theorem coe_inter_internalSupport (G : BondGraph V E) (S : Set V) (F t : Finset E)
-    (ht : t ⊆ F) : ((t ∩ internalSupport G S F : Finset E) : Set E) =
-      (t : Set E) ∩ internalEdges G S := by
-  ext e
-  simp only [Finset.mem_coe,Finset.mem_inter,internalSupport,Finset.mem_filter,Set.mem_inter_iff]
-  exact ⟨fun h => ⟨h.1,h.2.2⟩,fun h => ⟨h.1,ht h.1,h.2⟩⟩
-
- 
-noncomputable def regionProbability (G : BondGraph V E) (Λ S : Set V)
-    (F : Finset E) (p : ℝ) : ℝ :=
-  bernoulliPolynomial F (fun t => if unexposed G Λ (t : Set E) = S then 1 else 0) p
-
- theorem regionProbability_nonneg (G : BondGraph V E) (Λ S : Set V)
-    (F : Finset E) (p : ℝ) (hp : p ∈ Set.Icc (0:ℝ) 1) :
-    0 ≤ regionProbability G Λ S F p := by
-  apply bernoulliPolynomial_nonneg _ _ _ hp
-  intro t _
-  split_ifs <;> norm_num
-
- 
-theorem bernoulliPolynomial_exposure_factor (G : BondGraph V E) (Λ S : Set V)
-    (F : Finset E) (f : Finset E → ℝ) (p : ℝ) :
-    bernoulliPolynomial F
-      (fun t => (if unexposed G Λ (t : Set E) = S then (1:ℝ) else 0) *
-        f (t ∩ internalSupport G S F)) p =
-      regionProbability G Λ S F p * bernoulliPolynomial (internalSupport G S F) f p := by
-  let s := internalSupport G S F
-  let t := F \ s
-  have hs : (s : Set E) ⊆ internalEdges G S := by
-    intro e he
-    exact (Finset.mem_filter.mp he).2
-  have ht : Disjoint (t : Set E) (internalEdges G S) := by
-    apply Set.disjoint_left.mpr
-    intro e he hi
-    have he' := Finset.mem_sdiff.mp he
-    exact he'.2 (Finset.mem_filter.mpr ⟨he'.1,hi⟩)
-  have he : s ∪ t = F := internalSupport_union_sdiff G S F
-  have hq : bernoulliPolynomial t
-      (fun b => if unexposed G Λ (b : Set E) = S then (1:ℝ) else 0) p =
-      regionProbability G Λ S F p := by
-    simpa only [he,mul_one,bernoulliPolynomial_const,regionProbability] using
-      (bernoulliPolynomial_exposure G Λ S s t hs ht (fun _ => 1) p).symm
-  simpa only [he,hq] using bernoulliPolynomial_exposure G Λ S s t hs ht f p
-
- 
-def crossingReach (G : BondGraph V E) (S : Set V) (x : V) (ω : Set E) (e : E) : Prop :=
-  ∃ a b, G e = s(a,b) ∧ a ∈ S ∧ b ∉ S ∧ Conn G ω x a
-
-noncomputable def crossingCount (G : BondGraph V E) (S : Set V) (x : V)
-    (F : Finset E) (ω : Set E) : ℝ :=
-  ∑ e ∈ F, if crossingReach G S x ω e then (1:ℝ) else 0
-
- 
-noncomputable def boundaryExpectation (G : BondGraph V E) (S : Set V) (x : V)
-    (F : Finset E) (p : ℝ) : ℝ :=
-  bernoulliPolynomial (internalSupport G S F)
-    (fun t => crossingCount G S x F (t : Set E)) p
-
-noncomputable def closedEscapeCount (G : BondGraph V E) (Λ : Set V) (x : V)
-    (F : Finset E) (ω : Set E) : ℝ :=
-  ∑ e ∈ F, if ω ∈ closedPivotal (escapeEvent G Λ x) e then (1:ℝ) else 0
-
-theorem finite_escape_russo (G : BondGraph V E) (Λ : Set V) (x : V)
-    (F : Finset E) (p : ℝ) :
-    (1-p)*deriv (finiteEventPolynomial F (escapeEvent G Λ x)) p =
-      bernoulliPolynomial F (fun t => closedEscapeCount G Λ x F (t : Set E)) p := by
-  unfold finiteEventPolynomial
-  rw [closed_difference_russo]
-  unfold closedEscapeCount
-  rw [bernoulliPolynomial_sum]
-  apply Finset.sum_congr rfl
-  intro e _
-  apply bernoulliPolynomial_congr
-  intro t _
-  exact (closedPivotal_difference _ (escapeEvent_isUpperSet G Λ x) e t).symm
-
- 
-theorem closedEscapeCount_expectation (G : BondGraph V E) (Λ : Finset V) (x : V)
-    (F : Finset E) (p : ℝ) :
-    bernoulliPolynomial F (fun t => closedEscapeCount G Λ x F (t : Set E)) p =
-      ∑ S ∈ Λ.powerset, if x ∈ S then
-        regionProbability G Λ S F p * boundaryExpectation G S x F p else 0 := by
-  rw [bernoulliPolynomial_partition_region G Λ F]
-  apply Finset.sum_congr rfl
-  intro S _
-  by_cases hx : x ∈ S
-  · rw [ite_eq_left hx]
-    unfold boundaryExpectation
-    rw [← bernoulliPolynomial_exposure_factor G Λ S F
-      (fun t => crossingCount G S x F (t : Set E)) p]
-    apply bernoulliPolynomial_congr
-    intro t ht
-    by_cases hreg : unexposed G (Λ : Set V) (t : Set E) = (S : Set V)
-    · simp only [hreg,ite_true,one_mul]
-      apply Finset.sum_congr rfl
-      intro e _
-      have hc : (t : Set E) ∈ closedPivotal (escapeEvent G Λ x) e ↔
-          crossingReach G S x ((t ∩ internalSupport G S F : Finset E) : Set E) e := by
-        simpa only [closedPivotal,Set.mem_ofPred_eq,hreg,Finset.mem_coe,hx,true_and,
-          coe_inter_internalSupport G S F t ht,crossingReach] using
-          escape_closed_pivotal_iff G Λ (t : Set E) x e
-      exact if_congr hc rfl rfl
-    · simp [hreg]
-  · rw [ite_eq_right hx,← bernoulliPolynomial_zero F p]
-    apply bernoulliPolynomial_congr
-    intro t _
-    by_cases hreg : unexposed G (Λ : Set V) (t : Set E) = (S : Set V)
-    · simp only [hreg,ite_true]
-      apply Finset.sum_eq_zero
-      intro e _
-      have hn : (t : Set E) ∉ closedPivotal (escapeEvent G Λ x) e := by
-        intro he
-        have hh := ((escape_closed_pivotal_iff G Λ (t : Set E) x e).mp he).1
-        apply hx
-        simpa only [hreg,Finset.mem_coe] using hh
-      simp [hn]
-    · simp [hreg]
-
- 
-theorem finite_escape_complement (G : BondGraph V E) (Λ : Finset V) (x : V)
-    (F : Finset E) (p : ℝ) :
-    1-finiteEventPolynomial F (escapeEvent G Λ x) p =
-      ∑ S ∈ Λ.powerset, if x ∈ S then regionProbability G Λ S F p else 0 := by
-  have he : 1-finiteEventPolynomial F (escapeEvent G Λ x) p =
-      bernoulliPolynomial F
-        (fun t => if x ∈ unexposed G (Λ : Set V) (t : Set E) then (1:ℝ) else 0) p := by
-    change 1-bernoulliPolynomial F (fun t => if (t : Set E) ∈ escapeEvent G Λ x then 1 else 0) p = _
-    rw [show (1:ℝ) = bernoulliPolynomial F (fun _ => 1) p from (bernoulliPolynomial_const F 1 p).symm]
-    rw [← bernoulliPolynomial_sub]
-    apply bernoulliPolynomial_congr
-    intro t _
-    by_cases hx : x ∈ unexposed G (Λ : Set V) (t : Set E) <;> simp [escapeEvent,hx,bernoulliPolynomial_const]
-  rw [he,bernoulliPolynomial_partition_region G Λ F]
-  apply Finset.sum_congr rfl
-  intro S _
-  by_cases hx : x ∈ S
-  · rw [ite_eq_left hx]
-    apply bernoulliPolynomial_congr
-    intro t _
-    by_cases hreg : unexposed G (Λ : Set V) (t : Set E) = (S : Set V) <;> simp [hreg,hx]
-  · rw [ite_eq_right hx,← bernoulliPolynomial_zero F p]
-    apply bernoulliPolynomial_congr
-    intro t _
-    by_cases hreg : unexposed G (Λ : Set V) (t : Set E) = (S : Set V) <;> simp [hreg,hx]
-
- 
-
-theorem finite_escape_differential_lower (G : BondGraph V E) (Λ : Finset V) (x : V)
-    (F : Finset E) (p : ℝ) (hp : p ∈ Set.Icc (0:ℝ) 1)
-    (hcost : ∀ S ⊆ Λ, x ∈ S → 1 ≤ p*boundaryExpectation G (S : Set V) x F p) :
-    1-finiteEventPolynomial F (escapeEvent G Λ x) p ≤
-      p*(1-p)*deriv (finiteEventPolynomial F (escapeEvent G Λ x)) p := by
-  rw [mul_assoc,finite_escape_russo,closedEscapeCount_expectation,finite_escape_complement,
-    Finset.mul_sum]
-  apply Finset.sum_le_sum
-  intro S hS
-  by_cases hx : x ∈ S
-  · simp only [ite_eq_left hx]
-    have hq := regionProbability_nonneg G Λ S F p hp
-    calc
-      regionProbability G Λ S F p ≤
-          regionProbability G Λ S F p * (p*boundaryExpectation G S x F p) :=
-        le_mul_of_one_le_right hq (hcost S (Finset.mem_powerset.mp hS) hx)
-      _ = p*(regionProbability G Λ S F p * boundaryExpectation G S x F p) := by ring
-  · simp [hx]
-
- theorem bernoulliPolynomial_monotoneOn (F : Finset E) (f : Finset E → ℝ)
-    (hf : ∀ a b, a ⊆ b → f a ≤ f b) :
-    MonotoneOn (bernoulliPolynomial F f) (Set.Icc (0:ℝ) 1) := by
-  have hd : Differentiable ℝ (bernoulliPolynomial F f) :=
-    fun q => (hasDerivAt_bernoulliPolynomial F f q).differentiableAt
-  apply monotoneOn_of_deriv_nonneg (convex_Icc 0 1) hd.continuous.continuousOn hd.differentiableOn
-  intro q hq
-  have hq' : q ∈ Set.Icc (0:ℝ) 1 := interior_subset hq
-  rw [(hasDerivAt_bernoulliPolynomial F f q).deriv]
-  apply Finset.sum_nonneg
-  intro e _
-  apply bernoulliPolynomial_nonneg _ _ _ hq'
-  intro t _
-  exact sub_nonneg.mpr (hf t (insert e t) (Finset.subset_insert e t))
-
- theorem crossingCount_mono (G : BondGraph V E) (S : Set V) (x : V)
-    (F : Finset E) {ω η : Set E} (h : ω ⊆ η) :
-    crossingCount G S x F ω ≤ crossingCount G S x F η := by
-  apply Finset.sum_le_sum
-  intro e _
-  have hh : crossingReach G S x ω e → crossingReach G S x η e := by
-    rintro ⟨a,b,he,ha,hb,hc⟩
-    exact ⟨a,b,he,ha,hb,conn_mono G h hc⟩
-  by_cases hω : crossingReach G S x ω e
-  · simp [hω,hh hω]
-  · simp only [ite_eq_right hω]
-    split_ifs <;> norm_num
-
- theorem boundaryExpectation_nonneg (G : BondGraph V E) (S : Set V) (x : V)
-    (F : Finset E) (p : ℝ) (hp : p ∈ Set.Icc (0:ℝ) 1) :
-    0 ≤ boundaryExpectation G S x F p := by
-  apply bernoulliPolynomial_nonneg _ _ _ hp
-  intro t _
-  apply Finset.sum_nonneg
-  intro e _
-  split_ifs <;> norm_num
-
- theorem boundaryExpectation_monotoneOn (G : BondGraph V E) (S : Set V) (x : V)
-    (F : Finset E) : MonotoneOn (boundaryExpectation G S x F) (Set.Icc (0:ℝ) 1) := by
-  apply bernoulliPolynomial_monotoneOn
-  intro a b hab
-  exact crossingCount_mono G S x F (by exact_mod_cast hab)
-
- theorem boundaryCost_monotoneOn (G : BondGraph V E) (S : Set V) (x : V)
-    (F : Finset E) : MonotoneOn (fun p => p*boundaryExpectation G S x F p)
-      (Set.Icc (0:ℝ) 1) := by
-  intro p hp q hq hpq
-  exact mul_le_mul hpq (boundaryExpectation_monotoneOn G S x F hp hq hpq)
-    (boundaryExpectation_nonneg G S x F p hp) hq.1
-
- theorem finiteEventPolynomial_mem_Icc (F : Finset E) (A : Set (Set E))
-    (p : ℝ) (hp : p ∈ Set.Icc (0:ℝ) 1) :
-    finiteEventPolynomial F A p ∈ Set.Icc (0:ℝ) 1 := by
-  constructor
-  · apply bernoulliPolynomial_nonneg _ _ _ hp
-    intro t _
-    split_ifs <;> norm_num
-  · calc
-      finiteEventPolynomial F A p ≤ bernoulliPolynomial F (fun _ => 1) p := by
-        apply bernoulliPolynomial_mono _ _ _ _ hp
-        intro t _
-        split_ifs <;> norm_num
-      _ = 1 := bernoulliPolynomial_const F 1 p
-
- 
-
-theorem differential_escape_growth {f : ℝ → ℝ} {r p : ℝ}
-    (hr : 0 < r) (hrp : r ≤ p) (hp : p < 1)
-    (hf : Differentiable ℝ f) (hbound : ∀ q ∈ Set.Icc r p, f q ∈ Set.Icc (0:ℝ) 1)
-    (hineq : ∀ q ∈ Set.Icc r p, 1-f q ≤ q*(1-q)*deriv f q) :
-    1-Real.exp (r-p) ≤ f p := by
-  have hd (q : ℝ) : HasDerivAt (fun u => Real.exp u*(1-f u))
-      (Real.exp q*(1-f q)+Real.exp q*(0-deriv f q)) q :=
-    (Real.hasDerivAt_exp q).mul ((hasDerivAt_const q 1).sub (hf q).hasDerivAt)
-  have hanti : AntitoneOn (fun q => Real.exp q*(1-f q)) (Set.Icc r p) := by
-    apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc r p)
-      (Real.continuous_exp.mul (continuous_const.sub hf.continuous)).continuousOn
-      (fun q _ => (hd q).hasDerivWithinAt)
-    intro q hq
-    have hqi : q ∈ Set.Icc r p := interior_subset hq
-    have hq0 : 0 < q := hr.trans_le hqi.1
-    have hq1 : q < 1 := hqi.2.trans_lt hp
-    have hcoef : 0 < q*(1-q) := mul_pos hq0 (sub_pos.mpr hq1)
-    have hder : 0 ≤ deriv f q := nonneg_of_mul_nonneg_right
-      ((sub_nonneg.mpr (hbound q hqi).2).trans (hineq q hqi)) hcoef
-    have hc : q*(1-q) ≤ 1 := by nlinarith [sq_nonneg q]
-    have hsmall : 1-f q ≤ deriv f q := (hineq q hqi).trans
-      (mul_le_of_le_one_left hder hc)
-    have hn := mul_nonpos_of_nonneg_of_nonpos (Real.exp_pos q).le
-      (show 1-f q-deriv f q ≤ 0 by linarith)
-    nlinarith only [hn]
-  have hle := hanti (show r ∈ Set.Icc r p from ⟨le_rfl,hrp⟩)
-    (show p ∈ Set.Icc r p from ⟨hrp,le_rfl⟩) hrp
-  have he : Real.exp p*(1-f p) ≤ Real.exp r := by
-    calc
-      Real.exp p*(1-f p) ≤ Real.exp r*(1-f r) := hle
-      _ ≤ Real.exp r := mul_le_of_le_one_right (Real.exp_pos r).le
-        (by linarith [(hbound r ⟨le_rfl,hrp⟩).1])
-  have hfin : 1-f p ≤ Real.exp (r-p) := by
-    rw [Real.exp_sub]
-    apply (le_div_iff₀ (Real.exp_pos p)).mpr
-    simpa only [mul_comm] using he
-  linarith
-
- 
-
-theorem finite_escape_probability_lower (G : BondGraph V E) (Λ : Finset V) (x : V)
-    (F : Finset E) (r p : ℝ) (hr : 0 < r) (hrp : r < p) (hp : p < 1)
-    (hcost : ∀ S ⊆ Λ, x ∈ S → 1 ≤ r*boundaryExpectation G (S : Set V) x F r) :
-    0 < 1-Real.exp (r-p) ∧
-      1-Real.exp (r-p) ≤ finiteEventPolynomial F (escapeEvent G Λ x) p := by
-  have hrI : r ∈ Set.Icc (0:ℝ) 1 := ⟨hr.le,(hrp.trans hp).le⟩
-  have hqI (q : ℝ) (hq : q ∈ Set.Icc r p) : q ∈ Set.Icc (0:ℝ) 1 :=
-    ⟨hr.le.trans hq.1,hq.2.trans hp.le⟩
-  refine ⟨sub_pos.mpr (Real.exp_lt_one_iff.mpr (sub_neg.mpr hrp)),?_⟩
-  apply differential_escape_growth hr hrp.le hp
-  · intro q
-    exact (hasDerivAt_bernoulliPolynomial F _ q).differentiableAt
-  · intro q hq
-    exact finiteEventPolynomial_mem_Icc F _ q (hqI q hq)
-  · intro q hq
-    apply finite_escape_differential_lower G Λ x F q (hqI q hq)
-    intro S hS hx
-    exact (hcost S hS hx).trans (boundaryCost_monotoneOn G S x F hrI (hqI q hq) hq.1)
-
-end ExteriorExposure
-
-section SharpnessBridge
-attribute [local instance] Classical.propDecidable Classical.decEq
-open Filter Topology
-
-theorem bernoulliPolynomial_map {F : Type*} (f : E ↪ F) (s : Finset E)
-    (h : Finset F → ℝ) (p : ℝ) :
-    bernoulliPolynomial (s.map f) h p =
-      bernoulliPolynomial s (fun t => h (t.map f)) p := by
-  induction s using Finset.induction_on generalizing h with
-  | empty => simp
-  | @insert e s he ih =>
-    rw [Finset.map_insert,bernoulliPolynomial_insert _ (f e) (by simpa using he),
-      bernoulliPolynomial_insert s e he,ih,ih]
-    simp only [Finset.map_insert]
-
-theorem finiteEventPolynomial_restrict (F : Finset E) (A : Set (Set E)) (p : unitInterval) :
-    finiteEventPolynomial F A (p : ℝ) = (law p).real {ω | ω ∩ (F : Set E) ∈ A} := by
-  let f : (F : Set E) ↪ E := ⟨Subtype.val,Subtype.val_injective⟩
-  have hm : (Finset.univ : Finset (F : Set E)).map f = F := by
-    ext e
-    simp [f]
-  have he : finiteEventPolynomial F A (p : ℝ) =
-      finiteEventPolynomial (Finset.univ : Finset (F : Set E))
-        {η | f '' η ∈ A} (p : ℝ) := by
-    unfold finiteEventPolynomial
-    calc
-      bernoulliPolynomial F (fun t => if (t : Set E) ∈ A then 1 else 0) (p : ℝ) =
-        bernoulliPolynomial (Finset.univ.map f) (fun t => if (t : Set E) ∈ A then 1 else 0) (p : ℝ) :=
-          congrArg (fun H => bernoulliPolynomial H (fun t => if (t : Set E) ∈ A then 1 else 0) (p : ℝ)) hm.symm
-      _ = _ := by
-        rw [bernoulliPolynomial_map]
-        apply bernoulliPolynomial_congr
-        intro t _
-        simp only [Finset.coe_map]
-        rfl
-  rw [he,finiteEventPolynomial_eq,finiteProbability_univ,← law_map_restrict p (F : Set E),
-    measureReal_def,Measure.map_apply (measurable_set_restrict _) .of_discrete]
-  congr 2
-  ext ω
-  have hh : f '' (Subtype.val ⁻¹' ω : Set (F : Set E)) = ω ∩ (F : Set E) := by
-    ext e
-    simp [f,and_comm]
-  simp only [Set.mem_preimage,Set.mem_ofPred_eq,hh]
-
-theorem bernoulliPolynomial_restrict (F H : Finset E) (hFH : F ⊆ H)
-    (f : Finset E → ℝ) (p : ℝ) :
-    bernoulliPolynomial H (fun t => f (t ∩ F)) p = bernoulliPolynomial F f p := by
-  rw [← Finset.union_sdiff_of_subset hFH,bernoulliPolynomial_union _ _ Finset.disjoint_sdiff]
-  trans bernoulliPolynomial F (fun a => f a) p
-  · apply bernoulliPolynomial_congr
-    intro a ha
-    trans bernoulliPolynomial (H \ F) (fun _ => f a) p
-    · apply bernoulliPolynomial_congr
-      intro b hb
-      have hdis : Disjoint b F := (Finset.disjoint_sdiff.symm).mono_left hb
-      rw [Finset.union_inter_distrib_right,Finset.inter_eq_left.mpr ha,
-        Finset.disjoint_iff_inter_eq_empty.mp hdis,Finset.union_empty]
-    · exact bernoulliPolynomial_const _ _ _
-  · rfl
-
- 
-noncomputable def incidentSupport (G : BondGraph V E) (hloc : LocallyFinite G)
-    (Λ : Finset V) : Finset E := Λ.biUnion (fun x => (hloc x).toFinset)
-
-@[simp] theorem mem_incidentSupport (G : BondGraph V E) (hloc : LocallyFinite G)
-    (Λ : Finset V) (e : E) : e ∈ incidentSupport G hloc Λ ↔ ∃ x ∈ Λ, x ∈ G e := by
-  simp [incidentSupport]
-
-theorem incidentSupport_mono (G : BondGraph V E) (hloc : LocallyFinite G)
-    {S Λ : Finset V} (hS : S ⊆ Λ) : incidentSupport G hloc S ⊆ incidentSupport G hloc Λ := by
-  intro e he
-  obtain ⟨x,hx,hxe⟩ := (mem_incidentSupport G hloc S e).mp he
-  exact (mem_incidentSupport G hloc Λ e).mpr ⟨x,hS hx,hxe⟩
-
- 
-theorem escapeEvent_restrict_incident (G : BondGraph V E) (hloc : LocallyFinite G)
-    (Λ : Finset V) (x : V) (ω : Set E) :
-    ω ∩ (incidentSupport G hloc Λ : Set E) ∈ escapeEvent G Λ x ↔
-      ω ∈ escapeEvent G Λ x := by
-  constructor
-  · exact escapeEvent_isUpperSet G Λ x Set.inter_subset_left
-  · intro hx
-    by_contra hn
-    have hx' : x ∈ unexposed G Λ (ω ∩ (incidentSupport G hloc Λ : Set E)) := by
-      simpa only [escapeEvent,Set.mem_ofPred_eq,not_not] using hn
-    apply hx
-    intro y hxy
-    have hc : Conn G (ω ∩ (incidentSupport G hloc Λ : Set E)) x y := by
-      induction hxy with
-      | refl => exact conn_refl _ _ _
-      | @tail a b hxa hab ih =>
-        obtain ⟨e,he,heab⟩ := hab
-        have ha : a ∈ Λ := hx' a ih
-        have hF : e ∈ incidentSupport G hloc Λ :=
-          (mem_incidentSupport G hloc Λ e).mpr ⟨a,ha,by simp [heab]⟩
-        exact ih.tail ⟨e,⟨he,hF⟩,heab⟩
-    exact hx' y hc
-
-theorem finite_escape_eq_law (G : BondGraph V E) (hloc : LocallyFinite G)
-    (Λ : Finset V) (x : V) (p : unitInterval) :
-    finiteEventPolynomial (incidentSupport G hloc Λ) (escapeEvent G Λ x) (p : ℝ) =
-      (law p).real (escapeEvent G Λ x) := by
-  rw [finiteEventPolynomial_restrict]
-  congr 1
-  ext ω
-  exact escapeEvent_restrict_incident G hloc Λ x ω
-
-theorem internal_edge_mem_incident (G : BondGraph V E) (hloc : LocallyFinite G)
-    (S : Finset V) {e : E} (he : e ∈ internalEdges G (S : Set V)) :
-    e ∈ incidentSupport G hloc S := by
-  obtain ⟨⟨a,b⟩,hab⟩ := Sym2.mk_surjective (G e)
-  have ha : a ∈ G e := by rw [← hab]; simp
-  exact (mem_incidentSupport G hloc S e).mpr ⟨a,he a ha,ha⟩
-
-theorem internalSupport_eq_of_incident_subset (G : BondGraph V E) (hloc : LocallyFinite G)
-    (S : Finset V) (F : Finset E) (hF : incidentSupport G hloc S ⊆ F) :
-    internalSupport G S F = internalSupport G S (incidentSupport G hloc S) := by
-  ext e
-  simp only [internalSupport,Finset.mem_filter]
-  exact ⟨fun h => ⟨internal_edge_mem_incident G hloc S h.2,h.2⟩,
-    fun h => ⟨hF h.1,h.2⟩⟩
-
-theorem crossingCount_eq_of_incident_subset (G : BondGraph V E) (hloc : LocallyFinite G)
-    (S : Finset V) (x : V) (F : Finset E) (hF : incidentSupport G hloc S ⊆ F) (ω : Set E) :
-    crossingCount G S x F ω = crossingCount G S x (incidentSupport G hloc S) ω := by
-  symm
-  apply Finset.sum_subset hF
-  intro e _ he
-  have hn : ¬ crossingReach G S x ω e := by
-    rintro ⟨a,b,hab,ha,_,_⟩
-    exact he ((mem_incidentSupport G hloc S e).mpr ⟨a,ha,by simp [hab]⟩)
-  simp [hn]
-
-theorem boundaryExpectation_eq_of_incident_subset (G : BondGraph V E) (hloc : LocallyFinite G)
-    (S : Finset V) (x : V) (F : Finset E) (hF : incidentSupport G hloc S ⊆ F) (p : ℝ) :
-    boundaryExpectation G S x F p =
-      boundaryExpectation G S x (incidentSupport G hloc S) p := by
-  unfold boundaryExpectation
-  rw [internalSupport_eq_of_incident_subset G hloc S F hF]
-  apply bernoulliPolynomial_congr
-  intro t _
-  exact crossingCount_eq_of_incident_subset G hloc S x F hF t
-
- 
-noncomputable def boundaryCost (G : BondGraph V E) (hloc : LocallyFinite G)
-    (S : Finset V) (x : V) (p : ℝ) : ℝ :=
-  p*boundaryExpectation G S x (incidentSupport G hloc S) p
-
-theorem measurableSet_escapeEvent (G : BondGraph V E) (hloc : LocallyFinite G)
-    (Λ : Finset V) (x : V) : MeasurableSet (escapeEvent G Λ x) := by
-  let F := incidentSupport G hloc Λ
-  have he : escapeEvent G Λ x =
-      (fun ω => (Subtype.val ⁻¹' ω : Set (F : Set E))) ⁻¹'
-        {η | Subtype.val '' η ∈ escapeEvent G Λ x} := by
-    ext ω
-    have hh : Subtype.val '' (Subtype.val ⁻¹' ω : Set (F : Set E)) = ω ∩ (F : Set E) := by
-      ext e
-      simp [and_comm]
-    simp only [Set.mem_preimage,Set.mem_ofPred_eq,hh]
-    exact (escapeEvent_restrict_incident G hloc Λ x ω).symm
-  rw [he]
-  exact MeasurableSet.preimage .of_discrete (measurable_set_restrict _)
-
-theorem escapeEvent_antitone_domain (G : BondGraph V E) {S Λ : Set V} (hS : S ⊆ Λ)
-    (x : V) : escapeEvent G Λ x ⊆ escapeEvent G S x := by
-  intro ω hω hn
-  apply hω
-  intro y hxy
-  exact hS (hn y hxy)
-
-noncomputable def vertexExhaustion [Encodable V] (n : ℕ) : Finset V :=
-  (Finset.range n).preimage Encodable.encode Encodable.encode_injective.injOn
-
-theorem vertexExhaustion_mono [Encodable V] :
-    Monotone (vertexExhaustion : ℕ → Finset V) := by
-  intro m n h x hx
-  simp only [vertexExhaustion,Finset.mem_preimage,Finset.mem_range] at hx ⊢
-  exact hx.trans_le h
-
-theorem subset_vertexExhaustion [Encodable V] (S : Finset V) :
-    ∃ n, S ⊆ vertexExhaustion n := by
-  refine ⟨S.sup Encodable.encode+1,?_⟩
-  intro x hx
-  simp only [vertexExhaustion,Finset.mem_preimage,Finset.mem_range]
-  exact Nat.lt_succ_of_le (Finset.le_sup hx)
-
-theorem infinite_cluster_eq_iInter_escape [Encodable V] (G : BondGraph V E) (x : V) :
-    {ω | (cluster G ω x).Infinite} = ⋂ n, escapeEvent G (vertexExhaustion (V := V) n : Set V) x := by
-  ext ω
-  simp only [Set.mem_ofPred_eq,Set.mem_iInter]
-  constructor
-  · intro hi n hn
-    exact hi ((vertexExhaustion n).finite_toSet.subset (by
-      intro y hy; exact hn y hy))
-  · intro he hf
-    obtain ⟨n,hn⟩ := subset_vertexExhaustion hf.toFinset
-    apply he n
-    intro y hy
-    exact hn (hf.mem_toFinset.mpr hy)
-
- 
-theorem infinite_cluster_probability_ge_of_escape [Countable V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) (p : unitInterval) (c : ℝ)
-    (hc : ∀ Λ : Finset V, c ≤ (law p).real (escapeEvent G Λ x)) :
-    c ≤ (law p).real {ω | (cluster G ω x).Infinite} := by
-  let := Encodable.ofCountable V
-  rw [infinite_cluster_eq_iInter_escape]
-  have hm : Antitone (fun n => escapeEvent G (vertexExhaustion (V := V) n : Set V) x) := by
-    intro m n h
-    exact escapeEvent_antitone_domain G (by exact_mod_cast vertexExhaustion_mono h) x
-  have ht := tendsto_measure_iInter_atTop (μ := law p)
-    (fun n => (measurableSet_escapeEvent G hloc (vertexExhaustion n) x).nullMeasurableSet)
-    hm ⟨0,measure_ne_top _ _⟩
-  have hr := (ENNReal.tendsto_toReal (measure_ne_top _ _)).comp ht
-  exact ge_of_tendsto hr (Eventually.of_forall (fun n => hc (vertexExhaustion n)))
-
- 
-
-theorem infinite_cluster_probability_pos_of_bad_cost [Countable V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) (r p : unitInterval)
-    (hr : 0 < r) (hrp : r < p) (hp : p < 1)
-    (hcost : ∀ S : Finset V, x ∈ S → 1 ≤ boundaryCost G hloc S x (r : ℝ)) :
-    0 < (law p).real {ω | (cluster G ω x).Infinite} := by
-  have hc : 0 < 1-Real.exp ((r:ℝ)-(p:ℝ)) :=
-    sub_pos.mpr (Real.exp_lt_one_iff.mpr (sub_neg.mpr hrp))
-  apply hc.trans_le
-  apply infinite_cluster_probability_ge_of_escape G hloc x p
-  intro Λ
-  rw [← finite_escape_eq_law G hloc Λ x p]
-  apply (finite_escape_probability_lower G Λ x (incidentSupport G hloc Λ)
-    (r:ℝ) (p:ℝ) hr hrp hp ?_).2
-  intro S hS hx
-  rw [boundaryExpectation_eq_of_incident_subset G hloc S x _ (incidentSupport_mono G hloc hS)]
-  exact hcost S hx
-
-theorem pc_le_of_infinite_cluster_probability_pos (G : BondGraph V E)
-    (x : V) (p : unitInterval) (hp : 0 < (law p).real {ω | (cluster G ω x).Infinite}) :
-    pc G ≤ p := by
-  apply sInf_le
-  have hh : 0 < law p {ω | (cluster G ω x).Infinite} := (ENNReal.toReal_pos_iff.mp hp).1
-  apply hh.trans_le
-  apply measure_mono
-  intro ω hω
-  exact ⟨cluster G ω x,hω,x,rfl⟩
-
- 
-theorem exists_subcritical_testSet [Countable V] (G : BondGraph V E)
-    (hloc : LocallyFinite G) (p : unitInterval) (hp : p < pc G) (x : V) :
-    ∃ S : Finset V, x ∈ S ∧ boundaryCost G hloc S x (p:ℝ) < 1 := by
-  by_cases hz : p = 0
-  · exact ⟨{x},by simp,by simp [hz,boundaryCost]⟩
-  · have hp0 : 0 < p := bot_lt_iff_ne_bot.mpr hz
-    by_contra! hn
-    obtain ⟨q,hpq,hqc⟩ := exists_between hp
-    have hq1 : q < 1 := hqc.trans_le (show pc G ≤ 1 from (pc G).2.2)
-    have hs := infinite_cluster_probability_pos_of_bad_cost G hloc x p q hp0 hpq hq1 hn
-    exact (not_le_of_gt hqc) (pc_le_of_infinite_cluster_probability_pos G x q hs)
-
-theorem internalEdges_aut_iff (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (h : IsAut G a b) (S : Set V) (e : E) :
-    b e ∈ internalEdges G (a '' S) ↔ e ∈ internalEdges G S := by
-  constructor
-  · intro he x hx
-    have hh := he (a x) (by rw [h e]; exact Sym2.mem_map.mpr ⟨x,hx,rfl⟩)
-    simpa using hh
-  · intro he y hy
-    rw [h e,Sym2.mem_map] at hy
-    obtain ⟨x,hx,rfl⟩ := hy
-    exact Set.mem_image_of_mem a (he x hx)
-
-theorem crossingReach_aut_iff (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (h : IsAut G a b) (S : Set V) (x : V) (ω : Set E) (e : E) :
-    crossingReach G (a '' S) (a x) (b '' ω) (b e) ↔ crossingReach G S x ω e := by
-  constructor
-  · rintro ⟨c,d,he,hc,hd,hconn⟩
-    have he' : G e = s(a.symm c,a.symm d) := by
-      have hh := congrArg (Sym2.map a.symm) he
-      rw [h e,Sym2.map_map,Equiv.symm_comp_self,Sym2.map_id,Sym2.map_mk] at hh
-      exact hh
-    refine ⟨a.symm c,a.symm d,he',?_,?_,?_⟩
-    · simpa using hc
-    · simpa using hd
-    · have hh := (conn_aut_iff G h (b '' ω) x (a.symm c)).mpr (by simpa using hconn)
-      simpa only [Set.preimage_image_eq _ b.injective] using hh
-  · rintro ⟨c,d,he,hc,hd,hconn⟩
-    refine ⟨a c,a d,?_,by simpa using hc,by simpa using hd,conn_aut G h hconn⟩
-    rw [h e,he,Sym2.map_mk]
-
-theorem internalSupport_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (h : IsAut G a b) (S : Set V) (F : Finset E) :
-    internalSupport G (a '' S) (F.map b.toEmbedding) =
-      (internalSupport G S F).map b.toEmbedding := by
-  ext e
-  obtain ⟨d,rfl⟩ := b.surjective e
-  simp [internalSupport,internalEdges_aut_iff G h]
-
-theorem crossingCount_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (h : IsAut G a b) (S : Set V) (x : V) (F : Finset E) (ω : Set E) :
-    crossingCount G (a '' S) (a x) (F.map b.toEmbedding) (b '' ω) =
-      crossingCount G S x F ω := by
-  simp only [crossingCount,Finset.sum_map,Equiv.toEmbedding_apply,crossingReach_aut_iff G h]
-
-theorem boundaryExpectation_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (h : IsAut G a b) (S : Set V) (x : V) (F : Finset E) (p : ℝ) :
-    boundaryExpectation G (a '' S) (a x) (F.map b.toEmbedding) p =
-      boundaryExpectation G S x F p := by
-  unfold boundaryExpectation
-  rw [internalSupport_aut G h,bernoulliPolynomial_map]
-  apply bernoulliPolynomial_congr
-  intro t _
-  simp only [Finset.coe_map,Equiv.coe_toEmbedding]
-  exact crossingCount_aut G h S x F t
-
-theorem incidentSupport_aut (G : BondGraph V E) (hloc : LocallyFinite G)
-    {a : V ≃ V} {b : E ≃ E} (h : IsAut G a b) (S : Finset V) :
-    incidentSupport G hloc (S.map a.toEmbedding) =
-      (incidentSupport G hloc S).map b.toEmbedding := by
-  ext e
-  obtain ⟨d,rfl⟩ := b.surjective e
-  simp only [mem_incidentSupport,Finset.mem_map,Equiv.toEmbedding_apply,
-    b.injective.eq_iff,exists_eq_right]
-  constructor
-  · rintro ⟨x,⟨y,hy,rfl⟩,hx⟩
-    rw [h d,Sym2.mem_map] at hx
-    obtain ⟨z,hz,hzy⟩ := hx
-    have hz' : z = y := a.injective hzy
-    exact ⟨y,hy,hz' ▸ hz⟩
-  · rintro ⟨x,hx,he⟩
-    exact ⟨a x,⟨x,hx,rfl⟩,by rw [h d]; exact Sym2.mem_map.mpr ⟨x,he,rfl⟩⟩
-
-theorem boundaryCost_aut (G : BondGraph V E) (hloc : LocallyFinite G)
-    {a : V ≃ V} {b : E ≃ E} (h : IsAut G a b) (S : Finset V) (x : V) (p : ℝ) :
-    boundaryCost G hloc (S.map a.toEmbedding) (a x) p = boundaryCost G hloc S x p := by
-  unfold boundaryCost
-  rw [incidentSupport_aut G hloc h,Finset.coe_map]
-  exact congrArg (p*·) (boundaryExpectation_aut G h S x (incidentSupport G hloc S) p)
-
- 
-
-theorem exists_uniform_subcritical_testSets [Countable V] [Nonempty V] (G : BondGraph V E)
-    (hloc : LocallyFinite G) (hqt : QuasiTransitive G) (p : unitInterval) (hp : p < pc G) :
-    ∃ (L : ℕ) (a : ℝ), 0 ≤ a ∧ a < 1 ∧ ∀ x, ∃ S : Finset V,
-      x ∈ S ∧ S.card ≤ L ∧ boundaryCost G hloc S x (p:ℝ) ≤ a := by
-  obtain ⟨O,hO⟩ := hqt
-  choose S hmem hcost using exists_subcritical_testSet G hloc p hp
-  have hOn : O.Nonempty := by
-    obtain ⟨o,ho,_⟩ := hO (Classical.arbitrary V)
-    exact ⟨o,ho⟩
-  let L := O.sup (fun x => (S x).card)
-  let a := max 0 (O.sup' hOn (fun x => boundaryCost G hloc (S x) x (p:ℝ)))
-  refine ⟨L,a,le_max_left _ _,?_,?_⟩
-  · apply max_lt zero_lt_one
-    exact (Finset.sup'_lt_iff hOn).mpr (fun x _ => hcost x)
-  · intro x
-    obtain ⟨o,ho,α,β,hαβ,rfl⟩ := hO x
-    refine ⟨(S o).map α.toEmbedding,?_,?_,?_⟩
-    · simp only [Finset.mem_map,Equiv.toEmbedding_apply]
-      exact ⟨o,hmem o,rfl⟩
-    · simpa only [Finset.card_map] using Finset.le_sup (f := fun x => (S x).card) ho
-    · rw [boundaryCost_aut G hloc hαβ]
-      exact (Finset.le_sup' (f := fun x => boundaryCost G hloc (S x) x (p:ℝ)) ho).trans
-        (le_max_right _ _)
-
-end SharpnessBridge
-
-section SharpnessSusceptibility
-open Filter Topology
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-noncomputable def domainTwoPoint (G : BondGraph V E) (S : Set V) (p : unitInterval)
-    (x y : V) : ℝ := (law p).real {ω | Conn G (ω ∩ internalEdges G S) x y}
-
-theorem conn_internal_endpoint (G : BondGraph V E) (S : Set V) {ω : Set E} {x y : V}
-    (hx : x ∈ S) (h : Conn G (ω ∩ internalEdges G S) x y) : y ∈ S := by
-  induction h with
-  | refl => exact hx
-  | @tail a b hxa hab ih =>
-    obtain ⟨e,he,heq⟩ := hab
-    exact ((mem_internalEdges_of_ends G S e a b heq).mp he.2).2
-
-def CutExists (G : BondGraph V E) (S : Set V) (e : E) : Prop :=
-  ∃ a b, G e = s(a,b) ∧ a ∈ S ∧ b ∉ S
-
-noncomputable def cutInside [Nonempty V] (G : BondGraph V E) (S : Set V) (e : E) : V :=
-  if h : CutExists G S e then h.choose else Classical.arbitrary V
-
-noncomputable def cutOutside [Nonempty V] (G : BondGraph V E) (S : Set V) (e : E) : V :=
-  if h : CutExists G S e then h.choose_spec.choose else Classical.arbitrary V
-
-theorem cut_spec [Nonempty V] (G : BondGraph V E) (S : Set V) {e : E}
-    (he : CutExists G S e) :
-    G e = s(cutInside G S e,cutOutside G S e) ∧
-      cutInside G S e ∈ S ∧ cutOutside G S e ∉ S := by
-  simpa only [cutInside,cutOutside,dite_eq_left he] using he.choose_spec.choose_spec
-
-theorem cut_unique [Nonempty V] (G : BondGraph V E) (S : Set V) {e : E} {a b : V}
-    (he : G e = s(a,b)) (ha : a ∈ S) (hb : b ∉ S) :
-    a = cutInside G S e ∧ b = cutOutside G S e := by
-  have hc := cut_spec G S (e := e) ⟨a,b,he,ha,hb⟩
-  have hh := Sym2.eq_iff.mp (he.symm.trans hc.1)
-  rcases hh with ⟨h₁,h₂⟩ | ⟨h₁,h₂⟩
-  · exact ⟨h₁,h₂⟩
-  · exact (hc.2.2 (h₁ ▸ ha)).elim
-
-theorem crossingReach_cut [Nonempty V] (G : BondGraph V E) (S : Set V)
-    (x : V) (ω : Set E) (e : E) :
-    crossingReach G S x ω e ↔ CutExists G S e ∧ Conn G ω x (cutInside G S e) := by
-  constructor
-  · rintro ⟨a,b,he,ha,hb,h⟩
-    exact ⟨⟨a,b,he,ha,hb⟩,(cut_unique G S he ha hb).1 ▸ h⟩
-  · rintro ⟨he,h⟩
-    have hs := cut_spec G S he
-    exact ⟨_,_,hs.1,hs.2.1,hs.2.2,h⟩
-
-theorem crossingReach_upper (G : BondGraph V E) (S : Set V) (x : V) (e : E) :
-    IsUpperSet {ω | crossingReach G S x (ω ∩ internalEdges G S) e} := by
-  rintro ω η h ⟨a,b,he,ha,hb,hconn⟩
-  exact ⟨a,b,he,ha,hb,conn_mono G (Set.inter_subset_inter_left _ h) hconn⟩
-
-theorem finiteWitness_crossingReach (G : BondGraph V E) (S : Set V) (x : V) (e : E) :
-    FiniteWitness {ω | crossingReach G S x (ω ∩ internalEdges G S) e} := by
-  rintro ω ⟨a,b,he,ha,hb,hconn⟩
-  obtain ⟨t,ht,hc⟩ := finiteWitness_inter {η | Conn G η x a} (finiteWitness_conn G x a)
-    (internalEdges G S) ω hconn
-  exact ⟨t,ht,a,b,he,ha,hb,hc⟩
-
- 
-theorem conn_first_exit [Nonempty V] (G : BondGraph V E) (hloc : LocallyFinite G)
-    (S : Finset V) (U : Set E) {ω : Set E} {x y : V} (hx : x ∈ S)
-    (hxy : Conn G (ω ∩ U) x y) :
-    y ∈ S ∨ ∃ e ∈ incidentSupport G hloc S,
-      ω ∈ DisjointOccurrence {η | crossingReach G S x (η ∩ internalEdges G S) e}
-        (DisjointOccurrence {η | e ∈ η} {η | Conn G (η ∩ U) (cutOutside G S e) y}) := by
-  obtain ⟨l,hl,hn,ho⟩ := conn_exists_bond_trail G hxy
-  rcases hl.first_outside (internalEdges G S) with hall |
-      ⟨a,b,e,l₁,l₂,heq,h₁,he,h₂,hfirst,hnot⟩
-  · exact Or.inl (conn_internal_endpoint G S hx (hl.conn fun e he => ⟨(ho e he).1,hall e he⟩))
-  · subst l
-    have hnod := List.nodup_append.mp hn
-    have hen : e ∉ l₂ := (List.nodup_cons.mp hnod.2.1).1
-    have ha : a ∈ S := conn_internal_endpoint G S hx
-      (h₁.conn (ω := ω ∩ internalEdges G S) fun d hd =>
-        ⟨(ho d (List.mem_append_left _ hd)).1,hfirst d hd⟩)
-    have hb : b ∉ S := fun hb => hnot ((mem_internalEdges_of_ends G S e a b he).mpr ⟨ha,hb⟩)
-    have hbe := (cut_unique G S he ha hb).2
-    refine Or.inr ⟨e,(mem_incidentSupport G hloc S e).mpr ⟨a,ha,by simp [he]⟩,
-      l₁.toFinset,(e::l₂).toFinset,?_,?_,?_,?_,?_⟩
-    · intro d hd
-      exact (ho d (List.mem_append_left _ (List.mem_toFinset.mp hd))).1
-    · intro d hd
-      exact (ho d (List.mem_append_right _ (List.mem_toFinset.mp hd))).1
-    · apply Finset.disjoint_left.mpr
-      intro d hd hd'
-      exact hnod.2.2 d (List.mem_toFinset.mp hd) d (List.mem_toFinset.mp hd') rfl
-    · exact ⟨a,b,he,ha,hb,h₁.conn (fun d hd => ⟨by simpa using hd,hfirst d hd⟩)⟩
-    · refine ⟨{e},l₂.toFinset,by simp,by simp,by simpa using hen,by simp,?_⟩
-      rw [← hbe]
-      exact h₂.conn fun d hd => ⟨by simpa using hd,
-        (ho d (List.mem_append_right _ (List.mem_cons_of_mem e hd))).2⟩
-
- 
-theorem boundaryExpectation_eq_sum (G : BondGraph V E) (hloc : LocallyFinite G)
-    (S : Finset V) (x : V) (p : unitInterval) :
-    boundaryExpectation G S x (incidentSupport G hloc S) (p:ℝ) =
-      ∑ e ∈ incidentSupport G hloc S,
-        (law p).real {ω | crossingReach G S x (ω ∩ internalEdges G S) e} := by
-  have hI : (internalSupport G S (incidentSupport G hloc S) : Set E) = internalEdges G S := by
-    ext e
-    simp only [Finset.mem_coe,internalSupport,Finset.mem_filter]
-    exact ⟨And.right,fun he => ⟨internal_edge_mem_incident G hloc S he,he⟩⟩
-  unfold boundaryExpectation crossingCount bernoulliPolynomial
-  simp only [Finset.sum_mul]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro e _
-  change finiteEventPolynomial (internalSupport G S (incidentSupport G hloc S))
-    {ω | crossingReach G S x ω e} (p:ℝ) = _
-  rw [finiteEventPolynomial_restrict,hI]
-  rfl
-
- 
-theorem domainTwoPoint_first_exit [Countable E] [Nonempty V] (G : BondGraph V E)
-    (hloc : LocallyFinite G) (S : Finset V) (Λ : Set V) (p : unitInterval)
-    (x y : V) (hx : x ∈ S) :
-    domainTwoPoint G Λ p x y ≤ (if y ∈ S then (1:ℝ) else 0) +
-      ∑ e ∈ incidentSupport G hloc S,
-        (law p).real {ω | crossingReach G S x (ω ∩ internalEdges G S) e} *
-          (p:ℝ) * domainTwoPoint G Λ p (cutOutside G S e) y := by
-  by_cases hy : y ∈ S
-  · have hnon : 0 ≤ ∑ e ∈ incidentSupport G hloc S,
-        (law p).real {ω | crossingReach G S x (ω ∩ internalEdges G S) e} *
-          (p:ℝ) * domainTwoPoint G Λ p (cutOutside G S e) y := by
-      apply Finset.sum_nonneg
-      intro e _
-      exact mul_nonneg (mul_nonneg (measureReal_nonneg) p.2.1) measureReal_nonneg
-    simpa only [ite_eq_left hy,domainTwoPoint] using (measureReal_le_one (μ := law p)
-      (s := {ω | Conn G (ω ∩ internalEdges G Λ) x y})).trans (le_add_of_nonneg_right hnon)
-  · simp only [ite_eq_right hy,zero_add]
-    let A := fun e => {ω | crossingReach G S x (ω ∩ internalEdges G S) e}
-    let B := fun e => {ω | Conn G (ω ∩ internalEdges G Λ) (cutOutside G S e) y}
-    let F := incidentSupport G hloc S
-    let C := fun e => DisjointOccurrence (A e) (DisjointOccurrence {ω | e ∈ ω} (B e))
-    have hsub : {ω | Conn G (ω ∩ internalEdges G Λ) x y} ⊆ ⋃ e ∈ F, C e := by
-      intro ω hω
-      rcases conn_first_exit G hloc S (internalEdges G Λ) hx hω with h | ⟨e,he,h⟩
-      · exact (hy h).elim
-      · exact Set.mem_iUnion.mpr ⟨e,Set.mem_iUnion.mpr ⟨he,h⟩⟩
-    calc
-      domainTwoPoint G Λ p x y ≤ (law p).real (⋃ e ∈ F, C e) := measureReal_mono (μ := law p) hsub (measure_ne_top _ _)
-      _ ≤ ∑ e ∈ F, (law p).real (C e) := measureReal_biUnion_finset_le _ _
-      _ ≤ _ := by
-        apply Finset.sum_le_sum
-        intro e _
-        have hB : IsUpperSet (B e) := fun _ _ h hc =>
-          conn_mono G (Set.inter_subset_inter_left _ h) hc
-        have hfB : FiniteWitness (B e) := finiteWitness_inter _
-          (finiteWitness_conn G _ _) (internalEdges G Λ)
-        have hb := bk p (A e) (DisjointOccurrence {ω | e ∈ ω} (B e))
-          (crossingReach_upper G S x e) (disjointOccurrence_upper _ _)
-          (finiteWitness_crossingReach G S x e) (finiteWitness_disjointOccurrence _ _)
-        have hb' := bk p {ω | e ∈ ω} (B e) (fun _ _ h he => h he) hB (finiteWitness_edge e) hfB
-        have heq : (law p).real {ω : Set E | e ∈ ω} = (p:ℝ) := by
-          simpa using finite_open_probability p {e}
-        rw [heq] at hb'
-        calc
-          (law p).real (C e) ≤ (law p).real (A e) * (law p).real
-              (DisjointOccurrence {ω | e ∈ ω} (B e)) := hb
-          _ ≤ (law p).real (A e) * ((p:ℝ) * (law p).real (B e)) :=
-            mul_le_mul_of_nonneg_left hb' measureReal_nonneg
-          _ = _ := (mul_assoc _ _ _).symm
-
-theorem conn_internal_eq_or_mem (G : BondGraph V E) (S : Set V) {ω : Set E} {x y : V}
-    (h : Conn G (ω ∩ internalEdges G S) x y) : x = y ∨ x ∈ S ∧ y ∈ S := by
-  induction h with
-  | refl => exact Or.inl rfl
-  | @tail a b hxa hab ih =>
-    obtain ⟨e,he,heq⟩ := hab
-    have he' := (mem_internalEdges_of_ends G S e a b heq).mp he.2
-    right
-    exact ⟨ih.elim (fun h => h.symm ▸ he'.1) And.left,he'.2⟩
-
-theorem domainTwoPoint_zero_of_outside (G : BondGraph V E) (S : Set V) (p : unitInterval)
-    {x y : V} (hx : x ∉ S) (hy : y ∈ S) : domainTwoPoint G S p x y = 0 := by
-  have he : {ω | Conn G (ω ∩ internalEdges G S) x y} = ∅ := by
-    apply Set.eq_empty_iff_forall_notMem.mpr
-    intro ω h
-    rcases conn_internal_eq_or_mem G S h with rfl | h
-    · exact hx hy
-    · exact hx h.1
-  simp only [domainTwoPoint,he,measureReal_empty]
-
-theorem domainRow_le_boundaryCost [Countable E] [Nonempty V] (G : BondGraph V E)
-    (hloc : LocallyFinite G) (S Λ : Finset V) (p : unitInterval) (x : V) (hx : x ∈ S)
-    (H : ℝ) (hH : ∀ z, ∑ y ∈ Λ, domainTwoPoint G Λ p z y ≤ H) :
-    (∑ y ∈ Λ, domainTwoPoint G Λ p x y) ≤ S.card + boundaryCost G hloc S x (p:ℝ) * H := by
-  have hind : (∑ y ∈ Λ, if y ∈ S then (1:ℝ) else 0) ≤ (S.card:ℝ) := by
-    simp only [← Finset.sum_filter]
-    simpa using (Nat.cast_le (α := ℝ)).mpr (Finset.card_le_card
-      (show Λ.filter (fun y => y ∈ S) ⊆ S from fun y hy => (Finset.mem_filter.mp hy).2))
-  calc
-    (∑ y ∈ Λ, domainTwoPoint G Λ p x y) ≤
-        ∑ y ∈ Λ, ((if y ∈ S then (1:ℝ) else 0) +
-          ∑ e ∈ incidentSupport G hloc S,
-            (law p).real {ω | crossingReach G S x (ω ∩ internalEdges G S) e} *
-              (p:ℝ) * domainTwoPoint G Λ p (cutOutside G S e) y) :=
-      Finset.sum_le_sum fun y _ => domainTwoPoint_first_exit G hloc S Λ p x y hx
-    _ = (∑ y ∈ Λ, if y ∈ S then (1:ℝ) else 0) +
-        ∑ e ∈ incidentSupport G hloc S,
-          (law p).real {ω | crossingReach G S x (ω ∩ internalEdges G S) e} *
-            (p:ℝ) * ∑ y ∈ Λ, domainTwoPoint G Λ p (cutOutside G S e) y := by
-      rw [Finset.sum_add_distrib,Finset.sum_comm]
-      simp only [← Finset.mul_sum]
-    _ ≤ S.card + ∑ e ∈ incidentSupport G hloc S,
-          (law p).real {ω | crossingReach G S x (ω ∩ internalEdges G S) e} * (p:ℝ) * H := by
-      apply add_le_add hind
-      apply Finset.sum_le_sum
-      intro e _
-      exact mul_le_mul_of_nonneg_left (hH _) (mul_nonneg measureReal_nonneg p.2.1)
-    _ = S.card + boundaryCost G hloc S x (p:ℝ) * H := by
-      unfold boundaryCost
-      rw [boundaryExpectation_eq_sum]
-      simp only [← Finset.sum_mul]
-      ring
-
-theorem domainRow_uniform_bound [Countable E] [Nonempty V] (G : BondGraph V E)
-    (hloc : LocallyFinite G) (p : unitInterval) (L : ℕ) (a : ℝ) (ha : a < 1)
-    (htest : ∀ x, ∃ S : Finset V, x ∈ S ∧ S.card ≤ L ∧ boundaryCost G hloc S x (p:ℝ) ≤ a)
-    (Λ : Finset V) (x : V) :
-    (∑ y ∈ Λ, domainTwoPoint G Λ p x y) ≤ (L:ℝ)/(1-a) := by
-  by_cases hΛ : Λ.Nonempty
-  · let row := fun z => ∑ y ∈ Λ, domainTwoPoint G Λ p z y
-    obtain ⟨z,hz,hmax⟩ := Λ.exists_max_image row hΛ
-    have hnon : 0 ≤ row z := Finset.sum_nonneg (fun _ _ => measureReal_nonneg)
-    have hH : ∀ w, row w ≤ row z := by
-      intro w
-      by_cases hw : w ∈ Λ
-      · exact hmax w hw
-      · have he : row w = 0 := by
-          apply Finset.sum_eq_zero
-          intro y hy
-          exact domainTwoPoint_zero_of_outside G Λ p hw hy
-        rwa [he]
-    obtain ⟨S,hS,hcard,hcost⟩ := htest z
-    have h := domainRow_le_boundaryCost G hloc S Λ p z hS (row z) hH
-    have hL : (S.card:ℝ) ≤ (L:ℝ) := Nat.cast_le.mpr hcard
-    have hc := mul_le_mul_of_nonneg_right hcost hnon
-    have hrow : row z ≤ (L:ℝ)/(1-a) := (le_div_iff₀ (sub_pos.mpr ha)).mpr (by nlinarith)
-    exact (hH x).trans hrow
-  · simp only [Finset.not_nonempty_iff_eq_empty.mp hΛ,Finset.sum_empty]
-    exact div_nonneg (Nat.cast_nonneg _) (sub_pos.mpr ha).le
-
-theorem internalEdges_mono (G : BondGraph V E) {S T : Set V} (h : S ⊆ T) :
-    internalEdges G S ⊆ internalEdges G T := fun _ he x hx => h (he x hx)
-
-theorem exists_internal_vertexExhaustion [Encodable V] (G : BondGraph V E) (e : E) :
-    ∃ n, e ∈ internalEdges G (vertexExhaustion (V := V) n : Set V) := by
-  obtain ⟨⟨a,b⟩,hab⟩ := Sym2.mk_surjective (G e)
-  obtain ⟨n,hn⟩ := subset_vertexExhaustion ({a,b} : Finset V)
-  refine ⟨n,?_⟩
-  rw [mem_internalEdges_of_ends G _ e a b hab.symm]
-  exact ⟨hn (by simp),hn (by simp)⟩
-
-theorem conn_domain_exhaustion [Encodable V] (G : BondGraph V E) {ω : Set E} {x y : V}
-    (h : Conn G ω x y) :
-    ∃ n, Conn G (ω ∩ internalEdges G (vertexExhaustion (V := V) n : Set V)) x y := by
-  induction h with
-  | refl => exact ⟨0,conn_refl _ _ _⟩
-  | @tail a b hxa hab ih =>
-    obtain ⟨n,hn⟩ := ih
-    obtain ⟨e,he,heq⟩ := hab
-    obtain ⟨m,hm⟩ := exists_internal_vertexExhaustion G e
-    refine ⟨max n m,?_⟩
-    have hnm := internalEdges_mono G (show (vertexExhaustion (V := V) n : Set V) ⊆
-        (vertexExhaustion (V := V) (max n m) : Set V) from vertexExhaustion_mono (le_max_left n m))
-    have hmm := internalEdges_mono G (show (vertexExhaustion (V := V) m : Set V) ⊆
-        (vertexExhaustion (V := V) (max n m) : Set V) from vertexExhaustion_mono (le_max_right n m))
-    exact (conn_mono G (Set.inter_subset_inter_right _ hnm) hn).tail ⟨e,⟨he,hmm hm⟩,heq⟩
-
-theorem tendsto_domainTwoPoint [Encodable V] (G : BondGraph V E) (p : unitInterval) (x y : V) :
-    Tendsto (fun n => domainTwoPoint G (vertexExhaustion (V := V) n : Set V) p x y)
-      atTop (𝓝 (twoPoint G p x y)) := by
-  have hm : Monotone (fun n => {ω | Conn G
-      (ω ∩ internalEdges G (vertexExhaustion (V := V) n : Set V)) x y}) := by
-    intro m n h ω hc
-    exact conn_mono G (Set.inter_subset_inter_right _ (internalEdges_mono G
-      (show (vertexExhaustion (V := V) m : Set V) ⊆ (vertexExhaustion (V := V) n : Set V) from
-        vertexExhaustion_mono h))) hc
-  have he : (⋃ n, {ω | Conn G (ω ∩ internalEdges G (vertexExhaustion (V := V) n : Set V)) x y}) =
-      {ω | Conn G ω x y} := by
-    ext ω
-    simp only [Set.mem_iUnion,Set.mem_ofPred_eq]
-    exact ⟨fun ⟨n,h⟩ => conn_mono G Set.inter_subset_left h,conn_domain_exhaustion G⟩
-  have h := tendsto_measure_iUnion_atTop (μ := law p) hm
-  rw [he] at h
-  exact (ENNReal.tendsto_toReal (measure_ne_top _ _)).comp h
-
- 
-theorem subcritical_row_bound [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) :
-    ∃ C : ℝ, 0 ≤ C ∧ ∀ x (F : Finset V), ∑ y ∈ F, twoPoint G p x y ≤ C := by
-  let : Encodable V := Encodable.ofCountable V
-  obtain ⟨L,a,_,ha,htest⟩ := exists_uniform_subcritical_testSets G hloc hqt p hp
-  refine ⟨(L:ℝ)/(1-a),div_nonneg (Nat.cast_nonneg _) (sub_pos.mpr ha).le,?_⟩
-  intro x F
-  apply le_of_tendsto (tendsto_finsetSum F (fun y _ => tendsto_domainTwoPoint G p x y))
-  obtain ⟨N,hN⟩ := subset_vertexExhaustion F
-  filter_upwards [eventually_ge_atTop N] with n hn
-  have hFn : F ⊆ vertexExhaustion n := hN.trans (vertexExhaustion_mono hn)
-  exact (Finset.sum_le_sum_of_subset_of_nonneg hFn (fun _ _ _ => measureReal_nonneg)).trans
-    (domainRow_uniform_bound G hloc p L a ha htest (vertexExhaustion n) x)
-
- 
-theorem operatorBounded_of_lt_pc [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) : OperatorBounded G p := by
-  obtain ⟨C,hC,hrow⟩ := subcritical_row_bound G hloc hqt p hp
-  obtain ⟨A,_,hA⟩ := exists_operator_of_row_column_bound (twoPoint G p) C hC
-    (twoPoint_nonneg G p) hrow (fun y F => by
-      simpa only [twoPoint_symm G p] using hrow y F)
-  exact ⟨A,hA⟩
-
-end SharpnessSusceptibility
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Kernel.Sprinkling
+
+namespace OAI
+
+/-! Finite Russo formulae, exterior exposure and subcritical sharpness. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+section FiniteRusso
+attribute [local instance] Classical.propDecidable
+attribute [local instance] Classical.decEq
+
+ 
+noncomputable def bernoulliPolynomial (s : Finset E) (f : Finset E → ℝ) (p : ℝ) : ℝ :=
+  ∑ t ∈ s.powerset, f t * p ^ t.card * (1-p) ^ (s.card-t.card)
+
+@[simp] theorem bernoulliPolynomial_empty (f : Finset E → ℝ) (p : ℝ) :
+    bernoulliPolynomial ∅ f p = f ∅ := by simp [bernoulliPolynomial]
+
+theorem bernoulliPolynomial_congr (s : Finset E) {f g : Finset E → ℝ}
+    (h : ∀ t ⊆ s, f t = g t) (p : ℝ) :
+    bernoulliPolynomial s f p = bernoulliPolynomial s g p := by
+  apply Finset.sum_congr rfl
+  intro t ht
+  rw [h t (Finset.mem_powerset.mp ht)]
+
+theorem bernoulliPolynomial_sub (s : Finset E) (f g : Finset E → ℝ) (p : ℝ) :
+    bernoulliPolynomial s (fun t => f t-g t) p =
+      bernoulliPolynomial s f p - bernoulliPolynomial s g p := by
+  simp only [bernoulliPolynomial, sub_mul, Finset.sum_sub_distrib]
+
+theorem bernoulliPolynomial_insert (s : Finset E) (e : E) (he : e ∉ s)
+    (f : Finset E → ℝ) (p : ℝ) :
+    bernoulliPolynomial (insert e s) f p =
+      (1-p)*bernoulliPolynomial s f p + p*bernoulliPolynomial s (fun t => f (insert e t)) p := by
+  have hd : Disjoint s.powerset (s.powerset.image (insert e)) := by
+    apply Finset.disjoint_left.mpr
+    intro t ht himg
+    obtain ⟨r, hr, rfl⟩ := Finset.mem_image.mp himg
+    exact he ((Finset.mem_powerset.mp ht) (by simp))
+  have hinj : Set.InjOn (insert e) (↑s.powerset : Set (Finset E)) := by
+    intro t ht r hr h
+    have ht' : e ∉ t := fun h => he (Finset.mem_powerset.mp ht h)
+    have hr' : e ∉ r := fun h => he (Finset.mem_powerset.mp hr h)
+    have hh := congrArg (fun t : Finset E => t.erase e) h
+    simpa [ht', hr'] using hh
+  simp only [bernoulliPolynomial, Finset.powerset_insert, Finset.sum_union hd,
+    Finset.sum_image hinj, Finset.mul_sum, Finset.card_insert_of_notMem he]
+  congr 1
+  · apply Finset.sum_congr rfl
+    intro t ht
+    have hcard := Finset.card_le_card (Finset.mem_powerset.mp ht)
+    have hexp : s.card+1-t.card = (s.card-t.card)+1 := by omega
+    rw [hexp, pow_succ]
+    ring
+  · apply Finset.sum_congr rfl
+    intro t ht
+    have het : e ∉ t := fun h => he (Finset.mem_powerset.mp ht h)
+    simp only [Finset.card_insert_of_notMem het, Nat.add_sub_add_right, pow_succ]
+    ring
+
+ 
+theorem hasDerivAt_bernoulliPolynomial (s : Finset E) (f : Finset E → ℝ) (p : ℝ) :
+    HasDerivAt (bernoulliPolynomial s f)
+      (∑ e ∈ s, bernoulliPolynomial (s.erase e) (fun t => f (insert e t)-f t) p) p := by
+  induction s using Finset.induction_on generalizing f with
+  | empty =>
+    convert hasDerivAt_const p (f ∅) using 1 <;> try rfl
+    · funext q
+      exact bernoulliPolynomial_empty f q
+  | @insert e s he ih =>
+    have hd := (((hasDerivAt_const p 1).sub (hasDerivAt_id p)).mul (ih f)).add
+      ((hasDerivAt_id p).mul (ih (fun t => f (insert e t))))
+    have hf : bernoulliPolynomial (insert e s) f =
+        fun q => (1-q)*bernoulliPolynomial s f q +
+          q*bernoulliPolynomial s (fun t => f (insert e t)) q :=
+      funext fun q => bernoulliPolynomial_insert s e he f q
+    rw [hf]
+    convert hd using 1 <;> try rfl
+    have hs : ∑ a ∈ s, bernoulliPolynomial ((insert e s).erase a)
+          (fun t => f (insert a t)-f t) p =
+        (1-p)*(∑ a ∈ s, bernoulliPolynomial (s.erase a) (fun t => f (insert a t)-f t) p) +
+        p*(∑ a ∈ s, bernoulliPolynomial (s.erase a)
+          (fun t => f (insert e (insert a t))-f (insert e t)) p) := by
+      simp only [Finset.mul_sum, ← Finset.sum_add_distrib]
+      apply Finset.sum_congr rfl
+      intro a ha
+      have hae : e ≠ a := (ne_of_mem_of_not_mem ha he).symm
+      rw [Finset.erase_insert_of_ne hae, bernoulliPolynomial_insert _ e
+        (fun hh => he (Finset.mem_of_mem_erase hh))]
+      congr 2
+      apply bernoulliPolynomial_congr
+      intro t _
+      rw [Finset.insert_comm e a]
+    simp only [Finset.sum_insert he, Finset.erase_insert he, zero_sub, neg_one_mul,
+      one_mul, Pi.sub_apply, id_eq]
+    rw [hs, bernoulliPolynomial_sub]
+    ring
+
+@[simp] theorem bernoulliPolynomial_zero (s : Finset E) (p : ℝ) :
+    bernoulliPolynomial s (fun _ => 0) p = 0 := by simp [bernoulliPolynomial]
+
+theorem bernoulliPolynomial_const (s : Finset E) (c p : ℝ) :
+    bernoulliPolynomial s (fun _ => c) p = c := by
+  induction s using Finset.induction_on with
+  | empty => simp
+  | @insert e s he ih => rw [bernoulliPolynomial_insert s e he]; simp only [ih]; ring
+
+theorem bernoulliPolynomial_nonneg (s : Finset E) (f : Finset E → ℝ) (p : ℝ)
+    (hp : p ∈ Set.Icc (0:ℝ) 1) (hf : ∀ t ⊆ s, 0 ≤ f t) :
+    0 ≤ bernoulliPolynomial s f p := by
+  apply Finset.sum_nonneg
+  intro t ht
+  exact mul_nonneg (mul_nonneg (hf t (Finset.mem_powerset.mp ht))
+    (pow_nonneg hp.1 _)) (pow_nonneg (sub_nonneg.mpr hp.2) _)
+
+theorem bernoulliPolynomial_mono (s : Finset E) (f g : Finset E → ℝ) (p : ℝ)
+    (hp : p ∈ Set.Icc (0:ℝ) 1) (hfg : ∀ t ⊆ s, f t ≤ g t) :
+    bernoulliPolynomial s f p ≤ bernoulliPolynomial s g p := by
+  apply Finset.sum_le_sum
+  intro t ht
+  exact mul_le_mul_of_nonneg_right
+    (mul_le_mul_of_nonneg_right (hfg t (Finset.mem_powerset.mp ht)) (pow_nonneg hp.1 _))
+    (pow_nonneg (sub_nonneg.mpr hp.2) _)
+
+ 
+theorem bernoulliPolynomial_union (s t : Finset E) (hdis : Disjoint s t)
+    (f : Finset E → ℝ) (p : ℝ) :
+    bernoulliPolynomial (s ∪ t) f p =
+      bernoulliPolynomial s (fun a => bernoulliPolynomial t (fun b => f (a ∪ b)) p) p := by
+  induction s using Finset.induction_on generalizing f with
+  | empty => simp
+  | @insert e s he ih =>
+    have hed : e ∉ t := (Finset.disjoint_insert_left.mp hdis).1
+    have hst : Disjoint s t := (Finset.disjoint_insert_left.mp hdis).2
+    rw [Finset.insert_union, bernoulliPolynomial_insert _ e (by simp [he, hed]),
+      bernoulliPolynomial_insert s e he, ih hst, ih hst]
+    congr 2
+    apply bernoulliPolynomial_congr
+    intro a _
+    apply bernoulliPolynomial_congr
+    intro b _
+    rw [Finset.insert_union]
+
+ 
+
+theorem bernoulliPolynomial_closed_difference (s : Finset E) (e : E) (he : e ∈ s)
+    (f : Finset E → ℝ) (p : ℝ) :
+    bernoulliPolynomial s (fun t => if e ∈ t then 0 else f (insert e t)-f t) p =
+      (1-p)*bernoulliPolynomial (s.erase e) (fun t => f (insert e t)-f t) p := by
+  have hes : e ∉ s.erase e := Finset.notMem_erase e s
+  conv_lhs => rw [← Finset.insert_erase he]
+  rw [bernoulliPolynomial_insert _ e hes]
+  have hc : bernoulliPolynomial (s.erase e)
+      (fun t => if e ∈ t then 0 else f (insert e t)-f t) p =
+      bernoulliPolynomial (s.erase e) (fun t => f (insert e t)-f t) p := by
+    apply bernoulliPolynomial_congr
+    intro t ht
+    simp [show e ∉ t from fun h => hes (ht h)]
+  rw [hc]
+  simp
+
+theorem closed_difference_russo (s : Finset E) (f : Finset E → ℝ) (p : ℝ) :
+    (1-p)*deriv (bernoulliPolynomial s f) p =
+      ∑ e ∈ s, bernoulliPolynomial s
+        (fun t => if e ∈ t then 0 else f (insert e t)-f t) p := by
+  rw [(hasDerivAt_bernoulliPolynomial s f p).deriv, Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro e he
+  exact (bernoulliPolynomial_closed_difference s e he f p).symm
+
+ 
+noncomputable def finiteEventPolynomial (s : Finset E) (A : Set (Set E)) (p : ℝ) : ℝ :=
+  bernoulliPolynomial s (fun t => if (t : Set E) ∈ A then 1 else 0) p
+
+theorem finiteEventPolynomial_eq (s : Finset E) (A : Set (Set E)) (p : unitInterval) :
+    finiteEventPolynomial s A (p : ℝ) = finiteProbability p s A := by
+  unfold finiteEventPolynomial bernoulliPolynomial finiteProbability
+  apply Finset.sum_congr rfl
+  intro t _
+  by_cases ht : (t : Set E) ∈ A <;> simp [ht]
+
+ 
+def closedPivotal (A : Set (Set E)) (e : E) : Set (Set E) :=
+  {ω | e ∉ ω ∧ ω ∉ A ∧ insert e ω ∈ A}
+
+theorem closedPivotal_difference (A : Set (Set E)) (hA : IsUpperSet A) (e : E)
+    (t : Finset E) :
+    (if (t : Set E) ∈ closedPivotal A e then (1:ℝ) else 0) =
+    (if e ∈ t then 0 else
+      (if ((insert e t : Finset E) : Set E) ∈ A then (1:ℝ) else 0) -
+      (if (t : Set E) ∈ A then 1 else 0)) := by
+  by_cases he : e ∈ t
+  · simp [closedPivotal, he]
+  · by_cases ha : (t : Set E) ∈ A
+    · have hi : insert e (t : Set E) ∈ A := hA (Set.subset_insert e _) ha
+      simp [closedPivotal, he, ha, hi]
+    · by_cases hi : insert e (t : Set E) ∈ A <;> simp [closedPivotal, he, ha, hi]
+
+ 
+theorem finite_russo_closed (s : Finset E) (A : Set (Set E)) (hA : IsUpperSet A)
+    (p : unitInterval) :
+    (1-(p:ℝ))*deriv (finiteEventPolynomial s A) (p:ℝ) =
+      ∑ e ∈ s, finiteProbability p s (closedPivotal A e) := by
+  rw [show finiteEventPolynomial s A =
+    bernoulliPolynomial s (fun t => if (t : Set E) ∈ A then 1 else 0) from rfl,
+    closed_difference_russo]
+  apply Finset.sum_congr rfl
+  intro e _
+  rw [← finiteEventPolynomial_eq]
+  apply bernoulliPolynomial_congr
+  intro t _
+  exact (closedPivotal_difference A hA e t).symm
+
+ 
+
+theorem finite_russo_closed_law [Fintype E] (A : Set (Set E)) (hA : IsUpperSet A)
+    (p : unitInterval) :
+    (1-(p:ℝ))*deriv (finiteEventPolynomial Finset.univ A) (p:ℝ) =
+      ∑ e : E, (law p).real (closedPivotal A e) := by
+  simpa only [finiteProbability_univ] using finite_russo_closed Finset.univ A hA p
+
+end FiniteRusso
+
+section ExteriorExposure
+attribute [local instance] Classical.propDecidable
+attribute [local instance] Classical.decEq
+
+ 
+theorem conn_insert_iff (G : BondGraph V E) (ω : Set E) (e : E) (a b x y : V)
+    (he : G e = s(a,b)) :
+    Conn G (insert e ω) x y ↔ Conn G ω x y ∨
+      (Conn G ω x a ∧ Conn G ω b y) ∨ (Conn G ω x b ∧ Conn G ω a y) := by
+  constructor
+  · intro hh
+    induction hh with
+    | refl => exact Or.inl (conn_refl G ω x)
+    | @tail u v huv huv' ih =>
+      obtain ⟨f,hf,hfuv⟩ := huv'
+      rcases Set.mem_insert_iff.mp hf with rfl | hf
+      · have hp : s(a,b) = s(u,v) := he.symm.trans hfuv
+        rcases Sym2.eq_iff.mp hp with ⟨rfl,rfl⟩ | ⟨rfl,rfl⟩
+        · rcases ih with h | ⟨hxa,hba⟩ | ⟨hxb,haa⟩
+          · exact Or.inr (Or.inl ⟨h,conn_refl G ω b⟩)
+          · exact Or.inr (Or.inl ⟨hxa,conn_refl G ω b⟩)
+          · exact Or.inl hxb
+        · rcases ih with h | ⟨hxa,hbb⟩ | ⟨hxb,hab⟩
+          · exact Or.inr (Or.inr ⟨h,conn_refl G ω a⟩)
+          · exact Or.inl hxa
+          · exact Or.inr (Or.inr ⟨hxb,conn_refl G ω a⟩)
+      · have huvω : Conn G ω u v := .single ⟨f,hf,hfuv⟩
+        rcases ih with h | ⟨hxa,hbu⟩ | ⟨hxb,hau⟩
+        · exact Or.inl (h.trans huvω)
+        · exact Or.inr (Or.inl ⟨hxa,hbu.trans huvω⟩)
+        · exact Or.inr (Or.inr ⟨hxb,hau.trans huvω⟩)
+  · intro hh
+    have hm : ω ⊆ insert e ω := Set.subset_insert e ω
+    have hab : Conn G (insert e ω) a b := .single ⟨e,Set.mem_insert e ω,he⟩
+    rcases hh with h | ⟨hxa,hby⟩ | ⟨hxb,hay⟩
+    · exact conn_mono G hm h
+    · exact (conn_mono G hm hxa).trans (hab.trans (conn_mono G hm hby))
+    · exact (conn_mono G hm hxb).trans ((conn_symm G _ hab).trans (conn_mono G hm hay))
+
+ 
+def internalEdges (G : BondGraph V E) (S : Set V) : Set E :=
+  {e | ∀ x ∈ G e, x ∈ S}
+
+@[simp] theorem mem_internalEdges_of_ends (G : BondGraph V E) (S : Set V)
+    (e : E) (a b : V) (he : G e = s(a,b)) :
+    e ∈ internalEdges G S ↔ a ∈ S ∧ b ∈ S := by
+  simp [internalEdges, he]
+
+ 
+def unexposed (G : BondGraph V E) (Λ : Set V) (ω : Set E) : Set V :=
+  {x | ∀ y, Conn G ω x y → y ∈ Λ}
+
+theorem unexposed_subset (G : BondGraph V E) (Λ : Set V) (ω : Set E) :
+    unexposed G Λ ω ⊆ Λ := fun x hx => hx x (conn_refl G ω x)
+
+theorem mem_unexposed_of_conn (G : BondGraph V E) (Λ : Set V) (ω : Set E)
+    {x y : V} (hx : x ∈ unexposed G Λ ω) (hxy : Conn G ω x y) :
+    y ∈ unexposed G Λ ω := by
+  intro z hyz
+  exact hx z (hxy.trans hyz)
+
+ 
+theorem conn_restrict_unexposed (G : BondGraph V E) (Λ : Set V) (ω : Set E)
+    {x y : V} (hx : x ∈ unexposed G Λ ω) (hxy : Conn G ω x y) :
+    Conn G (ω ∩ internalEdges G (unexposed G Λ ω)) x y := by
+  induction hxy with
+  | refl => exact conn_refl _ _ _
+  | @tail y z hxy hyz ih =>
+    obtain ⟨e,he,hends⟩ := hyz
+    have hy : y ∈ unexposed G Λ ω := mem_unexposed_of_conn G Λ ω hx hxy
+    have hz : z ∈ unexposed G Λ ω := mem_unexposed_of_conn G Λ ω hy (.single ⟨e,he,hends⟩)
+    exact ih.tail ⟨e,⟨he,(mem_internalEdges_of_ends G _ e y z hends).mpr ⟨hy,hz⟩⟩,hends⟩
+
+ 
+theorem unexposed_boundary_closed (G : BondGraph V E) (Λ : Set V) (ω : Set E)
+    {e : E} {a b : V} (he : G e = s(a,b))
+    (ha : a ∈ unexposed G Λ ω) (hb : b ∉ unexposed G Λ ω) : e ∉ ω := by
+  intro ho
+  exact hb (mem_unexposed_of_conn G Λ ω ha (.single ⟨e,ho,he⟩))
+
+ 
+
+theorem unexposed_insert_crossing (G : BondGraph V E) (Λ : Set V) (ω : Set E)
+    {e : E} {a b x : V} (he : G e = s(a,b))
+    (hx : x ∈ unexposed G Λ ω) (_ha : a ∈ unexposed G Λ ω)
+    (hb : b ∉ unexposed G Λ ω) :
+    x ∉ unexposed G Λ (insert e ω) ↔
+      Conn G (ω ∩ internalEdges G (unexposed G Λ ω)) x a := by
+  classical
+  constructor
+  · intro hn
+    obtain ⟨y,hxy,hy⟩ := by simpa [unexposed, not_forall, Classical.not_imp] using hn
+    rcases (conn_insert_iff G ω e a b x y he).mp hxy with h | h | h
+    · exact (hy (hx y h)).elim
+    · exact conn_restrict_unexposed G Λ ω hx h.1
+    · exact (hb (mem_unexposed_of_conn G Λ ω hx h.1)).elim
+  · intro hxa hn
+    have hxa' : Conn G ω x a := conn_mono G Set.inter_subset_left hxa
+    apply hb
+    intro y hby
+    exact hn y ((conn_insert_iff G ω e a b x y he).mpr (Or.inr (Or.inl ⟨hxa',hby⟩)))
+
+ 
+theorem not_mem_unexposed_of_conn (G : BondGraph V E) (Λ : Set V) (ω : Set E)
+    {x y : V} (hx : x ∉ unexposed G Λ ω) (hxy : Conn G ω x y) :
+    y ∉ unexposed G Λ ω := by
+  intro hy
+  exact hx (mem_unexposed_of_conn G Λ ω hy (conn_symm G ω hxy))
+
+ 
+theorem conn_sdiff_internal_unexposed (G : BondGraph V E) (Λ : Set V) (ω : Set E)
+    {x y : V} (hx : x ∉ unexposed G Λ ω) (hxy : Conn G ω x y) :
+    Conn G (ω \ internalEdges G (unexposed G Λ ω)) x y := by
+  induction hxy with
+  | refl => exact conn_refl _ _ _
+  | @tail y z hxy hyz ih =>
+    obtain ⟨e,he,hends⟩ := hyz
+    have hy : y ∉ unexposed G Λ ω := not_mem_unexposed_of_conn G Λ ω hx hxy
+    exact ih.tail ⟨e,⟨he,fun hi => hy ((mem_internalEdges_of_ends G _ e y z hends).mp hi).1⟩,hends⟩
+
+ 
+theorem unexposed_sdiff_internal (G : BondGraph V E) (Λ : Set V) (ω : Set E) :
+    unexposed G Λ (ω \ internalEdges G (unexposed G Λ ω)) = unexposed G Λ ω := by
+  ext x
+  constructor
+  · intro hx
+    by_contra hn
+    apply hn
+    intro y hxy
+    exact hx y (conn_sdiff_internal_unexposed G Λ ω hn hxy)
+  · intro hx y hxy
+    exact hx y (conn_mono G Set.sdiff_subset hxy)
+
+ 
+theorem unexposed_union_internal (G : BondGraph V E) (Λ : Set V) (ω η : Set E)
+    (hη : η ⊆ internalEdges G (unexposed G Λ ω)) :
+    unexposed G Λ (ω ∪ η) = unexposed G Λ ω := by
+  ext x
+  constructor
+  · intro hx y hxy
+    exact hx y (conn_mono G Set.subset_union_left hxy)
+  · intro hx y hxy
+    suffices hy : y ∈ unexposed G Λ ω from unexposed_subset G Λ ω hy
+    induction hxy with
+    | refl => exact hx
+    | @tail y z hxy hyz ih =>
+      obtain ⟨e,hωe,he⟩ := hyz
+      rcases hωe with ho | hηe
+      · exact mem_unexposed_of_conn G Λ ω ih (.single ⟨e,ho,he⟩)
+      · exact ((mem_internalEdges_of_ends G _ e y z he).mp (hη hηe)).2
+
+ 
+theorem unexposed_union_eq_iff (G : BondGraph V E) (Λ S : Set V) (a b : Set E)
+    (ha : a ⊆ internalEdges G S) (hb : Disjoint b (internalEdges G S)) :
+    unexposed G Λ (a ∪ b) = S ↔ unexposed G Λ b = S := by
+  constructor
+  · intro h
+    have he : (a ∪ b) \ internalEdges G S = b := by
+      ext e
+      constructor
+      · rintro ⟨he,hn⟩
+        exact he.resolve_left (fun he => hn (ha he))
+      · intro he
+        exact ⟨Or.inr he,fun hi => Set.disjoint_left.mp hb he hi⟩
+    simpa only [h, he] using unexposed_sdiff_internal G Λ (a ∪ b)
+  · intro h
+    have he := unexposed_union_internal G Λ b a (h ▸ ha)
+    simpa only [Set.union_comm b a,h] using he
+
+ 
+def escapeEvent (G : BondGraph V E) (Λ : Set V) (x : V) : Set (Set E) :=
+  {ω | x ∉ unexposed G Λ ω}
+
+theorem escapeEvent_isUpperSet (G : BondGraph V E) (Λ : Set V) (x : V) :
+    IsUpperSet (escapeEvent G Λ x) := by
+  intro ω η hωη hx hη
+  apply hx
+  intro y hxy
+  exact hη y (conn_mono G hωη hxy)
+
+ 
+theorem escape_closed_pivotal_iff (G : BondGraph V E) (Λ : Set V)
+    (ω : Set E) (x : V) (e : E) :
+    (e ∉ ω ∧ ω ∉ escapeEvent G Λ x ∧ insert e ω ∈ escapeEvent G Λ x) ↔
+      x ∈ unexposed G Λ ω ∧ ∃ a b,
+        G e = s(a,b) ∧ a ∈ unexposed G Λ ω ∧ b ∉ unexposed G Λ ω ∧
+        Conn G (ω ∩ internalEdges G (unexposed G Λ ω)) x a := by
+  classical
+  constructor
+  · rintro ⟨_,hx,hn⟩
+    have hx' : x ∈ unexposed G Λ ω := by simpa [escapeEvent] using hx
+    refine ⟨hx',?_⟩
+    obtain ⟨⟨a,b⟩,he'⟩ := Sym2.mk_surjective (G e)
+    have he : G e = s(a,b) := he'.symm
+    obtain ⟨y,hxy,hy⟩ := by simpa [escapeEvent,unexposed,not_forall,Classical.not_imp] using hn
+    rcases (conn_insert_iff G ω e a b x y he).mp hxy with h | h | h
+    · exact (hy (hx' y h)).elim
+    · exact ⟨a,b,he,mem_unexposed_of_conn G Λ ω hx' h.1,
+        fun hb => hy (hb y h.2),conn_restrict_unexposed G Λ ω hx' h.1⟩
+    · exact ⟨b,a,he.trans (Sym2.eq_swap ..),mem_unexposed_of_conn G Λ ω hx' h.1,
+        fun ha => hy (ha y h.2),conn_restrict_unexposed G Λ ω hx' h.1⟩
+  · rintro ⟨hx,a,b,he,ha,hb,hxa⟩
+    exact ⟨unexposed_boundary_closed G Λ ω he ha hb,by simpa [escapeEvent] using hx,
+      (unexposed_insert_crossing G Λ ω he hx ha hb).mpr hxa⟩
+
+attribute [local instance] Classical.decEq
+
+ theorem bernoulliPolynomial_add (s : Finset E) (f g : Finset E → ℝ) (p : ℝ) :
+    bernoulliPolynomial s (fun t => f t+g t) p =
+      bernoulliPolynomial s f p + bernoulliPolynomial s g p := by
+  simp only [bernoulliPolynomial, add_mul, Finset.sum_add_distrib]
+
+ theorem bernoulliPolynomial_mul_const (s : Finset E) (f : Finset E → ℝ) (c p : ℝ) :
+    bernoulliPolynomial s (fun t => f t*c) p = bernoulliPolynomial s f p*c := by
+  simp only [bernoulliPolynomial, Finset.sum_mul]
+  apply Finset.sum_congr rfl
+  intro t _
+  ring
+
+ theorem bernoulliPolynomial_const_mul (s : Finset E) (f : Finset E → ℝ) (c p : ℝ) :
+    bernoulliPolynomial s (fun t => c*f t) p = c*bernoulliPolynomial s f p := by
+  simpa only [mul_comm c] using bernoulliPolynomial_mul_const s f c p
+
+ theorem bernoulliPolynomial_sum {ι : Type*} (s : Finset E) (I : Finset ι)
+    (f : ι → Finset E → ℝ) (p : ℝ) :
+    bernoulliPolynomial s (fun t => ∑ i ∈ I, f i t) p =
+      ∑ i ∈ I, bernoulliPolynomial s (f i) p := by
+  classical
+  simp only [bernoulliPolynomial, Finset.sum_mul]
+  exact Finset.sum_comm
+
+ 
+theorem bernoulliPolynomial_blocks (s t : Finset E) (hdis : Disjoint s t)
+    (f g : Finset E → ℝ) (p : ℝ) :
+    bernoulliPolynomial (s ∪ t) (fun a => f (a ∩ s)*g (a ∩ t)) p =
+      bernoulliPolynomial s f p * bernoulliPolynomial t g p := by
+  rw [bernoulliPolynomial_union s t hdis]
+  trans bernoulliPolynomial s (fun a => f a*bernoulliPolynomial t g p) p
+  · apply bernoulliPolynomial_congr
+    intro a ha
+    rw [← bernoulliPolynomial_const_mul]
+    apply bernoulliPolynomial_congr
+    intro b hb
+    have has : a ∩ s = a := Finset.inter_eq_left.mpr ha
+    have hbt : b ∩ t = b := Finset.inter_eq_left.mpr hb
+    have hbs : b ∩ s = ∅ := Finset.disjoint_iff_inter_eq_empty.mp
+      (hdis.symm.mono_left hb)
+    have hat : a ∩ t = ∅ := Finset.disjoint_iff_inter_eq_empty.mp
+      (hdis.mono_left ha)
+    simp [Finset.union_inter_distrib_right,has,hbt,hbs,hat]
+  · exact bernoulliPolynomial_mul_const s f _ p
+
+ 
+
+theorem bernoulliPolynomial_exposure (G : BondGraph V E) (Λ S : Set V)
+    (s t : Finset E) (hs : (s : Set E) ⊆ internalEdges G S)
+    (ht : Disjoint (t : Set E) (internalEdges G S)) (f : Finset E → ℝ) (p : ℝ) :
+    bernoulliPolynomial (s ∪ t)
+      (fun a => (if unexposed G Λ (a : Set E) = S then (1:ℝ) else 0) * f (a ∩ s)) p =
+    bernoulliPolynomial t (fun b => if unexposed G Λ (b : Set E) = S then (1:ℝ) else 0) p *
+      bernoulliPolynomial s f p := by
+  have hdis : Disjoint s t := by
+    apply Finset.disjoint_left.mpr
+    intro e he hf
+    exact Set.disjoint_left.mp ht hf (hs he)
+  rw [bernoulliPolynomial_union s t hdis]
+  trans bernoulliPolynomial s (fun a =>
+    bernoulliPolynomial t (fun b => if unexposed G Λ (b : Set E) = S then (1:ℝ) else 0) p * f a) p
+  · apply bernoulliPolynomial_congr
+    intro a ha
+    rw [← bernoulliPolynomial_mul_const]
+    apply bernoulliPolynomial_congr
+    intro b hb
+    have hab : unexposed G Λ ((a ∪ b : Finset E) : Set E) = S ↔
+        unexposed G Λ (b : Set E) = S := by
+      rw [Finset.coe_union]
+      exact unexposed_union_eq_iff G Λ S _ _ (fun _ he => hs (ha he))
+        (ht.mono_left (by exact_mod_cast hb))
+    have hbs : b ∩ s = ∅ :=
+      Finset.disjoint_iff_inter_eq_empty.mp (hdis.symm.mono_left hb)
+    rw [Finset.union_inter_distrib_right,Finset.inter_eq_left.mpr ha,hbs,Finset.union_empty]
+    simp only [hab]
+  · exact bernoulliPolynomial_const_mul s f _ p
+
+ 
+noncomputable def unexposedRegion (G : BondGraph V E) (Λ : Finset V) (ω : Set E) : Finset V :=
+  ((Set.toFinite (Λ : Set V)).subset (unexposed_subset G (Λ : Set V) ω)).toFinset
+
+@[simp] theorem coe_unexposedRegion (G : BondGraph V E) (Λ : Finset V) (ω : Set E) :
+    (unexposedRegion G Λ ω : Set V) = unexposed G (Λ : Set V) ω := by
+  simp [unexposedRegion]
+
+theorem unexposedRegion_subset (G : BondGraph V E) (Λ : Finset V) (ω : Set E) :
+    unexposedRegion G Λ ω ⊆ Λ := by
+  intro x hx
+  apply unexposed_subset G (Λ : Set V) ω
+  simpa only [← coe_unexposedRegion,Finset.mem_coe] using hx
+
+ 
+theorem bernoulliPolynomial_partition_region (G : BondGraph V E) (Λ : Finset V)
+    (F : Finset E) (f : Finset E → ℝ) (p : ℝ) :
+    bernoulliPolynomial F f p = ∑ S ∈ Λ.powerset,
+      bernoulliPolynomial F
+        (fun t => if unexposed G (Λ : Set V) (t : Set E) = (S : Set V) then f t else 0) p := by
+  rw [← bernoulliPolynomial_sum]
+  apply bernoulliPolynomial_congr
+  intro t _
+  symm
+  rw [Finset.sum_eq_single (unexposedRegion G Λ (t : Set E))]
+  · simp
+  · intro S _ hne
+    have hh : unexposed G (Λ : Set V) (t : Set E) ≠ (S : Set V) := by
+      intro he
+      apply hne
+      apply Finset.coe_injective
+      exact he.symm.trans (coe_unexposedRegion G Λ (t : Set E)).symm
+    simp [hh]
+  · intro hn
+    exact (hn (Finset.mem_powerset.mpr (unexposedRegion_subset G Λ (t : Set E)))).elim
+
+ 
+noncomputable def internalSupport (G : BondGraph V E) (S : Set V) (F : Finset E) : Finset E :=
+  F.filter (fun e => e ∈ internalEdges G S)
+
+ theorem internalSupport_subset (G : BondGraph V E) (S : Set V) (F : Finset E) :
+    internalSupport G S F ⊆ F := Finset.filter_subset _ _
+
+ theorem internalSupport_union_sdiff (G : BondGraph V E) (S : Set V) (F : Finset E) :
+    internalSupport G S F ∪ (F \ internalSupport G S F) = F :=
+  Finset.union_sdiff_of_subset (internalSupport_subset G S F)
+
+ theorem coe_inter_internalSupport (G : BondGraph V E) (S : Set V) (F t : Finset E)
+    (ht : t ⊆ F) : ((t ∩ internalSupport G S F : Finset E) : Set E) =
+      (t : Set E) ∩ internalEdges G S := by
+  ext e
+  simp only [Finset.mem_coe,Finset.mem_inter,internalSupport,Finset.mem_filter,Set.mem_inter_iff]
+  exact ⟨fun h => ⟨h.1,h.2.2⟩,fun h => ⟨h.1,ht h.1,h.2⟩⟩
+
+ 
+noncomputable def regionProbability (G : BondGraph V E) (Λ S : Set V)
+    (F : Finset E) (p : ℝ) : ℝ :=
+  bernoulliPolynomial F (fun t => if unexposed G Λ (t : Set E) = S then 1 else 0) p
+
+ theorem regionProbability_nonneg (G : BondGraph V E) (Λ S : Set V)
+    (F : Finset E) (p : ℝ) (hp : p ∈ Set.Icc (0:ℝ) 1) :
+    0 ≤ regionProbability G Λ S F p := by
+  apply bernoulliPolynomial_nonneg _ _ _ hp
+  intro t _
+  split_ifs <;> norm_num
+
+ 
+theorem bernoulliPolynomial_exposure_factor (G : BondGraph V E) (Λ S : Set V)
+    (F : Finset E) (f : Finset E → ℝ) (p : ℝ) :
+    bernoulliPolynomial F
+      (fun t => (if unexposed G Λ (t : Set E) = S then (1:ℝ) else 0) *
+        f (t ∩ internalSupport G S F)) p =
+      regionProbability G Λ S F p * bernoulliPolynomial (internalSupport G S F) f p := by
+  let s := internalSupport G S F
+  let t := F \ s
+  have hs : (s : Set E) ⊆ internalEdges G S := by
+    intro e he
+    exact (Finset.mem_filter.mp he).2
+  have ht : Disjoint (t : Set E) (internalEdges G S) := by
+    apply Set.disjoint_left.mpr
+    intro e he hi
+    have he' := Finset.mem_sdiff.mp he
+    exact he'.2 (Finset.mem_filter.mpr ⟨he'.1,hi⟩)
+  have he : s ∪ t = F := internalSupport_union_sdiff G S F
+  have hq : bernoulliPolynomial t
+      (fun b => if unexposed G Λ (b : Set E) = S then (1:ℝ) else 0) p =
+      regionProbability G Λ S F p := by
+    simpa only [he,mul_one,bernoulliPolynomial_const,regionProbability] using
+      (bernoulliPolynomial_exposure G Λ S s t hs ht (fun _ => 1) p).symm
+  simpa only [he,hq] using bernoulliPolynomial_exposure G Λ S s t hs ht f p
+
+ 
+def crossingReach (G : BondGraph V E) (S : Set V) (x : V) (ω : Set E) (e : E) : Prop :=
+  ∃ a b, G e = s(a,b) ∧ a ∈ S ∧ b ∉ S ∧ Conn G ω x a
+
+noncomputable def crossingCount (G : BondGraph V E) (S : Set V) (x : V)
+    (F : Finset E) (ω : Set E) : ℝ :=
+  ∑ e ∈ F, if crossingReach G S x ω e then (1:ℝ) else 0
+
+ 
+noncomputable def boundaryExpectation (G : BondGraph V E) (S : Set V) (x : V)
+    (F : Finset E) (p : ℝ) : ℝ :=
+  bernoulliPolynomial (internalSupport G S F)
+    (fun t => crossingCount G S x F (t : Set E)) p
+
+noncomputable def closedEscapeCount (G : BondGraph V E) (Λ : Set V) (x : V)
+    (F : Finset E) (ω : Set E) : ℝ :=
+  ∑ e ∈ F, if ω ∈ closedPivotal (escapeEvent G Λ x) e then (1:ℝ) else 0
+
+theorem finite_escape_russo (G : BondGraph V E) (Λ : Set V) (x : V)
+    (F : Finset E) (p : ℝ) :
+    (1-p)*deriv (finiteEventPolynomial F (escapeEvent G Λ x)) p =
+      bernoulliPolynomial F (fun t => closedEscapeCount G Λ x F (t : Set E)) p := by
+  unfold finiteEventPolynomial
+  rw [closed_difference_russo]
+  unfold closedEscapeCount
+  rw [bernoulliPolynomial_sum]
+  apply Finset.sum_congr rfl
+  intro e _
+  apply bernoulliPolynomial_congr
+  intro t _
+  exact (closedPivotal_difference _ (escapeEvent_isUpperSet G Λ x) e t).symm
+
+ 
+theorem closedEscapeCount_expectation (G : BondGraph V E) (Λ : Finset V) (x : V)
+    (F : Finset E) (p : ℝ) :
+    bernoulliPolynomial F (fun t => closedEscapeCount G Λ x F (t : Set E)) p =
+      ∑ S ∈ Λ.powerset, if x ∈ S then
+        regionProbability G Λ S F p * boundaryExpectation G S x F p else 0 := by
+  rw [bernoulliPolynomial_partition_region G Λ F]
+  apply Finset.sum_congr rfl
+  intro S _
+  by_cases hx : x ∈ S
+  · rw [ite_eq_left hx]
+    unfold boundaryExpectation
+    rw [← bernoulliPolynomial_exposure_factor G Λ S F
+      (fun t => crossingCount G S x F (t : Set E)) p]
+    apply bernoulliPolynomial_congr
+    intro t ht
+    by_cases hreg : unexposed G (Λ : Set V) (t : Set E) = (S : Set V)
+    · simp only [hreg,ite_true,one_mul]
+      apply Finset.sum_congr rfl
+      intro e _
+      have hc : (t : Set E) ∈ closedPivotal (escapeEvent G Λ x) e ↔
+          crossingReach G S x ((t ∩ internalSupport G S F : Finset E) : Set E) e := by
+        simpa only [closedPivotal,Set.mem_ofPred_eq,hreg,Finset.mem_coe,hx,true_and,
+          coe_inter_internalSupport G S F t ht,crossingReach] using
+          escape_closed_pivotal_iff G Λ (t : Set E) x e
+      exact if_congr hc rfl rfl
+    · simp [hreg]
+  · rw [ite_eq_right hx,← bernoulliPolynomial_zero F p]
+    apply bernoulliPolynomial_congr
+    intro t _
+    by_cases hreg : unexposed G (Λ : Set V) (t : Set E) = (S : Set V)
+    · simp only [hreg,ite_true]
+      apply Finset.sum_eq_zero
+      intro e _
+      have hn : (t : Set E) ∉ closedPivotal (escapeEvent G Λ x) e := by
+        intro he
+        have hh := ((escape_closed_pivotal_iff G Λ (t : Set E) x e).mp he).1
+        apply hx
+        simpa only [hreg,Finset.mem_coe] using hh
+      simp [hn]
+    · simp [hreg]
+
+ 
+theorem finite_escape_complement (G : BondGraph V E) (Λ : Finset V) (x : V)
+    (F : Finset E) (p : ℝ) :
+    1-finiteEventPolynomial F (escapeEvent G Λ x) p =
+      ∑ S ∈ Λ.powerset, if x ∈ S then regionProbability G Λ S F p else 0 := by
+  have he : 1-finiteEventPolynomial F (escapeEvent G Λ x) p =
+      bernoulliPolynomial F
+        (fun t => if x ∈ unexposed G (Λ : Set V) (t : Set E) then (1:ℝ) else 0) p := by
+    change 1-bernoulliPolynomial F (fun t => if (t : Set E) ∈ escapeEvent G Λ x then 1 else 0) p = _
+    rw [show (1:ℝ) = bernoulliPolynomial F (fun _ => 1) p from (bernoulliPolynomial_const F 1 p).symm]
+    rw [← bernoulliPolynomial_sub]
+    apply bernoulliPolynomial_congr
+    intro t _
+    by_cases hx : x ∈ unexposed G (Λ : Set V) (t : Set E) <;> simp [escapeEvent,hx,bernoulliPolynomial_const]
+  rw [he,bernoulliPolynomial_partition_region G Λ F]
+  apply Finset.sum_congr rfl
+  intro S _
+  by_cases hx : x ∈ S
+  · rw [ite_eq_left hx]
+    apply bernoulliPolynomial_congr
+    intro t _
+    by_cases hreg : unexposed G (Λ : Set V) (t : Set E) = (S : Set V) <;> simp [hreg,hx]
+  · rw [ite_eq_right hx,← bernoulliPolynomial_zero F p]
+    apply bernoulliPolynomial_congr
+    intro t _
+    by_cases hreg : unexposed G (Λ : Set V) (t : Set E) = (S : Set V) <;> simp [hreg,hx]
+
+ 
+
+theorem finite_escape_differential_lower (G : BondGraph V E) (Λ : Finset V) (x : V)
+    (F : Finset E) (p : ℝ) (hp : p ∈ Set.Icc (0:ℝ) 1)
+    (hcost : ∀ S ⊆ Λ, x ∈ S → 1 ≤ p*boundaryExpectation G (S : Set V) x F p) :
+    1-finiteEventPolynomial F (escapeEvent G Λ x) p ≤
+      p*(1-p)*deriv (finiteEventPolynomial F (escapeEvent G Λ x)) p := by
+  rw [mul_assoc,finite_escape_russo,closedEscapeCount_expectation,finite_escape_complement,
+    Finset.mul_sum]
+  apply Finset.sum_le_sum
+  intro S hS
+  by_cases hx : x ∈ S
+  · simp only [ite_eq_left hx]
+    have hq := regionProbability_nonneg G Λ S F p hp
+    calc
+      regionProbability G Λ S F p ≤
+          regionProbability G Λ S F p * (p*boundaryExpectation G S x F p) :=
+        le_mul_of_one_le_right hq (hcost S (Finset.mem_powerset.mp hS) hx)
+      _ = p*(regionProbability G Λ S F p * boundaryExpectation G S x F p) := by ring
+  · simp [hx]
+
+ theorem bernoulliPolynomial_monotoneOn (F : Finset E) (f : Finset E → ℝ)
+    (hf : ∀ a b, a ⊆ b → f a ≤ f b) :
+    MonotoneOn (bernoulliPolynomial F f) (Set.Icc (0:ℝ) 1) := by
+  have hd : Differentiable ℝ (bernoulliPolynomial F f) :=
+    fun q => (hasDerivAt_bernoulliPolynomial F f q).differentiableAt
+  apply monotoneOn_of_deriv_nonneg (convex_Icc 0 1) hd.continuous.continuousOn hd.differentiableOn
+  intro q hq
+  have hq' : q ∈ Set.Icc (0:ℝ) 1 := interior_subset hq
+  rw [(hasDerivAt_bernoulliPolynomial F f q).deriv]
+  apply Finset.sum_nonneg
+  intro e _
+  apply bernoulliPolynomial_nonneg _ _ _ hq'
+  intro t _
+  exact sub_nonneg.mpr (hf t (insert e t) (Finset.subset_insert e t))
+
+ theorem crossingCount_mono (G : BondGraph V E) (S : Set V) (x : V)
+    (F : Finset E) {ω η : Set E} (h : ω ⊆ η) :
+    crossingCount G S x F ω ≤ crossingCount G S x F η := by
+  apply Finset.sum_le_sum
+  intro e _
+  have hh : crossingReach G S x ω e → crossingReach G S x η e := by
+    rintro ⟨a,b,he,ha,hb,hc⟩
+    exact ⟨a,b,he,ha,hb,conn_mono G h hc⟩
+  by_cases hω : crossingReach G S x ω e
+  · simp [hω,hh hω]
+  · simp only [ite_eq_right hω]
+    split_ifs <;> norm_num
+
+ theorem boundaryExpectation_nonneg (G : BondGraph V E) (S : Set V) (x : V)
+    (F : Finset E) (p : ℝ) (hp : p ∈ Set.Icc (0:ℝ) 1) :
+    0 ≤ boundaryExpectation G S x F p := by
+  apply bernoulliPolynomial_nonneg _ _ _ hp
+  intro t _
+  apply Finset.sum_nonneg
+  intro e _
+  split_ifs <;> norm_num
+
+ theorem boundaryExpectation_monotoneOn (G : BondGraph V E) (S : Set V) (x : V)
+    (F : Finset E) : MonotoneOn (boundaryExpectation G S x F) (Set.Icc (0:ℝ) 1) := by
+  apply bernoulliPolynomial_monotoneOn
+  intro a b hab
+  exact crossingCount_mono G S x F (by exact_mod_cast hab)
+
+ theorem boundaryCost_monotoneOn (G : BondGraph V E) (S : Set V) (x : V)
+    (F : Finset E) : MonotoneOn (fun p => p*boundaryExpectation G S x F p)
+      (Set.Icc (0:ℝ) 1) := by
+  intro p hp q hq hpq
+  exact mul_le_mul hpq (boundaryExpectation_monotoneOn G S x F hp hq hpq)
+    (boundaryExpectation_nonneg G S x F p hp) hq.1
+
+ theorem finiteEventPolynomial_mem_Icc (F : Finset E) (A : Set (Set E))
+    (p : ℝ) (hp : p ∈ Set.Icc (0:ℝ) 1) :
+    finiteEventPolynomial F A p ∈ Set.Icc (0:ℝ) 1 := by
+  constructor
+  · apply bernoulliPolynomial_nonneg _ _ _ hp
+    intro t _
+    split_ifs <;> norm_num
+  · calc
+      finiteEventPolynomial F A p ≤ bernoulliPolynomial F (fun _ => 1) p := by
+        apply bernoulliPolynomial_mono _ _ _ _ hp
+        intro t _
+        split_ifs <;> norm_num
+      _ = 1 := bernoulliPolynomial_const F 1 p
+
+ 
+
+theorem differential_escape_growth {f : ℝ → ℝ} {r p : ℝ}
+    (hr : 0 < r) (hrp : r ≤ p) (hp : p < 1)
+    (hf : Differentiable ℝ f) (hbound : ∀ q ∈ Set.Icc r p, f q ∈ Set.Icc (0:ℝ) 1)
+    (hineq : ∀ q ∈ Set.Icc r p, 1-f q ≤ q*(1-q)*deriv f q) :
+    1-Real.exp (r-p) ≤ f p := by
+  have hd (q : ℝ) : HasDerivAt (fun u => Real.exp u*(1-f u))
+      (Real.exp q*(1-f q)+Real.exp q*(0-deriv f q)) q :=
+    (Real.hasDerivAt_exp q).mul ((hasDerivAt_const q 1).sub (hf q).hasDerivAt)
+  have hanti : AntitoneOn (fun q => Real.exp q*(1-f q)) (Set.Icc r p) := by
+    apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc r p)
+      (Real.continuous_exp.mul (continuous_const.sub hf.continuous)).continuousOn
+      (fun q _ => (hd q).hasDerivWithinAt)
+    intro q hq
+    have hqi : q ∈ Set.Icc r p := interior_subset hq
+    have hq0 : 0 < q := hr.trans_le hqi.1
+    have hq1 : q < 1 := hqi.2.trans_lt hp
+    have hcoef : 0 < q*(1-q) := mul_pos hq0 (sub_pos.mpr hq1)
+    have hder : 0 ≤ deriv f q := nonneg_of_mul_nonneg_right
+      ((sub_nonneg.mpr (hbound q hqi).2).trans (hineq q hqi)) hcoef
+    have hc : q*(1-q) ≤ 1 := by nlinarith [sq_nonneg q]
+    have hsmall : 1-f q ≤ deriv f q := (hineq q hqi).trans
+      (mul_le_of_le_one_left hder hc)
+    have hn := mul_nonpos_of_nonneg_of_nonpos (Real.exp_pos q).le
+      (show 1-f q-deriv f q ≤ 0 by linarith)
+    nlinarith only [hn]
+  have hle := hanti (show r ∈ Set.Icc r p from ⟨le_rfl,hrp⟩)
+    (show p ∈ Set.Icc r p from ⟨hrp,le_rfl⟩) hrp
+  have he : Real.exp p*(1-f p) ≤ Real.exp r := by
+    calc
+      Real.exp p*(1-f p) ≤ Real.exp r*(1-f r) := hle
+      _ ≤ Real.exp r := mul_le_of_le_one_right (Real.exp_pos r).le
+        (by linarith [(hbound r ⟨le_rfl,hrp⟩).1])
+  have hfin : 1-f p ≤ Real.exp (r-p) := by
+    rw [Real.exp_sub]
+    apply (le_div_iff₀ (Real.exp_pos p)).mpr
+    simpa only [mul_comm] using he
+  linarith
+
+ 
+
+theorem finite_escape_probability_lower (G : BondGraph V E) (Λ : Finset V) (x : V)
+    (F : Finset E) (r p : ℝ) (hr : 0 < r) (hrp : r < p) (hp : p < 1)
+    (hcost : ∀ S ⊆ Λ, x ∈ S → 1 ≤ r*boundaryExpectation G (S : Set V) x F r) :
+    0 < 1-Real.exp (r-p) ∧
+      1-Real.exp (r-p) ≤ finiteEventPolynomial F (escapeEvent G Λ x) p := by
+  have hrI : r ∈ Set.Icc (0:ℝ) 1 := ⟨hr.le,(hrp.trans hp).le⟩
+  have hqI (q : ℝ) (hq : q ∈ Set.Icc r p) : q ∈ Set.Icc (0:ℝ) 1 :=
+    ⟨hr.le.trans hq.1,hq.2.trans hp.le⟩
+  refine ⟨sub_pos.mpr (Real.exp_lt_one_iff.mpr (sub_neg.mpr hrp)),?_⟩
+  apply differential_escape_growth hr hrp.le hp
+  · intro q
+    exact (hasDerivAt_bernoulliPolynomial F _ q).differentiableAt
+  · intro q hq
+    exact finiteEventPolynomial_mem_Icc F _ q (hqI q hq)
+  · intro q hq
+    apply finite_escape_differential_lower G Λ x F q (hqI q hq)
+    intro S hS hx
+    exact (hcost S hS hx).trans (boundaryCost_monotoneOn G S x F hrI (hqI q hq) hq.1)
+
+end ExteriorExposure
+
+section SharpnessBridge
+attribute [local instance] Classical.propDecidable Classical.decEq
+open Filter Topology
+
+theorem bernoulliPolynomial_map {F : Type*} (f : E ↪ F) (s : Finset E)
+    (h : Finset F → ℝ) (p : ℝ) :
+    bernoulliPolynomial (s.map f) h p =
+      bernoulliPolynomial s (fun t => h (t.map f)) p := by
+  induction s using Finset.induction_on generalizing h with
+  | empty => simp
+  | @insert e s he ih =>
+    rw [Finset.map_insert,bernoulliPolynomial_insert _ (f e) (by simpa using he),
+      bernoulliPolynomial_insert s e he,ih,ih]
+    simp only [Finset.map_insert]
+
+theorem finiteEventPolynomial_restrict (F : Finset E) (A : Set (Set E)) (p : unitInterval) :
+    finiteEventPolynomial F A (p : ℝ) = (law p).real {ω | ω ∩ (F : Set E) ∈ A} := by
+  let f : (F : Set E) ↪ E := ⟨Subtype.val,Subtype.val_injective⟩
+  have hm : (Finset.univ : Finset (F : Set E)).map f = F := by
+    ext e
+    simp [f]
+  have he : finiteEventPolynomial F A (p : ℝ) =
+      finiteEventPolynomial (Finset.univ : Finset (F : Set E))
+        {η | f '' η ∈ A} (p : ℝ) := by
+    unfold finiteEventPolynomial
+    calc
+      bernoulliPolynomial F (fun t => if (t : Set E) ∈ A then 1 else 0) (p : ℝ) =
+        bernoulliPolynomial (Finset.univ.map f) (fun t => if (t : Set E) ∈ A then 1 else 0) (p : ℝ) :=
+          congrArg (fun H => bernoulliPolynomial H (fun t => if (t : Set E) ∈ A then 1 else 0) (p : ℝ)) hm.symm
+      _ = _ := by
+        rw [bernoulliPolynomial_map]
+        apply bernoulliPolynomial_congr
+        intro t _
+        simp only [Finset.coe_map]
+        rfl
+  rw [he,finiteEventPolynomial_eq,finiteProbability_univ,← law_map_restrict p (F : Set E),
+    measureReal_def,Measure.map_apply (measurable_set_restrict _) .of_discrete]
+  congr 2
+  ext ω
+  have hh : f '' (Subtype.val ⁻¹' ω : Set (F : Set E)) = ω ∩ (F : Set E) := by
+    ext e
+    simp [f,and_comm]
+  simp only [Set.mem_preimage,Set.mem_ofPred_eq,hh]
+
+theorem bernoulliPolynomial_restrict (F H : Finset E) (hFH : F ⊆ H)
+    (f : Finset E → ℝ) (p : ℝ) :
+    bernoulliPolynomial H (fun t => f (t ∩ F)) p = bernoulliPolynomial F f p := by
+  rw [← Finset.union_sdiff_of_subset hFH,bernoulliPolynomial_union _ _ Finset.disjoint_sdiff]
+  trans bernoulliPolynomial F (fun a => f a) p
+  · apply bernoulliPolynomial_congr
+    intro a ha
+    trans bernoulliPolynomial (H \ F) (fun _ => f a) p
+    · apply bernoulliPolynomial_congr
+      intro b hb
+      have hdis : Disjoint b F := (Finset.disjoint_sdiff.symm).mono_left hb
+      rw [Finset.union_inter_distrib_right,Finset.inter_eq_left.mpr ha,
+        Finset.disjoint_iff_inter_eq_empty.mp hdis,Finset.union_empty]
+    · exact bernoulliPolynomial_const _ _ _
+  · rfl
+
+ 
+noncomputable def incidentSupport (G : BondGraph V E) (hloc : LocallyFinite G)
+    (Λ : Finset V) : Finset E := Λ.biUnion (fun x => (hloc x).toFinset)
+
+@[simp] theorem mem_incidentSupport (G : BondGraph V E) (hloc : LocallyFinite G)
+    (Λ : Finset V) (e : E) : e ∈ incidentSupport G hloc Λ ↔ ∃ x ∈ Λ, x ∈ G e := by
+  simp [incidentSupport]
+
+theorem incidentSupport_mono (G : BondGraph V E) (hloc : LocallyFinite G)
+    {S Λ : Finset V} (hS : S ⊆ Λ) : incidentSupport G hloc S ⊆ incidentSupport G hloc Λ := by
+  intro e he
+  obtain ⟨x,hx,hxe⟩ := (mem_incidentSupport G hloc S e).mp he
+  exact (mem_incidentSupport G hloc Λ e).mpr ⟨x,hS hx,hxe⟩
+
+ 
+theorem escapeEvent_restrict_incident (G : BondGraph V E) (hloc : LocallyFinite G)
+    (Λ : Finset V) (x : V) (ω : Set E) :
+    ω ∩ (incidentSupport G hloc Λ : Set E) ∈ escapeEvent G Λ x ↔
+      ω ∈ escapeEvent G Λ x := by
+  constructor
+  · exact escapeEvent_isUpperSet G Λ x Set.inter_subset_left
+  · intro hx
+    by_contra hn
+    have hx' : x ∈ unexposed G Λ (ω ∩ (incidentSupport G hloc Λ : Set E)) := by
+      simpa only [escapeEvent,Set.mem_ofPred_eq,not_not] using hn
+    apply hx
+    intro y hxy
+    have hc : Conn G (ω ∩ (incidentSupport G hloc Λ : Set E)) x y := by
+      induction hxy with
+      | refl => exact conn_refl _ _ _
+      | @tail a b hxa hab ih =>
+        obtain ⟨e,he,heab⟩ := hab
+        have ha : a ∈ Λ := hx' a ih
+        have hF : e ∈ incidentSupport G hloc Λ :=
+          (mem_incidentSupport G hloc Λ e).mpr ⟨a,ha,by simp [heab]⟩
+        exact ih.tail ⟨e,⟨he,hF⟩,heab⟩
+    exact hx' y hc
+
+theorem finite_escape_eq_law (G : BondGraph V E) (hloc : LocallyFinite G)
+    (Λ : Finset V) (x : V) (p : unitInterval) :
+    finiteEventPolynomial (incidentSupport G hloc Λ) (escapeEvent G Λ x) (p : ℝ) =
+      (law p).real (escapeEvent G Λ x) := by
+  rw [finiteEventPolynomial_restrict]
+  congr 1
+  ext ω
+  exact escapeEvent_restrict_incident G hloc Λ x ω
+
+theorem internal_edge_mem_incident (G : BondGraph V E) (hloc : LocallyFinite G)
+    (S : Finset V) {e : E} (he : e ∈ internalEdges G (S : Set V)) :
+    e ∈ incidentSupport G hloc S := by
+  obtain ⟨⟨a,b⟩,hab⟩ := Sym2.mk_surjective (G e)
+  have ha : a ∈ G e := by rw [← hab]; simp
+  exact (mem_incidentSupport G hloc S e).mpr ⟨a,he a ha,ha⟩
+
+theorem internalSupport_eq_of_incident_subset (G : BondGraph V E) (hloc : LocallyFinite G)
+    (S : Finset V) (F : Finset E) (hF : incidentSupport G hloc S ⊆ F) :
+    internalSupport G S F = internalSupport G S (incidentSupport G hloc S) := by
+  ext e
+  simp only [internalSupport,Finset.mem_filter]
+  exact ⟨fun h => ⟨internal_edge_mem_incident G hloc S h.2,h.2⟩,
+    fun h => ⟨hF h.1,h.2⟩⟩
+
+theorem crossingCount_eq_of_incident_subset (G : BondGraph V E) (hloc : LocallyFinite G)
+    (S : Finset V) (x : V) (F : Finset E) (hF : incidentSupport G hloc S ⊆ F) (ω : Set E) :
+    crossingCount G S x F ω = crossingCount G S x (incidentSupport G hloc S) ω := by
+  symm
+  apply Finset.sum_subset hF
+  intro e _ he
+  have hn : ¬ crossingReach G S x ω e := by
+    rintro ⟨a,b,hab,ha,_,_⟩
+    exact he ((mem_incidentSupport G hloc S e).mpr ⟨a,ha,by simp [hab]⟩)
+  simp [hn]
+
+theorem boundaryExpectation_eq_of_incident_subset (G : BondGraph V E) (hloc : LocallyFinite G)
+    (S : Finset V) (x : V) (F : Finset E) (hF : incidentSupport G hloc S ⊆ F) (p : ℝ) :
+    boundaryExpectation G S x F p =
+      boundaryExpectation G S x (incidentSupport G hloc S) p := by
+  unfold boundaryExpectation
+  rw [internalSupport_eq_of_incident_subset G hloc S F hF]
+  apply bernoulliPolynomial_congr
+  intro t _
+  exact crossingCount_eq_of_incident_subset G hloc S x F hF t
+
+ 
+noncomputable def boundaryCost (G : BondGraph V E) (hloc : LocallyFinite G)
+    (S : Finset V) (x : V) (p : ℝ) : ℝ :=
+  p*boundaryExpectation G S x (incidentSupport G hloc S) p
+
+theorem measurableSet_escapeEvent (G : BondGraph V E) (hloc : LocallyFinite G)
+    (Λ : Finset V) (x : V) : MeasurableSet (escapeEvent G Λ x) := by
+  let F := incidentSupport G hloc Λ
+  have he : escapeEvent G Λ x =
+      (fun ω => (Subtype.val ⁻¹' ω : Set (F : Set E))) ⁻¹'
+        {η | Subtype.val '' η ∈ escapeEvent G Λ x} := by
+    ext ω
+    have hh : Subtype.val '' (Subtype.val ⁻¹' ω : Set (F : Set E)) = ω ∩ (F : Set E) := by
+      ext e
+      simp [and_comm]
+    simp only [Set.mem_preimage,Set.mem_ofPred_eq,hh]
+    exact (escapeEvent_restrict_incident G hloc Λ x ω).symm
+  rw [he]
+  exact MeasurableSet.preimage .of_discrete (measurable_set_restrict _)
+
+theorem escapeEvent_antitone_domain (G : BondGraph V E) {S Λ : Set V} (hS : S ⊆ Λ)
+    (x : V) : escapeEvent G Λ x ⊆ escapeEvent G S x := by
+  intro ω hω hn
+  apply hω
+  intro y hxy
+  exact hS (hn y hxy)
+
+noncomputable def vertexExhaustion [Encodable V] (n : ℕ) : Finset V :=
+  (Finset.range n).preimage Encodable.encode Encodable.encode_injective.injOn
+
+theorem vertexExhaustion_mono [Encodable V] :
+    Monotone (vertexExhaustion : ℕ → Finset V) := by
+  intro m n h x hx
+  simp only [vertexExhaustion,Finset.mem_preimage,Finset.mem_range] at hx ⊢
+  exact hx.trans_le h
+
+theorem subset_vertexExhaustion [Encodable V] (S : Finset V) :
+    ∃ n, S ⊆ vertexExhaustion n := by
+  refine ⟨S.sup Encodable.encode+1,?_⟩
+  intro x hx
+  simp only [vertexExhaustion,Finset.mem_preimage,Finset.mem_range]
+  exact Nat.lt_succ_of_le (Finset.le_sup hx)
+
+theorem infinite_cluster_eq_iInter_escape [Encodable V] (G : BondGraph V E) (x : V) :
+    {ω | (cluster G ω x).Infinite} = ⋂ n, escapeEvent G (vertexExhaustion (V := V) n : Set V) x := by
+  ext ω
+  simp only [Set.mem_ofPred_eq,Set.mem_iInter]
+  constructor
+  · intro hi n hn
+    exact hi ((vertexExhaustion n).finite_toSet.subset (by
+      intro y hy; exact hn y hy))
+  · intro he hf
+    obtain ⟨n,hn⟩ := subset_vertexExhaustion hf.toFinset
+    apply he n
+    intro y hy
+    exact hn (hf.mem_toFinset.mpr hy)
+
+ 
+theorem infinite_cluster_probability_ge_of_escape [Countable V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) (p : unitInterval) (c : ℝ)
+    (hc : ∀ Λ : Finset V, c ≤ (law p).real (escapeEvent G Λ x)) :
+    c ≤ (law p).real {ω | (cluster G ω x).Infinite} := by
+  let := Encodable.ofCountable V
+  rw [infinite_cluster_eq_iInter_escape]
+  have hm : Antitone (fun n => escapeEvent G (vertexExhaustion (V := V) n : Set V) x) := by
+    intro m n h
+    exact escapeEvent_antitone_domain G (by exact_mod_cast vertexExhaustion_mono h) x
+  have ht := tendsto_measure_iInter_atTop (μ := law p)
+    (fun n => (measurableSet_escapeEvent G hloc (vertexExhaustion n) x).nullMeasurableSet)
+    hm ⟨0,measure_ne_top _ _⟩
+  have hr := (ENNReal.tendsto_toReal (measure_ne_top _ _)).comp ht
+  exact ge_of_tendsto hr (Eventually.of_forall (fun n => hc (vertexExhaustion n)))
+
+ 
+
+theorem infinite_cluster_probability_pos_of_bad_cost [Countable V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) (r p : unitInterval)
+    (hr : 0 < r) (hrp : r < p) (hp : p < 1)
+    (hcost : ∀ S : Finset V, x ∈ S → 1 ≤ boundaryCost G hloc S x (r : ℝ)) :
+    0 < (law p).real {ω | (cluster G ω x).Infinite} := by
+  have hc : 0 < 1-Real.exp ((r:ℝ)-(p:ℝ)) :=
+    sub_pos.mpr (Real.exp_lt_one_iff.mpr (sub_neg.mpr hrp))
+  apply hc.trans_le
+  apply infinite_cluster_probability_ge_of_escape G hloc x p
+  intro Λ
+  rw [← finite_escape_eq_law G hloc Λ x p]
+  apply (finite_escape_probability_lower G Λ x (incidentSupport G hloc Λ)
+    (r:ℝ) (p:ℝ) hr hrp hp ?_).2
+  intro S hS hx
+  rw [boundaryExpectation_eq_of_incident_subset G hloc S x _ (incidentSupport_mono G hloc hS)]
+  exact hcost S hx
+
+theorem pc_le_of_infinite_cluster_probability_pos (G : BondGraph V E)
+    (x : V) (p : unitInterval) (hp : 0 < (law p).real {ω | (cluster G ω x).Infinite}) :
+    pc G ≤ p := by
+  apply sInf_le
+  have hh : 0 < law p {ω | (cluster G ω x).Infinite} := (ENNReal.toReal_pos_iff.mp hp).1
+  apply hh.trans_le
+  apply measure_mono
+  intro ω hω
+  exact ⟨cluster G ω x,hω,x,rfl⟩
+
+ 
+theorem exists_subcritical_testSet [Countable V] (G : BondGraph V E)
+    (hloc : LocallyFinite G) (p : unitInterval) (hp : p < pc G) (x : V) :
+    ∃ S : Finset V, x ∈ S ∧ boundaryCost G hloc S x (p:ℝ) < 1 := by
+  by_cases hz : p = 0
+  · exact ⟨{x},by simp,by simp [hz,boundaryCost]⟩
+  · have hp0 : 0 < p := bot_lt_iff_ne_bot.mpr hz
+    by_contra! hn
+    obtain ⟨q,hpq,hqc⟩ := exists_between hp
+    have hq1 : q < 1 := hqc.trans_le (show pc G ≤ 1 from (pc G).2.2)
+    have hs := infinite_cluster_probability_pos_of_bad_cost G hloc x p q hp0 hpq hq1 hn
+    exact (not_le_of_gt hqc) (pc_le_of_infinite_cluster_probability_pos G x q hs)
+
+theorem internalEdges_aut_iff (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (h : IsAut G a b) (S : Set V) (e : E) :
+    b e ∈ internalEdges G (a '' S) ↔ e ∈ internalEdges G S := by
+  constructor
+  · intro he x hx
+    have hh := he (a x) (by rw [h e]; exact Sym2.mem_map.mpr ⟨x,hx,rfl⟩)
+    simpa using hh
+  · intro he y hy
+    rw [h e,Sym2.mem_map] at hy
+    obtain ⟨x,hx,rfl⟩ := hy
+    exact Set.mem_image_of_mem a (he x hx)
+
+theorem crossingReach_aut_iff (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (h : IsAut G a b) (S : Set V) (x : V) (ω : Set E) (e : E) :
+    crossingReach G (a '' S) (a x) (b '' ω) (b e) ↔ crossingReach G S x ω e := by
+  constructor
+  · rintro ⟨c,d,he,hc,hd,hconn⟩
+    have he' : G e = s(a.symm c,a.symm d) := by
+      have hh := congrArg (Sym2.map a.symm) he
+      rw [h e,Sym2.map_map,Equiv.symm_comp_self,Sym2.map_id,Sym2.map_mk] at hh
+      exact hh
+    refine ⟨a.symm c,a.symm d,he',?_,?_,?_⟩
+    · simpa using hc
+    · simpa using hd
+    · have hh := (conn_aut_iff G h (b '' ω) x (a.symm c)).mpr (by simpa using hconn)
+      simpa only [Set.preimage_image_eq _ b.injective] using hh
+  · rintro ⟨c,d,he,hc,hd,hconn⟩
+    refine ⟨a c,a d,?_,by simpa using hc,by simpa using hd,conn_aut G h hconn⟩
+    rw [h e,he,Sym2.map_mk]
+
+theorem internalSupport_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (h : IsAut G a b) (S : Set V) (F : Finset E) :
+    internalSupport G (a '' S) (F.map b.toEmbedding) =
+      (internalSupport G S F).map b.toEmbedding := by
+  ext e
+  obtain ⟨d,rfl⟩ := b.surjective e
+  simp [internalSupport,internalEdges_aut_iff G h]
+
+theorem crossingCount_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (h : IsAut G a b) (S : Set V) (x : V) (F : Finset E) (ω : Set E) :
+    crossingCount G (a '' S) (a x) (F.map b.toEmbedding) (b '' ω) =
+      crossingCount G S x F ω := by
+  simp only [crossingCount,Finset.sum_map,Equiv.toEmbedding_apply,crossingReach_aut_iff G h]
+
+theorem boundaryExpectation_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (h : IsAut G a b) (S : Set V) (x : V) (F : Finset E) (p : ℝ) :
+    boundaryExpectation G (a '' S) (a x) (F.map b.toEmbedding) p =
+      boundaryExpectation G S x F p := by
+  unfold boundaryExpectation
+  rw [internalSupport_aut G h,bernoulliPolynomial_map]
+  apply bernoulliPolynomial_congr
+  intro t _
+  simp only [Finset.coe_map,Equiv.coe_toEmbedding]
+  exact crossingCount_aut G h S x F t
+
+theorem incidentSupport_aut (G : BondGraph V E) (hloc : LocallyFinite G)
+    {a : V ≃ V} {b : E ≃ E} (h : IsAut G a b) (S : Finset V) :
+    incidentSupport G hloc (S.map a.toEmbedding) =
+      (incidentSupport G hloc S).map b.toEmbedding := by
+  ext e
+  obtain ⟨d,rfl⟩ := b.surjective e
+  simp only [mem_incidentSupport,Finset.mem_map,Equiv.toEmbedding_apply,
+    b.injective.eq_iff,exists_eq_right]
+  constructor
+  · rintro ⟨x,⟨y,hy,rfl⟩,hx⟩
+    rw [h d,Sym2.mem_map] at hx
+    obtain ⟨z,hz,hzy⟩ := hx
+    have hz' : z = y := a.injective hzy
+    exact ⟨y,hy,hz' ▸ hz⟩
+  · rintro ⟨x,hx,he⟩
+    exact ⟨a x,⟨x,hx,rfl⟩,by rw [h d]; exact Sym2.mem_map.mpr ⟨x,he,rfl⟩⟩
+
+theorem boundaryCost_aut (G : BondGraph V E) (hloc : LocallyFinite G)
+    {a : V ≃ V} {b : E ≃ E} (h : IsAut G a b) (S : Finset V) (x : V) (p : ℝ) :
+    boundaryCost G hloc (S.map a.toEmbedding) (a x) p = boundaryCost G hloc S x p := by
+  unfold boundaryCost
+  rw [incidentSupport_aut G hloc h,Finset.coe_map]
+  exact congrArg (p*·) (boundaryExpectation_aut G h S x (incidentSupport G hloc S) p)
+
+ 
+
+theorem exists_uniform_subcritical_testSets [Countable V] [Nonempty V] (G : BondGraph V E)
+    (hloc : LocallyFinite G) (hqt : QuasiTransitive G) (p : unitInterval) (hp : p < pc G) :
+    ∃ (L : ℕ) (a : ℝ), 0 ≤ a ∧ a < 1 ∧ ∀ x, ∃ S : Finset V,
+      x ∈ S ∧ S.card ≤ L ∧ boundaryCost G hloc S x (p:ℝ) ≤ a := by
+  obtain ⟨O,hO⟩ := hqt
+  choose S hmem hcost using exists_subcritical_testSet G hloc p hp
+  have hOn : O.Nonempty := by
+    obtain ⟨o,ho,_⟩ := hO (Classical.arbitrary V)
+    exact ⟨o,ho⟩
+  let L := O.sup (fun x => (S x).card)
+  let a := max 0 (O.sup' hOn (fun x => boundaryCost G hloc (S x) x (p:ℝ)))
+  refine ⟨L,a,le_max_left _ _,?_,?_⟩
+  · apply max_lt zero_lt_one
+    exact (Finset.sup'_lt_iff hOn).mpr (fun x _ => hcost x)
+  · intro x
+    obtain ⟨o,ho,α,β,hαβ,rfl⟩ := hO x
+    refine ⟨(S o).map α.toEmbedding,?_,?_,?_⟩
+    · simp only [Finset.mem_map,Equiv.toEmbedding_apply]
+      exact ⟨o,hmem o,rfl⟩
+    · simpa only [Finset.card_map] using Finset.le_sup (f := fun x => (S x).card) ho
+    · rw [boundaryCost_aut G hloc hαβ]
+      exact (Finset.le_sup' (f := fun x => boundaryCost G hloc (S x) x (p:ℝ)) ho).trans
+        (le_max_right _ _)
+
+end SharpnessBridge
+
+section SharpnessSusceptibility
+open Filter Topology
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+noncomputable def domainTwoPoint (G : BondGraph V E) (S : Set V) (p : unitInterval)
+    (x y : V) : ℝ := (law p).real {ω | Conn G (ω ∩ internalEdges G S) x y}
+
+theorem conn_internal_endpoint (G : BondGraph V E) (S : Set V) {ω : Set E} {x y : V}
+    (hx : x ∈ S) (h : Conn G (ω ∩ internalEdges G S) x y) : y ∈ S := by
+  induction h with
+  | refl => exact hx
+  | @tail a b hxa hab ih =>
+    obtain ⟨e,he,heq⟩ := hab
+    exact ((mem_internalEdges_of_ends G S e a b heq).mp he.2).2
+
+def CutExists (G : BondGraph V E) (S : Set V) (e : E) : Prop :=
+  ∃ a b, G e = s(a,b) ∧ a ∈ S ∧ b ∉ S
+
+noncomputable def cutInside [Nonempty V] (G : BondGraph V E) (S : Set V) (e : E) : V :=
+  if h : CutExists G S e then h.choose else Classical.arbitrary V
+
+noncomputable def cutOutside [Nonempty V] (G : BondGraph V E) (S : Set V) (e : E) : V :=
+  if h : CutExists G S e then h.choose_spec.choose else Classical.arbitrary V
+
+theorem cut_spec [Nonempty V] (G : BondGraph V E) (S : Set V) {e : E}
+    (he : CutExists G S e) :
+    G e = s(cutInside G S e,cutOutside G S e) ∧
+      cutInside G S e ∈ S ∧ cutOutside G S e ∉ S := by
+  simpa only [cutInside,cutOutside,dite_eq_left he] using he.choose_spec.choose_spec
+
+theorem cut_unique [Nonempty V] (G : BondGraph V E) (S : Set V) {e : E} {a b : V}
+    (he : G e = s(a,b)) (ha : a ∈ S) (hb : b ∉ S) :
+    a = cutInside G S e ∧ b = cutOutside G S e := by
+  have hc := cut_spec G S (e := e) ⟨a,b,he,ha,hb⟩
+  have hh := Sym2.eq_iff.mp (he.symm.trans hc.1)
+  rcases hh with ⟨h₁,h₂⟩ | ⟨h₁,h₂⟩
+  · exact ⟨h₁,h₂⟩
+  · exact (hc.2.2 (h₁ ▸ ha)).elim
+
+theorem crossingReach_cut [Nonempty V] (G : BondGraph V E) (S : Set V)
+    (x : V) (ω : Set E) (e : E) :
+    crossingReach G S x ω e ↔ CutExists G S e ∧ Conn G ω x (cutInside G S e) := by
+  constructor
+  · rintro ⟨a,b,he,ha,hb,h⟩
+    exact ⟨⟨a,b,he,ha,hb⟩,(cut_unique G S he ha hb).1 ▸ h⟩
+  · rintro ⟨he,h⟩
+    have hs := cut_spec G S he
+    exact ⟨_,_,hs.1,hs.2.1,hs.2.2,h⟩
+
+theorem crossingReach_upper (G : BondGraph V E) (S : Set V) (x : V) (e : E) :
+    IsUpperSet {ω | crossingReach G S x (ω ∩ internalEdges G S) e} := by
+  rintro ω η h ⟨a,b,he,ha,hb,hconn⟩
+  exact ⟨a,b,he,ha,hb,conn_mono G (Set.inter_subset_inter_left _ h) hconn⟩
+
+theorem finiteWitness_crossingReach (G : BondGraph V E) (S : Set V) (x : V) (e : E) :
+    FiniteWitness {ω | crossingReach G S x (ω ∩ internalEdges G S) e} := by
+  rintro ω ⟨a,b,he,ha,hb,hconn⟩
+  obtain ⟨t,ht,hc⟩ := finiteWitness_inter {η | Conn G η x a} (finiteWitness_conn G x a)
+    (internalEdges G S) ω hconn
+  exact ⟨t,ht,a,b,he,ha,hb,hc⟩
+
+ 
+theorem conn_first_exit [Nonempty V] (G : BondGraph V E) (hloc : LocallyFinite G)
+    (S : Finset V) (U : Set E) {ω : Set E} {x y : V} (hx : x ∈ S)
+    (hxy : Conn G (ω ∩ U) x y) :
+    y ∈ S ∨ ∃ e ∈ incidentSupport G hloc S,
+      ω ∈ DisjointOccurrence {η | crossingReach G S x (η ∩ internalEdges G S) e}
+        (DisjointOccurrence {η | e ∈ η} {η | Conn G (η ∩ U) (cutOutside G S e) y}) := by
+  obtain ⟨l,hl,hn,ho⟩ := conn_exists_bond_trail G hxy
+  rcases hl.first_outside (internalEdges G S) with hall |
+      ⟨a,b,e,l₁,l₂,heq,h₁,he,h₂,hfirst,hnot⟩
+  · exact Or.inl (conn_internal_endpoint G S hx (hl.conn fun e he => ⟨(ho e he).1,hall e he⟩))
+  · subst l
+    have hnod := List.nodup_append.mp hn
+    have hen : e ∉ l₂ := (List.nodup_cons.mp hnod.2.1).1
+    have ha : a ∈ S := conn_internal_endpoint G S hx
+      (h₁.conn (ω := ω ∩ internalEdges G S) fun d hd =>
+        ⟨(ho d (List.mem_append_left _ hd)).1,hfirst d hd⟩)
+    have hb : b ∉ S := fun hb => hnot ((mem_internalEdges_of_ends G S e a b he).mpr ⟨ha,hb⟩)
+    have hbe := (cut_unique G S he ha hb).2
+    refine Or.inr ⟨e,(mem_incidentSupport G hloc S e).mpr ⟨a,ha,by simp [he]⟩,
+      l₁.toFinset,(e::l₂).toFinset,?_,?_,?_,?_,?_⟩
+    · intro d hd
+      exact (ho d (List.mem_append_left _ (List.mem_toFinset.mp hd))).1
+    · intro d hd
+      exact (ho d (List.mem_append_right _ (List.mem_toFinset.mp hd))).1
+    · apply Finset.disjoint_left.mpr
+      intro d hd hd'
+      exact hnod.2.2 d (List.mem_toFinset.mp hd) d (List.mem_toFinset.mp hd') rfl
+    · exact ⟨a,b,he,ha,hb,h₁.conn (fun d hd => ⟨by simpa using hd,hfirst d hd⟩)⟩
+    · refine ⟨{e},l₂.toFinset,by simp,by simp,by simpa using hen,by simp,?_⟩
+      rw [← hbe]
+      exact h₂.conn fun d hd => ⟨by simpa using hd,
+        (ho d (List.mem_append_right _ (List.mem_cons_of_mem e hd))).2⟩
+
+ 
+theorem boundaryExpectation_eq_sum (G : BondGraph V E) (hloc : LocallyFinite G)
+    (S : Finset V) (x : V) (p : unitInterval) :
+    boundaryExpectation G S x (incidentSupport G hloc S) (p:ℝ) =
+      ∑ e ∈ incidentSupport G hloc S,
+        (law p).real {ω | crossingReach G S x (ω ∩ internalEdges G S) e} := by
+  have hI : (internalSupport G S (incidentSupport G hloc S) : Set E) = internalEdges G S := by
+    ext e
+    simp only [Finset.mem_coe,internalSupport,Finset.mem_filter]
+    exact ⟨And.right,fun he => ⟨internal_edge_mem_incident G hloc S he,he⟩⟩
+  unfold boundaryExpectation crossingCount bernoulliPolynomial
+  simp only [Finset.sum_mul]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro e _
+  change finiteEventPolynomial (internalSupport G S (incidentSupport G hloc S))
+    {ω | crossingReach G S x ω e} (p:ℝ) = _
+  rw [finiteEventPolynomial_restrict,hI]
+  rfl
+
+ 
+theorem domainTwoPoint_first_exit [Countable E] [Nonempty V] (G : BondGraph V E)
+    (hloc : LocallyFinite G) (S : Finset V) (Λ : Set V) (p : unitInterval)
+    (x y : V) (hx : x ∈ S) :
+    domainTwoPoint G Λ p x y ≤ (if y ∈ S then (1:ℝ) else 0) +
+      ∑ e ∈ incidentSupport G hloc S,
+        (law p).real {ω | crossingReach G S x (ω ∩ internalEdges G S) e} *
+          (p:ℝ) * domainTwoPoint G Λ p (cutOutside G S e) y := by
+  by_cases hy : y ∈ S
+  · have hnon : 0 ≤ ∑ e ∈ incidentSupport G hloc S,
+        (law p).real {ω | crossingReach G S x (ω ∩ internalEdges G S) e} *
+          (p:ℝ) * domainTwoPoint G Λ p (cutOutside G S e) y := by
+      apply Finset.sum_nonneg
+      intro e _
+      exact mul_nonneg (mul_nonneg (measureReal_nonneg) p.2.1) measureReal_nonneg
+    simpa only [ite_eq_left hy,domainTwoPoint] using (measureReal_le_one (μ := law p)
+      (s := {ω | Conn G (ω ∩ internalEdges G Λ) x y})).trans (le_add_of_nonneg_right hnon)
+  · simp only [ite_eq_right hy,zero_add]
+    let A := fun e => {ω | crossingReach G S x (ω ∩ internalEdges G S) e}
+    let B := fun e => {ω | Conn G (ω ∩ internalEdges G Λ) (cutOutside G S e) y}
+    let F := incidentSupport G hloc S
+    let C := fun e => DisjointOccurrence (A e) (DisjointOccurrence {ω | e ∈ ω} (B e))
+    have hsub : {ω | Conn G (ω ∩ internalEdges G Λ) x y} ⊆ ⋃ e ∈ F, C e := by
+      intro ω hω
+      rcases conn_first_exit G hloc S (internalEdges G Λ) hx hω with h | ⟨e,he,h⟩
+      · exact (hy h).elim
+      · exact Set.mem_iUnion.mpr ⟨e,Set.mem_iUnion.mpr ⟨he,h⟩⟩
+    calc
+      domainTwoPoint G Λ p x y ≤ (law p).real (⋃ e ∈ F, C e) := measureReal_mono (μ := law p) hsub (measure_ne_top _ _)
+      _ ≤ ∑ e ∈ F, (law p).real (C e) := measureReal_biUnion_finset_le _ _
+      _ ≤ _ := by
+        apply Finset.sum_le_sum
+        intro e _
+        have hB : IsUpperSet (B e) := fun _ _ h hc =>
+          conn_mono G (Set.inter_subset_inter_left _ h) hc
+        have hfB : FiniteWitness (B e) := finiteWitness_inter _
+          (finiteWitness_conn G _ _) (internalEdges G Λ)
+        have hb := bk p (A e) (DisjointOccurrence {ω | e ∈ ω} (B e))
+          (crossingReach_upper G S x e) (disjointOccurrence_upper _ _)
+          (finiteWitness_crossingReach G S x e) (finiteWitness_disjointOccurrence _ _)
+        have hb' := bk p {ω | e ∈ ω} (B e) (fun _ _ h he => h he) hB (finiteWitness_edge e) hfB
+        have heq : (law p).real {ω : Set E | e ∈ ω} = (p:ℝ) := by
+          simpa using finite_open_probability p {e}
+        rw [heq] at hb'
+        calc
+          (law p).real (C e) ≤ (law p).real (A e) * (law p).real
+              (DisjointOccurrence {ω | e ∈ ω} (B e)) := hb
+          _ ≤ (law p).real (A e) * ((p:ℝ) * (law p).real (B e)) :=
+            mul_le_mul_of_nonneg_left hb' measureReal_nonneg
+          _ = _ := (mul_assoc _ _ _).symm
+
+theorem conn_internal_eq_or_mem (G : BondGraph V E) (S : Set V) {ω : Set E} {x y : V}
+    (h : Conn G (ω ∩ internalEdges G S) x y) : x = y ∨ x ∈ S ∧ y ∈ S := by
+  induction h with
+  | refl => exact Or.inl rfl
+  | @tail a b hxa hab ih =>
+    obtain ⟨e,he,heq⟩ := hab
+    have he' := (mem_internalEdges_of_ends G S e a b heq).mp he.2
+    right
+    exact ⟨ih.elim (fun h => h.symm ▸ he'.1) And.left,he'.2⟩
+
+theorem domainTwoPoint_zero_of_outside (G : BondGraph V E) (S : Set V) (p : unitInterval)
+    {x y : V} (hx : x ∉ S) (hy : y ∈ S) : domainTwoPoint G S p x y = 0 := by
+  have he : {ω | Conn G (ω ∩ internalEdges G S) x y} = ∅ := by
+    apply Set.eq_empty_iff_forall_notMem.mpr
+    intro ω h
+    rcases conn_internal_eq_or_mem G S h with rfl | h
+    · exact hx hy
+    · exact hx h.1
+  simp only [domainTwoPoint,he,measureReal_empty]
+
+theorem domainRow_le_boundaryCost [Countable E] [Nonempty V] (G : BondGraph V E)
+    (hloc : LocallyFinite G) (S Λ : Finset V) (p : unitInterval) (x : V) (hx : x ∈ S)
+    (H : ℝ) (hH : ∀ z, ∑ y ∈ Λ, domainTwoPoint G Λ p z y ≤ H) :
+    (∑ y ∈ Λ, domainTwoPoint G Λ p x y) ≤ S.card + boundaryCost G hloc S x (p:ℝ) * H := by
+  have hind : (∑ y ∈ Λ, if y ∈ S then (1:ℝ) else 0) ≤ (S.card:ℝ) := by
+    simp only [← Finset.sum_filter]
+    simpa using (Nat.cast_le (α := ℝ)).mpr (Finset.card_le_card
+      (show Λ.filter (fun y => y ∈ S) ⊆ S from fun y hy => (Finset.mem_filter.mp hy).2))
+  calc
+    (∑ y ∈ Λ, domainTwoPoint G Λ p x y) ≤
+        ∑ y ∈ Λ, ((if y ∈ S then (1:ℝ) else 0) +
+          ∑ e ∈ incidentSupport G hloc S,
+            (law p).real {ω | crossingReach G S x (ω ∩ internalEdges G S) e} *
+              (p:ℝ) * domainTwoPoint G Λ p (cutOutside G S e) y) :=
+      Finset.sum_le_sum fun y _ => domainTwoPoint_first_exit G hloc S Λ p x y hx
+    _ = (∑ y ∈ Λ, if y ∈ S then (1:ℝ) else 0) +
+        ∑ e ∈ incidentSupport G hloc S,
+          (law p).real {ω | crossingReach G S x (ω ∩ internalEdges G S) e} *
+            (p:ℝ) * ∑ y ∈ Λ, domainTwoPoint G Λ p (cutOutside G S e) y := by
+      rw [Finset.sum_add_distrib,Finset.sum_comm]
+      simp only [← Finset.mul_sum]
+    _ ≤ S.card + ∑ e ∈ incidentSupport G hloc S,
+          (law p).real {ω | crossingReach G S x (ω ∩ internalEdges G S) e} * (p:ℝ) * H := by
+      apply add_le_add hind
+      apply Finset.sum_le_sum
+      intro e _
+      exact mul_le_mul_of_nonneg_left (hH _) (mul_nonneg measureReal_nonneg p.2.1)
+    _ = S.card + boundaryCost G hloc S x (p:ℝ) * H := by
+      unfold boundaryCost
+      rw [boundaryExpectation_eq_sum]
+      simp only [← Finset.sum_mul]
+      ring
+
+theorem domainRow_uniform_bound [Countable E] [Nonempty V] (G : BondGraph V E)
+    (hloc : LocallyFinite G) (p : unitInterval) (L : ℕ) (a : ℝ) (ha : a < 1)
+    (htest : ∀ x, ∃ S : Finset V, x ∈ S ∧ S.card ≤ L ∧ boundaryCost G hloc S x (p:ℝ) ≤ a)
+    (Λ : Finset V) (x : V) :
+    (∑ y ∈ Λ, domainTwoPoint G Λ p x y) ≤ (L:ℝ)/(1-a) := by
+  by_cases hΛ : Λ.Nonempty
+  · let row := fun z => ∑ y ∈ Λ, domainTwoPoint G Λ p z y
+    obtain ⟨z,hz,hmax⟩ := Λ.exists_max_image row hΛ
+    have hnon : 0 ≤ row z := Finset.sum_nonneg (fun _ _ => measureReal_nonneg)
+    have hH : ∀ w, row w ≤ row z := by
+      intro w
+      by_cases hw : w ∈ Λ
+      · exact hmax w hw
+      · have he : row w = 0 := by
+          apply Finset.sum_eq_zero
+          intro y hy
+          exact domainTwoPoint_zero_of_outside G Λ p hw hy
+        rwa [he]
+    obtain ⟨S,hS,hcard,hcost⟩ := htest z
+    have h := domainRow_le_boundaryCost G hloc S Λ p z hS (row z) hH
+    have hL : (S.card:ℝ) ≤ (L:ℝ) := Nat.cast_le.mpr hcard
+    have hc := mul_le_mul_of_nonneg_right hcost hnon
+    have hrow : row z ≤ (L:ℝ)/(1-a) := (le_div_iff₀ (sub_pos.mpr ha)).mpr (by nlinarith)
+    exact (hH x).trans hrow
+  · simp only [Finset.not_nonempty_iff_eq_empty.mp hΛ,Finset.sum_empty]
+    exact div_nonneg (Nat.cast_nonneg _) (sub_pos.mpr ha).le
+
+theorem internalEdges_mono (G : BondGraph V E) {S T : Set V} (h : S ⊆ T) :
+    internalEdges G S ⊆ internalEdges G T := fun _ he x hx => h (he x hx)
+
+theorem exists_internal_vertexExhaustion [Encodable V] (G : BondGraph V E) (e : E) :
+    ∃ n, e ∈ internalEdges G (vertexExhaustion (V := V) n : Set V) := by
+  obtain ⟨⟨a,b⟩,hab⟩ := Sym2.mk_surjective (G e)
+  obtain ⟨n,hn⟩ := subset_vertexExhaustion ({a,b} : Finset V)
+  refine ⟨n,?_⟩
+  rw [mem_internalEdges_of_ends G _ e a b hab.symm]
+  exact ⟨hn (by simp),hn (by simp)⟩
+
+theorem conn_domain_exhaustion [Encodable V] (G : BondGraph V E) {ω : Set E} {x y : V}
+    (h : Conn G ω x y) :
+    ∃ n, Conn G (ω ∩ internalEdges G (vertexExhaustion (V := V) n : Set V)) x y := by
+  induction h with
+  | refl => exact ⟨0,conn_refl _ _ _⟩
+  | @tail a b hxa hab ih =>
+    obtain ⟨n,hn⟩ := ih
+    obtain ⟨e,he,heq⟩ := hab
+    obtain ⟨m,hm⟩ := exists_internal_vertexExhaustion G e
+    refine ⟨max n m,?_⟩
+    have hnm := internalEdges_mono G (show (vertexExhaustion (V := V) n : Set V) ⊆
+        (vertexExhaustion (V := V) (max n m) : Set V) from vertexExhaustion_mono (le_max_left n m))
+    have hmm := internalEdges_mono G (show (vertexExhaustion (V := V) m : Set V) ⊆
+        (vertexExhaustion (V := V) (max n m) : Set V) from vertexExhaustion_mono (le_max_right n m))
+    exact (conn_mono G (Set.inter_subset_inter_right _ hnm) hn).tail ⟨e,⟨he,hmm hm⟩,heq⟩
+
+theorem tendsto_domainTwoPoint [Encodable V] (G : BondGraph V E) (p : unitInterval) (x y : V) :
+    Tendsto (fun n => domainTwoPoint G (vertexExhaustion (V := V) n : Set V) p x y)
+      atTop (𝓝 (twoPoint G p x y)) := by
+  have hm : Monotone (fun n => {ω | Conn G
+      (ω ∩ internalEdges G (vertexExhaustion (V := V) n : Set V)) x y}) := by
+    intro m n h ω hc
+    exact conn_mono G (Set.inter_subset_inter_right _ (internalEdges_mono G
+      (show (vertexExhaustion (V := V) m : Set V) ⊆ (vertexExhaustion (V := V) n : Set V) from
+        vertexExhaustion_mono h))) hc
+  have he : (⋃ n, {ω | Conn G (ω ∩ internalEdges G (vertexExhaustion (V := V) n : Set V)) x y}) =
+      {ω | Conn G ω x y} := by
+    ext ω
+    simp only [Set.mem_iUnion,Set.mem_ofPred_eq]
+    exact ⟨fun ⟨n,h⟩ => conn_mono G Set.inter_subset_left h,conn_domain_exhaustion G⟩
+  have h := tendsto_measure_iUnion_atTop (μ := law p) hm
+  rw [he] at h
+  exact (ENNReal.tendsto_toReal (measure_ne_top _ _)).comp h
+
+ 
+theorem subcritical_row_bound [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ x (F : Finset V), ∑ y ∈ F, twoPoint G p x y ≤ C := by
+  let : Encodable V := Encodable.ofCountable V
+  obtain ⟨L,a,_,ha,htest⟩ := exists_uniform_subcritical_testSets G hloc hqt p hp
+  refine ⟨(L:ℝ)/(1-a),div_nonneg (Nat.cast_nonneg _) (sub_pos.mpr ha).le,?_⟩
+  intro x F
+  apply le_of_tendsto (tendsto_finsetSum F (fun y _ => tendsto_domainTwoPoint G p x y))
+  obtain ⟨N,hN⟩ := subset_vertexExhaustion F
+  filter_upwards [eventually_ge_atTop N] with n hn
+  have hFn : F ⊆ vertexExhaustion n := hN.trans (vertexExhaustion_mono hn)
+  exact (Finset.sum_le_sum_of_subset_of_nonneg hFn (fun _ _ _ => measureReal_nonneg)).trans
+    (domainRow_uniform_bound G hloc p L a ha htest (vertexExhaustion n) x)
+
+ 
+theorem operatorBounded_of_lt_pc [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) : OperatorBounded G p := by
+  obtain ⟨C,hC,hrow⟩ := subcritical_row_bound G hloc hqt p hp
+  obtain ⟨A,_,hA⟩ := exists_operator_of_row_column_bound (twoPoint G p) C hC
+    (twoPoint_nonneg G p) hrow (fun y F => by
+      simpa only [twoPoint_symm G p] using hrow y F)
+  exact ⟨A,hA⟩
+
+end SharpnessSusceptibility
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Criticality/Susceptibility.lean b/lean/OAI/Probability/UnimodularPercolation/Criticality/Susceptibility.lean
index 35ae2cb0f581ceaecaca768a231dfea655cd9601..969977d20946e81f1160c6536b6aa29b07ebd2f6 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Criticality/Susceptibility.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Criticality/Susceptibility.lean
@@ -1,1196 +1,1196 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Kernel.DistinctMoments
-
-namespace OAI
-
-/-! Susceptibility for bond percolation. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section SusceptibilityPivotalGain
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem sprinkleParameter_ge_left (p a : unitInterval) : p ≤ sprinkleParameter p a := by
-  change (p:ℝ) ≤ (p:ℝ)+(1-(p:ℝ))*(a:ℝ)
-  exact le_add_of_nonneg_right (mul_nonneg (sub_nonneg.mpr p.2.2) a.2.1)
-
- 
-
-theorem susceptibility_sprinkling_gain [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p a : unitInterval) (hq : sprinkleParameter p a < pc G) (x : V)
-    (Q : Finset E) (F : Finset V) :
-    (a:ℝ)*(1-(a:ℝ))^(Q.card-1) *
-      (∑ y ∈ F, ∑ e ∈ Q, (law p).real (closedPivotal {ω | Conn G ω x y} e)) ≤
-      susceptibility G (sprinkleParameter p a) x - susceptibility G p x := by
-  have hpa := sprinkleParameter_ge_left p a
-  have hp : p < pc G := lt_of_le_of_lt hpa hq
-  have hsum := (summable_twoPoint_subcritical G hloc hqt (sprinkleParameter p a) hq x).sub
-    (summable_twoPoint_subcritical G hloc hqt p hp x)
-  calc
-    _ = ∑ y ∈ F, (a:ℝ)*(1-(a:ℝ))^(Q.card-1)*
-      (∑ e ∈ Q, (law p).real (closedPivotal {ω | Conn G ω x y} e)) := Finset.mul_sum _ _ _
-    _ ≤ ∑ y ∈ F, (twoPoint G (sprinkleParameter p a) x y - twoPoint G p x y) := by
-      apply Finset.sum_le_sum
-      intro y _
-      exact event_sprinkling_gain p a Q {ω | Conn G ω x y} (measurableSet_conn G x y)
-        (fun _ _ h hc => conn_mono G h hc)
-    _ ≤ ∑' y, (twoPoint G (sprinkleParameter p a) x y - twoPoint G p x y) :=
-      hsum.sum_le_tsum F (fun y _ => sub_nonneg.mpr (twoPoint_mono G x y hpa))
-    _ = _ := (summable_twoPoint_subcritical G hloc hqt (sprinkleParameter p a) hq x).tsum_sub
-      (summable_twoPoint_subcritical G hloc hqt p hp x)
-
- 
-theorem rootSusceptibility_sprinkling_gain [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 ≤ weight i)
-    (p a : unitInterval) (hq : sprinkleParameter p a < pc G)
-    (Q : Finset E) (F : Finset V) :
-    (a:ℝ)*(1-(a:ℝ))^(Q.card-1) *
-      (∑ i, weight i*(∑ y ∈ F, ∑ e ∈ Q, (law p).real (closedPivotal {ω | Conn G ω (root i) y} e))) ≤
-      rootSusceptibility G root weight (sprinkleParameter p a) - rootSusceptibility G root weight p := by
-  rw [rootSusceptibility,rootSusceptibility,←Finset.sum_sub_distrib,Finset.mul_sum]
-  apply Finset.sum_le_sum
-  intro i _
-  have h := mul_le_mul_of_nonneg_left
-    (susceptibility_sprinkling_gain G hloc hqt p a hq (root i) Q F) (hw i)
-  convert h using 1 <;> ring
-
-end SusceptibilityPivotalGain
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section OrientedPivotalConnections
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-def orientedPivotalEvent (G : BondGraph V E) (x y a b : V) : Set (Set E) :=
-  {ω | Conn G ω x a ∧ Conn G ω b y ∧ ¬ Conn G ω a b}
-
-theorem measurableSet_orientedPivotalEvent [Countable V] [Countable E]
-    (G : BondGraph V E) (x y a b : V) : MeasurableSet (orientedPivotalEvent G x y a b) :=
-  (measurableSet_conn G x a).inter ((measurableSet_conn G b y).inter (measurableSet_conn G a b).compl)
-
-theorem orientedPivotalEvent_disjoint (G : BondGraph V E) (x y a b : V) :
-    Disjoint (orientedPivotalEvent G x y a b) (orientedPivotalEvent G x y b a) := by
-  apply Set.disjoint_left.mpr
-  intro ω h h'
-  exact h.2.2 ((conn_symm G ω h.1).trans h'.1)
-
-theorem closedPivotal_conn (G : BondGraph V E) (e : E) (a b x y : V)
-    (he : G e = s(a,b)) :
-    closedPivotal {ω | Conn G ω x y} e =
-      orientedPivotalEvent G x y a b ∪ orientedPivotalEvent G x y b a := by
-  ext ω
-  constructor
-  · rintro ⟨_,hn,hc⟩
-    rcases (conn_insert_iff G ω e a b x y he).mp hc with h | h | h
-    · exact (hn h).elim
-    · exact Or.inl ⟨h.1,h.2,fun hab => hn (h.1.trans (hab.trans h.2))⟩
-    · exact Or.inr ⟨h.1,h.2,fun hba => hn (h.1.trans (hba.trans h.2))⟩
-  · intro h
-    have hi : ∀ a b : V, G e = s(a,b) → ω ∈ orientedPivotalEvent G x y a b →
-        ω ∈ closedPivotal {ω | Conn G ω x y} e := by
-      intro a b he h
-      refine ⟨?_,?_,?_⟩
-      · intro heω
-        exact h.2.2 (.single ⟨e,heω,he⟩)
-      · intro hxy
-        exact h.2.2 ((conn_symm G ω h.1).trans (hxy.trans (conn_symm G ω h.2.1)))
-      · exact (conn_insert_iff G ω e a b x y he).mpr (Or.inr (Or.inl ⟨h.1,h.2.1⟩))
-    exact h.elim (hi a b he) (hi b a (he.trans (Sym2.eq_swap ..)))
-
-theorem closedPivotal_conn_probability [Countable V] [Countable E]
-    (G : BondGraph V E) (p : unitInterval) (e : E) (a b x y : V) (he : G e = s(a,b)) :
-    (law p).real (closedPivotal {ω | Conn G ω x y} e) =
-      (law p).real (orientedPivotalEvent G x y a b) +
-      (law p).real (orientedPivotalEvent G x y b a) := by
-  rw [closedPivotal_conn G e a b x y he]
-  exact measureReal_union (orientedPivotalEvent_disjoint G x y a b)
-    (measurableSet_orientedPivotalEvent G x y b a)
-
- 
-theorem orientedPivotalEvent_subset_disjoint (G : BondGraph V E) (x y a b : V) :
-    orientedPivotalEvent G x y a b ⊆
-      DisjointOccurrence {ω | Conn G ω x a} {ω | Conn G ω b y} := by
-  intro ω hω
-  have hx : x ∈ cluster G ω a := conn_symm G ω hω.1
-  have hb : b ∉ cluster G ω a := hω.2.2
-  obtain ⟨Q,hQ,hQa⟩ := finiteWitness_conn G x a _ (conn_restrict_cluster G ω a hx hω.1)
-  obtain ⟨R,hR,hRb⟩ := finiteWitness_conn G b y _ (conn_restrict_cluster_compl G ω a hb hω.2.1)
-  refine ⟨Q,R,fun e he => (hQ he).1,fun e he => (hR he).1,?_,hQa,hRb⟩
-  apply Finset.disjoint_left.mpr
-  intro e heQ heR
-  obtain ⟨⟨z,w⟩,he⟩ := Quot.exists_rep (G e)
-  have hz : z ∈ G e := by rw [←he]; exact Sym2.mem_mk_left z w
-  exact (hR heR).2 z hz ((hQ heQ).2 z hz)
-
-end OrientedPivotalConnections
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-open scoped ENNReal
-section PivotalMomentIdentity
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem distinctClusterMass_eq_pivotal (G : BondGraph V E) (ω : Set E) (a b : V) :
-    (if ¬Conn G ω a b then clusterMass G ω a * clusterMass G ω b else 0) =
-      ∑' x, ∑' y, if ω ∈ orientedPivotalEvent G x y a b then (1:ℝ≥0∞) else 0 := by
-  by_cases hab : Conn G ω a b
-  · simp only [hab,not_true_eq_false,ite_false,orientedPivotalEvent,Set.mem_ofPred_eq,
-       and_false,ite_false,tsum_zero]
-  · rw [ite_eq_left hab,clusterMass_mul]
-    apply tsum_congr
-    intro x
-    apply tsum_congr
-    intro y
-    apply if_congr _ rfl rfl
-    simp only [orientedPivotalEvent,Set.mem_ofPred_eq,hab,not_false_eq_true,and_true]
-    exact and_congr (show Conn G ω a x ↔ Conn G ω x a from ⟨conn_symm G ω,conn_symm G ω⟩) Iff.rfl
-
-theorem lintegral_distinctClusterMass [Countable V] [Countable E]
-    (G : BondGraph V E) (p : unitInterval) (a b : V) :
-    (∫⁻ ω, (if ¬Conn G ω a b then clusterMass G ω a * clusterMass G ω b else 0) ∂law p) =
-      ∑' x, ∑' y, law p (orientedPivotalEvent G x y a b) := by
-  have hm (x y : V) : Measurable (fun ω : Set E =>
-      if ω ∈ orientedPivotalEvent G x y a b then (1:ℝ≥0∞) else 0) :=
-    Measurable.ite (measurableSet_orientedPivotalEvent G x y a b) measurable_const measurable_const
-  simp_rw [distinctClusterMass_eq_pivotal]
-  rw [lintegral_tsum (fun x => (Measurable.tsum (fun y => hm x y)).aemeasurable)]
-  apply tsum_congr
-  intro x
-  rw [lintegral_tsum (fun y => (hm x y).aemeasurable)]
-  apply tsum_congr
-  intro y
-  simpa only [Set.indicator,Pi.one_apply] using
-    (lintegral_indicator_one (μ := law p) (measurableSet_orientedPivotalEvent G x y a b))
-
-theorem ennreal_distinctClusterMoment [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (a b : V) :
-    ENNReal.ofReal (distinctClusterMoment G p a b) =
-      ∑' x, ∑' y, law p (orientedPivotalEvent G x y a b) := by
-  have hi := integrable_distinct_cluster_product G hloc hqt p hp a b
-  have hn : 0 ≤ᵐ[law p] (fun ω => if ¬Conn G ω a b
-      then ((cluster G ω a).ncard:ℝ)*((cluster G ω b).ncard:ℝ) else 0) := by
-    exact Filter.Eventually.of_forall (fun ω => by dsimp only; split_ifs <;> positivity)
-  have he : (fun ω => ENNReal.ofReal (if ¬Conn G ω a b
-      then ((cluster G ω a).ncard:ℝ)*((cluster G ω b).ncard:ℝ) else 0)) =ᵐ[law p]
-      (fun ω => if ¬Conn G ω a b then clusterMass G ω a * clusterMass G ω b else 0) := by
-    filter_upwards [subcritical_cluster_finite_ae G p hp a,
-      subcritical_cluster_finite_ae G p hp b] with ω ha hb
-    by_cases hab : Conn G ω a b
-    · simp only [hab,not_true_eq_false,ite_false,ENNReal.ofReal_zero]
-    · simp only [hab,not_false_eq_true,ite_true]
-      rw [clusterMass_eq_ncard G ω a ha,clusterMass_eq_ncard G ω b hb,
-        ENNReal.ofReal_mul (Nat.cast_nonneg _)]
-      simp only [ENNReal.ofReal_natCast]
-  rw [←lintegral_distinctClusterMass,←lintegral_congr_ae he]
-  exact ofReal_integral_eq_lintegral_ofReal hi hn
-
-theorem summable_orientedPivotal_pairs [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (a b : V) :
-    Summable (fun xy : V × V => (law p).real (orientedPivotalEvent G xy.1 xy.2 a b)) := by
-  apply ENNReal.summable_toReal
-  rw [ENNReal.tsum_prod',←ennreal_distinctClusterMoment G hloc hqt p hp a b]
-  exact ENNReal.ofReal_ne_top
-
-theorem distinctClusterMoment_eq_pivotal [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (a b : V) :
-    distinctClusterMoment G p a b =
-      ∑' x, ∑' y, (law p).real (orientedPivotalEvent G x y a b) := by
-  rw [←(summable_orientedPivotal_pairs G hloc hqt p hp a b).tsum_prod]
-  change distinctClusterMoment G p a b =
-    ∑' xy : V × V, (law p (orientedPivotalEvent G xy.1 xy.2 a b)).toReal
-  rw [←ENNReal.tsum_toReal_eq (f := fun xy : V × V => law p (orientedPivotalEvent G xy.1 xy.2 a b))
-    (fun _ => measure_ne_top _ _), ENNReal.tsum_prod',
-    ←ennreal_distinctClusterMoment G hloc hqt p hp a b,
-    ENNReal.toReal_ofReal (distinctClusterMoment_nonneg G p a b)]
-
-end PivotalMomentIdentity
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section PivotalTransport
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem orientedPivotal_probability_covariant [Countable V] [Countable E]
-    (G : BondGraph V E) (p : unitInterval) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (x y z w : V) :
-    (law p).real (orientedPivotalEvent G (a x) (a y) (a z) (a w)) =
-      (law p).real (orientedPivotalEvent G x y z w) := by
-  have he : orientedPivotalEvent G (a x) (a y) (a z) (a w) =
-      (fun ω => b ⁻¹' ω) ⁻¹' orientedPivotalEvent G x y z w := by
-    ext ω
-    simp only [orientedPivotalEvent,Set.mem_ofPred_eq,Set.mem_preimage,conn_aut_iff G hab]
-  unfold Measure.real
-  rw [he,←Measure.map_apply (measurable_set_preimage b)
-    (measurableSet_orientedPivotalEvent G x y z w),law_map_preimage_equiv]
-
-noncomputable def pivotalTransportMass (G : BondGraph V E) (hloc : LocallyFinite G)
-    (p : unitInterval) (x a : V) : ℝ :=
-  ∑ b ∈ neighbors G hloc a, ∑' y, (law p).real (orientedPivotalEvent G x y a b)
-
-theorem pivotalTransportMass_nonneg (G : BondGraph V E) (hloc : LocallyFinite G)
-    (p : unitInterval) (x a : V) : 0 ≤ pivotalTransportMass G hloc p x a :=
-  Finset.sum_nonneg (fun _ _ => tsum_nonneg (fun _ => measureReal_nonneg))
-
-theorem pivotalTransportMass_covariant [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (p : unitInterval)
-    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (x z : V) :
-    pivotalTransportMass G hloc p (a x) (a z) = pivotalTransportMass G hloc p x z := by
-  simp only [pivotalTransportMass,neighbors_covariant G hloc hab,Finset.sum_map,Equiv.coe_toEmbedding]
-  apply Finset.sum_congr rfl
-  intro w _
-  rw [←a.tsum_eq (fun y => (law p).real (orientedPivotalEvent G (a x) y (a z) (a w)))]
-  exact tsum_congr (fun y => orientedPivotal_probability_covariant G p hab x y z w)
-
-theorem pivotalTransportMass_incoming_summable [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (a : V) :
-    Summable (fun x => pivotalTransportMass G hloc p x a) := by
-  apply summable_sum
-  intro b _
-  exact (summable_orientedPivotal_pairs G hloc hqt p hp a b).prod
-
-theorem pivotalTransportMass_incoming [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (a : V) :
-    (∑' x, pivotalTransportMass G hloc p x a) = neighborClusterMoment G hloc p a := by
-  unfold pivotalTransportMass neighborClusterMoment
-  rw [Summable.tsum_finsetSum (fun b _ =>
-    (summable_orientedPivotal_pairs G hloc hqt p hp a b).prod)]
-  exact Finset.sum_congr rfl (fun b _ => (distinctClusterMoment_eq_pivotal G hloc hqt p hp a b).symm)
-
-end PivotalTransport
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section PivotalTransportTotal
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem pivotalTransportMass_outgoing_summable
-    [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
-    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (p : unitInterval) (hp : p < pc G) (x : V) :
-    Summable (pivotalTransportMass G hloc p x) := by
-  have hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i) := by
-    intro x
-    obtain ⟨i,a,ha,hax⟩ := hcov x
-    obtain ⟨b,hab⟩ := hΓ ha
-    exact ⟨i,a,b,hab,hax⟩
-  exact (transported_column_of_row_bound G hconn hloc root weight Γ hΓ htr hcov hw hw1 c hc hmin
-    (fun a x => pivotalTransportMass G hloc p x a)
-    (fun a x => pivotalTransportMass_nonneg G hloc p x a)
-    (fun a ha x y => by obtain ⟨b,hab⟩ := hΓ ha; exact pivotalTransportMass_covariant G hloc p hab y x)
-    (pivotalTransportMass_incoming_summable G hloc hqt p hp)
-    (c*rootNeighborMoment G hloc root weight p)
-    (fun a => by
-      rw [pivotalTransportMass_incoming G hloc hqt p hp]
-      exact neighborMoment_le_rootMoment G hloc root weight hw hcover c hc hmin p a) x).1
-
- 
-
-theorem rootPivotalMass_eq_neighborMoment
-    [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
-    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (p : unitInterval) (hp : p < pc G) :
-    (∑ i, weight i * ∑' a, pivotalTransportMass G hloc p (root i) a) =
-      rootNeighborMoment G hloc root weight p := by
-  rw [htr.real root weight Γ (pivotalTransportMass G hloc p)
-    (fun a ha x y => by obtain ⟨b,hab⟩ := hΓ ha; exact pivotalTransportMass_covariant G hloc p hab x y)
-    (pivotalTransportMass_outgoing_summable G hconn hloc hqt root weight Γ hΓ htr hcov hw hw1 c hc hmin p hp)
-    (pivotalTransportMass_incoming_summable G hloc hqt p hp)]
-  exact Finset.sum_congr rfl (fun i _ => congrArg (fun t => weight i*t)
-    (pivotalTransportMass_incoming G hloc hqt p hp (root i)))
-
-end PivotalTransportTotal
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section FinitePivotalComparison
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem orientedPivotal_probability_le_closed [Countable V] [Countable E]
-    (G : BondGraph V E) (p : unitInterval) (e : E) (x y a b : V) (he : G e = s(a,b)) :
-    (law p).real (orientedPivotalEvent G x y a b) ≤
-      (law p).real (closedPivotal {ω | Conn G ω x y} e) := by
-  rw [closedPivotal_conn_probability G p e a b x y he]
-  exact le_add_of_nonneg_right measureReal_nonneg
-
- 
-
-theorem finite_neighbor_pivotal_le_incident [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (p : unitInterval) (x y a : V) :
-    (∑ b ∈ neighbors G hloc a, (law p).real (orientedPivotalEvent G x y a b)) ≤
-      ∑ e ∈ (hloc a).toFinset, (law p).real (closedPivotal {ω | Conn G ω x y} e) := by
-  let B := neighbors G hloc a
-  have he (b : B) : ∃ e, G e = s(a,(b:V)) := (mem_neighbors G hloc a b).mp b.property
-  choose edge hedge using he
-  have hinj : Function.Injective edge := by
-    intro b c h
-    apply Subtype.ext
-    have hh := (hedge b).symm.trans ((congrArg G h).trans (hedge c))
-    rcases Sym2.eq_iff.mp hh with h | h
-    · exact h.2
-    · exact h.2.trans h.1
-  calc
-    _ = ∑ b : B, (law p).real (orientedPivotalEvent G x y a b) := (Finset.sum_coe_sort B _).symm
-    _ ≤ ∑ b : B, (law p).real (closedPivotal {ω | Conn G ω x y} (edge b)) :=
-      Finset.sum_le_sum (fun b _ => orientedPivotal_probability_le_closed G p (edge b) x y a b (hedge b))
-    _ = ∑ e ∈ Finset.univ.image edge, (law p).real (closedPivotal {ω | Conn G ω x y} e) :=
-      (Finset.sum_image (f := fun e => (law p).real (closedPivotal {ω | Conn G ω x y} e))
-        (s := Finset.univ) (fun _ _ _ _ h => hinj h)).symm
-    _ ≤ _ := by
-      apply Finset.sum_le_sum_of_subset_of_nonneg
-      · intro e he
-        obtain ⟨b,_,rfl⟩ := Finset.mem_image.mp he
-        rw [Set.Finite.mem_toFinset]
-        change a ∈ G (edge b)
-        rw [hedge b]
-        exact Sym2.mem_mk_left _ _
-      · intro _ _ _
-        exact measureReal_nonneg
-
-theorem finite_incident_sum_le_twice (G : BondGraph V E) (hloc : LocallyFinite G)
-    (A : Finset V) (f : E → ℝ) (hf : ∀ e, 0 ≤ f e) :
-    (∑ a ∈ A, ∑ e ∈ (hloc a).toFinset, f e) ≤ 2*∑ e ∈ incidentSupport G hloc A, f e := by
-  let Q := incidentSupport G hloc A
-  have heq (a : V) (ha : a ∈ A) :
-      (∑ e ∈ (hloc a).toFinset, f e) = ∑ e ∈ Q, if a ∈ G e then f e else 0 := by
-    rw [←Finset.sum_filter]
-    apply Finset.sum_congr _ (fun _ _ => rfl)
-    ext e
-    simp only [Set.Finite.mem_toFinset,Finset.mem_filter]
-    exact ⟨fun he => ⟨(mem_incidentSupport G hloc A e).mpr ⟨a,ha,he⟩,he⟩,fun h => h.2⟩
-  calc
-    _ = ∑ a ∈ A, ∑ e ∈ Q, if a ∈ G e then f e else 0 := Finset.sum_congr rfl heq
-    _ = ∑ e ∈ Q, ∑ a ∈ A, if a ∈ G e then f e else 0 := Finset.sum_comm
-    _ ≤ ∑ e ∈ Q, 2*f e := by
-      apply Finset.sum_le_sum
-      intro e _
-      have hsub : A.filter (fun a => a ∈ G e) ⊆ (G e).toFinset := by
-        intro a ha
-        exact Sym2.mem_toFinset.mpr (Finset.mem_filter.mp ha).2
-      rw [←Finset.sum_filter,Finset.sum_const, nsmul_eq_mul]
-      have hc : (A.filter (fun a => a ∈ G e)).card ≤ 2 := (Finset.card_le_card hsub).trans (by
-        rw [Sym2.card_toFinset]; split_ifs <;> omega)
-      exact mul_le_mul_of_nonneg_right (by exact_mod_cast hc) (hf e)
-    _ = _ := (Finset.mul_sum _ _ _).symm
-
-theorem finite_oriented_pivotal_sum_le [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (p : unitInterval) (x y : V) (A : Finset V) :
-    (∑ a ∈ A, ∑ b ∈ neighbors G hloc a, (law p).real (orientedPivotalEvent G x y a b)) ≤
-      2*∑ e ∈ incidentSupport G hloc A, (law p).real (closedPivotal {ω | Conn G ω x y} e) := by
-  exact (Finset.sum_le_sum (fun a _ => finite_neighbor_pivotal_le_incident G hloc p x y a)).trans
-    (finite_incident_sum_le_twice G hloc A _ (fun _ => measureReal_nonneg))
-
-end FinitePivotalComparison
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section SubcriticalSusceptibilityContinuity
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem eventually_cluster_eq_at_finite (G : BondGraph V E) (hloc : LocallyFinite G)
-    (labels : E → unitInterval) (p : unitInterval) (x : V)
-    (hneq : ∀ e, labels e ≠ p) (hfin : (cluster G (openEdges labels p) x).Finite) :
-    ∀ᶠ q in nhds p, cluster G (openEdges labels q) x = cluster G (openEdges labels p) x := by
-  let Q := incidentSupport G hloc hfin.toFinset
-  have he (e : E) : ∀ᶠ q in nhds p, (e ∈ openEdges labels q ↔ e ∈ openEdges labels p) := by
-    change ∀ᶠ q in nhds p, (labels e ≤ q ↔ labels e ≤ p)
-    by_cases hp : labels e ≤ p
-    · have hlt : labels e < p := lt_of_le_of_ne hp (hneq e)
-      filter_upwards [eventually_gt_nhds hlt] with q hq
-      exact iff_of_true hq.le hp
-    · have hlt : p < labels e := lt_of_not_ge hp
-      filter_upwards [eventually_lt_nhds hlt] with q hq
-      exact iff_of_false (not_le.mpr hq) hp
-  filter_upwards [(eventually_all_finset Q).mpr (fun e _ => he e)] with q hq
-  apply cluster_eq_of_incident_agreement G
-  intro e he
-  apply (hq e _).symm
-  obtain ⟨v,hv,hve⟩ := he
-  exact (mem_incidentSupport G hloc hfin.toFinset e).mpr ⟨v,hfin.mem_toFinset.mpr hv,hve⟩
-
-theorem label_cluster_finite_ae_subcritical [Countable E]
-    (G : BondGraph V E) (p : unitInterval) (hp : p < pc G) (x : V) :
-    ∀ᵐ labels ∂(labelLaw : Measure (E → unitInterval)), (cluster G (openEdges labels p) x).Finite := by
-  apply ae_of_ae_map (μ := (labelLaw : Measure (E → unitInterval)))
-    (p := fun ω : Set E => (cluster G ω x).Finite) (measurable_openEdges p).aemeasurable
-  rw [map_openEdges]
-  exact subcritical_cluster_finite_ae G p hp x
-
-theorem susceptibility_eq_label_integral [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (x : V) :
-    Integrable (fun labels : E → unitInterval => ((cluster G (openEdges labels p) x).ncard:ℝ)) labelLaw ∧
-    susceptibility G p x = ∫ labels : E → unitInterval, ((cluster G (openEdges labels p) x).ncard:ℝ) ∂labelLaw := by
-  obtain ⟨hi,he⟩ := susceptibility_eq_integral G hloc hqt p hp x
-  have hi' : Integrable (fun ω : Set E => ((cluster G ω x).ncard:ℝ))
-      ((labelLaw : Measure (E → unitInterval)).map (fun labels => openEdges labels p)) := by
-    rwa [map_openEdges]
-  refine ⟨hi'.comp_measurable (measurable_openEdges p),?_⟩
-  rw [he,←integral_map (measurable_openEdges p).aemeasurable hi'.aestronglyMeasurable,map_openEdges]
-
- 
-
-theorem continuousAt_susceptibility_subcritical [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (x : V) : ContinuousAt (fun q => susceptibility G q x) p := by
-  obtain ⟨r,hpr,hr⟩ := exists_between hp
-  let F (q : unitInterval) (labels : E → unitInterval) : ℝ := ((cluster G (openEdges labels q) x).ncard:ℝ)
-  have hm (q : unitInterval) : Measurable (F q) :=
-    (measurable_of_countable (fun n : ℕ => (n:ℝ))).comp
-      ((measurable_cluster_ncard G hloc x).comp (measurable_openEdges q))
-  have hcont : ContinuousAt (fun q => ∫ labels, F q labels ∂labelLaw) p := by
-    apply continuousAt_of_dominated
-      (bound := F r) (Filter.Eventually.of_forall (fun q => (hm q).aestronglyMeasurable))
-    · filter_upwards [eventually_lt_nhds hpr] with q hqr
-      filter_upwards [label_cluster_finite_ae_subcritical G r hr x] with labels hfin
-      rw [Real.norm_of_nonneg (Nat.cast_nonneg _)]
-      dsimp only [F]
-      exact_mod_cast Set.ncard_le_ncard (show cluster G (openEdges labels q) x ⊆ cluster G (openEdges labels r) x from
-        fun y hy => conn_mono G (openEdges_mono labels hqr.le) hy) hfin
-    · exact (susceptibility_eq_label_integral G hloc hqt r hr x).1
-    · filter_upwards [ae_labels_ne (E := E) p,label_cluster_finite_ae_subcritical G p hp x] with labels hneq hfin
-      apply Filter.EventuallyEq.continuousAt
-      filter_upwards [eventually_cluster_eq_at_finite G hloc labels p x hneq hfin] with q hq
-      exact congrArg (fun S : Set V => (S.ncard:ℝ)) hq
-  apply (continuousAt_congr _).mpr hcont
-  filter_upwards [eventually_lt_nhds hp] with q hq
-  exact (susceptibility_eq_label_integral G hloc hqt q hq x).2
-
-theorem continuousAt_rootSusceptibility_subcritical [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (p : unitInterval) (hp : p < pc G) :
-    ContinuousAt (rootSusceptibility G root weight) p := by
-  exact tendsto_finsetSum Finset.univ (fun i _ => continuousAt_const.mul
-    (continuousAt_susceptibility_subcritical G hloc hqt p hp (root i)))
-
-end SubcriticalSusceptibilityContinuity
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section FiniteRootPivotalApproximation
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem summable_orientedPivotal_targets [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (x a b : V) :
-    Summable (fun y => (law p).real (orientedPivotalEvent G x y a b)) := by
-  apply (summable_twoPoint_subcritical G hloc hqt p hp b).of_nonneg_of_le
-  · exact fun _ => measureReal_nonneg
-  · intro y
-    exact measureReal_mono (fun _ h => h.2.1)
-
-noncomputable def finiteRootPivotalMass (G : BondGraph V E)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (p : unitInterval) (Q : Finset E) (F : Finset V) : ℝ :=
-  ∑ i, weight i*(∑ y ∈ F, ∑ e ∈ Q, (law p).real (closedPivotal {ω | Conn G ω (root i) y} e))
-
-theorem finiteRootPivotalMass_nonneg (G : BondGraph V E)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 ≤ weight i)
-    (p : unitInterval) (Q : Finset E) (F : Finset V) :
-    0 ≤ finiteRootPivotalMass G root weight p Q F :=
-  Finset.sum_nonneg (fun i _ => mul_nonneg (hw i)
-    (Finset.sum_nonneg (fun _ _ => Finset.sum_nonneg (fun _ _ => measureReal_nonneg))))
-
- 
-
-theorem exists_finiteRootPivotalMass_gt
-    [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
-    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (p : unitInterval) (hp : p < pc G) (B : ℝ)
-    (hB : B < rootNeighborMoment G hloc root weight p / 2) :
-    ∃ Q : Finset E, ∃ F : Finset V, B < finiteRootPivotalMass G root weight p Q F := by
-  have hs (x : V) := pivotalTransportMass_outgoing_summable G hconn hloc hqt root weight Γ hΓ htr hcov hw hw1 c hc hmin p hp x
-  let f (a : V) := ∑ i, weight i*pivotalTransportMass G hloc p (root i) a
-  have hsf : Summable f := summable_sum (fun i _ => (hs (root i)).mul_left (weight i))
-  have htf : (∑' a, f a) = rootNeighborMoment G hloc root weight p := by
-    rw [Summable.tsum_finsetSum (fun i _ => (hs (root i)).mul_left (weight i))]
-    simp_rw [tsum_mul_left]
-    exact rootPivotalMass_eq_neighborMoment G hconn hloc hqt root weight Γ hΓ htr hcov hw hw1 c hc hmin p hp
-  have hB' : 2*B < ∑' a, f a := by rw [htf]; linarith
-  obtain ⟨A,hA⟩ := ((tendsto_order.mp hsf.hasSum).1 (2*B) hB').exists
-  let g (y : V) := ∑ i, weight i*(∑ a ∈ A, ∑ b ∈ neighbors G hloc a,
-    (law p).real (orientedPivotalEvent G (root i) y a b))
-  have hst (i : ι) (a b : V) := summable_orientedPivotal_targets G hloc hqt p hp (root i) a b
-  have hsg : Summable g := summable_sum (fun i _ =>
-    (summable_sum (fun a _ => summable_sum (fun b _ => hst i a b))).mul_left (weight i))
-  have htg : (∑' y, g y) = ∑ a ∈ A, f a := by
-    rw [Summable.tsum_finsetSum (fun i _ =>
-      (summable_sum (fun a _ => summable_sum (fun b _ => hst i a b))).mul_left (weight i))]
-    simp_rw [tsum_mul_left]
-    simp_rw [Summable.tsum_finsetSum (fun a _ => summable_sum (fun b _ => hst _ a b))]
-    simp_rw [Summable.tsum_finsetSum (fun b _ => hst _ _ b)]
-    simp only [f,Finset.mul_sum,pivotalTransportMass]
-    rw [Finset.sum_comm]
-  obtain ⟨F,hF⟩ := ((tendsto_order.mp hsg.hasSum).1 (2*B) (by rwa [htg])).exists
-  refine ⟨incidentSupport G hloc A,F,?_⟩
-  have hb : (∑ y ∈ F, g y) ≤ 2*finiteRootPivotalMass G root weight p (incidentSupport G hloc A) F := by
-    unfold g finiteRootPivotalMass
-    rw [Finset.sum_comm, Finset.mul_sum]
-    apply Finset.sum_le_sum
-    intro i _
-    rw [← Finset.mul_sum, mul_left_comm]
-    apply mul_le_mul_of_nonneg_left _ (hw i)
-    rw [Finset.mul_sum]
-    exact Finset.sum_le_sum (fun y _ => finite_oriented_pivotal_sum_le G hloc p (root i) y A)
-  linarith
-
-end FiniteRootPivotalApproximation
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section VanishingSprinkles
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def vanishingSprinkle (n : ℕ) : unitInterval :=
-  ⟨1/((n:ℝ)+1), by
-    constructor
-    · positivity
-    · apply (div_le_one (by positivity : 0 < (n:ℝ)+1)).mpr
-      linarith [show (0:ℝ) ≤ (n:ℝ) by positivity]
-  ⟩
-
-theorem vanishingSprinkle_pos (n : ℕ) : 0 < (vanishingSprinkle n : ℝ) := by
-  dsimp [vanishingSprinkle]
-  positivity
-
-theorem tendsto_vanishingSprinkle :
-    Tendsto (fun n => (vanishingSprinkle n:ℝ)) atTop (nhds 0) :=
-  tendsto_one_div_add_atTop_nhds_zero_nat
-
-theorem sprinkle_sequence_gt (p : unitInterval) (hp : (p:ℝ) < 1) (n : ℕ) :
-    p < sprinkleParameter p (vanishingSprinkle n) := by
-  change (p:ℝ) < (p:ℝ)+(1-(p:ℝ))*(vanishingSprinkle n:ℝ)
-  exact lt_add_of_pos_right _ (mul_pos (sub_pos.mpr hp) (vanishingSprinkle_pos n))
-
-theorem tendsto_sprinkle_sequence (p : unitInterval) :
-    Tendsto (fun n => sprinkleParameter p (vanishingSprinkle n)) atTop (nhds p) := by
-  apply tendsto_subtype_rng.mpr
-  change Tendsto (fun n => (p:ℝ)+(1-(p:ℝ))*(vanishingSprinkle n:ℝ)) atTop (nhds (p:ℝ))
-  convert tendsto_const_nhds.add (tendsto_const_nhds.mul tendsto_vanishingSprinkle) using 1; simp
-
-theorem tendsto_sprinkle_sequence_right (p : unitInterval) (hp : (p:ℝ) < 1) :
-    Tendsto (fun n => (sprinkleParameter p (vanishingSprinkle n):ℝ)) atTop (nhdsWithin (p:ℝ) (Set.Ioi (p:ℝ))) := by
-  apply tendsto_nhdsWithin_iff.mpr
-  exact ⟨continuous_subtype_val.continuousAt.tendsto.comp (tendsto_sprinkle_sequence p),
-    Eventually.of_forall (sprinkle_sequence_gt p hp)⟩
-
- 
-
-theorem rootSusceptibility_sequence_slope_lower
-    [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 ≤ weight i)
-    (p : unitInterval) (hp : p < pc G) (Q : Finset E) (F : Finset V) (d : ℝ)
-    (hd : d < finiteRootPivotalMass G root weight p Q F) :
-    ∀ᶠ n in atTop,
-      ((sprinkleParameter p (vanishingSprinkle n):ℝ)-(p:ℝ))*d ≤
-      rootSusceptibility G root weight (sprinkleParameter p (vanishingSprinkle n)) -
-      rootSusceptibility G root weight p := by
-  have ht : Tendsto (fun n => (1-(vanishingSprinkle n:ℝ))^(Q.card-1) *
-      finiteRootPivotalMass G root weight p Q F) atTop
-      (nhds (finiteRootPivotalMass G root weight p Q F)) := by
-    convert ((tendsto_const_nhds.sub tendsto_vanishingSprinkle).pow (Q.card-1)).mul tendsto_const_nhds using 1; simp
-  have hnn := finiteRootPivotalMass_nonneg G root weight hw p Q F
-  filter_upwards [(tendsto_sprinkle_sequence p).eventually (eventually_lt_nhds hp),
-      ht.eventually (eventually_gt_nhds hd)] with n hn hdn
-  have hgain := rootSusceptibility_sprinkling_gain G hloc hqt root weight hw
-    p (vanishingSprinkle n) hn Q F
-  change (vanishingSprinkle n:ℝ)*(1-(vanishingSprinkle n:ℝ))^(Q.card-1)*
-    finiteRootPivotalMass G root weight p Q F ≤ _ at hgain
-  by_cases hd0 : 0 ≤ d
-  · have hdp : (1-(p:ℝ))*d ≤ (1-(vanishingSprinkle n:ℝ))^(Q.card-1)*
-        finiteRootPivotalMass G root weight p Q F := by
-      calc
-        _ ≤ d := by nlinarith [p.2.1]
-        _ ≤ _ := hdn.le
-    have hm := mul_le_mul_of_nonneg_left hdp (vanishingSprinkle n).2.1
-    calc
-      _ = (vanishingSprinkle n:ℝ)*((1-(p:ℝ))*d) := by
-        change ((p:ℝ)+(1-(p:ℝ))*(vanishingSprinkle n:ℝ)-(p:ℝ))*d = _
-        ring
-      _ ≤ (vanishingSprinkle n:ℝ)*((1-(vanishingSprinkle n:ℝ))^(Q.card-1)*
-          finiteRootPivotalMass G root weight p Q F) := hm
-      _ ≤ _ := by simpa only [mul_assoc] using hgain
-  · have hmono : 0 ≤ rootSusceptibility G root weight (sprinkleParameter p (vanishingSprinkle n)) - rootSusceptibility G root weight p :=
-      (mul_nonneg (mul_nonneg (vanishingSprinkle n).2.1 (pow_nonneg (sub_nonneg.mpr (vanishingSprinkle n).2.2) _)) hnn).trans hgain
-    have hdp : ((sprinkleParameter p (vanishingSprinkle n):ℝ)-(p:ℝ))*d ≤ 0 :=
-      mul_nonpos_of_nonneg_of_nonpos (sub_nonneg.mpr (sprinkleParameter_ge_left _ _)) (le_of_not_ge hd0)
-    linarith
-
-end VanishingSprinkles
-end Percolation
-
- 
-namespace Percolation
-open Filter
-section NegativePowerRightSlopes
-
- 
-
-theorem frequently_rpow_slope_le_of_sequence
-    (f : ℝ → ℝ) (x d e : ℝ) (hx : 0 < f x) (hd : 0 < d) (he : e < 0)
-    (z : ℕ → ℝ) (hz : Tendsto z atTop (nhdsWithin x (Set.Ioi x)))
-    (hgain : ∀ᶠ n in atTop, (z n-x)*d ≤ f (z n)-f x) :
-    ∀ r, e*(f x)^(e-1)*d < r →
-      ∃ᶠ y in nhdsWithin x (Set.Ioi x), slope (fun t => (f t)^e) x y < r := by
-  let A (t : ℝ) := f x+(t-x)*d
-  have hA : HasDerivAt A d x := by
-    simpa only [zero_add, one_mul] using! (hasDerivAt_const x (f x)).add (((hasDerivAt_id x).sub_const x).mul_const d)
-  have hAx : A x = f x := by simp [A]
-  have hR : HasDerivAt (fun t => (A t)^e) (e*(f x)^(e-1)*d) x := by
-    simpa only [hAx] using! (Real.hasDerivAt_rpow_const (p := e) (Or.inl (show A x ≠ 0 by rw [hAx]; exact hx.ne'))).comp x hA
-  intro r hr
-  have hlim := hR.tendsto_slope.comp (hz.mono_right (nhdsWithin_mono x (show Set.Ioi x ⊆ {x}ᶜ from fun y hy => by simpa using ne_of_gt hy)))
-  have hright : ∀ᶠ n in atTop, x < z n := (tendsto_nhdsWithin_iff.mp hz).2
-  apply hz.frequently
-  apply Filter.Eventually.frequently
-  filter_upwards [hgain, hright, hlim.eventually (eventually_lt_nhds hr)] with n hn hzn hnr
-  have hApos : 0 < A (z n) := by dsimp [A]; positivity
-  have hAr : A (z n) ≤ f (z n) := by dsimp [A]; linarith
-  have hp := Real.rpow_le_rpow_of_nonpos hApos hAr he.le
-  apply lt_of_le_of_lt _ hnr
-  simp only [Function.comp_apply, slope_def_field, hAx]
-  exact div_le_div_of_nonneg_right (sub_le_sub_right hp _) (sub_pos.mpr hzn).le
-
-end NegativePowerRightSlopes
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section ActualSusceptibilityRightSlope
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def realRootSusceptibility (G : BondGraph V E)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (t : ℝ) : ℝ :=
-  rootSusceptibility G root weight (Set.projIcc 0 1 zero_le_one t)
-
-@[simp] theorem realRootSusceptibility_coe (G : BondGraph V E)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (p : unitInterval) :
-    realRootSusceptibility G root weight (p:ℝ) = rootSusceptibility G root weight p := by
-  simp only [realRootSusceptibility, Set.projIcc_val]
-
-theorem rootSusceptibility_one_le [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
-    (p : unitInterval) (hp : p < pc G) : 1 ≤ rootSusceptibility G root weight p := by
-  calc
-    1 = ∑ i, weight i*1 := by simp only [mul_one, hw1]
-    _ ≤ _ := Finset.sum_le_sum (fun i _ => mul_le_mul_of_nonneg_left
-      (susceptibility_one_le G hloc hqt p hp (root i)) (hw i))
-
-theorem continuousAt_realRootSusceptibility [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (p : unitInterval) (hp : p < pc G) :
-    ContinuousAt (realRootSusceptibility G root weight) (p:ℝ) := by
-  apply ContinuousAt.comp (f := Set.projIcc 0 1 zero_le_one)
-  · simpa only [Set.projIcc_val] using continuousAt_rootSusceptibility_subcritical G hloc hqt root weight p hp
-  · exact continuous_projIcc.continuousAt
-
- 
-
-theorem rootSusceptibility_power_right_slope
-    [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
-    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (p : unitInterval) (hp : p < pc G) (b γ : ℝ) (hb : 0 < b) (hγ : γ < 1)
-    (hD : b*(rootSusceptibility G root weight p)^(2-γ) ≤ rootNeighborMoment G hloc root weight p) :
-    ∀ r, -(1-γ)*b/4 < r → ∃ᶠ y in nhdsWithin (p:ℝ) (Set.Ioi (p:ℝ)),
-      slope (fun t => (realRootSusceptibility G root weight t)^(γ-1)) (p:ℝ) y < r := by
-  let χ := rootSusceptibility G root weight p
-  have hχ : 0 < χ := lt_of_lt_of_le zero_lt_one (rootSusceptibility_one_le G hloc hqt root weight hw hw1 p hp)
-  let d := (b/4)*χ^(2-γ)
-  have hd : 0 < d := mul_pos (by positivity) (Real.rpow_pos_of_pos hχ _)
-  have hBd : d < rootNeighborMoment G hloc root weight p/2 := by
-    dsimp [d, χ] at *
-    nlinarith [Real.rpow_pos_of_pos hχ (2-γ)]
-  obtain ⟨Q,F,hQ⟩ := exists_finiteRootPivotalMass_gt G hconn hloc hqt root weight Γ hΓ htr hcov hw hw1 c hc hmin p hp d hBd
-  have hg := rootSusceptibility_sequence_slope_lower G hloc hqt root weight hw p hp Q F d hQ
-  have hp1 : (p:ℝ) < 1 := lt_of_lt_of_le hp (pc G).2.2
-  have hsl := frequently_rpow_slope_le_of_sequence (realRootSusceptibility G root weight)
-    (p:ℝ) d (γ-1) (by simpa only [realRootSusceptibility_coe] using hχ) hd (by linarith)
-    (fun n => (sprinkleParameter p (vanishingSprinkle n):ℝ)) (tendsto_sprinkle_sequence_right p hp1)
-    (by simpa only [realRootSusceptibility_coe] using hg)
-  have heq : (γ-1)*(realRootSusceptibility G root weight (p:ℝ))^(γ-1-1)*d = -(1-γ)*b/4 := by
-    rw [realRootSusceptibility_coe]
-    change (γ-1)*χ^(γ-1-1)*((b/4)*χ^(2-γ)) = _
-    rw [show γ-1-1 = γ-2 by ring]
-    calc
-      _ = ((γ-1)*b/4)*(χ^(γ-2)*χ^(2-γ)) := by ring
-      _ = _ := by rw [←Real.rpow_add hχ, sub_add_sub_cancel, sub_self, Real.rpow_zero]; ring
-  rwa [heq] at hsl
-
-end ActualSusceptibilityRightSlope
-end Percolation
-
- 
-namespace Percolation
-open Filter
-section IntegratedRightSlopes
-
- 
-
-theorem interval_decrease_of_right_slope (f : ℝ → ℝ) (a b k : ℝ)
-    (hab : a ≤ b) (hf : ContinuousOn f (Set.Icc a b))
-    (hs : ∀ x ∈ Set.Ico a b, ∀ r, -k < r →
-      ∃ᶠ y in nhdsWithin x (Set.Ioi x), slope f x y < r) :
-    k*(b-a) ≤ f a-f b := by
-  let B (t : ℝ) := f a-k*(t-a)
-  have hB : ContinuousOn B (Set.Icc a b) := by fun_prop
-  have hB' (x : ℝ) : HasDerivAt B (-k) x := by
-    simpa only [one_mul, sub_zero, mul_one, zero_sub] using!
-      (hasDerivAt_const x (f a)).sub (((hasDerivAt_id x).sub_const a).const_mul k)
-  have h := image_le_of_liminf_slope_right_le_deriv_boundary hf
-    (show f a ≤ B a by simp [B]) hB (fun x _ => (hB' x).hasDerivWithinAt) hs ⟨hab,le_rfl⟩
-  dsimp [B] at h
-  linarith
-
- 
-
-theorem right_slope_distance_bound (f : ℝ → ℝ) (a b k : ℝ)
-    (hab : a < b) (hk : 0 < k)
-    (hpos : ∀ x ∈ Set.Ico a b, 0 ≤ f x)
-    (hc : ContinuousOn f (Set.Ico a b))
-    (hs : ∀ x ∈ Set.Ico a b, ∀ r, -k < r →
-      ∃ᶠ y in nhdsWithin x (Set.Ioi x), slope f x y < r) :
-    k*(b-a) ≤ f a := by
-  have hbound (r : ℝ) (hr : r ∈ Set.Ico a b) : k*(r-a) ≤ f a := by
-    have hsub : Set.Icc a r ⊆ Set.Ico a b := fun t ht => ⟨ht.1,lt_of_le_of_lt ht.2 hr.2⟩
-    have h := interval_decrease_of_right_slope f a r k hr.1 (hc.mono hsub)
-      (fun x hx => hs x ⟨hx.1,lt_trans hx.2 hr.2⟩)
-    linarith [hpos r hr]
-  by_contra h
-  have hd : a+f a/k < b := by
-    have hh : f a < k*(b-a) := lt_of_not_ge h
-    have hv : f a/k < b-a := (div_lt_iff₀ hk).mpr (by nlinarith)
-    linarith
-  obtain ⟨r,hr1,hr2⟩ := exists_between hd
-  have ha : a ≤ r := by
-    have hf := hpos a ⟨le_rfl,hab⟩
-    have hd0 := div_nonneg hf hk.le
-    linarith
-  have hb := hbound r ⟨ha,hr2⟩
-  have hr : f a < (r-a)*k := (div_lt_iff₀ hk).mp (by linarith : f a/k < r-a)
-  nlinarith
-
-end IntegratedRightSlopes
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section ActualIntegratedSusceptibility
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem susceptibility_bound_of_neighborMoment
-    [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
-    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (p : unitInterval) (hp : p < pc G) (b γ : ℝ) (hb : 0 < b) (hγ : γ < 1)
-    (hD : ∀ q, p ≤ q → q < pc G → b*(rootSusceptibility G root weight q)^(2-γ) ≤
-      rootNeighborMoment G hloc root weight q) :
-    rootSusceptibility G root weight p ≤
-      (((1-γ)*b/4)*((pc G:ℝ)-(p:ℝ)))^(-1/(1-γ)) := by
-  let k := (1-γ)*b/4
-  have hk : 0 < k := by dsimp [k]; positivity
-  let f (t : ℝ) := (realRootSusceptibility G root weight t)^(γ-1)
-  have hparam (t : ℝ) (ht : t ∈ Set.Ico (p:ℝ) (pc G:ℝ)) : t ∈ Set.Icc (0:ℝ) 1 :=
-    ⟨le_trans p.2.1 ht.1, le_trans ht.2.le (pc G).2.2⟩
-  have hcp (t : ℝ) (ht : t ∈ Set.Ico (p:ℝ) (pc G:ℝ)) :
-      ContinuousAt (realRootSusceptibility G root weight) t :=
-    continuousAt_realRootSusceptibility G hloc hqt root weight ⟨t,hparam t ht⟩ ht.2
-  have hpos (t : ℝ) (ht : t ∈ Set.Ico (p:ℝ) (pc G:ℝ)) :
-      0 < realRootSusceptibility G root weight t := by
-    have hh := rootSusceptibility_one_le G hloc hqt root weight hw hw1 ⟨t,hparam t ht⟩ ht.2
-    rw [←realRootSusceptibility_coe] at hh
-    exact lt_of_lt_of_le zero_lt_one hh
-  have hcont : ContinuousOn f (Set.Ico (p:ℝ) (pc G:ℝ)) := by
-    intro t ht
-    exact ((hcp t ht).rpow_const (Or.inl (hpos t ht).ne')).continuousWithinAt
-  have hs (t : ℝ) (ht : t ∈ Set.Ico (p:ℝ) (pc G:ℝ)) :=
-    rootSusceptibility_power_right_slope G hconn hloc hqt root weight Γ hΓ htr hcov hw hw1 c hc hmin
-      ⟨t,hparam t ht⟩ ht.2 b γ hb hγ (hD ⟨t,hparam t ht⟩ ht.1 ht.2)
-  have hdist : k*((pc G:ℝ)-(p:ℝ)) ≤ f (p:ℝ) := by
-    apply right_slope_distance_bound f (p:ℝ) (pc G:ℝ) k hp hk
-      (fun t ht => (Real.rpow_pos_of_pos (hpos t ht) _).le) hcont
-    intro t ht r hr
-    apply hs t ht r
-    dsimp [k] at hr
-    linarith
-  have hχ := lt_of_lt_of_le zero_lt_one (rootSusceptibility_one_le G hloc hqt root weight hw hw1 p hp)
-  have hd : 0 < k*((pc G:ℝ)-(p:ℝ)) := mul_pos hk (sub_pos.mpr hp)
-  have he : -1/(1-γ) = 1/(γ-1) := by rw [show γ-1=-(1-γ) by ring, div_neg, neg_div]
-  have he0 : 1/(γ-1) < 0 := one_div_neg.mpr (by linarith)
-  have hinv := Real.rpow_le_rpow_of_nonpos hd hdist he0.le
-  rw [he]
-  apply le_trans _ hinv
-  change rootSusceptibility G root weight p ≤
-    ((realRootSusceptibility G root weight (p:ℝ))^(γ-1))^(1/(γ-1))
-  rw [realRootSusceptibility_coe,←Real.rpow_mul hχ.le]
-  rw [mul_one_div_cancel (by linarith : γ-1 ≠ 0),Real.rpow_one]
-
-end ActualIntegratedSusceptibility
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section SusceptibilityGrowthConsequence
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem eventually_upper_interval_unit (p₀ : unitInterval) (P : unitInterval → Prop)
-    (h : ∀ᶠ p in nhds p₀, P p) :
-    ∀ᶠ p in nhds p₀, p < p₀ → ∀ q, p ≤ q → q < p₀ → P q := by
-  obtain ⟨ε,hε,hball⟩ := Metric.eventually_nhds_iff.mp h
-  apply Metric.eventually_nhds_iff.mpr
-  refine ⟨ε,hε,?_⟩
-  intro p hp hpp q hpq hq
-  apply hball
-  have hd : dist q p₀ ≤ dist p p₀ := by
-    change |(q:ℝ)-(p₀:ℝ)| ≤ |(p:ℝ)-(p₀:ℝ)|
-    rw [abs_of_nonpos (sub_nonpos.mpr (show (q:ℝ) ≤ (p₀:ℝ) from hq.le)),abs_of_nonpos (sub_nonpos.mpr (show (p:ℝ) ≤ (p₀:ℝ) from hpp.le))]
-    have hpqR : (p:ℝ) ≤ (q:ℝ) := hpq
-    linarith
-  exact lt_of_le_of_lt hd hp
-
- 
-
-theorem susceptibility_growth_of_divergent_moment
-    [Countable V] [Countable E] [Nonempty V] [Nonempty E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hconn : Connected G)
-    (hqt : QuasiTransitive G) (hh : 0 < hV G)
-    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
-    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
-    (hp₀ : 0 < (pc G:ℝ)) (hp₁ : (pc G:ℝ) < 1)
-    (hfin : ∀ i, ∀ᵐ ω ∂law (pc G), (cluster G ω (root i)).Finite)
-    (α β ε : ℝ) (hα : 0 < α) (hαβ : α < β)
-    (hε : 0 < ε) (hε1 : ε < 1/2) (hsmall : β+(1/2+ε) < 1)
-    (hdiv : ∃ i, ¬ Integrable (fun ω => ((cluster G ω (root i)).ncard:ℝ)^α) (law (pc G))) :
-    ∀ γ : ℝ, 0 < γ → γ < 1 → ∃ C : ℝ, 0 < C ∧ ∀ᶠ p in nhds (pc G), p < pc G →
-      rootSusceptibility G root weight p ≤ C*((pc G:ℝ)-(p:ℝ))^(-1/(1-γ)) := by
-  intro γ hγ hγ1
-  obtain ⟨b,hb,hnear⟩ := neighborMoment_growth_of_divergent_moment G hloc hconn hqt hh root weight Γ hΓ htr
-    hcov hw hw1 (pc G) hp₀ hp₁ hfin α β ε hα hαβ hε hε1 hsmall hdiv γ hγ
-  let c : ℝ := ∑ i, 1/weight i
-  have hc : 0 ≤ c := Finset.sum_nonneg (fun i _ => (one_div_pos.mpr (hw i)).le)
-  have hmin (i : ι) : 1 ≤ c*weight i := by
-    apply (div_le_iff₀ (hw i)).mp
-    exact Finset.single_le_sum (fun j _ => (one_div_pos.mpr (hw j)).le) (Finset.mem_univ i)
-  let k := (1-γ)*b/4
-  have hk : 0 < k := by dsimp [k]; positivity
-  refine ⟨k^(-1/(1-γ)),Real.rpow_pos_of_pos hk _,?_⟩
-  filter_upwards [eventually_upper_interval_unit (pc G) _ hnear] with p hp
-  intro hpc
-  have hD (q : unitInterval) (hpq : p ≤ q) (hq : q < pc G) := hp hpc q hpq hq hq
-  have hs := susceptibility_bound_of_neighborMoment G hconn hloc hqt root weight Γ hΓ htr hcov
-    (fun i => (hw i).le) hw1 c hc hmin p hpc b γ hb hγ1 hD
-  rwa [Real.mul_rpow hk.le (sub_pos.mpr (show (p:ℝ) < (pc G:ℝ) from hpc)).le] at hs
-
-end SusceptibilityGrowthConsequence
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section MarkedSearchTransfer
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem exponential_transfer_of_deriv_bound (H : ℝ → ℝ) (a b K : ℝ)
-    (hH : Differentiable ℝ H) (hab : a ≤ b)
-    (hb : ∀ t ∈ Set.Icc a b, deriv H t ≤ K*H t) :
-    H b ≤ Real.exp (K*(b-a))*H a := by
-  let F (t : ℝ) := Real.exp (-K*(t-a))*H t
-  have hF (t : ℝ) : HasDerivAt F (Real.exp (-K*(t-a))*(deriv H t-K*H t)) t := by
-    have he := (((hasDerivAt_id t).sub_const a).const_mul (-K)).exp
-    convert! he.mul (hH t).hasDerivAt using 1
-    simp only [id_eq]
-    ring
-  have hanti : AntitoneOn F (Set.Icc a b) := by
-    apply antitoneOn_of_deriv_nonpos (convex_Icc a b)
-      (fun t _ => (hF t).continuousAt.continuousWithinAt)
-      (fun t _ => (hF t).differentiableAt.differentiableWithinAt)
-    intro t ht
-    rw [(hF t).deriv]
-    exact mul_nonpos_of_nonneg_of_nonpos (Real.exp_pos _).le (sub_nonpos.mpr (hb t (interior_subset ht)))
-  have h := hanti ⟨le_rfl,hab⟩ ⟨hab,le_rfl⟩ hab
-  have hm := mul_le_mul_of_nonneg_left h (Real.exp_pos (K*(b-a))).le
-  simpa only [F, sub_self,mul_zero,Real.exp_zero,one_mul,←mul_assoc,←Real.exp_add,
-    neg_mul,add_neg_cancel,Real.exp_zero,one_mul,neg_zero,add_zero] using hm
-
- 
-
-theorem markedConnection_transfer (G : BondGraph V E) (Q : Finset E) (x : V)
-    (d : ℕ) (hd : 0 < d) (hdeg : ∀ v, (Q.filter (fun e => v ∈ G e)).card ≤ d)
-    (p q h : unitInterval) (hp : 0 < (p:ℝ)) (hpq : p ≤ q) (hq : (q:ℝ) < 1)
-    (hh : 0 < (h:ℝ)) :
-    markedConnectionPolynomial G Q x h q ≤
-      Real.exp ((Real.sqrt ((d:ℝ)/2)/((p:ℝ)*(1-(q:ℝ)))) *
-        Real.sqrt (Real.log (1/(h:ℝ))/(h:ℝ))*((q:ℝ)-(p:ℝ))) *
-      markedConnectionPolynomial G Q x h p := by
-  let H := markedConnectionPolynomial G Q x h
-  have hdiff : Differentiable ℝ H := fun t => (hasDerivAt_bernoulliPolynomial Q _ t).differentiableAt
-  apply exponential_transfer_of_deriv_bound H p q _ hdiff hpq
-  intro t ht
-  let r : unitInterval := ⟨t,⟨le_trans p.2.1 ht.1,le_trans ht.2 q.2.2⟩⟩
-  have hr0 : 0 < t := lt_of_lt_of_le hp ht.1
-  have hr1 : t < 1 := lt_of_le_of_lt ht.2 hq
-  have HB := markedConnection_derivative_bound G Q x r h hr1 hh d hd hdeg
-  have hH : 0 ≤ H t := by
-    change 0 ≤ markedConnectionPolynomial G Q x h r
-    rw [markedConnectionPolynomial_eq_tree]
-    exact ProbeTree.successMass_nonneg r h _
-  have hs : 0 ≤ Real.sqrt (Real.log (1/(h:ℝ))/(h:ℝ)) := Real.sqrt_nonneg _
-  have hden : (p:ℝ)*(1-(q:ℝ)) ≤ t*(1-t) := by
-    exact mul_le_mul ht.1 (by linarith [ht.2]) (sub_pos.mpr hq).le (by positivity)
-  have hratio : Real.sqrt ((d:ℝ)/2)/(t*(1-t)) ≤
-      Real.sqrt ((d:ℝ)/2)/((p:ℝ)*(1-(q:ℝ))) :=
-    div_le_div_of_nonneg_left (Real.sqrt_nonneg _) (mul_pos hp (sub_pos.mpr hq)) hden
-  have hle : deriv H t ≤ (Real.sqrt ((d:ℝ)/2)/(t*(1-t)))*
-      Real.sqrt (Real.log (1/(h:ℝ))/(h:ℝ))*H t := by
-    have HBr : deriv H t ≤ ((Real.sqrt ((d:ℝ)/2)/(1-t))*
-        Real.sqrt (Real.log (1/(h:ℝ))/(h:ℝ))*H t)/t := by
-      apply (le_div_iff₀ hr0).mpr
-      simpa only [mul_comm (deriv H t) t] using HB
-    convert HBr using 1
-    field_simp
-  exact hle.trans (mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hratio hs) hH)
-
-end MarkedSearchTransfer
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section MarkedComponentIdentity
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem markedConnection_component_polynomial (G : BondGraph V E)
-    (Q : Finset E) (x : V) (h p : ℝ) :
-    markedConnectionPolynomial G Q x h p =
-      bernoulliPolynomial Q (fun ω => 1-(1-h)^(finiteRootComponent G ω x).card) p := by
-  apply bernoulliPolynomial_congr
-  intro ω hω
-  let C := finiteRootComponent G ω x
-  let F := insert x (searchVertices G Q)
-  have hCF : C ⊆ F := finiteRootComponent_subset G Q F x ω hω le_rfl
-  have he (m : Finset V) : (∃ y ∈ m, Conn G (ω:Set E) x y) ↔ m ∩ C ≠ ∅ := by
-    rw [Finset.nonempty_iff_ne_empty.symm]
-    constructor
-    · rintro ⟨y,hy,hxy⟩
-      exact ⟨y,Finset.mem_inter.mpr ⟨hy,by change y ∈ (C:Set V); simpa only [C,coe_finiteRootComponent,cluster,Set.mem_ofPred_eq] using hxy⟩⟩
-    · rintro ⟨y,hy⟩
-      refine ⟨y,(Finset.mem_inter.mp hy).1,?_⟩
-      have hm : y ∈ (C:Set V) := (Finset.mem_inter.mp hy).2
-      simpa only [C,coe_finiteRootComponent,cluster,Set.mem_ofPred_eq] using hm
-  have hid : bernoulliPolynomial F (fun m => if ∃ y ∈ m, Conn G (ω:Set E) x y then 1 else 0) h =
-      1-bernoulliPolynomial F (fun m => if m ∩ C = ∅ then 1 else 0) h := by
-    calc
-      _ = bernoulliPolynomial F (fun m => 1-(if m ∩ C = ∅ then 1 else 0)) h := by
-        apply bernoulliPolynomial_congr
-        intro m _
-        by_cases hm : m ∩ C = ∅
-        · have hn : ¬∃ y ∈ m, Conn G (ω:Set E) x y := fun hc => (he m).mp hc hm
-          simp only [hm,hn,ite_false,ite_true,sub_self]
-        · have hy := (he m).mpr hm
-          simp only [hm,hy,ite_false,ite_true,sub_zero]
-      _ = _ := by rw [bernoulliPolynomial_sub,bernoulliPolynomial_const]
-  rw [hid,bernoulliPolynomial_restrict C F hCF (fun m => if m = ∅ then 1 else 0) h,
-    bernoulliPolynomial_empty_indicator]
-
-theorem one_sub_pow_one_sub_le (h : unitInterval) (n : ℕ) :
-    1-(1-(h:ℝ))^n ≤ (h:ℝ)*n := by
-  induction n with
-  | zero => simp
-  | succ n ih =>
-    rw [pow_succ,Nat.cast_succ]
-    have hm := mul_le_mul_of_nonneg_right ih (sub_nonneg.mpr h.2.2)
-    have hn : 0 ≤ (h:ℝ)^2*n := by positivity
-    nlinarith
-
-end MarkedComponentIdentity
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section MarkedConnectionLaw
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem markedConnection_integral (G : BondGraph V E) (Q : Finset E) (x : V)
-    (h p : unitInterval) :
-    markedConnectionPolynomial G Q x h p =
-      ∫ ω, (1-(1-(h:ℝ))^(cluster G (ω ∩ (Q:Set E)) x).ncard) ∂law p := by
-  rw [integral_eq_bernoulliPolynomial Q p _ (by intro ω; simp only [Set.inter_assoc,Set.inter_self]),
-    markedConnection_component_polynomial]
-  apply bernoulliPolynomial_congr
-  intro ω hω
-  rw [Set.inter_eq_left.mpr (show (ω:Set E) ⊆ Q from hω)]
-  congr 2
-  rw [←coe_finiteRootComponent,Set.ncard_coe_finset]
-
-theorem markedConnection_le_susceptibility [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (Q : Finset E) (x : V) (h p : unitInterval) (hp : p < pc G) :
-    markedConnectionPolynomial G Q x h p ≤ (h:ℝ)*susceptibility G p x := by
-  obtain ⟨hi,he⟩ := susceptibility_eq_integral G hloc hqt p hp x
-  rw [markedConnection_integral,he,←integral_const_mul]
-  apply integral_mono_ae
-  · exact integrable_finite_local Q p _ (by intro ω; simp only [Set.inter_assoc,Set.inter_self])
-  · exact hi.const_mul _
-  · filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
-    apply (one_sub_pow_one_sub_le h _).trans
-    apply mul_le_mul_of_nonneg_left _ h.2.1
-    exact_mod_cast Set.ncard_le_ncard (fun y hy => conn_mono G Set.inter_subset_left hy) hω
-
- 
-
-theorem one_sub_pow_one_sub_ge_half (h : unitInterval) (n : ℕ)
-    (hn : 1 ≤ (h:ℝ)*n) : (1:ℝ)/2 ≤ 1-(1-(h:ℝ))^n := by
-  have hb (m : ℕ) : (1+(h:ℝ)*m)*(1-(h:ℝ))^m ≤ 1 := by
-    induction m with
-    | zero => simp
-    | succ m ih =>
-      have hp : 0 ≤ (1-(h:ℝ))^m := pow_nonneg (sub_nonneg.mpr h.2.2) _
-      have ha : (1+(h:ℝ)*(m+1))*(1-(h:ℝ)) ≤ 1+(h:ℝ)*m := by
-        have hh : 0 ≤ (h:ℝ)^2*((m:ℝ)+1) := by positivity
-        nlinarith
-      have hm := mul_le_mul_of_nonneg_right ha hp
-      rw [Nat.cast_succ,pow_succ]
-      nlinarith
-  have hp : 0 ≤ (1-(h:ℝ))^n := pow_nonneg (sub_nonneg.mpr h.2.2) _
-  have hm := mul_le_mul_of_nonneg_right hn hp
-  have hb' := hb n
-  nlinarith
-
- 
-
-theorem clusterWindow_le_markedConnection (G : BondGraph V E) (hloc : LocallyFinite G)
-    (x : V) (lo hi : ℕ) (p h : unitInterval) (hh : 1 ≤ (h:ℝ)*lo) :
-    clusterWindowProbability G p x lo hi ≤ 2*markedConnectionPolynomial G
-      (incidentSupport G hloc (finite_reachBall hloc x hi).toFinset) x h p := by
-  let Q := incidentSupport G hloc (finite_reachBall hloc x hi).toFinset
-  let f (ω : Set E) := 1-(1-(h:ℝ))^(cluster G (ω ∩ (Q:Set E)) x).ncard
-  let A := {ω | ClusterWindow G x lo hi ω}
-  have hA : MeasurableSet A := measurableSet_clusterWindow G hloc x lo hi
-  have hf : Integrable f (law p) := integrable_finite_local Q p _
-    (by intro ω; simp only [f,Set.inter_assoc,Set.inter_self])
-  have hnon (ω : Set E) : 0 ≤ f ω := by
-    apply sub_nonneg.mpr
-    exact pow_le_one₀ (sub_nonneg.mpr h.2.2) (by linarith [h.2.1])
-  have hbound (ω : Set E) : A.indicator (fun _ => (1:ℝ)/2) ω ≤ f ω := by
-    by_cases hω : ω ∈ A
-    · rw [Set.indicator_of_mem hω]
-      have hs : cluster G ω x ⊆ (finite_reachBall hloc x hi).toFinset := by
-        simpa only [Set.Finite.coe_toFinset] using finite_cluster_subset_reachBall G ω x hω.1 hi hω.2.2.le
-      have he := cluster_restrict_incident_of_full G hloc _ ω x hs
-      change 1/2 ≤ 1-(1-(h:ℝ))^(cluster G (ω ∩ (Q:Set E)) x).ncard
-      rw [he]
-      apply one_sub_pow_one_sub_ge_half
-      exact hh.trans (mul_le_mul_of_nonneg_left (by exact_mod_cast hω.2.1) h.2.1)
-    · rw [Set.indicator_of_notMem hω]
-      exact hnon ω
-  have hib := integral_mono ((integrable_const _).indicator hA) hf hbound
-  rw [integral_indicator_const _ hA] at hib
-  have he := markedConnection_integral G Q x h p
-  change markedConnectionPolynomial G Q x h p = ∫ ω, f ω ∂law p at he
-  rw [←he] at hib
-  change (law p).real A ≤ 2*markedConnectionPolynomial G Q x h p
-  simp only [smul_eq_mul] at hib
-  linarith
-
-end MarkedConnectionLaw
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Kernel.DistinctMoments
+
+namespace OAI
+
+/-! Susceptibility for bond percolation. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section SusceptibilityPivotalGain
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem sprinkleParameter_ge_left (p a : unitInterval) : p ≤ sprinkleParameter p a := by
+  change (p:ℝ) ≤ (p:ℝ)+(1-(p:ℝ))*(a:ℝ)
+  exact le_add_of_nonneg_right (mul_nonneg (sub_nonneg.mpr p.2.2) a.2.1)
+
+ 
+
+theorem susceptibility_sprinkling_gain [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p a : unitInterval) (hq : sprinkleParameter p a < pc G) (x : V)
+    (Q : Finset E) (F : Finset V) :
+    (a:ℝ)*(1-(a:ℝ))^(Q.card-1) *
+      (∑ y ∈ F, ∑ e ∈ Q, (law p).real (closedPivotal {ω | Conn G ω x y} e)) ≤
+      susceptibility G (sprinkleParameter p a) x - susceptibility G p x := by
+  have hpa := sprinkleParameter_ge_left p a
+  have hp : p < pc G := lt_of_le_of_lt hpa hq
+  have hsum := (summable_twoPoint_subcritical G hloc hqt (sprinkleParameter p a) hq x).sub
+    (summable_twoPoint_subcritical G hloc hqt p hp x)
+  calc
+    _ = ∑ y ∈ F, (a:ℝ)*(1-(a:ℝ))^(Q.card-1)*
+      (∑ e ∈ Q, (law p).real (closedPivotal {ω | Conn G ω x y} e)) := Finset.mul_sum _ _ _
+    _ ≤ ∑ y ∈ F, (twoPoint G (sprinkleParameter p a) x y - twoPoint G p x y) := by
+      apply Finset.sum_le_sum
+      intro y _
+      exact event_sprinkling_gain p a Q {ω | Conn G ω x y} (measurableSet_conn G x y)
+        (fun _ _ h hc => conn_mono G h hc)
+    _ ≤ ∑' y, (twoPoint G (sprinkleParameter p a) x y - twoPoint G p x y) :=
+      hsum.sum_le_tsum F (fun y _ => sub_nonneg.mpr (twoPoint_mono G x y hpa))
+    _ = _ := (summable_twoPoint_subcritical G hloc hqt (sprinkleParameter p a) hq x).tsum_sub
+      (summable_twoPoint_subcritical G hloc hqt p hp x)
+
+ 
+theorem rootSusceptibility_sprinkling_gain [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 ≤ weight i)
+    (p a : unitInterval) (hq : sprinkleParameter p a < pc G)
+    (Q : Finset E) (F : Finset V) :
+    (a:ℝ)*(1-(a:ℝ))^(Q.card-1) *
+      (∑ i, weight i*(∑ y ∈ F, ∑ e ∈ Q, (law p).real (closedPivotal {ω | Conn G ω (root i) y} e))) ≤
+      rootSusceptibility G root weight (sprinkleParameter p a) - rootSusceptibility G root weight p := by
+  rw [rootSusceptibility,rootSusceptibility,←Finset.sum_sub_distrib,Finset.mul_sum]
+  apply Finset.sum_le_sum
+  intro i _
+  have h := mul_le_mul_of_nonneg_left
+    (susceptibility_sprinkling_gain G hloc hqt p a hq (root i) Q F) (hw i)
+  convert h using 1 <;> ring
+
+end SusceptibilityPivotalGain
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section OrientedPivotalConnections
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+def orientedPivotalEvent (G : BondGraph V E) (x y a b : V) : Set (Set E) :=
+  {ω | Conn G ω x a ∧ Conn G ω b y ∧ ¬ Conn G ω a b}
+
+theorem measurableSet_orientedPivotalEvent [Countable V] [Countable E]
+    (G : BondGraph V E) (x y a b : V) : MeasurableSet (orientedPivotalEvent G x y a b) :=
+  (measurableSet_conn G x a).inter ((measurableSet_conn G b y).inter (measurableSet_conn G a b).compl)
+
+theorem orientedPivotalEvent_disjoint (G : BondGraph V E) (x y a b : V) :
+    Disjoint (orientedPivotalEvent G x y a b) (orientedPivotalEvent G x y b a) := by
+  apply Set.disjoint_left.mpr
+  intro ω h h'
+  exact h.2.2 ((conn_symm G ω h.1).trans h'.1)
+
+theorem closedPivotal_conn (G : BondGraph V E) (e : E) (a b x y : V)
+    (he : G e = s(a,b)) :
+    closedPivotal {ω | Conn G ω x y} e =
+      orientedPivotalEvent G x y a b ∪ orientedPivotalEvent G x y b a := by
+  ext ω
+  constructor
+  · rintro ⟨_,hn,hc⟩
+    rcases (conn_insert_iff G ω e a b x y he).mp hc with h | h | h
+    · exact (hn h).elim
+    · exact Or.inl ⟨h.1,h.2,fun hab => hn (h.1.trans (hab.trans h.2))⟩
+    · exact Or.inr ⟨h.1,h.2,fun hba => hn (h.1.trans (hba.trans h.2))⟩
+  · intro h
+    have hi : ∀ a b : V, G e = s(a,b) → ω ∈ orientedPivotalEvent G x y a b →
+        ω ∈ closedPivotal {ω | Conn G ω x y} e := by
+      intro a b he h
+      refine ⟨?_,?_,?_⟩
+      · intro heω
+        exact h.2.2 (.single ⟨e,heω,he⟩)
+      · intro hxy
+        exact h.2.2 ((conn_symm G ω h.1).trans (hxy.trans (conn_symm G ω h.2.1)))
+      · exact (conn_insert_iff G ω e a b x y he).mpr (Or.inr (Or.inl ⟨h.1,h.2.1⟩))
+    exact h.elim (hi a b he) (hi b a (he.trans (Sym2.eq_swap ..)))
+
+theorem closedPivotal_conn_probability [Countable V] [Countable E]
+    (G : BondGraph V E) (p : unitInterval) (e : E) (a b x y : V) (he : G e = s(a,b)) :
+    (law p).real (closedPivotal {ω | Conn G ω x y} e) =
+      (law p).real (orientedPivotalEvent G x y a b) +
+      (law p).real (orientedPivotalEvent G x y b a) := by
+  rw [closedPivotal_conn G e a b x y he]
+  exact measureReal_union (orientedPivotalEvent_disjoint G x y a b)
+    (measurableSet_orientedPivotalEvent G x y b a)
+
+ 
+theorem orientedPivotalEvent_subset_disjoint (G : BondGraph V E) (x y a b : V) :
+    orientedPivotalEvent G x y a b ⊆
+      DisjointOccurrence {ω | Conn G ω x a} {ω | Conn G ω b y} := by
+  intro ω hω
+  have hx : x ∈ cluster G ω a := conn_symm G ω hω.1
+  have hb : b ∉ cluster G ω a := hω.2.2
+  obtain ⟨Q,hQ,hQa⟩ := finiteWitness_conn G x a _ (conn_restrict_cluster G ω a hx hω.1)
+  obtain ⟨R,hR,hRb⟩ := finiteWitness_conn G b y _ (conn_restrict_cluster_compl G ω a hb hω.2.1)
+  refine ⟨Q,R,fun e he => (hQ he).1,fun e he => (hR he).1,?_,hQa,hRb⟩
+  apply Finset.disjoint_left.mpr
+  intro e heQ heR
+  obtain ⟨⟨z,w⟩,he⟩ := Quot.exists_rep (G e)
+  have hz : z ∈ G e := by rw [←he]; exact Sym2.mem_mk_left z w
+  exact (hR heR).2 z hz ((hQ heQ).2 z hz)
+
+end OrientedPivotalConnections
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+open scoped ENNReal
+section PivotalMomentIdentity
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem distinctClusterMass_eq_pivotal (G : BondGraph V E) (ω : Set E) (a b : V) :
+    (if ¬Conn G ω a b then clusterMass G ω a * clusterMass G ω b else 0) =
+      ∑' x, ∑' y, if ω ∈ orientedPivotalEvent G x y a b then (1:ℝ≥0∞) else 0 := by
+  by_cases hab : Conn G ω a b
+  · simp only [hab,not_true_eq_false,ite_false,orientedPivotalEvent,Set.mem_ofPred_eq,
+       and_false,ite_false,tsum_zero]
+  · rw [ite_eq_left hab,clusterMass_mul]
+    apply tsum_congr
+    intro x
+    apply tsum_congr
+    intro y
+    apply if_congr _ rfl rfl
+    simp only [orientedPivotalEvent,Set.mem_ofPred_eq,hab,not_false_eq_true,and_true]
+    exact and_congr (show Conn G ω a x ↔ Conn G ω x a from ⟨conn_symm G ω,conn_symm G ω⟩) Iff.rfl
+
+theorem lintegral_distinctClusterMass [Countable V] [Countable E]
+    (G : BondGraph V E) (p : unitInterval) (a b : V) :
+    (∫⁻ ω, (if ¬Conn G ω a b then clusterMass G ω a * clusterMass G ω b else 0) ∂law p) =
+      ∑' x, ∑' y, law p (orientedPivotalEvent G x y a b) := by
+  have hm (x y : V) : Measurable (fun ω : Set E =>
+      if ω ∈ orientedPivotalEvent G x y a b then (1:ℝ≥0∞) else 0) :=
+    Measurable.ite (measurableSet_orientedPivotalEvent G x y a b) measurable_const measurable_const
+  simp_rw [distinctClusterMass_eq_pivotal]
+  rw [lintegral_tsum (fun x => (Measurable.tsum (fun y => hm x y)).aemeasurable)]
+  apply tsum_congr
+  intro x
+  rw [lintegral_tsum (fun y => (hm x y).aemeasurable)]
+  apply tsum_congr
+  intro y
+  simpa only [Set.indicator,Pi.one_apply] using
+    (lintegral_indicator_one (μ := law p) (measurableSet_orientedPivotalEvent G x y a b))
+
+theorem ennreal_distinctClusterMoment [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (a b : V) :
+    ENNReal.ofReal (distinctClusterMoment G p a b) =
+      ∑' x, ∑' y, law p (orientedPivotalEvent G x y a b) := by
+  have hi := integrable_distinct_cluster_product G hloc hqt p hp a b
+  have hn : 0 ≤ᵐ[law p] (fun ω => if ¬Conn G ω a b
+      then ((cluster G ω a).ncard:ℝ)*((cluster G ω b).ncard:ℝ) else 0) := by
+    exact Filter.Eventually.of_forall (fun ω => by dsimp only; split_ifs <;> positivity)
+  have he : (fun ω => ENNReal.ofReal (if ¬Conn G ω a b
+      then ((cluster G ω a).ncard:ℝ)*((cluster G ω b).ncard:ℝ) else 0)) =ᵐ[law p]
+      (fun ω => if ¬Conn G ω a b then clusterMass G ω a * clusterMass G ω b else 0) := by
+    filter_upwards [subcritical_cluster_finite_ae G p hp a,
+      subcritical_cluster_finite_ae G p hp b] with ω ha hb
+    by_cases hab : Conn G ω a b
+    · simp only [hab,not_true_eq_false,ite_false,ENNReal.ofReal_zero]
+    · simp only [hab,not_false_eq_true,ite_true]
+      rw [clusterMass_eq_ncard G ω a ha,clusterMass_eq_ncard G ω b hb,
+        ENNReal.ofReal_mul (Nat.cast_nonneg _)]
+      simp only [ENNReal.ofReal_natCast]
+  rw [←lintegral_distinctClusterMass,←lintegral_congr_ae he]
+  exact ofReal_integral_eq_lintegral_ofReal hi hn
+
+theorem summable_orientedPivotal_pairs [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (a b : V) :
+    Summable (fun xy : V × V => (law p).real (orientedPivotalEvent G xy.1 xy.2 a b)) := by
+  apply ENNReal.summable_toReal
+  rw [ENNReal.tsum_prod',←ennreal_distinctClusterMoment G hloc hqt p hp a b]
+  exact ENNReal.ofReal_ne_top
+
+theorem distinctClusterMoment_eq_pivotal [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (a b : V) :
+    distinctClusterMoment G p a b =
+      ∑' x, ∑' y, (law p).real (orientedPivotalEvent G x y a b) := by
+  rw [←(summable_orientedPivotal_pairs G hloc hqt p hp a b).tsum_prod]
+  change distinctClusterMoment G p a b =
+    ∑' xy : V × V, (law p (orientedPivotalEvent G xy.1 xy.2 a b)).toReal
+  rw [←ENNReal.tsum_toReal_eq (f := fun xy : V × V => law p (orientedPivotalEvent G xy.1 xy.2 a b))
+    (fun _ => measure_ne_top _ _), ENNReal.tsum_prod',
+    ←ennreal_distinctClusterMoment G hloc hqt p hp a b,
+    ENNReal.toReal_ofReal (distinctClusterMoment_nonneg G p a b)]
+
+end PivotalMomentIdentity
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section PivotalTransport
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem orientedPivotal_probability_covariant [Countable V] [Countable E]
+    (G : BondGraph V E) (p : unitInterval) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (x y z w : V) :
+    (law p).real (orientedPivotalEvent G (a x) (a y) (a z) (a w)) =
+      (law p).real (orientedPivotalEvent G x y z w) := by
+  have he : orientedPivotalEvent G (a x) (a y) (a z) (a w) =
+      (fun ω => b ⁻¹' ω) ⁻¹' orientedPivotalEvent G x y z w := by
+    ext ω
+    simp only [orientedPivotalEvent,Set.mem_ofPred_eq,Set.mem_preimage,conn_aut_iff G hab]
+  unfold Measure.real
+  rw [he,←Measure.map_apply (measurable_set_preimage b)
+    (measurableSet_orientedPivotalEvent G x y z w),law_map_preimage_equiv]
+
+noncomputable def pivotalTransportMass (G : BondGraph V E) (hloc : LocallyFinite G)
+    (p : unitInterval) (x a : V) : ℝ :=
+  ∑ b ∈ neighbors G hloc a, ∑' y, (law p).real (orientedPivotalEvent G x y a b)
+
+theorem pivotalTransportMass_nonneg (G : BondGraph V E) (hloc : LocallyFinite G)
+    (p : unitInterval) (x a : V) : 0 ≤ pivotalTransportMass G hloc p x a :=
+  Finset.sum_nonneg (fun _ _ => tsum_nonneg (fun _ => measureReal_nonneg))
+
+theorem pivotalTransportMass_covariant [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (p : unitInterval)
+    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (x z : V) :
+    pivotalTransportMass G hloc p (a x) (a z) = pivotalTransportMass G hloc p x z := by
+  simp only [pivotalTransportMass,neighbors_covariant G hloc hab,Finset.sum_map,Equiv.coe_toEmbedding]
+  apply Finset.sum_congr rfl
+  intro w _
+  rw [←a.tsum_eq (fun y => (law p).real (orientedPivotalEvent G (a x) y (a z) (a w)))]
+  exact tsum_congr (fun y => orientedPivotal_probability_covariant G p hab x y z w)
+
+theorem pivotalTransportMass_incoming_summable [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (a : V) :
+    Summable (fun x => pivotalTransportMass G hloc p x a) := by
+  apply summable_sum
+  intro b _
+  exact (summable_orientedPivotal_pairs G hloc hqt p hp a b).prod
+
+theorem pivotalTransportMass_incoming [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (a : V) :
+    (∑' x, pivotalTransportMass G hloc p x a) = neighborClusterMoment G hloc p a := by
+  unfold pivotalTransportMass neighborClusterMoment
+  rw [Summable.tsum_finsetSum (fun b _ =>
+    (summable_orientedPivotal_pairs G hloc hqt p hp a b).prod)]
+  exact Finset.sum_congr rfl (fun b _ => (distinctClusterMoment_eq_pivotal G hloc hqt p hp a b).symm)
+
+end PivotalTransport
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section PivotalTransportTotal
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem pivotalTransportMass_outgoing_summable
+    [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
+    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (p : unitInterval) (hp : p < pc G) (x : V) :
+    Summable (pivotalTransportMass G hloc p x) := by
+  have hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i) := by
+    intro x
+    obtain ⟨i,a,ha,hax⟩ := hcov x
+    obtain ⟨b,hab⟩ := hΓ ha
+    exact ⟨i,a,b,hab,hax⟩
+  exact (transported_column_of_row_bound G hconn hloc root weight Γ hΓ htr hcov hw hw1 c hc hmin
+    (fun a x => pivotalTransportMass G hloc p x a)
+    (fun a x => pivotalTransportMass_nonneg G hloc p x a)
+    (fun a ha x y => by obtain ⟨b,hab⟩ := hΓ ha; exact pivotalTransportMass_covariant G hloc p hab y x)
+    (pivotalTransportMass_incoming_summable G hloc hqt p hp)
+    (c*rootNeighborMoment G hloc root weight p)
+    (fun a => by
+      rw [pivotalTransportMass_incoming G hloc hqt p hp]
+      exact neighborMoment_le_rootMoment G hloc root weight hw hcover c hc hmin p a) x).1
+
+ 
+
+theorem rootPivotalMass_eq_neighborMoment
+    [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
+    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (p : unitInterval) (hp : p < pc G) :
+    (∑ i, weight i * ∑' a, pivotalTransportMass G hloc p (root i) a) =
+      rootNeighborMoment G hloc root weight p := by
+  rw [htr.real root weight Γ (pivotalTransportMass G hloc p)
+    (fun a ha x y => by obtain ⟨b,hab⟩ := hΓ ha; exact pivotalTransportMass_covariant G hloc p hab x y)
+    (pivotalTransportMass_outgoing_summable G hconn hloc hqt root weight Γ hΓ htr hcov hw hw1 c hc hmin p hp)
+    (pivotalTransportMass_incoming_summable G hloc hqt p hp)]
+  exact Finset.sum_congr rfl (fun i _ => congrArg (fun t => weight i*t)
+    (pivotalTransportMass_incoming G hloc hqt p hp (root i)))
+
+end PivotalTransportTotal
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section FinitePivotalComparison
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem orientedPivotal_probability_le_closed [Countable V] [Countable E]
+    (G : BondGraph V E) (p : unitInterval) (e : E) (x y a b : V) (he : G e = s(a,b)) :
+    (law p).real (orientedPivotalEvent G x y a b) ≤
+      (law p).real (closedPivotal {ω | Conn G ω x y} e) := by
+  rw [closedPivotal_conn_probability G p e a b x y he]
+  exact le_add_of_nonneg_right measureReal_nonneg
+
+ 
+
+theorem finite_neighbor_pivotal_le_incident [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (p : unitInterval) (x y a : V) :
+    (∑ b ∈ neighbors G hloc a, (law p).real (orientedPivotalEvent G x y a b)) ≤
+      ∑ e ∈ (hloc a).toFinset, (law p).real (closedPivotal {ω | Conn G ω x y} e) := by
+  let B := neighbors G hloc a
+  have he (b : B) : ∃ e, G e = s(a,(b:V)) := (mem_neighbors G hloc a b).mp b.property
+  choose edge hedge using he
+  have hinj : Function.Injective edge := by
+    intro b c h
+    apply Subtype.ext
+    have hh := (hedge b).symm.trans ((congrArg G h).trans (hedge c))
+    rcases Sym2.eq_iff.mp hh with h | h
+    · exact h.2
+    · exact h.2.trans h.1
+  calc
+    _ = ∑ b : B, (law p).real (orientedPivotalEvent G x y a b) := (Finset.sum_coe_sort B _).symm
+    _ ≤ ∑ b : B, (law p).real (closedPivotal {ω | Conn G ω x y} (edge b)) :=
+      Finset.sum_le_sum (fun b _ => orientedPivotal_probability_le_closed G p (edge b) x y a b (hedge b))
+    _ = ∑ e ∈ Finset.univ.image edge, (law p).real (closedPivotal {ω | Conn G ω x y} e) :=
+      (Finset.sum_image (f := fun e => (law p).real (closedPivotal {ω | Conn G ω x y} e))
+        (s := Finset.univ) (fun _ _ _ _ h => hinj h)).symm
+    _ ≤ _ := by
+      apply Finset.sum_le_sum_of_subset_of_nonneg
+      · intro e he
+        obtain ⟨b,_,rfl⟩ := Finset.mem_image.mp he
+        rw [Set.Finite.mem_toFinset]
+        change a ∈ G (edge b)
+        rw [hedge b]
+        exact Sym2.mem_mk_left _ _
+      · intro _ _ _
+        exact measureReal_nonneg
+
+theorem finite_incident_sum_le_twice (G : BondGraph V E) (hloc : LocallyFinite G)
+    (A : Finset V) (f : E → ℝ) (hf : ∀ e, 0 ≤ f e) :
+    (∑ a ∈ A, ∑ e ∈ (hloc a).toFinset, f e) ≤ 2*∑ e ∈ incidentSupport G hloc A, f e := by
+  let Q := incidentSupport G hloc A
+  have heq (a : V) (ha : a ∈ A) :
+      (∑ e ∈ (hloc a).toFinset, f e) = ∑ e ∈ Q, if a ∈ G e then f e else 0 := by
+    rw [←Finset.sum_filter]
+    apply Finset.sum_congr _ (fun _ _ => rfl)
+    ext e
+    simp only [Set.Finite.mem_toFinset,Finset.mem_filter]
+    exact ⟨fun he => ⟨(mem_incidentSupport G hloc A e).mpr ⟨a,ha,he⟩,he⟩,fun h => h.2⟩
+  calc
+    _ = ∑ a ∈ A, ∑ e ∈ Q, if a ∈ G e then f e else 0 := Finset.sum_congr rfl heq
+    _ = ∑ e ∈ Q, ∑ a ∈ A, if a ∈ G e then f e else 0 := Finset.sum_comm
+    _ ≤ ∑ e ∈ Q, 2*f e := by
+      apply Finset.sum_le_sum
+      intro e _
+      have hsub : A.filter (fun a => a ∈ G e) ⊆ (G e).toFinset := by
+        intro a ha
+        exact Sym2.mem_toFinset.mpr (Finset.mem_filter.mp ha).2
+      rw [←Finset.sum_filter,Finset.sum_const, nsmul_eq_mul]
+      have hc : (A.filter (fun a => a ∈ G e)).card ≤ 2 := (Finset.card_le_card hsub).trans (by
+        rw [Sym2.card_toFinset]; split_ifs <;> omega)
+      exact mul_le_mul_of_nonneg_right (by exact_mod_cast hc) (hf e)
+    _ = _ := (Finset.mul_sum _ _ _).symm
+
+theorem finite_oriented_pivotal_sum_le [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (p : unitInterval) (x y : V) (A : Finset V) :
+    (∑ a ∈ A, ∑ b ∈ neighbors G hloc a, (law p).real (orientedPivotalEvent G x y a b)) ≤
+      2*∑ e ∈ incidentSupport G hloc A, (law p).real (closedPivotal {ω | Conn G ω x y} e) := by
+  exact (Finset.sum_le_sum (fun a _ => finite_neighbor_pivotal_le_incident G hloc p x y a)).trans
+    (finite_incident_sum_le_twice G hloc A _ (fun _ => measureReal_nonneg))
+
+end FinitePivotalComparison
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section SubcriticalSusceptibilityContinuity
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem eventually_cluster_eq_at_finite (G : BondGraph V E) (hloc : LocallyFinite G)
+    (labels : E → unitInterval) (p : unitInterval) (x : V)
+    (hneq : ∀ e, labels e ≠ p) (hfin : (cluster G (openEdges labels p) x).Finite) :
+    ∀ᶠ q in nhds p, cluster G (openEdges labels q) x = cluster G (openEdges labels p) x := by
+  let Q := incidentSupport G hloc hfin.toFinset
+  have he (e : E) : ∀ᶠ q in nhds p, (e ∈ openEdges labels q ↔ e ∈ openEdges labels p) := by
+    change ∀ᶠ q in nhds p, (labels e ≤ q ↔ labels e ≤ p)
+    by_cases hp : labels e ≤ p
+    · have hlt : labels e < p := lt_of_le_of_ne hp (hneq e)
+      filter_upwards [eventually_gt_nhds hlt] with q hq
+      exact iff_of_true hq.le hp
+    · have hlt : p < labels e := lt_of_not_ge hp
+      filter_upwards [eventually_lt_nhds hlt] with q hq
+      exact iff_of_false (not_le.mpr hq) hp
+  filter_upwards [(eventually_all_finset Q).mpr (fun e _ => he e)] with q hq
+  apply cluster_eq_of_incident_agreement G
+  intro e he
+  apply (hq e _).symm
+  obtain ⟨v,hv,hve⟩ := he
+  exact (mem_incidentSupport G hloc hfin.toFinset e).mpr ⟨v,hfin.mem_toFinset.mpr hv,hve⟩
+
+theorem label_cluster_finite_ae_subcritical [Countable E]
+    (G : BondGraph V E) (p : unitInterval) (hp : p < pc G) (x : V) :
+    ∀ᵐ labels ∂(labelLaw : Measure (E → unitInterval)), (cluster G (openEdges labels p) x).Finite := by
+  apply ae_of_ae_map (μ := (labelLaw : Measure (E → unitInterval)))
+    (p := fun ω : Set E => (cluster G ω x).Finite) (measurable_openEdges p).aemeasurable
+  rw [map_openEdges]
+  exact subcritical_cluster_finite_ae G p hp x
+
+theorem susceptibility_eq_label_integral [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (x : V) :
+    Integrable (fun labels : E → unitInterval => ((cluster G (openEdges labels p) x).ncard:ℝ)) labelLaw ∧
+    susceptibility G p x = ∫ labels : E → unitInterval, ((cluster G (openEdges labels p) x).ncard:ℝ) ∂labelLaw := by
+  obtain ⟨hi,he⟩ := susceptibility_eq_integral G hloc hqt p hp x
+  have hi' : Integrable (fun ω : Set E => ((cluster G ω x).ncard:ℝ))
+      ((labelLaw : Measure (E → unitInterval)).map (fun labels => openEdges labels p)) := by
+    rwa [map_openEdges]
+  refine ⟨hi'.comp_measurable (measurable_openEdges p),?_⟩
+  rw [he,←integral_map (measurable_openEdges p).aemeasurable hi'.aestronglyMeasurable,map_openEdges]
+
+ 
+
+theorem continuousAt_susceptibility_subcritical [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (x : V) : ContinuousAt (fun q => susceptibility G q x) p := by
+  obtain ⟨r,hpr,hr⟩ := exists_between hp
+  let F (q : unitInterval) (labels : E → unitInterval) : ℝ := ((cluster G (openEdges labels q) x).ncard:ℝ)
+  have hm (q : unitInterval) : Measurable (F q) :=
+    (measurable_of_countable (fun n : ℕ => (n:ℝ))).comp
+      ((measurable_cluster_ncard G hloc x).comp (measurable_openEdges q))
+  have hcont : ContinuousAt (fun q => ∫ labels, F q labels ∂labelLaw) p := by
+    apply continuousAt_of_dominated
+      (bound := F r) (Filter.Eventually.of_forall (fun q => (hm q).aestronglyMeasurable))
+    · filter_upwards [eventually_lt_nhds hpr] with q hqr
+      filter_upwards [label_cluster_finite_ae_subcritical G r hr x] with labels hfin
+      rw [Real.norm_of_nonneg (Nat.cast_nonneg _)]
+      dsimp only [F]
+      exact_mod_cast Set.ncard_le_ncard (show cluster G (openEdges labels q) x ⊆ cluster G (openEdges labels r) x from
+        fun y hy => conn_mono G (openEdges_mono labels hqr.le) hy) hfin
+    · exact (susceptibility_eq_label_integral G hloc hqt r hr x).1
+    · filter_upwards [ae_labels_ne (E := E) p,label_cluster_finite_ae_subcritical G p hp x] with labels hneq hfin
+      apply Filter.EventuallyEq.continuousAt
+      filter_upwards [eventually_cluster_eq_at_finite G hloc labels p x hneq hfin] with q hq
+      exact congrArg (fun S : Set V => (S.ncard:ℝ)) hq
+  apply (continuousAt_congr _).mpr hcont
+  filter_upwards [eventually_lt_nhds hp] with q hq
+  exact (susceptibility_eq_label_integral G hloc hqt q hq x).2
+
+theorem continuousAt_rootSusceptibility_subcritical [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (p : unitInterval) (hp : p < pc G) :
+    ContinuousAt (rootSusceptibility G root weight) p := by
+  exact tendsto_finsetSum Finset.univ (fun i _ => continuousAt_const.mul
+    (continuousAt_susceptibility_subcritical G hloc hqt p hp (root i)))
+
+end SubcriticalSusceptibilityContinuity
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section FiniteRootPivotalApproximation
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem summable_orientedPivotal_targets [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (x a b : V) :
+    Summable (fun y => (law p).real (orientedPivotalEvent G x y a b)) := by
+  apply (summable_twoPoint_subcritical G hloc hqt p hp b).of_nonneg_of_le
+  · exact fun _ => measureReal_nonneg
+  · intro y
+    exact measureReal_mono (fun _ h => h.2.1)
+
+noncomputable def finiteRootPivotalMass (G : BondGraph V E)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (p : unitInterval) (Q : Finset E) (F : Finset V) : ℝ :=
+  ∑ i, weight i*(∑ y ∈ F, ∑ e ∈ Q, (law p).real (closedPivotal {ω | Conn G ω (root i) y} e))
+
+theorem finiteRootPivotalMass_nonneg (G : BondGraph V E)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 ≤ weight i)
+    (p : unitInterval) (Q : Finset E) (F : Finset V) :
+    0 ≤ finiteRootPivotalMass G root weight p Q F :=
+  Finset.sum_nonneg (fun i _ => mul_nonneg (hw i)
+    (Finset.sum_nonneg (fun _ _ => Finset.sum_nonneg (fun _ _ => measureReal_nonneg))))
+
+ 
+
+theorem exists_finiteRootPivotalMass_gt
+    [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
+    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (p : unitInterval) (hp : p < pc G) (B : ℝ)
+    (hB : B < rootNeighborMoment G hloc root weight p / 2) :
+    ∃ Q : Finset E, ∃ F : Finset V, B < finiteRootPivotalMass G root weight p Q F := by
+  have hs (x : V) := pivotalTransportMass_outgoing_summable G hconn hloc hqt root weight Γ hΓ htr hcov hw hw1 c hc hmin p hp x
+  let f (a : V) := ∑ i, weight i*pivotalTransportMass G hloc p (root i) a
+  have hsf : Summable f := summable_sum (fun i _ => (hs (root i)).mul_left (weight i))
+  have htf : (∑' a, f a) = rootNeighborMoment G hloc root weight p := by
+    rw [Summable.tsum_finsetSum (fun i _ => (hs (root i)).mul_left (weight i))]
+    simp_rw [tsum_mul_left]
+    exact rootPivotalMass_eq_neighborMoment G hconn hloc hqt root weight Γ hΓ htr hcov hw hw1 c hc hmin p hp
+  have hB' : 2*B < ∑' a, f a := by rw [htf]; linarith
+  obtain ⟨A,hA⟩ := ((tendsto_order.mp hsf.hasSum).1 (2*B) hB').exists
+  let g (y : V) := ∑ i, weight i*(∑ a ∈ A, ∑ b ∈ neighbors G hloc a,
+    (law p).real (orientedPivotalEvent G (root i) y a b))
+  have hst (i : ι) (a b : V) := summable_orientedPivotal_targets G hloc hqt p hp (root i) a b
+  have hsg : Summable g := summable_sum (fun i _ =>
+    (summable_sum (fun a _ => summable_sum (fun b _ => hst i a b))).mul_left (weight i))
+  have htg : (∑' y, g y) = ∑ a ∈ A, f a := by
+    rw [Summable.tsum_finsetSum (fun i _ =>
+      (summable_sum (fun a _ => summable_sum (fun b _ => hst i a b))).mul_left (weight i))]
+    simp_rw [tsum_mul_left]
+    simp_rw [Summable.tsum_finsetSum (fun a _ => summable_sum (fun b _ => hst _ a b))]
+    simp_rw [Summable.tsum_finsetSum (fun b _ => hst _ _ b)]
+    simp only [f,Finset.mul_sum,pivotalTransportMass]
+    rw [Finset.sum_comm]
+  obtain ⟨F,hF⟩ := ((tendsto_order.mp hsg.hasSum).1 (2*B) (by rwa [htg])).exists
+  refine ⟨incidentSupport G hloc A,F,?_⟩
+  have hb : (∑ y ∈ F, g y) ≤ 2*finiteRootPivotalMass G root weight p (incidentSupport G hloc A) F := by
+    unfold g finiteRootPivotalMass
+    rw [Finset.sum_comm, Finset.mul_sum]
+    apply Finset.sum_le_sum
+    intro i _
+    rw [← Finset.mul_sum, mul_left_comm]
+    apply mul_le_mul_of_nonneg_left _ (hw i)
+    rw [Finset.mul_sum]
+    exact Finset.sum_le_sum (fun y _ => finite_oriented_pivotal_sum_le G hloc p (root i) y A)
+  linarith
+
+end FiniteRootPivotalApproximation
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section VanishingSprinkles
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def vanishingSprinkle (n : ℕ) : unitInterval :=
+  ⟨1/((n:ℝ)+1), by
+    constructor
+    · positivity
+    · apply (div_le_one (by positivity : 0 < (n:ℝ)+1)).mpr
+      linarith [show (0:ℝ) ≤ (n:ℝ) by positivity]
+  ⟩
+
+theorem vanishingSprinkle_pos (n : ℕ) : 0 < (vanishingSprinkle n : ℝ) := by
+  dsimp [vanishingSprinkle]
+  positivity
+
+theorem tendsto_vanishingSprinkle :
+    Tendsto (fun n => (vanishingSprinkle n:ℝ)) atTop (nhds 0) :=
+  tendsto_one_div_add_atTop_nhds_zero_nat
+
+theorem sprinkle_sequence_gt (p : unitInterval) (hp : (p:ℝ) < 1) (n : ℕ) :
+    p < sprinkleParameter p (vanishingSprinkle n) := by
+  change (p:ℝ) < (p:ℝ)+(1-(p:ℝ))*(vanishingSprinkle n:ℝ)
+  exact lt_add_of_pos_right _ (mul_pos (sub_pos.mpr hp) (vanishingSprinkle_pos n))
+
+theorem tendsto_sprinkle_sequence (p : unitInterval) :
+    Tendsto (fun n => sprinkleParameter p (vanishingSprinkle n)) atTop (nhds p) := by
+  apply tendsto_subtype_rng.mpr
+  change Tendsto (fun n => (p:ℝ)+(1-(p:ℝ))*(vanishingSprinkle n:ℝ)) atTop (nhds (p:ℝ))
+  convert tendsto_const_nhds.add (tendsto_const_nhds.mul tendsto_vanishingSprinkle) using 1; simp
+
+theorem tendsto_sprinkle_sequence_right (p : unitInterval) (hp : (p:ℝ) < 1) :
+    Tendsto (fun n => (sprinkleParameter p (vanishingSprinkle n):ℝ)) atTop (nhdsWithin (p:ℝ) (Set.Ioi (p:ℝ))) := by
+  apply tendsto_nhdsWithin_iff.mpr
+  exact ⟨continuous_subtype_val.continuousAt.tendsto.comp (tendsto_sprinkle_sequence p),
+    Eventually.of_forall (sprinkle_sequence_gt p hp)⟩
+
+ 
+
+theorem rootSusceptibility_sequence_slope_lower
+    [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 ≤ weight i)
+    (p : unitInterval) (hp : p < pc G) (Q : Finset E) (F : Finset V) (d : ℝ)
+    (hd : d < finiteRootPivotalMass G root weight p Q F) :
+    ∀ᶠ n in atTop,
+      ((sprinkleParameter p (vanishingSprinkle n):ℝ)-(p:ℝ))*d ≤
+      rootSusceptibility G root weight (sprinkleParameter p (vanishingSprinkle n)) -
+      rootSusceptibility G root weight p := by
+  have ht : Tendsto (fun n => (1-(vanishingSprinkle n:ℝ))^(Q.card-1) *
+      finiteRootPivotalMass G root weight p Q F) atTop
+      (nhds (finiteRootPivotalMass G root weight p Q F)) := by
+    convert ((tendsto_const_nhds.sub tendsto_vanishingSprinkle).pow (Q.card-1)).mul tendsto_const_nhds using 1; simp
+  have hnn := finiteRootPivotalMass_nonneg G root weight hw p Q F
+  filter_upwards [(tendsto_sprinkle_sequence p).eventually (eventually_lt_nhds hp),
+      ht.eventually (eventually_gt_nhds hd)] with n hn hdn
+  have hgain := rootSusceptibility_sprinkling_gain G hloc hqt root weight hw
+    p (vanishingSprinkle n) hn Q F
+  change (vanishingSprinkle n:ℝ)*(1-(vanishingSprinkle n:ℝ))^(Q.card-1)*
+    finiteRootPivotalMass G root weight p Q F ≤ _ at hgain
+  by_cases hd0 : 0 ≤ d
+  · have hdp : (1-(p:ℝ))*d ≤ (1-(vanishingSprinkle n:ℝ))^(Q.card-1)*
+        finiteRootPivotalMass G root weight p Q F := by
+      calc
+        _ ≤ d := by nlinarith [p.2.1]
+        _ ≤ _ := hdn.le
+    have hm := mul_le_mul_of_nonneg_left hdp (vanishingSprinkle n).2.1
+    calc
+      _ = (vanishingSprinkle n:ℝ)*((1-(p:ℝ))*d) := by
+        change ((p:ℝ)+(1-(p:ℝ))*(vanishingSprinkle n:ℝ)-(p:ℝ))*d = _
+        ring
+      _ ≤ (vanishingSprinkle n:ℝ)*((1-(vanishingSprinkle n:ℝ))^(Q.card-1)*
+          finiteRootPivotalMass G root weight p Q F) := hm
+      _ ≤ _ := by simpa only [mul_assoc] using hgain
+  · have hmono : 0 ≤ rootSusceptibility G root weight (sprinkleParameter p (vanishingSprinkle n)) - rootSusceptibility G root weight p :=
+      (mul_nonneg (mul_nonneg (vanishingSprinkle n).2.1 (pow_nonneg (sub_nonneg.mpr (vanishingSprinkle n).2.2) _)) hnn).trans hgain
+    have hdp : ((sprinkleParameter p (vanishingSprinkle n):ℝ)-(p:ℝ))*d ≤ 0 :=
+      mul_nonpos_of_nonneg_of_nonpos (sub_nonneg.mpr (sprinkleParameter_ge_left _ _)) (le_of_not_ge hd0)
+    linarith
+
+end VanishingSprinkles
+end Percolation
+
+ 
+namespace Percolation
+open Filter
+section NegativePowerRightSlopes
+
+ 
+
+theorem frequently_rpow_slope_le_of_sequence
+    (f : ℝ → ℝ) (x d e : ℝ) (hx : 0 < f x) (hd : 0 < d) (he : e < 0)
+    (z : ℕ → ℝ) (hz : Tendsto z atTop (nhdsWithin x (Set.Ioi x)))
+    (hgain : ∀ᶠ n in atTop, (z n-x)*d ≤ f (z n)-f x) :
+    ∀ r, e*(f x)^(e-1)*d < r →
+      ∃ᶠ y in nhdsWithin x (Set.Ioi x), slope (fun t => (f t)^e) x y < r := by
+  let A (t : ℝ) := f x+(t-x)*d
+  have hA : HasDerivAt A d x := by
+    simpa only [zero_add, one_mul] using! (hasDerivAt_const x (f x)).add (((hasDerivAt_id x).sub_const x).mul_const d)
+  have hAx : A x = f x := by simp [A]
+  have hR : HasDerivAt (fun t => (A t)^e) (e*(f x)^(e-1)*d) x := by
+    simpa only [hAx] using! (Real.hasDerivAt_rpow_const (p := e) (Or.inl (show A x ≠ 0 by rw [hAx]; exact hx.ne'))).comp x hA
+  intro r hr
+  have hlim := hR.tendsto_slope.comp (hz.mono_right (nhdsWithin_mono x (show Set.Ioi x ⊆ {x}ᶜ from fun y hy => by simpa using ne_of_gt hy)))
+  have hright : ∀ᶠ n in atTop, x < z n := (tendsto_nhdsWithin_iff.mp hz).2
+  apply hz.frequently
+  apply Filter.Eventually.frequently
+  filter_upwards [hgain, hright, hlim.eventually (eventually_lt_nhds hr)] with n hn hzn hnr
+  have hApos : 0 < A (z n) := by dsimp [A]; positivity
+  have hAr : A (z n) ≤ f (z n) := by dsimp [A]; linarith
+  have hp := Real.rpow_le_rpow_of_nonpos hApos hAr he.le
+  apply lt_of_le_of_lt _ hnr
+  simp only [Function.comp_apply, slope_def_field, hAx]
+  exact div_le_div_of_nonneg_right (sub_le_sub_right hp _) (sub_pos.mpr hzn).le
+
+end NegativePowerRightSlopes
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section ActualSusceptibilityRightSlope
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def realRootSusceptibility (G : BondGraph V E)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (t : ℝ) : ℝ :=
+  rootSusceptibility G root weight (Set.projIcc 0 1 zero_le_one t)
+
+@[simp] theorem realRootSusceptibility_coe (G : BondGraph V E)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (p : unitInterval) :
+    realRootSusceptibility G root weight (p:ℝ) = rootSusceptibility G root weight p := by
+  simp only [realRootSusceptibility, Set.projIcc_val]
+
+theorem rootSusceptibility_one_le [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
+    (p : unitInterval) (hp : p < pc G) : 1 ≤ rootSusceptibility G root weight p := by
+  calc
+    1 = ∑ i, weight i*1 := by simp only [mul_one, hw1]
+    _ ≤ _ := Finset.sum_le_sum (fun i _ => mul_le_mul_of_nonneg_left
+      (susceptibility_one_le G hloc hqt p hp (root i)) (hw i))
+
+theorem continuousAt_realRootSusceptibility [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (p : unitInterval) (hp : p < pc G) :
+    ContinuousAt (realRootSusceptibility G root weight) (p:ℝ) := by
+  apply ContinuousAt.comp (f := Set.projIcc 0 1 zero_le_one)
+  · simpa only [Set.projIcc_val] using continuousAt_rootSusceptibility_subcritical G hloc hqt root weight p hp
+  · exact continuous_projIcc.continuousAt
+
+ 
+
+theorem rootSusceptibility_power_right_slope
+    [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
+    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (p : unitInterval) (hp : p < pc G) (b γ : ℝ) (hb : 0 < b) (hγ : γ < 1)
+    (hD : b*(rootSusceptibility G root weight p)^(2-γ) ≤ rootNeighborMoment G hloc root weight p) :
+    ∀ r, -(1-γ)*b/4 < r → ∃ᶠ y in nhdsWithin (p:ℝ) (Set.Ioi (p:ℝ)),
+      slope (fun t => (realRootSusceptibility G root weight t)^(γ-1)) (p:ℝ) y < r := by
+  let χ := rootSusceptibility G root weight p
+  have hχ : 0 < χ := lt_of_lt_of_le zero_lt_one (rootSusceptibility_one_le G hloc hqt root weight hw hw1 p hp)
+  let d := (b/4)*χ^(2-γ)
+  have hd : 0 < d := mul_pos (by positivity) (Real.rpow_pos_of_pos hχ _)
+  have hBd : d < rootNeighborMoment G hloc root weight p/2 := by
+    dsimp [d, χ] at *
+    nlinarith [Real.rpow_pos_of_pos hχ (2-γ)]
+  obtain ⟨Q,F,hQ⟩ := exists_finiteRootPivotalMass_gt G hconn hloc hqt root weight Γ hΓ htr hcov hw hw1 c hc hmin p hp d hBd
+  have hg := rootSusceptibility_sequence_slope_lower G hloc hqt root weight hw p hp Q F d hQ
+  have hp1 : (p:ℝ) < 1 := lt_of_lt_of_le hp (pc G).2.2
+  have hsl := frequently_rpow_slope_le_of_sequence (realRootSusceptibility G root weight)
+    (p:ℝ) d (γ-1) (by simpa only [realRootSusceptibility_coe] using hχ) hd (by linarith)
+    (fun n => (sprinkleParameter p (vanishingSprinkle n):ℝ)) (tendsto_sprinkle_sequence_right p hp1)
+    (by simpa only [realRootSusceptibility_coe] using hg)
+  have heq : (γ-1)*(realRootSusceptibility G root weight (p:ℝ))^(γ-1-1)*d = -(1-γ)*b/4 := by
+    rw [realRootSusceptibility_coe]
+    change (γ-1)*χ^(γ-1-1)*((b/4)*χ^(2-γ)) = _
+    rw [show γ-1-1 = γ-2 by ring]
+    calc
+      _ = ((γ-1)*b/4)*(χ^(γ-2)*χ^(2-γ)) := by ring
+      _ = _ := by rw [←Real.rpow_add hχ, sub_add_sub_cancel, sub_self, Real.rpow_zero]; ring
+  rwa [heq] at hsl
+
+end ActualSusceptibilityRightSlope
+end Percolation
+
+ 
+namespace Percolation
+open Filter
+section IntegratedRightSlopes
+
+ 
+
+theorem interval_decrease_of_right_slope (f : ℝ → ℝ) (a b k : ℝ)
+    (hab : a ≤ b) (hf : ContinuousOn f (Set.Icc a b))
+    (hs : ∀ x ∈ Set.Ico a b, ∀ r, -k < r →
+      ∃ᶠ y in nhdsWithin x (Set.Ioi x), slope f x y < r) :
+    k*(b-a) ≤ f a-f b := by
+  let B (t : ℝ) := f a-k*(t-a)
+  have hB : ContinuousOn B (Set.Icc a b) := by fun_prop
+  have hB' (x : ℝ) : HasDerivAt B (-k) x := by
+    simpa only [one_mul, sub_zero, mul_one, zero_sub] using!
+      (hasDerivAt_const x (f a)).sub (((hasDerivAt_id x).sub_const a).const_mul k)
+  have h := image_le_of_liminf_slope_right_le_deriv_boundary hf
+    (show f a ≤ B a by simp [B]) hB (fun x _ => (hB' x).hasDerivWithinAt) hs ⟨hab,le_rfl⟩
+  dsimp [B] at h
+  linarith
+
+ 
+
+theorem right_slope_distance_bound (f : ℝ → ℝ) (a b k : ℝ)
+    (hab : a < b) (hk : 0 < k)
+    (hpos : ∀ x ∈ Set.Ico a b, 0 ≤ f x)
+    (hc : ContinuousOn f (Set.Ico a b))
+    (hs : ∀ x ∈ Set.Ico a b, ∀ r, -k < r →
+      ∃ᶠ y in nhdsWithin x (Set.Ioi x), slope f x y < r) :
+    k*(b-a) ≤ f a := by
+  have hbound (r : ℝ) (hr : r ∈ Set.Ico a b) : k*(r-a) ≤ f a := by
+    have hsub : Set.Icc a r ⊆ Set.Ico a b := fun t ht => ⟨ht.1,lt_of_le_of_lt ht.2 hr.2⟩
+    have h := interval_decrease_of_right_slope f a r k hr.1 (hc.mono hsub)
+      (fun x hx => hs x ⟨hx.1,lt_trans hx.2 hr.2⟩)
+    linarith [hpos r hr]
+  by_contra h
+  have hd : a+f a/k < b := by
+    have hh : f a < k*(b-a) := lt_of_not_ge h
+    have hv : f a/k < b-a := (div_lt_iff₀ hk).mpr (by nlinarith)
+    linarith
+  obtain ⟨r,hr1,hr2⟩ := exists_between hd
+  have ha : a ≤ r := by
+    have hf := hpos a ⟨le_rfl,hab⟩
+    have hd0 := div_nonneg hf hk.le
+    linarith
+  have hb := hbound r ⟨ha,hr2⟩
+  have hr : f a < (r-a)*k := (div_lt_iff₀ hk).mp (by linarith : f a/k < r-a)
+  nlinarith
+
+end IntegratedRightSlopes
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section ActualIntegratedSusceptibility
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem susceptibility_bound_of_neighborMoment
+    [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
+    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (p : unitInterval) (hp : p < pc G) (b γ : ℝ) (hb : 0 < b) (hγ : γ < 1)
+    (hD : ∀ q, p ≤ q → q < pc G → b*(rootSusceptibility G root weight q)^(2-γ) ≤
+      rootNeighborMoment G hloc root weight q) :
+    rootSusceptibility G root weight p ≤
+      (((1-γ)*b/4)*((pc G:ℝ)-(p:ℝ)))^(-1/(1-γ)) := by
+  let k := (1-γ)*b/4
+  have hk : 0 < k := by dsimp [k]; positivity
+  let f (t : ℝ) := (realRootSusceptibility G root weight t)^(γ-1)
+  have hparam (t : ℝ) (ht : t ∈ Set.Ico (p:ℝ) (pc G:ℝ)) : t ∈ Set.Icc (0:ℝ) 1 :=
+    ⟨le_trans p.2.1 ht.1, le_trans ht.2.le (pc G).2.2⟩
+  have hcp (t : ℝ) (ht : t ∈ Set.Ico (p:ℝ) (pc G:ℝ)) :
+      ContinuousAt (realRootSusceptibility G root weight) t :=
+    continuousAt_realRootSusceptibility G hloc hqt root weight ⟨t,hparam t ht⟩ ht.2
+  have hpos (t : ℝ) (ht : t ∈ Set.Ico (p:ℝ) (pc G:ℝ)) :
+      0 < realRootSusceptibility G root weight t := by
+    have hh := rootSusceptibility_one_le G hloc hqt root weight hw hw1 ⟨t,hparam t ht⟩ ht.2
+    rw [←realRootSusceptibility_coe] at hh
+    exact lt_of_lt_of_le zero_lt_one hh
+  have hcont : ContinuousOn f (Set.Ico (p:ℝ) (pc G:ℝ)) := by
+    intro t ht
+    exact ((hcp t ht).rpow_const (Or.inl (hpos t ht).ne')).continuousWithinAt
+  have hs (t : ℝ) (ht : t ∈ Set.Ico (p:ℝ) (pc G:ℝ)) :=
+    rootSusceptibility_power_right_slope G hconn hloc hqt root weight Γ hΓ htr hcov hw hw1 c hc hmin
+      ⟨t,hparam t ht⟩ ht.2 b γ hb hγ (hD ⟨t,hparam t ht⟩ ht.1 ht.2)
+  have hdist : k*((pc G:ℝ)-(p:ℝ)) ≤ f (p:ℝ) := by
+    apply right_slope_distance_bound f (p:ℝ) (pc G:ℝ) k hp hk
+      (fun t ht => (Real.rpow_pos_of_pos (hpos t ht) _).le) hcont
+    intro t ht r hr
+    apply hs t ht r
+    dsimp [k] at hr
+    linarith
+  have hχ := lt_of_lt_of_le zero_lt_one (rootSusceptibility_one_le G hloc hqt root weight hw hw1 p hp)
+  have hd : 0 < k*((pc G:ℝ)-(p:ℝ)) := mul_pos hk (sub_pos.mpr hp)
+  have he : -1/(1-γ) = 1/(γ-1) := by rw [show γ-1=-(1-γ) by ring, div_neg, neg_div]
+  have he0 : 1/(γ-1) < 0 := one_div_neg.mpr (by linarith)
+  have hinv := Real.rpow_le_rpow_of_nonpos hd hdist he0.le
+  rw [he]
+  apply le_trans _ hinv
+  change rootSusceptibility G root weight p ≤
+    ((realRootSusceptibility G root weight (p:ℝ))^(γ-1))^(1/(γ-1))
+  rw [realRootSusceptibility_coe,←Real.rpow_mul hχ.le]
+  rw [mul_one_div_cancel (by linarith : γ-1 ≠ 0),Real.rpow_one]
+
+end ActualIntegratedSusceptibility
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section SusceptibilityGrowthConsequence
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem eventually_upper_interval_unit (p₀ : unitInterval) (P : unitInterval → Prop)
+    (h : ∀ᶠ p in nhds p₀, P p) :
+    ∀ᶠ p in nhds p₀, p < p₀ → ∀ q, p ≤ q → q < p₀ → P q := by
+  obtain ⟨ε,hε,hball⟩ := Metric.eventually_nhds_iff.mp h
+  apply Metric.eventually_nhds_iff.mpr
+  refine ⟨ε,hε,?_⟩
+  intro p hp hpp q hpq hq
+  apply hball
+  have hd : dist q p₀ ≤ dist p p₀ := by
+    change |(q:ℝ)-(p₀:ℝ)| ≤ |(p:ℝ)-(p₀:ℝ)|
+    rw [abs_of_nonpos (sub_nonpos.mpr (show (q:ℝ) ≤ (p₀:ℝ) from hq.le)),abs_of_nonpos (sub_nonpos.mpr (show (p:ℝ) ≤ (p₀:ℝ) from hpp.le))]
+    have hpqR : (p:ℝ) ≤ (q:ℝ) := hpq
+    linarith
+  exact lt_of_le_of_lt hd hp
+
+ 
+
+theorem susceptibility_growth_of_divergent_moment
+    [Countable V] [Countable E] [Nonempty V] [Nonempty E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hconn : Connected G)
+    (hqt : QuasiTransitive G) (hh : 0 < hV G)
+    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
+    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
+    (hp₀ : 0 < (pc G:ℝ)) (hp₁ : (pc G:ℝ) < 1)
+    (hfin : ∀ i, ∀ᵐ ω ∂law (pc G), (cluster G ω (root i)).Finite)
+    (α β ε : ℝ) (hα : 0 < α) (hαβ : α < β)
+    (hε : 0 < ε) (hε1 : ε < 1/2) (hsmall : β+(1/2+ε) < 1)
+    (hdiv : ∃ i, ¬ Integrable (fun ω => ((cluster G ω (root i)).ncard:ℝ)^α) (law (pc G))) :
+    ∀ γ : ℝ, 0 < γ → γ < 1 → ∃ C : ℝ, 0 < C ∧ ∀ᶠ p in nhds (pc G), p < pc G →
+      rootSusceptibility G root weight p ≤ C*((pc G:ℝ)-(p:ℝ))^(-1/(1-γ)) := by
+  intro γ hγ hγ1
+  obtain ⟨b,hb,hnear⟩ := neighborMoment_growth_of_divergent_moment G hloc hconn hqt hh root weight Γ hΓ htr
+    hcov hw hw1 (pc G) hp₀ hp₁ hfin α β ε hα hαβ hε hε1 hsmall hdiv γ hγ
+  let c : ℝ := ∑ i, 1/weight i
+  have hc : 0 ≤ c := Finset.sum_nonneg (fun i _ => (one_div_pos.mpr (hw i)).le)
+  have hmin (i : ι) : 1 ≤ c*weight i := by
+    apply (div_le_iff₀ (hw i)).mp
+    exact Finset.single_le_sum (fun j _ => (one_div_pos.mpr (hw j)).le) (Finset.mem_univ i)
+  let k := (1-γ)*b/4
+  have hk : 0 < k := by dsimp [k]; positivity
+  refine ⟨k^(-1/(1-γ)),Real.rpow_pos_of_pos hk _,?_⟩
+  filter_upwards [eventually_upper_interval_unit (pc G) _ hnear] with p hp
+  intro hpc
+  have hD (q : unitInterval) (hpq : p ≤ q) (hq : q < pc G) := hp hpc q hpq hq hq
+  have hs := susceptibility_bound_of_neighborMoment G hconn hloc hqt root weight Γ hΓ htr hcov
+    (fun i => (hw i).le) hw1 c hc hmin p hpc b γ hb hγ1 hD
+  rwa [Real.mul_rpow hk.le (sub_pos.mpr (show (p:ℝ) < (pc G:ℝ) from hpc)).le] at hs
+
+end SusceptibilityGrowthConsequence
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section MarkedSearchTransfer
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem exponential_transfer_of_deriv_bound (H : ℝ → ℝ) (a b K : ℝ)
+    (hH : Differentiable ℝ H) (hab : a ≤ b)
+    (hb : ∀ t ∈ Set.Icc a b, deriv H t ≤ K*H t) :
+    H b ≤ Real.exp (K*(b-a))*H a := by
+  let F (t : ℝ) := Real.exp (-K*(t-a))*H t
+  have hF (t : ℝ) : HasDerivAt F (Real.exp (-K*(t-a))*(deriv H t-K*H t)) t := by
+    have he := (((hasDerivAt_id t).sub_const a).const_mul (-K)).exp
+    convert! he.mul (hH t).hasDerivAt using 1
+    simp only [id_eq]
+    ring
+  have hanti : AntitoneOn F (Set.Icc a b) := by
+    apply antitoneOn_of_deriv_nonpos (convex_Icc a b)
+      (fun t _ => (hF t).continuousAt.continuousWithinAt)
+      (fun t _ => (hF t).differentiableAt.differentiableWithinAt)
+    intro t ht
+    rw [(hF t).deriv]
+    exact mul_nonpos_of_nonneg_of_nonpos (Real.exp_pos _).le (sub_nonpos.mpr (hb t (interior_subset ht)))
+  have h := hanti ⟨le_rfl,hab⟩ ⟨hab,le_rfl⟩ hab
+  have hm := mul_le_mul_of_nonneg_left h (Real.exp_pos (K*(b-a))).le
+  simpa only [F, sub_self,mul_zero,Real.exp_zero,one_mul,←mul_assoc,←Real.exp_add,
+    neg_mul,add_neg_cancel,Real.exp_zero,one_mul,neg_zero,add_zero] using hm
+
+ 
+
+theorem markedConnection_transfer (G : BondGraph V E) (Q : Finset E) (x : V)
+    (d : ℕ) (hd : 0 < d) (hdeg : ∀ v, (Q.filter (fun e => v ∈ G e)).card ≤ d)
+    (p q h : unitInterval) (hp : 0 < (p:ℝ)) (hpq : p ≤ q) (hq : (q:ℝ) < 1)
+    (hh : 0 < (h:ℝ)) :
+    markedConnectionPolynomial G Q x h q ≤
+      Real.exp ((Real.sqrt ((d:ℝ)/2)/((p:ℝ)*(1-(q:ℝ)))) *
+        Real.sqrt (Real.log (1/(h:ℝ))/(h:ℝ))*((q:ℝ)-(p:ℝ))) *
+      markedConnectionPolynomial G Q x h p := by
+  let H := markedConnectionPolynomial G Q x h
+  have hdiff : Differentiable ℝ H := fun t => (hasDerivAt_bernoulliPolynomial Q _ t).differentiableAt
+  apply exponential_transfer_of_deriv_bound H p q _ hdiff hpq
+  intro t ht
+  let r : unitInterval := ⟨t,⟨le_trans p.2.1 ht.1,le_trans ht.2 q.2.2⟩⟩
+  have hr0 : 0 < t := lt_of_lt_of_le hp ht.1
+  have hr1 : t < 1 := lt_of_le_of_lt ht.2 hq
+  have HB := markedConnection_derivative_bound G Q x r h hr1 hh d hd hdeg
+  have hH : 0 ≤ H t := by
+    change 0 ≤ markedConnectionPolynomial G Q x h r
+    rw [markedConnectionPolynomial_eq_tree]
+    exact ProbeTree.successMass_nonneg r h _
+  have hs : 0 ≤ Real.sqrt (Real.log (1/(h:ℝ))/(h:ℝ)) := Real.sqrt_nonneg _
+  have hden : (p:ℝ)*(1-(q:ℝ)) ≤ t*(1-t) := by
+    exact mul_le_mul ht.1 (by linarith [ht.2]) (sub_pos.mpr hq).le (by positivity)
+  have hratio : Real.sqrt ((d:ℝ)/2)/(t*(1-t)) ≤
+      Real.sqrt ((d:ℝ)/2)/((p:ℝ)*(1-(q:ℝ))) :=
+    div_le_div_of_nonneg_left (Real.sqrt_nonneg _) (mul_pos hp (sub_pos.mpr hq)) hden
+  have hle : deriv H t ≤ (Real.sqrt ((d:ℝ)/2)/(t*(1-t)))*
+      Real.sqrt (Real.log (1/(h:ℝ))/(h:ℝ))*H t := by
+    have HBr : deriv H t ≤ ((Real.sqrt ((d:ℝ)/2)/(1-t))*
+        Real.sqrt (Real.log (1/(h:ℝ))/(h:ℝ))*H t)/t := by
+      apply (le_div_iff₀ hr0).mpr
+      simpa only [mul_comm (deriv H t) t] using HB
+    convert HBr using 1
+    field_simp
+  exact hle.trans (mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hratio hs) hH)
+
+end MarkedSearchTransfer
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section MarkedComponentIdentity
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem markedConnection_component_polynomial (G : BondGraph V E)
+    (Q : Finset E) (x : V) (h p : ℝ) :
+    markedConnectionPolynomial G Q x h p =
+      bernoulliPolynomial Q (fun ω => 1-(1-h)^(finiteRootComponent G ω x).card) p := by
+  apply bernoulliPolynomial_congr
+  intro ω hω
+  let C := finiteRootComponent G ω x
+  let F := insert x (searchVertices G Q)
+  have hCF : C ⊆ F := finiteRootComponent_subset G Q F x ω hω le_rfl
+  have he (m : Finset V) : (∃ y ∈ m, Conn G (ω:Set E) x y) ↔ m ∩ C ≠ ∅ := by
+    rw [Finset.nonempty_iff_ne_empty.symm]
+    constructor
+    · rintro ⟨y,hy,hxy⟩
+      exact ⟨y,Finset.mem_inter.mpr ⟨hy,by change y ∈ (C:Set V); simpa only [C,coe_finiteRootComponent,cluster,Set.mem_ofPred_eq] using hxy⟩⟩
+    · rintro ⟨y,hy⟩
+      refine ⟨y,(Finset.mem_inter.mp hy).1,?_⟩
+      have hm : y ∈ (C:Set V) := (Finset.mem_inter.mp hy).2
+      simpa only [C,coe_finiteRootComponent,cluster,Set.mem_ofPred_eq] using hm
+  have hid : bernoulliPolynomial F (fun m => if ∃ y ∈ m, Conn G (ω:Set E) x y then 1 else 0) h =
+      1-bernoulliPolynomial F (fun m => if m ∩ C = ∅ then 1 else 0) h := by
+    calc
+      _ = bernoulliPolynomial F (fun m => 1-(if m ∩ C = ∅ then 1 else 0)) h := by
+        apply bernoulliPolynomial_congr
+        intro m _
+        by_cases hm : m ∩ C = ∅
+        · have hn : ¬∃ y ∈ m, Conn G (ω:Set E) x y := fun hc => (he m).mp hc hm
+          simp only [hm,hn,ite_false,ite_true,sub_self]
+        · have hy := (he m).mpr hm
+          simp only [hm,hy,ite_false,ite_true,sub_zero]
+      _ = _ := by rw [bernoulliPolynomial_sub,bernoulliPolynomial_const]
+  rw [hid,bernoulliPolynomial_restrict C F hCF (fun m => if m = ∅ then 1 else 0) h,
+    bernoulliPolynomial_empty_indicator]
+
+theorem one_sub_pow_one_sub_le (h : unitInterval) (n : ℕ) :
+    1-(1-(h:ℝ))^n ≤ (h:ℝ)*n := by
+  induction n with
+  | zero => simp
+  | succ n ih =>
+    rw [pow_succ,Nat.cast_succ]
+    have hm := mul_le_mul_of_nonneg_right ih (sub_nonneg.mpr h.2.2)
+    have hn : 0 ≤ (h:ℝ)^2*n := by positivity
+    nlinarith
+
+end MarkedComponentIdentity
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section MarkedConnectionLaw
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem markedConnection_integral (G : BondGraph V E) (Q : Finset E) (x : V)
+    (h p : unitInterval) :
+    markedConnectionPolynomial G Q x h p =
+      ∫ ω, (1-(1-(h:ℝ))^(cluster G (ω ∩ (Q:Set E)) x).ncard) ∂law p := by
+  rw [integral_eq_bernoulliPolynomial Q p _ (by intro ω; simp only [Set.inter_assoc,Set.inter_self]),
+    markedConnection_component_polynomial]
+  apply bernoulliPolynomial_congr
+  intro ω hω
+  rw [Set.inter_eq_left.mpr (show (ω:Set E) ⊆ Q from hω)]
+  congr 2
+  rw [←coe_finiteRootComponent,Set.ncard_coe_finset]
+
+theorem markedConnection_le_susceptibility [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (Q : Finset E) (x : V) (h p : unitInterval) (hp : p < pc G) :
+    markedConnectionPolynomial G Q x h p ≤ (h:ℝ)*susceptibility G p x := by
+  obtain ⟨hi,he⟩ := susceptibility_eq_integral G hloc hqt p hp x
+  rw [markedConnection_integral,he,←integral_const_mul]
+  apply integral_mono_ae
+  · exact integrable_finite_local Q p _ (by intro ω; simp only [Set.inter_assoc,Set.inter_self])
+  · exact hi.const_mul _
+  · filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
+    apply (one_sub_pow_one_sub_le h _).trans
+    apply mul_le_mul_of_nonneg_left _ h.2.1
+    exact_mod_cast Set.ncard_le_ncard (fun y hy => conn_mono G Set.inter_subset_left hy) hω
+
+ 
+
+theorem one_sub_pow_one_sub_ge_half (h : unitInterval) (n : ℕ)
+    (hn : 1 ≤ (h:ℝ)*n) : (1:ℝ)/2 ≤ 1-(1-(h:ℝ))^n := by
+  have hb (m : ℕ) : (1+(h:ℝ)*m)*(1-(h:ℝ))^m ≤ 1 := by
+    induction m with
+    | zero => simp
+    | succ m ih =>
+      have hp : 0 ≤ (1-(h:ℝ))^m := pow_nonneg (sub_nonneg.mpr h.2.2) _
+      have ha : (1+(h:ℝ)*(m+1))*(1-(h:ℝ)) ≤ 1+(h:ℝ)*m := by
+        have hh : 0 ≤ (h:ℝ)^2*((m:ℝ)+1) := by positivity
+        nlinarith
+      have hm := mul_le_mul_of_nonneg_right ha hp
+      rw [Nat.cast_succ,pow_succ]
+      nlinarith
+  have hp : 0 ≤ (1-(h:ℝ))^n := pow_nonneg (sub_nonneg.mpr h.2.2) _
+  have hm := mul_le_mul_of_nonneg_right hn hp
+  have hb' := hb n
+  nlinarith
+
+ 
+
+theorem clusterWindow_le_markedConnection (G : BondGraph V E) (hloc : LocallyFinite G)
+    (x : V) (lo hi : ℕ) (p h : unitInterval) (hh : 1 ≤ (h:ℝ)*lo) :
+    clusterWindowProbability G p x lo hi ≤ 2*markedConnectionPolynomial G
+      (incidentSupport G hloc (finite_reachBall hloc x hi).toFinset) x h p := by
+  let Q := incidentSupport G hloc (finite_reachBall hloc x hi).toFinset
+  let f (ω : Set E) := 1-(1-(h:ℝ))^(cluster G (ω ∩ (Q:Set E)) x).ncard
+  let A := {ω | ClusterWindow G x lo hi ω}
+  have hA : MeasurableSet A := measurableSet_clusterWindow G hloc x lo hi
+  have hf : Integrable f (law p) := integrable_finite_local Q p _
+    (by intro ω; simp only [f,Set.inter_assoc,Set.inter_self])
+  have hnon (ω : Set E) : 0 ≤ f ω := by
+    apply sub_nonneg.mpr
+    exact pow_le_one₀ (sub_nonneg.mpr h.2.2) (by linarith [h.2.1])
+  have hbound (ω : Set E) : A.indicator (fun _ => (1:ℝ)/2) ω ≤ f ω := by
+    by_cases hω : ω ∈ A
+    · rw [Set.indicator_of_mem hω]
+      have hs : cluster G ω x ⊆ (finite_reachBall hloc x hi).toFinset := by
+        simpa only [Set.Finite.coe_toFinset] using finite_cluster_subset_reachBall G ω x hω.1 hi hω.2.2.le
+      have he := cluster_restrict_incident_of_full G hloc _ ω x hs
+      change 1/2 ≤ 1-(1-(h:ℝ))^(cluster G (ω ∩ (Q:Set E)) x).ncard
+      rw [he]
+      apply one_sub_pow_one_sub_ge_half
+      exact hh.trans (mul_le_mul_of_nonneg_left (by exact_mod_cast hω.2.1) h.2.1)
+    · rw [Set.indicator_of_notMem hω]
+      exact hnon ω
+  have hib := integral_mono ((integrable_const _).indicator hA) hf hbound
+  rw [integral_indicator_const _ hA] at hib
+  have he := markedConnection_integral G Q x h p
+  change markedConnectionPolynomial G Q x h p = ∫ ω, f ω ∂law p at he
+  rw [←he] at hib
+  change (law p).real A ≤ 2*markedConnectionPolynomial G Q x h p
+  simp only [smul_eq_mul] at hib
+  linarith
+
+end MarkedConnectionLaw
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Criticality/UnimodularMain.lean b/lean/OAI/Probability/UnimodularPercolation/Criticality/UnimodularMain.lean
index 84463baeb074b4e398f1361ac190a4df93269df2..d33df847fabae53139c31d4d053d341231bac16b 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Criticality/UnimodularMain.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Criticality/UnimodularMain.lean
@@ -1,28 +1,28 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Main
-import OAI.Probability.UnimodularPercolation.Criticality.FullOperator
-
-namespace OAI
-
-/-! Threshold separation and simultaneous nonuniqueness under unimodularity. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-theorem main_unimodular [Infinite V] (G : BondGraph V E)
-    (hinj : Function.Injective G)
-    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (hh : 0 < hV G) (x₀ : V)
-    (hu : FullAutUnimodular G hinj hloc hconn x₀) : MainConclusion G := by
-  let : TopologicalSpace V := ⊥
-  let : DiscreteTopology V := ⟨rfl⟩
-  apply mainConclusion_of_critical_bounded G hconn hloc hqt
-  exact critical_bounded_of_graphUnimodular G hinj hconn hloc hqt hh x₀ hu
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Main
+import OAI.Probability.UnimodularPercolation.Criticality.FullOperator
+
+namespace OAI
+
+/-! Threshold separation and simultaneous nonuniqueness under unimodularity. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+theorem main_unimodular [Infinite V] (G : BondGraph V E)
+    (hinj : Function.Injective G)
+    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (hh : 0 < hV G) (x₀ : V)
+    (hu : FullAutUnimodular G hinj hloc hconn x₀) : MainConclusion G := by
+  let : TopologicalSpace V := ⊥
+  let : DiscreteTopology V := ⟨rfl⟩
+  apply mainConclusion_of_critical_bounded G hconn hloc hqt
+  exact critical_bounded_of_graphUnimodular G hinj hconn hloc hqt hh x₀ hu
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Deformation/AveragedDecrease.lean b/lean/OAI/Probability/UnimodularPercolation/Deformation/AveragedDecrease.lean
index 8e0cd6756c4f11024d84dbb4d0e0a4dee52fd5ee..196c89bea2201062e6000c89fc5bd8a2edf98a27 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Deformation/AveragedDecrease.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Deformation/AveragedDecrease.lean
@@ -1,1209 +1,1209 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Deformation.Calibration
-
-namespace OAI
-
-/-! Averaged Decrease for bond percolation. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section CountedCovariance
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem preimage_diff_singleton (b : E ≃ E) (ω : Set E) (e : E) :
-    (b ⁻¹' ω) \ {e} = b ⁻¹' (ω \ {b e}) := by
-  ext f
-  simp only [Set.mem_sdiff,Set.mem_preimage,Set.mem_singleton_iff,b.injective.eq_iff]
-
-theorem corridorCounted_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (S B : Set E) (D : Set V) (ω : Set E) (x y : V) :
-    corridorCounted G (b ⁻¹' S) (b ⁻¹' B) (a ⁻¹' D) (b ⁻¹' ω) x y ↔
-      corridorCounted G S B D ω (a x) (a y) := by
-  have he : (b ⁻¹' ω) ∩ internalEdges G (a ⁻¹' D) = b ⁻¹' (ω ∩ internalEdges G D) := by
-    rw [internalEdges_aut_preimage G hab,Set.preimage_inter]
-  simp only [corridorCounted,he,conn_aut_iff G hab,preimage_diff_singleton,
-    Set.mem_preimage]
-  apply and_congr_right
-  intro _
-  constructor
-  · rintro ⟨hp,hB⟩
-    constructor
-    · intro e he
-      obtain ⟨f,rfl⟩ := b.surjective e
-      exact hp f he
-    · apply Set.disjoint_left.mpr
-      intro e he hw
-      obtain ⟨f,rfl⟩ := b.surjective e
-      exact Set.disjoint_left.mp hB he hw
-  · rintro ⟨hp,hB⟩
-    exact ⟨fun e he => hp (b e) he,Set.disjoint_left.mpr (fun e he hw => Set.disjoint_left.mp hB he hw)⟩
-
-namespace RawCorridor
-variable {G : BondGraph V E} {k : ℕ}
-
-theorem aut_counted {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b)
-    (w : RawCorridor G k) (D : Set V) (ω : Set E) (x y : V) :
-    w.counted (a ⁻¹' D) (b ⁻¹' ω) x y ↔ (autEquiv hab w).counted D ω (a x) (a y) := by
-  have hr : w.retained = b ⁻¹' (autEquiv hab w).retained := by
-    ext e
-    exact (aut_retained hab w e).symm
-  have hb : w.blockers = b ⁻¹' (autEquiv hab w).blockers := by
-    ext e
-    exact (aut_blockers hab w e).symm
-  rw [counted_iff,counted_iff,aut_path_length,hr,hb,corridorCounted_aut G hab]
-
-end RawCorridor
-
-theorem corridorPenaltyTerm_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (P : V → V → ℝ) (hP : ∀ x y, P (a x) (a y) = P x y)
-    (p : ℝ) {k : ℕ} (w : RawCorridor G k) (D : Set V) (ω : Set E) (x y : V) :
-    corridorPenaltyTerm G P p (a ⁻¹' D) (b ⁻¹' ω) x y w =
-      corridorPenaltyTerm G P p D ω (a x) (a y) (RawCorridor.autEquiv hab w) := by
-  simp only [corridorPenaltyTerm,w.aut_counted hab D ω x y,RawCorridor.aut_weight hab P hP]
-
-theorem corridorPenalty_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (P : V → V → ℝ) (hP : ∀ x y, P (a x) (a y) = P x y)
-    (p : ℝ) (k : ℕ) (D : Set V) (ω : Set E) (x y : V) :
-    corridorPenalty G P p k (a ⁻¹' D) (b ⁻¹' ω) x y =
-      corridorPenalty G P p k D ω (a x) (a y) := by
-  unfold corridorPenalty
-  rw [←(RawCorridor.autEquiv hab).tsum_eq (corridorPenaltyTerm G P p D ω (a x) (a y))]
-  exact tsum_congr (fun w => corridorPenaltyTerm_aut G hab P hP p w D ω x y)
-
-end CountedCovariance
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section DeformationCovariance
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem deformedConnection_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (P : V → V → ℝ) (hP : ∀ x y, P (a x) (a y) = P x y)
-    (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x y : V) :
-    deformedConnection G P p k t (a ⁻¹' D) (b ⁻¹' ω) x y =
-      deformedConnection G P p k t D ω (a x) (a y) := by
-  simp only [deformedConnection,corridorPenalty_aut G hab P hP,Set.mem_preimage,
-    internalEdges_aut_preimage G hab,←Set.preimage_inter,conn_aut_iff G hab]
-
-theorem deformedKernel_aut [Countable V] [Countable E]
-    (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (P : V → V → ℝ) (hP : ∀ x y, P (a x) (a y) = P x y)
-    (p : unitInterval) (k : ℕ) (t : ℝ) (D : Set V) (x y : V) :
-    deformedKernel G P p k t D (a x) (a y) = deformedKernel G P p k t (a ⁻¹' D) x y := by
-  have h := integral_map (μ := law p) (measurable_set_preimage b).aemeasurable
-    (measurable_deformedConnection G P p k t (a ⁻¹' D) x y).aestronglyMeasurable
-  rw [law_map_preimage_equiv p b] at h
-  simpa only [deformedKernel,deformedConnection_aut G hab P hP] using h.symm
-
-theorem deformedSusceptibility_aut [Countable V] [Countable E]
-    (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (P : V → V → ℝ) (hP : ∀ x y, P (a x) (a y) = P x y)
-    (p : unitInterval) (k : ℕ) (t : ℝ) (D : Set V) (x : V) :
-    deformedSusceptibility G P p k t D (a x) = deformedSusceptibility G P p k t (a ⁻¹' D) x := by
-  unfold deformedSusceptibility
-  rw [←a.tsum_eq (deformedKernel G P p k t D (a x))]
-  exact tsum_congr (fun y => deformedKernel_aut G hab P hP p k t D x y)
-
- 
-
-noncomputable def corridorDerivativeContribution (G : BondGraph V E) (P : V → V → ℝ)
-    (p : unitInterval) {k : ℕ} (t : ℝ) (w : RawCorridor G k) (x y : V) : ℝ :=
-  ∫ ω, corridorPenaltyTerm G P p Set.univ ω x y w *
-    deformedConnection G P p k t Set.univ ω x y ∂law p
-
-theorem corridorDerivativeContribution_nonneg (G : BondGraph V E) (P : V → V → ℝ)
-    (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) {k : ℕ} (t : ℝ) (w : RawCorridor G k) (x y : V) :
-    0 ≤ corridorDerivativeContribution G P p t w x y :=
-  integral_corridorDerivativeTerm_nonneg G P hP p k t w x y
-
-theorem corridorDerivativeContribution_aut [Countable V] [Countable E]
-    (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (P : V → V → ℝ) (hP : ∀ x y, P (a x) (a y) = P x y)
-    (p : unitInterval) {k : ℕ} (t : ℝ) (w : RawCorridor G k) (x y : V) :
-    corridorDerivativeContribution G P p t (RawCorridor.autEquiv hab w) (a x) (a y) =
-      corridorDerivativeContribution G P p t w x y := by
-  have hm := ((measurable_corridorPenaltyTerm G P p (a ⁻¹' Set.univ) x y w).mul
-    (measurable_deformedConnection G P p k t (a ⁻¹' Set.univ) x y))
-  have h := integral_map (μ := law p) (measurable_set_preimage b).aemeasurable hm.aestronglyMeasurable
-  rw [law_map_preimage_equiv p b] at h
-  have hc (ω : Set E) := corridorPenaltyTerm_aut G hab P hP p w Set.univ ω x y
-  have hd (ω : Set E) := deformedConnection_aut G hab P hP p k t Set.univ ω x y
-  simp only [Set.preimage_univ] at hc hd
-  simpa only [Pi.mul_apply,corridorDerivativeContribution,Set.preimage_univ,hc,hd] using h.symm
-
-end DeformationCovariance
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section DerivativeTonelli
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem derivative_field_summable (G : BondGraph V E) (hloc : LocallyFinite G)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval)
-    (k : ℕ) (t : ℝ) (ω : Set E) (x : V) (hf : (cluster G ω x).Finite) :
-    Summable (fun yw : V × RawCorridor G k => corridorPenaltyTerm G P p Set.univ ω x yw.1 yw.2 *
-      deformedConnection G P p k t Set.univ ω x yw.1) := by
-  have hs (y : V) : Summable (fun w : RawCorridor G k =>
-      corridorPenaltyTerm G P p Set.univ ω x y w * deformedConnection G P p k t Set.univ ω x y) :=
-    (corridorPenalty_summable G hloc P p k Set.univ ω x y).mul_right
-      (deformedConnection G P p k t Set.univ ω x y)
-  have hr : Summable (fun y => corridorPenalty G P p k Set.univ ω x y *
-      deformedConnection G P p k t Set.univ ω x y) := by
-    apply summable_of_hasFiniteSupport
-    apply hf.subset
-    intro y hy
-    by_contra hn
-    apply hy
-    change corridorPenalty G P p k Set.univ ω x y * deformedConnection G P p k t Set.univ ω x y = 0
-    rw [deformedConnection_eq_zero_of_not_conn G P p k t Set.univ ω x y hn,mul_zero]
-  have hn : 0 ≤ (fun yw : V × RawCorridor G k =>
-      corridorPenaltyTerm G P p Set.univ ω x yw.1 yw.2 * deformedConnection G P p k t Set.univ ω x yw.1) :=
-    fun yw => mul_nonneg (corridorPenaltyTerm_nonneg G P hP p Set.univ ω x yw.1 yw.2)
-      (deformedConnection_nonneg G P p k t Set.univ ω x yw.1)
-  apply (summable_prod_of_nonneg hn).mpr
-  refine ⟨hs,?_⟩
-  simpa only [tsum_mul_right,corridorPenalty] using hr
-
-theorem derivative_field_sum (G : BondGraph V E) (hloc : LocallyFinite G)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval)
-    (k : ℕ) (t : ℝ) (ω : Set E) (x : V) (hf : (cluster G ω x).Finite) :
-    (∑' yw : V × RawCorridor G k, corridorPenaltyTerm G P p Set.univ ω x yw.1 yw.2 *
-      deformedConnection G P p k t Set.univ ω x yw.1) =
-        deformedDerivativeMass G P p k t Set.univ ω x := by
-  rw [(derivative_field_summable G hloc P hP p k t ω x hf).tsum_prod]
-  simp only [tsum_mul_right,corridorPenalty,deformedDerivativeMass]
-
- 
-
-theorem derivative_contribution_sum [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
-    (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y) (p : unitInterval)
-    (hp0 : 0 < (p:ℝ)) (hp : p < pc G) (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (x : V) :
-    Summable (fun yw : V × RawCorridor G k => corridorDerivativeContribution G (walkKernel G hloc a) p t yw.2 x yw.1) ∧
-    (∑' yw : V × RawCorridor G k, corridorDerivativeContribution G (walkKernel G hloc a) p t yw.2 x yw.1) =
-      ∫ ω, deformedDerivativeMass G (walkKernel G hloc a) p k t Set.univ ω x ∂law p := by
-  let F := fun yw : V × RawCorridor G k => fun ω =>
-    corridorPenaltyTerm G (walkKernel G hloc a) p Set.univ ω x yw.1 yw.2 *
-      deformedConnection G (walkKernel G hloc a) p k t Set.univ ω x yw.1
-  have hs : ∀ᵐ ω ∂law p, Summable (fun yw => F yw ω) := by
-    filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
-    exact derivative_field_summable G hloc _ hP p k t ω x hω
-  have he : (fun ω => ∑' yw, F yw ω) =ᵐ[law p]
-      fun ω => deformedDerivativeMass G (walkKernel G hloc a) p k t Set.univ ω x := by
-    filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
-    exact derivative_field_sum G hloc _ hP p k t ω x hω
-  have hh := real_tonelli_of_integrable_sum (law p) F
-    (fun yw => integrable_corridorDerivativeTerm G _ hP p k t ht Set.univ x yw.1 yw.2)
-    (fun yw ω => mul_nonneg (corridorPenaltyTerm_nonneg G _ hP p Set.univ ω x yw.1 yw.2)
-      (deformedConnection_nonneg G _ p k t Set.univ ω x yw.1)) hs
-    ((integrable_deformedDerivativeMass G hloc hqt d hd a hP p hp0 hp k t ht Set.univ x).congr he.symm)
-  exact ⟨hh.1,hh.2.trans (integral_congr_ae he)⟩
-
-end DerivativeTonelli
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section DerivativeTransport
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem raw_from_sum_eq_masked (G : BondGraph V E) (hloc : LocallyFinite G)
-    (k : ℕ) (u : V) (f : RawCorridor G k → ℝ) :
-    (∑' w : RawFrom (openSimpleGraph G Set.univ) k u, f ⟨u,w⟩) =
-      ∑' w : RawCorridor G k, if w.base = u then f w else 0 := by
-  have hs := summable_raw_selection G hloc k {u} (Set.finite_singleton u)
-    (fun w : RawCorridor G k => w.base = u) (fun _ h => h) f
-  rw [hs.tsum_sigma]
-  have he (x : V) : (∑' w : RawFrom (openSimpleGraph G Set.univ) k x,
-      if (RawCorridor.base ⟨x,w⟩) = u then f ⟨x,w⟩ else 0) =
-      if x = u then (∑' w : RawFrom (openSimpleGraph G Set.univ) k u, f ⟨u,w⟩) else 0 := by
-    by_cases hx : x = u
-    · subst x; simp only [RawCorridor.base,ite_true]
-    · simp only [RawCorridor.base,hx,ite_false,tsum_zero]
-  simp only [he,tsum_ite_eq]
-
-noncomputable def derivativeTransportMass (G : BondGraph V E) (P : V → V → ℝ)
-    (p : unitInterval) (k : ℕ) (t : ℝ) (x u : V) : ℝ :=
-  ∑' w : RawFrom (openSimpleGraph G Set.univ) k u,
-    ∑' y, corridorDerivativeContribution G P p t ⟨u,w⟩ x y
-
-theorem derivativeTransportMass_nonneg (G : BondGraph V E) (P : V → V → ℝ)
-    (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (k : ℕ) (t : ℝ) (x u : V) :
-    0 ≤ derivativeTransportMass G P p k t x u :=
-  tsum_nonneg (fun w => tsum_nonneg (fun y => corridorDerivativeContribution_nonneg G P hP p t ⟨u,w⟩ x y))
-
-theorem derivativeTransportMass_aut [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (P : V → V → ℝ) (hP : ∀ x y, P (a x) (a y) = P x y)
-    (p : unitInterval) (k : ℕ) (t : ℝ) (x u : V) :
-    derivativeTransportMass G P p k t (a x) (a u) = derivativeTransportMass G P p k t x u := by
-  unfold derivativeTransportMass
-  rw [raw_from_sum_eq_masked G hloc k (a u) (fun w => ∑' y, corridorDerivativeContribution G P p t w (a x) y),
-    raw_from_sum_eq_masked G hloc k u (fun w => ∑' y, corridorDerivativeContribution G P p t w x y)]
-  rw [←(RawCorridor.autEquiv hab).tsum_eq]
-  apply tsum_congr
-  intro w
-  simp only [RawCorridor.aut_base,a.injective.eq_iff]
-  by_cases hw : w.base = u
-  · simp only [hw,ite_true]
-    rw [←a.tsum_eq]
-    exact tsum_congr (fun y => corridorDerivativeContribution_aut G hab P hP p t w x y)
-  · simp only [hw,ite_false]
-
-end DerivativeTransport
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section DerivativeTransportSums
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Countable V] [Countable E] [Nonempty V]
-
-theorem summable_swapped_real {α β : Type*} (f : α → β → ℝ)
-    (hs : Summable (fun ab : α × β => f ab.1 ab.2)) :
-    Summable (fun ba : β × α => f ba.2 ba.1) := by
-  exact hs.comp_injective (Equiv.prodComm β α).injective
-
-theorem derivativeTransportMass_row (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
-    (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y) (p : unitInterval)
-    (hp0 : 0 < (p:ℝ)) (hp : p < pc G) (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (x : V) :
-    Summable (derivativeTransportMass G (walkKernel G hloc a) p k t x) ∧
-    (∑' u, derivativeTransportMass G (walkKernel G hloc a) p k t x u) =
-      ∫ ω, deformedDerivativeMass G (walkKernel G hloc a) p k t Set.univ ω x ∂law p := by
-  obtain ⟨hs,he⟩ := derivative_contribution_sum G hloc hqt d hd a hP p hp0 hp k t ht x
-  have hs' : Summable (fun wy : RawCorridor G k × V =>
-      corridorDerivativeContribution G (walkKernel G hloc a) p t wy.1 x wy.2) := by
-    exact summable_swapped_real (fun y w => corridorDerivativeContribution G (walkKernel G hloc a) p t w x y) hs
-  have hr := hs'.prod.sigma
-  refine ⟨hr,?_⟩
-  change (∑' u, ∑' w : RawFrom (openSimpleGraph G Set.univ) k u, ∑' y,
-    corridorDerivativeContribution G (walkKernel G hloc a) p t ⟨u,w⟩ x y) = _
-  rw [←hs'.prod.tsum_sigma,←hs'.tsum_prod]
-  exact ((Equiv.prodComm (RawCorridor G k) V).tsum_eq
-    (fun yw : V × RawCorridor G k => corridorDerivativeContribution G (walkKernel G hloc a) p t yw.2 x yw.1)).trans he
-
-theorem derivativeTransportMass_column (G : BondGraph V E) (hG : Function.Injective G)
-    (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y)
-    (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp : p < pc G)
-    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (u : V) :
-    Summable (fun x => derivativeTransportMass G P p k t x u) ∧
-    (∑' x, derivativeTransportMass G P p k t x u) =
-      ∑' w : RawFrom (openSimpleGraph G Set.univ) k u,
-        ∑' xy : V × V, corridorDerivativeContribution G P p t ⟨u,w⟩ xy.1 xy.2 := by
-  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
-  let := Fintype.ofFinite (RawFrom (openSimpleGraph G Set.univ) k u)
-  have hs (w : RawFrom (openSimpleGraph G Set.univ) k u) :
-      Summable (fun xy : V × V => corridorDerivativeContribution G P p t ⟨u,w⟩ xy.1 xy.2) :=
-    summable_corridorDerivativeTerm_pairs G hG hloc hqt P hP p hp0 hp t ht ⟨u,w⟩
-  have hh : Summable (fun x => ∑ w : RawFrom (openSimpleGraph G Set.univ) k u,
-      ∑' y, corridorDerivativeContribution G P p t ⟨u,w⟩ x y) :=
-    summable_sum (fun w _ => (hs w).prod)
-  refine ⟨by simpa only [derivativeTransportMass,tsum_fintype] using hh,?_⟩
-  simp only [derivativeTransportMass,tsum_fintype]
-  rw [Summable.tsum_finsetSum (fun w _ => (hs w).prod)]
-  exact Finset.sum_congr rfl (fun w _ => (hs w).tsum_prod.symm)
-
-end DerivativeTransportSums
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section AveragedDerivative
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Countable V] [Countable E] [Nonempty V]
-variable {ι : Type*} [Fintype ι]
-
- 
-
-theorem averaged_derivative_transport (G : BondGraph V E) (hG : Function.Injective G)
-    (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
-    (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y) (p : unitInterval)
-    (hp0 : 0 < (p:ℝ)) (hp : p < pc G) (k : ℕ) (t : ℝ) (ht : 0 ≤ t) :
-    (∑ i, weight i * ∫ ω, deformedDerivativeMass G (walkKernel G hloc a) p k t Set.univ ω (root i) ∂law p) =
-      ∑ i, weight i * ∑' w : RawFrom (openSimpleGraph G Set.univ) k (root i),
-        ∑' xy : V × V, corridorDerivativeContribution G (walkKernel G hloc a) p t ⟨root i,w⟩ xy.1 xy.2 := by
-  have hh := htr.real root weight Γ (derivativeTransportMass G (walkKernel G hloc a) p k t)
-    (fun g hg x u => by
-      obtain ⟨b,hb⟩ := hΓ hg
-      exact derivativeTransportMass_aut G hloc hb _ (walkKernel_covariant G hloc a hb) p k t x u)
-    (fun x => (derivativeTransportMass_row G hloc hqt d hd a hP p hp0 hp k t ht x).1)
-    (fun u => (derivativeTransportMass_column G hG hloc hqt _ hP p hp0 hp k t ht u).1)
-  simpa only [(derivativeTransportMass_row G hloc hqt d hd a hP p hp0 hp k t ht _).2,
-    (derivativeTransportMass_column G hG hloc hqt _ hP p hp0 hp k t ht _).2] using hh
-
-noncomputable def averagedDeformedSusceptibility (G : BondGraph V E) (P : V → V → ℝ)
-    (p : unitInterval) (k : ℕ) (root : ι → V) (weight : ι → ℝ) (t : ℝ) : ℝ :=
-  ∑ i, weight i * deformedSusceptibility G P p k t Set.univ (root i)
-
-theorem hasDerivAt_averagedDeformedSusceptibility (G : BondGraph V E)
-    (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (root : ι → V) (weight : ι → ℝ)
-    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
-    (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y) (p : unitInterval)
-    (hp0 : 0 < (p:ℝ)) (hp : p < pc G) (k : ℕ) (t : ℝ) (ht : 0 < t) :
-    HasDerivAt (averagedDeformedSusceptibility G (walkKernel G hloc a) p k root weight)
-      (-(∑ i, weight i * ∫ ω, deformedDerivativeMass G (walkKernel G hloc a) p k t Set.univ ω (root i) ∂law p)) t := by
-  have he : -(∑ i, weight i * ∫ ω, deformedDerivativeMass G (walkKernel G hloc a) p k t Set.univ ω (root i) ∂law p) =
-      ∑ i, weight i * (-(∫ ω, deformedDerivativeMass G (walkKernel G hloc a) p k t Set.univ ω (root i) ∂law p)) := by
-    simp only [mul_neg,Finset.sum_neg_distrib]
-  rw [he]
-  exact HasDerivAt.fun_sum (u := Finset.univ) (fun i _ =>
-    (hasDerivAt_deformedSusceptibility G hloc hqt d hd a hP p hp0 hp k t ht Set.univ (root i)).const_mul (weight i))
-
-end AveragedDerivative
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section PathAverage
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {ι : Type*} [Fintype ι]
-
-noncomputable def rawPathAverage (G : BondGraph V E) (P : V → V → ℝ) (k : ℕ)
-    (root : ι → V) (weight : ι → ℝ) (f : RawCorridor G k → ℝ) : ℝ :=
-  ∑ i, weight i * ∑' w : RawFrom (openSimpleGraph G Set.univ) k (root i),
-    w.2.mass P * f ⟨root i,w⟩
-
-theorem rawPathAverage_mono (G : BondGraph V E) (hloc : LocallyFinite G)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (k : ℕ)
-    (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 ≤ weight i)
-    (f g : RawCorridor G k → ℝ) (hfg : ∀ w, f w ≤ g w) :
-    rawPathAverage G P k root weight f ≤ rawPathAverage G P k root weight g := by
-  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
-  unfold rawPathAverage
-  apply Finset.sum_le_sum
-  intro i _
-  apply mul_le_mul_of_nonneg_left _ (hw i)
-  let := Fintype.ofFinite (RawFrom (openSimpleGraph G Set.univ) k (root i))
-  simp only [tsum_fintype]
-  exact Finset.sum_le_sum (fun w _ => mul_le_mul_of_nonneg_left (hfg _) (w.2.mass_nonneg P hP))
-
-theorem rawPathAverage_sub (G : BondGraph V E) (hloc : LocallyFinite G)
-    (P : V → V → ℝ) (k : ℕ) (root : ι → V) (weight : ι → ℝ)
-    (f g : RawCorridor G k → ℝ) :
-    rawPathAverage G P k root weight (fun w => f w-g w) =
-      rawPathAverage G P k root weight f - rawPathAverage G P k root weight g := by
-  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
-  have he (i : ι) : (∑' w : RawFrom (openSimpleGraph G Set.univ) k (root i),
-      w.2.mass P * (f ⟨root i,w⟩-g ⟨root i,w⟩)) =
-      (∑' w : RawFrom (openSimpleGraph G Set.univ) k (root i), w.2.mass P*f ⟨root i,w⟩) -
-      (∑' w : RawFrom (openSimpleGraph G Set.univ) k (root i), w.2.mass P*g ⟨root i,w⟩) := by
-    let := Fintype.ofFinite (RawFrom (openSimpleGraph G Set.univ) k (root i))
-    simp only [tsum_fintype,mul_sub,Finset.sum_sub_distrib]
-  simp only [rawPathAverage]
-  simp only [he]
-  simp only [mul_sub,Finset.sum_sub_distrib]
-
-theorem rawPathAverage_const_mul (G : BondGraph V E) (P : V → V → ℝ) (k : ℕ)
-    (root : ι → V) (weight : ι → ℝ) (c : ℝ) (f : RawCorridor G k → ℝ) :
-    rawPathAverage G P k root weight (fun w => c*f w) = c*rawPathAverage G P k root weight f := by
-  unfold rawPathAverage
-  simp only [←mul_left_comm c,tsum_mul_left,Finset.mul_sum]
-
-noncomputable def retainedSusceptibilityProduct (G : BondGraph V E) (P : V → V → ℝ)
-    (p : unitInterval) {k : ℕ} (t : ℝ) (w : RawCorridor G k) : ℝ :=
-  if 3 ≤ w.path.length then
-    deformedSusceptibility G P p k t (w.interior:Set V)ᶜ w.base *
-      deformedSusceptibility G P p k t (w.interior:Set V)ᶜ w.endpoint
-  else 0
-
-end PathAverage
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section AveragedCalibratedDecrease
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Countable V] [Countable E] [Nonempty V]
-variable {ι : Type*} [Fintype ι]
-
-theorem raw_derivative_template_row_lower (G : BondGraph V E) (hG : Function.Injective G)
-    (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (d : ℕ) (hd : ∀ x, degree G x ≤ d)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (hpcrit : 0 < (pc G:ℝ))
-    (p : unitInterval) (hp : p < pc G) (hpnear : (pc G:ℝ)/2 ≤ (p:ℝ))
-    (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T)
-    (k : ℕ) (w : RawCorridor G k)
-    (c : ℝ) (hc : (∑ e ∈ (w.retained_finite hG).toFinset, ∑' r : RawCorridor G k,
-        if e ∈ r.retained then r.weight P p else 0) ≤ c)
-    (t : ℝ) (ht : 0 ≤ t) :
-    Real.exp (-t*c) * (w.probability P *
-      (retainedSusceptibilityProduct G P p t w -
-        (1+2*(d:ℝ)/(pc G:ℝ))*(maxSusceptibility G p)^2*(matrixCoeff (T^3) w.base w.endpoint).re)) ≤
-      ∑' xy : V × V, corridorDerivativeContribution G P p t w xy.1 xy.2 := by
-  by_cases hlen : 3 ≤ w.path.length
-  · have hh := calibrated_derivative_template_row_lower G hG hloc hqt d
-      (fun x => (neighbors_card_le_degree G hloc x).trans (hd x)) P hP hpcrit p hp hpnear T hT k w hlen c hc t ht
-    simpa only [retainedSusceptibilityProduct,hlen,ite_true,corridorDerivativeContribution,
-      mul_assoc,mul_left_comm (Real.exp (-t*c))] using hh
-  · have hC : 0 ≤ 1+2*(d:ℝ)/(pc G:ℝ) := by positivity
-    have hA := (((hT.positiveRealMatrix G p T).pow 3) w.base w.endpoint).2
-    have hpr := w.2.2.mass_nonneg P hP
-    have hr : 0 ≤ ∑' xy : V × V, corridorDerivativeContribution G P p t w xy.1 xy.2 :=
-      tsum_nonneg (fun xy => corridorDerivativeContribution_nonneg G P hP p t w xy.1 xy.2)
-    apply le_trans _ hr
-    simp only [retainedSusceptibilityProduct,hlen,ite_false,zero_sub,mul_neg]
-    exact neg_nonpos.mpr (mul_nonneg (Real.exp_pos _).le (mul_nonneg hpr
-      (mul_nonneg (mul_nonneg hC (sq_nonneg _)) hA)))
-
- 
-theorem averaged_derivative_raw_lower (G : BondGraph V E) (hG : Function.Injective G)
-    (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 ≤ weight i)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
-    (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y) (hpcrit : 0 < (pc G:ℝ))
-    (p : unitInterval) (hp : p < pc G) (hpnear : (pc G:ℝ)/2 ≤ (p:ℝ))
-    (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T)
-    (k : ℕ) (c : ℝ) (hc : ∀ w : RawCorridor G k,
-      (∑ e ∈ (w.retained_finite hG).toFinset, ∑' r : RawCorridor G k,
-        if e ∈ r.retained then r.weight (walkKernel G hloc a) p else 0) ≤ c)
-    (t : ℝ) (ht : 0 ≤ t) :
-    Real.exp (-t*c) *
-      (rawPathAverage G (walkKernel G hloc a) k root weight (retainedSusceptibilityProduct G (walkKernel G hloc a) p t) -
-       (1+2*(d:ℝ)/(pc G:ℝ))*(maxSusceptibility G p)^2 *
-         rawPathAverage G (walkKernel G hloc a) k root weight (fun w => (matrixCoeff (T^3) w.base w.endpoint).re)) ≤
-      ∑ i, weight i * ∫ ω, deformedDerivativeMass G (walkKernel G hloc a) p k t Set.univ ω (root i) ∂law p := by
-  rw [averaged_derivative_transport G hG hloc hqt root weight Γ hΓ htr d hd a hP p
-      ((half_pos hpcrit).trans_le hpnear) hp k t ht]
-  rw [←rawPathAverage_const_mul G _ k root weight _ (fun w => (matrixCoeff (T^3) w.base w.endpoint).re),
-    ←rawPathAverage_sub G hloc]
-  unfold rawPathAverage
-  rw [Finset.mul_sum]
-  apply Finset.sum_le_sum
-  intro i _
-  rw [mul_left_comm]
-  apply mul_le_mul_of_nonneg_left _ (hw i)
-  rw [←tsum_mul_left]
-  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
-  let := Fintype.ofFinite (RawFrom (openSimpleGraph G Set.univ) k (root i))
-  simp only [tsum_fintype]
-  exact Finset.sum_le_sum (fun w _ => raw_derivative_template_row_lower G hG hloc hqt d hd _ hP
-    hpcrit p hp hpnear T hT k ⟨root i,w⟩ c (hc _) t ht)
-
-end AveragedCalibratedDecrease
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section PathErrorTrace
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {ι : Type*} [Fintype ι]
-
-theorem rawPathAverage_operator_trace (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (A : Hilbert V →L[ℂ] Hilbert V) (hAs : IsSelfAdjoint A)
-    (k : ℕ) (root : ι → V) (weight : ι → ℝ) :
-    rawPathAverage G (walkKernel G hloc a) k root weight
-      (fun w => (matrixCoeff A w.base w.endpoint).re) = (rootTrace root weight (P^k*A)).re := by
-  have he (u : V) : (∑' w : RawFrom (openSimpleGraph G Set.univ) k u,
-      w.2.mass (walkKernel G hloc a) * (matrixCoeff A u w.1).re) = (matrixCoeff (P^k*A) u u).re := by
-    change RawFrom.integral (H := openSimpleGraph G Set.univ) (walkKernel G hloc a) k u
-      (fun v => (matrixCoeff A u v).re) = _
-    rw [raw_integral_operator G hloc a P hpos hP]
-    simp only [matrixCoeff_real_symm A hAs u]
-    exact ((hpos.pow k).mul_hasSum A u u).tsum_eq
-  simp only [rawPathAverage,RawCorridor.base,RawCorridor.endpoint,he,rootTrace,
-    Complex.re_sum,Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero]
-
-open scoped ComplexOrder
-attribute [local instance] spectralRealOperatorScalarTower spectralRealOperatorSMulCommClass
-local instance averagedDecreasePathNonUnitalContinuousFunctionalCalculus : NonUnitalContinuousFunctionalCalculus ℝ
-    (Hilbert V →L[ℂ] Hilbert V) IsSelfAdjoint :=
-  IsSelfAdjoint.instNonUnitalContinuousFunctionalCalculus (A := Hilbert V →L[ℂ] Hilbert V)
-local instance averagedDecreasePathContinuousFunctionalCalculus : ContinuousFunctionalCalculus ℝ
-    (Hilbert V →L[ℂ] Hilbert V) IsSelfAdjoint :=
-  IsSelfAdjoint.instContinuousFunctionalCalculus (A := Hilbert V →L[ℂ] Hilbert V)
-
- 
-
-theorem rawPathAverage_operator_le_norm (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 ≤ weight i)
-    (Γ : Set (V ≃ V)) (htr : RootTransport root weight Γ)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (hPi : KernelInvariant Γ P) (hPs : IsSelfAdjoint P)
-    (A : Hilbert V →L[ℂ] Hilbert V) (hAi : KernelInvariant Γ A) (hA : 0 ≤ A) (k : ℕ) :
-    rawPathAverage G (walkKernel G hloc a) k root weight
-      (fun w => (matrixCoeff A w.base w.endpoint).re) ≤
-      ‖P‖^k * (rootTrace root weight A).re := by
-  rw [rawPathAverage_operator_trace G hloc a P hpos hP A (IsSelfAdjoint.of_nonneg hA)]
-  have hh := rootTrace_positive_mul_le root weight htr hw A (P^k) hAi (hPi.pow k) hA (hPs.pow k)
-  have hr := (Complex.le_def.mp hh).1
-  simp only [Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero] at hr
-  have hn : ‖P^k‖ ≤ ‖P‖^k := by
-    cases k with
-    | zero =>
-      simp only [pow_zero]
-      have he : (1 : Hilbert V →L[ℂ] Hilbert V) = ContinuousLinearMap.id ℂ (Hilbert V) := by ext f; rfl
-      rw [he]
-      exact ContinuousLinearMap.norm_id_le
-    | succ j => exact norm_pow_le' P (Nat.succ_pos j)
-  exact hr.trans (mul_le_mul_of_nonneg_right hn
-    (Complex.nonneg_iff.mp (rootTrace_nonneg root weight hw A hA)).1)
-
-end PathErrorTrace
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-open scoped ComplexOrder
-section UniformAveragedDecrease
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Countable V] [Countable E] [Nonempty V]
-variable {ι : Type*} [Fintype ι]
-attribute [local instance] spectralRealOperatorScalarTower spectralRealOperatorSMulCommClass
-local instance averagedDecreaseUniformNonUnitalContinuousFunctionalCalculus : NonUnitalContinuousFunctionalCalculus ℝ
-    (Hilbert V →L[ℂ] Hilbert V) IsSelfAdjoint :=
-  IsSelfAdjoint.instNonUnitalContinuousFunctionalCalculus (A := Hilbert V →L[ℂ] Hilbert V)
-local instance averagedDecreaseUniformContinuousFunctionalCalculus : ContinuousFunctionalCalculus ℝ
-    (Hilbert V →L[ℂ] Hilbert V) IsSelfAdjoint :=
-  IsSelfAdjoint.instContinuousFunctionalCalculus (A := Hilbert V →L[ℂ] Hilbert V)
-
- 
-
-theorem exists_uniform_averaged_decrease (G : BondGraph V E) (hG : Function.Injective G)
-    (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 ≤ weight i)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
-    (hK : ∀ x y, 0 ≤ walkKernel G hloc a x y)
-    (hpcrit : 0 < (pc G:ℝ)) (hpcrit₁ : (pc G:ℝ) < 1)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (hPs : IsSelfAdjoint P) :
-    ∃ C : ℝ, 0 < C ∧ ∀ (p : unitInterval), (pc G:ℝ)/2 ≤ (p:ℝ) → p < pc G →
-      ∀ (T : Hilbert V →L[ℂ] Hilbert V), HasKernel G p T → ∀ (k : ℕ), 3 ≤ k →
-      ∀ (t : ℝ), 0 ≤ t →
-      Real.exp (-t*Real.exp (C*(k:ℝ))) *
-        (rawPathAverage G (walkKernel G hloc a) k root weight
-          (retainedSusceptibilityProduct G (walkKernel G hloc a) p t) -
-          (1+2*(d:ℝ)/(pc G:ℝ))*(maxSusceptibility G p)^2 *
-            (‖P‖^k * (rootTrace root weight (T^3)).re)) ≤
-        ∑ i, weight i * ∫ ω, deformedDerivativeMass G (walkKernel G hloc a) p k t Set.univ ω (root i) ∂law p := by
-  obtain ⟨C,hC,hcost⟩ := exists_uniform_corridor_leg_cost G hG hloc d hd a hK hpcrit hpcrit₁
-  refine ⟨C,hC,?_⟩
-  intro p hpnear hp T hT k hk t ht
-  have hpositive : 0 ≤ T := ContinuousLinearMap.nonneg_iff_isPositive.mpr (hT.isPositive G p T)
-  have hTi : KernelInvariant Γ T := fun g hg => hT.invariant G p T g (hΓ hg)
-  have hPi : KernelInvariant Γ P := by
-    intro g hg x y
-    obtain ⟨b,hb⟩ := hΓ hg
-    simp only [hP,walkKernel_covariant G hloc a hb]
-  have herr := rawPathAverage_operator_le_norm G hloc a root weight hw Γ htr P hpos hP hPi hPs
-    (T^3) (hTi.pow 3) (CStarAlgebra.pow_nonneg T 3 hpositive) k
-  have hCnonneg : 0 ≤ (1+2*(d:ℝ)/(pc G:ℝ))*(maxSusceptibility G p)^2 := by positivity
-  apply le_trans _ (averaged_derivative_raw_lower G hG hloc hqt root weight hw Γ hΓ htr
-    d hd a hK hpcrit p hp hpnear T hT k (Real.exp (C*(k:ℝ))) (hcost p hpnear hp k hk) t ht)
-  exact mul_le_mul_of_nonneg_left (sub_le_sub_left (mul_le_mul_of_nonneg_left herr hCnonneg) _)
-    (Real.exp_pos _).le
-
-end UniformAveragedDecrease
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section DeformedRowComparison
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Countable V] [Countable E] [Nonempty V]
-variable {ι : Type*} [Fintype ι]
-
- 
-
-theorem exists_uniform_deformed_row_comparison (G : BondGraph V E)
-    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
-    (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i))
-    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
-    (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y)
-    (hpcrit : 0 < (pc G:ℝ)) (hpcrit₁ : (pc G:ℝ) < 1) :
-    ∃ C cf : ℝ, 0 < C ∧ 0 < cf ∧ ∀ (p : unitInterval),
-      (pc G:ℝ)/2 ≤ (p:ℝ) → p < pc G → ∀ (k : ℕ), 3 ≤ k → ∀ (t : ℝ),
-      0 ≤ t → t*Real.exp (C*(k:ℝ)) ≤ 1 → ∀ x,
-      cf*averagedDeformedSusceptibility G (walkKernel G hloc a) p k root weight t ≤
-        deformedSusceptibility G (walkKernel G hloc a) p k t Set.univ x := by
-  obtain ⟨C,hC,hmove⟩ := exists_uniform_deformed_endpoint_comparison G hloc hqt d hd a hP hpcrit hpcrit₁
-  have hex (ij : ι × ι) : ∃ l : List E, Along G (root ij.1) l (root ij.2) := by
-    obtain ⟨l,hl,_,_⟩ := conn_exists_bond_trail G (conn_univ_of_connected G hconn (root ij.1) (root ij.2))
-    exact ⟨l,hl⟩
-  choose path hpath using hex
-  let N := ∑ ij : ι × ι, (path ij).length
-  have hlen (i j : ι) : (path (i,j)).length ≤ N :=
-    Finset.single_le_sum (f := fun ij : ι × ι => (path ij).length) (fun _ _ => Nat.zero_le _) (Finset.mem_univ (i,j))
-  refine ⟨C,Real.exp (-(N:ℝ))*((pc G:ℝ)/2)^N,hC,mul_pos (Real.exp_pos _) (pow_pos (half_pos hpcrit) N),?_⟩
-  intro p hpnear hp k hk t ht htc x
-  obtain ⟨j,g,b,hgb,hgx⟩ := hcover x
-  have hb (i : ι) : (Real.exp (-(N:ℝ))*((pc G:ℝ)/2)^N)*
-      deformedSusceptibility G (walkKernel G hloc a) p k t Set.univ (root i) ≤
-      deformedSusceptibility G (walkKernel G hloc a) p k t Set.univ (root j) := by
-    have hh := hmove p hpnear hp k hk t ht Set.univ (root j) (root i) (path (j,i))
-      (Set.mem_univ _) (hpath (j,i)) (fun _ _ _ _ => Set.mem_univ _)
-    have hpow : ((pc G:ℝ)/2)^N ≤ (p:ℝ)^(path (j,i)).length :=
-      (pow_le_pow_left₀ (half_pos hpcrit).le hpnear N).trans
-        (pow_le_pow_of_le_one p.2.1 p.2.2 (hlen j i))
-    have he : Real.exp (-(N:ℝ)) ≤ Real.exp (-t*((path (j,i)).length*Real.exp (C*(k:ℝ)))) := by
-      apply Real.exp_le_exp.mpr
-      have hh := mul_le_mul_of_nonneg_right htc (Nat.cast_nonneg (path (j,i)).length : (0:ℝ) ≤ _)
-      have hn : ((path (j,i)).length:ℝ) ≤ (N:ℝ) := Nat.cast_le.mpr (hlen j i)
-      nlinarith
-    exact (mul_le_mul_of_nonneg_right (mul_le_mul he hpow (pow_nonneg (half_pos hpcrit).le _) (Real.exp_pos _).le)
-      (deformedSusceptibility_nonneg G _ p k t Set.univ (root i))).trans hh
-  rw [←hgx,deformedSusceptibility_aut G hgb _ (walkKernel_covariant G hloc a hgb)]
-  simp only [Set.preimage_univ]
-  calc
-    _ = ∑ i, weight i*((Real.exp (-(N:ℝ))*((pc G:ℝ)/2)^N)*
-        deformedSusceptibility G (walkKernel G hloc a) p k t Set.univ (root i)) := by
-      simp only [averagedDeformedSusceptibility,Finset.mul_sum]
-      exact Finset.sum_congr rfl (fun _ _ => by ring)
-    _ ≤ ∑ i, weight i*deformedSusceptibility G (walkKernel G hloc a) p k t Set.univ (root j) :=
-      Finset.sum_le_sum (fun i _ => mul_le_mul_of_nonneg_left (hb i) (hw i))
-    _ = _ := by rw [←Finset.sum_mul,hw1,one_mul]
-
-end DeformedRowComparison
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section AnchoredSegments
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {I : Type*} [Fintype I]
-
- 
-def segmentOrbit (G : BondGraph V E) (γ : I → V) : Set (I → V) :=
-  {s | ∃ a : V ≃ V, ∃ b : E ≃ E, IsAut G a b ∧ s = a ∘ γ}
-
-omit [Fintype I] in
-theorem segmentOrbit_aut_iff (G : BondGraph V E) (γ s : I → V)
-    {a : V ≃ V} {b : E ≃ E} (h : IsAut G a b) :
-    a ∘ s ∈ segmentOrbit G γ ↔ s ∈ segmentOrbit G γ := by
-  constructor
-  · rintro ⟨c,d,hcd,he⟩
-    refine ⟨c.trans a.symm,d.trans b.symm,isAut_trans G hcd (isAut_symm G h),?_⟩
-    funext i
-    have hh := congrFun he i
-    simpa only [Function.comp_apply,Equiv.trans_apply,Equiv.symm_apply_apply] using congrArg a.symm hh
-  · rintro ⟨c,d,hcd,rfl⟩
-    exact ⟨c.trans a,d.trans b,isAut_trans G hcd h,rfl⟩
-
-theorem finite_segmentAnchors (G : BondGraph V E) (hloc : LocallyFinite G)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i : I) (x : V) :
-    {s ∈ segmentOrbit G γ | s i = x}.Finite := by
-  apply (Set.Finite.pi' (fun _ : I => finite_reachBall hloc x n)).subset
-  rintro s ⟨⟨a,b,hab,rfl⟩,hx⟩ j
-  have hh := mem_reachBall_aut G hab (hdiam i j)
-  change a (γ i) = x at hx
-  simpa only [hx,Function.comp_apply] using hh
-
-noncomputable def segmentAnchors (G : BondGraph V E) (hloc : LocallyFinite G)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i : I) (x : V) : Finset (I → V) :=
-  (finite_segmentAnchors G hloc γ n hdiam i x).toFinset
-
-theorem mem_segmentAnchors (G : BondGraph V E) (hloc : LocallyFinite G)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i : I) (x : V) (s : I → V) :
-    s ∈ segmentAnchors G hloc γ n hdiam i x ↔ s ∈ segmentOrbit G γ ∧ s i = x := by
-  exact Set.Finite.mem_toFinset _
-
-theorem segmentAnchors_nonempty_iff (G : BondGraph V E) (hloc : LocallyFinite G)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i : I) (x : V) :
-    (segmentAnchors G hloc γ n hdiam i x).Nonempty ↔ x ∈ vertexOrbit G (γ i) := by
-  constructor
-  · rintro ⟨s,hs⟩
-    obtain ⟨⟨a,b,h,rfl⟩,hx⟩ := (mem_segmentAnchors G hloc γ n hdiam i x s).mp hs
-    exact ⟨a,b,h,hx⟩
-  · rintro ⟨a,b,h,hx⟩
-    exact ⟨a ∘ γ,(mem_segmentAnchors G hloc γ n hdiam i x _).mpr ⟨⟨a,b,h,rfl⟩,hx⟩⟩
-
-end AnchoredSegments
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section AnchoredPositionKernels
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {I : Type*} [Fintype I]
-
-def segmentAut (a : V ≃ V) : (I → V) ≃ (I → V) where
-  toFun s := a ∘ s
-  invFun s := a.symm ∘ s
-  left_inv s := by funext i; exact a.symm_apply_apply (s i)
-  right_inv s := by funext i; exact a.apply_symm_apply (s i)
-
-theorem segmentAnchors_map (G : BondGraph V E) (hloc : LocallyFinite G)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i : I) (x : V)
-    {a : V ≃ V} {b : E ≃ E} (h : IsAut G a b) :
-    segmentAnchors G hloc γ n hdiam i (a x) =
-      (segmentAnchors G hloc γ n hdiam i x).map (segmentAut a).toEmbedding := by
-  ext s
-  rw [Finset.mem_map,mem_segmentAnchors]
-  constructor
-  · rintro ⟨hs,hx⟩
-    refine ⟨a.symm ∘ s,(mem_segmentAnchors G hloc γ n hdiam i x _).mpr ⟨?_,?_⟩,?_⟩
-    · exact (segmentOrbit_aut_iff G γ s (isAut_symm G h)).mpr hs
-    · simp only [Function.comp_apply,hx,Equiv.symm_apply_apply]
-    · funext j; exact a.apply_symm_apply (s j)
-  · rintro ⟨t,ht,rfl⟩
-    obtain ⟨ht,hx⟩ := (mem_segmentAnchors G hloc γ n hdiam i x t).mp ht
-    exact ⟨(segmentOrbit_aut_iff G γ t h).mpr ht,congrArg a hx⟩
-
-theorem segmentAnchors_card_aut (G : BondGraph V E) (hloc : LocallyFinite G)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i : I) (x : V)
-    {a : V ≃ V} {b : E ≃ E} (h : IsAut G a b) :
-    (segmentAnchors G hloc γ n hdiam i (a x)).card = (segmentAnchors G hloc γ n hdiam i x).card := by
-  rw [segmentAnchors_map G hloc γ n hdiam i x h,Finset.card_map]
-
- 
-
-noncomputable def anchoredPositionKernel (G : BondGraph V E) (hloc : LocallyFinite G)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i j : I) (x y : V) : ℝ :=
-  ∑ s ∈ segmentAnchors G hloc γ n hdiam i x,
-    if y = s j then ((segmentAnchors G hloc γ n hdiam i x).card : ℝ)⁻¹ else 0
-
-theorem anchoredPositionKernel_nonneg (G : BondGraph V E) (hloc : LocallyFinite G)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i j : I) (x y : V) :
-    0 ≤ anchoredPositionKernel G hloc γ n hdiam i j x y := by
-  apply Finset.sum_nonneg
-  intro s _
-  split_ifs <;> positivity
-
-theorem anchoredPositionKernel_aut (G : BondGraph V E) (hloc : LocallyFinite G)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i j : I) (x y : V)
-    {a : V ≃ V} {b : E ≃ E} (h : IsAut G a b) :
-    anchoredPositionKernel G hloc γ n hdiam i j (a x) (a y) =
-      anchoredPositionKernel G hloc γ n hdiam i j x y := by
-  simp only [anchoredPositionKernel,segmentAnchors_map G hloc γ n hdiam i x h,
-    Finset.card_map,Finset.sum_map,Equiv.toEmbedding_apply,
-    segmentAut,Equiv.coe_fn_mk,Function.comp_apply,Equiv.apply_eq_iff_eq]
-
-end AnchoredPositionKernels
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section AnchoredPositionRows
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {I : Type*} [Fintype I]
-
-theorem anchoredPositionKernel_hasSum (G : BondGraph V E) (hloc : LocallyFinite G)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i j : I) (x : V) :
-    HasSum (anchoredPositionKernel G hloc γ n hdiam i j x)
-      (if x ∈ vertexOrbit G (γ i) then (1:ℝ) else 0) := by
-  have hh := hasSum_sum (s := segmentAnchors G hloc γ n hdiam i x)
-    (fun s _ => hasSum_ite_eq (s j) (((segmentAnchors G hloc γ n hdiam i x).card:ℝ)⁻¹))
-  have he : (∑ s ∈ segmentAnchors G hloc γ n hdiam i x,
-      ((segmentAnchors G hloc γ n hdiam i x).card:ℝ)⁻¹) =
-      if x ∈ vertexOrbit G (γ i) then (1:ℝ) else 0 := by
-    simp only [Finset.sum_const,nsmul_eq_mul]
-    by_cases hx : x ∈ vertexOrbit G (γ i)
-    · have hn := Finset.card_pos.mpr ((segmentAnchors_nonempty_iff G hloc γ n hdiam i x).mpr hx)
-      rw [ite_eq_left hx,mul_inv_cancel₀ (Nat.cast_ne_zero.mpr (Nat.ne_of_gt hn))]
-    · have he : segmentAnchors G hloc γ n hdiam i x = ∅ := Finset.not_nonempty_iff_eq_empty.mp
-        (fun h => hx ((segmentAnchors_nonempty_iff G hloc γ n hdiam i x).mp h))
-      simp only [he,Finset.card_empty,Nat.cast_zero,zero_mul,ite_eq_right hx]
-  rw [he] at hh
-  exact hh
-
-theorem anchoredPositionKernel_eq_zero_off_ball (G : BondGraph V E) (hloc : LocallyFinite G)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i j : I) (x y : V)
-    (hxy : y ∉ reachBall G x n) : anchoredPositionKernel G hloc γ n hdiam i j x y = 0 := by
-  apply Finset.sum_eq_zero
-  intro s hs
-  apply ite_eq_right
-  intro hy
-  obtain ⟨⟨a,b,hab,rfl⟩,hx⟩ := (mem_segmentAnchors G hloc γ n hdiam i x s).mp hs
-  change a (γ i) = x at hx
-  change y = a (γ j) at hy
-  exact hxy (by simpa only [hx,←hy] using mem_reachBall_aut G hab (hdiam i j))
-
-theorem anchoredPositionKernel_eq_zero_off_orbits (G : BondGraph V E) (hloc : LocallyFinite G)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i j : I) (x y : V) :
-    x ∉ vertexOrbit G (γ i) ∨ y ∉ vertexOrbit G (γ j) →
-      anchoredPositionKernel G hloc γ n hdiam i j x y = 0 := by
-  intro h
-  apply Finset.sum_eq_zero
-  intro s hs
-  apply ite_eq_right
-  intro hy
-  obtain ⟨⟨a,b,hab,rfl⟩,hx⟩ := (mem_segmentAnchors G hloc γ n hdiam i x s).mp hs
-  rcases h with h | h
-  · exact h ⟨a,b,hab,hx⟩
-  · exact h ⟨a,b,hab,hy.symm⟩
-
-end AnchoredPositionRows
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section AnchoredPositionColumns
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {I : Type*} [Fintype I]
-
-theorem anchoredPositionKernel_columnSummable (G : BondGraph V E) (hloc : LocallyFinite G)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i j : I) (y : V) :
-    Summable (fun x => anchoredPositionKernel G hloc γ n hdiam i j x y) := by
-  apply summable_of_ne_finset_zero (s := (finite_reachBall hloc y n).toFinset)
-  intro x hx
-  simp only [Set.Finite.mem_toFinset] at hx
-  apply Finset.sum_eq_zero
-  intro s hs
-  apply ite_eq_right
-  intro hy
-  obtain ⟨⟨a,b,hab,rfl⟩,he⟩ := (mem_segmentAnchors G hloc γ n hdiam i x s).mp hs
-  change a (γ i) = x at he
-  change y = a (γ j) at hy
-  exact hx (by simpa only [he,←hy] using mem_reachBall_aut G hab (hdiam j i))
-
-theorem anchoredPositionKernel_column_orbit (G : BondGraph V E) (hloc : LocallyFinite G)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i j : I) (y : V) :
-    (∑' x, anchoredPositionKernel G hloc γ n hdiam i j x y) =
-      if y ∈ vertexOrbit G (γ j) then
-        ∑' x, anchoredPositionKernel G hloc γ n hdiam i j x (γ j) else 0 := by
-  by_cases hy : y ∈ vertexOrbit G (γ j)
-  · rw [ite_eq_left hy]
-    obtain ⟨a,b,hab,rfl⟩ := hy
-    rw [←a.tsum_eq]
-    exact tsum_congr (fun x => anchoredPositionKernel_aut G hloc γ n hdiam i j x (γ j) hab)
-  · rw [ite_eq_right hy]
-    calc
-      _ = ∑' _ : V, (0:ℝ) := tsum_congr (fun x =>
-        anchoredPositionKernel_eq_zero_off_orbits G hloc γ n hdiam i j x y (Or.inr hy))
-      _ = 0 := tsum_zero
-
- 
-
-theorem anchoredPositionKernel_column_mass {ι : Type*} [Fintype ι]
-    (G : BondGraph V E) (hloc : LocallyFinite G)
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G)
-    (htr : RootTransport root weight Γ)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i j : I) :
-    (∑ r, if root r ∈ vertexOrbit G (γ i) then weight r else 0) =
-      (∑ r, if root r ∈ vertexOrbit G (γ j) then weight r else 0) *
-      ∑' x, anchoredPositionKernel G hloc γ n hdiam i j x (γ j) := by
-  have ht := htr.real root weight Γ (anchoredPositionKernel G hloc γ n hdiam i j)
-    (fun a ha x y => by
-      obtain ⟨b,hb⟩ := hΓ ha
-      exact anchoredPositionKernel_aut G hloc γ n hdiam i j x y hb)
-    (fun x => (anchoredPositionKernel_hasSum G hloc γ n hdiam i j x).summable)
-    (anchoredPositionKernel_columnSummable G hloc γ n hdiam i j)
-  calc
-    _ = ∑ r, weight r * ∑' y, anchoredPositionKernel G hloc γ n hdiam i j (root r) y := by
-      apply Finset.sum_congr rfl
-      intro r _
-      rw [(anchoredPositionKernel_hasSum G hloc γ n hdiam i j (root r)).tsum_eq]
-      split_ifs <;> simp only [mul_one,mul_zero]
-    _ = _ := ht
-    _ = _ := by
-      rw [Finset.sum_mul]
-      apply Finset.sum_congr rfl
-      intro r _
-      rw [anchoredPositionKernel_column_orbit]
-      split_ifs <;> simp only [mul_zero,zero_mul]
-
-theorem anchoredPositionKernel_column_ratio {ι : Type*} [Fintype ι]
-    (G : BondGraph V E) (hloc : LocallyFinite G)
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G)
-    (htr : RootTransport root weight Γ)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i j : I)
-    (hj : 0 < ∑ r, if root r ∈ vertexOrbit G (γ j) then weight r else 0) (y : V) :
-    (∑' x, anchoredPositionKernel G hloc γ n hdiam i j x y) =
-      if y ∈ vertexOrbit G (γ j) then
-        (∑ r, if root r ∈ vertexOrbit G (γ i) then weight r else 0) /
-        (∑ r, if root r ∈ vertexOrbit G (γ j) then weight r else 0) else 0 := by
-  have he := anchoredPositionKernel_column_mass G hloc root weight Γ hΓ htr γ n hdiam i j
-  rw [anchoredPositionKernel_column_orbit]
-  congr 1
-  exact (eq_div_iff (ne_of_gt hj)).mpr (by simpa only [mul_comm] using he.symm)
-
-end AnchoredPositionColumns
-end Percolation
-
- 
-namespace Percolation
-universe u
-variable {V : Type u}
-section AsymmetricSchur
-open scoped ENNReal
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem exists_operator_of_unit_row_column_bound (U : V → V → ℝ) (D : ℝ)
-    (hD : 0 ≤ D) (hU : ∀ x y, 0 ≤ U x y)
-    (hrow : ∀ x (s : Finset V), ∑ y ∈ s, U x y ≤ 1)
-    (hcol : ∀ y (s : Finset V), ∑ x ∈ s, U x y ≤ D) :
-    ∃ A : Hilbert V →L[ℂ] Hilbert V, ‖A‖ ≤ Real.sqrt D ∧
-      ∀ x y, matrixCoeff A x y = (U x y : ℂ) := by
-  have hentry (x y : V) : U x y ≤ 1 := by simpa only [Finset.sum_singleton] using hrow x {y}
-  have hcm (y : V) : Memℓp (fun x => (U x y : ℂ)) 2 := by
-    apply (memℓp_gen_iff'' (by norm_num : 0 < (2 : ℝ≥0∞).toReal)).mpr
-    refine ⟨D,hD,fun s => ?_⟩
-    simp only [ENNReal.toReal_ofNat,Real.rpow_two,Complex.norm_real,Real.norm_eq_abs,sq_abs]
-    exact (Finset.sum_le_sum (fun x _ => by nlinarith [hentry x y,hU x y])).trans (hcol y s)
-  let col (y : V) : Hilbert V := ⟨fun x => (U x y : ℂ),hcm y⟩
-  let L : (V →₀ ℂ) →ₗ[ℂ] Hilbert V := Finsupp.linearCombination ℂ col
-  have hn (f : V →₀ ℂ) : ‖L f‖ ≤ Real.sqrt D * ‖finiteEmbedding f‖ := by
-    apply lp.norm_le_of_forall_sum_le (by norm_num)
-      (mul_nonneg (Real.sqrt_nonneg _) (norm_nonneg _))
-    intro s
-    simp only [ENNReal.toReal_ofNat,Real.rpow_two]
-    have hpoint (x : V) : ‖L f x‖^2 ≤ ∑ y ∈ f.support, ‖f y‖^2 * U x y := by
-      have hn0 : 0 ≤ ∑ y ∈ f.support, ‖f y‖*U x y :=
-        Finset.sum_nonneg (fun y _ => mul_nonneg (norm_nonneg _) (hU x y))
-      have hn1 : ‖L f x‖ ≤ ∑ y ∈ f.support, ‖f y‖*U x y := by
-        simp only [L,Finsupp.linearCombination_apply,Finsupp.sum,lp.coeFn_sum,
-          Finset.sum_apply,lp.coeFn_smul,Pi.smul_apply,smul_eq_mul]
-        apply (norm_sum_le _ _).trans
-        apply Finset.sum_le_sum
-        intro y _
-        simp [col,abs_of_nonneg (hU x y)]
-      have hcs := Finset.sum_sq_le_sum_mul_sum_of_sq_le_mul f.support
-        (f := fun y => U x y) (g := fun y => ‖f y‖^2*U x y)
-        (r := fun y => ‖f y‖*U x y)
-        (fun y _ => hU x y)
-        (fun y _ => mul_nonneg (sq_nonneg _) (hU x y)) (fun y _ => by nlinarith)
-      have hn2 : 0 ≤ ∑ y ∈ f.support, ‖f y‖^2*U x y :=
-        Finset.sum_nonneg (fun y _ => mul_nonneg (sq_nonneg _) (hU x y))
-      have hr := mul_le_mul_of_nonneg_right (hrow x f.support) hn2
-      rw [one_mul] at hr
-      exact (sq_le_sq₀ (norm_nonneg _) hn0 |>.mpr hn1).trans (hcs.trans hr)
-    calc
-      ∑ x ∈ s, ‖L f x‖^2 ≤ ∑ x ∈ s, ∑ y ∈ f.support, ‖f y‖^2*U x y :=
-        Finset.sum_le_sum (fun x _ => hpoint x)
-      _ = ∑ y ∈ f.support, ‖f y‖^2*∑ x ∈ s, U x y := by
-        rw [Finset.sum_comm]
-        simp_rw [Finset.mul_sum]
-      _ ≤ ∑ y ∈ f.support, ‖f y‖^2*D :=
-        Finset.sum_le_sum (fun y _ => mul_le_mul_of_nonneg_left (hcol y s) (sq_nonneg _))
-      _ = D*∑ y ∈ f.support, ‖finiteEmbedding f y‖^2 := by
-        simp only [finiteEmbedding_apply]
-        rw [←Finset.sum_mul,mul_comm]
-      _ ≤ D*‖finiteEmbedding f‖^2 := by
-        apply mul_le_mul_of_nonneg_left _ hD
-        simpa only [ENNReal.toReal_ofNat,Real.rpow_two] using
-          lp.sum_rpow_le_norm_rpow (by norm_num : 0 < (2:ℝ≥0∞).toReal) (finiteEmbedding f) f.support
-      _ = (Real.sqrt D*‖finiteEmbedding f‖)^2 := by rw [mul_pow,Real.sq_sqrt hD]
-  refine ⟨L.extendOfNorm finiteEmbedding,
-    LinearMap.opNorm_extendOfNorm_le finiteEmbedding_dense (Real.sqrt_nonneg _) hn,?_⟩
-  intro x y
-  have he := LinearMap.extendOfNorm_eq (f := L) finiteEmbedding_dense ⟨Real.sqrt D,hn⟩
-    (Finsupp.single y (1:ℂ))
-  simp only [finiteEmbedding_single] at he
-  change (L.extendOfNorm finiteEmbedding) (lp.single 2 y (1:ℂ)) x = _
-  rw [he]
-  simp [L,col]
-
-end AsymmetricSchur
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section AnchoredPositionOperators
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {I ι : Type*} [Fintype I] [Fintype ι]
-
- 
-
-theorem exists_anchoredPositionOperator (G : BondGraph V E) (hloc : LocallyFinite G)
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G)
-    (htr : RootTransport root weight Γ) (hw : ∀ r, 0 ≤ weight r)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i j : I)
-    (hj : 0 < ∑ r, if root r ∈ vertexOrbit G (γ j) then weight r else 0) :
-    ∃ H : Hilbert V →L[ℂ] Hilbert V,
-      (∀ x y, matrixCoeff H x y = (anchoredPositionKernel G hloc γ n hdiam i j x y : ℂ)) ∧
-      ‖H‖ ≤ Real.sqrt ((∑ r, if root r ∈ vertexOrbit G (γ i) then weight r else 0) /
-        (∑ r, if root r ∈ vertexOrbit G (γ j) then weight r else 0)) := by
-  have hnum : 0 ≤ ∑ r, if root r ∈ vertexOrbit G (γ i) then weight r else 0 :=
-    Finset.sum_nonneg (fun r _ => by split_ifs; exact hw r; exact le_refl 0)
-  have hratio := div_nonneg hnum hj.le
-  obtain ⟨H,hn,hH⟩ := exists_operator_of_unit_row_column_bound
-    (anchoredPositionKernel G hloc γ n hdiam i j) _ hratio
-    (anchoredPositionKernel_nonneg G hloc γ n hdiam i j)
-    (fun x s => ((anchoredPositionKernel_hasSum G hloc γ n hdiam i j x).summable.sum_le_tsum s
-      (fun y _ => anchoredPositionKernel_nonneg G hloc γ n hdiam i j x y)).trans (by
-        rw [(anchoredPositionKernel_hasSum G hloc γ n hdiam i j x).tsum_eq]
-        split_ifs <;> norm_num))
-    (fun y s => ((anchoredPositionKernel_columnSummable G hloc γ n hdiam i j y).sum_le_tsum s
-      (fun x _ => anchoredPositionKernel_nonneg G hloc γ n hdiam i j x y)).trans (by
-        rw [anchoredPositionKernel_column_ratio G hloc root weight Γ hΓ htr γ n hdiam i j hj y]
-        split_ifs; exact le_refl _; exact hratio))
-  exact ⟨H,hH,hn⟩
-
- 
-theorem exists_anchoredPositionOperator_uniform (G : BondGraph V E) (hloc : LocallyFinite G)
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G)
-    (htr : RootTransport root weight Γ) (hw : ∀ r, 0 ≤ weight r) (hw1 : ∑ r, weight r = 1)
-    (hcov : ∀ x, ∃ r, ∃ a ∈ Γ, a (root r) = x)
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ r, 1 ≤ c*weight r)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i j : I) :
-    ∃ H : Hilbert V →L[ℂ] Hilbert V,
-      (∀ x y, matrixCoeff H x y = (anchoredPositionKernel G hloc γ n hdiam i j x y : ℂ)) ∧
-      ‖H‖ ≤ Real.sqrt c := by
-  let U := anchoredPositionKernel G hloc γ n hdiam i j
-  have hn : ∀ x y, 0 ≤ U x y := anchoredPositionKernel_nonneg G hloc γ n hdiam i j
-  have hr : ∀ x, Summable (U x) := fun x => (anchoredPositionKernel_hasSum G hloc γ n hdiam i j x).summable
-  have hcol := anchoredPositionKernel_columnSummable G hloc γ n hdiam i j
-  have hb (x : V) : (∑' y, U x y) ≤ 1 := by
-    rw [(anchoredPositionKernel_hasSum G hloc γ n hdiam i j x).tsum_eq]
-    split_ifs <;> norm_num
-  have hi : ∀ a ∈ Γ, ∀ x y, U (a x) (a y) = U x y := by
-    intro a ha x y
-    obtain ⟨b,hb⟩ := hΓ ha
-    exact anchoredPositionKernel_aut G hloc γ n hdiam i j x y hb
-  have hbcol (y : V) : (∑' x, U x y) ≤ c := by
-    simpa only [mul_one] using transported_column_le root weight Γ htr hcov hw hw1 c hc hmin U hn hi hr hcol 1 hb y
-  obtain ⟨H,hHn,hH⟩ := exists_operator_of_unit_row_column_bound U c hc hn
-    (fun x s => ((hr x).sum_le_tsum s (fun y _ => hn x y)).trans (hb x))
-    (fun y s => ((hcol y).sum_le_tsum s (fun x _ => hn x y)).trans (hbcol y))
-  exact ⟨H,hH,hHn⟩
-
-end AnchoredPositionOperators
-end Percolation
-
- 
-namespace Percolation
-universe u
-variable {V : Type u}
-section RawReversal
-namespace HoldingWalk
-variable {H : SimpleGraph V}
-
- 
-def snocHold : {k : ℕ} → {x y : V} → HoldingWalk H k x y → HoldingWalk H (k+1) x y
-  | _,_,_,.nil x => .hold (.nil x)
-  | _,_,_,.hold w => .hold (snocHold w)
-  | _,_,_,.step h w => .step h (snocHold w)
-
- 
-def snocStep : {k : ℕ} → {x y z : V} → HoldingWalk H k x y → H.Adj y z → HoldingWalk H (k+1) x z
-  | _,_,_,z,.nil _,h => .step h (.nil z)
-  | _,_,_,_,.hold w,h => .hold (snocStep w h)
-  | _,_,_,_,.step h' w,h => .step h' (snocStep w h)
-
- 
-
-def reverse : {k : ℕ} → {x y : V} → HoldingWalk H k x y → HoldingWalk H k y x
-  | _,_,_,.nil x => .nil x
-  | _,_,_,.hold w => snocHold (reverse w)
-  | _,_,_,.step h w => snocStep (reverse w) h.symm
-
-theorem reverse_snocHold {k : ℕ} {x y : V} (w : HoldingWalk H k x y) :
-    w.snocHold.reverse = w.reverse.hold := by
-  induction w with
-  | nil => rfl
-  | hold w ih => simp only [snocHold,reverse,ih]
-  | step h w ih => simp only [snocHold,reverse,ih,snocStep]
-
-theorem reverse_snocStep {k : ℕ} {x y z : V} (w : HoldingWalk H k x y) (h : H.Adj y z) :
-    (w.snocStep h).reverse = .step h.symm w.reverse := by
-  induction w with
-  | nil => rfl
-  | hold w ih => simp only [snocStep,reverse,ih,snocHold]
-  | step h' w ih => simp only [snocStep,reverse,ih]
-
-theorem reverse_reverse {k : ℕ} {x y : V} (w : HoldingWalk H k x y) : w.reverse.reverse = w := by
-  induction w with
-  | nil => rfl
-  | hold w ih => simp only [reverse,reverse_snocHold,ih]
-  | step h w ih => simp only [reverse,reverse_snocStep,ih]
-
-def reverseEquiv (k : ℕ) (x y : V) : HoldingWalk H k x y ≃ HoldingWalk H k y x where
-  toFun := reverse
-  invFun := reverse
-  left_inv := reverse_reverse
-  right_inv := reverse_reverse
-
-theorem mass_snocHold (P : V → V → ℝ) {k : ℕ} {x y : V} (w : HoldingWalk H k x y) :
-    w.snocHold.mass P = w.mass P * P y y := by
-  induction w with
-  | nil => simp only [snocHold,mass,mul_one,one_mul]
-  | hold w ih => simp only [snocHold,mass,ih,mul_assoc]
-  | step h w ih => simp only [snocHold,mass,ih,mul_assoc]
-
-theorem mass_snocStep (P : V → V → ℝ) {k : ℕ} {x y z : V} (w : HoldingWalk H k x y) (h : H.Adj y z) :
-    (w.snocStep h).mass P = w.mass P * P y z := by
-  induction w with
-  | nil => simp only [snocStep,mass,mul_one,one_mul]
-  | hold w ih => simp only [snocStep,mass,ih,mul_assoc]
-  | step h' w ih => simp only [snocStep,mass,ih,mul_assoc]
-
-theorem mass_reverse (P : V → V → ℝ) (hP : ∀ x y, P x y = P y x)
-    {k : ℕ} {x y : V} (w : HoldingWalk H k x y) : w.reverse.mass P = w.mass P := by
-  induction w with
-  | nil => rfl
-  | hold w ih => simp only [reverse,mass_snocHold,mass,ih,mul_comm]
-  | step h w ih => simp only [reverse,mass_snocStep,mass,ih,hP,mul_comm]
-
-theorem vertices_snocHold {k : ℕ} {x y : V} (w : HoldingWalk H k x y) :
-    w.snocHold.vertices = w.vertices ++ [y] := by
-  induction w with
-  | nil => rfl
-  | hold w ih => simp only [snocHold,vertices,ih,List.cons_append]
-  | step h w ih => simp only [snocHold,vertices,ih,List.cons_append]
-
-theorem vertices_snocStep {k : ℕ} {x y z : V} (w : HoldingWalk H k x y) (h : H.Adj y z) :
-    (w.snocStep h).vertices = w.vertices ++ [z] := by
-  induction w with
-  | nil => rfl
-  | hold w ih => simp only [snocStep,vertices,ih,List.cons_append]
-  | step h' w ih => simp only [snocStep,vertices,ih,List.cons_append]
-
-theorem vertices_reverse {k : ℕ} {x y : V} (w : HoldingWalk H k x y) :
-    w.reverse.vertices = w.vertices.reverse := by
-  induction w with
-  | nil => rfl
-  | hold w ih => simp only [reverse,vertices_snocHold,vertices,ih,List.reverse_cons]
-  | step h w ih => simp only [reverse,vertices_snocStep,vertices,ih,List.reverse_cons]
-
-end HoldingWalk
-end RawReversal
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Deformation.Calibration
+
+namespace OAI
+
+/-! Averaged Decrease for bond percolation. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section CountedCovariance
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem preimage_diff_singleton (b : E ≃ E) (ω : Set E) (e : E) :
+    (b ⁻¹' ω) \ {e} = b ⁻¹' (ω \ {b e}) := by
+  ext f
+  simp only [Set.mem_sdiff,Set.mem_preimage,Set.mem_singleton_iff,b.injective.eq_iff]
+
+theorem corridorCounted_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (S B : Set E) (D : Set V) (ω : Set E) (x y : V) :
+    corridorCounted G (b ⁻¹' S) (b ⁻¹' B) (a ⁻¹' D) (b ⁻¹' ω) x y ↔
+      corridorCounted G S B D ω (a x) (a y) := by
+  have he : (b ⁻¹' ω) ∩ internalEdges G (a ⁻¹' D) = b ⁻¹' (ω ∩ internalEdges G D) := by
+    rw [internalEdges_aut_preimage G hab,Set.preimage_inter]
+  simp only [corridorCounted,he,conn_aut_iff G hab,preimage_diff_singleton,
+    Set.mem_preimage]
+  apply and_congr_right
+  intro _
+  constructor
+  · rintro ⟨hp,hB⟩
+    constructor
+    · intro e he
+      obtain ⟨f,rfl⟩ := b.surjective e
+      exact hp f he
+    · apply Set.disjoint_left.mpr
+      intro e he hw
+      obtain ⟨f,rfl⟩ := b.surjective e
+      exact Set.disjoint_left.mp hB he hw
+  · rintro ⟨hp,hB⟩
+    exact ⟨fun e he => hp (b e) he,Set.disjoint_left.mpr (fun e he hw => Set.disjoint_left.mp hB he hw)⟩
+
+namespace RawCorridor
+variable {G : BondGraph V E} {k : ℕ}
+
+theorem aut_counted {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b)
+    (w : RawCorridor G k) (D : Set V) (ω : Set E) (x y : V) :
+    w.counted (a ⁻¹' D) (b ⁻¹' ω) x y ↔ (autEquiv hab w).counted D ω (a x) (a y) := by
+  have hr : w.retained = b ⁻¹' (autEquiv hab w).retained := by
+    ext e
+    exact (aut_retained hab w e).symm
+  have hb : w.blockers = b ⁻¹' (autEquiv hab w).blockers := by
+    ext e
+    exact (aut_blockers hab w e).symm
+  rw [counted_iff,counted_iff,aut_path_length,hr,hb,corridorCounted_aut G hab]
+
+end RawCorridor
+
+theorem corridorPenaltyTerm_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (P : V → V → ℝ) (hP : ∀ x y, P (a x) (a y) = P x y)
+    (p : ℝ) {k : ℕ} (w : RawCorridor G k) (D : Set V) (ω : Set E) (x y : V) :
+    corridorPenaltyTerm G P p (a ⁻¹' D) (b ⁻¹' ω) x y w =
+      corridorPenaltyTerm G P p D ω (a x) (a y) (RawCorridor.autEquiv hab w) := by
+  simp only [corridorPenaltyTerm,w.aut_counted hab D ω x y,RawCorridor.aut_weight hab P hP]
+
+theorem corridorPenalty_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (P : V → V → ℝ) (hP : ∀ x y, P (a x) (a y) = P x y)
+    (p : ℝ) (k : ℕ) (D : Set V) (ω : Set E) (x y : V) :
+    corridorPenalty G P p k (a ⁻¹' D) (b ⁻¹' ω) x y =
+      corridorPenalty G P p k D ω (a x) (a y) := by
+  unfold corridorPenalty
+  rw [←(RawCorridor.autEquiv hab).tsum_eq (corridorPenaltyTerm G P p D ω (a x) (a y))]
+  exact tsum_congr (fun w => corridorPenaltyTerm_aut G hab P hP p w D ω x y)
+
+end CountedCovariance
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section DeformationCovariance
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem deformedConnection_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (P : V → V → ℝ) (hP : ∀ x y, P (a x) (a y) = P x y)
+    (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x y : V) :
+    deformedConnection G P p k t (a ⁻¹' D) (b ⁻¹' ω) x y =
+      deformedConnection G P p k t D ω (a x) (a y) := by
+  simp only [deformedConnection,corridorPenalty_aut G hab P hP,Set.mem_preimage,
+    internalEdges_aut_preimage G hab,←Set.preimage_inter,conn_aut_iff G hab]
+
+theorem deformedKernel_aut [Countable V] [Countable E]
+    (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (P : V → V → ℝ) (hP : ∀ x y, P (a x) (a y) = P x y)
+    (p : unitInterval) (k : ℕ) (t : ℝ) (D : Set V) (x y : V) :
+    deformedKernel G P p k t D (a x) (a y) = deformedKernel G P p k t (a ⁻¹' D) x y := by
+  have h := integral_map (μ := law p) (measurable_set_preimage b).aemeasurable
+    (measurable_deformedConnection G P p k t (a ⁻¹' D) x y).aestronglyMeasurable
+  rw [law_map_preimage_equiv p b] at h
+  simpa only [deformedKernel,deformedConnection_aut G hab P hP] using h.symm
+
+theorem deformedSusceptibility_aut [Countable V] [Countable E]
+    (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (P : V → V → ℝ) (hP : ∀ x y, P (a x) (a y) = P x y)
+    (p : unitInterval) (k : ℕ) (t : ℝ) (D : Set V) (x : V) :
+    deformedSusceptibility G P p k t D (a x) = deformedSusceptibility G P p k t (a ⁻¹' D) x := by
+  unfold deformedSusceptibility
+  rw [←a.tsum_eq (deformedKernel G P p k t D (a x))]
+  exact tsum_congr (fun y => deformedKernel_aut G hab P hP p k t D x y)
+
+ 
+
+noncomputable def corridorDerivativeContribution (G : BondGraph V E) (P : V → V → ℝ)
+    (p : unitInterval) {k : ℕ} (t : ℝ) (w : RawCorridor G k) (x y : V) : ℝ :=
+  ∫ ω, corridorPenaltyTerm G P p Set.univ ω x y w *
+    deformedConnection G P p k t Set.univ ω x y ∂law p
+
+theorem corridorDerivativeContribution_nonneg (G : BondGraph V E) (P : V → V → ℝ)
+    (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) {k : ℕ} (t : ℝ) (w : RawCorridor G k) (x y : V) :
+    0 ≤ corridorDerivativeContribution G P p t w x y :=
+  integral_corridorDerivativeTerm_nonneg G P hP p k t w x y
+
+theorem corridorDerivativeContribution_aut [Countable V] [Countable E]
+    (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (P : V → V → ℝ) (hP : ∀ x y, P (a x) (a y) = P x y)
+    (p : unitInterval) {k : ℕ} (t : ℝ) (w : RawCorridor G k) (x y : V) :
+    corridorDerivativeContribution G P p t (RawCorridor.autEquiv hab w) (a x) (a y) =
+      corridorDerivativeContribution G P p t w x y := by
+  have hm := ((measurable_corridorPenaltyTerm G P p (a ⁻¹' Set.univ) x y w).mul
+    (measurable_deformedConnection G P p k t (a ⁻¹' Set.univ) x y))
+  have h := integral_map (μ := law p) (measurable_set_preimage b).aemeasurable hm.aestronglyMeasurable
+  rw [law_map_preimage_equiv p b] at h
+  have hc (ω : Set E) := corridorPenaltyTerm_aut G hab P hP p w Set.univ ω x y
+  have hd (ω : Set E) := deformedConnection_aut G hab P hP p k t Set.univ ω x y
+  simp only [Set.preimage_univ] at hc hd
+  simpa only [Pi.mul_apply,corridorDerivativeContribution,Set.preimage_univ,hc,hd] using h.symm
+
+end DeformationCovariance
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section DerivativeTonelli
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem derivative_field_summable (G : BondGraph V E) (hloc : LocallyFinite G)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval)
+    (k : ℕ) (t : ℝ) (ω : Set E) (x : V) (hf : (cluster G ω x).Finite) :
+    Summable (fun yw : V × RawCorridor G k => corridorPenaltyTerm G P p Set.univ ω x yw.1 yw.2 *
+      deformedConnection G P p k t Set.univ ω x yw.1) := by
+  have hs (y : V) : Summable (fun w : RawCorridor G k =>
+      corridorPenaltyTerm G P p Set.univ ω x y w * deformedConnection G P p k t Set.univ ω x y) :=
+    (corridorPenalty_summable G hloc P p k Set.univ ω x y).mul_right
+      (deformedConnection G P p k t Set.univ ω x y)
+  have hr : Summable (fun y => corridorPenalty G P p k Set.univ ω x y *
+      deformedConnection G P p k t Set.univ ω x y) := by
+    apply summable_of_hasFiniteSupport
+    apply hf.subset
+    intro y hy
+    by_contra hn
+    apply hy
+    change corridorPenalty G P p k Set.univ ω x y * deformedConnection G P p k t Set.univ ω x y = 0
+    rw [deformedConnection_eq_zero_of_not_conn G P p k t Set.univ ω x y hn,mul_zero]
+  have hn : 0 ≤ (fun yw : V × RawCorridor G k =>
+      corridorPenaltyTerm G P p Set.univ ω x yw.1 yw.2 * deformedConnection G P p k t Set.univ ω x yw.1) :=
+    fun yw => mul_nonneg (corridorPenaltyTerm_nonneg G P hP p Set.univ ω x yw.1 yw.2)
+      (deformedConnection_nonneg G P p k t Set.univ ω x yw.1)
+  apply (summable_prod_of_nonneg hn).mpr
+  refine ⟨hs,?_⟩
+  simpa only [tsum_mul_right,corridorPenalty] using hr
+
+theorem derivative_field_sum (G : BondGraph V E) (hloc : LocallyFinite G)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval)
+    (k : ℕ) (t : ℝ) (ω : Set E) (x : V) (hf : (cluster G ω x).Finite) :
+    (∑' yw : V × RawCorridor G k, corridorPenaltyTerm G P p Set.univ ω x yw.1 yw.2 *
+      deformedConnection G P p k t Set.univ ω x yw.1) =
+        deformedDerivativeMass G P p k t Set.univ ω x := by
+  rw [(derivative_field_summable G hloc P hP p k t ω x hf).tsum_prod]
+  simp only [tsum_mul_right,corridorPenalty,deformedDerivativeMass]
+
+ 
+
+theorem derivative_contribution_sum [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
+    (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y) (p : unitInterval)
+    (hp0 : 0 < (p:ℝ)) (hp : p < pc G) (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (x : V) :
+    Summable (fun yw : V × RawCorridor G k => corridorDerivativeContribution G (walkKernel G hloc a) p t yw.2 x yw.1) ∧
+    (∑' yw : V × RawCorridor G k, corridorDerivativeContribution G (walkKernel G hloc a) p t yw.2 x yw.1) =
+      ∫ ω, deformedDerivativeMass G (walkKernel G hloc a) p k t Set.univ ω x ∂law p := by
+  let F := fun yw : V × RawCorridor G k => fun ω =>
+    corridorPenaltyTerm G (walkKernel G hloc a) p Set.univ ω x yw.1 yw.2 *
+      deformedConnection G (walkKernel G hloc a) p k t Set.univ ω x yw.1
+  have hs : ∀ᵐ ω ∂law p, Summable (fun yw => F yw ω) := by
+    filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
+    exact derivative_field_summable G hloc _ hP p k t ω x hω
+  have he : (fun ω => ∑' yw, F yw ω) =ᵐ[law p]
+      fun ω => deformedDerivativeMass G (walkKernel G hloc a) p k t Set.univ ω x := by
+    filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
+    exact derivative_field_sum G hloc _ hP p k t ω x hω
+  have hh := real_tonelli_of_integrable_sum (law p) F
+    (fun yw => integrable_corridorDerivativeTerm G _ hP p k t ht Set.univ x yw.1 yw.2)
+    (fun yw ω => mul_nonneg (corridorPenaltyTerm_nonneg G _ hP p Set.univ ω x yw.1 yw.2)
+      (deformedConnection_nonneg G _ p k t Set.univ ω x yw.1)) hs
+    ((integrable_deformedDerivativeMass G hloc hqt d hd a hP p hp0 hp k t ht Set.univ x).congr he.symm)
+  exact ⟨hh.1,hh.2.trans (integral_congr_ae he)⟩
+
+end DerivativeTonelli
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section DerivativeTransport
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem raw_from_sum_eq_masked (G : BondGraph V E) (hloc : LocallyFinite G)
+    (k : ℕ) (u : V) (f : RawCorridor G k → ℝ) :
+    (∑' w : RawFrom (openSimpleGraph G Set.univ) k u, f ⟨u,w⟩) =
+      ∑' w : RawCorridor G k, if w.base = u then f w else 0 := by
+  have hs := summable_raw_selection G hloc k {u} (Set.finite_singleton u)
+    (fun w : RawCorridor G k => w.base = u) (fun _ h => h) f
+  rw [hs.tsum_sigma]
+  have he (x : V) : (∑' w : RawFrom (openSimpleGraph G Set.univ) k x,
+      if (RawCorridor.base ⟨x,w⟩) = u then f ⟨x,w⟩ else 0) =
+      if x = u then (∑' w : RawFrom (openSimpleGraph G Set.univ) k u, f ⟨u,w⟩) else 0 := by
+    by_cases hx : x = u
+    · subst x; simp only [RawCorridor.base,ite_true]
+    · simp only [RawCorridor.base,hx,ite_false,tsum_zero]
+  simp only [he,tsum_ite_eq]
+
+noncomputable def derivativeTransportMass (G : BondGraph V E) (P : V → V → ℝ)
+    (p : unitInterval) (k : ℕ) (t : ℝ) (x u : V) : ℝ :=
+  ∑' w : RawFrom (openSimpleGraph G Set.univ) k u,
+    ∑' y, corridorDerivativeContribution G P p t ⟨u,w⟩ x y
+
+theorem derivativeTransportMass_nonneg (G : BondGraph V E) (P : V → V → ℝ)
+    (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (k : ℕ) (t : ℝ) (x u : V) :
+    0 ≤ derivativeTransportMass G P p k t x u :=
+  tsum_nonneg (fun w => tsum_nonneg (fun y => corridorDerivativeContribution_nonneg G P hP p t ⟨u,w⟩ x y))
+
+theorem derivativeTransportMass_aut [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (P : V → V → ℝ) (hP : ∀ x y, P (a x) (a y) = P x y)
+    (p : unitInterval) (k : ℕ) (t : ℝ) (x u : V) :
+    derivativeTransportMass G P p k t (a x) (a u) = derivativeTransportMass G P p k t x u := by
+  unfold derivativeTransportMass
+  rw [raw_from_sum_eq_masked G hloc k (a u) (fun w => ∑' y, corridorDerivativeContribution G P p t w (a x) y),
+    raw_from_sum_eq_masked G hloc k u (fun w => ∑' y, corridorDerivativeContribution G P p t w x y)]
+  rw [←(RawCorridor.autEquiv hab).tsum_eq]
+  apply tsum_congr
+  intro w
+  simp only [RawCorridor.aut_base,a.injective.eq_iff]
+  by_cases hw : w.base = u
+  · simp only [hw,ite_true]
+    rw [←a.tsum_eq]
+    exact tsum_congr (fun y => corridorDerivativeContribution_aut G hab P hP p t w x y)
+  · simp only [hw,ite_false]
+
+end DerivativeTransport
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section DerivativeTransportSums
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Countable V] [Countable E] [Nonempty V]
+
+theorem summable_swapped_real {α β : Type*} (f : α → β → ℝ)
+    (hs : Summable (fun ab : α × β => f ab.1 ab.2)) :
+    Summable (fun ba : β × α => f ba.2 ba.1) := by
+  exact hs.comp_injective (Equiv.prodComm β α).injective
+
+theorem derivativeTransportMass_row (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
+    (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y) (p : unitInterval)
+    (hp0 : 0 < (p:ℝ)) (hp : p < pc G) (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (x : V) :
+    Summable (derivativeTransportMass G (walkKernel G hloc a) p k t x) ∧
+    (∑' u, derivativeTransportMass G (walkKernel G hloc a) p k t x u) =
+      ∫ ω, deformedDerivativeMass G (walkKernel G hloc a) p k t Set.univ ω x ∂law p := by
+  obtain ⟨hs,he⟩ := derivative_contribution_sum G hloc hqt d hd a hP p hp0 hp k t ht x
+  have hs' : Summable (fun wy : RawCorridor G k × V =>
+      corridorDerivativeContribution G (walkKernel G hloc a) p t wy.1 x wy.2) := by
+    exact summable_swapped_real (fun y w => corridorDerivativeContribution G (walkKernel G hloc a) p t w x y) hs
+  have hr := hs'.prod.sigma
+  refine ⟨hr,?_⟩
+  change (∑' u, ∑' w : RawFrom (openSimpleGraph G Set.univ) k u, ∑' y,
+    corridorDerivativeContribution G (walkKernel G hloc a) p t ⟨u,w⟩ x y) = _
+  rw [←hs'.prod.tsum_sigma,←hs'.tsum_prod]
+  exact ((Equiv.prodComm (RawCorridor G k) V).tsum_eq
+    (fun yw : V × RawCorridor G k => corridorDerivativeContribution G (walkKernel G hloc a) p t yw.2 x yw.1)).trans he
+
+theorem derivativeTransportMass_column (G : BondGraph V E) (hG : Function.Injective G)
+    (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y)
+    (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp : p < pc G)
+    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (u : V) :
+    Summable (fun x => derivativeTransportMass G P p k t x u) ∧
+    (∑' x, derivativeTransportMass G P p k t x u) =
+      ∑' w : RawFrom (openSimpleGraph G Set.univ) k u,
+        ∑' xy : V × V, corridorDerivativeContribution G P p t ⟨u,w⟩ xy.1 xy.2 := by
+  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
+  let := Fintype.ofFinite (RawFrom (openSimpleGraph G Set.univ) k u)
+  have hs (w : RawFrom (openSimpleGraph G Set.univ) k u) :
+      Summable (fun xy : V × V => corridorDerivativeContribution G P p t ⟨u,w⟩ xy.1 xy.2) :=
+    summable_corridorDerivativeTerm_pairs G hG hloc hqt P hP p hp0 hp t ht ⟨u,w⟩
+  have hh : Summable (fun x => ∑ w : RawFrom (openSimpleGraph G Set.univ) k u,
+      ∑' y, corridorDerivativeContribution G P p t ⟨u,w⟩ x y) :=
+    summable_sum (fun w _ => (hs w).prod)
+  refine ⟨by simpa only [derivativeTransportMass,tsum_fintype] using hh,?_⟩
+  simp only [derivativeTransportMass,tsum_fintype]
+  rw [Summable.tsum_finsetSum (fun w _ => (hs w).prod)]
+  exact Finset.sum_congr rfl (fun w _ => (hs w).tsum_prod.symm)
+
+end DerivativeTransportSums
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section AveragedDerivative
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Countable V] [Countable E] [Nonempty V]
+variable {ι : Type*} [Fintype ι]
+
+ 
+
+theorem averaged_derivative_transport (G : BondGraph V E) (hG : Function.Injective G)
+    (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
+    (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y) (p : unitInterval)
+    (hp0 : 0 < (p:ℝ)) (hp : p < pc G) (k : ℕ) (t : ℝ) (ht : 0 ≤ t) :
+    (∑ i, weight i * ∫ ω, deformedDerivativeMass G (walkKernel G hloc a) p k t Set.univ ω (root i) ∂law p) =
+      ∑ i, weight i * ∑' w : RawFrom (openSimpleGraph G Set.univ) k (root i),
+        ∑' xy : V × V, corridorDerivativeContribution G (walkKernel G hloc a) p t ⟨root i,w⟩ xy.1 xy.2 := by
+  have hh := htr.real root weight Γ (derivativeTransportMass G (walkKernel G hloc a) p k t)
+    (fun g hg x u => by
+      obtain ⟨b,hb⟩ := hΓ hg
+      exact derivativeTransportMass_aut G hloc hb _ (walkKernel_covariant G hloc a hb) p k t x u)
+    (fun x => (derivativeTransportMass_row G hloc hqt d hd a hP p hp0 hp k t ht x).1)
+    (fun u => (derivativeTransportMass_column G hG hloc hqt _ hP p hp0 hp k t ht u).1)
+  simpa only [(derivativeTransportMass_row G hloc hqt d hd a hP p hp0 hp k t ht _).2,
+    (derivativeTransportMass_column G hG hloc hqt _ hP p hp0 hp k t ht _).2] using hh
+
+noncomputable def averagedDeformedSusceptibility (G : BondGraph V E) (P : V → V → ℝ)
+    (p : unitInterval) (k : ℕ) (root : ι → V) (weight : ι → ℝ) (t : ℝ) : ℝ :=
+  ∑ i, weight i * deformedSusceptibility G P p k t Set.univ (root i)
+
+theorem hasDerivAt_averagedDeformedSusceptibility (G : BondGraph V E)
+    (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (root : ι → V) (weight : ι → ℝ)
+    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
+    (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y) (p : unitInterval)
+    (hp0 : 0 < (p:ℝ)) (hp : p < pc G) (k : ℕ) (t : ℝ) (ht : 0 < t) :
+    HasDerivAt (averagedDeformedSusceptibility G (walkKernel G hloc a) p k root weight)
+      (-(∑ i, weight i * ∫ ω, deformedDerivativeMass G (walkKernel G hloc a) p k t Set.univ ω (root i) ∂law p)) t := by
+  have he : -(∑ i, weight i * ∫ ω, deformedDerivativeMass G (walkKernel G hloc a) p k t Set.univ ω (root i) ∂law p) =
+      ∑ i, weight i * (-(∫ ω, deformedDerivativeMass G (walkKernel G hloc a) p k t Set.univ ω (root i) ∂law p)) := by
+    simp only [mul_neg,Finset.sum_neg_distrib]
+  rw [he]
+  exact HasDerivAt.fun_sum (u := Finset.univ) (fun i _ =>
+    (hasDerivAt_deformedSusceptibility G hloc hqt d hd a hP p hp0 hp k t ht Set.univ (root i)).const_mul (weight i))
+
+end AveragedDerivative
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section PathAverage
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {ι : Type*} [Fintype ι]
+
+noncomputable def rawPathAverage (G : BondGraph V E) (P : V → V → ℝ) (k : ℕ)
+    (root : ι → V) (weight : ι → ℝ) (f : RawCorridor G k → ℝ) : ℝ :=
+  ∑ i, weight i * ∑' w : RawFrom (openSimpleGraph G Set.univ) k (root i),
+    w.2.mass P * f ⟨root i,w⟩
+
+theorem rawPathAverage_mono (G : BondGraph V E) (hloc : LocallyFinite G)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (k : ℕ)
+    (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 ≤ weight i)
+    (f g : RawCorridor G k → ℝ) (hfg : ∀ w, f w ≤ g w) :
+    rawPathAverage G P k root weight f ≤ rawPathAverage G P k root weight g := by
+  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
+  unfold rawPathAverage
+  apply Finset.sum_le_sum
+  intro i _
+  apply mul_le_mul_of_nonneg_left _ (hw i)
+  let := Fintype.ofFinite (RawFrom (openSimpleGraph G Set.univ) k (root i))
+  simp only [tsum_fintype]
+  exact Finset.sum_le_sum (fun w _ => mul_le_mul_of_nonneg_left (hfg _) (w.2.mass_nonneg P hP))
+
+theorem rawPathAverage_sub (G : BondGraph V E) (hloc : LocallyFinite G)
+    (P : V → V → ℝ) (k : ℕ) (root : ι → V) (weight : ι → ℝ)
+    (f g : RawCorridor G k → ℝ) :
+    rawPathAverage G P k root weight (fun w => f w-g w) =
+      rawPathAverage G P k root weight f - rawPathAverage G P k root weight g := by
+  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
+  have he (i : ι) : (∑' w : RawFrom (openSimpleGraph G Set.univ) k (root i),
+      w.2.mass P * (f ⟨root i,w⟩-g ⟨root i,w⟩)) =
+      (∑' w : RawFrom (openSimpleGraph G Set.univ) k (root i), w.2.mass P*f ⟨root i,w⟩) -
+      (∑' w : RawFrom (openSimpleGraph G Set.univ) k (root i), w.2.mass P*g ⟨root i,w⟩) := by
+    let := Fintype.ofFinite (RawFrom (openSimpleGraph G Set.univ) k (root i))
+    simp only [tsum_fintype,mul_sub,Finset.sum_sub_distrib]
+  simp only [rawPathAverage]
+  simp only [he]
+  simp only [mul_sub,Finset.sum_sub_distrib]
+
+theorem rawPathAverage_const_mul (G : BondGraph V E) (P : V → V → ℝ) (k : ℕ)
+    (root : ι → V) (weight : ι → ℝ) (c : ℝ) (f : RawCorridor G k → ℝ) :
+    rawPathAverage G P k root weight (fun w => c*f w) = c*rawPathAverage G P k root weight f := by
+  unfold rawPathAverage
+  simp only [←mul_left_comm c,tsum_mul_left,Finset.mul_sum]
+
+noncomputable def retainedSusceptibilityProduct (G : BondGraph V E) (P : V → V → ℝ)
+    (p : unitInterval) {k : ℕ} (t : ℝ) (w : RawCorridor G k) : ℝ :=
+  if 3 ≤ w.path.length then
+    deformedSusceptibility G P p k t (w.interior:Set V)ᶜ w.base *
+      deformedSusceptibility G P p k t (w.interior:Set V)ᶜ w.endpoint
+  else 0
+
+end PathAverage
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section AveragedCalibratedDecrease
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Countable V] [Countable E] [Nonempty V]
+variable {ι : Type*} [Fintype ι]
+
+theorem raw_derivative_template_row_lower (G : BondGraph V E) (hG : Function.Injective G)
+    (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (d : ℕ) (hd : ∀ x, degree G x ≤ d)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (hpcrit : 0 < (pc G:ℝ))
+    (p : unitInterval) (hp : p < pc G) (hpnear : (pc G:ℝ)/2 ≤ (p:ℝ))
+    (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T)
+    (k : ℕ) (w : RawCorridor G k)
+    (c : ℝ) (hc : (∑ e ∈ (w.retained_finite hG).toFinset, ∑' r : RawCorridor G k,
+        if e ∈ r.retained then r.weight P p else 0) ≤ c)
+    (t : ℝ) (ht : 0 ≤ t) :
+    Real.exp (-t*c) * (w.probability P *
+      (retainedSusceptibilityProduct G P p t w -
+        (1+2*(d:ℝ)/(pc G:ℝ))*(maxSusceptibility G p)^2*(matrixCoeff (T^3) w.base w.endpoint).re)) ≤
+      ∑' xy : V × V, corridorDerivativeContribution G P p t w xy.1 xy.2 := by
+  by_cases hlen : 3 ≤ w.path.length
+  · have hh := calibrated_derivative_template_row_lower G hG hloc hqt d
+      (fun x => (neighbors_card_le_degree G hloc x).trans (hd x)) P hP hpcrit p hp hpnear T hT k w hlen c hc t ht
+    simpa only [retainedSusceptibilityProduct,hlen,ite_true,corridorDerivativeContribution,
+      mul_assoc,mul_left_comm (Real.exp (-t*c))] using hh
+  · have hC : 0 ≤ 1+2*(d:ℝ)/(pc G:ℝ) := by positivity
+    have hA := (((hT.positiveRealMatrix G p T).pow 3) w.base w.endpoint).2
+    have hpr := w.2.2.mass_nonneg P hP
+    have hr : 0 ≤ ∑' xy : V × V, corridorDerivativeContribution G P p t w xy.1 xy.2 :=
+      tsum_nonneg (fun xy => corridorDerivativeContribution_nonneg G P hP p t w xy.1 xy.2)
+    apply le_trans _ hr
+    simp only [retainedSusceptibilityProduct,hlen,ite_false,zero_sub,mul_neg]
+    exact neg_nonpos.mpr (mul_nonneg (Real.exp_pos _).le (mul_nonneg hpr
+      (mul_nonneg (mul_nonneg hC (sq_nonneg _)) hA)))
+
+ 
+theorem averaged_derivative_raw_lower (G : BondGraph V E) (hG : Function.Injective G)
+    (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 ≤ weight i)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
+    (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y) (hpcrit : 0 < (pc G:ℝ))
+    (p : unitInterval) (hp : p < pc G) (hpnear : (pc G:ℝ)/2 ≤ (p:ℝ))
+    (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T)
+    (k : ℕ) (c : ℝ) (hc : ∀ w : RawCorridor G k,
+      (∑ e ∈ (w.retained_finite hG).toFinset, ∑' r : RawCorridor G k,
+        if e ∈ r.retained then r.weight (walkKernel G hloc a) p else 0) ≤ c)
+    (t : ℝ) (ht : 0 ≤ t) :
+    Real.exp (-t*c) *
+      (rawPathAverage G (walkKernel G hloc a) k root weight (retainedSusceptibilityProduct G (walkKernel G hloc a) p t) -
+       (1+2*(d:ℝ)/(pc G:ℝ))*(maxSusceptibility G p)^2 *
+         rawPathAverage G (walkKernel G hloc a) k root weight (fun w => (matrixCoeff (T^3) w.base w.endpoint).re)) ≤
+      ∑ i, weight i * ∫ ω, deformedDerivativeMass G (walkKernel G hloc a) p k t Set.univ ω (root i) ∂law p := by
+  rw [averaged_derivative_transport G hG hloc hqt root weight Γ hΓ htr d hd a hP p
+      ((half_pos hpcrit).trans_le hpnear) hp k t ht]
+  rw [←rawPathAverage_const_mul G _ k root weight _ (fun w => (matrixCoeff (T^3) w.base w.endpoint).re),
+    ←rawPathAverage_sub G hloc]
+  unfold rawPathAverage
+  rw [Finset.mul_sum]
+  apply Finset.sum_le_sum
+  intro i _
+  rw [mul_left_comm]
+  apply mul_le_mul_of_nonneg_left _ (hw i)
+  rw [←tsum_mul_left]
+  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
+  let := Fintype.ofFinite (RawFrom (openSimpleGraph G Set.univ) k (root i))
+  simp only [tsum_fintype]
+  exact Finset.sum_le_sum (fun w _ => raw_derivative_template_row_lower G hG hloc hqt d hd _ hP
+    hpcrit p hp hpnear T hT k ⟨root i,w⟩ c (hc _) t ht)
+
+end AveragedCalibratedDecrease
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section PathErrorTrace
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {ι : Type*} [Fintype ι]
+
+theorem rawPathAverage_operator_trace (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (A : Hilbert V →L[ℂ] Hilbert V) (hAs : IsSelfAdjoint A)
+    (k : ℕ) (root : ι → V) (weight : ι → ℝ) :
+    rawPathAverage G (walkKernel G hloc a) k root weight
+      (fun w => (matrixCoeff A w.base w.endpoint).re) = (rootTrace root weight (P^k*A)).re := by
+  have he (u : V) : (∑' w : RawFrom (openSimpleGraph G Set.univ) k u,
+      w.2.mass (walkKernel G hloc a) * (matrixCoeff A u w.1).re) = (matrixCoeff (P^k*A) u u).re := by
+    change RawFrom.integral (H := openSimpleGraph G Set.univ) (walkKernel G hloc a) k u
+      (fun v => (matrixCoeff A u v).re) = _
+    rw [raw_integral_operator G hloc a P hpos hP]
+    simp only [matrixCoeff_real_symm A hAs u]
+    exact ((hpos.pow k).mul_hasSum A u u).tsum_eq
+  simp only [rawPathAverage,RawCorridor.base,RawCorridor.endpoint,he,rootTrace,
+    Complex.re_sum,Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero]
+
+open scoped ComplexOrder
+attribute [local instance] spectralRealOperatorScalarTower spectralRealOperatorSMulCommClass
+local instance averagedDecreasePathNonUnitalContinuousFunctionalCalculus : NonUnitalContinuousFunctionalCalculus ℝ
+    (Hilbert V →L[ℂ] Hilbert V) IsSelfAdjoint :=
+  IsSelfAdjoint.instNonUnitalContinuousFunctionalCalculus (A := Hilbert V →L[ℂ] Hilbert V)
+local instance averagedDecreasePathContinuousFunctionalCalculus : ContinuousFunctionalCalculus ℝ
+    (Hilbert V →L[ℂ] Hilbert V) IsSelfAdjoint :=
+  IsSelfAdjoint.instContinuousFunctionalCalculus (A := Hilbert V →L[ℂ] Hilbert V)
+
+ 
+
+theorem rawPathAverage_operator_le_norm (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 ≤ weight i)
+    (Γ : Set (V ≃ V)) (htr : RootTransport root weight Γ)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (hPi : KernelInvariant Γ P) (hPs : IsSelfAdjoint P)
+    (A : Hilbert V →L[ℂ] Hilbert V) (hAi : KernelInvariant Γ A) (hA : 0 ≤ A) (k : ℕ) :
+    rawPathAverage G (walkKernel G hloc a) k root weight
+      (fun w => (matrixCoeff A w.base w.endpoint).re) ≤
+      ‖P‖^k * (rootTrace root weight A).re := by
+  rw [rawPathAverage_operator_trace G hloc a P hpos hP A (IsSelfAdjoint.of_nonneg hA)]
+  have hh := rootTrace_positive_mul_le root weight htr hw A (P^k) hAi (hPi.pow k) hA (hPs.pow k)
+  have hr := (Complex.le_def.mp hh).1
+  simp only [Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero] at hr
+  have hn : ‖P^k‖ ≤ ‖P‖^k := by
+    cases k with
+    | zero =>
+      simp only [pow_zero]
+      have he : (1 : Hilbert V →L[ℂ] Hilbert V) = ContinuousLinearMap.id ℂ (Hilbert V) := by ext f; rfl
+      rw [he]
+      exact ContinuousLinearMap.norm_id_le
+    | succ j => exact norm_pow_le' P (Nat.succ_pos j)
+  exact hr.trans (mul_le_mul_of_nonneg_right hn
+    (Complex.nonneg_iff.mp (rootTrace_nonneg root weight hw A hA)).1)
+
+end PathErrorTrace
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+open scoped ComplexOrder
+section UniformAveragedDecrease
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Countable V] [Countable E] [Nonempty V]
+variable {ι : Type*} [Fintype ι]
+attribute [local instance] spectralRealOperatorScalarTower spectralRealOperatorSMulCommClass
+local instance averagedDecreaseUniformNonUnitalContinuousFunctionalCalculus : NonUnitalContinuousFunctionalCalculus ℝ
+    (Hilbert V →L[ℂ] Hilbert V) IsSelfAdjoint :=
+  IsSelfAdjoint.instNonUnitalContinuousFunctionalCalculus (A := Hilbert V →L[ℂ] Hilbert V)
+local instance averagedDecreaseUniformContinuousFunctionalCalculus : ContinuousFunctionalCalculus ℝ
+    (Hilbert V →L[ℂ] Hilbert V) IsSelfAdjoint :=
+  IsSelfAdjoint.instContinuousFunctionalCalculus (A := Hilbert V →L[ℂ] Hilbert V)
+
+ 
+
+theorem exists_uniform_averaged_decrease (G : BondGraph V E) (hG : Function.Injective G)
+    (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 ≤ weight i)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
+    (hK : ∀ x y, 0 ≤ walkKernel G hloc a x y)
+    (hpcrit : 0 < (pc G:ℝ)) (hpcrit₁ : (pc G:ℝ) < 1)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (hPs : IsSelfAdjoint P) :
+    ∃ C : ℝ, 0 < C ∧ ∀ (p : unitInterval), (pc G:ℝ)/2 ≤ (p:ℝ) → p < pc G →
+      ∀ (T : Hilbert V →L[ℂ] Hilbert V), HasKernel G p T → ∀ (k : ℕ), 3 ≤ k →
+      ∀ (t : ℝ), 0 ≤ t →
+      Real.exp (-t*Real.exp (C*(k:ℝ))) *
+        (rawPathAverage G (walkKernel G hloc a) k root weight
+          (retainedSusceptibilityProduct G (walkKernel G hloc a) p t) -
+          (1+2*(d:ℝ)/(pc G:ℝ))*(maxSusceptibility G p)^2 *
+            (‖P‖^k * (rootTrace root weight (T^3)).re)) ≤
+        ∑ i, weight i * ∫ ω, deformedDerivativeMass G (walkKernel G hloc a) p k t Set.univ ω (root i) ∂law p := by
+  obtain ⟨C,hC,hcost⟩ := exists_uniform_corridor_leg_cost G hG hloc d hd a hK hpcrit hpcrit₁
+  refine ⟨C,hC,?_⟩
+  intro p hpnear hp T hT k hk t ht
+  have hpositive : 0 ≤ T := ContinuousLinearMap.nonneg_iff_isPositive.mpr (hT.isPositive G p T)
+  have hTi : KernelInvariant Γ T := fun g hg => hT.invariant G p T g (hΓ hg)
+  have hPi : KernelInvariant Γ P := by
+    intro g hg x y
+    obtain ⟨b,hb⟩ := hΓ hg
+    simp only [hP,walkKernel_covariant G hloc a hb]
+  have herr := rawPathAverage_operator_le_norm G hloc a root weight hw Γ htr P hpos hP hPi hPs
+    (T^3) (hTi.pow 3) (CStarAlgebra.pow_nonneg T 3 hpositive) k
+  have hCnonneg : 0 ≤ (1+2*(d:ℝ)/(pc G:ℝ))*(maxSusceptibility G p)^2 := by positivity
+  apply le_trans _ (averaged_derivative_raw_lower G hG hloc hqt root weight hw Γ hΓ htr
+    d hd a hK hpcrit p hp hpnear T hT k (Real.exp (C*(k:ℝ))) (hcost p hpnear hp k hk) t ht)
+  exact mul_le_mul_of_nonneg_left (sub_le_sub_left (mul_le_mul_of_nonneg_left herr hCnonneg) _)
+    (Real.exp_pos _).le
+
+end UniformAveragedDecrease
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section DeformedRowComparison
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Countable V] [Countable E] [Nonempty V]
+variable {ι : Type*} [Fintype ι]
+
+ 
+
+theorem exists_uniform_deformed_row_comparison (G : BondGraph V E)
+    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
+    (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i))
+    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
+    (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y)
+    (hpcrit : 0 < (pc G:ℝ)) (hpcrit₁ : (pc G:ℝ) < 1) :
+    ∃ C cf : ℝ, 0 < C ∧ 0 < cf ∧ ∀ (p : unitInterval),
+      (pc G:ℝ)/2 ≤ (p:ℝ) → p < pc G → ∀ (k : ℕ), 3 ≤ k → ∀ (t : ℝ),
+      0 ≤ t → t*Real.exp (C*(k:ℝ)) ≤ 1 → ∀ x,
+      cf*averagedDeformedSusceptibility G (walkKernel G hloc a) p k root weight t ≤
+        deformedSusceptibility G (walkKernel G hloc a) p k t Set.univ x := by
+  obtain ⟨C,hC,hmove⟩ := exists_uniform_deformed_endpoint_comparison G hloc hqt d hd a hP hpcrit hpcrit₁
+  have hex (ij : ι × ι) : ∃ l : List E, Along G (root ij.1) l (root ij.2) := by
+    obtain ⟨l,hl,_,_⟩ := conn_exists_bond_trail G (conn_univ_of_connected G hconn (root ij.1) (root ij.2))
+    exact ⟨l,hl⟩
+  choose path hpath using hex
+  let N := ∑ ij : ι × ι, (path ij).length
+  have hlen (i j : ι) : (path (i,j)).length ≤ N :=
+    Finset.single_le_sum (f := fun ij : ι × ι => (path ij).length) (fun _ _ => Nat.zero_le _) (Finset.mem_univ (i,j))
+  refine ⟨C,Real.exp (-(N:ℝ))*((pc G:ℝ)/2)^N,hC,mul_pos (Real.exp_pos _) (pow_pos (half_pos hpcrit) N),?_⟩
+  intro p hpnear hp k hk t ht htc x
+  obtain ⟨j,g,b,hgb,hgx⟩ := hcover x
+  have hb (i : ι) : (Real.exp (-(N:ℝ))*((pc G:ℝ)/2)^N)*
+      deformedSusceptibility G (walkKernel G hloc a) p k t Set.univ (root i) ≤
+      deformedSusceptibility G (walkKernel G hloc a) p k t Set.univ (root j) := by
+    have hh := hmove p hpnear hp k hk t ht Set.univ (root j) (root i) (path (j,i))
+      (Set.mem_univ _) (hpath (j,i)) (fun _ _ _ _ => Set.mem_univ _)
+    have hpow : ((pc G:ℝ)/2)^N ≤ (p:ℝ)^(path (j,i)).length :=
+      (pow_le_pow_left₀ (half_pos hpcrit).le hpnear N).trans
+        (pow_le_pow_of_le_one p.2.1 p.2.2 (hlen j i))
+    have he : Real.exp (-(N:ℝ)) ≤ Real.exp (-t*((path (j,i)).length*Real.exp (C*(k:ℝ)))) := by
+      apply Real.exp_le_exp.mpr
+      have hh := mul_le_mul_of_nonneg_right htc (Nat.cast_nonneg (path (j,i)).length : (0:ℝ) ≤ _)
+      have hn : ((path (j,i)).length:ℝ) ≤ (N:ℝ) := Nat.cast_le.mpr (hlen j i)
+      nlinarith
+    exact (mul_le_mul_of_nonneg_right (mul_le_mul he hpow (pow_nonneg (half_pos hpcrit).le _) (Real.exp_pos _).le)
+      (deformedSusceptibility_nonneg G _ p k t Set.univ (root i))).trans hh
+  rw [←hgx,deformedSusceptibility_aut G hgb _ (walkKernel_covariant G hloc a hgb)]
+  simp only [Set.preimage_univ]
+  calc
+    _ = ∑ i, weight i*((Real.exp (-(N:ℝ))*((pc G:ℝ)/2)^N)*
+        deformedSusceptibility G (walkKernel G hloc a) p k t Set.univ (root i)) := by
+      simp only [averagedDeformedSusceptibility,Finset.mul_sum]
+      exact Finset.sum_congr rfl (fun _ _ => by ring)
+    _ ≤ ∑ i, weight i*deformedSusceptibility G (walkKernel G hloc a) p k t Set.univ (root j) :=
+      Finset.sum_le_sum (fun i _ => mul_le_mul_of_nonneg_left (hb i) (hw i))
+    _ = _ := by rw [←Finset.sum_mul,hw1,one_mul]
+
+end DeformedRowComparison
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section AnchoredSegments
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {I : Type*} [Fintype I]
+
+ 
+def segmentOrbit (G : BondGraph V E) (γ : I → V) : Set (I → V) :=
+  {s | ∃ a : V ≃ V, ∃ b : E ≃ E, IsAut G a b ∧ s = a ∘ γ}
+
+omit [Fintype I] in
+theorem segmentOrbit_aut_iff (G : BondGraph V E) (γ s : I → V)
+    {a : V ≃ V} {b : E ≃ E} (h : IsAut G a b) :
+    a ∘ s ∈ segmentOrbit G γ ↔ s ∈ segmentOrbit G γ := by
+  constructor
+  · rintro ⟨c,d,hcd,he⟩
+    refine ⟨c.trans a.symm,d.trans b.symm,isAut_trans G hcd (isAut_symm G h),?_⟩
+    funext i
+    have hh := congrFun he i
+    simpa only [Function.comp_apply,Equiv.trans_apply,Equiv.symm_apply_apply] using congrArg a.symm hh
+  · rintro ⟨c,d,hcd,rfl⟩
+    exact ⟨c.trans a,d.trans b,isAut_trans G hcd h,rfl⟩
+
+theorem finite_segmentAnchors (G : BondGraph V E) (hloc : LocallyFinite G)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i : I) (x : V) :
+    {s ∈ segmentOrbit G γ | s i = x}.Finite := by
+  apply (Set.Finite.pi' (fun _ : I => finite_reachBall hloc x n)).subset
+  rintro s ⟨⟨a,b,hab,rfl⟩,hx⟩ j
+  have hh := mem_reachBall_aut G hab (hdiam i j)
+  change a (γ i) = x at hx
+  simpa only [hx,Function.comp_apply] using hh
+
+noncomputable def segmentAnchors (G : BondGraph V E) (hloc : LocallyFinite G)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i : I) (x : V) : Finset (I → V) :=
+  (finite_segmentAnchors G hloc γ n hdiam i x).toFinset
+
+theorem mem_segmentAnchors (G : BondGraph V E) (hloc : LocallyFinite G)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i : I) (x : V) (s : I → V) :
+    s ∈ segmentAnchors G hloc γ n hdiam i x ↔ s ∈ segmentOrbit G γ ∧ s i = x := by
+  exact Set.Finite.mem_toFinset _
+
+theorem segmentAnchors_nonempty_iff (G : BondGraph V E) (hloc : LocallyFinite G)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i : I) (x : V) :
+    (segmentAnchors G hloc γ n hdiam i x).Nonempty ↔ x ∈ vertexOrbit G (γ i) := by
+  constructor
+  · rintro ⟨s,hs⟩
+    obtain ⟨⟨a,b,h,rfl⟩,hx⟩ := (mem_segmentAnchors G hloc γ n hdiam i x s).mp hs
+    exact ⟨a,b,h,hx⟩
+  · rintro ⟨a,b,h,hx⟩
+    exact ⟨a ∘ γ,(mem_segmentAnchors G hloc γ n hdiam i x _).mpr ⟨⟨a,b,h,rfl⟩,hx⟩⟩
+
+end AnchoredSegments
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section AnchoredPositionKernels
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {I : Type*} [Fintype I]
+
+def segmentAut (a : V ≃ V) : (I → V) ≃ (I → V) where
+  toFun s := a ∘ s
+  invFun s := a.symm ∘ s
+  left_inv s := by funext i; exact a.symm_apply_apply (s i)
+  right_inv s := by funext i; exact a.apply_symm_apply (s i)
+
+theorem segmentAnchors_map (G : BondGraph V E) (hloc : LocallyFinite G)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i : I) (x : V)
+    {a : V ≃ V} {b : E ≃ E} (h : IsAut G a b) :
+    segmentAnchors G hloc γ n hdiam i (a x) =
+      (segmentAnchors G hloc γ n hdiam i x).map (segmentAut a).toEmbedding := by
+  ext s
+  rw [Finset.mem_map,mem_segmentAnchors]
+  constructor
+  · rintro ⟨hs,hx⟩
+    refine ⟨a.symm ∘ s,(mem_segmentAnchors G hloc γ n hdiam i x _).mpr ⟨?_,?_⟩,?_⟩
+    · exact (segmentOrbit_aut_iff G γ s (isAut_symm G h)).mpr hs
+    · simp only [Function.comp_apply,hx,Equiv.symm_apply_apply]
+    · funext j; exact a.apply_symm_apply (s j)
+  · rintro ⟨t,ht,rfl⟩
+    obtain ⟨ht,hx⟩ := (mem_segmentAnchors G hloc γ n hdiam i x t).mp ht
+    exact ⟨(segmentOrbit_aut_iff G γ t h).mpr ht,congrArg a hx⟩
+
+theorem segmentAnchors_card_aut (G : BondGraph V E) (hloc : LocallyFinite G)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i : I) (x : V)
+    {a : V ≃ V} {b : E ≃ E} (h : IsAut G a b) :
+    (segmentAnchors G hloc γ n hdiam i (a x)).card = (segmentAnchors G hloc γ n hdiam i x).card := by
+  rw [segmentAnchors_map G hloc γ n hdiam i x h,Finset.card_map]
+
+ 
+
+noncomputable def anchoredPositionKernel (G : BondGraph V E) (hloc : LocallyFinite G)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i j : I) (x y : V) : ℝ :=
+  ∑ s ∈ segmentAnchors G hloc γ n hdiam i x,
+    if y = s j then ((segmentAnchors G hloc γ n hdiam i x).card : ℝ)⁻¹ else 0
+
+theorem anchoredPositionKernel_nonneg (G : BondGraph V E) (hloc : LocallyFinite G)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i j : I) (x y : V) :
+    0 ≤ anchoredPositionKernel G hloc γ n hdiam i j x y := by
+  apply Finset.sum_nonneg
+  intro s _
+  split_ifs <;> positivity
+
+theorem anchoredPositionKernel_aut (G : BondGraph V E) (hloc : LocallyFinite G)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i j : I) (x y : V)
+    {a : V ≃ V} {b : E ≃ E} (h : IsAut G a b) :
+    anchoredPositionKernel G hloc γ n hdiam i j (a x) (a y) =
+      anchoredPositionKernel G hloc γ n hdiam i j x y := by
+  simp only [anchoredPositionKernel,segmentAnchors_map G hloc γ n hdiam i x h,
+    Finset.card_map,Finset.sum_map,Equiv.toEmbedding_apply,
+    segmentAut,Equiv.coe_fn_mk,Function.comp_apply,Equiv.apply_eq_iff_eq]
+
+end AnchoredPositionKernels
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section AnchoredPositionRows
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {I : Type*} [Fintype I]
+
+theorem anchoredPositionKernel_hasSum (G : BondGraph V E) (hloc : LocallyFinite G)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i j : I) (x : V) :
+    HasSum (anchoredPositionKernel G hloc γ n hdiam i j x)
+      (if x ∈ vertexOrbit G (γ i) then (1:ℝ) else 0) := by
+  have hh := hasSum_sum (s := segmentAnchors G hloc γ n hdiam i x)
+    (fun s _ => hasSum_ite_eq (s j) (((segmentAnchors G hloc γ n hdiam i x).card:ℝ)⁻¹))
+  have he : (∑ s ∈ segmentAnchors G hloc γ n hdiam i x,
+      ((segmentAnchors G hloc γ n hdiam i x).card:ℝ)⁻¹) =
+      if x ∈ vertexOrbit G (γ i) then (1:ℝ) else 0 := by
+    simp only [Finset.sum_const,nsmul_eq_mul]
+    by_cases hx : x ∈ vertexOrbit G (γ i)
+    · have hn := Finset.card_pos.mpr ((segmentAnchors_nonempty_iff G hloc γ n hdiam i x).mpr hx)
+      rw [ite_eq_left hx,mul_inv_cancel₀ (Nat.cast_ne_zero.mpr (Nat.ne_of_gt hn))]
+    · have he : segmentAnchors G hloc γ n hdiam i x = ∅ := Finset.not_nonempty_iff_eq_empty.mp
+        (fun h => hx ((segmentAnchors_nonempty_iff G hloc γ n hdiam i x).mp h))
+      simp only [he,Finset.card_empty,Nat.cast_zero,zero_mul,ite_eq_right hx]
+  rw [he] at hh
+  exact hh
+
+theorem anchoredPositionKernel_eq_zero_off_ball (G : BondGraph V E) (hloc : LocallyFinite G)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i j : I) (x y : V)
+    (hxy : y ∉ reachBall G x n) : anchoredPositionKernel G hloc γ n hdiam i j x y = 0 := by
+  apply Finset.sum_eq_zero
+  intro s hs
+  apply ite_eq_right
+  intro hy
+  obtain ⟨⟨a,b,hab,rfl⟩,hx⟩ := (mem_segmentAnchors G hloc γ n hdiam i x s).mp hs
+  change a (γ i) = x at hx
+  change y = a (γ j) at hy
+  exact hxy (by simpa only [hx,←hy] using mem_reachBall_aut G hab (hdiam i j))
+
+theorem anchoredPositionKernel_eq_zero_off_orbits (G : BondGraph V E) (hloc : LocallyFinite G)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i j : I) (x y : V) :
+    x ∉ vertexOrbit G (γ i) ∨ y ∉ vertexOrbit G (γ j) →
+      anchoredPositionKernel G hloc γ n hdiam i j x y = 0 := by
+  intro h
+  apply Finset.sum_eq_zero
+  intro s hs
+  apply ite_eq_right
+  intro hy
+  obtain ⟨⟨a,b,hab,rfl⟩,hx⟩ := (mem_segmentAnchors G hloc γ n hdiam i x s).mp hs
+  rcases h with h | h
+  · exact h ⟨a,b,hab,hx⟩
+  · exact h ⟨a,b,hab,hy.symm⟩
+
+end AnchoredPositionRows
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section AnchoredPositionColumns
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {I : Type*} [Fintype I]
+
+theorem anchoredPositionKernel_columnSummable (G : BondGraph V E) (hloc : LocallyFinite G)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i j : I) (y : V) :
+    Summable (fun x => anchoredPositionKernel G hloc γ n hdiam i j x y) := by
+  apply summable_of_ne_finset_zero (s := (finite_reachBall hloc y n).toFinset)
+  intro x hx
+  simp only [Set.Finite.mem_toFinset] at hx
+  apply Finset.sum_eq_zero
+  intro s hs
+  apply ite_eq_right
+  intro hy
+  obtain ⟨⟨a,b,hab,rfl⟩,he⟩ := (mem_segmentAnchors G hloc γ n hdiam i x s).mp hs
+  change a (γ i) = x at he
+  change y = a (γ j) at hy
+  exact hx (by simpa only [he,←hy] using mem_reachBall_aut G hab (hdiam j i))
+
+theorem anchoredPositionKernel_column_orbit (G : BondGraph V E) (hloc : LocallyFinite G)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i j : I) (y : V) :
+    (∑' x, anchoredPositionKernel G hloc γ n hdiam i j x y) =
+      if y ∈ vertexOrbit G (γ j) then
+        ∑' x, anchoredPositionKernel G hloc γ n hdiam i j x (γ j) else 0 := by
+  by_cases hy : y ∈ vertexOrbit G (γ j)
+  · rw [ite_eq_left hy]
+    obtain ⟨a,b,hab,rfl⟩ := hy
+    rw [←a.tsum_eq]
+    exact tsum_congr (fun x => anchoredPositionKernel_aut G hloc γ n hdiam i j x (γ j) hab)
+  · rw [ite_eq_right hy]
+    calc
+      _ = ∑' _ : V, (0:ℝ) := tsum_congr (fun x =>
+        anchoredPositionKernel_eq_zero_off_orbits G hloc γ n hdiam i j x y (Or.inr hy))
+      _ = 0 := tsum_zero
+
+ 
+
+theorem anchoredPositionKernel_column_mass {ι : Type*} [Fintype ι]
+    (G : BondGraph V E) (hloc : LocallyFinite G)
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G)
+    (htr : RootTransport root weight Γ)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i j : I) :
+    (∑ r, if root r ∈ vertexOrbit G (γ i) then weight r else 0) =
+      (∑ r, if root r ∈ vertexOrbit G (γ j) then weight r else 0) *
+      ∑' x, anchoredPositionKernel G hloc γ n hdiam i j x (γ j) := by
+  have ht := htr.real root weight Γ (anchoredPositionKernel G hloc γ n hdiam i j)
+    (fun a ha x y => by
+      obtain ⟨b,hb⟩ := hΓ ha
+      exact anchoredPositionKernel_aut G hloc γ n hdiam i j x y hb)
+    (fun x => (anchoredPositionKernel_hasSum G hloc γ n hdiam i j x).summable)
+    (anchoredPositionKernel_columnSummable G hloc γ n hdiam i j)
+  calc
+    _ = ∑ r, weight r * ∑' y, anchoredPositionKernel G hloc γ n hdiam i j (root r) y := by
+      apply Finset.sum_congr rfl
+      intro r _
+      rw [(anchoredPositionKernel_hasSum G hloc γ n hdiam i j (root r)).tsum_eq]
+      split_ifs <;> simp only [mul_one,mul_zero]
+    _ = _ := ht
+    _ = _ := by
+      rw [Finset.sum_mul]
+      apply Finset.sum_congr rfl
+      intro r _
+      rw [anchoredPositionKernel_column_orbit]
+      split_ifs <;> simp only [mul_zero,zero_mul]
+
+theorem anchoredPositionKernel_column_ratio {ι : Type*} [Fintype ι]
+    (G : BondGraph V E) (hloc : LocallyFinite G)
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G)
+    (htr : RootTransport root weight Γ)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i j : I)
+    (hj : 0 < ∑ r, if root r ∈ vertexOrbit G (γ j) then weight r else 0) (y : V) :
+    (∑' x, anchoredPositionKernel G hloc γ n hdiam i j x y) =
+      if y ∈ vertexOrbit G (γ j) then
+        (∑ r, if root r ∈ vertexOrbit G (γ i) then weight r else 0) /
+        (∑ r, if root r ∈ vertexOrbit G (γ j) then weight r else 0) else 0 := by
+  have he := anchoredPositionKernel_column_mass G hloc root weight Γ hΓ htr γ n hdiam i j
+  rw [anchoredPositionKernel_column_orbit]
+  congr 1
+  exact (eq_div_iff (ne_of_gt hj)).mpr (by simpa only [mul_comm] using he.symm)
+
+end AnchoredPositionColumns
+end Percolation
+
+ 
+namespace Percolation
+universe u
+variable {V : Type u}
+section AsymmetricSchur
+open scoped ENNReal
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem exists_operator_of_unit_row_column_bound (U : V → V → ℝ) (D : ℝ)
+    (hD : 0 ≤ D) (hU : ∀ x y, 0 ≤ U x y)
+    (hrow : ∀ x (s : Finset V), ∑ y ∈ s, U x y ≤ 1)
+    (hcol : ∀ y (s : Finset V), ∑ x ∈ s, U x y ≤ D) :
+    ∃ A : Hilbert V →L[ℂ] Hilbert V, ‖A‖ ≤ Real.sqrt D ∧
+      ∀ x y, matrixCoeff A x y = (U x y : ℂ) := by
+  have hentry (x y : V) : U x y ≤ 1 := by simpa only [Finset.sum_singleton] using hrow x {y}
+  have hcm (y : V) : Memℓp (fun x => (U x y : ℂ)) 2 := by
+    apply (memℓp_gen_iff'' (by norm_num : 0 < (2 : ℝ≥0∞).toReal)).mpr
+    refine ⟨D,hD,fun s => ?_⟩
+    simp only [ENNReal.toReal_ofNat,Real.rpow_two,Complex.norm_real,Real.norm_eq_abs,sq_abs]
+    exact (Finset.sum_le_sum (fun x _ => by nlinarith [hentry x y,hU x y])).trans (hcol y s)
+  let col (y : V) : Hilbert V := ⟨fun x => (U x y : ℂ),hcm y⟩
+  let L : (V →₀ ℂ) →ₗ[ℂ] Hilbert V := Finsupp.linearCombination ℂ col
+  have hn (f : V →₀ ℂ) : ‖L f‖ ≤ Real.sqrt D * ‖finiteEmbedding f‖ := by
+    apply lp.norm_le_of_forall_sum_le (by norm_num)
+      (mul_nonneg (Real.sqrt_nonneg _) (norm_nonneg _))
+    intro s
+    simp only [ENNReal.toReal_ofNat,Real.rpow_two]
+    have hpoint (x : V) : ‖L f x‖^2 ≤ ∑ y ∈ f.support, ‖f y‖^2 * U x y := by
+      have hn0 : 0 ≤ ∑ y ∈ f.support, ‖f y‖*U x y :=
+        Finset.sum_nonneg (fun y _ => mul_nonneg (norm_nonneg _) (hU x y))
+      have hn1 : ‖L f x‖ ≤ ∑ y ∈ f.support, ‖f y‖*U x y := by
+        simp only [L,Finsupp.linearCombination_apply,Finsupp.sum,lp.coeFn_sum,
+          Finset.sum_apply,lp.coeFn_smul,Pi.smul_apply,smul_eq_mul]
+        apply (norm_sum_le _ _).trans
+        apply Finset.sum_le_sum
+        intro y _
+        simp [col,abs_of_nonneg (hU x y)]
+      have hcs := Finset.sum_sq_le_sum_mul_sum_of_sq_le_mul f.support
+        (f := fun y => U x y) (g := fun y => ‖f y‖^2*U x y)
+        (r := fun y => ‖f y‖*U x y)
+        (fun y _ => hU x y)
+        (fun y _ => mul_nonneg (sq_nonneg _) (hU x y)) (fun y _ => by nlinarith)
+      have hn2 : 0 ≤ ∑ y ∈ f.support, ‖f y‖^2*U x y :=
+        Finset.sum_nonneg (fun y _ => mul_nonneg (sq_nonneg _) (hU x y))
+      have hr := mul_le_mul_of_nonneg_right (hrow x f.support) hn2
+      rw [one_mul] at hr
+      exact (sq_le_sq₀ (norm_nonneg _) hn0 |>.mpr hn1).trans (hcs.trans hr)
+    calc
+      ∑ x ∈ s, ‖L f x‖^2 ≤ ∑ x ∈ s, ∑ y ∈ f.support, ‖f y‖^2*U x y :=
+        Finset.sum_le_sum (fun x _ => hpoint x)
+      _ = ∑ y ∈ f.support, ‖f y‖^2*∑ x ∈ s, U x y := by
+        rw [Finset.sum_comm]
+        simp_rw [Finset.mul_sum]
+      _ ≤ ∑ y ∈ f.support, ‖f y‖^2*D :=
+        Finset.sum_le_sum (fun y _ => mul_le_mul_of_nonneg_left (hcol y s) (sq_nonneg _))
+      _ = D*∑ y ∈ f.support, ‖finiteEmbedding f y‖^2 := by
+        simp only [finiteEmbedding_apply]
+        rw [←Finset.sum_mul,mul_comm]
+      _ ≤ D*‖finiteEmbedding f‖^2 := by
+        apply mul_le_mul_of_nonneg_left _ hD
+        simpa only [ENNReal.toReal_ofNat,Real.rpow_two] using
+          lp.sum_rpow_le_norm_rpow (by norm_num : 0 < (2:ℝ≥0∞).toReal) (finiteEmbedding f) f.support
+      _ = (Real.sqrt D*‖finiteEmbedding f‖)^2 := by rw [mul_pow,Real.sq_sqrt hD]
+  refine ⟨L.extendOfNorm finiteEmbedding,
+    LinearMap.opNorm_extendOfNorm_le finiteEmbedding_dense (Real.sqrt_nonneg _) hn,?_⟩
+  intro x y
+  have he := LinearMap.extendOfNorm_eq (f := L) finiteEmbedding_dense ⟨Real.sqrt D,hn⟩
+    (Finsupp.single y (1:ℂ))
+  simp only [finiteEmbedding_single] at he
+  change (L.extendOfNorm finiteEmbedding) (lp.single 2 y (1:ℂ)) x = _
+  rw [he]
+  simp [L,col]
+
+end AsymmetricSchur
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section AnchoredPositionOperators
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {I ι : Type*} [Fintype I] [Fintype ι]
+
+ 
+
+theorem exists_anchoredPositionOperator (G : BondGraph V E) (hloc : LocallyFinite G)
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G)
+    (htr : RootTransport root weight Γ) (hw : ∀ r, 0 ≤ weight r)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i j : I)
+    (hj : 0 < ∑ r, if root r ∈ vertexOrbit G (γ j) then weight r else 0) :
+    ∃ H : Hilbert V →L[ℂ] Hilbert V,
+      (∀ x y, matrixCoeff H x y = (anchoredPositionKernel G hloc γ n hdiam i j x y : ℂ)) ∧
+      ‖H‖ ≤ Real.sqrt ((∑ r, if root r ∈ vertexOrbit G (γ i) then weight r else 0) /
+        (∑ r, if root r ∈ vertexOrbit G (γ j) then weight r else 0)) := by
+  have hnum : 0 ≤ ∑ r, if root r ∈ vertexOrbit G (γ i) then weight r else 0 :=
+    Finset.sum_nonneg (fun r _ => by split_ifs; exact hw r; exact le_refl 0)
+  have hratio := div_nonneg hnum hj.le
+  obtain ⟨H,hn,hH⟩ := exists_operator_of_unit_row_column_bound
+    (anchoredPositionKernel G hloc γ n hdiam i j) _ hratio
+    (anchoredPositionKernel_nonneg G hloc γ n hdiam i j)
+    (fun x s => ((anchoredPositionKernel_hasSum G hloc γ n hdiam i j x).summable.sum_le_tsum s
+      (fun y _ => anchoredPositionKernel_nonneg G hloc γ n hdiam i j x y)).trans (by
+        rw [(anchoredPositionKernel_hasSum G hloc γ n hdiam i j x).tsum_eq]
+        split_ifs <;> norm_num))
+    (fun y s => ((anchoredPositionKernel_columnSummable G hloc γ n hdiam i j y).sum_le_tsum s
+      (fun x _ => anchoredPositionKernel_nonneg G hloc γ n hdiam i j x y)).trans (by
+        rw [anchoredPositionKernel_column_ratio G hloc root weight Γ hΓ htr γ n hdiam i j hj y]
+        split_ifs; exact le_refl _; exact hratio))
+  exact ⟨H,hH,hn⟩
+
+ 
+theorem exists_anchoredPositionOperator_uniform (G : BondGraph V E) (hloc : LocallyFinite G)
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G)
+    (htr : RootTransport root weight Γ) (hw : ∀ r, 0 ≤ weight r) (hw1 : ∑ r, weight r = 1)
+    (hcov : ∀ x, ∃ r, ∃ a ∈ Γ, a (root r) = x)
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ r, 1 ≤ c*weight r)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i j : I) :
+    ∃ H : Hilbert V →L[ℂ] Hilbert V,
+      (∀ x y, matrixCoeff H x y = (anchoredPositionKernel G hloc γ n hdiam i j x y : ℂ)) ∧
+      ‖H‖ ≤ Real.sqrt c := by
+  let U := anchoredPositionKernel G hloc γ n hdiam i j
+  have hn : ∀ x y, 0 ≤ U x y := anchoredPositionKernel_nonneg G hloc γ n hdiam i j
+  have hr : ∀ x, Summable (U x) := fun x => (anchoredPositionKernel_hasSum G hloc γ n hdiam i j x).summable
+  have hcol := anchoredPositionKernel_columnSummable G hloc γ n hdiam i j
+  have hb (x : V) : (∑' y, U x y) ≤ 1 := by
+    rw [(anchoredPositionKernel_hasSum G hloc γ n hdiam i j x).tsum_eq]
+    split_ifs <;> norm_num
+  have hi : ∀ a ∈ Γ, ∀ x y, U (a x) (a y) = U x y := by
+    intro a ha x y
+    obtain ⟨b,hb⟩ := hΓ ha
+    exact anchoredPositionKernel_aut G hloc γ n hdiam i j x y hb
+  have hbcol (y : V) : (∑' x, U x y) ≤ c := by
+    simpa only [mul_one] using transported_column_le root weight Γ htr hcov hw hw1 c hc hmin U hn hi hr hcol 1 hb y
+  obtain ⟨H,hHn,hH⟩ := exists_operator_of_unit_row_column_bound U c hc hn
+    (fun x s => ((hr x).sum_le_tsum s (fun y _ => hn x y)).trans (hb x))
+    (fun y s => ((hcol y).sum_le_tsum s (fun x _ => hn x y)).trans (hbcol y))
+  exact ⟨H,hH,hHn⟩
+
+end AnchoredPositionOperators
+end Percolation
+
+ 
+namespace Percolation
+universe u
+variable {V : Type u}
+section RawReversal
+namespace HoldingWalk
+variable {H : SimpleGraph V}
+
+ 
+def snocHold : {k : ℕ} → {x y : V} → HoldingWalk H k x y → HoldingWalk H (k+1) x y
+  | _,_,_,.nil x => .hold (.nil x)
+  | _,_,_,.hold w => .hold (snocHold w)
+  | _,_,_,.step h w => .step h (snocHold w)
+
+ 
+def snocStep : {k : ℕ} → {x y z : V} → HoldingWalk H k x y → H.Adj y z → HoldingWalk H (k+1) x z
+  | _,_,_,z,.nil _,h => .step h (.nil z)
+  | _,_,_,_,.hold w,h => .hold (snocStep w h)
+  | _,_,_,_,.step h' w,h => .step h' (snocStep w h)
+
+ 
+
+def reverse : {k : ℕ} → {x y : V} → HoldingWalk H k x y → HoldingWalk H k y x
+  | _,_,_,.nil x => .nil x
+  | _,_,_,.hold w => snocHold (reverse w)
+  | _,_,_,.step h w => snocStep (reverse w) h.symm
+
+theorem reverse_snocHold {k : ℕ} {x y : V} (w : HoldingWalk H k x y) :
+    w.snocHold.reverse = w.reverse.hold := by
+  induction w with
+  | nil => rfl
+  | hold w ih => simp only [snocHold,reverse,ih]
+  | step h w ih => simp only [snocHold,reverse,ih,snocStep]
+
+theorem reverse_snocStep {k : ℕ} {x y z : V} (w : HoldingWalk H k x y) (h : H.Adj y z) :
+    (w.snocStep h).reverse = .step h.symm w.reverse := by
+  induction w with
+  | nil => rfl
+  | hold w ih => simp only [snocStep,reverse,ih,snocHold]
+  | step h' w ih => simp only [snocStep,reverse,ih]
+
+theorem reverse_reverse {k : ℕ} {x y : V} (w : HoldingWalk H k x y) : w.reverse.reverse = w := by
+  induction w with
+  | nil => rfl
+  | hold w ih => simp only [reverse,reverse_snocHold,ih]
+  | step h w ih => simp only [reverse,reverse_snocStep,ih]
+
+def reverseEquiv (k : ℕ) (x y : V) : HoldingWalk H k x y ≃ HoldingWalk H k y x where
+  toFun := reverse
+  invFun := reverse
+  left_inv := reverse_reverse
+  right_inv := reverse_reverse
+
+theorem mass_snocHold (P : V → V → ℝ) {k : ℕ} {x y : V} (w : HoldingWalk H k x y) :
+    w.snocHold.mass P = w.mass P * P y y := by
+  induction w with
+  | nil => simp only [snocHold,mass,mul_one,one_mul]
+  | hold w ih => simp only [snocHold,mass,ih,mul_assoc]
+  | step h w ih => simp only [snocHold,mass,ih,mul_assoc]
+
+theorem mass_snocStep (P : V → V → ℝ) {k : ℕ} {x y z : V} (w : HoldingWalk H k x y) (h : H.Adj y z) :
+    (w.snocStep h).mass P = w.mass P * P y z := by
+  induction w with
+  | nil => simp only [snocStep,mass,mul_one,one_mul]
+  | hold w ih => simp only [snocStep,mass,ih,mul_assoc]
+  | step h' w ih => simp only [snocStep,mass,ih,mul_assoc]
+
+theorem mass_reverse (P : V → V → ℝ) (hP : ∀ x y, P x y = P y x)
+    {k : ℕ} {x y : V} (w : HoldingWalk H k x y) : w.reverse.mass P = w.mass P := by
+  induction w with
+  | nil => rfl
+  | hold w ih => simp only [reverse,mass_snocHold,mass,ih,mul_comm]
+  | step h w ih => simp only [reverse,mass_snocStep,mass,ih,hP,mul_comm]
+
+theorem vertices_snocHold {k : ℕ} {x y : V} (w : HoldingWalk H k x y) :
+    w.snocHold.vertices = w.vertices ++ [y] := by
+  induction w with
+  | nil => rfl
+  | hold w ih => simp only [snocHold,vertices,ih,List.cons_append]
+  | step h w ih => simp only [snocHold,vertices,ih,List.cons_append]
+
+theorem vertices_snocStep {k : ℕ} {x y z : V} (w : HoldingWalk H k x y) (h : H.Adj y z) :
+    (w.snocStep h).vertices = w.vertices ++ [z] := by
+  induction w with
+  | nil => rfl
+  | hold w ih => simp only [snocStep,vertices,ih,List.cons_append]
+  | step h' w ih => simp only [snocStep,vertices,ih,List.cons_append]
+
+theorem vertices_reverse {k : ℕ} {x y : V} (w : HoldingWalk H k x y) :
+    w.reverse.vertices = w.vertices.reverse := by
+  induction w with
+  | nil => rfl
+  | hold w ih => simp only [reverse,vertices_snocHold,vertices,ih,List.reverse_cons]
+  | step h w ih => simp only [reverse,vertices_snocStep,vertices,ih,List.reverse_cons]
+
+end HoldingWalk
+end RawReversal
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Deformation/Calibration.lean b/lean/OAI/Probability/UnimodularPercolation/Deformation/Calibration.lean
index 7b7983e0d30f3958b244ec88db58bdd86f7b03c7..791e378349d3f3b35d103a756138f070c6b71480 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Deformation/Calibration.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Deformation/Calibration.lean
@@ -1,1247 +1,1247 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Deformation.Derivatives
-
-namespace OAI
-
-/-! Calibration for bond percolation. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section UniformDeformedEndpoint
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem exists_uniform_deformed_endpoint_comparison [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
-    (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y)
-    (hpcrit : 0 < (pc G:ℝ)) (hpcrit₁ : (pc G:ℝ) < 1) :
-    ∃ C : ℝ, 0 < C ∧ ∀ (p : unitInterval), (pc G:ℝ)/2 ≤ (p:ℝ) → p < pc G →
-      ∀ (k : ℕ), 3 ≤ k → ∀ (t : ℝ), 0 ≤ t → ∀ (D : Set V) (b z : V) (l : List E),
-      b ∈ D → Along G b l z → (∀ e ∈ l, e ∈ internalEdges G D) →
-      Real.exp (-t*((l.length:ℝ)*Real.exp (C*(k:ℝ)))) * (p:ℝ)^l.length *
-        deformedSusceptibility G (walkKernel G hloc a) p k t D z ≤
-        deformedSusceptibility G (walkKernel G hloc a) p k t D b := by
-  obtain ⟨C,hC,hcost⟩ := exists_uniform_corridor_edge_cost G hloc d hd a hP hpcrit hpcrit₁
-  refine ⟨C,hC,?_⟩
-  intro p hpnear hp k hk t ht D b z l hb hl hD
-  have hc : ∀ e, (∑' w : RawCorridor G k,
-      if e ∈ w.retained then w.weight (walkKernel G hloc a) p else 0) ≤ Real.exp (C*(k:ℝ)) := by
-    intro e
-    exact (hcost p hpnear hp k hk e).trans (div_le_self (Real.exp_pos _).le (by exact_mod_cast (show 1 ≤ k by omega)))
-  have hh := deformedSusceptibility_move_lower G hloc hqt (walkKernel G hloc a) hP p hp k t ht D
-    l.toFinset (fun e he => hD e (List.mem_toFinset.mp he)) b z hb
-    (hl.conn (fun e he => List.mem_toFinset.mpr he)) (Real.exp (C*(k:ℝ))) hc
-  have hcard : (l.toFinset.card:ℝ) ≤ (l.length:ℝ) := Nat.cast_le.mpr l.toFinset_card_le
-  have he : Real.exp (-t*((l.length:ℝ)*Real.exp (C*(k:ℝ)))) ≤
-      Real.exp (-t*((l.toFinset.card:ℝ)*Real.exp (C*(k:ℝ)))) := by
-    apply Real.exp_le_exp.mpr
-    exact mul_le_mul_of_nonpos_left (mul_le_mul_of_nonneg_right hcard (Real.exp_pos _).le) (neg_nonpos.mpr ht)
-  have hpw : (p:ℝ)^l.length ≤ (p:ℝ)^l.toFinset.card :=
-    pow_le_pow_of_le_one p.2.1 p.2.2 l.toFinset_card_le
-  exact (mul_le_mul_of_nonneg_right (mul_le_mul he hpw (pow_nonneg p.2.1 _) (Real.exp_pos _).le)
-    (tsum_nonneg (deformedKernel_nonneg G (walkKernel G hloc a) p k t D z))).trans hh
-
-end UniformDeformedEndpoint
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section PivotalLegGeometry
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem pivotal_endpoint_connected (G : BondGraph V E) (S : Set E) (x y : V)
-    (hc : Conn G S x y) (e : E) (hp : ¬Conn G (S \ {e}) x y) (z : V) (hz : z ∈ G e) :
-    Conn G S x z := by
-  obtain ⟨q⟩ := (conn_iff_reachable G S x y).mp hc
-  have hzs : z ∈ q.support := SimpleGraph.Walk.mem_support_of_mem_edges
-    (pivotal_image_mem_edges G S q e hp) hz
-  exact (conn_iff_reachable G S x z).mpr ⟨q.takeUntil z hzs⟩
-
-theorem conn_delete_of_unconnected_endpoint (G : BondGraph V E) (S : Set E) (x y z : V)
-    (hc : Conn G S x y) (e : E) (hz : z ∈ G e) (hn : ¬Conn G S x z) :
-    Conn G (S \ {e}) x y := by
-  by_contra hh
-  exact hn (pivotal_endpoint_connected G S x y hc e hh z hz)
-
-namespace RawCorridor
-variable {G : BondGraph V E} {k : ℕ}
-
-theorem retained_conn_base (w : RawCorridor G k) (z : V) (hz : z ∈ w.path.support) :
-    Conn G w.retained w.base z := by
-  obtain ⟨l,hl,hm,_⟩ := walk_lift_bonds G Set.univ (w.path.takeUntil z hz)
-  apply hl.conn
-  intro e he
-  exact w.path.edges_takeUntil_subset_edges hz (hm ▸ List.mem_map_of_mem he)
-
-theorem retained_conn (w : RawCorridor G k) : Conn G w.retained w.base w.endpoint :=
-  w.retained_conn_base w.endpoint w.path.end_mem_support
-
-theorem retained_connects_support (w : RawCorridor G k) (x y : V)
-    (hx : x ∈ w.path.support) (hy : y ∈ w.path.support) : Conn G w.retained x y :=
-  conn_trans G _ (conn_symm G _ (w.retained_conn_base x hx)) (w.retained_conn_base y hy)
-
-theorem retained_nonempty (w : RawCorridor G k) (hl : 0 < w.path.length) : w.retained.Nonempty := by
-  have hp : w.path.edges ≠ [] := by
-    intro he
-    have hh := congrArg List.length he
-    simp only [SimpleGraph.Walk.length_edges,List.length_nil] at hh
-    omega
-  obtain ⟨s,hs⟩ := List.exists_mem_of_ne_nil _ hp
-  obtain ⟨e,he,_⟩ := w.edge_has_label s hs
-  exact ⟨e,he⟩
-
-end RawCorridor
-end PivotalLegGeometry
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section OneLegGeometry
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace RawCorridor
-variable {G : BondGraph V E} {k : ℕ}
-
- 
-
-theorem counted_one_leg (r : RawCorridor G k) (D : Set V) (ω B : Set E)
-    (x y u v : V) (hc : r.counted Set.univ ω x y)
-    (hr : r.retained ⊆ ω ∩ internalEdges G D) (hB : B ⊆ ω)
-    (hd : Disjoint B r.retained) (huv : Conn G B u v)
-    (hxu : Conn G (ω ∩ internalEdges G D) x u)
-    (hvy : Conn G (ω ∩ internalEdges G D) v y)
-    (hne : ¬Conn G (ω ∩ internalEdges G D) u v)
-    (e : E) (he : e ∈ r.retained)
-    (hep : ¬Conn G ((ω ∩ internalEdges G D) \ {e}) x u) : r.counted D ω x u := by
-  let S := ω ∩ internalEdges G D
-  have huniv : ω ∩ internalEdges G Set.univ = ω := by
-    apply Set.Subset.antisymm Set.inter_subset_left
-    exact fun _ he => ⟨he,fun _ _ => trivial⟩
-  obtain ⟨z,z',hzz⟩ : ∃ z z', G e = s(z,z') := by
-    obtain ⟨⟨z,z'⟩,hh⟩ := Quot.exists_rep (G e)
-    exact ⟨z,z',hh.symm⟩
-  have hz : z ∈ G e := hzz.symm ▸ Sym2.mem_mk_left z z'
-  have hxz : Conn G S x z := pivotal_endpoint_connected G S x u hxu e hep z hz
-  have hzsupport : z ∈ r.path.support := SimpleGraph.Walk.mem_support_of_mem_edges he hz
-  apply (r.counted_iff D ω x u).mpr
-  refine ⟨hc.1,hxu,?_,hc.2.2.2.2.mono_right ?_⟩
-  · intro f hf
-    refine ⟨hr hf,?_⟩
-    intro hxf
-    obtain ⟨a,b,hab⟩ : ∃ a b, G f = s(a,b) := by
-      obtain ⟨⟨a,b⟩,hh⟩ := Quot.exists_rep (G f)
-      exact ⟨a,b,hh.symm⟩
-    have ha : a ∈ G f := hab.symm ▸ Sym2.mem_mk_left a b
-    have haz : Conn G S z a := conn_mono G hr (r.retained_connects_support z a hzsupport
-      (SimpleGraph.Walk.mem_support_of_mem_edges hf ha))
-    have hua : Conn G S u a := conn_trans G S (conn_symm G S hxu) (conn_trans G S hxz haz)
-    have hvf : Conn G (S \ {f}) v y := conn_delete_of_unconnected_endpoint G S v y a hvy f ha
-      (fun hva => hne (conn_trans G S hua (conn_symm G S hva)))
-    have hBf : Conn G (ω \ {f}) u v := conn_mono G (show B ⊆ ω \ {f} from fun g hg => ⟨hB hg,fun hgf =>
-      Set.disjoint_left.mp hd hg ((Set.mem_singleton_iff.mp hgf).symm ▸ hf)⟩) huv
-    apply (hc.2.2.2.1 f hf).2
-    rw [huniv]
-    exact conn_trans G _ (conn_mono G (Set.sdiff_subset_sdiff_left Set.inter_subset_left) hxf)
-      (conn_trans G _ hBf (conn_mono G (Set.sdiff_subset_sdiff_left Set.inter_subset_left) hvf))
-  · rw [huniv]
-    exact Set.inter_subset_left
-
-end RawCorridor
-end OneLegGeometry
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section TwoLegGeometry
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace RawCorridor
-variable {G : BondGraph V E} {k : ℕ}
-
-theorem counted_split_legs (r : RawCorridor G k) (D : Set V) (ω B : Set E)
-    (x y u v : V) (hc : r.counted Set.univ ω x y)
-    (hr : r.retained ⊆ ω ∩ internalEdges G D) (hB : B ⊆ ω)
-    (hd : Disjoint B r.retained) (huv : Conn G B u v)
-    (hxu : Conn G (ω ∩ internalEdges G D) x u)
-    (hvy : Conn G (ω ∩ internalEdges G D) v y)
-    (hne : ¬Conn G (ω ∩ internalEdges G D) u v) :
-    r.counted D ω x u ∨ r.counted D ω v y := by
-  obtain ⟨e,he⟩ := r.retained_nonempty (by have := hc.1; omega)
-  by_cases hp : ¬Conn G ((ω ∩ internalEdges G D) \ {e}) x u
-  · exact Or.inl (r.counted_one_leg D ω B x y u v hc hr hB hd huv hxu hvy hne e he hp)
-  · have hxu' : Conn G ((ω ∩ internalEdges G D) \ {e}) x u := not_not.mp hp
-    have hp' : ¬Conn G ((ω ∩ internalEdges G D) \ {e}) y v := by
-      intro hyv
-      apply (hc.2.2.2.1 e he).2
-      have huniv : ω ∩ internalEdges G Set.univ = ω := by
-        apply Set.Subset.antisymm Set.inter_subset_left
-        exact fun _ he => ⟨he,fun _ _ => trivial⟩
-      rw [huniv]
-      have hB' : Conn G (ω \ {e}) u v := conn_mono G (show B ⊆ ω \ {e} from fun f hf => ⟨hB hf,fun hfe =>
-        Set.disjoint_left.mp hd hf ((Set.mem_singleton_iff.mp hfe).symm ▸ he)⟩) huv
-      exact conn_trans G _ (conn_mono G (Set.sdiff_subset_sdiff_left Set.inter_subset_left) hxu')
-        (conn_trans G _ hB' (conn_mono G (Set.sdiff_subset_sdiff_left Set.inter_subset_left) (conn_symm G _ hyv)))
-    exact Or.inr ((r.counted_symm D ω y v).mp (r.counted_one_leg D ω B y x v u
-      ((r.counted_symm Set.univ ω x y).mp hc) hr hB hd (conn_symm G B huv)
-      (conn_symm G _ hvy) (conn_symm G _ hxu) (fun h => hne (conn_symm G _ h)) e he hp'))
-
- 
-theorem counted_legs_of_disjoint (w r : RawCorridor G k) (ω : Set E)
-    (hq : ω ∈ w.calibrationEvent) (x y : V)
-    (hxu : Conn G (w.exterior ω) x w.base)
-    (hvy : Conn G (w.exterior ω) w.endpoint y)
-    (hne : ¬Conn G (w.exterior ω) w.base w.endpoint)
-    (hc : r.counted Set.univ ω x y) (hd : Disjoint w.retained r.retained) :
-    r.counted (w.interior : Set V)ᶜ ω x w.base ∨ r.counted (w.interior : Set V)ᶜ ω w.endpoint y := by
-  apply r.counted_split_legs (w.interior : Set V)ᶜ ω w.retained x y w.base w.endpoint
-    hc ?_ hq.1 hd w.retained_conn hxu hvy hne
-  intro e he
-  have heω := (hc.2.2.2.1 e he).1.1
-  exact (w.open_subset_exterior_retained ω hq heω).resolve_right
-    (fun hw => Set.disjoint_left.mp hd hw he)
-
-end RawCorridor
-end TwoLegGeometry
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section TwoLegPenaltyTerms
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem corridorPenaltyTerm_legs (G : BondGraph V E) (hG : Function.Injective G)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (k : ℕ)
-    (w r : RawCorridor G k) (ω : Set E) (hq : ω ∈ w.calibrationEvent) (x y : V)
-    (hxu : Conn G (w.exterior ω) x w.base) (hvy : Conn G (w.exterior ω) w.endpoint y)
-    (hne : ¬Conn G (w.exterior ω) w.base w.endpoint) :
-    corridorPenaltyTerm G P p Set.univ ω x y r ≤
-      (∑ e ∈ (w.retained_finite hG).toFinset, if e ∈ r.retained then r.weight P p else 0) +
-        corridorPenaltyTerm G P p (w.interior : Set V)ᶜ ω x w.base r +
-        corridorPenaltyTerm G P p (w.interior : Set V)ᶜ ω w.endpoint y r := by
-  have hr := r.weight_nonneg P hP p
-  have hs : 0 ≤ ∑ e ∈ (w.retained_finite hG).toFinset, if e ∈ r.retained then r.weight P p else 0 :=
-    Finset.sum_nonneg (fun _ _ => ite_nonneg hr (le_refl _))
-  have hl := corridorPenaltyTerm_nonneg G P hP p (w.interior : Set V)ᶜ ω x w.base r
-  have ht := corridorPenaltyTerm_nonneg G P hP p (w.interior : Set V)ᶜ ω w.endpoint y r
-  by_cases hc : r.counted Set.univ ω x y
-  · rw [corridorPenaltyTerm,ite_eq_left hc]
-    by_cases hd : Disjoint w.retained r.retained
-    · rcases w.counted_legs_of_disjoint r ω hq x y hxu hvy hne hc hd with hh | hh
-      · have he : corridorPenaltyTerm G P p (w.interior : Set V)ᶜ ω x w.base r = r.weight P p := ite_eq_left hh
-        linarith
-      · have he : corridorPenaltyTerm G P p (w.interior : Set V)ᶜ ω w.endpoint y r = r.weight P p := ite_eq_left hh
-        linarith
-    · obtain ⟨e,hew,her⟩ := Set.not_disjoint_iff.mp hd
-      have he := Finset.single_le_sum (f := fun e => if e ∈ r.retained then r.weight P p else 0)
-        (s := (w.retained_finite hG).toFinset) (fun _ _ => ite_nonneg hr (le_refl _))
-        ((w.retained_finite hG).mem_toFinset.mpr hew)
-      rw [ite_eq_left her] at he
-      linarith
-  · rw [corridorPenaltyTerm,ite_eq_right hc]
-    linarith
-
- 
-
-theorem corridorPenalty_legs (G : BondGraph V E) (hG : Function.Injective G)
-    (hloc : LocallyFinite G) (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y)
-    (p : unitInterval) (k : ℕ) (w : RawCorridor G k) (ω : Set E)
-    (hq : ω ∈ w.calibrationEvent) (x y : V)
-    (hxu : Conn G (w.exterior ω) x w.base) (hvy : Conn G (w.exterior ω) w.endpoint y)
-    (hne : ¬Conn G (w.exterior ω) w.base w.endpoint) :
-    corridorPenalty G P p k Set.univ ω x y ≤
-      (∑ e ∈ (w.retained_finite hG).toFinset, ∑' r : RawCorridor G k,
-        if e ∈ r.retained then r.weight P p else 0) +
-      corridorPenalty G P p k (w.interior : Set V)ᶜ ω x w.base +
-      corridorPenalty G P p k (w.interior : Set V)ᶜ ω w.endpoint y := by
-  have hs : Summable (fun r : RawCorridor G k => ∑ e ∈ (w.retained_finite hG).toFinset,
-      if e ∈ r.retained then r.weight P p else 0) :=
-    (hasSum_sum (fun e (_ : e ∈ (w.retained_finite hG).toFinset) =>
-      (summable_raw_edge_weights G hloc P p k e).hasSum)).summable
-  have hl := corridorPenalty_summable G hloc P p k (w.interior : Set V)ᶜ ω x w.base
-  have hr := corridorPenalty_summable G hloc P p k (w.interior : Set V)ᶜ ω w.endpoint y
-  have hh := (corridorPenalty_summable G hloc P p k Set.univ ω x y).tsum_le_tsum
-    (fun r => corridorPenaltyTerm_legs G hG P hP p k w r ω hq x y hxu hvy hne) ((hs.add hl).add hr)
-  rw [(hs.add hl).tsum_add hr,hs.tsum_add hl,
-    Summable.tsum_finsetSum (fun e _ => summable_raw_edge_weights G hloc P p k e)] at hh
-  exact hh
-
-end TwoLegPenaltyTerms
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section WeightedLegComparison
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-namespace RawCorridor
-variable {G : BondGraph V E} {k : ℕ}
-theorem base_not_interior (w : RawCorridor G k) : w.base ∉ w.interior := by
-  simp only [interior,Finset.mem_sdiff,Finset.mem_insert,true_or,not_true_eq_false,and_false,not_false_eq_true]
-theorem endpoint_not_interior (w : RawCorridor G k) : w.endpoint ∉ w.interior := by
-  simp only [interior,Finset.mem_sdiff,Finset.mem_insert,Finset.mem_singleton,or_true,not_true_eq_false,and_false,not_false_eq_true]
-end RawCorridor
-
- 
-theorem deformedConnection_legs_lower (G : BondGraph V E) (hG : Function.Injective G)
-    (hloc : LocallyFinite G) (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y)
-    (p : unitInterval) (k : ℕ) (w : RawCorridor G k) (ω : Set E)
-    (hq : ω ∈ w.calibrationEvent) (x y : V)
-    (hxu : Conn G (w.exterior ω) x w.base) (hvy : Conn G (w.exterior ω) w.endpoint y)
-    (hne : ¬Conn G (w.exterior ω) w.base w.endpoint) (c : ℝ)
-    (hc : (∑ e ∈ (w.retained_finite hG).toFinset, ∑' r : RawCorridor G k,
-        if e ∈ r.retained then r.weight P p else 0) ≤ c) (t : ℝ) (ht : 0 ≤ t) :
-    Real.exp (-t*c) * deformedConnection G P p k t (w.interior : Set V)ᶜ ω x w.base *
-      deformedConnection G P p k t (w.interior : Set V)ᶜ ω w.endpoint y ≤
-        deformedConnection G P p k t Set.univ ω x y := by
-  have hx : x ∈ (w.interior : Set V)ᶜ := by
-    rcases conn_internal_eq_or_mem G (w.interior : Set V)ᶜ hxu with h | h
-    · rw [h]; exact w.base_not_interior
-    · exact h.1
-  have hy : y ∈ (w.interior : Set V)ᶜ := by
-    rcases conn_internal_eq_or_mem G (w.interior : Set V)ᶜ hvy with h | h
-    · rw [←h]; exact w.endpoint_not_interior
-    · exact h.2
-  have hxy : Conn G (ω ∩ internalEdges G Set.univ) x y := by
-    apply conn_mono G (show ω ⊆ ω ∩ internalEdges G Set.univ from fun _ h => ⟨h,fun _ _ => trivial⟩)
-    exact conn_trans G ω (conn_mono G Set.inter_subset_left hxu)
-      (conn_trans G ω (conn_mono G hq.1 w.retained_conn) (conn_mono G Set.inter_subset_left hvy))
-  have hleft : x ∈ (w.interior : Set V)ᶜ ∧ w.base ∈ (w.interior : Set V)ᶜ ∧
-      Conn G (ω ∩ internalEdges G (w.interior : Set V)ᶜ) x w.base := ⟨hx,w.base_not_interior,hxu⟩
-  have hright : w.endpoint ∈ (w.interior : Set V)ᶜ ∧ y ∈ (w.interior : Set V)ᶜ ∧
-      Conn G (ω ∩ internalEdges G (w.interior : Set V)ᶜ) w.endpoint y := ⟨w.endpoint_not_interior,hy,hvy⟩
-  simp only [deformedConnection,ite_eq_left hleft,ite_eq_left hright,Set.mem_univ,hxy,and_self,ite_true]
-  rw [←Real.exp_add,←Real.exp_add]
-  apply Real.exp_le_exp.mpr
-  have hh := (corridorPenalty_legs G hG hloc P hP p k w ω hq x y hxu hvy hne).trans
-    (add_le_add (add_le_add hc le_rfl) le_rfl)
-  simpa only [mul_add] using mul_le_mul_of_nonpos_left hh (neg_nonpos.mpr ht)
-
-end WeightedLegComparison
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section UniformLegCost
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem exists_uniform_corridor_leg_cost (G : BondGraph V E) (hG : Function.Injective G)
-    (hloc : LocallyFinite G) (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
-    (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y)
-    (hpcrit : 0 < (pc G:ℝ)) (hpcrit₁ : (pc G:ℝ) < 1) :
-    ∃ C : ℝ, 0 < C ∧ ∀ (p : unitInterval), (pc G:ℝ)/2 ≤ (p:ℝ) → p < pc G →
-      ∀ (k : ℕ), 3 ≤ k → ∀ w : RawCorridor G k,
-      (∑ e ∈ (w.retained_finite hG).toFinset, ∑' r : RawCorridor G k,
-        if e ∈ r.retained then r.weight (walkKernel G hloc a) p else 0) ≤ Real.exp (C*(k:ℝ)) := by
-  obtain ⟨C,hC,hcost⟩ := exists_uniform_corridor_edge_cost G hloc d hd a hP hpcrit hpcrit₁
-  refine ⟨C,hC,?_⟩
-  intro p hpnear hp k hk w
-  have hh := Finset.sum_le_sum (s := (w.retained_finite hG).toFinset)
-    (fun e _ => hcost p hpnear hp k hk e)
-  have hcard : ((w.retained_finite hG).toFinset.card:ℝ) ≤ (k:ℝ) := by
-    rw [←Set.ncard_eq_toFinset_card w.retained (w.retained_finite hG),w.retained_ncard hG]
-    exact Nat.cast_le.mpr w.path_length
-  have hk0 : (0:ℝ) < k := Nat.cast_pos.mpr (by omega)
-  calc
-    _ ≤ ((w.retained_finite hG).toFinset.card:ℝ)*(Real.exp (C*(k:ℝ))/(k:ℝ)) := by
-      simpa only [Finset.sum_const,nsmul_eq_mul] using hh
-    _ ≤ (k:ℝ)*(Real.exp (C*(k:ℝ))/(k:ℝ)) := mul_le_mul_of_nonneg_right hcard (div_nonneg (Real.exp_pos _).le hk0.le)
-    _ = _ := by field_simp
-
-end UniformLegCost
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section SimplePathCuts
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem simplePath_edge_cut {H : SimpleGraph V} {u v : V} (q : H.Walk u v)
-    (hq : q.IsPath) (e : Sym2 V) (he : e ∈ q.edges) :
-    ∃ A : Set V, u ∈ A ∧ v ∉ A ∧ (∀ z ∈ A, z ∈ q.support) ∧
-      ∀ f ∈ q.edges, f ≠ e → ∀ a ∈ f, ∀ b ∈ f, a ∈ A ↔ b ∈ A := by
-  induction q with
-  | nil => simp only [SimpleGraph.Walk.edges_nil,List.not_mem_nil] at he
-  | @cons u a v h q ih =>
-    have hu : u ∉ q.support := ((SimpleGraph.Walk.cons_isPath_iff _ _).mp hq).2
-    have huv : v ≠ u := fun hvu => hu (hvu ▸ q.end_mem_support)
-    by_cases hh : e = s(u,a)
-    · refine ⟨{u},rfl,huv,?_,?_⟩
-      · intro z hz; rw [Set.mem_singleton_iff] at hz; subst z; simp
-      · intro f hf hfe x hx y hy
-        have hfq : f ∈ q.edges := (List.mem_cons.mp hf).resolve_left (fun h' => hfe (h'.trans hh.symm))
-        have hxu : x ≠ u := fun hxu => hu (hxu ▸ SimpleGraph.Walk.mem_support_of_mem_edges hfq hx)
-        have hyu : y ≠ u := fun hyu => hu (hyu ▸ SimpleGraph.Walk.mem_support_of_mem_edges hfq hy)
-        simp only [Set.mem_singleton_iff,hxu,hyu]
-    · have heq : e ∈ q.edges := (List.mem_cons.mp he).resolve_left hh
-      obtain ⟨A,ha,hv,hA,hcut⟩ := ih hq.of_cons heq
-      refine ⟨insert u A,Set.mem_insert u A,?_,?_,?_⟩
-      · intro hvA
-        rcases hvA with h' | h'
-        · exact huv h'
-        · exact hv h'
-      · intro z hz
-        rcases hz with rfl | hz
-        · simp
-        · exact List.mem_cons_of_mem u (hA z hz)
-      · intro f hf hfe x hx y hy
-        rcases List.mem_cons.mp hf with rfl | hf
-        · have hx' : x = u ∨ x = a := by simpa only [SimpleGraph.Dart.edge,Sym2.mem_iff] using hx
-          have hy' : y = u ∨ y = a := by simpa only [SimpleGraph.Dart.edge,Sym2.mem_iff] using hy
-          have hxm : x ∈ insert u A := hx'.elim (fun h' => h' ▸ Set.mem_insert u A)
-            (fun h' => h' ▸ Set.mem_insert_of_mem u ha)
-          have hym : y ∈ insert u A := hy'.elim (fun h' => h' ▸ Set.mem_insert u A)
-            (fun h' => h' ▸ Set.mem_insert_of_mem u ha)
-          exact iff_of_true hxm hym
-        · have hxu : x ≠ u := fun hxu => hu (hxu ▸ SimpleGraph.Walk.mem_support_of_mem_edges hf hx)
-          have hyu : y ≠ u := fun hyu => hu (hyu ▸ SimpleGraph.Walk.mem_support_of_mem_edges hf hy)
-          simpa only [Set.mem_insert_iff,hxu,hyu,false_or] using hcut f hf hfe x hx y hy
-
-theorem conn_closed_vertex_set (G : BondGraph V E) (S : Set E) (A : Set V)
-    (hA : ∀ e ∈ S, ∀ a ∈ G e, ∀ b ∈ G e, a ∈ A → b ∈ A)
-    {x y : V} (hc : Conn G S x y) (hx : x ∈ A) : y ∈ A := by
-  induction hc with
-  | refl => exact hx
-  | @tail a b _ hab ih =>
-    obtain ⟨e,he,heq⟩ := hab
-    exact hA e he a (heq.symm ▸ Sym2.mem_mk_left a b) b (heq.symm ▸ Sym2.mem_mk_right a b) ih
-
-end SimplePathCuts
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section CalibratedCut
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem conn_edge_members (G : BondGraph V E) (S : Set E) (e : E) (he : e ∈ S)
-    (a b : V) (ha : a ∈ G e) (hb : b ∈ G e) : Conn G S a b := by
-  obtain ⟨x,y,hxy⟩ : ∃ x y, G e = s(x,y) := by
-    obtain ⟨⟨x,y⟩,hh⟩ := Quot.exists_rep (G e)
-    exact ⟨x,y,hh.symm⟩
-  rw [hxy,Sym2.mem_iff] at ha hb
-  rcases ha with rfl | rfl <;> rcases hb with rfl | rfl
-  · exact conn_refl G S _
-  · exact Relation.ReflTransGen.single ⟨e,he,hxy⟩
-  · exact conn_symm G S (Relation.ReflTransGen.single ⟨e,he,hxy⟩)
-  · exact conn_refl G S _
-
-namespace RawCorridor
-variable {G : BondGraph V E} {k : ℕ}
-
- 
-
-theorem calibration_pivotal_ports (hG : Function.Injective G) (w : RawCorridor G k)
-    (ω : Set E) (hq : ω ∈ w.calibrationEvent)
-    (hne : ¬Conn G (w.exterior ω) w.base w.endpoint)
-    (e : E) (he : e ∈ w.retained) : ¬Conn G (ω \ {e}) w.base w.endpoint := by
-  obtain ⟨A,hu,hv,hA,hcut⟩ := simplePath_edge_cut w.path w.path_isPath (G e) he
-  let B := A ∪ {z | Conn G (w.exterior ω) w.base z}
-  have hAport (z : V) (hz : z ∈ A) (ho : z ∉ w.interior) : z = w.base := by
-    rcases w.support_exterior_ports (hA z hz) ho with h | h
-    · exact h
-    · exact (hv (h ▸ hz)).elim
-  have hConnOut (z : V) (hz : Conn G (w.exterior ω) w.base z) : z ∉ w.interior := by
-    rcases conn_internal_eq_or_mem G (w.interior : Set V)ᶜ hz with h | h
-    · rw [←h]; exact w.base_not_interior
-    · exact h.2
-  have hBclosed : ∀ f ∈ ω \ {e}, ∀ a ∈ G f, ∀ b ∈ G f, a ∈ B → b ∈ B := by
-    intro f hf a ha b hb hba
-    rcases w.open_subset_exterior_retained ω hq hf.1 with hfs | hfr
-    · have hab := conn_edge_members G (w.exterior ω) f hfs a b ha hb
-      rcases hba with haa | hua
-      · have hau : a = w.base := hAport a haa (hfs.2 a ha)
-        exact Or.inr (hau ▸ hab)
-      · exact Or.inr (conn_trans G _ hua hab)
-    · have hfe : G f ≠ G e := fun h => hf.2 (hG h)
-      have haa : a ∈ A := by
-        rcases hba with h | h
-        · exact h
-        · rcases w.support_exterior_ports (SimpleGraph.Walk.mem_support_of_mem_edges hfr ha)
-            (hConnOut a h) with hau | hav
-          · exact hau ▸ hu
-          · exact (hne (hav ▸ h)).elim
-      exact Or.inl ((hcut (G f) hfr hfe a ha b hb).mp haa)
-  intro hc
-  have hh := conn_closed_vertex_set G (ω \ {e}) B hBclosed hc (Or.inl hu)
-  exact hh.elim hv hne
-
-end RawCorridor
-end CalibratedCut
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section ReverseCalibration
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace RawCorridor
-variable {G : BondGraph V E} {k : ℕ}
-
-theorem calibration_counted_of_orientation (hG : Function.Injective G) (w : RawCorridor G k)
-    (hlen : 3 ≤ w.path.length) (ω : Set E) (hq : ω ∈ w.calibrationEvent) (x y : V)
-    (hxu : Conn G (w.exterior ω) x w.base) (hvy : Conn G (w.exterior ω) w.endpoint y)
-    (hne : ¬Conn G (w.exterior ω) w.base w.endpoint) : w.counted Set.univ ω x y := by
-  have huniv : ω ∩ internalEdges G Set.univ = ω := by
-    apply Set.Subset.antisymm Set.inter_subset_left
-    exact fun _ he => ⟨he,fun _ _ => trivial⟩
-  apply (w.counted_iff Set.univ ω x y).mpr
-  refine ⟨hlen,?_⟩
-  unfold corridorCounted
-  rw [huniv]
-  refine ⟨conn_trans G ω (conn_mono G Set.inter_subset_left hxu)
-    (conn_trans G ω (conn_mono G hq.1 w.retained_conn) (conn_mono G Set.inter_subset_left hvy)),?_,hq.2⟩
-  intro e he
-  refine ⟨hq.1 he,?_⟩
-  intro hxy
-  have hS : w.exterior ω ⊆ ω \ {e} := by
-    intro f hf
-    refine ⟨hf.1,?_⟩
-    intro hfe
-    have hfe' : f = e := hfe
-    subst f
-    exact Set.disjoint_left.mp (w.specified_disjoint_exterior hlen) (Or.inl he) hf.2
-  exact w.calibration_pivotal_ports hG ω hq hne e he
-    (conn_trans G _ (conn_mono G hS (conn_symm G _ hxu))
-      (conn_trans G _ hxy (conn_mono G hS (conn_symm G _ hvy))))
-
- 
-
-theorem calibration_counted_iff (hG : Function.Injective G) (w : RawCorridor G k)
-    (hlen : 3 ≤ w.path.length) (ω : Set E) (hq : ω ∈ w.calibrationEvent) (x y : V) :
-    w.counted Set.univ ω x y ↔ ¬Conn G (w.exterior ω) w.base w.endpoint ∧
-      ((Conn G (w.exterior ω) w.base x ∧ Conn G (w.exterior ω) w.endpoint y) ∨
-       (Conn G (w.exterior ω) w.endpoint x ∧ Conn G (w.exterior ω) w.base y)) := by
-  constructor
-  · intro hc
-    exact w.counted_exterior_orientation hlen ω hq x y hc.2.2
-  · rintro ⟨hne,ho⟩
-    rcases ho with ⟨hx,hy⟩ | ⟨hx,hy⟩
-    · exact w.calibration_counted_of_orientation hG hlen ω hq x y (conn_symm G _ hx) hy hne
-    · exact (w.counted_symm Set.univ ω y x).mp
-        (w.calibration_counted_of_orientation hG hlen ω hq y x (conn_symm G _ hy) hx hne)
-
-end RawCorridor
-end ReverseCalibration
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section ExteriorLegObservable
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem deformedConnection_restrict_config (G : BondGraph V E) (P : V → V → ℝ)
-    (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x y : V) :
-    deformedConnection G P p k t D (ω ∩ internalEdges G D) x y =
-      deformedConnection G P p k t D ω x y := by
-  have he : (ω ∩ internalEdges G D) ∩ internalEdges G D = ω ∩ internalEdges G D :=
-    Set.inter_assoc _ _ _ |>.trans (congrArg (ω ∩ ·) (Set.inter_self _))
-  have hc (w : RawCorridor G k) : w.counted D (ω ∩ internalEdges G D) x y ↔ w.counted D ω x y := by
-    simp only [RawCorridor.counted,corridorCounted,he]
-  simp only [deformedConnection,corridorPenalty,corridorPenaltyTerm,hc,he]
-
-noncomputable def exteriorLegProduct (G : BondGraph V E) (P : V → V → ℝ)
-    (p : ℝ) {k : ℕ} (t : ℝ) (w : RawCorridor G k) (ω : Set E) (x y : V) : ℝ :=
-  if ¬Conn G (w.exterior ω) w.base w.endpoint then
-    deformedConnection G P p k t (w.interior : Set V)ᶜ ω x w.base *
-      deformedConnection G P p k t (w.interior : Set V)ᶜ ω w.endpoint y else 0
-
-theorem exteriorLegProduct_nonneg (G : BondGraph V E) (P : V → V → ℝ)
-    (p : ℝ) {k : ℕ} (t : ℝ) (w : RawCorridor G k) (ω : Set E) (x y : V) :
-    0 ≤ exteriorLegProduct G P p t w ω x y := by
-  unfold exteriorLegProduct
-  by_cases h : ¬Conn G (w.exterior ω) w.base w.endpoint
-  · rw [ite_eq_left h]
-    exact mul_nonneg (deformedConnection_nonneg _ _ _ _ _ _ _ _ _)
-      (deformedConnection_nonneg _ _ _ _ _ _ _ _ _)
-  · rw [ite_eq_right h]
-
-theorem exteriorLegProduct_le_one (G : BondGraph V E) (P : V → V → ℝ)
-    (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) {k : ℕ} (t : ℝ) (ht : 0 ≤ t)
-    (w : RawCorridor G k) (ω : Set E) (x y : V) :
-    exteriorLegProduct G P p t w ω x y ≤ 1 := by
-  unfold exteriorLegProduct
-  by_cases h : ¬Conn G (w.exterior ω) w.base w.endpoint
-  · rw [ite_eq_left h]
-    exact (mul_le_of_le_one_left (deformedConnection_nonneg _ _ _ _ _ _ _ _ _)
-      (deformedConnection_le_one G P hP p k t ht _ _ _ _)).trans
-      (deformedConnection_le_one G P hP p k t ht _ _ _ _)
-  · rw [ite_eq_right h]
-    exact zero_le_one
-
-theorem measurable_exteriorLegProduct [Countable V] [Countable E]
-    (G : BondGraph V E) (P : V → V → ℝ) (p : ℝ) {k : ℕ} (t : ℝ)
-    (w : RawCorridor G k) (x y : V) :
-    Measurable (fun ω => exteriorLegProduct G P p t w ω x y) := by
-  apply Measurable.ite
-    ((measurableSet_restrictedConn G (internalEdges G (w.interior:Set V)ᶜ) w.base w.endpoint).compl)
-  · exact (measurable_deformedConnection G P p k t _ x w.base).mul
-      (measurable_deformedConnection G P p k t _ w.endpoint y)
-  · exact measurable_const
-
-theorem exteriorLegProduct_restrict (G : BondGraph V E) (P : V → V → ℝ)
-    (p : ℝ) {k : ℕ} (t : ℝ) (w : RawCorridor G k) (ω : Set E) (x y : V) :
-    exteriorLegProduct G P p t w ω x y =
-      exteriorLegProduct G P p t w (ω ∩ internalEdges G (w.interior:Set V)ᶜ) x y := by
-  simp only [exteriorLegProduct,RawCorridor.exterior,Set.inter_assoc,Set.inter_self,
-    deformedConnection_restrict_config]
-  rfl
-
-end ExteriorLegObservable
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section CalibrationFactor
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace RawCorridor
-variable {G : BondGraph V E} {k : ℕ}
-
-theorem measurableSet_calibration (hG : Function.Injective G) (hloc : LocallyFinite G)
-    (w : RawCorridor G k) : MeasurableSet w.calibrationEvent := by
-  rw [w.calibrationEvent_eq hG hloc]
-  apply measurableSet_edgePattern
-  apply Finset.disjoint_left.mpr
-  intro e he hb
-  exact Set.disjoint_left.mp w.retained_disjoint_blockers
-    ((w.retained_finite hG).mem_toFinset.mp he) ((w.blockers_finite hloc).mem_toFinset.mp hb)
-
- 
-
-theorem calibration_integral (hG : Function.Injective G) (hloc : LocallyFinite G)
-    (w : RawCorridor G k) (hlen : 3 ≤ w.path.length) (p : unitInterval)
-    (f : Set E → ℝ) (hf : Measurable f)
-    (hfr : ∀ ω, f ω = f (ω ∩ internalEdges G (w.interior:Set V)ᶜ)) :
-    (∫ ω, if ω ∈ w.calibrationEvent then f ω else 0 ∂law p) =
-      (law p).real w.calibrationEvent * ∫ ω, f ω ∂law p := by
-  have hh := integral_mul_disjoint_coordinates p _ _ (w.specified_disjoint_exterior hlen)
-    (fun ω => if ω ∈ w.calibrationEvent then (1:ℝ) else 0) f
-    (Measurable.ite (w.measurableSet_calibration hG hloc) measurable_const measurable_const) hf
-    (fun ω => by rw [w.calibration_restriction ω]) hfr
-  simpa only [ite_mul,one_mul,zero_mul,integral_event_indicator p _ (w.measurableSet_calibration hG hloc)] using hh
-
-theorem calibration_weighted_integral (hG : Function.Injective G) (hloc : LocallyFinite G)
-    (w : RawCorridor G k) (hlen : 3 ≤ w.path.length) (P : V → V → ℝ)
-    (p : unitInterval) (hp : 0 < (p:ℝ)) (hp₁ : (p:ℝ) < 1)
-    (f : Set E → ℝ) (hf : Measurable f)
-    (hfr : ∀ ω, f ω = f (ω ∩ internalEdges G (w.interior:Set V)ᶜ)) :
-    (∫ ω, if ω ∈ w.calibrationEvent then w.weight P p * f ω else 0 ∂law p) =
-      w.probability P * ∫ ω, f ω ∂law p := by
-  have he : (fun ω => if ω ∈ w.calibrationEvent then w.weight P p * f ω else 0) =
-      (fun ω => w.weight P p * (if ω ∈ w.calibrationEvent then f ω else 0)) := by
-    funext ω
-    by_cases h : ω ∈ w.calibrationEvent <;> simp only [h,ite_true,ite_false,mul_zero]
-  rw [he,integral_const_mul,w.calibration_integral hG hloc hlen p f hf hfr,
-    ←mul_assoc,w.calibration_identity hG hloc P p hp hp₁]
-
-end RawCorridor
-end CalibrationFactor
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section CalibratedDerivativeTerm
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem calibrated_leg_product_le_term (G : BondGraph V E) (hG : Function.Injective G)
-    (hloc : LocallyFinite G) (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y)
-    (p : unitInterval) (k : ℕ) (w : RawCorridor G k) (hlen : 3 ≤ w.path.length)
-    (c : ℝ) (hc : (∑ e ∈ (w.retained_finite hG).toFinset, ∑' r : RawCorridor G k,
-        if e ∈ r.retained then r.weight P p else 0) ≤ c)
-    (t : ℝ) (ht : 0 ≤ t) (ω : Set E) (x y : V) :
-    (if ω ∈ w.calibrationEvent then w.weight P p * (Real.exp (-t*c) *
-      exteriorLegProduct G P p t w ω x y) else 0) ≤
-        corridorPenaltyTerm G P p Set.univ ω x y w * deformedConnection G P p k t Set.univ ω x y := by
-  have hr : 0 ≤ corridorPenaltyTerm G P p Set.univ ω x y w * deformedConnection G P p k t Set.univ ω x y :=
-    mul_nonneg (corridorPenaltyTerm_nonneg G P hP p Set.univ ω x y w)
-      (deformedConnection_nonneg G P p k t Set.univ ω x y)
-  by_cases hq : ω ∈ w.calibrationEvent
-  · rw [ite_eq_left hq]
-    by_cases hn : ¬Conn G (w.exterior ω) w.base w.endpoint
-    · rw [exteriorLegProduct,ite_eq_left hn]
-      by_cases hx : Conn G (w.exterior ω) x w.base
-      · by_cases hy : Conn G (w.exterior ω) w.endpoint y
-        · have hj := w.calibration_counted_of_orientation hG hlen ω hq x y hx hy hn
-          rw [corridorPenaltyTerm,ite_eq_left hj]
-          have hh := mul_le_mul_of_nonneg_left
-            (deformedConnection_legs_lower G hG hloc P hP p k w ω hq x y hx hy hn c hc t ht)
-            (w.weight_nonneg P hP p)
-          simpa only [mul_assoc] using hh
-        · have hz : deformedConnection G P p k t (w.interior:Set V)ᶜ ω w.endpoint y = 0 := by
-            simp only [deformedConnection,show ¬Conn G (ω ∩ internalEdges G (w.interior:Set V)ᶜ) w.endpoint y from hy,and_false,ite_false]
-          simpa only [hz,mul_zero] using hr
-      · have hz : deformedConnection G P p k t (w.interior:Set V)ᶜ ω x w.base = 0 := by
-          simp only [deformedConnection,show ¬Conn G (ω ∩ internalEdges G (w.interior:Set V)ᶜ) x w.base from hx,and_false,ite_false]
-        simpa only [hz,zero_mul,mul_zero] using hr
-    · simpa only [exteriorLegProduct,ite_eq_right hn,mul_zero] using hr
-  · simpa only [ite_eq_right hq] using hr
-
-theorem integrable_corridorDerivativeTerm [Countable V] [Countable E]
-    (G : BondGraph V E) (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y)
-    (p : unitInterval) (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D : Set V) (x y : V)
-    (w : RawCorridor G k) :
-    Integrable (fun ω => corridorPenaltyTerm G P p D ω x y w *
-      deformedConnection G P p k t D ω x y) (law p) := by
-  have hh := ((integrable_deformedConnection G P hP p k t ht D x y).const_mul
-    (w.weight P p)).indicator (w.measurableSet_counted D x y)
-  have he : (fun ω => corridorPenaltyTerm G P p D ω x y w *
-      deformedConnection G P p k t D ω x y) =
-      {ω | w.counted D ω x y}.indicator (fun ω => w.weight P p * deformedConnection G P p k t D ω x y) := by
-    funext ω
-    simp only [Set.indicator,Set.mem_ofPred_eq,corridorPenaltyTerm,ite_mul,zero_mul]
-  rw [he]
-  exact hh
-
-end CalibratedDerivativeTerm
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section CalibratedDecreaseKernel
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem integrable_exteriorLegProduct [Countable V] [Countable E]
-    (G : BondGraph V E) (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y)
-    (p : unitInterval) {k : ℕ} (t : ℝ) (ht : 0 ≤ t) (w : RawCorridor G k) (x y : V) :
-    Integrable (fun ω => exteriorLegProduct G P p t w ω x y) (law p) := by
-  apply Integrable.mono' (integrable_const (1:ℝ))
-    (measurable_exteriorLegProduct G P p t w x y).aestronglyMeasurable
-  filter_upwards [] with ω
-  rw [Real.norm_of_nonneg (exteriorLegProduct_nonneg G P p t w ω x y)]
-  exact exteriorLegProduct_le_one G P hP p t ht w ω x y
-
- 
-
-theorem calibrated_derivative_kernel_lower [Countable V] [Countable E]
-    (G : BondGraph V E) (hG : Function.Injective G) (hloc : LocallyFinite G)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y)
-    (p : unitInterval) (hp : 0 < (p:ℝ)) (hp₁ : (p:ℝ) < 1)
-    (k : ℕ) (w : RawCorridor G k) (hlen : 3 ≤ w.path.length)
-    (c : ℝ) (hc : (∑ e ∈ (w.retained_finite hG).toFinset, ∑' r : RawCorridor G k,
-        if e ∈ r.retained then r.weight P p else 0) ≤ c)
-    (t : ℝ) (ht : 0 ≤ t) (x y : V) :
-    w.probability P * Real.exp (-t*c) * (∫ ω, exteriorLegProduct G P p t w ω x y ∂law p) ≤
-      ∫ ω, corridorPenaltyTerm G P p Set.univ ω x y w *
-        deformedConnection G P p k t Set.univ ω x y ∂law p := by
-  have hf := integrable_exteriorLegProduct G P hP p t ht w x y
-  have hi : Integrable (fun ω => if ω ∈ w.calibrationEvent then w.weight P p *
-      (Real.exp (-t*c) * exteriorLegProduct G P p t w ω x y) else 0) (law p) :=
-    ((hf.const_mul (Real.exp (-t*c))).const_mul (w.weight P p)).indicator
-      (w.measurableSet_calibration hG hloc)
-  have hh := integral_mono hi (integrable_corridorDerivativeTerm G P hP p k t ht Set.univ x y w)
-    (fun ω => calibrated_leg_product_le_term G hG hloc P hP p k w hlen c hc t ht ω x y)
-  rw [w.calibration_weighted_integral hG hloc hlen P p hp hp₁
-    (fun ω => Real.exp (-t*c)*exteriorLegProduct G P p t w ω x y)
-    ((measurable_exteriorLegProduct G P p t w x y).const_mul _)
-    (fun ω => congrArg (Real.exp (-t*c)*·) (exteriorLegProduct_restrict G P p t w ω x y)),
-    integral_const_mul] at hh
-  simpa only [mul_assoc] using hh
-
-end CalibratedDecreaseKernel
-end Percolation
-
- 
-namespace Percolation
-open MeasureTheory
-section NonnegativeRealTonelli
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem real_tonelli_of_integrable_sum {Ω I : Type*} [MeasurableSpace Ω] [Countable I]
-    (μ : Measure Ω) (F : I → Ω → ℝ) (hF : ∀ i, Integrable (F i) μ)
-    (hn : ∀ i ω, 0 ≤ F i ω) (hs : ∀ᵐ ω ∂μ, Summable (fun i => F i ω))
-    (hi : Integrable (fun ω => ∑' i, F i ω) μ) :
-    Summable (fun i => ∫ ω, F i ω ∂μ) ∧
-      (∑' i, ∫ ω, F i ω ∂μ) = ∫ ω, ∑' i, F i ω ∂μ := by
-  have he : (∑' i, ENNReal.ofReal (∫ ω, F i ω ∂μ)) =
-      ∫⁻ ω, ENNReal.ofReal (∑' i, F i ω) ∂μ := by
-    simp_rw [ofReal_integral_eq_lintegral_ofReal (hF _) (Filter.Eventually.of_forall (hn _))]
-    rw [←lintegral_tsum (fun i => (hF i).aestronglyMeasurable.aemeasurable.ennreal_ofReal)]
-    apply lintegral_congr_ae
-    filter_upwards [hs] with ω hω
-    exact (ENNReal.ofReal_tsum_of_nonneg (fun i => hn i ω) hω).symm
-  have ht : (∑' i, ENNReal.ofReal (∫ ω, F i ω ∂μ)) ≠ ⊤ := by
-    rw [he,←ofReal_integral_eq_lintegral_ofReal hi
-      (Filter.Eventually.of_forall (fun ω => tsum_nonneg (fun i => hn i ω)))]
-    exact ENNReal.ofReal_ne_top
-  have hh : Summable (fun i => ∫ ω, F i ω ∂μ) := by
-    simpa only [ENNReal.toReal_ofReal (integral_nonneg (hn _))] using ENNReal.summable_toReal ht
-  refine ⟨hh,?_⟩
-  apply integral_tsum_of_summable_integral_norm hF
-  simpa only [Real.norm_of_nonneg (hn _ _)] using hh
-
-end NonnegativeRealTonelli
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section TotalExteriorLegs
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem exteriorLegProduct_sum (G : BondGraph V E) (P : V → V → ℝ)
-    (p : ℝ) {k : ℕ} (t : ℝ) (w : RawCorridor G k) (ω : Set E)
-    (hu : (cluster G ω w.base).Finite) (hv : (cluster G ω w.endpoint).Finite) :
-    Summable (fun xy : V × V => exteriorLegProduct G P p t w ω xy.1 xy.2) ∧
-    (∑' xy : V × V, exteriorLegProduct G P p t w ω xy.1 xy.2) =
-      if ¬Conn G (w.exterior ω) w.base w.endpoint then
-        deformedClusterMass G P p k t (w.interior:Set V)ᶜ ω w.base *
-          deformedClusterMass G P p k t (w.interior:Set V)ᶜ ω w.endpoint else 0 := by
-  by_cases hn : ¬Conn G (w.exterior ω) w.base w.endpoint
-  · have hsu : Summable (deformedConnection G P p k t (w.interior:Set V)ᶜ ω w.base) :=
-      summable_of_hasFiniteSupport (deformedConnection_finite_support G P p k t (w.interior:Set V)ᶜ ω w.base hu)
-    have hsv : Summable (deformedConnection G P p k t (w.interior:Set V)ᶜ ω w.endpoint) :=
-      summable_of_hasFiniteSupport (deformedConnection_finite_support G P p k t (w.interior:Set V)ᶜ ω w.endpoint hv)
-    have hsx : Summable (fun x => deformedConnection G P p k t (w.interior:Set V)ᶜ ω x w.base) := by
-      simpa only [deformedConnection_symm G P p k t (w.interior:Set V)ᶜ ω _ w.base] using hsu
-    have hs := hsx.mul_of_nonneg hsv
-      (fun x => deformedConnection_nonneg G P p k t _ ω x w.base)
-      (fun y => deformedConnection_nonneg G P p k t _ ω w.endpoint y)
-    simp only [exteriorLegProduct,ite_eq_left hn]
-    refine ⟨hs,?_⟩
-    rw [←hsx.tsum_mul_tsum hsv hs]
-    simp only [deformedClusterMass,deformedConnection_symm G P p k t (w.interior:Set V)ᶜ ω _ w.base]
-  · simp only [exteriorLegProduct,ite_eq_right hn,summable_zero,tsum_zero,and_self]
-
- 
-
-theorem exteriorLegProduct_integral_sum [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (hp : p < pc G)
-    {k : ℕ} (t : ℝ) (ht : 0 ≤ t) (w : RawCorridor G k) :
-    Summable (fun xy : V × V => ∫ ω, exteriorLegProduct G P p t w ω xy.1 xy.2 ∂law p) ∧
-    (∑' xy : V × V, ∫ ω, exteriorLegProduct G P p t w ω xy.1 xy.2 ∂law p) =
-      ∫ ω, (if ¬Conn G (w.exterior ω) w.base w.endpoint then
-        deformedClusterMass G P p k t (w.interior:Set V)ᶜ ω w.base *
-          deformedClusterMass G P p k t (w.interior:Set V)ᶜ ω w.endpoint else 0) ∂law p := by
-  have hae : ∀ᵐ ω ∂law p, Summable (fun xy : V × V => exteriorLegProduct G P p t w ω xy.1 xy.2) ∧
-    (∑' xy : V × V, exteriorLegProduct G P p t w ω xy.1 xy.2) =
-      if ¬Conn G (w.exterior ω) w.base w.endpoint then
-        deformedClusterMass G P p k t (w.interior:Set V)ᶜ ω w.base *
-          deformedClusterMass G P p k t (w.interior:Set V)ᶜ ω w.endpoint else 0 := by
-    filter_upwards [subcritical_cluster_finite_ae G p hp w.base,
-      subcritical_cluster_finite_ae G p hp w.endpoint] with ω hu hv
-    exact exteriorLegProduct_sum G P p t w ω hu hv
-  have hi := (integrable_distinct_deformed_product G hloc hqt P hP p hp k t ht
-    (w.interior:Set V)ᶜ w.base w.endpoint).congr (hae.mono (fun _ h => h.2.symm))
-  obtain ⟨hs,he⟩ := real_tonelli_of_integrable_sum (law p)
-    (fun xy : V × V => fun ω => exteriorLegProduct G P p t w ω xy.1 xy.2)
-    (fun xy => integrable_exteriorLegProduct G P hP p t ht w xy.1 xy.2)
-    (fun xy ω => exteriorLegProduct_nonneg G P p t w ω xy.1 xy.2)
-    (hae.mono (fun _ h => h.1)) hi
-  exact ⟨hs,he.trans (integral_congr_ae (hae.mono (fun _ h => h.2)))⟩
-
-end TotalExteriorLegs
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section SummableDerivativeTerm
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Countable V] [Countable E] [Nonempty V]
-
-omit [Nonempty V] in
-theorem integral_corridorDerivativeTerm_le (G : BondGraph V E) (hG : Function.Injective G)
-    (hloc : LocallyFinite G) (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y)
-    (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp1 : (p:ℝ) < 1)
-    {k : ℕ} (t : ℝ) (ht : 0 ≤ t) (w : RawCorridor G k) (x y : V) :
-    (∫ ω, corridorPenaltyTerm G P p Set.univ ω x y w *
-      deformedConnection G P p k t Set.univ ω x y ∂law p) ≤
-      w.probability P * (twoPoint G p x w.base * twoPoint G p w.endpoint y +
-        twoPoint G p x w.endpoint * twoPoint G p w.base y) := by
-  apply le_trans (integral_mono (integrable_corridorDerivativeTerm G P hP p k t ht Set.univ x y w)
-    (integrable_corridorPenaltyTerm G P p Set.univ x y w) (fun ω =>
-      mul_le_of_le_one_right (corridorPenaltyTerm_nonneg G P hP p Set.univ ω x y w)
-        (deformedConnection_le_one G P hP p k t ht Set.univ ω x y)))
-  exact integral_corridorPenaltyTerm_le G hG hloc P hP p hp0 hp1 x y w
-
-theorem summable_twoPoint_pairs (G : BondGraph V E)
-    (hloc : LocallyFinite G) (hqt : QuasiTransitive G) (p : unitInterval)
-    (hp : p < pc G) (u v : V) :
-    Summable (fun xy : V × V => twoPoint G p xy.1 u * twoPoint G p v xy.2) := by
-  have hs : Summable (fun x => twoPoint G p x u) := by
-    simpa only [twoPoint_symm G p _ u] using summable_twoPoint_subcritical G hloc hqt p hp u
-  exact hs.mul_of_nonneg (summable_twoPoint_subcritical G hloc hqt p hp v)
-    (fun x => twoPoint_nonneg G p x u) (fun y => twoPoint_nonneg G p v y)
-
-omit [Nonempty V] [Countable V] [Countable E] in
- theorem integral_corridorDerivativeTerm_nonneg (G : BondGraph V E)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (k : ℕ)
-    (t : ℝ) (w : RawCorridor G k) (x y : V) :
-    0 ≤ ∫ ω, corridorPenaltyTerm G P p Set.univ ω x y w *
-      deformedConnection G P p k t Set.univ ω x y ∂law p :=
-  integral_nonneg (fun ω => mul_nonneg (corridorPenaltyTerm_nonneg G P hP p Set.univ ω x y w)
-    (deformedConnection_nonneg G P p k t Set.univ ω x y))
-
-theorem summable_corridorDerivativeTerm_pairs (G : BondGraph V E) (hG : Function.Injective G)
-    (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y)
-    (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp : p < pc G)
-    {k : ℕ} (t : ℝ) (ht : 0 ≤ t) (w : RawCorridor G k) :
-    Summable (fun xy : V × V => ∫ ω, corridorPenaltyTerm G P p Set.univ ω xy.1 xy.2 w *
-      deformedConnection G P p k t Set.univ ω xy.1 xy.2 ∂law p) := by
-  have hs : Summable (fun xy : V × V => w.probability P *
-      (twoPoint G p xy.1 w.base * twoPoint G p w.endpoint xy.2 +
-       twoPoint G p xy.1 w.endpoint * twoPoint G p w.base xy.2)) :=
-    ((summable_twoPoint_pairs G hloc hqt p hp w.base w.endpoint).add
-      (summable_twoPoint_pairs G hloc hqt p hp w.endpoint w.base)).mul_left (w.probability P)
-  exact Summable.of_nonneg_of_le
-    (fun (xy : V × V) => integral_corridorDerivativeTerm_nonneg G P hP p k t w xy.1 xy.2)
-    (fun (xy : V × V) => integral_corridorDerivativeTerm_le G hG hloc P hP p hp0
-      ((show (p:ℝ)<(pc G:ℝ) from hp).trans_le (pc G).2.2) t ht w xy.1 xy.2) hs
-
-end SummableDerivativeTerm
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section SingleTemplateDecrease
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Countable V] [Countable E] [Nonempty V]
-
- 
-
-theorem calibrated_derivative_template_lower (G : BondGraph V E) (hG : Function.Injective G)
-    (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y)
-    (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp : p < pc G)
-    (k : ℕ) (w : RawCorridor G k) (hlen : 3 ≤ w.path.length)
-    (c : ℝ) (hc : (∑ e ∈ (w.retained_finite hG).toFinset, ∑' r : RawCorridor G k,
-        if e ∈ r.retained then r.weight P p else 0) ≤ c)
-    (t : ℝ) (ht : 0 ≤ t) :
-    w.probability P * Real.exp (-t*c) *
-      (∫ ω, (if ¬Conn G (w.exterior ω) w.base w.endpoint then
-        deformedClusterMass G P p k t (w.interior:Set V)ᶜ ω w.base *
-          deformedClusterMass G P p k t (w.interior:Set V)ᶜ ω w.endpoint else 0) ∂law p) ≤
-      ∑' xy : V × V, ∫ ω, corridorPenaltyTerm G P p Set.univ ω xy.1 xy.2 w *
-        deformedConnection G P p k t Set.univ ω xy.1 xy.2 ∂law p := by
-  obtain ⟨hs,he⟩ := exteriorLegProduct_integral_sum G hloc hqt P hP p hp t ht w
-  have hh := (hs.mul_left (w.probability P * Real.exp (-t*c))).tsum_le_tsum
-    (fun (xy : V × V) => calibrated_derivative_kernel_lower G hG hloc P hP p hp0
-      ((show (p:ℝ)<(pc G:ℝ) from hp).trans_le (pc G).2.2) k w hlen c hc t ht xy.1 xy.2)
-    (summable_corridorDerivativeTerm_pairs G hG hloc hqt P hP p hp0 hp t ht w)
-  rwa [tsum_mul_left,he] at hh
-
- 
-
-theorem calibrated_derivative_template_row_lower (G : BondGraph V E) (hG : Function.Injective G)
-    (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (d : ℕ) (hd : ∀ a, (neighbors G hloc a).card ≤ d)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (hpcrit : 0 < (pc G:ℝ))
-    (p : unitInterval) (hp : p < pc G) (hpnear : (pc G:ℝ)/2 ≤ (p:ℝ))
-    (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T)
-    (k : ℕ) (w : RawCorridor G k) (hlen : 3 ≤ w.path.length)
-    (c : ℝ) (hc : (∑ e ∈ (w.retained_finite hG).toFinset, ∑' r : RawCorridor G k,
-        if e ∈ r.retained then r.weight P p else 0) ≤ c)
-    (t : ℝ) (ht : 0 ≤ t) :
-    w.probability P * Real.exp (-t*c) *
-      (deformedSusceptibility G P p k t (w.interior:Set V)ᶜ w.base *
-        deformedSusceptibility G P p k t (w.interior:Set V)ᶜ w.endpoint -
-        (1+2*(d:ℝ)/(pc G:ℝ))*(maxSusceptibility G p)^2*(matrixCoeff (T^3) w.base w.endpoint).re) ≤
-      ∑' xy : V × V, ∫ ω, corridorPenaltyTerm G P p Set.univ ω xy.1 xy.2 w *
-        deformedConnection G P p k t Set.univ ω xy.1 xy.2 ∂law p := by
-  have hm := deformed_distinct_correlation_uniform G hloc hqt d hd P hP hpcrit p hp hpnear T hT
-    k t ht (w.interior:Set V)ᶜ w.base w.endpoint
-  exact (mul_le_mul_of_nonneg_left hm (mul_nonneg (w.2.2.mass_nonneg P hP) (Real.exp_pos _).le)).trans
-    (calibrated_derivative_template_lower G hG hloc hqt P hP p ((half_pos hpcrit).trans_le hpnear)
-      hp k w hlen c hc t ht)
-
-end SingleTemplateDecrease
-end Percolation
-
- 
-namespace Percolation
-universe u u' v
-section ErasureNaturality
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {V : Type u} {V' : Type u'} {H : SimpleGraph V} {H' : SimpleGraph V'}
-
-theorem walk_map_dropUntil (f : H →g H') (hf : Function.Injective f)
-    {x y : V} (w : H.Walk x y) (z : V) (hz : z ∈ w.support) :
-    (w.dropUntil z hz).map f = (w.map f).dropUntil (f z)
-      (by simpa only [SimpleGraph.Walk.support_map] using List.mem_map_of_mem hz) := by
-  induction w with
-  | @nil x =>
-    have he : z = x := by simpa using hz
-    subst z
-    rfl
-  | @cons x a y h w ih =>
-    by_cases he : x = z
-    · subst z
-      simp only [SimpleGraph.Walk.dropUntil_first]
-    · have hfe : f x ≠ f z := fun h => he (hf h)
-      simp only [SimpleGraph.Walk.dropUntil,he,hfe,SimpleGraph.Walk.map_cons]
-      exact ih _
-
- 
-
-theorem walk_map_bypass (f : H →g H') (hf : Function.Injective f)
-    {x y : V} (w : H.Walk x y) : w.bypass.map f = (w.map f).bypass := by
-  induction w with
-  | nil => rfl
-  | @cons x a y h w ih =>
-    have hm : f x ∈ (w.map f).bypass.support ↔ x ∈ w.bypass.support := by
-      rw [←ih,SimpleGraph.Walk.support_map]
-      exact List.mem_map_of_injective hf
-    by_cases hx : x ∈ w.bypass.support
-    · have hfx := hm.mpr hx
-      simp only [SimpleGraph.Walk.bypass,SimpleGraph.Walk.map_cons,dite_eq_left hx,dite_eq_left hfx]
-      simp only [walk_map_dropUntil f hf,ih]
-    · have hfx := mt hm.mp hx
-      simp only [SimpleGraph.Walk.bypass,SimpleGraph.Walk.map_cons,dite_eq_right hx,dite_eq_right hfx,ih]
-
-namespace HoldingWalk
-
-def map (f : H →g H') : {k : ℕ} → {x y : V} → HoldingWalk H k x y → HoldingWalk H' k (f x) (f y)
-  | _,_,_,.nil _ => .nil _
-  | _,_,_,.hold w => .hold (w.map f)
-  | _,_,_,.step h w => .step (f.map_rel h) (w.map f)
-
-theorem map_vertices (f : H →g H') {k : ℕ} {x y : V} (w : HoldingWalk H k x y) :
-    (w.map f).vertices = w.vertices.map f := by
-  induction w with
-  | nil => rfl
-  | hold w ih => simp only [map,vertices,List.map_cons,ih]
-  | step h w ih => simp only [map,vertices,List.map_cons,ih]
-
-theorem map_withoutHolds (f : H →g H') {k : ℕ} {x y : V} (w : HoldingWalk H k x y) :
-    (w.map f).withoutHolds = w.withoutHolds.map f := by
-  induction w with
-  | nil => rfl
-  | hold w ih => exact ih
-  | step h w ih => simp only [map,withoutHolds,SimpleGraph.Walk.map_cons,ih]
-
-theorem map_erase (f : H →g H') (hf : Function.Injective f)
-    {k : ℕ} {x y : V} (w : HoldingWalk H k x y) :
-    (w.map f).erase = w.erase.map f := by
-  simp only [erase,map_withoutHolds,SimpleGraph.Walk.reverse_map,
-    ←walk_map_bypass f hf]
-
-end HoldingWalk
-end ErasureNaturality
-end Percolation
-
- 
-namespace Percolation
-universe u u' u'' v
-section RawWalkMaps
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {V : Type u} {V' : Type u'} {V'' : Type u''}
-  {H : SimpleGraph V} {H' : SimpleGraph V'} {H'' : SimpleGraph V''}
-namespace HoldingWalk
-
-theorem map_map (f : H →g H') (g : H' →g H'')
-    {k : ℕ} {x y : V} (w : HoldingWalk H k x y) : (w.map f).map g = w.map (g.comp f) := by
-  induction w with
-  | nil => rfl
-  | hold w ih => exact congrArg hold ih
-  | step h w ih => exact congrArg (step _) ih
-
-theorem map_id {k : ℕ} {x y : V} (w : HoldingWalk H k x y) : w.map ((SimpleGraph.Hom.id : H →g H)) = w := by
-  induction w with
-  | nil => rfl
-  | hold w ih => exact congrArg hold ih
-  | step h w ih => exact congrArg (step _) ih
-
-theorem mass_map (f : H →g H') (P : V → V → ℝ) (Q : V' → V' → ℝ)
-    (hPQ : ∀ x y, Q (f x) (f y) = P x y)
-    {k : ℕ} {x y : V} (w : HoldingWalk H k x y) : (w.map f).mass Q = w.mass P := by
-  induction w with
-  | nil => rfl
-  | hold w ih => simp only [map,mass,hPQ,ih]
-  | step h w ih => simp only [map,mass,hPQ,ih]
-end HoldingWalk
-
-def rawWalkMap (f : H →g H') (k : ℕ) (w : Σ x y, HoldingWalk H k x y) :
-    Σ x y, HoldingWalk H' k x y := ⟨f w.1,f w.2.1,w.2.2.map f⟩
-
-theorem rawWalkMap_comp (f : H →g H') (g : H' →g H'') (k : ℕ)
-    (w : Σ x y, HoldingWalk H k x y) : rawWalkMap g k (rawWalkMap f k w) = rawWalkMap (g.comp f) k w := by
-  cases w with
-  | mk x w =>
-    cases w with
-    | mk y w =>
-      simp only [rawWalkMap,HoldingWalk.map_map]
-      rfl
-
-theorem rawWalkMap_id (k : ℕ) (w : Σ x y, HoldingWalk H k x y) :
-    rawWalkMap ((SimpleGraph.Hom.id : H →g H)) k w = w := by
-  cases w with
-  | mk x w =>
-    cases w with
-    | mk y w =>
-      simp only [rawWalkMap,HoldingWalk.map_id]
-      rfl
-
-def rawWalkEquiv (f : H ≃g H') (k : ℕ) :
-    (Σ x y, HoldingWalk H k x y) ≃ (Σ x y, HoldingWalk H' k x y) where
-  toFun := rawWalkMap f.toHom k
-  invFun := rawWalkMap f.symm.toHom k
-  left_inv w := by
-    rw [rawWalkMap_comp]
-    have he : f.symm.toHom.comp f.toHom = (SimpleGraph.Hom.id : H →g H) := by
-      ext x
-      exact f.symm_apply_apply x
-    rw [he,rawWalkMap_id]
-  right_inv w := by
-    rw [rawWalkMap_comp]
-    have he : f.toHom.comp f.symm.toHom = (SimpleGraph.Hom.id : H' →g H') := by
-      ext x
-      exact f.apply_symm_apply x
-    rw [he,rawWalkMap_id]
-
-end RawWalkMaps
-section GraphAutWalks
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {V : Type u} {E : Type v}
-
-theorem openSimpleGraph_univ_adj (G : BondGraph V E) (x y : V) :
-    (openSimpleGraph G Set.univ).Adj x y ↔ x ≠ y ∧ Adj G x y := by
-  simp only [openSimpleGraph,Adj,Set.mem_univ,true_and]
-
-def autSimpleGraph (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) :
-    openSimpleGraph G Set.univ ≃g openSimpleGraph G Set.univ where
-  toEquiv := a
-  map_rel_iff' := by
-    intro x y
-    simp only [openSimpleGraph_univ_adj,a.injective.ne_iff]
-    exact and_congr_right (fun _ => ⟨fun h => by simpa using adj_aut G (isAut_symm G hab) h,
-      adj_aut G hab⟩)
-
-namespace RawCorridor
-variable {G : BondGraph V E} {k : ℕ}
-def autEquiv {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) : RawCorridor G k ≃ RawCorridor G k :=
-  rawWalkEquiv (autSimpleGraph G hab) k
-
-theorem aut_path {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (w : RawCorridor G k) :
-    (autEquiv hab w).path = w.path.map (autSimpleGraph G hab).toHom :=
-  w.2.2.map_erase _ a.injective
-
-theorem aut_path_length {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (w : RawCorridor G k) :
-    (autEquiv hab w).path.length = w.path.length := by
-  have hh := congrArg (fun q => q.length) (aut_path hab w)
-  exact hh.trans (SimpleGraph.Walk.length_map _ _)
-
-end RawCorridor
-end GraphAutWalks
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section TemplateCovariance
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace RawCorridor
-variable {G : BondGraph V E} {k : ℕ}
-
-theorem aut_base {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (w : RawCorridor G k) :
-    (autEquiv hab w).base = a w.base := rfl
-
-theorem aut_endpoint {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (w : RawCorridor G k) :
-    (autEquiv hab w).endpoint = a w.endpoint := rfl
-
-theorem aut_support {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (w : RawCorridor G k) :
-    (autEquiv hab w).path.support = w.path.support.map a := by
-  exact (congrArg (fun q => q.support) (aut_path hab w)).trans
-    (SimpleGraph.Walk.support_map _ _)
-
-theorem aut_edges {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (w : RawCorridor G k) :
-    (autEquiv hab w).path.edges = w.path.edges.map (Sym2.map a) := by
-  exact (congrArg (fun q => q.edges) (aut_path hab w)).trans
-    (SimpleGraph.Walk.edges_map _ _)
-
-theorem aut_interior {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (w : RawCorridor G k) (z : V) :
-    a z ∈ (autEquiv hab w).interior ↔ z ∈ w.interior := by
-  simp only [interior,Finset.mem_sdiff,List.mem_toFinset]
-  rw [aut_support hab w]
-  simp only [aut_base,aut_endpoint,Finset.mem_insert,Finset.mem_singleton,
-    List.mem_map_of_injective a.injective,a.injective.eq_iff]
-
-theorem aut_retained {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (w : RawCorridor G k) (e : E) :
-    b e ∈ (autEquiv hab w).retained ↔ e ∈ w.retained := by
-  simp only [retained,Set.mem_ofPred_eq,aut_edges hab w,hab e]
-  exact List.mem_map_of_injective (Sym2.map.injective a.injective)
-
-theorem aut_blockers {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (w : RawCorridor G k) (e : E) :
-    b e ∈ (autEquiv hab w).blockers ↔ e ∈ w.blockers := by
-  simp only [blockers,Set.mem_sdiff,Set.mem_ofPred_eq,aut_retained]
-  apply and_congr_left
-  intro _
-  constructor
-  · rintro ⟨z,hz,he⟩
-    rw [hab e] at he
-    obtain ⟨y,hy,rfl⟩ := Sym2.mem_map.mp he
-    exact ⟨y,(aut_interior hab w y).mp hz,hy⟩
-  · rintro ⟨z,hz,he⟩
-    refine ⟨a z,(aut_interior hab w z).mpr hz,?_⟩
-    rw [hab e]
-    exact Sym2.mem_map.mpr ⟨z,he,rfl⟩
-
-theorem aut_blockers_ncard {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (w : RawCorridor G k) :
-    (autEquiv hab w).blockers.ncard = w.blockers.ncard := by
-  have he : w.blockers = b ⁻¹' (autEquiv hab w).blockers := by
-    ext e
-    exact (aut_blockers hab w e).symm
-  rw [he,Set.ncard_preimage_of_injective_subset_range b.injective]
-  exact fun e _ => b.surjective e
-
-theorem aut_probability {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b)
-    (P : V → V → ℝ) (hP : ∀ x y, P (a x) (a y) = P x y) (w : RawCorridor G k) :
-    (autEquiv hab w).probability P = w.probability P :=
-  w.2.2.mass_map (autSimpleGraph G hab).toHom P P hP
-
-theorem aut_weight {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b)
-    (P : V → V → ℝ) (hP : ∀ x y, P (a x) (a y) = P x y) (p : ℝ) (w : RawCorridor G k) :
-    (autEquiv hab w).weight P p = w.weight P p := by
-  simp only [weight,aut_probability hab P hP,aut_path_length,aut_blockers_ncard]
-
-end RawCorridor
-end TemplateCovariance
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Deformation.Derivatives
+
+namespace OAI
+
+/-! Calibration for bond percolation. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section UniformDeformedEndpoint
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem exists_uniform_deformed_endpoint_comparison [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
+    (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y)
+    (hpcrit : 0 < (pc G:ℝ)) (hpcrit₁ : (pc G:ℝ) < 1) :
+    ∃ C : ℝ, 0 < C ∧ ∀ (p : unitInterval), (pc G:ℝ)/2 ≤ (p:ℝ) → p < pc G →
+      ∀ (k : ℕ), 3 ≤ k → ∀ (t : ℝ), 0 ≤ t → ∀ (D : Set V) (b z : V) (l : List E),
+      b ∈ D → Along G b l z → (∀ e ∈ l, e ∈ internalEdges G D) →
+      Real.exp (-t*((l.length:ℝ)*Real.exp (C*(k:ℝ)))) * (p:ℝ)^l.length *
+        deformedSusceptibility G (walkKernel G hloc a) p k t D z ≤
+        deformedSusceptibility G (walkKernel G hloc a) p k t D b := by
+  obtain ⟨C,hC,hcost⟩ := exists_uniform_corridor_edge_cost G hloc d hd a hP hpcrit hpcrit₁
+  refine ⟨C,hC,?_⟩
+  intro p hpnear hp k hk t ht D b z l hb hl hD
+  have hc : ∀ e, (∑' w : RawCorridor G k,
+      if e ∈ w.retained then w.weight (walkKernel G hloc a) p else 0) ≤ Real.exp (C*(k:ℝ)) := by
+    intro e
+    exact (hcost p hpnear hp k hk e).trans (div_le_self (Real.exp_pos _).le (by exact_mod_cast (show 1 ≤ k by omega)))
+  have hh := deformedSusceptibility_move_lower G hloc hqt (walkKernel G hloc a) hP p hp k t ht D
+    l.toFinset (fun e he => hD e (List.mem_toFinset.mp he)) b z hb
+    (hl.conn (fun e he => List.mem_toFinset.mpr he)) (Real.exp (C*(k:ℝ))) hc
+  have hcard : (l.toFinset.card:ℝ) ≤ (l.length:ℝ) := Nat.cast_le.mpr l.toFinset_card_le
+  have he : Real.exp (-t*((l.length:ℝ)*Real.exp (C*(k:ℝ)))) ≤
+      Real.exp (-t*((l.toFinset.card:ℝ)*Real.exp (C*(k:ℝ)))) := by
+    apply Real.exp_le_exp.mpr
+    exact mul_le_mul_of_nonpos_left (mul_le_mul_of_nonneg_right hcard (Real.exp_pos _).le) (neg_nonpos.mpr ht)
+  have hpw : (p:ℝ)^l.length ≤ (p:ℝ)^l.toFinset.card :=
+    pow_le_pow_of_le_one p.2.1 p.2.2 l.toFinset_card_le
+  exact (mul_le_mul_of_nonneg_right (mul_le_mul he hpw (pow_nonneg p.2.1 _) (Real.exp_pos _).le)
+    (tsum_nonneg (deformedKernel_nonneg G (walkKernel G hloc a) p k t D z))).trans hh
+
+end UniformDeformedEndpoint
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section PivotalLegGeometry
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem pivotal_endpoint_connected (G : BondGraph V E) (S : Set E) (x y : V)
+    (hc : Conn G S x y) (e : E) (hp : ¬Conn G (S \ {e}) x y) (z : V) (hz : z ∈ G e) :
+    Conn G S x z := by
+  obtain ⟨q⟩ := (conn_iff_reachable G S x y).mp hc
+  have hzs : z ∈ q.support := SimpleGraph.Walk.mem_support_of_mem_edges
+    (pivotal_image_mem_edges G S q e hp) hz
+  exact (conn_iff_reachable G S x z).mpr ⟨q.takeUntil z hzs⟩
+
+theorem conn_delete_of_unconnected_endpoint (G : BondGraph V E) (S : Set E) (x y z : V)
+    (hc : Conn G S x y) (e : E) (hz : z ∈ G e) (hn : ¬Conn G S x z) :
+    Conn G (S \ {e}) x y := by
+  by_contra hh
+  exact hn (pivotal_endpoint_connected G S x y hc e hh z hz)
+
+namespace RawCorridor
+variable {G : BondGraph V E} {k : ℕ}
+
+theorem retained_conn_base (w : RawCorridor G k) (z : V) (hz : z ∈ w.path.support) :
+    Conn G w.retained w.base z := by
+  obtain ⟨l,hl,hm,_⟩ := walk_lift_bonds G Set.univ (w.path.takeUntil z hz)
+  apply hl.conn
+  intro e he
+  exact w.path.edges_takeUntil_subset_edges hz (hm ▸ List.mem_map_of_mem he)
+
+theorem retained_conn (w : RawCorridor G k) : Conn G w.retained w.base w.endpoint :=
+  w.retained_conn_base w.endpoint w.path.end_mem_support
+
+theorem retained_connects_support (w : RawCorridor G k) (x y : V)
+    (hx : x ∈ w.path.support) (hy : y ∈ w.path.support) : Conn G w.retained x y :=
+  conn_trans G _ (conn_symm G _ (w.retained_conn_base x hx)) (w.retained_conn_base y hy)
+
+theorem retained_nonempty (w : RawCorridor G k) (hl : 0 < w.path.length) : w.retained.Nonempty := by
+  have hp : w.path.edges ≠ [] := by
+    intro he
+    have hh := congrArg List.length he
+    simp only [SimpleGraph.Walk.length_edges,List.length_nil] at hh
+    omega
+  obtain ⟨s,hs⟩ := List.exists_mem_of_ne_nil _ hp
+  obtain ⟨e,he,_⟩ := w.edge_has_label s hs
+  exact ⟨e,he⟩
+
+end RawCorridor
+end PivotalLegGeometry
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section OneLegGeometry
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace RawCorridor
+variable {G : BondGraph V E} {k : ℕ}
+
+ 
+
+theorem counted_one_leg (r : RawCorridor G k) (D : Set V) (ω B : Set E)
+    (x y u v : V) (hc : r.counted Set.univ ω x y)
+    (hr : r.retained ⊆ ω ∩ internalEdges G D) (hB : B ⊆ ω)
+    (hd : Disjoint B r.retained) (huv : Conn G B u v)
+    (hxu : Conn G (ω ∩ internalEdges G D) x u)
+    (hvy : Conn G (ω ∩ internalEdges G D) v y)
+    (hne : ¬Conn G (ω ∩ internalEdges G D) u v)
+    (e : E) (he : e ∈ r.retained)
+    (hep : ¬Conn G ((ω ∩ internalEdges G D) \ {e}) x u) : r.counted D ω x u := by
+  let S := ω ∩ internalEdges G D
+  have huniv : ω ∩ internalEdges G Set.univ = ω := by
+    apply Set.Subset.antisymm Set.inter_subset_left
+    exact fun _ he => ⟨he,fun _ _ => trivial⟩
+  obtain ⟨z,z',hzz⟩ : ∃ z z', G e = s(z,z') := by
+    obtain ⟨⟨z,z'⟩,hh⟩ := Quot.exists_rep (G e)
+    exact ⟨z,z',hh.symm⟩
+  have hz : z ∈ G e := hzz.symm ▸ Sym2.mem_mk_left z z'
+  have hxz : Conn G S x z := pivotal_endpoint_connected G S x u hxu e hep z hz
+  have hzsupport : z ∈ r.path.support := SimpleGraph.Walk.mem_support_of_mem_edges he hz
+  apply (r.counted_iff D ω x u).mpr
+  refine ⟨hc.1,hxu,?_,hc.2.2.2.2.mono_right ?_⟩
+  · intro f hf
+    refine ⟨hr hf,?_⟩
+    intro hxf
+    obtain ⟨a,b,hab⟩ : ∃ a b, G f = s(a,b) := by
+      obtain ⟨⟨a,b⟩,hh⟩ := Quot.exists_rep (G f)
+      exact ⟨a,b,hh.symm⟩
+    have ha : a ∈ G f := hab.symm ▸ Sym2.mem_mk_left a b
+    have haz : Conn G S z a := conn_mono G hr (r.retained_connects_support z a hzsupport
+      (SimpleGraph.Walk.mem_support_of_mem_edges hf ha))
+    have hua : Conn G S u a := conn_trans G S (conn_symm G S hxu) (conn_trans G S hxz haz)
+    have hvf : Conn G (S \ {f}) v y := conn_delete_of_unconnected_endpoint G S v y a hvy f ha
+      (fun hva => hne (conn_trans G S hua (conn_symm G S hva)))
+    have hBf : Conn G (ω \ {f}) u v := conn_mono G (show B ⊆ ω \ {f} from fun g hg => ⟨hB hg,fun hgf =>
+      Set.disjoint_left.mp hd hg ((Set.mem_singleton_iff.mp hgf).symm ▸ hf)⟩) huv
+    apply (hc.2.2.2.1 f hf).2
+    rw [huniv]
+    exact conn_trans G _ (conn_mono G (Set.sdiff_subset_sdiff_left Set.inter_subset_left) hxf)
+      (conn_trans G _ hBf (conn_mono G (Set.sdiff_subset_sdiff_left Set.inter_subset_left) hvf))
+  · rw [huniv]
+    exact Set.inter_subset_left
+
+end RawCorridor
+end OneLegGeometry
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section TwoLegGeometry
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace RawCorridor
+variable {G : BondGraph V E} {k : ℕ}
+
+theorem counted_split_legs (r : RawCorridor G k) (D : Set V) (ω B : Set E)
+    (x y u v : V) (hc : r.counted Set.univ ω x y)
+    (hr : r.retained ⊆ ω ∩ internalEdges G D) (hB : B ⊆ ω)
+    (hd : Disjoint B r.retained) (huv : Conn G B u v)
+    (hxu : Conn G (ω ∩ internalEdges G D) x u)
+    (hvy : Conn G (ω ∩ internalEdges G D) v y)
+    (hne : ¬Conn G (ω ∩ internalEdges G D) u v) :
+    r.counted D ω x u ∨ r.counted D ω v y := by
+  obtain ⟨e,he⟩ := r.retained_nonempty (by have := hc.1; omega)
+  by_cases hp : ¬Conn G ((ω ∩ internalEdges G D) \ {e}) x u
+  · exact Or.inl (r.counted_one_leg D ω B x y u v hc hr hB hd huv hxu hvy hne e he hp)
+  · have hxu' : Conn G ((ω ∩ internalEdges G D) \ {e}) x u := not_not.mp hp
+    have hp' : ¬Conn G ((ω ∩ internalEdges G D) \ {e}) y v := by
+      intro hyv
+      apply (hc.2.2.2.1 e he).2
+      have huniv : ω ∩ internalEdges G Set.univ = ω := by
+        apply Set.Subset.antisymm Set.inter_subset_left
+        exact fun _ he => ⟨he,fun _ _ => trivial⟩
+      rw [huniv]
+      have hB' : Conn G (ω \ {e}) u v := conn_mono G (show B ⊆ ω \ {e} from fun f hf => ⟨hB hf,fun hfe =>
+        Set.disjoint_left.mp hd hf ((Set.mem_singleton_iff.mp hfe).symm ▸ he)⟩) huv
+      exact conn_trans G _ (conn_mono G (Set.sdiff_subset_sdiff_left Set.inter_subset_left) hxu')
+        (conn_trans G _ hB' (conn_mono G (Set.sdiff_subset_sdiff_left Set.inter_subset_left) (conn_symm G _ hyv)))
+    exact Or.inr ((r.counted_symm D ω y v).mp (r.counted_one_leg D ω B y x v u
+      ((r.counted_symm Set.univ ω x y).mp hc) hr hB hd (conn_symm G B huv)
+      (conn_symm G _ hvy) (conn_symm G _ hxu) (fun h => hne (conn_symm G _ h)) e he hp'))
+
+ 
+theorem counted_legs_of_disjoint (w r : RawCorridor G k) (ω : Set E)
+    (hq : ω ∈ w.calibrationEvent) (x y : V)
+    (hxu : Conn G (w.exterior ω) x w.base)
+    (hvy : Conn G (w.exterior ω) w.endpoint y)
+    (hne : ¬Conn G (w.exterior ω) w.base w.endpoint)
+    (hc : r.counted Set.univ ω x y) (hd : Disjoint w.retained r.retained) :
+    r.counted (w.interior : Set V)ᶜ ω x w.base ∨ r.counted (w.interior : Set V)ᶜ ω w.endpoint y := by
+  apply r.counted_split_legs (w.interior : Set V)ᶜ ω w.retained x y w.base w.endpoint
+    hc ?_ hq.1 hd w.retained_conn hxu hvy hne
+  intro e he
+  have heω := (hc.2.2.2.1 e he).1.1
+  exact (w.open_subset_exterior_retained ω hq heω).resolve_right
+    (fun hw => Set.disjoint_left.mp hd hw he)
+
+end RawCorridor
+end TwoLegGeometry
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section TwoLegPenaltyTerms
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem corridorPenaltyTerm_legs (G : BondGraph V E) (hG : Function.Injective G)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (k : ℕ)
+    (w r : RawCorridor G k) (ω : Set E) (hq : ω ∈ w.calibrationEvent) (x y : V)
+    (hxu : Conn G (w.exterior ω) x w.base) (hvy : Conn G (w.exterior ω) w.endpoint y)
+    (hne : ¬Conn G (w.exterior ω) w.base w.endpoint) :
+    corridorPenaltyTerm G P p Set.univ ω x y r ≤
+      (∑ e ∈ (w.retained_finite hG).toFinset, if e ∈ r.retained then r.weight P p else 0) +
+        corridorPenaltyTerm G P p (w.interior : Set V)ᶜ ω x w.base r +
+        corridorPenaltyTerm G P p (w.interior : Set V)ᶜ ω w.endpoint y r := by
+  have hr := r.weight_nonneg P hP p
+  have hs : 0 ≤ ∑ e ∈ (w.retained_finite hG).toFinset, if e ∈ r.retained then r.weight P p else 0 :=
+    Finset.sum_nonneg (fun _ _ => ite_nonneg hr (le_refl _))
+  have hl := corridorPenaltyTerm_nonneg G P hP p (w.interior : Set V)ᶜ ω x w.base r
+  have ht := corridorPenaltyTerm_nonneg G P hP p (w.interior : Set V)ᶜ ω w.endpoint y r
+  by_cases hc : r.counted Set.univ ω x y
+  · rw [corridorPenaltyTerm,ite_eq_left hc]
+    by_cases hd : Disjoint w.retained r.retained
+    · rcases w.counted_legs_of_disjoint r ω hq x y hxu hvy hne hc hd with hh | hh
+      · have he : corridorPenaltyTerm G P p (w.interior : Set V)ᶜ ω x w.base r = r.weight P p := ite_eq_left hh
+        linarith
+      · have he : corridorPenaltyTerm G P p (w.interior : Set V)ᶜ ω w.endpoint y r = r.weight P p := ite_eq_left hh
+        linarith
+    · obtain ⟨e,hew,her⟩ := Set.not_disjoint_iff.mp hd
+      have he := Finset.single_le_sum (f := fun e => if e ∈ r.retained then r.weight P p else 0)
+        (s := (w.retained_finite hG).toFinset) (fun _ _ => ite_nonneg hr (le_refl _))
+        ((w.retained_finite hG).mem_toFinset.mpr hew)
+      rw [ite_eq_left her] at he
+      linarith
+  · rw [corridorPenaltyTerm,ite_eq_right hc]
+    linarith
+
+ 
+
+theorem corridorPenalty_legs (G : BondGraph V E) (hG : Function.Injective G)
+    (hloc : LocallyFinite G) (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y)
+    (p : unitInterval) (k : ℕ) (w : RawCorridor G k) (ω : Set E)
+    (hq : ω ∈ w.calibrationEvent) (x y : V)
+    (hxu : Conn G (w.exterior ω) x w.base) (hvy : Conn G (w.exterior ω) w.endpoint y)
+    (hne : ¬Conn G (w.exterior ω) w.base w.endpoint) :
+    corridorPenalty G P p k Set.univ ω x y ≤
+      (∑ e ∈ (w.retained_finite hG).toFinset, ∑' r : RawCorridor G k,
+        if e ∈ r.retained then r.weight P p else 0) +
+      corridorPenalty G P p k (w.interior : Set V)ᶜ ω x w.base +
+      corridorPenalty G P p k (w.interior : Set V)ᶜ ω w.endpoint y := by
+  have hs : Summable (fun r : RawCorridor G k => ∑ e ∈ (w.retained_finite hG).toFinset,
+      if e ∈ r.retained then r.weight P p else 0) :=
+    (hasSum_sum (fun e (_ : e ∈ (w.retained_finite hG).toFinset) =>
+      (summable_raw_edge_weights G hloc P p k e).hasSum)).summable
+  have hl := corridorPenalty_summable G hloc P p k (w.interior : Set V)ᶜ ω x w.base
+  have hr := corridorPenalty_summable G hloc P p k (w.interior : Set V)ᶜ ω w.endpoint y
+  have hh := (corridorPenalty_summable G hloc P p k Set.univ ω x y).tsum_le_tsum
+    (fun r => corridorPenaltyTerm_legs G hG P hP p k w r ω hq x y hxu hvy hne) ((hs.add hl).add hr)
+  rw [(hs.add hl).tsum_add hr,hs.tsum_add hl,
+    Summable.tsum_finsetSum (fun e _ => summable_raw_edge_weights G hloc P p k e)] at hh
+  exact hh
+
+end TwoLegPenaltyTerms
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section WeightedLegComparison
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+namespace RawCorridor
+variable {G : BondGraph V E} {k : ℕ}
+theorem base_not_interior (w : RawCorridor G k) : w.base ∉ w.interior := by
+  simp only [interior,Finset.mem_sdiff,Finset.mem_insert,true_or,not_true_eq_false,and_false,not_false_eq_true]
+theorem endpoint_not_interior (w : RawCorridor G k) : w.endpoint ∉ w.interior := by
+  simp only [interior,Finset.mem_sdiff,Finset.mem_insert,Finset.mem_singleton,or_true,not_true_eq_false,and_false,not_false_eq_true]
+end RawCorridor
+
+ 
+theorem deformedConnection_legs_lower (G : BondGraph V E) (hG : Function.Injective G)
+    (hloc : LocallyFinite G) (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y)
+    (p : unitInterval) (k : ℕ) (w : RawCorridor G k) (ω : Set E)
+    (hq : ω ∈ w.calibrationEvent) (x y : V)
+    (hxu : Conn G (w.exterior ω) x w.base) (hvy : Conn G (w.exterior ω) w.endpoint y)
+    (hne : ¬Conn G (w.exterior ω) w.base w.endpoint) (c : ℝ)
+    (hc : (∑ e ∈ (w.retained_finite hG).toFinset, ∑' r : RawCorridor G k,
+        if e ∈ r.retained then r.weight P p else 0) ≤ c) (t : ℝ) (ht : 0 ≤ t) :
+    Real.exp (-t*c) * deformedConnection G P p k t (w.interior : Set V)ᶜ ω x w.base *
+      deformedConnection G P p k t (w.interior : Set V)ᶜ ω w.endpoint y ≤
+        deformedConnection G P p k t Set.univ ω x y := by
+  have hx : x ∈ (w.interior : Set V)ᶜ := by
+    rcases conn_internal_eq_or_mem G (w.interior : Set V)ᶜ hxu with h | h
+    · rw [h]; exact w.base_not_interior
+    · exact h.1
+  have hy : y ∈ (w.interior : Set V)ᶜ := by
+    rcases conn_internal_eq_or_mem G (w.interior : Set V)ᶜ hvy with h | h
+    · rw [←h]; exact w.endpoint_not_interior
+    · exact h.2
+  have hxy : Conn G (ω ∩ internalEdges G Set.univ) x y := by
+    apply conn_mono G (show ω ⊆ ω ∩ internalEdges G Set.univ from fun _ h => ⟨h,fun _ _ => trivial⟩)
+    exact conn_trans G ω (conn_mono G Set.inter_subset_left hxu)
+      (conn_trans G ω (conn_mono G hq.1 w.retained_conn) (conn_mono G Set.inter_subset_left hvy))
+  have hleft : x ∈ (w.interior : Set V)ᶜ ∧ w.base ∈ (w.interior : Set V)ᶜ ∧
+      Conn G (ω ∩ internalEdges G (w.interior : Set V)ᶜ) x w.base := ⟨hx,w.base_not_interior,hxu⟩
+  have hright : w.endpoint ∈ (w.interior : Set V)ᶜ ∧ y ∈ (w.interior : Set V)ᶜ ∧
+      Conn G (ω ∩ internalEdges G (w.interior : Set V)ᶜ) w.endpoint y := ⟨w.endpoint_not_interior,hy,hvy⟩
+  simp only [deformedConnection,ite_eq_left hleft,ite_eq_left hright,Set.mem_univ,hxy,and_self,ite_true]
+  rw [←Real.exp_add,←Real.exp_add]
+  apply Real.exp_le_exp.mpr
+  have hh := (corridorPenalty_legs G hG hloc P hP p k w ω hq x y hxu hvy hne).trans
+    (add_le_add (add_le_add hc le_rfl) le_rfl)
+  simpa only [mul_add] using mul_le_mul_of_nonpos_left hh (neg_nonpos.mpr ht)
+
+end WeightedLegComparison
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section UniformLegCost
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem exists_uniform_corridor_leg_cost (G : BondGraph V E) (hG : Function.Injective G)
+    (hloc : LocallyFinite G) (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
+    (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y)
+    (hpcrit : 0 < (pc G:ℝ)) (hpcrit₁ : (pc G:ℝ) < 1) :
+    ∃ C : ℝ, 0 < C ∧ ∀ (p : unitInterval), (pc G:ℝ)/2 ≤ (p:ℝ) → p < pc G →
+      ∀ (k : ℕ), 3 ≤ k → ∀ w : RawCorridor G k,
+      (∑ e ∈ (w.retained_finite hG).toFinset, ∑' r : RawCorridor G k,
+        if e ∈ r.retained then r.weight (walkKernel G hloc a) p else 0) ≤ Real.exp (C*(k:ℝ)) := by
+  obtain ⟨C,hC,hcost⟩ := exists_uniform_corridor_edge_cost G hloc d hd a hP hpcrit hpcrit₁
+  refine ⟨C,hC,?_⟩
+  intro p hpnear hp k hk w
+  have hh := Finset.sum_le_sum (s := (w.retained_finite hG).toFinset)
+    (fun e _ => hcost p hpnear hp k hk e)
+  have hcard : ((w.retained_finite hG).toFinset.card:ℝ) ≤ (k:ℝ) := by
+    rw [←Set.ncard_eq_toFinset_card w.retained (w.retained_finite hG),w.retained_ncard hG]
+    exact Nat.cast_le.mpr w.path_length
+  have hk0 : (0:ℝ) < k := Nat.cast_pos.mpr (by omega)
+  calc
+    _ ≤ ((w.retained_finite hG).toFinset.card:ℝ)*(Real.exp (C*(k:ℝ))/(k:ℝ)) := by
+      simpa only [Finset.sum_const,nsmul_eq_mul] using hh
+    _ ≤ (k:ℝ)*(Real.exp (C*(k:ℝ))/(k:ℝ)) := mul_le_mul_of_nonneg_right hcard (div_nonneg (Real.exp_pos _).le hk0.le)
+    _ = _ := by field_simp
+
+end UniformLegCost
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section SimplePathCuts
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem simplePath_edge_cut {H : SimpleGraph V} {u v : V} (q : H.Walk u v)
+    (hq : q.IsPath) (e : Sym2 V) (he : e ∈ q.edges) :
+    ∃ A : Set V, u ∈ A ∧ v ∉ A ∧ (∀ z ∈ A, z ∈ q.support) ∧
+      ∀ f ∈ q.edges, f ≠ e → ∀ a ∈ f, ∀ b ∈ f, a ∈ A ↔ b ∈ A := by
+  induction q with
+  | nil => simp only [SimpleGraph.Walk.edges_nil,List.not_mem_nil] at he
+  | @cons u a v h q ih =>
+    have hu : u ∉ q.support := ((SimpleGraph.Walk.cons_isPath_iff _ _).mp hq).2
+    have huv : v ≠ u := fun hvu => hu (hvu ▸ q.end_mem_support)
+    by_cases hh : e = s(u,a)
+    · refine ⟨{u},rfl,huv,?_,?_⟩
+      · intro z hz; rw [Set.mem_singleton_iff] at hz; subst z; simp
+      · intro f hf hfe x hx y hy
+        have hfq : f ∈ q.edges := (List.mem_cons.mp hf).resolve_left (fun h' => hfe (h'.trans hh.symm))
+        have hxu : x ≠ u := fun hxu => hu (hxu ▸ SimpleGraph.Walk.mem_support_of_mem_edges hfq hx)
+        have hyu : y ≠ u := fun hyu => hu (hyu ▸ SimpleGraph.Walk.mem_support_of_mem_edges hfq hy)
+        simp only [Set.mem_singleton_iff,hxu,hyu]
+    · have heq : e ∈ q.edges := (List.mem_cons.mp he).resolve_left hh
+      obtain ⟨A,ha,hv,hA,hcut⟩ := ih hq.of_cons heq
+      refine ⟨insert u A,Set.mem_insert u A,?_,?_,?_⟩
+      · intro hvA
+        rcases hvA with h' | h'
+        · exact huv h'
+        · exact hv h'
+      · intro z hz
+        rcases hz with rfl | hz
+        · simp
+        · exact List.mem_cons_of_mem u (hA z hz)
+      · intro f hf hfe x hx y hy
+        rcases List.mem_cons.mp hf with rfl | hf
+        · have hx' : x = u ∨ x = a := by simpa only [SimpleGraph.Dart.edge,Sym2.mem_iff] using hx
+          have hy' : y = u ∨ y = a := by simpa only [SimpleGraph.Dart.edge,Sym2.mem_iff] using hy
+          have hxm : x ∈ insert u A := hx'.elim (fun h' => h' ▸ Set.mem_insert u A)
+            (fun h' => h' ▸ Set.mem_insert_of_mem u ha)
+          have hym : y ∈ insert u A := hy'.elim (fun h' => h' ▸ Set.mem_insert u A)
+            (fun h' => h' ▸ Set.mem_insert_of_mem u ha)
+          exact iff_of_true hxm hym
+        · have hxu : x ≠ u := fun hxu => hu (hxu ▸ SimpleGraph.Walk.mem_support_of_mem_edges hf hx)
+          have hyu : y ≠ u := fun hyu => hu (hyu ▸ SimpleGraph.Walk.mem_support_of_mem_edges hf hy)
+          simpa only [Set.mem_insert_iff,hxu,hyu,false_or] using hcut f hf hfe x hx y hy
+
+theorem conn_closed_vertex_set (G : BondGraph V E) (S : Set E) (A : Set V)
+    (hA : ∀ e ∈ S, ∀ a ∈ G e, ∀ b ∈ G e, a ∈ A → b ∈ A)
+    {x y : V} (hc : Conn G S x y) (hx : x ∈ A) : y ∈ A := by
+  induction hc with
+  | refl => exact hx
+  | @tail a b _ hab ih =>
+    obtain ⟨e,he,heq⟩ := hab
+    exact hA e he a (heq.symm ▸ Sym2.mem_mk_left a b) b (heq.symm ▸ Sym2.mem_mk_right a b) ih
+
+end SimplePathCuts
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section CalibratedCut
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem conn_edge_members (G : BondGraph V E) (S : Set E) (e : E) (he : e ∈ S)
+    (a b : V) (ha : a ∈ G e) (hb : b ∈ G e) : Conn G S a b := by
+  obtain ⟨x,y,hxy⟩ : ∃ x y, G e = s(x,y) := by
+    obtain ⟨⟨x,y⟩,hh⟩ := Quot.exists_rep (G e)
+    exact ⟨x,y,hh.symm⟩
+  rw [hxy,Sym2.mem_iff] at ha hb
+  rcases ha with rfl | rfl <;> rcases hb with rfl | rfl
+  · exact conn_refl G S _
+  · exact Relation.ReflTransGen.single ⟨e,he,hxy⟩
+  · exact conn_symm G S (Relation.ReflTransGen.single ⟨e,he,hxy⟩)
+  · exact conn_refl G S _
+
+namespace RawCorridor
+variable {G : BondGraph V E} {k : ℕ}
+
+ 
+
+theorem calibration_pivotal_ports (hG : Function.Injective G) (w : RawCorridor G k)
+    (ω : Set E) (hq : ω ∈ w.calibrationEvent)
+    (hne : ¬Conn G (w.exterior ω) w.base w.endpoint)
+    (e : E) (he : e ∈ w.retained) : ¬Conn G (ω \ {e}) w.base w.endpoint := by
+  obtain ⟨A,hu,hv,hA,hcut⟩ := simplePath_edge_cut w.path w.path_isPath (G e) he
+  let B := A ∪ {z | Conn G (w.exterior ω) w.base z}
+  have hAport (z : V) (hz : z ∈ A) (ho : z ∉ w.interior) : z = w.base := by
+    rcases w.support_exterior_ports (hA z hz) ho with h | h
+    · exact h
+    · exact (hv (h ▸ hz)).elim
+  have hConnOut (z : V) (hz : Conn G (w.exterior ω) w.base z) : z ∉ w.interior := by
+    rcases conn_internal_eq_or_mem G (w.interior : Set V)ᶜ hz with h | h
+    · rw [←h]; exact w.base_not_interior
+    · exact h.2
+  have hBclosed : ∀ f ∈ ω \ {e}, ∀ a ∈ G f, ∀ b ∈ G f, a ∈ B → b ∈ B := by
+    intro f hf a ha b hb hba
+    rcases w.open_subset_exterior_retained ω hq hf.1 with hfs | hfr
+    · have hab := conn_edge_members G (w.exterior ω) f hfs a b ha hb
+      rcases hba with haa | hua
+      · have hau : a = w.base := hAport a haa (hfs.2 a ha)
+        exact Or.inr (hau ▸ hab)
+      · exact Or.inr (conn_trans G _ hua hab)
+    · have hfe : G f ≠ G e := fun h => hf.2 (hG h)
+      have haa : a ∈ A := by
+        rcases hba with h | h
+        · exact h
+        · rcases w.support_exterior_ports (SimpleGraph.Walk.mem_support_of_mem_edges hfr ha)
+            (hConnOut a h) with hau | hav
+          · exact hau ▸ hu
+          · exact (hne (hav ▸ h)).elim
+      exact Or.inl ((hcut (G f) hfr hfe a ha b hb).mp haa)
+  intro hc
+  have hh := conn_closed_vertex_set G (ω \ {e}) B hBclosed hc (Or.inl hu)
+  exact hh.elim hv hne
+
+end RawCorridor
+end CalibratedCut
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section ReverseCalibration
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace RawCorridor
+variable {G : BondGraph V E} {k : ℕ}
+
+theorem calibration_counted_of_orientation (hG : Function.Injective G) (w : RawCorridor G k)
+    (hlen : 3 ≤ w.path.length) (ω : Set E) (hq : ω ∈ w.calibrationEvent) (x y : V)
+    (hxu : Conn G (w.exterior ω) x w.base) (hvy : Conn G (w.exterior ω) w.endpoint y)
+    (hne : ¬Conn G (w.exterior ω) w.base w.endpoint) : w.counted Set.univ ω x y := by
+  have huniv : ω ∩ internalEdges G Set.univ = ω := by
+    apply Set.Subset.antisymm Set.inter_subset_left
+    exact fun _ he => ⟨he,fun _ _ => trivial⟩
+  apply (w.counted_iff Set.univ ω x y).mpr
+  refine ⟨hlen,?_⟩
+  unfold corridorCounted
+  rw [huniv]
+  refine ⟨conn_trans G ω (conn_mono G Set.inter_subset_left hxu)
+    (conn_trans G ω (conn_mono G hq.1 w.retained_conn) (conn_mono G Set.inter_subset_left hvy)),?_,hq.2⟩
+  intro e he
+  refine ⟨hq.1 he,?_⟩
+  intro hxy
+  have hS : w.exterior ω ⊆ ω \ {e} := by
+    intro f hf
+    refine ⟨hf.1,?_⟩
+    intro hfe
+    have hfe' : f = e := hfe
+    subst f
+    exact Set.disjoint_left.mp (w.specified_disjoint_exterior hlen) (Or.inl he) hf.2
+  exact w.calibration_pivotal_ports hG ω hq hne e he
+    (conn_trans G _ (conn_mono G hS (conn_symm G _ hxu))
+      (conn_trans G _ hxy (conn_mono G hS (conn_symm G _ hvy))))
+
+ 
+
+theorem calibration_counted_iff (hG : Function.Injective G) (w : RawCorridor G k)
+    (hlen : 3 ≤ w.path.length) (ω : Set E) (hq : ω ∈ w.calibrationEvent) (x y : V) :
+    w.counted Set.univ ω x y ↔ ¬Conn G (w.exterior ω) w.base w.endpoint ∧
+      ((Conn G (w.exterior ω) w.base x ∧ Conn G (w.exterior ω) w.endpoint y) ∨
+       (Conn G (w.exterior ω) w.endpoint x ∧ Conn G (w.exterior ω) w.base y)) := by
+  constructor
+  · intro hc
+    exact w.counted_exterior_orientation hlen ω hq x y hc.2.2
+  · rintro ⟨hne,ho⟩
+    rcases ho with ⟨hx,hy⟩ | ⟨hx,hy⟩
+    · exact w.calibration_counted_of_orientation hG hlen ω hq x y (conn_symm G _ hx) hy hne
+    · exact (w.counted_symm Set.univ ω y x).mp
+        (w.calibration_counted_of_orientation hG hlen ω hq y x (conn_symm G _ hy) hx hne)
+
+end RawCorridor
+end ReverseCalibration
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section ExteriorLegObservable
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem deformedConnection_restrict_config (G : BondGraph V E) (P : V → V → ℝ)
+    (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x y : V) :
+    deformedConnection G P p k t D (ω ∩ internalEdges G D) x y =
+      deformedConnection G P p k t D ω x y := by
+  have he : (ω ∩ internalEdges G D) ∩ internalEdges G D = ω ∩ internalEdges G D :=
+    Set.inter_assoc _ _ _ |>.trans (congrArg (ω ∩ ·) (Set.inter_self _))
+  have hc (w : RawCorridor G k) : w.counted D (ω ∩ internalEdges G D) x y ↔ w.counted D ω x y := by
+    simp only [RawCorridor.counted,corridorCounted,he]
+  simp only [deformedConnection,corridorPenalty,corridorPenaltyTerm,hc,he]
+
+noncomputable def exteriorLegProduct (G : BondGraph V E) (P : V → V → ℝ)
+    (p : ℝ) {k : ℕ} (t : ℝ) (w : RawCorridor G k) (ω : Set E) (x y : V) : ℝ :=
+  if ¬Conn G (w.exterior ω) w.base w.endpoint then
+    deformedConnection G P p k t (w.interior : Set V)ᶜ ω x w.base *
+      deformedConnection G P p k t (w.interior : Set V)ᶜ ω w.endpoint y else 0
+
+theorem exteriorLegProduct_nonneg (G : BondGraph V E) (P : V → V → ℝ)
+    (p : ℝ) {k : ℕ} (t : ℝ) (w : RawCorridor G k) (ω : Set E) (x y : V) :
+    0 ≤ exteriorLegProduct G P p t w ω x y := by
+  unfold exteriorLegProduct
+  by_cases h : ¬Conn G (w.exterior ω) w.base w.endpoint
+  · rw [ite_eq_left h]
+    exact mul_nonneg (deformedConnection_nonneg _ _ _ _ _ _ _ _ _)
+      (deformedConnection_nonneg _ _ _ _ _ _ _ _ _)
+  · rw [ite_eq_right h]
+
+theorem exteriorLegProduct_le_one (G : BondGraph V E) (P : V → V → ℝ)
+    (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) {k : ℕ} (t : ℝ) (ht : 0 ≤ t)
+    (w : RawCorridor G k) (ω : Set E) (x y : V) :
+    exteriorLegProduct G P p t w ω x y ≤ 1 := by
+  unfold exteriorLegProduct
+  by_cases h : ¬Conn G (w.exterior ω) w.base w.endpoint
+  · rw [ite_eq_left h]
+    exact (mul_le_of_le_one_left (deformedConnection_nonneg _ _ _ _ _ _ _ _ _)
+      (deformedConnection_le_one G P hP p k t ht _ _ _ _)).trans
+      (deformedConnection_le_one G P hP p k t ht _ _ _ _)
+  · rw [ite_eq_right h]
+    exact zero_le_one
+
+theorem measurable_exteriorLegProduct [Countable V] [Countable E]
+    (G : BondGraph V E) (P : V → V → ℝ) (p : ℝ) {k : ℕ} (t : ℝ)
+    (w : RawCorridor G k) (x y : V) :
+    Measurable (fun ω => exteriorLegProduct G P p t w ω x y) := by
+  apply Measurable.ite
+    ((measurableSet_restrictedConn G (internalEdges G (w.interior:Set V)ᶜ) w.base w.endpoint).compl)
+  · exact (measurable_deformedConnection G P p k t _ x w.base).mul
+      (measurable_deformedConnection G P p k t _ w.endpoint y)
+  · exact measurable_const
+
+theorem exteriorLegProduct_restrict (G : BondGraph V E) (P : V → V → ℝ)
+    (p : ℝ) {k : ℕ} (t : ℝ) (w : RawCorridor G k) (ω : Set E) (x y : V) :
+    exteriorLegProduct G P p t w ω x y =
+      exteriorLegProduct G P p t w (ω ∩ internalEdges G (w.interior:Set V)ᶜ) x y := by
+  simp only [exteriorLegProduct,RawCorridor.exterior,Set.inter_assoc,Set.inter_self,
+    deformedConnection_restrict_config]
+  rfl
+
+end ExteriorLegObservable
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section CalibrationFactor
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace RawCorridor
+variable {G : BondGraph V E} {k : ℕ}
+
+theorem measurableSet_calibration (hG : Function.Injective G) (hloc : LocallyFinite G)
+    (w : RawCorridor G k) : MeasurableSet w.calibrationEvent := by
+  rw [w.calibrationEvent_eq hG hloc]
+  apply measurableSet_edgePattern
+  apply Finset.disjoint_left.mpr
+  intro e he hb
+  exact Set.disjoint_left.mp w.retained_disjoint_blockers
+    ((w.retained_finite hG).mem_toFinset.mp he) ((w.blockers_finite hloc).mem_toFinset.mp hb)
+
+ 
+
+theorem calibration_integral (hG : Function.Injective G) (hloc : LocallyFinite G)
+    (w : RawCorridor G k) (hlen : 3 ≤ w.path.length) (p : unitInterval)
+    (f : Set E → ℝ) (hf : Measurable f)
+    (hfr : ∀ ω, f ω = f (ω ∩ internalEdges G (w.interior:Set V)ᶜ)) :
+    (∫ ω, if ω ∈ w.calibrationEvent then f ω else 0 ∂law p) =
+      (law p).real w.calibrationEvent * ∫ ω, f ω ∂law p := by
+  have hh := integral_mul_disjoint_coordinates p _ _ (w.specified_disjoint_exterior hlen)
+    (fun ω => if ω ∈ w.calibrationEvent then (1:ℝ) else 0) f
+    (Measurable.ite (w.measurableSet_calibration hG hloc) measurable_const measurable_const) hf
+    (fun ω => by rw [w.calibration_restriction ω]) hfr
+  simpa only [ite_mul,one_mul,zero_mul,integral_event_indicator p _ (w.measurableSet_calibration hG hloc)] using hh
+
+theorem calibration_weighted_integral (hG : Function.Injective G) (hloc : LocallyFinite G)
+    (w : RawCorridor G k) (hlen : 3 ≤ w.path.length) (P : V → V → ℝ)
+    (p : unitInterval) (hp : 0 < (p:ℝ)) (hp₁ : (p:ℝ) < 1)
+    (f : Set E → ℝ) (hf : Measurable f)
+    (hfr : ∀ ω, f ω = f (ω ∩ internalEdges G (w.interior:Set V)ᶜ)) :
+    (∫ ω, if ω ∈ w.calibrationEvent then w.weight P p * f ω else 0 ∂law p) =
+      w.probability P * ∫ ω, f ω ∂law p := by
+  have he : (fun ω => if ω ∈ w.calibrationEvent then w.weight P p * f ω else 0) =
+      (fun ω => w.weight P p * (if ω ∈ w.calibrationEvent then f ω else 0)) := by
+    funext ω
+    by_cases h : ω ∈ w.calibrationEvent <;> simp only [h,ite_true,ite_false,mul_zero]
+  rw [he,integral_const_mul,w.calibration_integral hG hloc hlen p f hf hfr,
+    ←mul_assoc,w.calibration_identity hG hloc P p hp hp₁]
+
+end RawCorridor
+end CalibrationFactor
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section CalibratedDerivativeTerm
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem calibrated_leg_product_le_term (G : BondGraph V E) (hG : Function.Injective G)
+    (hloc : LocallyFinite G) (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y)
+    (p : unitInterval) (k : ℕ) (w : RawCorridor G k) (hlen : 3 ≤ w.path.length)
+    (c : ℝ) (hc : (∑ e ∈ (w.retained_finite hG).toFinset, ∑' r : RawCorridor G k,
+        if e ∈ r.retained then r.weight P p else 0) ≤ c)
+    (t : ℝ) (ht : 0 ≤ t) (ω : Set E) (x y : V) :
+    (if ω ∈ w.calibrationEvent then w.weight P p * (Real.exp (-t*c) *
+      exteriorLegProduct G P p t w ω x y) else 0) ≤
+        corridorPenaltyTerm G P p Set.univ ω x y w * deformedConnection G P p k t Set.univ ω x y := by
+  have hr : 0 ≤ corridorPenaltyTerm G P p Set.univ ω x y w * deformedConnection G P p k t Set.univ ω x y :=
+    mul_nonneg (corridorPenaltyTerm_nonneg G P hP p Set.univ ω x y w)
+      (deformedConnection_nonneg G P p k t Set.univ ω x y)
+  by_cases hq : ω ∈ w.calibrationEvent
+  · rw [ite_eq_left hq]
+    by_cases hn : ¬Conn G (w.exterior ω) w.base w.endpoint
+    · rw [exteriorLegProduct,ite_eq_left hn]
+      by_cases hx : Conn G (w.exterior ω) x w.base
+      · by_cases hy : Conn G (w.exterior ω) w.endpoint y
+        · have hj := w.calibration_counted_of_orientation hG hlen ω hq x y hx hy hn
+          rw [corridorPenaltyTerm,ite_eq_left hj]
+          have hh := mul_le_mul_of_nonneg_left
+            (deformedConnection_legs_lower G hG hloc P hP p k w ω hq x y hx hy hn c hc t ht)
+            (w.weight_nonneg P hP p)
+          simpa only [mul_assoc] using hh
+        · have hz : deformedConnection G P p k t (w.interior:Set V)ᶜ ω w.endpoint y = 0 := by
+            simp only [deformedConnection,show ¬Conn G (ω ∩ internalEdges G (w.interior:Set V)ᶜ) w.endpoint y from hy,and_false,ite_false]
+          simpa only [hz,mul_zero] using hr
+      · have hz : deformedConnection G P p k t (w.interior:Set V)ᶜ ω x w.base = 0 := by
+          simp only [deformedConnection,show ¬Conn G (ω ∩ internalEdges G (w.interior:Set V)ᶜ) x w.base from hx,and_false,ite_false]
+        simpa only [hz,zero_mul,mul_zero] using hr
+    · simpa only [exteriorLegProduct,ite_eq_right hn,mul_zero] using hr
+  · simpa only [ite_eq_right hq] using hr
+
+theorem integrable_corridorDerivativeTerm [Countable V] [Countable E]
+    (G : BondGraph V E) (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y)
+    (p : unitInterval) (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D : Set V) (x y : V)
+    (w : RawCorridor G k) :
+    Integrable (fun ω => corridorPenaltyTerm G P p D ω x y w *
+      deformedConnection G P p k t D ω x y) (law p) := by
+  have hh := ((integrable_deformedConnection G P hP p k t ht D x y).const_mul
+    (w.weight P p)).indicator (w.measurableSet_counted D x y)
+  have he : (fun ω => corridorPenaltyTerm G P p D ω x y w *
+      deformedConnection G P p k t D ω x y) =
+      {ω | w.counted D ω x y}.indicator (fun ω => w.weight P p * deformedConnection G P p k t D ω x y) := by
+    funext ω
+    simp only [Set.indicator,Set.mem_ofPred_eq,corridorPenaltyTerm,ite_mul,zero_mul]
+  rw [he]
+  exact hh
+
+end CalibratedDerivativeTerm
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section CalibratedDecreaseKernel
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem integrable_exteriorLegProduct [Countable V] [Countable E]
+    (G : BondGraph V E) (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y)
+    (p : unitInterval) {k : ℕ} (t : ℝ) (ht : 0 ≤ t) (w : RawCorridor G k) (x y : V) :
+    Integrable (fun ω => exteriorLegProduct G P p t w ω x y) (law p) := by
+  apply Integrable.mono' (integrable_const (1:ℝ))
+    (measurable_exteriorLegProduct G P p t w x y).aestronglyMeasurable
+  filter_upwards [] with ω
+  rw [Real.norm_of_nonneg (exteriorLegProduct_nonneg G P p t w ω x y)]
+  exact exteriorLegProduct_le_one G P hP p t ht w ω x y
+
+ 
+
+theorem calibrated_derivative_kernel_lower [Countable V] [Countable E]
+    (G : BondGraph V E) (hG : Function.Injective G) (hloc : LocallyFinite G)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y)
+    (p : unitInterval) (hp : 0 < (p:ℝ)) (hp₁ : (p:ℝ) < 1)
+    (k : ℕ) (w : RawCorridor G k) (hlen : 3 ≤ w.path.length)
+    (c : ℝ) (hc : (∑ e ∈ (w.retained_finite hG).toFinset, ∑' r : RawCorridor G k,
+        if e ∈ r.retained then r.weight P p else 0) ≤ c)
+    (t : ℝ) (ht : 0 ≤ t) (x y : V) :
+    w.probability P * Real.exp (-t*c) * (∫ ω, exteriorLegProduct G P p t w ω x y ∂law p) ≤
+      ∫ ω, corridorPenaltyTerm G P p Set.univ ω x y w *
+        deformedConnection G P p k t Set.univ ω x y ∂law p := by
+  have hf := integrable_exteriorLegProduct G P hP p t ht w x y
+  have hi : Integrable (fun ω => if ω ∈ w.calibrationEvent then w.weight P p *
+      (Real.exp (-t*c) * exteriorLegProduct G P p t w ω x y) else 0) (law p) :=
+    ((hf.const_mul (Real.exp (-t*c))).const_mul (w.weight P p)).indicator
+      (w.measurableSet_calibration hG hloc)
+  have hh := integral_mono hi (integrable_corridorDerivativeTerm G P hP p k t ht Set.univ x y w)
+    (fun ω => calibrated_leg_product_le_term G hG hloc P hP p k w hlen c hc t ht ω x y)
+  rw [w.calibration_weighted_integral hG hloc hlen P p hp hp₁
+    (fun ω => Real.exp (-t*c)*exteriorLegProduct G P p t w ω x y)
+    ((measurable_exteriorLegProduct G P p t w x y).const_mul _)
+    (fun ω => congrArg (Real.exp (-t*c)*·) (exteriorLegProduct_restrict G P p t w ω x y)),
+    integral_const_mul] at hh
+  simpa only [mul_assoc] using hh
+
+end CalibratedDecreaseKernel
+end Percolation
+
+ 
+namespace Percolation
+open MeasureTheory
+section NonnegativeRealTonelli
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem real_tonelli_of_integrable_sum {Ω I : Type*} [MeasurableSpace Ω] [Countable I]
+    (μ : Measure Ω) (F : I → Ω → ℝ) (hF : ∀ i, Integrable (F i) μ)
+    (hn : ∀ i ω, 0 ≤ F i ω) (hs : ∀ᵐ ω ∂μ, Summable (fun i => F i ω))
+    (hi : Integrable (fun ω => ∑' i, F i ω) μ) :
+    Summable (fun i => ∫ ω, F i ω ∂μ) ∧
+      (∑' i, ∫ ω, F i ω ∂μ) = ∫ ω, ∑' i, F i ω ∂μ := by
+  have he : (∑' i, ENNReal.ofReal (∫ ω, F i ω ∂μ)) =
+      ∫⁻ ω, ENNReal.ofReal (∑' i, F i ω) ∂μ := by
+    simp_rw [ofReal_integral_eq_lintegral_ofReal (hF _) (Filter.Eventually.of_forall (hn _))]
+    rw [←lintegral_tsum (fun i => (hF i).aestronglyMeasurable.aemeasurable.ennreal_ofReal)]
+    apply lintegral_congr_ae
+    filter_upwards [hs] with ω hω
+    exact (ENNReal.ofReal_tsum_of_nonneg (fun i => hn i ω) hω).symm
+  have ht : (∑' i, ENNReal.ofReal (∫ ω, F i ω ∂μ)) ≠ ⊤ := by
+    rw [he,←ofReal_integral_eq_lintegral_ofReal hi
+      (Filter.Eventually.of_forall (fun ω => tsum_nonneg (fun i => hn i ω)))]
+    exact ENNReal.ofReal_ne_top
+  have hh : Summable (fun i => ∫ ω, F i ω ∂μ) := by
+    simpa only [ENNReal.toReal_ofReal (integral_nonneg (hn _))] using ENNReal.summable_toReal ht
+  refine ⟨hh,?_⟩
+  apply integral_tsum_of_summable_integral_norm hF
+  simpa only [Real.norm_of_nonneg (hn _ _)] using hh
+
+end NonnegativeRealTonelli
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section TotalExteriorLegs
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem exteriorLegProduct_sum (G : BondGraph V E) (P : V → V → ℝ)
+    (p : ℝ) {k : ℕ} (t : ℝ) (w : RawCorridor G k) (ω : Set E)
+    (hu : (cluster G ω w.base).Finite) (hv : (cluster G ω w.endpoint).Finite) :
+    Summable (fun xy : V × V => exteriorLegProduct G P p t w ω xy.1 xy.2) ∧
+    (∑' xy : V × V, exteriorLegProduct G P p t w ω xy.1 xy.2) =
+      if ¬Conn G (w.exterior ω) w.base w.endpoint then
+        deformedClusterMass G P p k t (w.interior:Set V)ᶜ ω w.base *
+          deformedClusterMass G P p k t (w.interior:Set V)ᶜ ω w.endpoint else 0 := by
+  by_cases hn : ¬Conn G (w.exterior ω) w.base w.endpoint
+  · have hsu : Summable (deformedConnection G P p k t (w.interior:Set V)ᶜ ω w.base) :=
+      summable_of_hasFiniteSupport (deformedConnection_finite_support G P p k t (w.interior:Set V)ᶜ ω w.base hu)
+    have hsv : Summable (deformedConnection G P p k t (w.interior:Set V)ᶜ ω w.endpoint) :=
+      summable_of_hasFiniteSupport (deformedConnection_finite_support G P p k t (w.interior:Set V)ᶜ ω w.endpoint hv)
+    have hsx : Summable (fun x => deformedConnection G P p k t (w.interior:Set V)ᶜ ω x w.base) := by
+      simpa only [deformedConnection_symm G P p k t (w.interior:Set V)ᶜ ω _ w.base] using hsu
+    have hs := hsx.mul_of_nonneg hsv
+      (fun x => deformedConnection_nonneg G P p k t _ ω x w.base)
+      (fun y => deformedConnection_nonneg G P p k t _ ω w.endpoint y)
+    simp only [exteriorLegProduct,ite_eq_left hn]
+    refine ⟨hs,?_⟩
+    rw [←hsx.tsum_mul_tsum hsv hs]
+    simp only [deformedClusterMass,deformedConnection_symm G P p k t (w.interior:Set V)ᶜ ω _ w.base]
+  · simp only [exteriorLegProduct,ite_eq_right hn,summable_zero,tsum_zero,and_self]
+
+ 
+
+theorem exteriorLegProduct_integral_sum [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (hp : p < pc G)
+    {k : ℕ} (t : ℝ) (ht : 0 ≤ t) (w : RawCorridor G k) :
+    Summable (fun xy : V × V => ∫ ω, exteriorLegProduct G P p t w ω xy.1 xy.2 ∂law p) ∧
+    (∑' xy : V × V, ∫ ω, exteriorLegProduct G P p t w ω xy.1 xy.2 ∂law p) =
+      ∫ ω, (if ¬Conn G (w.exterior ω) w.base w.endpoint then
+        deformedClusterMass G P p k t (w.interior:Set V)ᶜ ω w.base *
+          deformedClusterMass G P p k t (w.interior:Set V)ᶜ ω w.endpoint else 0) ∂law p := by
+  have hae : ∀ᵐ ω ∂law p, Summable (fun xy : V × V => exteriorLegProduct G P p t w ω xy.1 xy.2) ∧
+    (∑' xy : V × V, exteriorLegProduct G P p t w ω xy.1 xy.2) =
+      if ¬Conn G (w.exterior ω) w.base w.endpoint then
+        deformedClusterMass G P p k t (w.interior:Set V)ᶜ ω w.base *
+          deformedClusterMass G P p k t (w.interior:Set V)ᶜ ω w.endpoint else 0 := by
+    filter_upwards [subcritical_cluster_finite_ae G p hp w.base,
+      subcritical_cluster_finite_ae G p hp w.endpoint] with ω hu hv
+    exact exteriorLegProduct_sum G P p t w ω hu hv
+  have hi := (integrable_distinct_deformed_product G hloc hqt P hP p hp k t ht
+    (w.interior:Set V)ᶜ w.base w.endpoint).congr (hae.mono (fun _ h => h.2.symm))
+  obtain ⟨hs,he⟩ := real_tonelli_of_integrable_sum (law p)
+    (fun xy : V × V => fun ω => exteriorLegProduct G P p t w ω xy.1 xy.2)
+    (fun xy => integrable_exteriorLegProduct G P hP p t ht w xy.1 xy.2)
+    (fun xy ω => exteriorLegProduct_nonneg G P p t w ω xy.1 xy.2)
+    (hae.mono (fun _ h => h.1)) hi
+  exact ⟨hs,he.trans (integral_congr_ae (hae.mono (fun _ h => h.2)))⟩
+
+end TotalExteriorLegs
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section SummableDerivativeTerm
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Countable V] [Countable E] [Nonempty V]
+
+omit [Nonempty V] in
+theorem integral_corridorDerivativeTerm_le (G : BondGraph V E) (hG : Function.Injective G)
+    (hloc : LocallyFinite G) (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y)
+    (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp1 : (p:ℝ) < 1)
+    {k : ℕ} (t : ℝ) (ht : 0 ≤ t) (w : RawCorridor G k) (x y : V) :
+    (∫ ω, corridorPenaltyTerm G P p Set.univ ω x y w *
+      deformedConnection G P p k t Set.univ ω x y ∂law p) ≤
+      w.probability P * (twoPoint G p x w.base * twoPoint G p w.endpoint y +
+        twoPoint G p x w.endpoint * twoPoint G p w.base y) := by
+  apply le_trans (integral_mono (integrable_corridorDerivativeTerm G P hP p k t ht Set.univ x y w)
+    (integrable_corridorPenaltyTerm G P p Set.univ x y w) (fun ω =>
+      mul_le_of_le_one_right (corridorPenaltyTerm_nonneg G P hP p Set.univ ω x y w)
+        (deformedConnection_le_one G P hP p k t ht Set.univ ω x y)))
+  exact integral_corridorPenaltyTerm_le G hG hloc P hP p hp0 hp1 x y w
+
+theorem summable_twoPoint_pairs (G : BondGraph V E)
+    (hloc : LocallyFinite G) (hqt : QuasiTransitive G) (p : unitInterval)
+    (hp : p < pc G) (u v : V) :
+    Summable (fun xy : V × V => twoPoint G p xy.1 u * twoPoint G p v xy.2) := by
+  have hs : Summable (fun x => twoPoint G p x u) := by
+    simpa only [twoPoint_symm G p _ u] using summable_twoPoint_subcritical G hloc hqt p hp u
+  exact hs.mul_of_nonneg (summable_twoPoint_subcritical G hloc hqt p hp v)
+    (fun x => twoPoint_nonneg G p x u) (fun y => twoPoint_nonneg G p v y)
+
+omit [Nonempty V] [Countable V] [Countable E] in
+ theorem integral_corridorDerivativeTerm_nonneg (G : BondGraph V E)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (k : ℕ)
+    (t : ℝ) (w : RawCorridor G k) (x y : V) :
+    0 ≤ ∫ ω, corridorPenaltyTerm G P p Set.univ ω x y w *
+      deformedConnection G P p k t Set.univ ω x y ∂law p :=
+  integral_nonneg (fun ω => mul_nonneg (corridorPenaltyTerm_nonneg G P hP p Set.univ ω x y w)
+    (deformedConnection_nonneg G P p k t Set.univ ω x y))
+
+theorem summable_corridorDerivativeTerm_pairs (G : BondGraph V E) (hG : Function.Injective G)
+    (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y)
+    (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp : p < pc G)
+    {k : ℕ} (t : ℝ) (ht : 0 ≤ t) (w : RawCorridor G k) :
+    Summable (fun xy : V × V => ∫ ω, corridorPenaltyTerm G P p Set.univ ω xy.1 xy.2 w *
+      deformedConnection G P p k t Set.univ ω xy.1 xy.2 ∂law p) := by
+  have hs : Summable (fun xy : V × V => w.probability P *
+      (twoPoint G p xy.1 w.base * twoPoint G p w.endpoint xy.2 +
+       twoPoint G p xy.1 w.endpoint * twoPoint G p w.base xy.2)) :=
+    ((summable_twoPoint_pairs G hloc hqt p hp w.base w.endpoint).add
+      (summable_twoPoint_pairs G hloc hqt p hp w.endpoint w.base)).mul_left (w.probability P)
+  exact Summable.of_nonneg_of_le
+    (fun (xy : V × V) => integral_corridorDerivativeTerm_nonneg G P hP p k t w xy.1 xy.2)
+    (fun (xy : V × V) => integral_corridorDerivativeTerm_le G hG hloc P hP p hp0
+      ((show (p:ℝ)<(pc G:ℝ) from hp).trans_le (pc G).2.2) t ht w xy.1 xy.2) hs
+
+end SummableDerivativeTerm
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section SingleTemplateDecrease
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Countable V] [Countable E] [Nonempty V]
+
+ 
+
+theorem calibrated_derivative_template_lower (G : BondGraph V E) (hG : Function.Injective G)
+    (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y)
+    (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp : p < pc G)
+    (k : ℕ) (w : RawCorridor G k) (hlen : 3 ≤ w.path.length)
+    (c : ℝ) (hc : (∑ e ∈ (w.retained_finite hG).toFinset, ∑' r : RawCorridor G k,
+        if e ∈ r.retained then r.weight P p else 0) ≤ c)
+    (t : ℝ) (ht : 0 ≤ t) :
+    w.probability P * Real.exp (-t*c) *
+      (∫ ω, (if ¬Conn G (w.exterior ω) w.base w.endpoint then
+        deformedClusterMass G P p k t (w.interior:Set V)ᶜ ω w.base *
+          deformedClusterMass G P p k t (w.interior:Set V)ᶜ ω w.endpoint else 0) ∂law p) ≤
+      ∑' xy : V × V, ∫ ω, corridorPenaltyTerm G P p Set.univ ω xy.1 xy.2 w *
+        deformedConnection G P p k t Set.univ ω xy.1 xy.2 ∂law p := by
+  obtain ⟨hs,he⟩ := exteriorLegProduct_integral_sum G hloc hqt P hP p hp t ht w
+  have hh := (hs.mul_left (w.probability P * Real.exp (-t*c))).tsum_le_tsum
+    (fun (xy : V × V) => calibrated_derivative_kernel_lower G hG hloc P hP p hp0
+      ((show (p:ℝ)<(pc G:ℝ) from hp).trans_le (pc G).2.2) k w hlen c hc t ht xy.1 xy.2)
+    (summable_corridorDerivativeTerm_pairs G hG hloc hqt P hP p hp0 hp t ht w)
+  rwa [tsum_mul_left,he] at hh
+
+ 
+
+theorem calibrated_derivative_template_row_lower (G : BondGraph V E) (hG : Function.Injective G)
+    (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (d : ℕ) (hd : ∀ a, (neighbors G hloc a).card ≤ d)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (hpcrit : 0 < (pc G:ℝ))
+    (p : unitInterval) (hp : p < pc G) (hpnear : (pc G:ℝ)/2 ≤ (p:ℝ))
+    (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T)
+    (k : ℕ) (w : RawCorridor G k) (hlen : 3 ≤ w.path.length)
+    (c : ℝ) (hc : (∑ e ∈ (w.retained_finite hG).toFinset, ∑' r : RawCorridor G k,
+        if e ∈ r.retained then r.weight P p else 0) ≤ c)
+    (t : ℝ) (ht : 0 ≤ t) :
+    w.probability P * Real.exp (-t*c) *
+      (deformedSusceptibility G P p k t (w.interior:Set V)ᶜ w.base *
+        deformedSusceptibility G P p k t (w.interior:Set V)ᶜ w.endpoint -
+        (1+2*(d:ℝ)/(pc G:ℝ))*(maxSusceptibility G p)^2*(matrixCoeff (T^3) w.base w.endpoint).re) ≤
+      ∑' xy : V × V, ∫ ω, corridorPenaltyTerm G P p Set.univ ω xy.1 xy.2 w *
+        deformedConnection G P p k t Set.univ ω xy.1 xy.2 ∂law p := by
+  have hm := deformed_distinct_correlation_uniform G hloc hqt d hd P hP hpcrit p hp hpnear T hT
+    k t ht (w.interior:Set V)ᶜ w.base w.endpoint
+  exact (mul_le_mul_of_nonneg_left hm (mul_nonneg (w.2.2.mass_nonneg P hP) (Real.exp_pos _).le)).trans
+    (calibrated_derivative_template_lower G hG hloc hqt P hP p ((half_pos hpcrit).trans_le hpnear)
+      hp k w hlen c hc t ht)
+
+end SingleTemplateDecrease
+end Percolation
+
+ 
+namespace Percolation
+universe u u' v
+section ErasureNaturality
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {V : Type u} {V' : Type u'} {H : SimpleGraph V} {H' : SimpleGraph V'}
+
+theorem walk_map_dropUntil (f : H →g H') (hf : Function.Injective f)
+    {x y : V} (w : H.Walk x y) (z : V) (hz : z ∈ w.support) :
+    (w.dropUntil z hz).map f = (w.map f).dropUntil (f z)
+      (by simpa only [SimpleGraph.Walk.support_map] using List.mem_map_of_mem hz) := by
+  induction w with
+  | @nil x =>
+    have he : z = x := by simpa using hz
+    subst z
+    rfl
+  | @cons x a y h w ih =>
+    by_cases he : x = z
+    · subst z
+      simp only [SimpleGraph.Walk.dropUntil_first]
+    · have hfe : f x ≠ f z := fun h => he (hf h)
+      simp only [SimpleGraph.Walk.dropUntil,he,hfe,SimpleGraph.Walk.map_cons]
+      exact ih _
+
+ 
+
+theorem walk_map_bypass (f : H →g H') (hf : Function.Injective f)
+    {x y : V} (w : H.Walk x y) : w.bypass.map f = (w.map f).bypass := by
+  induction w with
+  | nil => rfl
+  | @cons x a y h w ih =>
+    have hm : f x ∈ (w.map f).bypass.support ↔ x ∈ w.bypass.support := by
+      rw [←ih,SimpleGraph.Walk.support_map]
+      exact List.mem_map_of_injective hf
+    by_cases hx : x ∈ w.bypass.support
+    · have hfx := hm.mpr hx
+      simp only [SimpleGraph.Walk.bypass,SimpleGraph.Walk.map_cons,dite_eq_left hx,dite_eq_left hfx]
+      simp only [walk_map_dropUntil f hf,ih]
+    · have hfx := mt hm.mp hx
+      simp only [SimpleGraph.Walk.bypass,SimpleGraph.Walk.map_cons,dite_eq_right hx,dite_eq_right hfx,ih]
+
+namespace HoldingWalk
+
+def map (f : H →g H') : {k : ℕ} → {x y : V} → HoldingWalk H k x y → HoldingWalk H' k (f x) (f y)
+  | _,_,_,.nil _ => .nil _
+  | _,_,_,.hold w => .hold (w.map f)
+  | _,_,_,.step h w => .step (f.map_rel h) (w.map f)
+
+theorem map_vertices (f : H →g H') {k : ℕ} {x y : V} (w : HoldingWalk H k x y) :
+    (w.map f).vertices = w.vertices.map f := by
+  induction w with
+  | nil => rfl
+  | hold w ih => simp only [map,vertices,List.map_cons,ih]
+  | step h w ih => simp only [map,vertices,List.map_cons,ih]
+
+theorem map_withoutHolds (f : H →g H') {k : ℕ} {x y : V} (w : HoldingWalk H k x y) :
+    (w.map f).withoutHolds = w.withoutHolds.map f := by
+  induction w with
+  | nil => rfl
+  | hold w ih => exact ih
+  | step h w ih => simp only [map,withoutHolds,SimpleGraph.Walk.map_cons,ih]
+
+theorem map_erase (f : H →g H') (hf : Function.Injective f)
+    {k : ℕ} {x y : V} (w : HoldingWalk H k x y) :
+    (w.map f).erase = w.erase.map f := by
+  simp only [erase,map_withoutHolds,SimpleGraph.Walk.reverse_map,
+    ←walk_map_bypass f hf]
+
+end HoldingWalk
+end ErasureNaturality
+end Percolation
+
+ 
+namespace Percolation
+universe u u' u'' v
+section RawWalkMaps
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {V : Type u} {V' : Type u'} {V'' : Type u''}
+  {H : SimpleGraph V} {H' : SimpleGraph V'} {H'' : SimpleGraph V''}
+namespace HoldingWalk
+
+theorem map_map (f : H →g H') (g : H' →g H'')
+    {k : ℕ} {x y : V} (w : HoldingWalk H k x y) : (w.map f).map g = w.map (g.comp f) := by
+  induction w with
+  | nil => rfl
+  | hold w ih => exact congrArg hold ih
+  | step h w ih => exact congrArg (step _) ih
+
+theorem map_id {k : ℕ} {x y : V} (w : HoldingWalk H k x y) : w.map ((SimpleGraph.Hom.id : H →g H)) = w := by
+  induction w with
+  | nil => rfl
+  | hold w ih => exact congrArg hold ih
+  | step h w ih => exact congrArg (step _) ih
+
+theorem mass_map (f : H →g H') (P : V → V → ℝ) (Q : V' → V' → ℝ)
+    (hPQ : ∀ x y, Q (f x) (f y) = P x y)
+    {k : ℕ} {x y : V} (w : HoldingWalk H k x y) : (w.map f).mass Q = w.mass P := by
+  induction w with
+  | nil => rfl
+  | hold w ih => simp only [map,mass,hPQ,ih]
+  | step h w ih => simp only [map,mass,hPQ,ih]
+end HoldingWalk
+
+def rawWalkMap (f : H →g H') (k : ℕ) (w : Σ x y, HoldingWalk H k x y) :
+    Σ x y, HoldingWalk H' k x y := ⟨f w.1,f w.2.1,w.2.2.map f⟩
+
+theorem rawWalkMap_comp (f : H →g H') (g : H' →g H'') (k : ℕ)
+    (w : Σ x y, HoldingWalk H k x y) : rawWalkMap g k (rawWalkMap f k w) = rawWalkMap (g.comp f) k w := by
+  cases w with
+  | mk x w =>
+    cases w with
+    | mk y w =>
+      simp only [rawWalkMap,HoldingWalk.map_map]
+      rfl
+
+theorem rawWalkMap_id (k : ℕ) (w : Σ x y, HoldingWalk H k x y) :
+    rawWalkMap ((SimpleGraph.Hom.id : H →g H)) k w = w := by
+  cases w with
+  | mk x w =>
+    cases w with
+    | mk y w =>
+      simp only [rawWalkMap,HoldingWalk.map_id]
+      rfl
+
+def rawWalkEquiv (f : H ≃g H') (k : ℕ) :
+    (Σ x y, HoldingWalk H k x y) ≃ (Σ x y, HoldingWalk H' k x y) where
+  toFun := rawWalkMap f.toHom k
+  invFun := rawWalkMap f.symm.toHom k
+  left_inv w := by
+    rw [rawWalkMap_comp]
+    have he : f.symm.toHom.comp f.toHom = (SimpleGraph.Hom.id : H →g H) := by
+      ext x
+      exact f.symm_apply_apply x
+    rw [he,rawWalkMap_id]
+  right_inv w := by
+    rw [rawWalkMap_comp]
+    have he : f.toHom.comp f.symm.toHom = (SimpleGraph.Hom.id : H' →g H') := by
+      ext x
+      exact f.apply_symm_apply x
+    rw [he,rawWalkMap_id]
+
+end RawWalkMaps
+section GraphAutWalks
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {V : Type u} {E : Type v}
+
+theorem openSimpleGraph_univ_adj (G : BondGraph V E) (x y : V) :
+    (openSimpleGraph G Set.univ).Adj x y ↔ x ≠ y ∧ Adj G x y := by
+  simp only [openSimpleGraph,Adj,Set.mem_univ,true_and]
+
+def autSimpleGraph (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) :
+    openSimpleGraph G Set.univ ≃g openSimpleGraph G Set.univ where
+  toEquiv := a
+  map_rel_iff' := by
+    intro x y
+    simp only [openSimpleGraph_univ_adj,a.injective.ne_iff]
+    exact and_congr_right (fun _ => ⟨fun h => by simpa using adj_aut G (isAut_symm G hab) h,
+      adj_aut G hab⟩)
+
+namespace RawCorridor
+variable {G : BondGraph V E} {k : ℕ}
+def autEquiv {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) : RawCorridor G k ≃ RawCorridor G k :=
+  rawWalkEquiv (autSimpleGraph G hab) k
+
+theorem aut_path {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (w : RawCorridor G k) :
+    (autEquiv hab w).path = w.path.map (autSimpleGraph G hab).toHom :=
+  w.2.2.map_erase _ a.injective
+
+theorem aut_path_length {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (w : RawCorridor G k) :
+    (autEquiv hab w).path.length = w.path.length := by
+  have hh := congrArg (fun q => q.length) (aut_path hab w)
+  exact hh.trans (SimpleGraph.Walk.length_map _ _)
+
+end RawCorridor
+end GraphAutWalks
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section TemplateCovariance
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace RawCorridor
+variable {G : BondGraph V E} {k : ℕ}
+
+theorem aut_base {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (w : RawCorridor G k) :
+    (autEquiv hab w).base = a w.base := rfl
+
+theorem aut_endpoint {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (w : RawCorridor G k) :
+    (autEquiv hab w).endpoint = a w.endpoint := rfl
+
+theorem aut_support {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (w : RawCorridor G k) :
+    (autEquiv hab w).path.support = w.path.support.map a := by
+  exact (congrArg (fun q => q.support) (aut_path hab w)).trans
+    (SimpleGraph.Walk.support_map _ _)
+
+theorem aut_edges {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (w : RawCorridor G k) :
+    (autEquiv hab w).path.edges = w.path.edges.map (Sym2.map a) := by
+  exact (congrArg (fun q => q.edges) (aut_path hab w)).trans
+    (SimpleGraph.Walk.edges_map _ _)
+
+theorem aut_interior {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (w : RawCorridor G k) (z : V) :
+    a z ∈ (autEquiv hab w).interior ↔ z ∈ w.interior := by
+  simp only [interior,Finset.mem_sdiff,List.mem_toFinset]
+  rw [aut_support hab w]
+  simp only [aut_base,aut_endpoint,Finset.mem_insert,Finset.mem_singleton,
+    List.mem_map_of_injective a.injective,a.injective.eq_iff]
+
+theorem aut_retained {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (w : RawCorridor G k) (e : E) :
+    b e ∈ (autEquiv hab w).retained ↔ e ∈ w.retained := by
+  simp only [retained,Set.mem_ofPred_eq,aut_edges hab w,hab e]
+  exact List.mem_map_of_injective (Sym2.map.injective a.injective)
+
+theorem aut_blockers {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (w : RawCorridor G k) (e : E) :
+    b e ∈ (autEquiv hab w).blockers ↔ e ∈ w.blockers := by
+  simp only [blockers,Set.mem_sdiff,Set.mem_ofPred_eq,aut_retained]
+  apply and_congr_left
+  intro _
+  constructor
+  · rintro ⟨z,hz,he⟩
+    rw [hab e] at he
+    obtain ⟨y,hy,rfl⟩ := Sym2.mem_map.mp he
+    exact ⟨y,(aut_interior hab w y).mp hz,hy⟩
+  · rintro ⟨z,hz,he⟩
+    refine ⟨a z,(aut_interior hab w z).mpr hz,?_⟩
+    rw [hab e]
+    exact Sym2.mem_map.mpr ⟨z,he,rfl⟩
+
+theorem aut_blockers_ncard {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (w : RawCorridor G k) :
+    (autEquiv hab w).blockers.ncard = w.blockers.ncard := by
+  have he : w.blockers = b ⁻¹' (autEquiv hab w).blockers := by
+    ext e
+    exact (aut_blockers hab w e).symm
+  rw [he,Set.ncard_preimage_of_injective_subset_range b.injective]
+  exact fun e _ => b.surjective e
+
+theorem aut_probability {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b)
+    (P : V → V → ℝ) (hP : ∀ x y, P (a x) (a y) = P x y) (w : RawCorridor G k) :
+    (autEquiv hab w).probability P = w.probability P :=
+  w.2.2.mass_map (autSimpleGraph G hab).toHom P P hP
+
+theorem aut_weight {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b)
+    (P : V → V → ℝ) (hP : ∀ x y, P (a x) (a y) = P x y) (p : ℝ) (w : RawCorridor G k) :
+    (autEquiv hab w).weight P p = w.weight P p := by
+  simp only [weight,aut_probability hab P hP,aut_path_length,aut_blockers_ncard]
+
+end RawCorridor
+end TemplateCovariance
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Deformation/ClusterProducts.lean b/lean/OAI/Probability/UnimodularPercolation/Deformation/ClusterProducts.lean
index bc040b5b14e2bf92200a63bed18acc26315d1082..349943265271290de2186f851e473dc90229510c 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Deformation/ClusterProducts.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Deformation/ClusterProducts.lean
@@ -1,1223 +1,1223 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Deformation.Corridors
-
-namespace OAI
-
-/-! Cluster Products for bond percolation. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section SmallDeformationOperators
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Countable V] [Countable E]
-
- 
-
-theorem exists_deformedOperator_small_change (G : BondGraph V E)
-    (hG : Function.Injective G) (hloc : LocallyFinite G) (p : unitInterval)
-    (hp : 0 < (p:ℝ)) (hp₁ : (p:ℝ) < 1) (a : ℝ)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hPs : IsSelfAdjoint P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (A : Hilbert V →L[ℂ] Hilbert V) (hA : PositiveRealMatrix A) (hAs : IsSelfAdjoint A)
-    (hAT : ∀ x y, matrixCoeff A x y = (twoPoint G p x y : ℂ))
-    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) :
-    ∃ B : Hilbert V →L[ℂ] Hilbert V,
-      (∀ x y, matrixCoeff B x y = (deformedKernel G (walkKernel G hloc a) p k t Set.univ x y : ℂ)) ∧
-      ‖B‖ ≤ ‖A‖ ∧ ‖A-B‖ ≤ 2*t*‖P‖^k*‖A‖^2 ∧ ‖A‖-2*t*‖P‖^k*‖A‖^2 ≤ ‖B‖ := by
-  let K := deformedKernel G (walkKernel G hloc a) p k t Set.univ
-  have hKn (x y : V) : 0 ≤ K x y := deformedKernel_nonneg G _ p k t _ x y
-  have hb (x y : V) := deformedKernel_change_bound G hG hloc p hp hp₁ a P hpos hPs hP A hA hAs hAT k t ht x y
-  obtain ⟨B,hBn,hBK⟩ := exists_operator_of_matrix_domination (twoPoint G p) K (twoPoint_nonneg G p)
-    (fun x y => by rw [abs_of_nonneg (hKn x y)]; exact sub_nonneg.mp (hb x y).1) A hAT
-  have hBK' (x y : V) : matrixCoeff B x y = (K x y : ℂ) := hBK x y
-  let C := A*(P^k*A)
-  have hCn : PositiveRealMatrix C := hA.mul ((hpos.pow k).mul hA)
-  have hscale (x y : V) : matrixCoeff (((2*t:ℝ):ℂ) • C) x y =
-      (2*t*(matrixCoeff C x y).re : ℝ) := by
-    change ((2*t:ℝ):ℂ)*matrixCoeff C x y = _
-    rw [hCn.real]
-    simp only [Complex.ofReal_re,Complex.ofReal_mul]
-  obtain ⟨D,hDn,hDK⟩ := exists_operator_of_matrix_domination
-    (fun x y => 2*t*(matrixCoeff C x y).re) (fun x y => twoPoint G p x y - K x y)
-    (fun x y => mul_nonneg (mul_nonneg (by norm_num) ht) (hCn x y).2)
-    (fun x y => by rw [abs_of_nonneg (hb x y).1]; exact (hb x y).2)
-    (((2*t:ℝ):ℂ) • C) hscale
-  have he : D = A-B := kernel_extension_unique hDK (by
-    intro x y
-    change matrixCoeff A x y - matrixCoeff B x y = _
-    rw [hAT,hBK',Complex.ofReal_sub])
-  rw [he] at hDn
-  have hCnorm : ‖C‖ ≤ ‖A‖*‖P‖^k*‖A‖ := by
-    calc
-      _ ≤ ‖A‖*‖P^k*A‖ := norm_mul_le _ _
-      _ ≤ ‖A‖*(‖P^k‖*‖A‖) := mul_le_mul_of_nonneg_left (norm_mul_le _ _) (norm_nonneg _)
-      _ ≤ ‖A‖*(‖P‖^k*‖A‖) := by
-        gcongr
-        cases k with
-        | zero =>
-          simp only [pow_zero]
-          have he : (1 : Hilbert V →L[ℂ] Hilbert V) = ContinuousLinearMap.id ℂ (Hilbert V) := by ext f; rfl
-          rw [he]
-          exact ContinuousLinearMap.norm_id_le
-        | succ j => exact norm_pow_le' P (Nat.succ_pos j)
-      _ = _ := by ring
-  have hd : ‖A-B‖ ≤ 2*t*‖P‖^k*‖A‖^2 := by
-    calc
-      _ ≤ ‖((2*t:ℝ):ℂ) • C‖ := hDn
-      _ = 2*t*‖C‖ := by rw [norm_smul,Complex.norm_of_nonneg (mul_nonneg (by norm_num) ht)]
-      _ ≤ 2*t*(‖A‖*‖P‖^k*‖A‖) := mul_le_mul_of_nonneg_left hCnorm (mul_nonneg (by norm_num) ht)
-      _ = _ := by ring
-  refine ⟨B,hBK,hBn,hd,?_⟩
-  have hh : ‖A‖ ≤ ‖A-B‖+‖B‖ := by simpa only [sub_add_cancel] using norm_add_le (A-B) B
-  linarith
-
-end SmallDeformationOperators
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section DeletionGeometry
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-def closedBondNeighborhood (G : BondGraph V E) (A : Set V) : Set V :=
-  {z | z ∈ A ∨ ∃ e, z ∈ G e ∧ ∃ a ∈ G e, a ∈ A}
-
-def SimpleVisit (G : BondGraph V E) (S : Set E) (A : Set V) (x y : V) : Prop :=
-  ∃ q : (openSimpleGraph G S).Walk x y, q.IsPath ∧ ∃ z ∈ q.support, z ∈ A
-
-theorem SimpleVisit.mono (G : BondGraph V E) {S T : Set E} {A B : Set V}
-    (hST : S ⊆ T) (hAB : A ⊆ B) {x y : V} (h : SimpleVisit G S A x y) :
-    SimpleVisit G T B x y := by
-  obtain ⟨q,hq,z,hz,ha⟩ := h
-  have hg : openSimpleGraph G S ≤ openSimpleGraph G T := by
-    rintro a b ⟨hne,e,he,hGe⟩
-    exact ⟨hne,e,hST he,hGe⟩
-  exact ⟨q.mapLe hg,hq.mapLe hg,z,by simpa only [SimpleGraph.Walk.support_mapLe_eq_support] using hz,hAB ha⟩
-
-theorem walk_conn_restrict_vertices (G : BondGraph V E) (S : Set E) {x y : V}
-    (q : (openSimpleGraph G S).Walk x y) (D : Set V)
-    (hD : ∀ z ∈ q.support, z ∈ D) : Conn G (S ∩ internalEdges G D) x y := by
-  obtain ⟨l,hl,hm,ho⟩ := walk_lift_bonds G S q
-  apply hl.conn
-  intro e he
-  refine ⟨ho e he,?_⟩
-  intro z hz
-  exact hD z (SimpleGraph.Walk.mem_support_of_mem_edges (hm ▸ List.mem_map_of_mem he) hz)
-
-theorem conn_restrict_of_no_simpleVisit (G : BondGraph V E) (S : Set E)
-    (A : Set V) (x y : V) (hc : Conn G S x y) (hn : ¬SimpleVisit G S A x y) :
-    Conn G (S ∩ internalEdges G Aᶜ) x y := by
-  obtain ⟨q⟩ := (conn_iff_reachable G S x y).mp hc
-  exact walk_conn_restrict_vertices G S q.bypass Aᶜ
-    (fun z hz ha => hn ⟨q.bypass,q.bypass_isPath,z,hz,ha⟩)
-
-namespace RawCorridor
-variable {G : BondGraph V E} {k : ℕ}
-
-theorem counted_support_subset_walk (w : RawCorridor G k) (D : Set V) (ω : Set E)
-    (x y : V) (hc : w.counted D ω x y)
-    (q : (openSimpleGraph G (ω ∩ internalEdges G D)).Walk x y) :
-    ∀ z ∈ w.path.support, z ∈ q.support := by
-  intro z hz
-  have hl : 0 < w.path.length := by have := hc.1; omega
-  obtain ⟨s,hs,hzs⟩ := (SimpleGraph.Walk.mem_support_iff_exists_mem_edges_of_not_nil
-    (SimpleGraph.Walk.not_nil_iff_lt_length.mpr hl)).mp hz
-  obtain ⟨e,he,hGe⟩ := w.edge_has_label s hs
-  exact SimpleGraph.Walk.mem_support_of_mem_edges
-    (pivotal_image_mem_edges G _ q e (hc.2.2.2.1 e he).2) (hGe.symm ▸ hzs)
-
-end RawCorridor
-
- 
-
-theorem simplePath_disjoint_at (G : BondGraph V E) (S : Set E) {x y z : V}
-    (q : (openSimpleGraph G S).Walk x y) (hq : q.IsPath) (hz : z ∈ q.support) :
-    S ∈ DisjointOccurrence {η | Conn G η x z} {η | Conn G η z y} := by
-  obtain ⟨a,ha,hea,hoa⟩ := walk_lift_bonds G S (q.takeUntil z hz)
-  obtain ⟨b,hb,heb,hob⟩ := walk_lift_bonds G S (q.dropUntil z hz)
-  refine ⟨a.toFinset,b.toFinset,?_,?_,?_,?_,?_⟩
-  · intro e he; exact hoa e (List.mem_toFinset.mp he)
-  · intro e he; exact hob e (List.mem_toFinset.mp he)
-  · apply Finset.disjoint_left.mpr
-    intro e hae hbe
-    exact List.disjoint_left.mp (hq.isTrail.disjoint_edges_takeUntil_dropUntil hz)
-      (hea ▸ List.mem_map_of_mem (List.mem_toFinset.mp hae))
-      (heb ▸ List.mem_map_of_mem (List.mem_toFinset.mp hbe))
-  · exact ha.conn (fun e he => List.mem_toFinset.mpr he)
-  · exact hb.conn (fun e he => List.mem_toFinset.mpr he)
-
-end DeletionGeometry
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section DeletionCounted
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem conn_delete_of_no_visit (G : BondGraph V E) (ω : Set E) (D A : Set V)
-    (x y : V) (hn : ¬SimpleVisit G (ω ∩ internalEdges G D) (closedBondNeighborhood G A) x y)
-    (S : Set E) (hS : S ⊆ ω ∩ internalEdges G D) (hc : Conn G S x y) :
-    Conn G (S ∩ internalEdges G (D \ A)) x y := by
-  have hno : ¬SimpleVisit G S A x y := fun h => hn (h.mono G hS (fun _ => Or.inl))
-  have h := conn_restrict_of_no_simpleVisit G S A x y hc hno
-  apply conn_mono G ?_ h
-  rintro e ⟨he,ha⟩
-  exact ⟨he,fun z hz => ⟨(hS he).2 z hz,ha z hz⟩⟩
-
-namespace RawCorridor
-variable {G : BondGraph V E} {k : ℕ}
-
- 
-
-theorem counted_before_deletion_of_no_visit (w : RawCorridor G k)
-    (D A : Set V) (ω : Set E) (x y : V)
-    (hn : ¬SimpleVisit G (ω ∩ internalEdges G D) (closedBondNeighborhood G A) x y)
-    (hc : w.counted (D \ A) ω x y) : w.counted D ω x y := by
-  have hs : ω ∩ internalEdges G (D \ A) ⊆ ω ∩ internalEdges G D :=
-    Set.inter_subset_inter_right _ (internalEdges_mono G Set.sdiff_subset)
-  apply (w.counted_iff D ω x y).mpr
-  refine ⟨hc.1,conn_mono G hs hc.2.2.1,?_,?_⟩
-  · intro e he
-    refine ⟨hs (hc.2.2.2.1 e he).1,?_⟩
-    intro hbad
-    have hd := conn_delete_of_no_visit G ω D A x y hn
-      ((ω ∩ internalEdges G D) \ {e}) Set.sdiff_subset hbad
-    apply (hc.2.2.2.1 e he).2
-    apply conn_mono G ?_ hd
-    rintro f ⟨⟨⟨hfω,hfD⟩,hfne⟩,hfnew⟩
-    exact ⟨⟨hfω,hfnew⟩,hfne⟩
-  · apply Set.disjoint_left.mpr
-    intro e he ho
-    have henew : e ∉ internalEdges G (D \ A) := fun he' =>
-      Set.disjoint_left.mp hc.2.2.2.2 he ⟨ho.1,he'⟩
-    obtain ⟨a,ha,hna⟩ : ∃ a ∈ G e, a ∉ D \ A := by
-      change ¬(∀ a ∈ G e, a ∈ D \ A) at henew
-      push Not at henew
-      exact henew
-    have haA : a ∈ A := by
-      by_contra ha'
-      exact hna ⟨ho.2 a ha,ha'⟩
-    obtain ⟨z,hzi,hze⟩ := he.1
-    obtain ⟨q⟩ := (conn_iff_reachable G _ x y).mp hc.2.2.1
-    have hz : z ∈ q.bypass.support := w.counted_support_subset_walk (D \ A) ω x y hc q.bypass z
-      (List.mem_toFinset.mp (Finset.mem_sdiff.mp hzi).1)
-    have hv : SimpleVisit G (ω ∩ internalEdges G (D \ A)) (closedBondNeighborhood G A) x y :=
-      ⟨q.bypass,q.bypass_isPath,z,hz,Or.inr ⟨e,hze,a,ha,haA⟩⟩
-    exact hn (hv.mono G hs (Set.Subset.refl _))
-
-end RawCorridor
-end DeletionCounted
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section DeletionWeightedLoss
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem deformedConnection_no_visit (G : BondGraph V E) (hloc : LocallyFinite G)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval)
-    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D A : Set V) (ω : Set E) (x y : V)
-    (hn : ¬SimpleVisit G (ω ∩ internalEdges G D) (closedBondNeighborhood G A) x y) :
-    deformedConnection G P p k t D ω x y ≤ deformedConnection G P p k t (D \ A) ω x y := by
-  by_cases hc : x ∈ D ∧ y ∈ D ∧ Conn G (ω ∩ internalEdges G D) x y
-  · obtain ⟨q⟩ := (conn_iff_reachable G _ x y).mp hc.2.2
-    have hx : x ∉ A := fun ha => hn ⟨q.bypass,q.bypass_isPath,x,q.bypass.start_mem_support,Or.inl ha⟩
-    have hy : y ∉ A := fun ha => hn ⟨q.bypass,q.bypass_isPath,y,q.bypass.end_mem_support,Or.inl ha⟩
-    have hc' : x ∈ D \ A ∧ y ∈ D \ A ∧ Conn G (ω ∩ internalEdges G (D \ A)) x y := by
-      refine ⟨⟨hc.1,hx⟩,⟨hc.2.1,hy⟩,?_⟩
-      apply conn_mono G ?_ (conn_delete_of_no_visit G ω D A x y hn _ (Set.Subset.refl _) hc.2.2)
-      rintro e ⟨⟨heω,heD⟩,henew⟩
-      exact ⟨heω,henew⟩
-    simp only [deformedConnection,hc,hc']
-    apply Real.exp_le_exp.mpr
-    apply mul_le_mul_of_nonpos_left ?_ (neg_nonpos.mpr ht)
-    apply (corridorPenalty_summable G hloc P p k (D \ A) ω x y).tsum_le_tsum ?_
-      (corridorPenalty_summable G hloc P p k D ω x y)
-    intro w
-    by_cases hw : w.counted (D \ A) ω x y
-    · have hw' := w.counted_before_deletion_of_no_visit D A ω x y hn hw
-      simp only [corridorPenaltyTerm,hw,hw',ite_true,le_refl]
-    · simpa only [corridorPenaltyTerm,hw,ite_false] using corridorPenaltyTerm_nonneg G P hP p D ω x y w
-  · simpa only [deformedConnection,hc,ite_false] using deformedConnection_nonneg G P p k t (D \ A) ω x y
-
- 
-
-theorem deformedConnection_deletion_pointwise (G : BondGraph V E) (hloc : LocallyFinite G)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval)
-    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D A : Set V) (ω : Set E) (x y : V) :
-    deformedConnection G P p k t D ω x y - deformedConnection G P p k t (D \ A) ω x y ≤
-      if ∃ z ∈ closedBondNeighborhood G A,
-        ω ∈ DisjointOccurrence {η | Conn G η x z} {η | Conn G η z y} then 1 else 0 := by
-  split_ifs with h
-  · linarith [deformedConnection_le_one G P hP p k t ht D ω x y,
-      deformedConnection_nonneg G P p k t (D \ A) ω x y]
-  · apply sub_nonpos.mpr
-    apply deformedConnection_no_visit G hloc P hP p k t ht D A ω x y
-    rintro ⟨q,hq,z,hz,hza⟩
-    apply h
-    exact ⟨z,hza,disjointOccurrence_upper _ _ Set.inter_subset_left (simplePath_disjoint_at G _ q hq hz)⟩
-
-end DeletionWeightedLoss
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section DeletionKernelBound
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Countable V] [Countable E]
-
- 
-
-theorem deformedKernel_deletion_bound (G : BondGraph V E) (hloc : LocallyFinite G)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval)
-    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D A : Set V) (x y : V)
-    (hs : Summable (fun z : closedBondNeighborhood G A => twoPoint G p x z * twoPoint G p z y)) :
-    deformedKernel G P p k t D x y - deformedKernel G P p k t (D \ A) x y ≤
-      ∑' z : closedBondNeighborhood G A, twoPoint G p x z * twoPoint G p z y := by
-  let B := fun z : closedBondNeighborhood G A =>
-    DisjointOccurrence {η | Conn G η x z} {η | Conn G η z y}
-  let U := ⋃ z, B z
-  have hBm (z : closedBondNeighborhood G A) : MeasurableSet (B z) :=
-    measurableSet_finiteWitness _ (disjointOccurrence_upper _ _) (finiteWitness_disjointOccurrence _ _)
-  have hUm : MeasurableSet U := MeasurableSet.iUnion hBm
-  have hBn (z : closedBondNeighborhood G A) : 0 ≤ twoPoint G p x z * twoPoint G p z y :=
-    mul_nonneg (twoPoint_nonneg G p x z) (twoPoint_nonneg G p z y)
-  have hBK (z : closedBondNeighborhood G A) : law p (B z) ≤ ENNReal.ofReal (twoPoint G p x z * twoPoint G p z y) := by
-    rw [←ENNReal.ofReal_toReal (measure_ne_top (law p) (B z))]
-    exact ENNReal.ofReal_le_ofReal (bk p _ _ (fun _ _ h hc => conn_mono G h hc)
-      (fun _ _ h hc => conn_mono G h hc) (finiteWitness_conn G x z) (finiteWitness_conn G z y))
-  have hU : (law p).real U ≤ ∑' z : closedBondNeighborhood G A, twoPoint G p x z * twoPoint G p z y := by
-    have hh : law p U ≤ ENNReal.ofReal (∑' z : closedBondNeighborhood G A, twoPoint G p x z * twoPoint G p z y) := by
-      calc
-        _ ≤ ∑' z, law p (B z) := measure_iUnion_le B
-        _ ≤ ∑' z : closedBondNeighborhood G A, ENNReal.ofReal (twoPoint G p x z * twoPoint G p z y) := ENNReal.tsum_le_tsum hBK
-        _ = _ := (ENNReal.ofReal_tsum_of_nonneg hBn hs).symm
-    exact (ENNReal.toReal_mono ENNReal.ofReal_ne_top hh).trans_eq (ENNReal.toReal_ofReal (tsum_nonneg hBn))
-  have hUi : Integrable (U.indicator (fun _ : Set E => (1:ℝ))) (law p) := (integrable_const _).indicator hUm
-  have hpoint (ω : Set E) : deformedConnection G P p k t D ω x y - deformedConnection G P p k t (D \ A) ω x y ≤
-      U.indicator (fun _ : Set E => (1:ℝ)) ω := by
-    have he : ω ∈ U ↔ ∃ z ∈ closedBondNeighborhood G A, ω ∈ DisjointOccurrence {η | Conn G η x z} {η | Conn G η z y} := by
-      simp [U,B]
-    simpa only [Set.indicator_apply,he] using deformedConnection_deletion_pointwise G hloc P hP p k t ht D A ω x y
-  have hi := integrable_deformedConnection G P hP p k t ht D x y
-  have hi' := integrable_deformedConnection G P hP p k t ht (D \ A) x y
-  calc
-    _ = ∫ ω, deformedConnection G P p k t D ω x y - deformedConnection G P p k t (D \ A) ω x y ∂law p := (integral_sub hi hi').symm
-    _ ≤ ∫ ω, U.indicator (fun _ : Set E => (1:ℝ)) ω ∂law p := integral_mono (hi.sub hi') hUi hpoint
-    _ = (law p).real U := integral_indicator_one hUm
-    _ ≤ _ := hU
-
-end DeletionKernelBound
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section DeformedSusceptibility
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def deformedSusceptibility (G : BondGraph V E) (P : V → V → ℝ)
-    (p : unitInterval) (k : ℕ) (t : ℝ) (D : Set V) (x : V) : ℝ :=
-  ∑' y, deformedKernel G P p k t D x y
-
-theorem deformedKernel_le_twoPoint [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval)
-    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D : Set V) (x y : V) :
-    deformedKernel G P p k t D x y ≤ twoPoint G p x y := by
-  rw [←deformedKernel_zero G P p k x y]
-  apply integral_mono (integrable_deformedConnection G P hP p k t ht D x y)
-    (integrable_deformedConnection G P hP p k 0 (le_refl _) Set.univ x y)
-  intro ω
-  exact (deformedConnection_mono G hloc P hP p k t ht (Set.subset_univ D)
-    (Set.Subset.refl _) x y).trans (sub_nonneg.mp (deformedConnection_change G P hP p k t ht Set.univ ω x y).1)
-
-theorem summable_deformedKernel [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (hp : p < pc G)
-    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D : Set V) (x : V) :
-    Summable (deformedKernel G P p k t D x) :=
-  Summable.of_nonneg_of_le (deformedKernel_nonneg G P p k t D x)
-    (deformedKernel_le_twoPoint G hloc P hP p k t ht D x)
-    (summable_twoPoint_subcritical G hloc hqt p hp x)
-
-theorem deformedSusceptibility_le [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (hp : p < pc G)
-    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D : Set V) (x : V) :
-    deformedSusceptibility G P p k t D x ≤ susceptibility G p x :=
-  (summable_deformedKernel G hloc hqt P hP p hp k t ht D x).tsum_le_tsum
-    (deformedKernel_le_twoPoint G hloc P hP p k t ht D x)
-    (summable_twoPoint_subcritical G hloc hqt p hp x)
-
-end DeformedSusceptibility
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section DeletionSusceptibilityEstimate
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Countable V] [Countable E] [Nonempty V]
-
-theorem summable_twoPoint_restricted_pair (G : BondGraph V E) (hloc : LocallyFinite G)
-    (hqt : QuasiTransitive G) (p : unitInterval) (hp : p < pc G) (x : V) (N : Set V) :
-    Summable (fun zy : N×V => twoPoint G p x zy.1 * twoPoint G p zy.1 zy.2) := by
-  apply (summable_twoPoint_pair G hloc hqt p hp x).comp_injective (i := fun zy : N×V => (zy.1.val,zy.2))
-  intro a b hab
-  exact Prod.ext (Subtype.ext (congrArg Prod.fst hab)) (congrArg (fun z : V×V => z.2) hab)
-
-theorem twoPoint_double_restricted_bound (G : BondGraph V E) (hloc : LocallyFinite G)
-    (hqt : QuasiTransitive G) (p : unitInterval) (hp : p < pc G) (x : V) (N : Set V) :
-    (∑' y : V, ∑' z : N, twoPoint G p x z * twoPoint G p z y) ≤
-      maxSusceptibility G p * (∑' z : N, twoPoint G p x z) := by
-  have hpairs := summable_twoPoint_restricted_pair G hloc hqt p hp x N
-  have hcomm : (∑' y : V, ∑' z : N, twoPoint G p x z * twoPoint G p z y) =
-      ∑' z : N, twoPoint G p x z * susceptibility G p z := by
-    calc
-      _ = ∑' z : N, ∑' y : V, twoPoint G p x z * twoPoint G p z y := hpairs.tsum_comm
-      _ = _ := by simp only [tsum_mul_left, susceptibility]
-  have hprod : Summable (fun z : N => twoPoint G p x z * susceptibility G p z) := by
-    simpa only [tsum_mul_left, susceptibility] using hpairs.prod
-  rw [hcomm]
-  calc
-    _ ≤ ∑' z : N, twoPoint G p x z * maxSusceptibility G p :=
-      hprod.tsum_le_tsum
-        (fun z => mul_le_mul_of_nonneg_left (susceptibility_le_max G hloc hqt p hp z) (twoPoint_nonneg G p x z))
-        ((summable_twoPoint_subcritical G hloc hqt p hp x).subtype N |>.mul_right _)
-    _ = _ := by rw [tsum_mul_right,mul_comm]
-
- 
-
-theorem deformedSusceptibility_deletion (G : BondGraph V E) (hloc : LocallyFinite G)
-    (hqt : QuasiTransitive G) (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y)
-    (p : unitInterval) (hp : p < pc G) (k : ℕ) (t : ℝ) (ht : 0 ≤ t)
-    (D A : Set V) (x : V) :
-    deformedSusceptibility G P p k t D x -
-      maxSusceptibility G p * (∑' z : closedBondNeighborhood G A, twoPoint G p x z) ≤
-      deformedSusceptibility G P p k t (D \ A) x := by
-  let N := closedBondNeighborhood G A
-  have hpairs := summable_twoPoint_restricted_pair G hloc hqt p hp x N
-  have hdiag (y : V) := hpairs.prod_symm.prod_factor y
-  have hi := summable_deformedKernel G hloc hqt P hP p hp k t ht D x
-  have hi' := summable_deformedKernel G hloc hqt P hP p hp k t ht (D \ A) x
-  have hb := (hi.sub hi').tsum_le_tsum
-    (fun y => deformedKernel_deletion_bound G hloc P hP p k t ht D A x y (hdiag y)) hpairs.prod_symm.prod
-  rw [hi.tsum_sub hi'] at hb
-  change deformedSusceptibility G P p k t D x - deformedSusceptibility G P p k t (D \ A) x ≤ _ at hb
-  have herr := twoPoint_double_restricted_bound G hloc hqt p hp x N
-  linarith
-
-end DeletionSusceptibilityEstimate
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section ExteriorCorridorGeometry
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem conn_of_open_incident (G : BondGraph V E) (ω : Set E) (e : E)
-    (he : e ∈ ω) (a b : V) (ha : a ∈ G e) (hb : b ∈ G e) : Conn G ω a b := by
-  obtain ⟨c,d,hcd⟩ : ∃ c d, G e = s(c,d) := by
-    obtain ⟨⟨c,d⟩,hh⟩ := Quot.exists_rep (G e)
-    exact ⟨c,d,hh.symm⟩
-  rw [hcd,Sym2.mem_iff] at ha hb
-  rcases ha with rfl | rfl <;> rcases hb with rfl | rfl
-  · exact conn_refl G ω _
-  · exact .single ⟨e,he,hcd⟩
-  · exact conn_symm G ω (.single ⟨e,he,hcd⟩)
-  · exact conn_refl G ω _
-
- 
-
-theorem conn_restrict_away_from_cluster (G : BondGraph V E) (ω S : Set E)
-    (hS : S ⊆ ω) (u x y : V) (hx : x ∉ cluster G ω u) (hc : Conn G S x y) :
-    Conn G (S ∩ internalEdges G (cluster G ω u)ᶜ) x y := by
-  apply conn_restrict_of_no_simpleVisit G S (cluster G ω u) x y hc
-  rintro ⟨q,_,z,hz,hzu⟩
-  have hxz : Conn G ω x z := conn_mono G hS
-    ((conn_iff_reachable G S x z).mpr ⟨q.takeUntil z hz⟩)
-  exact hx (hzu.trans (conn_symm G ω hxz))
-
-theorem conn_delete_cluster (G : BondGraph V E) (D : Set V) (ω : Set E)
-    (u x y : V) (hx : x ∉ cluster G (ω ∩ internalEdges G D) u)
-    (S : Set E) (hS : S ⊆ ω ∩ internalEdges G D) (hc : Conn G S x y) :
-    Conn G (S ∩ internalEdges G (D \ cluster G (ω ∩ internalEdges G D) u)) x y := by
-  apply conn_mono G ?_ (conn_restrict_away_from_cluster G _ S hS u x y hx hc)
-  rintro e ⟨he,heA⟩
-  exact ⟨he,fun z hz => ⟨(hS he).2 z hz,heA z hz⟩⟩
-
-namespace RawCorridor
-variable {G : BondGraph V E} {k : ℕ}
-
- 
-
-theorem counted_delete_other_cluster (w : RawCorridor G k) (D : Set V) (ω : Set E)
-    (u x y : V) (hx : x ∉ cluster G (ω ∩ internalEdges G D) u) :
-    w.counted (D \ cluster G (ω ∩ internalEdges G D) u) ω x y ↔ w.counted D ω x y := by
-  let A := cluster G (ω ∩ internalEdges G D) u
-  have hs : ω ∩ internalEdges G (D \ A) ⊆ ω ∩ internalEdges G D :=
-    Set.inter_subset_inter_right _ (internalEdges_mono G Set.sdiff_subset)
-  constructor
-  · intro hc
-    apply (w.counted_iff D ω x y).mpr
-    refine ⟨hc.1,conn_mono G hs hc.2.2.1,?_,?_⟩
-    · intro e he
-      refine ⟨hs (hc.2.2.2.1 e he).1,?_⟩
-      intro hbad
-      have hd := conn_delete_cluster G D ω u x y hx
-        ((ω ∩ internalEdges G D) \ {e}) Set.sdiff_subset hbad
-      apply (hc.2.2.2.1 e he).2
-      apply conn_mono G ?_ hd
-      rintro f ⟨⟨⟨hfω,hfD⟩,hfne⟩,hfnew⟩
-      exact ⟨⟨hfω,hfnew⟩,hfne⟩
-    · apply Set.disjoint_left.mpr
-      intro e he ho
-      obtain ⟨z,hzi,hze⟩ := he.1
-      obtain ⟨q⟩ := (conn_iff_reachable G _ x y).mp hc.2.2.1
-      have hz : z ∈ q.support := w.counted_support_subset_walk (D \ A) ω x y hc q z
-        (List.mem_toFinset.mp (Finset.mem_sdiff.mp hzi).1)
-      have hxz : Conn G (ω ∩ internalEdges G D) x z := conn_mono G hs
-        ((conn_iff_reachable G _ x z).mpr ⟨q.takeUntil z hz⟩)
-      have heA : e ∈ internalEdges G (D \ A) := by
-        intro a ha
-        refine ⟨ho.2 a ha,?_⟩
-        intro hau
-        have hza := conn_of_open_incident G _ e ho z a hze ha
-        exact hx (hau.trans (conn_symm G _ (hxz.trans hza)))
-      exact Set.disjoint_left.mp hc.2.2.2.2 he ⟨ho.1,heA⟩
-  · intro hc
-    have hnew := conn_delete_cluster G D ω u x y hx _ (Set.Subset.refl _) hc.2.2.1
-    have hc' : Conn G (ω ∩ internalEdges G (D \ A)) x y := by
-      apply conn_mono G ?_ hnew
-      rintro e ⟨⟨heω,_⟩,heA⟩
-      exact ⟨heω,heA⟩
-    exact (w.counted_iff _ ω x y).mpr ⟨hc.1,
-      corridorCounted_restrict G _ _ Set.sdiff_subset (Set.Subset.refl _) x y hc' hc.2.2⟩
-
-end RawCorridor
-end ExteriorCorridorGeometry
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section ExteriorWeightedConnections
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem corridorPenalty_delete_other_cluster (G : BondGraph V E) (P : V → V → ℝ)
-    (p : ℝ) (k : ℕ) (D : Set V) (ω : Set E) (u x y : V)
-    (hx : x ∉ cluster G (ω ∩ internalEdges G D) u) :
-    corridorPenalty G P p k (D \ cluster G (ω ∩ internalEdges G D) u) ω x y =
-      corridorPenalty G P p k D ω x y := by
-  unfold corridorPenalty
-  apply tsum_congr
-  intro w
-  simp only [corridorPenaltyTerm,w.counted_delete_other_cluster D ω u x y hx]
-
- 
-
-theorem deformedConnection_delete_other_cluster (G : BondGraph V E) (P : V → V → ℝ)
-    (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (u x y : V)
-    (hx : x ∉ cluster G (ω ∩ internalEdges G D) u) :
-    deformedConnection G P p k t (D \ cluster G (ω ∩ internalEdges G D) u) ω x y =
-      deformedConnection G P p k t D ω x y := by
-  let A := cluster G (ω ∩ internalEdges G D) u
-  have hs : ω ∩ internalEdges G (D \ A) ⊆ ω ∩ internalEdges G D :=
-    Set.inter_subset_inter_right _ (internalEdges_mono G Set.sdiff_subset)
-  have hg : (x ∈ D \ A ∧ y ∈ D \ A ∧ Conn G (ω ∩ internalEdges G (D \ A)) x y) ↔
-      (x ∈ D ∧ y ∈ D ∧ Conn G (ω ∩ internalEdges G D) x y) := by
-    constructor
-    · exact fun h => ⟨h.1.1,h.2.1.1,conn_mono G hs h.2.2⟩
-    · rintro ⟨hxD,hyD,hc⟩
-      have hy : y ∉ A := fun hy => hx (hy.trans (conn_symm G _ hc))
-      refine ⟨⟨hxD,hx⟩,⟨hyD,hy⟩,?_⟩
-      apply conn_mono G ?_ (conn_delete_cluster G D ω u x y hx _ (Set.Subset.refl _) hc)
-      rintro e ⟨⟨heω,_⟩,heA⟩
-      exact ⟨heω,heA⟩
-  dsimp only [A] at hg
-  simp only [deformedConnection,hg,corridorPenalty_delete_other_cluster G P p k D ω u x y hx]
-
-noncomputable def deformedClusterMass (G : BondGraph V E) (P : V → V → ℝ)
-    (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x : V) : ℝ :=
-  ∑' y, deformedConnection G P p k t D ω x y
-
-theorem deformedClusterMass_delete_other_cluster (G : BondGraph V E) (P : V → V → ℝ)
-    (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (u x : V)
-    (hx : x ∉ cluster G (ω ∩ internalEdges G D) u) :
-    deformedClusterMass G P p k t (D \ cluster G (ω ∩ internalEdges G D) u) ω x =
-      deformedClusterMass G P p k t D ω x := by
-  exact tsum_congr (fun y => deformedConnection_delete_other_cluster G P p k t D ω u x y hx)
-
-theorem deformedConnection_le_conn (G : BondGraph V E) (P : V → V → ℝ)
-    (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (k : ℕ) (t : ℝ) (ht : 0 ≤ t)
-    (D : Set V) (ω : Set E) (x y : V) : deformedConnection G P p k t D ω x y ≤
-      if Conn G ω x y then 1 else 0 := by
-  by_cases hc : Conn G ω x y
-  · simpa only [hc,ite_true] using deformedConnection_le_one G P hP p k t ht D ω x y
-  · have hn : ¬Conn G (ω ∩ internalEdges G D) x y := fun h => hc (conn_mono G Set.inter_subset_left h)
-    simp only [deformedConnection,hn,hc,and_false,ite_false,le_refl]
-
-theorem measurable_deformedClusterMass [Countable V] [Countable E] (G : BondGraph V E)
-    (P : V → V → ℝ) (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (x : V) :
-    Measurable (fun ω => deformedClusterMass G P p k t D ω x) :=
-  Measurable.tsum (fun y => measurable_deformedConnection G P p k t D x y)
-
-end ExteriorWeightedConnections
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section DeformedMassIntegrals
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem deformedConnection_finite_support (G : BondGraph V E) (P : V → V → ℝ)
-    (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x : V)
-    (hf : (cluster G ω x).Finite) :
-    (deformedConnection G P p k t D ω x).HasFiniteSupport := by
-  apply hf.subset
-  intro y hy
-  by_contra hn
-  have hnc : ¬Conn G (ω ∩ internalEdges G D) x y := fun h => hn (conn_mono G Set.inter_subset_left h)
-  exact hy (by simp only [deformedConnection,hnc,and_false,ite_false])
-
-theorem deformedClusterMass_nonneg (G : BondGraph V E) (P : V → V → ℝ)
-    (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x : V) :
-    0 ≤ deformedClusterMass G P p k t D ω x :=
-  tsum_nonneg (deformedConnection_nonneg G P p k t D ω x)
-
-theorem deformedClusterMass_le_ncard (G : BondGraph V E) (P : V → V → ℝ)
-    (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (k : ℕ) (t : ℝ) (ht : 0 ≤ t)
-    (D : Set V) (ω : Set E) (x : V) (hf : (cluster G ω x).Finite) :
-    deformedClusterMass G P p k t D ω x ≤ ((cluster G ω x).ncard:ℝ) := by
-  unfold deformedClusterMass
-  rw [tsum_eq_sum (s := hf.toFinset) (fun y hy => by
-    have hn : ¬Conn G ω x y := fun hc => hy (hf.mem_toFinset.mpr hc)
-    have hn' : ¬Conn G (ω ∩ internalEdges G D) x y := fun hc => hn (conn_mono G Set.inter_subset_left hc)
-    simp only [deformedConnection,hn',and_false,ite_false])]
-  calc
-    _ ≤ ∑ _y ∈ hf.toFinset, (1:ℝ) := Finset.sum_le_sum (fun y _ => deformedConnection_le_one G P hP p k t ht D ω x y)
-    _ = _ := by simp only [Finset.sum_const,nsmul_eq_mul,mul_one,←Set.ncard_eq_toFinset_card _ hf]
-
-theorem integrable_deformedClusterMass [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (hp : p < pc G)
-    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D : Set V) (x : V) :
-    Integrable (fun ω => deformedClusterMass G P p k t D ω x) (law p) := by
-  apply Integrable.mono' (susceptibility_eq_integral G hloc hqt p hp x).1
-    (measurable_deformedClusterMass G P p k t D x).aestronglyMeasurable
-  filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
-  rw [Real.norm_of_nonneg (deformedClusterMass_nonneg G P p k t D ω x)]
-  exact deformedClusterMass_le_ncard G P hP p k t ht D ω x hω
-
- 
-
-theorem deformedSusceptibility_eq_integral [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (hp : p < pc G)
-    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D : Set V) (x : V) :
-    deformedSusceptibility G P p k t D x = ∫ ω, deformedClusterMass G P p k t D ω x ∂law p := by
-  have hi := integrable_deformedConnection G P hP p k t ht D x
-  have hs : Summable (fun y => ∫ ω, ‖deformedConnection G P p k t D ω x y‖ ∂law p) := by
-    simp only [Real.norm_of_nonneg (deformedConnection_nonneg G P p k t D _ x _)]
-    exact summable_deformedKernel G hloc hqt P hP p hp k t ht D x
-  exact integral_tsum_of_summable_integral_norm hi hs
-
-end DeformedMassIntegrals
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section ExactComponentRestriction
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-def OpenClosed (G : BondGraph V E) (ω : Set E) (A : Set V) : Prop :=
-  ∀ e ∈ ω, ∀ z ∈ G e, z ∈ A → ∀ a ∈ G e, a ∈ A
-
-theorem openClosed_cluster (G : BondGraph V E) (ω : Set E) (x : V) :
-    OpenClosed G ω (cluster G ω x) := by
-  intro e he z hz hx a ha
-  exact hx.trans (conn_of_open_incident G ω e he z a hz ha)
-
-theorem conn_restrict_openClosed (G : BondGraph V E) (ω S : Set E) (hS : S ⊆ ω)
-    (A : Set V) (hA : OpenClosed G ω A) (x y : V) (hx : x ∈ A) (hc : Conn G S x y) :
-    y ∈ A ∧ Conn G (S ∩ internalEdges G A) x y := by
-  induction hc with
-  | refl => exact ⟨hx,conn_refl G _ x⟩
-  | @tail a b hxa hab ih =>
-    obtain ⟨e,he,heq⟩ := hab
-    have ha : a ∈ G e := by simp [heq]
-    have hb : b ∈ G e := by simp [heq]
-    have hei : e ∈ internalEdges G A := hA e (hS he) a ha ih.1
-    exact ⟨hei b hb,ih.2.tail ⟨e,⟨he,hei⟩,heq⟩⟩
-
-namespace RawCorridor
-variable {G : BondGraph V E} {k : ℕ}
-
- 
-
-theorem counted_restrict_openClosed (w : RawCorridor G k) (D A : Set V) (ω : Set E)
-    (hA : OpenClosed G (ω ∩ internalEdges G D) A) (x y : V) (hx : x ∈ A) :
-    w.counted (D ∩ A) ω x y ↔ w.counted D ω x y := by
-  have hs : ω ∩ internalEdges G (D ∩ A) ⊆ ω ∩ internalEdges G D :=
-    Set.inter_subset_inter_right _ (internalEdges_mono G Set.inter_subset_left)
-  have hrest (S : Set E) (hS : S ⊆ ω ∩ internalEdges G D) (hc : Conn G S x y) :
-      Conn G (S ∩ internalEdges G (D ∩ A)) x y := by
-    apply conn_mono G ?_ (conn_restrict_openClosed G _ S hS A hA x y hx hc).2
-    rintro e ⟨he,heA⟩
-    exact ⟨he,fun z hz => ⟨(hS he).2 z hz,heA z hz⟩⟩
-  constructor
-  · intro hc
-    apply (w.counted_iff D ω x y).mpr
-    refine ⟨hc.1,conn_mono G hs hc.2.2.1,?_,?_⟩
-    · intro e he
-      refine ⟨hs (hc.2.2.2.1 e he).1,?_⟩
-      intro hbad
-      apply (hc.2.2.2.1 e he).2
-      apply conn_mono G ?_ (hrest _ Set.sdiff_subset hbad)
-      rintro f ⟨⟨⟨hfω,_⟩,hfne⟩,hfnew⟩
-      exact ⟨⟨hfω,hfnew⟩,hfne⟩
-    · apply Set.disjoint_left.mpr
-      intro e he ho
-      obtain ⟨z,hzi,hze⟩ := he.1
-      have hz : z ∈ A := (hc.2.1 z (List.mem_toFinset.mp (Finset.mem_sdiff.mp hzi).1)).2
-      have heA : e ∈ internalEdges G (D ∩ A) := fun a ha => ⟨ho.2 a ha,hA e ho z hze hz a ha⟩
-      exact Set.disjoint_left.mp hc.2.2.2.2 he ⟨ho.1,heA⟩
-  · intro hc
-    have hc' : Conn G (ω ∩ internalEdges G (D ∩ A)) x y := by
-      apply conn_mono G ?_ (hrest _ (Set.Subset.refl _) hc.2.2.1)
-      rintro e ⟨⟨heω,_⟩,heA⟩
-      exact ⟨heω,heA⟩
-    exact (w.counted_iff _ ω x y).mpr ⟨hc.1,
-      corridorCounted_restrict G _ _ Set.inter_subset_left (Set.Subset.refl _) x y hc' hc.2.2⟩
-
-end RawCorridor
-end ExactComponentRestriction
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section InternalWeightedRestriction
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem corridorPenalty_restrict_openClosed (G : BondGraph V E) (P : V → V → ℝ)
-    (p : ℝ) (k : ℕ) (D A : Set V) (ω : Set E)
-    (hA : OpenClosed G (ω ∩ internalEdges G D) A) (x y : V) (hx : x ∈ A) :
-    corridorPenalty G P p k (D ∩ A) ω x y = corridorPenalty G P p k D ω x y := by
-  unfold corridorPenalty
-  apply tsum_congr
-  intro w
-  simp only [corridorPenaltyTerm,w.counted_restrict_openClosed D A ω hA x y hx]
-
-theorem deformedConnection_restrict_openClosed (G : BondGraph V E) (P : V → V → ℝ)
-    (p : ℝ) (k : ℕ) (t : ℝ) (D A : Set V) (ω : Set E)
-    (hA : OpenClosed G (ω ∩ internalEdges G D) A) (x y : V) (hx : x ∈ A) :
-    deformedConnection G P p k t (D ∩ A) ω x y = deformedConnection G P p k t D ω x y := by
-  have hg : (x ∈ D ∩ A ∧ y ∈ D ∩ A ∧ Conn G (ω ∩ internalEdges G (D ∩ A)) x y) ↔
-      (x ∈ D ∧ y ∈ D ∧ Conn G (ω ∩ internalEdges G D) x y) := by
-    constructor
-    · exact fun h => ⟨h.1.1,h.2.1.1,conn_mono G
-        (Set.inter_subset_inter_right _ (internalEdges_mono G Set.inter_subset_left)) h.2.2⟩
-    · rintro ⟨hxD,hyD,hc⟩
-      have hh := conn_restrict_openClosed G _ _ (Set.Subset.refl _) A hA x y hx hc
-      refine ⟨⟨hxD,hx⟩,⟨hyD,hh.1⟩,conn_mono G ?_ hh.2⟩
-      rintro e ⟨⟨heω,heD⟩,heA⟩
-      exact ⟨heω,fun z hz => ⟨heD z hz,heA z hz⟩⟩
-  simp only [deformedConnection,hg,corridorPenalty_restrict_openClosed G P p k D A ω hA x y hx]
-
-theorem deformedClusterMass_restrict_openClosed (G : BondGraph V E) (P : V → V → ℝ)
-    (p : ℝ) (k : ℕ) (t : ℝ) (D A : Set V) (ω : Set E)
-    (hA : OpenClosed G (ω ∩ internalEdges G D) A) (x : V) (hx : x ∈ A) :
-    deformedClusterMass G P p k t (D ∩ A) ω x = deformedClusterMass G P p k t D ω x :=
-  tsum_congr (fun y => deformedConnection_restrict_openClosed G P p k t D A ω hA x y hx)
-
- 
-theorem deformedClusterMass_restrict_config (G : BondGraph V E) (P : V → V → ℝ)
-    (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x : V) (S : Set E)
-    (hS : internalEdges G D ⊆ S) :
-    deformedClusterMass G P p k t D (ω ∩ S) x = deformedClusterMass G P p k t D ω x := by
-  have he : (ω ∩ S) ∩ internalEdges G D = ω ∩ internalEdges G D := by
-    ext e
-    exact ⟨fun h => ⟨h.1.1,h.2⟩,fun h => ⟨⟨h.1,hS h.2⟩,h.2⟩⟩
-  apply tsum_congr
-  intro y
-  have hc (w : RawCorridor G k) : w.counted D (ω ∩ S) x y ↔ w.counted D ω x y := by
-    simp only [RawCorridor.counted,corridorCounted,he]
-  simp only [deformedConnection,corridorPenalty,corridorPenaltyTerm,hc,he]
-
-end InternalWeightedRestriction
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory ProbabilityTheory
-section DisjointCoordinateObservables
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem restriction_functions_independent (p : unitInterval) (S T : Set E)
-    (hST : Disjoint S T) (f g : Set E → ℝ) (hf : Measurable f) (hg : Measurable g)
-    (hfS : ∀ ω, f ω = f (ω ∩ S)) (hgT : ∀ ω, g ω = g (ω ∩ T)) :
-    IndepFun f g (law p) := by
-  rw [indepFun_iff_measure_inter_preimage_eq_mul]
-  intro A B hA hB
-  have he (h : Set E → ℝ) (U : Set E) (hU : ∀ ω, h ω = h (ω ∩ U)) (C : Set ℝ) :
-      (fun ω => ω ∩ U) ⁻¹' (h ⁻¹' C) = h ⁻¹' C := by
-    ext ω
-    simp only [Set.mem_preimage,←hU ω]
-  simpa only [he f S hfS A,he g T hgT B] using
-    restriction_events_independent p S T hST (f ⁻¹' A) (g ⁻¹' B)
-      (hA.preimage hf) (hB.preimage hg)
-
-theorem integral_mul_disjoint_coordinates (p : unitInterval) (S T : Set E)
-    (hST : Disjoint S T) (f g : Set E → ℝ) (hf : Measurable f) (hg : Measurable g)
-    (hfS : ∀ ω, f ω = f (ω ∩ S)) (hgT : ∀ ω, g ω = g (ω ∩ T)) :
-    (∫ ω, f ω*g ω ∂law p) = (∫ ω, f ω ∂law p)*(∫ ω, g ω ∂law p) :=
-  (restriction_functions_independent p S T hST f g hf hg hfS hgT).integral_mul_eq_mul_integral
-    hf.aestronglyMeasurable hg.aestronglyMeasurable
-
-end DisjointCoordinateObservables
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section ExactWeightedClusterCell
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem induced_cluster_restrict_incident_iff (G : BondGraph V E) (hloc : LocallyFinite G)
-    (W : Finset V) (D : Set V) (ω : Set E) (x : V) :
-    cluster G ((ω ∩ (incidentSupport G hloc W:Set E)) ∩ internalEdges G D) x = (W:Set V) ↔
-      cluster G (ω ∩ internalEdges G D) x = (W:Set V) := by
-  rw [Set.inter_right_comm]
-  exact cluster_restrict_incident_eq_iff G hloc W _ (Finset.Subset.refl _) _ x
-
-theorem weighted_cluster_cell_local (G : BondGraph V E) (hloc : LocallyFinite G)
-    (P : V → V → ℝ) (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (W : Finset V) (x : V)
-    (ω : Set E) :
-    (if cluster G (ω ∩ internalEdges G D) x = (W:Set V)
-      then deformedClusterMass G P p k t D ω x else 0) =
-    (if cluster G ((ω ∩ (incidentSupport G hloc W:Set E)) ∩ internalEdges G D) x = (W:Set V)
-      then deformedClusterMass G P p k t D (ω ∩ (incidentSupport G hloc W:Set E)) x else 0) := by
-  simp only [induced_cluster_restrict_incident_iff]
-  split_ifs with hW
-  · have hx : x ∈ (W:Set V) := hW ▸ conn_refl G _ x
-    have hA : OpenClosed G (ω ∩ internalEdges G D) (W:Set V) := hW ▸ openClosed_cluster G _ x
-    have hW' := (induced_cluster_restrict_incident_iff G hloc W D ω x).mpr hW
-    have hA' : OpenClosed G ((ω ∩ (incidentSupport G hloc W:Set E)) ∩ internalEdges G D) (W:Set V) :=
-      hW' ▸ openClosed_cluster G _ x
-    rw [←deformedClusterMass_restrict_openClosed G P p k t D W ω hA x hx,
-      ←deformedClusterMass_restrict_openClosed G P p k t D W _ hA' x hx]
-    symm
-    apply deformedClusterMass_restrict_config
-    intro e he
-    exact (mem_incidentSupport G hloc W e).mpr
-      ⟨(G e).out.1,(he _ (Sym2.out_fst_mem _)).2,Sym2.out_fst_mem _⟩
-  · rfl
-
- 
-
-theorem weighted_cluster_cell_exposure [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (hp : p < pc G)
-    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D : Set V) (W : Finset V) (x y : V) :
-    (∫ ω, (if cluster G (ω ∩ internalEdges G D) x = (W:Set V)
-      then deformedClusterMass G P p k t D ω x else 0) *
-        deformedClusterMass G P p k t (D \ (W:Set V)) ω y ∂law p) =
-    (∫ ω, (if cluster G (ω ∩ internalEdges G D) x = (W:Set V)
-      then deformedClusterMass G P p k t D ω x else 0) ∂law p) *
-        deformedSusceptibility G P p k t (D \ (W:Set V)) y := by
-  rw [deformedSusceptibility_eq_integral G hloc hqt P hP p hp k t ht]
-  apply integral_mul_disjoint_coordinates p (incidentSupport G hloc W:Set E)
-    (incidentSupport G hloc W:Set E)ᶜ disjoint_compl_right
-  · exact Measurable.ite ((measurableSet_cluster_eq G x W).preimage (measurable_inter_const _))
-      (measurable_deformedClusterMass G P p k t D x) measurable_const
-  · exact measurable_deformedClusterMass G P p k t _ y
-  · exact weighted_cluster_cell_local G hloc P p k t D W x
-  · intro ω
-    symm
-    apply deformedClusterMass_restrict_config
-    rw [complement_incidentSupport]
-    exact fun e he z hz => (he z hz).2
-
-end ExactWeightedClusterCell
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section InducedClusterPartition
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem integral_induced_cluster_finite_partition [Countable V] [Countable E]
-    (G : BondGraph V E) (p : unitInterval) (hp : p < pc G) (D : Set V) (x : V)
-    (f : Set E → ℝ) (hf : Integrable f (law p)) :
-    (∫ ω, f ω ∂law p) = ∑' W : Finset V,
-      ∫ ω, (if cluster G (ω ∩ internalEdges G D) x = (W:Set V) then f ω else 0) ∂law p := by
-  let F (W : Finset V) (ω : Set E) := if cluster G (ω ∩ internalEdges G D) x = (W:Set V) then f ω else 0
-  have hi (W : Finset V) : Integrable (F W) (law p) := by
-    have he : F W = {ω : Set E | cluster G (ω ∩ internalEdges G D) x = (W:Set V)}.indicator f := by
-      funext ω
-      simp only [F,Set.indicator,Set.mem_ofPred_eq]
-      split_ifs <;> rfl
-    rw [he]
-    exact hf.indicator ((measurableSet_cluster_eq G x W).preimage (measurable_inter_const _))
-  have hb (S : Finset (Finset V)) (ω : Set E) : (∑ W ∈ S, ‖F W ω‖) ≤ ‖f ω‖ := by
-    by_cases hex : ∃ W ∈ S, cluster G (ω ∩ internalEdges G D) x = (W:Set V)
-    · obtain ⟨W,hWS,hW⟩ := hex
-      rw [Finset.sum_eq_single W]
-      · simp only [F,hW,ite_true,le_refl]
-      · intro U _ hU
-        have hu : cluster G (ω ∩ internalEdges G D) x ≠ (U:Set V) := fun h => hU (Finset.coe_injective (h.symm.trans hW))
-        simp only [F,ite_eq_right hu,norm_zero]
-      · exact fun h => (h hWS).elim
-    · have he (W : Finset V) (hW : W ∈ S) : cluster G (ω ∩ internalEdges G D) x ≠ (W:Set V) := fun h => hex ⟨W,hW,h⟩
-      calc
-        _ = 0 := Finset.sum_eq_zero (fun W hW => by simp only [F,ite_eq_right (he W hW),norm_zero])
-        _ ≤ _ := norm_nonneg _
-  have hn : Summable (fun W => ∫ ω, ‖F W ω‖ ∂law p) := by
-    apply summable_of_sum_le (fun _ => integral_nonneg (fun _ => norm_nonneg _))
-      (c := ∫ ω, ‖f ω‖ ∂law p)
-    intro S
-    rw [←integral_finsetSum _ (fun W _ => (hi W).norm)]
-    exact integral_mono (integrable_finsetSum _ (fun W _ => (hi W).norm)) hf.norm (hb S)
-  calc
-    _ = ∫ ω, ∑' W : Finset V, F W ω ∂law p := by
-      apply integral_congr_ae
-      filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
-      exact (tsum_cluster_cells G (ω ∩ internalEdges G D) x
-        (hω.subset (fun _ h => conn_mono G Set.inter_subset_left h)) (f ω)).symm
-    _ = _ := (integral_tsum_of_summable_integral_norm hi hn).symm
-
-end InducedClusterPartition
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section DeformedClusterProducts
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem deformedClusterMass_eq_zero_of_not_mem (G : BondGraph V E) (P : V → V → ℝ)
-    (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x : V) (hx : x ∉ D) :
-    deformedClusterMass G P p k t D ω x = 0 := by
-  simp only [deformedClusterMass,deformedConnection,hx,false_and,ite_false,tsum_zero]
-
-theorem deformed_distinct_product_eq_exterior (G : BondGraph V E) (P : V → V → ℝ)
-    (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x y : V) :
-    (if ¬Conn G (ω ∩ internalEdges G D) x y then
-      deformedClusterMass G P p k t D ω x * deformedClusterMass G P p k t D ω y else 0) =
-    deformedClusterMass G P p k t D ω x *
-      deformedClusterMass G P p k t (D \ cluster G (ω ∩ internalEdges G D) x) ω y := by
-  by_cases hc : Conn G (ω ∩ internalEdges G D) x y
-  · rw [ite_eq_right (not_not_intro hc),deformedClusterMass_eq_zero_of_not_mem G P p k t _ ω y
-      (fun h => h.2 hc),mul_zero]
-  · rw [ite_eq_left hc,deformedClusterMass_delete_other_cluster G P p k t D ω x y hc]
-
-theorem integrable_deformed_product [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (hp : p < pc G)
-    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D : Set V) (x y : V) :
-    Integrable (fun ω => deformedClusterMass G P p k t D ω x *
-      deformedClusterMass G P p k t D ω y) (law p) := by
-  apply Integrable.mono' (subcritical_cluster_association G hloc hqt p hp x y).1
-    ((measurable_deformedClusterMass G P p k t D x).mul
-      (measurable_deformedClusterMass G P p k t D y)).aestronglyMeasurable
-  filter_upwards [subcritical_cluster_finite_ae G p hp x,subcritical_cluster_finite_ae G p hp y] with ω hx hy
-  dsimp only [Pi.mul_apply]
-  rw [Real.norm_of_nonneg (mul_nonneg (deformedClusterMass_nonneg G P p k t D ω x)
-    (deformedClusterMass_nonneg G P p k t D ω y))]
-  exact mul_le_mul (deformedClusterMass_le_ncard G P hP p k t ht D ω x hx)
-    (deformedClusterMass_le_ncard G P hP p k t ht D ω y hy)
-    (deformedClusterMass_nonneg G P p k t D ω y) (Nat.cast_nonneg _)
-
-theorem integrable_distinct_deformed_product [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (hp : p < pc G)
-    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D : Set V) (x y : V) :
-    Integrable (fun ω => if ¬Conn G (ω ∩ internalEdges G D) x y then
-      deformedClusterMass G P p k t D ω x * deformedClusterMass G P p k t D ω y else 0) (law p) := by
-  have he : (fun ω => if ¬Conn G (ω ∩ internalEdges G D) x y then
-      deformedClusterMass G P p k t D ω x * deformedClusterMass G P p k t D ω y else 0) =
-      {ω | Conn G (ω ∩ internalEdges G D) x y}ᶜ.indicator
-        (fun ω => deformedClusterMass G P p k t D ω x * deformedClusterMass G P p k t D ω y) := by
-    funext ω
-    simp only [Set.indicator,Set.mem_compl_iff,Set.mem_ofPred_eq]
-  rw [he]
-  exact (integrable_deformed_product G hloc hqt P hP p hp k t ht D x y).indicator
-    (((measurableSet_conn G x y).preimage (measurable_inter_const (internalEdges G D))).compl)
-
-end DeformedClusterProducts
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section FreshWeightedMass
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def deformedFreshExteriorMoment (G : BondGraph V E) (P : V → V → ℝ)
-    (p : unitInterval) (k : ℕ) (t : ℝ) (D : Set V) (x y : V) (ω : Set E) : ℝ :=
-  deformedClusterMass G P p k t D ω x *
-    deformedSusceptibility G P p k t (D \ cluster G (ω ∩ internalEdges G D) x) y
-
-theorem deformedSusceptibility_nonneg (G : BondGraph V E) (P : V → V → ℝ)
-    (p : unitInterval) (k : ℕ) (t : ℝ) (D : Set V) (x : V) :
-    0 ≤ deformedSusceptibility G P p k t D x :=
-  tsum_nonneg (deformedKernel_nonneg G P p k t D x)
-
-theorem deformedFreshExteriorMoment_eq_tsum (G : BondGraph V E) (P : V → V → ℝ)
-    (p : unitInterval) (k : ℕ) (t : ℝ) (D : Set V) (x y : V) (ω : Set E)
-    (hf : (cluster G (ω ∩ internalEdges G D) x).Finite) :
-    deformedFreshExteriorMoment G P p k t D x y ω = ∑' W : Finset V,
-      if cluster G (ω ∩ internalEdges G D) x = (W:Set V) then
-        deformedClusterMass G P p k t D ω x * deformedSusceptibility G P p k t (D \ (W:Set V)) y else 0 := by
-  rw [←tsum_cluster_cells G (ω ∩ internalEdges G D) x hf (deformedFreshExteriorMoment G P p k t D x y ω)]
-  apply tsum_congr
-  intro W
-  split_ifs with hW
-  · simp only [deformedFreshExteriorMoment,hW]
-  · rfl
-
-theorem integrable_deformedFreshExteriorMoment [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (hp : p < pc G)
-    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D : Set V) (x y : V) :
-    Integrable (deformedFreshExteriorMoment G P p k t D x y) (law p) := by
-  have hm : AEStronglyMeasurable (deformedFreshExteriorMoment G P p k t D x y) (law p) := by
-    have hh : Measurable (fun ω => ∑' W : Finset V,
-        if cluster G (ω ∩ internalEdges G D) x = (W:Set V) then
-          deformedClusterMass G P p k t D ω x * deformedSusceptibility G P p k t (D \ (W:Set V)) y else 0) := by
-      apply Measurable.tsum
-      intro W
-      exact Measurable.ite ((measurableSet_cluster_eq G x W).preimage (measurable_inter_const _))
-        ((measurable_deformedClusterMass G P p k t D x).mul_const _) measurable_const
-    apply hh.aestronglyMeasurable.congr
-    filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
-    exact (deformedFreshExteriorMoment_eq_tsum G P p k t D x y ω
-      (hω.subset (fun _ h => conn_mono G Set.inter_subset_left h))).symm
-  apply Integrable.mono' ((integrable_deformedClusterMass G hloc hqt P hP p hp k t ht D x).mul_const
-    (susceptibility G p y)) hm
-  exact Filter.Eventually.of_forall (fun ω => by
-    dsimp only [deformedFreshExteriorMoment]
-    rw [Real.norm_of_nonneg (mul_nonneg (deformedClusterMass_nonneg G P p k t D ω x)
-      (deformedSusceptibility_nonneg G P p k t _ y))]
-    exact mul_le_mul_of_nonneg_left (deformedSusceptibility_le G hloc hqt P hP p hp k t ht _ y)
-      (deformedClusterMass_nonneg G P p k t D ω x))
-
-end FreshWeightedMass
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section WeightedExposureIdentity
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem integral_deformedFreshExteriorMoment [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (hp : p < pc G)
-    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D : Set V) (x y : V) :
-    (∫ ω, deformedFreshExteriorMoment G P p k t D x y ω ∂law p) =
-      ∫ ω, (if ¬Conn G (ω ∩ internalEdges G D) x y then
-        deformedClusterMass G P p k t D ω x * deformedClusterMass G P p k t D ω y else 0) ∂law p := by
-  rw [integral_induced_cluster_finite_partition G p hp D x _
-      (integrable_deformedFreshExteriorMoment G hloc hqt P hP p hp k t ht D x y),
-    integral_induced_cluster_finite_partition G p hp D x _
-      (integrable_distinct_deformed_product G hloc hqt P hP p hp k t ht D x y)]
-  apply tsum_congr
-  intro W
-  have hl : (∫ ω, (if cluster G (ω ∩ internalEdges G D) x = (W:Set V) then
-      deformedFreshExteriorMoment G P p k t D x y ω else 0) ∂law p) =
-      (∫ ω, (if cluster G (ω ∩ internalEdges G D) x = (W:Set V) then
-        deformedClusterMass G P p k t D ω x else 0) ∂law p) *
-      deformedSusceptibility G P p k t (D \ (W:Set V)) y := by
-    rw [←integral_mul_const]
-    apply integral_congr_ae
-    exact Filter.Eventually.of_forall (fun ω => by
-      by_cases hW : cluster G (ω ∩ internalEdges G D) x = (W:Set V)
-      · simp only [hW,ite_true,deformedFreshExteriorMoment]
-      · simp only [hW,ite_false,zero_mul])
-  rw [hl,←weighted_cluster_cell_exposure G hloc hqt P hP p hp k t ht D W x y]
-  apply integral_congr_ae
-  exact Filter.Eventually.of_forall (fun ω => by
-    by_cases hW : cluster G (ω ∩ internalEdges G D) x = (W:Set V)
-    · simp only [hW,ite_true]
-      rw [deformed_distinct_product_eq_exterior,hW]
-    · simp only [hW,ite_false,zero_mul])
-
-end WeightedExposureIdentity
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section WeightedThreeArmEstimate
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Countable V] [Countable E] [Nonempty V]
-
-theorem summable_three_arm_pair (G : BondGraph V E) (hloc : LocallyFinite G)
-    (hqt : QuasiTransitive G) (p : unitInterval) (hp : p < pc G) (x a : V) :
-    Summable (fun bz : V×V => twoPoint G p x bz.1 * twoPoint G p bz.1 a * twoPoint G p bz.1 bz.2) := by
-  apply (summable_twoPoint_pair G hloc hqt p hp x).of_nonneg_of_le
-  · exact fun bz => mul_nonneg (mul_nonneg (twoPoint_nonneg G p _ _) (twoPoint_nonneg G p _ _))
-      (twoPoint_nonneg G p _ _)
-  · intro bz
-    exact mul_le_mul_of_nonneg_right (mul_le_of_le_one_right (twoPoint_nonneg G p _ _) (twoPoint_le_one G p _ _))
-      (twoPoint_nonneg G p _ _)
-
-theorem three_arm_real_bound (G : BondGraph V E) (hloc : LocallyFinite G)
-    (hqt : QuasiTransitive G) (p : unitInterval) (hp : p < pc G) (x a z : V) :
-    (law p).real {ω | Conn G ω x a ∧ Conn G ω x z} ≤
-      ∑' b, twoPoint G p x b * twoPoint G p b a * twoPoint G p b z := by
-  have hs := (summable_three_arm_pair G hloc hqt p hp x a).prod_symm.prod_factor z
-  have hn (b : V) : 0 ≤ twoPoint G p x b * twoPoint G p b a * twoPoint G p b z :=
-    mul_nonneg (mul_nonneg (twoPoint_nonneg G p _ _) (twoPoint_nonneg G p _ _)) (twoPoint_nonneg G p _ _)
-  have he : (∑' b, law p {ω | Conn G ω x b} * law p {ω | Conn G ω b a} * law p {ω | Conn G ω b z}) =
-      ENNReal.ofReal (∑' b, twoPoint G p x b * twoPoint G p b a * twoPoint G p b z) := by
-    rw [ENNReal.ofReal_tsum_of_nonneg hn hs]
-    apply tsum_congr
-    intro b
-    rw [ENNReal.ofReal_mul (mul_nonneg (twoPoint_nonneg G p _ _) (twoPoint_nonneg G p _ _)),
-      ENNReal.ofReal_mul (twoPoint_nonneg G p _ _)]
-    simp only [twoPoint,ENNReal.ofReal_toReal (measure_ne_top _ _)]
-  have hh := (three_arm_bound G p x a z).trans_eq he
-  exact (ENNReal.toReal_mono ENNReal.ofReal_ne_top hh).trans_eq (ENNReal.toReal_ofReal (tsum_nonneg hn))
-
- 
-
-theorem integral_size_membership_le (G : BondGraph V E) (hloc : LocallyFinite G)
-    (hqt : QuasiTransitive G) (p : unitInterval) (hp : p < pc G) (x a : V) :
-    (∫ ω, (if Conn G ω x a then ((cluster G ω x).ncard:ℝ) else 0) ∂law p) ≤
-      maxSusceptibility G p * (∑' b, twoPoint G p x b * twoPoint G p b a) := by
-  have he := integral_cluster_size_event G hloc hqt p hp x _ (measurableSet_conn G x a)
-  simp only [Set.mem_ofPred_eq] at he
-  rw [he]
-  have hs := summable_three_arm_pair G hloc hqt p hp x a
-  have hl : Summable (fun z => (law p).real {ω | Conn G ω x a ∧ Conn G ω x z}) :=
-    (summable_twoPoint_subcritical G hloc hqt p hp x).of_nonneg_of_le (fun _ => measureReal_nonneg)
-      (fun _ => measureReal_mono (fun _ h => h.2))
-  have hb : Summable (fun b => twoPoint G p x b * twoPoint G p b a) :=
-    (summable_twoPoint_subcritical G hloc hqt p hp x).of_nonneg_of_le
-      (fun b => mul_nonneg (twoPoint_nonneg G p _ _) (twoPoint_nonneg G p _ _))
-      (fun b => mul_le_of_le_one_right (twoPoint_nonneg G p _ _) (twoPoint_le_one G p _ _))
-  have hj : Summable (fun b => twoPoint G p x b * twoPoint G p b a * susceptibility G p b) := by
-    simpa only [tsum_mul_left,susceptibility] using hs.prod
-  calc
-    _ ≤ ∑' z, ∑' b, twoPoint G p x b * twoPoint G p b a * twoPoint G p b z :=
-      hl.tsum_le_tsum (three_arm_real_bound G hloc hqt p hp x a) hs.prod_symm.prod
-    _ = ∑' b, ∑' z, twoPoint G p x b * twoPoint G p b a * twoPoint G p b z := hs.tsum_comm
-    _ = ∑' b, twoPoint G p x b * twoPoint G p b a * susceptibility G p b := by
-      simp only [tsum_mul_left,susceptibility]
-    _ ≤ ∑' b, twoPoint G p x b * twoPoint G p b a * maxSusceptibility G p := by
-      apply hj.tsum_le_tsum
-        (fun b => mul_le_mul_of_nonneg_left (susceptibility_le_max G hloc hqt p hp b)
-          (mul_nonneg (twoPoint_nonneg G p _ _) (twoPoint_nonneg G p _ _))) (hb.mul_right _)
-    _ = _ := by rw [tsum_mul_right,mul_comm]
-
-end WeightedThreeArmEstimate
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section FiniteNeighborhoodMass
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem finset_sum_union_le (f : V → ℝ) (hf : ∀ x, 0 ≤ f x) (S T : Finset V) :
-    (∑ x ∈ S ∪ T, f x) ≤ (∑ x ∈ S, f x) + ∑ x ∈ T, f x := by
-  have he := Finset.sum_union_inter (f := f) (s₁ := S) (s₂ := T)
-  have hn := Finset.sum_nonneg (s := S ∩ T) (fun x _ => hf x)
-  linarith
-
-theorem finset_sum_biUnion_le (f : V → ℝ) (hf : ∀ x, 0 ≤ f x)
-    (W : Finset V) (N : V → Finset V) :
-    (∑ z ∈ W.biUnion N, f z) ≤ ∑ a ∈ W, ∑ z ∈ N a, f z := by
-  induction W using Finset.induction_on with
-  | empty => simp
-  | @insert a W ha ih =>
-    rw [Finset.biUnion_insert,Finset.sum_insert ha]
-    exact (finset_sum_union_le f hf _ _).trans (add_le_add (le_refl _) ih)
-
-theorem closedBondNeighborhood_finset (G : BondGraph V E) (hloc : LocallyFinite G) (W : Finset V) :
-    closedBondNeighborhood G (W:Set V) = (W ∪ W.biUnion (neighbors G hloc) : Finset V) := by
-  ext z
-  simp only [closedBondNeighborhood,Set.mem_ofPred_eq,Finset.mem_coe,Finset.mem_union,Finset.mem_biUnion]
-  constructor
-  · rintro (hz | ⟨e,hz,a,ha,haW⟩)
-    · exact Or.inl hz
-    · by_cases hza : a = z
-      · exact Or.inl (hza ▸ haW)
-      · exact Or.inr ⟨a,haW,(mem_neighbors G hloc a z).mpr
-          ⟨e,(Sym2.mem_and_mem_iff hza).mp ⟨ha,hz⟩⟩⟩
-  · rintro (hz | ⟨a,ha,haz⟩)
-    · exact Or.inl hz
-    · obtain ⟨e,he⟩ := (mem_neighbors G hloc a z).mp haz
-      exact Or.inr ⟨e,he.symm ▸ Sym2.mem_mk_right a z,a,he.symm ▸ Sym2.mem_mk_left a z,ha⟩
-
-theorem neighborhood_mass_le (G : BondGraph V E) (hloc : LocallyFinite G)
-    (W : Finset V) (f : V → ℝ) (hf : ∀ x, 0 ≤ f x) :
-    (∑' z : closedBondNeighborhood G (W:Set V), f z) ≤
-      ∑ a ∈ W, (f a + ∑ z ∈ neighbors G hloc a, f z) := by
-  rw [closedBondNeighborhood_finset G hloc W,Finset.tsum_subtype']
-  exact (finset_sum_union_le f hf _ _).trans
-    ((add_le_add (le_refl _) (finset_sum_biUnion_le f hf W (neighbors G hloc))).trans_eq
-      (Finset.sum_add_distrib).symm)
-
-end FiniteNeighborhoodMass
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Deformation.Corridors
+
+namespace OAI
+
+/-! Cluster Products for bond percolation. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section SmallDeformationOperators
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Countable V] [Countable E]
+
+ 
+
+theorem exists_deformedOperator_small_change (G : BondGraph V E)
+    (hG : Function.Injective G) (hloc : LocallyFinite G) (p : unitInterval)
+    (hp : 0 < (p:ℝ)) (hp₁ : (p:ℝ) < 1) (a : ℝ)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hPs : IsSelfAdjoint P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (A : Hilbert V →L[ℂ] Hilbert V) (hA : PositiveRealMatrix A) (hAs : IsSelfAdjoint A)
+    (hAT : ∀ x y, matrixCoeff A x y = (twoPoint G p x y : ℂ))
+    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) :
+    ∃ B : Hilbert V →L[ℂ] Hilbert V,
+      (∀ x y, matrixCoeff B x y = (deformedKernel G (walkKernel G hloc a) p k t Set.univ x y : ℂ)) ∧
+      ‖B‖ ≤ ‖A‖ ∧ ‖A-B‖ ≤ 2*t*‖P‖^k*‖A‖^2 ∧ ‖A‖-2*t*‖P‖^k*‖A‖^2 ≤ ‖B‖ := by
+  let K := deformedKernel G (walkKernel G hloc a) p k t Set.univ
+  have hKn (x y : V) : 0 ≤ K x y := deformedKernel_nonneg G _ p k t _ x y
+  have hb (x y : V) := deformedKernel_change_bound G hG hloc p hp hp₁ a P hpos hPs hP A hA hAs hAT k t ht x y
+  obtain ⟨B,hBn,hBK⟩ := exists_operator_of_matrix_domination (twoPoint G p) K (twoPoint_nonneg G p)
+    (fun x y => by rw [abs_of_nonneg (hKn x y)]; exact sub_nonneg.mp (hb x y).1) A hAT
+  have hBK' (x y : V) : matrixCoeff B x y = (K x y : ℂ) := hBK x y
+  let C := A*(P^k*A)
+  have hCn : PositiveRealMatrix C := hA.mul ((hpos.pow k).mul hA)
+  have hscale (x y : V) : matrixCoeff (((2*t:ℝ):ℂ) • C) x y =
+      (2*t*(matrixCoeff C x y).re : ℝ) := by
+    change ((2*t:ℝ):ℂ)*matrixCoeff C x y = _
+    rw [hCn.real]
+    simp only [Complex.ofReal_re,Complex.ofReal_mul]
+  obtain ⟨D,hDn,hDK⟩ := exists_operator_of_matrix_domination
+    (fun x y => 2*t*(matrixCoeff C x y).re) (fun x y => twoPoint G p x y - K x y)
+    (fun x y => mul_nonneg (mul_nonneg (by norm_num) ht) (hCn x y).2)
+    (fun x y => by rw [abs_of_nonneg (hb x y).1]; exact (hb x y).2)
+    (((2*t:ℝ):ℂ) • C) hscale
+  have he : D = A-B := kernel_extension_unique hDK (by
+    intro x y
+    change matrixCoeff A x y - matrixCoeff B x y = _
+    rw [hAT,hBK',Complex.ofReal_sub])
+  rw [he] at hDn
+  have hCnorm : ‖C‖ ≤ ‖A‖*‖P‖^k*‖A‖ := by
+    calc
+      _ ≤ ‖A‖*‖P^k*A‖ := norm_mul_le _ _
+      _ ≤ ‖A‖*(‖P^k‖*‖A‖) := mul_le_mul_of_nonneg_left (norm_mul_le _ _) (norm_nonneg _)
+      _ ≤ ‖A‖*(‖P‖^k*‖A‖) := by
+        gcongr
+        cases k with
+        | zero =>
+          simp only [pow_zero]
+          have he : (1 : Hilbert V →L[ℂ] Hilbert V) = ContinuousLinearMap.id ℂ (Hilbert V) := by ext f; rfl
+          rw [he]
+          exact ContinuousLinearMap.norm_id_le
+        | succ j => exact norm_pow_le' P (Nat.succ_pos j)
+      _ = _ := by ring
+  have hd : ‖A-B‖ ≤ 2*t*‖P‖^k*‖A‖^2 := by
+    calc
+      _ ≤ ‖((2*t:ℝ):ℂ) • C‖ := hDn
+      _ = 2*t*‖C‖ := by rw [norm_smul,Complex.norm_of_nonneg (mul_nonneg (by norm_num) ht)]
+      _ ≤ 2*t*(‖A‖*‖P‖^k*‖A‖) := mul_le_mul_of_nonneg_left hCnorm (mul_nonneg (by norm_num) ht)
+      _ = _ := by ring
+  refine ⟨B,hBK,hBn,hd,?_⟩
+  have hh : ‖A‖ ≤ ‖A-B‖+‖B‖ := by simpa only [sub_add_cancel] using norm_add_le (A-B) B
+  linarith
+
+end SmallDeformationOperators
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section DeletionGeometry
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+def closedBondNeighborhood (G : BondGraph V E) (A : Set V) : Set V :=
+  {z | z ∈ A ∨ ∃ e, z ∈ G e ∧ ∃ a ∈ G e, a ∈ A}
+
+def SimpleVisit (G : BondGraph V E) (S : Set E) (A : Set V) (x y : V) : Prop :=
+  ∃ q : (openSimpleGraph G S).Walk x y, q.IsPath ∧ ∃ z ∈ q.support, z ∈ A
+
+theorem SimpleVisit.mono (G : BondGraph V E) {S T : Set E} {A B : Set V}
+    (hST : S ⊆ T) (hAB : A ⊆ B) {x y : V} (h : SimpleVisit G S A x y) :
+    SimpleVisit G T B x y := by
+  obtain ⟨q,hq,z,hz,ha⟩ := h
+  have hg : openSimpleGraph G S ≤ openSimpleGraph G T := by
+    rintro a b ⟨hne,e,he,hGe⟩
+    exact ⟨hne,e,hST he,hGe⟩
+  exact ⟨q.mapLe hg,hq.mapLe hg,z,by simpa only [SimpleGraph.Walk.support_mapLe_eq_support] using hz,hAB ha⟩
+
+theorem walk_conn_restrict_vertices (G : BondGraph V E) (S : Set E) {x y : V}
+    (q : (openSimpleGraph G S).Walk x y) (D : Set V)
+    (hD : ∀ z ∈ q.support, z ∈ D) : Conn G (S ∩ internalEdges G D) x y := by
+  obtain ⟨l,hl,hm,ho⟩ := walk_lift_bonds G S q
+  apply hl.conn
+  intro e he
+  refine ⟨ho e he,?_⟩
+  intro z hz
+  exact hD z (SimpleGraph.Walk.mem_support_of_mem_edges (hm ▸ List.mem_map_of_mem he) hz)
+
+theorem conn_restrict_of_no_simpleVisit (G : BondGraph V E) (S : Set E)
+    (A : Set V) (x y : V) (hc : Conn G S x y) (hn : ¬SimpleVisit G S A x y) :
+    Conn G (S ∩ internalEdges G Aᶜ) x y := by
+  obtain ⟨q⟩ := (conn_iff_reachable G S x y).mp hc
+  exact walk_conn_restrict_vertices G S q.bypass Aᶜ
+    (fun z hz ha => hn ⟨q.bypass,q.bypass_isPath,z,hz,ha⟩)
+
+namespace RawCorridor
+variable {G : BondGraph V E} {k : ℕ}
+
+theorem counted_support_subset_walk (w : RawCorridor G k) (D : Set V) (ω : Set E)
+    (x y : V) (hc : w.counted D ω x y)
+    (q : (openSimpleGraph G (ω ∩ internalEdges G D)).Walk x y) :
+    ∀ z ∈ w.path.support, z ∈ q.support := by
+  intro z hz
+  have hl : 0 < w.path.length := by have := hc.1; omega
+  obtain ⟨s,hs,hzs⟩ := (SimpleGraph.Walk.mem_support_iff_exists_mem_edges_of_not_nil
+    (SimpleGraph.Walk.not_nil_iff_lt_length.mpr hl)).mp hz
+  obtain ⟨e,he,hGe⟩ := w.edge_has_label s hs
+  exact SimpleGraph.Walk.mem_support_of_mem_edges
+    (pivotal_image_mem_edges G _ q e (hc.2.2.2.1 e he).2) (hGe.symm ▸ hzs)
+
+end RawCorridor
+
+ 
+
+theorem simplePath_disjoint_at (G : BondGraph V E) (S : Set E) {x y z : V}
+    (q : (openSimpleGraph G S).Walk x y) (hq : q.IsPath) (hz : z ∈ q.support) :
+    S ∈ DisjointOccurrence {η | Conn G η x z} {η | Conn G η z y} := by
+  obtain ⟨a,ha,hea,hoa⟩ := walk_lift_bonds G S (q.takeUntil z hz)
+  obtain ⟨b,hb,heb,hob⟩ := walk_lift_bonds G S (q.dropUntil z hz)
+  refine ⟨a.toFinset,b.toFinset,?_,?_,?_,?_,?_⟩
+  · intro e he; exact hoa e (List.mem_toFinset.mp he)
+  · intro e he; exact hob e (List.mem_toFinset.mp he)
+  · apply Finset.disjoint_left.mpr
+    intro e hae hbe
+    exact List.disjoint_left.mp (hq.isTrail.disjoint_edges_takeUntil_dropUntil hz)
+      (hea ▸ List.mem_map_of_mem (List.mem_toFinset.mp hae))
+      (heb ▸ List.mem_map_of_mem (List.mem_toFinset.mp hbe))
+  · exact ha.conn (fun e he => List.mem_toFinset.mpr he)
+  · exact hb.conn (fun e he => List.mem_toFinset.mpr he)
+
+end DeletionGeometry
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section DeletionCounted
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem conn_delete_of_no_visit (G : BondGraph V E) (ω : Set E) (D A : Set V)
+    (x y : V) (hn : ¬SimpleVisit G (ω ∩ internalEdges G D) (closedBondNeighborhood G A) x y)
+    (S : Set E) (hS : S ⊆ ω ∩ internalEdges G D) (hc : Conn G S x y) :
+    Conn G (S ∩ internalEdges G (D \ A)) x y := by
+  have hno : ¬SimpleVisit G S A x y := fun h => hn (h.mono G hS (fun _ => Or.inl))
+  have h := conn_restrict_of_no_simpleVisit G S A x y hc hno
+  apply conn_mono G ?_ h
+  rintro e ⟨he,ha⟩
+  exact ⟨he,fun z hz => ⟨(hS he).2 z hz,ha z hz⟩⟩
+
+namespace RawCorridor
+variable {G : BondGraph V E} {k : ℕ}
+
+ 
+
+theorem counted_before_deletion_of_no_visit (w : RawCorridor G k)
+    (D A : Set V) (ω : Set E) (x y : V)
+    (hn : ¬SimpleVisit G (ω ∩ internalEdges G D) (closedBondNeighborhood G A) x y)
+    (hc : w.counted (D \ A) ω x y) : w.counted D ω x y := by
+  have hs : ω ∩ internalEdges G (D \ A) ⊆ ω ∩ internalEdges G D :=
+    Set.inter_subset_inter_right _ (internalEdges_mono G Set.sdiff_subset)
+  apply (w.counted_iff D ω x y).mpr
+  refine ⟨hc.1,conn_mono G hs hc.2.2.1,?_,?_⟩
+  · intro e he
+    refine ⟨hs (hc.2.2.2.1 e he).1,?_⟩
+    intro hbad
+    have hd := conn_delete_of_no_visit G ω D A x y hn
+      ((ω ∩ internalEdges G D) \ {e}) Set.sdiff_subset hbad
+    apply (hc.2.2.2.1 e he).2
+    apply conn_mono G ?_ hd
+    rintro f ⟨⟨⟨hfω,hfD⟩,hfne⟩,hfnew⟩
+    exact ⟨⟨hfω,hfnew⟩,hfne⟩
+  · apply Set.disjoint_left.mpr
+    intro e he ho
+    have henew : e ∉ internalEdges G (D \ A) := fun he' =>
+      Set.disjoint_left.mp hc.2.2.2.2 he ⟨ho.1,he'⟩
+    obtain ⟨a,ha,hna⟩ : ∃ a ∈ G e, a ∉ D \ A := by
+      change ¬(∀ a ∈ G e, a ∈ D \ A) at henew
+      push Not at henew
+      exact henew
+    have haA : a ∈ A := by
+      by_contra ha'
+      exact hna ⟨ho.2 a ha,ha'⟩
+    obtain ⟨z,hzi,hze⟩ := he.1
+    obtain ⟨q⟩ := (conn_iff_reachable G _ x y).mp hc.2.2.1
+    have hz : z ∈ q.bypass.support := w.counted_support_subset_walk (D \ A) ω x y hc q.bypass z
+      (List.mem_toFinset.mp (Finset.mem_sdiff.mp hzi).1)
+    have hv : SimpleVisit G (ω ∩ internalEdges G (D \ A)) (closedBondNeighborhood G A) x y :=
+      ⟨q.bypass,q.bypass_isPath,z,hz,Or.inr ⟨e,hze,a,ha,haA⟩⟩
+    exact hn (hv.mono G hs (Set.Subset.refl _))
+
+end RawCorridor
+end DeletionCounted
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section DeletionWeightedLoss
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem deformedConnection_no_visit (G : BondGraph V E) (hloc : LocallyFinite G)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval)
+    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D A : Set V) (ω : Set E) (x y : V)
+    (hn : ¬SimpleVisit G (ω ∩ internalEdges G D) (closedBondNeighborhood G A) x y) :
+    deformedConnection G P p k t D ω x y ≤ deformedConnection G P p k t (D \ A) ω x y := by
+  by_cases hc : x ∈ D ∧ y ∈ D ∧ Conn G (ω ∩ internalEdges G D) x y
+  · obtain ⟨q⟩ := (conn_iff_reachable G _ x y).mp hc.2.2
+    have hx : x ∉ A := fun ha => hn ⟨q.bypass,q.bypass_isPath,x,q.bypass.start_mem_support,Or.inl ha⟩
+    have hy : y ∉ A := fun ha => hn ⟨q.bypass,q.bypass_isPath,y,q.bypass.end_mem_support,Or.inl ha⟩
+    have hc' : x ∈ D \ A ∧ y ∈ D \ A ∧ Conn G (ω ∩ internalEdges G (D \ A)) x y := by
+      refine ⟨⟨hc.1,hx⟩,⟨hc.2.1,hy⟩,?_⟩
+      apply conn_mono G ?_ (conn_delete_of_no_visit G ω D A x y hn _ (Set.Subset.refl _) hc.2.2)
+      rintro e ⟨⟨heω,heD⟩,henew⟩
+      exact ⟨heω,henew⟩
+    simp only [deformedConnection,hc,hc']
+    apply Real.exp_le_exp.mpr
+    apply mul_le_mul_of_nonpos_left ?_ (neg_nonpos.mpr ht)
+    apply (corridorPenalty_summable G hloc P p k (D \ A) ω x y).tsum_le_tsum ?_
+      (corridorPenalty_summable G hloc P p k D ω x y)
+    intro w
+    by_cases hw : w.counted (D \ A) ω x y
+    · have hw' := w.counted_before_deletion_of_no_visit D A ω x y hn hw
+      simp only [corridorPenaltyTerm,hw,hw',ite_true,le_refl]
+    · simpa only [corridorPenaltyTerm,hw,ite_false] using corridorPenaltyTerm_nonneg G P hP p D ω x y w
+  · simpa only [deformedConnection,hc,ite_false] using deformedConnection_nonneg G P p k t (D \ A) ω x y
+
+ 
+
+theorem deformedConnection_deletion_pointwise (G : BondGraph V E) (hloc : LocallyFinite G)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval)
+    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D A : Set V) (ω : Set E) (x y : V) :
+    deformedConnection G P p k t D ω x y - deformedConnection G P p k t (D \ A) ω x y ≤
+      if ∃ z ∈ closedBondNeighborhood G A,
+        ω ∈ DisjointOccurrence {η | Conn G η x z} {η | Conn G η z y} then 1 else 0 := by
+  split_ifs with h
+  · linarith [deformedConnection_le_one G P hP p k t ht D ω x y,
+      deformedConnection_nonneg G P p k t (D \ A) ω x y]
+  · apply sub_nonpos.mpr
+    apply deformedConnection_no_visit G hloc P hP p k t ht D A ω x y
+    rintro ⟨q,hq,z,hz,hza⟩
+    apply h
+    exact ⟨z,hza,disjointOccurrence_upper _ _ Set.inter_subset_left (simplePath_disjoint_at G _ q hq hz)⟩
+
+end DeletionWeightedLoss
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section DeletionKernelBound
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Countable V] [Countable E]
+
+ 
+
+theorem deformedKernel_deletion_bound (G : BondGraph V E) (hloc : LocallyFinite G)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval)
+    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D A : Set V) (x y : V)
+    (hs : Summable (fun z : closedBondNeighborhood G A => twoPoint G p x z * twoPoint G p z y)) :
+    deformedKernel G P p k t D x y - deformedKernel G P p k t (D \ A) x y ≤
+      ∑' z : closedBondNeighborhood G A, twoPoint G p x z * twoPoint G p z y := by
+  let B := fun z : closedBondNeighborhood G A =>
+    DisjointOccurrence {η | Conn G η x z} {η | Conn G η z y}
+  let U := ⋃ z, B z
+  have hBm (z : closedBondNeighborhood G A) : MeasurableSet (B z) :=
+    measurableSet_finiteWitness _ (disjointOccurrence_upper _ _) (finiteWitness_disjointOccurrence _ _)
+  have hUm : MeasurableSet U := MeasurableSet.iUnion hBm
+  have hBn (z : closedBondNeighborhood G A) : 0 ≤ twoPoint G p x z * twoPoint G p z y :=
+    mul_nonneg (twoPoint_nonneg G p x z) (twoPoint_nonneg G p z y)
+  have hBK (z : closedBondNeighborhood G A) : law p (B z) ≤ ENNReal.ofReal (twoPoint G p x z * twoPoint G p z y) := by
+    rw [←ENNReal.ofReal_toReal (measure_ne_top (law p) (B z))]
+    exact ENNReal.ofReal_le_ofReal (bk p _ _ (fun _ _ h hc => conn_mono G h hc)
+      (fun _ _ h hc => conn_mono G h hc) (finiteWitness_conn G x z) (finiteWitness_conn G z y))
+  have hU : (law p).real U ≤ ∑' z : closedBondNeighborhood G A, twoPoint G p x z * twoPoint G p z y := by
+    have hh : law p U ≤ ENNReal.ofReal (∑' z : closedBondNeighborhood G A, twoPoint G p x z * twoPoint G p z y) := by
+      calc
+        _ ≤ ∑' z, law p (B z) := measure_iUnion_le B
+        _ ≤ ∑' z : closedBondNeighborhood G A, ENNReal.ofReal (twoPoint G p x z * twoPoint G p z y) := ENNReal.tsum_le_tsum hBK
+        _ = _ := (ENNReal.ofReal_tsum_of_nonneg hBn hs).symm
+    exact (ENNReal.toReal_mono ENNReal.ofReal_ne_top hh).trans_eq (ENNReal.toReal_ofReal (tsum_nonneg hBn))
+  have hUi : Integrable (U.indicator (fun _ : Set E => (1:ℝ))) (law p) := (integrable_const _).indicator hUm
+  have hpoint (ω : Set E) : deformedConnection G P p k t D ω x y - deformedConnection G P p k t (D \ A) ω x y ≤
+      U.indicator (fun _ : Set E => (1:ℝ)) ω := by
+    have he : ω ∈ U ↔ ∃ z ∈ closedBondNeighborhood G A, ω ∈ DisjointOccurrence {η | Conn G η x z} {η | Conn G η z y} := by
+      simp [U,B]
+    simpa only [Set.indicator_apply,he] using deformedConnection_deletion_pointwise G hloc P hP p k t ht D A ω x y
+  have hi := integrable_deformedConnection G P hP p k t ht D x y
+  have hi' := integrable_deformedConnection G P hP p k t ht (D \ A) x y
+  calc
+    _ = ∫ ω, deformedConnection G P p k t D ω x y - deformedConnection G P p k t (D \ A) ω x y ∂law p := (integral_sub hi hi').symm
+    _ ≤ ∫ ω, U.indicator (fun _ : Set E => (1:ℝ)) ω ∂law p := integral_mono (hi.sub hi') hUi hpoint
+    _ = (law p).real U := integral_indicator_one hUm
+    _ ≤ _ := hU
+
+end DeletionKernelBound
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section DeformedSusceptibility
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def deformedSusceptibility (G : BondGraph V E) (P : V → V → ℝ)
+    (p : unitInterval) (k : ℕ) (t : ℝ) (D : Set V) (x : V) : ℝ :=
+  ∑' y, deformedKernel G P p k t D x y
+
+theorem deformedKernel_le_twoPoint [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval)
+    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D : Set V) (x y : V) :
+    deformedKernel G P p k t D x y ≤ twoPoint G p x y := by
+  rw [←deformedKernel_zero G P p k x y]
+  apply integral_mono (integrable_deformedConnection G P hP p k t ht D x y)
+    (integrable_deformedConnection G P hP p k 0 (le_refl _) Set.univ x y)
+  intro ω
+  exact (deformedConnection_mono G hloc P hP p k t ht (Set.subset_univ D)
+    (Set.Subset.refl _) x y).trans (sub_nonneg.mp (deformedConnection_change G P hP p k t ht Set.univ ω x y).1)
+
+theorem summable_deformedKernel [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (hp : p < pc G)
+    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D : Set V) (x : V) :
+    Summable (deformedKernel G P p k t D x) :=
+  Summable.of_nonneg_of_le (deformedKernel_nonneg G P p k t D x)
+    (deformedKernel_le_twoPoint G hloc P hP p k t ht D x)
+    (summable_twoPoint_subcritical G hloc hqt p hp x)
+
+theorem deformedSusceptibility_le [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (hp : p < pc G)
+    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D : Set V) (x : V) :
+    deformedSusceptibility G P p k t D x ≤ susceptibility G p x :=
+  (summable_deformedKernel G hloc hqt P hP p hp k t ht D x).tsum_le_tsum
+    (deformedKernel_le_twoPoint G hloc P hP p k t ht D x)
+    (summable_twoPoint_subcritical G hloc hqt p hp x)
+
+end DeformedSusceptibility
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section DeletionSusceptibilityEstimate
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Countable V] [Countable E] [Nonempty V]
+
+theorem summable_twoPoint_restricted_pair (G : BondGraph V E) (hloc : LocallyFinite G)
+    (hqt : QuasiTransitive G) (p : unitInterval) (hp : p < pc G) (x : V) (N : Set V) :
+    Summable (fun zy : N×V => twoPoint G p x zy.1 * twoPoint G p zy.1 zy.2) := by
+  apply (summable_twoPoint_pair G hloc hqt p hp x).comp_injective (i := fun zy : N×V => (zy.1.val,zy.2))
+  intro a b hab
+  exact Prod.ext (Subtype.ext (congrArg Prod.fst hab)) (congrArg (fun z : V×V => z.2) hab)
+
+theorem twoPoint_double_restricted_bound (G : BondGraph V E) (hloc : LocallyFinite G)
+    (hqt : QuasiTransitive G) (p : unitInterval) (hp : p < pc G) (x : V) (N : Set V) :
+    (∑' y : V, ∑' z : N, twoPoint G p x z * twoPoint G p z y) ≤
+      maxSusceptibility G p * (∑' z : N, twoPoint G p x z) := by
+  have hpairs := summable_twoPoint_restricted_pair G hloc hqt p hp x N
+  have hcomm : (∑' y : V, ∑' z : N, twoPoint G p x z * twoPoint G p z y) =
+      ∑' z : N, twoPoint G p x z * susceptibility G p z := by
+    calc
+      _ = ∑' z : N, ∑' y : V, twoPoint G p x z * twoPoint G p z y := hpairs.tsum_comm
+      _ = _ := by simp only [tsum_mul_left, susceptibility]
+  have hprod : Summable (fun z : N => twoPoint G p x z * susceptibility G p z) := by
+    simpa only [tsum_mul_left, susceptibility] using hpairs.prod
+  rw [hcomm]
+  calc
+    _ ≤ ∑' z : N, twoPoint G p x z * maxSusceptibility G p :=
+      hprod.tsum_le_tsum
+        (fun z => mul_le_mul_of_nonneg_left (susceptibility_le_max G hloc hqt p hp z) (twoPoint_nonneg G p x z))
+        ((summable_twoPoint_subcritical G hloc hqt p hp x).subtype N |>.mul_right _)
+    _ = _ := by rw [tsum_mul_right,mul_comm]
+
+ 
+
+theorem deformedSusceptibility_deletion (G : BondGraph V E) (hloc : LocallyFinite G)
+    (hqt : QuasiTransitive G) (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y)
+    (p : unitInterval) (hp : p < pc G) (k : ℕ) (t : ℝ) (ht : 0 ≤ t)
+    (D A : Set V) (x : V) :
+    deformedSusceptibility G P p k t D x -
+      maxSusceptibility G p * (∑' z : closedBondNeighborhood G A, twoPoint G p x z) ≤
+      deformedSusceptibility G P p k t (D \ A) x := by
+  let N := closedBondNeighborhood G A
+  have hpairs := summable_twoPoint_restricted_pair G hloc hqt p hp x N
+  have hdiag (y : V) := hpairs.prod_symm.prod_factor y
+  have hi := summable_deformedKernel G hloc hqt P hP p hp k t ht D x
+  have hi' := summable_deformedKernel G hloc hqt P hP p hp k t ht (D \ A) x
+  have hb := (hi.sub hi').tsum_le_tsum
+    (fun y => deformedKernel_deletion_bound G hloc P hP p k t ht D A x y (hdiag y)) hpairs.prod_symm.prod
+  rw [hi.tsum_sub hi'] at hb
+  change deformedSusceptibility G P p k t D x - deformedSusceptibility G P p k t (D \ A) x ≤ _ at hb
+  have herr := twoPoint_double_restricted_bound G hloc hqt p hp x N
+  linarith
+
+end DeletionSusceptibilityEstimate
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section ExteriorCorridorGeometry
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem conn_of_open_incident (G : BondGraph V E) (ω : Set E) (e : E)
+    (he : e ∈ ω) (a b : V) (ha : a ∈ G e) (hb : b ∈ G e) : Conn G ω a b := by
+  obtain ⟨c,d,hcd⟩ : ∃ c d, G e = s(c,d) := by
+    obtain ⟨⟨c,d⟩,hh⟩ := Quot.exists_rep (G e)
+    exact ⟨c,d,hh.symm⟩
+  rw [hcd,Sym2.mem_iff] at ha hb
+  rcases ha with rfl | rfl <;> rcases hb with rfl | rfl
+  · exact conn_refl G ω _
+  · exact .single ⟨e,he,hcd⟩
+  · exact conn_symm G ω (.single ⟨e,he,hcd⟩)
+  · exact conn_refl G ω _
+
+ 
+
+theorem conn_restrict_away_from_cluster (G : BondGraph V E) (ω S : Set E)
+    (hS : S ⊆ ω) (u x y : V) (hx : x ∉ cluster G ω u) (hc : Conn G S x y) :
+    Conn G (S ∩ internalEdges G (cluster G ω u)ᶜ) x y := by
+  apply conn_restrict_of_no_simpleVisit G S (cluster G ω u) x y hc
+  rintro ⟨q,_,z,hz,hzu⟩
+  have hxz : Conn G ω x z := conn_mono G hS
+    ((conn_iff_reachable G S x z).mpr ⟨q.takeUntil z hz⟩)
+  exact hx (hzu.trans (conn_symm G ω hxz))
+
+theorem conn_delete_cluster (G : BondGraph V E) (D : Set V) (ω : Set E)
+    (u x y : V) (hx : x ∉ cluster G (ω ∩ internalEdges G D) u)
+    (S : Set E) (hS : S ⊆ ω ∩ internalEdges G D) (hc : Conn G S x y) :
+    Conn G (S ∩ internalEdges G (D \ cluster G (ω ∩ internalEdges G D) u)) x y := by
+  apply conn_mono G ?_ (conn_restrict_away_from_cluster G _ S hS u x y hx hc)
+  rintro e ⟨he,heA⟩
+  exact ⟨he,fun z hz => ⟨(hS he).2 z hz,heA z hz⟩⟩
+
+namespace RawCorridor
+variable {G : BondGraph V E} {k : ℕ}
+
+ 
+
+theorem counted_delete_other_cluster (w : RawCorridor G k) (D : Set V) (ω : Set E)
+    (u x y : V) (hx : x ∉ cluster G (ω ∩ internalEdges G D) u) :
+    w.counted (D \ cluster G (ω ∩ internalEdges G D) u) ω x y ↔ w.counted D ω x y := by
+  let A := cluster G (ω ∩ internalEdges G D) u
+  have hs : ω ∩ internalEdges G (D \ A) ⊆ ω ∩ internalEdges G D :=
+    Set.inter_subset_inter_right _ (internalEdges_mono G Set.sdiff_subset)
+  constructor
+  · intro hc
+    apply (w.counted_iff D ω x y).mpr
+    refine ⟨hc.1,conn_mono G hs hc.2.2.1,?_,?_⟩
+    · intro e he
+      refine ⟨hs (hc.2.2.2.1 e he).1,?_⟩
+      intro hbad
+      have hd := conn_delete_cluster G D ω u x y hx
+        ((ω ∩ internalEdges G D) \ {e}) Set.sdiff_subset hbad
+      apply (hc.2.2.2.1 e he).2
+      apply conn_mono G ?_ hd
+      rintro f ⟨⟨⟨hfω,hfD⟩,hfne⟩,hfnew⟩
+      exact ⟨⟨hfω,hfnew⟩,hfne⟩
+    · apply Set.disjoint_left.mpr
+      intro e he ho
+      obtain ⟨z,hzi,hze⟩ := he.1
+      obtain ⟨q⟩ := (conn_iff_reachable G _ x y).mp hc.2.2.1
+      have hz : z ∈ q.support := w.counted_support_subset_walk (D \ A) ω x y hc q z
+        (List.mem_toFinset.mp (Finset.mem_sdiff.mp hzi).1)
+      have hxz : Conn G (ω ∩ internalEdges G D) x z := conn_mono G hs
+        ((conn_iff_reachable G _ x z).mpr ⟨q.takeUntil z hz⟩)
+      have heA : e ∈ internalEdges G (D \ A) := by
+        intro a ha
+        refine ⟨ho.2 a ha,?_⟩
+        intro hau
+        have hza := conn_of_open_incident G _ e ho z a hze ha
+        exact hx (hau.trans (conn_symm G _ (hxz.trans hza)))
+      exact Set.disjoint_left.mp hc.2.2.2.2 he ⟨ho.1,heA⟩
+  · intro hc
+    have hnew := conn_delete_cluster G D ω u x y hx _ (Set.Subset.refl _) hc.2.2.1
+    have hc' : Conn G (ω ∩ internalEdges G (D \ A)) x y := by
+      apply conn_mono G ?_ hnew
+      rintro e ⟨⟨heω,_⟩,heA⟩
+      exact ⟨heω,heA⟩
+    exact (w.counted_iff _ ω x y).mpr ⟨hc.1,
+      corridorCounted_restrict G _ _ Set.sdiff_subset (Set.Subset.refl _) x y hc' hc.2.2⟩
+
+end RawCorridor
+end ExteriorCorridorGeometry
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section ExteriorWeightedConnections
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem corridorPenalty_delete_other_cluster (G : BondGraph V E) (P : V → V → ℝ)
+    (p : ℝ) (k : ℕ) (D : Set V) (ω : Set E) (u x y : V)
+    (hx : x ∉ cluster G (ω ∩ internalEdges G D) u) :
+    corridorPenalty G P p k (D \ cluster G (ω ∩ internalEdges G D) u) ω x y =
+      corridorPenalty G P p k D ω x y := by
+  unfold corridorPenalty
+  apply tsum_congr
+  intro w
+  simp only [corridorPenaltyTerm,w.counted_delete_other_cluster D ω u x y hx]
+
+ 
+
+theorem deformedConnection_delete_other_cluster (G : BondGraph V E) (P : V → V → ℝ)
+    (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (u x y : V)
+    (hx : x ∉ cluster G (ω ∩ internalEdges G D) u) :
+    deformedConnection G P p k t (D \ cluster G (ω ∩ internalEdges G D) u) ω x y =
+      deformedConnection G P p k t D ω x y := by
+  let A := cluster G (ω ∩ internalEdges G D) u
+  have hs : ω ∩ internalEdges G (D \ A) ⊆ ω ∩ internalEdges G D :=
+    Set.inter_subset_inter_right _ (internalEdges_mono G Set.sdiff_subset)
+  have hg : (x ∈ D \ A ∧ y ∈ D \ A ∧ Conn G (ω ∩ internalEdges G (D \ A)) x y) ↔
+      (x ∈ D ∧ y ∈ D ∧ Conn G (ω ∩ internalEdges G D) x y) := by
+    constructor
+    · exact fun h => ⟨h.1.1,h.2.1.1,conn_mono G hs h.2.2⟩
+    · rintro ⟨hxD,hyD,hc⟩
+      have hy : y ∉ A := fun hy => hx (hy.trans (conn_symm G _ hc))
+      refine ⟨⟨hxD,hx⟩,⟨hyD,hy⟩,?_⟩
+      apply conn_mono G ?_ (conn_delete_cluster G D ω u x y hx _ (Set.Subset.refl _) hc)
+      rintro e ⟨⟨heω,_⟩,heA⟩
+      exact ⟨heω,heA⟩
+  dsimp only [A] at hg
+  simp only [deformedConnection,hg,corridorPenalty_delete_other_cluster G P p k D ω u x y hx]
+
+noncomputable def deformedClusterMass (G : BondGraph V E) (P : V → V → ℝ)
+    (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x : V) : ℝ :=
+  ∑' y, deformedConnection G P p k t D ω x y
+
+theorem deformedClusterMass_delete_other_cluster (G : BondGraph V E) (P : V → V → ℝ)
+    (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (u x : V)
+    (hx : x ∉ cluster G (ω ∩ internalEdges G D) u) :
+    deformedClusterMass G P p k t (D \ cluster G (ω ∩ internalEdges G D) u) ω x =
+      deformedClusterMass G P p k t D ω x := by
+  exact tsum_congr (fun y => deformedConnection_delete_other_cluster G P p k t D ω u x y hx)
+
+theorem deformedConnection_le_conn (G : BondGraph V E) (P : V → V → ℝ)
+    (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (k : ℕ) (t : ℝ) (ht : 0 ≤ t)
+    (D : Set V) (ω : Set E) (x y : V) : deformedConnection G P p k t D ω x y ≤
+      if Conn G ω x y then 1 else 0 := by
+  by_cases hc : Conn G ω x y
+  · simpa only [hc,ite_true] using deformedConnection_le_one G P hP p k t ht D ω x y
+  · have hn : ¬Conn G (ω ∩ internalEdges G D) x y := fun h => hc (conn_mono G Set.inter_subset_left h)
+    simp only [deformedConnection,hn,hc,and_false,ite_false,le_refl]
+
+theorem measurable_deformedClusterMass [Countable V] [Countable E] (G : BondGraph V E)
+    (P : V → V → ℝ) (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (x : V) :
+    Measurable (fun ω => deformedClusterMass G P p k t D ω x) :=
+  Measurable.tsum (fun y => measurable_deformedConnection G P p k t D x y)
+
+end ExteriorWeightedConnections
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section DeformedMassIntegrals
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem deformedConnection_finite_support (G : BondGraph V E) (P : V → V → ℝ)
+    (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x : V)
+    (hf : (cluster G ω x).Finite) :
+    (deformedConnection G P p k t D ω x).HasFiniteSupport := by
+  apply hf.subset
+  intro y hy
+  by_contra hn
+  have hnc : ¬Conn G (ω ∩ internalEdges G D) x y := fun h => hn (conn_mono G Set.inter_subset_left h)
+  exact hy (by simp only [deformedConnection,hnc,and_false,ite_false])
+
+theorem deformedClusterMass_nonneg (G : BondGraph V E) (P : V → V → ℝ)
+    (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x : V) :
+    0 ≤ deformedClusterMass G P p k t D ω x :=
+  tsum_nonneg (deformedConnection_nonneg G P p k t D ω x)
+
+theorem deformedClusterMass_le_ncard (G : BondGraph V E) (P : V → V → ℝ)
+    (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (k : ℕ) (t : ℝ) (ht : 0 ≤ t)
+    (D : Set V) (ω : Set E) (x : V) (hf : (cluster G ω x).Finite) :
+    deformedClusterMass G P p k t D ω x ≤ ((cluster G ω x).ncard:ℝ) := by
+  unfold deformedClusterMass
+  rw [tsum_eq_sum (s := hf.toFinset) (fun y hy => by
+    have hn : ¬Conn G ω x y := fun hc => hy (hf.mem_toFinset.mpr hc)
+    have hn' : ¬Conn G (ω ∩ internalEdges G D) x y := fun hc => hn (conn_mono G Set.inter_subset_left hc)
+    simp only [deformedConnection,hn',and_false,ite_false])]
+  calc
+    _ ≤ ∑ _y ∈ hf.toFinset, (1:ℝ) := Finset.sum_le_sum (fun y _ => deformedConnection_le_one G P hP p k t ht D ω x y)
+    _ = _ := by simp only [Finset.sum_const,nsmul_eq_mul,mul_one,←Set.ncard_eq_toFinset_card _ hf]
+
+theorem integrable_deformedClusterMass [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (hp : p < pc G)
+    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D : Set V) (x : V) :
+    Integrable (fun ω => deformedClusterMass G P p k t D ω x) (law p) := by
+  apply Integrable.mono' (susceptibility_eq_integral G hloc hqt p hp x).1
+    (measurable_deformedClusterMass G P p k t D x).aestronglyMeasurable
+  filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
+  rw [Real.norm_of_nonneg (deformedClusterMass_nonneg G P p k t D ω x)]
+  exact deformedClusterMass_le_ncard G P hP p k t ht D ω x hω
+
+ 
+
+theorem deformedSusceptibility_eq_integral [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (hp : p < pc G)
+    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D : Set V) (x : V) :
+    deformedSusceptibility G P p k t D x = ∫ ω, deformedClusterMass G P p k t D ω x ∂law p := by
+  have hi := integrable_deformedConnection G P hP p k t ht D x
+  have hs : Summable (fun y => ∫ ω, ‖deformedConnection G P p k t D ω x y‖ ∂law p) := by
+    simp only [Real.norm_of_nonneg (deformedConnection_nonneg G P p k t D _ x _)]
+    exact summable_deformedKernel G hloc hqt P hP p hp k t ht D x
+  exact integral_tsum_of_summable_integral_norm hi hs
+
+end DeformedMassIntegrals
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section ExactComponentRestriction
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+def OpenClosed (G : BondGraph V E) (ω : Set E) (A : Set V) : Prop :=
+  ∀ e ∈ ω, ∀ z ∈ G e, z ∈ A → ∀ a ∈ G e, a ∈ A
+
+theorem openClosed_cluster (G : BondGraph V E) (ω : Set E) (x : V) :
+    OpenClosed G ω (cluster G ω x) := by
+  intro e he z hz hx a ha
+  exact hx.trans (conn_of_open_incident G ω e he z a hz ha)
+
+theorem conn_restrict_openClosed (G : BondGraph V E) (ω S : Set E) (hS : S ⊆ ω)
+    (A : Set V) (hA : OpenClosed G ω A) (x y : V) (hx : x ∈ A) (hc : Conn G S x y) :
+    y ∈ A ∧ Conn G (S ∩ internalEdges G A) x y := by
+  induction hc with
+  | refl => exact ⟨hx,conn_refl G _ x⟩
+  | @tail a b hxa hab ih =>
+    obtain ⟨e,he,heq⟩ := hab
+    have ha : a ∈ G e := by simp [heq]
+    have hb : b ∈ G e := by simp [heq]
+    have hei : e ∈ internalEdges G A := hA e (hS he) a ha ih.1
+    exact ⟨hei b hb,ih.2.tail ⟨e,⟨he,hei⟩,heq⟩⟩
+
+namespace RawCorridor
+variable {G : BondGraph V E} {k : ℕ}
+
+ 
+
+theorem counted_restrict_openClosed (w : RawCorridor G k) (D A : Set V) (ω : Set E)
+    (hA : OpenClosed G (ω ∩ internalEdges G D) A) (x y : V) (hx : x ∈ A) :
+    w.counted (D ∩ A) ω x y ↔ w.counted D ω x y := by
+  have hs : ω ∩ internalEdges G (D ∩ A) ⊆ ω ∩ internalEdges G D :=
+    Set.inter_subset_inter_right _ (internalEdges_mono G Set.inter_subset_left)
+  have hrest (S : Set E) (hS : S ⊆ ω ∩ internalEdges G D) (hc : Conn G S x y) :
+      Conn G (S ∩ internalEdges G (D ∩ A)) x y := by
+    apply conn_mono G ?_ (conn_restrict_openClosed G _ S hS A hA x y hx hc).2
+    rintro e ⟨he,heA⟩
+    exact ⟨he,fun z hz => ⟨(hS he).2 z hz,heA z hz⟩⟩
+  constructor
+  · intro hc
+    apply (w.counted_iff D ω x y).mpr
+    refine ⟨hc.1,conn_mono G hs hc.2.2.1,?_,?_⟩
+    · intro e he
+      refine ⟨hs (hc.2.2.2.1 e he).1,?_⟩
+      intro hbad
+      apply (hc.2.2.2.1 e he).2
+      apply conn_mono G ?_ (hrest _ Set.sdiff_subset hbad)
+      rintro f ⟨⟨⟨hfω,_⟩,hfne⟩,hfnew⟩
+      exact ⟨⟨hfω,hfnew⟩,hfne⟩
+    · apply Set.disjoint_left.mpr
+      intro e he ho
+      obtain ⟨z,hzi,hze⟩ := he.1
+      have hz : z ∈ A := (hc.2.1 z (List.mem_toFinset.mp (Finset.mem_sdiff.mp hzi).1)).2
+      have heA : e ∈ internalEdges G (D ∩ A) := fun a ha => ⟨ho.2 a ha,hA e ho z hze hz a ha⟩
+      exact Set.disjoint_left.mp hc.2.2.2.2 he ⟨ho.1,heA⟩
+  · intro hc
+    have hc' : Conn G (ω ∩ internalEdges G (D ∩ A)) x y := by
+      apply conn_mono G ?_ (hrest _ (Set.Subset.refl _) hc.2.2.1)
+      rintro e ⟨⟨heω,_⟩,heA⟩
+      exact ⟨heω,heA⟩
+    exact (w.counted_iff _ ω x y).mpr ⟨hc.1,
+      corridorCounted_restrict G _ _ Set.inter_subset_left (Set.Subset.refl _) x y hc' hc.2.2⟩
+
+end RawCorridor
+end ExactComponentRestriction
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section InternalWeightedRestriction
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem corridorPenalty_restrict_openClosed (G : BondGraph V E) (P : V → V → ℝ)
+    (p : ℝ) (k : ℕ) (D A : Set V) (ω : Set E)
+    (hA : OpenClosed G (ω ∩ internalEdges G D) A) (x y : V) (hx : x ∈ A) :
+    corridorPenalty G P p k (D ∩ A) ω x y = corridorPenalty G P p k D ω x y := by
+  unfold corridorPenalty
+  apply tsum_congr
+  intro w
+  simp only [corridorPenaltyTerm,w.counted_restrict_openClosed D A ω hA x y hx]
+
+theorem deformedConnection_restrict_openClosed (G : BondGraph V E) (P : V → V → ℝ)
+    (p : ℝ) (k : ℕ) (t : ℝ) (D A : Set V) (ω : Set E)
+    (hA : OpenClosed G (ω ∩ internalEdges G D) A) (x y : V) (hx : x ∈ A) :
+    deformedConnection G P p k t (D ∩ A) ω x y = deformedConnection G P p k t D ω x y := by
+  have hg : (x ∈ D ∩ A ∧ y ∈ D ∩ A ∧ Conn G (ω ∩ internalEdges G (D ∩ A)) x y) ↔
+      (x ∈ D ∧ y ∈ D ∧ Conn G (ω ∩ internalEdges G D) x y) := by
+    constructor
+    · exact fun h => ⟨h.1.1,h.2.1.1,conn_mono G
+        (Set.inter_subset_inter_right _ (internalEdges_mono G Set.inter_subset_left)) h.2.2⟩
+    · rintro ⟨hxD,hyD,hc⟩
+      have hh := conn_restrict_openClosed G _ _ (Set.Subset.refl _) A hA x y hx hc
+      refine ⟨⟨hxD,hx⟩,⟨hyD,hh.1⟩,conn_mono G ?_ hh.2⟩
+      rintro e ⟨⟨heω,heD⟩,heA⟩
+      exact ⟨heω,fun z hz => ⟨heD z hz,heA z hz⟩⟩
+  simp only [deformedConnection,hg,corridorPenalty_restrict_openClosed G P p k D A ω hA x y hx]
+
+theorem deformedClusterMass_restrict_openClosed (G : BondGraph V E) (P : V → V → ℝ)
+    (p : ℝ) (k : ℕ) (t : ℝ) (D A : Set V) (ω : Set E)
+    (hA : OpenClosed G (ω ∩ internalEdges G D) A) (x : V) (hx : x ∈ A) :
+    deformedClusterMass G P p k t (D ∩ A) ω x = deformedClusterMass G P p k t D ω x :=
+  tsum_congr (fun y => deformedConnection_restrict_openClosed G P p k t D A ω hA x y hx)
+
+ 
+theorem deformedClusterMass_restrict_config (G : BondGraph V E) (P : V → V → ℝ)
+    (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x : V) (S : Set E)
+    (hS : internalEdges G D ⊆ S) :
+    deformedClusterMass G P p k t D (ω ∩ S) x = deformedClusterMass G P p k t D ω x := by
+  have he : (ω ∩ S) ∩ internalEdges G D = ω ∩ internalEdges G D := by
+    ext e
+    exact ⟨fun h => ⟨h.1.1,h.2⟩,fun h => ⟨⟨h.1,hS h.2⟩,h.2⟩⟩
+  apply tsum_congr
+  intro y
+  have hc (w : RawCorridor G k) : w.counted D (ω ∩ S) x y ↔ w.counted D ω x y := by
+    simp only [RawCorridor.counted,corridorCounted,he]
+  simp only [deformedConnection,corridorPenalty,corridorPenaltyTerm,hc,he]
+
+end InternalWeightedRestriction
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory ProbabilityTheory
+section DisjointCoordinateObservables
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem restriction_functions_independent (p : unitInterval) (S T : Set E)
+    (hST : Disjoint S T) (f g : Set E → ℝ) (hf : Measurable f) (hg : Measurable g)
+    (hfS : ∀ ω, f ω = f (ω ∩ S)) (hgT : ∀ ω, g ω = g (ω ∩ T)) :
+    IndepFun f g (law p) := by
+  rw [indepFun_iff_measure_inter_preimage_eq_mul]
+  intro A B hA hB
+  have he (h : Set E → ℝ) (U : Set E) (hU : ∀ ω, h ω = h (ω ∩ U)) (C : Set ℝ) :
+      (fun ω => ω ∩ U) ⁻¹' (h ⁻¹' C) = h ⁻¹' C := by
+    ext ω
+    simp only [Set.mem_preimage,←hU ω]
+  simpa only [he f S hfS A,he g T hgT B] using
+    restriction_events_independent p S T hST (f ⁻¹' A) (g ⁻¹' B)
+      (hA.preimage hf) (hB.preimage hg)
+
+theorem integral_mul_disjoint_coordinates (p : unitInterval) (S T : Set E)
+    (hST : Disjoint S T) (f g : Set E → ℝ) (hf : Measurable f) (hg : Measurable g)
+    (hfS : ∀ ω, f ω = f (ω ∩ S)) (hgT : ∀ ω, g ω = g (ω ∩ T)) :
+    (∫ ω, f ω*g ω ∂law p) = (∫ ω, f ω ∂law p)*(∫ ω, g ω ∂law p) :=
+  (restriction_functions_independent p S T hST f g hf hg hfS hgT).integral_mul_eq_mul_integral
+    hf.aestronglyMeasurable hg.aestronglyMeasurable
+
+end DisjointCoordinateObservables
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section ExactWeightedClusterCell
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem induced_cluster_restrict_incident_iff (G : BondGraph V E) (hloc : LocallyFinite G)
+    (W : Finset V) (D : Set V) (ω : Set E) (x : V) :
+    cluster G ((ω ∩ (incidentSupport G hloc W:Set E)) ∩ internalEdges G D) x = (W:Set V) ↔
+      cluster G (ω ∩ internalEdges G D) x = (W:Set V) := by
+  rw [Set.inter_right_comm]
+  exact cluster_restrict_incident_eq_iff G hloc W _ (Finset.Subset.refl _) _ x
+
+theorem weighted_cluster_cell_local (G : BondGraph V E) (hloc : LocallyFinite G)
+    (P : V → V → ℝ) (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (W : Finset V) (x : V)
+    (ω : Set E) :
+    (if cluster G (ω ∩ internalEdges G D) x = (W:Set V)
+      then deformedClusterMass G P p k t D ω x else 0) =
+    (if cluster G ((ω ∩ (incidentSupport G hloc W:Set E)) ∩ internalEdges G D) x = (W:Set V)
+      then deformedClusterMass G P p k t D (ω ∩ (incidentSupport G hloc W:Set E)) x else 0) := by
+  simp only [induced_cluster_restrict_incident_iff]
+  split_ifs with hW
+  · have hx : x ∈ (W:Set V) := hW ▸ conn_refl G _ x
+    have hA : OpenClosed G (ω ∩ internalEdges G D) (W:Set V) := hW ▸ openClosed_cluster G _ x
+    have hW' := (induced_cluster_restrict_incident_iff G hloc W D ω x).mpr hW
+    have hA' : OpenClosed G ((ω ∩ (incidentSupport G hloc W:Set E)) ∩ internalEdges G D) (W:Set V) :=
+      hW' ▸ openClosed_cluster G _ x
+    rw [←deformedClusterMass_restrict_openClosed G P p k t D W ω hA x hx,
+      ←deformedClusterMass_restrict_openClosed G P p k t D W _ hA' x hx]
+    symm
+    apply deformedClusterMass_restrict_config
+    intro e he
+    exact (mem_incidentSupport G hloc W e).mpr
+      ⟨(G e).out.1,(he _ (Sym2.out_fst_mem _)).2,Sym2.out_fst_mem _⟩
+  · rfl
+
+ 
+
+theorem weighted_cluster_cell_exposure [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (hp : p < pc G)
+    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D : Set V) (W : Finset V) (x y : V) :
+    (∫ ω, (if cluster G (ω ∩ internalEdges G D) x = (W:Set V)
+      then deformedClusterMass G P p k t D ω x else 0) *
+        deformedClusterMass G P p k t (D \ (W:Set V)) ω y ∂law p) =
+    (∫ ω, (if cluster G (ω ∩ internalEdges G D) x = (W:Set V)
+      then deformedClusterMass G P p k t D ω x else 0) ∂law p) *
+        deformedSusceptibility G P p k t (D \ (W:Set V)) y := by
+  rw [deformedSusceptibility_eq_integral G hloc hqt P hP p hp k t ht]
+  apply integral_mul_disjoint_coordinates p (incidentSupport G hloc W:Set E)
+    (incidentSupport G hloc W:Set E)ᶜ disjoint_compl_right
+  · exact Measurable.ite ((measurableSet_cluster_eq G x W).preimage (measurable_inter_const _))
+      (measurable_deformedClusterMass G P p k t D x) measurable_const
+  · exact measurable_deformedClusterMass G P p k t _ y
+  · exact weighted_cluster_cell_local G hloc P p k t D W x
+  · intro ω
+    symm
+    apply deformedClusterMass_restrict_config
+    rw [complement_incidentSupport]
+    exact fun e he z hz => (he z hz).2
+
+end ExactWeightedClusterCell
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section InducedClusterPartition
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem integral_induced_cluster_finite_partition [Countable V] [Countable E]
+    (G : BondGraph V E) (p : unitInterval) (hp : p < pc G) (D : Set V) (x : V)
+    (f : Set E → ℝ) (hf : Integrable f (law p)) :
+    (∫ ω, f ω ∂law p) = ∑' W : Finset V,
+      ∫ ω, (if cluster G (ω ∩ internalEdges G D) x = (W:Set V) then f ω else 0) ∂law p := by
+  let F (W : Finset V) (ω : Set E) := if cluster G (ω ∩ internalEdges G D) x = (W:Set V) then f ω else 0
+  have hi (W : Finset V) : Integrable (F W) (law p) := by
+    have he : F W = {ω : Set E | cluster G (ω ∩ internalEdges G D) x = (W:Set V)}.indicator f := by
+      funext ω
+      simp only [F,Set.indicator,Set.mem_ofPred_eq]
+      split_ifs <;> rfl
+    rw [he]
+    exact hf.indicator ((measurableSet_cluster_eq G x W).preimage (measurable_inter_const _))
+  have hb (S : Finset (Finset V)) (ω : Set E) : (∑ W ∈ S, ‖F W ω‖) ≤ ‖f ω‖ := by
+    by_cases hex : ∃ W ∈ S, cluster G (ω ∩ internalEdges G D) x = (W:Set V)
+    · obtain ⟨W,hWS,hW⟩ := hex
+      rw [Finset.sum_eq_single W]
+      · simp only [F,hW,ite_true,le_refl]
+      · intro U _ hU
+        have hu : cluster G (ω ∩ internalEdges G D) x ≠ (U:Set V) := fun h => hU (Finset.coe_injective (h.symm.trans hW))
+        simp only [F,ite_eq_right hu,norm_zero]
+      · exact fun h => (h hWS).elim
+    · have he (W : Finset V) (hW : W ∈ S) : cluster G (ω ∩ internalEdges G D) x ≠ (W:Set V) := fun h => hex ⟨W,hW,h⟩
+      calc
+        _ = 0 := Finset.sum_eq_zero (fun W hW => by simp only [F,ite_eq_right (he W hW),norm_zero])
+        _ ≤ _ := norm_nonneg _
+  have hn : Summable (fun W => ∫ ω, ‖F W ω‖ ∂law p) := by
+    apply summable_of_sum_le (fun _ => integral_nonneg (fun _ => norm_nonneg _))
+      (c := ∫ ω, ‖f ω‖ ∂law p)
+    intro S
+    rw [←integral_finsetSum _ (fun W _ => (hi W).norm)]
+    exact integral_mono (integrable_finsetSum _ (fun W _ => (hi W).norm)) hf.norm (hb S)
+  calc
+    _ = ∫ ω, ∑' W : Finset V, F W ω ∂law p := by
+      apply integral_congr_ae
+      filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
+      exact (tsum_cluster_cells G (ω ∩ internalEdges G D) x
+        (hω.subset (fun _ h => conn_mono G Set.inter_subset_left h)) (f ω)).symm
+    _ = _ := (integral_tsum_of_summable_integral_norm hi hn).symm
+
+end InducedClusterPartition
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section DeformedClusterProducts
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem deformedClusterMass_eq_zero_of_not_mem (G : BondGraph V E) (P : V → V → ℝ)
+    (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x : V) (hx : x ∉ D) :
+    deformedClusterMass G P p k t D ω x = 0 := by
+  simp only [deformedClusterMass,deformedConnection,hx,false_and,ite_false,tsum_zero]
+
+theorem deformed_distinct_product_eq_exterior (G : BondGraph V E) (P : V → V → ℝ)
+    (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x y : V) :
+    (if ¬Conn G (ω ∩ internalEdges G D) x y then
+      deformedClusterMass G P p k t D ω x * deformedClusterMass G P p k t D ω y else 0) =
+    deformedClusterMass G P p k t D ω x *
+      deformedClusterMass G P p k t (D \ cluster G (ω ∩ internalEdges G D) x) ω y := by
+  by_cases hc : Conn G (ω ∩ internalEdges G D) x y
+  · rw [ite_eq_right (not_not_intro hc),deformedClusterMass_eq_zero_of_not_mem G P p k t _ ω y
+      (fun h => h.2 hc),mul_zero]
+  · rw [ite_eq_left hc,deformedClusterMass_delete_other_cluster G P p k t D ω x y hc]
+
+theorem integrable_deformed_product [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (hp : p < pc G)
+    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D : Set V) (x y : V) :
+    Integrable (fun ω => deformedClusterMass G P p k t D ω x *
+      deformedClusterMass G P p k t D ω y) (law p) := by
+  apply Integrable.mono' (subcritical_cluster_association G hloc hqt p hp x y).1
+    ((measurable_deformedClusterMass G P p k t D x).mul
+      (measurable_deformedClusterMass G P p k t D y)).aestronglyMeasurable
+  filter_upwards [subcritical_cluster_finite_ae G p hp x,subcritical_cluster_finite_ae G p hp y] with ω hx hy
+  dsimp only [Pi.mul_apply]
+  rw [Real.norm_of_nonneg (mul_nonneg (deformedClusterMass_nonneg G P p k t D ω x)
+    (deformedClusterMass_nonneg G P p k t D ω y))]
+  exact mul_le_mul (deformedClusterMass_le_ncard G P hP p k t ht D ω x hx)
+    (deformedClusterMass_le_ncard G P hP p k t ht D ω y hy)
+    (deformedClusterMass_nonneg G P p k t D ω y) (Nat.cast_nonneg _)
+
+theorem integrable_distinct_deformed_product [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (hp : p < pc G)
+    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D : Set V) (x y : V) :
+    Integrable (fun ω => if ¬Conn G (ω ∩ internalEdges G D) x y then
+      deformedClusterMass G P p k t D ω x * deformedClusterMass G P p k t D ω y else 0) (law p) := by
+  have he : (fun ω => if ¬Conn G (ω ∩ internalEdges G D) x y then
+      deformedClusterMass G P p k t D ω x * deformedClusterMass G P p k t D ω y else 0) =
+      {ω | Conn G (ω ∩ internalEdges G D) x y}ᶜ.indicator
+        (fun ω => deformedClusterMass G P p k t D ω x * deformedClusterMass G P p k t D ω y) := by
+    funext ω
+    simp only [Set.indicator,Set.mem_compl_iff,Set.mem_ofPred_eq]
+  rw [he]
+  exact (integrable_deformed_product G hloc hqt P hP p hp k t ht D x y).indicator
+    (((measurableSet_conn G x y).preimage (measurable_inter_const (internalEdges G D))).compl)
+
+end DeformedClusterProducts
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section FreshWeightedMass
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def deformedFreshExteriorMoment (G : BondGraph V E) (P : V → V → ℝ)
+    (p : unitInterval) (k : ℕ) (t : ℝ) (D : Set V) (x y : V) (ω : Set E) : ℝ :=
+  deformedClusterMass G P p k t D ω x *
+    deformedSusceptibility G P p k t (D \ cluster G (ω ∩ internalEdges G D) x) y
+
+theorem deformedSusceptibility_nonneg (G : BondGraph V E) (P : V → V → ℝ)
+    (p : unitInterval) (k : ℕ) (t : ℝ) (D : Set V) (x : V) :
+    0 ≤ deformedSusceptibility G P p k t D x :=
+  tsum_nonneg (deformedKernel_nonneg G P p k t D x)
+
+theorem deformedFreshExteriorMoment_eq_tsum (G : BondGraph V E) (P : V → V → ℝ)
+    (p : unitInterval) (k : ℕ) (t : ℝ) (D : Set V) (x y : V) (ω : Set E)
+    (hf : (cluster G (ω ∩ internalEdges G D) x).Finite) :
+    deformedFreshExteriorMoment G P p k t D x y ω = ∑' W : Finset V,
+      if cluster G (ω ∩ internalEdges G D) x = (W:Set V) then
+        deformedClusterMass G P p k t D ω x * deformedSusceptibility G P p k t (D \ (W:Set V)) y else 0 := by
+  rw [←tsum_cluster_cells G (ω ∩ internalEdges G D) x hf (deformedFreshExteriorMoment G P p k t D x y ω)]
+  apply tsum_congr
+  intro W
+  split_ifs with hW
+  · simp only [deformedFreshExteriorMoment,hW]
+  · rfl
+
+theorem integrable_deformedFreshExteriorMoment [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (hp : p < pc G)
+    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D : Set V) (x y : V) :
+    Integrable (deformedFreshExteriorMoment G P p k t D x y) (law p) := by
+  have hm : AEStronglyMeasurable (deformedFreshExteriorMoment G P p k t D x y) (law p) := by
+    have hh : Measurable (fun ω => ∑' W : Finset V,
+        if cluster G (ω ∩ internalEdges G D) x = (W:Set V) then
+          deformedClusterMass G P p k t D ω x * deformedSusceptibility G P p k t (D \ (W:Set V)) y else 0) := by
+      apply Measurable.tsum
+      intro W
+      exact Measurable.ite ((measurableSet_cluster_eq G x W).preimage (measurable_inter_const _))
+        ((measurable_deformedClusterMass G P p k t D x).mul_const _) measurable_const
+    apply hh.aestronglyMeasurable.congr
+    filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
+    exact (deformedFreshExteriorMoment_eq_tsum G P p k t D x y ω
+      (hω.subset (fun _ h => conn_mono G Set.inter_subset_left h))).symm
+  apply Integrable.mono' ((integrable_deformedClusterMass G hloc hqt P hP p hp k t ht D x).mul_const
+    (susceptibility G p y)) hm
+  exact Filter.Eventually.of_forall (fun ω => by
+    dsimp only [deformedFreshExteriorMoment]
+    rw [Real.norm_of_nonneg (mul_nonneg (deformedClusterMass_nonneg G P p k t D ω x)
+      (deformedSusceptibility_nonneg G P p k t _ y))]
+    exact mul_le_mul_of_nonneg_left (deformedSusceptibility_le G hloc hqt P hP p hp k t ht _ y)
+      (deformedClusterMass_nonneg G P p k t D ω x))
+
+end FreshWeightedMass
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section WeightedExposureIdentity
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem integral_deformedFreshExteriorMoment [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (hp : p < pc G)
+    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D : Set V) (x y : V) :
+    (∫ ω, deformedFreshExteriorMoment G P p k t D x y ω ∂law p) =
+      ∫ ω, (if ¬Conn G (ω ∩ internalEdges G D) x y then
+        deformedClusterMass G P p k t D ω x * deformedClusterMass G P p k t D ω y else 0) ∂law p := by
+  rw [integral_induced_cluster_finite_partition G p hp D x _
+      (integrable_deformedFreshExteriorMoment G hloc hqt P hP p hp k t ht D x y),
+    integral_induced_cluster_finite_partition G p hp D x _
+      (integrable_distinct_deformed_product G hloc hqt P hP p hp k t ht D x y)]
+  apply tsum_congr
+  intro W
+  have hl : (∫ ω, (if cluster G (ω ∩ internalEdges G D) x = (W:Set V) then
+      deformedFreshExteriorMoment G P p k t D x y ω else 0) ∂law p) =
+      (∫ ω, (if cluster G (ω ∩ internalEdges G D) x = (W:Set V) then
+        deformedClusterMass G P p k t D ω x else 0) ∂law p) *
+      deformedSusceptibility G P p k t (D \ (W:Set V)) y := by
+    rw [←integral_mul_const]
+    apply integral_congr_ae
+    exact Filter.Eventually.of_forall (fun ω => by
+      by_cases hW : cluster G (ω ∩ internalEdges G D) x = (W:Set V)
+      · simp only [hW,ite_true,deformedFreshExteriorMoment]
+      · simp only [hW,ite_false,zero_mul])
+  rw [hl,←weighted_cluster_cell_exposure G hloc hqt P hP p hp k t ht D W x y]
+  apply integral_congr_ae
+  exact Filter.Eventually.of_forall (fun ω => by
+    by_cases hW : cluster G (ω ∩ internalEdges G D) x = (W:Set V)
+    · simp only [hW,ite_true]
+      rw [deformed_distinct_product_eq_exterior,hW]
+    · simp only [hW,ite_false,zero_mul])
+
+end WeightedExposureIdentity
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section WeightedThreeArmEstimate
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Countable V] [Countable E] [Nonempty V]
+
+theorem summable_three_arm_pair (G : BondGraph V E) (hloc : LocallyFinite G)
+    (hqt : QuasiTransitive G) (p : unitInterval) (hp : p < pc G) (x a : V) :
+    Summable (fun bz : V×V => twoPoint G p x bz.1 * twoPoint G p bz.1 a * twoPoint G p bz.1 bz.2) := by
+  apply (summable_twoPoint_pair G hloc hqt p hp x).of_nonneg_of_le
+  · exact fun bz => mul_nonneg (mul_nonneg (twoPoint_nonneg G p _ _) (twoPoint_nonneg G p _ _))
+      (twoPoint_nonneg G p _ _)
+  · intro bz
+    exact mul_le_mul_of_nonneg_right (mul_le_of_le_one_right (twoPoint_nonneg G p _ _) (twoPoint_le_one G p _ _))
+      (twoPoint_nonneg G p _ _)
+
+theorem three_arm_real_bound (G : BondGraph V E) (hloc : LocallyFinite G)
+    (hqt : QuasiTransitive G) (p : unitInterval) (hp : p < pc G) (x a z : V) :
+    (law p).real {ω | Conn G ω x a ∧ Conn G ω x z} ≤
+      ∑' b, twoPoint G p x b * twoPoint G p b a * twoPoint G p b z := by
+  have hs := (summable_three_arm_pair G hloc hqt p hp x a).prod_symm.prod_factor z
+  have hn (b : V) : 0 ≤ twoPoint G p x b * twoPoint G p b a * twoPoint G p b z :=
+    mul_nonneg (mul_nonneg (twoPoint_nonneg G p _ _) (twoPoint_nonneg G p _ _)) (twoPoint_nonneg G p _ _)
+  have he : (∑' b, law p {ω | Conn G ω x b} * law p {ω | Conn G ω b a} * law p {ω | Conn G ω b z}) =
+      ENNReal.ofReal (∑' b, twoPoint G p x b * twoPoint G p b a * twoPoint G p b z) := by
+    rw [ENNReal.ofReal_tsum_of_nonneg hn hs]
+    apply tsum_congr
+    intro b
+    rw [ENNReal.ofReal_mul (mul_nonneg (twoPoint_nonneg G p _ _) (twoPoint_nonneg G p _ _)),
+      ENNReal.ofReal_mul (twoPoint_nonneg G p _ _)]
+    simp only [twoPoint,ENNReal.ofReal_toReal (measure_ne_top _ _)]
+  have hh := (three_arm_bound G p x a z).trans_eq he
+  exact (ENNReal.toReal_mono ENNReal.ofReal_ne_top hh).trans_eq (ENNReal.toReal_ofReal (tsum_nonneg hn))
+
+ 
+
+theorem integral_size_membership_le (G : BondGraph V E) (hloc : LocallyFinite G)
+    (hqt : QuasiTransitive G) (p : unitInterval) (hp : p < pc G) (x a : V) :
+    (∫ ω, (if Conn G ω x a then ((cluster G ω x).ncard:ℝ) else 0) ∂law p) ≤
+      maxSusceptibility G p * (∑' b, twoPoint G p x b * twoPoint G p b a) := by
+  have he := integral_cluster_size_event G hloc hqt p hp x _ (measurableSet_conn G x a)
+  simp only [Set.mem_ofPred_eq] at he
+  rw [he]
+  have hs := summable_three_arm_pair G hloc hqt p hp x a
+  have hl : Summable (fun z => (law p).real {ω | Conn G ω x a ∧ Conn G ω x z}) :=
+    (summable_twoPoint_subcritical G hloc hqt p hp x).of_nonneg_of_le (fun _ => measureReal_nonneg)
+      (fun _ => measureReal_mono (fun _ h => h.2))
+  have hb : Summable (fun b => twoPoint G p x b * twoPoint G p b a) :=
+    (summable_twoPoint_subcritical G hloc hqt p hp x).of_nonneg_of_le
+      (fun b => mul_nonneg (twoPoint_nonneg G p _ _) (twoPoint_nonneg G p _ _))
+      (fun b => mul_le_of_le_one_right (twoPoint_nonneg G p _ _) (twoPoint_le_one G p _ _))
+  have hj : Summable (fun b => twoPoint G p x b * twoPoint G p b a * susceptibility G p b) := by
+    simpa only [tsum_mul_left,susceptibility] using hs.prod
+  calc
+    _ ≤ ∑' z, ∑' b, twoPoint G p x b * twoPoint G p b a * twoPoint G p b z :=
+      hl.tsum_le_tsum (three_arm_real_bound G hloc hqt p hp x a) hs.prod_symm.prod
+    _ = ∑' b, ∑' z, twoPoint G p x b * twoPoint G p b a * twoPoint G p b z := hs.tsum_comm
+    _ = ∑' b, twoPoint G p x b * twoPoint G p b a * susceptibility G p b := by
+      simp only [tsum_mul_left,susceptibility]
+    _ ≤ ∑' b, twoPoint G p x b * twoPoint G p b a * maxSusceptibility G p := by
+      apply hj.tsum_le_tsum
+        (fun b => mul_le_mul_of_nonneg_left (susceptibility_le_max G hloc hqt p hp b)
+          (mul_nonneg (twoPoint_nonneg G p _ _) (twoPoint_nonneg G p _ _))) (hb.mul_right _)
+    _ = _ := by rw [tsum_mul_right,mul_comm]
+
+end WeightedThreeArmEstimate
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section FiniteNeighborhoodMass
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem finset_sum_union_le (f : V → ℝ) (hf : ∀ x, 0 ≤ f x) (S T : Finset V) :
+    (∑ x ∈ S ∪ T, f x) ≤ (∑ x ∈ S, f x) + ∑ x ∈ T, f x := by
+  have he := Finset.sum_union_inter (f := f) (s₁ := S) (s₂ := T)
+  have hn := Finset.sum_nonneg (s := S ∩ T) (fun x _ => hf x)
+  linarith
+
+theorem finset_sum_biUnion_le (f : V → ℝ) (hf : ∀ x, 0 ≤ f x)
+    (W : Finset V) (N : V → Finset V) :
+    (∑ z ∈ W.biUnion N, f z) ≤ ∑ a ∈ W, ∑ z ∈ N a, f z := by
+  induction W using Finset.induction_on with
+  | empty => simp
+  | @insert a W ha ih =>
+    rw [Finset.biUnion_insert,Finset.sum_insert ha]
+    exact (finset_sum_union_le f hf _ _).trans (add_le_add (le_refl _) ih)
+
+theorem closedBondNeighborhood_finset (G : BondGraph V E) (hloc : LocallyFinite G) (W : Finset V) :
+    closedBondNeighborhood G (W:Set V) = (W ∪ W.biUnion (neighbors G hloc) : Finset V) := by
+  ext z
+  simp only [closedBondNeighborhood,Set.mem_ofPred_eq,Finset.mem_coe,Finset.mem_union,Finset.mem_biUnion]
+  constructor
+  · rintro (hz | ⟨e,hz,a,ha,haW⟩)
+    · exact Or.inl hz
+    · by_cases hza : a = z
+      · exact Or.inl (hza ▸ haW)
+      · exact Or.inr ⟨a,haW,(mem_neighbors G hloc a z).mpr
+          ⟨e,(Sym2.mem_and_mem_iff hza).mp ⟨ha,hz⟩⟩⟩
+  · rintro (hz | ⟨a,ha,haz⟩)
+    · exact Or.inl hz
+    · obtain ⟨e,he⟩ := (mem_neighbors G hloc a z).mp haz
+      exact Or.inr ⟨e,he.symm ▸ Sym2.mem_mk_right a z,a,he.symm ▸ Sym2.mem_mk_left a z,ha⟩
+
+theorem neighborhood_mass_le (G : BondGraph V E) (hloc : LocallyFinite G)
+    (W : Finset V) (f : V → ℝ) (hf : ∀ x, 0 ≤ f x) :
+    (∑' z : closedBondNeighborhood G (W:Set V), f z) ≤
+      ∑ a ∈ W, (f a + ∑ z ∈ neighbors G hloc a, f z) := by
+  rw [closedBondNeighborhood_finset G hloc W,Finset.tsum_subtype']
+  exact (finset_sum_union_le f hf _ _).trans
+    ((add_le_add (le_refl _) (finset_sum_biUnion_le f hf W (neighbors G hloc))).trans_eq
+      (Finset.sum_add_distrib).symm)
+
+end FiniteNeighborhoodMass
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Deformation/Corridors.lean b/lean/OAI/Probability/UnimodularPercolation/Deformation/Corridors.lean
index e8d78b96cccd5eb9f6ab9ba39ace4e57b572aa6a..3c72c97786a2a8b0a8104cd9affb96af933a3c53 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Deformation/Corridors.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Deformation/Corridors.lean
@@ -1,1230 +1,1230 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Kernel.TraceBounds
-
-namespace OAI
-
-/-! Corridors for bond percolation. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section CorridorEligibility
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace RawCorridor
-variable {G : BondGraph V E} {k : ℕ}
-
-theorem edge_has_label (w : RawCorridor G k) (s : Sym2 V) (hs : s ∈ w.path.edges) :
-    ∃ e ∈ w.retained, G e = s := by
-  induction s using Sym2.inductionOn with
-  | _ x y =>
-    obtain ⟨_,e,_,he⟩ := w.path.adj_of_mem_edges hs
-    exact ⟨e,by simpa only [retained,Set.mem_ofPred_eq,he] using hs,he⟩
-
-theorem image_retained (w : RawCorridor G k) : G '' w.retained = {s | s ∈ w.path.edges} := by
-  ext s
-  constructor
-  · rintro ⟨e,he,rfl⟩; exact he
-  · exact w.edge_has_label s
-
-theorem retained_ncard (hG : Function.Injective G) (w : RawCorridor G k) :
-    w.retained.ncard = w.path.length := by
-  rw [←hG.injOn.ncard_image (s := w.retained),w.image_retained]
-  have hh := List.toFinset_card_of_nodup w.path_isPath.isTrail.edges_nodup
-  simpa only [←List.coe_toFinset,Set.ncard_coe_finset,SimpleGraph.Walk.length_edges] using hh
-
- 
-
-theorem support_subset_of_retained (w : RawCorridor G k) (hlen : 0 < w.path.length)
-    (D : Set V) (he : w.retained ⊆ internalEdges G D) :
-    ∀ z ∈ w.path.support, z ∈ D := by
-  intro z hz
-  obtain ⟨s,hs,hzs⟩ := (SimpleGraph.Walk.mem_support_iff_exists_mem_edges_of_not_nil
-    (SimpleGraph.Walk.not_nil_iff_lt_length.mpr hlen)).mp hz
-  obtain ⟨e,he',hGe⟩ := w.edge_has_label s hs
-  exact he he' z (hGe.symm ▸ hzs)
-
-theorem retained_subset_of_support (w : RawCorridor G k) (D : Set V)
-    (hs : ∀ z ∈ w.path.support, z ∈ D) : w.retained ⊆ internalEdges G D := by
-  intro e he z hz
-  exact hs z (SimpleGraph.Walk.mem_support_of_mem_edges he hz)
-
-theorem support_iff_retained (w : RawCorridor G k) (hlen : 0 < w.path.length) (D : Set V) :
-    (∀ z ∈ w.path.support, z ∈ D) ↔ w.retained ⊆ internalEdges G D :=
-  ⟨w.retained_subset_of_support D,w.support_subset_of_retained hlen D⟩
-
-theorem retained_incident_interior (w : RawCorridor G k) (hlen : 3 ≤ w.path.length)
-    (e : E) (he : e ∈ w.retained) : ∃ z ∈ w.interior, z ∈ G e := by
-  change G e ∈ w.path.edges at he
-  generalize hab : G e = s at he
-  induction s using Sym2.inductionOn with
-  | _ a b =>
-    have hedge : s(a,b) ∈ w.path.edges := he
-    have ha := w.path.fst_mem_support_of_mem_edges hedge
-    have hb := w.path.snd_mem_support_of_mem_edges hedge
-    have habne : a ≠ b := (w.path.adj_of_mem_edges hedge).1
-    by_contra hn
-    have hna : a ∉ w.interior := fun hi => hn ⟨a,hi,by simp⟩
-    have hnb : b ∉ w.interior := fun hi => hn ⟨b,hi,by simp⟩
-    have hax : a = w.base ∨ a = w.endpoint := by simpa only [interior,Finset.mem_sdiff,List.mem_toFinset,ha,true_and,Finset.mem_insert,Finset.mem_singleton,not_not] using hna
-    have hbx : b = w.base ∨ b = w.endpoint := by simpa only [interior,Finset.mem_sdiff,List.mem_toFinset,hb,true_and,Finset.mem_insert,Finset.mem_singleton,not_not] using hnb
-    have hedge' : s(w.base,w.endpoint) ∈ w.path.edges := by
-      rcases hax with rfl | rfl <;> rcases hbx with rfl | rfl
-      · exact (habne rfl).elim
-      · exact hedge
-      · simpa only [Sym2.eq_swap] using hedge
-      · exact (habne rfl).elim
-    have hh := w.path_isPath.length_eq_one_of_mem_edges hedge'
-    omega
-
-theorem specified_disjoint_exterior (w : RawCorridor G k) (hlen : 3 ≤ w.path.length) :
-    Disjoint (w.retained ∪ w.blockers) (internalEdges G (w.interior : Set V)ᶜ) := by
-  apply Set.disjoint_left.mpr
-  intro e he hx
-  obtain ⟨z,hz,he⟩ := he.elim (w.retained_incident_interior hlen e) (fun hb => hb.1)
-  exact hx z he hz
-
-end RawCorridor
-end CorridorEligibility
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section ExactCorridorPattern
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-def edgePattern (A B : Finset E) : Set (Set E) :=
-  {ω | (A : Set E) ⊆ ω ∧ Disjoint (B : Set E) ω}
-
-theorem edgePattern_eq_restriction (A B : Finset E) (hAB : Disjoint A B) :
-    edgePattern A B = {ω | ω ∩ ((A ∪ B : Finset E) : Set E) = (A : Set E)} := by
-  ext ω
-  constructor
-  · rintro ⟨ha,hb⟩
-    apply Set.Subset.antisymm
-    · rintro e ⟨he,heQ⟩
-      rcases Finset.mem_union.mp heQ with h | h
-      · exact h
-      · exact (Set.disjoint_left.mp hb h he).elim
-    · intro e he
-      exact ⟨ha he,Finset.mem_union_left _ he⟩
-  · intro h
-    constructor
-    · intro e he
-      exact ((Set.ext_iff.mp h e).mpr he).1
-    · apply Set.disjoint_left.mpr
-      intro e he hw
-      have ha : e ∈ A := (Set.ext_iff.mp h e).mp ⟨hw,Finset.mem_union_right _ he⟩
-      exact Finset.disjoint_left.mp hAB ha he
-
-theorem measurableSet_edgePattern (A B : Finset E) (hAB : Disjoint A B) :
-    MeasurableSet (edgePattern A B) := by
-  rw [edgePattern_eq_restriction A B hAB]
-  exact measurableSet_finite_restriction (A∪B) {(A:Set E)}
-
- 
-theorem edgePattern_probability (A B : Finset E) (hAB : Disjoint A B) (p : unitInterval) :
-    (law p).real (edgePattern A B) = (p:ℝ)^A.card*(1-(p:ℝ))^B.card := by
-  let f : Set E → ℝ := fun ω => if ω ∈ edgePattern A B then 1 else 0
-  have hf (ω : Set E) : f ω = f (ω ∩ ((A∪B:Finset E):Set E)) := by
-    simp only [f,edgePattern_eq_restriction A B hAB,Set.mem_ofPred_eq,Set.inter_assoc,Set.inter_self]
-  rw [←integral_event_indicator p _ (measurableSet_edgePattern A B hAB)]
-  change (∫ ω, f ω ∂law p) = _
-  rw [integral_eq_bernoulliPolynomial (A∪B) p f hf]
-  trans bernoulliPolynomial (A∪B) (fun η => if η=A then 1 else 0) p
-  · apply bernoulliPolynomial_congr
-    intro η hη
-    have he : (η:Set E) ∩ ((A∪B:Finset E):Set E) = (η:Set E) :=
-      Set.inter_eq_left.mpr hη
-    simp only [f,edgePattern_eq_restriction A B hAB,Set.mem_ofPred_eq,he,Finset.coe_inj]
-  · simp only [bernoulliPolynomial,ite_mul,zero_mul,one_mul]
-    rw [Finset.sum_ite_eq']
-    simp only [Finset.mem_powerset,Finset.subset_union_left,ite_true,Finset.card_union_of_disjoint hAB,
-      Nat.add_sub_cancel_left]
-
-namespace RawCorridor
-variable {G : BondGraph V E} {k : ℕ}
-
-def calibrationEvent (w : RawCorridor G k) : Set (Set E) :=
-  {ω | w.retained ⊆ ω ∧ Disjoint w.blockers ω}
-
-theorem calibrationEvent_eq (hG : Function.Injective G) (hloc : LocallyFinite G)
-    (w : RawCorridor G k) : w.calibrationEvent =
-      edgePattern (w.retained_finite hG).toFinset (w.blockers_finite hloc).toFinset := by
-  simp only [calibrationEvent,edgePattern,Set.Finite.coe_toFinset]
-
-theorem calibration_probability (hG : Function.Injective G) (hloc : LocallyFinite G)
-    (w : RawCorridor G k) (p : unitInterval) :
-    (law p).real w.calibrationEvent = (p:ℝ)^w.path.length*(1-(p:ℝ))^w.blockers.ncard := by
-  rw [w.calibrationEvent_eq hG hloc,edgePattern_probability]
-  · rw [←Set.ncard_eq_toFinset_card w.retained (w.retained_finite hG),←Set.ncard_eq_toFinset_card w.blockers (w.blockers_finite hloc),w.retained_ncard hG]
-  · apply Finset.disjoint_left.mpr
-    intro e he hb
-    exact Set.disjoint_left.mp w.retained_disjoint_blockers
-      ((w.retained_finite hG).mem_toFinset.mp he) ((w.blockers_finite hloc).mem_toFinset.mp hb)
-
- 
-
-theorem calibration_identity (hG : Function.Injective G) (hloc : LocallyFinite G)
-    (w : RawCorridor G k) (P : V → V → ℝ) (p : unitInterval)
-    (hp : 0 < (p:ℝ)) (hp₁ : (p:ℝ) < 1) :
-    w.weight P p * (law p).real w.calibrationEvent = w.probability P := by
-  rw [w.calibration_probability hG hloc p]
-  unfold weight
-  have h₁ := pow_ne_zero w.path.length hp.ne'
-  have h₂ := pow_ne_zero w.blockers.ncard (sub_pos.mpr hp₁).ne'
-  field_simp
-
-end RawCorridor
-end ExactCorridorPattern
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section CorridorTerminalGeometry
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem path_internal_two_neighbors {H : SimpleGraph V} {u v z : V}
-    (q : H.Walk u v) (hq : q.IsPath) (hz : z ∈ q.support) (hzu : z ≠ u) (hzv : z ≠ v) :
-    ∃ a b : V, a ≠ b ∧ s(z,a) ∈ q.edges ∧ s(z,b) ∈ q.edges := by
-  induction q with
-  | nil => simp only [SimpleGraph.Walk.support_nil,List.mem_singleton] at hz; exact (hzu hz).elim
-  | @cons u a v h q ih =>
-    have htail : q.IsPath := hq.of_cons
-    have huz : u ∉ q.support := ((SimpleGraph.Walk.cons_isPath_iff _ _).mp hq).2
-    have hzq : z ∈ q.support := (List.mem_cons.mp hz).resolve_left hzu
-    by_cases hza : z = a
-    · subst z
-      cases q with
-      | nil => exact (hzv rfl).elim
-      | @cons a b v h' q =>
-        refine ⟨u,b,?_,?_,?_⟩
-        · intro hub
-          apply huz
-          subst b
-          simp
-        · simp only [SimpleGraph.Walk.edges_cons,List.mem_cons,Sym2.eq_swap]; exact Or.inl trivial
-        · simp only [SimpleGraph.Walk.edges_cons,List.mem_cons]; exact Or.inr (Or.inl trivial)
-    · obtain ⟨b,c,hbc,hb,hc⟩ := ih htail hzq hza hzv
-      exact ⟨b,c,hbc,List.mem_cons_of_mem _ hb,List.mem_cons_of_mem _ hc⟩
-
- 
-
-theorem pivotal_image_mem_edges (G : BondGraph V E) (ω : Set E) {x y : V}
-    (q : (openSimpleGraph G ω).Walk x y) (e : E)
-    (hp : ¬Conn G (ω \ {e}) x y) : G e ∈ q.edges := by
-  obtain ⟨l,hl,hmap,ho⟩ := walk_lift_bonds G ω q
-  by_contra he
-  apply hp
-  apply hl.conn
-  intro f hf
-  refine ⟨ho f hf,?_⟩
-  intro hfe
-  have hfe' : f = e := Set.mem_singleton_iff.mp hfe
-  subst f
-  apply he
-  rw [←hmap]
-  exact List.mem_map_of_mem hf
-
-namespace RawCorridor
-variable {G : BondGraph V E} {k : ℕ}
-
-theorem counted_terminal_outside (w : RawCorridor G k) (D : Set V) (ω : Set E) (x y : V)
-    (hc : corridorCounted G w.retained w.blockers D ω x y) : x ∉ w.interior ∧ y ∉ w.interior := by
-  have hleft (x y : V) (hc : corridorCounted G w.retained w.blockers D ω x y) : x ∉ w.interior := by
-    intro hx
-    have hx' := Finset.mem_sdiff.mp hx
-    have hxs : x ∈ w.path.support := List.mem_toFinset.mp hx'.1
-    have hxne : x ≠ w.base ∧ x ≠ w.endpoint := by
-      simpa only [Finset.mem_insert,Finset.mem_singleton,not_or] using hx'.2
-    have hxu := hxne.1
-    have hxv := hxne.2
-    obtain ⟨a,b,hab,ha,hb⟩ := path_internal_two_neighbors w.path w.path_isPath hxs hxu hxv
-    obtain ⟨q⟩ := (conn_iff_reachable G _ x y).mp hc.1
-    let qp := q.bypass
-    have hqp : qp.IsPath := q.bypass_isPath
-    obtain ⟨e,he,hGe⟩ := w.edge_has_label _ ha
-    obtain ⟨f,hf,hGf⟩ := w.edge_has_label _ hb
-    have hea := pivotal_image_mem_edges G _ qp e (hc.2.1 e he).2
-    have heb := pivotal_image_mem_edges G _ qp f (hc.2.1 f hf).2
-    rw [hGe] at hea
-    rw [hGf] at heb
-    exact hab ((hqp.eq_snd_of_mem_edges hea).trans (hqp.eq_snd_of_mem_edges heb).symm)
-  exact ⟨hleft x y hc,hleft y x ((corridorCounted_symm G _ _ D ω x y).mp hc)⟩
-
-end RawCorridor
-end CorridorTerminalGeometry
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory ProbabilityTheory
-section CorridorIndependence
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem measurable_inter_const (S : Set E) : Measurable (fun ω : Set E => ω ∩ S) := by
-  apply measurable_setOfPred.comp
-  apply Measurable.of_eval
-  intro e
-  exact ((measurable_pi_apply e).comp MeasurableEquiv.setOfPred.symm.measurable).and measurable_const
-
- 
-theorem restriction_events_independent (p : unitInterval) (S T : Set E)
-    (hST : Disjoint S T) (A B : Set (Set E)) (hA : MeasurableSet A) (hB : MeasurableSet B) :
-    (law p) ((fun ω => ω ∩ S) ⁻¹' A ∩ (fun ω => ω ∩ T) ⁻¹' B) =
-      (law p) ((fun ω => ω ∩ S) ⁻¹' A) * (law p) ((fun ω => ω ∩ T) ⁻¹' B) := by
-  let μ : Measure (E → Prop) := Measure.infinitePi (fun _ => bitLaw p)
-  let σ : E → MeasurableSpace (E → Prop) := fun e =>
-    MeasurableSpace.comap (fun f : E → Prop => f e) inferInstance
-  have hi : iIndep σ μ := iIndepFun_infinitePi (X := fun (_ : E) (b : Prop) => b)
-    (fun _ => measurable_id)
-  have hle (e : E) : σ e ≤ (inferInstance : MeasurableSpace (E → Prop)) :=
-    (measurable_pi_apply e).comap_le
-  have hr (S : Set E) : @Measurable (E → Prop) (Set E) (⨆ e ∈ S, σ e) inferInstance
-      (fun f => {e | f e} ∩ S) := by
-    let : MeasurableSpace (E → Prop) := ⨆ e ∈ S, σ e
-    apply measurable_setOfPred.comp
-    apply Measurable.of_eval
-    intro e
-    change @Measurable (E → Prop) Prop (⨆ e ∈ S, σ e) inferInstance (fun f => f e ∧ e ∈ S)
-    by_cases he : e ∈ S
-    · have hm : @Measurable (E → Prop) Prop (⨆ e ∈ S, σ e) inferInstance (fun f => f e) :=
-        Measurable.of_comap_le (le_iSup_of_le e (le_iSup_of_le he le_rfl))
-      simpa only [Set.mem_inter_iff,Set.mem_ofPred_eq,he,and_true] using hm
-    · simpa only [Set.mem_inter_iff,Set.mem_ofPred_eq,he,and_false] using
-        (measurable_const : @Measurable (E → Prop) Prop (⨆ e ∈ S, σ e) inferInstance (fun _ => False))
-  have hm : (law p : Measure (Set E)) = μ.map (fun f => {e | f e}) := by
-    simp only [law,setBernoulli_eq_map,Set.mem_univ,μ,bitLaw,bernoulliMeasure_def]
-  rw [hm,Measure.map_apply measurable_setOfPred ((hA.preimage (measurable_inter_const S)).inter
-    (hB.preimage (measurable_inter_const T))),
-    Measure.map_apply measurable_setOfPred (hA.preimage (measurable_inter_const S)),
-    Measure.map_apply measurable_setOfPred (hB.preimage (measurable_inter_const T))]
-  exact (Indep_iff _ _ _).mp (indep_iSup_of_disjoint hle hi hST) _ _
-    (hA.preimage (hr S)) (hB.preimage (hr T))
-
-namespace RawCorridor
-variable {G : BondGraph V E} {k : ℕ}
-
-theorem calibration_restriction (w : RawCorridor G k) (ω : Set E) :
-    ω ∈ w.calibrationEvent ↔ ω ∩ (w.retained ∪ w.blockers) ∈ w.calibrationEvent := by
-  constructor
-  · rintro ⟨ha,hb⟩
-    exact ⟨fun e he => ⟨ha he,Or.inl he⟩,hb.mono_right Set.inter_subset_left⟩
-  · rintro ⟨ha,hb⟩
-    refine ⟨fun e he => (ha he).1,Set.disjoint_left.mpr ?_⟩
-    intro e he hw
-    exact Set.disjoint_left.mp hb he ⟨hw,Or.inr he⟩
-
- 
-
-theorem calibration_independent (hG : Function.Injective G) (hloc : LocallyFinite G)
-    (w : RawCorridor G k) (hlen : 3 ≤ w.path.length) (p : unitInterval)
-    (A : Set (Set E)) (hA : MeasurableSet A) :
-    (law p) (w.calibrationEvent ∩ (fun ω => ω ∩ internalEdges G (w.interior : Set V)ᶜ) ⁻¹' A) =
-      (law p) w.calibrationEvent *
-        (law p) ((fun ω => ω ∩ internalEdges G (w.interior : Set V)ᶜ) ⁻¹' A) := by
-  have hQ : MeasurableSet w.calibrationEvent := by
-    rw [w.calibrationEvent_eq hG hloc]
-    apply measurableSet_edgePattern
-    apply Finset.disjoint_left.mpr
-    intro e he hb
-    exact Set.disjoint_left.mp w.retained_disjoint_blockers
-      ((w.retained_finite hG).mem_toFinset.mp he) ((w.blockers_finite hloc).mem_toFinset.mp hb)
-  have heq : (fun ω => ω ∩ (w.retained ∪ w.blockers)) ⁻¹' w.calibrationEvent = w.calibrationEvent := by
-    ext ω; exact (w.calibration_restriction ω).symm
-  have hh := restriction_events_independent p _ _ (w.specified_disjoint_exterior hlen)
-    w.calibrationEvent A hQ hA
-  simpa only [heq] using hh
-
-end RawCorridor
-end CorridorIndependence
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section CorridorContraction
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem conn_union_attachment (G : BondGraph V E) (S T : Set E) (A : Set V)
-    (hclosed : ∀ a b, a ∈ A → Conn G S a b → b ∈ A)
-    (hT : ∀ e ∈ T, ∀ z ∈ G e, z ∈ A) {x y : V} (hc : Conn G (S ∪ T) x y) :
-    Conn G S x y ∨ x ∈ A ∧ y ∈ A := by
-  induction hc with
-  | refl => exact Or.inl (conn_refl G S x)
-  | @tail a b hxa hab ih =>
-    obtain ⟨e,he,heq⟩ := hab
-    rcases he with he | he
-    · have hab : Conn G S a b := Relation.ReflTransGen.single ⟨e,he,heq⟩
-      exact ih.elim (fun h => Or.inl (conn_trans G S h hab))
-        (fun h => Or.inr ⟨h.1,hclosed a b h.2 hab⟩)
-    · have ha := hT e he a (heq.symm ▸ (by simp : a ∈ s(a,b)))
-      have hb := hT e he b (heq.symm ▸ (by simp : b ∈ s(a,b)))
-      exact Or.inr ⟨ih.elim (fun h => hclosed a x ha (conn_symm G S h)) And.left,hb⟩
-
-namespace RawCorridor
-variable {G : BondGraph V E} {k : ℕ}
-
-def exterior (w : RawCorridor G k) (ω : Set E) : Set E :=
-  ω ∩ internalEdges G (w.interior : Set V)ᶜ
-
-def attachment (w : RawCorridor G k) (ω : Set E) : Set V :=
-  {z | z ∈ w.path.support ∨ Conn G (w.exterior ω) w.base z ∨ Conn G (w.exterior ω) w.endpoint z}
-
-theorem support_exterior_ports (w : RawCorridor G k) {z : V}
-    (hz : z ∈ w.path.support) (ho : z ∉ w.interior) : z = w.base ∨ z = w.endpoint := by
-  simpa only [interior,Finset.mem_sdiff,List.mem_toFinset,hz,true_and,
-    Finset.mem_insert,Finset.mem_singleton,not_not] using ho
-
-theorem attachment_closed (w : RawCorridor G k) (ω : Set E) (a b : V)
-    (ha : a ∈ w.attachment ω) (hc : Conn G (w.exterior ω) a b) : b ∈ w.attachment ω := by
-  rcases ha with ha | ha | ha
-  · rcases conn_internal_eq_or_mem G (w.interior : Set V)ᶜ hc with rfl | ho
-    · exact Or.inl ha
-    · rcases w.support_exterior_ports ha ho.1 with rfl | rfl
-      · exact Or.inr (Or.inl hc)
-      · exact Or.inr (Or.inr hc)
-  · exact Or.inr (Or.inl (conn_trans G _ ha hc))
-  · exact Or.inr (Or.inr (conn_trans G _ ha hc))
-
-theorem open_subset_exterior_retained (w : RawCorridor G k) (ω : Set E)
-    (hq : ω ∈ w.calibrationEvent) : ω ⊆ w.exterior ω ∪ w.retained := by
-  intro e he
-  by_cases hr : e ∈ w.retained
-  · exact Or.inr hr
-  · left
-    refine ⟨he,?_⟩
-    intro z hz hi
-    exact Set.disjoint_left.mp hq.2 ⟨⟨z,hi,hz⟩,hr⟩ he
-
-theorem connection_exterior_or_attachment (w : RawCorridor G k) (ω : Set E)
-    (hq : ω ∈ w.calibrationEvent) {x y : V} (hc : Conn G ω x y) :
-    Conn G (w.exterior ω) x y ∨ x ∈ w.attachment ω ∧ y ∈ w.attachment ω := by
-  apply conn_union_attachment G _ w.retained (w.attachment ω) (w.attachment_closed ω)
-    (fun e he z hz => Or.inl (SimpleGraph.Walk.mem_support_of_mem_edges he hz))
-  exact conn_mono G (w.open_subset_exterior_retained ω hq) hc
-
-theorem attachment_outside (w : RawCorridor G k) (ω : Set E) {z : V}
-    (ha : z ∈ w.attachment ω) (hz : z ∉ w.interior) :
-    Conn G (w.exterior ω) w.base z ∨ Conn G (w.exterior ω) w.endpoint z := by
-  rcases ha with ha | ha | ha
-  · rcases w.support_exterior_ports ha hz with rfl | rfl
-    · exact Or.inl (conn_refl G _ _)
-    · exact Or.inr (conn_refl G _ _)
-  · exact Or.inl ha
-  · exact Or.inr ha
-
- 
-theorem counted_exterior_orientation (w : RawCorridor G k) (hlen : 3 ≤ w.path.length)
-    (ω : Set E) (hq : ω ∈ w.calibrationEvent) (x y : V)
-    (hc : corridorCounted G w.retained w.blockers Set.univ ω x y) :
-    ¬Conn G (w.exterior ω) w.base w.endpoint ∧
-      ((Conn G (w.exterior ω) w.base x ∧ Conn G (w.exterior ω) w.endpoint y) ∨
-       (Conn G (w.exterior ω) w.endpoint x ∧ Conn G (w.exterior ω) w.base y)) := by
-  have hi : internalEdges G Set.univ = Set.univ := Set.eq_univ_of_forall (fun _ _ _ => trivial)
-  have hc' : Conn G ω x y := by simpa only [hi,Set.inter_univ] using hc.1
-  have hs : w.retained.Nonempty := by
-    have hp : w.path.edges ≠ [] := by
-      intro he
-      have hh := congrArg List.length he
-      simp only [SimpleGraph.Walk.length_edges,List.length_nil] at hh
-      omega
-    obtain ⟨s,hs⟩ := List.exists_mem_of_ne_nil _ hp
-    obtain ⟨e,he,_⟩ := w.edge_has_label s hs
-    exact ⟨e,he⟩
-  have hnot : ¬Conn G (w.exterior ω) x y := by
-    intro h
-    obtain ⟨e,he⟩ := hs
-    apply (hc.2.1 e he).2
-    apply conn_mono G ?_ h
-    intro f hf
-    refine ⟨?_,?_⟩
-    · simpa only [hi,Set.inter_univ] using hf.1
-    · intro hfe
-      have hfe' := Set.mem_singleton_iff.mp hfe
-      subst f
-      exact Set.disjoint_left.mp (w.specified_disjoint_exterior hlen) (Or.inl he) hf.2
-  have hatt := (w.connection_exterior_or_attachment ω hq hc').resolve_left hnot
-  have hout := w.counted_terminal_outside Set.univ ω x y hc
-  have hx := w.attachment_outside ω hatt.1 hout.1
-  have hy := w.attachment_outside ω hatt.2 hout.2
-  have hne : ¬Conn G (w.exterior ω) w.base w.endpoint := by
-    intro h
-    apply hnot
-    have hx' : Conn G (w.exterior ω) w.base x := hx.elim id (conn_trans G _ h)
-    have hy' : Conn G (w.exterior ω) w.base y := hy.elim id (conn_trans G _ h)
-    exact conn_trans G _ (conn_symm G _ hx') hy'
-  refine ⟨hne,?_⟩
-  rcases hx with hx | hx <;> rcases hy with hy | hy
-  · exact (hnot (conn_trans G _ (conn_symm G _ hx) hy)).elim
-  · exact Or.inl ⟨hx,hy⟩
-  · exact Or.inr ⟨hx,hy⟩
-  · exact (hnot (conn_trans G _ (conn_symm G _ hx) hy)).elim
-
-end RawCorridor
-end CorridorContraction
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section CorridorBKCalibration
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Countable V] [Countable E]
-namespace RawCorridor
-variable {G : BondGraph V E} {k : ℕ}
-
-def exteriorOrientations (w : RawCorridor G k) (x y : V) : Set (Set E) :=
-  orientedPivotalEvent G x y w.base w.endpoint ∪ orientedPivotalEvent G x y w.endpoint w.base
-
-theorem measurableSet_exteriorOrientations (w : RawCorridor G k) (x y : V) :
-    MeasurableSet (w.exteriorOrientations x y) :=
-  (measurableSet_orientedPivotalEvent G x y w.base w.endpoint).union
-    (measurableSet_orientedPivotalEvent G x y w.endpoint w.base)
-
-omit [Countable V] [Countable E] in
-theorem counted_subset_calibration_exterior (w : RawCorridor G k)
-    (hlen : 3 ≤ w.path.length) (x y : V) :
-    {ω | corridorCounted G w.retained w.blockers Set.univ ω x y} ⊆
-      w.calibrationEvent ∩ (w.exterior ⁻¹' w.exteriorOrientations x y) := by
-  intro ω hc
-  have hi : internalEdges G Set.univ = Set.univ := Set.eq_univ_of_forall (fun _ _ _ => trivial)
-  have hq : ω ∈ w.calibrationEvent := by
-    refine ⟨fun e he => ?_,?_⟩
-    · simpa only [hi,Set.inter_univ] using (hc.2.1 e he).1
-    · simpa only [hi,Set.inter_univ] using hc.2.2
-  refine ⟨hq,?_⟩
-  obtain ⟨hne,ho⟩ := w.counted_exterior_orientation hlen ω hq x y hc
-  rcases ho with ho | ho
-  · exact Or.inl ⟨conn_symm G _ ho.1,ho.2,hne⟩
-  · exact Or.inr ⟨conn_symm G _ ho.1,ho.2,fun h => hne (conn_symm G _ h)⟩
-
-omit [Countable V] in
-theorem exterior_orientation_BK (w : RawCorridor G k) (p : unitInterval) (x y a b : V) :
-    (law p).real (w.exterior ⁻¹' orientedPivotalEvent G x y a b) ≤
-      twoPoint G p x a * twoPoint G p b y := by
-  apply le_trans (measureReal_mono ?_) (bk p {ω | Conn G ω x a} {ω | Conn G ω b y}
-    (fun _ _ h hc => conn_mono G h hc) (fun _ _ h hc => conn_mono G h hc) (finiteWitness_conn G x a) (finiteWitness_conn G b y))
-  intro ω h
-  exact disjointOccurrence_upper _ _ Set.inter_subset_left
-    (orientedPivotalEvent_subset_disjoint G x y a b h)
-
- 
-
-theorem counted_weight_bound (hG : Function.Injective G) (hloc : LocallyFinite G)
-    (w : RawCorridor G k) (hlen : 3 ≤ w.path.length)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval)
-    (hp : 0 < (p:ℝ)) (hp₁ : (p:ℝ) < 1) (x y : V) :
-    w.weight P p * (law p).real {ω | corridorCounted G w.retained w.blockers Set.univ ω x y} ≤
-      w.probability P * (twoPoint G p x w.base * twoPoint G p w.endpoint y +
-        twoPoint G p x w.endpoint * twoPoint G p w.base y) := by
-  have hi := w.calibration_independent hG hloc hlen p (w.exteriorOrientations x y)
-    (w.measurableSet_exteriorOrientations x y)
-  have hir : (law p).real (w.calibrationEvent ∩ w.exterior ⁻¹' w.exteriorOrientations x y) =
-      (law p).real w.calibrationEvent * (law p).real (w.exterior ⁻¹' w.exteriorOrientations x y) := by
-    exact congrArg ENNReal.toReal hi |>.trans ENNReal.toReal_mul
-  have hb : (law p).real (w.exterior ⁻¹' w.exteriorOrientations x y) ≤
-      twoPoint G p x w.base * twoPoint G p w.endpoint y +
-        twoPoint G p x w.endpoint * twoPoint G p w.base y := by
-    apply le_trans (measureReal_union_le _ _)
-    exact add_le_add (w.exterior_orientation_BK p x y w.base w.endpoint)
-      (w.exterior_orientation_BK p x y w.endpoint w.base)
-  calc
-    _ ≤ w.weight P p * (law p).real (w.calibrationEvent ∩ w.exterior ⁻¹' w.exteriorOrientations x y) :=
-      mul_le_mul_of_nonneg_left (measureReal_mono (w.counted_subset_calibration_exterior hlen x y))
-        (w.weight_nonneg P hP p)
-    _ = w.probability P * (law p).real (w.exterior ⁻¹' w.exteriorOrientations x y) := by
-      rw [hir,←mul_assoc,w.calibration_identity hG hloc P p hp hp₁]
-    _ ≤ _ := mul_le_mul_of_nonneg_left hb (w.2.2.mass_nonneg P hP)
-
-end RawCorridor
-end CorridorBKCalibration
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section RawPathIntegration
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-abbrev RawFrom (H : SimpleGraph V) (k : ℕ) (x : V) := Σ y : V, HoldingWalk H k x y
-
-namespace RawFrom
-variable {H : SimpleGraph V}
-
-def zeroEquiv (x : V) : RawFrom H 0 x ≃ Unit where
-  toFun _ := ()
-  invFun _ := ⟨x,.nil x⟩
-  left_inv w := by rcases w with ⟨y,w⟩; cases w; rfl
-  right_inv _ := rfl
-
-def succEquiv (k : ℕ) (x : V) : RawFrom H (k+1) x ≃
-    RawFrom H k x ⊕ (Σ z : H.neighborSet x, RawFrom H k z.val) where
-  toFun
-    | ⟨y,.hold w⟩ => .inl ⟨y,w⟩
-    | ⟨y,.step (z := z) h w⟩ => .inr ⟨⟨z,h⟩,y,w⟩
-  invFun
-    | .inl ⟨y,w⟩ => ⟨y,.hold w⟩
-    | .inr ⟨⟨z,h⟩,y,w⟩ => ⟨y,.step h w⟩
-  left_inv w := by rcases w with ⟨y,w⟩; cases w <;> rfl
-  right_inv w := by rcases w with ⟨y,w⟩ | ⟨⟨z,h⟩,y,w⟩ <;> rfl
-
-instance finite [∀ x, Finite (H.neighborSet x)] (k : ℕ) (x : V) : Finite (RawFrom H k x) := by
-  induction k generalizing x with
-  | zero => exact Finite.of_equiv Unit (zeroEquiv x).symm
-  | succ k ih =>
-    let (z : V) : Finite (RawFrom H k z) := ih z
-    exact Finite.of_equiv _ (succEquiv k x).symm
-
-noncomputable def integral (P : V → V → ℝ) (k : ℕ) (x : V) (f : V → ℝ) : ℝ :=
-  ∑' w : RawFrom H k x, w.2.mass P * f w.1
-
-theorem integral_zero (P : V → V → ℝ) (x : V) (f : V → ℝ) :
-    integral (H := H) P 0 x f = f x := by
-  unfold integral
-  rw [←(zeroEquiv x).symm.tsum_eq]
-  simp only [zeroEquiv,Equiv.symm_mk,Equiv.coe_fn_mk,HoldingWalk.mass,one_mul,tsum_fintype,Fintype.sum_unique]
-
-theorem integral_succ [∀ x, Finite (H.neighborSet x)] (P : V → V → ℝ)
-    (k : ℕ) (x : V) (f : V → ℝ) :
-    integral (H := H) P (k+1) x f =
-      P x x * integral (H := H) P k x f +
-        ∑' z : H.neighborSet x, P x z * integral (H := H) P k z f := by
-  unfold integral
-  rw [←(succEquiv k x).symm.tsum_eq]
-  let := Fintype.ofFinite (RawFrom H k x)
-  let := Fintype.ofFinite (H.neighborSet x)
-  let (z : H.neighborSet x) := Fintype.ofFinite (RawFrom H k z)
-  simp only [tsum_fintype,Fintype.sum_sum_type,Fintype.sum_sigma,succEquiv,
-    Equiv.symm_mk,Equiv.coe_fn_mk,HoldingWalk.mass,mul_assoc,Finset.mul_sum]
-
-end RawFrom
-
-theorem finite_openSimpleGraph_neighbors (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) :
-    Finite ((openSimpleGraph G Set.univ).neighborSet x) := by
-  apply Set.Finite.to_subtype
-  apply (finite_neighbors hloc x).subset
-  intro y hy
-  obtain ⟨_,e,_,he⟩ := hy
-  exact ⟨e,he⟩
-
-end RawPathIntegration
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section RawPathOperator
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- theorem sum_step_decomposition (H : SimpleGraph V) (P : V → V → ℝ)
-    (hs : ∀ x, (Function.support (P x)).Finite)
-    (hstep : ∀ x y, x ≠ y → ¬H.Adj x y → P x y = 0) (x : V) (f : V → ℝ) :
-    (∑' y, P x y * f y) = P x x * f x + ∑' y : H.neighborSet x, P x y * f y := by
-  rw [(finiteKernel_summable_mul P hs x f).tsum_eq_add_tsum_ite x]
-  congr 1
-  have he : (∑' y : H.neighborSet x, if (y:V) = x then (0:ℝ) else P x y * f y) =
-      ∑' y, if y = x then (0:ℝ) else P x y * f y := by
-    apply tsum_subtype_eq_of_support_subset (f := fun y : V => if y = x then (0:ℝ) else P x y * f y)
-    intro y hy
-    by_contra hn
-    apply hy
-    by_cases h : y = x
-    · simp only [h,ite_true]
-    · simp only [ite_eq_right h,hstep x y (Ne.symm h) hn,zero_mul]
-  rw [←he]
-  apply tsum_congr
-  intro y
-  have hn : (y:V) ≠ x := by
-    intro h
-    have hy : H.Adj x y := y.property
-    rw [h] at hy
-    exact H.loopless.irrefl x hy
-  rw [ite_eq_right hn]
-
- theorem raw_integral_operator (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (k : ℕ) (x : V) (f : V → ℝ) :
-    RawFrom.integral (H := openSimpleGraph G Set.univ) (walkKernel G hloc a) k x f =
-      ∑' y, (matrixCoeff (P^k) x y).re * f y := by
-  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) :=
-    finite_openSimpleGraph_neighbors G hloc z
-  have hs (x : V) : (Function.support (walkKernel G hloc a x)).Finite := by
-    simpa only [pow_one,hP,Complex.ofReal_re] using walkPower_finite_support G hloc a P hpos hP 1 x
-  have hsP (x : V) : (Function.support (fun y => (matrixCoeff P x y).re)).Finite := by
-    simpa only [hP,Complex.ofReal_re] using hs x
-  induction k generalizing x with
-  | zero => rw [RawFrom.integral_zero,pow_zero,positive_identity_action]
-  | succ k ih =>
-    rw [RawFrom.integral_succ,pow_succ',finite_positive_product_action P (P^k) hpos hsP
-      (finite_positive_pow_support P hpos hsP k)]
-    simp only [hP,Complex.ofReal_re]
-    rw [sum_step_decomposition (openSimpleGraph G Set.univ) (walkKernel G hloc a) hs]
-    · simp only [ih]
-    · intro u v huv hn
-      apply walkKernel_zero_of_not_step G hloc a u v huv
-      intro ha
-      apply hn
-      obtain ⟨e,he⟩ := ha
-      exact ⟨huv,e,Set.mem_univ e,he⟩
-
-end RawPathOperator
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section CorridorSum
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace RawCorridor
-variable {G : BondGraph V E} {k : ℕ}
-
- 
-
-def counted (w : RawCorridor G k) (D : Set V) (ω : Set E) (x y : V) : Prop :=
-  3 ≤ w.path.length ∧ (∀ z ∈ w.path.support, z ∈ D) ∧
-    corridorCounted G w.retained w.blockers D ω x y
-
-theorem counted_iff (w : RawCorridor G k) (D : Set V) (ω : Set E) (x y : V) :
-    w.counted D ω x y ↔ 3 ≤ w.path.length ∧ corridorCounted G w.retained w.blockers D ω x y := by
-  constructor
-  · exact fun h => ⟨h.1,h.2.2⟩
-  · rintro ⟨hlen,hc⟩
-    exact ⟨hlen,w.support_subset_of_retained (by omega) D (fun e he => (hc.2.1 e he).1.2),hc⟩
-
-theorem counted_base_mem_walk (w : RawCorridor G k) (D : Set V) (ω : Set E) (x y : V)
-    (hc : w.counted D ω x y) (q : (openSimpleGraph G (ω ∩ internalEdges G D)).Walk x y) :
-    w.base ∈ q.support := by
-  have hl : 0 < w.path.length := by have := hc.1; omega
-  obtain ⟨s,hs,hbs⟩ := (SimpleGraph.Walk.mem_support_iff_exists_mem_edges_of_not_nil
-    (SimpleGraph.Walk.not_nil_iff_lt_length.mpr hl)).mp w.path.start_mem_support
-  obtain ⟨e,he,hGe⟩ := w.edge_has_label s hs
-  have heq := pivotal_image_mem_edges G _ q e (hc.2.2.2.1 e he).2
-  exact SimpleGraph.Walk.mem_support_of_mem_edges heq (hGe.symm ▸ hbs)
-
-theorem counted_symm (w : RawCorridor G k) (D : Set V) (ω : Set E) (x y : V) :
-    w.counted D ω x y ↔ w.counted D ω y x := by
-  rw [counted_iff,counted_iff,corridorCounted_symm]
-
-theorem measurableSet_counted [Countable E] (w : RawCorridor G k) (D : Set V) (x y : V) :
-    MeasurableSet {ω | w.counted D ω x y} := by
-  simp_rw [w.counted_iff]
-  by_cases h : 3 ≤ w.path.length
-  · simpa only [h,true_and] using measurableSet_corridorCounted G w.retained w.blockers D x y
-  · simp only [h,false_and,Set.ofPred_false]; exact MeasurableSet.empty
-
-end RawCorridor
-
- theorem finite_corridors_on_bases (G : BondGraph V E) (hloc : LocallyFinite G) (k : ℕ)
-    (S : Set V) (hS : S.Finite) : {w : RawCorridor G k | w.base ∈ S}.Finite := by
-  let (x : V) : Finite ((openSimpleGraph G Set.univ).neighborSet x) :=
-    finite_openSimpleGraph_neighbors G hloc x
-  let : Finite S := hS.to_subtype
-  let f : (Σ x : S, RawFrom (openSimpleGraph G Set.univ) k x.val) → RawCorridor G k :=
-    fun w => ⟨w.1.val,w.2⟩
-  have hr : {w : RawCorridor G k | w.base ∈ S} = Set.range f := by
-    ext w
-    constructor
-    · intro hw
-      exact ⟨⟨⟨w.base,hw⟩,w.2⟩,rfl⟩
-    · rintro ⟨w,rfl⟩
-      exact w.1.property
-  rw [hr]
-  exact Set.finite_range f
-
- 
-
-theorem finite_counted_corridors (G : BondGraph V E) (hloc : LocallyFinite G) (k : ℕ)
-    (D : Set V) (ω : Set E) (x y : V) : {w : RawCorridor G k | w.counted D ω x y}.Finite := by
-  by_cases hc : Conn G (ω ∩ internalEdges G D) x y
-  · obtain ⟨q⟩ := (conn_iff_reachable G _ x y).mp hc
-    apply (finite_corridors_on_bases G hloc k {z | z ∈ q.support} (List.finite_toSet _)).subset
-    exact fun w hw => w.counted_base_mem_walk D ω x y hw q
-  · have he : {w : RawCorridor G k | w.counted D ω x y} = ∅ := by
-      ext w
-      exact ⟨fun h => hc h.2.2.1,False.elim⟩
-    rw [he]
-    exact Set.finite_empty
-
-end CorridorSum
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section ActualCorridorPenalty
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def corridorPenaltyTerm (G : BondGraph V E) (P : V → V → ℝ) (p : ℝ)
-    {k : ℕ} (D : Set V) (ω : Set E) (x y : V) (w : RawCorridor G k) : ℝ :=
-  if w.counted D ω x y then w.weight P p else 0
-
-noncomputable def corridorPenalty (G : BondGraph V E) (P : V → V → ℝ) (p : ℝ)
-    (k : ℕ) (D : Set V) (ω : Set E) (x y : V) : ℝ :=
-  ∑' w : RawCorridor G k, corridorPenaltyTerm G P p D ω x y w
-
-theorem corridorPenaltyTerm_nonneg (G : BondGraph V E) (P : V → V → ℝ)
-    (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) {k : ℕ} (D : Set V)
-    (ω : Set E) (x y : V) (w : RawCorridor G k) :
-    0 ≤ corridorPenaltyTerm G P p D ω x y w := by
-  unfold corridorPenaltyTerm
-  split_ifs
-  · exact w.weight_nonneg P hP p
-  · exact le_rfl
-
-theorem corridorPenalty_finite_support (G : BondGraph V E) (hloc : LocallyFinite G)
-    (P : V → V → ℝ) (p : ℝ) (k : ℕ) (D : Set V) (ω : Set E) (x y : V) :
-    (Function.support (corridorPenaltyTerm G P p (k := k) D ω x y)).Finite := by
-  apply (finite_counted_corridors G hloc k D ω x y).subset
-  intro w hw
-  by_contra hn
-  exact hw (ite_eq_right hn)
-
-theorem corridorPenalty_summable (G : BondGraph V E) (hloc : LocallyFinite G)
-    (P : V → V → ℝ) (p : ℝ) (k : ℕ) (D : Set V) (ω : Set E) (x y : V) :
-    Summable (corridorPenaltyTerm G P p (k := k) D ω x y) :=
-  summable_of_hasFiniteSupport (corridorPenalty_finite_support G hloc P p k D ω x y)
-
-theorem corridorPenalty_nonneg (G : BondGraph V E) (P : V → V → ℝ)
-    (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (k : ℕ) (D : Set V)
-    (ω : Set E) (x y : V) : 0 ≤ corridorPenalty G P p k D ω x y :=
-  tsum_nonneg (corridorPenaltyTerm_nonneg G P hP p D ω x y)
-
-theorem corridorPenalty_symm (G : BondGraph V E) (P : V → V → ℝ) (p : ℝ)
-    (k : ℕ) (D : Set V) (ω : Set E) (x y : V) :
-    corridorPenalty G P p k D ω x y = corridorPenalty G P p k D ω y x := by
-  unfold corridorPenalty
-  apply tsum_congr
-  intro w
-  simp only [corridorPenaltyTerm,w.counted_symm D ω x y]
-
-theorem corridorPenalty_restrict (G : BondGraph V E) (hloc : LocallyFinite G)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (k : ℕ)
-    {D D' : Set V} {ω ω' : Set E} (hD : D ⊆ D') (hω : ω ⊆ ω')
-    (x y : V) (hc : Conn G (ω ∩ internalEdges G D) x y) :
-    corridorPenalty G P p k D' ω' x y ≤ corridorPenalty G P p k D ω x y := by
-  apply (corridorPenalty_summable G hloc P p k D' ω' x y).tsum_le_tsum ?_
-    (corridorPenalty_summable G hloc P p k D ω x y)
-  intro w
-  by_cases hn : w.counted D' ω' x y
-  · have ho : w.counted D ω x y := (w.counted_iff D ω x y).mpr
-      ⟨hn.1,corridorCounted_restrict G _ _ hD hω x y hc hn.2.2⟩
-    simp only [corridorPenaltyTerm,hn,ho,ite_true,le_refl]
-  · simpa only [corridorPenaltyTerm,hn,ite_false] using corridorPenaltyTerm_nonneg G P hP p D ω x y w
-
-theorem measurable_corridorPenaltyTerm [Countable E] (G : BondGraph V E)
-    (P : V → V → ℝ) (p : ℝ) {k : ℕ} (D : Set V) (x y : V) (w : RawCorridor G k) :
-    Measurable (fun ω => corridorPenaltyTerm G P p D ω x y w) := by
-  exact Measurable.ite (w.measurableSet_counted D x y) measurable_const measurable_const
-
-theorem measurable_corridorPenalty [Countable V] [Countable E] (G : BondGraph V E)
-    (P : V → V → ℝ) (p : ℝ) (k : ℕ) (D : Set V) (x y : V) :
-    Measurable (fun ω => corridorPenalty G P p k D ω x y) := by
-  exact Measurable.tsum (fun w => measurable_corridorPenaltyTerm G P p D x y w)
-
-end ActualCorridorPenalty
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section ActualWeightedConnections
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-noncomputable def deformedConnection (G : BondGraph V E) (P : V → V → ℝ) (p : ℝ)
-    (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x y : V) : ℝ :=
-  if x ∈ D ∧ y ∈ D ∧ Conn G (ω ∩ internalEdges G D) x y then
-    Real.exp (-t * corridorPenalty G P p k D ω x y) else 0
-
-noncomputable def deformedKernel (G : BondGraph V E) (P : V → V → ℝ) (p : unitInterval)
-    (k : ℕ) (t : ℝ) (D : Set V) (x y : V) : ℝ :=
-  ∫ ω, deformedConnection G P p k t D ω x y ∂law p
-
-theorem deformedConnection_nonneg (G : BondGraph V E) (P : V → V → ℝ) (p : ℝ)
-    (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x y : V) :
-    0 ≤ deformedConnection G P p k t D ω x y := by
-  unfold deformedConnection
-  split_ifs
-  · exact (Real.exp_pos _).le
-  · exact le_rfl
-
-theorem deformedConnection_le_one (G : BondGraph V E) (P : V → V → ℝ)
-    (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (k : ℕ) (t : ℝ) (ht : 0 ≤ t)
-    (D : Set V) (ω : Set E) (x y : V) : deformedConnection G P p k t D ω x y ≤ 1 := by
-  unfold deformedConnection
-  split_ifs
-  · apply Real.exp_le_one_iff.mpr
-    exact mul_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr ht) (corridorPenalty_nonneg G P hP p k D ω x y)
-  · exact zero_le_one
-
-theorem deformedConnection_symm (G : BondGraph V E) (P : V → V → ℝ) (p : ℝ)
-    (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x y : V) :
-    deformedConnection G P p k t D ω x y = deformedConnection G P p k t D ω y x := by
-  have hc : Conn G (ω ∩ internalEdges G D) x y ↔ Conn G (ω ∩ internalEdges G D) y x :=
-    ⟨conn_symm G _,conn_symm G _⟩
-  simp only [deformedConnection,corridorPenalty_symm G P p k D ω x y,hc]
-  congr 1
-  apply propext; tauto
-
-theorem deformedConnection_mono (G : BondGraph V E) (hloc : LocallyFinite G)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (k : ℕ)
-    (t : ℝ) (ht : 0 ≤ t) {D D' : Set V} {ω ω' : Set E} (hD : D ⊆ D') (hω : ω ⊆ ω')
-    (x y : V) : deformedConnection G P p k t D ω x y ≤ deformedConnection G P p k t D' ω' x y := by
-  by_cases hc : x ∈ D ∧ y ∈ D ∧ Conn G (ω ∩ internalEdges G D) x y
-  · have hc' : x ∈ D' ∧ y ∈ D' ∧ Conn G (ω' ∩ internalEdges G D') x y :=
-      ⟨hD hc.1,hD hc.2.1,conn_mono G (Set.inter_subset_inter hω (internalEdges_mono G hD)) hc.2.2⟩
-    simp only [deformedConnection,hc,hc']
-    apply Real.exp_le_exp.mpr
-    exact mul_le_mul_of_nonpos_left (corridorPenalty_restrict G hloc P hP p k hD hω x y hc.2.2)
-      (neg_nonpos.mpr ht)
-  · simpa only [deformedConnection,hc,ite_false] using deformedConnection_nonneg G P p k t D' ω' x y
-
-theorem measurable_deformedConnection [Countable V] [Countable E] (G : BondGraph V E)
-    (P : V → V → ℝ) (p : ℝ) (k : ℕ) (t : ℝ)
-    (D : Set V) (x y : V) : Measurable (fun ω => deformedConnection G P p k t D ω x y) := by
-  have hs : MeasurableSet {ω : Set E | x ∈ D ∧ y ∈ D ∧ Conn G (ω ∩ internalEdges G D) x y} := by
-    by_cases hx : x ∈ D <;> by_cases hy : y ∈ D
-    · simpa only [hx,hy,true_and] using measurableSet_restrictedConn G (internalEdges G D) x y
-    all_goals simp only [hx,hy,false_and,and_false,Set.ofPred_false]; exact MeasurableSet.empty
-  exact Measurable.ite hs ((measurable_corridorPenalty G P p k D x y).const_mul (-t)).exp measurable_const
-
-theorem integrable_deformedConnection [Countable V] [Countable E] (G : BondGraph V E)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y)
-    (p : unitInterval) (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D : Set V) (x y : V) :
-    Integrable (fun ω => deformedConnection G P p k t D ω x y) (law p) := by
-  apply Integrable.mono' (integrable_const (1:ℝ)) (measurable_deformedConnection G P p k t D x y).aestronglyMeasurable
-  filter_upwards [] with ω
-  rw [Real.norm_eq_abs,abs_of_nonneg (deformedConnection_nonneg G P p k t D ω x y)]
-  exact deformedConnection_le_one G P hP p k t ht D ω x y
-
-end ActualWeightedConnections
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section WeightedConnectionEndpoints
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem corridorPenalty_diag (G : BondGraph V E) (P : V → V → ℝ) (p : ℝ)
-    (k : ℕ) (D : Set V) (ω : Set E) (x : V) : corridorPenalty G P p k D ω x x = 0 := by
-  unfold corridorPenalty
-  rw [←tsum_zero]
-  apply tsum_congr
-  intro w
-  have hn : ¬w.counted D ω x x := by
-    intro hc
-    have hl : 0 < w.path.length := by have := hc.1; omega
-    obtain ⟨s,hs,_⟩ := (SimpleGraph.Walk.mem_support_iff_exists_mem_edges_of_not_nil
-      (SimpleGraph.Walk.not_nil_iff_lt_length.mpr hl)).mp w.path.start_mem_support
-    obtain ⟨e,he,_⟩ := w.edge_has_label s hs
-    exact (hc.2.2.2.1 e he).2 (conn_refl G _ x)
-  exact ite_eq_right hn
-
-theorem deformedConnection_diag (G : BondGraph V E) (P : V → V → ℝ) (p : ℝ)
-    (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x : V) (hx : x ∈ D) :
-    deformedConnection G P p k t D ω x x = 1 := by
-  simp only [deformedConnection,hx,conn_refl G _ x,and_self,ite_true,
-    corridorPenalty_diag,mul_zero,Real.exp_zero]
-
-theorem deformedConnection_zero (G : BondGraph V E) (P : V → V → ℝ) (p : ℝ)
-    (k : ℕ) (D : Set V) (ω : Set E) (x y : V) :
-    deformedConnection G P p k 0 D ω x y =
-      if x ∈ D ∧ y ∈ D ∧ Conn G (ω ∩ internalEdges G D) x y then 1 else 0 := by
-  simp only [deformedConnection,neg_zero,zero_mul,Real.exp_zero]
-
-theorem deformedConnection_full_zero (G : BondGraph V E) (P : V → V → ℝ) (p : ℝ)
-    (k : ℕ) (ω : Set E) (x y : V) : deformedConnection G P p k 0 Set.univ ω x y =
-      if Conn G ω x y then 1 else 0 := by
-  have hi : internalEdges G Set.univ = Set.univ := Set.eq_univ_of_forall (fun _ _ _ => trivial)
-  simp only [deformedConnection_zero,Set.mem_univ,true_and,hi,Set.inter_univ]
-
-theorem deformedConnection_change (G : BondGraph V E) (P : V → V → ℝ)
-    (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (k : ℕ) (t : ℝ) (ht : 0 ≤ t)
-    (D : Set V) (ω : Set E) (x y : V) :
-    0 ≤ deformedConnection G P p k 0 D ω x y - deformedConnection G P p k t D ω x y ∧
-    deformedConnection G P p k 0 D ω x y - deformedConnection G P p k t D ω x y ≤
-      t*corridorPenalty G P p k D ω x y := by
-  have hN := corridorPenalty_nonneg G P hP p k D ω x y
-  by_cases hc : x ∈ D ∧ y ∈ D ∧ Conn G (ω ∩ internalEdges G D) x y
-  · simp only [deformedConnection,hc,ite_true,neg_zero,zero_mul,Real.exp_zero,and_self]
-    constructor
-    · exact sub_nonneg.mpr ((Real.exp_le_one_iff).mpr (mul_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr ht) hN))
-    · have h := Real.one_sub_le_exp_neg (t*corridorPenalty G P p k D ω x y)
-      rw [←neg_mul] at h
-      linarith
-  · simp only [deformedConnection,hc,ite_false,sub_self]
-    exact ⟨le_rfl,mul_nonneg ht hN⟩
-
-theorem deformedKernel_zero [Countable V] [Countable E] (G : BondGraph V E)
-    (P : V → V → ℝ) (p : unitInterval) (k : ℕ) (x y : V) :
-    deformedKernel G P p k 0 Set.univ x y = twoPoint G p x y := by
-  unfold deformedKernel
-  simp_rw [deformedConnection_full_zero]
-  have hi := integral_indicator_const (μ := law p) (1:ℝ) (measurableSet_conn G x y)
-  simpa only [Set.indicator,Set.mem_ofPred_eq,smul_eq_mul,mul_one,Measure.real,twoPoint] using hi
-
-end WeightedConnectionEndpoints
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section RawWalkDiagrams
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem raw_diagram_hasSum (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (hK : ∀ x y, 0 ≤ walkKernel G hloc a x y)
-    (A : Hilbert V →L[ℂ] Hilbert V) (hA : PositiveRealMatrix A) (k : ℕ) (x y : V) :
-    HasSum (fun w : RawCorridor G k => w.probability (walkKernel G hloc a) *
-      ((matrixCoeff A x w.base).re * (matrixCoeff A w.endpoint y).re))
-      (matrixCoeff (A*(P^k*A)) x y).re := by
-  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) :=
-    finite_openSimpleGraph_neighbors G hloc z
-  let F := fun w : RawCorridor G k => w.probability (walkKernel G hloc a) *
-    ((matrixCoeff A x w.base).re * (matrixCoeff A w.endpoint y).re)
-  have hFn (w : RawCorridor G k) : 0 ≤ F w :=
-    mul_nonneg (w.2.2.mass_nonneg _ hK) (mul_nonneg (hA _ _).2 (hA _ _).2)
-  have hr (z : V) : (∑' w : RawFrom (openSimpleGraph G Set.univ) k z, F ⟨z,w⟩) =
-      (matrixCoeff A x z).re * (matrixCoeff (P^k*A) z y).re := by
-    have h := raw_integral_operator G hloc a P hpos hP k z (fun u => (matrixCoeff A u y).re)
-    have he : (∑' w : RawFrom (openSimpleGraph G Set.univ) k z, F ⟨z,w⟩) =
-        (matrixCoeff A x z).re * RawFrom.integral (H := openSimpleGraph G Set.univ)
-          (walkKernel G hloc a) k z (fun u => (matrixCoeff A u y).re) := by
-      rw [RawFrom.integral,←tsum_mul_left]
-      apply tsum_congr
-      intro w
-      dsimp only [F,RawCorridor.probability,RawCorridor.base,RawCorridor.endpoint]
-      change w.2.mass _ * (_ * _) = _ * (w.2.mass _ * _)
-      ring
-    rw [he,h,((hpos.pow k).mul_hasSum A z y).tsum_eq]
-  have ho := hA.mul_hasSum (P^k*A) x y
-  have hs : Summable F := (summable_sigma_of_nonneg hFn).mpr ⟨fun z => by
-    let := Fintype.ofFinite (RawFrom (openSimpleGraph G Set.univ) k z)
-    exact (hasSum_fintype _).summable,by simpa only [hr] using ho.summable⟩
-  have he : (∑' w, F w) = (matrixCoeff (A*(P^k*A)) x y).re := by
-    rw [hs.tsum_sigma]
-    simpa only [hr] using ho.tsum_eq
-  exact he ▸ hs.hasSum
-
- theorem sandwich_selfAdjoint (A P : Hilbert V →L[ℂ] Hilbert V)
-    (hA : IsSelfAdjoint A) (hP : IsSelfAdjoint P) (k : ℕ) : IsSelfAdjoint (A*(P^k*A)) := by
-  change star (A*(P^k*A)) = A*(P^k*A)
-  simp only [star_mul,hA.star_eq,(hP.pow k).star_eq,mul_assoc]
-
-end RawWalkDiagrams
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section RawWalkSymmetricDiagrams
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem raw_two_diagrams_hasSum (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hPs : IsSelfAdjoint P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (hK : ∀ x y, 0 ≤ walkKernel G hloc a x y)
-    (A : Hilbert V →L[ℂ] Hilbert V) (hA : PositiveRealMatrix A) (hAs : IsSelfAdjoint A)
-    (k : ℕ) (x y : V) :
-    HasSum (fun w : RawCorridor G k => w.probability (walkKernel G hloc a) *
-      ((matrixCoeff A x w.base).re * (matrixCoeff A w.endpoint y).re +
-       (matrixCoeff A x w.endpoint).re * (matrixCoeff A w.base y).re))
-      (2*(matrixCoeff (A*(P^k*A)) x y).re) := by
-  have h₁ := raw_diagram_hasSum G hloc a P hpos hP hK A hA k x y
-  have h₂ := raw_diagram_hasSum G hloc a P hpos hP hK A hA k y x
-  have hh := h₁.add h₂
-  have hC := matrixCoeff_real_symm (A*(P^k*A)) (sandwich_selfAdjoint A P hAs hPs k) y x
-  rw [hC,←two_mul] at hh
-  convert hh using 1
-  ext w
-  rw [matrixCoeff_real_symm A hAs y w.base,matrixCoeff_real_symm A hAs w.endpoint x]
-  ring
-
-end RawWalkSymmetricDiagrams
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section PenaltyTermExpectation
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- theorem integrable_corridorPenaltyTerm [Countable E] (G : BondGraph V E)
-    (P : V → V → ℝ) (p : unitInterval) {k : ℕ} (D : Set V) (x y : V) (w : RawCorridor G k) :
-    Integrable (fun ω => corridorPenaltyTerm G P p D ω x y w) (law p) := by
-  exact (integrable_const (w.weight P p)).indicator (w.measurableSet_counted D x y)
-
-theorem integral_corridorPenaltyTerm [Countable E] (G : BondGraph V E)
-    (P : V → V → ℝ) (p : unitInterval) {k : ℕ} (D : Set V) (x y : V) (w : RawCorridor G k) :
-    (∫ ω, corridorPenaltyTerm G P p D ω x y w ∂law p) =
-      w.weight P p * (law p).real {ω | w.counted D ω x y} := by
-  have h := integral_indicator_const (μ := law p) (w.weight P p) (w.measurableSet_counted D x y)
-  simpa only [corridorPenaltyTerm,Set.indicator,Set.mem_ofPred_eq,smul_eq_mul,mul_comm] using h
-
- 
-theorem integral_corridorPenaltyTerm_le [Countable V] [Countable E]
-    (G : BondGraph V E) (hG : Function.Injective G) (hloc : LocallyFinite G)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval)
-    (hp : 0 < (p:ℝ)) (hp₁ : (p:ℝ) < 1) {k : ℕ} (x y : V) (w : RawCorridor G k) :
-    (∫ ω, corridorPenaltyTerm G P p Set.univ ω x y w ∂law p) ≤
-      w.probability P * (twoPoint G p x w.base * twoPoint G p w.endpoint y +
-        twoPoint G p x w.endpoint * twoPoint G p w.base y) := by
-  rw [integral_corridorPenaltyTerm]
-  by_cases hl : 3 ≤ w.path.length
-  · have he : {ω | w.counted Set.univ ω x y} =
-        {ω | corridorCounted G w.retained w.blockers Set.univ ω x y} := by
-      ext ω
-      simp only [Set.mem_ofPred_eq,w.counted_iff,hl,true_and]
-    rw [he]
-    exact w.counted_weight_bound hG hloc hl P hP p hp hp₁ x y
-  · have he : {ω | w.counted Set.univ ω x y} = ∅ := by
-      ext ω
-      simp only [Set.mem_ofPred_eq,w.counted_iff,hl,false_and,Set.mem_empty_iff_false]
-    rw [he,measureReal_empty,mul_zero]
-    exact mul_nonneg (w.2.2.mass_nonneg P hP) (add_nonneg
-      (mul_nonneg (twoPoint_nonneg G p x w.base) (twoPoint_nonneg G p w.endpoint y))
-      (mul_nonneg (twoPoint_nonneg G p x w.endpoint) (twoPoint_nonneg G p w.base y)))
-
-end PenaltyTermExpectation
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section FullCorridorPenaltyExpectation
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Countable V] [Countable E]
-
-theorem corridorPenalty_integrable_bound (G : BondGraph V E)
-    (hG : Function.Injective G) (hloc : LocallyFinite G) (p : unitInterval)
-    (hp : 0 < (p:ℝ)) (hp₁ : (p:ℝ) < 1) (a : ℝ)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hPs : IsSelfAdjoint P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (A : Hilbert V →L[ℂ] Hilbert V) (hA : PositiveRealMatrix A) (hAs : IsSelfAdjoint A)
-    (hAT : ∀ x y, matrixCoeff A x y = (twoPoint G p x y : ℂ)) (k : ℕ) (x y : V) :
-    Integrable (fun ω => corridorPenalty G (walkKernel G hloc a) p k Set.univ ω x y) (law p) ∧
-    (∫ ω, corridorPenalty G (walkKernel G hloc a) p k Set.univ ω x y ∂law p) ≤
-      2*(matrixCoeff (A*(P^k*A)) x y).re := by
-  let K := walkKernel G hloc a
-  have hK (u v : V) : 0 ≤ K u v := by simpa only [hP,Complex.ofReal_re] using (hpos u v).2
-  let F := fun w : RawCorridor G k => fun ω : Set E => corridorPenaltyTerm G K p Set.univ ω x y w
-  have hFi (w : RawCorridor G k) : Integrable (F w) (law p) := integrable_corridorPenaltyTerm G K p Set.univ x y w
-  have hFn (w : RawCorridor G k) (ω : Set E) : 0 ≤ F w ω := corridorPenaltyTerm_nonneg G K hK p _ ω x y w
-  have hd := raw_two_diagrams_hasSum G hloc a P hpos hPs hP hK A hA hAs k x y
-  simp only [hAT,Complex.ofReal_re] at hd
-  have hb (w : RawCorridor G k) := integral_corridorPenaltyTerm_le G hG hloc K hK p hp hp₁ x y w
-  have hs : Summable (fun w => ∫ ω, F w ω ∂law p) := Summable.of_nonneg_of_le
-    (fun w => integral_nonneg (hFn w)) hb hd.summable
-  have hsN (ω : Set E) := corridorPenalty_summable G hloc K p k Set.univ ω x y
-  have hlin : (∫⁻ ω, ENNReal.ofReal (corridorPenalty G K p k Set.univ ω x y) ∂law p) =
-      ENNReal.ofReal (∑' w, ∫ ω, F w ω ∂law p) := by
-    have he (ω : Set E) : ENNReal.ofReal (corridorPenalty G K p k Set.univ ω x y) =
-        ∑' w, ENNReal.ofReal (F w ω) := ENNReal.ofReal_tsum_of_nonneg (fun w => hFn w ω) (hsN ω)
-    simp_rw [he]
-    rw [lintegral_tsum (fun w => (hFi w).aestronglyMeasurable.aemeasurable.ennreal_ofReal)]
-    simp_rw [←ofReal_integral_eq_lintegral_ofReal (hFi _) (Filter.Eventually.of_forall (hFn _))]
-    exact (ENNReal.ofReal_tsum_of_nonneg (fun w => integral_nonneg (hFn w)) hs).symm
-  have hNn (ω : Set E) := corridorPenalty_nonneg G K hK p k Set.univ ω x y
-  have hi : Integrable (fun ω => corridorPenalty G K p k Set.univ ω x y) (law p) :=
-    ⟨(measurable_corridorPenalty G K p k Set.univ x y).aestronglyMeasurable,
-      (hasFiniteIntegral_iff_ofReal (Filter.Eventually.of_forall hNn)).mpr (by rw [hlin]; exact ENNReal.ofReal_lt_top)⟩
-  refine ⟨hi,?_⟩
-  have hn : Summable (fun w => ∫ ω, ‖F w ω‖ ∂law p) := by
-    simpa only [Real.norm_of_nonneg (hFn _ _)] using hs
-  change (∫ ω, ∑' w, F w ω ∂law p) ≤ _
-  rw [←integral_tsum_of_summable_integral_norm hFi hn]
-  exact (hs.tsum_le_tsum hb hd.summable).trans_eq hd.tsum_eq
-
-end FullCorridorPenaltyExpectation
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section SmallDeformationEntrywise
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Countable V] [Countable E]
-
- 
-
-theorem deformedKernel_change_bound (G : BondGraph V E)
-    (hG : Function.Injective G) (hloc : LocallyFinite G) (p : unitInterval)
-    (hp : 0 < (p:ℝ)) (hp₁ : (p:ℝ) < 1) (a : ℝ)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hPs : IsSelfAdjoint P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (A : Hilbert V →L[ℂ] Hilbert V) (hA : PositiveRealMatrix A) (hAs : IsSelfAdjoint A)
-    (hAT : ∀ x y, matrixCoeff A x y = (twoPoint G p x y : ℂ))
-    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (x y : V) :
-    0 ≤ twoPoint G p x y - deformedKernel G (walkKernel G hloc a) p k t Set.univ x y ∧
-    twoPoint G p x y - deformedKernel G (walkKernel G hloc a) p k t Set.univ x y ≤
-      2*t*(matrixCoeff (A*(P^k*A)) x y).re := by
-  let K := walkKernel G hloc a
-  have hK (u v : V) : 0 ≤ K u v := by simpa only [hP,Complex.ofReal_re] using (hpos u v).2
-  have hi₀ := integrable_deformedConnection G K hK p k 0 (le_refl _) Set.univ x y
-  have hi := integrable_deformedConnection G K hK p k t ht Set.univ x y
-  have he : twoPoint G p x y - deformedKernel G K p k t Set.univ x y =
-      ∫ ω, deformedConnection G K p k 0 Set.univ ω x y - deformedConnection G K p k t Set.univ ω x y ∂law p := by
-    rw [integral_sub hi₀ hi]
-    change _ = deformedKernel G K p k 0 Set.univ x y - _
-    rw [deformedKernel_zero]
-    rfl
-  rw [he]
-  refine ⟨integral_nonneg (fun ω => (deformedConnection_change G K hK p k t ht Set.univ ω x y).1),?_⟩
-  obtain ⟨hNi,hNb⟩ := corridorPenalty_integrable_bound G hG hloc p hp hp₁ a P hpos hPs hP A hA hAs hAT k x y
-  calc
-    _ ≤ ∫ ω, t*corridorPenalty G K p k Set.univ ω x y ∂law p :=
-      integral_mono (hi₀.sub hi) (hNi.const_mul t)
-        (fun ω => (deformedConnection_change G K hK p k t ht Set.univ ω x y).2)
-    _ = t*∫ ω, corridorPenalty G K p k Set.univ ω x y ∂law p := integral_const_mul _ _
-    _ ≤ t*(2*(matrixCoeff (A*(P^k*A)) x y).re) := mul_le_mul_of_nonneg_left hNb ht
-    _ = _ := by ring
-
-omit [Countable V] [Countable E] in
-theorem deformedKernel_nonneg (G : BondGraph V E) (P : V → V → ℝ) (p : unitInterval)
-    (k : ℕ) (t : ℝ) (D : Set V) (x y : V) : 0 ≤ deformedKernel G P p k t D x y :=
-  integral_nonneg (fun ω => deformedConnection_nonneg G P p k t D ω x y)
-
-end SmallDeformationEntrywise
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Kernel.TraceBounds
+
+namespace OAI
+
+/-! Corridors for bond percolation. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section CorridorEligibility
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace RawCorridor
+variable {G : BondGraph V E} {k : ℕ}
+
+theorem edge_has_label (w : RawCorridor G k) (s : Sym2 V) (hs : s ∈ w.path.edges) :
+    ∃ e ∈ w.retained, G e = s := by
+  induction s using Sym2.inductionOn with
+  | _ x y =>
+    obtain ⟨_,e,_,he⟩ := w.path.adj_of_mem_edges hs
+    exact ⟨e,by simpa only [retained,Set.mem_ofPred_eq,he] using hs,he⟩
+
+theorem image_retained (w : RawCorridor G k) : G '' w.retained = {s | s ∈ w.path.edges} := by
+  ext s
+  constructor
+  · rintro ⟨e,he,rfl⟩; exact he
+  · exact w.edge_has_label s
+
+theorem retained_ncard (hG : Function.Injective G) (w : RawCorridor G k) :
+    w.retained.ncard = w.path.length := by
+  rw [←hG.injOn.ncard_image (s := w.retained),w.image_retained]
+  have hh := List.toFinset_card_of_nodup w.path_isPath.isTrail.edges_nodup
+  simpa only [←List.coe_toFinset,Set.ncard_coe_finset,SimpleGraph.Walk.length_edges] using hh
+
+ 
+
+theorem support_subset_of_retained (w : RawCorridor G k) (hlen : 0 < w.path.length)
+    (D : Set V) (he : w.retained ⊆ internalEdges G D) :
+    ∀ z ∈ w.path.support, z ∈ D := by
+  intro z hz
+  obtain ⟨s,hs,hzs⟩ := (SimpleGraph.Walk.mem_support_iff_exists_mem_edges_of_not_nil
+    (SimpleGraph.Walk.not_nil_iff_lt_length.mpr hlen)).mp hz
+  obtain ⟨e,he',hGe⟩ := w.edge_has_label s hs
+  exact he he' z (hGe.symm ▸ hzs)
+
+theorem retained_subset_of_support (w : RawCorridor G k) (D : Set V)
+    (hs : ∀ z ∈ w.path.support, z ∈ D) : w.retained ⊆ internalEdges G D := by
+  intro e he z hz
+  exact hs z (SimpleGraph.Walk.mem_support_of_mem_edges he hz)
+
+theorem support_iff_retained (w : RawCorridor G k) (hlen : 0 < w.path.length) (D : Set V) :
+    (∀ z ∈ w.path.support, z ∈ D) ↔ w.retained ⊆ internalEdges G D :=
+  ⟨w.retained_subset_of_support D,w.support_subset_of_retained hlen D⟩
+
+theorem retained_incident_interior (w : RawCorridor G k) (hlen : 3 ≤ w.path.length)
+    (e : E) (he : e ∈ w.retained) : ∃ z ∈ w.interior, z ∈ G e := by
+  change G e ∈ w.path.edges at he
+  generalize hab : G e = s at he
+  induction s using Sym2.inductionOn with
+  | _ a b =>
+    have hedge : s(a,b) ∈ w.path.edges := he
+    have ha := w.path.fst_mem_support_of_mem_edges hedge
+    have hb := w.path.snd_mem_support_of_mem_edges hedge
+    have habne : a ≠ b := (w.path.adj_of_mem_edges hedge).1
+    by_contra hn
+    have hna : a ∉ w.interior := fun hi => hn ⟨a,hi,by simp⟩
+    have hnb : b ∉ w.interior := fun hi => hn ⟨b,hi,by simp⟩
+    have hax : a = w.base ∨ a = w.endpoint := by simpa only [interior,Finset.mem_sdiff,List.mem_toFinset,ha,true_and,Finset.mem_insert,Finset.mem_singleton,not_not] using hna
+    have hbx : b = w.base ∨ b = w.endpoint := by simpa only [interior,Finset.mem_sdiff,List.mem_toFinset,hb,true_and,Finset.mem_insert,Finset.mem_singleton,not_not] using hnb
+    have hedge' : s(w.base,w.endpoint) ∈ w.path.edges := by
+      rcases hax with rfl | rfl <;> rcases hbx with rfl | rfl
+      · exact (habne rfl).elim
+      · exact hedge
+      · simpa only [Sym2.eq_swap] using hedge
+      · exact (habne rfl).elim
+    have hh := w.path_isPath.length_eq_one_of_mem_edges hedge'
+    omega
+
+theorem specified_disjoint_exterior (w : RawCorridor G k) (hlen : 3 ≤ w.path.length) :
+    Disjoint (w.retained ∪ w.blockers) (internalEdges G (w.interior : Set V)ᶜ) := by
+  apply Set.disjoint_left.mpr
+  intro e he hx
+  obtain ⟨z,hz,he⟩ := he.elim (w.retained_incident_interior hlen e) (fun hb => hb.1)
+  exact hx z he hz
+
+end RawCorridor
+end CorridorEligibility
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section ExactCorridorPattern
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+def edgePattern (A B : Finset E) : Set (Set E) :=
+  {ω | (A : Set E) ⊆ ω ∧ Disjoint (B : Set E) ω}
+
+theorem edgePattern_eq_restriction (A B : Finset E) (hAB : Disjoint A B) :
+    edgePattern A B = {ω | ω ∩ ((A ∪ B : Finset E) : Set E) = (A : Set E)} := by
+  ext ω
+  constructor
+  · rintro ⟨ha,hb⟩
+    apply Set.Subset.antisymm
+    · rintro e ⟨he,heQ⟩
+      rcases Finset.mem_union.mp heQ with h | h
+      · exact h
+      · exact (Set.disjoint_left.mp hb h he).elim
+    · intro e he
+      exact ⟨ha he,Finset.mem_union_left _ he⟩
+  · intro h
+    constructor
+    · intro e he
+      exact ((Set.ext_iff.mp h e).mpr he).1
+    · apply Set.disjoint_left.mpr
+      intro e he hw
+      have ha : e ∈ A := (Set.ext_iff.mp h e).mp ⟨hw,Finset.mem_union_right _ he⟩
+      exact Finset.disjoint_left.mp hAB ha he
+
+theorem measurableSet_edgePattern (A B : Finset E) (hAB : Disjoint A B) :
+    MeasurableSet (edgePattern A B) := by
+  rw [edgePattern_eq_restriction A B hAB]
+  exact measurableSet_finite_restriction (A∪B) {(A:Set E)}
+
+ 
+theorem edgePattern_probability (A B : Finset E) (hAB : Disjoint A B) (p : unitInterval) :
+    (law p).real (edgePattern A B) = (p:ℝ)^A.card*(1-(p:ℝ))^B.card := by
+  let f : Set E → ℝ := fun ω => if ω ∈ edgePattern A B then 1 else 0
+  have hf (ω : Set E) : f ω = f (ω ∩ ((A∪B:Finset E):Set E)) := by
+    simp only [f,edgePattern_eq_restriction A B hAB,Set.mem_ofPred_eq,Set.inter_assoc,Set.inter_self]
+  rw [←integral_event_indicator p _ (measurableSet_edgePattern A B hAB)]
+  change (∫ ω, f ω ∂law p) = _
+  rw [integral_eq_bernoulliPolynomial (A∪B) p f hf]
+  trans bernoulliPolynomial (A∪B) (fun η => if η=A then 1 else 0) p
+  · apply bernoulliPolynomial_congr
+    intro η hη
+    have he : (η:Set E) ∩ ((A∪B:Finset E):Set E) = (η:Set E) :=
+      Set.inter_eq_left.mpr hη
+    simp only [f,edgePattern_eq_restriction A B hAB,Set.mem_ofPred_eq,he,Finset.coe_inj]
+  · simp only [bernoulliPolynomial,ite_mul,zero_mul,one_mul]
+    rw [Finset.sum_ite_eq']
+    simp only [Finset.mem_powerset,Finset.subset_union_left,ite_true,Finset.card_union_of_disjoint hAB,
+      Nat.add_sub_cancel_left]
+
+namespace RawCorridor
+variable {G : BondGraph V E} {k : ℕ}
+
+def calibrationEvent (w : RawCorridor G k) : Set (Set E) :=
+  {ω | w.retained ⊆ ω ∧ Disjoint w.blockers ω}
+
+theorem calibrationEvent_eq (hG : Function.Injective G) (hloc : LocallyFinite G)
+    (w : RawCorridor G k) : w.calibrationEvent =
+      edgePattern (w.retained_finite hG).toFinset (w.blockers_finite hloc).toFinset := by
+  simp only [calibrationEvent,edgePattern,Set.Finite.coe_toFinset]
+
+theorem calibration_probability (hG : Function.Injective G) (hloc : LocallyFinite G)
+    (w : RawCorridor G k) (p : unitInterval) :
+    (law p).real w.calibrationEvent = (p:ℝ)^w.path.length*(1-(p:ℝ))^w.blockers.ncard := by
+  rw [w.calibrationEvent_eq hG hloc,edgePattern_probability]
+  · rw [←Set.ncard_eq_toFinset_card w.retained (w.retained_finite hG),←Set.ncard_eq_toFinset_card w.blockers (w.blockers_finite hloc),w.retained_ncard hG]
+  · apply Finset.disjoint_left.mpr
+    intro e he hb
+    exact Set.disjoint_left.mp w.retained_disjoint_blockers
+      ((w.retained_finite hG).mem_toFinset.mp he) ((w.blockers_finite hloc).mem_toFinset.mp hb)
+
+ 
+
+theorem calibration_identity (hG : Function.Injective G) (hloc : LocallyFinite G)
+    (w : RawCorridor G k) (P : V → V → ℝ) (p : unitInterval)
+    (hp : 0 < (p:ℝ)) (hp₁ : (p:ℝ) < 1) :
+    w.weight P p * (law p).real w.calibrationEvent = w.probability P := by
+  rw [w.calibration_probability hG hloc p]
+  unfold weight
+  have h₁ := pow_ne_zero w.path.length hp.ne'
+  have h₂ := pow_ne_zero w.blockers.ncard (sub_pos.mpr hp₁).ne'
+  field_simp
+
+end RawCorridor
+end ExactCorridorPattern
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section CorridorTerminalGeometry
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem path_internal_two_neighbors {H : SimpleGraph V} {u v z : V}
+    (q : H.Walk u v) (hq : q.IsPath) (hz : z ∈ q.support) (hzu : z ≠ u) (hzv : z ≠ v) :
+    ∃ a b : V, a ≠ b ∧ s(z,a) ∈ q.edges ∧ s(z,b) ∈ q.edges := by
+  induction q with
+  | nil => simp only [SimpleGraph.Walk.support_nil,List.mem_singleton] at hz; exact (hzu hz).elim
+  | @cons u a v h q ih =>
+    have htail : q.IsPath := hq.of_cons
+    have huz : u ∉ q.support := ((SimpleGraph.Walk.cons_isPath_iff _ _).mp hq).2
+    have hzq : z ∈ q.support := (List.mem_cons.mp hz).resolve_left hzu
+    by_cases hza : z = a
+    · subst z
+      cases q with
+      | nil => exact (hzv rfl).elim
+      | @cons a b v h' q =>
+        refine ⟨u,b,?_,?_,?_⟩
+        · intro hub
+          apply huz
+          subst b
+          simp
+        · simp only [SimpleGraph.Walk.edges_cons,List.mem_cons,Sym2.eq_swap]; exact Or.inl trivial
+        · simp only [SimpleGraph.Walk.edges_cons,List.mem_cons]; exact Or.inr (Or.inl trivial)
+    · obtain ⟨b,c,hbc,hb,hc⟩ := ih htail hzq hza hzv
+      exact ⟨b,c,hbc,List.mem_cons_of_mem _ hb,List.mem_cons_of_mem _ hc⟩
+
+ 
+
+theorem pivotal_image_mem_edges (G : BondGraph V E) (ω : Set E) {x y : V}
+    (q : (openSimpleGraph G ω).Walk x y) (e : E)
+    (hp : ¬Conn G (ω \ {e}) x y) : G e ∈ q.edges := by
+  obtain ⟨l,hl,hmap,ho⟩ := walk_lift_bonds G ω q
+  by_contra he
+  apply hp
+  apply hl.conn
+  intro f hf
+  refine ⟨ho f hf,?_⟩
+  intro hfe
+  have hfe' : f = e := Set.mem_singleton_iff.mp hfe
+  subst f
+  apply he
+  rw [←hmap]
+  exact List.mem_map_of_mem hf
+
+namespace RawCorridor
+variable {G : BondGraph V E} {k : ℕ}
+
+theorem counted_terminal_outside (w : RawCorridor G k) (D : Set V) (ω : Set E) (x y : V)
+    (hc : corridorCounted G w.retained w.blockers D ω x y) : x ∉ w.interior ∧ y ∉ w.interior := by
+  have hleft (x y : V) (hc : corridorCounted G w.retained w.blockers D ω x y) : x ∉ w.interior := by
+    intro hx
+    have hx' := Finset.mem_sdiff.mp hx
+    have hxs : x ∈ w.path.support := List.mem_toFinset.mp hx'.1
+    have hxne : x ≠ w.base ∧ x ≠ w.endpoint := by
+      simpa only [Finset.mem_insert,Finset.mem_singleton,not_or] using hx'.2
+    have hxu := hxne.1
+    have hxv := hxne.2
+    obtain ⟨a,b,hab,ha,hb⟩ := path_internal_two_neighbors w.path w.path_isPath hxs hxu hxv
+    obtain ⟨q⟩ := (conn_iff_reachable G _ x y).mp hc.1
+    let qp := q.bypass
+    have hqp : qp.IsPath := q.bypass_isPath
+    obtain ⟨e,he,hGe⟩ := w.edge_has_label _ ha
+    obtain ⟨f,hf,hGf⟩ := w.edge_has_label _ hb
+    have hea := pivotal_image_mem_edges G _ qp e (hc.2.1 e he).2
+    have heb := pivotal_image_mem_edges G _ qp f (hc.2.1 f hf).2
+    rw [hGe] at hea
+    rw [hGf] at heb
+    exact hab ((hqp.eq_snd_of_mem_edges hea).trans (hqp.eq_snd_of_mem_edges heb).symm)
+  exact ⟨hleft x y hc,hleft y x ((corridorCounted_symm G _ _ D ω x y).mp hc)⟩
+
+end RawCorridor
+end CorridorTerminalGeometry
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory ProbabilityTheory
+section CorridorIndependence
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem measurable_inter_const (S : Set E) : Measurable (fun ω : Set E => ω ∩ S) := by
+  apply measurable_setOfPred.comp
+  apply Measurable.of_eval
+  intro e
+  exact ((measurable_pi_apply e).comp MeasurableEquiv.setOfPred.symm.measurable).and measurable_const
+
+ 
+theorem restriction_events_independent (p : unitInterval) (S T : Set E)
+    (hST : Disjoint S T) (A B : Set (Set E)) (hA : MeasurableSet A) (hB : MeasurableSet B) :
+    (law p) ((fun ω => ω ∩ S) ⁻¹' A ∩ (fun ω => ω ∩ T) ⁻¹' B) =
+      (law p) ((fun ω => ω ∩ S) ⁻¹' A) * (law p) ((fun ω => ω ∩ T) ⁻¹' B) := by
+  let μ : Measure (E → Prop) := Measure.infinitePi (fun _ => bitLaw p)
+  let σ : E → MeasurableSpace (E → Prop) := fun e =>
+    MeasurableSpace.comap (fun f : E → Prop => f e) inferInstance
+  have hi : iIndep σ μ := iIndepFun_infinitePi (X := fun (_ : E) (b : Prop) => b)
+    (fun _ => measurable_id)
+  have hle (e : E) : σ e ≤ (inferInstance : MeasurableSpace (E → Prop)) :=
+    (measurable_pi_apply e).comap_le
+  have hr (S : Set E) : @Measurable (E → Prop) (Set E) (⨆ e ∈ S, σ e) inferInstance
+      (fun f => {e | f e} ∩ S) := by
+    let : MeasurableSpace (E → Prop) := ⨆ e ∈ S, σ e
+    apply measurable_setOfPred.comp
+    apply Measurable.of_eval
+    intro e
+    change @Measurable (E → Prop) Prop (⨆ e ∈ S, σ e) inferInstance (fun f => f e ∧ e ∈ S)
+    by_cases he : e ∈ S
+    · have hm : @Measurable (E → Prop) Prop (⨆ e ∈ S, σ e) inferInstance (fun f => f e) :=
+        Measurable.of_comap_le (le_iSup_of_le e (le_iSup_of_le he le_rfl))
+      simpa only [Set.mem_inter_iff,Set.mem_ofPred_eq,he,and_true] using hm
+    · simpa only [Set.mem_inter_iff,Set.mem_ofPred_eq,he,and_false] using
+        (measurable_const : @Measurable (E → Prop) Prop (⨆ e ∈ S, σ e) inferInstance (fun _ => False))
+  have hm : (law p : Measure (Set E)) = μ.map (fun f => {e | f e}) := by
+    simp only [law,setBernoulli_eq_map,Set.mem_univ,μ,bitLaw,bernoulliMeasure_def]
+  rw [hm,Measure.map_apply measurable_setOfPred ((hA.preimage (measurable_inter_const S)).inter
+    (hB.preimage (measurable_inter_const T))),
+    Measure.map_apply measurable_setOfPred (hA.preimage (measurable_inter_const S)),
+    Measure.map_apply measurable_setOfPred (hB.preimage (measurable_inter_const T))]
+  exact (Indep_iff _ _ _).mp (indep_iSup_of_disjoint hle hi hST) _ _
+    (hA.preimage (hr S)) (hB.preimage (hr T))
+
+namespace RawCorridor
+variable {G : BondGraph V E} {k : ℕ}
+
+theorem calibration_restriction (w : RawCorridor G k) (ω : Set E) :
+    ω ∈ w.calibrationEvent ↔ ω ∩ (w.retained ∪ w.blockers) ∈ w.calibrationEvent := by
+  constructor
+  · rintro ⟨ha,hb⟩
+    exact ⟨fun e he => ⟨ha he,Or.inl he⟩,hb.mono_right Set.inter_subset_left⟩
+  · rintro ⟨ha,hb⟩
+    refine ⟨fun e he => (ha he).1,Set.disjoint_left.mpr ?_⟩
+    intro e he hw
+    exact Set.disjoint_left.mp hb he ⟨hw,Or.inr he⟩
+
+ 
+
+theorem calibration_independent (hG : Function.Injective G) (hloc : LocallyFinite G)
+    (w : RawCorridor G k) (hlen : 3 ≤ w.path.length) (p : unitInterval)
+    (A : Set (Set E)) (hA : MeasurableSet A) :
+    (law p) (w.calibrationEvent ∩ (fun ω => ω ∩ internalEdges G (w.interior : Set V)ᶜ) ⁻¹' A) =
+      (law p) w.calibrationEvent *
+        (law p) ((fun ω => ω ∩ internalEdges G (w.interior : Set V)ᶜ) ⁻¹' A) := by
+  have hQ : MeasurableSet w.calibrationEvent := by
+    rw [w.calibrationEvent_eq hG hloc]
+    apply measurableSet_edgePattern
+    apply Finset.disjoint_left.mpr
+    intro e he hb
+    exact Set.disjoint_left.mp w.retained_disjoint_blockers
+      ((w.retained_finite hG).mem_toFinset.mp he) ((w.blockers_finite hloc).mem_toFinset.mp hb)
+  have heq : (fun ω => ω ∩ (w.retained ∪ w.blockers)) ⁻¹' w.calibrationEvent = w.calibrationEvent := by
+    ext ω; exact (w.calibration_restriction ω).symm
+  have hh := restriction_events_independent p _ _ (w.specified_disjoint_exterior hlen)
+    w.calibrationEvent A hQ hA
+  simpa only [heq] using hh
+
+end RawCorridor
+end CorridorIndependence
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section CorridorContraction
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem conn_union_attachment (G : BondGraph V E) (S T : Set E) (A : Set V)
+    (hclosed : ∀ a b, a ∈ A → Conn G S a b → b ∈ A)
+    (hT : ∀ e ∈ T, ∀ z ∈ G e, z ∈ A) {x y : V} (hc : Conn G (S ∪ T) x y) :
+    Conn G S x y ∨ x ∈ A ∧ y ∈ A := by
+  induction hc with
+  | refl => exact Or.inl (conn_refl G S x)
+  | @tail a b hxa hab ih =>
+    obtain ⟨e,he,heq⟩ := hab
+    rcases he with he | he
+    · have hab : Conn G S a b := Relation.ReflTransGen.single ⟨e,he,heq⟩
+      exact ih.elim (fun h => Or.inl (conn_trans G S h hab))
+        (fun h => Or.inr ⟨h.1,hclosed a b h.2 hab⟩)
+    · have ha := hT e he a (heq.symm ▸ (by simp : a ∈ s(a,b)))
+      have hb := hT e he b (heq.symm ▸ (by simp : b ∈ s(a,b)))
+      exact Or.inr ⟨ih.elim (fun h => hclosed a x ha (conn_symm G S h)) And.left,hb⟩
+
+namespace RawCorridor
+variable {G : BondGraph V E} {k : ℕ}
+
+def exterior (w : RawCorridor G k) (ω : Set E) : Set E :=
+  ω ∩ internalEdges G (w.interior : Set V)ᶜ
+
+def attachment (w : RawCorridor G k) (ω : Set E) : Set V :=
+  {z | z ∈ w.path.support ∨ Conn G (w.exterior ω) w.base z ∨ Conn G (w.exterior ω) w.endpoint z}
+
+theorem support_exterior_ports (w : RawCorridor G k) {z : V}
+    (hz : z ∈ w.path.support) (ho : z ∉ w.interior) : z = w.base ∨ z = w.endpoint := by
+  simpa only [interior,Finset.mem_sdiff,List.mem_toFinset,hz,true_and,
+    Finset.mem_insert,Finset.mem_singleton,not_not] using ho
+
+theorem attachment_closed (w : RawCorridor G k) (ω : Set E) (a b : V)
+    (ha : a ∈ w.attachment ω) (hc : Conn G (w.exterior ω) a b) : b ∈ w.attachment ω := by
+  rcases ha with ha | ha | ha
+  · rcases conn_internal_eq_or_mem G (w.interior : Set V)ᶜ hc with rfl | ho
+    · exact Or.inl ha
+    · rcases w.support_exterior_ports ha ho.1 with rfl | rfl
+      · exact Or.inr (Or.inl hc)
+      · exact Or.inr (Or.inr hc)
+  · exact Or.inr (Or.inl (conn_trans G _ ha hc))
+  · exact Or.inr (Or.inr (conn_trans G _ ha hc))
+
+theorem open_subset_exterior_retained (w : RawCorridor G k) (ω : Set E)
+    (hq : ω ∈ w.calibrationEvent) : ω ⊆ w.exterior ω ∪ w.retained := by
+  intro e he
+  by_cases hr : e ∈ w.retained
+  · exact Or.inr hr
+  · left
+    refine ⟨he,?_⟩
+    intro z hz hi
+    exact Set.disjoint_left.mp hq.2 ⟨⟨z,hi,hz⟩,hr⟩ he
+
+theorem connection_exterior_or_attachment (w : RawCorridor G k) (ω : Set E)
+    (hq : ω ∈ w.calibrationEvent) {x y : V} (hc : Conn G ω x y) :
+    Conn G (w.exterior ω) x y ∨ x ∈ w.attachment ω ∧ y ∈ w.attachment ω := by
+  apply conn_union_attachment G _ w.retained (w.attachment ω) (w.attachment_closed ω)
+    (fun e he z hz => Or.inl (SimpleGraph.Walk.mem_support_of_mem_edges he hz))
+  exact conn_mono G (w.open_subset_exterior_retained ω hq) hc
+
+theorem attachment_outside (w : RawCorridor G k) (ω : Set E) {z : V}
+    (ha : z ∈ w.attachment ω) (hz : z ∉ w.interior) :
+    Conn G (w.exterior ω) w.base z ∨ Conn G (w.exterior ω) w.endpoint z := by
+  rcases ha with ha | ha | ha
+  · rcases w.support_exterior_ports ha hz with rfl | rfl
+    · exact Or.inl (conn_refl G _ _)
+    · exact Or.inr (conn_refl G _ _)
+  · exact Or.inl ha
+  · exact Or.inr ha
+
+ 
+theorem counted_exterior_orientation (w : RawCorridor G k) (hlen : 3 ≤ w.path.length)
+    (ω : Set E) (hq : ω ∈ w.calibrationEvent) (x y : V)
+    (hc : corridorCounted G w.retained w.blockers Set.univ ω x y) :
+    ¬Conn G (w.exterior ω) w.base w.endpoint ∧
+      ((Conn G (w.exterior ω) w.base x ∧ Conn G (w.exterior ω) w.endpoint y) ∨
+       (Conn G (w.exterior ω) w.endpoint x ∧ Conn G (w.exterior ω) w.base y)) := by
+  have hi : internalEdges G Set.univ = Set.univ := Set.eq_univ_of_forall (fun _ _ _ => trivial)
+  have hc' : Conn G ω x y := by simpa only [hi,Set.inter_univ] using hc.1
+  have hs : w.retained.Nonempty := by
+    have hp : w.path.edges ≠ [] := by
+      intro he
+      have hh := congrArg List.length he
+      simp only [SimpleGraph.Walk.length_edges,List.length_nil] at hh
+      omega
+    obtain ⟨s,hs⟩ := List.exists_mem_of_ne_nil _ hp
+    obtain ⟨e,he,_⟩ := w.edge_has_label s hs
+    exact ⟨e,he⟩
+  have hnot : ¬Conn G (w.exterior ω) x y := by
+    intro h
+    obtain ⟨e,he⟩ := hs
+    apply (hc.2.1 e he).2
+    apply conn_mono G ?_ h
+    intro f hf
+    refine ⟨?_,?_⟩
+    · simpa only [hi,Set.inter_univ] using hf.1
+    · intro hfe
+      have hfe' := Set.mem_singleton_iff.mp hfe
+      subst f
+      exact Set.disjoint_left.mp (w.specified_disjoint_exterior hlen) (Or.inl he) hf.2
+  have hatt := (w.connection_exterior_or_attachment ω hq hc').resolve_left hnot
+  have hout := w.counted_terminal_outside Set.univ ω x y hc
+  have hx := w.attachment_outside ω hatt.1 hout.1
+  have hy := w.attachment_outside ω hatt.2 hout.2
+  have hne : ¬Conn G (w.exterior ω) w.base w.endpoint := by
+    intro h
+    apply hnot
+    have hx' : Conn G (w.exterior ω) w.base x := hx.elim id (conn_trans G _ h)
+    have hy' : Conn G (w.exterior ω) w.base y := hy.elim id (conn_trans G _ h)
+    exact conn_trans G _ (conn_symm G _ hx') hy'
+  refine ⟨hne,?_⟩
+  rcases hx with hx | hx <;> rcases hy with hy | hy
+  · exact (hnot (conn_trans G _ (conn_symm G _ hx) hy)).elim
+  · exact Or.inl ⟨hx,hy⟩
+  · exact Or.inr ⟨hx,hy⟩
+  · exact (hnot (conn_trans G _ (conn_symm G _ hx) hy)).elim
+
+end RawCorridor
+end CorridorContraction
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section CorridorBKCalibration
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Countable V] [Countable E]
+namespace RawCorridor
+variable {G : BondGraph V E} {k : ℕ}
+
+def exteriorOrientations (w : RawCorridor G k) (x y : V) : Set (Set E) :=
+  orientedPivotalEvent G x y w.base w.endpoint ∪ orientedPivotalEvent G x y w.endpoint w.base
+
+theorem measurableSet_exteriorOrientations (w : RawCorridor G k) (x y : V) :
+    MeasurableSet (w.exteriorOrientations x y) :=
+  (measurableSet_orientedPivotalEvent G x y w.base w.endpoint).union
+    (measurableSet_orientedPivotalEvent G x y w.endpoint w.base)
+
+omit [Countable V] [Countable E] in
+theorem counted_subset_calibration_exterior (w : RawCorridor G k)
+    (hlen : 3 ≤ w.path.length) (x y : V) :
+    {ω | corridorCounted G w.retained w.blockers Set.univ ω x y} ⊆
+      w.calibrationEvent ∩ (w.exterior ⁻¹' w.exteriorOrientations x y) := by
+  intro ω hc
+  have hi : internalEdges G Set.univ = Set.univ := Set.eq_univ_of_forall (fun _ _ _ => trivial)
+  have hq : ω ∈ w.calibrationEvent := by
+    refine ⟨fun e he => ?_,?_⟩
+    · simpa only [hi,Set.inter_univ] using (hc.2.1 e he).1
+    · simpa only [hi,Set.inter_univ] using hc.2.2
+  refine ⟨hq,?_⟩
+  obtain ⟨hne,ho⟩ := w.counted_exterior_orientation hlen ω hq x y hc
+  rcases ho with ho | ho
+  · exact Or.inl ⟨conn_symm G _ ho.1,ho.2,hne⟩
+  · exact Or.inr ⟨conn_symm G _ ho.1,ho.2,fun h => hne (conn_symm G _ h)⟩
+
+omit [Countable V] in
+theorem exterior_orientation_BK (w : RawCorridor G k) (p : unitInterval) (x y a b : V) :
+    (law p).real (w.exterior ⁻¹' orientedPivotalEvent G x y a b) ≤
+      twoPoint G p x a * twoPoint G p b y := by
+  apply le_trans (measureReal_mono ?_) (bk p {ω | Conn G ω x a} {ω | Conn G ω b y}
+    (fun _ _ h hc => conn_mono G h hc) (fun _ _ h hc => conn_mono G h hc) (finiteWitness_conn G x a) (finiteWitness_conn G b y))
+  intro ω h
+  exact disjointOccurrence_upper _ _ Set.inter_subset_left
+    (orientedPivotalEvent_subset_disjoint G x y a b h)
+
+ 
+
+theorem counted_weight_bound (hG : Function.Injective G) (hloc : LocallyFinite G)
+    (w : RawCorridor G k) (hlen : 3 ≤ w.path.length)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval)
+    (hp : 0 < (p:ℝ)) (hp₁ : (p:ℝ) < 1) (x y : V) :
+    w.weight P p * (law p).real {ω | corridorCounted G w.retained w.blockers Set.univ ω x y} ≤
+      w.probability P * (twoPoint G p x w.base * twoPoint G p w.endpoint y +
+        twoPoint G p x w.endpoint * twoPoint G p w.base y) := by
+  have hi := w.calibration_independent hG hloc hlen p (w.exteriorOrientations x y)
+    (w.measurableSet_exteriorOrientations x y)
+  have hir : (law p).real (w.calibrationEvent ∩ w.exterior ⁻¹' w.exteriorOrientations x y) =
+      (law p).real w.calibrationEvent * (law p).real (w.exterior ⁻¹' w.exteriorOrientations x y) := by
+    exact congrArg ENNReal.toReal hi |>.trans ENNReal.toReal_mul
+  have hb : (law p).real (w.exterior ⁻¹' w.exteriorOrientations x y) ≤
+      twoPoint G p x w.base * twoPoint G p w.endpoint y +
+        twoPoint G p x w.endpoint * twoPoint G p w.base y := by
+    apply le_trans (measureReal_union_le _ _)
+    exact add_le_add (w.exterior_orientation_BK p x y w.base w.endpoint)
+      (w.exterior_orientation_BK p x y w.endpoint w.base)
+  calc
+    _ ≤ w.weight P p * (law p).real (w.calibrationEvent ∩ w.exterior ⁻¹' w.exteriorOrientations x y) :=
+      mul_le_mul_of_nonneg_left (measureReal_mono (w.counted_subset_calibration_exterior hlen x y))
+        (w.weight_nonneg P hP p)
+    _ = w.probability P * (law p).real (w.exterior ⁻¹' w.exteriorOrientations x y) := by
+      rw [hir,←mul_assoc,w.calibration_identity hG hloc P p hp hp₁]
+    _ ≤ _ := mul_le_mul_of_nonneg_left hb (w.2.2.mass_nonneg P hP)
+
+end RawCorridor
+end CorridorBKCalibration
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section RawPathIntegration
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+abbrev RawFrom (H : SimpleGraph V) (k : ℕ) (x : V) := Σ y : V, HoldingWalk H k x y
+
+namespace RawFrom
+variable {H : SimpleGraph V}
+
+def zeroEquiv (x : V) : RawFrom H 0 x ≃ Unit where
+  toFun _ := ()
+  invFun _ := ⟨x,.nil x⟩
+  left_inv w := by rcases w with ⟨y,w⟩; cases w; rfl
+  right_inv _ := rfl
+
+def succEquiv (k : ℕ) (x : V) : RawFrom H (k+1) x ≃
+    RawFrom H k x ⊕ (Σ z : H.neighborSet x, RawFrom H k z.val) where
+  toFun
+    | ⟨y,.hold w⟩ => .inl ⟨y,w⟩
+    | ⟨y,.step (z := z) h w⟩ => .inr ⟨⟨z,h⟩,y,w⟩
+  invFun
+    | .inl ⟨y,w⟩ => ⟨y,.hold w⟩
+    | .inr ⟨⟨z,h⟩,y,w⟩ => ⟨y,.step h w⟩
+  left_inv w := by rcases w with ⟨y,w⟩; cases w <;> rfl
+  right_inv w := by rcases w with ⟨y,w⟩ | ⟨⟨z,h⟩,y,w⟩ <;> rfl
+
+instance finite [∀ x, Finite (H.neighborSet x)] (k : ℕ) (x : V) : Finite (RawFrom H k x) := by
+  induction k generalizing x with
+  | zero => exact Finite.of_equiv Unit (zeroEquiv x).symm
+  | succ k ih =>
+    let (z : V) : Finite (RawFrom H k z) := ih z
+    exact Finite.of_equiv _ (succEquiv k x).symm
+
+noncomputable def integral (P : V → V → ℝ) (k : ℕ) (x : V) (f : V → ℝ) : ℝ :=
+  ∑' w : RawFrom H k x, w.2.mass P * f w.1
+
+theorem integral_zero (P : V → V → ℝ) (x : V) (f : V → ℝ) :
+    integral (H := H) P 0 x f = f x := by
+  unfold integral
+  rw [←(zeroEquiv x).symm.tsum_eq]
+  simp only [zeroEquiv,Equiv.symm_mk,Equiv.coe_fn_mk,HoldingWalk.mass,one_mul,tsum_fintype,Fintype.sum_unique]
+
+theorem integral_succ [∀ x, Finite (H.neighborSet x)] (P : V → V → ℝ)
+    (k : ℕ) (x : V) (f : V → ℝ) :
+    integral (H := H) P (k+1) x f =
+      P x x * integral (H := H) P k x f +
+        ∑' z : H.neighborSet x, P x z * integral (H := H) P k z f := by
+  unfold integral
+  rw [←(succEquiv k x).symm.tsum_eq]
+  let := Fintype.ofFinite (RawFrom H k x)
+  let := Fintype.ofFinite (H.neighborSet x)
+  let (z : H.neighborSet x) := Fintype.ofFinite (RawFrom H k z)
+  simp only [tsum_fintype,Fintype.sum_sum_type,Fintype.sum_sigma,succEquiv,
+    Equiv.symm_mk,Equiv.coe_fn_mk,HoldingWalk.mass,mul_assoc,Finset.mul_sum]
+
+end RawFrom
+
+theorem finite_openSimpleGraph_neighbors (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) :
+    Finite ((openSimpleGraph G Set.univ).neighborSet x) := by
+  apply Set.Finite.to_subtype
+  apply (finite_neighbors hloc x).subset
+  intro y hy
+  obtain ⟨_,e,_,he⟩ := hy
+  exact ⟨e,he⟩
+
+end RawPathIntegration
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section RawPathOperator
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ theorem sum_step_decomposition (H : SimpleGraph V) (P : V → V → ℝ)
+    (hs : ∀ x, (Function.support (P x)).Finite)
+    (hstep : ∀ x y, x ≠ y → ¬H.Adj x y → P x y = 0) (x : V) (f : V → ℝ) :
+    (∑' y, P x y * f y) = P x x * f x + ∑' y : H.neighborSet x, P x y * f y := by
+  rw [(finiteKernel_summable_mul P hs x f).tsum_eq_add_tsum_ite x]
+  congr 1
+  have he : (∑' y : H.neighborSet x, if (y:V) = x then (0:ℝ) else P x y * f y) =
+      ∑' y, if y = x then (0:ℝ) else P x y * f y := by
+    apply tsum_subtype_eq_of_support_subset (f := fun y : V => if y = x then (0:ℝ) else P x y * f y)
+    intro y hy
+    by_contra hn
+    apply hy
+    by_cases h : y = x
+    · simp only [h,ite_true]
+    · simp only [ite_eq_right h,hstep x y (Ne.symm h) hn,zero_mul]
+  rw [←he]
+  apply tsum_congr
+  intro y
+  have hn : (y:V) ≠ x := by
+    intro h
+    have hy : H.Adj x y := y.property
+    rw [h] at hy
+    exact H.loopless.irrefl x hy
+  rw [ite_eq_right hn]
+
+ theorem raw_integral_operator (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (k : ℕ) (x : V) (f : V → ℝ) :
+    RawFrom.integral (H := openSimpleGraph G Set.univ) (walkKernel G hloc a) k x f =
+      ∑' y, (matrixCoeff (P^k) x y).re * f y := by
+  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) :=
+    finite_openSimpleGraph_neighbors G hloc z
+  have hs (x : V) : (Function.support (walkKernel G hloc a x)).Finite := by
+    simpa only [pow_one,hP,Complex.ofReal_re] using walkPower_finite_support G hloc a P hpos hP 1 x
+  have hsP (x : V) : (Function.support (fun y => (matrixCoeff P x y).re)).Finite := by
+    simpa only [hP,Complex.ofReal_re] using hs x
+  induction k generalizing x with
+  | zero => rw [RawFrom.integral_zero,pow_zero,positive_identity_action]
+  | succ k ih =>
+    rw [RawFrom.integral_succ,pow_succ',finite_positive_product_action P (P^k) hpos hsP
+      (finite_positive_pow_support P hpos hsP k)]
+    simp only [hP,Complex.ofReal_re]
+    rw [sum_step_decomposition (openSimpleGraph G Set.univ) (walkKernel G hloc a) hs]
+    · simp only [ih]
+    · intro u v huv hn
+      apply walkKernel_zero_of_not_step G hloc a u v huv
+      intro ha
+      apply hn
+      obtain ⟨e,he⟩ := ha
+      exact ⟨huv,e,Set.mem_univ e,he⟩
+
+end RawPathOperator
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section CorridorSum
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace RawCorridor
+variable {G : BondGraph V E} {k : ℕ}
+
+ 
+
+def counted (w : RawCorridor G k) (D : Set V) (ω : Set E) (x y : V) : Prop :=
+  3 ≤ w.path.length ∧ (∀ z ∈ w.path.support, z ∈ D) ∧
+    corridorCounted G w.retained w.blockers D ω x y
+
+theorem counted_iff (w : RawCorridor G k) (D : Set V) (ω : Set E) (x y : V) :
+    w.counted D ω x y ↔ 3 ≤ w.path.length ∧ corridorCounted G w.retained w.blockers D ω x y := by
+  constructor
+  · exact fun h => ⟨h.1,h.2.2⟩
+  · rintro ⟨hlen,hc⟩
+    exact ⟨hlen,w.support_subset_of_retained (by omega) D (fun e he => (hc.2.1 e he).1.2),hc⟩
+
+theorem counted_base_mem_walk (w : RawCorridor G k) (D : Set V) (ω : Set E) (x y : V)
+    (hc : w.counted D ω x y) (q : (openSimpleGraph G (ω ∩ internalEdges G D)).Walk x y) :
+    w.base ∈ q.support := by
+  have hl : 0 < w.path.length := by have := hc.1; omega
+  obtain ⟨s,hs,hbs⟩ := (SimpleGraph.Walk.mem_support_iff_exists_mem_edges_of_not_nil
+    (SimpleGraph.Walk.not_nil_iff_lt_length.mpr hl)).mp w.path.start_mem_support
+  obtain ⟨e,he,hGe⟩ := w.edge_has_label s hs
+  have heq := pivotal_image_mem_edges G _ q e (hc.2.2.2.1 e he).2
+  exact SimpleGraph.Walk.mem_support_of_mem_edges heq (hGe.symm ▸ hbs)
+
+theorem counted_symm (w : RawCorridor G k) (D : Set V) (ω : Set E) (x y : V) :
+    w.counted D ω x y ↔ w.counted D ω y x := by
+  rw [counted_iff,counted_iff,corridorCounted_symm]
+
+theorem measurableSet_counted [Countable E] (w : RawCorridor G k) (D : Set V) (x y : V) :
+    MeasurableSet {ω | w.counted D ω x y} := by
+  simp_rw [w.counted_iff]
+  by_cases h : 3 ≤ w.path.length
+  · simpa only [h,true_and] using measurableSet_corridorCounted G w.retained w.blockers D x y
+  · simp only [h,false_and,Set.ofPred_false]; exact MeasurableSet.empty
+
+end RawCorridor
+
+ theorem finite_corridors_on_bases (G : BondGraph V E) (hloc : LocallyFinite G) (k : ℕ)
+    (S : Set V) (hS : S.Finite) : {w : RawCorridor G k | w.base ∈ S}.Finite := by
+  let (x : V) : Finite ((openSimpleGraph G Set.univ).neighborSet x) :=
+    finite_openSimpleGraph_neighbors G hloc x
+  let : Finite S := hS.to_subtype
+  let f : (Σ x : S, RawFrom (openSimpleGraph G Set.univ) k x.val) → RawCorridor G k :=
+    fun w => ⟨w.1.val,w.2⟩
+  have hr : {w : RawCorridor G k | w.base ∈ S} = Set.range f := by
+    ext w
+    constructor
+    · intro hw
+      exact ⟨⟨⟨w.base,hw⟩,w.2⟩,rfl⟩
+    · rintro ⟨w,rfl⟩
+      exact w.1.property
+  rw [hr]
+  exact Set.finite_range f
+
+ 
+
+theorem finite_counted_corridors (G : BondGraph V E) (hloc : LocallyFinite G) (k : ℕ)
+    (D : Set V) (ω : Set E) (x y : V) : {w : RawCorridor G k | w.counted D ω x y}.Finite := by
+  by_cases hc : Conn G (ω ∩ internalEdges G D) x y
+  · obtain ⟨q⟩ := (conn_iff_reachable G _ x y).mp hc
+    apply (finite_corridors_on_bases G hloc k {z | z ∈ q.support} (List.finite_toSet _)).subset
+    exact fun w hw => w.counted_base_mem_walk D ω x y hw q
+  · have he : {w : RawCorridor G k | w.counted D ω x y} = ∅ := by
+      ext w
+      exact ⟨fun h => hc h.2.2.1,False.elim⟩
+    rw [he]
+    exact Set.finite_empty
+
+end CorridorSum
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section ActualCorridorPenalty
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def corridorPenaltyTerm (G : BondGraph V E) (P : V → V → ℝ) (p : ℝ)
+    {k : ℕ} (D : Set V) (ω : Set E) (x y : V) (w : RawCorridor G k) : ℝ :=
+  if w.counted D ω x y then w.weight P p else 0
+
+noncomputable def corridorPenalty (G : BondGraph V E) (P : V → V → ℝ) (p : ℝ)
+    (k : ℕ) (D : Set V) (ω : Set E) (x y : V) : ℝ :=
+  ∑' w : RawCorridor G k, corridorPenaltyTerm G P p D ω x y w
+
+theorem corridorPenaltyTerm_nonneg (G : BondGraph V E) (P : V → V → ℝ)
+    (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) {k : ℕ} (D : Set V)
+    (ω : Set E) (x y : V) (w : RawCorridor G k) :
+    0 ≤ corridorPenaltyTerm G P p D ω x y w := by
+  unfold corridorPenaltyTerm
+  split_ifs
+  · exact w.weight_nonneg P hP p
+  · exact le_rfl
+
+theorem corridorPenalty_finite_support (G : BondGraph V E) (hloc : LocallyFinite G)
+    (P : V → V → ℝ) (p : ℝ) (k : ℕ) (D : Set V) (ω : Set E) (x y : V) :
+    (Function.support (corridorPenaltyTerm G P p (k := k) D ω x y)).Finite := by
+  apply (finite_counted_corridors G hloc k D ω x y).subset
+  intro w hw
+  by_contra hn
+  exact hw (ite_eq_right hn)
+
+theorem corridorPenalty_summable (G : BondGraph V E) (hloc : LocallyFinite G)
+    (P : V → V → ℝ) (p : ℝ) (k : ℕ) (D : Set V) (ω : Set E) (x y : V) :
+    Summable (corridorPenaltyTerm G P p (k := k) D ω x y) :=
+  summable_of_hasFiniteSupport (corridorPenalty_finite_support G hloc P p k D ω x y)
+
+theorem corridorPenalty_nonneg (G : BondGraph V E) (P : V → V → ℝ)
+    (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (k : ℕ) (D : Set V)
+    (ω : Set E) (x y : V) : 0 ≤ corridorPenalty G P p k D ω x y :=
+  tsum_nonneg (corridorPenaltyTerm_nonneg G P hP p D ω x y)
+
+theorem corridorPenalty_symm (G : BondGraph V E) (P : V → V → ℝ) (p : ℝ)
+    (k : ℕ) (D : Set V) (ω : Set E) (x y : V) :
+    corridorPenalty G P p k D ω x y = corridorPenalty G P p k D ω y x := by
+  unfold corridorPenalty
+  apply tsum_congr
+  intro w
+  simp only [corridorPenaltyTerm,w.counted_symm D ω x y]
+
+theorem corridorPenalty_restrict (G : BondGraph V E) (hloc : LocallyFinite G)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (k : ℕ)
+    {D D' : Set V} {ω ω' : Set E} (hD : D ⊆ D') (hω : ω ⊆ ω')
+    (x y : V) (hc : Conn G (ω ∩ internalEdges G D) x y) :
+    corridorPenalty G P p k D' ω' x y ≤ corridorPenalty G P p k D ω x y := by
+  apply (corridorPenalty_summable G hloc P p k D' ω' x y).tsum_le_tsum ?_
+    (corridorPenalty_summable G hloc P p k D ω x y)
+  intro w
+  by_cases hn : w.counted D' ω' x y
+  · have ho : w.counted D ω x y := (w.counted_iff D ω x y).mpr
+      ⟨hn.1,corridorCounted_restrict G _ _ hD hω x y hc hn.2.2⟩
+    simp only [corridorPenaltyTerm,hn,ho,ite_true,le_refl]
+  · simpa only [corridorPenaltyTerm,hn,ite_false] using corridorPenaltyTerm_nonneg G P hP p D ω x y w
+
+theorem measurable_corridorPenaltyTerm [Countable E] (G : BondGraph V E)
+    (P : V → V → ℝ) (p : ℝ) {k : ℕ} (D : Set V) (x y : V) (w : RawCorridor G k) :
+    Measurable (fun ω => corridorPenaltyTerm G P p D ω x y w) := by
+  exact Measurable.ite (w.measurableSet_counted D x y) measurable_const measurable_const
+
+theorem measurable_corridorPenalty [Countable V] [Countable E] (G : BondGraph V E)
+    (P : V → V → ℝ) (p : ℝ) (k : ℕ) (D : Set V) (x y : V) :
+    Measurable (fun ω => corridorPenalty G P p k D ω x y) := by
+  exact Measurable.tsum (fun w => measurable_corridorPenaltyTerm G P p D x y w)
+
+end ActualCorridorPenalty
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section ActualWeightedConnections
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+noncomputable def deformedConnection (G : BondGraph V E) (P : V → V → ℝ) (p : ℝ)
+    (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x y : V) : ℝ :=
+  if x ∈ D ∧ y ∈ D ∧ Conn G (ω ∩ internalEdges G D) x y then
+    Real.exp (-t * corridorPenalty G P p k D ω x y) else 0
+
+noncomputable def deformedKernel (G : BondGraph V E) (P : V → V → ℝ) (p : unitInterval)
+    (k : ℕ) (t : ℝ) (D : Set V) (x y : V) : ℝ :=
+  ∫ ω, deformedConnection G P p k t D ω x y ∂law p
+
+theorem deformedConnection_nonneg (G : BondGraph V E) (P : V → V → ℝ) (p : ℝ)
+    (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x y : V) :
+    0 ≤ deformedConnection G P p k t D ω x y := by
+  unfold deformedConnection
+  split_ifs
+  · exact (Real.exp_pos _).le
+  · exact le_rfl
+
+theorem deformedConnection_le_one (G : BondGraph V E) (P : V → V → ℝ)
+    (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (k : ℕ) (t : ℝ) (ht : 0 ≤ t)
+    (D : Set V) (ω : Set E) (x y : V) : deformedConnection G P p k t D ω x y ≤ 1 := by
+  unfold deformedConnection
+  split_ifs
+  · apply Real.exp_le_one_iff.mpr
+    exact mul_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr ht) (corridorPenalty_nonneg G P hP p k D ω x y)
+  · exact zero_le_one
+
+theorem deformedConnection_symm (G : BondGraph V E) (P : V → V → ℝ) (p : ℝ)
+    (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x y : V) :
+    deformedConnection G P p k t D ω x y = deformedConnection G P p k t D ω y x := by
+  have hc : Conn G (ω ∩ internalEdges G D) x y ↔ Conn G (ω ∩ internalEdges G D) y x :=
+    ⟨conn_symm G _,conn_symm G _⟩
+  simp only [deformedConnection,corridorPenalty_symm G P p k D ω x y,hc]
+  congr 1
+  apply propext; tauto
+
+theorem deformedConnection_mono (G : BondGraph V E) (hloc : LocallyFinite G)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (k : ℕ)
+    (t : ℝ) (ht : 0 ≤ t) {D D' : Set V} {ω ω' : Set E} (hD : D ⊆ D') (hω : ω ⊆ ω')
+    (x y : V) : deformedConnection G P p k t D ω x y ≤ deformedConnection G P p k t D' ω' x y := by
+  by_cases hc : x ∈ D ∧ y ∈ D ∧ Conn G (ω ∩ internalEdges G D) x y
+  · have hc' : x ∈ D' ∧ y ∈ D' ∧ Conn G (ω' ∩ internalEdges G D') x y :=
+      ⟨hD hc.1,hD hc.2.1,conn_mono G (Set.inter_subset_inter hω (internalEdges_mono G hD)) hc.2.2⟩
+    simp only [deformedConnection,hc,hc']
+    apply Real.exp_le_exp.mpr
+    exact mul_le_mul_of_nonpos_left (corridorPenalty_restrict G hloc P hP p k hD hω x y hc.2.2)
+      (neg_nonpos.mpr ht)
+  · simpa only [deformedConnection,hc,ite_false] using deformedConnection_nonneg G P p k t D' ω' x y
+
+theorem measurable_deformedConnection [Countable V] [Countable E] (G : BondGraph V E)
+    (P : V → V → ℝ) (p : ℝ) (k : ℕ) (t : ℝ)
+    (D : Set V) (x y : V) : Measurable (fun ω => deformedConnection G P p k t D ω x y) := by
+  have hs : MeasurableSet {ω : Set E | x ∈ D ∧ y ∈ D ∧ Conn G (ω ∩ internalEdges G D) x y} := by
+    by_cases hx : x ∈ D <;> by_cases hy : y ∈ D
+    · simpa only [hx,hy,true_and] using measurableSet_restrictedConn G (internalEdges G D) x y
+    all_goals simp only [hx,hy,false_and,and_false,Set.ofPred_false]; exact MeasurableSet.empty
+  exact Measurable.ite hs ((measurable_corridorPenalty G P p k D x y).const_mul (-t)).exp measurable_const
+
+theorem integrable_deformedConnection [Countable V] [Countable E] (G : BondGraph V E)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y)
+    (p : unitInterval) (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D : Set V) (x y : V) :
+    Integrable (fun ω => deformedConnection G P p k t D ω x y) (law p) := by
+  apply Integrable.mono' (integrable_const (1:ℝ)) (measurable_deformedConnection G P p k t D x y).aestronglyMeasurable
+  filter_upwards [] with ω
+  rw [Real.norm_eq_abs,abs_of_nonneg (deformedConnection_nonneg G P p k t D ω x y)]
+  exact deformedConnection_le_one G P hP p k t ht D ω x y
+
+end ActualWeightedConnections
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section WeightedConnectionEndpoints
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem corridorPenalty_diag (G : BondGraph V E) (P : V → V → ℝ) (p : ℝ)
+    (k : ℕ) (D : Set V) (ω : Set E) (x : V) : corridorPenalty G P p k D ω x x = 0 := by
+  unfold corridorPenalty
+  rw [←tsum_zero]
+  apply tsum_congr
+  intro w
+  have hn : ¬w.counted D ω x x := by
+    intro hc
+    have hl : 0 < w.path.length := by have := hc.1; omega
+    obtain ⟨s,hs,_⟩ := (SimpleGraph.Walk.mem_support_iff_exists_mem_edges_of_not_nil
+      (SimpleGraph.Walk.not_nil_iff_lt_length.mpr hl)).mp w.path.start_mem_support
+    obtain ⟨e,he,_⟩ := w.edge_has_label s hs
+    exact (hc.2.2.2.1 e he).2 (conn_refl G _ x)
+  exact ite_eq_right hn
+
+theorem deformedConnection_diag (G : BondGraph V E) (P : V → V → ℝ) (p : ℝ)
+    (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x : V) (hx : x ∈ D) :
+    deformedConnection G P p k t D ω x x = 1 := by
+  simp only [deformedConnection,hx,conn_refl G _ x,and_self,ite_true,
+    corridorPenalty_diag,mul_zero,Real.exp_zero]
+
+theorem deformedConnection_zero (G : BondGraph V E) (P : V → V → ℝ) (p : ℝ)
+    (k : ℕ) (D : Set V) (ω : Set E) (x y : V) :
+    deformedConnection G P p k 0 D ω x y =
+      if x ∈ D ∧ y ∈ D ∧ Conn G (ω ∩ internalEdges G D) x y then 1 else 0 := by
+  simp only [deformedConnection,neg_zero,zero_mul,Real.exp_zero]
+
+theorem deformedConnection_full_zero (G : BondGraph V E) (P : V → V → ℝ) (p : ℝ)
+    (k : ℕ) (ω : Set E) (x y : V) : deformedConnection G P p k 0 Set.univ ω x y =
+      if Conn G ω x y then 1 else 0 := by
+  have hi : internalEdges G Set.univ = Set.univ := Set.eq_univ_of_forall (fun _ _ _ => trivial)
+  simp only [deformedConnection_zero,Set.mem_univ,true_and,hi,Set.inter_univ]
+
+theorem deformedConnection_change (G : BondGraph V E) (P : V → V → ℝ)
+    (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (k : ℕ) (t : ℝ) (ht : 0 ≤ t)
+    (D : Set V) (ω : Set E) (x y : V) :
+    0 ≤ deformedConnection G P p k 0 D ω x y - deformedConnection G P p k t D ω x y ∧
+    deformedConnection G P p k 0 D ω x y - deformedConnection G P p k t D ω x y ≤
+      t*corridorPenalty G P p k D ω x y := by
+  have hN := corridorPenalty_nonneg G P hP p k D ω x y
+  by_cases hc : x ∈ D ∧ y ∈ D ∧ Conn G (ω ∩ internalEdges G D) x y
+  · simp only [deformedConnection,hc,ite_true,neg_zero,zero_mul,Real.exp_zero,and_self]
+    constructor
+    · exact sub_nonneg.mpr ((Real.exp_le_one_iff).mpr (mul_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr ht) hN))
+    · have h := Real.one_sub_le_exp_neg (t*corridorPenalty G P p k D ω x y)
+      rw [←neg_mul] at h
+      linarith
+  · simp only [deformedConnection,hc,ite_false,sub_self]
+    exact ⟨le_rfl,mul_nonneg ht hN⟩
+
+theorem deformedKernel_zero [Countable V] [Countable E] (G : BondGraph V E)
+    (P : V → V → ℝ) (p : unitInterval) (k : ℕ) (x y : V) :
+    deformedKernel G P p k 0 Set.univ x y = twoPoint G p x y := by
+  unfold deformedKernel
+  simp_rw [deformedConnection_full_zero]
+  have hi := integral_indicator_const (μ := law p) (1:ℝ) (measurableSet_conn G x y)
+  simpa only [Set.indicator,Set.mem_ofPred_eq,smul_eq_mul,mul_one,Measure.real,twoPoint] using hi
+
+end WeightedConnectionEndpoints
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section RawWalkDiagrams
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem raw_diagram_hasSum (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (hK : ∀ x y, 0 ≤ walkKernel G hloc a x y)
+    (A : Hilbert V →L[ℂ] Hilbert V) (hA : PositiveRealMatrix A) (k : ℕ) (x y : V) :
+    HasSum (fun w : RawCorridor G k => w.probability (walkKernel G hloc a) *
+      ((matrixCoeff A x w.base).re * (matrixCoeff A w.endpoint y).re))
+      (matrixCoeff (A*(P^k*A)) x y).re := by
+  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) :=
+    finite_openSimpleGraph_neighbors G hloc z
+  let F := fun w : RawCorridor G k => w.probability (walkKernel G hloc a) *
+    ((matrixCoeff A x w.base).re * (matrixCoeff A w.endpoint y).re)
+  have hFn (w : RawCorridor G k) : 0 ≤ F w :=
+    mul_nonneg (w.2.2.mass_nonneg _ hK) (mul_nonneg (hA _ _).2 (hA _ _).2)
+  have hr (z : V) : (∑' w : RawFrom (openSimpleGraph G Set.univ) k z, F ⟨z,w⟩) =
+      (matrixCoeff A x z).re * (matrixCoeff (P^k*A) z y).re := by
+    have h := raw_integral_operator G hloc a P hpos hP k z (fun u => (matrixCoeff A u y).re)
+    have he : (∑' w : RawFrom (openSimpleGraph G Set.univ) k z, F ⟨z,w⟩) =
+        (matrixCoeff A x z).re * RawFrom.integral (H := openSimpleGraph G Set.univ)
+          (walkKernel G hloc a) k z (fun u => (matrixCoeff A u y).re) := by
+      rw [RawFrom.integral,←tsum_mul_left]
+      apply tsum_congr
+      intro w
+      dsimp only [F,RawCorridor.probability,RawCorridor.base,RawCorridor.endpoint]
+      change w.2.mass _ * (_ * _) = _ * (w.2.mass _ * _)
+      ring
+    rw [he,h,((hpos.pow k).mul_hasSum A z y).tsum_eq]
+  have ho := hA.mul_hasSum (P^k*A) x y
+  have hs : Summable F := (summable_sigma_of_nonneg hFn).mpr ⟨fun z => by
+    let := Fintype.ofFinite (RawFrom (openSimpleGraph G Set.univ) k z)
+    exact (hasSum_fintype _).summable,by simpa only [hr] using ho.summable⟩
+  have he : (∑' w, F w) = (matrixCoeff (A*(P^k*A)) x y).re := by
+    rw [hs.tsum_sigma]
+    simpa only [hr] using ho.tsum_eq
+  exact he ▸ hs.hasSum
+
+ theorem sandwich_selfAdjoint (A P : Hilbert V →L[ℂ] Hilbert V)
+    (hA : IsSelfAdjoint A) (hP : IsSelfAdjoint P) (k : ℕ) : IsSelfAdjoint (A*(P^k*A)) := by
+  change star (A*(P^k*A)) = A*(P^k*A)
+  simp only [star_mul,hA.star_eq,(hP.pow k).star_eq,mul_assoc]
+
+end RawWalkDiagrams
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section RawWalkSymmetricDiagrams
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem raw_two_diagrams_hasSum (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hPs : IsSelfAdjoint P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (hK : ∀ x y, 0 ≤ walkKernel G hloc a x y)
+    (A : Hilbert V →L[ℂ] Hilbert V) (hA : PositiveRealMatrix A) (hAs : IsSelfAdjoint A)
+    (k : ℕ) (x y : V) :
+    HasSum (fun w : RawCorridor G k => w.probability (walkKernel G hloc a) *
+      ((matrixCoeff A x w.base).re * (matrixCoeff A w.endpoint y).re +
+       (matrixCoeff A x w.endpoint).re * (matrixCoeff A w.base y).re))
+      (2*(matrixCoeff (A*(P^k*A)) x y).re) := by
+  have h₁ := raw_diagram_hasSum G hloc a P hpos hP hK A hA k x y
+  have h₂ := raw_diagram_hasSum G hloc a P hpos hP hK A hA k y x
+  have hh := h₁.add h₂
+  have hC := matrixCoeff_real_symm (A*(P^k*A)) (sandwich_selfAdjoint A P hAs hPs k) y x
+  rw [hC,←two_mul] at hh
+  convert hh using 1
+  ext w
+  rw [matrixCoeff_real_symm A hAs y w.base,matrixCoeff_real_symm A hAs w.endpoint x]
+  ring
+
+end RawWalkSymmetricDiagrams
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section PenaltyTermExpectation
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ theorem integrable_corridorPenaltyTerm [Countable E] (G : BondGraph V E)
+    (P : V → V → ℝ) (p : unitInterval) {k : ℕ} (D : Set V) (x y : V) (w : RawCorridor G k) :
+    Integrable (fun ω => corridorPenaltyTerm G P p D ω x y w) (law p) := by
+  exact (integrable_const (w.weight P p)).indicator (w.measurableSet_counted D x y)
+
+theorem integral_corridorPenaltyTerm [Countable E] (G : BondGraph V E)
+    (P : V → V → ℝ) (p : unitInterval) {k : ℕ} (D : Set V) (x y : V) (w : RawCorridor G k) :
+    (∫ ω, corridorPenaltyTerm G P p D ω x y w ∂law p) =
+      w.weight P p * (law p).real {ω | w.counted D ω x y} := by
+  have h := integral_indicator_const (μ := law p) (w.weight P p) (w.measurableSet_counted D x y)
+  simpa only [corridorPenaltyTerm,Set.indicator,Set.mem_ofPred_eq,smul_eq_mul,mul_comm] using h
+
+ 
+theorem integral_corridorPenaltyTerm_le [Countable V] [Countable E]
+    (G : BondGraph V E) (hG : Function.Injective G) (hloc : LocallyFinite G)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval)
+    (hp : 0 < (p:ℝ)) (hp₁ : (p:ℝ) < 1) {k : ℕ} (x y : V) (w : RawCorridor G k) :
+    (∫ ω, corridorPenaltyTerm G P p Set.univ ω x y w ∂law p) ≤
+      w.probability P * (twoPoint G p x w.base * twoPoint G p w.endpoint y +
+        twoPoint G p x w.endpoint * twoPoint G p w.base y) := by
+  rw [integral_corridorPenaltyTerm]
+  by_cases hl : 3 ≤ w.path.length
+  · have he : {ω | w.counted Set.univ ω x y} =
+        {ω | corridorCounted G w.retained w.blockers Set.univ ω x y} := by
+      ext ω
+      simp only [Set.mem_ofPred_eq,w.counted_iff,hl,true_and]
+    rw [he]
+    exact w.counted_weight_bound hG hloc hl P hP p hp hp₁ x y
+  · have he : {ω | w.counted Set.univ ω x y} = ∅ := by
+      ext ω
+      simp only [Set.mem_ofPred_eq,w.counted_iff,hl,false_and,Set.mem_empty_iff_false]
+    rw [he,measureReal_empty,mul_zero]
+    exact mul_nonneg (w.2.2.mass_nonneg P hP) (add_nonneg
+      (mul_nonneg (twoPoint_nonneg G p x w.base) (twoPoint_nonneg G p w.endpoint y))
+      (mul_nonneg (twoPoint_nonneg G p x w.endpoint) (twoPoint_nonneg G p w.base y)))
+
+end PenaltyTermExpectation
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section FullCorridorPenaltyExpectation
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Countable V] [Countable E]
+
+theorem corridorPenalty_integrable_bound (G : BondGraph V E)
+    (hG : Function.Injective G) (hloc : LocallyFinite G) (p : unitInterval)
+    (hp : 0 < (p:ℝ)) (hp₁ : (p:ℝ) < 1) (a : ℝ)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hPs : IsSelfAdjoint P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (A : Hilbert V →L[ℂ] Hilbert V) (hA : PositiveRealMatrix A) (hAs : IsSelfAdjoint A)
+    (hAT : ∀ x y, matrixCoeff A x y = (twoPoint G p x y : ℂ)) (k : ℕ) (x y : V) :
+    Integrable (fun ω => corridorPenalty G (walkKernel G hloc a) p k Set.univ ω x y) (law p) ∧
+    (∫ ω, corridorPenalty G (walkKernel G hloc a) p k Set.univ ω x y ∂law p) ≤
+      2*(matrixCoeff (A*(P^k*A)) x y).re := by
+  let K := walkKernel G hloc a
+  have hK (u v : V) : 0 ≤ K u v := by simpa only [hP,Complex.ofReal_re] using (hpos u v).2
+  let F := fun w : RawCorridor G k => fun ω : Set E => corridorPenaltyTerm G K p Set.univ ω x y w
+  have hFi (w : RawCorridor G k) : Integrable (F w) (law p) := integrable_corridorPenaltyTerm G K p Set.univ x y w
+  have hFn (w : RawCorridor G k) (ω : Set E) : 0 ≤ F w ω := corridorPenaltyTerm_nonneg G K hK p _ ω x y w
+  have hd := raw_two_diagrams_hasSum G hloc a P hpos hPs hP hK A hA hAs k x y
+  simp only [hAT,Complex.ofReal_re] at hd
+  have hb (w : RawCorridor G k) := integral_corridorPenaltyTerm_le G hG hloc K hK p hp hp₁ x y w
+  have hs : Summable (fun w => ∫ ω, F w ω ∂law p) := Summable.of_nonneg_of_le
+    (fun w => integral_nonneg (hFn w)) hb hd.summable
+  have hsN (ω : Set E) := corridorPenalty_summable G hloc K p k Set.univ ω x y
+  have hlin : (∫⁻ ω, ENNReal.ofReal (corridorPenalty G K p k Set.univ ω x y) ∂law p) =
+      ENNReal.ofReal (∑' w, ∫ ω, F w ω ∂law p) := by
+    have he (ω : Set E) : ENNReal.ofReal (corridorPenalty G K p k Set.univ ω x y) =
+        ∑' w, ENNReal.ofReal (F w ω) := ENNReal.ofReal_tsum_of_nonneg (fun w => hFn w ω) (hsN ω)
+    simp_rw [he]
+    rw [lintegral_tsum (fun w => (hFi w).aestronglyMeasurable.aemeasurable.ennreal_ofReal)]
+    simp_rw [←ofReal_integral_eq_lintegral_ofReal (hFi _) (Filter.Eventually.of_forall (hFn _))]
+    exact (ENNReal.ofReal_tsum_of_nonneg (fun w => integral_nonneg (hFn w)) hs).symm
+  have hNn (ω : Set E) := corridorPenalty_nonneg G K hK p k Set.univ ω x y
+  have hi : Integrable (fun ω => corridorPenalty G K p k Set.univ ω x y) (law p) :=
+    ⟨(measurable_corridorPenalty G K p k Set.univ x y).aestronglyMeasurable,
+      (hasFiniteIntegral_iff_ofReal (Filter.Eventually.of_forall hNn)).mpr (by rw [hlin]; exact ENNReal.ofReal_lt_top)⟩
+  refine ⟨hi,?_⟩
+  have hn : Summable (fun w => ∫ ω, ‖F w ω‖ ∂law p) := by
+    simpa only [Real.norm_of_nonneg (hFn _ _)] using hs
+  change (∫ ω, ∑' w, F w ω ∂law p) ≤ _
+  rw [←integral_tsum_of_summable_integral_norm hFi hn]
+  exact (hs.tsum_le_tsum hb hd.summable).trans_eq hd.tsum_eq
+
+end FullCorridorPenaltyExpectation
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section SmallDeformationEntrywise
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Countable V] [Countable E]
+
+ 
+
+theorem deformedKernel_change_bound (G : BondGraph V E)
+    (hG : Function.Injective G) (hloc : LocallyFinite G) (p : unitInterval)
+    (hp : 0 < (p:ℝ)) (hp₁ : (p:ℝ) < 1) (a : ℝ)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hPs : IsSelfAdjoint P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (A : Hilbert V →L[ℂ] Hilbert V) (hA : PositiveRealMatrix A) (hAs : IsSelfAdjoint A)
+    (hAT : ∀ x y, matrixCoeff A x y = (twoPoint G p x y : ℂ))
+    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (x y : V) :
+    0 ≤ twoPoint G p x y - deformedKernel G (walkKernel G hloc a) p k t Set.univ x y ∧
+    twoPoint G p x y - deformedKernel G (walkKernel G hloc a) p k t Set.univ x y ≤
+      2*t*(matrixCoeff (A*(P^k*A)) x y).re := by
+  let K := walkKernel G hloc a
+  have hK (u v : V) : 0 ≤ K u v := by simpa only [hP,Complex.ofReal_re] using (hpos u v).2
+  have hi₀ := integrable_deformedConnection G K hK p k 0 (le_refl _) Set.univ x y
+  have hi := integrable_deformedConnection G K hK p k t ht Set.univ x y
+  have he : twoPoint G p x y - deformedKernel G K p k t Set.univ x y =
+      ∫ ω, deformedConnection G K p k 0 Set.univ ω x y - deformedConnection G K p k t Set.univ ω x y ∂law p := by
+    rw [integral_sub hi₀ hi]
+    change _ = deformedKernel G K p k 0 Set.univ x y - _
+    rw [deformedKernel_zero]
+    rfl
+  rw [he]
+  refine ⟨integral_nonneg (fun ω => (deformedConnection_change G K hK p k t ht Set.univ ω x y).1),?_⟩
+  obtain ⟨hNi,hNb⟩ := corridorPenalty_integrable_bound G hG hloc p hp hp₁ a P hpos hPs hP A hA hAs hAT k x y
+  calc
+    _ ≤ ∫ ω, t*corridorPenalty G K p k Set.univ ω x y ∂law p :=
+      integral_mono (hi₀.sub hi) (hNi.const_mul t)
+        (fun ω => (deformedConnection_change G K hK p k t ht Set.univ ω x y).2)
+    _ = t*∫ ω, corridorPenalty G K p k Set.univ ω x y ∂law p := integral_const_mul _ _
+    _ ≤ t*(2*(matrixCoeff (A*(P^k*A)) x y).re) := mul_le_mul_of_nonneg_left hNb ht
+    _ = _ := by ring
+
+omit [Countable V] [Countable E] in
+theorem deformedKernel_nonneg (G : BondGraph V E) (P : V → V → ℝ) (p : unitInterval)
+    (k : ℕ) (t : ℝ) (D : Set V) (x y : V) : 0 ≤ deformedKernel G P p k t D x y :=
+  integral_nonneg (fun ω => deformedConnection_nonneg G P p k t D ω x y)
+
+end SmallDeformationEntrywise
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Deformation/Derivatives.lean b/lean/OAI/Probability/UnimodularPercolation/Deformation/Derivatives.lean
index 47f1556f17ce5fd710511deefc89bd0ba508a172..278ecf37c0ee3b054c32c42129eebdd6908c6d87 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Deformation/Derivatives.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Deformation/Derivatives.lean
@@ -1,1225 +1,1225 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Deformation.ClusterProducts
-
-namespace OAI
-
-/-! Derivatives for bond percolation. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section WeightedNeighborhoodEstimate
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem twoPoint_neighbor_div_bound [Countable E] (G : BondGraph V E) (hloc : LocallyFinite G)
-    (p : unitInterval) (hp : 0 < (p:ℝ)) (v a z : V) (hz : z ∈ neighbors G hloc a) :
-    twoPoint G p v z ≤ twoPoint G p v a / (p:ℝ) := by
-  obtain ⟨e,he⟩ := (mem_neighbors G hloc a z).mp hz
-  have hh := restrictedTwoPoint_adj_lower G Set.univ p a z v e he (Set.mem_univ _)
-  simp only [Set.inter_univ] at hh
-  change (p:ℝ)*twoPoint G p z v ≤ twoPoint G p a v at hh
-  rw [twoPoint_symm G p z v,twoPoint_symm G p a v] at hh
-  exact (le_div_iff₀ hp).mpr (by simpa only [mul_comm] using hh)
-
- 
-
-theorem neighborhood_twoPoint_bound [Countable E] (G : BondGraph V E) (hloc : LocallyFinite G)
-    (d : ℕ) (hd : ∀ a, (neighbors G hloc a).card ≤ d) (p : unitInterval) (hp : 0 < (p:ℝ))
-    (W : Finset V) (v : V) :
-    (∑' z : closedBondNeighborhood G (W:Set V), twoPoint G p v z) ≤
-      (1+(d:ℝ)/(p:ℝ)) * (∑ a ∈ W, twoPoint G p v a) := by
-  apply (neighborhood_mass_le G hloc W (twoPoint G p v) (twoPoint_nonneg G p v)).trans
-  rw [Finset.mul_sum]
-  apply Finset.sum_le_sum
-  intro a _
-  have hsum : (∑ z ∈ neighbors G hloc a, twoPoint G p v z) ≤
-      (d:ℝ)*(twoPoint G p v a/(p:ℝ)) := by
-    calc
-      _ ≤ ∑ _z ∈ neighbors G hloc a, twoPoint G p v a/(p:ℝ) :=
-        Finset.sum_le_sum (fun z hz => twoPoint_neighbor_div_bound G hloc p hp v a z hz)
-      _ = ((neighbors G hloc a).card:ℝ)*(twoPoint G p v a/(p:ℝ)) := by simp only [Finset.sum_const,nsmul_eq_mul]
-      _ ≤ _ := mul_le_mul_of_nonneg_right (Nat.cast_le.mpr (hd a)) (div_nonneg (twoPoint_nonneg G p _ _) hp.le)
-  calc
-    _ ≤ twoPoint G p v a + (d:ℝ)*(twoPoint G p v a/(p:ℝ)) := add_le_add (le_refl _) hsum
-    _ = _ := by ring
-
-end WeightedNeighborhoodEstimate
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section SizeWeightedPotential
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def sizeMembership (G : BondGraph V E) (x a : V) (ω : Set E) : ℝ :=
-  if Conn G ω x a then ((cluster G ω x).ncard:ℝ) else 0
-
-noncomputable def sizeWeightedConnectionPotential (G : BondGraph V E) (p : unitInterval)
-    (x y : V) (ω : Set E) : ℝ := ∑' a, sizeMembership G x a ω * twoPoint G p y a
-
-theorem sizeMembership_nonneg (G : BondGraph V E) (x a : V) (ω : Set E) :
-    0 ≤ sizeMembership G x a ω := by
-  unfold sizeMembership
-  split_ifs
-  · exact Nat.cast_nonneg _
-  · rfl
-
-theorem sizeMembership_le (G : BondGraph V E) (x a : V) (ω : Set E) :
-    sizeMembership G x a ω ≤ ((cluster G ω x).ncard:ℝ) := by
-  unfold sizeMembership
-  split_ifs
-  · rfl
-  · exact Nat.cast_nonneg _
-
-theorem measurable_sizeMembership [Countable V] [Countable E] (G : BondGraph V E)
-    (hloc : LocallyFinite G) (x a : V) : Measurable (sizeMembership G x a) := by
-  apply Measurable.ite (measurableSet_conn G x a)
-    ((measurable_of_countable (fun n : ℕ => (n:ℝ))).comp (measurable_cluster_ncard G hloc x)) measurable_const
-
-theorem integrable_sizeMembership [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (x a : V) : Integrable (sizeMembership G x a) (law p) := by
-  apply Integrable.mono' (susceptibility_eq_integral G hloc hqt p hp x).1
-    (measurable_sizeMembership G hloc x a).aestronglyMeasurable
-  exact Filter.Eventually.of_forall (fun ω => by
-    rw [Real.norm_of_nonneg (sizeMembership_nonneg G x a ω)]
-    exact sizeMembership_le G x a ω)
-
-theorem integral_sizeMembership_le_susceptibility [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (x a : V) :
-    (∫ ω, sizeMembership G x a ω ∂law p) ≤ susceptibility G p x := by
-  rw [(susceptibility_eq_integral G hloc hqt p hp x).2]
-  exact integral_mono (integrable_sizeMembership G hloc hqt p hp x a)
-    (susceptibility_eq_integral G hloc hqt p hp x).1 (sizeMembership_le G x a)
-
-theorem sizeWeightedConnectionPotential_eq_finite (G : BondGraph V E) (p : unitInterval)
-    (x y : V) (ω : Set E) (hf : (cluster G ω x).Finite) :
-    sizeWeightedConnectionPotential G p x y ω =
-      ((cluster G ω x).ncard:ℝ) * ∑ a ∈ hf.toFinset, twoPoint G p y a := by
-  unfold sizeWeightedConnectionPotential
-  rw [tsum_eq_sum (s := hf.toFinset)]
-  · rw [Finset.mul_sum]
-    apply Finset.sum_congr rfl
-    intro a ha
-    have hc : Conn G ω x a := hf.mem_toFinset.mp ha
-    simp only [sizeMembership,ite_eq_left hc]
-  · intro a ha
-    have hn : ¬Conn G ω x a := fun hc => ha (hf.mem_toFinset.mpr hc)
-    simp only [sizeMembership,ite_eq_right hn,zero_mul]
-
-end SizeWeightedPotential
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section SizePotentialIntegral
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Countable V] [Countable E] [Nonempty V]
-
-theorem summable_sizePotential_integral_norm (G : BondGraph V E) (hloc : LocallyFinite G)
-    (hqt : QuasiTransitive G) (p : unitInterval) (hp : p < pc G) (x y : V) :
-    Summable (fun a => ∫ ω, ‖sizeMembership G x a ω * twoPoint G p y a‖ ∂law p) := by
-  apply ((summable_twoPoint_subcritical G hloc hqt p hp y).mul_left (susceptibility G p x)).of_nonneg_of_le
-  · exact fun a => integral_nonneg (fun _ => norm_nonneg _)
-  · intro a
-    simp only [Real.norm_of_nonneg (mul_nonneg (sizeMembership_nonneg G x a _) (twoPoint_nonneg G p y a)),
-      integral_mul_const]
-    exact mul_le_mul_of_nonneg_right (integral_sizeMembership_le_susceptibility G hloc hqt p hp x a)
-      (twoPoint_nonneg G p y a)
-
-theorem integrable_sizeWeightedConnectionPotential (G : BondGraph V E) (hloc : LocallyFinite G)
-    (hqt : QuasiTransitive G) (p : unitInterval) (hp : p < pc G) (x y : V) :
-    Integrable (sizeWeightedConnectionPotential G p x y) (law p) := by
-  apply Integrable.mono' ((susceptibility_eq_integral G hloc hqt p hp x).1.mul_const
-    (susceptibility G p y))
-    (Measurable.tsum (fun a => (measurable_sizeMembership G hloc x a).mul_const (twoPoint G p y a))).aestronglyMeasurable
-  exact Filter.Eventually.of_forall (fun ω => by
-    have hb (a : V) : sizeMembership G x a ω * twoPoint G p y a ≤
-        ((cluster G ω x).ncard:ℝ)*twoPoint G p y a :=
-      mul_le_mul_of_nonneg_right (sizeMembership_le G x a ω) (twoPoint_nonneg G p y a)
-    have hn (a : V) : 0 ≤ sizeMembership G x a ω * twoPoint G p y a :=
-      mul_nonneg (sizeMembership_nonneg G x a ω) (twoPoint_nonneg G p y a)
-    have hs := (summable_twoPoint_subcritical G hloc hqt p hp y).mul_left ((cluster G ω x).ncard:ℝ)
-    change ‖∑' a, sizeMembership G x a ω * twoPoint G p y a‖ ≤ _
-    rw [Real.norm_of_nonneg (tsum_nonneg hn)]
-    exact ((hs.of_nonneg_of_le hn hb).tsum_le_tsum hb hs).trans_eq (tsum_mul_left))
-
-omit [Countable V] [Countable E] [Nonempty V] in
-theorem sizeWeightedConnectionPotential_nonneg (G : BondGraph V E) (p : unitInterval)
-    (x y : V) (ω : Set E) : 0 ≤ sizeWeightedConnectionPotential G p x y ω :=
-  tsum_nonneg (fun a => mul_nonneg (sizeMembership_nonneg G x a ω) (twoPoint_nonneg G p y a))
-
-theorem summable_twoPoint_cube_pair (G : BondGraph V E) (hloc : LocallyFinite G)
-    (hqt : QuasiTransitive G) (p : unitInterval) (hp : p < pc G) (x y : V) :
-    Summable (fun ba : V×V => twoPoint G p x ba.1 * twoPoint G p ba.1 ba.2 * twoPoint G p y ba.2) := by
-  exact (summable_twoPoint_pair G hloc hqt p hp x).of_nonneg_of_le
-    (fun ba => mul_nonneg (mul_nonneg (twoPoint_nonneg G p _ _) (twoPoint_nonneg G p _ _))
-      (twoPoint_nonneg G p _ _))
-    (fun ba => mul_le_of_le_one_right
-      (mul_nonneg (twoPoint_nonneg G p _ _) (twoPoint_nonneg G p _ _)) (twoPoint_le_one G p _ _))
-
- 
-theorem integral_sizeWeightedConnectionPotential_le (G : BondGraph V E) (hloc : LocallyFinite G)
-    (hqt : QuasiTransitive G) (p : unitInterval) (hp : p < pc G) (x y : V) :
-    (∫ ω, sizeWeightedConnectionPotential G p x y ω ∂law p) ≤
-      maxSusceptibility G p * (∑' a, ∑' b, twoPoint G p x b * twoPoint G p b a * twoPoint G p y a) := by
-  have hi (a : V) := (integrable_sizeMembership G hloc hqt p hp x a).mul_const (twoPoint G p y a)
-  have hs := summable_sizePotential_integral_norm G hloc hqt p hp x y
-  have hc : Summable (fun a => ∑' b, twoPoint G p x b * twoPoint G p b a * twoPoint G p y a) :=
-    (summable_twoPoint_cube_pair G hloc hqt p hp x y).prod_symm.prod
-  change (∫ ω, ∑' a, sizeMembership G x a ω * twoPoint G p y a ∂law p) ≤ _
-  rw [←integral_tsum_of_summable_integral_norm hi hs,←tsum_mul_left]
-  have hs' : Summable (fun a => ∫ ω, sizeMembership G x a ω * twoPoint G p y a ∂law p) := by
-    simpa only [Real.norm_of_nonneg (mul_nonneg (sizeMembership_nonneg G x _ _) (twoPoint_nonneg G p y _))] using hs
-  apply hs'.tsum_le_tsum (fun a => ?_) (hc.mul_left _)
-  rw [integral_mul_const,tsum_mul_right]
-  calc
-    _ ≤ (maxSusceptibility G p * (∑' b, twoPoint G p x b * twoPoint G p b a)) * twoPoint G p y a :=
-      mul_le_mul_of_nonneg_right (integral_size_membership_le G hloc hqt p hp x a) (twoPoint_nonneg G p y a)
-    _ = _ := mul_assoc _ _ _
-
-end SizePotentialIntegral
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section WeightedCorrelationPointwise
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Countable V] [Countable E] [Nonempty V]
-
- 
-
-theorem deformedFreshExteriorMoment_lower (G : BondGraph V E) (hloc : LocallyFinite G)
-    (hqt : QuasiTransitive G) (d : ℕ) (hd : ∀ a, (neighbors G hloc a).card ≤ d)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval)
-    (hp : p < pc G) (hp₀ : 0 < (p:ℝ)) (k : ℕ) (t : ℝ) (ht : 0 ≤ t)
-    (D : Set V) (x y : V) (ω : Set E) (hf : (cluster G ω x).Finite) :
-    deformedClusterMass G P p k t D ω x * deformedSusceptibility G P p k t D y -
-      (1+(d:ℝ)/(p:ℝ))*maxSusceptibility G p * sizeWeightedConnectionPotential G p x y ω ≤
-      deformedFreshExteriorMoment G P p k t D x y ω := by
-  let A := cluster G (ω ∩ internalEdges G D) x
-  have hsub : A ⊆ cluster G ω x := fun _ h => conn_mono G Set.inter_subset_left h
-  have hA : A.Finite := hf.subset hsub
-  have hn : (∑' z : closedBondNeighborhood G A, twoPoint G p y z) ≤
-      (1+(d:ℝ)/(p:ℝ))*(∑ a ∈ hf.toFinset, twoPoint G p y a) := by
-    have hh := neighborhood_twoPoint_bound G hloc d hd p hp₀ hA.toFinset y
-    rw [hA.coe_toFinset] at hh
-    exact hh.trans (mul_le_mul_of_nonneg_left
-      (Finset.sum_le_sum_of_subset_of_nonneg
-        (fun a ha => hf.mem_toFinset.mpr (hsub (hA.mem_toFinset.mp ha)))
-        (fun a _ _ => twoPoint_nonneg G p y a))
-      (add_nonneg zero_le_one (div_nonneg (Nat.cast_nonneg _) hp₀.le)))
-  have hm := deformedClusterMass_le_ncard G P hP p k t ht D ω x hf
-  have hm₀ := deformedClusterMass_nonneg G P p k t D ω x
-  have hχ : 0 ≤ maxSusceptibility G p := zero_le_one.trans (maxSusceptibility_one_le G hloc hqt p hp)
-  have hh := mul_le_mul_of_nonneg_left
-    (deformedSusceptibility_deletion G hloc hqt P hP p hp k t ht D A y) hm₀
-  have hsum : 0 ≤ ∑ a ∈ hf.toFinset, twoPoint G p y a :=
-    Finset.sum_nonneg (fun a _ => twoPoint_nonneg G p y a)
-  have hC : 0 ≤ 1+(d:ℝ)/(p:ℝ) := add_nonneg zero_le_one (div_nonneg (Nat.cast_nonneg _) hp₀.le)
-  have herr : deformedClusterMass G P p k t D ω x *
-      (maxSusceptibility G p * (∑' z : closedBondNeighborhood G A, twoPoint G p y z)) ≤
-      (1+(d:ℝ)/(p:ℝ))*maxSusceptibility G p * sizeWeightedConnectionPotential G p x y ω := by
-    calc
-      _ ≤ deformedClusterMass G P p k t D ω x *
-          (maxSusceptibility G p * ((1+(d:ℝ)/(p:ℝ))*(∑ a ∈ hf.toFinset, twoPoint G p y a))) :=
-        mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_left hn hχ) hm₀
-      _ ≤ ((cluster G ω x).ncard:ℝ)*
-          (maxSusceptibility G p * ((1+(d:ℝ)/(p:ℝ))*(∑ a ∈ hf.toFinset, twoPoint G p y a))) :=
-        mul_le_mul_of_nonneg_right hm (mul_nonneg hχ (mul_nonneg hC hsum))
-      _ = _ := by rw [sizeWeightedConnectionPotential_eq_finite G p x y ω hf]; ring
-  dsimp only [deformedFreshExteriorMoment]
-  dsimp only [A] at hh
-  nlinarith
-
-end WeightedCorrelationPointwise
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section WeightedCorrelationIntegral
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Countable V] [Countable E] [Nonempty V]
-
- 
-
-theorem deformed_distinct_correlation (G : BondGraph V E) (hloc : LocallyFinite G)
-    (hqt : QuasiTransitive G) (d : ℕ) (hd : ∀ a, (neighbors G hloc a).card ≤ d)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval)
-    (hp : p < pc G) (hp₀ : 0 < (p:ℝ)) (k : ℕ) (t : ℝ) (ht : 0 ≤ t)
-    (D : Set V) (x y : V) :
-    deformedSusceptibility G P p k t D x * deformedSusceptibility G P p k t D y -
-      (1+(d:ℝ)/(p:ℝ))*(maxSusceptibility G p)^2 *
-        (∑' a, ∑' b, twoPoint G p x b * twoPoint G p b a * twoPoint G p y a) ≤
-      ∫ ω, (if ¬Conn G (ω ∩ internalEdges G D) x y then
-        deformedClusterMass G P p k t D ω x * deformedClusterMass G P p k t D ω y else 0) ∂law p := by
-  rw [←integral_deformedFreshExteriorMoment G hloc hqt P hP p hp k t ht D x y]
-  have hi := integrable_deformedClusterMass G hloc hqt P hP p hp k t ht D x
-  have hj := integrable_sizeWeightedConnectionPotential G hloc hqt p hp x y
-  have hb := integral_mono_ae
-    ((hi.mul_const (deformedSusceptibility G P p k t D y)).sub
-      (hj.const_mul ((1+(d:ℝ)/(p:ℝ))*maxSusceptibility G p)))
-    (integrable_deformedFreshExteriorMoment G hloc hqt P hP p hp k t ht D x y)
-    (by
-      filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
-      exact deformedFreshExteriorMoment_lower G hloc hqt d hd P hP p hp hp₀ k t ht D x y ω hω)
-  simp only [Pi.sub_apply] at hb
-  rw [integral_sub (hi.mul_const _) (hj.const_mul _),integral_mul_const,integral_const_mul,
-    ←deformedSusceptibility_eq_integral G hloc hqt P hP p hp k t ht D x] at hb
-  have hC : 0 ≤ (1+(d:ℝ)/(p:ℝ))*maxSusceptibility G p :=
-    mul_nonneg (add_nonneg zero_le_one (div_nonneg (Nat.cast_nonneg _) hp₀.le))
-      (zero_le_one.trans (maxSusceptibility_one_le G hloc hqt p hp))
-  have herr := mul_le_mul_of_nonneg_left
-    (integral_sizeWeightedConnectionPotential_le G hloc hqt p hp x y) hC
-  nlinarith
-
-end WeightedCorrelationIntegral
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section ExactCubeKernel
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem HasKernel.square_real_kernel (G : BondGraph V E) (p : unitInterval)
-    (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T) (x y : V) :
-    (∑' b, twoPoint G p x b * twoPoint G p b y) = (matrixCoeff (T^2) x y).re := by
-  have hh := ((hT.positiveRealMatrix G p T).mul_hasSum T x y).tsum_eq
-  simpa only [hT.matrixCoeff_re G p T,pow_two] using hh
-
-theorem HasKernel.cube_real_kernel (G : BondGraph V E) (p : unitInterval)
-    (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T) (x y : V) :
-    (∑' a, ∑' b, twoPoint G p x b * twoPoint G p b a * twoPoint G p y a) =
-      (matrixCoeff (T^3) x y).re := by
-  have hh := (((hT.positiveRealMatrix G p T).pow 2).mul_hasSum T x y).tsum_eq
-  calc
-    _ = ∑' a, (matrixCoeff (T^2) x a).re * (matrixCoeff T a y).re := by
-      apply tsum_congr
-      intro a
-      rw [tsum_mul_right,hT.square_real_kernel G p T x a,hT.matrixCoeff_re G p T a y,twoPoint_symm G p y a]
-    _ = _ := by simpa only [show T^2*T = T^3 by rw [←pow_succ]] using hh
-
-end ExactCubeKernel
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section UniformWeightedCorrelation
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Countable V] [Countable E] [Nonempty V]
-
- 
-
-theorem deformed_distinct_correlation_uniform (G : BondGraph V E) (hloc : LocallyFinite G)
-    (hqt : QuasiTransitive G) (d : ℕ) (hd : ∀ a, (neighbors G hloc a).card ≤ d)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (hpcrit : 0 < (pc G:ℝ))
-    (p : unitInterval) (hp : p < pc G) (hpnear : (pc G:ℝ)/2 ≤ (p:ℝ))
-    (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T)
-    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D : Set V) (x y : V) :
-    deformedSusceptibility G P p k t D x * deformedSusceptibility G P p k t D y -
-      (1+2*(d:ℝ)/(pc G:ℝ))*(maxSusceptibility G p)^2 * (matrixCoeff (T^3) x y).re ≤
-      ∫ ω, (if ¬Conn G (ω ∩ internalEdges G D) x y then
-        deformedClusterMass G P p k t D ω x * deformedClusterMass G P p k t D ω y else 0) ∂law p := by
-  have hp₀ : 0 < (p:ℝ) := (half_pos hpcrit).trans_le hpnear
-  have hb := deformed_distinct_correlation G hloc hqt d hd P hP p hp hp₀ k t ht D x y
-  rw [hT.cube_real_kernel G p T x y] at hb
-  have hc : (d:ℝ)/(p:ℝ) ≤ 2*(d:ℝ)/(pc G:ℝ) := by
-    apply (div_le_div_iff₀ hp₀ hpcrit).mpr
-    nlinarith [Nat.cast_nonneg (α := ℝ) d]
-  have he := mul_le_mul_of_nonneg_right (add_le_add (le_refl (1:ℝ)) hc)
-    (mul_nonneg (sq_nonneg (maxSusceptibility G p)) (((hT.positiveRealMatrix G p T).pow 3) x y).2)
-  nlinarith
-
-end UniformWeightedCorrelation
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section CorridorSizeBounds
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem incidentSupport_card_le (G : BondGraph V E) (hloc : LocallyFinite G)
-    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (W : Finset V) :
-    (incidentSupport G hloc W).card ≤ W.card*d := by
-  apply Finset.card_biUnion_le_card_mul
-  intro x _
-  rw [←Set.ncard_eq_toFinset_card _ (hloc x)]
-  exact (incident_ncard_le_degree G hloc x).trans (hd x)
-
-theorem reachBall_ncard_le_pow (G : BondGraph V E) (hloc : LocallyFinite G)
-    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (x : V) (k : ℕ) :
-    (reachBall G x k).ncard ≤ (d+1)^k := by
-  induction k with
-  | zero => simp only [reachBall,Set.ncard_singleton,pow_zero,le_refl]
-  | succ k ih =>
-    let W := (finite_reachBall hloc x k).toFinset
-    have he : reachBall G x (k+1) = (W ∪ W.biUnion (neighbors G hloc) : Finset V) := by
-      ext z
-      simp only [reachBall,Set.mem_union,Set.mem_iUnion,Set.mem_ofPred_eq,Finset.mem_coe,
-        Finset.mem_union,Finset.mem_biUnion,mem_neighbors,Set.Finite.mem_toFinset,W,Adj,exists_prop]
-    rw [he,Set.ncard_coe_finset]
-    have hb : (W.biUnion (neighbors G hloc)).card ≤ W.card*d :=
-      Finset.card_biUnion_le_card_mul _ _ _ (fun a _ => (neighbors_card_le_degree G hloc a).trans (hd a))
-    have hi : W.card ≤ (d+1)^k := by simpa only [W,←Set.ncard_eq_toFinset_card] using ih
-    calc
-      _ ≤ W.card + (W.biUnion (neighbors G hloc)).card := Finset.card_union_le _ _
-      _ ≤ W.card + W.card*d := Nat.add_le_add_left hb _
-      _ = W.card*(d+1) := by ring
-      _ ≤ (d+1)^k*(d+1) := Nat.mul_le_mul_right _ hi
-      _ = _ := (pow_succ _ _).symm
-
-namespace RawCorridor
-variable {G : BondGraph V E} {k : ℕ}
-
-theorem interior_card_le (w : RawCorridor G k) : w.interior.card ≤ w.path.length := by
-  have hs : w.interior ⊂ w.path.support.toFinset := by
-    apply Finset.ssubset_iff_subset_ne.mpr
-    refine ⟨Finset.sdiff_subset,?_⟩
-    intro he
-    have hb : w.base ∈ w.path.support.toFinset := List.mem_toFinset.mpr w.path.start_mem_support
-    rw [←he] at hb
-    have hn : w.base ∉ w.interior := by simp only [interior,Finset.mem_sdiff,Finset.mem_insert,
-      true_or,not_true_eq_false,and_false,not_false_eq_true]
-    exact hn hb
-  have hh := Finset.card_lt_card hs
-  rw [List.toFinset_card_of_nodup w.path_isPath.support_nodup,SimpleGraph.Walk.length_support] at hh
-  omega
-
-theorem blockers_ncard_le (hloc : LocallyFinite G) (d : ℕ) (hd : ∀ x, degree G x ≤ d)
-    (w : RawCorridor G k) : w.blockers.ncard ≤ d*k := by
-  have hs : w.blockers ⊆ (incidentSupport G hloc w.interior : Set E) :=
-    fun e he => (mem_incidentSupport G hloc w.interior e).mpr he.1
-  calc
-    _ ≤ (incidentSupport G hloc w.interior : Set E).ncard := Set.ncard_le_ncard hs (Finset.finite_toSet _)
-    _ = (incidentSupport G hloc w.interior).card := Set.ncard_coe_finset _
-    _ ≤ w.interior.card*d := incidentSupport_card_le G hloc d hd _
-    _ ≤ k*d := Nat.mul_le_mul_right _ (w.interior_card_le.trans w.path_length)
-    _ = _ := Nat.mul_comm _ _
-
-end RawCorridor
-end CorridorSizeBounds
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section RawWalkTotalMass
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem walkKernel_finite_support (G : BondGraph V E) (hloc : LocallyFinite G)
-    (a : ℝ) (x : V) : (Function.support (walkKernel G hloc a x)).Finite := by
-  apply (Finset.finite_toSet (insert x (neighbors G hloc x))).subset
-  intro y hy
-  by_contra hn
-  have hn' : y ≠ x ∧ ¬Adj G x y := by simpa only [Finset.mem_coe,Finset.mem_insert,mem_neighbors,not_or] using hn
-  exact hy (by simp only [walkKernel,ite_eq_right hn'.1.symm,ite_eq_right hn'.2,add_zero])
-
-theorem walkKernel_tsum (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ) (x : V) :
-    (∑' y, walkKernel G hloc a x y) = 1 := by
-  rw [tsum_eq_sum (s := insert x (neighbors G hloc x))]
-  · exact walkKernel_row_sum G hloc a x
-  · intro y hy
-    have hn : y ≠ x ∧ ¬Adj G x y := by simpa only [Finset.mem_insert,mem_neighbors,not_or] using hy
-    simp only [walkKernel,ite_eq_right hn.1.symm,ite_eq_right hn.2,add_zero]
-
-theorem raw_integral_one (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (k : ℕ) (x : V) :
-    RawFrom.integral (H := openSimpleGraph G Set.univ) (walkKernel G hloc a) k x (fun _ => 1) = 1 := by
-  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
-  induction k generalizing x with
-  | zero => exact RawFrom.integral_zero _ _ _
-  | succ k ih =>
-    rw [RawFrom.integral_succ]
-    simp only [ih,mul_one]
-    have he := sum_step_decomposition (openSimpleGraph G Set.univ) (walkKernel G hloc a)
-      (walkKernel_finite_support G hloc a) (fun x y hxy hn => by
-        apply walkKernel_zero_of_not_step G hloc a x y hxy
-        intro h
-        obtain ⟨e,he⟩ := h
-        exact hn ⟨hxy,e,Set.mem_univ e,he⟩) x (fun _ => 1)
-    simpa only [mul_one,walkKernel_tsum G hloc a] using he.symm
-
- 
-theorem rawFrom_probability_sum (G : BondGraph V E) (hloc : LocallyFinite G)
-    (a : ℝ) (k : ℕ) (x : V) :
-    (∑' w : RawFrom (openSimpleGraph G Set.univ) k x, w.2.mass (walkKernel G hloc a)) = 1 := by
-  simpa only [RawFrom.integral,mul_one] using raw_integral_one G hloc a k x
-
-end RawWalkTotalMass
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section RawWalkFiniteBases
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem raw_probability_on_bases_sum (G : BondGraph V E) (hloc : LocallyFinite G)
-    (a : ℝ) (k : ℕ) (S : Set V) (hS : S.Finite) :
-    (∑' w : RawCorridor G k, if w.base ∈ S then w.probability (walkKernel G hloc a) else 0) = (S.ncard:ℝ) := by
-  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
-  have hs : Summable (fun w : RawCorridor G k => if w.base ∈ S then w.probability (walkKernel G hloc a) else 0) := by
-    apply summable_of_hasFiniteSupport
-    apply (finite_corridors_on_bases G hloc k S hS).subset
-    intro w hw
-    by_contra hn
-    exact hw (ite_eq_right hn)
-  rw [hs.tsum_sigma]
-  have he (x : V) : (∑' w : RawFrom (openSimpleGraph G Set.univ) k x,
-      if (RawCorridor.base ⟨x,w⟩) ∈ S then (RawCorridor.probability (walkKernel G hloc a) ⟨x,w⟩) else 0) =
-      if x ∈ S then 1 else 0 := by
-    by_cases hx : x ∈ S
-    · simp only [RawCorridor.base,RawCorridor.probability,hx,ite_true,rawFrom_probability_sum G hloc a k x]
-    · simp only [RawCorridor.base,hx,ite_false,tsum_zero]
-  simp only [he]
-  rw [tsum_eq_sum (s := hS.toFinset)]
-  · rw [Finset.sum_congr rfl (fun x hx => ite_eq_left (hS.mem_toFinset.mp hx))]
-    simp only [Finset.sum_const,nsmul_eq_mul,mul_one]
-    rw [←Set.ncard_eq_toFinset_card _ hS]
-  · intro x hx
-    exact ite_eq_right (fun h => hx (hS.mem_toFinset.mpr h))
-
-namespace RawCorridor
-variable {G : BondGraph V E} {k : ℕ}
-
-theorem base_mem_reachBall_of_retained (w : RawCorridor G k) (e : E)
-    (he : e ∈ w.retained) (z : V) (hz : z ∈ G e) : w.base ∈ reachBall G z k := by
-  have hs : z ∈ w.path.support := SimpleGraph.Walk.mem_support_of_mem_edges he hz
-  let q := (w.path.takeUntil z hs).reverse
-  have hq : q.length ≤ k := by
-    change (w.path.takeUntil z hs).reverse.length ≤ k
-    rw [SimpleGraph.Walk.length_reverse]
-    exact (w.path.length_takeUntil_le_length hs).trans w.path_length
-  have hh := walk_mem_reachBall G Set.univ (x := z) q 0 (by simp only [reachBall,Set.mem_singleton_iff])
-  simp only [Nat.zero_add] at hh
-  exact reachBall_mono_local G z hq hh
-
-theorem counted_base_mem_cluster (w : RawCorridor G k) (D : Set V) (ω : Set E) (x y : V)
-    (hc : w.counted D ω x y) : w.base ∈ cluster G ω x := by
-  obtain ⟨q⟩ := (conn_iff_reachable G _ x y).mp hc.2.2.1
-  have hm := w.counted_base_mem_walk D ω x y hc q
-  exact conn_mono G Set.inter_subset_left ((conn_iff_reachable G _ x w.base).mpr ⟨q.takeUntil w.base hm⟩)
-
-end RawCorridor
-end RawWalkFiniteBases
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section UniformRawWeight
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-noncomputable def rawWeightBase (r q : ℝ) (d : ℕ) : ℝ := r⁻¹ * (q⁻¹)^d
-
-theorem rawWeightBase_one_le {r q : ℝ} (hr : 0 < r) (hr₁ : r ≤ 1)
-    (hq : 0 < q) (hq₁ : q ≤ 1) (d : ℕ) : 1 ≤ rawWeightBase r q d := by
-  exact one_le_mul_of_one_le_of_one_le ((one_le_inv₀ hr).mpr hr₁)
-    (one_le_pow₀ ((one_le_inv₀ hq).mpr hq₁))
-
-namespace RawCorridor
-variable {G : BondGraph V E} {k : ℕ}
-
-theorem weight_le_uniform (hloc : LocallyFinite G) (d : ℕ) (hd : ∀ x, degree G x ≤ d)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval)
-    {r q : ℝ} (hr : 0 < r) (hq : 0 < q) (hrp : r ≤ (p:ℝ)) (hqp : q ≤ 1-(p:ℝ))
-    (w : RawCorridor G k) : w.weight P p ≤ w.probability P * (rawWeightBase r q d)^k := by
-  have hp₀ : 0 < (p:ℝ) := hr.trans_le hrp
-  have hp₁ : 0 < 1-(p:ℝ) := hq.trans_le hqp
-  have hr₁ : r ≤ 1 := hrp.trans p.2.2
-  have hq₁ : q ≤ 1 := hqp.trans (sub_le_self _ p.2.1)
-  have hi : (p:ℝ)⁻¹ ≤ r⁻¹ := inv_anti₀ hr hrp
-  have hj : (1-(p:ℝ))⁻¹ ≤ q⁻¹ := inv_anti₀ hq hqp
-  have hL : ((p:ℝ)^w.path.length)⁻¹ ≤ (r⁻¹)^k := by
-    simp only [←inv_pow]
-    exact (pow_le_pow_left₀ (inv_nonneg.mpr p.2.1) hi _).trans
-      (pow_le_pow_right₀ ((one_le_inv₀ hr).mpr hr₁) w.path_length)
-  have hB : ((1-(p:ℝ))^w.blockers.ncard)⁻¹ ≤ ((q⁻¹)^d)^k := by
-    simp only [←inv_pow,←pow_mul]
-    exact (pow_le_pow_left₀ (inv_nonneg.mpr hp₁.le) hj _).trans
-      (pow_le_pow_right₀ ((one_le_inv₀ hq).mpr hq₁) (w.blockers_ncard_le hloc d hd))
-  unfold weight
-  calc
-    _ = w.probability P * (((p:ℝ)^w.path.length)⁻¹ * ((1-(p:ℝ))^w.blockers.ncard)⁻¹) := mul_assoc _ _ _
-    _ ≤ w.probability P * ((r⁻¹)^k * ((q⁻¹)^d)^k) :=
-      mul_le_mul_of_nonneg_left (mul_le_mul hL hB
-        (inv_nonneg.mpr (pow_nonneg hp₁.le _)) (pow_nonneg (inv_nonneg.mpr hr.le) _))
-        (w.2.2.mass_nonneg P hP)
-    _ = _ := by rw [rawWeightBase,mul_pow]
-
-end RawCorridor
-end UniformRawWeight
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section UniformPenaltyDomination
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem summable_raw_selection (G : BondGraph V E) (hloc : LocallyFinite G)
-    (k : ℕ) (S : Set V) (hS : S.Finite) (Q : RawCorridor G k → Prop)
-    (hQ : ∀ w, Q w → w.base ∈ S) (f : RawCorridor G k → ℝ) :
-    Summable (fun w => if Q w then f w else 0) := by
-  apply summable_of_hasFiniteSupport
-  apply (finite_corridors_on_bases G hloc k S hS).subset
-  intro w hw
-  by_contra hn
-  exact hw (ite_eq_right (fun h => hn (hQ w h)))
-
-theorem mask_le_mask_mul {U W : Prop} [Decidable U] [Decidable W] (hUW : U → W)
-    {f g B : ℝ} (hg : 0 ≤ g) (hB : 0 ≤ B) (hfg : f ≤ g*B) :
-    (if U then f else 0) ≤ (if W then g else 0)*B := by
-  by_cases hU : U
-  · simp only [hU,hUW hU,ite_true]; exact hfg
-  · rw [ite_eq_right hU]
-    by_cases hW : W
-    · rw [ite_eq_left hW]; exact mul_nonneg hg hB
-    · rw [ite_eq_right hW,zero_mul]
-
-theorem raw_weight_sum_le_on_bases (G : BondGraph V E) (hloc : LocallyFinite G)
-    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
-    (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y) (p : unitInterval)
-    {r q : ℝ} (hr : 0 < r) (hq : 0 < q) (hrp : r ≤ (p:ℝ)) (hqp : q ≤ 1-(p:ℝ))
-    (k : ℕ) (S : Set V) (hS : S.Finite) (Q : RawCorridor G k → Prop)
-    (hQ : ∀ w, Q w → w.base ∈ S) :
-    (∑' w : RawCorridor G k, if Q w then w.weight (walkKernel G hloc a) p else 0) ≤
-      (S.ncard:ℝ)*(rawWeightBase r q d)^k := by
-  have hf := summable_raw_selection G hloc k S hS Q hQ
-    (fun w => w.weight (walkKernel G hloc a) p)
-  have hg := summable_raw_selection G hloc k S hS (fun w => w.base ∈ S)
-    (fun _ h => h) (fun w => w.probability (walkKernel G hloc a))
-  have hB : 0 ≤ (rawWeightBase r q d)^k := pow_nonneg
-    (zero_le_one.trans (rawWeightBase_one_le hr (hrp.trans p.2.2) hq (hqp.trans (sub_le_self _ p.2.1)) d)) _
-  have hpoint (w : RawCorridor G k) :
-      (if Q w then w.weight (walkKernel G hloc a) p else 0) ≤
-        (if w.base ∈ S then w.probability (walkKernel G hloc a) else 0) * (rawWeightBase r q d)^k :=
-    mask_le_mask_mul (hQ w) (w.2.2.mass_nonneg _ hP) hB
-      (w.weight_le_uniform hloc d hd (walkKernel G hloc a) hP p hr hq hrp hqp)
-  have hb := hf.tsum_le_tsum hpoint (hg.mul_right ((rawWeightBase r q d)^k))
-  rw [tsum_mul_right,raw_probability_on_bases_sum G hloc a k S hS] at hb
-  exact hb
-
- 
-
-theorem corridorPenalty_le_cluster (G : BondGraph V E) (hloc : LocallyFinite G)
-    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
-    (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y) (p : unitInterval)
-    {r q : ℝ} (hr : 0 < r) (hq : 0 < q) (hrp : r ≤ (p:ℝ)) (hqp : q ≤ 1-(p:ℝ))
-    (k : ℕ) (D : Set V) (ω : Set E) (x y : V) (hf : (cluster G ω x).Finite) :
-    corridorPenalty G (walkKernel G hloc a) p k D ω x y ≤
-      ((cluster G ω x).ncard:ℝ)*(rawWeightBase r q d)^k := by
-  exact raw_weight_sum_le_on_bases G hloc d hd a hP p hr hq hrp hqp k _ hf
-    (fun w => w.counted D ω x y) (fun w hw => w.counted_base_mem_cluster D ω x y hw)
-
- 
-
-theorem raw_edge_weight_bound (G : BondGraph V E) (hloc : LocallyFinite G)
-    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
-    (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y) (p : unitInterval)
-    {r q : ℝ} (hr : 0 < r) (hq : 0 < q) (hrp : r ≤ (p:ℝ)) (hqp : q ≤ 1-(p:ℝ))
-    (k : ℕ) (e : E) (z : V) (hz : z ∈ G e) :
-    (∑' w : RawCorridor G k, if e ∈ w.retained then w.weight (walkKernel G hloc a) p else 0) ≤
-      (((d:ℝ)+1)*rawWeightBase r q d)^k := by
-  have hb := raw_weight_sum_le_on_bases G hloc d hd a hP p hr hq hrp hqp k
-    (reachBall G z k) (finite_reachBall hloc z k) (fun w => e ∈ w.retained)
-    (fun w hw => w.base_mem_reachBall_of_retained e hw z hz)
-  have hcard : ((reachBall G z k).ncard:ℝ) ≤ ((d:ℝ)+1)^k := by
-    exact_mod_cast reachBall_ncard_le_pow G hloc d hd z k
-  have hB : 0 ≤ (rawWeightBase r q d)^k := pow_nonneg
-    (zero_le_one.trans (rawWeightBase_one_le hr (hrp.trans p.2.2) hq (hqp.trans (sub_le_self _ p.2.1)) d)) _
-  exact hb.trans ((mul_le_mul_of_nonneg_right hcard hB).trans_eq (mul_pow _ _ _).symm)
-
-end UniformPenaltyDomination
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section ExponentialCorridorCost
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem pow_le_exp_div_self {B : ℝ} (hB : 1 ≤ B) {k : ℕ} (hk : 0 < k) :
-    B^k ≤ Real.exp ((Real.log B + 1)*(k:ℝ))/(k:ℝ) := by
-  have hB₀ : 0 < B := zero_lt_one.trans_le hB
-  apply (le_div_iff₀ (Nat.cast_pos.mpr hk)).mpr
-  have hh : (k:ℝ) ≤ Real.exp (k:ℝ) := by linarith [Real.add_one_le_exp (k:ℝ)]
-  calc
-    _ ≤ B^k * Real.exp (k:ℝ) := mul_le_mul_of_nonneg_left hh (pow_nonneg hB₀.le _)
-    _ = Real.exp ((k:ℝ)*Real.log B) * Real.exp (k:ℝ) := by rw [Real.exp_nat_mul,Real.exp_log hB₀]
-    _ = _ := by rw [←Real.exp_add]; congr 1; ring
-
- 
-
-theorem exists_uniform_corridor_edge_cost (G : BondGraph V E) (hloc : LocallyFinite G)
-    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
-    (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y)
-    (hpcrit : 0 < (pc G:ℝ)) (hpcrit₁ : (pc G:ℝ) < 1) :
-    ∃ C : ℝ, 0 < C ∧ ∀ (p : unitInterval), (pc G:ℝ)/2 ≤ (p:ℝ) → p < pc G →
-      ∀ (k : ℕ), 3 ≤ k → ∀ e : E,
-      (∑' w : RawCorridor G k, if e ∈ w.retained then w.weight (walkKernel G hloc a) p else 0) ≤
-        Real.exp (C*(k:ℝ))/(k:ℝ) := by
-  let B := ((d:ℝ)+1)*rawWeightBase ((pc G:ℝ)/2) (1-(pc G:ℝ)) d
-  have hB : 1 ≤ B := one_le_mul_of_one_le_of_one_le (by have := Nat.cast_nonneg (α := ℝ) d; linarith)
-    (rawWeightBase_one_le (half_pos hpcrit) (by have := (pc G).2.2; linarith)
-      (sub_pos.mpr hpcrit₁) (sub_le_self _ (pc G).2.1) d)
-  refine ⟨Real.log B + 1,by linarith [Real.log_nonneg hB],?_⟩
-  intro p hpnear hp k hk e
-  obtain ⟨z,z',hzz⟩ : ∃ z z', G e = s(z,z') := by
-    obtain ⟨⟨z,z'⟩,hh⟩ := Quot.exists_rep (G e)
-    exact ⟨z,z',hh.symm⟩
-  have he : z ∈ G e := hzz.symm ▸ Sym2.mem_mk_left z z'
-  have hq : 1-(pc G:ℝ) ≤ 1-(p:ℝ) := sub_le_sub_left (show (p:ℝ) ≤ (pc G:ℝ) from hp.le) (1:ℝ)
-  exact (raw_edge_weight_bound G hloc d hd a hP p (half_pos hpcrit)
-    (sub_pos.mpr hpcrit₁) hpnear hq k e z he).trans (pow_le_exp_div_self hB (by omega))
-
-end ExponentialCorridorCost
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section LiteralDeformationDerivative
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def deformedDerivativeMass (G : BondGraph V E) (P : V → V → ℝ)
-    (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x : V) : ℝ :=
-  ∑' y, corridorPenalty G P p k D ω x y * deformedConnection G P p k t D ω x y
-
-theorem deformedConnection_eq_zero_of_not_conn (G : BondGraph V E) (P : V → V → ℝ)
-    (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x y : V) (hn : ¬Conn G ω x y) :
-    deformedConnection G P p k t D ω x y = 0 := by
-  have hnc : ¬Conn G (ω ∩ internalEdges G D) x y := fun h => hn (conn_mono G Set.inter_subset_left h)
-  simp only [deformedConnection,hnc,and_false,ite_false]
-
-theorem deformedClusterMass_eq_sum (G : BondGraph V E) (P : V → V → ℝ)
-    (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x : V)
-    (hf : (cluster G ω x).Finite) : deformedClusterMass G P p k t D ω x =
-      ∑ y ∈ hf.toFinset, deformedConnection G P p k t D ω x y := by
-  apply tsum_eq_sum
-  intro y hy
-  exact deformedConnection_eq_zero_of_not_conn G P p k t D ω x y (fun h => hy (hf.mem_toFinset.mpr h))
-
-theorem deformedDerivativeMass_eq_sum (G : BondGraph V E) (P : V → V → ℝ)
-    (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x : V)
-    (hf : (cluster G ω x).Finite) : deformedDerivativeMass G P p k t D ω x =
-      ∑ y ∈ hf.toFinset, corridorPenalty G P p k D ω x y * deformedConnection G P p k t D ω x y := by
-  apply tsum_eq_sum
-  intro y hy
-  rw [deformedConnection_eq_zero_of_not_conn G P p k t D ω x y (fun h => hy (hf.mem_toFinset.mpr h)),mul_zero]
-
-theorem measurable_deformedDerivativeMass [Countable V] [Countable E]
-    (G : BondGraph V E) (P : V → V → ℝ) (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (x : V) :
-    Measurable (fun ω => deformedDerivativeMass G P p k t D ω x) :=
-  Measurable.tsum (fun y => (measurable_corridorPenalty G P p k D x y).mul
-    (measurable_deformedConnection G P p k t D x y))
-
-theorem deformedDerivativeMass_nonneg (G : BondGraph V E) (P : V → V → ℝ)
-    (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (k : ℕ) (t : ℝ)
-    (D : Set V) (ω : Set E) (x : V) : 0 ≤ deformedDerivativeMass G P p k t D ω x :=
-  tsum_nonneg (fun y => mul_nonneg (corridorPenalty_nonneg G P hP p k D ω x y)
-    (deformedConnection_nonneg G P p k t D ω x y))
-
-theorem hasDerivAt_deformedConnection (G : BondGraph V E) (P : V → V → ℝ)
-    (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x y : V) :
-    HasDerivAt (fun s => deformedConnection G P p k s D ω x y)
-      (-(corridorPenalty G P p k D ω x y * deformedConnection G P p k t D ω x y)) t := by
-  by_cases hc : x ∈ D ∧ y ∈ D ∧ Conn G (ω ∩ internalEdges G D) x y
-  · simp only [deformedConnection,hc,and_self,ite_true]
-    simpa only [Pi.neg_apply,id_eq,neg_mul,one_mul,mul_neg,mul_comm] using
-      (((hasDerivAt_id t).neg.mul_const (corridorPenalty G P p k D ω x y)).exp)
-  · simpa only [deformedConnection,hc,ite_false,mul_zero,neg_zero] using hasDerivAt_const t (0:ℝ)
-
- 
-
-theorem hasDerivAt_deformedClusterMass (G : BondGraph V E) (P : V → V → ℝ)
-    (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x : V)
-    (hf : (cluster G ω x).Finite) :
-    HasDerivAt (fun s => deformedClusterMass G P p k s D ω x)
-      (-deformedDerivativeMass G P p k t D ω x) t := by
-  have hh := HasDerivAt.fun_sum (u := hf.toFinset) (fun y _ => hasDerivAt_deformedConnection G P p k t D ω x y)
-  simpa only [deformedClusterMass_eq_sum G P p k _ D ω x hf,
-    deformedDerivativeMass_eq_sum G P p k t D ω x hf,Finset.sum_neg_distrib] using hh
-
-end LiteralDeformationDerivative
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section DeformationDerivativeDomination
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem deformedDerivativeMass_le_square (G : BondGraph V E) (hloc : LocallyFinite G)
-    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
-    (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y) (p : unitInterval)
-    (hp0 : 0 < (p:ℝ)) (hp1 : (p:ℝ) < 1) (k : ℕ) (t : ℝ) (ht : 0 ≤ t)
-    (D : Set V) (ω : Set E) (x : V) (hf : (cluster G ω x).Finite) :
-    deformedDerivativeMass G (walkKernel G hloc a) p k t D ω x ≤
-      ((cluster G ω x).ncard:ℝ)^2 * (rawWeightBase p (1-(p:ℝ)) d)^k := by
-  rw [deformedDerivativeMass_eq_sum G (walkKernel G hloc a) p k t D ω x hf]
-  calc
-    _ ≤ ∑ _y ∈ hf.toFinset, ((cluster G ω x).ncard:ℝ)*(rawWeightBase p (1-(p:ℝ)) d)^k := by
-      apply Finset.sum_le_sum
-      intro y _
-      have hn := corridorPenalty_nonneg G (walkKernel G hloc a) hP p k D ω x y
-      exact (mul_le_of_le_one_right hn (deformedConnection_le_one G (walkKernel G hloc a) hP p k t ht D ω x y)).trans
-        (corridorPenalty_le_cluster G hloc d hd a hP p hp0 (sub_pos.mpr hp1) le_rfl le_rfl k D ω x y hf)
-    _ = _ := by
-      simp only [Finset.sum_const,nsmul_eq_mul,←Set.ncard_eq_toFinset_card _ hf]
-      ring
-
-theorem integrable_deformedDerivativeMass [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
-    (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y) (p : unitInterval)
-    (hp0 : 0 < (p:ℝ)) (hp : p < pc G) (k : ℕ) (t : ℝ) (ht : 0 ≤ t)
-    (D : Set V) (x : V) :
-    Integrable (fun ω => deformedDerivativeMass G (walkKernel G hloc a) p k t D ω x) (law p) := by
-  obtain ⟨_,_,hs⟩ := subcritical_second_moment G hloc hqt p hp
-  apply Integrable.mono' ((hs x).1.mul_const ((rawWeightBase p (1-(p:ℝ)) d)^k))
-    (measurable_deformedDerivativeMass G (walkKernel G hloc a) p k t D x).aestronglyMeasurable
-  filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
-  rw [Real.norm_of_nonneg (deformedDerivativeMass_nonneg G (walkKernel G hloc a) hP p k t D ω x)]
-  exact deformedDerivativeMass_le_square G hloc d hd a hP p hp0
-    ((show (p:ℝ) < (pc G:ℝ) from hp).trans_le (pc G).2.2) k t ht D ω x hω
-
-end DeformationDerivativeDomination
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter Topology
-section DeformationDifferentiation
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem hasDerivAt_deformedSusceptibility [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
-    (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y) (p : unitInterval)
-    (hp0 : 0 < (p:ℝ)) (hp : p < pc G) (k : ℕ) (t : ℝ) (ht : 0 < t)
-    (D : Set V) (x : V) :
-    HasDerivAt (fun s => deformedSusceptibility G (walkKernel G hloc a) p k s D x)
-      (-(∫ ω, deformedDerivativeMass G (walkKernel G hloc a) p k t D ω x ∂law p)) t := by
-  obtain ⟨_,_,hs⟩ := subcritical_second_moment G hloc hqt p hp
-  have hp1 : (p:ℝ) < 1 := (show (p:ℝ) < (pc G:ℝ) from hp).trans_le (pc G).2.2
-  have hder := hasDerivAt_integral_of_dominated_loc_of_deriv_le
-    (μ := law p) (s := Set.Ioi 0) (x₀ := t)
-    (F := fun s ω => deformedClusterMass G (walkKernel G hloc a) p k s D ω x)
-    (F' := fun s ω => -deformedDerivativeMass G (walkKernel G hloc a) p k s D ω x)
-    (bound := fun ω => ((cluster G ω x).ncard:ℝ)^2*(rawWeightBase p (1-(p:ℝ)) d)^k)
-    (Ioi_mem_nhds ht)
-    (Filter.Eventually.of_forall (fun s => (measurable_deformedClusterMass G (walkKernel G hloc a) p k s D x).aestronglyMeasurable))
-    (integrable_deformedClusterMass G hloc hqt (walkKernel G hloc a) hP p hp k t ht.le D x)
-    (measurable_deformedDerivativeMass G (walkKernel G hloc a) p k t D x).neg.aestronglyMeasurable
-    (by
-      filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
-      intro s hs
-      rw [norm_neg,Real.norm_of_nonneg (deformedDerivativeMass_nonneg G (walkKernel G hloc a) hP p k s D ω x)]
-      exact deformedDerivativeMass_le_square G hloc d hd a hP p hp0 hp1 k s hs.le D ω x hω)
-    ((hs x).1.mul_const _)
-    (by
-      filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
-      intro s _
-      exact hasDerivAt_deformedClusterMass G (walkKernel G hloc a) p k s D ω x hω)
-  rw [integral_neg] at hder
-  apply hder.2.congr_of_eventuallyEq
-  filter_upwards [Ioi_mem_nhds ht] with s hs
-  exact deformedSusceptibility_eq_integral G hloc hqt (walkKernel G hloc a) hP p hp k s hs.le D x
-
-end DeformationDifferentiation
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter Topology
-section ContinuousDeformationDerivative
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem continuous_deformedConnection (G : BondGraph V E) (P : V → V → ℝ)
-    (p : ℝ) (k : ℕ) (D : Set V) (ω : Set E) (x y : V) :
-    Continuous (fun s => deformedConnection G P p k s D ω x y) :=
-  continuous_iff_continuousAt.mpr (fun t => (hasDerivAt_deformedConnection G P p k t D ω x y).continuousAt)
-
-theorem continuous_deformedClusterMass (G : BondGraph V E) (P : V → V → ℝ)
-    (p : ℝ) (k : ℕ) (D : Set V) (ω : Set E) (x : V) (hf : (cluster G ω x).Finite) :
-    Continuous (fun s => deformedClusterMass G P p k s D ω x) :=
-  continuous_iff_continuousAt.mpr (fun t => (hasDerivAt_deformedClusterMass G P p k t D ω x hf).continuousAt)
-
-theorem continuous_deformedDerivativeMass (G : BondGraph V E) (P : V → V → ℝ)
-    (p : ℝ) (k : ℕ) (D : Set V) (ω : Set E) (x : V) (hf : (cluster G ω x).Finite) :
-    Continuous (fun s => deformedDerivativeMass G P p k s D ω x) := by
-  simp_rw [deformedDerivativeMass_eq_sum G P p k _ D ω x hf]
-  exact continuous_finsetSum _ (fun y _ => continuous_const.mul (continuous_deformedConnection G P p k D ω x y))
-
- 
-
-theorem continuousOn_integral_deformedDerivativeMass [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
-    (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y) (p : unitInterval)
-    (hp0 : 0 < (p:ℝ)) (hp : p < pc G) (k : ℕ) (D : Set V) (x : V) :
-    ContinuousOn (fun t => ∫ ω, deformedDerivativeMass G (walkKernel G hloc a) p k t D ω x ∂law p) (Set.Ici 0) := by
-  obtain ⟨_,_,hs⟩ := subcritical_second_moment G hloc hqt p hp
-  apply continuousOn_of_dominated
-    (bound := fun ω => ((cluster G ω x).ncard:ℝ)^2*(rawWeightBase p (1-(p:ℝ)) d)^k)
-  · intro t _
-    exact (measurable_deformedDerivativeMass G (walkKernel G hloc a) p k t D x).aestronglyMeasurable
-  · intro t ht
-    filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
-    rw [Real.norm_of_nonneg (deformedDerivativeMass_nonneg G (walkKernel G hloc a) hP p k t D ω x)]
-    exact deformedDerivativeMass_le_square G hloc d hd a hP p hp0
-      ((show (p:ℝ) < (pc G:ℝ) from hp).trans_le (pc G).2.2) k t ht D ω x hω
-  · exact (hs x).1.mul_const _
-  · filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
-    exact (continuous_deformedDerivativeMass G (walkKernel G hloc a) p k D ω x hω).continuousOn
-
-end ContinuousDeformationDerivative
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter Topology
-section RightDeformationDerivative
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem continuousOn_deformedSusceptibility [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (hp : p < pc G)
-    (k : ℕ) (D : Set V) (x : V) :
-    ContinuousOn (fun t => deformedSusceptibility G P p k t D x) (Set.Ici 0) := by
-  have hc : ContinuousOn (fun t => ∫ ω, deformedClusterMass G P p k t D ω x ∂law p) (Set.Ici 0) := by
-    apply continuousOn_of_dominated (bound := fun ω => ((cluster G ω x).ncard:ℝ))
-    · intro t _
-      exact (measurable_deformedClusterMass G P p k t D x).aestronglyMeasurable
-    · intro t ht
-      filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
-      rw [Real.norm_of_nonneg (deformedClusterMass_nonneg G P p k t D ω x)]
-      exact deformedClusterMass_le_ncard G P hP p k t ht D ω x hω
-    · exact (susceptibility_eq_integral G hloc hqt p hp x).1
-    · filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
-      exact (continuous_deformedClusterMass G P p k D ω x hω).continuousOn
-  exact hc.congr (fun t ht => deformedSusceptibility_eq_integral G hloc hqt P hP p hp k t ht D x)
-
- 
-
-theorem hasDerivWithinAt_deformedSusceptibility_zero [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
-    (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y) (p : unitInterval)
-    (hp0 : 0 < (p:ℝ)) (hp : p < pc G) (k : ℕ) (D : Set V) (x : V) :
-    HasDerivWithinAt (fun t => deformedSusceptibility G (walkKernel G hloc a) p k t D x)
-      (-(∫ ω, deformedDerivativeMass G (walkKernel G hloc a) p k 0 D ω x ∂law p)) (Set.Ici 0) 0 := by
-  apply hasDerivWithinAt_Ici_of_tendsto_deriv (s := Set.Ioi 0)
-  · intro t ht
-    exact (hasDerivAt_deformedSusceptibility G hloc hqt d hd a hP p hp0 hp k t ht D x).differentiableAt.differentiableWithinAt
-  · exact (continuousOn_deformedSusceptibility G hloc hqt (walkKernel G hloc a) hP p hp k D x 0 (by simp)).mono Set.Ioi_subset_Ici_self
-  · exact self_mem_nhdsWithin
-  · have hc := (continuousOn_integral_deformedDerivativeMass G hloc hqt d hd a hP p hp0 hp k D x 0 (by simp)).neg
-    apply (hc.mono Set.Ioi_subset_Ici_self).congr'
-    filter_upwards [self_mem_nhdsWithin (s := Set.Ioi (0:ℝ)) (a := 0)] with t ht
-    exact (hasDerivAt_deformedSusceptibility G hloc hqt d hd a hP p hp0 hp k t ht D x).deriv.symm
-
-end RightDeformationDerivative
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section TerminalMoveGeometry
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-namespace RawCorridor
-variable {G : BondGraph V E} {k : ℕ}
-
- 
-
-theorem counted_move_terminal (w : RawCorridor G k) (D : Set V) (ω : Set E)
-    (F : Set E) (hF : F ⊆ ω ∩ internalEdges G D) (b z y : V)
-    (hbz : Conn G F b z) (hc : w.counted D ω b y) (hdis : Disjoint F w.retained) :
-    w.counted D ω z y := by
-  apply (w.counted_iff D ω z y).mpr
-  refine ⟨hc.1,conn_trans G _ (conn_symm G _ (conn_mono G hF hbz)) hc.2.2.1,?_,hc.2.2.2.2⟩
-  intro e he
-  refine ⟨(hc.2.2.2.1 e he).1,?_⟩
-  intro hzy
-  apply (hc.2.2.2.1 e he).2
-  apply conn_trans G _ (conn_mono G ?_ hbz) hzy
-  intro f hf
-  refine ⟨hF hf,?_⟩
-  intro hfe
-  have hh : f = e := Set.mem_singleton_iff.mp hfe
-  exact Set.disjoint_left.mp hdis hf (hh.symm ▸ he)
-
-end RawCorridor
-
- 
-
-theorem corridorPenaltyTerm_move_bound (G : BondGraph V E) (P : V → V → ℝ)
-    (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (k : ℕ) (D : Set V) (ω : Set E)
-    (F : Finset E) (hF : (F:Set E) ⊆ ω ∩ internalEdges G D) (b z y : V)
-    (hbz : Conn G (F:Set E) b z) (w : RawCorridor G k) :
-    corridorPenaltyTerm G P p D ω b y w ≤
-      (∑ e ∈ F, if e ∈ w.retained then w.weight P p else 0) + corridorPenaltyTerm G P p D ω z y w := by
-  have hn := w.weight_nonneg P hP p
-  have hs : 0 ≤ ∑ e ∈ F, if e ∈ w.retained then w.weight P p else 0 :=
-    Finset.sum_nonneg (fun e _ => by split_ifs; exact hn; exact le_rfl)
-  by_cases hc : w.counted D ω b y
-  · rw [corridorPenaltyTerm,ite_eq_left hc]
-    by_cases hd : Disjoint (F:Set E) w.retained
-    · have hc' := w.counted_move_terminal D ω F hF b z y hbz hc hd
-      simpa only [corridorPenaltyTerm,hc',ite_true] using le_add_of_nonneg_left hs
-    · obtain ⟨e,he,hew⟩ := Set.not_disjoint_iff.mp hd
-      have he' : (if e ∈ w.retained then w.weight P p else 0) ≤
-          ∑ f ∈ F, if f ∈ w.retained then w.weight P p else 0 :=
-        Finset.single_le_sum (f := fun f => if f ∈ w.retained then w.weight P p else 0) (fun f _ => by split_ifs; exact hn; exact le_rfl) he
-      rw [ite_eq_left hew] at he'
-      exact he'.trans (le_add_of_nonneg_right (corridorPenaltyTerm_nonneg G P hP p D ω z y w))
-  · rw [corridorPenaltyTerm,ite_eq_right hc]
-    exact add_nonneg hs (corridorPenaltyTerm_nonneg G P hP p D ω z y w)
-
-end TerminalMoveGeometry
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section TerminalMovePenalty
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem summable_raw_edge_weights (G : BondGraph V E) (hloc : LocallyFinite G)
-    (P : V → V → ℝ) (p : ℝ) (k : ℕ) (e : E) :
-    Summable (fun w : RawCorridor G k => if e ∈ w.retained then w.weight P p else 0) := by
-  obtain ⟨z,z',hzz⟩ : ∃ z z', G e = s(z,z') := by
-    obtain ⟨⟨z,z'⟩,hh⟩ := Quot.exists_rep (G e)
-    exact ⟨z,z',hh.symm⟩
-  have he : z ∈ G e := hzz.symm ▸ Sym2.mem_mk_left z z'
-  exact summable_raw_selection G hloc k (reachBall G z k) (finite_reachBall hloc z k)
-    (fun w => e ∈ w.retained) (fun w hw => w.base_mem_reachBall_of_retained e hw z he)
-    (fun w => w.weight P p)
-
- 
-
-theorem corridorPenalty_move_bound (G : BondGraph V E) (hloc : LocallyFinite G)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval)
-    (k : ℕ) (D : Set V) (ω : Set E) (F : Finset E)
-    (hF : (F:Set E) ⊆ ω ∩ internalEdges G D) (b z y : V) (hbz : Conn G (F:Set E) b z) :
-    corridorPenalty G P p k D ω b y ≤
-      (∑ e ∈ F, ∑' w : RawCorridor G k, if e ∈ w.retained then w.weight P p else 0) +
-        corridorPenalty G P p k D ω z y := by
-  have hs : Summable (fun w : RawCorridor G k => ∑ e ∈ F, if e ∈ w.retained then w.weight P p else 0) :=
-    (hasSum_sum (fun e (_ : e ∈ F) => (summable_raw_edge_weights G hloc P p k e).hasSum)).summable
-  have hr := corridorPenalty_summable G hloc P p k D ω z y
-  have hl := (corridorPenalty_summable G hloc P p k D ω b y).tsum_le_tsum
-    (corridorPenaltyTerm_move_bound G P hP p k D ω F hF b z y hbz) (hs.add hr)
-  rw [hs.tsum_add hr,Summable.tsum_finsetSum (fun e _ => summable_raw_edge_weights G hloc P p k e)] at hl
-  exact hl
-
-end TerminalMovePenalty
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section FiniteOpeningAssociation
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem measurable_union_const (S : Set E) : Measurable (fun ω : Set E => ω ∪ S) := by
-  apply measurable_setOfPred.comp
-  apply Measurable.of_eval
-  intro e
-  exact ((measurable_pi_apply e).comp MeasurableEquiv.setOfPred.symm.measurable).or measurable_const
-
-theorem measurableSet_finite_open (F : Finset E) : MeasurableSet {ω : Set E | (F:Set E) ⊆ ω} := by
-  have he : {ω : Set E | (F:Set E) ⊆ ω} = ⋂ e ∈ F, {ω : Set E | e ∈ ω} := by
-    ext ω
-    simp only [Set.mem_ofPred_eq,Set.mem_iInter,Finset.mem_coe,Set.subset_def]
-  rw [he]
-  exact F.measurableSet_biInter (fun e _ => measurableSet_setOfPred.mpr ((measurable_pi_apply e).comp MeasurableEquiv.setOfPred.symm.measurable))
-
- 
-
-theorem finite_open_integral_lower (p : unitInterval) (F : Finset E) (f : Set E → ℝ)
-    (hf : Measurable f) (hn : ∀ ω, 0 ≤ f ω) (hb : ∀ ω, f ω ≤ 1)
-    (hm : Monotone f) :
-    (p:ℝ)^F.card*(∫ ω, f ω ∂law p) ≤ ∫ ω, if (F:Set E) ⊆ ω then f ω else 0 ∂law p := by
-  let J : Set E → ℝ := fun ω => if (F:Set E) ⊆ ω then 1 else 0
-  let g : Set E → ℝ := fun ω => f (ω ∪ (F:Set E))
-  have hJ : Measurable J := Measurable.ite (measurableSet_finite_open F) measurable_const measurable_const
-  have hg : Measurable g := hf.comp (measurable_union_const (F:Set E))
-  have hi (f : Set E → ℝ) (hfm : Measurable f) (hfn : ∀ ω, 0 ≤ f ω) (hfb : ∀ ω, f ω ≤ 1) : Integrable f (law p) := by
-    exact Integrable.mono' (integrable_const (1:ℝ)) hfm.aestronglyMeasurable
-      (Filter.Eventually.of_forall (fun ω => by rw [Real.norm_of_nonneg (hfn ω)]; exact hfb ω))
-  have hfi := hi f hf hn hb
-  have hgi := hi g hg (fun ω => hn _) (fun ω => hb _)
-  have hJint : (∫ ω, J ω ∂law p) = (p:ℝ)^F.card := by
-    have hh := integral_indicator_const (μ := law p) (1:ℝ) (measurableSet_finite_open F)
-    simpa only [Set.indicator,Set.mem_ofPred_eq,smul_eq_mul,mul_one,finite_open_probability,J] using hh
-  have he := integral_mul_disjoint_coordinates p (F:Set E) (F:Set E)ᶜ disjoint_compl_right J g hJ hg
-    (fun ω => by simp only [J,Set.subset_inter_iff,Set.Subset.refl,and_true])
-    (fun ω => by
-      dsimp only [g]
-      congr 1
-      ext e
-      simp only [Set.mem_union,Set.mem_inter_iff,Set.mem_compl_iff]
-      tauto)
-  calc
-    _ ≤ (p:ℝ)^F.card*(∫ ω, g ω ∂law p) := mul_le_mul_of_nonneg_left
-      (integral_mono hfi hgi (fun ω => hm Set.subset_union_left)) (pow_nonneg p.2.1 _)
-    _ = (∫ ω, J ω ∂law p)*(∫ ω, g ω ∂law p) := by rw [hJint]
-    _ = ∫ ω, J ω*g ω ∂law p := he.symm
-    _ = _ := by
-      apply integral_congr_ae
-      exact Filter.Eventually.of_forall (fun ω => by
-        by_cases hF : (F:Set E) ⊆ ω
-        · simp only [J,g,hF,ite_true,one_mul,Set.union_eq_left.mpr hF]
-        · simp only [J,hF,ite_false,zero_mul])
-
-end FiniteOpeningAssociation
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section TerminalConnectionComparison
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem corridorPenalty_move_edge_cost (G : BondGraph V E) (hloc : LocallyFinite G)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval)
-    (k : ℕ) (D : Set V) (ω : Set E) (F : Finset E)
-    (hF : (F:Set E) ⊆ ω ∩ internalEdges G D) (b z y : V) (hbz : Conn G (F:Set E) b z)
-    (c : ℝ) (hc : ∀ e, (∑' w : RawCorridor G k, if e ∈ w.retained then w.weight P p else 0) ≤ c) :
-    corridorPenalty G P p k D ω b y ≤ (F.card:ℝ)*c+corridorPenalty G P p k D ω z y := by
-  apply (corridorPenalty_move_bound G hloc P hP p k D ω F hF b z y hbz).trans
-  apply add_le_add _ le_rfl
-  exact (Finset.sum_le_sum (fun e _ => hc e)).trans_eq (by simp only [Finset.sum_const,nsmul_eq_mul])
-
- 
-
-theorem deformedConnection_move_lower (G : BondGraph V E) (hloc : LocallyFinite G)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval)
-    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D : Set V) (ω : Set E) (F : Finset E)
-    (hF : (F:Set E) ⊆ internalEdges G D) (b z y : V) (hb : b ∈ D)
-    (hbz : Conn G (F:Set E) b z) (c : ℝ)
-    (hc : ∀ e, (∑' w : RawCorridor G k, if e ∈ w.retained then w.weight P p else 0) ≤ c) :
-    Real.exp (-t*((F.card:ℝ)*c)) * (if (F:Set E) ⊆ ω then deformedConnection G P p k t D ω z y else 0) ≤
-      deformedConnection G P p k t D ω b y := by
-  by_cases ho : (F:Set E) ⊆ ω
-  · rw [ite_eq_left ho]
-    by_cases hz : z ∈ D ∧ y ∈ D ∧ Conn G (ω ∩ internalEdges G D) z y
-    · have hFa : (F:Set E) ⊆ ω ∩ internalEdges G D := fun e he => ⟨ho he,hF he⟩
-      have hb' : b ∈ D ∧ y ∈ D ∧ Conn G (ω ∩ internalEdges G D) b y :=
-        ⟨hb,hz.2.1,conn_trans G _ (conn_mono G hFa hbz) hz.2.2⟩
-      simp only [deformedConnection,hz,hb',and_self,ite_true]
-      rw [←Real.exp_add]
-      apply Real.exp_le_exp.mpr
-      have hh := mul_le_mul_of_nonpos_left
-        (corridorPenalty_move_edge_cost G hloc P hP p k D ω F hFa b z y hbz c hc) (neg_nonpos.mpr ht)
-      simpa only [mul_add] using hh
-    · rw [deformedConnection,ite_eq_right hz,mul_zero]
-      exact deformedConnection_nonneg G P p k t D ω b y
-  · rw [ite_eq_right ho,mul_zero]
-    exact deformedConnection_nonneg G P p k t D ω b y
-
-end TerminalConnectionComparison
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section EndpointSusceptibilityComparison
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem deformedKernel_move_lower [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval)
-    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D : Set V) (F : Finset E)
-    (hF : (F:Set E) ⊆ internalEdges G D) (b z y : V) (hb : b ∈ D)
-    (hbz : Conn G (F:Set E) b z) (c : ℝ)
-    (hc : ∀ e, (∑' w : RawCorridor G k, if e ∈ w.retained then w.weight P p else 0) ≤ c) :
-    Real.exp (-t*((F.card:ℝ)*c)) * (p:ℝ)^F.card * deformedKernel G P p k t D z y ≤
-      deformedKernel G P p k t D b y := by
-  have hi := integrable_deformedConnection G P hP p k t ht D z y
-  have hIi := hi.indicator (measurableSet_finite_open F)
-  have has := finite_open_integral_lower p F (fun ω => deformedConnection G P p k t D ω z y)
-    (measurable_deformedConnection G P p k t D z y)
-    (fun ω => deformedConnection_nonneg G P p k t D ω z y)
-    (fun ω => deformedConnection_le_one G P hP p k t ht D ω z y)
-    (fun _ _ hω => deformedConnection_mono G hloc P hP p k t ht (Set.Subset.refl _) hω z y)
-  have hb' := integral_mono (hIi.const_mul (Real.exp (-t*((F.card:ℝ)*c))))
-    (integrable_deformedConnection G P hP p k t ht D b y)
-    (fun ω => deformedConnection_move_lower G hloc P hP p k t ht D ω F hF b z y hb hbz c hc)
-  simp only [Set.indicator,Set.mem_ofPred_eq,integral_const_mul] at hb'
-  have hh := mul_le_mul_of_nonneg_left has (Real.exp_pos (-t*((F.card:ℝ)*c))).le
-  simp only [mul_assoc] at hh ⊢
-  exact hh.trans hb'
-
- 
-theorem deformedSusceptibility_move_lower [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (hp : p < pc G)
-    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D : Set V) (F : Finset E)
-    (hF : (F:Set E) ⊆ internalEdges G D) (b z : V) (hb : b ∈ D)
-    (hbz : Conn G (F:Set E) b z) (c : ℝ)
-    (hc : ∀ e, (∑' w : RawCorridor G k, if e ∈ w.retained then w.weight P p else 0) ≤ c) :
-    Real.exp (-t*((F.card:ℝ)*c)) * (p:ℝ)^F.card * deformedSusceptibility G P p k t D z ≤
-      deformedSusceptibility G P p k t D b := by
-  simpa only [deformedSusceptibility,tsum_mul_left] using
-    ((summable_deformedKernel G hloc hqt P hP p hp k t ht D z).mul_left
-      (Real.exp (-t*((F.card:ℝ)*c)) * (p:ℝ)^F.card)).tsum_le_tsum
-      (fun y => deformedKernel_move_lower G hloc P hP p k t ht D F hF b z y hb hbz c hc)
-      (summable_deformedKernel G hloc hqt P hP p hp k t ht D b)
-
-end EndpointSusceptibilityComparison
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Deformation.ClusterProducts
+
+namespace OAI
+
+/-! Derivatives for bond percolation. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section WeightedNeighborhoodEstimate
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem twoPoint_neighbor_div_bound [Countable E] (G : BondGraph V E) (hloc : LocallyFinite G)
+    (p : unitInterval) (hp : 0 < (p:ℝ)) (v a z : V) (hz : z ∈ neighbors G hloc a) :
+    twoPoint G p v z ≤ twoPoint G p v a / (p:ℝ) := by
+  obtain ⟨e,he⟩ := (mem_neighbors G hloc a z).mp hz
+  have hh := restrictedTwoPoint_adj_lower G Set.univ p a z v e he (Set.mem_univ _)
+  simp only [Set.inter_univ] at hh
+  change (p:ℝ)*twoPoint G p z v ≤ twoPoint G p a v at hh
+  rw [twoPoint_symm G p z v,twoPoint_symm G p a v] at hh
+  exact (le_div_iff₀ hp).mpr (by simpa only [mul_comm] using hh)
+
+ 
+
+theorem neighborhood_twoPoint_bound [Countable E] (G : BondGraph V E) (hloc : LocallyFinite G)
+    (d : ℕ) (hd : ∀ a, (neighbors G hloc a).card ≤ d) (p : unitInterval) (hp : 0 < (p:ℝ))
+    (W : Finset V) (v : V) :
+    (∑' z : closedBondNeighborhood G (W:Set V), twoPoint G p v z) ≤
+      (1+(d:ℝ)/(p:ℝ)) * (∑ a ∈ W, twoPoint G p v a) := by
+  apply (neighborhood_mass_le G hloc W (twoPoint G p v) (twoPoint_nonneg G p v)).trans
+  rw [Finset.mul_sum]
+  apply Finset.sum_le_sum
+  intro a _
+  have hsum : (∑ z ∈ neighbors G hloc a, twoPoint G p v z) ≤
+      (d:ℝ)*(twoPoint G p v a/(p:ℝ)) := by
+    calc
+      _ ≤ ∑ _z ∈ neighbors G hloc a, twoPoint G p v a/(p:ℝ) :=
+        Finset.sum_le_sum (fun z hz => twoPoint_neighbor_div_bound G hloc p hp v a z hz)
+      _ = ((neighbors G hloc a).card:ℝ)*(twoPoint G p v a/(p:ℝ)) := by simp only [Finset.sum_const,nsmul_eq_mul]
+      _ ≤ _ := mul_le_mul_of_nonneg_right (Nat.cast_le.mpr (hd a)) (div_nonneg (twoPoint_nonneg G p _ _) hp.le)
+  calc
+    _ ≤ twoPoint G p v a + (d:ℝ)*(twoPoint G p v a/(p:ℝ)) := add_le_add (le_refl _) hsum
+    _ = _ := by ring
+
+end WeightedNeighborhoodEstimate
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section SizeWeightedPotential
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def sizeMembership (G : BondGraph V E) (x a : V) (ω : Set E) : ℝ :=
+  if Conn G ω x a then ((cluster G ω x).ncard:ℝ) else 0
+
+noncomputable def sizeWeightedConnectionPotential (G : BondGraph V E) (p : unitInterval)
+    (x y : V) (ω : Set E) : ℝ := ∑' a, sizeMembership G x a ω * twoPoint G p y a
+
+theorem sizeMembership_nonneg (G : BondGraph V E) (x a : V) (ω : Set E) :
+    0 ≤ sizeMembership G x a ω := by
+  unfold sizeMembership
+  split_ifs
+  · exact Nat.cast_nonneg _
+  · rfl
+
+theorem sizeMembership_le (G : BondGraph V E) (x a : V) (ω : Set E) :
+    sizeMembership G x a ω ≤ ((cluster G ω x).ncard:ℝ) := by
+  unfold sizeMembership
+  split_ifs
+  · rfl
+  · exact Nat.cast_nonneg _
+
+theorem measurable_sizeMembership [Countable V] [Countable E] (G : BondGraph V E)
+    (hloc : LocallyFinite G) (x a : V) : Measurable (sizeMembership G x a) := by
+  apply Measurable.ite (measurableSet_conn G x a)
+    ((measurable_of_countable (fun n : ℕ => (n:ℝ))).comp (measurable_cluster_ncard G hloc x)) measurable_const
+
+theorem integrable_sizeMembership [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (x a : V) : Integrable (sizeMembership G x a) (law p) := by
+  apply Integrable.mono' (susceptibility_eq_integral G hloc hqt p hp x).1
+    (measurable_sizeMembership G hloc x a).aestronglyMeasurable
+  exact Filter.Eventually.of_forall (fun ω => by
+    rw [Real.norm_of_nonneg (sizeMembership_nonneg G x a ω)]
+    exact sizeMembership_le G x a ω)
+
+theorem integral_sizeMembership_le_susceptibility [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (x a : V) :
+    (∫ ω, sizeMembership G x a ω ∂law p) ≤ susceptibility G p x := by
+  rw [(susceptibility_eq_integral G hloc hqt p hp x).2]
+  exact integral_mono (integrable_sizeMembership G hloc hqt p hp x a)
+    (susceptibility_eq_integral G hloc hqt p hp x).1 (sizeMembership_le G x a)
+
+theorem sizeWeightedConnectionPotential_eq_finite (G : BondGraph V E) (p : unitInterval)
+    (x y : V) (ω : Set E) (hf : (cluster G ω x).Finite) :
+    sizeWeightedConnectionPotential G p x y ω =
+      ((cluster G ω x).ncard:ℝ) * ∑ a ∈ hf.toFinset, twoPoint G p y a := by
+  unfold sizeWeightedConnectionPotential
+  rw [tsum_eq_sum (s := hf.toFinset)]
+  · rw [Finset.mul_sum]
+    apply Finset.sum_congr rfl
+    intro a ha
+    have hc : Conn G ω x a := hf.mem_toFinset.mp ha
+    simp only [sizeMembership,ite_eq_left hc]
+  · intro a ha
+    have hn : ¬Conn G ω x a := fun hc => ha (hf.mem_toFinset.mpr hc)
+    simp only [sizeMembership,ite_eq_right hn,zero_mul]
+
+end SizeWeightedPotential
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section SizePotentialIntegral
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Countable V] [Countable E] [Nonempty V]
+
+theorem summable_sizePotential_integral_norm (G : BondGraph V E) (hloc : LocallyFinite G)
+    (hqt : QuasiTransitive G) (p : unitInterval) (hp : p < pc G) (x y : V) :
+    Summable (fun a => ∫ ω, ‖sizeMembership G x a ω * twoPoint G p y a‖ ∂law p) := by
+  apply ((summable_twoPoint_subcritical G hloc hqt p hp y).mul_left (susceptibility G p x)).of_nonneg_of_le
+  · exact fun a => integral_nonneg (fun _ => norm_nonneg _)
+  · intro a
+    simp only [Real.norm_of_nonneg (mul_nonneg (sizeMembership_nonneg G x a _) (twoPoint_nonneg G p y a)),
+      integral_mul_const]
+    exact mul_le_mul_of_nonneg_right (integral_sizeMembership_le_susceptibility G hloc hqt p hp x a)
+      (twoPoint_nonneg G p y a)
+
+theorem integrable_sizeWeightedConnectionPotential (G : BondGraph V E) (hloc : LocallyFinite G)
+    (hqt : QuasiTransitive G) (p : unitInterval) (hp : p < pc G) (x y : V) :
+    Integrable (sizeWeightedConnectionPotential G p x y) (law p) := by
+  apply Integrable.mono' ((susceptibility_eq_integral G hloc hqt p hp x).1.mul_const
+    (susceptibility G p y))
+    (Measurable.tsum (fun a => (measurable_sizeMembership G hloc x a).mul_const (twoPoint G p y a))).aestronglyMeasurable
+  exact Filter.Eventually.of_forall (fun ω => by
+    have hb (a : V) : sizeMembership G x a ω * twoPoint G p y a ≤
+        ((cluster G ω x).ncard:ℝ)*twoPoint G p y a :=
+      mul_le_mul_of_nonneg_right (sizeMembership_le G x a ω) (twoPoint_nonneg G p y a)
+    have hn (a : V) : 0 ≤ sizeMembership G x a ω * twoPoint G p y a :=
+      mul_nonneg (sizeMembership_nonneg G x a ω) (twoPoint_nonneg G p y a)
+    have hs := (summable_twoPoint_subcritical G hloc hqt p hp y).mul_left ((cluster G ω x).ncard:ℝ)
+    change ‖∑' a, sizeMembership G x a ω * twoPoint G p y a‖ ≤ _
+    rw [Real.norm_of_nonneg (tsum_nonneg hn)]
+    exact ((hs.of_nonneg_of_le hn hb).tsum_le_tsum hb hs).trans_eq (tsum_mul_left))
+
+omit [Countable V] [Countable E] [Nonempty V] in
+theorem sizeWeightedConnectionPotential_nonneg (G : BondGraph V E) (p : unitInterval)
+    (x y : V) (ω : Set E) : 0 ≤ sizeWeightedConnectionPotential G p x y ω :=
+  tsum_nonneg (fun a => mul_nonneg (sizeMembership_nonneg G x a ω) (twoPoint_nonneg G p y a))
+
+theorem summable_twoPoint_cube_pair (G : BondGraph V E) (hloc : LocallyFinite G)
+    (hqt : QuasiTransitive G) (p : unitInterval) (hp : p < pc G) (x y : V) :
+    Summable (fun ba : V×V => twoPoint G p x ba.1 * twoPoint G p ba.1 ba.2 * twoPoint G p y ba.2) := by
+  exact (summable_twoPoint_pair G hloc hqt p hp x).of_nonneg_of_le
+    (fun ba => mul_nonneg (mul_nonneg (twoPoint_nonneg G p _ _) (twoPoint_nonneg G p _ _))
+      (twoPoint_nonneg G p _ _))
+    (fun ba => mul_le_of_le_one_right
+      (mul_nonneg (twoPoint_nonneg G p _ _) (twoPoint_nonneg G p _ _)) (twoPoint_le_one G p _ _))
+
+ 
+theorem integral_sizeWeightedConnectionPotential_le (G : BondGraph V E) (hloc : LocallyFinite G)
+    (hqt : QuasiTransitive G) (p : unitInterval) (hp : p < pc G) (x y : V) :
+    (∫ ω, sizeWeightedConnectionPotential G p x y ω ∂law p) ≤
+      maxSusceptibility G p * (∑' a, ∑' b, twoPoint G p x b * twoPoint G p b a * twoPoint G p y a) := by
+  have hi (a : V) := (integrable_sizeMembership G hloc hqt p hp x a).mul_const (twoPoint G p y a)
+  have hs := summable_sizePotential_integral_norm G hloc hqt p hp x y
+  have hc : Summable (fun a => ∑' b, twoPoint G p x b * twoPoint G p b a * twoPoint G p y a) :=
+    (summable_twoPoint_cube_pair G hloc hqt p hp x y).prod_symm.prod
+  change (∫ ω, ∑' a, sizeMembership G x a ω * twoPoint G p y a ∂law p) ≤ _
+  rw [←integral_tsum_of_summable_integral_norm hi hs,←tsum_mul_left]
+  have hs' : Summable (fun a => ∫ ω, sizeMembership G x a ω * twoPoint G p y a ∂law p) := by
+    simpa only [Real.norm_of_nonneg (mul_nonneg (sizeMembership_nonneg G x _ _) (twoPoint_nonneg G p y _))] using hs
+  apply hs'.tsum_le_tsum (fun a => ?_) (hc.mul_left _)
+  rw [integral_mul_const,tsum_mul_right]
+  calc
+    _ ≤ (maxSusceptibility G p * (∑' b, twoPoint G p x b * twoPoint G p b a)) * twoPoint G p y a :=
+      mul_le_mul_of_nonneg_right (integral_size_membership_le G hloc hqt p hp x a) (twoPoint_nonneg G p y a)
+    _ = _ := mul_assoc _ _ _
+
+end SizePotentialIntegral
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section WeightedCorrelationPointwise
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Countable V] [Countable E] [Nonempty V]
+
+ 
+
+theorem deformedFreshExteriorMoment_lower (G : BondGraph V E) (hloc : LocallyFinite G)
+    (hqt : QuasiTransitive G) (d : ℕ) (hd : ∀ a, (neighbors G hloc a).card ≤ d)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval)
+    (hp : p < pc G) (hp₀ : 0 < (p:ℝ)) (k : ℕ) (t : ℝ) (ht : 0 ≤ t)
+    (D : Set V) (x y : V) (ω : Set E) (hf : (cluster G ω x).Finite) :
+    deformedClusterMass G P p k t D ω x * deformedSusceptibility G P p k t D y -
+      (1+(d:ℝ)/(p:ℝ))*maxSusceptibility G p * sizeWeightedConnectionPotential G p x y ω ≤
+      deformedFreshExteriorMoment G P p k t D x y ω := by
+  let A := cluster G (ω ∩ internalEdges G D) x
+  have hsub : A ⊆ cluster G ω x := fun _ h => conn_mono G Set.inter_subset_left h
+  have hA : A.Finite := hf.subset hsub
+  have hn : (∑' z : closedBondNeighborhood G A, twoPoint G p y z) ≤
+      (1+(d:ℝ)/(p:ℝ))*(∑ a ∈ hf.toFinset, twoPoint G p y a) := by
+    have hh := neighborhood_twoPoint_bound G hloc d hd p hp₀ hA.toFinset y
+    rw [hA.coe_toFinset] at hh
+    exact hh.trans (mul_le_mul_of_nonneg_left
+      (Finset.sum_le_sum_of_subset_of_nonneg
+        (fun a ha => hf.mem_toFinset.mpr (hsub (hA.mem_toFinset.mp ha)))
+        (fun a _ _ => twoPoint_nonneg G p y a))
+      (add_nonneg zero_le_one (div_nonneg (Nat.cast_nonneg _) hp₀.le)))
+  have hm := deformedClusterMass_le_ncard G P hP p k t ht D ω x hf
+  have hm₀ := deformedClusterMass_nonneg G P p k t D ω x
+  have hχ : 0 ≤ maxSusceptibility G p := zero_le_one.trans (maxSusceptibility_one_le G hloc hqt p hp)
+  have hh := mul_le_mul_of_nonneg_left
+    (deformedSusceptibility_deletion G hloc hqt P hP p hp k t ht D A y) hm₀
+  have hsum : 0 ≤ ∑ a ∈ hf.toFinset, twoPoint G p y a :=
+    Finset.sum_nonneg (fun a _ => twoPoint_nonneg G p y a)
+  have hC : 0 ≤ 1+(d:ℝ)/(p:ℝ) := add_nonneg zero_le_one (div_nonneg (Nat.cast_nonneg _) hp₀.le)
+  have herr : deformedClusterMass G P p k t D ω x *
+      (maxSusceptibility G p * (∑' z : closedBondNeighborhood G A, twoPoint G p y z)) ≤
+      (1+(d:ℝ)/(p:ℝ))*maxSusceptibility G p * sizeWeightedConnectionPotential G p x y ω := by
+    calc
+      _ ≤ deformedClusterMass G P p k t D ω x *
+          (maxSusceptibility G p * ((1+(d:ℝ)/(p:ℝ))*(∑ a ∈ hf.toFinset, twoPoint G p y a))) :=
+        mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_left hn hχ) hm₀
+      _ ≤ ((cluster G ω x).ncard:ℝ)*
+          (maxSusceptibility G p * ((1+(d:ℝ)/(p:ℝ))*(∑ a ∈ hf.toFinset, twoPoint G p y a))) :=
+        mul_le_mul_of_nonneg_right hm (mul_nonneg hχ (mul_nonneg hC hsum))
+      _ = _ := by rw [sizeWeightedConnectionPotential_eq_finite G p x y ω hf]; ring
+  dsimp only [deformedFreshExteriorMoment]
+  dsimp only [A] at hh
+  nlinarith
+
+end WeightedCorrelationPointwise
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section WeightedCorrelationIntegral
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Countable V] [Countable E] [Nonempty V]
+
+ 
+
+theorem deformed_distinct_correlation (G : BondGraph V E) (hloc : LocallyFinite G)
+    (hqt : QuasiTransitive G) (d : ℕ) (hd : ∀ a, (neighbors G hloc a).card ≤ d)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval)
+    (hp : p < pc G) (hp₀ : 0 < (p:ℝ)) (k : ℕ) (t : ℝ) (ht : 0 ≤ t)
+    (D : Set V) (x y : V) :
+    deformedSusceptibility G P p k t D x * deformedSusceptibility G P p k t D y -
+      (1+(d:ℝ)/(p:ℝ))*(maxSusceptibility G p)^2 *
+        (∑' a, ∑' b, twoPoint G p x b * twoPoint G p b a * twoPoint G p y a) ≤
+      ∫ ω, (if ¬Conn G (ω ∩ internalEdges G D) x y then
+        deformedClusterMass G P p k t D ω x * deformedClusterMass G P p k t D ω y else 0) ∂law p := by
+  rw [←integral_deformedFreshExteriorMoment G hloc hqt P hP p hp k t ht D x y]
+  have hi := integrable_deformedClusterMass G hloc hqt P hP p hp k t ht D x
+  have hj := integrable_sizeWeightedConnectionPotential G hloc hqt p hp x y
+  have hb := integral_mono_ae
+    ((hi.mul_const (deformedSusceptibility G P p k t D y)).sub
+      (hj.const_mul ((1+(d:ℝ)/(p:ℝ))*maxSusceptibility G p)))
+    (integrable_deformedFreshExteriorMoment G hloc hqt P hP p hp k t ht D x y)
+    (by
+      filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
+      exact deformedFreshExteriorMoment_lower G hloc hqt d hd P hP p hp hp₀ k t ht D x y ω hω)
+  simp only [Pi.sub_apply] at hb
+  rw [integral_sub (hi.mul_const _) (hj.const_mul _),integral_mul_const,integral_const_mul,
+    ←deformedSusceptibility_eq_integral G hloc hqt P hP p hp k t ht D x] at hb
+  have hC : 0 ≤ (1+(d:ℝ)/(p:ℝ))*maxSusceptibility G p :=
+    mul_nonneg (add_nonneg zero_le_one (div_nonneg (Nat.cast_nonneg _) hp₀.le))
+      (zero_le_one.trans (maxSusceptibility_one_le G hloc hqt p hp))
+  have herr := mul_le_mul_of_nonneg_left
+    (integral_sizeWeightedConnectionPotential_le G hloc hqt p hp x y) hC
+  nlinarith
+
+end WeightedCorrelationIntegral
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section ExactCubeKernel
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem HasKernel.square_real_kernel (G : BondGraph V E) (p : unitInterval)
+    (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T) (x y : V) :
+    (∑' b, twoPoint G p x b * twoPoint G p b y) = (matrixCoeff (T^2) x y).re := by
+  have hh := ((hT.positiveRealMatrix G p T).mul_hasSum T x y).tsum_eq
+  simpa only [hT.matrixCoeff_re G p T,pow_two] using hh
+
+theorem HasKernel.cube_real_kernel (G : BondGraph V E) (p : unitInterval)
+    (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T) (x y : V) :
+    (∑' a, ∑' b, twoPoint G p x b * twoPoint G p b a * twoPoint G p y a) =
+      (matrixCoeff (T^3) x y).re := by
+  have hh := (((hT.positiveRealMatrix G p T).pow 2).mul_hasSum T x y).tsum_eq
+  calc
+    _ = ∑' a, (matrixCoeff (T^2) x a).re * (matrixCoeff T a y).re := by
+      apply tsum_congr
+      intro a
+      rw [tsum_mul_right,hT.square_real_kernel G p T x a,hT.matrixCoeff_re G p T a y,twoPoint_symm G p y a]
+    _ = _ := by simpa only [show T^2*T = T^3 by rw [←pow_succ]] using hh
+
+end ExactCubeKernel
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section UniformWeightedCorrelation
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Countable V] [Countable E] [Nonempty V]
+
+ 
+
+theorem deformed_distinct_correlation_uniform (G : BondGraph V E) (hloc : LocallyFinite G)
+    (hqt : QuasiTransitive G) (d : ℕ) (hd : ∀ a, (neighbors G hloc a).card ≤ d)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (hpcrit : 0 < (pc G:ℝ))
+    (p : unitInterval) (hp : p < pc G) (hpnear : (pc G:ℝ)/2 ≤ (p:ℝ))
+    (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T)
+    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D : Set V) (x y : V) :
+    deformedSusceptibility G P p k t D x * deformedSusceptibility G P p k t D y -
+      (1+2*(d:ℝ)/(pc G:ℝ))*(maxSusceptibility G p)^2 * (matrixCoeff (T^3) x y).re ≤
+      ∫ ω, (if ¬Conn G (ω ∩ internalEdges G D) x y then
+        deformedClusterMass G P p k t D ω x * deformedClusterMass G P p k t D ω y else 0) ∂law p := by
+  have hp₀ : 0 < (p:ℝ) := (half_pos hpcrit).trans_le hpnear
+  have hb := deformed_distinct_correlation G hloc hqt d hd P hP p hp hp₀ k t ht D x y
+  rw [hT.cube_real_kernel G p T x y] at hb
+  have hc : (d:ℝ)/(p:ℝ) ≤ 2*(d:ℝ)/(pc G:ℝ) := by
+    apply (div_le_div_iff₀ hp₀ hpcrit).mpr
+    nlinarith [Nat.cast_nonneg (α := ℝ) d]
+  have he := mul_le_mul_of_nonneg_right (add_le_add (le_refl (1:ℝ)) hc)
+    (mul_nonneg (sq_nonneg (maxSusceptibility G p)) (((hT.positiveRealMatrix G p T).pow 3) x y).2)
+  nlinarith
+
+end UniformWeightedCorrelation
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section CorridorSizeBounds
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem incidentSupport_card_le (G : BondGraph V E) (hloc : LocallyFinite G)
+    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (W : Finset V) :
+    (incidentSupport G hloc W).card ≤ W.card*d := by
+  apply Finset.card_biUnion_le_card_mul
+  intro x _
+  rw [←Set.ncard_eq_toFinset_card _ (hloc x)]
+  exact (incident_ncard_le_degree G hloc x).trans (hd x)
+
+theorem reachBall_ncard_le_pow (G : BondGraph V E) (hloc : LocallyFinite G)
+    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (x : V) (k : ℕ) :
+    (reachBall G x k).ncard ≤ (d+1)^k := by
+  induction k with
+  | zero => simp only [reachBall,Set.ncard_singleton,pow_zero,le_refl]
+  | succ k ih =>
+    let W := (finite_reachBall hloc x k).toFinset
+    have he : reachBall G x (k+1) = (W ∪ W.biUnion (neighbors G hloc) : Finset V) := by
+      ext z
+      simp only [reachBall,Set.mem_union,Set.mem_iUnion,Set.mem_ofPred_eq,Finset.mem_coe,
+        Finset.mem_union,Finset.mem_biUnion,mem_neighbors,Set.Finite.mem_toFinset,W,Adj,exists_prop]
+    rw [he,Set.ncard_coe_finset]
+    have hb : (W.biUnion (neighbors G hloc)).card ≤ W.card*d :=
+      Finset.card_biUnion_le_card_mul _ _ _ (fun a _ => (neighbors_card_le_degree G hloc a).trans (hd a))
+    have hi : W.card ≤ (d+1)^k := by simpa only [W,←Set.ncard_eq_toFinset_card] using ih
+    calc
+      _ ≤ W.card + (W.biUnion (neighbors G hloc)).card := Finset.card_union_le _ _
+      _ ≤ W.card + W.card*d := Nat.add_le_add_left hb _
+      _ = W.card*(d+1) := by ring
+      _ ≤ (d+1)^k*(d+1) := Nat.mul_le_mul_right _ hi
+      _ = _ := (pow_succ _ _).symm
+
+namespace RawCorridor
+variable {G : BondGraph V E} {k : ℕ}
+
+theorem interior_card_le (w : RawCorridor G k) : w.interior.card ≤ w.path.length := by
+  have hs : w.interior ⊂ w.path.support.toFinset := by
+    apply Finset.ssubset_iff_subset_ne.mpr
+    refine ⟨Finset.sdiff_subset,?_⟩
+    intro he
+    have hb : w.base ∈ w.path.support.toFinset := List.mem_toFinset.mpr w.path.start_mem_support
+    rw [←he] at hb
+    have hn : w.base ∉ w.interior := by simp only [interior,Finset.mem_sdiff,Finset.mem_insert,
+      true_or,not_true_eq_false,and_false,not_false_eq_true]
+    exact hn hb
+  have hh := Finset.card_lt_card hs
+  rw [List.toFinset_card_of_nodup w.path_isPath.support_nodup,SimpleGraph.Walk.length_support] at hh
+  omega
+
+theorem blockers_ncard_le (hloc : LocallyFinite G) (d : ℕ) (hd : ∀ x, degree G x ≤ d)
+    (w : RawCorridor G k) : w.blockers.ncard ≤ d*k := by
+  have hs : w.blockers ⊆ (incidentSupport G hloc w.interior : Set E) :=
+    fun e he => (mem_incidentSupport G hloc w.interior e).mpr he.1
+  calc
+    _ ≤ (incidentSupport G hloc w.interior : Set E).ncard := Set.ncard_le_ncard hs (Finset.finite_toSet _)
+    _ = (incidentSupport G hloc w.interior).card := Set.ncard_coe_finset _
+    _ ≤ w.interior.card*d := incidentSupport_card_le G hloc d hd _
+    _ ≤ k*d := Nat.mul_le_mul_right _ (w.interior_card_le.trans w.path_length)
+    _ = _ := Nat.mul_comm _ _
+
+end RawCorridor
+end CorridorSizeBounds
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section RawWalkTotalMass
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem walkKernel_finite_support (G : BondGraph V E) (hloc : LocallyFinite G)
+    (a : ℝ) (x : V) : (Function.support (walkKernel G hloc a x)).Finite := by
+  apply (Finset.finite_toSet (insert x (neighbors G hloc x))).subset
+  intro y hy
+  by_contra hn
+  have hn' : y ≠ x ∧ ¬Adj G x y := by simpa only [Finset.mem_coe,Finset.mem_insert,mem_neighbors,not_or] using hn
+  exact hy (by simp only [walkKernel,ite_eq_right hn'.1.symm,ite_eq_right hn'.2,add_zero])
+
+theorem walkKernel_tsum (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ) (x : V) :
+    (∑' y, walkKernel G hloc a x y) = 1 := by
+  rw [tsum_eq_sum (s := insert x (neighbors G hloc x))]
+  · exact walkKernel_row_sum G hloc a x
+  · intro y hy
+    have hn : y ≠ x ∧ ¬Adj G x y := by simpa only [Finset.mem_insert,mem_neighbors,not_or] using hy
+    simp only [walkKernel,ite_eq_right hn.1.symm,ite_eq_right hn.2,add_zero]
+
+theorem raw_integral_one (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (k : ℕ) (x : V) :
+    RawFrom.integral (H := openSimpleGraph G Set.univ) (walkKernel G hloc a) k x (fun _ => 1) = 1 := by
+  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
+  induction k generalizing x with
+  | zero => exact RawFrom.integral_zero _ _ _
+  | succ k ih =>
+    rw [RawFrom.integral_succ]
+    simp only [ih,mul_one]
+    have he := sum_step_decomposition (openSimpleGraph G Set.univ) (walkKernel G hloc a)
+      (walkKernel_finite_support G hloc a) (fun x y hxy hn => by
+        apply walkKernel_zero_of_not_step G hloc a x y hxy
+        intro h
+        obtain ⟨e,he⟩ := h
+        exact hn ⟨hxy,e,Set.mem_univ e,he⟩) x (fun _ => 1)
+    simpa only [mul_one,walkKernel_tsum G hloc a] using he.symm
+
+ 
+theorem rawFrom_probability_sum (G : BondGraph V E) (hloc : LocallyFinite G)
+    (a : ℝ) (k : ℕ) (x : V) :
+    (∑' w : RawFrom (openSimpleGraph G Set.univ) k x, w.2.mass (walkKernel G hloc a)) = 1 := by
+  simpa only [RawFrom.integral,mul_one] using raw_integral_one G hloc a k x
+
+end RawWalkTotalMass
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section RawWalkFiniteBases
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem raw_probability_on_bases_sum (G : BondGraph V E) (hloc : LocallyFinite G)
+    (a : ℝ) (k : ℕ) (S : Set V) (hS : S.Finite) :
+    (∑' w : RawCorridor G k, if w.base ∈ S then w.probability (walkKernel G hloc a) else 0) = (S.ncard:ℝ) := by
+  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
+  have hs : Summable (fun w : RawCorridor G k => if w.base ∈ S then w.probability (walkKernel G hloc a) else 0) := by
+    apply summable_of_hasFiniteSupport
+    apply (finite_corridors_on_bases G hloc k S hS).subset
+    intro w hw
+    by_contra hn
+    exact hw (ite_eq_right hn)
+  rw [hs.tsum_sigma]
+  have he (x : V) : (∑' w : RawFrom (openSimpleGraph G Set.univ) k x,
+      if (RawCorridor.base ⟨x,w⟩) ∈ S then (RawCorridor.probability (walkKernel G hloc a) ⟨x,w⟩) else 0) =
+      if x ∈ S then 1 else 0 := by
+    by_cases hx : x ∈ S
+    · simp only [RawCorridor.base,RawCorridor.probability,hx,ite_true,rawFrom_probability_sum G hloc a k x]
+    · simp only [RawCorridor.base,hx,ite_false,tsum_zero]
+  simp only [he]
+  rw [tsum_eq_sum (s := hS.toFinset)]
+  · rw [Finset.sum_congr rfl (fun x hx => ite_eq_left (hS.mem_toFinset.mp hx))]
+    simp only [Finset.sum_const,nsmul_eq_mul,mul_one]
+    rw [←Set.ncard_eq_toFinset_card _ hS]
+  · intro x hx
+    exact ite_eq_right (fun h => hx (hS.mem_toFinset.mpr h))
+
+namespace RawCorridor
+variable {G : BondGraph V E} {k : ℕ}
+
+theorem base_mem_reachBall_of_retained (w : RawCorridor G k) (e : E)
+    (he : e ∈ w.retained) (z : V) (hz : z ∈ G e) : w.base ∈ reachBall G z k := by
+  have hs : z ∈ w.path.support := SimpleGraph.Walk.mem_support_of_mem_edges he hz
+  let q := (w.path.takeUntil z hs).reverse
+  have hq : q.length ≤ k := by
+    change (w.path.takeUntil z hs).reverse.length ≤ k
+    rw [SimpleGraph.Walk.length_reverse]
+    exact (w.path.length_takeUntil_le_length hs).trans w.path_length
+  have hh := walk_mem_reachBall G Set.univ (x := z) q 0 (by simp only [reachBall,Set.mem_singleton_iff])
+  simp only [Nat.zero_add] at hh
+  exact reachBall_mono_local G z hq hh
+
+theorem counted_base_mem_cluster (w : RawCorridor G k) (D : Set V) (ω : Set E) (x y : V)
+    (hc : w.counted D ω x y) : w.base ∈ cluster G ω x := by
+  obtain ⟨q⟩ := (conn_iff_reachable G _ x y).mp hc.2.2.1
+  have hm := w.counted_base_mem_walk D ω x y hc q
+  exact conn_mono G Set.inter_subset_left ((conn_iff_reachable G _ x w.base).mpr ⟨q.takeUntil w.base hm⟩)
+
+end RawCorridor
+end RawWalkFiniteBases
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section UniformRawWeight
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+noncomputable def rawWeightBase (r q : ℝ) (d : ℕ) : ℝ := r⁻¹ * (q⁻¹)^d
+
+theorem rawWeightBase_one_le {r q : ℝ} (hr : 0 < r) (hr₁ : r ≤ 1)
+    (hq : 0 < q) (hq₁ : q ≤ 1) (d : ℕ) : 1 ≤ rawWeightBase r q d := by
+  exact one_le_mul_of_one_le_of_one_le ((one_le_inv₀ hr).mpr hr₁)
+    (one_le_pow₀ ((one_le_inv₀ hq).mpr hq₁))
+
+namespace RawCorridor
+variable {G : BondGraph V E} {k : ℕ}
+
+theorem weight_le_uniform (hloc : LocallyFinite G) (d : ℕ) (hd : ∀ x, degree G x ≤ d)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval)
+    {r q : ℝ} (hr : 0 < r) (hq : 0 < q) (hrp : r ≤ (p:ℝ)) (hqp : q ≤ 1-(p:ℝ))
+    (w : RawCorridor G k) : w.weight P p ≤ w.probability P * (rawWeightBase r q d)^k := by
+  have hp₀ : 0 < (p:ℝ) := hr.trans_le hrp
+  have hp₁ : 0 < 1-(p:ℝ) := hq.trans_le hqp
+  have hr₁ : r ≤ 1 := hrp.trans p.2.2
+  have hq₁ : q ≤ 1 := hqp.trans (sub_le_self _ p.2.1)
+  have hi : (p:ℝ)⁻¹ ≤ r⁻¹ := inv_anti₀ hr hrp
+  have hj : (1-(p:ℝ))⁻¹ ≤ q⁻¹ := inv_anti₀ hq hqp
+  have hL : ((p:ℝ)^w.path.length)⁻¹ ≤ (r⁻¹)^k := by
+    simp only [←inv_pow]
+    exact (pow_le_pow_left₀ (inv_nonneg.mpr p.2.1) hi _).trans
+      (pow_le_pow_right₀ ((one_le_inv₀ hr).mpr hr₁) w.path_length)
+  have hB : ((1-(p:ℝ))^w.blockers.ncard)⁻¹ ≤ ((q⁻¹)^d)^k := by
+    simp only [←inv_pow,←pow_mul]
+    exact (pow_le_pow_left₀ (inv_nonneg.mpr hp₁.le) hj _).trans
+      (pow_le_pow_right₀ ((one_le_inv₀ hq).mpr hq₁) (w.blockers_ncard_le hloc d hd))
+  unfold weight
+  calc
+    _ = w.probability P * (((p:ℝ)^w.path.length)⁻¹ * ((1-(p:ℝ))^w.blockers.ncard)⁻¹) := mul_assoc _ _ _
+    _ ≤ w.probability P * ((r⁻¹)^k * ((q⁻¹)^d)^k) :=
+      mul_le_mul_of_nonneg_left (mul_le_mul hL hB
+        (inv_nonneg.mpr (pow_nonneg hp₁.le _)) (pow_nonneg (inv_nonneg.mpr hr.le) _))
+        (w.2.2.mass_nonneg P hP)
+    _ = _ := by rw [rawWeightBase,mul_pow]
+
+end RawCorridor
+end UniformRawWeight
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section UniformPenaltyDomination
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem summable_raw_selection (G : BondGraph V E) (hloc : LocallyFinite G)
+    (k : ℕ) (S : Set V) (hS : S.Finite) (Q : RawCorridor G k → Prop)
+    (hQ : ∀ w, Q w → w.base ∈ S) (f : RawCorridor G k → ℝ) :
+    Summable (fun w => if Q w then f w else 0) := by
+  apply summable_of_hasFiniteSupport
+  apply (finite_corridors_on_bases G hloc k S hS).subset
+  intro w hw
+  by_contra hn
+  exact hw (ite_eq_right (fun h => hn (hQ w h)))
+
+theorem mask_le_mask_mul {U W : Prop} [Decidable U] [Decidable W] (hUW : U → W)
+    {f g B : ℝ} (hg : 0 ≤ g) (hB : 0 ≤ B) (hfg : f ≤ g*B) :
+    (if U then f else 0) ≤ (if W then g else 0)*B := by
+  by_cases hU : U
+  · simp only [hU,hUW hU,ite_true]; exact hfg
+  · rw [ite_eq_right hU]
+    by_cases hW : W
+    · rw [ite_eq_left hW]; exact mul_nonneg hg hB
+    · rw [ite_eq_right hW,zero_mul]
+
+theorem raw_weight_sum_le_on_bases (G : BondGraph V E) (hloc : LocallyFinite G)
+    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
+    (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y) (p : unitInterval)
+    {r q : ℝ} (hr : 0 < r) (hq : 0 < q) (hrp : r ≤ (p:ℝ)) (hqp : q ≤ 1-(p:ℝ))
+    (k : ℕ) (S : Set V) (hS : S.Finite) (Q : RawCorridor G k → Prop)
+    (hQ : ∀ w, Q w → w.base ∈ S) :
+    (∑' w : RawCorridor G k, if Q w then w.weight (walkKernel G hloc a) p else 0) ≤
+      (S.ncard:ℝ)*(rawWeightBase r q d)^k := by
+  have hf := summable_raw_selection G hloc k S hS Q hQ
+    (fun w => w.weight (walkKernel G hloc a) p)
+  have hg := summable_raw_selection G hloc k S hS (fun w => w.base ∈ S)
+    (fun _ h => h) (fun w => w.probability (walkKernel G hloc a))
+  have hB : 0 ≤ (rawWeightBase r q d)^k := pow_nonneg
+    (zero_le_one.trans (rawWeightBase_one_le hr (hrp.trans p.2.2) hq (hqp.trans (sub_le_self _ p.2.1)) d)) _
+  have hpoint (w : RawCorridor G k) :
+      (if Q w then w.weight (walkKernel G hloc a) p else 0) ≤
+        (if w.base ∈ S then w.probability (walkKernel G hloc a) else 0) * (rawWeightBase r q d)^k :=
+    mask_le_mask_mul (hQ w) (w.2.2.mass_nonneg _ hP) hB
+      (w.weight_le_uniform hloc d hd (walkKernel G hloc a) hP p hr hq hrp hqp)
+  have hb := hf.tsum_le_tsum hpoint (hg.mul_right ((rawWeightBase r q d)^k))
+  rw [tsum_mul_right,raw_probability_on_bases_sum G hloc a k S hS] at hb
+  exact hb
+
+ 
+
+theorem corridorPenalty_le_cluster (G : BondGraph V E) (hloc : LocallyFinite G)
+    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
+    (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y) (p : unitInterval)
+    {r q : ℝ} (hr : 0 < r) (hq : 0 < q) (hrp : r ≤ (p:ℝ)) (hqp : q ≤ 1-(p:ℝ))
+    (k : ℕ) (D : Set V) (ω : Set E) (x y : V) (hf : (cluster G ω x).Finite) :
+    corridorPenalty G (walkKernel G hloc a) p k D ω x y ≤
+      ((cluster G ω x).ncard:ℝ)*(rawWeightBase r q d)^k := by
+  exact raw_weight_sum_le_on_bases G hloc d hd a hP p hr hq hrp hqp k _ hf
+    (fun w => w.counted D ω x y) (fun w hw => w.counted_base_mem_cluster D ω x y hw)
+
+ 
+
+theorem raw_edge_weight_bound (G : BondGraph V E) (hloc : LocallyFinite G)
+    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
+    (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y) (p : unitInterval)
+    {r q : ℝ} (hr : 0 < r) (hq : 0 < q) (hrp : r ≤ (p:ℝ)) (hqp : q ≤ 1-(p:ℝ))
+    (k : ℕ) (e : E) (z : V) (hz : z ∈ G e) :
+    (∑' w : RawCorridor G k, if e ∈ w.retained then w.weight (walkKernel G hloc a) p else 0) ≤
+      (((d:ℝ)+1)*rawWeightBase r q d)^k := by
+  have hb := raw_weight_sum_le_on_bases G hloc d hd a hP p hr hq hrp hqp k
+    (reachBall G z k) (finite_reachBall hloc z k) (fun w => e ∈ w.retained)
+    (fun w hw => w.base_mem_reachBall_of_retained e hw z hz)
+  have hcard : ((reachBall G z k).ncard:ℝ) ≤ ((d:ℝ)+1)^k := by
+    exact_mod_cast reachBall_ncard_le_pow G hloc d hd z k
+  have hB : 0 ≤ (rawWeightBase r q d)^k := pow_nonneg
+    (zero_le_one.trans (rawWeightBase_one_le hr (hrp.trans p.2.2) hq (hqp.trans (sub_le_self _ p.2.1)) d)) _
+  exact hb.trans ((mul_le_mul_of_nonneg_right hcard hB).trans_eq (mul_pow _ _ _).symm)
+
+end UniformPenaltyDomination
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section ExponentialCorridorCost
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem pow_le_exp_div_self {B : ℝ} (hB : 1 ≤ B) {k : ℕ} (hk : 0 < k) :
+    B^k ≤ Real.exp ((Real.log B + 1)*(k:ℝ))/(k:ℝ) := by
+  have hB₀ : 0 < B := zero_lt_one.trans_le hB
+  apply (le_div_iff₀ (Nat.cast_pos.mpr hk)).mpr
+  have hh : (k:ℝ) ≤ Real.exp (k:ℝ) := by linarith [Real.add_one_le_exp (k:ℝ)]
+  calc
+    _ ≤ B^k * Real.exp (k:ℝ) := mul_le_mul_of_nonneg_left hh (pow_nonneg hB₀.le _)
+    _ = Real.exp ((k:ℝ)*Real.log B) * Real.exp (k:ℝ) := by rw [Real.exp_nat_mul,Real.exp_log hB₀]
+    _ = _ := by rw [←Real.exp_add]; congr 1; ring
+
+ 
+
+theorem exists_uniform_corridor_edge_cost (G : BondGraph V E) (hloc : LocallyFinite G)
+    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
+    (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y)
+    (hpcrit : 0 < (pc G:ℝ)) (hpcrit₁ : (pc G:ℝ) < 1) :
+    ∃ C : ℝ, 0 < C ∧ ∀ (p : unitInterval), (pc G:ℝ)/2 ≤ (p:ℝ) → p < pc G →
+      ∀ (k : ℕ), 3 ≤ k → ∀ e : E,
+      (∑' w : RawCorridor G k, if e ∈ w.retained then w.weight (walkKernel G hloc a) p else 0) ≤
+        Real.exp (C*(k:ℝ))/(k:ℝ) := by
+  let B := ((d:ℝ)+1)*rawWeightBase ((pc G:ℝ)/2) (1-(pc G:ℝ)) d
+  have hB : 1 ≤ B := one_le_mul_of_one_le_of_one_le (by have := Nat.cast_nonneg (α := ℝ) d; linarith)
+    (rawWeightBase_one_le (half_pos hpcrit) (by have := (pc G).2.2; linarith)
+      (sub_pos.mpr hpcrit₁) (sub_le_self _ (pc G).2.1) d)
+  refine ⟨Real.log B + 1,by linarith [Real.log_nonneg hB],?_⟩
+  intro p hpnear hp k hk e
+  obtain ⟨z,z',hzz⟩ : ∃ z z', G e = s(z,z') := by
+    obtain ⟨⟨z,z'⟩,hh⟩ := Quot.exists_rep (G e)
+    exact ⟨z,z',hh.symm⟩
+  have he : z ∈ G e := hzz.symm ▸ Sym2.mem_mk_left z z'
+  have hq : 1-(pc G:ℝ) ≤ 1-(p:ℝ) := sub_le_sub_left (show (p:ℝ) ≤ (pc G:ℝ) from hp.le) (1:ℝ)
+  exact (raw_edge_weight_bound G hloc d hd a hP p (half_pos hpcrit)
+    (sub_pos.mpr hpcrit₁) hpnear hq k e z he).trans (pow_le_exp_div_self hB (by omega))
+
+end ExponentialCorridorCost
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section LiteralDeformationDerivative
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def deformedDerivativeMass (G : BondGraph V E) (P : V → V → ℝ)
+    (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x : V) : ℝ :=
+  ∑' y, corridorPenalty G P p k D ω x y * deformedConnection G P p k t D ω x y
+
+theorem deformedConnection_eq_zero_of_not_conn (G : BondGraph V E) (P : V → V → ℝ)
+    (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x y : V) (hn : ¬Conn G ω x y) :
+    deformedConnection G P p k t D ω x y = 0 := by
+  have hnc : ¬Conn G (ω ∩ internalEdges G D) x y := fun h => hn (conn_mono G Set.inter_subset_left h)
+  simp only [deformedConnection,hnc,and_false,ite_false]
+
+theorem deformedClusterMass_eq_sum (G : BondGraph V E) (P : V → V → ℝ)
+    (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x : V)
+    (hf : (cluster G ω x).Finite) : deformedClusterMass G P p k t D ω x =
+      ∑ y ∈ hf.toFinset, deformedConnection G P p k t D ω x y := by
+  apply tsum_eq_sum
+  intro y hy
+  exact deformedConnection_eq_zero_of_not_conn G P p k t D ω x y (fun h => hy (hf.mem_toFinset.mpr h))
+
+theorem deformedDerivativeMass_eq_sum (G : BondGraph V E) (P : V → V → ℝ)
+    (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x : V)
+    (hf : (cluster G ω x).Finite) : deformedDerivativeMass G P p k t D ω x =
+      ∑ y ∈ hf.toFinset, corridorPenalty G P p k D ω x y * deformedConnection G P p k t D ω x y := by
+  apply tsum_eq_sum
+  intro y hy
+  rw [deformedConnection_eq_zero_of_not_conn G P p k t D ω x y (fun h => hy (hf.mem_toFinset.mpr h)),mul_zero]
+
+theorem measurable_deformedDerivativeMass [Countable V] [Countable E]
+    (G : BondGraph V E) (P : V → V → ℝ) (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (x : V) :
+    Measurable (fun ω => deformedDerivativeMass G P p k t D ω x) :=
+  Measurable.tsum (fun y => (measurable_corridorPenalty G P p k D x y).mul
+    (measurable_deformedConnection G P p k t D x y))
+
+theorem deformedDerivativeMass_nonneg (G : BondGraph V E) (P : V → V → ℝ)
+    (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (k : ℕ) (t : ℝ)
+    (D : Set V) (ω : Set E) (x : V) : 0 ≤ deformedDerivativeMass G P p k t D ω x :=
+  tsum_nonneg (fun y => mul_nonneg (corridorPenalty_nonneg G P hP p k D ω x y)
+    (deformedConnection_nonneg G P p k t D ω x y))
+
+theorem hasDerivAt_deformedConnection (G : BondGraph V E) (P : V → V → ℝ)
+    (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x y : V) :
+    HasDerivAt (fun s => deformedConnection G P p k s D ω x y)
+      (-(corridorPenalty G P p k D ω x y * deformedConnection G P p k t D ω x y)) t := by
+  by_cases hc : x ∈ D ∧ y ∈ D ∧ Conn G (ω ∩ internalEdges G D) x y
+  · simp only [deformedConnection,hc,and_self,ite_true]
+    simpa only [Pi.neg_apply,id_eq,neg_mul,one_mul,mul_neg,mul_comm] using
+      (((hasDerivAt_id t).neg.mul_const (corridorPenalty G P p k D ω x y)).exp)
+  · simpa only [deformedConnection,hc,ite_false,mul_zero,neg_zero] using hasDerivAt_const t (0:ℝ)
+
+ 
+
+theorem hasDerivAt_deformedClusterMass (G : BondGraph V E) (P : V → V → ℝ)
+    (p : ℝ) (k : ℕ) (t : ℝ) (D : Set V) (ω : Set E) (x : V)
+    (hf : (cluster G ω x).Finite) :
+    HasDerivAt (fun s => deformedClusterMass G P p k s D ω x)
+      (-deformedDerivativeMass G P p k t D ω x) t := by
+  have hh := HasDerivAt.fun_sum (u := hf.toFinset) (fun y _ => hasDerivAt_deformedConnection G P p k t D ω x y)
+  simpa only [deformedClusterMass_eq_sum G P p k _ D ω x hf,
+    deformedDerivativeMass_eq_sum G P p k t D ω x hf,Finset.sum_neg_distrib] using hh
+
+end LiteralDeformationDerivative
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section DeformationDerivativeDomination
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem deformedDerivativeMass_le_square (G : BondGraph V E) (hloc : LocallyFinite G)
+    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
+    (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y) (p : unitInterval)
+    (hp0 : 0 < (p:ℝ)) (hp1 : (p:ℝ) < 1) (k : ℕ) (t : ℝ) (ht : 0 ≤ t)
+    (D : Set V) (ω : Set E) (x : V) (hf : (cluster G ω x).Finite) :
+    deformedDerivativeMass G (walkKernel G hloc a) p k t D ω x ≤
+      ((cluster G ω x).ncard:ℝ)^2 * (rawWeightBase p (1-(p:ℝ)) d)^k := by
+  rw [deformedDerivativeMass_eq_sum G (walkKernel G hloc a) p k t D ω x hf]
+  calc
+    _ ≤ ∑ _y ∈ hf.toFinset, ((cluster G ω x).ncard:ℝ)*(rawWeightBase p (1-(p:ℝ)) d)^k := by
+      apply Finset.sum_le_sum
+      intro y _
+      have hn := corridorPenalty_nonneg G (walkKernel G hloc a) hP p k D ω x y
+      exact (mul_le_of_le_one_right hn (deformedConnection_le_one G (walkKernel G hloc a) hP p k t ht D ω x y)).trans
+        (corridorPenalty_le_cluster G hloc d hd a hP p hp0 (sub_pos.mpr hp1) le_rfl le_rfl k D ω x y hf)
+    _ = _ := by
+      simp only [Finset.sum_const,nsmul_eq_mul,←Set.ncard_eq_toFinset_card _ hf]
+      ring
+
+theorem integrable_deformedDerivativeMass [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
+    (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y) (p : unitInterval)
+    (hp0 : 0 < (p:ℝ)) (hp : p < pc G) (k : ℕ) (t : ℝ) (ht : 0 ≤ t)
+    (D : Set V) (x : V) :
+    Integrable (fun ω => deformedDerivativeMass G (walkKernel G hloc a) p k t D ω x) (law p) := by
+  obtain ⟨_,_,hs⟩ := subcritical_second_moment G hloc hqt p hp
+  apply Integrable.mono' ((hs x).1.mul_const ((rawWeightBase p (1-(p:ℝ)) d)^k))
+    (measurable_deformedDerivativeMass G (walkKernel G hloc a) p k t D x).aestronglyMeasurable
+  filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
+  rw [Real.norm_of_nonneg (deformedDerivativeMass_nonneg G (walkKernel G hloc a) hP p k t D ω x)]
+  exact deformedDerivativeMass_le_square G hloc d hd a hP p hp0
+    ((show (p:ℝ) < (pc G:ℝ) from hp).trans_le (pc G).2.2) k t ht D ω x hω
+
+end DeformationDerivativeDomination
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter Topology
+section DeformationDifferentiation
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem hasDerivAt_deformedSusceptibility [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
+    (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y) (p : unitInterval)
+    (hp0 : 0 < (p:ℝ)) (hp : p < pc G) (k : ℕ) (t : ℝ) (ht : 0 < t)
+    (D : Set V) (x : V) :
+    HasDerivAt (fun s => deformedSusceptibility G (walkKernel G hloc a) p k s D x)
+      (-(∫ ω, deformedDerivativeMass G (walkKernel G hloc a) p k t D ω x ∂law p)) t := by
+  obtain ⟨_,_,hs⟩ := subcritical_second_moment G hloc hqt p hp
+  have hp1 : (p:ℝ) < 1 := (show (p:ℝ) < (pc G:ℝ) from hp).trans_le (pc G).2.2
+  have hder := hasDerivAt_integral_of_dominated_loc_of_deriv_le
+    (μ := law p) (s := Set.Ioi 0) (x₀ := t)
+    (F := fun s ω => deformedClusterMass G (walkKernel G hloc a) p k s D ω x)
+    (F' := fun s ω => -deformedDerivativeMass G (walkKernel G hloc a) p k s D ω x)
+    (bound := fun ω => ((cluster G ω x).ncard:ℝ)^2*(rawWeightBase p (1-(p:ℝ)) d)^k)
+    (Ioi_mem_nhds ht)
+    (Filter.Eventually.of_forall (fun s => (measurable_deformedClusterMass G (walkKernel G hloc a) p k s D x).aestronglyMeasurable))
+    (integrable_deformedClusterMass G hloc hqt (walkKernel G hloc a) hP p hp k t ht.le D x)
+    (measurable_deformedDerivativeMass G (walkKernel G hloc a) p k t D x).neg.aestronglyMeasurable
+    (by
+      filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
+      intro s hs
+      rw [norm_neg,Real.norm_of_nonneg (deformedDerivativeMass_nonneg G (walkKernel G hloc a) hP p k s D ω x)]
+      exact deformedDerivativeMass_le_square G hloc d hd a hP p hp0 hp1 k s hs.le D ω x hω)
+    ((hs x).1.mul_const _)
+    (by
+      filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
+      intro s _
+      exact hasDerivAt_deformedClusterMass G (walkKernel G hloc a) p k s D ω x hω)
+  rw [integral_neg] at hder
+  apply hder.2.congr_of_eventuallyEq
+  filter_upwards [Ioi_mem_nhds ht] with s hs
+  exact deformedSusceptibility_eq_integral G hloc hqt (walkKernel G hloc a) hP p hp k s hs.le D x
+
+end DeformationDifferentiation
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter Topology
+section ContinuousDeformationDerivative
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem continuous_deformedConnection (G : BondGraph V E) (P : V → V → ℝ)
+    (p : ℝ) (k : ℕ) (D : Set V) (ω : Set E) (x y : V) :
+    Continuous (fun s => deformedConnection G P p k s D ω x y) :=
+  continuous_iff_continuousAt.mpr (fun t => (hasDerivAt_deformedConnection G P p k t D ω x y).continuousAt)
+
+theorem continuous_deformedClusterMass (G : BondGraph V E) (P : V → V → ℝ)
+    (p : ℝ) (k : ℕ) (D : Set V) (ω : Set E) (x : V) (hf : (cluster G ω x).Finite) :
+    Continuous (fun s => deformedClusterMass G P p k s D ω x) :=
+  continuous_iff_continuousAt.mpr (fun t => (hasDerivAt_deformedClusterMass G P p k t D ω x hf).continuousAt)
+
+theorem continuous_deformedDerivativeMass (G : BondGraph V E) (P : V → V → ℝ)
+    (p : ℝ) (k : ℕ) (D : Set V) (ω : Set E) (x : V) (hf : (cluster G ω x).Finite) :
+    Continuous (fun s => deformedDerivativeMass G P p k s D ω x) := by
+  simp_rw [deformedDerivativeMass_eq_sum G P p k _ D ω x hf]
+  exact continuous_finsetSum _ (fun y _ => continuous_const.mul (continuous_deformedConnection G P p k D ω x y))
+
+ 
+
+theorem continuousOn_integral_deformedDerivativeMass [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
+    (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y) (p : unitInterval)
+    (hp0 : 0 < (p:ℝ)) (hp : p < pc G) (k : ℕ) (D : Set V) (x : V) :
+    ContinuousOn (fun t => ∫ ω, deformedDerivativeMass G (walkKernel G hloc a) p k t D ω x ∂law p) (Set.Ici 0) := by
+  obtain ⟨_,_,hs⟩ := subcritical_second_moment G hloc hqt p hp
+  apply continuousOn_of_dominated
+    (bound := fun ω => ((cluster G ω x).ncard:ℝ)^2*(rawWeightBase p (1-(p:ℝ)) d)^k)
+  · intro t _
+    exact (measurable_deformedDerivativeMass G (walkKernel G hloc a) p k t D x).aestronglyMeasurable
+  · intro t ht
+    filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
+    rw [Real.norm_of_nonneg (deformedDerivativeMass_nonneg G (walkKernel G hloc a) hP p k t D ω x)]
+    exact deformedDerivativeMass_le_square G hloc d hd a hP p hp0
+      ((show (p:ℝ) < (pc G:ℝ) from hp).trans_le (pc G).2.2) k t ht D ω x hω
+  · exact (hs x).1.mul_const _
+  · filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
+    exact (continuous_deformedDerivativeMass G (walkKernel G hloc a) p k D ω x hω).continuousOn
+
+end ContinuousDeformationDerivative
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter Topology
+section RightDeformationDerivative
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem continuousOn_deformedSusceptibility [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (hp : p < pc G)
+    (k : ℕ) (D : Set V) (x : V) :
+    ContinuousOn (fun t => deformedSusceptibility G P p k t D x) (Set.Ici 0) := by
+  have hc : ContinuousOn (fun t => ∫ ω, deformedClusterMass G P p k t D ω x ∂law p) (Set.Ici 0) := by
+    apply continuousOn_of_dominated (bound := fun ω => ((cluster G ω x).ncard:ℝ))
+    · intro t _
+      exact (measurable_deformedClusterMass G P p k t D x).aestronglyMeasurable
+    · intro t ht
+      filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
+      rw [Real.norm_of_nonneg (deformedClusterMass_nonneg G P p k t D ω x)]
+      exact deformedClusterMass_le_ncard G P hP p k t ht D ω x hω
+    · exact (susceptibility_eq_integral G hloc hqt p hp x).1
+    · filter_upwards [subcritical_cluster_finite_ae G p hp x] with ω hω
+      exact (continuous_deformedClusterMass G P p k D ω x hω).continuousOn
+  exact hc.congr (fun t ht => deformedSusceptibility_eq_integral G hloc hqt P hP p hp k t ht D x)
+
+ 
+
+theorem hasDerivWithinAt_deformedSusceptibility_zero [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
+    (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y) (p : unitInterval)
+    (hp0 : 0 < (p:ℝ)) (hp : p < pc G) (k : ℕ) (D : Set V) (x : V) :
+    HasDerivWithinAt (fun t => deformedSusceptibility G (walkKernel G hloc a) p k t D x)
+      (-(∫ ω, deformedDerivativeMass G (walkKernel G hloc a) p k 0 D ω x ∂law p)) (Set.Ici 0) 0 := by
+  apply hasDerivWithinAt_Ici_of_tendsto_deriv (s := Set.Ioi 0)
+  · intro t ht
+    exact (hasDerivAt_deformedSusceptibility G hloc hqt d hd a hP p hp0 hp k t ht D x).differentiableAt.differentiableWithinAt
+  · exact (continuousOn_deformedSusceptibility G hloc hqt (walkKernel G hloc a) hP p hp k D x 0 (by simp)).mono Set.Ioi_subset_Ici_self
+  · exact self_mem_nhdsWithin
+  · have hc := (continuousOn_integral_deformedDerivativeMass G hloc hqt d hd a hP p hp0 hp k D x 0 (by simp)).neg
+    apply (hc.mono Set.Ioi_subset_Ici_self).congr'
+    filter_upwards [self_mem_nhdsWithin (s := Set.Ioi (0:ℝ)) (a := 0)] with t ht
+    exact (hasDerivAt_deformedSusceptibility G hloc hqt d hd a hP p hp0 hp k t ht D x).deriv.symm
+
+end RightDeformationDerivative
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section TerminalMoveGeometry
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+namespace RawCorridor
+variable {G : BondGraph V E} {k : ℕ}
+
+ 
+
+theorem counted_move_terminal (w : RawCorridor G k) (D : Set V) (ω : Set E)
+    (F : Set E) (hF : F ⊆ ω ∩ internalEdges G D) (b z y : V)
+    (hbz : Conn G F b z) (hc : w.counted D ω b y) (hdis : Disjoint F w.retained) :
+    w.counted D ω z y := by
+  apply (w.counted_iff D ω z y).mpr
+  refine ⟨hc.1,conn_trans G _ (conn_symm G _ (conn_mono G hF hbz)) hc.2.2.1,?_,hc.2.2.2.2⟩
+  intro e he
+  refine ⟨(hc.2.2.2.1 e he).1,?_⟩
+  intro hzy
+  apply (hc.2.2.2.1 e he).2
+  apply conn_trans G _ (conn_mono G ?_ hbz) hzy
+  intro f hf
+  refine ⟨hF hf,?_⟩
+  intro hfe
+  have hh : f = e := Set.mem_singleton_iff.mp hfe
+  exact Set.disjoint_left.mp hdis hf (hh.symm ▸ he)
+
+end RawCorridor
+
+ 
+
+theorem corridorPenaltyTerm_move_bound (G : BondGraph V E) (P : V → V → ℝ)
+    (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (k : ℕ) (D : Set V) (ω : Set E)
+    (F : Finset E) (hF : (F:Set E) ⊆ ω ∩ internalEdges G D) (b z y : V)
+    (hbz : Conn G (F:Set E) b z) (w : RawCorridor G k) :
+    corridorPenaltyTerm G P p D ω b y w ≤
+      (∑ e ∈ F, if e ∈ w.retained then w.weight P p else 0) + corridorPenaltyTerm G P p D ω z y w := by
+  have hn := w.weight_nonneg P hP p
+  have hs : 0 ≤ ∑ e ∈ F, if e ∈ w.retained then w.weight P p else 0 :=
+    Finset.sum_nonneg (fun e _ => by split_ifs; exact hn; exact le_rfl)
+  by_cases hc : w.counted D ω b y
+  · rw [corridorPenaltyTerm,ite_eq_left hc]
+    by_cases hd : Disjoint (F:Set E) w.retained
+    · have hc' := w.counted_move_terminal D ω F hF b z y hbz hc hd
+      simpa only [corridorPenaltyTerm,hc',ite_true] using le_add_of_nonneg_left hs
+    · obtain ⟨e,he,hew⟩ := Set.not_disjoint_iff.mp hd
+      have he' : (if e ∈ w.retained then w.weight P p else 0) ≤
+          ∑ f ∈ F, if f ∈ w.retained then w.weight P p else 0 :=
+        Finset.single_le_sum (f := fun f => if f ∈ w.retained then w.weight P p else 0) (fun f _ => by split_ifs; exact hn; exact le_rfl) he
+      rw [ite_eq_left hew] at he'
+      exact he'.trans (le_add_of_nonneg_right (corridorPenaltyTerm_nonneg G P hP p D ω z y w))
+  · rw [corridorPenaltyTerm,ite_eq_right hc]
+    exact add_nonneg hs (corridorPenaltyTerm_nonneg G P hP p D ω z y w)
+
+end TerminalMoveGeometry
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section TerminalMovePenalty
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem summable_raw_edge_weights (G : BondGraph V E) (hloc : LocallyFinite G)
+    (P : V → V → ℝ) (p : ℝ) (k : ℕ) (e : E) :
+    Summable (fun w : RawCorridor G k => if e ∈ w.retained then w.weight P p else 0) := by
+  obtain ⟨z,z',hzz⟩ : ∃ z z', G e = s(z,z') := by
+    obtain ⟨⟨z,z'⟩,hh⟩ := Quot.exists_rep (G e)
+    exact ⟨z,z',hh.symm⟩
+  have he : z ∈ G e := hzz.symm ▸ Sym2.mem_mk_left z z'
+  exact summable_raw_selection G hloc k (reachBall G z k) (finite_reachBall hloc z k)
+    (fun w => e ∈ w.retained) (fun w hw => w.base_mem_reachBall_of_retained e hw z he)
+    (fun w => w.weight P p)
+
+ 
+
+theorem corridorPenalty_move_bound (G : BondGraph V E) (hloc : LocallyFinite G)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval)
+    (k : ℕ) (D : Set V) (ω : Set E) (F : Finset E)
+    (hF : (F:Set E) ⊆ ω ∩ internalEdges G D) (b z y : V) (hbz : Conn G (F:Set E) b z) :
+    corridorPenalty G P p k D ω b y ≤
+      (∑ e ∈ F, ∑' w : RawCorridor G k, if e ∈ w.retained then w.weight P p else 0) +
+        corridorPenalty G P p k D ω z y := by
+  have hs : Summable (fun w : RawCorridor G k => ∑ e ∈ F, if e ∈ w.retained then w.weight P p else 0) :=
+    (hasSum_sum (fun e (_ : e ∈ F) => (summable_raw_edge_weights G hloc P p k e).hasSum)).summable
+  have hr := corridorPenalty_summable G hloc P p k D ω z y
+  have hl := (corridorPenalty_summable G hloc P p k D ω b y).tsum_le_tsum
+    (corridorPenaltyTerm_move_bound G P hP p k D ω F hF b z y hbz) (hs.add hr)
+  rw [hs.tsum_add hr,Summable.tsum_finsetSum (fun e _ => summable_raw_edge_weights G hloc P p k e)] at hl
+  exact hl
+
+end TerminalMovePenalty
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section FiniteOpeningAssociation
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem measurable_union_const (S : Set E) : Measurable (fun ω : Set E => ω ∪ S) := by
+  apply measurable_setOfPred.comp
+  apply Measurable.of_eval
+  intro e
+  exact ((measurable_pi_apply e).comp MeasurableEquiv.setOfPred.symm.measurable).or measurable_const
+
+theorem measurableSet_finite_open (F : Finset E) : MeasurableSet {ω : Set E | (F:Set E) ⊆ ω} := by
+  have he : {ω : Set E | (F:Set E) ⊆ ω} = ⋂ e ∈ F, {ω : Set E | e ∈ ω} := by
+    ext ω
+    simp only [Set.mem_ofPred_eq,Set.mem_iInter,Finset.mem_coe,Set.subset_def]
+  rw [he]
+  exact F.measurableSet_biInter (fun e _ => measurableSet_setOfPred.mpr ((measurable_pi_apply e).comp MeasurableEquiv.setOfPred.symm.measurable))
+
+ 
+
+theorem finite_open_integral_lower (p : unitInterval) (F : Finset E) (f : Set E → ℝ)
+    (hf : Measurable f) (hn : ∀ ω, 0 ≤ f ω) (hb : ∀ ω, f ω ≤ 1)
+    (hm : Monotone f) :
+    (p:ℝ)^F.card*(∫ ω, f ω ∂law p) ≤ ∫ ω, if (F:Set E) ⊆ ω then f ω else 0 ∂law p := by
+  let J : Set E → ℝ := fun ω => if (F:Set E) ⊆ ω then 1 else 0
+  let g : Set E → ℝ := fun ω => f (ω ∪ (F:Set E))
+  have hJ : Measurable J := Measurable.ite (measurableSet_finite_open F) measurable_const measurable_const
+  have hg : Measurable g := hf.comp (measurable_union_const (F:Set E))
+  have hi (f : Set E → ℝ) (hfm : Measurable f) (hfn : ∀ ω, 0 ≤ f ω) (hfb : ∀ ω, f ω ≤ 1) : Integrable f (law p) := by
+    exact Integrable.mono' (integrable_const (1:ℝ)) hfm.aestronglyMeasurable
+      (Filter.Eventually.of_forall (fun ω => by rw [Real.norm_of_nonneg (hfn ω)]; exact hfb ω))
+  have hfi := hi f hf hn hb
+  have hgi := hi g hg (fun ω => hn _) (fun ω => hb _)
+  have hJint : (∫ ω, J ω ∂law p) = (p:ℝ)^F.card := by
+    have hh := integral_indicator_const (μ := law p) (1:ℝ) (measurableSet_finite_open F)
+    simpa only [Set.indicator,Set.mem_ofPred_eq,smul_eq_mul,mul_one,finite_open_probability,J] using hh
+  have he := integral_mul_disjoint_coordinates p (F:Set E) (F:Set E)ᶜ disjoint_compl_right J g hJ hg
+    (fun ω => by simp only [J,Set.subset_inter_iff,Set.Subset.refl,and_true])
+    (fun ω => by
+      dsimp only [g]
+      congr 1
+      ext e
+      simp only [Set.mem_union,Set.mem_inter_iff,Set.mem_compl_iff]
+      tauto)
+  calc
+    _ ≤ (p:ℝ)^F.card*(∫ ω, g ω ∂law p) := mul_le_mul_of_nonneg_left
+      (integral_mono hfi hgi (fun ω => hm Set.subset_union_left)) (pow_nonneg p.2.1 _)
+    _ = (∫ ω, J ω ∂law p)*(∫ ω, g ω ∂law p) := by rw [hJint]
+    _ = ∫ ω, J ω*g ω ∂law p := he.symm
+    _ = _ := by
+      apply integral_congr_ae
+      exact Filter.Eventually.of_forall (fun ω => by
+        by_cases hF : (F:Set E) ⊆ ω
+        · simp only [J,g,hF,ite_true,one_mul,Set.union_eq_left.mpr hF]
+        · simp only [J,hF,ite_false,zero_mul])
+
+end FiniteOpeningAssociation
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section TerminalConnectionComparison
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem corridorPenalty_move_edge_cost (G : BondGraph V E) (hloc : LocallyFinite G)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval)
+    (k : ℕ) (D : Set V) (ω : Set E) (F : Finset E)
+    (hF : (F:Set E) ⊆ ω ∩ internalEdges G D) (b z y : V) (hbz : Conn G (F:Set E) b z)
+    (c : ℝ) (hc : ∀ e, (∑' w : RawCorridor G k, if e ∈ w.retained then w.weight P p else 0) ≤ c) :
+    corridorPenalty G P p k D ω b y ≤ (F.card:ℝ)*c+corridorPenalty G P p k D ω z y := by
+  apply (corridorPenalty_move_bound G hloc P hP p k D ω F hF b z y hbz).trans
+  apply add_le_add _ le_rfl
+  exact (Finset.sum_le_sum (fun e _ => hc e)).trans_eq (by simp only [Finset.sum_const,nsmul_eq_mul])
+
+ 
+
+theorem deformedConnection_move_lower (G : BondGraph V E) (hloc : LocallyFinite G)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval)
+    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D : Set V) (ω : Set E) (F : Finset E)
+    (hF : (F:Set E) ⊆ internalEdges G D) (b z y : V) (hb : b ∈ D)
+    (hbz : Conn G (F:Set E) b z) (c : ℝ)
+    (hc : ∀ e, (∑' w : RawCorridor G k, if e ∈ w.retained then w.weight P p else 0) ≤ c) :
+    Real.exp (-t*((F.card:ℝ)*c)) * (if (F:Set E) ⊆ ω then deformedConnection G P p k t D ω z y else 0) ≤
+      deformedConnection G P p k t D ω b y := by
+  by_cases ho : (F:Set E) ⊆ ω
+  · rw [ite_eq_left ho]
+    by_cases hz : z ∈ D ∧ y ∈ D ∧ Conn G (ω ∩ internalEdges G D) z y
+    · have hFa : (F:Set E) ⊆ ω ∩ internalEdges G D := fun e he => ⟨ho he,hF he⟩
+      have hb' : b ∈ D ∧ y ∈ D ∧ Conn G (ω ∩ internalEdges G D) b y :=
+        ⟨hb,hz.2.1,conn_trans G _ (conn_mono G hFa hbz) hz.2.2⟩
+      simp only [deformedConnection,hz,hb',and_self,ite_true]
+      rw [←Real.exp_add]
+      apply Real.exp_le_exp.mpr
+      have hh := mul_le_mul_of_nonpos_left
+        (corridorPenalty_move_edge_cost G hloc P hP p k D ω F hFa b z y hbz c hc) (neg_nonpos.mpr ht)
+      simpa only [mul_add] using hh
+    · rw [deformedConnection,ite_eq_right hz,mul_zero]
+      exact deformedConnection_nonneg G P p k t D ω b y
+  · rw [ite_eq_right ho,mul_zero]
+    exact deformedConnection_nonneg G P p k t D ω b y
+
+end TerminalConnectionComparison
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section EndpointSusceptibilityComparison
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem deformedKernel_move_lower [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval)
+    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D : Set V) (F : Finset E)
+    (hF : (F:Set E) ⊆ internalEdges G D) (b z y : V) (hb : b ∈ D)
+    (hbz : Conn G (F:Set E) b z) (c : ℝ)
+    (hc : ∀ e, (∑' w : RawCorridor G k, if e ∈ w.retained then w.weight P p else 0) ≤ c) :
+    Real.exp (-t*((F.card:ℝ)*c)) * (p:ℝ)^F.card * deformedKernel G P p k t D z y ≤
+      deformedKernel G P p k t D b y := by
+  have hi := integrable_deformedConnection G P hP p k t ht D z y
+  have hIi := hi.indicator (measurableSet_finite_open F)
+  have has := finite_open_integral_lower p F (fun ω => deformedConnection G P p k t D ω z y)
+    (measurable_deformedConnection G P p k t D z y)
+    (fun ω => deformedConnection_nonneg G P p k t D ω z y)
+    (fun ω => deformedConnection_le_one G P hP p k t ht D ω z y)
+    (fun _ _ hω => deformedConnection_mono G hloc P hP p k t ht (Set.Subset.refl _) hω z y)
+  have hb' := integral_mono (hIi.const_mul (Real.exp (-t*((F.card:ℝ)*c))))
+    (integrable_deformedConnection G P hP p k t ht D b y)
+    (fun ω => deformedConnection_move_lower G hloc P hP p k t ht D ω F hF b z y hb hbz c hc)
+  simp only [Set.indicator,Set.mem_ofPred_eq,integral_const_mul] at hb'
+  have hh := mul_le_mul_of_nonneg_left has (Real.exp_pos (-t*((F.card:ℝ)*c))).le
+  simp only [mul_assoc] at hh ⊢
+  exact hh.trans hb'
+
+ 
+theorem deformedSusceptibility_move_lower [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (hp : p < pc G)
+    (k : ℕ) (t : ℝ) (ht : 0 ≤ t) (D : Set V) (F : Finset E)
+    (hF : (F:Set E) ⊆ internalEdges G D) (b z : V) (hb : b ∈ D)
+    (hbz : Conn G (F:Set E) b z) (c : ℝ)
+    (hc : ∀ e, (∑' w : RawCorridor G k, if e ∈ w.retained then w.weight P p else 0) ≤ c) :
+    Real.exp (-t*((F.card:ℝ)*c)) * (p:ℝ)^F.card * deformedSusceptibility G P p k t D z ≤
+      deformedSusceptibility G P p k t D b := by
+  simpa only [deformedSusceptibility,tsum_mul_left] using
+    ((summable_deformedKernel G hloc hqt P hP p hp k t ht D z).mul_left
+      (Real.exp (-t*((F.card:ℝ)*c)) * (p:ℝ)^F.card)).tsum_le_tsum
+      (fun y => deformedKernel_move_lower G hloc P hP p k t ht D F hF b z y hb hbz c hc)
+      (summable_deformedKernel G hloc hqt P hP p hp k t ht D b)
+
+end EndpointSusceptibilityComparison
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Deformation/FiniteScale.lean b/lean/OAI/Probability/UnimodularPercolation/Deformation/FiniteScale.lean
index 8ec4f8d4a02d423271df06a44e261bf3432b366a..1314fd60a28a734e94936275eb7ab745e4f28fe8 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Deformation/FiniteScale.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Deformation/FiniteScale.lean
@@ -1,454 +1,454 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Escape.PathSelection
-
-namespace OAI
-
-/-! Deformed susceptibility, finite-scale estimates and the Riccati contradiction. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section EscapeSusceptibility
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Countable V] [Countable E] [Nonempty V]
-
-omit [Countable V] [Countable E] [Nonempty V] in
- 
-
-theorem deformed_move_of_walk (G : BondGraph V E) (hloc : LocallyFinite G)
-    (a C : ℝ) (p : unitInterval) (k : ℕ) (t : ℝ) (ht : 0 ≤ t)
-    (hmove : ∀ (D : Set V) (b z : V) (l : List E),
-      b ∈ D → Along G b l z → (∀ e ∈ l, e ∈ internalEdges G D) →
-      Real.exp (-t*((l.length:ℝ)*Real.exp (C*(k:ℝ)))) * (p:ℝ)^l.length *
-        deformedSusceptibility G (walkKernel G hloc a) p k t D z ≤
-        deformedSusceptibility G (walkKernel G hloc a) p k t D b)
-    (D : Set V) (b z : V) (q : (openSimpleGraph G Set.univ).Walk b z)
-    (j : ℕ) (hj : q.length ≤ j) (hD : ∀ x ∈ q.support, x ∈ D) :
-    Real.exp (-t*((j:ℝ)*Real.exp (C*(k:ℝ)))) * (p:ℝ)^j *
-      deformedSusceptibility G (walkKernel G hloc a) p k t D z ≤
-      deformedSusceptibility G (walkKernel G hloc a) p k t D b := by
-  obtain ⟨l,hl,he,_⟩ := walk_lift_bonds G Set.univ q
-  have hlen : l.length = q.length := by
-    have hh := congrArg List.length he
-    simpa only [List.length_map,SimpleGraph.Walk.length_edges] using hh
-  have hle : l.length ≤ j := hlen.trans_le hj
-  have hcomp := hmove D b z l (hD b q.start_mem_support) hl (by
-    intro e hel x hxe
-    exact hD x (SimpleGraph.Walk.mem_support_of_mem_edges
-      (he ▸ List.mem_map_of_mem hel) hxe))
-  have he' : Real.exp (-t*((j:ℝ)*Real.exp (C*(k:ℝ)))) ≤
-      Real.exp (-t*((l.length:ℝ)*Real.exp (C*(k:ℝ)))) := by
-    apply Real.exp_le_exp.mpr
-    exact mul_le_mul_of_nonpos_left
-      (mul_le_mul_of_nonneg_right (Nat.cast_le.mpr hle) (Real.exp_pos _).le) (neg_nonpos.mpr ht)
-  have hpw : (p:ℝ)^j ≤ (p:ℝ)^l.length := pow_le_pow_of_le_one p.2.1 p.2.2 hle
-  exact (mul_le_mul_of_nonneg_right (mul_le_mul he' hpw (pow_nonneg p.2.1 _) (Real.exp_pos _).le)
-    (deformedSusceptibility_nonneg G _ p k t D z)).trans hcomp
-
- 
-theorem deformedSusceptibility_delete_interior (G : BondGraph V E) (hloc : LocallyFinite G)
-    (hqt : QuasiTransitive G) (a : ℝ) (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y)
-    (p : unitInterval) (hp : p < pc G) (k : ℕ) (t : ℝ) (ht : 0 ≤ t)
-    (w : RawCorridor G k) (x : V) :
-    deformedSusceptibility G (walkKernel G hloc a) p k t Set.univ x -
-      maxSusceptibility G p * (∑ z ∈ w.neighborhood hloc, twoPoint G p x z) ≤
-      deformedSusceptibility G (walkKernel G hloc a) p k t (w.interior : Set V)ᶜ x := by
-  have hh := deformedSusceptibility_deletion G hloc hqt _ hP p hp k t ht Set.univ (w.interior : Set V) x
-  rw [←w.coe_neighborhood hloc,Finset.tsum_subtype'] at hh
-  have he : Set.univ \ (w.interior : Set V) = (w.interior : Set V)ᶜ := by ext z; simp
-  rw [he] at hh
-  exact hh
-
- 
-
-theorem selected_retained_product_lower (G : BondGraph V E) (hloc : LocallyFinite G)
-    (hqt : QuasiTransitive G) (a : ℝ) (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y)
-    (p : unitInterval) (hp : p < pc G) (k : ℕ) (t : ℝ) (ht : 0 ≤ t)
-    (j : ℕ) (s C cf χ : ℝ) (hcf : 0 ≤ cf) (hχ : 0 ≤ χ)
-    (hrow : ∀ x, cf*χ ≤ deformedSusceptibility G (walkKernel G hloc a) p k t Set.univ x)
-    (hcost : maxSusceptibility G p*s ≤ (cf/2)*χ)
-    (hmove : ∀ (D : Set V) (b z : V) (l : List E),
-      b ∈ D → Along G b l z → (∀ e ∈ l, e ∈ internalEdges G D) →
-      Real.exp (-t*((l.length:ℝ)*Real.exp (C*(k:ℝ)))) * (p:ℝ)^l.length *
-        deformedSusceptibility G (walkKernel G hloc a) p k t D z ≤
-        deformedSusceptibility G (walkKernel G hloc a) p k t D b)
-    (w : RawCorridor G k) (hw : w.Selected hloc p j s) :
-    Real.exp (-2*t*(j:ℝ)*Real.exp (C*(k:ℝ))) * (p:ℝ)^(2*j) * ((cf/2)*χ)^2 ≤
-      retainedSusceptibilityProduct G (walkKernel G hloc a) p t w := by
-  have hx (x : V) (he : w.HasEscape hloc p j s x) :
-      Real.exp (-t*((j:ℝ)*Real.exp (C*(k:ℝ))))*(p:ℝ)^j*((cf/2)*χ) ≤
-        deformedSusceptibility G (walkKernel G hloc a) p k t (w.interior : Set V)ᶜ x := by
-    obtain ⟨z,q,_hq,hj,havoid,hs⟩ := he
-    have hd := deformedSusceptibility_delete_interior G hloc hqt a hP p hp k t ht w z
-    have hδ : (cf/2)*χ ≤
-        deformedSusceptibility G (walkKernel G hloc a) p k t (w.interior : Set V)ᶜ z := by
-      have hb := mul_le_mul_of_nonneg_left hs (zero_le_one.trans (maxSusceptibility_one_le G hloc hqt p hp))
-      have hr := hrow z
-      linarith
-    exact (mul_le_mul_of_nonneg_left hδ
-      (mul_nonneg (Real.exp_pos _).le (pow_nonneg p.2.1 _))).trans
-        (deformed_move_of_walk G hloc a C p k t ht hmove (w.interior : Set V)ᶜ x z q j hj havoid)
-  have h₁ := hx w.base hw.2.1
-  have h₂ := hx w.endpoint hw.2.2
-  have hfactor : 0 ≤ Real.exp (-t*((j:ℝ)*Real.exp (C*(k:ℝ))))*(p:ℝ)^j*((cf/2)*χ) :=
-    mul_nonneg (mul_nonneg (Real.exp_pos _).le (pow_nonneg p.2.1 _))
-      (mul_nonneg (div_nonneg hcf (by norm_num)) hχ)
-  have hh := mul_le_mul h₁ h₂ hfactor
-    (deformedSusceptibility_nonneg G _ p k t (w.interior : Set V)ᶜ w.base)
-  rw [retainedSusceptibilityProduct,ite_eq_left hw.1]
-  apply le_trans (le_of_eq ?_) hh
-  have he : -2*t*(j:ℝ)*Real.exp (C*(k:ℝ)) =
-      -t*((j:ℝ)*Real.exp (C*(k:ℝ))) + -t*((j:ℝ)*Real.exp (C*(k:ℝ))) := by ring
-  rw [he,Real.exp_add,show 2*j=j+j by omega,pow_add]
-  ring
-
-end EscapeSusceptibility
-end Percolation
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section AveragedRetention
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Countable V] [Countable E] [Nonempty V]
-variable {ι : Type*} [Fintype ι]
-
-omit [Countable V] [Countable E] [Nonempty V] in
-theorem averagedDeformedSusceptibility_nonneg (G : BondGraph V E) (P : V → V → ℝ)
-    (p : unitInterval) (k : ℕ) (root : ι → V) (weight : ι → ℝ)
-    (hw : ∀ i, 0 ≤ weight i) (t : ℝ) :
-    0 ≤ averagedDeformedSusceptibility G P p k root weight t :=
-  Finset.sum_nonneg (fun i _ => mul_nonneg (hw i) (deformedSusceptibility_nonneg G P p k t Set.univ (root i)))
-
-omit [Countable V] [Countable E] [Nonempty V] in
-theorem deformedKernel_symm (G : BondGraph V E) (P : V → V → ℝ)
-    (p : unitInterval) (k : ℕ) (t : ℝ) (D : Set V) (x y : V) :
-    deformedKernel G P p k t D x y = deformedKernel G P p k t D y x := by
-  simp only [deformedKernel,deformedConnection_symm G P p k t D _ x y]
-
-omit [Nonempty V] in
- 
-
-theorem deformed_row_le_root_sum (G : BondGraph V E) (hloc : LocallyFinite G)
-    (a : ℝ) (p : unitInterval) (k : ℕ) (t : ℝ)
-    (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 ≤ weight i)
-    (hcov : ∀ x, ∃ i, ∃ g ∈ vertexAut G, g (root i) = x)
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i) (x : V) :
-    deformedSusceptibility G (walkKernel G hloc a) p k t Set.univ x ≤
-      c*averagedDeformedSusceptibility G (walkKernel G hloc a) p k root weight t := by
-  obtain ⟨i,g,⟨e,he⟩,rfl⟩ := hcov x
-  rw [deformedSusceptibility_aut G he _ (walkKernel_covariant G hloc a he)]
-  simp only [Set.preimage_univ]
-  have hs := deformedSusceptibility_nonneg G (walkKernel G hloc a) p k t Set.univ
-  have hi := Finset.single_le_sum
-    (s := Finset.univ)
-    (f := fun r => weight r * deformedSusceptibility G (walkKernel G hloc a) p k t Set.univ (root r))
-    (fun r _ => mul_nonneg (hw r) (hs (root r))) (Finset.mem_univ i)
-  have hi' := mul_le_mul_of_nonneg_left hi hc
-  have hh := mul_le_mul_of_nonneg_right (hmin i) (hs (root i))
-  change _ ≤ c*∑ r, weight r * deformedSusceptibility G (walkKernel G hloc a) p k t Set.univ (root r)
-  nlinarith
-
- 
-
-theorem deformed_norm_le_root_sum (G : BondGraph V E) (hloc : LocallyFinite G)
-    (hqt : QuasiTransitive G) (a : ℝ) (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y)
-    (p : unitInterval) (hp : p < pc G) (k : ℕ) (t : ℝ) (ht : 0 ≤ t)
-    (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 ≤ weight i)
-    (hcov : ∀ x, ∃ i, ∃ g ∈ vertexAut G, g (root i) = x)
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (B : Hilbert V →L[ℂ] Hilbert V)
-    (hB : ∀ x y, matrixCoeff B x y = (deformedKernel G (walkKernel G hloc a) p k t Set.univ x y : ℂ)) :
-    ‖B‖ ≤ c*averagedDeformedSusceptibility G (walkKernel G hloc a) p k root weight t := by
-  have hs (x : V) (F : Finset V) :
-      (∑ y ∈ F, |deformedKernel G (walkKernel G hloc a) p k t Set.univ x y|) ≤
-        c*averagedDeformedSusceptibility G (walkKernel G hloc a) p k root weight t := by
-    simp only [abs_of_nonneg (deformedKernel_nonneg G _ p k t _ _ _)]
-    exact ((summable_deformedKernel G hloc hqt _ hP p hp k t ht Set.univ x).sum_le_tsum F
-      (fun y _ => deformedKernel_nonneg G _ p k t _ x y)).trans
-        (deformed_row_le_root_sum G hloc a p k t root weight hw hcov c hc hmin x)
-  exact operator_norm_le_absolute_sums _ _
-    (mul_nonneg hc (averagedDeformedSusceptibility_nonneg G _ p k root weight hw t)) hs
-    (fun y F => by simpa only [deformedKernel_symm G _ p k t Set.univ] using hs y F) B hB
-
- 
-
-theorem deformed_root_sum_retained (G : BondGraph V E) (hG : Function.Injective G)
-    (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (a : ℝ) (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp : p < pc G)
-    (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 ≤ weight i)
-    (hcov : ∀ x, ∃ i, ∃ g ∈ vertexAut G, g (root i) = x)
-    (c : ℝ) (hc : 0 < c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (P A : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hPs : IsSelfAdjoint P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (hA : HasKernel G p A) (k : ℕ) (t B : ℝ) (ht : 0 ≤ t)
-    (htB : t*‖A‖ ≤ B) (hB : B*‖P‖^k ≤ 1/4) :
-    ‖A‖/(2*c) ≤ averagedDeformedSusceptibility G (walkKernel G hloc a) p k root weight t := by
-  have hK (x y : V) : 0 ≤ walkKernel G hloc a x y := by
-    simpa only [hP,Complex.ofReal_re] using (hpos x y).2
-  obtain ⟨T,hT,_,_,hret⟩ := exists_deformedOperator_small_change G hG hloc p hp0
-    ((show (p:ℝ)<(pc G:ℝ) from hp).trans_le (pc G).2.2) a P hpos hPs hP
-    A (hA.positiveRealMatrix G p A) (hA.isPositive G p A).isSelfAdjoint hA k t ht
-  have hn := deformed_norm_le_root_sum G hloc hqt a hK p hp k t ht root weight hw hcov c hc.le hmin T hT
-  have hsmall := (mul_le_mul_of_nonneg_right htB (pow_nonneg (norm_nonneg P) k)).trans hB
-  have hh := mul_le_mul_of_nonneg_right hsmall (norm_nonneg A)
-  apply (div_le_iff₀ (by positivity : 0 < 2*c)).mpr
-  nlinarith
-
-end AveragedRetention
-end Percolation
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section AveragePositiveTerm
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Countable V] [Countable E] [Nonempty V]
-variable {ι : Type*} [Fintype ι]
-
-omit [Countable V] [Countable E] [Nonempty V] in
- 
-theorem selected_average_lower (G : BondGraph V E) (hloc : LocallyFinite G)
-    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (k : ℕ) (t : ℝ)
-    (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 ≤ weight i)
-    (j : ℕ) (s b F : ℝ) (hF : 0 ≤ F)
-    (hm : b ≤ rawPathAverage G P k root weight
-      (fun w => if w.Selected hloc p j s then 1 else 0))
-    (hgood : ∀ w : RawCorridor G k, w.Selected hloc p j s →
-      F ≤ retainedSusceptibilityProduct G P p t w) :
-    F*b ≤ rawPathAverage G P k root weight (retainedSusceptibilityProduct G P p t) := by
-  have hh := rawPathAverage_mono G hloc P hP k root weight hw
-    (fun w => F*(if w.Selected hloc p j s then 1 else 0))
-    (retainedSusceptibilityProduct G P p t) (by
-      intro w
-      by_cases hs : w.Selected hloc p j s
-      · simpa only [hs,ite_true,mul_one] using hgood w hs
-      · simp only [hs,ite_false,mul_zero,retainedSusceptibilityProduct]
-        split_ifs
-        · exact mul_nonneg (deformedSusceptibility_nonneg G P p k t _ w.base)
-            (deformedSusceptibility_nonneg G P p k t _ w.endpoint)
-        · exact le_rfl)
-  rw [rawPathAverage_const_mul] at hh
-  exact (mul_le_mul_of_nonneg_left hm hF).trans hh
-end AveragePositiveTerm
-
-section RiccatiContradiction
- 
-
-theorem riccati_retention_bound (χ D : ℝ → ℝ) (T m c h : ℝ)
-    (hT : 0 < T) (hm : 0 < m) (hc : 0 < c) (_hh : 0 ≤ h)
-    (hret : ∀ t ∈ Set.Icc (T/2) T, m/(2*c) ≤ χ t)
-    (hder : ∀ t ∈ Set.Icc (T/2) T, HasDerivAt χ (-D t) t)
-    (hdec : ∀ t ∈ Set.Icc (T/2) T, h*(χ t)^2 ≤ D t) :
-    h*T*m ≤ 4*c := by
-  have hpos (t) (ht : t ∈ Set.Icc (T/2) T) : 0 < χ t :=
-    (div_pos hm (by positivity)).trans_le (hret t ht)
-  have hd (t) (ht : t ∈ Set.Icc (T/2) T) :
-      HasDerivAt (fun s => (χ s)⁻¹) (D t/(χ t)^2) t := by
-    convert! (hder t ht).inv (hpos t ht).ne' using 1
-    simp only [neg_neg]
-  have hcχ : ContinuousOn (fun s => (χ s)⁻¹) (Set.Icc (T/2) T) :=
-    fun t ht => (hd t ht).continuousAt.continuousWithinAt
-  have hdiff : DifferentiableOn ℝ (fun s => (χ s)⁻¹) (interior (Set.Icc (T/2) T)) :=
-    fun t ht => (hd t (interior_subset ht)).differentiableAt.differentiableWithinAt
-  have hdl (t) (ht : t ∈ interior (Set.Icc (T/2) T)) :
-      h ≤ deriv (fun s => (χ s)⁻¹) t := by
-    rw [(hd t (interior_subset ht)).deriv]
-    exact (le_div_iff₀ (sq_pos_of_pos (hpos t (interior_subset ht)))).mpr (hdec t (interior_subset ht))
-  have hs := (convex_Icc (T/2) T).mul_sub_le_image_sub_of_le_deriv hcχ hdiff hdl
-    (T/2) ⟨le_rfl,by linarith⟩ T ⟨by linarith,le_rfl⟩ (by linarith)
-  have hstart : 0 ≤ (χ (T/2))⁻¹ := (inv_pos.mpr (hpos _ ⟨le_rfl,by linarith⟩)).le
-  have hend : (χ T)⁻¹ ≤ 2*c/m := by
-    have hb := one_div_le_one_div_of_le (div_pos hm (by positivity)) (hret T ⟨by linarith,le_rfl⟩)
-    simpa only [one_div,inv_div] using hb
-  have hs' : h*(T/2) ≤ 2*c/m := by linarith
-  have hs'' := (le_div_iff₀ hm).mp hs'
-  nlinarith
-end RiccatiContradiction
-end Percolation
-
- 
-
- 
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section FiniteScale
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Countable V] [Countable E] [Infinite V]
-variable {ι : Type*} [Fintype ι]
-
- 
-
-theorem exists_finite_scale_obstruction
-    (G : BondGraph V E) (hG : Function.Injective G)
-    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (a : ℝ) (ha : 0 < a) (root : ι → V) (weight : ι → ℝ)
-    (htr : RootTransport root weight (vertexAut G))
-    (hcov : ∀ x, ∃ r, ∃ g ∈ vertexAut G, g (root r) = x)
-    (hw : ∀ r, 0 < weight r) (hw1 : ∑ r, weight r = 1)
-    (c : ℝ) (hc : 0 < c) (hmin : ∀ r, 1 ≤ c*weight r)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re) (hρ : ‖P‖ < 1)
-    (d : ℕ) (hd : ∀ z, degree G z ≤ d)
-    (hpcrit : 0 < (pc G:ℝ)) (hpcrit₁ : (pc G:ℝ) < 1) :
-    ∃ C : ℝ, 0 < C ∧ ∃ J : ℕ, 0 < J ∧ ∃ C₁ : ℝ, 0 < C₁ ∧
-      ∃ k₀ : ℕ, 3 ≤ k₀ ∧ ∃ ε M : ℝ, 0 < ε ∧ 0 < M ∧
-      ∀ (p : unitInterval), (pc G:ℝ)/2 ≤ (p:ℝ) → p < pc G →
-      ∀ (A : Hilbert V →L[ℂ] Hilbert V), HasKernel G p A →
-      ∀ k : ℕ, k₀ ≤ k → ∀ B : ℝ, 0 < B →
-      let R := maxSusceptibility G p/‖A‖
-      let j := J+⌈C₁*Real.log (((k:ℝ)+1)*R)⌉₊
-      B*‖P‖^k ≤ 1/4 →
-      B*(j:ℝ)*Real.exp (C*(k:ℝ)) ≤ ‖A‖ →
-      R^2*rootTriangle G root weight (p:ℝ)*‖P‖^k ≤ ε*((pc G:ℝ)/2)^(2*j) →
-      M < B*((pc G:ℝ)/2)^(2*j) → False := by
-  have hK (x y : V) : 0 ≤ walkKernel G hloc a x y := by
-    simpa only [hP,Complex.ofReal_re] using (hpos x y).2
-  have hw₀ (i) := (hw i).le
-  obtain ⟨Cr,cf,hCr,hcf,hrow⟩ := exists_uniform_deformed_row_comparison G hconn hloc hqt
-    root weight hw₀ hw1 (fun x => by
-      obtain ⟨i,g,⟨e,he⟩,hx⟩ := hcov x
-      exact ⟨i,g,e,he,hx⟩) d hd a hK hpcrit hpcrit₁
-  obtain ⟨Cm,hCm,hmove⟩ := exists_uniform_deformed_endpoint_comparison G hloc hqt d hd a hK hpcrit hpcrit₁
-  obtain ⟨Cd,hCd,hdecrease⟩ := exists_uniform_averaged_decrease G hG hloc hqt
-    root weight hw₀ (vertexAut G) (Set.Subset.refl _) htr d hd a hK hpcrit hpcrit₁ P hpos hP hsym
-  let τ := cf/(8*c)
-  have hτ : 0 < τ := div_pos hcf (by positivity)
-  obtain ⟨J,C₁,k₀,b,hJ,hC₁,hk₀,hb,hselected⟩ := exists_selected_path_mass G hconn hloc hqt
-    a ha root weight htr hcov hw hw1 c hc hmin P hpos hsym hP hdiag hρ d hd τ hτ
-  let C := Cr+Cm+Cd
-  let q := b*Real.exp (-2)*(cf/2)^2
-  let H := (1+2*(d:ℝ)/(pc G:ℝ))*(2*c)^2
-  let ε := q/(2*H)
-  let h₀ := Real.exp (-1)*q/2
-  let M := 4*c/h₀
-  have hq : 0 < q := by dsimp [q]; positivity
-  have hH : 0 < H := by dsimp [H]; positivity
-  have hε : 0 < ε := div_pos hq (by positivity)
-  have hh₀ : 0 < h₀ := by dsimp [h₀]; positivity
-  have hM : 0 < M := div_pos (by positivity) hh₀
-  refine ⟨C,by dsimp [C]; positivity,J,hJ,C₁,hC₁,k₀,hk₀,ε,M,hε,hM,?_⟩
-  intro p hpnear hp A hA k hk B hB
-  dsimp only
-  let m := ‖A‖
-  let R := maxSusceptibility G p/m
-  let j := J+⌈C₁*Real.log (((k:ℝ)+1)*R)⌉₊
-  let β := ((pc G:ℝ)/2)^(2*j)
-  let T := B/m
-  let χ := averagedDeformedSusceptibility G (walkKernel G hloc a) p k root weight
-  let D := fun t => ∑ i, weight i * ∫ ω, deformedDerivativeMass G (walkKernel G hloc a) p k t Set.univ ω (root i) ∂law p
-  change B*‖P‖^k ≤ 1/4 → B*(j:ℝ)*Real.exp (C*(k:ℝ)) ≤ m →
-    R^2*rootTriangle G root weight (p:ℝ)*‖P‖^k ≤ ε*β → M < B*β → False
-  intro hsmall htime herrscale hlarge
-  have hm : 0 < m := zero_lt_one.trans_le (hA.norm_one_le G p A)
-  have hR : 1 ≤ R := (one_le_div hm).mpr (hA.norm_le_maxSusceptibility G hloc hqt p hp A)
-  have hβ : 0 < β := pow_pos (half_pos hpcrit) _
-  have hj : 1 ≤ j := by dsimp [j]; omega
-  have hjr : (1:ℝ) ≤ (j:ℝ) := Nat.one_le_cast.mpr hj
-  have hk3 : 3 ≤ k := hk₀.trans hk
-  have hpzero : 0 < (p:ℝ) := (half_pos hpcrit).trans_le hpnear
-  have hT : 0 < T := div_pos hB hm
-  have hTb (t : ℝ) (ht : t ≤ T) : t*m ≤ B := (le_div_iff₀ hm).mp ht
-  have htim (t : ℝ) (ht : t ≤ T) : t*(j:ℝ)*Real.exp (C*(k:ℝ)) ≤ 1 := by
-    have hm' := mul_le_mul_of_nonneg_right (hTb t ht)
-      (mul_nonneg (Nat.cast_nonneg j) (Real.exp_pos (C*(k:ℝ))).le)
-    nlinarith only [hm',htime,hm]
-  have hCbound (C' : ℝ) (hC' : C' ≤ C) : Real.exp (C'*(k:ℝ)) ≤ Real.exp (C*(k:ℝ)) :=
-    Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right hC' (Nat.cast_nonneg k))
-  have htim' (t : ℝ) (ht : 0 ≤ t) (htT : t ≤ T) (C' : ℝ) (hC' : C' ≤ C) :
-      t*Real.exp (C'*(k:ℝ)) ≤ 1 := by
-    have he := mul_le_mul_of_nonneg_left (hCbound C' hC') ht
-    have hj' := mul_le_mul_of_nonneg_left hjr (mul_nonneg ht (Real.exp_pos (C*(k:ℝ))).le)
-    nlinarith only [he,hj',htim t htT]
-  have hret (t : ℝ) (ht : 0 ≤ t) (htT : t ≤ T) : m/(2*c) ≤ χ t :=
-    deformed_root_sum_retained G hG hloc hqt a p hpzero hp root weight hw₀ hcov c hc hmin
-      P A hpos hsym hP hA k t B ht (hTb t htT) hsmall
-  have hdec (t : ℝ) (ht : 0 ≤ t) (htT : t ≤ T) : (h₀*β)*(χ t)^2 ≤ D t := by
-    have hχ : 0 ≤ χ t := averagedDeformedSusceptibility_nonneg G _ p k root weight hw₀ t
-    have hret' := (div_le_iff₀ (by positivity : 0 < 2*c)).mp (hret t ht htT)
-    have hrow' := hrow p hpnear hp k hk3 t ht (htim' t ht htT Cr (by dsimp [C]; linarith only [hCr,hCm,hCd]))
-    have hcost : maxSusceptibility G p*(τ/R) ≤ (cf/2)*χ t := by
-      have hmax : maxSusceptibility G p ≠ 0 :=
-        (zero_lt_one.trans_le (maxSusceptibility_one_le G hloc hqt p hp)).ne'
-      have he : maxSusceptibility G p*(τ/R) = τ*m := by dsimp [R]; field_simp [hmax]
-      rw [he]
-      have hh := mul_le_mul_of_nonneg_left hret' hτ.le
-      have hτcf : τ*(2*c) = cf/4 := by dsimp [τ]; field_simp; ring
-      rw [mul_left_comm τ (χ t) (2*c),hτcf,mul_comm (χ t)] at hh
-      nlinarith only [hh,hcf,hχ]
-    have hprod (w : RawCorridor G k) (hwsel : w.Selected hloc p j (τ/R)) :
-        Real.exp (-2)*β*((cf/2)*χ t)^2 ≤ retainedSusceptibilityProduct G (walkKernel G hloc a) p t w := by
-      have hbnd := selected_retained_product_lower G hloc hqt a hK p hp k t ht j (τ/R) Cm cf (χ t)
-        hcf.le hχ hrow' hcost (hmove p hpnear hp k hk3 t ht) w hwsel
-      have he : Real.exp (-2) ≤ Real.exp (-2*t*(j:ℝ)*Real.exp (Cm*(k:ℝ))) := by
-        apply Real.exp_le_exp.mpr
-        have he' := mul_le_mul_of_nonneg_left (hCbound Cm (by dsimp [C]; linarith only [hCr,hCm,hCd]))
-          (mul_nonneg ht (Nat.cast_nonneg j))
-        linarith only [he',htim t htT]
-      have hpw : β ≤ (p:ℝ)^(2*j) := pow_le_pow_left₀ (half_pos hpcrit).le hpnear _
-      exact (mul_le_mul_of_nonneg_right
-        (mul_le_mul he hpw hβ.le (Real.exp_pos _).le) (sq_nonneg _)).trans hbnd
-    have hsel := hselected p hpnear hp k hk R hR
-    have hpositive := selected_average_lower G hloc _ hK p k t root weight hw₀ j (τ/R) b
-      (Real.exp (-2)*β*((cf/2)*χ t)^2) (by positivity) hsel hprod
-    have hpositive' : q*β*(χ t)^2 ≤ rawPathAverage G (walkKernel G hloc a) k root weight
-        (retainedSusceptibilityProduct G (walkKernel G hloc a) p t) := by
-      convert hpositive using 1
-      dsimp [q]
-      ring
-    have ha₃ : 0 ≤ rootTriangle G root weight (p:ℝ) :=
-      zero_le_one.trans (rootTriangle_one_le G hloc hqt root weight hw₀ hw1 p hp)
-    have herror : (1+2*(d:ℝ)/(pc G:ℝ))*(maxSusceptibility G p)^2 *
-        (‖P‖^k*(rootTrace root weight (A^3)).re) ≤ (q/2)*β*(χ t)^2 := by
-      rw [←rootTriangle_eq_trace G p root weight A hA]
-      have hx : maxSusceptibility G p = R*m := by dsimp [R]; field_simp
-      rw [hx]
-      have hcoeff : 0 ≤ (1+2*(d:ℝ)/(pc G:ℝ)) := by positivity
-      have hm2 : m^2 ≤ (2*c)^2*(χ t)^2 := by
-        calc
-          m^2 ≤ (χ t*(2*c))^2 := pow_le_pow_left₀ hm.le hret' 2
-          _ = _ := by ring
-      have h₁ := mul_le_mul_of_nonneg_left hm2
-        (mul_nonneg (mul_nonneg hcoeff (sq_nonneg R)) (mul_nonneg ha₃ (pow_nonneg (norm_nonneg P) k)))
-      have h₂ := mul_le_mul_of_nonneg_left herrscale (mul_nonneg hH.le (sq_nonneg (χ t)))
-      have he : H*ε = q/2 := by dsimp [ε]; field_simp
-      calc
-        _ ≤ H*(R^2*rootTriangle G root weight (p:ℝ)*‖P‖^k)*(χ t)^2 := by dsimp [H]; nlinarith only [h₁]
-        _ ≤ H*(ε*β)*(χ t)^2 := by nlinarith only [h₂]
-        _ = _ := by rw [←mul_assoc H ε β,he]
-    have hdiff : (q/2)*β*(χ t)^2 ≤
-        rawPathAverage G (walkKernel G hloc a) k root weight
-          (retainedSusceptibilityProduct G (walkKernel G hloc a) p t) -
-        (1+2*(d:ℝ)/(pc G:ℝ))*(maxSusceptibility G p)^2 *
-          (‖P‖^k*(rootTrace root weight (A^3)).re) := by linarith only [hpositive',herror]
-    have hexp : Real.exp (-1) ≤ Real.exp (-t*Real.exp (Cd*(k:ℝ))) :=
-      Real.exp_le_exp.mpr (by linarith only [htim' t ht htT Cd (by dsimp [C]; linarith only [hCr,hCm]) ])
-    have hh := mul_le_mul hexp hdiff (by positivity) (Real.exp_pos _).le
-    have hdecr := hdecrease p hpnear hp A hA k hk3 t ht
-    apply le_trans _ hdecr
-    convert hh using 1
-    dsimp [h₀]
-    ring
-  have hb := riccati_retention_bound χ D T m c (h₀*β) hT hm hc (mul_pos hh₀ hβ).le
-    (fun t ht => hret t (by linarith only [ht.1,hT]) ht.2)
-    (fun t ht => hasDerivAt_averagedDeformedSusceptibility G hloc hqt root weight d hd a hK p hpzero hp k t
-      (by linarith only [ht.1,hT]))
-    (fun t ht => hdec t (by linarith only [ht.1,hT]) ht.2)
-  have he : h₀*β*T*m = h₀*(B*β) := by dsimp [T]; field_simp
-  rw [he] at hb
-  have hg := (div_lt_iff₀ hh₀).mp hlarge
-  nlinarith only [hg,hb]
-
-end FiniteScale
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Escape.PathSelection
+
+namespace OAI
+
+/-! Deformed susceptibility, finite-scale estimates and the Riccati contradiction. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section EscapeSusceptibility
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Countable V] [Countable E] [Nonempty V]
+
+omit [Countable V] [Countable E] [Nonempty V] in
+ 
+
+theorem deformed_move_of_walk (G : BondGraph V E) (hloc : LocallyFinite G)
+    (a C : ℝ) (p : unitInterval) (k : ℕ) (t : ℝ) (ht : 0 ≤ t)
+    (hmove : ∀ (D : Set V) (b z : V) (l : List E),
+      b ∈ D → Along G b l z → (∀ e ∈ l, e ∈ internalEdges G D) →
+      Real.exp (-t*((l.length:ℝ)*Real.exp (C*(k:ℝ)))) * (p:ℝ)^l.length *
+        deformedSusceptibility G (walkKernel G hloc a) p k t D z ≤
+        deformedSusceptibility G (walkKernel G hloc a) p k t D b)
+    (D : Set V) (b z : V) (q : (openSimpleGraph G Set.univ).Walk b z)
+    (j : ℕ) (hj : q.length ≤ j) (hD : ∀ x ∈ q.support, x ∈ D) :
+    Real.exp (-t*((j:ℝ)*Real.exp (C*(k:ℝ)))) * (p:ℝ)^j *
+      deformedSusceptibility G (walkKernel G hloc a) p k t D z ≤
+      deformedSusceptibility G (walkKernel G hloc a) p k t D b := by
+  obtain ⟨l,hl,he,_⟩ := walk_lift_bonds G Set.univ q
+  have hlen : l.length = q.length := by
+    have hh := congrArg List.length he
+    simpa only [List.length_map,SimpleGraph.Walk.length_edges] using hh
+  have hle : l.length ≤ j := hlen.trans_le hj
+  have hcomp := hmove D b z l (hD b q.start_mem_support) hl (by
+    intro e hel x hxe
+    exact hD x (SimpleGraph.Walk.mem_support_of_mem_edges
+      (he ▸ List.mem_map_of_mem hel) hxe))
+  have he' : Real.exp (-t*((j:ℝ)*Real.exp (C*(k:ℝ)))) ≤
+      Real.exp (-t*((l.length:ℝ)*Real.exp (C*(k:ℝ)))) := by
+    apply Real.exp_le_exp.mpr
+    exact mul_le_mul_of_nonpos_left
+      (mul_le_mul_of_nonneg_right (Nat.cast_le.mpr hle) (Real.exp_pos _).le) (neg_nonpos.mpr ht)
+  have hpw : (p:ℝ)^j ≤ (p:ℝ)^l.length := pow_le_pow_of_le_one p.2.1 p.2.2 hle
+  exact (mul_le_mul_of_nonneg_right (mul_le_mul he' hpw (pow_nonneg p.2.1 _) (Real.exp_pos _).le)
+    (deformedSusceptibility_nonneg G _ p k t D z)).trans hcomp
+
+ 
+theorem deformedSusceptibility_delete_interior (G : BondGraph V E) (hloc : LocallyFinite G)
+    (hqt : QuasiTransitive G) (a : ℝ) (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y)
+    (p : unitInterval) (hp : p < pc G) (k : ℕ) (t : ℝ) (ht : 0 ≤ t)
+    (w : RawCorridor G k) (x : V) :
+    deformedSusceptibility G (walkKernel G hloc a) p k t Set.univ x -
+      maxSusceptibility G p * (∑ z ∈ w.neighborhood hloc, twoPoint G p x z) ≤
+      deformedSusceptibility G (walkKernel G hloc a) p k t (w.interior : Set V)ᶜ x := by
+  have hh := deformedSusceptibility_deletion G hloc hqt _ hP p hp k t ht Set.univ (w.interior : Set V) x
+  rw [←w.coe_neighborhood hloc,Finset.tsum_subtype'] at hh
+  have he : Set.univ \ (w.interior : Set V) = (w.interior : Set V)ᶜ := by ext z; simp
+  rw [he] at hh
+  exact hh
+
+ 
+
+theorem selected_retained_product_lower (G : BondGraph V E) (hloc : LocallyFinite G)
+    (hqt : QuasiTransitive G) (a : ℝ) (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y)
+    (p : unitInterval) (hp : p < pc G) (k : ℕ) (t : ℝ) (ht : 0 ≤ t)
+    (j : ℕ) (s C cf χ : ℝ) (hcf : 0 ≤ cf) (hχ : 0 ≤ χ)
+    (hrow : ∀ x, cf*χ ≤ deformedSusceptibility G (walkKernel G hloc a) p k t Set.univ x)
+    (hcost : maxSusceptibility G p*s ≤ (cf/2)*χ)
+    (hmove : ∀ (D : Set V) (b z : V) (l : List E),
+      b ∈ D → Along G b l z → (∀ e ∈ l, e ∈ internalEdges G D) →
+      Real.exp (-t*((l.length:ℝ)*Real.exp (C*(k:ℝ)))) * (p:ℝ)^l.length *
+        deformedSusceptibility G (walkKernel G hloc a) p k t D z ≤
+        deformedSusceptibility G (walkKernel G hloc a) p k t D b)
+    (w : RawCorridor G k) (hw : w.Selected hloc p j s) :
+    Real.exp (-2*t*(j:ℝ)*Real.exp (C*(k:ℝ))) * (p:ℝ)^(2*j) * ((cf/2)*χ)^2 ≤
+      retainedSusceptibilityProduct G (walkKernel G hloc a) p t w := by
+  have hx (x : V) (he : w.HasEscape hloc p j s x) :
+      Real.exp (-t*((j:ℝ)*Real.exp (C*(k:ℝ))))*(p:ℝ)^j*((cf/2)*χ) ≤
+        deformedSusceptibility G (walkKernel G hloc a) p k t (w.interior : Set V)ᶜ x := by
+    obtain ⟨z,q,_hq,hj,havoid,hs⟩ := he
+    have hd := deformedSusceptibility_delete_interior G hloc hqt a hP p hp k t ht w z
+    have hδ : (cf/2)*χ ≤
+        deformedSusceptibility G (walkKernel G hloc a) p k t (w.interior : Set V)ᶜ z := by
+      have hb := mul_le_mul_of_nonneg_left hs (zero_le_one.trans (maxSusceptibility_one_le G hloc hqt p hp))
+      have hr := hrow z
+      linarith
+    exact (mul_le_mul_of_nonneg_left hδ
+      (mul_nonneg (Real.exp_pos _).le (pow_nonneg p.2.1 _))).trans
+        (deformed_move_of_walk G hloc a C p k t ht hmove (w.interior : Set V)ᶜ x z q j hj havoid)
+  have h₁ := hx w.base hw.2.1
+  have h₂ := hx w.endpoint hw.2.2
+  have hfactor : 0 ≤ Real.exp (-t*((j:ℝ)*Real.exp (C*(k:ℝ))))*(p:ℝ)^j*((cf/2)*χ) :=
+    mul_nonneg (mul_nonneg (Real.exp_pos _).le (pow_nonneg p.2.1 _))
+      (mul_nonneg (div_nonneg hcf (by norm_num)) hχ)
+  have hh := mul_le_mul h₁ h₂ hfactor
+    (deformedSusceptibility_nonneg G _ p k t (w.interior : Set V)ᶜ w.base)
+  rw [retainedSusceptibilityProduct,ite_eq_left hw.1]
+  apply le_trans (le_of_eq ?_) hh
+  have he : -2*t*(j:ℝ)*Real.exp (C*(k:ℝ)) =
+      -t*((j:ℝ)*Real.exp (C*(k:ℝ))) + -t*((j:ℝ)*Real.exp (C*(k:ℝ))) := by ring
+  rw [he,Real.exp_add,show 2*j=j+j by omega,pow_add]
+  ring
+
+end EscapeSusceptibility
+end Percolation
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section AveragedRetention
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Countable V] [Countable E] [Nonempty V]
+variable {ι : Type*} [Fintype ι]
+
+omit [Countable V] [Countable E] [Nonempty V] in
+theorem averagedDeformedSusceptibility_nonneg (G : BondGraph V E) (P : V → V → ℝ)
+    (p : unitInterval) (k : ℕ) (root : ι → V) (weight : ι → ℝ)
+    (hw : ∀ i, 0 ≤ weight i) (t : ℝ) :
+    0 ≤ averagedDeformedSusceptibility G P p k root weight t :=
+  Finset.sum_nonneg (fun i _ => mul_nonneg (hw i) (deformedSusceptibility_nonneg G P p k t Set.univ (root i)))
+
+omit [Countable V] [Countable E] [Nonempty V] in
+theorem deformedKernel_symm (G : BondGraph V E) (P : V → V → ℝ)
+    (p : unitInterval) (k : ℕ) (t : ℝ) (D : Set V) (x y : V) :
+    deformedKernel G P p k t D x y = deformedKernel G P p k t D y x := by
+  simp only [deformedKernel,deformedConnection_symm G P p k t D _ x y]
+
+omit [Nonempty V] in
+ 
+
+theorem deformed_row_le_root_sum (G : BondGraph V E) (hloc : LocallyFinite G)
+    (a : ℝ) (p : unitInterval) (k : ℕ) (t : ℝ)
+    (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 ≤ weight i)
+    (hcov : ∀ x, ∃ i, ∃ g ∈ vertexAut G, g (root i) = x)
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i) (x : V) :
+    deformedSusceptibility G (walkKernel G hloc a) p k t Set.univ x ≤
+      c*averagedDeformedSusceptibility G (walkKernel G hloc a) p k root weight t := by
+  obtain ⟨i,g,⟨e,he⟩,rfl⟩ := hcov x
+  rw [deformedSusceptibility_aut G he _ (walkKernel_covariant G hloc a he)]
+  simp only [Set.preimage_univ]
+  have hs := deformedSusceptibility_nonneg G (walkKernel G hloc a) p k t Set.univ
+  have hi := Finset.single_le_sum
+    (s := Finset.univ)
+    (f := fun r => weight r * deformedSusceptibility G (walkKernel G hloc a) p k t Set.univ (root r))
+    (fun r _ => mul_nonneg (hw r) (hs (root r))) (Finset.mem_univ i)
+  have hi' := mul_le_mul_of_nonneg_left hi hc
+  have hh := mul_le_mul_of_nonneg_right (hmin i) (hs (root i))
+  change _ ≤ c*∑ r, weight r * deformedSusceptibility G (walkKernel G hloc a) p k t Set.univ (root r)
+  nlinarith
+
+ 
+
+theorem deformed_norm_le_root_sum (G : BondGraph V E) (hloc : LocallyFinite G)
+    (hqt : QuasiTransitive G) (a : ℝ) (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y)
+    (p : unitInterval) (hp : p < pc G) (k : ℕ) (t : ℝ) (ht : 0 ≤ t)
+    (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 ≤ weight i)
+    (hcov : ∀ x, ∃ i, ∃ g ∈ vertexAut G, g (root i) = x)
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (B : Hilbert V →L[ℂ] Hilbert V)
+    (hB : ∀ x y, matrixCoeff B x y = (deformedKernel G (walkKernel G hloc a) p k t Set.univ x y : ℂ)) :
+    ‖B‖ ≤ c*averagedDeformedSusceptibility G (walkKernel G hloc a) p k root weight t := by
+  have hs (x : V) (F : Finset V) :
+      (∑ y ∈ F, |deformedKernel G (walkKernel G hloc a) p k t Set.univ x y|) ≤
+        c*averagedDeformedSusceptibility G (walkKernel G hloc a) p k root weight t := by
+    simp only [abs_of_nonneg (deformedKernel_nonneg G _ p k t _ _ _)]
+    exact ((summable_deformedKernel G hloc hqt _ hP p hp k t ht Set.univ x).sum_le_tsum F
+      (fun y _ => deformedKernel_nonneg G _ p k t _ x y)).trans
+        (deformed_row_le_root_sum G hloc a p k t root weight hw hcov c hc hmin x)
+  exact operator_norm_le_absolute_sums _ _
+    (mul_nonneg hc (averagedDeformedSusceptibility_nonneg G _ p k root weight hw t)) hs
+    (fun y F => by simpa only [deformedKernel_symm G _ p k t Set.univ] using hs y F) B hB
+
+ 
+
+theorem deformed_root_sum_retained (G : BondGraph V E) (hG : Function.Injective G)
+    (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (a : ℝ) (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp : p < pc G)
+    (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 ≤ weight i)
+    (hcov : ∀ x, ∃ i, ∃ g ∈ vertexAut G, g (root i) = x)
+    (c : ℝ) (hc : 0 < c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (P A : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hPs : IsSelfAdjoint P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (hA : HasKernel G p A) (k : ℕ) (t B : ℝ) (ht : 0 ≤ t)
+    (htB : t*‖A‖ ≤ B) (hB : B*‖P‖^k ≤ 1/4) :
+    ‖A‖/(2*c) ≤ averagedDeformedSusceptibility G (walkKernel G hloc a) p k root weight t := by
+  have hK (x y : V) : 0 ≤ walkKernel G hloc a x y := by
+    simpa only [hP,Complex.ofReal_re] using (hpos x y).2
+  obtain ⟨T,hT,_,_,hret⟩ := exists_deformedOperator_small_change G hG hloc p hp0
+    ((show (p:ℝ)<(pc G:ℝ) from hp).trans_le (pc G).2.2) a P hpos hPs hP
+    A (hA.positiveRealMatrix G p A) (hA.isPositive G p A).isSelfAdjoint hA k t ht
+  have hn := deformed_norm_le_root_sum G hloc hqt a hK p hp k t ht root weight hw hcov c hc.le hmin T hT
+  have hsmall := (mul_le_mul_of_nonneg_right htB (pow_nonneg (norm_nonneg P) k)).trans hB
+  have hh := mul_le_mul_of_nonneg_right hsmall (norm_nonneg A)
+  apply (div_le_iff₀ (by positivity : 0 < 2*c)).mpr
+  nlinarith
+
+end AveragedRetention
+end Percolation
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section AveragePositiveTerm
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Countable V] [Countable E] [Nonempty V]
+variable {ι : Type*} [Fintype ι]
+
+omit [Countable V] [Countable E] [Nonempty V] in
+ 
+theorem selected_average_lower (G : BondGraph V E) (hloc : LocallyFinite G)
+    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (p : unitInterval) (k : ℕ) (t : ℝ)
+    (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 ≤ weight i)
+    (j : ℕ) (s b F : ℝ) (hF : 0 ≤ F)
+    (hm : b ≤ rawPathAverage G P k root weight
+      (fun w => if w.Selected hloc p j s then 1 else 0))
+    (hgood : ∀ w : RawCorridor G k, w.Selected hloc p j s →
+      F ≤ retainedSusceptibilityProduct G P p t w) :
+    F*b ≤ rawPathAverage G P k root weight (retainedSusceptibilityProduct G P p t) := by
+  have hh := rawPathAverage_mono G hloc P hP k root weight hw
+    (fun w => F*(if w.Selected hloc p j s then 1 else 0))
+    (retainedSusceptibilityProduct G P p t) (by
+      intro w
+      by_cases hs : w.Selected hloc p j s
+      · simpa only [hs,ite_true,mul_one] using hgood w hs
+      · simp only [hs,ite_false,mul_zero,retainedSusceptibilityProduct]
+        split_ifs
+        · exact mul_nonneg (deformedSusceptibility_nonneg G P p k t _ w.base)
+            (deformedSusceptibility_nonneg G P p k t _ w.endpoint)
+        · exact le_rfl)
+  rw [rawPathAverage_const_mul] at hh
+  exact (mul_le_mul_of_nonneg_left hm hF).trans hh
+end AveragePositiveTerm
+
+section RiccatiContradiction
+ 
+
+theorem riccati_retention_bound (χ D : ℝ → ℝ) (T m c h : ℝ)
+    (hT : 0 < T) (hm : 0 < m) (hc : 0 < c) (_hh : 0 ≤ h)
+    (hret : ∀ t ∈ Set.Icc (T/2) T, m/(2*c) ≤ χ t)
+    (hder : ∀ t ∈ Set.Icc (T/2) T, HasDerivAt χ (-D t) t)
+    (hdec : ∀ t ∈ Set.Icc (T/2) T, h*(χ t)^2 ≤ D t) :
+    h*T*m ≤ 4*c := by
+  have hpos (t) (ht : t ∈ Set.Icc (T/2) T) : 0 < χ t :=
+    (div_pos hm (by positivity)).trans_le (hret t ht)
+  have hd (t) (ht : t ∈ Set.Icc (T/2) T) :
+      HasDerivAt (fun s => (χ s)⁻¹) (D t/(χ t)^2) t := by
+    convert! (hder t ht).inv (hpos t ht).ne' using 1
+    simp only [neg_neg]
+  have hcχ : ContinuousOn (fun s => (χ s)⁻¹) (Set.Icc (T/2) T) :=
+    fun t ht => (hd t ht).continuousAt.continuousWithinAt
+  have hdiff : DifferentiableOn ℝ (fun s => (χ s)⁻¹) (interior (Set.Icc (T/2) T)) :=
+    fun t ht => (hd t (interior_subset ht)).differentiableAt.differentiableWithinAt
+  have hdl (t) (ht : t ∈ interior (Set.Icc (T/2) T)) :
+      h ≤ deriv (fun s => (χ s)⁻¹) t := by
+    rw [(hd t (interior_subset ht)).deriv]
+    exact (le_div_iff₀ (sq_pos_of_pos (hpos t (interior_subset ht)))).mpr (hdec t (interior_subset ht))
+  have hs := (convex_Icc (T/2) T).mul_sub_le_image_sub_of_le_deriv hcχ hdiff hdl
+    (T/2) ⟨le_rfl,by linarith⟩ T ⟨by linarith,le_rfl⟩ (by linarith)
+  have hstart : 0 ≤ (χ (T/2))⁻¹ := (inv_pos.mpr (hpos _ ⟨le_rfl,by linarith⟩)).le
+  have hend : (χ T)⁻¹ ≤ 2*c/m := by
+    have hb := one_div_le_one_div_of_le (div_pos hm (by positivity)) (hret T ⟨by linarith,le_rfl⟩)
+    simpa only [one_div,inv_div] using hb
+  have hs' : h*(T/2) ≤ 2*c/m := by linarith
+  have hs'' := (le_div_iff₀ hm).mp hs'
+  nlinarith
+end RiccatiContradiction
+end Percolation
+
+ 
+
+ 
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section FiniteScale
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Countable V] [Countable E] [Infinite V]
+variable {ι : Type*} [Fintype ι]
+
+ 
+
+theorem exists_finite_scale_obstruction
+    (G : BondGraph V E) (hG : Function.Injective G)
+    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (a : ℝ) (ha : 0 < a) (root : ι → V) (weight : ι → ℝ)
+    (htr : RootTransport root weight (vertexAut G))
+    (hcov : ∀ x, ∃ r, ∃ g ∈ vertexAut G, g (root r) = x)
+    (hw : ∀ r, 0 < weight r) (hw1 : ∑ r, weight r = 1)
+    (c : ℝ) (hc : 0 < c) (hmin : ∀ r, 1 ≤ c*weight r)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re) (hρ : ‖P‖ < 1)
+    (d : ℕ) (hd : ∀ z, degree G z ≤ d)
+    (hpcrit : 0 < (pc G:ℝ)) (hpcrit₁ : (pc G:ℝ) < 1) :
+    ∃ C : ℝ, 0 < C ∧ ∃ J : ℕ, 0 < J ∧ ∃ C₁ : ℝ, 0 < C₁ ∧
+      ∃ k₀ : ℕ, 3 ≤ k₀ ∧ ∃ ε M : ℝ, 0 < ε ∧ 0 < M ∧
+      ∀ (p : unitInterval), (pc G:ℝ)/2 ≤ (p:ℝ) → p < pc G →
+      ∀ (A : Hilbert V →L[ℂ] Hilbert V), HasKernel G p A →
+      ∀ k : ℕ, k₀ ≤ k → ∀ B : ℝ, 0 < B →
+      let R := maxSusceptibility G p/‖A‖
+      let j := J+⌈C₁*Real.log (((k:ℝ)+1)*R)⌉₊
+      B*‖P‖^k ≤ 1/4 →
+      B*(j:ℝ)*Real.exp (C*(k:ℝ)) ≤ ‖A‖ →
+      R^2*rootTriangle G root weight (p:ℝ)*‖P‖^k ≤ ε*((pc G:ℝ)/2)^(2*j) →
+      M < B*((pc G:ℝ)/2)^(2*j) → False := by
+  have hK (x y : V) : 0 ≤ walkKernel G hloc a x y := by
+    simpa only [hP,Complex.ofReal_re] using (hpos x y).2
+  have hw₀ (i) := (hw i).le
+  obtain ⟨Cr,cf,hCr,hcf,hrow⟩ := exists_uniform_deformed_row_comparison G hconn hloc hqt
+    root weight hw₀ hw1 (fun x => by
+      obtain ⟨i,g,⟨e,he⟩,hx⟩ := hcov x
+      exact ⟨i,g,e,he,hx⟩) d hd a hK hpcrit hpcrit₁
+  obtain ⟨Cm,hCm,hmove⟩ := exists_uniform_deformed_endpoint_comparison G hloc hqt d hd a hK hpcrit hpcrit₁
+  obtain ⟨Cd,hCd,hdecrease⟩ := exists_uniform_averaged_decrease G hG hloc hqt
+    root weight hw₀ (vertexAut G) (Set.Subset.refl _) htr d hd a hK hpcrit hpcrit₁ P hpos hP hsym
+  let τ := cf/(8*c)
+  have hτ : 0 < τ := div_pos hcf (by positivity)
+  obtain ⟨J,C₁,k₀,b,hJ,hC₁,hk₀,hb,hselected⟩ := exists_selected_path_mass G hconn hloc hqt
+    a ha root weight htr hcov hw hw1 c hc hmin P hpos hsym hP hdiag hρ d hd τ hτ
+  let C := Cr+Cm+Cd
+  let q := b*Real.exp (-2)*(cf/2)^2
+  let H := (1+2*(d:ℝ)/(pc G:ℝ))*(2*c)^2
+  let ε := q/(2*H)
+  let h₀ := Real.exp (-1)*q/2
+  let M := 4*c/h₀
+  have hq : 0 < q := by dsimp [q]; positivity
+  have hH : 0 < H := by dsimp [H]; positivity
+  have hε : 0 < ε := div_pos hq (by positivity)
+  have hh₀ : 0 < h₀ := by dsimp [h₀]; positivity
+  have hM : 0 < M := div_pos (by positivity) hh₀
+  refine ⟨C,by dsimp [C]; positivity,J,hJ,C₁,hC₁,k₀,hk₀,ε,M,hε,hM,?_⟩
+  intro p hpnear hp A hA k hk B hB
+  dsimp only
+  let m := ‖A‖
+  let R := maxSusceptibility G p/m
+  let j := J+⌈C₁*Real.log (((k:ℝ)+1)*R)⌉₊
+  let β := ((pc G:ℝ)/2)^(2*j)
+  let T := B/m
+  let χ := averagedDeformedSusceptibility G (walkKernel G hloc a) p k root weight
+  let D := fun t => ∑ i, weight i * ∫ ω, deformedDerivativeMass G (walkKernel G hloc a) p k t Set.univ ω (root i) ∂law p
+  change B*‖P‖^k ≤ 1/4 → B*(j:ℝ)*Real.exp (C*(k:ℝ)) ≤ m →
+    R^2*rootTriangle G root weight (p:ℝ)*‖P‖^k ≤ ε*β → M < B*β → False
+  intro hsmall htime herrscale hlarge
+  have hm : 0 < m := zero_lt_one.trans_le (hA.norm_one_le G p A)
+  have hR : 1 ≤ R := (one_le_div hm).mpr (hA.norm_le_maxSusceptibility G hloc hqt p hp A)
+  have hβ : 0 < β := pow_pos (half_pos hpcrit) _
+  have hj : 1 ≤ j := by dsimp [j]; omega
+  have hjr : (1:ℝ) ≤ (j:ℝ) := Nat.one_le_cast.mpr hj
+  have hk3 : 3 ≤ k := hk₀.trans hk
+  have hpzero : 0 < (p:ℝ) := (half_pos hpcrit).trans_le hpnear
+  have hT : 0 < T := div_pos hB hm
+  have hTb (t : ℝ) (ht : t ≤ T) : t*m ≤ B := (le_div_iff₀ hm).mp ht
+  have htim (t : ℝ) (ht : t ≤ T) : t*(j:ℝ)*Real.exp (C*(k:ℝ)) ≤ 1 := by
+    have hm' := mul_le_mul_of_nonneg_right (hTb t ht)
+      (mul_nonneg (Nat.cast_nonneg j) (Real.exp_pos (C*(k:ℝ))).le)
+    nlinarith only [hm',htime,hm]
+  have hCbound (C' : ℝ) (hC' : C' ≤ C) : Real.exp (C'*(k:ℝ)) ≤ Real.exp (C*(k:ℝ)) :=
+    Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right hC' (Nat.cast_nonneg k))
+  have htim' (t : ℝ) (ht : 0 ≤ t) (htT : t ≤ T) (C' : ℝ) (hC' : C' ≤ C) :
+      t*Real.exp (C'*(k:ℝ)) ≤ 1 := by
+    have he := mul_le_mul_of_nonneg_left (hCbound C' hC') ht
+    have hj' := mul_le_mul_of_nonneg_left hjr (mul_nonneg ht (Real.exp_pos (C*(k:ℝ))).le)
+    nlinarith only [he,hj',htim t htT]
+  have hret (t : ℝ) (ht : 0 ≤ t) (htT : t ≤ T) : m/(2*c) ≤ χ t :=
+    deformed_root_sum_retained G hG hloc hqt a p hpzero hp root weight hw₀ hcov c hc hmin
+      P A hpos hsym hP hA k t B ht (hTb t htT) hsmall
+  have hdec (t : ℝ) (ht : 0 ≤ t) (htT : t ≤ T) : (h₀*β)*(χ t)^2 ≤ D t := by
+    have hχ : 0 ≤ χ t := averagedDeformedSusceptibility_nonneg G _ p k root weight hw₀ t
+    have hret' := (div_le_iff₀ (by positivity : 0 < 2*c)).mp (hret t ht htT)
+    have hrow' := hrow p hpnear hp k hk3 t ht (htim' t ht htT Cr (by dsimp [C]; linarith only [hCr,hCm,hCd]))
+    have hcost : maxSusceptibility G p*(τ/R) ≤ (cf/2)*χ t := by
+      have hmax : maxSusceptibility G p ≠ 0 :=
+        (zero_lt_one.trans_le (maxSusceptibility_one_le G hloc hqt p hp)).ne'
+      have he : maxSusceptibility G p*(τ/R) = τ*m := by dsimp [R]; field_simp [hmax]
+      rw [he]
+      have hh := mul_le_mul_of_nonneg_left hret' hτ.le
+      have hτcf : τ*(2*c) = cf/4 := by dsimp [τ]; field_simp; ring
+      rw [mul_left_comm τ (χ t) (2*c),hτcf,mul_comm (χ t)] at hh
+      nlinarith only [hh,hcf,hχ]
+    have hprod (w : RawCorridor G k) (hwsel : w.Selected hloc p j (τ/R)) :
+        Real.exp (-2)*β*((cf/2)*χ t)^2 ≤ retainedSusceptibilityProduct G (walkKernel G hloc a) p t w := by
+      have hbnd := selected_retained_product_lower G hloc hqt a hK p hp k t ht j (τ/R) Cm cf (χ t)
+        hcf.le hχ hrow' hcost (hmove p hpnear hp k hk3 t ht) w hwsel
+      have he : Real.exp (-2) ≤ Real.exp (-2*t*(j:ℝ)*Real.exp (Cm*(k:ℝ))) := by
+        apply Real.exp_le_exp.mpr
+        have he' := mul_le_mul_of_nonneg_left (hCbound Cm (by dsimp [C]; linarith only [hCr,hCm,hCd]))
+          (mul_nonneg ht (Nat.cast_nonneg j))
+        linarith only [he',htim t htT]
+      have hpw : β ≤ (p:ℝ)^(2*j) := pow_le_pow_left₀ (half_pos hpcrit).le hpnear _
+      exact (mul_le_mul_of_nonneg_right
+        (mul_le_mul he hpw hβ.le (Real.exp_pos _).le) (sq_nonneg _)).trans hbnd
+    have hsel := hselected p hpnear hp k hk R hR
+    have hpositive := selected_average_lower G hloc _ hK p k t root weight hw₀ j (τ/R) b
+      (Real.exp (-2)*β*((cf/2)*χ t)^2) (by positivity) hsel hprod
+    have hpositive' : q*β*(χ t)^2 ≤ rawPathAverage G (walkKernel G hloc a) k root weight
+        (retainedSusceptibilityProduct G (walkKernel G hloc a) p t) := by
+      convert hpositive using 1
+      dsimp [q]
+      ring
+    have ha₃ : 0 ≤ rootTriangle G root weight (p:ℝ) :=
+      zero_le_one.trans (rootTriangle_one_le G hloc hqt root weight hw₀ hw1 p hp)
+    have herror : (1+2*(d:ℝ)/(pc G:ℝ))*(maxSusceptibility G p)^2 *
+        (‖P‖^k*(rootTrace root weight (A^3)).re) ≤ (q/2)*β*(χ t)^2 := by
+      rw [←rootTriangle_eq_trace G p root weight A hA]
+      have hx : maxSusceptibility G p = R*m := by dsimp [R]; field_simp
+      rw [hx]
+      have hcoeff : 0 ≤ (1+2*(d:ℝ)/(pc G:ℝ)) := by positivity
+      have hm2 : m^2 ≤ (2*c)^2*(χ t)^2 := by
+        calc
+          m^2 ≤ (χ t*(2*c))^2 := pow_le_pow_left₀ hm.le hret' 2
+          _ = _ := by ring
+      have h₁ := mul_le_mul_of_nonneg_left hm2
+        (mul_nonneg (mul_nonneg hcoeff (sq_nonneg R)) (mul_nonneg ha₃ (pow_nonneg (norm_nonneg P) k)))
+      have h₂ := mul_le_mul_of_nonneg_left herrscale (mul_nonneg hH.le (sq_nonneg (χ t)))
+      have he : H*ε = q/2 := by dsimp [ε]; field_simp
+      calc
+        _ ≤ H*(R^2*rootTriangle G root weight (p:ℝ)*‖P‖^k)*(χ t)^2 := by dsimp [H]; nlinarith only [h₁]
+        _ ≤ H*(ε*β)*(χ t)^2 := by nlinarith only [h₂]
+        _ = _ := by rw [←mul_assoc H ε β,he]
+    have hdiff : (q/2)*β*(χ t)^2 ≤
+        rawPathAverage G (walkKernel G hloc a) k root weight
+          (retainedSusceptibilityProduct G (walkKernel G hloc a) p t) -
+        (1+2*(d:ℝ)/(pc G:ℝ))*(maxSusceptibility G p)^2 *
+          (‖P‖^k*(rootTrace root weight (A^3)).re) := by linarith only [hpositive',herror]
+    have hexp : Real.exp (-1) ≤ Real.exp (-t*Real.exp (Cd*(k:ℝ))) :=
+      Real.exp_le_exp.mpr (by linarith only [htim' t ht htT Cd (by dsimp [C]; linarith only [hCr,hCm]) ])
+    have hh := mul_le_mul hexp hdiff (by positivity) (Real.exp_pos _).le
+    have hdecr := hdecrease p hpnear hp A hA k hk3 t ht
+    apply le_trans _ hdecr
+    convert hh using 1
+    dsimp [h₀]
+    ring
+  have hb := riccati_retention_bound χ D T m c (h₀*β) hT hm hc (mul_pos hh₀ hβ).le
+    (fun t ht => hret t (by linarith only [ht.1,hT]) ht.2)
+    (fun t ht => hasDerivAt_averagedDeformedSusceptibility G hloc hqt root weight d hd a hK p hpzero hp k t
+      (by linarith only [ht.1,hT]))
+    (fun t ht => hdec t (by linarith only [ht.1,hT]) ht.2)
+  have he : h₀*β*T*m = h₀*(B*β) := by dsimp [T]; field_simp
+  rw [he] at hb
+  have hg := (div_lt_iff₀ hh₀).mp hlarge
+  nlinarith only [hg,hb]
+
+end FiniteScale
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Encounter/Counts.lean b/lean/OAI/Probability/UnimodularPercolation/Encounter/Counts.lean
index 73f6cc95365d08faf75d3fdb414d7237c876abc1..bbd92e41d53231a21c6175a6b3a3d597372641d0 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Encounter/Counts.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Encounter/Counts.lean
@@ -1,224 +1,224 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Main
-import OAI.Probability.UnimodularPercolation.Encounter.Measurability
-
-namespace OAI
-
-/-! Counting encounter vertices by mass transport. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-open MeasureTheory SimpleGraph
-open scoped ENNReal
-universe u v
-variable {V : Type u} {E : Type v}
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-def cutIso {W : Type*} {H : SimpleGraph V} {K : SimpleGraph W}
-    (f : H ≃g K) (x : V) : cutVertex H x ≃g cutVertex K (f x) where
-  toEquiv := f.toEquiv
-  map_rel_iff' := by
-    intro a b
-    change (K.Adj (f a) (f b) ∧ f a ≠ f x ∧ f b ≠ f x) ↔
-      (H.Adj a b ∧ a ≠ x ∧ b ≠ x)
-    simp only [f.map_adj_iff,f.injective.ne_iff]
-
-theorem branchCount_iso {W : Type*} {H : SimpleGraph V} {K : SimpleGraph W}
-    (f : H ≃g K) (S : Finset V) (x : V) :
-    branchCount K (S.image f) (f x) = branchCount H S x := by
-  unfold branchCount
-  rw [←Finset.image_erase f.injective,Finset.image_image]
-  let e := (cutIso f x).connectedComponentEquiv
-  have he : ((S.erase x).image (fun a => (cutVertex K (f x)).connectedComponentMk (f a))) =
-      (((S.erase x).image (cutVertex H x).connectedComponentMk).image e) := by
-    rw [Finset.image_image]
-    rfl
-  change ((S.erase x).image (fun a => (cutVertex K (f x)).connectedComponentMk (f a))).card = _
-  rw [he]
-  exact Finset.card_image_iff.mpr e.injective.injOn
-
-theorem clusterFinset_aut (G : BondGraph V E) (a : V ≃ V) (b : E ≃ E)
-    (hab : IsAut G a b) (ω : Set E) (x : V) :
-    (clusterFinset G (b ⁻¹' ω) x).image a = clusterFinset G ω (a x) := by
-  have hi : (cluster G (b ⁻¹' ω) x).Infinite ↔ (cluster G ω (a x)).Infinite := by
-    rw [←cluster_aut G hab]
-    exact (Set.infinite_image_iff a.injective.injOn).symm
-  by_cases hf : (cluster G (b ⁻¹' ω) x).Finite
-  · have hf' : (cluster G ω (a x)).Finite := by
-      rw [←cluster_aut G hab]
-      exact hf.image a
-    apply Finset.coe_injective
-    simp only [Finset.coe_image,clusterFinset_coe G _ _ hf,clusterFinset_coe G _ _ hf']
-    exact cluster_aut G hab ω x
-  · have hf' : ¬(cluster G ω (a x)).Finite := hi.mp hf
-    simp only [clusterFinset,dite_eq_right hf,dite_eq_right hf',Finset.image_empty]
-
-theorem measurable_branchCount_pair [Countable V] [Countable E]
-    (G : BondGraph V E) (x : V) :
-    Measurable (fun z : Set E × Finset V => branchCount (openSimpleGraph G z.1) z.2 x) :=
-  measurable_from_prod_countable_left (fun S => measurable_branchCount G S x)
-
-noncomputable def markedCluster (G : BondGraph V E) (ω η : Set E) (x : V) : Finset V :=
-  (clusterFinset G η x).filter (IsEncounter G ω)
-
-noncomputable def encounterCount (G : BondGraph V E) (ω η : Set E) (x : V) : ℕ :=
-  if IsEncounter G ω x then branchCount (openSimpleGraph G ω) (markedCluster G ω η x) x else 0
-
-theorem markedCluster_aut (G : BondGraph V E) (a : V ≃ V) (b : E ≃ E)
-    (hab : IsAut G a b) (ω η : Set E) (x : V) :
-    (markedCluster G (b ⁻¹' ω) (b ⁻¹' η) x).image a = markedCluster G ω η (a x) := by
-  unfold markedCluster
-  rw [←clusterFinset_aut G a b hab η x,Finset.filter_image]
-  simp only [isEncounter_aut_iff G a b hab]
-
-theorem encounterCount_aut (G : BondGraph V E) (a : V ≃ V) (b : E ≃ E)
-    (hab : IsAut G a b) (ω η : Set E) (x : V) :
-    encounterCount G (b ⁻¹' ω) (b ⁻¹' η) x = encounterCount G ω η (a x) := by
-  simp only [encounterCount,isEncounter_aut_iff G a b hab]
-  congr 1
-  rw [←markedCluster_aut G a b hab]
-  exact (branchCount_iso (openAutIso G a b hab ω) _ x).symm
-
-theorem measurable_markedCluster [Countable V] [Countable E]
-    (G : BondGraph V E) (p q : unitInterval) (x : V) :
-    Measurable (fun l : E → unitInterval => markedCluster G (openEdges l p) (openEdges l q) x) :=
-  measurable_filter_finset _ ((measurable_clusterFinset G x).comp (measurable_openEdges q)) _
-    (fun y => (measurableSet_isEncounter G y).preimage (measurable_openEdges p))
-
-theorem measurable_encounterCount [Countable V] [Countable E]
-    (G : BondGraph V E) (p q : unitInterval) (x : V) :
-    Measurable (fun l : E → unitInterval => encounterCount G (openEdges l p) (openEdges l q) x) := by
-  apply Measurable.ite ((measurableSet_isEncounter G x).preimage (measurable_openEdges p)) _ measurable_const
-  exact (measurable_branchCount_pair G x).comp
-    ((measurable_openEdges p).prodMk (measurable_markedCluster G p q x))
-
-theorem cluster_induce_connected (G : BondGraph V E) (ω η : Set E) (hsub : η ⊆ ω) (x : V) :
-    ((openSimpleGraph G ω).induce (cluster G η x)).Connected := by
-  let H := openSimpleGraph G η
-  have he : cluster G η x = (H.connectedComponentMk x).supp := by
-    ext y
-    simp only [cluster,Set.mem_ofPred_eq,ConnectedComponent.mem_supp_iff,ConnectedComponent.eq]
-    exact (conn_iff_reachable G η x y).trans SimpleGraph.reachable_comm
-  rw [he]
-  apply SimpleGraph.Connected.mono (G := H.induce (H.connectedComponentMk x).supp)
-  · intro a b hab
-    exact ⟨hab.1,by obtain ⟨e,he,hGe⟩ := hab.2; exact ⟨e,hsub he,hGe⟩⟩
-  · exact (H.connectedComponentMk x).connected_toSimpleGraph
-
-theorem markedCluster_eq_of_conn (G : BondGraph V E) (ω η : Set E) {x y : V}
-    (hxy : Conn G η x y) : markedCluster G ω η x = markedCluster G ω η y := by
-  unfold markedCluster clusterFinset
-  simp only [cluster_eq_of_conn G η hxy]
-
-theorem sum_encounterCount_le (G : BondGraph V E) (ω η : Set E) (hsub : η ⊆ ω) (x : V)
-    (hf : (cluster G η x).Finite) :
-    (∑ y ∈ clusterFinset G η x, encounterCount G ω η y) ≤
-      2*(markedCluster G ω η x).card := by
-  let D := clusterFinset G η x
-  let S := markedCluster G ω η x
-  have hSD : S ⊆ D := Finset.filter_subset _ _
-  have hDeq : (D:Set V) = cluster G η x := clusterFinset_coe G η x hf
-  have hDconn : ((openSimpleGraph G ω).induce (D:Set V)).Connected :=
-    hDeq ▸ cluster_induce_connected G ω η hsub x
-  have he : (∑ y ∈ D, encounterCount G ω η y) =
-      ∑ y ∈ S, branchCount (openSimpleGraph G ω) S y := by
-    calc
-      _ = ∑ y ∈ D, if IsEncounter G ω y then branchCount (openSimpleGraph G ω) S y else 0 := by
-        apply Finset.sum_congr rfl
-        intro y hy
-        have hxy : Conn G η x y := by
-          change y ∈ cluster G η x
-          rw [←hDeq]
-          exact hy
-        rw [encounterCount,←markedCluster_eq_of_conn G ω η hxy]
-      _ = _ := (Finset.sum_filter _ _).symm
-  rw [he]
-  by_cases hs : S.Nonempty
-  · exact (Nat.le_add_right _ 2).trans (sum_branchCount_le (openSimpleGraph G ω) S D hs hSD hDconn)
-  · rw [Finset.not_nonempty_iff_eq_empty.mp hs]
-    simp
-
-end Percolation
-
-namespace Percolation
-open MeasureTheory SimpleGraph
-open scoped ENNReal
-universe u v w
-variable {V : Type u} {E : Type v} {ι : Type w}
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem cluster_average_eq_sum (G : BondGraph V E) (η : Set E) (x : V)
-    (hf : (cluster G η x).Finite) (f : V → ℝ≥0∞) :
-    (∑' y, clusterAveragingKernel G η x y * f y) =
-      ((cluster G η x).ncard : ℝ≥0∞)⁻¹ * ∑ y ∈ clusterFinset G η x, f y := by
-  rw [tsum_eq_sum (s := clusterFinset G η x)]
-  · rw [Finset.mul_sum]
-    apply Finset.sum_congr rfl
-    intro y hy
-    have hc : Conn G η x y := by
-      change y ∈ cluster G η x
-      rw [←clusterFinset_coe G η x hf]
-      exact hy
-    simp only [clusterAveragingKernel,ite_eq_left hc]
-  · intro y hy
-    have hc : ¬Conn G η x y := by
-      intro h
-      apply hy
-      have hm : y ∈ cluster G η x := h
-      rwa [←clusterFinset_coe G η x hf] at hm
-    simp only [clusterAveragingKernel,ite_eq_right hc,zero_mul]
-
-theorem average_encounterCount_le (G : BondGraph V E) (ω η : Set E) (hsub : η ⊆ ω)
-    (x : V) (hf : (cluster G η x).Finite) :
-    (∑' y, clusterAveragingKernel G η x y * (encounterCount G ω η y : ℝ≥0∞)) ≤
-      2 * ∑' y, clusterAveragingKernel G η x y * (if IsEncounter G ω y then 1 else 0) := by
-  rw [cluster_average_eq_sum G η x hf,cluster_average_eq_sum G η x hf]
-  have hc : (∑ y ∈ clusterFinset G η x, (encounterCount G ω η y : ℝ≥0∞)) ≤
-      2 * (markedCluster G ω η x).card := by
-    exact_mod_cast sum_encounterCount_le G ω η hsub x hf
-  have he : (∑ y ∈ clusterFinset G η x, (if IsEncounter G ω y then (1:ℝ≥0∞) else 0)) =
-      (markedCluster G ω η x).card := by
-    rw [←Finset.sum_filter]
-    simp only [markedCluster,Finset.sum_const,nsmul_one]
-  rw [he,←mul_assoc, mul_comm (2:ℝ≥0∞),mul_assoc]
-  exact mul_le_mul_right hc _
-
-theorem expected_encounterCount_le [Countable V] [Countable E] [Fintype ι]
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
-    (root : ι → V) (weight : ι → ℝ)
-    (htr : RootTransport root weight (vertexAut G)) (hw : ∀ i, 0 ≤ weight i)
-    (p q : unitInterval) (hqp : q ≤ p)
-    (hfin : ∀ x, ∀ᵐ ω ∂law q, (cluster G ω x).Finite) :
-    (∑ i, ENNReal.ofReal (weight i) * ∫⁻ l,
-      (encounterCount G (openEdges l p) (openEdges l q) (root i) : ℝ≥0∞) ∂labelLaw) ≤
-    2 * (∑ i, ENNReal.ofReal (weight i) * ∫⁻ l,
-      (if IsEncounter G (openEdges l p) (root i) then 1 else 0) ∂labelLaw) := by
-  have ha := label_root_cluster_average G hconn hloc root weight htr hw q hfin
-    (fun l x => (encounterCount G (openEdges l p) (openEdges l q) x : ℝ≥0∞))
-    (fun x => (measurable_of_countable (fun n : ℕ => (n:ℝ≥0∞))).comp (measurable_encounterCount G p q x))
-    (fun a b hab l x => by exact_mod_cast encounterCount_aut G a b hab (openEdges l p) (openEdges l q) x)
-  have hb := label_root_cluster_average G hconn hloc root weight htr hw q hfin
-    (fun l x => if IsEncounter G (openEdges l p) x then (1:ℝ≥0∞) else 0)
-    (fun x => Measurable.ite ((measurableSet_isEncounter G x).preimage (measurable_openEdges p))
-      measurable_const measurable_const)
-    (fun a b hab l x => by simp only [show openEdges (l ∘ b) p = b ⁻¹' openEdges l p from rfl,
-      isEncounter_aut_iff G a b hab])
-  rw [←ha,←hb,Finset.mul_sum]
-  apply Finset.sum_le_sum
-  intro i _
-  rw [←mul_assoc,mul_comm (2:ℝ≥0∞),mul_assoc]
-  apply mul_le_mul_right
-  rw [←lintegral_const_mul' _ _ ENNReal.ofNat_ne_top]
-  apply lintegral_mono_ae
-  have hfl : ∀ᵐ l ∂labelLaw, (cluster G (openEdges l q) (root i)).Finite := by
-    apply ae_of_ae_map (μ := (labelLaw : Measure (E → unitInterval)))
-      (p := fun ω : Set E => (cluster G ω (root i)).Finite) (measurable_openEdges q).aemeasurable
-    rw [map_openEdges]
-    exact hfin (root i)
-  filter_upwards [hfl] with l hl
-  exact average_encounterCount_le G _ _ (openEdges_mono l hqp) (root i) hl
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Main
+import OAI.Probability.UnimodularPercolation.Encounter.Measurability
+
+namespace OAI
+
+/-! Counting encounter vertices by mass transport. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+open MeasureTheory SimpleGraph
+open scoped ENNReal
+universe u v
+variable {V : Type u} {E : Type v}
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+def cutIso {W : Type*} {H : SimpleGraph V} {K : SimpleGraph W}
+    (f : H ≃g K) (x : V) : cutVertex H x ≃g cutVertex K (f x) where
+  toEquiv := f.toEquiv
+  map_rel_iff' := by
+    intro a b
+    change (K.Adj (f a) (f b) ∧ f a ≠ f x ∧ f b ≠ f x) ↔
+      (H.Adj a b ∧ a ≠ x ∧ b ≠ x)
+    simp only [f.map_adj_iff,f.injective.ne_iff]
+
+theorem branchCount_iso {W : Type*} {H : SimpleGraph V} {K : SimpleGraph W}
+    (f : H ≃g K) (S : Finset V) (x : V) :
+    branchCount K (S.image f) (f x) = branchCount H S x := by
+  unfold branchCount
+  rw [←Finset.image_erase f.injective,Finset.image_image]
+  let e := (cutIso f x).connectedComponentEquiv
+  have he : ((S.erase x).image (fun a => (cutVertex K (f x)).connectedComponentMk (f a))) =
+      (((S.erase x).image (cutVertex H x).connectedComponentMk).image e) := by
+    rw [Finset.image_image]
+    rfl
+  change ((S.erase x).image (fun a => (cutVertex K (f x)).connectedComponentMk (f a))).card = _
+  rw [he]
+  exact Finset.card_image_iff.mpr e.injective.injOn
+
+theorem clusterFinset_aut (G : BondGraph V E) (a : V ≃ V) (b : E ≃ E)
+    (hab : IsAut G a b) (ω : Set E) (x : V) :
+    (clusterFinset G (b ⁻¹' ω) x).image a = clusterFinset G ω (a x) := by
+  have hi : (cluster G (b ⁻¹' ω) x).Infinite ↔ (cluster G ω (a x)).Infinite := by
+    rw [←cluster_aut G hab]
+    exact (Set.infinite_image_iff a.injective.injOn).symm
+  by_cases hf : (cluster G (b ⁻¹' ω) x).Finite
+  · have hf' : (cluster G ω (a x)).Finite := by
+      rw [←cluster_aut G hab]
+      exact hf.image a
+    apply Finset.coe_injective
+    simp only [Finset.coe_image,clusterFinset_coe G _ _ hf,clusterFinset_coe G _ _ hf']
+    exact cluster_aut G hab ω x
+  · have hf' : ¬(cluster G ω (a x)).Finite := hi.mp hf
+    simp only [clusterFinset,dite_eq_right hf,dite_eq_right hf',Finset.image_empty]
+
+theorem measurable_branchCount_pair [Countable V] [Countable E]
+    (G : BondGraph V E) (x : V) :
+    Measurable (fun z : Set E × Finset V => branchCount (openSimpleGraph G z.1) z.2 x) :=
+  measurable_from_prod_countable_left (fun S => measurable_branchCount G S x)
+
+noncomputable def markedCluster (G : BondGraph V E) (ω η : Set E) (x : V) : Finset V :=
+  (clusterFinset G η x).filter (IsEncounter G ω)
+
+noncomputable def encounterCount (G : BondGraph V E) (ω η : Set E) (x : V) : ℕ :=
+  if IsEncounter G ω x then branchCount (openSimpleGraph G ω) (markedCluster G ω η x) x else 0
+
+theorem markedCluster_aut (G : BondGraph V E) (a : V ≃ V) (b : E ≃ E)
+    (hab : IsAut G a b) (ω η : Set E) (x : V) :
+    (markedCluster G (b ⁻¹' ω) (b ⁻¹' η) x).image a = markedCluster G ω η (a x) := by
+  unfold markedCluster
+  rw [←clusterFinset_aut G a b hab η x,Finset.filter_image]
+  simp only [isEncounter_aut_iff G a b hab]
+
+theorem encounterCount_aut (G : BondGraph V E) (a : V ≃ V) (b : E ≃ E)
+    (hab : IsAut G a b) (ω η : Set E) (x : V) :
+    encounterCount G (b ⁻¹' ω) (b ⁻¹' η) x = encounterCount G ω η (a x) := by
+  simp only [encounterCount,isEncounter_aut_iff G a b hab]
+  congr 1
+  rw [←markedCluster_aut G a b hab]
+  exact (branchCount_iso (openAutIso G a b hab ω) _ x).symm
+
+theorem measurable_markedCluster [Countable V] [Countable E]
+    (G : BondGraph V E) (p q : unitInterval) (x : V) :
+    Measurable (fun l : E → unitInterval => markedCluster G (openEdges l p) (openEdges l q) x) :=
+  measurable_filter_finset _ ((measurable_clusterFinset G x).comp (measurable_openEdges q)) _
+    (fun y => (measurableSet_isEncounter G y).preimage (measurable_openEdges p))
+
+theorem measurable_encounterCount [Countable V] [Countable E]
+    (G : BondGraph V E) (p q : unitInterval) (x : V) :
+    Measurable (fun l : E → unitInterval => encounterCount G (openEdges l p) (openEdges l q) x) := by
+  apply Measurable.ite ((measurableSet_isEncounter G x).preimage (measurable_openEdges p)) _ measurable_const
+  exact (measurable_branchCount_pair G x).comp
+    ((measurable_openEdges p).prodMk (measurable_markedCluster G p q x))
+
+theorem cluster_induce_connected (G : BondGraph V E) (ω η : Set E) (hsub : η ⊆ ω) (x : V) :
+    ((openSimpleGraph G ω).induce (cluster G η x)).Connected := by
+  let H := openSimpleGraph G η
+  have he : cluster G η x = (H.connectedComponentMk x).supp := by
+    ext y
+    simp only [cluster,Set.mem_ofPred_eq,ConnectedComponent.mem_supp_iff,ConnectedComponent.eq]
+    exact (conn_iff_reachable G η x y).trans SimpleGraph.reachable_comm
+  rw [he]
+  apply SimpleGraph.Connected.mono (G := H.induce (H.connectedComponentMk x).supp)
+  · intro a b hab
+    exact ⟨hab.1,by obtain ⟨e,he,hGe⟩ := hab.2; exact ⟨e,hsub he,hGe⟩⟩
+  · exact (H.connectedComponentMk x).connected_toSimpleGraph
+
+theorem markedCluster_eq_of_conn (G : BondGraph V E) (ω η : Set E) {x y : V}
+    (hxy : Conn G η x y) : markedCluster G ω η x = markedCluster G ω η y := by
+  unfold markedCluster clusterFinset
+  simp only [cluster_eq_of_conn G η hxy]
+
+theorem sum_encounterCount_le (G : BondGraph V E) (ω η : Set E) (hsub : η ⊆ ω) (x : V)
+    (hf : (cluster G η x).Finite) :
+    (∑ y ∈ clusterFinset G η x, encounterCount G ω η y) ≤
+      2*(markedCluster G ω η x).card := by
+  let D := clusterFinset G η x
+  let S := markedCluster G ω η x
+  have hSD : S ⊆ D := Finset.filter_subset _ _
+  have hDeq : (D:Set V) = cluster G η x := clusterFinset_coe G η x hf
+  have hDconn : ((openSimpleGraph G ω).induce (D:Set V)).Connected :=
+    hDeq ▸ cluster_induce_connected G ω η hsub x
+  have he : (∑ y ∈ D, encounterCount G ω η y) =
+      ∑ y ∈ S, branchCount (openSimpleGraph G ω) S y := by
+    calc
+      _ = ∑ y ∈ D, if IsEncounter G ω y then branchCount (openSimpleGraph G ω) S y else 0 := by
+        apply Finset.sum_congr rfl
+        intro y hy
+        have hxy : Conn G η x y := by
+          change y ∈ cluster G η x
+          rw [←hDeq]
+          exact hy
+        rw [encounterCount,←markedCluster_eq_of_conn G ω η hxy]
+      _ = _ := (Finset.sum_filter _ _).symm
+  rw [he]
+  by_cases hs : S.Nonempty
+  · exact (Nat.le_add_right _ 2).trans (sum_branchCount_le (openSimpleGraph G ω) S D hs hSD hDconn)
+  · rw [Finset.not_nonempty_iff_eq_empty.mp hs]
+    simp
+
+end Percolation
+
+namespace Percolation
+open MeasureTheory SimpleGraph
+open scoped ENNReal
+universe u v w
+variable {V : Type u} {E : Type v} {ι : Type w}
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem cluster_average_eq_sum (G : BondGraph V E) (η : Set E) (x : V)
+    (hf : (cluster G η x).Finite) (f : V → ℝ≥0∞) :
+    (∑' y, clusterAveragingKernel G η x y * f y) =
+      ((cluster G η x).ncard : ℝ≥0∞)⁻¹ * ∑ y ∈ clusterFinset G η x, f y := by
+  rw [tsum_eq_sum (s := clusterFinset G η x)]
+  · rw [Finset.mul_sum]
+    apply Finset.sum_congr rfl
+    intro y hy
+    have hc : Conn G η x y := by
+      change y ∈ cluster G η x
+      rw [←clusterFinset_coe G η x hf]
+      exact hy
+    simp only [clusterAveragingKernel,ite_eq_left hc]
+  · intro y hy
+    have hc : ¬Conn G η x y := by
+      intro h
+      apply hy
+      have hm : y ∈ cluster G η x := h
+      rwa [←clusterFinset_coe G η x hf] at hm
+    simp only [clusterAveragingKernel,ite_eq_right hc,zero_mul]
+
+theorem average_encounterCount_le (G : BondGraph V E) (ω η : Set E) (hsub : η ⊆ ω)
+    (x : V) (hf : (cluster G η x).Finite) :
+    (∑' y, clusterAveragingKernel G η x y * (encounterCount G ω η y : ℝ≥0∞)) ≤
+      2 * ∑' y, clusterAveragingKernel G η x y * (if IsEncounter G ω y then 1 else 0) := by
+  rw [cluster_average_eq_sum G η x hf,cluster_average_eq_sum G η x hf]
+  have hc : (∑ y ∈ clusterFinset G η x, (encounterCount G ω η y : ℝ≥0∞)) ≤
+      2 * (markedCluster G ω η x).card := by
+    exact_mod_cast sum_encounterCount_le G ω η hsub x hf
+  have he : (∑ y ∈ clusterFinset G η x, (if IsEncounter G ω y then (1:ℝ≥0∞) else 0)) =
+      (markedCluster G ω η x).card := by
+    rw [←Finset.sum_filter]
+    simp only [markedCluster,Finset.sum_const,nsmul_one]
+  rw [he,←mul_assoc, mul_comm (2:ℝ≥0∞),mul_assoc]
+  exact mul_le_mul_right hc _
+
+theorem expected_encounterCount_le [Countable V] [Countable E] [Fintype ι]
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
+    (root : ι → V) (weight : ι → ℝ)
+    (htr : RootTransport root weight (vertexAut G)) (hw : ∀ i, 0 ≤ weight i)
+    (p q : unitInterval) (hqp : q ≤ p)
+    (hfin : ∀ x, ∀ᵐ ω ∂law q, (cluster G ω x).Finite) :
+    (∑ i, ENNReal.ofReal (weight i) * ∫⁻ l,
+      (encounterCount G (openEdges l p) (openEdges l q) (root i) : ℝ≥0∞) ∂labelLaw) ≤
+    2 * (∑ i, ENNReal.ofReal (weight i) * ∫⁻ l,
+      (if IsEncounter G (openEdges l p) (root i) then 1 else 0) ∂labelLaw) := by
+  have ha := label_root_cluster_average G hconn hloc root weight htr hw q hfin
+    (fun l x => (encounterCount G (openEdges l p) (openEdges l q) x : ℝ≥0∞))
+    (fun x => (measurable_of_countable (fun n : ℕ => (n:ℝ≥0∞))).comp (measurable_encounterCount G p q x))
+    (fun a b hab l x => by exact_mod_cast encounterCount_aut G a b hab (openEdges l p) (openEdges l q) x)
+  have hb := label_root_cluster_average G hconn hloc root weight htr hw q hfin
+    (fun l x => if IsEncounter G (openEdges l p) x then (1:ℝ≥0∞) else 0)
+    (fun x => Measurable.ite ((measurableSet_isEncounter G x).preimage (measurable_openEdges p))
+      measurable_const measurable_const)
+    (fun a b hab l x => by simp only [show openEdges (l ∘ b) p = b ⁻¹' openEdges l p from rfl,
+      isEncounter_aut_iff G a b hab])
+  rw [←ha,←hb,Finset.mul_sum]
+  apply Finset.sum_le_sum
+  intro i _
+  rw [←mul_assoc,mul_comm (2:ℝ≥0∞),mul_assoc]
+  apply mul_le_mul_right
+  rw [←lintegral_const_mul' _ _ ENNReal.ofNat_ne_top]
+  apply lintegral_mono_ae
+  have hfl : ∀ᵐ l ∂labelLaw, (cluster G (openEdges l q) (root i)).Finite := by
+    apply ae_of_ae_map (μ := (labelLaw : Measure (E → unitInterval)))
+      (p := fun ω : Set E => (cluster G ω (root i)).Finite) (measurable_openEdges q).aemeasurable
+    rw [map_openEdges]
+    exact hfin (root i)
+  filter_upwards [hfl] with l hl
+  exact average_encounterCount_le G _ _ (openEdges_mono l hqp) (root i) hl
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Encounter/Geometry.lean b/lean/OAI/Probability/UnimodularPercolation/Encounter/Geometry.lean
index 33685eed8abfdd5587c442b2c43cb50db02689ce..d22680c2aaf764807e9859ee28756270dada826f 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Encounter/Geometry.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Encounter/Geometry.lean
@@ -1,144 +1,144 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Main
-
-namespace OAI
-
-/-! Cut vertices and connected branches. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-open SimpleGraph
-universe u w
-variable {V : Type u} {W : Type w}
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-def cutVertex (H : SimpleGraph V) (x : V) : SimpleGraph V where
-  Adj a b := H.Adj a b ∧ a ≠ x ∧ b ≠ x
-  symm := ⟨fun _ _ h => ⟨h.1.symm,h.2.2,h.2.1⟩⟩
-  loopless := ⟨fun a h => H.loopless.irrefl a h.1⟩
-
-noncomputable def branchCount (H : SimpleGraph V) (S : Finset V) (x : V) : ℕ :=
-  ((S.erase x).image (cutVertex H x).connectedComponentMk).card
-
-theorem walk_reachable_cut (H : SimpleGraph V) (T : SimpleGraph W)
-    (f : W → V) (hf : ∀ {a b}, T.Adj a b → H.Adj (f a) (f b))
-    (x : V) {a b : W} (p : T.Walk a b)
-    (hp : ∀ z ∈ p.support, f z ≠ x) :
-    (cutVertex H x).Reachable (f a) (f b) := by
-  induction p with
-  | nil => exact .refl _
-  | @cons a b c hab p ih =>
-      apply Reachable.trans (Adj.reachable (G := cutVertex H x) ?_) (ih ?_)
-      · exact ⟨hf hab,hp a (by simp),hp b (by simp [p.start_mem_support])⟩
-      · intro z hz
-        exact hp z (by simp [hz])
-
-theorem branchCount_le_degree [Fintype W] (H : SimpleGraph V) (T : SimpleGraph W)
-    (f : W ↪ V) (hf : ∀ {a b}, T.Adj a b → H.Adj (f a) (f b))
-    (hT : T.Connected) (S : Finset V) (hS : ∀ z ∈ S, ∃ y, f y = z) (x : W) :
-    branchCount H S (f x) ≤ T.degree x := by
-  classical
-  let g : W → (cutVertex H (f x)).ConnectedComponent :=
-    fun y => (cutVertex H (f x)).connectedComponentMk (f y)
-  have hsub : (S.erase (f x)).image (cutVertex H (f x)).connectedComponentMk ⊆
-      (T.neighborFinset x).image g := by
-    intro c hc
-    obtain ⟨z,hz,rfl⟩ := Finset.mem_image.mp hc
-    obtain ⟨hyx,hzS⟩ := Finset.mem_erase.mp hz
-    obtain ⟨y,rfl⟩ := hS z hzS
-    obtain ⟨p,hp⟩ := hT.exists_isPath x y
-    cases p with
-    | nil => exact False.elim (hyx rfl)
-    | @cons x a y hab p =>
-        refine Finset.mem_image.mpr ⟨a,(T.mem_neighborFinset x a).mpr hab,?_⟩
-        apply ConnectedComponent.sound
-        apply walk_reachable_cut H T f hf (f x) p
-        intro z hz he
-        have hzx := f.injective he
-        subst z
-        exact (List.nodup_cons.mp hp.support_nodup).1 hz
-  exact (Finset.card_le_card hsub).trans ((Finset.card_image_le).trans (by simp))
-
-theorem connected_erase_leaf (H : SimpleGraph V) (D : Finset V)
-    (T : SimpleGraph (D:Set V)) (hTH : T ≤ H.induce (D:Set V))
-    (hT : T.Connected) (z : (D:Set V)) (hz : T.degree z = 1) :
-    (H.induce ((D.erase z.val):Set V)).Connected := by
-  let f : ({z}ᶜ : Set (D:Set V)) → ((D.erase z.val):Set V) :=
-    fun y => ⟨y.val.val,Finset.mem_erase.mpr ⟨by
-      intro h
-      apply y.property
-      exact Subtype.ext h,y.val.property⟩⟩
-  have hf : Function.Surjective f := by
-    intro y
-    refine ⟨⟨⟨y.val,(Finset.mem_erase.mp y.property).2⟩,?_⟩,rfl⟩
-    intro h
-    exact (Finset.mem_erase.mp y.property).1 (congrArg Subtype.val h)
-  apply (hT.induce_compl_singleton_of_degree_eq_one hz).map (f := ⟨f,?_⟩) hf
-  intro a b hab
-  exact hTH hab
-
-theorem exists_pruned_connector (H : SimpleGraph V) (S D : Finset V)
-    (hS : S.Nonempty) (hSD : S ⊆ D) (hD : (H.induce (D:Set V)).Connected) :
-    ∃ D' : Finset V, D' ⊆ D ∧ S ⊆ D' ∧
-      ∃ T : SimpleGraph (D':Set V), T ≤ H.induce (D':Set V) ∧ T.IsTree ∧
-        ∀ z : (D':Set V), z.val ∉ S → 2 ≤ T.degree z := by
-  classical
-  induction hn : D.card using Nat.strong_induction_on generalizing D with
-  | h n ih =>
-      obtain ⟨T,hTH,hT⟩ := hD.exists_isTree_le
-      by_cases hp : ∀ z : (D:Set V), z.val ∉ S → 2 ≤ T.degree z
-      · exact ⟨D,Finset.Subset.refl _,hSD,T,hTH,hT,hp⟩
-      push Not at hp
-      obtain ⟨z,hzS,hzdeg⟩ := hp
-      obtain ⟨s,hs⟩ := hS
-      have hsz : s ≠ z.val := fun h => hzS (h ▸ hs)
-      have hnon : Nontrivial (D:Set V) := ⟨⟨⟨s,hSD hs⟩,z,fun he => hsz (congrArg Subtype.val he)⟩⟩
-      let := hnon
-      have hdeg : T.degree z = 1 := by
-        have hh := hT.connected.preconnected.minDegree_pos_of_nontrivial
-        have hl := T.minDegree_le_degree z
-        omega
-      have hDel := connected_erase_leaf H D T hTH hT.connected z hdeg
-      have hSD' : S ⊆ D.erase z.val := by
-        intro a ha
-        exact Finset.mem_erase.mpr ⟨fun he => hzS (he ▸ ha),hSD ha⟩
-      obtain ⟨D',hD'D,hSD',T',hT'H,hT',hpr⟩ :=
-        ih (D.erase z.val).card (by rw [←hn]; exact Finset.card_erase_lt_of_mem z.property)
-          (D.erase z.val) hSD' hDel rfl
-      exact ⟨D',hD'D.trans (Finset.erase_subset _ _),hSD',T',hT'H,hT',hpr⟩
-
-theorem sum_branchCount_le (H : SimpleGraph V) (S D : Finset V)
-    (hS : S.Nonempty) (hSD : S ⊆ D) (hD : (H.induce (D:Set V)).Connected) :
-    (∑ x ∈ S, branchCount H S x) + 2 ≤ 2*S.card := by
-  classical
-  obtain ⟨D',_,hSD',T,hTH,hT,hpr⟩ := exists_pruned_connector H S D hS hSD hD
-  let f : (D':Set V) ↪ V := ⟨Subtype.val,Subtype.val_injective⟩
-  have hb (z : (D':Set V)) : branchCount H S z.val ≤ T.degree z :=
-    branchCount_le_degree H T f (fun h => hTH h) hT.connected S
-      (fun y hy => ⟨⟨y,hSD' hy⟩,rfl⟩) z
-  have hsum : (∑ z : (D':Set V), if z.val ∈ S then branchCount H S z.val else 2) ≤
-      ∑ z : (D':Set V), T.degree z := by
-    apply Finset.sum_le_sum
-    intro z _
-    split_ifs with hz
-    · exact hb z
-    · exact hpr z hz
-  have hsplit : (∑ z : (D':Set V), if z.val ∈ S then branchCount H S z.val else 2) =
-      (∑ z ∈ S, branchCount H S z) + 2*(D'.card-S.card) := by
-    rw [Finset.sum_finset_coe (fun z => if z ∈ S then branchCount H S z else 2) D']
-    rw [Finset.sum_ite]
-    simp only [Finset.filter_mem_eq_inter,Finset.inter_eq_right.mpr hSD',
-      ←Finset.sdiff_eq_filter]
-    simp only [Finset.sum_const,smul_eq_mul,Finset.card_sdiff_of_subset hSD',mul_comm]
-  rw [hsplit,T.sum_degrees_eq_twice_card_edges] at hsum
-  have hcard := hT.card_edgeFinset
-  have hDcard : Fintype.card (D':Set V) = D'.card := by simp
-  rw [hDcard] at hcard
-  have hle := Finset.card_le_card hSD'
-  omega
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Main
+
+namespace OAI
+
+/-! Cut vertices and connected branches. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+open SimpleGraph
+universe u w
+variable {V : Type u} {W : Type w}
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+def cutVertex (H : SimpleGraph V) (x : V) : SimpleGraph V where
+  Adj a b := H.Adj a b ∧ a ≠ x ∧ b ≠ x
+  symm := ⟨fun _ _ h => ⟨h.1.symm,h.2.2,h.2.1⟩⟩
+  loopless := ⟨fun a h => H.loopless.irrefl a h.1⟩
+
+noncomputable def branchCount (H : SimpleGraph V) (S : Finset V) (x : V) : ℕ :=
+  ((S.erase x).image (cutVertex H x).connectedComponentMk).card
+
+theorem walk_reachable_cut (H : SimpleGraph V) (T : SimpleGraph W)
+    (f : W → V) (hf : ∀ {a b}, T.Adj a b → H.Adj (f a) (f b))
+    (x : V) {a b : W} (p : T.Walk a b)
+    (hp : ∀ z ∈ p.support, f z ≠ x) :
+    (cutVertex H x).Reachable (f a) (f b) := by
+  induction p with
+  | nil => exact .refl _
+  | @cons a b c hab p ih =>
+      apply Reachable.trans (Adj.reachable (G := cutVertex H x) ?_) (ih ?_)
+      · exact ⟨hf hab,hp a (by simp),hp b (by simp [p.start_mem_support])⟩
+      · intro z hz
+        exact hp z (by simp [hz])
+
+theorem branchCount_le_degree [Fintype W] (H : SimpleGraph V) (T : SimpleGraph W)
+    (f : W ↪ V) (hf : ∀ {a b}, T.Adj a b → H.Adj (f a) (f b))
+    (hT : T.Connected) (S : Finset V) (hS : ∀ z ∈ S, ∃ y, f y = z) (x : W) :
+    branchCount H S (f x) ≤ T.degree x := by
+  classical
+  let g : W → (cutVertex H (f x)).ConnectedComponent :=
+    fun y => (cutVertex H (f x)).connectedComponentMk (f y)
+  have hsub : (S.erase (f x)).image (cutVertex H (f x)).connectedComponentMk ⊆
+      (T.neighborFinset x).image g := by
+    intro c hc
+    obtain ⟨z,hz,rfl⟩ := Finset.mem_image.mp hc
+    obtain ⟨hyx,hzS⟩ := Finset.mem_erase.mp hz
+    obtain ⟨y,rfl⟩ := hS z hzS
+    obtain ⟨p,hp⟩ := hT.exists_isPath x y
+    cases p with
+    | nil => exact False.elim (hyx rfl)
+    | @cons x a y hab p =>
+        refine Finset.mem_image.mpr ⟨a,(T.mem_neighborFinset x a).mpr hab,?_⟩
+        apply ConnectedComponent.sound
+        apply walk_reachable_cut H T f hf (f x) p
+        intro z hz he
+        have hzx := f.injective he
+        subst z
+        exact (List.nodup_cons.mp hp.support_nodup).1 hz
+  exact (Finset.card_le_card hsub).trans ((Finset.card_image_le).trans (by simp))
+
+theorem connected_erase_leaf (H : SimpleGraph V) (D : Finset V)
+    (T : SimpleGraph (D:Set V)) (hTH : T ≤ H.induce (D:Set V))
+    (hT : T.Connected) (z : (D:Set V)) (hz : T.degree z = 1) :
+    (H.induce ((D.erase z.val):Set V)).Connected := by
+  let f : ({z}ᶜ : Set (D:Set V)) → ((D.erase z.val):Set V) :=
+    fun y => ⟨y.val.val,Finset.mem_erase.mpr ⟨by
+      intro h
+      apply y.property
+      exact Subtype.ext h,y.val.property⟩⟩
+  have hf : Function.Surjective f := by
+    intro y
+    refine ⟨⟨⟨y.val,(Finset.mem_erase.mp y.property).2⟩,?_⟩,rfl⟩
+    intro h
+    exact (Finset.mem_erase.mp y.property).1 (congrArg Subtype.val h)
+  apply (hT.induce_compl_singleton_of_degree_eq_one hz).map (f := ⟨f,?_⟩) hf
+  intro a b hab
+  exact hTH hab
+
+theorem exists_pruned_connector (H : SimpleGraph V) (S D : Finset V)
+    (hS : S.Nonempty) (hSD : S ⊆ D) (hD : (H.induce (D:Set V)).Connected) :
+    ∃ D' : Finset V, D' ⊆ D ∧ S ⊆ D' ∧
+      ∃ T : SimpleGraph (D':Set V), T ≤ H.induce (D':Set V) ∧ T.IsTree ∧
+        ∀ z : (D':Set V), z.val ∉ S → 2 ≤ T.degree z := by
+  classical
+  induction hn : D.card using Nat.strong_induction_on generalizing D with
+  | h n ih =>
+      obtain ⟨T,hTH,hT⟩ := hD.exists_isTree_le
+      by_cases hp : ∀ z : (D:Set V), z.val ∉ S → 2 ≤ T.degree z
+      · exact ⟨D,Finset.Subset.refl _,hSD,T,hTH,hT,hp⟩
+      push Not at hp
+      obtain ⟨z,hzS,hzdeg⟩ := hp
+      obtain ⟨s,hs⟩ := hS
+      have hsz : s ≠ z.val := fun h => hzS (h ▸ hs)
+      have hnon : Nontrivial (D:Set V) := ⟨⟨⟨s,hSD hs⟩,z,fun he => hsz (congrArg Subtype.val he)⟩⟩
+      let := hnon
+      have hdeg : T.degree z = 1 := by
+        have hh := hT.connected.preconnected.minDegree_pos_of_nontrivial
+        have hl := T.minDegree_le_degree z
+        omega
+      have hDel := connected_erase_leaf H D T hTH hT.connected z hdeg
+      have hSD' : S ⊆ D.erase z.val := by
+        intro a ha
+        exact Finset.mem_erase.mpr ⟨fun he => hzS (he ▸ ha),hSD ha⟩
+      obtain ⟨D',hD'D,hSD',T',hT'H,hT',hpr⟩ :=
+        ih (D.erase z.val).card (by rw [←hn]; exact Finset.card_erase_lt_of_mem z.property)
+          (D.erase z.val) hSD' hDel rfl
+      exact ⟨D',hD'D.trans (Finset.erase_subset _ _),hSD',T',hT'H,hT',hpr⟩
+
+theorem sum_branchCount_le (H : SimpleGraph V) (S D : Finset V)
+    (hS : S.Nonempty) (hSD : S ⊆ D) (hD : (H.induce (D:Set V)).Connected) :
+    (∑ x ∈ S, branchCount H S x) + 2 ≤ 2*S.card := by
+  classical
+  obtain ⟨D',_,hSD',T,hTH,hT,hpr⟩ := exists_pruned_connector H S D hS hSD hD
+  let f : (D':Set V) ↪ V := ⟨Subtype.val,Subtype.val_injective⟩
+  have hb (z : (D':Set V)) : branchCount H S z.val ≤ T.degree z :=
+    branchCount_le_degree H T f (fun h => hTH h) hT.connected S
+      (fun y hy => ⟨⟨y,hSD' hy⟩,rfl⟩) z
+  have hsum : (∑ z : (D':Set V), if z.val ∈ S then branchCount H S z.val else 2) ≤
+      ∑ z : (D':Set V), T.degree z := by
+    apply Finset.sum_le_sum
+    intro z _
+    split_ifs with hz
+    · exact hb z
+    · exact hpr z hz
+  have hsplit : (∑ z : (D':Set V), if z.val ∈ S then branchCount H S z.val else 2) =
+      (∑ z ∈ S, branchCount H S z) + 2*(D'.card-S.card) := by
+    rw [Finset.sum_finset_coe (fun z => if z ∈ S then branchCount H S z else 2) D']
+    rw [Finset.sum_ite]
+    simp only [Finset.filter_mem_eq_inter,Finset.inter_eq_right.mpr hSD',
+      ←Finset.sdiff_eq_filter]
+    simp only [Finset.sum_const,smul_eq_mul,Finset.card_sdiff_of_subset hSD',mul_comm]
+  rw [hsplit,T.sum_degrees_eq_twice_card_edges] at hsum
+  have hcard := hT.card_edgeFinset
+  have hDcard : Fintype.card (D':Set V) = D'.card := by simp
+  rw [hDcard] at hcard
+  have hle := Finset.card_le_card hSD'
+  omega
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Encounter/IntrinsicDistance.lean b/lean/OAI/Probability/UnimodularPercolation/Encounter/IntrinsicDistance.lean
index 39bf58926ce9c7df554507f7670dfc9ac30083a8..d99fec8e948a812387db543196443c8ab636b538 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Encounter/IntrinsicDistance.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Encounter/IntrinsicDistance.lean
@@ -1,208 +1,208 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Main
-import OAI.Probability.UnimodularPercolation.Encounter.Geometry
-
-namespace OAI
-
-/-! Intrinsic distance and closest vertices in percolation clusters. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-open MeasureTheory SimpleGraph
-universe u v
-variable {V : Type u} {E : Type v}
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem Along.exists_open_walk_length_le {G : BondGraph V E} {ω : Set E}
-    {x y : V} {l : List E} (h : Along G x l y) (ho : ∀ e ∈ l, e ∈ ω) :
-    ∃ w : (openSimpleGraph G ω).Walk x y, w.length ≤ l.length := by
-  induction h with
-  | nil => exact ⟨.nil,le_rfl⟩
-  | @cons x z y l e he ht ih =>
-      obtain ⟨w,hw⟩ := ih (fun a ha => ho a (by simp [ha]))
-      by_cases hxz : x = z
-      · subst z
-        exact ⟨w,hw.trans (by simp)⟩
-      · exact ⟨.cons ⟨hxz,e,ho e (by simp),he⟩ w,by change w.length + 1 ≤ l.length + 1; omega⟩
-
-theorem mem_intrinsicBall_iff (G : BondGraph V E) (ω : Set E) (x y : V) (n : ℕ) :
-    y ∈ intrinsicBall G ω x n ↔ Conn G ω x y ∧ (openSimpleGraph G ω).dist x y ≤ n := by
-  constructor
-  · rintro ⟨l,hl,hlen,ho⟩
-    obtain ⟨w,hw⟩ := hl.exists_open_walk_length_le ho
-    exact ⟨hl.conn ho,(SimpleGraph.dist_le w).trans (hw.trans hlen)⟩
-  · rintro ⟨h,hn⟩
-    obtain ⟨w,hw⟩ := ((conn_iff_reachable G ω x y).mp h).exists_walk_length_eq_dist
-    obtain ⟨l,hl,hm,ho⟩ := walk_lift_bonds G ω w
-    refine ⟨l,hl,?_,ho⟩
-    have he := congrArg List.length hm
-    have hh : l.length = (openSimpleGraph G ω).dist x y := by
-      simpa only [List.length_map,Walk.length_edges,hw] using he
-    exact hh ▸ hn
-
-theorem measurableSet_mem_intrinsicBall [Countable E] (G : BondGraph V E)
-    (x y : V) (n : ℕ) : MeasurableSet {ω | y ∈ intrinsicBall G ω x n} := by
-  have he : {ω | y ∈ intrinsicBall G ω x n} =
-      ⋃ (l : List E), {ω | Along G x l y ∧ l.length ≤ n ∧ ∀ e ∈ l, e ∈ ω} := by
-    ext ω; simp [intrinsicBall]
-  rw [he]
-  apply MeasurableSet.iUnion
-  intro l
-  by_cases hl : Along G x l y ∧ l.length ≤ n
-  · have hm : MeasurableSet {ω : Set E | ∀ e ∈ l, e ∈ ω} := by
-      simpa only [Set.ofPred_forall] using MeasurableSet.iInter (fun e =>
-        MeasurableSet.iInter (fun _ : e ∈ l => (measurableSet_setOfPred.mpr (measurable_set_mem e))))
-    simpa [hl.1,hl.2] using hm
-  · have he : {ω : Set E | Along G x l y ∧ l.length ≤ n ∧ ∀ e ∈ l, e ∈ ω} = ∅ := by
-      ext ω; simp only [Set.mem_ofPred_eq,Set.mem_empty_iff_false,iff_false]
-      intro h; exact hl ⟨h.1,h.2.1⟩
-    rw [he]; exact MeasurableSet.empty
-
-def closest (H : SimpleGraph V) (A : Set V) (x : V) : Set V :=
-  {y | y ∈ A ∧ H.Reachable x y ∧ ∀ z ∈ A, H.Reachable x z → H.dist x y ≤ H.dist x z}
-
-theorem closest_finite (G : BondGraph V E) (hloc : LocallyFinite G) (ω : Set E)
-    (A : Set V) (x : V) : (closest (openSimpleGraph G ω) A x).Finite := by
-  by_cases he : (closest (openSimpleGraph G ω) A x).Nonempty
-  · obtain ⟨y,hy⟩ := he
-    apply (finite_reachBall hloc x ((openSimpleGraph G ω).dist x y)).subset
-    intro z hz
-    apply intrinsicBall_subset_reachBall G ω x _
-    exact (mem_intrinsicBall_iff G ω x z _).mpr
-      ⟨(conn_iff_reachable G ω x z).mpr hz.2.1,hz.2.2 y hy.1 hy.2.1⟩
-  · exact (Set.not_nonempty_iff_eq_empty.mp he) ▸ Set.finite_empty
-
-theorem closest_nonempty (H : SimpleGraph V) (A : Set V) (x : V)
-    (he : ∃ y ∈ A, H.Reachable x y) : (closest H A x).Nonempty := by
-  let P : ℕ → Prop := fun n => ∃ y ∈ A, H.Reachable x y ∧ H.dist x y = n
-  have hP : ∃ n, P n := by
-    obtain ⟨y,hy,hr⟩ := he
-    exact ⟨H.dist x y,y,hy,hr,rfl⟩
-  obtain ⟨y,hy,hr,hd⟩ := Nat.find_spec hP
-  refine ⟨y,hy,hr,?_⟩
-  intro z hz hzr
-  rw [hd]
-  exact Nat.find_min' hP ⟨z,hz,hzr,rfl⟩
-
-theorem mem_closest_iff_balls (G : BondGraph V E) (ω : Set E) (A : Set V) (x y : V) :
-    y ∈ closest (openSimpleGraph G ω) A x ↔ y ∈ A ∧ Conn G ω x y ∧
-      ∀ n z, z ∈ A → z ∈ intrinsicBall G ω x n → y ∈ intrinsicBall G ω x n := by
-  constructor
-  · rintro ⟨hy,hr,hmin⟩
-    refine ⟨hy,(conn_iff_reachable G ω x y).mpr hr,?_⟩
-    intro n z hz hzb
-    obtain ⟨hzc,hzd⟩ := (mem_intrinsicBall_iff G ω x z n).mp hzb
-    exact (mem_intrinsicBall_iff G ω x y n).mpr
-      ⟨(conn_iff_reachable G ω x y).mpr hr,
-        (hmin z hz ((conn_iff_reachable G ω x z).mp hzc)).trans hzd⟩
-  · rintro ⟨hy,hc,hmin⟩
-    refine ⟨hy,(conn_iff_reachable G ω x y).mp hc,?_⟩
-    intro z hz hzr
-    have hzb := (mem_intrinsicBall_iff G ω x z ((openSimpleGraph G ω).dist x z)).mpr
-      ⟨(conn_iff_reachable G ω x z).mpr hzr,le_rfl⟩
-    exact ((mem_intrinsicBall_iff G ω x y _).mp (hmin _ z hz hzb)).2
-
-theorem measurableSet_mem_closest [Countable V] [Countable E]
-    (G : BondGraph V E) (A : Set E → Set V)
-    (hA : ∀ y, MeasurableSet {ω | y ∈ A ω}) (x y : V) :
-    MeasurableSet {ω | y ∈ closest (openSimpleGraph G ω) (A ω) x} := by
-  simp only [mem_closest_iff_balls,Set.ofPred_and,Set.ofPred_forall]
-  apply (hA y).inter
-  apply (measurableSet_conn G x y).inter
-  apply MeasurableSet.iInter
-  intro n
-  apply MeasurableSet.iInter
-  intro z
-  have he : {ω | z ∈ A ω → z ∈ intrinsicBall G ω x n → y ∈ intrinsicBall G ω x n} =
-      {ω | z ∈ A ω}ᶜ ∪ {ω | z ∈ intrinsicBall G ω x n}ᶜ ∪
-        {ω | y ∈ intrinsicBall G ω x n} := by ext ω; simp; tauto
-  rw [he]
-  exact ((hA z).compl.union (measurableSet_mem_intrinsicBall G x z n).compl).union
-    (measurableSet_mem_intrinsicBall G x y n)
-
-def openAutIso (G : BondGraph V E) (a : V ≃ V) (b : E ≃ E)
-    (h : IsAut G a b) (ω : Set E) :
-    openSimpleGraph G (b ⁻¹' ω) ≃g openSimpleGraph G ω where
-  toEquiv := a
-  map_rel_iff' := by
-    intro x y
-    constructor
-    · rintro ⟨hxy,e,he,heq⟩
-      refine ⟨fun hh => hxy (congrArg a hh),b.symm e,by simpa using he,?_⟩
-      have hh := (isAut_symm G h) e
-      rw [heq,Sym2.map_mk] at hh
-      simpa only [Equiv.symm_apply_apply] using hh
-    · rintro ⟨hxy,e,he,heq⟩
-      exact ⟨fun hh => hxy (a.injective hh),b e,he,by rw [h,heq,Sym2.map_mk]⟩
-
-theorem iso_dist {W : Type*} {H : SimpleGraph V} {K : SimpleGraph W}
-    (f : H ≃g K) (x y : V) : K.dist (f x) (f y) = H.dist x y := by
-  by_cases hr : H.Reachable x y
-  · obtain ⟨w,hw⟩ := hr.exists_walk_length_eq_dist
-    have hr' := hr.map f.toHom
-    obtain ⟨w',hw'⟩ := hr'.exists_walk_length_eq_dist
-    apply le_antisymm
-    · have hh := SimpleGraph.dist_le (w.map f.toHom)
-      rw [Walk.length_map,hw] at hh
-      exact hh
-    · have hh := SimpleGraph.dist_le (w'.map f.symm.toHom)
-      rw [Walk.length_map,hw'] at hh
-      simpa using hh
-  · have hr' : ¬K.Reachable (f x) (f y) := fun hh => hr (Iso.reachable_iff.mp hh)
-    simp [SimpleGraph.dist_eq_zero_iff_eq_or_not_reachable.mpr (Or.inr hr),
-      SimpleGraph.dist_eq_zero_iff_eq_or_not_reachable.mpr (Or.inr hr')]
-
-theorem closest_iso {W : Type*} {H : SimpleGraph V} {K : SimpleGraph W}
-    (f : H ≃g K) (A : Set W) (x y : V) :
-    y ∈ closest H (f ⁻¹' A) x ↔ f y ∈ closest K A (f x) := by
-  constructor
-  · rintro ⟨hy,hr,hmin⟩
-    refine ⟨hy,hr.map f.toHom,?_⟩
-    intro z hz hzr
-    obtain ⟨z,rfl⟩ := f.surjective z
-    simpa only [iso_dist] using hmin z hz (Iso.reachable_iff.mp hzr)
-  · rintro ⟨hy,hr,hmin⟩
-    refine ⟨hy,Iso.reachable_iff.mp hr,?_⟩
-    intro z hz hzr
-    simpa only [iso_dist] using hmin (f z) hz (hzr.map f.toHom)
-
-theorem dist_lt_of_cut_separates (H : SimpleGraph V) {x y z : V}
-    (hyr : H.Reachable y z) (hcut : ¬ (cutVertex H x).Reachable y z)
-    (hxz : x ≠ z) : H.dist y x < H.dist y z := by
-  obtain ⟨w,hw⟩ := hyr.exists_walk_length_eq_dist
-  have hmem : x ∈ w.support := by
-    by_contra hnot
-    apply hcut
-    exact walk_reachable_cut H H id (fun h => h) x w
-      (fun a ha he => hnot (he ▸ ha))
-  exact (SimpleGraph.dist_le (w.takeUntil x hmem)).trans_lt
-    (hw ▸ w.length_takeUntil_lt_length hmem hxz)
-
-theorem closest_eq_singleton_of_branch (H : SimpleGraph V) (A : Set V)
-    {x y : V} (hx : x ∈ A) (hy : H.Reachable y x)
-    (hbranch : ∀ z ∈ A, z ≠ x → ¬ (cutVertex H x).Reachable y z) :
-    closest H A y = {x} := by
-  have hstrict (z : V) (hz : z ∈ A) (hne : z ≠ x) (hr : H.Reachable y z) :
-      H.dist y x < H.dist y z :=
-    dist_lt_of_cut_separates H hr (hbranch z hz hne) hne.symm
-  ext z
-  constructor
-  · intro hz
-    have he : z = x := by
-      by_contra hn
-      exact (not_le_of_gt (hstrict z hz.1 hn hz.2.1)) (hz.2.2 x hx hy)
-    exact he
-  · intro hz
-    have he : z = x := hz
-    subst z
-    refine ⟨hx,hy,?_⟩
-    intro a ha hr
-    by_cases he : a = x
-    · subst a; exact le_rfl
-    · exact (hstrict a ha he hr).le
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Main
+import OAI.Probability.UnimodularPercolation.Encounter.Geometry
+
+namespace OAI
+
+/-! Intrinsic distance and closest vertices in percolation clusters. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+open MeasureTheory SimpleGraph
+universe u v
+variable {V : Type u} {E : Type v}
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem Along.exists_open_walk_length_le {G : BondGraph V E} {ω : Set E}
+    {x y : V} {l : List E} (h : Along G x l y) (ho : ∀ e ∈ l, e ∈ ω) :
+    ∃ w : (openSimpleGraph G ω).Walk x y, w.length ≤ l.length := by
+  induction h with
+  | nil => exact ⟨.nil,le_rfl⟩
+  | @cons x z y l e he ht ih =>
+      obtain ⟨w,hw⟩ := ih (fun a ha => ho a (by simp [ha]))
+      by_cases hxz : x = z
+      · subst z
+        exact ⟨w,hw.trans (by simp)⟩
+      · exact ⟨.cons ⟨hxz,e,ho e (by simp),he⟩ w,by change w.length + 1 ≤ l.length + 1; omega⟩
+
+theorem mem_intrinsicBall_iff (G : BondGraph V E) (ω : Set E) (x y : V) (n : ℕ) :
+    y ∈ intrinsicBall G ω x n ↔ Conn G ω x y ∧ (openSimpleGraph G ω).dist x y ≤ n := by
+  constructor
+  · rintro ⟨l,hl,hlen,ho⟩
+    obtain ⟨w,hw⟩ := hl.exists_open_walk_length_le ho
+    exact ⟨hl.conn ho,(SimpleGraph.dist_le w).trans (hw.trans hlen)⟩
+  · rintro ⟨h,hn⟩
+    obtain ⟨w,hw⟩ := ((conn_iff_reachable G ω x y).mp h).exists_walk_length_eq_dist
+    obtain ⟨l,hl,hm,ho⟩ := walk_lift_bonds G ω w
+    refine ⟨l,hl,?_,ho⟩
+    have he := congrArg List.length hm
+    have hh : l.length = (openSimpleGraph G ω).dist x y := by
+      simpa only [List.length_map,Walk.length_edges,hw] using he
+    exact hh ▸ hn
+
+theorem measurableSet_mem_intrinsicBall [Countable E] (G : BondGraph V E)
+    (x y : V) (n : ℕ) : MeasurableSet {ω | y ∈ intrinsicBall G ω x n} := by
+  have he : {ω | y ∈ intrinsicBall G ω x n} =
+      ⋃ (l : List E), {ω | Along G x l y ∧ l.length ≤ n ∧ ∀ e ∈ l, e ∈ ω} := by
+    ext ω; simp [intrinsicBall]
+  rw [he]
+  apply MeasurableSet.iUnion
+  intro l
+  by_cases hl : Along G x l y ∧ l.length ≤ n
+  · have hm : MeasurableSet {ω : Set E | ∀ e ∈ l, e ∈ ω} := by
+      simpa only [Set.ofPred_forall] using MeasurableSet.iInter (fun e =>
+        MeasurableSet.iInter (fun _ : e ∈ l => (measurableSet_setOfPred.mpr (measurable_set_mem e))))
+    simpa [hl.1,hl.2] using hm
+  · have he : {ω : Set E | Along G x l y ∧ l.length ≤ n ∧ ∀ e ∈ l, e ∈ ω} = ∅ := by
+      ext ω; simp only [Set.mem_ofPred_eq,Set.mem_empty_iff_false,iff_false]
+      intro h; exact hl ⟨h.1,h.2.1⟩
+    rw [he]; exact MeasurableSet.empty
+
+def closest (H : SimpleGraph V) (A : Set V) (x : V) : Set V :=
+  {y | y ∈ A ∧ H.Reachable x y ∧ ∀ z ∈ A, H.Reachable x z → H.dist x y ≤ H.dist x z}
+
+theorem closest_finite (G : BondGraph V E) (hloc : LocallyFinite G) (ω : Set E)
+    (A : Set V) (x : V) : (closest (openSimpleGraph G ω) A x).Finite := by
+  by_cases he : (closest (openSimpleGraph G ω) A x).Nonempty
+  · obtain ⟨y,hy⟩ := he
+    apply (finite_reachBall hloc x ((openSimpleGraph G ω).dist x y)).subset
+    intro z hz
+    apply intrinsicBall_subset_reachBall G ω x _
+    exact (mem_intrinsicBall_iff G ω x z _).mpr
+      ⟨(conn_iff_reachable G ω x z).mpr hz.2.1,hz.2.2 y hy.1 hy.2.1⟩
+  · exact (Set.not_nonempty_iff_eq_empty.mp he) ▸ Set.finite_empty
+
+theorem closest_nonempty (H : SimpleGraph V) (A : Set V) (x : V)
+    (he : ∃ y ∈ A, H.Reachable x y) : (closest H A x).Nonempty := by
+  let P : ℕ → Prop := fun n => ∃ y ∈ A, H.Reachable x y ∧ H.dist x y = n
+  have hP : ∃ n, P n := by
+    obtain ⟨y,hy,hr⟩ := he
+    exact ⟨H.dist x y,y,hy,hr,rfl⟩
+  obtain ⟨y,hy,hr,hd⟩ := Nat.find_spec hP
+  refine ⟨y,hy,hr,?_⟩
+  intro z hz hzr
+  rw [hd]
+  exact Nat.find_min' hP ⟨z,hz,hzr,rfl⟩
+
+theorem mem_closest_iff_balls (G : BondGraph V E) (ω : Set E) (A : Set V) (x y : V) :
+    y ∈ closest (openSimpleGraph G ω) A x ↔ y ∈ A ∧ Conn G ω x y ∧
+      ∀ n z, z ∈ A → z ∈ intrinsicBall G ω x n → y ∈ intrinsicBall G ω x n := by
+  constructor
+  · rintro ⟨hy,hr,hmin⟩
+    refine ⟨hy,(conn_iff_reachable G ω x y).mpr hr,?_⟩
+    intro n z hz hzb
+    obtain ⟨hzc,hzd⟩ := (mem_intrinsicBall_iff G ω x z n).mp hzb
+    exact (mem_intrinsicBall_iff G ω x y n).mpr
+      ⟨(conn_iff_reachable G ω x y).mpr hr,
+        (hmin z hz ((conn_iff_reachable G ω x z).mp hzc)).trans hzd⟩
+  · rintro ⟨hy,hc,hmin⟩
+    refine ⟨hy,(conn_iff_reachable G ω x y).mp hc,?_⟩
+    intro z hz hzr
+    have hzb := (mem_intrinsicBall_iff G ω x z ((openSimpleGraph G ω).dist x z)).mpr
+      ⟨(conn_iff_reachable G ω x z).mpr hzr,le_rfl⟩
+    exact ((mem_intrinsicBall_iff G ω x y _).mp (hmin _ z hz hzb)).2
+
+theorem measurableSet_mem_closest [Countable V] [Countable E]
+    (G : BondGraph V E) (A : Set E → Set V)
+    (hA : ∀ y, MeasurableSet {ω | y ∈ A ω}) (x y : V) :
+    MeasurableSet {ω | y ∈ closest (openSimpleGraph G ω) (A ω) x} := by
+  simp only [mem_closest_iff_balls,Set.ofPred_and,Set.ofPred_forall]
+  apply (hA y).inter
+  apply (measurableSet_conn G x y).inter
+  apply MeasurableSet.iInter
+  intro n
+  apply MeasurableSet.iInter
+  intro z
+  have he : {ω | z ∈ A ω → z ∈ intrinsicBall G ω x n → y ∈ intrinsicBall G ω x n} =
+      {ω | z ∈ A ω}ᶜ ∪ {ω | z ∈ intrinsicBall G ω x n}ᶜ ∪
+        {ω | y ∈ intrinsicBall G ω x n} := by ext ω; simp; tauto
+  rw [he]
+  exact ((hA z).compl.union (measurableSet_mem_intrinsicBall G x z n).compl).union
+    (measurableSet_mem_intrinsicBall G x y n)
+
+def openAutIso (G : BondGraph V E) (a : V ≃ V) (b : E ≃ E)
+    (h : IsAut G a b) (ω : Set E) :
+    openSimpleGraph G (b ⁻¹' ω) ≃g openSimpleGraph G ω where
+  toEquiv := a
+  map_rel_iff' := by
+    intro x y
+    constructor
+    · rintro ⟨hxy,e,he,heq⟩
+      refine ⟨fun hh => hxy (congrArg a hh),b.symm e,by simpa using he,?_⟩
+      have hh := (isAut_symm G h) e
+      rw [heq,Sym2.map_mk] at hh
+      simpa only [Equiv.symm_apply_apply] using hh
+    · rintro ⟨hxy,e,he,heq⟩
+      exact ⟨fun hh => hxy (a.injective hh),b e,he,by rw [h,heq,Sym2.map_mk]⟩
+
+theorem iso_dist {W : Type*} {H : SimpleGraph V} {K : SimpleGraph W}
+    (f : H ≃g K) (x y : V) : K.dist (f x) (f y) = H.dist x y := by
+  by_cases hr : H.Reachable x y
+  · obtain ⟨w,hw⟩ := hr.exists_walk_length_eq_dist
+    have hr' := hr.map f.toHom
+    obtain ⟨w',hw'⟩ := hr'.exists_walk_length_eq_dist
+    apply le_antisymm
+    · have hh := SimpleGraph.dist_le (w.map f.toHom)
+      rw [Walk.length_map,hw] at hh
+      exact hh
+    · have hh := SimpleGraph.dist_le (w'.map f.symm.toHom)
+      rw [Walk.length_map,hw'] at hh
+      simpa using hh
+  · have hr' : ¬K.Reachable (f x) (f y) := fun hh => hr (Iso.reachable_iff.mp hh)
+    simp [SimpleGraph.dist_eq_zero_iff_eq_or_not_reachable.mpr (Or.inr hr),
+      SimpleGraph.dist_eq_zero_iff_eq_or_not_reachable.mpr (Or.inr hr')]
+
+theorem closest_iso {W : Type*} {H : SimpleGraph V} {K : SimpleGraph W}
+    (f : H ≃g K) (A : Set W) (x y : V) :
+    y ∈ closest H (f ⁻¹' A) x ↔ f y ∈ closest K A (f x) := by
+  constructor
+  · rintro ⟨hy,hr,hmin⟩
+    refine ⟨hy,hr.map f.toHom,?_⟩
+    intro z hz hzr
+    obtain ⟨z,rfl⟩ := f.surjective z
+    simpa only [iso_dist] using hmin z hz (Iso.reachable_iff.mp hzr)
+  · rintro ⟨hy,hr,hmin⟩
+    refine ⟨hy,Iso.reachable_iff.mp hr,?_⟩
+    intro z hz hzr
+    simpa only [iso_dist] using hmin (f z) hz (hzr.map f.toHom)
+
+theorem dist_lt_of_cut_separates (H : SimpleGraph V) {x y z : V}
+    (hyr : H.Reachable y z) (hcut : ¬ (cutVertex H x).Reachable y z)
+    (hxz : x ≠ z) : H.dist y x < H.dist y z := by
+  obtain ⟨w,hw⟩ := hyr.exists_walk_length_eq_dist
+  have hmem : x ∈ w.support := by
+    by_contra hnot
+    apply hcut
+    exact walk_reachable_cut H H id (fun h => h) x w
+      (fun a ha he => hnot (he ▸ ha))
+  exact (SimpleGraph.dist_le (w.takeUntil x hmem)).trans_lt
+    (hw ▸ w.length_takeUntil_lt_length hmem hxz)
+
+theorem closest_eq_singleton_of_branch (H : SimpleGraph V) (A : Set V)
+    {x y : V} (hx : x ∈ A) (hy : H.Reachable y x)
+    (hbranch : ∀ z ∈ A, z ≠ x → ¬ (cutVertex H x).Reachable y z) :
+    closest H A y = {x} := by
+  have hstrict (z : V) (hz : z ∈ A) (hne : z ≠ x) (hr : H.Reachable y z) :
+      H.dist y x < H.dist y z :=
+    dist_lt_of_cut_separates H hr (hbranch z hz hne) hne.symm
+  ext z
+  constructor
+  · intro hz
+    have he : z = x := by
+      by_contra hn
+      exact (not_le_of_gt (hstrict z hz.1 hn hz.2.1)) (hz.2.2 x hx hy)
+    exact he
+  · intro hz
+    have he : z = x := hz
+    subst z
+    refine ⟨hx,hy,?_⟩
+    intro a ha hr
+    by_cases he : a = x
+    · subst a; exact le_rfl
+    · exact (hstrict a ha he hr).le
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Encounter/Measurability.lean b/lean/OAI/Probability/UnimodularPercolation/Encounter/Measurability.lean
index 87be26baf5b8776900b0079974dd19f376bbf9f5..75c395c24ce89b3273c41ea331a2e71527dd3db9 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Encounter/Measurability.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Encounter/Measurability.lean
@@ -1,123 +1,123 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Main
-import OAI.Probability.UnimodularPercolation.Encounter.Vertices
-
-namespace OAI
-
-/-! Measurability of branches and finite clusters. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-open MeasureTheory SimpleGraph
-universe u v
-variable {V : Type u} {E : Type v}
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem card_image_eq_minima [Encodable V] {W : Type*} (S : Finset V) (f : V → W) :
-    (S.image f).card =
-      (S.filter (fun y => ∀ z ∈ S, f z = f y → Encodable.encode y ≤ Encodable.encode z)).card := by
-  let R := S.filter (fun y => ∀ z ∈ S, f z = f y → Encodable.encode y ≤ Encodable.encode z)
-  have hRS : R ⊆ S := Finset.filter_subset _ _
-  have hinj : Set.InjOn f (R:Set V) := by
-    intro y hy z hz he
-    apply Encodable.encode_injective
-    exact le_antisymm ((Finset.mem_filter.mp hy).2 z (hRS hz) he.symm)
-      ((Finset.mem_filter.mp hz).2 y (hRS hy) he)
-  have he : R.image f = S.image f := by
-    apply Finset.Subset.antisymm (Finset.image_subset_image hRS)
-    intro c hc
-    obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hc
-    obtain ⟨z,hz,hmin⟩ := (S.filter (fun z => f z = f y)).exists_min_image
-      Encodable.encode ⟨y,Finset.mem_filter.mpr ⟨hy,rfl⟩⟩
-    refine Finset.mem_image.mpr ⟨z,Finset.mem_filter.mpr ⟨(Finset.mem_filter.mp hz).1,?_⟩,
-      (Finset.mem_filter.mp hz).2⟩
-    intro w hw hwe
-    exact hmin w (Finset.mem_filter.mpr ⟨hw,hwe.trans (Finset.mem_filter.mp hz).2⟩)
-  rw [←he]
-  exact Finset.card_image_iff.mpr hinj
-
-theorem measurable_branchCount [Countable V] [Countable E]
-    (G : BondGraph V E) (S : Finset V) (x : V) :
-    Measurable (fun ω => branchCount (openSimpleGraph G ω) S x) := by
-  let : Encodable V := Encodable.ofCountable V
-  have he (ω : Set E) : branchCount (openSimpleGraph G ω) S x =
-      ∑ y ∈ S.erase x, if (∀ z ∈ S.erase x, Conn G (cutConfig G ω x) z y →
-        Encodable.encode y ≤ Encodable.encode z) then 1 else 0 := by
-    rw [branchCount,card_image_eq_minima,Finset.card_filter]
-    apply Finset.sum_congr rfl
-    intro y _
-    simp only [ConnectedComponent.eq,cutVertex_open,←conn_iff_reachable]
-  simp_rw [he]
-  apply Finset.measurable_sum
-  intro y hy
-  apply Measurable.ite _ measurable_const measurable_const
-  have he' : {ω : Set E | ∀ z ∈ S.erase x, Conn G (cutConfig G ω x) z y →
-      Encodable.encode y ≤ Encodable.encode z} =
-      ⋂ z ∈ S.erase x, ({ω | Conn G (cutConfig G ω x) z y}ᶜ ∪
-        {ω : Set E | Encodable.encode y ≤ Encodable.encode z}) := by ext ω; simp [imp_iff_not_or]
-  rw [he']
-  apply MeasurableSet.iInter
-  intro z
-  apply MeasurableSet.iInter
-  intro _
-  exact ((measurableSet_conn G z y).preimage (measurable_cutConfig G x)).compl.union
-    (by by_cases hh : Encodable.encode y ≤ Encodable.encode z <;> simp [hh])
-
-noncomputable def clusterFinset (G : BondGraph V E) (ω : Set E) (x : V) : Finset V :=
-  if h : (cluster G ω x).Finite then h.toFinset else ∅
-
-theorem clusterFinset_coe (G : BondGraph V E) (ω : Set E) (x : V)
-    (h : (cluster G ω x).Finite) : (clusterFinset G ω x : Set V) = cluster G ω x := by
-  simp [clusterFinset,h]
-
-theorem measurable_clusterFinset [Countable V] [Countable E]
-    (G : BondGraph V E) (x : V) : Measurable (fun ω => clusterFinset G ω x) := by
-  apply measurable_to_countable
-  intro ω₀
-  let S := clusterFinset G ω₀ x
-  have he : {ω | clusterFinset G ω x = S} =
-      {ω | cluster G ω x = (S:Set V)} ∪
-        ({ω | (cluster G ω x).Infinite} ∩ {ω : Set E | S = ∅}) := by
-    ext ω
-    by_cases h : (cluster G ω x).Finite
-    · simp only [Set.mem_ofPred,Set.mem_union,Set.mem_inter_iff,clusterFinset,dite_eq_left h]
-      simp only [Set.Infinite,h,not_true_eq_false,false_and,or_false]
-      constructor
-      · intro he; rw [←he,h.coe_toFinset]
-      · intro he; exact Finset.coe_injective (h.coe_toFinset.trans he)
-    · simp only [Set.mem_ofPred,Set.mem_union,Set.mem_inter_iff,clusterFinset,dite_eq_right h]
-      have hn : cluster G ω x ≠ (S:Set V) := fun he => h (he ▸ S.finite_toSet)
-      simp [hn,Set.Infinite,h,eq_comm]
-  change MeasurableSet {ω | clusterFinset G ω x = S}
-  rw [he]
-  exact (measurableSet_cluster_eq G x (S:Set V)).union
-    ((measurableSet_infinite_cluster G x).inter (by by_cases hs : S = ∅ <;> simp [hs]))
-
-theorem measurable_filter_finset {Ω : Type*} [MeasurableSpace Ω] [Countable V]
-    (S : Ω → Finset V) (hS : Measurable S) (P : Ω → V → Prop)
-    (hP : ∀ x, MeasurableSet {ω | P ω x}) :
-    Measurable (fun ω => (S ω).filter (P ω)) := by
-  have hf (D : Finset V) : Measurable (fun ω => D.filter (P ω)) := by
-    apply measurable_to_countable
-    intro ω₀
-    have he : {ω | D.filter (P ω) = D.filter (P ω₀)} =
-        ⋂ y, {ω | (y ∈ D ∧ P ω y) ↔ y ∈ D.filter (P ω₀)} := by
-      ext ω; simp only [Set.mem_ofPred_eq,Set.mem_iInter,Finset.ext_iff,Finset.mem_filter]
-    change MeasurableSet {ω | D.filter (P ω) = D.filter (P ω₀)}
-    rw [he]
-    apply MeasurableSet.iInter
-    intro y
-    by_cases hy : y ∈ D
-    · by_cases hp : P ω₀ y <;> simp only [hy,hp,Finset.mem_filter,true_and,iff_true,iff_false]
-      · exact hP y
-      · exact (hP y).compl
-    · simp [hy]
-  have hm : Measurable (fun z : Ω × Finset V => z.2.filter (P z.1)) :=
-    measurable_from_prod_countable_left hf
-  exact hm.comp (measurable_id.prodMk hS)
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Main
+import OAI.Probability.UnimodularPercolation.Encounter.Vertices
+
+namespace OAI
+
+/-! Measurability of branches and finite clusters. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+open MeasureTheory SimpleGraph
+universe u v
+variable {V : Type u} {E : Type v}
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem card_image_eq_minima [Encodable V] {W : Type*} (S : Finset V) (f : V → W) :
+    (S.image f).card =
+      (S.filter (fun y => ∀ z ∈ S, f z = f y → Encodable.encode y ≤ Encodable.encode z)).card := by
+  let R := S.filter (fun y => ∀ z ∈ S, f z = f y → Encodable.encode y ≤ Encodable.encode z)
+  have hRS : R ⊆ S := Finset.filter_subset _ _
+  have hinj : Set.InjOn f (R:Set V) := by
+    intro y hy z hz he
+    apply Encodable.encode_injective
+    exact le_antisymm ((Finset.mem_filter.mp hy).2 z (hRS hz) he.symm)
+      ((Finset.mem_filter.mp hz).2 y (hRS hy) he)
+  have he : R.image f = S.image f := by
+    apply Finset.Subset.antisymm (Finset.image_subset_image hRS)
+    intro c hc
+    obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hc
+    obtain ⟨z,hz,hmin⟩ := (S.filter (fun z => f z = f y)).exists_min_image
+      Encodable.encode ⟨y,Finset.mem_filter.mpr ⟨hy,rfl⟩⟩
+    refine Finset.mem_image.mpr ⟨z,Finset.mem_filter.mpr ⟨(Finset.mem_filter.mp hz).1,?_⟩,
+      (Finset.mem_filter.mp hz).2⟩
+    intro w hw hwe
+    exact hmin w (Finset.mem_filter.mpr ⟨hw,hwe.trans (Finset.mem_filter.mp hz).2⟩)
+  rw [←he]
+  exact Finset.card_image_iff.mpr hinj
+
+theorem measurable_branchCount [Countable V] [Countable E]
+    (G : BondGraph V E) (S : Finset V) (x : V) :
+    Measurable (fun ω => branchCount (openSimpleGraph G ω) S x) := by
+  let : Encodable V := Encodable.ofCountable V
+  have he (ω : Set E) : branchCount (openSimpleGraph G ω) S x =
+      ∑ y ∈ S.erase x, if (∀ z ∈ S.erase x, Conn G (cutConfig G ω x) z y →
+        Encodable.encode y ≤ Encodable.encode z) then 1 else 0 := by
+    rw [branchCount,card_image_eq_minima,Finset.card_filter]
+    apply Finset.sum_congr rfl
+    intro y _
+    simp only [ConnectedComponent.eq,cutVertex_open,←conn_iff_reachable]
+  simp_rw [he]
+  apply Finset.measurable_sum
+  intro y hy
+  apply Measurable.ite _ measurable_const measurable_const
+  have he' : {ω : Set E | ∀ z ∈ S.erase x, Conn G (cutConfig G ω x) z y →
+      Encodable.encode y ≤ Encodable.encode z} =
+      ⋂ z ∈ S.erase x, ({ω | Conn G (cutConfig G ω x) z y}ᶜ ∪
+        {ω : Set E | Encodable.encode y ≤ Encodable.encode z}) := by ext ω; simp [imp_iff_not_or]
+  rw [he']
+  apply MeasurableSet.iInter
+  intro z
+  apply MeasurableSet.iInter
+  intro _
+  exact ((measurableSet_conn G z y).preimage (measurable_cutConfig G x)).compl.union
+    (by by_cases hh : Encodable.encode y ≤ Encodable.encode z <;> simp [hh])
+
+noncomputable def clusterFinset (G : BondGraph V E) (ω : Set E) (x : V) : Finset V :=
+  if h : (cluster G ω x).Finite then h.toFinset else ∅
+
+theorem clusterFinset_coe (G : BondGraph V E) (ω : Set E) (x : V)
+    (h : (cluster G ω x).Finite) : (clusterFinset G ω x : Set V) = cluster G ω x := by
+  simp [clusterFinset,h]
+
+theorem measurable_clusterFinset [Countable V] [Countable E]
+    (G : BondGraph V E) (x : V) : Measurable (fun ω => clusterFinset G ω x) := by
+  apply measurable_to_countable
+  intro ω₀
+  let S := clusterFinset G ω₀ x
+  have he : {ω | clusterFinset G ω x = S} =
+      {ω | cluster G ω x = (S:Set V)} ∪
+        ({ω | (cluster G ω x).Infinite} ∩ {ω : Set E | S = ∅}) := by
+    ext ω
+    by_cases h : (cluster G ω x).Finite
+    · simp only [Set.mem_ofPred,Set.mem_union,Set.mem_inter_iff,clusterFinset,dite_eq_left h]
+      simp only [Set.Infinite,h,not_true_eq_false,false_and,or_false]
+      constructor
+      · intro he; rw [←he,h.coe_toFinset]
+      · intro he; exact Finset.coe_injective (h.coe_toFinset.trans he)
+    · simp only [Set.mem_ofPred,Set.mem_union,Set.mem_inter_iff,clusterFinset,dite_eq_right h]
+      have hn : cluster G ω x ≠ (S:Set V) := fun he => h (he ▸ S.finite_toSet)
+      simp [hn,Set.Infinite,h,eq_comm]
+  change MeasurableSet {ω | clusterFinset G ω x = S}
+  rw [he]
+  exact (measurableSet_cluster_eq G x (S:Set V)).union
+    ((measurableSet_infinite_cluster G x).inter (by by_cases hs : S = ∅ <;> simp [hs]))
+
+theorem measurable_filter_finset {Ω : Type*} [MeasurableSpace Ω] [Countable V]
+    (S : Ω → Finset V) (hS : Measurable S) (P : Ω → V → Prop)
+    (hP : ∀ x, MeasurableSet {ω | P ω x}) :
+    Measurable (fun ω => (S ω).filter (P ω)) := by
+  have hf (D : Finset V) : Measurable (fun ω => D.filter (P ω)) := by
+    apply measurable_to_countable
+    intro ω₀
+    have he : {ω | D.filter (P ω) = D.filter (P ω₀)} =
+        ⋂ y, {ω | (y ∈ D ∧ P ω y) ↔ y ∈ D.filter (P ω₀)} := by
+      ext ω; simp only [Set.mem_ofPred_eq,Set.mem_iInter,Finset.ext_iff,Finset.mem_filter]
+    change MeasurableSet {ω | D.filter (P ω) = D.filter (P ω₀)}
+    rw [he]
+    apply MeasurableSet.iInter
+    intro y
+    by_cases hy : y ∈ D
+    · by_cases hp : P ω₀ y <;> simp only [hy,hp,Finset.mem_filter,true_and,iff_true,iff_false]
+      · exact hP y
+      · exact (hP y).compl
+    · simp [hy]
+  have hm : Measurable (fun z : Ω × Finset V => z.2.filter (P z.1)) :=
+    measurable_from_prod_countable_left hf
+  exact hm.comp (measurable_id.prodMk hS)
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Encounter/NearestMass.lean b/lean/OAI/Probability/UnimodularPercolation/Encounter/NearestMass.lean
index 6028555f4369b3dd11c42d0c4f6706095347c744..5c86ae45579c00023d85c4a474bbbe46e5cf61e8 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Encounter/NearestMass.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Encounter/NearestMass.lean
@@ -1,88 +1,88 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Main
-import OAI.Probability.UnimodularPercolation.Encounter.IntrinsicDistance
-
-namespace OAI
-
-/-! Nearest-vertex transport and its finite incoming mass. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-open MeasureTheory SimpleGraph
-open scoped ENNReal
-universe u v w
-variable {V : Type u} {E : Type v} {ι : Type w}
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def nearestMass (G : BondGraph V E) (A : Set E → Set V)
-    (ω : Set E) (x y : V) : ℝ≥0∞ :=
-  if y ∈ closest (openSimpleGraph G ω) (A ω) x then
-    ((closest (openSimpleGraph G ω) (A ω) x).ncard : ℝ≥0∞)⁻¹ else 0
-
-theorem measurable_nearestMass [Countable V] [Countable E]
-    (G : BondGraph V E) (A : Set E → Set V)
-    (hA : ∀ y, MeasurableSet {ω | y ∈ A ω}) (x y : V) :
-    Measurable (fun ω => nearestMass G A ω x y) := by
-  have hm : Measurable (fun ω => closest (openSimpleGraph G ω) (A ω) x) := by
-    apply measurable_setOfPred.comp
-    apply Measurable.of_eval
-    intro z
-    exact measurableSet_setOfPred.mp (measurableSet_mem_closest G A hA x z)
-  exact Measurable.ite (measurableSet_mem_closest G A hA x y)
-    (((measurable_of_countable (fun n : ℕ => (n:ℝ≥0∞))).comp (measurable_ncard.comp hm)).inv) measurable_const
-
-theorem nearestMass_row_le (G : BondGraph V E) (hloc : LocallyFinite G)
-    (A : Set E → Set V) (ω : Set E) (x : V) :
-    ∑' y, nearestMass G A ω x y ≤ 1 := by
-  let S := closest (openSimpleGraph G ω) (A ω) x
-  have hf := closest_finite G hloc ω (A ω) x
-  rw [tsum_eq_sum (s := hf.toFinset) (fun y hy => by
-    simp only [nearestMass,ite_eq_right (fun hh => hy (hf.mem_toFinset.mpr hh))])]
-  simp only [nearestMass]
-  have he : (∑ y ∈ hf.toFinset, if y ∈ S then (S.ncard : ℝ≥0∞)⁻¹ else 0) =
-      (S.ncard : ℝ≥0∞) * (S.ncard : ℝ≥0∞)⁻¹ := by
-    calc
-      _ = ∑ _y ∈ hf.toFinset, (S.ncard : ℝ≥0∞)⁻¹ := by
-        apply Finset.sum_congr rfl
-        intro y hy
-        rw [ite_eq_left (hf.mem_toFinset.mp hy)]
-      _ = _ := by simp only [Finset.sum_const,nsmul_eq_mul,←Set.ncard_eq_toFinset_card _ hf,S]
-  change (∑ y ∈ hf.toFinset, if y ∈ S then (S.ncard : ℝ≥0∞)⁻¹ else 0) ≤ 1
-  rw [he]
-  exact ENNReal.mul_inv_le_one _
-
-theorem nearestMass_aut (G : BondGraph V E) (A : Set E → Set V)
-    (hA : ∀ a : V ≃ V, ∀ b : E ≃ E, IsAut G a b → ∀ ω x,
-      x ∈ A (b ⁻¹' ω) ↔ a x ∈ A ω)
-    (a : V ≃ V) (b : E ≃ E) (hab : IsAut G a b) (ω : Set E) (x y : V) :
-    nearestMass G A (b ⁻¹' ω) x y = nearestMass G A ω (a x) (a y) := by
-  have heA : A (b ⁻¹' ω) = a ⁻¹' A ω := by ext z; exact hA a b hab ω z
-  have heS : closest (openSimpleGraph G (b ⁻¹' ω)) (A (b ⁻¹' ω)) x =
-      a ⁻¹' closest (openSimpleGraph G ω) (A ω) (a x) := by
-    rw [heA]
-    ext z
-    exact closest_iso (openAutIso G a b hab ω) (A ω) x z
-  simp only [nearestMass,heS,Set.mem_preimage]
-  rw [Set.ncard_preimage_of_injective_subset_range a.injective (by simp)]
-
-theorem nearestMass_eq_one (G : BondGraph V E) (A : Set E → Set V)
-    (ω : Set E) (x y : V)
-    (h : closest (openSimpleGraph G ω) (A ω) x = {y}) :
-    nearestMass G A ω x y = 1 := by simp [nearestMass,h]
-
-theorem incoming_top_of_infinite (m : V → ℝ≥0∞) (B : Set V)
-    (hB : B.Infinite) (hm : ∀ x ∈ B, m x = 1) : ∑' x, m x = ⊤ := by
-  let : Infinite B := hB.to_subtype
-  apply top_unique
-  calc
-    ⊤ = ∑' x : B, (1:ℝ≥0∞) := (ENNReal.tsum_const_eq_top_of_ne_zero one_ne_zero).symm
-    _ = ∑' x : B, m x.val := tsum_congr (fun x => (hm x.val x.property).symm)
-    _ = ∑' x, B.indicator m x := tsum_subtype B m
-    _ ≤ ∑' x, m x := ENNReal.tsum_le_tsum (fun x => by
-      by_cases hx : x ∈ B <;> simp [Set.indicator,hx])
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Main
+import OAI.Probability.UnimodularPercolation.Encounter.IntrinsicDistance
+
+namespace OAI
+
+/-! Nearest-vertex transport and its finite incoming mass. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+open MeasureTheory SimpleGraph
+open scoped ENNReal
+universe u v w
+variable {V : Type u} {E : Type v} {ι : Type w}
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def nearestMass (G : BondGraph V E) (A : Set E → Set V)
+    (ω : Set E) (x y : V) : ℝ≥0∞ :=
+  if y ∈ closest (openSimpleGraph G ω) (A ω) x then
+    ((closest (openSimpleGraph G ω) (A ω) x).ncard : ℝ≥0∞)⁻¹ else 0
+
+theorem measurable_nearestMass [Countable V] [Countable E]
+    (G : BondGraph V E) (A : Set E → Set V)
+    (hA : ∀ y, MeasurableSet {ω | y ∈ A ω}) (x y : V) :
+    Measurable (fun ω => nearestMass G A ω x y) := by
+  have hm : Measurable (fun ω => closest (openSimpleGraph G ω) (A ω) x) := by
+    apply measurable_setOfPred.comp
+    apply Measurable.of_eval
+    intro z
+    exact measurableSet_setOfPred.mp (measurableSet_mem_closest G A hA x z)
+  exact Measurable.ite (measurableSet_mem_closest G A hA x y)
+    (((measurable_of_countable (fun n : ℕ => (n:ℝ≥0∞))).comp (measurable_ncard.comp hm)).inv) measurable_const
+
+theorem nearestMass_row_le (G : BondGraph V E) (hloc : LocallyFinite G)
+    (A : Set E → Set V) (ω : Set E) (x : V) :
+    ∑' y, nearestMass G A ω x y ≤ 1 := by
+  let S := closest (openSimpleGraph G ω) (A ω) x
+  have hf := closest_finite G hloc ω (A ω) x
+  rw [tsum_eq_sum (s := hf.toFinset) (fun y hy => by
+    simp only [nearestMass,ite_eq_right (fun hh => hy (hf.mem_toFinset.mpr hh))])]
+  simp only [nearestMass]
+  have he : (∑ y ∈ hf.toFinset, if y ∈ S then (S.ncard : ℝ≥0∞)⁻¹ else 0) =
+      (S.ncard : ℝ≥0∞) * (S.ncard : ℝ≥0∞)⁻¹ := by
+    calc
+      _ = ∑ _y ∈ hf.toFinset, (S.ncard : ℝ≥0∞)⁻¹ := by
+        apply Finset.sum_congr rfl
+        intro y hy
+        rw [ite_eq_left (hf.mem_toFinset.mp hy)]
+      _ = _ := by simp only [Finset.sum_const,nsmul_eq_mul,←Set.ncard_eq_toFinset_card _ hf,S]
+  change (∑ y ∈ hf.toFinset, if y ∈ S then (S.ncard : ℝ≥0∞)⁻¹ else 0) ≤ 1
+  rw [he]
+  exact ENNReal.mul_inv_le_one _
+
+theorem nearestMass_aut (G : BondGraph V E) (A : Set E → Set V)
+    (hA : ∀ a : V ≃ V, ∀ b : E ≃ E, IsAut G a b → ∀ ω x,
+      x ∈ A (b ⁻¹' ω) ↔ a x ∈ A ω)
+    (a : V ≃ V) (b : E ≃ E) (hab : IsAut G a b) (ω : Set E) (x y : V) :
+    nearestMass G A (b ⁻¹' ω) x y = nearestMass G A ω (a x) (a y) := by
+  have heA : A (b ⁻¹' ω) = a ⁻¹' A ω := by ext z; exact hA a b hab ω z
+  have heS : closest (openSimpleGraph G (b ⁻¹' ω)) (A (b ⁻¹' ω)) x =
+      a ⁻¹' closest (openSimpleGraph G ω) (A ω) (a x) := by
+    rw [heA]
+    ext z
+    exact closest_iso (openAutIso G a b hab ω) (A ω) x z
+  simp only [nearestMass,heS,Set.mem_preimage]
+  rw [Set.ncard_preimage_of_injective_subset_range a.injective (by simp)]
+
+theorem nearestMass_eq_one (G : BondGraph V E) (A : Set E → Set V)
+    (ω : Set E) (x y : V)
+    (h : closest (openSimpleGraph G ω) (A ω) x = {y}) :
+    nearestMass G A ω x y = 1 := by simp [nearestMass,h]
+
+theorem incoming_top_of_infinite (m : V → ℝ≥0∞) (B : Set V)
+    (hB : B.Infinite) (hm : ∀ x ∈ B, m x = 1) : ∑' x, m x = ⊤ := by
+  let : Infinite B := hB.to_subtype
+  apply top_unique
+  calc
+    ⊤ = ∑' x : B, (1:ℝ≥0∞) := (ENNReal.tsum_const_eq_top_of_ne_zero one_ne_zero).symm
+    _ = ∑' x : B, m x.val := tsum_congr (fun x => (hm x.val x.property).symm)
+    _ = ∑' x, B.indicator m x := tsum_subtype B m
+    _ ≤ ∑' x, m x := ENNReal.tsum_le_tsum (fun x => by
+      by_cases hx : x ∈ B <;> simp [Set.indicator,hx])
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Encounter/Surgery.lean b/lean/OAI/Probability/UnimodularPercolation/Encounter/Surgery.lean
index 07321b98a53989004d6d2a7f6f69f45eefbc50c0..c1871864974a8709b6ce9b59802e5207083ed04f 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Encounter/Surgery.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Encounter/Surgery.lean
@@ -1,231 +1,231 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Main
-import OAI.Probability.UnimodularPercolation.Criticality.FiniteChanges
-import OAI.Probability.UnimodularPercolation.Encounter.Counts
-import OAI.Probability.UnimodularPercolation.Encounter.TreeBranches
-
-namespace OAI
-
-/-! Exterior branches and finite encounter surgery. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-open MeasureTheory SimpleGraph
-universe u v
-variable {V : Type u} {E : Type v}
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem infinite_cluster_sdiff_contains (G : BondGraph V E) (ω : Set E)
-    (F : Set E) (hF : F.Finite) (x : V) (hx : (cluster G ω x).Infinite) :
-    ∃ z, Conn G ω x z ∧ (cluster G (ω \ F) z).Infinite := by
-  have he : (ω \ F) ∪ (ω ∩ F) = ω := by ext e; simp only [Set.mem_union,Set.mem_sdiff,Set.mem_inter_iff]; tauto
-  obtain ⟨z,hz,hzi⟩ := infinite_cluster_union_contains G (ω \ F) (ω ∩ F) (hF.inter_of_right ω) x (by rw [he]; exact hx)
-  exact ⟨z,by simpa only [he] using hz,hzi⟩
-
-theorem conn_eq_of_no_incident (G : BondGraph V E) (ω : Set E) (x y : V)
-    (hx : ∀ e ∈ ω, x ∉ G e) (hxy : Conn G ω x y) : y = x := by
-  induction hxy with
-  | refl => rfl
-  | @tail y z hy hed ih =>
-    obtain ⟨e,he,hGe⟩ := hed
-    exact False.elim (hx e he (by rw [hGe,ih]; simp))
-
-theorem cluster_singleton_of_no_incident (G : BondGraph V E) (ω : Set E) (x : V)
-    (hx : ∀ e ∈ ω, x ∉ G e) : cluster G ω x = {x} := by
-  ext y
-  constructor
-  · intro h; exact conn_eq_of_no_incident G ω x y hx h
-  · rintro rfl; exact mem_cluster_self G ω _
-
-theorem deleted_cluster_singleton (G : BondGraph V E) (hloc : LocallyFinite G)
-    (ω : Set E) (A : Finset V) (x : V) (hx : x ∈ A) :
-    cluster G (ω \ (incidentSupport G hloc A:Set E)) x = {x} := by
-  apply cluster_singleton_of_no_incident
-  intro e he hxe
-  exact he.2 ((mem_incidentSupport G hloc A e).mpr ⟨x,hx,hxe⟩)
-
-theorem conn_exterior_attachment (G : BondGraph V E) (hloc : LocallyFinite G)
-    (ω : Set E) (A : Finset V) (x : V) (hx : x ∈ A) {z : V} (hxz : Conn G ω x z) :
-    z ∈ A ∨ ∃ v ∈ A, ∃ w, w ∉ A ∧ ∃ e ∈ ω, G e = s(v,w) ∧
-      Conn G (ω \ (incidentSupport G hloc A:Set E)) w z := by
-  induction hxz with
-  | refl => exact Or.inl hx
-  | @tail y z hy hed ih =>
-    by_cases hz : z ∈ A
-    · exact Or.inl hz
-    right
-    obtain ⟨e,he,hGe⟩ := hed
-    rcases ih with hyA | ⟨v,hv,w,hw,f,hf,hGf,hwy⟩
-    · exact ⟨y,hyA,z,hz,e,he,hGe,conn_refl G _ z⟩
-    · have hyA : y ∉ A := by
-        intro hya
-        have hs := deleted_cluster_singleton G hloc ω A y hya
-        have hyy : w ∈ cluster G (ω \ (incidentSupport G hloc A:Set E)) y := conn_symm G _ hwy
-        rw [hs] at hyy
-        exact hw (hyy ▸ hya)
-      refine ⟨v,hv,w,hw,f,hf,hGf,hwy.tail ⟨e,⟨he,?_⟩,hGe⟩⟩
-      intro hei
-      obtain ⟨a,ha,hage⟩ := (mem_incidentSupport G hloc A e).mp hei
-      rw [hGe,Sym2.mem_iff] at hage
-      rcases hage with rfl | rfl
-      · exact hyA ha
-      · exact hz ha
-
-theorem infinite_exterior_attachment (G : BondGraph V E) (hloc : LocallyFinite G)
-    (ω : Set E) (A : Finset V) (x : V) (hx : x ∈ A) (hi : (cluster G ω x).Infinite) :
-    ∃ v ∈ A, ∃ w, w ∉ A ∧ ∃ e ∈ ω, G e = s(v,w) ∧ Conn G ω x w ∧
-      (cluster G (ω \ (incidentSupport G hloc A:Set E)) w).Infinite := by
-  obtain ⟨z,hxz,hz⟩ := infinite_cluster_sdiff_contains G ω _ (incidentSupport G hloc A).finite_toSet x hi
-  have hn : z ∉ A := by
-    intro hzA
-    rw [deleted_cluster_singleton G hloc ω A z hzA] at hz
-    exact hz (Set.finite_singleton z)
-  obtain ⟨v,hv,w,hw,e,he,hGe,hwz⟩ := (conn_exterior_attachment G hloc ω A x hx hxz).resolve_left hn
-  refine ⟨v,hv,w,hw,e,he,hGe,conn_trans G ω hxz (conn_symm G ω (conn_mono G Set.sdiff_subset hwz)),?_⟩
-  rwa [cluster_eq_of_conn G _ hwz]
-
-theorem exists_finite_connected_envelope (G : BondGraph V E) (hconn : Connected G)
-    (x : V) (R : Finset V) : ∃ A : Finset V, R ⊆ A ∧
-      ((openSimpleGraph G Set.univ).induce (A:Set V)).Connected := by
-  obtain ⟨F,hF⟩ := finite_connecting_edges G hconn x R
-  have hf := cluster_finite_of_finite_edges G F.finite_toSet x
-  refine ⟨hf.toFinset,?_,?_⟩
-  · intro y hy
-    exact hf.mem_toFinset.mpr (hF y hy)
-  · rw [hf.coe_toFinset]
-    exact cluster_induce_connected G Set.univ (F:Set E) (Set.subset_univ _) x
-end Percolation
-
-namespace Percolation
-open SimpleGraph
-universe u v
-variable {V : Type u} {E : Type v}
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem deleted_envelope_separated (G : BondGraph V E) (hloc : LocallyFinite G)
-    (ω : Set E) (A : Finset V) (w : Fin 3 → V)
-    (hsep : ∀ i j, Conn G (ω \ (incidentSupport G hloc A:Set E)) (w i) (w j) → i = j)
-    (a b : V) (ha : a ∈ A ∪ Finset.univ.image w) (hb : b ∈ A ∪ Finset.univ.image w)
-    (hab : Conn G (ω \ (incidentSupport G hloc A:Set E)) a b) : a = b := by
-  rcases Finset.mem_union.mp ha with ha | ha
-  · have hh := deleted_cluster_singleton G hloc ω A a ha
-    have hm : b ∈ cluster G (ω \ (incidentSupport G hloc A:Set E)) a := hab
-    rw [hh] at hm
-    exact hm.symm
-  rcases Finset.mem_union.mp hb with hb | hb
-  · have hh := deleted_cluster_singleton G hloc ω A b hb
-    have hm : a ∈ cluster G (ω \ (incidentSupport G hloc A:Set E)) b := conn_symm G _ hab
-    rw [hh] at hm
-    exact hm
-  obtain ⟨i,_,rfl⟩ := Finset.mem_image.mp ha
-  obtain ⟨j,_,rfl⟩ := Finset.mem_image.mp hb
-  exact congrArg w (hsep i j hab)
-
-theorem encounter_surgery_of_attachments (G : BondGraph V E) (hloc : LocallyFinite G)
-    (ω : Set E) (A : Finset V) (v w : Fin 3 → V)
-    (hv : ∀ i, v i ∈ A) (hw : ∀ i, w i ∉ A) (hwi : Function.Injective w)
-    (hadj : ∀ i, (openSimpleGraph G Set.univ).Adj (v i) (w i))
-    (hA : ((openSimpleGraph G Set.univ).induce (A:Set V)).Connected)
-    (hsep : ∀ i j, Conn G (ω \ (incidentSupport G hloc A:Set E)) (w i) (w j) → i = j)
-    (hinf : ∀ i, (cluster G (ω \ (incidentSupport G hloc A:Set E)) (w i)).Infinite) :
-    ∃ P : Finset E, P ⊆ incidentSupport G hloc A ∧
-      ∃ x, IsEncounter G (finiteOverwrite (incidentSupport G hloc A) P ω) x := by
-  obtain ⟨D,hD,hSD,T,hTK,hT,x,a,b,c,hx,ha,hb,hc,hab,hac,hbc⟩ :=
-    exists_terminal_tree (openSimpleGraph G Set.univ) A v w hv hw hwi hadj hA
-  let F := incidentSupport G hloc A
-  let η := ω \ (F:Set E)
-  let f : (D:Set V) ↪ V := ⟨Subtype.val,Subtype.val_injective⟩
-  let P := F.filter (fun e => ∃ a b : (D:Set V), T.Adj a b ∧ G e = s(a.val,b.val))
-  have hPF : P ⊆ F := Finset.filter_subset _ _
-  have hP (e : E) (he : e ∈ P) : ∃ a b, T.Adj a b ∧ G e = s(f a,f b) :=
-    (Finset.mem_filter.mp he).2
-  have hPres (a b : (D:Set V)) (hab : T.Adj a b) : ∃ e ∈ (P:Set E), G e = s(f a,f b) := by
-    have hk := hTK hab
-    obtain ⟨e,_,he⟩ := hk.1.2
-    refine ⟨e,Finset.mem_filter.mpr ⟨?_,⟨a,b,hab,he⟩⟩,he⟩
-    apply (mem_incidentSupport G hloc A e).mpr
-    rcases terminalGraph_touch _ A v w hv hk with ha | hb
-    · exact ⟨a.val,ha,by rw [he]; simp⟩
-    · exact ⟨b.val,hb,by rw [he]; simp⟩
-  have hsepD (a b : (D:Set V)) (h : Conn G η (f a) (f b)) : a = b := by
-    apply Subtype.ext
-    exact deleted_envelope_separated G hloc ω A w hsep a.val b.val (hD a.property) (hD b.property) h
-  obtain ⟨r,hr,hrη⟩ := exists_cluster_retraction G η f hsepD x
-  have hr' (a : (D:Set V)) : r a.val = a := hr a
-  have hcut : η ⊆ cutConfig G (η ∪ (P:Set E)) x.val := by
-    intro e he
-    refine ⟨Or.inl he,?_⟩
-    intro hxe
-    exact he.2 ((mem_incidentSupport G hloc A e).mpr ⟨x.val,hx,hxe⟩)
-  have hInf (z : (D:Set V)) (hz : z.val ∈ Finset.univ.image w) :
-      (cluster G (cutConfig G (η ∪ (P:Set E)) x.val) z.val).Infinite := by
-    obtain ⟨i,_,hi⟩ := Finset.mem_image.mp hz
-    apply (hinf i).mono
-    intro y hy
-    rw [hi] at hy
-    exact conn_mono G hcut hy
-  have hnew : finiteOverwrite F P ω = η ∪ (P:Set E) := by rfl
-  refine ⟨P,hPF,x.val,?_⟩
-  rw [hnew]
-  refine ⟨a.val,b.val,c.val,?_,?_,?_,hInf a ha,hInf b hb,hInf c hc,?_,?_,?_⟩
-  · exact glued_conn G _ T f hT.connected
-      (fun a b h => let ⟨e,he,hGe⟩ := hPres a b h; ⟨e,Or.inr he,hGe⟩) x a
-  · exact glued_conn G _ T f hT.connected
-      (fun a b h => let ⟨e,he,hGe⟩ := hPres a b h; ⟨e,Or.inr he,hGe⟩) x b
-  · exact glued_conn G _ T f hT.connected
-      (fun a b h => let ⟨e,he,hGe⟩ := hPres a b h; ⟨e,Or.inr he,hGe⟩) x c
-  · intro h
-    have hh := glued_cut_conn G η (P:Set E) T f r hr hrη hP x h
-    exact hab (by simpa only [hr'] using hh)
-  · intro h
-    have hh := glued_cut_conn G η (P:Set E) T f r hr hrη hP x h
-    exact hac (by simpa only [hr'] using hh)
-  · intro h
-    have hh := glued_cut_conn G η (P:Set E) T f r hr hrη hP x h
-    exact hbc (by simpa only [hr'] using hh)
-
-theorem encounter_surgery_of_three (G : BondGraph V E) (hconn : Connected G)
-    (hloc : LocallyFinite G) (ω : Set E) (z : Fin 3 → V)
-    (hinf : ∀ i, (cluster G ω (z i)).Infinite)
-    (hsep : ∀ i j, Conn G ω (z i) (z j) → i = j) :
-    ∃ F P : Finset E, P ⊆ F ∧ ∃ x, IsEncounter G (finiteOverwrite F P ω) x := by
-  obtain ⟨A,hA,hAc⟩ := exists_finite_connected_envelope G hconn (z 0) (Finset.univ.image z)
-  have hattach (i : Fin 3) := infinite_exterior_attachment G hloc ω A (z i)
-    (hA (Finset.mem_image.mpr ⟨i,Finset.mem_univ _,rfl⟩)) (hinf i)
-  choose v hv w hw e he hGe hzw hInf using hattach
-  have hext (i j : Fin 3)
-      (h : Conn G (ω \ (incidentSupport G hloc A:Set E)) (w i) (w j)) : i = j :=
-    hsep i j (conn_trans G ω (hzw i)
-      (conn_trans G ω (conn_mono G Set.sdiff_subset h) (conn_symm G ω (hzw j))))
-  have hwi : Function.Injective w := by
-    intro i j h
-    apply hext
-    rw [h]
-    exact conn_refl G _ _
-  have hadj (i : Fin 3) : (openSimpleGraph G Set.univ).Adj (v i) (w i) := by
-    refine ⟨?_,e i,Set.mem_univ _,hGe i⟩
-    intro h
-    exact hw i (h ▸ hv i)
-  obtain ⟨P,hPF,x,hx⟩ := encounter_surgery_of_attachments G hloc ω A v w hv hw hwi hadj hAc hext hInf
-  exact ⟨incidentSupport G hloc A,P,hPF,x,hx⟩
-
-theorem three_of_infinite_count (G : BondGraph V E) (ω : Set E)
-    (h : (infiniteClusters G ω).Infinite) :
-    ∃ z : Fin 3 → V, (∀ i, (cluster G ω (z i)).Infinite) ∧
-      ∀ i j, Conn G ω (z i) (z j) → i = j := by
-  let e := Set.Infinite.natEmbedding (infiniteClusters G ω) h
-  let C (i : Fin 3) := e i.val
-  choose z hz using (fun i : Fin 3 => (C i).property.2)
-  refine ⟨z,?_,?_⟩
-  · intro i; rw [←hz i]; exact (C i).property.1
-  · intro i j hij
-    have hc : C i = C j := by
-      apply Subtype.ext
-      rw [hz i,hz j]
-      exact cluster_eq_of_conn G ω hij
-    exact Fin.ext (e.injective hc)
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Main
+import OAI.Probability.UnimodularPercolation.Criticality.FiniteChanges
+import OAI.Probability.UnimodularPercolation.Encounter.Counts
+import OAI.Probability.UnimodularPercolation.Encounter.TreeBranches
+
+namespace OAI
+
+/-! Exterior branches and finite encounter surgery. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+open MeasureTheory SimpleGraph
+universe u v
+variable {V : Type u} {E : Type v}
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem infinite_cluster_sdiff_contains (G : BondGraph V E) (ω : Set E)
+    (F : Set E) (hF : F.Finite) (x : V) (hx : (cluster G ω x).Infinite) :
+    ∃ z, Conn G ω x z ∧ (cluster G (ω \ F) z).Infinite := by
+  have he : (ω \ F) ∪ (ω ∩ F) = ω := by ext e; simp only [Set.mem_union,Set.mem_sdiff,Set.mem_inter_iff]; tauto
+  obtain ⟨z,hz,hzi⟩ := infinite_cluster_union_contains G (ω \ F) (ω ∩ F) (hF.inter_of_right ω) x (by rw [he]; exact hx)
+  exact ⟨z,by simpa only [he] using hz,hzi⟩
+
+theorem conn_eq_of_no_incident (G : BondGraph V E) (ω : Set E) (x y : V)
+    (hx : ∀ e ∈ ω, x ∉ G e) (hxy : Conn G ω x y) : y = x := by
+  induction hxy with
+  | refl => rfl
+  | @tail y z hy hed ih =>
+    obtain ⟨e,he,hGe⟩ := hed
+    exact False.elim (hx e he (by rw [hGe,ih]; simp))
+
+theorem cluster_singleton_of_no_incident (G : BondGraph V E) (ω : Set E) (x : V)
+    (hx : ∀ e ∈ ω, x ∉ G e) : cluster G ω x = {x} := by
+  ext y
+  constructor
+  · intro h; exact conn_eq_of_no_incident G ω x y hx h
+  · rintro rfl; exact mem_cluster_self G ω _
+
+theorem deleted_cluster_singleton (G : BondGraph V E) (hloc : LocallyFinite G)
+    (ω : Set E) (A : Finset V) (x : V) (hx : x ∈ A) :
+    cluster G (ω \ (incidentSupport G hloc A:Set E)) x = {x} := by
+  apply cluster_singleton_of_no_incident
+  intro e he hxe
+  exact he.2 ((mem_incidentSupport G hloc A e).mpr ⟨x,hx,hxe⟩)
+
+theorem conn_exterior_attachment (G : BondGraph V E) (hloc : LocallyFinite G)
+    (ω : Set E) (A : Finset V) (x : V) (hx : x ∈ A) {z : V} (hxz : Conn G ω x z) :
+    z ∈ A ∨ ∃ v ∈ A, ∃ w, w ∉ A ∧ ∃ e ∈ ω, G e = s(v,w) ∧
+      Conn G (ω \ (incidentSupport G hloc A:Set E)) w z := by
+  induction hxz with
+  | refl => exact Or.inl hx
+  | @tail y z hy hed ih =>
+    by_cases hz : z ∈ A
+    · exact Or.inl hz
+    right
+    obtain ⟨e,he,hGe⟩ := hed
+    rcases ih with hyA | ⟨v,hv,w,hw,f,hf,hGf,hwy⟩
+    · exact ⟨y,hyA,z,hz,e,he,hGe,conn_refl G _ z⟩
+    · have hyA : y ∉ A := by
+        intro hya
+        have hs := deleted_cluster_singleton G hloc ω A y hya
+        have hyy : w ∈ cluster G (ω \ (incidentSupport G hloc A:Set E)) y := conn_symm G _ hwy
+        rw [hs] at hyy
+        exact hw (hyy ▸ hya)
+      refine ⟨v,hv,w,hw,f,hf,hGf,hwy.tail ⟨e,⟨he,?_⟩,hGe⟩⟩
+      intro hei
+      obtain ⟨a,ha,hage⟩ := (mem_incidentSupport G hloc A e).mp hei
+      rw [hGe,Sym2.mem_iff] at hage
+      rcases hage with rfl | rfl
+      · exact hyA ha
+      · exact hz ha
+
+theorem infinite_exterior_attachment (G : BondGraph V E) (hloc : LocallyFinite G)
+    (ω : Set E) (A : Finset V) (x : V) (hx : x ∈ A) (hi : (cluster G ω x).Infinite) :
+    ∃ v ∈ A, ∃ w, w ∉ A ∧ ∃ e ∈ ω, G e = s(v,w) ∧ Conn G ω x w ∧
+      (cluster G (ω \ (incidentSupport G hloc A:Set E)) w).Infinite := by
+  obtain ⟨z,hxz,hz⟩ := infinite_cluster_sdiff_contains G ω _ (incidentSupport G hloc A).finite_toSet x hi
+  have hn : z ∉ A := by
+    intro hzA
+    rw [deleted_cluster_singleton G hloc ω A z hzA] at hz
+    exact hz (Set.finite_singleton z)
+  obtain ⟨v,hv,w,hw,e,he,hGe,hwz⟩ := (conn_exterior_attachment G hloc ω A x hx hxz).resolve_left hn
+  refine ⟨v,hv,w,hw,e,he,hGe,conn_trans G ω hxz (conn_symm G ω (conn_mono G Set.sdiff_subset hwz)),?_⟩
+  rwa [cluster_eq_of_conn G _ hwz]
+
+theorem exists_finite_connected_envelope (G : BondGraph V E) (hconn : Connected G)
+    (x : V) (R : Finset V) : ∃ A : Finset V, R ⊆ A ∧
+      ((openSimpleGraph G Set.univ).induce (A:Set V)).Connected := by
+  obtain ⟨F,hF⟩ := finite_connecting_edges G hconn x R
+  have hf := cluster_finite_of_finite_edges G F.finite_toSet x
+  refine ⟨hf.toFinset,?_,?_⟩
+  · intro y hy
+    exact hf.mem_toFinset.mpr (hF y hy)
+  · rw [hf.coe_toFinset]
+    exact cluster_induce_connected G Set.univ (F:Set E) (Set.subset_univ _) x
+end Percolation
+
+namespace Percolation
+open SimpleGraph
+universe u v
+variable {V : Type u} {E : Type v}
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem deleted_envelope_separated (G : BondGraph V E) (hloc : LocallyFinite G)
+    (ω : Set E) (A : Finset V) (w : Fin 3 → V)
+    (hsep : ∀ i j, Conn G (ω \ (incidentSupport G hloc A:Set E)) (w i) (w j) → i = j)
+    (a b : V) (ha : a ∈ A ∪ Finset.univ.image w) (hb : b ∈ A ∪ Finset.univ.image w)
+    (hab : Conn G (ω \ (incidentSupport G hloc A:Set E)) a b) : a = b := by
+  rcases Finset.mem_union.mp ha with ha | ha
+  · have hh := deleted_cluster_singleton G hloc ω A a ha
+    have hm : b ∈ cluster G (ω \ (incidentSupport G hloc A:Set E)) a := hab
+    rw [hh] at hm
+    exact hm.symm
+  rcases Finset.mem_union.mp hb with hb | hb
+  · have hh := deleted_cluster_singleton G hloc ω A b hb
+    have hm : a ∈ cluster G (ω \ (incidentSupport G hloc A:Set E)) b := conn_symm G _ hab
+    rw [hh] at hm
+    exact hm
+  obtain ⟨i,_,rfl⟩ := Finset.mem_image.mp ha
+  obtain ⟨j,_,rfl⟩ := Finset.mem_image.mp hb
+  exact congrArg w (hsep i j hab)
+
+theorem encounter_surgery_of_attachments (G : BondGraph V E) (hloc : LocallyFinite G)
+    (ω : Set E) (A : Finset V) (v w : Fin 3 → V)
+    (hv : ∀ i, v i ∈ A) (hw : ∀ i, w i ∉ A) (hwi : Function.Injective w)
+    (hadj : ∀ i, (openSimpleGraph G Set.univ).Adj (v i) (w i))
+    (hA : ((openSimpleGraph G Set.univ).induce (A:Set V)).Connected)
+    (hsep : ∀ i j, Conn G (ω \ (incidentSupport G hloc A:Set E)) (w i) (w j) → i = j)
+    (hinf : ∀ i, (cluster G (ω \ (incidentSupport G hloc A:Set E)) (w i)).Infinite) :
+    ∃ P : Finset E, P ⊆ incidentSupport G hloc A ∧
+      ∃ x, IsEncounter G (finiteOverwrite (incidentSupport G hloc A) P ω) x := by
+  obtain ⟨D,hD,hSD,T,hTK,hT,x,a,b,c,hx,ha,hb,hc,hab,hac,hbc⟩ :=
+    exists_terminal_tree (openSimpleGraph G Set.univ) A v w hv hw hwi hadj hA
+  let F := incidentSupport G hloc A
+  let η := ω \ (F:Set E)
+  let f : (D:Set V) ↪ V := ⟨Subtype.val,Subtype.val_injective⟩
+  let P := F.filter (fun e => ∃ a b : (D:Set V), T.Adj a b ∧ G e = s(a.val,b.val))
+  have hPF : P ⊆ F := Finset.filter_subset _ _
+  have hP (e : E) (he : e ∈ P) : ∃ a b, T.Adj a b ∧ G e = s(f a,f b) :=
+    (Finset.mem_filter.mp he).2
+  have hPres (a b : (D:Set V)) (hab : T.Adj a b) : ∃ e ∈ (P:Set E), G e = s(f a,f b) := by
+    have hk := hTK hab
+    obtain ⟨e,_,he⟩ := hk.1.2
+    refine ⟨e,Finset.mem_filter.mpr ⟨?_,⟨a,b,hab,he⟩⟩,he⟩
+    apply (mem_incidentSupport G hloc A e).mpr
+    rcases terminalGraph_touch _ A v w hv hk with ha | hb
+    · exact ⟨a.val,ha,by rw [he]; simp⟩
+    · exact ⟨b.val,hb,by rw [he]; simp⟩
+  have hsepD (a b : (D:Set V)) (h : Conn G η (f a) (f b)) : a = b := by
+    apply Subtype.ext
+    exact deleted_envelope_separated G hloc ω A w hsep a.val b.val (hD a.property) (hD b.property) h
+  obtain ⟨r,hr,hrη⟩ := exists_cluster_retraction G η f hsepD x
+  have hr' (a : (D:Set V)) : r a.val = a := hr a
+  have hcut : η ⊆ cutConfig G (η ∪ (P:Set E)) x.val := by
+    intro e he
+    refine ⟨Or.inl he,?_⟩
+    intro hxe
+    exact he.2 ((mem_incidentSupport G hloc A e).mpr ⟨x.val,hx,hxe⟩)
+  have hInf (z : (D:Set V)) (hz : z.val ∈ Finset.univ.image w) :
+      (cluster G (cutConfig G (η ∪ (P:Set E)) x.val) z.val).Infinite := by
+    obtain ⟨i,_,hi⟩ := Finset.mem_image.mp hz
+    apply (hinf i).mono
+    intro y hy
+    rw [hi] at hy
+    exact conn_mono G hcut hy
+  have hnew : finiteOverwrite F P ω = η ∪ (P:Set E) := by rfl
+  refine ⟨P,hPF,x.val,?_⟩
+  rw [hnew]
+  refine ⟨a.val,b.val,c.val,?_,?_,?_,hInf a ha,hInf b hb,hInf c hc,?_,?_,?_⟩
+  · exact glued_conn G _ T f hT.connected
+      (fun a b h => let ⟨e,he,hGe⟩ := hPres a b h; ⟨e,Or.inr he,hGe⟩) x a
+  · exact glued_conn G _ T f hT.connected
+      (fun a b h => let ⟨e,he,hGe⟩ := hPres a b h; ⟨e,Or.inr he,hGe⟩) x b
+  · exact glued_conn G _ T f hT.connected
+      (fun a b h => let ⟨e,he,hGe⟩ := hPres a b h; ⟨e,Or.inr he,hGe⟩) x c
+  · intro h
+    have hh := glued_cut_conn G η (P:Set E) T f r hr hrη hP x h
+    exact hab (by simpa only [hr'] using hh)
+  · intro h
+    have hh := glued_cut_conn G η (P:Set E) T f r hr hrη hP x h
+    exact hac (by simpa only [hr'] using hh)
+  · intro h
+    have hh := glued_cut_conn G η (P:Set E) T f r hr hrη hP x h
+    exact hbc (by simpa only [hr'] using hh)
+
+theorem encounter_surgery_of_three (G : BondGraph V E) (hconn : Connected G)
+    (hloc : LocallyFinite G) (ω : Set E) (z : Fin 3 → V)
+    (hinf : ∀ i, (cluster G ω (z i)).Infinite)
+    (hsep : ∀ i j, Conn G ω (z i) (z j) → i = j) :
+    ∃ F P : Finset E, P ⊆ F ∧ ∃ x, IsEncounter G (finiteOverwrite F P ω) x := by
+  obtain ⟨A,hA,hAc⟩ := exists_finite_connected_envelope G hconn (z 0) (Finset.univ.image z)
+  have hattach (i : Fin 3) := infinite_exterior_attachment G hloc ω A (z i)
+    (hA (Finset.mem_image.mpr ⟨i,Finset.mem_univ _,rfl⟩)) (hinf i)
+  choose v hv w hw e he hGe hzw hInf using hattach
+  have hext (i j : Fin 3)
+      (h : Conn G (ω \ (incidentSupport G hloc A:Set E)) (w i) (w j)) : i = j :=
+    hsep i j (conn_trans G ω (hzw i)
+      (conn_trans G ω (conn_mono G Set.sdiff_subset h) (conn_symm G ω (hzw j))))
+  have hwi : Function.Injective w := by
+    intro i j h
+    apply hext
+    rw [h]
+    exact conn_refl G _ _
+  have hadj (i : Fin 3) : (openSimpleGraph G Set.univ).Adj (v i) (w i) := by
+    refine ⟨?_,e i,Set.mem_univ _,hGe i⟩
+    intro h
+    exact hw i (h ▸ hv i)
+  obtain ⟨P,hPF,x,hx⟩ := encounter_surgery_of_attachments G hloc ω A v w hv hw hwi hadj hAc hext hInf
+  exact ⟨incidentSupport G hloc A,P,hPF,x,hx⟩
+
+theorem three_of_infinite_count (G : BondGraph V E) (ω : Set E)
+    (h : (infiniteClusters G ω).Infinite) :
+    ∃ z : Fin 3 → V, (∀ i, (cluster G ω (z i)).Infinite) ∧
+      ∀ i j, Conn G ω (z i) (z j) → i = j := by
+  let e := Set.Infinite.natEmbedding (infiniteClusters G ω) h
+  let C (i : Fin 3) := e i.val
+  choose z hz using (fun i : Fin 3 => (C i).property.2)
+  refine ⟨z,?_,?_⟩
+  · intro i; rw [←hz i]; exact (C i).property.1
+  · intro i j hij
+    have hc : C i = C j := by
+      apply Subtype.ext
+      rw [hz i,hz j]
+      exact cluster_eq_of_conn G ω hij
+    exact Fin.ext (e.injective hc)
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Encounter/Transport.lean b/lean/OAI/Probability/UnimodularPercolation/Encounter/Transport.lean
index 683ea80098cf2668dd602c891a86b1508d210ef0..2ffb386c3ef42c910130c7e7ed5861b1375f9f14 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Encounter/Transport.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Encounter/Transport.lean
@@ -1,163 +1,163 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Main
-
-namespace OAI
-
-/-! Expected mass transport and cluster averaging. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-open MeasureTheory
-open scoped ENNReal
-universe u v w
-variable {V : Type u} {E : Type v} {ι : Type w}
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-variable [Countable V] [Countable E] [Fintype ι]
-
-omit [Countable V] [Countable E] [Fintype ι] in
-theorem expected_mass_transport [Countable V] [Countable E] [Fintype ι]
-    (G : BondGraph V E) (hconn : Connected G)
-    (hloc : LocallyFinite G) (root : ι → V) (weight : ι → ℝ)
-    (htr : RootTransport root weight (vertexAut G)) (hw : ∀ i, 0 ≤ weight i)
-    (p : unitInterval) (m : Set E → V → V → ℝ≥0∞)
-    (hm : ∀ x y, Measurable (fun ω => m ω x y))
-    (hinv : ∀ a : V ≃ V, ∀ b : E ≃ E, IsAut G a b → ∀ ω x y,
-      m (b ⁻¹' ω) x y = m ω (a x) (a y)) :
-    (∑ i, ENNReal.ofReal (weight i) * ∫⁻ ω, ∑' y, m ω (root i) y ∂law p) =
-      (∑ i, ENNReal.ofReal (weight i) * ∫⁻ ω, ∑' x, m ω x (root i) ∂law p) := by
-  have hInv (a : V ≃ V) (ha : a ∈ vertexAut G) (x y : V) :
-      (∫⁻ ω, m ω (a x) (a y) ∂law p) = ∫⁻ ω, m ω x y ∂law p := by
-    obtain ⟨b,hb⟩ := ha
-    calc
-      _ = ∫⁻ ω, m (b ⁻¹' ω) x y ∂law p := lintegral_congr (fun ω => (hinv a b hb ω x y).symm)
-      _ = ∫⁻ ω, m ω x y ∂(law p).map (fun ω => b ⁻¹' ω) :=
-        (lintegral_map (hm x y) (measurable_set_preimage b)).symm
-      _ = _ := by rw [law_map_preimage_equiv]
-  have ht := htr.ennreal G hconn hloc root weight (vertexAut G) (Set.Subset.refl _) hw
-    (fun x y => ∫⁻ ω, m ω x y ∂law p) hInv
-  have hr (x : V) : (∫⁻ ω, ∑' y, m ω x y ∂law p) = ∑' y, ∫⁻ ω, m ω x y ∂law p :=
-    lintegral_tsum (fun y => (hm x y).aemeasurable)
-  have hc (y : V) : (∫⁻ ω, ∑' x, m ω x y ∂law p) = ∑' x, ∫⁻ ω, m ω x y ∂law p :=
-    lintegral_tsum (fun x => (hm x y).aemeasurable)
-  simp_rw [hr,hc]
-  exact ht
-
-theorem incoming_finite_ae (G : BondGraph V E) (hconn : Connected G)
-    (hloc : LocallyFinite G) (root : ι → V) (weight : ι → ℝ)
-    (htr : RootTransport root weight (vertexAut G)) (hw : ∀ i, 0 < weight i)
-    (hw1 : ∑ i, weight i = 1) (p : unitInterval) (m : Set E → V → V → ℝ≥0∞)
-    (hm : ∀ x y, Measurable (fun ω => m ω x y))
-    (hinv : ∀ a : V ≃ V, ∀ b : E ≃ E, IsAut G a b → ∀ ω x y,
-      m (b ⁻¹' ω) x y = m ω (a x) (a y))
-    (hrow : ∀ ω x, ∑' y, m ω x y ≤ 1) (i : ι) :
-    ∀ᵐ ω ∂law p, ∑' x, m ω x (root i) < ⊤ := by
-  have ht := expected_mass_transport G hconn hloc root weight htr (fun i => (hw i).le)
-    p m hm hinv
-  have hupper : (∑ j, ENNReal.ofReal (weight j) * ∫⁻ ω, ∑' x, m ω x (root j) ∂law p) ≤ 1 := by
-    rw [←ht]
-    calc
-      _ ≤ ∑ j, ENNReal.ofReal (weight j) * 1 := by
-        apply Finset.sum_le_sum
-        intro j _
-        gcongr
-        calc
-          _ ≤ ∫⁻ ω : Set E, (1:ℝ≥0∞) ∂law p := lintegral_mono (fun ω => hrow ω (root j))
-          _ = 1 := by simp
-      _ = 1 := by
-        simp only [mul_one,←ENNReal.ofReal_sum_of_nonneg (fun j _ => (hw j).le),hw1,ENNReal.ofReal_one]
-  have hterm := (Finset.single_le_sum (fun j _ => (show (0:ℝ≥0∞) ≤ ENNReal.ofReal (weight j) * ∫⁻ ω, ∑' x, m ω x (root j) ∂law p from zero_le))
-      (Finset.mem_univ i)).trans hupper
-  have hfinite : (∫⁻ ω, ∑' x, m ω x (root i) ∂law p) ≠ ⊤ := by
-    intro he
-    rw [he,ENNReal.mul_top ((ENNReal.ofReal_pos.mpr (hw i)).ne')] at hterm
-    exact (not_le_of_gt ENNReal.one_lt_top) hterm
-  exact ae_lt_top (Measurable.tsum (fun x => hm x (root i))) hfinite
-
-omit [Countable V] [Countable E] [Fintype ι] in
-theorem expected_label_mass_transport [Countable V] [Countable E] [Fintype ι]
-    (G : BondGraph V E) (hconn : Connected G)
-    (hloc : LocallyFinite G) (root : ι → V) (weight : ι → ℝ)
-    (htr : RootTransport root weight (vertexAut G)) (hw : ∀ i, 0 ≤ weight i)
-    (m : (E → unitInterval) → V → V → ℝ≥0∞)
-    (hm : ∀ x y, Measurable (fun l => m l x y))
-    (hinv : ∀ a : V ≃ V, ∀ b : E ≃ E, IsAut G a b → ∀ l x y,
-      m (l ∘ b) x y = m l (a x) (a y)) :
-    (∑ i, ENNReal.ofReal (weight i) * ∫⁻ l, ∑' y, m l (root i) y ∂labelLaw) =
-      (∑ i, ENNReal.ofReal (weight i) * ∫⁻ l, ∑' x, m l x (root i) ∂labelLaw) := by
-  have hInv (a : V ≃ V) (ha : a ∈ vertexAut G) (x y : V) :
-      (∫⁻ l, m l (a x) (a y) ∂labelLaw) = ∫⁻ l, m l x y ∂labelLaw := by
-    obtain ⟨b,hb⟩ := ha
-    calc
-      _ = ∫⁻ l, m (l ∘ b) x y ∂labelLaw := lintegral_congr (fun l => (hinv a b hb l x y).symm)
-      _ = ∫⁻ l, m l x y ∂labelLaw.map (fun l => l ∘ b) :=
-        (lintegral_map (hm x y) (by fun_prop)).symm
-      _ = _ := by rw [labelLaw_map_precomp]
-  have ht := htr.ennreal G hconn hloc root weight (vertexAut G) (Set.Subset.refl _) hw
-    (fun x y => ∫⁻ l, m l x y ∂labelLaw) hInv
-  have hr x : (∫⁻ l, ∑' y, m l x y ∂labelLaw) = ∑' y, ∫⁻ l, m l x y ∂labelLaw :=
-    lintegral_tsum (fun y => (hm x y).aemeasurable)
-  have hc y : (∫⁻ l, ∑' x, m l x y ∂labelLaw) = ∑' x, ∫⁻ l, m l x y ∂labelLaw :=
-    lintegral_tsum (fun x => (hm x y).aemeasurable)
-  simp_rw [hr,hc]
-  exact ht
-
-theorem label_root_cluster_average (G : BondGraph V E) (hconn : Connected G)
-    (hloc : LocallyFinite G) (root : ι → V) (weight : ι → ℝ)
-    (htr : RootTransport root weight (vertexAut G)) (hw : ∀ i, 0 ≤ weight i)
-    (q : unitInterval) (hfin : ∀ x, ∀ᵐ ω ∂law q, (cluster G ω x).Finite)
-    (f : (E → unitInterval) → V → ℝ≥0∞) (hf : ∀ x, Measurable (fun l => f l x))
-    (hi : ∀ a : V ≃ V, ∀ b : E ≃ E, IsAut G a b → ∀ l x,
-      f (l ∘ b) x = f l (a x)) :
-    (∑ i, ENNReal.ofReal (weight i) * ∫⁻ l, ∑' y,
-      clusterAveragingKernel G (openEdges l q) (root i) y * f l y ∂labelLaw) =
-    (∑ i, ENNReal.ofReal (weight i) * ∫⁻ l, f l (root i) ∂labelLaw) := by
-  have hm x y : Measurable (fun l => clusterAveragingKernel G (openEdges l q) x y * f l y) :=
-    ((measurable_clusterAveragingKernel G hloc x y).comp (measurable_openEdges q)).mul (hf y)
-  have ht := expected_label_mass_transport G hconn hloc root weight htr hw
-    (fun l x y => clusterAveragingKernel G (openEdges l q) x y * f l y) hm
-    (fun a b hab l x y => by
-      rw [hi a b hab]
-      exact congrArg (fun t => t * f l (a y)) (clusterAveragingKernel_aut G hab (openEdges l q) x y))
-  rw [ht]
-  apply Finset.sum_congr rfl
-  intro i _
-  congr 1
-  apply lintegral_congr_ae
-  have hfl : ∀ᵐ l ∂labelLaw, (cluster G (openEdges l q) (root i)).Finite := by
-    apply ae_of_ae_map (μ := (labelLaw : Measure (E → unitInterval)))
-      (p := fun ω : Set E => (cluster G ω (root i)).Finite) (measurable_openEdges q).aemeasurable
-    rw [map_openEdges]
-    exact hfin (root i)
-  filter_upwards [hfl] with l hl
-  rw [ENNReal.tsum_mul_right,clusterAveragingKernel_column G (openEdges l q) (root i) hl,one_mul]
-
-theorem incoming_finite_ae_all (G : BondGraph V E) (hconn : Connected G)
-    (hloc : LocallyFinite G) (root : ι → V) (weight : ι → ℝ)
-    (htr : RootTransport root weight (vertexAut G)) (hw : ∀ i, 0 < weight i)
-    (hw1 : ∑ i, weight i = 1)
-    (hcov : ∀ x, ∃ i, ∃ a ∈ vertexAut G, a (root i) = x)
-    (p : unitInterval) (m : Set E → V → V → ℝ≥0∞)
-    (hm : ∀ x y, Measurable (fun ω => m ω x y))
-    (hinv : ∀ a : V ≃ V, ∀ b : E ≃ E, IsAut G a b → ∀ ω x y,
-      m (b ⁻¹' ω) x y = m ω (a x) (a y))
-    (hrow : ∀ ω x, ∑' y, m ω x y ≤ 1) :
-    ∀ᵐ ω ∂law p, ∀ y, ∑' x, m ω x y < ⊤ := by
-  apply ae_all_iff.mpr
-  intro y
-  obtain ⟨i,a,⟨b,hab⟩,rfl⟩ := hcov y
-  have hi := incoming_finite_ae G hconn hloc root weight htr hw hw1 p m hm hinv hrow i
-  have hi' : ∀ᵐ ω ∂law p, ∑' x, m (b ⁻¹' ω) x (root i) < ⊤ := by
-    apply ae_of_ae_map (μ := law p)
-      (p := fun ω : Set E => ∑' x, m ω x (root i) < ⊤)
-      (measurable_set_preimage b).aemeasurable
-    rw [law_map_preimage_equiv]
-    exact hi
-  filter_upwards [hi'] with ω hω
-  simpa only [hinv a b hab,a.tsum_eq (fun x => m ω x (a (root i)))] using hω
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Main
+
+namespace OAI
+
+/-! Expected mass transport and cluster averaging. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+open MeasureTheory
+open scoped ENNReal
+universe u v w
+variable {V : Type u} {E : Type v} {ι : Type w}
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+variable [Countable V] [Countable E] [Fintype ι]
+
+omit [Countable V] [Countable E] [Fintype ι] in
+theorem expected_mass_transport [Countable V] [Countable E] [Fintype ι]
+    (G : BondGraph V E) (hconn : Connected G)
+    (hloc : LocallyFinite G) (root : ι → V) (weight : ι → ℝ)
+    (htr : RootTransport root weight (vertexAut G)) (hw : ∀ i, 0 ≤ weight i)
+    (p : unitInterval) (m : Set E → V → V → ℝ≥0∞)
+    (hm : ∀ x y, Measurable (fun ω => m ω x y))
+    (hinv : ∀ a : V ≃ V, ∀ b : E ≃ E, IsAut G a b → ∀ ω x y,
+      m (b ⁻¹' ω) x y = m ω (a x) (a y)) :
+    (∑ i, ENNReal.ofReal (weight i) * ∫⁻ ω, ∑' y, m ω (root i) y ∂law p) =
+      (∑ i, ENNReal.ofReal (weight i) * ∫⁻ ω, ∑' x, m ω x (root i) ∂law p) := by
+  have hInv (a : V ≃ V) (ha : a ∈ vertexAut G) (x y : V) :
+      (∫⁻ ω, m ω (a x) (a y) ∂law p) = ∫⁻ ω, m ω x y ∂law p := by
+    obtain ⟨b,hb⟩ := ha
+    calc
+      _ = ∫⁻ ω, m (b ⁻¹' ω) x y ∂law p := lintegral_congr (fun ω => (hinv a b hb ω x y).symm)
+      _ = ∫⁻ ω, m ω x y ∂(law p).map (fun ω => b ⁻¹' ω) :=
+        (lintegral_map (hm x y) (measurable_set_preimage b)).symm
+      _ = _ := by rw [law_map_preimage_equiv]
+  have ht := htr.ennreal G hconn hloc root weight (vertexAut G) (Set.Subset.refl _) hw
+    (fun x y => ∫⁻ ω, m ω x y ∂law p) hInv
+  have hr (x : V) : (∫⁻ ω, ∑' y, m ω x y ∂law p) = ∑' y, ∫⁻ ω, m ω x y ∂law p :=
+    lintegral_tsum (fun y => (hm x y).aemeasurable)
+  have hc (y : V) : (∫⁻ ω, ∑' x, m ω x y ∂law p) = ∑' x, ∫⁻ ω, m ω x y ∂law p :=
+    lintegral_tsum (fun x => (hm x y).aemeasurable)
+  simp_rw [hr,hc]
+  exact ht
+
+theorem incoming_finite_ae (G : BondGraph V E) (hconn : Connected G)
+    (hloc : LocallyFinite G) (root : ι → V) (weight : ι → ℝ)
+    (htr : RootTransport root weight (vertexAut G)) (hw : ∀ i, 0 < weight i)
+    (hw1 : ∑ i, weight i = 1) (p : unitInterval) (m : Set E → V → V → ℝ≥0∞)
+    (hm : ∀ x y, Measurable (fun ω => m ω x y))
+    (hinv : ∀ a : V ≃ V, ∀ b : E ≃ E, IsAut G a b → ∀ ω x y,
+      m (b ⁻¹' ω) x y = m ω (a x) (a y))
+    (hrow : ∀ ω x, ∑' y, m ω x y ≤ 1) (i : ι) :
+    ∀ᵐ ω ∂law p, ∑' x, m ω x (root i) < ⊤ := by
+  have ht := expected_mass_transport G hconn hloc root weight htr (fun i => (hw i).le)
+    p m hm hinv
+  have hupper : (∑ j, ENNReal.ofReal (weight j) * ∫⁻ ω, ∑' x, m ω x (root j) ∂law p) ≤ 1 := by
+    rw [←ht]
+    calc
+      _ ≤ ∑ j, ENNReal.ofReal (weight j) * 1 := by
+        apply Finset.sum_le_sum
+        intro j _
+        gcongr
+        calc
+          _ ≤ ∫⁻ ω : Set E, (1:ℝ≥0∞) ∂law p := lintegral_mono (fun ω => hrow ω (root j))
+          _ = 1 := by simp
+      _ = 1 := by
+        simp only [mul_one,←ENNReal.ofReal_sum_of_nonneg (fun j _ => (hw j).le),hw1,ENNReal.ofReal_one]
+  have hterm := (Finset.single_le_sum (fun j _ => (show (0:ℝ≥0∞) ≤ ENNReal.ofReal (weight j) * ∫⁻ ω, ∑' x, m ω x (root j) ∂law p from zero_le))
+      (Finset.mem_univ i)).trans hupper
+  have hfinite : (∫⁻ ω, ∑' x, m ω x (root i) ∂law p) ≠ ⊤ := by
+    intro he
+    rw [he,ENNReal.mul_top ((ENNReal.ofReal_pos.mpr (hw i)).ne')] at hterm
+    exact (not_le_of_gt ENNReal.one_lt_top) hterm
+  exact ae_lt_top (Measurable.tsum (fun x => hm x (root i))) hfinite
+
+omit [Countable V] [Countable E] [Fintype ι] in
+theorem expected_label_mass_transport [Countable V] [Countable E] [Fintype ι]
+    (G : BondGraph V E) (hconn : Connected G)
+    (hloc : LocallyFinite G) (root : ι → V) (weight : ι → ℝ)
+    (htr : RootTransport root weight (vertexAut G)) (hw : ∀ i, 0 ≤ weight i)
+    (m : (E → unitInterval) → V → V → ℝ≥0∞)
+    (hm : ∀ x y, Measurable (fun l => m l x y))
+    (hinv : ∀ a : V ≃ V, ∀ b : E ≃ E, IsAut G a b → ∀ l x y,
+      m (l ∘ b) x y = m l (a x) (a y)) :
+    (∑ i, ENNReal.ofReal (weight i) * ∫⁻ l, ∑' y, m l (root i) y ∂labelLaw) =
+      (∑ i, ENNReal.ofReal (weight i) * ∫⁻ l, ∑' x, m l x (root i) ∂labelLaw) := by
+  have hInv (a : V ≃ V) (ha : a ∈ vertexAut G) (x y : V) :
+      (∫⁻ l, m l (a x) (a y) ∂labelLaw) = ∫⁻ l, m l x y ∂labelLaw := by
+    obtain ⟨b,hb⟩ := ha
+    calc
+      _ = ∫⁻ l, m (l ∘ b) x y ∂labelLaw := lintegral_congr (fun l => (hinv a b hb l x y).symm)
+      _ = ∫⁻ l, m l x y ∂labelLaw.map (fun l => l ∘ b) :=
+        (lintegral_map (hm x y) (by fun_prop)).symm
+      _ = _ := by rw [labelLaw_map_precomp]
+  have ht := htr.ennreal G hconn hloc root weight (vertexAut G) (Set.Subset.refl _) hw
+    (fun x y => ∫⁻ l, m l x y ∂labelLaw) hInv
+  have hr x : (∫⁻ l, ∑' y, m l x y ∂labelLaw) = ∑' y, ∫⁻ l, m l x y ∂labelLaw :=
+    lintegral_tsum (fun y => (hm x y).aemeasurable)
+  have hc y : (∫⁻ l, ∑' x, m l x y ∂labelLaw) = ∑' x, ∫⁻ l, m l x y ∂labelLaw :=
+    lintegral_tsum (fun x => (hm x y).aemeasurable)
+  simp_rw [hr,hc]
+  exact ht
+
+theorem label_root_cluster_average (G : BondGraph V E) (hconn : Connected G)
+    (hloc : LocallyFinite G) (root : ι → V) (weight : ι → ℝ)
+    (htr : RootTransport root weight (vertexAut G)) (hw : ∀ i, 0 ≤ weight i)
+    (q : unitInterval) (hfin : ∀ x, ∀ᵐ ω ∂law q, (cluster G ω x).Finite)
+    (f : (E → unitInterval) → V → ℝ≥0∞) (hf : ∀ x, Measurable (fun l => f l x))
+    (hi : ∀ a : V ≃ V, ∀ b : E ≃ E, IsAut G a b → ∀ l x,
+      f (l ∘ b) x = f l (a x)) :
+    (∑ i, ENNReal.ofReal (weight i) * ∫⁻ l, ∑' y,
+      clusterAveragingKernel G (openEdges l q) (root i) y * f l y ∂labelLaw) =
+    (∑ i, ENNReal.ofReal (weight i) * ∫⁻ l, f l (root i) ∂labelLaw) := by
+  have hm x y : Measurable (fun l => clusterAveragingKernel G (openEdges l q) x y * f l y) :=
+    ((measurable_clusterAveragingKernel G hloc x y).comp (measurable_openEdges q)).mul (hf y)
+  have ht := expected_label_mass_transport G hconn hloc root weight htr hw
+    (fun l x y => clusterAveragingKernel G (openEdges l q) x y * f l y) hm
+    (fun a b hab l x y => by
+      rw [hi a b hab]
+      exact congrArg (fun t => t * f l (a y)) (clusterAveragingKernel_aut G hab (openEdges l q) x y))
+  rw [ht]
+  apply Finset.sum_congr rfl
+  intro i _
+  congr 1
+  apply lintegral_congr_ae
+  have hfl : ∀ᵐ l ∂labelLaw, (cluster G (openEdges l q) (root i)).Finite := by
+    apply ae_of_ae_map (μ := (labelLaw : Measure (E → unitInterval)))
+      (p := fun ω : Set E => (cluster G ω (root i)).Finite) (measurable_openEdges q).aemeasurable
+    rw [map_openEdges]
+    exact hfin (root i)
+  filter_upwards [hfl] with l hl
+  rw [ENNReal.tsum_mul_right,clusterAveragingKernel_column G (openEdges l q) (root i) hl,one_mul]
+
+theorem incoming_finite_ae_all (G : BondGraph V E) (hconn : Connected G)
+    (hloc : LocallyFinite G) (root : ι → V) (weight : ι → ℝ)
+    (htr : RootTransport root weight (vertexAut G)) (hw : ∀ i, 0 < weight i)
+    (hw1 : ∑ i, weight i = 1)
+    (hcov : ∀ x, ∃ i, ∃ a ∈ vertexAut G, a (root i) = x)
+    (p : unitInterval) (m : Set E → V → V → ℝ≥0∞)
+    (hm : ∀ x y, Measurable (fun ω => m ω x y))
+    (hinv : ∀ a : V ≃ V, ∀ b : E ≃ E, IsAut G a b → ∀ ω x y,
+      m (b ⁻¹' ω) x y = m ω (a x) (a y))
+    (hrow : ∀ ω x, ∑' y, m ω x y ≤ 1) :
+    ∀ᵐ ω ∂law p, ∀ y, ∑' x, m ω x y < ⊤ := by
+  apply ae_all_iff.mpr
+  intro y
+  obtain ⟨i,a,⟨b,hab⟩,rfl⟩ := hcov y
+  have hi := incoming_finite_ae G hconn hloc root weight htr hw hw1 p m hm hinv hrow i
+  have hi' : ∀ᵐ ω ∂law p, ∑' x, m (b ⁻¹' ω) x (root i) < ⊤ := by
+    apply ae_of_ae_map (μ := law p)
+      (p := fun ω : Set E => ∑' x, m ω x (root i) < ⊤)
+      (measurable_set_preimage b).aemeasurable
+    rw [law_map_preimage_equiv]
+    exact hi
+  filter_upwards [hi'] with ω hω
+  simpa only [hinv a b hab,a.tsum_eq (fun x => m ω x (a (root i)))] using hω
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Encounter/TreeBranches.lean b/lean/OAI/Probability/UnimodularPercolation/Encounter/TreeBranches.lean
index 0fc685c8f8d646cb1dedd89dd1206fd76c37b322..3e02afddf7fa2b0484b042fc12ab271ae03e2a4f 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Encounter/TreeBranches.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Encounter/TreeBranches.lean
@@ -1,336 +1,336 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Main
-import OAI.Probability.UnimodularPercolation.Encounter.Vertices
-
-namespace OAI
-
-/-! Tree branches, cluster retractions and terminal connecting trees. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-open SimpleGraph
-universe u
-variable {V : Type u}
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem cutVertex_le_deleteEdge (T : SimpleGraph V) (x y : V) :
-    cutVertex T x ≤ T.deleteEdges {s(x,y)} := by
-  intro a b hab
-  apply deleteEdges_adj.mpr
-  refine ⟨hab.1,?_⟩
-  intro he
-  have hh : x ∈ s(a,b) := (Set.mem_singleton_iff.mp he) ▸ (by simp : x ∈ s(x,y))
-  rcases Sym2.mem_iff.mp hh with hx | hx
-  · exact hab.2.1 hx.symm
-  · exact hab.2.2 hx.symm
-
-theorem tree_cut_neighbors (T : SimpleGraph V) (ht : T.IsAcyclic) (x y z : V)
-    (hy : T.Adj x y) (hz : T.Adj x z) (hyz : (cutVertex T x).Reachable y z) : y = z := by
-  by_contra hn
-  have hb := (isAcyclic_iff_forall_adj_isBridge.mp ht) hy
-  have hd : (T.deleteEdges {s(x,y)}).Adj x z := by
-    apply deleteEdges_adj.mpr
-    refine ⟨hz,?_⟩
-    intro he
-    have hh := Sym2.eq_iff.mp (Set.mem_singleton_iff.mp he)
-    rcases hh with ⟨_,hzy⟩ | ⟨hxy,hzx⟩
-    · exact hn hzy.symm
-    · exact hy.ne hxy
-  exact (isBridge_iff.mp hb) (hd.reachable.trans (hyz.mono (cutVertex_le_deleteEdge T x y)).symm)
-
-theorem cut_reachable_self_eq (T : SimpleGraph V) (x z : V)
-    (h : (cutVertex T x).Reachable x z) : z = x := by
-  induction h with
-  | intro p =>
-    cases p with
-    | nil => rfl
-    | cons hadj _ => exact False.elim (hadj.2.1 rfl)
-
-theorem tree_branch_contains_leaf [Fintype V] (T : SimpleGraph V) (ht : T.IsTree)
-    (x y : V) (hxy : T.Adj x y) :
-    ∃ z, z ≠ x ∧ (cutVertex T x).Reachable y z ∧ T.degree z = 1 := by
-  let C := ((cutVertex T x).connectedComponentMk y).supp
-  have hyC : y ∈ C := (ConnectedComponent.mem_supp_iff _ _).mpr rfl
-  have hxC : x ∉ C := by
-    intro h
-    have hr : (cutVertex T x).Reachable y x := by
-      exact (ConnectedComponent.eq.mp ((ConnectedComponent.mem_supp_iff _ _).mp h)).symm
-    exact hxy.ne (cut_reachable_self_eq T x y hr.symm).symm
-  have hr (z : V) (hz : z ∈ C) : (cutVertex T x).Reachable y z :=
-    (ConnectedComponent.eq.mp ((ConnectedComponent.mem_supp_iff _ _).mp hz)).symm
-  have hnbr (z : V) (hz : z ∈ C) (w : V) (hzw : T.Adj z w) : w = x ∨ w ∈ C := by
-    by_cases hw : w = x
-    · exact Or.inl hw
-    · right
-      apply (ConnectedComponent.mem_supp_iff _ _).mpr
-      apply ConnectedComponent.sound
-      exact (hr z hz).trans (Adj.reachable (G := cutVertex T x)
-        ⟨hzw,fun h => hxC (h ▸ hz),hw⟩) |>.symm
-  have hnx (z : V) (hz : z ∈ C) (hzx : T.Adj z x) : z = y :=
-    (tree_cut_neighbors T ht.isAcyclic x y z hxy hzx.symm (hr z hz)).symm
-  have hccon : (T.induce C).Connected := by
-    apply SimpleGraph.Connected.mono (G := (cutVertex T x).induce C)
-    · intro a b hab; exact hab.1
-    · exact ((cutVertex T x).connectedComponentMk y).connected_toSimpleGraph
-  have hc : (T.induce C).IsTree := ⟨hccon,ht.isAcyclic.induce C⟩
-  by_cases he : ∃ z ∈ C, z ≠ y
-  · obtain ⟨z,hz,hzy⟩ := he
-    let : Nontrivial C := ⟨⟨⟨z,hz⟩,⟨y,hyC⟩,fun h => hzy (congrArg Subtype.val h)⟩⟩
-    obtain ⟨a,b,hab,ha,hb⟩ := hc.exists_ne_and_degree_eq_one
-    have hex : ∃ a : C, a.val ≠ y ∧ (T.induce C).degree a = 1 := by
-      by_cases hay : a.val = y
-      · refine ⟨b,?_,hb⟩
-        intro hby
-        exact hab (Subtype.ext (hay.trans hby.symm))
-      · exact ⟨a,hay,ha⟩
-    obtain ⟨a,hay,ha⟩ := hex
-    refine ⟨a.val,fun h => hxC (h ▸ a.property),hr _ a.property,?_⟩
-    have hn : T.neighborSet a.val ⊆ C := by
-      intro w hw
-      rcases hnbr a.val a.property w hw with rfl | hwc
-      · exact False.elim (hay (hnx a.val a.property hw))
-      · exact hwc
-    rwa [T.degree_induce_of_neighborSet_subset hn] at ha
-  · push Not at he
-    refine ⟨y,hxy.ne.symm,Reachable.refl y,?_⟩
-    apply degree_eq_one_iff_existsUnique_adj.mpr
-    refine ⟨x,hxy.symm,?_⟩
-    intro w hw
-    rcases hnbr y hyC w hw with hwx | hwC
-    · exact hwx
-    · exact False.elim (hw.ne (he w hwC).symm)
-
-theorem pruned_tree_branch_marked [Fintype V] (T : SimpleGraph V) (ht : T.IsTree)
-    (S : Finset V) (hpr : ∀ z, z ∉ S → 2 ≤ T.degree z)
-    (x y : V) (hxy : T.Adj x y) :
-    ∃ z ∈ S, z ≠ x ∧ (cutVertex T x).Reachable y z := by
-  obtain ⟨z,hzx,hz,hd⟩ := tree_branch_contains_leaf T ht x y hxy
-  refine ⟨z,?_,hzx,hz⟩
-  by_contra hn
-  have hh := hpr z hn
-  omega
-
-theorem tree_three_leaves_branch [Fintype V] (T : SimpleGraph V) (ht : T.IsTree)
-    (S : Finset V) (hcard : 3 ≤ S.card) (hleaf : ∀ z ∈ S, T.degree z = 1) :
-    ∃ x, 3 ≤ T.degree x := by
-  by_contra hn
-  have hd (x : V) : T.degree x ≤ 2 := by have := not_exists.mp hn x; omega
-  have hs : (∑ x : V, (T.degree x + (if x ∈ S then 1 else 0))) ≤
-      ∑ x : V, (2:ℕ) := by
-    apply Finset.sum_le_sum
-    intro x _
-    by_cases hx : x ∈ S
-    · simp only [hx,ite_true,hleaf x hx]; omega
-    · simpa only [hx,ite_false,add_zero] using hd x
-  have hsum : (∑ x : V, if x ∈ S then (1:ℕ) else 0) = S.card := by
-    simp
-  rw [Finset.sum_add_distrib,hsum,T.sum_degrees_eq_twice_card_edges] at hs
-  simp only [Finset.sum_const,Finset.card_univ,smul_eq_mul] at hs
-  have he := ht.card_edgeFinset
-  have hc : S.card ≤ Fintype.card V := Finset.card_le_univ S
-  omega
-
-theorem pruned_tree_three_branches [Fintype V] (T : SimpleGraph V) (ht : T.IsTree)
-    (S : Finset V) (hcard : 3 ≤ S.card) (hleaf : ∀ z ∈ S, T.degree z = 1)
-    (hpr : ∀ z, z ∉ S → 2 ≤ T.degree z) :
-    ∃ x a b c, x ∉ S ∧ a ∈ S ∧ b ∈ S ∧ c ∈ S ∧
-      a ≠ x ∧ b ≠ x ∧ c ≠ x ∧
-      ¬ (cutVertex T x).Reachable a b ∧
-      ¬ (cutVertex T x).Reachable a c ∧
-      ¬ (cutVertex T x).Reachable b c := by
-  obtain ⟨x,hx⟩ := tree_three_leaves_branch T ht S hcard hleaf
-  have hxS : x ∉ S := by intro h; have := hleaf x h; omega
-  have hnc : 2 < (T.neighborFinset x).card := by simpa using (show 2 < T.degree x by omega)
-  obtain ⟨u,v,w,hu,hv,hw,huv,huw,hvw⟩ := Finset.two_lt_card_iff.mp hnc
-  have hau := (T.mem_neighborFinset x u).mp hu
-  have hav := (T.mem_neighborFinset x v).mp hv
-  have haw := (T.mem_neighborFinset x w).mp hw
-  obtain ⟨a,ha,hax,hua⟩ := pruned_tree_branch_marked T ht S hpr x u hau
-  obtain ⟨b,hb,hbx,hvb⟩ := pruned_tree_branch_marked T ht S hpr x v hav
-  obtain ⟨c,hc,hcx,hwc⟩ := pruned_tree_branch_marked T ht S hpr x w haw
-  refine ⟨x,a,b,c,hxS,ha,hb,hc,hax,hbx,hcx,?_,?_,?_⟩
-  · intro h; exact huv (tree_cut_neighbors T ht.isAcyclic x u v hau hav
-      (hua.trans (h.trans hvb.symm)))
-  · intro h; exact huw (tree_cut_neighbors T ht.isAcyclic x u w hau haw
-      (hua.trans (h.trans hwc.symm)))
-  · intro h; exact hvw (tree_cut_neighbors T ht.isAcyclic x v w hav haw
-      (hvb.trans (h.trans hwc.symm)))
-end Percolation
-
-namespace Percolation
-open SimpleGraph
-universe u v w
-variable {V : Type u} {E : Type v} {W : Type w}
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem exists_cluster_retraction (G : BondGraph V E) (η : Set E) (f : W ↪ V)
-    (hsep : ∀ a b, Conn G η (f a) (f b) → a = b) (x : W) :
-    ∃ r : V → W, (∀ a, r (f a) = a) ∧
-      ∀ a b, Conn G η a b → r a = r b := by
-  let r : V → W := fun y => if h : ∃ z, Conn G η (f z) y then h.choose else x
-  have hval (y : V) (z : W) (hz : Conn G η (f z) y) : r y = z := by
-    have hex : ∃ z, Conn G η (f z) y := ⟨z,hz⟩
-    dsimp [r]
-    rw [dite_eq_left hex]
-    exact hsep _ _ (conn_trans G η hex.choose_spec (conn_symm G η hz))
-  refine ⟨r,fun a => hval _ _ (conn_refl G η _),?_⟩
-  intro a b hab
-  by_cases ha : ∃ z, Conn G η (f z) a
-  · obtain ⟨z,hz⟩ := ha
-    rw [hval a z hz,hval b z (conn_trans G η hz hab)]
-  · have hb : ¬ ∃ z, Conn G η (f z) b := by
-      rintro ⟨z,hz⟩
-      exact ha ⟨z,conn_trans G η hz (conn_symm G η hab)⟩
-    simp only [r,dite_eq_right ha,dite_eq_right hb]
-
-theorem glued_cut_conn (G : BondGraph V E) (η P : Set E) (T : SimpleGraph W)
-    (f : W ↪ V) (r : V → W) (hr : ∀ a, r (f a) = a)
-    (hη : ∀ a b, Conn G η a b → r a = r b)
-    (hP : ∀ e ∈ P, ∃ a b, T.Adj a b ∧ G e = s(f a,f b))
-    (x : W) {a b : V} (hab : Conn G (cutConfig G (η ∪ P) (f x)) a b) :
-    (cutVertex T x).Reachable (r a) (r b) := by
-  induction hab with
-  | refl => exact .refl _
-  | @tail a b hab hed ih =>
-    obtain ⟨e,⟨he,hx⟩,hGe⟩ := hed
-    rcases he with he | he
-    · have hh := hη a b ((conn_refl G η a).tail ⟨e,he,hGe⟩)
-      rwa [←hh]
-    · obtain ⟨u,v,huv,heq⟩ := hP e he
-      have hne : u ≠ x ∧ v ≠ x := by
-        rw [heq] at hx
-        simpa [Sym2.mem_iff,f.injective.eq_iff,eq_comm] using hx
-      have hkv : (cutVertex T x).Adj u v := ⟨huv,hne⟩
-      have habuv := Sym2.eq_iff.mp (hGe.symm.trans heq)
-      rcases habuv with ⟨rfl,rfl⟩ | ⟨rfl,rfl⟩
-      · rw [hr] at ih ⊢
-        exact ih.trans hkv.reachable
-      · rw [hr] at ih ⊢
-        exact ih.trans hkv.symm.reachable
-
-theorem glued_conn (G : BondGraph V E) (ω : Set E) (T : SimpleGraph W)
-    (f : W → V) (hT : T.Connected)
-    (he : ∀ a b, T.Adj a b → ∃ e ∈ ω, G e = s(f a,f b)) (a b : W) :
-    Conn G ω (f a) (f b) := by
-  obtain ⟨p⟩ := hT a b
-  induction p with
-  | nil => exact conn_refl G ω _
-  | @cons a b c hab p ih =>
-    exact conn_trans G ω ((conn_refl G ω _).tail (he a b hab)) ih
-
-end Percolation
-
-namespace Percolation
-open SimpleGraph
-universe u
-variable {V : Type u}
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-def terminalGraph (H : SimpleGraph V) (A : Finset V) (v w : Fin 3 → V) : SimpleGraph V where
-  Adj a b := H.Adj a b ∧ ((a ∈ A ∧ b ∈ A) ∨
-    ∃ i, (a = v i ∧ b = w i) ∨ (a = w i ∧ b = v i))
-  symm := ⟨by
-    intro a b h
-    refine ⟨h.1.symm,?_⟩
-    rcases h.2 with h | ⟨i,h|h⟩
-    · exact Or.inl h.symm
-    · exact Or.inr ⟨i,Or.inr h.symm⟩
-    · exact Or.inr ⟨i,Or.inl h.symm⟩⟩
-  loopless := ⟨fun a h => H.loopless.irrefl a h.1⟩
-
-theorem terminalGraph_touch (H : SimpleGraph V) (A : Finset V) (v w : Fin 3 → V)
-    (hv : ∀ i, v i ∈ A) {a b : V} (h : (terminalGraph H A v w).Adj a b) :
-    a ∈ A ∨ b ∈ A := by
-  rcases h.2 with h | ⟨i,h|h⟩
-  · exact Or.inl h.1
-  · exact Or.inl (h.1 ▸ hv i)
-  · exact Or.inr (h.2 ▸ hv i)
-
-theorem terminalGraph_leaf (H : SimpleGraph V) (A : Finset V) (v w : Fin 3 → V)
-    (hv : ∀ i, v i ∈ A) (hw : ∀ i, w i ∉ A) (hwi : Function.Injective w)
-    (i : Fin 3) {y : V} (h : (terminalGraph H A v w).Adj (w i) y) : y = v i := by
-  rcases h.2 with h | ⟨j,h|h⟩
-  · exact False.elim (hw i h.1)
-  · exact False.elim (hw i (h.1 ▸ hv j))
-  · have hij := hwi h.1
-    simpa only [hij] using h.2
-
-theorem terminalGraph_connected (H : SimpleGraph V) (A : Finset V) (v w : Fin 3 → V)
-    (hv : ∀ i, v i ∈ A) (hadj : ∀ i, H.Adj (v i) (w i))
-    (hA : (H.induce (A:Set V)).Connected) :
-    ((terminalGraph H A v w).induce ((A ∪ Finset.univ.image w : Finset V):Set V)).Connected := by
-  let D := A ∪ Finset.univ.image w
-  let K := terminalGraph H A v w
-  have hAD : A ⊆ D := Finset.subset_union_left
-  let f : (H.induce (A:Set V)) →g (K.induce (D:Set V)) :=
-    ⟨fun a => ⟨a.val,hAD a.property⟩,fun {a b} hab => ⟨hab,Or.inl ⟨a.property,b.property⟩⟩⟩
-  let z : (D:Set V) := ⟨v 0,hAD (hv 0)⟩
-  have connA (a : V) (ha : a ∈ A) : (K.induce (D:Set V)).Reachable ⟨a,hAD ha⟩ z :=
-    (hA ⟨a,ha⟩ ⟨v 0,hv 0⟩).map f
-  have connD (a : (D:Set V)) : (K.induce (D:Set V)).Reachable a z := by
-    rcases Finset.mem_union.mp a.property with ha | ha
-    · exact connA a.val ha
-    · obtain ⟨i,_,hi⟩ := Finset.mem_image.mp ha
-      have hK : (K.induce (D:Set V)).Adj a ⟨v i,hAD (hv i)⟩ := by
-        change K.Adj a.val (v i)
-        rw [←hi]
-        exact ⟨(hadj i).symm,Or.inr ⟨i,Or.inr ⟨rfl,rfl⟩⟩⟩
-      exact hK.reachable.trans (connA _ (hv i))
-  let : Nonempty (D:Set V) := ⟨z⟩
-  exact ⟨fun a b => (connD a).trans (connD b).symm⟩
-
-theorem exists_terminal_tree (H : SimpleGraph V) (A : Finset V) (v w : Fin 3 → V)
-    (hv : ∀ i, v i ∈ A) (hw : ∀ i, w i ∉ A) (hwi : Function.Injective w)
-    (hadj : ∀ i, H.Adj (v i) (w i))
-    (hA : (H.induce (A:Set V)).Connected) :
-    ∃ D : Finset V, D ⊆ A ∪ Finset.univ.image w ∧ Finset.univ.image w ⊆ D ∧
-      ∃ T : SimpleGraph (D:Set V), T ≤ (terminalGraph H A v w).induce (D:Set V) ∧
-        T.IsTree ∧ ∃ x a b c : (D:Set V), x.val ∈ A ∧
-          a.val ∈ Finset.univ.image w ∧ b.val ∈ Finset.univ.image w ∧
-          c.val ∈ Finset.univ.image w ∧
-          ¬ (cutVertex T x).Reachable a b ∧
-          ¬ (cutVertex T x).Reachable a c ∧
-          ¬ (cutVertex T x).Reachable b c := by
-  let S := Finset.univ.image w
-  have hS : S.Nonempty := ⟨w 0,Finset.mem_image.mpr ⟨0,Finset.mem_univ _,rfl⟩⟩
-  obtain ⟨D,hD,hSD,T,hTK,hT,hpr⟩ := exists_pruned_connector
-    (terminalGraph H A v w) S (A ∪ S) hS Finset.subset_union_right
-    (terminalGraph_connected H A v w hv hadj hA)
-  let s : Fin 3 → (D:Set V) := fun i => ⟨w i,hSD (Finset.mem_image.mpr ⟨i,Finset.mem_univ _,rfl⟩)⟩
-  have hsi : Function.Injective s := fun _ _ h => hwi (congrArg Subtype.val h)
-  let S' := Finset.univ.image s
-  have hS' (z : (D:Set V)) : z ∈ S' ↔ z.val ∈ S := by
-    simp only [S',S,Finset.mem_image,Finset.mem_univ,true_and]
-    constructor
-    · rintro ⟨i,rfl⟩; exact ⟨i,rfl⟩
-    · rintro ⟨i,hi⟩; exact ⟨i,Subtype.ext hi⟩
-  have hcard : S'.card = 3 := by
-    rw [Finset.card_image_of_injective _ hsi]
-    simp
-  let : Nontrivial (D:Set V) := ⟨⟨s 0,s 1,fun h => by have := hsi h; norm_num at this⟩⟩
-  have hleaf (z : (D:Set V)) (hz : z ∈ S') : T.degree z = 1 := by
-    obtain ⟨i,_,hi⟩ := Finset.mem_image.mp ((hS' z).mp hz)
-    have hpos : 0 < T.degree z := lt_of_lt_of_le
-      hT.connected.preconnected.minDegree_pos_of_nontrivial (T.minDegree_le_degree z)
-    obtain ⟨a,ha⟩ := (T.degree_pos_iff_exists_adj z).mp hpos
-    apply degree_eq_one_iff_existsUnique_adj.mpr
-    refine ⟨a,ha,?_⟩
-    intro b hb
-    apply Subtype.ext
-    have ha' : a.val = v i := terminalGraph_leaf H A v w hv hw hwi i (by rw [hi]; exact hTK ha)
-    have hb' : b.val = v i := terminalGraph_leaf H A v w hv hw hwi i (by rw [hi]; exact hTK hb)
-    exact hb'.trans ha'.symm
-  obtain ⟨x,a,b,c,hx,ha,hb,hc,_,_,_,hab,hac,hbc⟩ :=
-    pruned_tree_three_branches T hT S' (by omega) hleaf
-      (fun z hz => hpr z (fun h => hz ((hS' z).mpr h)))
-  have hxA : x.val ∈ A := by
-    rcases Finset.mem_union.mp (hD x.property) with h | h
-    · exact h
-    · exact False.elim (hx ((hS' x).mpr h))
-  exact ⟨D,hD,hSD,T,hTK,hT,x,a,b,c,hxA,(hS' a).mp ha,(hS' b).mp hb,
-    (hS' c).mp hc,hab,hac,hbc⟩
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Main
+import OAI.Probability.UnimodularPercolation.Encounter.Vertices
+
+namespace OAI
+
+/-! Tree branches, cluster retractions and terminal connecting trees. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+open SimpleGraph
+universe u
+variable {V : Type u}
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem cutVertex_le_deleteEdge (T : SimpleGraph V) (x y : V) :
+    cutVertex T x ≤ T.deleteEdges {s(x,y)} := by
+  intro a b hab
+  apply deleteEdges_adj.mpr
+  refine ⟨hab.1,?_⟩
+  intro he
+  have hh : x ∈ s(a,b) := (Set.mem_singleton_iff.mp he) ▸ (by simp : x ∈ s(x,y))
+  rcases Sym2.mem_iff.mp hh with hx | hx
+  · exact hab.2.1 hx.symm
+  · exact hab.2.2 hx.symm
+
+theorem tree_cut_neighbors (T : SimpleGraph V) (ht : T.IsAcyclic) (x y z : V)
+    (hy : T.Adj x y) (hz : T.Adj x z) (hyz : (cutVertex T x).Reachable y z) : y = z := by
+  by_contra hn
+  have hb := (isAcyclic_iff_forall_adj_isBridge.mp ht) hy
+  have hd : (T.deleteEdges {s(x,y)}).Adj x z := by
+    apply deleteEdges_adj.mpr
+    refine ⟨hz,?_⟩
+    intro he
+    have hh := Sym2.eq_iff.mp (Set.mem_singleton_iff.mp he)
+    rcases hh with ⟨_,hzy⟩ | ⟨hxy,hzx⟩
+    · exact hn hzy.symm
+    · exact hy.ne hxy
+  exact (isBridge_iff.mp hb) (hd.reachable.trans (hyz.mono (cutVertex_le_deleteEdge T x y)).symm)
+
+theorem cut_reachable_self_eq (T : SimpleGraph V) (x z : V)
+    (h : (cutVertex T x).Reachable x z) : z = x := by
+  induction h with
+  | intro p =>
+    cases p with
+    | nil => rfl
+    | cons hadj _ => exact False.elim (hadj.2.1 rfl)
+
+theorem tree_branch_contains_leaf [Fintype V] (T : SimpleGraph V) (ht : T.IsTree)
+    (x y : V) (hxy : T.Adj x y) :
+    ∃ z, z ≠ x ∧ (cutVertex T x).Reachable y z ∧ T.degree z = 1 := by
+  let C := ((cutVertex T x).connectedComponentMk y).supp
+  have hyC : y ∈ C := (ConnectedComponent.mem_supp_iff _ _).mpr rfl
+  have hxC : x ∉ C := by
+    intro h
+    have hr : (cutVertex T x).Reachable y x := by
+      exact (ConnectedComponent.eq.mp ((ConnectedComponent.mem_supp_iff _ _).mp h)).symm
+    exact hxy.ne (cut_reachable_self_eq T x y hr.symm).symm
+  have hr (z : V) (hz : z ∈ C) : (cutVertex T x).Reachable y z :=
+    (ConnectedComponent.eq.mp ((ConnectedComponent.mem_supp_iff _ _).mp hz)).symm
+  have hnbr (z : V) (hz : z ∈ C) (w : V) (hzw : T.Adj z w) : w = x ∨ w ∈ C := by
+    by_cases hw : w = x
+    · exact Or.inl hw
+    · right
+      apply (ConnectedComponent.mem_supp_iff _ _).mpr
+      apply ConnectedComponent.sound
+      exact (hr z hz).trans (Adj.reachable (G := cutVertex T x)
+        ⟨hzw,fun h => hxC (h ▸ hz),hw⟩) |>.symm
+  have hnx (z : V) (hz : z ∈ C) (hzx : T.Adj z x) : z = y :=
+    (tree_cut_neighbors T ht.isAcyclic x y z hxy hzx.symm (hr z hz)).symm
+  have hccon : (T.induce C).Connected := by
+    apply SimpleGraph.Connected.mono (G := (cutVertex T x).induce C)
+    · intro a b hab; exact hab.1
+    · exact ((cutVertex T x).connectedComponentMk y).connected_toSimpleGraph
+  have hc : (T.induce C).IsTree := ⟨hccon,ht.isAcyclic.induce C⟩
+  by_cases he : ∃ z ∈ C, z ≠ y
+  · obtain ⟨z,hz,hzy⟩ := he
+    let : Nontrivial C := ⟨⟨⟨z,hz⟩,⟨y,hyC⟩,fun h => hzy (congrArg Subtype.val h)⟩⟩
+    obtain ⟨a,b,hab,ha,hb⟩ := hc.exists_ne_and_degree_eq_one
+    have hex : ∃ a : C, a.val ≠ y ∧ (T.induce C).degree a = 1 := by
+      by_cases hay : a.val = y
+      · refine ⟨b,?_,hb⟩
+        intro hby
+        exact hab (Subtype.ext (hay.trans hby.symm))
+      · exact ⟨a,hay,ha⟩
+    obtain ⟨a,hay,ha⟩ := hex
+    refine ⟨a.val,fun h => hxC (h ▸ a.property),hr _ a.property,?_⟩
+    have hn : T.neighborSet a.val ⊆ C := by
+      intro w hw
+      rcases hnbr a.val a.property w hw with rfl | hwc
+      · exact False.elim (hay (hnx a.val a.property hw))
+      · exact hwc
+    rwa [T.degree_induce_of_neighborSet_subset hn] at ha
+  · push Not at he
+    refine ⟨y,hxy.ne.symm,Reachable.refl y,?_⟩
+    apply degree_eq_one_iff_existsUnique_adj.mpr
+    refine ⟨x,hxy.symm,?_⟩
+    intro w hw
+    rcases hnbr y hyC w hw with hwx | hwC
+    · exact hwx
+    · exact False.elim (hw.ne (he w hwC).symm)
+
+theorem pruned_tree_branch_marked [Fintype V] (T : SimpleGraph V) (ht : T.IsTree)
+    (S : Finset V) (hpr : ∀ z, z ∉ S → 2 ≤ T.degree z)
+    (x y : V) (hxy : T.Adj x y) :
+    ∃ z ∈ S, z ≠ x ∧ (cutVertex T x).Reachable y z := by
+  obtain ⟨z,hzx,hz,hd⟩ := tree_branch_contains_leaf T ht x y hxy
+  refine ⟨z,?_,hzx,hz⟩
+  by_contra hn
+  have hh := hpr z hn
+  omega
+
+theorem tree_three_leaves_branch [Fintype V] (T : SimpleGraph V) (ht : T.IsTree)
+    (S : Finset V) (hcard : 3 ≤ S.card) (hleaf : ∀ z ∈ S, T.degree z = 1) :
+    ∃ x, 3 ≤ T.degree x := by
+  by_contra hn
+  have hd (x : V) : T.degree x ≤ 2 := by have := not_exists.mp hn x; omega
+  have hs : (∑ x : V, (T.degree x + (if x ∈ S then 1 else 0))) ≤
+      ∑ x : V, (2:ℕ) := by
+    apply Finset.sum_le_sum
+    intro x _
+    by_cases hx : x ∈ S
+    · simp only [hx,ite_true,hleaf x hx]; omega
+    · simpa only [hx,ite_false,add_zero] using hd x
+  have hsum : (∑ x : V, if x ∈ S then (1:ℕ) else 0) = S.card := by
+    simp
+  rw [Finset.sum_add_distrib,hsum,T.sum_degrees_eq_twice_card_edges] at hs
+  simp only [Finset.sum_const,Finset.card_univ,smul_eq_mul] at hs
+  have he := ht.card_edgeFinset
+  have hc : S.card ≤ Fintype.card V := Finset.card_le_univ S
+  omega
+
+theorem pruned_tree_three_branches [Fintype V] (T : SimpleGraph V) (ht : T.IsTree)
+    (S : Finset V) (hcard : 3 ≤ S.card) (hleaf : ∀ z ∈ S, T.degree z = 1)
+    (hpr : ∀ z, z ∉ S → 2 ≤ T.degree z) :
+    ∃ x a b c, x ∉ S ∧ a ∈ S ∧ b ∈ S ∧ c ∈ S ∧
+      a ≠ x ∧ b ≠ x ∧ c ≠ x ∧
+      ¬ (cutVertex T x).Reachable a b ∧
+      ¬ (cutVertex T x).Reachable a c ∧
+      ¬ (cutVertex T x).Reachable b c := by
+  obtain ⟨x,hx⟩ := tree_three_leaves_branch T ht S hcard hleaf
+  have hxS : x ∉ S := by intro h; have := hleaf x h; omega
+  have hnc : 2 < (T.neighborFinset x).card := by simpa using (show 2 < T.degree x by omega)
+  obtain ⟨u,v,w,hu,hv,hw,huv,huw,hvw⟩ := Finset.two_lt_card_iff.mp hnc
+  have hau := (T.mem_neighborFinset x u).mp hu
+  have hav := (T.mem_neighborFinset x v).mp hv
+  have haw := (T.mem_neighborFinset x w).mp hw
+  obtain ⟨a,ha,hax,hua⟩ := pruned_tree_branch_marked T ht S hpr x u hau
+  obtain ⟨b,hb,hbx,hvb⟩ := pruned_tree_branch_marked T ht S hpr x v hav
+  obtain ⟨c,hc,hcx,hwc⟩ := pruned_tree_branch_marked T ht S hpr x w haw
+  refine ⟨x,a,b,c,hxS,ha,hb,hc,hax,hbx,hcx,?_,?_,?_⟩
+  · intro h; exact huv (tree_cut_neighbors T ht.isAcyclic x u v hau hav
+      (hua.trans (h.trans hvb.symm)))
+  · intro h; exact huw (tree_cut_neighbors T ht.isAcyclic x u w hau haw
+      (hua.trans (h.trans hwc.symm)))
+  · intro h; exact hvw (tree_cut_neighbors T ht.isAcyclic x v w hav haw
+      (hvb.trans (h.trans hwc.symm)))
+end Percolation
+
+namespace Percolation
+open SimpleGraph
+universe u v w
+variable {V : Type u} {E : Type v} {W : Type w}
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem exists_cluster_retraction (G : BondGraph V E) (η : Set E) (f : W ↪ V)
+    (hsep : ∀ a b, Conn G η (f a) (f b) → a = b) (x : W) :
+    ∃ r : V → W, (∀ a, r (f a) = a) ∧
+      ∀ a b, Conn G η a b → r a = r b := by
+  let r : V → W := fun y => if h : ∃ z, Conn G η (f z) y then h.choose else x
+  have hval (y : V) (z : W) (hz : Conn G η (f z) y) : r y = z := by
+    have hex : ∃ z, Conn G η (f z) y := ⟨z,hz⟩
+    dsimp [r]
+    rw [dite_eq_left hex]
+    exact hsep _ _ (conn_trans G η hex.choose_spec (conn_symm G η hz))
+  refine ⟨r,fun a => hval _ _ (conn_refl G η _),?_⟩
+  intro a b hab
+  by_cases ha : ∃ z, Conn G η (f z) a
+  · obtain ⟨z,hz⟩ := ha
+    rw [hval a z hz,hval b z (conn_trans G η hz hab)]
+  · have hb : ¬ ∃ z, Conn G η (f z) b := by
+      rintro ⟨z,hz⟩
+      exact ha ⟨z,conn_trans G η hz (conn_symm G η hab)⟩
+    simp only [r,dite_eq_right ha,dite_eq_right hb]
+
+theorem glued_cut_conn (G : BondGraph V E) (η P : Set E) (T : SimpleGraph W)
+    (f : W ↪ V) (r : V → W) (hr : ∀ a, r (f a) = a)
+    (hη : ∀ a b, Conn G η a b → r a = r b)
+    (hP : ∀ e ∈ P, ∃ a b, T.Adj a b ∧ G e = s(f a,f b))
+    (x : W) {a b : V} (hab : Conn G (cutConfig G (η ∪ P) (f x)) a b) :
+    (cutVertex T x).Reachable (r a) (r b) := by
+  induction hab with
+  | refl => exact .refl _
+  | @tail a b hab hed ih =>
+    obtain ⟨e,⟨he,hx⟩,hGe⟩ := hed
+    rcases he with he | he
+    · have hh := hη a b ((conn_refl G η a).tail ⟨e,he,hGe⟩)
+      rwa [←hh]
+    · obtain ⟨u,v,huv,heq⟩ := hP e he
+      have hne : u ≠ x ∧ v ≠ x := by
+        rw [heq] at hx
+        simpa [Sym2.mem_iff,f.injective.eq_iff,eq_comm] using hx
+      have hkv : (cutVertex T x).Adj u v := ⟨huv,hne⟩
+      have habuv := Sym2.eq_iff.mp (hGe.symm.trans heq)
+      rcases habuv with ⟨rfl,rfl⟩ | ⟨rfl,rfl⟩
+      · rw [hr] at ih ⊢
+        exact ih.trans hkv.reachable
+      · rw [hr] at ih ⊢
+        exact ih.trans hkv.symm.reachable
+
+theorem glued_conn (G : BondGraph V E) (ω : Set E) (T : SimpleGraph W)
+    (f : W → V) (hT : T.Connected)
+    (he : ∀ a b, T.Adj a b → ∃ e ∈ ω, G e = s(f a,f b)) (a b : W) :
+    Conn G ω (f a) (f b) := by
+  obtain ⟨p⟩ := hT a b
+  induction p with
+  | nil => exact conn_refl G ω _
+  | @cons a b c hab p ih =>
+    exact conn_trans G ω ((conn_refl G ω _).tail (he a b hab)) ih
+
+end Percolation
+
+namespace Percolation
+open SimpleGraph
+universe u
+variable {V : Type u}
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+def terminalGraph (H : SimpleGraph V) (A : Finset V) (v w : Fin 3 → V) : SimpleGraph V where
+  Adj a b := H.Adj a b ∧ ((a ∈ A ∧ b ∈ A) ∨
+    ∃ i, (a = v i ∧ b = w i) ∨ (a = w i ∧ b = v i))
+  symm := ⟨by
+    intro a b h
+    refine ⟨h.1.symm,?_⟩
+    rcases h.2 with h | ⟨i,h|h⟩
+    · exact Or.inl h.symm
+    · exact Or.inr ⟨i,Or.inr h.symm⟩
+    · exact Or.inr ⟨i,Or.inl h.symm⟩⟩
+  loopless := ⟨fun a h => H.loopless.irrefl a h.1⟩
+
+theorem terminalGraph_touch (H : SimpleGraph V) (A : Finset V) (v w : Fin 3 → V)
+    (hv : ∀ i, v i ∈ A) {a b : V} (h : (terminalGraph H A v w).Adj a b) :
+    a ∈ A ∨ b ∈ A := by
+  rcases h.2 with h | ⟨i,h|h⟩
+  · exact Or.inl h.1
+  · exact Or.inl (h.1 ▸ hv i)
+  · exact Or.inr (h.2 ▸ hv i)
+
+theorem terminalGraph_leaf (H : SimpleGraph V) (A : Finset V) (v w : Fin 3 → V)
+    (hv : ∀ i, v i ∈ A) (hw : ∀ i, w i ∉ A) (hwi : Function.Injective w)
+    (i : Fin 3) {y : V} (h : (terminalGraph H A v w).Adj (w i) y) : y = v i := by
+  rcases h.2 with h | ⟨j,h|h⟩
+  · exact False.elim (hw i h.1)
+  · exact False.elim (hw i (h.1 ▸ hv j))
+  · have hij := hwi h.1
+    simpa only [hij] using h.2
+
+theorem terminalGraph_connected (H : SimpleGraph V) (A : Finset V) (v w : Fin 3 → V)
+    (hv : ∀ i, v i ∈ A) (hadj : ∀ i, H.Adj (v i) (w i))
+    (hA : (H.induce (A:Set V)).Connected) :
+    ((terminalGraph H A v w).induce ((A ∪ Finset.univ.image w : Finset V):Set V)).Connected := by
+  let D := A ∪ Finset.univ.image w
+  let K := terminalGraph H A v w
+  have hAD : A ⊆ D := Finset.subset_union_left
+  let f : (H.induce (A:Set V)) →g (K.induce (D:Set V)) :=
+    ⟨fun a => ⟨a.val,hAD a.property⟩,fun {a b} hab => ⟨hab,Or.inl ⟨a.property,b.property⟩⟩⟩
+  let z : (D:Set V) := ⟨v 0,hAD (hv 0)⟩
+  have connA (a : V) (ha : a ∈ A) : (K.induce (D:Set V)).Reachable ⟨a,hAD ha⟩ z :=
+    (hA ⟨a,ha⟩ ⟨v 0,hv 0⟩).map f
+  have connD (a : (D:Set V)) : (K.induce (D:Set V)).Reachable a z := by
+    rcases Finset.mem_union.mp a.property with ha | ha
+    · exact connA a.val ha
+    · obtain ⟨i,_,hi⟩ := Finset.mem_image.mp ha
+      have hK : (K.induce (D:Set V)).Adj a ⟨v i,hAD (hv i)⟩ := by
+        change K.Adj a.val (v i)
+        rw [←hi]
+        exact ⟨(hadj i).symm,Or.inr ⟨i,Or.inr ⟨rfl,rfl⟩⟩⟩
+      exact hK.reachable.trans (connA _ (hv i))
+  let : Nonempty (D:Set V) := ⟨z⟩
+  exact ⟨fun a b => (connD a).trans (connD b).symm⟩
+
+theorem exists_terminal_tree (H : SimpleGraph V) (A : Finset V) (v w : Fin 3 → V)
+    (hv : ∀ i, v i ∈ A) (hw : ∀ i, w i ∉ A) (hwi : Function.Injective w)
+    (hadj : ∀ i, H.Adj (v i) (w i))
+    (hA : (H.induce (A:Set V)).Connected) :
+    ∃ D : Finset V, D ⊆ A ∪ Finset.univ.image w ∧ Finset.univ.image w ⊆ D ∧
+      ∃ T : SimpleGraph (D:Set V), T ≤ (terminalGraph H A v w).induce (D:Set V) ∧
+        T.IsTree ∧ ∃ x a b c : (D:Set V), x.val ∈ A ∧
+          a.val ∈ Finset.univ.image w ∧ b.val ∈ Finset.univ.image w ∧
+          c.val ∈ Finset.univ.image w ∧
+          ¬ (cutVertex T x).Reachable a b ∧
+          ¬ (cutVertex T x).Reachable a c ∧
+          ¬ (cutVertex T x).Reachable b c := by
+  let S := Finset.univ.image w
+  have hS : S.Nonempty := ⟨w 0,Finset.mem_image.mpr ⟨0,Finset.mem_univ _,rfl⟩⟩
+  obtain ⟨D,hD,hSD,T,hTK,hT,hpr⟩ := exists_pruned_connector
+    (terminalGraph H A v w) S (A ∪ S) hS Finset.subset_union_right
+    (terminalGraph_connected H A v w hv hadj hA)
+  let s : Fin 3 → (D:Set V) := fun i => ⟨w i,hSD (Finset.mem_image.mpr ⟨i,Finset.mem_univ _,rfl⟩)⟩
+  have hsi : Function.Injective s := fun _ _ h => hwi (congrArg Subtype.val h)
+  let S' := Finset.univ.image s
+  have hS' (z : (D:Set V)) : z ∈ S' ↔ z.val ∈ S := by
+    simp only [S',S,Finset.mem_image,Finset.mem_univ,true_and]
+    constructor
+    · rintro ⟨i,rfl⟩; exact ⟨i,rfl⟩
+    · rintro ⟨i,hi⟩; exact ⟨i,Subtype.ext hi⟩
+  have hcard : S'.card = 3 := by
+    rw [Finset.card_image_of_injective _ hsi]
+    simp
+  let : Nontrivial (D:Set V) := ⟨⟨s 0,s 1,fun h => by have := hsi h; norm_num at this⟩⟩
+  have hleaf (z : (D:Set V)) (hz : z ∈ S') : T.degree z = 1 := by
+    obtain ⟨i,_,hi⟩ := Finset.mem_image.mp ((hS' z).mp hz)
+    have hpos : 0 < T.degree z := lt_of_lt_of_le
+      hT.connected.preconnected.minDegree_pos_of_nontrivial (T.minDegree_le_degree z)
+    obtain ⟨a,ha⟩ := (T.degree_pos_iff_exists_adj z).mp hpos
+    apply degree_eq_one_iff_existsUnique_adj.mpr
+    refine ⟨a,ha,?_⟩
+    intro b hb
+    apply Subtype.ext
+    have ha' : a.val = v i := terminalGraph_leaf H A v w hv hw hwi i (by rw [hi]; exact hTK ha)
+    have hb' : b.val = v i := terminalGraph_leaf H A v w hv hw hwi i (by rw [hi]; exact hTK hb)
+    exact hb'.trans ha'.symm
+  obtain ⟨x,a,b,c,hx,ha,hb,hc,_,_,_,hab,hac,hbc⟩ :=
+    pruned_tree_three_branches T hT S' (by omega) hleaf
+      (fun z hz => hpr z (fun h => hz ((hS' z).mpr h)))
+  have hxA : x.val ∈ A := by
+    rcases Finset.mem_union.mp (hD x.property) with h | h
+    · exact h
+    · exact False.elim (hx ((hS' x).mpr h))
+  exact ⟨D,hD,hSD,T,hTK,hT,x,a,b,c,hxA,(hS' a).mp ha,(hS' b).mp hb,
+    (hS' c).mp hc,hab,hac,hbc⟩
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Encounter/Vertices.lean b/lean/OAI/Probability/UnimodularPercolation/Encounter/Vertices.lean
index b1c507254a603155ce57fdcc0cccdbca81314f2e..0d07a4ec7944eb3bad847d8079bf03ce187be218 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Encounter/Vertices.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Encounter/Vertices.lean
@@ -1,151 +1,151 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Main
-import OAI.Probability.UnimodularPercolation.Encounter.NearestMass
-import OAI.Probability.UnimodularPercolation.Encounter.Transport
-
-namespace OAI
-
-/-! Encounter vertices and infinite open branches. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-open MeasureTheory SimpleGraph
-open scoped ENNReal
-universe u v w
-variable {V : Type u} {E : Type v} {ι : Type w}
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-def cutConfig (G : BondGraph V E) (ω : Set E) (x : V) : Set E :=
-  {e | e ∈ ω ∧ x ∉ G e}
-
-theorem measurable_cutConfig (G : BondGraph V E) (x : V) :
-    Measurable (fun ω => cutConfig G ω x) := by
-  apply measurable_setOfPred.comp (Measurable.of_eval ?_)
-  intro e
-  exact (measurable_set_mem e).and measurable_const
-
-theorem cutVertex_open (G : BondGraph V E) (ω : Set E) (x : V) :
-    cutVertex (openSimpleGraph G ω) x = openSimpleGraph G (cutConfig G ω x) := by
-  ext a b
-  change ((a ≠ b ∧ ∃ e ∈ ω, G e = s(a,b)) ∧ a ≠ x ∧ b ≠ x) ↔
-    (a ≠ b ∧ ∃ e, (e ∈ ω ∧ x ∉ G e) ∧ G e = s(a,b))
-  constructor
-  · rintro ⟨⟨hab,e,he,hGe⟩,hax,hbx⟩
-    exact ⟨hab,e,⟨he,by rw [hGe]; simpa [Sym2.mem_iff,eq_comm] using And.intro hax hbx⟩,hGe⟩
-  · rintro ⟨hab,e,⟨he,hx⟩,hGe⟩
-    rw [hGe] at hx
-    have hh : a ≠ x ∧ b ≠ x := by simpa [Sym2.mem_iff,eq_comm] using hx
-    exact ⟨⟨hab,e,he,hGe⟩,hh⟩
-
-theorem cutConfig_aut (G : BondGraph V E) (a : V ≃ V) (b : E ≃ E)
-    (hab : IsAut G a b) (ω : Set E) (x : V) :
-    cutConfig G (b ⁻¹' ω) x = b ⁻¹' cutConfig G ω (a x) := by
-  ext e
-  simp only [cutConfig,Set.mem_ofPred_eq,Set.mem_preimage,hab e]
-  rw [Sym2.mem_map]
-  simp only [a.injective.eq_iff,exists_eq_right]
-
-def IsEncounter (G : BondGraph V E) (ω : Set E) (x : V) : Prop :=
-  ∃ a b c : V,
-    Conn G ω x a ∧ Conn G ω x b ∧ Conn G ω x c ∧
-    (cluster G (cutConfig G ω x) a).Infinite ∧
-    (cluster G (cutConfig G ω x) b).Infinite ∧
-    (cluster G (cutConfig G ω x) c).Infinite ∧
-    ¬ Conn G (cutConfig G ω x) a b ∧
-    ¬ Conn G (cutConfig G ω x) a c ∧
-    ¬ Conn G (cutConfig G ω x) b c
-
-def encounters (G : BondGraph V E) (ω : Set E) : Set V := {x | IsEncounter G ω x}
-
-theorem measurableSet_isEncounter [Countable V] [Countable E]
-    (G : BondGraph V E) (x : V) : MeasurableSet {ω | IsEncounter G ω x} := by
-  simp only [IsEncounter,Set.ofPred_exists,Set.ofPred_and]
-  apply MeasurableSet.iUnion
-  intro a
-  apply MeasurableSet.iUnion
-  intro b
-  apply MeasurableSet.iUnion
-  intro c
-  exact (measurableSet_conn G x a).inter ((measurableSet_conn G x b).inter
-    ((measurableSet_conn G x c).inter
-      (((measurableSet_infinite_cluster G a).preimage (measurable_cutConfig G x)).inter
-      (((measurableSet_infinite_cluster G b).preimage (measurable_cutConfig G x)).inter
-      (((measurableSet_infinite_cluster G c).preimage (measurable_cutConfig G x)).inter
-      (((measurableSet_conn G a b).preimage (measurable_cutConfig G x)).compl.inter
-      (((measurableSet_conn G a c).preimage (measurable_cutConfig G x)).compl.inter
-      ((measurableSet_conn G b c).preimage (measurable_cutConfig G x)).compl)))))))
-
-theorem isEncounter_aut_iff (G : BondGraph V E) (a : V ≃ V) (b : E ≃ E)
-    (hab : IsAut G a b) (ω : Set E) (x : V) :
-    IsEncounter G (b ⁻¹' ω) x ↔ IsEncounter G ω (a x) := by
-  have hc y z : Conn G (cutConfig G (b ⁻¹' ω) x) y z ↔
-      Conn G (cutConfig G ω (a x)) (a y) (a z) := by
-    rw [cutConfig_aut G a b hab]
-    exact conn_aut_iff G hab _ y z
-  have hi y : (cluster G (cutConfig G (b ⁻¹' ω) x) y).Infinite ↔
-      (cluster G (cutConfig G ω (a x)) (a y)).Infinite := by
-    rw [cutConfig_aut G a b hab,←cluster_aut G hab]
-    exact (Set.infinite_image_iff a.injective.injOn).symm
-  constructor
-  · rintro ⟨u,v,w,hxu,hxv,hxw,hu,hv,hw,huv,huw,hvw⟩
-    exact ⟨a u,a v,a w,(conn_aut_iff G hab ω x u).mp hxu,
-      (conn_aut_iff G hab ω x v).mp hxv,(conn_aut_iff G hab ω x w).mp hxw,
-      (hi u).mp hu,(hi v).mp hv,(hi w).mp hw,
-      fun h => huv ((hc u v).mpr h),fun h => huw ((hc u w).mpr h),fun h => hvw ((hc v w).mpr h)⟩
-  · rintro ⟨u,v,w,hxu,hxv,hxw,hu,hv,hw,huv,huw,hvw⟩
-    refine ⟨a.symm u,a.symm v,a.symm w,?_,?_,?_,?_,?_,?_,?_,?_,?_⟩
-    · exact (conn_aut_iff G hab ω x _).mpr (by simpa using hxu)
-    · exact (conn_aut_iff G hab ω x _).mpr (by simpa using hxv)
-    · exact (conn_aut_iff G hab ω x _).mpr (by simpa using hxw)
-    · exact (hi _).mpr (by simpa using hu)
-    · exact (hi _).mpr (by simpa using hv)
-    · exact (hi _).mpr (by simpa using hw)
-    · intro h; exact huv (by simpa using (hc _ _).mp h)
-    · intro h; exact huw (by simpa using (hc _ _).mp h)
-    · intro h; exact hvw (by simpa using (hc _ _).mp h)
-
-theorem encounter_branch_contains_of_finite_incoming (G : BondGraph V E)
-    (ω : Set E) (x a : V) (hx : IsEncounter G ω x) (hxa : Conn G ω x a)
-    (hInf : (cluster G (cutConfig G ω x) a).Infinite)
-    (hin : ∑' y, nearestMass G (encounters G) ω y x < ⊤) :
-    ∃ z ∈ cluster G (cutConfig G ω x) a, z ≠ x ∧ IsEncounter G ω z := by
-  by_contra hno
-  have hm (y : V) (hy : y ∈ cluster G (cutConfig G ω x) a) :
-      nearestMass G (encounters G) ω y x = 1 := by
-    apply nearestMass_eq_one
-    apply closest_eq_singleton_of_branch (openSimpleGraph G ω) (encounters G ω) hx
-    · apply (conn_iff_reachable G ω y x).mp
-      exact conn_symm G ω (conn_trans G ω hxa (conn_mono G (fun e he => he.1) hy))
-    · intro z hz hzx hcut
-      apply hno
-      refine ⟨z,?_,hzx,hz⟩
-      rw [cutVertex_open] at hcut
-      exact conn_trans G _ hy ((conn_iff_reachable G _ y z).mpr hcut)
-  have he := incoming_top_of_infinite (fun y => nearestMass G (encounters G) ω y x)
-    (cluster G (cutConfig G ω x) a) hInf hm
-  rw [he] at hin
-  exact lt_irrefl _ hin
-
-theorem encounter_branches_ae [Countable V] [Countable E] [Fintype ι]
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
-    (root : ι → V) (weight : ι → ℝ)
-    (htr : RootTransport root weight (vertexAut G)) (hw : ∀ i, 0 < weight i)
-    (hw1 : ∑ i, weight i = 1)
-    (hcov : ∀ x, ∃ i, ∃ a ∈ vertexAut G, a (root i) = x) (p : unitInterval) :
-    ∀ᵐ ω ∂law p, ∀ x, IsEncounter G ω x → ∀ a, Conn G ω x a →
-      (cluster G (cutConfig G ω x) a).Infinite →
-      ∃ z ∈ cluster G (cutConfig G ω x) a, z ≠ x ∧ IsEncounter G ω z := by
-  have hin := incoming_finite_ae_all G hconn hloc root weight htr hw hw1 hcov p
-    (nearestMass G (encounters G))
-    (measurable_nearestMass G (encounters G) (measurableSet_isEncounter G))
-    (nearestMass_aut G (encounters G) (isEncounter_aut_iff G))
-    (nearestMass_row_le G hloc (encounters G))
-  filter_upwards [hin] with ω hω
-  intro x hx a hxa hi
-  exact encounter_branch_contains_of_finite_incoming G ω x a hx hxa hi (hω x)
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Main
+import OAI.Probability.UnimodularPercolation.Encounter.NearestMass
+import OAI.Probability.UnimodularPercolation.Encounter.Transport
+
+namespace OAI
+
+/-! Encounter vertices and infinite open branches. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+open MeasureTheory SimpleGraph
+open scoped ENNReal
+universe u v w
+variable {V : Type u} {E : Type v} {ι : Type w}
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+def cutConfig (G : BondGraph V E) (ω : Set E) (x : V) : Set E :=
+  {e | e ∈ ω ∧ x ∉ G e}
+
+theorem measurable_cutConfig (G : BondGraph V E) (x : V) :
+    Measurable (fun ω => cutConfig G ω x) := by
+  apply measurable_setOfPred.comp (Measurable.of_eval ?_)
+  intro e
+  exact (measurable_set_mem e).and measurable_const
+
+theorem cutVertex_open (G : BondGraph V E) (ω : Set E) (x : V) :
+    cutVertex (openSimpleGraph G ω) x = openSimpleGraph G (cutConfig G ω x) := by
+  ext a b
+  change ((a ≠ b ∧ ∃ e ∈ ω, G e = s(a,b)) ∧ a ≠ x ∧ b ≠ x) ↔
+    (a ≠ b ∧ ∃ e, (e ∈ ω ∧ x ∉ G e) ∧ G e = s(a,b))
+  constructor
+  · rintro ⟨⟨hab,e,he,hGe⟩,hax,hbx⟩
+    exact ⟨hab,e,⟨he,by rw [hGe]; simpa [Sym2.mem_iff,eq_comm] using And.intro hax hbx⟩,hGe⟩
+  · rintro ⟨hab,e,⟨he,hx⟩,hGe⟩
+    rw [hGe] at hx
+    have hh : a ≠ x ∧ b ≠ x := by simpa [Sym2.mem_iff,eq_comm] using hx
+    exact ⟨⟨hab,e,he,hGe⟩,hh⟩
+
+theorem cutConfig_aut (G : BondGraph V E) (a : V ≃ V) (b : E ≃ E)
+    (hab : IsAut G a b) (ω : Set E) (x : V) :
+    cutConfig G (b ⁻¹' ω) x = b ⁻¹' cutConfig G ω (a x) := by
+  ext e
+  simp only [cutConfig,Set.mem_ofPred_eq,Set.mem_preimage,hab e]
+  rw [Sym2.mem_map]
+  simp only [a.injective.eq_iff,exists_eq_right]
+
+def IsEncounter (G : BondGraph V E) (ω : Set E) (x : V) : Prop :=
+  ∃ a b c : V,
+    Conn G ω x a ∧ Conn G ω x b ∧ Conn G ω x c ∧
+    (cluster G (cutConfig G ω x) a).Infinite ∧
+    (cluster G (cutConfig G ω x) b).Infinite ∧
+    (cluster G (cutConfig G ω x) c).Infinite ∧
+    ¬ Conn G (cutConfig G ω x) a b ∧
+    ¬ Conn G (cutConfig G ω x) a c ∧
+    ¬ Conn G (cutConfig G ω x) b c
+
+def encounters (G : BondGraph V E) (ω : Set E) : Set V := {x | IsEncounter G ω x}
+
+theorem measurableSet_isEncounter [Countable V] [Countable E]
+    (G : BondGraph V E) (x : V) : MeasurableSet {ω | IsEncounter G ω x} := by
+  simp only [IsEncounter,Set.ofPred_exists,Set.ofPred_and]
+  apply MeasurableSet.iUnion
+  intro a
+  apply MeasurableSet.iUnion
+  intro b
+  apply MeasurableSet.iUnion
+  intro c
+  exact (measurableSet_conn G x a).inter ((measurableSet_conn G x b).inter
+    ((measurableSet_conn G x c).inter
+      (((measurableSet_infinite_cluster G a).preimage (measurable_cutConfig G x)).inter
+      (((measurableSet_infinite_cluster G b).preimage (measurable_cutConfig G x)).inter
+      (((measurableSet_infinite_cluster G c).preimage (measurable_cutConfig G x)).inter
+      (((measurableSet_conn G a b).preimage (measurable_cutConfig G x)).compl.inter
+      (((measurableSet_conn G a c).preimage (measurable_cutConfig G x)).compl.inter
+      ((measurableSet_conn G b c).preimage (measurable_cutConfig G x)).compl)))))))
+
+theorem isEncounter_aut_iff (G : BondGraph V E) (a : V ≃ V) (b : E ≃ E)
+    (hab : IsAut G a b) (ω : Set E) (x : V) :
+    IsEncounter G (b ⁻¹' ω) x ↔ IsEncounter G ω (a x) := by
+  have hc y z : Conn G (cutConfig G (b ⁻¹' ω) x) y z ↔
+      Conn G (cutConfig G ω (a x)) (a y) (a z) := by
+    rw [cutConfig_aut G a b hab]
+    exact conn_aut_iff G hab _ y z
+  have hi y : (cluster G (cutConfig G (b ⁻¹' ω) x) y).Infinite ↔
+      (cluster G (cutConfig G ω (a x)) (a y)).Infinite := by
+    rw [cutConfig_aut G a b hab,←cluster_aut G hab]
+    exact (Set.infinite_image_iff a.injective.injOn).symm
+  constructor
+  · rintro ⟨u,v,w,hxu,hxv,hxw,hu,hv,hw,huv,huw,hvw⟩
+    exact ⟨a u,a v,a w,(conn_aut_iff G hab ω x u).mp hxu,
+      (conn_aut_iff G hab ω x v).mp hxv,(conn_aut_iff G hab ω x w).mp hxw,
+      (hi u).mp hu,(hi v).mp hv,(hi w).mp hw,
+      fun h => huv ((hc u v).mpr h),fun h => huw ((hc u w).mpr h),fun h => hvw ((hc v w).mpr h)⟩
+  · rintro ⟨u,v,w,hxu,hxv,hxw,hu,hv,hw,huv,huw,hvw⟩
+    refine ⟨a.symm u,a.symm v,a.symm w,?_,?_,?_,?_,?_,?_,?_,?_,?_⟩
+    · exact (conn_aut_iff G hab ω x _).mpr (by simpa using hxu)
+    · exact (conn_aut_iff G hab ω x _).mpr (by simpa using hxv)
+    · exact (conn_aut_iff G hab ω x _).mpr (by simpa using hxw)
+    · exact (hi _).mpr (by simpa using hu)
+    · exact (hi _).mpr (by simpa using hv)
+    · exact (hi _).mpr (by simpa using hw)
+    · intro h; exact huv (by simpa using (hc _ _).mp h)
+    · intro h; exact huw (by simpa using (hc _ _).mp h)
+    · intro h; exact hvw (by simpa using (hc _ _).mp h)
+
+theorem encounter_branch_contains_of_finite_incoming (G : BondGraph V E)
+    (ω : Set E) (x a : V) (hx : IsEncounter G ω x) (hxa : Conn G ω x a)
+    (hInf : (cluster G (cutConfig G ω x) a).Infinite)
+    (hin : ∑' y, nearestMass G (encounters G) ω y x < ⊤) :
+    ∃ z ∈ cluster G (cutConfig G ω x) a, z ≠ x ∧ IsEncounter G ω z := by
+  by_contra hno
+  have hm (y : V) (hy : y ∈ cluster G (cutConfig G ω x) a) :
+      nearestMass G (encounters G) ω y x = 1 := by
+    apply nearestMass_eq_one
+    apply closest_eq_singleton_of_branch (openSimpleGraph G ω) (encounters G ω) hx
+    · apply (conn_iff_reachable G ω y x).mp
+      exact conn_symm G ω (conn_trans G ω hxa (conn_mono G (fun e he => he.1) hy))
+    · intro z hz hzx hcut
+      apply hno
+      refine ⟨z,?_,hzx,hz⟩
+      rw [cutVertex_open] at hcut
+      exact conn_trans G _ hy ((conn_iff_reachable G _ y z).mpr hcut)
+  have he := incoming_top_of_infinite (fun y => nearestMass G (encounters G) ω y x)
+    (cluster G (cutConfig G ω x) a) hInf hm
+  rw [he] at hin
+  exact lt_irrefl _ hin
+
+theorem encounter_branches_ae [Countable V] [Countable E] [Fintype ι]
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
+    (root : ι → V) (weight : ι → ℝ)
+    (htr : RootTransport root weight (vertexAut G)) (hw : ∀ i, 0 < weight i)
+    (hw1 : ∑ i, weight i = 1)
+    (hcov : ∀ x, ∃ i, ∃ a ∈ vertexAut G, a (root i) = x) (p : unitInterval) :
+    ∀ᵐ ω ∂law p, ∀ x, IsEncounter G ω x → ∀ a, Conn G ω x a →
+      (cluster G (cutConfig G ω x) a).Infinite →
+      ∃ z ∈ cluster G (cutConfig G ω x) a, z ≠ x ∧ IsEncounter G ω z := by
+  have hin := incoming_finite_ae_all G hconn hloc root weight htr hw hw1 hcov p
+    (nearestMass G (encounters G))
+    (measurable_nearestMass G (encounters G) (measurableSet_isEncounter G))
+    (nearestMass_aut G (encounters G) (isEncounter_aut_iff G))
+    (nearestMass_row_le G hloc (encounters G))
+  filter_upwards [hin] with ω hω
+  intro x hx a hxa hi
+  exact encounter_branch_contains_of_finite_incoming G ω x a hx hxa hi (hω x)
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Encounter/ZeroIntensity.lean b/lean/OAI/Probability/UnimodularPercolation/Encounter/ZeroIntensity.lean
index bb9e0c10230fa6c080b6e08767f08f70627dc3ca..0231a29486b68fb0c0dc2e2fd6161bdcc15cb7a8 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Encounter/ZeroIntensity.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Encounter/ZeroIntensity.lean
@@ -1,234 +1,234 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Main
-import OAI.Probability.UnimodularPercolation.Encounter.Counts
-
-namespace OAI
-
-/-! Critical limits and zero encounter intensity. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-open MeasureTheory SimpleGraph Filter Topology
-open scoped ENNReal
-universe u v w
-variable {V : Type u} {E : Type v} {ι : Type w}
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem branchCount_three (H : SimpleGraph V) (S : Finset V) (x a b c : V)
-    (ha : a ∈ S) (hb : b ∈ S) (hc : c ∈ S)
-    (hax : a ≠ x) (hbx : b ≠ x) (hcx : c ≠ x)
-    (hab : ¬(cutVertex H x).Reachable a b)
-    (hac : ¬(cutVertex H x).Reachable a c)
-    (hbc : ¬(cutVertex H x).Reachable b c) : 3 ≤ branchCount H S x := by
-  let f := (cutVertex H x).connectedComponentMk
-  have hab' : f a ≠ f b := fun h => hab (ConnectedComponent.exact h)
-  have hac' : f a ≠ f c := fun h => hac (ConnectedComponent.exact h)
-  have hbc' : f b ≠ f c := fun h => hbc (ConnectedComponent.exact h)
-  have hs : ({f a,f b,f c}:Finset _) ⊆ (S.erase x).image f := by
-    intro y hy
-    simp only [Finset.mem_insert,Finset.mem_singleton] at hy
-    rcases hy with rfl | rfl | rfl
-    · exact Finset.mem_image.mpr ⟨a,Finset.mem_erase.mpr ⟨hax,ha⟩,rfl⟩
-    · exact Finset.mem_image.mpr ⟨b,Finset.mem_erase.mpr ⟨hbx,hb⟩,rfl⟩
-    · exact Finset.mem_image.mpr ⟨c,Finset.mem_erase.mpr ⟨hcx,hc⟩,rfl⟩
-  have hh := Finset.card_le_card hs
-  have hcard : ({f a,f b,f c}:Finset _).card = 3 := by simp [hab',hac',hbc']
-  rw [hcard] at hh
-  exact hh
-
-theorem eventually_conn_of_limit (G : BondGraph V E) (l : E → unitInterval)
-    (p : unitInterval) (q : ℕ → unitInterval) (hmono : Monotone q)
-    (hq : ∀ n, q n ≤ p) (hlim : Tendsto q atTop (𝓝 p)) (hne : ∀ e, l e ≠ p)
-    {x y : V} (hxy : Conn G (openEdges l p) x y) :
-    ∀ᶠ n in atTop, Conn G (openEdges l (q n)) x y := by
-  rw [openEdges_eq_iUnion_of_tendsto l p hne q hq hlim,conn_iUnion_iff G _
-    (fun m n hmn => openEdges_mono l (hmono hmn))] at hxy
-  obtain ⟨n,hn⟩ := hxy
-  exact eventually_atTop.mpr ⟨n,fun m hm => conn_mono G (openEdges_mono l (hmono hm)) hn⟩
-
-theorem eventually_encounterCount_three (G : BondGraph V E) (l : E → unitInterval)
-    (p : unitInterval) (q : ℕ → unitInterval) (hmono : Monotone q)
-    (hq : ∀ n, q n ≤ p) (hlim : Tendsto q atTop (𝓝 p)) (hne : ∀ e, l e ≠ p)
-    (hfin : ∀ n x, (cluster G (openEdges l (q n)) x).Finite)
-    (hbranch : ∀ x, IsEncounter G (openEdges l p) x → ∀ a,
-      Conn G (openEdges l p) x a → (cluster G (cutConfig G (openEdges l p) x) a).Infinite →
-      ∃ z ∈ cluster G (cutConfig G (openEdges l p) x) a, z ≠ x ∧ IsEncounter G (openEdges l p) z)
-    (x : V) (hx : IsEncounter G (openEdges l p) x) :
-    ∀ᶠ n in atTop, 3 ≤ encounterCount G (openEdges l p) (openEdges l (q n)) x := by
-  obtain ⟨a,b,c,hxa,hxb,hxc,ha,hb,hc,hab,hac,hbc⟩ := hx
-  obtain ⟨a',haa',hax,ha'⟩ := hbranch x ⟨a,b,c,hxa,hxb,hxc,ha,hb,hc,hab,hac,hbc⟩ a hxa ha
-  obtain ⟨b',hbb',hbx,hb'⟩ := hbranch x ⟨a,b,c,hxa,hxb,hxc,ha,hb,hc,hab,hac,hbc⟩ b hxb hb
-  obtain ⟨c',hcc',hcx,hc'⟩ := hbranch x ⟨a,b,c,hxa,hxb,hxc,ha,hb,hc,hab,hac,hbc⟩ c hxc hc
-  have hx' : IsEncounter G (openEdges l p) x := ⟨a,b,c,hxa,hxb,hxc,ha,hb,hc,hab,hac,hbc⟩
-  have hac' : Conn G (openEdges l p) x a' := conn_trans G _ hxa (conn_mono G (fun _ h => h.1) haa')
-  have hbc' : Conn G (openEdges l p) x b' := conn_trans G _ hxb (conn_mono G (fun _ h => h.1) hbb')
-  have hxc' : Conn G (openEdges l p) x c' := conn_trans G _ hxc (conn_mono G (fun _ h => h.1) hcc')
-  filter_upwards [eventually_conn_of_limit G l p q hmono hq hlim hne hac',
-    eventually_conn_of_limit G l p q hmono hq hlim hne hbc',
-    eventually_conn_of_limit G l p q hmono hq hlim hne hxc'] with n han hbn hcn
-  rw [encounterCount,ite_eq_left hx']
-  have hm (z : V) (hz : Conn G (openEdges l (q n)) x z) (he : IsEncounter G (openEdges l p) z) :
-      z ∈ markedCluster G (openEdges l p) (openEdges l (q n)) x := by
-    apply Finset.mem_filter.mpr
-    refine ⟨?_,he⟩
-    have ht : z ∈ cluster G (openEdges l (q n)) x := hz
-    rwa [←clusterFinset_coe G _ x (hfin n x)] at ht
-  apply branchCount_three _ _ x a' b' c' (hm _ han ha') (hm _ hbn hb') (hm _ hcn hc') hax hbx hcx
-  · intro ht
-    rw [cutVertex_open] at ht
-    exact hab (conn_trans G _ haa' (conn_trans G _ ((conn_iff_reachable G _ a' b').mpr ht) (conn_symm G _ hbb')))
-  · intro ht
-    rw [cutVertex_open] at ht
-    exact hac (conn_trans G _ haa' (conn_trans G _ ((conn_iff_reachable G _ a' c').mpr ht) (conn_symm G _ hcc')))
-  · intro ht
-    rw [cutVertex_open] at ht
-    exact hbc (conn_trans G _ hbb' (conn_trans G _ ((conn_iff_reachable G _ b' c').mpr ht) (conn_symm G _ hcc')))
-
-end Percolation
-
-namespace Percolation
-open MeasureTheory SimpleGraph Filter Topology
-open scoped ENNReal
-universe u v w
-variable {V : Type u} {E : Type v} {ι : Type w}
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Countable V] [Countable E] [Fintype ι]
-
-theorem critical_encounter_density_zero
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
-    (hp : 0 < pc G) (root : ι → V) (weight : ι → ℝ)
-    (htr : RootTransport root weight (vertexAut G)) (hw : ∀ i, 0 < weight i)
-    (hw1 : ∑ i, weight i = 1)
-    (hcov : ∀ x, ∃ i, ∃ a ∈ vertexAut G, a (root i) = x) :
-    (∑ i, ENNReal.ofReal (weight i) * ∫⁻ l,
-      (if IsEncounter G (openEdges l (pc G)) (root i) then 1 else 0) ∂labelLaw) = 0 := by
-  obtain ⟨q,hmono,hq,hlim⟩ := exists_seq_strictMono_tendsto' hp
-  let d (l : E → unitInterval) : ℝ≥0∞ := ∑ i, ENNReal.ofReal (weight i) *
-    if IsEncounter G (openEdges l (pc G)) (root i) then 1 else 0
-  let f (n : ℕ) (l : E → unitInterval) : ℝ≥0∞ := ∑ i, ENNReal.ofReal (weight i) *
-    (encounterCount G (openEdges l (pc G)) (openEdges l (q n)) (root i) : ℝ≥0∞)
-  have hd : Measurable d := by
-    apply Finset.measurable_sum
-    intro i _
-    exact measurable_const.mul (Measurable.ite ((measurableSet_isEncounter G (root i)).preimage
-      (measurable_openEdges (pc G))) measurable_const measurable_const)
-  have hf n : Measurable (f n) := by
-    apply Finset.measurable_sum
-    intro i _
-    exact measurable_const.mul ((measurable_of_countable (fun n : ℕ => (n:ℝ≥0∞))).comp
-      (measurable_encounterCount G (pc G) (q n) (root i)))
-  have hdint : (∫⁻ l, d l ∂labelLaw) = ∑ i, ENNReal.ofReal (weight i) * ∫⁻ l,
-      (if IsEncounter G (openEdges l (pc G)) (root i) then 1 else 0) ∂labelLaw := by
-    rw [show d = _ from rfl,lintegral_finsetSum]
-    · apply Finset.sum_congr rfl
-      intro i _
-      exact lintegral_const_mul' _ _ ENNReal.ofReal_ne_top
-    · intro i _
-      exact measurable_const.mul (Measurable.ite ((measurableSet_isEncounter G (root i)).preimage
-        (measurable_openEdges (pc G))) measurable_const measurable_const)
-  have hbound n : (∫⁻ l, f n l ∂labelLaw) ≤ 2 * ∫⁻ l, d l ∂labelLaw := by
-    rw [hdint,show f n = _ from rfl,lintegral_finsetSum]
-    · simp_rw [lintegral_const_mul' _ _ ENNReal.ofReal_ne_top]
-      exact expected_encounterCount_le G hconn hloc root weight htr (fun i => (hw i).le)
-        (pc G) (q n) (hq n).2.le (fun x => subcritical_cluster_finite_ae G (q n) (hq n).2 x)
-    · intro i _
-      exact measurable_const.mul ((measurable_of_countable (fun n : ℕ => (n:ℝ≥0∞))).comp
-        (measurable_encounterCount G (pc G) (q n) (root i)))
-  have hfl : ∀ᵐ l ∂labelLaw, ∀ n x, (cluster G (openEdges l (q n)) x).Finite :=
-    ae_all_iff.mpr fun n => ae_all_iff.mpr fun x => label_cluster_finite_ae_subcritical G (q n) (hq n).2 x
-  have hbl : ∀ᵐ l ∂labelLaw, ∀ x, IsEncounter G (openEdges l (pc G)) x → ∀ a,
-      Conn G (openEdges l (pc G)) x a → (cluster G (cutConfig G (openEdges l (pc G)) x) a).Infinite →
-      ∃ z ∈ cluster G (cutConfig G (openEdges l (pc G)) x) a, z ≠ x ∧ IsEncounter G (openEdges l (pc G)) z := by
-    apply ae_of_ae_map (μ := (labelLaw : Measure (E → unitInterval)))
-      (p := fun ω : Set E => ∀ x, IsEncounter G ω x → ∀ a,
-        Conn G ω x a → (cluster G (cutConfig G ω x) a).Infinite →
-        ∃ z ∈ cluster G (cutConfig G ω x) a, z ≠ x ∧ IsEncounter G ω z)
-      (measurable_openEdges (pc G)).aemeasurable
-    rw [map_openEdges]
-    exact encounter_branches_ae G hconn hloc root weight htr hw hw1 hcov (pc G)
-  have hlow : ∀ᵐ l ∂labelLaw, 3 * d l ≤ liminf (fun n => f n l) atTop := by
-    filter_upwards [hfl,hbl,ae_labels_ne (E := E) (pc G)] with l hl hb hn
-    refine le_liminf_of_le (by isBoundedDefault) ?_
-    have he (i : ι) : ∀ᶠ n in atTop,
-        (3:ℝ≥0∞) * (if IsEncounter G (openEdges l (pc G)) (root i) then 1 else 0) ≤
-        (encounterCount G (openEdges l (pc G)) (openEdges l (q n)) (root i) : ℝ≥0∞) := by
-      by_cases hi : IsEncounter G (openEdges l (pc G)) (root i)
-      · filter_upwards [eventually_encounterCount_three G l (pc G) q hmono.monotone
-          (fun n => (hq n).2.le) hlim hn hl hb (root i) hi] with n hnc
-        simpa only [ite_eq_left hi,mul_one] using (show (3:ℝ≥0∞) ≤ (encounterCount G (openEdges l (pc G)) (openEdges l (q n)) (root i):ℝ≥0∞) from by exact_mod_cast hnc)
-      · filter_upwards [] with n
-        simp only [ite_eq_right hi,mul_zero,zero_le]
-    filter_upwards [(eventually_all_finset (Finset.univ : Finset ι)).mpr (fun i _ => he i)] with n hnn
-    dsimp [d,f]
-    rw [Finset.mul_sum]
-    apply Finset.sum_le_sum
-    intro i hi
-    rw [←mul_assoc,mul_comm (3:ℝ≥0∞),mul_assoc]
-    exact mul_le_mul_right (hnn i hi) _
-  have hineq : 3 * (∫⁻ l, d l ∂labelLaw) ≤ 2 * ∫⁻ l, d l ∂labelLaw := by
-    calc
-      _ = ∫⁻ l, 3 * d l ∂labelLaw := (lintegral_const_mul' _ _ ENNReal.ofNat_ne_top).symm
-      _ ≤ ∫⁻ l, liminf (fun n => f n l) atTop ∂labelLaw := lintegral_mono_ae hlow
-      _ ≤ liminf (fun n => ∫⁻ l, f n l ∂labelLaw) atTop := lintegral_liminf_le hf
-      _ ≤ _ := by
-        refine liminf_le_of_le (by isBoundedDefault) ?_
-        intro b hb
-        obtain ⟨n,hn⟩ := hb.exists
-        exact hn.trans (hbound n)
-  have hdle (l : E → unitInterval) : d l ≤ 1 := by
-    calc
-      _ ≤ ∑ i, ENNReal.ofReal (weight i) * 1 := by
-        apply Finset.sum_le_sum
-        intro i _
-        apply mul_le_mul_right
-        split_ifs <;> simp
-      _ = 1 := by
-        simp only [mul_one,←ENNReal.ofReal_sum_of_nonneg (fun i _ => (hw i).le),hw1,ENNReal.ofReal_one]
-  have hdf : (∫⁻ l, d l ∂labelLaw) ≠ ⊤ := by
-    have he : (∫⁻ l, d l ∂labelLaw) ≤ 1 := (lintegral_mono hdle).trans_eq (by simp)
-    exact ne_of_lt (he.trans_lt ENNReal.one_lt_top)
-  rw [←hdint]
-  by_contra hne
-  exact (not_lt_of_ge hineq) (ENNReal.mul_lt_mul_left hne hdf (by norm_num))
-
-theorem critical_encounter_root_zero
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
-    (hp : 0 < pc G) (root : ι → V) (weight : ι → ℝ)
-    (htr : RootTransport root weight (vertexAut G)) (hw : ∀ i, 0 < weight i)
-    (hw1 : ∑ i, weight i = 1)
-    (hcov : ∀ x, ∃ i, ∃ a ∈ vertexAut G, a (root i) = x) (i : ι) :
-    law (pc G) {ω | IsEncounter G ω (root i)} = 0 := by
-  have hd := critical_encounter_density_zero G hconn hloc hp root weight htr hw hw1 hcov
-  have hi : ENNReal.ofReal (weight i) * ∫⁻ l,
-      (if IsEncounter G (openEdges l (pc G)) (root i) then (1:ℝ≥0∞) else 0) ∂labelLaw = 0 := by
-    exact (Finset.sum_eq_zero_iff.mp hd) i (Finset.mem_univ i)
-  have hz : (∫⁻ l, (if IsEncounter G (openEdges l (pc G)) (root i) then (1:ℝ≥0∞) else 0) ∂labelLaw) = 0 :=
-    (mul_eq_zero.mp hi).resolve_left ((ENNReal.ofReal_pos.mpr (hw i)).ne')
-  rw [←map_openEdges (pc G),Measure.map_apply (measurable_openEdges (pc G))
-    (measurableSet_isEncounter G (root i))]
-  have he : (fun l : E → unitInterval => if IsEncounter G (openEdges l (pc G)) (root i) then (1:ℝ≥0∞) else 0) =
-      ((fun l : E → unitInterval => openEdges l (pc G)) ⁻¹' {ω | IsEncounter G ω (root i)}).indicator (fun _ => 1) := rfl
-  rw [he,lintegral_indicator ((measurableSet_isEncounter G (root i)).preimage (measurable_openEdges (pc G)))] at hz
-  simpa only [lintegral_const,Measure.restrict_apply_univ,one_mul] using hz
-
-theorem critical_encounter_zero
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
-    (hp : 0 < pc G) (root : ι → V) (weight : ι → ℝ)
-    (htr : RootTransport root weight (vertexAut G)) (hw : ∀ i, 0 < weight i)
-    (hw1 : ∑ i, weight i = 1)
-    (hcov : ∀ x, ∃ i, ∃ a ∈ vertexAut G, a (root i) = x) (x : V) :
-    law (pc G) {ω | IsEncounter G ω x} = 0 := by
-  obtain ⟨i,a,⟨b,hab⟩,rfl⟩ := hcov x
-  have he : {ω | IsEncounter G ω (a (root i))} =
-      (fun ω : Set E => b ⁻¹' ω) ⁻¹' {ω | IsEncounter G ω (root i)} := by
-    ext ω
-    exact (isEncounter_aut_iff G a b hab ω (root i)).symm
-  rw [he,←Measure.map_apply (measurable_set_preimage b) (measurableSet_isEncounter G (root i)),
-    law_map_preimage_equiv]
-  exact critical_encounter_root_zero G hconn hloc hp root weight htr hw hw1 hcov i
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Main
+import OAI.Probability.UnimodularPercolation.Encounter.Counts
+
+namespace OAI
+
+/-! Critical limits and zero encounter intensity. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+open MeasureTheory SimpleGraph Filter Topology
+open scoped ENNReal
+universe u v w
+variable {V : Type u} {E : Type v} {ι : Type w}
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem branchCount_three (H : SimpleGraph V) (S : Finset V) (x a b c : V)
+    (ha : a ∈ S) (hb : b ∈ S) (hc : c ∈ S)
+    (hax : a ≠ x) (hbx : b ≠ x) (hcx : c ≠ x)
+    (hab : ¬(cutVertex H x).Reachable a b)
+    (hac : ¬(cutVertex H x).Reachable a c)
+    (hbc : ¬(cutVertex H x).Reachable b c) : 3 ≤ branchCount H S x := by
+  let f := (cutVertex H x).connectedComponentMk
+  have hab' : f a ≠ f b := fun h => hab (ConnectedComponent.exact h)
+  have hac' : f a ≠ f c := fun h => hac (ConnectedComponent.exact h)
+  have hbc' : f b ≠ f c := fun h => hbc (ConnectedComponent.exact h)
+  have hs : ({f a,f b,f c}:Finset _) ⊆ (S.erase x).image f := by
+    intro y hy
+    simp only [Finset.mem_insert,Finset.mem_singleton] at hy
+    rcases hy with rfl | rfl | rfl
+    · exact Finset.mem_image.mpr ⟨a,Finset.mem_erase.mpr ⟨hax,ha⟩,rfl⟩
+    · exact Finset.mem_image.mpr ⟨b,Finset.mem_erase.mpr ⟨hbx,hb⟩,rfl⟩
+    · exact Finset.mem_image.mpr ⟨c,Finset.mem_erase.mpr ⟨hcx,hc⟩,rfl⟩
+  have hh := Finset.card_le_card hs
+  have hcard : ({f a,f b,f c}:Finset _).card = 3 := by simp [hab',hac',hbc']
+  rw [hcard] at hh
+  exact hh
+
+theorem eventually_conn_of_limit (G : BondGraph V E) (l : E → unitInterval)
+    (p : unitInterval) (q : ℕ → unitInterval) (hmono : Monotone q)
+    (hq : ∀ n, q n ≤ p) (hlim : Tendsto q atTop (𝓝 p)) (hne : ∀ e, l e ≠ p)
+    {x y : V} (hxy : Conn G (openEdges l p) x y) :
+    ∀ᶠ n in atTop, Conn G (openEdges l (q n)) x y := by
+  rw [openEdges_eq_iUnion_of_tendsto l p hne q hq hlim,conn_iUnion_iff G _
+    (fun m n hmn => openEdges_mono l (hmono hmn))] at hxy
+  obtain ⟨n,hn⟩ := hxy
+  exact eventually_atTop.mpr ⟨n,fun m hm => conn_mono G (openEdges_mono l (hmono hm)) hn⟩
+
+theorem eventually_encounterCount_three (G : BondGraph V E) (l : E → unitInterval)
+    (p : unitInterval) (q : ℕ → unitInterval) (hmono : Monotone q)
+    (hq : ∀ n, q n ≤ p) (hlim : Tendsto q atTop (𝓝 p)) (hne : ∀ e, l e ≠ p)
+    (hfin : ∀ n x, (cluster G (openEdges l (q n)) x).Finite)
+    (hbranch : ∀ x, IsEncounter G (openEdges l p) x → ∀ a,
+      Conn G (openEdges l p) x a → (cluster G (cutConfig G (openEdges l p) x) a).Infinite →
+      ∃ z ∈ cluster G (cutConfig G (openEdges l p) x) a, z ≠ x ∧ IsEncounter G (openEdges l p) z)
+    (x : V) (hx : IsEncounter G (openEdges l p) x) :
+    ∀ᶠ n in atTop, 3 ≤ encounterCount G (openEdges l p) (openEdges l (q n)) x := by
+  obtain ⟨a,b,c,hxa,hxb,hxc,ha,hb,hc,hab,hac,hbc⟩ := hx
+  obtain ⟨a',haa',hax,ha'⟩ := hbranch x ⟨a,b,c,hxa,hxb,hxc,ha,hb,hc,hab,hac,hbc⟩ a hxa ha
+  obtain ⟨b',hbb',hbx,hb'⟩ := hbranch x ⟨a,b,c,hxa,hxb,hxc,ha,hb,hc,hab,hac,hbc⟩ b hxb hb
+  obtain ⟨c',hcc',hcx,hc'⟩ := hbranch x ⟨a,b,c,hxa,hxb,hxc,ha,hb,hc,hab,hac,hbc⟩ c hxc hc
+  have hx' : IsEncounter G (openEdges l p) x := ⟨a,b,c,hxa,hxb,hxc,ha,hb,hc,hab,hac,hbc⟩
+  have hac' : Conn G (openEdges l p) x a' := conn_trans G _ hxa (conn_mono G (fun _ h => h.1) haa')
+  have hbc' : Conn G (openEdges l p) x b' := conn_trans G _ hxb (conn_mono G (fun _ h => h.1) hbb')
+  have hxc' : Conn G (openEdges l p) x c' := conn_trans G _ hxc (conn_mono G (fun _ h => h.1) hcc')
+  filter_upwards [eventually_conn_of_limit G l p q hmono hq hlim hne hac',
+    eventually_conn_of_limit G l p q hmono hq hlim hne hbc',
+    eventually_conn_of_limit G l p q hmono hq hlim hne hxc'] with n han hbn hcn
+  rw [encounterCount,ite_eq_left hx']
+  have hm (z : V) (hz : Conn G (openEdges l (q n)) x z) (he : IsEncounter G (openEdges l p) z) :
+      z ∈ markedCluster G (openEdges l p) (openEdges l (q n)) x := by
+    apply Finset.mem_filter.mpr
+    refine ⟨?_,he⟩
+    have ht : z ∈ cluster G (openEdges l (q n)) x := hz
+    rwa [←clusterFinset_coe G _ x (hfin n x)] at ht
+  apply branchCount_three _ _ x a' b' c' (hm _ han ha') (hm _ hbn hb') (hm _ hcn hc') hax hbx hcx
+  · intro ht
+    rw [cutVertex_open] at ht
+    exact hab (conn_trans G _ haa' (conn_trans G _ ((conn_iff_reachable G _ a' b').mpr ht) (conn_symm G _ hbb')))
+  · intro ht
+    rw [cutVertex_open] at ht
+    exact hac (conn_trans G _ haa' (conn_trans G _ ((conn_iff_reachable G _ a' c').mpr ht) (conn_symm G _ hcc')))
+  · intro ht
+    rw [cutVertex_open] at ht
+    exact hbc (conn_trans G _ hbb' (conn_trans G _ ((conn_iff_reachable G _ b' c').mpr ht) (conn_symm G _ hcc')))
+
+end Percolation
+
+namespace Percolation
+open MeasureTheory SimpleGraph Filter Topology
+open scoped ENNReal
+universe u v w
+variable {V : Type u} {E : Type v} {ι : Type w}
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Countable V] [Countable E] [Fintype ι]
+
+theorem critical_encounter_density_zero
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
+    (hp : 0 < pc G) (root : ι → V) (weight : ι → ℝ)
+    (htr : RootTransport root weight (vertexAut G)) (hw : ∀ i, 0 < weight i)
+    (hw1 : ∑ i, weight i = 1)
+    (hcov : ∀ x, ∃ i, ∃ a ∈ vertexAut G, a (root i) = x) :
+    (∑ i, ENNReal.ofReal (weight i) * ∫⁻ l,
+      (if IsEncounter G (openEdges l (pc G)) (root i) then 1 else 0) ∂labelLaw) = 0 := by
+  obtain ⟨q,hmono,hq,hlim⟩ := exists_seq_strictMono_tendsto' hp
+  let d (l : E → unitInterval) : ℝ≥0∞ := ∑ i, ENNReal.ofReal (weight i) *
+    if IsEncounter G (openEdges l (pc G)) (root i) then 1 else 0
+  let f (n : ℕ) (l : E → unitInterval) : ℝ≥0∞ := ∑ i, ENNReal.ofReal (weight i) *
+    (encounterCount G (openEdges l (pc G)) (openEdges l (q n)) (root i) : ℝ≥0∞)
+  have hd : Measurable d := by
+    apply Finset.measurable_sum
+    intro i _
+    exact measurable_const.mul (Measurable.ite ((measurableSet_isEncounter G (root i)).preimage
+      (measurable_openEdges (pc G))) measurable_const measurable_const)
+  have hf n : Measurable (f n) := by
+    apply Finset.measurable_sum
+    intro i _
+    exact measurable_const.mul ((measurable_of_countable (fun n : ℕ => (n:ℝ≥0∞))).comp
+      (measurable_encounterCount G (pc G) (q n) (root i)))
+  have hdint : (∫⁻ l, d l ∂labelLaw) = ∑ i, ENNReal.ofReal (weight i) * ∫⁻ l,
+      (if IsEncounter G (openEdges l (pc G)) (root i) then 1 else 0) ∂labelLaw := by
+    rw [show d = _ from rfl,lintegral_finsetSum]
+    · apply Finset.sum_congr rfl
+      intro i _
+      exact lintegral_const_mul' _ _ ENNReal.ofReal_ne_top
+    · intro i _
+      exact measurable_const.mul (Measurable.ite ((measurableSet_isEncounter G (root i)).preimage
+        (measurable_openEdges (pc G))) measurable_const measurable_const)
+  have hbound n : (∫⁻ l, f n l ∂labelLaw) ≤ 2 * ∫⁻ l, d l ∂labelLaw := by
+    rw [hdint,show f n = _ from rfl,lintegral_finsetSum]
+    · simp_rw [lintegral_const_mul' _ _ ENNReal.ofReal_ne_top]
+      exact expected_encounterCount_le G hconn hloc root weight htr (fun i => (hw i).le)
+        (pc G) (q n) (hq n).2.le (fun x => subcritical_cluster_finite_ae G (q n) (hq n).2 x)
+    · intro i _
+      exact measurable_const.mul ((measurable_of_countable (fun n : ℕ => (n:ℝ≥0∞))).comp
+        (measurable_encounterCount G (pc G) (q n) (root i)))
+  have hfl : ∀ᵐ l ∂labelLaw, ∀ n x, (cluster G (openEdges l (q n)) x).Finite :=
+    ae_all_iff.mpr fun n => ae_all_iff.mpr fun x => label_cluster_finite_ae_subcritical G (q n) (hq n).2 x
+  have hbl : ∀ᵐ l ∂labelLaw, ∀ x, IsEncounter G (openEdges l (pc G)) x → ∀ a,
+      Conn G (openEdges l (pc G)) x a → (cluster G (cutConfig G (openEdges l (pc G)) x) a).Infinite →
+      ∃ z ∈ cluster G (cutConfig G (openEdges l (pc G)) x) a, z ≠ x ∧ IsEncounter G (openEdges l (pc G)) z := by
+    apply ae_of_ae_map (μ := (labelLaw : Measure (E → unitInterval)))
+      (p := fun ω : Set E => ∀ x, IsEncounter G ω x → ∀ a,
+        Conn G ω x a → (cluster G (cutConfig G ω x) a).Infinite →
+        ∃ z ∈ cluster G (cutConfig G ω x) a, z ≠ x ∧ IsEncounter G ω z)
+      (measurable_openEdges (pc G)).aemeasurable
+    rw [map_openEdges]
+    exact encounter_branches_ae G hconn hloc root weight htr hw hw1 hcov (pc G)
+  have hlow : ∀ᵐ l ∂labelLaw, 3 * d l ≤ liminf (fun n => f n l) atTop := by
+    filter_upwards [hfl,hbl,ae_labels_ne (E := E) (pc G)] with l hl hb hn
+    refine le_liminf_of_le (by isBoundedDefault) ?_
+    have he (i : ι) : ∀ᶠ n in atTop,
+        (3:ℝ≥0∞) * (if IsEncounter G (openEdges l (pc G)) (root i) then 1 else 0) ≤
+        (encounterCount G (openEdges l (pc G)) (openEdges l (q n)) (root i) : ℝ≥0∞) := by
+      by_cases hi : IsEncounter G (openEdges l (pc G)) (root i)
+      · filter_upwards [eventually_encounterCount_three G l (pc G) q hmono.monotone
+          (fun n => (hq n).2.le) hlim hn hl hb (root i) hi] with n hnc
+        simpa only [ite_eq_left hi,mul_one] using (show (3:ℝ≥0∞) ≤ (encounterCount G (openEdges l (pc G)) (openEdges l (q n)) (root i):ℝ≥0∞) from by exact_mod_cast hnc)
+      · filter_upwards [] with n
+        simp only [ite_eq_right hi,mul_zero,zero_le]
+    filter_upwards [(eventually_all_finset (Finset.univ : Finset ι)).mpr (fun i _ => he i)] with n hnn
+    dsimp [d,f]
+    rw [Finset.mul_sum]
+    apply Finset.sum_le_sum
+    intro i hi
+    rw [←mul_assoc,mul_comm (3:ℝ≥0∞),mul_assoc]
+    exact mul_le_mul_right (hnn i hi) _
+  have hineq : 3 * (∫⁻ l, d l ∂labelLaw) ≤ 2 * ∫⁻ l, d l ∂labelLaw := by
+    calc
+      _ = ∫⁻ l, 3 * d l ∂labelLaw := (lintegral_const_mul' _ _ ENNReal.ofNat_ne_top).symm
+      _ ≤ ∫⁻ l, liminf (fun n => f n l) atTop ∂labelLaw := lintegral_mono_ae hlow
+      _ ≤ liminf (fun n => ∫⁻ l, f n l ∂labelLaw) atTop := lintegral_liminf_le hf
+      _ ≤ _ := by
+        refine liminf_le_of_le (by isBoundedDefault) ?_
+        intro b hb
+        obtain ⟨n,hn⟩ := hb.exists
+        exact hn.trans (hbound n)
+  have hdle (l : E → unitInterval) : d l ≤ 1 := by
+    calc
+      _ ≤ ∑ i, ENNReal.ofReal (weight i) * 1 := by
+        apply Finset.sum_le_sum
+        intro i _
+        apply mul_le_mul_right
+        split_ifs <;> simp
+      _ = 1 := by
+        simp only [mul_one,←ENNReal.ofReal_sum_of_nonneg (fun i _ => (hw i).le),hw1,ENNReal.ofReal_one]
+  have hdf : (∫⁻ l, d l ∂labelLaw) ≠ ⊤ := by
+    have he : (∫⁻ l, d l ∂labelLaw) ≤ 1 := (lintegral_mono hdle).trans_eq (by simp)
+    exact ne_of_lt (he.trans_lt ENNReal.one_lt_top)
+  rw [←hdint]
+  by_contra hne
+  exact (not_lt_of_ge hineq) (ENNReal.mul_lt_mul_left hne hdf (by norm_num))
+
+theorem critical_encounter_root_zero
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
+    (hp : 0 < pc G) (root : ι → V) (weight : ι → ℝ)
+    (htr : RootTransport root weight (vertexAut G)) (hw : ∀ i, 0 < weight i)
+    (hw1 : ∑ i, weight i = 1)
+    (hcov : ∀ x, ∃ i, ∃ a ∈ vertexAut G, a (root i) = x) (i : ι) :
+    law (pc G) {ω | IsEncounter G ω (root i)} = 0 := by
+  have hd := critical_encounter_density_zero G hconn hloc hp root weight htr hw hw1 hcov
+  have hi : ENNReal.ofReal (weight i) * ∫⁻ l,
+      (if IsEncounter G (openEdges l (pc G)) (root i) then (1:ℝ≥0∞) else 0) ∂labelLaw = 0 := by
+    exact (Finset.sum_eq_zero_iff.mp hd) i (Finset.mem_univ i)
+  have hz : (∫⁻ l, (if IsEncounter G (openEdges l (pc G)) (root i) then (1:ℝ≥0∞) else 0) ∂labelLaw) = 0 :=
+    (mul_eq_zero.mp hi).resolve_left ((ENNReal.ofReal_pos.mpr (hw i)).ne')
+  rw [←map_openEdges (pc G),Measure.map_apply (measurable_openEdges (pc G))
+    (measurableSet_isEncounter G (root i))]
+  have he : (fun l : E → unitInterval => if IsEncounter G (openEdges l (pc G)) (root i) then (1:ℝ≥0∞) else 0) =
+      ((fun l : E → unitInterval => openEdges l (pc G)) ⁻¹' {ω | IsEncounter G ω (root i)}).indicator (fun _ => 1) := rfl
+  rw [he,lintegral_indicator ((measurableSet_isEncounter G (root i)).preimage (measurable_openEdges (pc G)))] at hz
+  simpa only [lintegral_const,Measure.restrict_apply_univ,one_mul] using hz
+
+theorem critical_encounter_zero
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
+    (hp : 0 < pc G) (root : ι → V) (weight : ι → ℝ)
+    (htr : RootTransport root weight (vertexAut G)) (hw : ∀ i, 0 < weight i)
+    (hw1 : ∑ i, weight i = 1)
+    (hcov : ∀ x, ∃ i, ∃ a ∈ vertexAut G, a (root i) = x) (x : V) :
+    law (pc G) {ω | IsEncounter G ω x} = 0 := by
+  obtain ⟨i,a,⟨b,hab⟩,rfl⟩ := hcov x
+  have he : {ω | IsEncounter G ω (a (root i))} =
+      (fun ω : Set E => b ⁻¹' ω) ⁻¹' {ω | IsEncounter G ω (root i)} := by
+    ext ω
+    exact (isEncounter_aut_iff G a b hab ω (root i)).symm
+  rw [he,←Measure.map_apply (measurable_set_preimage b) (measurableSet_isEncounter G (root i)),
+    law_map_preimage_equiv]
+  exact critical_encounter_root_zero G hconn hloc hp root weight htr hw hw1 hcov i
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Escape/CollisionTails.lean b/lean/OAI/Probability/UnimodularPercolation/Escape/CollisionTails.lean
index b7b1220f5548f4325339bbd1952699c2e08fdf19..72f592ce4bbf3c107fdc483c031b817ccda0f13c 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Escape/CollisionTails.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Escape/CollisionTails.lean
@@ -1,1204 +1,1204 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Escape.Labels
-
-namespace OAI
-
-/-! Collision Tails for bond percolation. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section RawTimeMarginals
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem raw_position_integral (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (i m : ℕ) (him : i ≤ m) (x : V) (f : V → ℝ) :
-    (∑' w : RawFrom (openSimpleGraph G Set.univ) m x,
-      w.2.mass (walkKernel G hloc a) * f (w.2.position i)) =
-      RawFrom.integral (H := openSimpleGraph G Set.univ) (walkKernel G hloc a) i x f := by
-  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
-  induction i generalizing m x with
-  | zero =>
-    simp only [HoldingWalk.position_zero,tsum_mul_right,rawFrom_probability_sum G hloc a,one_mul,RawFrom.integral_zero]
-  | succ i ih =>
-    cases m with
-    | zero => omega
-    | succ m =>
-      rw [←(RawFrom.succEquiv m x).symm.tsum_eq,RawFrom.integral_succ]
-      let (z : V) := Fintype.ofFinite (RawFrom (openSimpleGraph G Set.univ) m z)
-      let := Fintype.ofFinite ((openSimpleGraph G Set.univ).neighborSet x)
-      have hh (z : V) := ih m (Nat.le_of_succ_le_succ him) z
-      simp only [tsum_fintype,Fintype.sum_sum_type,Fintype.sum_sigma,RawFrom.succEquiv,
-        Equiv.symm_mk,Equiv.coe_fn_mk,HoldingWalk.mass,HoldingWalk.position,mul_assoc]
-      simp only [tsum_fintype] at hh
-      simp only [←Finset.mul_sum,hh]
-
-theorem raw_position_probability (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (i m : ℕ) (him : i ≤ m) (x z : V) :
-    (∑' w : RawFrom (openSimpleGraph G Set.univ) m x,
-      if w.2.position i = z then w.2.mass (walkKernel G hloc a) else 0) =
-      (matrixCoeff (P^i) x z).re := by
-  have hh := raw_position_integral G hloc a i m him x (fun y => if y = z then (1:ℝ) else 0)
-  simp only [mul_ite,mul_one,mul_zero] at hh
-  rw [hh,raw_integral_operator G hloc a P hpos hP]
-  simp only [mul_ite,mul_one,mul_zero,tsum_ite_eq]
-
-end RawTimeMarginals
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section IndependentRawCollisions
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem raw_two_time_collision (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (i m j n : ℕ) (him : i ≤ m) (hjn : j ≤ n) (x y : V) :
-    (∑' w : RawFrom (openSimpleGraph G Set.univ) m x × RawFrom (openSimpleGraph G Set.univ) n y,
-      if w.1.2.position i = w.2.2.position j then
-        w.1.2.mass (walkKernel G hloc a)*w.2.2.mass (walkKernel G hloc a) else 0) =
-      (matrixCoeff (P^(i+j)) x y).re := by
-  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
-  let := Fintype.ofFinite (RawFrom (openSimpleGraph G Set.univ) m x)
-  let := Fintype.ofFinite (RawFrom (openSimpleGraph G Set.univ) n y)
-  have hi (w : RawFrom (openSimpleGraph G Set.univ) m x) :
-      (∑' v : RawFrom (openSimpleGraph G Set.univ) n y,
-        if w.2.position i = v.2.position j then w.2.mass (walkKernel G hloc a)*v.2.mass (walkKernel G hloc a) else 0) =
-      w.2.mass (walkKernel G hloc a)*(matrixCoeff (P^j) y (w.2.position i)).re := by
-    rw [← raw_position_probability G hloc a P hpos hP j n hjn y (w.2.position i),←tsum_mul_left]
-    apply tsum_congr
-    intro v
-    by_cases hh : w.2.position i = v.2.position j
-    · simp only [ite_eq_left hh,ite_eq_left hh.symm]
-    · simp only [ite_eq_right hh,ite_eq_right (Ne.symm hh),mul_zero]
-  rw [(summable_of_hasFiniteSupport (Set.toFinite _)).tsum_prod]
-  simp only [hi]
-  rw [raw_position_integral G hloc a i m him x (fun z => (matrixCoeff (P^j) y z).re),raw_integral_operator G hloc a P hpos hP]
-  simp only [matrixCoeff_real_symm (P^j) (hsym.pow j) y]
-  rw [((hpos.pow i).mul_hasSum (P^j) x y).tsum_eq,←pow_add]
-
-theorem matrixCoeff_norm_le (A : Hilbert V →L[ℂ] Hilbert V) (x y : V) :
-    ‖matrixCoeff A x y‖ ≤ ‖A‖ := by
-  apply (lp.norm_apply_le_norm (by norm_num : (2:ENNReal) ≠ 0) (A (lp.single 2 y (1:ℂ))) x).trans
-  simpa using A.le_opNorm (lp.single 2 y (1:ℂ))
-
-theorem matrixCoeff_power_le (P : Hilbert V →L[ℂ] Hilbert V) (k : ℕ) (x y : V) :
-    (matrixCoeff (P^k) x y).re ≤ ‖P‖^k := by
-  have hn (k : ℕ) : ‖P^k‖ ≤ ‖P‖^k := by
-    induction k with
-    | zero => simpa only [pow_zero,ContinuousLinearMap.one_def] using
-        (ContinuousLinearMap.norm_id_le (𝕜 := ℂ) (E := Hilbert V))
-    | succ k ih =>
-      rw [pow_succ,pow_succ]
-      exact (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right ih (norm_nonneg P))
-  exact (Complex.re_le_norm _).trans ((matrixCoeff_norm_le (P^k) x y).trans (hn k))
-
-end IndependentRawCollisions
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section FinitePropagation
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem reachBall_exists_short_walk (G : BondGraph V E) (x : V) (k : ℕ) {y : V}
-    (hy : y ∈ reachBall G x k) :
-    ∃ q : (openSimpleGraph G Set.univ).Walk x y, q.length ≤ k := by
-  induction k generalizing y with
-  | zero =>
-    have he : y = x := hy
-    subst y
-    exact ⟨.nil,by simp⟩
-  | succ k ih =>
-    rcases hy with hy | hy
-    · obtain ⟨q,hq⟩ := ih hy
-      exact ⟨q,hq.trans (Nat.le_succ _)⟩
-    · simp only [Set.mem_iUnion,Set.mem_ofPred_eq] at hy
-      obtain ⟨z,hz,hzy⟩ := hy
-      obtain ⟨q,hq⟩ := ih hz
-      by_cases he : z = y
-      · subst z
-        exact ⟨q,hq.trans (Nat.le_succ _)⟩
-      · let h : (openSimpleGraph G Set.univ).Adj z y :=
-          (openSimpleGraph_univ_adj G z y).mpr ⟨he,hzy⟩
-        refine ⟨q.append (.cons h .nil),?_⟩
-        simpa only [SimpleGraph.Walk.length_append,SimpleGraph.Walk.length_cons,
-          SimpleGraph.Walk.length_nil,zero_add] using Nat.add_le_add_right hq 1
-
-theorem dist_le_of_mem_reachBall (G : BondGraph V E) (x y : V) (k : ℕ)
-    (hy : y ∈ reachBall G x k) : (openSimpleGraph G Set.univ).dist x y ≤ k := by
-  obtain ⟨q,hq⟩ := reachBall_exists_short_walk G x k hy
-  exact (SimpleGraph.dist_le q).trans hq
-
-theorem walkPower_zero_of_lt_dist (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (k : ℕ) (x y : V) (hk : k < (openSimpleGraph G Set.univ).dist x y) :
-    (matrixCoeff (P^k) x y).re = 0 := by
-  apply walkPower_support G hloc a P hpos hP
-  intro hy
-  exact Nat.not_le_of_gt hk (dist_le_of_mem_reachBall G x y k hy)
-
- 
-
-theorem raw_two_time_collision_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (i m j n : ℕ) (him : i ≤ m) (hjn : j ≤ n) (x y : V) :
-    (∑' w : RawFrom (openSimpleGraph G Set.univ) m x × RawFrom (openSimpleGraph G Set.univ) n y,
-      if w.1.2.position i = w.2.2.position j then
-        w.1.2.mass (walkKernel G hloc a)*w.2.2.mass (walkKernel G hloc a) else 0) ≤
-      if i+j < (openSimpleGraph G Set.univ).dist x y then 0 else ‖P‖^(i+j) := by
-  rw [raw_two_time_collision G hloc a P hpos hsym hP i m j n him hjn x y]
-  split_ifs with hh
-  · exact (walkPower_zero_of_lt_dist G hloc a P hpos hP (i+j) x y hh).le
-  · exact matrixCoeff_power_le P (i+j) x y
-
-end FinitePropagation
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section ActualCollisionUnion
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem finite_nonnegative_event_union {α β : Type*} [Fintype α] [Fintype β]
-    (μ : α → ℝ) (hμ : ∀ a, 0 ≤ μ a) (B : α → β → Prop) :
-    (∑ a, if ∃ b, B a b then μ a else 0) ≤ ∑ b, ∑ a, if B a b then μ a else 0 := by
-  rw [Finset.sum_comm]
-  apply Finset.sum_le_sum
-  intro a _
-  by_cases hh : ∃ b, B a b
-  · obtain ⟨b,hb⟩ := hh
-    rw [ite_eq_left ⟨b,hb⟩]
-    calc
-      μ a = (if B a b then μ a else 0) := (ite_eq_left hb).symm
-      _ ≤ ∑ b, if B a b then μ a else 0 :=
-        Finset.single_le_sum (f := fun b => if B a b then μ a else 0)
-          (fun b _ => by split_ifs; exact hμ a; exact le_rfl) (Finset.mem_univ b)
-  · rw [ite_eq_right hh]
-    exact Finset.sum_nonneg (fun b _ => by split_ifs; exact hμ a; exact le_rfl)
-
- 
-
-theorem raw_intersection_probability_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (m n : ℕ) (x y : V) :
-    (∑' w : RawFrom (openSimpleGraph G Set.univ) m x × RawFrom (openSimpleGraph G Set.univ) n y,
-      if ∃ z, z ∈ w.1.2.vertices ∧ z ∈ w.2.2.vertices then
-        w.1.2.mass (walkKernel G hloc a)*w.2.2.mass (walkKernel G hloc a) else 0) ≤
-    ∑ i : Fin (m+1), ∑ j : Fin (n+1),
-      if i.val+j.val < (openSimpleGraph G Set.univ).dist x y then 0 else ‖P‖^(i.val+j.val) := by
-  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
-  let := Fintype.ofFinite (RawFrom (openSimpleGraph G Set.univ) m x)
-  let := Fintype.ofFinite (RawFrom (openSimpleGraph G Set.univ) n y)
-  have hK (x y : V) : 0 ≤ walkKernel G hloc a x y := by
-    simpa only [hP,Complex.ofReal_re] using (hpos x y).2
-  have he (w : RawFrom (openSimpleGraph G Set.univ) m x × RawFrom (openSimpleGraph G Set.univ) n y) :
-      (∃ z, z ∈ w.1.2.vertices ∧ z ∈ w.2.2.vertices) ↔
-        ∃ ij : Fin (m+1) × Fin (n+1), w.1.2.position ij.1 = w.2.2.position ij.2 := by
-    constructor
-    · rintro ⟨z,hz,hz'⟩
-      obtain ⟨i,hi,he⟩ := w.1.2.mem_vertices_iff_position.mp hz
-      obtain ⟨j,hj,he'⟩ := w.2.2.mem_vertices_iff_position.mp hz'
-      exact ⟨(⟨i,by omega⟩,⟨j,by omega⟩),he.trans he'.symm⟩
-    · rintro ⟨⟨i,j⟩,hij⟩
-      exact ⟨w.1.2.position i,w.1.2.position_mem_vertices i,hij ▸ w.2.2.position_mem_vertices j⟩
-  simp only [he,tsum_fintype]
-  have hu := finite_nonnegative_event_union
-    (fun w : RawFrom (openSimpleGraph G Set.univ) m x × RawFrom (openSimpleGraph G Set.univ) n y =>
-      w.1.2.mass (walkKernel G hloc a)*w.2.2.mass (walkKernel G hloc a))
-    (fun w => mul_nonneg (w.1.2.mass_nonneg _ hK) (w.2.2.mass_nonneg _ hK))
-    (fun w (ij : Fin (m+1) × Fin (n+1)) => w.1.2.position ij.1 = w.2.2.position ij.2)
-  apply hu.trans
-  rw [Fintype.sum_prod_type]
-  apply Finset.sum_le_sum
-  intro i _
-  apply Finset.sum_le_sum
-  intro j _
-  simpa only [tsum_fintype] using
-    raw_two_time_collision_bound G hloc a P hpos hsym hP i m j n (by omega) (by omega) x y
-
-end ActualCollisionUnion
-end Percolation
-
- 
-namespace Percolation
-section UniformCollisionTails
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem geometric_cutoff_summable (ρ : ℝ) (hρ : 0 ≤ ρ) (hρ1 : ρ < 1) (J : ℕ) :
-    Summable (fun i : ℕ => if J ≤ i then ρ^i else 0) :=
-  Summable.of_nonneg_of_le (fun i => by split_ifs <;> positivity)
-    (fun i => by split_ifs <;> first | exact le_rfl | exact pow_nonneg hρ i)
-    (summable_geometric_of_lt_one hρ hρ1)
-
-theorem geometric_cutoff_tsum (ρ : ℝ) (hρ : 0 ≤ ρ) (hρ1 : ρ < 1) (J : ℕ) :
-    (∑' i : ℕ, if J ≤ i then ρ^i else 0) = ρ^J/(1-ρ) := by
-  have hs := geometric_cutoff_summable ρ hρ hρ1 J
-  rw [←hs.sum_add_tsum_nat_add J]
-  have hz : (∑ i ∈ Finset.range J, if J ≤ i then ρ^i else 0) = 0 := by
-    apply Finset.sum_eq_zero
-    intro i hi
-    exact ite_eq_right (Nat.not_le_of_gt (Finset.mem_range.mp hi))
-  rw [hz,zero_add]
-  simp only [Nat.le_add_left,ite_true,pow_add,tsum_mul_right,
-    tsum_geometric_of_lt_one hρ hρ1,div_eq_mul_inv]
-  exact mul_comm _ _
-
-theorem finite_geometric_cutoff_bound (ρ : ℝ) (hρ : 0 ≤ ρ) (hρ1 : ρ < 1) (J M : ℕ) :
-    (∑ i : Fin M, if J ≤ i.val then ρ^i.val else 0) ≤ ρ^J/(1-ρ) := by
-  change (∑ i : Fin M, (fun n : ℕ => if J ≤ n then ρ^n else 0) i.val) ≤ _
-  rw [Fin.sum_univ_eq_sum_range (fun n : ℕ => if J ≤ n then ρ^n else 0) M,←geometric_cutoff_tsum ρ hρ hρ1 J]
-  exact Summable.sum_le_tsum (s := Finset.range M) (fun i _ => by split_ifs <;> positivity)
-    (geometric_cutoff_summable ρ hρ hρ1 J)
-
- 
-
-theorem finite_double_geometric_cutoff_bound (ρ : ℝ) (hρ : 0 ≤ ρ) (hρ1 : ρ < 1)
-    (J M N D : ℕ) (hD : 2*J ≤ D) :
-    (∑ i : Fin M, ∑ j : Fin N, if i.val+j.val < D then 0 else ρ^(i.val+j.val)) ≤
-      2*ρ^J/(1-ρ)^2 := by
-  let q : ℕ → ℝ := fun i => if J ≤ i then ρ^i else 0
-  have hq (i : ℕ) : 0 ≤ q i := by dsimp [q]; split_ifs <;> positivity
-  have hp (i : ℕ) : 0 ≤ ρ^i := pow_nonneg hρ i
-  have hQ := finite_geometric_cutoff_bound ρ hρ hρ1 J M
-  have hQ' := finite_geometric_cutoff_bound ρ hρ hρ1 J N
-  have hS := finite_geometric_cutoff_bound ρ hρ hρ1 0 M
-  have hS' := finite_geometric_cutoff_bound ρ hρ hρ1 0 N
-  simp only [Nat.zero_le,ite_true,pow_zero,one_div] at hS hS'
-  have hd : 0 < 1-ρ := sub_pos.mpr hρ1
-  calc
-    _ ≤ ∑ i : Fin M, ∑ j : Fin N, (q i.val*ρ^j.val + ρ^i.val*q j.val) := by
-      apply Finset.sum_le_sum
-      intro i _
-      apply Finset.sum_le_sum
-      intro j _
-      by_cases hh : i.val+j.val < D
-      · rw [ite_eq_left hh]
-        exact add_nonneg (mul_nonneg (hq _) (hp _)) (mul_nonneg (hp _) (hq _))
-      · rw [ite_eq_right hh,pow_add]
-        have hj : J ≤ i.val ∨ J ≤ j.val := by omega
-        rcases hj with hi | hj
-        · simp only [q,ite_eq_left hi]
-          exact le_add_of_nonneg_right (mul_nonneg (hp _) (by split_ifs <;> positivity))
-        · simp only [q,ite_eq_left hj]
-          exact le_add_of_nonneg_left (mul_nonneg (by split_ifs <;> positivity) (hp _))
-    _ = (∑ i : Fin M, q i.val)*(∑ j : Fin N, ρ^j.val) +
-        (∑ i : Fin M, ρ^i.val)*(∑ j : Fin N, q j.val) := by
-      simp only [Finset.sum_add_distrib,←Finset.mul_sum,←Finset.sum_mul]
-    _ ≤ (ρ^J/(1-ρ))*(1-ρ)⁻¹ + (1-ρ)⁻¹*(ρ^J/(1-ρ)) := by
-      apply add_le_add
-      · exact mul_le_mul hQ hS' (Finset.sum_nonneg (fun i _ => hp _)) (by positivity)
-      · exact mul_le_mul hS hQ' (Finset.sum_nonneg (fun i _ => hq _)) (by positivity)
-    _ = _ := by field_simp; ring
-
-end UniformCollisionTails
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section UniformRawCollision
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem raw_intersection_uniform_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ)) (hcontract : ‖P‖ < 1)
-    (m n J : ℕ) (x y : V) (hD : 2*J ≤ (openSimpleGraph G Set.univ).dist x y) :
-    (∑' w : RawFrom (openSimpleGraph G Set.univ) m x × RawFrom (openSimpleGraph G Set.univ) n y,
-      if ∃ z, z ∈ w.1.2.vertices ∧ z ∈ w.2.2.vertices then
-        w.1.2.mass (walkKernel G hloc a)*w.2.2.mass (walkKernel G hloc a) else 0) ≤
-      2*‖P‖^J/(1-‖P‖)^2 :=
-  (raw_intersection_probability_bound G hloc a P hpos hsym hP m n x y).trans
-    (finite_double_geometric_cutoff_bound ‖P‖ (norm_nonneg _) hcontract J (m+1) (n+1) _ hD)
-
- 
-theorem exists_uniform_raw_collision_separation (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ)) (hcontract : ‖P‖ < 1)
-    (ε : ℝ) (hε : 0 < ε) :
-    ∃ J₀ : ℕ, ∀ J : ℕ, J₀ ≤ J → ∀ m n : ℕ, ∀ x y : V,
-      2*J ≤ (openSimpleGraph G Set.univ).dist x y →
-      (∑' w : RawFrom (openSimpleGraph G Set.univ) m x × RawFrom (openSimpleGraph G Set.univ) n y,
-        if ∃ z, z ∈ w.1.2.vertices ∧ z ∈ w.2.2.vertices then
-          w.1.2.mass (walkKernel G hloc a)*w.2.2.mass (walkKernel G hloc a) else 0) < ε := by
-  have ht : Filter.Tendsto (fun J : ℕ => 2*‖P‖^J/(1-‖P‖)^2) Filter.atTop (nhds 0) := by
-    simpa only [mul_zero,zero_div] using
-      ((tendsto_pow_atTop_nhds_zero_of_lt_one (norm_nonneg P) hcontract).const_mul 2).div_const ((1-‖P‖)^2)
-  obtain ⟨J₀,hJ⟩ := Filter.eventually_atTop.mp (ht.eventually (gt_mem_nhds hε))
-  refine ⟨J₀,?_⟩
-  intro J hJJ m n x y hd
-  exact (raw_intersection_uniform_bound G hloc a P hpos hsym hP hcontract m n J x y hd).trans_lt (hJ J hJJ)
-
-end UniformRawCollision
-end Percolation
-
- 
-namespace Percolation
-universe u
-variable {V : Type u}
-namespace HoldingWalk
-variable {H : SimpleGraph V}
-
- 
-
-def fromPositions : (k : ℕ) → (f : Fin (k+1) → V) →
-    (∀ i : Fin k, H.Adj (f i.castSucc) (f i.succ)) →
-    HoldingWalk H k (f 0) (f (Fin.last k))
-  | 0,f,_ => .nil (f 0)
-  | k+1,f,h => .step (h 0) (fromPositions k (fun i => f i.succ) (fun i => h i.succ))
-
-theorem mass_fromPositions (P : V → V → ℝ) (a : ℝ)
-    (hP : ∀ x y, H.Adj x y → P x y = a)
-    (k : ℕ) (f : Fin (k+1) → V) (h : ∀ i : Fin k, H.Adj (f i.castSucc) (f i.succ)) :
-    (fromPositions k f h).mass P = a^k := by
-  induction k with
-  | zero => rfl
-  | succ k ih =>
-    change P (f 0) (f 1) * (fromPositions k (fun i => f i.succ) (fun i => h i.succ)).mass P = a^(k+1)
-    have he : P (f 0) (f 1) = a := by simpa using hP _ _ (h 0)
-    rw [he,ih,pow_succ']
-
-theorem position_fromPositions (k : ℕ) (f : Fin (k+1) → V)
-    (h : ∀ i : Fin k, H.Adj (f i.castSucc) (f i.succ)) (i : ℕ) (hi : i ≤ k) :
-    (fromPositions k f h).position i = f ⟨i,by omega⟩ := by
-  induction k generalizing i with
-  | zero =>
-    obtain rfl : i = 0 := by omega
-    rfl
-  | succ k ih =>
-    cases i with
-    | zero => rfl
-    | succ i =>
-      exact ih (fun j => f j.succ) (fun j => h j.succ) i (by omega)
-
-end HoldingWalk
-end Percolation
-
- 
-namespace Percolation
-universe u
-variable {V : Type u}
-namespace HoldingWalk
-variable {H : SimpleGraph V}
-
-theorem position_append_left {k l : ℕ} {x y z : V}
-    (w : HoldingWalk H k x y) (v : HoldingWalk H l y z) (i : ℕ) (hi : i ≤ k) :
-    (w.append v).position i = w.position i := by
-  induction w generalizing i with
-  | nil =>
-    obtain rfl : i = 0 := by omega
-    exact v.position_zero
-  | hold w ih => cases i with
-    | zero => rfl
-    | succ i => exact ih v i (by omega)
-  | step h w ih => cases i with
-    | zero => rfl
-    | succ i => exact ih v i (by omega)
-
-theorem position_append_right {k l : ℕ} {x y z : V}
-    (w : HoldingWalk H k x y) (v : HoldingWalk H l y z) (i : ℕ) :
-    (w.append v).position (k+i) = v.position i := by
-  induction w with
-  | nil => simp only [Nat.zero_add,append]
-  | hold w ih => simpa only [Nat.succ_add,append,position] using ih v
-  | step h w ih => simpa only [Nat.succ_add,append,position] using ih v
-
- 
-
-theorem ext_positions {k : ℕ} {x y : V} (w v : HoldingWalk H k x y)
-    (h : ∀ i : ℕ, i ≤ k → w.position i = v.position i) : w = v := by
-  induction k generalizing x y with
-  | zero => cases w; cases v; rfl
-  | succ k ih =>
-    cases w with
-    | hold w =>
-      cases v with
-      | hold v =>
-        congr 1
-        exact ih w v (fun i hi => h (i+1) (Nat.succ_le_succ hi))
-      | step hv v =>
-        have he := h 1 (Nat.succ_le_succ (Nat.zero_le k))
-        simp only [position,position_zero] at he
-        exact (hv.ne he).elim
-    | step hw w =>
-      cases v with
-      | hold v =>
-        have he := h 1 (Nat.succ_le_succ (Nat.zero_le k))
-        simp only [position,position_zero] at he
-        exact (hw.ne he.symm).elim
-      | step hv v =>
-        have he := h 1 (Nat.succ_le_succ (Nat.zero_le k))
-        simp only [position,position_zero] at he
-        cases he
-        congr 1
-        exact ih w v (fun i hi => h (i+1) (Nat.succ_le_succ hi))
-
-end HoldingWalk
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section IntrinsicVisibleBlocks
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-def segmentCenter (J : ℕ) : Fin (J+J+1) := ⟨J,by omega⟩
-
-def segmentPositivePositions (J : ℕ) (s : Fin (J+J+1) → V) : Fin (J+1) → V :=
-  fun i => s ⟨J+i.val,by omega⟩
-
-def positiveStem (H : SimpleGraph V) (J : ℕ) (s : Fin (J+J+1) → V)
-    (hs : ∀ i : Fin (J+J), H.Adj (s i.castSucc) (s i.succ)) :
-    HoldingWalk H J (s (segmentCenter J)) (s (Fin.last (J+J))) :=
-  HoldingWalk.fromPositions J (segmentPositivePositions J s) (fun i => hs ⟨J+i.val,by omega⟩)
-
-theorem positiveStem_mass (H : SimpleGraph V) (J : ℕ) (s : Fin (J+J+1) → V)
-    (hs : ∀ i : Fin (J+J), H.Adj (s i.castSucc) (s i.succ))
-    (P : V → V → ℝ) (a : ℝ) (hP : ∀ x y, H.Adj x y → P x y = a) :
-    (positiveStem H J s hs).mass P = a^J :=
-  HoldingWalk.mass_fromPositions P a hP J _ _
-
-theorem orbitSegment_consecutive (G : BondGraph V E) (J : ℕ) (γ s : Fin (J+J+1) → V)
-    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
-    (hs : s ∈ segmentOrbit G γ) :
-    ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (s i.castSucc) (s i.succ) :=
-  fun i => segmentOrbit_adj G γ s hs _ _ (hγ i)
-
- 
-
-def escapeLabelRaw (G : BondGraph V E) (J L m : ℕ) (γ : Fin (J+J+1) → V)
-    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
-    (ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m) :
-    HoldingWalk (openSimpleGraph G Set.univ) ((J+L)+J)
-      (ξ.first (segmentCenter J)) (ξ.last (segmentCenter J)) :=
-  (positiveStem _ J ξ.first (orbitSegment_consecutive G J γ ξ.first hγ ξ.first_mem)).append
-    (ξ.middle.append (positiveStem _ J ξ.last
-      (orbitSegment_consecutive G J γ ξ.last hγ ξ.last_mem)).reverse)
-
-theorem escapeLabelRaw_mass (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (J L m : ℕ) (γ : Fin (J+J+1) → V)
-    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
-    (ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m) :
-    (escapeLabelRaw G J L m γ hγ ξ).mass (walkKernel G hloc a) =
-      a^(J+J)*ξ.middle.mass (walkKernel G hloc a) := by
-  simp only [escapeLabelRaw,HoldingWalk.mass_append,HoldingWalk.mass_reverse _ (walkKernel_symmetric G hloc a),
-    positiveStem_mass _ _ _ _ _ a (fun _ _ h => walkKernel_step G hloc a h),pow_add]
-  ring
-
-theorem escapeLabelRaw_middle_position (G : BondGraph V E) (J L m : ℕ) (γ : Fin (J+J+1) → V)
-    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
-    (ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m) (i : ℕ) (hi : i ≤ L) :
-    (escapeLabelRaw G J L m γ hγ ξ).position (J+i) = ξ.middle.position i := by
-  rw [escapeLabelRaw,HoldingWalk.position_append_right,HoldingWalk.position_append_left _ _ i hi]
-
-end IntrinsicVisibleBlocks
-end Percolation
-
- 
-namespace Percolation
-universe u
-variable {V : Type u}
-namespace RawFrom
-variable {H : SimpleGraph V}
-
- 
-theorem ext_positions {k : ℕ} {x : V} (w v : RawFrom H k x)
-    (h : ∀ i : ℕ, i ≤ k → w.2.position i = v.2.position i) : w = v := by
-  rcases w with ⟨y,w⟩
-  rcases v with ⟨z,v⟩
-  have he : y = z := by simpa only [HoldingWalk.position_length] using h k le_rfl
-  subst z
-  exact Sigma.ext rfl (heq_of_eq (HoldingWalk.ext_positions w v h))
-
-theorem pairwise_disjoint_position_indicators {k : ℕ} {x : V} (f : ℕ → V)
-    {w v : RawFrom H k x}
-    (hw : ∀ i : ℕ, i ≤ k → w.2.position i = f i)
-    (hv : ∀ i : ℕ, i ≤ k → v.2.position i = f i) : w = v :=
-  ext_positions w v (fun i hi => (hw i hi).trans (hv i hi).symm)
-
-end RawFrom
-
- 
-
-theorem finite_single_fiber_weight_bound {α : Type*} [Fintype α]
-    (μ : α → ℝ) (B : α → Prop) [DecidablePred B] (C : ℝ) (hC : 0 ≤ C)
-    (hunique : ∀ a b, B a → B b → a = b) (hbound : ∀ a, B a → μ a ≤ C) :
-    (∑ a, if B a then μ a else 0) ≤ C := by
-  classical
-  by_cases hh : ∃ a, B a
-  · obtain ⟨a,ha⟩ := hh
-    calc
-      _ = μ a := by
-        rw [Finset.sum_eq_single a]
-        · exact ite_eq_left ha
-        · intro b _ hba
-          exact ite_eq_right (fun hb => hba (hunique b a hb ha))
-        · simp
-      _ ≤ C := hbound a ha
-  · simpa only [not_exists.mp hh,ite_false,Finset.sum_const_zero] using hC
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section RootedRawProjection
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-def rootedEscapeLabelRaw (G : BondGraph V E) (J L m : ℕ) (γ : Fin (J+J+1) → V)
-    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
-    (x : V) (ξ : {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m // ξ.first (segmentCenter J) = x}) :
-    RawFrom (openSimpleGraph G Set.univ) ((J+L)+J) x := by
-  rcases ξ with ⟨ξ,hξ⟩
-  subst x
-  exact ⟨ξ.last (segmentCenter J),escapeLabelRaw G J L m γ hγ ξ⟩
-
-theorem rootedEscapeLabelRaw_mass (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (J L m : ℕ) (γ : Fin (J+J+1) → V)
-    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
-    (x : V) (ξ : {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m // ξ.first (segmentCenter J) = x}) :
-    (rootedEscapeLabelRaw G J L m γ hγ x ξ).2.mass (walkKernel G hloc a) =
-      a^(J+J)*ξ.1.middle.mass (walkKernel G hloc a) := by
-  rcases ξ with ⟨ξ,hξ⟩
-  subst x
-  exact escapeLabelRaw_mass G hloc a J L m γ hγ ξ
-
-theorem rootedEscapeLabelRaw_middle_position (G : BondGraph V E) (J L m : ℕ) (γ : Fin (J+J+1) → V)
-    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
-    (x : V) (ξ : {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m // ξ.first (segmentCenter J) = x})
-    (i : ℕ) (hi : i ≤ L) :
-    (rootedEscapeLabelRaw G J L m γ hγ x ξ).2.position (J+i) = ξ.1.middle.position i := by
-  rcases ξ with ⟨ξ,hξ⟩
-  subst x
-  exact escapeLabelRaw_middle_position G J L m γ hγ ξ i hi
-
-theorem rootedEscapeLabelRaw_fiber_middle_mass (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (ha : 0 < a) (J L m : ℕ) (γ : Fin (J+J+1) → V)
-    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
-    (x : V) (ξ : {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m // ξ.first (segmentCenter J) = x})
-    (w : RawFrom (openSimpleGraph G Set.univ) ((J+L)+J) x)
-    (hw : rootedEscapeLabelRaw G J L m γ hγ x ξ = w) :
-    ξ.1.middle.mass (walkKernel G hloc a) = w.2.mass (walkKernel G hloc a)/a^(J+J) := by
-  apply (eq_div_iff (ne_of_gt (pow_pos ha _))).mpr
-  rw [mul_comm,←rootedEscapeLabelRaw_mass G hloc a J L m γ hγ x ξ,hw]
-
-theorem rootedEscapeLabelRaw_fiber_middle_positions (G : BondGraph V E) (J L m : ℕ) (γ : Fin (J+J+1) → V)
-    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
-    (x : V) (ξ : {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m // ξ.first (segmentCenter J) = x})
-    (w : RawFrom (openSimpleGraph G Set.univ) ((J+L)+J) x)
-    (hw : rootedEscapeLabelRaw G J L m γ hγ x ξ = w) :
-    ∀ i : ℕ, i ≤ L → ξ.1.middle.position i = w.2.position (J+i) := by
-  intro i hi
-  rw [←rootedEscapeLabelRaw_middle_position G J L m γ hγ x ξ i hi,hw]
-
-end RootedRawProjection
-end Percolation
-
- 
-namespace Percolation
-section FiniteFiberDomination
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem finite_auxiliary_event_bound {α : Type*} [Fintype α]
-    (μ : α → ℝ) (hμ : ∀ a, 0 ≤ μ a) (hs : ∑ a, μ a = 1)
-    (B : α → Prop) (c : ℝ) (hc : 0 ≤ c) :
-    (∑ a, if B a then c*μ a else 0) ≤ c := by
-  calc
-    _ ≤ ∑ a, c*μ a := Finset.sum_le_sum (fun a _ => by
-      split_ifs; exact le_rfl; exact mul_nonneg hc (hμ a))
-    _ = c := by rw [←Finset.mul_sum,hs,mul_one]
-
- 
-
-theorem finite_nested_unique_fiber_bound {β : Type*} [Fintype β]
-    {γ : β → Type*} [∀ b, Fintype (γ b)]
-    {T : (b : β) → γ b → Type*} [∀ b c, Fintype (T b c)]
-    (μ : β → ℝ) (hμ : ∀ b, 0 ≤ μ b)
-    (ν : (b : β) → (c : γ b) → T b c → ℝ) (hν : ∀ b c t, 0 ≤ ν b c t)
-    (hs : ∀ b c, ∑ t, ν b c t = 1)
-    (B : (b : β) → (c : γ b) → T b c → Prop)
-    (N C : ℝ) (hN : 0 ≤ N) (hC : 0 ≤ C)
-    (hcard : ∀ b, (Fintype.card (γ b) : ℝ) ≤ N)
-    (hunique : ∀ b c t b' c' t', B b c t → B b' c' t' → b = b')
-    (hbound : ∀ b c t, B b c t → μ b ≤ C) :
-    (∑ b, ∑ c, ∑ t, if B b c t then μ b*ν b c t else 0) ≤ N*C := by
-  by_cases hh : ∃ b c t, B b c t
-  · obtain ⟨b,c,t,hb⟩ := hh
-    have he : (∑ b', ∑ c', ∑ t', if B b' c' t' then μ b'*ν b' c' t' else 0) =
-        ∑ c', ∑ t', if B b c' t' then μ b*ν b c' t' else 0 := by
-      apply Finset.sum_eq_single b
-      · intro b' _ hne
-        apply Finset.sum_eq_zero
-        intro c' _
-        apply Finset.sum_eq_zero
-        intro t' _
-        exact ite_eq_right (fun hb' => hne (hunique b' c' t' b c t hb' hb))
-      · simp
-    rw [he]
-    calc
-      _ ≤ ∑ _c : γ b, μ b := Finset.sum_le_sum (fun c' _ =>
-        finite_auxiliary_event_bound (ν b c') (hν b c') (hs b c') (B b c') (μ b) (hμ b))
-      _ = (Fintype.card (γ b) : ℝ)*μ b := by simp
-      _ ≤ N*C := mul_le_mul (hcard b) (hbound b c t hb) (hμ b) hN
-  · have hz : ∀ b c t, ¬B b c t := by simpa only [not_exists] using hh
-    simp only [hz,ite_false,Finset.sum_const_zero]
-    exact mul_nonneg hN hC
-
-end FiniteFiberDomination
-end Percolation
-
- 
-namespace Percolation
-universe u v w
-variable {V : Type u} {E : Type v} {I : Type w}
-section CodeMiddleIdentities
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Fintype I]
-
-theorem labelFromCode_middle_position (G : BondGraph V E) (hloc : LocallyFinite G)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
-    (center ep neg : I) (L m : ℕ) (x : V)
-    (c : RootedLabelCode G hloc γ n hdiam center ep neg L m x) (i : ℕ) :
-    (labelFromCode G hloc γ n hdiam center ep neg L m x c).1.middle.position i =
-      c.2.1.2.position i := by
-  rcases c with ⟨⟨s,hs⟩,⟨z,mid⟩,⟨h,hh⟩,α,β⟩
-  obtain ⟨hhm,hhz⟩ := (mem_segmentAnchors G hloc γ n hdiam ep z h).mp hh
-  subst z
-  rfl
-
-theorem labelFromCode_middle_mass (G : BondGraph V E) (hloc : LocallyFinite G)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
-    (center ep neg : I) (L m : ℕ) (x : V) (P : V → V → ℝ)
-    (c : RootedLabelCode G hloc γ n hdiam center ep neg L m x) :
-    (labelFromCode G hloc γ n hdiam center ep neg L m x c).1.middle.mass P =
-      c.2.1.2.mass P := by
-  rcases c with ⟨⟨s,hs⟩,⟨z,mid⟩,⟨h,hh⟩,α,β⟩
-  obtain ⟨hhm,hhz⟩ := (mem_segmentAnchors G hloc γ n hdiam ep z h).mp hh
-  subst z
-  rfl
-
-end CodeMiddleIdentities
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section LiteralFiberEstimate
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem rooted_code_single_anchor_fiber_bound (G : BondGraph V E) (hloc : LocallyFinite G)
-    (a : ℝ) (ha : 0 < a) (hK : ∀ x y, 0 ≤ walkKernel G hloc a x y)
-    (J L m : ℕ) (γ : Fin (J+J+1) → V)
-    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
-    (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (x : V)
-    (s : segmentAnchors G hloc γ n hdiam (segmentCenter J) x)
-    (w : RawFrom (openSimpleGraph G Set.univ) ((J+L)+J) x) :
-    (∑' mid : RawFrom (openSimpleGraph G Set.univ) L (s.val (Fin.last (J+J))),
-      ∑' h : segmentAnchors G hloc γ n hdiam (Fin.last (J+J)) mid.1,
-        ∑' tails : RawFrom (openSimpleGraph G Set.univ) m (s.val 0) ×
-            RawFrom (openSimpleGraph G Set.univ) m (h.val 0),
-          if rootedEscapeLabelRaw G J L m γ hγ x
-              (labelFromCode G hloc γ n hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m x
-                ⟨s,mid,h,tails⟩) = w then
-            mid.2.mass (walkKernel G hloc a) * tails.1.2.mass (walkKernel G hloc a) *
-              tails.2.2.mass (walkKernel G hloc a) else 0) ≤
-      ((segmentAnchors G hloc γ n hdiam (Fin.last (J+J)) (γ (Fin.last (J+J)))).card : ℝ) *
-        (w.2.mass (walkKernel G hloc a)/a^(J+J)) := by
-  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
-  let (k : ℕ) (z : V) : Fintype (RawFrom (openSimpleGraph G Set.univ) k z) := Fintype.ofFinite _
-  simp only [tsum_fintype,mul_assoc]
-  convert finite_nested_unique_fiber_bound
-    (β := RawFrom (openSimpleGraph G Set.univ) L (s.val (Fin.last (J+J))))
-    (γ := fun mid => segmentAnchors G hloc γ n hdiam (Fin.last (J+J)) mid.1)
-    (T := fun _ h => RawFrom (openSimpleGraph G Set.univ) m (s.val 0) ×
-      RawFrom (openSimpleGraph G Set.univ) m (h.val 0))
-    (fun mid => mid.2.mass (walkKernel G hloc a))
-    (fun mid => mid.2.mass_nonneg _ hK)
-    (fun _ _ tails => tails.1.2.mass (walkKernel G hloc a)*tails.2.2.mass (walkKernel G hloc a))
-    (fun _ _ tails => mul_nonneg (tails.1.2.mass_nonneg _ hK) (tails.2.2.mass_nonneg _ hK)) ?_
-    (fun mid h tails => rootedEscapeLabelRaw G J L m γ hγ x
-      (labelFromCode G hloc γ n hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m x ⟨s,mid,h,tails⟩) = w)
-    (((segmentAnchors G hloc γ n hdiam (Fin.last (J+J)) (γ (Fin.last (J+J)))).card : ℝ))
-    (w.2.mass (walkKernel G hloc a)/a^(J+J)) ?_ ?_ ?_ ?_ ?_ using 1
-  · apply Finset.sum_congr rfl
-    intro mid _
-    apply Finset.sum_congr rfl
-    intro h _
-    apply Finset.sum_congr rfl
-    intro tails _
-    split_ifs <;> rfl
-  · intro mid h
-    rw [Fintype.sum_prod_type]
-    simp only [←Finset.mul_sum,←Finset.sum_mul]
-    simpa only [tsum_fintype,mul_one] using congrArg₂ (fun r s : ℝ => r*s)
-      (rawFrom_probability_sum G hloc a m (s.val 0))
-      (rawFrom_probability_sum G hloc a m (h.val 0))
-  · positivity
-  · exact div_nonneg (w.2.mass_nonneg _ hK) (pow_nonneg ha.le _)
-  · intro mid
-    rw [Fintype.card_coe,segmentAnchors_card_eq_indicator G hloc γ n hdiam (Fin.last (J+J)) mid.1]
-    split_ifs
-    · exact le_rfl
-    · exact_mod_cast Nat.zero_le _
-  · intro mid h tails mid' h' tails' hh hh'
-    apply RawFrom.ext_positions
-    intro i hi
-    have he := rootedEscapeLabelRaw_fiber_middle_positions G J L m γ hγ x
-      (labelFromCode G hloc γ n hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m x ⟨s,mid,h,tails⟩) w hh i hi
-    have he' := rootedEscapeLabelRaw_fiber_middle_positions G J L m γ hγ x
-      (labelFromCode G hloc γ n hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m x ⟨s,mid',h',tails'⟩) w hh' i hi
-    have hm := labelFromCode_middle_position G hloc γ n hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m x ⟨s,mid,h,tails⟩ i
-    have hm' := labelFromCode_middle_position G hloc γ n hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m x ⟨s,mid',h',tails'⟩ i
-    exact hm.symm.trans (he.trans (he'.symm.trans hm'))
-  · intro mid h tails hh
-    have he := rootedEscapeLabelRaw_fiber_middle_mass G hloc a ha J L m γ hγ x
-      (labelFromCode G hloc γ n hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m x ⟨s,mid,h,tails⟩) w hh
-    have hm := labelFromCode_middle_mass G hloc γ n hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m x (walkKernel G hloc a) ⟨s,mid,h,tails⟩
-    exact (hm.symm.trans he).le
-
-end LiteralFiberEstimate
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section LiteralProjectionDomination
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem rooted_label_raw_fiber_mass_bound (G : BondGraph V E) (hloc : LocallyFinite G)
-    (a : ℝ) (ha : 0 < a) (hK : ∀ x y, 0 ≤ walkKernel G hloc a x y)
-    (J L m : ℕ) (γ : Fin (J+J+1) → V)
-    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
-    (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (x : V)
-    (w : RawFrom (openSimpleGraph G Set.univ) ((J+L)+J) x) :
-    (∑' ξ : {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m // ξ.first (segmentCenter J) = x},
-      if rootedEscapeLabelRaw G J L m γ hγ x ξ = w then ξ.1.mass (walkKernel G hloc a) else 0) ≤
-      ((segmentAnchors G hloc γ n hdiam (segmentCenter J) x).card : ℝ) *
-      ((segmentAnchors G hloc γ n hdiam (Fin.last (J+J)) (γ (Fin.last (J+J)))).card : ℝ) *
-        (w.2.mass (walkKernel G hloc a)/a^(J+J)) := by
-  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
-  let := finite_rootedEscapeLabels G hloc γ n hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m x
-  let R := rootedLabelEquiv G hloc γ n hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m x
-  rw [←R.symm.tsum_eq]
-  simp only [R,rootedLabelEquiv,Equiv.coe_fn_symm_mk,labelFromCode_mass]
-  unfold RootedLabelCode at *
-  rw [(summable_of_hasFiniteSupport (Set.toFinite _)).tsum_sigma]
-  calc
-    _ ≤ ∑' _s : segmentAnchors G hloc γ n hdiam (segmentCenter J) x,
-        ((segmentAnchors G hloc γ n hdiam (Fin.last (J+J)) (γ (Fin.last (J+J)))).card : ℝ) *
-          (w.2.mass (walkKernel G hloc a)/a^(J+J)) := by
-      apply (summable_of_hasFiniteSupport (Set.toFinite _)).tsum_le_tsum _
-        (summable_of_hasFiniteSupport (Set.toFinite _))
-      intro s
-      rw [(summable_of_hasFiniteSupport (Set.toFinite _)).tsum_sigma]
-      conv_lhs => arg 1; ext mid; rw [(summable_of_hasFiniteSupport (Set.toFinite _)).tsum_sigma]
-      exact rooted_code_single_anchor_fiber_bound G hloc a ha hK J L m γ hγ n hdiam x s w
-    _ = _ := by
-      simp only [tsum_fintype,Finset.sum_const,Finset.card_univ,Fintype.card_coe,nsmul_eq_mul]
-      ring
-
-end LiteralProjectionDomination
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section NormalizedProjection
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem tsum_indicator_div {α : Type*} (f : α → ℝ) (B : α → Prop) [DecidablePred B] (d : ℝ) :
-    (∑' a, if B a then f a/d else 0) = (∑' a, if B a then f a else 0)/d := by
-  rw [←tsum_div_const]
-  apply tsum_congr
-  intro a
-  split_ifs <;> simp
-
-theorem rooted_label_raw_projection_bound (G : BondGraph V E) (hloc : LocallyFinite G)
-    (a : ℝ) (ha : 0 < a) (hK : ∀ x y, 0 ≤ walkKernel G hloc a x y)
-    (J L m : ℕ) (γ : Fin (J+J+1) → V)
-    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
-    (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (x : V)
-    (hx : x ∈ vertexOrbit G (γ (segmentCenter J)))
-    (q : ℝ) (hq : 0 < q) (hqle : q ≤ segmentReturnMass G hloc a γ (Fin.last (J+J)) L (γ (Fin.last (J+J))))
-    (w : RawFrom (openSimpleGraph G Set.univ) ((J+L)+J) x) :
-    (∑' ξ : {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m // ξ.first (segmentCenter J) = x},
-      if rootedEscapeLabelRaw G J L m γ hγ x ξ = w then
-        escapeLabelProbability G hloc a γ n hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m ξ.1 else 0) ≤
-      w.2.mass (walkKernel G hloc a)/(a^(J+J)*q) := by
-  have hN (i : Fin (J+J+1)) : 0 < ((segmentAnchors G hloc γ n hdiam i (γ i)).card : ℝ) := by
-    exact_mod_cast Finset.card_pos.mpr ((segmentAnchors_nonempty_iff G hloc γ n hdiam i (γ i)).mpr
-      (by exact ⟨Equiv.refl V,Equiv.refl E,(fun e => by simp),rfl⟩))
-  have hQ := hq.trans_le hqle
-  have hA := pow_pos ha (J+J)
-  have hm := w.2.mass_nonneg (walkKernel G hloc a) hK
-  simp only [escapeLabelProbability,tsum_indicator_div]
-  have hb := rooted_label_raw_fiber_mass_bound G hloc a ha hK J L m γ hγ n hdiam x w
-  rw [segmentAnchors_card_on_orbit G hloc γ n hdiam (segmentCenter J) x hx] at hb
-  apply (div_le_div_of_nonneg_right hb (mul_nonneg (mul_nonneg (hN (segmentCenter J)).le (hN (Fin.last (J+J))).le) hQ.le)).trans
-  calc
-    _ = w.2.mass (walkKernel G hloc a)/
-        (a^(J+J)*segmentReturnMass G hloc a γ (Fin.last (J+J)) L (γ (Fin.last (J+J)))) := by
-      field_simp [(hN (segmentCenter J)).ne', (hN (Fin.last (J+J))).ne']
-    _ ≤ _ := div_le_div_of_nonneg_left hm (mul_pos hA hq) (mul_le_mul_of_nonneg_left hqle hA.le)
-
-end NormalizedProjection
-end Percolation
-
- 
-namespace Percolation
-universe u
-variable {V : Type u}
-section SuffixCollisionOperator
-
-theorem operator_pow_norm_le (P : Hilbert V →L[ℂ] Hilbert V) (k : ℕ) : ‖P^k‖ ≤ ‖P‖^k := by
-  cases k with
-  | zero => simpa only [pow_zero,ContinuousLinearMap.one_def] using
-      (ContinuousLinearMap.norm_id_le (𝕜 := ℂ) (E := Hilbert V))
-  | succ k => exact norm_pow_le' P (Nat.succ_pos k)
-
- 
-
-theorem positive_rows_collision_hasSum (A B : Hilbert V →L[ℂ] Hilbert V)
-    (hA : PositiveRealMatrix A) (x y : V) :
-    HasSum (fun z => (matrixCoeff A x z).re*(matrixCoeff B y z).re)
-      (matrixCoeff (A*star B) x y).re := by
-  simpa only [matrixCoeff_adjoint,Complex.star_def,Complex.conj_re] using
-    hA.mul_hasSum (star B) x y
-
-theorem positive_rows_collision_bound (A B : Hilbert V →L[ℂ] Hilbert V)
-    (hA : PositiveRealMatrix A) (x y : V) :
-    (∑' z, (matrixCoeff A x z).re*(matrixCoeff B y z).re) ≤ ‖A‖*‖B‖ := by
-  rw [(positive_rows_collision_hasSum A B hA x y).tsum_eq]
-  apply (Complex.re_le_norm _).trans ((matrixCoeff_norm_le (A*star B) x y).trans _)
-  have hs : ‖star B‖ = ‖B‖ := ContinuousLinearMap.adjoint.norm_map B
-  simpa only [hs] using norm_mul_le A (star B)
-
-theorem walk_suffix_collision_operator_bound (P H : Hilbert V →L[ℂ] Hilbert V)
-    (hP : PositiveRealMatrix P) (C : ℝ) (hH : ‖H‖ ≤ C) (l L : ℕ) (x y : V) :
-    (∑' z, (matrixCoeff (P^l) x z).re*(matrixCoeff (P^L*H) y z).re) ≤
-      C*‖P‖^(l+L) := by
-  calc
-    _ ≤ ‖P^l‖*‖P^L*H‖ := positive_rows_collision_bound _ _ (hP.pow l) x y
-    _ ≤ ‖P‖^l*(‖P‖^L*C) := mul_le_mul (operator_pow_norm_le P l)
-      ((norm_mul_le _ _).trans (mul_le_mul (operator_pow_norm_le P L) hH (norm_nonneg H)
-        (pow_nonneg (norm_nonneg P) L))) (norm_nonneg _) (pow_nonneg (norm_nonneg P) l)
-    _ = _ := by rw [pow_add]; ring
-
-end SuffixCollisionOperator
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section ActualSuffixCollision
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {I : Type*} [Fintype I]
-
- 
-
-theorem raw_anchored_position_expectation (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (ep j : I)
-    (P H : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (hH : ∀ x y, matrixCoeff H x y = (anchoredPositionKernel G hloc γ n hdiam ep j x y : ℂ))
-    (L : ℕ) (x z : V) :
-    (∑' mid : RawFrom (openSimpleGraph G Set.univ) L x,
-      mid.2.mass (walkKernel G hloc a)*anchoredPositionKernel G hloc γ n hdiam ep j mid.1 z) =
-      (matrixCoeff (P^L*H) x z).re := by
-  change RawFrom.integral (walkKernel G hloc a) L x
-    (fun y => anchoredPositionKernel G hloc γ n hdiam ep j y z) = _
-  rw [raw_integral_operator G hloc a P hpos hP]
-  simpa only [hH,Complex.ofReal_re] using ((hpos.pow L).mul_hasSum H x z).tsum_eq
-
- 
-
-theorem raw_tail_anchored_collision_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (ep j : I)
-    (P H : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (hH : ∀ x y, matrixCoeff H x y = (anchoredPositionKernel G hloc γ n hdiam ep j x y : ℂ))
-    (C : ℝ) (hHC : ‖H‖ ≤ C) (l m L : ℕ) (hlm : l ≤ m) (x y : V) :
-    (∑' tail : RawFrom (openSimpleGraph G Set.univ) m x,
-      tail.2.mass (walkKernel G hloc a) *
-        ∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
-          mid.2.mass (walkKernel G hloc a)*
-            anchoredPositionKernel G hloc γ n hdiam ep j mid.1 (tail.2.position l)) ≤
-      C*‖P‖^(l+L) := by
-  simp only [raw_anchored_position_expectation G hloc a γ n hdiam ep j P H hpos hP hH]
-  rw [raw_position_integral G hloc a l m hlm x (fun z => (matrixCoeff (P^L*H) y z).re),
-    raw_integral_operator G hloc a P hpos hP]
-  exact walk_suffix_collision_operator_bound P H hpos C hHC l L x y
-
-end ActualSuffixCollision
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section UniformSuffixCollision
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {I ι : Type*} [Fintype I] [Fintype ι]
-
- 
-
-theorem raw_tail_anchored_collision_uniform (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G)
-    (htr : RootTransport root weight Γ) (hw : ∀ r, 0 ≤ weight r) (hw1 : ∑ r, weight r = 1)
-    (hcov : ∀ x, ∃ r, ∃ g ∈ Γ, g (root r) = x)
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ r, 1 ≤ c*weight r)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (ep j : I)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (l m L : ℕ) (hlm : l ≤ m) (x y : V) :
-    (∑' tail : RawFrom (openSimpleGraph G Set.univ) m x,
-      tail.2.mass (walkKernel G hloc a) *
-        ∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
-          mid.2.mass (walkKernel G hloc a)*
-            anchoredPositionKernel G hloc γ n hdiam ep j mid.1 (tail.2.position l)) ≤
-      Real.sqrt c*‖P‖^(l+L) := by
-  obtain ⟨H,hH,hHn⟩ := exists_anchoredPositionOperator_uniform G hloc root weight Γ hΓ htr hw hw1 hcov
-    c hc hmin γ n hdiam ep j
-  exact raw_tail_anchored_collision_bound G hloc a γ n hdiam ep j P H hpos hP hH
-    (Real.sqrt c) hHn l m L hlm x y
-
-end UniformSuffixCollision
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section GlobalAnchorDenominator
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {I : Type*} [Fintype I]
-
- 
-
-theorem anchoredPositionKernel_global_denominator (G : BondGraph V E) (hloc : LocallyFinite G)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (ep j : I) (x z : V) :
-    (∑ h : segmentAnchors G hloc γ n hdiam ep x,
-      if h.val j = z then ((segmentAnchors G hloc γ n hdiam ep (γ ep)).card : ℝ)⁻¹ else 0) =
-      anchoredPositionKernel G hloc γ n hdiam ep j x z := by
-  rw [Finset.sum_coe_sort _ (fun h : I → V => if h j = z then
-    ((segmentAnchors G hloc γ n hdiam ep (γ ep)).card : ℝ)⁻¹ else 0)]
-  by_cases hx : x ∈ vertexOrbit G (γ ep)
-  · unfold anchoredPositionKernel
-    rw [segmentAnchors_card_on_orbit G hloc γ n hdiam ep x hx]
-    apply Finset.sum_congr rfl
-    intro h _
-    simp only [eq_comm]
-  · have hz := ((segmentAnchors_nonempty_iff G hloc γ n hdiam ep x).not.mpr hx)
-    have he := Finset.not_nonempty_iff_eq_empty.mp hz
-    simp only [anchoredPositionKernel,he,Finset.sum_empty]
-
-theorem finite_geometric_sum_bound (ρ : ℝ) (hρ : 0 ≤ ρ) (hρ1 : ρ < 1) (M : ℕ) :
-    (∑ i : Fin M, ρ^i.val) ≤ 1/(1-ρ) := by
-  simpa only [Nat.zero_le,ite_true,pow_zero] using finite_geometric_cutoff_bound ρ hρ hρ1 0 M
-
-end GlobalAnchorDenominator
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section LiteralSuffixExperiment
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {I : Type*} [Fintype I]
-
- 
-
-theorem raw_tail_anchor_collision_identity (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (ep j : I)
-    (l m L : ℕ) (x y : V) :
-    (∑' tail : RawFrom (openSimpleGraph G Set.univ) m x,
-      ∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
-        ∑' h : segmentAnchors G hloc γ n hdiam ep mid.1,
-          if tail.2.position l = h.val j then
-            tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a)/
-              (segmentAnchors G hloc γ n hdiam ep (γ ep)).card else 0) =
-    (∑' tail : RawFrom (openSimpleGraph G Set.univ) m x,
-      tail.2.mass (walkKernel G hloc a) *
-        ∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
-          mid.2.mass (walkKernel G hloc a)*
-            anchoredPositionKernel G hloc γ n hdiam ep j mid.1 (tail.2.position l)) := by
-  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
-  let (k : ℕ) (z : V) : Fintype (RawFrom (openSimpleGraph G Set.univ) k z) := Fintype.ofFinite _
-  simp only [tsum_fintype]
-  apply Finset.sum_congr rfl
-  intro tail _
-  rw [Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro mid _
-  rw [←anchoredPositionKernel_global_denominator G hloc γ n hdiam ep j mid.1 (tail.2.position l)]
-  simp only [Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro h _
-  by_cases hh : tail.2.position l = h.val j
-  · simp only [ite_eq_left hh,ite_eq_left hh.symm,div_eq_mul_inv,mul_assoc]
-  · simp only [ite_eq_right hh,ite_eq_right (Ne.symm hh),mul_zero]
-
-end LiteralSuffixExperiment
-end Percolation
-
- 
-namespace Percolation
-section NestedFiniteUnion
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem finite_threefold_event_union {α β δ : Type*} [Fintype α] [Fintype β] [Fintype δ]
-    {γ : β → Type*} [∀ b, Fintype (γ b)]
-    (μ : (a : α) → (b : β) → γ b → ℝ) (hμ : ∀ a b c, 0 ≤ μ a b c)
-    (B : (a : α) → (b : β) → γ b → δ → Prop) :
-    (∑ a, ∑ b, ∑ c, if ∃ d, B a b c d then μ a b c else 0) ≤
-      ∑ d, ∑ a, ∑ b, ∑ c, if B a b c d then μ a b c else 0 := by
-  have hh := finite_nonnegative_event_union
-    (fun ξ : α × Sigma γ => μ ξ.1 ξ.2.1 ξ.2.2)
-    (fun ξ => hμ ξ.1 ξ.2.1 ξ.2.2)
-    (fun ξ : α × Sigma γ => B ξ.1 ξ.2.1 ξ.2.2)
-  simpa only [Fintype.sum_prod_type,Fintype.sum_sigma] using hh
-
-end NestedFiniteUnion
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section WholeSuffixCollision
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {I ι S : Type*} [Fintype I] [Fintype ι] [Fintype S]
-
- 
-
-theorem raw_tail_whole_suffix_collision_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G)
-    (htr : RootTransport root weight Γ) (hw : ∀ r, 0 ≤ weight r) (hw1 : ∑ r, weight r = 1)
-    (hcov : ∀ x, ∃ r, ∃ g ∈ Γ, g (root r) = x)
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ r, 1 ≤ c*weight r)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (ep : I) (j : S → I)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (hρ : ‖P‖ < 1) (m L : ℕ) (x y : V) :
-    (∑' tail : RawFrom (openSimpleGraph G Set.univ) m x,
-      ∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
-        ∑' h : segmentAnchors G hloc γ n hdiam ep mid.1,
-          if ∃ ij : Fin (m+1) × S, tail.2.position ij.1.val = h.val (j ij.2) then
-            tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a)/
-              (segmentAnchors G hloc γ n hdiam ep (γ ep)).card else 0) ≤
-      (Fintype.card S : ℝ)*Real.sqrt c*‖P‖^L/(1-‖P‖) := by
-  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
-  let (k : ℕ) (z : V) : Fintype (RawFrom (openSimpleGraph G Set.univ) k z) := Fintype.ofFinite _
-  have hK (x y : V) : 0 ≤ walkKernel G hloc a x y := by
-    simpa only [hP,Complex.ofReal_re] using (hpos x y).2
-  simp only [tsum_fintype]
-  have hu := finite_threefold_event_union
-    (α := RawFrom (openSimpleGraph G Set.univ) m x)
-    (β := RawFrom (openSimpleGraph G Set.univ) L y)
-    (γ := fun mid => segmentAnchors G hloc γ n hdiam ep mid.1)
-    (δ := Fin (m+1) × S)
-    (fun tail mid _ => tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a)/
-      (segmentAnchors G hloc γ n hdiam ep (γ ep)).card)
-    (fun tail mid _ => div_nonneg (mul_nonneg (tail.2.mass_nonneg _ hK) (mid.2.mass_nonneg _ hK)) (Nat.cast_nonneg _))
-    (fun tail _ h ij => tail.2.position ij.1.val = h.val (j ij.2))
-  apply hu.trans
-  rw [Fintype.sum_prod_type]
-  calc
-    _ ≤ ∑ i : Fin (m+1), ∑ _s : S, Real.sqrt c*‖P‖^(i.val+L) := by
-      apply Finset.sum_le_sum
-      intro i _
-      apply Finset.sum_le_sum
-      intro s _
-      have hh := raw_tail_anchored_collision_uniform G hloc a root weight Γ hΓ htr hw hw1 hcov c hc hmin
-        γ n hdiam ep (j s) P hpos hP i.val m L (by omega) x y
-      rw [←raw_tail_anchor_collision_identity G hloc a γ n hdiam ep (j s) i.val m L x y] at hh
-      simpa only [tsum_fintype] using hh
-    _ = ((Fintype.card S : ℝ)*Real.sqrt c*‖P‖^L)*(∑ i : Fin (m+1), ‖P‖^i.val) := by
-      simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,pow_add,←Finset.mul_sum,←Finset.sum_mul]
-      ring
-    _ ≤ ((Fintype.card S : ℝ)*Real.sqrt c*‖P‖^L)*(1/(1-‖P‖)) :=
-      mul_le_mul_of_nonneg_left (finite_geometric_sum_bound ‖P‖ (norm_nonneg P) hρ (m+1)) (by positivity)
-    _ = _ := by ring
-
-end WholeSuffixCollision
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section ActualAnchorPointTest
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {I : Type*} [Fintype I]
-
-theorem raw_anchor_position_identity (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (ep j : I)
-    (L : ℕ) (y z : V) :
-    (∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
-      ∑' h : segmentAnchors G hloc γ n hdiam ep mid.1,
-        if h.val j = z then mid.2.mass (walkKernel G hloc a)/
-          (segmentAnchors G hloc γ n hdiam ep (γ ep)).card else 0) =
-    (∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
-      mid.2.mass (walkKernel G hloc a)*anchoredPositionKernel G hloc γ n hdiam ep j mid.1 z) := by
-  apply tsum_congr
-  intro mid
-  rw [←anchoredPositionKernel_global_denominator G hloc γ n hdiam ep j mid.1 z,tsum_fintype,Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro h _
-  split_ifs <;> simp only [div_eq_mul_inv,mul_zero]
-
-theorem raw_anchor_point_collision_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (ep j : I)
-    (P H : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (hH : ∀ x y, matrixCoeff H x y = (anchoredPositionKernel G hloc γ n hdiam ep j x y : ℂ))
-    (C : ℝ) (hC : ‖H‖ ≤ C) (L : ℕ) (y z : V) :
-    (∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
-      ∑' h : segmentAnchors G hloc γ n hdiam ep mid.1,
-        if h.val j = z then mid.2.mass (walkKernel G hloc a)/
-          (segmentAnchors G hloc γ n hdiam ep (γ ep)).card else 0) ≤ C*‖P‖^L := by
-  rw [raw_anchor_position_identity G hloc a γ n hdiam ep j L y z,
-    raw_anchored_position_expectation G hloc a γ n hdiam ep j P H hpos hP hH L y z]
-  apply (Complex.re_le_norm _).trans ((matrixCoeff_norm_le (P^L*H) y z).trans _)
-  calc
-    _ ≤ ‖P^L‖*‖H‖ := norm_mul_le _ _
-    _ ≤ ‖P‖^L*C := mul_le_mul (operator_pow_norm_le P L) hC (norm_nonneg _) (pow_nonneg (norm_nonneg P) L)
-    _ = _ := mul_comm _ _
-
-end ActualAnchorPointTest
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Escape.Labels
+
+namespace OAI
+
+/-! Collision Tails for bond percolation. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section RawTimeMarginals
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem raw_position_integral (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (i m : ℕ) (him : i ≤ m) (x : V) (f : V → ℝ) :
+    (∑' w : RawFrom (openSimpleGraph G Set.univ) m x,
+      w.2.mass (walkKernel G hloc a) * f (w.2.position i)) =
+      RawFrom.integral (H := openSimpleGraph G Set.univ) (walkKernel G hloc a) i x f := by
+  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
+  induction i generalizing m x with
+  | zero =>
+    simp only [HoldingWalk.position_zero,tsum_mul_right,rawFrom_probability_sum G hloc a,one_mul,RawFrom.integral_zero]
+  | succ i ih =>
+    cases m with
+    | zero => omega
+    | succ m =>
+      rw [←(RawFrom.succEquiv m x).symm.tsum_eq,RawFrom.integral_succ]
+      let (z : V) := Fintype.ofFinite (RawFrom (openSimpleGraph G Set.univ) m z)
+      let := Fintype.ofFinite ((openSimpleGraph G Set.univ).neighborSet x)
+      have hh (z : V) := ih m (Nat.le_of_succ_le_succ him) z
+      simp only [tsum_fintype,Fintype.sum_sum_type,Fintype.sum_sigma,RawFrom.succEquiv,
+        Equiv.symm_mk,Equiv.coe_fn_mk,HoldingWalk.mass,HoldingWalk.position,mul_assoc]
+      simp only [tsum_fintype] at hh
+      simp only [←Finset.mul_sum,hh]
+
+theorem raw_position_probability (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (i m : ℕ) (him : i ≤ m) (x z : V) :
+    (∑' w : RawFrom (openSimpleGraph G Set.univ) m x,
+      if w.2.position i = z then w.2.mass (walkKernel G hloc a) else 0) =
+      (matrixCoeff (P^i) x z).re := by
+  have hh := raw_position_integral G hloc a i m him x (fun y => if y = z then (1:ℝ) else 0)
+  simp only [mul_ite,mul_one,mul_zero] at hh
+  rw [hh,raw_integral_operator G hloc a P hpos hP]
+  simp only [mul_ite,mul_one,mul_zero,tsum_ite_eq]
+
+end RawTimeMarginals
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section IndependentRawCollisions
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem raw_two_time_collision (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (i m j n : ℕ) (him : i ≤ m) (hjn : j ≤ n) (x y : V) :
+    (∑' w : RawFrom (openSimpleGraph G Set.univ) m x × RawFrom (openSimpleGraph G Set.univ) n y,
+      if w.1.2.position i = w.2.2.position j then
+        w.1.2.mass (walkKernel G hloc a)*w.2.2.mass (walkKernel G hloc a) else 0) =
+      (matrixCoeff (P^(i+j)) x y).re := by
+  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
+  let := Fintype.ofFinite (RawFrom (openSimpleGraph G Set.univ) m x)
+  let := Fintype.ofFinite (RawFrom (openSimpleGraph G Set.univ) n y)
+  have hi (w : RawFrom (openSimpleGraph G Set.univ) m x) :
+      (∑' v : RawFrom (openSimpleGraph G Set.univ) n y,
+        if w.2.position i = v.2.position j then w.2.mass (walkKernel G hloc a)*v.2.mass (walkKernel G hloc a) else 0) =
+      w.2.mass (walkKernel G hloc a)*(matrixCoeff (P^j) y (w.2.position i)).re := by
+    rw [← raw_position_probability G hloc a P hpos hP j n hjn y (w.2.position i),←tsum_mul_left]
+    apply tsum_congr
+    intro v
+    by_cases hh : w.2.position i = v.2.position j
+    · simp only [ite_eq_left hh,ite_eq_left hh.symm]
+    · simp only [ite_eq_right hh,ite_eq_right (Ne.symm hh),mul_zero]
+  rw [(summable_of_hasFiniteSupport (Set.toFinite _)).tsum_prod]
+  simp only [hi]
+  rw [raw_position_integral G hloc a i m him x (fun z => (matrixCoeff (P^j) y z).re),raw_integral_operator G hloc a P hpos hP]
+  simp only [matrixCoeff_real_symm (P^j) (hsym.pow j) y]
+  rw [((hpos.pow i).mul_hasSum (P^j) x y).tsum_eq,←pow_add]
+
+theorem matrixCoeff_norm_le (A : Hilbert V →L[ℂ] Hilbert V) (x y : V) :
+    ‖matrixCoeff A x y‖ ≤ ‖A‖ := by
+  apply (lp.norm_apply_le_norm (by norm_num : (2:ENNReal) ≠ 0) (A (lp.single 2 y (1:ℂ))) x).trans
+  simpa using A.le_opNorm (lp.single 2 y (1:ℂ))
+
+theorem matrixCoeff_power_le (P : Hilbert V →L[ℂ] Hilbert V) (k : ℕ) (x y : V) :
+    (matrixCoeff (P^k) x y).re ≤ ‖P‖^k := by
+  have hn (k : ℕ) : ‖P^k‖ ≤ ‖P‖^k := by
+    induction k with
+    | zero => simpa only [pow_zero,ContinuousLinearMap.one_def] using
+        (ContinuousLinearMap.norm_id_le (𝕜 := ℂ) (E := Hilbert V))
+    | succ k ih =>
+      rw [pow_succ,pow_succ]
+      exact (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right ih (norm_nonneg P))
+  exact (Complex.re_le_norm _).trans ((matrixCoeff_norm_le (P^k) x y).trans (hn k))
+
+end IndependentRawCollisions
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section FinitePropagation
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem reachBall_exists_short_walk (G : BondGraph V E) (x : V) (k : ℕ) {y : V}
+    (hy : y ∈ reachBall G x k) :
+    ∃ q : (openSimpleGraph G Set.univ).Walk x y, q.length ≤ k := by
+  induction k generalizing y with
+  | zero =>
+    have he : y = x := hy
+    subst y
+    exact ⟨.nil,by simp⟩
+  | succ k ih =>
+    rcases hy with hy | hy
+    · obtain ⟨q,hq⟩ := ih hy
+      exact ⟨q,hq.trans (Nat.le_succ _)⟩
+    · simp only [Set.mem_iUnion,Set.mem_ofPred_eq] at hy
+      obtain ⟨z,hz,hzy⟩ := hy
+      obtain ⟨q,hq⟩ := ih hz
+      by_cases he : z = y
+      · subst z
+        exact ⟨q,hq.trans (Nat.le_succ _)⟩
+      · let h : (openSimpleGraph G Set.univ).Adj z y :=
+          (openSimpleGraph_univ_adj G z y).mpr ⟨he,hzy⟩
+        refine ⟨q.append (.cons h .nil),?_⟩
+        simpa only [SimpleGraph.Walk.length_append,SimpleGraph.Walk.length_cons,
+          SimpleGraph.Walk.length_nil,zero_add] using Nat.add_le_add_right hq 1
+
+theorem dist_le_of_mem_reachBall (G : BondGraph V E) (x y : V) (k : ℕ)
+    (hy : y ∈ reachBall G x k) : (openSimpleGraph G Set.univ).dist x y ≤ k := by
+  obtain ⟨q,hq⟩ := reachBall_exists_short_walk G x k hy
+  exact (SimpleGraph.dist_le q).trans hq
+
+theorem walkPower_zero_of_lt_dist (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (k : ℕ) (x y : V) (hk : k < (openSimpleGraph G Set.univ).dist x y) :
+    (matrixCoeff (P^k) x y).re = 0 := by
+  apply walkPower_support G hloc a P hpos hP
+  intro hy
+  exact Nat.not_le_of_gt hk (dist_le_of_mem_reachBall G x y k hy)
+
+ 
+
+theorem raw_two_time_collision_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (i m j n : ℕ) (him : i ≤ m) (hjn : j ≤ n) (x y : V) :
+    (∑' w : RawFrom (openSimpleGraph G Set.univ) m x × RawFrom (openSimpleGraph G Set.univ) n y,
+      if w.1.2.position i = w.2.2.position j then
+        w.1.2.mass (walkKernel G hloc a)*w.2.2.mass (walkKernel G hloc a) else 0) ≤
+      if i+j < (openSimpleGraph G Set.univ).dist x y then 0 else ‖P‖^(i+j) := by
+  rw [raw_two_time_collision G hloc a P hpos hsym hP i m j n him hjn x y]
+  split_ifs with hh
+  · exact (walkPower_zero_of_lt_dist G hloc a P hpos hP (i+j) x y hh).le
+  · exact matrixCoeff_power_le P (i+j) x y
+
+end FinitePropagation
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section ActualCollisionUnion
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem finite_nonnegative_event_union {α β : Type*} [Fintype α] [Fintype β]
+    (μ : α → ℝ) (hμ : ∀ a, 0 ≤ μ a) (B : α → β → Prop) :
+    (∑ a, if ∃ b, B a b then μ a else 0) ≤ ∑ b, ∑ a, if B a b then μ a else 0 := by
+  rw [Finset.sum_comm]
+  apply Finset.sum_le_sum
+  intro a _
+  by_cases hh : ∃ b, B a b
+  · obtain ⟨b,hb⟩ := hh
+    rw [ite_eq_left ⟨b,hb⟩]
+    calc
+      μ a = (if B a b then μ a else 0) := (ite_eq_left hb).symm
+      _ ≤ ∑ b, if B a b then μ a else 0 :=
+        Finset.single_le_sum (f := fun b => if B a b then μ a else 0)
+          (fun b _ => by split_ifs; exact hμ a; exact le_rfl) (Finset.mem_univ b)
+  · rw [ite_eq_right hh]
+    exact Finset.sum_nonneg (fun b _ => by split_ifs; exact hμ a; exact le_rfl)
+
+ 
+
+theorem raw_intersection_probability_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (m n : ℕ) (x y : V) :
+    (∑' w : RawFrom (openSimpleGraph G Set.univ) m x × RawFrom (openSimpleGraph G Set.univ) n y,
+      if ∃ z, z ∈ w.1.2.vertices ∧ z ∈ w.2.2.vertices then
+        w.1.2.mass (walkKernel G hloc a)*w.2.2.mass (walkKernel G hloc a) else 0) ≤
+    ∑ i : Fin (m+1), ∑ j : Fin (n+1),
+      if i.val+j.val < (openSimpleGraph G Set.univ).dist x y then 0 else ‖P‖^(i.val+j.val) := by
+  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
+  let := Fintype.ofFinite (RawFrom (openSimpleGraph G Set.univ) m x)
+  let := Fintype.ofFinite (RawFrom (openSimpleGraph G Set.univ) n y)
+  have hK (x y : V) : 0 ≤ walkKernel G hloc a x y := by
+    simpa only [hP,Complex.ofReal_re] using (hpos x y).2
+  have he (w : RawFrom (openSimpleGraph G Set.univ) m x × RawFrom (openSimpleGraph G Set.univ) n y) :
+      (∃ z, z ∈ w.1.2.vertices ∧ z ∈ w.2.2.vertices) ↔
+        ∃ ij : Fin (m+1) × Fin (n+1), w.1.2.position ij.1 = w.2.2.position ij.2 := by
+    constructor
+    · rintro ⟨z,hz,hz'⟩
+      obtain ⟨i,hi,he⟩ := w.1.2.mem_vertices_iff_position.mp hz
+      obtain ⟨j,hj,he'⟩ := w.2.2.mem_vertices_iff_position.mp hz'
+      exact ⟨(⟨i,by omega⟩,⟨j,by omega⟩),he.trans he'.symm⟩
+    · rintro ⟨⟨i,j⟩,hij⟩
+      exact ⟨w.1.2.position i,w.1.2.position_mem_vertices i,hij ▸ w.2.2.position_mem_vertices j⟩
+  simp only [he,tsum_fintype]
+  have hu := finite_nonnegative_event_union
+    (fun w : RawFrom (openSimpleGraph G Set.univ) m x × RawFrom (openSimpleGraph G Set.univ) n y =>
+      w.1.2.mass (walkKernel G hloc a)*w.2.2.mass (walkKernel G hloc a))
+    (fun w => mul_nonneg (w.1.2.mass_nonneg _ hK) (w.2.2.mass_nonneg _ hK))
+    (fun w (ij : Fin (m+1) × Fin (n+1)) => w.1.2.position ij.1 = w.2.2.position ij.2)
+  apply hu.trans
+  rw [Fintype.sum_prod_type]
+  apply Finset.sum_le_sum
+  intro i _
+  apply Finset.sum_le_sum
+  intro j _
+  simpa only [tsum_fintype] using
+    raw_two_time_collision_bound G hloc a P hpos hsym hP i m j n (by omega) (by omega) x y
+
+end ActualCollisionUnion
+end Percolation
+
+ 
+namespace Percolation
+section UniformCollisionTails
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem geometric_cutoff_summable (ρ : ℝ) (hρ : 0 ≤ ρ) (hρ1 : ρ < 1) (J : ℕ) :
+    Summable (fun i : ℕ => if J ≤ i then ρ^i else 0) :=
+  Summable.of_nonneg_of_le (fun i => by split_ifs <;> positivity)
+    (fun i => by split_ifs <;> first | exact le_rfl | exact pow_nonneg hρ i)
+    (summable_geometric_of_lt_one hρ hρ1)
+
+theorem geometric_cutoff_tsum (ρ : ℝ) (hρ : 0 ≤ ρ) (hρ1 : ρ < 1) (J : ℕ) :
+    (∑' i : ℕ, if J ≤ i then ρ^i else 0) = ρ^J/(1-ρ) := by
+  have hs := geometric_cutoff_summable ρ hρ hρ1 J
+  rw [←hs.sum_add_tsum_nat_add J]
+  have hz : (∑ i ∈ Finset.range J, if J ≤ i then ρ^i else 0) = 0 := by
+    apply Finset.sum_eq_zero
+    intro i hi
+    exact ite_eq_right (Nat.not_le_of_gt (Finset.mem_range.mp hi))
+  rw [hz,zero_add]
+  simp only [Nat.le_add_left,ite_true,pow_add,tsum_mul_right,
+    tsum_geometric_of_lt_one hρ hρ1,div_eq_mul_inv]
+  exact mul_comm _ _
+
+theorem finite_geometric_cutoff_bound (ρ : ℝ) (hρ : 0 ≤ ρ) (hρ1 : ρ < 1) (J M : ℕ) :
+    (∑ i : Fin M, if J ≤ i.val then ρ^i.val else 0) ≤ ρ^J/(1-ρ) := by
+  change (∑ i : Fin M, (fun n : ℕ => if J ≤ n then ρ^n else 0) i.val) ≤ _
+  rw [Fin.sum_univ_eq_sum_range (fun n : ℕ => if J ≤ n then ρ^n else 0) M,←geometric_cutoff_tsum ρ hρ hρ1 J]
+  exact Summable.sum_le_tsum (s := Finset.range M) (fun i _ => by split_ifs <;> positivity)
+    (geometric_cutoff_summable ρ hρ hρ1 J)
+
+ 
+
+theorem finite_double_geometric_cutoff_bound (ρ : ℝ) (hρ : 0 ≤ ρ) (hρ1 : ρ < 1)
+    (J M N D : ℕ) (hD : 2*J ≤ D) :
+    (∑ i : Fin M, ∑ j : Fin N, if i.val+j.val < D then 0 else ρ^(i.val+j.val)) ≤
+      2*ρ^J/(1-ρ)^2 := by
+  let q : ℕ → ℝ := fun i => if J ≤ i then ρ^i else 0
+  have hq (i : ℕ) : 0 ≤ q i := by dsimp [q]; split_ifs <;> positivity
+  have hp (i : ℕ) : 0 ≤ ρ^i := pow_nonneg hρ i
+  have hQ := finite_geometric_cutoff_bound ρ hρ hρ1 J M
+  have hQ' := finite_geometric_cutoff_bound ρ hρ hρ1 J N
+  have hS := finite_geometric_cutoff_bound ρ hρ hρ1 0 M
+  have hS' := finite_geometric_cutoff_bound ρ hρ hρ1 0 N
+  simp only [Nat.zero_le,ite_true,pow_zero,one_div] at hS hS'
+  have hd : 0 < 1-ρ := sub_pos.mpr hρ1
+  calc
+    _ ≤ ∑ i : Fin M, ∑ j : Fin N, (q i.val*ρ^j.val + ρ^i.val*q j.val) := by
+      apply Finset.sum_le_sum
+      intro i _
+      apply Finset.sum_le_sum
+      intro j _
+      by_cases hh : i.val+j.val < D
+      · rw [ite_eq_left hh]
+        exact add_nonneg (mul_nonneg (hq _) (hp _)) (mul_nonneg (hp _) (hq _))
+      · rw [ite_eq_right hh,pow_add]
+        have hj : J ≤ i.val ∨ J ≤ j.val := by omega
+        rcases hj with hi | hj
+        · simp only [q,ite_eq_left hi]
+          exact le_add_of_nonneg_right (mul_nonneg (hp _) (by split_ifs <;> positivity))
+        · simp only [q,ite_eq_left hj]
+          exact le_add_of_nonneg_left (mul_nonneg (by split_ifs <;> positivity) (hp _))
+    _ = (∑ i : Fin M, q i.val)*(∑ j : Fin N, ρ^j.val) +
+        (∑ i : Fin M, ρ^i.val)*(∑ j : Fin N, q j.val) := by
+      simp only [Finset.sum_add_distrib,←Finset.mul_sum,←Finset.sum_mul]
+    _ ≤ (ρ^J/(1-ρ))*(1-ρ)⁻¹ + (1-ρ)⁻¹*(ρ^J/(1-ρ)) := by
+      apply add_le_add
+      · exact mul_le_mul hQ hS' (Finset.sum_nonneg (fun i _ => hp _)) (by positivity)
+      · exact mul_le_mul hS hQ' (Finset.sum_nonneg (fun i _ => hq _)) (by positivity)
+    _ = _ := by field_simp; ring
+
+end UniformCollisionTails
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section UniformRawCollision
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem raw_intersection_uniform_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ)) (hcontract : ‖P‖ < 1)
+    (m n J : ℕ) (x y : V) (hD : 2*J ≤ (openSimpleGraph G Set.univ).dist x y) :
+    (∑' w : RawFrom (openSimpleGraph G Set.univ) m x × RawFrom (openSimpleGraph G Set.univ) n y,
+      if ∃ z, z ∈ w.1.2.vertices ∧ z ∈ w.2.2.vertices then
+        w.1.2.mass (walkKernel G hloc a)*w.2.2.mass (walkKernel G hloc a) else 0) ≤
+      2*‖P‖^J/(1-‖P‖)^2 :=
+  (raw_intersection_probability_bound G hloc a P hpos hsym hP m n x y).trans
+    (finite_double_geometric_cutoff_bound ‖P‖ (norm_nonneg _) hcontract J (m+1) (n+1) _ hD)
+
+ 
+theorem exists_uniform_raw_collision_separation (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ)) (hcontract : ‖P‖ < 1)
+    (ε : ℝ) (hε : 0 < ε) :
+    ∃ J₀ : ℕ, ∀ J : ℕ, J₀ ≤ J → ∀ m n : ℕ, ∀ x y : V,
+      2*J ≤ (openSimpleGraph G Set.univ).dist x y →
+      (∑' w : RawFrom (openSimpleGraph G Set.univ) m x × RawFrom (openSimpleGraph G Set.univ) n y,
+        if ∃ z, z ∈ w.1.2.vertices ∧ z ∈ w.2.2.vertices then
+          w.1.2.mass (walkKernel G hloc a)*w.2.2.mass (walkKernel G hloc a) else 0) < ε := by
+  have ht : Filter.Tendsto (fun J : ℕ => 2*‖P‖^J/(1-‖P‖)^2) Filter.atTop (nhds 0) := by
+    simpa only [mul_zero,zero_div] using
+      ((tendsto_pow_atTop_nhds_zero_of_lt_one (norm_nonneg P) hcontract).const_mul 2).div_const ((1-‖P‖)^2)
+  obtain ⟨J₀,hJ⟩ := Filter.eventually_atTop.mp (ht.eventually (gt_mem_nhds hε))
+  refine ⟨J₀,?_⟩
+  intro J hJJ m n x y hd
+  exact (raw_intersection_uniform_bound G hloc a P hpos hsym hP hcontract m n J x y hd).trans_lt (hJ J hJJ)
+
+end UniformRawCollision
+end Percolation
+
+ 
+namespace Percolation
+universe u
+variable {V : Type u}
+namespace HoldingWalk
+variable {H : SimpleGraph V}
+
+ 
+
+def fromPositions : (k : ℕ) → (f : Fin (k+1) → V) →
+    (∀ i : Fin k, H.Adj (f i.castSucc) (f i.succ)) →
+    HoldingWalk H k (f 0) (f (Fin.last k))
+  | 0,f,_ => .nil (f 0)
+  | k+1,f,h => .step (h 0) (fromPositions k (fun i => f i.succ) (fun i => h i.succ))
+
+theorem mass_fromPositions (P : V → V → ℝ) (a : ℝ)
+    (hP : ∀ x y, H.Adj x y → P x y = a)
+    (k : ℕ) (f : Fin (k+1) → V) (h : ∀ i : Fin k, H.Adj (f i.castSucc) (f i.succ)) :
+    (fromPositions k f h).mass P = a^k := by
+  induction k with
+  | zero => rfl
+  | succ k ih =>
+    change P (f 0) (f 1) * (fromPositions k (fun i => f i.succ) (fun i => h i.succ)).mass P = a^(k+1)
+    have he : P (f 0) (f 1) = a := by simpa using hP _ _ (h 0)
+    rw [he,ih,pow_succ']
+
+theorem position_fromPositions (k : ℕ) (f : Fin (k+1) → V)
+    (h : ∀ i : Fin k, H.Adj (f i.castSucc) (f i.succ)) (i : ℕ) (hi : i ≤ k) :
+    (fromPositions k f h).position i = f ⟨i,by omega⟩ := by
+  induction k generalizing i with
+  | zero =>
+    obtain rfl : i = 0 := by omega
+    rfl
+  | succ k ih =>
+    cases i with
+    | zero => rfl
+    | succ i =>
+      exact ih (fun j => f j.succ) (fun j => h j.succ) i (by omega)
+
+end HoldingWalk
+end Percolation
+
+ 
+namespace Percolation
+universe u
+variable {V : Type u}
+namespace HoldingWalk
+variable {H : SimpleGraph V}
+
+theorem position_append_left {k l : ℕ} {x y z : V}
+    (w : HoldingWalk H k x y) (v : HoldingWalk H l y z) (i : ℕ) (hi : i ≤ k) :
+    (w.append v).position i = w.position i := by
+  induction w generalizing i with
+  | nil =>
+    obtain rfl : i = 0 := by omega
+    exact v.position_zero
+  | hold w ih => cases i with
+    | zero => rfl
+    | succ i => exact ih v i (by omega)
+  | step h w ih => cases i with
+    | zero => rfl
+    | succ i => exact ih v i (by omega)
+
+theorem position_append_right {k l : ℕ} {x y z : V}
+    (w : HoldingWalk H k x y) (v : HoldingWalk H l y z) (i : ℕ) :
+    (w.append v).position (k+i) = v.position i := by
+  induction w with
+  | nil => simp only [Nat.zero_add,append]
+  | hold w ih => simpa only [Nat.succ_add,append,position] using ih v
+  | step h w ih => simpa only [Nat.succ_add,append,position] using ih v
+
+ 
+
+theorem ext_positions {k : ℕ} {x y : V} (w v : HoldingWalk H k x y)
+    (h : ∀ i : ℕ, i ≤ k → w.position i = v.position i) : w = v := by
+  induction k generalizing x y with
+  | zero => cases w; cases v; rfl
+  | succ k ih =>
+    cases w with
+    | hold w =>
+      cases v with
+      | hold v =>
+        congr 1
+        exact ih w v (fun i hi => h (i+1) (Nat.succ_le_succ hi))
+      | step hv v =>
+        have he := h 1 (Nat.succ_le_succ (Nat.zero_le k))
+        simp only [position,position_zero] at he
+        exact (hv.ne he).elim
+    | step hw w =>
+      cases v with
+      | hold v =>
+        have he := h 1 (Nat.succ_le_succ (Nat.zero_le k))
+        simp only [position,position_zero] at he
+        exact (hw.ne he.symm).elim
+      | step hv v =>
+        have he := h 1 (Nat.succ_le_succ (Nat.zero_le k))
+        simp only [position,position_zero] at he
+        cases he
+        congr 1
+        exact ih w v (fun i hi => h (i+1) (Nat.succ_le_succ hi))
+
+end HoldingWalk
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section IntrinsicVisibleBlocks
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+def segmentCenter (J : ℕ) : Fin (J+J+1) := ⟨J,by omega⟩
+
+def segmentPositivePositions (J : ℕ) (s : Fin (J+J+1) → V) : Fin (J+1) → V :=
+  fun i => s ⟨J+i.val,by omega⟩
+
+def positiveStem (H : SimpleGraph V) (J : ℕ) (s : Fin (J+J+1) → V)
+    (hs : ∀ i : Fin (J+J), H.Adj (s i.castSucc) (s i.succ)) :
+    HoldingWalk H J (s (segmentCenter J)) (s (Fin.last (J+J))) :=
+  HoldingWalk.fromPositions J (segmentPositivePositions J s) (fun i => hs ⟨J+i.val,by omega⟩)
+
+theorem positiveStem_mass (H : SimpleGraph V) (J : ℕ) (s : Fin (J+J+1) → V)
+    (hs : ∀ i : Fin (J+J), H.Adj (s i.castSucc) (s i.succ))
+    (P : V → V → ℝ) (a : ℝ) (hP : ∀ x y, H.Adj x y → P x y = a) :
+    (positiveStem H J s hs).mass P = a^J :=
+  HoldingWalk.mass_fromPositions P a hP J _ _
+
+theorem orbitSegment_consecutive (G : BondGraph V E) (J : ℕ) (γ s : Fin (J+J+1) → V)
+    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
+    (hs : s ∈ segmentOrbit G γ) :
+    ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (s i.castSucc) (s i.succ) :=
+  fun i => segmentOrbit_adj G γ s hs _ _ (hγ i)
+
+ 
+
+def escapeLabelRaw (G : BondGraph V E) (J L m : ℕ) (γ : Fin (J+J+1) → V)
+    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
+    (ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m) :
+    HoldingWalk (openSimpleGraph G Set.univ) ((J+L)+J)
+      (ξ.first (segmentCenter J)) (ξ.last (segmentCenter J)) :=
+  (positiveStem _ J ξ.first (orbitSegment_consecutive G J γ ξ.first hγ ξ.first_mem)).append
+    (ξ.middle.append (positiveStem _ J ξ.last
+      (orbitSegment_consecutive G J γ ξ.last hγ ξ.last_mem)).reverse)
+
+theorem escapeLabelRaw_mass (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (J L m : ℕ) (γ : Fin (J+J+1) → V)
+    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
+    (ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m) :
+    (escapeLabelRaw G J L m γ hγ ξ).mass (walkKernel G hloc a) =
+      a^(J+J)*ξ.middle.mass (walkKernel G hloc a) := by
+  simp only [escapeLabelRaw,HoldingWalk.mass_append,HoldingWalk.mass_reverse _ (walkKernel_symmetric G hloc a),
+    positiveStem_mass _ _ _ _ _ a (fun _ _ h => walkKernel_step G hloc a h),pow_add]
+  ring
+
+theorem escapeLabelRaw_middle_position (G : BondGraph V E) (J L m : ℕ) (γ : Fin (J+J+1) → V)
+    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
+    (ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m) (i : ℕ) (hi : i ≤ L) :
+    (escapeLabelRaw G J L m γ hγ ξ).position (J+i) = ξ.middle.position i := by
+  rw [escapeLabelRaw,HoldingWalk.position_append_right,HoldingWalk.position_append_left _ _ i hi]
+
+end IntrinsicVisibleBlocks
+end Percolation
+
+ 
+namespace Percolation
+universe u
+variable {V : Type u}
+namespace RawFrom
+variable {H : SimpleGraph V}
+
+ 
+theorem ext_positions {k : ℕ} {x : V} (w v : RawFrom H k x)
+    (h : ∀ i : ℕ, i ≤ k → w.2.position i = v.2.position i) : w = v := by
+  rcases w with ⟨y,w⟩
+  rcases v with ⟨z,v⟩
+  have he : y = z := by simpa only [HoldingWalk.position_length] using h k le_rfl
+  subst z
+  exact Sigma.ext rfl (heq_of_eq (HoldingWalk.ext_positions w v h))
+
+theorem pairwise_disjoint_position_indicators {k : ℕ} {x : V} (f : ℕ → V)
+    {w v : RawFrom H k x}
+    (hw : ∀ i : ℕ, i ≤ k → w.2.position i = f i)
+    (hv : ∀ i : ℕ, i ≤ k → v.2.position i = f i) : w = v :=
+  ext_positions w v (fun i hi => (hw i hi).trans (hv i hi).symm)
+
+end RawFrom
+
+ 
+
+theorem finite_single_fiber_weight_bound {α : Type*} [Fintype α]
+    (μ : α → ℝ) (B : α → Prop) [DecidablePred B] (C : ℝ) (hC : 0 ≤ C)
+    (hunique : ∀ a b, B a → B b → a = b) (hbound : ∀ a, B a → μ a ≤ C) :
+    (∑ a, if B a then μ a else 0) ≤ C := by
+  classical
+  by_cases hh : ∃ a, B a
+  · obtain ⟨a,ha⟩ := hh
+    calc
+      _ = μ a := by
+        rw [Finset.sum_eq_single a]
+        · exact ite_eq_left ha
+        · intro b _ hba
+          exact ite_eq_right (fun hb => hba (hunique b a hb ha))
+        · simp
+      _ ≤ C := hbound a ha
+  · simpa only [not_exists.mp hh,ite_false,Finset.sum_const_zero] using hC
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section RootedRawProjection
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+def rootedEscapeLabelRaw (G : BondGraph V E) (J L m : ℕ) (γ : Fin (J+J+1) → V)
+    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
+    (x : V) (ξ : {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m // ξ.first (segmentCenter J) = x}) :
+    RawFrom (openSimpleGraph G Set.univ) ((J+L)+J) x := by
+  rcases ξ with ⟨ξ,hξ⟩
+  subst x
+  exact ⟨ξ.last (segmentCenter J),escapeLabelRaw G J L m γ hγ ξ⟩
+
+theorem rootedEscapeLabelRaw_mass (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (J L m : ℕ) (γ : Fin (J+J+1) → V)
+    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
+    (x : V) (ξ : {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m // ξ.first (segmentCenter J) = x}) :
+    (rootedEscapeLabelRaw G J L m γ hγ x ξ).2.mass (walkKernel G hloc a) =
+      a^(J+J)*ξ.1.middle.mass (walkKernel G hloc a) := by
+  rcases ξ with ⟨ξ,hξ⟩
+  subst x
+  exact escapeLabelRaw_mass G hloc a J L m γ hγ ξ
+
+theorem rootedEscapeLabelRaw_middle_position (G : BondGraph V E) (J L m : ℕ) (γ : Fin (J+J+1) → V)
+    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
+    (x : V) (ξ : {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m // ξ.first (segmentCenter J) = x})
+    (i : ℕ) (hi : i ≤ L) :
+    (rootedEscapeLabelRaw G J L m γ hγ x ξ).2.position (J+i) = ξ.1.middle.position i := by
+  rcases ξ with ⟨ξ,hξ⟩
+  subst x
+  exact escapeLabelRaw_middle_position G J L m γ hγ ξ i hi
+
+theorem rootedEscapeLabelRaw_fiber_middle_mass (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (ha : 0 < a) (J L m : ℕ) (γ : Fin (J+J+1) → V)
+    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
+    (x : V) (ξ : {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m // ξ.first (segmentCenter J) = x})
+    (w : RawFrom (openSimpleGraph G Set.univ) ((J+L)+J) x)
+    (hw : rootedEscapeLabelRaw G J L m γ hγ x ξ = w) :
+    ξ.1.middle.mass (walkKernel G hloc a) = w.2.mass (walkKernel G hloc a)/a^(J+J) := by
+  apply (eq_div_iff (ne_of_gt (pow_pos ha _))).mpr
+  rw [mul_comm,←rootedEscapeLabelRaw_mass G hloc a J L m γ hγ x ξ,hw]
+
+theorem rootedEscapeLabelRaw_fiber_middle_positions (G : BondGraph V E) (J L m : ℕ) (γ : Fin (J+J+1) → V)
+    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
+    (x : V) (ξ : {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m // ξ.first (segmentCenter J) = x})
+    (w : RawFrom (openSimpleGraph G Set.univ) ((J+L)+J) x)
+    (hw : rootedEscapeLabelRaw G J L m γ hγ x ξ = w) :
+    ∀ i : ℕ, i ≤ L → ξ.1.middle.position i = w.2.position (J+i) := by
+  intro i hi
+  rw [←rootedEscapeLabelRaw_middle_position G J L m γ hγ x ξ i hi,hw]
+
+end RootedRawProjection
+end Percolation
+
+ 
+namespace Percolation
+section FiniteFiberDomination
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem finite_auxiliary_event_bound {α : Type*} [Fintype α]
+    (μ : α → ℝ) (hμ : ∀ a, 0 ≤ μ a) (hs : ∑ a, μ a = 1)
+    (B : α → Prop) (c : ℝ) (hc : 0 ≤ c) :
+    (∑ a, if B a then c*μ a else 0) ≤ c := by
+  calc
+    _ ≤ ∑ a, c*μ a := Finset.sum_le_sum (fun a _ => by
+      split_ifs; exact le_rfl; exact mul_nonneg hc (hμ a))
+    _ = c := by rw [←Finset.mul_sum,hs,mul_one]
+
+ 
+
+theorem finite_nested_unique_fiber_bound {β : Type*} [Fintype β]
+    {γ : β → Type*} [∀ b, Fintype (γ b)]
+    {T : (b : β) → γ b → Type*} [∀ b c, Fintype (T b c)]
+    (μ : β → ℝ) (hμ : ∀ b, 0 ≤ μ b)
+    (ν : (b : β) → (c : γ b) → T b c → ℝ) (hν : ∀ b c t, 0 ≤ ν b c t)
+    (hs : ∀ b c, ∑ t, ν b c t = 1)
+    (B : (b : β) → (c : γ b) → T b c → Prop)
+    (N C : ℝ) (hN : 0 ≤ N) (hC : 0 ≤ C)
+    (hcard : ∀ b, (Fintype.card (γ b) : ℝ) ≤ N)
+    (hunique : ∀ b c t b' c' t', B b c t → B b' c' t' → b = b')
+    (hbound : ∀ b c t, B b c t → μ b ≤ C) :
+    (∑ b, ∑ c, ∑ t, if B b c t then μ b*ν b c t else 0) ≤ N*C := by
+  by_cases hh : ∃ b c t, B b c t
+  · obtain ⟨b,c,t,hb⟩ := hh
+    have he : (∑ b', ∑ c', ∑ t', if B b' c' t' then μ b'*ν b' c' t' else 0) =
+        ∑ c', ∑ t', if B b c' t' then μ b*ν b c' t' else 0 := by
+      apply Finset.sum_eq_single b
+      · intro b' _ hne
+        apply Finset.sum_eq_zero
+        intro c' _
+        apply Finset.sum_eq_zero
+        intro t' _
+        exact ite_eq_right (fun hb' => hne (hunique b' c' t' b c t hb' hb))
+      · simp
+    rw [he]
+    calc
+      _ ≤ ∑ _c : γ b, μ b := Finset.sum_le_sum (fun c' _ =>
+        finite_auxiliary_event_bound (ν b c') (hν b c') (hs b c') (B b c') (μ b) (hμ b))
+      _ = (Fintype.card (γ b) : ℝ)*μ b := by simp
+      _ ≤ N*C := mul_le_mul (hcard b) (hbound b c t hb) (hμ b) hN
+  · have hz : ∀ b c t, ¬B b c t := by simpa only [not_exists] using hh
+    simp only [hz,ite_false,Finset.sum_const_zero]
+    exact mul_nonneg hN hC
+
+end FiniteFiberDomination
+end Percolation
+
+ 
+namespace Percolation
+universe u v w
+variable {V : Type u} {E : Type v} {I : Type w}
+section CodeMiddleIdentities
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Fintype I]
+
+theorem labelFromCode_middle_position (G : BondGraph V E) (hloc : LocallyFinite G)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
+    (center ep neg : I) (L m : ℕ) (x : V)
+    (c : RootedLabelCode G hloc γ n hdiam center ep neg L m x) (i : ℕ) :
+    (labelFromCode G hloc γ n hdiam center ep neg L m x c).1.middle.position i =
+      c.2.1.2.position i := by
+  rcases c with ⟨⟨s,hs⟩,⟨z,mid⟩,⟨h,hh⟩,α,β⟩
+  obtain ⟨hhm,hhz⟩ := (mem_segmentAnchors G hloc γ n hdiam ep z h).mp hh
+  subst z
+  rfl
+
+theorem labelFromCode_middle_mass (G : BondGraph V E) (hloc : LocallyFinite G)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
+    (center ep neg : I) (L m : ℕ) (x : V) (P : V → V → ℝ)
+    (c : RootedLabelCode G hloc γ n hdiam center ep neg L m x) :
+    (labelFromCode G hloc γ n hdiam center ep neg L m x c).1.middle.mass P =
+      c.2.1.2.mass P := by
+  rcases c with ⟨⟨s,hs⟩,⟨z,mid⟩,⟨h,hh⟩,α,β⟩
+  obtain ⟨hhm,hhz⟩ := (mem_segmentAnchors G hloc γ n hdiam ep z h).mp hh
+  subst z
+  rfl
+
+end CodeMiddleIdentities
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section LiteralFiberEstimate
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem rooted_code_single_anchor_fiber_bound (G : BondGraph V E) (hloc : LocallyFinite G)
+    (a : ℝ) (ha : 0 < a) (hK : ∀ x y, 0 ≤ walkKernel G hloc a x y)
+    (J L m : ℕ) (γ : Fin (J+J+1) → V)
+    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
+    (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (x : V)
+    (s : segmentAnchors G hloc γ n hdiam (segmentCenter J) x)
+    (w : RawFrom (openSimpleGraph G Set.univ) ((J+L)+J) x) :
+    (∑' mid : RawFrom (openSimpleGraph G Set.univ) L (s.val (Fin.last (J+J))),
+      ∑' h : segmentAnchors G hloc γ n hdiam (Fin.last (J+J)) mid.1,
+        ∑' tails : RawFrom (openSimpleGraph G Set.univ) m (s.val 0) ×
+            RawFrom (openSimpleGraph G Set.univ) m (h.val 0),
+          if rootedEscapeLabelRaw G J L m γ hγ x
+              (labelFromCode G hloc γ n hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m x
+                ⟨s,mid,h,tails⟩) = w then
+            mid.2.mass (walkKernel G hloc a) * tails.1.2.mass (walkKernel G hloc a) *
+              tails.2.2.mass (walkKernel G hloc a) else 0) ≤
+      ((segmentAnchors G hloc γ n hdiam (Fin.last (J+J)) (γ (Fin.last (J+J)))).card : ℝ) *
+        (w.2.mass (walkKernel G hloc a)/a^(J+J)) := by
+  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
+  let (k : ℕ) (z : V) : Fintype (RawFrom (openSimpleGraph G Set.univ) k z) := Fintype.ofFinite _
+  simp only [tsum_fintype,mul_assoc]
+  convert finite_nested_unique_fiber_bound
+    (β := RawFrom (openSimpleGraph G Set.univ) L (s.val (Fin.last (J+J))))
+    (γ := fun mid => segmentAnchors G hloc γ n hdiam (Fin.last (J+J)) mid.1)
+    (T := fun _ h => RawFrom (openSimpleGraph G Set.univ) m (s.val 0) ×
+      RawFrom (openSimpleGraph G Set.univ) m (h.val 0))
+    (fun mid => mid.2.mass (walkKernel G hloc a))
+    (fun mid => mid.2.mass_nonneg _ hK)
+    (fun _ _ tails => tails.1.2.mass (walkKernel G hloc a)*tails.2.2.mass (walkKernel G hloc a))
+    (fun _ _ tails => mul_nonneg (tails.1.2.mass_nonneg _ hK) (tails.2.2.mass_nonneg _ hK)) ?_
+    (fun mid h tails => rootedEscapeLabelRaw G J L m γ hγ x
+      (labelFromCode G hloc γ n hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m x ⟨s,mid,h,tails⟩) = w)
+    (((segmentAnchors G hloc γ n hdiam (Fin.last (J+J)) (γ (Fin.last (J+J)))).card : ℝ))
+    (w.2.mass (walkKernel G hloc a)/a^(J+J)) ?_ ?_ ?_ ?_ ?_ using 1
+  · apply Finset.sum_congr rfl
+    intro mid _
+    apply Finset.sum_congr rfl
+    intro h _
+    apply Finset.sum_congr rfl
+    intro tails _
+    split_ifs <;> rfl
+  · intro mid h
+    rw [Fintype.sum_prod_type]
+    simp only [←Finset.mul_sum,←Finset.sum_mul]
+    simpa only [tsum_fintype,mul_one] using congrArg₂ (fun r s : ℝ => r*s)
+      (rawFrom_probability_sum G hloc a m (s.val 0))
+      (rawFrom_probability_sum G hloc a m (h.val 0))
+  · positivity
+  · exact div_nonneg (w.2.mass_nonneg _ hK) (pow_nonneg ha.le _)
+  · intro mid
+    rw [Fintype.card_coe,segmentAnchors_card_eq_indicator G hloc γ n hdiam (Fin.last (J+J)) mid.1]
+    split_ifs
+    · exact le_rfl
+    · exact_mod_cast Nat.zero_le _
+  · intro mid h tails mid' h' tails' hh hh'
+    apply RawFrom.ext_positions
+    intro i hi
+    have he := rootedEscapeLabelRaw_fiber_middle_positions G J L m γ hγ x
+      (labelFromCode G hloc γ n hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m x ⟨s,mid,h,tails⟩) w hh i hi
+    have he' := rootedEscapeLabelRaw_fiber_middle_positions G J L m γ hγ x
+      (labelFromCode G hloc γ n hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m x ⟨s,mid',h',tails'⟩) w hh' i hi
+    have hm := labelFromCode_middle_position G hloc γ n hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m x ⟨s,mid,h,tails⟩ i
+    have hm' := labelFromCode_middle_position G hloc γ n hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m x ⟨s,mid',h',tails'⟩ i
+    exact hm.symm.trans (he.trans (he'.symm.trans hm'))
+  · intro mid h tails hh
+    have he := rootedEscapeLabelRaw_fiber_middle_mass G hloc a ha J L m γ hγ x
+      (labelFromCode G hloc γ n hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m x ⟨s,mid,h,tails⟩) w hh
+    have hm := labelFromCode_middle_mass G hloc γ n hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m x (walkKernel G hloc a) ⟨s,mid,h,tails⟩
+    exact (hm.symm.trans he).le
+
+end LiteralFiberEstimate
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section LiteralProjectionDomination
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem rooted_label_raw_fiber_mass_bound (G : BondGraph V E) (hloc : LocallyFinite G)
+    (a : ℝ) (ha : 0 < a) (hK : ∀ x y, 0 ≤ walkKernel G hloc a x y)
+    (J L m : ℕ) (γ : Fin (J+J+1) → V)
+    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
+    (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (x : V)
+    (w : RawFrom (openSimpleGraph G Set.univ) ((J+L)+J) x) :
+    (∑' ξ : {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m // ξ.first (segmentCenter J) = x},
+      if rootedEscapeLabelRaw G J L m γ hγ x ξ = w then ξ.1.mass (walkKernel G hloc a) else 0) ≤
+      ((segmentAnchors G hloc γ n hdiam (segmentCenter J) x).card : ℝ) *
+      ((segmentAnchors G hloc γ n hdiam (Fin.last (J+J)) (γ (Fin.last (J+J)))).card : ℝ) *
+        (w.2.mass (walkKernel G hloc a)/a^(J+J)) := by
+  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
+  let := finite_rootedEscapeLabels G hloc γ n hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m x
+  let R := rootedLabelEquiv G hloc γ n hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m x
+  rw [←R.symm.tsum_eq]
+  simp only [R,rootedLabelEquiv,Equiv.coe_fn_symm_mk,labelFromCode_mass]
+  unfold RootedLabelCode at *
+  rw [(summable_of_hasFiniteSupport (Set.toFinite _)).tsum_sigma]
+  calc
+    _ ≤ ∑' _s : segmentAnchors G hloc γ n hdiam (segmentCenter J) x,
+        ((segmentAnchors G hloc γ n hdiam (Fin.last (J+J)) (γ (Fin.last (J+J)))).card : ℝ) *
+          (w.2.mass (walkKernel G hloc a)/a^(J+J)) := by
+      apply (summable_of_hasFiniteSupport (Set.toFinite _)).tsum_le_tsum _
+        (summable_of_hasFiniteSupport (Set.toFinite _))
+      intro s
+      rw [(summable_of_hasFiniteSupport (Set.toFinite _)).tsum_sigma]
+      conv_lhs => arg 1; ext mid; rw [(summable_of_hasFiniteSupport (Set.toFinite _)).tsum_sigma]
+      exact rooted_code_single_anchor_fiber_bound G hloc a ha hK J L m γ hγ n hdiam x s w
+    _ = _ := by
+      simp only [tsum_fintype,Finset.sum_const,Finset.card_univ,Fintype.card_coe,nsmul_eq_mul]
+      ring
+
+end LiteralProjectionDomination
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section NormalizedProjection
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem tsum_indicator_div {α : Type*} (f : α → ℝ) (B : α → Prop) [DecidablePred B] (d : ℝ) :
+    (∑' a, if B a then f a/d else 0) = (∑' a, if B a then f a else 0)/d := by
+  rw [←tsum_div_const]
+  apply tsum_congr
+  intro a
+  split_ifs <;> simp
+
+theorem rooted_label_raw_projection_bound (G : BondGraph V E) (hloc : LocallyFinite G)
+    (a : ℝ) (ha : 0 < a) (hK : ∀ x y, 0 ≤ walkKernel G hloc a x y)
+    (J L m : ℕ) (γ : Fin (J+J+1) → V)
+    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
+    (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (x : V)
+    (hx : x ∈ vertexOrbit G (γ (segmentCenter J)))
+    (q : ℝ) (hq : 0 < q) (hqle : q ≤ segmentReturnMass G hloc a γ (Fin.last (J+J)) L (γ (Fin.last (J+J))))
+    (w : RawFrom (openSimpleGraph G Set.univ) ((J+L)+J) x) :
+    (∑' ξ : {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m // ξ.first (segmentCenter J) = x},
+      if rootedEscapeLabelRaw G J L m γ hγ x ξ = w then
+        escapeLabelProbability G hloc a γ n hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m ξ.1 else 0) ≤
+      w.2.mass (walkKernel G hloc a)/(a^(J+J)*q) := by
+  have hN (i : Fin (J+J+1)) : 0 < ((segmentAnchors G hloc γ n hdiam i (γ i)).card : ℝ) := by
+    exact_mod_cast Finset.card_pos.mpr ((segmentAnchors_nonempty_iff G hloc γ n hdiam i (γ i)).mpr
+      (by exact ⟨Equiv.refl V,Equiv.refl E,(fun e => by simp),rfl⟩))
+  have hQ := hq.trans_le hqle
+  have hA := pow_pos ha (J+J)
+  have hm := w.2.mass_nonneg (walkKernel G hloc a) hK
+  simp only [escapeLabelProbability,tsum_indicator_div]
+  have hb := rooted_label_raw_fiber_mass_bound G hloc a ha hK J L m γ hγ n hdiam x w
+  rw [segmentAnchors_card_on_orbit G hloc γ n hdiam (segmentCenter J) x hx] at hb
+  apply (div_le_div_of_nonneg_right hb (mul_nonneg (mul_nonneg (hN (segmentCenter J)).le (hN (Fin.last (J+J))).le) hQ.le)).trans
+  calc
+    _ = w.2.mass (walkKernel G hloc a)/
+        (a^(J+J)*segmentReturnMass G hloc a γ (Fin.last (J+J)) L (γ (Fin.last (J+J)))) := by
+      field_simp [(hN (segmentCenter J)).ne', (hN (Fin.last (J+J))).ne']
+    _ ≤ _ := div_le_div_of_nonneg_left hm (mul_pos hA hq) (mul_le_mul_of_nonneg_left hqle hA.le)
+
+end NormalizedProjection
+end Percolation
+
+ 
+namespace Percolation
+universe u
+variable {V : Type u}
+section SuffixCollisionOperator
+
+theorem operator_pow_norm_le (P : Hilbert V →L[ℂ] Hilbert V) (k : ℕ) : ‖P^k‖ ≤ ‖P‖^k := by
+  cases k with
+  | zero => simpa only [pow_zero,ContinuousLinearMap.one_def] using
+      (ContinuousLinearMap.norm_id_le (𝕜 := ℂ) (E := Hilbert V))
+  | succ k => exact norm_pow_le' P (Nat.succ_pos k)
+
+ 
+
+theorem positive_rows_collision_hasSum (A B : Hilbert V →L[ℂ] Hilbert V)
+    (hA : PositiveRealMatrix A) (x y : V) :
+    HasSum (fun z => (matrixCoeff A x z).re*(matrixCoeff B y z).re)
+      (matrixCoeff (A*star B) x y).re := by
+  simpa only [matrixCoeff_adjoint,Complex.star_def,Complex.conj_re] using
+    hA.mul_hasSum (star B) x y
+
+theorem positive_rows_collision_bound (A B : Hilbert V →L[ℂ] Hilbert V)
+    (hA : PositiveRealMatrix A) (x y : V) :
+    (∑' z, (matrixCoeff A x z).re*(matrixCoeff B y z).re) ≤ ‖A‖*‖B‖ := by
+  rw [(positive_rows_collision_hasSum A B hA x y).tsum_eq]
+  apply (Complex.re_le_norm _).trans ((matrixCoeff_norm_le (A*star B) x y).trans _)
+  have hs : ‖star B‖ = ‖B‖ := ContinuousLinearMap.adjoint.norm_map B
+  simpa only [hs] using norm_mul_le A (star B)
+
+theorem walk_suffix_collision_operator_bound (P H : Hilbert V →L[ℂ] Hilbert V)
+    (hP : PositiveRealMatrix P) (C : ℝ) (hH : ‖H‖ ≤ C) (l L : ℕ) (x y : V) :
+    (∑' z, (matrixCoeff (P^l) x z).re*(matrixCoeff (P^L*H) y z).re) ≤
+      C*‖P‖^(l+L) := by
+  calc
+    _ ≤ ‖P^l‖*‖P^L*H‖ := positive_rows_collision_bound _ _ (hP.pow l) x y
+    _ ≤ ‖P‖^l*(‖P‖^L*C) := mul_le_mul (operator_pow_norm_le P l)
+      ((norm_mul_le _ _).trans (mul_le_mul (operator_pow_norm_le P L) hH (norm_nonneg H)
+        (pow_nonneg (norm_nonneg P) L))) (norm_nonneg _) (pow_nonneg (norm_nonneg P) l)
+    _ = _ := by rw [pow_add]; ring
+
+end SuffixCollisionOperator
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section ActualSuffixCollision
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {I : Type*} [Fintype I]
+
+ 
+
+theorem raw_anchored_position_expectation (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (ep j : I)
+    (P H : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (hH : ∀ x y, matrixCoeff H x y = (anchoredPositionKernel G hloc γ n hdiam ep j x y : ℂ))
+    (L : ℕ) (x z : V) :
+    (∑' mid : RawFrom (openSimpleGraph G Set.univ) L x,
+      mid.2.mass (walkKernel G hloc a)*anchoredPositionKernel G hloc γ n hdiam ep j mid.1 z) =
+      (matrixCoeff (P^L*H) x z).re := by
+  change RawFrom.integral (walkKernel G hloc a) L x
+    (fun y => anchoredPositionKernel G hloc γ n hdiam ep j y z) = _
+  rw [raw_integral_operator G hloc a P hpos hP]
+  simpa only [hH,Complex.ofReal_re] using ((hpos.pow L).mul_hasSum H x z).tsum_eq
+
+ 
+
+theorem raw_tail_anchored_collision_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (ep j : I)
+    (P H : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (hH : ∀ x y, matrixCoeff H x y = (anchoredPositionKernel G hloc γ n hdiam ep j x y : ℂ))
+    (C : ℝ) (hHC : ‖H‖ ≤ C) (l m L : ℕ) (hlm : l ≤ m) (x y : V) :
+    (∑' tail : RawFrom (openSimpleGraph G Set.univ) m x,
+      tail.2.mass (walkKernel G hloc a) *
+        ∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
+          mid.2.mass (walkKernel G hloc a)*
+            anchoredPositionKernel G hloc γ n hdiam ep j mid.1 (tail.2.position l)) ≤
+      C*‖P‖^(l+L) := by
+  simp only [raw_anchored_position_expectation G hloc a γ n hdiam ep j P H hpos hP hH]
+  rw [raw_position_integral G hloc a l m hlm x (fun z => (matrixCoeff (P^L*H) y z).re),
+    raw_integral_operator G hloc a P hpos hP]
+  exact walk_suffix_collision_operator_bound P H hpos C hHC l L x y
+
+end ActualSuffixCollision
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section UniformSuffixCollision
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {I ι : Type*} [Fintype I] [Fintype ι]
+
+ 
+
+theorem raw_tail_anchored_collision_uniform (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G)
+    (htr : RootTransport root weight Γ) (hw : ∀ r, 0 ≤ weight r) (hw1 : ∑ r, weight r = 1)
+    (hcov : ∀ x, ∃ r, ∃ g ∈ Γ, g (root r) = x)
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ r, 1 ≤ c*weight r)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (ep j : I)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (l m L : ℕ) (hlm : l ≤ m) (x y : V) :
+    (∑' tail : RawFrom (openSimpleGraph G Set.univ) m x,
+      tail.2.mass (walkKernel G hloc a) *
+        ∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
+          mid.2.mass (walkKernel G hloc a)*
+            anchoredPositionKernel G hloc γ n hdiam ep j mid.1 (tail.2.position l)) ≤
+      Real.sqrt c*‖P‖^(l+L) := by
+  obtain ⟨H,hH,hHn⟩ := exists_anchoredPositionOperator_uniform G hloc root weight Γ hΓ htr hw hw1 hcov
+    c hc hmin γ n hdiam ep j
+  exact raw_tail_anchored_collision_bound G hloc a γ n hdiam ep j P H hpos hP hH
+    (Real.sqrt c) hHn l m L hlm x y
+
+end UniformSuffixCollision
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section GlobalAnchorDenominator
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {I : Type*} [Fintype I]
+
+ 
+
+theorem anchoredPositionKernel_global_denominator (G : BondGraph V E) (hloc : LocallyFinite G)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (ep j : I) (x z : V) :
+    (∑ h : segmentAnchors G hloc γ n hdiam ep x,
+      if h.val j = z then ((segmentAnchors G hloc γ n hdiam ep (γ ep)).card : ℝ)⁻¹ else 0) =
+      anchoredPositionKernel G hloc γ n hdiam ep j x z := by
+  rw [Finset.sum_coe_sort _ (fun h : I → V => if h j = z then
+    ((segmentAnchors G hloc γ n hdiam ep (γ ep)).card : ℝ)⁻¹ else 0)]
+  by_cases hx : x ∈ vertexOrbit G (γ ep)
+  · unfold anchoredPositionKernel
+    rw [segmentAnchors_card_on_orbit G hloc γ n hdiam ep x hx]
+    apply Finset.sum_congr rfl
+    intro h _
+    simp only [eq_comm]
+  · have hz := ((segmentAnchors_nonempty_iff G hloc γ n hdiam ep x).not.mpr hx)
+    have he := Finset.not_nonempty_iff_eq_empty.mp hz
+    simp only [anchoredPositionKernel,he,Finset.sum_empty]
+
+theorem finite_geometric_sum_bound (ρ : ℝ) (hρ : 0 ≤ ρ) (hρ1 : ρ < 1) (M : ℕ) :
+    (∑ i : Fin M, ρ^i.val) ≤ 1/(1-ρ) := by
+  simpa only [Nat.zero_le,ite_true,pow_zero] using finite_geometric_cutoff_bound ρ hρ hρ1 0 M
+
+end GlobalAnchorDenominator
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section LiteralSuffixExperiment
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {I : Type*} [Fintype I]
+
+ 
+
+theorem raw_tail_anchor_collision_identity (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (ep j : I)
+    (l m L : ℕ) (x y : V) :
+    (∑' tail : RawFrom (openSimpleGraph G Set.univ) m x,
+      ∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
+        ∑' h : segmentAnchors G hloc γ n hdiam ep mid.1,
+          if tail.2.position l = h.val j then
+            tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a)/
+              (segmentAnchors G hloc γ n hdiam ep (γ ep)).card else 0) =
+    (∑' tail : RawFrom (openSimpleGraph G Set.univ) m x,
+      tail.2.mass (walkKernel G hloc a) *
+        ∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
+          mid.2.mass (walkKernel G hloc a)*
+            anchoredPositionKernel G hloc γ n hdiam ep j mid.1 (tail.2.position l)) := by
+  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
+  let (k : ℕ) (z : V) : Fintype (RawFrom (openSimpleGraph G Set.univ) k z) := Fintype.ofFinite _
+  simp only [tsum_fintype]
+  apply Finset.sum_congr rfl
+  intro tail _
+  rw [Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro mid _
+  rw [←anchoredPositionKernel_global_denominator G hloc γ n hdiam ep j mid.1 (tail.2.position l)]
+  simp only [Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro h _
+  by_cases hh : tail.2.position l = h.val j
+  · simp only [ite_eq_left hh,ite_eq_left hh.symm,div_eq_mul_inv,mul_assoc]
+  · simp only [ite_eq_right hh,ite_eq_right (Ne.symm hh),mul_zero]
+
+end LiteralSuffixExperiment
+end Percolation
+
+ 
+namespace Percolation
+section NestedFiniteUnion
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem finite_threefold_event_union {α β δ : Type*} [Fintype α] [Fintype β] [Fintype δ]
+    {γ : β → Type*} [∀ b, Fintype (γ b)]
+    (μ : (a : α) → (b : β) → γ b → ℝ) (hμ : ∀ a b c, 0 ≤ μ a b c)
+    (B : (a : α) → (b : β) → γ b → δ → Prop) :
+    (∑ a, ∑ b, ∑ c, if ∃ d, B a b c d then μ a b c else 0) ≤
+      ∑ d, ∑ a, ∑ b, ∑ c, if B a b c d then μ a b c else 0 := by
+  have hh := finite_nonnegative_event_union
+    (fun ξ : α × Sigma γ => μ ξ.1 ξ.2.1 ξ.2.2)
+    (fun ξ => hμ ξ.1 ξ.2.1 ξ.2.2)
+    (fun ξ : α × Sigma γ => B ξ.1 ξ.2.1 ξ.2.2)
+  simpa only [Fintype.sum_prod_type,Fintype.sum_sigma] using hh
+
+end NestedFiniteUnion
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section WholeSuffixCollision
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {I ι S : Type*} [Fintype I] [Fintype ι] [Fintype S]
+
+ 
+
+theorem raw_tail_whole_suffix_collision_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G)
+    (htr : RootTransport root weight Γ) (hw : ∀ r, 0 ≤ weight r) (hw1 : ∑ r, weight r = 1)
+    (hcov : ∀ x, ∃ r, ∃ g ∈ Γ, g (root r) = x)
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ r, 1 ≤ c*weight r)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (ep : I) (j : S → I)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (hρ : ‖P‖ < 1) (m L : ℕ) (x y : V) :
+    (∑' tail : RawFrom (openSimpleGraph G Set.univ) m x,
+      ∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
+        ∑' h : segmentAnchors G hloc γ n hdiam ep mid.1,
+          if ∃ ij : Fin (m+1) × S, tail.2.position ij.1.val = h.val (j ij.2) then
+            tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a)/
+              (segmentAnchors G hloc γ n hdiam ep (γ ep)).card else 0) ≤
+      (Fintype.card S : ℝ)*Real.sqrt c*‖P‖^L/(1-‖P‖) := by
+  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
+  let (k : ℕ) (z : V) : Fintype (RawFrom (openSimpleGraph G Set.univ) k z) := Fintype.ofFinite _
+  have hK (x y : V) : 0 ≤ walkKernel G hloc a x y := by
+    simpa only [hP,Complex.ofReal_re] using (hpos x y).2
+  simp only [tsum_fintype]
+  have hu := finite_threefold_event_union
+    (α := RawFrom (openSimpleGraph G Set.univ) m x)
+    (β := RawFrom (openSimpleGraph G Set.univ) L y)
+    (γ := fun mid => segmentAnchors G hloc γ n hdiam ep mid.1)
+    (δ := Fin (m+1) × S)
+    (fun tail mid _ => tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a)/
+      (segmentAnchors G hloc γ n hdiam ep (γ ep)).card)
+    (fun tail mid _ => div_nonneg (mul_nonneg (tail.2.mass_nonneg _ hK) (mid.2.mass_nonneg _ hK)) (Nat.cast_nonneg _))
+    (fun tail _ h ij => tail.2.position ij.1.val = h.val (j ij.2))
+  apply hu.trans
+  rw [Fintype.sum_prod_type]
+  calc
+    _ ≤ ∑ i : Fin (m+1), ∑ _s : S, Real.sqrt c*‖P‖^(i.val+L) := by
+      apply Finset.sum_le_sum
+      intro i _
+      apply Finset.sum_le_sum
+      intro s _
+      have hh := raw_tail_anchored_collision_uniform G hloc a root weight Γ hΓ htr hw hw1 hcov c hc hmin
+        γ n hdiam ep (j s) P hpos hP i.val m L (by omega) x y
+      rw [←raw_tail_anchor_collision_identity G hloc a γ n hdiam ep (j s) i.val m L x y] at hh
+      simpa only [tsum_fintype] using hh
+    _ = ((Fintype.card S : ℝ)*Real.sqrt c*‖P‖^L)*(∑ i : Fin (m+1), ‖P‖^i.val) := by
+      simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,pow_add,←Finset.mul_sum,←Finset.sum_mul]
+      ring
+    _ ≤ ((Fintype.card S : ℝ)*Real.sqrt c*‖P‖^L)*(1/(1-‖P‖)) :=
+      mul_le_mul_of_nonneg_left (finite_geometric_sum_bound ‖P‖ (norm_nonneg P) hρ (m+1)) (by positivity)
+    _ = _ := by ring
+
+end WholeSuffixCollision
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section ActualAnchorPointTest
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {I : Type*} [Fintype I]
+
+theorem raw_anchor_position_identity (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (ep j : I)
+    (L : ℕ) (y z : V) :
+    (∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
+      ∑' h : segmentAnchors G hloc γ n hdiam ep mid.1,
+        if h.val j = z then mid.2.mass (walkKernel G hloc a)/
+          (segmentAnchors G hloc γ n hdiam ep (γ ep)).card else 0) =
+    (∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
+      mid.2.mass (walkKernel G hloc a)*anchoredPositionKernel G hloc γ n hdiam ep j mid.1 z) := by
+  apply tsum_congr
+  intro mid
+  rw [←anchoredPositionKernel_global_denominator G hloc γ n hdiam ep j mid.1 z,tsum_fintype,Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro h _
+  split_ifs <;> simp only [div_eq_mul_inv,mul_zero]
+
+theorem raw_anchor_point_collision_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (ep j : I)
+    (P H : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (hH : ∀ x y, matrixCoeff H x y = (anchoredPositionKernel G hloc γ n hdiam ep j x y : ℂ))
+    (C : ℝ) (hC : ‖H‖ ≤ C) (L : ℕ) (y z : V) :
+    (∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
+      ∑' h : segmentAnchors G hloc γ n hdiam ep mid.1,
+        if h.val j = z then mid.2.mass (walkKernel G hloc a)/
+          (segmentAnchors G hloc γ n hdiam ep (γ ep)).card else 0) ≤ C*‖P‖^L := by
+  rw [raw_anchor_position_identity G hloc a γ n hdiam ep j L y z,
+    raw_anchored_position_expectation G hloc a γ n hdiam ep j P H hpos hP hH L y z]
+  apply (Complex.re_le_norm _).trans ((matrixCoeff_norm_le (P^L*H) y z).trans _)
+  calc
+    _ ≤ ‖P^L‖*‖H‖ := norm_mul_le _ _
+    _ ≤ ‖P‖^L*C := mul_le_mul (operator_pow_norm_le P L) hC (norm_nonneg _) (pow_nonneg (norm_nonneg P) L)
+    _ = _ := mul_comm _ _
+
+end ActualAnchorPointTest
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Escape/Labels.lean b/lean/OAI/Probability/UnimodularPercolation/Escape/Labels.lean
index 754c35764ab8bf74561289de683454e85796bb11..d5eb73e2ab00b8ba3b92be5634c7f924e6ca1d58 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Escape/Labels.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Escape/Labels.lean
@@ -1,1216 +1,1216 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Deformation.AveragedDecrease
-
-namespace OAI
-
-/-! Labels for bond percolation. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v w
-variable {V : Type u} {E : Type v} {I : Type w}
-section LabeledEscapeExperiment
-
- 
-
-structure EscapeLabel (G : BondGraph V E) (γ : I → V) (endpoint negative : I) (L m : ℕ) where
-  first : I → V
-  first_mem : first ∈ segmentOrbit G γ
-  last : I → V
-  last_mem : last ∈ segmentOrbit G γ
-  middle : HoldingWalk (openSimpleGraph G Set.univ) L (first endpoint) (last endpoint)
-  leftTail : RawFrom (openSimpleGraph G Set.univ) m (first negative)
-  rightTail : RawFrom (openSimpleGraph G Set.univ) m (last negative)
-
-namespace EscapeLabel
-variable {G : BondGraph V E} {γ : I → V} {ep neg : I} {L m : ℕ}
-
-def reverse (ξ : EscapeLabel G γ ep neg L m) : EscapeLabel G γ ep neg L m where
-  first := ξ.last
-  first_mem := ξ.last_mem
-  last := ξ.first
-  last_mem := ξ.first_mem
-  middle := ξ.middle.reverse
-  leftTail := ξ.rightTail
-  rightTail := ξ.leftTail
-
-theorem reverse_reverse (ξ : EscapeLabel G γ ep neg L m) : ξ.reverse.reverse = ξ := by
-  cases ξ
-  simp only [reverse,HoldingWalk.reverse_reverse]
-
-def reverseEquiv : EscapeLabel G γ ep neg L m ≃ EscapeLabel G γ ep neg L m where
-  toFun := reverse
-  invFun := reverse
-  left_inv := reverse_reverse
-  right_inv := reverse_reverse
-
-noncomputable def mass (P : V → V → ℝ) (ξ : EscapeLabel G γ ep neg L m) : ℝ :=
-  ξ.middle.mass P * ξ.leftTail.2.mass P * ξ.rightTail.2.mass P
-
-theorem mass_nonneg (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y)
-    (ξ : EscapeLabel G γ ep neg L m) : 0 ≤ ξ.mass P :=
-  mul_nonneg (mul_nonneg (ξ.middle.mass_nonneg P hP) (ξ.leftTail.2.mass_nonneg P hP))
-    (ξ.rightTail.2.mass_nonneg P hP)
-
-theorem mass_reverse (P : V → V → ℝ) (hP : ∀ x y, P x y = P y x)
-    (ξ : EscapeLabel G γ ep neg L m) : ξ.reverse.mass P = ξ.mass P := by
-  simp only [mass,reverse,HoldingWalk.mass_reverse P hP]
-  ring
-
- 
-
-def reverseRootedEquiv (center : I) (x y : V) :
-    {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x ∧ ξ.last center = y} ≃
-    {ξ : EscapeLabel G γ ep neg L m // ξ.first center = y ∧ ξ.last center = x} where
-  toFun ξ := ⟨ξ.1.reverse,ξ.2.2,ξ.2.1⟩
-  invFun ξ := ⟨ξ.1.reverse,ξ.2.2,ξ.2.1⟩
-  left_inv ξ := Subtype.ext (reverse_reverse ξ.1)
-  right_inv ξ := Subtype.ext (reverse_reverse ξ.1)
-
-end EscapeLabel
-end LabeledEscapeExperiment
-end Percolation
-
- 
-namespace Percolation
-universe u v w
-variable {V : Type u} {E : Type v} {I : Type w}
-section FiniteRootedLabels
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Fintype I]
-
-noncomputable def RootedLabelCode (G : BondGraph V E) (hloc : LocallyFinite G)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
-    (center ep neg : I) (L m : ℕ) (x : V) : Type (max u w) :=
-  Σ s : (segmentAnchors G hloc γ n hdiam center x),
-    Σ mid : RawFrom (openSimpleGraph G Set.univ) L (s.val ep),
-      Σ h : (segmentAnchors G hloc γ n hdiam ep mid.1),
-        RawFrom (openSimpleGraph G Set.univ) m (s.val neg) ×
-          RawFrom (openSimpleGraph G Set.univ) m (h.val neg)
-
-instance finite_rootedLabelCode (G : BondGraph V E) (hloc : LocallyFinite G)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
-    (center ep neg : I) (L m : ℕ) (x : V) :
-    Finite (RootedLabelCode G hloc γ n hdiam center ep neg L m x) := by
-  let (z : V) := finite_openSimpleGraph_neighbors G hloc z
-  unfold RootedLabelCode
-  infer_instance
-
-noncomputable def labelFromCode (G : BondGraph V E) (hloc : LocallyFinite G)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
-    (center ep neg : I) (L m : ℕ) (x : V)
-    (c : RootedLabelCode G hloc γ n hdiam center ep neg L m x) :
-    {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x} := by
-  rcases c with ⟨⟨s,hs⟩,⟨z,mid⟩,⟨h,hh⟩,α,β⟩
-  have hsm := ((mem_segmentAnchors G hloc γ n hdiam center x s).mp hs).1
-  have hsx := ((mem_segmentAnchors G hloc γ n hdiam center x s).mp hs).2
-  have hhm := ((mem_segmentAnchors G hloc γ n hdiam ep z h).mp hh).1
-  have hhz := ((mem_segmentAnchors G hloc γ n hdiam ep z h).mp hh).2
-  subst z
-  exact ⟨⟨s,hsm,h,hhm,mid,α,β⟩,hsx⟩
-
-noncomputable def labelToCode (G : BondGraph V E) (hloc : LocallyFinite G)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
-    (center ep neg : I) (L m : ℕ) (x : V)
-    (ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x}) :
-    RootedLabelCode G hloc γ n hdiam center ep neg L m x :=
-  ⟨⟨ξ.1.first,(mem_segmentAnchors G hloc γ n hdiam center x ξ.1.first).mpr ⟨ξ.1.first_mem,ξ.2⟩⟩,
-    ⟨ξ.1.last ep,ξ.1.middle⟩,
-    ⟨ξ.1.last,(mem_segmentAnchors G hloc γ n hdiam ep (ξ.1.last ep) ξ.1.last).mpr ⟨ξ.1.last_mem,rfl⟩⟩,
-    ξ.1.leftTail,ξ.1.rightTail⟩
-
-noncomputable def rootedLabelEquiv (G : BondGraph V E) (hloc : LocallyFinite G)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
-    (center ep neg : I) (L m : ℕ) (x : V) :
-    {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x} ≃
-      RootedLabelCode G hloc γ n hdiam center ep neg L m x where
-  toFun := labelToCode G hloc γ n hdiam center ep neg L m x
-  invFun := labelFromCode G hloc γ n hdiam center ep neg L m x
-  left_inv ξ := by
-    cases ξ with
-    | mk ξ hξ => cases ξ; rfl
-  right_inv c := by
-    rcases c with ⟨⟨s,hs⟩,⟨z,mid⟩,⟨h,hh⟩,α,β⟩
-    have hh' := (mem_segmentAnchors G hloc γ n hdiam ep z h).mp hh
-    obtain ⟨hhm,hhz⟩ := hh'
-    subst z
-    rfl
-
-theorem finite_rootedEscapeLabels (G : BondGraph V E) (hloc : LocallyFinite G)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
-    (center ep neg : I) (L m : ℕ) (x : V) :
-    Finite {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x} :=
-  Finite.of_equiv _ (rootedLabelEquiv G hloc γ n hdiam center ep neg L m x).symm
-
-end FiniteRootedLabels
-end Percolation
-
- 
-namespace Percolation
-universe u v w
-variable {V : Type u} {E : Type v} {I : Type w}
-section LabeledMass
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Fintype I]
-
-theorem segmentAnchors_card_on_orbit (G : BondGraph V E) (hloc : LocallyFinite G)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i : I) (x : V)
-    (hx : x ∈ vertexOrbit G (γ i)) :
-    (segmentAnchors G hloc γ n hdiam i x).card = (segmentAnchors G hloc γ n hdiam i (γ i)).card := by
-  obtain ⟨a,b,h,rfl⟩ := hx
-  exact segmentAnchors_card_aut G hloc γ n hdiam i (γ i) h
-
-theorem segmentAnchors_card_eq_indicator (G : BondGraph V E) (hloc : LocallyFinite G)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i : I) (x : V) :
-    ((segmentAnchors G hloc γ n hdiam i x).card : ℝ) =
-      if x ∈ vertexOrbit G (γ i) then (segmentAnchors G hloc γ n hdiam i (γ i)).card else 0 := by
-  split_ifs with hx
-  · exact congrArg Nat.cast (segmentAnchors_card_on_orbit G hloc γ n hdiam i x hx)
-  · have hh := (segmentAnchors_nonempty_iff G hloc γ n hdiam i x).not.mpr hx
-    rw [Finset.not_nonempty_iff_eq_empty.mp hh]
-    rfl
-
-noncomputable def segmentReturnMass (G : BondGraph V E) (hloc : LocallyFinite G)
-    (a : ℝ) (γ : I → V) (ep : I) (L : ℕ) (x : V) : ℝ :=
-  ∑' mid : RawFrom (openSimpleGraph G Set.univ) L x,
-    if mid.1 ∈ vertexOrbit G (γ ep) then mid.2.mass (walkKernel G hloc a) else 0
-
-theorem labelFromCode_mass (G : BondGraph V E) (hloc : LocallyFinite G)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
-    (center ep neg : I) (L m : ℕ) (x : V) (P : V → V → ℝ)
-    (c : RootedLabelCode G hloc γ n hdiam center ep neg L m x) :
-    (labelFromCode G hloc γ n hdiam center ep neg L m x c).1.mass P =
-      c.2.1.2.mass P * c.2.2.2.1.2.mass P * c.2.2.2.2.2.mass P := by
-  rcases c with ⟨⟨s,hs⟩,⟨z,mid⟩,⟨h,hh⟩,α,β⟩
-  obtain ⟨hhm,hhz⟩ := (mem_segmentAnchors G hloc γ n hdiam ep z h).mp hh
-  subst z
-  rfl
-
- 
-
-theorem rooted_label_mass_sum (G : BondGraph V E) (hloc : LocallyFinite G)
-    (a : ℝ) (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
-    (center ep neg : I) (L m : ℕ) (x : V) :
-    (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
-      ξ.1.mass (walkKernel G hloc a)) =
-    ∑ s ∈ segmentAnchors G hloc γ n hdiam center x,
-      ((segmentAnchors G hloc γ n hdiam ep (γ ep)).card : ℝ) *
-        segmentReturnMass G hloc a γ ep L (s ep) := by
-  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
-  let := finite_rootedEscapeLabels G hloc γ n hdiam center ep neg L m x
-  let R := rootedLabelEquiv G hloc γ n hdiam center ep neg L m x
-  rw [←R.symm.tsum_eq]
-  simp only [R,rootedLabelEquiv,Equiv.coe_fn_symm_mk,labelFromCode_mass]
-  unfold RootedLabelCode at *
-  rw [(summable_of_hasFiniteSupport (Set.toFinite _)).tsum_sigma]
-  change (∑' s : (segmentAnchors G hloc γ n hdiam center x), _) = _
-  rw [tsum_fintype]
-  conv_rhs => rw [←Finset.sum_coe_sort]
-  apply Finset.sum_congr rfl
-  intro s _
-  rw [(summable_of_hasFiniteSupport (Set.toFinite _)).tsum_sigma]
-  have he (mid : RawFrom (openSimpleGraph G Set.univ) L (s.val ep)) :
-      (∑' h : (segmentAnchors G hloc γ n hdiam ep mid.1),
-        ∑' tails : RawFrom (openSimpleGraph G Set.univ) m (s.val neg) ×
-            RawFrom (openSimpleGraph G Set.univ) m (h.val neg),
-          mid.2.mass (walkKernel G hloc a) * tails.1.2.mass (walkKernel G hloc a) *
-            tails.2.2.mass (walkKernel G hloc a)) =
-      ((segmentAnchors G hloc γ n hdiam ep (γ ep)).card : ℝ) *
-        (if mid.1 ∈ vertexOrbit G (γ ep) then mid.2.mass (walkKernel G hloc a) else 0) := by
-    have ht (h : (segmentAnchors G hloc γ n hdiam ep mid.1)) :
-        (∑' tails : RawFrom (openSimpleGraph G Set.univ) m (s.val neg) ×
-            RawFrom (openSimpleGraph G Set.univ) m (h.val neg),
-          mid.2.mass (walkKernel G hloc a) * tails.1.2.mass (walkKernel G hloc a) *
-            tails.2.2.mass (walkKernel G hloc a)) = mid.2.mass (walkKernel G hloc a) := by
-      rw [(summable_of_hasFiniteSupport (Set.toFinite _)).tsum_prod]
-      simp only [tsum_mul_left,rawFrom_probability_sum G hloc a,mul_one]
-    simp only [ht,tsum_fintype,Finset.sum_const,Finset.card_univ,Fintype.card_coe,nsmul_eq_mul]
-    rw [segmentAnchors_card_eq_indicator G hloc γ n hdiam ep mid.1]
-    split_ifs <;> ring
-  calc
-    _ = ∑' mid : RawFrom (openSimpleGraph G Set.univ) L (s.val ep),
-        ((segmentAnchors G hloc γ n hdiam ep (γ ep)).card : ℝ) *
-          (if mid.1 ∈ vertexOrbit G (γ ep) then mid.2.mass (walkKernel G hloc a) else 0) := by
-      apply tsum_congr
-      intro mid
-      rw [(summable_of_hasFiniteSupport (Set.toFinite _)).tsum_sigma]
-      exact he mid
-    _ = _ := tsum_mul_left
-
-end LabeledMass
-end Percolation
-
- 
-namespace Percolation
-universe u v w
-variable {V : Type u} {E : Type v} {I : Type w}
-section LabeledNormalization
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Fintype I]
-
-omit [Fintype I] in
-theorem segmentReturnMass_operator (G : BondGraph V E) (hloc : LocallyFinite G)
-    (a : ℝ) (γ : I → V) (ep : I) (L : ℕ) (x : V)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ)) :
-    segmentReturnMass G hloc a γ ep L x =
-      ∑' y, if y ∈ vertexOrbit G (γ ep) then (matrixCoeff (P^L) x y).re else 0 := by
-  have hh := raw_integral_operator G hloc a P hpos hP L x
-    (fun y => if y ∈ vertexOrbit G (γ ep) then 1 else 0)
-  simpa only [RawFrom.integral,mul_ite,mul_one,mul_zero,segmentReturnMass] using hh
-
-omit [Fintype I] in
-theorem segmentReturnMass_aut (G : BondGraph V E) (hloc : LocallyFinite G)
-    (a : ℝ) (γ : I → V) (ep : I) (L : ℕ) (x : V)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    {g : V ≃ V} {b : E ≃ E} (hg : IsAut G g b) :
-    segmentReturnMass G hloc a γ ep L (g x) = segmentReturnMass G hloc a γ ep L x := by
-  rw [segmentReturnMass_operator G hloc a γ ep L (g x) P hpos hP,
-    segmentReturnMass_operator G hloc a γ ep L x P hpos hP,←g.tsum_eq]
-  apply tsum_congr
-  intro y
-  rw [mem_vertexOrbit_aut_iff G (γ ep) hg]
-  have hh := walkPower_invariant G hloc a P hP L g ⟨b,hg⟩ x y
-  rw [hh]
-
-omit [Fintype I] in
-theorem segmentReturnMass_on_orbit (G : BondGraph V E) (hloc : LocallyFinite G)
-    (a : ℝ) (γ : I → V) (ep : I) (L : ℕ) (x : V)
-    (hx : x ∈ vertexOrbit G (γ ep))
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ)) :
-    segmentReturnMass G hloc a γ ep L x = segmentReturnMass G hloc a γ ep L (γ ep) := by
-  obtain ⟨g,b,hg,rfl⟩ := hx
-  exact segmentReturnMass_aut G hloc a γ ep L (γ ep) P hpos hP hg
-
- 
-
-noncomputable def escapeLabelProbability (G : BondGraph V E) (hloc : LocallyFinite G)
-    (a : ℝ) (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
-    (center ep neg : I) (L m : ℕ) (ξ : EscapeLabel G γ ep neg L m) : ℝ :=
-  ξ.mass (walkKernel G hloc a) /
-    (((segmentAnchors G hloc γ n hdiam center (γ center)).card : ℝ) *
-      ((segmentAnchors G hloc γ n hdiam ep (γ ep)).card : ℝ) *
-      segmentReturnMass G hloc a γ ep L (γ ep))
-
-theorem rooted_label_probability_sum (G : BondGraph V E) (hloc : LocallyFinite G)
-    (a : ℝ) (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
-    (center ep neg : I) (L m : ℕ) (x : V) (hx : x ∈ vertexOrbit G (γ center))
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (hq : segmentReturnMass G hloc a γ ep L (γ ep) ≠ 0) :
-    (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
-      escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1) = 1 := by
-  have hN (i : I) : ((segmentAnchors G hloc γ n hdiam i (γ i)).card : ℝ) ≠ 0 := by
-    apply ne_of_gt
-    exact_mod_cast Finset.card_pos.mpr ((segmentAnchors_nonempty_iff G hloc γ n hdiam i (γ i)).mpr
-      (by exact ⟨Equiv.refl V,Equiv.refl E,(fun e => by simp),rfl⟩))
-  simp only [escapeLabelProbability,tsum_div_const]
-  rw [rooted_label_mass_sum G hloc a γ n hdiam center ep neg L m x]
-  have he (s : I → V) (hs : s ∈ segmentAnchors G hloc γ n hdiam center x) :
-      segmentReturnMass G hloc a γ ep L (s ep) = segmentReturnMass G hloc a γ ep L (γ ep) := by
-    obtain ⟨⟨g,b,hg,rfl⟩,_⟩ := (mem_segmentAnchors G hloc γ n hdiam center x s).mp hs
-    exact segmentReturnMass_aut G hloc a γ ep L (γ ep) P hpos hP hg
-  rw [Finset.sum_congr rfl (fun s hs => congrArg (fun q => _ * q) (he s hs))]
-  simp only [Finset.sum_const,nsmul_eq_mul]
-  rw [segmentAnchors_card_on_orbit G hloc γ n hdiam center x hx,←mul_assoc]
-  exact div_self (mul_ne_zero (mul_ne_zero (hN center) (hN ep)) hq)
-
-theorem escapeLabelProbability_reverse (G : BondGraph V E) (hloc : LocallyFinite G)
-    (a : ℝ) (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
-    (center ep neg : I) (L m : ℕ) (ξ : EscapeLabel G γ ep neg L m) :
-    escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.reverse =
-      escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ := by
-  simp only [escapeLabelProbability,EscapeLabel.mass_reverse _ (walkKernel_symmetric G hloc a)]
-
-end LabeledNormalization
-end Percolation
-
- 
-namespace Percolation
-universe u v w
-variable {V : Type u} {E : Type v} {I : Type w}
-theorem isAut_refl (G : BondGraph V E) : IsAut G (Equiv.refl V) (Equiv.refl E) := by
-  intro e
-  simp
-
-namespace EscapeLabel
-variable {G : BondGraph V E} {γ : I → V} {ep neg : I} {L m : ℕ}
-
-def autMap {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b)
-    (ξ : EscapeLabel G γ ep neg L m) : EscapeLabel G γ ep neg L m where
-  first := a ∘ ξ.first
-  first_mem := (segmentOrbit_aut_iff G γ ξ.first hab).mpr ξ.first_mem
-  last := a ∘ ξ.last
-  last_mem := (segmentOrbit_aut_iff G γ ξ.last hab).mpr ξ.last_mem
-  middle := ξ.middle.map (autSimpleGraph G hab).toHom
-  leftTail := ⟨a ξ.leftTail.1,ξ.leftTail.2.map (autSimpleGraph G hab).toHom⟩
-  rightTail := ⟨a ξ.rightTail.1,ξ.rightTail.2.map (autSimpleGraph G hab).toHom⟩
-
-theorem autMap_trans {a c : V ≃ V} {b d : E ≃ E} (hab : IsAut G a b) (hcd : IsAut G c d)
-    (ξ : EscapeLabel G γ ep neg L m) :
-    autMap hcd (autMap hab ξ) = autMap (isAut_trans G hab hcd) ξ := by
-  cases ξ
-  dsimp only [autMap,autSimpleGraph]
-  simp only [HoldingWalk.map_map]
-  rfl
-
-theorem autMap_refl (ξ : EscapeLabel G γ ep neg L m) :
-    autMap (isAut_refl G) ξ = ξ := by
-  cases ξ with
-  | mk f hf l hl mid α β =>
-    change EscapeLabel.mk f hf l hl (mid.map SimpleGraph.Hom.id)
-      ⟨α.1,α.2.map SimpleGraph.Hom.id⟩ ⟨β.1,β.2.map SimpleGraph.Hom.id⟩ = _
-    simp only [HoldingWalk.map_id]
-
-theorem autMap_symm {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b)
-    (ξ : EscapeLabel G γ ep neg L m) :
-    autMap (isAut_symm G hab) (autMap hab ξ) = ξ := by
-  rw [autMap_trans]
-  have ha : a.trans a.symm = Equiv.refl V := by ext x; exact a.symm_apply_apply x
-  have hb : b.trans b.symm = Equiv.refl E := by ext x; exact b.symm_apply_apply x
-  generalize hp : isAut_trans G hab (isAut_symm G hab) = h
-  clear hp
-  revert h
-  rw [ha,hb]
-  intro h
-  exact autMap_refl ξ
-
-def autEquiv {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) :
-    EscapeLabel G γ ep neg L m ≃ EscapeLabel G γ ep neg L m where
-  toFun := autMap hab
-  invFun := autMap (isAut_symm G hab)
-  left_inv := autMap_symm hab
-  right_inv := autMap_symm (isAut_symm G hab)
-
-def autRootedEquiv {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (center : I) (x : V) :
-    {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x} ≃
-    {ξ : EscapeLabel G γ ep neg L m // ξ.first center = a x} where
-  toFun ξ := ⟨autMap hab ξ.1,congrArg a ξ.2⟩
-  invFun ξ := ⟨autMap (isAut_symm G hab) ξ.1,by
-    change a.symm (ξ.1.first center) = x
-    rw [ξ.2,a.symm_apply_apply]⟩
-  left_inv ξ := Subtype.ext (autMap_symm hab ξ.1)
-  right_inv ξ := Subtype.ext (autMap_symm (isAut_symm G hab) ξ.1)
-
-theorem mass_autMap (hloc : LocallyFinite G) (a₀ : ℝ)
-    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b)
-    (ξ : EscapeLabel G γ ep neg L m) :
-    (autMap hab ξ).mass (walkKernel G hloc a₀) = ξ.mass (walkKernel G hloc a₀) := by
-  change (ξ.middle.map (autSimpleGraph G hab).toHom).mass (walkKernel G hloc a₀) *
-    (ξ.leftTail.2.map (autSimpleGraph G hab).toHom).mass (walkKernel G hloc a₀) *
-    (ξ.rightTail.2.map (autSimpleGraph G hab).toHom).mass (walkKernel G hloc a₀) = _
-  simp only [HoldingWalk.mass_map (autSimpleGraph G hab).toHom _ _ (walkKernel_covariant G hloc a₀ hab)]
-  rfl
-
-end EscapeLabel
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section LazyOrbitReturn
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem walkKernel_lazy (G : BondGraph V E) (hloc : LocallyFinite G)
-    (D : ℕ) (hD : ∀ x, degree G x ≤ D) (a : ℝ) (ha : 0 ≤ a) (haD : a*D ≤ 1/2)
-    (x : V) : 1/2 ≤ walkKernel G hloc a x x := by
-  have hn : ((neighbors G hloc x).card : ℝ) ≤ D := by
-    exact_mod_cast (neighbors_card_le_degree G hloc x).trans (hD x)
-  have hh := mul_le_mul_of_nonneg_left hn ha
-  simp only [walkKernel,ite_true]
-  split_ifs <;> linarith
-
-theorem positive_lazy_power_step (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
-    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re) (k : ℕ) (x y : V) :
-    (1/2) * (matrixCoeff (P^k) x y).re ≤ (matrixCoeff (P^(k+1)) x y).re := by
-  have hs := hpos.mul_hasSum (P^k) x y
-  have hh := hs.summable.sum_le_tsum ({x} : Finset V)
-    (fun z _ => mul_nonneg (hpos x z).2 ((hpos.pow k) z y).2)
-  simp only [Finset.sum_singleton,hs.tsum_eq,←pow_succ'] at hh
-  exact (mul_le_mul_of_nonneg_right (hdiag x) ((hpos.pow k) x y).2).trans hh
-
- 
-
-theorem walkPower_orbit_return_lower (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
-    (i₀ : ι) (hw₀ : 0 < weight i₀)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re) (L : ℕ) :
-    (1/2) * (∑ i, if root i ∈ vertexOrbit G (root i₀) then weight i else 0) ≤
-      ∑' y, if y ∈ vertexOrbit G (root i₀) then (matrixCoeff (P^L) (root i₀) y).re else 0 := by
-  have hsum (k : ℕ) : Summable (fun y =>
-      if y ∈ vertexOrbit G (root i₀) then (matrixCoeff (P^k) (root i₀) y).re else 0) := by
-    simpa only [mul_ite,mul_one,mul_zero] using finiteKernel_summable_mul
-      (fun x y => (matrixCoeff (P^k) x y).re) (walkPower_finite_support G hloc a P hpos hP k)
-      (root i₀) (fun y => if y ∈ vertexOrbit G (root i₀) then 1 else 0)
-  obtain ⟨k,hk | hk⟩ := Nat.even_or_odd' L
-  · rw [hk]
-    have hh := walkPower_even_orbit_return G hloc a root weight Γ hΓ htr hw hw1 i₀ hw₀ P hpos hsym hP k
-    have hn : 0 ≤ ∑ i, if root i ∈ vertexOrbit G (root i₀) then weight i else 0 :=
-      Finset.sum_nonneg (fun i _ => by split_ifs <;> first | exact hw i | rfl)
-    linarith
-  · rw [hk]
-    have hh := mul_le_mul_of_nonneg_left
-      (walkPower_even_orbit_return G hloc a root weight Γ hΓ htr hw hw1 i₀ hw₀ P hpos hsym hP k)
-      (show (0:ℝ) ≤ 1/2 by norm_num)
-    apply hh.trans
-    rw [←tsum_mul_left]
-    apply ((hsum (2*k)).mul_left (1/2)).tsum_le_tsum _ (hsum (2*k+1))
-    intro y
-    split_ifs
-    · exact positive_lazy_power_step P hpos hdiag (2*k) (root i₀) y
-    · simp only [mul_zero,le_refl]
-
-end LazyOrbitReturn
-end Percolation
-
- 
-namespace Percolation
-universe u v w
-variable {V : Type u} {E : Type v} {I : Type w}
-section UniformLabeledLaw
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Fintype I]
-
-omit [Fintype I] in
- 
-
-theorem segmentReturnMass_uniform_lower (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ g ∈ Γ, g (root i) = x)
-    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
-    (c : ℝ) (hc : 0 < c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re) (γ : I → V) (ep : I) (L : ℕ) :
-    1/(2*c) ≤ segmentReturnMass G hloc a γ ep L (γ ep) := by
-  obtain ⟨i,g,hg,hx⟩ := hcov (γ ep)
-  obtain ⟨b,hb⟩ := hΓ hg
-  have hO := vertexOrbit_eq_of_mem G (root i) (γ ep) ⟨g,b,hb,hx⟩
-  rw [←hx,segmentReturnMass_aut G hloc a γ ep L (root i) P hpos hP hb]
-  rw [segmentReturnMass_operator G hloc a γ ep L (root i) P hpos hP]
-  rw [hO]
-  have hh := walkPower_orbit_return_lower G hloc a root weight Γ hΓ htr
-    (fun i => (hw i).le) hw1 i (hw i) P hpos hsym hP hdiag L
-  have hi : weight i ≤ ∑ r, if root r ∈ vertexOrbit G (root i) then weight r else 0 := by
-    have hx' : root i ∈ vertexOrbit G (root i) := ⟨Equiv.refl _,Equiv.refl _,isAut_refl G,rfl⟩
-    simpa only [ite_eq_left hx'] using Finset.single_le_sum
-      (f := fun r => if root r ∈ vertexOrbit G (root i) then weight r else 0)
-      (fun r _ => by split_ifs <;> first | exact (hw r).le | rfl) (Finset.mem_univ i)
-  apply le_trans _ hh
-  have hct : 0 < 2*c := mul_pos (by norm_num) hc
-  apply (div_le_iff₀ hct).mpr
-  have hh' := mul_le_mul_of_nonneg_left hi hc.le
-  nlinarith [hmin i]
-
- 
-
-theorem rooted_label_probability_hasSum (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ g ∈ Γ, g (root i) = x)
-    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
-    (c : ℝ) (hc : 0 < c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
-    (center ep neg : I) (L m : ℕ) (x : V) (hx : x ∈ vertexOrbit G (γ center)) :
-    HasSum (fun ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x} =>
-      escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1) 1 := by
-  let := finite_rootedEscapeLabels G hloc γ n hdiam center ep neg L m x
-  have hq := segmentReturnMass_uniform_lower G hloc a root weight Γ hΓ htr hcov hw hw1 c hc hmin
-    P hpos hsym hP hdiag γ ep L
-  have hp : 0 < segmentReturnMass G hloc a γ ep L (γ ep) :=
-    (one_div_pos.mpr (mul_pos (by norm_num) hc)).trans_le hq
-  have hs : Summable (fun ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x} =>
-      escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1) :=
-    summable_of_hasFiniteSupport (Set.toFinite _)
-  rw [←rooted_label_probability_sum G hloc a γ n hdiam center ep neg L m x hx P hpos hP (ne_of_gt hp)]
-  exact hs.hasSum
-
-end UniformLabeledLaw
-end Percolation
-
- 
-namespace Percolation
-universe u v w
-variable {V : Type u} {E : Type v} {I : Type w}
-section LabeledTransport
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {G : BondGraph V E} {γ : I → V} {ep neg : I} {L m : ℕ}
-
-def labelEndpointEquiv (center : I) (x : V) :
-    (Σ y : V, {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x ∧ ξ.last center = y}) ≃
-      {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x} where
-  toFun ξ := ⟨ξ.2.1,ξ.2.2.1⟩
-  invFun ξ := ⟨ξ.1.last center,ξ.1,ξ.2,rfl⟩
-  left_inv ξ := by rcases ξ with ⟨y,ξ,hx,hy⟩; subst y; rfl
-  right_inv ξ := rfl
-
-def labelAutPairEquiv {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (center : I) (x y : V) :
-    {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x ∧ ξ.last center = y} ≃
-    {ξ : EscapeLabel G γ ep neg L m // ξ.first center = a x ∧ ξ.last center = a y} where
-  toFun ξ := ⟨EscapeLabel.autMap hab ξ.1,congrArg a ξ.2.1,congrArg a ξ.2.2⟩
-  invFun ξ := ⟨EscapeLabel.autMap (isAut_symm G hab) ξ.1,by
-    change a.symm (ξ.1.first center) = x ∧ a.symm (ξ.1.last center) = y
-    rw [ξ.2.1,ξ.2.2,a.symm_apply_apply,a.symm_apply_apply]
-    exact ⟨rfl,rfl⟩⟩
-  left_inv ξ := Subtype.ext (EscapeLabel.autMap_symm hab ξ.1)
-  right_inv ξ := Subtype.ext (EscapeLabel.autMap_symm (isAut_symm G hab) ξ.1)
-
-noncomputable def escapeLabelTransport (center : I) (B : EscapeLabel G γ ep neg L m → ℝ) (x y : V) : ℝ :=
-  ∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x ∧ ξ.last center = y}, B ξ.1
-
-theorem escapeLabelTransport_reverse (center : I) (B : EscapeLabel G γ ep neg L m → ℝ) (x y : V) :
-    escapeLabelTransport center B x y =
-      escapeLabelTransport center (fun ξ => B ξ.reverse) y x := by
-  unfold escapeLabelTransport
-  exact (EscapeLabel.reverseRootedEquiv center y x).tsum_eq _ |>.symm
-
-theorem escapeLabelTransport_aut (center : I) (B : EscapeLabel G γ ep neg L m → ℝ)
-    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b)
-    (hB : ∀ ξ, B (EscapeLabel.autMap hab ξ) = B ξ) (x y : V) :
-    escapeLabelTransport center B (a x) (a y) = escapeLabelTransport center B x y := by
-  unfold escapeLabelTransport
-  rw [←(labelAutPairEquiv hab center x y).tsum_eq]
-  exact tsum_congr (fun ξ => hB ξ.1)
-
-variable [Fintype I]
-
-theorem finite_pair_label_sigma (hloc : LocallyFinite G) (n : ℕ)
-    (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (center : I) (x : V) :
-    Finite (Σ y : V, {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x ∧ ξ.last center = y}) := by
-  let := finite_rootedEscapeLabels G hloc γ n hdiam center ep neg L m x
-  exact Finite.of_equiv _ (labelEndpointEquiv center x).symm
-
-theorem escapeLabelTransport_row_sum (hloc : LocallyFinite G) (n : ℕ)
-    (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (center : I)
-    (B : EscapeLabel G γ ep neg L m → ℝ) (x : V) :
-    (∑' y, escapeLabelTransport center B x y) =
-      ∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x}, B ξ.1 := by
-  let := finite_pair_label_sigma hloc n hdiam center (ep := ep) (neg := neg) (L := L) (m := m) x
-  have hs : Summable (fun ξ : Σ y : V, {ξ : EscapeLabel G γ ep neg L m //
-      ξ.first center = x ∧ ξ.last center = y} => B ξ.2.1) :=
-    summable_of_hasFiniteSupport (Set.toFinite _)
-  unfold escapeLabelTransport
-  rw [←hs.tsum_sigma]
-  exact (labelEndpointEquiv (G := G) (γ := γ) (ep := ep) (neg := neg) (L := L) (m := m) center x).tsum_eq (fun ξ => B ξ.1)
-
-theorem escapeLabelTransport_rowSummable (hloc : LocallyFinite G) (n : ℕ)
-    (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (center : I)
-    (B : EscapeLabel G γ ep neg L m → ℝ) (x : V) :
-    Summable (escapeLabelTransport center B x) := by
-  let := finite_pair_label_sigma hloc n hdiam center (ep := ep) (neg := neg) (L := L) (m := m) x
-  have hs : Summable (fun ξ : Σ y : V, {ξ : EscapeLabel G γ ep neg L m //
-      ξ.first center = x ∧ ξ.last center = y} => B ξ.2.1) :=
-    summable_of_hasFiniteSupport (Set.toFinite _)
-  exact hs.sigma
-
-theorem escapeLabelTransport_columnSummable (hloc : LocallyFinite G) (n : ℕ)
-    (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (center : I)
-    (B : EscapeLabel G γ ep neg L m → ℝ) (y : V) :
-    Summable (fun x => escapeLabelTransport center B x y) := by
-  simp only [escapeLabelTransport_reverse center B]
-  exact escapeLabelTransport_rowSummable hloc n hdiam center (fun ξ => B ξ.reverse) y
-
-end LabeledTransport
-end Percolation
-
- 
-namespace Percolation
-universe u v w
-variable {V : Type u} {E : Type v} {I : Type w}
-section LabeledReRooting
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {G : BondGraph V E} {γ : I → V} {ep neg : I} {L m : ℕ}
-
-theorem EscapeLabel.first_orbit (ξ : EscapeLabel G γ ep neg L m) (i : I) :
-    ξ.first i ∈ vertexOrbit G (γ i) := by
-  obtain ⟨a,b,h,he⟩ := ξ.first_mem
-  exact ⟨a,b,h,(congrFun he i).symm⟩
-
-theorem EscapeLabel.last_orbit (ξ : EscapeLabel G γ ep neg L m) (i : I) :
-    ξ.last i ∈ vertexOrbit G (γ i) := by
-  obtain ⟨a,b,h,he⟩ := ξ.last_mem
-  exact ⟨a,b,h,(congrFun he i).symm⟩
-
-theorem escapeLabelTransport_zero (center : I) (B : EscapeLabel G γ ep neg L m → ℝ)
-    (x y : V) (hxy : x ∉ vertexOrbit G (γ center) ∨ y ∉ vertexOrbit G (γ center)) :
-    escapeLabelTransport center B x y = 0 := by
-  let : IsEmpty {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x ∧ ξ.last center = y} :=
-    ⟨fun ξ => by
-      rcases hxy with hx | hy
-      · exact hx (ξ.2.1 ▸ ξ.1.first_orbit center)
-      · exact hy (ξ.2.2 ▸ ξ.1.last_orbit center)⟩
-  exact tsum_empty
-
-variable [Fintype I]
-
- 
-
-theorem rooted_label_weight_reversal (hloc : LocallyFinite G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (center : I)
-    (hθ : 0 < ∑ i, if root i ∈ vertexOrbit G (γ center) then weight i else 0)
-    (B : EscapeLabel G γ ep neg L m → ℝ)
-    (hB : ∀ (a : V ≃ V) (b : E ≃ E) (hab : IsAut G a b) ξ, B (EscapeLabel.autMap hab ξ) = B ξ)
-    (x : V) (hx : x ∈ vertexOrbit G (γ center)) :
-    (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x}, B ξ.1) =
-      ∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x}, B ξ.1.reverse := by
-  let M := escapeLabelTransport center B
-  have hM (a : V ≃ V) (b : E ≃ E) (hab : IsAut G a b) (u v : V) : M (a u) (a v) = M u v :=
-    escapeLabelTransport_aut center B hab (hB a b hab) u v
-  have hr (u : V) : (∑' v, M u v) = if u ∈ vertexOrbit G (γ center) then ∑' v, M (γ center) v else 0 := by
-    split_ifs with hu
-    · obtain ⟨a,b,hab,rfl⟩ := hu
-      rw [←a.tsum_eq]
-      exact tsum_congr (fun v => hM a b hab (γ center) v)
-    · calc
-        _ = ∑' _ : V, (0:ℝ) := tsum_congr (fun v => escapeLabelTransport_zero center B u v (Or.inl hu))
-        _ = 0 := tsum_zero
-  have hc (v : V) : (∑' u, M u v) = if v ∈ vertexOrbit G (γ center) then ∑' u, M u (γ center) else 0 := by
-    split_ifs with hv
-    · obtain ⟨a,b,hab,rfl⟩ := hv
-      rw [←a.tsum_eq]
-      exact tsum_congr (fun u => hM a b hab u (γ center))
-    · calc
-        _ = ∑' _ : V, (0:ℝ) := tsum_congr (fun u => escapeLabelTransport_zero center B u v (Or.inr hv))
-        _ = 0 := tsum_zero
-  have ht := htr.real root weight Γ M
-    (fun a ha u v => by obtain ⟨b,hb⟩ := hΓ ha; exact hM a b hb u v)
-    (escapeLabelTransport_rowSummable hloc n hdiam center B)
-    (escapeLabelTransport_columnSummable hloc n hdiam center B)
-  have hfactor (f : V → ℝ) (q : ℝ) (he : ∀ u, f u = if u ∈ vertexOrbit G (γ center) then q else 0) :
-      (∑ i, weight i*f (root i)) = (∑ i, if root i ∈ vertexOrbit G (γ center) then weight i else 0)*q := by
-    rw [Finset.sum_mul]
-    apply Finset.sum_congr rfl
-    intro i _
-    rw [he]
-    split_ifs <;> simp only [mul_zero,zero_mul]
-  rw [hfactor _ _ hr,hfactor _ _ hc] at ht
-  have hrc := (mul_left_cancel₀ (ne_of_gt hθ)) ht
-  have hxe : (∑' v, M x v) = ∑' u, M u x := by
-    rw [hr x,hc x,ite_eq_left hx,ite_eq_left hx]
-    exact hrc
-  calc
-    _ = ∑' v, M x v := (escapeLabelTransport_row_sum hloc n hdiam center B x).symm
-    _ = ∑' u, M u x := hxe
-    _ = ∑' u, escapeLabelTransport center (fun ξ => B ξ.reverse) x u :=
-      tsum_congr (fun u => escapeLabelTransport_reverse center B u x)
-    _ = _ := escapeLabelTransport_row_sum hloc n hdiam center (fun ξ => B ξ.reverse) x
-
-end LabeledReRooting
-end Percolation
-
- 
-namespace Percolation
-universe u v w
-variable {V : Type u} {E : Type v} {I : Type w}
-section ExactLabeledReversal
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Fintype I]
-
-theorem escapeLabelProbability_aut (G : BondGraph V E) (hloc : LocallyFinite G)
-    (a₀ : ℝ) (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
-    (center ep neg : I) (L m : ℕ) {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b)
-    (ξ : EscapeLabel G γ ep neg L m) :
-    escapeLabelProbability G hloc a₀ γ n hdiam center ep neg L m (EscapeLabel.autMap hab ξ) =
-      escapeLabelProbability G hloc a₀ γ n hdiam center ep neg L m ξ := by
-  simp only [escapeLabelProbability,EscapeLabel.mass_autMap]
-
- 
-
-theorem rooted_escape_label_reversal (G : BondGraph V E) (hloc : LocallyFinite G) (a₀ : ℝ)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (center ep neg : I) (L m : ℕ)
-    (hθ : 0 < ∑ i, if root i ∈ vertexOrbit G (γ center) then weight i else 0)
-    (F : EscapeLabel G γ ep neg L m → ℝ)
-    (hF : ∀ (a : V ≃ V) (b : E ≃ E) (hab : IsAut G a b) ξ, F (EscapeLabel.autMap hab ξ) = F ξ)
-    (x : V) (hx : x ∈ vertexOrbit G (γ center)) :
-    (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
-      escapeLabelProbability G hloc a₀ γ n hdiam center ep neg L m ξ.1 * F ξ.1) =
-    ∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
-      escapeLabelProbability G hloc a₀ γ n hdiam center ep neg L m ξ.1 * F ξ.1.reverse := by
-  have hh := rooted_label_weight_reversal hloc root weight Γ hΓ htr n hdiam center hθ
-    (fun ξ => escapeLabelProbability G hloc a₀ γ n hdiam center ep neg L m ξ * F ξ)
-    (fun a b hab ξ => by rw [escapeLabelProbability_aut G hloc a₀ γ n hdiam center ep neg L m hab,hF a b hab]) x hx
-  simpa only [escapeLabelProbability_reverse] using hh
-
-theorem escapeLabelProbability_nonneg (G : BondGraph V E) (hloc : LocallyFinite G)
-    (a : ℝ) (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
-    (center ep neg : I) (L m : ℕ) (ξ : EscapeLabel G γ ep neg L m) :
-    0 ≤ escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ := by
-  have hq : 0 ≤ segmentReturnMass G hloc a γ ep L (γ ep) := by
-    apply tsum_nonneg
-    intro mid
-    split_ifs
-    · exact mid.2.mass_nonneg _ hP
-    · rfl
-  exact div_nonneg (ξ.mass_nonneg _ hP) (mul_nonneg (mul_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _)) hq)
-
-end ExactLabeledReversal
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section LongGeodesics
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem exists_geodesic_length (G : BondGraph V E) [Infinite V]
-    (hloc : LocallyFinite G) (hconn : Connected G) (x : V) (N : ℕ) :
-    ∃ y : V, ∃ q : (openSimpleGraph G Set.univ).Walk x y,
-      q.length = N ∧ q.length = (openSimpleGraph G Set.univ).dist x y := by
-  obtain ⟨z,hz⟩ := (finite_reachBall hloc x N).exists_notMem
-  have hc := (conn_iff_reachable G Set.univ x z).mp (conn_univ_of_connected G hconn x z)
-  obtain ⟨q,hq⟩ := hc.exists_walk_length_eq_dist
-  have hN : N < q.length := by
-    by_contra hh
-    have hm := walk_mem_reachBall G Set.univ q 0 (show x ∈ reachBall G x 0 from rfl)
-    have hm' : z ∈ reachBall G x q.length := by simpa only [Nat.zero_add] using hm
-    exact hz (reachBall_mono_local G x (Nat.le_of_not_gt hh) hm')
-  refine ⟨q.getVert N,q.take N,?_,?_⟩
-  · simp only [SimpleGraph.Walk.take_length,inf_eq_left.mpr hN.le]
-  · exact SimpleGraph.length_eq_dist_of_subwalk hq (q.isSubwalk_take N)
-
- 
-
-theorem geodesic_getVert_dist (H : SimpleGraph V) {x y : V} (q : H.Walk x y)
-    (hq : q.length = H.dist x y) (i j : ℕ) (hij : i ≤ j) (hj : j ≤ q.length) :
-    H.dist (q.getVert i) (q.getVert j) = j-i := by
-  have hs := SimpleGraph.length_eq_dist_of_subwalk hq
-    (((q.drop i).isSubwalk_take (j-i)).trans (q.isSubwalk_drop i))
-  have he : (q.drop i).getVert (j-i) = q.getVert j := by
-    rw [SimpleGraph.Walk.drop_getVert,Nat.add_sub_of_le hij]
-  rw [SimpleGraph.Walk.take_length,SimpleGraph.Walk.drop_length,he,
-    min_eq_left (Nat.sub_le_sub_right hj i)] at hs
-  exact hs.symm
-
-end LongGeodesics
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-namespace HoldingWalk
-variable {H : SimpleGraph V}
-
- 
-
-def append : {k l : ℕ} → {x y z : V} → HoldingWalk H k x y → HoldingWalk H l y z → HoldingWalk H (l+k) x z
-  | _,_,_,_,_,.nil _,v => v
-  | _,_,_,_,_,.hold w,v => .hold (w.append v)
-  | _,_,_,_,_,.step h w,v => .step h (w.append v)
-
-def ofWalk : {x y : V} → (w : H.Walk x y) → HoldingWalk H w.length x y
-  | _,_,.nil => .nil _
-  | _,_,.cons h w => .step h (ofWalk w)
-
-theorem mass_append (P : V → V → ℝ) {k l : ℕ} {x y z : V}
-    (w : HoldingWalk H k x y) (v : HoldingWalk H l y z) :
-    (w.append v).mass P = w.mass P*v.mass P := by
-  induction w with
-  | nil => simp only [append,mass,one_mul]
-  | hold w ih => simp only [append,mass,ih,mul_assoc]
-  | step h w ih => simp only [append,mass,ih,mul_assoc]
-
-theorem mass_ofWalk (P : V → V → ℝ) (a : ℝ) (hP : ∀ x y, H.Adj x y → P x y = a)
-    {x y : V} (w : H.Walk x y) : (ofWalk w).mass P = a^w.length := by
-  induction w with
-  | nil => rfl
-  | @cons x z y h w ih => simp only [ofWalk,mass,hP x z h,ih,SimpleGraph.Walk.length_cons,pow_succ'];
-
-theorem ofWalk_vertices {x y : V} (w : H.Walk x y) : (ofWalk w).vertices = w.support := by
-  induction w with
-  | nil => rfl
-  | cons h w ih => simp only [ofWalk,vertices,SimpleGraph.Walk.support_cons,ih]
-
-theorem append_vertices {k l : ℕ} {x y z : V}
-    (w : HoldingWalk H k x y) (v : HoldingWalk H l y z) :
-    (w.append v).vertices = w.vertices.dropLast ++ v.vertices := by
-  induction w with
-  | nil => rfl
-  | hold w ih =>
-    have hn : w.vertices ≠ [] := by cases w <;> simp [vertices]
-    simp only [append,vertices,ih,List.dropLast_cons_of_ne_nil hn,List.cons_append]
-  | step h w ih =>
-    have hn : w.vertices ≠ [] := by cases w <;> simp [vertices]
-    simp only [append,vertices,ih,List.dropLast_cons_of_ne_nil hn,List.cons_append]
-
-end HoldingWalk
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section GeodesicSegments
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-def geodesicSegment (G : BondGraph V E) {x y : V}
-    (q : (openSimpleGraph G Set.univ).Walk x y) (N : ℕ) : Fin (N+1) → V :=
-  fun i => q.getVert i
-
-theorem geodesicSegment_adj (G : BondGraph V E) {x y : V}
-    (q : (openSimpleGraph G Set.univ).Walk x y) (N : ℕ) (hN : q.length = N)
-    (i : ℕ) (hi : i < N) :
-    (openSimpleGraph G Set.univ).Adj
-      (geodesicSegment G q N ⟨i,by omega⟩) (geodesicSegment G q N ⟨i+1,by omega⟩) := by
-  change (openSimpleGraph G Set.univ).Adj (q.getVert i) (q.getVert (i+1))
-  exact q.adj_getVert_succ (by omega)
-
-theorem geodesicSegment_injective (G : BondGraph V E) {x y : V}
-    (q : (openSimpleGraph G Set.univ).Walk x y) (N : ℕ) (hN : q.length = N)
-    (hq : q.length = (openSimpleGraph G Set.univ).dist x y) :
-    Function.Injective (geodesicSegment G q N) := by
-  intro i j hh
-  apply Fin.ext
-  exact (q.isPath_of_length_eq_dist hq).getVert_injOn
-    (by change i.val ≤ q.length; omega) (by change j.val ≤ q.length; omega) hh
-
-theorem geodesicSegment_diameter (G : BondGraph V E) {x y : V}
-    (q : (openSimpleGraph G Set.univ).Walk x y) (N : ℕ) (hN : q.length = N) :
-    ∀ i j, geodesicSegment G q N j ∈ reachBall G (geodesicSegment G q N i) N := by
-  intro i j
-  have hc : (openSimpleGraph G Set.univ).Reachable (q.getVert i) (q.getVert j) :=
-    (q.take i).reachable.symm.trans (q.take j).reachable
-  obtain ⟨r,hr⟩ := hc.exists_walk_length_eq_dist
-  have hd : (openSimpleGraph G Set.univ).dist (q.getVert i) (q.getVert j) ≤ N := by
-    rcases le_total i.val j.val with hij | hji
-    · have hle := SimpleGraph.dist_le ((q.drop i).take (j-i))
-      rw [SimpleGraph.Walk.take_length,SimpleGraph.Walk.drop_length,
-        SimpleGraph.Walk.drop_getVert,Nat.add_sub_of_le hij] at hle
-      exact hle.trans (by omega)
-    · have hle := SimpleGraph.dist_le ((q.drop j).take (i-j))
-      rw [SimpleGraph.Walk.take_length,SimpleGraph.Walk.drop_length,
-        SimpleGraph.Walk.drop_getVert,Nat.add_sub_of_le hji] at hle
-      rw [SimpleGraph.dist_comm] at hle
-      exact hle.trans (by omega)
-  have hm := walk_mem_reachBall G Set.univ r 0 (show q.getVert i ∈ reachBall G (q.getVert i) 0 from rfl)
-  apply reachBall_mono_local G (q.getVert i) (a := r.length) (by omega)
-  simpa only [Nat.zero_add, geodesicSegment] using hm
-
-theorem segmentOrbit_adj (G : BondGraph V E) {I : Type*} (γ s : I → V)
-    (hs : s ∈ segmentOrbit G γ) (i j : I) (hij : (openSimpleGraph G Set.univ).Adj (γ i) (γ j)) :
-    (openSimpleGraph G Set.univ).Adj (s i) (s j) := by
-  obtain ⟨a,b,hab,rfl⟩ := hs
-  exact (autSimpleGraph G hab).map_rel_iff.mpr hij
-
-theorem walkKernel_step (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    {x y : V} (h : (openSimpleGraph G Set.univ).Adj x y) : walkKernel G hloc a x y = a := by
-  have hh := (openSimpleGraph_univ_adj G x y).mp h
-  simp only [walkKernel,ite_eq_right hh.1,ite_eq_left hh.2,zero_add]
-
-theorem walkKernel_ofWalk_mass (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    {x y : V} (q : (openSimpleGraph G Set.univ).Walk x y) :
-    (HoldingWalk.ofWalk q).mass (walkKernel G hloc a) = a^q.length :=
-  HoldingWalk.mass_ofWalk _ a (fun _ _ h => walkKernel_step G hloc a h) q
-
-end GeodesicSegments
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section ActualTailPotentials
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem raw_tail_twoPoint_expectation (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (P A : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (p : unitInterval) (hA : HasKernel G p A) (m : ℕ) (x b : V) :
-    (∑' w : RawFrom (openSimpleGraph G Set.univ) m x,
-      w.2.mass (walkKernel G hloc a) * twoPoint G p w.1 b) =
-      (matrixCoeff (P^m*A) x b).re := by
-  have he : (fun y => twoPoint G p y b) = (fun y => (matrixCoeff A y b).re) := by
-    funext y
-    rw [matrixCoeff,hA y b,Complex.ofReal_re]
-  change RawFrom.integral (H := openSimpleGraph G Set.univ) (walkKernel G hloc a) m x (twoPoint G p · b) = _
-  rw [he,raw_integral_operator G hloc a P hpos hP]
-  exact ((hpos.pow m).mul_hasSum A x b).tsum_eq
-
- 
-
-theorem raw_tail_potential_bound [Countable V] [Countable E] [Nonempty V]
-    {ι : Type*} [Fintype ι]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (root : ι → V) (weight : ι → ℝ) (htr : RootTransport root weight (vertexAut G))
-    (hcov : ∀ x, ∃ i, ∃ e ∈ vertexAut G, e (root i) = x)
-    (hw : ∀ i, 0 ≤ weight i) (hsum : ∑ i, weight i = 1)
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c * weight i)
-    (P A : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (p : unitInterval) (hA : HasKernel G p A) (m : ℕ) (x : V) (S : Finset V) :
-    (∑' w : RawFrom (openSimpleGraph G Set.univ) m x,
-      w.2.mass (walkKernel G hloc a) * ∑ b ∈ S, twoPoint G p w.1 b) ≤
-      (S.card:ℝ) * Real.sqrt c * ‖P‖^m := by
-  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
-  let := Fintype.ofFinite (RawFrom (openSimpleGraph G Set.univ) m x)
-  have hPinv : KernelInvariant (vertexAut G) P := by
-    intro e he x y
-    obtain ⟨f,hf⟩ := he
-    rw [hP,hP,walkKernel_covariant G hloc a hf]
-  calc
-    _ = ∑ b ∈ S, (∑' w : RawFrom (openSimpleGraph G Set.univ) m x,
-        w.2.mass (walkKernel G hloc a) * twoPoint G p w.1 b) := by
-      simp only [tsum_fintype,Finset.mul_sum]
-      exact Finset.sum_comm
-    _ = ∑ b ∈ S, (matrixCoeff (P^m*A) x b).re := by
-      apply Finset.sum_congr rfl
-      intro b _
-      exact raw_tail_twoPoint_expectation G hloc a P A hpos hP p hA m x b
-    _ ≤ ∑ _b ∈ S, Real.sqrt c * ‖P‖^m := by
-      apply Finset.sum_le_sum
-      intro b _
-      exact (Complex.re_le_norm _).trans (twoPoint_operator_smoothing G p root weight htr hcov hw hsum c hc hmin A P hA hPinv m x b)
-    _ = _ := by simp only [Finset.sum_const,nsmul_eq_mul,mul_assoc]
-
-end ActualTailPotentials
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section AuxiliaryTailMarkov
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem finite_weighted_markov {α : Type*} [Fintype α] (μ F : α → ℝ)
-    (hμ : ∀ x, 0 ≤ μ x) (hF : ∀ x, 0 ≤ F x) (t : ℝ) (ht : 0 < t) :
-    (∑ x, if t < F x then μ x else 0) ≤ (∑ x, μ x*F x)/t := by
-  apply (le_div_iff₀ ht).mpr
-  rw [Finset.sum_mul]
-  apply Finset.sum_le_sum
-  intro x _
-  split_ifs with hx
-  · exact mul_le_mul_of_nonneg_left hx.le (hμ x)
-  · exact (by simpa only [zero_mul] using mul_nonneg (hμ x) (hF x))
-
-theorem raw_tail_potential_failure [Countable V] [Countable E] [Nonempty V]
-    {ι : Type*} [Fintype ι]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (root : ι → V) (weight : ι → ℝ) (htr : RootTransport root weight (vertexAut G))
-    (hcov : ∀ x, ∃ i, ∃ e ∈ vertexAut G, e (root i) = x)
-    (hw : ∀ i, 0 ≤ weight i) (hsum : ∑ i, weight i = 1)
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c * weight i)
-    (P A : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (p : unitInterval) (hA : HasKernel G p A) (m : ℕ) (x : V) (S : Finset V)
-    (t : ℝ) (ht : 0 < t) :
-    (∑' w : RawFrom (openSimpleGraph G Set.univ) m x,
-      if t < ∑ b ∈ S, twoPoint G p w.1 b then w.2.mass (walkKernel G hloc a) else 0) ≤
-      ((S.card:ℝ) * Real.sqrt c * ‖P‖^m)/t := by
-  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
-  let := Fintype.ofFinite (RawFrom (openSimpleGraph G Set.univ) m x)
-  have hK (x y : V) : 0 ≤ walkKernel G hloc a x y := by
-    simpa only [hP,Complex.ofReal_re] using (hpos x y).2
-  have hM := finite_weighted_markov
-    (fun w : RawFrom (openSimpleGraph G Set.univ) m x => w.2.mass (walkKernel G hloc a))
-    (fun w => ∑ b ∈ S, twoPoint G p w.1 b)
-    (fun w => w.2.mass_nonneg _ hK)
-    (fun w => Finset.sum_nonneg (fun b _ => twoPoint_nonneg G p w.1 b)) t ht
-  have hB := raw_tail_potential_bound G hloc a root weight htr hcov hw hsum c hc hmin P A hpos hP p hA m x S
-  simp only [tsum_fintype] at hB ⊢
-  exact hM.trans (div_le_div_of_nonneg_right hB ht.le)
-
- 
-
-theorem finite_product_failure_union {α β : Type*} [Fintype α] [Fintype β]
-    (μ : α → ℝ) (ν : β → ℝ) (hμ : ∀ x, 0 ≤ μ x) (hν : ∀ y, 0 ≤ ν y)
-    (hμ1 : ∑ x, μ x = 1) (hν1 : ∑ y, ν y = 1) (B : α → Prop) (C : β → Prop) :
-    (∑ z : α × β, if B z.1 ∨ C z.2 then μ z.1*ν z.2 else 0) ≤
-      (∑ x, if B x then μ x else 0) + ∑ y, if C y then ν y else 0 := by
-  calc
-    _ ≤ ∑ z : α × β, ((if B z.1 then μ z.1 else 0)*ν z.2 + μ z.1*(if C z.2 then ν z.2 else 0)) := by
-      apply Finset.sum_le_sum
-      intro z _
-      by_cases hB : B z.1 <;> by_cases hC : C z.2 <;>
-        simp [hB,hC]
-      exact mul_nonneg (hμ _) (hν _)
-    _ = _ := by
-      rw [Fintype.sum_prod_type]
-      simp only [Finset.sum_add_distrib,←Finset.mul_sum,←Finset.sum_mul,hμ1,hν1,mul_one,one_mul]
-
-end AuxiliaryTailMarkov
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section IndependentTailPotentialTests
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem two_raw_tail_potential_failure [Countable V] [Countable E] [Nonempty V]
-    {ι : Type*} [Fintype ι]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (root : ι → V) (weight : ι → ℝ) (htr : RootTransport root weight (vertexAut G))
-    (hcov : ∀ x, ∃ i, ∃ e ∈ vertexAut G, e (root i) = x)
-    (hw : ∀ i, 0 ≤ weight i) (hsum : ∑ i, weight i = 1)
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c * weight i)
-    (P A : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (p : unitInterval) (hA : HasKernel G p A) (m : ℕ) (x y : V) (S : Finset V)
-    (t : ℝ) (ht : 0 < t) :
-    (∑' w : RawFrom (openSimpleGraph G Set.univ) m x × RawFrom (openSimpleGraph G Set.univ) m y,
-      if (t < ∑ b ∈ S, twoPoint G p w.1.1 b) ∨ (t < ∑ b ∈ S, twoPoint G p w.2.1 b)
-      then w.1.2.mass (walkKernel G hloc a) * w.2.2.mass (walkKernel G hloc a) else 0) ≤
-      2*((S.card:ℝ) * Real.sqrt c * ‖P‖^m)/t := by
-  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
-  let := Fintype.ofFinite (RawFrom (openSimpleGraph G Set.univ) m x)
-  let := Fintype.ofFinite (RawFrom (openSimpleGraph G Set.univ) m y)
-  have hK (u v : V) : 0 ≤ walkKernel G hloc a u v := by
-    simpa only [hP,Complex.ofReal_re] using (hpos u v).2
-  have hμ := rawFrom_probability_sum G hloc a m x
-  have hν := rawFrom_probability_sum G hloc a m y
-  simp only [tsum_fintype] at hμ hν
-  have hU := finite_product_failure_union
-    (fun w : RawFrom (openSimpleGraph G Set.univ) m x => w.2.mass (walkKernel G hloc a))
-    (fun w : RawFrom (openSimpleGraph G Set.univ) m y => w.2.mass (walkKernel G hloc a))
-    (fun w => w.2.mass_nonneg _ hK) (fun w => w.2.mass_nonneg _ hK) hμ hν
-    (fun w => t < ∑ b ∈ S, twoPoint G p w.1 b)
-    (fun w => t < ∑ b ∈ S, twoPoint G p w.1 b)
-  have hx := raw_tail_potential_failure G hloc a root weight htr hcov hw hsum c hc hmin P A hpos hP p hA m x S t ht
-  have hy := raw_tail_potential_failure G hloc a root weight htr hcov hw hsum c hc hmin P A hpos hP p hA m y S t ht
-  simp only [tsum_fintype] at hx hy ⊢
-  exact hU.trans ((add_le_add hx hy).trans_eq (by ring))
-
-end IndependentTailPotentialTests
-end Percolation
-
- 
-namespace Percolation
-section LogarithmicTailLength
-
- 
-
-theorem logarithmic_tail_power (ρ X : ℝ) (hρ : 0 < ρ) (hρ1 : ρ < 1) (hX : 0 < X) :
-    ρ ^ ⌈(2/(-Real.log ρ))*Real.log X⌉₊ ≤ X^(-2:ℝ) := by
-  have hl : Real.log ρ ≠ 0 := ne_of_lt (Real.log_neg hρ hρ1)
-  calc
-    _ = ρ ^ (⌈(2/(-Real.log ρ))*Real.log X⌉₊ : ℝ) := (Real.rpow_natCast _ _).symm
-    _ ≤ ρ ^ ((2/(-Real.log ρ))*Real.log X) :=
-      Real.rpow_le_rpow_of_exponent_ge hρ hρ1.le (Nat.le_ceil _)
-    _ = X^(-2:ℝ) := by
-      rw [Real.rpow_def_of_pos hρ,Real.rpow_def_of_pos hX]
-      congr 1
-      field_simp [hl]
-
- 
-
-theorem exists_uniform_logarithmic_tail_cutoff (ρ A : ℝ)
-    (hρ : 0 < ρ) (hρ1 : ρ < 1) (hA : 0 ≤ A) (kmin : ℕ) :
-    ∃ C₁ : ℝ, 0 < C₁ ∧ 1 < (-Real.log ρ)*C₁ ∧
-      ∃ k₀ : ℕ, kmin ≤ k₀ ∧ ∀ k : ℕ, k₀ ≤ k → ∀ R : ℝ, 1 ≤ R →
-        A*((k:ℝ)+1)*R*ρ^⌈C₁*Real.log (((k:ℝ)+1)*R)⌉₊ ≤ 1/4 := by
-  have hlambda : 0 < -Real.log ρ := neg_pos.mpr (Real.log_neg hρ hρ1)
-  refine ⟨2/(-Real.log ρ),div_pos (by norm_num) hlambda,?_,max kmin ⌈4*A⌉₊,le_max_left _ _,?_⟩
-  · rw [mul_div_cancel₀ _ (ne_of_gt hlambda)]
-    norm_num
-  · intro k hk R hR
-    have hk' : ⌈4*A⌉₊ ≤ k := (le_max_right _ _).trans hk
-    have hAk : 4*A ≤ (k:ℝ) := (Nat.le_ceil (4*A)).trans (by exact_mod_cast hk')
-    have hkpos : 0 < (k:ℝ)+1 := by positivity
-    have hRp : 0 < R := lt_of_lt_of_le (by norm_num) hR
-    let X : ℝ := ((k:ℝ)+1)*R
-    have hX : 0 < X := mul_pos hkpos hRp
-    have hXk : (k:ℝ)+1 ≤ X := by dsimp [X]; exact le_mul_of_one_le_right hkpos.le hR
-    have hp := logarithmic_tail_power ρ X hρ hρ1 hX
-    have hneg : X^(-2:ℝ) = (X^2)⁻¹ := by rw [Real.rpow_neg,show (2:ℝ) = (2:ℕ) by norm_num,Real.rpow_natCast]; exact hX.le
-    calc
-      _ = A*X*ρ^⌈(2/(-Real.log ρ))*Real.log X⌉₊ := by dsimp [X]; ring
-      _ ≤ A*X*X^(-2:ℝ) := mul_le_mul_of_nonneg_left hp (mul_nonneg hA hX.le)
-      _ = A/X := by rw [hneg]; field_simp [ne_of_gt hX]
-      _ ≤ 1/4 := (div_le_iff₀ hX).mpr (by linarith)
-
-end LogarithmicTailLength
-end Percolation
-
- 
-namespace Percolation
-universe u
-variable {V : Type u}
-namespace HoldingWalk
-variable {H : SimpleGraph V}
-
- 
-
-def position : {k : ℕ} → {x y : V} → HoldingWalk H k x y → ℕ → V
-  | _,x,_,.nil _,_ => x
-  | _,x,_,.hold _,0 => x
-  | _,_,_,.hold w,n+1 => w.position n
-  | _,x,_,.step _ _,0 => x
-  | _,_,_,.step _ w,n+1 => w.position n
-
-@[simp] theorem position_zero {k : ℕ} {x y : V} (w : HoldingWalk H k x y) : w.position 0 = x := by
-  cases w <;> rfl
-
-@[simp] theorem position_length {k : ℕ} {x y : V} (w : HoldingWalk H k x y) : w.position k = y := by
-  induction w with
-  | nil => rfl
-  | hold w ih => exact ih
-  | step h w ih => exact ih
-
-theorem position_mem_vertices {k : ℕ} {x y : V} (w : HoldingWalk H k x y) (i : ℕ) :
-    w.position i ∈ w.vertices := by
-  induction w generalizing i with
-  | nil => simp only [position,vertices,List.mem_singleton]
-  | hold w ih => cases i <;> simp only [position,vertices,List.mem_cons,true_or,ih,or_true]
-  | step h w ih => cases i <;> simp only [position,vertices,List.mem_cons,true_or,ih,or_true]
-
-theorem mem_vertices_iff_position {k : ℕ} {x y z : V} (w : HoldingWalk H k x y) :
-    z ∈ w.vertices ↔ ∃ i : ℕ, i ≤ k ∧ w.position i = z := by
-  induction w with
-  | nil => simp only [vertices,List.mem_singleton,Nat.le_zero,exists_eq_left,position,eq_comm]
-  | @hold k x y w ih =>
-    constructor
-    · intro hz
-      rcases List.mem_cons.mp hz with rfl | hz
-      · exact ⟨0,Nat.zero_le _,rfl⟩
-      · obtain ⟨i,hi,he⟩ := ih.mp hz
-        exact ⟨i+1,Nat.succ_le_succ hi,he⟩
-    · rintro ⟨i,hi,he⟩
-      cases i with
-      | zero => exact List.mem_cons.mpr (Or.inl he.symm)
-      | succ i => exact List.mem_cons.mpr (Or.inr (ih.mpr ⟨i,Nat.le_of_succ_le_succ hi,he⟩))
-  | @step k x u y h w ih =>
-    constructor
-    · intro hz
-      rcases List.mem_cons.mp hz with rfl | hz
-      · exact ⟨0,Nat.zero_le _,rfl⟩
-      · obtain ⟨i,hi,he⟩ := ih.mp hz
-        exact ⟨i+1,Nat.succ_le_succ hi,he⟩
-    · rintro ⟨i,hi,he⟩
-      cases i with
-      | zero => exact List.mem_cons.mpr (Or.inl he.symm)
-      | succ i => exact List.mem_cons.mpr (Or.inr (ih.mpr ⟨i,Nat.le_of_succ_le_succ hi,he⟩))
-
-end HoldingWalk
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Deformation.AveragedDecrease
+
+namespace OAI
+
+/-! Labels for bond percolation. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v w
+variable {V : Type u} {E : Type v} {I : Type w}
+section LabeledEscapeExperiment
+
+ 
+
+structure EscapeLabel (G : BondGraph V E) (γ : I → V) (endpoint negative : I) (L m : ℕ) where
+  first : I → V
+  first_mem : first ∈ segmentOrbit G γ
+  last : I → V
+  last_mem : last ∈ segmentOrbit G γ
+  middle : HoldingWalk (openSimpleGraph G Set.univ) L (first endpoint) (last endpoint)
+  leftTail : RawFrom (openSimpleGraph G Set.univ) m (first negative)
+  rightTail : RawFrom (openSimpleGraph G Set.univ) m (last negative)
+
+namespace EscapeLabel
+variable {G : BondGraph V E} {γ : I → V} {ep neg : I} {L m : ℕ}
+
+def reverse (ξ : EscapeLabel G γ ep neg L m) : EscapeLabel G γ ep neg L m where
+  first := ξ.last
+  first_mem := ξ.last_mem
+  last := ξ.first
+  last_mem := ξ.first_mem
+  middle := ξ.middle.reverse
+  leftTail := ξ.rightTail
+  rightTail := ξ.leftTail
+
+theorem reverse_reverse (ξ : EscapeLabel G γ ep neg L m) : ξ.reverse.reverse = ξ := by
+  cases ξ
+  simp only [reverse,HoldingWalk.reverse_reverse]
+
+def reverseEquiv : EscapeLabel G γ ep neg L m ≃ EscapeLabel G γ ep neg L m where
+  toFun := reverse
+  invFun := reverse
+  left_inv := reverse_reverse
+  right_inv := reverse_reverse
+
+noncomputable def mass (P : V → V → ℝ) (ξ : EscapeLabel G γ ep neg L m) : ℝ :=
+  ξ.middle.mass P * ξ.leftTail.2.mass P * ξ.rightTail.2.mass P
+
+theorem mass_nonneg (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y)
+    (ξ : EscapeLabel G γ ep neg L m) : 0 ≤ ξ.mass P :=
+  mul_nonneg (mul_nonneg (ξ.middle.mass_nonneg P hP) (ξ.leftTail.2.mass_nonneg P hP))
+    (ξ.rightTail.2.mass_nonneg P hP)
+
+theorem mass_reverse (P : V → V → ℝ) (hP : ∀ x y, P x y = P y x)
+    (ξ : EscapeLabel G γ ep neg L m) : ξ.reverse.mass P = ξ.mass P := by
+  simp only [mass,reverse,HoldingWalk.mass_reverse P hP]
+  ring
+
+ 
+
+def reverseRootedEquiv (center : I) (x y : V) :
+    {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x ∧ ξ.last center = y} ≃
+    {ξ : EscapeLabel G γ ep neg L m // ξ.first center = y ∧ ξ.last center = x} where
+  toFun ξ := ⟨ξ.1.reverse,ξ.2.2,ξ.2.1⟩
+  invFun ξ := ⟨ξ.1.reverse,ξ.2.2,ξ.2.1⟩
+  left_inv ξ := Subtype.ext (reverse_reverse ξ.1)
+  right_inv ξ := Subtype.ext (reverse_reverse ξ.1)
+
+end EscapeLabel
+end LabeledEscapeExperiment
+end Percolation
+
+ 
+namespace Percolation
+universe u v w
+variable {V : Type u} {E : Type v} {I : Type w}
+section FiniteRootedLabels
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Fintype I]
+
+noncomputable def RootedLabelCode (G : BondGraph V E) (hloc : LocallyFinite G)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
+    (center ep neg : I) (L m : ℕ) (x : V) : Type (max u w) :=
+  Σ s : (segmentAnchors G hloc γ n hdiam center x),
+    Σ mid : RawFrom (openSimpleGraph G Set.univ) L (s.val ep),
+      Σ h : (segmentAnchors G hloc γ n hdiam ep mid.1),
+        RawFrom (openSimpleGraph G Set.univ) m (s.val neg) ×
+          RawFrom (openSimpleGraph G Set.univ) m (h.val neg)
+
+instance finite_rootedLabelCode (G : BondGraph V E) (hloc : LocallyFinite G)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
+    (center ep neg : I) (L m : ℕ) (x : V) :
+    Finite (RootedLabelCode G hloc γ n hdiam center ep neg L m x) := by
+  let (z : V) := finite_openSimpleGraph_neighbors G hloc z
+  unfold RootedLabelCode
+  infer_instance
+
+noncomputable def labelFromCode (G : BondGraph V E) (hloc : LocallyFinite G)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
+    (center ep neg : I) (L m : ℕ) (x : V)
+    (c : RootedLabelCode G hloc γ n hdiam center ep neg L m x) :
+    {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x} := by
+  rcases c with ⟨⟨s,hs⟩,⟨z,mid⟩,⟨h,hh⟩,α,β⟩
+  have hsm := ((mem_segmentAnchors G hloc γ n hdiam center x s).mp hs).1
+  have hsx := ((mem_segmentAnchors G hloc γ n hdiam center x s).mp hs).2
+  have hhm := ((mem_segmentAnchors G hloc γ n hdiam ep z h).mp hh).1
+  have hhz := ((mem_segmentAnchors G hloc γ n hdiam ep z h).mp hh).2
+  subst z
+  exact ⟨⟨s,hsm,h,hhm,mid,α,β⟩,hsx⟩
+
+noncomputable def labelToCode (G : BondGraph V E) (hloc : LocallyFinite G)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
+    (center ep neg : I) (L m : ℕ) (x : V)
+    (ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x}) :
+    RootedLabelCode G hloc γ n hdiam center ep neg L m x :=
+  ⟨⟨ξ.1.first,(mem_segmentAnchors G hloc γ n hdiam center x ξ.1.first).mpr ⟨ξ.1.first_mem,ξ.2⟩⟩,
+    ⟨ξ.1.last ep,ξ.1.middle⟩,
+    ⟨ξ.1.last,(mem_segmentAnchors G hloc γ n hdiam ep (ξ.1.last ep) ξ.1.last).mpr ⟨ξ.1.last_mem,rfl⟩⟩,
+    ξ.1.leftTail,ξ.1.rightTail⟩
+
+noncomputable def rootedLabelEquiv (G : BondGraph V E) (hloc : LocallyFinite G)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
+    (center ep neg : I) (L m : ℕ) (x : V) :
+    {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x} ≃
+      RootedLabelCode G hloc γ n hdiam center ep neg L m x where
+  toFun := labelToCode G hloc γ n hdiam center ep neg L m x
+  invFun := labelFromCode G hloc γ n hdiam center ep neg L m x
+  left_inv ξ := by
+    cases ξ with
+    | mk ξ hξ => cases ξ; rfl
+  right_inv c := by
+    rcases c with ⟨⟨s,hs⟩,⟨z,mid⟩,⟨h,hh⟩,α,β⟩
+    have hh' := (mem_segmentAnchors G hloc γ n hdiam ep z h).mp hh
+    obtain ⟨hhm,hhz⟩ := hh'
+    subst z
+    rfl
+
+theorem finite_rootedEscapeLabels (G : BondGraph V E) (hloc : LocallyFinite G)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
+    (center ep neg : I) (L m : ℕ) (x : V) :
+    Finite {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x} :=
+  Finite.of_equiv _ (rootedLabelEquiv G hloc γ n hdiam center ep neg L m x).symm
+
+end FiniteRootedLabels
+end Percolation
+
+ 
+namespace Percolation
+universe u v w
+variable {V : Type u} {E : Type v} {I : Type w}
+section LabeledMass
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Fintype I]
+
+theorem segmentAnchors_card_on_orbit (G : BondGraph V E) (hloc : LocallyFinite G)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i : I) (x : V)
+    (hx : x ∈ vertexOrbit G (γ i)) :
+    (segmentAnchors G hloc γ n hdiam i x).card = (segmentAnchors G hloc γ n hdiam i (γ i)).card := by
+  obtain ⟨a,b,h,rfl⟩ := hx
+  exact segmentAnchors_card_aut G hloc γ n hdiam i (γ i) h
+
+theorem segmentAnchors_card_eq_indicator (G : BondGraph V E) (hloc : LocallyFinite G)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (i : I) (x : V) :
+    ((segmentAnchors G hloc γ n hdiam i x).card : ℝ) =
+      if x ∈ vertexOrbit G (γ i) then (segmentAnchors G hloc γ n hdiam i (γ i)).card else 0 := by
+  split_ifs with hx
+  · exact congrArg Nat.cast (segmentAnchors_card_on_orbit G hloc γ n hdiam i x hx)
+  · have hh := (segmentAnchors_nonempty_iff G hloc γ n hdiam i x).not.mpr hx
+    rw [Finset.not_nonempty_iff_eq_empty.mp hh]
+    rfl
+
+noncomputable def segmentReturnMass (G : BondGraph V E) (hloc : LocallyFinite G)
+    (a : ℝ) (γ : I → V) (ep : I) (L : ℕ) (x : V) : ℝ :=
+  ∑' mid : RawFrom (openSimpleGraph G Set.univ) L x,
+    if mid.1 ∈ vertexOrbit G (γ ep) then mid.2.mass (walkKernel G hloc a) else 0
+
+theorem labelFromCode_mass (G : BondGraph V E) (hloc : LocallyFinite G)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
+    (center ep neg : I) (L m : ℕ) (x : V) (P : V → V → ℝ)
+    (c : RootedLabelCode G hloc γ n hdiam center ep neg L m x) :
+    (labelFromCode G hloc γ n hdiam center ep neg L m x c).1.mass P =
+      c.2.1.2.mass P * c.2.2.2.1.2.mass P * c.2.2.2.2.2.mass P := by
+  rcases c with ⟨⟨s,hs⟩,⟨z,mid⟩,⟨h,hh⟩,α,β⟩
+  obtain ⟨hhm,hhz⟩ := (mem_segmentAnchors G hloc γ n hdiam ep z h).mp hh
+  subst z
+  rfl
+
+ 
+
+theorem rooted_label_mass_sum (G : BondGraph V E) (hloc : LocallyFinite G)
+    (a : ℝ) (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
+    (center ep neg : I) (L m : ℕ) (x : V) :
+    (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
+      ξ.1.mass (walkKernel G hloc a)) =
+    ∑ s ∈ segmentAnchors G hloc γ n hdiam center x,
+      ((segmentAnchors G hloc γ n hdiam ep (γ ep)).card : ℝ) *
+        segmentReturnMass G hloc a γ ep L (s ep) := by
+  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
+  let := finite_rootedEscapeLabels G hloc γ n hdiam center ep neg L m x
+  let R := rootedLabelEquiv G hloc γ n hdiam center ep neg L m x
+  rw [←R.symm.tsum_eq]
+  simp only [R,rootedLabelEquiv,Equiv.coe_fn_symm_mk,labelFromCode_mass]
+  unfold RootedLabelCode at *
+  rw [(summable_of_hasFiniteSupport (Set.toFinite _)).tsum_sigma]
+  change (∑' s : (segmentAnchors G hloc γ n hdiam center x), _) = _
+  rw [tsum_fintype]
+  conv_rhs => rw [←Finset.sum_coe_sort]
+  apply Finset.sum_congr rfl
+  intro s _
+  rw [(summable_of_hasFiniteSupport (Set.toFinite _)).tsum_sigma]
+  have he (mid : RawFrom (openSimpleGraph G Set.univ) L (s.val ep)) :
+      (∑' h : (segmentAnchors G hloc γ n hdiam ep mid.1),
+        ∑' tails : RawFrom (openSimpleGraph G Set.univ) m (s.val neg) ×
+            RawFrom (openSimpleGraph G Set.univ) m (h.val neg),
+          mid.2.mass (walkKernel G hloc a) * tails.1.2.mass (walkKernel G hloc a) *
+            tails.2.2.mass (walkKernel G hloc a)) =
+      ((segmentAnchors G hloc γ n hdiam ep (γ ep)).card : ℝ) *
+        (if mid.1 ∈ vertexOrbit G (γ ep) then mid.2.mass (walkKernel G hloc a) else 0) := by
+    have ht (h : (segmentAnchors G hloc γ n hdiam ep mid.1)) :
+        (∑' tails : RawFrom (openSimpleGraph G Set.univ) m (s.val neg) ×
+            RawFrom (openSimpleGraph G Set.univ) m (h.val neg),
+          mid.2.mass (walkKernel G hloc a) * tails.1.2.mass (walkKernel G hloc a) *
+            tails.2.2.mass (walkKernel G hloc a)) = mid.2.mass (walkKernel G hloc a) := by
+      rw [(summable_of_hasFiniteSupport (Set.toFinite _)).tsum_prod]
+      simp only [tsum_mul_left,rawFrom_probability_sum G hloc a,mul_one]
+    simp only [ht,tsum_fintype,Finset.sum_const,Finset.card_univ,Fintype.card_coe,nsmul_eq_mul]
+    rw [segmentAnchors_card_eq_indicator G hloc γ n hdiam ep mid.1]
+    split_ifs <;> ring
+  calc
+    _ = ∑' mid : RawFrom (openSimpleGraph G Set.univ) L (s.val ep),
+        ((segmentAnchors G hloc γ n hdiam ep (γ ep)).card : ℝ) *
+          (if mid.1 ∈ vertexOrbit G (γ ep) then mid.2.mass (walkKernel G hloc a) else 0) := by
+      apply tsum_congr
+      intro mid
+      rw [(summable_of_hasFiniteSupport (Set.toFinite _)).tsum_sigma]
+      exact he mid
+    _ = _ := tsum_mul_left
+
+end LabeledMass
+end Percolation
+
+ 
+namespace Percolation
+universe u v w
+variable {V : Type u} {E : Type v} {I : Type w}
+section LabeledNormalization
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Fintype I]
+
+omit [Fintype I] in
+theorem segmentReturnMass_operator (G : BondGraph V E) (hloc : LocallyFinite G)
+    (a : ℝ) (γ : I → V) (ep : I) (L : ℕ) (x : V)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ)) :
+    segmentReturnMass G hloc a γ ep L x =
+      ∑' y, if y ∈ vertexOrbit G (γ ep) then (matrixCoeff (P^L) x y).re else 0 := by
+  have hh := raw_integral_operator G hloc a P hpos hP L x
+    (fun y => if y ∈ vertexOrbit G (γ ep) then 1 else 0)
+  simpa only [RawFrom.integral,mul_ite,mul_one,mul_zero,segmentReturnMass] using hh
+
+omit [Fintype I] in
+theorem segmentReturnMass_aut (G : BondGraph V E) (hloc : LocallyFinite G)
+    (a : ℝ) (γ : I → V) (ep : I) (L : ℕ) (x : V)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    {g : V ≃ V} {b : E ≃ E} (hg : IsAut G g b) :
+    segmentReturnMass G hloc a γ ep L (g x) = segmentReturnMass G hloc a γ ep L x := by
+  rw [segmentReturnMass_operator G hloc a γ ep L (g x) P hpos hP,
+    segmentReturnMass_operator G hloc a γ ep L x P hpos hP,←g.tsum_eq]
+  apply tsum_congr
+  intro y
+  rw [mem_vertexOrbit_aut_iff G (γ ep) hg]
+  have hh := walkPower_invariant G hloc a P hP L g ⟨b,hg⟩ x y
+  rw [hh]
+
+omit [Fintype I] in
+theorem segmentReturnMass_on_orbit (G : BondGraph V E) (hloc : LocallyFinite G)
+    (a : ℝ) (γ : I → V) (ep : I) (L : ℕ) (x : V)
+    (hx : x ∈ vertexOrbit G (γ ep))
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ)) :
+    segmentReturnMass G hloc a γ ep L x = segmentReturnMass G hloc a γ ep L (γ ep) := by
+  obtain ⟨g,b,hg,rfl⟩ := hx
+  exact segmentReturnMass_aut G hloc a γ ep L (γ ep) P hpos hP hg
+
+ 
+
+noncomputable def escapeLabelProbability (G : BondGraph V E) (hloc : LocallyFinite G)
+    (a : ℝ) (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
+    (center ep neg : I) (L m : ℕ) (ξ : EscapeLabel G γ ep neg L m) : ℝ :=
+  ξ.mass (walkKernel G hloc a) /
+    (((segmentAnchors G hloc γ n hdiam center (γ center)).card : ℝ) *
+      ((segmentAnchors G hloc γ n hdiam ep (γ ep)).card : ℝ) *
+      segmentReturnMass G hloc a γ ep L (γ ep))
+
+theorem rooted_label_probability_sum (G : BondGraph V E) (hloc : LocallyFinite G)
+    (a : ℝ) (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
+    (center ep neg : I) (L m : ℕ) (x : V) (hx : x ∈ vertexOrbit G (γ center))
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (hq : segmentReturnMass G hloc a γ ep L (γ ep) ≠ 0) :
+    (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
+      escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1) = 1 := by
+  have hN (i : I) : ((segmentAnchors G hloc γ n hdiam i (γ i)).card : ℝ) ≠ 0 := by
+    apply ne_of_gt
+    exact_mod_cast Finset.card_pos.mpr ((segmentAnchors_nonempty_iff G hloc γ n hdiam i (γ i)).mpr
+      (by exact ⟨Equiv.refl V,Equiv.refl E,(fun e => by simp),rfl⟩))
+  simp only [escapeLabelProbability,tsum_div_const]
+  rw [rooted_label_mass_sum G hloc a γ n hdiam center ep neg L m x]
+  have he (s : I → V) (hs : s ∈ segmentAnchors G hloc γ n hdiam center x) :
+      segmentReturnMass G hloc a γ ep L (s ep) = segmentReturnMass G hloc a γ ep L (γ ep) := by
+    obtain ⟨⟨g,b,hg,rfl⟩,_⟩ := (mem_segmentAnchors G hloc γ n hdiam center x s).mp hs
+    exact segmentReturnMass_aut G hloc a γ ep L (γ ep) P hpos hP hg
+  rw [Finset.sum_congr rfl (fun s hs => congrArg (fun q => _ * q) (he s hs))]
+  simp only [Finset.sum_const,nsmul_eq_mul]
+  rw [segmentAnchors_card_on_orbit G hloc γ n hdiam center x hx,←mul_assoc]
+  exact div_self (mul_ne_zero (mul_ne_zero (hN center) (hN ep)) hq)
+
+theorem escapeLabelProbability_reverse (G : BondGraph V E) (hloc : LocallyFinite G)
+    (a : ℝ) (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
+    (center ep neg : I) (L m : ℕ) (ξ : EscapeLabel G γ ep neg L m) :
+    escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.reverse =
+      escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ := by
+  simp only [escapeLabelProbability,EscapeLabel.mass_reverse _ (walkKernel_symmetric G hloc a)]
+
+end LabeledNormalization
+end Percolation
+
+ 
+namespace Percolation
+universe u v w
+variable {V : Type u} {E : Type v} {I : Type w}
+theorem isAut_refl (G : BondGraph V E) : IsAut G (Equiv.refl V) (Equiv.refl E) := by
+  intro e
+  simp
+
+namespace EscapeLabel
+variable {G : BondGraph V E} {γ : I → V} {ep neg : I} {L m : ℕ}
+
+def autMap {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b)
+    (ξ : EscapeLabel G γ ep neg L m) : EscapeLabel G γ ep neg L m where
+  first := a ∘ ξ.first
+  first_mem := (segmentOrbit_aut_iff G γ ξ.first hab).mpr ξ.first_mem
+  last := a ∘ ξ.last
+  last_mem := (segmentOrbit_aut_iff G γ ξ.last hab).mpr ξ.last_mem
+  middle := ξ.middle.map (autSimpleGraph G hab).toHom
+  leftTail := ⟨a ξ.leftTail.1,ξ.leftTail.2.map (autSimpleGraph G hab).toHom⟩
+  rightTail := ⟨a ξ.rightTail.1,ξ.rightTail.2.map (autSimpleGraph G hab).toHom⟩
+
+theorem autMap_trans {a c : V ≃ V} {b d : E ≃ E} (hab : IsAut G a b) (hcd : IsAut G c d)
+    (ξ : EscapeLabel G γ ep neg L m) :
+    autMap hcd (autMap hab ξ) = autMap (isAut_trans G hab hcd) ξ := by
+  cases ξ
+  dsimp only [autMap,autSimpleGraph]
+  simp only [HoldingWalk.map_map]
+  rfl
+
+theorem autMap_refl (ξ : EscapeLabel G γ ep neg L m) :
+    autMap (isAut_refl G) ξ = ξ := by
+  cases ξ with
+  | mk f hf l hl mid α β =>
+    change EscapeLabel.mk f hf l hl (mid.map SimpleGraph.Hom.id)
+      ⟨α.1,α.2.map SimpleGraph.Hom.id⟩ ⟨β.1,β.2.map SimpleGraph.Hom.id⟩ = _
+    simp only [HoldingWalk.map_id]
+
+theorem autMap_symm {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b)
+    (ξ : EscapeLabel G γ ep neg L m) :
+    autMap (isAut_symm G hab) (autMap hab ξ) = ξ := by
+  rw [autMap_trans]
+  have ha : a.trans a.symm = Equiv.refl V := by ext x; exact a.symm_apply_apply x
+  have hb : b.trans b.symm = Equiv.refl E := by ext x; exact b.symm_apply_apply x
+  generalize hp : isAut_trans G hab (isAut_symm G hab) = h
+  clear hp
+  revert h
+  rw [ha,hb]
+  intro h
+  exact autMap_refl ξ
+
+def autEquiv {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) :
+    EscapeLabel G γ ep neg L m ≃ EscapeLabel G γ ep neg L m where
+  toFun := autMap hab
+  invFun := autMap (isAut_symm G hab)
+  left_inv := autMap_symm hab
+  right_inv := autMap_symm (isAut_symm G hab)
+
+def autRootedEquiv {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (center : I) (x : V) :
+    {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x} ≃
+    {ξ : EscapeLabel G γ ep neg L m // ξ.first center = a x} where
+  toFun ξ := ⟨autMap hab ξ.1,congrArg a ξ.2⟩
+  invFun ξ := ⟨autMap (isAut_symm G hab) ξ.1,by
+    change a.symm (ξ.1.first center) = x
+    rw [ξ.2,a.symm_apply_apply]⟩
+  left_inv ξ := Subtype.ext (autMap_symm hab ξ.1)
+  right_inv ξ := Subtype.ext (autMap_symm (isAut_symm G hab) ξ.1)
+
+theorem mass_autMap (hloc : LocallyFinite G) (a₀ : ℝ)
+    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b)
+    (ξ : EscapeLabel G γ ep neg L m) :
+    (autMap hab ξ).mass (walkKernel G hloc a₀) = ξ.mass (walkKernel G hloc a₀) := by
+  change (ξ.middle.map (autSimpleGraph G hab).toHom).mass (walkKernel G hloc a₀) *
+    (ξ.leftTail.2.map (autSimpleGraph G hab).toHom).mass (walkKernel G hloc a₀) *
+    (ξ.rightTail.2.map (autSimpleGraph G hab).toHom).mass (walkKernel G hloc a₀) = _
+  simp only [HoldingWalk.mass_map (autSimpleGraph G hab).toHom _ _ (walkKernel_covariant G hloc a₀ hab)]
+  rfl
+
+end EscapeLabel
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section LazyOrbitReturn
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem walkKernel_lazy (G : BondGraph V E) (hloc : LocallyFinite G)
+    (D : ℕ) (hD : ∀ x, degree G x ≤ D) (a : ℝ) (ha : 0 ≤ a) (haD : a*D ≤ 1/2)
+    (x : V) : 1/2 ≤ walkKernel G hloc a x x := by
+  have hn : ((neighbors G hloc x).card : ℝ) ≤ D := by
+    exact_mod_cast (neighbors_card_le_degree G hloc x).trans (hD x)
+  have hh := mul_le_mul_of_nonneg_left hn ha
+  simp only [walkKernel,ite_true]
+  split_ifs <;> linarith
+
+theorem positive_lazy_power_step (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
+    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re) (k : ℕ) (x y : V) :
+    (1/2) * (matrixCoeff (P^k) x y).re ≤ (matrixCoeff (P^(k+1)) x y).re := by
+  have hs := hpos.mul_hasSum (P^k) x y
+  have hh := hs.summable.sum_le_tsum ({x} : Finset V)
+    (fun z _ => mul_nonneg (hpos x z).2 ((hpos.pow k) z y).2)
+  simp only [Finset.sum_singleton,hs.tsum_eq,←pow_succ'] at hh
+  exact (mul_le_mul_of_nonneg_right (hdiag x) ((hpos.pow k) x y).2).trans hh
+
+ 
+
+theorem walkPower_orbit_return_lower (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
+    (i₀ : ι) (hw₀ : 0 < weight i₀)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re) (L : ℕ) :
+    (1/2) * (∑ i, if root i ∈ vertexOrbit G (root i₀) then weight i else 0) ≤
+      ∑' y, if y ∈ vertexOrbit G (root i₀) then (matrixCoeff (P^L) (root i₀) y).re else 0 := by
+  have hsum (k : ℕ) : Summable (fun y =>
+      if y ∈ vertexOrbit G (root i₀) then (matrixCoeff (P^k) (root i₀) y).re else 0) := by
+    simpa only [mul_ite,mul_one,mul_zero] using finiteKernel_summable_mul
+      (fun x y => (matrixCoeff (P^k) x y).re) (walkPower_finite_support G hloc a P hpos hP k)
+      (root i₀) (fun y => if y ∈ vertexOrbit G (root i₀) then 1 else 0)
+  obtain ⟨k,hk | hk⟩ := Nat.even_or_odd' L
+  · rw [hk]
+    have hh := walkPower_even_orbit_return G hloc a root weight Γ hΓ htr hw hw1 i₀ hw₀ P hpos hsym hP k
+    have hn : 0 ≤ ∑ i, if root i ∈ vertexOrbit G (root i₀) then weight i else 0 :=
+      Finset.sum_nonneg (fun i _ => by split_ifs <;> first | exact hw i | rfl)
+    linarith
+  · rw [hk]
+    have hh := mul_le_mul_of_nonneg_left
+      (walkPower_even_orbit_return G hloc a root weight Γ hΓ htr hw hw1 i₀ hw₀ P hpos hsym hP k)
+      (show (0:ℝ) ≤ 1/2 by norm_num)
+    apply hh.trans
+    rw [←tsum_mul_left]
+    apply ((hsum (2*k)).mul_left (1/2)).tsum_le_tsum _ (hsum (2*k+1))
+    intro y
+    split_ifs
+    · exact positive_lazy_power_step P hpos hdiag (2*k) (root i₀) y
+    · simp only [mul_zero,le_refl]
+
+end LazyOrbitReturn
+end Percolation
+
+ 
+namespace Percolation
+universe u v w
+variable {V : Type u} {E : Type v} {I : Type w}
+section UniformLabeledLaw
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Fintype I]
+
+omit [Fintype I] in
+ 
+
+theorem segmentReturnMass_uniform_lower (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ g ∈ Γ, g (root i) = x)
+    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
+    (c : ℝ) (hc : 0 < c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re) (γ : I → V) (ep : I) (L : ℕ) :
+    1/(2*c) ≤ segmentReturnMass G hloc a γ ep L (γ ep) := by
+  obtain ⟨i,g,hg,hx⟩ := hcov (γ ep)
+  obtain ⟨b,hb⟩ := hΓ hg
+  have hO := vertexOrbit_eq_of_mem G (root i) (γ ep) ⟨g,b,hb,hx⟩
+  rw [←hx,segmentReturnMass_aut G hloc a γ ep L (root i) P hpos hP hb]
+  rw [segmentReturnMass_operator G hloc a γ ep L (root i) P hpos hP]
+  rw [hO]
+  have hh := walkPower_orbit_return_lower G hloc a root weight Γ hΓ htr
+    (fun i => (hw i).le) hw1 i (hw i) P hpos hsym hP hdiag L
+  have hi : weight i ≤ ∑ r, if root r ∈ vertexOrbit G (root i) then weight r else 0 := by
+    have hx' : root i ∈ vertexOrbit G (root i) := ⟨Equiv.refl _,Equiv.refl _,isAut_refl G,rfl⟩
+    simpa only [ite_eq_left hx'] using Finset.single_le_sum
+      (f := fun r => if root r ∈ vertexOrbit G (root i) then weight r else 0)
+      (fun r _ => by split_ifs <;> first | exact (hw r).le | rfl) (Finset.mem_univ i)
+  apply le_trans _ hh
+  have hct : 0 < 2*c := mul_pos (by norm_num) hc
+  apply (div_le_iff₀ hct).mpr
+  have hh' := mul_le_mul_of_nonneg_left hi hc.le
+  nlinarith [hmin i]
+
+ 
+
+theorem rooted_label_probability_hasSum (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ g ∈ Γ, g (root i) = x)
+    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
+    (c : ℝ) (hc : 0 < c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
+    (center ep neg : I) (L m : ℕ) (x : V) (hx : x ∈ vertexOrbit G (γ center)) :
+    HasSum (fun ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x} =>
+      escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1) 1 := by
+  let := finite_rootedEscapeLabels G hloc γ n hdiam center ep neg L m x
+  have hq := segmentReturnMass_uniform_lower G hloc a root weight Γ hΓ htr hcov hw hw1 c hc hmin
+    P hpos hsym hP hdiag γ ep L
+  have hp : 0 < segmentReturnMass G hloc a γ ep L (γ ep) :=
+    (one_div_pos.mpr (mul_pos (by norm_num) hc)).trans_le hq
+  have hs : Summable (fun ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x} =>
+      escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1) :=
+    summable_of_hasFiniteSupport (Set.toFinite _)
+  rw [←rooted_label_probability_sum G hloc a γ n hdiam center ep neg L m x hx P hpos hP (ne_of_gt hp)]
+  exact hs.hasSum
+
+end UniformLabeledLaw
+end Percolation
+
+ 
+namespace Percolation
+universe u v w
+variable {V : Type u} {E : Type v} {I : Type w}
+section LabeledTransport
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {G : BondGraph V E} {γ : I → V} {ep neg : I} {L m : ℕ}
+
+def labelEndpointEquiv (center : I) (x : V) :
+    (Σ y : V, {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x ∧ ξ.last center = y}) ≃
+      {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x} where
+  toFun ξ := ⟨ξ.2.1,ξ.2.2.1⟩
+  invFun ξ := ⟨ξ.1.last center,ξ.1,ξ.2,rfl⟩
+  left_inv ξ := by rcases ξ with ⟨y,ξ,hx,hy⟩; subst y; rfl
+  right_inv ξ := rfl
+
+def labelAutPairEquiv {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (center : I) (x y : V) :
+    {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x ∧ ξ.last center = y} ≃
+    {ξ : EscapeLabel G γ ep neg L m // ξ.first center = a x ∧ ξ.last center = a y} where
+  toFun ξ := ⟨EscapeLabel.autMap hab ξ.1,congrArg a ξ.2.1,congrArg a ξ.2.2⟩
+  invFun ξ := ⟨EscapeLabel.autMap (isAut_symm G hab) ξ.1,by
+    change a.symm (ξ.1.first center) = x ∧ a.symm (ξ.1.last center) = y
+    rw [ξ.2.1,ξ.2.2,a.symm_apply_apply,a.symm_apply_apply]
+    exact ⟨rfl,rfl⟩⟩
+  left_inv ξ := Subtype.ext (EscapeLabel.autMap_symm hab ξ.1)
+  right_inv ξ := Subtype.ext (EscapeLabel.autMap_symm (isAut_symm G hab) ξ.1)
+
+noncomputable def escapeLabelTransport (center : I) (B : EscapeLabel G γ ep neg L m → ℝ) (x y : V) : ℝ :=
+  ∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x ∧ ξ.last center = y}, B ξ.1
+
+theorem escapeLabelTransport_reverse (center : I) (B : EscapeLabel G γ ep neg L m → ℝ) (x y : V) :
+    escapeLabelTransport center B x y =
+      escapeLabelTransport center (fun ξ => B ξ.reverse) y x := by
+  unfold escapeLabelTransport
+  exact (EscapeLabel.reverseRootedEquiv center y x).tsum_eq _ |>.symm
+
+theorem escapeLabelTransport_aut (center : I) (B : EscapeLabel G γ ep neg L m → ℝ)
+    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b)
+    (hB : ∀ ξ, B (EscapeLabel.autMap hab ξ) = B ξ) (x y : V) :
+    escapeLabelTransport center B (a x) (a y) = escapeLabelTransport center B x y := by
+  unfold escapeLabelTransport
+  rw [←(labelAutPairEquiv hab center x y).tsum_eq]
+  exact tsum_congr (fun ξ => hB ξ.1)
+
+variable [Fintype I]
+
+theorem finite_pair_label_sigma (hloc : LocallyFinite G) (n : ℕ)
+    (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (center : I) (x : V) :
+    Finite (Σ y : V, {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x ∧ ξ.last center = y}) := by
+  let := finite_rootedEscapeLabels G hloc γ n hdiam center ep neg L m x
+  exact Finite.of_equiv _ (labelEndpointEquiv center x).symm
+
+theorem escapeLabelTransport_row_sum (hloc : LocallyFinite G) (n : ℕ)
+    (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (center : I)
+    (B : EscapeLabel G γ ep neg L m → ℝ) (x : V) :
+    (∑' y, escapeLabelTransport center B x y) =
+      ∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x}, B ξ.1 := by
+  let := finite_pair_label_sigma hloc n hdiam center (ep := ep) (neg := neg) (L := L) (m := m) x
+  have hs : Summable (fun ξ : Σ y : V, {ξ : EscapeLabel G γ ep neg L m //
+      ξ.first center = x ∧ ξ.last center = y} => B ξ.2.1) :=
+    summable_of_hasFiniteSupport (Set.toFinite _)
+  unfold escapeLabelTransport
+  rw [←hs.tsum_sigma]
+  exact (labelEndpointEquiv (G := G) (γ := γ) (ep := ep) (neg := neg) (L := L) (m := m) center x).tsum_eq (fun ξ => B ξ.1)
+
+theorem escapeLabelTransport_rowSummable (hloc : LocallyFinite G) (n : ℕ)
+    (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (center : I)
+    (B : EscapeLabel G γ ep neg L m → ℝ) (x : V) :
+    Summable (escapeLabelTransport center B x) := by
+  let := finite_pair_label_sigma hloc n hdiam center (ep := ep) (neg := neg) (L := L) (m := m) x
+  have hs : Summable (fun ξ : Σ y : V, {ξ : EscapeLabel G γ ep neg L m //
+      ξ.first center = x ∧ ξ.last center = y} => B ξ.2.1) :=
+    summable_of_hasFiniteSupport (Set.toFinite _)
+  exact hs.sigma
+
+theorem escapeLabelTransport_columnSummable (hloc : LocallyFinite G) (n : ℕ)
+    (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (center : I)
+    (B : EscapeLabel G γ ep neg L m → ℝ) (y : V) :
+    Summable (fun x => escapeLabelTransport center B x y) := by
+  simp only [escapeLabelTransport_reverse center B]
+  exact escapeLabelTransport_rowSummable hloc n hdiam center (fun ξ => B ξ.reverse) y
+
+end LabeledTransport
+end Percolation
+
+ 
+namespace Percolation
+universe u v w
+variable {V : Type u} {E : Type v} {I : Type w}
+section LabeledReRooting
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {G : BondGraph V E} {γ : I → V} {ep neg : I} {L m : ℕ}
+
+theorem EscapeLabel.first_orbit (ξ : EscapeLabel G γ ep neg L m) (i : I) :
+    ξ.first i ∈ vertexOrbit G (γ i) := by
+  obtain ⟨a,b,h,he⟩ := ξ.first_mem
+  exact ⟨a,b,h,(congrFun he i).symm⟩
+
+theorem EscapeLabel.last_orbit (ξ : EscapeLabel G γ ep neg L m) (i : I) :
+    ξ.last i ∈ vertexOrbit G (γ i) := by
+  obtain ⟨a,b,h,he⟩ := ξ.last_mem
+  exact ⟨a,b,h,(congrFun he i).symm⟩
+
+theorem escapeLabelTransport_zero (center : I) (B : EscapeLabel G γ ep neg L m → ℝ)
+    (x y : V) (hxy : x ∉ vertexOrbit G (γ center) ∨ y ∉ vertexOrbit G (γ center)) :
+    escapeLabelTransport center B x y = 0 := by
+  let : IsEmpty {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x ∧ ξ.last center = y} :=
+    ⟨fun ξ => by
+      rcases hxy with hx | hy
+      · exact hx (ξ.2.1 ▸ ξ.1.first_orbit center)
+      · exact hy (ξ.2.2 ▸ ξ.1.last_orbit center)⟩
+  exact tsum_empty
+
+variable [Fintype I]
+
+ 
+
+theorem rooted_label_weight_reversal (hloc : LocallyFinite G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (center : I)
+    (hθ : 0 < ∑ i, if root i ∈ vertexOrbit G (γ center) then weight i else 0)
+    (B : EscapeLabel G γ ep neg L m → ℝ)
+    (hB : ∀ (a : V ≃ V) (b : E ≃ E) (hab : IsAut G a b) ξ, B (EscapeLabel.autMap hab ξ) = B ξ)
+    (x : V) (hx : x ∈ vertexOrbit G (γ center)) :
+    (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x}, B ξ.1) =
+      ∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x}, B ξ.1.reverse := by
+  let M := escapeLabelTransport center B
+  have hM (a : V ≃ V) (b : E ≃ E) (hab : IsAut G a b) (u v : V) : M (a u) (a v) = M u v :=
+    escapeLabelTransport_aut center B hab (hB a b hab) u v
+  have hr (u : V) : (∑' v, M u v) = if u ∈ vertexOrbit G (γ center) then ∑' v, M (γ center) v else 0 := by
+    split_ifs with hu
+    · obtain ⟨a,b,hab,rfl⟩ := hu
+      rw [←a.tsum_eq]
+      exact tsum_congr (fun v => hM a b hab (γ center) v)
+    · calc
+        _ = ∑' _ : V, (0:ℝ) := tsum_congr (fun v => escapeLabelTransport_zero center B u v (Or.inl hu))
+        _ = 0 := tsum_zero
+  have hc (v : V) : (∑' u, M u v) = if v ∈ vertexOrbit G (γ center) then ∑' u, M u (γ center) else 0 := by
+    split_ifs with hv
+    · obtain ⟨a,b,hab,rfl⟩ := hv
+      rw [←a.tsum_eq]
+      exact tsum_congr (fun u => hM a b hab u (γ center))
+    · calc
+        _ = ∑' _ : V, (0:ℝ) := tsum_congr (fun u => escapeLabelTransport_zero center B u v (Or.inr hv))
+        _ = 0 := tsum_zero
+  have ht := htr.real root weight Γ M
+    (fun a ha u v => by obtain ⟨b,hb⟩ := hΓ ha; exact hM a b hb u v)
+    (escapeLabelTransport_rowSummable hloc n hdiam center B)
+    (escapeLabelTransport_columnSummable hloc n hdiam center B)
+  have hfactor (f : V → ℝ) (q : ℝ) (he : ∀ u, f u = if u ∈ vertexOrbit G (γ center) then q else 0) :
+      (∑ i, weight i*f (root i)) = (∑ i, if root i ∈ vertexOrbit G (γ center) then weight i else 0)*q := by
+    rw [Finset.sum_mul]
+    apply Finset.sum_congr rfl
+    intro i _
+    rw [he]
+    split_ifs <;> simp only [mul_zero,zero_mul]
+  rw [hfactor _ _ hr,hfactor _ _ hc] at ht
+  have hrc := (mul_left_cancel₀ (ne_of_gt hθ)) ht
+  have hxe : (∑' v, M x v) = ∑' u, M u x := by
+    rw [hr x,hc x,ite_eq_left hx,ite_eq_left hx]
+    exact hrc
+  calc
+    _ = ∑' v, M x v := (escapeLabelTransport_row_sum hloc n hdiam center B x).symm
+    _ = ∑' u, M u x := hxe
+    _ = ∑' u, escapeLabelTransport center (fun ξ => B ξ.reverse) x u :=
+      tsum_congr (fun u => escapeLabelTransport_reverse center B u x)
+    _ = _ := escapeLabelTransport_row_sum hloc n hdiam center (fun ξ => B ξ.reverse) x
+
+end LabeledReRooting
+end Percolation
+
+ 
+namespace Percolation
+universe u v w
+variable {V : Type u} {E : Type v} {I : Type w}
+section ExactLabeledReversal
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Fintype I]
+
+theorem escapeLabelProbability_aut (G : BondGraph V E) (hloc : LocallyFinite G)
+    (a₀ : ℝ) (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
+    (center ep neg : I) (L m : ℕ) {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b)
+    (ξ : EscapeLabel G γ ep neg L m) :
+    escapeLabelProbability G hloc a₀ γ n hdiam center ep neg L m (EscapeLabel.autMap hab ξ) =
+      escapeLabelProbability G hloc a₀ γ n hdiam center ep neg L m ξ := by
+  simp only [escapeLabelProbability,EscapeLabel.mass_autMap]
+
+ 
+
+theorem rooted_escape_label_reversal (G : BondGraph V E) (hloc : LocallyFinite G) (a₀ : ℝ)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (center ep neg : I) (L m : ℕ)
+    (hθ : 0 < ∑ i, if root i ∈ vertexOrbit G (γ center) then weight i else 0)
+    (F : EscapeLabel G γ ep neg L m → ℝ)
+    (hF : ∀ (a : V ≃ V) (b : E ≃ E) (hab : IsAut G a b) ξ, F (EscapeLabel.autMap hab ξ) = F ξ)
+    (x : V) (hx : x ∈ vertexOrbit G (γ center)) :
+    (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
+      escapeLabelProbability G hloc a₀ γ n hdiam center ep neg L m ξ.1 * F ξ.1) =
+    ∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
+      escapeLabelProbability G hloc a₀ γ n hdiam center ep neg L m ξ.1 * F ξ.1.reverse := by
+  have hh := rooted_label_weight_reversal hloc root weight Γ hΓ htr n hdiam center hθ
+    (fun ξ => escapeLabelProbability G hloc a₀ γ n hdiam center ep neg L m ξ * F ξ)
+    (fun a b hab ξ => by rw [escapeLabelProbability_aut G hloc a₀ γ n hdiam center ep neg L m hab,hF a b hab]) x hx
+  simpa only [escapeLabelProbability_reverse] using hh
+
+theorem escapeLabelProbability_nonneg (G : BondGraph V E) (hloc : LocallyFinite G)
+    (a : ℝ) (hP : ∀ x y, 0 ≤ walkKernel G hloc a x y)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
+    (center ep neg : I) (L m : ℕ) (ξ : EscapeLabel G γ ep neg L m) :
+    0 ≤ escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ := by
+  have hq : 0 ≤ segmentReturnMass G hloc a γ ep L (γ ep) := by
+    apply tsum_nonneg
+    intro mid
+    split_ifs
+    · exact mid.2.mass_nonneg _ hP
+    · rfl
+  exact div_nonneg (ξ.mass_nonneg _ hP) (mul_nonneg (mul_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _)) hq)
+
+end ExactLabeledReversal
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section LongGeodesics
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem exists_geodesic_length (G : BondGraph V E) [Infinite V]
+    (hloc : LocallyFinite G) (hconn : Connected G) (x : V) (N : ℕ) :
+    ∃ y : V, ∃ q : (openSimpleGraph G Set.univ).Walk x y,
+      q.length = N ∧ q.length = (openSimpleGraph G Set.univ).dist x y := by
+  obtain ⟨z,hz⟩ := (finite_reachBall hloc x N).exists_notMem
+  have hc := (conn_iff_reachable G Set.univ x z).mp (conn_univ_of_connected G hconn x z)
+  obtain ⟨q,hq⟩ := hc.exists_walk_length_eq_dist
+  have hN : N < q.length := by
+    by_contra hh
+    have hm := walk_mem_reachBall G Set.univ q 0 (show x ∈ reachBall G x 0 from rfl)
+    have hm' : z ∈ reachBall G x q.length := by simpa only [Nat.zero_add] using hm
+    exact hz (reachBall_mono_local G x (Nat.le_of_not_gt hh) hm')
+  refine ⟨q.getVert N,q.take N,?_,?_⟩
+  · simp only [SimpleGraph.Walk.take_length,inf_eq_left.mpr hN.le]
+  · exact SimpleGraph.length_eq_dist_of_subwalk hq (q.isSubwalk_take N)
+
+ 
+
+theorem geodesic_getVert_dist (H : SimpleGraph V) {x y : V} (q : H.Walk x y)
+    (hq : q.length = H.dist x y) (i j : ℕ) (hij : i ≤ j) (hj : j ≤ q.length) :
+    H.dist (q.getVert i) (q.getVert j) = j-i := by
+  have hs := SimpleGraph.length_eq_dist_of_subwalk hq
+    (((q.drop i).isSubwalk_take (j-i)).trans (q.isSubwalk_drop i))
+  have he : (q.drop i).getVert (j-i) = q.getVert j := by
+    rw [SimpleGraph.Walk.drop_getVert,Nat.add_sub_of_le hij]
+  rw [SimpleGraph.Walk.take_length,SimpleGraph.Walk.drop_length,he,
+    min_eq_left (Nat.sub_le_sub_right hj i)] at hs
+  exact hs.symm
+
+end LongGeodesics
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+namespace HoldingWalk
+variable {H : SimpleGraph V}
+
+ 
+
+def append : {k l : ℕ} → {x y z : V} → HoldingWalk H k x y → HoldingWalk H l y z → HoldingWalk H (l+k) x z
+  | _,_,_,_,_,.nil _,v => v
+  | _,_,_,_,_,.hold w,v => .hold (w.append v)
+  | _,_,_,_,_,.step h w,v => .step h (w.append v)
+
+def ofWalk : {x y : V} → (w : H.Walk x y) → HoldingWalk H w.length x y
+  | _,_,.nil => .nil _
+  | _,_,.cons h w => .step h (ofWalk w)
+
+theorem mass_append (P : V → V → ℝ) {k l : ℕ} {x y z : V}
+    (w : HoldingWalk H k x y) (v : HoldingWalk H l y z) :
+    (w.append v).mass P = w.mass P*v.mass P := by
+  induction w with
+  | nil => simp only [append,mass,one_mul]
+  | hold w ih => simp only [append,mass,ih,mul_assoc]
+  | step h w ih => simp only [append,mass,ih,mul_assoc]
+
+theorem mass_ofWalk (P : V → V → ℝ) (a : ℝ) (hP : ∀ x y, H.Adj x y → P x y = a)
+    {x y : V} (w : H.Walk x y) : (ofWalk w).mass P = a^w.length := by
+  induction w with
+  | nil => rfl
+  | @cons x z y h w ih => simp only [ofWalk,mass,hP x z h,ih,SimpleGraph.Walk.length_cons,pow_succ'];
+
+theorem ofWalk_vertices {x y : V} (w : H.Walk x y) : (ofWalk w).vertices = w.support := by
+  induction w with
+  | nil => rfl
+  | cons h w ih => simp only [ofWalk,vertices,SimpleGraph.Walk.support_cons,ih]
+
+theorem append_vertices {k l : ℕ} {x y z : V}
+    (w : HoldingWalk H k x y) (v : HoldingWalk H l y z) :
+    (w.append v).vertices = w.vertices.dropLast ++ v.vertices := by
+  induction w with
+  | nil => rfl
+  | hold w ih =>
+    have hn : w.vertices ≠ [] := by cases w <;> simp [vertices]
+    simp only [append,vertices,ih,List.dropLast_cons_of_ne_nil hn,List.cons_append]
+  | step h w ih =>
+    have hn : w.vertices ≠ [] := by cases w <;> simp [vertices]
+    simp only [append,vertices,ih,List.dropLast_cons_of_ne_nil hn,List.cons_append]
+
+end HoldingWalk
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section GeodesicSegments
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+def geodesicSegment (G : BondGraph V E) {x y : V}
+    (q : (openSimpleGraph G Set.univ).Walk x y) (N : ℕ) : Fin (N+1) → V :=
+  fun i => q.getVert i
+
+theorem geodesicSegment_adj (G : BondGraph V E) {x y : V}
+    (q : (openSimpleGraph G Set.univ).Walk x y) (N : ℕ) (hN : q.length = N)
+    (i : ℕ) (hi : i < N) :
+    (openSimpleGraph G Set.univ).Adj
+      (geodesicSegment G q N ⟨i,by omega⟩) (geodesicSegment G q N ⟨i+1,by omega⟩) := by
+  change (openSimpleGraph G Set.univ).Adj (q.getVert i) (q.getVert (i+1))
+  exact q.adj_getVert_succ (by omega)
+
+theorem geodesicSegment_injective (G : BondGraph V E) {x y : V}
+    (q : (openSimpleGraph G Set.univ).Walk x y) (N : ℕ) (hN : q.length = N)
+    (hq : q.length = (openSimpleGraph G Set.univ).dist x y) :
+    Function.Injective (geodesicSegment G q N) := by
+  intro i j hh
+  apply Fin.ext
+  exact (q.isPath_of_length_eq_dist hq).getVert_injOn
+    (by change i.val ≤ q.length; omega) (by change j.val ≤ q.length; omega) hh
+
+theorem geodesicSegment_diameter (G : BondGraph V E) {x y : V}
+    (q : (openSimpleGraph G Set.univ).Walk x y) (N : ℕ) (hN : q.length = N) :
+    ∀ i j, geodesicSegment G q N j ∈ reachBall G (geodesicSegment G q N i) N := by
+  intro i j
+  have hc : (openSimpleGraph G Set.univ).Reachable (q.getVert i) (q.getVert j) :=
+    (q.take i).reachable.symm.trans (q.take j).reachable
+  obtain ⟨r,hr⟩ := hc.exists_walk_length_eq_dist
+  have hd : (openSimpleGraph G Set.univ).dist (q.getVert i) (q.getVert j) ≤ N := by
+    rcases le_total i.val j.val with hij | hji
+    · have hle := SimpleGraph.dist_le ((q.drop i).take (j-i))
+      rw [SimpleGraph.Walk.take_length,SimpleGraph.Walk.drop_length,
+        SimpleGraph.Walk.drop_getVert,Nat.add_sub_of_le hij] at hle
+      exact hle.trans (by omega)
+    · have hle := SimpleGraph.dist_le ((q.drop j).take (i-j))
+      rw [SimpleGraph.Walk.take_length,SimpleGraph.Walk.drop_length,
+        SimpleGraph.Walk.drop_getVert,Nat.add_sub_of_le hji] at hle
+      rw [SimpleGraph.dist_comm] at hle
+      exact hle.trans (by omega)
+  have hm := walk_mem_reachBall G Set.univ r 0 (show q.getVert i ∈ reachBall G (q.getVert i) 0 from rfl)
+  apply reachBall_mono_local G (q.getVert i) (a := r.length) (by omega)
+  simpa only [Nat.zero_add, geodesicSegment] using hm
+
+theorem segmentOrbit_adj (G : BondGraph V E) {I : Type*} (γ s : I → V)
+    (hs : s ∈ segmentOrbit G γ) (i j : I) (hij : (openSimpleGraph G Set.univ).Adj (γ i) (γ j)) :
+    (openSimpleGraph G Set.univ).Adj (s i) (s j) := by
+  obtain ⟨a,b,hab,rfl⟩ := hs
+  exact (autSimpleGraph G hab).map_rel_iff.mpr hij
+
+theorem walkKernel_step (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    {x y : V} (h : (openSimpleGraph G Set.univ).Adj x y) : walkKernel G hloc a x y = a := by
+  have hh := (openSimpleGraph_univ_adj G x y).mp h
+  simp only [walkKernel,ite_eq_right hh.1,ite_eq_left hh.2,zero_add]
+
+theorem walkKernel_ofWalk_mass (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    {x y : V} (q : (openSimpleGraph G Set.univ).Walk x y) :
+    (HoldingWalk.ofWalk q).mass (walkKernel G hloc a) = a^q.length :=
+  HoldingWalk.mass_ofWalk _ a (fun _ _ h => walkKernel_step G hloc a h) q
+
+end GeodesicSegments
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section ActualTailPotentials
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem raw_tail_twoPoint_expectation (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (P A : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (p : unitInterval) (hA : HasKernel G p A) (m : ℕ) (x b : V) :
+    (∑' w : RawFrom (openSimpleGraph G Set.univ) m x,
+      w.2.mass (walkKernel G hloc a) * twoPoint G p w.1 b) =
+      (matrixCoeff (P^m*A) x b).re := by
+  have he : (fun y => twoPoint G p y b) = (fun y => (matrixCoeff A y b).re) := by
+    funext y
+    rw [matrixCoeff,hA y b,Complex.ofReal_re]
+  change RawFrom.integral (H := openSimpleGraph G Set.univ) (walkKernel G hloc a) m x (twoPoint G p · b) = _
+  rw [he,raw_integral_operator G hloc a P hpos hP]
+  exact ((hpos.pow m).mul_hasSum A x b).tsum_eq
+
+ 
+
+theorem raw_tail_potential_bound [Countable V] [Countable E] [Nonempty V]
+    {ι : Type*} [Fintype ι]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (root : ι → V) (weight : ι → ℝ) (htr : RootTransport root weight (vertexAut G))
+    (hcov : ∀ x, ∃ i, ∃ e ∈ vertexAut G, e (root i) = x)
+    (hw : ∀ i, 0 ≤ weight i) (hsum : ∑ i, weight i = 1)
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c * weight i)
+    (P A : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (p : unitInterval) (hA : HasKernel G p A) (m : ℕ) (x : V) (S : Finset V) :
+    (∑' w : RawFrom (openSimpleGraph G Set.univ) m x,
+      w.2.mass (walkKernel G hloc a) * ∑ b ∈ S, twoPoint G p w.1 b) ≤
+      (S.card:ℝ) * Real.sqrt c * ‖P‖^m := by
+  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
+  let := Fintype.ofFinite (RawFrom (openSimpleGraph G Set.univ) m x)
+  have hPinv : KernelInvariant (vertexAut G) P := by
+    intro e he x y
+    obtain ⟨f,hf⟩ := he
+    rw [hP,hP,walkKernel_covariant G hloc a hf]
+  calc
+    _ = ∑ b ∈ S, (∑' w : RawFrom (openSimpleGraph G Set.univ) m x,
+        w.2.mass (walkKernel G hloc a) * twoPoint G p w.1 b) := by
+      simp only [tsum_fintype,Finset.mul_sum]
+      exact Finset.sum_comm
+    _ = ∑ b ∈ S, (matrixCoeff (P^m*A) x b).re := by
+      apply Finset.sum_congr rfl
+      intro b _
+      exact raw_tail_twoPoint_expectation G hloc a P A hpos hP p hA m x b
+    _ ≤ ∑ _b ∈ S, Real.sqrt c * ‖P‖^m := by
+      apply Finset.sum_le_sum
+      intro b _
+      exact (Complex.re_le_norm _).trans (twoPoint_operator_smoothing G p root weight htr hcov hw hsum c hc hmin A P hA hPinv m x b)
+    _ = _ := by simp only [Finset.sum_const,nsmul_eq_mul,mul_assoc]
+
+end ActualTailPotentials
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section AuxiliaryTailMarkov
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem finite_weighted_markov {α : Type*} [Fintype α] (μ F : α → ℝ)
+    (hμ : ∀ x, 0 ≤ μ x) (hF : ∀ x, 0 ≤ F x) (t : ℝ) (ht : 0 < t) :
+    (∑ x, if t < F x then μ x else 0) ≤ (∑ x, μ x*F x)/t := by
+  apply (le_div_iff₀ ht).mpr
+  rw [Finset.sum_mul]
+  apply Finset.sum_le_sum
+  intro x _
+  split_ifs with hx
+  · exact mul_le_mul_of_nonneg_left hx.le (hμ x)
+  · exact (by simpa only [zero_mul] using mul_nonneg (hμ x) (hF x))
+
+theorem raw_tail_potential_failure [Countable V] [Countable E] [Nonempty V]
+    {ι : Type*} [Fintype ι]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (root : ι → V) (weight : ι → ℝ) (htr : RootTransport root weight (vertexAut G))
+    (hcov : ∀ x, ∃ i, ∃ e ∈ vertexAut G, e (root i) = x)
+    (hw : ∀ i, 0 ≤ weight i) (hsum : ∑ i, weight i = 1)
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c * weight i)
+    (P A : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (p : unitInterval) (hA : HasKernel G p A) (m : ℕ) (x : V) (S : Finset V)
+    (t : ℝ) (ht : 0 < t) :
+    (∑' w : RawFrom (openSimpleGraph G Set.univ) m x,
+      if t < ∑ b ∈ S, twoPoint G p w.1 b then w.2.mass (walkKernel G hloc a) else 0) ≤
+      ((S.card:ℝ) * Real.sqrt c * ‖P‖^m)/t := by
+  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
+  let := Fintype.ofFinite (RawFrom (openSimpleGraph G Set.univ) m x)
+  have hK (x y : V) : 0 ≤ walkKernel G hloc a x y := by
+    simpa only [hP,Complex.ofReal_re] using (hpos x y).2
+  have hM := finite_weighted_markov
+    (fun w : RawFrom (openSimpleGraph G Set.univ) m x => w.2.mass (walkKernel G hloc a))
+    (fun w => ∑ b ∈ S, twoPoint G p w.1 b)
+    (fun w => w.2.mass_nonneg _ hK)
+    (fun w => Finset.sum_nonneg (fun b _ => twoPoint_nonneg G p w.1 b)) t ht
+  have hB := raw_tail_potential_bound G hloc a root weight htr hcov hw hsum c hc hmin P A hpos hP p hA m x S
+  simp only [tsum_fintype] at hB ⊢
+  exact hM.trans (div_le_div_of_nonneg_right hB ht.le)
+
+ 
+
+theorem finite_product_failure_union {α β : Type*} [Fintype α] [Fintype β]
+    (μ : α → ℝ) (ν : β → ℝ) (hμ : ∀ x, 0 ≤ μ x) (hν : ∀ y, 0 ≤ ν y)
+    (hμ1 : ∑ x, μ x = 1) (hν1 : ∑ y, ν y = 1) (B : α → Prop) (C : β → Prop) :
+    (∑ z : α × β, if B z.1 ∨ C z.2 then μ z.1*ν z.2 else 0) ≤
+      (∑ x, if B x then μ x else 0) + ∑ y, if C y then ν y else 0 := by
+  calc
+    _ ≤ ∑ z : α × β, ((if B z.1 then μ z.1 else 0)*ν z.2 + μ z.1*(if C z.2 then ν z.2 else 0)) := by
+      apply Finset.sum_le_sum
+      intro z _
+      by_cases hB : B z.1 <;> by_cases hC : C z.2 <;>
+        simp [hB,hC]
+      exact mul_nonneg (hμ _) (hν _)
+    _ = _ := by
+      rw [Fintype.sum_prod_type]
+      simp only [Finset.sum_add_distrib,←Finset.mul_sum,←Finset.sum_mul,hμ1,hν1,mul_one,one_mul]
+
+end AuxiliaryTailMarkov
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section IndependentTailPotentialTests
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem two_raw_tail_potential_failure [Countable V] [Countable E] [Nonempty V]
+    {ι : Type*} [Fintype ι]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (root : ι → V) (weight : ι → ℝ) (htr : RootTransport root weight (vertexAut G))
+    (hcov : ∀ x, ∃ i, ∃ e ∈ vertexAut G, e (root i) = x)
+    (hw : ∀ i, 0 ≤ weight i) (hsum : ∑ i, weight i = 1)
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c * weight i)
+    (P A : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (p : unitInterval) (hA : HasKernel G p A) (m : ℕ) (x y : V) (S : Finset V)
+    (t : ℝ) (ht : 0 < t) :
+    (∑' w : RawFrom (openSimpleGraph G Set.univ) m x × RawFrom (openSimpleGraph G Set.univ) m y,
+      if (t < ∑ b ∈ S, twoPoint G p w.1.1 b) ∨ (t < ∑ b ∈ S, twoPoint G p w.2.1 b)
+      then w.1.2.mass (walkKernel G hloc a) * w.2.2.mass (walkKernel G hloc a) else 0) ≤
+      2*((S.card:ℝ) * Real.sqrt c * ‖P‖^m)/t := by
+  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
+  let := Fintype.ofFinite (RawFrom (openSimpleGraph G Set.univ) m x)
+  let := Fintype.ofFinite (RawFrom (openSimpleGraph G Set.univ) m y)
+  have hK (u v : V) : 0 ≤ walkKernel G hloc a u v := by
+    simpa only [hP,Complex.ofReal_re] using (hpos u v).2
+  have hμ := rawFrom_probability_sum G hloc a m x
+  have hν := rawFrom_probability_sum G hloc a m y
+  simp only [tsum_fintype] at hμ hν
+  have hU := finite_product_failure_union
+    (fun w : RawFrom (openSimpleGraph G Set.univ) m x => w.2.mass (walkKernel G hloc a))
+    (fun w : RawFrom (openSimpleGraph G Set.univ) m y => w.2.mass (walkKernel G hloc a))
+    (fun w => w.2.mass_nonneg _ hK) (fun w => w.2.mass_nonneg _ hK) hμ hν
+    (fun w => t < ∑ b ∈ S, twoPoint G p w.1 b)
+    (fun w => t < ∑ b ∈ S, twoPoint G p w.1 b)
+  have hx := raw_tail_potential_failure G hloc a root weight htr hcov hw hsum c hc hmin P A hpos hP p hA m x S t ht
+  have hy := raw_tail_potential_failure G hloc a root weight htr hcov hw hsum c hc hmin P A hpos hP p hA m y S t ht
+  simp only [tsum_fintype] at hx hy ⊢
+  exact hU.trans ((add_le_add hx hy).trans_eq (by ring))
+
+end IndependentTailPotentialTests
+end Percolation
+
+ 
+namespace Percolation
+section LogarithmicTailLength
+
+ 
+
+theorem logarithmic_tail_power (ρ X : ℝ) (hρ : 0 < ρ) (hρ1 : ρ < 1) (hX : 0 < X) :
+    ρ ^ ⌈(2/(-Real.log ρ))*Real.log X⌉₊ ≤ X^(-2:ℝ) := by
+  have hl : Real.log ρ ≠ 0 := ne_of_lt (Real.log_neg hρ hρ1)
+  calc
+    _ = ρ ^ (⌈(2/(-Real.log ρ))*Real.log X⌉₊ : ℝ) := (Real.rpow_natCast _ _).symm
+    _ ≤ ρ ^ ((2/(-Real.log ρ))*Real.log X) :=
+      Real.rpow_le_rpow_of_exponent_ge hρ hρ1.le (Nat.le_ceil _)
+    _ = X^(-2:ℝ) := by
+      rw [Real.rpow_def_of_pos hρ,Real.rpow_def_of_pos hX]
+      congr 1
+      field_simp [hl]
+
+ 
+
+theorem exists_uniform_logarithmic_tail_cutoff (ρ A : ℝ)
+    (hρ : 0 < ρ) (hρ1 : ρ < 1) (hA : 0 ≤ A) (kmin : ℕ) :
+    ∃ C₁ : ℝ, 0 < C₁ ∧ 1 < (-Real.log ρ)*C₁ ∧
+      ∃ k₀ : ℕ, kmin ≤ k₀ ∧ ∀ k : ℕ, k₀ ≤ k → ∀ R : ℝ, 1 ≤ R →
+        A*((k:ℝ)+1)*R*ρ^⌈C₁*Real.log (((k:ℝ)+1)*R)⌉₊ ≤ 1/4 := by
+  have hlambda : 0 < -Real.log ρ := neg_pos.mpr (Real.log_neg hρ hρ1)
+  refine ⟨2/(-Real.log ρ),div_pos (by norm_num) hlambda,?_,max kmin ⌈4*A⌉₊,le_max_left _ _,?_⟩
+  · rw [mul_div_cancel₀ _ (ne_of_gt hlambda)]
+    norm_num
+  · intro k hk R hR
+    have hk' : ⌈4*A⌉₊ ≤ k := (le_max_right _ _).trans hk
+    have hAk : 4*A ≤ (k:ℝ) := (Nat.le_ceil (4*A)).trans (by exact_mod_cast hk')
+    have hkpos : 0 < (k:ℝ)+1 := by positivity
+    have hRp : 0 < R := lt_of_lt_of_le (by norm_num) hR
+    let X : ℝ := ((k:ℝ)+1)*R
+    have hX : 0 < X := mul_pos hkpos hRp
+    have hXk : (k:ℝ)+1 ≤ X := by dsimp [X]; exact le_mul_of_one_le_right hkpos.le hR
+    have hp := logarithmic_tail_power ρ X hρ hρ1 hX
+    have hneg : X^(-2:ℝ) = (X^2)⁻¹ := by rw [Real.rpow_neg,show (2:ℝ) = (2:ℕ) by norm_num,Real.rpow_natCast]; exact hX.le
+    calc
+      _ = A*X*ρ^⌈(2/(-Real.log ρ))*Real.log X⌉₊ := by dsimp [X]; ring
+      _ ≤ A*X*X^(-2:ℝ) := mul_le_mul_of_nonneg_left hp (mul_nonneg hA hX.le)
+      _ = A/X := by rw [hneg]; field_simp [ne_of_gt hX]
+      _ ≤ 1/4 := (div_le_iff₀ hX).mpr (by linarith)
+
+end LogarithmicTailLength
+end Percolation
+
+ 
+namespace Percolation
+universe u
+variable {V : Type u}
+namespace HoldingWalk
+variable {H : SimpleGraph V}
+
+ 
+
+def position : {k : ℕ} → {x y : V} → HoldingWalk H k x y → ℕ → V
+  | _,x,_,.nil _,_ => x
+  | _,x,_,.hold _,0 => x
+  | _,_,_,.hold w,n+1 => w.position n
+  | _,x,_,.step _ _,0 => x
+  | _,_,_,.step _ w,n+1 => w.position n
+
+@[simp] theorem position_zero {k : ℕ} {x y : V} (w : HoldingWalk H k x y) : w.position 0 = x := by
+  cases w <;> rfl
+
+@[simp] theorem position_length {k : ℕ} {x y : V} (w : HoldingWalk H k x y) : w.position k = y := by
+  induction w with
+  | nil => rfl
+  | hold w ih => exact ih
+  | step h w ih => exact ih
+
+theorem position_mem_vertices {k : ℕ} {x y : V} (w : HoldingWalk H k x y) (i : ℕ) :
+    w.position i ∈ w.vertices := by
+  induction w generalizing i with
+  | nil => simp only [position,vertices,List.mem_singleton]
+  | hold w ih => cases i <;> simp only [position,vertices,List.mem_cons,true_or,ih,or_true]
+  | step h w ih => cases i <;> simp only [position,vertices,List.mem_cons,true_or,ih,or_true]
+
+theorem mem_vertices_iff_position {k : ℕ} {x y z : V} (w : HoldingWalk H k x y) :
+    z ∈ w.vertices ↔ ∃ i : ℕ, i ≤ k ∧ w.position i = z := by
+  induction w with
+  | nil => simp only [vertices,List.mem_singleton,Nat.le_zero,exists_eq_left,position,eq_comm]
+  | @hold k x y w ih =>
+    constructor
+    · intro hz
+      rcases List.mem_cons.mp hz with rfl | hz
+      · exact ⟨0,Nat.zero_le _,rfl⟩
+      · obtain ⟨i,hi,he⟩ := ih.mp hz
+        exact ⟨i+1,Nat.succ_le_succ hi,he⟩
+    · rintro ⟨i,hi,he⟩
+      cases i with
+      | zero => exact List.mem_cons.mpr (Or.inl he.symm)
+      | succ i => exact List.mem_cons.mpr (Or.inr (ih.mpr ⟨i,Nat.le_of_succ_le_succ hi,he⟩))
+  | @step k x u y h w ih =>
+    constructor
+    · intro hz
+      rcases List.mem_cons.mp hz with rfl | hz
+      · exact ⟨0,Nat.zero_le _,rfl⟩
+      · obtain ⟨i,hi,he⟩ := ih.mp hz
+        exact ⟨i+1,Nat.succ_le_succ hi,he⟩
+    · rintro ⟨i,hi,he⟩
+      cases i with
+      | zero => exact List.mem_cons.mpr (Or.inl he.symm)
+      | succ i => exact List.mem_cons.mpr (Or.inr (ih.mpr ⟨i,Nat.le_of_succ_le_succ hi,he⟩))
+
+end HoldingWalk
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Escape/PathSelection.lean b/lean/OAI/Probability/UnimodularPercolation/Escape/PathSelection.lean
index b1756ad27bcf378bce02258d541be95f1a50056d..696641b6be8b3458a3b55c67a0a55941b7d99b28 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Escape/PathSelection.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Escape/PathSelection.lean
@@ -1,563 +1,563 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Escape.SelectionBounds
-
-namespace OAI
-
-/-! Selection of geometrically separated paths from labelled escape events. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-section EscapeGeometry
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-namespace HoldingWalk
-variable {H : SimpleGraph V}
-
-theorem mem_vertices_append {k l : ℕ} {x y z b : V}
-    (w : HoldingWalk H k x y) (v : HoldingWalk H l y z) :
-    b ∈ (w.append v).vertices ↔ b ∈ w.vertices ∨ b ∈ v.vertices := by
-  induction w with
-  | nil x =>
-    have hx : x ∈ v.vertices := by simpa using v.position_mem_vertices 0
-    simp only [append, vertices, List.mem_singleton]
-    exact ⟨fun h => Or.inr h, fun h => h.elim (fun he => he ▸ hx) id⟩
-  | hold w ih => simp only [append, vertices, List.mem_cons, ih, or_assoc]
-  | step h w ih => simp only [append, vertices, List.mem_cons, ih, or_assoc]
-
-theorem mem_vertices_fromPositions (k : ℕ) (f : Fin (k+1) → V)
-    (h : ∀ i : Fin k, H.Adj (f i.castSucc) (f i.succ)) (b : V) :
-    b ∈ (fromPositions k f h).vertices ↔ ∃ i, f i = b := by
-  rw [mem_vertices_iff_position]
-  constructor
-  · rintro ⟨i, hi, he⟩
-    exact ⟨⟨i,by omega⟩, (position_fromPositions k f h i hi).symm.trans he⟩
-  · rintro ⟨i,hi⟩
-    exact ⟨i.val,by omega,(position_fromPositions k f h i.val (by omega)).trans hi⟩
-
-end HoldingWalk
-
- 
-def negativeStem (H : SimpleGraph V) (J : ℕ) (s : Fin (J+J+1) → V)
-    (hs : ∀ i : Fin (J+J), H.Adj (s i.castSucc) (s i.succ)) :
-    HoldingWalk H J (s (segmentCenter J)) (s 0) :=
-  (HoldingWalk.fromPositions J (fun i => s ⟨i.val,by omega⟩)
-    (fun i => hs ⟨i.val,by omega⟩)).reverse
-
-theorem positiveStem_vertices (H : SimpleGraph V) (J : ℕ) (s : Fin (J+J+1) → V)
-    (hs : ∀ i : Fin (J+J), H.Adj (s i.castSucc) (s i.succ)) (z : V) :
-    z ∈ (positiveStem H J s hs).vertices ↔ ∃ i : Fin (J+1), s ⟨J+i.val,by omega⟩ = z := by
-  exact HoldingWalk.mem_vertices_fromPositions J _ _ z
-
-theorem negativeStem_vertices (H : SimpleGraph V) (J : ℕ) (s : Fin (J+J+1) → V)
-    (hs : ∀ i : Fin (J+J), H.Adj (s i.castSucc) (s i.succ)) (z : V) :
-    z ∈ (negativeStem H J s hs).vertices ↔ ∃ i : Fin (J+1), s ⟨i.val,by omega⟩ = z := by
-  let w := HoldingWalk.fromPositions J (fun i : Fin (J+1) => s ⟨i.val,by omega⟩)
-    (fun i : Fin J => hs ⟨i.val,by omega⟩)
-  have he : z ∈ w.reverse.vertices ↔ z ∈ w.vertices := by
-    rw [HoldingWalk.vertices_reverse, List.mem_reverse]
-  exact he.trans (HoldingWalk.mem_vertices_fromPositions J _ _ z)
-
- 
-def labelLeftEscape (G : BondGraph V E) (J L m : ℕ) (γ : Fin (J+J+1) → V)
-    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
-    (ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m) :
-    HoldingWalk (openSimpleGraph G Set.univ) (m+J)
-      (ξ.first (segmentCenter J)) ξ.leftTail.1 :=
-  (negativeStem _ J ξ.first (orbitSegment_consecutive G J γ ξ.first hγ ξ.first_mem)).append ξ.leftTail.2
-
- 
-def labelCorridor (G : BondGraph V E) (J L m : ℕ) (γ : Fin (J+J+1) → V)
-    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
-    (ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m) : RawCorridor G ((J+L)+J) :=
-  ⟨ξ.first (segmentCenter J), ξ.last (segmentCenter J), escapeLabelRaw G J L m γ hγ ξ⟩
-
-def segmentPositiveIndex (J : ℕ) : Fin (J+1) → Fin (J+J+1) :=
-  fun i => ⟨J+i.val,by omega⟩
-def segmentNegativeIndex (J : ℕ) : Fin (J+1) → Fin (J+J+1) :=
-  fun i => ⟨i.val,by omega⟩
-
-theorem segmentOrbit_injective {I : Type*} (G : BondGraph V E) (γ s : I → V)
-    (hγ : Function.Injective γ) (hs : s ∈ segmentOrbit G γ) : Function.Injective s := by
-  obtain ⟨a,b,hab,rfl⟩ := hs
-  exact a.injective.comp hγ
-
- 
-
-theorem labelLeftEscape_raw_intersection (G : BondGraph V E) (J L m : ℕ)
-    (γ : Fin (J+J+1) → V) (hinj : Function.Injective γ)
-    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
-    (ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m)
-    (hgood : ¬labelFirstEscapeCollision G γ (Fin.last (J+J)) 0 J L m
-      (segmentPositiveIndex J) (segmentNegativeIndex J) (segmentPositiveIndex J) ξ)
-    (z : V) (hz : z ∈ (labelLeftEscape G J L m γ hγ ξ).vertices)
-    (hr : z ∈ (escapeLabelRaw G J L m γ hγ ξ).vertices) :
-    z = ξ.first (segmentCenter J) := by
-  have hfirst := segmentOrbit_injective G γ ξ.first hinj ξ.first_mem
-  have hfail := hgood
-  unfold labelFirstEscapeCollision firstEscapeCollision at hfail
-  have hraw : (∃ i : Fin (J+1), ξ.first (segmentPositiveIndex J i) = z) ∨
-      z ∈ ξ.middle.vertices ∨ (∃ i : Fin (J+1), ξ.last (segmentPositiveIndex J i) = z) := by
-    simpa only [escapeLabelRaw, HoldingWalk.mem_vertices_append,
-      HoldingWalk.vertices_reverse, List.mem_reverse, positiveStem_vertices,
-      segmentPositiveIndex] using hr
-  have hleft : (∃ i : Fin (J+1), ξ.first (segmentNegativeIndex J i) = z) ∨
-      z ∈ ξ.leftTail.2.vertices := by
-    simpa only [labelLeftEscape, HoldingWalk.mem_vertices_append,
-      negativeStem_vertices, segmentNegativeIndex] using hz
-  rcases hleft with ⟨i,hi⟩ | ht
-  · rcases hraw with ⟨j,hj⟩ | hm | ⟨j,hj⟩
-    · have he := congrArg Fin.val (hfirst (hi.trans hj.symm))
-      change i.val = J+j.val at he
-      have hiJ : i.val = J := by omega
-      rw [←hi]
-      congr 1
-      apply Fin.ext
-      exact hiJ
-    · exact False.elim (hfail (Or.inl (Or.inl (Or.inr ⟨i,by simpa only [hi] using hm⟩))))
-    · exact False.elim (hfail (Or.inr (Or.inl ⟨(j,i),hj.trans hi.symm⟩)))
-  · rcases hraw with ⟨i,hi⟩ | hm | ⟨i,hi⟩
-    · exact False.elim (hfail (Or.inl (Or.inl (Or.inl ⟨i,by simpa only [hi] using ht⟩))))
-    · exact False.elim (hfail (Or.inl (Or.inr ⟨z,ht,hm⟩)))
-    · obtain ⟨l,hl,he⟩ := ξ.leftTail.2.mem_vertices_iff_position.mp ht
-      exact False.elim (hfail (Or.inr (Or.inr ⟨(⟨l,by omega⟩,i),he.trans hi.symm⟩)))
-
- 
-theorem labelRaw_reverse_vertices (G : BondGraph V E) (J L m : ℕ)
-    (γ : Fin (J+J+1) → V)
-    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
-    (ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m) (z : V) :
-    z ∈ (escapeLabelRaw G J L m γ hγ ξ.reverse).vertices ↔
-      z ∈ (escapeLabelRaw G J L m γ hγ ξ).vertices := by
-  simp only [escapeLabelRaw, HoldingWalk.mem_vertices_append,
-    HoldingWalk.vertices_reverse, List.mem_reverse, EscapeLabel.reverse]
-  tauto
-
-namespace RawCorridor
-variable {G : BondGraph V E} {k : ℕ}
-
-theorem interior_mem_raw (w : RawCorridor G k) {z : V} (hz : z ∈ w.interior) :
-    z ∈ w.2.2.vertices := by
-  have hp : z ∈ w.path.support := List.mem_toFinset.mp (Finset.mem_sdiff.mp hz).1
-  exact w.2.2.erase_support.subset hp
-
- 
-noncomputable def neighborhood (hloc : LocallyFinite G) (w : RawCorridor G k) : Finset V :=
-  w.interior ∪ w.interior.biUnion (neighbors G hloc)
-
-theorem coe_neighborhood (hloc : LocallyFinite G) (w : RawCorridor G k) :
-    (w.neighborhood hloc : Set V) = closedBondNeighborhood G (w.interior : Set V) :=
-  (closedBondNeighborhood_finset G hloc w.interior).symm
-
- 
-def HasEscape (hloc : LocallyFinite G) (p : unitInterval) (j : ℕ) (t : ℝ)
-    (w : RawCorridor G k) (x : V) : Prop :=
-  ∃ z, ∃ q : (openSimpleGraph G Set.univ).Walk x z,
-    q.IsPath ∧ q.length ≤ j ∧ (∀ v ∈ q.support, v ∉ w.interior) ∧
-      (∑ b ∈ w.neighborhood hloc, twoPoint G p z b) ≤ t
-
- 
-def Selected (hloc : LocallyFinite G) (p : unitInterval) (j : ℕ) (t : ℝ)
-    (w : RawCorridor G k) : Prop :=
-  3 ≤ w.path.length ∧ w.HasEscape hloc p j t w.base ∧ w.HasEscape hloc p j t w.endpoint
-
-end RawCorridor
-
- 
-
-noncomputable def spineVertices (G : BondGraph V E) (J L : ℕ)
-    (s : Fin (J+J+1) → V)
-    (mid : RawFrom (openSimpleGraph G Set.univ) L (s (Fin.last (J+J))))
-    (h : Fin (J+J+1) → V) : Finset V :=
-  Finset.univ.image (s ∘ segmentPositiveIndex J) ∪ mid.2.vertices.toFinset ∪
-    Finset.univ.image (h ∘ segmentPositiveIndex J)
-
-noncomputable def spineNeighborhood (G : BondGraph V E) (hloc : LocallyFinite G) (J L : ℕ)
-    (s : Fin (J+J+1) → V)
-    (mid : RawFrom (openSimpleGraph G Set.univ) L (s (Fin.last (J+J))))
-    (h : Fin (J+J+1) → V) : Finset V :=
-  let W := spineVertices G J L s mid h
-  W ∪ W.biUnion (neighbors G hloc)
-
-theorem mem_spineVertices (G : BondGraph V E) (J L m : ℕ) (γ : Fin (J+J+1) → V)
-    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
-    (ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m) (z : V) :
-    z ∈ spineVertices G J L ξ.first ⟨ξ.last (Fin.last (J+J)),ξ.middle⟩ ξ.last ↔
-      z ∈ (escapeLabelRaw G J L m γ hγ ξ).vertices := by
-  simp only [spineVertices, Finset.mem_union, Finset.mem_image, Finset.mem_univ,
-    true_and, Function.comp_apply, List.mem_toFinset, escapeLabelRaw,
-    HoldingWalk.mem_vertices_append, HoldingWalk.vertices_reverse, List.mem_reverse,
-    positiveStem_vertices, segmentPositiveIndex]
-  tauto
-
-theorem labelCorridor_neighborhood_subset (G : BondGraph V E) (hloc : LocallyFinite G)
-    (J L m : ℕ) (γ : Fin (J+J+1) → V)
-    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
-    (ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m) :
-    (labelCorridor G J L m γ hγ ξ).neighborhood hloc ⊆
-      spineNeighborhood G hloc J L ξ.first ⟨ξ.last (Fin.last (J+J)),ξ.middle⟩ ξ.last := by
-  have hi : (labelCorridor G J L m γ hγ ξ).interior ⊆
-      spineVertices G J L ξ.first ⟨ξ.last (Fin.last (J+J)),ξ.middle⟩ ξ.last := by
-    intro z hz
-    exact (mem_spineVertices G J L m γ hγ ξ z).mpr
-      ((labelCorridor G J L m γ hγ ξ).interior_mem_raw hz)
-  exact Finset.union_subset_union hi (Finset.biUnion_subset_biUnion_of_subset_left _ hi)
-
-theorem neighborhood_card_le (G : BondGraph V E) (hloc : LocallyFinite G)
-    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (W : Finset V) :
-    (W ∪ W.biUnion (neighbors G hloc)).card ≤ (d+1)*W.card := by
-  have hb : (W.biUnion (neighbors G hloc)).card ≤ W.card*d :=
-    Finset.card_biUnion_le_card_mul _ _ _
-      (fun x _ => (neighbors_card_le_degree G hloc x).trans (hd x))
-  calc
-    _ ≤ W.card + (W.biUnion (neighbors G hloc)).card := Finset.card_union_le _ _
-    _ ≤ W.card + W.card*d := Nat.add_le_add_left hb _
-    _ = _ := by ring
-
-theorem spineVertices_card_le (G : BondGraph V E) (J L : ℕ)
-    (s : Fin (J+J+1) → V)
-    (mid : RawFrom (openSimpleGraph G Set.univ) L (s (Fin.last (J+J))))
-    (h : Fin (J+J+1) → V) :
-    (spineVertices G J L s mid h).card ≤ ((J+L)+J)+3 := by
-  have h₁ : (Finset.univ.image (s ∘ segmentPositiveIndex J)).card ≤ J+1 :=
-    (Finset.card_image_le).trans (by simp)
-  have h₂ : mid.2.vertices.toFinset.card ≤ L+1 :=
-    (List.toFinset_card_le _).trans (le_of_eq mid.2.length_vertices)
-  have h₃ : (Finset.univ.image (h ∘ segmentPositiveIndex J)).card ≤ J+1 :=
-    (Finset.card_image_le).trans (by simp)
-  have hauto := (Finset.card_union_le
-    (Finset.univ.image (s ∘ segmentPositiveIndex J)) mid.2.vertices.toFinset)
-  have hauto' := (Finset.card_union_le
-    (Finset.univ.image (s ∘ segmentPositiveIndex J) ∪ mid.2.vertices.toFinset)
-    (Finset.univ.image (h ∘ segmentPositiveIndex J)))
-  unfold spineVertices
-  omega
-
-theorem spineNeighborhood_card_le (G : BondGraph V E) (hloc : LocallyFinite G)
-    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (J L : ℕ)
-    (s : Fin (J+J+1) → V)
-    (mid : RawFrom (openSimpleGraph G Set.univ) L (s (Fin.last (J+J))))
-    (h : Fin (J+J+1) → V) :
-    ((spineNeighborhood G hloc J L s mid h).card : ℝ) ≤
-      (3*(d:ℝ)+3)*((((J+L)+J):ℕ)+1) := by
-  have hn := neighborhood_card_le G hloc d hd (spineVertices G J L s mid h)
-  have hs := spineVertices_card_le G J L s mid h
-  have hb : (spineNeighborhood G hloc J L s mid h).card ≤ (d+1)*(((J+L)+J)+3) :=
-    hn.trans (Nat.mul_le_mul_left _ hs)
-  have hb' : ((spineNeighborhood G hloc J L s mid h).card : ℝ) ≤
-      ((d:ℝ)+1)*((((J+L)+J):ℕ)+3) := by exact_mod_cast hb
-  have hk : (0:ℝ) ≤ (((J+L)+J):ℕ) := Nat.cast_nonneg _
-  nlinarith [show (0:ℝ) ≤ (d:ℝ) from Nat.cast_nonneg d]
-
- 
-
-theorem selected_of_good_label (G : BondGraph V E) (hloc : LocallyFinite G)
-    (J L m : ℕ) (γ : Fin (J+J+1) → V) (hinj : Function.Injective γ)
-    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
-    (ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m) (p : unitInterval) (t : ℝ)
-    (hleft : ¬labelFirstEscapeCollision G γ (Fin.last (J+J)) 0 J L m
-      (segmentPositiveIndex J) (segmentNegativeIndex J) (segmentPositiveIndex J) ξ)
-    (hright : ¬labelFirstEscapeCollision G γ (Fin.last (J+J)) 0 J L m
-      (segmentPositiveIndex J) (segmentNegativeIndex J) (segmentPositiveIndex J) ξ.reverse)
-    (hdist : ξ.last (segmentCenter J) ∉ reachBall G (ξ.first (segmentCenter J)) 2)
-    (hpot : ¬labelPotentialFailure G p γ (Fin.last (J+J)) 0 L m
-      (spineNeighborhood G hloc J L) t ξ) :
-    (labelCorridor G J L m γ hγ ξ).Selected hloc p (J+m) t := by
-  let w := labelCorridor G J L m γ hγ ξ
-  have hlength : 3 ≤ w.path.length := by
-    by_contra h
-    have hx : w.endpoint ∈ reachBall G w.base w.path.length := by
-      simpa only [Nat.zero_add] using walk_mem_reachBall G Set.univ w.path 0 rfl
-    exact hdist (reachBall_mono_local G w.base (by omega : w.path.length ≤ 2) hx)
-  have hn := labelCorridor_neighborhood_subset G hloc J L m γ hγ ξ
-  have hp := not_or.mp hpot
-  have hp₁ : (∑ b ∈ w.neighborhood hloc, twoPoint G p ξ.leftTail.1 b) ≤ t := by
-    apply le_trans (Finset.sum_le_sum_of_subset_of_nonneg hn (fun b _ _ => twoPoint_nonneg G p _ b))
-    exact le_of_not_gt hp.1
-  have hp₂ : (∑ b ∈ w.neighborhood hloc, twoPoint G p ξ.rightTail.1 b) ≤ t := by
-    apply le_trans (Finset.sum_le_sum_of_subset_of_nonneg hn (fun b _ _ => twoPoint_nonneg G p _ b))
-    exact le_of_not_gt hp.2
-  refine ⟨hlength, ?_, ?_⟩
-  · let e := labelLeftEscape G J L m γ hγ ξ
-    refine ⟨ξ.leftTail.1, e.erase, e.erase_isPath, ?_, ?_, hp₁⟩
-    · exact e.erase_length.trans (by omega)
-    · intro z hz hzI
-      have hr := w.interior_mem_raw hzI
-      have he := labelLeftEscape_raw_intersection G J L m γ hinj hγ ξ hleft z
-        (e.erase_support.subset hz) hr
-      change z = w.base at he
-      apply w.base_not_interior
-      simpa only [he] using hzI
-  · let e := labelLeftEscape G J L m γ hγ ξ.reverse
-    refine ⟨ξ.rightTail.1, e.erase, e.erase_isPath, ?_, ?_, hp₂⟩
-    · exact e.erase_length.trans (by omega)
-    · intro z hz hzI
-      have hr := w.interior_mem_raw hzI
-      have he := labelLeftEscape_raw_intersection G J L m γ hinj hγ ξ.reverse hright z
-        (e.erase_support.subset hz) ((labelRaw_reverse_vertices G J L m γ hγ ξ z).mpr hr)
-      change z = w.endpoint at he
-      apply w.endpoint_not_interior
-      simpa only [he] using hzI
-
-end EscapeGeometry
-end Percolation
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section SelectedLabelMass
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {ι : Type*} [Fintype ι]
-
- 
-
-theorem rooted_selected_label_mass_lower [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (root : ι → V) (weight : ι → ℝ)
-    (htr : RootTransport root weight (vertexAut G))
-    (hcov : ∀ x, ∃ r, ∃ g ∈ vertexAut G, g (root r) = x)
-    (hw : ∀ r, 0 < weight r) (hw1 : ∑ r, weight r = 1)
-    (c : ℝ) (hc : 0 < c) (hmin : ∀ r, 1 ≤ c*weight r)
-    (P A : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re)
-    (p : unitInterval) (hA : HasKernel G p A)
-    (J L m : ℕ) (γ : Fin (J+J+1) → V) (hinj : Function.Injective γ)
-    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
-    (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) (J+J))
-    (x : V) (hx : x ∈ vertexOrbit G (γ (segmentCenter J))) (hρ : ‖P‖ < 1)
-    (hsep : ∀ s ∈ segmentOrbit G γ,
-      (∀ i, J ≤ (openSimpleGraph G Set.univ).dist (s 0) (s (segmentPositiveIndex J i))) ∧
-      (∀ i, J ≤ (openSimpleGraph G Set.univ).dist (s (Fin.last (J+J))) (s (segmentNegativeIndex J i))) ∧
-      2*J ≤ (openSimpleGraph G Set.univ).dist (s 0) (s (Fin.last (J+J))))
-    (d : ℕ) (hd : ∀ z, degree G z ≤ d) (t : ℝ) (ht : 0 < t) :
-    1 - (4*c*(earlyCollisionCost ‖P‖ J + suffixCollisionCost ‖P‖ c J*‖P‖^L) +
-      2*((3*(d:ℝ)+3)*((((J+L)+J):ℕ)+1)*Real.sqrt c*‖P‖^m)/t +
-      2*c*(((d+1)^2:ℕ):ℝ)*Real.sqrt c*‖P‖^L) ≤
-    ∑' ξ : {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m // ξ.first (segmentCenter J) = x},
-      if (labelCorridor G J L m γ hγ ξ.1).Selected hloc p (J+m) t then
-        escapeLabelProbability G hloc a γ (J+J) hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m ξ.1 else 0 := by
-  let F := spineNeighborhood G hloc J L
-  let μ := fun ξ : {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m // ξ.first (segmentCenter J) = x} =>
-    escapeLabelProbability G hloc a γ (J+J) hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m ξ.1
-  let B₁ := fun ξ : {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m // ξ.first (segmentCenter J) = x} =>
-    (labelFirstEscapeCollision G γ (Fin.last (J+J)) 0 J L m
-        (segmentPositiveIndex J) (segmentNegativeIndex J) (segmentPositiveIndex J) ξ.1 ∨
-      labelFirstEscapeCollision G γ (Fin.last (J+J)) 0 J L m
-        (segmentPositiveIndex J) (segmentNegativeIndex J) (segmentPositiveIndex J) ξ.1.reverse) ∨
-      labelPotentialFailure G p γ (Fin.last (J+J)) 0 L m F t ξ.1
-  let B₂ := fun ξ : {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m // ξ.first (segmentCenter J) = x} =>
-    ξ.1.last (segmentCenter J) ∈ reachBall G x 2
-  have hb := rooted_label_collision_potential_failure_bound G hloc a root weight htr hcov hw hw1 c hc hmin
-    P A hpos hsym hP hdiag p hA γ (J+J) hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m x hx hρ
-    J (segmentPositiveIndex J) (segmentNegativeIndex J) (segmentPositiveIndex J) hsep F
-    ((3*(d:ℝ)+3)*((((J+L)+J):ℕ)+1)) (spineNeighborhood_card_le G hloc d hd J L) t ht
-  have he := rooted_label_endpoint_ball_bound G hloc a root weight (vertexAut G) (Set.Subset.refl _)
-    htr hw hw1 hcov c hc hmin γ (J+J) hdiam (segmentCenter J) (Fin.last (J+J)) 0
-    P hpos hsym hP hdiag m L x hx d hd 2
-  have hs := (rooted_label_probability_hasSum G hloc a root weight (vertexAut G) (Set.Subset.refl _)
-    htr hcov hw hw1 c hc hmin P hpos hsym hP hdiag γ (J+J) hdiam
-    (segmentCenter J) (Fin.last (J+J)) 0 L m x hx).tsum_eq
-  have hK (x y : V) : 0 ≤ walkKernel G hloc a x y := by
-    simpa only [hP,Complex.ofReal_re] using (hpos x y).2
-  let := finite_rootedEscapeLabels G hloc γ (J+J) hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m x
-  let := Fintype.ofFinite {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m // ξ.first (segmentCenter J) = x}
-  have hμ (ξ) : 0 ≤ μ ξ := escapeLabelProbability_nonneg G hloc a hK γ (J+J) hdiam
-    (segmentCenter J) (Fin.last (J+J)) 0 L m ξ.1
-  simp only [tsum_fintype] at hs hb he ⊢
-  have hu := (finite_weighted_event_or μ hμ B₁ B₂).trans (add_le_add hb he)
-  have hl := finite_weighted_success_lower μ hs (fun ξ => B₁ ξ ∨ B₂ ξ) _ hu
-  apply hl.trans
-  apply Finset.sum_le_sum
-  intro ξ _
-  by_cases h : B₁ ξ ∨ B₂ ξ
-  · simp only [h,not_true_eq_false,ite_false]
-    split_ifs <;> first | exact hμ ξ | rfl
-  · have h₁ := not_or.mp (not_or.mp h).1
-    have h₂ := not_or.mp h₁.1
-    have hdist : ξ.1.last (segmentCenter J) ∉ reachBall G (ξ.1.first (segmentCenter J)) 2 := by
-      simpa only [ξ.2] using (not_or.mp h).2
-    have hsel := selected_of_good_label G hloc J L m γ hinj hγ ξ.1 p t h₂.1 h₂.2 hdist h₁.2
-    simp only [h,not_false_eq_true,hsel,ite_true]
-    exact le_rfl
-
-end SelectedLabelMass
-end Percolation
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section SelectedRawMass
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem rootedEscapeLabelRaw_corridor (G : BondGraph V E) (J L m : ℕ)
-    (γ : Fin (J+J+1) → V)
-    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
-    (x : V) (ξ : {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m // ξ.first (segmentCenter J) = x}) :
-    (⟨x,rootedEscapeLabelRaw G J L m γ hγ x ξ⟩ : RawCorridor G ((J+L)+J)) =
-      labelCorridor G J L m γ hγ ξ.1 := by
-  rcases ξ with ⟨ξ,hξ⟩
-  subst x
-  rfl
-
- 
-
-theorem selected_raw_mass_of_label_mass (G : BondGraph V E) (hloc : LocallyFinite G)
-    (a : ℝ) (ha : 0 < a) (hK : ∀ x y, 0 ≤ walkKernel G hloc a x y)
-    (J L m : ℕ) (γ : Fin (J+J+1) → V)
-    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
-    (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) (J+J)) (x : V)
-    (hx : x ∈ vertexOrbit G (γ (segmentCenter J)))
-    (q : ℝ) (hq : 0 < q) (hqle : q ≤ segmentReturnMass G hloc a γ (Fin.last (J+J)) L (γ (Fin.last (J+J))))
-    (p : unitInterval) (t b : ℝ)
-    (hb : b ≤ ∑' ξ : {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m // ξ.first (segmentCenter J) = x},
-      if (labelCorridor G J L m γ hγ ξ.1).Selected hloc p (J+m) t then
-        escapeLabelProbability G hloc a γ (J+J) hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m ξ.1 else 0) :
-    b*(a^(J+J)*q) ≤ ∑' w : RawFrom (openSimpleGraph G Set.univ) ((J+L)+J) x,
-      if RawCorridor.Selected hloc p (J+m) t (⟨x,w⟩ : RawCorridor G ((J+L)+J)) then
-        w.2.mass (walkKernel G hloc a) else 0 := by
-  have hh := rooted_label_raw_event_bound G hloc a ha hK J L m γ hγ (J+J) hdiam x hx
-    q hq hqle (fun w => RawCorridor.Selected hloc p (J+m) t (⟨x,w⟩ : RawCorridor G ((J+L)+J)))
-  simp only [rootedEscapeLabelRaw_corridor] at hh
-  exact (le_div_iff₀ (mul_pos (pow_pos ha _) hq)).mp (hb.trans hh)
-
-end SelectedRawMass
-
-section FullSelectionCutoffs
- 
-
-theorem exists_full_selection_cutoffs (ρ c d τ : ℝ)
-    (hρ : 0 < ρ) (hρ1 : ρ < 1) (hc : 0 < c) (hd : 0 ≤ d) (hτ : 0 < τ) :
-    ∃ J : ℕ, 0 < J ∧ ∃ C₁ : ℝ, 0 < C₁ ∧ ∃ k₀ : ℕ, J+J ≤ k₀ ∧
-      ∀ k : ℕ, k₀ ≤ k → ∀ R : ℝ, 1 ≤ R →
-        4*c*(earlyCollisionCost ρ J + suffixCollisionCost ρ c J*ρ^(k-(J+J))) +
-          2*((3*d+3)*((k:ℝ)+1)*Real.sqrt c*ρ^⌈C₁*Real.log (((k:ℝ)+1)*R)⌉₊)/(τ/R) +
-          2*c*(d+1)^2*Real.sqrt c*ρ^(k-(J+J)) ≤ 1/2 := by
-  obtain ⟨J,hJ,L₀,C₁,hC₁,_,k₀,hk₀,hk⟩ :=
-    exists_joint_collision_potential_cutoffs ρ c (3*d+2) τ hρ hρ1 hc (by linarith) hτ
-  have ht : Filter.Tendsto (fun L : ℕ => 2*c*(d+1)^2*Real.sqrt c*ρ^L) Filter.atTop (nhds 0) := by
-    simpa using (tendsto_pow_atTop_nhds_zero_of_lt_one hρ.le hρ1).const_mul
-      (2*c*(d+1)^2*Real.sqrt c)
-  obtain ⟨L₁,hL₁⟩ := Filter.eventually_atTop.mp (ht.eventually (gt_mem_nhds (show (0:ℝ) < 1/8 by norm_num)))
-  refine ⟨J,hJ,C₁,hC₁,max k₀ (J+J+L₁),by omega,fun k hk' R hR => ?_⟩
-  have h₁ := hk k (le_trans (le_max_left _ _) hk') R hR
-  have h₂ := hL₁ (k-(J+J)) (by omega)
-  have he : 3*d+2+1 = 3*d+3 := by ring
-  rw [he] at h₁
-  linarith
-end FullSelectionCutoffs
-end Percolation
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section FullPathSelection
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {ι : Type*} [Fintype ι]
-
- 
-
-theorem exists_selected_path_mass [Countable V] [Countable E] [Infinite V]
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (a : ℝ) (ha : 0 < a) (root : ι → V) (weight : ι → ℝ)
-    (htr : RootTransport root weight (vertexAut G))
-    (hcov : ∀ x, ∃ r, ∃ g ∈ vertexAut G, g (root r) = x)
-    (hw : ∀ r, 0 < weight r) (hw1 : ∑ r, weight r = 1)
-    (c : ℝ) (hc : 0 < c) (hmin : ∀ r, 1 ≤ c*weight r)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re) (hρ : ‖P‖ < 1)
-    (d : ℕ) (hd : ∀ z, degree G z ≤ d) (τ : ℝ) (hτ : 0 < τ) :
-    ∃ J : ℕ, ∃ C₁ : ℝ, ∃ k₀ : ℕ, ∃ b : ℝ,
-      0 < J ∧ 0 < C₁ ∧ 3 ≤ k₀ ∧ 0 < b ∧
-      ∀ p : unitInterval, (pc G:ℝ)/2 ≤ (p:ℝ) → p < pc G →
-      ∀ k : ℕ, k₀ ≤ k → ∀ R : ℝ, 1 ≤ R →
-        b ≤ rawPathAverage G (walkKernel G hloc a) k root weight
-          (fun w => if w.Selected hloc p (J+⌈C₁*Real.log (((k:ℝ)+1)*R)⌉₊) (τ/R) then 1 else 0) := by
-  let x₀ : V := Classical.arbitrary V
-  have hρpos : 0 < ‖P‖ := by
-    have hh := (hdiag x₀).trans ((Complex.re_le_norm _).trans (matrixCoeff_norm_le P x₀ x₀))
-    linarith
-  have hK (x y : V) : 0 ≤ walkKernel G hloc a x y := by
-    simpa only [hP,Complex.ofReal_re] using (hpos x y).2
-  obtain ⟨J,hJ,C₁,hC₁,k₀,hJk,hcut⟩ := exists_full_selection_cutoffs ‖P‖ c d τ hρpos hρ hc (Nat.cast_nonneg d) hτ
-  obtain ⟨y,q,hqN,hq⟩ := exists_geodesic_length G hloc hconn x₀ (J+J)
-  let γ := geodesicSegment G q (J+J)
-  have hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ) :=
-    fun i => geodesicSegment_adj G q (J+J) hqN i.val i.isLt
-  have hinj : Function.Injective γ := geodesicSegment_injective G q (J+J) hqN hq
-  have hdiam := geodesicSegment_diameter G q (J+J) hqN
-  have hsep : ∀ s ∈ segmentOrbit G γ,
-      (∀ i, J ≤ (openSimpleGraph G Set.univ).dist (s 0) (s (segmentPositiveIndex J i))) ∧
-      (∀ i, J ≤ (openSimpleGraph G Set.univ).dist (s (Fin.last (J+J))) (s (segmentNegativeIndex J i))) ∧
-      2*J ≤ (openSimpleGraph G Set.univ).dist (s 0) (s (Fin.last (J+J))) := by
-    rintro s ⟨g,e,hge,rfl⟩
-    constructor
-    · intro i
-      change J ≤ (openSimpleGraph G Set.univ).dist (g (q.getVert 0)) (g (q.getVert (J+i.val)))
-      rw [graph_dist_aut G hconn hge,geodesic_getVert_dist _ q hq 0 (J+i.val) (by omega) (by omega)]
-      omega
-    constructor
-    · intro i
-      change J ≤ (openSimpleGraph G Set.univ).dist (g (q.getVert (J+J))) (g (q.getVert i.val))
-      rw [graph_dist_aut G hconn hge,SimpleGraph.dist_comm,
-        geodesic_getVert_dist _ q hq i.val (J+J) (by omega) (by omega)]
-      omega
-    · change 2*J ≤ (openSimpleGraph G Set.univ).dist (g (q.getVert 0)) (g (q.getVert (J+J)))
-      rw [graph_dist_aut G hconn hge,geodesic_getVert_dist _ q hq 0 (J+J) (by omega) (by omega)]
-      omega
-  obtain ⟨i,g,hg,hgx⟩ := hcov (γ (segmentCenter J))
-  obtain ⟨e,he⟩ := hg
-  have hx : root i ∈ vertexOrbit G (γ (segmentCenter J)) :=
-    ⟨g.symm,e.symm,isAut_symm G he,by rw [←hgx,g.symm_apply_apply]⟩
-  let b : ℝ := weight i * ((1/2)*(a^(J+J)*(1/(2*c))))
-  have hb : 0 < b := mul_pos (hw i) (mul_pos (by norm_num)
-    (mul_pos (pow_pos ha _) (by positivity)))
-  refine ⟨J,C₁,max k₀ 3,b,hJ,hC₁,le_max_right _ _,hb,?_⟩
-  intro p _hpnear hp k hk R hR
-  have hk₀ : k₀ ≤ k := (le_max_left _ _).trans hk
-  have hJk' : J+J ≤ k := hJk.trans hk₀
-  obtain ⟨L,rfl⟩ : ∃ L : ℕ, (J+L)+J = k := ⟨k-(J+J),by omega⟩
-  let m' := ⌈C₁*Real.log ((((((J+L)+J):ℕ):ℝ)+1)*R)⌉₊
-  obtain ⟨A,hA⟩ := operatorBounded_of_lt_pc G hloc hqt p hp
-  have hcost := hcut ((J+L)+J) hk₀ R hR
-  have hL : (J+L)+J-(J+J) = L := by omega
-  rw [hL] at hcost
-  have hcost' :
-      4*c*(earlyCollisionCost ‖P‖ J + suffixCollisionCost ‖P‖ c J*‖P‖^L) +
-      2*((3*(d:ℝ)+3)*((((J+L)+J):ℕ)+1)*Real.sqrt c*‖P‖^m')/(τ/R) +
-      2*c*(((d+1)^2:ℕ):ℝ)*Real.sqrt c*‖P‖^L ≤ 1/2 := by
-    simpa only [m',Nat.cast_pow,Nat.cast_add,Nat.cast_one] using hcost
-  have hl := rooted_selected_label_mass_lower G hloc a root weight htr hcov hw hw1 c hc hmin
-    P A hpos hsym hP hdiag p hA J L m' γ hinj hγ hdiam (root i) hx hρ hsep d hd (τ/R)
-    (div_pos hτ (lt_of_lt_of_le zero_lt_one hR))
-  have hhalf : (1/2:ℝ) ≤ ∑' ξ : {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m' // ξ.first (segmentCenter J) = root i},
-      if (labelCorridor G J L m' γ hγ ξ.1).Selected hloc p (J+m') (τ/R) then
-        escapeLabelProbability G hloc a γ (J+J) hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m' ξ.1 else 0 := by
-    linarith
-  have hqreturn := segmentReturnMass_uniform_lower G hloc a root weight (vertexAut G) (Set.Subset.refl _)
-    htr hcov hw hw1 c hc hmin P hpos hsym hP hdiag γ (Fin.last (J+J)) L
-  have hmraw := selected_raw_mass_of_label_mass G hloc a ha hK J L m' γ hγ hdiam (root i) hx
-    (1/(2*c)) (by positivity) hqreturn p (τ/R) (1/2) hhalf
-  have hroot := mul_le_mul_of_nonneg_left hmraw (hw i).le
-  apply hroot.trans
-  unfold rawPathAverage
-  have hall (r : ι) : 0 ≤ weight r * ∑' w : RawFrom (openSimpleGraph G Set.univ) ((J+L)+J) (root r),
-      w.2.mass (walkKernel G hloc a) *
-        (if RawCorridor.Selected hloc p (J+m') (τ/R) ⟨root r,w⟩ then 1 else 0) := by
-    apply mul_nonneg (hw r).le
-    apply tsum_nonneg
-    intro w
-    apply mul_nonneg (w.2.mass_nonneg _ hK)
-    split_ifs <;> norm_num
-  have hi := Finset.single_le_sum (s := Finset.univ) (fun r _ => hall r) (Finset.mem_univ i)
-  simpa only [mul_ite,mul_one,mul_zero] using hi
-
-end FullPathSelection
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Escape.SelectionBounds
+
+namespace OAI
+
+/-! Selection of geometrically separated paths from labelled escape events. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+section EscapeGeometry
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+namespace HoldingWalk
+variable {H : SimpleGraph V}
+
+theorem mem_vertices_append {k l : ℕ} {x y z b : V}
+    (w : HoldingWalk H k x y) (v : HoldingWalk H l y z) :
+    b ∈ (w.append v).vertices ↔ b ∈ w.vertices ∨ b ∈ v.vertices := by
+  induction w with
+  | nil x =>
+    have hx : x ∈ v.vertices := by simpa using v.position_mem_vertices 0
+    simp only [append, vertices, List.mem_singleton]
+    exact ⟨fun h => Or.inr h, fun h => h.elim (fun he => he ▸ hx) id⟩
+  | hold w ih => simp only [append, vertices, List.mem_cons, ih, or_assoc]
+  | step h w ih => simp only [append, vertices, List.mem_cons, ih, or_assoc]
+
+theorem mem_vertices_fromPositions (k : ℕ) (f : Fin (k+1) → V)
+    (h : ∀ i : Fin k, H.Adj (f i.castSucc) (f i.succ)) (b : V) :
+    b ∈ (fromPositions k f h).vertices ↔ ∃ i, f i = b := by
+  rw [mem_vertices_iff_position]
+  constructor
+  · rintro ⟨i, hi, he⟩
+    exact ⟨⟨i,by omega⟩, (position_fromPositions k f h i hi).symm.trans he⟩
+  · rintro ⟨i,hi⟩
+    exact ⟨i.val,by omega,(position_fromPositions k f h i.val (by omega)).trans hi⟩
+
+end HoldingWalk
+
+ 
+def negativeStem (H : SimpleGraph V) (J : ℕ) (s : Fin (J+J+1) → V)
+    (hs : ∀ i : Fin (J+J), H.Adj (s i.castSucc) (s i.succ)) :
+    HoldingWalk H J (s (segmentCenter J)) (s 0) :=
+  (HoldingWalk.fromPositions J (fun i => s ⟨i.val,by omega⟩)
+    (fun i => hs ⟨i.val,by omega⟩)).reverse
+
+theorem positiveStem_vertices (H : SimpleGraph V) (J : ℕ) (s : Fin (J+J+1) → V)
+    (hs : ∀ i : Fin (J+J), H.Adj (s i.castSucc) (s i.succ)) (z : V) :
+    z ∈ (positiveStem H J s hs).vertices ↔ ∃ i : Fin (J+1), s ⟨J+i.val,by omega⟩ = z := by
+  exact HoldingWalk.mem_vertices_fromPositions J _ _ z
+
+theorem negativeStem_vertices (H : SimpleGraph V) (J : ℕ) (s : Fin (J+J+1) → V)
+    (hs : ∀ i : Fin (J+J), H.Adj (s i.castSucc) (s i.succ)) (z : V) :
+    z ∈ (negativeStem H J s hs).vertices ↔ ∃ i : Fin (J+1), s ⟨i.val,by omega⟩ = z := by
+  let w := HoldingWalk.fromPositions J (fun i : Fin (J+1) => s ⟨i.val,by omega⟩)
+    (fun i : Fin J => hs ⟨i.val,by omega⟩)
+  have he : z ∈ w.reverse.vertices ↔ z ∈ w.vertices := by
+    rw [HoldingWalk.vertices_reverse, List.mem_reverse]
+  exact he.trans (HoldingWalk.mem_vertices_fromPositions J _ _ z)
+
+ 
+def labelLeftEscape (G : BondGraph V E) (J L m : ℕ) (γ : Fin (J+J+1) → V)
+    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
+    (ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m) :
+    HoldingWalk (openSimpleGraph G Set.univ) (m+J)
+      (ξ.first (segmentCenter J)) ξ.leftTail.1 :=
+  (negativeStem _ J ξ.first (orbitSegment_consecutive G J γ ξ.first hγ ξ.first_mem)).append ξ.leftTail.2
+
+ 
+def labelCorridor (G : BondGraph V E) (J L m : ℕ) (γ : Fin (J+J+1) → V)
+    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
+    (ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m) : RawCorridor G ((J+L)+J) :=
+  ⟨ξ.first (segmentCenter J), ξ.last (segmentCenter J), escapeLabelRaw G J L m γ hγ ξ⟩
+
+def segmentPositiveIndex (J : ℕ) : Fin (J+1) → Fin (J+J+1) :=
+  fun i => ⟨J+i.val,by omega⟩
+def segmentNegativeIndex (J : ℕ) : Fin (J+1) → Fin (J+J+1) :=
+  fun i => ⟨i.val,by omega⟩
+
+theorem segmentOrbit_injective {I : Type*} (G : BondGraph V E) (γ s : I → V)
+    (hγ : Function.Injective γ) (hs : s ∈ segmentOrbit G γ) : Function.Injective s := by
+  obtain ⟨a,b,hab,rfl⟩ := hs
+  exact a.injective.comp hγ
+
+ 
+
+theorem labelLeftEscape_raw_intersection (G : BondGraph V E) (J L m : ℕ)
+    (γ : Fin (J+J+1) → V) (hinj : Function.Injective γ)
+    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
+    (ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m)
+    (hgood : ¬labelFirstEscapeCollision G γ (Fin.last (J+J)) 0 J L m
+      (segmentPositiveIndex J) (segmentNegativeIndex J) (segmentPositiveIndex J) ξ)
+    (z : V) (hz : z ∈ (labelLeftEscape G J L m γ hγ ξ).vertices)
+    (hr : z ∈ (escapeLabelRaw G J L m γ hγ ξ).vertices) :
+    z = ξ.first (segmentCenter J) := by
+  have hfirst := segmentOrbit_injective G γ ξ.first hinj ξ.first_mem
+  have hfail := hgood
+  unfold labelFirstEscapeCollision firstEscapeCollision at hfail
+  have hraw : (∃ i : Fin (J+1), ξ.first (segmentPositiveIndex J i) = z) ∨
+      z ∈ ξ.middle.vertices ∨ (∃ i : Fin (J+1), ξ.last (segmentPositiveIndex J i) = z) := by
+    simpa only [escapeLabelRaw, HoldingWalk.mem_vertices_append,
+      HoldingWalk.vertices_reverse, List.mem_reverse, positiveStem_vertices,
+      segmentPositiveIndex] using hr
+  have hleft : (∃ i : Fin (J+1), ξ.first (segmentNegativeIndex J i) = z) ∨
+      z ∈ ξ.leftTail.2.vertices := by
+    simpa only [labelLeftEscape, HoldingWalk.mem_vertices_append,
+      negativeStem_vertices, segmentNegativeIndex] using hz
+  rcases hleft with ⟨i,hi⟩ | ht
+  · rcases hraw with ⟨j,hj⟩ | hm | ⟨j,hj⟩
+    · have he := congrArg Fin.val (hfirst (hi.trans hj.symm))
+      change i.val = J+j.val at he
+      have hiJ : i.val = J := by omega
+      rw [←hi]
+      congr 1
+      apply Fin.ext
+      exact hiJ
+    · exact False.elim (hfail (Or.inl (Or.inl (Or.inr ⟨i,by simpa only [hi] using hm⟩))))
+    · exact False.elim (hfail (Or.inr (Or.inl ⟨(j,i),hj.trans hi.symm⟩)))
+  · rcases hraw with ⟨i,hi⟩ | hm | ⟨i,hi⟩
+    · exact False.elim (hfail (Or.inl (Or.inl (Or.inl ⟨i,by simpa only [hi] using ht⟩))))
+    · exact False.elim (hfail (Or.inl (Or.inr ⟨z,ht,hm⟩)))
+    · obtain ⟨l,hl,he⟩ := ξ.leftTail.2.mem_vertices_iff_position.mp ht
+      exact False.elim (hfail (Or.inr (Or.inr ⟨(⟨l,by omega⟩,i),he.trans hi.symm⟩)))
+
+ 
+theorem labelRaw_reverse_vertices (G : BondGraph V E) (J L m : ℕ)
+    (γ : Fin (J+J+1) → V)
+    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
+    (ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m) (z : V) :
+    z ∈ (escapeLabelRaw G J L m γ hγ ξ.reverse).vertices ↔
+      z ∈ (escapeLabelRaw G J L m γ hγ ξ).vertices := by
+  simp only [escapeLabelRaw, HoldingWalk.mem_vertices_append,
+    HoldingWalk.vertices_reverse, List.mem_reverse, EscapeLabel.reverse]
+  tauto
+
+namespace RawCorridor
+variable {G : BondGraph V E} {k : ℕ}
+
+theorem interior_mem_raw (w : RawCorridor G k) {z : V} (hz : z ∈ w.interior) :
+    z ∈ w.2.2.vertices := by
+  have hp : z ∈ w.path.support := List.mem_toFinset.mp (Finset.mem_sdiff.mp hz).1
+  exact w.2.2.erase_support.subset hp
+
+ 
+noncomputable def neighborhood (hloc : LocallyFinite G) (w : RawCorridor G k) : Finset V :=
+  w.interior ∪ w.interior.biUnion (neighbors G hloc)
+
+theorem coe_neighborhood (hloc : LocallyFinite G) (w : RawCorridor G k) :
+    (w.neighborhood hloc : Set V) = closedBondNeighborhood G (w.interior : Set V) :=
+  (closedBondNeighborhood_finset G hloc w.interior).symm
+
+ 
+def HasEscape (hloc : LocallyFinite G) (p : unitInterval) (j : ℕ) (t : ℝ)
+    (w : RawCorridor G k) (x : V) : Prop :=
+  ∃ z, ∃ q : (openSimpleGraph G Set.univ).Walk x z,
+    q.IsPath ∧ q.length ≤ j ∧ (∀ v ∈ q.support, v ∉ w.interior) ∧
+      (∑ b ∈ w.neighborhood hloc, twoPoint G p z b) ≤ t
+
+ 
+def Selected (hloc : LocallyFinite G) (p : unitInterval) (j : ℕ) (t : ℝ)
+    (w : RawCorridor G k) : Prop :=
+  3 ≤ w.path.length ∧ w.HasEscape hloc p j t w.base ∧ w.HasEscape hloc p j t w.endpoint
+
+end RawCorridor
+
+ 
+
+noncomputable def spineVertices (G : BondGraph V E) (J L : ℕ)
+    (s : Fin (J+J+1) → V)
+    (mid : RawFrom (openSimpleGraph G Set.univ) L (s (Fin.last (J+J))))
+    (h : Fin (J+J+1) → V) : Finset V :=
+  Finset.univ.image (s ∘ segmentPositiveIndex J) ∪ mid.2.vertices.toFinset ∪
+    Finset.univ.image (h ∘ segmentPositiveIndex J)
+
+noncomputable def spineNeighborhood (G : BondGraph V E) (hloc : LocallyFinite G) (J L : ℕ)
+    (s : Fin (J+J+1) → V)
+    (mid : RawFrom (openSimpleGraph G Set.univ) L (s (Fin.last (J+J))))
+    (h : Fin (J+J+1) → V) : Finset V :=
+  let W := spineVertices G J L s mid h
+  W ∪ W.biUnion (neighbors G hloc)
+
+theorem mem_spineVertices (G : BondGraph V E) (J L m : ℕ) (γ : Fin (J+J+1) → V)
+    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
+    (ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m) (z : V) :
+    z ∈ spineVertices G J L ξ.first ⟨ξ.last (Fin.last (J+J)),ξ.middle⟩ ξ.last ↔
+      z ∈ (escapeLabelRaw G J L m γ hγ ξ).vertices := by
+  simp only [spineVertices, Finset.mem_union, Finset.mem_image, Finset.mem_univ,
+    true_and, Function.comp_apply, List.mem_toFinset, escapeLabelRaw,
+    HoldingWalk.mem_vertices_append, HoldingWalk.vertices_reverse, List.mem_reverse,
+    positiveStem_vertices, segmentPositiveIndex]
+  tauto
+
+theorem labelCorridor_neighborhood_subset (G : BondGraph V E) (hloc : LocallyFinite G)
+    (J L m : ℕ) (γ : Fin (J+J+1) → V)
+    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
+    (ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m) :
+    (labelCorridor G J L m γ hγ ξ).neighborhood hloc ⊆
+      spineNeighborhood G hloc J L ξ.first ⟨ξ.last (Fin.last (J+J)),ξ.middle⟩ ξ.last := by
+  have hi : (labelCorridor G J L m γ hγ ξ).interior ⊆
+      spineVertices G J L ξ.first ⟨ξ.last (Fin.last (J+J)),ξ.middle⟩ ξ.last := by
+    intro z hz
+    exact (mem_spineVertices G J L m γ hγ ξ z).mpr
+      ((labelCorridor G J L m γ hγ ξ).interior_mem_raw hz)
+  exact Finset.union_subset_union hi (Finset.biUnion_subset_biUnion_of_subset_left _ hi)
+
+theorem neighborhood_card_le (G : BondGraph V E) (hloc : LocallyFinite G)
+    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (W : Finset V) :
+    (W ∪ W.biUnion (neighbors G hloc)).card ≤ (d+1)*W.card := by
+  have hb : (W.biUnion (neighbors G hloc)).card ≤ W.card*d :=
+    Finset.card_biUnion_le_card_mul _ _ _
+      (fun x _ => (neighbors_card_le_degree G hloc x).trans (hd x))
+  calc
+    _ ≤ W.card + (W.biUnion (neighbors G hloc)).card := Finset.card_union_le _ _
+    _ ≤ W.card + W.card*d := Nat.add_le_add_left hb _
+    _ = _ := by ring
+
+theorem spineVertices_card_le (G : BondGraph V E) (J L : ℕ)
+    (s : Fin (J+J+1) → V)
+    (mid : RawFrom (openSimpleGraph G Set.univ) L (s (Fin.last (J+J))))
+    (h : Fin (J+J+1) → V) :
+    (spineVertices G J L s mid h).card ≤ ((J+L)+J)+3 := by
+  have h₁ : (Finset.univ.image (s ∘ segmentPositiveIndex J)).card ≤ J+1 :=
+    (Finset.card_image_le).trans (by simp)
+  have h₂ : mid.2.vertices.toFinset.card ≤ L+1 :=
+    (List.toFinset_card_le _).trans (le_of_eq mid.2.length_vertices)
+  have h₃ : (Finset.univ.image (h ∘ segmentPositiveIndex J)).card ≤ J+1 :=
+    (Finset.card_image_le).trans (by simp)
+  have hauto := (Finset.card_union_le
+    (Finset.univ.image (s ∘ segmentPositiveIndex J)) mid.2.vertices.toFinset)
+  have hauto' := (Finset.card_union_le
+    (Finset.univ.image (s ∘ segmentPositiveIndex J) ∪ mid.2.vertices.toFinset)
+    (Finset.univ.image (h ∘ segmentPositiveIndex J)))
+  unfold spineVertices
+  omega
+
+theorem spineNeighborhood_card_le (G : BondGraph V E) (hloc : LocallyFinite G)
+    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (J L : ℕ)
+    (s : Fin (J+J+1) → V)
+    (mid : RawFrom (openSimpleGraph G Set.univ) L (s (Fin.last (J+J))))
+    (h : Fin (J+J+1) → V) :
+    ((spineNeighborhood G hloc J L s mid h).card : ℝ) ≤
+      (3*(d:ℝ)+3)*((((J+L)+J):ℕ)+1) := by
+  have hn := neighborhood_card_le G hloc d hd (spineVertices G J L s mid h)
+  have hs := spineVertices_card_le G J L s mid h
+  have hb : (spineNeighborhood G hloc J L s mid h).card ≤ (d+1)*(((J+L)+J)+3) :=
+    hn.trans (Nat.mul_le_mul_left _ hs)
+  have hb' : ((spineNeighborhood G hloc J L s mid h).card : ℝ) ≤
+      ((d:ℝ)+1)*((((J+L)+J):ℕ)+3) := by exact_mod_cast hb
+  have hk : (0:ℝ) ≤ (((J+L)+J):ℕ) := Nat.cast_nonneg _
+  nlinarith [show (0:ℝ) ≤ (d:ℝ) from Nat.cast_nonneg d]
+
+ 
+
+theorem selected_of_good_label (G : BondGraph V E) (hloc : LocallyFinite G)
+    (J L m : ℕ) (γ : Fin (J+J+1) → V) (hinj : Function.Injective γ)
+    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
+    (ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m) (p : unitInterval) (t : ℝ)
+    (hleft : ¬labelFirstEscapeCollision G γ (Fin.last (J+J)) 0 J L m
+      (segmentPositiveIndex J) (segmentNegativeIndex J) (segmentPositiveIndex J) ξ)
+    (hright : ¬labelFirstEscapeCollision G γ (Fin.last (J+J)) 0 J L m
+      (segmentPositiveIndex J) (segmentNegativeIndex J) (segmentPositiveIndex J) ξ.reverse)
+    (hdist : ξ.last (segmentCenter J) ∉ reachBall G (ξ.first (segmentCenter J)) 2)
+    (hpot : ¬labelPotentialFailure G p γ (Fin.last (J+J)) 0 L m
+      (spineNeighborhood G hloc J L) t ξ) :
+    (labelCorridor G J L m γ hγ ξ).Selected hloc p (J+m) t := by
+  let w := labelCorridor G J L m γ hγ ξ
+  have hlength : 3 ≤ w.path.length := by
+    by_contra h
+    have hx : w.endpoint ∈ reachBall G w.base w.path.length := by
+      simpa only [Nat.zero_add] using walk_mem_reachBall G Set.univ w.path 0 rfl
+    exact hdist (reachBall_mono_local G w.base (by omega : w.path.length ≤ 2) hx)
+  have hn := labelCorridor_neighborhood_subset G hloc J L m γ hγ ξ
+  have hp := not_or.mp hpot
+  have hp₁ : (∑ b ∈ w.neighborhood hloc, twoPoint G p ξ.leftTail.1 b) ≤ t := by
+    apply le_trans (Finset.sum_le_sum_of_subset_of_nonneg hn (fun b _ _ => twoPoint_nonneg G p _ b))
+    exact le_of_not_gt hp.1
+  have hp₂ : (∑ b ∈ w.neighborhood hloc, twoPoint G p ξ.rightTail.1 b) ≤ t := by
+    apply le_trans (Finset.sum_le_sum_of_subset_of_nonneg hn (fun b _ _ => twoPoint_nonneg G p _ b))
+    exact le_of_not_gt hp.2
+  refine ⟨hlength, ?_, ?_⟩
+  · let e := labelLeftEscape G J L m γ hγ ξ
+    refine ⟨ξ.leftTail.1, e.erase, e.erase_isPath, ?_, ?_, hp₁⟩
+    · exact e.erase_length.trans (by omega)
+    · intro z hz hzI
+      have hr := w.interior_mem_raw hzI
+      have he := labelLeftEscape_raw_intersection G J L m γ hinj hγ ξ hleft z
+        (e.erase_support.subset hz) hr
+      change z = w.base at he
+      apply w.base_not_interior
+      simpa only [he] using hzI
+  · let e := labelLeftEscape G J L m γ hγ ξ.reverse
+    refine ⟨ξ.rightTail.1, e.erase, e.erase_isPath, ?_, ?_, hp₂⟩
+    · exact e.erase_length.trans (by omega)
+    · intro z hz hzI
+      have hr := w.interior_mem_raw hzI
+      have he := labelLeftEscape_raw_intersection G J L m γ hinj hγ ξ.reverse hright z
+        (e.erase_support.subset hz) ((labelRaw_reverse_vertices G J L m γ hγ ξ z).mpr hr)
+      change z = w.endpoint at he
+      apply w.endpoint_not_interior
+      simpa only [he] using hzI
+
+end EscapeGeometry
+end Percolation
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section SelectedLabelMass
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {ι : Type*} [Fintype ι]
+
+ 
+
+theorem rooted_selected_label_mass_lower [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (root : ι → V) (weight : ι → ℝ)
+    (htr : RootTransport root weight (vertexAut G))
+    (hcov : ∀ x, ∃ r, ∃ g ∈ vertexAut G, g (root r) = x)
+    (hw : ∀ r, 0 < weight r) (hw1 : ∑ r, weight r = 1)
+    (c : ℝ) (hc : 0 < c) (hmin : ∀ r, 1 ≤ c*weight r)
+    (P A : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re)
+    (p : unitInterval) (hA : HasKernel G p A)
+    (J L m : ℕ) (γ : Fin (J+J+1) → V) (hinj : Function.Injective γ)
+    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
+    (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) (J+J))
+    (x : V) (hx : x ∈ vertexOrbit G (γ (segmentCenter J))) (hρ : ‖P‖ < 1)
+    (hsep : ∀ s ∈ segmentOrbit G γ,
+      (∀ i, J ≤ (openSimpleGraph G Set.univ).dist (s 0) (s (segmentPositiveIndex J i))) ∧
+      (∀ i, J ≤ (openSimpleGraph G Set.univ).dist (s (Fin.last (J+J))) (s (segmentNegativeIndex J i))) ∧
+      2*J ≤ (openSimpleGraph G Set.univ).dist (s 0) (s (Fin.last (J+J))))
+    (d : ℕ) (hd : ∀ z, degree G z ≤ d) (t : ℝ) (ht : 0 < t) :
+    1 - (4*c*(earlyCollisionCost ‖P‖ J + suffixCollisionCost ‖P‖ c J*‖P‖^L) +
+      2*((3*(d:ℝ)+3)*((((J+L)+J):ℕ)+1)*Real.sqrt c*‖P‖^m)/t +
+      2*c*(((d+1)^2:ℕ):ℝ)*Real.sqrt c*‖P‖^L) ≤
+    ∑' ξ : {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m // ξ.first (segmentCenter J) = x},
+      if (labelCorridor G J L m γ hγ ξ.1).Selected hloc p (J+m) t then
+        escapeLabelProbability G hloc a γ (J+J) hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m ξ.1 else 0 := by
+  let F := spineNeighborhood G hloc J L
+  let μ := fun ξ : {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m // ξ.first (segmentCenter J) = x} =>
+    escapeLabelProbability G hloc a γ (J+J) hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m ξ.1
+  let B₁ := fun ξ : {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m // ξ.first (segmentCenter J) = x} =>
+    (labelFirstEscapeCollision G γ (Fin.last (J+J)) 0 J L m
+        (segmentPositiveIndex J) (segmentNegativeIndex J) (segmentPositiveIndex J) ξ.1 ∨
+      labelFirstEscapeCollision G γ (Fin.last (J+J)) 0 J L m
+        (segmentPositiveIndex J) (segmentNegativeIndex J) (segmentPositiveIndex J) ξ.1.reverse) ∨
+      labelPotentialFailure G p γ (Fin.last (J+J)) 0 L m F t ξ.1
+  let B₂ := fun ξ : {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m // ξ.first (segmentCenter J) = x} =>
+    ξ.1.last (segmentCenter J) ∈ reachBall G x 2
+  have hb := rooted_label_collision_potential_failure_bound G hloc a root weight htr hcov hw hw1 c hc hmin
+    P A hpos hsym hP hdiag p hA γ (J+J) hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m x hx hρ
+    J (segmentPositiveIndex J) (segmentNegativeIndex J) (segmentPositiveIndex J) hsep F
+    ((3*(d:ℝ)+3)*((((J+L)+J):ℕ)+1)) (spineNeighborhood_card_le G hloc d hd J L) t ht
+  have he := rooted_label_endpoint_ball_bound G hloc a root weight (vertexAut G) (Set.Subset.refl _)
+    htr hw hw1 hcov c hc hmin γ (J+J) hdiam (segmentCenter J) (Fin.last (J+J)) 0
+    P hpos hsym hP hdiag m L x hx d hd 2
+  have hs := (rooted_label_probability_hasSum G hloc a root weight (vertexAut G) (Set.Subset.refl _)
+    htr hcov hw hw1 c hc hmin P hpos hsym hP hdiag γ (J+J) hdiam
+    (segmentCenter J) (Fin.last (J+J)) 0 L m x hx).tsum_eq
+  have hK (x y : V) : 0 ≤ walkKernel G hloc a x y := by
+    simpa only [hP,Complex.ofReal_re] using (hpos x y).2
+  let := finite_rootedEscapeLabels G hloc γ (J+J) hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m x
+  let := Fintype.ofFinite {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m // ξ.first (segmentCenter J) = x}
+  have hμ (ξ) : 0 ≤ μ ξ := escapeLabelProbability_nonneg G hloc a hK γ (J+J) hdiam
+    (segmentCenter J) (Fin.last (J+J)) 0 L m ξ.1
+  simp only [tsum_fintype] at hs hb he ⊢
+  have hu := (finite_weighted_event_or μ hμ B₁ B₂).trans (add_le_add hb he)
+  have hl := finite_weighted_success_lower μ hs (fun ξ => B₁ ξ ∨ B₂ ξ) _ hu
+  apply hl.trans
+  apply Finset.sum_le_sum
+  intro ξ _
+  by_cases h : B₁ ξ ∨ B₂ ξ
+  · simp only [h,not_true_eq_false,ite_false]
+    split_ifs <;> first | exact hμ ξ | rfl
+  · have h₁ := not_or.mp (not_or.mp h).1
+    have h₂ := not_or.mp h₁.1
+    have hdist : ξ.1.last (segmentCenter J) ∉ reachBall G (ξ.1.first (segmentCenter J)) 2 := by
+      simpa only [ξ.2] using (not_or.mp h).2
+    have hsel := selected_of_good_label G hloc J L m γ hinj hγ ξ.1 p t h₂.1 h₂.2 hdist h₁.2
+    simp only [h,not_false_eq_true,hsel,ite_true]
+    exact le_rfl
+
+end SelectedLabelMass
+end Percolation
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section SelectedRawMass
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem rootedEscapeLabelRaw_corridor (G : BondGraph V E) (J L m : ℕ)
+    (γ : Fin (J+J+1) → V)
+    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
+    (x : V) (ξ : {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m // ξ.first (segmentCenter J) = x}) :
+    (⟨x,rootedEscapeLabelRaw G J L m γ hγ x ξ⟩ : RawCorridor G ((J+L)+J)) =
+      labelCorridor G J L m γ hγ ξ.1 := by
+  rcases ξ with ⟨ξ,hξ⟩
+  subst x
+  rfl
+
+ 
+
+theorem selected_raw_mass_of_label_mass (G : BondGraph V E) (hloc : LocallyFinite G)
+    (a : ℝ) (ha : 0 < a) (hK : ∀ x y, 0 ≤ walkKernel G hloc a x y)
+    (J L m : ℕ) (γ : Fin (J+J+1) → V)
+    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
+    (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) (J+J)) (x : V)
+    (hx : x ∈ vertexOrbit G (γ (segmentCenter J)))
+    (q : ℝ) (hq : 0 < q) (hqle : q ≤ segmentReturnMass G hloc a γ (Fin.last (J+J)) L (γ (Fin.last (J+J))))
+    (p : unitInterval) (t b : ℝ)
+    (hb : b ≤ ∑' ξ : {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m // ξ.first (segmentCenter J) = x},
+      if (labelCorridor G J L m γ hγ ξ.1).Selected hloc p (J+m) t then
+        escapeLabelProbability G hloc a γ (J+J) hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m ξ.1 else 0) :
+    b*(a^(J+J)*q) ≤ ∑' w : RawFrom (openSimpleGraph G Set.univ) ((J+L)+J) x,
+      if RawCorridor.Selected hloc p (J+m) t (⟨x,w⟩ : RawCorridor G ((J+L)+J)) then
+        w.2.mass (walkKernel G hloc a) else 0 := by
+  have hh := rooted_label_raw_event_bound G hloc a ha hK J L m γ hγ (J+J) hdiam x hx
+    q hq hqle (fun w => RawCorridor.Selected hloc p (J+m) t (⟨x,w⟩ : RawCorridor G ((J+L)+J)))
+  simp only [rootedEscapeLabelRaw_corridor] at hh
+  exact (le_div_iff₀ (mul_pos (pow_pos ha _) hq)).mp (hb.trans hh)
+
+end SelectedRawMass
+
+section FullSelectionCutoffs
+ 
+
+theorem exists_full_selection_cutoffs (ρ c d τ : ℝ)
+    (hρ : 0 < ρ) (hρ1 : ρ < 1) (hc : 0 < c) (hd : 0 ≤ d) (hτ : 0 < τ) :
+    ∃ J : ℕ, 0 < J ∧ ∃ C₁ : ℝ, 0 < C₁ ∧ ∃ k₀ : ℕ, J+J ≤ k₀ ∧
+      ∀ k : ℕ, k₀ ≤ k → ∀ R : ℝ, 1 ≤ R →
+        4*c*(earlyCollisionCost ρ J + suffixCollisionCost ρ c J*ρ^(k-(J+J))) +
+          2*((3*d+3)*((k:ℝ)+1)*Real.sqrt c*ρ^⌈C₁*Real.log (((k:ℝ)+1)*R)⌉₊)/(τ/R) +
+          2*c*(d+1)^2*Real.sqrt c*ρ^(k-(J+J)) ≤ 1/2 := by
+  obtain ⟨J,hJ,L₀,C₁,hC₁,_,k₀,hk₀,hk⟩ :=
+    exists_joint_collision_potential_cutoffs ρ c (3*d+2) τ hρ hρ1 hc (by linarith) hτ
+  have ht : Filter.Tendsto (fun L : ℕ => 2*c*(d+1)^2*Real.sqrt c*ρ^L) Filter.atTop (nhds 0) := by
+    simpa using (tendsto_pow_atTop_nhds_zero_of_lt_one hρ.le hρ1).const_mul
+      (2*c*(d+1)^2*Real.sqrt c)
+  obtain ⟨L₁,hL₁⟩ := Filter.eventually_atTop.mp (ht.eventually (gt_mem_nhds (show (0:ℝ) < 1/8 by norm_num)))
+  refine ⟨J,hJ,C₁,hC₁,max k₀ (J+J+L₁),by omega,fun k hk' R hR => ?_⟩
+  have h₁ := hk k (le_trans (le_max_left _ _) hk') R hR
+  have h₂ := hL₁ (k-(J+J)) (by omega)
+  have he : 3*d+2+1 = 3*d+3 := by ring
+  rw [he] at h₁
+  linarith
+end FullSelectionCutoffs
+end Percolation
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section FullPathSelection
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {ι : Type*} [Fintype ι]
+
+ 
+
+theorem exists_selected_path_mass [Countable V] [Countable E] [Infinite V]
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (a : ℝ) (ha : 0 < a) (root : ι → V) (weight : ι → ℝ)
+    (htr : RootTransport root weight (vertexAut G))
+    (hcov : ∀ x, ∃ r, ∃ g ∈ vertexAut G, g (root r) = x)
+    (hw : ∀ r, 0 < weight r) (hw1 : ∑ r, weight r = 1)
+    (c : ℝ) (hc : 0 < c) (hmin : ∀ r, 1 ≤ c*weight r)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re) (hρ : ‖P‖ < 1)
+    (d : ℕ) (hd : ∀ z, degree G z ≤ d) (τ : ℝ) (hτ : 0 < τ) :
+    ∃ J : ℕ, ∃ C₁ : ℝ, ∃ k₀ : ℕ, ∃ b : ℝ,
+      0 < J ∧ 0 < C₁ ∧ 3 ≤ k₀ ∧ 0 < b ∧
+      ∀ p : unitInterval, (pc G:ℝ)/2 ≤ (p:ℝ) → p < pc G →
+      ∀ k : ℕ, k₀ ≤ k → ∀ R : ℝ, 1 ≤ R →
+        b ≤ rawPathAverage G (walkKernel G hloc a) k root weight
+          (fun w => if w.Selected hloc p (J+⌈C₁*Real.log (((k:ℝ)+1)*R)⌉₊) (τ/R) then 1 else 0) := by
+  let x₀ : V := Classical.arbitrary V
+  have hρpos : 0 < ‖P‖ := by
+    have hh := (hdiag x₀).trans ((Complex.re_le_norm _).trans (matrixCoeff_norm_le P x₀ x₀))
+    linarith
+  have hK (x y : V) : 0 ≤ walkKernel G hloc a x y := by
+    simpa only [hP,Complex.ofReal_re] using (hpos x y).2
+  obtain ⟨J,hJ,C₁,hC₁,k₀,hJk,hcut⟩ := exists_full_selection_cutoffs ‖P‖ c d τ hρpos hρ hc (Nat.cast_nonneg d) hτ
+  obtain ⟨y,q,hqN,hq⟩ := exists_geodesic_length G hloc hconn x₀ (J+J)
+  let γ := geodesicSegment G q (J+J)
+  have hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ) :=
+    fun i => geodesicSegment_adj G q (J+J) hqN i.val i.isLt
+  have hinj : Function.Injective γ := geodesicSegment_injective G q (J+J) hqN hq
+  have hdiam := geodesicSegment_diameter G q (J+J) hqN
+  have hsep : ∀ s ∈ segmentOrbit G γ,
+      (∀ i, J ≤ (openSimpleGraph G Set.univ).dist (s 0) (s (segmentPositiveIndex J i))) ∧
+      (∀ i, J ≤ (openSimpleGraph G Set.univ).dist (s (Fin.last (J+J))) (s (segmentNegativeIndex J i))) ∧
+      2*J ≤ (openSimpleGraph G Set.univ).dist (s 0) (s (Fin.last (J+J))) := by
+    rintro s ⟨g,e,hge,rfl⟩
+    constructor
+    · intro i
+      change J ≤ (openSimpleGraph G Set.univ).dist (g (q.getVert 0)) (g (q.getVert (J+i.val)))
+      rw [graph_dist_aut G hconn hge,geodesic_getVert_dist _ q hq 0 (J+i.val) (by omega) (by omega)]
+      omega
+    constructor
+    · intro i
+      change J ≤ (openSimpleGraph G Set.univ).dist (g (q.getVert (J+J))) (g (q.getVert i.val))
+      rw [graph_dist_aut G hconn hge,SimpleGraph.dist_comm,
+        geodesic_getVert_dist _ q hq i.val (J+J) (by omega) (by omega)]
+      omega
+    · change 2*J ≤ (openSimpleGraph G Set.univ).dist (g (q.getVert 0)) (g (q.getVert (J+J)))
+      rw [graph_dist_aut G hconn hge,geodesic_getVert_dist _ q hq 0 (J+J) (by omega) (by omega)]
+      omega
+  obtain ⟨i,g,hg,hgx⟩ := hcov (γ (segmentCenter J))
+  obtain ⟨e,he⟩ := hg
+  have hx : root i ∈ vertexOrbit G (γ (segmentCenter J)) :=
+    ⟨g.symm,e.symm,isAut_symm G he,by rw [←hgx,g.symm_apply_apply]⟩
+  let b : ℝ := weight i * ((1/2)*(a^(J+J)*(1/(2*c))))
+  have hb : 0 < b := mul_pos (hw i) (mul_pos (by norm_num)
+    (mul_pos (pow_pos ha _) (by positivity)))
+  refine ⟨J,C₁,max k₀ 3,b,hJ,hC₁,le_max_right _ _,hb,?_⟩
+  intro p _hpnear hp k hk R hR
+  have hk₀ : k₀ ≤ k := (le_max_left _ _).trans hk
+  have hJk' : J+J ≤ k := hJk.trans hk₀
+  obtain ⟨L,rfl⟩ : ∃ L : ℕ, (J+L)+J = k := ⟨k-(J+J),by omega⟩
+  let m' := ⌈C₁*Real.log ((((((J+L)+J):ℕ):ℝ)+1)*R)⌉₊
+  obtain ⟨A,hA⟩ := operatorBounded_of_lt_pc G hloc hqt p hp
+  have hcost := hcut ((J+L)+J) hk₀ R hR
+  have hL : (J+L)+J-(J+J) = L := by omega
+  rw [hL] at hcost
+  have hcost' :
+      4*c*(earlyCollisionCost ‖P‖ J + suffixCollisionCost ‖P‖ c J*‖P‖^L) +
+      2*((3*(d:ℝ)+3)*((((J+L)+J):ℕ)+1)*Real.sqrt c*‖P‖^m')/(τ/R) +
+      2*c*(((d+1)^2:ℕ):ℝ)*Real.sqrt c*‖P‖^L ≤ 1/2 := by
+    simpa only [m',Nat.cast_pow,Nat.cast_add,Nat.cast_one] using hcost
+  have hl := rooted_selected_label_mass_lower G hloc a root weight htr hcov hw hw1 c hc hmin
+    P A hpos hsym hP hdiag p hA J L m' γ hinj hγ hdiam (root i) hx hρ hsep d hd (τ/R)
+    (div_pos hτ (lt_of_lt_of_le zero_lt_one hR))
+  have hhalf : (1/2:ℝ) ≤ ∑' ξ : {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m' // ξ.first (segmentCenter J) = root i},
+      if (labelCorridor G J L m' γ hγ ξ.1).Selected hloc p (J+m') (τ/R) then
+        escapeLabelProbability G hloc a γ (J+J) hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m' ξ.1 else 0 := by
+    linarith
+  have hqreturn := segmentReturnMass_uniform_lower G hloc a root weight (vertexAut G) (Set.Subset.refl _)
+    htr hcov hw hw1 c hc hmin P hpos hsym hP hdiag γ (Fin.last (J+J)) L
+  have hmraw := selected_raw_mass_of_label_mass G hloc a ha hK J L m' γ hγ hdiam (root i) hx
+    (1/(2*c)) (by positivity) hqreturn p (τ/R) (1/2) hhalf
+  have hroot := mul_le_mul_of_nonneg_left hmraw (hw i).le
+  apply hroot.trans
+  unfold rawPathAverage
+  have hall (r : ι) : 0 ≤ weight r * ∑' w : RawFrom (openSimpleGraph G Set.univ) ((J+L)+J) (root r),
+      w.2.mass (walkKernel G hloc a) *
+        (if RawCorridor.Selected hloc p (J+m') (τ/R) ⟨root r,w⟩ then 1 else 0) := by
+    apply mul_nonneg (hw r).le
+    apply tsum_nonneg
+    intro w
+    apply mul_nonneg (w.2.mass_nonneg _ hK)
+    split_ifs <;> norm_num
+  have hi := Finset.single_le_sum (s := Finset.univ) (fun r _ => hall r) (Finset.mem_univ i)
+  simpa only [mul_ite,mul_one,mul_zero] using hi
+
+end FullPathSelection
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Escape/PositiveEscapes.lean b/lean/OAI/Probability/UnimodularPercolation/Escape/PositiveEscapes.lean
index 0d57d840b58800a484f9cc4bcd179d6ac6d61bfa..f3769f2fe8151c0a4fd94f4f35415dc0c0995f30 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Escape/PositiveEscapes.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Escape/PositiveEscapes.lean
@@ -1,1221 +1,1221 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Escape.CollisionTails
-
-namespace OAI
-
-/-! Positive Escapes for bond percolation. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-section DependentFiniteUnion
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem finite_dependent_event_union {β δ : Type*} [Fintype β] [Fintype δ]
-    {γ : β → Type*} [∀ b, Fintype (γ b)]
-    (μ : (b : β) → γ b → ℝ) (hμ : ∀ b c, 0 ≤ μ b c)
-    (B : (b : β) → γ b → δ → Prop) :
-    (∑ b, ∑ c, if ∃ d, B b c d then μ b c else 0) ≤
-      ∑ d, ∑ b, ∑ c, if B b c d then μ b c else 0 := by
-  have hh := finite_nonnegative_event_union
-    (fun ξ : Sigma γ => μ ξ.1 ξ.2)
-    (fun ξ => hμ ξ.1 ξ.2)
-    (fun ξ : Sigma γ => B ξ.1 ξ.2)
-  simpa only [Fintype.sum_sigma] using hh
-
-end DependentFiniteUnion
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section ActualFinitePositionHits
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {I ι S T : Type*} [Fintype I] [Fintype ι] [Fintype S] [Fintype T]
-
- 
-
-theorem raw_anchor_finite_collision_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G)
-    (htr : RootTransport root weight Γ) (hw : ∀ r, 0 ≤ weight r) (hw1 : ∑ r, weight r = 1)
-    (hcov : ∀ x, ∃ r, ∃ g ∈ Γ, g (root r) = x)
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ r, 1 ≤ c*weight r)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (ep : I)
-    (j : S → I) (z : T → V)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (L : ℕ) (y : V) :
-    (∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
-      ∑' h : segmentAnchors G hloc γ n hdiam ep mid.1,
-        if ∃ st : S × T, h.val (j st.1) = z st.2 then mid.2.mass (walkKernel G hloc a)/
-          (segmentAnchors G hloc γ n hdiam ep (γ ep)).card else 0) ≤
-      (Fintype.card S : ℝ)*(Fintype.card T : ℝ)*Real.sqrt c*‖P‖^L := by
-  let (v : V) : Finite ((openSimpleGraph G Set.univ).neighborSet v) := finite_openSimpleGraph_neighbors G hloc v
-  let : Fintype (RawFrom (openSimpleGraph G Set.univ) L y) := Fintype.ofFinite _
-  have hK (x y : V) : 0 ≤ walkKernel G hloc a x y := by
-    simpa only [hP,Complex.ofReal_re] using (hpos x y).2
-  simp only [tsum_fintype]
-  have hu := finite_dependent_event_union
-    (β := RawFrom (openSimpleGraph G Set.univ) L y)
-    (γ := fun mid => segmentAnchors G hloc γ n hdiam ep mid.1)
-    (δ := S × T)
-    (fun mid _ => mid.2.mass (walkKernel G hloc a)/(segmentAnchors G hloc γ n hdiam ep (γ ep)).card)
-    (fun mid _ => div_nonneg (mid.2.mass_nonneg _ hK) (Nat.cast_nonneg _))
-    (fun _ h st => h.val (j st.1) = z st.2)
-  apply hu.trans
-  calc
-    _ ≤ ∑ _st : S × T, Real.sqrt c*‖P‖^L := by
-      apply Finset.sum_le_sum
-      intro st _
-      obtain ⟨H,hH,hHn⟩ := exists_anchoredPositionOperator_uniform G hloc root weight Γ hΓ htr hw hw1 hcov
-        c hc hmin γ n hdiam ep (j st.1)
-      simpa only [tsum_fintype] using raw_anchor_point_collision_bound G hloc a γ n hdiam ep (j st.1)
-        P H hpos hP hH (Real.sqrt c) hHn L y (z st.2)
-    _ = _ := by simp only [Finset.sum_const,Finset.card_univ,Fintype.card_prod,Nat.cast_mul,nsmul_eq_mul]; ring
-
-end ActualFinitePositionHits
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section EarlyFiniteCollisions
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {S : Type*} [Fintype S]
-
- 
-
-theorem raw_hits_distant_set_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ)) (hρ : ‖P‖ < 1)
-    (m J : ℕ) (x : V) (z : S → V) (hD : ∀ s, J ≤ (openSimpleGraph G Set.univ).dist x (z s)) :
-    (∑' w : RawFrom (openSimpleGraph G Set.univ) m x,
-      if ∃ s, z s ∈ w.2.vertices then w.2.mass (walkKernel G hloc a) else 0) ≤
-      (Fintype.card S : ℝ)*‖P‖^J/(1-‖P‖) := by
-  let (v : V) : Finite ((openSimpleGraph G Set.univ).neighborSet v) := finite_openSimpleGraph_neighbors G hloc v
-  let : Fintype (RawFrom (openSimpleGraph G Set.univ) m x) := Fintype.ofFinite _
-  have hK (x y : V) : 0 ≤ walkKernel G hloc a x y := by
-    simpa only [hP,Complex.ofReal_re] using (hpos x y).2
-  have he (w : RawFrom (openSimpleGraph G Set.univ) m x) :
-      (∃ s, z s ∈ w.2.vertices) ↔ ∃ si : S × Fin (m+1), w.2.position si.2.val = z si.1 := by
-    constructor
-    · rintro ⟨s,hs⟩
-      obtain ⟨i,hi,he⟩ := w.2.mem_vertices_iff_position.mp hs
-      exact ⟨(s,⟨i,by omega⟩),he⟩
-    · rintro ⟨⟨s,i⟩,he⟩
-      exact ⟨s,he ▸ w.2.position_mem_vertices i.val⟩
-  simp only [he,tsum_fintype]
-  have hu := finite_nonnegative_event_union
-    (fun w : RawFrom (openSimpleGraph G Set.univ) m x => w.2.mass (walkKernel G hloc a))
-    (fun w => w.2.mass_nonneg _ hK)
-    (fun w (si : S × Fin (m+1)) => w.2.position si.2.val = z si.1)
-  apply hu.trans
-  rw [Fintype.sum_prod_type]
-  calc
-    _ ≤ ∑ _s : S, ∑ i : Fin (m+1), if J ≤ i.val then ‖P‖^i.val else 0 := by
-      apply Finset.sum_le_sum
-      intro s _
-      apply Finset.sum_le_sum
-      intro i _
-      have hh := raw_position_probability G hloc a P hpos hP i.val m (by omega) x (z s)
-      simp only [tsum_fintype] at hh
-      rw [hh]
-      split_ifs with hi
-      · exact matrixCoeff_power_le P i.val x (z s)
-      · exact (walkPower_zero_of_lt_dist G hloc a P hpos hP i.val x (z s) (by have hd := hD s; omega)).le
-    _ ≤ ∑ _s : S, ‖P‖^J/(1-‖P‖) := Finset.sum_le_sum (fun _ _ =>
-      finite_geometric_cutoff_bound ‖P‖ (norm_nonneg _) hρ J (m+1))
-    _ = _ := by simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]; ring
-
-end EarlyFiniteCollisions
-end Percolation
-
- 
-namespace Percolation
-section QuantitativeEarlyCutoff
-
- 
-
-noncomputable def earlyCollisionCost (ρ : ℝ) (J : ℕ) : ℝ :=
-  2*((J:ℝ)+1)*ρ^J/(1-ρ) + 2*ρ^J/(1-ρ)^2
-
-theorem earlyCollisionCost_tendsto (ρ : ℝ) (hρ : 0 ≤ ρ) (hρ1 : ρ < 1) :
-    Filter.Tendsto (earlyCollisionCost ρ) Filter.atTop (nhds 0) := by
-  have hn : ‖ρ‖ < 1 := by simpa only [Real.norm_eq_abs,abs_of_nonneg hρ] using hρ1
-  have ht : Filter.Tendsto (fun J : ℕ => (J:ℝ)*ρ^J) Filter.atTop (nhds 0) := by
-    simpa only [pow_one] using (summable_pow_mul_geometric_of_norm_lt_one 1 hn).tendsto_atTop_zero
-  have hp := tendsto_pow_atTop_nhds_zero_of_lt_one hρ hρ1
-  have hh := (((ht.add hp).const_mul 2).div_const (1-ρ)).add ((hp.const_mul 2).div_const ((1-ρ)^2))
-  have he : earlyCollisionCost ρ = (fun J : ℕ => 2*((J:ℝ)*ρ^J+ρ^J)/(1-ρ)+2*ρ^J/(1-ρ)^2) := by
-    funext J
-    unfold earlyCollisionCost
-    ring
-  rw [he]
-  simpa only [zero_add,mul_zero,zero_div] using hh
-
-theorem exists_earlyCollisionCutoff (ρ : ℝ) (hρ : 0 ≤ ρ) (hρ1 : ρ < 1) (ε : ℝ) (hε : 0 < ε) :
-    ∃ J₀ : ℕ, ∀ J : ℕ, J₀ ≤ J → earlyCollisionCost ρ J < ε := by
-  exact Filter.eventually_atTop.mp ((earlyCollisionCost_tendsto ρ hρ hρ1).eventually (gt_mem_nhds hε))
-
-end QuantitativeEarlyCutoff
-end Percolation
-
- 
-namespace Percolation
-section WeightedBinaryUnion
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem finite_weighted_event_or {α : Type*} [Fintype α]
-    (μ : α → ℝ) (hμ : ∀ a, 0 ≤ μ a) (B C : α → Prop) [DecidablePred B] [DecidablePred C] :
-    (∑ a, if B a ∨ C a then μ a else 0) ≤
-      (∑ a, if B a then μ a else 0) + ∑ a, if C a then μ a else 0 := by
-  rw [←Finset.sum_add_distrib]
-  apply Finset.sum_le_sum
-  intro a _
-  by_cases hb : B a <;> by_cases hc : C a <;> simp only [hb,hc,or_self,or_false,false_or,ite_true,ite_false,add_zero,zero_add,le_refl]
-  exact le_add_of_nonneg_right (hμ a)
-
-end WeightedBinaryUnion
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section ActualEarlyExperiment
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem raw_early_collision_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ)) (hρ : ‖P‖ < 1)
-    (m L J : ℕ) (x y : V) (pos neg : Fin (J+1) → V)
-    (hposD : ∀ s, J ≤ (openSimpleGraph G Set.univ).dist x (pos s))
-    (hnegD : ∀ s, J ≤ (openSimpleGraph G Set.univ).dist y (neg s))
-    (hxy : 2*J ≤ (openSimpleGraph G Set.univ).dist x y) :
-    (∑' w : RawFrom (openSimpleGraph G Set.univ) m x × RawFrom (openSimpleGraph G Set.univ) L y,
-      if ((∃ s, pos s ∈ w.1.2.vertices) ∨ (∃ s, neg s ∈ w.2.2.vertices)) ∨
-          (∃ z, z ∈ w.1.2.vertices ∧ z ∈ w.2.2.vertices) then
-        w.1.2.mass (walkKernel G hloc a)*w.2.2.mass (walkKernel G hloc a) else 0) ≤
-      earlyCollisionCost ‖P‖ J := by
-  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
-  let : Fintype (RawFrom (openSimpleGraph G Set.univ) m x) := Fintype.ofFinite _
-  let : Fintype (RawFrom (openSimpleGraph G Set.univ) L y) := Fintype.ofFinite _
-  have hK (x y : V) : 0 ≤ walkKernel G hloc a x y := by
-    simpa only [hP,Complex.ofReal_re] using (hpos x y).2
-  have hu := finite_weighted_event_or
-    (fun w : RawFrom (openSimpleGraph G Set.univ) m x × RawFrom (openSimpleGraph G Set.univ) L y =>
-      w.1.2.mass (walkKernel G hloc a)*w.2.2.mass (walkKernel G hloc a))
-    (fun w => mul_nonneg (w.1.2.mass_nonneg _ hK) (w.2.2.mass_nonneg _ hK))
-    (fun w => (∃ s, pos s ∈ w.1.2.vertices) ∨ (∃ s, neg s ∈ w.2.2.vertices))
-    (fun w => ∃ z, z ∈ w.1.2.vertices ∧ z ∈ w.2.2.vertices)
-  have hp := finite_product_failure_union
-    (fun w : RawFrom (openSimpleGraph G Set.univ) m x => w.2.mass (walkKernel G hloc a))
-    (fun w : RawFrom (openSimpleGraph G Set.univ) L y => w.2.mass (walkKernel G hloc a))
-    (fun w => w.2.mass_nonneg _ hK) (fun w => w.2.mass_nonneg _ hK)
-    (by simpa only [tsum_fintype] using rawFrom_probability_sum G hloc a m x)
-    (by simpa only [tsum_fintype] using rawFrom_probability_sum G hloc a L y)
-    (fun w => ∃ s, pos s ∈ w.2.vertices) (fun w => ∃ s, neg s ∈ w.2.vertices)
-  have hx := raw_hits_distant_set_bound G hloc a P hpos hP hρ m J x pos hposD
-  have hy := raw_hits_distant_set_bound G hloc a P hpos hP hρ L J y neg hnegD
-  have hi := raw_intersection_uniform_bound G hloc a P hpos hsym hP hρ m L J x y hxy
-  simp only [tsum_fintype,Fintype.card_fin,Nat.cast_add,Nat.cast_one] at hx hy hi ⊢
-  have hx' : (∑ w : RawFrom (openSimpleGraph G Set.univ) m x,
-      if ∃ s, pos s ∈ w.2.vertices then w.2.mass (walkKernel G hloc a) else 0) ≤
-      ((J:ℝ)+1)*‖P‖^J/(1-‖P‖) := by
-    convert hx using 1
-    apply Finset.sum_congr rfl
-    intro w _
-    split_ifs <;> rfl
-  have hy' : (∑ w : RawFrom (openSimpleGraph G Set.univ) L y,
-      if ∃ s, neg s ∈ w.2.vertices then w.2.mass (walkKernel G hloc a) else 0) ≤
-      ((J:ℝ)+1)*‖P‖^J/(1-‖P‖) := by
-    convert hy using 1
-    apply Finset.sum_congr rfl
-    intro w _
-    split_ifs <;> rfl
-  have hp' : (∑ w : RawFrom (openSimpleGraph G Set.univ) m x × RawFrom (openSimpleGraph G Set.univ) L y,
-      if (∃ s, pos s ∈ w.1.2.vertices) ∨ (∃ s, neg s ∈ w.2.2.vertices) then
-        w.1.2.mass (walkKernel G hloc a)*w.2.2.mass (walkKernel G hloc a) else 0) ≤
-      (∑ w : RawFrom (openSimpleGraph G Set.univ) m x,
-        if ∃ s, pos s ∈ w.2.vertices then w.2.mass (walkKernel G hloc a) else 0) +
-      (∑ w : RawFrom (openSimpleGraph G Set.univ) L y,
-        if ∃ s, neg s ∈ w.2.vertices then w.2.mass (walkKernel G hloc a) else 0) := by
-    convert hp using 1
-    · apply Finset.sum_congr rfl
-      intro w _
-      split_ifs <;> rfl
-    · congr 1 <;> apply Finset.sum_congr rfl <;> intro w _ <;> split_ifs <;> rfl
-  have hr : (((J:ℝ)+1)*‖P‖^J/(1-‖P‖) + ((J:ℝ)+1)*‖P‖^J/(1-‖P‖)) +
-      2*‖P‖^J/(1-‖P‖)^2 = earlyCollisionCost ‖P‖ J := by
-    unfold earlyCollisionCost
-    ring
-  have hh := hu.trans ((add_le_add (hp'.trans (add_le_add hx' hy')) hi).trans_eq hr)
-  exact hh
-
-end ActualEarlyExperiment
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section ConditioningCost
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {I : Type*} [Fintype I]
-
- 
-
-theorem raw_anchor_event_le_unconditioned (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (hK : ∀ x y, 0 ≤ walkKernel G hloc a x y)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (ep : I)
-    (m L : ℕ) (x y : V)
-    (B : RawFrom (openSimpleGraph G Set.univ) m x → RawFrom (openSimpleGraph G Set.univ) L y → Prop) [∀ tail mid, Decidable (B tail mid)] :
-    (∑' tail : RawFrom (openSimpleGraph G Set.univ) m x,
-      ∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
-        ∑' _h : segmentAnchors G hloc γ n hdiam ep mid.1,
-          if B tail mid then tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a)/
-            (segmentAnchors G hloc γ n hdiam ep (γ ep)).card else 0) ≤
-    (∑' w : RawFrom (openSimpleGraph G Set.univ) m x × RawFrom (openSimpleGraph G Set.univ) L y,
-      if B w.1 w.2 then w.1.2.mass (walkKernel G hloc a)*w.2.2.mass (walkKernel G hloc a) else 0) := by
-  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
-  let (k : ℕ) (z : V) : Fintype (RawFrom (openSimpleGraph G Set.univ) k z) := Fintype.ofFinite _
-  have hN : 0 < ((segmentAnchors G hloc γ n hdiam ep (γ ep)).card : ℝ) := by
-    exact_mod_cast Finset.card_pos.mpr ((segmentAnchors_nonempty_iff G hloc γ n hdiam ep (γ ep)).mpr
-      ⟨Equiv.refl V,Equiv.refl E,isAut_refl G,rfl⟩)
-  simp only [tsum_fintype,Fintype.sum_prod_type]
-  apply Finset.sum_le_sum
-  intro tail _
-  apply Finset.sum_le_sum
-  intro mid _
-  by_cases hb : B tail mid
-  · simp only [ite_eq_left hb,Finset.sum_const,Finset.card_univ,Fintype.card_coe,nsmul_eq_mul]
-    have hc : ((segmentAnchors G hloc γ n hdiam ep mid.1).card : ℝ) ≤
-        (segmentAnchors G hloc γ n hdiam ep (γ ep)).card := by
-      rw [segmentAnchors_card_eq_indicator G hloc γ n hdiam ep mid.1]
-      split_ifs
-      · exact le_rfl
-      · exact_mod_cast Nat.zero_le _
-    rw [←mul_div_assoc]
-    apply (div_le_iff₀ hN).mpr
-    have hh := mul_le_mul_of_nonneg_right hc
-      (mul_nonneg (tail.2.mass_nonneg _ hK) (mid.2.mass_nonneg _ hK))
-    nlinarith
-  · simp only [ite_eq_right hb,Finset.sum_const_zero]
-    exact le_rfl
-
-end ConditioningCost
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section EarlyAnchoredCollision
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {I : Type*} [Fintype I]
-
- 
-
-theorem raw_anchor_early_collision_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (ep : I)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ)) (hρ : ‖P‖ < 1)
-    (m L J : ℕ) (x y : V) (pos neg : Fin (J+1) → V)
-    (hposD : ∀ s, J ≤ (openSimpleGraph G Set.univ).dist x (pos s))
-    (hnegD : ∀ s, J ≤ (openSimpleGraph G Set.univ).dist y (neg s))
-    (hxy : 2*J ≤ (openSimpleGraph G Set.univ).dist x y) :
-    (∑' tail : RawFrom (openSimpleGraph G Set.univ) m x,
-      ∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
-        ∑' _h : segmentAnchors G hloc γ n hdiam ep mid.1,
-          if ((∃ s, pos s ∈ tail.2.vertices) ∨ (∃ s, neg s ∈ mid.2.vertices)) ∨
-              (∃ z, z ∈ tail.2.vertices ∧ z ∈ mid.2.vertices) then
-            tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a)/
-              (segmentAnchors G hloc γ n hdiam ep (γ ep)).card else 0) ≤
-      earlyCollisionCost ‖P‖ J := by
-  have hK (x y : V) : 0 ≤ walkKernel G hloc a x y := by
-    simpa only [hP,Complex.ofReal_re] using (hpos x y).2
-  exact (raw_anchor_event_le_unconditioned G hloc a hK γ n hdiam ep m L x y
-    (fun tail mid => ((∃ s, pos s ∈ tail.2.vertices) ∨ (∃ s, neg s ∈ mid.2.vertices)) ∨
-      (∃ z, z ∈ tail.2.vertices ∧ z ∈ mid.2.vertices))).trans
-    (raw_early_collision_bound G hloc a P hpos hsym hP hρ m L J x y pos neg hposD hnegD hxy)
-
- 
-
-theorem raw_tail_integrate_anchor_event (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (ep : I)
-    (m L : ℕ) (x y : V)
-    (B : (mid : RawFrom (openSimpleGraph G Set.univ) L y) →
-      segmentAnchors G hloc γ n hdiam ep mid.1 → Prop) [∀ mid h, Decidable (B mid h)] :
-    (∑' tail : RawFrom (openSimpleGraph G Set.univ) m x,
-      ∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
-        ∑' h : segmentAnchors G hloc γ n hdiam ep mid.1,
-          if B mid h then tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a)/
-            (segmentAnchors G hloc γ n hdiam ep (γ ep)).card else 0) =
-    (∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
-      ∑' h : segmentAnchors G hloc γ n hdiam ep mid.1,
-        if B mid h then mid.2.mass (walkKernel G hloc a)/
-          (segmentAnchors G hloc γ n hdiam ep (γ ep)).card else 0) := by
-  have he (tail : RawFrom (openSimpleGraph G Set.univ) m x)
-      (mid : RawFrom (openSimpleGraph G Set.univ) L y)
-      (h : segmentAnchors G hloc γ n hdiam ep mid.1) :
-      (if B mid h then tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a)/
-        (segmentAnchors G hloc γ n hdiam ep (γ ep)).card else 0) =
-      tail.2.mass (walkKernel G hloc a) *
-        (if B mid h then mid.2.mass (walkKernel G hloc a)/
-          (segmentAnchors G hloc γ n hdiam ep (γ ep)).card else 0) := by
-    split_ifs <;> ring
-  simp only [he,tsum_mul_left,tsum_mul_right,rawFrom_probability_sum G hloc a,one_mul]
-
-end EarlyAnchoredCollision
-end Percolation
-
- 
-namespace Percolation
-section DependentUnion
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem finite_threefold_event_or {α β : Type*} {γ : β → Type*}
-    [Fintype α] [Fintype β] [∀ b, Fintype (γ b)]
-    (μ : α → (b : β) → γ b → ℝ) (hμ : ∀ a b c, 0 ≤ μ a b c)
-    (B C : α → (b : β) → γ b → Prop)
-    [∀ a b c, Decidable (B a b c)] [∀ a b c, Decidable (C a b c)] :
-    (∑ a, ∑ b, ∑ c, if B a b c ∨ C a b c then μ a b c else 0) ≤
-      (∑ a, ∑ b, ∑ c, if B a b c then μ a b c else 0) +
-      (∑ a, ∑ b, ∑ c, if C a b c then μ a b c else 0) := by
-  simp only [←Finset.sum_add_distrib]
-  apply Finset.sum_le_sum
-  intro a _
-  apply Finset.sum_le_sum
-  intro b _
-  apply Finset.sum_le_sum
-  intro c _
-  by_cases hb : B a b c <;> by_cases hc : C a b c <;>
-    simp only [hb,hc,or_self,or_false,false_or,ite_true,ite_false,add_zero,zero_add,le_refl]
-  exact le_add_of_nonneg_right (hμ a b c)
-
-end DependentUnion
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section CombinedRawAvoidance
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {I ι : Type*} [Fintype I] [Fintype ι]
-
- 
-
-def firstEscapeCollision (G : BondGraph V E) (J m L : ℕ) (x y : V)
-    (pos neg : Fin (J+1) → V) (j : Fin (J+1) → I)
-    (tail : RawFrom (openSimpleGraph G Set.univ) m x)
-    (mid : RawFrom (openSimpleGraph G Set.univ) L y) (h : I → V) : Prop :=
-  (((∃ s, pos s ∈ tail.2.vertices) ∨ (∃ s, neg s ∈ mid.2.vertices)) ∨
-    (∃ z, z ∈ tail.2.vertices ∧ z ∈ mid.2.vertices)) ∨
-  ((∃ st : Fin (J+1) × Fin (J+1), h (j st.1) = neg st.2) ∨
-    (∃ ij : Fin (m+1) × Fin (J+1), tail.2.position ij.1.val = h (j ij.2)))
-
-noncomputable def suffixCollisionCost (ρ c : ℝ) (J : ℕ) : ℝ :=
-  ((J:ℝ)+1)^2*Real.sqrt c + ((J:ℝ)+1)*Real.sqrt c/(1-ρ)
-
- 
-
-theorem raw_first_escape_collision_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G)
-    (htr : RootTransport root weight Γ) (hw : ∀ r, 0 ≤ weight r) (hw1 : ∑ r, weight r = 1)
-    (hcov : ∀ x, ∃ r, ∃ g ∈ Γ, g (root r) = x)
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ r, 1 ≤ c*weight r)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (ep : I)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ)) (hρ : ‖P‖ < 1)
-    (m L J : ℕ) (x y : V) (pos neg : Fin (J+1) → V) (j : Fin (J+1) → I)
-    (hposD : ∀ s, J ≤ (openSimpleGraph G Set.univ).dist x (pos s))
-    (hnegD : ∀ s, J ≤ (openSimpleGraph G Set.univ).dist y (neg s))
-    (hxy : 2*J ≤ (openSimpleGraph G Set.univ).dist x y) :
-    (∑' tail : RawFrom (openSimpleGraph G Set.univ) m x,
-      ∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
-        ∑' h : segmentAnchors G hloc γ n hdiam ep mid.1,
-          if firstEscapeCollision G J m L x y pos neg j tail mid h.val then
-            tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a)/
-              (segmentAnchors G hloc γ n hdiam ep (γ ep)).card else 0) ≤
-      earlyCollisionCost ‖P‖ J + suffixCollisionCost ‖P‖ c J*‖P‖^L := by
-  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
-  let (k : ℕ) (z : V) : Fintype (RawFrom (openSimpleGraph G Set.univ) k z) := Fintype.ofFinite _
-  have hK (x y : V) : 0 ≤ walkKernel G hloc a x y := by
-    simpa only [hP,Complex.ofReal_re] using (hpos x y).2
-  let μ := fun (tail : RawFrom (openSimpleGraph G Set.univ) m x)
-      (mid : RawFrom (openSimpleGraph G Set.univ) L y)
-      (_h : segmentAnchors G hloc γ n hdiam ep mid.1) =>
-    tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a)/
-      (segmentAnchors G hloc γ n hdiam ep (γ ep)).card
-  have hμ : ∀ tail mid h, 0 ≤ μ tail mid h := fun tail mid _ =>
-    div_nonneg (mul_nonneg (tail.2.mass_nonneg _ hK) (mid.2.mass_nonneg _ hK)) (Nat.cast_nonneg _)
-  let B := fun (tail : RawFrom (openSimpleGraph G Set.univ) m x)
-      (mid : RawFrom (openSimpleGraph G Set.univ) L y)
-      (_h : segmentAnchors G hloc γ n hdiam ep mid.1) =>
-    ((∃ s, pos s ∈ tail.2.vertices) ∨ (∃ s, neg s ∈ mid.2.vertices)) ∨
-      (∃ z, z ∈ tail.2.vertices ∧ z ∈ mid.2.vertices)
-  let C := fun (_tail : RawFrom (openSimpleGraph G Set.univ) m x)
-      (mid : RawFrom (openSimpleGraph G Set.univ) L y)
-      (h : segmentAnchors G hloc γ n hdiam ep mid.1) =>
-    ∃ st : Fin (J+1) × Fin (J+1), h.val (j st.1) = neg st.2
-  let D := fun (tail : RawFrom (openSimpleGraph G Set.univ) m x)
-      (mid : RawFrom (openSimpleGraph G Set.univ) L y)
-      (h : segmentAnchors G hloc γ n hdiam ep mid.1) =>
-    ∃ ij : Fin (m+1) × Fin (J+1), tail.2.position ij.1.val = h.val (j ij.2)
-  have hu := finite_threefold_event_or μ hμ B (fun tail mid h => C tail mid h ∨ D tail mid h)
-  have hv := finite_threefold_event_or μ hμ C D
-  have he := raw_anchor_early_collision_bound G hloc a γ n hdiam ep P hpos hsym hP hρ
-    m L J x y pos neg hposD hnegD hxy
-  have hn := raw_anchor_finite_collision_bound G hloc a root weight Γ hΓ htr hw hw1 hcov
-    c hc hmin γ n hdiam ep j neg P hpos hP L y
-  rw [←raw_tail_integrate_anchor_event G hloc a γ n hdiam ep m L x y
-    (fun _ h => ∃ st : Fin (J+1) × Fin (J+1), h.val (j st.1) = neg st.2)] at hn
-  have ht := raw_tail_whole_suffix_collision_bound G hloc a root weight Γ hΓ htr hw hw1 hcov
-    c hc hmin γ n hdiam ep j P hpos hP hρ m L x y
-  simp only [tsum_fintype,Fintype.card_fin,Nat.cast_add,Nat.cast_one] at he hn ht ⊢
-  have hr : earlyCollisionCost ‖P‖ J +
-      ((((J:ℝ)+1)*((J:ℝ)+1)*Real.sqrt c*‖P‖^L) +
-        ((J:ℝ)+1)*Real.sqrt c*‖P‖^L/(1-‖P‖)) =
-      earlyCollisionCost ‖P‖ J + suffixCollisionCost ‖P‖ c J*‖P‖^L := by
-    unfold suffixCollisionCost
-    ring
-  have hh := hu.trans ((add_le_add he (hv.trans (add_le_add hn ht))).trans_eq hr)
-  apply le_trans ?_ hh
-  apply le_of_eq
-  apply Finset.sum_congr rfl
-  intro tail _
-  apply Finset.sum_congr rfl
-  intro mid _
-  apply Finset.sum_congr rfl
-  intro h _
-  unfold firstEscapeCollision B C D μ
-  split_ifs <;> rfl
-
-end CombinedRawAvoidance
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section NormalizedFirstEscape
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {I ι : Type*} [Fintype I] [Fintype ι]
-
- 
-
-theorem conditional_first_escape_collision_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G)
-    (htr : RootTransport root weight Γ) (hw : ∀ r, 0 < weight r) (hw1 : ∑ r, weight r = 1)
-    (hcov : ∀ x, ∃ r, ∃ g ∈ Γ, g (root r) = x)
-    (c : ℝ) (hc : 0 < c) (hmin : ∀ r, 1 ≤ c*weight r)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (ep : I)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re) (hρ : ‖P‖ < 1)
-    (m L J : ℕ) (x y : V) (hy : y ∈ vertexOrbit G (γ ep))
-    (pos neg : Fin (J+1) → V) (j : Fin (J+1) → I)
-    (hposD : ∀ s, J ≤ (openSimpleGraph G Set.univ).dist x (pos s))
-    (hnegD : ∀ s, J ≤ (openSimpleGraph G Set.univ).dist y (neg s))
-    (hxy : 2*J ≤ (openSimpleGraph G Set.univ).dist x y) :
-    (∑' tail : RawFrom (openSimpleGraph G Set.univ) m x,
-      ∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
-        ∑' h : segmentAnchors G hloc γ n hdiam ep mid.1,
-          if firstEscapeCollision G J m L x y pos neg j tail mid h.val then
-            tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a)/
-              (segmentAnchors G hloc γ n hdiam ep (γ ep)).card else 0) /
-        segmentReturnMass G hloc a γ ep L y ≤
-      2*c*(earlyCollisionCost ‖P‖ J + suffixCollisionCost ‖P‖ c J*‖P‖^L) := by
-  have hb := raw_first_escape_collision_bound G hloc a root weight Γ hΓ htr
-    (fun r => (hw r).le) hw1 hcov c hc.le hmin γ n hdiam ep P hpos hsym hP hρ
-    m L J x y pos neg j hposD hnegD hxy
-  have hq := segmentReturnMass_uniform_lower G hloc a root weight Γ hΓ htr hcov hw hw1 c hc hmin
-    P hpos hsym hP hdiag γ ep L
-  rw [←segmentReturnMass_on_orbit G hloc a γ ep L y hy P hpos hP] at hq
-  have htc : 0 < 2*c := mul_pos (by norm_num) hc
-  have hqp : 0 < segmentReturnMass G hloc a γ ep L y := (one_div_pos.mpr htc).trans_le hq
-  apply (div_le_iff₀ hqp).mpr
-  apply hb.trans
-  have hB : 0 ≤ earlyCollisionCost ‖P‖ J + suffixCollisionCost ‖P‖ c J*‖P‖^L := by
-    unfold earlyCollisionCost suffixCollisionCost
-    have : 0 < 1-‖P‖ := sub_pos.mpr hρ
-    positivity
-  have h1 : 1 ≤ (2*c)*segmentReturnMass G hloc a γ ep L y := by
-    exact (div_le_iff₀ htc).mp hq |>.trans_eq (mul_comm _ _)
-  have hh := mul_le_mul_of_nonneg_right h1 hB
-  nlinarith only [hh]
-
-end NormalizedFirstEscape
-end Percolation
-
- 
-namespace Percolation
-section FirstEscapeChoiceOrder
-
- 
-
-theorem exists_first_escape_uniform_cutoffs (ρ c ε : ℝ)
-    (hρ : 0 ≤ ρ) (hρ1 : ρ < 1) (hc : 0 < c) (hε : 0 < ε) :
-    ∃ J : ℕ, 0 < J ∧ ∃ L₀ : ℕ, ∀ L : ℕ, L₀ ≤ L →
-      2*c*(earlyCollisionCost ρ J + suffixCollisionCost ρ c J*ρ^L) < ε := by
-  have hδ : 0 < ε/(4*c) := div_pos hε (by positivity)
-  obtain ⟨J₀,hJ₀⟩ := exists_earlyCollisionCutoff ρ hρ hρ1 (ε/(4*c)) hδ
-  let J := max J₀ 1
-  have hJ : 0 < J := lt_of_lt_of_le (by omega : 0 < 1) (le_max_right _ _)
-  have he : earlyCollisionCost ρ J < ε/(4*c) := hJ₀ J (le_max_left _ _)
-  have hel : 2*c*earlyCollisionCost ρ J < ε := by
-    have hh := (lt_div_iff₀ (by positivity : 0 < 4*c)).mp he
-    nlinarith
-  have ht : Filter.Tendsto
-      (fun L : ℕ => 2*c*(earlyCollisionCost ρ J + suffixCollisionCost ρ c J*ρ^L))
-      Filter.atTop (nhds (2*c*earlyCollisionCost ρ J)) := by
-    have hh := ((tendsto_pow_atTop_nhds_zero_of_lt_one hρ hρ1).const_mul
-      (suffixCollisionCost ρ c J)).const_add (earlyCollisionCost ρ J)
-    have hh' := hh.const_mul (2*c)
-    simpa only [mul_zero,add_zero] using hh'
-  obtain ⟨L₀,hL₀⟩ := Filter.eventually_atTop.mp (ht.eventually (gt_mem_nhds hel))
-  exact ⟨J,hJ,L₀,hL₀⟩
-
-end FirstEscapeChoiceOrder
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section LiteralConditionalMass
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {I : Type*} [Fintype I]
-
- 
-
-theorem raw_tail_anchor_total_mass (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (ep : I)
-    (m L : ℕ) (x y : V) :
-    (∑' tail : RawFrom (openSimpleGraph G Set.univ) m x,
-      ∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
-        ∑' _h : segmentAnchors G hloc γ n hdiam ep mid.1,
-          tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a)/
-            (segmentAnchors G hloc γ n hdiam ep (γ ep)).card) =
-      segmentReturnMass G hloc a γ ep L y := by
-  have hN : ((segmentAnchors G hloc γ n hdiam ep (γ ep)).card : ℝ) ≠ 0 := by
-    apply ne_of_gt
-    exact_mod_cast Finset.card_pos.mpr ((segmentAnchors_nonempty_iff G hloc γ n hdiam ep (γ ep)).mpr
-      ⟨Equiv.refl V,Equiv.refl E,isAut_refl G,rfl⟩)
-  have hh := raw_tail_integrate_anchor_event G hloc a γ n hdiam ep m L x y (fun _ _ => True)
-  simp only [ite_true] at hh
-  rw [hh]
-  unfold segmentReturnMass
-  apply tsum_congr
-  intro mid
-  simp only [tsum_fintype,Finset.sum_const,Finset.card_univ,Fintype.card_coe,nsmul_eq_mul]
-  rw [segmentAnchors_card_eq_indicator G hloc γ n hdiam ep mid.1]
-  split_ifs
-  · field_simp
-  · simp only [Nat.cast_zero,zero_mul]
-
- 
-
-theorem raw_tail_anchor_conditional_mass (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (ep : I)
-    (m L : ℕ) (x y : V) (hq : segmentReturnMass G hloc a γ ep L y ≠ 0) :
-    (∑' tail : RawFrom (openSimpleGraph G Set.univ) m x,
-      ∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
-        ∑' _h : segmentAnchors G hloc γ n hdiam ep mid.1,
-          tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a)/
-            (segmentAnchors G hloc γ n hdiam ep (γ ep)).card /
-              segmentReturnMass G hloc a γ ep L y) = 1 := by
-  have hh := raw_tail_anchor_total_mass G hloc a γ n hdiam ep m L x y
-  simp only [tsum_div_const] at hh ⊢
-  rw [hh]
-  exact div_self hq
-
-end LiteralConditionalMass
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section LiteralGeodesicSeparation
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem graph_dist_aut (G : BondGraph V E) (hconn : Connected G)
-    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (x y : V) :
-    (openSimpleGraph G Set.univ).dist (a x) (a y) =
-      (openSimpleGraph G Set.univ).dist x y := by
-  have hreach (x y : V) : (openSimpleGraph G Set.univ).Reachable x y :=
-    (conn_iff_reachable G Set.univ x y).mp (conn_univ_of_connected G hconn x y)
-  obtain ⟨q,hq⟩ := (hreach x y).exists_walk_length_eq_dist
-  obtain ⟨r,hr⟩ := (hreach (a x) (a y)).exists_walk_length_eq_dist
-  have hq' := SimpleGraph.dist_le (q.map (autSimpleGraph G hab).toHom)
-  have hr' := SimpleGraph.dist_le (r.map (autSimpleGraph G (isAut_symm G hab)).toHom)
-  simp only [SimpleGraph.Walk.length_map,hq] at hq'
-  change (openSimpleGraph G Set.univ).dist (a.symm (a x)) (a.symm (a y)) ≤ _ at hr'
-  simp only [a.symm_apply_apply,SimpleGraph.Walk.length_map,hr] at hr'
-  exact le_antisymm hq' hr'
-
- 
-
-theorem geodesic_anchor_early_separations (G : BondGraph V E) (hconn : Connected G)
-    {x y : V} (q : (openSimpleGraph G Set.univ).Walk x y) (J : ℕ)
-    (hlen : q.length = 2*J) (hq : q.length = (openSimpleGraph G Set.univ).dist x y)
-    (s : Fin (2*J+1) → V) (hs : s ∈ segmentOrbit G (geodesicSegment G q (2*J))) :
-    (∀ i : Fin (J+1), J ≤ (openSimpleGraph G Set.univ).dist
-      (s ⟨0,by omega⟩) (s ⟨J+i.val,by omega⟩)) ∧
-    (∀ i : Fin (J+1), J ≤ (openSimpleGraph G Set.univ).dist
-      (s ⟨2*J,by omega⟩) (s ⟨i.val,by omega⟩)) ∧
-    (2*J ≤ (openSimpleGraph G Set.univ).dist (s ⟨0,by omega⟩) (s ⟨2*J,by omega⟩)) := by
-  obtain ⟨a,b,hab,rfl⟩ := hs
-  constructor
-  · intro i
-    dsimp only [Function.comp_apply,geodesicSegment]
-    rw [graph_dist_aut G hconn hab]
-    rw [geodesic_getVert_dist _ q hq 0 (J+i.val) (by omega) (by omega)]
-    omega
-  constructor
-  · intro i
-    dsimp only [Function.comp_apply,geodesicSegment]
-    rw [graph_dist_aut G hconn hab,SimpleGraph.dist_comm]
-    rw [geodesic_getVert_dist _ q hq i.val (2*J) (by omega) (by omega)]
-    omega
-  · dsimp only [Function.comp_apply,geodesicSegment]
-    rw [graph_dist_aut G hconn hab]
-    rw [geodesic_getVert_dist _ q hq 0 (2*J) (by omega) (by omega)]
-    omega
-
-end LiteralGeodesicSeparation
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section PositiveFirstEscape
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {I ι : Type*} [Fintype I] [Fintype ι]
-
- 
-
-theorem exists_positive_first_escape (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G)
-    (htr : RootTransport root weight Γ) (hw : ∀ r, 0 < weight r) (hw1 : ∑ r, weight r = 1)
-    (hcov : ∀ x, ∃ r, ∃ g ∈ Γ, g (root r) = x)
-    (c : ℝ) (hc : 0 < c) (hmin : ∀ r, 1 ≤ c*weight r)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (ep : I)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re) (hρ : ‖P‖ < 1)
-    (m L J : ℕ) (x y : V) (hy : y ∈ vertexOrbit G (γ ep))
-    (pos neg : Fin (J+1) → V) (j : Fin (J+1) → I)
-    (hposD : ∀ s, J ≤ (openSimpleGraph G Set.univ).dist x (pos s))
-    (hnegD : ∀ s, J ≤ (openSimpleGraph G Set.univ).dist y (neg s))
-    (hxy : 2*J ≤ (openSimpleGraph G Set.univ).dist x y)
-    (hsmall : 2*c*(earlyCollisionCost ‖P‖ J + suffixCollisionCost ‖P‖ c J*‖P‖^L) < 1) :
-    ∃ (tail : RawFrom (openSimpleGraph G Set.univ) m x)
-      (mid : RawFrom (openSimpleGraph G Set.univ) L y)
-      (h : segmentAnchors G hloc γ n hdiam ep mid.1),
-      ¬firstEscapeCollision G J m L x y pos neg j tail mid h.val ∧
-      0 < tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a) := by
-  have hb := conditional_first_escape_collision_bound G hloc a root weight Γ hΓ htr hw hw1 hcov
-    c hc hmin γ n hdiam ep P hpos hsym hP hdiag hρ m L J x y hy pos neg j hposD hnegD hxy
-  have hq := segmentReturnMass_uniform_lower G hloc a root weight Γ hΓ htr hcov hw hw1 c hc hmin
-    P hpos hsym hP hdiag γ ep L
-  rw [←segmentReturnMass_on_orbit G hloc a γ ep L y hy P hpos hP] at hq
-  have hqp : 0 < segmentReturnMass G hloc a γ ep L y :=
-    (one_div_pos.mpr (mul_pos (by norm_num) hc)).trans_le hq
-  have hK (x y : V) : 0 ≤ walkKernel G hloc a x y := by
-    simpa only [hP,Complex.ofReal_re] using (hpos x y).2
-  by_contra H
-  have heq :
-      (∑' tail : RawFrom (openSimpleGraph G Set.univ) m x,
-        ∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
-          ∑' h : segmentAnchors G hloc γ n hdiam ep mid.1,
-            if firstEscapeCollision G J m L x y pos neg j tail mid h.val then
-              tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a)/
-                (segmentAnchors G hloc γ n hdiam ep (γ ep)).card else 0) =
-      ∑' tail : RawFrom (openSimpleGraph G Set.univ) m x,
-        ∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
-          ∑' _h : segmentAnchors G hloc γ n hdiam ep mid.1,
-            tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a)/
-              (segmentAnchors G hloc γ n hdiam ep (γ ep)).card := by
-    apply tsum_congr
-    intro tail
-    apply tsum_congr
-    intro mid
-    apply tsum_congr
-    intro h
-    split_ifs with ht
-    · rfl
-    · have hz : tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a) = 0 := by
-        apply le_antisymm
-        · exact le_of_not_gt (fun hp => H ⟨tail,mid,h,ht,hp⟩)
-        · exact mul_nonneg (tail.2.mass_nonneg _ hK) (mid.2.mass_nonneg _ hK)
-      rw [hz,zero_div]
-  rw [heq,raw_tail_anchor_total_mass G hloc a γ n hdiam ep m L x y,div_self hqp.ne'] at hb
-  exact (not_lt_of_ge hb) hsmall
-
-end PositiveFirstEscape
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section ProjectionEvents
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem finite_event_fiber_sum {α β : Type*} [Fintype α] [Fintype β] [DecidableEq β]
-    (f : α → β) (μ : α → ℝ) (B : β → Prop) [DecidablePred B] :
-    (∑ a, if B (f a) then μ a else 0) =
-      ∑ b, if B b then ∑ a, if f a = b then μ a else 0 else 0 := by
-  calc
-    _ = ∑ a, ∑ b, if B b then (if f a = b then μ a else 0) else 0 := by
-      apply Finset.sum_congr rfl
-      intro a _
-      have hf : (fun b => if B b then (if f a = b then μ a else 0) else 0) =
-          fun b => if f a = b then (if B (f a) then μ a else 0) else 0 := by
-        funext b
-        by_cases h : f a = b
-        · subst b
-          simp
-        · simp [h]
-      rw [hf]
-      simp
-    _ = _ := by
-      rw [Finset.sum_comm]
-      apply Finset.sum_congr rfl
-      intro b _
-      by_cases h : B b <;> simp [h]
-
- 
-
-theorem rooted_label_raw_event_bound (G : BondGraph V E) (hloc : LocallyFinite G)
-    (a : ℝ) (ha : 0 < a) (hK : ∀ x y, 0 ≤ walkKernel G hloc a x y)
-    (J L m : ℕ) (γ : Fin (J+J+1) → V)
-    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
-    (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (x : V)
-    (hx : x ∈ vertexOrbit G (γ (segmentCenter J)))
-    (q : ℝ) (hq : 0 < q) (hqle : q ≤ segmentReturnMass G hloc a γ (Fin.last (J+J)) L (γ (Fin.last (J+J))))
-    (B : RawFrom (openSimpleGraph G Set.univ) ((J+L)+J) x → Prop) [DecidablePred B] :
-    (∑' ξ : {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m // ξ.first (segmentCenter J) = x},
-      if B (rootedEscapeLabelRaw G J L m γ hγ x ξ) then
-        escapeLabelProbability G hloc a γ n hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m ξ.1 else 0) ≤
-      (∑' w : RawFrom (openSimpleGraph G Set.univ) ((J+L)+J) x,
-        if B w then w.2.mass (walkKernel G hloc a) else 0)/(a^(J+J)*q) := by
-  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
-  let := finite_rootedEscapeLabels G hloc γ n hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m x
-  let : Fintype {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m // ξ.first (segmentCenter J) = x} := Fintype.ofFinite _
-  let : Fintype (RawFrom (openSimpleGraph G Set.univ) ((J+L)+J) x) := Fintype.ofFinite _
-  rw [←tsum_indicator_div]
-  simp only [tsum_fintype]
-  rw [finite_event_fiber_sum]
-  apply Finset.sum_le_sum
-  intro w _
-  by_cases hw : B w
-  · simp only [hw,ite_true]
-    simpa only [tsum_fintype] using
-      rooted_label_raw_projection_bound G hloc a ha hK J L m γ hγ n hdiam x hx q hq hqle w
-  · simp only [hw,ite_false,le_refl]
-
-end ProjectionEvents
-end Percolation
-
- 
-namespace Percolation
-universe u v w
-variable {V : Type u} {E : Type v} {I : Type w}
-section FirstLabeledMarginal
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Fintype I]
-
- 
-def labelFirstEscapeCollision (G : BondGraph V E) (γ : I → V) (ep neg : I)
-    (J L m : ℕ) (posIdx negIdx suffixIdx : Fin (J+1) → I)
-    (ξ : EscapeLabel G γ ep neg L m) : Prop :=
-  firstEscapeCollision G J m L (ξ.first neg) (ξ.first ep)
-    (fun t => ξ.first (posIdx t)) (fun t => ξ.first (negIdx t)) suffixIdx
-    ξ.leftTail ⟨ξ.last ep,ξ.middle⟩ ξ.last
-
-theorem labelFromCode_first_collision (G : BondGraph V E) (hloc : LocallyFinite G)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
-    (center ep neg : I) (J L m : ℕ) (x : V)
-    (posIdx negIdx suffixIdx : Fin (J+1) → I)
-    (c : RootedLabelCode G hloc γ n hdiam center ep neg L m x) :
-    labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx
-      (labelFromCode G hloc γ n hdiam center ep neg L m x c).1 =
-    firstEscapeCollision G J m L (c.1.val neg) (c.1.val ep)
-      (fun t => c.1.val (posIdx t)) (fun t => c.1.val (negIdx t)) suffixIdx
-      c.2.2.2.1 c.2.1 c.2.2.1.val := by
-  rcases c with ⟨⟨s,hs⟩,⟨z,mid⟩,⟨h,hh⟩,α,β⟩
-  obtain ⟨_,hhz⟩ := (mem_segmentAnchors G hloc γ n hdiam ep z h).mp hh
-  subst z
-  rfl
-
- 
-
-theorem raw_rooted_first_collision_marginal (G : BondGraph V E) (hloc : LocallyFinite G)
-    (a : ℝ) (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
-    (center ep neg : I) (J L m : ℕ) (x : V)
-    (posIdx negIdx suffixIdx : Fin (J+1) → I) :
-    (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
-      if labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.1 then
-        ξ.1.mass (walkKernel G hloc a) else 0) =
-    ∑' s : segmentAnchors G hloc γ n hdiam center x,
-      ∑' tail : RawFrom (openSimpleGraph G Set.univ) m (s.val neg),
-        ∑' mid : RawFrom (openSimpleGraph G Set.univ) L (s.val ep),
-          ∑' h : segmentAnchors G hloc γ n hdiam ep mid.1,
-            if firstEscapeCollision G J m L (s.val neg) (s.val ep)
-              (fun t => s.val (posIdx t)) (fun t => s.val (negIdx t)) suffixIdx tail mid h.val then
-              tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a) else 0 := by
-  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
-  let (k : ℕ) (z : V) : Fintype (RawFrom (openSimpleGraph G Set.univ) k z) := Fintype.ofFinite _
-  let := finite_rootedEscapeLabels G hloc γ n hdiam center ep neg L m x
-  let R := rootedLabelEquiv G hloc γ n hdiam center ep neg L m x
-  rw [←R.symm.tsum_eq]
-  simp only [R,rootedLabelEquiv,Equiv.coe_fn_symm_mk,labelFromCode_mass,labelFromCode_first_collision]
-  unfold RootedLabelCode at *
-  rw [(summable_of_hasFiniteSupport (Set.toFinite _)).tsum_sigma]
-  apply tsum_congr
-  intro s
-  rw [(summable_of_hasFiniteSupport (Set.toFinite _)).tsum_sigma]
-  have hh (mid : RawFrom (openSimpleGraph G Set.univ) L (s.val ep)) :
-      (∑' hh : Σ h : segmentAnchors G hloc γ n hdiam ep mid.1,
-          RawFrom (openSimpleGraph G Set.univ) m (s.val neg) × RawFrom (openSimpleGraph G Set.univ) m (h.val neg),
-        if firstEscapeCollision G J m L (s.val neg) (s.val ep)
-          (fun t => s.val (posIdx t)) (fun t => s.val (negIdx t)) suffixIdx hh.2.1 mid hh.1.val then
-          mid.2.mass (walkKernel G hloc a)*hh.2.1.2.mass (walkKernel G hloc a)*hh.2.2.2.mass (walkKernel G hloc a) else 0) =
-      ∑' h : segmentAnchors G hloc γ n hdiam ep mid.1,
-        ∑' tail : RawFrom (openSimpleGraph G Set.univ) m (s.val neg),
-          if firstEscapeCollision G J m L (s.val neg) (s.val ep)
-            (fun t => s.val (posIdx t)) (fun t => s.val (negIdx t)) suffixIdx tail mid h.val then
-            tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a) else 0 := by
-    rw [(summable_of_hasFiniteSupport (Set.toFinite _)).tsum_sigma]
-    apply tsum_congr
-    intro h
-    rw [(summable_of_hasFiniteSupport (Set.toFinite _)).tsum_prod]
-    apply tsum_congr
-    intro tail
-    split_ifs
-    · simp only [tsum_mul_left,rawFrom_probability_sum G hloc a,mul_one]
-      exact mul_comm _ _
-    · simp only [tsum_zero]
-  simp_rw [hh]
-  simp only [tsum_fintype]
-  conv_lhs =>
-    arg 2
-    intro mid
-    rw [Finset.sum_comm]
-  rw [Finset.sum_comm]
-
-end FirstLabeledMarginal
-end Percolation
-
- 
-namespace Percolation
-universe u v w
-variable {V : Type u} {E : Type v} {I : Type w}
-section NormalizedMarginal
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Fintype I]
-
- 
-
-theorem rooted_first_collision_probability_marginal (G : BondGraph V E) (hloc : LocallyFinite G)
-    (a : ℝ) (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
-    (center ep neg : I) (J L m : ℕ) (x : V)
-    (posIdx negIdx suffixIdx : Fin (J+1) → I) :
-    (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
-      if labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.1 then
-        escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1 else 0) =
-    (∑' s : segmentAnchors G hloc γ n hdiam center x,
-      (∑' tail : RawFrom (openSimpleGraph G Set.univ) m (s.val neg),
-        ∑' mid : RawFrom (openSimpleGraph G Set.univ) L (s.val ep),
-          ∑' h : segmentAnchors G hloc γ n hdiam ep mid.1,
-            if firstEscapeCollision G J m L (s.val neg) (s.val ep)
-              (fun t => s.val (posIdx t)) (fun t => s.val (negIdx t)) suffixIdx tail mid h.val then
-              tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a)/
-                (segmentAnchors G hloc γ n hdiam ep (γ ep)).card else 0) /
-          segmentReturnMass G hloc a γ ep L (γ ep)) /
-      (segmentAnchors G hloc γ n hdiam center (γ center)).card := by
-  simp only [escapeLabelProbability,tsum_indicator_div,tsum_div_const]
-  rw [raw_rooted_first_collision_marginal G hloc a γ n hdiam center ep neg J L m x posIdx negIdx suffixIdx]
-  ring
-
-end NormalizedMarginal
-end Percolation
-
- 
-namespace Percolation
-universe u v w
-variable {V : Type u} {E : Type v} {I : Type w}
-section FullFirstEscapeBound
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Fintype I] {ι : Type*} [Fintype ι]
-
- 
-
-theorem rooted_first_collision_probability_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G)
-    (htr : RootTransport root weight Γ) (hw : ∀ r, 0 < weight r) (hw1 : ∑ r, weight r = 1)
-    (hcov : ∀ x, ∃ r, ∃ g ∈ Γ, g (root r) = x)
-    (c : ℝ) (hc : 0 < c) (hmin : ∀ r, 1 ≤ c*weight r)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (center ep neg : I)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re) (hρ : ‖P‖ < 1)
-    (m L J : ℕ) (x : V) (hx : x ∈ vertexOrbit G (γ center))
-    (posIdx negIdx suffixIdx : Fin (J+1) → I)
-    (hsep : ∀ s ∈ segmentOrbit G γ,
-      (∀ i, J ≤ (openSimpleGraph G Set.univ).dist (s neg) (s (posIdx i))) ∧
-      (∀ i, J ≤ (openSimpleGraph G Set.univ).dist (s ep) (s (negIdx i))) ∧
-      2*J ≤ (openSimpleGraph G Set.univ).dist (s neg) (s ep)) :
-    (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
-      if labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.1 then
-        escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1 else 0) ≤
-      2*c*(earlyCollisionCost ‖P‖ J + suffixCollisionCost ‖P‖ c J*‖P‖^L) := by
-  rw [rooted_first_collision_probability_marginal G hloc a γ n hdiam center ep neg J L m x posIdx negIdx suffixIdx]
-  have hN : 0 < ((segmentAnchors G hloc γ n hdiam center (γ center)).card : ℝ) := by
-    exact_mod_cast Finset.card_pos.mpr ((segmentAnchors_nonempty_iff G hloc γ n hdiam center (γ center)).mpr
-      ⟨Equiv.refl V,Equiv.refl E,isAut_refl G,rfl⟩)
-  apply (div_le_iff₀ hN).mpr
-  simp only [tsum_fintype]
-  calc
-    _ ≤ ∑ s : segmentAnchors G hloc γ n hdiam center x,
-        2*c*(earlyCollisionCost ‖P‖ J + suffixCollisionCost ‖P‖ c J*‖P‖^L) := by
-      apply Finset.sum_le_sum
-      intro s _
-      have hs := ((mem_segmentAnchors G hloc γ n hdiam center x s.val).mp s.property).1
-      have hy : s.val ep ∈ vertexOrbit G (γ ep) := by
-        obtain ⟨g,b,hg,he⟩ := hs
-        rw [he]
-        exact ⟨g,b,hg,rfl⟩
-      obtain ⟨hp,hn,hxy⟩ := hsep s.val hs
-      have hh := conditional_first_escape_collision_bound G hloc a root weight Γ hΓ htr hw hw1 hcov
-        c hc hmin γ n hdiam ep P hpos hsym hP hdiag hρ m L J (s.val neg) (s.val ep) hy
-        (fun i => s.val (posIdx i)) (fun i => s.val (negIdx i)) suffixIdx hp hn hxy
-      rw [segmentReturnMass_on_orbit G hloc a γ ep L (s.val ep) hy P hpos hP] at hh
-      simpa only [tsum_fintype] using hh
-    _ = _ := by
-      simp only [Finset.sum_const,Finset.card_univ,Fintype.card_coe,nsmul_eq_mul]
-      rw [segmentAnchors_card_on_orbit G hloc γ n hdiam center x hx]
-      ring
-
-end FullFirstEscapeBound
-end Percolation
-
- 
-namespace Percolation
-universe u v w
-variable {V : Type u} {E : Type v} {I : Type w}
-section CollisionNaturality
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-namespace HoldingWalk
-variable {W : Type*} {H : SimpleGraph V} {H' : SimpleGraph W}
-theorem map_position (f : H →g H') {k : ℕ} {x y : V} (q : HoldingWalk H k x y) (i : ℕ) :
-    (q.map f).position i = f (q.position i) := by
-  induction q generalizing i with
-  | nil => rfl
-  | hold q ih => cases i <;> simp only [map,position,ih]
-  | step h q ih => cases i <;> simp only [map,position,ih]
-end HoldingWalk
-
-theorem list_common_vertex_map_iff {W : Type*} (f : V → W) (hf : Function.Injective f)
-    (l r : List V) :
-    (∃ z, z ∈ l.map f ∧ z ∈ r.map f) ↔ ∃ z, z ∈ l ∧ z ∈ r := by
-  constructor
-  · rintro ⟨z,hz,hz'⟩
-    obtain ⟨x,hx,rfl⟩ := List.mem_map.mp hz
-    obtain ⟨y,hy,he⟩ := List.mem_map.mp hz'
-    have : y = x := hf he
-    subst y
-    exact ⟨x,hx,hy⟩
-  · rintro ⟨z,hz,hz'⟩
-    exact ⟨f z,List.mem_map.mpr ⟨z,hz,rfl⟩,List.mem_map.mpr ⟨z,hz',rfl⟩⟩
-
- 
-theorem labelFirstEscapeCollision_aut (G : BondGraph V E) (γ : I → V) (ep neg : I)
-    (J L m : ℕ) (posIdx negIdx suffixIdx : Fin (J+1) → I)
-    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b)
-    (ξ : EscapeLabel G γ ep neg L m) :
-    labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx (EscapeLabel.autMap hab ξ) ↔
-      labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ := by
-  dsimp only [labelFirstEscapeCollision,firstEscapeCollision,EscapeLabel.autMap,Function.comp_apply,autSimpleGraph]
-  simp only [HoldingWalk.map_vertices,HoldingWalk.map_position]
-  change (((∃ s, a (ξ.first (posIdx s)) ∈ ξ.leftTail.2.vertices.map a) ∨
-    (∃ s, a (ξ.first (negIdx s)) ∈ ξ.middle.vertices.map a)) ∨
-    (∃ z, z ∈ ξ.leftTail.2.vertices.map a ∧ z ∈ ξ.middle.vertices.map a)) ∨
-    ((∃ st : Fin (J+1) × Fin (J+1), a (ξ.last (suffixIdx st.1)) = a (ξ.first (negIdx st.2))) ∨
-      (∃ ij : Fin (m+1) × Fin (J+1), a (ξ.leftTail.2.position ij.1.val) = a (ξ.last (suffixIdx ij.2)))) ↔ _
-  rw [list_common_vertex_map_iff a a.injective]
-  simp only [List.mem_map,Equiv.apply_eq_iff_eq]
-  simp
-
-end CollisionNaturality
-end Percolation
-
- 
-namespace Percolation
-universe u v w
-variable {V : Type u} {E : Type v} {I : Type w}
-section BothEscapeEvents
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Fintype I] {ι : Type*} [Fintype ι]
-
- 
-theorem rootOrbitWeight_pos (G : BondGraph V E)
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (hΓ : Γ ⊆ vertexAut G) (hw : ∀ i, 0 < weight i)
-    (hcov : ∀ x, ∃ r, ∃ g ∈ Γ, g (root r) = x) (x : V) :
-    0 < ∑ i, if root i ∈ vertexOrbit G x then weight i else 0 := by
-  obtain ⟨r,g,hg,he⟩ := hcov x
-  obtain ⟨b,hab⟩ := hΓ hg
-  have hr : root r ∈ vertexOrbit G x := by
-    refine ⟨g.symm,b.symm,isAut_symm G hab,?_⟩
-    rw [←he,Equiv.symm_apply_apply]
-  have hh : (if root r ∈ vertexOrbit G x then weight r else 0) ≤
-      ∑ i, if root i ∈ vertexOrbit G x then weight i else 0 :=
-    Finset.single_le_sum (f := fun i => if root i ∈ vertexOrbit G x then weight i else 0)
-      (fun i _ => by split_ifs <;> first | exact (hw i).le | rfl) (Finset.mem_univ r)
-  simpa only [ite_eq_left hr] using lt_of_lt_of_le (by simpa only [ite_eq_left hr] using hw r) hh
-
- 
-
-theorem rooted_escape_collision_reversal (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G)
-    (htr : RootTransport root weight Γ) (hw : ∀ r, 0 < weight r)
-    (hcov : ∀ x, ∃ r, ∃ g ∈ Γ, g (root r) = x)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (center ep neg : I)
-    (m L J : ℕ) (x : V) (hx : x ∈ vertexOrbit G (γ center))
-    (posIdx negIdx suffixIdx : Fin (J+1) → I) :
-    (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
-      if labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.1 then
-        escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1 else 0) =
-    ∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
-      if labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.1.reverse then
-        escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1 else 0 := by
-  have hh := rooted_escape_label_reversal G hloc a root weight Γ hΓ htr γ n hdiam center ep neg L m
-    (rootOrbitWeight_pos G root weight Γ hΓ hw hcov (γ center))
-    (fun ξ => if labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ then (1:ℝ) else 0)
-    (fun g b hab ξ => by rw [labelFirstEscapeCollision_aut G γ ep neg J L m posIdx negIdx suffixIdx hab ξ]) x hx
-  simpa only [mul_ite,mul_one,mul_zero] using hh
-
-end BothEscapeEvents
-end Percolation
-
- 
-namespace Percolation
-universe u v w
-variable {V : Type u} {E : Type v} {I : Type w}
-section BothEscapeBound
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Fintype I] {ι : Type*} [Fintype ι]
-
- 
-theorem rooted_both_collision_probability_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G)
-    (htr : RootTransport root weight Γ) (hw : ∀ r, 0 < weight r) (hw1 : ∑ r, weight r = 1)
-    (hcov : ∀ x, ∃ r, ∃ g ∈ Γ, g (root r) = x)
-    (c : ℝ) (hc : 0 < c) (hmin : ∀ r, 1 ≤ c*weight r)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (center ep neg : I)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re) (hρ : ‖P‖ < 1)
-    (m L J : ℕ) (x : V) (hx : x ∈ vertexOrbit G (γ center))
-    (posIdx negIdx suffixIdx : Fin (J+1) → I)
-    (hsep : ∀ s ∈ segmentOrbit G γ,
-      (∀ i, J ≤ (openSimpleGraph G Set.univ).dist (s neg) (s (posIdx i))) ∧
-      (∀ i, J ≤ (openSimpleGraph G Set.univ).dist (s ep) (s (negIdx i))) ∧
-      2*J ≤ (openSimpleGraph G Set.univ).dist (s neg) (s ep)) :
-    (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
-      if labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.1 ∨
-        labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.1.reverse then
-        escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1 else 0) ≤
-      4*c*(earlyCollisionCost ‖P‖ J + suffixCollisionCost ‖P‖ c J*‖P‖^L) := by
-  have hK (x y : V) : 0 ≤ walkKernel G hloc a x y := by
-    simpa only [hP,Complex.ofReal_re] using (hpos x y).2
-  have hb := rooted_first_collision_probability_bound G hloc a root weight Γ hΓ htr hw hw1 hcov
-    c hc hmin γ n hdiam center ep neg P hpos hsym hP hdiag hρ m L J x hx posIdx negIdx suffixIdx hsep
-  have he := rooted_escape_collision_reversal G hloc a root weight Γ hΓ htr hw hcov
-    γ n hdiam center ep neg m L J x hx posIdx negIdx suffixIdx
-  let := finite_rootedEscapeLabels G hloc γ n hdiam center ep neg L m x
-  let := Fintype.ofFinite {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x}
-  have hu := finite_weighted_event_or
-    (fun ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x} =>
-      escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1)
-    (fun ξ => escapeLabelProbability_nonneg G hloc a hK γ n hdiam center ep neg L m ξ.1)
-    (fun ξ => labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.1)
-    (fun ξ => labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.1.reverse)
-  simp only [tsum_fintype] at hb he ⊢
-  rw [←he] at hu
-  linarith
-
-end BothEscapeBound
-end Percolation
-
- 
-namespace Percolation
-universe u v w
-variable {V : Type u} {E : Type v} {I : Type w}
-section PositiveTwoEscapes
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Fintype I] {ι : Type*} [Fintype ι]
-
- 
-
-theorem exists_positive_two_escape_label (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G)
-    (htr : RootTransport root weight Γ) (hw : ∀ r, 0 < weight r) (hw1 : ∑ r, weight r = 1)
-    (hcov : ∀ x, ∃ r, ∃ g ∈ Γ, g (root r) = x)
-    (c : ℝ) (hc : 0 < c) (hmin : ∀ r, 1 ≤ c*weight r)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (center ep neg : I)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re) (hρ : ‖P‖ < 1)
-    (m L J : ℕ) (x : V) (hx : x ∈ vertexOrbit G (γ center))
-    (posIdx negIdx suffixIdx : Fin (J+1) → I)
-    (hsep : ∀ s ∈ segmentOrbit G γ,
-      (∀ i, J ≤ (openSimpleGraph G Set.univ).dist (s neg) (s (posIdx i))) ∧
-      (∀ i, J ≤ (openSimpleGraph G Set.univ).dist (s ep) (s (negIdx i))) ∧
-      2*J ≤ (openSimpleGraph G Set.univ).dist (s neg) (s ep)) (hsmall : 4*c*(earlyCollisionCost ‖P‖ J + suffixCollisionCost ‖P‖ c J*‖P‖^L) < 1) :
-    ∃ ξ : EscapeLabel G γ ep neg L m, ξ.first center = x ∧
-      ¬labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ ∧
-      ¬labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.reverse ∧
-      0 < escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ := by
-  have hb := rooted_both_collision_probability_bound G hloc a root weight Γ hΓ htr hw hw1 hcov
-    c hc hmin γ n hdiam center ep neg P hpos hsym hP hdiag hρ m L J x hx posIdx negIdx suffixIdx hsep
-  have hs := (rooted_label_probability_hasSum G hloc a root weight Γ hΓ htr hcov hw hw1 c hc hmin
-    P hpos hsym hP hdiag γ n hdiam center ep neg L m x hx).tsum_eq
-  have hK (x y : V) : 0 ≤ walkKernel G hloc a x y := by
-    simpa only [hP,Complex.ofReal_re] using (hpos x y).2
-  by_contra H
-  have heq :
-      (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
-        if labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.1 ∨
-          labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.1.reverse then
-          escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1 else 0) =
-      ∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
-        escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1 := by
-    apply tsum_congr
-    intro ξ
-    split_ifs with ht
-    · rfl
-    · apply Eq.symm
-      apply le_antisymm
-      · exact le_of_not_gt (fun hp => H ⟨ξ.1,ξ.2,(not_or.mp ht).1,(not_or.mp ht).2,hp⟩)
-      · exact escapeLabelProbability_nonneg G hloc a hK γ n hdiam center ep neg L m ξ.1
-  rw [heq,hs] at hb
-  exact (not_lt_of_ge hb) hsmall
-
-end PositiveTwoEscapes
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Escape.CollisionTails
+
+namespace OAI
+
+/-! Positive Escapes for bond percolation. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+section DependentFiniteUnion
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem finite_dependent_event_union {β δ : Type*} [Fintype β] [Fintype δ]
+    {γ : β → Type*} [∀ b, Fintype (γ b)]
+    (μ : (b : β) → γ b → ℝ) (hμ : ∀ b c, 0 ≤ μ b c)
+    (B : (b : β) → γ b → δ → Prop) :
+    (∑ b, ∑ c, if ∃ d, B b c d then μ b c else 0) ≤
+      ∑ d, ∑ b, ∑ c, if B b c d then μ b c else 0 := by
+  have hh := finite_nonnegative_event_union
+    (fun ξ : Sigma γ => μ ξ.1 ξ.2)
+    (fun ξ => hμ ξ.1 ξ.2)
+    (fun ξ : Sigma γ => B ξ.1 ξ.2)
+  simpa only [Fintype.sum_sigma] using hh
+
+end DependentFiniteUnion
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section ActualFinitePositionHits
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {I ι S T : Type*} [Fintype I] [Fintype ι] [Fintype S] [Fintype T]
+
+ 
+
+theorem raw_anchor_finite_collision_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G)
+    (htr : RootTransport root weight Γ) (hw : ∀ r, 0 ≤ weight r) (hw1 : ∑ r, weight r = 1)
+    (hcov : ∀ x, ∃ r, ∃ g ∈ Γ, g (root r) = x)
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ r, 1 ≤ c*weight r)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (ep : I)
+    (j : S → I) (z : T → V)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (L : ℕ) (y : V) :
+    (∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
+      ∑' h : segmentAnchors G hloc γ n hdiam ep mid.1,
+        if ∃ st : S × T, h.val (j st.1) = z st.2 then mid.2.mass (walkKernel G hloc a)/
+          (segmentAnchors G hloc γ n hdiam ep (γ ep)).card else 0) ≤
+      (Fintype.card S : ℝ)*(Fintype.card T : ℝ)*Real.sqrt c*‖P‖^L := by
+  let (v : V) : Finite ((openSimpleGraph G Set.univ).neighborSet v) := finite_openSimpleGraph_neighbors G hloc v
+  let : Fintype (RawFrom (openSimpleGraph G Set.univ) L y) := Fintype.ofFinite _
+  have hK (x y : V) : 0 ≤ walkKernel G hloc a x y := by
+    simpa only [hP,Complex.ofReal_re] using (hpos x y).2
+  simp only [tsum_fintype]
+  have hu := finite_dependent_event_union
+    (β := RawFrom (openSimpleGraph G Set.univ) L y)
+    (γ := fun mid => segmentAnchors G hloc γ n hdiam ep mid.1)
+    (δ := S × T)
+    (fun mid _ => mid.2.mass (walkKernel G hloc a)/(segmentAnchors G hloc γ n hdiam ep (γ ep)).card)
+    (fun mid _ => div_nonneg (mid.2.mass_nonneg _ hK) (Nat.cast_nonneg _))
+    (fun _ h st => h.val (j st.1) = z st.2)
+  apply hu.trans
+  calc
+    _ ≤ ∑ _st : S × T, Real.sqrt c*‖P‖^L := by
+      apply Finset.sum_le_sum
+      intro st _
+      obtain ⟨H,hH,hHn⟩ := exists_anchoredPositionOperator_uniform G hloc root weight Γ hΓ htr hw hw1 hcov
+        c hc hmin γ n hdiam ep (j st.1)
+      simpa only [tsum_fintype] using raw_anchor_point_collision_bound G hloc a γ n hdiam ep (j st.1)
+        P H hpos hP hH (Real.sqrt c) hHn L y (z st.2)
+    _ = _ := by simp only [Finset.sum_const,Finset.card_univ,Fintype.card_prod,Nat.cast_mul,nsmul_eq_mul]; ring
+
+end ActualFinitePositionHits
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section EarlyFiniteCollisions
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {S : Type*} [Fintype S]
+
+ 
+
+theorem raw_hits_distant_set_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ)) (hρ : ‖P‖ < 1)
+    (m J : ℕ) (x : V) (z : S → V) (hD : ∀ s, J ≤ (openSimpleGraph G Set.univ).dist x (z s)) :
+    (∑' w : RawFrom (openSimpleGraph G Set.univ) m x,
+      if ∃ s, z s ∈ w.2.vertices then w.2.mass (walkKernel G hloc a) else 0) ≤
+      (Fintype.card S : ℝ)*‖P‖^J/(1-‖P‖) := by
+  let (v : V) : Finite ((openSimpleGraph G Set.univ).neighborSet v) := finite_openSimpleGraph_neighbors G hloc v
+  let : Fintype (RawFrom (openSimpleGraph G Set.univ) m x) := Fintype.ofFinite _
+  have hK (x y : V) : 0 ≤ walkKernel G hloc a x y := by
+    simpa only [hP,Complex.ofReal_re] using (hpos x y).2
+  have he (w : RawFrom (openSimpleGraph G Set.univ) m x) :
+      (∃ s, z s ∈ w.2.vertices) ↔ ∃ si : S × Fin (m+1), w.2.position si.2.val = z si.1 := by
+    constructor
+    · rintro ⟨s,hs⟩
+      obtain ⟨i,hi,he⟩ := w.2.mem_vertices_iff_position.mp hs
+      exact ⟨(s,⟨i,by omega⟩),he⟩
+    · rintro ⟨⟨s,i⟩,he⟩
+      exact ⟨s,he ▸ w.2.position_mem_vertices i.val⟩
+  simp only [he,tsum_fintype]
+  have hu := finite_nonnegative_event_union
+    (fun w : RawFrom (openSimpleGraph G Set.univ) m x => w.2.mass (walkKernel G hloc a))
+    (fun w => w.2.mass_nonneg _ hK)
+    (fun w (si : S × Fin (m+1)) => w.2.position si.2.val = z si.1)
+  apply hu.trans
+  rw [Fintype.sum_prod_type]
+  calc
+    _ ≤ ∑ _s : S, ∑ i : Fin (m+1), if J ≤ i.val then ‖P‖^i.val else 0 := by
+      apply Finset.sum_le_sum
+      intro s _
+      apply Finset.sum_le_sum
+      intro i _
+      have hh := raw_position_probability G hloc a P hpos hP i.val m (by omega) x (z s)
+      simp only [tsum_fintype] at hh
+      rw [hh]
+      split_ifs with hi
+      · exact matrixCoeff_power_le P i.val x (z s)
+      · exact (walkPower_zero_of_lt_dist G hloc a P hpos hP i.val x (z s) (by have hd := hD s; omega)).le
+    _ ≤ ∑ _s : S, ‖P‖^J/(1-‖P‖) := Finset.sum_le_sum (fun _ _ =>
+      finite_geometric_cutoff_bound ‖P‖ (norm_nonneg _) hρ J (m+1))
+    _ = _ := by simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]; ring
+
+end EarlyFiniteCollisions
+end Percolation
+
+ 
+namespace Percolation
+section QuantitativeEarlyCutoff
+
+ 
+
+noncomputable def earlyCollisionCost (ρ : ℝ) (J : ℕ) : ℝ :=
+  2*((J:ℝ)+1)*ρ^J/(1-ρ) + 2*ρ^J/(1-ρ)^2
+
+theorem earlyCollisionCost_tendsto (ρ : ℝ) (hρ : 0 ≤ ρ) (hρ1 : ρ < 1) :
+    Filter.Tendsto (earlyCollisionCost ρ) Filter.atTop (nhds 0) := by
+  have hn : ‖ρ‖ < 1 := by simpa only [Real.norm_eq_abs,abs_of_nonneg hρ] using hρ1
+  have ht : Filter.Tendsto (fun J : ℕ => (J:ℝ)*ρ^J) Filter.atTop (nhds 0) := by
+    simpa only [pow_one] using (summable_pow_mul_geometric_of_norm_lt_one 1 hn).tendsto_atTop_zero
+  have hp := tendsto_pow_atTop_nhds_zero_of_lt_one hρ hρ1
+  have hh := (((ht.add hp).const_mul 2).div_const (1-ρ)).add ((hp.const_mul 2).div_const ((1-ρ)^2))
+  have he : earlyCollisionCost ρ = (fun J : ℕ => 2*((J:ℝ)*ρ^J+ρ^J)/(1-ρ)+2*ρ^J/(1-ρ)^2) := by
+    funext J
+    unfold earlyCollisionCost
+    ring
+  rw [he]
+  simpa only [zero_add,mul_zero,zero_div] using hh
+
+theorem exists_earlyCollisionCutoff (ρ : ℝ) (hρ : 0 ≤ ρ) (hρ1 : ρ < 1) (ε : ℝ) (hε : 0 < ε) :
+    ∃ J₀ : ℕ, ∀ J : ℕ, J₀ ≤ J → earlyCollisionCost ρ J < ε := by
+  exact Filter.eventually_atTop.mp ((earlyCollisionCost_tendsto ρ hρ hρ1).eventually (gt_mem_nhds hε))
+
+end QuantitativeEarlyCutoff
+end Percolation
+
+ 
+namespace Percolation
+section WeightedBinaryUnion
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem finite_weighted_event_or {α : Type*} [Fintype α]
+    (μ : α → ℝ) (hμ : ∀ a, 0 ≤ μ a) (B C : α → Prop) [DecidablePred B] [DecidablePred C] :
+    (∑ a, if B a ∨ C a then μ a else 0) ≤
+      (∑ a, if B a then μ a else 0) + ∑ a, if C a then μ a else 0 := by
+  rw [←Finset.sum_add_distrib]
+  apply Finset.sum_le_sum
+  intro a _
+  by_cases hb : B a <;> by_cases hc : C a <;> simp only [hb,hc,or_self,or_false,false_or,ite_true,ite_false,add_zero,zero_add,le_refl]
+  exact le_add_of_nonneg_right (hμ a)
+
+end WeightedBinaryUnion
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section ActualEarlyExperiment
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem raw_early_collision_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ)) (hρ : ‖P‖ < 1)
+    (m L J : ℕ) (x y : V) (pos neg : Fin (J+1) → V)
+    (hposD : ∀ s, J ≤ (openSimpleGraph G Set.univ).dist x (pos s))
+    (hnegD : ∀ s, J ≤ (openSimpleGraph G Set.univ).dist y (neg s))
+    (hxy : 2*J ≤ (openSimpleGraph G Set.univ).dist x y) :
+    (∑' w : RawFrom (openSimpleGraph G Set.univ) m x × RawFrom (openSimpleGraph G Set.univ) L y,
+      if ((∃ s, pos s ∈ w.1.2.vertices) ∨ (∃ s, neg s ∈ w.2.2.vertices)) ∨
+          (∃ z, z ∈ w.1.2.vertices ∧ z ∈ w.2.2.vertices) then
+        w.1.2.mass (walkKernel G hloc a)*w.2.2.mass (walkKernel G hloc a) else 0) ≤
+      earlyCollisionCost ‖P‖ J := by
+  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
+  let : Fintype (RawFrom (openSimpleGraph G Set.univ) m x) := Fintype.ofFinite _
+  let : Fintype (RawFrom (openSimpleGraph G Set.univ) L y) := Fintype.ofFinite _
+  have hK (x y : V) : 0 ≤ walkKernel G hloc a x y := by
+    simpa only [hP,Complex.ofReal_re] using (hpos x y).2
+  have hu := finite_weighted_event_or
+    (fun w : RawFrom (openSimpleGraph G Set.univ) m x × RawFrom (openSimpleGraph G Set.univ) L y =>
+      w.1.2.mass (walkKernel G hloc a)*w.2.2.mass (walkKernel G hloc a))
+    (fun w => mul_nonneg (w.1.2.mass_nonneg _ hK) (w.2.2.mass_nonneg _ hK))
+    (fun w => (∃ s, pos s ∈ w.1.2.vertices) ∨ (∃ s, neg s ∈ w.2.2.vertices))
+    (fun w => ∃ z, z ∈ w.1.2.vertices ∧ z ∈ w.2.2.vertices)
+  have hp := finite_product_failure_union
+    (fun w : RawFrom (openSimpleGraph G Set.univ) m x => w.2.mass (walkKernel G hloc a))
+    (fun w : RawFrom (openSimpleGraph G Set.univ) L y => w.2.mass (walkKernel G hloc a))
+    (fun w => w.2.mass_nonneg _ hK) (fun w => w.2.mass_nonneg _ hK)
+    (by simpa only [tsum_fintype] using rawFrom_probability_sum G hloc a m x)
+    (by simpa only [tsum_fintype] using rawFrom_probability_sum G hloc a L y)
+    (fun w => ∃ s, pos s ∈ w.2.vertices) (fun w => ∃ s, neg s ∈ w.2.vertices)
+  have hx := raw_hits_distant_set_bound G hloc a P hpos hP hρ m J x pos hposD
+  have hy := raw_hits_distant_set_bound G hloc a P hpos hP hρ L J y neg hnegD
+  have hi := raw_intersection_uniform_bound G hloc a P hpos hsym hP hρ m L J x y hxy
+  simp only [tsum_fintype,Fintype.card_fin,Nat.cast_add,Nat.cast_one] at hx hy hi ⊢
+  have hx' : (∑ w : RawFrom (openSimpleGraph G Set.univ) m x,
+      if ∃ s, pos s ∈ w.2.vertices then w.2.mass (walkKernel G hloc a) else 0) ≤
+      ((J:ℝ)+1)*‖P‖^J/(1-‖P‖) := by
+    convert hx using 1
+    apply Finset.sum_congr rfl
+    intro w _
+    split_ifs <;> rfl
+  have hy' : (∑ w : RawFrom (openSimpleGraph G Set.univ) L y,
+      if ∃ s, neg s ∈ w.2.vertices then w.2.mass (walkKernel G hloc a) else 0) ≤
+      ((J:ℝ)+1)*‖P‖^J/(1-‖P‖) := by
+    convert hy using 1
+    apply Finset.sum_congr rfl
+    intro w _
+    split_ifs <;> rfl
+  have hp' : (∑ w : RawFrom (openSimpleGraph G Set.univ) m x × RawFrom (openSimpleGraph G Set.univ) L y,
+      if (∃ s, pos s ∈ w.1.2.vertices) ∨ (∃ s, neg s ∈ w.2.2.vertices) then
+        w.1.2.mass (walkKernel G hloc a)*w.2.2.mass (walkKernel G hloc a) else 0) ≤
+      (∑ w : RawFrom (openSimpleGraph G Set.univ) m x,
+        if ∃ s, pos s ∈ w.2.vertices then w.2.mass (walkKernel G hloc a) else 0) +
+      (∑ w : RawFrom (openSimpleGraph G Set.univ) L y,
+        if ∃ s, neg s ∈ w.2.vertices then w.2.mass (walkKernel G hloc a) else 0) := by
+    convert hp using 1
+    · apply Finset.sum_congr rfl
+      intro w _
+      split_ifs <;> rfl
+    · congr 1 <;> apply Finset.sum_congr rfl <;> intro w _ <;> split_ifs <;> rfl
+  have hr : (((J:ℝ)+1)*‖P‖^J/(1-‖P‖) + ((J:ℝ)+1)*‖P‖^J/(1-‖P‖)) +
+      2*‖P‖^J/(1-‖P‖)^2 = earlyCollisionCost ‖P‖ J := by
+    unfold earlyCollisionCost
+    ring
+  have hh := hu.trans ((add_le_add (hp'.trans (add_le_add hx' hy')) hi).trans_eq hr)
+  exact hh
+
+end ActualEarlyExperiment
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section ConditioningCost
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {I : Type*} [Fintype I]
+
+ 
+
+theorem raw_anchor_event_le_unconditioned (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (hK : ∀ x y, 0 ≤ walkKernel G hloc a x y)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (ep : I)
+    (m L : ℕ) (x y : V)
+    (B : RawFrom (openSimpleGraph G Set.univ) m x → RawFrom (openSimpleGraph G Set.univ) L y → Prop) [∀ tail mid, Decidable (B tail mid)] :
+    (∑' tail : RawFrom (openSimpleGraph G Set.univ) m x,
+      ∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
+        ∑' _h : segmentAnchors G hloc γ n hdiam ep mid.1,
+          if B tail mid then tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a)/
+            (segmentAnchors G hloc γ n hdiam ep (γ ep)).card else 0) ≤
+    (∑' w : RawFrom (openSimpleGraph G Set.univ) m x × RawFrom (openSimpleGraph G Set.univ) L y,
+      if B w.1 w.2 then w.1.2.mass (walkKernel G hloc a)*w.2.2.mass (walkKernel G hloc a) else 0) := by
+  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
+  let (k : ℕ) (z : V) : Fintype (RawFrom (openSimpleGraph G Set.univ) k z) := Fintype.ofFinite _
+  have hN : 0 < ((segmentAnchors G hloc γ n hdiam ep (γ ep)).card : ℝ) := by
+    exact_mod_cast Finset.card_pos.mpr ((segmentAnchors_nonempty_iff G hloc γ n hdiam ep (γ ep)).mpr
+      ⟨Equiv.refl V,Equiv.refl E,isAut_refl G,rfl⟩)
+  simp only [tsum_fintype,Fintype.sum_prod_type]
+  apply Finset.sum_le_sum
+  intro tail _
+  apply Finset.sum_le_sum
+  intro mid _
+  by_cases hb : B tail mid
+  · simp only [ite_eq_left hb,Finset.sum_const,Finset.card_univ,Fintype.card_coe,nsmul_eq_mul]
+    have hc : ((segmentAnchors G hloc γ n hdiam ep mid.1).card : ℝ) ≤
+        (segmentAnchors G hloc γ n hdiam ep (γ ep)).card := by
+      rw [segmentAnchors_card_eq_indicator G hloc γ n hdiam ep mid.1]
+      split_ifs
+      · exact le_rfl
+      · exact_mod_cast Nat.zero_le _
+    rw [←mul_div_assoc]
+    apply (div_le_iff₀ hN).mpr
+    have hh := mul_le_mul_of_nonneg_right hc
+      (mul_nonneg (tail.2.mass_nonneg _ hK) (mid.2.mass_nonneg _ hK))
+    nlinarith
+  · simp only [ite_eq_right hb,Finset.sum_const_zero]
+    exact le_rfl
+
+end ConditioningCost
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section EarlyAnchoredCollision
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {I : Type*} [Fintype I]
+
+ 
+
+theorem raw_anchor_early_collision_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (ep : I)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ)) (hρ : ‖P‖ < 1)
+    (m L J : ℕ) (x y : V) (pos neg : Fin (J+1) → V)
+    (hposD : ∀ s, J ≤ (openSimpleGraph G Set.univ).dist x (pos s))
+    (hnegD : ∀ s, J ≤ (openSimpleGraph G Set.univ).dist y (neg s))
+    (hxy : 2*J ≤ (openSimpleGraph G Set.univ).dist x y) :
+    (∑' tail : RawFrom (openSimpleGraph G Set.univ) m x,
+      ∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
+        ∑' _h : segmentAnchors G hloc γ n hdiam ep mid.1,
+          if ((∃ s, pos s ∈ tail.2.vertices) ∨ (∃ s, neg s ∈ mid.2.vertices)) ∨
+              (∃ z, z ∈ tail.2.vertices ∧ z ∈ mid.2.vertices) then
+            tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a)/
+              (segmentAnchors G hloc γ n hdiam ep (γ ep)).card else 0) ≤
+      earlyCollisionCost ‖P‖ J := by
+  have hK (x y : V) : 0 ≤ walkKernel G hloc a x y := by
+    simpa only [hP,Complex.ofReal_re] using (hpos x y).2
+  exact (raw_anchor_event_le_unconditioned G hloc a hK γ n hdiam ep m L x y
+    (fun tail mid => ((∃ s, pos s ∈ tail.2.vertices) ∨ (∃ s, neg s ∈ mid.2.vertices)) ∨
+      (∃ z, z ∈ tail.2.vertices ∧ z ∈ mid.2.vertices))).trans
+    (raw_early_collision_bound G hloc a P hpos hsym hP hρ m L J x y pos neg hposD hnegD hxy)
+
+ 
+
+theorem raw_tail_integrate_anchor_event (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (ep : I)
+    (m L : ℕ) (x y : V)
+    (B : (mid : RawFrom (openSimpleGraph G Set.univ) L y) →
+      segmentAnchors G hloc γ n hdiam ep mid.1 → Prop) [∀ mid h, Decidable (B mid h)] :
+    (∑' tail : RawFrom (openSimpleGraph G Set.univ) m x,
+      ∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
+        ∑' h : segmentAnchors G hloc γ n hdiam ep mid.1,
+          if B mid h then tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a)/
+            (segmentAnchors G hloc γ n hdiam ep (γ ep)).card else 0) =
+    (∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
+      ∑' h : segmentAnchors G hloc γ n hdiam ep mid.1,
+        if B mid h then mid.2.mass (walkKernel G hloc a)/
+          (segmentAnchors G hloc γ n hdiam ep (γ ep)).card else 0) := by
+  have he (tail : RawFrom (openSimpleGraph G Set.univ) m x)
+      (mid : RawFrom (openSimpleGraph G Set.univ) L y)
+      (h : segmentAnchors G hloc γ n hdiam ep mid.1) :
+      (if B mid h then tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a)/
+        (segmentAnchors G hloc γ n hdiam ep (γ ep)).card else 0) =
+      tail.2.mass (walkKernel G hloc a) *
+        (if B mid h then mid.2.mass (walkKernel G hloc a)/
+          (segmentAnchors G hloc γ n hdiam ep (γ ep)).card else 0) := by
+    split_ifs <;> ring
+  simp only [he,tsum_mul_left,tsum_mul_right,rawFrom_probability_sum G hloc a,one_mul]
+
+end EarlyAnchoredCollision
+end Percolation
+
+ 
+namespace Percolation
+section DependentUnion
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem finite_threefold_event_or {α β : Type*} {γ : β → Type*}
+    [Fintype α] [Fintype β] [∀ b, Fintype (γ b)]
+    (μ : α → (b : β) → γ b → ℝ) (hμ : ∀ a b c, 0 ≤ μ a b c)
+    (B C : α → (b : β) → γ b → Prop)
+    [∀ a b c, Decidable (B a b c)] [∀ a b c, Decidable (C a b c)] :
+    (∑ a, ∑ b, ∑ c, if B a b c ∨ C a b c then μ a b c else 0) ≤
+      (∑ a, ∑ b, ∑ c, if B a b c then μ a b c else 0) +
+      (∑ a, ∑ b, ∑ c, if C a b c then μ a b c else 0) := by
+  simp only [←Finset.sum_add_distrib]
+  apply Finset.sum_le_sum
+  intro a _
+  apply Finset.sum_le_sum
+  intro b _
+  apply Finset.sum_le_sum
+  intro c _
+  by_cases hb : B a b c <;> by_cases hc : C a b c <;>
+    simp only [hb,hc,or_self,or_false,false_or,ite_true,ite_false,add_zero,zero_add,le_refl]
+  exact le_add_of_nonneg_right (hμ a b c)
+
+end DependentUnion
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section CombinedRawAvoidance
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {I ι : Type*} [Fintype I] [Fintype ι]
+
+ 
+
+def firstEscapeCollision (G : BondGraph V E) (J m L : ℕ) (x y : V)
+    (pos neg : Fin (J+1) → V) (j : Fin (J+1) → I)
+    (tail : RawFrom (openSimpleGraph G Set.univ) m x)
+    (mid : RawFrom (openSimpleGraph G Set.univ) L y) (h : I → V) : Prop :=
+  (((∃ s, pos s ∈ tail.2.vertices) ∨ (∃ s, neg s ∈ mid.2.vertices)) ∨
+    (∃ z, z ∈ tail.2.vertices ∧ z ∈ mid.2.vertices)) ∨
+  ((∃ st : Fin (J+1) × Fin (J+1), h (j st.1) = neg st.2) ∨
+    (∃ ij : Fin (m+1) × Fin (J+1), tail.2.position ij.1.val = h (j ij.2)))
+
+noncomputable def suffixCollisionCost (ρ c : ℝ) (J : ℕ) : ℝ :=
+  ((J:ℝ)+1)^2*Real.sqrt c + ((J:ℝ)+1)*Real.sqrt c/(1-ρ)
+
+ 
+
+theorem raw_first_escape_collision_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G)
+    (htr : RootTransport root weight Γ) (hw : ∀ r, 0 ≤ weight r) (hw1 : ∑ r, weight r = 1)
+    (hcov : ∀ x, ∃ r, ∃ g ∈ Γ, g (root r) = x)
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ r, 1 ≤ c*weight r)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (ep : I)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ)) (hρ : ‖P‖ < 1)
+    (m L J : ℕ) (x y : V) (pos neg : Fin (J+1) → V) (j : Fin (J+1) → I)
+    (hposD : ∀ s, J ≤ (openSimpleGraph G Set.univ).dist x (pos s))
+    (hnegD : ∀ s, J ≤ (openSimpleGraph G Set.univ).dist y (neg s))
+    (hxy : 2*J ≤ (openSimpleGraph G Set.univ).dist x y) :
+    (∑' tail : RawFrom (openSimpleGraph G Set.univ) m x,
+      ∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
+        ∑' h : segmentAnchors G hloc γ n hdiam ep mid.1,
+          if firstEscapeCollision G J m L x y pos neg j tail mid h.val then
+            tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a)/
+              (segmentAnchors G hloc γ n hdiam ep (γ ep)).card else 0) ≤
+      earlyCollisionCost ‖P‖ J + suffixCollisionCost ‖P‖ c J*‖P‖^L := by
+  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
+  let (k : ℕ) (z : V) : Fintype (RawFrom (openSimpleGraph G Set.univ) k z) := Fintype.ofFinite _
+  have hK (x y : V) : 0 ≤ walkKernel G hloc a x y := by
+    simpa only [hP,Complex.ofReal_re] using (hpos x y).2
+  let μ := fun (tail : RawFrom (openSimpleGraph G Set.univ) m x)
+      (mid : RawFrom (openSimpleGraph G Set.univ) L y)
+      (_h : segmentAnchors G hloc γ n hdiam ep mid.1) =>
+    tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a)/
+      (segmentAnchors G hloc γ n hdiam ep (γ ep)).card
+  have hμ : ∀ tail mid h, 0 ≤ μ tail mid h := fun tail mid _ =>
+    div_nonneg (mul_nonneg (tail.2.mass_nonneg _ hK) (mid.2.mass_nonneg _ hK)) (Nat.cast_nonneg _)
+  let B := fun (tail : RawFrom (openSimpleGraph G Set.univ) m x)
+      (mid : RawFrom (openSimpleGraph G Set.univ) L y)
+      (_h : segmentAnchors G hloc γ n hdiam ep mid.1) =>
+    ((∃ s, pos s ∈ tail.2.vertices) ∨ (∃ s, neg s ∈ mid.2.vertices)) ∨
+      (∃ z, z ∈ tail.2.vertices ∧ z ∈ mid.2.vertices)
+  let C := fun (_tail : RawFrom (openSimpleGraph G Set.univ) m x)
+      (mid : RawFrom (openSimpleGraph G Set.univ) L y)
+      (h : segmentAnchors G hloc γ n hdiam ep mid.1) =>
+    ∃ st : Fin (J+1) × Fin (J+1), h.val (j st.1) = neg st.2
+  let D := fun (tail : RawFrom (openSimpleGraph G Set.univ) m x)
+      (mid : RawFrom (openSimpleGraph G Set.univ) L y)
+      (h : segmentAnchors G hloc γ n hdiam ep mid.1) =>
+    ∃ ij : Fin (m+1) × Fin (J+1), tail.2.position ij.1.val = h.val (j ij.2)
+  have hu := finite_threefold_event_or μ hμ B (fun tail mid h => C tail mid h ∨ D tail mid h)
+  have hv := finite_threefold_event_or μ hμ C D
+  have he := raw_anchor_early_collision_bound G hloc a γ n hdiam ep P hpos hsym hP hρ
+    m L J x y pos neg hposD hnegD hxy
+  have hn := raw_anchor_finite_collision_bound G hloc a root weight Γ hΓ htr hw hw1 hcov
+    c hc hmin γ n hdiam ep j neg P hpos hP L y
+  rw [←raw_tail_integrate_anchor_event G hloc a γ n hdiam ep m L x y
+    (fun _ h => ∃ st : Fin (J+1) × Fin (J+1), h.val (j st.1) = neg st.2)] at hn
+  have ht := raw_tail_whole_suffix_collision_bound G hloc a root weight Γ hΓ htr hw hw1 hcov
+    c hc hmin γ n hdiam ep j P hpos hP hρ m L x y
+  simp only [tsum_fintype,Fintype.card_fin,Nat.cast_add,Nat.cast_one] at he hn ht ⊢
+  have hr : earlyCollisionCost ‖P‖ J +
+      ((((J:ℝ)+1)*((J:ℝ)+1)*Real.sqrt c*‖P‖^L) +
+        ((J:ℝ)+1)*Real.sqrt c*‖P‖^L/(1-‖P‖)) =
+      earlyCollisionCost ‖P‖ J + suffixCollisionCost ‖P‖ c J*‖P‖^L := by
+    unfold suffixCollisionCost
+    ring
+  have hh := hu.trans ((add_le_add he (hv.trans (add_le_add hn ht))).trans_eq hr)
+  apply le_trans ?_ hh
+  apply le_of_eq
+  apply Finset.sum_congr rfl
+  intro tail _
+  apply Finset.sum_congr rfl
+  intro mid _
+  apply Finset.sum_congr rfl
+  intro h _
+  unfold firstEscapeCollision B C D μ
+  split_ifs <;> rfl
+
+end CombinedRawAvoidance
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section NormalizedFirstEscape
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {I ι : Type*} [Fintype I] [Fintype ι]
+
+ 
+
+theorem conditional_first_escape_collision_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G)
+    (htr : RootTransport root weight Γ) (hw : ∀ r, 0 < weight r) (hw1 : ∑ r, weight r = 1)
+    (hcov : ∀ x, ∃ r, ∃ g ∈ Γ, g (root r) = x)
+    (c : ℝ) (hc : 0 < c) (hmin : ∀ r, 1 ≤ c*weight r)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (ep : I)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re) (hρ : ‖P‖ < 1)
+    (m L J : ℕ) (x y : V) (hy : y ∈ vertexOrbit G (γ ep))
+    (pos neg : Fin (J+1) → V) (j : Fin (J+1) → I)
+    (hposD : ∀ s, J ≤ (openSimpleGraph G Set.univ).dist x (pos s))
+    (hnegD : ∀ s, J ≤ (openSimpleGraph G Set.univ).dist y (neg s))
+    (hxy : 2*J ≤ (openSimpleGraph G Set.univ).dist x y) :
+    (∑' tail : RawFrom (openSimpleGraph G Set.univ) m x,
+      ∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
+        ∑' h : segmentAnchors G hloc γ n hdiam ep mid.1,
+          if firstEscapeCollision G J m L x y pos neg j tail mid h.val then
+            tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a)/
+              (segmentAnchors G hloc γ n hdiam ep (γ ep)).card else 0) /
+        segmentReturnMass G hloc a γ ep L y ≤
+      2*c*(earlyCollisionCost ‖P‖ J + suffixCollisionCost ‖P‖ c J*‖P‖^L) := by
+  have hb := raw_first_escape_collision_bound G hloc a root weight Γ hΓ htr
+    (fun r => (hw r).le) hw1 hcov c hc.le hmin γ n hdiam ep P hpos hsym hP hρ
+    m L J x y pos neg j hposD hnegD hxy
+  have hq := segmentReturnMass_uniform_lower G hloc a root weight Γ hΓ htr hcov hw hw1 c hc hmin
+    P hpos hsym hP hdiag γ ep L
+  rw [←segmentReturnMass_on_orbit G hloc a γ ep L y hy P hpos hP] at hq
+  have htc : 0 < 2*c := mul_pos (by norm_num) hc
+  have hqp : 0 < segmentReturnMass G hloc a γ ep L y := (one_div_pos.mpr htc).trans_le hq
+  apply (div_le_iff₀ hqp).mpr
+  apply hb.trans
+  have hB : 0 ≤ earlyCollisionCost ‖P‖ J + suffixCollisionCost ‖P‖ c J*‖P‖^L := by
+    unfold earlyCollisionCost suffixCollisionCost
+    have : 0 < 1-‖P‖ := sub_pos.mpr hρ
+    positivity
+  have h1 : 1 ≤ (2*c)*segmentReturnMass G hloc a γ ep L y := by
+    exact (div_le_iff₀ htc).mp hq |>.trans_eq (mul_comm _ _)
+  have hh := mul_le_mul_of_nonneg_right h1 hB
+  nlinarith only [hh]
+
+end NormalizedFirstEscape
+end Percolation
+
+ 
+namespace Percolation
+section FirstEscapeChoiceOrder
+
+ 
+
+theorem exists_first_escape_uniform_cutoffs (ρ c ε : ℝ)
+    (hρ : 0 ≤ ρ) (hρ1 : ρ < 1) (hc : 0 < c) (hε : 0 < ε) :
+    ∃ J : ℕ, 0 < J ∧ ∃ L₀ : ℕ, ∀ L : ℕ, L₀ ≤ L →
+      2*c*(earlyCollisionCost ρ J + suffixCollisionCost ρ c J*ρ^L) < ε := by
+  have hδ : 0 < ε/(4*c) := div_pos hε (by positivity)
+  obtain ⟨J₀,hJ₀⟩ := exists_earlyCollisionCutoff ρ hρ hρ1 (ε/(4*c)) hδ
+  let J := max J₀ 1
+  have hJ : 0 < J := lt_of_lt_of_le (by omega : 0 < 1) (le_max_right _ _)
+  have he : earlyCollisionCost ρ J < ε/(4*c) := hJ₀ J (le_max_left _ _)
+  have hel : 2*c*earlyCollisionCost ρ J < ε := by
+    have hh := (lt_div_iff₀ (by positivity : 0 < 4*c)).mp he
+    nlinarith
+  have ht : Filter.Tendsto
+      (fun L : ℕ => 2*c*(earlyCollisionCost ρ J + suffixCollisionCost ρ c J*ρ^L))
+      Filter.atTop (nhds (2*c*earlyCollisionCost ρ J)) := by
+    have hh := ((tendsto_pow_atTop_nhds_zero_of_lt_one hρ hρ1).const_mul
+      (suffixCollisionCost ρ c J)).const_add (earlyCollisionCost ρ J)
+    have hh' := hh.const_mul (2*c)
+    simpa only [mul_zero,add_zero] using hh'
+  obtain ⟨L₀,hL₀⟩ := Filter.eventually_atTop.mp (ht.eventually (gt_mem_nhds hel))
+  exact ⟨J,hJ,L₀,hL₀⟩
+
+end FirstEscapeChoiceOrder
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section LiteralConditionalMass
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {I : Type*} [Fintype I]
+
+ 
+
+theorem raw_tail_anchor_total_mass (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (ep : I)
+    (m L : ℕ) (x y : V) :
+    (∑' tail : RawFrom (openSimpleGraph G Set.univ) m x,
+      ∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
+        ∑' _h : segmentAnchors G hloc γ n hdiam ep mid.1,
+          tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a)/
+            (segmentAnchors G hloc γ n hdiam ep (γ ep)).card) =
+      segmentReturnMass G hloc a γ ep L y := by
+  have hN : ((segmentAnchors G hloc γ n hdiam ep (γ ep)).card : ℝ) ≠ 0 := by
+    apply ne_of_gt
+    exact_mod_cast Finset.card_pos.mpr ((segmentAnchors_nonempty_iff G hloc γ n hdiam ep (γ ep)).mpr
+      ⟨Equiv.refl V,Equiv.refl E,isAut_refl G,rfl⟩)
+  have hh := raw_tail_integrate_anchor_event G hloc a γ n hdiam ep m L x y (fun _ _ => True)
+  simp only [ite_true] at hh
+  rw [hh]
+  unfold segmentReturnMass
+  apply tsum_congr
+  intro mid
+  simp only [tsum_fintype,Finset.sum_const,Finset.card_univ,Fintype.card_coe,nsmul_eq_mul]
+  rw [segmentAnchors_card_eq_indicator G hloc γ n hdiam ep mid.1]
+  split_ifs
+  · field_simp
+  · simp only [Nat.cast_zero,zero_mul]
+
+ 
+
+theorem raw_tail_anchor_conditional_mass (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (ep : I)
+    (m L : ℕ) (x y : V) (hq : segmentReturnMass G hloc a γ ep L y ≠ 0) :
+    (∑' tail : RawFrom (openSimpleGraph G Set.univ) m x,
+      ∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
+        ∑' _h : segmentAnchors G hloc γ n hdiam ep mid.1,
+          tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a)/
+            (segmentAnchors G hloc γ n hdiam ep (γ ep)).card /
+              segmentReturnMass G hloc a γ ep L y) = 1 := by
+  have hh := raw_tail_anchor_total_mass G hloc a γ n hdiam ep m L x y
+  simp only [tsum_div_const] at hh ⊢
+  rw [hh]
+  exact div_self hq
+
+end LiteralConditionalMass
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section LiteralGeodesicSeparation
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem graph_dist_aut (G : BondGraph V E) (hconn : Connected G)
+    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (x y : V) :
+    (openSimpleGraph G Set.univ).dist (a x) (a y) =
+      (openSimpleGraph G Set.univ).dist x y := by
+  have hreach (x y : V) : (openSimpleGraph G Set.univ).Reachable x y :=
+    (conn_iff_reachable G Set.univ x y).mp (conn_univ_of_connected G hconn x y)
+  obtain ⟨q,hq⟩ := (hreach x y).exists_walk_length_eq_dist
+  obtain ⟨r,hr⟩ := (hreach (a x) (a y)).exists_walk_length_eq_dist
+  have hq' := SimpleGraph.dist_le (q.map (autSimpleGraph G hab).toHom)
+  have hr' := SimpleGraph.dist_le (r.map (autSimpleGraph G (isAut_symm G hab)).toHom)
+  simp only [SimpleGraph.Walk.length_map,hq] at hq'
+  change (openSimpleGraph G Set.univ).dist (a.symm (a x)) (a.symm (a y)) ≤ _ at hr'
+  simp only [a.symm_apply_apply,SimpleGraph.Walk.length_map,hr] at hr'
+  exact le_antisymm hq' hr'
+
+ 
+
+theorem geodesic_anchor_early_separations (G : BondGraph V E) (hconn : Connected G)
+    {x y : V} (q : (openSimpleGraph G Set.univ).Walk x y) (J : ℕ)
+    (hlen : q.length = 2*J) (hq : q.length = (openSimpleGraph G Set.univ).dist x y)
+    (s : Fin (2*J+1) → V) (hs : s ∈ segmentOrbit G (geodesicSegment G q (2*J))) :
+    (∀ i : Fin (J+1), J ≤ (openSimpleGraph G Set.univ).dist
+      (s ⟨0,by omega⟩) (s ⟨J+i.val,by omega⟩)) ∧
+    (∀ i : Fin (J+1), J ≤ (openSimpleGraph G Set.univ).dist
+      (s ⟨2*J,by omega⟩) (s ⟨i.val,by omega⟩)) ∧
+    (2*J ≤ (openSimpleGraph G Set.univ).dist (s ⟨0,by omega⟩) (s ⟨2*J,by omega⟩)) := by
+  obtain ⟨a,b,hab,rfl⟩ := hs
+  constructor
+  · intro i
+    dsimp only [Function.comp_apply,geodesicSegment]
+    rw [graph_dist_aut G hconn hab]
+    rw [geodesic_getVert_dist _ q hq 0 (J+i.val) (by omega) (by omega)]
+    omega
+  constructor
+  · intro i
+    dsimp only [Function.comp_apply,geodesicSegment]
+    rw [graph_dist_aut G hconn hab,SimpleGraph.dist_comm]
+    rw [geodesic_getVert_dist _ q hq i.val (2*J) (by omega) (by omega)]
+    omega
+  · dsimp only [Function.comp_apply,geodesicSegment]
+    rw [graph_dist_aut G hconn hab]
+    rw [geodesic_getVert_dist _ q hq 0 (2*J) (by omega) (by omega)]
+    omega
+
+end LiteralGeodesicSeparation
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section PositiveFirstEscape
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {I ι : Type*} [Fintype I] [Fintype ι]
+
+ 
+
+theorem exists_positive_first_escape (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G)
+    (htr : RootTransport root weight Γ) (hw : ∀ r, 0 < weight r) (hw1 : ∑ r, weight r = 1)
+    (hcov : ∀ x, ∃ r, ∃ g ∈ Γ, g (root r) = x)
+    (c : ℝ) (hc : 0 < c) (hmin : ∀ r, 1 ≤ c*weight r)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (ep : I)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re) (hρ : ‖P‖ < 1)
+    (m L J : ℕ) (x y : V) (hy : y ∈ vertexOrbit G (γ ep))
+    (pos neg : Fin (J+1) → V) (j : Fin (J+1) → I)
+    (hposD : ∀ s, J ≤ (openSimpleGraph G Set.univ).dist x (pos s))
+    (hnegD : ∀ s, J ≤ (openSimpleGraph G Set.univ).dist y (neg s))
+    (hxy : 2*J ≤ (openSimpleGraph G Set.univ).dist x y)
+    (hsmall : 2*c*(earlyCollisionCost ‖P‖ J + suffixCollisionCost ‖P‖ c J*‖P‖^L) < 1) :
+    ∃ (tail : RawFrom (openSimpleGraph G Set.univ) m x)
+      (mid : RawFrom (openSimpleGraph G Set.univ) L y)
+      (h : segmentAnchors G hloc γ n hdiam ep mid.1),
+      ¬firstEscapeCollision G J m L x y pos neg j tail mid h.val ∧
+      0 < tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a) := by
+  have hb := conditional_first_escape_collision_bound G hloc a root weight Γ hΓ htr hw hw1 hcov
+    c hc hmin γ n hdiam ep P hpos hsym hP hdiag hρ m L J x y hy pos neg j hposD hnegD hxy
+  have hq := segmentReturnMass_uniform_lower G hloc a root weight Γ hΓ htr hcov hw hw1 c hc hmin
+    P hpos hsym hP hdiag γ ep L
+  rw [←segmentReturnMass_on_orbit G hloc a γ ep L y hy P hpos hP] at hq
+  have hqp : 0 < segmentReturnMass G hloc a γ ep L y :=
+    (one_div_pos.mpr (mul_pos (by norm_num) hc)).trans_le hq
+  have hK (x y : V) : 0 ≤ walkKernel G hloc a x y := by
+    simpa only [hP,Complex.ofReal_re] using (hpos x y).2
+  by_contra H
+  have heq :
+      (∑' tail : RawFrom (openSimpleGraph G Set.univ) m x,
+        ∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
+          ∑' h : segmentAnchors G hloc γ n hdiam ep mid.1,
+            if firstEscapeCollision G J m L x y pos neg j tail mid h.val then
+              tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a)/
+                (segmentAnchors G hloc γ n hdiam ep (γ ep)).card else 0) =
+      ∑' tail : RawFrom (openSimpleGraph G Set.univ) m x,
+        ∑' mid : RawFrom (openSimpleGraph G Set.univ) L y,
+          ∑' _h : segmentAnchors G hloc γ n hdiam ep mid.1,
+            tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a)/
+              (segmentAnchors G hloc γ n hdiam ep (γ ep)).card := by
+    apply tsum_congr
+    intro tail
+    apply tsum_congr
+    intro mid
+    apply tsum_congr
+    intro h
+    split_ifs with ht
+    · rfl
+    · have hz : tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a) = 0 := by
+        apply le_antisymm
+        · exact le_of_not_gt (fun hp => H ⟨tail,mid,h,ht,hp⟩)
+        · exact mul_nonneg (tail.2.mass_nonneg _ hK) (mid.2.mass_nonneg _ hK)
+      rw [hz,zero_div]
+  rw [heq,raw_tail_anchor_total_mass G hloc a γ n hdiam ep m L x y,div_self hqp.ne'] at hb
+  exact (not_lt_of_ge hb) hsmall
+
+end PositiveFirstEscape
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section ProjectionEvents
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem finite_event_fiber_sum {α β : Type*} [Fintype α] [Fintype β] [DecidableEq β]
+    (f : α → β) (μ : α → ℝ) (B : β → Prop) [DecidablePred B] :
+    (∑ a, if B (f a) then μ a else 0) =
+      ∑ b, if B b then ∑ a, if f a = b then μ a else 0 else 0 := by
+  calc
+    _ = ∑ a, ∑ b, if B b then (if f a = b then μ a else 0) else 0 := by
+      apply Finset.sum_congr rfl
+      intro a _
+      have hf : (fun b => if B b then (if f a = b then μ a else 0) else 0) =
+          fun b => if f a = b then (if B (f a) then μ a else 0) else 0 := by
+        funext b
+        by_cases h : f a = b
+        · subst b
+          simp
+        · simp [h]
+      rw [hf]
+      simp
+    _ = _ := by
+      rw [Finset.sum_comm]
+      apply Finset.sum_congr rfl
+      intro b _
+      by_cases h : B b <;> simp [h]
+
+ 
+
+theorem rooted_label_raw_event_bound (G : BondGraph V E) (hloc : LocallyFinite G)
+    (a : ℝ) (ha : 0 < a) (hK : ∀ x y, 0 ≤ walkKernel G hloc a x y)
+    (J L m : ℕ) (γ : Fin (J+J+1) → V)
+    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
+    (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (x : V)
+    (hx : x ∈ vertexOrbit G (γ (segmentCenter J)))
+    (q : ℝ) (hq : 0 < q) (hqle : q ≤ segmentReturnMass G hloc a γ (Fin.last (J+J)) L (γ (Fin.last (J+J))))
+    (B : RawFrom (openSimpleGraph G Set.univ) ((J+L)+J) x → Prop) [DecidablePred B] :
+    (∑' ξ : {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m // ξ.first (segmentCenter J) = x},
+      if B (rootedEscapeLabelRaw G J L m γ hγ x ξ) then
+        escapeLabelProbability G hloc a γ n hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m ξ.1 else 0) ≤
+      (∑' w : RawFrom (openSimpleGraph G Set.univ) ((J+L)+J) x,
+        if B w then w.2.mass (walkKernel G hloc a) else 0)/(a^(J+J)*q) := by
+  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
+  let := finite_rootedEscapeLabels G hloc γ n hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m x
+  let : Fintype {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m // ξ.first (segmentCenter J) = x} := Fintype.ofFinite _
+  let : Fintype (RawFrom (openSimpleGraph G Set.univ) ((J+L)+J) x) := Fintype.ofFinite _
+  rw [←tsum_indicator_div]
+  simp only [tsum_fintype]
+  rw [finite_event_fiber_sum]
+  apply Finset.sum_le_sum
+  intro w _
+  by_cases hw : B w
+  · simp only [hw,ite_true]
+    simpa only [tsum_fintype] using
+      rooted_label_raw_projection_bound G hloc a ha hK J L m γ hγ n hdiam x hx q hq hqle w
+  · simp only [hw,ite_false,le_refl]
+
+end ProjectionEvents
+end Percolation
+
+ 
+namespace Percolation
+universe u v w
+variable {V : Type u} {E : Type v} {I : Type w}
+section FirstLabeledMarginal
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Fintype I]
+
+ 
+def labelFirstEscapeCollision (G : BondGraph V E) (γ : I → V) (ep neg : I)
+    (J L m : ℕ) (posIdx negIdx suffixIdx : Fin (J+1) → I)
+    (ξ : EscapeLabel G γ ep neg L m) : Prop :=
+  firstEscapeCollision G J m L (ξ.first neg) (ξ.first ep)
+    (fun t => ξ.first (posIdx t)) (fun t => ξ.first (negIdx t)) suffixIdx
+    ξ.leftTail ⟨ξ.last ep,ξ.middle⟩ ξ.last
+
+theorem labelFromCode_first_collision (G : BondGraph V E) (hloc : LocallyFinite G)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
+    (center ep neg : I) (J L m : ℕ) (x : V)
+    (posIdx negIdx suffixIdx : Fin (J+1) → I)
+    (c : RootedLabelCode G hloc γ n hdiam center ep neg L m x) :
+    labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx
+      (labelFromCode G hloc γ n hdiam center ep neg L m x c).1 =
+    firstEscapeCollision G J m L (c.1.val neg) (c.1.val ep)
+      (fun t => c.1.val (posIdx t)) (fun t => c.1.val (negIdx t)) suffixIdx
+      c.2.2.2.1 c.2.1 c.2.2.1.val := by
+  rcases c with ⟨⟨s,hs⟩,⟨z,mid⟩,⟨h,hh⟩,α,β⟩
+  obtain ⟨_,hhz⟩ := (mem_segmentAnchors G hloc γ n hdiam ep z h).mp hh
+  subst z
+  rfl
+
+ 
+
+theorem raw_rooted_first_collision_marginal (G : BondGraph V E) (hloc : LocallyFinite G)
+    (a : ℝ) (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
+    (center ep neg : I) (J L m : ℕ) (x : V)
+    (posIdx negIdx suffixIdx : Fin (J+1) → I) :
+    (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
+      if labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.1 then
+        ξ.1.mass (walkKernel G hloc a) else 0) =
+    ∑' s : segmentAnchors G hloc γ n hdiam center x,
+      ∑' tail : RawFrom (openSimpleGraph G Set.univ) m (s.val neg),
+        ∑' mid : RawFrom (openSimpleGraph G Set.univ) L (s.val ep),
+          ∑' h : segmentAnchors G hloc γ n hdiam ep mid.1,
+            if firstEscapeCollision G J m L (s.val neg) (s.val ep)
+              (fun t => s.val (posIdx t)) (fun t => s.val (negIdx t)) suffixIdx tail mid h.val then
+              tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a) else 0 := by
+  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
+  let (k : ℕ) (z : V) : Fintype (RawFrom (openSimpleGraph G Set.univ) k z) := Fintype.ofFinite _
+  let := finite_rootedEscapeLabels G hloc γ n hdiam center ep neg L m x
+  let R := rootedLabelEquiv G hloc γ n hdiam center ep neg L m x
+  rw [←R.symm.tsum_eq]
+  simp only [R,rootedLabelEquiv,Equiv.coe_fn_symm_mk,labelFromCode_mass,labelFromCode_first_collision]
+  unfold RootedLabelCode at *
+  rw [(summable_of_hasFiniteSupport (Set.toFinite _)).tsum_sigma]
+  apply tsum_congr
+  intro s
+  rw [(summable_of_hasFiniteSupport (Set.toFinite _)).tsum_sigma]
+  have hh (mid : RawFrom (openSimpleGraph G Set.univ) L (s.val ep)) :
+      (∑' hh : Σ h : segmentAnchors G hloc γ n hdiam ep mid.1,
+          RawFrom (openSimpleGraph G Set.univ) m (s.val neg) × RawFrom (openSimpleGraph G Set.univ) m (h.val neg),
+        if firstEscapeCollision G J m L (s.val neg) (s.val ep)
+          (fun t => s.val (posIdx t)) (fun t => s.val (negIdx t)) suffixIdx hh.2.1 mid hh.1.val then
+          mid.2.mass (walkKernel G hloc a)*hh.2.1.2.mass (walkKernel G hloc a)*hh.2.2.2.mass (walkKernel G hloc a) else 0) =
+      ∑' h : segmentAnchors G hloc γ n hdiam ep mid.1,
+        ∑' tail : RawFrom (openSimpleGraph G Set.univ) m (s.val neg),
+          if firstEscapeCollision G J m L (s.val neg) (s.val ep)
+            (fun t => s.val (posIdx t)) (fun t => s.val (negIdx t)) suffixIdx tail mid h.val then
+            tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a) else 0 := by
+    rw [(summable_of_hasFiniteSupport (Set.toFinite _)).tsum_sigma]
+    apply tsum_congr
+    intro h
+    rw [(summable_of_hasFiniteSupport (Set.toFinite _)).tsum_prod]
+    apply tsum_congr
+    intro tail
+    split_ifs
+    · simp only [tsum_mul_left,rawFrom_probability_sum G hloc a,mul_one]
+      exact mul_comm _ _
+    · simp only [tsum_zero]
+  simp_rw [hh]
+  simp only [tsum_fintype]
+  conv_lhs =>
+    arg 2
+    intro mid
+    rw [Finset.sum_comm]
+  rw [Finset.sum_comm]
+
+end FirstLabeledMarginal
+end Percolation
+
+ 
+namespace Percolation
+universe u v w
+variable {V : Type u} {E : Type v} {I : Type w}
+section NormalizedMarginal
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Fintype I]
+
+ 
+
+theorem rooted_first_collision_probability_marginal (G : BondGraph V E) (hloc : LocallyFinite G)
+    (a : ℝ) (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
+    (center ep neg : I) (J L m : ℕ) (x : V)
+    (posIdx negIdx suffixIdx : Fin (J+1) → I) :
+    (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
+      if labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.1 then
+        escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1 else 0) =
+    (∑' s : segmentAnchors G hloc γ n hdiam center x,
+      (∑' tail : RawFrom (openSimpleGraph G Set.univ) m (s.val neg),
+        ∑' mid : RawFrom (openSimpleGraph G Set.univ) L (s.val ep),
+          ∑' h : segmentAnchors G hloc γ n hdiam ep mid.1,
+            if firstEscapeCollision G J m L (s.val neg) (s.val ep)
+              (fun t => s.val (posIdx t)) (fun t => s.val (negIdx t)) suffixIdx tail mid h.val then
+              tail.2.mass (walkKernel G hloc a)*mid.2.mass (walkKernel G hloc a)/
+                (segmentAnchors G hloc γ n hdiam ep (γ ep)).card else 0) /
+          segmentReturnMass G hloc a γ ep L (γ ep)) /
+      (segmentAnchors G hloc γ n hdiam center (γ center)).card := by
+  simp only [escapeLabelProbability,tsum_indicator_div,tsum_div_const]
+  rw [raw_rooted_first_collision_marginal G hloc a γ n hdiam center ep neg J L m x posIdx negIdx suffixIdx]
+  ring
+
+end NormalizedMarginal
+end Percolation
+
+ 
+namespace Percolation
+universe u v w
+variable {V : Type u} {E : Type v} {I : Type w}
+section FullFirstEscapeBound
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Fintype I] {ι : Type*} [Fintype ι]
+
+ 
+
+theorem rooted_first_collision_probability_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G)
+    (htr : RootTransport root weight Γ) (hw : ∀ r, 0 < weight r) (hw1 : ∑ r, weight r = 1)
+    (hcov : ∀ x, ∃ r, ∃ g ∈ Γ, g (root r) = x)
+    (c : ℝ) (hc : 0 < c) (hmin : ∀ r, 1 ≤ c*weight r)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (center ep neg : I)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re) (hρ : ‖P‖ < 1)
+    (m L J : ℕ) (x : V) (hx : x ∈ vertexOrbit G (γ center))
+    (posIdx negIdx suffixIdx : Fin (J+1) → I)
+    (hsep : ∀ s ∈ segmentOrbit G γ,
+      (∀ i, J ≤ (openSimpleGraph G Set.univ).dist (s neg) (s (posIdx i))) ∧
+      (∀ i, J ≤ (openSimpleGraph G Set.univ).dist (s ep) (s (negIdx i))) ∧
+      2*J ≤ (openSimpleGraph G Set.univ).dist (s neg) (s ep)) :
+    (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
+      if labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.1 then
+        escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1 else 0) ≤
+      2*c*(earlyCollisionCost ‖P‖ J + suffixCollisionCost ‖P‖ c J*‖P‖^L) := by
+  rw [rooted_first_collision_probability_marginal G hloc a γ n hdiam center ep neg J L m x posIdx negIdx suffixIdx]
+  have hN : 0 < ((segmentAnchors G hloc γ n hdiam center (γ center)).card : ℝ) := by
+    exact_mod_cast Finset.card_pos.mpr ((segmentAnchors_nonempty_iff G hloc γ n hdiam center (γ center)).mpr
+      ⟨Equiv.refl V,Equiv.refl E,isAut_refl G,rfl⟩)
+  apply (div_le_iff₀ hN).mpr
+  simp only [tsum_fintype]
+  calc
+    _ ≤ ∑ s : segmentAnchors G hloc γ n hdiam center x,
+        2*c*(earlyCollisionCost ‖P‖ J + suffixCollisionCost ‖P‖ c J*‖P‖^L) := by
+      apply Finset.sum_le_sum
+      intro s _
+      have hs := ((mem_segmentAnchors G hloc γ n hdiam center x s.val).mp s.property).1
+      have hy : s.val ep ∈ vertexOrbit G (γ ep) := by
+        obtain ⟨g,b,hg,he⟩ := hs
+        rw [he]
+        exact ⟨g,b,hg,rfl⟩
+      obtain ⟨hp,hn,hxy⟩ := hsep s.val hs
+      have hh := conditional_first_escape_collision_bound G hloc a root weight Γ hΓ htr hw hw1 hcov
+        c hc hmin γ n hdiam ep P hpos hsym hP hdiag hρ m L J (s.val neg) (s.val ep) hy
+        (fun i => s.val (posIdx i)) (fun i => s.val (negIdx i)) suffixIdx hp hn hxy
+      rw [segmentReturnMass_on_orbit G hloc a γ ep L (s.val ep) hy P hpos hP] at hh
+      simpa only [tsum_fintype] using hh
+    _ = _ := by
+      simp only [Finset.sum_const,Finset.card_univ,Fintype.card_coe,nsmul_eq_mul]
+      rw [segmentAnchors_card_on_orbit G hloc γ n hdiam center x hx]
+      ring
+
+end FullFirstEscapeBound
+end Percolation
+
+ 
+namespace Percolation
+universe u v w
+variable {V : Type u} {E : Type v} {I : Type w}
+section CollisionNaturality
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+namespace HoldingWalk
+variable {W : Type*} {H : SimpleGraph V} {H' : SimpleGraph W}
+theorem map_position (f : H →g H') {k : ℕ} {x y : V} (q : HoldingWalk H k x y) (i : ℕ) :
+    (q.map f).position i = f (q.position i) := by
+  induction q generalizing i with
+  | nil => rfl
+  | hold q ih => cases i <;> simp only [map,position,ih]
+  | step h q ih => cases i <;> simp only [map,position,ih]
+end HoldingWalk
+
+theorem list_common_vertex_map_iff {W : Type*} (f : V → W) (hf : Function.Injective f)
+    (l r : List V) :
+    (∃ z, z ∈ l.map f ∧ z ∈ r.map f) ↔ ∃ z, z ∈ l ∧ z ∈ r := by
+  constructor
+  · rintro ⟨z,hz,hz'⟩
+    obtain ⟨x,hx,rfl⟩ := List.mem_map.mp hz
+    obtain ⟨y,hy,he⟩ := List.mem_map.mp hz'
+    have : y = x := hf he
+    subst y
+    exact ⟨x,hx,hy⟩
+  · rintro ⟨z,hz,hz'⟩
+    exact ⟨f z,List.mem_map.mpr ⟨z,hz,rfl⟩,List.mem_map.mpr ⟨z,hz',rfl⟩⟩
+
+ 
+theorem labelFirstEscapeCollision_aut (G : BondGraph V E) (γ : I → V) (ep neg : I)
+    (J L m : ℕ) (posIdx negIdx suffixIdx : Fin (J+1) → I)
+    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b)
+    (ξ : EscapeLabel G γ ep neg L m) :
+    labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx (EscapeLabel.autMap hab ξ) ↔
+      labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ := by
+  dsimp only [labelFirstEscapeCollision,firstEscapeCollision,EscapeLabel.autMap,Function.comp_apply,autSimpleGraph]
+  simp only [HoldingWalk.map_vertices,HoldingWalk.map_position]
+  change (((∃ s, a (ξ.first (posIdx s)) ∈ ξ.leftTail.2.vertices.map a) ∨
+    (∃ s, a (ξ.first (negIdx s)) ∈ ξ.middle.vertices.map a)) ∨
+    (∃ z, z ∈ ξ.leftTail.2.vertices.map a ∧ z ∈ ξ.middle.vertices.map a)) ∨
+    ((∃ st : Fin (J+1) × Fin (J+1), a (ξ.last (suffixIdx st.1)) = a (ξ.first (negIdx st.2))) ∨
+      (∃ ij : Fin (m+1) × Fin (J+1), a (ξ.leftTail.2.position ij.1.val) = a (ξ.last (suffixIdx ij.2)))) ↔ _
+  rw [list_common_vertex_map_iff a a.injective]
+  simp only [List.mem_map,Equiv.apply_eq_iff_eq]
+  simp
+
+end CollisionNaturality
+end Percolation
+
+ 
+namespace Percolation
+universe u v w
+variable {V : Type u} {E : Type v} {I : Type w}
+section BothEscapeEvents
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Fintype I] {ι : Type*} [Fintype ι]
+
+ 
+theorem rootOrbitWeight_pos (G : BondGraph V E)
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (hΓ : Γ ⊆ vertexAut G) (hw : ∀ i, 0 < weight i)
+    (hcov : ∀ x, ∃ r, ∃ g ∈ Γ, g (root r) = x) (x : V) :
+    0 < ∑ i, if root i ∈ vertexOrbit G x then weight i else 0 := by
+  obtain ⟨r,g,hg,he⟩ := hcov x
+  obtain ⟨b,hab⟩ := hΓ hg
+  have hr : root r ∈ vertexOrbit G x := by
+    refine ⟨g.symm,b.symm,isAut_symm G hab,?_⟩
+    rw [←he,Equiv.symm_apply_apply]
+  have hh : (if root r ∈ vertexOrbit G x then weight r else 0) ≤
+      ∑ i, if root i ∈ vertexOrbit G x then weight i else 0 :=
+    Finset.single_le_sum (f := fun i => if root i ∈ vertexOrbit G x then weight i else 0)
+      (fun i _ => by split_ifs <;> first | exact (hw i).le | rfl) (Finset.mem_univ r)
+  simpa only [ite_eq_left hr] using lt_of_lt_of_le (by simpa only [ite_eq_left hr] using hw r) hh
+
+ 
+
+theorem rooted_escape_collision_reversal (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G)
+    (htr : RootTransport root weight Γ) (hw : ∀ r, 0 < weight r)
+    (hcov : ∀ x, ∃ r, ∃ g ∈ Γ, g (root r) = x)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (center ep neg : I)
+    (m L J : ℕ) (x : V) (hx : x ∈ vertexOrbit G (γ center))
+    (posIdx negIdx suffixIdx : Fin (J+1) → I) :
+    (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
+      if labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.1 then
+        escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1 else 0) =
+    ∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
+      if labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.1.reverse then
+        escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1 else 0 := by
+  have hh := rooted_escape_label_reversal G hloc a root weight Γ hΓ htr γ n hdiam center ep neg L m
+    (rootOrbitWeight_pos G root weight Γ hΓ hw hcov (γ center))
+    (fun ξ => if labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ then (1:ℝ) else 0)
+    (fun g b hab ξ => by rw [labelFirstEscapeCollision_aut G γ ep neg J L m posIdx negIdx suffixIdx hab ξ]) x hx
+  simpa only [mul_ite,mul_one,mul_zero] using hh
+
+end BothEscapeEvents
+end Percolation
+
+ 
+namespace Percolation
+universe u v w
+variable {V : Type u} {E : Type v} {I : Type w}
+section BothEscapeBound
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Fintype I] {ι : Type*} [Fintype ι]
+
+ 
+theorem rooted_both_collision_probability_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G)
+    (htr : RootTransport root weight Γ) (hw : ∀ r, 0 < weight r) (hw1 : ∑ r, weight r = 1)
+    (hcov : ∀ x, ∃ r, ∃ g ∈ Γ, g (root r) = x)
+    (c : ℝ) (hc : 0 < c) (hmin : ∀ r, 1 ≤ c*weight r)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (center ep neg : I)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re) (hρ : ‖P‖ < 1)
+    (m L J : ℕ) (x : V) (hx : x ∈ vertexOrbit G (γ center))
+    (posIdx negIdx suffixIdx : Fin (J+1) → I)
+    (hsep : ∀ s ∈ segmentOrbit G γ,
+      (∀ i, J ≤ (openSimpleGraph G Set.univ).dist (s neg) (s (posIdx i))) ∧
+      (∀ i, J ≤ (openSimpleGraph G Set.univ).dist (s ep) (s (negIdx i))) ∧
+      2*J ≤ (openSimpleGraph G Set.univ).dist (s neg) (s ep)) :
+    (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
+      if labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.1 ∨
+        labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.1.reverse then
+        escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1 else 0) ≤
+      4*c*(earlyCollisionCost ‖P‖ J + suffixCollisionCost ‖P‖ c J*‖P‖^L) := by
+  have hK (x y : V) : 0 ≤ walkKernel G hloc a x y := by
+    simpa only [hP,Complex.ofReal_re] using (hpos x y).2
+  have hb := rooted_first_collision_probability_bound G hloc a root weight Γ hΓ htr hw hw1 hcov
+    c hc hmin γ n hdiam center ep neg P hpos hsym hP hdiag hρ m L J x hx posIdx negIdx suffixIdx hsep
+  have he := rooted_escape_collision_reversal G hloc a root weight Γ hΓ htr hw hcov
+    γ n hdiam center ep neg m L J x hx posIdx negIdx suffixIdx
+  let := finite_rootedEscapeLabels G hloc γ n hdiam center ep neg L m x
+  let := Fintype.ofFinite {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x}
+  have hu := finite_weighted_event_or
+    (fun ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x} =>
+      escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1)
+    (fun ξ => escapeLabelProbability_nonneg G hloc a hK γ n hdiam center ep neg L m ξ.1)
+    (fun ξ => labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.1)
+    (fun ξ => labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.1.reverse)
+  simp only [tsum_fintype] at hb he ⊢
+  rw [←he] at hu
+  linarith
+
+end BothEscapeBound
+end Percolation
+
+ 
+namespace Percolation
+universe u v w
+variable {V : Type u} {E : Type v} {I : Type w}
+section PositiveTwoEscapes
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Fintype I] {ι : Type*} [Fintype ι]
+
+ 
+
+theorem exists_positive_two_escape_label (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G)
+    (htr : RootTransport root weight Γ) (hw : ∀ r, 0 < weight r) (hw1 : ∑ r, weight r = 1)
+    (hcov : ∀ x, ∃ r, ∃ g ∈ Γ, g (root r) = x)
+    (c : ℝ) (hc : 0 < c) (hmin : ∀ r, 1 ≤ c*weight r)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (center ep neg : I)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re) (hρ : ‖P‖ < 1)
+    (m L J : ℕ) (x : V) (hx : x ∈ vertexOrbit G (γ center))
+    (posIdx negIdx suffixIdx : Fin (J+1) → I)
+    (hsep : ∀ s ∈ segmentOrbit G γ,
+      (∀ i, J ≤ (openSimpleGraph G Set.univ).dist (s neg) (s (posIdx i))) ∧
+      (∀ i, J ≤ (openSimpleGraph G Set.univ).dist (s ep) (s (negIdx i))) ∧
+      2*J ≤ (openSimpleGraph G Set.univ).dist (s neg) (s ep)) (hsmall : 4*c*(earlyCollisionCost ‖P‖ J + suffixCollisionCost ‖P‖ c J*‖P‖^L) < 1) :
+    ∃ ξ : EscapeLabel G γ ep neg L m, ξ.first center = x ∧
+      ¬labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ ∧
+      ¬labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.reverse ∧
+      0 < escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ := by
+  have hb := rooted_both_collision_probability_bound G hloc a root weight Γ hΓ htr hw hw1 hcov
+    c hc hmin γ n hdiam center ep neg P hpos hsym hP hdiag hρ m L J x hx posIdx negIdx suffixIdx hsep
+  have hs := (rooted_label_probability_hasSum G hloc a root weight Γ hΓ htr hcov hw hw1 c hc hmin
+    P hpos hsym hP hdiag γ n hdiam center ep neg L m x hx).tsum_eq
+  have hK (x y : V) : 0 ≤ walkKernel G hloc a x y := by
+    simpa only [hP,Complex.ofReal_re] using (hpos x y).2
+  by_contra H
+  have heq :
+      (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
+        if labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.1 ∨
+          labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.1.reverse then
+          escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1 else 0) =
+      ∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
+        escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1 := by
+    apply tsum_congr
+    intro ξ
+    split_ifs with ht
+    · rfl
+    · apply Eq.symm
+      apply le_antisymm
+      · exact le_of_not_gt (fun hp => H ⟨ξ.1,ξ.2,(not_or.mp ht).1,(not_or.mp ht).2,hp⟩)
+      · exact escapeLabelProbability_nonneg G hloc a hK γ n hdiam center ep neg L m ξ.1
+  rw [heq,hs] at hb
+  exact (not_lt_of_ge hb) hsmall
+
+end PositiveTwoEscapes
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Escape/SelectionBounds.lean b/lean/OAI/Probability/UnimodularPercolation/Escape/SelectionBounds.lean
index ee327c3c7cd97e24c9be4b23e7ec74d933138861..f6160315e5ac72428dd155faa73b2b88b8556899 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Escape/SelectionBounds.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Escape/SelectionBounds.lean
@@ -1,553 +1,553 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Escape.PositiveEscapes
-
-namespace OAI
-
-/-! Selection Bounds for bond percolation. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-section BothEscapeChoiceOrder
-
- 
-
-theorem exists_both_escape_uniform_cutoffs (ρ c ε : ℝ)
-    (hρ : 0 ≤ ρ) (hρ1 : ρ < 1) (hc : 0 < c) (hε : 0 < ε) :
-    ∃ J : ℕ, 0 < J ∧ ∃ L₀ : ℕ, ∀ L : ℕ, L₀ ≤ L →
-      4*c*(earlyCollisionCost ρ J + suffixCollisionCost ρ c J*ρ^L) < ε := by
-  obtain ⟨J,hJ,L₀,hL₀⟩ := exists_first_escape_uniform_cutoffs ρ c (ε/2) hρ hρ1 hc (by positivity)
-  refine ⟨J,hJ,L₀,fun L hL => ?_⟩
-  have hh := hL₀ L hL
-  linarith
-
-end BothEscapeChoiceOrder
-end Percolation
-
- 
-namespace Percolation
-universe u v w
-variable {V : Type u} {E : Type v} {I : Type w}
-section TailConditioning
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Fintype I]
-
- 
-
-theorem rooted_label_event_of_tail_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (hK : ∀ x y, 0 ≤ walkKernel G hloc a x y)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
-    (center ep neg : I) (L m : ℕ) (x : V)
-    (B : EscapeLabel G γ ep neg L m → Prop) (ε : ℝ)
-    (hB : ∀ (s : segmentAnchors G hloc γ n hdiam center x)
-      (mid : RawFrom (openSimpleGraph G Set.univ) L (s.val ep))
-      (h : segmentAnchors G hloc γ n hdiam ep mid.1),
-      (∑' tails : RawFrom (openSimpleGraph G Set.univ) m (s.val neg) ×
-          RawFrom (openSimpleGraph G Set.univ) m (h.val neg),
-        if B (labelFromCode G hloc γ n hdiam center ep neg L m x ⟨s,mid,h,tails⟩).1 then
-          tails.1.2.mass (walkKernel G hloc a)*tails.2.2.mass (walkKernel G hloc a) else 0) ≤ ε) :
-    (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
-      if B ξ.1 then ξ.1.mass (walkKernel G hloc a) else 0) ≤
-      ε * ∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
-        ξ.1.mass (walkKernel G hloc a) := by
-  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
-  let (k : ℕ) (z : V) : Fintype (RawFrom (openSimpleGraph G Set.univ) k z) := Fintype.ofFinite _
-  let := finite_rootedEscapeLabels G hloc γ n hdiam center ep neg L m x
-  let R := rootedLabelEquiv G hloc γ n hdiam center ep neg L m x
-  rw [←R.symm.tsum_eq,←R.symm.tsum_eq]
-  simp only [R,rootedLabelEquiv,Equiv.coe_fn_symm_mk,labelFromCode_mass]
-  unfold RootedLabelCode at *
-  simp only [tsum_fintype,Fintype.sum_sigma,Finset.mul_sum]
-  apply Finset.sum_le_sum
-  intro s _
-  apply Finset.sum_le_sum
-  intro mid _
-  apply Finset.sum_le_sum
-  intro h _
-  have hh := hB s mid h
-  simp only [tsum_fintype] at hh
-  have ht : (∑ tails : RawFrom (openSimpleGraph G Set.univ) m (s.val neg) ×
-      RawFrom (openSimpleGraph G Set.univ) m (h.val neg),
-        tails.1.2.mass (walkKernel G hloc a)*tails.2.2.mass (walkKernel G hloc a)) = 1 := by
-    rw [Fintype.sum_prod_type]
-    simp only [←Finset.mul_sum,←Finset.sum_mul]
-    simpa only [tsum_fintype,mul_one] using congrArg₂ (fun r t : ℝ => r*t)
-      (rawFrom_probability_sum G hloc a m (s.val neg))
-      (rawFrom_probability_sum G hloc a m (h.val neg))
-  calc
-    _ = mid.2.mass (walkKernel G hloc a) *
-        ∑ tails : RawFrom (openSimpleGraph G Set.univ) m (s.val neg) ×
-          RawFrom (openSimpleGraph G Set.univ) m (h.val neg),
-          if B (labelFromCode G hloc γ n hdiam center ep neg L m x ⟨s,mid,h,tails⟩).1 then
-            tails.1.2.mass (walkKernel G hloc a)*tails.2.2.mass (walkKernel G hloc a) else 0 := by
-      rw [Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro tails _
-      split_ifs <;> simp only [mul_zero,mul_assoc]
-    _ ≤ mid.2.mass (walkKernel G hloc a)*ε := mul_le_mul_of_nonneg_left hh (mid.2.mass_nonneg _ hK)
-    _ = _ := by
-      simp only [mul_assoc,←Finset.mul_sum]
-      rw [ht]
-      ring
-
-end TailConditioning
-end Percolation
-
- 
-namespace Percolation
-universe u v w
-variable {V : Type u} {E : Type v} {I : Type w}
-section NormalizedTailConditioning
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Fintype I]
-
- 
-
-theorem rooted_label_probability_of_tail_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ g ∈ Γ, g (root i) = x)
-    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
-    (c : ℝ) (hc : 0 < c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
-    (center ep neg : I) (L m : ℕ) (x : V) (hx : x ∈ vertexOrbit G (γ center)) 
-    (B : EscapeLabel G γ ep neg L m → Prop) (ε : ℝ)
-    (hB : ∀ (s : segmentAnchors G hloc γ n hdiam center x)
-      (mid : RawFrom (openSimpleGraph G Set.univ) L (s.val ep))
-      (h : segmentAnchors G hloc γ n hdiam ep mid.1),
-      (∑' tails : RawFrom (openSimpleGraph G Set.univ) m (s.val neg) ×
-          RawFrom (openSimpleGraph G Set.univ) m (h.val neg),
-        if B (labelFromCode G hloc γ n hdiam center ep neg L m x ⟨s,mid,h,tails⟩).1 then
-          tails.1.2.mass (walkKernel G hloc a)*tails.2.2.mass (walkKernel G hloc a) else 0) ≤ ε) :
-    (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
-      if B ξ.1 then escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1 else 0) ≤ ε := by
-  have hK (x y : V) : 0 ≤ walkKernel G hloc a x y := by
-    simpa only [hP,Complex.ofReal_re] using (hpos x y).2
-  have hq := segmentReturnMass_uniform_lower G hloc a root weight Γ hΓ htr hcov hw hw1 c hc hmin
-    P hpos hsym hP hdiag γ ep L
-  have hqp : 0 ≤ segmentReturnMass G hloc a γ ep L (γ ep) :=
-    (one_div_pos.mpr (mul_pos (by norm_num) hc)).le.trans hq
-  let D := ((segmentAnchors G hloc γ n hdiam center (γ center)).card : ℝ) *
-    ((segmentAnchors G hloc γ n hdiam ep (γ ep)).card : ℝ) *
-      segmentReturnMass G hloc a γ ep L (γ ep)
-  have hD : 0 ≤ D := mul_nonneg (mul_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _)) hqp
-  have hr := rooted_label_event_of_tail_bound G hloc a hK γ n hdiam center ep neg L m x B ε hB
-  have hs := (rooted_label_probability_hasSum G hloc a root weight Γ hΓ htr hcov hw hw1 c hc hmin
-    P hpos hsym hP hdiag γ n hdiam center ep neg L m x hx).tsum_eq
-  have hd := div_le_div_of_nonneg_right hr hD
-  have he : (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
-      ξ.1.mass (walkKernel G hloc a))/D = 1 := by
-    simpa only [escapeLabelProbability,tsum_div_const,D] using hs
-  simpa only [escapeLabelProbability,tsum_indicator_div,tsum_div_const,D,
-    mul_div_assoc,he,mul_one] using hd
-
-end NormalizedTailConditioning
-end Percolation
-
- 
-namespace Percolation
-universe u v w
-variable {V : Type u} {E : Type v} {I : Type w}
-section FullPotentialTests
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Fintype I]
-
- 
-def labelPotentialFailure (G : BondGraph V E) (p : unitInterval) (γ : I → V)
-    (ep neg : I) (L m : ℕ)
-    (F : (s : I → V) → RawFrom (openSimpleGraph G Set.univ) L (s ep) → (I → V) → Finset V)
-    (t : ℝ) (ξ : EscapeLabel G γ ep neg L m) : Prop :=
-  (t < ∑ b ∈ F ξ.first ⟨ξ.last ep,ξ.middle⟩ ξ.last, twoPoint G p ξ.leftTail.1 b) ∨
-  (t < ∑ b ∈ F ξ.first ⟨ξ.last ep,ξ.middle⟩ ξ.last, twoPoint G p ξ.rightTail.1 b)
-
-theorem labelFromCode_potential_failure (G : BondGraph V E) (hloc : LocallyFinite G)
-    (p : unitInterval) (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
-    (center ep neg : I) (L m : ℕ) (x : V)
-    (F : (s : I → V) → RawFrom (openSimpleGraph G Set.univ) L (s ep) → (I → V) → Finset V)
-    (t : ℝ) (s : segmentAnchors G hloc γ n hdiam center x)
-    (mid : RawFrom (openSimpleGraph G Set.univ) L (s.val ep))
-    (h : segmentAnchors G hloc γ n hdiam ep mid.1)
-    (tails : RawFrom (openSimpleGraph G Set.univ) m (s.val neg) ×
-      RawFrom (openSimpleGraph G Set.univ) m (h.val neg)) :
-    labelPotentialFailure G p γ ep neg L m F t
-      (labelFromCode G hloc γ n hdiam center ep neg L m x ⟨s,mid,h,tails⟩).1 ↔
-      (t < ∑ b ∈ F s.val mid h.val, twoPoint G p tails.1.1 b) ∨
-      (t < ∑ b ∈ F s.val mid h.val, twoPoint G p tails.2.1 b) := by
-  rcases s with ⟨s,hs⟩
-  rcases mid with ⟨z,mid⟩
-  rcases h with ⟨h,hh⟩
-  obtain ⟨_,hhz⟩ := (mem_segmentAnchors G hloc γ n hdiam ep z h).mp hh
-  subst z
-  rfl
-
- 
-
-theorem rooted_label_potential_failure_bound [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (htr : RootTransport root weight (vertexAut G))
-    (hcov : ∀ x, ∃ r, ∃ g ∈ vertexAut G, g (root r) = x)
-    (hw : ∀ r, 0 < weight r) (hw1 : ∑ r, weight r = 1)
-    (c : ℝ) (hc : 0 < c) (hmin : ∀ r, 1 ≤ c*weight r)
-    (P A : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re)
-    (p : unitInterval) (hA : HasKernel G p A)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
-    (center ep neg : I) (L m : ℕ) (x : V) (hx : x ∈ vertexOrbit G (γ center))
-    (F : (s : I → V) → RawFrom (openSimpleGraph G Set.univ) L (s ep) → (I → V) → Finset V)
-    (N : ℝ) (hN : ∀ s mid h, ((F s mid h).card : ℝ) ≤ N) (t : ℝ) (ht : 0 < t) :
-    (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
-      if labelPotentialFailure G p γ ep neg L m F t ξ.1 then
-        escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1 else 0) ≤
-      2*(N*Real.sqrt c*‖P‖^m)/t := by
-  apply rooted_label_probability_of_tail_bound G hloc a root weight (vertexAut G) (Set.Subset.refl _)
-    htr hcov hw hw1 c hc hmin P hpos hsym hP hdiag γ n hdiam center ep neg L m x hx
-  intro s mid h
-  simp only [labelFromCode_potential_failure]
-  have hh := two_raw_tail_potential_failure G hloc a root weight htr hcov (fun r => (hw r).le) hw1
-    c hc.le hmin P A hpos hP p hA m (s.val neg) (h.val neg) (F s.val mid h.val) t ht
-  apply hh.trans
-  gcongr
-  exact hN s.val mid h.val
-
-end FullPotentialTests
-end Percolation
-
- 
-namespace Percolation
-universe u v w
-variable {V : Type u} {E : Type v} {I : Type w}
-section CombinedSelectionEvents
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Fintype I]
-
- 
-
-theorem rooted_label_collision_potential_failure_bound [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (htr : RootTransport root weight (vertexAut G))
-    (hcov : ∀ x, ∃ r, ∃ g ∈ vertexAut G, g (root r) = x)
-    (hw : ∀ r, 0 < weight r) (hw1 : ∑ r, weight r = 1)
-    (c : ℝ) (hc : 0 < c) (hmin : ∀ r, 1 ≤ c*weight r)
-    (P A : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re)
-    (p : unitInterval) (hA : HasKernel G p A)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
-    (center ep neg : I) (L m : ℕ) (x : V) (hx : x ∈ vertexOrbit G (γ center))
-    (hρ : ‖P‖ < 1) (J : ℕ) (posIdx negIdx suffixIdx : Fin (J+1) → I)
-    (hsep : ∀ s ∈ segmentOrbit G γ,
-      (∀ i, J ≤ (openSimpleGraph G Set.univ).dist (s neg) (s (posIdx i))) ∧
-      (∀ i, J ≤ (openSimpleGraph G Set.univ).dist (s ep) (s (negIdx i))) ∧
-      2*J ≤ (openSimpleGraph G Set.univ).dist (s neg) (s ep))
-    (F : (s : I → V) → RawFrom (openSimpleGraph G Set.univ) L (s ep) → (I → V) → Finset V)
-    (N : ℝ) (hN : ∀ s mid h, ((F s mid h).card : ℝ) ≤ N) (t : ℝ) (ht : 0 < t) :
-    (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
-      if (labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.1 ∨
-        labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.1.reverse) ∨
-        labelPotentialFailure G p γ ep neg L m F t ξ.1 then
-        escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1 else 0) ≤
-      4*c*(earlyCollisionCost ‖P‖ J + suffixCollisionCost ‖P‖ c J*‖P‖^L) +
-        2*(N*Real.sqrt c*‖P‖^m)/t := by
-  have hK (x y : V) : 0 ≤ walkKernel G hloc a x y := by
-    simpa only [hP,Complex.ofReal_re] using (hpos x y).2
-  have hb := rooted_both_collision_probability_bound G hloc a root weight (vertexAut G) (Set.Subset.refl _)
-    htr hw hw1 hcov c hc hmin γ n hdiam center ep neg P hpos hsym hP hdiag hρ m L J x hx posIdx negIdx suffixIdx hsep
-  have hp := rooted_label_potential_failure_bound G hloc a root weight htr hcov hw hw1 c hc hmin
-    P A hpos hsym hP hdiag p hA γ n hdiam center ep neg L m x hx F N hN t ht
-  let := finite_rootedEscapeLabels G hloc γ n hdiam center ep neg L m x
-  let := Fintype.ofFinite {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x}
-  have hu := finite_weighted_event_or
-    (fun ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x} =>
-      escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1)
-    (fun ξ => escapeLabelProbability_nonneg G hloc a hK γ n hdiam center ep neg L m ξ.1)
-    (fun ξ => labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.1 ∨
-      labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.1.reverse)
-    (fun ξ => labelPotentialFailure G p γ ep neg L m F t ξ.1)
-  simp only [tsum_fintype] at hb hp ⊢
-  exact hu.trans (add_le_add hb hp)
-
-end CombinedSelectionEvents
-end Percolation
-
- 
-namespace Percolation
-universe u v w
-variable {V : Type u} {E : Type v} {I : Type w}
-section QuantitativeSelectionMass
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Fintype I]
-
-theorem finite_weighted_success_lower {α : Type*} [Fintype α]
-    (μ : α → ℝ) (hμ : ∑ a, μ a = 1) (B : α → Prop) [DecidablePred B] (ε : ℝ)
-    (hB : (∑ a, if B a then μ a else 0) ≤ ε) :
-    1-ε ≤ ∑ a, if ¬B a then μ a else 0 := by
-  have he : (∑ a, if ¬B a then μ a else 0) + (∑ a, if B a then μ a else 0) = 1 := by
-    rw [←Finset.sum_add_distrib,←hμ]
-    apply Finset.sum_congr rfl
-    intro a _
-    by_cases h : B a <;> simp only [h,not_true_eq_false,not_false_eq_true,ite_true,ite_false,zero_add,add_zero]
-  linarith
-
- 
-
-theorem rooted_label_collision_potential_success_lower [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (htr : RootTransport root weight (vertexAut G))
-    (hcov : ∀ x, ∃ r, ∃ g ∈ vertexAut G, g (root r) = x)
-    (hw : ∀ r, 0 < weight r) (hw1 : ∑ r, weight r = 1)
-    (c : ℝ) (hc : 0 < c) (hmin : ∀ r, 1 ≤ c*weight r)
-    (P A : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re)
-    (p : unitInterval) (hA : HasKernel G p A)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
-    (center ep neg : I) (L m : ℕ) (x : V) (hx : x ∈ vertexOrbit G (γ center))
-    (hρ : ‖P‖ < 1) (J : ℕ) (posIdx negIdx suffixIdx : Fin (J+1) → I)
-    (hsep : ∀ s ∈ segmentOrbit G γ,
-      (∀ i, J ≤ (openSimpleGraph G Set.univ).dist (s neg) (s (posIdx i))) ∧
-      (∀ i, J ≤ (openSimpleGraph G Set.univ).dist (s ep) (s (negIdx i))) ∧
-      2*J ≤ (openSimpleGraph G Set.univ).dist (s neg) (s ep))
-    (F : (s : I → V) → RawFrom (openSimpleGraph G Set.univ) L (s ep) → (I → V) → Finset V)
-    (N : ℝ) (hN : ∀ s mid h, ((F s mid h).card : ℝ) ≤ N) (t : ℝ) (ht : 0 < t) :
-    1 - (4*c*(earlyCollisionCost ‖P‖ J + suffixCollisionCost ‖P‖ c J*‖P‖^L) +
-      2*(N*Real.sqrt c*‖P‖^m)/t) ≤
-    ∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
-      if ¬((labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.1 ∨
-        labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.1.reverse) ∨
-        labelPotentialFailure G p γ ep neg L m F t ξ.1) then
-        escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1 else 0 := by
-  have hb := rooted_label_collision_potential_failure_bound G hloc a root weight htr hcov hw hw1 c hc hmin
-    P A hpos hsym hP hdiag p hA γ n hdiam center ep neg L m x hx hρ J posIdx negIdx suffixIdx hsep F N hN t ht
-  have hs := (rooted_label_probability_hasSum G hloc a root weight (vertexAut G) (Set.Subset.refl _)
-    htr hcov hw hw1 c hc hmin P hpos hsym hP hdiag γ n hdiam center ep neg L m x hx).tsum_eq
-  let := finite_rootedEscapeLabels G hloc γ n hdiam center ep neg L m x
-  let := Fintype.ofFinite {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x}
-  simp only [tsum_fintype] at hs hb ⊢
-  exact finite_weighted_success_lower _ hs _ _ hb
-
-end QuantitativeSelectionMass
-end Percolation
-
- 
-namespace Percolation
-section JointSelectionChoiceOrder
-
- 
-
-theorem exists_joint_collision_potential_cutoffs (ρ c d τ : ℝ)
-    (hρ : 0 < ρ) (hρ1 : ρ < 1) (hc : 0 < c) (hd : 0 ≤ d) (hτ : 0 < τ) :
-    ∃ J : ℕ, 0 < J ∧ ∃ L₀ : ℕ, ∃ C₁ : ℝ, 0 < C₁ ∧ 1 < (-Real.log ρ)*C₁ ∧
-      ∃ k₀ : ℕ, J+J+L₀ ≤ k₀ ∧ ∀ k : ℕ, k₀ ≤ k → ∀ R : ℝ, 1 ≤ R →
-        4*c*(earlyCollisionCost ρ J + suffixCollisionCost ρ c J*ρ^(k-(J+J))) +
-          2*((d+1)*((k:ℝ)+1)*Real.sqrt c*ρ^⌈C₁*Real.log (((k:ℝ)+1)*R)⌉₊)/(τ/R) ≤ 3/8 := by
-  obtain ⟨J,hJ,L₀,hL₀⟩ := exists_both_escape_uniform_cutoffs ρ c (1/8) hρ.le hρ1 hc (by norm_num)
-  let A : ℝ := 2*(d+1)*Real.sqrt c/τ
-  have hA : 0 ≤ A := div_nonneg (mul_nonneg (mul_nonneg (by norm_num) (by linarith)) (Real.sqrt_nonneg _)) hτ.le
-  obtain ⟨C₁,hC₁,hlam,k₀,hk₀,hk⟩ := exists_uniform_logarithmic_tail_cutoff ρ A hρ hρ1 hA (J+J+L₀)
-  refine ⟨J,hJ,L₀,C₁,hC₁,hlam,k₀,hk₀,fun k hk' R hR => ?_⟩
-  have hL : L₀ ≤ k-(J+J) := by omega
-  have hb := hL₀ (k-(J+J)) hL
-  have hp := hk k hk' R hR
-  have hRp : 0 < R := lt_of_lt_of_le (by norm_num) hR
-  have he : 2*((d+1)*((k:ℝ)+1)*Real.sqrt c*ρ^⌈C₁*Real.log (((k:ℝ)+1)*R)⌉₊)/(τ/R) =
-      A*((k:ℝ)+1)*R*ρ^⌈C₁*Real.log (((k:ℝ)+1)*R)⌉₊ := by
-    dsimp only [A]
-    field_simp [hτ.ne',hRp.ne']
-  rw [he]
-  linarith
-
-end JointSelectionChoiceOrder
-end Percolation
-
- 
-namespace Percolation
-universe u v w
-variable {V : Type u} {E : Type v} {I : Type w}
-section SpineMarginal
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Fintype I]
-
-theorem labelFromCode_anchor_event (G : BondGraph V E) (hloc : LocallyFinite G)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
-    (center ep neg : I) (L m : ℕ) (x : V) (B : (I → V) → (I → V) → Prop)
-    (s : segmentAnchors G hloc γ n hdiam center x)
-    (mid : RawFrom (openSimpleGraph G Set.univ) L (s.val ep))
-    (h : segmentAnchors G hloc γ n hdiam ep mid.1)
-    (tails : RawFrom (openSimpleGraph G Set.univ) m (s.val neg) ×
-      RawFrom (openSimpleGraph G Set.univ) m (h.val neg)) :
-    B (labelFromCode G hloc γ n hdiam center ep neg L m x ⟨s,mid,h,tails⟩).1.first
-      (labelFromCode G hloc γ n hdiam center ep neg L m x ⟨s,mid,h,tails⟩).1.last ↔ B s.val h.val := by
-  rcases s with ⟨s,hs⟩
-  rcases mid with ⟨z,mid⟩
-  rcases h with ⟨h,hh⟩
-  obtain ⟨_,hhz⟩ := (mem_segmentAnchors G hloc γ n hdiam ep z h).mp hh
-  subst z
-  rfl
-
- 
-theorem rooted_label_anchor_event_marginal (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
-    (center ep neg : I) (L m : ℕ) (x : V) (B : (I → V) → (I → V) → Prop)
-    [∀ s h, Decidable (B s h)] :
-    (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
-      if B ξ.1.first ξ.1.last then ξ.1.mass (walkKernel G hloc a) else 0) =
-    ∑' s : segmentAnchors G hloc γ n hdiam center x,
-      ∑' mid : RawFrom (openSimpleGraph G Set.univ) L (s.val ep),
-        ∑' h : segmentAnchors G hloc γ n hdiam ep mid.1,
-          if B s.val h.val then mid.2.mass (walkKernel G hloc a) else 0 := by
-  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
-  let (k : ℕ) (z : V) : Fintype (RawFrom (openSimpleGraph G Set.univ) k z) := Fintype.ofFinite _
-  let := finite_rootedEscapeLabels G hloc γ n hdiam center ep neg L m x
-  let R := rootedLabelEquiv G hloc γ n hdiam center ep neg L m x
-  rw [←R.symm.tsum_eq]
-  simp only [R,rootedLabelEquiv,Equiv.coe_fn_symm_mk,labelFromCode_mass]
-  unfold RootedLabelCode at *
-  simp only [tsum_fintype,Fintype.sum_sigma,labelFromCode_anchor_event]
-  apply Finset.sum_congr rfl
-  intro s _
-  apply Finset.sum_congr rfl
-  intro mid _
-  apply Finset.sum_congr rfl
-  intro h _
-  by_cases hb : B s.val h.val
-  · simp only [hb,ite_true,Fintype.sum_prod_type,mul_assoc,←Finset.mul_sum,←Finset.sum_mul]
-    have hl := rawFrom_probability_sum G hloc a m (s.val neg)
-    have hr := rawFrom_probability_sum G hloc a m (h.val neg)
-    simp only [tsum_fintype] at hl hr
-    rw [hl,hr]
-    ring
-  · simp only [hb,ite_false,Finset.sum_const_zero]
-
-end SpineMarginal
-end Percolation
-
- 
-namespace Percolation
-universe u v w
-variable {V : Type u} {E : Type v} {I : Type w}
-section EndpointMarginal
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Fintype I]
-
- 
-theorem rooted_label_endpoint_marginal (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
-    (center ep neg : I) (L m : ℕ) (x : V) (S : Finset V) :
-    (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
-      if ξ.1.last center ∈ S then escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1 else 0) =
-    (∑' s : segmentAnchors G hloc γ n hdiam center x,
-      ∑' mid : RawFrom (openSimpleGraph G Set.univ) L (s.val ep),
-        ∑' h : segmentAnchors G hloc γ n hdiam ep mid.1,
-          if h.val center ∈ S then mid.2.mass (walkKernel G hloc a)/
-            (segmentAnchors G hloc γ n hdiam ep (γ ep)).card else 0) /
-      (((segmentAnchors G hloc γ n hdiam center (γ center)).card : ℝ)*
-        segmentReturnMass G hloc a γ ep L (γ ep)) := by
-  simp only [escapeLabelProbability,tsum_indicator_div,tsum_div_const]
-  rw [rooted_label_anchor_event_marginal G hloc a γ n hdiam center ep neg L m x (fun _ h => h center ∈ S)]
-  ring
-
-end EndpointMarginal
-end Percolation
-
- 
-namespace Percolation
-universe u v w
-variable {V : Type u} {E : Type v} {I : Type w}
-section ActualEndpointSeparation
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Fintype I] {ι : Type*} [Fintype ι]
-
- 
-
-theorem rooted_label_endpoint_hit_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G)
-    (htr : RootTransport root weight Γ) (hw : ∀ r, 0 < weight r) (hw1 : ∑ r, weight r = 1)
-    (hcov : ∀ x, ∃ r, ∃ g ∈ Γ, g (root r) = x)
-    (c : ℝ) (hc : 0 < c) (hmin : ∀ r, 1 ≤ c*weight r)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (center ep neg : I)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re)
-    (m L : ℕ) (x : V) (hx : x ∈ vertexOrbit G (γ center))
-    (S : Finset V) :
-    (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
-      if ξ.1.last center ∈ S then escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1 else 0) ≤
-      2*c*(S.card:ℝ)*Real.sqrt c*‖P‖^L := by
-  rw [rooted_label_endpoint_marginal G hloc a γ n hdiam center ep neg L m x S]
-  have hN : 0 < ((segmentAnchors G hloc γ n hdiam center (γ center)).card : ℝ) := by
-    exact_mod_cast Finset.card_pos.mpr ((segmentAnchors_nonempty_iff G hloc γ n hdiam center (γ center)).mpr
-      ⟨Equiv.refl V,Equiv.refl E,isAut_refl G,rfl⟩)
-  have hq := segmentReturnMass_uniform_lower G hloc a root weight Γ hΓ htr hcov hw hw1 c hc hmin
-    P hpos hsym hP hdiag γ ep L
-  have hqp : 0 < segmentReturnMass G hloc a γ ep L (γ ep) :=
-    (one_div_pos.mpr (mul_pos (by norm_num) hc)).trans_le hq
-  apply (div_le_iff₀ (mul_pos hN hqp)).mpr
-  have hb (s : segmentAnchors G hloc γ n hdiam center x) :
-      (∑' mid : RawFrom (openSimpleGraph G Set.univ) L (s.val ep),
-        ∑' h : segmentAnchors G hloc γ n hdiam ep mid.1,
-          if h.val center ∈ S then mid.2.mass (walkKernel G hloc a)/
-            (segmentAnchors G hloc γ n hdiam ep (γ ep)).card else 0) ≤
-      (S.card:ℝ)*Real.sqrt c*‖P‖^L := by
-    have hh := raw_anchor_finite_collision_bound G hloc a root weight Γ hΓ htr (fun r => (hw r).le)
-      hw1 hcov c hc.le hmin γ n hdiam ep (fun _ : PUnit.{1} => center) (fun z : S => z.val)
-      P hpos hP L (s.val ep)
-    simpa [Subtype.exists] using hh
-  have hu := Finset.sum_le_sum (s := Finset.univ) (fun s _ => hb s)
-  simp only [tsum_fintype,Finset.sum_const,Finset.card_univ,Fintype.card_coe,nsmul_eq_mul] at hu ⊢
-  rw [segmentAnchors_card_on_orbit G hloc γ n hdiam center x hx] at hu
-  apply hu.trans
-  have he : 1 ≤ 2*c*segmentReturnMass G hloc a γ ep L (γ ep) := by
-    have hh := (div_le_iff₀ (by positivity : 0 < 2*c)).mp hq
-    nlinarith
-  have hp : 0 ≤ ((segmentAnchors G hloc γ n hdiam center (γ center)).card : ℝ)*
-      (S.card:ℝ)*Real.sqrt c*‖P‖^L := by positivity
-  nlinarith [mul_le_mul_of_nonneg_left he hp]
-
-end ActualEndpointSeparation
-end Percolation
-
- 
-namespace Percolation
-universe u v w
-variable {V : Type u} {E : Type v} {I : Type w}
-section EndpointBallBound
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Fintype I] {ι : Type*} [Fintype ι]
-
- 
-theorem rooted_label_endpoint_ball_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G)
-    (htr : RootTransport root weight Γ) (hw : ∀ r, 0 < weight r) (hw1 : ∑ r, weight r = 1)
-    (hcov : ∀ x, ∃ r, ∃ g ∈ Γ, g (root r) = x)
-    (c : ℝ) (hc : 0 < c) (hmin : ∀ r, 1 ≤ c*weight r)
-    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (center ep neg : I)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re)
-    (m L : ℕ) (x : V) (hx : x ∈ vertexOrbit G (γ center))
-    (d : ℕ) (hd : ∀ z, degree G z ≤ d) (r : ℕ) :
-    (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
-      if ξ.1.last center ∈ reachBall G x r then escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1 else 0) ≤
-      2*c*(((d+1)^r:ℕ):ℝ)*Real.sqrt c*‖P‖^L := by
-  have hh := rooted_label_endpoint_hit_bound G hloc a root weight Γ hΓ htr hw hw1 hcov c hc hmin
-    γ n hdiam center ep neg P hpos hsym hP hdiag m L x hx (finite_reachBall hloc x r).toFinset
-  have hb : ((finite_reachBall hloc x r).toFinset.card : ℝ) ≤ (((d+1)^r:ℕ):ℝ) := by
-    exact_mod_cast (by
-      simpa only [←Set.ncard_eq_toFinset_card] using reachBall_ncard_le_pow G hloc d hd x r)
-  calc
-    _ ≤ 2*c*((finite_reachBall hloc x r).toFinset.card:ℝ)*Real.sqrt c*‖P‖^L := by
-      simpa only [Set.Finite.mem_toFinset] using hh
-    _ ≤ _ := by gcongr
-
-end EndpointBallBound
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Escape.PositiveEscapes
+
+namespace OAI
+
+/-! Selection Bounds for bond percolation. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+section BothEscapeChoiceOrder
+
+ 
+
+theorem exists_both_escape_uniform_cutoffs (ρ c ε : ℝ)
+    (hρ : 0 ≤ ρ) (hρ1 : ρ < 1) (hc : 0 < c) (hε : 0 < ε) :
+    ∃ J : ℕ, 0 < J ∧ ∃ L₀ : ℕ, ∀ L : ℕ, L₀ ≤ L →
+      4*c*(earlyCollisionCost ρ J + suffixCollisionCost ρ c J*ρ^L) < ε := by
+  obtain ⟨J,hJ,L₀,hL₀⟩ := exists_first_escape_uniform_cutoffs ρ c (ε/2) hρ hρ1 hc (by positivity)
+  refine ⟨J,hJ,L₀,fun L hL => ?_⟩
+  have hh := hL₀ L hL
+  linarith
+
+end BothEscapeChoiceOrder
+end Percolation
+
+ 
+namespace Percolation
+universe u v w
+variable {V : Type u} {E : Type v} {I : Type w}
+section TailConditioning
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Fintype I]
+
+ 
+
+theorem rooted_label_event_of_tail_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (hK : ∀ x y, 0 ≤ walkKernel G hloc a x y)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
+    (center ep neg : I) (L m : ℕ) (x : V)
+    (B : EscapeLabel G γ ep neg L m → Prop) (ε : ℝ)
+    (hB : ∀ (s : segmentAnchors G hloc γ n hdiam center x)
+      (mid : RawFrom (openSimpleGraph G Set.univ) L (s.val ep))
+      (h : segmentAnchors G hloc γ n hdiam ep mid.1),
+      (∑' tails : RawFrom (openSimpleGraph G Set.univ) m (s.val neg) ×
+          RawFrom (openSimpleGraph G Set.univ) m (h.val neg),
+        if B (labelFromCode G hloc γ n hdiam center ep neg L m x ⟨s,mid,h,tails⟩).1 then
+          tails.1.2.mass (walkKernel G hloc a)*tails.2.2.mass (walkKernel G hloc a) else 0) ≤ ε) :
+    (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
+      if B ξ.1 then ξ.1.mass (walkKernel G hloc a) else 0) ≤
+      ε * ∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
+        ξ.1.mass (walkKernel G hloc a) := by
+  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
+  let (k : ℕ) (z : V) : Fintype (RawFrom (openSimpleGraph G Set.univ) k z) := Fintype.ofFinite _
+  let := finite_rootedEscapeLabels G hloc γ n hdiam center ep neg L m x
+  let R := rootedLabelEquiv G hloc γ n hdiam center ep neg L m x
+  rw [←R.symm.tsum_eq,←R.symm.tsum_eq]
+  simp only [R,rootedLabelEquiv,Equiv.coe_fn_symm_mk,labelFromCode_mass]
+  unfold RootedLabelCode at *
+  simp only [tsum_fintype,Fintype.sum_sigma,Finset.mul_sum]
+  apply Finset.sum_le_sum
+  intro s _
+  apply Finset.sum_le_sum
+  intro mid _
+  apply Finset.sum_le_sum
+  intro h _
+  have hh := hB s mid h
+  simp only [tsum_fintype] at hh
+  have ht : (∑ tails : RawFrom (openSimpleGraph G Set.univ) m (s.val neg) ×
+      RawFrom (openSimpleGraph G Set.univ) m (h.val neg),
+        tails.1.2.mass (walkKernel G hloc a)*tails.2.2.mass (walkKernel G hloc a)) = 1 := by
+    rw [Fintype.sum_prod_type]
+    simp only [←Finset.mul_sum,←Finset.sum_mul]
+    simpa only [tsum_fintype,mul_one] using congrArg₂ (fun r t : ℝ => r*t)
+      (rawFrom_probability_sum G hloc a m (s.val neg))
+      (rawFrom_probability_sum G hloc a m (h.val neg))
+  calc
+    _ = mid.2.mass (walkKernel G hloc a) *
+        ∑ tails : RawFrom (openSimpleGraph G Set.univ) m (s.val neg) ×
+          RawFrom (openSimpleGraph G Set.univ) m (h.val neg),
+          if B (labelFromCode G hloc γ n hdiam center ep neg L m x ⟨s,mid,h,tails⟩).1 then
+            tails.1.2.mass (walkKernel G hloc a)*tails.2.2.mass (walkKernel G hloc a) else 0 := by
+      rw [Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro tails _
+      split_ifs <;> simp only [mul_zero,mul_assoc]
+    _ ≤ mid.2.mass (walkKernel G hloc a)*ε := mul_le_mul_of_nonneg_left hh (mid.2.mass_nonneg _ hK)
+    _ = _ := by
+      simp only [mul_assoc,←Finset.mul_sum]
+      rw [ht]
+      ring
+
+end TailConditioning
+end Percolation
+
+ 
+namespace Percolation
+universe u v w
+variable {V : Type u} {E : Type v} {I : Type w}
+section NormalizedTailConditioning
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Fintype I]
+
+ 
+
+theorem rooted_label_probability_of_tail_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ g ∈ Γ, g (root i) = x)
+    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
+    (c : ℝ) (hc : 0 < c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
+    (center ep neg : I) (L m : ℕ) (x : V) (hx : x ∈ vertexOrbit G (γ center)) 
+    (B : EscapeLabel G γ ep neg L m → Prop) (ε : ℝ)
+    (hB : ∀ (s : segmentAnchors G hloc γ n hdiam center x)
+      (mid : RawFrom (openSimpleGraph G Set.univ) L (s.val ep))
+      (h : segmentAnchors G hloc γ n hdiam ep mid.1),
+      (∑' tails : RawFrom (openSimpleGraph G Set.univ) m (s.val neg) ×
+          RawFrom (openSimpleGraph G Set.univ) m (h.val neg),
+        if B (labelFromCode G hloc γ n hdiam center ep neg L m x ⟨s,mid,h,tails⟩).1 then
+          tails.1.2.mass (walkKernel G hloc a)*tails.2.2.mass (walkKernel G hloc a) else 0) ≤ ε) :
+    (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
+      if B ξ.1 then escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1 else 0) ≤ ε := by
+  have hK (x y : V) : 0 ≤ walkKernel G hloc a x y := by
+    simpa only [hP,Complex.ofReal_re] using (hpos x y).2
+  have hq := segmentReturnMass_uniform_lower G hloc a root weight Γ hΓ htr hcov hw hw1 c hc hmin
+    P hpos hsym hP hdiag γ ep L
+  have hqp : 0 ≤ segmentReturnMass G hloc a γ ep L (γ ep) :=
+    (one_div_pos.mpr (mul_pos (by norm_num) hc)).le.trans hq
+  let D := ((segmentAnchors G hloc γ n hdiam center (γ center)).card : ℝ) *
+    ((segmentAnchors G hloc γ n hdiam ep (γ ep)).card : ℝ) *
+      segmentReturnMass G hloc a γ ep L (γ ep)
+  have hD : 0 ≤ D := mul_nonneg (mul_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _)) hqp
+  have hr := rooted_label_event_of_tail_bound G hloc a hK γ n hdiam center ep neg L m x B ε hB
+  have hs := (rooted_label_probability_hasSum G hloc a root weight Γ hΓ htr hcov hw hw1 c hc hmin
+    P hpos hsym hP hdiag γ n hdiam center ep neg L m x hx).tsum_eq
+  have hd := div_le_div_of_nonneg_right hr hD
+  have he : (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
+      ξ.1.mass (walkKernel G hloc a))/D = 1 := by
+    simpa only [escapeLabelProbability,tsum_div_const,D] using hs
+  simpa only [escapeLabelProbability,tsum_indicator_div,tsum_div_const,D,
+    mul_div_assoc,he,mul_one] using hd
+
+end NormalizedTailConditioning
+end Percolation
+
+ 
+namespace Percolation
+universe u v w
+variable {V : Type u} {E : Type v} {I : Type w}
+section FullPotentialTests
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Fintype I]
+
+ 
+def labelPotentialFailure (G : BondGraph V E) (p : unitInterval) (γ : I → V)
+    (ep neg : I) (L m : ℕ)
+    (F : (s : I → V) → RawFrom (openSimpleGraph G Set.univ) L (s ep) → (I → V) → Finset V)
+    (t : ℝ) (ξ : EscapeLabel G γ ep neg L m) : Prop :=
+  (t < ∑ b ∈ F ξ.first ⟨ξ.last ep,ξ.middle⟩ ξ.last, twoPoint G p ξ.leftTail.1 b) ∨
+  (t < ∑ b ∈ F ξ.first ⟨ξ.last ep,ξ.middle⟩ ξ.last, twoPoint G p ξ.rightTail.1 b)
+
+theorem labelFromCode_potential_failure (G : BondGraph V E) (hloc : LocallyFinite G)
+    (p : unitInterval) (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
+    (center ep neg : I) (L m : ℕ) (x : V)
+    (F : (s : I → V) → RawFrom (openSimpleGraph G Set.univ) L (s ep) → (I → V) → Finset V)
+    (t : ℝ) (s : segmentAnchors G hloc γ n hdiam center x)
+    (mid : RawFrom (openSimpleGraph G Set.univ) L (s.val ep))
+    (h : segmentAnchors G hloc γ n hdiam ep mid.1)
+    (tails : RawFrom (openSimpleGraph G Set.univ) m (s.val neg) ×
+      RawFrom (openSimpleGraph G Set.univ) m (h.val neg)) :
+    labelPotentialFailure G p γ ep neg L m F t
+      (labelFromCode G hloc γ n hdiam center ep neg L m x ⟨s,mid,h,tails⟩).1 ↔
+      (t < ∑ b ∈ F s.val mid h.val, twoPoint G p tails.1.1 b) ∨
+      (t < ∑ b ∈ F s.val mid h.val, twoPoint G p tails.2.1 b) := by
+  rcases s with ⟨s,hs⟩
+  rcases mid with ⟨z,mid⟩
+  rcases h with ⟨h,hh⟩
+  obtain ⟨_,hhz⟩ := (mem_segmentAnchors G hloc γ n hdiam ep z h).mp hh
+  subst z
+  rfl
+
+ 
+
+theorem rooted_label_potential_failure_bound [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (htr : RootTransport root weight (vertexAut G))
+    (hcov : ∀ x, ∃ r, ∃ g ∈ vertexAut G, g (root r) = x)
+    (hw : ∀ r, 0 < weight r) (hw1 : ∑ r, weight r = 1)
+    (c : ℝ) (hc : 0 < c) (hmin : ∀ r, 1 ≤ c*weight r)
+    (P A : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re)
+    (p : unitInterval) (hA : HasKernel G p A)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
+    (center ep neg : I) (L m : ℕ) (x : V) (hx : x ∈ vertexOrbit G (γ center))
+    (F : (s : I → V) → RawFrom (openSimpleGraph G Set.univ) L (s ep) → (I → V) → Finset V)
+    (N : ℝ) (hN : ∀ s mid h, ((F s mid h).card : ℝ) ≤ N) (t : ℝ) (ht : 0 < t) :
+    (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
+      if labelPotentialFailure G p γ ep neg L m F t ξ.1 then
+        escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1 else 0) ≤
+      2*(N*Real.sqrt c*‖P‖^m)/t := by
+  apply rooted_label_probability_of_tail_bound G hloc a root weight (vertexAut G) (Set.Subset.refl _)
+    htr hcov hw hw1 c hc hmin P hpos hsym hP hdiag γ n hdiam center ep neg L m x hx
+  intro s mid h
+  simp only [labelFromCode_potential_failure]
+  have hh := two_raw_tail_potential_failure G hloc a root weight htr hcov (fun r => (hw r).le) hw1
+    c hc.le hmin P A hpos hP p hA m (s.val neg) (h.val neg) (F s.val mid h.val) t ht
+  apply hh.trans
+  gcongr
+  exact hN s.val mid h.val
+
+end FullPotentialTests
+end Percolation
+
+ 
+namespace Percolation
+universe u v w
+variable {V : Type u} {E : Type v} {I : Type w}
+section CombinedSelectionEvents
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Fintype I]
+
+ 
+
+theorem rooted_label_collision_potential_failure_bound [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (htr : RootTransport root weight (vertexAut G))
+    (hcov : ∀ x, ∃ r, ∃ g ∈ vertexAut G, g (root r) = x)
+    (hw : ∀ r, 0 < weight r) (hw1 : ∑ r, weight r = 1)
+    (c : ℝ) (hc : 0 < c) (hmin : ∀ r, 1 ≤ c*weight r)
+    (P A : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re)
+    (p : unitInterval) (hA : HasKernel G p A)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
+    (center ep neg : I) (L m : ℕ) (x : V) (hx : x ∈ vertexOrbit G (γ center))
+    (hρ : ‖P‖ < 1) (J : ℕ) (posIdx negIdx suffixIdx : Fin (J+1) → I)
+    (hsep : ∀ s ∈ segmentOrbit G γ,
+      (∀ i, J ≤ (openSimpleGraph G Set.univ).dist (s neg) (s (posIdx i))) ∧
+      (∀ i, J ≤ (openSimpleGraph G Set.univ).dist (s ep) (s (negIdx i))) ∧
+      2*J ≤ (openSimpleGraph G Set.univ).dist (s neg) (s ep))
+    (F : (s : I → V) → RawFrom (openSimpleGraph G Set.univ) L (s ep) → (I → V) → Finset V)
+    (N : ℝ) (hN : ∀ s mid h, ((F s mid h).card : ℝ) ≤ N) (t : ℝ) (ht : 0 < t) :
+    (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
+      if (labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.1 ∨
+        labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.1.reverse) ∨
+        labelPotentialFailure G p γ ep neg L m F t ξ.1 then
+        escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1 else 0) ≤
+      4*c*(earlyCollisionCost ‖P‖ J + suffixCollisionCost ‖P‖ c J*‖P‖^L) +
+        2*(N*Real.sqrt c*‖P‖^m)/t := by
+  have hK (x y : V) : 0 ≤ walkKernel G hloc a x y := by
+    simpa only [hP,Complex.ofReal_re] using (hpos x y).2
+  have hb := rooted_both_collision_probability_bound G hloc a root weight (vertexAut G) (Set.Subset.refl _)
+    htr hw hw1 hcov c hc hmin γ n hdiam center ep neg P hpos hsym hP hdiag hρ m L J x hx posIdx negIdx suffixIdx hsep
+  have hp := rooted_label_potential_failure_bound G hloc a root weight htr hcov hw hw1 c hc hmin
+    P A hpos hsym hP hdiag p hA γ n hdiam center ep neg L m x hx F N hN t ht
+  let := finite_rootedEscapeLabels G hloc γ n hdiam center ep neg L m x
+  let := Fintype.ofFinite {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x}
+  have hu := finite_weighted_event_or
+    (fun ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x} =>
+      escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1)
+    (fun ξ => escapeLabelProbability_nonneg G hloc a hK γ n hdiam center ep neg L m ξ.1)
+    (fun ξ => labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.1 ∨
+      labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.1.reverse)
+    (fun ξ => labelPotentialFailure G p γ ep neg L m F t ξ.1)
+  simp only [tsum_fintype] at hb hp ⊢
+  exact hu.trans (add_le_add hb hp)
+
+end CombinedSelectionEvents
+end Percolation
+
+ 
+namespace Percolation
+universe u v w
+variable {V : Type u} {E : Type v} {I : Type w}
+section QuantitativeSelectionMass
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Fintype I]
+
+theorem finite_weighted_success_lower {α : Type*} [Fintype α]
+    (μ : α → ℝ) (hμ : ∑ a, μ a = 1) (B : α → Prop) [DecidablePred B] (ε : ℝ)
+    (hB : (∑ a, if B a then μ a else 0) ≤ ε) :
+    1-ε ≤ ∑ a, if ¬B a then μ a else 0 := by
+  have he : (∑ a, if ¬B a then μ a else 0) + (∑ a, if B a then μ a else 0) = 1 := by
+    rw [←Finset.sum_add_distrib,←hμ]
+    apply Finset.sum_congr rfl
+    intro a _
+    by_cases h : B a <;> simp only [h,not_true_eq_false,not_false_eq_true,ite_true,ite_false,zero_add,add_zero]
+  linarith
+
+ 
+
+theorem rooted_label_collision_potential_success_lower [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (htr : RootTransport root weight (vertexAut G))
+    (hcov : ∀ x, ∃ r, ∃ g ∈ vertexAut G, g (root r) = x)
+    (hw : ∀ r, 0 < weight r) (hw1 : ∑ r, weight r = 1)
+    (c : ℝ) (hc : 0 < c) (hmin : ∀ r, 1 ≤ c*weight r)
+    (P A : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re)
+    (p : unitInterval) (hA : HasKernel G p A)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
+    (center ep neg : I) (L m : ℕ) (x : V) (hx : x ∈ vertexOrbit G (γ center))
+    (hρ : ‖P‖ < 1) (J : ℕ) (posIdx negIdx suffixIdx : Fin (J+1) → I)
+    (hsep : ∀ s ∈ segmentOrbit G γ,
+      (∀ i, J ≤ (openSimpleGraph G Set.univ).dist (s neg) (s (posIdx i))) ∧
+      (∀ i, J ≤ (openSimpleGraph G Set.univ).dist (s ep) (s (negIdx i))) ∧
+      2*J ≤ (openSimpleGraph G Set.univ).dist (s neg) (s ep))
+    (F : (s : I → V) → RawFrom (openSimpleGraph G Set.univ) L (s ep) → (I → V) → Finset V)
+    (N : ℝ) (hN : ∀ s mid h, ((F s mid h).card : ℝ) ≤ N) (t : ℝ) (ht : 0 < t) :
+    1 - (4*c*(earlyCollisionCost ‖P‖ J + suffixCollisionCost ‖P‖ c J*‖P‖^L) +
+      2*(N*Real.sqrt c*‖P‖^m)/t) ≤
+    ∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
+      if ¬((labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.1 ∨
+        labelFirstEscapeCollision G γ ep neg J L m posIdx negIdx suffixIdx ξ.1.reverse) ∨
+        labelPotentialFailure G p γ ep neg L m F t ξ.1) then
+        escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1 else 0 := by
+  have hb := rooted_label_collision_potential_failure_bound G hloc a root weight htr hcov hw hw1 c hc hmin
+    P A hpos hsym hP hdiag p hA γ n hdiam center ep neg L m x hx hρ J posIdx negIdx suffixIdx hsep F N hN t ht
+  have hs := (rooted_label_probability_hasSum G hloc a root weight (vertexAut G) (Set.Subset.refl _)
+    htr hcov hw hw1 c hc hmin P hpos hsym hP hdiag γ n hdiam center ep neg L m x hx).tsum_eq
+  let := finite_rootedEscapeLabels G hloc γ n hdiam center ep neg L m x
+  let := Fintype.ofFinite {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x}
+  simp only [tsum_fintype] at hs hb ⊢
+  exact finite_weighted_success_lower _ hs _ _ hb
+
+end QuantitativeSelectionMass
+end Percolation
+
+ 
+namespace Percolation
+section JointSelectionChoiceOrder
+
+ 
+
+theorem exists_joint_collision_potential_cutoffs (ρ c d τ : ℝ)
+    (hρ : 0 < ρ) (hρ1 : ρ < 1) (hc : 0 < c) (hd : 0 ≤ d) (hτ : 0 < τ) :
+    ∃ J : ℕ, 0 < J ∧ ∃ L₀ : ℕ, ∃ C₁ : ℝ, 0 < C₁ ∧ 1 < (-Real.log ρ)*C₁ ∧
+      ∃ k₀ : ℕ, J+J+L₀ ≤ k₀ ∧ ∀ k : ℕ, k₀ ≤ k → ∀ R : ℝ, 1 ≤ R →
+        4*c*(earlyCollisionCost ρ J + suffixCollisionCost ρ c J*ρ^(k-(J+J))) +
+          2*((d+1)*((k:ℝ)+1)*Real.sqrt c*ρ^⌈C₁*Real.log (((k:ℝ)+1)*R)⌉₊)/(τ/R) ≤ 3/8 := by
+  obtain ⟨J,hJ,L₀,hL₀⟩ := exists_both_escape_uniform_cutoffs ρ c (1/8) hρ.le hρ1 hc (by norm_num)
+  let A : ℝ := 2*(d+1)*Real.sqrt c/τ
+  have hA : 0 ≤ A := div_nonneg (mul_nonneg (mul_nonneg (by norm_num) (by linarith)) (Real.sqrt_nonneg _)) hτ.le
+  obtain ⟨C₁,hC₁,hlam,k₀,hk₀,hk⟩ := exists_uniform_logarithmic_tail_cutoff ρ A hρ hρ1 hA (J+J+L₀)
+  refine ⟨J,hJ,L₀,C₁,hC₁,hlam,k₀,hk₀,fun k hk' R hR => ?_⟩
+  have hL : L₀ ≤ k-(J+J) := by omega
+  have hb := hL₀ (k-(J+J)) hL
+  have hp := hk k hk' R hR
+  have hRp : 0 < R := lt_of_lt_of_le (by norm_num) hR
+  have he : 2*((d+1)*((k:ℝ)+1)*Real.sqrt c*ρ^⌈C₁*Real.log (((k:ℝ)+1)*R)⌉₊)/(τ/R) =
+      A*((k:ℝ)+1)*R*ρ^⌈C₁*Real.log (((k:ℝ)+1)*R)⌉₊ := by
+    dsimp only [A]
+    field_simp [hτ.ne',hRp.ne']
+  rw [he]
+  linarith
+
+end JointSelectionChoiceOrder
+end Percolation
+
+ 
+namespace Percolation
+universe u v w
+variable {V : Type u} {E : Type v} {I : Type w}
+section SpineMarginal
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Fintype I]
+
+theorem labelFromCode_anchor_event (G : BondGraph V E) (hloc : LocallyFinite G)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
+    (center ep neg : I) (L m : ℕ) (x : V) (B : (I → V) → (I → V) → Prop)
+    (s : segmentAnchors G hloc γ n hdiam center x)
+    (mid : RawFrom (openSimpleGraph G Set.univ) L (s.val ep))
+    (h : segmentAnchors G hloc γ n hdiam ep mid.1)
+    (tails : RawFrom (openSimpleGraph G Set.univ) m (s.val neg) ×
+      RawFrom (openSimpleGraph G Set.univ) m (h.val neg)) :
+    B (labelFromCode G hloc γ n hdiam center ep neg L m x ⟨s,mid,h,tails⟩).1.first
+      (labelFromCode G hloc γ n hdiam center ep neg L m x ⟨s,mid,h,tails⟩).1.last ↔ B s.val h.val := by
+  rcases s with ⟨s,hs⟩
+  rcases mid with ⟨z,mid⟩
+  rcases h with ⟨h,hh⟩
+  obtain ⟨_,hhz⟩ := (mem_segmentAnchors G hloc γ n hdiam ep z h).mp hh
+  subst z
+  rfl
+
+ 
+theorem rooted_label_anchor_event_marginal (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
+    (center ep neg : I) (L m : ℕ) (x : V) (B : (I → V) → (I → V) → Prop)
+    [∀ s h, Decidable (B s h)] :
+    (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
+      if B ξ.1.first ξ.1.last then ξ.1.mass (walkKernel G hloc a) else 0) =
+    ∑' s : segmentAnchors G hloc γ n hdiam center x,
+      ∑' mid : RawFrom (openSimpleGraph G Set.univ) L (s.val ep),
+        ∑' h : segmentAnchors G hloc γ n hdiam ep mid.1,
+          if B s.val h.val then mid.2.mass (walkKernel G hloc a) else 0 := by
+  let (z : V) : Finite ((openSimpleGraph G Set.univ).neighborSet z) := finite_openSimpleGraph_neighbors G hloc z
+  let (k : ℕ) (z : V) : Fintype (RawFrom (openSimpleGraph G Set.univ) k z) := Fintype.ofFinite _
+  let := finite_rootedEscapeLabels G hloc γ n hdiam center ep neg L m x
+  let R := rootedLabelEquiv G hloc γ n hdiam center ep neg L m x
+  rw [←R.symm.tsum_eq]
+  simp only [R,rootedLabelEquiv,Equiv.coe_fn_symm_mk,labelFromCode_mass]
+  unfold RootedLabelCode at *
+  simp only [tsum_fintype,Fintype.sum_sigma,labelFromCode_anchor_event]
+  apply Finset.sum_congr rfl
+  intro s _
+  apply Finset.sum_congr rfl
+  intro mid _
+  apply Finset.sum_congr rfl
+  intro h _
+  by_cases hb : B s.val h.val
+  · simp only [hb,ite_true,Fintype.sum_prod_type,mul_assoc,←Finset.mul_sum,←Finset.sum_mul]
+    have hl := rawFrom_probability_sum G hloc a m (s.val neg)
+    have hr := rawFrom_probability_sum G hloc a m (h.val neg)
+    simp only [tsum_fintype] at hl hr
+    rw [hl,hr]
+    ring
+  · simp only [hb,ite_false,Finset.sum_const_zero]
+
+end SpineMarginal
+end Percolation
+
+ 
+namespace Percolation
+universe u v w
+variable {V : Type u} {E : Type v} {I : Type w}
+section EndpointMarginal
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Fintype I]
+
+ 
+theorem rooted_label_endpoint_marginal (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n)
+    (center ep neg : I) (L m : ℕ) (x : V) (S : Finset V) :
+    (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
+      if ξ.1.last center ∈ S then escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1 else 0) =
+    (∑' s : segmentAnchors G hloc γ n hdiam center x,
+      ∑' mid : RawFrom (openSimpleGraph G Set.univ) L (s.val ep),
+        ∑' h : segmentAnchors G hloc γ n hdiam ep mid.1,
+          if h.val center ∈ S then mid.2.mass (walkKernel G hloc a)/
+            (segmentAnchors G hloc γ n hdiam ep (γ ep)).card else 0) /
+      (((segmentAnchors G hloc γ n hdiam center (γ center)).card : ℝ)*
+        segmentReturnMass G hloc a γ ep L (γ ep)) := by
+  simp only [escapeLabelProbability,tsum_indicator_div,tsum_div_const]
+  rw [rooted_label_anchor_event_marginal G hloc a γ n hdiam center ep neg L m x (fun _ h => h center ∈ S)]
+  ring
+
+end EndpointMarginal
+end Percolation
+
+ 
+namespace Percolation
+universe u v w
+variable {V : Type u} {E : Type v} {I : Type w}
+section ActualEndpointSeparation
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Fintype I] {ι : Type*} [Fintype ι]
+
+ 
+
+theorem rooted_label_endpoint_hit_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G)
+    (htr : RootTransport root weight Γ) (hw : ∀ r, 0 < weight r) (hw1 : ∑ r, weight r = 1)
+    (hcov : ∀ x, ∃ r, ∃ g ∈ Γ, g (root r) = x)
+    (c : ℝ) (hc : 0 < c) (hmin : ∀ r, 1 ≤ c*weight r)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (center ep neg : I)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re)
+    (m L : ℕ) (x : V) (hx : x ∈ vertexOrbit G (γ center))
+    (S : Finset V) :
+    (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
+      if ξ.1.last center ∈ S then escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1 else 0) ≤
+      2*c*(S.card:ℝ)*Real.sqrt c*‖P‖^L := by
+  rw [rooted_label_endpoint_marginal G hloc a γ n hdiam center ep neg L m x S]
+  have hN : 0 < ((segmentAnchors G hloc γ n hdiam center (γ center)).card : ℝ) := by
+    exact_mod_cast Finset.card_pos.mpr ((segmentAnchors_nonempty_iff G hloc γ n hdiam center (γ center)).mpr
+      ⟨Equiv.refl V,Equiv.refl E,isAut_refl G,rfl⟩)
+  have hq := segmentReturnMass_uniform_lower G hloc a root weight Γ hΓ htr hcov hw hw1 c hc hmin
+    P hpos hsym hP hdiag γ ep L
+  have hqp : 0 < segmentReturnMass G hloc a γ ep L (γ ep) :=
+    (one_div_pos.mpr (mul_pos (by norm_num) hc)).trans_le hq
+  apply (div_le_iff₀ (mul_pos hN hqp)).mpr
+  have hb (s : segmentAnchors G hloc γ n hdiam center x) :
+      (∑' mid : RawFrom (openSimpleGraph G Set.univ) L (s.val ep),
+        ∑' h : segmentAnchors G hloc γ n hdiam ep mid.1,
+          if h.val center ∈ S then mid.2.mass (walkKernel G hloc a)/
+            (segmentAnchors G hloc γ n hdiam ep (γ ep)).card else 0) ≤
+      (S.card:ℝ)*Real.sqrt c*‖P‖^L := by
+    have hh := raw_anchor_finite_collision_bound G hloc a root weight Γ hΓ htr (fun r => (hw r).le)
+      hw1 hcov c hc.le hmin γ n hdiam ep (fun _ : PUnit.{1} => center) (fun z : S => z.val)
+      P hpos hP L (s.val ep)
+    simpa [Subtype.exists] using hh
+  have hu := Finset.sum_le_sum (s := Finset.univ) (fun s _ => hb s)
+  simp only [tsum_fintype,Finset.sum_const,Finset.card_univ,Fintype.card_coe,nsmul_eq_mul] at hu ⊢
+  rw [segmentAnchors_card_on_orbit G hloc γ n hdiam center x hx] at hu
+  apply hu.trans
+  have he : 1 ≤ 2*c*segmentReturnMass G hloc a γ ep L (γ ep) := by
+    have hh := (div_le_iff₀ (by positivity : 0 < 2*c)).mp hq
+    nlinarith
+  have hp : 0 ≤ ((segmentAnchors G hloc γ n hdiam center (γ center)).card : ℝ)*
+      (S.card:ℝ)*Real.sqrt c*‖P‖^L := by positivity
+  nlinarith [mul_le_mul_of_nonneg_left he hp]
+
+end ActualEndpointSeparation
+end Percolation
+
+ 
+namespace Percolation
+universe u v w
+variable {V : Type u} {E : Type v} {I : Type w}
+section EndpointBallBound
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Fintype I] {ι : Type*} [Fintype ι]
+
+ 
+theorem rooted_label_endpoint_ball_bound (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G)
+    (htr : RootTransport root weight Γ) (hw : ∀ r, 0 < weight r) (hw1 : ∑ r, weight r = 1)
+    (hcov : ∀ x, ∃ r, ∃ g ∈ Γ, g (root r) = x)
+    (c : ℝ) (hc : 0 < c) (hmin : ∀ r, 1 ≤ c*weight r)
+    (γ : I → V) (n : ℕ) (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) n) (center ep neg : I)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re)
+    (m L : ℕ) (x : V) (hx : x ∈ vertexOrbit G (γ center))
+    (d : ℕ) (hd : ∀ z, degree G z ≤ d) (r : ℕ) :
+    (∑' ξ : {ξ : EscapeLabel G γ ep neg L m // ξ.first center = x},
+      if ξ.1.last center ∈ reachBall G x r then escapeLabelProbability G hloc a γ n hdiam center ep neg L m ξ.1 else 0) ≤
+      2*c*(((d+1)^r:ℕ):ℝ)*Real.sqrt c*‖P‖^L := by
+  have hh := rooted_label_endpoint_hit_bound G hloc a root weight Γ hΓ htr hw hw1 hcov c hc hmin
+    γ n hdiam center ep neg P hpos hsym hP hdiag m L x hx (finite_reachBall hloc x r).toFinset
+  have hb : ((finite_reachBall hloc x r).toFinset.card : ℝ) ≤ (((d+1)^r:ℕ):ℝ) := by
+    exact_mod_cast (by
+      simpa only [←Set.ncard_eq_toFinset_card] using reachBall_ncard_le_pow G hloc d hd x r)
+  calc
+    _ ≤ 2*c*((finite_reachBall hloc x r).toFinset.card:ℝ)*Real.sqrt c*‖P‖^L := by
+      simpa only [Set.Finite.mem_toFinset] using hh
+    _ ≤ _ := by gcongr
+
+end EndpointBallBound
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Exploration/FractionalMoments.lean b/lean/OAI/Probability/UnimodularPercolation/Exploration/FractionalMoments.lean
index cea62e429347915fef391347e72116bed67ffbab..82fc755eb4f74c5bad31dedce83803424ec55ece 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Exploration/FractionalMoments.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Exploration/FractionalMoments.lean
@@ -1,1219 +1,1219 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Criticality.Susceptibility
-
-namespace OAI
-
-/-! Fractional Moments for bond percolation. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section GhostPowerTransfer
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem sqrt_log_power_bound (h : unitInterval) (hh : 0 < (h:ℝ))
-    (a : ℝ) (ha : 1/2 < a) :
-    Real.sqrt (Real.log (1/(h:ℝ))/(h:ℝ))*(h:ℝ)^a ≤ Real.sqrt (1/(2*a-1)) := by
-  let ε := 2*a-1
-  have hε : 0 < ε := by dsimp [ε]; linarith
-  have hpow := Real.rpow_pos_of_pos hh ε
-  have hl := Real.log_le_rpow_div (show 0 ≤ 1/(h:ℝ) by positivity) hε
-  rw [Real.div_rpow (by norm_num) hh.le,Real.one_rpow] at hl
-  have hpowid : ((h:ℝ)^a)^2 = (h:ℝ)^ε*(h:ℝ) := by
-    rw [←Real.rpow_natCast,←Real.rpow_mul hh.le]
-    have he : a*(2:ℝ) = ε+1 := by dsimp [ε]; ring
-    norm_num only [Nat.cast_ofNat]
-    rw [he,Real.rpow_add hh,Real.rpow_one]
-  have hbase : (Real.log (1/(h:ℝ))/(h:ℝ))*((h:ℝ)^a)^2 ≤ 1/ε := by
-    rw [hpowid]
-    calc
-      _ = Real.log (1/(h:ℝ))*(h:ℝ)^ε := by field_simp
-      _ ≤ ((1/(h:ℝ)^ε)/ε)*(h:ℝ)^ε := mul_le_mul_of_nonneg_right hl hpow.le
-      _ = 1/ε := by field_simp
-  have hlog : 0 ≤ Real.log (1/(h:ℝ))/(h:ℝ) := div_nonneg
-    (Real.log_nonneg ((one_le_div hh).mpr h.2.2)) hh.le
-  have hsq := Real.sq_sqrt hlog
-  have htarget := Real.sq_sqrt (one_div_pos.mpr hε).le
-  have hn := Real.sqrt_nonneg (1/ε)
-  change _ ≤ Real.sqrt (1/ε)
-  nlinarith [sq_nonneg (Real.sqrt (1/ε)-Real.sqrt (Real.log (1/(h:ℝ))/(h:ℝ))*(h:ℝ)^a)]
-
- 
-
-theorem markedConnection_power_transfer (G : BondGraph V E) (Q : Finset E) (x : V)
-    (d : ℕ) (hd : 0 < d) (hdeg : ∀ v, (Q.filter (fun e => v ∈ G e)).card ≤ d)
-    (p q h : unitInterval) (hq0 : 0 < (q:ℝ)) (hq1 : (q:ℝ) < 1)
-    (hp : (q:ℝ)/2 ≤ (p:ℝ)) (hh : 0 < (h:ℝ))
-    (a : ℝ) (ha : 1/2 < a) (he : (q:ℝ)-(p:ℝ) = (h:ℝ)^a) :
-    markedConnectionPolynomial G Q x h q ≤
-      Real.exp ((Real.sqrt ((d:ℝ)/2)/(((q:ℝ)/2)*(1-(q:ℝ)))) *
-        Real.sqrt (1/(2*a-1))) * markedConnectionPolynomial G Q x h p := by
-  have hp0 : 0 < (p:ℝ) := lt_of_lt_of_le (half_pos hq0) hp
-  have hpq : p ≤ q := by change (p:ℝ) ≤ (q:ℝ); linarith [Real.rpow_pos_of_pos hh a]
-  have H := markedConnection_transfer G Q x d hd hdeg p q h hp0 hpq hq1 hh
-  apply H.trans
-  apply mul_le_mul_of_nonneg_right
-  · apply Real.exp_le_exp.mpr
-    rw [he]
-    have hr : Real.sqrt ((d:ℝ)/2)/((p:ℝ)*(1-(q:ℝ))) ≤
-        Real.sqrt ((d:ℝ)/2)/(((q:ℝ)/2)*(1-(q:ℝ))) :=
-      div_le_div_of_nonneg_left (Real.sqrt_nonneg _) (mul_pos (half_pos hq0) (sub_pos.mpr hq1))
-        (mul_le_mul_of_nonneg_right hp (sub_pos.mpr hq1).le)
-    calc
-      _ = (Real.sqrt ((d:ℝ)/2)/((p:ℝ)*(1-(q:ℝ)))) *
-        (Real.sqrt (Real.log (1/(h:ℝ))/(h:ℝ))*(h:ℝ)^a) := by ring
-      _ ≤ _ := mul_le_mul hr (sqrt_log_power_bound h hh a ha)
-        (mul_nonneg (Real.sqrt_nonneg _) (Real.rpow_nonneg h.2.1 _)) (by positivity)
-  · rw [markedConnectionPolynomial_eq_tree]
-    exact ProbeTree.successMass_nonneg p h _
-
-end GhostPowerTransfer
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section DyadicGhostParameters
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def dyadicGhost (j : ℕ) : unitInterval :=
-  ⟨(1/2:ℝ)^j,⟨pow_nonneg (by norm_num) _,pow_le_one₀ (by norm_num) (by norm_num)⟩⟩
-
-theorem dyadicGhost_pos (j : ℕ) : 0 < (dyadicGhost j:ℝ) := pow_pos (by norm_num) _
-
-theorem dyadicGhost_volume (j : ℕ) : (dyadicGhost j:ℝ)*(2^j:ℕ) = 1 := by
-  simp only [dyadicGhost,Nat.cast_pow,Nat.cast_ofNat,←mul_pow]
-  norm_num
-
-theorem tendsto_dyadicGhost : Tendsto (fun j => (dyadicGhost j:ℝ)) atTop (nhds 0) :=
-  tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num) (by norm_num)
-
-theorem dyadicGhost_rpow (j : ℕ) (t : ℝ) :
-    (dyadicGhost j:ℝ)^t = (2:ℝ)^(-t*(j:ℝ)) := by
-  change ((1/2:ℝ)^j)^t = _
-  rw [←Real.rpow_natCast,←Real.rpow_mul (by norm_num : (0:ℝ) ≤ 1/2)]
-  rw [show (1/2:ℝ) = (2:ℝ)^(-1:ℝ) by norm_num,
-    ←Real.rpow_mul (by norm_num : (0:ℝ) ≤ 2)]
-  congr 1
-  ring
-
- 
-
-theorem eventually_dyadic_backstep (q : unitInterval) (hq : 0 < (q:ℝ))
-    (a : ℝ) (ha : 0 < a) (P : unitInterval → Prop) (hP : ∀ᶠ p in nhds q, P p) :
-    ∀ᶠ j : ℕ in atTop, ∃ p : unitInterval,
-      (q:ℝ)/2 ≤ (p:ℝ) ∧ p < q ∧ (q:ℝ)-(p:ℝ) = (dyadicGhost j:ℝ)^a ∧ P p := by
-  obtain ⟨ε,hε,hball⟩ := Metric.eventually_nhds_iff.mp hP
-  have ht : Tendsto (fun j => (dyadicGhost j:ℝ)^a) atTop (nhds 0) := by
-    simpa only [Real.zero_rpow ha.ne', Function.comp_apply] using!
-      (Real.continuousAt_rpow_const 0 a (Or.inr ha.le)).tendsto.comp tendsto_dyadicGhost
-  filter_upwards [ht.eventually (eventually_lt_nhds (half_pos hq)),ht.eventually (eventually_lt_nhds hε)] with j hj hεj
-  have hz := Real.rpow_pos_of_pos (dyadicGhost_pos j) a
-  let p : unitInterval := ⟨(q:ℝ)-(dyadicGhost j:ℝ)^a,⟨by linarith,by linarith [q.2.2]⟩⟩
-  refine ⟨p,by dsimp [p]; linarith,by change (q:ℝ)-(dyadicGhost j:ℝ)^a < (q:ℝ); linarith,
-    by dsimp [p]; ring,?_⟩
-  apply hball
-  change |((q:ℝ)-(dyadicGhost j:ℝ)^a)-(q:ℝ)| < ε
-  rw [show (q:ℝ)-(dyadicGhost j:ℝ)^a-(q:ℝ) = -(dyadicGhost j:ℝ)^a by ring,abs_neg,abs_of_pos hz]
-  exact hεj
-
-end DyadicGhostParameters
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section CriticalDyadicTail
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem critical_dyadic_bound_of_susceptibility [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (hq0 : 0 < (pc G:ℝ)) (hq1 : (pc G:ℝ) < 1)
-    (C b a : ℝ) (hC : 0 < C) (ha : 1/2 < a)
-    (hχ : ∀ᶠ p in nhds (pc G), p < pc G → ∀ x,
-      susceptibility G p x ≤ C*((pc G:ℝ)-(p:ℝ))^(-b)) :
-    ∃ K : ℝ, 0 < K ∧ ∀ᶠ j : ℕ in atTop, ∀ x,
-      clusterWindowProbability G (pc G) x (2^j) (2^(j+1)) ≤
-        K*(dyadicGhost j:ℝ)^(1-a*b) := by
-  obtain ⟨d,hd,hdeg⟩ := exists_degree_bound G hqt
-  let B := Real.exp ((Real.sqrt ((d:ℝ)/2)/(((pc G:ℝ)/2)*(1-(pc G:ℝ)))) * Real.sqrt (1/(2*a-1)))
-  have hB : 0 < B := Real.exp_pos _
-  refine ⟨2*B*C,by positivity,?_⟩
-  filter_upwards [eventually_dyadic_backstep (pc G) hq0 a (by linarith) _ hχ] with j hj
-  obtain ⟨p,hp,hpc,he,hχp⟩ := hj
-  intro x
-  let Q := incidentSupport G hloc (finite_reachBall hloc x (2^(j+1))).toFinset
-  have hwin := clusterWindow_le_markedConnection G hloc x (2^j) (2^(j+1)) (pc G) (dyadicGhost j)
-    (by rw [dyadicGhost_volume])
-  have htra := markedConnection_power_transfer G Q x d hd
-    (fun v => (incident_filter_card_le_degree G hloc Q v).trans (hdeg v)) p (pc G) (dyadicGhost j)
-    hq0 hq1 hp (dyadicGhost_pos j) a ha he
-  have hsub := markedConnection_le_susceptibility G hloc hqt Q x (dyadicGhost j) p hpc
-  have hid : (dyadicGhost j:ℝ)*((dyadicGhost j:ℝ)^a)^(-b) = (dyadicGhost j:ℝ)^(1-a*b) := by
-    rw [←Real.rpow_mul (dyadicGhost j).2.1]
-    calc
-      _ = (dyadicGhost j:ℝ)^1*(dyadicGhost j:ℝ)^(a*(-b)) := by rw [Real.rpow_one]
-      _ = (dyadicGhost j:ℝ)^(1+a*(-b)) := (Real.rpow_add (dyadicGhost_pos j) _ _).symm
-      _ = _ := by congr 1; ring
-  calc
-    _ ≤ 2*markedConnectionPolynomial G Q x (dyadicGhost j) (pc G) := hwin
-    _ ≤ 2*(B*markedConnectionPolynomial G Q x (dyadicGhost j) p) :=
-      mul_le_mul_of_nonneg_left htra (by norm_num)
-    _ ≤ 2*(B*((dyadicGhost j:ℝ)*susceptibility G p x)) :=
-      mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_left hsub hB.le) (by norm_num)
-    _ ≤ 2*(B*((dyadicGhost j:ℝ)*(C*((pc G:ℝ)-(p:ℝ))^(-b)))) :=
-      mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_left
-        (mul_le_mul_of_nonneg_left (hχp hpc x) (dyadicGhost j).2.1) hB.le) (by norm_num)
-    _ = 2*B*C*((dyadicGhost j:ℝ)*((dyadicGhost j:ℝ)^a)^(-b)) := by rw [he]; ring
-    _ = _ := by rw [hid]
-
- 
-
-theorem critical_power_integrable_of_dyadic_bound (G : BondGraph V E) (hloc : LocallyFinite G)
-    (p : unitInterval) (α θ K : ℝ) (hα : 0 < α) (hθ : α < θ)
-    (hbound : ∀ᶠ j : ℕ in atTop, ∀ x,
-      clusterWindowProbability G p x (2^j) (2^(j+1)) ≤ K*(dyadicGhost j:ℝ)^θ) :
-    ∀ x, Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^α) (law p) := by
-  intro x
-  apply integrable_cluster_power_of_dyadic G hloc p x α hα
-  have hs : Summable (fun j : ℕ => K*((2:ℝ)^(α-θ))^j) := by
-    apply Summable.mul_left
-    apply summable_geometric_of_norm_lt_one
-    rw [Real.norm_of_nonneg (Real.rpow_nonneg (by norm_num) _)]
-    exact Real.rpow_lt_one_of_one_lt_of_neg (by norm_num) (sub_neg.mpr hθ)
-  apply hs.of_norm_bounded_eventually
-  rw [Nat.cofinite_eq_atTop]
-  filter_upwards [hbound] with j hj
-  rw [Real.norm_of_nonneg (mul_nonneg (Real.rpow_nonneg (by norm_num) _)
-    (clusterWindowProbability_nonneg G p x _ _))]
-  calc
-    _ ≤ (2:ℝ)^(α*(j:ℝ))*(K*(dyadicGhost j:ℝ)^θ) :=
-      mul_le_mul_of_nonneg_left (hj x) (Real.rpow_nonneg (by norm_num) _)
-    _ = K*((2:ℝ)^(α-θ))^j := by
-      rw [dyadicGhost_rpow,←Real.rpow_natCast,←Real.rpow_mul (by norm_num : (0:ℝ) ≤ 2)]
-      rw [show (α-θ)*(j:ℝ) = α*(j:ℝ)+(-θ*(j:ℝ)) by ring,Real.rpow_add (by norm_num : (0:ℝ)<2)]
-      ring
-
-end CriticalDyadicTail
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section MomentFromGrowth
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem critical_moment_of_susceptibility_growth [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (hq0 : 0 < (pc G:ℝ)) (hq1 : (pc G:ℝ) < 1)
-    (hgrowth : ∀ γ : ℝ, 0 < γ → γ < 1 → ∃ C : ℝ, 0 < C ∧
-      ∀ᶠ p in nhds (pc G), p < pc G → ∀ x,
-        susceptibility G p x ≤ C*((pc G:ℝ)-(p:ℝ))^(-1/(1-γ)))
-    (α : ℝ) (hα : 0 < α) (hα1 : α < 1/2) :
-    ∀ x, Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^α) (law (pc G)) := by
-  let γ := (1/2-α)/2
-  have hγ : 0 < γ := by dsimp [γ]; linarith
-  have hγ1 : γ < 1 := by dsimp [γ]; linarith
-  have hprod : (1/2:ℝ) < (1-γ)*(1-α) := by
-    have : 0 < γ*α := mul_pos hγ hα
-    dsimp [γ] at *
-    nlinarith
-  obtain ⟨a,ha,hap⟩ := exists_between hprod
-  have hden : 0 < 1-γ := sub_pos.mpr hγ1
-  obtain ⟨C,hC,hnear⟩ := hgrowth γ hγ hγ1
-  have he : -(1/(1-γ)) = -1/(1-γ) := by ring
-  obtain ⟨K,hK,hbound⟩ := critical_dyadic_bound_of_susceptibility G hloc hqt hq0 hq1
-    C (1/(1-γ)) a hC ha (by simpa only [he] using hnear)
-  apply critical_power_integrable_of_dyadic_bound G hloc (pc G) α (1-a*(1/(1-γ))) K hα _ hbound
-  have : a*(1/(1-γ)) < 1-α := by
-    rw [mul_one_div]
-    apply (div_lt_iff₀ hden).mpr
-    nlinarith [hap]
-  linarith
-
-end MomentFromGrowth
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section CriticalRootMoments
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem critical_root_power_integrable
-    [Countable V] [Countable E] [Nonempty V] [Nonempty E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hconn : Connected G)
-    (hqt : QuasiTransitive G) (hh : 0 < hV G)
-    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
-    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
-    (hp₀ : 0 < (pc G:ℝ)) (hp₁ : (pc G:ℝ) < 1)
-    (hfin : ∀ i, ∀ᵐ ω ∂law (pc G), (cluster G ω (root i)).Finite)
-    (α : ℝ) (hα : 0 < α) (hα1 : α < 1/2) :
-    ∀ i, Integrable (fun ω => ((cluster G ω (root i)).ncard:ℝ)^α) (law (pc G)) := by
-  by_contra hbad
-  have hdiv : ∃ i, ¬ Integrable (fun ω => ((cluster G ω (root i)).ncard:ℝ)^α) (law (pc G)) :=
-    not_forall.mp hbad
-  obtain ⟨β,hαβ,hβ⟩ := exists_between hα1
-  let ε := (1/2-β)/2
-  have hε : 0 < ε := by dsimp [ε]; linarith
-  have hε1 : ε < 1/2 := by dsimp [ε]; linarith
-  have hsmall : β+(1/2+ε) < 1 := by dsimp [ε]; linarith
-  have hg := susceptibility_growth_of_divergent_moment G hloc hconn hqt hh root weight Γ hΓ htr
-    hcov hw hw1 hp₀ hp₁ hfin α β ε hα hαβ hε hε1 hsmall hdiv
-  have hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i) := by
-    intro x
-    obtain ⟨i,a,ha,hax⟩ := hcov x
-    obtain ⟨b,hab⟩ := hΓ ha
-    exact ⟨i,a,b,hab,hax⟩
-  let c : ℝ := ∑ i, 1/weight i
-  have hc : 0 < c := Finset.sum_pos (fun i _ => one_div_pos.mpr (hw i)) Finset.univ_nonempty
-  have hmin (i : ι) : 1 ≤ c*weight i := by
-    apply (div_le_iff₀ (hw i)).mp
-    exact Finset.single_le_sum (fun j _ => (one_div_pos.mpr (hw j)).le) (Finset.mem_univ i)
-  have hall := critical_moment_of_susceptibility_growth G hloc hqt hp₀ hp₁ (by
-    intro γ hγ hγ1
-    obtain ⟨C,hC,hnear⟩ := hg γ hγ hγ1
-    refine ⟨c*C,mul_pos hc hC,?_⟩
-    filter_upwards [hnear] with p hp
-    intro hpc x
-    calc
-      _ ≤ c*rootSusceptibility G root weight p :=
-        susceptibility_le_rootSusceptibility G root weight (fun i => (hw i).le) hcover c hc.le hmin p x
-      _ ≤ c*(C*((pc G:ℝ)-(p:ℝ))^(-1/(1-γ))) := mul_le_mul_of_nonneg_left (hp hpc) hc.le
-      _ = _ := by ring) α hα hα1
-  obtain ⟨i,hi⟩ := hdiv
-  exact hi (hall (root i))
-
-end CriticalRootMoments
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section UniformFractionalMoments
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem measurable_cluster_power (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) (α : ℝ) :
-    Measurable (fun ω : Set E => ((cluster G ω x).ncard:ℝ)^α) :=
-  (measurable_of_countable (fun n : ℕ => (n:ℝ)^α)).comp (measurable_cluster_ncard G hloc x)
-
-theorem integrable_cluster_power_covariant [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (p : unitInterval)
-    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (x : V) (α : ℝ) :
-    Integrable (fun ω => ((cluster G ω (a x)).ncard:ℝ)^α) (law p) ↔
-      Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^α) (law p) := by
-  have h := integrable_map_measure (μ := law p)
-    (measurable_cluster_power G hloc x α).aestronglyMeasurable
-    (measurable_set_preimage b).aemeasurable
-  rw [law_map_preimage_equiv] at h
-  have he : ((fun ω : Set E => ((cluster G ω x).ncard:ℝ)^α) ∘ (fun ω => b ⁻¹' ω)) =
-      (fun ω : Set E => ((cluster G ω (a x)).ncard:ℝ)^α) := by
-    funext ω
-    dsimp only [Function.comp_apply]
-    rw [cluster_ncard_covariant G hab]
-  rw [he] at h
-  exact h.symm
-
-theorem integral_cluster_power_covariant [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (p : unitInterval)
-    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (x : V) (α : ℝ) :
-    (∫ ω, ((cluster G ω (a x)).ncard:ℝ)^α ∂law p) =
-      ∫ ω, ((cluster G ω x).ncard:ℝ)^α ∂law p := by
-  have h := integral_map (μ := law p) (measurable_set_preimage b).aemeasurable
-    (measurable_cluster_power G hloc x α).aestronglyMeasurable
-  rw [law_map_preimage_equiv] at h
-  simpa only [cluster_ncard_covariant G hab] using! h.symm
-
- 
-theorem uniform_cluster_power_of_root [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (p : unitInterval)
-    {ι : Type*} [Fintype ι] (root : ι → V)
-    (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i)) (α : ℝ)
-    (hi : ∀ i, Integrable (fun ω => ((cluster G ω (root i)).ncard:ℝ)^α) (law p)) :
-    ∃ C : ℝ, 0 < C ∧ ∀ x,
-      Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^α) (law p) ∧
-      (∫ ω, ((cluster G ω x).ncard:ℝ)^α ∂law p) ≤ C := by
-  let C := 1+∑ i, ∫ ω, ((cluster G ω (root i)).ncard:ℝ)^α ∂law p
-  have hn (x : V) : 0 ≤ ∫ ω, ((cluster G ω x).ncard:ℝ)^α ∂law p :=
-    integral_nonneg (fun _ => Real.rpow_nonneg (Nat.cast_nonneg _) _)
-  have hC : 0 < C := by dsimp [C]; linarith [Finset.sum_nonneg (fun i (_ : i ∈ Finset.univ) => hn (root i))]
-  refine ⟨C,hC,?_⟩
-  intro x
-  obtain ⟨i,a,b,hab,rfl⟩ := hcover x
-  refine ⟨(integrable_cluster_power_covariant G hloc p hab (root i) α).mpr (hi i),?_⟩
-  rw [integral_cluster_power_covariant G hloc p hab]
-  have hh := Finset.single_le_sum (fun j (_ : j ∈ Finset.univ) => hn (root j)) (Finset.mem_univ i)
-  dsimp [C]
-  linarith
-
-end UniformFractionalMoments
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section FractionalMomentCoupling
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem cluster_power_mono_integral [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) (p q : unitInterval) (hpq : p ≤ q)
-    (α : ℝ) (hα : 0 ≤ α)
-    (hfin : ∀ᵐ ω ∂law q, (cluster G ω x).Finite)
-    (hi : Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^α) (law q)) :
-    Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^α) (law p) ∧
-      (∫ ω, ((cluster G ω x).ncard:ℝ)^α ∂law p) ≤
-      ∫ ω, ((cluster G ω x).ncard:ℝ)^α ∂law q := by
-  let f (r : unitInterval) (labels : E → unitInterval) := ((cluster G (openEdges labels r) x).ncard:ℝ)^α
-  have hm (r : unitInterval) : Measurable (f r) :=
-    (measurable_cluster_power G hloc x α).comp (measurable_openEdges r)
-  have hmap (r : unitInterval) :
-      Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^α) (law r) ↔ Integrable (f r) labelLaw := by
-    rw [←map_openEdges r]
-    exact integrable_map_measure (measurable_cluster_power G hloc x α).aestronglyMeasurable
-      (measurable_openEdges r).aemeasurable
-  have hf : ∀ᵐ labels ∂(labelLaw : Measure (E → unitInterval)), (cluster G (openEdges labels q) x).Finite := by
-    apply ae_of_ae_map (μ := (labelLaw : Measure (E → unitInterval)))
-      (p := fun ω : Set E => (cluster G ω x).Finite) (measurable_openEdges q).aemeasurable
-    rwa [map_openEdges]
-  have hle : f p ≤ᵐ[labelLaw] f q := by
-    filter_upwards [hf] with labels hlabels
-    apply Real.rpow_le_rpow (Nat.cast_nonneg _) _ hα
-    exact_mod_cast Set.ncard_le_ncard (show cluster G (openEdges labels p) x ⊆ cluster G (openEdges labels q) x from
-      fun y hy => conn_mono G (openEdges_mono labels hpq) hy) hlabels
-  have hiq := (hmap q).mp hi
-  have hip : Integrable (f p) labelLaw := hiq.mono' (hm p).aestronglyMeasurable (by
-    filter_upwards [hle] with labels hl
-    rwa [Real.norm_of_nonneg (Real.rpow_nonneg (Nat.cast_nonneg _) _)] )
-  refine ⟨(hmap p).mpr hip,?_⟩
-  have he (r : unitInterval) : (∫ labels, f r labels ∂labelLaw) =
-      ∫ ω, ((cluster G ω x).ncard:ℝ)^α ∂law r := by
-    rw [←map_openEdges r]
-    exact (integral_map (measurable_openEdges r).aemeasurable
-      (measurable_cluster_power G hloc x α).aestronglyMeasurable).symm
-  rw [←he p,←he q]
-  exact integral_mono_ae hip hiq hle
-
-end FractionalMomentCoupling
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {E : Type v}
-section DecisionCovarianceAlgebra
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def finiteCovariance (Q : Finset E) (p : ℝ) (f g : Finset E → ℝ) : ℝ :=
-  bernoulliPolynomial Q (fun ω => f ω*g ω) p - bernoulliPolynomial Q f p*bernoulliPolynomial Q g p
-
-noncomputable def coordinateAverage (e : E) (p : ℝ) (f : Finset E → ℝ) (ω : Finset E) : ℝ :=
-  (1-p)*f ω+p*f (insert e ω)
-
-noncomputable def coordinateVariation (Q : Finset E) (p : ℝ) (f : Finset E → ℝ) (e : E) : ℝ :=
-  bernoulliPolynomial (Q.erase e) (fun ω => |f (insert e ω)-f ω|) p
-
-theorem bernoulliPolynomial_abs (Q : Finset E) (p : unitInterval) (f : Finset E → ℝ) :
-    |bernoulliPolynomial Q f p| ≤ bernoulliPolynomial Q (fun ω => |f ω|) p := by
-  rw [abs_le]
-  constructor
-  · have h := bernoulliPolynomial_mono Q (fun ω => -|f ω|) f p p.2
-      (fun _ _ => neg_abs_le _)
-    have he : bernoulliPolynomial Q (fun ω => -|f ω|) p =
-        -bernoulliPolynomial Q (fun ω => |f ω|) p := by
-      simp only [bernoulliPolynomial,neg_mul,Finset.sum_neg_distrib]
-    rwa [he] at h
-  · exact bernoulliPolynomial_mono Q f (fun ω => |f ω|) p p.2 (fun _ _ => le_abs_self _)
-
-theorem coordinateVariation_nonneg (Q : Finset E) (p : unitInterval) (f : Finset E → ℝ) (e : E) :
-    0 ≤ coordinateVariation Q p f e :=
-  bernoulliPolynomial_nonneg _ _ p p.2 (fun _ _ => abs_nonneg _)
-
- 
-theorem finiteCovariance_first_coordinate (S : Finset E) (e : E) (he : e ∉ S)
-    (p : ℝ) (f g : Finset E → ℝ) :
-    finiteCovariance (insert e S) p f g =
-      (1-p)*finiteCovariance S p (coordinateAverage e p f) g +
-      p*finiteCovariance S p (coordinateAverage e p f) (fun ω => g (insert e ω)) +
-      p*(1-p)*bernoulliPolynomial S
-        (fun ω => (f (insert e ω)-f ω)*(g (insert e ω)-g ω)) p := by
-  unfold finiteCovariance
-  rw [bernoulliPolynomial_insert S e he,bernoulliPolynomial_insert S e he,
-    bernoulliPolynomial_insert S e he]
-  have hav := bernoulliPolynomial_affine S f (fun ω => f (insert e ω)) (1-p) p p
-  change bernoulliPolynomial S (coordinateAverage e p f) p = _ at hav
-  rw [hav]
-  have h0 : bernoulliPolynomial S (fun ω => coordinateAverage e p f ω*g ω) p =
-      (1-p)*bernoulliPolynomial S (fun ω => f ω*g ω) p+
-      p*bernoulliPolynomial S (fun ω => f (insert e ω)*g ω) p := by
-    simpa only [coordinateAverage,add_mul,mul_assoc] using
-      bernoulliPolynomial_affine S (fun ω => f ω*g ω) (fun ω => f (insert e ω)*g ω) (1-p) p p
-  have h1 : bernoulliPolynomial S (fun ω => coordinateAverage e p f ω*g (insert e ω)) p =
-      (1-p)*bernoulliPolynomial S (fun ω => f ω*g (insert e ω)) p+
-      p*bernoulliPolynomial S (fun ω => f (insert e ω)*g (insert e ω)) p := by
-    simpa only [coordinateAverage,add_mul,mul_assoc] using
-      bernoulliPolynomial_affine S (fun ω => f ω*g (insert e ω)) (fun ω => f (insert e ω)*g (insert e ω)) (1-p) p p
-  rw [h0,h1]
-  have herr : bernoulliPolynomial S (fun ω => (f (insert e ω)-f ω)*(g (insert e ω)-g ω)) p =
-      bernoulliPolynomial S (fun ω => f (insert e ω)*g (insert e ω)) p-
-      bernoulliPolynomial S (fun ω => f (insert e ω)*g ω) p-
-      bernoulliPolynomial S (fun ω => f ω*g (insert e ω)) p+
-      bernoulliPolynomial S (fun ω => f ω*g ω) p := by
-    simp only [bernoulliPolynomial]
-    simp only [←Finset.sum_sub_distrib,←Finset.sum_add_distrib]
-    apply Finset.sum_congr rfl
-    intro ω _
-    ring
-  rw [herr]
-  ring
-
-end DecisionCovarianceAlgebra
-end Percolation
-
- 
-namespace Percolation
-universe v
-variable {E : Type v}
-section CoordinateAveragingBounds
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem coordinateVariation_average_le (S : Finset E) (e j : E) (he : e ∉ S) (hje : j ≠ e)
-    (p : unitInterval) (f : Finset E → ℝ) :
-    coordinateVariation S p (coordinateAverage e p f) j ≤ coordinateVariation (insert e S) p f j := by
-  unfold coordinateVariation
-  rw [Finset.erase_insert_of_ne hje.symm,bernoulliPolynomial_insert _ e (by simp [he])]
-  let f₀ := fun ω : Finset E => |f (insert j ω)-f ω|
-  let f₁ := fun ω : Finset E => |f (insert j (insert e ω))-f (insert e ω)|
-  change _ ≤ (1-(p:ℝ))*bernoulliPolynomial (S.erase j) f₀ p+
-    (p:ℝ)*bernoulliPolynomial (S.erase j) f₁ p
-  rw [←bernoulliPolynomial_affine]
-  apply bernoulliPolynomial_mono _ _ _ p p.2
-  intro ω _
-  have hid : coordinateAverage e p f (insert j ω)-coordinateAverage e p f ω =
-      (1-(p:ℝ))*(f (insert j ω)-f ω)+(p:ℝ)*(f (insert j (insert e ω))-f (insert e ω)) := by
-    simp only [coordinateAverage,Finset.insert_comm e j]
-    ring
-  rw [hid]
-  calc
-    _ ≤ |(1-(p:ℝ))*(f (insert j ω)-f ω)|+|(p:ℝ)*(f (insert j (insert e ω))-f (insert e ω))| := abs_add_le _ _
-    _ = _ := by rw [abs_mul,abs_mul,abs_of_nonneg (sub_nonneg.mpr p.2.2),abs_of_nonneg p.2.1]
-
- 
-
-theorem finiteCovariance_first_bound (S : Finset E) (e : E) (he : e ∉ S)
-    (p : unitInterval) (f g : Finset E → ℝ)
-    (hg : ∀ ω ⊆ insert e S, g ω ∈ Set.Icc (0:ℝ) 1) :
-    |finiteCovariance (insert e S) p f g| ≤
-      (1-(p:ℝ))*|finiteCovariance S p (coordinateAverage e p f) g|+
-      (p:ℝ)*|finiteCovariance S p (coordinateAverage e p f) (fun ω => g (insert e ω))|+
-      (p:ℝ)*(1-(p:ℝ))*coordinateVariation (insert e S) p f e := by
-  have hgood : |bernoulliPolynomial S (fun ω => (f (insert e ω)-f ω)*(g (insert e ω)-g ω)) p| ≤
-      coordinateVariation (insert e S) p f e := by
-    apply (bernoulliPolynomial_abs S p _).trans
-    unfold coordinateVariation
-    rw [Finset.erase_insert he]
-    apply bernoulliPolynomial_mono _ _ _ p p.2
-    intro ω hω
-    rw [abs_mul]
-    have h₀ := hg ω (hω.trans (Finset.subset_insert _ _))
-    have h₁ := hg (insert e ω) (Finset.insert_subset_insert e hω)
-    have hd : |g (insert e ω)-g ω| ≤ 1 := by rw [abs_le]; constructor <;> linarith [h₀.1,h₀.2,h₁.1,h₁.2]
-    simpa only [mul_one] using mul_le_mul_of_nonneg_left hd (abs_nonneg (f (insert e ω)-f ω))
-  rw [finiteCovariance_first_coordinate S e he]
-  calc
-    _ ≤ |(1-(p:ℝ))*finiteCovariance S p (coordinateAverage e p f) g+
-          (p:ℝ)*finiteCovariance S p (coordinateAverage e p f) (fun ω => g (insert e ω))|+
-        |(p:ℝ)*(1-(p:ℝ))*bernoulliPolynomial S (fun ω => (f (insert e ω)-f ω)*(g (insert e ω)-g ω)) p| := abs_add_le _ _
-    _ ≤ |(1-(p:ℝ))*finiteCovariance S p (coordinateAverage e p f) g|+
-          |(p:ℝ)*finiteCovariance S p (coordinateAverage e p f) (fun ω => g (insert e ω))|+
-        |(p:ℝ)*(1-(p:ℝ))*bernoulliPolynomial S (fun ω => (f (insert e ω)-f ω)*(g (insert e ω)-g ω)) p| :=
-      add_le_add (abs_add_le _ _) le_rfl
-    _ ≤ _ := by
-      rw [abs_mul,abs_mul,abs_mul,abs_of_nonneg (sub_nonneg.mpr p.2.2),abs_of_nonneg p.2.1,
-        abs_of_nonneg (mul_nonneg p.2.1 (sub_nonneg.mpr p.2.2))]
-      exact add_le_add le_rfl (mul_le_mul_of_nonneg_left hgood (mul_nonneg p.2.1 (sub_nonneg.mpr p.2.2)))
-
-end CoordinateAveragingBounds
-end Percolation
-
- 
-namespace Percolation
-universe v
-variable {E : Type v}
-section EdgeDecisionTrees
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-inductive EdgeQueryTree (E : Type v) where
-  | done (b : Bool)
-  | query (e : E) (closed opened : EdgeQueryTree E)
-
-namespace EdgeQueryTree
-
-def Valid (Q : Finset E) : EdgeQueryTree E → Prop
-  | .done _ => True
-  | .query e t₀ t₁ => e ∈ Q ∧ Valid (Q.erase e) t₀ ∧ Valid (Q.erase e) t₁
-
-noncomputable def eval (ω : Finset E) : EdgeQueryTree E → Bool
-  | .done b => b
-  | .query e t₀ t₁ => if e ∈ ω then eval ω t₁ else eval ω t₀
-
-noncomputable def value (t : EdgeQueryTree E) (ω : Finset E) : ℝ := if t.eval ω then 1 else 0
-
-noncomputable def reads (ω : Finset E) (i : E) : EdgeQueryTree E → Prop
-  | .done _ => False
-  | .query e t₀ t₁ => i = e ∨ if e ∈ ω then reads ω i t₁ else reads ω i t₀
-
-noncomputable def revealment (p : ℝ) (i : E) : EdgeQueryTree E → ℝ
-  | .done _ => 0
-  | .query e t₀ t₁ => if i = e then 1 else (1-p)*revealment p i t₀+p*revealment p i t₁
-
-theorem value_mem (t : EdgeQueryTree E) (ω : Finset E) : t.value ω ∈ Set.Icc (0:ℝ) 1 := by
-  unfold value
-  split <;> simp
-
-theorem revealment_nonneg (p : unitInterval) (i : E) (t : EdgeQueryTree E) : 0 ≤ t.revealment p i := by
-  induction t with
-  | done => rfl
-  | query e t₀ t₁ ih₀ ih₁ =>
-    simp only [revealment]
-    split_ifs
-    · norm_num
-    · exact add_nonneg (mul_nonneg (sub_nonneg.mpr p.2.2) ih₀) (mul_nonneg p.2.1 ih₁)
-
-theorem eval_congr (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
-    {ω η : Finset E} (h : ∀ e ∈ Q, e ∈ ω ↔ e ∈ η) : t.eval ω = t.eval η := by
-  induction t generalizing Q with
-  | done => rfl
-  | query e t₀ t₁ ih₀ ih₁ =>
-    simp only [eval,h e ht.1]
-    split
-    · exact ih₁ ht.2.2 (fun a ha => h a (Finset.mem_of_mem_erase ha))
-    · exact ih₀ ht.2.1 (fun a ha => h a (Finset.mem_of_mem_erase ha))
-
-theorem eval_insert (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
-    {e : E} (he : e ∉ Q) (ω : Finset E) : t.eval (insert e ω) = t.eval ω := by
-  apply eval_congr t ht
-  intro a ha
-  simp [show a ≠ e from fun h => he (h ▸ ha)]
-
-theorem value_insert (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
-    {e : E} (he : e ∉ Q) (ω : Finset E) : t.value (insert e ω) = t.value ω := by
-  simp only [value,eval_insert t ht he]
-
-theorem reads_congr (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
-    {ω η : Finset E} (h : ∀ e ∈ Q, e ∈ ω ↔ e ∈ η) (i : E) : t.reads ω i ↔ t.reads η i := by
-  induction t generalizing Q with
-  | done => rfl
-  | query e t₀ t₁ ih₀ ih₁ =>
-    simp only [reads,h e ht.1]
-    split
-    · rw [ih₁ ht.2.2 (fun a ha => h a (Finset.mem_of_mem_erase ha))]
-    · rw [ih₀ ht.2.1 (fun a ha => h a (Finset.mem_of_mem_erase ha))]
-
-theorem reads_mem (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
-    (ω : Finset E) (i : E) (hi : t.reads ω i) : i ∈ Q := by
-  induction t generalizing Q with
-  | done => exact hi.elim
-  | query e t₀ t₁ ih₀ ih₁ =>
-    rcases hi with rfl | hi
-    · exact ht.1
-    · split_ifs at hi
-      · exact Finset.mem_of_mem_erase (ih₁ ht.2.2 hi)
-      · exact Finset.mem_of_mem_erase (ih₀ ht.2.1 hi)
-
-end EdgeQueryTree
-end EdgeDecisionTrees
-end Percolation
-
- 
-namespace Percolation
-universe v
-variable {E : Type v}
-section EdgeTreeCovariance
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem finiteCovariance_congr_right (Q : Finset E) (p : ℝ) (f : Finset E → ℝ)
-    {g h : Finset E → ℝ} (he : ∀ ω ⊆ Q, g ω = h ω) :
-    finiteCovariance Q p f g = finiteCovariance Q p f h := by
-  unfold finiteCovariance
-  rw [bernoulliPolynomial_congr Q (fun ω hω => congrArg (f ω * ·) (he ω hω)),
-    bernoulliPolynomial_congr Q he]
-
-namespace EdgeQueryTree
-
- 
-
-theorem covariance_bound (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
-    (p : unitInterval) (f : Finset E → ℝ) :
-    |finiteCovariance Q p f t.value| ≤
-      ∑ i ∈ Q, t.revealment p i * ((p:ℝ)*(1-(p:ℝ))*coordinateVariation Q p f i) := by
-  induction t generalizing Q f with
-  | done b =>
-    have hv : (done b : EdgeQueryTree E).value = (fun _ => if b then (1:ℝ) else 0) := rfl
-    rw [hv]
-    cases b
-    · simp [finiteCovariance,revealment]
-    · simp [finiteCovariance,revealment,bernoulliPolynomial_const]
-  | query e t₀ t₁ ih₀ ih₁ =>
-    have he := ht.1
-    have heS : e ∉ Q.erase e := Finset.notMem_erase _ _
-    have hQ : insert e (Q.erase e) = Q := Finset.insert_erase he
-    let fav := coordinateAverage e (p:ℝ) f
-    have hcov₀ : finiteCovariance (Q.erase e) p fav (query e t₀ t₁).value =
-        finiteCovariance (Q.erase e) p fav t₀.value := by
-      apply finiteCovariance_congr_right
-      intro ω hω
-      have heω : e ∉ ω := fun h => heS (hω h)
-      simp only [value,eval,ite_eq_right heω]
-    have hcov₁ : finiteCovariance (Q.erase e) p fav (fun ω => (query e t₀ t₁).value (insert e ω)) =
-        finiteCovariance (Q.erase e) p fav t₁.value := by
-      apply finiteCovariance_congr_right
-      intro ω _
-      simp only [value,eval,Finset.mem_insert_self,ite_true,eval_insert t₁ ht.2.2 heS]
-    have hav (i : E) (hi : i ∈ Q.erase e) :
-        coordinateVariation (Q.erase e) p fav i ≤ coordinateVariation Q p f i := by
-      simpa only [hQ] using coordinateVariation_average_le (Q.erase e) e i heS
-        (Finset.ne_of_mem_erase hi) p f
-    have hb₀ : |finiteCovariance (Q.erase e) p fav t₀.value| ≤
-        ∑ i ∈ Q.erase e, t₀.revealment p i * ((p:ℝ)*(1-(p:ℝ))*coordinateVariation Q p f i) := by
-      apply (ih₀ ht.2.1 fav).trans
-      apply Finset.sum_le_sum
-      intro i hi
-      exact mul_le_mul_of_nonneg_left
-        (mul_le_mul_of_nonneg_left (hav i hi) (mul_nonneg p.2.1 (sub_nonneg.mpr p.2.2)))
-        (revealment_nonneg p i t₀)
-    have hb₁ : |finiteCovariance (Q.erase e) p fav t₁.value| ≤
-        ∑ i ∈ Q.erase e, t₁.revealment p i * ((p:ℝ)*(1-(p:ℝ))*coordinateVariation Q p f i) := by
-      apply (ih₁ ht.2.2 fav).trans
-      apply Finset.sum_le_sum
-      intro i hi
-      exact mul_le_mul_of_nonneg_left
-        (mul_le_mul_of_nonneg_left (hav i hi) (mul_nonneg p.2.1 (sub_nonneg.mpr p.2.2)))
-        (revealment_nonneg p i t₁)
-    have hfirst := finiteCovariance_first_bound (Q.erase e) e heS p f (query e t₀ t₁).value
-      (fun ω _ => value_mem _ ω)
-    change _ ≤ (1-(p:ℝ))*|finiteCovariance (Q.erase e) p fav (query e t₀ t₁).value|+
-      (p:ℝ)*|finiteCovariance (Q.erase e) p fav (fun ω => (query e t₀ t₁).value (insert e ω))|+_ at hfirst
-    rw [hQ,hcov₀,hcov₁] at hfirst
-    apply hfirst.trans
-    calc
-      _ ≤ (1-(p:ℝ))*(∑ i ∈ Q.erase e, t₀.revealment p i * ((p:ℝ)*(1-(p:ℝ))*coordinateVariation Q p f i))+
-          (p:ℝ)*(∑ i ∈ Q.erase e, t₁.revealment p i * ((p:ℝ)*(1-(p:ℝ))*coordinateVariation Q p f i))+
-          (p:ℝ)*(1-(p:ℝ))*coordinateVariation Q p f e :=
-        add_le_add (add_le_add (mul_le_mul_of_nonneg_left hb₀ (sub_nonneg.mpr p.2.2))
-          (mul_le_mul_of_nonneg_left hb₁ p.2.1)) le_rfl
-      _ = _ := by
-        rw [←Finset.sum_erase_add _ _ he]
-        congr 1
-        · rw [Finset.mul_sum,Finset.mul_sum,←Finset.sum_add_distrib]
-          apply Finset.sum_congr rfl
-          intro i hi
-          rw [revealment,ite_eq_right (Finset.ne_of_mem_erase hi)]
-          ring
-        · simp [revealment]
-
-end EdgeQueryTree
-end EdgeTreeCovariance
-end Percolation
-
- 
-namespace Percolation
-universe v
-variable {E : Type v}
-section RevealmentProbability
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace EdgeQueryTree
-
-theorem reads_insert (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
-    {e : E} (he : e ∉ Q) (ω : Finset E) (i : E) : t.reads (insert e ω) i ↔ t.reads ω i := by
-  apply reads_congr t ht
-  intro a ha
-  simp [show a ≠ e from fun h => he (h ▸ ha)]
-
- 
-theorem revealment_eq_probability (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
-    (p : ℝ) (i : E) :
-    t.revealment p i = bernoulliPolynomial Q (fun ω => if t.reads ω i then 1 else 0) p := by
-  induction t generalizing Q with
-  | done => simp [revealment,reads]
-  | query e t₀ t₁ ih₀ ih₁ =>
-    by_cases hie : i = e
-    · subst i
-      simp [revealment,reads,bernoulliPolynomial_const]
-    · have heS : e ∉ Q.erase e := Finset.notMem_erase _ _
-      conv_rhs => rw [←Finset.insert_erase ht.1]
-      rw [bernoulliPolynomial_insert _ e heS]
-      simp only [revealment,ite_eq_right hie]
-      rw [ih₀ ht.2.1,ih₁ ht.2.2]
-      congr 1
-      · congr 1
-        apply bernoulliPolynomial_congr
-        intro ω hω
-        have heω : e ∉ ω := fun h => heS (hω h)
-        simp only [reads,hie,false_or,ite_eq_right heω]
-      · congr 1
-        apply bernoulliPolynomial_congr
-        intro ω _
-        simp only [reads,hie,false_or,Finset.mem_insert_self,ite_true,reads_insert t₁ ht.2.2 heS]
-
-end EdgeQueryTree
-end RevealmentProbability
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section SeededEdgeSearch
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-noncomputable def seededEdgeSearch (G : BondGraph V E) (x : V) :
-    ℕ → Finset E → Finset V → EdgeQueryTree E
-  | 0, _, R => .done (decide (x ∈ R))
-  | n+1, Q, R =>
-    if hc : SearchCrossing G Q R then
-      let t := searchCrossing G Q R hc
-      .query t.1 (seededEdgeSearch G x n (Q.erase t.1) R)
-        (seededEdgeSearch G x n (Q.erase t.1) (insert t.2.2 R))
-    else .done (decide (x ∈ R))
-
-theorem seededEdgeSearch_valid (G : BondGraph V E) (x : V) (n : ℕ) (Q : Finset E) (R : Finset V) :
-    (seededEdgeSearch G x n Q R).Valid Q := by
-  induction n generalizing Q R with
-  | zero => trivial
-  | succ n ih =>
-    simp only [seededEdgeSearch]
-    split_ifs with hc
-    · exact ⟨(searchCrossing_spec G Q R hc).1,ih _ _,ih _ _⟩
-    · trivial
-
-theorem seedMarked_singleton_of_no_crossing (G : BondGraph V E) (Q : Finset E) (R : Finset V)
-    (x : V) (h : ¬ SearchCrossing G Q R) (ω : Finset E) (hω : ω ⊆ Q) :
-    SeedMarked G (ω : Set E) R {x} ↔ x ∈ R := by
-  constructor
-  · intro hx
-    by_contra hn
-    exact seedMarked_false_of_no_crossing G Q R {x} h (by simpa using hn) ω hω hx
-  · intro hx
-    exact ⟨x,hx,x,Finset.mem_singleton_self x,conn_refl G _ x⟩
-
- 
-theorem seededEdgeSearch_correct (G : BondGraph V E) (x : V) (n : ℕ) (Q : Finset E) (R : Finset V)
-    (hn : Q.card ≤ n) (ω : Finset E) (hω : ω ⊆ Q) :
-    (seededEdgeSearch G x n Q R).eval ω = true ↔ SeedMarked G (ω : Set E) R {x} := by
-  induction n generalizing Q R ω with
-  | zero =>
-    have hQ : Q = ∅ := Finset.card_eq_zero.mp (by omega)
-    simp only [seededEdgeSearch,EdgeQueryTree.eval,decide_eq_true_eq]
-    exact (seedMarked_singleton_of_no_crossing G Q R x (by simp [SearchCrossing,hQ]) ω hω).symm
-  | succ n ih =>
-    simp only [seededEdgeSearch]
-    split_ifs with hc
-    · obtain ⟨he,ha,hb,hends⟩ := searchCrossing_spec G Q R hc
-      have hcard : (Q.erase (searchCrossing G Q R hc).1).card ≤ n := by
-        rw [Finset.card_erase_of_mem he]; omega
-      simp only [EdgeQueryTree.eval]
-      by_cases heω : (searchCrossing G Q R hc).1 ∈ ω
-      · simp only [ite_eq_left heω]
-        rw [←Finset.insert_erase heω,Finset.coe_insert,seedMarked_insert_edge G _ R {x} _ _ _ ha hends,
-          EdgeQueryTree.eval_insert _ (seededEdgeSearch_valid G x n _ _) (Finset.notMem_erase _ _)]
-        exact ih _ _ hcard _ (Finset.erase_subset_erase _ hω)
-      · simp only [ite_eq_right heω]
-        exact ih _ _ hcard ω (Finset.subset_erase.mpr ⟨hω,heω⟩)
-    · simp only [EdgeQueryTree.eval,decide_eq_true_eq]
-      exact (seedMarked_singleton_of_no_crossing G Q R x hc ω hω).symm
-
- 
-
-theorem seededEdgeSearch_reads (G : BondGraph V E) (x : V) (n : ℕ) (Q : Finset E) (R : Finset V)
-    (ω : Finset E) (i : E) (hi : (seededEdgeSearch G x n Q R).reads ω i) :
-    ∃ a ∈ G i, ∃ r ∈ R, Conn G (ω : Set E) r a := by
-  induction n generalizing Q R with
-  | zero => exact hi.elim
-  | succ n ih =>
-    simp only [seededEdgeSearch] at hi
-    split_ifs at hi with hc
-    · obtain ⟨he,ha,hb,hends⟩ := searchCrossing_spec G Q R hc
-      rcases hi with rfl | hi
-      · refine ⟨(searchCrossing G Q R hc).2.1,?_,_,ha,conn_refl G _ _⟩
-        rw [hends]; simp
-      · split_ifs at hi with heω
-        · obtain ⟨a,hai,r,hr,hra⟩ := ih _ _ hi
-          rcases Finset.mem_insert.mp hr with rfl | hr
-          · refine ⟨a,hai,(searchCrossing G Q R hc).2.1,ha,?_⟩
-            exact (Relation.ReflTransGen.single ⟨(searchCrossing G Q R hc).1,heω,hends⟩).trans hra
-          · exact ⟨a,hai,r,hr,hra⟩
-        · exact ih _ _ hi
-    · exact hi.elim
-
-end SeededEdgeSearch
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section RandomizedDecisionCovariance
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem finiteCovariance_average_right (Q : Finset E) (F : Finset V) (p h : ℝ)
-    (f : Finset E → ℝ) (g : Finset V → Finset E → ℝ) :
-    finiteCovariance Q p f (fun ω => bernoulliPolynomial F (fun m => g m ω) h) =
-      bernoulliPolynomial F (fun m => finiteCovariance Q p f (g m)) h := by
-  unfold finiteCovariance
-  rw [bernoulliPolynomial_sub,bernoulliPolynomial_const_mul,
-    ←bernoulliPolynomial_commute Q F (fun ω m => g m ω) p h]
-  congr 1
-  calc
-    _ = bernoulliPolynomial Q (fun ω => bernoulliPolynomial F (fun m => f ω*g m ω) h) p := by
-      apply bernoulliPolynomial_congr
-      intro ω _
-      exact (bernoulliPolynomial_const_mul F (fun m => g m ω) (f ω) h).symm
-    _ = _ := bernoulliPolynomial_commute Q F _ p h
-
- 
-
-theorem EdgeQueryTree.randomized_covariance_bound (Q : Finset E) (F : Finset V)
-    (p h : unitInterval) (f : Finset E → ℝ) (trees : Finset V → EdgeQueryTree E)
-    (ht : ∀ m ⊆ F, (trees m).Valid Q) :
-    |finiteCovariance Q p f (fun ω => bernoulliPolynomial F (fun m => (trees m).value ω) h)| ≤
-      ∑ i ∈ Q, bernoulliPolynomial F (fun m => (trees m).revealment p i) h *
-        ((p:ℝ)*(1-(p:ℝ))*coordinateVariation Q p f i) := by
-  rw [finiteCovariance_average_right]
-  apply (bernoulliPolynomial_abs F h _).trans
-  calc
-    _ ≤ bernoulliPolynomial F (fun m => ∑ i ∈ Q, (trees m).revealment p i *
-        ((p:ℝ)*(1-(p:ℝ))*coordinateVariation Q p f i)) h := by
-      apply bernoulliPolynomial_mono _ _ _ h h.2
-      intro m hm
-      exact (trees m).covariance_bound (ht m hm) p f
-    _ = _ := by simp only [bernoulliPolynomial_sum,bernoulliPolynomial_mul_const]
-
-noncomputable def finiteSeedProbability (G : BondGraph V E) (F : Finset V)
-    (ω : Finset E) (x : V) (h : ℝ) : ℝ :=
-  bernoulliPolynomial F (fun m => if ∃ y ∈ m, Conn G (ω:Set E) x y then 1 else 0) h
-
-theorem seededEdgeSearch_value (G : BondGraph V E) (x : V) (Q : Finset E)
-    (m : Finset V) (ω : Finset E) (hω : ω ⊆ Q) :
-    (seededEdgeSearch G x Q.card Q m).value ω =
-      if ∃ y ∈ m, Conn G (ω:Set E) x y then 1 else 0 := by
-  have hc := seededEdgeSearch_correct G x Q.card Q m le_rfl ω hω
-  have he : SeedMarked G (ω:Set E) m {x} ↔ ∃ y ∈ m, Conn G (ω:Set E) x y := by
-    simp only [SeedMarked,Finset.mem_singleton,exists_eq_left]
-    exact exists_congr (fun y => and_congr_right (fun _ => ⟨conn_symm G _,conn_symm G _⟩))
-  unfold EdgeQueryTree.value
-  simp only [hc,he]
-
-theorem finiteSeedProbability_tree (G : BondGraph V E) (Q : Finset E) (F : Finset V)
-    (x : V) (h : ℝ) (ω : Finset E) (hω : ω ⊆ Q) :
-    finiteSeedProbability G F ω x h =
-      bernoulliPolynomial F (fun m => (seededEdgeSearch G x Q.card Q m).value ω) h := by
-  apply bernoulliPolynomial_congr
-  intro m _
-  exact (seededEdgeSearch_value G x Q m ω hω).symm
-
-end RandomizedDecisionCovariance
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section SeedRevealmentBounds
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem seededEdgeSearch_reads_endpoints (G : BondGraph V E) (x : V) (Q : Finset E)
-    (m : Finset V) (ω : Finset E) (i : E) (a b : V) (he : G i = s(a,b))
-    (hi : (seededEdgeSearch G x Q.card Q m).reads ω i) :
-    (∃ y ∈ m, Conn G (ω:Set E) a y) ∨ (∃ y ∈ m, Conn G (ω:Set E) b y) := by
-  obtain ⟨z,hz,r,hr,hrz⟩ := seededEdgeSearch_reads G x Q.card Q m ω i hi
-  rw [he] at hz
-  rcases (by simpa using hz : z = a ∨ z = b) with rfl | rfl
-  · exact Or.inl ⟨r,hr,conn_symm G _ hrz⟩
-  · exact Or.inr ⟨r,hr,conn_symm G _ hrz⟩
-
-theorem seededEdgeSearch_mean_revealment_le (G : BondGraph V E) (x : V)
-    (Q : Finset E) (F : Finset V) (p h : unitInterval) (i : E) (a b : V) (he : G i = s(a,b)) :
-    bernoulliPolynomial F (fun m => (seededEdgeSearch G x Q.card Q m).revealment p i) h ≤
-      bernoulliPolynomial Q (fun ω => finiteSeedProbability G F ω a h) p +
-      bernoulliPolynomial Q (fun ω => finiteSeedProbability G F ω b h) p := by
-  simp_rw [EdgeQueryTree.revealment_eq_probability _ (seededEdgeSearch_valid G x Q.card Q _)]
-  rw [←bernoulliPolynomial_commute,←bernoulliPolynomial_add]
-  apply bernoulliPolynomial_mono _ _ _ p p.2
-  intro ω _
-  unfold finiteSeedProbability
-  rw [←bernoulliPolynomial_add]
-  apply bernoulliPolynomial_mono _ _ _ h h.2
-  intro m _
-  have hr := seededEdgeSearch_reads_endpoints G x Q m ω i a b he
-  by_cases hread : (seededEdgeSearch G x Q.card Q m).reads ω i
-  · rcases hr hread with ha | hb
-    · simp only [ite_eq_left hread,ite_eq_left ha]
-      split_ifs <;> norm_num
-    · simp only [ite_eq_left hread,ite_eq_left hb]
-      split_ifs <;> norm_num
-  · simp only [ite_eq_right hread]
-    split_ifs <;> norm_num
-
- 
-
-theorem finiteSeedProbability_component (G : BondGraph V E) (Q : Finset E) (F : Finset V)
-    (x : V) (hF : insert x (searchVertices G Q) ⊆ F)
-    (ω : Finset E) (hω : ω ⊆ Q) (h : ℝ) :
-    finiteSeedProbability G F ω x h = 1-(1-h)^(finiteRootComponent G ω x).card := by
-  let C := finiteRootComponent G ω x
-  have hCF : C ⊆ F := finiteRootComponent_subset G Q F x ω hω hF
-  have he (m : Finset V) : (∃ y ∈ m, Conn G (ω:Set E) x y) ↔ m ∩ C ≠ ∅ := by
-    rw [Finset.nonempty_iff_ne_empty.symm]
-    constructor
-    · rintro ⟨y,hy,hxy⟩
-      exact ⟨y,Finset.mem_inter.mpr ⟨hy,by change y ∈ (C:Set V); simpa only [C,coe_finiteRootComponent,cluster,Set.mem_ofPred_eq] using hxy⟩⟩
-    · rintro ⟨y,hy⟩
-      refine ⟨y,(Finset.mem_inter.mp hy).1,?_⟩
-      have hm : y ∈ (C:Set V) := (Finset.mem_inter.mp hy).2
-      simpa only [C,coe_finiteRootComponent,cluster,Set.mem_ofPred_eq] using hm
-  calc
-    _ = bernoulliPolynomial F (fun m => 1-(if m ∩ C = ∅ then 1 else 0)) h := by
-      apply bernoulliPolynomial_congr
-      intro m _
-      by_cases hm : m ∩ C = ∅
-      · have hn : ¬∃ y ∈ m, Conn G (ω:Set E) x y := fun hc => (he m).mp hc hm
-        simp only [hm,hn,ite_false,ite_true,sub_self]
-      · have hy := (he m).mpr hm
-        simp only [hm,hy,ite_false,ite_true,sub_zero]
-    _ = 1-bernoulliPolynomial F (fun m => if m ∩ C = ∅ then 1 else 0) h := by
-      rw [bernoulliPolynomial_sub,bernoulliPolynomial_const]
-    _ = _ := by rw [bernoulliPolynomial_restrict C F hCF (fun m => if m = ∅ then 1 else 0) h,
-      bernoulliPolynomial_empty_indicator]
-
-end SeedRevealmentBounds
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section GhostFractionalPower
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem one_sub_pow_one_sub_le_power (h : unitInterval) (n : ℕ) (α : ℝ)
-    (hα : 0 ≤ α) (hα1 : α ≤ 1) :
-    1-(1-(h:ℝ))^n ≤ (h:ℝ)^α*(n:ℝ)^α := by
-  rw [←Real.mul_rpow h.2.1 (Nat.cast_nonneg _)]
-  by_cases ht : (h:ℝ)*n ≤ 1
-  · exact (one_sub_pow_one_sub_le h n).trans (Real.self_le_rpow_of_le_one (mul_nonneg h.2.1 (Nat.cast_nonneg _)) ht hα1)
-  · have ht' : 1 ≤ (h:ℝ)*n := (lt_of_not_ge ht).le
-    exact (sub_le_self _ (pow_nonneg (sub_nonneg.mpr h.2.2) _)).trans (Real.one_le_rpow ht' hα)
-
-theorem markedConnection_le_cluster_power (G : BondGraph V E) (Q : Finset E)
-    (x : V) (h p : unitInterval) (α : ℝ) (hα : 0 ≤ α) (hα1 : α ≤ 1)
-    (hfin : ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
-    (hi : Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^α) (law p)) :
-    markedConnectionPolynomial G Q x h p ≤ (h:ℝ)^α*(∫ ω, ((cluster G ω x).ncard:ℝ)^α ∂law p) := by
-  rw [markedConnection_integral,←integral_const_mul]
-  apply integral_mono_ae
-  · exact integrable_finite_local Q p _ (by intro ω; simp only [Set.inter_assoc,Set.inter_self])
-  · exact hi.const_mul _
-  · filter_upwards [hfin] with ω hω
-    apply (one_sub_pow_one_sub_le_power h _ α hα hα1).trans
-    apply mul_le_mul_of_nonneg_left _ (Real.rpow_nonneg h.2.1 _)
-    apply Real.rpow_le_rpow (Nat.cast_nonneg _) _ hα
-    exact_mod_cast Set.ncard_le_ncard (fun y hy => conn_mono G Set.inter_subset_left hy) hω
-
-theorem finiteSeedProbability_expectation (G : BondGraph V E) (Q : Finset E) (F : Finset V)
-    (x : V) (hF : insert x (searchVertices G Q) ⊆ F) (h p : ℝ) :
-    bernoulliPolynomial Q (fun ω => finiteSeedProbability G F ω x h) p =
-      markedConnectionPolynomial G Q x h p := by
-  rw [markedConnection_component_polynomial]
-  apply bernoulliPolynomial_congr
-  intro ω hω
-  exact finiteSeedProbability_component G Q F x hF ω hω h
-
-theorem finiteSeedProbability_le_cluster_power (G : BondGraph V E) (Q : Finset E) (F : Finset V)
-    (x : V) (hF : insert x (searchVertices G Q) ⊆ F) (h p : unitInterval)
-    (α : ℝ) (hα : 0 ≤ α) (hα1 : α ≤ 1)
-    (hfin : ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
-    (hi : Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^α) (law p)) :
-    bernoulliPolynomial Q (fun ω => finiteSeedProbability G F ω x h) p ≤
-      (h:ℝ)^α*(∫ ω, ((cluster G ω x).ncard:ℝ)^α ∂law p) := by
-  rw [finiteSeedProbability_expectation G Q F x hF]
-  exact markedConnection_le_cluster_power G Q x h p α hα hα1 hfin hi
-
-end GhostFractionalPower
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section MonotoneDecisionRusso
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem sum_coordinateVariation_eq_deriv (Q : Finset E) (f : Finset E → ℝ)
-    (hf : Monotone f) (p : ℝ) :
-    (∑ i ∈ Q, coordinateVariation Q p f i) = deriv (bernoulliPolynomial Q f) p := by
-  rw [(hasDerivAt_bernoulliPolynomial Q f p).deriv]
-  apply Finset.sum_congr rfl
-  intro i _
-  apply bernoulliPolynomial_congr
-  intro ω _
-  exact abs_of_nonneg (sub_nonneg.mpr (hf (Finset.subset_insert _ _)))
-
-theorem EdgeQueryTree.randomized_covariance_le_deriv (Q : Finset E) (F : Finset V)
-    (p h : unitInterval) (f : Finset E → ℝ) (hf : Monotone f)
-    (trees : Finset V → EdgeQueryTree E) (ht : ∀ m ⊆ F, (trees m).Valid Q)
-    (R : ℝ) (hR : ∀ i ∈ Q, bernoulliPolynomial F (fun m => (trees m).revealment p i) h ≤ R) :
-    |finiteCovariance Q p f (fun ω => bernoulliPolynomial F (fun m => (trees m).value ω) h)| ≤
-      R*((p:ℝ)*(1-(p:ℝ)))*deriv (bernoulliPolynomial Q f) p := by
-  apply (randomized_covariance_bound Q F p h f trees ht).trans
-  calc
-    _ ≤ ∑ i ∈ Q, R*((p:ℝ)*(1-(p:ℝ))*coordinateVariation Q p f i) := by
-      apply Finset.sum_le_sum
-      intro i hi
-      exact mul_le_mul_of_nonneg_right (hR i hi)
-        (mul_nonneg (mul_nonneg p.2.1 (sub_nonneg.mpr p.2.2)) (coordinateVariation_nonneg Q p f i))
-    _ = _ := by simp only [←mul_assoc,←Finset.mul_sum,sum_coordinateVariation_eq_deriv Q f hf]
-
- 
-
-theorem seededEdgeSearch_revealment_of_moments (G : BondGraph V E) (Q : Finset E)
-    (x : V) (p h : unitInterval) (α C : ℝ) (hα : 0 ≤ α) (hα1 : α ≤ 1)
-    (hfin : ∀ y, ∀ᵐ ω ∂law p, (cluster G ω y).Finite)
-    (hI : ∀ y, Integrable (fun ω => ((cluster G ω y).ncard:ℝ)^α) (law p))
-    (hC : ∀ y, (∫ ω, ((cluster G ω y).ncard:ℝ)^α ∂law p) ≤ C) :
-    ∀ i ∈ Q, bernoulliPolynomial (insert x (searchVertices G Q))
-      (fun m => (seededEdgeSearch G x Q.card Q m).revealment p i) h ≤ 2*C*(h:ℝ)^α := by
-  intro i hi
-  obtain ⟨⟨a,b⟩,he⟩ := Quot.exists_rep (G i)
-  have he' : G i = s(a,b) := he.symm
-  have hFa : insert a (searchVertices G Q) ⊆ insert x (searchVertices G Q) :=
-    Finset.insert_subset_iff.mpr ⟨Finset.mem_insert_of_mem (mem_searchVertices G hi (by rw [he']; simp)),Finset.subset_insert _ _⟩
-  have hFb : insert b (searchVertices G Q) ⊆ insert x (searchVertices G Q) :=
-    Finset.insert_subset_iff.mpr ⟨Finset.mem_insert_of_mem (mem_searchVertices G hi (by rw [he']; simp)),Finset.subset_insert _ _⟩
-  have ha := finiteSeedProbability_le_cluster_power G Q _ a hFa h p α hα hα1 (hfin a) (hI a)
-  have hb := finiteSeedProbability_le_cluster_power G Q _ b hFb h p α hα hα1 (hfin b) (hI b)
-  have hca := mul_le_mul_of_nonneg_left (hC a) (Real.rpow_nonneg h.2.1 α)
-  have hcb := mul_le_mul_of_nonneg_left (hC b) (Real.rpow_nonneg h.2.1 α)
-  have hr := seededEdgeSearch_mean_revealment_le G x Q (insert x (searchVertices G Q)) p h i a b he'
-  linarith
-
-theorem finiteSeed_covariance_upper (G : BondGraph V E) (Q : Finset E) (x : V)
-    (p h : unitInterval) (α C : ℝ) (hα : 0 ≤ α) (hα1 : α ≤ 1)
-    (hfin : ∀ y, ∀ᵐ ω ∂law p, (cluster G ω y).Finite)
-    (hI : ∀ y, Integrable (fun ω => ((cluster G ω y).ncard:ℝ)^α) (law p))
-    (hC : ∀ y, (∫ ω, ((cluster G ω y).ncard:ℝ)^α ∂law p) ≤ C)
-    (f : Finset E → ℝ) (hf : Monotone f) :
-    |finiteCovariance Q p f (fun ω => finiteSeedProbability G (insert x (searchVertices G Q)) ω x h)| ≤
-      (2*C*(h:ℝ)^α)*((p:ℝ)*(1-(p:ℝ)))*deriv (bernoulliPolynomial Q f) p := by
-  have hc := EdgeQueryTree.randomized_covariance_le_deriv Q (insert x (searchVertices G Q)) p h f hf
-    (fun m => seededEdgeSearch G x Q.card Q m) (fun _ _ => seededEdgeSearch_valid G x Q.card Q _)
-    (2*C*(h:ℝ)^α) (seededEdgeSearch_revealment_of_moments G Q x p h α C hα hα1 hfin hI hC)
-  convert hc using 2
-  apply finiteCovariance_congr_right
-  intro ω hω
-  exact finiteSeedProbability_tree G Q _ x h ω hω
-
-end MonotoneDecisionRusso
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section ClusterTailCovariance
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def finiteClusterTailIndicator (G : BondGraph V E) (x : V) (n : ℕ)
-    (ω : Finset E) : ℝ := if n ≤ (finiteRootComponent G ω x).card then 1 else 0
-
-noncomputable def finiteClusterTail (G : BondGraph V E) (Q : Finset E) (x : V) (n : ℕ) (p : ℝ) : ℝ :=
-  bernoulliPolynomial Q (finiteClusterTailIndicator G x n) p
-
-theorem finiteClusterTailIndicator_mono (G : BondGraph V E) (x : V) (n : ℕ) :
-    Monotone (finiteClusterTailIndicator G x n) := by
-  intro ω η hωη
-  have hc := Finset.card_le_card (finiteRootComponent_mono G hωη x)
-  unfold finiteClusterTailIndicator
-  split_ifs <;> norm_num
-  omega
-
-theorem finiteClusterTail_nonneg (G : BondGraph V E) (Q : Finset E) (x : V) (n : ℕ) (p : unitInterval) :
-    0 ≤ finiteClusterTail G Q x n p := by
-  apply bernoulliPolynomial_nonneg _ _ p p.2
-  intro ω _
-  unfold finiteClusterTailIndicator
-  split_ifs <;> norm_num
-
-theorem finiteClusterTail_deriv_nonneg (G : BondGraph V E) (Q : Finset E) (x : V) (n : ℕ) (p : unitInterval) :
-    0 ≤ deriv (finiteClusterTail G Q x n) p := by
-  change 0 ≤ deriv (bernoulliPolynomial Q (finiteClusterTailIndicator G x n)) p
-  rw [←sum_coordinateVariation_eq_deriv Q _ (finiteClusterTailIndicator_mono G x n)]
-  exact Finset.sum_nonneg (fun i _ => coordinateVariation_nonneg Q p _ i)
-
- 
-
-theorem finiteSeed_covariance_lower (G : BondGraph V E) (Q : Finset E) (x : V)
-    (n : ℕ) (p h : unitInterval) (hn : 1 ≤ (h:ℝ)*n) (B : ℝ)
-    (hB : markedConnectionPolynomial G Q x h p ≤ B) :
-    (1/2-B)*finiteClusterTail G Q x n p ≤ finiteCovariance Q p
-      (finiteClusterTailIndicator G x n)
-      (fun ω => finiteSeedProbability G (insert x (searchVertices G Q)) ω x h) := by
-  have hprod : (1:ℝ)/2*finiteClusterTail G Q x n p ≤
-      bernoulliPolynomial Q (fun ω => finiteClusterTailIndicator G x n ω*
-        finiteSeedProbability G (insert x (searchVertices G Q)) ω x h) p := by
-    rw [finiteClusterTail,←bernoulliPolynomial_const_mul]
-    apply bernoulliPolynomial_mono _ _ _ p p.2
-    intro ω hω
-    rw [finiteSeedProbability_component G Q _ x le_rfl ω hω]
-    unfold finiteClusterTailIndicator
-    by_cases hsize : n ≤ (finiteRootComponent G ω x).card
-    · simp only [ite_eq_left hsize,mul_one,one_mul]
-      apply one_sub_pow_one_sub_ge_half
-      exact hn.trans (mul_le_mul_of_nonneg_left (by exact_mod_cast hsize) h.2.1)
-    · simp only [ite_eq_right hsize,mul_zero,zero_mul,le_refl]
-  have hmean := finiteSeedProbability_expectation G Q (insert x (searchVertices G Q)) x le_rfl (h:ℝ) p
-  unfold finiteCovariance
-  rw [hmean]
-  change _ ≤ _-finiteClusterTail G Q x n p*markedConnectionPolynomial G Q x h p
-  have hm := mul_le_mul_of_nonneg_left hB (finiteClusterTail_nonneg G Q x n p)
-  nlinarith
-
-end ClusterTailCovariance
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Criticality.Susceptibility
+
+namespace OAI
+
+/-! Fractional Moments for bond percolation. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section GhostPowerTransfer
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem sqrt_log_power_bound (h : unitInterval) (hh : 0 < (h:ℝ))
+    (a : ℝ) (ha : 1/2 < a) :
+    Real.sqrt (Real.log (1/(h:ℝ))/(h:ℝ))*(h:ℝ)^a ≤ Real.sqrt (1/(2*a-1)) := by
+  let ε := 2*a-1
+  have hε : 0 < ε := by dsimp [ε]; linarith
+  have hpow := Real.rpow_pos_of_pos hh ε
+  have hl := Real.log_le_rpow_div (show 0 ≤ 1/(h:ℝ) by positivity) hε
+  rw [Real.div_rpow (by norm_num) hh.le,Real.one_rpow] at hl
+  have hpowid : ((h:ℝ)^a)^2 = (h:ℝ)^ε*(h:ℝ) := by
+    rw [←Real.rpow_natCast,←Real.rpow_mul hh.le]
+    have he : a*(2:ℝ) = ε+1 := by dsimp [ε]; ring
+    norm_num only [Nat.cast_ofNat]
+    rw [he,Real.rpow_add hh,Real.rpow_one]
+  have hbase : (Real.log (1/(h:ℝ))/(h:ℝ))*((h:ℝ)^a)^2 ≤ 1/ε := by
+    rw [hpowid]
+    calc
+      _ = Real.log (1/(h:ℝ))*(h:ℝ)^ε := by field_simp
+      _ ≤ ((1/(h:ℝ)^ε)/ε)*(h:ℝ)^ε := mul_le_mul_of_nonneg_right hl hpow.le
+      _ = 1/ε := by field_simp
+  have hlog : 0 ≤ Real.log (1/(h:ℝ))/(h:ℝ) := div_nonneg
+    (Real.log_nonneg ((one_le_div hh).mpr h.2.2)) hh.le
+  have hsq := Real.sq_sqrt hlog
+  have htarget := Real.sq_sqrt (one_div_pos.mpr hε).le
+  have hn := Real.sqrt_nonneg (1/ε)
+  change _ ≤ Real.sqrt (1/ε)
+  nlinarith [sq_nonneg (Real.sqrt (1/ε)-Real.sqrt (Real.log (1/(h:ℝ))/(h:ℝ))*(h:ℝ)^a)]
+
+ 
+
+theorem markedConnection_power_transfer (G : BondGraph V E) (Q : Finset E) (x : V)
+    (d : ℕ) (hd : 0 < d) (hdeg : ∀ v, (Q.filter (fun e => v ∈ G e)).card ≤ d)
+    (p q h : unitInterval) (hq0 : 0 < (q:ℝ)) (hq1 : (q:ℝ) < 1)
+    (hp : (q:ℝ)/2 ≤ (p:ℝ)) (hh : 0 < (h:ℝ))
+    (a : ℝ) (ha : 1/2 < a) (he : (q:ℝ)-(p:ℝ) = (h:ℝ)^a) :
+    markedConnectionPolynomial G Q x h q ≤
+      Real.exp ((Real.sqrt ((d:ℝ)/2)/(((q:ℝ)/2)*(1-(q:ℝ)))) *
+        Real.sqrt (1/(2*a-1))) * markedConnectionPolynomial G Q x h p := by
+  have hp0 : 0 < (p:ℝ) := lt_of_lt_of_le (half_pos hq0) hp
+  have hpq : p ≤ q := by change (p:ℝ) ≤ (q:ℝ); linarith [Real.rpow_pos_of_pos hh a]
+  have H := markedConnection_transfer G Q x d hd hdeg p q h hp0 hpq hq1 hh
+  apply H.trans
+  apply mul_le_mul_of_nonneg_right
+  · apply Real.exp_le_exp.mpr
+    rw [he]
+    have hr : Real.sqrt ((d:ℝ)/2)/((p:ℝ)*(1-(q:ℝ))) ≤
+        Real.sqrt ((d:ℝ)/2)/(((q:ℝ)/2)*(1-(q:ℝ))) :=
+      div_le_div_of_nonneg_left (Real.sqrt_nonneg _) (mul_pos (half_pos hq0) (sub_pos.mpr hq1))
+        (mul_le_mul_of_nonneg_right hp (sub_pos.mpr hq1).le)
+    calc
+      _ = (Real.sqrt ((d:ℝ)/2)/((p:ℝ)*(1-(q:ℝ)))) *
+        (Real.sqrt (Real.log (1/(h:ℝ))/(h:ℝ))*(h:ℝ)^a) := by ring
+      _ ≤ _ := mul_le_mul hr (sqrt_log_power_bound h hh a ha)
+        (mul_nonneg (Real.sqrt_nonneg _) (Real.rpow_nonneg h.2.1 _)) (by positivity)
+  · rw [markedConnectionPolynomial_eq_tree]
+    exact ProbeTree.successMass_nonneg p h _
+
+end GhostPowerTransfer
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section DyadicGhostParameters
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def dyadicGhost (j : ℕ) : unitInterval :=
+  ⟨(1/2:ℝ)^j,⟨pow_nonneg (by norm_num) _,pow_le_one₀ (by norm_num) (by norm_num)⟩⟩
+
+theorem dyadicGhost_pos (j : ℕ) : 0 < (dyadicGhost j:ℝ) := pow_pos (by norm_num) _
+
+theorem dyadicGhost_volume (j : ℕ) : (dyadicGhost j:ℝ)*(2^j:ℕ) = 1 := by
+  simp only [dyadicGhost,Nat.cast_pow,Nat.cast_ofNat,←mul_pow]
+  norm_num
+
+theorem tendsto_dyadicGhost : Tendsto (fun j => (dyadicGhost j:ℝ)) atTop (nhds 0) :=
+  tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num) (by norm_num)
+
+theorem dyadicGhost_rpow (j : ℕ) (t : ℝ) :
+    (dyadicGhost j:ℝ)^t = (2:ℝ)^(-t*(j:ℝ)) := by
+  change ((1/2:ℝ)^j)^t = _
+  rw [←Real.rpow_natCast,←Real.rpow_mul (by norm_num : (0:ℝ) ≤ 1/2)]
+  rw [show (1/2:ℝ) = (2:ℝ)^(-1:ℝ) by norm_num,
+    ←Real.rpow_mul (by norm_num : (0:ℝ) ≤ 2)]
+  congr 1
+  ring
+
+ 
+
+theorem eventually_dyadic_backstep (q : unitInterval) (hq : 0 < (q:ℝ))
+    (a : ℝ) (ha : 0 < a) (P : unitInterval → Prop) (hP : ∀ᶠ p in nhds q, P p) :
+    ∀ᶠ j : ℕ in atTop, ∃ p : unitInterval,
+      (q:ℝ)/2 ≤ (p:ℝ) ∧ p < q ∧ (q:ℝ)-(p:ℝ) = (dyadicGhost j:ℝ)^a ∧ P p := by
+  obtain ⟨ε,hε,hball⟩ := Metric.eventually_nhds_iff.mp hP
+  have ht : Tendsto (fun j => (dyadicGhost j:ℝ)^a) atTop (nhds 0) := by
+    simpa only [Real.zero_rpow ha.ne', Function.comp_apply] using!
+      (Real.continuousAt_rpow_const 0 a (Or.inr ha.le)).tendsto.comp tendsto_dyadicGhost
+  filter_upwards [ht.eventually (eventually_lt_nhds (half_pos hq)),ht.eventually (eventually_lt_nhds hε)] with j hj hεj
+  have hz := Real.rpow_pos_of_pos (dyadicGhost_pos j) a
+  let p : unitInterval := ⟨(q:ℝ)-(dyadicGhost j:ℝ)^a,⟨by linarith,by linarith [q.2.2]⟩⟩
+  refine ⟨p,by dsimp [p]; linarith,by change (q:ℝ)-(dyadicGhost j:ℝ)^a < (q:ℝ); linarith,
+    by dsimp [p]; ring,?_⟩
+  apply hball
+  change |((q:ℝ)-(dyadicGhost j:ℝ)^a)-(q:ℝ)| < ε
+  rw [show (q:ℝ)-(dyadicGhost j:ℝ)^a-(q:ℝ) = -(dyadicGhost j:ℝ)^a by ring,abs_neg,abs_of_pos hz]
+  exact hεj
+
+end DyadicGhostParameters
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section CriticalDyadicTail
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem critical_dyadic_bound_of_susceptibility [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (hq0 : 0 < (pc G:ℝ)) (hq1 : (pc G:ℝ) < 1)
+    (C b a : ℝ) (hC : 0 < C) (ha : 1/2 < a)
+    (hχ : ∀ᶠ p in nhds (pc G), p < pc G → ∀ x,
+      susceptibility G p x ≤ C*((pc G:ℝ)-(p:ℝ))^(-b)) :
+    ∃ K : ℝ, 0 < K ∧ ∀ᶠ j : ℕ in atTop, ∀ x,
+      clusterWindowProbability G (pc G) x (2^j) (2^(j+1)) ≤
+        K*(dyadicGhost j:ℝ)^(1-a*b) := by
+  obtain ⟨d,hd,hdeg⟩ := exists_degree_bound G hqt
+  let B := Real.exp ((Real.sqrt ((d:ℝ)/2)/(((pc G:ℝ)/2)*(1-(pc G:ℝ)))) * Real.sqrt (1/(2*a-1)))
+  have hB : 0 < B := Real.exp_pos _
+  refine ⟨2*B*C,by positivity,?_⟩
+  filter_upwards [eventually_dyadic_backstep (pc G) hq0 a (by linarith) _ hχ] with j hj
+  obtain ⟨p,hp,hpc,he,hχp⟩ := hj
+  intro x
+  let Q := incidentSupport G hloc (finite_reachBall hloc x (2^(j+1))).toFinset
+  have hwin := clusterWindow_le_markedConnection G hloc x (2^j) (2^(j+1)) (pc G) (dyadicGhost j)
+    (by rw [dyadicGhost_volume])
+  have htra := markedConnection_power_transfer G Q x d hd
+    (fun v => (incident_filter_card_le_degree G hloc Q v).trans (hdeg v)) p (pc G) (dyadicGhost j)
+    hq0 hq1 hp (dyadicGhost_pos j) a ha he
+  have hsub := markedConnection_le_susceptibility G hloc hqt Q x (dyadicGhost j) p hpc
+  have hid : (dyadicGhost j:ℝ)*((dyadicGhost j:ℝ)^a)^(-b) = (dyadicGhost j:ℝ)^(1-a*b) := by
+    rw [←Real.rpow_mul (dyadicGhost j).2.1]
+    calc
+      _ = (dyadicGhost j:ℝ)^1*(dyadicGhost j:ℝ)^(a*(-b)) := by rw [Real.rpow_one]
+      _ = (dyadicGhost j:ℝ)^(1+a*(-b)) := (Real.rpow_add (dyadicGhost_pos j) _ _).symm
+      _ = _ := by congr 1; ring
+  calc
+    _ ≤ 2*markedConnectionPolynomial G Q x (dyadicGhost j) (pc G) := hwin
+    _ ≤ 2*(B*markedConnectionPolynomial G Q x (dyadicGhost j) p) :=
+      mul_le_mul_of_nonneg_left htra (by norm_num)
+    _ ≤ 2*(B*((dyadicGhost j:ℝ)*susceptibility G p x)) :=
+      mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_left hsub hB.le) (by norm_num)
+    _ ≤ 2*(B*((dyadicGhost j:ℝ)*(C*((pc G:ℝ)-(p:ℝ))^(-b)))) :=
+      mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_left
+        (mul_le_mul_of_nonneg_left (hχp hpc x) (dyadicGhost j).2.1) hB.le) (by norm_num)
+    _ = 2*B*C*((dyadicGhost j:ℝ)*((dyadicGhost j:ℝ)^a)^(-b)) := by rw [he]; ring
+    _ = _ := by rw [hid]
+
+ 
+
+theorem critical_power_integrable_of_dyadic_bound (G : BondGraph V E) (hloc : LocallyFinite G)
+    (p : unitInterval) (α θ K : ℝ) (hα : 0 < α) (hθ : α < θ)
+    (hbound : ∀ᶠ j : ℕ in atTop, ∀ x,
+      clusterWindowProbability G p x (2^j) (2^(j+1)) ≤ K*(dyadicGhost j:ℝ)^θ) :
+    ∀ x, Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^α) (law p) := by
+  intro x
+  apply integrable_cluster_power_of_dyadic G hloc p x α hα
+  have hs : Summable (fun j : ℕ => K*((2:ℝ)^(α-θ))^j) := by
+    apply Summable.mul_left
+    apply summable_geometric_of_norm_lt_one
+    rw [Real.norm_of_nonneg (Real.rpow_nonneg (by norm_num) _)]
+    exact Real.rpow_lt_one_of_one_lt_of_neg (by norm_num) (sub_neg.mpr hθ)
+  apply hs.of_norm_bounded_eventually
+  rw [Nat.cofinite_eq_atTop]
+  filter_upwards [hbound] with j hj
+  rw [Real.norm_of_nonneg (mul_nonneg (Real.rpow_nonneg (by norm_num) _)
+    (clusterWindowProbability_nonneg G p x _ _))]
+  calc
+    _ ≤ (2:ℝ)^(α*(j:ℝ))*(K*(dyadicGhost j:ℝ)^θ) :=
+      mul_le_mul_of_nonneg_left (hj x) (Real.rpow_nonneg (by norm_num) _)
+    _ = K*((2:ℝ)^(α-θ))^j := by
+      rw [dyadicGhost_rpow,←Real.rpow_natCast,←Real.rpow_mul (by norm_num : (0:ℝ) ≤ 2)]
+      rw [show (α-θ)*(j:ℝ) = α*(j:ℝ)+(-θ*(j:ℝ)) by ring,Real.rpow_add (by norm_num : (0:ℝ)<2)]
+      ring
+
+end CriticalDyadicTail
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section MomentFromGrowth
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem critical_moment_of_susceptibility_growth [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (hq0 : 0 < (pc G:ℝ)) (hq1 : (pc G:ℝ) < 1)
+    (hgrowth : ∀ γ : ℝ, 0 < γ → γ < 1 → ∃ C : ℝ, 0 < C ∧
+      ∀ᶠ p in nhds (pc G), p < pc G → ∀ x,
+        susceptibility G p x ≤ C*((pc G:ℝ)-(p:ℝ))^(-1/(1-γ)))
+    (α : ℝ) (hα : 0 < α) (hα1 : α < 1/2) :
+    ∀ x, Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^α) (law (pc G)) := by
+  let γ := (1/2-α)/2
+  have hγ : 0 < γ := by dsimp [γ]; linarith
+  have hγ1 : γ < 1 := by dsimp [γ]; linarith
+  have hprod : (1/2:ℝ) < (1-γ)*(1-α) := by
+    have : 0 < γ*α := mul_pos hγ hα
+    dsimp [γ] at *
+    nlinarith
+  obtain ⟨a,ha,hap⟩ := exists_between hprod
+  have hden : 0 < 1-γ := sub_pos.mpr hγ1
+  obtain ⟨C,hC,hnear⟩ := hgrowth γ hγ hγ1
+  have he : -(1/(1-γ)) = -1/(1-γ) := by ring
+  obtain ⟨K,hK,hbound⟩ := critical_dyadic_bound_of_susceptibility G hloc hqt hq0 hq1
+    C (1/(1-γ)) a hC ha (by simpa only [he] using hnear)
+  apply critical_power_integrable_of_dyadic_bound G hloc (pc G) α (1-a*(1/(1-γ))) K hα _ hbound
+  have : a*(1/(1-γ)) < 1-α := by
+    rw [mul_one_div]
+    apply (div_lt_iff₀ hden).mpr
+    nlinarith [hap]
+  linarith
+
+end MomentFromGrowth
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section CriticalRootMoments
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem critical_root_power_integrable
+    [Countable V] [Countable E] [Nonempty V] [Nonempty E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hconn : Connected G)
+    (hqt : QuasiTransitive G) (hh : 0 < hV G)
+    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
+    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
+    (hp₀ : 0 < (pc G:ℝ)) (hp₁ : (pc G:ℝ) < 1)
+    (hfin : ∀ i, ∀ᵐ ω ∂law (pc G), (cluster G ω (root i)).Finite)
+    (α : ℝ) (hα : 0 < α) (hα1 : α < 1/2) :
+    ∀ i, Integrable (fun ω => ((cluster G ω (root i)).ncard:ℝ)^α) (law (pc G)) := by
+  by_contra hbad
+  have hdiv : ∃ i, ¬ Integrable (fun ω => ((cluster G ω (root i)).ncard:ℝ)^α) (law (pc G)) :=
+    not_forall.mp hbad
+  obtain ⟨β,hαβ,hβ⟩ := exists_between hα1
+  let ε := (1/2-β)/2
+  have hε : 0 < ε := by dsimp [ε]; linarith
+  have hε1 : ε < 1/2 := by dsimp [ε]; linarith
+  have hsmall : β+(1/2+ε) < 1 := by dsimp [ε]; linarith
+  have hg := susceptibility_growth_of_divergent_moment G hloc hconn hqt hh root weight Γ hΓ htr
+    hcov hw hw1 hp₀ hp₁ hfin α β ε hα hαβ hε hε1 hsmall hdiv
+  have hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i) := by
+    intro x
+    obtain ⟨i,a,ha,hax⟩ := hcov x
+    obtain ⟨b,hab⟩ := hΓ ha
+    exact ⟨i,a,b,hab,hax⟩
+  let c : ℝ := ∑ i, 1/weight i
+  have hc : 0 < c := Finset.sum_pos (fun i _ => one_div_pos.mpr (hw i)) Finset.univ_nonempty
+  have hmin (i : ι) : 1 ≤ c*weight i := by
+    apply (div_le_iff₀ (hw i)).mp
+    exact Finset.single_le_sum (fun j _ => (one_div_pos.mpr (hw j)).le) (Finset.mem_univ i)
+  have hall := critical_moment_of_susceptibility_growth G hloc hqt hp₀ hp₁ (by
+    intro γ hγ hγ1
+    obtain ⟨C,hC,hnear⟩ := hg γ hγ hγ1
+    refine ⟨c*C,mul_pos hc hC,?_⟩
+    filter_upwards [hnear] with p hp
+    intro hpc x
+    calc
+      _ ≤ c*rootSusceptibility G root weight p :=
+        susceptibility_le_rootSusceptibility G root weight (fun i => (hw i).le) hcover c hc.le hmin p x
+      _ ≤ c*(C*((pc G:ℝ)-(p:ℝ))^(-1/(1-γ))) := mul_le_mul_of_nonneg_left (hp hpc) hc.le
+      _ = _ := by ring) α hα hα1
+  obtain ⟨i,hi⟩ := hdiv
+  exact hi (hall (root i))
+
+end CriticalRootMoments
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section UniformFractionalMoments
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem measurable_cluster_power (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) (α : ℝ) :
+    Measurable (fun ω : Set E => ((cluster G ω x).ncard:ℝ)^α) :=
+  (measurable_of_countable (fun n : ℕ => (n:ℝ)^α)).comp (measurable_cluster_ncard G hloc x)
+
+theorem integrable_cluster_power_covariant [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (p : unitInterval)
+    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (x : V) (α : ℝ) :
+    Integrable (fun ω => ((cluster G ω (a x)).ncard:ℝ)^α) (law p) ↔
+      Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^α) (law p) := by
+  have h := integrable_map_measure (μ := law p)
+    (measurable_cluster_power G hloc x α).aestronglyMeasurable
+    (measurable_set_preimage b).aemeasurable
+  rw [law_map_preimage_equiv] at h
+  have he : ((fun ω : Set E => ((cluster G ω x).ncard:ℝ)^α) ∘ (fun ω => b ⁻¹' ω)) =
+      (fun ω : Set E => ((cluster G ω (a x)).ncard:ℝ)^α) := by
+    funext ω
+    dsimp only [Function.comp_apply]
+    rw [cluster_ncard_covariant G hab]
+  rw [he] at h
+  exact h.symm
+
+theorem integral_cluster_power_covariant [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (p : unitInterval)
+    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (x : V) (α : ℝ) :
+    (∫ ω, ((cluster G ω (a x)).ncard:ℝ)^α ∂law p) =
+      ∫ ω, ((cluster G ω x).ncard:ℝ)^α ∂law p := by
+  have h := integral_map (μ := law p) (measurable_set_preimage b).aemeasurable
+    (measurable_cluster_power G hloc x α).aestronglyMeasurable
+  rw [law_map_preimage_equiv] at h
+  simpa only [cluster_ncard_covariant G hab] using! h.symm
+
+ 
+theorem uniform_cluster_power_of_root [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (p : unitInterval)
+    {ι : Type*} [Fintype ι] (root : ι → V)
+    (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i)) (α : ℝ)
+    (hi : ∀ i, Integrable (fun ω => ((cluster G ω (root i)).ncard:ℝ)^α) (law p)) :
+    ∃ C : ℝ, 0 < C ∧ ∀ x,
+      Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^α) (law p) ∧
+      (∫ ω, ((cluster G ω x).ncard:ℝ)^α ∂law p) ≤ C := by
+  let C := 1+∑ i, ∫ ω, ((cluster G ω (root i)).ncard:ℝ)^α ∂law p
+  have hn (x : V) : 0 ≤ ∫ ω, ((cluster G ω x).ncard:ℝ)^α ∂law p :=
+    integral_nonneg (fun _ => Real.rpow_nonneg (Nat.cast_nonneg _) _)
+  have hC : 0 < C := by dsimp [C]; linarith [Finset.sum_nonneg (fun i (_ : i ∈ Finset.univ) => hn (root i))]
+  refine ⟨C,hC,?_⟩
+  intro x
+  obtain ⟨i,a,b,hab,rfl⟩ := hcover x
+  refine ⟨(integrable_cluster_power_covariant G hloc p hab (root i) α).mpr (hi i),?_⟩
+  rw [integral_cluster_power_covariant G hloc p hab]
+  have hh := Finset.single_le_sum (fun j (_ : j ∈ Finset.univ) => hn (root j)) (Finset.mem_univ i)
+  dsimp [C]
+  linarith
+
+end UniformFractionalMoments
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section FractionalMomentCoupling
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem cluster_power_mono_integral [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) (p q : unitInterval) (hpq : p ≤ q)
+    (α : ℝ) (hα : 0 ≤ α)
+    (hfin : ∀ᵐ ω ∂law q, (cluster G ω x).Finite)
+    (hi : Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^α) (law q)) :
+    Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^α) (law p) ∧
+      (∫ ω, ((cluster G ω x).ncard:ℝ)^α ∂law p) ≤
+      ∫ ω, ((cluster G ω x).ncard:ℝ)^α ∂law q := by
+  let f (r : unitInterval) (labels : E → unitInterval) := ((cluster G (openEdges labels r) x).ncard:ℝ)^α
+  have hm (r : unitInterval) : Measurable (f r) :=
+    (measurable_cluster_power G hloc x α).comp (measurable_openEdges r)
+  have hmap (r : unitInterval) :
+      Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^α) (law r) ↔ Integrable (f r) labelLaw := by
+    rw [←map_openEdges r]
+    exact integrable_map_measure (measurable_cluster_power G hloc x α).aestronglyMeasurable
+      (measurable_openEdges r).aemeasurable
+  have hf : ∀ᵐ labels ∂(labelLaw : Measure (E → unitInterval)), (cluster G (openEdges labels q) x).Finite := by
+    apply ae_of_ae_map (μ := (labelLaw : Measure (E → unitInterval)))
+      (p := fun ω : Set E => (cluster G ω x).Finite) (measurable_openEdges q).aemeasurable
+    rwa [map_openEdges]
+  have hle : f p ≤ᵐ[labelLaw] f q := by
+    filter_upwards [hf] with labels hlabels
+    apply Real.rpow_le_rpow (Nat.cast_nonneg _) _ hα
+    exact_mod_cast Set.ncard_le_ncard (show cluster G (openEdges labels p) x ⊆ cluster G (openEdges labels q) x from
+      fun y hy => conn_mono G (openEdges_mono labels hpq) hy) hlabels
+  have hiq := (hmap q).mp hi
+  have hip : Integrable (f p) labelLaw := hiq.mono' (hm p).aestronglyMeasurable (by
+    filter_upwards [hle] with labels hl
+    rwa [Real.norm_of_nonneg (Real.rpow_nonneg (Nat.cast_nonneg _) _)] )
+  refine ⟨(hmap p).mpr hip,?_⟩
+  have he (r : unitInterval) : (∫ labels, f r labels ∂labelLaw) =
+      ∫ ω, ((cluster G ω x).ncard:ℝ)^α ∂law r := by
+    rw [←map_openEdges r]
+    exact (integral_map (measurable_openEdges r).aemeasurable
+      (measurable_cluster_power G hloc x α).aestronglyMeasurable).symm
+  rw [←he p,←he q]
+  exact integral_mono_ae hip hiq hle
+
+end FractionalMomentCoupling
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {E : Type v}
+section DecisionCovarianceAlgebra
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def finiteCovariance (Q : Finset E) (p : ℝ) (f g : Finset E → ℝ) : ℝ :=
+  bernoulliPolynomial Q (fun ω => f ω*g ω) p - bernoulliPolynomial Q f p*bernoulliPolynomial Q g p
+
+noncomputable def coordinateAverage (e : E) (p : ℝ) (f : Finset E → ℝ) (ω : Finset E) : ℝ :=
+  (1-p)*f ω+p*f (insert e ω)
+
+noncomputable def coordinateVariation (Q : Finset E) (p : ℝ) (f : Finset E → ℝ) (e : E) : ℝ :=
+  bernoulliPolynomial (Q.erase e) (fun ω => |f (insert e ω)-f ω|) p
+
+theorem bernoulliPolynomial_abs (Q : Finset E) (p : unitInterval) (f : Finset E → ℝ) :
+    |bernoulliPolynomial Q f p| ≤ bernoulliPolynomial Q (fun ω => |f ω|) p := by
+  rw [abs_le]
+  constructor
+  · have h := bernoulliPolynomial_mono Q (fun ω => -|f ω|) f p p.2
+      (fun _ _ => neg_abs_le _)
+    have he : bernoulliPolynomial Q (fun ω => -|f ω|) p =
+        -bernoulliPolynomial Q (fun ω => |f ω|) p := by
+      simp only [bernoulliPolynomial,neg_mul,Finset.sum_neg_distrib]
+    rwa [he] at h
+  · exact bernoulliPolynomial_mono Q f (fun ω => |f ω|) p p.2 (fun _ _ => le_abs_self _)
+
+theorem coordinateVariation_nonneg (Q : Finset E) (p : unitInterval) (f : Finset E → ℝ) (e : E) :
+    0 ≤ coordinateVariation Q p f e :=
+  bernoulliPolynomial_nonneg _ _ p p.2 (fun _ _ => abs_nonneg _)
+
+ 
+theorem finiteCovariance_first_coordinate (S : Finset E) (e : E) (he : e ∉ S)
+    (p : ℝ) (f g : Finset E → ℝ) :
+    finiteCovariance (insert e S) p f g =
+      (1-p)*finiteCovariance S p (coordinateAverage e p f) g +
+      p*finiteCovariance S p (coordinateAverage e p f) (fun ω => g (insert e ω)) +
+      p*(1-p)*bernoulliPolynomial S
+        (fun ω => (f (insert e ω)-f ω)*(g (insert e ω)-g ω)) p := by
+  unfold finiteCovariance
+  rw [bernoulliPolynomial_insert S e he,bernoulliPolynomial_insert S e he,
+    bernoulliPolynomial_insert S e he]
+  have hav := bernoulliPolynomial_affine S f (fun ω => f (insert e ω)) (1-p) p p
+  change bernoulliPolynomial S (coordinateAverage e p f) p = _ at hav
+  rw [hav]
+  have h0 : bernoulliPolynomial S (fun ω => coordinateAverage e p f ω*g ω) p =
+      (1-p)*bernoulliPolynomial S (fun ω => f ω*g ω) p+
+      p*bernoulliPolynomial S (fun ω => f (insert e ω)*g ω) p := by
+    simpa only [coordinateAverage,add_mul,mul_assoc] using
+      bernoulliPolynomial_affine S (fun ω => f ω*g ω) (fun ω => f (insert e ω)*g ω) (1-p) p p
+  have h1 : bernoulliPolynomial S (fun ω => coordinateAverage e p f ω*g (insert e ω)) p =
+      (1-p)*bernoulliPolynomial S (fun ω => f ω*g (insert e ω)) p+
+      p*bernoulliPolynomial S (fun ω => f (insert e ω)*g (insert e ω)) p := by
+    simpa only [coordinateAverage,add_mul,mul_assoc] using
+      bernoulliPolynomial_affine S (fun ω => f ω*g (insert e ω)) (fun ω => f (insert e ω)*g (insert e ω)) (1-p) p p
+  rw [h0,h1]
+  have herr : bernoulliPolynomial S (fun ω => (f (insert e ω)-f ω)*(g (insert e ω)-g ω)) p =
+      bernoulliPolynomial S (fun ω => f (insert e ω)*g (insert e ω)) p-
+      bernoulliPolynomial S (fun ω => f (insert e ω)*g ω) p-
+      bernoulliPolynomial S (fun ω => f ω*g (insert e ω)) p+
+      bernoulliPolynomial S (fun ω => f ω*g ω) p := by
+    simp only [bernoulliPolynomial]
+    simp only [←Finset.sum_sub_distrib,←Finset.sum_add_distrib]
+    apply Finset.sum_congr rfl
+    intro ω _
+    ring
+  rw [herr]
+  ring
+
+end DecisionCovarianceAlgebra
+end Percolation
+
+ 
+namespace Percolation
+universe v
+variable {E : Type v}
+section CoordinateAveragingBounds
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem coordinateVariation_average_le (S : Finset E) (e j : E) (he : e ∉ S) (hje : j ≠ e)
+    (p : unitInterval) (f : Finset E → ℝ) :
+    coordinateVariation S p (coordinateAverage e p f) j ≤ coordinateVariation (insert e S) p f j := by
+  unfold coordinateVariation
+  rw [Finset.erase_insert_of_ne hje.symm,bernoulliPolynomial_insert _ e (by simp [he])]
+  let f₀ := fun ω : Finset E => |f (insert j ω)-f ω|
+  let f₁ := fun ω : Finset E => |f (insert j (insert e ω))-f (insert e ω)|
+  change _ ≤ (1-(p:ℝ))*bernoulliPolynomial (S.erase j) f₀ p+
+    (p:ℝ)*bernoulliPolynomial (S.erase j) f₁ p
+  rw [←bernoulliPolynomial_affine]
+  apply bernoulliPolynomial_mono _ _ _ p p.2
+  intro ω _
+  have hid : coordinateAverage e p f (insert j ω)-coordinateAverage e p f ω =
+      (1-(p:ℝ))*(f (insert j ω)-f ω)+(p:ℝ)*(f (insert j (insert e ω))-f (insert e ω)) := by
+    simp only [coordinateAverage,Finset.insert_comm e j]
+    ring
+  rw [hid]
+  calc
+    _ ≤ |(1-(p:ℝ))*(f (insert j ω)-f ω)|+|(p:ℝ)*(f (insert j (insert e ω))-f (insert e ω))| := abs_add_le _ _
+    _ = _ := by rw [abs_mul,abs_mul,abs_of_nonneg (sub_nonneg.mpr p.2.2),abs_of_nonneg p.2.1]
+
+ 
+
+theorem finiteCovariance_first_bound (S : Finset E) (e : E) (he : e ∉ S)
+    (p : unitInterval) (f g : Finset E → ℝ)
+    (hg : ∀ ω ⊆ insert e S, g ω ∈ Set.Icc (0:ℝ) 1) :
+    |finiteCovariance (insert e S) p f g| ≤
+      (1-(p:ℝ))*|finiteCovariance S p (coordinateAverage e p f) g|+
+      (p:ℝ)*|finiteCovariance S p (coordinateAverage e p f) (fun ω => g (insert e ω))|+
+      (p:ℝ)*(1-(p:ℝ))*coordinateVariation (insert e S) p f e := by
+  have hgood : |bernoulliPolynomial S (fun ω => (f (insert e ω)-f ω)*(g (insert e ω)-g ω)) p| ≤
+      coordinateVariation (insert e S) p f e := by
+    apply (bernoulliPolynomial_abs S p _).trans
+    unfold coordinateVariation
+    rw [Finset.erase_insert he]
+    apply bernoulliPolynomial_mono _ _ _ p p.2
+    intro ω hω
+    rw [abs_mul]
+    have h₀ := hg ω (hω.trans (Finset.subset_insert _ _))
+    have h₁ := hg (insert e ω) (Finset.insert_subset_insert e hω)
+    have hd : |g (insert e ω)-g ω| ≤ 1 := by rw [abs_le]; constructor <;> linarith [h₀.1,h₀.2,h₁.1,h₁.2]
+    simpa only [mul_one] using mul_le_mul_of_nonneg_left hd (abs_nonneg (f (insert e ω)-f ω))
+  rw [finiteCovariance_first_coordinate S e he]
+  calc
+    _ ≤ |(1-(p:ℝ))*finiteCovariance S p (coordinateAverage e p f) g+
+          (p:ℝ)*finiteCovariance S p (coordinateAverage e p f) (fun ω => g (insert e ω))|+
+        |(p:ℝ)*(1-(p:ℝ))*bernoulliPolynomial S (fun ω => (f (insert e ω)-f ω)*(g (insert e ω)-g ω)) p| := abs_add_le _ _
+    _ ≤ |(1-(p:ℝ))*finiteCovariance S p (coordinateAverage e p f) g|+
+          |(p:ℝ)*finiteCovariance S p (coordinateAverage e p f) (fun ω => g (insert e ω))|+
+        |(p:ℝ)*(1-(p:ℝ))*bernoulliPolynomial S (fun ω => (f (insert e ω)-f ω)*(g (insert e ω)-g ω)) p| :=
+      add_le_add (abs_add_le _ _) le_rfl
+    _ ≤ _ := by
+      rw [abs_mul,abs_mul,abs_mul,abs_of_nonneg (sub_nonneg.mpr p.2.2),abs_of_nonneg p.2.1,
+        abs_of_nonneg (mul_nonneg p.2.1 (sub_nonneg.mpr p.2.2))]
+      exact add_le_add le_rfl (mul_le_mul_of_nonneg_left hgood (mul_nonneg p.2.1 (sub_nonneg.mpr p.2.2)))
+
+end CoordinateAveragingBounds
+end Percolation
+
+ 
+namespace Percolation
+universe v
+variable {E : Type v}
+section EdgeDecisionTrees
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+inductive EdgeQueryTree (E : Type v) where
+  | done (b : Bool)
+  | query (e : E) (closed opened : EdgeQueryTree E)
+
+namespace EdgeQueryTree
+
+def Valid (Q : Finset E) : EdgeQueryTree E → Prop
+  | .done _ => True
+  | .query e t₀ t₁ => e ∈ Q ∧ Valid (Q.erase e) t₀ ∧ Valid (Q.erase e) t₁
+
+noncomputable def eval (ω : Finset E) : EdgeQueryTree E → Bool
+  | .done b => b
+  | .query e t₀ t₁ => if e ∈ ω then eval ω t₁ else eval ω t₀
+
+noncomputable def value (t : EdgeQueryTree E) (ω : Finset E) : ℝ := if t.eval ω then 1 else 0
+
+noncomputable def reads (ω : Finset E) (i : E) : EdgeQueryTree E → Prop
+  | .done _ => False
+  | .query e t₀ t₁ => i = e ∨ if e ∈ ω then reads ω i t₁ else reads ω i t₀
+
+noncomputable def revealment (p : ℝ) (i : E) : EdgeQueryTree E → ℝ
+  | .done _ => 0
+  | .query e t₀ t₁ => if i = e then 1 else (1-p)*revealment p i t₀+p*revealment p i t₁
+
+theorem value_mem (t : EdgeQueryTree E) (ω : Finset E) : t.value ω ∈ Set.Icc (0:ℝ) 1 := by
+  unfold value
+  split <;> simp
+
+theorem revealment_nonneg (p : unitInterval) (i : E) (t : EdgeQueryTree E) : 0 ≤ t.revealment p i := by
+  induction t with
+  | done => rfl
+  | query e t₀ t₁ ih₀ ih₁ =>
+    simp only [revealment]
+    split_ifs
+    · norm_num
+    · exact add_nonneg (mul_nonneg (sub_nonneg.mpr p.2.2) ih₀) (mul_nonneg p.2.1 ih₁)
+
+theorem eval_congr (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
+    {ω η : Finset E} (h : ∀ e ∈ Q, e ∈ ω ↔ e ∈ η) : t.eval ω = t.eval η := by
+  induction t generalizing Q with
+  | done => rfl
+  | query e t₀ t₁ ih₀ ih₁ =>
+    simp only [eval,h e ht.1]
+    split
+    · exact ih₁ ht.2.2 (fun a ha => h a (Finset.mem_of_mem_erase ha))
+    · exact ih₀ ht.2.1 (fun a ha => h a (Finset.mem_of_mem_erase ha))
+
+theorem eval_insert (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
+    {e : E} (he : e ∉ Q) (ω : Finset E) : t.eval (insert e ω) = t.eval ω := by
+  apply eval_congr t ht
+  intro a ha
+  simp [show a ≠ e from fun h => he (h ▸ ha)]
+
+theorem value_insert (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
+    {e : E} (he : e ∉ Q) (ω : Finset E) : t.value (insert e ω) = t.value ω := by
+  simp only [value,eval_insert t ht he]
+
+theorem reads_congr (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
+    {ω η : Finset E} (h : ∀ e ∈ Q, e ∈ ω ↔ e ∈ η) (i : E) : t.reads ω i ↔ t.reads η i := by
+  induction t generalizing Q with
+  | done => rfl
+  | query e t₀ t₁ ih₀ ih₁ =>
+    simp only [reads,h e ht.1]
+    split
+    · rw [ih₁ ht.2.2 (fun a ha => h a (Finset.mem_of_mem_erase ha))]
+    · rw [ih₀ ht.2.1 (fun a ha => h a (Finset.mem_of_mem_erase ha))]
+
+theorem reads_mem (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
+    (ω : Finset E) (i : E) (hi : t.reads ω i) : i ∈ Q := by
+  induction t generalizing Q with
+  | done => exact hi.elim
+  | query e t₀ t₁ ih₀ ih₁ =>
+    rcases hi with rfl | hi
+    · exact ht.1
+    · split_ifs at hi
+      · exact Finset.mem_of_mem_erase (ih₁ ht.2.2 hi)
+      · exact Finset.mem_of_mem_erase (ih₀ ht.2.1 hi)
+
+end EdgeQueryTree
+end EdgeDecisionTrees
+end Percolation
+
+ 
+namespace Percolation
+universe v
+variable {E : Type v}
+section EdgeTreeCovariance
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem finiteCovariance_congr_right (Q : Finset E) (p : ℝ) (f : Finset E → ℝ)
+    {g h : Finset E → ℝ} (he : ∀ ω ⊆ Q, g ω = h ω) :
+    finiteCovariance Q p f g = finiteCovariance Q p f h := by
+  unfold finiteCovariance
+  rw [bernoulliPolynomial_congr Q (fun ω hω => congrArg (f ω * ·) (he ω hω)),
+    bernoulliPolynomial_congr Q he]
+
+namespace EdgeQueryTree
+
+ 
+
+theorem covariance_bound (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
+    (p : unitInterval) (f : Finset E → ℝ) :
+    |finiteCovariance Q p f t.value| ≤
+      ∑ i ∈ Q, t.revealment p i * ((p:ℝ)*(1-(p:ℝ))*coordinateVariation Q p f i) := by
+  induction t generalizing Q f with
+  | done b =>
+    have hv : (done b : EdgeQueryTree E).value = (fun _ => if b then (1:ℝ) else 0) := rfl
+    rw [hv]
+    cases b
+    · simp [finiteCovariance,revealment]
+    · simp [finiteCovariance,revealment,bernoulliPolynomial_const]
+  | query e t₀ t₁ ih₀ ih₁ =>
+    have he := ht.1
+    have heS : e ∉ Q.erase e := Finset.notMem_erase _ _
+    have hQ : insert e (Q.erase e) = Q := Finset.insert_erase he
+    let fav := coordinateAverage e (p:ℝ) f
+    have hcov₀ : finiteCovariance (Q.erase e) p fav (query e t₀ t₁).value =
+        finiteCovariance (Q.erase e) p fav t₀.value := by
+      apply finiteCovariance_congr_right
+      intro ω hω
+      have heω : e ∉ ω := fun h => heS (hω h)
+      simp only [value,eval,ite_eq_right heω]
+    have hcov₁ : finiteCovariance (Q.erase e) p fav (fun ω => (query e t₀ t₁).value (insert e ω)) =
+        finiteCovariance (Q.erase e) p fav t₁.value := by
+      apply finiteCovariance_congr_right
+      intro ω _
+      simp only [value,eval,Finset.mem_insert_self,ite_true,eval_insert t₁ ht.2.2 heS]
+    have hav (i : E) (hi : i ∈ Q.erase e) :
+        coordinateVariation (Q.erase e) p fav i ≤ coordinateVariation Q p f i := by
+      simpa only [hQ] using coordinateVariation_average_le (Q.erase e) e i heS
+        (Finset.ne_of_mem_erase hi) p f
+    have hb₀ : |finiteCovariance (Q.erase e) p fav t₀.value| ≤
+        ∑ i ∈ Q.erase e, t₀.revealment p i * ((p:ℝ)*(1-(p:ℝ))*coordinateVariation Q p f i) := by
+      apply (ih₀ ht.2.1 fav).trans
+      apply Finset.sum_le_sum
+      intro i hi
+      exact mul_le_mul_of_nonneg_left
+        (mul_le_mul_of_nonneg_left (hav i hi) (mul_nonneg p.2.1 (sub_nonneg.mpr p.2.2)))
+        (revealment_nonneg p i t₀)
+    have hb₁ : |finiteCovariance (Q.erase e) p fav t₁.value| ≤
+        ∑ i ∈ Q.erase e, t₁.revealment p i * ((p:ℝ)*(1-(p:ℝ))*coordinateVariation Q p f i) := by
+      apply (ih₁ ht.2.2 fav).trans
+      apply Finset.sum_le_sum
+      intro i hi
+      exact mul_le_mul_of_nonneg_left
+        (mul_le_mul_of_nonneg_left (hav i hi) (mul_nonneg p.2.1 (sub_nonneg.mpr p.2.2)))
+        (revealment_nonneg p i t₁)
+    have hfirst := finiteCovariance_first_bound (Q.erase e) e heS p f (query e t₀ t₁).value
+      (fun ω _ => value_mem _ ω)
+    change _ ≤ (1-(p:ℝ))*|finiteCovariance (Q.erase e) p fav (query e t₀ t₁).value|+
+      (p:ℝ)*|finiteCovariance (Q.erase e) p fav (fun ω => (query e t₀ t₁).value (insert e ω))|+_ at hfirst
+    rw [hQ,hcov₀,hcov₁] at hfirst
+    apply hfirst.trans
+    calc
+      _ ≤ (1-(p:ℝ))*(∑ i ∈ Q.erase e, t₀.revealment p i * ((p:ℝ)*(1-(p:ℝ))*coordinateVariation Q p f i))+
+          (p:ℝ)*(∑ i ∈ Q.erase e, t₁.revealment p i * ((p:ℝ)*(1-(p:ℝ))*coordinateVariation Q p f i))+
+          (p:ℝ)*(1-(p:ℝ))*coordinateVariation Q p f e :=
+        add_le_add (add_le_add (mul_le_mul_of_nonneg_left hb₀ (sub_nonneg.mpr p.2.2))
+          (mul_le_mul_of_nonneg_left hb₁ p.2.1)) le_rfl
+      _ = _ := by
+        rw [←Finset.sum_erase_add _ _ he]
+        congr 1
+        · rw [Finset.mul_sum,Finset.mul_sum,←Finset.sum_add_distrib]
+          apply Finset.sum_congr rfl
+          intro i hi
+          rw [revealment,ite_eq_right (Finset.ne_of_mem_erase hi)]
+          ring
+        · simp [revealment]
+
+end EdgeQueryTree
+end EdgeTreeCovariance
+end Percolation
+
+ 
+namespace Percolation
+universe v
+variable {E : Type v}
+section RevealmentProbability
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace EdgeQueryTree
+
+theorem reads_insert (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
+    {e : E} (he : e ∉ Q) (ω : Finset E) (i : E) : t.reads (insert e ω) i ↔ t.reads ω i := by
+  apply reads_congr t ht
+  intro a ha
+  simp [show a ≠ e from fun h => he (h ▸ ha)]
+
+ 
+theorem revealment_eq_probability (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
+    (p : ℝ) (i : E) :
+    t.revealment p i = bernoulliPolynomial Q (fun ω => if t.reads ω i then 1 else 0) p := by
+  induction t generalizing Q with
+  | done => simp [revealment,reads]
+  | query e t₀ t₁ ih₀ ih₁ =>
+    by_cases hie : i = e
+    · subst i
+      simp [revealment,reads,bernoulliPolynomial_const]
+    · have heS : e ∉ Q.erase e := Finset.notMem_erase _ _
+      conv_rhs => rw [←Finset.insert_erase ht.1]
+      rw [bernoulliPolynomial_insert _ e heS]
+      simp only [revealment,ite_eq_right hie]
+      rw [ih₀ ht.2.1,ih₁ ht.2.2]
+      congr 1
+      · congr 1
+        apply bernoulliPolynomial_congr
+        intro ω hω
+        have heω : e ∉ ω := fun h => heS (hω h)
+        simp only [reads,hie,false_or,ite_eq_right heω]
+      · congr 1
+        apply bernoulliPolynomial_congr
+        intro ω _
+        simp only [reads,hie,false_or,Finset.mem_insert_self,ite_true,reads_insert t₁ ht.2.2 heS]
+
+end EdgeQueryTree
+end RevealmentProbability
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section SeededEdgeSearch
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+noncomputable def seededEdgeSearch (G : BondGraph V E) (x : V) :
+    ℕ → Finset E → Finset V → EdgeQueryTree E
+  | 0, _, R => .done (decide (x ∈ R))
+  | n+1, Q, R =>
+    if hc : SearchCrossing G Q R then
+      let t := searchCrossing G Q R hc
+      .query t.1 (seededEdgeSearch G x n (Q.erase t.1) R)
+        (seededEdgeSearch G x n (Q.erase t.1) (insert t.2.2 R))
+    else .done (decide (x ∈ R))
+
+theorem seededEdgeSearch_valid (G : BondGraph V E) (x : V) (n : ℕ) (Q : Finset E) (R : Finset V) :
+    (seededEdgeSearch G x n Q R).Valid Q := by
+  induction n generalizing Q R with
+  | zero => trivial
+  | succ n ih =>
+    simp only [seededEdgeSearch]
+    split_ifs with hc
+    · exact ⟨(searchCrossing_spec G Q R hc).1,ih _ _,ih _ _⟩
+    · trivial
+
+theorem seedMarked_singleton_of_no_crossing (G : BondGraph V E) (Q : Finset E) (R : Finset V)
+    (x : V) (h : ¬ SearchCrossing G Q R) (ω : Finset E) (hω : ω ⊆ Q) :
+    SeedMarked G (ω : Set E) R {x} ↔ x ∈ R := by
+  constructor
+  · intro hx
+    by_contra hn
+    exact seedMarked_false_of_no_crossing G Q R {x} h (by simpa using hn) ω hω hx
+  · intro hx
+    exact ⟨x,hx,x,Finset.mem_singleton_self x,conn_refl G _ x⟩
+
+ 
+theorem seededEdgeSearch_correct (G : BondGraph V E) (x : V) (n : ℕ) (Q : Finset E) (R : Finset V)
+    (hn : Q.card ≤ n) (ω : Finset E) (hω : ω ⊆ Q) :
+    (seededEdgeSearch G x n Q R).eval ω = true ↔ SeedMarked G (ω : Set E) R {x} := by
+  induction n generalizing Q R ω with
+  | zero =>
+    have hQ : Q = ∅ := Finset.card_eq_zero.mp (by omega)
+    simp only [seededEdgeSearch,EdgeQueryTree.eval,decide_eq_true_eq]
+    exact (seedMarked_singleton_of_no_crossing G Q R x (by simp [SearchCrossing,hQ]) ω hω).symm
+  | succ n ih =>
+    simp only [seededEdgeSearch]
+    split_ifs with hc
+    · obtain ⟨he,ha,hb,hends⟩ := searchCrossing_spec G Q R hc
+      have hcard : (Q.erase (searchCrossing G Q R hc).1).card ≤ n := by
+        rw [Finset.card_erase_of_mem he]; omega
+      simp only [EdgeQueryTree.eval]
+      by_cases heω : (searchCrossing G Q R hc).1 ∈ ω
+      · simp only [ite_eq_left heω]
+        rw [←Finset.insert_erase heω,Finset.coe_insert,seedMarked_insert_edge G _ R {x} _ _ _ ha hends,
+          EdgeQueryTree.eval_insert _ (seededEdgeSearch_valid G x n _ _) (Finset.notMem_erase _ _)]
+        exact ih _ _ hcard _ (Finset.erase_subset_erase _ hω)
+      · simp only [ite_eq_right heω]
+        exact ih _ _ hcard ω (Finset.subset_erase.mpr ⟨hω,heω⟩)
+    · simp only [EdgeQueryTree.eval,decide_eq_true_eq]
+      exact (seedMarked_singleton_of_no_crossing G Q R x hc ω hω).symm
+
+ 
+
+theorem seededEdgeSearch_reads (G : BondGraph V E) (x : V) (n : ℕ) (Q : Finset E) (R : Finset V)
+    (ω : Finset E) (i : E) (hi : (seededEdgeSearch G x n Q R).reads ω i) :
+    ∃ a ∈ G i, ∃ r ∈ R, Conn G (ω : Set E) r a := by
+  induction n generalizing Q R with
+  | zero => exact hi.elim
+  | succ n ih =>
+    simp only [seededEdgeSearch] at hi
+    split_ifs at hi with hc
+    · obtain ⟨he,ha,hb,hends⟩ := searchCrossing_spec G Q R hc
+      rcases hi with rfl | hi
+      · refine ⟨(searchCrossing G Q R hc).2.1,?_,_,ha,conn_refl G _ _⟩
+        rw [hends]; simp
+      · split_ifs at hi with heω
+        · obtain ⟨a,hai,r,hr,hra⟩ := ih _ _ hi
+          rcases Finset.mem_insert.mp hr with rfl | hr
+          · refine ⟨a,hai,(searchCrossing G Q R hc).2.1,ha,?_⟩
+            exact (Relation.ReflTransGen.single ⟨(searchCrossing G Q R hc).1,heω,hends⟩).trans hra
+          · exact ⟨a,hai,r,hr,hra⟩
+        · exact ih _ _ hi
+    · exact hi.elim
+
+end SeededEdgeSearch
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section RandomizedDecisionCovariance
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem finiteCovariance_average_right (Q : Finset E) (F : Finset V) (p h : ℝ)
+    (f : Finset E → ℝ) (g : Finset V → Finset E → ℝ) :
+    finiteCovariance Q p f (fun ω => bernoulliPolynomial F (fun m => g m ω) h) =
+      bernoulliPolynomial F (fun m => finiteCovariance Q p f (g m)) h := by
+  unfold finiteCovariance
+  rw [bernoulliPolynomial_sub,bernoulliPolynomial_const_mul,
+    ←bernoulliPolynomial_commute Q F (fun ω m => g m ω) p h]
+  congr 1
+  calc
+    _ = bernoulliPolynomial Q (fun ω => bernoulliPolynomial F (fun m => f ω*g m ω) h) p := by
+      apply bernoulliPolynomial_congr
+      intro ω _
+      exact (bernoulliPolynomial_const_mul F (fun m => g m ω) (f ω) h).symm
+    _ = _ := bernoulliPolynomial_commute Q F _ p h
+
+ 
+
+theorem EdgeQueryTree.randomized_covariance_bound (Q : Finset E) (F : Finset V)
+    (p h : unitInterval) (f : Finset E → ℝ) (trees : Finset V → EdgeQueryTree E)
+    (ht : ∀ m ⊆ F, (trees m).Valid Q) :
+    |finiteCovariance Q p f (fun ω => bernoulliPolynomial F (fun m => (trees m).value ω) h)| ≤
+      ∑ i ∈ Q, bernoulliPolynomial F (fun m => (trees m).revealment p i) h *
+        ((p:ℝ)*(1-(p:ℝ))*coordinateVariation Q p f i) := by
+  rw [finiteCovariance_average_right]
+  apply (bernoulliPolynomial_abs F h _).trans
+  calc
+    _ ≤ bernoulliPolynomial F (fun m => ∑ i ∈ Q, (trees m).revealment p i *
+        ((p:ℝ)*(1-(p:ℝ))*coordinateVariation Q p f i)) h := by
+      apply bernoulliPolynomial_mono _ _ _ h h.2
+      intro m hm
+      exact (trees m).covariance_bound (ht m hm) p f
+    _ = _ := by simp only [bernoulliPolynomial_sum,bernoulliPolynomial_mul_const]
+
+noncomputable def finiteSeedProbability (G : BondGraph V E) (F : Finset V)
+    (ω : Finset E) (x : V) (h : ℝ) : ℝ :=
+  bernoulliPolynomial F (fun m => if ∃ y ∈ m, Conn G (ω:Set E) x y then 1 else 0) h
+
+theorem seededEdgeSearch_value (G : BondGraph V E) (x : V) (Q : Finset E)
+    (m : Finset V) (ω : Finset E) (hω : ω ⊆ Q) :
+    (seededEdgeSearch G x Q.card Q m).value ω =
+      if ∃ y ∈ m, Conn G (ω:Set E) x y then 1 else 0 := by
+  have hc := seededEdgeSearch_correct G x Q.card Q m le_rfl ω hω
+  have he : SeedMarked G (ω:Set E) m {x} ↔ ∃ y ∈ m, Conn G (ω:Set E) x y := by
+    simp only [SeedMarked,Finset.mem_singleton,exists_eq_left]
+    exact exists_congr (fun y => and_congr_right (fun _ => ⟨conn_symm G _,conn_symm G _⟩))
+  unfold EdgeQueryTree.value
+  simp only [hc,he]
+
+theorem finiteSeedProbability_tree (G : BondGraph V E) (Q : Finset E) (F : Finset V)
+    (x : V) (h : ℝ) (ω : Finset E) (hω : ω ⊆ Q) :
+    finiteSeedProbability G F ω x h =
+      bernoulliPolynomial F (fun m => (seededEdgeSearch G x Q.card Q m).value ω) h := by
+  apply bernoulliPolynomial_congr
+  intro m _
+  exact (seededEdgeSearch_value G x Q m ω hω).symm
+
+end RandomizedDecisionCovariance
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section SeedRevealmentBounds
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem seededEdgeSearch_reads_endpoints (G : BondGraph V E) (x : V) (Q : Finset E)
+    (m : Finset V) (ω : Finset E) (i : E) (a b : V) (he : G i = s(a,b))
+    (hi : (seededEdgeSearch G x Q.card Q m).reads ω i) :
+    (∃ y ∈ m, Conn G (ω:Set E) a y) ∨ (∃ y ∈ m, Conn G (ω:Set E) b y) := by
+  obtain ⟨z,hz,r,hr,hrz⟩ := seededEdgeSearch_reads G x Q.card Q m ω i hi
+  rw [he] at hz
+  rcases (by simpa using hz : z = a ∨ z = b) with rfl | rfl
+  · exact Or.inl ⟨r,hr,conn_symm G _ hrz⟩
+  · exact Or.inr ⟨r,hr,conn_symm G _ hrz⟩
+
+theorem seededEdgeSearch_mean_revealment_le (G : BondGraph V E) (x : V)
+    (Q : Finset E) (F : Finset V) (p h : unitInterval) (i : E) (a b : V) (he : G i = s(a,b)) :
+    bernoulliPolynomial F (fun m => (seededEdgeSearch G x Q.card Q m).revealment p i) h ≤
+      bernoulliPolynomial Q (fun ω => finiteSeedProbability G F ω a h) p +
+      bernoulliPolynomial Q (fun ω => finiteSeedProbability G F ω b h) p := by
+  simp_rw [EdgeQueryTree.revealment_eq_probability _ (seededEdgeSearch_valid G x Q.card Q _)]
+  rw [←bernoulliPolynomial_commute,←bernoulliPolynomial_add]
+  apply bernoulliPolynomial_mono _ _ _ p p.2
+  intro ω _
+  unfold finiteSeedProbability
+  rw [←bernoulliPolynomial_add]
+  apply bernoulliPolynomial_mono _ _ _ h h.2
+  intro m _
+  have hr := seededEdgeSearch_reads_endpoints G x Q m ω i a b he
+  by_cases hread : (seededEdgeSearch G x Q.card Q m).reads ω i
+  · rcases hr hread with ha | hb
+    · simp only [ite_eq_left hread,ite_eq_left ha]
+      split_ifs <;> norm_num
+    · simp only [ite_eq_left hread,ite_eq_left hb]
+      split_ifs <;> norm_num
+  · simp only [ite_eq_right hread]
+    split_ifs <;> norm_num
+
+ 
+
+theorem finiteSeedProbability_component (G : BondGraph V E) (Q : Finset E) (F : Finset V)
+    (x : V) (hF : insert x (searchVertices G Q) ⊆ F)
+    (ω : Finset E) (hω : ω ⊆ Q) (h : ℝ) :
+    finiteSeedProbability G F ω x h = 1-(1-h)^(finiteRootComponent G ω x).card := by
+  let C := finiteRootComponent G ω x
+  have hCF : C ⊆ F := finiteRootComponent_subset G Q F x ω hω hF
+  have he (m : Finset V) : (∃ y ∈ m, Conn G (ω:Set E) x y) ↔ m ∩ C ≠ ∅ := by
+    rw [Finset.nonempty_iff_ne_empty.symm]
+    constructor
+    · rintro ⟨y,hy,hxy⟩
+      exact ⟨y,Finset.mem_inter.mpr ⟨hy,by change y ∈ (C:Set V); simpa only [C,coe_finiteRootComponent,cluster,Set.mem_ofPred_eq] using hxy⟩⟩
+    · rintro ⟨y,hy⟩
+      refine ⟨y,(Finset.mem_inter.mp hy).1,?_⟩
+      have hm : y ∈ (C:Set V) := (Finset.mem_inter.mp hy).2
+      simpa only [C,coe_finiteRootComponent,cluster,Set.mem_ofPred_eq] using hm
+  calc
+    _ = bernoulliPolynomial F (fun m => 1-(if m ∩ C = ∅ then 1 else 0)) h := by
+      apply bernoulliPolynomial_congr
+      intro m _
+      by_cases hm : m ∩ C = ∅
+      · have hn : ¬∃ y ∈ m, Conn G (ω:Set E) x y := fun hc => (he m).mp hc hm
+        simp only [hm,hn,ite_false,ite_true,sub_self]
+      · have hy := (he m).mpr hm
+        simp only [hm,hy,ite_false,ite_true,sub_zero]
+    _ = 1-bernoulliPolynomial F (fun m => if m ∩ C = ∅ then 1 else 0) h := by
+      rw [bernoulliPolynomial_sub,bernoulliPolynomial_const]
+    _ = _ := by rw [bernoulliPolynomial_restrict C F hCF (fun m => if m = ∅ then 1 else 0) h,
+      bernoulliPolynomial_empty_indicator]
+
+end SeedRevealmentBounds
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section GhostFractionalPower
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem one_sub_pow_one_sub_le_power (h : unitInterval) (n : ℕ) (α : ℝ)
+    (hα : 0 ≤ α) (hα1 : α ≤ 1) :
+    1-(1-(h:ℝ))^n ≤ (h:ℝ)^α*(n:ℝ)^α := by
+  rw [←Real.mul_rpow h.2.1 (Nat.cast_nonneg _)]
+  by_cases ht : (h:ℝ)*n ≤ 1
+  · exact (one_sub_pow_one_sub_le h n).trans (Real.self_le_rpow_of_le_one (mul_nonneg h.2.1 (Nat.cast_nonneg _)) ht hα1)
+  · have ht' : 1 ≤ (h:ℝ)*n := (lt_of_not_ge ht).le
+    exact (sub_le_self _ (pow_nonneg (sub_nonneg.mpr h.2.2) _)).trans (Real.one_le_rpow ht' hα)
+
+theorem markedConnection_le_cluster_power (G : BondGraph V E) (Q : Finset E)
+    (x : V) (h p : unitInterval) (α : ℝ) (hα : 0 ≤ α) (hα1 : α ≤ 1)
+    (hfin : ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
+    (hi : Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^α) (law p)) :
+    markedConnectionPolynomial G Q x h p ≤ (h:ℝ)^α*(∫ ω, ((cluster G ω x).ncard:ℝ)^α ∂law p) := by
+  rw [markedConnection_integral,←integral_const_mul]
+  apply integral_mono_ae
+  · exact integrable_finite_local Q p _ (by intro ω; simp only [Set.inter_assoc,Set.inter_self])
+  · exact hi.const_mul _
+  · filter_upwards [hfin] with ω hω
+    apply (one_sub_pow_one_sub_le_power h _ α hα hα1).trans
+    apply mul_le_mul_of_nonneg_left _ (Real.rpow_nonneg h.2.1 _)
+    apply Real.rpow_le_rpow (Nat.cast_nonneg _) _ hα
+    exact_mod_cast Set.ncard_le_ncard (fun y hy => conn_mono G Set.inter_subset_left hy) hω
+
+theorem finiteSeedProbability_expectation (G : BondGraph V E) (Q : Finset E) (F : Finset V)
+    (x : V) (hF : insert x (searchVertices G Q) ⊆ F) (h p : ℝ) :
+    bernoulliPolynomial Q (fun ω => finiteSeedProbability G F ω x h) p =
+      markedConnectionPolynomial G Q x h p := by
+  rw [markedConnection_component_polynomial]
+  apply bernoulliPolynomial_congr
+  intro ω hω
+  exact finiteSeedProbability_component G Q F x hF ω hω h
+
+theorem finiteSeedProbability_le_cluster_power (G : BondGraph V E) (Q : Finset E) (F : Finset V)
+    (x : V) (hF : insert x (searchVertices G Q) ⊆ F) (h p : unitInterval)
+    (α : ℝ) (hα : 0 ≤ α) (hα1 : α ≤ 1)
+    (hfin : ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
+    (hi : Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^α) (law p)) :
+    bernoulliPolynomial Q (fun ω => finiteSeedProbability G F ω x h) p ≤
+      (h:ℝ)^α*(∫ ω, ((cluster G ω x).ncard:ℝ)^α ∂law p) := by
+  rw [finiteSeedProbability_expectation G Q F x hF]
+  exact markedConnection_le_cluster_power G Q x h p α hα hα1 hfin hi
+
+end GhostFractionalPower
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section MonotoneDecisionRusso
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem sum_coordinateVariation_eq_deriv (Q : Finset E) (f : Finset E → ℝ)
+    (hf : Monotone f) (p : ℝ) :
+    (∑ i ∈ Q, coordinateVariation Q p f i) = deriv (bernoulliPolynomial Q f) p := by
+  rw [(hasDerivAt_bernoulliPolynomial Q f p).deriv]
+  apply Finset.sum_congr rfl
+  intro i _
+  apply bernoulliPolynomial_congr
+  intro ω _
+  exact abs_of_nonneg (sub_nonneg.mpr (hf (Finset.subset_insert _ _)))
+
+theorem EdgeQueryTree.randomized_covariance_le_deriv (Q : Finset E) (F : Finset V)
+    (p h : unitInterval) (f : Finset E → ℝ) (hf : Monotone f)
+    (trees : Finset V → EdgeQueryTree E) (ht : ∀ m ⊆ F, (trees m).Valid Q)
+    (R : ℝ) (hR : ∀ i ∈ Q, bernoulliPolynomial F (fun m => (trees m).revealment p i) h ≤ R) :
+    |finiteCovariance Q p f (fun ω => bernoulliPolynomial F (fun m => (trees m).value ω) h)| ≤
+      R*((p:ℝ)*(1-(p:ℝ)))*deriv (bernoulliPolynomial Q f) p := by
+  apply (randomized_covariance_bound Q F p h f trees ht).trans
+  calc
+    _ ≤ ∑ i ∈ Q, R*((p:ℝ)*(1-(p:ℝ))*coordinateVariation Q p f i) := by
+      apply Finset.sum_le_sum
+      intro i hi
+      exact mul_le_mul_of_nonneg_right (hR i hi)
+        (mul_nonneg (mul_nonneg p.2.1 (sub_nonneg.mpr p.2.2)) (coordinateVariation_nonneg Q p f i))
+    _ = _ := by simp only [←mul_assoc,←Finset.mul_sum,sum_coordinateVariation_eq_deriv Q f hf]
+
+ 
+
+theorem seededEdgeSearch_revealment_of_moments (G : BondGraph V E) (Q : Finset E)
+    (x : V) (p h : unitInterval) (α C : ℝ) (hα : 0 ≤ α) (hα1 : α ≤ 1)
+    (hfin : ∀ y, ∀ᵐ ω ∂law p, (cluster G ω y).Finite)
+    (hI : ∀ y, Integrable (fun ω => ((cluster G ω y).ncard:ℝ)^α) (law p))
+    (hC : ∀ y, (∫ ω, ((cluster G ω y).ncard:ℝ)^α ∂law p) ≤ C) :
+    ∀ i ∈ Q, bernoulliPolynomial (insert x (searchVertices G Q))
+      (fun m => (seededEdgeSearch G x Q.card Q m).revealment p i) h ≤ 2*C*(h:ℝ)^α := by
+  intro i hi
+  obtain ⟨⟨a,b⟩,he⟩ := Quot.exists_rep (G i)
+  have he' : G i = s(a,b) := he.symm
+  have hFa : insert a (searchVertices G Q) ⊆ insert x (searchVertices G Q) :=
+    Finset.insert_subset_iff.mpr ⟨Finset.mem_insert_of_mem (mem_searchVertices G hi (by rw [he']; simp)),Finset.subset_insert _ _⟩
+  have hFb : insert b (searchVertices G Q) ⊆ insert x (searchVertices G Q) :=
+    Finset.insert_subset_iff.mpr ⟨Finset.mem_insert_of_mem (mem_searchVertices G hi (by rw [he']; simp)),Finset.subset_insert _ _⟩
+  have ha := finiteSeedProbability_le_cluster_power G Q _ a hFa h p α hα hα1 (hfin a) (hI a)
+  have hb := finiteSeedProbability_le_cluster_power G Q _ b hFb h p α hα hα1 (hfin b) (hI b)
+  have hca := mul_le_mul_of_nonneg_left (hC a) (Real.rpow_nonneg h.2.1 α)
+  have hcb := mul_le_mul_of_nonneg_left (hC b) (Real.rpow_nonneg h.2.1 α)
+  have hr := seededEdgeSearch_mean_revealment_le G x Q (insert x (searchVertices G Q)) p h i a b he'
+  linarith
+
+theorem finiteSeed_covariance_upper (G : BondGraph V E) (Q : Finset E) (x : V)
+    (p h : unitInterval) (α C : ℝ) (hα : 0 ≤ α) (hα1 : α ≤ 1)
+    (hfin : ∀ y, ∀ᵐ ω ∂law p, (cluster G ω y).Finite)
+    (hI : ∀ y, Integrable (fun ω => ((cluster G ω y).ncard:ℝ)^α) (law p))
+    (hC : ∀ y, (∫ ω, ((cluster G ω y).ncard:ℝ)^α ∂law p) ≤ C)
+    (f : Finset E → ℝ) (hf : Monotone f) :
+    |finiteCovariance Q p f (fun ω => finiteSeedProbability G (insert x (searchVertices G Q)) ω x h)| ≤
+      (2*C*(h:ℝ)^α)*((p:ℝ)*(1-(p:ℝ)))*deriv (bernoulliPolynomial Q f) p := by
+  have hc := EdgeQueryTree.randomized_covariance_le_deriv Q (insert x (searchVertices G Q)) p h f hf
+    (fun m => seededEdgeSearch G x Q.card Q m) (fun _ _ => seededEdgeSearch_valid G x Q.card Q _)
+    (2*C*(h:ℝ)^α) (seededEdgeSearch_revealment_of_moments G Q x p h α C hα hα1 hfin hI hC)
+  convert hc using 2
+  apply finiteCovariance_congr_right
+  intro ω hω
+  exact finiteSeedProbability_tree G Q _ x h ω hω
+
+end MonotoneDecisionRusso
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section ClusterTailCovariance
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def finiteClusterTailIndicator (G : BondGraph V E) (x : V) (n : ℕ)
+    (ω : Finset E) : ℝ := if n ≤ (finiteRootComponent G ω x).card then 1 else 0
+
+noncomputable def finiteClusterTail (G : BondGraph V E) (Q : Finset E) (x : V) (n : ℕ) (p : ℝ) : ℝ :=
+  bernoulliPolynomial Q (finiteClusterTailIndicator G x n) p
+
+theorem finiteClusterTailIndicator_mono (G : BondGraph V E) (x : V) (n : ℕ) :
+    Monotone (finiteClusterTailIndicator G x n) := by
+  intro ω η hωη
+  have hc := Finset.card_le_card (finiteRootComponent_mono G hωη x)
+  unfold finiteClusterTailIndicator
+  split_ifs <;> norm_num
+  omega
+
+theorem finiteClusterTail_nonneg (G : BondGraph V E) (Q : Finset E) (x : V) (n : ℕ) (p : unitInterval) :
+    0 ≤ finiteClusterTail G Q x n p := by
+  apply bernoulliPolynomial_nonneg _ _ p p.2
+  intro ω _
+  unfold finiteClusterTailIndicator
+  split_ifs <;> norm_num
+
+theorem finiteClusterTail_deriv_nonneg (G : BondGraph V E) (Q : Finset E) (x : V) (n : ℕ) (p : unitInterval) :
+    0 ≤ deriv (finiteClusterTail G Q x n) p := by
+  change 0 ≤ deriv (bernoulliPolynomial Q (finiteClusterTailIndicator G x n)) p
+  rw [←sum_coordinateVariation_eq_deriv Q _ (finiteClusterTailIndicator_mono G x n)]
+  exact Finset.sum_nonneg (fun i _ => coordinateVariation_nonneg Q p _ i)
+
+ 
+
+theorem finiteSeed_covariance_lower (G : BondGraph V E) (Q : Finset E) (x : V)
+    (n : ℕ) (p h : unitInterval) (hn : 1 ≤ (h:ℝ)*n) (B : ℝ)
+    (hB : markedConnectionPolynomial G Q x h p ≤ B) :
+    (1/2-B)*finiteClusterTail G Q x n p ≤ finiteCovariance Q p
+      (finiteClusterTailIndicator G x n)
+      (fun ω => finiteSeedProbability G (insert x (searchVertices G Q)) ω x h) := by
+  have hprod : (1:ℝ)/2*finiteClusterTail G Q x n p ≤
+      bernoulliPolynomial Q (fun ω => finiteClusterTailIndicator G x n ω*
+        finiteSeedProbability G (insert x (searchVertices G Q)) ω x h) p := by
+    rw [finiteClusterTail,←bernoulliPolynomial_const_mul]
+    apply bernoulliPolynomial_mono _ _ _ p p.2
+    intro ω hω
+    rw [finiteSeedProbability_component G Q _ x le_rfl ω hω]
+    unfold finiteClusterTailIndicator
+    by_cases hsize : n ≤ (finiteRootComponent G ω x).card
+    · simp only [ite_eq_left hsize,mul_one,one_mul]
+      apply one_sub_pow_one_sub_ge_half
+      exact hn.trans (mul_le_mul_of_nonneg_left (by exact_mod_cast hsize) h.2.1)
+    · simp only [ite_eq_right hsize,mul_zero,zero_mul,le_refl]
+  have hmean := finiteSeedProbability_expectation G Q (insert x (searchVertices G Q)) x le_rfl (h:ℝ) p
+  unfold finiteCovariance
+  rw [hmean]
+  change _ ≤ _-finiteClusterTail G Q x n p*markedConnectionPolynomial G Q x h p
+  have hm := mul_le_mul_of_nonneg_left hB (finiteClusterTail_nonneg G Q x n p)
+  nlinarith
+
+end ClusterTailCovariance
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Exploration/Goodness.lean b/lean/OAI/Probability/UnimodularPercolation/Exploration/Goodness.lean
index 4dcf51caf407d855ac6c4582e708c6436e612671..7998b0b4906ce8ba4ff5aa3e5c9b4714c66c136e 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Exploration/Goodness.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Exploration/Goodness.lean
@@ -1,1316 +1,1316 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Exploration.Moments
-
-namespace OAI
-
-/-! Pruning, finite and infinite goodness events and uniform bounds. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-section FiniteCappedRadius
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-noncomputable def pairOpenPivotals (G : BondGraph V E) (Q : Finset E) (x y : V)
-    (ω : Finset E) : Finset E :=
-  Q.filter fun e => e ∈ ω ∧ Conn G (ω : Set E) x y ∧ ¬Conn G (ω.erase e : Set E) x y
-
- 
-def CappedRadiusEvent (G : BondGraph V E) (Q : Finset E) (x : V) (M r : ℕ)
-    (ω : Finset E) : Prop :=
-  (finiteRootComponent G ω x).card ≤ M ∧
-    ∃ y, Conn G (ω : Set E) x y ∧ r ≤ (pairOpenPivotals G Q x y ω).card
-
-noncomputable def finiteCappedRadiusTail (G : BondGraph V E) (Q : Finset E)
-    (x : V) (M r : ℕ) (p : ℝ) : ℝ :=
-  bernoulliPolynomial Q (fun ω => if CappedRadiusEvent G Q x M r ω then 1 else 0) p
-
- 
-theorem sole_root_mark (G : BondGraph V E) (Q : Finset E) (x y : V) (ω : Finset E)
-    (m : Finset V) (hxy : Conn G (ω : Set E) x y)
-    (hm : m ∩ finiteRootComponent G ω x = {y}) :
-    MarkedEvent G ω m x ∧ markedOpenPivotals G Q x ω m = pairOpenPivotals G Q x y ω := by
-  have hyW : y ∈ finiteRootComponent G ω x := by
-    change y ∈ (finiteRootComponent G ω x : Set V)
-    rwa [coe_finiteRootComponent]
-  have hym : y ∈ m := (Finset.mem_inter.mp (hm ▸ Finset.mem_singleton_self y)).1
-  have hmconn (η : Finset E) (hη : η ⊆ ω) :
-      MarkedEvent G η m x ↔ Conn G (η : Set E) x y := by
-    constructor
-    · rintro ⟨z,hzm,hxz⟩
-      have hzW : z ∈ finiteRootComponent G ω x := by
-        change z ∈ (finiteRootComponent G ω x : Set V)
-        rw [coe_finiteRootComponent]
-        exact conn_mono G (by exact_mod_cast hη) hxz
-      have hzy : z = y := Finset.mem_singleton.mp (hm ▸ Finset.mem_inter.mpr ⟨hzm,hzW⟩)
-      rwa [hzy] at hxz
-    · exact fun h => ⟨y,hym,h⟩
-  refine ⟨⟨y,hym,hxy⟩,?_⟩
-  ext e
-  simp only [markedOpenPivotals,pairOpenPivotals,Finset.mem_filter]
-  change (e ∈ Q ∧ e ∈ ω ∧ MarkedEvent G ω m x ∧ ¬MarkedEvent G (ω.erase e) m x) ↔ _
-  rw [hmconn ω Finset.Subset.rfl,hmconn (ω.erase e) (Finset.erase_subset e ω)]
-
- 
-theorem bernoulliPolynomial_exact_indicator (F S : Finset V) (hSF : S ⊆ F) (h : ℝ) :
-    bernoulliPolynomial F (fun m => if m = S then (1:ℝ) else 0) h =
-      h^S.card*(1-h)^(F.card-S.card) := by
-  unfold bernoulliPolynomial
-  rw [Finset.sum_eq_single_of_mem S (Finset.mem_powerset.mpr hSF)]
-  · simp
-  · intro m hm hne
-    simp [hne]
-
- 
-theorem bernoulliPolynomial_sole_root_mark (G : BondGraph V E) (Q : Finset E)
-    (F : Finset V) (x y : V) (ω : Finset E) (hω : ω ⊆ Q)
-    (hF : insert x (searchVertices G Q) ⊆ F) (hxy : Conn G (ω : Set E) x y) (h : ℝ) :
-    bernoulliPolynomial F (fun m => if m ∩ finiteRootComponent G ω x = {y} then (1:ℝ) else 0) h =
-      h*(1-h)^((finiteRootComponent G ω x).card-1) := by
-  rw [bernoulliPolynomial_restrict _ _ (finiteRootComponent_subset G Q F x ω hω hF)
-    (fun a => if a = {y} then (1:ℝ) else 0) h]
-  have hy : y ∈ finiteRootComponent G ω x := by
-    change y ∈ (finiteRootComponent G ω x : Set V)
-    rwa [coe_finiteRootComponent]
-  rw [bernoulliPolynomial_exact_indicator _ {y} (Finset.singleton_subset_iff.mpr hy)]
-  simp
-
- 
-theorem cappedRadius_mark_comparison (G : BondGraph V E) (Q : Finset E) (F : Finset V)
-    (x : V) (M r : ℕ) (p h : unitInterval)
-    (hF : insert x (searchVertices G Q) ⊆ F) :
-    (h:ℝ)*(1-h)^M*finiteCappedRadiusTail G Q x M r p ≤ finiteMarkedTail G Q F x r h p := by
-  unfold finiteCappedRadiusTail finiteMarkedTail
-  rw [← bernoulliPolynomial_const_mul]
-  apply bernoulliPolynomial_mono _ _ _ _ p.2
-  intro ω hω
-  by_cases he : CappedRadiusEvent G Q x M r ω
-  · obtain ⟨hcap,y,hxy,hr⟩ := he
-    have he : CappedRadiusEvent G Q x M r ω := ⟨hcap,y,hxy,hr⟩
-    simp only [he,ite_true,mul_one]
-    calc
-      (h:ℝ)*(1-h)^M ≤ (h:ℝ)*(1-h)^((finiteRootComponent G ω x).card-1) := by
-        apply mul_le_mul_of_nonneg_left _ h.2.1
-        exact pow_le_pow_of_le_one (sub_nonneg.mpr h.2.2) (by linarith [h.2.1])
-          (Nat.le_trans (Nat.sub_le _ _) hcap)
-      _ = bernoulliPolynomial F
-          (fun m => if m ∩ finiteRootComponent G ω x = {y} then (1:ℝ) else 0) h :=
-        (bernoulliPolynomial_sole_root_mark G Q F x y ω hω hF hxy h).symm
-      _ ≤ _ := by
-        apply bernoulliPolynomial_mono _ _ _ _ h.2
-        intro m hm
-        by_cases hs : m ∩ finiteRootComponent G ω x = {y}
-        · obtain ⟨heq,hpiv⟩ := sole_root_mark G Q x y ω m hxy hs
-          have hlarge : r ≤ (markedOpenPivotals G Q x ω m).card := hpiv ▸ hr
-          simp only [hs,heq,hlarge,and_self,ite_true,le_refl]
-        · simp only [hs,ite_false]
-          split_ifs <;> norm_num
-  · simp only [he,ite_false,mul_zero]
-    apply bernoulliPolynomial_nonneg _ _ _ h.2
-    intro m hm
-    split_ifs <;> norm_num
-
-theorem finiteMarkedMoment_zero_le_one (G : BondGraph V E) (Q : Finset E) (F : Finset V)
-    (x : V) (p h : unitInterval) : finiteMarkedMoment G Q F x 0 h p ≤ 1 := by
-  unfold finiteMarkedMoment
-  rw [← bernoulliPolynomial_const Q (1:ℝ) (p:ℝ)]
-  apply bernoulliPolynomial_mono _ _ _ _ p.2
-  intro ω hω
-  rw [← bernoulliPolynomial_const F (1:ℝ) (h:ℝ)]
-  apply bernoulliPolynomial_mono _ _ _ _ h.2
-  intro m hm
-  rw [markedMomentValue_zero]
-  split_ifs <;> norm_num
-
- 
-theorem finiteCappedRadiusTail_exponential [Nonempty V] (G : BondGraph V E)
-    (p h : unitInterval) (hp : (p:ℝ) < 1) (hh : 0 < (h:ℝ)) (hh1 : (h:ℝ) < 1)
-    (d : ℕ) (hd : 0 < d) (Q : Finset E) (x : V)
-    (hdeg : ∀ v, (Q.filter (fun e => v ∈ G e)).card ≤ d)
-    (B : ℝ) (hB : 1 ≤ B)
-    (hAB : (Real.sqrt ((d:ℝ)/2)/(1-p))*Real.sqrt (Real.log (1/(h:ℝ))/h) ≤ B) (M r : ℕ) :
-    finiteCappedRadiusTail G Q x M r p ≤
-      (2/((h:ℝ)*(1-h)^M))*Real.exp (-(r:ℝ)/(4*B)) := by
-  let F := insert x (searchVertices G Q)
-  have hcomp := cappedRadius_mark_comparison G Q F x M r p h Finset.Subset.rfl
-  have ht := finiteMarkedTail_exponential G p h hp hh d hd Q F x hdeg Finset.Subset.rfl B hB hAB r
-  have hm := finiteMarkedMoment_zero_le_one G Q F x p h
-  have hb : 0 < (h:ℝ)*(1-h)^M := mul_pos hh (pow_pos (sub_pos.mpr hh1) _)
-  have hbound : (h:ℝ)*(1-h)^M*finiteCappedRadiusTail G Q x M r p ≤
-      2*Real.exp (-(r:ℝ)/(4*B)) := by
-    have hpos : 0 ≤ 2*Real.exp (-(r:ℝ)/(4*B)) := by positivity
-    have hl := mul_le_mul_of_nonneg_left hm hpos
-    rw [mul_one] at hl
-    exact hcomp.trans (ht.trans hl)
-  calc
-    _ ≤ (2*Real.exp (-(r:ℝ)/(4*B)))/((h:ℝ)*(1-h)^M) :=
-      (le_div_iff₀ hb).mpr (by simpa only [mul_comm] using hbound)
-    _ = _ := by ring
-
-end FiniteCappedRadius
-
-section PruningGeometry
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem pairOpenPivotals_symm (G : BondGraph V E) (Q : Finset E) (ω : Finset E) (x y : V) :
-    pairOpenPivotals G Q x y ω = pairOpenPivotals G Q y x ω := by
-  ext e
-  simp only [pairOpenPivotals,Finset.mem_filter]
-  have h₁ : Conn G (ω : Set E) x y ↔ Conn G (ω : Set E) y x :=
-    ⟨conn_symm G _,conn_symm G _⟩
-  have h₂ : Conn G (ω.erase e : Set E) x y ↔ Conn G (ω.erase e : Set E) y x :=
-    ⟨conn_symm G _,conn_symm G _⟩
-  rw [h₁,h₂]
-
- 
-theorem pairOpenPivotals_triangle_subset (G : BondGraph V E) (Q : Finset E) (ω : Finset E)
-    (x y z : V) (hxy : Conn G (ω : Set E) x y) (hyz : Conn G (ω : Set E) y z) :
-    pairOpenPivotals G Q x z ω ⊆ pairOpenPivotals G Q x y ω ∪ pairOpenPivotals G Q y z ω := by
-  intro e he
-  obtain ⟨heQ,heω,_,hnxz⟩ := Finset.mem_filter.mp he
-  by_cases hxy' : Conn G (ω.erase e : Set E) x y
-  · exact Finset.mem_union_right _ (Finset.mem_filter.mpr
-      ⟨heQ,heω,hyz,fun hyz' => hnxz (hxy'.trans hyz')⟩)
-  · exact Finset.mem_union_left _ (Finset.mem_filter.mpr ⟨heQ,heω,hxy,hxy'⟩)
-
-theorem pairOpenPivotals_triangle (G : BondGraph V E) (Q : Finset E) (ω : Finset E)
-    (x y z : V) (hxy : Conn G (ω : Set E) x y) (hyz : Conn G (ω : Set E) y z) :
-    (pairOpenPivotals G Q x z ω).card ≤
-      (pairOpenPivotals G Q x y ω).card+(pairOpenPivotals G Q y z ω).card :=
-  (Finset.card_le_card (pairOpenPivotals_triangle_subset G Q ω x y z hxy hyz)).trans
-    (Finset.card_union_le _ _)
-
- 
-theorem conn_no_crossing_internal (G : BondGraph V E) (W : Set V) (ω : Set E)
-    (hc : ∀ e ∈ ω, ¬CutExists G W e) {x y : V} (hx : x ∈ W) (hxy : Conn G ω x y) :
-    Conn G (ω ∩ internalEdges G W) x y ∧ y ∈ W := by
-  induction hxy with
-  | refl => exact ⟨conn_refl _ _ _,hx⟩
-  | @tail a b hxa hab ih =>
-    obtain ⟨e,he,heq⟩ := hab
-    have hb : b ∈ W := by
-      by_contra hn
-      exact hc e he ⟨a,b,heq,ih.2,hn⟩
-    exact ⟨ih.1.tail ⟨e,⟨he,(mem_internalEdges_of_ends G W e a b heq).mpr ⟨ih.2,hb⟩⟩,heq⟩,hb⟩
-
- 
-theorem conn_one_attachment_internal (G : BondGraph V E) (W : Set V) (ω : Set E)
-    (e : E) (he : e ∈ ω) (hec : CutExists G W e)
-    (hone : ∀ f ∈ ω, CutExists G W f → f = e)
-    {x y : V} (hx : x ∈ W) (hy : y ∈ W) (hxy : Conn G ω x y) :
-    Conn G (ω ∩ internalEdges G W) x y := by
-  obtain ⟨a,b,hab,ha,hb⟩ := hec
-  let η := ω \ {e}
-  have hn : ∀ f ∈ η, ¬CutExists G W f := by
-    intro f hf hc
-    exact hf.2 (Set.mem_singleton_iff.mpr (hone f hf.1 hc))
-  have hη (u v : V) (hu : u ∈ W) (huv : Conn G η u v) :=
-    conn_no_crossing_internal G W η hn hu huv
-  have hω : insert e η = ω := by dsimp [η]; ext f; simp [he]
-  have hw : Conn G (insert e η) x y := hω ▸ hxy
-  rcases (conn_insert_iff G η e a b x y hab).mp hw with h | ⟨_,h⟩ | ⟨h,_⟩
-  · exact conn_mono G (show η ∩ internalEdges G W ⊆ ω ∩ internalEdges G W from
-      fun _ hf => ⟨hf.1.1,hf.2⟩) (hη x y hx h).1
-  · exact (hb (hη y b hy (conn_symm G η h)).2).elim
-  · exact (hb (hη x b hx h).2).elim
-
-noncomputable def internalBonds (G : BondGraph V E) (ω : Finset E) (W : Set V) : Finset E :=
-  ω.filter fun e => e ∈ internalEdges G W
-
-@[simp] theorem coe_internalBonds (G : BondGraph V E) (ω : Finset E) (W : Set V) :
-    (internalBonds G ω W : Set E) = (ω : Set E) ∩ internalEdges G W := by
-  ext e; simp [internalBonds]
-
- 
-theorem pairOpenPivotals_one_attachment (G : BondGraph V E) (Q ω : Finset E) (W : Set V)
-    (e : E) (he : e ∈ ω) (hec : CutExists G W e)
-    (hone : ∀ f ∈ ω, CutExists G W f → f = e)
-    {x y : V} (hx : x ∈ W) (hy : y ∈ W) :
-    pairOpenPivotals G Q x y (internalBonds G ω W) ⊆ pairOpenPivotals G Q x y ω := by
-  intro f hf
-  obtain ⟨hfQ,hfω,hxy,hnxy⟩ := Finset.mem_filter.mp hf
-  have hfint : f ∈ internalEdges G W := (Finset.mem_filter.mp hfω).2
-  have hfe : e ≠ f := by
-    intro hef
-    obtain ⟨a,b,hab,ha,hb⟩ := hec
-    have hh : e ∈ internalEdges G W := hef ▸ hfint
-    exact hb ((mem_internalEdges_of_ends G W e a b hab).mp hh).2
-  have he' : e ∈ ω.erase f := Finset.mem_erase.mpr ⟨hfe,he⟩
-  have hcoe : (internalBonds G ω W).erase f = internalBonds G (ω.erase f) W := by
-    ext g; simp [internalBonds]; tauto
-  refine Finset.mem_filter.mpr ⟨hfQ,(Finset.mem_filter.mp hfω).1,?_,?_⟩
-  · exact conn_mono G (by simpa only [coe_internalBonds] using Set.inter_subset_left) hxy
-  · intro h
-    have hh := conn_one_attachment_internal G W (ω.erase f : Set E) e he' hec
-      (fun g hg hc => hone g (Finset.mem_of_mem_erase hg) hc) hx hy h
-    apply hnxy
-    simpa only [hcoe,coe_internalBonds] using hh
-
- 
-theorem conn_first_entry (G : BondGraph V E) (W : Set V) (ω : Set E)
-    {x y : V} (hx : x ∉ W) (hy : y ∈ W) (hxy : Conn G ω x y) :
-    ∃ a b e, a ∉ W ∧ b ∈ W ∧ e ∈ ω ∧ G e = s(a,b) ∧
-      Conn G (ω ∩ internalEdges G Wᶜ) x a := by
-  let Entry : Prop := ∃ a b e, a ∉ W ∧ b ∈ W ∧ e ∈ ω ∧ G e = s(a,b) ∧
-      Conn G (ω ∩ internalEdges G Wᶜ) x a
-  have haux : Conn G (ω ∩ internalEdges G Wᶜ) x y ∨ Entry := by
-    clear hy
-    induction hxy with
-    | refl => exact Or.inl (conn_refl _ _ _)
-    | @tail a b hxa hab ih =>
-      rcases ih with ih | ih
-      · have ha : a ∉ W := conn_internal_endpoint G Wᶜ hx ih
-        obtain ⟨e,he,heq⟩ := hab
-        by_cases hb : b ∈ W
-        · exact Or.inr ⟨a,b,e,ha,hb,he,heq,ih⟩
-        · exact Or.inl (ih.tail ⟨e,⟨he,(mem_internalEdges_of_ends G Wᶜ e a b heq).mpr ⟨ha,hb⟩⟩,heq⟩)
-      · exact Or.inr ih
-  rcases haux with h | h
-  · exact ((conn_internal_endpoint G Wᶜ hx h) hy).elim
-  · exact h
-
- 
-noncomputable def pruneContacts (G : BondGraph V E) (ω : Finset E) (W : Set V) (e : E) : Finset E :=
-  ω.filter fun f => ¬CutExists G W f ∨ f = e
-
-theorem internal_not_cut (G : BondGraph V E) (W : Set V) {e : E}
-    (he : e ∈ internalEdges G W) : ¬CutExists G W e := by
-  rintro ⟨a,b,hab,ha,hb⟩
-  exact hb ((mem_internalEdges_of_ends G W e a b hab).mp he).2
-
-theorem exterior_not_cut (G : BondGraph V E) (W : Set V) {e : E}
-    (he : e ∈ internalEdges G Wᶜ) : ¬CutExists G W e := by
-  rintro ⟨a,b,hab,ha,hb⟩
-  exact ((mem_internalEdges_of_ends G Wᶜ e a b hab).mp he).1 ha
-
-theorem internal_pruneContacts (G : BondGraph V E) (ω : Finset E) (W : Set V) (e : E) :
-    internalBonds G (pruneContacts G ω W e) W = internalBonds G ω W := by
-  ext f
-  simp only [internalBonds,pruneContacts,Finset.mem_filter]
-  constructor
-  · exact fun h => ⟨h.1.1,h.2⟩
-  · exact fun h => ⟨⟨h.1,Or.inl (internal_not_cut G W h.2)⟩,h.2⟩
-
-theorem pruneContacts_only_attachment (G : BondGraph V E) (ω : Finset E) (W : Set V) (e : E) :
-    ∀ f ∈ pruneContacts G ω W e, CutExists G W f → f = e := by
-  intro f hf hc
-  exact (Finset.mem_filter.mp hf).2.resolve_left (not_not.mpr hc)
-
- 
-theorem exists_pruneContacts_root (G : BondGraph V E) (ω : Finset E) (W : Set V) (x : V)
-    (hWroot : W ⊆ cluster G ω x) (hWconn : ∀ a ∈ W, ∀ b ∈ W,
-      Conn G (ω ∩ internalEdges G W) a b)
-    (hcontact : ∃ e ∈ ω, CutExists G W e) :
-    ∃ e ∈ ω, CutExists G W e ∧ W ⊆ cluster G (pruneContacts G ω W e) x := by
-  by_cases hx : x ∈ W
-  · obtain ⟨e,he,hec⟩ := hcontact
-    refine ⟨e,he,hec,?_⟩
-    intro y hy
-    apply conn_mono G _ (hWconn x hx y hy)
-    intro f hf
-    exact Finset.mem_filter.mpr ⟨hf.1,Or.inl (internal_not_cut G W hf.2)⟩
-  · obtain ⟨e₀,he₀,a,b,hab,ha,hb⟩ := hcontact
-    obtain ⟨c,z,e,hc,hz,he,hez,hxc⟩ := conn_first_entry G W ω hx ha (hWroot ha)
-    have hec : CutExists G W e := ⟨z,c,by simpa only [Sym2.eq_swap] using hez,hz,hc⟩
-    have hep : e ∈ pruneContacts G ω W e := Finset.mem_filter.mpr ⟨he,Or.inr rfl⟩
-    have hxz : Conn G (pruneContacts G ω W e : Set E) x z :=
-      (conn_mono G (fun f hf => Finset.mem_filter.mpr ⟨hf.1,Or.inl (exterior_not_cut G W hf.2)⟩) hxc).tail
-        ⟨e,hep,hez⟩
-    refine ⟨e,he,hec,?_⟩
-    intro y hy
-    exact hxz.trans (conn_mono G (fun f hf =>
-      Finset.mem_filter.mpr ⟨hf.1,Or.inl (internal_not_cut G W hf.2)⟩) (hWconn z hz y hy))
-
-end PruningGeometry
-
-section GoodnessGeometry
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-@[simp] theorem mem_pairOpenPivotals (G : BondGraph V E) (Q : Finset E) (x y : V)
-    (ω : Finset E) (e : E) : e ∈ pairOpenPivotals G Q x y ω ↔
-      e ∈ Q ∧ OpenMarkedPivotal G (ω : Set E) {y} x e := by
-  simp only [pairOpenPivotals,Finset.mem_filter,OpenMarkedPivotal,MarkedEvent,
-    Set.mem_singleton_iff,exists_eq_left,Finset.coe_erase,Finset.mem_coe]
-
- 
-theorem pairOpenPivotals_card_le (G : BondGraph V E) (Q : Finset E) (x y : V) (ω : Finset E) :
-    (pairOpenPivotals G Q x y ω).card ≤ (finiteRootComponent G ω x).card-1 := by
-  let N := (finiteRootComponent G ω x).card
-  have hN : (cluster G (ω : Set E) x).ncard = N := by
-    rw [← coe_finiteRootComponent,Set.ncard_coe_finset]
-  have hm : Set.MapsTo (pivotalRank G ω x) (pairOpenPivotals G Q x y ω : Set E)
-      (Finset.Ico 1 N : Set ℕ) := by
-    intro e he
-    have he' := ((mem_pairOpenPivotals G Q x y ω e).mp he).2
-    have hfin := cluster_finite_of_finite_edges G (ω := (ω : Set E) \ {e}) (ω.finite_toSet.subset Set.sdiff_subset) x
-    have hpos : 0 < pivotalRank G ω x e :=
-      (Set.ncard_pos hfin).mpr ⟨x,conn_refl G _ x⟩
-    have hlt : pivotalRank G ω x e < N := by
-      rw [pivotalRank,← hN]
-      apply Set.ncard_lt_ncard _ (cluster_finite_of_finite_edges G ω.finite_toSet x)
-      refine Set.ssubset_iff_subset_ne.mpr ⟨fun z hz => conn_mono G Set.sdiff_subset hz,?_⟩
-      intro heq
-      obtain ⟨z,hz,hxz⟩ := he'.2.1
-      have hz' : z ∈ cluster G (ω : Set E) x := hxz
-      have hz'' : z ∈ cluster G ((ω : Set E) \ {e}) x := heq.symm ▸ hz'
-      exact he'.2.2 ⟨z,hz,hz''⟩
-    exact Finset.mem_Ico.mpr ⟨hpos,hlt⟩
-  have hi : Set.InjOn (pivotalRank G ω x) (pairOpenPivotals G Q x y ω : Set E) := by
-    intro e he f hf h
-    exact pivotalRank_injective G ω.finite_toSet {y} x
-      ((mem_pairOpenPivotals G Q x y ω e).mp he).2
-      ((mem_pairOpenPivotals G Q x y ω f).mp hf).2 h
-  simpa [N] using Finset.card_le_card_of_injOn (pivotalRank G ω x) hm hi
-
-theorem finiteRootComponent_mono (G : BondGraph V E) {ω η : Finset E} (h : ω ⊆ η) (x : V) :
-    finiteRootComponent G ω x ⊆ finiteRootComponent G η x := by
-  intro y hy
-  change y ∈ (finiteRootComponent G η x : Set V)
-  rw [coe_finiteRootComponent]
-  apply conn_mono G (show (ω : Set E) ⊆ η from h)
-  change y ∈ (finiteRootComponent G ω x : Set V) at hy
-  rwa [coe_finiteRootComponent] at hy
-
- 
-noncomputable def openContacts (G : BondGraph V E) (ω : Finset E) (W : Set V) : Finset E :=
-  ω.filter fun e => CutExists G W e
-
- 
-def BadPieceEvent (G : BondGraph V E) (Q : Finset E) (x : V) (M K : ℕ) (L : ℝ)
-    (ω : Finset E) : Prop :=
-  (finiteRootComponent G ω x).card ≤ M ∧ ∃ W : Finset V,
-    W ⊆ finiteRootComponent G ω x ∧
-    (∀ a ∈ W, ∀ b ∈ W, Conn G (internalBonds G ω (W : Set V) : Set E) a b) ∧
-    (openContacts G ω (W : Set V)).card = K ∧
-    ∃ a ∈ W, ∃ b ∈ W, (K:ℝ)*L < (pairOpenPivotals G Q a b (internalBonds G ω (W : Set V))).card
-
- 
-theorem badPieceEvent_contact_bound (G : BondGraph V E) (Q : Finset E) (x : V)
-    (M K : ℕ) (L : ℝ) (ω : Finset E) (h : BadPieceEvent G Q x M K L ω) :
-    (K:ℝ)*L < (M:ℝ) := by
-  obtain ⟨hcap,W,hW,_,_,a,ha,b,_,hbad⟩ := h
-  have hsub : finiteRootComponent G (internalBonds G ω (W : Set V)) a ⊆ W := by
-    intro z hz
-    apply conn_internal_endpoint G (W : Set V) (ω := (ω : Set E)) ha
-    change z ∈ (finiteRootComponent G (internalBonds G ω (W : Set V)) a : Set V) at hz
-    rw [coe_finiteRootComponent] at hz
-    simpa only [coe_internalBonds,cluster,Set.mem_ofPred_eq] using hz
-  have hab := pairOpenPivotals_card_le G Q a b (internalBonds G ω (W : Set V))
-  have hpos : 0 < (finiteRootComponent G (internalBonds G ω (W : Set V)) a).card := by
-    apply Finset.card_pos.mpr
-    refine ⟨a,?_⟩
-    change a ∈ (finiteRootComponent G (internalBonds G ω (W : Set V)) a : Set V)
-    rw [coe_finiteRootComponent]
-    exact conn_refl G _ a
-  have hn : (pairOpenPivotals G Q a b (internalBonds G ω (W : Set V))).card < M := by
-    have h₁ := Finset.card_le_card hsub
-    have h₂ := Finset.card_le_card hW
-    omega
-  exact hbad.trans (by exact_mod_cast hn)
-
- 
-noncomputable def incidentBonds (G : BondGraph V E) (Q : Finset E) (W : Finset V) : Finset E :=
-  W.biUnion fun v => Q.filter fun e => v ∈ G e
-
-@[simp] theorem mem_incidentBonds (G : BondGraph V E) (Q : Finset E) (W : Finset V) (e : E) :
-    e ∈ incidentBonds G Q W ↔ e ∈ Q ∧ ∃ v ∈ W, v ∈ G e := by
-  simp only [incidentBonds,Finset.mem_biUnion,Finset.mem_filter]
-  aesop
-
-theorem incidentBonds_card_le (G : BondGraph V E) (Q : Finset E) (W : Finset V) (d : ℕ)
-    (hd : ∀ v, (Q.filter fun e => v ∈ G e).card ≤ d) :
-    (incidentBonds G Q W).card ≤ d*W.card := by
-  calc
-    _ ≤ ∑ v ∈ W, (Q.filter fun e => v ∈ G e).card := Finset.card_biUnion_le
-    _ ≤ ∑ _v ∈ W, d := Finset.sum_le_sum fun _ _ => hd _
-    _ = _ := by simp [Nat.mul_comm]
-
- 
-
-theorem badPieceEvent_pruning (G : BondGraph V E) (Q : Finset E) (x : V) (M K r : ℕ)
-    (L : ℝ) (hK : 0 < K) (hr : 2*(r:ℝ) ≤ (K:ℝ)*L)
-    (ω : Finset E) (hω : ω ⊆ Q) (h : BadPieceEvent G Q x M K L ω) :
-    ∃ η : Finset E, η ⊆ ω ∧ CappedRadiusEvent G Q x M r η ∧
-      (ω \ η).card ≤ K ∧ ω \ η ⊆ incidentBonds G Q (finiteRootComponent G η x) := by
-  obtain ⟨hcap,W,hW,hWconn,hcontacts,a,ha,b,hb,hbad⟩ := h
-  have hcontact : ∃ e ∈ ω, CutExists G (W : Set V) e := by
-    have : (openContacts G ω (W : Set V)).Nonempty := Finset.card_pos.mp (hcontacts ▸ hK)
-    obtain ⟨e,he⟩ := this
-    exact ⟨e,(Finset.mem_filter.mp he).1,(Finset.mem_filter.mp he).2⟩
-  have hWroot : (W : Set V) ⊆ cluster G ω x := by
-    intro z hz
-    have hh : z ∈ (finiteRootComponent G ω x : Set V) := hW hz
-    rwa [coe_finiteRootComponent] at hh
-  obtain ⟨e,he,hec,hWη⟩ := exists_pruneContacts_root G ω (W : Set V) x hWroot
-    (by intro a ha b hb; simpa only [coe_internalBonds] using hWconn a ha b hb) hcontact
-  let η := pruneContacts G ω (W : Set V) e
-  have hη : η ⊆ ω := Finset.filter_subset _ _
-  have hep : e ∈ η := Finset.mem_filter.mpr ⟨he,Or.inr rfl⟩
-  have hpiv := pairOpenPivotals_one_attachment G Q η (W : Set V) e hep hec
-    (pruneContacts_only_attachment G ω (W : Set V) e) ha hb
-  rw [internal_pruneContacts] at hpiv
-  have hfar : (K:ℝ)*L < (pairOpenPivotals G Q a b η).card :=
-    hbad.trans_le (by exact_mod_cast Finset.card_le_card hpiv)
-  have hxa : Conn G (η : Set E) x a := hWη ha
-  have hxb : Conn G (η : Set E) x b := hWη hb
-  have htri := pairOpenPivotals_triangle G Q η a x b (conn_symm G η hxa) hxb
-  rw [pairOpenPivotals_symm G Q η a x] at htri
-  have hlarge : r ≤ (pairOpenPivotals G Q x a η).card ∨
-      r ≤ (pairOpenPivotals G Q x b η).card := by
-    by_contra hn
-    push Not at hn
-    have htri' : ((pairOpenPivotals G Q a b η).card:ℝ) ≤
-        (pairOpenPivotals G Q x a η).card+(pairOpenPivotals G Q x b η).card := by exact_mod_cast htri
-    have h₁ : ((pairOpenPivotals G Q x a η).card:ℝ) < r := by exact_mod_cast hn.1
-    have h₂ : ((pairOpenPivotals G Q x b η).card:ℝ) < r := by exact_mod_cast hn.2
-    linarith
-  have hchanges : ω \ η ⊆ openContacts G ω (W : Set V) := by
-    intro f hf
-    obtain ⟨hfω,hfη⟩ := Finset.mem_sdiff.mp hf
-    refine Finset.mem_filter.mpr ⟨hfω,?_⟩
-    by_contra hn
-    exact hfη (Finset.mem_filter.mpr ⟨hfω,Or.inl hn⟩)
-  refine ⟨η,hη,⟨(Finset.card_le_card (finiteRootComponent_mono G hη x)).trans hcap,?_⟩,
-    (Finset.card_le_card hchanges).trans (le_of_eq hcontacts),?_⟩
-  · rcases hlarge with h | h
-    · exact ⟨a,hxa,h⟩
-    · exact ⟨b,hxb,h⟩
-  · intro f hf
-    obtain ⟨hfω,c,z,hcz,hc,_⟩ := Finset.mem_filter.mp (hchanges hf)
-    refine (mem_incidentBonds G Q _ f).mpr ⟨hω hfω,c,?_,?_⟩
-    · change c ∈ (finiteRootComponent G η x : Set V)
-      rw [coe_finiteRootComponent]
-      exact hWη hc
-    · simp [hcz]
-
-end GoodnessGeometry
-
-section FiniteInverseCost
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-def bernoulliWeight (Q ω : Finset E) (p : ℝ) : ℝ :=
-  p^ω.card*(1-p)^(Q.card-ω.card)
-
-theorem bernoulliWeight_nonneg (Q ω : Finset E) (p : unitInterval) :
-    0 ≤ bernoulliWeight Q ω p := mul_nonneg (pow_nonneg p.2.1 _) (pow_nonneg (sub_nonneg.mpr p.2.2) _)
-
- 
-theorem bernoulliWeight_closure_ratio (Q ω η : Finset E) (hη : η ⊆ ω) (hω : ω ⊆ Q)
-    (p : ℝ) (hp : p < 1) :
-    bernoulliWeight Q ω p = bernoulliWeight Q η p*(p/(1-p))^((ω \ η).card) := by
-  have h₁ := Finset.card_le_card hη
-  have h₂ := Finset.card_le_card hω
-  have hj : (ω \ η).card = ω.card-η.card := Finset.card_sdiff_of_subset hη
-  have ha : ω.card = η.card+(ω \ η).card := by omega
-  have hb : Q.card-η.card = (Q.card-ω.card)+(ω \ η).card := by omega
-  have hq : (1-p) ≠ 0 := ne_of_gt (sub_pos.mpr hp)
-  unfold bernoulliWeight
-  rw [hb,pow_add,div_pow]
-  rw [← mul_div_assoc]
-  apply (eq_div_iff (pow_ne_zero _ hq)).mpr
-  rw [ha,pow_add]
-  ring
-
- 
-theorem bernoulliWeight_closure_le (Q ω η : Finset E) (hη : η ⊆ ω) (hω : ω ⊆ Q)
-    (p : unitInterval) (hp : (p:ℝ) < 1) (K : ℕ) (hj : (ω \ η).card ≤ K) :
-    bernoulliWeight Q ω p ≤ (max 1 ((p:ℝ)/(1-p)))^K*bernoulliWeight Q η p := by
-  rw [bernoulliWeight_closure_ratio Q ω η hη hω p hp,mul_comm]
-  apply mul_le_mul_of_nonneg_right _ (bernoulliWeight_nonneg Q η p)
-  calc
-    _ ≤ (max 1 ((p:ℝ)/(1-p)))^((ω \ η).card) :=
-      pow_le_pow_left₀ (div_nonneg p.2.1 (sub_nonneg.mpr p.2.2)) (le_max_right ..) _
-    _ ≤ _ := pow_le_pow_right₀ (le_max_left ..) hj
-
- 
-theorem card_small_subsets_le (S : Finset E) (K N : ℕ) (hS : S.card ≤ N) (hN : 1 ≤ N) :
-    (S.powerset.filter fun T => T.card ≤ K).card ≤ (K+1)*N^K := by
-  have hsub : S.powerset.filter (fun T => T.card ≤ K) ⊆
-      (Finset.range (K+1)).biUnion (fun j => S.powersetCard j) := by
-    intro T hT
-    obtain ⟨hTS,hTK⟩ := Finset.mem_filter.mp hT
-    exact Finset.mem_biUnion.mpr ⟨T.card,Finset.mem_range.mpr (Nat.lt_succ_of_le hTK),
-      Finset.mem_powersetCard.mpr ⟨Finset.mem_powerset.mp hTS,rfl⟩⟩
-  calc
-    _ ≤ _ := Finset.card_le_card hsub
-    _ ≤ ∑ j ∈ Finset.range (K+1), (S.powersetCard j).card := Finset.card_biUnion_le
-    _ ≤ ∑ _j ∈ Finset.range (K+1), N^K := by
-      apply Finset.sum_le_sum
-      intro j hj
-      rw [Finset.card_powersetCard]
-      exact (Nat.choose_le_pow _ _).trans ((Nat.pow_le_pow_left hS _).trans
-        (Nat.pow_le_pow_right hN (Nat.le_of_lt_succ (Finset.mem_range.mp hj))))
-    _ = _ := by simp
-
- 
-theorem closure_fiber_card_le (A : Finset (Finset E)) (f : Finset E → Finset E)
-    (S : Finset E) (η : Finset E) (K N : ℕ) (hN : 1 ≤ N) (hS : S.card ≤ N)
-    (hf : ∀ ω ∈ A, f ω = η → f ω ⊆ ω ∧ (ω \ f ω).card ≤ K ∧ ω \ f ω ⊆ S) :
-    (A.filter fun ω => f ω = η).card ≤ (K+1)*N^K := by
-  have hm : Set.MapsTo (fun ω : Finset E => ω \ η) (A.filter fun ω => f ω = η : Set (Finset E))
-      (S.powerset.filter fun T => T.card ≤ K : Set (Finset E)) := by
-    intro ω hω
-    obtain ⟨hωA,heq⟩ := Finset.mem_filter.mp hω
-    obtain ⟨_,hj,hs⟩ := hf ω hωA heq
-    rw [heq] at hj hs
-    exact Finset.mem_filter.mpr ⟨Finset.mem_powerset.mpr hs,hj⟩
-  have hi : Set.InjOn (fun ω : Finset E => ω \ η) (A.filter fun ω => f ω = η : Set (Finset E)) := by
-    intro ω hω ζ hζ heq
-    obtain ⟨hωA,hωη⟩ := Finset.mem_filter.mp hω
-    obtain ⟨hζA,hζη⟩ := Finset.mem_filter.mp hζ
-    have h₁ := (hf ω hωA hωη).1
-    have h₂ := (hf ζ hζA hζη).1
-    rw [hωη] at h₁
-    rw [hζη] at h₂
-    have : η ∪ (ω \ η) = η ∪ (ζ \ η) := congrArg (fun s => η ∪ s) heq
-    simpa only [Finset.union_sdiff_of_subset h₁,Finset.union_sdiff_of_subset h₂] using this
-  exact (Finset.card_le_card_of_injOn (fun ω => ω \ η) hm hi).trans (card_small_subsets_le S K N hS hN)
-
- 
-theorem finite_closure_comparison (Q : Finset E) (A B : Finset (Finset E))
-    (hA : A ⊆ Q.powerset) (f : Finset E → Finset E) (S : Finset E → Finset E)
-    (K N : ℕ) (hN : 1 ≤ N) (p : unitInterval) (hp : (p:ℝ) < 1)
-    (hf : ∀ ω ∈ A, f ω ∈ B ∧ f ω ⊆ ω ∧ (ω \ f ω).card ≤ K ∧ ω \ f ω ⊆ S (f ω))
-    (hS : ∀ η ∈ B, (S η).card ≤ N) :
-    (∑ ω ∈ A, bernoulliWeight Q ω p) ≤
-      (((K+1)*N^K : ℕ):ℝ)*(max 1 ((p:ℝ)/(1-p)))^K*∑ η ∈ B, bernoulliWeight Q η p := by
-  rw [← Finset.sum_fiberwise_of_maps_to (fun ω hω => (hf ω hω).1) (fun ω => bernoulliWeight Q ω p),
-    Finset.mul_sum]
-  apply Finset.sum_le_sum
-  intro η hη
-  let T := A.filter fun ω => f ω = η
-  have hc : T.card ≤ (K+1)*N^K := closure_fiber_card_le A f (S η) η K N hN (hS η hη) (by
-    intro ω hω heq
-    have hh := (hf ω hω).2
-    simpa only [heq] using hh)
-  have hweight : ∀ ω ∈ T, bernoulliWeight Q ω p ≤
-      (max 1 ((p:ℝ)/(1-p)))^K*bernoulliWeight Q η p := by
-    intro ω hω
-    obtain ⟨hωA,heq⟩ := Finset.mem_filter.mp hω
-    have hh := bernoulliWeight_closure_le Q ω (f ω) (hf ω hωA).2.1
-      (Finset.mem_powerset.mp (hA hωA)) p hp K (hf ω hωA).2.2.1
-    simpa only [heq] using hh
-  calc
-    _ ≤ ∑ _ω ∈ T, (max 1 ((p:ℝ)/(1-p)))^K*bernoulliWeight Q η p := Finset.sum_le_sum hweight
-    _ = (T.card:ℝ)*((max 1 ((p:ℝ)/(1-p)))^K*bernoulliWeight Q η p) := by simp
-    _ ≤ (((K+1)*N^K:ℕ):ℝ)*((max 1 ((p:ℝ)/(1-p)))^K*bernoulliWeight Q η p) :=
-      mul_le_mul_of_nonneg_right (by exact_mod_cast hc)
-        (mul_nonneg (pow_nonneg (le_trans zero_le_one (le_max_left ..)) _) (bernoulliWeight_nonneg Q η p))
-    _ = _ := by ring
-
-end FiniteInverseCost
-
-section FiniteGoodnessProbability
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def finiteBadPieceProbability (G : BondGraph V E) (Q : Finset E)
-    (x : V) (M K : ℕ) (L p : ℝ) : ℝ :=
-  bernoulliPolynomial Q (fun ω => if BadPieceEvent G Q x M K L ω then 1 else 0) p
-
-theorem bernoulliPolynomial_indicator_weight (Q : Finset E) (P : Finset E → Prop) (p : ℝ) :
-    bernoulliPolynomial Q (fun ω => if P ω then 1 else 0) p =
-      ∑ ω ∈ Q.powerset.filter P, bernoulliWeight Q ω p := by
-  rw [Finset.sum_filter]
-  unfold bernoulliPolynomial
-  apply Finset.sum_congr rfl
-  intro ω hω
-  by_cases h : P ω <;> simp [h,bernoulliWeight]
-
- 
-theorem finiteBadPieceProbability_le_radius (G : BondGraph V E) (Q : Finset E) (x : V)
-    (M K r d : ℕ) (L : ℝ) (hK : 0 < K) (hN : 1 ≤ d*M)
-    (hdeg : ∀ v, (Q.filter fun e => v ∈ G e).card ≤ d)
-    (hr : 2*(r:ℝ) ≤ (K:ℝ)*L) (p : unitInterval) (hp : (p:ℝ) < 1) :
-    finiteBadPieceProbability G Q x M K L p ≤
-      (((K+1)*(d*M)^K:ℕ):ℝ)*(max 1 ((p:ℝ)/(1-p)))^K*finiteCappedRadiusTail G Q x M r p := by
-  let A := Q.powerset.filter (BadPieceEvent G Q x M K L)
-  let B := Q.powerset.filter (CappedRadiusEvent G Q x M r)
-  let S := fun η => incidentBonds G Q (finiteRootComponent G η x)
-  have hex : ∀ ω : Finset E, ∃ η : Finset E, ω ∈ A →
-      η ∈ B ∧ η ⊆ ω ∧ (ω \ η).card ≤ K ∧ ω \ η ⊆ S η := by
-    intro ω
-    by_cases hω : ω ∈ A
-    · obtain ⟨hωQ,hωbad⟩ := Finset.mem_filter.mp hω
-      have hωsub := Finset.mem_powerset.mp hωQ
-      obtain ⟨η,hη,hcap,hcard,hS⟩ := badPieceEvent_pruning G Q x M K r L hK hr ω hωsub hωbad
-      exact ⟨η,fun _ => ⟨Finset.mem_filter.mpr ⟨Finset.mem_powerset.mpr (hη.trans hωsub),hcap⟩,
-        hη,hcard,hS⟩⟩
-    · exact ⟨∅,fun h => (hω h).elim⟩
-  choose f hf using hex
-  have hS : ∀ η ∈ B, (S η).card ≤ d*M := by
-    intro η hη
-    exact (incidentBonds_card_le G Q (finiteRootComponent G η x) d hdeg).trans
-      (Nat.mul_le_mul_left d (Finset.mem_filter.mp hη).2.1)
-  have hc := finite_closure_comparison Q A B (Finset.filter_subset _ _) f S K (d*M) hN p hp hf hS
-  simpa only [finiteBadPieceProbability,finiteCappedRadiusTail,bernoulliPolynomial_indicator_weight,
-    A,B] using hc
-
- 
-theorem finiteBadPieceProbability_exponential [Nonempty V] (G : BondGraph V E)
-    (Q : Finset E) (x : V) (M K d : ℕ) (L : ℝ) (hK : 0 < K) (hN : 1 ≤ d*M)
-    (hL : 4 ≤ (K:ℝ)*L) (hd : 0 < d)
-    (hdeg : ∀ v, (Q.filter fun e => v ∈ G e).card ≤ d)
-    (p h : unitInterval) (hp : (p:ℝ) < 1) (hh : 0 < (h:ℝ)) (hh1 : (h:ℝ) < 1)
-    (B : ℝ) (hB : 1 ≤ B)
-    (hAB : (Real.sqrt ((d:ℝ)/2)/(1-p))*Real.sqrt (Real.log (1/(h:ℝ))/h) ≤ B) :
-    finiteBadPieceProbability G Q x M K L p ≤
-      (((K+1)*(d*M)^K:ℕ):ℝ)*(max 1 ((p:ℝ)/(1-p)))^K*
-        (2/((h:ℝ)*(1-h)^M))*Real.exp (-((K:ℝ)*L)/(16*B)) := by
-  let r := ⌈(K:ℝ)*L/4⌉₊
-  have hpos : 0 ≤ (K:ℝ)*L/4 := by linarith
-  have hrlo : (K:ℝ)*L/4 ≤ r := Nat.le_ceil _
-  have hrhi : (r:ℝ) < (K:ℝ)*L/4+1 := Nat.ceil_lt_add_one hpos
-  have hr : 2*(r:ℝ) ≤ (K:ℝ)*L := by linarith
-  have hrad := finiteCappedRadiusTail_exponential G p h hp hh hh1 d hd Q x hdeg B hB hAB M r
-  have hcmp := finiteBadPieceProbability_le_radius G Q x M K r d L hK hN hdeg hr p hp
-  have hc : 0 ≤ (((K+1)*(d*M)^K:ℕ):ℝ)*(max 1 ((p:ℝ)/(1-p)))^K := by positivity
-  have hhfac : 0 ≤ 2/((h:ℝ)*(1-h)^M) := by positivity
-  have hBpos : 0 < B := lt_of_lt_of_le zero_lt_one hB
-  have hratio : -((r:ℝ)/(4*B)) ≤ -((K:ℝ)*L)/(16*B) := by
-    apply (le_div_iff₀ (by positivity : 0 < 16*B)).mpr
-    field_simp
-    nlinarith
-  exact hcmp.trans ((mul_le_mul_of_nonneg_left hrad hc).trans (by
-    rw [← mul_assoc]
-    exact mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr (by simpa only [neg_div] using hratio)) (mul_nonneg hc hhfac)))
-
- 
-
-def FiniteGoodForScale (G : BondGraph V E) (Q : Finset E) (x : V) (L : ℝ) (ω : Finset E) : Prop :=
-  ∀ W : Finset V, W ⊆ finiteRootComponent G ω x →
-    (∀ a ∈ W, ∀ b ∈ W, Conn G (internalBonds G ω (W : Set V) : Set E) a b) →
-    0 < (openContacts G ω (W : Set V)).card →
-    ∀ a ∈ W, ∀ b ∈ W,
-      (pairOpenPivotals G Q a b (internalBonds G ω (W : Set V))).card ≤
-        ((openContacts G ω (W : Set V)).card:ℝ)*L
-
-def CappedBadGoodnessEvent (G : BondGraph V E) (Q : Finset E) (x : V) (M : ℕ) (L : ℝ)
-    (ω : Finset E) : Prop :=
-  (finiteRootComponent G ω x).card ≤ M ∧ ¬FiniteGoodForScale G Q x L ω
-
-noncomputable def finiteBadGoodnessProbability (G : BondGraph V E) (Q : Finset E)
-    (x : V) (M : ℕ) (L p : ℝ) : ℝ :=
-  bernoulliPolynomial Q (fun ω => if CappedBadGoodnessEvent G Q x M L ω then 1 else 0) p
-
- 
-theorem cappedBadGoodness_iff (G : BondGraph V E) (Q : Finset E) (x : V) (M d : ℕ)
-    (L : ℝ) (ω : Finset E) (hω : ω ⊆ Q)
-    (hdeg : ∀ v, (Q.filter fun e => v ∈ G e).card ≤ d) :
-    CappedBadGoodnessEvent G Q x M L ω ↔
-      ∃ K ∈ Finset.Icc 1 (d*M), BadPieceEvent G Q x M K L ω := by
-  constructor
-  · rintro ⟨hcap,hbad⟩
-    simp only [FiniteGoodForScale,not_forall] at hbad
-    obtain ⟨W,hbad⟩ := hbad
-    push Not at hbad
-    obtain ⟨hW,hWconn,hK,a,ha,b,hb,hfar⟩ := hbad
-    have hinc : openContacts G ω (W : Set V) ⊆ incidentBonds G Q W := by
-      intro e he
-      obtain ⟨heω,c,z,hcz,hc,_⟩ := Finset.mem_filter.mp he
-      exact (mem_incidentBonds G Q W e).mpr ⟨hω heω,c,hc,by simp [hcz]⟩
-    have hbound := (Finset.card_le_card hinc).trans ((incidentBonds_card_le G Q W d hdeg).trans
-      (Nat.mul_le_mul_left d ((Finset.card_le_card hW).trans hcap)))
-    exact ⟨_,Finset.mem_Icc.mpr ⟨hK,hbound⟩,hcap,W,hW,hWconn,rfl,a,ha,b,hb,hfar⟩
-  · rintro ⟨K,hK,hcap,W,hW,hWconn,hcontacts,a,ha,b,hb,hfar⟩
-    refine ⟨hcap,fun hgood => ?_⟩
-    have hh := hgood W hW hWconn (by rw [hcontacts]; exact (Finset.mem_Icc.mp hK).1) a ha b hb
-    rw [hcontacts] at hh
-    exact not_lt_of_ge hh hfar
-
- 
-theorem finiteBadGoodnessProbability_le_sum (G : BondGraph V E) (Q : Finset E) (x : V)
-    (M d : ℕ) (L : ℝ) (p : unitInterval)
-    (hdeg : ∀ v, (Q.filter fun e => v ∈ G e).card ≤ d) :
-    finiteBadGoodnessProbability G Q x M L p ≤
-      ∑ K ∈ Finset.Icc 1 (d*M), finiteBadPieceProbability G Q x M K L p := by
-  unfold finiteBadPieceProbability finiteBadGoodnessProbability
-  rw [← bernoulliPolynomial_sum]
-  apply bernoulliPolynomial_mono _ _ _ _ p.2
-  intro ω hω
-  by_cases he : CappedBadGoodnessEvent G Q x M L ω
-  · obtain ⟨K,hK,hbad⟩ := (cappedBadGoodness_iff G Q x M d L ω hω hdeg).mp he
-    simp only [he,ite_true]
-    exact (by simp [hbad] : (1:ℝ) ≤ if BadPieceEvent G Q x M K L ω then 1 else 0).trans
-      (Finset.single_le_sum (f := fun j => if BadPieceEvent G Q x M j L ω then (1:ℝ) else 0)
-        (fun j _ => by split_ifs <;> norm_num) hK)
-  · simp only [he,ite_false]
-    exact Finset.sum_nonneg fun j _ => by split_ifs <;> norm_num
-
- 
-theorem finiteBadGoodnessProbability_geometric [Nonempty V] (G : BondGraph V E)
-    (Q : Finset E) (x : V) (M d : ℕ) (L : ℝ) (hN : 1 ≤ d*M) (hL : 4 ≤ L)
-    (hd : 0 < d) (hdeg : ∀ v, (Q.filter fun e => v ∈ G e).card ≤ d)
-    (p h : unitInterval) (hp : (p:ℝ) < 1) (hh : 0 < (h:ℝ)) (hh1 : (h:ℝ) < 1)
-    (B : ℝ) (hB : 1 ≤ B)
-    (hAB : (Real.sqrt ((d:ℝ)/2)/(1-p))*Real.sqrt (Real.log (1/(h:ℝ))/h) ≤ B)
-    (hq : ((d*M:ℕ):ℝ)*max 1 ((p:ℝ)/(1-p))*Real.exp (-L/(16*B)) ≤ 1) :
-    finiteBadGoodnessProbability G Q x M L p ≤
-      (2/((h:ℝ)*(1-h)^M))*((d*M:ℕ):ℝ)*((d*M:ℕ)+1)*
-        (((d*M:ℕ):ℝ)*max 1 ((p:ℝ)/(1-p))*Real.exp (-L/(16*B))) := by
-  let N := d*M
-  let a := max 1 ((p:ℝ)/(1-p))
-  let H := 2/((h:ℝ)*(1-h)^M)
-  let q := (N:ℝ)*a*Real.exp (-L/(16*B))
-  have ha : 0 ≤ a := le_trans zero_le_one (le_max_left ..)
-  have hq0 : 0 ≤ q := mul_nonneg (mul_nonneg (Nat.cast_nonneg _) ha) (Real.exp_nonneg _)
-  have hH : 0 ≤ H := by dsimp [H]; positivity
-  have hkbound : ∀ K ∈ Finset.Icc 1 N, finiteBadPieceProbability G Q x M K L p ≤ H*((N:ℝ)+1)*q := by
-    intro K hK
-    obtain ⟨hK1,hKN⟩ := Finset.mem_Icc.mp hK
-    have hKR : (1:ℝ) ≤ K := by exact_mod_cast hK1
-    have hKL : 4 ≤ (K:ℝ)*L := by nlinarith
-    have hexp := finiteBadPieceProbability_exponential G Q x M K d L hK1 hN hKL hd hdeg
-      p h hp hh hh1 B hB hAB
-    have hid : (((K+1)*(d*M)^K:ℕ):ℝ)*(max 1 ((p:ℝ)/(1-p)))^K*
-        (2/((h:ℝ)*(1-h)^M))*Real.exp (-((K:ℝ)*L)/(16*B)) = H*((K:ℝ)+1)*q^K := by
-      rw [show -((K:ℝ)*L)/(16*B) = (K:ℝ)*(-L/(16*B)) by ring,Real.exp_nat_mul]
-      simp only [q,H,N,a,mul_pow,Nat.cast_mul,Nat.cast_add,Nat.cast_pow,Nat.cast_one]
-      ring
-    rw [hid] at hexp
-    have hpow : q^K ≤ q := by
-      simpa only [pow_one] using pow_le_pow_of_le_one hq0 hq hK1
-    calc
-      _ ≤ H*((K:ℝ)+1)*q^K := hexp
-      _ ≤ H*((N:ℝ)+1)*q := mul_le_mul
-        (mul_le_mul_of_nonneg_left (by exact_mod_cast Nat.add_le_add_right hKN 1) hH)
-        hpow (pow_nonneg hq0 _) (mul_nonneg hH (by positivity))
-  calc
-    _ ≤ ∑ K ∈ Finset.Icc 1 N, finiteBadPieceProbability G Q x M K L p :=
-      finiteBadGoodnessProbability_le_sum G Q x M d L p hdeg
-    _ ≤ ∑ _K ∈ Finset.Icc 1 N, H*((N:ℝ)+1)*q := Finset.sum_le_sum hkbound
-    _ = (N:ℝ)*(H*((N:ℝ)+1)*q) := by simp
-    _ = _ := by dsimp [N,H,q,a]; push_cast; ring
-
-end FiniteGoodnessProbability
-
-section GoodnessAsymptotics
-
- 
-theorem inverse_mark_cost_le (h : ℝ) (hh : 0 ≤ h) (hh2 : h ≤ 1/2) (M : ℕ) :
-    ((1-h)^M)⁻¹ ≤ Real.exp (2*(M:ℝ)*h) := by
-  have hq : 0 < 1-h := by linarith
-  have hi : 1/(1-h) ≤ 1+2*h := (div_le_iff₀ hq).mpr (by nlinarith)
-  have hl := Real.one_sub_inv_le_log_of_pos hq
-  rw [one_div] at hi
-  have hlog : -2*h ≤ Real.log (1-h) := by linarith
-  calc
-    ((1-h)^M)⁻¹ = Real.exp (-((M:ℝ)*Real.log (1-h))) := by
-      rw [← Real.log_pow,Real.exp_neg,Real.exp_log (pow_pos hq _)]
-    _ ≤ _ := Real.exp_le_exp.mpr (by
-      have := mul_le_mul_of_nonneg_left hlog (Nat.cast_nonneg M)
-      nlinarith)
-
- 
-
-theorem ghost_scale_estimates (M : ℕ) (ε c a : ℝ) (d : ℕ)
-    (hM : 1 ≤ (M:ℝ)) (hε : 0 < ε) (hε1 : ε < 1/2) (hc : 1 ≤ c)
-    (ha : 1 ≤ a) (hd : 1 ≤ d)
-    (hsmall : (M:ℝ)^(ε-1) ≤ 1/2)
-    (hlog : 4*Real.log M ≤ (M:ℝ)^ε)
-    (hlog2 : Real.log M ≤ (M:ℝ)^(ε/2))
-    (hgrowth : 80*c ≤ (M:ℝ)^(ε/4))
-    (hconstant : Real.log (2*(d:ℝ)^2*((d:ℝ)+1)*a) ≤ (M:ℝ)^ε) :
-    let h := (M:ℝ)^(ε-1)
-    let B := c*(M:ℝ)^(1/2-ε/4)
-    let L := (M:ℝ)^(1/2+ε)
-    0 < h ∧ h ≤ 1/2 ∧ 1 ≤ B ∧
-      c*Real.sqrt (Real.log (1/h)/h) ≤ B ∧
-      (2/(h*(1-h)^M))*((d*M:ℕ):ℝ)*((d*M:ℕ)+1)*
-        (((d*M:ℕ):ℝ)*a*Real.exp (-L/(16*B))) ≤ Real.exp (-(M:ℝ)^ε) := by
-  dsimp only
-  let h := (M:ℝ)^(ε-1)
-  let B := c*(M:ℝ)^(1/2-ε/4)
-  let L := (M:ℝ)^(1/2+ε)
-  have hMp : 0 < (M:ℝ) := lt_of_lt_of_le zero_lt_one hM
-  have hcp : 0 < c := lt_of_lt_of_le zero_lt_one hc
-  have hap : 0 < a := lt_of_lt_of_le zero_lt_one ha
-  have hdp : 0 < (d:ℝ) := by exact_mod_cast (show 0 < d by omega)
-  have hhp : 0 < h := Real.rpow_pos_of_pos hMp _
-  have hq : 0 < 1-h := by dsimp [h] at *; linarith
-  have hlogM : 0 ≤ Real.log M := Real.log_nonneg hM
-  have hinv : 1/h = (M:ℝ)^(1-ε) := by
-    dsimp [h]
-    rw [show 1-ε = -(ε-1) by ring,Real.rpow_neg hMp.le,one_div]
-  have hmh : (M:ℝ)*h = (M:ℝ)^ε := by
-    dsimp [h]
-    calc
-      _ = (M:ℝ)^((1:ℝ)+(ε-1)) := by rw [Real.rpow_add hMp,Real.rpow_one]
-      _ = _ := by congr 1; ring
-  have hhM : 1/h ≤ (M:ℝ) := by
-    rw [hinv]
-    simpa only [Real.rpow_one] using
-      Real.rpow_le_rpow_of_exponent_le hM (show 1-ε ≤ (1:ℝ) by linarith)
-  have hB1 : 1 ≤ B := by
-    have hp := Real.one_le_rpow hM (show 0 ≤ 1/2-ε/4 by linarith)
-    dsimp [B]
-    nlinarith
-  have hBp : 0 < B := lt_of_lt_of_le zero_lt_one hB1
-  have hAB : c*Real.sqrt (Real.log (1/h)/h) ≤ B := by
-    apply mul_le_mul_of_nonneg_left _ hcp.le
-    have hl : Real.log (1/h) ≤ Real.log M := by
-      rw [hinv,Real.log_rpow hMp]
-      nlinarith
-    have hl2 : Real.log (1/h)/h ≤ (M:ℝ)^(1-ε/2) := by
-      calc
-        _ = Real.log (1/h)*(M:ℝ)^(1-ε) := by rw [div_eq_mul_one_div,hinv]
-        _ ≤ (M:ℝ)^(ε/2)*(M:ℝ)^(1-ε) :=
-          mul_le_mul_of_nonneg_right (hl.trans hlog2) (Real.rpow_nonneg hMp.le _)
-        _ = _ := by rw [← Real.rpow_add hMp]; congr 1; ring
-    calc
-      _ ≤ Real.sqrt ((M:ℝ)^(1-ε/2)) := Real.sqrt_le_sqrt hl2
-      _ = (M:ℝ)^(1/2-ε/4) := by
-        rw [Real.sqrt_eq_rpow,← Real.rpow_mul hMp.le]
-        congr 1
-        ring
-  have hratio : 5*(M:ℝ)^ε ≤ L/(16*B) := by
-    have hid : (M:ℝ)^ε*(M:ℝ)^(ε/4)*(M:ℝ)^(1/2-ε/4) = L := by
-      dsimp [L]
-      rw [← Real.rpow_add hMp,← Real.rpow_add hMp]
-      congr 1
-      ring
-    apply (le_div_iff₀ (by positivity : 0 < 16*B)).mpr
-    have hh := mul_le_mul_of_nonneg_left hgrowth
-      (mul_nonneg (Real.rpow_nonneg hMp.le ε) (Real.rpow_nonneg hMp.le (1/2-ε/4)))
-    dsimp [B]
-    nlinarith [hid]
-  refine ⟨hhp,hsmall,hB1,hAB,?_⟩
-  let A := 2*(d:ℝ)^2*((d:ℝ)+1)*a
-  have hAp : 0 < A := by dsimp [A]; positivity
-  have hcost : 2/(h*(1-h)^M) ≤ 2*(M:ℝ)*Real.exp (2*(M:ℝ)^ε) := by
-    calc
-      _ = 2*(1/h)*((1-h)^M)⁻¹ := by field_simp
-      _ ≤ 2*(M:ℝ)*Real.exp (2*(M:ℝ)*h) :=
-        mul_le_mul (mul_le_mul_of_nonneg_left hhM (by norm_num))
-          (inverse_mark_cost_le h hhp.le hsmall M) (inv_nonneg.mpr (pow_nonneg hq.le _)) (by positivity)
-      _ = _ := by rw [show 2*(M:ℝ)*h = 2*(M:ℝ)^ε by rw [mul_assoc,hmh]]
-  have hN1 : ((d*M:ℕ):ℝ)+1 ≤ ((d:ℝ)+1)*(M:ℝ) := by push_cast; nlinarith
-  have hpoint :
-      (2/(h*(1-h)^M))*((d*M:ℕ):ℝ)*((d*M:ℕ)+1)*
-        (((d*M:ℕ):ℝ)*a*Real.exp (-L/(16*B))) ≤
-      A*(M:ℝ)^4*Real.exp (2*(M:ℝ)^ε-L/(16*B)) := by
-    calc
-      _ ≤ (2*(M:ℝ)*Real.exp (2*(M:ℝ)^ε))*((d*M:ℕ):ℝ)*(((d:ℝ)+1)*(M:ℝ))*
-          (((d*M:ℕ):ℝ)*a*Real.exp (-L/(16*B))) := by
-        apply mul_le_mul_of_nonneg_right _ (by positivity)
-        exact mul_le_mul (mul_le_mul_of_nonneg_right hcost (by positivity)) hN1
-          (by positivity) (by positivity)
-      _ = _ := by
-        rw [show 2*(M:ℝ)^ε-L/(16*B) = 2*(M:ℝ)^ε+(-L/(16*B)) by ring,Real.exp_add]
-        dsimp [A]
-        push_cast
-        ring
-  have heq : A*(M:ℝ)^4*Real.exp (2*(M:ℝ)^ε-L/(16*B)) =
-      Real.exp (Real.log A+4*Real.log M+2*(M:ℝ)^ε-L/(16*B)) := by
-    rw [show Real.log A+4*Real.log M+2*(M:ℝ)^ε-L/(16*B) =
-      Real.log A+4*Real.log M+(2*(M:ℝ)^ε-L/(16*B)) by ring,Real.exp_add,
-      Real.exp_add,Real.exp_log hAp]
-    have hh : Real.exp (4*Real.log M) = (M:ℝ)^4 := by
-      simpa only [Nat.cast_ofNat,Real.exp_log hMp] using Real.exp_nat_mul (Real.log M) 4
-    rw [hh]
-  rw [heq] at hpoint
-  exact hpoint.trans (Real.exp_le_exp.mpr (by linarith))
-
- 
-
-theorem eventually_ghost_scale (ε c a : ℝ) (d : ℕ) (hε : 0 < ε) (hε1 : ε < 1/2)
-    (hc : 1 ≤ c) (ha : 1 ≤ a) (hd : 1 ≤ d) :
-    ∀ᶠ M : ℕ in Filter.atTop,
-      let h := (M:ℝ)^(ε-1)
-      let B := c*(M:ℝ)^(1/2-ε/4)
-      let L := (M:ℝ)^(1/2+ε)
-      1 ≤ M ∧ 4 ≤ L ∧ 0 < h ∧ h ≤ 1/2 ∧ 1 ≤ B ∧
-        c*Real.sqrt (Real.log (1/h)/h) ≤ B ∧
-        (2/(h*(1-h)^M))*((d*M:ℕ):ℝ)*((d*M:ℕ)+1)*
-          (((d*M:ℕ):ℝ)*a*Real.exp (-L/(16*B))) ≤ Real.exp (-(M:ℝ)^ε) := by
-  have hh : ∀ᶠ m : ℝ in Filter.atTop, m^(ε-1) ≤ 1/2 := by
-    have ht := tendsto_rpow_neg_atTop (show 0 < 1-ε by linarith)
-    filter_upwards [ht.eventually (Iio_mem_nhds (by norm_num : (0:ℝ) < 1/2))] with m hm
-    simpa only [neg_sub] using (show m^(-(1-ε)) < 1/2 from hm).le
-  have hl := (isLittleO_log_rpow_atTop hε).bound (show (0:ℝ) < 1/4 by norm_num)
-  have hl2 := (isLittleO_log_rpow_atTop (show 0 < ε/2 by linarith)).bound zero_lt_one
-  have hg := (tendsto_rpow_atTop (show 0 < ε/4 by linarith)).eventually (Filter.eventually_ge_atTop (80*c))
-  have hC := (tendsto_rpow_atTop hε).eventually
-    (Filter.eventually_ge_atTop (Real.log (2*(d:ℝ)^2*((d:ℝ)+1)*a)))
-  have hL := (tendsto_rpow_atTop (show 0 < 1/2+ε by linarith)).eventually (Filter.eventually_ge_atTop 4)
-  have hnum : ∀ᶠ m : ℝ in Filter.atTop, 1 ≤ m ∧ 4 ≤ m^(1/2+ε) ∧ m^(ε-1) ≤ 1/2 ∧
-      4*Real.log m ≤ m^ε ∧ Real.log m ≤ m^(ε/2) ∧ 80*c ≤ m^(ε/4) ∧
-      Real.log (2*(d:ℝ)^2*((d:ℝ)+1)*a) ≤ m^ε := by
-    filter_upwards [Filter.eventually_ge_atTop (1:ℝ),hh,hl,hl2,hg,hC,hL] with m hm hh hl hl2 hg hC hL
-    have hm0 : 0 ≤ m := le_trans zero_le_one hm
-    simp only [Real.norm_eq_abs,abs_of_nonneg (Real.log_nonneg hm),
-      abs_of_nonneg (Real.rpow_nonneg hm0 ε)] at hl
-    simp only [Real.norm_eq_abs,abs_of_nonneg (Real.log_nonneg hm),
-      abs_of_nonneg (Real.rpow_nonneg hm0 (ε/2)),one_mul] at hl2
-    exact ⟨hm,hL,hh,by linarith,hl2,hg,hC⟩
-  have hcast : Filter.Tendsto (fun M : ℕ => (M:ℝ)) Filter.atTop Filter.atTop := tendsto_natCast_atTop_atTop
-  filter_upwards [hcast.eventually hnum] with M hM
-  obtain ⟨hM,hL,hh,hl,hl2,hg,hC⟩ := hM
-  exact ⟨by exact_mod_cast hM,hL,ghost_scale_estimates M ε c a d hM hε hε1 hc ha hd hh hl hl2 hg hC⟩
-
- 
-theorem ghost_ratio_le_one (M d : ℕ) (hM : 1 ≤ M) (hd : 1 ≤ d)
-    (h B L a ε : ℝ) (hh : 0 < h) (hh2 : h ≤ 1/2) (ha : 0 ≤ a)
-    (hb : (2/(h*(1-h)^M))*((d*M:ℕ):ℝ)*((d*M:ℕ)+1)*
-      (((d*M:ℕ):ℝ)*a*Real.exp (-L/(16*B))) ≤ Real.exp (-(M:ℝ)^ε)) :
-    ((d*M:ℕ):ℝ)*a*Real.exp (-L/(16*B)) ≤ 1 := by
-  have hq : 0 < 1-h := by linarith
-  have hp : (1-h)^M ≤ 1 := pow_le_one₀ hq.le (by linarith)
-  have hden : 0 < h*(1-h)^M := mul_pos hh (pow_pos hq _)
-  have hH : 1 ≤ 2/(h*(1-h)^M) := by
-    apply (le_div_iff₀ hden).mpr
-    have := mul_le_mul_of_nonneg_left hp hh.le
-    nlinarith
-  have hN : (1:ℝ) ≤ (d*M:ℕ) := by exact_mod_cast Nat.mul_le_mul hd hM
-  have hprod : 1 ≤ (2/(h*(1-h)^M))*((d*M:ℕ):ℝ)*((d*M:ℕ)+1) :=
-    one_le_mul_of_one_le_of_one_le (one_le_mul_of_one_le_of_one_le hH hN) (by linarith)
-  have hq0 : 0 ≤ ((d*M:ℕ):ℝ)*a*Real.exp (-L/(16*B)) := by positivity
-  exact (le_mul_of_one_le_left hq0 hprod).trans (hb.trans
-    (Real.exp_le_one_iff.mpr (neg_nonpos.mpr (Real.rpow_nonneg (Nat.cast_nonneg M) ε))))
-
-end GoodnessAsymptotics
-
-section UniformFiniteGoodness
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem finite_bad_goodness_stretched_exp [Nonempty V] (G : BondGraph V E)
-    (d : ℕ) (hd : 0 < d) (pbar : unitInterval) (hpbar : (pbar:ℝ) < 1)
-    (ε : ℝ) (hε : 0 < ε) (hε1 : ε < 1/2) :
-    ∀ᶠ M : ℕ in Filter.atTop, ∀ (Q : Finset E) (x : V),
-      (∀ v, (Q.filter fun e => v ∈ G e).card ≤ d) →
-      ∀ p : unitInterval, p ≤ pbar →
-        finiteBadGoodnessProbability G Q x M ((M:ℝ)^(1/2+ε)) p ≤
-          Real.exp (-(M:ℝ)^ε) := by
-  let c := max 1 (Real.sqrt ((d:ℝ)/2)/(1-pbar))
-  let a := max 1 ((pbar:ℝ)/(1-pbar))
-  have hc : 1 ≤ c := le_max_left ..
-  have ha : 1 ≤ a := le_max_left ..
-  filter_upwards [eventually_ghost_scale ε c a d hε hε1 hc ha hd] with M hM
-  dsimp only at hM
-  obtain ⟨hM1,hL,hh,hh2,hB,hAB,hpref⟩ := hM
-  intro Q x hdeg p hpp
-  have hpp' : (p:ℝ) ≤ pbar := hpp
-  have hp : (p:ℝ) < 1 := hpp'.trans_lt hpbar
-  let h : unitInterval := ⟨(M:ℝ)^(ε-1),hh.le,by linarith⟩
-  let B := c*(M:ℝ)^(1/2-ε/4)
-  let L := (M:ℝ)^(1/2+ε)
-  have hodds : max 1 ((p:ℝ)/(1-p)) ≤ a := by
-    apply max_le (le_max_left ..)
-    have hr : (p:ℝ)/(1-p) ≤ (pbar:ℝ)/(1-pbar) :=
-      div_le_div₀ pbar.2.1 hpp' (by linarith) (by linarith)
-    exact hr.trans (le_max_right ..)
-  have hcoeff : Real.sqrt ((d:ℝ)/2)/(1-p) ≤ c := by
-    have hr : Real.sqrt ((d:ℝ)/2)/(1-p) ≤ Real.sqrt ((d:ℝ)/2)/(1-pbar) :=
-      div_le_div_of_nonneg_left (Real.sqrt_nonneg _) (by linarith) (by linarith)
-    exact hr.trans (le_max_right ..)
-  have hsqrt : 0 ≤ Real.sqrt (Real.log (1/(h:ℝ))/h) := Real.sqrt_nonneg _
-  have hqbar := ghost_ratio_le_one M d hM1 hd h B L a ε hh hh2
-    (le_trans zero_le_one ha) hpref
-  have hqcmp : ((d*M:ℕ):ℝ)*max 1 ((p:ℝ)/(1-p))*Real.exp (-L/(16*B)) ≤
-      ((d*M:ℕ):ℝ)*a*Real.exp (-L/(16*B)) :=
-    mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left hodds (Nat.cast_nonneg _)) (Real.exp_nonneg _)
-  have hg := finiteBadGoodnessProbability_geometric G Q x M d L (Nat.mul_le_mul hd hM1)
-    hL hd hdeg p h hp hh (by dsimp [h]; linarith) B hB
-    ((mul_le_mul_of_nonneg_right hcoeff hsqrt).trans hAB) (hqcmp.trans hqbar)
-  refine hg.trans ((mul_le_mul_of_nonneg_left hqcmp ?_).trans hpref)
-  have hnon : 0 ≤ 1-(h:ℝ) := sub_nonneg.mpr h.2.2
-  positivity
-
-end UniformFiniteGoodness
-
-section ClusterLocality
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem reachBall_le_step (G : BondGraph V E) (x : V) (n : ℕ) :
-    reachBall G x n ⊆ reachBall G x (n+1) := fun _ h => Or.inl h
-
-theorem reachBall_mono_local (G : BondGraph V E) (x : V) : Monotone (reachBall G x) :=
-  monotone_nat_of_le_succ (reachBall_le_step G x)
-
-theorem walk_mem_reachBall (G : BondGraph V E) (ω : Set E) {a b x : V}
-    (w : (openSimpleGraph G ω).Walk a b) (n : ℕ) (ha : a ∈ reachBall G x n) :
-    b ∈ reachBall G x (n+w.length) := by
-  induction w generalizing n with
-  | nil => simpa using ha
-  | @cons a z b hz w ih =>
-    have hz' : z ∈ reachBall G x (n+1) := by
-      apply Or.inr
-      simp only [Set.mem_iUnion,Set.mem_ofPred_eq]
-      obtain ⟨e,he,hend⟩ := hz.2
-      exact ⟨a,ha,e,hend⟩
-    simpa only [SimpleGraph.Walk.length_cons,Nat.add_assoc,Nat.add_left_comm,Nat.add_comm]
-      using ih (n+1) hz'
-
- 
-theorem finite_cluster_subset_reachBall (G : BondGraph V E) (ω : Set E) (x : V)
-    (hfin : (cluster G ω x).Finite) (M : ℕ) (hM : (cluster G ω x).ncard ≤ M) :
-    cluster G ω x ⊆ reachBall G x M := by
-  intro y hy
-  obtain ⟨w⟩ := (conn_iff_reachable G ω x y).mp hy
-  let q := w.toPath
-  have hs : q.val.support.toFinset ⊆ hfin.toFinset := by
-    intro z hz
-    have hz' : z ∈ q.val.support := List.mem_toFinset.mp hz
-    exact hfin.mem_toFinset.mpr ((conn_iff_reachable G ω x z).mpr ⟨q.val.takeUntil z hz'⟩)
-  have hc : q.val.length+1 ≤ (cluster G ω x).ncard := by
-    have hc := Finset.card_le_card hs
-    rwa [List.toFinset_card_of_nodup q.property.support_nodup,
-      SimpleGraph.Walk.length_support,← Set.ncard_eq_toFinset_card _ hfin] at hc
-  have hr := walk_mem_reachBall G ω q.val 0 (Set.mem_singleton x)
-  simp only [Nat.zero_add] at hr
-  exact reachBall_mono_local G x (by omega) hr
-
- 
-
-theorem cluster_restrict_incident_eq (G : BondGraph V E) (hloc : LocallyFinite G)
-    (Λ : Finset V) (ω : Set E) (x : V)
-    (hsub : cluster G (ω ∩ (incidentSupport G hloc Λ : Set E)) x ⊆ Λ) :
-    cluster G (ω ∩ (incidentSupport G hloc Λ : Set E)) x = cluster G ω x := by
-  apply Set.Subset.antisymm
-  · exact fun _ h => conn_mono G Set.inter_subset_left h
-  · intro y hy
-    induction hy with
-    | refl => exact conn_refl G _ x
-    | @tail a b _ hab ih =>
-      obtain ⟨e,he,hend⟩ := hab
-      have heQ : e ∈ incidentSupport G hloc Λ :=
-        (mem_incidentSupport G hloc Λ e).mpr ⟨a,hsub ih,by simp [hend]⟩
-      exact ih.tail ⟨e,⟨he,heQ⟩,hend⟩
-
-theorem cluster_restrict_incident_of_full (G : BondGraph V E) (hloc : LocallyFinite G)
-    (Λ : Finset V) (ω : Set E) (x : V) (hsub : cluster G ω x ⊆ Λ) :
-    cluster G (ω ∩ (incidentSupport G hloc Λ : Set E)) x = cluster G ω x := by
-  apply cluster_restrict_incident_eq G hloc Λ ω x
-  exact (fun _ h => hsub (conn_mono G Set.inter_subset_left h))
-
- 
-theorem capped_cluster_local (G : BondGraph V E) (hloc : LocallyFinite G)
-    (x : V) (M : ℕ) (ω : Set E) :
-    let Q := incidentSupport G hloc (finite_reachBall hloc x M).toFinset
-    ((cluster G ω x).Finite ∧ (cluster G ω x).ncard ≤ M) ↔
-      ((cluster G (ω ∩ (Q : Set E)) x).Finite ∧
-        (cluster G (ω ∩ (Q : Set E)) x).ncard ≤ M) := by
-  dsimp only
-  let Λ := (finite_reachBall hloc x M).toFinset
-  let Q := incidentSupport G hloc Λ
-  constructor
-  · rintro ⟨hfin,hcap⟩
-    have hsub : cluster G ω x ⊆ Λ := by
-      simpa only [Λ,Set.Finite.coe_toFinset] using finite_cluster_subset_reachBall G ω x hfin M hcap
-    rw [cluster_restrict_incident_of_full G hloc Λ ω x hsub]
-    exact ⟨hfin,hcap⟩
-  · rintro ⟨hfin,hcap⟩
-    have hsub : cluster G (ω ∩ (Q : Set E)) x ⊆ Λ := by
-      simpa only [Λ,Set.Finite.coe_toFinset] using
-        finite_cluster_subset_reachBall G (ω ∩ (Q : Set E)) x hfin M hcap
-    have heq := cluster_restrict_incident_eq G hloc Λ ω x hsub
-    rw [heq] at hfin hcap
-    exact ⟨hfin,hcap⟩
-
-end ClusterLocality
-
-section InfiniteGoodness
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-noncomputable def pivotalDistance (G : BondGraph V E) (ω : Set E) (a b : V) : ℕ :=
-  {e | OpenMarkedPivotal G ω {b} a e}.ncard
-
- 
-def contactSet (G : BondGraph V E) (ω : Set E) (W : Set V) : Set E :=
-  {e | e ∈ ω ∧ CutExists G W e}
-
- 
-def GoodForScale (G : BondGraph V E) (ω : Set E) (x : V) (L : ℝ) : Prop :=
-  ∀ W : Finset V, (W : Set V) ⊆ cluster G ω x →
-    (∀ a ∈ W, ∀ b ∈ W, Conn G (ω ∩ internalEdges G (W : Set V)) a b) →
-    0 < (contactSet G ω (W : Set V)).ncard →
-    ∀ a ∈ W, ∀ b ∈ W,
-      (pivotalDistance G (ω ∩ internalEdges G (W : Set V)) a b : ℝ) ≤
-        (contactSet G ω (W : Set V)).ncard*L
-
- 
-def CappedBadGoodness (G : BondGraph V E) (ω : Set E) (x : V) (M : ℕ) (L : ℝ) : Prop :=
-  (cluster G ω x).Finite ∧ (cluster G ω x).ncard ≤ M ∧ ¬GoodForScale G ω x L
-
-theorem pivotalDistance_finset (G : BondGraph V E) (Q ω : Finset E) (hω : ω ⊆ Q) (a b : V) :
-    pivotalDistance G (ω : Set E) a b = (pairOpenPivotals G Q a b ω).card := by
-  have he : {e | OpenMarkedPivotal G (ω : Set E) {b} a e} =
-      (pairOpenPivotals G Q a b ω : Set E) := by
-    ext e
-    simp only [Finset.mem_coe,mem_pairOpenPivotals,Set.mem_ofPred_eq]
-    exact ⟨fun h => ⟨hω h.1,h⟩,And.right⟩
-  simp only [pivotalDistance,he,Set.ncard_coe_finset]
-
-theorem contactSet_finset (G : BondGraph V E) (ω : Finset E) (W : Set V) :
-    contactSet G (ω : Set E) W = (openContacts G ω W : Set E) := by
-  ext e
-  simp only [contactSet,openContacts,Finset.mem_coe,Finset.mem_filter,Set.mem_ofPred_eq]
-
-theorem finiteGoodForScale_iff (G : BondGraph V E) (Q ω : Finset E) (hω : ω ⊆ Q)
-    (x : V) (L : ℝ) : FiniteGoodForScale G Q x L ω ↔ GoodForScale G (ω : Set E) x L := by
-  have hroot : ∀ W : Finset V, W ⊆ finiteRootComponent G ω x ↔ (W : Set V) ⊆ cluster G ω x := by
-    intro W
-    rw [← coe_finiteRootComponent]
-    exact Finset.coe_subset.symm
-  have hpiv (W : Finset V) (a b : V) := pivotalDistance_finset G Q
-    (internalBonds G ω (W : Set V)) ((Finset.filter_subset _ _).trans hω) a b
-  simp only [FiniteGoodForScale,GoodForScale,hroot,← coe_internalBonds,contactSet_finset,
-    Set.ncard_coe_finset,hpiv]
-
-theorem cappedBadGoodness_finset_iff (G : BondGraph V E) (Q ω : Finset E) (hω : ω ⊆ Q)
-    (x : V) (M : ℕ) (L : ℝ) :
-    CappedBadGoodness G (ω : Set E) x M L ↔ CappedBadGoodnessEvent G Q x M L ω := by
-  have hfin := cluster_finite_of_finite_edges G ω.finite_toSet x
-  simp only [CappedBadGoodness,CappedBadGoodnessEvent,hfin,true_and]
-  rw [← coe_finiteRootComponent,Set.ncard_coe_finset,finiteGoodForScale_iff G Q ω hω]
-
- 
-theorem internal_config_eq_of_incident (G : BondGraph V E) (ω η : Set E) (C W : Set V)
-    (hW : W ⊆ C) (he : ∀ e, (∃ v ∈ C, v ∈ G e) → (e ∈ ω ↔ e ∈ η)) :
-    ω ∩ internalEdges G W = η ∩ internalEdges G W := by
-  ext e
-  by_cases hi : e ∈ internalEdges G W
-  · have hinc : ∃ v ∈ C, v ∈ G e :=
-      ⟨(G e).out.1,hW (hi _ (Sym2.out_fst_mem _)),Sym2.out_fst_mem _⟩
-    simp only [Set.mem_inter_iff,hi,and_true,he e hinc]
-  · simp only [Set.mem_inter_iff,hi,and_false]
-
-theorem contacts_eq_of_incident (G : BondGraph V E) (ω η : Set E) (C W : Set V)
-    (hW : W ⊆ C) (he : ∀ e, (∃ v ∈ C, v ∈ G e) → (e ∈ ω ↔ e ∈ η)) :
-    contactSet G ω W = contactSet G η W := by
-  ext e
-  by_cases hi : CutExists G W e
-  · obtain ⟨a,b,hab,ha,_⟩ := hi
-    have hinc : ∃ v ∈ C, v ∈ G e := ⟨a,hW ha,by simp [hab]⟩
-    simp only [contactSet,Set.mem_ofPred_eq,he e hinc]
-  · simp only [contactSet,Set.mem_ofPred_eq,hi,and_false]
-
- 
-theorem goodForScale_congr (G : BondGraph V E) (ω η : Set E) (x : V) (L : ℝ)
-    (hC : cluster G ω x = cluster G η x)
-    (he : ∀ e, (∃ v ∈ cluster G ω x, v ∈ G e) → (e ∈ ω ↔ e ∈ η)) :
-    GoodForScale G ω x L ↔ GoodForScale G η x L := by
-  unfold GoodForScale
-  apply forall_congr'
-  intro W
-  rw [← hC]
-  apply forall_congr'
-  intro hW
-  rw [internal_config_eq_of_incident G ω η _ _ hW he,contacts_eq_of_incident G ω η _ _ hW he]
-
-theorem goodForScale_restrict_incident (G : BondGraph V E) (hloc : LocallyFinite G)
-    (Λ : Finset V) (ω : Set E) (x : V) (L : ℝ) (hsub : cluster G ω x ⊆ Λ) :
-    GoodForScale G ω x L ↔ GoodForScale G (ω ∩ (incidentSupport G hloc Λ : Set E)) x L := by
-  apply goodForScale_congr G ω _ x L
-    (cluster_restrict_incident_of_full G hloc Λ ω x hsub).symm
-  intro e he
-  obtain ⟨v,hv,hve⟩ := he
-  have hQ := (mem_incidentSupport G hloc Λ e).mpr ⟨v,hsub hv,hve⟩
-  simp only [Set.mem_inter_iff,Finset.mem_coe,hQ,and_true]
-
- 
-theorem cappedBadGoodness_local (G : BondGraph V E) (hloc : LocallyFinite G)
-    (x : V) (M : ℕ) (L : ℝ) (ω : Set E) :
-    let Q := incidentSupport G hloc (finite_reachBall hloc x M).toFinset
-    CappedBadGoodness G ω x M L ↔ CappedBadGoodness G (ω ∩ (Q : Set E)) x M L := by
-  let Λ := (finite_reachBall hloc x M).toFinset
-  let Q := incidentSupport G hloc Λ
-  change CappedBadGoodness G ω x M L ↔ CappedBadGoodness G (ω ∩ (Q : Set E)) x M L
-  have hcap := capped_cluster_local G hloc x M ω
-  change ((cluster G ω x).Finite ∧ (cluster G ω x).ncard ≤ M) ↔
-    ((cluster G (ω ∩ (Q : Set E)) x).Finite ∧ (cluster G (ω ∩ (Q : Set E)) x).ncard ≤ M) at hcap
-  have hg : ((cluster G ω x).Finite ∧ (cluster G ω x).ncard ≤ M) →
-      (GoodForScale G ω x L ↔ GoodForScale G (ω ∩ (Q : Set E)) x L) := by
-    rintro ⟨hfin,hM⟩
-    apply goodForScale_restrict_incident G hloc Λ ω x L
-    simpa only [Λ,Set.Finite.coe_toFinset] using finite_cluster_subset_reachBall G ω x hfin M hM
-  constructor
-  · rintro ⟨hfin,hM,hbad⟩
-    exact ⟨(hcap.mp ⟨hfin,hM⟩).1,(hcap.mp ⟨hfin,hM⟩).2,fun hh => hbad ((hg ⟨hfin,hM⟩).mpr hh)⟩
-  · rintro ⟨hfin,hM,hbad⟩
-    have hc := hcap.mpr ⟨hfin,hM⟩
-    exact ⟨hc.1,hc.2,fun hh => hbad ((hg hc).mp hh)⟩
-
-end InfiniteGoodness
-
-section GoodnessLaw
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem measurableSet_finite_restriction (Q : Finset E) (A : Set (Set E)) :
-    MeasurableSet {ω : Set E | ω ∩ (Q : Set E) ∈ A} := by
-  have hm : MeasurableSet {η : Set (Q : Set E) | (Subtype.val '' η : Set E) ∈ A} := .of_discrete
-  convert hm.preimage (measurable_set_restrict (Q : Set E)) using 1
-  ext ω
-  have he : (Subtype.val '' (Subtype.val ⁻¹' ω : Set (Q : Set E)) : Set E) = ω ∩ (Q : Set E) := by
-    ext e
-    simp [and_comm]
-  simp only [Set.mem_preimage,Set.mem_ofPred_eq,he]
-
-theorem measurableSet_cappedBadGoodness (G : BondGraph V E) (hloc : LocallyFinite G)
-    (x : V) (M : ℕ) (L : ℝ) : MeasurableSet {ω | CappedBadGoodness G ω x M L} := by
-  let Q := incidentSupport G hloc (finite_reachBall hloc x M).toFinset
-  have he : {ω | CappedBadGoodness G ω x M L} =
-      {ω | ω ∩ (Q : Set E) ∈ {η | CappedBadGoodness G η x M L}} := by
-    ext ω
-    exact cappedBadGoodness_local G hloc x M L ω
-  rw [he]
-  exact measurableSet_finite_restriction Q _
-
- 
-theorem bad_goodness_law_eq_finite (G : BondGraph V E) (hloc : LocallyFinite G)
-    (x : V) (M : ℕ) (L : ℝ) (p : unitInterval) :
-    (law p).real {ω | CappedBadGoodness G ω x M L} =
-      finiteBadGoodnessProbability G
-        (incidentSupport G hloc (finite_reachBall hloc x M).toFinset) x M L p := by
-  let Q := incidentSupport G hloc (finite_reachBall hloc x M).toFinset
-  have he : finiteBadGoodnessProbability G Q x M L p =
-      finiteEventPolynomial Q {ω | CappedBadGoodness G ω x M L} p := by
-    apply bernoulliPolynomial_congr
-    intro ω hω
-    simp only [Set.mem_ofPred_eq,cappedBadGoodness_finset_iff G Q ω hω]
-  change _ = finiteBadGoodnessProbability G Q x M L p
-  rw [he,finiteEventPolynomial_restrict]
-  congr 1
-  ext ω
-  exact cappedBadGoodness_local G hloc x M L ω
-
-theorem incident_filter_card_le_degree (G : BondGraph V E) (hloc : LocallyFinite G)
-    (Q : Finset E) (x : V) : (Q.filter fun e => x ∈ G e).card ≤ degree G x := by
-  let : Fintype (Step G x) := (finite_steps G hloc x).fintype
-  have hex (e : {e // e ∈ Q.filter fun e => x ∈ G e}) : ∃ y, G e.val = s(x,y) :=
-    Sym2.mem_iff_exists.mp (Finset.mem_filter.mp e.property).2
-  let f : {e // e ∈ Q.filter fun e => x ∈ G e} → Step G x :=
-    fun e => ⟨(e.val,(hex e).choose),(hex e).choose_spec⟩
-  have hf : Function.Injective f := by
-    intro a b hab
-    exact Subtype.ext (congrArg (fun z : Step G x => z.val.1) hab)
-  simpa only [degree,Nat.card_eq_fintype_card,Fintype.card_coe] using Fintype.card_le_of_injective f hf
-
- 
-
-theorem bad_goodness_stretched_exp [Nonempty V] (G : BondGraph V E)
-    (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (pbar : unitInterval) (hpbar : (pbar:ℝ) < 1) (ε : ℝ) (hε : 0 < ε) (hε1 : ε < 1/2) :
-    ∀ᶠ M : ℕ in Filter.atTop, ∀ x : V, ∀ p : unitInterval, p ≤ pbar →
-      (law p).real {ω | CappedBadGoodness G ω x M ((M:ℝ)^(1/2+ε))} ≤
-        Real.exp (-(M:ℝ)^ε) := by
-  obtain ⟨d,hd,hdeg⟩ := exists_degree_bound G hqt
-  filter_upwards [finite_bad_goodness_stretched_exp G d hd pbar hpbar ε hε hε1] with M hM
-  intro x p hp
-  rw [bad_goodness_law_eq_finite G hloc]
-  exact hM _ x (fun v => (incident_filter_card_le_degree G hloc _ v).trans (hdeg v)) p hp
-
-end GoodnessLaw
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Exploration.Moments
+
+namespace OAI
+
+/-! Pruning, finite and infinite goodness events and uniform bounds. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+section FiniteCappedRadius
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+noncomputable def pairOpenPivotals (G : BondGraph V E) (Q : Finset E) (x y : V)
+    (ω : Finset E) : Finset E :=
+  Q.filter fun e => e ∈ ω ∧ Conn G (ω : Set E) x y ∧ ¬Conn G (ω.erase e : Set E) x y
+
+ 
+def CappedRadiusEvent (G : BondGraph V E) (Q : Finset E) (x : V) (M r : ℕ)
+    (ω : Finset E) : Prop :=
+  (finiteRootComponent G ω x).card ≤ M ∧
+    ∃ y, Conn G (ω : Set E) x y ∧ r ≤ (pairOpenPivotals G Q x y ω).card
+
+noncomputable def finiteCappedRadiusTail (G : BondGraph V E) (Q : Finset E)
+    (x : V) (M r : ℕ) (p : ℝ) : ℝ :=
+  bernoulliPolynomial Q (fun ω => if CappedRadiusEvent G Q x M r ω then 1 else 0) p
+
+ 
+theorem sole_root_mark (G : BondGraph V E) (Q : Finset E) (x y : V) (ω : Finset E)
+    (m : Finset V) (hxy : Conn G (ω : Set E) x y)
+    (hm : m ∩ finiteRootComponent G ω x = {y}) :
+    MarkedEvent G ω m x ∧ markedOpenPivotals G Q x ω m = pairOpenPivotals G Q x y ω := by
+  have hyW : y ∈ finiteRootComponent G ω x := by
+    change y ∈ (finiteRootComponent G ω x : Set V)
+    rwa [coe_finiteRootComponent]
+  have hym : y ∈ m := (Finset.mem_inter.mp (hm ▸ Finset.mem_singleton_self y)).1
+  have hmconn (η : Finset E) (hη : η ⊆ ω) :
+      MarkedEvent G η m x ↔ Conn G (η : Set E) x y := by
+    constructor
+    · rintro ⟨z,hzm,hxz⟩
+      have hzW : z ∈ finiteRootComponent G ω x := by
+        change z ∈ (finiteRootComponent G ω x : Set V)
+        rw [coe_finiteRootComponent]
+        exact conn_mono G (by exact_mod_cast hη) hxz
+      have hzy : z = y := Finset.mem_singleton.mp (hm ▸ Finset.mem_inter.mpr ⟨hzm,hzW⟩)
+      rwa [hzy] at hxz
+    · exact fun h => ⟨y,hym,h⟩
+  refine ⟨⟨y,hym,hxy⟩,?_⟩
+  ext e
+  simp only [markedOpenPivotals,pairOpenPivotals,Finset.mem_filter]
+  change (e ∈ Q ∧ e ∈ ω ∧ MarkedEvent G ω m x ∧ ¬MarkedEvent G (ω.erase e) m x) ↔ _
+  rw [hmconn ω Finset.Subset.rfl,hmconn (ω.erase e) (Finset.erase_subset e ω)]
+
+ 
+theorem bernoulliPolynomial_exact_indicator (F S : Finset V) (hSF : S ⊆ F) (h : ℝ) :
+    bernoulliPolynomial F (fun m => if m = S then (1:ℝ) else 0) h =
+      h^S.card*(1-h)^(F.card-S.card) := by
+  unfold bernoulliPolynomial
+  rw [Finset.sum_eq_single_of_mem S (Finset.mem_powerset.mpr hSF)]
+  · simp
+  · intro m hm hne
+    simp [hne]
+
+ 
+theorem bernoulliPolynomial_sole_root_mark (G : BondGraph V E) (Q : Finset E)
+    (F : Finset V) (x y : V) (ω : Finset E) (hω : ω ⊆ Q)
+    (hF : insert x (searchVertices G Q) ⊆ F) (hxy : Conn G (ω : Set E) x y) (h : ℝ) :
+    bernoulliPolynomial F (fun m => if m ∩ finiteRootComponent G ω x = {y} then (1:ℝ) else 0) h =
+      h*(1-h)^((finiteRootComponent G ω x).card-1) := by
+  rw [bernoulliPolynomial_restrict _ _ (finiteRootComponent_subset G Q F x ω hω hF)
+    (fun a => if a = {y} then (1:ℝ) else 0) h]
+  have hy : y ∈ finiteRootComponent G ω x := by
+    change y ∈ (finiteRootComponent G ω x : Set V)
+    rwa [coe_finiteRootComponent]
+  rw [bernoulliPolynomial_exact_indicator _ {y} (Finset.singleton_subset_iff.mpr hy)]
+  simp
+
+ 
+theorem cappedRadius_mark_comparison (G : BondGraph V E) (Q : Finset E) (F : Finset V)
+    (x : V) (M r : ℕ) (p h : unitInterval)
+    (hF : insert x (searchVertices G Q) ⊆ F) :
+    (h:ℝ)*(1-h)^M*finiteCappedRadiusTail G Q x M r p ≤ finiteMarkedTail G Q F x r h p := by
+  unfold finiteCappedRadiusTail finiteMarkedTail
+  rw [← bernoulliPolynomial_const_mul]
+  apply bernoulliPolynomial_mono _ _ _ _ p.2
+  intro ω hω
+  by_cases he : CappedRadiusEvent G Q x M r ω
+  · obtain ⟨hcap,y,hxy,hr⟩ := he
+    have he : CappedRadiusEvent G Q x M r ω := ⟨hcap,y,hxy,hr⟩
+    simp only [he,ite_true,mul_one]
+    calc
+      (h:ℝ)*(1-h)^M ≤ (h:ℝ)*(1-h)^((finiteRootComponent G ω x).card-1) := by
+        apply mul_le_mul_of_nonneg_left _ h.2.1
+        exact pow_le_pow_of_le_one (sub_nonneg.mpr h.2.2) (by linarith [h.2.1])
+          (Nat.le_trans (Nat.sub_le _ _) hcap)
+      _ = bernoulliPolynomial F
+          (fun m => if m ∩ finiteRootComponent G ω x = {y} then (1:ℝ) else 0) h :=
+        (bernoulliPolynomial_sole_root_mark G Q F x y ω hω hF hxy h).symm
+      _ ≤ _ := by
+        apply bernoulliPolynomial_mono _ _ _ _ h.2
+        intro m hm
+        by_cases hs : m ∩ finiteRootComponent G ω x = {y}
+        · obtain ⟨heq,hpiv⟩ := sole_root_mark G Q x y ω m hxy hs
+          have hlarge : r ≤ (markedOpenPivotals G Q x ω m).card := hpiv ▸ hr
+          simp only [hs,heq,hlarge,and_self,ite_true,le_refl]
+        · simp only [hs,ite_false]
+          split_ifs <;> norm_num
+  · simp only [he,ite_false,mul_zero]
+    apply bernoulliPolynomial_nonneg _ _ _ h.2
+    intro m hm
+    split_ifs <;> norm_num
+
+theorem finiteMarkedMoment_zero_le_one (G : BondGraph V E) (Q : Finset E) (F : Finset V)
+    (x : V) (p h : unitInterval) : finiteMarkedMoment G Q F x 0 h p ≤ 1 := by
+  unfold finiteMarkedMoment
+  rw [← bernoulliPolynomial_const Q (1:ℝ) (p:ℝ)]
+  apply bernoulliPolynomial_mono _ _ _ _ p.2
+  intro ω hω
+  rw [← bernoulliPolynomial_const F (1:ℝ) (h:ℝ)]
+  apply bernoulliPolynomial_mono _ _ _ _ h.2
+  intro m hm
+  rw [markedMomentValue_zero]
+  split_ifs <;> norm_num
+
+ 
+theorem finiteCappedRadiusTail_exponential [Nonempty V] (G : BondGraph V E)
+    (p h : unitInterval) (hp : (p:ℝ) < 1) (hh : 0 < (h:ℝ)) (hh1 : (h:ℝ) < 1)
+    (d : ℕ) (hd : 0 < d) (Q : Finset E) (x : V)
+    (hdeg : ∀ v, (Q.filter (fun e => v ∈ G e)).card ≤ d)
+    (B : ℝ) (hB : 1 ≤ B)
+    (hAB : (Real.sqrt ((d:ℝ)/2)/(1-p))*Real.sqrt (Real.log (1/(h:ℝ))/h) ≤ B) (M r : ℕ) :
+    finiteCappedRadiusTail G Q x M r p ≤
+      (2/((h:ℝ)*(1-h)^M))*Real.exp (-(r:ℝ)/(4*B)) := by
+  let F := insert x (searchVertices G Q)
+  have hcomp := cappedRadius_mark_comparison G Q F x M r p h Finset.Subset.rfl
+  have ht := finiteMarkedTail_exponential G p h hp hh d hd Q F x hdeg Finset.Subset.rfl B hB hAB r
+  have hm := finiteMarkedMoment_zero_le_one G Q F x p h
+  have hb : 0 < (h:ℝ)*(1-h)^M := mul_pos hh (pow_pos (sub_pos.mpr hh1) _)
+  have hbound : (h:ℝ)*(1-h)^M*finiteCappedRadiusTail G Q x M r p ≤
+      2*Real.exp (-(r:ℝ)/(4*B)) := by
+    have hpos : 0 ≤ 2*Real.exp (-(r:ℝ)/(4*B)) := by positivity
+    have hl := mul_le_mul_of_nonneg_left hm hpos
+    rw [mul_one] at hl
+    exact hcomp.trans (ht.trans hl)
+  calc
+    _ ≤ (2*Real.exp (-(r:ℝ)/(4*B)))/((h:ℝ)*(1-h)^M) :=
+      (le_div_iff₀ hb).mpr (by simpa only [mul_comm] using hbound)
+    _ = _ := by ring
+
+end FiniteCappedRadius
+
+section PruningGeometry
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem pairOpenPivotals_symm (G : BondGraph V E) (Q : Finset E) (ω : Finset E) (x y : V) :
+    pairOpenPivotals G Q x y ω = pairOpenPivotals G Q y x ω := by
+  ext e
+  simp only [pairOpenPivotals,Finset.mem_filter]
+  have h₁ : Conn G (ω : Set E) x y ↔ Conn G (ω : Set E) y x :=
+    ⟨conn_symm G _,conn_symm G _⟩
+  have h₂ : Conn G (ω.erase e : Set E) x y ↔ Conn G (ω.erase e : Set E) y x :=
+    ⟨conn_symm G _,conn_symm G _⟩
+  rw [h₁,h₂]
+
+ 
+theorem pairOpenPivotals_triangle_subset (G : BondGraph V E) (Q : Finset E) (ω : Finset E)
+    (x y z : V) (hxy : Conn G (ω : Set E) x y) (hyz : Conn G (ω : Set E) y z) :
+    pairOpenPivotals G Q x z ω ⊆ pairOpenPivotals G Q x y ω ∪ pairOpenPivotals G Q y z ω := by
+  intro e he
+  obtain ⟨heQ,heω,_,hnxz⟩ := Finset.mem_filter.mp he
+  by_cases hxy' : Conn G (ω.erase e : Set E) x y
+  · exact Finset.mem_union_right _ (Finset.mem_filter.mpr
+      ⟨heQ,heω,hyz,fun hyz' => hnxz (hxy'.trans hyz')⟩)
+  · exact Finset.mem_union_left _ (Finset.mem_filter.mpr ⟨heQ,heω,hxy,hxy'⟩)
+
+theorem pairOpenPivotals_triangle (G : BondGraph V E) (Q : Finset E) (ω : Finset E)
+    (x y z : V) (hxy : Conn G (ω : Set E) x y) (hyz : Conn G (ω : Set E) y z) :
+    (pairOpenPivotals G Q x z ω).card ≤
+      (pairOpenPivotals G Q x y ω).card+(pairOpenPivotals G Q y z ω).card :=
+  (Finset.card_le_card (pairOpenPivotals_triangle_subset G Q ω x y z hxy hyz)).trans
+    (Finset.card_union_le _ _)
+
+ 
+theorem conn_no_crossing_internal (G : BondGraph V E) (W : Set V) (ω : Set E)
+    (hc : ∀ e ∈ ω, ¬CutExists G W e) {x y : V} (hx : x ∈ W) (hxy : Conn G ω x y) :
+    Conn G (ω ∩ internalEdges G W) x y ∧ y ∈ W := by
+  induction hxy with
+  | refl => exact ⟨conn_refl _ _ _,hx⟩
+  | @tail a b hxa hab ih =>
+    obtain ⟨e,he,heq⟩ := hab
+    have hb : b ∈ W := by
+      by_contra hn
+      exact hc e he ⟨a,b,heq,ih.2,hn⟩
+    exact ⟨ih.1.tail ⟨e,⟨he,(mem_internalEdges_of_ends G W e a b heq).mpr ⟨ih.2,hb⟩⟩,heq⟩,hb⟩
+
+ 
+theorem conn_one_attachment_internal (G : BondGraph V E) (W : Set V) (ω : Set E)
+    (e : E) (he : e ∈ ω) (hec : CutExists G W e)
+    (hone : ∀ f ∈ ω, CutExists G W f → f = e)
+    {x y : V} (hx : x ∈ W) (hy : y ∈ W) (hxy : Conn G ω x y) :
+    Conn G (ω ∩ internalEdges G W) x y := by
+  obtain ⟨a,b,hab,ha,hb⟩ := hec
+  let η := ω \ {e}
+  have hn : ∀ f ∈ η, ¬CutExists G W f := by
+    intro f hf hc
+    exact hf.2 (Set.mem_singleton_iff.mpr (hone f hf.1 hc))
+  have hη (u v : V) (hu : u ∈ W) (huv : Conn G η u v) :=
+    conn_no_crossing_internal G W η hn hu huv
+  have hω : insert e η = ω := by dsimp [η]; ext f; simp [he]
+  have hw : Conn G (insert e η) x y := hω ▸ hxy
+  rcases (conn_insert_iff G η e a b x y hab).mp hw with h | ⟨_,h⟩ | ⟨h,_⟩
+  · exact conn_mono G (show η ∩ internalEdges G W ⊆ ω ∩ internalEdges G W from
+      fun _ hf => ⟨hf.1.1,hf.2⟩) (hη x y hx h).1
+  · exact (hb (hη y b hy (conn_symm G η h)).2).elim
+  · exact (hb (hη x b hx h).2).elim
+
+noncomputable def internalBonds (G : BondGraph V E) (ω : Finset E) (W : Set V) : Finset E :=
+  ω.filter fun e => e ∈ internalEdges G W
+
+@[simp] theorem coe_internalBonds (G : BondGraph V E) (ω : Finset E) (W : Set V) :
+    (internalBonds G ω W : Set E) = (ω : Set E) ∩ internalEdges G W := by
+  ext e; simp [internalBonds]
+
+ 
+theorem pairOpenPivotals_one_attachment (G : BondGraph V E) (Q ω : Finset E) (W : Set V)
+    (e : E) (he : e ∈ ω) (hec : CutExists G W e)
+    (hone : ∀ f ∈ ω, CutExists G W f → f = e)
+    {x y : V} (hx : x ∈ W) (hy : y ∈ W) :
+    pairOpenPivotals G Q x y (internalBonds G ω W) ⊆ pairOpenPivotals G Q x y ω := by
+  intro f hf
+  obtain ⟨hfQ,hfω,hxy,hnxy⟩ := Finset.mem_filter.mp hf
+  have hfint : f ∈ internalEdges G W := (Finset.mem_filter.mp hfω).2
+  have hfe : e ≠ f := by
+    intro hef
+    obtain ⟨a,b,hab,ha,hb⟩ := hec
+    have hh : e ∈ internalEdges G W := hef ▸ hfint
+    exact hb ((mem_internalEdges_of_ends G W e a b hab).mp hh).2
+  have he' : e ∈ ω.erase f := Finset.mem_erase.mpr ⟨hfe,he⟩
+  have hcoe : (internalBonds G ω W).erase f = internalBonds G (ω.erase f) W := by
+    ext g; simp [internalBonds]; tauto
+  refine Finset.mem_filter.mpr ⟨hfQ,(Finset.mem_filter.mp hfω).1,?_,?_⟩
+  · exact conn_mono G (by simpa only [coe_internalBonds] using Set.inter_subset_left) hxy
+  · intro h
+    have hh := conn_one_attachment_internal G W (ω.erase f : Set E) e he' hec
+      (fun g hg hc => hone g (Finset.mem_of_mem_erase hg) hc) hx hy h
+    apply hnxy
+    simpa only [hcoe,coe_internalBonds] using hh
+
+ 
+theorem conn_first_entry (G : BondGraph V E) (W : Set V) (ω : Set E)
+    {x y : V} (hx : x ∉ W) (hy : y ∈ W) (hxy : Conn G ω x y) :
+    ∃ a b e, a ∉ W ∧ b ∈ W ∧ e ∈ ω ∧ G e = s(a,b) ∧
+      Conn G (ω ∩ internalEdges G Wᶜ) x a := by
+  let Entry : Prop := ∃ a b e, a ∉ W ∧ b ∈ W ∧ e ∈ ω ∧ G e = s(a,b) ∧
+      Conn G (ω ∩ internalEdges G Wᶜ) x a
+  have haux : Conn G (ω ∩ internalEdges G Wᶜ) x y ∨ Entry := by
+    clear hy
+    induction hxy with
+    | refl => exact Or.inl (conn_refl _ _ _)
+    | @tail a b hxa hab ih =>
+      rcases ih with ih | ih
+      · have ha : a ∉ W := conn_internal_endpoint G Wᶜ hx ih
+        obtain ⟨e,he,heq⟩ := hab
+        by_cases hb : b ∈ W
+        · exact Or.inr ⟨a,b,e,ha,hb,he,heq,ih⟩
+        · exact Or.inl (ih.tail ⟨e,⟨he,(mem_internalEdges_of_ends G Wᶜ e a b heq).mpr ⟨ha,hb⟩⟩,heq⟩)
+      · exact Or.inr ih
+  rcases haux with h | h
+  · exact ((conn_internal_endpoint G Wᶜ hx h) hy).elim
+  · exact h
+
+ 
+noncomputable def pruneContacts (G : BondGraph V E) (ω : Finset E) (W : Set V) (e : E) : Finset E :=
+  ω.filter fun f => ¬CutExists G W f ∨ f = e
+
+theorem internal_not_cut (G : BondGraph V E) (W : Set V) {e : E}
+    (he : e ∈ internalEdges G W) : ¬CutExists G W e := by
+  rintro ⟨a,b,hab,ha,hb⟩
+  exact hb ((mem_internalEdges_of_ends G W e a b hab).mp he).2
+
+theorem exterior_not_cut (G : BondGraph V E) (W : Set V) {e : E}
+    (he : e ∈ internalEdges G Wᶜ) : ¬CutExists G W e := by
+  rintro ⟨a,b,hab,ha,hb⟩
+  exact ((mem_internalEdges_of_ends G Wᶜ e a b hab).mp he).1 ha
+
+theorem internal_pruneContacts (G : BondGraph V E) (ω : Finset E) (W : Set V) (e : E) :
+    internalBonds G (pruneContacts G ω W e) W = internalBonds G ω W := by
+  ext f
+  simp only [internalBonds,pruneContacts,Finset.mem_filter]
+  constructor
+  · exact fun h => ⟨h.1.1,h.2⟩
+  · exact fun h => ⟨⟨h.1,Or.inl (internal_not_cut G W h.2)⟩,h.2⟩
+
+theorem pruneContacts_only_attachment (G : BondGraph V E) (ω : Finset E) (W : Set V) (e : E) :
+    ∀ f ∈ pruneContacts G ω W e, CutExists G W f → f = e := by
+  intro f hf hc
+  exact (Finset.mem_filter.mp hf).2.resolve_left (not_not.mpr hc)
+
+ 
+theorem exists_pruneContacts_root (G : BondGraph V E) (ω : Finset E) (W : Set V) (x : V)
+    (hWroot : W ⊆ cluster G ω x) (hWconn : ∀ a ∈ W, ∀ b ∈ W,
+      Conn G (ω ∩ internalEdges G W) a b)
+    (hcontact : ∃ e ∈ ω, CutExists G W e) :
+    ∃ e ∈ ω, CutExists G W e ∧ W ⊆ cluster G (pruneContacts G ω W e) x := by
+  by_cases hx : x ∈ W
+  · obtain ⟨e,he,hec⟩ := hcontact
+    refine ⟨e,he,hec,?_⟩
+    intro y hy
+    apply conn_mono G _ (hWconn x hx y hy)
+    intro f hf
+    exact Finset.mem_filter.mpr ⟨hf.1,Or.inl (internal_not_cut G W hf.2)⟩
+  · obtain ⟨e₀,he₀,a,b,hab,ha,hb⟩ := hcontact
+    obtain ⟨c,z,e,hc,hz,he,hez,hxc⟩ := conn_first_entry G W ω hx ha (hWroot ha)
+    have hec : CutExists G W e := ⟨z,c,by simpa only [Sym2.eq_swap] using hez,hz,hc⟩
+    have hep : e ∈ pruneContacts G ω W e := Finset.mem_filter.mpr ⟨he,Or.inr rfl⟩
+    have hxz : Conn G (pruneContacts G ω W e : Set E) x z :=
+      (conn_mono G (fun f hf => Finset.mem_filter.mpr ⟨hf.1,Or.inl (exterior_not_cut G W hf.2)⟩) hxc).tail
+        ⟨e,hep,hez⟩
+    refine ⟨e,he,hec,?_⟩
+    intro y hy
+    exact hxz.trans (conn_mono G (fun f hf =>
+      Finset.mem_filter.mpr ⟨hf.1,Or.inl (internal_not_cut G W hf.2)⟩) (hWconn z hz y hy))
+
+end PruningGeometry
+
+section GoodnessGeometry
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+@[simp] theorem mem_pairOpenPivotals (G : BondGraph V E) (Q : Finset E) (x y : V)
+    (ω : Finset E) (e : E) : e ∈ pairOpenPivotals G Q x y ω ↔
+      e ∈ Q ∧ OpenMarkedPivotal G (ω : Set E) {y} x e := by
+  simp only [pairOpenPivotals,Finset.mem_filter,OpenMarkedPivotal,MarkedEvent,
+    Set.mem_singleton_iff,exists_eq_left,Finset.coe_erase,Finset.mem_coe]
+
+ 
+theorem pairOpenPivotals_card_le (G : BondGraph V E) (Q : Finset E) (x y : V) (ω : Finset E) :
+    (pairOpenPivotals G Q x y ω).card ≤ (finiteRootComponent G ω x).card-1 := by
+  let N := (finiteRootComponent G ω x).card
+  have hN : (cluster G (ω : Set E) x).ncard = N := by
+    rw [← coe_finiteRootComponent,Set.ncard_coe_finset]
+  have hm : Set.MapsTo (pivotalRank G ω x) (pairOpenPivotals G Q x y ω : Set E)
+      (Finset.Ico 1 N : Set ℕ) := by
+    intro e he
+    have he' := ((mem_pairOpenPivotals G Q x y ω e).mp he).2
+    have hfin := cluster_finite_of_finite_edges G (ω := (ω : Set E) \ {e}) (ω.finite_toSet.subset Set.sdiff_subset) x
+    have hpos : 0 < pivotalRank G ω x e :=
+      (Set.ncard_pos hfin).mpr ⟨x,conn_refl G _ x⟩
+    have hlt : pivotalRank G ω x e < N := by
+      rw [pivotalRank,← hN]
+      apply Set.ncard_lt_ncard _ (cluster_finite_of_finite_edges G ω.finite_toSet x)
+      refine Set.ssubset_iff_subset_ne.mpr ⟨fun z hz => conn_mono G Set.sdiff_subset hz,?_⟩
+      intro heq
+      obtain ⟨z,hz,hxz⟩ := he'.2.1
+      have hz' : z ∈ cluster G (ω : Set E) x := hxz
+      have hz'' : z ∈ cluster G ((ω : Set E) \ {e}) x := heq.symm ▸ hz'
+      exact he'.2.2 ⟨z,hz,hz''⟩
+    exact Finset.mem_Ico.mpr ⟨hpos,hlt⟩
+  have hi : Set.InjOn (pivotalRank G ω x) (pairOpenPivotals G Q x y ω : Set E) := by
+    intro e he f hf h
+    exact pivotalRank_injective G ω.finite_toSet {y} x
+      ((mem_pairOpenPivotals G Q x y ω e).mp he).2
+      ((mem_pairOpenPivotals G Q x y ω f).mp hf).2 h
+  simpa [N] using Finset.card_le_card_of_injOn (pivotalRank G ω x) hm hi
+
+theorem finiteRootComponent_mono (G : BondGraph V E) {ω η : Finset E} (h : ω ⊆ η) (x : V) :
+    finiteRootComponent G ω x ⊆ finiteRootComponent G η x := by
+  intro y hy
+  change y ∈ (finiteRootComponent G η x : Set V)
+  rw [coe_finiteRootComponent]
+  apply conn_mono G (show (ω : Set E) ⊆ η from h)
+  change y ∈ (finiteRootComponent G ω x : Set V) at hy
+  rwa [coe_finiteRootComponent] at hy
+
+ 
+noncomputable def openContacts (G : BondGraph V E) (ω : Finset E) (W : Set V) : Finset E :=
+  ω.filter fun e => CutExists G W e
+
+ 
+def BadPieceEvent (G : BondGraph V E) (Q : Finset E) (x : V) (M K : ℕ) (L : ℝ)
+    (ω : Finset E) : Prop :=
+  (finiteRootComponent G ω x).card ≤ M ∧ ∃ W : Finset V,
+    W ⊆ finiteRootComponent G ω x ∧
+    (∀ a ∈ W, ∀ b ∈ W, Conn G (internalBonds G ω (W : Set V) : Set E) a b) ∧
+    (openContacts G ω (W : Set V)).card = K ∧
+    ∃ a ∈ W, ∃ b ∈ W, (K:ℝ)*L < (pairOpenPivotals G Q a b (internalBonds G ω (W : Set V))).card
+
+ 
+theorem badPieceEvent_contact_bound (G : BondGraph V E) (Q : Finset E) (x : V)
+    (M K : ℕ) (L : ℝ) (ω : Finset E) (h : BadPieceEvent G Q x M K L ω) :
+    (K:ℝ)*L < (M:ℝ) := by
+  obtain ⟨hcap,W,hW,_,_,a,ha,b,_,hbad⟩ := h
+  have hsub : finiteRootComponent G (internalBonds G ω (W : Set V)) a ⊆ W := by
+    intro z hz
+    apply conn_internal_endpoint G (W : Set V) (ω := (ω : Set E)) ha
+    change z ∈ (finiteRootComponent G (internalBonds G ω (W : Set V)) a : Set V) at hz
+    rw [coe_finiteRootComponent] at hz
+    simpa only [coe_internalBonds,cluster,Set.mem_ofPred_eq] using hz
+  have hab := pairOpenPivotals_card_le G Q a b (internalBonds G ω (W : Set V))
+  have hpos : 0 < (finiteRootComponent G (internalBonds G ω (W : Set V)) a).card := by
+    apply Finset.card_pos.mpr
+    refine ⟨a,?_⟩
+    change a ∈ (finiteRootComponent G (internalBonds G ω (W : Set V)) a : Set V)
+    rw [coe_finiteRootComponent]
+    exact conn_refl G _ a
+  have hn : (pairOpenPivotals G Q a b (internalBonds G ω (W : Set V))).card < M := by
+    have h₁ := Finset.card_le_card hsub
+    have h₂ := Finset.card_le_card hW
+    omega
+  exact hbad.trans (by exact_mod_cast hn)
+
+ 
+noncomputable def incidentBonds (G : BondGraph V E) (Q : Finset E) (W : Finset V) : Finset E :=
+  W.biUnion fun v => Q.filter fun e => v ∈ G e
+
+@[simp] theorem mem_incidentBonds (G : BondGraph V E) (Q : Finset E) (W : Finset V) (e : E) :
+    e ∈ incidentBonds G Q W ↔ e ∈ Q ∧ ∃ v ∈ W, v ∈ G e := by
+  simp only [incidentBonds,Finset.mem_biUnion,Finset.mem_filter]
+  aesop
+
+theorem incidentBonds_card_le (G : BondGraph V E) (Q : Finset E) (W : Finset V) (d : ℕ)
+    (hd : ∀ v, (Q.filter fun e => v ∈ G e).card ≤ d) :
+    (incidentBonds G Q W).card ≤ d*W.card := by
+  calc
+    _ ≤ ∑ v ∈ W, (Q.filter fun e => v ∈ G e).card := Finset.card_biUnion_le
+    _ ≤ ∑ _v ∈ W, d := Finset.sum_le_sum fun _ _ => hd _
+    _ = _ := by simp [Nat.mul_comm]
+
+ 
+
+theorem badPieceEvent_pruning (G : BondGraph V E) (Q : Finset E) (x : V) (M K r : ℕ)
+    (L : ℝ) (hK : 0 < K) (hr : 2*(r:ℝ) ≤ (K:ℝ)*L)
+    (ω : Finset E) (hω : ω ⊆ Q) (h : BadPieceEvent G Q x M K L ω) :
+    ∃ η : Finset E, η ⊆ ω ∧ CappedRadiusEvent G Q x M r η ∧
+      (ω \ η).card ≤ K ∧ ω \ η ⊆ incidentBonds G Q (finiteRootComponent G η x) := by
+  obtain ⟨hcap,W,hW,hWconn,hcontacts,a,ha,b,hb,hbad⟩ := h
+  have hcontact : ∃ e ∈ ω, CutExists G (W : Set V) e := by
+    have : (openContacts G ω (W : Set V)).Nonempty := Finset.card_pos.mp (hcontacts ▸ hK)
+    obtain ⟨e,he⟩ := this
+    exact ⟨e,(Finset.mem_filter.mp he).1,(Finset.mem_filter.mp he).2⟩
+  have hWroot : (W : Set V) ⊆ cluster G ω x := by
+    intro z hz
+    have hh : z ∈ (finiteRootComponent G ω x : Set V) := hW hz
+    rwa [coe_finiteRootComponent] at hh
+  obtain ⟨e,he,hec,hWη⟩ := exists_pruneContacts_root G ω (W : Set V) x hWroot
+    (by intro a ha b hb; simpa only [coe_internalBonds] using hWconn a ha b hb) hcontact
+  let η := pruneContacts G ω (W : Set V) e
+  have hη : η ⊆ ω := Finset.filter_subset _ _
+  have hep : e ∈ η := Finset.mem_filter.mpr ⟨he,Or.inr rfl⟩
+  have hpiv := pairOpenPivotals_one_attachment G Q η (W : Set V) e hep hec
+    (pruneContacts_only_attachment G ω (W : Set V) e) ha hb
+  rw [internal_pruneContacts] at hpiv
+  have hfar : (K:ℝ)*L < (pairOpenPivotals G Q a b η).card :=
+    hbad.trans_le (by exact_mod_cast Finset.card_le_card hpiv)
+  have hxa : Conn G (η : Set E) x a := hWη ha
+  have hxb : Conn G (η : Set E) x b := hWη hb
+  have htri := pairOpenPivotals_triangle G Q η a x b (conn_symm G η hxa) hxb
+  rw [pairOpenPivotals_symm G Q η a x] at htri
+  have hlarge : r ≤ (pairOpenPivotals G Q x a η).card ∨
+      r ≤ (pairOpenPivotals G Q x b η).card := by
+    by_contra hn
+    push Not at hn
+    have htri' : ((pairOpenPivotals G Q a b η).card:ℝ) ≤
+        (pairOpenPivotals G Q x a η).card+(pairOpenPivotals G Q x b η).card := by exact_mod_cast htri
+    have h₁ : ((pairOpenPivotals G Q x a η).card:ℝ) < r := by exact_mod_cast hn.1
+    have h₂ : ((pairOpenPivotals G Q x b η).card:ℝ) < r := by exact_mod_cast hn.2
+    linarith
+  have hchanges : ω \ η ⊆ openContacts G ω (W : Set V) := by
+    intro f hf
+    obtain ⟨hfω,hfη⟩ := Finset.mem_sdiff.mp hf
+    refine Finset.mem_filter.mpr ⟨hfω,?_⟩
+    by_contra hn
+    exact hfη (Finset.mem_filter.mpr ⟨hfω,Or.inl hn⟩)
+  refine ⟨η,hη,⟨(Finset.card_le_card (finiteRootComponent_mono G hη x)).trans hcap,?_⟩,
+    (Finset.card_le_card hchanges).trans (le_of_eq hcontacts),?_⟩
+  · rcases hlarge with h | h
+    · exact ⟨a,hxa,h⟩
+    · exact ⟨b,hxb,h⟩
+  · intro f hf
+    obtain ⟨hfω,c,z,hcz,hc,_⟩ := Finset.mem_filter.mp (hchanges hf)
+    refine (mem_incidentBonds G Q _ f).mpr ⟨hω hfω,c,?_,?_⟩
+    · change c ∈ (finiteRootComponent G η x : Set V)
+      rw [coe_finiteRootComponent]
+      exact hWη hc
+    · simp [hcz]
+
+end GoodnessGeometry
+
+section FiniteInverseCost
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+def bernoulliWeight (Q ω : Finset E) (p : ℝ) : ℝ :=
+  p^ω.card*(1-p)^(Q.card-ω.card)
+
+theorem bernoulliWeight_nonneg (Q ω : Finset E) (p : unitInterval) :
+    0 ≤ bernoulliWeight Q ω p := mul_nonneg (pow_nonneg p.2.1 _) (pow_nonneg (sub_nonneg.mpr p.2.2) _)
+
+ 
+theorem bernoulliWeight_closure_ratio (Q ω η : Finset E) (hη : η ⊆ ω) (hω : ω ⊆ Q)
+    (p : ℝ) (hp : p < 1) :
+    bernoulliWeight Q ω p = bernoulliWeight Q η p*(p/(1-p))^((ω \ η).card) := by
+  have h₁ := Finset.card_le_card hη
+  have h₂ := Finset.card_le_card hω
+  have hj : (ω \ η).card = ω.card-η.card := Finset.card_sdiff_of_subset hη
+  have ha : ω.card = η.card+(ω \ η).card := by omega
+  have hb : Q.card-η.card = (Q.card-ω.card)+(ω \ η).card := by omega
+  have hq : (1-p) ≠ 0 := ne_of_gt (sub_pos.mpr hp)
+  unfold bernoulliWeight
+  rw [hb,pow_add,div_pow]
+  rw [← mul_div_assoc]
+  apply (eq_div_iff (pow_ne_zero _ hq)).mpr
+  rw [ha,pow_add]
+  ring
+
+ 
+theorem bernoulliWeight_closure_le (Q ω η : Finset E) (hη : η ⊆ ω) (hω : ω ⊆ Q)
+    (p : unitInterval) (hp : (p:ℝ) < 1) (K : ℕ) (hj : (ω \ η).card ≤ K) :
+    bernoulliWeight Q ω p ≤ (max 1 ((p:ℝ)/(1-p)))^K*bernoulliWeight Q η p := by
+  rw [bernoulliWeight_closure_ratio Q ω η hη hω p hp,mul_comm]
+  apply mul_le_mul_of_nonneg_right _ (bernoulliWeight_nonneg Q η p)
+  calc
+    _ ≤ (max 1 ((p:ℝ)/(1-p)))^((ω \ η).card) :=
+      pow_le_pow_left₀ (div_nonneg p.2.1 (sub_nonneg.mpr p.2.2)) (le_max_right ..) _
+    _ ≤ _ := pow_le_pow_right₀ (le_max_left ..) hj
+
+ 
+theorem card_small_subsets_le (S : Finset E) (K N : ℕ) (hS : S.card ≤ N) (hN : 1 ≤ N) :
+    (S.powerset.filter fun T => T.card ≤ K).card ≤ (K+1)*N^K := by
+  have hsub : S.powerset.filter (fun T => T.card ≤ K) ⊆
+      (Finset.range (K+1)).biUnion (fun j => S.powersetCard j) := by
+    intro T hT
+    obtain ⟨hTS,hTK⟩ := Finset.mem_filter.mp hT
+    exact Finset.mem_biUnion.mpr ⟨T.card,Finset.mem_range.mpr (Nat.lt_succ_of_le hTK),
+      Finset.mem_powersetCard.mpr ⟨Finset.mem_powerset.mp hTS,rfl⟩⟩
+  calc
+    _ ≤ _ := Finset.card_le_card hsub
+    _ ≤ ∑ j ∈ Finset.range (K+1), (S.powersetCard j).card := Finset.card_biUnion_le
+    _ ≤ ∑ _j ∈ Finset.range (K+1), N^K := by
+      apply Finset.sum_le_sum
+      intro j hj
+      rw [Finset.card_powersetCard]
+      exact (Nat.choose_le_pow _ _).trans ((Nat.pow_le_pow_left hS _).trans
+        (Nat.pow_le_pow_right hN (Nat.le_of_lt_succ (Finset.mem_range.mp hj))))
+    _ = _ := by simp
+
+ 
+theorem closure_fiber_card_le (A : Finset (Finset E)) (f : Finset E → Finset E)
+    (S : Finset E) (η : Finset E) (K N : ℕ) (hN : 1 ≤ N) (hS : S.card ≤ N)
+    (hf : ∀ ω ∈ A, f ω = η → f ω ⊆ ω ∧ (ω \ f ω).card ≤ K ∧ ω \ f ω ⊆ S) :
+    (A.filter fun ω => f ω = η).card ≤ (K+1)*N^K := by
+  have hm : Set.MapsTo (fun ω : Finset E => ω \ η) (A.filter fun ω => f ω = η : Set (Finset E))
+      (S.powerset.filter fun T => T.card ≤ K : Set (Finset E)) := by
+    intro ω hω
+    obtain ⟨hωA,heq⟩ := Finset.mem_filter.mp hω
+    obtain ⟨_,hj,hs⟩ := hf ω hωA heq
+    rw [heq] at hj hs
+    exact Finset.mem_filter.mpr ⟨Finset.mem_powerset.mpr hs,hj⟩
+  have hi : Set.InjOn (fun ω : Finset E => ω \ η) (A.filter fun ω => f ω = η : Set (Finset E)) := by
+    intro ω hω ζ hζ heq
+    obtain ⟨hωA,hωη⟩ := Finset.mem_filter.mp hω
+    obtain ⟨hζA,hζη⟩ := Finset.mem_filter.mp hζ
+    have h₁ := (hf ω hωA hωη).1
+    have h₂ := (hf ζ hζA hζη).1
+    rw [hωη] at h₁
+    rw [hζη] at h₂
+    have : η ∪ (ω \ η) = η ∪ (ζ \ η) := congrArg (fun s => η ∪ s) heq
+    simpa only [Finset.union_sdiff_of_subset h₁,Finset.union_sdiff_of_subset h₂] using this
+  exact (Finset.card_le_card_of_injOn (fun ω => ω \ η) hm hi).trans (card_small_subsets_le S K N hS hN)
+
+ 
+theorem finite_closure_comparison (Q : Finset E) (A B : Finset (Finset E))
+    (hA : A ⊆ Q.powerset) (f : Finset E → Finset E) (S : Finset E → Finset E)
+    (K N : ℕ) (hN : 1 ≤ N) (p : unitInterval) (hp : (p:ℝ) < 1)
+    (hf : ∀ ω ∈ A, f ω ∈ B ∧ f ω ⊆ ω ∧ (ω \ f ω).card ≤ K ∧ ω \ f ω ⊆ S (f ω))
+    (hS : ∀ η ∈ B, (S η).card ≤ N) :
+    (∑ ω ∈ A, bernoulliWeight Q ω p) ≤
+      (((K+1)*N^K : ℕ):ℝ)*(max 1 ((p:ℝ)/(1-p)))^K*∑ η ∈ B, bernoulliWeight Q η p := by
+  rw [← Finset.sum_fiberwise_of_maps_to (fun ω hω => (hf ω hω).1) (fun ω => bernoulliWeight Q ω p),
+    Finset.mul_sum]
+  apply Finset.sum_le_sum
+  intro η hη
+  let T := A.filter fun ω => f ω = η
+  have hc : T.card ≤ (K+1)*N^K := closure_fiber_card_le A f (S η) η K N hN (hS η hη) (by
+    intro ω hω heq
+    have hh := (hf ω hω).2
+    simpa only [heq] using hh)
+  have hweight : ∀ ω ∈ T, bernoulliWeight Q ω p ≤
+      (max 1 ((p:ℝ)/(1-p)))^K*bernoulliWeight Q η p := by
+    intro ω hω
+    obtain ⟨hωA,heq⟩ := Finset.mem_filter.mp hω
+    have hh := bernoulliWeight_closure_le Q ω (f ω) (hf ω hωA).2.1
+      (Finset.mem_powerset.mp (hA hωA)) p hp K (hf ω hωA).2.2.1
+    simpa only [heq] using hh
+  calc
+    _ ≤ ∑ _ω ∈ T, (max 1 ((p:ℝ)/(1-p)))^K*bernoulliWeight Q η p := Finset.sum_le_sum hweight
+    _ = (T.card:ℝ)*((max 1 ((p:ℝ)/(1-p)))^K*bernoulliWeight Q η p) := by simp
+    _ ≤ (((K+1)*N^K:ℕ):ℝ)*((max 1 ((p:ℝ)/(1-p)))^K*bernoulliWeight Q η p) :=
+      mul_le_mul_of_nonneg_right (by exact_mod_cast hc)
+        (mul_nonneg (pow_nonneg (le_trans zero_le_one (le_max_left ..)) _) (bernoulliWeight_nonneg Q η p))
+    _ = _ := by ring
+
+end FiniteInverseCost
+
+section FiniteGoodnessProbability
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def finiteBadPieceProbability (G : BondGraph V E) (Q : Finset E)
+    (x : V) (M K : ℕ) (L p : ℝ) : ℝ :=
+  bernoulliPolynomial Q (fun ω => if BadPieceEvent G Q x M K L ω then 1 else 0) p
+
+theorem bernoulliPolynomial_indicator_weight (Q : Finset E) (P : Finset E → Prop) (p : ℝ) :
+    bernoulliPolynomial Q (fun ω => if P ω then 1 else 0) p =
+      ∑ ω ∈ Q.powerset.filter P, bernoulliWeight Q ω p := by
+  rw [Finset.sum_filter]
+  unfold bernoulliPolynomial
+  apply Finset.sum_congr rfl
+  intro ω hω
+  by_cases h : P ω <;> simp [h,bernoulliWeight]
+
+ 
+theorem finiteBadPieceProbability_le_radius (G : BondGraph V E) (Q : Finset E) (x : V)
+    (M K r d : ℕ) (L : ℝ) (hK : 0 < K) (hN : 1 ≤ d*M)
+    (hdeg : ∀ v, (Q.filter fun e => v ∈ G e).card ≤ d)
+    (hr : 2*(r:ℝ) ≤ (K:ℝ)*L) (p : unitInterval) (hp : (p:ℝ) < 1) :
+    finiteBadPieceProbability G Q x M K L p ≤
+      (((K+1)*(d*M)^K:ℕ):ℝ)*(max 1 ((p:ℝ)/(1-p)))^K*finiteCappedRadiusTail G Q x M r p := by
+  let A := Q.powerset.filter (BadPieceEvent G Q x M K L)
+  let B := Q.powerset.filter (CappedRadiusEvent G Q x M r)
+  let S := fun η => incidentBonds G Q (finiteRootComponent G η x)
+  have hex : ∀ ω : Finset E, ∃ η : Finset E, ω ∈ A →
+      η ∈ B ∧ η ⊆ ω ∧ (ω \ η).card ≤ K ∧ ω \ η ⊆ S η := by
+    intro ω
+    by_cases hω : ω ∈ A
+    · obtain ⟨hωQ,hωbad⟩ := Finset.mem_filter.mp hω
+      have hωsub := Finset.mem_powerset.mp hωQ
+      obtain ⟨η,hη,hcap,hcard,hS⟩ := badPieceEvent_pruning G Q x M K r L hK hr ω hωsub hωbad
+      exact ⟨η,fun _ => ⟨Finset.mem_filter.mpr ⟨Finset.mem_powerset.mpr (hη.trans hωsub),hcap⟩,
+        hη,hcard,hS⟩⟩
+    · exact ⟨∅,fun h => (hω h).elim⟩
+  choose f hf using hex
+  have hS : ∀ η ∈ B, (S η).card ≤ d*M := by
+    intro η hη
+    exact (incidentBonds_card_le G Q (finiteRootComponent G η x) d hdeg).trans
+      (Nat.mul_le_mul_left d (Finset.mem_filter.mp hη).2.1)
+  have hc := finite_closure_comparison Q A B (Finset.filter_subset _ _) f S K (d*M) hN p hp hf hS
+  simpa only [finiteBadPieceProbability,finiteCappedRadiusTail,bernoulliPolynomial_indicator_weight,
+    A,B] using hc
+
+ 
+theorem finiteBadPieceProbability_exponential [Nonempty V] (G : BondGraph V E)
+    (Q : Finset E) (x : V) (M K d : ℕ) (L : ℝ) (hK : 0 < K) (hN : 1 ≤ d*M)
+    (hL : 4 ≤ (K:ℝ)*L) (hd : 0 < d)
+    (hdeg : ∀ v, (Q.filter fun e => v ∈ G e).card ≤ d)
+    (p h : unitInterval) (hp : (p:ℝ) < 1) (hh : 0 < (h:ℝ)) (hh1 : (h:ℝ) < 1)
+    (B : ℝ) (hB : 1 ≤ B)
+    (hAB : (Real.sqrt ((d:ℝ)/2)/(1-p))*Real.sqrt (Real.log (1/(h:ℝ))/h) ≤ B) :
+    finiteBadPieceProbability G Q x M K L p ≤
+      (((K+1)*(d*M)^K:ℕ):ℝ)*(max 1 ((p:ℝ)/(1-p)))^K*
+        (2/((h:ℝ)*(1-h)^M))*Real.exp (-((K:ℝ)*L)/(16*B)) := by
+  let r := ⌈(K:ℝ)*L/4⌉₊
+  have hpos : 0 ≤ (K:ℝ)*L/4 := by linarith
+  have hrlo : (K:ℝ)*L/4 ≤ r := Nat.le_ceil _
+  have hrhi : (r:ℝ) < (K:ℝ)*L/4+1 := Nat.ceil_lt_add_one hpos
+  have hr : 2*(r:ℝ) ≤ (K:ℝ)*L := by linarith
+  have hrad := finiteCappedRadiusTail_exponential G p h hp hh hh1 d hd Q x hdeg B hB hAB M r
+  have hcmp := finiteBadPieceProbability_le_radius G Q x M K r d L hK hN hdeg hr p hp
+  have hc : 0 ≤ (((K+1)*(d*M)^K:ℕ):ℝ)*(max 1 ((p:ℝ)/(1-p)))^K := by positivity
+  have hhfac : 0 ≤ 2/((h:ℝ)*(1-h)^M) := by positivity
+  have hBpos : 0 < B := lt_of_lt_of_le zero_lt_one hB
+  have hratio : -((r:ℝ)/(4*B)) ≤ -((K:ℝ)*L)/(16*B) := by
+    apply (le_div_iff₀ (by positivity : 0 < 16*B)).mpr
+    field_simp
+    nlinarith
+  exact hcmp.trans ((mul_le_mul_of_nonneg_left hrad hc).trans (by
+    rw [← mul_assoc]
+    exact mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr (by simpa only [neg_div] using hratio)) (mul_nonneg hc hhfac)))
+
+ 
+
+def FiniteGoodForScale (G : BondGraph V E) (Q : Finset E) (x : V) (L : ℝ) (ω : Finset E) : Prop :=
+  ∀ W : Finset V, W ⊆ finiteRootComponent G ω x →
+    (∀ a ∈ W, ∀ b ∈ W, Conn G (internalBonds G ω (W : Set V) : Set E) a b) →
+    0 < (openContacts G ω (W : Set V)).card →
+    ∀ a ∈ W, ∀ b ∈ W,
+      (pairOpenPivotals G Q a b (internalBonds G ω (W : Set V))).card ≤
+        ((openContacts G ω (W : Set V)).card:ℝ)*L
+
+def CappedBadGoodnessEvent (G : BondGraph V E) (Q : Finset E) (x : V) (M : ℕ) (L : ℝ)
+    (ω : Finset E) : Prop :=
+  (finiteRootComponent G ω x).card ≤ M ∧ ¬FiniteGoodForScale G Q x L ω
+
+noncomputable def finiteBadGoodnessProbability (G : BondGraph V E) (Q : Finset E)
+    (x : V) (M : ℕ) (L p : ℝ) : ℝ :=
+  bernoulliPolynomial Q (fun ω => if CappedBadGoodnessEvent G Q x M L ω then 1 else 0) p
+
+ 
+theorem cappedBadGoodness_iff (G : BondGraph V E) (Q : Finset E) (x : V) (M d : ℕ)
+    (L : ℝ) (ω : Finset E) (hω : ω ⊆ Q)
+    (hdeg : ∀ v, (Q.filter fun e => v ∈ G e).card ≤ d) :
+    CappedBadGoodnessEvent G Q x M L ω ↔
+      ∃ K ∈ Finset.Icc 1 (d*M), BadPieceEvent G Q x M K L ω := by
+  constructor
+  · rintro ⟨hcap,hbad⟩
+    simp only [FiniteGoodForScale,not_forall] at hbad
+    obtain ⟨W,hbad⟩ := hbad
+    push Not at hbad
+    obtain ⟨hW,hWconn,hK,a,ha,b,hb,hfar⟩ := hbad
+    have hinc : openContacts G ω (W : Set V) ⊆ incidentBonds G Q W := by
+      intro e he
+      obtain ⟨heω,c,z,hcz,hc,_⟩ := Finset.mem_filter.mp he
+      exact (mem_incidentBonds G Q W e).mpr ⟨hω heω,c,hc,by simp [hcz]⟩
+    have hbound := (Finset.card_le_card hinc).trans ((incidentBonds_card_le G Q W d hdeg).trans
+      (Nat.mul_le_mul_left d ((Finset.card_le_card hW).trans hcap)))
+    exact ⟨_,Finset.mem_Icc.mpr ⟨hK,hbound⟩,hcap,W,hW,hWconn,rfl,a,ha,b,hb,hfar⟩
+  · rintro ⟨K,hK,hcap,W,hW,hWconn,hcontacts,a,ha,b,hb,hfar⟩
+    refine ⟨hcap,fun hgood => ?_⟩
+    have hh := hgood W hW hWconn (by rw [hcontacts]; exact (Finset.mem_Icc.mp hK).1) a ha b hb
+    rw [hcontacts] at hh
+    exact not_lt_of_ge hh hfar
+
+ 
+theorem finiteBadGoodnessProbability_le_sum (G : BondGraph V E) (Q : Finset E) (x : V)
+    (M d : ℕ) (L : ℝ) (p : unitInterval)
+    (hdeg : ∀ v, (Q.filter fun e => v ∈ G e).card ≤ d) :
+    finiteBadGoodnessProbability G Q x M L p ≤
+      ∑ K ∈ Finset.Icc 1 (d*M), finiteBadPieceProbability G Q x M K L p := by
+  unfold finiteBadPieceProbability finiteBadGoodnessProbability
+  rw [← bernoulliPolynomial_sum]
+  apply bernoulliPolynomial_mono _ _ _ _ p.2
+  intro ω hω
+  by_cases he : CappedBadGoodnessEvent G Q x M L ω
+  · obtain ⟨K,hK,hbad⟩ := (cappedBadGoodness_iff G Q x M d L ω hω hdeg).mp he
+    simp only [he,ite_true]
+    exact (by simp [hbad] : (1:ℝ) ≤ if BadPieceEvent G Q x M K L ω then 1 else 0).trans
+      (Finset.single_le_sum (f := fun j => if BadPieceEvent G Q x M j L ω then (1:ℝ) else 0)
+        (fun j _ => by split_ifs <;> norm_num) hK)
+  · simp only [he,ite_false]
+    exact Finset.sum_nonneg fun j _ => by split_ifs <;> norm_num
+
+ 
+theorem finiteBadGoodnessProbability_geometric [Nonempty V] (G : BondGraph V E)
+    (Q : Finset E) (x : V) (M d : ℕ) (L : ℝ) (hN : 1 ≤ d*M) (hL : 4 ≤ L)
+    (hd : 0 < d) (hdeg : ∀ v, (Q.filter fun e => v ∈ G e).card ≤ d)
+    (p h : unitInterval) (hp : (p:ℝ) < 1) (hh : 0 < (h:ℝ)) (hh1 : (h:ℝ) < 1)
+    (B : ℝ) (hB : 1 ≤ B)
+    (hAB : (Real.sqrt ((d:ℝ)/2)/(1-p))*Real.sqrt (Real.log (1/(h:ℝ))/h) ≤ B)
+    (hq : ((d*M:ℕ):ℝ)*max 1 ((p:ℝ)/(1-p))*Real.exp (-L/(16*B)) ≤ 1) :
+    finiteBadGoodnessProbability G Q x M L p ≤
+      (2/((h:ℝ)*(1-h)^M))*((d*M:ℕ):ℝ)*((d*M:ℕ)+1)*
+        (((d*M:ℕ):ℝ)*max 1 ((p:ℝ)/(1-p))*Real.exp (-L/(16*B))) := by
+  let N := d*M
+  let a := max 1 ((p:ℝ)/(1-p))
+  let H := 2/((h:ℝ)*(1-h)^M)
+  let q := (N:ℝ)*a*Real.exp (-L/(16*B))
+  have ha : 0 ≤ a := le_trans zero_le_one (le_max_left ..)
+  have hq0 : 0 ≤ q := mul_nonneg (mul_nonneg (Nat.cast_nonneg _) ha) (Real.exp_nonneg _)
+  have hH : 0 ≤ H := by dsimp [H]; positivity
+  have hkbound : ∀ K ∈ Finset.Icc 1 N, finiteBadPieceProbability G Q x M K L p ≤ H*((N:ℝ)+1)*q := by
+    intro K hK
+    obtain ⟨hK1,hKN⟩ := Finset.mem_Icc.mp hK
+    have hKR : (1:ℝ) ≤ K := by exact_mod_cast hK1
+    have hKL : 4 ≤ (K:ℝ)*L := by nlinarith
+    have hexp := finiteBadPieceProbability_exponential G Q x M K d L hK1 hN hKL hd hdeg
+      p h hp hh hh1 B hB hAB
+    have hid : (((K+1)*(d*M)^K:ℕ):ℝ)*(max 1 ((p:ℝ)/(1-p)))^K*
+        (2/((h:ℝ)*(1-h)^M))*Real.exp (-((K:ℝ)*L)/(16*B)) = H*((K:ℝ)+1)*q^K := by
+      rw [show -((K:ℝ)*L)/(16*B) = (K:ℝ)*(-L/(16*B)) by ring,Real.exp_nat_mul]
+      simp only [q,H,N,a,mul_pow,Nat.cast_mul,Nat.cast_add,Nat.cast_pow,Nat.cast_one]
+      ring
+    rw [hid] at hexp
+    have hpow : q^K ≤ q := by
+      simpa only [pow_one] using pow_le_pow_of_le_one hq0 hq hK1
+    calc
+      _ ≤ H*((K:ℝ)+1)*q^K := hexp
+      _ ≤ H*((N:ℝ)+1)*q := mul_le_mul
+        (mul_le_mul_of_nonneg_left (by exact_mod_cast Nat.add_le_add_right hKN 1) hH)
+        hpow (pow_nonneg hq0 _) (mul_nonneg hH (by positivity))
+  calc
+    _ ≤ ∑ K ∈ Finset.Icc 1 N, finiteBadPieceProbability G Q x M K L p :=
+      finiteBadGoodnessProbability_le_sum G Q x M d L p hdeg
+    _ ≤ ∑ _K ∈ Finset.Icc 1 N, H*((N:ℝ)+1)*q := Finset.sum_le_sum hkbound
+    _ = (N:ℝ)*(H*((N:ℝ)+1)*q) := by simp
+    _ = _ := by dsimp [N,H,q,a]; push_cast; ring
+
+end FiniteGoodnessProbability
+
+section GoodnessAsymptotics
+
+ 
+theorem inverse_mark_cost_le (h : ℝ) (hh : 0 ≤ h) (hh2 : h ≤ 1/2) (M : ℕ) :
+    ((1-h)^M)⁻¹ ≤ Real.exp (2*(M:ℝ)*h) := by
+  have hq : 0 < 1-h := by linarith
+  have hi : 1/(1-h) ≤ 1+2*h := (div_le_iff₀ hq).mpr (by nlinarith)
+  have hl := Real.one_sub_inv_le_log_of_pos hq
+  rw [one_div] at hi
+  have hlog : -2*h ≤ Real.log (1-h) := by linarith
+  calc
+    ((1-h)^M)⁻¹ = Real.exp (-((M:ℝ)*Real.log (1-h))) := by
+      rw [← Real.log_pow,Real.exp_neg,Real.exp_log (pow_pos hq _)]
+    _ ≤ _ := Real.exp_le_exp.mpr (by
+      have := mul_le_mul_of_nonneg_left hlog (Nat.cast_nonneg M)
+      nlinarith)
+
+ 
+
+theorem ghost_scale_estimates (M : ℕ) (ε c a : ℝ) (d : ℕ)
+    (hM : 1 ≤ (M:ℝ)) (hε : 0 < ε) (hε1 : ε < 1/2) (hc : 1 ≤ c)
+    (ha : 1 ≤ a) (hd : 1 ≤ d)
+    (hsmall : (M:ℝ)^(ε-1) ≤ 1/2)
+    (hlog : 4*Real.log M ≤ (M:ℝ)^ε)
+    (hlog2 : Real.log M ≤ (M:ℝ)^(ε/2))
+    (hgrowth : 80*c ≤ (M:ℝ)^(ε/4))
+    (hconstant : Real.log (2*(d:ℝ)^2*((d:ℝ)+1)*a) ≤ (M:ℝ)^ε) :
+    let h := (M:ℝ)^(ε-1)
+    let B := c*(M:ℝ)^(1/2-ε/4)
+    let L := (M:ℝ)^(1/2+ε)
+    0 < h ∧ h ≤ 1/2 ∧ 1 ≤ B ∧
+      c*Real.sqrt (Real.log (1/h)/h) ≤ B ∧
+      (2/(h*(1-h)^M))*((d*M:ℕ):ℝ)*((d*M:ℕ)+1)*
+        (((d*M:ℕ):ℝ)*a*Real.exp (-L/(16*B))) ≤ Real.exp (-(M:ℝ)^ε) := by
+  dsimp only
+  let h := (M:ℝ)^(ε-1)
+  let B := c*(M:ℝ)^(1/2-ε/4)
+  let L := (M:ℝ)^(1/2+ε)
+  have hMp : 0 < (M:ℝ) := lt_of_lt_of_le zero_lt_one hM
+  have hcp : 0 < c := lt_of_lt_of_le zero_lt_one hc
+  have hap : 0 < a := lt_of_lt_of_le zero_lt_one ha
+  have hdp : 0 < (d:ℝ) := by exact_mod_cast (show 0 < d by omega)
+  have hhp : 0 < h := Real.rpow_pos_of_pos hMp _
+  have hq : 0 < 1-h := by dsimp [h] at *; linarith
+  have hlogM : 0 ≤ Real.log M := Real.log_nonneg hM
+  have hinv : 1/h = (M:ℝ)^(1-ε) := by
+    dsimp [h]
+    rw [show 1-ε = -(ε-1) by ring,Real.rpow_neg hMp.le,one_div]
+  have hmh : (M:ℝ)*h = (M:ℝ)^ε := by
+    dsimp [h]
+    calc
+      _ = (M:ℝ)^((1:ℝ)+(ε-1)) := by rw [Real.rpow_add hMp,Real.rpow_one]
+      _ = _ := by congr 1; ring
+  have hhM : 1/h ≤ (M:ℝ) := by
+    rw [hinv]
+    simpa only [Real.rpow_one] using
+      Real.rpow_le_rpow_of_exponent_le hM (show 1-ε ≤ (1:ℝ) by linarith)
+  have hB1 : 1 ≤ B := by
+    have hp := Real.one_le_rpow hM (show 0 ≤ 1/2-ε/4 by linarith)
+    dsimp [B]
+    nlinarith
+  have hBp : 0 < B := lt_of_lt_of_le zero_lt_one hB1
+  have hAB : c*Real.sqrt (Real.log (1/h)/h) ≤ B := by
+    apply mul_le_mul_of_nonneg_left _ hcp.le
+    have hl : Real.log (1/h) ≤ Real.log M := by
+      rw [hinv,Real.log_rpow hMp]
+      nlinarith
+    have hl2 : Real.log (1/h)/h ≤ (M:ℝ)^(1-ε/2) := by
+      calc
+        _ = Real.log (1/h)*(M:ℝ)^(1-ε) := by rw [div_eq_mul_one_div,hinv]
+        _ ≤ (M:ℝ)^(ε/2)*(M:ℝ)^(1-ε) :=
+          mul_le_mul_of_nonneg_right (hl.trans hlog2) (Real.rpow_nonneg hMp.le _)
+        _ = _ := by rw [← Real.rpow_add hMp]; congr 1; ring
+    calc
+      _ ≤ Real.sqrt ((M:ℝ)^(1-ε/2)) := Real.sqrt_le_sqrt hl2
+      _ = (M:ℝ)^(1/2-ε/4) := by
+        rw [Real.sqrt_eq_rpow,← Real.rpow_mul hMp.le]
+        congr 1
+        ring
+  have hratio : 5*(M:ℝ)^ε ≤ L/(16*B) := by
+    have hid : (M:ℝ)^ε*(M:ℝ)^(ε/4)*(M:ℝ)^(1/2-ε/4) = L := by
+      dsimp [L]
+      rw [← Real.rpow_add hMp,← Real.rpow_add hMp]
+      congr 1
+      ring
+    apply (le_div_iff₀ (by positivity : 0 < 16*B)).mpr
+    have hh := mul_le_mul_of_nonneg_left hgrowth
+      (mul_nonneg (Real.rpow_nonneg hMp.le ε) (Real.rpow_nonneg hMp.le (1/2-ε/4)))
+    dsimp [B]
+    nlinarith [hid]
+  refine ⟨hhp,hsmall,hB1,hAB,?_⟩
+  let A := 2*(d:ℝ)^2*((d:ℝ)+1)*a
+  have hAp : 0 < A := by dsimp [A]; positivity
+  have hcost : 2/(h*(1-h)^M) ≤ 2*(M:ℝ)*Real.exp (2*(M:ℝ)^ε) := by
+    calc
+      _ = 2*(1/h)*((1-h)^M)⁻¹ := by field_simp
+      _ ≤ 2*(M:ℝ)*Real.exp (2*(M:ℝ)*h) :=
+        mul_le_mul (mul_le_mul_of_nonneg_left hhM (by norm_num))
+          (inverse_mark_cost_le h hhp.le hsmall M) (inv_nonneg.mpr (pow_nonneg hq.le _)) (by positivity)
+      _ = _ := by rw [show 2*(M:ℝ)*h = 2*(M:ℝ)^ε by rw [mul_assoc,hmh]]
+  have hN1 : ((d*M:ℕ):ℝ)+1 ≤ ((d:ℝ)+1)*(M:ℝ) := by push_cast; nlinarith
+  have hpoint :
+      (2/(h*(1-h)^M))*((d*M:ℕ):ℝ)*((d*M:ℕ)+1)*
+        (((d*M:ℕ):ℝ)*a*Real.exp (-L/(16*B))) ≤
+      A*(M:ℝ)^4*Real.exp (2*(M:ℝ)^ε-L/(16*B)) := by
+    calc
+      _ ≤ (2*(M:ℝ)*Real.exp (2*(M:ℝ)^ε))*((d*M:ℕ):ℝ)*(((d:ℝ)+1)*(M:ℝ))*
+          (((d*M:ℕ):ℝ)*a*Real.exp (-L/(16*B))) := by
+        apply mul_le_mul_of_nonneg_right _ (by positivity)
+        exact mul_le_mul (mul_le_mul_of_nonneg_right hcost (by positivity)) hN1
+          (by positivity) (by positivity)
+      _ = _ := by
+        rw [show 2*(M:ℝ)^ε-L/(16*B) = 2*(M:ℝ)^ε+(-L/(16*B)) by ring,Real.exp_add]
+        dsimp [A]
+        push_cast
+        ring
+  have heq : A*(M:ℝ)^4*Real.exp (2*(M:ℝ)^ε-L/(16*B)) =
+      Real.exp (Real.log A+4*Real.log M+2*(M:ℝ)^ε-L/(16*B)) := by
+    rw [show Real.log A+4*Real.log M+2*(M:ℝ)^ε-L/(16*B) =
+      Real.log A+4*Real.log M+(2*(M:ℝ)^ε-L/(16*B)) by ring,Real.exp_add,
+      Real.exp_add,Real.exp_log hAp]
+    have hh : Real.exp (4*Real.log M) = (M:ℝ)^4 := by
+      simpa only [Nat.cast_ofNat,Real.exp_log hMp] using Real.exp_nat_mul (Real.log M) 4
+    rw [hh]
+  rw [heq] at hpoint
+  exact hpoint.trans (Real.exp_le_exp.mpr (by linarith))
+
+ 
+
+theorem eventually_ghost_scale (ε c a : ℝ) (d : ℕ) (hε : 0 < ε) (hε1 : ε < 1/2)
+    (hc : 1 ≤ c) (ha : 1 ≤ a) (hd : 1 ≤ d) :
+    ∀ᶠ M : ℕ in Filter.atTop,
+      let h := (M:ℝ)^(ε-1)
+      let B := c*(M:ℝ)^(1/2-ε/4)
+      let L := (M:ℝ)^(1/2+ε)
+      1 ≤ M ∧ 4 ≤ L ∧ 0 < h ∧ h ≤ 1/2 ∧ 1 ≤ B ∧
+        c*Real.sqrt (Real.log (1/h)/h) ≤ B ∧
+        (2/(h*(1-h)^M))*((d*M:ℕ):ℝ)*((d*M:ℕ)+1)*
+          (((d*M:ℕ):ℝ)*a*Real.exp (-L/(16*B))) ≤ Real.exp (-(M:ℝ)^ε) := by
+  have hh : ∀ᶠ m : ℝ in Filter.atTop, m^(ε-1) ≤ 1/2 := by
+    have ht := tendsto_rpow_neg_atTop (show 0 < 1-ε by linarith)
+    filter_upwards [ht.eventually (Iio_mem_nhds (by norm_num : (0:ℝ) < 1/2))] with m hm
+    simpa only [neg_sub] using (show m^(-(1-ε)) < 1/2 from hm).le
+  have hl := (isLittleO_log_rpow_atTop hε).bound (show (0:ℝ) < 1/4 by norm_num)
+  have hl2 := (isLittleO_log_rpow_atTop (show 0 < ε/2 by linarith)).bound zero_lt_one
+  have hg := (tendsto_rpow_atTop (show 0 < ε/4 by linarith)).eventually (Filter.eventually_ge_atTop (80*c))
+  have hC := (tendsto_rpow_atTop hε).eventually
+    (Filter.eventually_ge_atTop (Real.log (2*(d:ℝ)^2*((d:ℝ)+1)*a)))
+  have hL := (tendsto_rpow_atTop (show 0 < 1/2+ε by linarith)).eventually (Filter.eventually_ge_atTop 4)
+  have hnum : ∀ᶠ m : ℝ in Filter.atTop, 1 ≤ m ∧ 4 ≤ m^(1/2+ε) ∧ m^(ε-1) ≤ 1/2 ∧
+      4*Real.log m ≤ m^ε ∧ Real.log m ≤ m^(ε/2) ∧ 80*c ≤ m^(ε/4) ∧
+      Real.log (2*(d:ℝ)^2*((d:ℝ)+1)*a) ≤ m^ε := by
+    filter_upwards [Filter.eventually_ge_atTop (1:ℝ),hh,hl,hl2,hg,hC,hL] with m hm hh hl hl2 hg hC hL
+    have hm0 : 0 ≤ m := le_trans zero_le_one hm
+    simp only [Real.norm_eq_abs,abs_of_nonneg (Real.log_nonneg hm),
+      abs_of_nonneg (Real.rpow_nonneg hm0 ε)] at hl
+    simp only [Real.norm_eq_abs,abs_of_nonneg (Real.log_nonneg hm),
+      abs_of_nonneg (Real.rpow_nonneg hm0 (ε/2)),one_mul] at hl2
+    exact ⟨hm,hL,hh,by linarith,hl2,hg,hC⟩
+  have hcast : Filter.Tendsto (fun M : ℕ => (M:ℝ)) Filter.atTop Filter.atTop := tendsto_natCast_atTop_atTop
+  filter_upwards [hcast.eventually hnum] with M hM
+  obtain ⟨hM,hL,hh,hl,hl2,hg,hC⟩ := hM
+  exact ⟨by exact_mod_cast hM,hL,ghost_scale_estimates M ε c a d hM hε hε1 hc ha hd hh hl hl2 hg hC⟩
+
+ 
+theorem ghost_ratio_le_one (M d : ℕ) (hM : 1 ≤ M) (hd : 1 ≤ d)
+    (h B L a ε : ℝ) (hh : 0 < h) (hh2 : h ≤ 1/2) (ha : 0 ≤ a)
+    (hb : (2/(h*(1-h)^M))*((d*M:ℕ):ℝ)*((d*M:ℕ)+1)*
+      (((d*M:ℕ):ℝ)*a*Real.exp (-L/(16*B))) ≤ Real.exp (-(M:ℝ)^ε)) :
+    ((d*M:ℕ):ℝ)*a*Real.exp (-L/(16*B)) ≤ 1 := by
+  have hq : 0 < 1-h := by linarith
+  have hp : (1-h)^M ≤ 1 := pow_le_one₀ hq.le (by linarith)
+  have hden : 0 < h*(1-h)^M := mul_pos hh (pow_pos hq _)
+  have hH : 1 ≤ 2/(h*(1-h)^M) := by
+    apply (le_div_iff₀ hden).mpr
+    have := mul_le_mul_of_nonneg_left hp hh.le
+    nlinarith
+  have hN : (1:ℝ) ≤ (d*M:ℕ) := by exact_mod_cast Nat.mul_le_mul hd hM
+  have hprod : 1 ≤ (2/(h*(1-h)^M))*((d*M:ℕ):ℝ)*((d*M:ℕ)+1) :=
+    one_le_mul_of_one_le_of_one_le (one_le_mul_of_one_le_of_one_le hH hN) (by linarith)
+  have hq0 : 0 ≤ ((d*M:ℕ):ℝ)*a*Real.exp (-L/(16*B)) := by positivity
+  exact (le_mul_of_one_le_left hq0 hprod).trans (hb.trans
+    (Real.exp_le_one_iff.mpr (neg_nonpos.mpr (Real.rpow_nonneg (Nat.cast_nonneg M) ε))))
+
+end GoodnessAsymptotics
+
+section UniformFiniteGoodness
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem finite_bad_goodness_stretched_exp [Nonempty V] (G : BondGraph V E)
+    (d : ℕ) (hd : 0 < d) (pbar : unitInterval) (hpbar : (pbar:ℝ) < 1)
+    (ε : ℝ) (hε : 0 < ε) (hε1 : ε < 1/2) :
+    ∀ᶠ M : ℕ in Filter.atTop, ∀ (Q : Finset E) (x : V),
+      (∀ v, (Q.filter fun e => v ∈ G e).card ≤ d) →
+      ∀ p : unitInterval, p ≤ pbar →
+        finiteBadGoodnessProbability G Q x M ((M:ℝ)^(1/2+ε)) p ≤
+          Real.exp (-(M:ℝ)^ε) := by
+  let c := max 1 (Real.sqrt ((d:ℝ)/2)/(1-pbar))
+  let a := max 1 ((pbar:ℝ)/(1-pbar))
+  have hc : 1 ≤ c := le_max_left ..
+  have ha : 1 ≤ a := le_max_left ..
+  filter_upwards [eventually_ghost_scale ε c a d hε hε1 hc ha hd] with M hM
+  dsimp only at hM
+  obtain ⟨hM1,hL,hh,hh2,hB,hAB,hpref⟩ := hM
+  intro Q x hdeg p hpp
+  have hpp' : (p:ℝ) ≤ pbar := hpp
+  have hp : (p:ℝ) < 1 := hpp'.trans_lt hpbar
+  let h : unitInterval := ⟨(M:ℝ)^(ε-1),hh.le,by linarith⟩
+  let B := c*(M:ℝ)^(1/2-ε/4)
+  let L := (M:ℝ)^(1/2+ε)
+  have hodds : max 1 ((p:ℝ)/(1-p)) ≤ a := by
+    apply max_le (le_max_left ..)
+    have hr : (p:ℝ)/(1-p) ≤ (pbar:ℝ)/(1-pbar) :=
+      div_le_div₀ pbar.2.1 hpp' (by linarith) (by linarith)
+    exact hr.trans (le_max_right ..)
+  have hcoeff : Real.sqrt ((d:ℝ)/2)/(1-p) ≤ c := by
+    have hr : Real.sqrt ((d:ℝ)/2)/(1-p) ≤ Real.sqrt ((d:ℝ)/2)/(1-pbar) :=
+      div_le_div_of_nonneg_left (Real.sqrt_nonneg _) (by linarith) (by linarith)
+    exact hr.trans (le_max_right ..)
+  have hsqrt : 0 ≤ Real.sqrt (Real.log (1/(h:ℝ))/h) := Real.sqrt_nonneg _
+  have hqbar := ghost_ratio_le_one M d hM1 hd h B L a ε hh hh2
+    (le_trans zero_le_one ha) hpref
+  have hqcmp : ((d*M:ℕ):ℝ)*max 1 ((p:ℝ)/(1-p))*Real.exp (-L/(16*B)) ≤
+      ((d*M:ℕ):ℝ)*a*Real.exp (-L/(16*B)) :=
+    mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left hodds (Nat.cast_nonneg _)) (Real.exp_nonneg _)
+  have hg := finiteBadGoodnessProbability_geometric G Q x M d L (Nat.mul_le_mul hd hM1)
+    hL hd hdeg p h hp hh (by dsimp [h]; linarith) B hB
+    ((mul_le_mul_of_nonneg_right hcoeff hsqrt).trans hAB) (hqcmp.trans hqbar)
+  refine hg.trans ((mul_le_mul_of_nonneg_left hqcmp ?_).trans hpref)
+  have hnon : 0 ≤ 1-(h:ℝ) := sub_nonneg.mpr h.2.2
+  positivity
+
+end UniformFiniteGoodness
+
+section ClusterLocality
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem reachBall_le_step (G : BondGraph V E) (x : V) (n : ℕ) :
+    reachBall G x n ⊆ reachBall G x (n+1) := fun _ h => Or.inl h
+
+theorem reachBall_mono_local (G : BondGraph V E) (x : V) : Monotone (reachBall G x) :=
+  monotone_nat_of_le_succ (reachBall_le_step G x)
+
+theorem walk_mem_reachBall (G : BondGraph V E) (ω : Set E) {a b x : V}
+    (w : (openSimpleGraph G ω).Walk a b) (n : ℕ) (ha : a ∈ reachBall G x n) :
+    b ∈ reachBall G x (n+w.length) := by
+  induction w generalizing n with
+  | nil => simpa using ha
+  | @cons a z b hz w ih =>
+    have hz' : z ∈ reachBall G x (n+1) := by
+      apply Or.inr
+      simp only [Set.mem_iUnion,Set.mem_ofPred_eq]
+      obtain ⟨e,he,hend⟩ := hz.2
+      exact ⟨a,ha,e,hend⟩
+    simpa only [SimpleGraph.Walk.length_cons,Nat.add_assoc,Nat.add_left_comm,Nat.add_comm]
+      using ih (n+1) hz'
+
+ 
+theorem finite_cluster_subset_reachBall (G : BondGraph V E) (ω : Set E) (x : V)
+    (hfin : (cluster G ω x).Finite) (M : ℕ) (hM : (cluster G ω x).ncard ≤ M) :
+    cluster G ω x ⊆ reachBall G x M := by
+  intro y hy
+  obtain ⟨w⟩ := (conn_iff_reachable G ω x y).mp hy
+  let q := w.toPath
+  have hs : q.val.support.toFinset ⊆ hfin.toFinset := by
+    intro z hz
+    have hz' : z ∈ q.val.support := List.mem_toFinset.mp hz
+    exact hfin.mem_toFinset.mpr ((conn_iff_reachable G ω x z).mpr ⟨q.val.takeUntil z hz'⟩)
+  have hc : q.val.length+1 ≤ (cluster G ω x).ncard := by
+    have hc := Finset.card_le_card hs
+    rwa [List.toFinset_card_of_nodup q.property.support_nodup,
+      SimpleGraph.Walk.length_support,← Set.ncard_eq_toFinset_card _ hfin] at hc
+  have hr := walk_mem_reachBall G ω q.val 0 (Set.mem_singleton x)
+  simp only [Nat.zero_add] at hr
+  exact reachBall_mono_local G x (by omega) hr
+
+ 
+
+theorem cluster_restrict_incident_eq (G : BondGraph V E) (hloc : LocallyFinite G)
+    (Λ : Finset V) (ω : Set E) (x : V)
+    (hsub : cluster G (ω ∩ (incidentSupport G hloc Λ : Set E)) x ⊆ Λ) :
+    cluster G (ω ∩ (incidentSupport G hloc Λ : Set E)) x = cluster G ω x := by
+  apply Set.Subset.antisymm
+  · exact fun _ h => conn_mono G Set.inter_subset_left h
+  · intro y hy
+    induction hy with
+    | refl => exact conn_refl G _ x
+    | @tail a b _ hab ih =>
+      obtain ⟨e,he,hend⟩ := hab
+      have heQ : e ∈ incidentSupport G hloc Λ :=
+        (mem_incidentSupport G hloc Λ e).mpr ⟨a,hsub ih,by simp [hend]⟩
+      exact ih.tail ⟨e,⟨he,heQ⟩,hend⟩
+
+theorem cluster_restrict_incident_of_full (G : BondGraph V E) (hloc : LocallyFinite G)
+    (Λ : Finset V) (ω : Set E) (x : V) (hsub : cluster G ω x ⊆ Λ) :
+    cluster G (ω ∩ (incidentSupport G hloc Λ : Set E)) x = cluster G ω x := by
+  apply cluster_restrict_incident_eq G hloc Λ ω x
+  exact (fun _ h => hsub (conn_mono G Set.inter_subset_left h))
+
+ 
+theorem capped_cluster_local (G : BondGraph V E) (hloc : LocallyFinite G)
+    (x : V) (M : ℕ) (ω : Set E) :
+    let Q := incidentSupport G hloc (finite_reachBall hloc x M).toFinset
+    ((cluster G ω x).Finite ∧ (cluster G ω x).ncard ≤ M) ↔
+      ((cluster G (ω ∩ (Q : Set E)) x).Finite ∧
+        (cluster G (ω ∩ (Q : Set E)) x).ncard ≤ M) := by
+  dsimp only
+  let Λ := (finite_reachBall hloc x M).toFinset
+  let Q := incidentSupport G hloc Λ
+  constructor
+  · rintro ⟨hfin,hcap⟩
+    have hsub : cluster G ω x ⊆ Λ := by
+      simpa only [Λ,Set.Finite.coe_toFinset] using finite_cluster_subset_reachBall G ω x hfin M hcap
+    rw [cluster_restrict_incident_of_full G hloc Λ ω x hsub]
+    exact ⟨hfin,hcap⟩
+  · rintro ⟨hfin,hcap⟩
+    have hsub : cluster G (ω ∩ (Q : Set E)) x ⊆ Λ := by
+      simpa only [Λ,Set.Finite.coe_toFinset] using
+        finite_cluster_subset_reachBall G (ω ∩ (Q : Set E)) x hfin M hcap
+    have heq := cluster_restrict_incident_eq G hloc Λ ω x hsub
+    rw [heq] at hfin hcap
+    exact ⟨hfin,hcap⟩
+
+end ClusterLocality
+
+section InfiniteGoodness
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+noncomputable def pivotalDistance (G : BondGraph V E) (ω : Set E) (a b : V) : ℕ :=
+  {e | OpenMarkedPivotal G ω {b} a e}.ncard
+
+ 
+def contactSet (G : BondGraph V E) (ω : Set E) (W : Set V) : Set E :=
+  {e | e ∈ ω ∧ CutExists G W e}
+
+ 
+def GoodForScale (G : BondGraph V E) (ω : Set E) (x : V) (L : ℝ) : Prop :=
+  ∀ W : Finset V, (W : Set V) ⊆ cluster G ω x →
+    (∀ a ∈ W, ∀ b ∈ W, Conn G (ω ∩ internalEdges G (W : Set V)) a b) →
+    0 < (contactSet G ω (W : Set V)).ncard →
+    ∀ a ∈ W, ∀ b ∈ W,
+      (pivotalDistance G (ω ∩ internalEdges G (W : Set V)) a b : ℝ) ≤
+        (contactSet G ω (W : Set V)).ncard*L
+
+ 
+def CappedBadGoodness (G : BondGraph V E) (ω : Set E) (x : V) (M : ℕ) (L : ℝ) : Prop :=
+  (cluster G ω x).Finite ∧ (cluster G ω x).ncard ≤ M ∧ ¬GoodForScale G ω x L
+
+theorem pivotalDistance_finset (G : BondGraph V E) (Q ω : Finset E) (hω : ω ⊆ Q) (a b : V) :
+    pivotalDistance G (ω : Set E) a b = (pairOpenPivotals G Q a b ω).card := by
+  have he : {e | OpenMarkedPivotal G (ω : Set E) {b} a e} =
+      (pairOpenPivotals G Q a b ω : Set E) := by
+    ext e
+    simp only [Finset.mem_coe,mem_pairOpenPivotals,Set.mem_ofPred_eq]
+    exact ⟨fun h => ⟨hω h.1,h⟩,And.right⟩
+  simp only [pivotalDistance,he,Set.ncard_coe_finset]
+
+theorem contactSet_finset (G : BondGraph V E) (ω : Finset E) (W : Set V) :
+    contactSet G (ω : Set E) W = (openContacts G ω W : Set E) := by
+  ext e
+  simp only [contactSet,openContacts,Finset.mem_coe,Finset.mem_filter,Set.mem_ofPred_eq]
+
+theorem finiteGoodForScale_iff (G : BondGraph V E) (Q ω : Finset E) (hω : ω ⊆ Q)
+    (x : V) (L : ℝ) : FiniteGoodForScale G Q x L ω ↔ GoodForScale G (ω : Set E) x L := by
+  have hroot : ∀ W : Finset V, W ⊆ finiteRootComponent G ω x ↔ (W : Set V) ⊆ cluster G ω x := by
+    intro W
+    rw [← coe_finiteRootComponent]
+    exact Finset.coe_subset.symm
+  have hpiv (W : Finset V) (a b : V) := pivotalDistance_finset G Q
+    (internalBonds G ω (W : Set V)) ((Finset.filter_subset _ _).trans hω) a b
+  simp only [FiniteGoodForScale,GoodForScale,hroot,← coe_internalBonds,contactSet_finset,
+    Set.ncard_coe_finset,hpiv]
+
+theorem cappedBadGoodness_finset_iff (G : BondGraph V E) (Q ω : Finset E) (hω : ω ⊆ Q)
+    (x : V) (M : ℕ) (L : ℝ) :
+    CappedBadGoodness G (ω : Set E) x M L ↔ CappedBadGoodnessEvent G Q x M L ω := by
+  have hfin := cluster_finite_of_finite_edges G ω.finite_toSet x
+  simp only [CappedBadGoodness,CappedBadGoodnessEvent,hfin,true_and]
+  rw [← coe_finiteRootComponent,Set.ncard_coe_finset,finiteGoodForScale_iff G Q ω hω]
+
+ 
+theorem internal_config_eq_of_incident (G : BondGraph V E) (ω η : Set E) (C W : Set V)
+    (hW : W ⊆ C) (he : ∀ e, (∃ v ∈ C, v ∈ G e) → (e ∈ ω ↔ e ∈ η)) :
+    ω ∩ internalEdges G W = η ∩ internalEdges G W := by
+  ext e
+  by_cases hi : e ∈ internalEdges G W
+  · have hinc : ∃ v ∈ C, v ∈ G e :=
+      ⟨(G e).out.1,hW (hi _ (Sym2.out_fst_mem _)),Sym2.out_fst_mem _⟩
+    simp only [Set.mem_inter_iff,hi,and_true,he e hinc]
+  · simp only [Set.mem_inter_iff,hi,and_false]
+
+theorem contacts_eq_of_incident (G : BondGraph V E) (ω η : Set E) (C W : Set V)
+    (hW : W ⊆ C) (he : ∀ e, (∃ v ∈ C, v ∈ G e) → (e ∈ ω ↔ e ∈ η)) :
+    contactSet G ω W = contactSet G η W := by
+  ext e
+  by_cases hi : CutExists G W e
+  · obtain ⟨a,b,hab,ha,_⟩ := hi
+    have hinc : ∃ v ∈ C, v ∈ G e := ⟨a,hW ha,by simp [hab]⟩
+    simp only [contactSet,Set.mem_ofPred_eq,he e hinc]
+  · simp only [contactSet,Set.mem_ofPred_eq,hi,and_false]
+
+ 
+theorem goodForScale_congr (G : BondGraph V E) (ω η : Set E) (x : V) (L : ℝ)
+    (hC : cluster G ω x = cluster G η x)
+    (he : ∀ e, (∃ v ∈ cluster G ω x, v ∈ G e) → (e ∈ ω ↔ e ∈ η)) :
+    GoodForScale G ω x L ↔ GoodForScale G η x L := by
+  unfold GoodForScale
+  apply forall_congr'
+  intro W
+  rw [← hC]
+  apply forall_congr'
+  intro hW
+  rw [internal_config_eq_of_incident G ω η _ _ hW he,contacts_eq_of_incident G ω η _ _ hW he]
+
+theorem goodForScale_restrict_incident (G : BondGraph V E) (hloc : LocallyFinite G)
+    (Λ : Finset V) (ω : Set E) (x : V) (L : ℝ) (hsub : cluster G ω x ⊆ Λ) :
+    GoodForScale G ω x L ↔ GoodForScale G (ω ∩ (incidentSupport G hloc Λ : Set E)) x L := by
+  apply goodForScale_congr G ω _ x L
+    (cluster_restrict_incident_of_full G hloc Λ ω x hsub).symm
+  intro e he
+  obtain ⟨v,hv,hve⟩ := he
+  have hQ := (mem_incidentSupport G hloc Λ e).mpr ⟨v,hsub hv,hve⟩
+  simp only [Set.mem_inter_iff,Finset.mem_coe,hQ,and_true]
+
+ 
+theorem cappedBadGoodness_local (G : BondGraph V E) (hloc : LocallyFinite G)
+    (x : V) (M : ℕ) (L : ℝ) (ω : Set E) :
+    let Q := incidentSupport G hloc (finite_reachBall hloc x M).toFinset
+    CappedBadGoodness G ω x M L ↔ CappedBadGoodness G (ω ∩ (Q : Set E)) x M L := by
+  let Λ := (finite_reachBall hloc x M).toFinset
+  let Q := incidentSupport G hloc Λ
+  change CappedBadGoodness G ω x M L ↔ CappedBadGoodness G (ω ∩ (Q : Set E)) x M L
+  have hcap := capped_cluster_local G hloc x M ω
+  change ((cluster G ω x).Finite ∧ (cluster G ω x).ncard ≤ M) ↔
+    ((cluster G (ω ∩ (Q : Set E)) x).Finite ∧ (cluster G (ω ∩ (Q : Set E)) x).ncard ≤ M) at hcap
+  have hg : ((cluster G ω x).Finite ∧ (cluster G ω x).ncard ≤ M) →
+      (GoodForScale G ω x L ↔ GoodForScale G (ω ∩ (Q : Set E)) x L) := by
+    rintro ⟨hfin,hM⟩
+    apply goodForScale_restrict_incident G hloc Λ ω x L
+    simpa only [Λ,Set.Finite.coe_toFinset] using finite_cluster_subset_reachBall G ω x hfin M hM
+  constructor
+  · rintro ⟨hfin,hM,hbad⟩
+    exact ⟨(hcap.mp ⟨hfin,hM⟩).1,(hcap.mp ⟨hfin,hM⟩).2,fun hh => hbad ((hg ⟨hfin,hM⟩).mpr hh)⟩
+  · rintro ⟨hfin,hM,hbad⟩
+    have hc := hcap.mpr ⟨hfin,hM⟩
+    exact ⟨hc.1,hc.2,fun hh => hbad ((hg hc).mp hh)⟩
+
+end InfiniteGoodness
+
+section GoodnessLaw
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem measurableSet_finite_restriction (Q : Finset E) (A : Set (Set E)) :
+    MeasurableSet {ω : Set E | ω ∩ (Q : Set E) ∈ A} := by
+  have hm : MeasurableSet {η : Set (Q : Set E) | (Subtype.val '' η : Set E) ∈ A} := .of_discrete
+  convert hm.preimage (measurable_set_restrict (Q : Set E)) using 1
+  ext ω
+  have he : (Subtype.val '' (Subtype.val ⁻¹' ω : Set (Q : Set E)) : Set E) = ω ∩ (Q : Set E) := by
+    ext e
+    simp [and_comm]
+  simp only [Set.mem_preimage,Set.mem_ofPred_eq,he]
+
+theorem measurableSet_cappedBadGoodness (G : BondGraph V E) (hloc : LocallyFinite G)
+    (x : V) (M : ℕ) (L : ℝ) : MeasurableSet {ω | CappedBadGoodness G ω x M L} := by
+  let Q := incidentSupport G hloc (finite_reachBall hloc x M).toFinset
+  have he : {ω | CappedBadGoodness G ω x M L} =
+      {ω | ω ∩ (Q : Set E) ∈ {η | CappedBadGoodness G η x M L}} := by
+    ext ω
+    exact cappedBadGoodness_local G hloc x M L ω
+  rw [he]
+  exact measurableSet_finite_restriction Q _
+
+ 
+theorem bad_goodness_law_eq_finite (G : BondGraph V E) (hloc : LocallyFinite G)
+    (x : V) (M : ℕ) (L : ℝ) (p : unitInterval) :
+    (law p).real {ω | CappedBadGoodness G ω x M L} =
+      finiteBadGoodnessProbability G
+        (incidentSupport G hloc (finite_reachBall hloc x M).toFinset) x M L p := by
+  let Q := incidentSupport G hloc (finite_reachBall hloc x M).toFinset
+  have he : finiteBadGoodnessProbability G Q x M L p =
+      finiteEventPolynomial Q {ω | CappedBadGoodness G ω x M L} p := by
+    apply bernoulliPolynomial_congr
+    intro ω hω
+    simp only [Set.mem_ofPred_eq,cappedBadGoodness_finset_iff G Q ω hω]
+  change _ = finiteBadGoodnessProbability G Q x M L p
+  rw [he,finiteEventPolynomial_restrict]
+  congr 1
+  ext ω
+  exact cappedBadGoodness_local G hloc x M L ω
+
+theorem incident_filter_card_le_degree (G : BondGraph V E) (hloc : LocallyFinite G)
+    (Q : Finset E) (x : V) : (Q.filter fun e => x ∈ G e).card ≤ degree G x := by
+  let : Fintype (Step G x) := (finite_steps G hloc x).fintype
+  have hex (e : {e // e ∈ Q.filter fun e => x ∈ G e}) : ∃ y, G e.val = s(x,y) :=
+    Sym2.mem_iff_exists.mp (Finset.mem_filter.mp e.property).2
+  let f : {e // e ∈ Q.filter fun e => x ∈ G e} → Step G x :=
+    fun e => ⟨(e.val,(hex e).choose),(hex e).choose_spec⟩
+  have hf : Function.Injective f := by
+    intro a b hab
+    exact Subtype.ext (congrArg (fun z : Step G x => z.val.1) hab)
+  simpa only [degree,Nat.card_eq_fintype_card,Fintype.card_coe] using Fintype.card_le_of_injective f hf
+
+ 
+
+theorem bad_goodness_stretched_exp [Nonempty V] (G : BondGraph V E)
+    (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (pbar : unitInterval) (hpbar : (pbar:ℝ) < 1) (ε : ℝ) (hε : 0 < ε) (hε1 : ε < 1/2) :
+    ∀ᶠ M : ℕ in Filter.atTop, ∀ x : V, ∀ p : unitInterval, p ≤ pbar →
+      (law p).real {ω | CappedBadGoodness G ω x M ((M:ℝ)^(1/2+ε))} ≤
+        Real.exp (-(M:ℝ)^ε) := by
+  obtain ⟨d,hd,hdeg⟩ := exists_degree_bound G hqt
+  filter_upwards [finite_bad_goodness_stretched_exp G d hd pbar hpbar ε hε hε1] with M hM
+  intro x p hp
+  rw [bad_goodness_law_eq_finite G hloc]
+  exact hM _ x (fun v => (incident_filter_card_le_degree G hloc _ v).trans (hdeg v)) p hp
+
+end GoodnessLaw
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Exploration/Marked.lean b/lean/OAI/Probability/UnimodularPercolation/Exploration/Marked.lean
index 11f6c1836b4f2f6831ccc2786ba797ddbbd9827b..cd25034ecb04e701bb7304fb46ae53851b136285 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Exploration/Marked.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Exploration/Marked.lean
@@ -1,1226 +1,1226 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Criticality.Sharpness
-
-namespace OAI
-
-/-! Marked decision trees, graph searches and pivotal geometry. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-section MarkedExploration
-
- 
-theorem centered_bernoulli_mgf_le (p : unitInterval) (lam : ℝ) :
-    (1-(p:ℝ))*Real.exp (-lam*p) + p*Real.exp (lam*(1-p)) ≤ Real.exp (lam^2/8) := by
-  let X : Bool → ℝ := fun b => if b then 1-(p:ℝ) else -(p:ℝ)
-  have hb : ∀ᵐ b ∂bernoulliMeasure true false p, X b ∈ Set.Icc (-(p:ℝ)) (1-p) := by
-    filter_upwards with b
-    cases b <;> simp [X]
-  have hc : ∫ b, X b ∂bernoulliMeasure true false p = 0 := by
-    rw [integral_bernoulliMeasure]
-    simp [X]
-    ring
-  have hg := hasSubgaussianMGF_of_mem_Icc_of_integral_eq_zero
-    (μ := bernoulliMeasure true false p) (X := X)
-    (Measurable.of_discrete.aemeasurable) hb hc
-  have hh := hg.mgf_le lam
-  simp only [mgf,integral_bernoulliMeasure] at hh
-  norm_num [X,sub_neg_eq_add,sub_add_cancel,smul_eq_mul] at hh
-  rw [show (1/4:ℝ)*lam^2/2 = lam^2/8 by ring] at hh
-  simpa only [neg_mul,add_comm] using hh
-
- 
-inductive ProbeTree where
-  | done (success : Bool)
-  | edge (closed open_ : ProbeTree)
-  | mark (unmarked marked : ProbeTree)
-
-namespace ProbeTree
-
-@[reducible] def Leaf : ProbeTree → Type
-  | .done _ => Unit
-  | .edge t₀ t₁ => Leaf t₀ ⊕ Leaf t₁
-  | .mark t₀ t₁ => Leaf t₀ ⊕ Leaf t₁
-
-instance leafFintype : (t : ProbeTree) → Fintype t.Leaf
-  | .done _ => inferInstanceAs (Fintype Unit)
-  | .edge t₀ t₁ => @instFintypeSum _ _ (leafFintype t₀) (leafFintype t₁)
-  | .mark t₀ t₁ => @instFintypeSum _ _ (leafFintype t₀) (leafFintype t₁)
-
-noncomputable def weight (p h : ℝ) : (t : ProbeTree) → t.Leaf → ℝ
-  | .done _, _ => 1
-  | .edge t₀ _, .inl l => (1-p)*weight p h t₀ l
-  | .edge _ t₁, .inr l => p*weight p h t₁ l
-  | .mark t₀ _, .inl l => (1-h)*weight p h t₀ l
-  | .mark _ t₁, .inr l => h*weight p h t₁ l
-
-noncomputable def score (p : ℝ) : (t : ProbeTree) → t.Leaf → ℝ
-  | .done _, _ => 0
-  | .edge t₀ _, .inl l => score p t₀ l-p
-  | .edge _ t₁, .inr l => score p t₁ l+(1-p)
-  | .mark t₀ _, .inl l => score p t₀ l
-  | .mark _ t₁, .inr l => score p t₁ l
-
-def queries : (t : ProbeTree) → t.Leaf → ℕ
-  | .done _, _ => 0
-  | .edge t₀ _, .inl l => queries t₀ l+1
-  | .edge _ t₁, .inr l => queries t₁ l+1
-  | .mark t₀ _, .inl l => queries t₀ l
-  | .mark _ t₁, .inr l => queries t₁ l
-
-@[reducible] def outcome : (t : ProbeTree) → t.Leaf → Bool
-  | .done b, _ => b
-  | .edge t₀ _, .inl l => outcome t₀ l
-  | .edge _ t₁, .inr l => outcome t₁ l
-  | .mark t₀ _, .inl l => outcome t₀ l
-  | .mark _ t₁, .inr l => outcome t₁ l
-
-lemma weight_nonneg (p h : unitInterval) (t : ProbeTree) (l : t.Leaf) :
-    0 ≤ weight p h t l := by
-  induction t with
-  | done b => exact zero_le_one
-  | edge t₀ t₁ ih₀ ih₁ =>
-    cases l with
-    | inl l => exact mul_nonneg (sub_nonneg.mpr p.2.2) (ih₀ l)
-    | inr l => exact mul_nonneg p.2.1 (ih₁ l)
-  | mark t₀ t₁ ih₀ ih₁ =>
-    cases l with
-    | inl l => exact mul_nonneg (sub_nonneg.mpr h.2.2) (ih₀ l)
-    | inr l => exact mul_nonneg h.2.1 (ih₁ l)
-
-lemma sum_weight (p h : ℝ) (t : ProbeTree) : ∑ l, weight p h t l = 1 := by
-  induction t with
-  | done b => change (∑ _ : Unit, (1:ℝ)) = 1; simp
-  | edge t₀ t₁ ih₀ ih₁ =>
-    change (∑ l : t₀.Leaf ⊕ t₁.Leaf, _) = 1
-    simp only [Fintype.sum_sum_type,weight,← Finset.mul_sum,ih₀,ih₁]
-    ring
-  | mark t₀ t₁ ih₀ ih₁ =>
-    change (∑ l : t₀.Leaf ⊕ t₁.Leaf, _) = 1
-    simp only [Fintype.sum_sum_type,weight,← Finset.mul_sum,ih₀,ih₁]
-    ring
-
- 
-theorem score_mgf (p h : unitInterval) (lam : ℝ) (t : ProbeTree) :
-    ∑ l, weight p h t l * Real.exp (lam*score p t l-lam^2*(queries t l : ℝ)/8) ≤ 1 := by
-  induction t with
-  | done b => change (∑ l : Unit, _) ≤ (1:ℝ); simp [weight,score,queries]
-  | edge t₀ t₁ ih₀ ih₁ =>
-    change (∑ l : t₀.Leaf ⊕ t₁.Leaf, _) ≤ (1:ℝ)
-    rw [Fintype.sum_sum_type]
-    have h₀ (l : t₀.Leaf) :
-        weight p h (.edge t₀ t₁) (.inl l) *
-          Real.exp (lam*score p (.edge t₀ t₁) (.inl l)-lam^2*(queries (.edge t₀ t₁) (.inl l):ℝ)/8) =
-        ((1-(p:ℝ))*Real.exp (-lam*p-lam^2/8)) *
-          (weight p h t₀ l * Real.exp (lam*score p t₀ l-lam^2*(queries t₀ l:ℝ)/8)) := by
-      simp only [weight,score,queries,Nat.cast_add,Nat.cast_one]
-      rw [show lam*(score p t₀ l-p)-lam^2*((queries t₀ l:ℝ)+1)/8 =
-        (-lam*p-lam^2/8)+(lam*score p t₀ l-lam^2*(queries t₀ l:ℝ)/8) by ring,Real.exp_add]
-      ring
-    have h₁ (l : t₁.Leaf) :
-        weight p h (.edge t₀ t₁) (.inr l) *
-          Real.exp (lam*score p (.edge t₀ t₁) (.inr l)-lam^2*(queries (.edge t₀ t₁) (.inr l):ℝ)/8) =
-        ((p:ℝ)*Real.exp (lam*(1-p)-lam^2/8)) *
-          (weight p h t₁ l * Real.exp (lam*score p t₁ l-lam^2*(queries t₁ l:ℝ)/8)) := by
-      simp only [weight,score,queries,Nat.cast_add,Nat.cast_one]
-      rw [show lam*(score p t₁ l+(1-p))-lam^2*((queries t₁ l:ℝ)+1)/8 =
-        (lam*(1-p)-lam^2/8)+(lam*score p t₁ l-lam^2*(queries t₁ l:ℝ)/8) by ring,Real.exp_add]
-      ring
-    simp only [h₀,h₁,← Finset.mul_sum]
-    have hc₀ : 0 ≤ (1-(p:ℝ))*Real.exp (-lam*p-lam^2/8) :=
-      mul_nonneg (sub_nonneg.mpr p.2.2) (Real.exp_pos _).le
-    have hc₁ : 0 ≤ (p:ℝ)*Real.exp (lam*(1-p)-lam^2/8) :=
-      mul_nonneg p.2.1 (Real.exp_pos _).le
-    calc
-      _ ≤ (1-(p:ℝ))*Real.exp (-lam*p-lam^2/8) + p*Real.exp (lam*(1-p)-lam^2/8) := by
-        simpa using add_le_add (mul_le_mul_of_nonneg_left ih₀ hc₀) (mul_le_mul_of_nonneg_left ih₁ hc₁)
-      _ = ((1-(p:ℝ))*Real.exp (-lam*p)+p*Real.exp (lam*(1-p))) / Real.exp (lam^2/8) := by
-        simp only [Real.exp_sub]; ring
-      _ ≤ 1 := (div_le_one (Real.exp_pos _)).mpr (centered_bernoulli_mgf_le p lam)
-  | mark t₀ t₁ ih₀ ih₁ =>
-    change (∑ l : t₀.Leaf ⊕ t₁.Leaf, _) ≤ (1:ℝ)
-    simp only [Fintype.sum_sum_type,weight,score,queries,mul_assoc,← Finset.mul_sum]
-    calc
-      _ ≤ (1-(h:ℝ))*1+(h:ℝ)*1 :=
-        add_le_add (mul_le_mul_of_nonneg_left ih₀ (sub_nonneg.mpr h.2.2))
-          (mul_le_mul_of_nonneg_left ih₁ h.2.1)
-      _ = 1 := by ring
-
-noncomputable def successWeight (p h : ℝ) (t : ProbeTree) (l : t.Leaf) : ℝ :=
-  if outcome t l then weight p h t l else 0
-
-noncomputable def successMass (p h : ℝ) (t : ProbeTree) : ℝ :=
-  ∑ l, successWeight p h t l
-
-noncomputable def successScore (p h : ℝ) (t : ProbeTree) : ℝ :=
-  ∑ l, successWeight p h t l * score p t l
-
-noncomputable def successQueries (p h : ℝ) (t : ProbeTree) : ℝ :=
-  ∑ l, successWeight p h t l * queries t l
-
-lemma successWeight_nonneg (p h : unitInterval) (t : ProbeTree) (l : t.Leaf) :
-    0 ≤ successWeight p h t l := by
-  unfold successWeight
-  split <;> simp_all [weight_nonneg]
-
-lemma successWeight_le (p h : unitInterval) (t : ProbeTree) (l : t.Leaf) :
-    successWeight p h t l ≤ weight p h t l := by
-  unfold successWeight
-  split
-  · exact le_rfl
-  · exact weight_nonneg p h t l
-
-lemma successMass_nonneg (p h : unitInterval) (t : ProbeTree) : 0 ≤ successMass p h t :=
-  Finset.sum_nonneg fun _ _ => successWeight_nonneg p h t _
-
-lemma successMass_le_one (p h : unitInterval) (t : ProbeTree) : successMass p h t ≤ 1 := by
-  rw [← sum_weight p h t]
-  exact Finset.sum_le_sum fun _ _ => successWeight_le p h t _
-
-lemma successQueries_nonneg (p h : unitInterval) (t : ProbeTree) : 0 ≤ successQueries p h t :=
-  Finset.sum_nonneg fun _ _ => mul_nonneg (successWeight_nonneg p h t _) (Nat.cast_nonneg _)
-
- 
-theorem conditional_score_bound (p h : unitInterval) (t : ProbeTree)
-    (hH : 0 < successMass p h t) (lam : ℝ) :
-    lam*(successScore p h t / successMass p h t) ≤
-      Real.log (1/successMass p h t) + lam^2/8*(successQueries p h t / successMass p h t) := by
-  let H := successMass p h t
-  let w : t.Leaf → ℝ := fun l => successWeight p h t l / H
-  let X : t.Leaf → ℝ := fun l => lam*score p t l-lam^2*(queries t l : ℝ)/8
-  have hw : ∀ l ∈ (Finset.univ : Finset t.Leaf), 0 ≤ w l :=
-    fun l _ => div_nonneg (successWeight_nonneg p h t l) hH.le
-  have hw1 : ∑ l, w l = 1 := by
-    simp only [w,← Finset.sum_div]
-    exact div_self hH.ne'
-  have hJ := convexOn_exp.map_sum_le hw hw1 (fun _ _ => Set.mem_univ _ : ∀ l ∈
-    (Finset.univ : Finset t.Leaf), X l ∈ Set.univ)
-  simp only [smul_eq_mul] at hJ
-  have hsum : ∑ l, w l * X l = lam*(successScore p h t/H)-
-      lam^2/8*(successQueries p h t/H) := by
-    simp only [w,X,successScore,successQueries,Finset.sum_div,Finset.mul_sum,← Finset.sum_sub_distrib]
-    apply Finset.sum_congr rfl
-    intro l _
-    ring
-  have he : ∑ l, w l * Real.exp (X l) ≤ 1/H := by
-    calc
-      _ = (∑ l, successWeight p h t l * Real.exp (X l))/H := by
-        simp only [Finset.sum_div,w]; apply Finset.sum_congr rfl; intro l _; ring
-      _ ≤ (∑ l, weight p h t l * Real.exp (X l))/H := by
-        apply div_le_div_of_nonneg_right _ hH.le
-        exact Finset.sum_le_sum fun l _ => mul_le_mul_of_nonneg_right (successWeight_le p h t l)
-          (Real.exp_pos _).le
-      _ ≤ 1/H := div_le_div_of_nonneg_right (score_mgf p h lam t) hH.le
-  rw [hsum] at hJ
-  have hb := (Real.le_log_iff_exp_le (one_div_pos.mpr hH)).mpr (hJ.trans he)
-  linarith
-
-noncomputable def dweight (p h : ℝ) : (t : ProbeTree) → t.Leaf → ℝ
-  | .done _, _ => 0
-  | .edge t₀ _, .inl l => -weight p h t₀ l+(1-p)*dweight p h t₀ l
-  | .edge _ t₁, .inr l => weight p h t₁ l+p*dweight p h t₁ l
-  | .mark t₀ _, .inl l => (1-h)*dweight p h t₀ l
-  | .mark _ t₁, .inr l => h*dweight p h t₁ l
-
-lemma hasDerivAt_weight (p h : ℝ) (t : ProbeTree) (l : t.Leaf) :
-    HasDerivAt (fun q => weight q h t l) (dweight p h t l) p := by
-  induction t with
-  | done b => exact hasDerivAt_const p 1
-  | edge t₀ t₁ ih₀ ih₁ =>
-    cases l with
-    | inl l => convert ((hasDerivAt_const p 1).sub (hasDerivAt_id p)).mul (ih₀ l) using 1 <;> try rfl
-               simp [dweight]
-    | inr l => convert (hasDerivAt_id p).mul (ih₁ l) using 1 <;> try rfl
-               simp [dweight]
-  | mark t₀ t₁ ih₀ ih₁ =>
-    cases l with
-    | inl l => exact (ih₀ l).const_mul (1-h)
-    | inr l => exact (ih₁ l).const_mul h
-
-lemma score_eq_derivative (p h : ℝ) (t : ProbeTree) (l : t.Leaf) :
-    p*(1-p)*dweight p h t l = weight p h t l * score p t l := by
-  induction t with
-  | done b => simp [dweight,weight,score]
-  | edge t₀ t₁ ih₀ ih₁ =>
-    cases l with
-    | inl l => dsimp [dweight,weight,score]; linear_combination (1-p)*(ih₀ l)
-    | inr l => dsimp [dweight,weight,score]; linear_combination p*(ih₁ l)
-  | mark t₀ t₁ ih₀ ih₁ =>
-    cases l with
-    | inl l => dsimp [dweight,weight,score]; linear_combination (1-h)*(ih₀ l)
-    | inr l => dsimp [dweight,weight,score]; linear_combination h*(ih₁ l)
-
- 
-theorem successScore_eq_deriv (p h : ℝ) (t : ProbeTree) :
-    successScore p h t = p*(1-p)*deriv (fun q => successMass q h t) p := by
-  have hd : HasDerivAt (fun q => successMass q h t)
-      (∑ l, if outcome t l then dweight p h t l else 0) p := by
-    have hh (l : t.Leaf) (_ : l ∈ (Finset.univ : Finset t.Leaf)) :
-        HasDerivAt (fun q => successWeight q h t l) (if outcome t l then dweight p h t l else 0) p := by
-      by_cases hl : outcome t l
-      · simpa [successWeight,hl] using hasDerivAt_weight p h t l
-      · simpa [successWeight,hl] using hasDerivAt_const p (0:ℝ)
-    simpa only [successMass,Finset.sum_apply] using HasDerivAt.fun_sum hh
-  rw [hd.deriv,Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro l _
-  by_cases hl : outcome t l
-  · simpa [successWeight,hl] using (score_eq_derivative p h t l).symm
-  · simp [successWeight,hl]
-
- 
-inductive IsMarkSearch : ProbeTree → Prop
-  | fail : IsMarkSearch (.done false)
-  | edge {t₀ t₁} : IsMarkSearch t₀ → IsMarkSearch t₁ → IsMarkSearch (.edge t₀ t₁)
-  | mark {t₀} : IsMarkSearch t₀ → IsMarkSearch (.mark t₀ (.done true))
-
-def marks : (t : ProbeTree) → t.Leaf → ℕ
-  | .done _, _ => 0
-  | .edge t₀ _, .inl l => marks t₀ l
-  | .edge _ t₁, .inr l => marks t₁ l
-  | .mark t₀ _, .inl l => marks t₀ l+1
-  | .mark _ t₁, .inr l => marks t₁ l+1
-
-noncomputable def successMarks (p h : ℝ) (t : ProbeTree) : ℝ :=
-  ∑ l, successWeight p h t l * marks t l
-
-lemma successMass_edge (p h : ℝ) (t₀ t₁ : ProbeTree) :
-    successMass p h (.edge t₀ t₁) = (1-p)*successMass p h t₀+p*successMass p h t₁ := by
-  simp only [successMass,Fintype.sum_sum_type,successWeight,outcome,weight,Finset.mul_sum]
-  congr 1 <;> apply Finset.sum_congr rfl <;> intro l _ <;> split <;> simp
-
-lemma successMass_mark (p h : ℝ) (t₀ t₁ : ProbeTree) :
-    successMass p h (.mark t₀ t₁) = (1-h)*successMass p h t₀+h*successMass p h t₁ := by
-  simp only [successMass,Fintype.sum_sum_type,successWeight,outcome,weight,Finset.mul_sum]
-  congr 1 <;> apply Finset.sum_congr rfl <;> intro l _ <;> split <;> simp
-
-@[simp] lemma successMass_done (p h : ℝ) (b : Bool) :
-    successMass p h (.done b) = if b then 1 else 0 := by
-  cases b <;> simp [successMass,successWeight,outcome,weight]
-
-lemma successMarks_edge (p h : ℝ) (t₀ t₁ : ProbeTree) :
-    successMarks p h (.edge t₀ t₁) = (1-p)*successMarks p h t₀+p*successMarks p h t₁ := by
-  simp only [successMarks,Fintype.sum_sum_type,successWeight,outcome,weight,marks,Finset.mul_sum]
-  congr 1 <;> apply Finset.sum_congr rfl <;> intro l _ <;> split <;> simp [mul_assoc]
-
-lemma successMarks_mark (p h : ℝ) (t₀ t₁ : ProbeTree) :
-    successMarks p h (.mark t₀ t₁) =
-      (1-h)*(successMarks p h t₀+successMass p h t₀) +
-        h*(successMarks p h t₁+successMass p h t₁) := by
-  simp only [successMarks,successMass,Fintype.sum_sum_type,successWeight,outcome,weight,marks,
-    Nat.cast_add,Nat.cast_one,← Finset.sum_add_distrib,Finset.mul_sum]
-  congr 1 <;> apply Finset.sum_congr rfl <;> intro l _ <;> split_ifs <;> ring
-
-@[simp] lemma successMarks_done (p h : ℝ) (b : Bool) : successMarks p h (.done b) = 0 := by
-  simp [successMarks,marks]
-
- 
-theorem marks_times_parameter_le (p h : unitInterval) {t : ProbeTree} (ht : IsMarkSearch t) :
-    (h:ℝ)*successMarks p h t ≤ successMass p h t := by
-  induction ht with
-  | fail => simp
-  | @edge t₀ t₁ _ _ ih₀ ih₁ =>
-    rw [successMarks_edge,successMass_edge]
-    nlinarith [mul_nonneg (sub_nonneg.mpr p.2.2) (sub_nonneg.mpr ih₀),
-      mul_nonneg p.2.1 (sub_nonneg.mpr ih₁)]
-  | @mark t₀ _ ih =>
-    rw [successMarks_mark,successMass_mark]
-    simp only [successMarks_done,successMass_done,ite_true,zero_add,mul_one]
-    have hn := mul_nonneg (mul_nonneg h.2.1 (sub_nonneg.mpr h.2.2))
-      (sub_nonneg.mpr (successMass_le_one p h t₀))
-    nlinarith [mul_nonneg (sub_nonneg.mpr h.2.2) (sub_nonneg.mpr ih)]
-
-theorem successQueries_times_parameter_le (p h : unitInterval) {t : ProbeTree}
-    (ht : IsMarkSearch t) (d : ℕ) (hcost : ∀ l, queries t l ≤ d*marks t l) :
-    (h:ℝ)*successQueries p h t ≤ d*successMass p h t := by
-  have hs : successQueries p h t ≤ d*successMarks p h t := by
-    simp only [successQueries,successMarks,Finset.mul_sum]
-    apply Finset.sum_le_sum
-    intro l _
-    calc
-      _ ≤ successWeight p h t l * ((d:ℝ)*marks t l) :=
-        mul_le_mul_of_nonneg_left (by exact_mod_cast hcost l) (successWeight_nonneg p h t l)
-      _ = _ := by ring
-  calc
-    _ ≤ (h:ℝ)*(d*successMarks p h t) := mul_le_mul_of_nonneg_left hs h.2.1
-    _ = d*((h:ℝ)*successMarks p h t) := by ring
-    _ ≤ d*successMass p h t := mul_le_mul_of_nonneg_left (marks_times_parameter_le p h ht)
-      (Nat.cast_nonneg _)
-
- 
-theorem marked_score_bound (p h : unitInterval) (hh : 0 < (h:ℝ))
-    {t : ProbeTree} (ht : IsMarkSearch t) (d : ℕ)
-    (hcost : ∀ l, queries t l ≤ d*marks t l) (hroot : (h:ℝ) ≤ successMass p h t)
-    (lam : ℝ) :
-    lam*(successScore p h t / successMass p h t) ≤
-      Real.log (1/(h:ℝ)) + lam^2*(d:ℝ)/(8*h) := by
-  have hH := hh.trans_le hroot
-  have hlog : Real.log (1/successMass p h t) ≤ Real.log (1/(h:ℝ)) :=
-    Real.log_le_log (one_div_pos.mpr hH) (one_div_le_one_div_of_le hh hroot)
-  have hQ : successQueries p h t / successMass p h t ≤ (d:ℝ)/h := by
-    apply (div_le_div_iff₀ hH hh).mpr
-    simpa [mul_comm] using successQueries_times_parameter_le p h ht d hcost
-  calc
-    _ ≤ Real.log (1/successMass p h t)+lam^2/8*(successQueries p h t/successMass p h t) :=
-      conditional_score_bound p h t hH lam
-    _ ≤ Real.log (1/(h:ℝ))+lam^2/8*((d:ℝ)/h) :=
-      add_le_add hlog (mul_le_mul_of_nonneg_left hQ (by positivity))
-    _ = _ := by ring
-
-end ProbeTree
-end MarkedExploration
-
-section LabelledExploration
-attribute [local instance] Classical.propDecidable
-attribute [local instance] Classical.decEq
-
-inductive LabelledTree (V E : Type*) where
-  | done (success : Bool)
-  | edge (e : E) (closed opened : LabelledTree V E)
-  | mark (v : V) (unmarked marked : LabelledTree V E)
-
-namespace LabelledTree
-
-def forget : LabelledTree V E → ProbeTree
-  | .done b => .done b
-  | .edge _ t₀ t₁ => .edge t₀.forget t₁.forget
-  | .mark _ t₀ t₁ => .mark t₀.forget t₁.forget
-
-@[reducible] noncomputable def eval (ω : Finset E) (m : Finset V) : LabelledTree V E → Bool
-  | .done b => b
-  | .edge e t₀ t₁ => if e ∈ ω then eval ω m t₁ else eval ω m t₀
-  | .mark v t₀ t₁ => if v ∈ m then eval ω m t₁ else eval ω m t₀
-
-def Valid (Q : Finset E) (F : Finset V) : LabelledTree V E → Prop
-  | .done _ => True
-  | .edge e t₀ t₁ => e ∈ Q ∧ Valid (Q.erase e) F t₀ ∧ Valid (Q.erase e) F t₁
-  | .mark v t₀ t₁ => v ∈ F ∧ Valid Q (F.erase v) t₀ ∧ Valid Q (F.erase v) t₁
-
-theorem eval_congr (t : LabelledTree V E) {Q : Finset E} {F : Finset V}
-    (ht : Valid Q F t) {ω ω' : Finset E} {m m' : Finset V}
-    (hω : ∀ e ∈ Q, e ∈ ω ↔ e ∈ ω') (hm : ∀ v ∈ F, v ∈ m ↔ v ∈ m') :
-    t.eval ω m = t.eval ω' m' := by
-  induction t generalizing Q F with
-  | done b => rfl
-  | edge e t₀ t₁ ih₀ ih₁ =>
-    obtain ⟨he,h₀,h₁⟩ := ht
-    simp only [eval,hω e he]
-    split
-    · exact ih₁ h₁ (fun a ha => hω a (Finset.mem_of_mem_erase ha)) hm
-    · exact ih₀ h₀ (fun a ha => hω a (Finset.mem_of_mem_erase ha)) hm
-  | mark v t₀ t₁ ih₀ ih₁ =>
-    obtain ⟨hv,h₀,h₁⟩ := ht
-    simp only [eval,hm v hv]
-    split
-    · exact ih₁ h₁ hω (fun a ha => hm a (Finset.mem_of_mem_erase ha))
-    · exact ih₀ h₀ hω (fun a ha => hm a (Finset.mem_of_mem_erase ha))
-
-lemma eval_insert_edge (t : LabelledTree V E) {Q : Finset E} {F : Finset V}
-    (ht : Valid Q F t) {e : E} (he : e ∉ Q) (ω : Finset E) (m : Finset V) :
-    t.eval (insert e ω) m = t.eval ω m := by
-  apply eval_congr t ht
-  · intro a ha
-    simp [show a ≠ e from fun hh => he (hh ▸ ha)]
-  · simp
-
-lemma eval_insert_mark (t : LabelledTree V E) {Q : Finset E} {F : Finset V}
-    (ht : Valid Q F t) {v : V} (hv : v ∉ F) (ω : Finset E) (m : Finset V) :
-    t.eval ω (insert v m) = t.eval ω m := by
-  apply eval_congr t ht
-  · simp
-  · intro a ha
-    simp [show a ≠ v from fun hh => hv (hh ▸ ha)]
-
-end LabelledTree
-
-theorem bernoulliPolynomial_affine (s : Finset E) (f g : Finset E → ℝ) (a b p : ℝ) :
-    bernoulliPolynomial s (fun t => a*f t+b*g t) p =
-      a*bernoulliPolynomial s f p+b*bernoulliPolynomial s g p := by
-  simp only [bernoulliPolynomial,add_mul,Finset.sum_add_distrib,Finset.mul_sum]
-  congr 1 <;> apply Finset.sum_congr rfl <;> intro t _ <;> ring
-
-namespace LabelledTree
-
- 
-theorem expectation_eq (t : LabelledTree V E) {Q : Finset E} {F : Finset V}
-    (ht : Valid Q F t) (p h : ℝ) :
-    bernoulliPolynomial Q (fun ω => bernoulliPolynomial F
-      (fun m => if t.eval ω m then 1 else 0) h) p =
-      ProbeTree.successMass p h t.forget := by
-  induction t generalizing Q F with
-  | done b =>
-    cases b <;> simp [eval,forget,ProbeTree.successMass_done,bernoulliPolynomial_const]
-  | edge e t₀ t₁ ih₀ ih₁ =>
-    obtain ⟨he,h₀,h₁⟩ := ht
-    conv_lhs => rw [← Finset.insert_erase he]
-    rw [bernoulliPolynomial_insert _ e (Finset.notMem_erase e Q)]
-    have hc : bernoulliPolynomial (Q.erase e) (fun ω => bernoulliPolynomial F
-        (fun m => if (edge e t₀ t₁).eval ω m then 1 else 0) h) p =
-        bernoulliPolynomial (Q.erase e) (fun ω => bernoulliPolynomial F
-        (fun m => if t₀.eval ω m then 1 else 0) h) p := by
-      apply bernoulliPolynomial_congr
-      intro ω hω
-      have heω : e ∉ ω := fun hh => Finset.notMem_erase e Q (hω hh)
-      simp only [eval,heω,ite_false]
-    have ho : bernoulliPolynomial (Q.erase e) (fun ω => bernoulliPolynomial F
-        (fun m => if (edge e t₀ t₁).eval (insert e ω) m then 1 else 0) h) p =
-        bernoulliPolynomial (Q.erase e) (fun ω => bernoulliPolynomial F
-        (fun m => if t₁.eval ω m then 1 else 0) h) p := by
-      apply bernoulliPolynomial_congr
-      intro ω _
-      simp only [eval,Finset.mem_insert_self,ite_true]
-      apply bernoulliPolynomial_congr
-      intro m _
-      rw [eval_insert_edge t₁ h₁ (Finset.notMem_erase e Q)]
-    rw [hc,ho,ih₀ h₀,ih₁ h₁,forget,ProbeTree.successMass_edge]
-  | mark v t₀ t₁ ih₀ ih₁ =>
-    obtain ⟨hv,h₀,h₁⟩ := ht
-    have hh : ∀ ω, bernoulliPolynomial F
-        (fun m => if (mark v t₀ t₁).eval ω m then 1 else 0) h =
-      (1-h)*bernoulliPolynomial (F.erase v) (fun m => if t₀.eval ω m then 1 else 0) h +
-        h*bernoulliPolynomial (F.erase v) (fun m => if t₁.eval ω m then 1 else 0) h := by
-      intro ω
-      conv_lhs => rw [← Finset.insert_erase hv]
-      rw [bernoulliPolynomial_insert _ v (Finset.notMem_erase v F)]
-      congr 1
-      · congr 1
-        apply bernoulliPolynomial_congr
-        intro m hm
-        have hvm : v ∉ m := fun hh => Finset.notMem_erase v F (hm hh)
-        simp only [eval,hvm,ite_false]
-      · congr 1
-        apply bernoulliPolynomial_congr
-        intro m _
-        simp only [eval,Finset.mem_insert_self,ite_true]
-        rw [eval_insert_mark t₁ h₁ (Finset.notMem_erase v F)]
-    simp_rw [hh]
-    rw [bernoulliPolynomial_affine,ih₀ h₀,ih₁ h₁,forget,ProbeTree.successMass_mark]
-
-end LabelledTree
-end LabelledExploration
-
-section GraphMarkedSearch
-attribute [local instance] Classical.propDecidable
-attribute [local instance] Classical.decEq
-
- 
-def SearchCrossing (G : BondGraph V E) (Q : Finset E) (R : Finset V) : Prop :=
-  ∃ t : E × V × V, t.1 ∈ Q ∧ t.2.1 ∈ R ∧ t.2.2 ∉ R ∧ G t.1 = s(t.2.1,t.2.2)
-
-noncomputable def searchCrossing (G : BondGraph V E) (Q : Finset E) (R : Finset V)
-    (h : SearchCrossing G Q R) : E × V × V := Classical.choose h
-
-theorem searchCrossing_spec (G : BondGraph V E) (Q : Finset E) (R : Finset V)
-    (h : SearchCrossing G Q R) :
-    (searchCrossing G Q R h).1 ∈ Q ∧ (searchCrossing G Q R h).2.1 ∈ R ∧
-      (searchCrossing G Q R h).2.2 ∉ R ∧
-      G (searchCrossing G Q R h).1 = s((searchCrossing G Q R h).2.1,(searchCrossing G Q R h).2.2) :=
-  Classical.choose_spec h
-
- 
-def SearchAvailable (G : BondGraph V E) (Q : Finset E) (R F : Finset V) : Prop :=
-  ∀ e ∈ Q, ∀ v ∈ G e, v ∈ R ∨ v ∈ F
-
-noncomputable def clusterSearch (G : BondGraph V E) : ℕ → Finset E → Finset V → LabelledTree V E
-  | 0, _, _ => .done false
-  | n+1, Q, R =>
-    if hc : SearchCrossing G Q R then
-      let t := searchCrossing G Q R hc
-      .edge t.1 (clusterSearch G n (Q.erase t.1) R)
-        (.mark t.2.2 (clusterSearch G n (Q.erase t.1) (insert t.2.2 R)) (.done true))
-    else .done false
-
-lemma searchAvailable_erase (G : BondGraph V E) {Q : Finset E} {R F : Finset V}
-    (h : SearchAvailable G Q R F) (e : E) : SearchAvailable G (Q.erase e) R F :=
-  fun _ ha _ hv => h _ (Finset.mem_of_mem_erase ha) _ hv
-
-lemma searchAvailable_discover (G : BondGraph V E) {Q : Finset E} {R F : Finset V}
-    (h : SearchAvailable G Q R F) (e : E) (v : V) :
-    SearchAvailable G (Q.erase e) (insert v R) (F.erase v) := by
-  intro a ha b hb
-  rcases h a (Finset.mem_of_mem_erase ha) b hb with hb | hb
-  · exact Or.inl (Finset.mem_insert_of_mem hb)
-  · by_cases hbe : b = v
-    · exact Or.inl (by simp [hbe])
-    · exact Or.inr (Finset.mem_erase.mpr ⟨hbe,hb⟩)
-
-theorem clusterSearch_valid (G : BondGraph V E) (n : ℕ) (Q : Finset E) (R F : Finset V)
-    (ha : SearchAvailable G Q R F) : LabelledTree.Valid Q F (clusterSearch G n Q R) := by
-  induction n generalizing Q R F with
-  | zero => trivial
-  | succ n ih =>
-    simp only [clusterSearch]
-    split_ifs with hc
-    · obtain ⟨he,hv,hw,hends⟩ := searchCrossing_spec G Q R hc
-      refine ⟨he,ih _ _ _ (searchAvailable_erase G ha _),?_,?_,trivial⟩
-      · exact (ha _ he _ (by rw [hends]; simp)).resolve_left hw
-      · exact ih _ _ _ (searchAvailable_discover G ha _ _)
-    · trivial
-
-theorem clusterSearch_markSearch (G : BondGraph V E) (n : ℕ) (Q : Finset E) (R : Finset V) :
-    ProbeTree.IsMarkSearch (clusterSearch G n Q R).forget := by
-  induction n generalizing Q R with
-  | zero => exact .fail
-  | succ n ih =>
-    simp only [clusterSearch]
-    split_ifs
-    · exact .edge (ih _ _) (.mark (ih _ _))
-    · exact .fail
-
- 
-def SeedMarked (G : BondGraph V E) (ω : Set E) (R m : Finset V) : Prop :=
-  ∃ a ∈ R, ∃ b ∈ m, Conn G ω a b
-
-theorem seedMarked_insert_edge (G : BondGraph V E) (ω : Set E) (R m : Finset V)
-    (e : E) (a b : V) (ha : a ∈ R) (he : G e = s(a,b)) :
-    SeedMarked G (insert e ω) R m ↔ SeedMarked G ω (insert b R) m := by
-  constructor
-  · rintro ⟨x,hx,y,hy,hxy⟩
-    rcases (conn_insert_iff G ω e a b x y he).mp hxy with h | ⟨_,h⟩ | ⟨_,h⟩
-    · exact ⟨x,Finset.mem_insert_of_mem hx,y,hy,h⟩
-    · exact ⟨b,by simp,y,hy,h⟩
-    · exact ⟨a,Finset.mem_insert_of_mem ha,y,hy,h⟩
-  · rintro ⟨x,hx,y,hy,hxy⟩
-    rcases Finset.mem_insert.mp hx with rfl | hx
-    · refine ⟨a,ha,y,hy,?_⟩
-      exact (Relation.ReflTransGen.single ⟨e,Set.mem_insert e ω,he⟩).trans
-        (conn_mono G (Set.subset_insert e ω) hxy)
-    · exact ⟨x,hx,y,hy,conn_mono G (Set.subset_insert e ω) hxy⟩
-
-theorem seedMarked_false_of_no_crossing (G : BondGraph V E) (Q : Finset E) (R m : Finset V)
-    (hc : ¬ SearchCrossing G Q R) (hm : Disjoint R m) (ω : Finset E) (hω : ω ⊆ Q) :
-    ¬ SeedMarked G (ω : Set E) R m := by
-  rintro ⟨a,ha,b,hb,hab⟩
-  have hclosed : ∀ x ∈ R, ∀ y, Conn G (ω : Set E) x y → y ∈ R := by
-    intro x hx y hxy
-    induction hxy with
-    | refl => exact hx
-    | @tail v w hv hw ih =>
-      by_contra hn
-      obtain ⟨e,he,hevw⟩ := hw
-      exact hc ⟨(e,v,w),hω he,ih,hn,hevw⟩
-  exact Finset.disjoint_left.mp hm (hclosed a ha b hab) hb
-
- 
-theorem clusterSearch_correct (G : BondGraph V E) (n : ℕ) (Q : Finset E) (R F m : Finset V)
-    (hn : Q.card ≤ n) (ha : SearchAvailable G Q R F) (hm : Disjoint R m)
-    (ω : Finset E) (hω : ω ⊆ Q) :
-    (clusterSearch G n Q R).eval ω m = true ↔ SeedMarked G (ω : Set E) R m := by
-  induction n generalizing Q R F ω with
-  | zero =>
-    have hQ : Q = ∅ := Finset.card_eq_zero.mp (by omega)
-    have hωe : ω = ∅ := Finset.eq_empty_iff_forall_notMem.mpr (by simp_all)
-    simp only [clusterSearch,LabelledTree.eval,Bool.false_eq_true,false_iff]
-    exact seedMarked_false_of_no_crossing G Q R m (by simp [SearchCrossing,hQ]) hm ω hω
-  | succ n ih =>
-    simp only [clusterSearch]
-    split_ifs with hc
-    · obtain ⟨he,hv,hw,hends⟩ := searchCrossing_spec G Q R hc
-      have hcard : (Q.erase (searchCrossing G Q R hc).1).card ≤ n := by
-        rw [Finset.card_erase_of_mem he]; omega
-      simp only [LabelledTree.eval]
-      by_cases heω : (searchCrossing G Q R hc).1 ∈ ω
-      · simp only [heω,ite_true]
-        rw [← Finset.insert_erase heω,Finset.coe_insert,
-          seedMarked_insert_edge G _ R m _ _ _ hv hends]
-        by_cases hwm : (searchCrossing G Q R hc).2.2 ∈ m
-        · simp only [hwm,ite_true,true_iff]
-          exact ⟨_,by simp,_,hwm,conn_refl G _ _⟩
-        · simp only [hwm,ite_false]
-          rw [LabelledTree.eval_insert_edge _
-            (clusterSearch_valid G n _ _ (F.erase (searchCrossing G Q R hc).2.2)
-              (searchAvailable_discover G ha _ _)) (Finset.notMem_erase _ _)]
-          exact ih _ _ _ hcard (searchAvailable_discover G ha _ _)
-            (Finset.disjoint_insert_left.mpr ⟨hwm,hm⟩) (ω.erase (searchCrossing G Q R hc).1)
-            (Finset.erase_subset_erase _ hω)
-      · simp only [heω,ite_false]
-        exact ih _ _ _ hcard (searchAvailable_erase G ha _ ) hm ω
-          (fun e he => Finset.mem_erase.mpr ⟨fun heq => heω (heq ▸ he),hω he⟩)
-    · simp only [false_iff]
-      exact seedMarked_false_of_no_crossing G Q R m hc hm ω hω
-
-end GraphMarkedSearch
-
-section GraphSearchCost
-attribute [local instance] Classical.propDecidable
-attribute [local instance] Classical.decEq
-
-namespace ProbeTree
-
-def Budget (t : ProbeTree) (K d : ℕ) : Prop := ∀ l, queries t l ≤ K+d*marks t l
-
-lemma budget_done (b : Bool) (K d : ℕ) : Budget (.done b) K d := by
-  intro l
-  simp [queries,marks]
-
-lemma budget_mono {t : ProbeTree} {K L d : ℕ} (h : Budget t K d) (hKL : K ≤ L) :
-    Budget t L d := fun l => (h l).trans (Nat.add_le_add_right hKL _)
-
-lemma budget_edge {t₀ t₁ : ProbeTree} {K d : ℕ} (h₀ : Budget t₀ K d) (h₁ : Budget t₁ K d) :
-    Budget (.edge t₀ t₁) (K+1) d := by
-  intro l
-  cases l with
-  | inl l => simpa [queries,marks,Nat.add_right_comm] using Nat.add_le_add_right (h₀ l) 1
-  | inr l => simpa [queries,marks,Nat.add_right_comm] using Nat.add_le_add_right (h₁ l) 1
-
-lemma budget_mark {t₀ t₁ : ProbeTree} {K d : ℕ}
-    (h₀ : Budget t₀ (K+d) d) (h₁ : Budget t₁ (K+d) d) :
-    Budget (.mark t₀ t₁) K d := by
-  intro l
-  cases l with
-  | inl l => simpa [queries,marks,Nat.mul_add,Nat.add_assoc,Nat.add_comm,Nat.add_left_comm] using h₀ l
-  | inr l => simpa [queries,marks,Nat.mul_add,Nat.add_assoc,Nat.add_comm,Nat.add_left_comm] using h₁ l
-
-end ProbeTree
-
- 
-noncomputable def queryPotential (G : BondGraph V E) (Q : Finset E) (R : Finset V) : Finset E :=
-  Q.filter (fun e => ∃ v ∈ R, v ∈ G e)
-
-lemma queryPotential_erase (G : BondGraph V E) (Q : Finset E) (R : Finset V) (e : E) :
-    queryPotential G (Q.erase e) R = (queryPotential G Q R).erase e := by
-  ext a
-  simp [queryPotential,and_assoc]
-
-lemma queryPotential_insert (G : BondGraph V E) (Q : Finset E) (R : Finset V) (v : V) :
-    queryPotential G Q (insert v R) = queryPotential G Q R ∪ Q.filter (fun e => v ∈ G e) := by
-  ext a
-  simp [queryPotential,and_or_left,or_comm]
-
-lemma filtered_degree_erase (G : BondGraph V E) (Q : Finset E) (d : ℕ)
-    (hd : ∀ v, (Q.filter (fun e => v ∈ G e)).card ≤ d) (a : E) :
-    ∀ v, ((Q.erase a).filter (fun e => v ∈ G e)).card ≤ d := by
-  intro v
-  apply (Finset.card_le_card ?_).trans (hd v)
-  intro e he
-  exact Finset.mem_filter.mpr ⟨Finset.mem_of_mem_erase (Finset.mem_filter.mp he).1,
-    (Finset.mem_filter.mp he).2⟩
-
-lemma queryPotential_discover_card (G : BondGraph V E) (Q : Finset E) (R : Finset V)
-    (e : E) (v : V) (d : ℕ) (hd : ∀ v, (Q.filter (fun e => v ∈ G e)).card ≤ d) :
-    (queryPotential G (Q.erase e) (insert v R)).card ≤
-      ((queryPotential G Q R).erase e).card+d := by
-  rw [queryPotential_insert,queryPotential_erase]
-  exact (Finset.card_union_le _ _).trans
-    (Nat.add_le_add_left (filtered_degree_erase G Q d hd e v) _)
-
- 
-theorem clusterSearch_budget (G : BondGraph V E) (n : ℕ) (Q : Finset E) (R : Finset V)
-    (d : ℕ) (hd : ∀ v, (Q.filter (fun e => v ∈ G e)).card ≤ d) :
-    ProbeTree.Budget (clusterSearch G n Q R).forget (queryPotential G Q R).card d := by
-  induction n generalizing Q R with
-  | zero => exact ProbeTree.budget_done _ _ _
-  | succ n ih =>
-    simp only [clusterSearch]
-    split_ifs with hc
-    · obtain ⟨he,hv,hw,hends⟩ := searchCrossing_spec G Q R hc
-      have hpot : (searchCrossing G Q R hc).1 ∈ queryPotential G Q R := by
-        exact Finset.mem_filter.mpr ⟨he,_,hv,by rw [hends]; simp⟩
-      have hpos : 0 < (queryPotential G Q R).card := Finset.card_pos.mpr ⟨_,hpot⟩
-      have hclosed := ih (Q.erase (searchCrossing G Q R hc).1) R
-        (filtered_degree_erase G Q d hd _)
-      have hopen := ih (Q.erase (searchCrossing G Q R hc).1)
-        (insert (searchCrossing G Q R hc).2.2 R) (filtered_degree_erase G Q d hd _)
-      rw [queryPotential_erase,Finset.card_erase_of_mem hpot] at hclosed
-      have hnew := queryPotential_discover_card G Q R (searchCrossing G Q R hc).1
-        (searchCrossing G Q R hc).2.2 d hd
-      rw [Finset.card_erase_of_mem hpot] at hnew
-      have hh := ProbeTree.budget_edge hclosed
-        (ProbeTree.budget_mark (ProbeTree.budget_mono hopen hnew) (ProbeTree.budget_done true _ _))
-      have hcadd : (queryPotential G Q R).card-1+1 = (queryPotential G Q R).card := by omega
-      simpa only [LabelledTree.forget,hcadd] using hh
-    · exact ProbeTree.budget_done _ _ _
-
-noncomputable def searchVertices (G : BondGraph V E) (Q : Finset E) : Finset V :=
-  Q.biUnion fun e => (G e).toFinset
-
-lemma mem_searchVertices (G : BondGraph V E) {Q : Finset E} {e : E} (he : e ∈ Q)
-    {v : V} (hv : v ∈ G e) : v ∈ searchVertices G Q :=
-  Finset.mem_biUnion.mpr ⟨e,he,Sym2.mem_toFinset.mpr hv⟩
-
-noncomputable def markedClusterTree (G : BondGraph V E) (Q : Finset E) (x : V) : LabelledTree V E :=
-  .mark x (clusterSearch G Q.card Q {x}) (.done true)
-
-lemma markedClusterTree_available (G : BondGraph V E) (Q : Finset E) (x : V) :
-    SearchAvailable G Q {x} ((insert x (searchVertices G Q)).erase x) := by
-  intro e he v hv
-  by_cases hx : v = x
-  · exact Or.inl (by simpa using hx)
-  · exact Or.inr (Finset.mem_erase.mpr ⟨hx,Finset.mem_insert_of_mem (mem_searchVertices G he hv)⟩)
-
-theorem markedClusterTree_valid (G : BondGraph V E) (Q : Finset E) (x : V) :
-    LabelledTree.Valid Q (insert x (searchVertices G Q)) (markedClusterTree G Q x) :=
-  ⟨by simp,clusterSearch_valid G _ _ _ _ (markedClusterTree_available G Q x),trivial⟩
-
-theorem markedClusterTree_budget (G : BondGraph V E) (Q : Finset E) (x : V)
-    (d : ℕ) (hd : ∀ v, (Q.filter (fun e => v ∈ G e)).card ≤ d) :
-    ProbeTree.Budget (markedClusterTree G Q x).forget 0 d := by
-  have hh := clusterSearch_budget G Q.card Q {x} d hd
-  have hcost : (queryPotential G Q {x}).card ≤ 0+d := by simpa [queryPotential] using hd x
-  exact ProbeTree.budget_mark (ProbeTree.budget_mono hh hcost) (ProbeTree.budget_done true _ _)
-
-theorem markedClusterTree_correct (G : BondGraph V E) (Q : Finset E) (x : V)
-    (ω : Finset E) (hω : ω ⊆ Q) (m : Finset V) :
-    (markedClusterTree G Q x).eval ω m = true ↔ ∃ y ∈ m, Conn G (ω : Set E) x y := by
-  simp only [markedClusterTree,LabelledTree.eval]
-  by_cases hxm : x ∈ m
-  · simp only [hxm,ite_true,true_iff]
-    exact ⟨x,hxm,conn_refl G _ x⟩
-  · simp only [hxm,ite_false]
-    rw [clusterSearch_correct G Q.card Q {x} ((insert x (searchVertices G Q)).erase x) m
-      le_rfl (markedClusterTree_available G Q x) (by simpa) ω hω]
-    simp only [SeedMarked,Finset.mem_singleton,exists_eq_left]
-
-end GraphSearchCost
-
-section MarkedFirstMoment
-attribute [local instance] Classical.propDecidable
-attribute [local instance] Classical.decEq
-
-theorem quadratic_test_bound {z L B : ℝ} (hL : 0 ≤ L) (hB : 0 < B)
-    (h : ∀ lam : ℝ, lam*z ≤ L+lam^2*B) : z ≤ Real.sqrt (4*B*L) := by
-  have hh := h (z/(2*B))
-  have hid : z/(2*B)*z-(z/(2*B))^2*B = z^2/(4*B) := by
-    field_simp
-    ring
-  have hs : z^2/(4*B) ≤ L := by rw [← hid]; linarith
-  have hs' := (div_le_iff₀ (by positivity : 0 < 4*B)).mp hs
-  have hn : 0 ≤ 4*B*L := by positivity
-  have hsq := Real.sq_sqrt hn
-  have hpos := Real.sqrt_nonneg (4*B*L)
-  nlinarith
-
- 
-noncomputable def markedConnectionPolynomial (G : BondGraph V E) (Q : Finset E)
-    (x : V) (h p : ℝ) : ℝ :=
-  bernoulliPolynomial Q (fun ω => bernoulliPolynomial (insert x (searchVertices G Q))
-    (fun m => if ∃ y ∈ m, Conn G (ω : Set E) x y then 1 else 0) h) p
-
-theorem markedConnectionPolynomial_eq_tree (G : BondGraph V E) (Q : Finset E)
-    (x : V) (h p : ℝ) :
-    markedConnectionPolynomial G Q x h p =
-      ProbeTree.successMass p h (markedClusterTree G Q x).forget := by
-  rw [← LabelledTree.expectation_eq _ (markedClusterTree_valid G Q x) p h]
-  apply bernoulliPolynomial_congr
-  intro ω hω
-  apply bernoulliPolynomial_congr
-  intro m _
-  simp only [markedClusterTree_correct G Q x ω hω m]
-
-theorem markedConnectionPolynomial_ge_mark (G : BondGraph V E) (Q : Finset E)
-    (x : V) (p h : unitInterval) : (h:ℝ) ≤ markedConnectionPolynomial G Q x h p := by
-  rw [markedConnectionPolynomial_eq_tree,markedClusterTree,LabelledTree.forget,
-    ProbeTree.successMass_mark]
-  simp only [LabelledTree.forget,ProbeTree.successMass_done,ite_true,mul_one]
-  exact le_add_of_nonneg_left (mul_nonneg (sub_nonneg.mpr h.2.2)
-    (ProbeTree.successMass_nonneg p h _))
-
- 
-theorem markedConnection_derivative_bound (G : BondGraph V E) (Q : Finset E)
-    (x : V) (p h : unitInterval) (hp : (p:ℝ) < 1) (hh : 0 < (h:ℝ))
-    (d : ℕ) (hd : 0 < d) (hdeg : ∀ v, (Q.filter (fun e => v ∈ G e)).card ≤ d) :
-    (p:ℝ)*deriv (markedConnectionPolynomial G Q x h) p ≤
-      (Real.sqrt ((d:ℝ)/2)/(1-p))*Real.sqrt (Real.log (1/(h:ℝ))/h)*
-        markedConnectionPolynomial G Q x h p := by
-  let t := (markedClusterTree G Q x).forget
-  have ht : ProbeTree.IsMarkSearch t := .mark (clusterSearch_markSearch G _ _ _)
-  have hcost : ∀ l, ProbeTree.queries t l ≤ d*ProbeTree.marks t l := by
-    simpa only [ProbeTree.Budget,zero_add] using markedClusterTree_budget G Q x d hdeg
-  have hroot : (h:ℝ) ≤ ProbeTree.successMass p h t := by
-    simpa only [markedConnectionPolynomial_eq_tree] using markedConnectionPolynomial_ge_mark G Q x p h
-  have hB : 0 < (d:ℝ)/(8*h) := div_pos (by exact_mod_cast hd) (by positivity)
-  have hL : 0 ≤ Real.log (1/(h:ℝ)) := Real.log_nonneg ((one_le_div hh).mpr h.2.2)
-  have htest (lam : ℝ) : lam*(ProbeTree.successScore p h t/ProbeTree.successMass p h t) ≤
-      Real.log (1/(h:ℝ))+lam^2*((d:ℝ)/(8*h)) := by
-    simpa only [mul_div_assoc] using ProbeTree.marked_score_bound p h hh ht d hcost hroot lam
-  have hz := quadratic_test_bound hL hB htest
-  have hsqrt : Real.sqrt (4*((d:ℝ)/(8*h))*Real.log (1/(h:ℝ))) =
-      Real.sqrt ((d:ℝ)/2)*Real.sqrt (Real.log (1/(h:ℝ))/h) := by
-    rw [show 4*((d:ℝ)/(8*h))*Real.log (1/(h:ℝ)) =
-      ((d:ℝ)/2)*(Real.log (1/(h:ℝ))/h) by ring,Real.sqrt_mul (by positivity)]
-  rw [hsqrt] at hz
-  have hH : 0 < ProbeTree.successMass p h t := hh.trans_le hroot
-  have hscore := (div_le_iff₀ hH).mp hz
-  rw [ProbeTree.successScore_eq_deriv] at hscore
-  have heq : markedConnectionPolynomial G Q x h = fun q => ProbeTree.successMass q h t :=
-    funext fun q => markedConnectionPolynomial_eq_tree G Q x h q
-  rw [heq]
-  have hq : 0 < 1-(p:ℝ) := sub_pos.mpr hp
-  calc
-    _ = ((p:ℝ)*(1-p)*deriv (fun q => ProbeTree.successMass q h t) p)/(1-p) := by
-      field_simp
-    _ ≤ (Real.sqrt ((d:ℝ)/2)*Real.sqrt (Real.log (1/(h:ℝ))/h)*
-        ProbeTree.successMass p h t)/(1-p) := div_le_div_of_nonneg_right hscore hq.le
-    _ = _ := by ring
-
-theorem bernoulliPolynomial_open_difference (s : Finset E) (e : E) (he : e ∈ s)
-    (f : Finset E → ℝ) (p : ℝ) :
-    bernoulliPolynomial s (fun t => if e ∈ t then f t-f (t.erase e) else 0) p =
-      p*bernoulliPolynomial (s.erase e) (fun t => f (insert e t)-f t) p := by
-  conv_lhs => rw [← Finset.insert_erase he]
-  rw [bernoulliPolynomial_insert _ e (Finset.notMem_erase e s)]
-  have hclosed : bernoulliPolynomial (s.erase e)
-      (fun t => if e ∈ t then f t-f (t.erase e) else 0) p = 0 := by
-    rw [← bernoulliPolynomial_zero (s.erase e) p]
-    apply bernoulliPolynomial_congr
-    intro t ht
-    simp [show e ∉ t from fun hh => Finset.notMem_erase e s (ht hh)]
-  rw [hclosed,mul_zero,zero_add]
-  congr 1
-  apply bernoulliPolynomial_congr
-  intro t ht
-  have het : e ∉ t := fun hh => Finset.notMem_erase e s (ht hh)
-  simp [het]
-
- 
-noncomputable def markedOpenPivotals (G : BondGraph V E) (Q : Finset E) (x : V)
-    (ω : Finset E) (m : Finset V) : Finset E :=
-  Q.filter fun e => e ∈ ω ∧ (∃ y ∈ m, Conn G (ω : Set E) x y) ∧
-      ¬ ∃ y ∈ m, Conn G (ω.erase e : Set E) x y
-
- 
-noncomputable def markedPivotalMomentOne (G : BondGraph V E) (Q : Finset E) (x : V)
-    (h p : ℝ) : ℝ :=
-  bernoulliPolynomial Q (fun ω => bernoulliPolynomial (insert x (searchVertices G Q))
-    (fun m => ((markedOpenPivotals G Q x ω m).card:ℝ)) h) p
-
-theorem marked_pivotal_russo (G : BondGraph V E) (Q : Finset E) (x : V) (h p : ℝ) :
-    markedPivotalMomentOne G Q x h p = p*deriv (markedConnectionPolynomial G Q x h) p := by
-  let F := insert x (searchVertices G Q)
-  let f : Finset E → Finset V → ℝ := fun ω m =>
-    if ∃ y ∈ m, Conn G (ω : Set E) x y then 1 else 0
-  have hcard (ω : Finset E) (m : Finset V) : ((markedOpenPivotals G Q x ω m).card:ℝ) =
-      ∑ e ∈ Q, if e ∈ ω then f ω m-f (ω.erase e) m else 0 := by
-    rw [markedOpenPivotals,Finset.card_filter,Nat.cast_sum]
-    apply Finset.sum_congr rfl
-    intro e _
-    have hmono : (∃ y ∈ m, Conn G (ω.erase e : Set E) x y) →
-        ∃ y ∈ m, Conn G (ω : Set E) x y := by
-      rintro ⟨y,hy,hxy⟩
-      exact ⟨y,hy,conn_mono G (by exact_mod_cast Finset.erase_subset e ω) hxy⟩
-    by_cases he : e ∈ ω <;> by_cases h₁ : ∃ y ∈ m, Conn G (ω : Set E) x y <;>
-      by_cases h₀ : ∃ y ∈ m, Conn G (ω.erase e : Set E) x y <;>
-      simp only [f,he,h₁,h₀,and_true,and_false,not_true_eq_false,
-        not_false_eq_true,ite_true,ite_false,Nat.cast_one,Nat.cast_zero,sub_self,sub_zero]
-    exact False.elim (h₁ (hmono h₀))
-  have heq : markedConnectionPolynomial G Q x h =
-      bernoulliPolynomial Q (fun ω => bernoulliPolynomial F (f ω) h) := rfl
-  rw [heq,(hasDerivAt_bernoulliPolynomial Q _ p).deriv,Finset.mul_sum]
-  simp only [markedPivotalMomentOne,hcard,bernoulliPolynomial_sum]
-  apply Finset.sum_congr rfl
-  intro e he
-  rw [← bernoulliPolynomial_open_difference Q e he (fun ω => bernoulliPolynomial F (f ω) h) p]
-  apply bernoulliPolynomial_congr
-  intro ω _
-  by_cases heω : e ∈ ω
-  · simp only [heω,ite_true,F]
-    exact bernoulliPolynomial_sub _ _ _ _
-  · simp only [heω,ite_false,bernoulliPolynomial_zero]
-
- 
-theorem marked_pivotal_first_moment (G : BondGraph V E) (Q : Finset E) (x : V)
-    (p h : unitInterval) (hp : (p:ℝ) < 1) (hh : 0 < (h:ℝ)) (d : ℕ) (hd : 0 < d)
-    (hdeg : ∀ v, (Q.filter (fun e => v ∈ G e)).card ≤ d) :
-    markedPivotalMomentOne G Q x h p ≤
-      (Real.sqrt ((d:ℝ)/2)/(1-p))*Real.sqrt (Real.log (1/(h:ℝ))/h)*
-        markedConnectionPolynomial G Q x h p := by
-  rw [marked_pivotal_russo]
-  exact markedConnection_derivative_bound G Q x p h hp hh d hd hdeg
-
-end MarkedFirstMoment
-
-section PivotalGeometry
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-def MarkedEvent (G : BondGraph V E) (ω : Set E) (m : Set V) (x : V) : Prop :=
-  ∃ y ∈ m, Conn G ω x y
-
- 
-def OpenMarkedPivotal (G : BondGraph V E) (ω : Set E) (m : Set V) (x : V)
-    (e : E) : Prop :=
-  e ∈ ω ∧ MarkedEvent G ω m x ∧ ¬MarkedEvent G (ω \ {e}) m x
-
-theorem markedEvent_mono (G : BondGraph V E) {ω η : Set E} (h : ω ⊆ η)
-    {m : Set V} {x : V} (hx : MarkedEvent G ω m x) : MarkedEvent G η m x := by
-  obtain ⟨y,hy,hxy⟩ := hx
-  exact ⟨y,hy,conn_mono G h hxy⟩
-
-theorem edge_cluster_sides (G : BondGraph V E) (ω : Set E) (x : V)
-    {e : E} {a b : V} (he : e ∈ ω) (hab : G e = s(a,b)) :
-    a ∈ cluster G ω x ↔ b ∈ cluster G ω x := by
-  have hc : Conn G ω a b := .single ⟨e,he,hab⟩
-  exact ⟨fun h => h.trans hc,fun h => h.trans (conn_symm G ω hc)⟩
-
- 
-theorem conn_restrict_cluster_compl (G : BondGraph V E) (ω : Set E) (x : V)
-    {a b : V} (ha : a ∉ cluster G ω x) (hab : Conn G ω a b) :
-    Conn G (ω ∩ internalEdges G (cluster G ω x)ᶜ) a b := by
-  induction hab with
-  | refl => exact conn_refl G _ a
-  | @tail u v hau huv ih =>
-    obtain ⟨e,he,heq⟩ := huv
-    have hu : u ∉ cluster G ω x := fun h => ha (h.trans (conn_symm G ω hau))
-    have hv : v ∉ cluster G ω x := fun h => hu ((edge_cluster_sides G ω x he heq).mpr h)
-    exact ih.tail ⟨e,⟨he,(mem_internalEdges_of_ends G _ e u v heq).mpr ⟨hu,hv⟩⟩,heq⟩
-
-theorem conn_restrict_cluster (G : BondGraph V E) (ω : Set E) (x : V)
-    {a b : V} (ha : a ∈ cluster G ω x) (hab : Conn G ω a b) :
-    Conn G (ω ∩ internalEdges G (cluster G ω x)) a b := by
-  induction hab with
-  | refl => exact conn_refl G _ a
-  | @tail u v hau huv ih =>
-    obtain ⟨e,he,heq⟩ := huv
-    have hu : u ∈ cluster G ω x := ha.trans hau
-    have hv : v ∈ cluster G ω x := (edge_cluster_sides G ω x he heq).mp hu
-    exact ih.tail ⟨e,⟨he,(mem_internalEdges_of_ends G _ e u v heq).mpr ⟨hu,hv⟩⟩,heq⟩
-
- 
-theorem openMarkedPivotal_oriented (G : BondGraph V E) (ω : Set E) (m : Set V)
-    (x : V) {e : E} (he : OpenMarkedPivotal G ω m x e) :
-    ∃ a b, G e = s(a,b) ∧ Conn G (ω \ {e}) x a ∧
-      ¬Conn G (ω \ {e}) x b ∧ MarkedEvent G (ω \ {e}) m b := by
-  obtain ⟨y,hy,hxy⟩ := he.2.1
-  obtain ⟨a,b,hab⟩ : ∃ a b, G e = s(a,b) := by
-    obtain ⟨⟨a,b⟩,h⟩ := Quot.exists_rep (G e)
-    exact ⟨a,b,h.symm⟩
-  have heq : insert e (ω \ {e}) = ω := by
-    have heω : e ∈ ω := he.1
-    ext d
-    simp only [Set.mem_insert_iff,Set.mem_sdiff,Set.mem_singleton_iff]
-    by_cases h : d = e <;> simp_all
-  rw [← heq,conn_insert_iff G _ e a b x y hab] at hxy
-  rcases hxy with hxy | ⟨hxa,hby⟩ | ⟨hxb,hay⟩
-  · exact (he.2.2 ⟨y,hy,hxy⟩).elim
-  · exact ⟨a,b,hab,hxa,fun hxb => he.2.2 ⟨y,hy,hxb.trans hby⟩,y,hy,hby⟩
-  · exact ⟨b,a,hab.trans (Sym2.eq_swap ..),hxb,
-      fun hxa => he.2.2 ⟨y,hy,hxa.trans hay⟩,y,hy,hay⟩
-
- 
-theorem pivotal_rootSide_disjoint_marks (G : BondGraph V E) (ω : Set E)
-    (m : Set V) (x : V) {e : E} (he : OpenMarkedPivotal G ω m x e) :
-    Disjoint (cluster G (ω \ {e}) x) m := by
-  rw [Set.disjoint_left]
-  exact fun y hy hm => he.2.2 ⟨y,hm,hy⟩
-
- 
-theorem conn_delete_internal_of_outside (G : BondGraph V E) (ω : Set E) (x : V)
-    {f : E} {a b : V} (hf : f ∈ internalEdges G (cluster G ω x))
-    (ha : a ∉ cluster G ω x) (hab : Conn G ω a b) : Conn G (ω \ {f}) a b := by
-  apply conn_mono G _ (conn_restrict_cluster_compl G ω x ha hab)
-  intro e he
-  refine ⟨he.1,?_⟩
-  intro hef
-  have hef' : e = f := Set.mem_singleton_iff.mp hef
-  subst e
-  obtain ⟨u,v,heq⟩ : ∃ u v, G f = s(u,v) := by
-    obtain ⟨⟨a,b⟩,h⟩ := Quot.exists_rep (G f)
-    exact ⟨a,b,h.symm⟩
-  exact ((mem_internalEdges_of_ends G _ f u v heq).mp he.2).1
-    ((mem_internalEdges_of_ends G _ f u v heq).mp hf).1
-
- 
-theorem pivotal_other_sides (G : BondGraph V E) (ω : Set E) (m : Set V) (x : V)
-    {e f : E} (_he : OpenMarkedPivotal G ω m x e)
-    (hf : OpenMarkedPivotal G ω m x f) (hne : f ≠ e) :
-    f ∈ internalEdges G (cluster G (ω \ {e}) x) ∨
-      f ∈ internalEdges G (cluster G (ω \ {e}) x)ᶜ := by
-  obtain ⟨a,b,hab⟩ : ∃ a b, G f = s(a,b) := by
-    obtain ⟨⟨a,b⟩,h⟩ := Quot.exists_rep (G f)
-    exact ⟨a,b,h.symm⟩
-  have hop : f ∈ ω \ {e} := ⟨hf.1,hne⟩
-  have hiff := edge_cluster_sides G (ω \ {e}) x hop hab
-  by_cases ha : a ∈ cluster G (ω \ {e}) x
-  · exact Or.inl ((mem_internalEdges_of_ends G _ f a b hab).mpr ⟨ha,hiff.mp ha⟩)
-  · exact Or.inr ((mem_internalEdges_of_ends G _ f a b hab).mpr ⟨ha,fun hb => ha (hiff.mpr hb)⟩)
-
- 
-theorem conn_delete_exterior_of_inside (G : BondGraph V E) (ω : Set E) (x : V)
-    {f : E} {a b : V} (hf : f ∈ internalEdges G (cluster G ω x)ᶜ)
-    (ha : a ∈ cluster G ω x) (hab : Conn G ω a b) : Conn G (ω \ {f}) a b := by
-  apply conn_mono G _ (conn_restrict_cluster G ω x ha hab)
-  intro e he
-  refine ⟨he.1,?_⟩
-  intro hef
-  have hef' : e = f := Set.mem_singleton_iff.mp hef
-  subst e
-  obtain ⟨⟨u,v⟩,heq⟩ := Quot.exists_rep (G f)
-  exact ((mem_internalEdges_of_ends G _ f u v heq.symm).mp hf).1
-    ((mem_internalEdges_of_ends G _ f u v heq.symm).mp he.2).1
-
- 
-theorem pivotal_rootSide_ssubset (G : BondGraph V E) (ω : Set E) (m : Set V) (x : V)
-    {e f : E} (he : OpenMarkedPivotal G ω m x e) (hne : e ≠ f)
-    (hf : f ∈ internalEdges G (cluster G (ω \ {e}) x)ᶜ) :
-    cluster G (ω \ {e}) x ⊂ cluster G (ω \ {f}) x := by
-  have hsub : cluster G (ω \ {e}) x ⊆ cluster G (ω \ {f}) x := by
-    intro y hy
-    exact conn_mono G (ω₁ := (ω \ {e}) \ {f}) (ω₂ := ω \ {f}) (fun _ h => ⟨h.1.1,h.2⟩)
-      (conn_delete_exterior_of_inside G (ω \ {e}) x hf (conn_refl G _ x) hy)
-  refine ⟨hsub,?_⟩
-  obtain ⟨a,b,hab,hxa,hxb,_⟩ := openMarkedPivotal_oriented G ω m x he
-  intro hback
-  apply hxb
-  apply hback
-  exact (hsub hxa).tail ⟨e,⟨he.1,hne⟩,hab⟩
-
- 
-theorem pivotal_rootSides_nested (G : BondGraph V E) (ω : Set E) (m : Set V) (x : V)
-    {e f : E} (he : OpenMarkedPivotal G ω m x e)
-    (hf : OpenMarkedPivotal G ω m x f) (hne : e ≠ f) :
-    cluster G (ω \ {e}) x ⊂ cluster G (ω \ {f}) x ∨
-      cluster G (ω \ {f}) x ⊂ cluster G (ω \ {e}) x := by
-  rcases pivotal_other_sides G ω m x he hf hne.symm with hinside | houtside
-  · obtain ⟨a,b,hab,_,hxb,hbm⟩ := openMarkedPivotal_oriented G ω m x he
-    obtain ⟨y,hy,hby⟩ := hbm
-    have hby' : Conn G (ω \ {f}) b y := conn_mono G (ω₁ := (ω \ {e}) \ {f}) (ω₂ := ω \ {f}) (fun _ h => ⟨h.1.1,h.2⟩)
-      (conn_delete_internal_of_outside G (ω \ {e}) x hinside hxb hby)
-    have hb : b ∉ cluster G (ω \ {f}) x := fun h => hf.2.2 ⟨y,hy,h.trans hby'⟩
-    have ha : a ∉ cluster G (ω \ {f}) x := fun h =>
-      hb ((edge_cluster_sides G (ω \ {f}) x ⟨he.1,hne⟩ hab).mp h)
-    exact Or.inr (pivotal_rootSide_ssubset G ω m x hf hne.symm
-      ((mem_internalEdges_of_ends G _ e a b hab).mpr ⟨ha,hb⟩))
-  · exact Or.inl (pivotal_rootSide_ssubset G ω m x he hne houtside)
-
- 
-theorem conn_restrict_cluster_compl_of_subset (G : BondGraph V E) (ω : Set E) (x : V)
-    {η : Set E} (hη : η ⊆ ω) {a b : V} (ha : a ∉ cluster G ω x)
-    (hab : Conn G η a b) : Conn G (η ∩ internalEdges G (cluster G ω x)ᶜ) a b := by
-  induction hab with
-  | refl => exact conn_refl G _ a
-  | @tail u v hau huv ih =>
-    obtain ⟨e,he,heq⟩ := huv
-    have hu : u ∉ cluster G ω x := fun h => ha (h.trans (conn_symm G ω (conn_mono G hη hau)))
-    have hv : v ∉ cluster G ω x := fun h => hu ((edge_cluster_sides G ω x (hη he) heq).mpr h)
-    exact ih.tail ⟨e,⟨he,(mem_internalEdges_of_ends G _ e u v heq).mpr ⟨hu,hv⟩⟩,heq⟩
-
- 
-
-theorem markedEvent_after_pivotal (G : BondGraph V E) (ω : Set E) (m : Set V) (x : V)
-    {e f : E} (he : OpenMarkedPivotal G ω m x e) (hne : e ≠ f)
-    {a b : V} (hab : G e = s(a,b)) (hxa : Conn G (ω \ {e}) x a)
-    (hxb : ¬Conn G (ω \ {e}) x b)
-    (hf : f ∈ internalEdges G (cluster G (ω \ {e}) x)ᶜ) :
-    MarkedEvent G (ω \ {f}) m x ↔
-      MarkedEvent G ((ω ∩ internalEdges G (cluster G (ω \ {e}) x)ᶜ) \ {f}) m b := by
-  let W := cluster G (ω \ {e}) x
-  have heq : insert e ((ω \ {e}) \ {f}) = ω \ {f} := by
-    ext d
-    simp only [Set.mem_insert_iff,Set.mem_sdiff,Set.mem_singleton_iff]
-    constructor
-    · rintro (rfl | ⟨⟨hd,hde⟩,hdf⟩)
-      · exact ⟨he.1,hne⟩
-      · exact ⟨hd,hdf⟩
-    · rintro ⟨hd,hdf⟩
-      by_cases hde : d = e
-      · exact Or.inl hde
-      · exact Or.inr ⟨⟨hd,hde⟩,hdf⟩
-  constructor
-  · rintro ⟨y,hy,hxy⟩
-    rw [← heq,conn_insert_iff G _ e a b x y hab] at hxy
-    rcases hxy with hxy | ⟨_,hby⟩ | ⟨hxb',_⟩
-    · exact (he.2.2 ⟨y,hy,conn_mono G Set.sdiff_subset hxy⟩).elim
-    · refine ⟨y,hy,conn_mono G ?_ (conn_restrict_cluster_compl_of_subset G (ω \ {e}) x
-        Set.sdiff_subset hxb hby)⟩
-      intro d hd
-      exact ⟨⟨hd.1.1.1,hd.2⟩,hd.1.2⟩
-    · exact (hxb (conn_mono G Set.sdiff_subset hxb')).elim
-  · rintro ⟨y,hy,hby⟩
-    have hxa' : Conn G (ω \ {f}) x a := conn_mono G (ω₁ := (ω \ {e}) \ {f}) (ω₂ := ω \ {f}) (fun _ h => ⟨h.1.1,h.2⟩)
-      (conn_delete_exterior_of_inside G (ω \ {e}) x hf (conn_refl G _ x) hxa)
-    refine ⟨y,hy,(hxa'.tail ⟨e,⟨he.1,hne⟩,hab⟩).trans (conn_mono G ?_ hby)⟩
-    exact fun _ h => ⟨h.1.1,h.2⟩
-
- 
-theorem openMarkedPivotal_exterior_iff (G : BondGraph V E) (ω : Set E) (m : Set V)
-    (x : V) {e : E} (he : OpenMarkedPivotal G ω m x e)
-    {a b : V} (hab : G e = s(a,b)) (hxa : Conn G (ω \ {e}) x a)
-    (hxb : ¬Conn G (ω \ {e}) x b) (hbm : MarkedEvent G (ω \ {e}) m b)
-    (f : E) :
-    OpenMarkedPivotal G (ω ∩ internalEdges G (cluster G (ω \ {e}) x)ᶜ) m b f ↔
-      OpenMarkedPivotal G ω m x f ∧ f ∈ internalEdges G (cluster G (ω \ {e}) x)ᶜ := by
-  have hbase : MarkedEvent G (ω ∩ internalEdges G (cluster G (ω \ {e}) x)ᶜ) m b := by
-    obtain ⟨y,hy,hby⟩ := hbm
-    refine ⟨y,hy,conn_mono G ?_ (conn_restrict_cluster_compl G (ω \ {e}) x hxb hby)⟩
-    exact fun _ h => ⟨h.1.1,h.2⟩
-  have hexterior (hf : f ∈ internalEdges G (cluster G (ω \ {e}) x)ᶜ) : e ≠ f := by
-    intro h; subst f
-    exact ((mem_internalEdges_of_ends G _ e a b hab).mp hf).1 hxa
-  constructor
-  · intro hf
-    have hiff := markedEvent_after_pivotal G ω m x he (hexterior hf.1.2) hab hxa hxb hf.1.2
-    exact ⟨⟨hf.1.1,he.2.1,fun h => hf.2.2 (hiff.mp h)⟩,hf.1.2⟩
-  · rintro ⟨hf,hout⟩
-    have hiff := markedEvent_after_pivotal G ω m x he (hexterior hout) hab hxa hxb hout
-    exact ⟨⟨hf.1,hout⟩,hbase,fun h => hf.2.2 (hiff.mpr h)⟩
-
- 
-theorem cluster_finite_of_finite_edges (G : BondGraph V E) {ω : Set E}
-    (hω : ω.Finite) (x : V) : (cluster G ω x).Finite := by
-  let F : Finset V := insert x (hω.toFinset.biUnion fun e => (G e).toFinset)
-  apply F.finite_toSet.subset
-  intro y hxy
-  induction hxy with
-  | refl => exact Finset.mem_insert_self x _
-  | @tail a b hxa hab ih =>
-    obtain ⟨e,he,heq⟩ := hab
-    apply Finset.mem_insert_of_mem
-    apply Finset.mem_biUnion.mpr
-    exact ⟨e,hω.mem_toFinset.mpr he,by simp [heq]⟩
-
- 
-noncomputable def pivotalRank (G : BondGraph V E) (ω : Set E) (x : V) (e : E) : ℕ :=
-  (cluster G (ω \ {e}) x).ncard
-
-theorem pivotalRank_injective (G : BondGraph V E) {ω : Set E} (hω : ω.Finite)
-    (m : Set V) (x : V) : Set.InjOn (pivotalRank G ω x) {e | OpenMarkedPivotal G ω m x e} := by
-  intro e he f hf hsame
-  by_contra hne
-  rcases pivotal_rootSides_nested G ω m x he hf hne with h | h
-  · have hh := Set.ncard_lt_ncard h (cluster_finite_of_finite_edges G (hω.subset Set.sdiff_subset) x)
-    exact (Nat.ne_of_lt hh) hsame
-  · have hh := Set.ncard_lt_ncard h (cluster_finite_of_finite_edges G (hω.subset Set.sdiff_subset) x)
-    exact (Nat.ne_of_lt hh) hsame.symm
-
- 
-theorem pivotalRank_lt_iff (G : BondGraph V E) {ω : Set E} (hω : ω.Finite)
-    (m : Set V) (x : V) {e f : E} (he : OpenMarkedPivotal G ω m x e)
-    (hf : OpenMarkedPivotal G ω m x f) :
-    pivotalRank G ω x e < pivotalRank G ω x f ↔
-      f ∈ internalEdges G (cluster G (ω \ {e}) x)ᶜ := by
-  have hexterior (hout : f ∈ internalEdges G (cluster G (ω \ {e}) x)ᶜ) : e ≠ f := by
-    obtain ⟨a,b,hab,hxa,_,_⟩ := openMarkedPivotal_oriented G ω m x he
-    intro h; subst f
-    exact ((mem_internalEdges_of_ends G _ e a b hab).mp hout).1 hxa
-  constructor
-  · intro hlt
-    have hne : e ≠ f := fun h => (Nat.ne_of_lt hlt) (congrArg (pivotalRank G ω x) h)
-    rcases pivotal_other_sides G ω m x he hf hne.symm with hinside | hout
-    · obtain ⟨a,b,hab,_,hxb,hbm⟩ := openMarkedPivotal_oriented G ω m x he
-      obtain ⟨y,hy,hby⟩ := hbm
-      have hby' : Conn G (ω \ {f}) b y := conn_mono G
-        (ω₁ := (ω \ {e}) \ {f}) (ω₂ := ω \ {f}) (fun _ h => ⟨h.1.1,h.2⟩)
-        (conn_delete_internal_of_outside G (ω \ {e}) x hinside hxb hby)
-      have hb : b ∉ cluster G (ω \ {f}) x := fun h => hf.2.2 ⟨y,hy,h.trans hby'⟩
-      have ha : a ∉ cluster G (ω \ {f}) x := fun h =>
-        hb ((edge_cluster_sides G (ω \ {f}) x ⟨he.1,hne⟩ hab).mp h)
-      have hs := pivotal_rootSide_ssubset G ω m x hf hne.symm
-        ((mem_internalEdges_of_ends G _ e a b hab).mpr ⟨ha,hb⟩)
-      have hrev := Set.ncard_lt_ncard hs
-        (cluster_finite_of_finite_edges G (hω.subset Set.sdiff_subset) x)
-      exact (Nat.not_lt_of_ge (Nat.le_of_lt hrev) hlt).elim
-    · exact hout
-  · intro hout
-    exact Set.ncard_lt_ncard (pivotal_rootSide_ssubset G ω m x he (hexterior hout) hout)
-      (cluster_finite_of_finite_edges G (hω.subset Set.sdiff_subset) x)
-
-end PivotalGeometry
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Criticality.Sharpness
+
+namespace OAI
+
+/-! Marked decision trees, graph searches and pivotal geometry. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+section MarkedExploration
+
+ 
+theorem centered_bernoulli_mgf_le (p : unitInterval) (lam : ℝ) :
+    (1-(p:ℝ))*Real.exp (-lam*p) + p*Real.exp (lam*(1-p)) ≤ Real.exp (lam^2/8) := by
+  let X : Bool → ℝ := fun b => if b then 1-(p:ℝ) else -(p:ℝ)
+  have hb : ∀ᵐ b ∂bernoulliMeasure true false p, X b ∈ Set.Icc (-(p:ℝ)) (1-p) := by
+    filter_upwards with b
+    cases b <;> simp [X]
+  have hc : ∫ b, X b ∂bernoulliMeasure true false p = 0 := by
+    rw [integral_bernoulliMeasure]
+    simp [X]
+    ring
+  have hg := hasSubgaussianMGF_of_mem_Icc_of_integral_eq_zero
+    (μ := bernoulliMeasure true false p) (X := X)
+    (Measurable.of_discrete.aemeasurable) hb hc
+  have hh := hg.mgf_le lam
+  simp only [mgf,integral_bernoulliMeasure] at hh
+  norm_num [X,sub_neg_eq_add,sub_add_cancel,smul_eq_mul] at hh
+  rw [show (1/4:ℝ)*lam^2/2 = lam^2/8 by ring] at hh
+  simpa only [neg_mul,add_comm] using hh
+
+ 
+inductive ProbeTree where
+  | done (success : Bool)
+  | edge (closed open_ : ProbeTree)
+  | mark (unmarked marked : ProbeTree)
+
+namespace ProbeTree
+
+@[reducible] def Leaf : ProbeTree → Type
+  | .done _ => Unit
+  | .edge t₀ t₁ => Leaf t₀ ⊕ Leaf t₁
+  | .mark t₀ t₁ => Leaf t₀ ⊕ Leaf t₁
+
+instance leafFintype : (t : ProbeTree) → Fintype t.Leaf
+  | .done _ => inferInstanceAs (Fintype Unit)
+  | .edge t₀ t₁ => @instFintypeSum _ _ (leafFintype t₀) (leafFintype t₁)
+  | .mark t₀ t₁ => @instFintypeSum _ _ (leafFintype t₀) (leafFintype t₁)
+
+noncomputable def weight (p h : ℝ) : (t : ProbeTree) → t.Leaf → ℝ
+  | .done _, _ => 1
+  | .edge t₀ _, .inl l => (1-p)*weight p h t₀ l
+  | .edge _ t₁, .inr l => p*weight p h t₁ l
+  | .mark t₀ _, .inl l => (1-h)*weight p h t₀ l
+  | .mark _ t₁, .inr l => h*weight p h t₁ l
+
+noncomputable def score (p : ℝ) : (t : ProbeTree) → t.Leaf → ℝ
+  | .done _, _ => 0
+  | .edge t₀ _, .inl l => score p t₀ l-p
+  | .edge _ t₁, .inr l => score p t₁ l+(1-p)
+  | .mark t₀ _, .inl l => score p t₀ l
+  | .mark _ t₁, .inr l => score p t₁ l
+
+def queries : (t : ProbeTree) → t.Leaf → ℕ
+  | .done _, _ => 0
+  | .edge t₀ _, .inl l => queries t₀ l+1
+  | .edge _ t₁, .inr l => queries t₁ l+1
+  | .mark t₀ _, .inl l => queries t₀ l
+  | .mark _ t₁, .inr l => queries t₁ l
+
+@[reducible] def outcome : (t : ProbeTree) → t.Leaf → Bool
+  | .done b, _ => b
+  | .edge t₀ _, .inl l => outcome t₀ l
+  | .edge _ t₁, .inr l => outcome t₁ l
+  | .mark t₀ _, .inl l => outcome t₀ l
+  | .mark _ t₁, .inr l => outcome t₁ l
+
+lemma weight_nonneg (p h : unitInterval) (t : ProbeTree) (l : t.Leaf) :
+    0 ≤ weight p h t l := by
+  induction t with
+  | done b => exact zero_le_one
+  | edge t₀ t₁ ih₀ ih₁ =>
+    cases l with
+    | inl l => exact mul_nonneg (sub_nonneg.mpr p.2.2) (ih₀ l)
+    | inr l => exact mul_nonneg p.2.1 (ih₁ l)
+  | mark t₀ t₁ ih₀ ih₁ =>
+    cases l with
+    | inl l => exact mul_nonneg (sub_nonneg.mpr h.2.2) (ih₀ l)
+    | inr l => exact mul_nonneg h.2.1 (ih₁ l)
+
+lemma sum_weight (p h : ℝ) (t : ProbeTree) : ∑ l, weight p h t l = 1 := by
+  induction t with
+  | done b => change (∑ _ : Unit, (1:ℝ)) = 1; simp
+  | edge t₀ t₁ ih₀ ih₁ =>
+    change (∑ l : t₀.Leaf ⊕ t₁.Leaf, _) = 1
+    simp only [Fintype.sum_sum_type,weight,← Finset.mul_sum,ih₀,ih₁]
+    ring
+  | mark t₀ t₁ ih₀ ih₁ =>
+    change (∑ l : t₀.Leaf ⊕ t₁.Leaf, _) = 1
+    simp only [Fintype.sum_sum_type,weight,← Finset.mul_sum,ih₀,ih₁]
+    ring
+
+ 
+theorem score_mgf (p h : unitInterval) (lam : ℝ) (t : ProbeTree) :
+    ∑ l, weight p h t l * Real.exp (lam*score p t l-lam^2*(queries t l : ℝ)/8) ≤ 1 := by
+  induction t with
+  | done b => change (∑ l : Unit, _) ≤ (1:ℝ); simp [weight,score,queries]
+  | edge t₀ t₁ ih₀ ih₁ =>
+    change (∑ l : t₀.Leaf ⊕ t₁.Leaf, _) ≤ (1:ℝ)
+    rw [Fintype.sum_sum_type]
+    have h₀ (l : t₀.Leaf) :
+        weight p h (.edge t₀ t₁) (.inl l) *
+          Real.exp (lam*score p (.edge t₀ t₁) (.inl l)-lam^2*(queries (.edge t₀ t₁) (.inl l):ℝ)/8) =
+        ((1-(p:ℝ))*Real.exp (-lam*p-lam^2/8)) *
+          (weight p h t₀ l * Real.exp (lam*score p t₀ l-lam^2*(queries t₀ l:ℝ)/8)) := by
+      simp only [weight,score,queries,Nat.cast_add,Nat.cast_one]
+      rw [show lam*(score p t₀ l-p)-lam^2*((queries t₀ l:ℝ)+1)/8 =
+        (-lam*p-lam^2/8)+(lam*score p t₀ l-lam^2*(queries t₀ l:ℝ)/8) by ring,Real.exp_add]
+      ring
+    have h₁ (l : t₁.Leaf) :
+        weight p h (.edge t₀ t₁) (.inr l) *
+          Real.exp (lam*score p (.edge t₀ t₁) (.inr l)-lam^2*(queries (.edge t₀ t₁) (.inr l):ℝ)/8) =
+        ((p:ℝ)*Real.exp (lam*(1-p)-lam^2/8)) *
+          (weight p h t₁ l * Real.exp (lam*score p t₁ l-lam^2*(queries t₁ l:ℝ)/8)) := by
+      simp only [weight,score,queries,Nat.cast_add,Nat.cast_one]
+      rw [show lam*(score p t₁ l+(1-p))-lam^2*((queries t₁ l:ℝ)+1)/8 =
+        (lam*(1-p)-lam^2/8)+(lam*score p t₁ l-lam^2*(queries t₁ l:ℝ)/8) by ring,Real.exp_add]
+      ring
+    simp only [h₀,h₁,← Finset.mul_sum]
+    have hc₀ : 0 ≤ (1-(p:ℝ))*Real.exp (-lam*p-lam^2/8) :=
+      mul_nonneg (sub_nonneg.mpr p.2.2) (Real.exp_pos _).le
+    have hc₁ : 0 ≤ (p:ℝ)*Real.exp (lam*(1-p)-lam^2/8) :=
+      mul_nonneg p.2.1 (Real.exp_pos _).le
+    calc
+      _ ≤ (1-(p:ℝ))*Real.exp (-lam*p-lam^2/8) + p*Real.exp (lam*(1-p)-lam^2/8) := by
+        simpa using add_le_add (mul_le_mul_of_nonneg_left ih₀ hc₀) (mul_le_mul_of_nonneg_left ih₁ hc₁)
+      _ = ((1-(p:ℝ))*Real.exp (-lam*p)+p*Real.exp (lam*(1-p))) / Real.exp (lam^2/8) := by
+        simp only [Real.exp_sub]; ring
+      _ ≤ 1 := (div_le_one (Real.exp_pos _)).mpr (centered_bernoulli_mgf_le p lam)
+  | mark t₀ t₁ ih₀ ih₁ =>
+    change (∑ l : t₀.Leaf ⊕ t₁.Leaf, _) ≤ (1:ℝ)
+    simp only [Fintype.sum_sum_type,weight,score,queries,mul_assoc,← Finset.mul_sum]
+    calc
+      _ ≤ (1-(h:ℝ))*1+(h:ℝ)*1 :=
+        add_le_add (mul_le_mul_of_nonneg_left ih₀ (sub_nonneg.mpr h.2.2))
+          (mul_le_mul_of_nonneg_left ih₁ h.2.1)
+      _ = 1 := by ring
+
+noncomputable def successWeight (p h : ℝ) (t : ProbeTree) (l : t.Leaf) : ℝ :=
+  if outcome t l then weight p h t l else 0
+
+noncomputable def successMass (p h : ℝ) (t : ProbeTree) : ℝ :=
+  ∑ l, successWeight p h t l
+
+noncomputable def successScore (p h : ℝ) (t : ProbeTree) : ℝ :=
+  ∑ l, successWeight p h t l * score p t l
+
+noncomputable def successQueries (p h : ℝ) (t : ProbeTree) : ℝ :=
+  ∑ l, successWeight p h t l * queries t l
+
+lemma successWeight_nonneg (p h : unitInterval) (t : ProbeTree) (l : t.Leaf) :
+    0 ≤ successWeight p h t l := by
+  unfold successWeight
+  split <;> simp_all [weight_nonneg]
+
+lemma successWeight_le (p h : unitInterval) (t : ProbeTree) (l : t.Leaf) :
+    successWeight p h t l ≤ weight p h t l := by
+  unfold successWeight
+  split
+  · exact le_rfl
+  · exact weight_nonneg p h t l
+
+lemma successMass_nonneg (p h : unitInterval) (t : ProbeTree) : 0 ≤ successMass p h t :=
+  Finset.sum_nonneg fun _ _ => successWeight_nonneg p h t _
+
+lemma successMass_le_one (p h : unitInterval) (t : ProbeTree) : successMass p h t ≤ 1 := by
+  rw [← sum_weight p h t]
+  exact Finset.sum_le_sum fun _ _ => successWeight_le p h t _
+
+lemma successQueries_nonneg (p h : unitInterval) (t : ProbeTree) : 0 ≤ successQueries p h t :=
+  Finset.sum_nonneg fun _ _ => mul_nonneg (successWeight_nonneg p h t _) (Nat.cast_nonneg _)
+
+ 
+theorem conditional_score_bound (p h : unitInterval) (t : ProbeTree)
+    (hH : 0 < successMass p h t) (lam : ℝ) :
+    lam*(successScore p h t / successMass p h t) ≤
+      Real.log (1/successMass p h t) + lam^2/8*(successQueries p h t / successMass p h t) := by
+  let H := successMass p h t
+  let w : t.Leaf → ℝ := fun l => successWeight p h t l / H
+  let X : t.Leaf → ℝ := fun l => lam*score p t l-lam^2*(queries t l : ℝ)/8
+  have hw : ∀ l ∈ (Finset.univ : Finset t.Leaf), 0 ≤ w l :=
+    fun l _ => div_nonneg (successWeight_nonneg p h t l) hH.le
+  have hw1 : ∑ l, w l = 1 := by
+    simp only [w,← Finset.sum_div]
+    exact div_self hH.ne'
+  have hJ := convexOn_exp.map_sum_le hw hw1 (fun _ _ => Set.mem_univ _ : ∀ l ∈
+    (Finset.univ : Finset t.Leaf), X l ∈ Set.univ)
+  simp only [smul_eq_mul] at hJ
+  have hsum : ∑ l, w l * X l = lam*(successScore p h t/H)-
+      lam^2/8*(successQueries p h t/H) := by
+    simp only [w,X,successScore,successQueries,Finset.sum_div,Finset.mul_sum,← Finset.sum_sub_distrib]
+    apply Finset.sum_congr rfl
+    intro l _
+    ring
+  have he : ∑ l, w l * Real.exp (X l) ≤ 1/H := by
+    calc
+      _ = (∑ l, successWeight p h t l * Real.exp (X l))/H := by
+        simp only [Finset.sum_div,w]; apply Finset.sum_congr rfl; intro l _; ring
+      _ ≤ (∑ l, weight p h t l * Real.exp (X l))/H := by
+        apply div_le_div_of_nonneg_right _ hH.le
+        exact Finset.sum_le_sum fun l _ => mul_le_mul_of_nonneg_right (successWeight_le p h t l)
+          (Real.exp_pos _).le
+      _ ≤ 1/H := div_le_div_of_nonneg_right (score_mgf p h lam t) hH.le
+  rw [hsum] at hJ
+  have hb := (Real.le_log_iff_exp_le (one_div_pos.mpr hH)).mpr (hJ.trans he)
+  linarith
+
+noncomputable def dweight (p h : ℝ) : (t : ProbeTree) → t.Leaf → ℝ
+  | .done _, _ => 0
+  | .edge t₀ _, .inl l => -weight p h t₀ l+(1-p)*dweight p h t₀ l
+  | .edge _ t₁, .inr l => weight p h t₁ l+p*dweight p h t₁ l
+  | .mark t₀ _, .inl l => (1-h)*dweight p h t₀ l
+  | .mark _ t₁, .inr l => h*dweight p h t₁ l
+
+lemma hasDerivAt_weight (p h : ℝ) (t : ProbeTree) (l : t.Leaf) :
+    HasDerivAt (fun q => weight q h t l) (dweight p h t l) p := by
+  induction t with
+  | done b => exact hasDerivAt_const p 1
+  | edge t₀ t₁ ih₀ ih₁ =>
+    cases l with
+    | inl l => convert ((hasDerivAt_const p 1).sub (hasDerivAt_id p)).mul (ih₀ l) using 1 <;> try rfl
+               simp [dweight]
+    | inr l => convert (hasDerivAt_id p).mul (ih₁ l) using 1 <;> try rfl
+               simp [dweight]
+  | mark t₀ t₁ ih₀ ih₁ =>
+    cases l with
+    | inl l => exact (ih₀ l).const_mul (1-h)
+    | inr l => exact (ih₁ l).const_mul h
+
+lemma score_eq_derivative (p h : ℝ) (t : ProbeTree) (l : t.Leaf) :
+    p*(1-p)*dweight p h t l = weight p h t l * score p t l := by
+  induction t with
+  | done b => simp [dweight,weight,score]
+  | edge t₀ t₁ ih₀ ih₁ =>
+    cases l with
+    | inl l => dsimp [dweight,weight,score]; linear_combination (1-p)*(ih₀ l)
+    | inr l => dsimp [dweight,weight,score]; linear_combination p*(ih₁ l)
+  | mark t₀ t₁ ih₀ ih₁ =>
+    cases l with
+    | inl l => dsimp [dweight,weight,score]; linear_combination (1-h)*(ih₀ l)
+    | inr l => dsimp [dweight,weight,score]; linear_combination h*(ih₁ l)
+
+ 
+theorem successScore_eq_deriv (p h : ℝ) (t : ProbeTree) :
+    successScore p h t = p*(1-p)*deriv (fun q => successMass q h t) p := by
+  have hd : HasDerivAt (fun q => successMass q h t)
+      (∑ l, if outcome t l then dweight p h t l else 0) p := by
+    have hh (l : t.Leaf) (_ : l ∈ (Finset.univ : Finset t.Leaf)) :
+        HasDerivAt (fun q => successWeight q h t l) (if outcome t l then dweight p h t l else 0) p := by
+      by_cases hl : outcome t l
+      · simpa [successWeight,hl] using hasDerivAt_weight p h t l
+      · simpa [successWeight,hl] using hasDerivAt_const p (0:ℝ)
+    simpa only [successMass,Finset.sum_apply] using HasDerivAt.fun_sum hh
+  rw [hd.deriv,Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro l _
+  by_cases hl : outcome t l
+  · simpa [successWeight,hl] using (score_eq_derivative p h t l).symm
+  · simp [successWeight,hl]
+
+ 
+inductive IsMarkSearch : ProbeTree → Prop
+  | fail : IsMarkSearch (.done false)
+  | edge {t₀ t₁} : IsMarkSearch t₀ → IsMarkSearch t₁ → IsMarkSearch (.edge t₀ t₁)
+  | mark {t₀} : IsMarkSearch t₀ → IsMarkSearch (.mark t₀ (.done true))
+
+def marks : (t : ProbeTree) → t.Leaf → ℕ
+  | .done _, _ => 0
+  | .edge t₀ _, .inl l => marks t₀ l
+  | .edge _ t₁, .inr l => marks t₁ l
+  | .mark t₀ _, .inl l => marks t₀ l+1
+  | .mark _ t₁, .inr l => marks t₁ l+1
+
+noncomputable def successMarks (p h : ℝ) (t : ProbeTree) : ℝ :=
+  ∑ l, successWeight p h t l * marks t l
+
+lemma successMass_edge (p h : ℝ) (t₀ t₁ : ProbeTree) :
+    successMass p h (.edge t₀ t₁) = (1-p)*successMass p h t₀+p*successMass p h t₁ := by
+  simp only [successMass,Fintype.sum_sum_type,successWeight,outcome,weight,Finset.mul_sum]
+  congr 1 <;> apply Finset.sum_congr rfl <;> intro l _ <;> split <;> simp
+
+lemma successMass_mark (p h : ℝ) (t₀ t₁ : ProbeTree) :
+    successMass p h (.mark t₀ t₁) = (1-h)*successMass p h t₀+h*successMass p h t₁ := by
+  simp only [successMass,Fintype.sum_sum_type,successWeight,outcome,weight,Finset.mul_sum]
+  congr 1 <;> apply Finset.sum_congr rfl <;> intro l _ <;> split <;> simp
+
+@[simp] lemma successMass_done (p h : ℝ) (b : Bool) :
+    successMass p h (.done b) = if b then 1 else 0 := by
+  cases b <;> simp [successMass,successWeight,outcome,weight]
+
+lemma successMarks_edge (p h : ℝ) (t₀ t₁ : ProbeTree) :
+    successMarks p h (.edge t₀ t₁) = (1-p)*successMarks p h t₀+p*successMarks p h t₁ := by
+  simp only [successMarks,Fintype.sum_sum_type,successWeight,outcome,weight,marks,Finset.mul_sum]
+  congr 1 <;> apply Finset.sum_congr rfl <;> intro l _ <;> split <;> simp [mul_assoc]
+
+lemma successMarks_mark (p h : ℝ) (t₀ t₁ : ProbeTree) :
+    successMarks p h (.mark t₀ t₁) =
+      (1-h)*(successMarks p h t₀+successMass p h t₀) +
+        h*(successMarks p h t₁+successMass p h t₁) := by
+  simp only [successMarks,successMass,Fintype.sum_sum_type,successWeight,outcome,weight,marks,
+    Nat.cast_add,Nat.cast_one,← Finset.sum_add_distrib,Finset.mul_sum]
+  congr 1 <;> apply Finset.sum_congr rfl <;> intro l _ <;> split_ifs <;> ring
+
+@[simp] lemma successMarks_done (p h : ℝ) (b : Bool) : successMarks p h (.done b) = 0 := by
+  simp [successMarks,marks]
+
+ 
+theorem marks_times_parameter_le (p h : unitInterval) {t : ProbeTree} (ht : IsMarkSearch t) :
+    (h:ℝ)*successMarks p h t ≤ successMass p h t := by
+  induction ht with
+  | fail => simp
+  | @edge t₀ t₁ _ _ ih₀ ih₁ =>
+    rw [successMarks_edge,successMass_edge]
+    nlinarith [mul_nonneg (sub_nonneg.mpr p.2.2) (sub_nonneg.mpr ih₀),
+      mul_nonneg p.2.1 (sub_nonneg.mpr ih₁)]
+  | @mark t₀ _ ih =>
+    rw [successMarks_mark,successMass_mark]
+    simp only [successMarks_done,successMass_done,ite_true,zero_add,mul_one]
+    have hn := mul_nonneg (mul_nonneg h.2.1 (sub_nonneg.mpr h.2.2))
+      (sub_nonneg.mpr (successMass_le_one p h t₀))
+    nlinarith [mul_nonneg (sub_nonneg.mpr h.2.2) (sub_nonneg.mpr ih)]
+
+theorem successQueries_times_parameter_le (p h : unitInterval) {t : ProbeTree}
+    (ht : IsMarkSearch t) (d : ℕ) (hcost : ∀ l, queries t l ≤ d*marks t l) :
+    (h:ℝ)*successQueries p h t ≤ d*successMass p h t := by
+  have hs : successQueries p h t ≤ d*successMarks p h t := by
+    simp only [successQueries,successMarks,Finset.mul_sum]
+    apply Finset.sum_le_sum
+    intro l _
+    calc
+      _ ≤ successWeight p h t l * ((d:ℝ)*marks t l) :=
+        mul_le_mul_of_nonneg_left (by exact_mod_cast hcost l) (successWeight_nonneg p h t l)
+      _ = _ := by ring
+  calc
+    _ ≤ (h:ℝ)*(d*successMarks p h t) := mul_le_mul_of_nonneg_left hs h.2.1
+    _ = d*((h:ℝ)*successMarks p h t) := by ring
+    _ ≤ d*successMass p h t := mul_le_mul_of_nonneg_left (marks_times_parameter_le p h ht)
+      (Nat.cast_nonneg _)
+
+ 
+theorem marked_score_bound (p h : unitInterval) (hh : 0 < (h:ℝ))
+    {t : ProbeTree} (ht : IsMarkSearch t) (d : ℕ)
+    (hcost : ∀ l, queries t l ≤ d*marks t l) (hroot : (h:ℝ) ≤ successMass p h t)
+    (lam : ℝ) :
+    lam*(successScore p h t / successMass p h t) ≤
+      Real.log (1/(h:ℝ)) + lam^2*(d:ℝ)/(8*h) := by
+  have hH := hh.trans_le hroot
+  have hlog : Real.log (1/successMass p h t) ≤ Real.log (1/(h:ℝ)) :=
+    Real.log_le_log (one_div_pos.mpr hH) (one_div_le_one_div_of_le hh hroot)
+  have hQ : successQueries p h t / successMass p h t ≤ (d:ℝ)/h := by
+    apply (div_le_div_iff₀ hH hh).mpr
+    simpa [mul_comm] using successQueries_times_parameter_le p h ht d hcost
+  calc
+    _ ≤ Real.log (1/successMass p h t)+lam^2/8*(successQueries p h t/successMass p h t) :=
+      conditional_score_bound p h t hH lam
+    _ ≤ Real.log (1/(h:ℝ))+lam^2/8*((d:ℝ)/h) :=
+      add_le_add hlog (mul_le_mul_of_nonneg_left hQ (by positivity))
+    _ = _ := by ring
+
+end ProbeTree
+end MarkedExploration
+
+section LabelledExploration
+attribute [local instance] Classical.propDecidable
+attribute [local instance] Classical.decEq
+
+inductive LabelledTree (V E : Type*) where
+  | done (success : Bool)
+  | edge (e : E) (closed opened : LabelledTree V E)
+  | mark (v : V) (unmarked marked : LabelledTree V E)
+
+namespace LabelledTree
+
+def forget : LabelledTree V E → ProbeTree
+  | .done b => .done b
+  | .edge _ t₀ t₁ => .edge t₀.forget t₁.forget
+  | .mark _ t₀ t₁ => .mark t₀.forget t₁.forget
+
+@[reducible] noncomputable def eval (ω : Finset E) (m : Finset V) : LabelledTree V E → Bool
+  | .done b => b
+  | .edge e t₀ t₁ => if e ∈ ω then eval ω m t₁ else eval ω m t₀
+  | .mark v t₀ t₁ => if v ∈ m then eval ω m t₁ else eval ω m t₀
+
+def Valid (Q : Finset E) (F : Finset V) : LabelledTree V E → Prop
+  | .done _ => True
+  | .edge e t₀ t₁ => e ∈ Q ∧ Valid (Q.erase e) F t₀ ∧ Valid (Q.erase e) F t₁
+  | .mark v t₀ t₁ => v ∈ F ∧ Valid Q (F.erase v) t₀ ∧ Valid Q (F.erase v) t₁
+
+theorem eval_congr (t : LabelledTree V E) {Q : Finset E} {F : Finset V}
+    (ht : Valid Q F t) {ω ω' : Finset E} {m m' : Finset V}
+    (hω : ∀ e ∈ Q, e ∈ ω ↔ e ∈ ω') (hm : ∀ v ∈ F, v ∈ m ↔ v ∈ m') :
+    t.eval ω m = t.eval ω' m' := by
+  induction t generalizing Q F with
+  | done b => rfl
+  | edge e t₀ t₁ ih₀ ih₁ =>
+    obtain ⟨he,h₀,h₁⟩ := ht
+    simp only [eval,hω e he]
+    split
+    · exact ih₁ h₁ (fun a ha => hω a (Finset.mem_of_mem_erase ha)) hm
+    · exact ih₀ h₀ (fun a ha => hω a (Finset.mem_of_mem_erase ha)) hm
+  | mark v t₀ t₁ ih₀ ih₁ =>
+    obtain ⟨hv,h₀,h₁⟩ := ht
+    simp only [eval,hm v hv]
+    split
+    · exact ih₁ h₁ hω (fun a ha => hm a (Finset.mem_of_mem_erase ha))
+    · exact ih₀ h₀ hω (fun a ha => hm a (Finset.mem_of_mem_erase ha))
+
+lemma eval_insert_edge (t : LabelledTree V E) {Q : Finset E} {F : Finset V}
+    (ht : Valid Q F t) {e : E} (he : e ∉ Q) (ω : Finset E) (m : Finset V) :
+    t.eval (insert e ω) m = t.eval ω m := by
+  apply eval_congr t ht
+  · intro a ha
+    simp [show a ≠ e from fun hh => he (hh ▸ ha)]
+  · simp
+
+lemma eval_insert_mark (t : LabelledTree V E) {Q : Finset E} {F : Finset V}
+    (ht : Valid Q F t) {v : V} (hv : v ∉ F) (ω : Finset E) (m : Finset V) :
+    t.eval ω (insert v m) = t.eval ω m := by
+  apply eval_congr t ht
+  · simp
+  · intro a ha
+    simp [show a ≠ v from fun hh => hv (hh ▸ ha)]
+
+end LabelledTree
+
+theorem bernoulliPolynomial_affine (s : Finset E) (f g : Finset E → ℝ) (a b p : ℝ) :
+    bernoulliPolynomial s (fun t => a*f t+b*g t) p =
+      a*bernoulliPolynomial s f p+b*bernoulliPolynomial s g p := by
+  simp only [bernoulliPolynomial,add_mul,Finset.sum_add_distrib,Finset.mul_sum]
+  congr 1 <;> apply Finset.sum_congr rfl <;> intro t _ <;> ring
+
+namespace LabelledTree
+
+ 
+theorem expectation_eq (t : LabelledTree V E) {Q : Finset E} {F : Finset V}
+    (ht : Valid Q F t) (p h : ℝ) :
+    bernoulliPolynomial Q (fun ω => bernoulliPolynomial F
+      (fun m => if t.eval ω m then 1 else 0) h) p =
+      ProbeTree.successMass p h t.forget := by
+  induction t generalizing Q F with
+  | done b =>
+    cases b <;> simp [eval,forget,ProbeTree.successMass_done,bernoulliPolynomial_const]
+  | edge e t₀ t₁ ih₀ ih₁ =>
+    obtain ⟨he,h₀,h₁⟩ := ht
+    conv_lhs => rw [← Finset.insert_erase he]
+    rw [bernoulliPolynomial_insert _ e (Finset.notMem_erase e Q)]
+    have hc : bernoulliPolynomial (Q.erase e) (fun ω => bernoulliPolynomial F
+        (fun m => if (edge e t₀ t₁).eval ω m then 1 else 0) h) p =
+        bernoulliPolynomial (Q.erase e) (fun ω => bernoulliPolynomial F
+        (fun m => if t₀.eval ω m then 1 else 0) h) p := by
+      apply bernoulliPolynomial_congr
+      intro ω hω
+      have heω : e ∉ ω := fun hh => Finset.notMem_erase e Q (hω hh)
+      simp only [eval,heω,ite_false]
+    have ho : bernoulliPolynomial (Q.erase e) (fun ω => bernoulliPolynomial F
+        (fun m => if (edge e t₀ t₁).eval (insert e ω) m then 1 else 0) h) p =
+        bernoulliPolynomial (Q.erase e) (fun ω => bernoulliPolynomial F
+        (fun m => if t₁.eval ω m then 1 else 0) h) p := by
+      apply bernoulliPolynomial_congr
+      intro ω _
+      simp only [eval,Finset.mem_insert_self,ite_true]
+      apply bernoulliPolynomial_congr
+      intro m _
+      rw [eval_insert_edge t₁ h₁ (Finset.notMem_erase e Q)]
+    rw [hc,ho,ih₀ h₀,ih₁ h₁,forget,ProbeTree.successMass_edge]
+  | mark v t₀ t₁ ih₀ ih₁ =>
+    obtain ⟨hv,h₀,h₁⟩ := ht
+    have hh : ∀ ω, bernoulliPolynomial F
+        (fun m => if (mark v t₀ t₁).eval ω m then 1 else 0) h =
+      (1-h)*bernoulliPolynomial (F.erase v) (fun m => if t₀.eval ω m then 1 else 0) h +
+        h*bernoulliPolynomial (F.erase v) (fun m => if t₁.eval ω m then 1 else 0) h := by
+      intro ω
+      conv_lhs => rw [← Finset.insert_erase hv]
+      rw [bernoulliPolynomial_insert _ v (Finset.notMem_erase v F)]
+      congr 1
+      · congr 1
+        apply bernoulliPolynomial_congr
+        intro m hm
+        have hvm : v ∉ m := fun hh => Finset.notMem_erase v F (hm hh)
+        simp only [eval,hvm,ite_false]
+      · congr 1
+        apply bernoulliPolynomial_congr
+        intro m _
+        simp only [eval,Finset.mem_insert_self,ite_true]
+        rw [eval_insert_mark t₁ h₁ (Finset.notMem_erase v F)]
+    simp_rw [hh]
+    rw [bernoulliPolynomial_affine,ih₀ h₀,ih₁ h₁,forget,ProbeTree.successMass_mark]
+
+end LabelledTree
+end LabelledExploration
+
+section GraphMarkedSearch
+attribute [local instance] Classical.propDecidable
+attribute [local instance] Classical.decEq
+
+ 
+def SearchCrossing (G : BondGraph V E) (Q : Finset E) (R : Finset V) : Prop :=
+  ∃ t : E × V × V, t.1 ∈ Q ∧ t.2.1 ∈ R ∧ t.2.2 ∉ R ∧ G t.1 = s(t.2.1,t.2.2)
+
+noncomputable def searchCrossing (G : BondGraph V E) (Q : Finset E) (R : Finset V)
+    (h : SearchCrossing G Q R) : E × V × V := Classical.choose h
+
+theorem searchCrossing_spec (G : BondGraph V E) (Q : Finset E) (R : Finset V)
+    (h : SearchCrossing G Q R) :
+    (searchCrossing G Q R h).1 ∈ Q ∧ (searchCrossing G Q R h).2.1 ∈ R ∧
+      (searchCrossing G Q R h).2.2 ∉ R ∧
+      G (searchCrossing G Q R h).1 = s((searchCrossing G Q R h).2.1,(searchCrossing G Q R h).2.2) :=
+  Classical.choose_spec h
+
+ 
+def SearchAvailable (G : BondGraph V E) (Q : Finset E) (R F : Finset V) : Prop :=
+  ∀ e ∈ Q, ∀ v ∈ G e, v ∈ R ∨ v ∈ F
+
+noncomputable def clusterSearch (G : BondGraph V E) : ℕ → Finset E → Finset V → LabelledTree V E
+  | 0, _, _ => .done false
+  | n+1, Q, R =>
+    if hc : SearchCrossing G Q R then
+      let t := searchCrossing G Q R hc
+      .edge t.1 (clusterSearch G n (Q.erase t.1) R)
+        (.mark t.2.2 (clusterSearch G n (Q.erase t.1) (insert t.2.2 R)) (.done true))
+    else .done false
+
+lemma searchAvailable_erase (G : BondGraph V E) {Q : Finset E} {R F : Finset V}
+    (h : SearchAvailable G Q R F) (e : E) : SearchAvailable G (Q.erase e) R F :=
+  fun _ ha _ hv => h _ (Finset.mem_of_mem_erase ha) _ hv
+
+lemma searchAvailable_discover (G : BondGraph V E) {Q : Finset E} {R F : Finset V}
+    (h : SearchAvailable G Q R F) (e : E) (v : V) :
+    SearchAvailable G (Q.erase e) (insert v R) (F.erase v) := by
+  intro a ha b hb
+  rcases h a (Finset.mem_of_mem_erase ha) b hb with hb | hb
+  · exact Or.inl (Finset.mem_insert_of_mem hb)
+  · by_cases hbe : b = v
+    · exact Or.inl (by simp [hbe])
+    · exact Or.inr (Finset.mem_erase.mpr ⟨hbe,hb⟩)
+
+theorem clusterSearch_valid (G : BondGraph V E) (n : ℕ) (Q : Finset E) (R F : Finset V)
+    (ha : SearchAvailable G Q R F) : LabelledTree.Valid Q F (clusterSearch G n Q R) := by
+  induction n generalizing Q R F with
+  | zero => trivial
+  | succ n ih =>
+    simp only [clusterSearch]
+    split_ifs with hc
+    · obtain ⟨he,hv,hw,hends⟩ := searchCrossing_spec G Q R hc
+      refine ⟨he,ih _ _ _ (searchAvailable_erase G ha _),?_,?_,trivial⟩
+      · exact (ha _ he _ (by rw [hends]; simp)).resolve_left hw
+      · exact ih _ _ _ (searchAvailable_discover G ha _ _)
+    · trivial
+
+theorem clusterSearch_markSearch (G : BondGraph V E) (n : ℕ) (Q : Finset E) (R : Finset V) :
+    ProbeTree.IsMarkSearch (clusterSearch G n Q R).forget := by
+  induction n generalizing Q R with
+  | zero => exact .fail
+  | succ n ih =>
+    simp only [clusterSearch]
+    split_ifs
+    · exact .edge (ih _ _) (.mark (ih _ _))
+    · exact .fail
+
+ 
+def SeedMarked (G : BondGraph V E) (ω : Set E) (R m : Finset V) : Prop :=
+  ∃ a ∈ R, ∃ b ∈ m, Conn G ω a b
+
+theorem seedMarked_insert_edge (G : BondGraph V E) (ω : Set E) (R m : Finset V)
+    (e : E) (a b : V) (ha : a ∈ R) (he : G e = s(a,b)) :
+    SeedMarked G (insert e ω) R m ↔ SeedMarked G ω (insert b R) m := by
+  constructor
+  · rintro ⟨x,hx,y,hy,hxy⟩
+    rcases (conn_insert_iff G ω e a b x y he).mp hxy with h | ⟨_,h⟩ | ⟨_,h⟩
+    · exact ⟨x,Finset.mem_insert_of_mem hx,y,hy,h⟩
+    · exact ⟨b,by simp,y,hy,h⟩
+    · exact ⟨a,Finset.mem_insert_of_mem ha,y,hy,h⟩
+  · rintro ⟨x,hx,y,hy,hxy⟩
+    rcases Finset.mem_insert.mp hx with rfl | hx
+    · refine ⟨a,ha,y,hy,?_⟩
+      exact (Relation.ReflTransGen.single ⟨e,Set.mem_insert e ω,he⟩).trans
+        (conn_mono G (Set.subset_insert e ω) hxy)
+    · exact ⟨x,hx,y,hy,conn_mono G (Set.subset_insert e ω) hxy⟩
+
+theorem seedMarked_false_of_no_crossing (G : BondGraph V E) (Q : Finset E) (R m : Finset V)
+    (hc : ¬ SearchCrossing G Q R) (hm : Disjoint R m) (ω : Finset E) (hω : ω ⊆ Q) :
+    ¬ SeedMarked G (ω : Set E) R m := by
+  rintro ⟨a,ha,b,hb,hab⟩
+  have hclosed : ∀ x ∈ R, ∀ y, Conn G (ω : Set E) x y → y ∈ R := by
+    intro x hx y hxy
+    induction hxy with
+    | refl => exact hx
+    | @tail v w hv hw ih =>
+      by_contra hn
+      obtain ⟨e,he,hevw⟩ := hw
+      exact hc ⟨(e,v,w),hω he,ih,hn,hevw⟩
+  exact Finset.disjoint_left.mp hm (hclosed a ha b hab) hb
+
+ 
+theorem clusterSearch_correct (G : BondGraph V E) (n : ℕ) (Q : Finset E) (R F m : Finset V)
+    (hn : Q.card ≤ n) (ha : SearchAvailable G Q R F) (hm : Disjoint R m)
+    (ω : Finset E) (hω : ω ⊆ Q) :
+    (clusterSearch G n Q R).eval ω m = true ↔ SeedMarked G (ω : Set E) R m := by
+  induction n generalizing Q R F ω with
+  | zero =>
+    have hQ : Q = ∅ := Finset.card_eq_zero.mp (by omega)
+    have hωe : ω = ∅ := Finset.eq_empty_iff_forall_notMem.mpr (by simp_all)
+    simp only [clusterSearch,LabelledTree.eval,Bool.false_eq_true,false_iff]
+    exact seedMarked_false_of_no_crossing G Q R m (by simp [SearchCrossing,hQ]) hm ω hω
+  | succ n ih =>
+    simp only [clusterSearch]
+    split_ifs with hc
+    · obtain ⟨he,hv,hw,hends⟩ := searchCrossing_spec G Q R hc
+      have hcard : (Q.erase (searchCrossing G Q R hc).1).card ≤ n := by
+        rw [Finset.card_erase_of_mem he]; omega
+      simp only [LabelledTree.eval]
+      by_cases heω : (searchCrossing G Q R hc).1 ∈ ω
+      · simp only [heω,ite_true]
+        rw [← Finset.insert_erase heω,Finset.coe_insert,
+          seedMarked_insert_edge G _ R m _ _ _ hv hends]
+        by_cases hwm : (searchCrossing G Q R hc).2.2 ∈ m
+        · simp only [hwm,ite_true,true_iff]
+          exact ⟨_,by simp,_,hwm,conn_refl G _ _⟩
+        · simp only [hwm,ite_false]
+          rw [LabelledTree.eval_insert_edge _
+            (clusterSearch_valid G n _ _ (F.erase (searchCrossing G Q R hc).2.2)
+              (searchAvailable_discover G ha _ _)) (Finset.notMem_erase _ _)]
+          exact ih _ _ _ hcard (searchAvailable_discover G ha _ _)
+            (Finset.disjoint_insert_left.mpr ⟨hwm,hm⟩) (ω.erase (searchCrossing G Q R hc).1)
+            (Finset.erase_subset_erase _ hω)
+      · simp only [heω,ite_false]
+        exact ih _ _ _ hcard (searchAvailable_erase G ha _ ) hm ω
+          (fun e he => Finset.mem_erase.mpr ⟨fun heq => heω (heq ▸ he),hω he⟩)
+    · simp only [false_iff]
+      exact seedMarked_false_of_no_crossing G Q R m hc hm ω hω
+
+end GraphMarkedSearch
+
+section GraphSearchCost
+attribute [local instance] Classical.propDecidable
+attribute [local instance] Classical.decEq
+
+namespace ProbeTree
+
+def Budget (t : ProbeTree) (K d : ℕ) : Prop := ∀ l, queries t l ≤ K+d*marks t l
+
+lemma budget_done (b : Bool) (K d : ℕ) : Budget (.done b) K d := by
+  intro l
+  simp [queries,marks]
+
+lemma budget_mono {t : ProbeTree} {K L d : ℕ} (h : Budget t K d) (hKL : K ≤ L) :
+    Budget t L d := fun l => (h l).trans (Nat.add_le_add_right hKL _)
+
+lemma budget_edge {t₀ t₁ : ProbeTree} {K d : ℕ} (h₀ : Budget t₀ K d) (h₁ : Budget t₁ K d) :
+    Budget (.edge t₀ t₁) (K+1) d := by
+  intro l
+  cases l with
+  | inl l => simpa [queries,marks,Nat.add_right_comm] using Nat.add_le_add_right (h₀ l) 1
+  | inr l => simpa [queries,marks,Nat.add_right_comm] using Nat.add_le_add_right (h₁ l) 1
+
+lemma budget_mark {t₀ t₁ : ProbeTree} {K d : ℕ}
+    (h₀ : Budget t₀ (K+d) d) (h₁ : Budget t₁ (K+d) d) :
+    Budget (.mark t₀ t₁) K d := by
+  intro l
+  cases l with
+  | inl l => simpa [queries,marks,Nat.mul_add,Nat.add_assoc,Nat.add_comm,Nat.add_left_comm] using h₀ l
+  | inr l => simpa [queries,marks,Nat.mul_add,Nat.add_assoc,Nat.add_comm,Nat.add_left_comm] using h₁ l
+
+end ProbeTree
+
+ 
+noncomputable def queryPotential (G : BondGraph V E) (Q : Finset E) (R : Finset V) : Finset E :=
+  Q.filter (fun e => ∃ v ∈ R, v ∈ G e)
+
+lemma queryPotential_erase (G : BondGraph V E) (Q : Finset E) (R : Finset V) (e : E) :
+    queryPotential G (Q.erase e) R = (queryPotential G Q R).erase e := by
+  ext a
+  simp [queryPotential,and_assoc]
+
+lemma queryPotential_insert (G : BondGraph V E) (Q : Finset E) (R : Finset V) (v : V) :
+    queryPotential G Q (insert v R) = queryPotential G Q R ∪ Q.filter (fun e => v ∈ G e) := by
+  ext a
+  simp [queryPotential,and_or_left,or_comm]
+
+lemma filtered_degree_erase (G : BondGraph V E) (Q : Finset E) (d : ℕ)
+    (hd : ∀ v, (Q.filter (fun e => v ∈ G e)).card ≤ d) (a : E) :
+    ∀ v, ((Q.erase a).filter (fun e => v ∈ G e)).card ≤ d := by
+  intro v
+  apply (Finset.card_le_card ?_).trans (hd v)
+  intro e he
+  exact Finset.mem_filter.mpr ⟨Finset.mem_of_mem_erase (Finset.mem_filter.mp he).1,
+    (Finset.mem_filter.mp he).2⟩
+
+lemma queryPotential_discover_card (G : BondGraph V E) (Q : Finset E) (R : Finset V)
+    (e : E) (v : V) (d : ℕ) (hd : ∀ v, (Q.filter (fun e => v ∈ G e)).card ≤ d) :
+    (queryPotential G (Q.erase e) (insert v R)).card ≤
+      ((queryPotential G Q R).erase e).card+d := by
+  rw [queryPotential_insert,queryPotential_erase]
+  exact (Finset.card_union_le _ _).trans
+    (Nat.add_le_add_left (filtered_degree_erase G Q d hd e v) _)
+
+ 
+theorem clusterSearch_budget (G : BondGraph V E) (n : ℕ) (Q : Finset E) (R : Finset V)
+    (d : ℕ) (hd : ∀ v, (Q.filter (fun e => v ∈ G e)).card ≤ d) :
+    ProbeTree.Budget (clusterSearch G n Q R).forget (queryPotential G Q R).card d := by
+  induction n generalizing Q R with
+  | zero => exact ProbeTree.budget_done _ _ _
+  | succ n ih =>
+    simp only [clusterSearch]
+    split_ifs with hc
+    · obtain ⟨he,hv,hw,hends⟩ := searchCrossing_spec G Q R hc
+      have hpot : (searchCrossing G Q R hc).1 ∈ queryPotential G Q R := by
+        exact Finset.mem_filter.mpr ⟨he,_,hv,by rw [hends]; simp⟩
+      have hpos : 0 < (queryPotential G Q R).card := Finset.card_pos.mpr ⟨_,hpot⟩
+      have hclosed := ih (Q.erase (searchCrossing G Q R hc).1) R
+        (filtered_degree_erase G Q d hd _)
+      have hopen := ih (Q.erase (searchCrossing G Q R hc).1)
+        (insert (searchCrossing G Q R hc).2.2 R) (filtered_degree_erase G Q d hd _)
+      rw [queryPotential_erase,Finset.card_erase_of_mem hpot] at hclosed
+      have hnew := queryPotential_discover_card G Q R (searchCrossing G Q R hc).1
+        (searchCrossing G Q R hc).2.2 d hd
+      rw [Finset.card_erase_of_mem hpot] at hnew
+      have hh := ProbeTree.budget_edge hclosed
+        (ProbeTree.budget_mark (ProbeTree.budget_mono hopen hnew) (ProbeTree.budget_done true _ _))
+      have hcadd : (queryPotential G Q R).card-1+1 = (queryPotential G Q R).card := by omega
+      simpa only [LabelledTree.forget,hcadd] using hh
+    · exact ProbeTree.budget_done _ _ _
+
+noncomputable def searchVertices (G : BondGraph V E) (Q : Finset E) : Finset V :=
+  Q.biUnion fun e => (G e).toFinset
+
+lemma mem_searchVertices (G : BondGraph V E) {Q : Finset E} {e : E} (he : e ∈ Q)
+    {v : V} (hv : v ∈ G e) : v ∈ searchVertices G Q :=
+  Finset.mem_biUnion.mpr ⟨e,he,Sym2.mem_toFinset.mpr hv⟩
+
+noncomputable def markedClusterTree (G : BondGraph V E) (Q : Finset E) (x : V) : LabelledTree V E :=
+  .mark x (clusterSearch G Q.card Q {x}) (.done true)
+
+lemma markedClusterTree_available (G : BondGraph V E) (Q : Finset E) (x : V) :
+    SearchAvailable G Q {x} ((insert x (searchVertices G Q)).erase x) := by
+  intro e he v hv
+  by_cases hx : v = x
+  · exact Or.inl (by simpa using hx)
+  · exact Or.inr (Finset.mem_erase.mpr ⟨hx,Finset.mem_insert_of_mem (mem_searchVertices G he hv)⟩)
+
+theorem markedClusterTree_valid (G : BondGraph V E) (Q : Finset E) (x : V) :
+    LabelledTree.Valid Q (insert x (searchVertices G Q)) (markedClusterTree G Q x) :=
+  ⟨by simp,clusterSearch_valid G _ _ _ _ (markedClusterTree_available G Q x),trivial⟩
+
+theorem markedClusterTree_budget (G : BondGraph V E) (Q : Finset E) (x : V)
+    (d : ℕ) (hd : ∀ v, (Q.filter (fun e => v ∈ G e)).card ≤ d) :
+    ProbeTree.Budget (markedClusterTree G Q x).forget 0 d := by
+  have hh := clusterSearch_budget G Q.card Q {x} d hd
+  have hcost : (queryPotential G Q {x}).card ≤ 0+d := by simpa [queryPotential] using hd x
+  exact ProbeTree.budget_mark (ProbeTree.budget_mono hh hcost) (ProbeTree.budget_done true _ _)
+
+theorem markedClusterTree_correct (G : BondGraph V E) (Q : Finset E) (x : V)
+    (ω : Finset E) (hω : ω ⊆ Q) (m : Finset V) :
+    (markedClusterTree G Q x).eval ω m = true ↔ ∃ y ∈ m, Conn G (ω : Set E) x y := by
+  simp only [markedClusterTree,LabelledTree.eval]
+  by_cases hxm : x ∈ m
+  · simp only [hxm,ite_true,true_iff]
+    exact ⟨x,hxm,conn_refl G _ x⟩
+  · simp only [hxm,ite_false]
+    rw [clusterSearch_correct G Q.card Q {x} ((insert x (searchVertices G Q)).erase x) m
+      le_rfl (markedClusterTree_available G Q x) (by simpa) ω hω]
+    simp only [SeedMarked,Finset.mem_singleton,exists_eq_left]
+
+end GraphSearchCost
+
+section MarkedFirstMoment
+attribute [local instance] Classical.propDecidable
+attribute [local instance] Classical.decEq
+
+theorem quadratic_test_bound {z L B : ℝ} (hL : 0 ≤ L) (hB : 0 < B)
+    (h : ∀ lam : ℝ, lam*z ≤ L+lam^2*B) : z ≤ Real.sqrt (4*B*L) := by
+  have hh := h (z/(2*B))
+  have hid : z/(2*B)*z-(z/(2*B))^2*B = z^2/(4*B) := by
+    field_simp
+    ring
+  have hs : z^2/(4*B) ≤ L := by rw [← hid]; linarith
+  have hs' := (div_le_iff₀ (by positivity : 0 < 4*B)).mp hs
+  have hn : 0 ≤ 4*B*L := by positivity
+  have hsq := Real.sq_sqrt hn
+  have hpos := Real.sqrt_nonneg (4*B*L)
+  nlinarith
+
+ 
+noncomputable def markedConnectionPolynomial (G : BondGraph V E) (Q : Finset E)
+    (x : V) (h p : ℝ) : ℝ :=
+  bernoulliPolynomial Q (fun ω => bernoulliPolynomial (insert x (searchVertices G Q))
+    (fun m => if ∃ y ∈ m, Conn G (ω : Set E) x y then 1 else 0) h) p
+
+theorem markedConnectionPolynomial_eq_tree (G : BondGraph V E) (Q : Finset E)
+    (x : V) (h p : ℝ) :
+    markedConnectionPolynomial G Q x h p =
+      ProbeTree.successMass p h (markedClusterTree G Q x).forget := by
+  rw [← LabelledTree.expectation_eq _ (markedClusterTree_valid G Q x) p h]
+  apply bernoulliPolynomial_congr
+  intro ω hω
+  apply bernoulliPolynomial_congr
+  intro m _
+  simp only [markedClusterTree_correct G Q x ω hω m]
+
+theorem markedConnectionPolynomial_ge_mark (G : BondGraph V E) (Q : Finset E)
+    (x : V) (p h : unitInterval) : (h:ℝ) ≤ markedConnectionPolynomial G Q x h p := by
+  rw [markedConnectionPolynomial_eq_tree,markedClusterTree,LabelledTree.forget,
+    ProbeTree.successMass_mark]
+  simp only [LabelledTree.forget,ProbeTree.successMass_done,ite_true,mul_one]
+  exact le_add_of_nonneg_left (mul_nonneg (sub_nonneg.mpr h.2.2)
+    (ProbeTree.successMass_nonneg p h _))
+
+ 
+theorem markedConnection_derivative_bound (G : BondGraph V E) (Q : Finset E)
+    (x : V) (p h : unitInterval) (hp : (p:ℝ) < 1) (hh : 0 < (h:ℝ))
+    (d : ℕ) (hd : 0 < d) (hdeg : ∀ v, (Q.filter (fun e => v ∈ G e)).card ≤ d) :
+    (p:ℝ)*deriv (markedConnectionPolynomial G Q x h) p ≤
+      (Real.sqrt ((d:ℝ)/2)/(1-p))*Real.sqrt (Real.log (1/(h:ℝ))/h)*
+        markedConnectionPolynomial G Q x h p := by
+  let t := (markedClusterTree G Q x).forget
+  have ht : ProbeTree.IsMarkSearch t := .mark (clusterSearch_markSearch G _ _ _)
+  have hcost : ∀ l, ProbeTree.queries t l ≤ d*ProbeTree.marks t l := by
+    simpa only [ProbeTree.Budget,zero_add] using markedClusterTree_budget G Q x d hdeg
+  have hroot : (h:ℝ) ≤ ProbeTree.successMass p h t := by
+    simpa only [markedConnectionPolynomial_eq_tree] using markedConnectionPolynomial_ge_mark G Q x p h
+  have hB : 0 < (d:ℝ)/(8*h) := div_pos (by exact_mod_cast hd) (by positivity)
+  have hL : 0 ≤ Real.log (1/(h:ℝ)) := Real.log_nonneg ((one_le_div hh).mpr h.2.2)
+  have htest (lam : ℝ) : lam*(ProbeTree.successScore p h t/ProbeTree.successMass p h t) ≤
+      Real.log (1/(h:ℝ))+lam^2*((d:ℝ)/(8*h)) := by
+    simpa only [mul_div_assoc] using ProbeTree.marked_score_bound p h hh ht d hcost hroot lam
+  have hz := quadratic_test_bound hL hB htest
+  have hsqrt : Real.sqrt (4*((d:ℝ)/(8*h))*Real.log (1/(h:ℝ))) =
+      Real.sqrt ((d:ℝ)/2)*Real.sqrt (Real.log (1/(h:ℝ))/h) := by
+    rw [show 4*((d:ℝ)/(8*h))*Real.log (1/(h:ℝ)) =
+      ((d:ℝ)/2)*(Real.log (1/(h:ℝ))/h) by ring,Real.sqrt_mul (by positivity)]
+  rw [hsqrt] at hz
+  have hH : 0 < ProbeTree.successMass p h t := hh.trans_le hroot
+  have hscore := (div_le_iff₀ hH).mp hz
+  rw [ProbeTree.successScore_eq_deriv] at hscore
+  have heq : markedConnectionPolynomial G Q x h = fun q => ProbeTree.successMass q h t :=
+    funext fun q => markedConnectionPolynomial_eq_tree G Q x h q
+  rw [heq]
+  have hq : 0 < 1-(p:ℝ) := sub_pos.mpr hp
+  calc
+    _ = ((p:ℝ)*(1-p)*deriv (fun q => ProbeTree.successMass q h t) p)/(1-p) := by
+      field_simp
+    _ ≤ (Real.sqrt ((d:ℝ)/2)*Real.sqrt (Real.log (1/(h:ℝ))/h)*
+        ProbeTree.successMass p h t)/(1-p) := div_le_div_of_nonneg_right hscore hq.le
+    _ = _ := by ring
+
+theorem bernoulliPolynomial_open_difference (s : Finset E) (e : E) (he : e ∈ s)
+    (f : Finset E → ℝ) (p : ℝ) :
+    bernoulliPolynomial s (fun t => if e ∈ t then f t-f (t.erase e) else 0) p =
+      p*bernoulliPolynomial (s.erase e) (fun t => f (insert e t)-f t) p := by
+  conv_lhs => rw [← Finset.insert_erase he]
+  rw [bernoulliPolynomial_insert _ e (Finset.notMem_erase e s)]
+  have hclosed : bernoulliPolynomial (s.erase e)
+      (fun t => if e ∈ t then f t-f (t.erase e) else 0) p = 0 := by
+    rw [← bernoulliPolynomial_zero (s.erase e) p]
+    apply bernoulliPolynomial_congr
+    intro t ht
+    simp [show e ∉ t from fun hh => Finset.notMem_erase e s (ht hh)]
+  rw [hclosed,mul_zero,zero_add]
+  congr 1
+  apply bernoulliPolynomial_congr
+  intro t ht
+  have het : e ∉ t := fun hh => Finset.notMem_erase e s (ht hh)
+  simp [het]
+
+ 
+noncomputable def markedOpenPivotals (G : BondGraph V E) (Q : Finset E) (x : V)
+    (ω : Finset E) (m : Finset V) : Finset E :=
+  Q.filter fun e => e ∈ ω ∧ (∃ y ∈ m, Conn G (ω : Set E) x y) ∧
+      ¬ ∃ y ∈ m, Conn G (ω.erase e : Set E) x y
+
+ 
+noncomputable def markedPivotalMomentOne (G : BondGraph V E) (Q : Finset E) (x : V)
+    (h p : ℝ) : ℝ :=
+  bernoulliPolynomial Q (fun ω => bernoulliPolynomial (insert x (searchVertices G Q))
+    (fun m => ((markedOpenPivotals G Q x ω m).card:ℝ)) h) p
+
+theorem marked_pivotal_russo (G : BondGraph V E) (Q : Finset E) (x : V) (h p : ℝ) :
+    markedPivotalMomentOne G Q x h p = p*deriv (markedConnectionPolynomial G Q x h) p := by
+  let F := insert x (searchVertices G Q)
+  let f : Finset E → Finset V → ℝ := fun ω m =>
+    if ∃ y ∈ m, Conn G (ω : Set E) x y then 1 else 0
+  have hcard (ω : Finset E) (m : Finset V) : ((markedOpenPivotals G Q x ω m).card:ℝ) =
+      ∑ e ∈ Q, if e ∈ ω then f ω m-f (ω.erase e) m else 0 := by
+    rw [markedOpenPivotals,Finset.card_filter,Nat.cast_sum]
+    apply Finset.sum_congr rfl
+    intro e _
+    have hmono : (∃ y ∈ m, Conn G (ω.erase e : Set E) x y) →
+        ∃ y ∈ m, Conn G (ω : Set E) x y := by
+      rintro ⟨y,hy,hxy⟩
+      exact ⟨y,hy,conn_mono G (by exact_mod_cast Finset.erase_subset e ω) hxy⟩
+    by_cases he : e ∈ ω <;> by_cases h₁ : ∃ y ∈ m, Conn G (ω : Set E) x y <;>
+      by_cases h₀ : ∃ y ∈ m, Conn G (ω.erase e : Set E) x y <;>
+      simp only [f,he,h₁,h₀,and_true,and_false,not_true_eq_false,
+        not_false_eq_true,ite_true,ite_false,Nat.cast_one,Nat.cast_zero,sub_self,sub_zero]
+    exact False.elim (h₁ (hmono h₀))
+  have heq : markedConnectionPolynomial G Q x h =
+      bernoulliPolynomial Q (fun ω => bernoulliPolynomial F (f ω) h) := rfl
+  rw [heq,(hasDerivAt_bernoulliPolynomial Q _ p).deriv,Finset.mul_sum]
+  simp only [markedPivotalMomentOne,hcard,bernoulliPolynomial_sum]
+  apply Finset.sum_congr rfl
+  intro e he
+  rw [← bernoulliPolynomial_open_difference Q e he (fun ω => bernoulliPolynomial F (f ω) h) p]
+  apply bernoulliPolynomial_congr
+  intro ω _
+  by_cases heω : e ∈ ω
+  · simp only [heω,ite_true,F]
+    exact bernoulliPolynomial_sub _ _ _ _
+  · simp only [heω,ite_false,bernoulliPolynomial_zero]
+
+ 
+theorem marked_pivotal_first_moment (G : BondGraph V E) (Q : Finset E) (x : V)
+    (p h : unitInterval) (hp : (p:ℝ) < 1) (hh : 0 < (h:ℝ)) (d : ℕ) (hd : 0 < d)
+    (hdeg : ∀ v, (Q.filter (fun e => v ∈ G e)).card ≤ d) :
+    markedPivotalMomentOne G Q x h p ≤
+      (Real.sqrt ((d:ℝ)/2)/(1-p))*Real.sqrt (Real.log (1/(h:ℝ))/h)*
+        markedConnectionPolynomial G Q x h p := by
+  rw [marked_pivotal_russo]
+  exact markedConnection_derivative_bound G Q x p h hp hh d hd hdeg
+
+end MarkedFirstMoment
+
+section PivotalGeometry
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+def MarkedEvent (G : BondGraph V E) (ω : Set E) (m : Set V) (x : V) : Prop :=
+  ∃ y ∈ m, Conn G ω x y
+
+ 
+def OpenMarkedPivotal (G : BondGraph V E) (ω : Set E) (m : Set V) (x : V)
+    (e : E) : Prop :=
+  e ∈ ω ∧ MarkedEvent G ω m x ∧ ¬MarkedEvent G (ω \ {e}) m x
+
+theorem markedEvent_mono (G : BondGraph V E) {ω η : Set E} (h : ω ⊆ η)
+    {m : Set V} {x : V} (hx : MarkedEvent G ω m x) : MarkedEvent G η m x := by
+  obtain ⟨y,hy,hxy⟩ := hx
+  exact ⟨y,hy,conn_mono G h hxy⟩
+
+theorem edge_cluster_sides (G : BondGraph V E) (ω : Set E) (x : V)
+    {e : E} {a b : V} (he : e ∈ ω) (hab : G e = s(a,b)) :
+    a ∈ cluster G ω x ↔ b ∈ cluster G ω x := by
+  have hc : Conn G ω a b := .single ⟨e,he,hab⟩
+  exact ⟨fun h => h.trans hc,fun h => h.trans (conn_symm G ω hc)⟩
+
+ 
+theorem conn_restrict_cluster_compl (G : BondGraph V E) (ω : Set E) (x : V)
+    {a b : V} (ha : a ∉ cluster G ω x) (hab : Conn G ω a b) :
+    Conn G (ω ∩ internalEdges G (cluster G ω x)ᶜ) a b := by
+  induction hab with
+  | refl => exact conn_refl G _ a
+  | @tail u v hau huv ih =>
+    obtain ⟨e,he,heq⟩ := huv
+    have hu : u ∉ cluster G ω x := fun h => ha (h.trans (conn_symm G ω hau))
+    have hv : v ∉ cluster G ω x := fun h => hu ((edge_cluster_sides G ω x he heq).mpr h)
+    exact ih.tail ⟨e,⟨he,(mem_internalEdges_of_ends G _ e u v heq).mpr ⟨hu,hv⟩⟩,heq⟩
+
+theorem conn_restrict_cluster (G : BondGraph V E) (ω : Set E) (x : V)
+    {a b : V} (ha : a ∈ cluster G ω x) (hab : Conn G ω a b) :
+    Conn G (ω ∩ internalEdges G (cluster G ω x)) a b := by
+  induction hab with
+  | refl => exact conn_refl G _ a
+  | @tail u v hau huv ih =>
+    obtain ⟨e,he,heq⟩ := huv
+    have hu : u ∈ cluster G ω x := ha.trans hau
+    have hv : v ∈ cluster G ω x := (edge_cluster_sides G ω x he heq).mp hu
+    exact ih.tail ⟨e,⟨he,(mem_internalEdges_of_ends G _ e u v heq).mpr ⟨hu,hv⟩⟩,heq⟩
+
+ 
+theorem openMarkedPivotal_oriented (G : BondGraph V E) (ω : Set E) (m : Set V)
+    (x : V) {e : E} (he : OpenMarkedPivotal G ω m x e) :
+    ∃ a b, G e = s(a,b) ∧ Conn G (ω \ {e}) x a ∧
+      ¬Conn G (ω \ {e}) x b ∧ MarkedEvent G (ω \ {e}) m b := by
+  obtain ⟨y,hy,hxy⟩ := he.2.1
+  obtain ⟨a,b,hab⟩ : ∃ a b, G e = s(a,b) := by
+    obtain ⟨⟨a,b⟩,h⟩ := Quot.exists_rep (G e)
+    exact ⟨a,b,h.symm⟩
+  have heq : insert e (ω \ {e}) = ω := by
+    have heω : e ∈ ω := he.1
+    ext d
+    simp only [Set.mem_insert_iff,Set.mem_sdiff,Set.mem_singleton_iff]
+    by_cases h : d = e <;> simp_all
+  rw [← heq,conn_insert_iff G _ e a b x y hab] at hxy
+  rcases hxy with hxy | ⟨hxa,hby⟩ | ⟨hxb,hay⟩
+  · exact (he.2.2 ⟨y,hy,hxy⟩).elim
+  · exact ⟨a,b,hab,hxa,fun hxb => he.2.2 ⟨y,hy,hxb.trans hby⟩,y,hy,hby⟩
+  · exact ⟨b,a,hab.trans (Sym2.eq_swap ..),hxb,
+      fun hxa => he.2.2 ⟨y,hy,hxa.trans hay⟩,y,hy,hay⟩
+
+ 
+theorem pivotal_rootSide_disjoint_marks (G : BondGraph V E) (ω : Set E)
+    (m : Set V) (x : V) {e : E} (he : OpenMarkedPivotal G ω m x e) :
+    Disjoint (cluster G (ω \ {e}) x) m := by
+  rw [Set.disjoint_left]
+  exact fun y hy hm => he.2.2 ⟨y,hm,hy⟩
+
+ 
+theorem conn_delete_internal_of_outside (G : BondGraph V E) (ω : Set E) (x : V)
+    {f : E} {a b : V} (hf : f ∈ internalEdges G (cluster G ω x))
+    (ha : a ∉ cluster G ω x) (hab : Conn G ω a b) : Conn G (ω \ {f}) a b := by
+  apply conn_mono G _ (conn_restrict_cluster_compl G ω x ha hab)
+  intro e he
+  refine ⟨he.1,?_⟩
+  intro hef
+  have hef' : e = f := Set.mem_singleton_iff.mp hef
+  subst e
+  obtain ⟨u,v,heq⟩ : ∃ u v, G f = s(u,v) := by
+    obtain ⟨⟨a,b⟩,h⟩ := Quot.exists_rep (G f)
+    exact ⟨a,b,h.symm⟩
+  exact ((mem_internalEdges_of_ends G _ f u v heq).mp he.2).1
+    ((mem_internalEdges_of_ends G _ f u v heq).mp hf).1
+
+ 
+theorem pivotal_other_sides (G : BondGraph V E) (ω : Set E) (m : Set V) (x : V)
+    {e f : E} (_he : OpenMarkedPivotal G ω m x e)
+    (hf : OpenMarkedPivotal G ω m x f) (hne : f ≠ e) :
+    f ∈ internalEdges G (cluster G (ω \ {e}) x) ∨
+      f ∈ internalEdges G (cluster G (ω \ {e}) x)ᶜ := by
+  obtain ⟨a,b,hab⟩ : ∃ a b, G f = s(a,b) := by
+    obtain ⟨⟨a,b⟩,h⟩ := Quot.exists_rep (G f)
+    exact ⟨a,b,h.symm⟩
+  have hop : f ∈ ω \ {e} := ⟨hf.1,hne⟩
+  have hiff := edge_cluster_sides G (ω \ {e}) x hop hab
+  by_cases ha : a ∈ cluster G (ω \ {e}) x
+  · exact Or.inl ((mem_internalEdges_of_ends G _ f a b hab).mpr ⟨ha,hiff.mp ha⟩)
+  · exact Or.inr ((mem_internalEdges_of_ends G _ f a b hab).mpr ⟨ha,fun hb => ha (hiff.mpr hb)⟩)
+
+ 
+theorem conn_delete_exterior_of_inside (G : BondGraph V E) (ω : Set E) (x : V)
+    {f : E} {a b : V} (hf : f ∈ internalEdges G (cluster G ω x)ᶜ)
+    (ha : a ∈ cluster G ω x) (hab : Conn G ω a b) : Conn G (ω \ {f}) a b := by
+  apply conn_mono G _ (conn_restrict_cluster G ω x ha hab)
+  intro e he
+  refine ⟨he.1,?_⟩
+  intro hef
+  have hef' : e = f := Set.mem_singleton_iff.mp hef
+  subst e
+  obtain ⟨⟨u,v⟩,heq⟩ := Quot.exists_rep (G f)
+  exact ((mem_internalEdges_of_ends G _ f u v heq.symm).mp hf).1
+    ((mem_internalEdges_of_ends G _ f u v heq.symm).mp he.2).1
+
+ 
+theorem pivotal_rootSide_ssubset (G : BondGraph V E) (ω : Set E) (m : Set V) (x : V)
+    {e f : E} (he : OpenMarkedPivotal G ω m x e) (hne : e ≠ f)
+    (hf : f ∈ internalEdges G (cluster G (ω \ {e}) x)ᶜ) :
+    cluster G (ω \ {e}) x ⊂ cluster G (ω \ {f}) x := by
+  have hsub : cluster G (ω \ {e}) x ⊆ cluster G (ω \ {f}) x := by
+    intro y hy
+    exact conn_mono G (ω₁ := (ω \ {e}) \ {f}) (ω₂ := ω \ {f}) (fun _ h => ⟨h.1.1,h.2⟩)
+      (conn_delete_exterior_of_inside G (ω \ {e}) x hf (conn_refl G _ x) hy)
+  refine ⟨hsub,?_⟩
+  obtain ⟨a,b,hab,hxa,hxb,_⟩ := openMarkedPivotal_oriented G ω m x he
+  intro hback
+  apply hxb
+  apply hback
+  exact (hsub hxa).tail ⟨e,⟨he.1,hne⟩,hab⟩
+
+ 
+theorem pivotal_rootSides_nested (G : BondGraph V E) (ω : Set E) (m : Set V) (x : V)
+    {e f : E} (he : OpenMarkedPivotal G ω m x e)
+    (hf : OpenMarkedPivotal G ω m x f) (hne : e ≠ f) :
+    cluster G (ω \ {e}) x ⊂ cluster G (ω \ {f}) x ∨
+      cluster G (ω \ {f}) x ⊂ cluster G (ω \ {e}) x := by
+  rcases pivotal_other_sides G ω m x he hf hne.symm with hinside | houtside
+  · obtain ⟨a,b,hab,_,hxb,hbm⟩ := openMarkedPivotal_oriented G ω m x he
+    obtain ⟨y,hy,hby⟩ := hbm
+    have hby' : Conn G (ω \ {f}) b y := conn_mono G (ω₁ := (ω \ {e}) \ {f}) (ω₂ := ω \ {f}) (fun _ h => ⟨h.1.1,h.2⟩)
+      (conn_delete_internal_of_outside G (ω \ {e}) x hinside hxb hby)
+    have hb : b ∉ cluster G (ω \ {f}) x := fun h => hf.2.2 ⟨y,hy,h.trans hby'⟩
+    have ha : a ∉ cluster G (ω \ {f}) x := fun h =>
+      hb ((edge_cluster_sides G (ω \ {f}) x ⟨he.1,hne⟩ hab).mp h)
+    exact Or.inr (pivotal_rootSide_ssubset G ω m x hf hne.symm
+      ((mem_internalEdges_of_ends G _ e a b hab).mpr ⟨ha,hb⟩))
+  · exact Or.inl (pivotal_rootSide_ssubset G ω m x he hne houtside)
+
+ 
+theorem conn_restrict_cluster_compl_of_subset (G : BondGraph V E) (ω : Set E) (x : V)
+    {η : Set E} (hη : η ⊆ ω) {a b : V} (ha : a ∉ cluster G ω x)
+    (hab : Conn G η a b) : Conn G (η ∩ internalEdges G (cluster G ω x)ᶜ) a b := by
+  induction hab with
+  | refl => exact conn_refl G _ a
+  | @tail u v hau huv ih =>
+    obtain ⟨e,he,heq⟩ := huv
+    have hu : u ∉ cluster G ω x := fun h => ha (h.trans (conn_symm G ω (conn_mono G hη hau)))
+    have hv : v ∉ cluster G ω x := fun h => hu ((edge_cluster_sides G ω x (hη he) heq).mpr h)
+    exact ih.tail ⟨e,⟨he,(mem_internalEdges_of_ends G _ e u v heq).mpr ⟨hu,hv⟩⟩,heq⟩
+
+ 
+
+theorem markedEvent_after_pivotal (G : BondGraph V E) (ω : Set E) (m : Set V) (x : V)
+    {e f : E} (he : OpenMarkedPivotal G ω m x e) (hne : e ≠ f)
+    {a b : V} (hab : G e = s(a,b)) (hxa : Conn G (ω \ {e}) x a)
+    (hxb : ¬Conn G (ω \ {e}) x b)
+    (hf : f ∈ internalEdges G (cluster G (ω \ {e}) x)ᶜ) :
+    MarkedEvent G (ω \ {f}) m x ↔
+      MarkedEvent G ((ω ∩ internalEdges G (cluster G (ω \ {e}) x)ᶜ) \ {f}) m b := by
+  let W := cluster G (ω \ {e}) x
+  have heq : insert e ((ω \ {e}) \ {f}) = ω \ {f} := by
+    ext d
+    simp only [Set.mem_insert_iff,Set.mem_sdiff,Set.mem_singleton_iff]
+    constructor
+    · rintro (rfl | ⟨⟨hd,hde⟩,hdf⟩)
+      · exact ⟨he.1,hne⟩
+      · exact ⟨hd,hdf⟩
+    · rintro ⟨hd,hdf⟩
+      by_cases hde : d = e
+      · exact Or.inl hde
+      · exact Or.inr ⟨⟨hd,hde⟩,hdf⟩
+  constructor
+  · rintro ⟨y,hy,hxy⟩
+    rw [← heq,conn_insert_iff G _ e a b x y hab] at hxy
+    rcases hxy with hxy | ⟨_,hby⟩ | ⟨hxb',_⟩
+    · exact (he.2.2 ⟨y,hy,conn_mono G Set.sdiff_subset hxy⟩).elim
+    · refine ⟨y,hy,conn_mono G ?_ (conn_restrict_cluster_compl_of_subset G (ω \ {e}) x
+        Set.sdiff_subset hxb hby)⟩
+      intro d hd
+      exact ⟨⟨hd.1.1.1,hd.2⟩,hd.1.2⟩
+    · exact (hxb (conn_mono G Set.sdiff_subset hxb')).elim
+  · rintro ⟨y,hy,hby⟩
+    have hxa' : Conn G (ω \ {f}) x a := conn_mono G (ω₁ := (ω \ {e}) \ {f}) (ω₂ := ω \ {f}) (fun _ h => ⟨h.1.1,h.2⟩)
+      (conn_delete_exterior_of_inside G (ω \ {e}) x hf (conn_refl G _ x) hxa)
+    refine ⟨y,hy,(hxa'.tail ⟨e,⟨he.1,hne⟩,hab⟩).trans (conn_mono G ?_ hby)⟩
+    exact fun _ h => ⟨h.1.1,h.2⟩
+
+ 
+theorem openMarkedPivotal_exterior_iff (G : BondGraph V E) (ω : Set E) (m : Set V)
+    (x : V) {e : E} (he : OpenMarkedPivotal G ω m x e)
+    {a b : V} (hab : G e = s(a,b)) (hxa : Conn G (ω \ {e}) x a)
+    (hxb : ¬Conn G (ω \ {e}) x b) (hbm : MarkedEvent G (ω \ {e}) m b)
+    (f : E) :
+    OpenMarkedPivotal G (ω ∩ internalEdges G (cluster G (ω \ {e}) x)ᶜ) m b f ↔
+      OpenMarkedPivotal G ω m x f ∧ f ∈ internalEdges G (cluster G (ω \ {e}) x)ᶜ := by
+  have hbase : MarkedEvent G (ω ∩ internalEdges G (cluster G (ω \ {e}) x)ᶜ) m b := by
+    obtain ⟨y,hy,hby⟩ := hbm
+    refine ⟨y,hy,conn_mono G ?_ (conn_restrict_cluster_compl G (ω \ {e}) x hxb hby)⟩
+    exact fun _ h => ⟨h.1.1,h.2⟩
+  have hexterior (hf : f ∈ internalEdges G (cluster G (ω \ {e}) x)ᶜ) : e ≠ f := by
+    intro h; subst f
+    exact ((mem_internalEdges_of_ends G _ e a b hab).mp hf).1 hxa
+  constructor
+  · intro hf
+    have hiff := markedEvent_after_pivotal G ω m x he (hexterior hf.1.2) hab hxa hxb hf.1.2
+    exact ⟨⟨hf.1.1,he.2.1,fun h => hf.2.2 (hiff.mp h)⟩,hf.1.2⟩
+  · rintro ⟨hf,hout⟩
+    have hiff := markedEvent_after_pivotal G ω m x he (hexterior hout) hab hxa hxb hout
+    exact ⟨⟨hf.1,hout⟩,hbase,fun h => hf.2.2 (hiff.mpr h)⟩
+
+ 
+theorem cluster_finite_of_finite_edges (G : BondGraph V E) {ω : Set E}
+    (hω : ω.Finite) (x : V) : (cluster G ω x).Finite := by
+  let F : Finset V := insert x (hω.toFinset.biUnion fun e => (G e).toFinset)
+  apply F.finite_toSet.subset
+  intro y hxy
+  induction hxy with
+  | refl => exact Finset.mem_insert_self x _
+  | @tail a b hxa hab ih =>
+    obtain ⟨e,he,heq⟩ := hab
+    apply Finset.mem_insert_of_mem
+    apply Finset.mem_biUnion.mpr
+    exact ⟨e,hω.mem_toFinset.mpr he,by simp [heq]⟩
+
+ 
+noncomputable def pivotalRank (G : BondGraph V E) (ω : Set E) (x : V) (e : E) : ℕ :=
+  (cluster G (ω \ {e}) x).ncard
+
+theorem pivotalRank_injective (G : BondGraph V E) {ω : Set E} (hω : ω.Finite)
+    (m : Set V) (x : V) : Set.InjOn (pivotalRank G ω x) {e | OpenMarkedPivotal G ω m x e} := by
+  intro e he f hf hsame
+  by_contra hne
+  rcases pivotal_rootSides_nested G ω m x he hf hne with h | h
+  · have hh := Set.ncard_lt_ncard h (cluster_finite_of_finite_edges G (hω.subset Set.sdiff_subset) x)
+    exact (Nat.ne_of_lt hh) hsame
+  · have hh := Set.ncard_lt_ncard h (cluster_finite_of_finite_edges G (hω.subset Set.sdiff_subset) x)
+    exact (Nat.ne_of_lt hh) hsame.symm
+
+ 
+theorem pivotalRank_lt_iff (G : BondGraph V E) {ω : Set E} (hω : ω.Finite)
+    (m : Set V) (x : V) {e f : E} (he : OpenMarkedPivotal G ω m x e)
+    (hf : OpenMarkedPivotal G ω m x f) :
+    pivotalRank G ω x e < pivotalRank G ω x f ↔
+      f ∈ internalEdges G (cluster G (ω \ {e}) x)ᶜ := by
+  have hexterior (hout : f ∈ internalEdges G (cluster G (ω \ {e}) x)ᶜ) : e ≠ f := by
+    obtain ⟨a,b,hab,hxa,_,_⟩ := openMarkedPivotal_oriented G ω m x he
+    intro h; subst f
+    exact ((mem_internalEdges_of_ends G _ e a b hab).mp hout).1 hxa
+  constructor
+  · intro hlt
+    have hne : e ≠ f := fun h => (Nat.ne_of_lt hlt) (congrArg (pivotalRank G ω x) h)
+    rcases pivotal_other_sides G ω m x he hf hne.symm with hinside | hout
+    · obtain ⟨a,b,hab,_,hxb,hbm⟩ := openMarkedPivotal_oriented G ω m x he
+      obtain ⟨y,hy,hby⟩ := hbm
+      have hby' : Conn G (ω \ {f}) b y := conn_mono G
+        (ω₁ := (ω \ {e}) \ {f}) (ω₂ := ω \ {f}) (fun _ h => ⟨h.1.1,h.2⟩)
+        (conn_delete_internal_of_outside G (ω \ {e}) x hinside hxb hby)
+      have hb : b ∉ cluster G (ω \ {f}) x := fun h => hf.2.2 ⟨y,hy,h.trans hby'⟩
+      have ha : a ∉ cluster G (ω \ {f}) x := fun h =>
+        hb ((edge_cluster_sides G (ω \ {f}) x ⟨he.1,hne⟩ hab).mp h)
+      have hs := pivotal_rootSide_ssubset G ω m x hf hne.symm
+        ((mem_internalEdges_of_ends G _ e a b hab).mpr ⟨ha,hb⟩)
+      have hrev := Set.ncard_lt_ncard hs
+        (cluster_finite_of_finite_edges G (hω.subset Set.sdiff_subset) x)
+      exact (Nat.not_lt_of_ge (Nat.le_of_lt hrev) hlt).elim
+    · exact hout
+  · intro hout
+    exact Set.ncard_lt_ncard (pivotal_rootSide_ssubset G ω m x he (hexterior hout) hout)
+      (cluster_finite_of_finite_edges G (hω.subset Set.sdiff_subset) x)
+
+end PivotalGeometry
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Exploration/Moments.lean b/lean/OAI/Probability/UnimodularPercolation/Exploration/Moments.lean
index 266cf36c0c6ec555502b96fdd02df2d6f9cc5b3a..58f20a82f835bf42c2fd84704a4ed2f07cc01087 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Exploration/Moments.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Exploration/Moments.lean
@@ -1,928 +1,928 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Exploration.Marked
-
-namespace OAI
-
-/-! Recursive marked moments and exponential estimates. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-section OrderedPivotalMoments
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem sum_choose_rank_suffix {I : Type*} (P : Finset I) (rank : I → ℕ)
-    (hinj : Set.InjOn rank (P : Set I)) (i : ℕ) :
-    (∑ e ∈ P, ((P.filter fun f => rank e < rank f).card).choose i) =
-      P.card.choose (i+1) := by
-  induction P using Finset.strongInductionOn with
-  | _ P ih =>
-    rcases P.eq_empty_or_nonempty with rfl | hP
-    · simp
-    obtain ⟨e,he,hmin⟩ := Finset.exists_min_image P rank hP
-    have hT : P.erase e ⊂ P := Finset.erase_ssubset he
-    have hinjT : Set.InjOn rank (↑(P.erase e) : Set I) := fun a ha b hb =>
-      hinj (Finset.mem_of_mem_erase ha) (Finset.mem_of_mem_erase hb)
-    have hroot : P.filter (fun f => rank e < rank f) = P.erase e := by
-      ext f
-      simp only [Finset.mem_filter,Finset.mem_erase]
-      constructor
-      · rintro ⟨hf,hlt⟩
-        exact ⟨fun h => (Nat.ne_of_lt hlt) (congrArg rank h.symm),hf⟩
-      · rintro ⟨hne,hf⟩
-        exact ⟨hf,lt_of_le_of_ne (hmin f hf) (fun h => hne (hinj he hf h).symm)⟩
-    have hrest (f : I) (hf : f ∈ P.erase e) :
-        P.filter (fun g => rank f < rank g) = (P.erase e).filter (fun g => rank f < rank g) := by
-      ext g
-      simp only [Finset.mem_filter,Finset.mem_erase]
-      constructor
-      · rintro ⟨hg,hfg⟩
-        refine ⟨⟨?_,hg⟩,hfg⟩
-        intro h; subst g
-        exact (not_lt_of_ge (hmin f (Finset.mem_of_mem_erase hf))) hfg
-      · exact fun h => ⟨h.1.2,h.2⟩
-    calc
-      _ = (∑ f ∈ P.erase e, ((P.filter fun g => rank f < rank g).card).choose i) +
-          ((P.filter fun f => rank e < rank f).card).choose i :=
-        (Finset.sum_erase_add _ _ he).symm
-      _ = (∑ f ∈ P.erase e, (((P.erase e).filter fun g => rank f < rank g).card).choose i) +
-          (P.erase e).card.choose i := by
-        rw [hroot]
-        congr 1
-        exact Finset.sum_congr rfl fun f hf => congrArg (fun s : Finset I => s.card.choose i) (hrest f hf)
-      _ = (P.erase e).card.choose (i+1)+(P.erase e).card.choose i := by rw [ih _ hT hinjT]
-      _ = P.card.choose (i+1) := by
-        have hc : (P.erase e).card+1=P.card := by
-          rw [Finset.card_erase_of_mem he]
-          have hp := Finset.card_pos.mpr hP
-          omega
-        rw [← hc,Nat.choose_succ_succ]
-        simp [Nat.succ_eq_add_one,Nat.add_comm]
-
-end OrderedPivotalMoments
-
-section MarkedRecursion
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem cluster_union_external (G : BondGraph V E) (ω η : Set E) (x : V) (W : Set V)
-    (hω : cluster G ω x ⊆ W) (hη : η ⊆ internalEdges G Wᶜ) :
-    cluster G (ω ∪ η) x = cluster G ω x := by
-  apply Set.Subset.antisymm
-  · intro y hy
-    induction hy with
-    | refl => exact conn_refl G ω x
-    | @tail a b hxa hab ih =>
-      obtain ⟨e,he,heq⟩ := hab
-      rcases he with he | he
-      · exact ih.tail ⟨e,he,heq⟩
-      · exact (((mem_internalEdges_of_ends G _ e a b heq).mp (hη he)).1 (hω ih)).elim
-  · exact fun _ h => conn_mono G Set.subset_union_left h
-
- 
-theorem cluster_union_external_eq_iff (G : BondGraph V E) (ω η : Set E)
-    (x : V) (W : Set V) (hη : η ⊆ internalEdges G Wᶜ) :
-    cluster G (ω ∪ η) x = W ↔ cluster G ω x = W := by
-  constructor
-  · intro h
-    have hω : cluster G ω x ⊆ W := fun y hy => h ▸ conn_mono G Set.subset_union_left hy
-    rwa [cluster_union_external G ω η x W hω hη] at h
-  · intro h
-    rw [cluster_union_external G ω η x W (h ▸ Set.Subset.rfl) hη,h]
-
- 
-theorem markedEvent_internal_marks (G : BondGraph V E) (ω : Set E) (m W : Set V)
-    (x : V) (hx : x ∈ W) (hω : ω ⊆ internalEdges G W) :
-    MarkedEvent G ω m x ↔ MarkedEvent G ω (m ∩ W) x := by
-  constructor
-  · rintro ⟨y,hy,hxy⟩
-    exact ⟨y,⟨hy,conn_internal_endpoint G W hx (conn_mono G
-      (ω₁ := ω) (ω₂ := ω ∩ internalEdges G W) (fun _ h => ⟨h,hω h⟩) hxy)⟩,hxy⟩
-  · rintro ⟨y,hy,hxy⟩
-    exact ⟨y,hy.1,hxy⟩
-
-theorem openMarkedPivotal_internal_marks (G : BondGraph V E) (ω : Set E)
-    (m W : Set V) (x : V) (hx : x ∈ W) (hω : ω ⊆ internalEdges G W) (e : E) :
-    OpenMarkedPivotal G ω m x e ↔ OpenMarkedPivotal G ω (m ∩ W) x e := by
-  simp only [OpenMarkedPivotal,markedEvent_internal_marks G ω m W x hx hω,
-    markedEvent_internal_marks G (ω \ {e}) m W x hx (Set.Subset.trans Set.sdiff_subset hω)]
-
- 
-theorem openMarkedPivotal_insert_cut (G : BondGraph V E) (ω : Set E) (m W : Set V)
-    (x : V) {e : E} {a b : V} (hW : cluster G ω x = W)
-    (hab : G e = s(a,b)) (ha : a ∈ W) (hb : b ∉ W) :
-    OpenMarkedPivotal G (insert e ω) m x e ↔
-      Disjoint W m ∧ MarkedEvent G (ω ∩ internalEdges G Wᶜ) (m ∩ Wᶜ) b := by
-  have he : e ∉ ω := fun he => hb (hW ▸ ((edge_cluster_sides G ω x he hab).mp (hW ▸ ha)))
-  have hdel : insert e ω \ {e} = ω := by ext f; simp [he]
-  have hxa : Conn G ω x a := by change a ∈ cluster G ω x; rwa [hW]
-  constructor
-  · intro hp
-    have hdis := pivotal_rootSide_disjoint_marks G (insert e ω) m x hp
-    rw [hdel,hW] at hdis
-    refine ⟨hdis,?_⟩
-    obtain ⟨y,hy,hxy⟩ := hp.2.1
-    rw [conn_insert_iff G ω e a b x y hab] at hxy
-    rcases hxy with hxy | ⟨_,hby⟩ | ⟨hxb,_⟩
-    · exact (Set.disjoint_left.mp hdis (hW ▸ hxy) hy).elim
-    · have hby' := conn_restrict_cluster_compl G ω x (a := b) (hW ▸ hb) hby
-      rw [hW] at hby'
-      exact (markedEvent_internal_marks G _ m Wᶜ b hb Set.inter_subset_right).mp ⟨y,hy,hby'⟩
-    · exact (hb (hW ▸ hxb)).elim
-  · rintro ⟨hdis,y,hy,hby⟩
-    refine ⟨Set.mem_insert _ _,?_,?_⟩
-    · exact ⟨y,hy.1,(conn_mono G (Set.subset_insert e ω) hxa).tail
-        ⟨e,Set.mem_insert _ _,hab⟩ |>.trans (conn_mono G
-          (fun _ h => Set.mem_insert_of_mem _ h.1) hby)⟩
-    · rw [hdel]
-      rintro ⟨z,hz,hxz⟩
-      exact Set.disjoint_left.mp hdis (hW ▸ hxz) hz
-
-@[simp] theorem mem_markedOpenPivotals (G : BondGraph V E) (Q : Finset E) (x : V)
-    (ω : Finset E) (m : Finset V) (e : E) :
-    e ∈ markedOpenPivotals G Q x ω m ↔ e ∈ Q ∧ OpenMarkedPivotal G ω m x e := by
-  simp only [markedOpenPivotals,Finset.mem_filter,OpenMarkedPivotal,MarkedEvent,Finset.coe_erase,
-    Finset.mem_coe]
-
- 
-noncomputable def markedMomentValue (G : BondGraph V E) (Q : Finset E) (x : V)
-    (i : ℕ) (ω : Finset E) (m : Finset V) : ℝ :=
-  if MarkedEvent G ω m x then ((markedOpenPivotals G Q x ω m).card.choose i : ℝ) else 0
-
- 
-theorem markedMomentValue_succ (G : BondGraph V E) (Q : Finset E) (x : V)
-    (i : ℕ) (ω : Finset E) (m : Finset V) :
-    markedMomentValue G Q x (i+1) ω m =
-      ∑ e ∈ markedOpenPivotals G Q x ω m,
-        (((markedOpenPivotals G Q x ω m).filter
-          (fun f => f ∈ internalEdges G (cluster G (↑ω \ {e}) x)ᶜ)).card.choose i : ℝ) := by
-  have hinj : Set.InjOn (pivotalRank G ω x) (↑(markedOpenPivotals G Q x ω m) : Set E) := by
-    intro e he f hf
-    exact pivotalRank_injective G ω.finite_toSet m x
-      ((mem_markedOpenPivotals G Q x ω m e).mp he).2
-      ((mem_markedOpenPivotals G Q x ω m f).mp hf).2
-  have hs := sum_choose_rank_suffix (markedOpenPivotals G Q x ω m) (pivotalRank G ω x) hinj i
-  have heq (e : E) (he : e ∈ markedOpenPivotals G Q x ω m) :
-      (markedOpenPivotals G Q x ω m).filter (fun f => f ∈ internalEdges G (cluster G (↑ω \ {e}) x)ᶜ) =
-      (markedOpenPivotals G Q x ω m).filter (fun f => pivotalRank G ω x e < pivotalRank G ω x f) := by
-    apply Finset.filter_congr
-    intro f hf
-    exact (pivotalRank_lt_iff G ω.finite_toSet m x
-      ((mem_markedOpenPivotals G Q x ω m e).mp he).2
-      ((mem_markedOpenPivotals G Q x ω m f).mp hf).2).symm
-  by_cases h : MarkedEvent G ω m x
-  · simp only [markedMomentValue,h,ite_true]
-    rw [← hs,Nat.cast_sum]
-    apply Finset.sum_congr rfl
-    intro e he
-    rw [heq e he]
-  · have hempty : markedOpenPivotals G Q x ω m = ∅ := by
-      apply Finset.eq_empty_iff_forall_notMem.mpr
-      intro e he
-      exact h ((mem_markedOpenPivotals G Q x ω m e).mp he).2.2.1
-    simp [markedMomentValue,h,hempty]
-
- 
-theorem conn_mem_searchSupport (G : BondGraph V E) (Q : Finset E) (x : V)
-    {ω : Set E} (hω : ω ⊆ Q) {y : V} (hxy : Conn G ω x y) :
-    y ∈ insert x (searchVertices G Q) := by
-  induction hxy with
-  | refl => exact Finset.mem_insert_self _ _
-  | @tail a b hxa hab ih =>
-    obtain ⟨e,he,heq⟩ := hab
-    exact Finset.mem_insert_of_mem (mem_searchVertices G (hω he) (by simp [heq]))
-
-theorem markedEvent_searchSupport (G : BondGraph V E) (Q : Finset E) (x : V)
-    {ω : Set E} (hω : ω ⊆ Q) (m : Set V) :
-    MarkedEvent G ω m x ↔ MarkedEvent G ω (m ∩ ↑(insert x (searchVertices G Q))) x := by
-  constructor
-  · rintro ⟨y,hy,hxy⟩
-    exact ⟨y,⟨hy,conn_mem_searchSupport G Q x hω hxy⟩,hxy⟩
-  · rintro ⟨y,hy,hxy⟩
-    exact ⟨y,hy.1,hxy⟩
-
-theorem markedOpenPivotals_searchSupport (G : BondGraph V E) (Q : Finset E) (x : V)
-    (ω : Finset E) (hω : ω ⊆ Q) (m : Finset V) :
-    markedOpenPivotals G Q x ω m =
-      markedOpenPivotals G Q x ω (m ∩ insert x (searchVertices G Q)) := by
-  ext e
-  simp only [mem_markedOpenPivotals,OpenMarkedPivotal,Finset.coe_inter]
-  rw [markedEvent_searchSupport G Q x (by exact_mod_cast hω),
-    markedEvent_searchSupport G Q x (ω := (ω : Set E) \ {e})
-      (fun _ h => hω h.1)]
-
-theorem markedMomentValue_searchSupport (G : BondGraph V E) (Q : Finset E) (x : V)
-    (i : ℕ) (ω : Finset E) (hω : ω ⊆ Q) (m : Finset V) :
-    markedMomentValue G Q x i ω m =
-      markedMomentValue G Q x i ω (m ∩ insert x (searchVertices G Q)) := by
-  unfold markedMomentValue
-  rw [markedOpenPivotals_searchSupport G Q x ω hω m]
-  simp only [Finset.coe_inter]
-  rw [markedEvent_searchSupport G Q x (by exact_mod_cast hω)]
-
-@[simp] theorem markedMomentValue_zero (G : BondGraph V E) (Q : Finset E) (x : V)
-    (ω : Finset E) (m : Finset V) :
-    markedMomentValue G Q x 0 ω m = if MarkedEvent G ω m x then 1 else 0 := by
-  simp only [markedMomentValue,Nat.choose_zero_right,Nat.cast_one]
-
-@[simp] theorem markedMomentValue_one (G : BondGraph V E) (Q : Finset E) (x : V)
-    (ω : Finset E) (m : Finset V) :
-    markedMomentValue G Q x 1 ω m = ((markedOpenPivotals G Q x ω m).card : ℝ) := by
-  by_cases h : MarkedEvent G ω m x
-  · simp [markedMomentValue,h]
-  · have he : markedOpenPivotals G Q x ω m = ∅ := by
-      apply Finset.eq_empty_iff_forall_notMem.mpr
-      intro e he
-      exact h ((mem_markedOpenPivotals G Q x ω m e).mp he).2.2.1
-    simp [markedMomentValue,h,he]
-
- 
-noncomputable def finiteMarkedMoment (G : BondGraph V E) (Q : Finset E) (F : Finset V)
-    (x : V) (i : ℕ) (h p : ℝ) : ℝ :=
-  bernoulliPolynomial Q (fun ω => bernoulliPolynomial F (markedMomentValue G Q x i ω) h) p
-
-theorem finiteMarkedMoment_support (G : BondGraph V E) (Q : Finset E) (F : Finset V)
-    (x : V) (i : ℕ) (h p : ℝ) (hF : insert x (searchVertices G Q) ⊆ F) :
-    finiteMarkedMoment G Q F x i h p =
-      finiteMarkedMoment G Q (insert x (searchVertices G Q)) x i h p := by
-  apply bernoulliPolynomial_congr
-  intro ω hω
-  rw [← bernoulliPolynomial_restrict _ F hF]
-  apply bernoulliPolynomial_congr
-  intro m _
-  exact markedMomentValue_searchSupport G Q x i ω hω m
-
- 
-theorem finiteMarkedMoment_first_bound (G : BondGraph V E) (Q : Finset E) (F : Finset V)
-    (x : V) (p h : unitInterval) (hp : (p:ℝ) < 1) (hh : 0 < (h:ℝ)) (d : ℕ) (hd : 0 < d)
-    (hdeg : ∀ v, (Q.filter (fun e => v ∈ G e)).card ≤ d)
-    (hF : insert x (searchVertices G Q) ⊆ F) :
-    finiteMarkedMoment G Q F x 1 h p ≤
-      (Real.sqrt ((d:ℝ)/2)/(1-p))*Real.sqrt (Real.log (1/(h:ℝ))/h)*
-        finiteMarkedMoment G Q F x 0 h p := by
-  rw [finiteMarkedMoment_support G Q F x 1 h p hF,
-    finiteMarkedMoment_support G Q F x 0 h p hF]
-  simpa only [finiteMarkedMoment,markedMomentValue_one,markedMomentValue_zero,MarkedEvent,
-    markedPivotalMomentOne,markedConnectionPolynomial,bernoulliPolynomial,Finset.mem_coe] using
-    marked_pivotal_first_moment G Q x p h hp hh d hd hdeg
-
-noncomputable def exteriorBonds (G : BondGraph V E) (Q : Finset E) (W : Set V) : Finset E :=
-  Q.filter fun e => e ∈ internalEdges G Wᶜ
-
-noncomputable def exteriorMarks (m : Finset V) (W : Set V) : Finset V :=
-  m.filter fun x => x ∉ W
-
-@[simp] theorem coe_exteriorBonds (G : BondGraph V E) (Q : Finset E) (W : Set V) :
-    (exteriorBonds G Q W : Set E) = (Q : Set E) ∩ internalEdges G Wᶜ := by
-  ext e; simp [exteriorBonds]
-
-@[simp] theorem coe_exteriorMarks (m : Finset V) (W : Set V) :
-    (exteriorMarks m W : Set V) = (m : Set V) ∩ Wᶜ := by
-  ext x; simp [exteriorMarks]
-
- 
-theorem markedOpenPivotals_exterior [Nonempty V] (G : BondGraph V E) (Q : Finset E)
-    (x : V) (ω : Finset E) (m : Finset V) {e : E}
-    (he : OpenMarkedPivotal G ω m x e) :
-    markedOpenPivotals G (exteriorBonds G Q (cluster G (↑ω \ {e}) x))
-      (cutOutside G (cluster G (↑ω \ {e}) x) e)
-      (exteriorBonds G ω (cluster G (↑ω \ {e}) x))
-      (exteriorMarks m (cluster G (↑ω \ {e}) x)) =
-      (markedOpenPivotals G Q x ω m).filter
-        (fun f => f ∈ internalEdges G (cluster G (↑ω \ {e}) x)ᶜ) := by
-  obtain ⟨a,b,hab,hxa,hxb,hbm⟩ := openMarkedPivotal_oriented G ω m x he
-  have hb := (cut_unique G (cluster G (↑ω \ {e}) x) hab hxa hxb).2
-  rw [← hb]
-  ext f
-  simp only [mem_markedOpenPivotals,exteriorBonds,Finset.mem_filter,coe_exteriorMarks,
-    Finset.coe_filter]
-  have hmarks := openMarkedPivotal_internal_marks G
-    ((ω : Set E) ∩ internalEdges G (cluster G (↑ω \ {e}) x)ᶜ)
-    m (cluster G (↑ω \ {e}) x)ᶜ b hxb Set.inter_subset_right f
-  have hiff := openMarkedPivotal_exterior_iff G ω m x he hab hxa hxb hbm f
-  change (f ∈ Q ∧ f ∈ internalEdges G (cluster G (↑ω \ {e}) x)ᶜ) ∧
-    OpenMarkedPivotal G ((ω : Set E) ∩ internalEdges G (cluster G (↑ω \ {e}) x)ᶜ)
-      ((m : Set V) ∩ (cluster G (↑ω \ {e}) x)ᶜ) b f ↔ _
-  rw [← hmarks,hiff]
-  tauto
-
- 
-theorem markedMomentValue_exterior [Nonempty V] (G : BondGraph V E) (Q : Finset E)
-    (x : V) (i : ℕ) (ω : Finset E) (m : Finset V) {e : E}
-    (he : OpenMarkedPivotal G ω m x e) :
-    markedMomentValue G (exteriorBonds G Q (cluster G (↑ω \ {e}) x))
-      (cutOutside G (cluster G (↑ω \ {e}) x) e) i
-      (exteriorBonds G ω (cluster G (↑ω \ {e}) x))
-      (exteriorMarks m (cluster G (↑ω \ {e}) x)) =
-      (((markedOpenPivotals G Q x ω m).filter
-        (fun f => f ∈ internalEdges G (cluster G (↑ω \ {e}) x)ᶜ)).card.choose i : ℝ) := by
-  have hsuccess : MarkedEvent G (exteriorBonds G ω (cluster G (↑ω \ {e}) x))
-      (exteriorMarks m (cluster G (↑ω \ {e}) x))
-      (cutOutside G (cluster G (↑ω \ {e}) x) e) := by
-    obtain ⟨a,b,hab,hxa,hxb,y,hy,hby⟩ := openMarkedPivotal_oriented G ω m x he
-    rw [← (cut_unique G (cluster G (↑ω \ {e}) x) hab hxa hxb).2,
-      coe_exteriorBonds,coe_exteriorMarks]
-    apply (markedEvent_internal_marks G _ m (cluster G (↑ω \ {e}) x)ᶜ b hxb
-      Set.inter_subset_right).mp
-    refine ⟨y,hy,conn_mono G ?_ (conn_restrict_cluster_compl G (↑ω \ {e}) x hxb hby)⟩
-    exact fun _ h => ⟨h.1.1,h.2⟩
-  rw [markedMomentValue,ite_eq_left hsuccess,markedOpenPivotals_exterior G Q x ω m he]
-
- 
-theorem markedMomentValue_succ_exterior [Nonempty V] (G : BondGraph V E) (Q : Finset E)
-    (x : V) (i : ℕ) (ω : Finset E) (m : Finset V) :
-    markedMomentValue G Q x (i+1) ω m = ∑ e ∈ Q,
-      if OpenMarkedPivotal G ω m x e then
-        markedMomentValue G (exteriorBonds G Q (cluster G (↑ω \ {e}) x))
-          (cutOutside G (cluster G (↑ω \ {e}) x) e) i
-          (exteriorBonds G ω (cluster G (↑ω \ {e}) x))
-          (exteriorMarks m (cluster G (↑ω \ {e}) x)) else 0 := by
-  rw [markedMomentValue_succ]
-  trans ∑ e ∈ markedOpenPivotals G Q x ω m,
-    markedMomentValue G (exteriorBonds G Q (cluster G (↑ω \ {e}) x))
-      (cutOutside G (cluster G (↑ω \ {e}) x) e) i
-      (exteriorBonds G ω (cluster G (↑ω \ {e}) x))
-      (exteriorMarks m (cluster G (↑ω \ {e}) x))
-  · apply Finset.sum_congr rfl
-    intro e he
-    exact (markedMomentValue_exterior G Q x i ω m
-      ((mem_markedOpenPivotals G Q x ω m e).mp he).2).symm
-  · have heq : markedOpenPivotals G Q x ω m = Q.filter (fun e => OpenMarkedPivotal G ω m x e) := by
-      ext e; simp only [mem_markedOpenPivotals,Finset.mem_filter]
-    rw [heq,Finset.sum_filter]
-
-end MarkedRecursion
-
-section CutFactor
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Nonempty V]
-
-noncomputable def cutMomentValue (G : BondGraph V E) (Q : Finset E) (x : V)
-    (e : E) (W : Finset V) (i : ℕ) (ω : Finset E) (m : Finset V) : ℝ :=
-  if cluster G ω x = (W : Set V) then
-    if Disjoint (W : Set V) (m : Set V) then
-      markedMomentValue G (exteriorBonds G Q W) (cutOutside G W e) i
-        (exteriorBonds G ω W) (exteriorMarks m W)
-    else 0
-  else 0
-
-omit [Nonempty V] in
-theorem cut_not_exterior (G : BondGraph V E) (W : Set V) {e : E}
-    (hc : CutExists G W e) : e ∉ internalEdges G Wᶜ := by
-  obtain ⟨a,b,hab,ha,hb⟩ := hc
-  intro he
-  exact ((mem_internalEdges_of_ends G Wᶜ e a b hab).mp he).1 ha
-
-omit [Nonempty V] in
-theorem exteriorBonds_subset_erase (G : BondGraph V E) (Q : Finset E)
-    (W : Set V) {e : E} (hc : CutExists G W e) : exteriorBonds G Q W ⊆ Q.erase e := by
-  intro f hf
-  obtain ⟨hf,hout⟩ := Finset.mem_filter.mp hf
-  exact Finset.mem_erase.mpr ⟨fun h => cut_not_exterior G W hc (h ▸ hout),hf⟩
-
- 
-theorem cutMomentValue_blocks (G : BondGraph V E) (Q : Finset E) (F W : Finset V)
-    (x : V) (e : E) (i : ℕ) (a b : Finset E) (u v : Finset V)
-    (ha : a ⊆ (Q.erase e) \ exteriorBonds G Q W)
-    (hb : b ⊆ exteriorBonds G Q W) (hu : u ⊆ W) (hv : v ⊆ F \ W) :
-    cutMomentValue G Q x e W i (a ∪ b) (u ∪ v) =
-      (if cluster G a x = (W : Set V) then (1:ℝ) else 0) *
-      (if u = ∅ then (1:ℝ) else 0) *
-      markedMomentValue G (exteriorBonds G Q W) (cutOutside G W e) i b v := by
-  have hbout : (b : Set E) ⊆ internalEdges G (W : Set V)ᶜ :=
-    fun f hf => (Finset.mem_filter.mp (hb hf)).2
-  have hc := cluster_union_external_eq_iff G (a : Set E) b x W hbout
-  have he : exteriorBonds G (a ∪ b) W = b := by
-    ext f
-    simp only [exteriorBonds,Finset.mem_filter,Finset.mem_union]
-    constructor
-    · rintro ⟨hf,hout⟩
-      rcases hf with hf | hf
-      · have hfa := Finset.mem_sdiff.mp (ha hf)
-        exact (hfa.2 (Finset.mem_filter.mpr ⟨Finset.mem_of_mem_erase hfa.1,hout⟩)).elim
-      · exact hf
-    · exact fun hf => ⟨Or.inr hf,hbout hf⟩
-  have hm : exteriorMarks (u ∪ v) W = v := by
-    ext y
-    simp only [exteriorMarks,Finset.mem_filter,Finset.mem_union,Finset.mem_coe]
-    constructor
-    · rintro ⟨hy,hyW⟩
-      rcases hy with hy | hy
-      · exact (hyW (hu hy)).elim
-      · exact hy
-    · intro hy
-      exact ⟨Or.inr hy,(Finset.mem_sdiff.mp (hv hy)).2⟩
-  have hd : Disjoint (W : Set V) ((u ∪ v : Finset V) : Set V) ↔ u = ∅ := by
-    constructor
-    · intro h
-      apply Finset.eq_empty_iff_forall_notMem.mpr
-      intro y hy
-      exact Set.disjoint_left.mp h (hu hy) (Finset.mem_union_left v hy)
-    · rintro rfl
-      apply Set.disjoint_left.mpr
-      intro y hy hyv
-      have hyv' : y ∈ v := by simpa using hyv
-      exact (Finset.mem_sdiff.mp (hv hyv')).2 hy
-  rw [Finset.coe_union] at hd
-  simp only [cutMomentValue,Finset.coe_union,hc,hd,he,hm]
-  split_ifs <;> simp_all
-
-omit [Nonempty V] in
- 
-theorem bernoulliPolynomial_empty_indicator (F : Finset V) (h : ℝ) :
-    bernoulliPolynomial F (fun m => if m = ∅ then (1:ℝ) else 0) h = (1-h)^F.card := by
-  unfold bernoulliPolynomial
-  rw [Finset.sum_eq_single ∅]
-  · simp
-  · intro m hm hne
-    simp [hne]
-  · simp
-
- 
-theorem cutMoment_factor (G : BondGraph V E) (Q : Finset E) (F W : Finset V)
-    (x : V) (e : E) (i : ℕ) (h p : ℝ) (hW : W ⊆ F) (hc : CutExists G W e) :
-    bernoulliPolynomial (Q.erase e) (fun ω => bernoulliPolynomial F
-      (cutMomentValue G Q x e W i ω) h) p =
-      bernoulliPolynomial ((Q.erase e) \ exteriorBonds G Q W)
-        (fun a => if cluster G a x = (W : Set V) then (1:ℝ) else 0) p *
-      (1-h)^W.card * finiteMarkedMoment G (exteriorBonds G Q W) (F \ W)
-        (cutOutside G W e) i h p := by
-  let R := exteriorBonds G Q W
-  let A := (Q.erase e) \ R
-  let B := F \ W
-  have hR : R ⊆ Q.erase e := exteriorBonds_subset_erase G Q W hc
-  have hQR : Q.erase e = A ∪ R := by
-    dsimp [A]; ext f; simp only [Finset.mem_union,Finset.mem_sdiff]
-    tauto
-  have hAR : Disjoint A R := Finset.disjoint_sdiff.symm
-  have hFB : F = W ∪ B := (Finset.union_sdiff_of_subset hW).symm
-  have hWB : Disjoint W B := Finset.disjoint_sdiff
-  conv_lhs => rw [hQR,bernoulliPolynomial_union A R hAR]
-  trans bernoulliPolynomial A (fun a =>
-    (if cluster G a x = (W : Set V) then (1:ℝ) else 0) *
-      ((1-h)^W.card * finiteMarkedMoment G R B (cutOutside G W e) i h p)) p
-  · apply bernoulliPolynomial_congr
-    intro a ha
-    trans bernoulliPolynomial R (fun b =>
-      (if cluster G a x = (W : Set V) then (1:ℝ) else 0) * (1-h)^W.card *
-        bernoulliPolynomial B (markedMomentValue G R (cutOutside G W e) i b) h) p
-    · apply bernoulliPolynomial_congr
-      intro b hb
-      rw [hFB,bernoulliPolynomial_union W B hWB]
-      trans bernoulliPolynomial W (fun u =>
-        (if cluster G a x = (W : Set V) then (1:ℝ) else 0) *
-        (if u = ∅ then (1:ℝ) else 0) *
-        bernoulliPolynomial B (markedMomentValue G R (cutOutside G W e) i b) h) h
-      · apply bernoulliPolynomial_congr
-        intro u hu
-        rw [← bernoulliPolynomial_const_mul]
-        apply bernoulliPolynomial_congr
-        intro v hv
-        exact cutMomentValue_blocks G Q F W x e i a b u v ha hb hu hv
-      · rw [bernoulliPolynomial_mul_const,bernoulliPolynomial_const_mul,
-          bernoulliPolynomial_empty_indicator]
-    · rw [bernoulliPolynomial_const_mul]
-      exact mul_assoc _ _ _
-  · rw [bernoulliPolynomial_mul_const]
-    exact (mul_assoc _ _ _).symm
-
-end CutFactor
-
-section PointwiseRecursion
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Nonempty V]
-
-noncomputable def pivotalSuffixValue (G : BondGraph V E) (Q : Finset E) (x : V)
-    (e : E) (i : ℕ) (ω : Finset E) (m : Finset V) : ℝ :=
-  if OpenMarkedPivotal G ω m x e then
-    markedMomentValue G (exteriorBonds G Q (cluster G (↑ω \ {e}) x))
-      (cutOutside G (cluster G (↑ω \ {e}) x) e) i
-      (exteriorBonds G ω (cluster G (↑ω \ {e}) x))
-      (exteriorMarks m (cluster G (↑ω \ {e}) x)) else 0
-
- 
-theorem pivotalSuffixValue_insert (G : BondGraph V E) (Q : Finset E) (x : V)
-    (e : E) (i : ℕ) (ω : Finset E) (he : e ∉ ω) (m : Finset V) :
-    pivotalSuffixValue G Q x e i (insert e ω) m =
-      if CutExists G (cluster G ω x) e then
-        if Disjoint (cluster G ω x) (m : Set V) then
-          markedMomentValue G (exteriorBonds G Q (cluster G ω x))
-            (cutOutside G (cluster G ω x) e) i
-            (exteriorBonds G ω (cluster G ω x)) (exteriorMarks m (cluster G ω x))
-        else 0
-      else 0 := by
-  have hdel : ((insert e ω : Finset E) : Set E) \ {e} = (ω : Set E) := by
-    simp [he]
-  by_cases hc : CutExists G (cluster G ω x) e
-  · obtain ⟨hab,ha,hb⟩ := cut_spec G (cluster G ω x) hc
-    have heout := cut_not_exterior G (cluster G ω x) hc
-    have hbe : exteriorBonds G (insert e ω) (cluster G ω x) = exteriorBonds G ω (cluster G ω x) := by
-      ext f
-      simp only [exteriorBonds,Finset.mem_filter,Finset.mem_insert]
-      constructor
-      · rintro ⟨hf,hout⟩
-        rcases hf with rfl | hf
-        · exact (heout hout).elim
-        · exact ⟨hf,hout⟩
-      · exact fun h => ⟨Or.inr h.1,h.2⟩
-    have hp := openMarkedPivotal_insert_cut G (ω : Set E) m (cluster G ω x) x rfl hab ha hb
-    have hps : OpenMarkedPivotal G (insert e ω : Finset E) m x e ↔
-        Disjoint (cluster G ω x) (m : Set V) ∧
-        MarkedEvent G (exteriorBonds G ω (cluster G ω x))
-          (exteriorMarks m (cluster G ω x)) (cutOutside G (cluster G ω x) e) := by
-      simpa only [Finset.coe_insert,coe_exteriorBonds,coe_exteriorMarks] using hp
-    simp only [pivotalSuffixValue,hdel,hbe,hc,ite_true,hps]
-    by_cases hd : Disjoint (cluster G ω x) (m : Set V) <;>
-      by_cases hm : MarkedEvent G (exteriorBonds G ω (cluster G ω x))
-        (exteriorMarks m (cluster G ω x)) (cutOutside G (cluster G ω x) e) <;>
-      simp only [markedMomentValue,hd,hm,and_true,and_false,ite_true,ite_false]
-  · have hn : ¬OpenMarkedPivotal G (insert e ω : Finset E) m x e := by
-      intro hp
-      obtain ⟨a,b,hab,hxa,hxb,_⟩ := openMarkedPivotal_oriented G (insert e ω : Finset E) m x hp
-      rw [hdel] at hxa hxb
-      exact hc ⟨a,b,hab,hxa,hxb⟩
-    simp only [pivotalSuffixValue,hn,hc,ite_false]
-
-noncomputable def finiteRootComponent (G : BondGraph V E) (ω : Finset E) (x : V) : Finset V :=
-  (cluster_finite_of_finite_edges G ω.finite_toSet x).toFinset
-
-omit [Nonempty V] in
-@[simp]
-theorem coe_finiteRootComponent (G : BondGraph V E) (ω : Finset E) (x : V) :
-    (finiteRootComponent G ω x : Set V) = cluster G ω x := by simp [finiteRootComponent]
-
-omit [Nonempty V] in
-theorem finiteRootComponent_subset (G : BondGraph V E) (Q : Finset E) (F : Finset V)
-    (x : V) (ω : Finset E) (hω : ω ⊆ Q) (hF : insert x (searchVertices G Q) ⊆ F) :
-    finiteRootComponent G ω x ⊆ F := by
-  intro y hy
-  have hxy : Conn G (ω : Set E) x y := by
-    change y ∈ (finiteRootComponent G ω x : Set V) at hy
-    rwa [coe_finiteRootComponent] at hy
-  exact hF (conn_mem_searchSupport G Q x (by exact_mod_cast hω) hxy)
-
-omit [Nonempty V] in
- 
-theorem sum_component_partition (G : BondGraph V E) (Q : Finset E) (F : Finset V)
-    (x : V) (ω : Finset E) (hω : ω ⊆ Q) (hF : insert x (searchVertices G Q) ⊆ F)
-    (f : Finset V → ℝ) :
-    (∑ W ∈ F.powerset, if cluster G ω x = (W : Set V) then f W else 0) =
-      f (finiteRootComponent G ω x) := by
-  have hmem : finiteRootComponent G ω x ∈ F.powerset :=
-    Finset.mem_powerset.mpr (finiteRootComponent_subset G Q F x ω hω hF)
-  rw [Finset.sum_eq_single_of_mem _ hmem]
-  · simp
-  · intro W hW hne
-    have hn : cluster G ω x ≠ (W : Set V) := by
-      intro h
-      apply hne
-      apply Finset.coe_injective
-      exact h.symm.trans (coe_finiteRootComponent G ω x).symm
-    simp [hn]
-
- 
-theorem pivotalSuffixValue_component_sum (G : BondGraph V E) (Q : Finset E) (F : Finset V)
-    (x : V) (e : E) (i : ℕ) (ω : Finset E) (he : e ∉ ω) (hω : ω ⊆ Q)
-    (hF : insert x (searchVertices G Q) ⊆ F) (m : Finset V) :
-    pivotalSuffixValue G Q x e i (insert e ω) m = ∑ W ∈ F.powerset,
-      if CutExists G W e then cutMomentValue G Q x e W i ω m else 0 := by
-  rw [pivotalSuffixValue_insert G Q x e i ω he m]
-  symm
-  trans ∑ W ∈ F.powerset, if cluster G ω x = (W : Set V) then
-    (if CutExists G W e then
-      if Disjoint (W : Set V) (m : Set V) then
-        markedMomentValue G (exteriorBonds G Q W) (cutOutside G W e) i
-          (exteriorBonds G ω W) (exteriorMarks m W) else 0
-      else 0) else 0
-  · apply Finset.sum_congr rfl
-    intro W _
-    by_cases hc : CutExists G W e <;> by_cases hw : cluster G ω x = (W : Set V) <;>
-      simp [cutMomentValue,hc,hw]
-  · rw [sum_component_partition G Q F x ω hω hF]
-    simp only [coe_finiteRootComponent]
-
-omit [Nonempty V] in
- 
-theorem bernoulliPolynomial_open_gate (Q : Finset E) (e : E) (he : e ∈ Q)
-    (f : Finset E → ℝ) (p : ℝ) (hf : ∀ ω ⊆ Q, e ∉ ω → f ω = 0) :
-    bernoulliPolynomial Q f p =
-      p*bernoulliPolynomial (Q.erase e) (fun ω => f (insert e ω)) p := by
-  conv_lhs => rw [← Finset.insert_erase he]
-  rw [bernoulliPolynomial_insert _ e (Finset.notMem_erase e Q)]
-  have hz : bernoulliPolynomial (Q.erase e) f p = 0 := by
-    rw [← bernoulliPolynomial_zero (Q.erase e) p]
-    apply bernoulliPolynomial_congr
-    intro ω hω
-    exact hf ω (Finset.Subset.trans hω (Finset.erase_subset e Q))
-      (fun h => Finset.notMem_erase e Q (hω h))
-  rw [hz,mul_zero,zero_add]
-
-end PointwiseRecursion
-
-section FiniteMomentRecursion
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Nonempty V]
-
- 
-noncomputable def cutCoefficient (G : BondGraph V E) (Q : Finset E) (x : V)
-    (e : E) (W : Finset V) (h p : ℝ) : ℝ :=
-  bernoulliPolynomial ((Q.erase e) \ exteriorBonds G Q W)
-    (fun a => if cluster G a x = (W : Set V) then (1:ℝ) else 0) p * (1-h)^W.card
-
- 
-theorem finiteMarkedMoment_recursion (G : BondGraph V E) (Q : Finset E) (F : Finset V)
-    (x : V) (i : ℕ) (h p : ℝ) (hF : insert x (searchVertices G Q) ⊆ F) :
-    finiteMarkedMoment G Q F x (i+1) h p = p * ∑ e ∈ Q, ∑ W ∈ F.powerset,
-      if CutExists G W e then cutCoefficient G Q x e W h p *
-        finiteMarkedMoment G (exteriorBonds G Q W) (F \ W) (cutOutside G W e) i h p else 0 := by
-  have hsum : finiteMarkedMoment G Q F x (i+1) h p =
-      ∑ e ∈ Q, bernoulliPolynomial Q (fun ω => bernoulliPolynomial F
-        (pivotalSuffixValue G Q x e i ω) h) p := by
-    change bernoulliPolynomial Q (fun ω => bernoulliPolynomial F
-      (fun m => markedMomentValue G Q x (i+1) ω m) h) p = _
-    simp only [markedMomentValue_succ_exterior,bernoulliPolynomial_sum]
-    rfl
-  rw [hsum,Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro e he
-  rw [bernoulliPolynomial_open_gate Q e he]
-  · congr 1
-    trans bernoulliPolynomial (Q.erase e) (fun ω => bernoulliPolynomial F
-      (fun m => ∑ W ∈ F.powerset, if CutExists G W e then cutMomentValue G Q x e W i ω m else 0) h) p
-    · apply bernoulliPolynomial_congr
-      intro ω hω
-      apply bernoulliPolynomial_congr
-      intro m _
-      exact pivotalSuffixValue_component_sum G Q F x e i ω
-        (fun h => Finset.notMem_erase e Q (hω h))
-        (Finset.Subset.trans hω (Finset.erase_subset e Q)) hF m
-    · simp only [bernoulliPolynomial_sum]
-      apply Finset.sum_congr rfl
-      intro W hW
-      by_cases hc : CutExists G W e
-      · simp only [hc,ite_true]
-        exact cutMoment_factor G Q F W x e i h p (Finset.mem_powerset.mp hW) hc
-      · simp only [hc,ite_false,bernoulliPolynomial_zero]
-  · intro ω hω heω
-    rw [← bernoulliPolynomial_zero F h]
-    apply bernoulliPolynomial_congr
-    intro m _
-    have hpiv : ¬OpenMarkedPivotal G ω m x e := fun h => heω h.1
-    simp only [pivotalSuffixValue,hpiv,ite_false]
-
-omit [Nonempty V] in
-theorem cutCoefficient_nonneg (G : BondGraph V E) (Q : Finset E) (x : V)
-    (e : E) (W : Finset V) (h p : unitInterval) : 0 ≤ cutCoefficient G Q x e W h p := by
-  apply mul_nonneg
-  · exact bernoulliPolynomial_nonneg _ _ _ p.2 (fun _ _ => by split_ifs <;> norm_num)
-  · exact pow_nonneg (sub_nonneg.mpr h.2.2) _
-
-omit [Nonempty V] in
-theorem finiteMarkedMoment_nonneg (G : BondGraph V E) (Q : Finset E) (F : Finset V)
-    (x : V) (i : ℕ) (h p : unitInterval) : 0 ≤ finiteMarkedMoment G Q F x i h p := by
-  apply bernoulliPolynomial_nonneg _ _ _ p.2
-  intro ω _
-  apply bernoulliPolynomial_nonneg _ _ _ h.2
-  intro m _
-  unfold markedMomentValue
-  split_ifs <;> positivity
-
- 
-theorem exterior_mark_support (G : BondGraph V E) (Q : Finset E) (F W : Finset V)
-    (x : V) (e : E) (he : e ∈ Q) (hc : CutExists G W e)
-    (hF : insert x (searchVertices G Q) ⊆ F) :
-    insert (cutOutside G W e) (searchVertices G (exteriorBonds G Q W)) ⊆ F \ W := by
-  have hs := cut_spec G W hc
-  intro y hy
-  rcases Finset.mem_insert.mp hy with rfl | hy
-  · exact Finset.mem_sdiff.mpr ⟨hF (Finset.mem_insert_of_mem
-      (mem_searchVertices G he (by simp [hs.1]))),hs.2.2⟩
-  · obtain ⟨f,hf,hyf⟩ := Finset.mem_biUnion.mp hy
-    have hfQ := (Finset.mem_filter.mp hf).1
-    have hfo := (Finset.mem_filter.mp hf).2
-    have hyf' := Sym2.mem_toFinset.mp hyf
-    exact Finset.mem_sdiff.mpr ⟨hF (Finset.mem_insert_of_mem
-      (mem_searchVertices G hfQ hyf')),hfo y hyf'⟩
-
-omit [Nonempty V] in
- 
-theorem exterior_degree_bound (G : BondGraph V E) (Q : Finset E) (W : Set V)
-    (d : ℕ) (hd : ∀ v, (Q.filter (fun e => v ∈ G e)).card ≤ d) :
-    ∀ v, ((exteriorBonds G Q W).filter (fun e => v ∈ G e)).card ≤ d := by
-  intro v
-  apply le_trans (Finset.card_le_card ?_) (hd v)
-  intro e he
-  exact Finset.mem_filter.mpr ⟨(Finset.mem_filter.mp (Finset.mem_filter.mp he).1).1,
-    (Finset.mem_filter.mp he).2⟩
-
- 
-theorem finiteMarkedMoment_geometric (G : BondGraph V E) (p h : unitInterval)
-    (hp : (p:ℝ) < 1) (hh : 0 < (h:ℝ)) (d : ℕ) (hd : 0 < d) (i : ℕ)
-    (Q : Finset E) (F : Finset V) (x : V)
-    (hdeg : ∀ v, (Q.filter (fun e => v ∈ G e)).card ≤ d)
-    (hF : insert x (searchVertices G Q) ⊆ F) :
-    finiteMarkedMoment G Q F x i h p ≤
-      ((Real.sqrt ((d:ℝ)/2)/(1-p))*Real.sqrt (Real.log (1/(h:ℝ))/h))^i *
-        finiteMarkedMoment G Q F x 0 h p := by
-  let A : ℝ := (Real.sqrt ((d:ℝ)/2)/(1-p))*Real.sqrt (Real.log (1/(h:ℝ))/h)
-  have hA : 0 ≤ A := mul_nonneg (div_nonneg (Real.sqrt_nonneg _) (sub_nonneg.mpr p.2.2))
-    (Real.sqrt_nonneg _)
-  change _ ≤ A^i * _
-  induction i generalizing Q F x with
-  | zero => simp
-  | succ i ih =>
-    have hpow := pow_nonneg hA i
-    calc
-      finiteMarkedMoment G Q F x (i+1) h p =
-          (p:ℝ) * ∑ e ∈ Q, ∑ W ∈ F.powerset,
-            if CutExists G W e then cutCoefficient G Q x e W h p *
-              finiteMarkedMoment G (exteriorBonds G Q W) (F \ W) (cutOutside G W e) i h p else 0 :=
-        finiteMarkedMoment_recursion G Q F x i h p hF
-      _ ≤ (p:ℝ) * ∑ e ∈ Q, ∑ W ∈ F.powerset,
-            if CutExists G W e then cutCoefficient G Q x e W h p *
-              (A^i * finiteMarkedMoment G (exteriorBonds G Q W) (F \ W) (cutOutside G W e) 0 h p)
-              else 0 := by
-        apply mul_le_mul_of_nonneg_left _ p.2.1
-        apply Finset.sum_le_sum
-        intro e he
-        apply Finset.sum_le_sum
-        intro W hW
-        by_cases hc : CutExists G W e
-        · simp only [hc,ite_true]
-          apply mul_le_mul_of_nonneg_left _ (cutCoefficient_nonneg G Q x e W h p)
-          exact ih _ _ _ (exterior_degree_bound G Q W d hdeg) (exterior_mark_support G Q F W x e he hc hF)
-        · simp only [hc,ite_false,le_refl]
-      _ = A^i * finiteMarkedMoment G Q F x 1 h p := by
-        rw [show (1:ℕ) = 0+1 from rfl,finiteMarkedMoment_recursion G Q F x 0 h p hF]
-        simp only [Finset.mul_sum]
-        apply Finset.sum_congr rfl
-        intro e he
-        apply Finset.sum_congr rfl
-        intro W hW
-        by_cases hc : CutExists G W e <;> simp only [hc,ite_true,ite_false] <;> ring
-      _ ≤ A^i * (A * finiteMarkedMoment G Q F x 0 h p) :=
-        mul_le_mul_of_nonneg_left (finiteMarkedMoment_first_bound G Q F x p h hp hh d hd hdeg hF) hpow
-      _ = A^(i+1) * finiteMarkedMoment G Q F x 0 h p := by rw [pow_succ]; ring
-
-end FiniteMomentRecursion
-
-section MarkedExponential
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem choose_generating_sum (n N : ℕ) (hn : n ≤ N) (z : ℝ) :
-    (1+z)^n = ∑ i ∈ Finset.range (N+1), z^i*(n.choose i : ℝ) := by
-  rw [add_comm (1:ℝ),add_pow]
-  simp only [one_pow,mul_one]
-  apply Finset.sum_subset (Finset.range_mono (Nat.succ_le_succ hn))
-  intro i hi hin
-  have hni : n < i := by simp only [Finset.mem_range,not_lt] at hin; omega
-  simp [Nat.choose_eq_zero_of_lt hni]
-
- 
-theorem markedMomentValue_generating (G : BondGraph V E) (Q : Finset E) (x : V)
-    (z : ℝ) (ω : Finset E) (m : Finset V) :
-    (if MarkedEvent G ω m x then (1+z)^(markedOpenPivotals G Q x ω m).card else 0) =
-      ∑ i ∈ Finset.range (Q.card+1), z^i*markedMomentValue G Q x i ω m := by
-  by_cases hm : MarkedEvent G ω m x
-  · simp only [markedMomentValue,hm,ite_true]
-    exact choose_generating_sum _ _ (Finset.card_filter_le ..) z
-  · simp only [markedMomentValue,hm,ite_false,mul_zero,Finset.sum_const_zero]
-
-noncomputable def finiteMarkedGenerating (G : BondGraph V E) (Q : Finset E) (F : Finset V)
-    (x : V) (z h p : ℝ) : ℝ :=
-  bernoulliPolynomial Q (fun ω => bernoulliPolynomial F (fun m =>
-    if MarkedEvent G ω m x then (1+z)^(markedOpenPivotals G Q x ω m).card else 0) h) p
-
-noncomputable def finiteMarkedTail (G : BondGraph V E) (Q : Finset E) (F : Finset V)
-    (x : V) (r : ℕ) (h p : ℝ) : ℝ :=
-  bernoulliPolynomial Q (fun ω => bernoulliPolynomial F (fun m =>
-    if MarkedEvent G ω m x ∧ r ≤ (markedOpenPivotals G Q x ω m).card then (1:ℝ) else 0) h) p
-
-theorem finiteMarkedTail_nonneg (G : BondGraph V E) (Q : Finset E) (F : Finset V)
-    (x : V) (r : ℕ) (h p : unitInterval) : 0 ≤ finiteMarkedTail G Q F x r h p := by
-  apply bernoulliPolynomial_nonneg _ _ _ p.2
-  intro ω _
-  apply bernoulliPolynomial_nonneg _ _ _ h.2
-  intro m _
-  split_ifs <;> norm_num
-
-theorem finiteMarkedGenerating_eq (G : BondGraph V E) (Q : Finset E) (F : Finset V)
-    (x : V) (z h p : ℝ) :
-    finiteMarkedGenerating G Q F x z h p =
-      ∑ i ∈ Finset.range (Q.card+1), z^i*finiteMarkedMoment G Q F x i h p := by
-  simp only [finiteMarkedGenerating,markedMomentValue_generating,bernoulliPolynomial_sum,
-    bernoulliPolynomial_const_mul]
-  rfl
-
-theorem half_geometric_sum_le (N : ℕ) :
-    (∑ i ∈ Finset.range N, (1/2:ℝ)^i) ≤ 2 := by
-  have heq : (∑ i ∈ Finset.range N, (1/2:ℝ)^i) = 2*(1-(1/2:ℝ)^N) := by
-    induction N with
-    | zero => simp
-    | succ N ih => rw [Finset.sum_range_succ,ih,pow_succ]; ring
-  rw [heq]
-  have := pow_nonneg (by norm_num : (0:ℝ) ≤ 1/2) N
-  linarith
-
- 
-theorem finiteMarkedGenerating_le [Nonempty V] (G : BondGraph V E) (p h : unitInterval)
-    (hp : (p:ℝ) < 1) (hh : 0 < (h:ℝ)) (d : ℕ) (hd : 0 < d)
-    (Q : Finset E) (F : Finset V) (x : V)
-    (hdeg : ∀ v, (Q.filter (fun e => v ∈ G e)).card ≤ d)
-    (hF : insert x (searchVertices G Q) ⊆ F) (B : ℝ) (hB : 0 < B)
-    (hAB : (Real.sqrt ((d:ℝ)/2)/(1-p))*Real.sqrt (Real.log (1/(h:ℝ))/h) ≤ B) :
-    finiteMarkedGenerating G Q F x (1/(2*B)) h p ≤ 2*finiteMarkedMoment G Q F x 0 h p := by
-  let A : ℝ := (Real.sqrt ((d:ℝ)/2)/(1-p))*Real.sqrt (Real.log (1/(h:ℝ))/h)
-  have hA : 0 ≤ A := mul_nonneg (div_nonneg (Real.sqrt_nonneg _) (sub_nonneg.mpr p.2.2))
-    (Real.sqrt_nonneg _)
-  have hb0 := finiteMarkedMoment_nonneg G Q F x 0 h p
-  rw [finiteMarkedGenerating_eq]
-  calc
-    _ ≤ ∑ i ∈ Finset.range (Q.card+1), (1/(2*B))^i*(B^i*finiteMarkedMoment G Q F x 0 h p) := by
-      apply Finset.sum_le_sum
-      intro i hi
-      apply mul_le_mul_of_nonneg_left _ (pow_nonneg (by positivity) i)
-      apply (finiteMarkedMoment_geometric G p h hp hh d hd i Q F x hdeg hF).trans
-      exact mul_le_mul_of_nonneg_right (pow_le_pow_left₀ hA hAB i) hb0
-    _ = (∑ i ∈ Finset.range (Q.card+1), (1/2:ℝ)^i)*finiteMarkedMoment G Q F x 0 h p := by
-      rw [Finset.sum_mul]
-      apply Finset.sum_congr rfl
-      intro i hi
-      rw [← mul_assoc,← mul_pow]
-      congr 2
-      field_simp
-    _ ≤ 2*finiteMarkedMoment G Q F x 0 h p := mul_le_mul_of_nonneg_right (half_geometric_sum_le _) hb0
-
- 
-theorem finiteMarkedTail_markov (G : BondGraph V E) (p h : unitInterval) (Q : Finset E)
-    (F : Finset V) (x : V) (z : ℝ) (hz : 0 ≤ z) (r : ℕ) :
-    (1+z)^r * finiteMarkedTail G Q F x r h p ≤ finiteMarkedGenerating G Q F x z h p := by
-  unfold finiteMarkedTail finiteMarkedGenerating
-  rw [← bernoulliPolynomial_const_mul]
-  apply bernoulliPolynomial_mono _ _ _ _ p.2
-  intro ω hω
-  rw [← bernoulliPolynomial_const_mul]
-  apply bernoulliPolynomial_mono _ _ _ _ h.2
-  intro m hm
-  by_cases he : MarkedEvent G ω m x
-  · by_cases hc : r ≤ (markedOpenPivotals G Q x ω m).card
-    · simp only [he,hc,and_self,ite_true,mul_one]
-      exact pow_le_pow_right₀ (by linarith : (1:ℝ) ≤ 1+z) hc
-    · simp only [he,hc,and_false,ite_true,ite_false,mul_zero]
-      exact pow_nonneg (by linarith) _
-  · simp only [he,false_and,ite_false,mul_zero,le_refl]
-
- 
-theorem log_one_add_half_inv_ge (B : ℝ) (hB : 1 ≤ B) :
-    1/(4*B) ≤ Real.log (1+1/(2*B)) := by
-  have hBp : 0 < B := by linarith
-  let z : ℝ := 1/(2*B)
-  have hz : 0 ≤ z := by dsimp [z]; positivity
-  have hzle : z ≤ 2 := by dsimp [z]; apply (div_le_iff₀ (by positivity)).mpr; linarith
-  calc
-    1/(4*B) = z/2 := by dsimp [z]; ring
-    _ ≤ 2*z/(z+2) := by
-      apply (le_div_iff₀ (by linarith : 0 < z+2)).mpr
-      have hs := mul_le_mul_of_nonneg_right hzle hz
-      nlinarith
-    _ ≤ Real.log (1+z) := Real.le_log_one_add_of_nonneg hz
-
- 
-theorem finiteMarkedTail_exponential [Nonempty V] (G : BondGraph V E) (p h : unitInterval)
-    (hp : (p:ℝ) < 1) (hh : 0 < (h:ℝ)) (d : ℕ) (hd : 0 < d)
-    (Q : Finset E) (F : Finset V) (x : V)
-    (hdeg : ∀ v, (Q.filter (fun e => v ∈ G e)).card ≤ d)
-    (hF : insert x (searchVertices G Q) ⊆ F) (B : ℝ) (hB : 1 ≤ B)
-    (hAB : (Real.sqrt ((d:ℝ)/2)/(1-p))*Real.sqrt (Real.log (1/(h:ℝ))/h) ≤ B) (r : ℕ) :
-    finiteMarkedTail G Q F x r h p ≤
-      2*Real.exp (-(r:ℝ)/(4*B))*finiteMarkedMoment G Q F x 0 h p := by
-  have hBp : 0 < B := by linarith
-  have hz : 0 ≤ 1/(2*B) := by positivity
-  have hbase : 0 < 1+1/(2*B) := by positivity
-  have hpow : Real.exp ((r:ℝ)/(4*B)) ≤ (1+1/(2*B))^r := by
-    rw [← Real.exp_log (pow_pos hbase r),Real.log_pow]
-    apply Real.exp_le_exp.mpr
-    have hlog := mul_le_mul_of_nonneg_left (log_one_add_half_inv_ge B hB) (Nat.cast_nonneg r : (0:ℝ) ≤ r)
-    simpa only [mul_one_div] using hlog
-  have hm := (finiteMarkedTail_markov G p h Q F x (1/(2*B)) hz r).trans
-    (finiteMarkedGenerating_le G p h hp hh d hd Q F x hdeg hF B hBp hAB)
-  have ht := finiteMarkedTail_nonneg G Q F x r h p
-  have he := (mul_le_mul_of_nonneg_right hpow ht).trans hm
-  calc
-    finiteMarkedTail G Q F x r h p ≤ (2*finiteMarkedMoment G Q F x 0 h p)/Real.exp ((r:ℝ)/(4*B)) :=
-      (le_div_iff₀ (Real.exp_pos _)).mpr (by simpa only [mul_comm] using he)
-    _ = 2*Real.exp (-(r:ℝ)/(4*B))*finiteMarkedMoment G Q F x 0 h p := by
-      rw [neg_div,Real.exp_neg,div_eq_mul_inv]
-      ring
-
-end MarkedExponential
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Exploration.Marked
+
+namespace OAI
+
+/-! Recursive marked moments and exponential estimates. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+section OrderedPivotalMoments
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem sum_choose_rank_suffix {I : Type*} (P : Finset I) (rank : I → ℕ)
+    (hinj : Set.InjOn rank (P : Set I)) (i : ℕ) :
+    (∑ e ∈ P, ((P.filter fun f => rank e < rank f).card).choose i) =
+      P.card.choose (i+1) := by
+  induction P using Finset.strongInductionOn with
+  | _ P ih =>
+    rcases P.eq_empty_or_nonempty with rfl | hP
+    · simp
+    obtain ⟨e,he,hmin⟩ := Finset.exists_min_image P rank hP
+    have hT : P.erase e ⊂ P := Finset.erase_ssubset he
+    have hinjT : Set.InjOn rank (↑(P.erase e) : Set I) := fun a ha b hb =>
+      hinj (Finset.mem_of_mem_erase ha) (Finset.mem_of_mem_erase hb)
+    have hroot : P.filter (fun f => rank e < rank f) = P.erase e := by
+      ext f
+      simp only [Finset.mem_filter,Finset.mem_erase]
+      constructor
+      · rintro ⟨hf,hlt⟩
+        exact ⟨fun h => (Nat.ne_of_lt hlt) (congrArg rank h.symm),hf⟩
+      · rintro ⟨hne,hf⟩
+        exact ⟨hf,lt_of_le_of_ne (hmin f hf) (fun h => hne (hinj he hf h).symm)⟩
+    have hrest (f : I) (hf : f ∈ P.erase e) :
+        P.filter (fun g => rank f < rank g) = (P.erase e).filter (fun g => rank f < rank g) := by
+      ext g
+      simp only [Finset.mem_filter,Finset.mem_erase]
+      constructor
+      · rintro ⟨hg,hfg⟩
+        refine ⟨⟨?_,hg⟩,hfg⟩
+        intro h; subst g
+        exact (not_lt_of_ge (hmin f (Finset.mem_of_mem_erase hf))) hfg
+      · exact fun h => ⟨h.1.2,h.2⟩
+    calc
+      _ = (∑ f ∈ P.erase e, ((P.filter fun g => rank f < rank g).card).choose i) +
+          ((P.filter fun f => rank e < rank f).card).choose i :=
+        (Finset.sum_erase_add _ _ he).symm
+      _ = (∑ f ∈ P.erase e, (((P.erase e).filter fun g => rank f < rank g).card).choose i) +
+          (P.erase e).card.choose i := by
+        rw [hroot]
+        congr 1
+        exact Finset.sum_congr rfl fun f hf => congrArg (fun s : Finset I => s.card.choose i) (hrest f hf)
+      _ = (P.erase e).card.choose (i+1)+(P.erase e).card.choose i := by rw [ih _ hT hinjT]
+      _ = P.card.choose (i+1) := by
+        have hc : (P.erase e).card+1=P.card := by
+          rw [Finset.card_erase_of_mem he]
+          have hp := Finset.card_pos.mpr hP
+          omega
+        rw [← hc,Nat.choose_succ_succ]
+        simp [Nat.succ_eq_add_one,Nat.add_comm]
+
+end OrderedPivotalMoments
+
+section MarkedRecursion
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem cluster_union_external (G : BondGraph V E) (ω η : Set E) (x : V) (W : Set V)
+    (hω : cluster G ω x ⊆ W) (hη : η ⊆ internalEdges G Wᶜ) :
+    cluster G (ω ∪ η) x = cluster G ω x := by
+  apply Set.Subset.antisymm
+  · intro y hy
+    induction hy with
+    | refl => exact conn_refl G ω x
+    | @tail a b hxa hab ih =>
+      obtain ⟨e,he,heq⟩ := hab
+      rcases he with he | he
+      · exact ih.tail ⟨e,he,heq⟩
+      · exact (((mem_internalEdges_of_ends G _ e a b heq).mp (hη he)).1 (hω ih)).elim
+  · exact fun _ h => conn_mono G Set.subset_union_left h
+
+ 
+theorem cluster_union_external_eq_iff (G : BondGraph V E) (ω η : Set E)
+    (x : V) (W : Set V) (hη : η ⊆ internalEdges G Wᶜ) :
+    cluster G (ω ∪ η) x = W ↔ cluster G ω x = W := by
+  constructor
+  · intro h
+    have hω : cluster G ω x ⊆ W := fun y hy => h ▸ conn_mono G Set.subset_union_left hy
+    rwa [cluster_union_external G ω η x W hω hη] at h
+  · intro h
+    rw [cluster_union_external G ω η x W (h ▸ Set.Subset.rfl) hη,h]
+
+ 
+theorem markedEvent_internal_marks (G : BondGraph V E) (ω : Set E) (m W : Set V)
+    (x : V) (hx : x ∈ W) (hω : ω ⊆ internalEdges G W) :
+    MarkedEvent G ω m x ↔ MarkedEvent G ω (m ∩ W) x := by
+  constructor
+  · rintro ⟨y,hy,hxy⟩
+    exact ⟨y,⟨hy,conn_internal_endpoint G W hx (conn_mono G
+      (ω₁ := ω) (ω₂ := ω ∩ internalEdges G W) (fun _ h => ⟨h,hω h⟩) hxy)⟩,hxy⟩
+  · rintro ⟨y,hy,hxy⟩
+    exact ⟨y,hy.1,hxy⟩
+
+theorem openMarkedPivotal_internal_marks (G : BondGraph V E) (ω : Set E)
+    (m W : Set V) (x : V) (hx : x ∈ W) (hω : ω ⊆ internalEdges G W) (e : E) :
+    OpenMarkedPivotal G ω m x e ↔ OpenMarkedPivotal G ω (m ∩ W) x e := by
+  simp only [OpenMarkedPivotal,markedEvent_internal_marks G ω m W x hx hω,
+    markedEvent_internal_marks G (ω \ {e}) m W x hx (Set.Subset.trans Set.sdiff_subset hω)]
+
+ 
+theorem openMarkedPivotal_insert_cut (G : BondGraph V E) (ω : Set E) (m W : Set V)
+    (x : V) {e : E} {a b : V} (hW : cluster G ω x = W)
+    (hab : G e = s(a,b)) (ha : a ∈ W) (hb : b ∉ W) :
+    OpenMarkedPivotal G (insert e ω) m x e ↔
+      Disjoint W m ∧ MarkedEvent G (ω ∩ internalEdges G Wᶜ) (m ∩ Wᶜ) b := by
+  have he : e ∉ ω := fun he => hb (hW ▸ ((edge_cluster_sides G ω x he hab).mp (hW ▸ ha)))
+  have hdel : insert e ω \ {e} = ω := by ext f; simp [he]
+  have hxa : Conn G ω x a := by change a ∈ cluster G ω x; rwa [hW]
+  constructor
+  · intro hp
+    have hdis := pivotal_rootSide_disjoint_marks G (insert e ω) m x hp
+    rw [hdel,hW] at hdis
+    refine ⟨hdis,?_⟩
+    obtain ⟨y,hy,hxy⟩ := hp.2.1
+    rw [conn_insert_iff G ω e a b x y hab] at hxy
+    rcases hxy with hxy | ⟨_,hby⟩ | ⟨hxb,_⟩
+    · exact (Set.disjoint_left.mp hdis (hW ▸ hxy) hy).elim
+    · have hby' := conn_restrict_cluster_compl G ω x (a := b) (hW ▸ hb) hby
+      rw [hW] at hby'
+      exact (markedEvent_internal_marks G _ m Wᶜ b hb Set.inter_subset_right).mp ⟨y,hy,hby'⟩
+    · exact (hb (hW ▸ hxb)).elim
+  · rintro ⟨hdis,y,hy,hby⟩
+    refine ⟨Set.mem_insert _ _,?_,?_⟩
+    · exact ⟨y,hy.1,(conn_mono G (Set.subset_insert e ω) hxa).tail
+        ⟨e,Set.mem_insert _ _,hab⟩ |>.trans (conn_mono G
+          (fun _ h => Set.mem_insert_of_mem _ h.1) hby)⟩
+    · rw [hdel]
+      rintro ⟨z,hz,hxz⟩
+      exact Set.disjoint_left.mp hdis (hW ▸ hxz) hz
+
+@[simp] theorem mem_markedOpenPivotals (G : BondGraph V E) (Q : Finset E) (x : V)
+    (ω : Finset E) (m : Finset V) (e : E) :
+    e ∈ markedOpenPivotals G Q x ω m ↔ e ∈ Q ∧ OpenMarkedPivotal G ω m x e := by
+  simp only [markedOpenPivotals,Finset.mem_filter,OpenMarkedPivotal,MarkedEvent,Finset.coe_erase,
+    Finset.mem_coe]
+
+ 
+noncomputable def markedMomentValue (G : BondGraph V E) (Q : Finset E) (x : V)
+    (i : ℕ) (ω : Finset E) (m : Finset V) : ℝ :=
+  if MarkedEvent G ω m x then ((markedOpenPivotals G Q x ω m).card.choose i : ℝ) else 0
+
+ 
+theorem markedMomentValue_succ (G : BondGraph V E) (Q : Finset E) (x : V)
+    (i : ℕ) (ω : Finset E) (m : Finset V) :
+    markedMomentValue G Q x (i+1) ω m =
+      ∑ e ∈ markedOpenPivotals G Q x ω m,
+        (((markedOpenPivotals G Q x ω m).filter
+          (fun f => f ∈ internalEdges G (cluster G (↑ω \ {e}) x)ᶜ)).card.choose i : ℝ) := by
+  have hinj : Set.InjOn (pivotalRank G ω x) (↑(markedOpenPivotals G Q x ω m) : Set E) := by
+    intro e he f hf
+    exact pivotalRank_injective G ω.finite_toSet m x
+      ((mem_markedOpenPivotals G Q x ω m e).mp he).2
+      ((mem_markedOpenPivotals G Q x ω m f).mp hf).2
+  have hs := sum_choose_rank_suffix (markedOpenPivotals G Q x ω m) (pivotalRank G ω x) hinj i
+  have heq (e : E) (he : e ∈ markedOpenPivotals G Q x ω m) :
+      (markedOpenPivotals G Q x ω m).filter (fun f => f ∈ internalEdges G (cluster G (↑ω \ {e}) x)ᶜ) =
+      (markedOpenPivotals G Q x ω m).filter (fun f => pivotalRank G ω x e < pivotalRank G ω x f) := by
+    apply Finset.filter_congr
+    intro f hf
+    exact (pivotalRank_lt_iff G ω.finite_toSet m x
+      ((mem_markedOpenPivotals G Q x ω m e).mp he).2
+      ((mem_markedOpenPivotals G Q x ω m f).mp hf).2).symm
+  by_cases h : MarkedEvent G ω m x
+  · simp only [markedMomentValue,h,ite_true]
+    rw [← hs,Nat.cast_sum]
+    apply Finset.sum_congr rfl
+    intro e he
+    rw [heq e he]
+  · have hempty : markedOpenPivotals G Q x ω m = ∅ := by
+      apply Finset.eq_empty_iff_forall_notMem.mpr
+      intro e he
+      exact h ((mem_markedOpenPivotals G Q x ω m e).mp he).2.2.1
+    simp [markedMomentValue,h,hempty]
+
+ 
+theorem conn_mem_searchSupport (G : BondGraph V E) (Q : Finset E) (x : V)
+    {ω : Set E} (hω : ω ⊆ Q) {y : V} (hxy : Conn G ω x y) :
+    y ∈ insert x (searchVertices G Q) := by
+  induction hxy with
+  | refl => exact Finset.mem_insert_self _ _
+  | @tail a b hxa hab ih =>
+    obtain ⟨e,he,heq⟩ := hab
+    exact Finset.mem_insert_of_mem (mem_searchVertices G (hω he) (by simp [heq]))
+
+theorem markedEvent_searchSupport (G : BondGraph V E) (Q : Finset E) (x : V)
+    {ω : Set E} (hω : ω ⊆ Q) (m : Set V) :
+    MarkedEvent G ω m x ↔ MarkedEvent G ω (m ∩ ↑(insert x (searchVertices G Q))) x := by
+  constructor
+  · rintro ⟨y,hy,hxy⟩
+    exact ⟨y,⟨hy,conn_mem_searchSupport G Q x hω hxy⟩,hxy⟩
+  · rintro ⟨y,hy,hxy⟩
+    exact ⟨y,hy.1,hxy⟩
+
+theorem markedOpenPivotals_searchSupport (G : BondGraph V E) (Q : Finset E) (x : V)
+    (ω : Finset E) (hω : ω ⊆ Q) (m : Finset V) :
+    markedOpenPivotals G Q x ω m =
+      markedOpenPivotals G Q x ω (m ∩ insert x (searchVertices G Q)) := by
+  ext e
+  simp only [mem_markedOpenPivotals,OpenMarkedPivotal,Finset.coe_inter]
+  rw [markedEvent_searchSupport G Q x (by exact_mod_cast hω),
+    markedEvent_searchSupport G Q x (ω := (ω : Set E) \ {e})
+      (fun _ h => hω h.1)]
+
+theorem markedMomentValue_searchSupport (G : BondGraph V E) (Q : Finset E) (x : V)
+    (i : ℕ) (ω : Finset E) (hω : ω ⊆ Q) (m : Finset V) :
+    markedMomentValue G Q x i ω m =
+      markedMomentValue G Q x i ω (m ∩ insert x (searchVertices G Q)) := by
+  unfold markedMomentValue
+  rw [markedOpenPivotals_searchSupport G Q x ω hω m]
+  simp only [Finset.coe_inter]
+  rw [markedEvent_searchSupport G Q x (by exact_mod_cast hω)]
+
+@[simp] theorem markedMomentValue_zero (G : BondGraph V E) (Q : Finset E) (x : V)
+    (ω : Finset E) (m : Finset V) :
+    markedMomentValue G Q x 0 ω m = if MarkedEvent G ω m x then 1 else 0 := by
+  simp only [markedMomentValue,Nat.choose_zero_right,Nat.cast_one]
+
+@[simp] theorem markedMomentValue_one (G : BondGraph V E) (Q : Finset E) (x : V)
+    (ω : Finset E) (m : Finset V) :
+    markedMomentValue G Q x 1 ω m = ((markedOpenPivotals G Q x ω m).card : ℝ) := by
+  by_cases h : MarkedEvent G ω m x
+  · simp [markedMomentValue,h]
+  · have he : markedOpenPivotals G Q x ω m = ∅ := by
+      apply Finset.eq_empty_iff_forall_notMem.mpr
+      intro e he
+      exact h ((mem_markedOpenPivotals G Q x ω m e).mp he).2.2.1
+    simp [markedMomentValue,h,he]
+
+ 
+noncomputable def finiteMarkedMoment (G : BondGraph V E) (Q : Finset E) (F : Finset V)
+    (x : V) (i : ℕ) (h p : ℝ) : ℝ :=
+  bernoulliPolynomial Q (fun ω => bernoulliPolynomial F (markedMomentValue G Q x i ω) h) p
+
+theorem finiteMarkedMoment_support (G : BondGraph V E) (Q : Finset E) (F : Finset V)
+    (x : V) (i : ℕ) (h p : ℝ) (hF : insert x (searchVertices G Q) ⊆ F) :
+    finiteMarkedMoment G Q F x i h p =
+      finiteMarkedMoment G Q (insert x (searchVertices G Q)) x i h p := by
+  apply bernoulliPolynomial_congr
+  intro ω hω
+  rw [← bernoulliPolynomial_restrict _ F hF]
+  apply bernoulliPolynomial_congr
+  intro m _
+  exact markedMomentValue_searchSupport G Q x i ω hω m
+
+ 
+theorem finiteMarkedMoment_first_bound (G : BondGraph V E) (Q : Finset E) (F : Finset V)
+    (x : V) (p h : unitInterval) (hp : (p:ℝ) < 1) (hh : 0 < (h:ℝ)) (d : ℕ) (hd : 0 < d)
+    (hdeg : ∀ v, (Q.filter (fun e => v ∈ G e)).card ≤ d)
+    (hF : insert x (searchVertices G Q) ⊆ F) :
+    finiteMarkedMoment G Q F x 1 h p ≤
+      (Real.sqrt ((d:ℝ)/2)/(1-p))*Real.sqrt (Real.log (1/(h:ℝ))/h)*
+        finiteMarkedMoment G Q F x 0 h p := by
+  rw [finiteMarkedMoment_support G Q F x 1 h p hF,
+    finiteMarkedMoment_support G Q F x 0 h p hF]
+  simpa only [finiteMarkedMoment,markedMomentValue_one,markedMomentValue_zero,MarkedEvent,
+    markedPivotalMomentOne,markedConnectionPolynomial,bernoulliPolynomial,Finset.mem_coe] using
+    marked_pivotal_first_moment G Q x p h hp hh d hd hdeg
+
+noncomputable def exteriorBonds (G : BondGraph V E) (Q : Finset E) (W : Set V) : Finset E :=
+  Q.filter fun e => e ∈ internalEdges G Wᶜ
+
+noncomputable def exteriorMarks (m : Finset V) (W : Set V) : Finset V :=
+  m.filter fun x => x ∉ W
+
+@[simp] theorem coe_exteriorBonds (G : BondGraph V E) (Q : Finset E) (W : Set V) :
+    (exteriorBonds G Q W : Set E) = (Q : Set E) ∩ internalEdges G Wᶜ := by
+  ext e; simp [exteriorBonds]
+
+@[simp] theorem coe_exteriorMarks (m : Finset V) (W : Set V) :
+    (exteriorMarks m W : Set V) = (m : Set V) ∩ Wᶜ := by
+  ext x; simp [exteriorMarks]
+
+ 
+theorem markedOpenPivotals_exterior [Nonempty V] (G : BondGraph V E) (Q : Finset E)
+    (x : V) (ω : Finset E) (m : Finset V) {e : E}
+    (he : OpenMarkedPivotal G ω m x e) :
+    markedOpenPivotals G (exteriorBonds G Q (cluster G (↑ω \ {e}) x))
+      (cutOutside G (cluster G (↑ω \ {e}) x) e)
+      (exteriorBonds G ω (cluster G (↑ω \ {e}) x))
+      (exteriorMarks m (cluster G (↑ω \ {e}) x)) =
+      (markedOpenPivotals G Q x ω m).filter
+        (fun f => f ∈ internalEdges G (cluster G (↑ω \ {e}) x)ᶜ) := by
+  obtain ⟨a,b,hab,hxa,hxb,hbm⟩ := openMarkedPivotal_oriented G ω m x he
+  have hb := (cut_unique G (cluster G (↑ω \ {e}) x) hab hxa hxb).2
+  rw [← hb]
+  ext f
+  simp only [mem_markedOpenPivotals,exteriorBonds,Finset.mem_filter,coe_exteriorMarks,
+    Finset.coe_filter]
+  have hmarks := openMarkedPivotal_internal_marks G
+    ((ω : Set E) ∩ internalEdges G (cluster G (↑ω \ {e}) x)ᶜ)
+    m (cluster G (↑ω \ {e}) x)ᶜ b hxb Set.inter_subset_right f
+  have hiff := openMarkedPivotal_exterior_iff G ω m x he hab hxa hxb hbm f
+  change (f ∈ Q ∧ f ∈ internalEdges G (cluster G (↑ω \ {e}) x)ᶜ) ∧
+    OpenMarkedPivotal G ((ω : Set E) ∩ internalEdges G (cluster G (↑ω \ {e}) x)ᶜ)
+      ((m : Set V) ∩ (cluster G (↑ω \ {e}) x)ᶜ) b f ↔ _
+  rw [← hmarks,hiff]
+  tauto
+
+ 
+theorem markedMomentValue_exterior [Nonempty V] (G : BondGraph V E) (Q : Finset E)
+    (x : V) (i : ℕ) (ω : Finset E) (m : Finset V) {e : E}
+    (he : OpenMarkedPivotal G ω m x e) :
+    markedMomentValue G (exteriorBonds G Q (cluster G (↑ω \ {e}) x))
+      (cutOutside G (cluster G (↑ω \ {e}) x) e) i
+      (exteriorBonds G ω (cluster G (↑ω \ {e}) x))
+      (exteriorMarks m (cluster G (↑ω \ {e}) x)) =
+      (((markedOpenPivotals G Q x ω m).filter
+        (fun f => f ∈ internalEdges G (cluster G (↑ω \ {e}) x)ᶜ)).card.choose i : ℝ) := by
+  have hsuccess : MarkedEvent G (exteriorBonds G ω (cluster G (↑ω \ {e}) x))
+      (exteriorMarks m (cluster G (↑ω \ {e}) x))
+      (cutOutside G (cluster G (↑ω \ {e}) x) e) := by
+    obtain ⟨a,b,hab,hxa,hxb,y,hy,hby⟩ := openMarkedPivotal_oriented G ω m x he
+    rw [← (cut_unique G (cluster G (↑ω \ {e}) x) hab hxa hxb).2,
+      coe_exteriorBonds,coe_exteriorMarks]
+    apply (markedEvent_internal_marks G _ m (cluster G (↑ω \ {e}) x)ᶜ b hxb
+      Set.inter_subset_right).mp
+    refine ⟨y,hy,conn_mono G ?_ (conn_restrict_cluster_compl G (↑ω \ {e}) x hxb hby)⟩
+    exact fun _ h => ⟨h.1.1,h.2⟩
+  rw [markedMomentValue,ite_eq_left hsuccess,markedOpenPivotals_exterior G Q x ω m he]
+
+ 
+theorem markedMomentValue_succ_exterior [Nonempty V] (G : BondGraph V E) (Q : Finset E)
+    (x : V) (i : ℕ) (ω : Finset E) (m : Finset V) :
+    markedMomentValue G Q x (i+1) ω m = ∑ e ∈ Q,
+      if OpenMarkedPivotal G ω m x e then
+        markedMomentValue G (exteriorBonds G Q (cluster G (↑ω \ {e}) x))
+          (cutOutside G (cluster G (↑ω \ {e}) x) e) i
+          (exteriorBonds G ω (cluster G (↑ω \ {e}) x))
+          (exteriorMarks m (cluster G (↑ω \ {e}) x)) else 0 := by
+  rw [markedMomentValue_succ]
+  trans ∑ e ∈ markedOpenPivotals G Q x ω m,
+    markedMomentValue G (exteriorBonds G Q (cluster G (↑ω \ {e}) x))
+      (cutOutside G (cluster G (↑ω \ {e}) x) e) i
+      (exteriorBonds G ω (cluster G (↑ω \ {e}) x))
+      (exteriorMarks m (cluster G (↑ω \ {e}) x))
+  · apply Finset.sum_congr rfl
+    intro e he
+    exact (markedMomentValue_exterior G Q x i ω m
+      ((mem_markedOpenPivotals G Q x ω m e).mp he).2).symm
+  · have heq : markedOpenPivotals G Q x ω m = Q.filter (fun e => OpenMarkedPivotal G ω m x e) := by
+      ext e; simp only [mem_markedOpenPivotals,Finset.mem_filter]
+    rw [heq,Finset.sum_filter]
+
+end MarkedRecursion
+
+section CutFactor
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Nonempty V]
+
+noncomputable def cutMomentValue (G : BondGraph V E) (Q : Finset E) (x : V)
+    (e : E) (W : Finset V) (i : ℕ) (ω : Finset E) (m : Finset V) : ℝ :=
+  if cluster G ω x = (W : Set V) then
+    if Disjoint (W : Set V) (m : Set V) then
+      markedMomentValue G (exteriorBonds G Q W) (cutOutside G W e) i
+        (exteriorBonds G ω W) (exteriorMarks m W)
+    else 0
+  else 0
+
+omit [Nonempty V] in
+theorem cut_not_exterior (G : BondGraph V E) (W : Set V) {e : E}
+    (hc : CutExists G W e) : e ∉ internalEdges G Wᶜ := by
+  obtain ⟨a,b,hab,ha,hb⟩ := hc
+  intro he
+  exact ((mem_internalEdges_of_ends G Wᶜ e a b hab).mp he).1 ha
+
+omit [Nonempty V] in
+theorem exteriorBonds_subset_erase (G : BondGraph V E) (Q : Finset E)
+    (W : Set V) {e : E} (hc : CutExists G W e) : exteriorBonds G Q W ⊆ Q.erase e := by
+  intro f hf
+  obtain ⟨hf,hout⟩ := Finset.mem_filter.mp hf
+  exact Finset.mem_erase.mpr ⟨fun h => cut_not_exterior G W hc (h ▸ hout),hf⟩
+
+ 
+theorem cutMomentValue_blocks (G : BondGraph V E) (Q : Finset E) (F W : Finset V)
+    (x : V) (e : E) (i : ℕ) (a b : Finset E) (u v : Finset V)
+    (ha : a ⊆ (Q.erase e) \ exteriorBonds G Q W)
+    (hb : b ⊆ exteriorBonds G Q W) (hu : u ⊆ W) (hv : v ⊆ F \ W) :
+    cutMomentValue G Q x e W i (a ∪ b) (u ∪ v) =
+      (if cluster G a x = (W : Set V) then (1:ℝ) else 0) *
+      (if u = ∅ then (1:ℝ) else 0) *
+      markedMomentValue G (exteriorBonds G Q W) (cutOutside G W e) i b v := by
+  have hbout : (b : Set E) ⊆ internalEdges G (W : Set V)ᶜ :=
+    fun f hf => (Finset.mem_filter.mp (hb hf)).2
+  have hc := cluster_union_external_eq_iff G (a : Set E) b x W hbout
+  have he : exteriorBonds G (a ∪ b) W = b := by
+    ext f
+    simp only [exteriorBonds,Finset.mem_filter,Finset.mem_union]
+    constructor
+    · rintro ⟨hf,hout⟩
+      rcases hf with hf | hf
+      · have hfa := Finset.mem_sdiff.mp (ha hf)
+        exact (hfa.2 (Finset.mem_filter.mpr ⟨Finset.mem_of_mem_erase hfa.1,hout⟩)).elim
+      · exact hf
+    · exact fun hf => ⟨Or.inr hf,hbout hf⟩
+  have hm : exteriorMarks (u ∪ v) W = v := by
+    ext y
+    simp only [exteriorMarks,Finset.mem_filter,Finset.mem_union,Finset.mem_coe]
+    constructor
+    · rintro ⟨hy,hyW⟩
+      rcases hy with hy | hy
+      · exact (hyW (hu hy)).elim
+      · exact hy
+    · intro hy
+      exact ⟨Or.inr hy,(Finset.mem_sdiff.mp (hv hy)).2⟩
+  have hd : Disjoint (W : Set V) ((u ∪ v : Finset V) : Set V) ↔ u = ∅ := by
+    constructor
+    · intro h
+      apply Finset.eq_empty_iff_forall_notMem.mpr
+      intro y hy
+      exact Set.disjoint_left.mp h (hu hy) (Finset.mem_union_left v hy)
+    · rintro rfl
+      apply Set.disjoint_left.mpr
+      intro y hy hyv
+      have hyv' : y ∈ v := by simpa using hyv
+      exact (Finset.mem_sdiff.mp (hv hyv')).2 hy
+  rw [Finset.coe_union] at hd
+  simp only [cutMomentValue,Finset.coe_union,hc,hd,he,hm]
+  split_ifs <;> simp_all
+
+omit [Nonempty V] in
+ 
+theorem bernoulliPolynomial_empty_indicator (F : Finset V) (h : ℝ) :
+    bernoulliPolynomial F (fun m => if m = ∅ then (1:ℝ) else 0) h = (1-h)^F.card := by
+  unfold bernoulliPolynomial
+  rw [Finset.sum_eq_single ∅]
+  · simp
+  · intro m hm hne
+    simp [hne]
+  · simp
+
+ 
+theorem cutMoment_factor (G : BondGraph V E) (Q : Finset E) (F W : Finset V)
+    (x : V) (e : E) (i : ℕ) (h p : ℝ) (hW : W ⊆ F) (hc : CutExists G W e) :
+    bernoulliPolynomial (Q.erase e) (fun ω => bernoulliPolynomial F
+      (cutMomentValue G Q x e W i ω) h) p =
+      bernoulliPolynomial ((Q.erase e) \ exteriorBonds G Q W)
+        (fun a => if cluster G a x = (W : Set V) then (1:ℝ) else 0) p *
+      (1-h)^W.card * finiteMarkedMoment G (exteriorBonds G Q W) (F \ W)
+        (cutOutside G W e) i h p := by
+  let R := exteriorBonds G Q W
+  let A := (Q.erase e) \ R
+  let B := F \ W
+  have hR : R ⊆ Q.erase e := exteriorBonds_subset_erase G Q W hc
+  have hQR : Q.erase e = A ∪ R := by
+    dsimp [A]; ext f; simp only [Finset.mem_union,Finset.mem_sdiff]
+    tauto
+  have hAR : Disjoint A R := Finset.disjoint_sdiff.symm
+  have hFB : F = W ∪ B := (Finset.union_sdiff_of_subset hW).symm
+  have hWB : Disjoint W B := Finset.disjoint_sdiff
+  conv_lhs => rw [hQR,bernoulliPolynomial_union A R hAR]
+  trans bernoulliPolynomial A (fun a =>
+    (if cluster G a x = (W : Set V) then (1:ℝ) else 0) *
+      ((1-h)^W.card * finiteMarkedMoment G R B (cutOutside G W e) i h p)) p
+  · apply bernoulliPolynomial_congr
+    intro a ha
+    trans bernoulliPolynomial R (fun b =>
+      (if cluster G a x = (W : Set V) then (1:ℝ) else 0) * (1-h)^W.card *
+        bernoulliPolynomial B (markedMomentValue G R (cutOutside G W e) i b) h) p
+    · apply bernoulliPolynomial_congr
+      intro b hb
+      rw [hFB,bernoulliPolynomial_union W B hWB]
+      trans bernoulliPolynomial W (fun u =>
+        (if cluster G a x = (W : Set V) then (1:ℝ) else 0) *
+        (if u = ∅ then (1:ℝ) else 0) *
+        bernoulliPolynomial B (markedMomentValue G R (cutOutside G W e) i b) h) h
+      · apply bernoulliPolynomial_congr
+        intro u hu
+        rw [← bernoulliPolynomial_const_mul]
+        apply bernoulliPolynomial_congr
+        intro v hv
+        exact cutMomentValue_blocks G Q F W x e i a b u v ha hb hu hv
+      · rw [bernoulliPolynomial_mul_const,bernoulliPolynomial_const_mul,
+          bernoulliPolynomial_empty_indicator]
+    · rw [bernoulliPolynomial_const_mul]
+      exact mul_assoc _ _ _
+  · rw [bernoulliPolynomial_mul_const]
+    exact (mul_assoc _ _ _).symm
+
+end CutFactor
+
+section PointwiseRecursion
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Nonempty V]
+
+noncomputable def pivotalSuffixValue (G : BondGraph V E) (Q : Finset E) (x : V)
+    (e : E) (i : ℕ) (ω : Finset E) (m : Finset V) : ℝ :=
+  if OpenMarkedPivotal G ω m x e then
+    markedMomentValue G (exteriorBonds G Q (cluster G (↑ω \ {e}) x))
+      (cutOutside G (cluster G (↑ω \ {e}) x) e) i
+      (exteriorBonds G ω (cluster G (↑ω \ {e}) x))
+      (exteriorMarks m (cluster G (↑ω \ {e}) x)) else 0
+
+ 
+theorem pivotalSuffixValue_insert (G : BondGraph V E) (Q : Finset E) (x : V)
+    (e : E) (i : ℕ) (ω : Finset E) (he : e ∉ ω) (m : Finset V) :
+    pivotalSuffixValue G Q x e i (insert e ω) m =
+      if CutExists G (cluster G ω x) e then
+        if Disjoint (cluster G ω x) (m : Set V) then
+          markedMomentValue G (exteriorBonds G Q (cluster G ω x))
+            (cutOutside G (cluster G ω x) e) i
+            (exteriorBonds G ω (cluster G ω x)) (exteriorMarks m (cluster G ω x))
+        else 0
+      else 0 := by
+  have hdel : ((insert e ω : Finset E) : Set E) \ {e} = (ω : Set E) := by
+    simp [he]
+  by_cases hc : CutExists G (cluster G ω x) e
+  · obtain ⟨hab,ha,hb⟩ := cut_spec G (cluster G ω x) hc
+    have heout := cut_not_exterior G (cluster G ω x) hc
+    have hbe : exteriorBonds G (insert e ω) (cluster G ω x) = exteriorBonds G ω (cluster G ω x) := by
+      ext f
+      simp only [exteriorBonds,Finset.mem_filter,Finset.mem_insert]
+      constructor
+      · rintro ⟨hf,hout⟩
+        rcases hf with rfl | hf
+        · exact (heout hout).elim
+        · exact ⟨hf,hout⟩
+      · exact fun h => ⟨Or.inr h.1,h.2⟩
+    have hp := openMarkedPivotal_insert_cut G (ω : Set E) m (cluster G ω x) x rfl hab ha hb
+    have hps : OpenMarkedPivotal G (insert e ω : Finset E) m x e ↔
+        Disjoint (cluster G ω x) (m : Set V) ∧
+        MarkedEvent G (exteriorBonds G ω (cluster G ω x))
+          (exteriorMarks m (cluster G ω x)) (cutOutside G (cluster G ω x) e) := by
+      simpa only [Finset.coe_insert,coe_exteriorBonds,coe_exteriorMarks] using hp
+    simp only [pivotalSuffixValue,hdel,hbe,hc,ite_true,hps]
+    by_cases hd : Disjoint (cluster G ω x) (m : Set V) <;>
+      by_cases hm : MarkedEvent G (exteriorBonds G ω (cluster G ω x))
+        (exteriorMarks m (cluster G ω x)) (cutOutside G (cluster G ω x) e) <;>
+      simp only [markedMomentValue,hd,hm,and_true,and_false,ite_true,ite_false]
+  · have hn : ¬OpenMarkedPivotal G (insert e ω : Finset E) m x e := by
+      intro hp
+      obtain ⟨a,b,hab,hxa,hxb,_⟩ := openMarkedPivotal_oriented G (insert e ω : Finset E) m x hp
+      rw [hdel] at hxa hxb
+      exact hc ⟨a,b,hab,hxa,hxb⟩
+    simp only [pivotalSuffixValue,hn,hc,ite_false]
+
+noncomputable def finiteRootComponent (G : BondGraph V E) (ω : Finset E) (x : V) : Finset V :=
+  (cluster_finite_of_finite_edges G ω.finite_toSet x).toFinset
+
+omit [Nonempty V] in
+@[simp]
+theorem coe_finiteRootComponent (G : BondGraph V E) (ω : Finset E) (x : V) :
+    (finiteRootComponent G ω x : Set V) = cluster G ω x := by simp [finiteRootComponent]
+
+omit [Nonempty V] in
+theorem finiteRootComponent_subset (G : BondGraph V E) (Q : Finset E) (F : Finset V)
+    (x : V) (ω : Finset E) (hω : ω ⊆ Q) (hF : insert x (searchVertices G Q) ⊆ F) :
+    finiteRootComponent G ω x ⊆ F := by
+  intro y hy
+  have hxy : Conn G (ω : Set E) x y := by
+    change y ∈ (finiteRootComponent G ω x : Set V) at hy
+    rwa [coe_finiteRootComponent] at hy
+  exact hF (conn_mem_searchSupport G Q x (by exact_mod_cast hω) hxy)
+
+omit [Nonempty V] in
+ 
+theorem sum_component_partition (G : BondGraph V E) (Q : Finset E) (F : Finset V)
+    (x : V) (ω : Finset E) (hω : ω ⊆ Q) (hF : insert x (searchVertices G Q) ⊆ F)
+    (f : Finset V → ℝ) :
+    (∑ W ∈ F.powerset, if cluster G ω x = (W : Set V) then f W else 0) =
+      f (finiteRootComponent G ω x) := by
+  have hmem : finiteRootComponent G ω x ∈ F.powerset :=
+    Finset.mem_powerset.mpr (finiteRootComponent_subset G Q F x ω hω hF)
+  rw [Finset.sum_eq_single_of_mem _ hmem]
+  · simp
+  · intro W hW hne
+    have hn : cluster G ω x ≠ (W : Set V) := by
+      intro h
+      apply hne
+      apply Finset.coe_injective
+      exact h.symm.trans (coe_finiteRootComponent G ω x).symm
+    simp [hn]
+
+ 
+theorem pivotalSuffixValue_component_sum (G : BondGraph V E) (Q : Finset E) (F : Finset V)
+    (x : V) (e : E) (i : ℕ) (ω : Finset E) (he : e ∉ ω) (hω : ω ⊆ Q)
+    (hF : insert x (searchVertices G Q) ⊆ F) (m : Finset V) :
+    pivotalSuffixValue G Q x e i (insert e ω) m = ∑ W ∈ F.powerset,
+      if CutExists G W e then cutMomentValue G Q x e W i ω m else 0 := by
+  rw [pivotalSuffixValue_insert G Q x e i ω he m]
+  symm
+  trans ∑ W ∈ F.powerset, if cluster G ω x = (W : Set V) then
+    (if CutExists G W e then
+      if Disjoint (W : Set V) (m : Set V) then
+        markedMomentValue G (exteriorBonds G Q W) (cutOutside G W e) i
+          (exteriorBonds G ω W) (exteriorMarks m W) else 0
+      else 0) else 0
+  · apply Finset.sum_congr rfl
+    intro W _
+    by_cases hc : CutExists G W e <;> by_cases hw : cluster G ω x = (W : Set V) <;>
+      simp [cutMomentValue,hc,hw]
+  · rw [sum_component_partition G Q F x ω hω hF]
+    simp only [coe_finiteRootComponent]
+
+omit [Nonempty V] in
+ 
+theorem bernoulliPolynomial_open_gate (Q : Finset E) (e : E) (he : e ∈ Q)
+    (f : Finset E → ℝ) (p : ℝ) (hf : ∀ ω ⊆ Q, e ∉ ω → f ω = 0) :
+    bernoulliPolynomial Q f p =
+      p*bernoulliPolynomial (Q.erase e) (fun ω => f (insert e ω)) p := by
+  conv_lhs => rw [← Finset.insert_erase he]
+  rw [bernoulliPolynomial_insert _ e (Finset.notMem_erase e Q)]
+  have hz : bernoulliPolynomial (Q.erase e) f p = 0 := by
+    rw [← bernoulliPolynomial_zero (Q.erase e) p]
+    apply bernoulliPolynomial_congr
+    intro ω hω
+    exact hf ω (Finset.Subset.trans hω (Finset.erase_subset e Q))
+      (fun h => Finset.notMem_erase e Q (hω h))
+  rw [hz,mul_zero,zero_add]
+
+end PointwiseRecursion
+
+section FiniteMomentRecursion
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Nonempty V]
+
+ 
+noncomputable def cutCoefficient (G : BondGraph V E) (Q : Finset E) (x : V)
+    (e : E) (W : Finset V) (h p : ℝ) : ℝ :=
+  bernoulliPolynomial ((Q.erase e) \ exteriorBonds G Q W)
+    (fun a => if cluster G a x = (W : Set V) then (1:ℝ) else 0) p * (1-h)^W.card
+
+ 
+theorem finiteMarkedMoment_recursion (G : BondGraph V E) (Q : Finset E) (F : Finset V)
+    (x : V) (i : ℕ) (h p : ℝ) (hF : insert x (searchVertices G Q) ⊆ F) :
+    finiteMarkedMoment G Q F x (i+1) h p = p * ∑ e ∈ Q, ∑ W ∈ F.powerset,
+      if CutExists G W e then cutCoefficient G Q x e W h p *
+        finiteMarkedMoment G (exteriorBonds G Q W) (F \ W) (cutOutside G W e) i h p else 0 := by
+  have hsum : finiteMarkedMoment G Q F x (i+1) h p =
+      ∑ e ∈ Q, bernoulliPolynomial Q (fun ω => bernoulliPolynomial F
+        (pivotalSuffixValue G Q x e i ω) h) p := by
+    change bernoulliPolynomial Q (fun ω => bernoulliPolynomial F
+      (fun m => markedMomentValue G Q x (i+1) ω m) h) p = _
+    simp only [markedMomentValue_succ_exterior,bernoulliPolynomial_sum]
+    rfl
+  rw [hsum,Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro e he
+  rw [bernoulliPolynomial_open_gate Q e he]
+  · congr 1
+    trans bernoulliPolynomial (Q.erase e) (fun ω => bernoulliPolynomial F
+      (fun m => ∑ W ∈ F.powerset, if CutExists G W e then cutMomentValue G Q x e W i ω m else 0) h) p
+    · apply bernoulliPolynomial_congr
+      intro ω hω
+      apply bernoulliPolynomial_congr
+      intro m _
+      exact pivotalSuffixValue_component_sum G Q F x e i ω
+        (fun h => Finset.notMem_erase e Q (hω h))
+        (Finset.Subset.trans hω (Finset.erase_subset e Q)) hF m
+    · simp only [bernoulliPolynomial_sum]
+      apply Finset.sum_congr rfl
+      intro W hW
+      by_cases hc : CutExists G W e
+      · simp only [hc,ite_true]
+        exact cutMoment_factor G Q F W x e i h p (Finset.mem_powerset.mp hW) hc
+      · simp only [hc,ite_false,bernoulliPolynomial_zero]
+  · intro ω hω heω
+    rw [← bernoulliPolynomial_zero F h]
+    apply bernoulliPolynomial_congr
+    intro m _
+    have hpiv : ¬OpenMarkedPivotal G ω m x e := fun h => heω h.1
+    simp only [pivotalSuffixValue,hpiv,ite_false]
+
+omit [Nonempty V] in
+theorem cutCoefficient_nonneg (G : BondGraph V E) (Q : Finset E) (x : V)
+    (e : E) (W : Finset V) (h p : unitInterval) : 0 ≤ cutCoefficient G Q x e W h p := by
+  apply mul_nonneg
+  · exact bernoulliPolynomial_nonneg _ _ _ p.2 (fun _ _ => by split_ifs <;> norm_num)
+  · exact pow_nonneg (sub_nonneg.mpr h.2.2) _
+
+omit [Nonempty V] in
+theorem finiteMarkedMoment_nonneg (G : BondGraph V E) (Q : Finset E) (F : Finset V)
+    (x : V) (i : ℕ) (h p : unitInterval) : 0 ≤ finiteMarkedMoment G Q F x i h p := by
+  apply bernoulliPolynomial_nonneg _ _ _ p.2
+  intro ω _
+  apply bernoulliPolynomial_nonneg _ _ _ h.2
+  intro m _
+  unfold markedMomentValue
+  split_ifs <;> positivity
+
+ 
+theorem exterior_mark_support (G : BondGraph V E) (Q : Finset E) (F W : Finset V)
+    (x : V) (e : E) (he : e ∈ Q) (hc : CutExists G W e)
+    (hF : insert x (searchVertices G Q) ⊆ F) :
+    insert (cutOutside G W e) (searchVertices G (exteriorBonds G Q W)) ⊆ F \ W := by
+  have hs := cut_spec G W hc
+  intro y hy
+  rcases Finset.mem_insert.mp hy with rfl | hy
+  · exact Finset.mem_sdiff.mpr ⟨hF (Finset.mem_insert_of_mem
+      (mem_searchVertices G he (by simp [hs.1]))),hs.2.2⟩
+  · obtain ⟨f,hf,hyf⟩ := Finset.mem_biUnion.mp hy
+    have hfQ := (Finset.mem_filter.mp hf).1
+    have hfo := (Finset.mem_filter.mp hf).2
+    have hyf' := Sym2.mem_toFinset.mp hyf
+    exact Finset.mem_sdiff.mpr ⟨hF (Finset.mem_insert_of_mem
+      (mem_searchVertices G hfQ hyf')),hfo y hyf'⟩
+
+omit [Nonempty V] in
+ 
+theorem exterior_degree_bound (G : BondGraph V E) (Q : Finset E) (W : Set V)
+    (d : ℕ) (hd : ∀ v, (Q.filter (fun e => v ∈ G e)).card ≤ d) :
+    ∀ v, ((exteriorBonds G Q W).filter (fun e => v ∈ G e)).card ≤ d := by
+  intro v
+  apply le_trans (Finset.card_le_card ?_) (hd v)
+  intro e he
+  exact Finset.mem_filter.mpr ⟨(Finset.mem_filter.mp (Finset.mem_filter.mp he).1).1,
+    (Finset.mem_filter.mp he).2⟩
+
+ 
+theorem finiteMarkedMoment_geometric (G : BondGraph V E) (p h : unitInterval)
+    (hp : (p:ℝ) < 1) (hh : 0 < (h:ℝ)) (d : ℕ) (hd : 0 < d) (i : ℕ)
+    (Q : Finset E) (F : Finset V) (x : V)
+    (hdeg : ∀ v, (Q.filter (fun e => v ∈ G e)).card ≤ d)
+    (hF : insert x (searchVertices G Q) ⊆ F) :
+    finiteMarkedMoment G Q F x i h p ≤
+      ((Real.sqrt ((d:ℝ)/2)/(1-p))*Real.sqrt (Real.log (1/(h:ℝ))/h))^i *
+        finiteMarkedMoment G Q F x 0 h p := by
+  let A : ℝ := (Real.sqrt ((d:ℝ)/2)/(1-p))*Real.sqrt (Real.log (1/(h:ℝ))/h)
+  have hA : 0 ≤ A := mul_nonneg (div_nonneg (Real.sqrt_nonneg _) (sub_nonneg.mpr p.2.2))
+    (Real.sqrt_nonneg _)
+  change _ ≤ A^i * _
+  induction i generalizing Q F x with
+  | zero => simp
+  | succ i ih =>
+    have hpow := pow_nonneg hA i
+    calc
+      finiteMarkedMoment G Q F x (i+1) h p =
+          (p:ℝ) * ∑ e ∈ Q, ∑ W ∈ F.powerset,
+            if CutExists G W e then cutCoefficient G Q x e W h p *
+              finiteMarkedMoment G (exteriorBonds G Q W) (F \ W) (cutOutside G W e) i h p else 0 :=
+        finiteMarkedMoment_recursion G Q F x i h p hF
+      _ ≤ (p:ℝ) * ∑ e ∈ Q, ∑ W ∈ F.powerset,
+            if CutExists G W e then cutCoefficient G Q x e W h p *
+              (A^i * finiteMarkedMoment G (exteriorBonds G Q W) (F \ W) (cutOutside G W e) 0 h p)
+              else 0 := by
+        apply mul_le_mul_of_nonneg_left _ p.2.1
+        apply Finset.sum_le_sum
+        intro e he
+        apply Finset.sum_le_sum
+        intro W hW
+        by_cases hc : CutExists G W e
+        · simp only [hc,ite_true]
+          apply mul_le_mul_of_nonneg_left _ (cutCoefficient_nonneg G Q x e W h p)
+          exact ih _ _ _ (exterior_degree_bound G Q W d hdeg) (exterior_mark_support G Q F W x e he hc hF)
+        · simp only [hc,ite_false,le_refl]
+      _ = A^i * finiteMarkedMoment G Q F x 1 h p := by
+        rw [show (1:ℕ) = 0+1 from rfl,finiteMarkedMoment_recursion G Q F x 0 h p hF]
+        simp only [Finset.mul_sum]
+        apply Finset.sum_congr rfl
+        intro e he
+        apply Finset.sum_congr rfl
+        intro W hW
+        by_cases hc : CutExists G W e <;> simp only [hc,ite_true,ite_false] <;> ring
+      _ ≤ A^i * (A * finiteMarkedMoment G Q F x 0 h p) :=
+        mul_le_mul_of_nonneg_left (finiteMarkedMoment_first_bound G Q F x p h hp hh d hd hdeg hF) hpow
+      _ = A^(i+1) * finiteMarkedMoment G Q F x 0 h p := by rw [pow_succ]; ring
+
+end FiniteMomentRecursion
+
+section MarkedExponential
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem choose_generating_sum (n N : ℕ) (hn : n ≤ N) (z : ℝ) :
+    (1+z)^n = ∑ i ∈ Finset.range (N+1), z^i*(n.choose i : ℝ) := by
+  rw [add_comm (1:ℝ),add_pow]
+  simp only [one_pow,mul_one]
+  apply Finset.sum_subset (Finset.range_mono (Nat.succ_le_succ hn))
+  intro i hi hin
+  have hni : n < i := by simp only [Finset.mem_range,not_lt] at hin; omega
+  simp [Nat.choose_eq_zero_of_lt hni]
+
+ 
+theorem markedMomentValue_generating (G : BondGraph V E) (Q : Finset E) (x : V)
+    (z : ℝ) (ω : Finset E) (m : Finset V) :
+    (if MarkedEvent G ω m x then (1+z)^(markedOpenPivotals G Q x ω m).card else 0) =
+      ∑ i ∈ Finset.range (Q.card+1), z^i*markedMomentValue G Q x i ω m := by
+  by_cases hm : MarkedEvent G ω m x
+  · simp only [markedMomentValue,hm,ite_true]
+    exact choose_generating_sum _ _ (Finset.card_filter_le ..) z
+  · simp only [markedMomentValue,hm,ite_false,mul_zero,Finset.sum_const_zero]
+
+noncomputable def finiteMarkedGenerating (G : BondGraph V E) (Q : Finset E) (F : Finset V)
+    (x : V) (z h p : ℝ) : ℝ :=
+  bernoulliPolynomial Q (fun ω => bernoulliPolynomial F (fun m =>
+    if MarkedEvent G ω m x then (1+z)^(markedOpenPivotals G Q x ω m).card else 0) h) p
+
+noncomputable def finiteMarkedTail (G : BondGraph V E) (Q : Finset E) (F : Finset V)
+    (x : V) (r : ℕ) (h p : ℝ) : ℝ :=
+  bernoulliPolynomial Q (fun ω => bernoulliPolynomial F (fun m =>
+    if MarkedEvent G ω m x ∧ r ≤ (markedOpenPivotals G Q x ω m).card then (1:ℝ) else 0) h) p
+
+theorem finiteMarkedTail_nonneg (G : BondGraph V E) (Q : Finset E) (F : Finset V)
+    (x : V) (r : ℕ) (h p : unitInterval) : 0 ≤ finiteMarkedTail G Q F x r h p := by
+  apply bernoulliPolynomial_nonneg _ _ _ p.2
+  intro ω _
+  apply bernoulliPolynomial_nonneg _ _ _ h.2
+  intro m _
+  split_ifs <;> norm_num
+
+theorem finiteMarkedGenerating_eq (G : BondGraph V E) (Q : Finset E) (F : Finset V)
+    (x : V) (z h p : ℝ) :
+    finiteMarkedGenerating G Q F x z h p =
+      ∑ i ∈ Finset.range (Q.card+1), z^i*finiteMarkedMoment G Q F x i h p := by
+  simp only [finiteMarkedGenerating,markedMomentValue_generating,bernoulliPolynomial_sum,
+    bernoulliPolynomial_const_mul]
+  rfl
+
+theorem half_geometric_sum_le (N : ℕ) :
+    (∑ i ∈ Finset.range N, (1/2:ℝ)^i) ≤ 2 := by
+  have heq : (∑ i ∈ Finset.range N, (1/2:ℝ)^i) = 2*(1-(1/2:ℝ)^N) := by
+    induction N with
+    | zero => simp
+    | succ N ih => rw [Finset.sum_range_succ,ih,pow_succ]; ring
+  rw [heq]
+  have := pow_nonneg (by norm_num : (0:ℝ) ≤ 1/2) N
+  linarith
+
+ 
+theorem finiteMarkedGenerating_le [Nonempty V] (G : BondGraph V E) (p h : unitInterval)
+    (hp : (p:ℝ) < 1) (hh : 0 < (h:ℝ)) (d : ℕ) (hd : 0 < d)
+    (Q : Finset E) (F : Finset V) (x : V)
+    (hdeg : ∀ v, (Q.filter (fun e => v ∈ G e)).card ≤ d)
+    (hF : insert x (searchVertices G Q) ⊆ F) (B : ℝ) (hB : 0 < B)
+    (hAB : (Real.sqrt ((d:ℝ)/2)/(1-p))*Real.sqrt (Real.log (1/(h:ℝ))/h) ≤ B) :
+    finiteMarkedGenerating G Q F x (1/(2*B)) h p ≤ 2*finiteMarkedMoment G Q F x 0 h p := by
+  let A : ℝ := (Real.sqrt ((d:ℝ)/2)/(1-p))*Real.sqrt (Real.log (1/(h:ℝ))/h)
+  have hA : 0 ≤ A := mul_nonneg (div_nonneg (Real.sqrt_nonneg _) (sub_nonneg.mpr p.2.2))
+    (Real.sqrt_nonneg _)
+  have hb0 := finiteMarkedMoment_nonneg G Q F x 0 h p
+  rw [finiteMarkedGenerating_eq]
+  calc
+    _ ≤ ∑ i ∈ Finset.range (Q.card+1), (1/(2*B))^i*(B^i*finiteMarkedMoment G Q F x 0 h p) := by
+      apply Finset.sum_le_sum
+      intro i hi
+      apply mul_le_mul_of_nonneg_left _ (pow_nonneg (by positivity) i)
+      apply (finiteMarkedMoment_geometric G p h hp hh d hd i Q F x hdeg hF).trans
+      exact mul_le_mul_of_nonneg_right (pow_le_pow_left₀ hA hAB i) hb0
+    _ = (∑ i ∈ Finset.range (Q.card+1), (1/2:ℝ)^i)*finiteMarkedMoment G Q F x 0 h p := by
+      rw [Finset.sum_mul]
+      apply Finset.sum_congr rfl
+      intro i hi
+      rw [← mul_assoc,← mul_pow]
+      congr 2
+      field_simp
+    _ ≤ 2*finiteMarkedMoment G Q F x 0 h p := mul_le_mul_of_nonneg_right (half_geometric_sum_le _) hb0
+
+ 
+theorem finiteMarkedTail_markov (G : BondGraph V E) (p h : unitInterval) (Q : Finset E)
+    (F : Finset V) (x : V) (z : ℝ) (hz : 0 ≤ z) (r : ℕ) :
+    (1+z)^r * finiteMarkedTail G Q F x r h p ≤ finiteMarkedGenerating G Q F x z h p := by
+  unfold finiteMarkedTail finiteMarkedGenerating
+  rw [← bernoulliPolynomial_const_mul]
+  apply bernoulliPolynomial_mono _ _ _ _ p.2
+  intro ω hω
+  rw [← bernoulliPolynomial_const_mul]
+  apply bernoulliPolynomial_mono _ _ _ _ h.2
+  intro m hm
+  by_cases he : MarkedEvent G ω m x
+  · by_cases hc : r ≤ (markedOpenPivotals G Q x ω m).card
+    · simp only [he,hc,and_self,ite_true,mul_one]
+      exact pow_le_pow_right₀ (by linarith : (1:ℝ) ≤ 1+z) hc
+    · simp only [he,hc,and_false,ite_true,ite_false,mul_zero]
+      exact pow_nonneg (by linarith) _
+  · simp only [he,false_and,ite_false,mul_zero,le_refl]
+
+ 
+theorem log_one_add_half_inv_ge (B : ℝ) (hB : 1 ≤ B) :
+    1/(4*B) ≤ Real.log (1+1/(2*B)) := by
+  have hBp : 0 < B := by linarith
+  let z : ℝ := 1/(2*B)
+  have hz : 0 ≤ z := by dsimp [z]; positivity
+  have hzle : z ≤ 2 := by dsimp [z]; apply (div_le_iff₀ (by positivity)).mpr; linarith
+  calc
+    1/(4*B) = z/2 := by dsimp [z]; ring
+    _ ≤ 2*z/(z+2) := by
+      apply (le_div_iff₀ (by linarith : 0 < z+2)).mpr
+      have hs := mul_le_mul_of_nonneg_right hzle hz
+      nlinarith
+    _ ≤ Real.log (1+z) := Real.le_log_one_add_of_nonneg hz
+
+ 
+theorem finiteMarkedTail_exponential [Nonempty V] (G : BondGraph V E) (p h : unitInterval)
+    (hp : (p:ℝ) < 1) (hh : 0 < (h:ℝ)) (d : ℕ) (hd : 0 < d)
+    (Q : Finset E) (F : Finset V) (x : V)
+    (hdeg : ∀ v, (Q.filter (fun e => v ∈ G e)).card ≤ d)
+    (hF : insert x (searchVertices G Q) ⊆ F) (B : ℝ) (hB : 1 ≤ B)
+    (hAB : (Real.sqrt ((d:ℝ)/2)/(1-p))*Real.sqrt (Real.log (1/(h:ℝ))/h) ≤ B) (r : ℕ) :
+    finiteMarkedTail G Q F x r h p ≤
+      2*Real.exp (-(r:ℝ)/(4*B))*finiteMarkedMoment G Q F x 0 h p := by
+  have hBp : 0 < B := by linarith
+  have hz : 0 ≤ 1/(2*B) := by positivity
+  have hbase : 0 < 1+1/(2*B) := by positivity
+  have hpow : Real.exp ((r:ℝ)/(4*B)) ≤ (1+1/(2*B))^r := by
+    rw [← Real.exp_log (pow_pos hbase r),Real.log_pow]
+    apply Real.exp_le_exp.mpr
+    have hlog := mul_le_mul_of_nonneg_left (log_one_add_half_inv_ge B hB) (Nat.cast_nonneg r : (0:ℝ) ≤ r)
+    simpa only [mul_one_div] using hlog
+  have hm := (finiteMarkedTail_markov G p h Q F x (1/(2*B)) hz r).trans
+    (finiteMarkedGenerating_le G p h hp hh d hd Q F x hdeg hF B hBp hAB)
+  have ht := finiteMarkedTail_nonneg G Q F x r h p
+  have he := (mul_le_mul_of_nonneg_right hpow ht).trans hm
+  calc
+    finiteMarkedTail G Q F x r h p ≤ (2*finiteMarkedMoment G Q F x 0 h p)/Real.exp ((r:ℝ)/(4*B)) :=
+      (le_div_iff₀ (Real.exp_pos _)).mpr (by simpa only [mul_comm] using he)
+    _ = 2*Real.exp (-(r:ℝ)/(4*B))*finiteMarkedMoment G Q F x 0 h p := by
+      rw [neg_div,Real.exp_neg,div_eq_mul_inv]
+      ring
+
+end MarkedExponential
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Exploration/TailBounds.lean b/lean/OAI/Probability/UnimodularPercolation/Exploration/TailBounds.lean
index d49958a65a19a64223700800c3ec4584cd6926fa..7139dcbdbdba552db159d7b1f43af6a7457c62c7 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Exploration/TailBounds.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Exploration/TailBounds.lean
@@ -1,1241 +1,1241 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Exploration.FractionalMoments
-
-namespace OAI
-
-/-! Tail Bounds for bond percolation. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section ClusterTailDerivative
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem finiteClusterTail_derivative_bound (G : BondGraph V E) (Q : Finset E) (x : V)
-    (n : ℕ) (p h : unitInterval) (hn : 1 ≤ (h:ℝ)*n) (α C : ℝ)
-    (hα : 0 ≤ α) (hα1 : α ≤ 1) (hC0 : 0 < C) (hsmall : C*(h:ℝ)^α ≤ 1/4)
-    (hfin : ∀ y, ∀ᵐ ω ∂law p, (cluster G ω y).Finite)
-    (hI : ∀ y, Integrable (fun ω => ((cluster G ω y).ncard:ℝ)^α) (law p))
-    (hC : ∀ y, (∫ ω, ((cluster G ω y).ncard:ℝ)^α ∂law p) ≤ C) :
-    finiteClusterTail G Q x n p ≤ 8*C*(h:ℝ)^α*deriv (finiteClusterTail G Q x n) p := by
-  have hmark := (markedConnection_le_cluster_power G Q x h p α hα hα1 (hfin x) (hI x)).trans
-    (mul_le_mul_of_nonneg_left (hC x) (Real.rpow_nonneg h.2.1 α))
-  rw [mul_comm ((h:ℝ)^α) C] at hmark
-  have hlow := finiteSeed_covariance_lower G Q x n p h hn (C*(h:ℝ)^α) hmark
-  have hupp := finiteSeed_covariance_upper G Q x p h α C hα hα1 hfin hI hC
-    (finiteClusterTailIndicator G x n) (finiteClusterTailIndicator_mono G x n)
-  have hF := finiteClusterTail_nonneg G Q x n p
-  have hd := finiteClusterTail_deriv_nonneg G Q x n p
-  have hp : (p:ℝ)*(1-(p:ℝ)) ≤ 1 := by nlinarith [p.2.1,p.2.2]
-  have hprod := mul_le_mul_of_nonneg_right hp hd
-  have hR : 0 ≤ 2*C*(h:ℝ)^α := mul_nonneg (by positivity) (Real.rpow_nonneg h.2.1 α)
-  have hh := mul_le_mul_of_nonneg_left hprod hR
-  have ha := le_abs_self (finiteCovariance Q p (finiteClusterTailIndicator G x n)
-    (fun ω => finiteSeedProbability G (insert x (searchVertices G Q)) ω x h))
-  have hs := mul_le_mul_of_nonneg_right hsmall hF
-  change |_| ≤ _*deriv (finiteClusterTail G Q x n) p at hupp
-  nlinarith
-
-theorem exponential_transfer_of_deriv_lower (H : ℝ → ℝ) (a b K : ℝ)
-    (hH : Differentiable ℝ H) (hab : a ≤ b)
-    (hb : ∀ t ∈ Set.Icc a b, K*H t ≤ deriv H t) :
-    H a ≤ Real.exp (-K*(b-a))*H b := by
-  let F (t : ℝ) := Real.exp (-K*(t-a))*H t
-  have hF (t : ℝ) : HasDerivAt F (Real.exp (-K*(t-a))*(deriv H t-K*H t)) t := by
-    have he := (((hasDerivAt_id t).sub_const a).const_mul (-K)).exp
-    convert! he.mul (hH t).hasDerivAt using 1
-    simp only [id_eq]
-    ring
-  have hmono : MonotoneOn F (Set.Icc a b) := by
-    apply monotoneOn_of_deriv_nonneg (convex_Icc a b)
-      (fun t _ => (hF t).continuousAt.continuousWithinAt)
-      (fun t _ => (hF t).differentiableAt.differentiableWithinAt)
-    intro t ht
-    rw [(hF t).deriv]
-    exact mul_nonneg (Real.exp_pos _).le (sub_nonneg.mpr (hb t (interior_subset ht)))
-  have hh := hmono ⟨le_rfl,hab⟩ ⟨hab,le_rfl⟩ hab
-  simpa only [F,sub_self,mul_zero,Real.exp_zero,one_mul] using hh
-
-end ClusterTailDerivative
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section FiniteStretchedTail
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem cluster_finite_ae_mono [Countable E] (G : BondGraph V E)
-    (x : V) (p q : unitInterval) (hpq : p ≤ q)
-    (hq : ∀ᵐ ω ∂law q, (cluster G ω x).Finite) :
-    ∀ᵐ ω ∂law p, (cluster G ω x).Finite := by
-  have hlabels : ∀ᵐ labels ∂(labelLaw : Measure (E → unitInterval)),
-      (cluster G (openEdges labels q) x).Finite := by
-    apply ae_of_ae_map (μ := (labelLaw : Measure (E → unitInterval)))
-      (p := fun ω : Set E => (cluster G ω x).Finite) (measurable_openEdges q).aemeasurable
-    rwa [map_openEdges]
-  rw [←map_openEdges p]
-  apply (ae_map_iff (measurable_openEdges p).aemeasurable
-    (by simpa only [Set.compl_ofPred,Set.not_infinite] using (measurableSet_infinite_cluster G x).compl)).mpr
-  filter_upwards [hlabels] with labels hl
-  exact hl.subset (fun y h => conn_mono G (openEdges_mono labels hpq) h)
-
-theorem finiteClusterTail_le_markedConnection (G : BondGraph V E) (Q : Finset E)
-    (x : V) (n : ℕ) (p h : unitInterval) (hn : 1 ≤ (h:ℝ)*n) :
-    finiteClusterTail G Q x n p ≤ 2*markedConnectionPolynomial G Q x h p := by
-  have hh : (1:ℝ)/2*finiteClusterTail G Q x n p ≤ markedConnectionPolynomial G Q x h p := by
-    rw [finiteClusterTail,←bernoulliPolynomial_const_mul,markedConnection_component_polynomial]
-    apply bernoulliPolynomial_mono _ _ _ p p.2
-    intro ω _
-    unfold finiteClusterTailIndicator
-    by_cases hs : n ≤ (finiteRootComponent G ω x).card
-    · simp only [ite_eq_left hs,mul_one]
-      apply one_sub_pow_one_sub_ge_half
-      exact hn.trans (mul_le_mul_of_nonneg_left (by exact_mod_cast hs) h.2.1)
-    · simp only [ite_eq_right hs,mul_zero]
-      apply sub_nonneg.mpr
-      exact pow_le_one₀ (sub_nonneg.mpr h.2.2) (by linarith [h.2.1])
-  linarith
-
- 
-
-theorem finiteClusterTail_stretched [Countable E] (G : BondGraph V E) (hloc : LocallyFinite G)
-    (Q : Finset E) (x : V) (n : ℕ) (p q h : unitInterval) (hpq : p ≤ q)
-    (hh : 0 < (h:ℝ)) (hn : 1 ≤ (h:ℝ)*n) (α C : ℝ)
-    (hα : 0 ≤ α) (hα1 : α ≤ 1) (hC0 : 0 < C) (hsmall : C*(h:ℝ)^α ≤ 1/4)
-    (hfin : ∀ y, ∀ᵐ ω ∂law q, (cluster G ω y).Finite)
-    (hI : ∀ y, Integrable (fun ω => ((cluster G ω y).ncard:ℝ)^α) (law q))
-    (hC : ∀ y, (∫ ω, ((cluster G ω y).ncard:ℝ)^α ∂law q) ≤ C) :
-    finiteClusterTail G Q x n p ≤
-      Real.exp (-((q:ℝ)-(p:ℝ))/(8*C*(h:ℝ)^α))*(2*C*(h:ℝ)^α) := by
-  have hden : 0 < 8*C*(h:ℝ)^α := by positivity
-  have hexp := exponential_transfer_of_deriv_lower (finiteClusterTail G Q x n) p q
-    (1/(8*C*(h:ℝ)^α)) (fun t => (hasDerivAt_bernoulliPolynomial Q _ t).differentiableAt) hpq
-  have hder : ∀ t ∈ Set.Icc (p:ℝ) (q:ℝ),
-      (1/(8*C*(h:ℝ)^α))*finiteClusterTail G Q x n t ≤ deriv (finiteClusterTail G Q x n) t := by
-    intro t ht
-    let r : unitInterval := ⟨t,⟨p.2.1.trans ht.1,ht.2.trans q.2.2⟩⟩
-    have hrq : r ≤ q := ht.2
-    have hmoment (y : V) := cluster_power_mono_integral G hloc y r q hrq α hα (hfin y) (hI y)
-    have he := finiteClusterTail_derivative_bound G Q x n r h hn α C hα hα1 hC0 hsmall
-      (fun y => cluster_finite_ae_mono G y r q hrq (hfin y)) (fun y => (hmoment y).1)
-      (fun y => (hmoment y).2.trans (hC y))
-    rw [one_div,mul_comm,←div_eq_mul_inv]
-    apply (div_le_iff₀ hden).mpr
-    simpa only [mul_comm (deriv (finiteClusterTail G Q x n) t)] using he
-  have he := hexp hder
-  have hb := (finiteClusterTail_le_markedConnection G Q x n q h hn).trans
-    (mul_le_mul_of_nonneg_left ((markedConnection_le_cluster_power G Q x h q α hα hα1 (hfin x) (hI x)).trans
-      (mul_le_mul_of_nonneg_left (hC x) (Real.rpow_nonneg hh.le α))) (by norm_num : (0:ℝ) ≤ 2))
-  have hm := mul_le_mul_of_nonneg_left hb (Real.exp_pos (-((q:ℝ)-(p:ℝ))/(8*C*(h:ℝ)^α))).le
-  have heq : -(1/(8*C*(h:ℝ)^α))*((q:ℝ)-(p:ℝ)) = -((q:ℝ)-(p:ℝ))/(8*C*(h:ℝ)^α) := by ring
-  rw [heq] at he
-  have hb' : Real.exp (-((q:ℝ)-(p:ℝ))/(8*C*(h:ℝ)^α))*(2*((h:ℝ)^α*C)) =
-      Real.exp (-((q:ℝ)-(p:ℝ))/(8*C*(h:ℝ)^α))*(2*C*(h:ℝ)^α) := by ring
-  exact he.trans (hm.trans_eq hb')
-
-end FiniteStretchedTail
-end Percolation
-
- 
-namespace Percolation
-open MeasureTheory Filter
-section ExponentialPowerBound
-
- 
-
-theorem exists_exp_neg_le_rpow (r : ℝ) (hr : 0 ≤ r) :
-    ∃ K : ℝ, 0 < K ∧ ∀ t : ℝ, 0 < t → Real.exp (-t) ≤ K*t^(-r) := by
-  obtain ⟨m,hm⟩ := exists_nat_ge r
-  let K : ℝ := m.factorial
-  have hK1 : 1 ≤ K := by
-    dsimp [K]
-    exact_mod_cast (Nat.succ_le_of_lt (Nat.factorial_pos m))
-  refine ⟨K,by linarith,?_⟩
-  intro t ht
-  rw [Real.rpow_neg ht.le,←div_eq_mul_inv]
-  apply (le_div_iff₀ (Real.rpow_pos_of_pos ht r)).mpr
-  by_cases ht1 : t ≤ 1
-  · have hp := Real.rpow_le_one ht.le ht1 hr
-    have he : Real.exp (-t) ≤ 1 := by rw [Real.exp_le_one_iff]; linarith
-    exact (mul_le_of_le_one_left (Real.rpow_nonneg ht.le r) he).trans (hp.trans hK1)
-  · have hpow : t^r ≤ t^m := by
-      simpa only [Real.rpow_natCast] using Real.rpow_le_rpow_of_exponent_le (le_of_not_ge ht1) hm
-    have hTaylor : t^m ≤ K*Real.exp t := by
-      have hh := Real.pow_div_factorial_le_exp t ht.le m
-      exact (div_le_iff₀ (show (0:ℝ) < (m.factorial:ℝ) by positivity)).mp hh |>.trans_eq (mul_comm _ _)
-    calc
-      _ ≤ Real.exp (-t)*(K*Real.exp t) :=
-        mul_le_mul_of_nonneg_left (hpow.trans hTaylor) (Real.exp_pos _).le
-      _ = K := by rw [Real.exp_neg]; field_simp
-
-end ExponentialPowerBound
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section PolynomialTails
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem exp_stretched_power_bound (r K δ C z : ℝ) (hδ : 0 < δ) (hC : 0 < C)
-    (hz : 0 < z) (hK : ∀ t : ℝ, 0 < t → Real.exp (-t) ≤ K*t^(-r)) :
-    Real.exp (-δ/(8*C*z))*(2*C*z) ≤
-      (2*C*K*(8*C)^r)*δ^(-r)*z^(1+r) := by
-  have he := mul_le_mul_of_nonneg_right (hK (δ/(8*C*z)) (by positivity))
-    (by positivity : 0 ≤ 2*C*z)
-  have hid : (δ/(8*C*z))^(-r)*(2*C*z) =
-      (2*C*(8*C)^r)*δ^(-r)*z^(1+r) := by
-    rw [Real.div_rpow hδ.le (by positivity),Real.mul_rpow (by positivity : 0 ≤ 8*C) hz.le,
-      Real.rpow_neg (by positivity : 0 ≤ 8*C),Real.rpow_neg hz.le,
-      Real.rpow_add hz,Real.rpow_one]
-    field_simp
-  calc
-    _ ≤ K*(δ/(8*C*z))^(-r)*(2*C*z) := by simpa only [neg_div] using he
-    _ = K*((δ/(8*C*z))^(-r)*(2*C*z)) := by ring
-    _ = _ := by rw [hid]; ring
-
- 
-
-theorem finiteClusterTail_polynomial_large [Countable E] (G : BondGraph V E) (hloc : LocallyFinite G)
-    (q : unitInterval) (α C r : ℝ) (hα : 0 ≤ α) (hα1 : α ≤ 1) (hC0 : 0 < C)
-    (hr : 0 ≤ r)
-    (hfin : ∀ y, ∀ᵐ ω ∂law q, (cluster G ω y).Finite)
-    (hI : ∀ y, Integrable (fun ω => ((cluster G ω y).ncard:ℝ)^α) (law q))
-    (hC : ∀ y, (∫ ω, ((cluster G ω y).ncard:ℝ)^α ∂law q) ≤ C) :
-    ∃ A : ℝ, 0 < A ∧ ∀ (Q : Finset E) (x : V) (p : unitInterval), p < q →
-      ∀ n : ℕ, 1 ≤ n → C*(n:ℝ)^(-α) ≤ 1/4 →
-      finiteClusterTail G Q x n p ≤ A*((q:ℝ)-(p:ℝ))^(-r)*(n:ℝ)^(-α*(1+r)) := by
-  obtain ⟨K,hK,hKb⟩ := exists_exp_neg_le_rpow r hr
-  refine ⟨2*C*K*(8*C)^r,by positivity,?_⟩
-  intro Q x p hp n hn hsmall
-  have hn0 : (0:ℝ) < n := by exact_mod_cast (by omega : 0 < n)
-  let h : unitInterval := ⟨1/(n:ℝ),⟨by positivity,by apply (div_le_one hn0).mpr; exact_mod_cast hn⟩⟩
-  have hh : 0 < (h:ℝ) := by dsimp [h]; positivity
-  have hhpow : (h:ℝ)^α = (n:ℝ)^(-α) := by
-    dsimp [h]
-    rw [one_div,Real.inv_rpow hn0.le,Real.rpow_neg hn0.le]
-  have hhprod : 1 ≤ (h:ℝ)*n := by dsimp [h]; rw [one_div,inv_mul_cancel₀ hn0.ne']
-  have hst := finiteClusterTail_stretched G hloc Q x n p q h hp.le hh hhprod α C hα hα1 hC0
-    (by rwa [hhpow]) hfin hI hC
-  have he := exp_stretched_power_bound r K ((q:ℝ)-(p:ℝ)) C ((h:ℝ)^α)
-    (sub_pos.mpr hp) hC0 (Real.rpow_pos_of_pos hh α) hKb
-  apply hst.trans
-  convert! he using 1
-  rw [hhpow,←Real.rpow_mul hn0.le]
-
-end PolynomialTails
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section UniformPolynomialTails
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem finiteClusterTail_le_one (G : BondGraph V E) (Q : Finset E) (x : V)
-    (n : ℕ) (p : unitInterval) : finiteClusterTail G Q x n p ≤ 1 := by
-  rw [finiteClusterTail,←bernoulliPolynomial_const Q 1 p]
-  apply bernoulliPolynomial_mono _ _ _ p p.2
-  intro ω _
-  unfold finiteClusterTailIndicator
-  split_ifs <;> norm_num
-
- 
-
-theorem finiteClusterTail_polynomial [Countable E] (G : BondGraph V E) (hloc : LocallyFinite G)
-    (q : unitInterval) (α C r : ℝ) (hα : 0 < α) (hα1 : α ≤ 1) (hC0 : 0 < C)
-    (hr : 0 ≤ r)
-    (hfin : ∀ y, ∀ᵐ ω ∂law q, (cluster G ω y).Finite)
-    (hI : ∀ y, Integrable (fun ω => ((cluster G ω y).ncard:ℝ)^α) (law q))
-    (hC : ∀ y, (∫ ω, ((cluster G ω y).ncard:ℝ)^α ∂law q) ≤ C) :
-    ∃ A : ℝ, 0 < A ∧ ∀ (Q : Finset E) (x : V) (p : unitInterval), p < q →
-      ∀ n : ℕ, 1 ≤ n →
-      finiteClusterTail G Q x n p ≤ A*((q:ℝ)-(p:ℝ))^(-r)*(n:ℝ)^(-α*(1+r)) := by
-  obtain ⟨A,hA,hbig⟩ := finiteClusterTail_polynomial_large G hloc q α C r hα.le hα1 hC0 hr hfin hI hC
-  have ht : Tendsto (fun n : ℕ => C*(n:ℝ)^(-α)) atTop (nhds 0) := by
-    simpa only [mul_zero,Function.comp_apply] using tendsto_const_nhds.mul
-      ((tendsto_rpow_neg_atTop hα).comp tendsto_natCast_atTop_atTop)
-  obtain ⟨N,hN⟩ := eventually_atTop.mp (ht.eventually (gt_mem_nhds (by norm_num : (0:ℝ) < 1/4)))
-  let s := α*(1+r)
-  have hs : 0 ≤ s := by dsimp [s]; positivity
-  let B : ℝ := ((N+1:ℕ):ℝ)^s+A
-  have hB : 0 < B := by dsimp [B]; positivity
-  refine ⟨B,hB,?_⟩
-  intro Q x p hp n hn
-  have hn0 : (0:ℝ) < n := by exact_mod_cast (by omega : 0 < n)
-  have hδ : 0 < (q:ℝ)-(p:ℝ) := sub_pos.mpr hp
-  have hhead : 1 ≤ ((q:ℝ)-(p:ℝ))^(-r) :=
-    Real.one_le_rpow_of_pos_of_le_one_of_nonpos hδ (by linarith [p.2.1,q.2.2]) (by linarith)
-  by_cases hnN : N ≤ n
-  · apply (hbig Q x p hp n hn (hN n hnN).le).trans
-    exact mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right
-      (by dsimp [B]; linarith [Real.rpow_nonneg (Nat.cast_nonneg (N+1)) s]) (Real.rpow_nonneg hδ.le _)) (Real.rpow_nonneg hn0.le _)
-  · have hnB : (n:ℝ)^s ≤ B := by
-      apply (Real.rpow_le_rpow hn0.le (show (n:ℝ) ≤ ((N+1:ℕ):ℝ) by exact_mod_cast (by omega : n ≤ N+1)) hs).trans
-      dsimp [B]
-      linarith
-    have hBδ : B ≤ B*((q:ℝ)-(p:ℝ))^(-r) := by nlinarith
-    have hbound : 1 ≤ B*((q:ℝ)-(p:ℝ))^(-r)*(n:ℝ)^(-s) := by
-      rw [Real.rpow_neg hn0.le,←div_eq_mul_inv]
-      exact (one_le_div (Real.rpow_pos_of_pos hn0 s)).mpr (hnB.trans hBδ)
-    exact (finiteClusterTail_le_one G Q x n p).trans (by simpa only [s,neg_mul] using hbound)
-
-end UniformPolynomialTails
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section TailSummation
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem natCast_eq_sum_tail (k M : ℕ) (hk : k ≤ M) :
-    (k:ℝ) = ∑ i ∈ Finset.range M, if i+1 ≤ k then (1:ℝ) else 0 := by
-  rw [Finset.sum_boole]
-  congr 1
-  have hh : (Finset.range M).filter (fun i => i+1 ≤ k) = Finset.range k := by
-    ext i
-    simp only [Finset.mem_filter,Finset.mem_range]
-    omega
-  rw [hh,Finset.card_range]
-
-theorem finiteClusterSize_eq_sum_tail (G : BondGraph V E) (Q : Finset E) (x : V) (p : ℝ) :
-    bernoulliPolynomial Q (fun ω => ((finiteRootComponent G ω x).card:ℝ)) p =
-      ∑ n ∈ Finset.range (insert x (searchVertices G Q)).card, finiteClusterTail G Q x (n+1) p := by
-  simp only [finiteClusterTail]
-  rw [←bernoulliPolynomial_sum]
-  apply bernoulliPolynomial_congr
-  intro ω hω
-  exact natCast_eq_sum_tail _ _ (Finset.card_le_card (finiteRootComponent_subset G Q _ x ω hω (by rfl)))
-
-theorem summable_shifted_rpow (s : ℝ) (hs : 1 < s) :
-    Summable (fun n : ℕ => ((n+1:ℕ):ℝ)^(-s)) := by
-  exact (summable_nat_add_iff 1).mpr (Real.summable_nat_rpow.mpr (by linarith))
-
- 
-theorem finiteClusterSize_bound_of_tail (G : BondGraph V E) (Q : Finset E) (x : V)
-    (p : unitInterval) (A s : ℝ) (hA : 0 ≤ A) (hs : 1 < s)
-    (hf : ∀ n : ℕ, 1 ≤ n → finiteClusterTail G Q x n p ≤ A*(n:ℝ)^(-s)) :
-    bernoulliPolynomial Q (fun ω => ((finiteRootComponent G ω x).card:ℝ)) p ≤
-      A*(∑' n : ℕ, ((n+1:ℕ):ℝ)^(-s)) := by
-  rw [finiteClusterSize_eq_sum_tail,←tsum_mul_left]
-  have hsumm := (summable_shifted_rpow s hs).mul_left A
-  apply (Finset.sum_le_sum (fun n _ => hf (n+1) (by omega))).trans
-  exact hsumm.sum_le_tsum _ (fun n _ => mul_nonneg hA (Real.rpow_nonneg (Nat.cast_nonneg _) _))
-
-end TailSummation
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section SusceptibilityExhaustion
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem finite_connection_row_le_size (G : BondGraph V E) (Q : Finset E)
-    (x : V) (K : Finset V) (p : unitInterval) :
-    (∑ y ∈ K, bernoulliPolynomial Q (fun ω => if Conn G ω x y then (1:ℝ) else 0) p) ≤
-      bernoulliPolynomial Q (fun ω => ((finiteRootComponent G ω x).card:ℝ)) p := by
-  rw [←bernoulliPolynomial_sum]
-  apply bernoulliPolynomial_mono _ _ _ p p.2
-  intro ω _
-  rw [Finset.sum_boole]
-  apply Nat.cast_le.mpr
-  apply Finset.card_le_card
-  intro y hy
-  change y ∈ (finiteRootComponent G ω x : Set V)
-  rw [coe_finiteRootComponent]
-  exact (Finset.mem_filter.mp hy).2
-
- 
-
-theorem susceptibility_bound_of_finite_size [Countable E] (G : BondGraph V E)
-    (x : V) (p : unitInterval) (B : ℝ)
-    (hB : ∀ Q : Finset E, bernoulliPolynomial Q (fun ω => ((finiteRootComponent G ω x).card:ℝ)) p ≤ B) :
-    Summable (twoPoint G p x) ∧ susceptibility G p x ≤ B := by
-  let : Encodable E := Encodable.ofCountable E
-  let Q := vertexExhaustion (V := E)
-  have hrow (K : Finset V) : (∑ y ∈ K, twoPoint G p x y) ≤ B := by
-    have ht (y : V) : Tendsto (fun n => bernoulliPolynomial (Q n)
-        (fun ω => if Conn G ω x y then (1:ℝ) else 0) p) atTop (nhds (twoPoint G p x y)) := by
-      have hc (e : E) (_ : e ∈ (Set.univ : Set E)) : ∃ n, e ∈ Q n := by
-        obtain ⟨n,hn⟩ := subset_vertexExhaustion ({e} : Finset E)
-        exact ⟨n,hn (by simp)⟩
-      simpa only [Set.inter_univ,twoPoint,Measure.real] using! finite_connection_polynomial_tendsto G Set.univ Q
-        vertexExhaustion_mono (fun _ => Set.subset_univ _) hc p x y
-    apply le_of_tendsto (tendsto_finsetSum K (fun y _ => ht y))
-    exact Filter.Eventually.of_forall (fun n => (finite_connection_row_le_size G (Q n) x K p).trans (hB (Q n)))
-  exact ⟨summable_of_sum_le (twoPoint_nonneg G p x) hrow,
-    Real.tsum_le_of_sum_le (twoPoint_nonneg G p x) hrow⟩
-
-end SusceptibilityExhaustion
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section SusceptibilityPowerFromMoment
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem susceptibility_power_of_uniform_moment [Countable E] (G : BondGraph V E) (hloc : LocallyFinite G)
-    (q : unitInterval) (α C r : ℝ) (hα : 0 < α) (hα1 : α ≤ 1) (hC0 : 0 < C)
-    (hr : 0 ≤ r) (hs : 1 < α*(1+r))
-    (hfin : ∀ y, ∀ᵐ ω ∂law q, (cluster G ω y).Finite)
-    (hI : ∀ y, Integrable (fun ω => ((cluster G ω y).ncard:ℝ)^α) (law q))
-    (hC : ∀ y, (∫ ω, ((cluster G ω y).ncard:ℝ)^α ∂law q) ≤ C) :
-    ∃ D : ℝ, 0 < D ∧ ∀ (p : unitInterval), p < q → ∀ x,
-      Summable (twoPoint G p x) ∧ susceptibility G p x ≤ D*((q:ℝ)-(p:ℝ))^(-r) := by
-  obtain ⟨A,hA,hbound⟩ := finiteClusterTail_polynomial G hloc q α C r hα hα1 hC0 hr hfin hI hC
-  let S := ∑' n : ℕ, ((n+1:ℕ):ℝ)^(-(α*(1+r)))
-  have hS : 0 ≤ S := tsum_nonneg (fun _ => Real.rpow_nonneg (Nat.cast_nonneg _) _)
-  refine ⟨1+A*S,by positivity,?_⟩
-  intro p hp x
-  have hδ : 0 < (q:ℝ)-(p:ℝ) := sub_pos.mpr (show (p:ℝ)<(q:ℝ) from hp)
-  apply susceptibility_bound_of_finite_size
-  intro Q
-  have hh := finiteClusterSize_bound_of_tail G Q x p (A*((q:ℝ)-(p:ℝ))^(-r)) (α*(1+r))
-    (mul_nonneg hA.le (Real.rpow_nonneg hδ.le _)) hs (by
-      intro n hn
-      simpa only [neg_mul] using hbound Q x p hp n hn)
-  calc
-    _ ≤ (A*((q:ℝ)-(p:ℝ))^(-r))*S := hh
-    _ = (A*S)*((q:ℝ)-(p:ℝ))^(-r) := by ring
-    _ ≤ _ := mul_le_mul_of_nonneg_right (by linarith) (Real.rpow_nonneg hδ.le _)
-
- 
-
-theorem almost_linear_susceptibility_of_moments [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (q : unitInterval)
-    (hfin : ∀ y, ∀ᵐ ω ∂law q, (cluster G ω y).Finite)
-    (hmom : ∀ α : ℝ, 0 < α → α < 1/2 → ∃ C : ℝ, 0 < C ∧ ∀ y,
-      Integrable (fun ω => ((cluster G ω y).ncard:ℝ)^α) (law q) ∧
-      (∫ ω, ((cluster G ω y).ncard:ℝ)^α ∂law q) ≤ C)
-    (η : ℝ) (hη : 0 < η) :
-    ∃ D : ℝ, 0 < D ∧ ∀ (p : unitInterval), p < q → ∀ x,
-      Summable (twoPoint G p x) ∧ susceptibility G p x ≤ D*((q:ℝ)-(p:ℝ))^(-1-η) := by
-  let α : ℝ := 1/(2+η/2)
-  have hden : 0 < 2+η/2 := by linarith
-  have hα : 0 < α := one_div_pos.mpr hden
-  have hαhalf : α < 1/2 := by
-    apply (div_lt_div_iff₀ hden (by norm_num : (0:ℝ) < 2)).mpr
-    linarith
-  have hs : 1 < α*(1+(1+η)) := by
-    dsimp [α]
-    rw [one_div,mul_comm,←div_eq_mul_inv]
-    apply (one_lt_div hden).mpr
-    linarith
-  obtain ⟨C,hC,hm⟩ := hmom α hα hαhalf
-  obtain ⟨D,hD,hbound⟩ := susceptibility_power_of_uniform_moment G hloc q α C (1+η) hα
-    (by linarith) hC (by linarith) hs hfin (fun y => (hm y).1) (fun y => (hm y).2)
-  refine ⟨D,hD,?_⟩
-  intro p hp x
-  simpa only [neg_add_rev,sub_eq_add_neg,add_comm] using hbound p hp x
-
-end SusceptibilityPowerFromMoment
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section SourceAlmostLinear
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem cluster_finite_ae_covariant [Countable E] (G : BondGraph V E) (p : unitInterval)
-    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (x : V)
-    (hx : ∀ᵐ ω ∂law p, (cluster G ω x).Finite) :
-    ∀ᵐ ω ∂law p, (cluster G ω (a x)).Finite := by
-  have hh : ∀ᵐ ω ∂law p, (cluster G (b ⁻¹' ω) x).Finite := by
-    apply ae_of_ae_map (μ := law p) (p := fun ω : Set E => (cluster G ω x).Finite)
-      (measurable_set_preimage b).aemeasurable
-    rwa [law_map_preimage_equiv]
-  filter_upwards [hh] with ω hω
-  rw [←cluster_aut G hab ω x]
-  exact hω.image a
-
- 
-
-theorem almost_linear_susceptibility
-    [Countable V] [Countable E] [Nonempty V] [Nonempty E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hconn : Connected G)
-    (hqt : QuasiTransitive G) (hh : 0 < hV G)
-    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
-    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
-    (hp₀ : 0 < (pc G:ℝ)) (hp₁ : (pc G:ℝ) < 1)
-    (hfin : ∀ i, ∀ᵐ ω ∂law (pc G), (cluster G ω (root i)).Finite)
-    (η : ℝ) (hη : 0 < η) :
-    ∃ D : ℝ, 0 < D ∧ ∀ (p : unitInterval), p < pc G → ∀ x,
-      susceptibility G p x ≤ D*((pc G:ℝ)-(p:ℝ))^(-1-η) := by
-  have hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i) := by
-    intro x
-    obtain ⟨i,a,ha,hax⟩ := hcov x
-    obtain ⟨b,hab⟩ := hΓ ha
-    exact ⟨i,a,b,hab,hax⟩
-  have hfinite : ∀ x, ∀ᵐ ω ∂law (pc G), (cluster G ω x).Finite := by
-    intro x
-    obtain ⟨i,a,b,hab,rfl⟩ := hcover x
-    exact cluster_finite_ae_covariant G (pc G) hab (root i) (hfin i)
-  have hmom (α : ℝ) (hα : 0 < α) (hα1 : α < 1/2) :=
-    uniform_cluster_power_of_root G hloc (pc G) root hcover α
-      (critical_root_power_integrable G hloc hconn hqt hh root weight Γ hΓ htr hcov hw hw1 hp₀ hp₁ hfin α hα hα1)
-  obtain ⟨D,hD,hb⟩ := almost_linear_susceptibility_of_moments G hloc (pc G) hfinite hmom η hη
-  exact ⟨D,hD,fun p hp x => (hb p hp x).2⟩
-
-end SourceAlmostLinear
-end Percolation
-
- 
-namespace Percolation
-universe v
-variable {E : Type v}
-section QueryTranscriptLaw
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace EdgeQueryTree
-
-@[reducible] def forget : EdgeQueryTree E → ProbeTree
-  | .done b => .done b
-  | .query _ t₀ t₁ => .edge t₀.forget t₁.forget
-
-noncomputable def leaf (ω : Finset E) : (t : EdgeQueryTree E) → t.forget.Leaf
-  | .done _ => ()
-  | .query e t₀ t₁ => if e ∈ ω then Sum.inr (t₁.leaf ω) else Sum.inl (t₀.leaf ω)
-
-theorem leaf_congr (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
-    {ω η : Finset E} (h : ∀ e ∈ Q, e ∈ ω ↔ e ∈ η) : t.leaf ω = t.leaf η := by
-  induction t generalizing Q with
-  | done => rfl
-  | query e t₀ t₁ ih₀ ih₁ =>
-    simp only [leaf,h e ht.1]
-    split
-    · rw [ih₁ ht.2.2 (fun a ha => h a (Finset.mem_of_mem_erase ha))]
-    · rw [ih₀ ht.2.1 (fun a ha => h a (Finset.mem_of_mem_erase ha))]
-
-theorem leaf_insert (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
-    {e : E} (he : e ∉ Q) (ω : Finset E) : t.leaf (insert e ω) = t.leaf ω := by
-  apply leaf_congr t ht
-  intro a ha
-  simp [show a ≠ e from fun h => he (h ▸ ha)]
-
- 
-theorem leaf_expectation (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
-    (p : ℝ) (f : t.forget.Leaf → ℝ) :
-    bernoulliPolynomial Q (fun ω => f (t.leaf ω)) p =
-      ∑ l, ProbeTree.weight p 0 t.forget l*f l := by
-  induction t generalizing Q with
-  | done b =>
-    change bernoulliPolynomial Q (fun _ => f ()) p = ∑ l : Unit, 1*f l
-    simp [bernoulliPolynomial_const,Finset.sum_singleton]
-  | query e t₀ t₁ ih₀ ih₁ =>
-    obtain ⟨he,h₀,h₁⟩ := ht
-    conv_lhs => rw [←Finset.insert_erase he]
-    rw [bernoulliPolynomial_insert _ e (Finset.notMem_erase e Q)]
-    have hc : bernoulliPolynomial (Q.erase e) (fun ω => f ((query e t₀ t₁).leaf ω)) p =
-        bernoulliPolynomial (Q.erase e) (fun ω => f (Sum.inl (t₀.leaf ω))) p := by
-      apply bernoulliPolynomial_congr
-      intro ω hω
-      simp only [leaf,show e ∉ ω from fun h => Finset.notMem_erase e Q (hω h),ite_false]
-    have ho : bernoulliPolynomial (Q.erase e) (fun ω => f ((query e t₀ t₁).leaf (insert e ω))) p =
-        bernoulliPolynomial (Q.erase e) (fun ω => f (Sum.inr (t₁.leaf ω))) p := by
-      apply bernoulliPolynomial_congr
-      intro ω _
-      simp only [leaf,Finset.mem_insert_self,ite_true,
-        leaf_insert t₁ h₁ (Finset.notMem_erase e Q)]
-    rw [hc,ho,ih₀ h₀ (fun l => f (Sum.inl l)),ih₁ h₁ (fun l => f (Sum.inr l))]
-    change _ = ∑ l : t₀.forget.Leaf ⊕ t₁.forget.Leaf, _
-    simp only [Fintype.sum_sum_type,ProbeTree.weight,Finset.mul_sum,mul_assoc]
-
-noncomputable def score (p : ℝ) (t : EdgeQueryTree E) (ω : Finset E) : ℝ :=
-  ProbeTree.score p t.forget (t.leaf ω)
-
-noncomputable def queryCount (t : EdgeQueryTree E) (ω : Finset E) : ℕ :=
-  ProbeTree.queries t.forget (t.leaf ω)
-
- 
-theorem score_mgf (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
-    (p : unitInterval) (lam : ℝ) :
-    bernoulliPolynomial Q (fun ω => Real.exp (lam*t.score p ω-lam^2*(t.queryCount ω:ℝ)/8)) p ≤ 1 := by
-  change bernoulliPolynomial Q (fun ω => Real.exp (lam*ProbeTree.score p t.forget (t.leaf ω)-
-    lam^2*(ProbeTree.queries t.forget (t.leaf ω):ℝ)/8)) p ≤ 1
-  rw [leaf_expectation t ht p (fun l => Real.exp (lam*ProbeTree.score p t.forget l-
-    lam^2*(ProbeTree.queries t.forget l:ℝ)/8))]
-  exact ProbeTree.score_mgf p 0 lam t.forget
-
-end EdgeQueryTree
-end QueryTranscriptLaw
-end Percolation
-
- 
-namespace Percolation
-universe v
-variable {E : Type v}
-section CappedQueryConcentration
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace EdgeQueryTree
-
- 
-
-theorem capped_score_mgf (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
-    (p : unitInterval) (lam : ℝ) (N : ℕ) (hN : ∀ ω ⊆ Q, t.queryCount ω ≤ N) :
-    bernoulliPolynomial Q (fun ω => Real.exp (lam*t.score p ω)) p ≤ Real.exp (lam^2*(N:ℝ)/8) := by
-  calc
-    _ ≤ bernoulliPolynomial Q (fun ω => Real.exp (lam^2*(N:ℝ)/8)*
-        Real.exp (lam*t.score p ω-lam^2*(t.queryCount ω:ℝ)/8)) p := by
-      apply bernoulliPolynomial_mono _ _ _ p p.2
-      intro ω hω
-      rw [←Real.exp_add]
-      apply Real.exp_le_exp.mpr
-      have hn : (t.queryCount ω:ℝ) ≤ (N:ℝ) := Nat.cast_le.mpr (hN ω hω)
-      nlinarith [sq_nonneg lam]
-    _ = Real.exp (lam^2*(N:ℝ)/8)*bernoulliPolynomial Q
-        (fun ω => Real.exp (lam*t.score p ω-lam^2*(t.queryCount ω:ℝ)/8)) p :=
-      bernoulliPolynomial_const_mul _ _ _ _
-    _ ≤ _ := by simpa using mul_le_mul_of_nonneg_left (t.score_mgf ht p lam) (Real.exp_pos _).le
-
-theorem capped_score_chernoff (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
-    (p : unitInterval) (lam a : ℝ) (hlam : 0 ≤ lam) (N : ℕ)
-    (hN : ∀ ω ⊆ Q, t.queryCount ω ≤ N) :
-    bernoulliPolynomial Q (fun ω => if a ≤ t.score p ω then 1 else 0) p ≤
-      Real.exp (-lam*a+lam^2*(N:ℝ)/8) := by
-  calc
-    _ ≤ bernoulliPolynomial Q (fun ω => Real.exp (-lam*a)*Real.exp (lam*t.score p ω)) p := by
-      apply bernoulliPolynomial_mono _ _ _ p p.2
-      intro ω _
-      split_ifs with h
-      · rw [←Real.exp_add]
-        exact Real.one_le_exp (by nlinarith)
-      · positivity
-    _ = Real.exp (-lam*a)*bernoulliPolynomial Q (fun ω => Real.exp (lam*t.score p ω)) p :=
-      bernoulliPolynomial_const_mul _ _ _ _
-    _ ≤ Real.exp (-lam*a)*Real.exp (lam^2*(N:ℝ)/8) :=
-      mul_le_mul_of_nonneg_left (t.capped_score_mgf ht p lam N hN) (Real.exp_pos _).le
-    _ = _ := (Real.exp_add _ _).symm
-
-theorem capped_neg_score_chernoff (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
-    (p : unitInterval) (lam a : ℝ) (hlam : 0 ≤ lam) (N : ℕ)
-    (hN : ∀ ω ⊆ Q, t.queryCount ω ≤ N) :
-    bernoulliPolynomial Q (fun ω => if a ≤ -t.score p ω then 1 else 0) p ≤
-      Real.exp (-lam*a+lam^2*(N:ℝ)/8) := by
-  calc
-    _ ≤ bernoulliPolynomial Q (fun ω => Real.exp (-lam*a)*Real.exp ((-lam)*t.score p ω)) p := by
-      apply bernoulliPolynomial_mono _ _ _ p p.2
-      intro ω _
-      split_ifs with h
-      · rw [←Real.exp_add]
-        exact Real.one_le_exp (by nlinarith)
-      · positivity
-    _ = Real.exp (-lam*a)*bernoulliPolynomial Q (fun ω => Real.exp ((-lam)*t.score p ω)) p :=
-      bernoulliPolynomial_const_mul _ _ _ _
-    _ ≤ Real.exp (-lam*a)*Real.exp ((-lam)^2*(N:ℝ)/8) :=
-      mul_le_mul_of_nonneg_left (t.capped_score_mgf ht p (-lam) N hN) (Real.exp_pos _).le
-    _ = _ := by rw [←Real.exp_add]; congr 1; ring
-
- 
-
-theorem capped_score_tail (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
-    (p : unitInterval) (a : ℝ) (ha : 0 ≤ a) (N : ℕ) (hN0 : 0 < N)
-    (hN : ∀ ω ⊆ Q, t.queryCount ω ≤ N) :
-    bernoulliPolynomial Q (fun ω => if a ≤ |t.score p ω| then 1 else 0) p ≤
-      2*Real.exp (-2*a^2/(N:ℝ)) := by
-  let lam : ℝ := 4*a/(N:ℝ)
-  have hNr : 0 < (N:ℝ) := Nat.cast_pos.mpr hN0
-  have hlam : 0 ≤ lam := div_nonneg (by positivity) hNr.le
-  have he : -lam*a+lam^2*(N:ℝ)/8 = -2*a^2/(N:ℝ) := by
-    dsimp [lam]
-    field_simp
-    ring
-  calc
-    _ ≤ bernoulliPolynomial Q (fun ω => (if a ≤ t.score p ω then 1 else 0)+
-        (if a ≤ -t.score p ω then 1 else 0)) p := by
-      apply bernoulliPolynomial_mono _ _ _ p p.2
-      intro ω _
-      by_cases h : a ≤ |t.score p ω|
-      · rcases le_abs.mp h with h₁ | h₂
-        · simp only [ite_eq_left h,ite_eq_left h₁]; split_ifs <;> norm_num
-        · simp only [ite_eq_left h,ite_eq_left h₂]; split_ifs <;> norm_num
-      · simp only [ite_eq_right h]; split_ifs <;> norm_num
-    _ = bernoulliPolynomial Q (fun ω => if a ≤ t.score p ω then 1 else 0) p+
-        bernoulliPolynomial Q (fun ω => if a ≤ -t.score p ω then 1 else 0) p :=
-      bernoulliPolynomial_add _ _ _ _
-    _ ≤ Real.exp (-lam*a+lam^2*(N:ℝ)/8)+Real.exp (-lam*a+lam^2*(N:ℝ)/8) :=
-      add_le_add (t.capped_score_chernoff ht p lam a hlam N hN)
-        (t.capped_neg_score_chernoff ht p lam a hlam N hN)
-    _ = _ := by rw [he]; ring
-
-end EdgeQueryTree
-end CappedQueryConcentration
-end Percolation
-
- 
-namespace Percolation
-universe v
-variable {E : Type v}
-section QueryScoreSets
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace EdgeQueryTree
-
-noncomputable def readSet (t : EdgeQueryTree E) (Q ω : Finset E) : Finset E :=
-  Q.filter (fun e => t.reads ω e)
-
-@[simp] theorem mem_readSet (t : EdgeQueryTree E) (Q ω : Finset E) (e : E) :
-    e ∈ t.readSet Q ω ↔ e ∈ Q ∧ t.reads ω e := Finset.mem_filter
-
-theorem readSet_query (e : E) (t₀ t₁ : EdgeQueryTree E) {Q : Finset E}
-    (ht : (query e t₀ t₁).Valid Q) (ω : Finset E) :
-    (query e t₀ t₁).readSet Q ω = insert e
-      (if e ∈ ω then t₁.readSet (Q.erase e) ω else t₀.readSet (Q.erase e) ω) := by
-  ext i
-  by_cases hie : i = e
-  · subst i
-    simp only [mem_readSet,reads,true_or,and_true,Finset.mem_insert_self,iff_true]
-    exact ht.1
-  · by_cases heω : e ∈ ω <;> simp [mem_readSet,reads,hie,heω]
-
-theorem score_query (e : E) (t₀ t₁ : EdgeQueryTree E) (ω : Finset E) (p : ℝ) :
-    (query e t₀ t₁).score p ω =
-      (if e ∈ ω then t₁.score p ω else t₀.score p ω)+(if e ∈ ω then 1 else 0)-p := by
-  by_cases h : e ∈ ω
-  · simp only [score,leaf,ite_eq_left h,ProbeTree.score]; ring
-  · simp only [score,leaf,ite_eq_right h,ProbeTree.score]; ring
-
-theorem queryCount_query (e : E) (t₀ t₁ : EdgeQueryTree E) (ω : Finset E) :
-    (query e t₀ t₁).queryCount ω =
-      (if e ∈ ω then t₁.queryCount ω else t₀.queryCount ω)+1 := by
-  by_cases h : e ∈ ω <;> simp [queryCount,leaf,h,ProbeTree.queries]
-
-theorem queryCount_eq_card (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
-    (ω : Finset E) : t.queryCount ω = (t.readSet Q ω).card := by
-  induction t generalizing Q with
-  | done b =>
-    have hs : (done b : EdgeQueryTree E).readSet Q ω = ∅ := by
-      ext e
-      simp only [mem_readSet,reads,and_false,Finset.notMem_empty]
-    rw [hs]
-    rfl
-  | query e t₀ t₁ ih₀ ih₁ =>
-    rw [queryCount_query,readSet_query _ _ _ ht]
-    have h₀ : e ∉ t₀.readSet (Q.erase e) ω := by simp [readSet]
-    have h₁ : e ∉ t₁.readSet (Q.erase e) ω := by simp [readSet]
-    by_cases heω : e ∈ ω
-    · simp only [ite_eq_left heω,Finset.card_insert_of_notMem h₁,ih₁ ht.2.2]
-    · simp only [ite_eq_right heω,Finset.card_insert_of_notMem h₀,ih₀ ht.2.1]
-
- 
-theorem score_eq_sum (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
-    (ω : Finset E) (p : ℝ) :
-    t.score p ω = ∑ e ∈ t.readSet Q ω, ((if e ∈ ω then 1 else 0)-p) := by
-  induction t generalizing Q with
-  | done b =>
-    have hs : (done b : EdgeQueryTree E).readSet Q ω = ∅ := by
-      ext e
-      simp only [mem_readSet,reads,and_false,Finset.notMem_empty]
-    rw [hs,Finset.sum_empty]
-    rfl
-  | query e t₀ t₁ ih₀ ih₁ =>
-    rw [score_query,readSet_query _ _ _ ht]
-    have h₀ : e ∉ t₀.readSet (Q.erase e) ω := by simp [readSet]
-    have h₁ : e ∉ t₁.readSet (Q.erase e) ω := by simp [readSet]
-    by_cases heω : e ∈ ω
-    · simp only [ite_eq_left heω,Finset.sum_insert h₁,ih₁ ht.2.2]; ring
-    · simp only [ite_eq_right heω,Finset.sum_insert h₀,ih₀ ht.2.1]; ring
-
-end EdgeQueryTree
-end QueryScoreSets
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section IncidenceExploration
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-def IncidencePending (G : BondGraph V E) (Q : Finset E) (R : Finset V) : Prop :=
-  ∃ t : E × V × V, t.1 ∈ Q ∧ t.2.1 ∈ R ∧ G t.1 = s(t.2.1,t.2.2)
-
-noncomputable def incidencePick (G : BondGraph V E) (Q : Finset E) (R : Finset V)
-    (h : IncidencePending G Q R) : E × V × V := Classical.choose h
-
-theorem incidencePick_spec (G : BondGraph V E) (Q : Finset E) (R : Finset V)
-    (h : IncidencePending G Q R) :
-    (incidencePick G Q R h).1 ∈ Q ∧ (incidencePick G Q R h).2.1 ∈ R ∧
-      G (incidencePick G Q R h).1 = s((incidencePick G Q R h).2.1,(incidencePick G Q R h).2.2) :=
-  Classical.choose_spec h
-
-noncomputable def incidenceSearch (G : BondGraph V E) :
-    ℕ → Finset E → Finset V → EdgeQueryTree E
-  | 0, _, _ => .done false
-  | n+1, Q, R =>
-    if hc : IncidencePending G Q R then
-      let t := incidencePick G Q R hc
-      .query t.1 (incidenceSearch G n (Q.erase t.1) R)
-        (incidenceSearch G n (Q.erase t.1) (insert t.2.2 R))
-    else .done false
-
-theorem incidenceSearch_valid (G : BondGraph V E) (n : ℕ) (Q : Finset E) (R : Finset V) :
-    (incidenceSearch G n Q R).Valid Q := by
-  induction n generalizing Q R with
-  | zero => trivial
-  | succ n ih =>
-    simp only [incidenceSearch]
-    split_ifs with hc
-    · exact ⟨(incidencePick_spec G Q R hc).1,ih _ _,ih _ _⟩
-    · trivial
-
-theorem incidenceSearch_cap (G : BondGraph V E) (n : ℕ) (Q : Finset E) (R : Finset V)
-    (ω : Finset E) : (incidenceSearch G n Q R).queryCount ω ≤ n := by
-  induction n generalizing Q R with
-  | zero => exact le_rfl
-  | succ n ih =>
-    simp only [incidenceSearch]
-    split_ifs
-    · rw [EdgeQueryTree.queryCount_query]
-      split <;> exact Nat.add_le_add_right (ih _ _) 1
-    · exact Nat.zero_le _
-
- 
-
-structure IncidenceInvariant (G : BondGraph V E) (Q ω : Finset E) (R C : Finset V) : Prop where
-  discovered : R ⊆ C
-  closed : ∀ a ∈ C, ∀ e ∈ ω, ∀ b, G e = s(a,b) → b ∈ C
-  reachable : ∀ y ∈ C, ∃ r ∈ R, Conn G ((ω ∩ Q : Finset E):Set E) r y
-
-theorem incidentBonds_erase (G : BondGraph V E) (Q : Finset E) (C : Finset V) (e : E) :
-    incidentBonds G (Q.erase e) C = (incidentBonds G Q C).erase e := by
-  ext f
-  simp only [mem_incidentBonds,Finset.mem_erase]
-  tauto
-
-theorem incidencePending_of_mem (G : BondGraph V E) {Q : Finset E} {R : Finset V}
-    {e : E} (h : e ∈ incidentBonds G Q R) : IncidencePending G Q R := by
-  obtain ⟨he,a,ha,hae⟩ := (mem_incidentBonds G Q R e).mp h
-  obtain ⟨b,hab⟩ := Sym2.mem_iff_exists.mp hae
-  exact ⟨(e,a,b),he,ha,hab⟩
-
-end IncidenceExploration
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section IncidenceInvariantSteps
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-namespace IncidenceInvariant
-
-theorem closed_step {G : BondGraph V E} {Q ω : Finset E} {R C : Finset V}
-    (h : IncidenceInvariant G Q ω R C) (e : E) (he : e ∉ ω) :
-    IncidenceInvariant G (Q.erase e) ω R C := by
-  refine ⟨h.discovered,h.closed,?_⟩
-  have heq : ω ∩ Q.erase e = ω ∩ Q := by ext f; simp only [Finset.mem_inter,Finset.mem_erase]; grind
-  simpa only [heq] using h.reachable
-
-theorem open_step {G : BondGraph V E} {Q ω : Finset E} {R C : Finset V}
-    (h : IncidenceInvariant G Q ω R C) {e : E} (heQ : e ∈ Q) (heω : e ∈ ω)
-    {a b : V} (ha : a ∈ R) (hab : G e = s(a,b)) :
-    IncidenceInvariant G (Q.erase e) ω (insert b R) C := by
-  have hb : b ∈ C := h.closed a (h.discovered ha) e heω b hab
-  refine ⟨Finset.insert_subset hb h.discovered,h.closed,?_⟩
-  have heq : ((ω ∩ Q : Finset E):Set E) = insert e ((ω ∩ Q.erase e : Finset E):Set E) := by
-    ext f
-    simp only [Finset.mem_coe,Finset.mem_inter,Finset.mem_erase,Set.mem_insert_iff]
-    grind
-  intro y hy
-  obtain ⟨r,hr,hry⟩ := h.reachable y hy
-  rw [heq] at hry
-  rcases (conn_insert_iff G _ e a b r y hab).mp hry with h₀ | h₁ | h₁
-  · exact ⟨r,Finset.mem_insert_of_mem hr,h₀⟩
-  · exact ⟨b,Finset.mem_insert_self _ _,h₁.2⟩
-  · exact ⟨a,Finset.mem_insert_of_mem ha,h₁.2⟩
-
-theorem no_pending {G : BondGraph V E} {Q ω : Finset E} {R C : Finset V}
-    (h : IncidenceInvariant G Q ω R C) (hp : ¬ IncidencePending G Q R) :
-    incidentBonds G Q C = ∅ := by
-  have hCR : C ⊆ R := by
-    intro y hy
-    obtain ⟨r,hr,hry⟩ := h.reachable y hy
-    clear hy
-    induction hry with
-    | refl => exact hr
-    | @tail a b _ hab ih =>
-      obtain ⟨e,he,heab⟩ := hab
-      exact (hp ⟨(e,a,b),(Finset.mem_inter.mp he).2,ih,heab⟩).elim
-  apply Finset.eq_empty_iff_forall_notMem.mpr
-  intro e he
-  apply hp
-  apply incidencePending_of_mem G (e := e)
-  obtain ⟨heQ,a,ha,hae⟩ := (mem_incidentBonds G Q C e).mp he
-  exact (mem_incidentBonds G Q R e).mpr ⟨heQ,a,hCR ha,hae⟩
-
-end IncidenceInvariant
-end IncidenceInvariantSteps
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section IncidenceCompletion
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem incidenceSearch_readSet (G : BondGraph V E) (n : ℕ) (Q ω : Finset E) (R C : Finset V)
-    (hinv : IncidenceInvariant G Q ω R C) (hn : (incidentBonds G Q C).card ≤ n) :
-    (incidenceSearch G n Q R).readSet Q ω = incidentBonds G Q C := by
-  induction n generalizing Q R with
-  | zero =>
-    have he : incidentBonds G Q C = ∅ := Finset.card_eq_zero.mp (Nat.eq_zero_of_le_zero hn)
-    simp [incidenceSearch,EdgeQueryTree.readSet,EdgeQueryTree.reads,he]
-  | succ n ih =>
-    simp only [incidenceSearch]
-    split_ifs with hc
-    · obtain ⟨he,ha,hab⟩ := incidencePick_spec G Q R hc
-      have heC : (incidencePick G Q R hc).1 ∈ incidentBonds G Q C :=
-        (mem_incidentBonds G Q C _).mpr ⟨he,_,hinv.discovered ha,by rw [hab]; simp⟩
-      have hn' : (incidentBonds G (Q.erase (incidencePick G Q R hc).1) C).card ≤ n := by
-        rw [incidentBonds_erase,Finset.card_erase_of_mem heC]
-        have hh := Finset.card_pos.mpr ⟨_,heC⟩
-        omega
-      rw [EdgeQueryTree.readSet_query _ _ _ ⟨he,incidenceSearch_valid ..,incidenceSearch_valid ..⟩]
-      by_cases heω : (incidencePick G Q R hc).1 ∈ ω
-      · rw [ite_eq_left heω,ih _ _ (hinv.open_step he heω ha hab) hn',incidentBonds_erase,
-          Finset.insert_erase heC]
-      · rw [ite_eq_right heω,ih _ _ (hinv.closed_step _ heω) hn',incidentBonds_erase,
-          Finset.insert_erase heC]
-    · rw [hinv.no_pending hc]
-      ext e
-      simp only [EdgeQueryTree.mem_readSet,EdgeQueryTree.reads,
-        Finset.notMem_empty,and_false]
-
-theorem incidenceInvariant_root (G : BondGraph V E) (Q ω : Finset E) (hω : ω ⊆ Q) (x : V) :
-    IncidenceInvariant G Q ω {x} (finiteRootComponent G ω x) := by
-  have hmem (y : V) : y ∈ finiteRootComponent G ω x ↔ Conn G (ω:Set E) x y := by
-    change y ∈ ((finiteRootComponent G ω x:Set V)) ↔ _
-    rw [coe_finiteRootComponent]
-    rfl
-  refine ⟨?_,?_,?_⟩
-  · intro y hy
-    rw [Finset.mem_singleton] at hy
-    subst y
-    exact (hmem x).mpr (conn_refl G _ _)
-  · intro a ha e he b hab
-    exact (hmem b).mpr ((hmem a).mp ha |>.tail ⟨e,he,hab⟩)
-  · intro y hy
-    refine ⟨x,Finset.mem_singleton_self _,?_⟩
-    rw [Finset.inter_eq_left.mpr hω]
-    exact (hmem y).mp hy
-
- 
-
-theorem incidenceSearch_completed_score (G : BondGraph V E) (n : ℕ) (Q ω : Finset E)
-    (hω : ω ⊆ Q) (x : V) (p : ℝ)
-    (hn : (incidentBonds G Q (finiteRootComponent G ω x)).card ≤ n) :
-    (incidenceSearch G n Q {x}).score p ω =
-      ∑ e ∈ incidentBonds G Q (finiteRootComponent G ω x), ((if e ∈ ω then 1 else 0)-p) := by
-  rw [EdgeQueryTree.score_eq_sum _ (incidenceSearch_valid ..),
-    incidenceSearch_readSet G n Q ω {x} _ (incidenceInvariant_root G Q ω hω x) hn]
-
-theorem incidenceSearch_score_tail (G : BondGraph V E) (n : ℕ) (hn : 0 < n)
-    (Q : Finset E) (R : Finset V) (p : unitInterval) (a : ℝ) (ha : 0 ≤ a) :
-    bernoulliPolynomial Q (fun ω => if a ≤ |(incidenceSearch G n Q R).score p ω| then 1 else 0) p ≤
-      2*Real.exp (-2*a^2/(n:ℝ)) :=
-  EdgeQueryTree.capped_score_tail _ (incidenceSearch_valid ..) p a ha n hn
-    (fun ω _ => incidenceSearch_cap G n Q R ω)
-
-end IncidenceCompletion
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section ExteriorExplorationLaw
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem cluster_exterior_expectation_bound (G : BondGraph V E) (Q : Finset E)
-    (x : V) (p : unitInterval) (f : Finset V → Finset E → ℝ) (K : ℝ)
-    (hf : ∀ W : Finset V, bernoulliPolynomial (exteriorBonds G Q (W:Set V)) (f W) p ≤ K) :
-    bernoulliPolynomial Q (fun ω => f (finiteRootComponent G ω x)
-      (exteriorBonds G ω (finiteRootComponent G ω x:Set V))) p ≤ K := by
-  let F := insert x (searchVertices G Q)
-  have hpoint (ω : Finset E) (hω : ω ⊆ Q) :
-      f (finiteRootComponent G ω x) (exteriorBonds G ω (finiteRootComponent G ω x:Set V)) =
-      ∑ W ∈ F.powerset, (if cluster G ω x = (W:Set V) then (1:ℝ) else 0)*
-        f W (exteriorBonds G ω (W:Set V)) := by
-    rw [←sum_component_partition G Q F x ω hω (by rfl)
-      (fun W => f W (exteriorBonds G ω (W:Set V)))]
-    apply Finset.sum_congr rfl
-    intro W _
-    split_ifs <;> simp
-  rw [bernoulliPolynomial_congr Q hpoint,bernoulliPolynomial_sum]
-  calc
-    _ = ∑ W ∈ F.powerset,
-        bernoulliPolynomial Q (fun ω => if cluster G ω x = (W:Set V) then (1:ℝ) else 0) p*
-        bernoulliPolynomial (exteriorBonds G Q (W:Set V)) (f W) p := by
-      apply Finset.sum_congr rfl
-      intro W _
-      rw [cluster_cell_exterior_factor,←cluster_cell_incident_probability G Q x (W:Set V) p]
-    _ ≤ ∑ W ∈ F.powerset,
-        bernoulliPolynomial Q (fun ω => if cluster G ω x = (W:Set V) then (1:ℝ) else 0) p*K := by
-      apply Finset.sum_le_sum
-      intro W _
-      apply mul_le_mul_of_nonneg_left (hf W)
-      exact bernoulliPolynomial_nonneg _ _ p p.2 (fun _ _ => by split_ifs <;> norm_num)
-    _ = K := by
-      rw [←Finset.sum_mul,←bernoulliPolynomial_sum]
-      have hh : bernoulliPolynomial Q (fun ω => ∑ W ∈ F.powerset,
-          if cluster G ω x = (W:Set V) then (1:ℝ) else 0) p = 1 := by
-        trans bernoulliPolynomial Q (fun _ => (1:ℝ)) p
-        · apply bernoulliPolynomial_congr
-          intro ω hω
-          exact sum_component_partition G Q F x ω hω (by rfl) (fun _ => 1)
-        · exact bernoulliPolynomial_const Q 1 p
-      rw [hh,one_mul]
-
-noncomputable def exteriorSearchScore (G : BondGraph V E) (n : ℕ) (Q : Finset E)
-    (x v : V) (p : ℝ) (ω : Finset E) : ℝ :=
-  let W := finiteRootComponent G ω x
-  if v ∈ W then 0 else
-    (incidenceSearch G n (exteriorBonds G Q (W:Set V)) {v}).score p (exteriorBonds G ω (W:Set V))
-
-theorem exteriorSearchScore_tail (G : BondGraph V E) (n : ℕ) (hn : 0 < n)
-    (Q : Finset E) (x v : V) (p : unitInterval) (a : ℝ) (ha : 0 < a) :
-    bernoulliPolynomial Q (fun ω => if a ≤ |exteriorSearchScore G n Q x v p ω| then 1 else 0) p ≤
-      2*Real.exp (-2*a^2/(n:ℝ)) := by
-  apply cluster_exterior_expectation_bound G Q x p
-    (fun W η => if a ≤ |if v ∈ W then 0 else
-      (incidenceSearch G n (exteriorBonds G Q (W:Set V)) {v}).score p η| then 1 else 0)
-  intro W
-  by_cases hv : v ∈ W
-  · simp only [ite_eq_left hv,abs_zero,ite_eq_right (not_le.mpr ha),bernoulliPolynomial_zero]
-    positivity
-  · simp only [ite_eq_right hv]
-    exact incidenceSearch_score_tail G n hn _ _ p a ha.le
-
-end ExteriorExplorationLaw
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section ClosedClusterContacts
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-noncomputable def contactBonds (G : BondGraph V E) (Q : Finset E) (A B : Finset V) : Finset E :=
-  Q.filter (fun e => ∃ a ∈ A, ∃ b ∈ B, G e = s(a,b))
-
-theorem incident_exterior_subset (G : BondGraph V E) (Q : Finset E) (A B : Finset V) :
-    incidentBonds G (exteriorBonds G Q (A:Set V)) B ⊆ incidentBonds G Q B := by
-  intro e he
-  obtain ⟨he,b,hb,hbe⟩ := (mem_incidentBonds G _ B e).mp he
-  exact (mem_incidentBonds G Q B e).mpr ⟨(Finset.mem_filter.mp he).1,b,hb,hbe⟩
-
- 
-
-theorem incident_exterior_difference (G : BondGraph V E) (Q : Finset E) (A B : Finset V)
-    (hAB : Disjoint A B) :
-    incidentBonds G Q B \ incidentBonds G (exteriorBonds G Q (A:Set V)) B =
-      contactBonds G Q A B := by
-  ext e
-  simp only [Finset.mem_sdiff,mem_incidentBonds,contactBonds,Finset.mem_filter]
-  constructor
-  · rintro ⟨⟨he,b,hb,hbe⟩,hn⟩
-    obtain ⟨a,hea⟩ := Sym2.mem_iff_exists.mp hbe
-    have hbA : b ∉ A := fun hh => Finset.disjoint_left.mp hAB hh hb
-    have ha : a ∈ A := by
-      by_contra ha
-      apply hn
-      refine ⟨Finset.mem_filter.mpr ⟨he,?_⟩,b,hb,hbe⟩
-      exact (mem_internalEdges_of_ends G (A:Set V)ᶜ e b a hea).mpr ⟨hbA,ha⟩
-    exact ⟨he,a,ha,b,hb,hea.trans (Sym2.eq_swap ..)⟩
-  · rintro ⟨he,a,ha,b,hb,hab⟩
-    refine ⟨⟨he,b,hb,by rw [hab]; simp⟩,?_⟩
-    rintro ⟨he',_⟩
-    exact (Finset.mem_filter.mp he').2 a (by rw [hab]; simp) ha
-
-theorem finiteRootComponent_disjoint (G : BondGraph V E) (ω : Finset E) (x v : V)
-    (h : ¬Conn G (ω:Set E) x v) :
-    Disjoint (finiteRootComponent G ω x) (finiteRootComponent G ω v) := by
-  apply Finset.disjoint_left.mpr
-  intro z hz hz'
-  have h₁ : Conn G (ω:Set E) x z := by
-    change z ∈ (finiteRootComponent G ω x:Set V) at hz
-    rwa [coe_finiteRootComponent] at hz
-  have h₂ : Conn G (ω:Set E) v z := by
-    change z ∈ (finiteRootComponent G ω v:Set V) at hz'
-    rwa [coe_finiteRootComponent] at hz'
-  exact h (h₁.trans (conn_symm G _ h₂))
-
-theorem distinct_contact_closed (G : BondGraph V E) (Q ω : Finset E) (x v : V)
-    (h : ¬Conn G (ω:Set E) x v) {e : E}
-    (he : e ∈ contactBonds G Q (finiteRootComponent G ω x) (finiteRootComponent G ω v)) : e ∉ ω := by
-  obtain ⟨_,a,ha,b,hb,hab⟩ := Finset.mem_filter.mp he
-  have h₁ : Conn G (ω:Set E) x a := by
-    change a ∈ (finiteRootComponent G ω x:Set V) at ha
-    rwa [coe_finiteRootComponent] at ha
-  have h₂ : Conn G (ω:Set E) v b := by
-    change b ∈ (finiteRootComponent G ω v:Set V) at hb
-    rwa [coe_finiteRootComponent] at hb
-  intro heω
-  exact h ((h₁.tail ⟨e,heω,hab⟩).trans (conn_symm G _ h₂))
-
-theorem finiteRootComponent_exterior (G : BondGraph V E) (ω : Finset E) (x v : V)
-    (h : ¬Conn G (ω:Set E) x v) :
-    finiteRootComponent G (exteriorBonds G ω (finiteRootComponent G ω x:Set V)) v =
-      finiteRootComponent G ω v := by
-  apply Finset.coe_injective
-  simp only [coe_finiteRootComponent,coe_exteriorBonds]
-  ext z
-  exact conn_exterior_of_cluster_eq G ω x v z (cluster G ω x) rfl h
-
-end ClosedClusterContacts
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section ExplorationScoreDifference
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem incidence_exterior_score_difference (G : BondGraph V E) (n : ℕ) (Q ω : Finset E)
-    (hω : ω ⊆ Q) (x v : V) (p : ℝ) (h : ¬Conn G (ω:Set E) x v)
-    (hn : (incidentBonds G Q (finiteRootComponent G ω v)).card ≤ n) :
-    (incidenceSearch G n Q {v}).score p ω - exteriorSearchScore G n Q x v p ω =
-      -p*(contactBonds G Q (finiteRootComponent G ω x) (finiteRootComponent G ω v)).card := by
-  let A := finiteRootComponent G ω x
-  let B := finiteRootComponent G ω v
-  let R := exteriorBonds G Q (A:Set V)
-  let η := exteriorBonds G ω (A:Set V)
-  have hv : v ∉ A := by
-    intro hv
-    apply h
-    change v ∈ (A:Set V) at hv
-    simpa only [A,coe_finiteRootComponent,cluster,Set.mem_ofPred_eq] using hv
-  have hη : η ⊆ R := by
-    intro e he
-    obtain ⟨he,heA⟩ := Finset.mem_filter.mp he
-    exact Finset.mem_filter.mpr ⟨hω he,heA⟩
-  have hB : finiteRootComponent G η v = B := finiteRootComponent_exterior G ω x v h
-  have hsub : incidentBonds G R B ⊆ incidentBonds G Q B := incident_exterior_subset G Q A B
-  have hn' : (incidentBonds G R (finiteRootComponent G η v)).card ≤ n := by
-    rw [hB]
-    exact (Finset.card_le_card hsub).trans hn
-  have hs₁ := incidenceSearch_completed_score G n Q ω hω v p hn
-  have hs₂ := incidenceSearch_completed_score G n R η hη v p hn'
-  rw [hB] at hs₂
-  have hb (e : E) (he : e ∈ incidentBonds G R B) : (e ∈ η ↔ e ∈ ω) := by
-    have heR := ((mem_incidentBonds G R B e).mp he).1
-    change e ∈ exteriorBonds G ω (A:Set V) ↔ e ∈ ω
-    simp only [exteriorBonds,Finset.mem_filter]
-    exact and_iff_left (Finset.mem_filter.mp heR).2
-  have hs₂' : (incidenceSearch G n R {v}).score p η =
-      ∑ e ∈ incidentBonds G R B, ((if e ∈ ω then 1 else 0)-p) := by
-    rw [hs₂]
-    exact Finset.sum_congr rfl (fun e he => by simp only [hb e he])
-  change (incidenceSearch G n Q {v}).score p ω -
-    (if v ∈ A then 0 else (incidenceSearch G n R {v}).score p η) = _
-  rw [ite_eq_right hv,hs₁,hs₂',←Finset.sum_sdiff hsub,add_sub_cancel_right]
-  change (∑ e ∈ incidentBonds G Q B \ incidentBonds G (exteriorBonds G Q (A:Set V)) B,
-    ((if e ∈ ω then (1:ℝ) else 0)-p)) = _
-  rw [incident_exterior_difference G Q A B (finiteRootComponent_disjoint G ω x v h)]
-  have hc (e : E) (he : e ∈ contactBonds G Q A B) : e ∉ ω := distinct_contact_closed G Q ω x v h he
-  rw [show (∑ e ∈ contactBonds G Q A B, ((if e ∈ ω then (1:ℝ) else 0)-p)) =
-      ∑ _e ∈ contactBonds G Q A B, -p from Finset.sum_congr rfl (fun e he => by rw [ite_eq_right (hc e he)]; ring)]
-  simp only [Finset.sum_const,nsmul_eq_mul]
-  ring
-
-end ExplorationScoreDifference
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section ContactConcentration
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem contactBonds_comm (G : BondGraph V E) (Q : Finset E) (A B : Finset V) :
-    contactBonds G Q A B = contactBonds G Q B A := by
-  ext e
-  simp only [contactBonds,Finset.mem_filter]
-  constructor <;> rintro ⟨he,a,ha,b,hb,hab⟩
-  · exact ⟨he,b,hb,a,ha,hab.trans (Sym2.eq_swap ..)⟩
-  · exact ⟨he,b,hb,a,ha,hab.trans (Sym2.eq_swap ..)⟩
-
- 
-
-theorem ordered_cluster_contact_tail (G : BondGraph V E) (Q : Finset E)
-    (x v : V) (p : unitInterval) (N : ℕ) (hN : 0 < N) (a : ℝ) (ha : 0 < a) :
-    bernoulliPolynomial Q (fun ω => if ¬Conn G (ω:Set E) x v ∧
-      (incidentBonds G Q (finiteRootComponent G ω v)).card ≤ N ∧
-      2*a ≤ (p:ℝ)*(contactBonds G Q (finiteRootComponent G ω x)
-        (finiteRootComponent G ω v)).card then 1 else 0) p ≤
-      4*Real.exp (-2*a^2/(N:ℝ)) := by
-  calc
-    _ ≤ bernoulliPolynomial Q (fun ω =>
-        (if a ≤ |(incidenceSearch G N Q {v}).score p ω| then 1 else 0)+
-        (if a ≤ |exteriorSearchScore G N Q x v p ω| then 1 else 0)) p := by
-      apply bernoulliPolynomial_mono _ _ _ p p.2
-      intro ω hω
-      by_cases h : ¬Conn G (ω:Set E) x v ∧
-          (incidentBonds G Q (finiteRootComponent G ω v)).card ≤ N ∧
-          2*a ≤ (p:ℝ)*(contactBonds G Q (finiteRootComponent G ω x)
-            (finiteRootComponent G ω v)).card
-      · have hd := incidence_exterior_score_difference G N Q ω hω x v p h.1 h.2.1
-        have huo : a ≤ |(incidenceSearch G N Q {v}).score p ω| ∨
-            a ≤ |exteriorSearchScore G N Q x v p ω| := by
-          by_contra hh
-          push Not at hh
-          have h₁ := (abs_lt.mp hh.1).1
-          have h₂ := (abs_lt.mp hh.2).2
-          nlinarith [h.2.2]
-        simp only [ite_eq_left h]
-        rcases huo with huo | huo <;> simp only [ite_eq_left huo] <;> split_ifs <;> norm_num
-      · simp only [ite_eq_right h]; split_ifs <;> norm_num
-    _ = bernoulliPolynomial Q (fun ω =>
-        if a ≤ |(incidenceSearch G N Q {v}).score p ω| then 1 else 0) p+
-        bernoulliPolynomial Q (fun ω =>
-        if a ≤ |exteriorSearchScore G N Q x v p ω| then 1 else 0) p :=
-      bernoulliPolynomial_add _ _ _ _
-    _ ≤ 2*Real.exp (-2*a^2/(N:ℝ))+2*Real.exp (-2*a^2/(N:ℝ)) :=
-      add_le_add (incidenceSearch_score_tail G N hN Q {v} p a ha.le)
-        (exteriorSearchScore_tail G N hN Q x v p a ha)
-    _ = _ := by ring
-
-end ContactConcentration
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Exploration.FractionalMoments
+
+namespace OAI
+
+/-! Tail Bounds for bond percolation. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section ClusterTailDerivative
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem finiteClusterTail_derivative_bound (G : BondGraph V E) (Q : Finset E) (x : V)
+    (n : ℕ) (p h : unitInterval) (hn : 1 ≤ (h:ℝ)*n) (α C : ℝ)
+    (hα : 0 ≤ α) (hα1 : α ≤ 1) (hC0 : 0 < C) (hsmall : C*(h:ℝ)^α ≤ 1/4)
+    (hfin : ∀ y, ∀ᵐ ω ∂law p, (cluster G ω y).Finite)
+    (hI : ∀ y, Integrable (fun ω => ((cluster G ω y).ncard:ℝ)^α) (law p))
+    (hC : ∀ y, (∫ ω, ((cluster G ω y).ncard:ℝ)^α ∂law p) ≤ C) :
+    finiteClusterTail G Q x n p ≤ 8*C*(h:ℝ)^α*deriv (finiteClusterTail G Q x n) p := by
+  have hmark := (markedConnection_le_cluster_power G Q x h p α hα hα1 (hfin x) (hI x)).trans
+    (mul_le_mul_of_nonneg_left (hC x) (Real.rpow_nonneg h.2.1 α))
+  rw [mul_comm ((h:ℝ)^α) C] at hmark
+  have hlow := finiteSeed_covariance_lower G Q x n p h hn (C*(h:ℝ)^α) hmark
+  have hupp := finiteSeed_covariance_upper G Q x p h α C hα hα1 hfin hI hC
+    (finiteClusterTailIndicator G x n) (finiteClusterTailIndicator_mono G x n)
+  have hF := finiteClusterTail_nonneg G Q x n p
+  have hd := finiteClusterTail_deriv_nonneg G Q x n p
+  have hp : (p:ℝ)*(1-(p:ℝ)) ≤ 1 := by nlinarith [p.2.1,p.2.2]
+  have hprod := mul_le_mul_of_nonneg_right hp hd
+  have hR : 0 ≤ 2*C*(h:ℝ)^α := mul_nonneg (by positivity) (Real.rpow_nonneg h.2.1 α)
+  have hh := mul_le_mul_of_nonneg_left hprod hR
+  have ha := le_abs_self (finiteCovariance Q p (finiteClusterTailIndicator G x n)
+    (fun ω => finiteSeedProbability G (insert x (searchVertices G Q)) ω x h))
+  have hs := mul_le_mul_of_nonneg_right hsmall hF
+  change |_| ≤ _*deriv (finiteClusterTail G Q x n) p at hupp
+  nlinarith
+
+theorem exponential_transfer_of_deriv_lower (H : ℝ → ℝ) (a b K : ℝ)
+    (hH : Differentiable ℝ H) (hab : a ≤ b)
+    (hb : ∀ t ∈ Set.Icc a b, K*H t ≤ deriv H t) :
+    H a ≤ Real.exp (-K*(b-a))*H b := by
+  let F (t : ℝ) := Real.exp (-K*(t-a))*H t
+  have hF (t : ℝ) : HasDerivAt F (Real.exp (-K*(t-a))*(deriv H t-K*H t)) t := by
+    have he := (((hasDerivAt_id t).sub_const a).const_mul (-K)).exp
+    convert! he.mul (hH t).hasDerivAt using 1
+    simp only [id_eq]
+    ring
+  have hmono : MonotoneOn F (Set.Icc a b) := by
+    apply monotoneOn_of_deriv_nonneg (convex_Icc a b)
+      (fun t _ => (hF t).continuousAt.continuousWithinAt)
+      (fun t _ => (hF t).differentiableAt.differentiableWithinAt)
+    intro t ht
+    rw [(hF t).deriv]
+    exact mul_nonneg (Real.exp_pos _).le (sub_nonneg.mpr (hb t (interior_subset ht)))
+  have hh := hmono ⟨le_rfl,hab⟩ ⟨hab,le_rfl⟩ hab
+  simpa only [F,sub_self,mul_zero,Real.exp_zero,one_mul] using hh
+
+end ClusterTailDerivative
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section FiniteStretchedTail
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem cluster_finite_ae_mono [Countable E] (G : BondGraph V E)
+    (x : V) (p q : unitInterval) (hpq : p ≤ q)
+    (hq : ∀ᵐ ω ∂law q, (cluster G ω x).Finite) :
+    ∀ᵐ ω ∂law p, (cluster G ω x).Finite := by
+  have hlabels : ∀ᵐ labels ∂(labelLaw : Measure (E → unitInterval)),
+      (cluster G (openEdges labels q) x).Finite := by
+    apply ae_of_ae_map (μ := (labelLaw : Measure (E → unitInterval)))
+      (p := fun ω : Set E => (cluster G ω x).Finite) (measurable_openEdges q).aemeasurable
+    rwa [map_openEdges]
+  rw [←map_openEdges p]
+  apply (ae_map_iff (measurable_openEdges p).aemeasurable
+    (by simpa only [Set.compl_ofPred,Set.not_infinite] using (measurableSet_infinite_cluster G x).compl)).mpr
+  filter_upwards [hlabels] with labels hl
+  exact hl.subset (fun y h => conn_mono G (openEdges_mono labels hpq) h)
+
+theorem finiteClusterTail_le_markedConnection (G : BondGraph V E) (Q : Finset E)
+    (x : V) (n : ℕ) (p h : unitInterval) (hn : 1 ≤ (h:ℝ)*n) :
+    finiteClusterTail G Q x n p ≤ 2*markedConnectionPolynomial G Q x h p := by
+  have hh : (1:ℝ)/2*finiteClusterTail G Q x n p ≤ markedConnectionPolynomial G Q x h p := by
+    rw [finiteClusterTail,←bernoulliPolynomial_const_mul,markedConnection_component_polynomial]
+    apply bernoulliPolynomial_mono _ _ _ p p.2
+    intro ω _
+    unfold finiteClusterTailIndicator
+    by_cases hs : n ≤ (finiteRootComponent G ω x).card
+    · simp only [ite_eq_left hs,mul_one]
+      apply one_sub_pow_one_sub_ge_half
+      exact hn.trans (mul_le_mul_of_nonneg_left (by exact_mod_cast hs) h.2.1)
+    · simp only [ite_eq_right hs,mul_zero]
+      apply sub_nonneg.mpr
+      exact pow_le_one₀ (sub_nonneg.mpr h.2.2) (by linarith [h.2.1])
+  linarith
+
+ 
+
+theorem finiteClusterTail_stretched [Countable E] (G : BondGraph V E) (hloc : LocallyFinite G)
+    (Q : Finset E) (x : V) (n : ℕ) (p q h : unitInterval) (hpq : p ≤ q)
+    (hh : 0 < (h:ℝ)) (hn : 1 ≤ (h:ℝ)*n) (α C : ℝ)
+    (hα : 0 ≤ α) (hα1 : α ≤ 1) (hC0 : 0 < C) (hsmall : C*(h:ℝ)^α ≤ 1/4)
+    (hfin : ∀ y, ∀ᵐ ω ∂law q, (cluster G ω y).Finite)
+    (hI : ∀ y, Integrable (fun ω => ((cluster G ω y).ncard:ℝ)^α) (law q))
+    (hC : ∀ y, (∫ ω, ((cluster G ω y).ncard:ℝ)^α ∂law q) ≤ C) :
+    finiteClusterTail G Q x n p ≤
+      Real.exp (-((q:ℝ)-(p:ℝ))/(8*C*(h:ℝ)^α))*(2*C*(h:ℝ)^α) := by
+  have hden : 0 < 8*C*(h:ℝ)^α := by positivity
+  have hexp := exponential_transfer_of_deriv_lower (finiteClusterTail G Q x n) p q
+    (1/(8*C*(h:ℝ)^α)) (fun t => (hasDerivAt_bernoulliPolynomial Q _ t).differentiableAt) hpq
+  have hder : ∀ t ∈ Set.Icc (p:ℝ) (q:ℝ),
+      (1/(8*C*(h:ℝ)^α))*finiteClusterTail G Q x n t ≤ deriv (finiteClusterTail G Q x n) t := by
+    intro t ht
+    let r : unitInterval := ⟨t,⟨p.2.1.trans ht.1,ht.2.trans q.2.2⟩⟩
+    have hrq : r ≤ q := ht.2
+    have hmoment (y : V) := cluster_power_mono_integral G hloc y r q hrq α hα (hfin y) (hI y)
+    have he := finiteClusterTail_derivative_bound G Q x n r h hn α C hα hα1 hC0 hsmall
+      (fun y => cluster_finite_ae_mono G y r q hrq (hfin y)) (fun y => (hmoment y).1)
+      (fun y => (hmoment y).2.trans (hC y))
+    rw [one_div,mul_comm,←div_eq_mul_inv]
+    apply (div_le_iff₀ hden).mpr
+    simpa only [mul_comm (deriv (finiteClusterTail G Q x n) t)] using he
+  have he := hexp hder
+  have hb := (finiteClusterTail_le_markedConnection G Q x n q h hn).trans
+    (mul_le_mul_of_nonneg_left ((markedConnection_le_cluster_power G Q x h q α hα hα1 (hfin x) (hI x)).trans
+      (mul_le_mul_of_nonneg_left (hC x) (Real.rpow_nonneg hh.le α))) (by norm_num : (0:ℝ) ≤ 2))
+  have hm := mul_le_mul_of_nonneg_left hb (Real.exp_pos (-((q:ℝ)-(p:ℝ))/(8*C*(h:ℝ)^α))).le
+  have heq : -(1/(8*C*(h:ℝ)^α))*((q:ℝ)-(p:ℝ)) = -((q:ℝ)-(p:ℝ))/(8*C*(h:ℝ)^α) := by ring
+  rw [heq] at he
+  have hb' : Real.exp (-((q:ℝ)-(p:ℝ))/(8*C*(h:ℝ)^α))*(2*((h:ℝ)^α*C)) =
+      Real.exp (-((q:ℝ)-(p:ℝ))/(8*C*(h:ℝ)^α))*(2*C*(h:ℝ)^α) := by ring
+  exact he.trans (hm.trans_eq hb')
+
+end FiniteStretchedTail
+end Percolation
+
+ 
+namespace Percolation
+open MeasureTheory Filter
+section ExponentialPowerBound
+
+ 
+
+theorem exists_exp_neg_le_rpow (r : ℝ) (hr : 0 ≤ r) :
+    ∃ K : ℝ, 0 < K ∧ ∀ t : ℝ, 0 < t → Real.exp (-t) ≤ K*t^(-r) := by
+  obtain ⟨m,hm⟩ := exists_nat_ge r
+  let K : ℝ := m.factorial
+  have hK1 : 1 ≤ K := by
+    dsimp [K]
+    exact_mod_cast (Nat.succ_le_of_lt (Nat.factorial_pos m))
+  refine ⟨K,by linarith,?_⟩
+  intro t ht
+  rw [Real.rpow_neg ht.le,←div_eq_mul_inv]
+  apply (le_div_iff₀ (Real.rpow_pos_of_pos ht r)).mpr
+  by_cases ht1 : t ≤ 1
+  · have hp := Real.rpow_le_one ht.le ht1 hr
+    have he : Real.exp (-t) ≤ 1 := by rw [Real.exp_le_one_iff]; linarith
+    exact (mul_le_of_le_one_left (Real.rpow_nonneg ht.le r) he).trans (hp.trans hK1)
+  · have hpow : t^r ≤ t^m := by
+      simpa only [Real.rpow_natCast] using Real.rpow_le_rpow_of_exponent_le (le_of_not_ge ht1) hm
+    have hTaylor : t^m ≤ K*Real.exp t := by
+      have hh := Real.pow_div_factorial_le_exp t ht.le m
+      exact (div_le_iff₀ (show (0:ℝ) < (m.factorial:ℝ) by positivity)).mp hh |>.trans_eq (mul_comm _ _)
+    calc
+      _ ≤ Real.exp (-t)*(K*Real.exp t) :=
+        mul_le_mul_of_nonneg_left (hpow.trans hTaylor) (Real.exp_pos _).le
+      _ = K := by rw [Real.exp_neg]; field_simp
+
+end ExponentialPowerBound
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section PolynomialTails
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem exp_stretched_power_bound (r K δ C z : ℝ) (hδ : 0 < δ) (hC : 0 < C)
+    (hz : 0 < z) (hK : ∀ t : ℝ, 0 < t → Real.exp (-t) ≤ K*t^(-r)) :
+    Real.exp (-δ/(8*C*z))*(2*C*z) ≤
+      (2*C*K*(8*C)^r)*δ^(-r)*z^(1+r) := by
+  have he := mul_le_mul_of_nonneg_right (hK (δ/(8*C*z)) (by positivity))
+    (by positivity : 0 ≤ 2*C*z)
+  have hid : (δ/(8*C*z))^(-r)*(2*C*z) =
+      (2*C*(8*C)^r)*δ^(-r)*z^(1+r) := by
+    rw [Real.div_rpow hδ.le (by positivity),Real.mul_rpow (by positivity : 0 ≤ 8*C) hz.le,
+      Real.rpow_neg (by positivity : 0 ≤ 8*C),Real.rpow_neg hz.le,
+      Real.rpow_add hz,Real.rpow_one]
+    field_simp
+  calc
+    _ ≤ K*(δ/(8*C*z))^(-r)*(2*C*z) := by simpa only [neg_div] using he
+    _ = K*((δ/(8*C*z))^(-r)*(2*C*z)) := by ring
+    _ = _ := by rw [hid]; ring
+
+ 
+
+theorem finiteClusterTail_polynomial_large [Countable E] (G : BondGraph V E) (hloc : LocallyFinite G)
+    (q : unitInterval) (α C r : ℝ) (hα : 0 ≤ α) (hα1 : α ≤ 1) (hC0 : 0 < C)
+    (hr : 0 ≤ r)
+    (hfin : ∀ y, ∀ᵐ ω ∂law q, (cluster G ω y).Finite)
+    (hI : ∀ y, Integrable (fun ω => ((cluster G ω y).ncard:ℝ)^α) (law q))
+    (hC : ∀ y, (∫ ω, ((cluster G ω y).ncard:ℝ)^α ∂law q) ≤ C) :
+    ∃ A : ℝ, 0 < A ∧ ∀ (Q : Finset E) (x : V) (p : unitInterval), p < q →
+      ∀ n : ℕ, 1 ≤ n → C*(n:ℝ)^(-α) ≤ 1/4 →
+      finiteClusterTail G Q x n p ≤ A*((q:ℝ)-(p:ℝ))^(-r)*(n:ℝ)^(-α*(1+r)) := by
+  obtain ⟨K,hK,hKb⟩ := exists_exp_neg_le_rpow r hr
+  refine ⟨2*C*K*(8*C)^r,by positivity,?_⟩
+  intro Q x p hp n hn hsmall
+  have hn0 : (0:ℝ) < n := by exact_mod_cast (by omega : 0 < n)
+  let h : unitInterval := ⟨1/(n:ℝ),⟨by positivity,by apply (div_le_one hn0).mpr; exact_mod_cast hn⟩⟩
+  have hh : 0 < (h:ℝ) := by dsimp [h]; positivity
+  have hhpow : (h:ℝ)^α = (n:ℝ)^(-α) := by
+    dsimp [h]
+    rw [one_div,Real.inv_rpow hn0.le,Real.rpow_neg hn0.le]
+  have hhprod : 1 ≤ (h:ℝ)*n := by dsimp [h]; rw [one_div,inv_mul_cancel₀ hn0.ne']
+  have hst := finiteClusterTail_stretched G hloc Q x n p q h hp.le hh hhprod α C hα hα1 hC0
+    (by rwa [hhpow]) hfin hI hC
+  have he := exp_stretched_power_bound r K ((q:ℝ)-(p:ℝ)) C ((h:ℝ)^α)
+    (sub_pos.mpr hp) hC0 (Real.rpow_pos_of_pos hh α) hKb
+  apply hst.trans
+  convert! he using 1
+  rw [hhpow,←Real.rpow_mul hn0.le]
+
+end PolynomialTails
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section UniformPolynomialTails
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem finiteClusterTail_le_one (G : BondGraph V E) (Q : Finset E) (x : V)
+    (n : ℕ) (p : unitInterval) : finiteClusterTail G Q x n p ≤ 1 := by
+  rw [finiteClusterTail,←bernoulliPolynomial_const Q 1 p]
+  apply bernoulliPolynomial_mono _ _ _ p p.2
+  intro ω _
+  unfold finiteClusterTailIndicator
+  split_ifs <;> norm_num
+
+ 
+
+theorem finiteClusterTail_polynomial [Countable E] (G : BondGraph V E) (hloc : LocallyFinite G)
+    (q : unitInterval) (α C r : ℝ) (hα : 0 < α) (hα1 : α ≤ 1) (hC0 : 0 < C)
+    (hr : 0 ≤ r)
+    (hfin : ∀ y, ∀ᵐ ω ∂law q, (cluster G ω y).Finite)
+    (hI : ∀ y, Integrable (fun ω => ((cluster G ω y).ncard:ℝ)^α) (law q))
+    (hC : ∀ y, (∫ ω, ((cluster G ω y).ncard:ℝ)^α ∂law q) ≤ C) :
+    ∃ A : ℝ, 0 < A ∧ ∀ (Q : Finset E) (x : V) (p : unitInterval), p < q →
+      ∀ n : ℕ, 1 ≤ n →
+      finiteClusterTail G Q x n p ≤ A*((q:ℝ)-(p:ℝ))^(-r)*(n:ℝ)^(-α*(1+r)) := by
+  obtain ⟨A,hA,hbig⟩ := finiteClusterTail_polynomial_large G hloc q α C r hα.le hα1 hC0 hr hfin hI hC
+  have ht : Tendsto (fun n : ℕ => C*(n:ℝ)^(-α)) atTop (nhds 0) := by
+    simpa only [mul_zero,Function.comp_apply] using tendsto_const_nhds.mul
+      ((tendsto_rpow_neg_atTop hα).comp tendsto_natCast_atTop_atTop)
+  obtain ⟨N,hN⟩ := eventually_atTop.mp (ht.eventually (gt_mem_nhds (by norm_num : (0:ℝ) < 1/4)))
+  let s := α*(1+r)
+  have hs : 0 ≤ s := by dsimp [s]; positivity
+  let B : ℝ := ((N+1:ℕ):ℝ)^s+A
+  have hB : 0 < B := by dsimp [B]; positivity
+  refine ⟨B,hB,?_⟩
+  intro Q x p hp n hn
+  have hn0 : (0:ℝ) < n := by exact_mod_cast (by omega : 0 < n)
+  have hδ : 0 < (q:ℝ)-(p:ℝ) := sub_pos.mpr hp
+  have hhead : 1 ≤ ((q:ℝ)-(p:ℝ))^(-r) :=
+    Real.one_le_rpow_of_pos_of_le_one_of_nonpos hδ (by linarith [p.2.1,q.2.2]) (by linarith)
+  by_cases hnN : N ≤ n
+  · apply (hbig Q x p hp n hn (hN n hnN).le).trans
+    exact mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right
+      (by dsimp [B]; linarith [Real.rpow_nonneg (Nat.cast_nonneg (N+1)) s]) (Real.rpow_nonneg hδ.le _)) (Real.rpow_nonneg hn0.le _)
+  · have hnB : (n:ℝ)^s ≤ B := by
+      apply (Real.rpow_le_rpow hn0.le (show (n:ℝ) ≤ ((N+1:ℕ):ℝ) by exact_mod_cast (by omega : n ≤ N+1)) hs).trans
+      dsimp [B]
+      linarith
+    have hBδ : B ≤ B*((q:ℝ)-(p:ℝ))^(-r) := by nlinarith
+    have hbound : 1 ≤ B*((q:ℝ)-(p:ℝ))^(-r)*(n:ℝ)^(-s) := by
+      rw [Real.rpow_neg hn0.le,←div_eq_mul_inv]
+      exact (one_le_div (Real.rpow_pos_of_pos hn0 s)).mpr (hnB.trans hBδ)
+    exact (finiteClusterTail_le_one G Q x n p).trans (by simpa only [s,neg_mul] using hbound)
+
+end UniformPolynomialTails
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section TailSummation
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem natCast_eq_sum_tail (k M : ℕ) (hk : k ≤ M) :
+    (k:ℝ) = ∑ i ∈ Finset.range M, if i+1 ≤ k then (1:ℝ) else 0 := by
+  rw [Finset.sum_boole]
+  congr 1
+  have hh : (Finset.range M).filter (fun i => i+1 ≤ k) = Finset.range k := by
+    ext i
+    simp only [Finset.mem_filter,Finset.mem_range]
+    omega
+  rw [hh,Finset.card_range]
+
+theorem finiteClusterSize_eq_sum_tail (G : BondGraph V E) (Q : Finset E) (x : V) (p : ℝ) :
+    bernoulliPolynomial Q (fun ω => ((finiteRootComponent G ω x).card:ℝ)) p =
+      ∑ n ∈ Finset.range (insert x (searchVertices G Q)).card, finiteClusterTail G Q x (n+1) p := by
+  simp only [finiteClusterTail]
+  rw [←bernoulliPolynomial_sum]
+  apply bernoulliPolynomial_congr
+  intro ω hω
+  exact natCast_eq_sum_tail _ _ (Finset.card_le_card (finiteRootComponent_subset G Q _ x ω hω (by rfl)))
+
+theorem summable_shifted_rpow (s : ℝ) (hs : 1 < s) :
+    Summable (fun n : ℕ => ((n+1:ℕ):ℝ)^(-s)) := by
+  exact (summable_nat_add_iff 1).mpr (Real.summable_nat_rpow.mpr (by linarith))
+
+ 
+theorem finiteClusterSize_bound_of_tail (G : BondGraph V E) (Q : Finset E) (x : V)
+    (p : unitInterval) (A s : ℝ) (hA : 0 ≤ A) (hs : 1 < s)
+    (hf : ∀ n : ℕ, 1 ≤ n → finiteClusterTail G Q x n p ≤ A*(n:ℝ)^(-s)) :
+    bernoulliPolynomial Q (fun ω => ((finiteRootComponent G ω x).card:ℝ)) p ≤
+      A*(∑' n : ℕ, ((n+1:ℕ):ℝ)^(-s)) := by
+  rw [finiteClusterSize_eq_sum_tail,←tsum_mul_left]
+  have hsumm := (summable_shifted_rpow s hs).mul_left A
+  apply (Finset.sum_le_sum (fun n _ => hf (n+1) (by omega))).trans
+  exact hsumm.sum_le_tsum _ (fun n _ => mul_nonneg hA (Real.rpow_nonneg (Nat.cast_nonneg _) _))
+
+end TailSummation
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section SusceptibilityExhaustion
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem finite_connection_row_le_size (G : BondGraph V E) (Q : Finset E)
+    (x : V) (K : Finset V) (p : unitInterval) :
+    (∑ y ∈ K, bernoulliPolynomial Q (fun ω => if Conn G ω x y then (1:ℝ) else 0) p) ≤
+      bernoulliPolynomial Q (fun ω => ((finiteRootComponent G ω x).card:ℝ)) p := by
+  rw [←bernoulliPolynomial_sum]
+  apply bernoulliPolynomial_mono _ _ _ p p.2
+  intro ω _
+  rw [Finset.sum_boole]
+  apply Nat.cast_le.mpr
+  apply Finset.card_le_card
+  intro y hy
+  change y ∈ (finiteRootComponent G ω x : Set V)
+  rw [coe_finiteRootComponent]
+  exact (Finset.mem_filter.mp hy).2
+
+ 
+
+theorem susceptibility_bound_of_finite_size [Countable E] (G : BondGraph V E)
+    (x : V) (p : unitInterval) (B : ℝ)
+    (hB : ∀ Q : Finset E, bernoulliPolynomial Q (fun ω => ((finiteRootComponent G ω x).card:ℝ)) p ≤ B) :
+    Summable (twoPoint G p x) ∧ susceptibility G p x ≤ B := by
+  let : Encodable E := Encodable.ofCountable E
+  let Q := vertexExhaustion (V := E)
+  have hrow (K : Finset V) : (∑ y ∈ K, twoPoint G p x y) ≤ B := by
+    have ht (y : V) : Tendsto (fun n => bernoulliPolynomial (Q n)
+        (fun ω => if Conn G ω x y then (1:ℝ) else 0) p) atTop (nhds (twoPoint G p x y)) := by
+      have hc (e : E) (_ : e ∈ (Set.univ : Set E)) : ∃ n, e ∈ Q n := by
+        obtain ⟨n,hn⟩ := subset_vertexExhaustion ({e} : Finset E)
+        exact ⟨n,hn (by simp)⟩
+      simpa only [Set.inter_univ,twoPoint,Measure.real] using! finite_connection_polynomial_tendsto G Set.univ Q
+        vertexExhaustion_mono (fun _ => Set.subset_univ _) hc p x y
+    apply le_of_tendsto (tendsto_finsetSum K (fun y _ => ht y))
+    exact Filter.Eventually.of_forall (fun n => (finite_connection_row_le_size G (Q n) x K p).trans (hB (Q n)))
+  exact ⟨summable_of_sum_le (twoPoint_nonneg G p x) hrow,
+    Real.tsum_le_of_sum_le (twoPoint_nonneg G p x) hrow⟩
+
+end SusceptibilityExhaustion
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section SusceptibilityPowerFromMoment
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem susceptibility_power_of_uniform_moment [Countable E] (G : BondGraph V E) (hloc : LocallyFinite G)
+    (q : unitInterval) (α C r : ℝ) (hα : 0 < α) (hα1 : α ≤ 1) (hC0 : 0 < C)
+    (hr : 0 ≤ r) (hs : 1 < α*(1+r))
+    (hfin : ∀ y, ∀ᵐ ω ∂law q, (cluster G ω y).Finite)
+    (hI : ∀ y, Integrable (fun ω => ((cluster G ω y).ncard:ℝ)^α) (law q))
+    (hC : ∀ y, (∫ ω, ((cluster G ω y).ncard:ℝ)^α ∂law q) ≤ C) :
+    ∃ D : ℝ, 0 < D ∧ ∀ (p : unitInterval), p < q → ∀ x,
+      Summable (twoPoint G p x) ∧ susceptibility G p x ≤ D*((q:ℝ)-(p:ℝ))^(-r) := by
+  obtain ⟨A,hA,hbound⟩ := finiteClusterTail_polynomial G hloc q α C r hα hα1 hC0 hr hfin hI hC
+  let S := ∑' n : ℕ, ((n+1:ℕ):ℝ)^(-(α*(1+r)))
+  have hS : 0 ≤ S := tsum_nonneg (fun _ => Real.rpow_nonneg (Nat.cast_nonneg _) _)
+  refine ⟨1+A*S,by positivity,?_⟩
+  intro p hp x
+  have hδ : 0 < (q:ℝ)-(p:ℝ) := sub_pos.mpr (show (p:ℝ)<(q:ℝ) from hp)
+  apply susceptibility_bound_of_finite_size
+  intro Q
+  have hh := finiteClusterSize_bound_of_tail G Q x p (A*((q:ℝ)-(p:ℝ))^(-r)) (α*(1+r))
+    (mul_nonneg hA.le (Real.rpow_nonneg hδ.le _)) hs (by
+      intro n hn
+      simpa only [neg_mul] using hbound Q x p hp n hn)
+  calc
+    _ ≤ (A*((q:ℝ)-(p:ℝ))^(-r))*S := hh
+    _ = (A*S)*((q:ℝ)-(p:ℝ))^(-r) := by ring
+    _ ≤ _ := mul_le_mul_of_nonneg_right (by linarith) (Real.rpow_nonneg hδ.le _)
+
+ 
+
+theorem almost_linear_susceptibility_of_moments [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (q : unitInterval)
+    (hfin : ∀ y, ∀ᵐ ω ∂law q, (cluster G ω y).Finite)
+    (hmom : ∀ α : ℝ, 0 < α → α < 1/2 → ∃ C : ℝ, 0 < C ∧ ∀ y,
+      Integrable (fun ω => ((cluster G ω y).ncard:ℝ)^α) (law q) ∧
+      (∫ ω, ((cluster G ω y).ncard:ℝ)^α ∂law q) ≤ C)
+    (η : ℝ) (hη : 0 < η) :
+    ∃ D : ℝ, 0 < D ∧ ∀ (p : unitInterval), p < q → ∀ x,
+      Summable (twoPoint G p x) ∧ susceptibility G p x ≤ D*((q:ℝ)-(p:ℝ))^(-1-η) := by
+  let α : ℝ := 1/(2+η/2)
+  have hden : 0 < 2+η/2 := by linarith
+  have hα : 0 < α := one_div_pos.mpr hden
+  have hαhalf : α < 1/2 := by
+    apply (div_lt_div_iff₀ hden (by norm_num : (0:ℝ) < 2)).mpr
+    linarith
+  have hs : 1 < α*(1+(1+η)) := by
+    dsimp [α]
+    rw [one_div,mul_comm,←div_eq_mul_inv]
+    apply (one_lt_div hden).mpr
+    linarith
+  obtain ⟨C,hC,hm⟩ := hmom α hα hαhalf
+  obtain ⟨D,hD,hbound⟩ := susceptibility_power_of_uniform_moment G hloc q α C (1+η) hα
+    (by linarith) hC (by linarith) hs hfin (fun y => (hm y).1) (fun y => (hm y).2)
+  refine ⟨D,hD,?_⟩
+  intro p hp x
+  simpa only [neg_add_rev,sub_eq_add_neg,add_comm] using hbound p hp x
+
+end SusceptibilityPowerFromMoment
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section SourceAlmostLinear
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem cluster_finite_ae_covariant [Countable E] (G : BondGraph V E) (p : unitInterval)
+    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (x : V)
+    (hx : ∀ᵐ ω ∂law p, (cluster G ω x).Finite) :
+    ∀ᵐ ω ∂law p, (cluster G ω (a x)).Finite := by
+  have hh : ∀ᵐ ω ∂law p, (cluster G (b ⁻¹' ω) x).Finite := by
+    apply ae_of_ae_map (μ := law p) (p := fun ω : Set E => (cluster G ω x).Finite)
+      (measurable_set_preimage b).aemeasurable
+    rwa [law_map_preimage_equiv]
+  filter_upwards [hh] with ω hω
+  rw [←cluster_aut G hab ω x]
+  exact hω.image a
+
+ 
+
+theorem almost_linear_susceptibility
+    [Countable V] [Countable E] [Nonempty V] [Nonempty E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hconn : Connected G)
+    (hqt : QuasiTransitive G) (hh : 0 < hV G)
+    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
+    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
+    (hp₀ : 0 < (pc G:ℝ)) (hp₁ : (pc G:ℝ) < 1)
+    (hfin : ∀ i, ∀ᵐ ω ∂law (pc G), (cluster G ω (root i)).Finite)
+    (η : ℝ) (hη : 0 < η) :
+    ∃ D : ℝ, 0 < D ∧ ∀ (p : unitInterval), p < pc G → ∀ x,
+      susceptibility G p x ≤ D*((pc G:ℝ)-(p:ℝ))^(-1-η) := by
+  have hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i) := by
+    intro x
+    obtain ⟨i,a,ha,hax⟩ := hcov x
+    obtain ⟨b,hab⟩ := hΓ ha
+    exact ⟨i,a,b,hab,hax⟩
+  have hfinite : ∀ x, ∀ᵐ ω ∂law (pc G), (cluster G ω x).Finite := by
+    intro x
+    obtain ⟨i,a,b,hab,rfl⟩ := hcover x
+    exact cluster_finite_ae_covariant G (pc G) hab (root i) (hfin i)
+  have hmom (α : ℝ) (hα : 0 < α) (hα1 : α < 1/2) :=
+    uniform_cluster_power_of_root G hloc (pc G) root hcover α
+      (critical_root_power_integrable G hloc hconn hqt hh root weight Γ hΓ htr hcov hw hw1 hp₀ hp₁ hfin α hα hα1)
+  obtain ⟨D,hD,hb⟩ := almost_linear_susceptibility_of_moments G hloc (pc G) hfinite hmom η hη
+  exact ⟨D,hD,fun p hp x => (hb p hp x).2⟩
+
+end SourceAlmostLinear
+end Percolation
+
+ 
+namespace Percolation
+universe v
+variable {E : Type v}
+section QueryTranscriptLaw
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace EdgeQueryTree
+
+@[reducible] def forget : EdgeQueryTree E → ProbeTree
+  | .done b => .done b
+  | .query _ t₀ t₁ => .edge t₀.forget t₁.forget
+
+noncomputable def leaf (ω : Finset E) : (t : EdgeQueryTree E) → t.forget.Leaf
+  | .done _ => ()
+  | .query e t₀ t₁ => if e ∈ ω then Sum.inr (t₁.leaf ω) else Sum.inl (t₀.leaf ω)
+
+theorem leaf_congr (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
+    {ω η : Finset E} (h : ∀ e ∈ Q, e ∈ ω ↔ e ∈ η) : t.leaf ω = t.leaf η := by
+  induction t generalizing Q with
+  | done => rfl
+  | query e t₀ t₁ ih₀ ih₁ =>
+    simp only [leaf,h e ht.1]
+    split
+    · rw [ih₁ ht.2.2 (fun a ha => h a (Finset.mem_of_mem_erase ha))]
+    · rw [ih₀ ht.2.1 (fun a ha => h a (Finset.mem_of_mem_erase ha))]
+
+theorem leaf_insert (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
+    {e : E} (he : e ∉ Q) (ω : Finset E) : t.leaf (insert e ω) = t.leaf ω := by
+  apply leaf_congr t ht
+  intro a ha
+  simp [show a ≠ e from fun h => he (h ▸ ha)]
+
+ 
+theorem leaf_expectation (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
+    (p : ℝ) (f : t.forget.Leaf → ℝ) :
+    bernoulliPolynomial Q (fun ω => f (t.leaf ω)) p =
+      ∑ l, ProbeTree.weight p 0 t.forget l*f l := by
+  induction t generalizing Q with
+  | done b =>
+    change bernoulliPolynomial Q (fun _ => f ()) p = ∑ l : Unit, 1*f l
+    simp [bernoulliPolynomial_const,Finset.sum_singleton]
+  | query e t₀ t₁ ih₀ ih₁ =>
+    obtain ⟨he,h₀,h₁⟩ := ht
+    conv_lhs => rw [←Finset.insert_erase he]
+    rw [bernoulliPolynomial_insert _ e (Finset.notMem_erase e Q)]
+    have hc : bernoulliPolynomial (Q.erase e) (fun ω => f ((query e t₀ t₁).leaf ω)) p =
+        bernoulliPolynomial (Q.erase e) (fun ω => f (Sum.inl (t₀.leaf ω))) p := by
+      apply bernoulliPolynomial_congr
+      intro ω hω
+      simp only [leaf,show e ∉ ω from fun h => Finset.notMem_erase e Q (hω h),ite_false]
+    have ho : bernoulliPolynomial (Q.erase e) (fun ω => f ((query e t₀ t₁).leaf (insert e ω))) p =
+        bernoulliPolynomial (Q.erase e) (fun ω => f (Sum.inr (t₁.leaf ω))) p := by
+      apply bernoulliPolynomial_congr
+      intro ω _
+      simp only [leaf,Finset.mem_insert_self,ite_true,
+        leaf_insert t₁ h₁ (Finset.notMem_erase e Q)]
+    rw [hc,ho,ih₀ h₀ (fun l => f (Sum.inl l)),ih₁ h₁ (fun l => f (Sum.inr l))]
+    change _ = ∑ l : t₀.forget.Leaf ⊕ t₁.forget.Leaf, _
+    simp only [Fintype.sum_sum_type,ProbeTree.weight,Finset.mul_sum,mul_assoc]
+
+noncomputable def score (p : ℝ) (t : EdgeQueryTree E) (ω : Finset E) : ℝ :=
+  ProbeTree.score p t.forget (t.leaf ω)
+
+noncomputable def queryCount (t : EdgeQueryTree E) (ω : Finset E) : ℕ :=
+  ProbeTree.queries t.forget (t.leaf ω)
+
+ 
+theorem score_mgf (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
+    (p : unitInterval) (lam : ℝ) :
+    bernoulliPolynomial Q (fun ω => Real.exp (lam*t.score p ω-lam^2*(t.queryCount ω:ℝ)/8)) p ≤ 1 := by
+  change bernoulliPolynomial Q (fun ω => Real.exp (lam*ProbeTree.score p t.forget (t.leaf ω)-
+    lam^2*(ProbeTree.queries t.forget (t.leaf ω):ℝ)/8)) p ≤ 1
+  rw [leaf_expectation t ht p (fun l => Real.exp (lam*ProbeTree.score p t.forget l-
+    lam^2*(ProbeTree.queries t.forget l:ℝ)/8))]
+  exact ProbeTree.score_mgf p 0 lam t.forget
+
+end EdgeQueryTree
+end QueryTranscriptLaw
+end Percolation
+
+ 
+namespace Percolation
+universe v
+variable {E : Type v}
+section CappedQueryConcentration
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace EdgeQueryTree
+
+ 
+
+theorem capped_score_mgf (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
+    (p : unitInterval) (lam : ℝ) (N : ℕ) (hN : ∀ ω ⊆ Q, t.queryCount ω ≤ N) :
+    bernoulliPolynomial Q (fun ω => Real.exp (lam*t.score p ω)) p ≤ Real.exp (lam^2*(N:ℝ)/8) := by
+  calc
+    _ ≤ bernoulliPolynomial Q (fun ω => Real.exp (lam^2*(N:ℝ)/8)*
+        Real.exp (lam*t.score p ω-lam^2*(t.queryCount ω:ℝ)/8)) p := by
+      apply bernoulliPolynomial_mono _ _ _ p p.2
+      intro ω hω
+      rw [←Real.exp_add]
+      apply Real.exp_le_exp.mpr
+      have hn : (t.queryCount ω:ℝ) ≤ (N:ℝ) := Nat.cast_le.mpr (hN ω hω)
+      nlinarith [sq_nonneg lam]
+    _ = Real.exp (lam^2*(N:ℝ)/8)*bernoulliPolynomial Q
+        (fun ω => Real.exp (lam*t.score p ω-lam^2*(t.queryCount ω:ℝ)/8)) p :=
+      bernoulliPolynomial_const_mul _ _ _ _
+    _ ≤ _ := by simpa using mul_le_mul_of_nonneg_left (t.score_mgf ht p lam) (Real.exp_pos _).le
+
+theorem capped_score_chernoff (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
+    (p : unitInterval) (lam a : ℝ) (hlam : 0 ≤ lam) (N : ℕ)
+    (hN : ∀ ω ⊆ Q, t.queryCount ω ≤ N) :
+    bernoulliPolynomial Q (fun ω => if a ≤ t.score p ω then 1 else 0) p ≤
+      Real.exp (-lam*a+lam^2*(N:ℝ)/8) := by
+  calc
+    _ ≤ bernoulliPolynomial Q (fun ω => Real.exp (-lam*a)*Real.exp (lam*t.score p ω)) p := by
+      apply bernoulliPolynomial_mono _ _ _ p p.2
+      intro ω _
+      split_ifs with h
+      · rw [←Real.exp_add]
+        exact Real.one_le_exp (by nlinarith)
+      · positivity
+    _ = Real.exp (-lam*a)*bernoulliPolynomial Q (fun ω => Real.exp (lam*t.score p ω)) p :=
+      bernoulliPolynomial_const_mul _ _ _ _
+    _ ≤ Real.exp (-lam*a)*Real.exp (lam^2*(N:ℝ)/8) :=
+      mul_le_mul_of_nonneg_left (t.capped_score_mgf ht p lam N hN) (Real.exp_pos _).le
+    _ = _ := (Real.exp_add _ _).symm
+
+theorem capped_neg_score_chernoff (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
+    (p : unitInterval) (lam a : ℝ) (hlam : 0 ≤ lam) (N : ℕ)
+    (hN : ∀ ω ⊆ Q, t.queryCount ω ≤ N) :
+    bernoulliPolynomial Q (fun ω => if a ≤ -t.score p ω then 1 else 0) p ≤
+      Real.exp (-lam*a+lam^2*(N:ℝ)/8) := by
+  calc
+    _ ≤ bernoulliPolynomial Q (fun ω => Real.exp (-lam*a)*Real.exp ((-lam)*t.score p ω)) p := by
+      apply bernoulliPolynomial_mono _ _ _ p p.2
+      intro ω _
+      split_ifs with h
+      · rw [←Real.exp_add]
+        exact Real.one_le_exp (by nlinarith)
+      · positivity
+    _ = Real.exp (-lam*a)*bernoulliPolynomial Q (fun ω => Real.exp ((-lam)*t.score p ω)) p :=
+      bernoulliPolynomial_const_mul _ _ _ _
+    _ ≤ Real.exp (-lam*a)*Real.exp ((-lam)^2*(N:ℝ)/8) :=
+      mul_le_mul_of_nonneg_left (t.capped_score_mgf ht p (-lam) N hN) (Real.exp_pos _).le
+    _ = _ := by rw [←Real.exp_add]; congr 1; ring
+
+ 
+
+theorem capped_score_tail (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
+    (p : unitInterval) (a : ℝ) (ha : 0 ≤ a) (N : ℕ) (hN0 : 0 < N)
+    (hN : ∀ ω ⊆ Q, t.queryCount ω ≤ N) :
+    bernoulliPolynomial Q (fun ω => if a ≤ |t.score p ω| then 1 else 0) p ≤
+      2*Real.exp (-2*a^2/(N:ℝ)) := by
+  let lam : ℝ := 4*a/(N:ℝ)
+  have hNr : 0 < (N:ℝ) := Nat.cast_pos.mpr hN0
+  have hlam : 0 ≤ lam := div_nonneg (by positivity) hNr.le
+  have he : -lam*a+lam^2*(N:ℝ)/8 = -2*a^2/(N:ℝ) := by
+    dsimp [lam]
+    field_simp
+    ring
+  calc
+    _ ≤ bernoulliPolynomial Q (fun ω => (if a ≤ t.score p ω then 1 else 0)+
+        (if a ≤ -t.score p ω then 1 else 0)) p := by
+      apply bernoulliPolynomial_mono _ _ _ p p.2
+      intro ω _
+      by_cases h : a ≤ |t.score p ω|
+      · rcases le_abs.mp h with h₁ | h₂
+        · simp only [ite_eq_left h,ite_eq_left h₁]; split_ifs <;> norm_num
+        · simp only [ite_eq_left h,ite_eq_left h₂]; split_ifs <;> norm_num
+      · simp only [ite_eq_right h]; split_ifs <;> norm_num
+    _ = bernoulliPolynomial Q (fun ω => if a ≤ t.score p ω then 1 else 0) p+
+        bernoulliPolynomial Q (fun ω => if a ≤ -t.score p ω then 1 else 0) p :=
+      bernoulliPolynomial_add _ _ _ _
+    _ ≤ Real.exp (-lam*a+lam^2*(N:ℝ)/8)+Real.exp (-lam*a+lam^2*(N:ℝ)/8) :=
+      add_le_add (t.capped_score_chernoff ht p lam a hlam N hN)
+        (t.capped_neg_score_chernoff ht p lam a hlam N hN)
+    _ = _ := by rw [he]; ring
+
+end EdgeQueryTree
+end CappedQueryConcentration
+end Percolation
+
+ 
+namespace Percolation
+universe v
+variable {E : Type v}
+section QueryScoreSets
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace EdgeQueryTree
+
+noncomputable def readSet (t : EdgeQueryTree E) (Q ω : Finset E) : Finset E :=
+  Q.filter (fun e => t.reads ω e)
+
+@[simp] theorem mem_readSet (t : EdgeQueryTree E) (Q ω : Finset E) (e : E) :
+    e ∈ t.readSet Q ω ↔ e ∈ Q ∧ t.reads ω e := Finset.mem_filter
+
+theorem readSet_query (e : E) (t₀ t₁ : EdgeQueryTree E) {Q : Finset E}
+    (ht : (query e t₀ t₁).Valid Q) (ω : Finset E) :
+    (query e t₀ t₁).readSet Q ω = insert e
+      (if e ∈ ω then t₁.readSet (Q.erase e) ω else t₀.readSet (Q.erase e) ω) := by
+  ext i
+  by_cases hie : i = e
+  · subst i
+    simp only [mem_readSet,reads,true_or,and_true,Finset.mem_insert_self,iff_true]
+    exact ht.1
+  · by_cases heω : e ∈ ω <;> simp [mem_readSet,reads,hie,heω]
+
+theorem score_query (e : E) (t₀ t₁ : EdgeQueryTree E) (ω : Finset E) (p : ℝ) :
+    (query e t₀ t₁).score p ω =
+      (if e ∈ ω then t₁.score p ω else t₀.score p ω)+(if e ∈ ω then 1 else 0)-p := by
+  by_cases h : e ∈ ω
+  · simp only [score,leaf,ite_eq_left h,ProbeTree.score]; ring
+  · simp only [score,leaf,ite_eq_right h,ProbeTree.score]; ring
+
+theorem queryCount_query (e : E) (t₀ t₁ : EdgeQueryTree E) (ω : Finset E) :
+    (query e t₀ t₁).queryCount ω =
+      (if e ∈ ω then t₁.queryCount ω else t₀.queryCount ω)+1 := by
+  by_cases h : e ∈ ω <;> simp [queryCount,leaf,h,ProbeTree.queries]
+
+theorem queryCount_eq_card (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
+    (ω : Finset E) : t.queryCount ω = (t.readSet Q ω).card := by
+  induction t generalizing Q with
+  | done b =>
+    have hs : (done b : EdgeQueryTree E).readSet Q ω = ∅ := by
+      ext e
+      simp only [mem_readSet,reads,and_false,Finset.notMem_empty]
+    rw [hs]
+    rfl
+  | query e t₀ t₁ ih₀ ih₁ =>
+    rw [queryCount_query,readSet_query _ _ _ ht]
+    have h₀ : e ∉ t₀.readSet (Q.erase e) ω := by simp [readSet]
+    have h₁ : e ∉ t₁.readSet (Q.erase e) ω := by simp [readSet]
+    by_cases heω : e ∈ ω
+    · simp only [ite_eq_left heω,Finset.card_insert_of_notMem h₁,ih₁ ht.2.2]
+    · simp only [ite_eq_right heω,Finset.card_insert_of_notMem h₀,ih₀ ht.2.1]
+
+ 
+theorem score_eq_sum (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
+    (ω : Finset E) (p : ℝ) :
+    t.score p ω = ∑ e ∈ t.readSet Q ω, ((if e ∈ ω then 1 else 0)-p) := by
+  induction t generalizing Q with
+  | done b =>
+    have hs : (done b : EdgeQueryTree E).readSet Q ω = ∅ := by
+      ext e
+      simp only [mem_readSet,reads,and_false,Finset.notMem_empty]
+    rw [hs,Finset.sum_empty]
+    rfl
+  | query e t₀ t₁ ih₀ ih₁ =>
+    rw [score_query,readSet_query _ _ _ ht]
+    have h₀ : e ∉ t₀.readSet (Q.erase e) ω := by simp [readSet]
+    have h₁ : e ∉ t₁.readSet (Q.erase e) ω := by simp [readSet]
+    by_cases heω : e ∈ ω
+    · simp only [ite_eq_left heω,Finset.sum_insert h₁,ih₁ ht.2.2]; ring
+    · simp only [ite_eq_right heω,Finset.sum_insert h₀,ih₀ ht.2.1]; ring
+
+end EdgeQueryTree
+end QueryScoreSets
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section IncidenceExploration
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+def IncidencePending (G : BondGraph V E) (Q : Finset E) (R : Finset V) : Prop :=
+  ∃ t : E × V × V, t.1 ∈ Q ∧ t.2.1 ∈ R ∧ G t.1 = s(t.2.1,t.2.2)
+
+noncomputable def incidencePick (G : BondGraph V E) (Q : Finset E) (R : Finset V)
+    (h : IncidencePending G Q R) : E × V × V := Classical.choose h
+
+theorem incidencePick_spec (G : BondGraph V E) (Q : Finset E) (R : Finset V)
+    (h : IncidencePending G Q R) :
+    (incidencePick G Q R h).1 ∈ Q ∧ (incidencePick G Q R h).2.1 ∈ R ∧
+      G (incidencePick G Q R h).1 = s((incidencePick G Q R h).2.1,(incidencePick G Q R h).2.2) :=
+  Classical.choose_spec h
+
+noncomputable def incidenceSearch (G : BondGraph V E) :
+    ℕ → Finset E → Finset V → EdgeQueryTree E
+  | 0, _, _ => .done false
+  | n+1, Q, R =>
+    if hc : IncidencePending G Q R then
+      let t := incidencePick G Q R hc
+      .query t.1 (incidenceSearch G n (Q.erase t.1) R)
+        (incidenceSearch G n (Q.erase t.1) (insert t.2.2 R))
+    else .done false
+
+theorem incidenceSearch_valid (G : BondGraph V E) (n : ℕ) (Q : Finset E) (R : Finset V) :
+    (incidenceSearch G n Q R).Valid Q := by
+  induction n generalizing Q R with
+  | zero => trivial
+  | succ n ih =>
+    simp only [incidenceSearch]
+    split_ifs with hc
+    · exact ⟨(incidencePick_spec G Q R hc).1,ih _ _,ih _ _⟩
+    · trivial
+
+theorem incidenceSearch_cap (G : BondGraph V E) (n : ℕ) (Q : Finset E) (R : Finset V)
+    (ω : Finset E) : (incidenceSearch G n Q R).queryCount ω ≤ n := by
+  induction n generalizing Q R with
+  | zero => exact le_rfl
+  | succ n ih =>
+    simp only [incidenceSearch]
+    split_ifs
+    · rw [EdgeQueryTree.queryCount_query]
+      split <;> exact Nat.add_le_add_right (ih _ _) 1
+    · exact Nat.zero_le _
+
+ 
+
+structure IncidenceInvariant (G : BondGraph V E) (Q ω : Finset E) (R C : Finset V) : Prop where
+  discovered : R ⊆ C
+  closed : ∀ a ∈ C, ∀ e ∈ ω, ∀ b, G e = s(a,b) → b ∈ C
+  reachable : ∀ y ∈ C, ∃ r ∈ R, Conn G ((ω ∩ Q : Finset E):Set E) r y
+
+theorem incidentBonds_erase (G : BondGraph V E) (Q : Finset E) (C : Finset V) (e : E) :
+    incidentBonds G (Q.erase e) C = (incidentBonds G Q C).erase e := by
+  ext f
+  simp only [mem_incidentBonds,Finset.mem_erase]
+  tauto
+
+theorem incidencePending_of_mem (G : BondGraph V E) {Q : Finset E} {R : Finset V}
+    {e : E} (h : e ∈ incidentBonds G Q R) : IncidencePending G Q R := by
+  obtain ⟨he,a,ha,hae⟩ := (mem_incidentBonds G Q R e).mp h
+  obtain ⟨b,hab⟩ := Sym2.mem_iff_exists.mp hae
+  exact ⟨(e,a,b),he,ha,hab⟩
+
+end IncidenceExploration
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section IncidenceInvariantSteps
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+namespace IncidenceInvariant
+
+theorem closed_step {G : BondGraph V E} {Q ω : Finset E} {R C : Finset V}
+    (h : IncidenceInvariant G Q ω R C) (e : E) (he : e ∉ ω) :
+    IncidenceInvariant G (Q.erase e) ω R C := by
+  refine ⟨h.discovered,h.closed,?_⟩
+  have heq : ω ∩ Q.erase e = ω ∩ Q := by ext f; simp only [Finset.mem_inter,Finset.mem_erase]; grind
+  simpa only [heq] using h.reachable
+
+theorem open_step {G : BondGraph V E} {Q ω : Finset E} {R C : Finset V}
+    (h : IncidenceInvariant G Q ω R C) {e : E} (heQ : e ∈ Q) (heω : e ∈ ω)
+    {a b : V} (ha : a ∈ R) (hab : G e = s(a,b)) :
+    IncidenceInvariant G (Q.erase e) ω (insert b R) C := by
+  have hb : b ∈ C := h.closed a (h.discovered ha) e heω b hab
+  refine ⟨Finset.insert_subset hb h.discovered,h.closed,?_⟩
+  have heq : ((ω ∩ Q : Finset E):Set E) = insert e ((ω ∩ Q.erase e : Finset E):Set E) := by
+    ext f
+    simp only [Finset.mem_coe,Finset.mem_inter,Finset.mem_erase,Set.mem_insert_iff]
+    grind
+  intro y hy
+  obtain ⟨r,hr,hry⟩ := h.reachable y hy
+  rw [heq] at hry
+  rcases (conn_insert_iff G _ e a b r y hab).mp hry with h₀ | h₁ | h₁
+  · exact ⟨r,Finset.mem_insert_of_mem hr,h₀⟩
+  · exact ⟨b,Finset.mem_insert_self _ _,h₁.2⟩
+  · exact ⟨a,Finset.mem_insert_of_mem ha,h₁.2⟩
+
+theorem no_pending {G : BondGraph V E} {Q ω : Finset E} {R C : Finset V}
+    (h : IncidenceInvariant G Q ω R C) (hp : ¬ IncidencePending G Q R) :
+    incidentBonds G Q C = ∅ := by
+  have hCR : C ⊆ R := by
+    intro y hy
+    obtain ⟨r,hr,hry⟩ := h.reachable y hy
+    clear hy
+    induction hry with
+    | refl => exact hr
+    | @tail a b _ hab ih =>
+      obtain ⟨e,he,heab⟩ := hab
+      exact (hp ⟨(e,a,b),(Finset.mem_inter.mp he).2,ih,heab⟩).elim
+  apply Finset.eq_empty_iff_forall_notMem.mpr
+  intro e he
+  apply hp
+  apply incidencePending_of_mem G (e := e)
+  obtain ⟨heQ,a,ha,hae⟩ := (mem_incidentBonds G Q C e).mp he
+  exact (mem_incidentBonds G Q R e).mpr ⟨heQ,a,hCR ha,hae⟩
+
+end IncidenceInvariant
+end IncidenceInvariantSteps
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section IncidenceCompletion
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem incidenceSearch_readSet (G : BondGraph V E) (n : ℕ) (Q ω : Finset E) (R C : Finset V)
+    (hinv : IncidenceInvariant G Q ω R C) (hn : (incidentBonds G Q C).card ≤ n) :
+    (incidenceSearch G n Q R).readSet Q ω = incidentBonds G Q C := by
+  induction n generalizing Q R with
+  | zero =>
+    have he : incidentBonds G Q C = ∅ := Finset.card_eq_zero.mp (Nat.eq_zero_of_le_zero hn)
+    simp [incidenceSearch,EdgeQueryTree.readSet,EdgeQueryTree.reads,he]
+  | succ n ih =>
+    simp only [incidenceSearch]
+    split_ifs with hc
+    · obtain ⟨he,ha,hab⟩ := incidencePick_spec G Q R hc
+      have heC : (incidencePick G Q R hc).1 ∈ incidentBonds G Q C :=
+        (mem_incidentBonds G Q C _).mpr ⟨he,_,hinv.discovered ha,by rw [hab]; simp⟩
+      have hn' : (incidentBonds G (Q.erase (incidencePick G Q R hc).1) C).card ≤ n := by
+        rw [incidentBonds_erase,Finset.card_erase_of_mem heC]
+        have hh := Finset.card_pos.mpr ⟨_,heC⟩
+        omega
+      rw [EdgeQueryTree.readSet_query _ _ _ ⟨he,incidenceSearch_valid ..,incidenceSearch_valid ..⟩]
+      by_cases heω : (incidencePick G Q R hc).1 ∈ ω
+      · rw [ite_eq_left heω,ih _ _ (hinv.open_step he heω ha hab) hn',incidentBonds_erase,
+          Finset.insert_erase heC]
+      · rw [ite_eq_right heω,ih _ _ (hinv.closed_step _ heω) hn',incidentBonds_erase,
+          Finset.insert_erase heC]
+    · rw [hinv.no_pending hc]
+      ext e
+      simp only [EdgeQueryTree.mem_readSet,EdgeQueryTree.reads,
+        Finset.notMem_empty,and_false]
+
+theorem incidenceInvariant_root (G : BondGraph V E) (Q ω : Finset E) (hω : ω ⊆ Q) (x : V) :
+    IncidenceInvariant G Q ω {x} (finiteRootComponent G ω x) := by
+  have hmem (y : V) : y ∈ finiteRootComponent G ω x ↔ Conn G (ω:Set E) x y := by
+    change y ∈ ((finiteRootComponent G ω x:Set V)) ↔ _
+    rw [coe_finiteRootComponent]
+    rfl
+  refine ⟨?_,?_,?_⟩
+  · intro y hy
+    rw [Finset.mem_singleton] at hy
+    subst y
+    exact (hmem x).mpr (conn_refl G _ _)
+  · intro a ha e he b hab
+    exact (hmem b).mpr ((hmem a).mp ha |>.tail ⟨e,he,hab⟩)
+  · intro y hy
+    refine ⟨x,Finset.mem_singleton_self _,?_⟩
+    rw [Finset.inter_eq_left.mpr hω]
+    exact (hmem y).mp hy
+
+ 
+
+theorem incidenceSearch_completed_score (G : BondGraph V E) (n : ℕ) (Q ω : Finset E)
+    (hω : ω ⊆ Q) (x : V) (p : ℝ)
+    (hn : (incidentBonds G Q (finiteRootComponent G ω x)).card ≤ n) :
+    (incidenceSearch G n Q {x}).score p ω =
+      ∑ e ∈ incidentBonds G Q (finiteRootComponent G ω x), ((if e ∈ ω then 1 else 0)-p) := by
+  rw [EdgeQueryTree.score_eq_sum _ (incidenceSearch_valid ..),
+    incidenceSearch_readSet G n Q ω {x} _ (incidenceInvariant_root G Q ω hω x) hn]
+
+theorem incidenceSearch_score_tail (G : BondGraph V E) (n : ℕ) (hn : 0 < n)
+    (Q : Finset E) (R : Finset V) (p : unitInterval) (a : ℝ) (ha : 0 ≤ a) :
+    bernoulliPolynomial Q (fun ω => if a ≤ |(incidenceSearch G n Q R).score p ω| then 1 else 0) p ≤
+      2*Real.exp (-2*a^2/(n:ℝ)) :=
+  EdgeQueryTree.capped_score_tail _ (incidenceSearch_valid ..) p a ha n hn
+    (fun ω _ => incidenceSearch_cap G n Q R ω)
+
+end IncidenceCompletion
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section ExteriorExplorationLaw
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem cluster_exterior_expectation_bound (G : BondGraph V E) (Q : Finset E)
+    (x : V) (p : unitInterval) (f : Finset V → Finset E → ℝ) (K : ℝ)
+    (hf : ∀ W : Finset V, bernoulliPolynomial (exteriorBonds G Q (W:Set V)) (f W) p ≤ K) :
+    bernoulliPolynomial Q (fun ω => f (finiteRootComponent G ω x)
+      (exteriorBonds G ω (finiteRootComponent G ω x:Set V))) p ≤ K := by
+  let F := insert x (searchVertices G Q)
+  have hpoint (ω : Finset E) (hω : ω ⊆ Q) :
+      f (finiteRootComponent G ω x) (exteriorBonds G ω (finiteRootComponent G ω x:Set V)) =
+      ∑ W ∈ F.powerset, (if cluster G ω x = (W:Set V) then (1:ℝ) else 0)*
+        f W (exteriorBonds G ω (W:Set V)) := by
+    rw [←sum_component_partition G Q F x ω hω (by rfl)
+      (fun W => f W (exteriorBonds G ω (W:Set V)))]
+    apply Finset.sum_congr rfl
+    intro W _
+    split_ifs <;> simp
+  rw [bernoulliPolynomial_congr Q hpoint,bernoulliPolynomial_sum]
+  calc
+    _ = ∑ W ∈ F.powerset,
+        bernoulliPolynomial Q (fun ω => if cluster G ω x = (W:Set V) then (1:ℝ) else 0) p*
+        bernoulliPolynomial (exteriorBonds G Q (W:Set V)) (f W) p := by
+      apply Finset.sum_congr rfl
+      intro W _
+      rw [cluster_cell_exterior_factor,←cluster_cell_incident_probability G Q x (W:Set V) p]
+    _ ≤ ∑ W ∈ F.powerset,
+        bernoulliPolynomial Q (fun ω => if cluster G ω x = (W:Set V) then (1:ℝ) else 0) p*K := by
+      apply Finset.sum_le_sum
+      intro W _
+      apply mul_le_mul_of_nonneg_left (hf W)
+      exact bernoulliPolynomial_nonneg _ _ p p.2 (fun _ _ => by split_ifs <;> norm_num)
+    _ = K := by
+      rw [←Finset.sum_mul,←bernoulliPolynomial_sum]
+      have hh : bernoulliPolynomial Q (fun ω => ∑ W ∈ F.powerset,
+          if cluster G ω x = (W:Set V) then (1:ℝ) else 0) p = 1 := by
+        trans bernoulliPolynomial Q (fun _ => (1:ℝ)) p
+        · apply bernoulliPolynomial_congr
+          intro ω hω
+          exact sum_component_partition G Q F x ω hω (by rfl) (fun _ => 1)
+        · exact bernoulliPolynomial_const Q 1 p
+      rw [hh,one_mul]
+
+noncomputable def exteriorSearchScore (G : BondGraph V E) (n : ℕ) (Q : Finset E)
+    (x v : V) (p : ℝ) (ω : Finset E) : ℝ :=
+  let W := finiteRootComponent G ω x
+  if v ∈ W then 0 else
+    (incidenceSearch G n (exteriorBonds G Q (W:Set V)) {v}).score p (exteriorBonds G ω (W:Set V))
+
+theorem exteriorSearchScore_tail (G : BondGraph V E) (n : ℕ) (hn : 0 < n)
+    (Q : Finset E) (x v : V) (p : unitInterval) (a : ℝ) (ha : 0 < a) :
+    bernoulliPolynomial Q (fun ω => if a ≤ |exteriorSearchScore G n Q x v p ω| then 1 else 0) p ≤
+      2*Real.exp (-2*a^2/(n:ℝ)) := by
+  apply cluster_exterior_expectation_bound G Q x p
+    (fun W η => if a ≤ |if v ∈ W then 0 else
+      (incidenceSearch G n (exteriorBonds G Q (W:Set V)) {v}).score p η| then 1 else 0)
+  intro W
+  by_cases hv : v ∈ W
+  · simp only [ite_eq_left hv,abs_zero,ite_eq_right (not_le.mpr ha),bernoulliPolynomial_zero]
+    positivity
+  · simp only [ite_eq_right hv]
+    exact incidenceSearch_score_tail G n hn _ _ p a ha.le
+
+end ExteriorExplorationLaw
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section ClosedClusterContacts
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+noncomputable def contactBonds (G : BondGraph V E) (Q : Finset E) (A B : Finset V) : Finset E :=
+  Q.filter (fun e => ∃ a ∈ A, ∃ b ∈ B, G e = s(a,b))
+
+theorem incident_exterior_subset (G : BondGraph V E) (Q : Finset E) (A B : Finset V) :
+    incidentBonds G (exteriorBonds G Q (A:Set V)) B ⊆ incidentBonds G Q B := by
+  intro e he
+  obtain ⟨he,b,hb,hbe⟩ := (mem_incidentBonds G _ B e).mp he
+  exact (mem_incidentBonds G Q B e).mpr ⟨(Finset.mem_filter.mp he).1,b,hb,hbe⟩
+
+ 
+
+theorem incident_exterior_difference (G : BondGraph V E) (Q : Finset E) (A B : Finset V)
+    (hAB : Disjoint A B) :
+    incidentBonds G Q B \ incidentBonds G (exteriorBonds G Q (A:Set V)) B =
+      contactBonds G Q A B := by
+  ext e
+  simp only [Finset.mem_sdiff,mem_incidentBonds,contactBonds,Finset.mem_filter]
+  constructor
+  · rintro ⟨⟨he,b,hb,hbe⟩,hn⟩
+    obtain ⟨a,hea⟩ := Sym2.mem_iff_exists.mp hbe
+    have hbA : b ∉ A := fun hh => Finset.disjoint_left.mp hAB hh hb
+    have ha : a ∈ A := by
+      by_contra ha
+      apply hn
+      refine ⟨Finset.mem_filter.mpr ⟨he,?_⟩,b,hb,hbe⟩
+      exact (mem_internalEdges_of_ends G (A:Set V)ᶜ e b a hea).mpr ⟨hbA,ha⟩
+    exact ⟨he,a,ha,b,hb,hea.trans (Sym2.eq_swap ..)⟩
+  · rintro ⟨he,a,ha,b,hb,hab⟩
+    refine ⟨⟨he,b,hb,by rw [hab]; simp⟩,?_⟩
+    rintro ⟨he',_⟩
+    exact (Finset.mem_filter.mp he').2 a (by rw [hab]; simp) ha
+
+theorem finiteRootComponent_disjoint (G : BondGraph V E) (ω : Finset E) (x v : V)
+    (h : ¬Conn G (ω:Set E) x v) :
+    Disjoint (finiteRootComponent G ω x) (finiteRootComponent G ω v) := by
+  apply Finset.disjoint_left.mpr
+  intro z hz hz'
+  have h₁ : Conn G (ω:Set E) x z := by
+    change z ∈ (finiteRootComponent G ω x:Set V) at hz
+    rwa [coe_finiteRootComponent] at hz
+  have h₂ : Conn G (ω:Set E) v z := by
+    change z ∈ (finiteRootComponent G ω v:Set V) at hz'
+    rwa [coe_finiteRootComponent] at hz'
+  exact h (h₁.trans (conn_symm G _ h₂))
+
+theorem distinct_contact_closed (G : BondGraph V E) (Q ω : Finset E) (x v : V)
+    (h : ¬Conn G (ω:Set E) x v) {e : E}
+    (he : e ∈ contactBonds G Q (finiteRootComponent G ω x) (finiteRootComponent G ω v)) : e ∉ ω := by
+  obtain ⟨_,a,ha,b,hb,hab⟩ := Finset.mem_filter.mp he
+  have h₁ : Conn G (ω:Set E) x a := by
+    change a ∈ (finiteRootComponent G ω x:Set V) at ha
+    rwa [coe_finiteRootComponent] at ha
+  have h₂ : Conn G (ω:Set E) v b := by
+    change b ∈ (finiteRootComponent G ω v:Set V) at hb
+    rwa [coe_finiteRootComponent] at hb
+  intro heω
+  exact h ((h₁.tail ⟨e,heω,hab⟩).trans (conn_symm G _ h₂))
+
+theorem finiteRootComponent_exterior (G : BondGraph V E) (ω : Finset E) (x v : V)
+    (h : ¬Conn G (ω:Set E) x v) :
+    finiteRootComponent G (exteriorBonds G ω (finiteRootComponent G ω x:Set V)) v =
+      finiteRootComponent G ω v := by
+  apply Finset.coe_injective
+  simp only [coe_finiteRootComponent,coe_exteriorBonds]
+  ext z
+  exact conn_exterior_of_cluster_eq G ω x v z (cluster G ω x) rfl h
+
+end ClosedClusterContacts
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section ExplorationScoreDifference
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem incidence_exterior_score_difference (G : BondGraph V E) (n : ℕ) (Q ω : Finset E)
+    (hω : ω ⊆ Q) (x v : V) (p : ℝ) (h : ¬Conn G (ω:Set E) x v)
+    (hn : (incidentBonds G Q (finiteRootComponent G ω v)).card ≤ n) :
+    (incidenceSearch G n Q {v}).score p ω - exteriorSearchScore G n Q x v p ω =
+      -p*(contactBonds G Q (finiteRootComponent G ω x) (finiteRootComponent G ω v)).card := by
+  let A := finiteRootComponent G ω x
+  let B := finiteRootComponent G ω v
+  let R := exteriorBonds G Q (A:Set V)
+  let η := exteriorBonds G ω (A:Set V)
+  have hv : v ∉ A := by
+    intro hv
+    apply h
+    change v ∈ (A:Set V) at hv
+    simpa only [A,coe_finiteRootComponent,cluster,Set.mem_ofPred_eq] using hv
+  have hη : η ⊆ R := by
+    intro e he
+    obtain ⟨he,heA⟩ := Finset.mem_filter.mp he
+    exact Finset.mem_filter.mpr ⟨hω he,heA⟩
+  have hB : finiteRootComponent G η v = B := finiteRootComponent_exterior G ω x v h
+  have hsub : incidentBonds G R B ⊆ incidentBonds G Q B := incident_exterior_subset G Q A B
+  have hn' : (incidentBonds G R (finiteRootComponent G η v)).card ≤ n := by
+    rw [hB]
+    exact (Finset.card_le_card hsub).trans hn
+  have hs₁ := incidenceSearch_completed_score G n Q ω hω v p hn
+  have hs₂ := incidenceSearch_completed_score G n R η hη v p hn'
+  rw [hB] at hs₂
+  have hb (e : E) (he : e ∈ incidentBonds G R B) : (e ∈ η ↔ e ∈ ω) := by
+    have heR := ((mem_incidentBonds G R B e).mp he).1
+    change e ∈ exteriorBonds G ω (A:Set V) ↔ e ∈ ω
+    simp only [exteriorBonds,Finset.mem_filter]
+    exact and_iff_left (Finset.mem_filter.mp heR).2
+  have hs₂' : (incidenceSearch G n R {v}).score p η =
+      ∑ e ∈ incidentBonds G R B, ((if e ∈ ω then 1 else 0)-p) := by
+    rw [hs₂]
+    exact Finset.sum_congr rfl (fun e he => by simp only [hb e he])
+  change (incidenceSearch G n Q {v}).score p ω -
+    (if v ∈ A then 0 else (incidenceSearch G n R {v}).score p η) = _
+  rw [ite_eq_right hv,hs₁,hs₂',←Finset.sum_sdiff hsub,add_sub_cancel_right]
+  change (∑ e ∈ incidentBonds G Q B \ incidentBonds G (exteriorBonds G Q (A:Set V)) B,
+    ((if e ∈ ω then (1:ℝ) else 0)-p)) = _
+  rw [incident_exterior_difference G Q A B (finiteRootComponent_disjoint G ω x v h)]
+  have hc (e : E) (he : e ∈ contactBonds G Q A B) : e ∉ ω := distinct_contact_closed G Q ω x v h he
+  rw [show (∑ e ∈ contactBonds G Q A B, ((if e ∈ ω then (1:ℝ) else 0)-p)) =
+      ∑ _e ∈ contactBonds G Q A B, -p from Finset.sum_congr rfl (fun e he => by rw [ite_eq_right (hc e he)]; ring)]
+  simp only [Finset.sum_const,nsmul_eq_mul]
+  ring
+
+end ExplorationScoreDifference
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section ContactConcentration
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem contactBonds_comm (G : BondGraph V E) (Q : Finset E) (A B : Finset V) :
+    contactBonds G Q A B = contactBonds G Q B A := by
+  ext e
+  simp only [contactBonds,Finset.mem_filter]
+  constructor <;> rintro ⟨he,a,ha,b,hb,hab⟩
+  · exact ⟨he,b,hb,a,ha,hab.trans (Sym2.eq_swap ..)⟩
+  · exact ⟨he,b,hb,a,ha,hab.trans (Sym2.eq_swap ..)⟩
+
+ 
+
+theorem ordered_cluster_contact_tail (G : BondGraph V E) (Q : Finset E)
+    (x v : V) (p : unitInterval) (N : ℕ) (hN : 0 < N) (a : ℝ) (ha : 0 < a) :
+    bernoulliPolynomial Q (fun ω => if ¬Conn G (ω:Set E) x v ∧
+      (incidentBonds G Q (finiteRootComponent G ω v)).card ≤ N ∧
+      2*a ≤ (p:ℝ)*(contactBonds G Q (finiteRootComponent G ω x)
+        (finiteRootComponent G ω v)).card then 1 else 0) p ≤
+      4*Real.exp (-2*a^2/(N:ℝ)) := by
+  calc
+    _ ≤ bernoulliPolynomial Q (fun ω =>
+        (if a ≤ |(incidenceSearch G N Q {v}).score p ω| then 1 else 0)+
+        (if a ≤ |exteriorSearchScore G N Q x v p ω| then 1 else 0)) p := by
+      apply bernoulliPolynomial_mono _ _ _ p p.2
+      intro ω hω
+      by_cases h : ¬Conn G (ω:Set E) x v ∧
+          (incidentBonds G Q (finiteRootComponent G ω v)).card ≤ N ∧
+          2*a ≤ (p:ℝ)*(contactBonds G Q (finiteRootComponent G ω x)
+            (finiteRootComponent G ω v)).card
+      · have hd := incidence_exterior_score_difference G N Q ω hω x v p h.1 h.2.1
+        have huo : a ≤ |(incidenceSearch G N Q {v}).score p ω| ∨
+            a ≤ |exteriorSearchScore G N Q x v p ω| := by
+          by_contra hh
+          push Not at hh
+          have h₁ := (abs_lt.mp hh.1).1
+          have h₂ := (abs_lt.mp hh.2).2
+          nlinarith [h.2.2]
+        simp only [ite_eq_left h]
+        rcases huo with huo | huo <;> simp only [ite_eq_left huo] <;> split_ifs <;> norm_num
+      · simp only [ite_eq_right h]; split_ifs <;> norm_num
+    _ = bernoulliPolynomial Q (fun ω =>
+        if a ≤ |(incidenceSearch G N Q {v}).score p ω| then 1 else 0) p+
+        bernoulliPolynomial Q (fun ω =>
+        if a ≤ |exteriorSearchScore G N Q x v p ω| then 1 else 0) p :=
+      bernoulliPolynomial_add _ _ _ _
+    _ ≤ 2*Real.exp (-2*a^2/(N:ℝ))+2*Real.exp (-2*a^2/(N:ℝ)) :=
+      add_le_add (incidenceSearch_score_tail G N hN Q {v} p a ha.le)
+        (exteriorSearchScore_tail G N hN Q x v p a ha)
+    _ = _ := by ring
+
+end ContactConcentration
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Graph/Automorphisms.lean b/lean/OAI/Probability/UnimodularPercolation/Graph/Automorphisms.lean
index 46cb1ed728f0044a9cce385f1fe7edf4ee541180..cf432cdd3d21687df2843ebbdecfc6a6dd2823a5 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Graph/Automorphisms.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Graph/Automorphisms.lean
@@ -1,272 +1,272 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Escape.Labels
-import OAI.Probability.UnimodularPercolation.Graph.Topology
-import OAI.Probability.UnimodularPercolation.Transport.Haar
-
-namespace OAI
-
-/-! Locally compact graph automorphism groups and unimodular root data. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-open MeasureTheory Topology
-open scoped ENNReal
-universe u v
-variable {V : Type u} {E : Type v}
-
- 
-def graphAut (G : BondGraph V E) : Subgroup (Equiv.Perm V) where
-  carrier := vertexAut G
-  one_mem' := ⟨Equiv.refl E, isAut_refl G⟩
-  mul_mem' := by
-    rintro a c ⟨b,hb⟩ ⟨d,hd⟩
-    exact ⟨d.trans b, isAut_trans G hd hb⟩
-  inv_mem' := by
-    rintro a ⟨b,hb⟩
-    exact ⟨b.symm,isAut_symm G hb⟩
-
- 
-
-theorem mem_vertexAut_iff (G : BondGraph V E) (hG : Function.Injective G)
-    (a : Equiv.Perm V) :
-    a ∈ vertexAut G ↔ ∀ x y, Adj G (a x) (a y) ↔ Adj G x y := by
-  constructor
-  · rintro ⟨b,hb⟩ x y
-    exact ⟨fun h => by simpa using adj_aut G (isAut_symm G hb) h,adj_aut G hb⟩
-  · intro ha
-    have he (s : Sym2 V) : s ∈ Set.range G ↔ s.map a ∈ Set.range G := by
-      induction s using Sym2.inductionOn with
-      | _ x y => exact (ha x y).symm
-    let a₂ : Equiv.Perm (Sym2 V) :=
-      ⟨Sym2.map a,Sym2.map a.symm,
-        fun s => by simp only [Sym2.map_map,Equiv.symm_comp_self,Sym2.map_id',id_eq],
-        fun s => by simp only [Sym2.map_map,Equiv.self_comp_symm,Sym2.map_id',id_eq]⟩
-    let e : E ≃ Set.range G := Equiv.ofInjective G hG
-    let b : E ≃ E := (e.trans (a₂.subtypeEquiv he)).trans e.symm
-    refine ⟨b,?_⟩
-    intro z
-    have hb : e (b z) = a₂.subtypeEquiv he (e z) := by
-      exact e.apply_symm_apply _
-    exact congrArg Subtype.val hb
-
- 
-
-abbrev graphOrbitType (G : BondGraph V E) := Quotient (MulAction.orbitRel (graphAut G) V)
-
-instance graphOrbitType_nonempty [Nonempty V] (G : BondGraph V E) :
-    Nonempty (graphOrbitType G) := ⟨Quotient.mk _ (Classical.arbitrary V)⟩
-
-noncomputable def graphOrbitRoot (G : BondGraph V E) : graphOrbitType G → V := Quotient.out
-
-theorem graphOrbitRoot_cover (G : BondGraph V E) (x : V) :
-    ∃ i : graphOrbitType G, x ∈ actionOrbit (H := graphAut G) (graphOrbitRoot G i) := by
-  refine ⟨Quotient.mk _ x,?_⟩
-  apply (MulAction.mem_orbit_iff).mp
-  apply MulAction.orbitRel_apply.mp
-  apply Quotient.exact
-  exact (Quotient.out_eq (Quotient.mk (MulAction.orbitRel (graphAut G) V) x)).symm
-
-theorem graphOrbitRoot_separate (G : BondGraph V E) (i j : graphOrbitType G)
-    (h : graphOrbitRoot G j ∈ actionOrbit (H := graphAut G) (graphOrbitRoot G i)) : i = j := by
-  have he : (Quotient.mk _ (graphOrbitRoot G j) : graphOrbitType G) =
-      Quotient.mk _ (graphOrbitRoot G i) :=
-    Quotient.sound (MulAction.orbitRel_apply.mpr (MulAction.mem_orbit_iff.mpr h))
-  simpa only [graphOrbitRoot,Quotient.out_eq] using he.symm
-
-theorem finite_graphOrbitType (G : BondGraph V E) (hqt : QuasiTransitive G) :
-    Finite (graphOrbitType G) := by
-  classical
-  obtain ⟨R,hR⟩ := hqt
-  apply Finite.of_surjective (fun r : R => (Quotient.mk _ r.val : graphOrbitType G))
-  intro i
-  induction i using Quotient.inductionOn with
-  | _ x =>
-    obtain ⟨r,hr,a,b,hab,hax⟩ := hR x
-    refine ⟨⟨r,hr⟩,?_⟩
-    symm
-    apply Quotient.sound
-    apply MulAction.orbitRel_apply.mpr
-    apply MulAction.mem_orbit_iff.mpr
-    exact ⟨⟨a,⟨b,hab⟩⟩,hax⟩
-
-section Topology
-variable [TopologicalSpace V] [DiscreteTopology V]
-
-theorem isClosed_vertexAut (G : BondGraph V E) (hG : Function.Injective G) :
-    IsClosed (vertexAut G) := by
-  have h (x y : V) : IsClosed {a : Equiv.Perm V | Adj G (a x) (a y) ↔ Adj G x y} := by
-    have hf : Continuous (fun a : Equiv.Perm V => (a x,a y)) :=
-      (PointwisePerm.continuous_apply x).prodMk (PointwisePerm.continuous_apply y)
-    exact (isClosed_discrete {p : V × V | Adj G p.1 p.2 ↔ Adj G x y}).preimage hf
-  convert isClosed_iInter (fun x => isClosed_iInter (h x)) using 1
-  ext a
-  simp only [Set.mem_iInter,Set.mem_ofPred_eq,mem_vertexAut_iff G hG]
-
- 
-theorem isCompact_vertexStabilizer (G : BondGraph V E) (hG : Function.Injective G)
-    (hconn : Connected G) (hloc : LocallyFinite G) (x : V) :
-    IsCompact {a : Equiv.Perm V | a ∈ vertexAut G ∧ a x = x} := by
-  apply PointwisePerm.isCompact_of_finite_images _
-    ((isClosed_vertexAut G hG).inter (PointwisePerm.isClosed_fixed x x))
-  · intro y
-    obtain ⟨n,hn⟩ := exists_mem_reachBall (hconn x y)
-    apply (finite_reachBall hloc x n).subset
-    rintro z ⟨a,⟨⟨b,hb⟩,hx⟩,rfl⟩
-    change a x = x at hx
-    simpa only [hx] using mem_reachBall_aut G hb hn
-  · intro y
-    obtain ⟨n,hn⟩ := exists_mem_reachBall (hconn x y)
-    apply (finite_reachBall hloc x n).subset
-    rintro z ⟨a,⟨⟨b,hb⟩,hx⟩,rfl⟩
-    have hx' : a.symm x = x := by rw [Equiv.symm_apply_eq,hx]
-    simpa only [hx'] using mem_reachBall_aut G (isAut_symm G hb) hn
-
- 
-omit [DiscreteTopology V] in
-theorem graphAut_continuous_apply (G : BondGraph V E) (x : V) :
-    Continuous (fun g : graphAut G => g • x) :=
-  (PointwisePerm.continuous_apply x).comp continuous_subtype_val
-
-theorem graphAut_isOpen_stabilizer (G : BondGraph V E) (x : V) :
-    IsOpen (actionStabilizer (H := graphAut G) x) :=
-  (isOpen_discrete ({x} : Set V)).preimage (graphAut_continuous_apply G x)
-
-theorem graphAut_isCompact_stabilizer (G : BondGraph V E) (hG : Function.Injective G)
-    (hconn : Connected G) (hloc : LocallyFinite G) (x : V) :
-    IsCompact (actionStabilizer (H := graphAut G) x) := by
-  apply Topology.IsEmbedding.subtypeVal.isCompact_iff.mpr
-  convert isCompact_vertexStabilizer G hG hconn hloc x using 1
-  ext a
-  constructor
-  · rintro ⟨g,hg,rfl⟩
-    exact ⟨g.property,hg⟩
-  · rintro ⟨ha,hax⟩
-    exact ⟨⟨a,ha⟩,hax,rfl⟩
-
- 
-
-theorem graphAut_locallyCompact [Nonempty V] (G : BondGraph V E)
-    (hG : Function.Injective G) (hconn : Connected G) (hloc : LocallyFinite G) :
-    LocallyCompactSpace (graphAut G) := by
-  let x : V := Classical.arbitrary V
-  have hK := graphAut_isCompact_stabilizer G hG hconn hloc x
-  have hKo := graphAut_isOpen_stabilizer G x
-  have hw : WeaklyLocallyCompactSpace (graphAut G) := by
-    refine ⟨fun g => ?_⟩
-    refine ⟨(fun h : graphAut G => g * h) '' actionStabilizer (H := graphAut G) x,
-      hK.image (continuous_const.mul continuous_id),?_⟩
-    have h1 : actionStabilizer (H := graphAut G) x ∈ nhds (1 : graphAut G) :=
-      hKo.mem_nhds (one_smul _ _)
-    have he := (Homeomorph.mulLeft g).map_nhds_eq (1 : graphAut G)
-    change Filter.map (fun h : graphAut G => g * h) (nhds 1) = nhds (g * 1) at he
-    rw [mul_one] at he
-    rw [←he]
-    exact Filter.image_mem_map h1
-  let := hw
-  infer_instance
-
-instance graphAutSecondCountable [Countable V] (G : BondGraph V E) :
-    SecondCountableTopology (graphAut G) :=
-  Topology.IsEmbedding.subtypeVal.secondCountableTopology
-
-instance graphAutMeasurableSpace (G : BondGraph V E) : MeasurableSpace (graphAut G) :=
-  borel (graphAut G)
-
-instance graphAutBorelSpace (G : BondGraph V E) : BorelSpace (graphAut G) := ⟨rfl⟩
-
- 
-noncomputable def graphHaar [Nonempty V] (G : BondGraph V E)
-    (hG : Function.Injective G) (hconn : Connected G) (hloc : LocallyFinite G) :
-    Measure (graphAut G) := by
-  letI := graphAut_locallyCompact G hG hconn hloc
-  exact Measure.haar
-
-theorem graphHaar_isHaar [Nonempty V] (G : BondGraph V E)
-    (hG : Function.Injective G) (hconn : Connected G) (hloc : LocallyFinite G) :
-    (graphHaar G hG hconn hloc).IsHaarMeasure := by
-  let := graphAut_locallyCompact G hG hconn hloc
-  exact inferInstanceAs ((Measure.haar : Measure (graphAut G)).IsHaarMeasure)
-
-theorem graphHaar_regular [Nonempty V] (G : BondGraph V E)
-    (hG : Function.Injective G) (hconn : Connected G) (hloc : LocallyFinite G) :
-    (graphHaar G hG hconn hloc).Regular := by
-  let := graphAut_locallyCompact G hG hconn hloc
-  exact inferInstanceAs ((Measure.haar : Measure (graphAut G)).Regular)
-
- 
-theorem graphStabilizer_volume_pos (G : BondGraph V E) (hG : Function.Injective G)
-    (hconn : Connected G) (hloc : LocallyFinite G)
-    (μ : Measure (graphAut G)) [μ.IsHaarMeasure] (x : V) :
-    0 < μ.real (actionStabilizer (H := graphAut G) x) := by
-  apply ENNReal.toReal_pos
-  · exact ((graphAut_isOpen_stabilizer G x).measure_pos μ ⟨1,one_smul _ _⟩).ne'
-  · exact (graphAut_isCompact_stabilizer G hG hconn hloc x).measure_lt_top.ne
-
-section Countable
-variable [Countable V] [MeasurableSpace V] [BorelSpace V]
-
- 
-
-theorem graphHaar_rootTransport (G : BondGraph V E) (hG : Function.Injective G)
-    (hconn : Connected G) (hloc : LocallyFinite G)
-    [Fintype (graphOrbitType G)] [Nonempty V]
-    (μ : Measure (graphAut G)) [μ.IsHaarMeasure] [μ.IsInvInvariant] :
-    RootTransport (graphOrbitRoot G) (inverseStabilizerWeight μ (graphOrbitRoot G)) (vertexAut G) := by
-  apply rootTransport_of_invariant_measure μ
-    (fun x => (graphAut_continuous_apply G x).measurable)
-    (graphOrbitRoot G)
-    (fun i => graphStabilizer_volume_pos G hG hconn hloc μ (graphOrbitRoot G i))
-    (fun i => (graphAut_isCompact_stabilizer G hG hconn hloc (graphOrbitRoot G i)).measure_lt_top.ne)
-    (graphOrbitRoot_cover G) (graphOrbitRoot_separate G) (vertexAut G)
-  intro g
-  exact g.property
-
- 
-
-theorem exists_graphHaar_rootData [Nonempty V] (G : BondGraph V E)
-    (hG : Function.Injective G) (hconn : Connected G) (hloc : LocallyFinite G)
-    (hqt : QuasiTransitive G)
-    (μ : Measure (graphAut G)) [μ.IsHaarMeasure] [μ.IsInvInvariant] :
-    ∃ (ι : Type u) (_ : Fintype ι) (_ : Nonempty ι) (root : ι → V) (weight : ι → ℝ),
-      RootTransport root weight (vertexAut G) ∧
-      (∀ x, ∃ i, ∃ g ∈ vertexAut G, g (root i) = x) ∧
-      (∀ i, 0 < weight i) ∧ ∑ i, weight i = 1 := by
-  let := finite_graphOrbitType G hqt
-  let := Fintype.ofFinite (graphOrbitType G)
-  refine ⟨graphOrbitType G,inferInstance,inferInstance,graphOrbitRoot G,
-    inverseStabilizerWeight μ (graphOrbitRoot G),
-    graphHaar_rootTransport G hG hconn hloc μ,?_,?_,?_⟩
-  · intro x
-    obtain ⟨i,g,hg⟩ := graphOrbitRoot_cover G x
-    exact ⟨i,g.val,g.property,hg⟩
-  · exact inverseStabilizerWeight_pos μ (graphOrbitRoot G)
-      (fun i => graphStabilizer_volume_pos G hG hconn hloc μ (graphOrbitRoot G i))
-  · exact inverseStabilizerWeight_sum μ (graphOrbitRoot G)
-      (fun i => graphStabilizer_volume_pos G hG hconn hloc μ (graphOrbitRoot G i))
-
- 
-
-theorem exists_rootData_of_unimodular [Nonempty V] (G : BondGraph V E)
-    (hG : Function.Injective G) (hconn : Connected G) (hloc : LocallyFinite G)
-    (hqt : QuasiTransitive G)
-    (huni : (graphHaar G hG hconn hloc).IsMulRightInvariant) :
-    ∃ (ι : Type u) (_ : Fintype ι) (_ : Nonempty ι) (root : ι → V) (weight : ι → ℝ),
-      RootTransport root weight (vertexAut G) ∧
-      (∀ x, ∃ i, ∃ g ∈ vertexAut G, g (root i) = x) ∧
-      (∀ i, 0 < weight i) ∧ ∑ i, weight i = 1 := by
-  let := graphAut_locallyCompact G hG hconn hloc
-  let μ := graphHaar G hG hconn hloc
-  let : μ.IsHaarMeasure := graphHaar_isHaar G hG hconn hloc
-  let : μ.Regular := graphHaar_regular G hG hconn hloc
-  let : μ.IsMulRightInvariant := huni
-  let : μ.IsInvInvariant := haar_invInvariant_of_rightInvariant μ
-  exact exists_graphHaar_rootData G hG hconn hloc hqt μ
-
-end Countable
-
-end Topology
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Escape.Labels
+import OAI.Probability.UnimodularPercolation.Graph.Topology
+import OAI.Probability.UnimodularPercolation.Transport.Haar
+
+namespace OAI
+
+/-! Locally compact graph automorphism groups and unimodular root data. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+open MeasureTheory Topology
+open scoped ENNReal
+universe u v
+variable {V : Type u} {E : Type v}
+
+ 
+def graphAut (G : BondGraph V E) : Subgroup (Equiv.Perm V) where
+  carrier := vertexAut G
+  one_mem' := ⟨Equiv.refl E, isAut_refl G⟩
+  mul_mem' := by
+    rintro a c ⟨b,hb⟩ ⟨d,hd⟩
+    exact ⟨d.trans b, isAut_trans G hd hb⟩
+  inv_mem' := by
+    rintro a ⟨b,hb⟩
+    exact ⟨b.symm,isAut_symm G hb⟩
+
+ 
+
+theorem mem_vertexAut_iff (G : BondGraph V E) (hG : Function.Injective G)
+    (a : Equiv.Perm V) :
+    a ∈ vertexAut G ↔ ∀ x y, Adj G (a x) (a y) ↔ Adj G x y := by
+  constructor
+  · rintro ⟨b,hb⟩ x y
+    exact ⟨fun h => by simpa using adj_aut G (isAut_symm G hb) h,adj_aut G hb⟩
+  · intro ha
+    have he (s : Sym2 V) : s ∈ Set.range G ↔ s.map a ∈ Set.range G := by
+      induction s using Sym2.inductionOn with
+      | _ x y => exact (ha x y).symm
+    let a₂ : Equiv.Perm (Sym2 V) :=
+      ⟨Sym2.map a,Sym2.map a.symm,
+        fun s => by simp only [Sym2.map_map,Equiv.symm_comp_self,Sym2.map_id',id_eq],
+        fun s => by simp only [Sym2.map_map,Equiv.self_comp_symm,Sym2.map_id',id_eq]⟩
+    let e : E ≃ Set.range G := Equiv.ofInjective G hG
+    let b : E ≃ E := (e.trans (a₂.subtypeEquiv he)).trans e.symm
+    refine ⟨b,?_⟩
+    intro z
+    have hb : e (b z) = a₂.subtypeEquiv he (e z) := by
+      exact e.apply_symm_apply _
+    exact congrArg Subtype.val hb
+
+ 
+
+abbrev graphOrbitType (G : BondGraph V E) := Quotient (MulAction.orbitRel (graphAut G) V)
+
+instance graphOrbitType_nonempty [Nonempty V] (G : BondGraph V E) :
+    Nonempty (graphOrbitType G) := ⟨Quotient.mk _ (Classical.arbitrary V)⟩
+
+noncomputable def graphOrbitRoot (G : BondGraph V E) : graphOrbitType G → V := Quotient.out
+
+theorem graphOrbitRoot_cover (G : BondGraph V E) (x : V) :
+    ∃ i : graphOrbitType G, x ∈ actionOrbit (H := graphAut G) (graphOrbitRoot G i) := by
+  refine ⟨Quotient.mk _ x,?_⟩
+  apply (MulAction.mem_orbit_iff).mp
+  apply MulAction.orbitRel_apply.mp
+  apply Quotient.exact
+  exact (Quotient.out_eq (Quotient.mk (MulAction.orbitRel (graphAut G) V) x)).symm
+
+theorem graphOrbitRoot_separate (G : BondGraph V E) (i j : graphOrbitType G)
+    (h : graphOrbitRoot G j ∈ actionOrbit (H := graphAut G) (graphOrbitRoot G i)) : i = j := by
+  have he : (Quotient.mk _ (graphOrbitRoot G j) : graphOrbitType G) =
+      Quotient.mk _ (graphOrbitRoot G i) :=
+    Quotient.sound (MulAction.orbitRel_apply.mpr (MulAction.mem_orbit_iff.mpr h))
+  simpa only [graphOrbitRoot,Quotient.out_eq] using he.symm
+
+theorem finite_graphOrbitType (G : BondGraph V E) (hqt : QuasiTransitive G) :
+    Finite (graphOrbitType G) := by
+  classical
+  obtain ⟨R,hR⟩ := hqt
+  apply Finite.of_surjective (fun r : R => (Quotient.mk _ r.val : graphOrbitType G))
+  intro i
+  induction i using Quotient.inductionOn with
+  | _ x =>
+    obtain ⟨r,hr,a,b,hab,hax⟩ := hR x
+    refine ⟨⟨r,hr⟩,?_⟩
+    symm
+    apply Quotient.sound
+    apply MulAction.orbitRel_apply.mpr
+    apply MulAction.mem_orbit_iff.mpr
+    exact ⟨⟨a,⟨b,hab⟩⟩,hax⟩
+
+section Topology
+variable [TopologicalSpace V] [DiscreteTopology V]
+
+theorem isClosed_vertexAut (G : BondGraph V E) (hG : Function.Injective G) :
+    IsClosed (vertexAut G) := by
+  have h (x y : V) : IsClosed {a : Equiv.Perm V | Adj G (a x) (a y) ↔ Adj G x y} := by
+    have hf : Continuous (fun a : Equiv.Perm V => (a x,a y)) :=
+      (PointwisePerm.continuous_apply x).prodMk (PointwisePerm.continuous_apply y)
+    exact (isClosed_discrete {p : V × V | Adj G p.1 p.2 ↔ Adj G x y}).preimage hf
+  convert isClosed_iInter (fun x => isClosed_iInter (h x)) using 1
+  ext a
+  simp only [Set.mem_iInter,Set.mem_ofPred_eq,mem_vertexAut_iff G hG]
+
+ 
+theorem isCompact_vertexStabilizer (G : BondGraph V E) (hG : Function.Injective G)
+    (hconn : Connected G) (hloc : LocallyFinite G) (x : V) :
+    IsCompact {a : Equiv.Perm V | a ∈ vertexAut G ∧ a x = x} := by
+  apply PointwisePerm.isCompact_of_finite_images _
+    ((isClosed_vertexAut G hG).inter (PointwisePerm.isClosed_fixed x x))
+  · intro y
+    obtain ⟨n,hn⟩ := exists_mem_reachBall (hconn x y)
+    apply (finite_reachBall hloc x n).subset
+    rintro z ⟨a,⟨⟨b,hb⟩,hx⟩,rfl⟩
+    change a x = x at hx
+    simpa only [hx] using mem_reachBall_aut G hb hn
+  · intro y
+    obtain ⟨n,hn⟩ := exists_mem_reachBall (hconn x y)
+    apply (finite_reachBall hloc x n).subset
+    rintro z ⟨a,⟨⟨b,hb⟩,hx⟩,rfl⟩
+    have hx' : a.symm x = x := by rw [Equiv.symm_apply_eq,hx]
+    simpa only [hx'] using mem_reachBall_aut G (isAut_symm G hb) hn
+
+ 
+omit [DiscreteTopology V] in
+theorem graphAut_continuous_apply (G : BondGraph V E) (x : V) :
+    Continuous (fun g : graphAut G => g • x) :=
+  (PointwisePerm.continuous_apply x).comp continuous_subtype_val
+
+theorem graphAut_isOpen_stabilizer (G : BondGraph V E) (x : V) :
+    IsOpen (actionStabilizer (H := graphAut G) x) :=
+  (isOpen_discrete ({x} : Set V)).preimage (graphAut_continuous_apply G x)
+
+theorem graphAut_isCompact_stabilizer (G : BondGraph V E) (hG : Function.Injective G)
+    (hconn : Connected G) (hloc : LocallyFinite G) (x : V) :
+    IsCompact (actionStabilizer (H := graphAut G) x) := by
+  apply Topology.IsEmbedding.subtypeVal.isCompact_iff.mpr
+  convert isCompact_vertexStabilizer G hG hconn hloc x using 1
+  ext a
+  constructor
+  · rintro ⟨g,hg,rfl⟩
+    exact ⟨g.property,hg⟩
+  · rintro ⟨ha,hax⟩
+    exact ⟨⟨a,ha⟩,hax,rfl⟩
+
+ 
+
+theorem graphAut_locallyCompact [Nonempty V] (G : BondGraph V E)
+    (hG : Function.Injective G) (hconn : Connected G) (hloc : LocallyFinite G) :
+    LocallyCompactSpace (graphAut G) := by
+  let x : V := Classical.arbitrary V
+  have hK := graphAut_isCompact_stabilizer G hG hconn hloc x
+  have hKo := graphAut_isOpen_stabilizer G x
+  have hw : WeaklyLocallyCompactSpace (graphAut G) := by
+    refine ⟨fun g => ?_⟩
+    refine ⟨(fun h : graphAut G => g * h) '' actionStabilizer (H := graphAut G) x,
+      hK.image (continuous_const.mul continuous_id),?_⟩
+    have h1 : actionStabilizer (H := graphAut G) x ∈ nhds (1 : graphAut G) :=
+      hKo.mem_nhds (one_smul _ _)
+    have he := (Homeomorph.mulLeft g).map_nhds_eq (1 : graphAut G)
+    change Filter.map (fun h : graphAut G => g * h) (nhds 1) = nhds (g * 1) at he
+    rw [mul_one] at he
+    rw [←he]
+    exact Filter.image_mem_map h1
+  let := hw
+  infer_instance
+
+instance graphAutSecondCountable [Countable V] (G : BondGraph V E) :
+    SecondCountableTopology (graphAut G) :=
+  Topology.IsEmbedding.subtypeVal.secondCountableTopology
+
+instance graphAutMeasurableSpace (G : BondGraph V E) : MeasurableSpace (graphAut G) :=
+  borel (graphAut G)
+
+instance graphAutBorelSpace (G : BondGraph V E) : BorelSpace (graphAut G) := ⟨rfl⟩
+
+ 
+noncomputable def graphHaar [Nonempty V] (G : BondGraph V E)
+    (hG : Function.Injective G) (hconn : Connected G) (hloc : LocallyFinite G) :
+    Measure (graphAut G) := by
+  letI := graphAut_locallyCompact G hG hconn hloc
+  exact Measure.haar
+
+theorem graphHaar_isHaar [Nonempty V] (G : BondGraph V E)
+    (hG : Function.Injective G) (hconn : Connected G) (hloc : LocallyFinite G) :
+    (graphHaar G hG hconn hloc).IsHaarMeasure := by
+  let := graphAut_locallyCompact G hG hconn hloc
+  exact inferInstanceAs ((Measure.haar : Measure (graphAut G)).IsHaarMeasure)
+
+theorem graphHaar_regular [Nonempty V] (G : BondGraph V E)
+    (hG : Function.Injective G) (hconn : Connected G) (hloc : LocallyFinite G) :
+    (graphHaar G hG hconn hloc).Regular := by
+  let := graphAut_locallyCompact G hG hconn hloc
+  exact inferInstanceAs ((Measure.haar : Measure (graphAut G)).Regular)
+
+ 
+theorem graphStabilizer_volume_pos (G : BondGraph V E) (hG : Function.Injective G)
+    (hconn : Connected G) (hloc : LocallyFinite G)
+    (μ : Measure (graphAut G)) [μ.IsHaarMeasure] (x : V) :
+    0 < μ.real (actionStabilizer (H := graphAut G) x) := by
+  apply ENNReal.toReal_pos
+  · exact ((graphAut_isOpen_stabilizer G x).measure_pos μ ⟨1,one_smul _ _⟩).ne'
+  · exact (graphAut_isCompact_stabilizer G hG hconn hloc x).measure_lt_top.ne
+
+section Countable
+variable [Countable V] [MeasurableSpace V] [BorelSpace V]
+
+ 
+
+theorem graphHaar_rootTransport (G : BondGraph V E) (hG : Function.Injective G)
+    (hconn : Connected G) (hloc : LocallyFinite G)
+    [Fintype (graphOrbitType G)] [Nonempty V]
+    (μ : Measure (graphAut G)) [μ.IsHaarMeasure] [μ.IsInvInvariant] :
+    RootTransport (graphOrbitRoot G) (inverseStabilizerWeight μ (graphOrbitRoot G)) (vertexAut G) := by
+  apply rootTransport_of_invariant_measure μ
+    (fun x => (graphAut_continuous_apply G x).measurable)
+    (graphOrbitRoot G)
+    (fun i => graphStabilizer_volume_pos G hG hconn hloc μ (graphOrbitRoot G i))
+    (fun i => (graphAut_isCompact_stabilizer G hG hconn hloc (graphOrbitRoot G i)).measure_lt_top.ne)
+    (graphOrbitRoot_cover G) (graphOrbitRoot_separate G) (vertexAut G)
+  intro g
+  exact g.property
+
+ 
+
+theorem exists_graphHaar_rootData [Nonempty V] (G : BondGraph V E)
+    (hG : Function.Injective G) (hconn : Connected G) (hloc : LocallyFinite G)
+    (hqt : QuasiTransitive G)
+    (μ : Measure (graphAut G)) [μ.IsHaarMeasure] [μ.IsInvInvariant] :
+    ∃ (ι : Type u) (_ : Fintype ι) (_ : Nonempty ι) (root : ι → V) (weight : ι → ℝ),
+      RootTransport root weight (vertexAut G) ∧
+      (∀ x, ∃ i, ∃ g ∈ vertexAut G, g (root i) = x) ∧
+      (∀ i, 0 < weight i) ∧ ∑ i, weight i = 1 := by
+  let := finite_graphOrbitType G hqt
+  let := Fintype.ofFinite (graphOrbitType G)
+  refine ⟨graphOrbitType G,inferInstance,inferInstance,graphOrbitRoot G,
+    inverseStabilizerWeight μ (graphOrbitRoot G),
+    graphHaar_rootTransport G hG hconn hloc μ,?_,?_,?_⟩
+  · intro x
+    obtain ⟨i,g,hg⟩ := graphOrbitRoot_cover G x
+    exact ⟨i,g.val,g.property,hg⟩
+  · exact inverseStabilizerWeight_pos μ (graphOrbitRoot G)
+      (fun i => graphStabilizer_volume_pos G hG hconn hloc μ (graphOrbitRoot G i))
+  · exact inverseStabilizerWeight_sum μ (graphOrbitRoot G)
+      (fun i => graphStabilizer_volume_pos G hG hconn hloc μ (graphOrbitRoot G i))
+
+ 
+
+theorem exists_rootData_of_unimodular [Nonempty V] (G : BondGraph V E)
+    (hG : Function.Injective G) (hconn : Connected G) (hloc : LocallyFinite G)
+    (hqt : QuasiTransitive G)
+    (huni : (graphHaar G hG hconn hloc).IsMulRightInvariant) :
+    ∃ (ι : Type u) (_ : Fintype ι) (_ : Nonempty ι) (root : ι → V) (weight : ι → ℝ),
+      RootTransport root weight (vertexAut G) ∧
+      (∀ x, ∃ i, ∃ g ∈ vertexAut G, g (root i) = x) ∧
+      (∀ i, 0 < weight i) ∧ ∑ i, weight i = 1 := by
+  let := graphAut_locallyCompact G hG hconn hloc
+  let μ := graphHaar G hG hconn hloc
+  let : μ.IsHaarMeasure := graphHaar_isHaar G hG hconn hloc
+  let : μ.Regular := graphHaar_regular G hG hconn hloc
+  let : μ.IsMulRightInvariant := huni
+  let : μ.IsInvInvariant := haar_invInvariant_of_rightInvariant μ
+  exact exists_graphHaar_rootData G hG hconn hloc hqt μ
+
+end Countable
+
+end Topology
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Graph/Basic.lean b/lean/OAI/Probability/UnimodularPercolation/Graph/Basic.lean
index 4b3152b8d6d690fbea4b15a7f93f63b0a0518b3f..179886547fde03a342c523531703285ccd697f5b 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Graph/Basic.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Graph/Basic.lean
@@ -1,381 +1,381 @@
-import Mathlib
-
-namespace OAI
-
-/-! Indexed bond graphs, connectivity, percolation laws, kernels and countability. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-abbrev BondGraph (V : Type u) (E : Type v) := E → Sym2 V
-
- 
-def Adj (G : BondGraph V E) (x y : V) : Prop := ∃ e, G e = s(x, y)
-
- 
-def LocallyFinite (G : BondGraph V E) : Prop := ∀ x, {e | x ∈ G e}.Finite
-
- 
-def Connected (G : BondGraph V E) : Prop :=
-  ∀ x y, Relation.ReflTransGen (Adj G) x y
-
- 
-def IsAut (G : BondGraph V E) (a : V ≃ V) (b : E ≃ E) : Prop :=
-  ∀ e, G (b e) = (G e).map a
-
- 
-def QuasiTransitive (G : BondGraph V E) : Prop :=
-  ∃ representatives : Finset V, ∀ x, ∃ o ∈ representatives,
-    ∃ a : V ≃ V, ∃ b : E ≃ E, IsAut G a b ∧ a o = x
-
- 
-def vertexBoundary (G : BondGraph V E) (A : Finset V) : Set V :=
-  {y | y ∉ A ∧ ∃ x ∈ A, Adj G x y}
-
- 
-noncomputable def hV (G : BondGraph V E) : ℝ :=
-  sInf {r | ∃ A : Finset V, A.Nonempty ∧
-    r = (vertexBoundary G A).ncard / (A.card : ℝ)}
-
- 
-def Conn (G : BondGraph V E) (ω : Set E) (x y : V) : Prop :=
-  Relation.ReflTransGen (fun a b => ∃ e ∈ ω, G e = s(a, b)) x y
-
-def cluster (G : BondGraph V E) (ω : Set E) (x : V) : Set V :=
-  {y | Conn G ω x y}
-
- 
-def infiniteClusters (G : BondGraph V E) (ω : Set E) : Set (Set V) :=
-  {C | C.Infinite ∧ ∃ x, C = cluster G ω x}
-
- 
-noncomputable def law (p : unitInterval) : Measure (Set E) :=
-  setBernoulli Set.univ p
-
-noncomputable def pc (G : BondGraph V E) : unitInterval :=
-  sInf {p | 0 < law p {ω | (infiniteClusters G ω).Nonempty}}
-
-noncomputable def pu (G : BondGraph V E) : unitInterval :=
-  sInf {p | law p {ω | ∃! C, C ∈ infiniteClusters G ω} = 1}
-
-noncomputable def twoPoint (G : BondGraph V E) (p : unitInterval) (x y : V) : ℝ :=
-  (law p {ω | Conn G ω x y}).toReal
-
-noncomputable abbrev Hilbert (V : Type u) := lp (fun _ : V => ℂ) 2
-
- 
-noncomputable def HasKernel [DecidableEq V]
-    (G : BondGraph V E) (p : unitInterval)
-    (A : Hilbert V →L[ℂ] Hilbert V) : Prop :=
-  ∀ x y, (A (lp.single 2 y (1 : ℂ))) x = (twoPoint G p x y : ℂ)
-
-noncomputable def OperatorBounded (G : BondGraph V E) (p : unitInterval) : Prop := by
-  classical
-  exact ∃ A : Hilbert V →L[ℂ] Hilbert V, HasKernel G p A
-
- 
-noncomputable def operatorNorm (G : BondGraph V E) (p : unitInterval) : ℝ≥0∞ := by
-  classical
-  exact sInf {r | ∃ A : Hilbert V →L[ℂ] Hilbert V,
-    HasKernel G p A ∧ r = ENNReal.ofReal ‖A‖}
-
-noncomputable def ptwo (G : BondGraph V E) : unitInterval :=
-  sSup {p | OperatorBounded G p}
-
- 
-noncomputable def labelLaw : Measure (E → unitInterval) :=
-  Measure.infinitePi (fun _ => (volume : Measure unitInterval))
-
-def openEdges (labels : E → unitInterval) (p : unitInterval) : Set E :=
-  {e | labels e ≤ p}
-
- 
-def MainConclusion (G : BondGraph V E) : Prop :=
-  (sSup {r | ∃ p : unitInterval, p < pc G ∧ r = operatorNorm G p} =
-    operatorNorm G (pc G)) ∧
-  operatorNorm G (pc G) < ⊤ ∧
-  pc G < ptwo G ∧ ptwo G ≤ pu G ∧
-  ∃ p₁ p₂ : unitInterval, pc G < p₁ ∧ p₁ < p₂ ∧ p₂ < 1 ∧
-    ∀ᵐ labels ∂(labelLaw : Measure (E → unitInterval)),
-      ∀ p ∈ Set.Icc p₁ p₂, (infiniteClusters G (openEdges labels p)).Infinite
-
- 
-
-section Deterministic
-variable (G : BondGraph V E) (ω : Set E)
-
-@[simp] theorem conn_refl (x : V) : Conn G ω x x :=
-  Relation.ReflTransGen.refl
-
-theorem conn_trans {x y z : V} (hxy : Conn G ω x y) (hyz : Conn G ω y z) :
-    Conn G ω x z := hxy.trans hyz
-
-theorem conn_symm {x y : V} (h : Conn G ω x y) : Conn G ω y x := by
-  induction h with
-  | refl => exact conn_refl G ω _
-  | @tail y z _ hyz ih =>
-    rcases hyz with ⟨e, he, hlink⟩
-    exact (Relation.ReflTransGen.single ⟨e, he, hlink.trans (Sym2.eq_swap ..)⟩).trans ih
-
-theorem conn_mono {ω₁ ω₂ : Set E} (hω : ω₁ ⊆ ω₂) {x y : V}
-    (h : Conn G ω₁ x y) : Conn G ω₂ x y := by
-  apply Relation.ReflTransGen.mono (r := fun a b => ∃ e ∈ ω₁, G e = s(a, b)) ?_ x y h
-  rintro a b ⟨e, he, hab⟩
-  exact ⟨e, hω he, hab⟩
-
-theorem cluster_eq_of_conn {x y : V} (h : Conn G ω x y) :
-    cluster G ω x = cluster G ω y := by
-  ext z
-  exact ⟨fun hxz => conn_trans G ω (conn_symm G ω h) hxz,
-    fun hyz => conn_trans G ω h hyz⟩
-
-@[simp] theorem mem_cluster_self (x : V) : x ∈ cluster G ω x := conn_refl G ω x
-
-theorem clusters_eq_or_disjoint (x y : V) :
-    cluster G ω x = cluster G ω y ∨ Disjoint (cluster G ω x) (cluster G ω y) := by
-  by_cases h : Conn G ω x y
-  · exact Or.inl (cluster_eq_of_conn G ω h)
-  · right
-    apply Set.disjoint_left.mpr
-    intro z hxz hyz
-    exact h (conn_trans G ω hxz (conn_symm G ω hyz))
-
-@[simp] theorem conn_empty_iff (x y : V) : Conn G ∅ x y ↔ x = y := by
-  constructor
-  · intro h
-    induction h with
-    | refl => rfl
-    | tail _ he _ => rcases he with ⟨e, he, _⟩; exact he.elim
-  · rintro rfl
-    exact conn_refl G ∅ _
-
-@[simp] theorem cluster_empty (x : V) : cluster G ∅ x = {x} := by
-  ext y
-  simp [cluster, eq_comm]
-
-@[simp] theorem infiniteClusters_empty : infiniteClusters G ∅ = ∅ := by
-  ext C
-  simp only [infiniteClusters, Set.mem_ofPred_eq, Set.mem_empty_iff_false, iff_false,
-    not_and, not_exists]
-  intro h x hx
-  subst C
-  simp at h
-
-theorem conn_univ_of_connected (h : Connected G) (x y : V) : Conn G Set.univ x y := by
-  apply Relation.ReflTransGen.mono (r := Adj G) ?_ x y (h x y)
-  rintro a b ⟨e, he⟩
-  exact ⟨e, Set.mem_univ _, he⟩
-
-theorem cluster_univ_of_connected (h : Connected G) (x : V) :
-    cluster G Set.univ x = Set.univ := by
-  ext y
-  simp only [cluster, Set.mem_ofPred_eq, Set.mem_univ, iff_true]
-  exact conn_univ_of_connected G h x y
-
-theorem infiniteClusters_univ [Infinite V] (h : Connected G) :
-    infiniteClusters G Set.univ = {Set.univ} := by
-  ext C
-  constructor
-  · rintro ⟨_, x, rfl⟩
-    simp [cluster_univ_of_connected G h]
-  · intro hC
-    have hC' : C = Set.univ := Set.mem_singleton_iff.mp hC
-    subst C
-    exact ⟨Set.infinite_univ, Classical.arbitrary V,
-      (cluster_univ_of_connected G h _).symm⟩
-
-end Deterministic
-
-section Probability
-variable (G : BondGraph V E)
-
-instance law_isProbabilityMeasure (p : unitInterval) :
-    IsProbabilityMeasure (law p : Measure (Set E)) := by
-  dsimp [law]
-  infer_instance
-
-@[simp] theorem law_zero : (law 0 : Measure (Set E)) = Measure.dirac ∅ := by
-  simp [law]
-
-@[simp] theorem law_one : (law 1 : Measure (Set E)) = Measure.dirac Set.univ := by
-  simp [law]
-
-@[simp] theorem twoPoint_self (p : unitInterval) (x : V) : twoPoint G p x x = 1 := by
-  simp [twoPoint]
-
-theorem twoPoint_nonneg (p : unitInterval) (x y : V) : 0 ≤ twoPoint G p x y :=
-  ENNReal.toReal_nonneg
-
-theorem twoPoint_le_one (p : unitInterval) (x y : V) : twoPoint G p x y ≤ 1 := by
-  exact ENNReal.toReal_le_of_le_ofReal (by norm_num) (by simpa using
-    (prob_le_one (μ := law p) (s := {ω | Conn G ω x y})))
-
-theorem twoPoint_symm (p : unitInterval) (x y : V) : twoPoint G p x y = twoPoint G p y x := by
-  have he : {ω | Conn G ω x y} = {ω | Conn G ω y x} := by
-    ext ω
-    exact ⟨conn_symm G ω, conn_symm G ω⟩
-  simp only [twoPoint, he]
-
-@[simp] theorem twoPoint_zero [Countable E] [DecidableEq V] (x y : V) :
-    twoPoint G 0 x y = if x = y then 1 else 0 := by
-  rw [twoPoint, law_zero, Measure.dirac_apply]
-  by_cases h : x = y
-  · simp [h]
-  · rw [Set.indicator_of_notMem (by simpa using h)]
-    simp [h]
-
-theorem twoPoint_one (h : Connected G) (x y : V) : twoPoint G 1 x y = 1 := by
-  have hv : law (E := E) 1 {ω | Conn G ω x y} = 1 := by
-    rw [law_one]
-    exact Measure.dirac_apply_of_mem (conn_univ_of_connected G h x y)
-  change (law 1 {ω | Conn G ω x y}).toReal = 1
-  rw [hv]
-  norm_num
-
-end Probability
-
-section Kernel
-attribute [local instance] Classical.propDecidable
-
- 
-theorem kernel_extension_unique {K : V → V → ℂ}
-    {A B : Hilbert V →L[ℂ] Hilbert V}
-    (hA : ∀ x y, (A (lp.single 2 y (1 : ℂ))) x = K x y)
-    (hB : ∀ x y, (B (lp.single 2 y (1 : ℂ))) x = K x y) : A = B := by
-  apply lp.ext_continuousLinearMap (by norm_num : (2 : ℝ≥0∞) ≠ ⊤)
-  intro y
-  apply ContinuousLinearMap.ext
-  intro z
-  apply lp.ext
-  funext x
-  have hz : lp.single (E := fun _ : V => ℂ) 2 y z =
-      z • lp.single (E := fun _ : V => ℂ) 2 y (1 : ℂ) := by
-    simpa using (lp.single_smul (E := fun _ : V => ℂ) 2 y z (1 : ℂ))
-  simp only [ContinuousLinearMap.coe_comp, Function.comp_apply,
-    lp.singleContinuousLinearMap_apply, hz, map_smul, lp.coeFn_smul, Pi.smul_apply,
-    hA, hB]
-
-theorem HasKernel.unique {G : BondGraph V E} {p : unitInterval}
-    {A B : Hilbert V →L[ℂ] Hilbert V} (hA : HasKernel G p A) (hB : HasKernel G p B) :
-    A = B := kernel_extension_unique hA hB
-
- 
-theorem operatorNorm_of_hasKernel {G : BondGraph V E} {p : unitInterval}
-    {A : Hilbert V →L[ℂ] Hilbert V} (hA : HasKernel G p A) :
-    operatorNorm G p = ENNReal.ofReal ‖A‖ := by
-  have he : {r | ∃ B : Hilbert V →L[ℂ] Hilbert V,
-      HasKernel G p B ∧ r = ENNReal.ofReal ‖B‖} = {ENNReal.ofReal ‖A‖} := by
-    ext r
-    constructor
-    · rintro ⟨B, hB, rfl⟩
-      rw [hB.unique hA]
-      exact Set.mem_singleton _
-    · intro hr
-      exact ⟨A, hA, Set.mem_singleton_iff.mp hr⟩
-  change sInf _ = _
-  rw [he, sInf_singleton]
-
-theorem operatorNorm_eq_top_iff (G : BondGraph V E) (p : unitInterval) :
-    operatorNorm G p = ⊤ ↔ ¬ OperatorBounded G p := by
-  constructor
-  · intro h ⟨A, hA⟩
-    rw [operatorNorm_of_hasKernel hA] at h
-    exact ENNReal.ofReal_ne_top h
-  · intro h
-    unfold operatorNorm
-    have he : {r | ∃ A : Hilbert V →L[ℂ] Hilbert V,
-        HasKernel G p A ∧ r = ENNReal.ofReal ‖A‖} = ∅ := by
-      ext r
-      simp only [Set.mem_ofPred_eq, Set.mem_empty_iff_false, iff_false]
-      rintro ⟨A, hA, _⟩
-      exact h ⟨A, hA⟩
-    rw [he, sInf_empty]
-
-theorem operatorNorm_lt_top_iff (G : BondGraph V E) (p : unitInterval) :
-    operatorNorm G p < ⊤ ↔ OperatorBounded G p := by
-  rw [lt_top_iff_ne_top, ne_eq, operatorNorm_eq_top_iff, not_not]
-
-theorem hasKernel_zero [Countable E] (G : BondGraph V E) :
-    HasKernel G 0 (ContinuousLinearMap.id ℂ (Hilbert V)) := by
-  intro x y
-  simp only [ContinuousLinearMap.id_apply, lp.single_apply, twoPoint_zero]
-  by_cases h : x = y
-  · subst y
-    simp
-  · simp [h]
-
-@[simp] theorem operatorBounded_zero [Countable E] (G : BondGraph V E) :
-    OperatorBounded G 0 := ⟨_, hasKernel_zero G⟩
-
-end Kernel
-
-section Countability
-
- 
-theorem finite_neighbors {G : BondGraph V E} (hG : LocallyFinite G) (x : V) :
-    {y | Adj G x y}.Finite := by
-  classical
-  have hends (e : E) : {y | y ∈ G e}.Finite := by
-    apply (G e).toFinset.finite_toSet.subset
-    intro y hy
-    exact Sym2.mem_toFinset.mpr hy
-  have hs : (⋃ e ∈ {e | x ∈ G e}, {y | y ∈ G e}).Finite :=
-    (hG x).biUnion (fun e _ => hends e)
-  apply hs.subset
-  rintro y ⟨e, he⟩
-  simp only [Set.mem_iUnion, Set.mem_ofPred_eq]
-  exact ⟨e, he ▸ by simp, he ▸ by simp⟩
-
- 
-def reachBall (G : BondGraph V E) (x : V) : ℕ → Set V
-  | 0 => {x}
-  | n + 1 => reachBall G x n ∪ ⋃ y ∈ reachBall G x n, {z | Adj G y z}
-
-theorem finite_reachBall {G : BondGraph V E} (hG : LocallyFinite G) (x : V) :
-    ∀ n, (reachBall G x n).Finite := by
-  intro n
-  induction n with
-  | zero => exact Set.finite_singleton x
-  | succ n hn =>
-    exact hn.union (hn.biUnion (fun y _ => finite_neighbors hG y))
-
-theorem exists_mem_reachBall {G : BondGraph V E} {x y : V}
-    (h : Relation.ReflTransGen (Adj G) x y) : ∃ n, y ∈ reachBall G x n := by
-  induction h with
-  | refl => exact ⟨0, Set.mem_singleton _⟩
-  | @tail y z _ hyz ih =>
-    rcases ih with ⟨n, hn⟩
-    refine ⟨n + 1, Or.inr ?_⟩
-    simp only [Set.mem_iUnion, Set.mem_ofPred_eq]
-    exact ⟨y, hn, hyz⟩
-
-theorem countable_vertices [Nonempty V] {G : BondGraph V E}
-    (hG : LocallyFinite G) (hconn : Connected G) : Countable V := by
-  let x : V := Classical.arbitrary V
-  have he : (Set.univ : Set V) = ⋃ n, reachBall G x n := by
-    ext y
-    simp only [Set.mem_univ, Set.mem_iUnion, true_iff]
-    exact exists_mem_reachBall (hconn x y)
-  apply Set.countable_univ_iff.mp
-  rw [he]
-  exact Set.countable_iUnion (fun n => (finite_reachBall hG x n).countable)
-
-theorem countable_edges [Countable V] {G : BondGraph V E}
-    (hG : LocallyFinite G) : Countable E := by
-  have he : (Set.univ : Set E) = ⋃ x, {e | x ∈ G e} := by
-    ext e
-    simp only [Set.mem_univ, Set.mem_iUnion, Set.mem_ofPred_eq, true_iff]
-    exact ⟨(G e).out.1, Sym2.out_fst_mem _⟩
-  apply Set.countable_univ_iff.mp
-  rw [he]
-  exact Set.countable_iUnion (fun x => (hG x).countable)
-
-end Countability
-
-end Percolation
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-! Indexed bond graphs, connectivity, percolation laws, kernels and countability. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+abbrev BondGraph (V : Type u) (E : Type v) := E → Sym2 V
+
+ 
+def Adj (G : BondGraph V E) (x y : V) : Prop := ∃ e, G e = s(x, y)
+
+ 
+def LocallyFinite (G : BondGraph V E) : Prop := ∀ x, {e | x ∈ G e}.Finite
+
+ 
+def Connected (G : BondGraph V E) : Prop :=
+  ∀ x y, Relation.ReflTransGen (Adj G) x y
+
+ 
+def IsAut (G : BondGraph V E) (a : V ≃ V) (b : E ≃ E) : Prop :=
+  ∀ e, G (b e) = (G e).map a
+
+ 
+def QuasiTransitive (G : BondGraph V E) : Prop :=
+  ∃ representatives : Finset V, ∀ x, ∃ o ∈ representatives,
+    ∃ a : V ≃ V, ∃ b : E ≃ E, IsAut G a b ∧ a o = x
+
+ 
+def vertexBoundary (G : BondGraph V E) (A : Finset V) : Set V :=
+  {y | y ∉ A ∧ ∃ x ∈ A, Adj G x y}
+
+ 
+noncomputable def hV (G : BondGraph V E) : ℝ :=
+  sInf {r | ∃ A : Finset V, A.Nonempty ∧
+    r = (vertexBoundary G A).ncard / (A.card : ℝ)}
+
+ 
+def Conn (G : BondGraph V E) (ω : Set E) (x y : V) : Prop :=
+  Relation.ReflTransGen (fun a b => ∃ e ∈ ω, G e = s(a, b)) x y
+
+def cluster (G : BondGraph V E) (ω : Set E) (x : V) : Set V :=
+  {y | Conn G ω x y}
+
+ 
+def infiniteClusters (G : BondGraph V E) (ω : Set E) : Set (Set V) :=
+  {C | C.Infinite ∧ ∃ x, C = cluster G ω x}
+
+ 
+noncomputable def law (p : unitInterval) : Measure (Set E) :=
+  setBernoulli Set.univ p
+
+noncomputable def pc (G : BondGraph V E) : unitInterval :=
+  sInf {p | 0 < law p {ω | (infiniteClusters G ω).Nonempty}}
+
+noncomputable def pu (G : BondGraph V E) : unitInterval :=
+  sInf {p | law p {ω | ∃! C, C ∈ infiniteClusters G ω} = 1}
+
+noncomputable def twoPoint (G : BondGraph V E) (p : unitInterval) (x y : V) : ℝ :=
+  (law p {ω | Conn G ω x y}).toReal
+
+noncomputable abbrev Hilbert (V : Type u) := lp (fun _ : V => ℂ) 2
+
+ 
+noncomputable def HasKernel [DecidableEq V]
+    (G : BondGraph V E) (p : unitInterval)
+    (A : Hilbert V →L[ℂ] Hilbert V) : Prop :=
+  ∀ x y, (A (lp.single 2 y (1 : ℂ))) x = (twoPoint G p x y : ℂ)
+
+noncomputable def OperatorBounded (G : BondGraph V E) (p : unitInterval) : Prop := by
+  classical
+  exact ∃ A : Hilbert V →L[ℂ] Hilbert V, HasKernel G p A
+
+ 
+noncomputable def operatorNorm (G : BondGraph V E) (p : unitInterval) : ℝ≥0∞ := by
+  classical
+  exact sInf {r | ∃ A : Hilbert V →L[ℂ] Hilbert V,
+    HasKernel G p A ∧ r = ENNReal.ofReal ‖A‖}
+
+noncomputable def ptwo (G : BondGraph V E) : unitInterval :=
+  sSup {p | OperatorBounded G p}
+
+ 
+noncomputable def labelLaw : Measure (E → unitInterval) :=
+  Measure.infinitePi (fun _ => (volume : Measure unitInterval))
+
+def openEdges (labels : E → unitInterval) (p : unitInterval) : Set E :=
+  {e | labels e ≤ p}
+
+ 
+def MainConclusion (G : BondGraph V E) : Prop :=
+  (sSup {r | ∃ p : unitInterval, p < pc G ∧ r = operatorNorm G p} =
+    operatorNorm G (pc G)) ∧
+  operatorNorm G (pc G) < ⊤ ∧
+  pc G < ptwo G ∧ ptwo G ≤ pu G ∧
+  ∃ p₁ p₂ : unitInterval, pc G < p₁ ∧ p₁ < p₂ ∧ p₂ < 1 ∧
+    ∀ᵐ labels ∂(labelLaw : Measure (E → unitInterval)),
+      ∀ p ∈ Set.Icc p₁ p₂, (infiniteClusters G (openEdges labels p)).Infinite
+
+ 
+
+section Deterministic
+variable (G : BondGraph V E) (ω : Set E)
+
+@[simp] theorem conn_refl (x : V) : Conn G ω x x :=
+  Relation.ReflTransGen.refl
+
+theorem conn_trans {x y z : V} (hxy : Conn G ω x y) (hyz : Conn G ω y z) :
+    Conn G ω x z := hxy.trans hyz
+
+theorem conn_symm {x y : V} (h : Conn G ω x y) : Conn G ω y x := by
+  induction h with
+  | refl => exact conn_refl G ω _
+  | @tail y z _ hyz ih =>
+    rcases hyz with ⟨e, he, hlink⟩
+    exact (Relation.ReflTransGen.single ⟨e, he, hlink.trans (Sym2.eq_swap ..)⟩).trans ih
+
+theorem conn_mono {ω₁ ω₂ : Set E} (hω : ω₁ ⊆ ω₂) {x y : V}
+    (h : Conn G ω₁ x y) : Conn G ω₂ x y := by
+  apply Relation.ReflTransGen.mono (r := fun a b => ∃ e ∈ ω₁, G e = s(a, b)) ?_ x y h
+  rintro a b ⟨e, he, hab⟩
+  exact ⟨e, hω he, hab⟩
+
+theorem cluster_eq_of_conn {x y : V} (h : Conn G ω x y) :
+    cluster G ω x = cluster G ω y := by
+  ext z
+  exact ⟨fun hxz => conn_trans G ω (conn_symm G ω h) hxz,
+    fun hyz => conn_trans G ω h hyz⟩
+
+@[simp] theorem mem_cluster_self (x : V) : x ∈ cluster G ω x := conn_refl G ω x
+
+theorem clusters_eq_or_disjoint (x y : V) :
+    cluster G ω x = cluster G ω y ∨ Disjoint (cluster G ω x) (cluster G ω y) := by
+  by_cases h : Conn G ω x y
+  · exact Or.inl (cluster_eq_of_conn G ω h)
+  · right
+    apply Set.disjoint_left.mpr
+    intro z hxz hyz
+    exact h (conn_trans G ω hxz (conn_symm G ω hyz))
+
+@[simp] theorem conn_empty_iff (x y : V) : Conn G ∅ x y ↔ x = y := by
+  constructor
+  · intro h
+    induction h with
+    | refl => rfl
+    | tail _ he _ => rcases he with ⟨e, he, _⟩; exact he.elim
+  · rintro rfl
+    exact conn_refl G ∅ _
+
+@[simp] theorem cluster_empty (x : V) : cluster G ∅ x = {x} := by
+  ext y
+  simp [cluster, eq_comm]
+
+@[simp] theorem infiniteClusters_empty : infiniteClusters G ∅ = ∅ := by
+  ext C
+  simp only [infiniteClusters, Set.mem_ofPred_eq, Set.mem_empty_iff_false, iff_false,
+    not_and, not_exists]
+  intro h x hx
+  subst C
+  simp at h
+
+theorem conn_univ_of_connected (h : Connected G) (x y : V) : Conn G Set.univ x y := by
+  apply Relation.ReflTransGen.mono (r := Adj G) ?_ x y (h x y)
+  rintro a b ⟨e, he⟩
+  exact ⟨e, Set.mem_univ _, he⟩
+
+theorem cluster_univ_of_connected (h : Connected G) (x : V) :
+    cluster G Set.univ x = Set.univ := by
+  ext y
+  simp only [cluster, Set.mem_ofPred_eq, Set.mem_univ, iff_true]
+  exact conn_univ_of_connected G h x y
+
+theorem infiniteClusters_univ [Infinite V] (h : Connected G) :
+    infiniteClusters G Set.univ = {Set.univ} := by
+  ext C
+  constructor
+  · rintro ⟨_, x, rfl⟩
+    simp [cluster_univ_of_connected G h]
+  · intro hC
+    have hC' : C = Set.univ := Set.mem_singleton_iff.mp hC
+    subst C
+    exact ⟨Set.infinite_univ, Classical.arbitrary V,
+      (cluster_univ_of_connected G h _).symm⟩
+
+end Deterministic
+
+section Probability
+variable (G : BondGraph V E)
+
+instance law_isProbabilityMeasure (p : unitInterval) :
+    IsProbabilityMeasure (law p : Measure (Set E)) := by
+  dsimp [law]
+  infer_instance
+
+@[simp] theorem law_zero : (law 0 : Measure (Set E)) = Measure.dirac ∅ := by
+  simp [law]
+
+@[simp] theorem law_one : (law 1 : Measure (Set E)) = Measure.dirac Set.univ := by
+  simp [law]
+
+@[simp] theorem twoPoint_self (p : unitInterval) (x : V) : twoPoint G p x x = 1 := by
+  simp [twoPoint]
+
+theorem twoPoint_nonneg (p : unitInterval) (x y : V) : 0 ≤ twoPoint G p x y :=
+  ENNReal.toReal_nonneg
+
+theorem twoPoint_le_one (p : unitInterval) (x y : V) : twoPoint G p x y ≤ 1 := by
+  exact ENNReal.toReal_le_of_le_ofReal (by norm_num) (by simpa using
+    (prob_le_one (μ := law p) (s := {ω | Conn G ω x y})))
+
+theorem twoPoint_symm (p : unitInterval) (x y : V) : twoPoint G p x y = twoPoint G p y x := by
+  have he : {ω | Conn G ω x y} = {ω | Conn G ω y x} := by
+    ext ω
+    exact ⟨conn_symm G ω, conn_symm G ω⟩
+  simp only [twoPoint, he]
+
+@[simp] theorem twoPoint_zero [Countable E] [DecidableEq V] (x y : V) :
+    twoPoint G 0 x y = if x = y then 1 else 0 := by
+  rw [twoPoint, law_zero, Measure.dirac_apply]
+  by_cases h : x = y
+  · simp [h]
+  · rw [Set.indicator_of_notMem (by simpa using h)]
+    simp [h]
+
+theorem twoPoint_one (h : Connected G) (x y : V) : twoPoint G 1 x y = 1 := by
+  have hv : law (E := E) 1 {ω | Conn G ω x y} = 1 := by
+    rw [law_one]
+    exact Measure.dirac_apply_of_mem (conn_univ_of_connected G h x y)
+  change (law 1 {ω | Conn G ω x y}).toReal = 1
+  rw [hv]
+  norm_num
+
+end Probability
+
+section Kernel
+attribute [local instance] Classical.propDecidable
+
+ 
+theorem kernel_extension_unique {K : V → V → ℂ}
+    {A B : Hilbert V →L[ℂ] Hilbert V}
+    (hA : ∀ x y, (A (lp.single 2 y (1 : ℂ))) x = K x y)
+    (hB : ∀ x y, (B (lp.single 2 y (1 : ℂ))) x = K x y) : A = B := by
+  apply lp.ext_continuousLinearMap (by norm_num : (2 : ℝ≥0∞) ≠ ⊤)
+  intro y
+  apply ContinuousLinearMap.ext
+  intro z
+  apply lp.ext
+  funext x
+  have hz : lp.single (E := fun _ : V => ℂ) 2 y z =
+      z • lp.single (E := fun _ : V => ℂ) 2 y (1 : ℂ) := by
+    simpa using (lp.single_smul (E := fun _ : V => ℂ) 2 y z (1 : ℂ))
+  simp only [ContinuousLinearMap.coe_comp, Function.comp_apply,
+    lp.singleContinuousLinearMap_apply, hz, map_smul, lp.coeFn_smul, Pi.smul_apply,
+    hA, hB]
+
+theorem HasKernel.unique {G : BondGraph V E} {p : unitInterval}
+    {A B : Hilbert V →L[ℂ] Hilbert V} (hA : HasKernel G p A) (hB : HasKernel G p B) :
+    A = B := kernel_extension_unique hA hB
+
+ 
+theorem operatorNorm_of_hasKernel {G : BondGraph V E} {p : unitInterval}
+    {A : Hilbert V →L[ℂ] Hilbert V} (hA : HasKernel G p A) :
+    operatorNorm G p = ENNReal.ofReal ‖A‖ := by
+  have he : {r | ∃ B : Hilbert V →L[ℂ] Hilbert V,
+      HasKernel G p B ∧ r = ENNReal.ofReal ‖B‖} = {ENNReal.ofReal ‖A‖} := by
+    ext r
+    constructor
+    · rintro ⟨B, hB, rfl⟩
+      rw [hB.unique hA]
+      exact Set.mem_singleton _
+    · intro hr
+      exact ⟨A, hA, Set.mem_singleton_iff.mp hr⟩
+  change sInf _ = _
+  rw [he, sInf_singleton]
+
+theorem operatorNorm_eq_top_iff (G : BondGraph V E) (p : unitInterval) :
+    operatorNorm G p = ⊤ ↔ ¬ OperatorBounded G p := by
+  constructor
+  · intro h ⟨A, hA⟩
+    rw [operatorNorm_of_hasKernel hA] at h
+    exact ENNReal.ofReal_ne_top h
+  · intro h
+    unfold operatorNorm
+    have he : {r | ∃ A : Hilbert V →L[ℂ] Hilbert V,
+        HasKernel G p A ∧ r = ENNReal.ofReal ‖A‖} = ∅ := by
+      ext r
+      simp only [Set.mem_ofPred_eq, Set.mem_empty_iff_false, iff_false]
+      rintro ⟨A, hA, _⟩
+      exact h ⟨A, hA⟩
+    rw [he, sInf_empty]
+
+theorem operatorNorm_lt_top_iff (G : BondGraph V E) (p : unitInterval) :
+    operatorNorm G p < ⊤ ↔ OperatorBounded G p := by
+  rw [lt_top_iff_ne_top, ne_eq, operatorNorm_eq_top_iff, not_not]
+
+theorem hasKernel_zero [Countable E] (G : BondGraph V E) :
+    HasKernel G 0 (ContinuousLinearMap.id ℂ (Hilbert V)) := by
+  intro x y
+  simp only [ContinuousLinearMap.id_apply, lp.single_apply, twoPoint_zero]
+  by_cases h : x = y
+  · subst y
+    simp
+  · simp [h]
+
+@[simp] theorem operatorBounded_zero [Countable E] (G : BondGraph V E) :
+    OperatorBounded G 0 := ⟨_, hasKernel_zero G⟩
+
+end Kernel
+
+section Countability
+
+ 
+theorem finite_neighbors {G : BondGraph V E} (hG : LocallyFinite G) (x : V) :
+    {y | Adj G x y}.Finite := by
+  classical
+  have hends (e : E) : {y | y ∈ G e}.Finite := by
+    apply (G e).toFinset.finite_toSet.subset
+    intro y hy
+    exact Sym2.mem_toFinset.mpr hy
+  have hs : (⋃ e ∈ {e | x ∈ G e}, {y | y ∈ G e}).Finite :=
+    (hG x).biUnion (fun e _ => hends e)
+  apply hs.subset
+  rintro y ⟨e, he⟩
+  simp only [Set.mem_iUnion, Set.mem_ofPred_eq]
+  exact ⟨e, he ▸ by simp, he ▸ by simp⟩
+
+ 
+def reachBall (G : BondGraph V E) (x : V) : ℕ → Set V
+  | 0 => {x}
+  | n + 1 => reachBall G x n ∪ ⋃ y ∈ reachBall G x n, {z | Adj G y z}
+
+theorem finite_reachBall {G : BondGraph V E} (hG : LocallyFinite G) (x : V) :
+    ∀ n, (reachBall G x n).Finite := by
+  intro n
+  induction n with
+  | zero => exact Set.finite_singleton x
+  | succ n hn =>
+    exact hn.union (hn.biUnion (fun y _ => finite_neighbors hG y))
+
+theorem exists_mem_reachBall {G : BondGraph V E} {x y : V}
+    (h : Relation.ReflTransGen (Adj G) x y) : ∃ n, y ∈ reachBall G x n := by
+  induction h with
+  | refl => exact ⟨0, Set.mem_singleton _⟩
+  | @tail y z _ hyz ih =>
+    rcases ih with ⟨n, hn⟩
+    refine ⟨n + 1, Or.inr ?_⟩
+    simp only [Set.mem_iUnion, Set.mem_ofPred_eq]
+    exact ⟨y, hn, hyz⟩
+
+theorem countable_vertices [Nonempty V] {G : BondGraph V E}
+    (hG : LocallyFinite G) (hconn : Connected G) : Countable V := by
+  let x : V := Classical.arbitrary V
+  have he : (Set.univ : Set V) = ⋃ n, reachBall G x n := by
+    ext y
+    simp only [Set.mem_univ, Set.mem_iUnion, true_iff]
+    exact exists_mem_reachBall (hconn x y)
+  apply Set.countable_univ_iff.mp
+  rw [he]
+  exact Set.countable_iUnion (fun n => (finite_reachBall hG x n).countable)
+
+theorem countable_edges [Countable V] {G : BondGraph V E}
+    (hG : LocallyFinite G) : Countable E := by
+  have he : (Set.univ : Set E) = ⋃ x, {e | x ∈ G e} := by
+    ext e
+    simp only [Set.mem_univ, Set.mem_iUnion, Set.mem_ofPred_eq, true_iff]
+    exact ⟨(G e).out.1, Sym2.out_fst_mem _⟩
+  apply Set.countable_univ_iff.mp
+  rw [he]
+  exact Set.countable_iUnion (fun x => (hG x).countable)
+
+end Countability
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Graph/FullTopology.lean b/lean/OAI/Probability/UnimodularPercolation/Graph/FullTopology.lean
index 96c5b8402440fca91be45ab5434671559e598aae..9b7acc7fcfe8483c6ef5debfb36e3cace85d2051 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Graph/FullTopology.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Graph/FullTopology.lean
@@ -1,310 +1,310 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Main
-
-namespace OAI
-
-/-! The automorphism group with its topology of pointwise convergence. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-open Topology
-universe u v
-variable {V : Type u} {E : Type v}
-
-noncomputable def graphAutSubgroup (G : BondGraph V E) : Subgroup (Equiv.Perm V) where
-  carrier := vertexAut G
-  one_mem' := ⟨Equiv.refl E, isAut_refl G⟩
-  mul_mem' := by
-    rintro a b ⟨c,hc⟩ ⟨d,hd⟩
-    exact ⟨d.trans c,isAut_trans G hd hc⟩
-  inv_mem' := by
-    rintro a ⟨b,hb⟩
-    exact ⟨b.symm,isAut_symm G hb⟩
-
-def GraphAut (G : BondGraph V E) := ↥(graphAutSubgroup G)
-
-noncomputable instance (G : BondGraph V E) : Group (GraphAut G) :=
-  inferInstanceAs (Group ↥(graphAutSubgroup G))
-
-noncomputable instance (G : BondGraph V E) : MulAction (GraphAut G) V :=
-  inferInstanceAs (MulAction ↥(graphAutSubgroup G) V)
-
-noncomputable def graphAutPerm (G : BondGraph V E) : GraphAut G →* Equiv.Perm V :=
-  (graphAutSubgroup G).subtype
-
-theorem graphAutPerm_mem (G : BondGraph V E) (g : GraphAut G) :
-    graphAutPerm G g ∈ vertexAut G := g.property
-
-theorem mem_vertexAut_iff_adj (G : BondGraph V E) (hG : Function.Injective G)
-    (a : V ≃ V) : a ∈ vertexAut G ↔ ∀ x y, Adj G (a x) (a y) ↔ Adj G x y := by
-  classical
-  constructor
-  · rintro ⟨b,hb⟩ x y
-    refine ⟨fun h => ?_,adj_aut G hb⟩
-    simpa only [Equiv.symm_apply_apply] using adj_aut G (isAut_symm G hb) h
-  · intro ha
-    have hex (e : E) : ∃ f : E, G f = (G e).map a := by
-      obtain ⟨x,y,hxy⟩ := Sym2.exists.mp (show ∃ t : Sym2 V, t = G e from ⟨_,rfl⟩)
-      have h := (ha x y).mpr ⟨e,hxy.symm⟩
-      simpa only [←hxy,Sym2.map_mk] using (show ∃ f, G f = s(a x,a y) from h)
-    let f : E → E := fun e => Classical.choose (hex e)
-    have hf (e : E) : G (f e) = (G e).map a := Classical.choose_spec (hex e)
-    have hfi : Function.Injective f := by
-      intro e d hed
-      apply hG
-      have heq := congrArg (fun z => z.map a.symm) (congrArg G hed)
-      simpa only [hf,Sym2.map_map,Function.comp_def,Equiv.symm_apply_apply,Sym2.map_id',id_eq] using heq
-    have hfs : Function.Surjective f := by
-      intro e
-      obtain ⟨x,y,hxy⟩ := Sym2.exists.mp (show ∃ t : Sym2 V, t = G e from ⟨_,rfl⟩)
-      have h : Adj G (a.symm x) (a.symm y) := (ha _ _).mp (by
-        simpa only [Equiv.apply_symm_apply] using (show Adj G x y from ⟨e,hxy.symm⟩))
-      obtain ⟨d,hd⟩ := h
-      refine ⟨d,hG ?_⟩
-      rw [hf,hd,Sym2.map_mk,Equiv.apply_symm_apply,Equiv.apply_symm_apply,hxy]
-    exact ⟨Equiv.ofBijective f ⟨hfi,hfs⟩,hf⟩
-
-section Topology
-variable [TopologicalSpace V] [DiscreteTopology V]
-
-noncomputable def autCoordinates (G : BondGraph V E) (g : GraphAut G) :
-    (V → V) × (V → V) := (graphAutPerm G g,graphAutPerm G (g⁻¹))
-
-noncomputable instance graphAutTopology (G : BondGraph V E) : TopologicalSpace (GraphAut G) :=
-  TopologicalSpace.induced (autCoordinates G) inferInstance
-
-omit [DiscreteTopology V] in
-theorem inducing_autCoordinates (G : BondGraph V E) :
-    IsInducing (autCoordinates G) := ⟨rfl⟩
-
-omit [DiscreteTopology V] in
-theorem embedding_autCoordinates (G : BondGraph V E) :
-    IsEmbedding (autCoordinates G) := by
-  refine ⟨inducing_autCoordinates G,?_⟩
-  intro f g h
-  apply Subtype.ext
-  apply Equiv.ext
-  intro x
-  exact congrFun (congrArg Prod.fst h) x
-
-noncomputable instance graphAutT2 (G : BondGraph V E) : T2Space (GraphAut G) :=
-  (embedding_autCoordinates G).t2Space
-
-omit [DiscreteTopology V] in
-theorem continuous_aut_apply (G : BondGraph V E) (x : V) :
-    Continuous (fun g : GraphAut G => graphAutPerm G g x) :=
-  (continuous_apply x).comp ((inducing_autCoordinates G).continuous.fst)
-
-omit [DiscreteTopology V] in
-theorem continuous_aut_inv_apply (G : BondGraph V E) (x : V) :
-    Continuous (fun g : GraphAut G => graphAutPerm G (g⁻¹) x) :=
-  (continuous_apply x).comp ((inducing_autCoordinates G).continuous.snd)
-
-theorem continuous_discrete_eval : Continuous (fun p : (V → V) × V => p.1 p.2) :=
-  continuous_prod_of_discrete_right.mpr fun x => continuous_apply x
-
-noncomputable instance graphAutTopologicalGroup (G : BondGraph V E) :
-    IsTopologicalGroup (GraphAut G) where
-  continuous_mul := by
-    apply (inducing_autCoordinates G).continuous_iff.mpr
-    apply Continuous.prodMk
-    · apply continuous_pi
-      intro x
-      exact continuous_discrete_eval.comp
-        (((inducing_autCoordinates G).continuous.fst.comp continuous_fst).prodMk
-          ((continuous_aut_apply G x).comp continuous_snd))
-    · apply continuous_pi
-      intro x
-      exact continuous_discrete_eval.comp
-        (((inducing_autCoordinates G).continuous.snd.comp continuous_snd).prodMk
-          ((continuous_aut_inv_apply G x).comp continuous_fst))
-  continuous_inv := by
-    apply (inducing_autCoordinates G).continuous_iff.mpr
-    simpa only [Function.comp_def,autCoordinates,inv_inv] using ((inducing_autCoordinates G).continuous.snd.prodMk (inducing_autCoordinates G).continuous.fst)
-
-def autCoordinateSet (G : BondGraph V E) : Set ((V → V) × (V → V)) :=
-  {p | (∀ x, p.1 (p.2 x) = x) ∧ (∀ x, p.2 (p.1 x) = x) ∧
-    ∀ x y, Adj G (p.1 x) (p.1 y) ↔ Adj G x y}
-
-omit [TopologicalSpace V] [DiscreteTopology V] in
-theorem range_autCoordinates (G : BondGraph V E) (hG : Function.Injective G) :
-    Set.range (autCoordinates G) = autCoordinateSet G := by
-  classical
-  ext p
-  constructor
-  · rintro ⟨g,rfl⟩
-    refine ⟨fun x => ?_,fun x => ?_,?_⟩
-    · exact Equiv.apply_symm_apply (graphAutPerm G g) x
-    · exact Equiv.symm_apply_apply (graphAutPerm G g) x
-    · exact (mem_vertexAut_iff_adj G hG _).mp (graphAutPerm_mem G g)
-  · rintro ⟨h₁,h₂,hA⟩
-    let a : Equiv.Perm V := ⟨p.1,p.2,h₂,h₁⟩
-    let g : GraphAut G := ⟨a,(mem_vertexAut_iff_adj G hG a).mpr hA⟩
-    exact ⟨g,rfl⟩
-
-theorem continuous_coordinate_forward_inverse (x : V) :
-    Continuous (fun p : (V → V) × (V → V) => p.1 (p.2 x)) := by
-  have h₁ : Continuous (fun p : (V → V) × (V → V) => p.1) := continuous_fst
-  have h₂ : Continuous (fun p : (V → V) × (V → V) => p.2 x) :=
-    (continuous_apply x).comp continuous_snd
-  exact (continuous_discrete_eval (V := V)).comp (h₁.prodMk h₂)
-
-theorem continuous_coordinate_inverse_forward (x : V) :
-    Continuous (fun p : (V → V) × (V → V) => p.2 (p.1 x)) := by
-  have h₁ : Continuous (fun p : (V → V) × (V → V) => p.2) := continuous_snd
-  have h₂ : Continuous (fun p : (V → V) × (V → V) => p.1 x) :=
-    (continuous_apply x).comp continuous_fst
-  exact (continuous_discrete_eval (V := V)).comp (h₁.prodMk h₂)
-
-theorem isClosed_coordinate_forward_inverse :
-    IsClosed {p : (V → V) × (V → V) | ∀ x, p.1 (p.2 x) = x} := by
-  have h (x : V) : IsClosed {p : (V → V) × (V → V) | p.1 (p.2 x) = x} :=
-    isClosed_eq (continuous_coordinate_forward_inverse x) continuous_const
-  convert isClosed_iInter h using 1
-  ext p
-  simp only [Set.mem_iInter,Set.mem_ofPred_eq]
-
-theorem isClosed_coordinate_inverse_forward :
-    IsClosed {p : (V → V) × (V → V) | ∀ x, p.2 (p.1 x) = x} := by
-  have h (x : V) : IsClosed {p : (V → V) × (V → V) | p.2 (p.1 x) = x} :=
-    isClosed_eq (continuous_coordinate_inverse_forward x) continuous_const
-  convert isClosed_iInter h using 1
-  ext p
-  simp only [Set.mem_iInter,Set.mem_ofPred_eq]
-
-theorem isClosed_coordinate_adjacency (G : BondGraph V E) :
-    IsClosed {p : (V → V) × (V → V) |
-      ∀ x y, Adj G (p.1 x) (p.1 y) ↔ Adj G x y} := by
-  have h (x y : V) : IsClosed {p : (V → V) × (V → V) |
-      Adj G (p.1 x) (p.1 y) ↔ Adj G x y} := by
-    have hx : Continuous (fun p : (V → V) × (V → V) => p.1 x) :=
-      (continuous_apply x).comp continuous_fst
-    have hy : Continuous (fun p : (V → V) × (V → V) => p.1 y) :=
-      (continuous_apply y).comp continuous_fst
-    exact (isClosed_discrete {q : V × V | Adj G q.1 q.2 ↔ Adj G x y}).preimage
-      (hx.prodMk hy)
-  convert isClosed_iInter (fun x => isClosed_iInter (h x)) using 1
-  ext p
-  simp only [Set.mem_iInter,Set.mem_ofPred_eq]
-
-theorem isClosed_autCoordinateSet (G : BondGraph V E) :
-    IsClosed (autCoordinateSet G) :=
-  isClosed_coordinate_forward_inverse.inter
-    (isClosed_coordinate_inverse_forward.inter (isClosed_coordinate_adjacency G))
-
-def graphStabilizer (G : BondGraph V E) (x : V) : Set (GraphAut G) :=
-  {g | graphAutPerm G g x = x}
-
-theorem isOpen_graphStabilizer (G : BondGraph V E) (x : V) :
-    IsOpen (graphStabilizer G x) :=
-  (isOpen_discrete {x}).preimage (continuous_aut_apply G x)
-
-theorem isClosed_stabilizer_coordinates (G : BondGraph V E)
-    (hG : Function.Injective G) (x : V) :
-    IsClosed (autCoordinates G '' graphStabilizer G x) := by
-  have he : autCoordinates G '' graphStabilizer G x =
-      autCoordinateSet G ∩ {p | p.1 x = x} := by
-    ext p
-    constructor
-    · rintro ⟨g,hg,rfl⟩
-      refine ⟨?_,hg⟩
-      rw [←range_autCoordinates G hG]
-      exact ⟨g,rfl⟩
-    · rintro ⟨hp,hpx⟩
-      rw [←range_autCoordinates G hG] at hp
-      obtain ⟨g,rfl⟩ := hp
-      exact ⟨g,hpx,rfl⟩
-  rw [he]
-  exact (isClosed_autCoordinateSet G).inter
-    (isClosed_eq ((continuous_apply x).comp continuous_fst) continuous_const)
-
-theorem isCompact_graphStabilizer (G : BondGraph V E) (hG : Function.Injective G)
-    (hloc : LocallyFinite G) (hconn : Connected G) (x : V) :
-    IsCompact (graphStabilizer G x) := by
-  classical
-  choose n hn using fun y => exists_mem_reachBall (hconn x y)
-  let K : Set (V → V) := {f | ∀ y, f y ∈ reachBall G x (n y)}
-  have hK : IsCompact K := isCompact_pi_infinite fun y =>
-    (finite_reachBall hloc x (n y)).isCompact
-  apply (embedding_autCoordinates G).isCompact_iff.mpr
-  apply (hK.prod hK).of_isClosed_subset (isClosed_stabilizer_coordinates G hG x)
-  rintro p ⟨g,hg,rfl⟩
-  have hball (g : GraphAut G) (hg : graphAutPerm G g x = x) (y : V) :
-      graphAutPerm G g y ∈ reachBall G x (n y) := by
-    obtain ⟨b,hb⟩ := graphAutPerm_mem G g
-    simpa only [hg] using mem_reachBall_aut G hb (hn y)
-  refine ⟨fun y => hball g hg y,fun y => hball g⁻¹ ?_ y⟩
-  have he := congrArg (graphAutPerm G (g⁻¹)) hg
-  exact he.symm.trans (Equiv.symm_apply_apply (graphAutPerm G g) x)
-
-theorem locallyCompact_graphAut (G : BondGraph V E) (hG : Function.Injective G)
-    (hloc : LocallyFinite G) (hconn : Connected G) (x : V) :
-    LocallyCompactSpace (GraphAut G) :=
-  (isCompact_graphStabilizer G hG hloc hconn x).locallyCompactSpace_of_mem_nhds_of_group
-    ((isOpen_graphStabilizer G x).mem_nhds (show (1 : GraphAut G) ∈ graphStabilizer G x from rfl))
-
-theorem graphAutTopology_eq_pointwise (G : BondGraph V E) :
-    graphAutTopology G = TopologicalSpace.induced
-      (fun g : GraphAut G => (graphAutPerm G g : V → V)) inferInstance := by
-  apply le_antisymm
-  · apply continuous_iff_le_induced.mp
-    exact continuous_pi fun x => continuous_aut_apply G x
-  · let : TopologicalSpace (GraphAut G) := TopologicalSpace.induced
-      (fun g : GraphAut G => (graphAutPerm G g : V → V)) inferInstance
-    have hf : Continuous (fun g : GraphAut G => (graphAutPerm G g : V → V)) :=
-      continuous_induced_dom
-    have hi : Continuous (fun g : GraphAut G => (graphAutPerm G (g⁻¹) : V → V)) := by
-      apply continuous_pi
-      intro x
-      apply continuous_discrete_rng.mpr
-      intro y
-      have he : (fun g : GraphAut G => graphAutPerm G (g⁻¹) x) ⁻¹' {y} =
-          (fun g : GraphAut G => graphAutPerm G g y) ⁻¹' {x} := by
-        ext g
-        change (graphAutPerm G g).symm x = y ↔ graphAutPerm G g y = x
-        rw [Equiv.symm_apply_eq,eq_comm]
-      rw [he]
-      exact (isOpen_discrete {x}).preimage ((continuous_apply y).comp hf)
-    exact continuous_iff_le_induced.mp (hf.prodMk hi)
-
-instance fullGraphAutSecondCountable (G : BondGraph V E) [Countable V] :
-    SecondCountableTopology (GraphAut G) :=
-  (inducing_autCoordinates G).secondCountableTopology
-
-end Topology
-
-abbrev GraphOrbit (G : BondGraph V E) := MulAction.orbitRel.Quotient (GraphAut G) V
-
-theorem finite_graphOrbit (G : BondGraph V E) (hqt : QuasiTransitive G) :
-    Finite (GraphOrbit G) := by
-  classical
-  obtain ⟨R,hR⟩ := hqt
-  apply Finite.of_surjective (fun o : (R : Set V) =>
-    (Quotient.mk'' o.1 : GraphOrbit G))
-  intro q
-  induction q using Quotient.inductionOn with
-  | h x =>
-    obtain ⟨o,ho,a,b,hab,hao⟩ := hR x
-    refine ⟨⟨o,ho⟩,?_⟩
-    symm
-    exact Quotient.sound (show MulAction.orbitRel (GraphAut G) V x o from
-      ⟨(⟨a,⟨b,hab⟩⟩ : GraphAut G),hao⟩)
-
-noncomputable def fullGraphOrbitRoot (G : BondGraph V E) (q : GraphOrbit G) : V := q.out
-
-theorem graphOrbitRoot_covers (G : BondGraph V E) (x : V) :
-    ∃ i : GraphOrbit G, ∃ g : GraphAut G, g • fullGraphOrbitRoot G i = x := by
-  refine ⟨Quotient.mk'' x,?_⟩
-  exact Quotient.exact (Quotient.out_eq' (Quotient.mk'' x : GraphOrbit G)).symm
-
-theorem graphOrbitRoot_separates (G : BondGraph V E) (i j : GraphOrbit G)
-    (h : ∃ g : GraphAut G, g • fullGraphOrbitRoot G i = fullGraphOrbitRoot G j) : i = j := by
-  have he : (Quotient.mk'' (fullGraphOrbitRoot G j) : GraphOrbit G) =
-      Quotient.mk'' (fullGraphOrbitRoot G i) := Quotient.sound h
-  simpa only [fullGraphOrbitRoot,Quotient.out_eq'] using he.symm
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Main
+
+namespace OAI
+
+/-! The automorphism group with its topology of pointwise convergence. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+open Topology
+universe u v
+variable {V : Type u} {E : Type v}
+
+noncomputable def graphAutSubgroup (G : BondGraph V E) : Subgroup (Equiv.Perm V) where
+  carrier := vertexAut G
+  one_mem' := ⟨Equiv.refl E, isAut_refl G⟩
+  mul_mem' := by
+    rintro a b ⟨c,hc⟩ ⟨d,hd⟩
+    exact ⟨d.trans c,isAut_trans G hd hc⟩
+  inv_mem' := by
+    rintro a ⟨b,hb⟩
+    exact ⟨b.symm,isAut_symm G hb⟩
+
+def GraphAut (G : BondGraph V E) := ↥(graphAutSubgroup G)
+
+noncomputable instance (G : BondGraph V E) : Group (GraphAut G) :=
+  inferInstanceAs (Group ↥(graphAutSubgroup G))
+
+noncomputable instance (G : BondGraph V E) : MulAction (GraphAut G) V :=
+  inferInstanceAs (MulAction ↥(graphAutSubgroup G) V)
+
+noncomputable def graphAutPerm (G : BondGraph V E) : GraphAut G →* Equiv.Perm V :=
+  (graphAutSubgroup G).subtype
+
+theorem graphAutPerm_mem (G : BondGraph V E) (g : GraphAut G) :
+    graphAutPerm G g ∈ vertexAut G := g.property
+
+theorem mem_vertexAut_iff_adj (G : BondGraph V E) (hG : Function.Injective G)
+    (a : V ≃ V) : a ∈ vertexAut G ↔ ∀ x y, Adj G (a x) (a y) ↔ Adj G x y := by
+  classical
+  constructor
+  · rintro ⟨b,hb⟩ x y
+    refine ⟨fun h => ?_,adj_aut G hb⟩
+    simpa only [Equiv.symm_apply_apply] using adj_aut G (isAut_symm G hb) h
+  · intro ha
+    have hex (e : E) : ∃ f : E, G f = (G e).map a := by
+      obtain ⟨x,y,hxy⟩ := Sym2.exists.mp (show ∃ t : Sym2 V, t = G e from ⟨_,rfl⟩)
+      have h := (ha x y).mpr ⟨e,hxy.symm⟩
+      simpa only [←hxy,Sym2.map_mk] using (show ∃ f, G f = s(a x,a y) from h)
+    let f : E → E := fun e => Classical.choose (hex e)
+    have hf (e : E) : G (f e) = (G e).map a := Classical.choose_spec (hex e)
+    have hfi : Function.Injective f := by
+      intro e d hed
+      apply hG
+      have heq := congrArg (fun z => z.map a.symm) (congrArg G hed)
+      simpa only [hf,Sym2.map_map,Function.comp_def,Equiv.symm_apply_apply,Sym2.map_id',id_eq] using heq
+    have hfs : Function.Surjective f := by
+      intro e
+      obtain ⟨x,y,hxy⟩ := Sym2.exists.mp (show ∃ t : Sym2 V, t = G e from ⟨_,rfl⟩)
+      have h : Adj G (a.symm x) (a.symm y) := (ha _ _).mp (by
+        simpa only [Equiv.apply_symm_apply] using (show Adj G x y from ⟨e,hxy.symm⟩))
+      obtain ⟨d,hd⟩ := h
+      refine ⟨d,hG ?_⟩
+      rw [hf,hd,Sym2.map_mk,Equiv.apply_symm_apply,Equiv.apply_symm_apply,hxy]
+    exact ⟨Equiv.ofBijective f ⟨hfi,hfs⟩,hf⟩
+
+section Topology
+variable [TopologicalSpace V] [DiscreteTopology V]
+
+noncomputable def autCoordinates (G : BondGraph V E) (g : GraphAut G) :
+    (V → V) × (V → V) := (graphAutPerm G g,graphAutPerm G (g⁻¹))
+
+noncomputable instance graphAutTopology (G : BondGraph V E) : TopologicalSpace (GraphAut G) :=
+  TopologicalSpace.induced (autCoordinates G) inferInstance
+
+omit [DiscreteTopology V] in
+theorem inducing_autCoordinates (G : BondGraph V E) :
+    IsInducing (autCoordinates G) := ⟨rfl⟩
+
+omit [DiscreteTopology V] in
+theorem embedding_autCoordinates (G : BondGraph V E) :
+    IsEmbedding (autCoordinates G) := by
+  refine ⟨inducing_autCoordinates G,?_⟩
+  intro f g h
+  apply Subtype.ext
+  apply Equiv.ext
+  intro x
+  exact congrFun (congrArg Prod.fst h) x
+
+noncomputable instance graphAutT2 (G : BondGraph V E) : T2Space (GraphAut G) :=
+  (embedding_autCoordinates G).t2Space
+
+omit [DiscreteTopology V] in
+theorem continuous_aut_apply (G : BondGraph V E) (x : V) :
+    Continuous (fun g : GraphAut G => graphAutPerm G g x) :=
+  (continuous_apply x).comp ((inducing_autCoordinates G).continuous.fst)
+
+omit [DiscreteTopology V] in
+theorem continuous_aut_inv_apply (G : BondGraph V E) (x : V) :
+    Continuous (fun g : GraphAut G => graphAutPerm G (g⁻¹) x) :=
+  (continuous_apply x).comp ((inducing_autCoordinates G).continuous.snd)
+
+theorem continuous_discrete_eval : Continuous (fun p : (V → V) × V => p.1 p.2) :=
+  continuous_prod_of_discrete_right.mpr fun x => continuous_apply x
+
+noncomputable instance graphAutTopologicalGroup (G : BondGraph V E) :
+    IsTopologicalGroup (GraphAut G) where
+  continuous_mul := by
+    apply (inducing_autCoordinates G).continuous_iff.mpr
+    apply Continuous.prodMk
+    · apply continuous_pi
+      intro x
+      exact continuous_discrete_eval.comp
+        (((inducing_autCoordinates G).continuous.fst.comp continuous_fst).prodMk
+          ((continuous_aut_apply G x).comp continuous_snd))
+    · apply continuous_pi
+      intro x
+      exact continuous_discrete_eval.comp
+        (((inducing_autCoordinates G).continuous.snd.comp continuous_snd).prodMk
+          ((continuous_aut_inv_apply G x).comp continuous_fst))
+  continuous_inv := by
+    apply (inducing_autCoordinates G).continuous_iff.mpr
+    simpa only [Function.comp_def,autCoordinates,inv_inv] using ((inducing_autCoordinates G).continuous.snd.prodMk (inducing_autCoordinates G).continuous.fst)
+
+def autCoordinateSet (G : BondGraph V E) : Set ((V → V) × (V → V)) :=
+  {p | (∀ x, p.1 (p.2 x) = x) ∧ (∀ x, p.2 (p.1 x) = x) ∧
+    ∀ x y, Adj G (p.1 x) (p.1 y) ↔ Adj G x y}
+
+omit [TopologicalSpace V] [DiscreteTopology V] in
+theorem range_autCoordinates (G : BondGraph V E) (hG : Function.Injective G) :
+    Set.range (autCoordinates G) = autCoordinateSet G := by
+  classical
+  ext p
+  constructor
+  · rintro ⟨g,rfl⟩
+    refine ⟨fun x => ?_,fun x => ?_,?_⟩
+    · exact Equiv.apply_symm_apply (graphAutPerm G g) x
+    · exact Equiv.symm_apply_apply (graphAutPerm G g) x
+    · exact (mem_vertexAut_iff_adj G hG _).mp (graphAutPerm_mem G g)
+  · rintro ⟨h₁,h₂,hA⟩
+    let a : Equiv.Perm V := ⟨p.1,p.2,h₂,h₁⟩
+    let g : GraphAut G := ⟨a,(mem_vertexAut_iff_adj G hG a).mpr hA⟩
+    exact ⟨g,rfl⟩
+
+theorem continuous_coordinate_forward_inverse (x : V) :
+    Continuous (fun p : (V → V) × (V → V) => p.1 (p.2 x)) := by
+  have h₁ : Continuous (fun p : (V → V) × (V → V) => p.1) := continuous_fst
+  have h₂ : Continuous (fun p : (V → V) × (V → V) => p.2 x) :=
+    (continuous_apply x).comp continuous_snd
+  exact (continuous_discrete_eval (V := V)).comp (h₁.prodMk h₂)
+
+theorem continuous_coordinate_inverse_forward (x : V) :
+    Continuous (fun p : (V → V) × (V → V) => p.2 (p.1 x)) := by
+  have h₁ : Continuous (fun p : (V → V) × (V → V) => p.2) := continuous_snd
+  have h₂ : Continuous (fun p : (V → V) × (V → V) => p.1 x) :=
+    (continuous_apply x).comp continuous_fst
+  exact (continuous_discrete_eval (V := V)).comp (h₁.prodMk h₂)
+
+theorem isClosed_coordinate_forward_inverse :
+    IsClosed {p : (V → V) × (V → V) | ∀ x, p.1 (p.2 x) = x} := by
+  have h (x : V) : IsClosed {p : (V → V) × (V → V) | p.1 (p.2 x) = x} :=
+    isClosed_eq (continuous_coordinate_forward_inverse x) continuous_const
+  convert isClosed_iInter h using 1
+  ext p
+  simp only [Set.mem_iInter,Set.mem_ofPred_eq]
+
+theorem isClosed_coordinate_inverse_forward :
+    IsClosed {p : (V → V) × (V → V) | ∀ x, p.2 (p.1 x) = x} := by
+  have h (x : V) : IsClosed {p : (V → V) × (V → V) | p.2 (p.1 x) = x} :=
+    isClosed_eq (continuous_coordinate_inverse_forward x) continuous_const
+  convert isClosed_iInter h using 1
+  ext p
+  simp only [Set.mem_iInter,Set.mem_ofPred_eq]
+
+theorem isClosed_coordinate_adjacency (G : BondGraph V E) :
+    IsClosed {p : (V → V) × (V → V) |
+      ∀ x y, Adj G (p.1 x) (p.1 y) ↔ Adj G x y} := by
+  have h (x y : V) : IsClosed {p : (V → V) × (V → V) |
+      Adj G (p.1 x) (p.1 y) ↔ Adj G x y} := by
+    have hx : Continuous (fun p : (V → V) × (V → V) => p.1 x) :=
+      (continuous_apply x).comp continuous_fst
+    have hy : Continuous (fun p : (V → V) × (V → V) => p.1 y) :=
+      (continuous_apply y).comp continuous_fst
+    exact (isClosed_discrete {q : V × V | Adj G q.1 q.2 ↔ Adj G x y}).preimage
+      (hx.prodMk hy)
+  convert isClosed_iInter (fun x => isClosed_iInter (h x)) using 1
+  ext p
+  simp only [Set.mem_iInter,Set.mem_ofPred_eq]
+
+theorem isClosed_autCoordinateSet (G : BondGraph V E) :
+    IsClosed (autCoordinateSet G) :=
+  isClosed_coordinate_forward_inverse.inter
+    (isClosed_coordinate_inverse_forward.inter (isClosed_coordinate_adjacency G))
+
+def graphStabilizer (G : BondGraph V E) (x : V) : Set (GraphAut G) :=
+  {g | graphAutPerm G g x = x}
+
+theorem isOpen_graphStabilizer (G : BondGraph V E) (x : V) :
+    IsOpen (graphStabilizer G x) :=
+  (isOpen_discrete {x}).preimage (continuous_aut_apply G x)
+
+theorem isClosed_stabilizer_coordinates (G : BondGraph V E)
+    (hG : Function.Injective G) (x : V) :
+    IsClosed (autCoordinates G '' graphStabilizer G x) := by
+  have he : autCoordinates G '' graphStabilizer G x =
+      autCoordinateSet G ∩ {p | p.1 x = x} := by
+    ext p
+    constructor
+    · rintro ⟨g,hg,rfl⟩
+      refine ⟨?_,hg⟩
+      rw [←range_autCoordinates G hG]
+      exact ⟨g,rfl⟩
+    · rintro ⟨hp,hpx⟩
+      rw [←range_autCoordinates G hG] at hp
+      obtain ⟨g,rfl⟩ := hp
+      exact ⟨g,hpx,rfl⟩
+  rw [he]
+  exact (isClosed_autCoordinateSet G).inter
+    (isClosed_eq ((continuous_apply x).comp continuous_fst) continuous_const)
+
+theorem isCompact_graphStabilizer (G : BondGraph V E) (hG : Function.Injective G)
+    (hloc : LocallyFinite G) (hconn : Connected G) (x : V) :
+    IsCompact (graphStabilizer G x) := by
+  classical
+  choose n hn using fun y => exists_mem_reachBall (hconn x y)
+  let K : Set (V → V) := {f | ∀ y, f y ∈ reachBall G x (n y)}
+  have hK : IsCompact K := isCompact_pi_infinite fun y =>
+    (finite_reachBall hloc x (n y)).isCompact
+  apply (embedding_autCoordinates G).isCompact_iff.mpr
+  apply (hK.prod hK).of_isClosed_subset (isClosed_stabilizer_coordinates G hG x)
+  rintro p ⟨g,hg,rfl⟩
+  have hball (g : GraphAut G) (hg : graphAutPerm G g x = x) (y : V) :
+      graphAutPerm G g y ∈ reachBall G x (n y) := by
+    obtain ⟨b,hb⟩ := graphAutPerm_mem G g
+    simpa only [hg] using mem_reachBall_aut G hb (hn y)
+  refine ⟨fun y => hball g hg y,fun y => hball g⁻¹ ?_ y⟩
+  have he := congrArg (graphAutPerm G (g⁻¹)) hg
+  exact he.symm.trans (Equiv.symm_apply_apply (graphAutPerm G g) x)
+
+theorem locallyCompact_graphAut (G : BondGraph V E) (hG : Function.Injective G)
+    (hloc : LocallyFinite G) (hconn : Connected G) (x : V) :
+    LocallyCompactSpace (GraphAut G) :=
+  (isCompact_graphStabilizer G hG hloc hconn x).locallyCompactSpace_of_mem_nhds_of_group
+    ((isOpen_graphStabilizer G x).mem_nhds (show (1 : GraphAut G) ∈ graphStabilizer G x from rfl))
+
+theorem graphAutTopology_eq_pointwise (G : BondGraph V E) :
+    graphAutTopology G = TopologicalSpace.induced
+      (fun g : GraphAut G => (graphAutPerm G g : V → V)) inferInstance := by
+  apply le_antisymm
+  · apply continuous_iff_le_induced.mp
+    exact continuous_pi fun x => continuous_aut_apply G x
+  · let : TopologicalSpace (GraphAut G) := TopologicalSpace.induced
+      (fun g : GraphAut G => (graphAutPerm G g : V → V)) inferInstance
+    have hf : Continuous (fun g : GraphAut G => (graphAutPerm G g : V → V)) :=
+      continuous_induced_dom
+    have hi : Continuous (fun g : GraphAut G => (graphAutPerm G (g⁻¹) : V → V)) := by
+      apply continuous_pi
+      intro x
+      apply continuous_discrete_rng.mpr
+      intro y
+      have he : (fun g : GraphAut G => graphAutPerm G (g⁻¹) x) ⁻¹' {y} =
+          (fun g : GraphAut G => graphAutPerm G g y) ⁻¹' {x} := by
+        ext g
+        change (graphAutPerm G g).symm x = y ↔ graphAutPerm G g y = x
+        rw [Equiv.symm_apply_eq,eq_comm]
+      rw [he]
+      exact (isOpen_discrete {x}).preimage ((continuous_apply y).comp hf)
+    exact continuous_iff_le_induced.mp (hf.prodMk hi)
+
+instance fullGraphAutSecondCountable (G : BondGraph V E) [Countable V] :
+    SecondCountableTopology (GraphAut G) :=
+  (inducing_autCoordinates G).secondCountableTopology
+
+end Topology
+
+abbrev GraphOrbit (G : BondGraph V E) := MulAction.orbitRel.Quotient (GraphAut G) V
+
+theorem finite_graphOrbit (G : BondGraph V E) (hqt : QuasiTransitive G) :
+    Finite (GraphOrbit G) := by
+  classical
+  obtain ⟨R,hR⟩ := hqt
+  apply Finite.of_surjective (fun o : (R : Set V) =>
+    (Quotient.mk'' o.1 : GraphOrbit G))
+  intro q
+  induction q using Quotient.inductionOn with
+  | h x =>
+    obtain ⟨o,ho,a,b,hab,hao⟩ := hR x
+    refine ⟨⟨o,ho⟩,?_⟩
+    symm
+    exact Quotient.sound (show MulAction.orbitRel (GraphAut G) V x o from
+      ⟨(⟨a,⟨b,hab⟩⟩ : GraphAut G),hao⟩)
+
+noncomputable def fullGraphOrbitRoot (G : BondGraph V E) (q : GraphOrbit G) : V := q.out
+
+theorem graphOrbitRoot_covers (G : BondGraph V E) (x : V) :
+    ∃ i : GraphOrbit G, ∃ g : GraphAut G, g • fullGraphOrbitRoot G i = x := by
+  refine ⟨Quotient.mk'' x,?_⟩
+  exact Quotient.exact (Quotient.out_eq' (Quotient.mk'' x : GraphOrbit G)).symm
+
+theorem graphOrbitRoot_separates (G : BondGraph V E) (i j : GraphOrbit G)
+    (h : ∃ g : GraphAut G, g • fullGraphOrbitRoot G i = fullGraphOrbitRoot G j) : i = j := by
+  have he : (Quotient.mk'' (fullGraphOrbitRoot G j) : GraphOrbit G) =
+      Quotient.mk'' (fullGraphOrbitRoot G i) := Quotient.sound h
+  simpa only [fullGraphOrbitRoot,Quotient.out_eq'] using he.symm
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Graph/Paths.lean b/lean/OAI/Probability/UnimodularPercolation/Graph/Paths.lean
index 841551406451569844569ca72d18402f81a61cd2..d6ebcbc96f4e8f263b94e2591a6f6df5fa788414 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Graph/Paths.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Graph/Paths.lean
@@ -1,314 +1,314 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Clusters.Uniqueness
-
-namespace OAI
-
-/-! Bond paths, degree bounds and positivity of the critical parameter. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-section BondPaths
-attribute [local instance] Classical.propDecidable
-
- 
-inductive Along (G : BondGraph V E) : V → List E → V → Prop
-  | nil (x : V) : Along G x [] x
-  | cons {x z y : V} {l : List E} (e : E) (he : G e = s(x,z))
-      (tail : Along G z l y) : Along G x (e :: l) y
-
- theorem Along.conn {G : BondGraph V E} {x y : V} {l : List E} (h : Along G x l y)
-    {ω : Set E} (ho : ∀ e ∈ l, e ∈ ω) : Conn G ω x y := by
-  induction h with
-  | nil => exact conn_refl G ω _
-  | @cons x z y l e he ht ih =>
-    exact (Relation.ReflTransGen.single ⟨e, ho e (by simp), he⟩).trans
-      (ih (fun e' he' => ho e' (by simp [he'])))
-
- def openSimpleGraph (G : BondGraph V E) (ω : Set E) : SimpleGraph V where
-  Adj x y := x ≠ y ∧ ∃ e ∈ ω, G e = s(x,y)
-  symm := ⟨by
-    rintro x y ⟨hxy, e, he, h⟩
-    exact ⟨hxy.symm, e, he, h.trans (Sym2.eq_swap ..)⟩⟩
-  loopless := ⟨by simp⟩
-
- theorem conn_iff_reachable (G : BondGraph V E) (ω : Set E) (x y : V) :
-    Conn G ω x y ↔ (openSimpleGraph G ω).Reachable x y := by
-  constructor
-  · intro h
-    induction h with
-    | refl => exact ⟨.nil⟩
-    | @tail z y h hy ih =>
-      obtain ⟨w⟩ := ih
-      by_cases hzy : z = y
-      · subst y; exact ⟨w⟩
-      · exact ⟨w.concat ⟨hzy, hy⟩⟩
-  · rintro ⟨w⟩
-    induction w with
-    | nil => exact conn_refl G ω _
-    | @cons x z y hadj w ih =>
-      exact (Relation.ReflTransGen.single hadj.2).trans ih
-
- theorem walk_lift_bonds (G : BondGraph V E) (ω : Set E) {x y : V}
-    (w : (openSimpleGraph G ω).Walk x y) :
-    ∃ l : List E, Along G x l y ∧ l.map G = w.edges ∧ ∀ e ∈ l, e ∈ ω := by
-  induction w with
-  | nil => exact ⟨[], .nil _, rfl, by simp⟩
-  | @cons x z y hadj w ih =>
-    obtain ⟨e, he, heq⟩ := hadj.2
-    obtain ⟨l, hl, hmap, hopen⟩ := ih
-    refine ⟨e :: l, .cons e heq hl, ?_, ?_⟩
-    · simp [hmap, heq]
-    · intro e' he'
-      rcases List.mem_cons.mp he' with rfl | he'
-      · exact he
-      · exact hopen e' he'
-
- 
- theorem conn_exists_bond_trail (G : BondGraph V E) {ω : Set E} {x y : V}
-    (h : Conn G ω x y) :
-    ∃ l : List E, Along G x l y ∧ l.Nodup ∧ ∀ e ∈ l, e ∈ ω := by
-  obtain ⟨w⟩ := (conn_iff_reachable G ω x y).mp h
-  obtain ⟨l, hl, hmap, hopen⟩ := walk_lift_bonds G ω (w.toPath : (openSimpleGraph G ω).Walk x y)
-  refine ⟨l, hl, ?_, hopen⟩
-  apply List.Nodup.of_map (f := G)
-  rw [hmap]
-  exact w.toPath.property.isTrail.edges_nodup
-
- theorem Along.append {G : BondGraph V E} {x y z : V} {l k : List E}
-    (hl : Along G x l y) (hk : Along G y k z) : Along G x (l ++ k) z := by
-  induction hl with
-  | nil => exact hk
-  | cons e he ht ih => exact .cons e he (ih hk)
-
-end BondPaths
-
-section DegreeBound
-attribute [local instance] Classical.propDecidable
-
- def Step (G : BondGraph V E) (x : V) := {ey : E × V // G ey.1 = s(x,ey.2)}
-
- theorem finite_steps (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) :
-    Set.Finite {ey : E × V | G ey.1 = s(x,ey.2)} := by
-  apply ((hloc x).prod (finite_neighbors hloc x)).subset
-  rintro ⟨e,y⟩ he
-  change G e = s(x,y) at he
-  exact ⟨by change x ∈ G e; rw [he]; simp, e, he⟩
-
- noncomputable def stepEquiv (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (h : IsAut G a b) (x : V) : Step G x ≃ Step G (a x) where
-  toFun d := ⟨(b d.val.1, a d.val.2), by rw [h, d.property, Sym2.map_mk]⟩
-  invFun d := ⟨(b.symm d.val.1, a.symm d.val.2), by
-    rw [isAut_symm G h, d.property, Sym2.map_mk, Equiv.symm_apply_apply]⟩
-  left_inv d := by apply Subtype.ext; exact Prod.ext (b.symm_apply_apply _) (a.symm_apply_apply _)
-  right_inv d := by apply Subtype.ext; exact Prod.ext (b.apply_symm_apply _) (a.apply_symm_apply _)
-
- noncomputable def degree (G : BondGraph V E) (x : V) : ℕ := Nat.card (Step G x)
-
- theorem degree_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (h : IsAut G a b) (x : V) : degree G (a x) = degree G x :=
-  (Nat.card_congr (stepEquiv G h x)).symm
-
- theorem exists_degree_bound (G : BondGraph V E) (hqt : QuasiTransitive G) :
-    ∃ D : ℕ, 0 < D ∧ ∀ x, degree G x ≤ D := by
-  obtain ⟨F, hF⟩ := hqt
-  refine ⟨(∑ x ∈ F, degree G x) + 1, Nat.zero_lt_succ _, ?_⟩
-  intro x
-  obtain ⟨o, ho, a, b, hab, rfl⟩ := hF x
-  rw [degree_aut G hab]
-  exact (Finset.single_le_sum (fun _ _ => Nat.zero_le _) ho).trans (Nat.le_succ _)
-
-end DegreeBound
-
-section SmallParameter
-attribute [local instance] Classical.propDecidable
-
-def TrailLength (G : BondGraph V E) (x : V) (n : ℕ) : Set (Set E) :=
-  {ω | ∃ y l, Along G x l y ∧ l.Nodup ∧ l.length = n ∧ ∀ e ∈ l, e ∈ ω}
-
-theorem trailLength_upper (G : BondGraph V E) (x : V) (n : ℕ) :
-    IsUpperSet (TrailLength G x n) := by
-  rintro ω η h ⟨y,l,hl,hn,hlen,ho⟩
-  exact ⟨y,l,hl,hn,hlen,fun e he => h (ho e he)⟩
-
-theorem finiteWitness_trailLength (G : BondGraph V E) (x : V) (n : ℕ) :
-    FiniteWitness (TrailLength G x n) := by
-  rintro ω ⟨y,l,hl,hn,hlen,ho⟩
-  exact ⟨l.toFinset, fun e he => ho e (List.mem_toFinset.mp he),
-    y,l,hl,hn,hlen, fun e he => List.mem_toFinset.mpr he⟩
-
-theorem finiteWitness_edge (e : E) : FiniteWitness {ω : Set E | e ∈ ω} := by
-  intro ω he
-  exact ⟨{e}, by simpa using he, by simp⟩
-
-theorem trailLength_succ_subset (G : BondGraph V E) (x : V) (n : ℕ) :
-    TrailLength G x (n+1) ⊆ ⋃ d : Step G x,
-      DisjointOccurrence {ω | d.val.1 ∈ ω} (TrailLength G d.val.2 n) := by
-  rintro ω ⟨y,l,hl,hn,hlen,ho⟩
-  cases hl with
-  | nil => simp at hlen
-  | @cons x z y l e he tail =>
-    refine Set.mem_iUnion.mpr ⟨⟨(e,z),he⟩, {e}, l.toFinset, ?_, ?_, ?_, ?_, ?_⟩
-    · simpa using ho e (by simp)
-    · intro e' he'; exact ho e' (by simpa using Or.inr (List.mem_toFinset.mp he'))
-    · simpa using (List.nodup_cons.mp hn).1
-    · simp
-    · exact ⟨y,l,tail,(List.nodup_cons.mp hn).2, by simpa using hlen,
-        fun e' he' => List.mem_toFinset.mpr he'⟩
-
-theorem trailLength_probability_le [Countable E] (G : BondGraph V E)
-    (hloc : LocallyFinite G) (D : ℕ) (hD : ∀ x, degree G x ≤ D)
-    (p : unitInterval) (x : V) (n : ℕ) :
-    (law p).real (TrailLength G x n) ≤ ((D : ℝ) * p) ^ n := by
-  induction n generalizing x with
-  | zero => simpa only [pow_zero] using (measureReal_le_one (μ := law p) (s := TrailLength G x 0))
-  | succ n ih =>
-    let : Fintype (Step G x) := (finite_steps G hloc x).fintype
-    let F := fun d : Step G x =>
-      DisjointOccurrence {ω : Set E | d.val.1 ∈ ω} (TrailLength G d.val.2 n)
-    calc
-      (law p).real (TrailLength G x (n+1)) ≤ (law p).real (⋃ d, F d) :=
-        measureReal_mono (trailLength_succ_subset G x n)
-      _ ≤ ∑ d, (law p).real (F d) := measureReal_iUnion_fintype_le _
-      _ ≤ ∑ _d : Step G x, (p : ℝ) * ((D : ℝ) * p) ^ n := by
-        apply Finset.sum_le_sum
-        intro d _
-        have hb := bk p {ω : Set E | d.val.1 ∈ ω} (TrailLength G d.val.2 n)
-          (fun _ _ hω he => hω he) (trailLength_upper G d.val.2 n)
-          (finiteWitness_edge _) (finiteWitness_trailLength G d.val.2 n)
-        have heq : (law p).real {ω : Set E | d.val.1 ∈ ω} = (p : ℝ) := by
-          simpa using finite_open_probability p {d.val.1}
-        rw [heq] at hb
-        exact hb.trans (mul_le_mul_of_nonneg_left (ih d.val.2) p.2.1)
-      _ ≤ (D : ℝ) * ((p : ℝ) * ((D : ℝ) * p) ^ n) := by
-        simp only [Finset.sum_const, Finset.card_univ, nsmul_eq_mul]
-        apply mul_le_mul_of_nonneg_right _ (mul_nonneg p.2.1 (pow_nonneg (mul_nonneg (Nat.cast_nonneg D) p.2.1) n))
-        exact_mod_cast (show Fintype.card (Step G x) ≤ D by
-          simpa only [degree, Nat.card_eq_fintype_card] using hD x)
-      _ = ((D : ℝ) * p) ^ (n+1) := by ring
-
- theorem Along.mem_reachBall {G : BondGraph V E} {x y : V} {l : List E}
-    (h : Along G x l y) : y ∈ reachBall G x l.length := by
-  induction h with
-  | nil => exact Set.mem_singleton _
-  | @cons x z y l e he ht ih =>
-    have hadj : Adj G x z := ⟨e,he⟩
-    have hstep : ∀ n, reachBall G z n ⊆ reachBall G x (n+1) := by
-      intro n
-      induction n with
-      | zero =>
-        rintro w rfl
-        exact Or.inr (Set.mem_iUnion.mpr ⟨x, Set.mem_iUnion.mpr ⟨Set.mem_singleton _, hadj⟩⟩)
-      | succ n ih =>
-        rintro w (hw | hw)
-        · exact Or.inl (ih hw)
-        · right
-          simp only [Set.mem_iUnion] at hw ⊢
-          obtain ⟨v,hv,hvw⟩ := hw
-          exact ⟨v,ih hv,hvw⟩
-    exact hstep _ ih
-
- theorem Along.take {G : BondGraph V E} {x y : V} {l : List E}
-    (h : Along G x l y) (n : ℕ) : ∃ z, Along G x (l.take n) z := by
-  induction h generalizing n with
-  | @nil z => exact ⟨z, by simpa using (Along.nil (G := G) z)⟩
-  | @cons x z y l e he ht ih =>
-    cases n with
-    | zero => exact ⟨x,.nil x⟩
-    | succ n =>
-      obtain ⟨v,hv⟩ := ih n
-      exact ⟨v,.cons e he hv⟩
-
- theorem reachBall_mono (G : BondGraph V E) (x : V) : Monotone (reachBall G x) := by
-  apply monotone_nat_of_le_succ
-  intro n
-  exact Set.subset_union_left
-
- theorem infinite_cluster_subset_trailLength (G : BondGraph V E) (hloc : LocallyFinite G)
-    (x : V) (n : ℕ) : {ω | (cluster G ω x).Infinite} ⊆ TrailLength G x n := by
-  intro ω hω
-  obtain ⟨y,hy,hyball⟩ := hω.exists_notMem_finite (finite_reachBall hloc x n)
-  obtain ⟨l,hl,hn,ho⟩ := conn_exists_bond_trail G hy
-  have hlen : n ≤ l.length := by
-    by_contra h
-    exact hyball ((reachBall_mono G x (le_of_lt (Nat.lt_of_not_ge h))) hl.mem_reachBall)
-  obtain ⟨z,hz⟩ := hl.take n
-  exact ⟨z,l.take n,hz,hn.take,List.length_take_of_le hlen,
-    fun e he => ho e (List.mem_of_mem_take he)⟩
-
- theorem infinite_cluster_probability_eq_zero_of_small [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (D : ℕ) (hD : ∀ x, degree G x ≤ D)
-    (p : unitInterval) (hp : (D : ℝ) * p < 1) (x : V) :
-    law p {ω | (cluster G ω x).Infinite} = 0 := by
-  apply (measureReal_eq_zero_iff).mp
-  apply le_antisymm _ measureReal_nonneg
-  apply ge_of_tendsto (tendsto_pow_atTop_nhds_zero_of_lt_one (mul_nonneg (Nat.cast_nonneg D) p.2.1) hp)
-  filter_upwards [] with n
-  exact (measureReal_mono (infinite_cluster_subset_trailLength G hloc x n)).trans
-    (trailLength_probability_le G hloc D hD p x n)
-
- theorem exists_infinite_cluster_probability_eq_zero_of_small [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (D : ℕ) (hD : ∀ x, degree G x ≤ D)
-    (p : unitInterval) (hp : (D : ℝ) * p < 1) :
-    law p {ω | (infiniteClusters G ω).Nonempty} = 0 := by
-  have heq : {ω | (infiniteClusters G ω).Nonempty} =
-      ⋃ x : V, {ω | (cluster G ω x).Infinite} := by
-    ext ω
-    simp only [Set.mem_ofPred_eq, infiniteClusters_nonempty_iff, Set.mem_iUnion]
-  rw [heq, measure_iUnion_null_iff]
-  exact infinite_cluster_probability_eq_zero_of_small G hloc D hD p hp
-
- theorem pc_pos [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G) :
-    0 < pc G := by
-  obtain ⟨D, hDpos, hD⟩ := exists_degree_bound G hqt
-  let t : unitInterval := ⟨1 / ((D : ℝ) + 1), by
-    constructor
-    · positivity
-    · apply (div_le_one (by positivity)).mpr
-      linarith [Nat.cast_nonneg (α := ℝ) D]⟩
-  have ht : (0 : unitInterval) < t := by
-    change (0 : ℝ) < 1 / ((D : ℝ) + 1)
-    positivity
-  apply ht.trans_le
-  apply le_sInf
-  intro p hp
-  by_contra h
-  have hp' : (p : ℝ) < 1 / ((D : ℝ) + 1) := lt_of_not_ge h
-  have hsmall : (D : ℝ) * p < 1 := by
-    have hmul := (lt_div_iff₀ (show (0 : ℝ) < D + 1 by positivity)).mp hp'
-    nlinarith [p.2.1]
-  have hz := exists_infinite_cluster_probability_eq_zero_of_small G hloc D hD p hsmall
-  exact (ne_of_gt hp) hz
-
-end SmallParameter
-
-section MainBridge
-attribute [local instance] Classical.propDecidable
-
- 
-
-theorem mainConclusion_of_strict_gap [Infinite V] (G : BondGraph V E)
-    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (hgap : pc G < ptwo G) : MainConclusion G := by
-  let : Countable V := countable_vertices hloc hconn
-  let : Countable E := countable_edges hloc
-  obtain ⟨q, hq, hpq⟩ := lt_sSup_iff.mp hgap
-  obtain ⟨p₂, hp₂, hp₂q⟩ := exists_between hpq
-  obtain ⟨p₁, hp₁, hp₁₂⟩ := exists_between hp₂
-  have hc : OperatorBounded G (pc G) := operatorBounded_of_le G hpq.le hq
-  refine ⟨(operatorNorm_eq_sup_below G (pc G) (pc_pos G hloc hqt)).symm,
-    (operatorNorm_lt_top_iff G (pc G)).mpr hc, hgap, ptwo_le_pu G hconn hqt, ?_⟩
-  refine ⟨p₁, p₂, hp₁, hp₁₂, hp₂q.trans_le (show q ≤ 1 from q.2.2), ?_⟩
-  exact coupled_interval G hconn hloc hqt p₁ p₂ hp₁
-    (operatorBounded_of_le G hp₂q.le hq)
-
-end MainBridge
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Clusters.Uniqueness
+
+namespace OAI
+
+/-! Bond paths, degree bounds and positivity of the critical parameter. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+section BondPaths
+attribute [local instance] Classical.propDecidable
+
+ 
+inductive Along (G : BondGraph V E) : V → List E → V → Prop
+  | nil (x : V) : Along G x [] x
+  | cons {x z y : V} {l : List E} (e : E) (he : G e = s(x,z))
+      (tail : Along G z l y) : Along G x (e :: l) y
+
+ theorem Along.conn {G : BondGraph V E} {x y : V} {l : List E} (h : Along G x l y)
+    {ω : Set E} (ho : ∀ e ∈ l, e ∈ ω) : Conn G ω x y := by
+  induction h with
+  | nil => exact conn_refl G ω _
+  | @cons x z y l e he ht ih =>
+    exact (Relation.ReflTransGen.single ⟨e, ho e (by simp), he⟩).trans
+      (ih (fun e' he' => ho e' (by simp [he'])))
+
+ def openSimpleGraph (G : BondGraph V E) (ω : Set E) : SimpleGraph V where
+  Adj x y := x ≠ y ∧ ∃ e ∈ ω, G e = s(x,y)
+  symm := ⟨by
+    rintro x y ⟨hxy, e, he, h⟩
+    exact ⟨hxy.symm, e, he, h.trans (Sym2.eq_swap ..)⟩⟩
+  loopless := ⟨by simp⟩
+
+ theorem conn_iff_reachable (G : BondGraph V E) (ω : Set E) (x y : V) :
+    Conn G ω x y ↔ (openSimpleGraph G ω).Reachable x y := by
+  constructor
+  · intro h
+    induction h with
+    | refl => exact ⟨.nil⟩
+    | @tail z y h hy ih =>
+      obtain ⟨w⟩ := ih
+      by_cases hzy : z = y
+      · subst y; exact ⟨w⟩
+      · exact ⟨w.concat ⟨hzy, hy⟩⟩
+  · rintro ⟨w⟩
+    induction w with
+    | nil => exact conn_refl G ω _
+    | @cons x z y hadj w ih =>
+      exact (Relation.ReflTransGen.single hadj.2).trans ih
+
+ theorem walk_lift_bonds (G : BondGraph V E) (ω : Set E) {x y : V}
+    (w : (openSimpleGraph G ω).Walk x y) :
+    ∃ l : List E, Along G x l y ∧ l.map G = w.edges ∧ ∀ e ∈ l, e ∈ ω := by
+  induction w with
+  | nil => exact ⟨[], .nil _, rfl, by simp⟩
+  | @cons x z y hadj w ih =>
+    obtain ⟨e, he, heq⟩ := hadj.2
+    obtain ⟨l, hl, hmap, hopen⟩ := ih
+    refine ⟨e :: l, .cons e heq hl, ?_, ?_⟩
+    · simp [hmap, heq]
+    · intro e' he'
+      rcases List.mem_cons.mp he' with rfl | he'
+      · exact he
+      · exact hopen e' he'
+
+ 
+ theorem conn_exists_bond_trail (G : BondGraph V E) {ω : Set E} {x y : V}
+    (h : Conn G ω x y) :
+    ∃ l : List E, Along G x l y ∧ l.Nodup ∧ ∀ e ∈ l, e ∈ ω := by
+  obtain ⟨w⟩ := (conn_iff_reachable G ω x y).mp h
+  obtain ⟨l, hl, hmap, hopen⟩ := walk_lift_bonds G ω (w.toPath : (openSimpleGraph G ω).Walk x y)
+  refine ⟨l, hl, ?_, hopen⟩
+  apply List.Nodup.of_map (f := G)
+  rw [hmap]
+  exact w.toPath.property.isTrail.edges_nodup
+
+ theorem Along.append {G : BondGraph V E} {x y z : V} {l k : List E}
+    (hl : Along G x l y) (hk : Along G y k z) : Along G x (l ++ k) z := by
+  induction hl with
+  | nil => exact hk
+  | cons e he ht ih => exact .cons e he (ih hk)
+
+end BondPaths
+
+section DegreeBound
+attribute [local instance] Classical.propDecidable
+
+ def Step (G : BondGraph V E) (x : V) := {ey : E × V // G ey.1 = s(x,ey.2)}
+
+ theorem finite_steps (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) :
+    Set.Finite {ey : E × V | G ey.1 = s(x,ey.2)} := by
+  apply ((hloc x).prod (finite_neighbors hloc x)).subset
+  rintro ⟨e,y⟩ he
+  change G e = s(x,y) at he
+  exact ⟨by change x ∈ G e; rw [he]; simp, e, he⟩
+
+ noncomputable def stepEquiv (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (h : IsAut G a b) (x : V) : Step G x ≃ Step G (a x) where
+  toFun d := ⟨(b d.val.1, a d.val.2), by rw [h, d.property, Sym2.map_mk]⟩
+  invFun d := ⟨(b.symm d.val.1, a.symm d.val.2), by
+    rw [isAut_symm G h, d.property, Sym2.map_mk, Equiv.symm_apply_apply]⟩
+  left_inv d := by apply Subtype.ext; exact Prod.ext (b.symm_apply_apply _) (a.symm_apply_apply _)
+  right_inv d := by apply Subtype.ext; exact Prod.ext (b.apply_symm_apply _) (a.apply_symm_apply _)
+
+ noncomputable def degree (G : BondGraph V E) (x : V) : ℕ := Nat.card (Step G x)
+
+ theorem degree_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (h : IsAut G a b) (x : V) : degree G (a x) = degree G x :=
+  (Nat.card_congr (stepEquiv G h x)).symm
+
+ theorem exists_degree_bound (G : BondGraph V E) (hqt : QuasiTransitive G) :
+    ∃ D : ℕ, 0 < D ∧ ∀ x, degree G x ≤ D := by
+  obtain ⟨F, hF⟩ := hqt
+  refine ⟨(∑ x ∈ F, degree G x) + 1, Nat.zero_lt_succ _, ?_⟩
+  intro x
+  obtain ⟨o, ho, a, b, hab, rfl⟩ := hF x
+  rw [degree_aut G hab]
+  exact (Finset.single_le_sum (fun _ _ => Nat.zero_le _) ho).trans (Nat.le_succ _)
+
+end DegreeBound
+
+section SmallParameter
+attribute [local instance] Classical.propDecidable
+
+def TrailLength (G : BondGraph V E) (x : V) (n : ℕ) : Set (Set E) :=
+  {ω | ∃ y l, Along G x l y ∧ l.Nodup ∧ l.length = n ∧ ∀ e ∈ l, e ∈ ω}
+
+theorem trailLength_upper (G : BondGraph V E) (x : V) (n : ℕ) :
+    IsUpperSet (TrailLength G x n) := by
+  rintro ω η h ⟨y,l,hl,hn,hlen,ho⟩
+  exact ⟨y,l,hl,hn,hlen,fun e he => h (ho e he)⟩
+
+theorem finiteWitness_trailLength (G : BondGraph V E) (x : V) (n : ℕ) :
+    FiniteWitness (TrailLength G x n) := by
+  rintro ω ⟨y,l,hl,hn,hlen,ho⟩
+  exact ⟨l.toFinset, fun e he => ho e (List.mem_toFinset.mp he),
+    y,l,hl,hn,hlen, fun e he => List.mem_toFinset.mpr he⟩
+
+theorem finiteWitness_edge (e : E) : FiniteWitness {ω : Set E | e ∈ ω} := by
+  intro ω he
+  exact ⟨{e}, by simpa using he, by simp⟩
+
+theorem trailLength_succ_subset (G : BondGraph V E) (x : V) (n : ℕ) :
+    TrailLength G x (n+1) ⊆ ⋃ d : Step G x,
+      DisjointOccurrence {ω | d.val.1 ∈ ω} (TrailLength G d.val.2 n) := by
+  rintro ω ⟨y,l,hl,hn,hlen,ho⟩
+  cases hl with
+  | nil => simp at hlen
+  | @cons x z y l e he tail =>
+    refine Set.mem_iUnion.mpr ⟨⟨(e,z),he⟩, {e}, l.toFinset, ?_, ?_, ?_, ?_, ?_⟩
+    · simpa using ho e (by simp)
+    · intro e' he'; exact ho e' (by simpa using Or.inr (List.mem_toFinset.mp he'))
+    · simpa using (List.nodup_cons.mp hn).1
+    · simp
+    · exact ⟨y,l,tail,(List.nodup_cons.mp hn).2, by simpa using hlen,
+        fun e' he' => List.mem_toFinset.mpr he'⟩
+
+theorem trailLength_probability_le [Countable E] (G : BondGraph V E)
+    (hloc : LocallyFinite G) (D : ℕ) (hD : ∀ x, degree G x ≤ D)
+    (p : unitInterval) (x : V) (n : ℕ) :
+    (law p).real (TrailLength G x n) ≤ ((D : ℝ) * p) ^ n := by
+  induction n generalizing x with
+  | zero => simpa only [pow_zero] using (measureReal_le_one (μ := law p) (s := TrailLength G x 0))
+  | succ n ih =>
+    let : Fintype (Step G x) := (finite_steps G hloc x).fintype
+    let F := fun d : Step G x =>
+      DisjointOccurrence {ω : Set E | d.val.1 ∈ ω} (TrailLength G d.val.2 n)
+    calc
+      (law p).real (TrailLength G x (n+1)) ≤ (law p).real (⋃ d, F d) :=
+        measureReal_mono (trailLength_succ_subset G x n)
+      _ ≤ ∑ d, (law p).real (F d) := measureReal_iUnion_fintype_le _
+      _ ≤ ∑ _d : Step G x, (p : ℝ) * ((D : ℝ) * p) ^ n := by
+        apply Finset.sum_le_sum
+        intro d _
+        have hb := bk p {ω : Set E | d.val.1 ∈ ω} (TrailLength G d.val.2 n)
+          (fun _ _ hω he => hω he) (trailLength_upper G d.val.2 n)
+          (finiteWitness_edge _) (finiteWitness_trailLength G d.val.2 n)
+        have heq : (law p).real {ω : Set E | d.val.1 ∈ ω} = (p : ℝ) := by
+          simpa using finite_open_probability p {d.val.1}
+        rw [heq] at hb
+        exact hb.trans (mul_le_mul_of_nonneg_left (ih d.val.2) p.2.1)
+      _ ≤ (D : ℝ) * ((p : ℝ) * ((D : ℝ) * p) ^ n) := by
+        simp only [Finset.sum_const, Finset.card_univ, nsmul_eq_mul]
+        apply mul_le_mul_of_nonneg_right _ (mul_nonneg p.2.1 (pow_nonneg (mul_nonneg (Nat.cast_nonneg D) p.2.1) n))
+        exact_mod_cast (show Fintype.card (Step G x) ≤ D by
+          simpa only [degree, Nat.card_eq_fintype_card] using hD x)
+      _ = ((D : ℝ) * p) ^ (n+1) := by ring
+
+ theorem Along.mem_reachBall {G : BondGraph V E} {x y : V} {l : List E}
+    (h : Along G x l y) : y ∈ reachBall G x l.length := by
+  induction h with
+  | nil => exact Set.mem_singleton _
+  | @cons x z y l e he ht ih =>
+    have hadj : Adj G x z := ⟨e,he⟩
+    have hstep : ∀ n, reachBall G z n ⊆ reachBall G x (n+1) := by
+      intro n
+      induction n with
+      | zero =>
+        rintro w rfl
+        exact Or.inr (Set.mem_iUnion.mpr ⟨x, Set.mem_iUnion.mpr ⟨Set.mem_singleton _, hadj⟩⟩)
+      | succ n ih =>
+        rintro w (hw | hw)
+        · exact Or.inl (ih hw)
+        · right
+          simp only [Set.mem_iUnion] at hw ⊢
+          obtain ⟨v,hv,hvw⟩ := hw
+          exact ⟨v,ih hv,hvw⟩
+    exact hstep _ ih
+
+ theorem Along.take {G : BondGraph V E} {x y : V} {l : List E}
+    (h : Along G x l y) (n : ℕ) : ∃ z, Along G x (l.take n) z := by
+  induction h generalizing n with
+  | @nil z => exact ⟨z, by simpa using (Along.nil (G := G) z)⟩
+  | @cons x z y l e he ht ih =>
+    cases n with
+    | zero => exact ⟨x,.nil x⟩
+    | succ n =>
+      obtain ⟨v,hv⟩ := ih n
+      exact ⟨v,.cons e he hv⟩
+
+ theorem reachBall_mono (G : BondGraph V E) (x : V) : Monotone (reachBall G x) := by
+  apply monotone_nat_of_le_succ
+  intro n
+  exact Set.subset_union_left
+
+ theorem infinite_cluster_subset_trailLength (G : BondGraph V E) (hloc : LocallyFinite G)
+    (x : V) (n : ℕ) : {ω | (cluster G ω x).Infinite} ⊆ TrailLength G x n := by
+  intro ω hω
+  obtain ⟨y,hy,hyball⟩ := hω.exists_notMem_finite (finite_reachBall hloc x n)
+  obtain ⟨l,hl,hn,ho⟩ := conn_exists_bond_trail G hy
+  have hlen : n ≤ l.length := by
+    by_contra h
+    exact hyball ((reachBall_mono G x (le_of_lt (Nat.lt_of_not_ge h))) hl.mem_reachBall)
+  obtain ⟨z,hz⟩ := hl.take n
+  exact ⟨z,l.take n,hz,hn.take,List.length_take_of_le hlen,
+    fun e he => ho e (List.mem_of_mem_take he)⟩
+
+ theorem infinite_cluster_probability_eq_zero_of_small [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (D : ℕ) (hD : ∀ x, degree G x ≤ D)
+    (p : unitInterval) (hp : (D : ℝ) * p < 1) (x : V) :
+    law p {ω | (cluster G ω x).Infinite} = 0 := by
+  apply (measureReal_eq_zero_iff).mp
+  apply le_antisymm _ measureReal_nonneg
+  apply ge_of_tendsto (tendsto_pow_atTop_nhds_zero_of_lt_one (mul_nonneg (Nat.cast_nonneg D) p.2.1) hp)
+  filter_upwards [] with n
+  exact (measureReal_mono (infinite_cluster_subset_trailLength G hloc x n)).trans
+    (trailLength_probability_le G hloc D hD p x n)
+
+ theorem exists_infinite_cluster_probability_eq_zero_of_small [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (D : ℕ) (hD : ∀ x, degree G x ≤ D)
+    (p : unitInterval) (hp : (D : ℝ) * p < 1) :
+    law p {ω | (infiniteClusters G ω).Nonempty} = 0 := by
+  have heq : {ω | (infiniteClusters G ω).Nonempty} =
+      ⋃ x : V, {ω | (cluster G ω x).Infinite} := by
+    ext ω
+    simp only [Set.mem_ofPred_eq, infiniteClusters_nonempty_iff, Set.mem_iUnion]
+  rw [heq, measure_iUnion_null_iff]
+  exact infinite_cluster_probability_eq_zero_of_small G hloc D hD p hp
+
+ theorem pc_pos [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G) :
+    0 < pc G := by
+  obtain ⟨D, hDpos, hD⟩ := exists_degree_bound G hqt
+  let t : unitInterval := ⟨1 / ((D : ℝ) + 1), by
+    constructor
+    · positivity
+    · apply (div_le_one (by positivity)).mpr
+      linarith [Nat.cast_nonneg (α := ℝ) D]⟩
+  have ht : (0 : unitInterval) < t := by
+    change (0 : ℝ) < 1 / ((D : ℝ) + 1)
+    positivity
+  apply ht.trans_le
+  apply le_sInf
+  intro p hp
+  by_contra h
+  have hp' : (p : ℝ) < 1 / ((D : ℝ) + 1) := lt_of_not_ge h
+  have hsmall : (D : ℝ) * p < 1 := by
+    have hmul := (lt_div_iff₀ (show (0 : ℝ) < D + 1 by positivity)).mp hp'
+    nlinarith [p.2.1]
+  have hz := exists_infinite_cluster_probability_eq_zero_of_small G hloc D hD p hsmall
+  exact (ne_of_gt hp) hz
+
+end SmallParameter
+
+section MainBridge
+attribute [local instance] Classical.propDecidable
+
+ 
+
+theorem mainConclusion_of_strict_gap [Infinite V] (G : BondGraph V E)
+    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (hgap : pc G < ptwo G) : MainConclusion G := by
+  let : Countable V := countable_vertices hloc hconn
+  let : Countable E := countable_edges hloc
+  obtain ⟨q, hq, hpq⟩ := lt_sSup_iff.mp hgap
+  obtain ⟨p₂, hp₂, hp₂q⟩ := exists_between hpq
+  obtain ⟨p₁, hp₁, hp₁₂⟩ := exists_between hp₂
+  have hc : OperatorBounded G (pc G) := operatorBounded_of_le G hpq.le hq
+  refine ⟨(operatorNorm_eq_sup_below G (pc G) (pc_pos G hloc hqt)).symm,
+    (operatorNorm_lt_top_iff G (pc G)).mpr hc, hgap, ptwo_le_pu G hconn hqt, ?_⟩
+  refine ⟨p₁, p₂, hp₁, hp₁₂, hp₂q.trans_le (show q ≤ 1 from q.2.2), ?_⟩
+  exact coupled_interval G hconn hloc hqt p₁ p₂ hp₁
+    (operatorBounded_of_le G hp₂q.le hq)
+
+end MainBridge
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Graph/Topology.lean b/lean/OAI/Probability/UnimodularPercolation/Graph/Topology.lean
index 0ad89d50b4f4dad7dd5934b5ae1ce8fd63aa4dae..09d91fcafddd5a20f8a94f496cf5662ebc6c3e8e 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Graph/Topology.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Graph/Topology.lean
@@ -1,140 +1,140 @@
-import Mathlib
-
-namespace OAI
-
-/-! Pointwise topology on permutation groups. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-namespace PointwisePerm
-open Topology Set
-universe u
-variable {X : Type u} [TopologicalSpace X] [DiscreteTopology X]
-
- 
-def coordinates (g : Equiv.Perm X) : (X → X) × (X → X) := (g, g.symm)
-
-instance topology : TopologicalSpace (Equiv.Perm X) :=
-  TopologicalSpace.induced coordinates inferInstance
-
-omit [DiscreteTopology X] in
-theorem isEmbedding_coordinates : Topology.IsEmbedding (coordinates (X := X)) := by
-  refine ⟨⟨rfl⟩, ?_⟩
-  intro g h he
-  exact Equiv.ext (congrFun (congrArg Prod.fst he))
-
-omit [DiscreteTopology X] in
-theorem continuous_apply (x : X) : Continuous (fun g : Equiv.Perm X => g x) :=
-  (_root_.continuous_apply x).comp (continuous_fst.comp isEmbedding_coordinates.continuous)
-
-omit [DiscreteTopology X] in
-theorem continuous_symm_apply (x : X) : Continuous (fun g : Equiv.Perm X => g.symm x) :=
-  (_root_.continuous_apply x).comp (continuous_snd.comp isEmbedding_coordinates.continuous)
-
-theorem continuous_eval : Continuous (fun p : (X → X) × X => p.1 p.2) :=
-  continuous_prod_of_discrete_right.mpr fun x => _root_.continuous_apply x
-
-instance topologicalGroup : IsTopologicalGroup (Equiv.Perm X) where
-  continuous_mul := by
-    apply continuous_induced_rng.mpr
-    apply Continuous.prodMk
-    · apply continuous_pi
-      intro x
-      exact (continuous_eval (X := X)).comp
-        ((continuous_fst.comp (isEmbedding_coordinates.continuous.comp continuous_fst)).prodMk
-          ((continuous_apply x).comp continuous_snd))
-    · apply continuous_pi
-      intro x
-      exact (continuous_eval (X := X)).comp
-        ((continuous_snd.comp (isEmbedding_coordinates.continuous.comp continuous_snd)).prodMk
-          ((continuous_symm_apply x).comp continuous_fst))
-  continuous_inv := by
-    apply continuous_induced_rng.mpr
-    exact (continuous_snd.comp isEmbedding_coordinates.continuous).prodMk
-      (continuous_fst.comp isEmbedding_coordinates.continuous)
-
-instance hausdorff : T2Space (Equiv.Perm X) :=
-  isEmbedding_coordinates.t2Space
-
-instance secondCountable [Countable X] : SecondCountableTopology (Equiv.Perm X) :=
-  isEmbedding_coordinates.secondCountableTopology
-
- 
-
-theorem topology_eq_pointwise : topology (X := X) =
-    TopologicalSpace.induced (fun g : Equiv.Perm X => (g : X → X)) inferInstance := by
-  apply le_antisymm
-  · exact continuous_iff_le_induced.mp
-      (continuous_fst.comp isEmbedding_coordinates.continuous)
-  · let t : TopologicalSpace (Equiv.Perm X) :=
-      TopologicalSpace.induced (fun g : Equiv.Perm X => (g : X → X)) inferInstance
-    have hc : @Continuous (Equiv.Perm X) ((X → X) × (X → X)) t _ coordinates := by
-      let := t
-      have hf : Continuous (fun g : Equiv.Perm X => (g : X → X)) := continuous_induced_dom
-      have hi : Continuous (fun g : Equiv.Perm X => (g.symm : X → X)) := by
-        apply continuous_pi
-        intro x
-        apply continuous_discrete_rng.mpr
-        intro y
-        have he : (fun g : Equiv.Perm X => g.symm x) ⁻¹' {y} =
-            (fun g : Equiv.Perm X => g y) ⁻¹' {x} := by
-          ext g
-          simp only [Set.mem_preimage,Set.mem_singleton_iff]
-          exact (Equiv.symm_apply_eq g).trans eq_comm
-        rw [he]
-        exact (isOpen_discrete _).preimage ((_root_.continuous_apply y).comp hf)
-      exact hf.prodMk hi
-    exact continuous_iff_le_induced.mp hc
-
- 
-omit [TopologicalSpace X] [DiscreteTopology X] in
-theorem range_coordinates : Set.range (coordinates (X := X)) =
-    {p | (∀ x, p.1 (p.2 x) = x) ∧ (∀ x, p.2 (p.1 x) = x)} := by
-  ext p
-  constructor
-  · rintro ⟨g,rfl⟩
-    exact ⟨g.apply_symm_apply, g.symm_apply_apply⟩
-  · rintro ⟨hfg,hgf⟩
-    exact ⟨⟨p.1,p.2,hgf,hfg⟩,rfl⟩
-
-theorem isClosed_range_coordinates : IsClosed (Set.range (coordinates (X := X))) := by
-  rw [range_coordinates]
-  have hf (x : X) : IsClosed {p : (X → X) × (X → X) | p.1 (p.2 x) = x} :=
-    isClosed_eq
-      ((continuous_eval (X := X)).comp (continuous_fst.prodMk ((_root_.continuous_apply x).comp continuous_snd)))
-      continuous_const
-  have hg (x : X) : IsClosed {p : (X → X) × (X → X) | p.2 (p.1 x) = x} :=
-    isClosed_eq
-      ((continuous_eval (X := X)).comp (continuous_snd.prodMk ((_root_.continuous_apply x).comp continuous_fst)))
-      continuous_const
-  convert (isClosed_iInter hf).inter (isClosed_iInter hg) using 1
-  ext p
-  simp only [Set.mem_inter_iff, Set.mem_iInter, Set.mem_ofPred_eq]
-
- 
-
-theorem isCompact_of_finite_images (S : Set (Equiv.Perm X)) (hS : IsClosed S)
-    (hf : ∀ x, ((fun g : Equiv.Perm X => g x) '' S).Finite)
-    (hg : ∀ x, ((fun g : Equiv.Perm X => g.symm x) '' S).Finite) :
-    IsCompact S := by
-  have he : Topology.IsClosedEmbedding (coordinates (X := X)) :=
-    ⟨isEmbedding_coordinates, isClosed_range_coordinates⟩
-  apply he.isEmbedding.isCompact_iff.mpr
-  apply ((isCompact_pi_infinite fun x => (hf x).isCompact).prod
-    (isCompact_pi_infinite fun x => (hg x).isCompact)).of_isClosed_subset
-      (he.isClosedMap S hS)
-  rintro _ ⟨g,hgS,rfl⟩
-  exact ⟨fun x => ⟨g,hgS,rfl⟩, fun x => ⟨g,hgS,rfl⟩⟩
-
-theorem isOpen_fixed (x y : X) : IsOpen {g : Equiv.Perm X | g x = y} :=
-  (isOpen_discrete ({y} : Set X)).preimage (continuous_apply x)
-
-theorem isClosed_fixed (x y : X) : IsClosed {g : Equiv.Perm X | g x = y} :=
-  isClosed_eq (continuous_apply x) continuous_const
-
-end PointwisePerm
-end Percolation
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-! Pointwise topology on permutation groups. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+namespace PointwisePerm
+open Topology Set
+universe u
+variable {X : Type u} [TopologicalSpace X] [DiscreteTopology X]
+
+ 
+def coordinates (g : Equiv.Perm X) : (X → X) × (X → X) := (g, g.symm)
+
+instance topology : TopologicalSpace (Equiv.Perm X) :=
+  TopologicalSpace.induced coordinates inferInstance
+
+omit [DiscreteTopology X] in
+theorem isEmbedding_coordinates : Topology.IsEmbedding (coordinates (X := X)) := by
+  refine ⟨⟨rfl⟩, ?_⟩
+  intro g h he
+  exact Equiv.ext (congrFun (congrArg Prod.fst he))
+
+omit [DiscreteTopology X] in
+theorem continuous_apply (x : X) : Continuous (fun g : Equiv.Perm X => g x) :=
+  (_root_.continuous_apply x).comp (continuous_fst.comp isEmbedding_coordinates.continuous)
+
+omit [DiscreteTopology X] in
+theorem continuous_symm_apply (x : X) : Continuous (fun g : Equiv.Perm X => g.symm x) :=
+  (_root_.continuous_apply x).comp (continuous_snd.comp isEmbedding_coordinates.continuous)
+
+theorem continuous_eval : Continuous (fun p : (X → X) × X => p.1 p.2) :=
+  continuous_prod_of_discrete_right.mpr fun x => _root_.continuous_apply x
+
+instance topologicalGroup : IsTopologicalGroup (Equiv.Perm X) where
+  continuous_mul := by
+    apply continuous_induced_rng.mpr
+    apply Continuous.prodMk
+    · apply continuous_pi
+      intro x
+      exact (continuous_eval (X := X)).comp
+        ((continuous_fst.comp (isEmbedding_coordinates.continuous.comp continuous_fst)).prodMk
+          ((continuous_apply x).comp continuous_snd))
+    · apply continuous_pi
+      intro x
+      exact (continuous_eval (X := X)).comp
+        ((continuous_snd.comp (isEmbedding_coordinates.continuous.comp continuous_snd)).prodMk
+          ((continuous_symm_apply x).comp continuous_fst))
+  continuous_inv := by
+    apply continuous_induced_rng.mpr
+    exact (continuous_snd.comp isEmbedding_coordinates.continuous).prodMk
+      (continuous_fst.comp isEmbedding_coordinates.continuous)
+
+instance hausdorff : T2Space (Equiv.Perm X) :=
+  isEmbedding_coordinates.t2Space
+
+instance secondCountable [Countable X] : SecondCountableTopology (Equiv.Perm X) :=
+  isEmbedding_coordinates.secondCountableTopology
+
+ 
+
+theorem topology_eq_pointwise : topology (X := X) =
+    TopologicalSpace.induced (fun g : Equiv.Perm X => (g : X → X)) inferInstance := by
+  apply le_antisymm
+  · exact continuous_iff_le_induced.mp
+      (continuous_fst.comp isEmbedding_coordinates.continuous)
+  · let t : TopologicalSpace (Equiv.Perm X) :=
+      TopologicalSpace.induced (fun g : Equiv.Perm X => (g : X → X)) inferInstance
+    have hc : @Continuous (Equiv.Perm X) ((X → X) × (X → X)) t _ coordinates := by
+      let := t
+      have hf : Continuous (fun g : Equiv.Perm X => (g : X → X)) := continuous_induced_dom
+      have hi : Continuous (fun g : Equiv.Perm X => (g.symm : X → X)) := by
+        apply continuous_pi
+        intro x
+        apply continuous_discrete_rng.mpr
+        intro y
+        have he : (fun g : Equiv.Perm X => g.symm x) ⁻¹' {y} =
+            (fun g : Equiv.Perm X => g y) ⁻¹' {x} := by
+          ext g
+          simp only [Set.mem_preimage,Set.mem_singleton_iff]
+          exact (Equiv.symm_apply_eq g).trans eq_comm
+        rw [he]
+        exact (isOpen_discrete _).preimage ((_root_.continuous_apply y).comp hf)
+      exact hf.prodMk hi
+    exact continuous_iff_le_induced.mp hc
+
+ 
+omit [TopologicalSpace X] [DiscreteTopology X] in
+theorem range_coordinates : Set.range (coordinates (X := X)) =
+    {p | (∀ x, p.1 (p.2 x) = x) ∧ (∀ x, p.2 (p.1 x) = x)} := by
+  ext p
+  constructor
+  · rintro ⟨g,rfl⟩
+    exact ⟨g.apply_symm_apply, g.symm_apply_apply⟩
+  · rintro ⟨hfg,hgf⟩
+    exact ⟨⟨p.1,p.2,hgf,hfg⟩,rfl⟩
+
+theorem isClosed_range_coordinates : IsClosed (Set.range (coordinates (X := X))) := by
+  rw [range_coordinates]
+  have hf (x : X) : IsClosed {p : (X → X) × (X → X) | p.1 (p.2 x) = x} :=
+    isClosed_eq
+      ((continuous_eval (X := X)).comp (continuous_fst.prodMk ((_root_.continuous_apply x).comp continuous_snd)))
+      continuous_const
+  have hg (x : X) : IsClosed {p : (X → X) × (X → X) | p.2 (p.1 x) = x} :=
+    isClosed_eq
+      ((continuous_eval (X := X)).comp (continuous_snd.prodMk ((_root_.continuous_apply x).comp continuous_fst)))
+      continuous_const
+  convert (isClosed_iInter hf).inter (isClosed_iInter hg) using 1
+  ext p
+  simp only [Set.mem_inter_iff, Set.mem_iInter, Set.mem_ofPred_eq]
+
+ 
+
+theorem isCompact_of_finite_images (S : Set (Equiv.Perm X)) (hS : IsClosed S)
+    (hf : ∀ x, ((fun g : Equiv.Perm X => g x) '' S).Finite)
+    (hg : ∀ x, ((fun g : Equiv.Perm X => g.symm x) '' S).Finite) :
+    IsCompact S := by
+  have he : Topology.IsClosedEmbedding (coordinates (X := X)) :=
+    ⟨isEmbedding_coordinates, isClosed_range_coordinates⟩
+  apply he.isEmbedding.isCompact_iff.mpr
+  apply ((isCompact_pi_infinite fun x => (hf x).isCompact).prod
+    (isCompact_pi_infinite fun x => (hg x).isCompact)).of_isClosed_subset
+      (he.isClosedMap S hS)
+  rintro _ ⟨g,hgS,rfl⟩
+  exact ⟨fun x => ⟨g,hgS,rfl⟩, fun x => ⟨g,hgS,rfl⟩⟩
+
+theorem isOpen_fixed (x y : X) : IsOpen {g : Equiv.Perm X | g x = y} :=
+  (isOpen_discrete ({y} : Set X)).preimage (continuous_apply x)
+
+theorem isClosed_fixed (x y : X) : IsClosed {g : Equiv.Perm X | g x = y} :=
+  isClosed_eq (continuous_apply x) continuous_const
+
+end PointwisePerm
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Kernel/Amplification.lean b/lean/OAI/Probability/UnimodularPercolation/Kernel/Amplification.lean
index 6b5e8d89ac7c17e6cf2308050917e099a6ec75ad..1e7f20f32c87085c3d9cd3c99c7a4bdd2f543c90 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Kernel/Amplification.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Kernel/Amplification.lean
@@ -1,1240 +1,1240 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Transport.EndpointKernels
-
-namespace OAI
-
-/-! Amplification for bond percolation. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section AmplificationMarginal
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def restrictFiniteField (Q : Finset E) (ω : Set E) : FiniteField Q :=
-  ⟨domainBits Q ω,Finset.mem_powerset.mpr (domainBits_subset Q ω)⟩
-
-@[simp] theorem finiteFieldSet_restrict (Q : Finset E) (ω : Set E) :
-    finiteFieldSet Q (restrictFiniteField Q ω) = ω ∩ (Q : Set E) := by
-  ext e
-  simp [restrictFiniteField,domainBits,finiteFieldSet,and_comm]
-
-@[simp] theorem restrictFiniteField_fieldSet (Q : Finset E) (ω : FiniteField Q) :
-    restrictFiniteField Q (finiteFieldSet Q ω) = ω := by
-  apply Subtype.ext
-  exact domainBits_finiteFieldSet Q ω
-
-theorem cluster_restrict_of_cap (G : BondGraph V E) (hloc : LocallyFinite G)
-    (Q : Finset E) (ω : Set E) (x : V) (C : ℕ)
-    (hQ : incidentSupport G hloc (finite_reachBall hloc x C).toFinset ⊆ Q)
-    (hf : (cluster G (ω ∩ (Q : Set E)) x).Finite)
-    (hC : (cluster G (ω ∩ (Q : Set E)) x).ncard ≤ C) :
-    cluster G (ω ∩ (Q : Set E)) x = cluster G ω x := by
-  apply Set.Subset.antisymm
-  · exact fun _ h => conn_mono G Set.inter_subset_left h
-  · intro y hy
-    induction hy with
-    | refl => exact conn_refl G _ x
-    | @tail a b _ hab ih =>
-      obtain ⟨e,he,hab⟩ := hab
-      have ha := finite_cluster_subset_reachBall G (ω ∩ (Q : Set E)) x hf C hC ih
-      have heQ := hQ ((mem_incidentSupport G hloc _ e).mpr
-        ⟨a,by simpa only [Set.Finite.mem_toFinset] using ha,by simp [hab]⟩)
-      exact ih.tail ⟨e,⟨he,heQ⟩,hab⟩
-
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
-
- 
-
-noncomputable def amplificationMarginal (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (z : ↥B) (p : unitInterval) (ω : Set E) : ℝ :=
-  finiteRootLoad Q (Finset.Icc 1 k) p
-    (finiteChoiceForwardLoad G x k Q (q.amplificationInputs Q B x k n z)
-      (fun r _ => amplificationVertices B k z r) (q.amplificationWeight B k z)) (restrictFiniteField Q ω)
-
-theorem amplificationMarginal_local (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (z : ↥B) (p : unitInterval) (ω : Set E) :
-    q.amplificationMarginal Q B x k n z p ω =
-      q.amplificationMarginal Q B x k n z p (ω ∩ (Q : Set E)) := by
-  unfold amplificationMarginal
-  congr 1
-  apply (finiteFieldSet_injective Q)
-  simp only [finiteFieldSet_restrict,Set.inter_assoc,Set.inter_self]
-
-theorem amplificationMarginal_bounds (q : LocalEndpointRule G hloc O M L) (hL : 1 ≤ L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (z : ↥B) (p : unitInterval) (ω : Set E) :
-    0 ≤ q.amplificationMarginal Q B x k n z p ω ∧
-      q.amplificationMarginal Q B x k n z p ω ≤ inverseCodeMass M L ^ k := by
-  have hb := finiteChoiceForwardLoad_bounds G hloc x z.val k M L hL Q
-    (q.amplificationInputs Q B x k n z) (fun r _ => amplificationVertices B k z r)
-    (q.amplificationWeight B k z) (q.amplificationWeight_nonneg B k z)
-    (q.amplificationWeight_row B k z) (q.amplificationInputs_subset Q B x k n z)
-    (fun _ _ _ => congrArg Subtype.val (oldRunVertices_last ..))
-    (fun r s hs i hi hik => (q.amplificationInputs_chain Q B x k n z r s hs i hi hik).2)
-  exact ⟨finiteRootLoad_nonneg _ _ p _ (fun s _ => (hb s).1) _,
-    finiteRootLoad_le _ _ p _ _ (fun s _ => (hb s).2) _⟩
-
-theorem amplificationMarginal_support (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (z : ↥B) (p : unitInterval) (ω : Set E)
-    (h : q.amplificationMarginal Q B x k n z p ω ≠ 0) :
-    x ∈ O ∧ z.val ∈ O ∧ Conn G (ω ∩ (Q : Set E)) x z.val ∧
-      (cluster G (ω ∩ (Q : Set E)) x).Finite ∧
-      (cluster G (ω ∩ (Q : Set E)) x).ncard ≤ 2*n+k*M := by
-  let A := q.amplificationInputs Q B x k n z
-  let y := fun r (_ : SpliceState E) => amplificationVertices B k z r
-  let I := FiniteChoiceTraceIndex G x k Q A y
-  let out := choiceTraceOutput G x k Q A y
-  have ho (a : I) : out a ∈ finiteSpliceDomain Q (Finset.Icc 1 k) :=
-    traceIndexOutput_mem G x k Q (A a.1) (y a.1) (q.amplificationInputs_subset Q B x k n z a.1) a.2
-  have he := finiteRootLoad_eq_sum Q (Finset.Icc 1 k) p out
-    (choiceTraceMass G x k Q A y (q.amplificationWeight B k z)) ho (restrictFiniteField Q ω)
-  have ha : ∃ a : I, (out a).1 = ω ∩ (Q : Set E) := by
-    by_contra hn
-    push Not at hn
-    apply h
-    change finiteRootLoad Q (Finset.Icc 1 k) p (finiteInverseLoad out _) _ = 0
-    rw [he,finiteFieldSet_restrict]
-    simp only [hn,ite_false,Finset.sum_const_zero]
-  obtain ⟨a,ha⟩ := ha
-  have ht := (traceIndex_valid G x k Q (A a.1) (y a.1) a.2).1
-  have hs := q.amplificationTrace_support Q B x k n z a.1 a.2.1.val a.2.1.property
-    (choiceTraceCode G x k Q A y a) (out a).1 ht
-  rwa [ha] at hs
-
-theorem amplificationMarginal_physical_support (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (z : ↥B) (p : unitInterval) (ω : Set E)
-    (hQ : incidentSupport G hloc (finite_reachBall hloc x (2*n+k*M)).toFinset ⊆ Q)
-    (h : q.amplificationMarginal Q B x k n z p ω ≠ 0) :
-    x ∈ O ∧ z.val ∈ O ∧ Conn G ω x z.val ∧ (cluster G ω x).Finite ∧
-      (cluster G ω x).ncard ≤ 2*n+k*M := by
-  obtain ⟨hx,hz,hc,hf,hcap⟩ := q.amplificationMarginal_support Q B x k n z p ω h
-  have he := cluster_restrict_of_cap G hloc Q ω x (2*n+k*M) hQ hf hcap
-  exact ⟨hx,hz,conn_mono G Set.inter_subset_left hc,he ▸ hf,he ▸ hcap⟩
-
- 
-
-theorem amplificationMarginal_mean [Nonempty E] (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (z : ↥B) (p : unitInterval) :
-    (∫ ω, q.amplificationMarginal Q B x k n z p ω ∂law p) =
-      ∑ r, ∑ s ∈ q.amplificationInputs Q B x k n z r,
-        spliceTupleWeight Q (Finset.Icc 1 k) p s * q.amplificationWeight B k z s r := by
-  rw [integral_finite_local Q p _ (q.amplificationMarginal_local Q B x k n z p)]
-  simp only [amplificationMarginal,restrictFiniteField_fieldSet]
-  rw [finiteRootLoad_mean]
-  apply finiteChoiceForwardLoad_mean G hloc x k M Q p
-  · exact q.amplificationInputs_subset Q B x k n z
-  · intro r s hs
-    exact ((q.mem_amplificationInputs Q B x k n z r s).mp hs).2.2.2.2.2.1
-  · intro r s hs i hi
-    exact (q.amplificationInputs_chain Q B x k n z r s hs i
-      (Finset.mem_Icc.mp hi).1 (Finset.mem_Icc.mp hi).2).1
-  · intro r s hs i hi
-    have hc := q.amplificationInputs_chain Q B x k n z r s hs i
-      (Finset.mem_Icc.mp hi).1 (Finset.mem_Icc.mp hi).2
-    exact ⟨hc.2.1,hc.2.2.1⟩
-
-end LocalEndpointRule
-end AmplificationMarginal
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section AmplificationMeanKernel
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem finiteSpliceDomain_integral_factor (Q : Finset E) (J : Finset ℕ) (p : ℝ)
-    (f : Set E → ℝ) (g : (ℕ → Set E) → ℝ) :
-    (∑ s ∈ finiteSpliceDomain Q J, spliceTupleWeight Q J p s * f s.1 * g s.2) =
-      finiteFieldIntegral Q p f * finiteFamilyIntegral Q J p g := by
-  have hh := finiteRootLoad_mean Q J p (fun s => f s.1 * g s.2)
-  simp only [← mul_assoc] at hh
-  rw [← hh]
-  simp only [finiteRootLoad,finiteFieldIntegral,finiteFamilyIntegral,Finset.sum_mul]
-  apply Finset.sum_congr rfl
-  intro ω _
-  simp only [Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro η _
-  ring
-
- 
-noncomputable def finiteBinConnection (G : BondGraph V E) (Q : Finset E) (p : ℝ)
-    (x : V) (n : ℕ) (y : V) : ℝ := finiteFieldIntegral Q p (fun ω =>
-      if n ≤ (cluster G ω x).ncard ∧ (cluster G ω x).ncard < 2*n ∧ y ∈ cluster G ω x then 1 else 0)
-
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
-
-theorem finiteOldKernel_eq_kernel (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (p : unitInterval) (a b : V)
-    (hQ : incidentSupport G hloc (finite_reachBall hloc a q.radius).toFinset ⊆ Q) :
-    finiteOldKernel Q p q.mass a b = q.kernel p a b := by
-  symm
-  exact integral_finite_local Q p _ (finite_local_of_subset _ Q hQ _ (q.locality · a b))
-
- 
-
-theorem amplificationMarginal_mean_kernel [Nonempty E] (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (z : ↥B) (p : unitInterval)
-    (hx : x ∈ O) (hz : z.val ∈ O)
-    (hQ : ∀ a ∈ B, incidentSupport G hloc (finite_reachBall hloc a q.radius).toFinset ⊆ Q) :
-    (∫ ω, q.amplificationMarginal Q B x k n z p ω ∂law p) =
-      ∑ r : Fin k → ↥B, finiteBinConnection G Q p x n (amplificationVertices B k z r 0) *
-        oldRunWeight (fun _ (a b : ↥B) => q.kernel p a.val b.val) k z r := by
-  rw [q.amplificationMarginal_mean Q B x k n z p]
-  apply Finset.sum_congr rfl
-  intro r _
-  let f : Set E → ℝ := fun ω => if n ≤ (cluster G ω x).ncard ∧
-      (cluster G ω x).ncard < 2*n ∧ amplificationVertices B k z r 0 ∈ cluster G ω x then 1 else 0
-  let g : (ℕ → Set E) → ℝ := fun η =>
-    oldRunWeight (fun i (a b : ↥B) => q.mass (η i) a.val b.val) k z r
-  have he : (∑ s ∈ q.amplificationInputs Q B x k n z r,
-      spliceTupleWeight Q (Finset.Icc 1 k) p s * q.amplificationWeight B k z s r) =
-      ∑ s ∈ finiteSpliceDomain Q (Finset.Icc 1 k), spliceTupleWeight Q (Finset.Icc 1 k) p s * f s.1 * g s.2 := by
-    rw [amplificationInputs,Finset.sum_filter]
-    apply Finset.sum_congr rfl
-    intro s _
-    change (if _ then _ else _) = _
-    by_cases hw : q.amplificationWeight B k z s r = 0
-    · have hg : g s.2 = 0 := hw
-      simp [hx,hz,hw,hg]
-    · simp only [hx,hz,true_and,hw,ne_eq,not_false_eq_true,and_true]
-      by_cases hbin : n ≤ (cluster G s.1 x).ncard ∧ (cluster G s.1 x).ncard < 2*n ∧
-          amplificationVertices B k z r 0 ∈ cluster G s.1 x
-      · simp [f,hbin,g,amplificationWeight]
-      · simp only [hbin,ite_false,f,mul_zero,zero_mul]
-  rw [he,finiteSpliceDomain_integral_factor]
-  dsimp only [g]
-  rw [oldRunWeight_integral Q p k (fun ω (a b : ↥B) => q.mass ω a.val b.val) z r]
-  change finiteBinConnection G Q p x n _ * _ = _
-  congr 1
-  apply congrArg (fun K : ℕ → ↥B → ↥B → ℝ => oldRunWeight K k z r)
-  funext i a b
-  exact q.finiteOldKernel_eq_kernel Q p a.val b.val (hQ a.val a.property)
-
-end LocalEndpointRule
-end AmplificationMeanKernel
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section BinConnection
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-noncomputable def binConnectionMass (G : BondGraph V E) (n : ℕ) (ω : Set E) (x y : V) : ℝ :=
-  if (cluster G ω x).Finite ∧ n ≤ (cluster G ω x).ncard ∧
-    (cluster G ω x).ncard < 2*n ∧ Conn G ω x y then 1 else 0
-
-theorem binConnectionMass_nonneg (G : BondGraph V E) (n : ℕ) (ω : Set E) (x y : V) :
-    0 ≤ binConnectionMass G n ω x y := by unfold binConnectionMass; split_ifs <;> norm_num
-
-theorem binConnectionMass_symm (G : BondGraph V E) (n : ℕ) (ω : Set E) (x y : V) :
-    binConnectionMass G n ω x y = binConnectionMass G n ω y x := by
-  by_cases hc : Conn G ω x y
-  · have he := cluster_eq_of_conn G ω hc
-    simp only [binConnectionMass,hc,conn_symm G ω hc,and_true,he]
-  · have hc' : ¬Conn G ω y x := fun h => hc (conn_symm G ω h)
-    simp only [binConnectionMass,hc,hc',and_false,ite_false]
-
-theorem binConnectionMass_local (G : BondGraph V E) (hloc : LocallyFinite G)
-    (n : ℕ) (ω : Set E) (x y : V) :
-    let Q := incidentSupport G hloc (finite_reachBall hloc x (2*n)).toFinset
-    binConnectionMass G n ω x y = binConnectionMass G n (ω ∩ (Q : Set E)) x y := by
-  dsimp only
-  let Q := incidentSupport G hloc (finite_reachBall hloc x (2*n)).toFinset
-  have hp : ((cluster G ω x).Finite ∧ n ≤ (cluster G ω x).ncard ∧
-      (cluster G ω x).ncard < 2*n ∧ Conn G ω x y) ↔
-      ((cluster G (ω ∩ (Q : Set E)) x).Finite ∧ n ≤ (cluster G (ω ∩ (Q : Set E)) x).ncard ∧
-      (cluster G (ω ∩ (Q : Set E)) x).ncard < 2*n ∧ Conn G (ω ∩ (Q : Set E)) x y) := by
-    constructor
-    · rintro ⟨hf,hn,hN,hc⟩
-      have hs : cluster G ω x ⊆ (finite_reachBall hloc x (2*n)).toFinset := by
-        simpa only [Set.Finite.coe_toFinset] using finite_cluster_subset_reachBall G ω x hf (2*n) hN.le
-      have he := cluster_restrict_incident_of_full G hloc _ ω x hs
-      change cluster G (ω ∩ (Q : Set E)) x = cluster G ω x at he
-      refine ⟨he.symm ▸ hf,he.symm ▸ hn,he.symm ▸ hN,?_⟩
-      change y ∈ cluster G (ω ∩ (Q : Set E)) x
-      rw [he]
-      exact hc
-    · rintro ⟨hf,hn,hN,hc⟩
-      have he := cluster_restrict_of_cap G hloc Q ω x (2*n) (Finset.Subset.refl _) hf hN.le
-      exact ⟨he ▸ hf,he ▸ hn,he ▸ hN,conn_mono G Set.inter_subset_left hc⟩
-  exact if_congr hp rfl rfl
-
-theorem binConnectionMass_support (G : BondGraph V E) (n : ℕ) (ω : Set E) (x y : V)
-    (h : binConnectionMass G n ω x y ≠ 0) :
-    (cluster G ω x).Finite ∧ (cluster G ω x).ncard < 2*n ∧ Conn G ω x y := by
-  unfold binConnectionMass at h
-  split_ifs at h with hp
-  · exact ⟨hp.1,hp.2.2.1,hp.2.2.2⟩
-  · exact (h rfl).elim
-
-theorem binConnectionMass_outside (G : BondGraph V E) (n : ℕ) (ω : Set E) (x y : V)
-    (hy : y ∉ reachBall G x (2*n)) : binConnectionMass G n ω x y = 0 := by
-  by_contra h
-  obtain ⟨hf,hN,hc⟩ := binConnectionMass_support G n ω x y h
-  exact hy (finite_cluster_subset_reachBall G ω x hf (2*n) hN.le hc)
-
-theorem binConnectionMass_row (G : BondGraph V E) (n : ℕ) (ω : Set E) (x : V) :
-    (∑' y, binConnectionMass G n ω x y) ≤ 2*n := by
-  by_cases hb : (cluster G ω x).Finite ∧ n ≤ (cluster G ω x).ncard ∧ (cluster G ω x).ncard < 2*n
-  · have hs : (∑' y, binConnectionMass G n ω x y) = (cluster G ω x).ncard := by
-      rw [tsum_eq_sum (s := hb.1.toFinset) (fun y hy => by
-        have hc : ¬Conn G ω x y := fun h => hy (hb.1.mem_toFinset.mpr h)
-        simp only [binConnectionMass,hc,and_false,ite_false])]
-      have he : (∑ y ∈ hb.1.toFinset, binConnectionMass G n ω x y) =
-          ∑ _y ∈ hb.1.toFinset, (1 : ℝ) := by
-        apply Finset.sum_congr rfl
-        intro y hy
-        exact ite_eq_left ⟨hb.1,hb.2.1,hb.2.2,hb.1.mem_toFinset.mp hy⟩
-      rw [he]
-      simp only [Finset.sum_const,nsmul_eq_mul,mul_one]
-      exact_mod_cast (Set.ncard_eq_toFinset_card _ hb.1).symm
-    rw [hs]
-    exact_mod_cast hb.2.2.le
-  · have hz (y : V) : binConnectionMass G n ω x y = 0 := by
-      apply ite_eq_right
-      exact fun h => hb ⟨h.1,h.2.1,h.2.2.1⟩
-    simp only [hz,tsum_zero]
-    positivity
-
-noncomputable def binConnection (G : BondGraph V E) (p : unitInterval) (n : ℕ) (x y : V) : ℝ :=
-  ∫ ω, binConnectionMass G n ω x y ∂law p
-
-theorem binConnection_nonneg (G : BondGraph V E) (p : unitInterval) (n : ℕ) (x y : V) :
-    0 ≤ binConnection G p n x y := integral_nonneg fun ω => binConnectionMass_nonneg G n ω x y
-
-theorem binConnection_symm (G : BondGraph V E) (p : unitInterval) (n : ℕ) (x y : V) :
-    binConnection G p n x y = binConnection G p n y x :=
-  integral_congr_ae (ae_of_all _ fun ω => binConnectionMass_symm G n ω x y)
-
-theorem binConnection_outside (G : BondGraph V E) (p : unitInterval) (n : ℕ) (x y : V)
-    (hy : y ∉ reachBall G x (2*n)) : binConnection G p n x y = 0 := by
-  simp only [binConnection,fun ω => binConnectionMass_outside G n ω x y hy,integral_zero]
-
-theorem binConnection_row (G : BondGraph V E) (hloc : LocallyFinite G)
-    (p : unitInterval) (n : ℕ) (x : V) : (∑' y, binConnection G p n x y) ≤ 2*n := by
-  let B := (finite_reachBall hloc x (2*n)).toFinset
-  let Q := incidentSupport G hloc B
-  have hi (y : V) : Integrable (fun ω => binConnectionMass G n ω x y) (law p) :=
-    integrable_finite_local Q p _ (fun ω => binConnectionMass_local G hloc n ω x y)
-  have he (ω : Set E) : (∑' y, binConnectionMass G n ω x y) = ∑ y ∈ B, binConnectionMass G n ω x y :=
-    tsum_eq_sum fun y hy => binConnectionMass_outside G n ω x y (by simpa only [B,Set.Finite.mem_toFinset] using hy)
-  have hs : (∑' y, binConnection G p n x y) = ∑ y ∈ B, binConnection G p n x y :=
-    tsum_eq_sum fun y hy => binConnection_outside G p n x y (by simpa only [B,Set.Finite.mem_toFinset] using hy)
-  rw [hs]
-  simp only [binConnection]
-  rw [← integral_finsetSum B (fun y _ => hi y)]
-  calc
-    _ ≤ ∫ _ω, (2*n : ℝ) ∂law p := integral_mono (integrable_finsetSum B (fun y _ => hi y))
-      (integrable_const _) (fun ω => (he ω).symm ▸ binConnectionMass_row G n ω x)
-    _ = 2*n := by simp
-
- 
-
-theorem exists_binConnection_operator (G : BondGraph V E) (hloc : LocallyFinite G)
-    (p : unitInterval) (n : ℕ) :
-    ∃ A : Hilbert V →L[ℂ] Hilbert V, ‖A‖ ≤ 2*n ∧
-      ∀ x y, A (lp.single 2 y (1:ℂ)) x = (binConnection G p n x y : ℂ) := by
-  have hs (x : V) : Summable (binConnection G p n x) := by
-    apply summable_of_ne_finset_zero (s := (finite_reachBall hloc x (2*n)).toFinset)
-    intro y hy
-    exact binConnection_outside G p n x y (by simpa only [Set.Finite.mem_toFinset] using hy)
-  have hr (x : V) (S : Finset V) : (∑ y ∈ S, binConnection G p n x y) ≤ 2*n :=
-    ((hs x).sum_le_tsum S (fun y _ => binConnection_nonneg G p n x y)).trans (binConnection_row G hloc p n x)
-  apply exists_operator_of_row_column_bound (binConnection G p n) (2*n) (by positivity) (binConnection_nonneg G p n) hr
-  intro y S
-  simpa only [binConnection_symm G p n] using hr y S
-
-end BinConnection
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section FiniteBinConnection
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem finiteBinConnection_eq_binConnection (G : BondGraph V E) (hloc : LocallyFinite G)
-    (Q : Finset E) (p : unitInterval) (x : V) (n : ℕ) (y : V)
-    (hQ : incidentSupport G hloc (finite_reachBall hloc x (2*n)).toFinset ⊆ Q) :
-    finiteBinConnection G Q p x n y = binConnection G p n x y := by
-  rw [binConnection, integral_finite_local Q p _
-    (finite_local_of_subset _ Q hQ _ (fun ω => binConnectionMass_local G hloc n ω x y))]
-  unfold finiteBinConnection finiteFieldIntegral
-  apply Finset.sum_congr rfl
-  intro ω _
-  congr 1
-  have hf : (cluster G (finiteFieldSet Q ω) x).Finite :=
-    cluster_finite_of_finite_edges G ω.val.finite_toSet x
-  unfold cluster at hf
-  simp only [binConnectionMass,cluster,Set.mem_ofPred_eq,hf,true_and]
-  exact if_congr Iff.rfl rfl rfl
-
-end FiniteBinConnection
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section PositivePathOperator
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-def PositiveRealMatrix (A : Hilbert V →L[ℂ] Hilbert V) : Prop :=
-  ∀ x y, (matrixCoeff A x y).im = 0 ∧ 0 ≤ (matrixCoeff A x y).re
-
-namespace PositiveRealMatrix
-variable {A H : Hilbert V →L[ℂ] Hilbert V}
-
-theorem real (hA : PositiveRealMatrix A) (x y : V) :
-    matrixCoeff A x y = ((matrixCoeff A x y).re : ℂ) := by
-  apply Complex.ext
-  · rfl
-  · exact (hA x y).1
-
-theorem mul_hasSum (hA : PositiveRealMatrix A) (H : Hilbert V →L[ℂ] Hilbert V) (x y : V) :
-    HasSum (fun z => (matrixCoeff A x z).re * (matrixCoeff H z y).re)
-      (matrixCoeff (A*H) x y).re := by
-  have h := Complex.reCLM.hasSum (operator_kernel_hasSum A (H (lp.single 2 y (1:ℂ))) x)
-  change HasSum (fun z => (matrixCoeff A x z * matrixCoeff H z y).re)
-    (matrixCoeff (A*H) x y).re at h
-  simpa only [Complex.mul_re,(hA _ _).1,zero_mul,sub_zero] using h
-
-theorem mul (hA : PositiveRealMatrix A) (hH : PositiveRealMatrix H) :
-    PositiveRealMatrix (A*H) := by
-  intro x y
-  constructor
-  · have hs := (operator_kernel_hasSum A (H (lp.single 2 y (1:ℂ))) x).summable
-    change Summable (fun z => matrixCoeff A x z * matrixCoeff H z y) at hs
-    rw [matrixCoeff_mul,Complex.im_tsum hs]
-    simp only [Complex.mul_im,(hA _ _).1,(hH _ _).1,mul_zero,zero_mul,add_zero,tsum_zero]
-  · rw [← (hA.mul_hasSum H x y).tsum_eq]
-    exact tsum_nonneg fun z => mul_nonneg (hA x z).2 (hH z y).2
-
-theorem one : PositiveRealMatrix (1 : Hilbert V →L[ℂ] Hilbert V) := by
-  intro x y
-  simp only [matrixCoeff,one_apply_eq_self,lp.single_apply,Pi.single_apply]
-  split_ifs <;> norm_num
-
-theorem pow (hA : PositiveRealMatrix A) (k : ℕ) : PositiveRealMatrix (A^k) := by
-  induction k with
-  | zero => simpa only [pow_zero] using (one (V := V))
-  | succ k ih => rw [pow_succ]; exact ih.mul hA
-
-theorem star (hA : PositiveRealMatrix A) : PositiveRealMatrix (star A) := by
-  intro x y
-  simpa only [matrixCoeff_adjoint,Complex.star_def,Complex.conj_im,Complex.conj_re,neg_eq_zero] using hA y x
-
-end PositiveRealMatrix
-
- 
-
-theorem oldRun_test_succ {W : Type*} [Fintype W]
-    (K : ℕ → W → W → ℝ) (F : W → ℝ) (k : ℕ) (z : W) :
-    (∑ r : Fin (k+1) → W, oldRunWeight K (k+1) z r * F (oldRunVertices (k+1) z r 0)) =
-      ∑ a : W, K (k+1) z a *
-        ∑ r : Fin k → W, oldRunWeight K k a r * F (oldRunVertices k a r 0) := by
-  rw [Fintype.sum_equiv (oldRunConsEquiv k) _
-    (fun r => K (k+1) z r.1 * oldRunWeight K k r.1 r.2 * F (oldRunVertices k r.1 r.2 0))
-      (by intro r; simp only [oldRunWeight,oldRunConsEquiv,Equiv.coe_fn_mk,
-        oldRunVertices_succ_of_le k z r 0 (Nat.zero_le k)]), Fintype.sum_prod_type]
-  simp only [Finset.mul_sum,mul_assoc]
-
- 
-
-theorem finite_oldRun_operator_domination (A H : Hilbert V →L[ℂ] Hilbert V)
-    (hA : PositiveRealMatrix A) (hH : PositiveRealMatrix H)
-    (B : Finset V) (k : ℕ) (z : ↥B) (x : V) :
-    (∑ r : Fin k → ↥B,
-      oldRunWeight (fun _ (a b : ↥B) => (matrixCoeff A a.val b.val).re) k z r *
-        (matrixCoeff H (oldRunVertices k z r 0).val x).re) ≤
-      (matrixCoeff (A^k*H) z.val x).re := by
-  induction k generalizing z with
-  | zero => simp [oldRunWeight,oldRunVertices]
-  | succ k ih =>
-    rw [oldRun_test_succ (fun _ (a b : ↥B) => (matrixCoeff A a.val b.val).re)
-      (fun a : ↥B => (matrixCoeff H a.val x).re) k z]
-    calc
-      _ ≤ ∑ a : ↥B, (matrixCoeff A z.val a.val).re * (matrixCoeff (A^k*H) a.val x).re :=
-        Finset.sum_le_sum fun a _ => mul_le_mul_of_nonneg_left (ih a) (hA z.val a.val).2
-      _ ≤ ∑' a, (matrixCoeff A z.val a).re * (matrixCoeff (A^k*H) a x).re := by
-        rw [Finset.sum_coe_sort B (fun a => (matrixCoeff A z.val a).re * (matrixCoeff (A^k*H) a x).re)]
-        exact (hA.mul_hasSum (A^k*H) z.val x).summable.sum_le_tsum B
-          (fun a _ => mul_nonneg (hA z.val a).2 (((hA.pow k).mul hH) a x).2)
-      _ = (matrixCoeff (A^(k+1)*H) z.val x).re := by
-        rw [(hA.mul_hasSum (A^k*H) z.val x).tsum_eq,pow_succ',mul_assoc]
-
-end PositivePathOperator
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section OrbitBinOperator
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem isAut_trans (G : BondGraph V E) {a c : V ≃ V} {b d : E ≃ E}
-    (hab : IsAut G a b) (hcd : IsAut G c d) : IsAut G (a.trans c) (b.trans d) := by
-  intro e
-  change G (d (b e)) = (G e).map (c ∘ a)
-  rw [hcd,hab,Sym2.map_map]
-
-theorem mem_vertexOrbit_aut_iff (G : BondGraph V E) (o : V) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (x : V) : a x ∈ vertexOrbit G o ↔ x ∈ vertexOrbit G o := by
-  constructor
-  · rintro ⟨c,d,hcd,hc⟩
-    refine ⟨c.trans a.symm,d.trans b.symm,isAut_trans G hcd (isAut_symm G hab),?_⟩
-    simp only [Equiv.trans_apply,hc,Equiv.symm_apply_apply]
-  · rintro ⟨c,d,hcd,rfl⟩
-    exact ⟨c.trans a,d.trans b,isAut_trans G hcd hab,rfl⟩
-
-theorem binConnectionMass_covariant (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (n : ℕ) (ω : Set E) (x y : V) :
-    binConnectionMass G n (b ⁻¹' ω) x y = binConnectionMass G n ω (a x) (a y) := by
-  have hf : (cluster G (b ⁻¹' ω) x).Finite ↔ (cluster G ω (a x)).Finite := by
-    rw [← cluster_aut G hab ω x]
-    exact (Set.finite_image_iff a.injective.injOn).symm
-  have hn : (cluster G (b ⁻¹' ω) x).ncard = (cluster G ω (a x)).ncard := by
-    rw [← cluster_aut G hab ω x,Set.ncard_image_of_injective _ a.injective]
-  simp only [binConnectionMass,hf,hn,conn_aut_iff G hab]
-
-theorem binConnection_covariant (G : BondGraph V E) (hloc : LocallyFinite G)
-    (p : unitInterval) {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (n : ℕ) (x y : V) :
-    binConnection G p n (a x) (a y) = binConnection G p n x y := by
-  have hm : Measurable (fun ω => binConnectionMass G n ω x y) :=
-    measurable_finite_local _ _ (fun ω => binConnectionMass_local G hloc n ω x y)
-  have h := integral_map (μ := law p) (measurable_set_preimage b).aemeasurable hm.aestronglyMeasurable
-  rw [law_map_preimage_equiv p b] at h
-  simpa only [binConnection,binConnectionMass_covariant G hab] using h.symm
-
- 
-noncomputable def orbitBinConnection (G : BondGraph V E) (p : unitInterval) (o : V)
-    (n : ℕ) (x y : V) : ℝ :=
-  if x ∈ vertexOrbit G o ∧ y ∈ vertexOrbit G o then binConnection G p n x y else 0
-
-theorem orbitBinConnection_nonneg (G : BondGraph V E) (p : unitInterval) (o : V)
-    (n : ℕ) (x y : V) : 0 ≤ orbitBinConnection G p o n x y := by
-  unfold orbitBinConnection
-  split_ifs
-  · exact binConnection_nonneg G p n x y
-  · exact le_rfl
-
-theorem orbitBinConnection_symm (G : BondGraph V E) (p : unitInterval) (o : V)
-    (n : ℕ) (x y : V) : orbitBinConnection G p o n x y = orbitBinConnection G p o n y x := by
-  simp only [orbitBinConnection,and_comm,binConnection_symm G p n]
-
-theorem orbitBinConnection_covariant (G : BondGraph V E) (hloc : LocallyFinite G)
-    (p : unitInterval) (o : V) {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (n : ℕ) (x y : V) :
-    orbitBinConnection G p o n (a x) (a y) = orbitBinConnection G p o n x y := by
-  simp only [orbitBinConnection,mem_vertexOrbit_aut_iff G o hab,binConnection_covariant G hloc p hab n]
-
-theorem orbitBinConnection_summable (G : BondGraph V E) (hloc : LocallyFinite G)
-    (p : unitInterval) (o : V) (n : ℕ) (x : V) : Summable (orbitBinConnection G p o n x) := by
-  apply summable_of_ne_finset_zero (s := (finite_reachBall hloc x (2*n)).toFinset)
-  intro y hy
-  have hz := binConnection_outside G p n x y (by simpa only [Set.Finite.mem_toFinset] using hy)
-  simp only [orbitBinConnection,hz,ite_self]
-
-noncomputable def binOrbitRow (G : BondGraph V E) (p : unitInterval) (o : V) (n : ℕ) : ℝ :=
-  ∑' y, orbitBinConnection G p o n o y
-
-theorem orbitBinConnection_row (G : BondGraph V E) (hloc : LocallyFinite G)
-    (p : unitInterval) (o : V) (n : ℕ) (x : V) (hx : x ∈ vertexOrbit G o) :
-    (∑' y, orbitBinConnection G p o n x y) = binOrbitRow G p o n := by
-  obtain ⟨a,b,hab,rfl⟩ := hx
-  rw [← a.tsum_eq (fun y => orbitBinConnection G p o n (a o) y)]
-  exact tsum_congr (fun y => orbitBinConnection_covariant G hloc p o hab n o y)
-
-theorem orbitBinConnection_row_le (G : BondGraph V E) (hloc : LocallyFinite G)
-    (p : unitInterval) (o : V) (n : ℕ) (x : V) :
-    (∑' y, orbitBinConnection G p o n x y) ≤ binOrbitRow G p o n := by
-  by_cases hx : x ∈ vertexOrbit G o
-  · exact (orbitBinConnection_row G hloc p o n x hx).le
-  · simp only [orbitBinConnection,hx,false_and,ite_false,tsum_zero]
-    exact tsum_nonneg (orbitBinConnection_nonneg G p o n o)
-
- 
-
-theorem exists_orbitBinConnection_operator (G : BondGraph V E) (hloc : LocallyFinite G)
-    (p : unitInterval) (o : V) (n : ℕ) :
-    ∃ H : Hilbert V →L[ℂ] Hilbert V, ‖H‖ ≤ binOrbitRow G p o n ∧
-      ∀ x y, matrixCoeff H x y = (orbitBinConnection G p o n x y : ℂ) := by
-  have hr (x : V) (S : Finset V) : (∑ y ∈ S, orbitBinConnection G p o n x y) ≤ binOrbitRow G p o n :=
-    ((orbitBinConnection_summable G hloc p o n x).sum_le_tsum S
-      (fun y _ => orbitBinConnection_nonneg G p o n x y)).trans (orbitBinConnection_row_le G hloc p o n x)
-  exact exists_operator_of_row_column_bound (orbitBinConnection G p o n) (binOrbitRow G p o n)
-    (tsum_nonneg (orbitBinConnection_nonneg G p o n o)) (orbitBinConnection_nonneg G p o n) hr
-    (fun y S => by simpa only [orbitBinConnection_symm G p o n] using hr y S)
-
-end OrbitBinOperator
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-open MeasureTheory
-section AmplificationOperatorDomination
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem oldRun_endpoint_mem {W : Type*} (P : W → Prop) (K : ℕ → W → W → ℝ)
-    (hK : ∀ i a b, K i a b ≠ 0 → P b) (k : ℕ) (z : W) (hz : P z)
-    (r : Fin k → W) (hr : oldRunWeight K k z r ≠ 0) : P (oldRunVertices k z r 0) := by
-  induction k generalizing z with
-  | zero => exact hz
-  | succ k ih =>
-    have hm := mul_ne_zero_iff.mp hr
-    rw [oldRunVertices_succ_of_le _ _ _ _ (Nat.zero_le k)]
-    exact ih (r 0) (hK _ _ _ hm.1) (Fin.tail r) hm.2
-
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
-
- 
-
-theorem amplificationMarginal_operator_le [Nonempty E] (o : V)
-    (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (z : ↥B) (p : unitInterval)
-    (hx : x ∈ vertexOrbit G o) (hz : z.val ∈ vertexOrbit G o)
-    (hroot : incidentSupport G hloc (finite_reachBall hloc x (2*n)).toFinset ⊆ Q)
-    (hquery : ∀ a ∈ B, incidentSupport G hloc (finite_reachBall hloc a q.radius).toFinset ⊆ Q)
-    (A H : Hilbert V →L[ℂ] Hilbert V)
-    (hA : ∀ a b, matrixCoeff A a b = (q.kernel p a b : ℂ))
-    (hH : ∀ a b, matrixCoeff H a b = (orbitBinConnection G p o n a b : ℂ)) :
-    (∫ ω, q.amplificationMarginal Q B x k n z p ω ∂law p) ≤
-      (matrixCoeff (A^k*H) z.val x).re := by
-  have hposA : PositiveRealMatrix A := fun a b => by
-    rw [hA a b]
-    exact ⟨rfl,q.kernel_nonneg p a b⟩
-  have hposH : PositiveRealMatrix H := fun a b => by
-    rw [hH a b]
-    exact ⟨rfl,orbitBinConnection_nonneg G p o n a b⟩
-  rw [q.amplificationMarginal_mean_kernel Q B x k n z p hx hz hquery]
-  have he : (∑ r : Fin k → ↥B, finiteBinConnection G Q p x n (amplificationVertices B k z r 0) *
-      oldRunWeight (fun _ (a b : ↥B) => q.kernel p a.val b.val) k z r) =
-      ∑ r : Fin k → ↥B,
-        oldRunWeight (fun _ (a b : ↥B) => (matrixCoeff A a.val b.val).re) k z r *
-          (matrixCoeff H (oldRunVertices k z r 0).val x).re := by
-    apply Finset.sum_congr rfl
-    intro r _
-    simp only [hA,hH,Complex.ofReal_re,
-      finiteBinConnection_eq_binConnection G hloc Q p x n _ hroot]
-    by_cases hr : oldRunWeight (fun _ (a b : ↥B) => q.kernel p a.val b.val) k z r = 0
-    · simp only [hr,mul_zero,zero_mul]
-    · have hy : (oldRunVertices k z r 0).val ∈ vertexOrbit G o :=
-        oldRun_endpoint_mem (fun a : ↥B => a.val ∈ vertexOrbit G o)
-          (fun _ (a b : ↥B) => q.kernel p a.val b.val)
-          (fun _ a b hab => by
-            by_contra hb
-            exact hab (q.kernel_zero_of_end_not_mem p a.val b.val hb)) k z hz r hr
-      rw [orbitBinConnection,ite_eq_left ⟨hy,hx⟩,binConnection_symm G p n]
-      exact mul_comm _ _
-  rw [he]
-  exact finite_oldRun_operator_domination A H hposA hposH B k z x
-
-end LocalEndpointRule
-
- 
-
-theorem exists_operator_of_amplification_domination
-    (G : BondGraph V E) (hloc : LocallyFinite G) (p : unitInterval) (o : V) (n k : ℕ)
-    (A : Hilbert V →L[ℂ] Hilbert V) (hA : PositiveRealMatrix A)
-    (F : V → V → ℝ) (hF : ∀ x z, 0 ≤ F x z)
-    (hdom : ∀ (H : Hilbert V →L[ℂ] Hilbert V),
-      (∀ x y, matrixCoeff H x y = (orbitBinConnection G p o n x y : ℂ)) →
-      ∀ x z, F x z ≤ (matrixCoeff (A^k*H) z x).re) :
-    ∃ T : Hilbert V →L[ℂ] Hilbert V, ‖T‖ ≤ binOrbitRow G p o n * ‖A‖^k ∧
-      ∀ x z, matrixCoeff T x z = (F x z : ℂ) := by
-  obtain ⟨H,hHnorm,hH⟩ := exists_orbitBinConnection_operator G hloc p o n
-  have hposH : PositiveRealMatrix H := fun x y => by
-    rw [hH x y]
-    exact ⟨rfl,orbitBinConnection_nonneg G p o n x y⟩
-  have hpos := ((hA.pow k).mul hposH).star
-  have htr (x z : V) : (matrixCoeff (star (A^k*H)) x z).re = (matrixCoeff (A^k*H) z x).re := by
-    simp only [matrixCoeff_adjoint,Complex.star_def,Complex.conj_re]
-  obtain ⟨T,hTnorm,hT⟩ := exists_operator_of_matrix_domination
-    (fun x z => (matrixCoeff (star (A^k*H)) x z).re) F (fun x z => (hpos x z).2)
-    (fun x z => by rw [abs_of_nonneg (hF x z),htr]; exact hdom H hH x z)
-    (star (A^k*H)) (fun x z => hpos.real x z)
-  refine ⟨T,hTnorm.trans ?_,hT⟩
-  have hpow : ∀ j : ℕ, ‖A^j‖ ≤ ‖A‖^j := by
-    intro j
-    induction j with
-    | zero => simpa only [pow_zero, ContinuousLinearMap.one_def] using (ContinuousLinearMap.norm_id_le (𝕜 := ℂ) (E := Hilbert V))
-    | succ j ih =>
-      rw [pow_succ, pow_succ]
-      exact (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right ih (norm_nonneg _))
-  calc
-    ‖star (A^k*H)‖ = ‖A^k*H‖ := by
-      change ‖ContinuousLinearMap.adjoint (A^k*H)‖ = _
-      exact LinearIsometryEquiv.norm_map _ _
-    _ ≤ ‖A^k‖ * ‖H‖ := norm_mul_le _ _
-    _ ≤ ‖A‖^k * binOrbitRow G p o n := mul_le_mul (hpow k) hHnorm
-      (norm_nonneg _) (pow_nonneg (norm_nonneg _) _)
-    _ = binOrbitRow G p o n * ‖A‖^k := mul_comm _ _
-
-end AmplificationOperatorDomination
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section ClusterVisitKernel
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
-
-noncomputable def conditionalSuccess (q : LocalEndpointRule G hloc O M L) (ω : Set E) (x : V) : ℝ :=
-  ∑' y, q.mass ω x y
-
-theorem conditionalSuccess_nonneg (q : LocalEndpointRule G hloc O M L) (ω : Set E) (x : V) :
-    0 ≤ q.conditionalSuccess ω x := tsum_nonneg (q.nonneg ω x)
-
-theorem conditionalSuccess_support (q : LocalEndpointRule G hloc O M L) (ω : Set E) (x : V)
-    (h : q.conditionalSuccess ω x ≠ 0) :
-    x ∈ O ∧ (cluster G ω x).Finite ∧ (cluster G ω x).ncard ≤ M := by
-  have he : ∃ y, q.mass ω x y ≠ 0 := by
-    by_contra hn
-    push Not at hn
-    exact h (by simp only [conditionalSuccess,hn,tsum_zero])
-  obtain ⟨y,hy⟩ := he
-  obtain ⟨hx,_,_,hf,hc,_⟩ := q.support ω x y hy
-  exact ⟨hx,hf,hc⟩
-
-theorem conditionalSuccess_local (q : LocalEndpointRule G hloc O M L) (ω : Set E) (x : V) :
-    q.conditionalSuccess ω x = q.conditionalSuccess
-      (ω ∩ (incidentSupport G hloc (finite_reachBall hloc x q.radius).toFinset : Set E)) x :=
-  tsum_congr (q.locality ω x)
-
-theorem conditionalSuccess_integrable (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x : V) :
-    Integrable (fun ω => q.conditionalSuccess ω x) (law p) :=
-  integrable_finite_local _ p _ (fun ω => q.conditionalSuccess_local ω x)
-
-theorem conditionalSuccess_mean (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x : V) :
-    (∫ ω, q.conditionalSuccess ω x ∂law p) = q.successProbability p x :=
-  (q.kernel_row_integral p x).symm
-
-theorem conditionalSuccess_covariant (q : LocalEndpointRule G hloc O M L)
-    (a : V ≃ V) (b : E ≃ E) (hab : IsAut G a b) (ω : Set E) (x : V) :
-    q.conditionalSuccess (b ⁻¹' ω) x = q.conditionalSuccess ω (a x) := by
-  unfold conditionalSuccess
-  simp_rw [q.covariance a b hab]
-  exact a.tsum_eq _
-
- 
-
-noncomputable def clusterVisitMass (q : LocalEndpointRule G hloc O M L) (ω : Set E) (x a : V) : ℝ :=
-  if a ∈ cluster G ω x then q.conditionalSuccess ω x else 0
-
-theorem clusterVisitMass_nonneg (q : LocalEndpointRule G hloc O M L) (ω : Set E) (x a : V) :
-    0 ≤ q.clusterVisitMass ω x a := by
-  unfold clusterVisitMass
-  split_ifs
-  · exact q.conditionalSuccess_nonneg ω x
-  · exact le_refl _
-
-theorem clusterVisitMass_outside (q : LocalEndpointRule G hloc O M L) (ω : Set E) (x a : V)
-    (ha : a ∉ reachBall G x M) : q.clusterVisitMass ω x a = 0 := by
-  by_cases h : q.conditionalSuccess ω x = 0
-  · simp only [clusterVisitMass,h,ite_self]
-  · obtain ⟨_,hf,hc⟩ := q.conditionalSuccess_support ω x h
-    exact ite_eq_right (fun hmem => ha (finite_cluster_subset_reachBall G ω x hf M hc hmem))
-
-theorem clusterVisitMass_local (q : LocalEndpointRule G hloc O M L) (ω : Set E) (x a : V) :
-    let Q := incidentSupport G hloc (finite_reachBall hloc x (max q.radius M)).toFinset
-    q.clusterVisitMass ω x a = q.clusterVisitMass (ω ∩ (Q : Set E)) x a := by
-  dsimp only
-  let Q := incidentSupport G hloc (finite_reachBall hloc x (max q.radius M)).toFinset
-  change q.clusterVisitMass ω x a = q.clusterVisitMass (ω ∩ (Q:Set E)) x a
-  have he := finite_local_of_subset _ Q
-    (incidentSupport_ball_mono G hloc x (le_max_left q.radius M))
-    (fun ω => q.conditionalSuccess ω x) (fun ω => q.conditionalSuccess_local ω x) ω
-  by_cases h : q.conditionalSuccess ω x = 0
-  · have h' : q.conditionalSuccess (ω ∩ (Q:Set E)) x = 0 := he.symm.trans h
-    simp only [clusterVisitMass,h,h',ite_self]
-  · obtain ⟨_,hf,hc⟩ := q.conditionalSuccess_support ω x h
-    have hs : cluster G ω x ⊆ (finite_reachBall hloc x (max q.radius M)).toFinset := by
-      simpa only [Set.Finite.coe_toFinset] using
-        finite_cluster_subset_reachBall G ω x hf _ (hc.trans (le_max_right _ _))
-    have hc' := cluster_restrict_incident_of_full G hloc _ ω x hs
-    change cluster G (ω ∩ (Q:Set E)) x = cluster G ω x at hc'
-    change (if a ∈ cluster G ω x then _ else _) = (if a ∈ cluster G (ω ∩ (Q:Set E)) x then _ else _)
-    rw [hc',he]
-
-theorem clusterVisitMass_integrable (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x a : V) :
-    Integrable (fun ω => q.clusterVisitMass ω x a) (law p) :=
-  integrable_finite_local _ p _ (fun ω => q.clusterVisitMass_local ω x a)
-
-theorem clusterVisitMass_covariant (q : LocalEndpointRule G hloc O M L)
-    (a : V ≃ V) (b : E ≃ E) (hab : IsAut G a b) (ω : Set E) (x y : V) :
-    q.clusterVisitMass (b ⁻¹' ω) x y = q.clusterVisitMass ω (a x) (a y) := by
-  have hc : y ∈ cluster G (b ⁻¹' ω) x ↔ a y ∈ cluster G ω (a x) := by
-    rw [← cluster_aut G hab ω x]
-    constructor
-    · intro hy; exact ⟨y,hy,rfl⟩
-    · rintro ⟨z,hz,he⟩
-      exact (a.injective he) ▸ hz
-  simp only [clusterVisitMass,hc,q.conditionalSuccess_covariant a b hab]
-
-theorem clusterVisitMass_row_le (q : LocalEndpointRule G hloc O M L) (ω : Set E) (x : V) :
-    (∑' a, q.clusterVisitMass ω x a) ≤ M * q.conditionalSuccess ω x := by
-  by_cases h : q.conditionalSuccess ω x = 0
-  · simp only [clusterVisitMass,h,ite_self,tsum_zero,mul_zero,le_refl]
-  obtain ⟨_,hf,hc⟩ := q.conditionalSuccess_support ω x h
-  rw [tsum_eq_sum (f := q.clusterVisitMass ω x) (s := hf.toFinset) (fun a ha => ite_eq_right (by simpa only [Set.Finite.mem_toFinset] using ha))]
-  have he : (∑ a ∈ hf.toFinset, q.clusterVisitMass ω x a) =
-      (hf.toFinset.card:ℝ) * q.conditionalSuccess ω x := by
-    calc
-      _ = ∑ _a ∈ hf.toFinset, q.conditionalSuccess ω x := Finset.sum_congr rfl (fun a ha => ite_eq_left (hf.mem_toFinset.mp ha))
-      _ = _ := by simp only [Finset.sum_const,nsmul_eq_mul]
-  rw [he]
-  apply mul_le_mul_of_nonneg_right _ (q.conditionalSuccess_nonneg ω x)
-  exact_mod_cast (by simpa only [← Set.ncard_eq_toFinset_card _ hf] using hc)
-
-noncomputable def clusterVisitKernel (q : LocalEndpointRule G hloc O M L)
-    (p : unitInterval) (x a : V) : ℝ := ∫ ω, q.clusterVisitMass ω x a ∂law p
-
-theorem clusterVisitKernel_nonneg (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x a : V) :
-    0 ≤ q.clusterVisitKernel p x a := integral_nonneg (fun ω => q.clusterVisitMass_nonneg ω x a)
-
-theorem clusterVisitKernel_outside (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x a : V)
-    (ha : a ∉ reachBall G x M) : q.clusterVisitKernel p x a = 0 := by
-  simp only [clusterVisitKernel,q.clusterVisitMass_outside _ x a ha,integral_zero]
-
-theorem clusterVisitKernel_summable (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x : V) :
-    Summable (q.clusterVisitKernel p x) := by
-  apply summable_of_ne_finset_zero (s := (finite_reachBall hloc x M).toFinset)
-  intro a ha
-  exact q.clusterVisitKernel_outside p x a (by simpa only [Set.Finite.mem_toFinset] using ha)
-
-theorem clusterVisitKernel_row_le (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x : V) :
-    (∑' a, q.clusterVisitKernel p x a) ≤ M * q.successProbability p x := by
-  let B := (finite_reachBall hloc x M).toFinset
-  have he (ω : Set E) : (∑' a, q.clusterVisitMass ω x a) = ∑ a ∈ B, q.clusterVisitMass ω x a :=
-    tsum_eq_sum (fun a ha => q.clusterVisitMass_outside ω x a (by simpa only [B,Set.Finite.mem_toFinset] using ha))
-  rw [tsum_eq_sum (s := B) (fun a ha => q.clusterVisitKernel_outside p x a (by simpa only [B,Set.Finite.mem_toFinset] using ha))]
-  simp only [clusterVisitKernel]
-  rw [← integral_finsetSum B (fun a _ => q.clusterVisitMass_integrable p x a)]
-  calc
-    _ ≤ ∫ ω, (M:ℝ) * q.conditionalSuccess ω x ∂law p :=
-      integral_mono (integrable_finsetSum B (fun a _ => q.clusterVisitMass_integrable p x a))
-        ((q.conditionalSuccess_integrable p x).const_mul _) (fun ω => by rw [← he]; exact q.clusterVisitMass_row_le ω x)
-    _ = _ := by rw [integral_const_mul,q.conditionalSuccess_mean]
-
-theorem clusterVisitKernel_covariant (q : LocalEndpointRule G hloc O M L) (p : unitInterval)
-    (a : V ≃ V) (b : E ≃ E) (hab : IsAut G a b) (x y : V) :
-    q.clusterVisitKernel p (a x) (a y) = q.clusterVisitKernel p x y := by
-  have hm : Measurable (fun ω => q.clusterVisitMass ω x y) :=
-    measurable_finite_local _ _ (fun ω => q.clusterVisitMass_local ω x y)
-  have h := integral_map (μ := law p) (measurable_set_preimage b).aemeasurable hm.aestronglyMeasurable
-  rw [law_map_preimage_equiv p b] at h
-  simpa only [clusterVisitKernel,q.clusterVisitMass_covariant a b hab] using h.symm
-
-end LocalEndpointRule
-end ClusterVisitKernel
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section VisitEnvelope
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
-
- 
-
-noncomputable def visitEnvelope (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
-    (p : unitInterval) : ℕ → V → V → ℝ
-  | 0, _, _ => 0
-  | k+1, x, a => q.successProbability p o ^ k * q.clusterVisitKernel p x a +
-      ∑' y, q.kernel p x y * q.visitEnvelope o p k y a
-
-theorem visitEnvelope_nonneg (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
-    (p : unitInterval) (k : ℕ) (x a : V) : 0 ≤ q.visitEnvelope o p k x a := by
-  induction k generalizing x with
-  | zero => exact le_refl _
-  | succ k ih =>
-    exact add_nonneg (mul_nonneg (pow_nonneg (q.successProbability_nonneg p o) _) (q.clusterVisitKernel_nonneg p x a))
-      (tsum_nonneg (fun y => mul_nonneg (q.kernel_nonneg p x y) (ih y)))
-
-theorem visitEnvelope_covariant (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
-    (p : unitInterval) (k : ℕ) (a : V ≃ V) (b : E ≃ E) (hab : IsAut G a b) (x y : V) :
-    q.visitEnvelope o p k (a x) (a y) = q.visitEnvelope o p k x y := by
-  induction k generalizing x y with
-  | zero => rfl
-  | succ k ih =>
-    simp only [visitEnvelope,q.clusterVisitKernel_covariant p a b hab]
-    congr 1
-    rw [← a.tsum_eq (fun z => q.kernel p (a x) z * q.visitEnvelope o p k z (a y))]
-    exact tsum_congr fun z => by rw [q.kernel_covariant p a b hab,ih]
-
- 
-
-theorem visitEnvelope_row_bound (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
-    (p : unitInterval) (k : ℕ) (x : V) :
-    Summable (q.visitEnvelope o p k x) ∧
-      (∑' a, q.visitEnvelope o p k x a) ≤ (k*M:ℕ) * q.successProbability p o ^ k := by
-  let u := q.successProbability p o
-  have hu : 0 ≤ u := q.successProbability_nonneg p o
-  induction k generalizing x with
-  | zero => simp only [visitEnvelope,zero_mul,Nat.cast_zero,pow_zero,mul_one,tsum_zero,le_refl,and_true]; exact summable_zero
-  | succ k ih =>
-    let B := (finite_reachBall hloc x M).toFinset
-    have he (a : V) : (∑' y, q.kernel p x y * q.visitEnvelope o p k y a) =
-        ∑ y ∈ B, q.kernel p x y * q.visitEnvelope o p k y a :=
-      tsum_eq_sum (fun y hy => by
-        rw [q.kernel_outside_ball_zero p x y (by simpa only [B,Set.Finite.mem_toFinset] using hy),zero_mul])
-    have ht (a : V) : q.visitEnvelope o p (k+1) x a =
-        u^k * q.clusterVisitKernel p x a + ∑ y ∈ B, q.kernel p x y * q.visitEnvelope o p k y a := by
-      exact congrArg (_ + ·) (he a)
-    have hsum : Summable (fun a => ∑ y ∈ B, q.kernel p x y * q.visitEnvelope o p k y a) :=
-      summable_sum (s := B) (fun y _ => (ih y).1.mul_left _)
-    have hs := ((q.clusterVisitKernel_summable p x).mul_left (u^k)).add hsum
-    have hs' : Summable (q.visitEnvelope o p (k+1) x) := by
-      change Summable (fun a => q.visitEnvelope o p (k+1) x a)
-      simpa only [ht] using hs
-    refine ⟨hs',?_⟩
-    simp_rw [ht]
-    rw [Summable.tsum_add ((q.clusterVisitKernel_summable p x).mul_left _) hsum,tsum_mul_left,
-      Summable.tsum_finsetSum (fun y _ => (ih y).1.mul_left _)]
-    simp_rw [tsum_mul_left]
-    have hr : (∑ y ∈ B, q.kernel p x y) ≤ u :=
-      ((q.summable_kernel p x).sum_le_tsum B (fun y _ => q.kernel_nonneg p x y)).trans
-        (q.kernel_row_le_success o p x)
-    calc
-      _ ≤ u^k * (M*u) + ∑ y ∈ B, q.kernel p x y * ((k*M:ℕ)*u^k) := by
-        apply add_le_add
-        · apply mul_le_mul_of_nonneg_left _ (pow_nonneg hu _)
-          exact (q.clusterVisitKernel_row_le p x).trans
-            (mul_le_mul_of_nonneg_left (q.kernel_row_le_success o p x) (Nat.cast_nonneg _))
-        · exact Finset.sum_le_sum fun y _ => mul_le_mul_of_nonneg_left (ih y).2 (q.kernel_nonneg p x y)
-      _ ≤ u^k * (M*u) + u*((k*M:ℕ)*u^k) := by
-        rw [← Finset.sum_mul]
-        exact add_le_add (le_refl _) (mul_le_mul_of_nonneg_right hr (mul_nonneg (Nat.cast_nonneg _) (pow_nonneg hu _)))
-      _ = ((k+1)*M:ℕ) * q.successProbability p o ^ (k+1) := by
-        change _ = ((k+1)*M:ℕ) * u^(k+1)
-        simp only [Nat.cast_mul,Nat.cast_add,Nat.cast_one,pow_succ]
-        ring
-
-end LocalEndpointRule
-end VisitEnvelope
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section MarkedOldRun
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {W : Type*}
-
-theorem oldRunWeight_congr (K K' : ℕ → W → W → ℝ) (k : ℕ)
-    (h : ∀ i, 1 ≤ i → i ≤ k → ∀ a b, K i a b = K' i a b) (z : W) (r : Fin k → W) :
-    oldRunWeight K k z r = oldRunWeight K' k z r := by
-  induction k generalizing z with
-  | zero => rfl
-  | succ k ih =>
-    simp only [oldRunWeight]
-    rw [h (k+1) (by omega) (by omega),ih (fun i hi hik => h i hi (by omega))]
-
-theorem oldRunWeight_prod (K : ℕ → W → W → ℝ) (k : ℕ) (z : W) (r : Fin k → W) :
-    oldRunWeight K k z r = ∏ i ∈ Finset.Icc 1 k,
-      K i (oldRunVertices k z r i) (oldRunVertices k z r (i-1)) := by
-  induction k generalizing z with
-  | zero => simp [oldRunWeight]
-  | succ k ih =>
-    rw [Finset.prod_Icc_succ_top (by omega)]
-    have he : (∏ i ∈ Finset.Icc 1 k, K i (oldRunVertices (k+1) z r i)
-        (oldRunVertices (k+1) z r (i-1))) = oldRunWeight K k (r 0) (Fin.tail r) := by
-      rw [ih]
-      apply Finset.prod_congr rfl
-      intro i hi
-      have hi' := Finset.mem_Icc.mp hi
-      rw [oldRunVertices_succ_of_le _ _ _ _ hi'.2,
-        oldRunVertices_succ_of_le _ _ _ _ (by omega)]
-    rw [he,oldRunVertices_last,Nat.add_sub_cancel,
-      oldRunVertices_succ_of_le _ _ _ _ (le_refl _),oldRunVertices_last]
-    exact mul_comm _ _
-
- 
-
-noncomputable def markedOldRun (K C : ℕ → W → W → ℝ) (k : ℕ) (z : W) (r : Fin k → W) : ℝ :=
-  ∑ i ∈ Finset.Icc 1 k, oldRunWeight (fun j a b => if j = i then C j a b else K j a b) k z r
-
-theorem markedOldRun_nonneg (K C : ℕ → W → W → ℝ)
-    (hK : ∀ i a b, 0 ≤ K i a b) (hC : ∀ i a b, 0 ≤ C i a b)
-    (k : ℕ) (z : W) (r : Fin k → W) : 0 ≤ markedOldRun K C k z r := by
-  apply Finset.sum_nonneg
-  intro i _
-  apply oldRunWeight_nonneg
-  intro j a b
-  split_ifs
-  · exact hC j a b
-  · exact hK j a b
-
-theorem markedOldRun_succ (K C : ℕ → W → W → ℝ) (k : ℕ) (z : W) (r : Fin (k+1) → W) :
-    markedOldRun K C (k+1) z r =
-      K (k+1) z (r 0) * markedOldRun K C k (r 0) (Fin.tail r) +
-      C (k+1) z (r 0) * oldRunWeight K k (r 0) (Fin.tail r) := by
-  unfold markedOldRun
-  rw [Finset.sum_Icc_succ_top (by omega),Finset.mul_sum]
-  congr 1
-  · apply Finset.sum_congr rfl
-    intro i hi
-    have hne : k+1 ≠ i := by have := Finset.mem_Icc.mp hi; omega
-    simp only [oldRunWeight,ite_eq_right hne]
-  · simp only [oldRunWeight,ite_true]
-    congr 1
-    apply oldRunWeight_congr
-    intro i hi hik a b
-    exact ite_eq_right (by omega)
-
-theorem sum_markedOldRun_succ [Fintype W] (K C : ℕ → W → W → ℝ) (k : ℕ) (z : W) :
-    (∑ r : Fin (k+1) → W, markedOldRun K C (k+1) z r) =
-      ∑ b : W, (K (k+1) z b * (∑ r : Fin k → W, markedOldRun K C k b r) +
-        C (k+1) z b * (∑ r : Fin k → W, oldRunWeight K k b r)) := by
-  rw [Fintype.sum_equiv (oldRunConsEquiv k) _
-    (fun r => K (k+1) z r.1 * markedOldRun K C k r.1 r.2 + C (k+1) z r.1 * oldRunWeight K k r.1 r.2)
-      (by intro r; exact markedOldRun_succ K C k z r),Fintype.sum_prod_type]
-  simp only [Finset.sum_add_distrib,Finset.mul_sum]
-
- 
-theorem oldRunWeight_integral_indexed (Q : Finset E) (p : ℝ)
-    (q : ℕ → Set E → W → W → ℝ) (k : ℕ) (z : W) (r : Fin k → W) :
-    finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η => oldRunWeight (fun i => q i (η i)) k z r) =
-      oldRunWeight (fun i a b => finiteFieldIntegral Q p (fun ω => q i ω a b)) k z r := by
-  calc
-    _ = finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η =>
-        ∏ i : ↥(Finset.Icc 1 k), q i (η i) (oldRunVertices k z r i)
-          (oldRunVertices k z r (i.val-1))) := by
-      congr 1
-      funext η
-      rw [oldRunWeight_prod]
-      exact (Finset.prod_coe_sort (Finset.Icc 1 k)
-        (fun i => q i (η i) (oldRunVertices k z r i) (oldRunVertices k z r (i-1)))).symm
-    _ = ∏ i : ↥(Finset.Icc 1 k), finiteFieldIntegral Q p (fun ω =>
-        q i ω (oldRunVertices k z r i) (oldRunVertices k z r (i.val-1))) :=
-      finiteFamilyIntegral_prod Q (Finset.Icc 1 k) p
-        (fun i ω => q i ω (oldRunVertices k z r i) (oldRunVertices k z r (i.val-1)))
-    _ = _ := by
-      rw [oldRunWeight_prod]
-      exact Finset.prod_coe_sort (Finset.Icc 1 k) (fun i => finiteFieldIntegral Q p
-        (fun ω => q i ω (oldRunVertices k z r i) (oldRunVertices k z r (i-1))))
-
-theorem markedOldRun_integral (Q : Finset E) (p : ℝ)
-    (q c : ℕ → Set E → W → W → ℝ) (k : ℕ) (z : W) (r : Fin k → W) :
-    finiteFamilyIntegral Q (Finset.Icc 1 k) p
-      (fun η => markedOldRun (fun i => q i (η i)) (fun i => c i (η i)) k z r) =
-      markedOldRun (fun i a b => finiteFieldIntegral Q p (fun ω => q i ω a b))
-        (fun i a b => finiteFieldIntegral Q p (fun ω => c i ω a b)) k z r := by
-  unfold markedOldRun
-  rw [finiteFamilyIntegral_sum]
-  apply Finset.sum_congr rfl
-  intro i _
-  change finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η =>
-      oldRunWeight (fun j => (fun ω a b => if j=i then c j ω a b else q j ω a b) (η j)) k z r) = _
-  rw [oldRunWeight_integral_indexed Q p (fun j ω a b => if j=i then c j ω a b else q j ω a b) k z r]
-  congr 1
-  funext j a b
-  by_cases h : j = i <;> simp only [h,ite_true,ite_false]
-
- 
-
-theorem trajectory_indicator_le_marked (G : BondGraph V E) (e : W → V)
-    (q : Set E → W → W → ℝ) (hq : ∀ ω x y, 0 ≤ q ω x y)
-    (k : ℕ) (η : ℕ → Set E) (z : W) (r : Fin k → W) (a : V) :
-    (if a ∈ trajectoryUnion G η (e ∘ oldRunVertices k z r) k then
-      oldRunWeight (fun i => q (η i)) k z r else 0) ≤
-      markedOldRun (fun i => q (η i))
-        (fun i x y => if a ∈ cluster G (η i) (e x) then q (η i) x y else 0) k z r := by
-  have hC : ∀ i x y, 0 ≤ (if a ∈ cluster G (η i) (e x) then q (η i) x y else 0) := by
-    intro i x y; split_ifs; exact hq _ _ _; exact le_refl _
-  by_cases ha : a ∈ trajectoryUnion G η (e ∘ oldRunVertices k z r) k
-  · rw [ite_eq_left ha]
-    obtain ⟨i,hi,hik,hai⟩ := (mem_trajectoryUnion G η (e ∘ oldRunVertices k z r) k a).mp ha
-    have he : oldRunWeight (fun j x y => if j=i then
-        (if a ∈ cluster G (η j) (e x) then q (η j) x y else 0) else q (η j) x y) k z r =
-        oldRunWeight (fun j => q (η j)) k z r := by
-      simp_rw [oldRunWeight_prod]
-      apply Finset.prod_congr rfl
-      intro j _
-      by_cases hji : j = i
-      · subst j
-        simp only [ite_true,Function.comp_apply] at hai ⊢
-        exact ite_eq_left hai
-      · exact ite_eq_right hji
-    rw [← he]
-    unfold markedOldRun
-    apply Finset.single_le_sum _ (Finset.mem_Icc.mpr ⟨hi,hik⟩)
-    intro j _
-    apply oldRunWeight_nonneg
-    intro l x y
-    split_ifs
-    · exact hq _ _ _
-    · exact le_refl _
-    · exact hq _ _ _
-  · rw [ite_eq_right ha]
-    exact markedOldRun_nonneg _ _ (fun _ => hq _) hC k z r
-
-end MarkedOldRun
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section TrajectoryEnvelopeDomination
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem finiteTrajectoryMass_le_marked {W : Type*} [Fintype W]
-    (G : BondGraph V E) (e : W → V) (Q : Finset E) (p : unitInterval) (k : ℕ)
-    (q : Set E → W → W → ℝ) (hq : ∀ ω x y, 0 ≤ q ω x y) (z : W) (a : V) :
-    finiteTrajectoryMass G e Q p k q z a ≤
-      ∑ r : Fin k → W, markedOldRun (fun _ => finiteOldKernel Q p q)
-        (fun _ x y => finiteFieldIntegral Q p (fun ω => if a ∈ cluster G ω (e x) then q ω x y else 0)) k z r := by
-  unfold finiteTrajectoryMass
-  calc
-    _ ≤ finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η => ∑ r : Fin k → W,
-        markedOldRun (fun i => q (η i))
-          (fun i x y => if a ∈ cluster G (η i) (e x) then q (η i) x y else 0) k z r) :=
-      finiteFamilyIntegral_mono Q (Finset.Icc 1 k) p _ _ (fun η => Finset.sum_le_sum (fun r _ =>
-        trajectory_indicator_le_marked G e q hq k η z r a))
-    _ = _ := by
-      rw [finiteFamilyIntegral_sum]
-      apply Finset.sum_congr rfl
-      intro r _
-      exact markedOldRun_integral Q p (fun _ => q)
-        (fun _ ω x y => if a ∈ cluster G ω (e x) then q ω x y else 0) k z r
-
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
-
- 
-
-theorem finite_marked_step_row_le (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
-    (Q : Finset E) (B : Finset V) (p : unitInterval) (x a : V)
-    (hQ : incidentSupport G hloc (finite_reachBall hloc x (max q.radius M)).toFinset ⊆ Q) :
-    (∑ y : ↥B, finiteFieldIntegral Q p (fun ω =>
-      if a ∈ cluster G ω x then q.mass ω x y.val else 0)) ≤ q.clusterVisitKernel p x a := by
-  rw [← finiteFieldIntegral_sum]
-  have he : q.clusterVisitKernel p x a = finiteFieldIntegral Q p (fun ω => q.clusterVisitMass ω x a) :=
-    integral_finite_local Q p _ (finite_local_of_subset _ Q hQ _ (q.clusterVisitMass_local · x a))
-  rw [he]
-  apply Finset.sum_le_sum
-  intro ω _
-  apply mul_le_mul_of_nonneg_left _ (finiteFieldWeight_nonneg Q p ω)
-  dsimp only
-  by_cases ha : a ∈ cluster G (finiteFieldSet Q ω) x
-  · simp only [ha,ite_true,clusterVisitMass]
-    rw [Finset.sum_coe_sort B (fun y => q.mass (finiteFieldSet Q ω) x y)]
-    exact (q.summable_mass _ x).sum_le_tsum B (fun y _ => q.nonneg _ x y)
-  · simp only [ha,ite_false,clusterVisitMass,Finset.sum_const_zero,le_refl]
-
- 
-
-theorem sum_markedOldRun_le_envelope (o : V)
-    (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
-    (Q : Finset E) (B : Finset V) (p : unitInterval) (k : ℕ) (z : ↥B) (a : V)
-    (hQ : ∀ x ∈ B, incidentSupport G hloc (finite_reachBall hloc x (max q.radius M)).toFinset ⊆ Q) :
-    (∑ r : Fin k → ↥B, markedOldRun
-      (fun _ => finiteOldKernel Q p (fun ω (x y : ↥B) => q.mass ω x.val y.val))
-      (fun _ (x y : ↥B) => finiteFieldIntegral Q p (fun ω =>
-        if a ∈ cluster G ω x.val then q.mass ω x.val y.val else 0)) k z r) ≤
-      q.visitEnvelope o p k z.val a := by
-  let u := q.successProbability p o
-  let K : ↥B → ↥B → ℝ := finiteOldKernel Q p (fun ω (x y : ↥B) => q.mass ω x.val y.val)
-  let C : ↥B → ↥B → ℝ := fun x y => finiteFieldIntegral Q p (fun ω =>
-    if a ∈ cluster G ω x.val then q.mass ω x.val y.val else 0)
-  have hK (x y : ↥B) : K x y = q.kernel p x.val y.val :=
-    q.finiteOldKernel_eq_kernel Q p x.val y.val
-      ((incidentSupport_ball_mono G hloc x.val (le_max_left _ _)).trans (hQ x.val x.property))
-  have hKn (x y : ↥B) : 0 ≤ K x y := by rw [hK]; exact q.kernel_nonneg p _ _
-  have hCn (x y : ↥B) : 0 ≤ C x y := by
-    apply finiteFieldIntegral_nonneg
-    intro ω; split_ifs; exact q.nonneg ω _ _; exact le_refl _
-  have hKr (x : ↥B) : (∑ y, K x y) ≤ u := by
-    simp only [hK]
-    rw [Finset.sum_coe_sort B (fun y => q.kernel p x.val y)]
-    exact ((q.summable_kernel p x.val).sum_le_tsum B (fun y _ => q.kernel_nonneg p x.val y)).trans
-      (q.kernel_row_le_success o p x.val)
-  have hCr (x : ↥B) : (∑ y, C x y) ≤ q.clusterVisitKernel p x.val a :=
-    q.finite_marked_step_row_le o Q B p x.val a (hQ x.val x.property)
-  change (∑ r : Fin k → ↥B, markedOldRun (fun _ => K) (fun _ => C) k z r) ≤ _
-  induction k generalizing z with
-  | zero => simp [markedOldRun,visitEnvelope]
-  | succ k ih =>
-    rw [sum_markedOldRun_succ]
-    have hs : Summable (fun y => q.kernel p z.val y * q.visitEnvelope o p k y a) := by
-      apply summable_of_ne_finset_zero (s := (finite_reachBall hloc z.val M).toFinset)
-      intro y hy
-      rw [q.kernel_outside_ball_zero p z.val y (by simpa only [Set.Finite.mem_toFinset] using hy),zero_mul]
-    calc
-      _ ≤ ∑ y : ↥B, (K z y * q.visitEnvelope o p k y.val a + C z y * u^k) :=
-        Finset.sum_le_sum fun y _ => add_le_add
-          (mul_le_mul_of_nonneg_left (ih y) (hKn z y))
-          (mul_le_mul_of_nonneg_left (sum_oldRunWeight_le (fun _ => K) (fun _ => hKn)
-            u (q.successProbability_nonneg p o) (fun _ => hKr) k y) (hCn z y))
-      _ = (∑ y : ↥B, K z y * q.visitEnvelope o p k y.val a) + (∑ y : ↥B, C z y)*u^k := by
-        rw [Finset.sum_add_distrib,Finset.sum_mul]
-      _ ≤ (∑' y, q.kernel p z.val y * q.visitEnvelope o p k y a) + q.clusterVisitKernel p z.val a * u^k := by
-        apply add_le_add
-        · simp only [hK]
-          rw [Finset.sum_coe_sort B (fun y => q.kernel p z.val y * q.visitEnvelope o p k y a)]
-          exact hs.sum_le_tsum B (fun y _ => mul_nonneg (q.kernel_nonneg p z.val y) (q.visitEnvelope_nonneg o p k y a))
-        · exact mul_le_mul_of_nonneg_right (hCr z) (pow_nonneg (q.successProbability_nonneg p o) _)
-      _ = q.visitEnvelope o p (k+1) z.val a := by rw [visitEnvelope]; ring
-
-theorem finiteTrajectoryMass_le_envelope (o : V)
-    (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
-    (Q : Finset E) (B : Finset V) (p : unitInterval) (k : ℕ) (z : ↥B) (a : V)
-    (hQ : ∀ x ∈ B, incidentSupport G hloc (finite_reachBall hloc x (max q.radius M)).toFinset ⊆ Q) :
-    finiteTrajectoryMass G Subtype.val Q p k (fun ω (x y : ↥B) => q.mass ω x.val y.val) z a ≤
-      q.visitEnvelope o p k z.val a :=
-  (finiteTrajectoryMass_le_marked G Subtype.val Q p k _ (fun ω x y => q.nonneg ω x.val y.val) z a).trans
-    (q.sum_markedOldRun_le_envelope o Q B p k z a hQ)
-
-end LocalEndpointRule
-end TrajectoryEnvelopeDomination
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Transport.EndpointKernels
+
+namespace OAI
+
+/-! Amplification for bond percolation. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section AmplificationMarginal
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def restrictFiniteField (Q : Finset E) (ω : Set E) : FiniteField Q :=
+  ⟨domainBits Q ω,Finset.mem_powerset.mpr (domainBits_subset Q ω)⟩
+
+@[simp] theorem finiteFieldSet_restrict (Q : Finset E) (ω : Set E) :
+    finiteFieldSet Q (restrictFiniteField Q ω) = ω ∩ (Q : Set E) := by
+  ext e
+  simp [restrictFiniteField,domainBits,finiteFieldSet,and_comm]
+
+@[simp] theorem restrictFiniteField_fieldSet (Q : Finset E) (ω : FiniteField Q) :
+    restrictFiniteField Q (finiteFieldSet Q ω) = ω := by
+  apply Subtype.ext
+  exact domainBits_finiteFieldSet Q ω
+
+theorem cluster_restrict_of_cap (G : BondGraph V E) (hloc : LocallyFinite G)
+    (Q : Finset E) (ω : Set E) (x : V) (C : ℕ)
+    (hQ : incidentSupport G hloc (finite_reachBall hloc x C).toFinset ⊆ Q)
+    (hf : (cluster G (ω ∩ (Q : Set E)) x).Finite)
+    (hC : (cluster G (ω ∩ (Q : Set E)) x).ncard ≤ C) :
+    cluster G (ω ∩ (Q : Set E)) x = cluster G ω x := by
+  apply Set.Subset.antisymm
+  · exact fun _ h => conn_mono G Set.inter_subset_left h
+  · intro y hy
+    induction hy with
+    | refl => exact conn_refl G _ x
+    | @tail a b _ hab ih =>
+      obtain ⟨e,he,hab⟩ := hab
+      have ha := finite_cluster_subset_reachBall G (ω ∩ (Q : Set E)) x hf C hC ih
+      have heQ := hQ ((mem_incidentSupport G hloc _ e).mpr
+        ⟨a,by simpa only [Set.Finite.mem_toFinset] using ha,by simp [hab]⟩)
+      exact ih.tail ⟨e,⟨he,heQ⟩,hab⟩
+
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
+
+ 
+
+noncomputable def amplificationMarginal (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (z : ↥B) (p : unitInterval) (ω : Set E) : ℝ :=
+  finiteRootLoad Q (Finset.Icc 1 k) p
+    (finiteChoiceForwardLoad G x k Q (q.amplificationInputs Q B x k n z)
+      (fun r _ => amplificationVertices B k z r) (q.amplificationWeight B k z)) (restrictFiniteField Q ω)
+
+theorem amplificationMarginal_local (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (z : ↥B) (p : unitInterval) (ω : Set E) :
+    q.amplificationMarginal Q B x k n z p ω =
+      q.amplificationMarginal Q B x k n z p (ω ∩ (Q : Set E)) := by
+  unfold amplificationMarginal
+  congr 1
+  apply (finiteFieldSet_injective Q)
+  simp only [finiteFieldSet_restrict,Set.inter_assoc,Set.inter_self]
+
+theorem amplificationMarginal_bounds (q : LocalEndpointRule G hloc O M L) (hL : 1 ≤ L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (z : ↥B) (p : unitInterval) (ω : Set E) :
+    0 ≤ q.amplificationMarginal Q B x k n z p ω ∧
+      q.amplificationMarginal Q B x k n z p ω ≤ inverseCodeMass M L ^ k := by
+  have hb := finiteChoiceForwardLoad_bounds G hloc x z.val k M L hL Q
+    (q.amplificationInputs Q B x k n z) (fun r _ => amplificationVertices B k z r)
+    (q.amplificationWeight B k z) (q.amplificationWeight_nonneg B k z)
+    (q.amplificationWeight_row B k z) (q.amplificationInputs_subset Q B x k n z)
+    (fun _ _ _ => congrArg Subtype.val (oldRunVertices_last ..))
+    (fun r s hs i hi hik => (q.amplificationInputs_chain Q B x k n z r s hs i hi hik).2)
+  exact ⟨finiteRootLoad_nonneg _ _ p _ (fun s _ => (hb s).1) _,
+    finiteRootLoad_le _ _ p _ _ (fun s _ => (hb s).2) _⟩
+
+theorem amplificationMarginal_support (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (z : ↥B) (p : unitInterval) (ω : Set E)
+    (h : q.amplificationMarginal Q B x k n z p ω ≠ 0) :
+    x ∈ O ∧ z.val ∈ O ∧ Conn G (ω ∩ (Q : Set E)) x z.val ∧
+      (cluster G (ω ∩ (Q : Set E)) x).Finite ∧
+      (cluster G (ω ∩ (Q : Set E)) x).ncard ≤ 2*n+k*M := by
+  let A := q.amplificationInputs Q B x k n z
+  let y := fun r (_ : SpliceState E) => amplificationVertices B k z r
+  let I := FiniteChoiceTraceIndex G x k Q A y
+  let out := choiceTraceOutput G x k Q A y
+  have ho (a : I) : out a ∈ finiteSpliceDomain Q (Finset.Icc 1 k) :=
+    traceIndexOutput_mem G x k Q (A a.1) (y a.1) (q.amplificationInputs_subset Q B x k n z a.1) a.2
+  have he := finiteRootLoad_eq_sum Q (Finset.Icc 1 k) p out
+    (choiceTraceMass G x k Q A y (q.amplificationWeight B k z)) ho (restrictFiniteField Q ω)
+  have ha : ∃ a : I, (out a).1 = ω ∩ (Q : Set E) := by
+    by_contra hn
+    push Not at hn
+    apply h
+    change finiteRootLoad Q (Finset.Icc 1 k) p (finiteInverseLoad out _) _ = 0
+    rw [he,finiteFieldSet_restrict]
+    simp only [hn,ite_false,Finset.sum_const_zero]
+  obtain ⟨a,ha⟩ := ha
+  have ht := (traceIndex_valid G x k Q (A a.1) (y a.1) a.2).1
+  have hs := q.amplificationTrace_support Q B x k n z a.1 a.2.1.val a.2.1.property
+    (choiceTraceCode G x k Q A y a) (out a).1 ht
+  rwa [ha] at hs
+
+theorem amplificationMarginal_physical_support (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (z : ↥B) (p : unitInterval) (ω : Set E)
+    (hQ : incidentSupport G hloc (finite_reachBall hloc x (2*n+k*M)).toFinset ⊆ Q)
+    (h : q.amplificationMarginal Q B x k n z p ω ≠ 0) :
+    x ∈ O ∧ z.val ∈ O ∧ Conn G ω x z.val ∧ (cluster G ω x).Finite ∧
+      (cluster G ω x).ncard ≤ 2*n+k*M := by
+  obtain ⟨hx,hz,hc,hf,hcap⟩ := q.amplificationMarginal_support Q B x k n z p ω h
+  have he := cluster_restrict_of_cap G hloc Q ω x (2*n+k*M) hQ hf hcap
+  exact ⟨hx,hz,conn_mono G Set.inter_subset_left hc,he ▸ hf,he ▸ hcap⟩
+
+ 
+
+theorem amplificationMarginal_mean [Nonempty E] (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (z : ↥B) (p : unitInterval) :
+    (∫ ω, q.amplificationMarginal Q B x k n z p ω ∂law p) =
+      ∑ r, ∑ s ∈ q.amplificationInputs Q B x k n z r,
+        spliceTupleWeight Q (Finset.Icc 1 k) p s * q.amplificationWeight B k z s r := by
+  rw [integral_finite_local Q p _ (q.amplificationMarginal_local Q B x k n z p)]
+  simp only [amplificationMarginal,restrictFiniteField_fieldSet]
+  rw [finiteRootLoad_mean]
+  apply finiteChoiceForwardLoad_mean G hloc x k M Q p
+  · exact q.amplificationInputs_subset Q B x k n z
+  · intro r s hs
+    exact ((q.mem_amplificationInputs Q B x k n z r s).mp hs).2.2.2.2.2.1
+  · intro r s hs i hi
+    exact (q.amplificationInputs_chain Q B x k n z r s hs i
+      (Finset.mem_Icc.mp hi).1 (Finset.mem_Icc.mp hi).2).1
+  · intro r s hs i hi
+    have hc := q.amplificationInputs_chain Q B x k n z r s hs i
+      (Finset.mem_Icc.mp hi).1 (Finset.mem_Icc.mp hi).2
+    exact ⟨hc.2.1,hc.2.2.1⟩
+
+end LocalEndpointRule
+end AmplificationMarginal
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section AmplificationMeanKernel
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem finiteSpliceDomain_integral_factor (Q : Finset E) (J : Finset ℕ) (p : ℝ)
+    (f : Set E → ℝ) (g : (ℕ → Set E) → ℝ) :
+    (∑ s ∈ finiteSpliceDomain Q J, spliceTupleWeight Q J p s * f s.1 * g s.2) =
+      finiteFieldIntegral Q p f * finiteFamilyIntegral Q J p g := by
+  have hh := finiteRootLoad_mean Q J p (fun s => f s.1 * g s.2)
+  simp only [← mul_assoc] at hh
+  rw [← hh]
+  simp only [finiteRootLoad,finiteFieldIntegral,finiteFamilyIntegral,Finset.sum_mul]
+  apply Finset.sum_congr rfl
+  intro ω _
+  simp only [Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro η _
+  ring
+
+ 
+noncomputable def finiteBinConnection (G : BondGraph V E) (Q : Finset E) (p : ℝ)
+    (x : V) (n : ℕ) (y : V) : ℝ := finiteFieldIntegral Q p (fun ω =>
+      if n ≤ (cluster G ω x).ncard ∧ (cluster G ω x).ncard < 2*n ∧ y ∈ cluster G ω x then 1 else 0)
+
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
+
+theorem finiteOldKernel_eq_kernel (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (p : unitInterval) (a b : V)
+    (hQ : incidentSupport G hloc (finite_reachBall hloc a q.radius).toFinset ⊆ Q) :
+    finiteOldKernel Q p q.mass a b = q.kernel p a b := by
+  symm
+  exact integral_finite_local Q p _ (finite_local_of_subset _ Q hQ _ (q.locality · a b))
+
+ 
+
+theorem amplificationMarginal_mean_kernel [Nonempty E] (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (z : ↥B) (p : unitInterval)
+    (hx : x ∈ O) (hz : z.val ∈ O)
+    (hQ : ∀ a ∈ B, incidentSupport G hloc (finite_reachBall hloc a q.radius).toFinset ⊆ Q) :
+    (∫ ω, q.amplificationMarginal Q B x k n z p ω ∂law p) =
+      ∑ r : Fin k → ↥B, finiteBinConnection G Q p x n (amplificationVertices B k z r 0) *
+        oldRunWeight (fun _ (a b : ↥B) => q.kernel p a.val b.val) k z r := by
+  rw [q.amplificationMarginal_mean Q B x k n z p]
+  apply Finset.sum_congr rfl
+  intro r _
+  let f : Set E → ℝ := fun ω => if n ≤ (cluster G ω x).ncard ∧
+      (cluster G ω x).ncard < 2*n ∧ amplificationVertices B k z r 0 ∈ cluster G ω x then 1 else 0
+  let g : (ℕ → Set E) → ℝ := fun η =>
+    oldRunWeight (fun i (a b : ↥B) => q.mass (η i) a.val b.val) k z r
+  have he : (∑ s ∈ q.amplificationInputs Q B x k n z r,
+      spliceTupleWeight Q (Finset.Icc 1 k) p s * q.amplificationWeight B k z s r) =
+      ∑ s ∈ finiteSpliceDomain Q (Finset.Icc 1 k), spliceTupleWeight Q (Finset.Icc 1 k) p s * f s.1 * g s.2 := by
+    rw [amplificationInputs,Finset.sum_filter]
+    apply Finset.sum_congr rfl
+    intro s _
+    change (if _ then _ else _) = _
+    by_cases hw : q.amplificationWeight B k z s r = 0
+    · have hg : g s.2 = 0 := hw
+      simp [hx,hz,hw,hg]
+    · simp only [hx,hz,true_and,hw,ne_eq,not_false_eq_true,and_true]
+      by_cases hbin : n ≤ (cluster G s.1 x).ncard ∧ (cluster G s.1 x).ncard < 2*n ∧
+          amplificationVertices B k z r 0 ∈ cluster G s.1 x
+      · simp [f,hbin,g,amplificationWeight]
+      · simp only [hbin,ite_false,f,mul_zero,zero_mul]
+  rw [he,finiteSpliceDomain_integral_factor]
+  dsimp only [g]
+  rw [oldRunWeight_integral Q p k (fun ω (a b : ↥B) => q.mass ω a.val b.val) z r]
+  change finiteBinConnection G Q p x n _ * _ = _
+  congr 1
+  apply congrArg (fun K : ℕ → ↥B → ↥B → ℝ => oldRunWeight K k z r)
+  funext i a b
+  exact q.finiteOldKernel_eq_kernel Q p a.val b.val (hQ a.val a.property)
+
+end LocalEndpointRule
+end AmplificationMeanKernel
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section BinConnection
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+noncomputable def binConnectionMass (G : BondGraph V E) (n : ℕ) (ω : Set E) (x y : V) : ℝ :=
+  if (cluster G ω x).Finite ∧ n ≤ (cluster G ω x).ncard ∧
+    (cluster G ω x).ncard < 2*n ∧ Conn G ω x y then 1 else 0
+
+theorem binConnectionMass_nonneg (G : BondGraph V E) (n : ℕ) (ω : Set E) (x y : V) :
+    0 ≤ binConnectionMass G n ω x y := by unfold binConnectionMass; split_ifs <;> norm_num
+
+theorem binConnectionMass_symm (G : BondGraph V E) (n : ℕ) (ω : Set E) (x y : V) :
+    binConnectionMass G n ω x y = binConnectionMass G n ω y x := by
+  by_cases hc : Conn G ω x y
+  · have he := cluster_eq_of_conn G ω hc
+    simp only [binConnectionMass,hc,conn_symm G ω hc,and_true,he]
+  · have hc' : ¬Conn G ω y x := fun h => hc (conn_symm G ω h)
+    simp only [binConnectionMass,hc,hc',and_false,ite_false]
+
+theorem binConnectionMass_local (G : BondGraph V E) (hloc : LocallyFinite G)
+    (n : ℕ) (ω : Set E) (x y : V) :
+    let Q := incidentSupport G hloc (finite_reachBall hloc x (2*n)).toFinset
+    binConnectionMass G n ω x y = binConnectionMass G n (ω ∩ (Q : Set E)) x y := by
+  dsimp only
+  let Q := incidentSupport G hloc (finite_reachBall hloc x (2*n)).toFinset
+  have hp : ((cluster G ω x).Finite ∧ n ≤ (cluster G ω x).ncard ∧
+      (cluster G ω x).ncard < 2*n ∧ Conn G ω x y) ↔
+      ((cluster G (ω ∩ (Q : Set E)) x).Finite ∧ n ≤ (cluster G (ω ∩ (Q : Set E)) x).ncard ∧
+      (cluster G (ω ∩ (Q : Set E)) x).ncard < 2*n ∧ Conn G (ω ∩ (Q : Set E)) x y) := by
+    constructor
+    · rintro ⟨hf,hn,hN,hc⟩
+      have hs : cluster G ω x ⊆ (finite_reachBall hloc x (2*n)).toFinset := by
+        simpa only [Set.Finite.coe_toFinset] using finite_cluster_subset_reachBall G ω x hf (2*n) hN.le
+      have he := cluster_restrict_incident_of_full G hloc _ ω x hs
+      change cluster G (ω ∩ (Q : Set E)) x = cluster G ω x at he
+      refine ⟨he.symm ▸ hf,he.symm ▸ hn,he.symm ▸ hN,?_⟩
+      change y ∈ cluster G (ω ∩ (Q : Set E)) x
+      rw [he]
+      exact hc
+    · rintro ⟨hf,hn,hN,hc⟩
+      have he := cluster_restrict_of_cap G hloc Q ω x (2*n) (Finset.Subset.refl _) hf hN.le
+      exact ⟨he ▸ hf,he ▸ hn,he ▸ hN,conn_mono G Set.inter_subset_left hc⟩
+  exact if_congr hp rfl rfl
+
+theorem binConnectionMass_support (G : BondGraph V E) (n : ℕ) (ω : Set E) (x y : V)
+    (h : binConnectionMass G n ω x y ≠ 0) :
+    (cluster G ω x).Finite ∧ (cluster G ω x).ncard < 2*n ∧ Conn G ω x y := by
+  unfold binConnectionMass at h
+  split_ifs at h with hp
+  · exact ⟨hp.1,hp.2.2.1,hp.2.2.2⟩
+  · exact (h rfl).elim
+
+theorem binConnectionMass_outside (G : BondGraph V E) (n : ℕ) (ω : Set E) (x y : V)
+    (hy : y ∉ reachBall G x (2*n)) : binConnectionMass G n ω x y = 0 := by
+  by_contra h
+  obtain ⟨hf,hN,hc⟩ := binConnectionMass_support G n ω x y h
+  exact hy (finite_cluster_subset_reachBall G ω x hf (2*n) hN.le hc)
+
+theorem binConnectionMass_row (G : BondGraph V E) (n : ℕ) (ω : Set E) (x : V) :
+    (∑' y, binConnectionMass G n ω x y) ≤ 2*n := by
+  by_cases hb : (cluster G ω x).Finite ∧ n ≤ (cluster G ω x).ncard ∧ (cluster G ω x).ncard < 2*n
+  · have hs : (∑' y, binConnectionMass G n ω x y) = (cluster G ω x).ncard := by
+      rw [tsum_eq_sum (s := hb.1.toFinset) (fun y hy => by
+        have hc : ¬Conn G ω x y := fun h => hy (hb.1.mem_toFinset.mpr h)
+        simp only [binConnectionMass,hc,and_false,ite_false])]
+      have he : (∑ y ∈ hb.1.toFinset, binConnectionMass G n ω x y) =
+          ∑ _y ∈ hb.1.toFinset, (1 : ℝ) := by
+        apply Finset.sum_congr rfl
+        intro y hy
+        exact ite_eq_left ⟨hb.1,hb.2.1,hb.2.2,hb.1.mem_toFinset.mp hy⟩
+      rw [he]
+      simp only [Finset.sum_const,nsmul_eq_mul,mul_one]
+      exact_mod_cast (Set.ncard_eq_toFinset_card _ hb.1).symm
+    rw [hs]
+    exact_mod_cast hb.2.2.le
+  · have hz (y : V) : binConnectionMass G n ω x y = 0 := by
+      apply ite_eq_right
+      exact fun h => hb ⟨h.1,h.2.1,h.2.2.1⟩
+    simp only [hz,tsum_zero]
+    positivity
+
+noncomputable def binConnection (G : BondGraph V E) (p : unitInterval) (n : ℕ) (x y : V) : ℝ :=
+  ∫ ω, binConnectionMass G n ω x y ∂law p
+
+theorem binConnection_nonneg (G : BondGraph V E) (p : unitInterval) (n : ℕ) (x y : V) :
+    0 ≤ binConnection G p n x y := integral_nonneg fun ω => binConnectionMass_nonneg G n ω x y
+
+theorem binConnection_symm (G : BondGraph V E) (p : unitInterval) (n : ℕ) (x y : V) :
+    binConnection G p n x y = binConnection G p n y x :=
+  integral_congr_ae (ae_of_all _ fun ω => binConnectionMass_symm G n ω x y)
+
+theorem binConnection_outside (G : BondGraph V E) (p : unitInterval) (n : ℕ) (x y : V)
+    (hy : y ∉ reachBall G x (2*n)) : binConnection G p n x y = 0 := by
+  simp only [binConnection,fun ω => binConnectionMass_outside G n ω x y hy,integral_zero]
+
+theorem binConnection_row (G : BondGraph V E) (hloc : LocallyFinite G)
+    (p : unitInterval) (n : ℕ) (x : V) : (∑' y, binConnection G p n x y) ≤ 2*n := by
+  let B := (finite_reachBall hloc x (2*n)).toFinset
+  let Q := incidentSupport G hloc B
+  have hi (y : V) : Integrable (fun ω => binConnectionMass G n ω x y) (law p) :=
+    integrable_finite_local Q p _ (fun ω => binConnectionMass_local G hloc n ω x y)
+  have he (ω : Set E) : (∑' y, binConnectionMass G n ω x y) = ∑ y ∈ B, binConnectionMass G n ω x y :=
+    tsum_eq_sum fun y hy => binConnectionMass_outside G n ω x y (by simpa only [B,Set.Finite.mem_toFinset] using hy)
+  have hs : (∑' y, binConnection G p n x y) = ∑ y ∈ B, binConnection G p n x y :=
+    tsum_eq_sum fun y hy => binConnection_outside G p n x y (by simpa only [B,Set.Finite.mem_toFinset] using hy)
+  rw [hs]
+  simp only [binConnection]
+  rw [← integral_finsetSum B (fun y _ => hi y)]
+  calc
+    _ ≤ ∫ _ω, (2*n : ℝ) ∂law p := integral_mono (integrable_finsetSum B (fun y _ => hi y))
+      (integrable_const _) (fun ω => (he ω).symm ▸ binConnectionMass_row G n ω x)
+    _ = 2*n := by simp
+
+ 
+
+theorem exists_binConnection_operator (G : BondGraph V E) (hloc : LocallyFinite G)
+    (p : unitInterval) (n : ℕ) :
+    ∃ A : Hilbert V →L[ℂ] Hilbert V, ‖A‖ ≤ 2*n ∧
+      ∀ x y, A (lp.single 2 y (1:ℂ)) x = (binConnection G p n x y : ℂ) := by
+  have hs (x : V) : Summable (binConnection G p n x) := by
+    apply summable_of_ne_finset_zero (s := (finite_reachBall hloc x (2*n)).toFinset)
+    intro y hy
+    exact binConnection_outside G p n x y (by simpa only [Set.Finite.mem_toFinset] using hy)
+  have hr (x : V) (S : Finset V) : (∑ y ∈ S, binConnection G p n x y) ≤ 2*n :=
+    ((hs x).sum_le_tsum S (fun y _ => binConnection_nonneg G p n x y)).trans (binConnection_row G hloc p n x)
+  apply exists_operator_of_row_column_bound (binConnection G p n) (2*n) (by positivity) (binConnection_nonneg G p n) hr
+  intro y S
+  simpa only [binConnection_symm G p n] using hr y S
+
+end BinConnection
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section FiniteBinConnection
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem finiteBinConnection_eq_binConnection (G : BondGraph V E) (hloc : LocallyFinite G)
+    (Q : Finset E) (p : unitInterval) (x : V) (n : ℕ) (y : V)
+    (hQ : incidentSupport G hloc (finite_reachBall hloc x (2*n)).toFinset ⊆ Q) :
+    finiteBinConnection G Q p x n y = binConnection G p n x y := by
+  rw [binConnection, integral_finite_local Q p _
+    (finite_local_of_subset _ Q hQ _ (fun ω => binConnectionMass_local G hloc n ω x y))]
+  unfold finiteBinConnection finiteFieldIntegral
+  apply Finset.sum_congr rfl
+  intro ω _
+  congr 1
+  have hf : (cluster G (finiteFieldSet Q ω) x).Finite :=
+    cluster_finite_of_finite_edges G ω.val.finite_toSet x
+  unfold cluster at hf
+  simp only [binConnectionMass,cluster,Set.mem_ofPred_eq,hf,true_and]
+  exact if_congr Iff.rfl rfl rfl
+
+end FiniteBinConnection
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section PositivePathOperator
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+def PositiveRealMatrix (A : Hilbert V →L[ℂ] Hilbert V) : Prop :=
+  ∀ x y, (matrixCoeff A x y).im = 0 ∧ 0 ≤ (matrixCoeff A x y).re
+
+namespace PositiveRealMatrix
+variable {A H : Hilbert V →L[ℂ] Hilbert V}
+
+theorem real (hA : PositiveRealMatrix A) (x y : V) :
+    matrixCoeff A x y = ((matrixCoeff A x y).re : ℂ) := by
+  apply Complex.ext
+  · rfl
+  · exact (hA x y).1
+
+theorem mul_hasSum (hA : PositiveRealMatrix A) (H : Hilbert V →L[ℂ] Hilbert V) (x y : V) :
+    HasSum (fun z => (matrixCoeff A x z).re * (matrixCoeff H z y).re)
+      (matrixCoeff (A*H) x y).re := by
+  have h := Complex.reCLM.hasSum (operator_kernel_hasSum A (H (lp.single 2 y (1:ℂ))) x)
+  change HasSum (fun z => (matrixCoeff A x z * matrixCoeff H z y).re)
+    (matrixCoeff (A*H) x y).re at h
+  simpa only [Complex.mul_re,(hA _ _).1,zero_mul,sub_zero] using h
+
+theorem mul (hA : PositiveRealMatrix A) (hH : PositiveRealMatrix H) :
+    PositiveRealMatrix (A*H) := by
+  intro x y
+  constructor
+  · have hs := (operator_kernel_hasSum A (H (lp.single 2 y (1:ℂ))) x).summable
+    change Summable (fun z => matrixCoeff A x z * matrixCoeff H z y) at hs
+    rw [matrixCoeff_mul,Complex.im_tsum hs]
+    simp only [Complex.mul_im,(hA _ _).1,(hH _ _).1,mul_zero,zero_mul,add_zero,tsum_zero]
+  · rw [← (hA.mul_hasSum H x y).tsum_eq]
+    exact tsum_nonneg fun z => mul_nonneg (hA x z).2 (hH z y).2
+
+theorem one : PositiveRealMatrix (1 : Hilbert V →L[ℂ] Hilbert V) := by
+  intro x y
+  simp only [matrixCoeff,one_apply_eq_self,lp.single_apply,Pi.single_apply]
+  split_ifs <;> norm_num
+
+theorem pow (hA : PositiveRealMatrix A) (k : ℕ) : PositiveRealMatrix (A^k) := by
+  induction k with
+  | zero => simpa only [pow_zero] using (one (V := V))
+  | succ k ih => rw [pow_succ]; exact ih.mul hA
+
+theorem star (hA : PositiveRealMatrix A) : PositiveRealMatrix (star A) := by
+  intro x y
+  simpa only [matrixCoeff_adjoint,Complex.star_def,Complex.conj_im,Complex.conj_re,neg_eq_zero] using hA y x
+
+end PositiveRealMatrix
+
+ 
+
+theorem oldRun_test_succ {W : Type*} [Fintype W]
+    (K : ℕ → W → W → ℝ) (F : W → ℝ) (k : ℕ) (z : W) :
+    (∑ r : Fin (k+1) → W, oldRunWeight K (k+1) z r * F (oldRunVertices (k+1) z r 0)) =
+      ∑ a : W, K (k+1) z a *
+        ∑ r : Fin k → W, oldRunWeight K k a r * F (oldRunVertices k a r 0) := by
+  rw [Fintype.sum_equiv (oldRunConsEquiv k) _
+    (fun r => K (k+1) z r.1 * oldRunWeight K k r.1 r.2 * F (oldRunVertices k r.1 r.2 0))
+      (by intro r; simp only [oldRunWeight,oldRunConsEquiv,Equiv.coe_fn_mk,
+        oldRunVertices_succ_of_le k z r 0 (Nat.zero_le k)]), Fintype.sum_prod_type]
+  simp only [Finset.mul_sum,mul_assoc]
+
+ 
+
+theorem finite_oldRun_operator_domination (A H : Hilbert V →L[ℂ] Hilbert V)
+    (hA : PositiveRealMatrix A) (hH : PositiveRealMatrix H)
+    (B : Finset V) (k : ℕ) (z : ↥B) (x : V) :
+    (∑ r : Fin k → ↥B,
+      oldRunWeight (fun _ (a b : ↥B) => (matrixCoeff A a.val b.val).re) k z r *
+        (matrixCoeff H (oldRunVertices k z r 0).val x).re) ≤
+      (matrixCoeff (A^k*H) z.val x).re := by
+  induction k generalizing z with
+  | zero => simp [oldRunWeight,oldRunVertices]
+  | succ k ih =>
+    rw [oldRun_test_succ (fun _ (a b : ↥B) => (matrixCoeff A a.val b.val).re)
+      (fun a : ↥B => (matrixCoeff H a.val x).re) k z]
+    calc
+      _ ≤ ∑ a : ↥B, (matrixCoeff A z.val a.val).re * (matrixCoeff (A^k*H) a.val x).re :=
+        Finset.sum_le_sum fun a _ => mul_le_mul_of_nonneg_left (ih a) (hA z.val a.val).2
+      _ ≤ ∑' a, (matrixCoeff A z.val a).re * (matrixCoeff (A^k*H) a x).re := by
+        rw [Finset.sum_coe_sort B (fun a => (matrixCoeff A z.val a).re * (matrixCoeff (A^k*H) a x).re)]
+        exact (hA.mul_hasSum (A^k*H) z.val x).summable.sum_le_tsum B
+          (fun a _ => mul_nonneg (hA z.val a).2 (((hA.pow k).mul hH) a x).2)
+      _ = (matrixCoeff (A^(k+1)*H) z.val x).re := by
+        rw [(hA.mul_hasSum (A^k*H) z.val x).tsum_eq,pow_succ',mul_assoc]
+
+end PositivePathOperator
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section OrbitBinOperator
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem isAut_trans (G : BondGraph V E) {a c : V ≃ V} {b d : E ≃ E}
+    (hab : IsAut G a b) (hcd : IsAut G c d) : IsAut G (a.trans c) (b.trans d) := by
+  intro e
+  change G (d (b e)) = (G e).map (c ∘ a)
+  rw [hcd,hab,Sym2.map_map]
+
+theorem mem_vertexOrbit_aut_iff (G : BondGraph V E) (o : V) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (x : V) : a x ∈ vertexOrbit G o ↔ x ∈ vertexOrbit G o := by
+  constructor
+  · rintro ⟨c,d,hcd,hc⟩
+    refine ⟨c.trans a.symm,d.trans b.symm,isAut_trans G hcd (isAut_symm G hab),?_⟩
+    simp only [Equiv.trans_apply,hc,Equiv.symm_apply_apply]
+  · rintro ⟨c,d,hcd,rfl⟩
+    exact ⟨c.trans a,d.trans b,isAut_trans G hcd hab,rfl⟩
+
+theorem binConnectionMass_covariant (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (n : ℕ) (ω : Set E) (x y : V) :
+    binConnectionMass G n (b ⁻¹' ω) x y = binConnectionMass G n ω (a x) (a y) := by
+  have hf : (cluster G (b ⁻¹' ω) x).Finite ↔ (cluster G ω (a x)).Finite := by
+    rw [← cluster_aut G hab ω x]
+    exact (Set.finite_image_iff a.injective.injOn).symm
+  have hn : (cluster G (b ⁻¹' ω) x).ncard = (cluster G ω (a x)).ncard := by
+    rw [← cluster_aut G hab ω x,Set.ncard_image_of_injective _ a.injective]
+  simp only [binConnectionMass,hf,hn,conn_aut_iff G hab]
+
+theorem binConnection_covariant (G : BondGraph V E) (hloc : LocallyFinite G)
+    (p : unitInterval) {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (n : ℕ) (x y : V) :
+    binConnection G p n (a x) (a y) = binConnection G p n x y := by
+  have hm : Measurable (fun ω => binConnectionMass G n ω x y) :=
+    measurable_finite_local _ _ (fun ω => binConnectionMass_local G hloc n ω x y)
+  have h := integral_map (μ := law p) (measurable_set_preimage b).aemeasurable hm.aestronglyMeasurable
+  rw [law_map_preimage_equiv p b] at h
+  simpa only [binConnection,binConnectionMass_covariant G hab] using h.symm
+
+ 
+noncomputable def orbitBinConnection (G : BondGraph V E) (p : unitInterval) (o : V)
+    (n : ℕ) (x y : V) : ℝ :=
+  if x ∈ vertexOrbit G o ∧ y ∈ vertexOrbit G o then binConnection G p n x y else 0
+
+theorem orbitBinConnection_nonneg (G : BondGraph V E) (p : unitInterval) (o : V)
+    (n : ℕ) (x y : V) : 0 ≤ orbitBinConnection G p o n x y := by
+  unfold orbitBinConnection
+  split_ifs
+  · exact binConnection_nonneg G p n x y
+  · exact le_rfl
+
+theorem orbitBinConnection_symm (G : BondGraph V E) (p : unitInterval) (o : V)
+    (n : ℕ) (x y : V) : orbitBinConnection G p o n x y = orbitBinConnection G p o n y x := by
+  simp only [orbitBinConnection,and_comm,binConnection_symm G p n]
+
+theorem orbitBinConnection_covariant (G : BondGraph V E) (hloc : LocallyFinite G)
+    (p : unitInterval) (o : V) {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (n : ℕ) (x y : V) :
+    orbitBinConnection G p o n (a x) (a y) = orbitBinConnection G p o n x y := by
+  simp only [orbitBinConnection,mem_vertexOrbit_aut_iff G o hab,binConnection_covariant G hloc p hab n]
+
+theorem orbitBinConnection_summable (G : BondGraph V E) (hloc : LocallyFinite G)
+    (p : unitInterval) (o : V) (n : ℕ) (x : V) : Summable (orbitBinConnection G p o n x) := by
+  apply summable_of_ne_finset_zero (s := (finite_reachBall hloc x (2*n)).toFinset)
+  intro y hy
+  have hz := binConnection_outside G p n x y (by simpa only [Set.Finite.mem_toFinset] using hy)
+  simp only [orbitBinConnection,hz,ite_self]
+
+noncomputable def binOrbitRow (G : BondGraph V E) (p : unitInterval) (o : V) (n : ℕ) : ℝ :=
+  ∑' y, orbitBinConnection G p o n o y
+
+theorem orbitBinConnection_row (G : BondGraph V E) (hloc : LocallyFinite G)
+    (p : unitInterval) (o : V) (n : ℕ) (x : V) (hx : x ∈ vertexOrbit G o) :
+    (∑' y, orbitBinConnection G p o n x y) = binOrbitRow G p o n := by
+  obtain ⟨a,b,hab,rfl⟩ := hx
+  rw [← a.tsum_eq (fun y => orbitBinConnection G p o n (a o) y)]
+  exact tsum_congr (fun y => orbitBinConnection_covariant G hloc p o hab n o y)
+
+theorem orbitBinConnection_row_le (G : BondGraph V E) (hloc : LocallyFinite G)
+    (p : unitInterval) (o : V) (n : ℕ) (x : V) :
+    (∑' y, orbitBinConnection G p o n x y) ≤ binOrbitRow G p o n := by
+  by_cases hx : x ∈ vertexOrbit G o
+  · exact (orbitBinConnection_row G hloc p o n x hx).le
+  · simp only [orbitBinConnection,hx,false_and,ite_false,tsum_zero]
+    exact tsum_nonneg (orbitBinConnection_nonneg G p o n o)
+
+ 
+
+theorem exists_orbitBinConnection_operator (G : BondGraph V E) (hloc : LocallyFinite G)
+    (p : unitInterval) (o : V) (n : ℕ) :
+    ∃ H : Hilbert V →L[ℂ] Hilbert V, ‖H‖ ≤ binOrbitRow G p o n ∧
+      ∀ x y, matrixCoeff H x y = (orbitBinConnection G p o n x y : ℂ) := by
+  have hr (x : V) (S : Finset V) : (∑ y ∈ S, orbitBinConnection G p o n x y) ≤ binOrbitRow G p o n :=
+    ((orbitBinConnection_summable G hloc p o n x).sum_le_tsum S
+      (fun y _ => orbitBinConnection_nonneg G p o n x y)).trans (orbitBinConnection_row_le G hloc p o n x)
+  exact exists_operator_of_row_column_bound (orbitBinConnection G p o n) (binOrbitRow G p o n)
+    (tsum_nonneg (orbitBinConnection_nonneg G p o n o)) (orbitBinConnection_nonneg G p o n) hr
+    (fun y S => by simpa only [orbitBinConnection_symm G p o n] using hr y S)
+
+end OrbitBinOperator
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+open MeasureTheory
+section AmplificationOperatorDomination
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem oldRun_endpoint_mem {W : Type*} (P : W → Prop) (K : ℕ → W → W → ℝ)
+    (hK : ∀ i a b, K i a b ≠ 0 → P b) (k : ℕ) (z : W) (hz : P z)
+    (r : Fin k → W) (hr : oldRunWeight K k z r ≠ 0) : P (oldRunVertices k z r 0) := by
+  induction k generalizing z with
+  | zero => exact hz
+  | succ k ih =>
+    have hm := mul_ne_zero_iff.mp hr
+    rw [oldRunVertices_succ_of_le _ _ _ _ (Nat.zero_le k)]
+    exact ih (r 0) (hK _ _ _ hm.1) (Fin.tail r) hm.2
+
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
+
+ 
+
+theorem amplificationMarginal_operator_le [Nonempty E] (o : V)
+    (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (z : ↥B) (p : unitInterval)
+    (hx : x ∈ vertexOrbit G o) (hz : z.val ∈ vertexOrbit G o)
+    (hroot : incidentSupport G hloc (finite_reachBall hloc x (2*n)).toFinset ⊆ Q)
+    (hquery : ∀ a ∈ B, incidentSupport G hloc (finite_reachBall hloc a q.radius).toFinset ⊆ Q)
+    (A H : Hilbert V →L[ℂ] Hilbert V)
+    (hA : ∀ a b, matrixCoeff A a b = (q.kernel p a b : ℂ))
+    (hH : ∀ a b, matrixCoeff H a b = (orbitBinConnection G p o n a b : ℂ)) :
+    (∫ ω, q.amplificationMarginal Q B x k n z p ω ∂law p) ≤
+      (matrixCoeff (A^k*H) z.val x).re := by
+  have hposA : PositiveRealMatrix A := fun a b => by
+    rw [hA a b]
+    exact ⟨rfl,q.kernel_nonneg p a b⟩
+  have hposH : PositiveRealMatrix H := fun a b => by
+    rw [hH a b]
+    exact ⟨rfl,orbitBinConnection_nonneg G p o n a b⟩
+  rw [q.amplificationMarginal_mean_kernel Q B x k n z p hx hz hquery]
+  have he : (∑ r : Fin k → ↥B, finiteBinConnection G Q p x n (amplificationVertices B k z r 0) *
+      oldRunWeight (fun _ (a b : ↥B) => q.kernel p a.val b.val) k z r) =
+      ∑ r : Fin k → ↥B,
+        oldRunWeight (fun _ (a b : ↥B) => (matrixCoeff A a.val b.val).re) k z r *
+          (matrixCoeff H (oldRunVertices k z r 0).val x).re := by
+    apply Finset.sum_congr rfl
+    intro r _
+    simp only [hA,hH,Complex.ofReal_re,
+      finiteBinConnection_eq_binConnection G hloc Q p x n _ hroot]
+    by_cases hr : oldRunWeight (fun _ (a b : ↥B) => q.kernel p a.val b.val) k z r = 0
+    · simp only [hr,mul_zero,zero_mul]
+    · have hy : (oldRunVertices k z r 0).val ∈ vertexOrbit G o :=
+        oldRun_endpoint_mem (fun a : ↥B => a.val ∈ vertexOrbit G o)
+          (fun _ (a b : ↥B) => q.kernel p a.val b.val)
+          (fun _ a b hab => by
+            by_contra hb
+            exact hab (q.kernel_zero_of_end_not_mem p a.val b.val hb)) k z hz r hr
+      rw [orbitBinConnection,ite_eq_left ⟨hy,hx⟩,binConnection_symm G p n]
+      exact mul_comm _ _
+  rw [he]
+  exact finite_oldRun_operator_domination A H hposA hposH B k z x
+
+end LocalEndpointRule
+
+ 
+
+theorem exists_operator_of_amplification_domination
+    (G : BondGraph V E) (hloc : LocallyFinite G) (p : unitInterval) (o : V) (n k : ℕ)
+    (A : Hilbert V →L[ℂ] Hilbert V) (hA : PositiveRealMatrix A)
+    (F : V → V → ℝ) (hF : ∀ x z, 0 ≤ F x z)
+    (hdom : ∀ (H : Hilbert V →L[ℂ] Hilbert V),
+      (∀ x y, matrixCoeff H x y = (orbitBinConnection G p o n x y : ℂ)) →
+      ∀ x z, F x z ≤ (matrixCoeff (A^k*H) z x).re) :
+    ∃ T : Hilbert V →L[ℂ] Hilbert V, ‖T‖ ≤ binOrbitRow G p o n * ‖A‖^k ∧
+      ∀ x z, matrixCoeff T x z = (F x z : ℂ) := by
+  obtain ⟨H,hHnorm,hH⟩ := exists_orbitBinConnection_operator G hloc p o n
+  have hposH : PositiveRealMatrix H := fun x y => by
+    rw [hH x y]
+    exact ⟨rfl,orbitBinConnection_nonneg G p o n x y⟩
+  have hpos := ((hA.pow k).mul hposH).star
+  have htr (x z : V) : (matrixCoeff (star (A^k*H)) x z).re = (matrixCoeff (A^k*H) z x).re := by
+    simp only [matrixCoeff_adjoint,Complex.star_def,Complex.conj_re]
+  obtain ⟨T,hTnorm,hT⟩ := exists_operator_of_matrix_domination
+    (fun x z => (matrixCoeff (star (A^k*H)) x z).re) F (fun x z => (hpos x z).2)
+    (fun x z => by rw [abs_of_nonneg (hF x z),htr]; exact hdom H hH x z)
+    (star (A^k*H)) (fun x z => hpos.real x z)
+  refine ⟨T,hTnorm.trans ?_,hT⟩
+  have hpow : ∀ j : ℕ, ‖A^j‖ ≤ ‖A‖^j := by
+    intro j
+    induction j with
+    | zero => simpa only [pow_zero, ContinuousLinearMap.one_def] using (ContinuousLinearMap.norm_id_le (𝕜 := ℂ) (E := Hilbert V))
+    | succ j ih =>
+      rw [pow_succ, pow_succ]
+      exact (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right ih (norm_nonneg _))
+  calc
+    ‖star (A^k*H)‖ = ‖A^k*H‖ := by
+      change ‖ContinuousLinearMap.adjoint (A^k*H)‖ = _
+      exact LinearIsometryEquiv.norm_map _ _
+    _ ≤ ‖A^k‖ * ‖H‖ := norm_mul_le _ _
+    _ ≤ ‖A‖^k * binOrbitRow G p o n := mul_le_mul (hpow k) hHnorm
+      (norm_nonneg _) (pow_nonneg (norm_nonneg _) _)
+    _ = binOrbitRow G p o n * ‖A‖^k := mul_comm _ _
+
+end AmplificationOperatorDomination
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section ClusterVisitKernel
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
+
+noncomputable def conditionalSuccess (q : LocalEndpointRule G hloc O M L) (ω : Set E) (x : V) : ℝ :=
+  ∑' y, q.mass ω x y
+
+theorem conditionalSuccess_nonneg (q : LocalEndpointRule G hloc O M L) (ω : Set E) (x : V) :
+    0 ≤ q.conditionalSuccess ω x := tsum_nonneg (q.nonneg ω x)
+
+theorem conditionalSuccess_support (q : LocalEndpointRule G hloc O M L) (ω : Set E) (x : V)
+    (h : q.conditionalSuccess ω x ≠ 0) :
+    x ∈ O ∧ (cluster G ω x).Finite ∧ (cluster G ω x).ncard ≤ M := by
+  have he : ∃ y, q.mass ω x y ≠ 0 := by
+    by_contra hn
+    push Not at hn
+    exact h (by simp only [conditionalSuccess,hn,tsum_zero])
+  obtain ⟨y,hy⟩ := he
+  obtain ⟨hx,_,_,hf,hc,_⟩ := q.support ω x y hy
+  exact ⟨hx,hf,hc⟩
+
+theorem conditionalSuccess_local (q : LocalEndpointRule G hloc O M L) (ω : Set E) (x : V) :
+    q.conditionalSuccess ω x = q.conditionalSuccess
+      (ω ∩ (incidentSupport G hloc (finite_reachBall hloc x q.radius).toFinset : Set E)) x :=
+  tsum_congr (q.locality ω x)
+
+theorem conditionalSuccess_integrable (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x : V) :
+    Integrable (fun ω => q.conditionalSuccess ω x) (law p) :=
+  integrable_finite_local _ p _ (fun ω => q.conditionalSuccess_local ω x)
+
+theorem conditionalSuccess_mean (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x : V) :
+    (∫ ω, q.conditionalSuccess ω x ∂law p) = q.successProbability p x :=
+  (q.kernel_row_integral p x).symm
+
+theorem conditionalSuccess_covariant (q : LocalEndpointRule G hloc O M L)
+    (a : V ≃ V) (b : E ≃ E) (hab : IsAut G a b) (ω : Set E) (x : V) :
+    q.conditionalSuccess (b ⁻¹' ω) x = q.conditionalSuccess ω (a x) := by
+  unfold conditionalSuccess
+  simp_rw [q.covariance a b hab]
+  exact a.tsum_eq _
+
+ 
+
+noncomputable def clusterVisitMass (q : LocalEndpointRule G hloc O M L) (ω : Set E) (x a : V) : ℝ :=
+  if a ∈ cluster G ω x then q.conditionalSuccess ω x else 0
+
+theorem clusterVisitMass_nonneg (q : LocalEndpointRule G hloc O M L) (ω : Set E) (x a : V) :
+    0 ≤ q.clusterVisitMass ω x a := by
+  unfold clusterVisitMass
+  split_ifs
+  · exact q.conditionalSuccess_nonneg ω x
+  · exact le_refl _
+
+theorem clusterVisitMass_outside (q : LocalEndpointRule G hloc O M L) (ω : Set E) (x a : V)
+    (ha : a ∉ reachBall G x M) : q.clusterVisitMass ω x a = 0 := by
+  by_cases h : q.conditionalSuccess ω x = 0
+  · simp only [clusterVisitMass,h,ite_self]
+  · obtain ⟨_,hf,hc⟩ := q.conditionalSuccess_support ω x h
+    exact ite_eq_right (fun hmem => ha (finite_cluster_subset_reachBall G ω x hf M hc hmem))
+
+theorem clusterVisitMass_local (q : LocalEndpointRule G hloc O M L) (ω : Set E) (x a : V) :
+    let Q := incidentSupport G hloc (finite_reachBall hloc x (max q.radius M)).toFinset
+    q.clusterVisitMass ω x a = q.clusterVisitMass (ω ∩ (Q : Set E)) x a := by
+  dsimp only
+  let Q := incidentSupport G hloc (finite_reachBall hloc x (max q.radius M)).toFinset
+  change q.clusterVisitMass ω x a = q.clusterVisitMass (ω ∩ (Q:Set E)) x a
+  have he := finite_local_of_subset _ Q
+    (incidentSupport_ball_mono G hloc x (le_max_left q.radius M))
+    (fun ω => q.conditionalSuccess ω x) (fun ω => q.conditionalSuccess_local ω x) ω
+  by_cases h : q.conditionalSuccess ω x = 0
+  · have h' : q.conditionalSuccess (ω ∩ (Q:Set E)) x = 0 := he.symm.trans h
+    simp only [clusterVisitMass,h,h',ite_self]
+  · obtain ⟨_,hf,hc⟩ := q.conditionalSuccess_support ω x h
+    have hs : cluster G ω x ⊆ (finite_reachBall hloc x (max q.radius M)).toFinset := by
+      simpa only [Set.Finite.coe_toFinset] using
+        finite_cluster_subset_reachBall G ω x hf _ (hc.trans (le_max_right _ _))
+    have hc' := cluster_restrict_incident_of_full G hloc _ ω x hs
+    change cluster G (ω ∩ (Q:Set E)) x = cluster G ω x at hc'
+    change (if a ∈ cluster G ω x then _ else _) = (if a ∈ cluster G (ω ∩ (Q:Set E)) x then _ else _)
+    rw [hc',he]
+
+theorem clusterVisitMass_integrable (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x a : V) :
+    Integrable (fun ω => q.clusterVisitMass ω x a) (law p) :=
+  integrable_finite_local _ p _ (fun ω => q.clusterVisitMass_local ω x a)
+
+theorem clusterVisitMass_covariant (q : LocalEndpointRule G hloc O M L)
+    (a : V ≃ V) (b : E ≃ E) (hab : IsAut G a b) (ω : Set E) (x y : V) :
+    q.clusterVisitMass (b ⁻¹' ω) x y = q.clusterVisitMass ω (a x) (a y) := by
+  have hc : y ∈ cluster G (b ⁻¹' ω) x ↔ a y ∈ cluster G ω (a x) := by
+    rw [← cluster_aut G hab ω x]
+    constructor
+    · intro hy; exact ⟨y,hy,rfl⟩
+    · rintro ⟨z,hz,he⟩
+      exact (a.injective he) ▸ hz
+  simp only [clusterVisitMass,hc,q.conditionalSuccess_covariant a b hab]
+
+theorem clusterVisitMass_row_le (q : LocalEndpointRule G hloc O M L) (ω : Set E) (x : V) :
+    (∑' a, q.clusterVisitMass ω x a) ≤ M * q.conditionalSuccess ω x := by
+  by_cases h : q.conditionalSuccess ω x = 0
+  · simp only [clusterVisitMass,h,ite_self,tsum_zero,mul_zero,le_refl]
+  obtain ⟨_,hf,hc⟩ := q.conditionalSuccess_support ω x h
+  rw [tsum_eq_sum (f := q.clusterVisitMass ω x) (s := hf.toFinset) (fun a ha => ite_eq_right (by simpa only [Set.Finite.mem_toFinset] using ha))]
+  have he : (∑ a ∈ hf.toFinset, q.clusterVisitMass ω x a) =
+      (hf.toFinset.card:ℝ) * q.conditionalSuccess ω x := by
+    calc
+      _ = ∑ _a ∈ hf.toFinset, q.conditionalSuccess ω x := Finset.sum_congr rfl (fun a ha => ite_eq_left (hf.mem_toFinset.mp ha))
+      _ = _ := by simp only [Finset.sum_const,nsmul_eq_mul]
+  rw [he]
+  apply mul_le_mul_of_nonneg_right _ (q.conditionalSuccess_nonneg ω x)
+  exact_mod_cast (by simpa only [← Set.ncard_eq_toFinset_card _ hf] using hc)
+
+noncomputable def clusterVisitKernel (q : LocalEndpointRule G hloc O M L)
+    (p : unitInterval) (x a : V) : ℝ := ∫ ω, q.clusterVisitMass ω x a ∂law p
+
+theorem clusterVisitKernel_nonneg (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x a : V) :
+    0 ≤ q.clusterVisitKernel p x a := integral_nonneg (fun ω => q.clusterVisitMass_nonneg ω x a)
+
+theorem clusterVisitKernel_outside (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x a : V)
+    (ha : a ∉ reachBall G x M) : q.clusterVisitKernel p x a = 0 := by
+  simp only [clusterVisitKernel,q.clusterVisitMass_outside _ x a ha,integral_zero]
+
+theorem clusterVisitKernel_summable (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x : V) :
+    Summable (q.clusterVisitKernel p x) := by
+  apply summable_of_ne_finset_zero (s := (finite_reachBall hloc x M).toFinset)
+  intro a ha
+  exact q.clusterVisitKernel_outside p x a (by simpa only [Set.Finite.mem_toFinset] using ha)
+
+theorem clusterVisitKernel_row_le (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x : V) :
+    (∑' a, q.clusterVisitKernel p x a) ≤ M * q.successProbability p x := by
+  let B := (finite_reachBall hloc x M).toFinset
+  have he (ω : Set E) : (∑' a, q.clusterVisitMass ω x a) = ∑ a ∈ B, q.clusterVisitMass ω x a :=
+    tsum_eq_sum (fun a ha => q.clusterVisitMass_outside ω x a (by simpa only [B,Set.Finite.mem_toFinset] using ha))
+  rw [tsum_eq_sum (s := B) (fun a ha => q.clusterVisitKernel_outside p x a (by simpa only [B,Set.Finite.mem_toFinset] using ha))]
+  simp only [clusterVisitKernel]
+  rw [← integral_finsetSum B (fun a _ => q.clusterVisitMass_integrable p x a)]
+  calc
+    _ ≤ ∫ ω, (M:ℝ) * q.conditionalSuccess ω x ∂law p :=
+      integral_mono (integrable_finsetSum B (fun a _ => q.clusterVisitMass_integrable p x a))
+        ((q.conditionalSuccess_integrable p x).const_mul _) (fun ω => by rw [← he]; exact q.clusterVisitMass_row_le ω x)
+    _ = _ := by rw [integral_const_mul,q.conditionalSuccess_mean]
+
+theorem clusterVisitKernel_covariant (q : LocalEndpointRule G hloc O M L) (p : unitInterval)
+    (a : V ≃ V) (b : E ≃ E) (hab : IsAut G a b) (x y : V) :
+    q.clusterVisitKernel p (a x) (a y) = q.clusterVisitKernel p x y := by
+  have hm : Measurable (fun ω => q.clusterVisitMass ω x y) :=
+    measurable_finite_local _ _ (fun ω => q.clusterVisitMass_local ω x y)
+  have h := integral_map (μ := law p) (measurable_set_preimage b).aemeasurable hm.aestronglyMeasurable
+  rw [law_map_preimage_equiv p b] at h
+  simpa only [clusterVisitKernel,q.clusterVisitMass_covariant a b hab] using h.symm
+
+end LocalEndpointRule
+end ClusterVisitKernel
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section VisitEnvelope
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
+
+ 
+
+noncomputable def visitEnvelope (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
+    (p : unitInterval) : ℕ → V → V → ℝ
+  | 0, _, _ => 0
+  | k+1, x, a => q.successProbability p o ^ k * q.clusterVisitKernel p x a +
+      ∑' y, q.kernel p x y * q.visitEnvelope o p k y a
+
+theorem visitEnvelope_nonneg (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
+    (p : unitInterval) (k : ℕ) (x a : V) : 0 ≤ q.visitEnvelope o p k x a := by
+  induction k generalizing x with
+  | zero => exact le_refl _
+  | succ k ih =>
+    exact add_nonneg (mul_nonneg (pow_nonneg (q.successProbability_nonneg p o) _) (q.clusterVisitKernel_nonneg p x a))
+      (tsum_nonneg (fun y => mul_nonneg (q.kernel_nonneg p x y) (ih y)))
+
+theorem visitEnvelope_covariant (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
+    (p : unitInterval) (k : ℕ) (a : V ≃ V) (b : E ≃ E) (hab : IsAut G a b) (x y : V) :
+    q.visitEnvelope o p k (a x) (a y) = q.visitEnvelope o p k x y := by
+  induction k generalizing x y with
+  | zero => rfl
+  | succ k ih =>
+    simp only [visitEnvelope,q.clusterVisitKernel_covariant p a b hab]
+    congr 1
+    rw [← a.tsum_eq (fun z => q.kernel p (a x) z * q.visitEnvelope o p k z (a y))]
+    exact tsum_congr fun z => by rw [q.kernel_covariant p a b hab,ih]
+
+ 
+
+theorem visitEnvelope_row_bound (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
+    (p : unitInterval) (k : ℕ) (x : V) :
+    Summable (q.visitEnvelope o p k x) ∧
+      (∑' a, q.visitEnvelope o p k x a) ≤ (k*M:ℕ) * q.successProbability p o ^ k := by
+  let u := q.successProbability p o
+  have hu : 0 ≤ u := q.successProbability_nonneg p o
+  induction k generalizing x with
+  | zero => simp only [visitEnvelope,zero_mul,Nat.cast_zero,pow_zero,mul_one,tsum_zero,le_refl,and_true]; exact summable_zero
+  | succ k ih =>
+    let B := (finite_reachBall hloc x M).toFinset
+    have he (a : V) : (∑' y, q.kernel p x y * q.visitEnvelope o p k y a) =
+        ∑ y ∈ B, q.kernel p x y * q.visitEnvelope o p k y a :=
+      tsum_eq_sum (fun y hy => by
+        rw [q.kernel_outside_ball_zero p x y (by simpa only [B,Set.Finite.mem_toFinset] using hy),zero_mul])
+    have ht (a : V) : q.visitEnvelope o p (k+1) x a =
+        u^k * q.clusterVisitKernel p x a + ∑ y ∈ B, q.kernel p x y * q.visitEnvelope o p k y a := by
+      exact congrArg (_ + ·) (he a)
+    have hsum : Summable (fun a => ∑ y ∈ B, q.kernel p x y * q.visitEnvelope o p k y a) :=
+      summable_sum (s := B) (fun y _ => (ih y).1.mul_left _)
+    have hs := ((q.clusterVisitKernel_summable p x).mul_left (u^k)).add hsum
+    have hs' : Summable (q.visitEnvelope o p (k+1) x) := by
+      change Summable (fun a => q.visitEnvelope o p (k+1) x a)
+      simpa only [ht] using hs
+    refine ⟨hs',?_⟩
+    simp_rw [ht]
+    rw [Summable.tsum_add ((q.clusterVisitKernel_summable p x).mul_left _) hsum,tsum_mul_left,
+      Summable.tsum_finsetSum (fun y _ => (ih y).1.mul_left _)]
+    simp_rw [tsum_mul_left]
+    have hr : (∑ y ∈ B, q.kernel p x y) ≤ u :=
+      ((q.summable_kernel p x).sum_le_tsum B (fun y _ => q.kernel_nonneg p x y)).trans
+        (q.kernel_row_le_success o p x)
+    calc
+      _ ≤ u^k * (M*u) + ∑ y ∈ B, q.kernel p x y * ((k*M:ℕ)*u^k) := by
+        apply add_le_add
+        · apply mul_le_mul_of_nonneg_left _ (pow_nonneg hu _)
+          exact (q.clusterVisitKernel_row_le p x).trans
+            (mul_le_mul_of_nonneg_left (q.kernel_row_le_success o p x) (Nat.cast_nonneg _))
+        · exact Finset.sum_le_sum fun y _ => mul_le_mul_of_nonneg_left (ih y).2 (q.kernel_nonneg p x y)
+      _ ≤ u^k * (M*u) + u*((k*M:ℕ)*u^k) := by
+        rw [← Finset.sum_mul]
+        exact add_le_add (le_refl _) (mul_le_mul_of_nonneg_right hr (mul_nonneg (Nat.cast_nonneg _) (pow_nonneg hu _)))
+      _ = ((k+1)*M:ℕ) * q.successProbability p o ^ (k+1) := by
+        change _ = ((k+1)*M:ℕ) * u^(k+1)
+        simp only [Nat.cast_mul,Nat.cast_add,Nat.cast_one,pow_succ]
+        ring
+
+end LocalEndpointRule
+end VisitEnvelope
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section MarkedOldRun
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {W : Type*}
+
+theorem oldRunWeight_congr (K K' : ℕ → W → W → ℝ) (k : ℕ)
+    (h : ∀ i, 1 ≤ i → i ≤ k → ∀ a b, K i a b = K' i a b) (z : W) (r : Fin k → W) :
+    oldRunWeight K k z r = oldRunWeight K' k z r := by
+  induction k generalizing z with
+  | zero => rfl
+  | succ k ih =>
+    simp only [oldRunWeight]
+    rw [h (k+1) (by omega) (by omega),ih (fun i hi hik => h i hi (by omega))]
+
+theorem oldRunWeight_prod (K : ℕ → W → W → ℝ) (k : ℕ) (z : W) (r : Fin k → W) :
+    oldRunWeight K k z r = ∏ i ∈ Finset.Icc 1 k,
+      K i (oldRunVertices k z r i) (oldRunVertices k z r (i-1)) := by
+  induction k generalizing z with
+  | zero => simp [oldRunWeight]
+  | succ k ih =>
+    rw [Finset.prod_Icc_succ_top (by omega)]
+    have he : (∏ i ∈ Finset.Icc 1 k, K i (oldRunVertices (k+1) z r i)
+        (oldRunVertices (k+1) z r (i-1))) = oldRunWeight K k (r 0) (Fin.tail r) := by
+      rw [ih]
+      apply Finset.prod_congr rfl
+      intro i hi
+      have hi' := Finset.mem_Icc.mp hi
+      rw [oldRunVertices_succ_of_le _ _ _ _ hi'.2,
+        oldRunVertices_succ_of_le _ _ _ _ (by omega)]
+    rw [he,oldRunVertices_last,Nat.add_sub_cancel,
+      oldRunVertices_succ_of_le _ _ _ _ (le_refl _),oldRunVertices_last]
+    exact mul_comm _ _
+
+ 
+
+noncomputable def markedOldRun (K C : ℕ → W → W → ℝ) (k : ℕ) (z : W) (r : Fin k → W) : ℝ :=
+  ∑ i ∈ Finset.Icc 1 k, oldRunWeight (fun j a b => if j = i then C j a b else K j a b) k z r
+
+theorem markedOldRun_nonneg (K C : ℕ → W → W → ℝ)
+    (hK : ∀ i a b, 0 ≤ K i a b) (hC : ∀ i a b, 0 ≤ C i a b)
+    (k : ℕ) (z : W) (r : Fin k → W) : 0 ≤ markedOldRun K C k z r := by
+  apply Finset.sum_nonneg
+  intro i _
+  apply oldRunWeight_nonneg
+  intro j a b
+  split_ifs
+  · exact hC j a b
+  · exact hK j a b
+
+theorem markedOldRun_succ (K C : ℕ → W → W → ℝ) (k : ℕ) (z : W) (r : Fin (k+1) → W) :
+    markedOldRun K C (k+1) z r =
+      K (k+1) z (r 0) * markedOldRun K C k (r 0) (Fin.tail r) +
+      C (k+1) z (r 0) * oldRunWeight K k (r 0) (Fin.tail r) := by
+  unfold markedOldRun
+  rw [Finset.sum_Icc_succ_top (by omega),Finset.mul_sum]
+  congr 1
+  · apply Finset.sum_congr rfl
+    intro i hi
+    have hne : k+1 ≠ i := by have := Finset.mem_Icc.mp hi; omega
+    simp only [oldRunWeight,ite_eq_right hne]
+  · simp only [oldRunWeight,ite_true]
+    congr 1
+    apply oldRunWeight_congr
+    intro i hi hik a b
+    exact ite_eq_right (by omega)
+
+theorem sum_markedOldRun_succ [Fintype W] (K C : ℕ → W → W → ℝ) (k : ℕ) (z : W) :
+    (∑ r : Fin (k+1) → W, markedOldRun K C (k+1) z r) =
+      ∑ b : W, (K (k+1) z b * (∑ r : Fin k → W, markedOldRun K C k b r) +
+        C (k+1) z b * (∑ r : Fin k → W, oldRunWeight K k b r)) := by
+  rw [Fintype.sum_equiv (oldRunConsEquiv k) _
+    (fun r => K (k+1) z r.1 * markedOldRun K C k r.1 r.2 + C (k+1) z r.1 * oldRunWeight K k r.1 r.2)
+      (by intro r; exact markedOldRun_succ K C k z r),Fintype.sum_prod_type]
+  simp only [Finset.sum_add_distrib,Finset.mul_sum]
+
+ 
+theorem oldRunWeight_integral_indexed (Q : Finset E) (p : ℝ)
+    (q : ℕ → Set E → W → W → ℝ) (k : ℕ) (z : W) (r : Fin k → W) :
+    finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η => oldRunWeight (fun i => q i (η i)) k z r) =
+      oldRunWeight (fun i a b => finiteFieldIntegral Q p (fun ω => q i ω a b)) k z r := by
+  calc
+    _ = finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η =>
+        ∏ i : ↥(Finset.Icc 1 k), q i (η i) (oldRunVertices k z r i)
+          (oldRunVertices k z r (i.val-1))) := by
+      congr 1
+      funext η
+      rw [oldRunWeight_prod]
+      exact (Finset.prod_coe_sort (Finset.Icc 1 k)
+        (fun i => q i (η i) (oldRunVertices k z r i) (oldRunVertices k z r (i-1)))).symm
+    _ = ∏ i : ↥(Finset.Icc 1 k), finiteFieldIntegral Q p (fun ω =>
+        q i ω (oldRunVertices k z r i) (oldRunVertices k z r (i.val-1))) :=
+      finiteFamilyIntegral_prod Q (Finset.Icc 1 k) p
+        (fun i ω => q i ω (oldRunVertices k z r i) (oldRunVertices k z r (i.val-1)))
+    _ = _ := by
+      rw [oldRunWeight_prod]
+      exact Finset.prod_coe_sort (Finset.Icc 1 k) (fun i => finiteFieldIntegral Q p
+        (fun ω => q i ω (oldRunVertices k z r i) (oldRunVertices k z r (i-1))))
+
+theorem markedOldRun_integral (Q : Finset E) (p : ℝ)
+    (q c : ℕ → Set E → W → W → ℝ) (k : ℕ) (z : W) (r : Fin k → W) :
+    finiteFamilyIntegral Q (Finset.Icc 1 k) p
+      (fun η => markedOldRun (fun i => q i (η i)) (fun i => c i (η i)) k z r) =
+      markedOldRun (fun i a b => finiteFieldIntegral Q p (fun ω => q i ω a b))
+        (fun i a b => finiteFieldIntegral Q p (fun ω => c i ω a b)) k z r := by
+  unfold markedOldRun
+  rw [finiteFamilyIntegral_sum]
+  apply Finset.sum_congr rfl
+  intro i _
+  change finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η =>
+      oldRunWeight (fun j => (fun ω a b => if j=i then c j ω a b else q j ω a b) (η j)) k z r) = _
+  rw [oldRunWeight_integral_indexed Q p (fun j ω a b => if j=i then c j ω a b else q j ω a b) k z r]
+  congr 1
+  funext j a b
+  by_cases h : j = i <;> simp only [h,ite_true,ite_false]
+
+ 
+
+theorem trajectory_indicator_le_marked (G : BondGraph V E) (e : W → V)
+    (q : Set E → W → W → ℝ) (hq : ∀ ω x y, 0 ≤ q ω x y)
+    (k : ℕ) (η : ℕ → Set E) (z : W) (r : Fin k → W) (a : V) :
+    (if a ∈ trajectoryUnion G η (e ∘ oldRunVertices k z r) k then
+      oldRunWeight (fun i => q (η i)) k z r else 0) ≤
+      markedOldRun (fun i => q (η i))
+        (fun i x y => if a ∈ cluster G (η i) (e x) then q (η i) x y else 0) k z r := by
+  have hC : ∀ i x y, 0 ≤ (if a ∈ cluster G (η i) (e x) then q (η i) x y else 0) := by
+    intro i x y; split_ifs; exact hq _ _ _; exact le_refl _
+  by_cases ha : a ∈ trajectoryUnion G η (e ∘ oldRunVertices k z r) k
+  · rw [ite_eq_left ha]
+    obtain ⟨i,hi,hik,hai⟩ := (mem_trajectoryUnion G η (e ∘ oldRunVertices k z r) k a).mp ha
+    have he : oldRunWeight (fun j x y => if j=i then
+        (if a ∈ cluster G (η j) (e x) then q (η j) x y else 0) else q (η j) x y) k z r =
+        oldRunWeight (fun j => q (η j)) k z r := by
+      simp_rw [oldRunWeight_prod]
+      apply Finset.prod_congr rfl
+      intro j _
+      by_cases hji : j = i
+      · subst j
+        simp only [ite_true,Function.comp_apply] at hai ⊢
+        exact ite_eq_left hai
+      · exact ite_eq_right hji
+    rw [← he]
+    unfold markedOldRun
+    apply Finset.single_le_sum _ (Finset.mem_Icc.mpr ⟨hi,hik⟩)
+    intro j _
+    apply oldRunWeight_nonneg
+    intro l x y
+    split_ifs
+    · exact hq _ _ _
+    · exact le_refl _
+    · exact hq _ _ _
+  · rw [ite_eq_right ha]
+    exact markedOldRun_nonneg _ _ (fun _ => hq _) hC k z r
+
+end MarkedOldRun
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section TrajectoryEnvelopeDomination
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem finiteTrajectoryMass_le_marked {W : Type*} [Fintype W]
+    (G : BondGraph V E) (e : W → V) (Q : Finset E) (p : unitInterval) (k : ℕ)
+    (q : Set E → W → W → ℝ) (hq : ∀ ω x y, 0 ≤ q ω x y) (z : W) (a : V) :
+    finiteTrajectoryMass G e Q p k q z a ≤
+      ∑ r : Fin k → W, markedOldRun (fun _ => finiteOldKernel Q p q)
+        (fun _ x y => finiteFieldIntegral Q p (fun ω => if a ∈ cluster G ω (e x) then q ω x y else 0)) k z r := by
+  unfold finiteTrajectoryMass
+  calc
+    _ ≤ finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η => ∑ r : Fin k → W,
+        markedOldRun (fun i => q (η i))
+          (fun i x y => if a ∈ cluster G (η i) (e x) then q (η i) x y else 0) k z r) :=
+      finiteFamilyIntegral_mono Q (Finset.Icc 1 k) p _ _ (fun η => Finset.sum_le_sum (fun r _ =>
+        trajectory_indicator_le_marked G e q hq k η z r a))
+    _ = _ := by
+      rw [finiteFamilyIntegral_sum]
+      apply Finset.sum_congr rfl
+      intro r _
+      exact markedOldRun_integral Q p (fun _ => q)
+        (fun _ ω x y => if a ∈ cluster G ω (e x) then q ω x y else 0) k z r
+
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
+
+ 
+
+theorem finite_marked_step_row_le (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
+    (Q : Finset E) (B : Finset V) (p : unitInterval) (x a : V)
+    (hQ : incidentSupport G hloc (finite_reachBall hloc x (max q.radius M)).toFinset ⊆ Q) :
+    (∑ y : ↥B, finiteFieldIntegral Q p (fun ω =>
+      if a ∈ cluster G ω x then q.mass ω x y.val else 0)) ≤ q.clusterVisitKernel p x a := by
+  rw [← finiteFieldIntegral_sum]
+  have he : q.clusterVisitKernel p x a = finiteFieldIntegral Q p (fun ω => q.clusterVisitMass ω x a) :=
+    integral_finite_local Q p _ (finite_local_of_subset _ Q hQ _ (q.clusterVisitMass_local · x a))
+  rw [he]
+  apply Finset.sum_le_sum
+  intro ω _
+  apply mul_le_mul_of_nonneg_left _ (finiteFieldWeight_nonneg Q p ω)
+  dsimp only
+  by_cases ha : a ∈ cluster G (finiteFieldSet Q ω) x
+  · simp only [ha,ite_true,clusterVisitMass]
+    rw [Finset.sum_coe_sort B (fun y => q.mass (finiteFieldSet Q ω) x y)]
+    exact (q.summable_mass _ x).sum_le_tsum B (fun y _ => q.nonneg _ x y)
+  · simp only [ha,ite_false,clusterVisitMass,Finset.sum_const_zero,le_refl]
+
+ 
+
+theorem sum_markedOldRun_le_envelope (o : V)
+    (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
+    (Q : Finset E) (B : Finset V) (p : unitInterval) (k : ℕ) (z : ↥B) (a : V)
+    (hQ : ∀ x ∈ B, incidentSupport G hloc (finite_reachBall hloc x (max q.radius M)).toFinset ⊆ Q) :
+    (∑ r : Fin k → ↥B, markedOldRun
+      (fun _ => finiteOldKernel Q p (fun ω (x y : ↥B) => q.mass ω x.val y.val))
+      (fun _ (x y : ↥B) => finiteFieldIntegral Q p (fun ω =>
+        if a ∈ cluster G ω x.val then q.mass ω x.val y.val else 0)) k z r) ≤
+      q.visitEnvelope o p k z.val a := by
+  let u := q.successProbability p o
+  let K : ↥B → ↥B → ℝ := finiteOldKernel Q p (fun ω (x y : ↥B) => q.mass ω x.val y.val)
+  let C : ↥B → ↥B → ℝ := fun x y => finiteFieldIntegral Q p (fun ω =>
+    if a ∈ cluster G ω x.val then q.mass ω x.val y.val else 0)
+  have hK (x y : ↥B) : K x y = q.kernel p x.val y.val :=
+    q.finiteOldKernel_eq_kernel Q p x.val y.val
+      ((incidentSupport_ball_mono G hloc x.val (le_max_left _ _)).trans (hQ x.val x.property))
+  have hKn (x y : ↥B) : 0 ≤ K x y := by rw [hK]; exact q.kernel_nonneg p _ _
+  have hCn (x y : ↥B) : 0 ≤ C x y := by
+    apply finiteFieldIntegral_nonneg
+    intro ω; split_ifs; exact q.nonneg ω _ _; exact le_refl _
+  have hKr (x : ↥B) : (∑ y, K x y) ≤ u := by
+    simp only [hK]
+    rw [Finset.sum_coe_sort B (fun y => q.kernel p x.val y)]
+    exact ((q.summable_kernel p x.val).sum_le_tsum B (fun y _ => q.kernel_nonneg p x.val y)).trans
+      (q.kernel_row_le_success o p x.val)
+  have hCr (x : ↥B) : (∑ y, C x y) ≤ q.clusterVisitKernel p x.val a :=
+    q.finite_marked_step_row_le o Q B p x.val a (hQ x.val x.property)
+  change (∑ r : Fin k → ↥B, markedOldRun (fun _ => K) (fun _ => C) k z r) ≤ _
+  induction k generalizing z with
+  | zero => simp [markedOldRun,visitEnvelope]
+  | succ k ih =>
+    rw [sum_markedOldRun_succ]
+    have hs : Summable (fun y => q.kernel p z.val y * q.visitEnvelope o p k y a) := by
+      apply summable_of_ne_finset_zero (s := (finite_reachBall hloc z.val M).toFinset)
+      intro y hy
+      rw [q.kernel_outside_ball_zero p z.val y (by simpa only [Set.Finite.mem_toFinset] using hy),zero_mul]
+    calc
+      _ ≤ ∑ y : ↥B, (K z y * q.visitEnvelope o p k y.val a + C z y * u^k) :=
+        Finset.sum_le_sum fun y _ => add_le_add
+          (mul_le_mul_of_nonneg_left (ih y) (hKn z y))
+          (mul_le_mul_of_nonneg_left (sum_oldRunWeight_le (fun _ => K) (fun _ => hKn)
+            u (q.successProbability_nonneg p o) (fun _ => hKr) k y) (hCn z y))
+      _ = (∑ y : ↥B, K z y * q.visitEnvelope o p k y.val a) + (∑ y : ↥B, C z y)*u^k := by
+        rw [Finset.sum_add_distrib,Finset.sum_mul]
+      _ ≤ (∑' y, q.kernel p z.val y * q.visitEnvelope o p k y a) + q.clusterVisitKernel p z.val a * u^k := by
+        apply add_le_add
+        · simp only [hK]
+          rw [Finset.sum_coe_sort B (fun y => q.kernel p z.val y * q.visitEnvelope o p k y a)]
+          exact hs.sum_le_tsum B (fun y _ => mul_nonneg (q.kernel_nonneg p z.val y) (q.visitEnvelope_nonneg o p k y a))
+        · exact mul_le_mul_of_nonneg_right (hCr z) (pow_nonneg (q.successProbability_nonneg p o) _)
+      _ = q.visitEnvelope o p (k+1) z.val a := by rw [visitEnvelope]; ring
+
+theorem finiteTrajectoryMass_le_envelope (o : V)
+    (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
+    (Q : Finset E) (B : Finset V) (p : unitInterval) (k : ℕ) (z : ↥B) (a : V)
+    (hQ : ∀ x ∈ B, incidentSupport G hloc (finite_reachBall hloc x (max q.radius M)).toFinset ⊆ Q) :
+    finiteTrajectoryMass G Subtype.val Q p k (fun ω (x y : ↥B) => q.mass ω x.val y.val) z a ≤
+      q.visitEnvelope o p k z.val a :=
+  (finiteTrajectoryMass_le_marked G Subtype.val Q p k _ (fun ω x y => q.nonneg ω x.val y.val) z a).trans
+    (q.sum_markedOldRun_le_envelope o Q B p k z a hQ)
+
+end LocalEndpointRule
+end TrajectoryEnvelopeDomination
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Kernel/Bounds.lean b/lean/OAI/Probability/UnimodularPercolation/Kernel/Bounds.lean
index 25acc230afee3f4010badc5083f4aba83837a44b..8e50a57b88d5b2a1d273a6ba0c6cab503b11fc6a 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Kernel/Bounds.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Kernel/Bounds.lean
@@ -1,482 +1,482 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Graph.Basic
-
-namespace OAI
-
-/-! Kernel domination, monotone couplings and endpoint bounds. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-section PositiveColumn
-attribute [local instance] Classical.propDecidable
-
- 
-
-theorem no_positive_lower_bound [Infinite V] (f : Hilbert V) {c : ℝ}
-    (hc : 0 < c) (hf : ∀ x, c ≤ ‖f x‖) : False := by
-  have hs : Summable (fun x => ‖f x‖ ^ (2 : ℕ)) := by
-    simpa using (lp.hasSum_norm (by norm_num : 0 < (2 : ℝ≥0∞).toReal) f).summable
-  have hconst : Summable (fun _ : V => c ^ 2) :=
-    Summable.of_nonneg_of_le (fun _ => sq_nonneg c)
-      (fun x => pow_le_pow_left₀ hc.le (hf x) 2) hs
-  have hz : c ^ 2 = 0 := (summable_const_iff _).mp hconst
-  nlinarith
-
- 
-
-theorem not_operatorBounded_of_positive_column [Infinite V]
-    (G : BondGraph V E) (p : unitInterval) (y : V) {c : ℝ}
-    (hc : 0 < c) (hcol : ∀ x, c ≤ twoPoint G p x y) :
-    ¬ OperatorBounded G p := by
-  rintro ⟨A, hA⟩
-  apply no_positive_lower_bound (A (lp.single 2 y (1 : ℂ))) hc
-  intro x
-  rw [hA x y, Complex.norm_real, Real.norm_eq_abs,
-    abs_of_nonneg (twoPoint_nonneg G p x y)]
-  exact hcol x
-
-@[simp] theorem not_operatorBounded_one [Infinite V] (G : BondGraph V E)
-    (h : Connected G) : ¬ OperatorBounded G 1 := by
-  apply not_operatorBounded_of_positive_column G 1 (Classical.arbitrary V)
-    (c := 1) (by norm_num)
-  intro x
-  rw [twoPoint_one G h]
-
-@[simp] theorem operatorNorm_one [Infinite V] (G : BondGraph V E)
-    (h : Connected G) : operatorNorm G 1 = ⊤ :=
-  (operatorNorm_eq_top_iff G 1).mpr (not_operatorBounded_one G h)
-
-end PositiveColumn
-
-section KernelLimits
-open scoped Topology
-open Filter
-attribute [local instance] Classical.propDecidable
-
- 
-noncomputable def finiteEmbedding : (V →₀ ℂ) →ₗ[ℂ] Hilbert V :=
-  Finsupp.linearCombination ℂ (fun y => lp.single 2 y (1 : ℂ))
-
-@[simp] theorem finiteEmbedding_single (y : V) (c : ℂ) :
-    finiteEmbedding (Finsupp.single y c) = lp.single 2 y c := by
-  simp [finiteEmbedding, ← lp.single_smul]
-
-theorem finiteEmbedding_dense : DenseRange (finiteEmbedding (V := V)) := by
-  intro f
-  apply isClosed_closure.mem_of_tendsto (lp.hasSum_single (by norm_num) f)
-  apply Eventually.of_forall
-  intro s
-  apply subset_closure
-  refine ⟨∑ y ∈ s, Finsupp.single y (f y), ?_⟩
-  simp
-
- 
-
-theorem exists_operator_of_matrix_tendsto
-    (A : ℕ → Hilbert V →L[ℂ] Hilbert V) (K : V → V → ℂ)
-    {M : ℝ} (hM : 0 ≤ M) (hA : ∀ n, ‖A n‖ ≤ M)
-    (hlim : ∀ x y, Tendsto (fun n => A n (lp.single 2 y (1 : ℂ)) x)
-      atTop (𝓝 (K x y))) :
-    ∃ B : Hilbert V →L[ℂ] Hilbert V,
-      ‖B‖ ≤ M ∧ ∀ x y, B (lp.single 2 y (1 : ℂ)) x = K x y := by
-  have hn (n : ℕ) (f : Hilbert V) : ‖A n f‖ ≤ M * ‖f‖ :=
-    (A n).le_opNorm f |>.trans (mul_le_mul_of_nonneg_right (hA n) (norm_nonneg _))
-  have hcol (y : V) : Memℓp (fun x => K x y) 2 := by
-    apply lp.memℓp_of_tendsto (l := atTop) (F := fun n => A n (lp.single 2 y (1 : ℂ)))
-    · apply isBounded_iff_forall_norm_le.mpr
-      refine ⟨M, ?_⟩
-      rintro _ ⟨n, rfl⟩
-      simpa [lp.norm_single] using hn n (lp.single 2 y (1 : ℂ))
-    · exact tendsto_pi_nhds.mpr (fun x => hlim x y)
-  let col (y : V) : Hilbert V := ⟨fun x => K x y, hcol y⟩
-  let L : (V →₀ ℂ) →ₗ[ℂ] Hilbert V := Finsupp.linearCombination ℂ col
-  have hLlim (f : V →₀ ℂ) :
-      Tendsto (fun n => (A n (finiteEmbedding f) : V → ℂ)) atTop
-        (𝓝 (L f : V → ℂ)) := by
-    apply tendsto_pi_nhds.mpr
-    intro x
-    simp only [finiteEmbedding, L, Finsupp.linearCombination_apply, Finsupp.sum,
-      map_sum, map_smul, lp.coeFn_sum, Finset.sum_apply, lp.coeFn_smul, Pi.smul_apply,
-      smul_eq_mul]
-    exact tendsto_finsetSum _ (fun y _ => (hlim x y).const_mul (f y))
-  have hLn (f : V →₀ ℂ) : ‖L f‖ ≤ M * ‖finiteEmbedding f‖ := by
-    exact lp.norm_le_of_tendsto (Eventually.of_forall (fun n => hn n _)) (hLlim f)
-  refine ⟨L.extendOfNorm finiteEmbedding,
-    LinearMap.opNorm_extendOfNorm_le finiteEmbedding_dense hM hLn, ?_⟩
-  intro x y
-  have h := LinearMap.extendOfNorm_eq (f := L) finiteEmbedding_dense ⟨M, hLn⟩
-    (Finsupp.single y (1 : ℂ))
-  simp only [finiteEmbedding_single] at h
-  rw [h]
-  simp [L, col]
-
-end KernelLimits
-
-section Measurability
-
- 
-
-theorem measurableSet_conn [Countable V] [Countable E]
-    (G : BondGraph V E) (x y : V) : MeasurableSet {ω | Conn G ω x y} := by
-  classical
-  have hedge (a b : V) : MeasurableSet {ω : Set E | ∃ e ∈ ω, G e = s(a, b)} := by
-    simp only [Set.ofPred_exists]
-    apply MeasurableSet.iUnion
-    intro e
-    by_cases he : G e = s(a, b)
-    · simpa [he] using (measurableSet_setOfPred.mpr (measurable_set_mem e))
-    · simp [he]
-  have hlist (l : List V) : MeasurableSet {ω : Set E |
-      l.IsChain (fun a b => ∃ e ∈ ω, G e = s(a, b))} := by
-    induction l with
-    | nil => simp
-    | cons a l ih =>
-      cases l with
-      | nil => simp
-      | cons b l =>
-        simpa only [List.isChain_cons_cons, Set.ofPred_and] using (hedge a b).inter ih
-  have he : {ω | Conn G ω x y} = ⋃ l : List V,
-      {ω | (x :: l).IsChain (fun a b => ∃ e ∈ ω, G e = s(a, b)) ∧
-        (x :: l).getLast (by simp) = y} := by
-    ext ω
-    simp only [Set.mem_ofPred_eq, Set.mem_iUnion]
-    exact ⟨List.exists_isChain_cons_of_relationReflTransGen,
-      fun ⟨l, hl, hy⟩ => List.relationReflTransGen_of_exists_isChain_cons l hl hy⟩
-  rw [he]
-  apply MeasurableSet.iUnion
-  intro l
-  by_cases hlast : (x :: l).getLast (by simp) = y
-  · simpa [hlast] using hlist (x :: l)
-  · simp [hlast]
-
-end Measurability
-
-section Coupling
-
-instance labelLaw_isProbabilityMeasure :
-    IsProbabilityMeasure (labelLaw : Measure (E → unitInterval)) := by
-  dsimp [labelLaw]
-  infer_instance
-
-theorem measurable_threshold (p : unitInterval) :
-    Measurable (fun t : unitInterval => t ≤ p) :=
-  measurableSet_setOfPred.mp measurableSet_Iic
-
- 
-theorem map_threshold (p : unitInterval) :
-    (volume : Measure unitInterval).map (fun t => t ≤ p) =
-      unitInterval.toNNReal p • Measure.dirac True +
-      unitInterval.toNNReal (unitInterval.symm p) • Measure.dirac False := by
-  apply Measure.ext_of_singleton
-  intro b
-  by_cases hb : b
-  · have he : b = True := propext (iff_true_intro hb)
-    subst b
-    simp [Measure.map_apply (measurable_threshold p), Set.preimage]
-    change volume (Set.Iic p) = _
-    rw [unitInterval.volume_Iic, ← unitInterval.coe_toNNReal p,
-      ENNReal.ofReal_coe_nnreal]
-  · have he : b = False := propext (iff_false_intro hb)
-    subst b
-    simp [Measure.map_apply (measurable_threshold p), Set.preimage]
-    change volume (Set.Ioi p) = _
-    rw [unitInterval.volume_Ioi, ← unitInterval.coe_symm_eq,
-      ← unitInterval.coe_toNNReal (unitInterval.symm p), ENNReal.ofReal_coe_nnreal]
-
- 
-theorem map_openEdges (p : unitInterval) :
-    (labelLaw : Measure (E → unitInterval)).map (fun labels => openEdges labels p) =
-      law p := by
-  unfold law
-  rw [setBernoulli_eq_map]
-  have hmeas : Measurable (fun labels : E → unitInterval => fun e => labels e ≤ p) :=
-    Measurable.of_eval (fun e => (measurable_threshold p).comp (measurable_pi_apply e))
-  have he : (fun labels : E → unitInterval => openEdges labels p) =
-      (fun f : E → Prop => {e | f e}) ∘ (fun labels e => labels e ≤ p) := rfl
-  rw [he, ← Measure.map_map measurable_setOfPred hmeas]
-  unfold labelLaw
-  rw [Measure.infinitePi_map_pi _ (fun _ => measurable_threshold p)]
-  simp only [map_threshold, Set.mem_univ, bernoulliMeasure_def]
-
-end Coupling
-
-section KernelDomination
-attribute [local instance] Classical.propDecidable
-
-@[simp] theorem finiteEmbedding_apply (f : V →₀ ℂ) (x : V) :
-    finiteEmbedding f x = f x := by
-  simp only [finiteEmbedding, Finsupp.linearCombination_apply, Finsupp.sum,
-    ← lp.single_smul, lp.coeFn_sum, lp.coeFn_single]
-  rw [Finset.sum_apply, Finset.sum_pi_single]
-  split_ifs with h
-  · simp
-  · exact (Finsupp.notMem_support_iff.mp h).symm
-
- 
-noncomputable def positivePart (f : Hilbert V) : Hilbert V :=
-  ⟨fun x => (‖f x‖ : ℂ), (lp.memℓp f).mono' (by intro x; simp)⟩
-
-@[simp] theorem positivePart_apply (f : Hilbert V) (x : V) :
-    positivePart f x = (‖f x‖ : ℂ) := rfl
-
-@[simp] theorem norm_positivePart (f : Hilbert V) : ‖positivePart f‖ = ‖f‖ := by
-  apply le_antisymm
-  · apply lp.norm_mono (by norm_num : (2 : ℝ≥0∞) ≠ 0)
-    intro x
-    simp
-  · apply lp.norm_mono (by norm_num : (2 : ℝ≥0∞) ≠ 0)
-    intro x
-    simp
-
- 
-
-theorem exists_operator_of_matrix_domination
-    (U K : V → V → ℝ) (hU : ∀ x y, 0 ≤ U x y)
-    (hKU : ∀ x y, |K x y| ≤ U x y)
-    (B : Hilbert V →L[ℂ] Hilbert V)
-    (hB : ∀ x y, B (lp.single 2 y (1 : ℂ)) x = (U x y : ℂ)) :
-    ∃ A : Hilbert V →L[ℂ] Hilbert V, ‖A‖ ≤ ‖B‖ ∧
-      ∀ x y, A (lp.single 2 y (1 : ℂ)) x = (K x y : ℂ) := by
-  have hcol (y : V) : Memℓp (fun x => (K x y : ℂ)) 2 := by
-    apply (lp.memℓp (B (lp.single 2 y (1 : ℂ)))).mono'
-    intro x
-    simpa [hB x y, Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg (hU x y)]
-      using hKU x y
-  let col (y : V) : Hilbert V := ⟨fun x => (K x y : ℂ), hcol y⟩
-  let L : (V →₀ ℂ) →ₗ[ℂ] Hilbert V := Finsupp.linearCombination ℂ col
-  have hLn (f : V →₀ ℂ) : ‖L f‖ ≤ ‖B‖ * ‖finiteEmbedding f‖ := by
-    let g : V →₀ ℂ := f.mapRange (fun z => (‖z‖ : ℂ)) (by simp)
-    have hg : finiteEmbedding g = positivePart (finiteEmbedding f) := by
-      apply lp.ext
-      funext x
-      simp [g]
-    have hpoint (x : V) : ‖L f x‖ ≤ ‖B (finiteEmbedding g) x‖ := by
-      have hright : B (finiteEmbedding g) x =
-          (∑ y ∈ f.support, ‖f y‖ * U x y : ℝ) := by
-        simp only [finiteEmbedding, g, Finsupp.linearCombination_apply]
-        rw [Finsupp.sum_mapRange_index (by simp)]
-        simp only [Finsupp.sum, map_sum, map_smul, lp.coeFn_sum, Finset.sum_apply,
-          lp.coeFn_smul, Pi.smul_apply, smul_eq_mul, hB, Complex.ofReal_sum,
-          Complex.ofReal_mul]
-      have hnn : 0 ≤ ∑ y ∈ f.support, ‖f y‖ * U x y :=
-        Finset.sum_nonneg (fun y _ => mul_nonneg (norm_nonneg _) (hU x y))
-      rw [hright, Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg hnn]
-      simp only [L, Finsupp.linearCombination_apply, Finsupp.sum,
-        lp.coeFn_sum, Finset.sum_apply, lp.coeFn_smul, Pi.smul_apply, smul_eq_mul]
-      calc
-        ‖∑ y ∈ f.support, f y * col y x‖ ≤
-            ∑ y ∈ f.support, ‖f y * col y x‖ := norm_sum_le _ _
-        _ ≤ ∑ y ∈ f.support, ‖f y‖ * U x y := by
-          apply Finset.sum_le_sum
-          intro y _
-          simpa [col, norm_mul, Complex.norm_real, Real.norm_eq_abs] using
-            mul_le_mul_of_nonneg_left (hKU x y) (norm_nonneg (f y))
-    calc
-      ‖L f‖ ≤ ‖B (finiteEmbedding g)‖ := lp.norm_mono (by norm_num) hpoint
-      _ ≤ ‖B‖ * ‖finiteEmbedding g‖ := B.le_opNorm _
-      _ = ‖B‖ * ‖finiteEmbedding f‖ := by rw [hg, norm_positivePart]
-  refine ⟨L.extendOfNorm finiteEmbedding,
-    LinearMap.opNorm_extendOfNorm_le finiteEmbedding_dense (norm_nonneg _) hLn, ?_⟩
-  intro x y
-  have h := LinearMap.extendOfNorm_eq (f := L) finiteEmbedding_dense ⟨‖B‖, hLn⟩
-    (Finsupp.single y (1 : ℂ))
-  simp only [finiteEmbedding_single] at h
-  rw [h]
-  simp [L, col]
-
-end KernelDomination
-
-section CouplingOrder
-variable [Countable V] [Countable E] (G : BondGraph V E)
-
-omit [Countable E] in
-theorem measurable_openEdges (p : unitInterval) :
-    Measurable (fun labels : E → unitInterval => openEdges labels p) :=
-  measurable_setOfPred.comp (Measurable.of_eval
-    (fun e => (measurable_threshold p).comp (measurable_pi_apply e)))
-
-theorem twoPoint_eq_labelLaw (p : unitInterval) (x y : V) :
-    twoPoint G p x y =
-      (labelLaw {labels : E → unitInterval | Conn G (openEdges labels p) x y}).toReal := by
-  unfold twoPoint
-  rw [← map_openEdges p, Measure.map_apply (measurable_openEdges p)
-    (measurableSet_conn G x y)]
-  rfl
-
-omit [Countable E] in
-theorem openEdges_mono (labels : E → unitInterval) : Monotone (openEdges labels) := by
-  intro p q hpq e he
-  exact he.trans hpq
-
-theorem twoPoint_mono (x y : V) : Monotone (fun p => twoPoint G p x y) := by
-  intro p q hpq
-  change twoPoint G p x y ≤ twoPoint G q x y
-  rw [twoPoint_eq_labelLaw, twoPoint_eq_labelLaw]
-  apply ENNReal.toReal_mono (measure_ne_top _ _)
-  apply measure_mono
-  intro labels h
-  exact conn_mono G (openEdges_mono labels hpq) h
-
-end CouplingOrder
-
-section NormOrder
-attribute [local instance] Classical.propDecidable
-variable [Countable V] [Countable E] (G : BondGraph V E)
-
-theorem operatorBounded_of_le {p q : unitInterval} (hpq : p ≤ q)
-    (hq : OperatorBounded G q) : OperatorBounded G p := by
-  obtain ⟨B, hB⟩ := hq
-  obtain ⟨A, _, hA⟩ := exists_operator_of_matrix_domination
-    (twoPoint G q) (twoPoint G p) (twoPoint_nonneg G q)
-    (fun x y => (abs_of_nonneg (twoPoint_nonneg G p x y)).le.trans
-      (twoPoint_mono G x y hpq)) B hB
-  exact ⟨A, hA⟩
-
-theorem operatorNorm_mono : Monotone (operatorNorm G) := by
-  intro p q hpq
-  by_cases hq : OperatorBounded G q
-  · obtain ⟨B, hB⟩ := hq
-    obtain ⟨A, hnorm, hA⟩ := exists_operator_of_matrix_domination
-      (twoPoint G q) (twoPoint G p) (twoPoint_nonneg G q)
-      (fun x y => (abs_of_nonneg (twoPoint_nonneg G p x y)).le.trans
-        (twoPoint_mono G x y hpq)) B hB
-    rw [operatorNorm_of_hasKernel hA, operatorNorm_of_hasKernel hB]
-    exact ENNReal.ofReal_le_ofReal hnorm
-  · rw [(operatorNorm_eq_top_iff G q).mpr hq]
-    exact le_top
-
-end NormOrder
-
-section CouplingLimits
-open Filter Topology
-
-theorem conn_iUnion_iff (G : BondGraph V E) (ω : ℕ → Set E) (hω : Monotone ω)
-    (x y : V) : Conn G (⋃ n, ω n) x y ↔ ∃ n, Conn G (ω n) x y := by
-  constructor
-  · intro h
-    induction h with
-    | refl => exact ⟨0, conn_refl G _ _⟩
-    | @tail y z _ hyz ih =>
-      obtain ⟨m, hm⟩ := ih
-      obtain ⟨e, he, heq⟩ := hyz
-      obtain ⟨n, hn⟩ := Set.mem_iUnion.mp he
-      exact ⟨max m n, (conn_mono G (hω (le_max_left _ _)) hm).tail
-        ⟨e, hω (le_max_right _ _) hn, heq⟩⟩
-  · rintro ⟨n, hn⟩
-    exact conn_mono G (Set.subset_iUnion ω n) hn
-
- 
-theorem ae_labels_ne [Countable E] (q : unitInterval) :
-    ∀ᵐ labels ∂(labelLaw : Measure (E → unitInterval)), ∀ e, labels e ≠ q := by
-  rw [ae_all_iff]
-  intro e
-  rw [ae_iff]
-  simp only [not_not]
-  change labelLaw ((fun labels : E → unitInterval => labels e) ⁻¹' {q}) = 0
-  rw [← Measure.map_apply (measurable_pi_apply e) (measurableSet_singleton q)]
-  change (Measure.infinitePi (fun _ : E => (volume : Measure unitInterval))).map
-    (fun labels => labels e) {q} = 0
-  rw [Measure.infinitePi_map_eval]
-  simp
-
-theorem openEdges_eq_iUnion_of_tendsto (labels : E → unitInterval)
-    (q : unitInterval) (hneq : ∀ e, labels e ≠ q) (p : ℕ → unitInterval)
-    (hle : ∀ n, p n ≤ q) (hlim : Tendsto p atTop (𝓝 q)) :
-    openEdges labels q = ⋃ n, openEdges labels (p n) := by
-  ext e
-  constructor
-  · intro he
-    have hlt : labels e < q := lt_of_le_of_ne he (hneq e)
-    obtain ⟨n, hn⟩ := ((tendsto_order.mp hlim).1 _ hlt).exists
-    exact Set.mem_iUnion.mpr ⟨n, hn.le⟩
-  · intro he
-    obtain ⟨n, hn⟩ := Set.mem_iUnion.mp he
-    exact hn.trans (hle n)
-
- 
-
-theorem twoPoint_tendsto_of_monotone [Countable V] [Countable E]
-    (G : BondGraph V E) (p : ℕ → unitInterval) (hp : Monotone p)
-    (q : unitInterval) (hle : ∀ n, p n ≤ q) (hlim : Tendsto p atTop (𝓝 q))
-    (x y : V) : Tendsto (fun n => twoPoint G (p n) x y) atTop (𝓝 (twoPoint G q x y)) := by
-  let s (n : ℕ) : Set (E → unitInterval) := {labels | Conn G (openEdges labels (p n)) x y}
-  have hs : Monotone s := by
-    intro m n hmn labels hl
-    exact conn_mono G (openEdges_mono labels (hp hmn)) hl
-  have he : {labels : E → unitInterval | Conn G (openEdges labels q) x y} =ᵐ[labelLaw]
-      ⋃ n, s n := by
-    filter_upwards [ae_labels_ne (E := E) q] with labels hlabels
-    apply propext
-    rw [openEdges_eq_iUnion_of_tendsto labels q hlabels p hle hlim, conn_iUnion_iff]
-    · simp [s]
-    · exact fun m n hmn => openEdges_mono labels (hp hmn)
-  rw [twoPoint_eq_labelLaw G q x y, measure_congr he]
-  have ht := (tendsto_measure_iUnion_atTop (μ := (labelLaw : Measure (E → unitInterval))) hs)
-  have hr := (ENNReal.tendsto_toReal (measure_ne_top _ _)).comp ht
-  simpa only [Function.comp_def, s, ← twoPoint_eq_labelLaw] using hr
-
-end CouplingLimits
-
-section Endpoint
-open Filter Topology
-attribute [local instance] Classical.propDecidable
-variable [Countable V] [Countable E] (G : BondGraph V E)
-
- 
-
-theorem operatorNorm_eq_sup_below (q : unitInterval) (hq : 0 < q) :
-    operatorNorm G q = sSup {r | ∃ p : unitInterval, p < q ∧ r = operatorNorm G p} := by
-  let S : ℝ≥0∞ := sSup {r | ∃ p : unitInterval, p < q ∧ r = operatorNorm G p}
-  change operatorNorm G q = S
-  apply le_antisymm
-  · by_cases hS : S = ⊤
-    · rw [hS]
-      exact le_top
-    obtain ⟨p, hp, hpq, hlim⟩ := exists_seq_strictMono_tendsto' hq
-    have hbound (n : ℕ) : operatorNorm G (p n) ≤ S :=
-      le_sSup ⟨p n, (hpq n).2, rfl⟩
-    have hbounded (n : ℕ) : OperatorBounded G (p n) :=
-      (operatorNorm_lt_top_iff G (p n)).mp ((hbound n).trans_lt (lt_top_iff_ne_top.mpr hS))
-    choose A hA using hbounded
-    have hnorm (n : ℕ) : ‖A n‖ ≤ S.toReal := by
-      apply (ENNReal.ofReal_le_iff_le_toReal hS).mp
-      rw [← operatorNorm_of_hasKernel (hA n)]
-      exact hbound n
-    have hmatrix (x y : V) : Tendsto (fun n => A n (lp.single 2 y (1 : ℂ)) x)
-        atTop (𝓝 (twoPoint G q x y : ℂ)) := by
-      have ht := (Complex.continuous_ofReal.tendsto (twoPoint G q x y)).comp
-        (twoPoint_tendsto_of_monotone G p hp.monotone q (fun n => (hpq n).2.le) hlim x y)
-      convert ht using 1
-      funext n
-      exact hA n x y
-    obtain ⟨B, hBn, hB⟩ := exists_operator_of_matrix_tendsto A
-      (fun x y => (twoPoint G q x y : ℂ)) ENNReal.toReal_nonneg hnorm hmatrix
-    rw [operatorNorm_of_hasKernel hB]
-    exact (ENNReal.ofReal_le_iff_le_toReal hS).mpr hBn
-  · apply sSup_le
-    rintro r ⟨p, hp, rfl⟩
-    exact operatorNorm_mono G hp.le
-
-theorem operatorBounded_iff_uniform_below (q : unitInterval) (hq : 0 < q) :
-    OperatorBounded G q ↔ ∃ M : ℝ≥0∞, M < ⊤ ∧ ∀ p < q, operatorNorm G p ≤ M := by
-  rw [← operatorNorm_lt_top_iff]
-  constructor
-  · intro h
-    exact ⟨operatorNorm G q, h, fun p hp => operatorNorm_mono G hp.le⟩
-  · rintro ⟨M, hM, h⟩
-    rw [operatorNorm_eq_sup_below G q hq]
-    apply lt_of_le_of_lt ?_ hM
-    apply sSup_le
-    rintro r ⟨p, hp, rfl⟩
-    exact h p hp
-
-end Endpoint
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Graph.Basic
+
+namespace OAI
+
+/-! Kernel domination, monotone couplings and endpoint bounds. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+section PositiveColumn
+attribute [local instance] Classical.propDecidable
+
+ 
+
+theorem no_positive_lower_bound [Infinite V] (f : Hilbert V) {c : ℝ}
+    (hc : 0 < c) (hf : ∀ x, c ≤ ‖f x‖) : False := by
+  have hs : Summable (fun x => ‖f x‖ ^ (2 : ℕ)) := by
+    simpa using (lp.hasSum_norm (by norm_num : 0 < (2 : ℝ≥0∞).toReal) f).summable
+  have hconst : Summable (fun _ : V => c ^ 2) :=
+    Summable.of_nonneg_of_le (fun _ => sq_nonneg c)
+      (fun x => pow_le_pow_left₀ hc.le (hf x) 2) hs
+  have hz : c ^ 2 = 0 := (summable_const_iff _).mp hconst
+  nlinarith
+
+ 
+
+theorem not_operatorBounded_of_positive_column [Infinite V]
+    (G : BondGraph V E) (p : unitInterval) (y : V) {c : ℝ}
+    (hc : 0 < c) (hcol : ∀ x, c ≤ twoPoint G p x y) :
+    ¬ OperatorBounded G p := by
+  rintro ⟨A, hA⟩
+  apply no_positive_lower_bound (A (lp.single 2 y (1 : ℂ))) hc
+  intro x
+  rw [hA x y, Complex.norm_real, Real.norm_eq_abs,
+    abs_of_nonneg (twoPoint_nonneg G p x y)]
+  exact hcol x
+
+@[simp] theorem not_operatorBounded_one [Infinite V] (G : BondGraph V E)
+    (h : Connected G) : ¬ OperatorBounded G 1 := by
+  apply not_operatorBounded_of_positive_column G 1 (Classical.arbitrary V)
+    (c := 1) (by norm_num)
+  intro x
+  rw [twoPoint_one G h]
+
+@[simp] theorem operatorNorm_one [Infinite V] (G : BondGraph V E)
+    (h : Connected G) : operatorNorm G 1 = ⊤ :=
+  (operatorNorm_eq_top_iff G 1).mpr (not_operatorBounded_one G h)
+
+end PositiveColumn
+
+section KernelLimits
+open scoped Topology
+open Filter
+attribute [local instance] Classical.propDecidable
+
+ 
+noncomputable def finiteEmbedding : (V →₀ ℂ) →ₗ[ℂ] Hilbert V :=
+  Finsupp.linearCombination ℂ (fun y => lp.single 2 y (1 : ℂ))
+
+@[simp] theorem finiteEmbedding_single (y : V) (c : ℂ) :
+    finiteEmbedding (Finsupp.single y c) = lp.single 2 y c := by
+  simp [finiteEmbedding, ← lp.single_smul]
+
+theorem finiteEmbedding_dense : DenseRange (finiteEmbedding (V := V)) := by
+  intro f
+  apply isClosed_closure.mem_of_tendsto (lp.hasSum_single (by norm_num) f)
+  apply Eventually.of_forall
+  intro s
+  apply subset_closure
+  refine ⟨∑ y ∈ s, Finsupp.single y (f y), ?_⟩
+  simp
+
+ 
+
+theorem exists_operator_of_matrix_tendsto
+    (A : ℕ → Hilbert V →L[ℂ] Hilbert V) (K : V → V → ℂ)
+    {M : ℝ} (hM : 0 ≤ M) (hA : ∀ n, ‖A n‖ ≤ M)
+    (hlim : ∀ x y, Tendsto (fun n => A n (lp.single 2 y (1 : ℂ)) x)
+      atTop (𝓝 (K x y))) :
+    ∃ B : Hilbert V →L[ℂ] Hilbert V,
+      ‖B‖ ≤ M ∧ ∀ x y, B (lp.single 2 y (1 : ℂ)) x = K x y := by
+  have hn (n : ℕ) (f : Hilbert V) : ‖A n f‖ ≤ M * ‖f‖ :=
+    (A n).le_opNorm f |>.trans (mul_le_mul_of_nonneg_right (hA n) (norm_nonneg _))
+  have hcol (y : V) : Memℓp (fun x => K x y) 2 := by
+    apply lp.memℓp_of_tendsto (l := atTop) (F := fun n => A n (lp.single 2 y (1 : ℂ)))
+    · apply isBounded_iff_forall_norm_le.mpr
+      refine ⟨M, ?_⟩
+      rintro _ ⟨n, rfl⟩
+      simpa [lp.norm_single] using hn n (lp.single 2 y (1 : ℂ))
+    · exact tendsto_pi_nhds.mpr (fun x => hlim x y)
+  let col (y : V) : Hilbert V := ⟨fun x => K x y, hcol y⟩
+  let L : (V →₀ ℂ) →ₗ[ℂ] Hilbert V := Finsupp.linearCombination ℂ col
+  have hLlim (f : V →₀ ℂ) :
+      Tendsto (fun n => (A n (finiteEmbedding f) : V → ℂ)) atTop
+        (𝓝 (L f : V → ℂ)) := by
+    apply tendsto_pi_nhds.mpr
+    intro x
+    simp only [finiteEmbedding, L, Finsupp.linearCombination_apply, Finsupp.sum,
+      map_sum, map_smul, lp.coeFn_sum, Finset.sum_apply, lp.coeFn_smul, Pi.smul_apply,
+      smul_eq_mul]
+    exact tendsto_finsetSum _ (fun y _ => (hlim x y).const_mul (f y))
+  have hLn (f : V →₀ ℂ) : ‖L f‖ ≤ M * ‖finiteEmbedding f‖ := by
+    exact lp.norm_le_of_tendsto (Eventually.of_forall (fun n => hn n _)) (hLlim f)
+  refine ⟨L.extendOfNorm finiteEmbedding,
+    LinearMap.opNorm_extendOfNorm_le finiteEmbedding_dense hM hLn, ?_⟩
+  intro x y
+  have h := LinearMap.extendOfNorm_eq (f := L) finiteEmbedding_dense ⟨M, hLn⟩
+    (Finsupp.single y (1 : ℂ))
+  simp only [finiteEmbedding_single] at h
+  rw [h]
+  simp [L, col]
+
+end KernelLimits
+
+section Measurability
+
+ 
+
+theorem measurableSet_conn [Countable V] [Countable E]
+    (G : BondGraph V E) (x y : V) : MeasurableSet {ω | Conn G ω x y} := by
+  classical
+  have hedge (a b : V) : MeasurableSet {ω : Set E | ∃ e ∈ ω, G e = s(a, b)} := by
+    simp only [Set.ofPred_exists]
+    apply MeasurableSet.iUnion
+    intro e
+    by_cases he : G e = s(a, b)
+    · simpa [he] using (measurableSet_setOfPred.mpr (measurable_set_mem e))
+    · simp [he]
+  have hlist (l : List V) : MeasurableSet {ω : Set E |
+      l.IsChain (fun a b => ∃ e ∈ ω, G e = s(a, b))} := by
+    induction l with
+    | nil => simp
+    | cons a l ih =>
+      cases l with
+      | nil => simp
+      | cons b l =>
+        simpa only [List.isChain_cons_cons, Set.ofPred_and] using (hedge a b).inter ih
+  have he : {ω | Conn G ω x y} = ⋃ l : List V,
+      {ω | (x :: l).IsChain (fun a b => ∃ e ∈ ω, G e = s(a, b)) ∧
+        (x :: l).getLast (by simp) = y} := by
+    ext ω
+    simp only [Set.mem_ofPred_eq, Set.mem_iUnion]
+    exact ⟨List.exists_isChain_cons_of_relationReflTransGen,
+      fun ⟨l, hl, hy⟩ => List.relationReflTransGen_of_exists_isChain_cons l hl hy⟩
+  rw [he]
+  apply MeasurableSet.iUnion
+  intro l
+  by_cases hlast : (x :: l).getLast (by simp) = y
+  · simpa [hlast] using hlist (x :: l)
+  · simp [hlast]
+
+end Measurability
+
+section Coupling
+
+instance labelLaw_isProbabilityMeasure :
+    IsProbabilityMeasure (labelLaw : Measure (E → unitInterval)) := by
+  dsimp [labelLaw]
+  infer_instance
+
+theorem measurable_threshold (p : unitInterval) :
+    Measurable (fun t : unitInterval => t ≤ p) :=
+  measurableSet_setOfPred.mp measurableSet_Iic
+
+ 
+theorem map_threshold (p : unitInterval) :
+    (volume : Measure unitInterval).map (fun t => t ≤ p) =
+      unitInterval.toNNReal p • Measure.dirac True +
+      unitInterval.toNNReal (unitInterval.symm p) • Measure.dirac False := by
+  apply Measure.ext_of_singleton
+  intro b
+  by_cases hb : b
+  · have he : b = True := propext (iff_true_intro hb)
+    subst b
+    simp [Measure.map_apply (measurable_threshold p), Set.preimage]
+    change volume (Set.Iic p) = _
+    rw [unitInterval.volume_Iic, ← unitInterval.coe_toNNReal p,
+      ENNReal.ofReal_coe_nnreal]
+  · have he : b = False := propext (iff_false_intro hb)
+    subst b
+    simp [Measure.map_apply (measurable_threshold p), Set.preimage]
+    change volume (Set.Ioi p) = _
+    rw [unitInterval.volume_Ioi, ← unitInterval.coe_symm_eq,
+      ← unitInterval.coe_toNNReal (unitInterval.symm p), ENNReal.ofReal_coe_nnreal]
+
+ 
+theorem map_openEdges (p : unitInterval) :
+    (labelLaw : Measure (E → unitInterval)).map (fun labels => openEdges labels p) =
+      law p := by
+  unfold law
+  rw [setBernoulli_eq_map]
+  have hmeas : Measurable (fun labels : E → unitInterval => fun e => labels e ≤ p) :=
+    Measurable.of_eval (fun e => (measurable_threshold p).comp (measurable_pi_apply e))
+  have he : (fun labels : E → unitInterval => openEdges labels p) =
+      (fun f : E → Prop => {e | f e}) ∘ (fun labels e => labels e ≤ p) := rfl
+  rw [he, ← Measure.map_map measurable_setOfPred hmeas]
+  unfold labelLaw
+  rw [Measure.infinitePi_map_pi _ (fun _ => measurable_threshold p)]
+  simp only [map_threshold, Set.mem_univ, bernoulliMeasure_def]
+
+end Coupling
+
+section KernelDomination
+attribute [local instance] Classical.propDecidable
+
+@[simp] theorem finiteEmbedding_apply (f : V →₀ ℂ) (x : V) :
+    finiteEmbedding f x = f x := by
+  simp only [finiteEmbedding, Finsupp.linearCombination_apply, Finsupp.sum,
+    ← lp.single_smul, lp.coeFn_sum, lp.coeFn_single]
+  rw [Finset.sum_apply, Finset.sum_pi_single]
+  split_ifs with h
+  · simp
+  · exact (Finsupp.notMem_support_iff.mp h).symm
+
+ 
+noncomputable def positivePart (f : Hilbert V) : Hilbert V :=
+  ⟨fun x => (‖f x‖ : ℂ), (lp.memℓp f).mono' (by intro x; simp)⟩
+
+@[simp] theorem positivePart_apply (f : Hilbert V) (x : V) :
+    positivePart f x = (‖f x‖ : ℂ) := rfl
+
+@[simp] theorem norm_positivePart (f : Hilbert V) : ‖positivePart f‖ = ‖f‖ := by
+  apply le_antisymm
+  · apply lp.norm_mono (by norm_num : (2 : ℝ≥0∞) ≠ 0)
+    intro x
+    simp
+  · apply lp.norm_mono (by norm_num : (2 : ℝ≥0∞) ≠ 0)
+    intro x
+    simp
+
+ 
+
+theorem exists_operator_of_matrix_domination
+    (U K : V → V → ℝ) (hU : ∀ x y, 0 ≤ U x y)
+    (hKU : ∀ x y, |K x y| ≤ U x y)
+    (B : Hilbert V →L[ℂ] Hilbert V)
+    (hB : ∀ x y, B (lp.single 2 y (1 : ℂ)) x = (U x y : ℂ)) :
+    ∃ A : Hilbert V →L[ℂ] Hilbert V, ‖A‖ ≤ ‖B‖ ∧
+      ∀ x y, A (lp.single 2 y (1 : ℂ)) x = (K x y : ℂ) := by
+  have hcol (y : V) : Memℓp (fun x => (K x y : ℂ)) 2 := by
+    apply (lp.memℓp (B (lp.single 2 y (1 : ℂ)))).mono'
+    intro x
+    simpa [hB x y, Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg (hU x y)]
+      using hKU x y
+  let col (y : V) : Hilbert V := ⟨fun x => (K x y : ℂ), hcol y⟩
+  let L : (V →₀ ℂ) →ₗ[ℂ] Hilbert V := Finsupp.linearCombination ℂ col
+  have hLn (f : V →₀ ℂ) : ‖L f‖ ≤ ‖B‖ * ‖finiteEmbedding f‖ := by
+    let g : V →₀ ℂ := f.mapRange (fun z => (‖z‖ : ℂ)) (by simp)
+    have hg : finiteEmbedding g = positivePart (finiteEmbedding f) := by
+      apply lp.ext
+      funext x
+      simp [g]
+    have hpoint (x : V) : ‖L f x‖ ≤ ‖B (finiteEmbedding g) x‖ := by
+      have hright : B (finiteEmbedding g) x =
+          (∑ y ∈ f.support, ‖f y‖ * U x y : ℝ) := by
+        simp only [finiteEmbedding, g, Finsupp.linearCombination_apply]
+        rw [Finsupp.sum_mapRange_index (by simp)]
+        simp only [Finsupp.sum, map_sum, map_smul, lp.coeFn_sum, Finset.sum_apply,
+          lp.coeFn_smul, Pi.smul_apply, smul_eq_mul, hB, Complex.ofReal_sum,
+          Complex.ofReal_mul]
+      have hnn : 0 ≤ ∑ y ∈ f.support, ‖f y‖ * U x y :=
+        Finset.sum_nonneg (fun y _ => mul_nonneg (norm_nonneg _) (hU x y))
+      rw [hright, Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg hnn]
+      simp only [L, Finsupp.linearCombination_apply, Finsupp.sum,
+        lp.coeFn_sum, Finset.sum_apply, lp.coeFn_smul, Pi.smul_apply, smul_eq_mul]
+      calc
+        ‖∑ y ∈ f.support, f y * col y x‖ ≤
+            ∑ y ∈ f.support, ‖f y * col y x‖ := norm_sum_le _ _
+        _ ≤ ∑ y ∈ f.support, ‖f y‖ * U x y := by
+          apply Finset.sum_le_sum
+          intro y _
+          simpa [col, norm_mul, Complex.norm_real, Real.norm_eq_abs] using
+            mul_le_mul_of_nonneg_left (hKU x y) (norm_nonneg (f y))
+    calc
+      ‖L f‖ ≤ ‖B (finiteEmbedding g)‖ := lp.norm_mono (by norm_num) hpoint
+      _ ≤ ‖B‖ * ‖finiteEmbedding g‖ := B.le_opNorm _
+      _ = ‖B‖ * ‖finiteEmbedding f‖ := by rw [hg, norm_positivePart]
+  refine ⟨L.extendOfNorm finiteEmbedding,
+    LinearMap.opNorm_extendOfNorm_le finiteEmbedding_dense (norm_nonneg _) hLn, ?_⟩
+  intro x y
+  have h := LinearMap.extendOfNorm_eq (f := L) finiteEmbedding_dense ⟨‖B‖, hLn⟩
+    (Finsupp.single y (1 : ℂ))
+  simp only [finiteEmbedding_single] at h
+  rw [h]
+  simp [L, col]
+
+end KernelDomination
+
+section CouplingOrder
+variable [Countable V] [Countable E] (G : BondGraph V E)
+
+omit [Countable E] in
+theorem measurable_openEdges (p : unitInterval) :
+    Measurable (fun labels : E → unitInterval => openEdges labels p) :=
+  measurable_setOfPred.comp (Measurable.of_eval
+    (fun e => (measurable_threshold p).comp (measurable_pi_apply e)))
+
+theorem twoPoint_eq_labelLaw (p : unitInterval) (x y : V) :
+    twoPoint G p x y =
+      (labelLaw {labels : E → unitInterval | Conn G (openEdges labels p) x y}).toReal := by
+  unfold twoPoint
+  rw [← map_openEdges p, Measure.map_apply (measurable_openEdges p)
+    (measurableSet_conn G x y)]
+  rfl
+
+omit [Countable E] in
+theorem openEdges_mono (labels : E → unitInterval) : Monotone (openEdges labels) := by
+  intro p q hpq e he
+  exact he.trans hpq
+
+theorem twoPoint_mono (x y : V) : Monotone (fun p => twoPoint G p x y) := by
+  intro p q hpq
+  change twoPoint G p x y ≤ twoPoint G q x y
+  rw [twoPoint_eq_labelLaw, twoPoint_eq_labelLaw]
+  apply ENNReal.toReal_mono (measure_ne_top _ _)
+  apply measure_mono
+  intro labels h
+  exact conn_mono G (openEdges_mono labels hpq) h
+
+end CouplingOrder
+
+section NormOrder
+attribute [local instance] Classical.propDecidable
+variable [Countable V] [Countable E] (G : BondGraph V E)
+
+theorem operatorBounded_of_le {p q : unitInterval} (hpq : p ≤ q)
+    (hq : OperatorBounded G q) : OperatorBounded G p := by
+  obtain ⟨B, hB⟩ := hq
+  obtain ⟨A, _, hA⟩ := exists_operator_of_matrix_domination
+    (twoPoint G q) (twoPoint G p) (twoPoint_nonneg G q)
+    (fun x y => (abs_of_nonneg (twoPoint_nonneg G p x y)).le.trans
+      (twoPoint_mono G x y hpq)) B hB
+  exact ⟨A, hA⟩
+
+theorem operatorNorm_mono : Monotone (operatorNorm G) := by
+  intro p q hpq
+  by_cases hq : OperatorBounded G q
+  · obtain ⟨B, hB⟩ := hq
+    obtain ⟨A, hnorm, hA⟩ := exists_operator_of_matrix_domination
+      (twoPoint G q) (twoPoint G p) (twoPoint_nonneg G q)
+      (fun x y => (abs_of_nonneg (twoPoint_nonneg G p x y)).le.trans
+        (twoPoint_mono G x y hpq)) B hB
+    rw [operatorNorm_of_hasKernel hA, operatorNorm_of_hasKernel hB]
+    exact ENNReal.ofReal_le_ofReal hnorm
+  · rw [(operatorNorm_eq_top_iff G q).mpr hq]
+    exact le_top
+
+end NormOrder
+
+section CouplingLimits
+open Filter Topology
+
+theorem conn_iUnion_iff (G : BondGraph V E) (ω : ℕ → Set E) (hω : Monotone ω)
+    (x y : V) : Conn G (⋃ n, ω n) x y ↔ ∃ n, Conn G (ω n) x y := by
+  constructor
+  · intro h
+    induction h with
+    | refl => exact ⟨0, conn_refl G _ _⟩
+    | @tail y z _ hyz ih =>
+      obtain ⟨m, hm⟩ := ih
+      obtain ⟨e, he, heq⟩ := hyz
+      obtain ⟨n, hn⟩ := Set.mem_iUnion.mp he
+      exact ⟨max m n, (conn_mono G (hω (le_max_left _ _)) hm).tail
+        ⟨e, hω (le_max_right _ _) hn, heq⟩⟩
+  · rintro ⟨n, hn⟩
+    exact conn_mono G (Set.subset_iUnion ω n) hn
+
+ 
+theorem ae_labels_ne [Countable E] (q : unitInterval) :
+    ∀ᵐ labels ∂(labelLaw : Measure (E → unitInterval)), ∀ e, labels e ≠ q := by
+  rw [ae_all_iff]
+  intro e
+  rw [ae_iff]
+  simp only [not_not]
+  change labelLaw ((fun labels : E → unitInterval => labels e) ⁻¹' {q}) = 0
+  rw [← Measure.map_apply (measurable_pi_apply e) (measurableSet_singleton q)]
+  change (Measure.infinitePi (fun _ : E => (volume : Measure unitInterval))).map
+    (fun labels => labels e) {q} = 0
+  rw [Measure.infinitePi_map_eval]
+  simp
+
+theorem openEdges_eq_iUnion_of_tendsto (labels : E → unitInterval)
+    (q : unitInterval) (hneq : ∀ e, labels e ≠ q) (p : ℕ → unitInterval)
+    (hle : ∀ n, p n ≤ q) (hlim : Tendsto p atTop (𝓝 q)) :
+    openEdges labels q = ⋃ n, openEdges labels (p n) := by
+  ext e
+  constructor
+  · intro he
+    have hlt : labels e < q := lt_of_le_of_ne he (hneq e)
+    obtain ⟨n, hn⟩ := ((tendsto_order.mp hlim).1 _ hlt).exists
+    exact Set.mem_iUnion.mpr ⟨n, hn.le⟩
+  · intro he
+    obtain ⟨n, hn⟩ := Set.mem_iUnion.mp he
+    exact hn.trans (hle n)
+
+ 
+
+theorem twoPoint_tendsto_of_monotone [Countable V] [Countable E]
+    (G : BondGraph V E) (p : ℕ → unitInterval) (hp : Monotone p)
+    (q : unitInterval) (hle : ∀ n, p n ≤ q) (hlim : Tendsto p atTop (𝓝 q))
+    (x y : V) : Tendsto (fun n => twoPoint G (p n) x y) atTop (𝓝 (twoPoint G q x y)) := by
+  let s (n : ℕ) : Set (E → unitInterval) := {labels | Conn G (openEdges labels (p n)) x y}
+  have hs : Monotone s := by
+    intro m n hmn labels hl
+    exact conn_mono G (openEdges_mono labels (hp hmn)) hl
+  have he : {labels : E → unitInterval | Conn G (openEdges labels q) x y} =ᵐ[labelLaw]
+      ⋃ n, s n := by
+    filter_upwards [ae_labels_ne (E := E) q] with labels hlabels
+    apply propext
+    rw [openEdges_eq_iUnion_of_tendsto labels q hlabels p hle hlim, conn_iUnion_iff]
+    · simp [s]
+    · exact fun m n hmn => openEdges_mono labels (hp hmn)
+  rw [twoPoint_eq_labelLaw G q x y, measure_congr he]
+  have ht := (tendsto_measure_iUnion_atTop (μ := (labelLaw : Measure (E → unitInterval))) hs)
+  have hr := (ENNReal.tendsto_toReal (measure_ne_top _ _)).comp ht
+  simpa only [Function.comp_def, s, ← twoPoint_eq_labelLaw] using hr
+
+end CouplingLimits
+
+section Endpoint
+open Filter Topology
+attribute [local instance] Classical.propDecidable
+variable [Countable V] [Countable E] (G : BondGraph V E)
+
+ 
+
+theorem operatorNorm_eq_sup_below (q : unitInterval) (hq : 0 < q) :
+    operatorNorm G q = sSup {r | ∃ p : unitInterval, p < q ∧ r = operatorNorm G p} := by
+  let S : ℝ≥0∞ := sSup {r | ∃ p : unitInterval, p < q ∧ r = operatorNorm G p}
+  change operatorNorm G q = S
+  apply le_antisymm
+  · by_cases hS : S = ⊤
+    · rw [hS]
+      exact le_top
+    obtain ⟨p, hp, hpq, hlim⟩ := exists_seq_strictMono_tendsto' hq
+    have hbound (n : ℕ) : operatorNorm G (p n) ≤ S :=
+      le_sSup ⟨p n, (hpq n).2, rfl⟩
+    have hbounded (n : ℕ) : OperatorBounded G (p n) :=
+      (operatorNorm_lt_top_iff G (p n)).mp ((hbound n).trans_lt (lt_top_iff_ne_top.mpr hS))
+    choose A hA using hbounded
+    have hnorm (n : ℕ) : ‖A n‖ ≤ S.toReal := by
+      apply (ENNReal.ofReal_le_iff_le_toReal hS).mp
+      rw [← operatorNorm_of_hasKernel (hA n)]
+      exact hbound n
+    have hmatrix (x y : V) : Tendsto (fun n => A n (lp.single 2 y (1 : ℂ)) x)
+        atTop (𝓝 (twoPoint G q x y : ℂ)) := by
+      have ht := (Complex.continuous_ofReal.tendsto (twoPoint G q x y)).comp
+        (twoPoint_tendsto_of_monotone G p hp.monotone q (fun n => (hpq n).2.le) hlim x y)
+      convert ht using 1
+      funext n
+      exact hA n x y
+    obtain ⟨B, hBn, hB⟩ := exists_operator_of_matrix_tendsto A
+      (fun x y => (twoPoint G q x y : ℂ)) ENNReal.toReal_nonneg hnorm hmatrix
+    rw [operatorNorm_of_hasKernel hB]
+    exact (ENNReal.ofReal_le_iff_le_toReal hS).mpr hBn
+  · apply sSup_le
+    rintro r ⟨p, hp, rfl⟩
+    exact operatorNorm_mono G hp.le
+
+theorem operatorBounded_iff_uniform_below (q : unitInterval) (hq : 0 < q) :
+    OperatorBounded G q ↔ ∃ M : ℝ≥0∞, M < ⊤ ∧ ∀ p < q, operatorNorm G p ≤ M := by
+  rw [← operatorNorm_lt_top_iff]
+  constructor
+  · intro h
+    exact ⟨operatorNorm G q, h, fun p hp => operatorNorm_mono G hp.le⟩
+  · rintro ⟨M, hM, h⟩
+    rw [operatorNorm_eq_sup_below G q hq]
+    apply lt_of_le_of_lt ?_ hM
+    apply sSup_le
+    rintro r ⟨p, hp, rfl⟩
+    exact h p hp
+
+end Endpoint
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Kernel/Connections.lean b/lean/OAI/Probability/UnimodularPercolation/Kernel/Connections.lean
index d9ac9257cae341bace336549f3d649ce0c865e8c..01fa6972fadbc1ebeb7453a6a0d88ac41a3f861b 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Kernel/Connections.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Kernel/Connections.lean
@@ -1,1264 +1,1264 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Kernel.Covariance
-
-namespace OAI
-
-/-! Connections for bond percolation. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section ClusterWindowLaw
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-def ClusterWindow (G : BondGraph V E) (x : V) (lo hi : ℕ) (ω : Set E) : Prop :=
-  (cluster G ω x).Finite ∧ lo ≤ (cluster G ω x).ncard ∧ (cluster G ω x).ncard < hi
-
-theorem clusterWindow_local (G : BondGraph V E) (hloc : LocallyFinite G)
-    (x : V) (lo hi : ℕ) (ω : Set E) :
-    let Q := incidentSupport G hloc (finite_reachBall hloc x hi).toFinset
-    ClusterWindow G x lo hi ω ↔ ClusterWindow G x lo hi (ω ∩ (Q : Set E)) := by
-  let Q := incidentSupport G hloc (finite_reachBall hloc x hi).toFinset
-  change ClusterWindow G x lo hi ω ↔ ClusterWindow G x lo hi (ω ∩ (Q : Set E))
-  constructor
-  · rintro ⟨hf,hl,hh⟩
-    have hs : cluster G ω x ⊆ (finite_reachBall hloc x hi).toFinset := by
-      simpa only [Set.Finite.coe_toFinset] using finite_cluster_subset_reachBall G ω x hf hi hh.le
-    have he := cluster_restrict_incident_of_full G hloc _ ω x hs
-    change cluster G (ω ∩ (Q : Set E)) x = cluster G ω x at he
-    exact ⟨he.symm ▸ hf,he.symm ▸ hl,he.symm ▸ hh⟩
-  · rintro ⟨hf,hl,hh⟩
-    have he := cluster_restrict_of_cap G hloc Q ω x hi (Finset.Subset.refl _) hf hh.le
-    exact ⟨he ▸ hf,he ▸ hl,he ▸ hh⟩
-
-theorem measurableSet_clusterWindow (G : BondGraph V E) (hloc : LocallyFinite G)
-    (x : V) (lo hi : ℕ) : MeasurableSet {ω | ClusterWindow G x lo hi ω} := by
-  let Q := incidentSupport G hloc (finite_reachBall hloc x hi).toFinset
-  have he : {ω | ClusterWindow G x lo hi ω} =
-      {ω | ω ∩ (Q : Set E) ∈ {η | ClusterWindow G x lo hi η}} := by
-    ext ω
-    exact clusterWindow_local G hloc x lo hi ω
-  rw [he]
-  exact measurableSet_finite_restriction Q _
-
-noncomputable def clusterWindowProbability (G : BondGraph V E) (p : unitInterval)
-    (x : V) (lo hi : ℕ) : ℝ := (law p).real {ω | ClusterWindow G x lo hi ω}
-
-theorem finiteClusterWindowMass_eq (G : BondGraph V E) (hloc : LocallyFinite G)
-    (Q : Finset E) (p : unitInterval) (x : V) (lo hi : ℕ)
-    (hQ : incidentSupport G hloc (finite_reachBall hloc x hi).toFinset ⊆ Q) :
-    finiteClusterWindowMass G Q p x lo hi = clusterWindowProbability G p x lo hi := by
-  let S := incidentSupport G hloc (finite_reachBall hloc x hi).toFinset
-  let f : Set E → ℝ := fun ω => if ClusterWindow G x lo hi ω then 1 else 0
-  have hf (ω) : f ω = f (ω ∩ (S : Set E)) :=
-    if_congr (clusterWindow_local G hloc x lo hi ω) rfl rfl
-  have he := integral_finite_local Q p f (finite_local_of_subset S Q hQ f hf)
-  trans ∫ ω, f ω ∂law p
-  · rw [he]
-    unfold finiteClusterWindowMass finiteFieldIntegral
-    apply Finset.sum_congr rfl
-    intro η _
-    congr 1
-    unfold f ClusterWindow
-    dsimp only
-    split_ifs <;> rfl
-  change (∫ ω, ({η | ClusterWindow G x lo hi η}.indicator (1 : Set E → ℝ)) ω ∂law p) = _
-  exact integral_indicator_one (measurableSet_clusterWindow G hloc x lo hi)
-
-theorem clusterWindowProbability_nonneg (G : BondGraph V E) (p : unitInterval)
-    (x : V) (lo hi : ℕ) : 0 ≤ clusterWindowProbability G p x lo hi := ENNReal.toReal_nonneg
-
-theorem clusterWindowProbability_mono (G : BondGraph V E) (p : unitInterval)
-    (x : V) {a b c d : ℕ} (hac : c ≤ a) (hbd : b ≤ d) :
-    clusterWindowProbability G p x a b ≤ clusterWindowProbability G p x c d := by
-  apply measureReal_mono
-  · intro ω h
-    exact ⟨h.1,hac.trans h.2.1,h.2.2.trans_le hbd⟩
-  · exact measure_ne_top _ _
-
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
-
-theorem finiteClusterWindowMass_amplification (o : V)
-    (q : LocalEndpointRule G hloc (vertexOrbit G o) M L) (k n : ℕ) (p : unitInterval) (x : V) :
-    finiteClusterWindowMass G (q.amplificationBonds k n x) p x (n-k*M) (2*n) =
-      clusterWindowProbability G p x (n-k*M) (2*n) := by
-  apply finiteClusterWindowMass_eq G hloc
-  exact incidentSupport_ball_mono G hloc x (by dsimp [amplificationRadius]; omega)
-
-end LocalEndpointRule
-end ClusterWindowLaw
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section QuantitativeAmplificationMoment
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-noncomputable def amplificationMomentFactor (k M d : ℕ) (c a t : ℝ) : ℝ :=
-  4 + (((k*M)+(k*M)^2+d*(k*M)^3:ℕ):ℝ) + (2*c*(k*M:ℕ))^2*t/a
-
-theorem amplificationMomentFactor_pos (k M d : ℕ) (c a t : ℝ) (ha : 0 < a) (ht : 0 ≤ t) :
-    0 < amplificationMomentFactor k M d c a t := by
-  unfold amplificationMomentFactor
-  positivity
-
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
-
-theorem amplificationLoad_moment_factor [Nonempty E] (hconn : Connected G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
-    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (i₀ : ι) (hw₀ : 0 < weight i₀)
-    (q : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) M L) (hL : 1 ≤ L)
-    (p : unitInterval) (k n d : ℕ) (hn : 0 < n) (hdeg : ∀ x, degree G x ≤ d)
-    (a t π : ℝ) (ha : 0 < a) (ht : 0 ≤ t) (hπ : 0 ≤ π)
-    (hbin : a*n*π ≤ binOrbitRow G p (root i₀) n)
-    (hwindow : clusterWindowProbability G p (root i₀) (n-k*M) (2*n) ≤ t*π)
-    (hcode : inverseCodeMass M L^k ≤ n*q.successProbability p (root i₀)^k) :
-    (∫ ω, (q.amplificationLoad (root i₀) hL k n p).row ω (root i₀)^2 ∂law p) ≤
-      amplificationMomentFactor k M d c a t * q.successProbability p (root i₀)^k *
-        (q.successProbability p (root i₀)^k * binOrbitRow G p (root i₀) n / n) := by
-  let u := q.successProbability p (root i₀)
-  let B := u^k * binOrbitRow G p (root i₀) n / n
-  let D : ℝ := (((k*M)+(k*M)^2+d*(k*M)^3:ℕ):ℝ)
-  have hu : 0 ≤ u^k := pow_nonneg (q.successProbability_nonneg p _) _
-  have hn' : (0:ℝ) < n := Nat.cast_pos.mpr hn
-  have hB : a*π*u^k ≤ B := by
-    apply (le_div_iff₀ hn').mpr
-    calc
-      _ = u^k*(a*n*π) := by ring
-      _ ≤ _ := mul_le_mul_of_nonneg_left hbin hu
-  have hB0 : 0 ≤ B := (mul_nonneg (mul_nonneg ha.le hπ) hu).trans hB
-  have hcoef : 4*u^k+D*inverseCodeMass M L^k/n ≤ (4+D)*u^k := by
-    have h := mul_le_mul_of_nonneg_left hcode (show 0 ≤ D by positivity)
-    have h' : D*inverseCodeMass M L^k/n ≤ D*u^k := by
-      apply (div_le_iff₀ hn').mpr
-      nlinarith only [h]
-    nlinarith only [h']
-  have hrem : (2*(c*((k*M:ℕ)*u^k)))^2*(t*π) ≤
-      ((2*c*(k*M:ℕ))^2*t/a)*u^k*B := by
-    have h := mul_le_mul_of_nonneg_left hB
-      (show 0 ≤ ((2*c*(k*M:ℕ))^2*t/a)*u^k from
-        mul_nonneg (div_nonneg (mul_nonneg (sq_nonneg _) ht) ha.le) hu)
-    calc
-      _ = (((2*c*(k*M:ℕ))^2*t/a)*u^k)*(a*π*u^k) := by
-        field_simp
-      _ ≤ _ := h
-  have hself : root i₀ ∈ vertexOrbit G (root i₀) := by
-    refine ⟨Equiv.refl V,Equiv.refl E,?_,rfl⟩
-    intro e
-    simp
-  have he := q.amplificationLoad_mean root weight Γ htr
-    (fun e he => hΓ he) hw i₀ hw₀ hL p k n (root i₀) hself
-  have hm := q.amplificationLoad_second_moment_le hconn root weight Γ hΓ htr hcov hw hw1
-    c hc hmin i₀ hw₀ hL p (root i₀) k n d hn hdeg
-  rw [he,q.finiteClusterWindowMass_amplification] at hm
-  change _ ≤ (4*u^k+D*inverseCodeMass M L^k/n)*B +
-    (2*(c*((k*M:ℕ)*u^k)))^2*clusterWindowProbability G p (root i₀) (n-k*M) (2*n) at hm
-  calc
-    _ ≤ _ := hm
-    _ ≤ (4+D)*u^k*B + (2*(c*((k*M:ℕ)*u^k)))^2*(t*π) :=
-      add_le_add (mul_le_mul_of_nonneg_right hcoef hB0)
-        (mul_le_mul_of_nonneg_left hwindow (sq_nonneg _))
-    _ ≤ (4+D)*u^k*B + ((2*c*(k*M:ℕ))^2*t/a)*u^k*B := add_le_add (le_refl _) hrem
-    _ = _ := by dsimp [amplificationMomentFactor,D,B,u]; ring
-
-end LocalEndpointRule
-end QuantitativeAmplificationMoment
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section QuantitativeAmplificationStep
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
-
- 
-
-theorem quantitative_amplification_step [Nonempty E] (hconn : Connected G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
-    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (i₀ : ι) (hw₀ : 0 < weight i₀)
-    (q : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) M L) (hL : 1 ≤ L)
-    (p : unitInterval) (k n d : ℕ) (hn : 0 < n) (hdeg : ∀ x, degree G x ≤ d)
-    (hu : 0 < q.successProbability p (root i₀))
-    (a t π L' : ℝ) (ha : 0 < a) (ht : 0 ≤ t) (hπ : 0 < π)
-    (hbin : a*n*π ≤ binOrbitRow G p (root i₀) n)
-    (hwindow : clusterWindowProbability G p (root i₀) (n-k*M) (2*n) ≤ t*π)
-    (hcode : inverseCodeMass M L^k ≤ n*q.successProbability p (root i₀)^k)
-    (hbad : 16*amplificationMomentFactor k M d c a t *
-      (law p).real {ω | CappedBadGoodness G ω (root i₀) (2*n+k*M) L'} ≤ a*π)
-    (A : Hilbert V →L[ℂ] Hilbert V)
-    (hA : ∀ x y, matrixCoeff A x y = (q.normalizedKernel (root i₀) p x y : ℂ)) :
-    ∃ q' : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) (2*n+k*M) L',
-      a*π/(8*amplificationMomentFactor k M d c a t) ≤ q'.successProbability p (root i₀) ∧
-      ∃ A' : Hilbert V →L[ℂ] Hilbert V, 0 < ‖A'‖ ∧ ‖A'‖ ≤ 2*‖A‖^k ∧
-        ∀ x y, matrixCoeff A' x y = (q'.normalizedKernel (root i₀) p x y : ℂ) := by
-  let Q := amplificationMomentFactor k M d c a t
-  let u := q.successProbability p (root i₀)
-  let B := u^k*binOrbitRow G p (root i₀) n / n
-  let b := 4*Q*u^k
-  have hQ : 0 < Q := amplificationMomentFactor_pos k M d c a t ha ht
-  have huk : 0 < u^k := pow_pos hu _
-  have hb : 0 < b := mul_pos (mul_pos (by norm_num) hQ) huk
-  have hn' : (0:ℝ) < n := Nat.cast_pos.mpr hn
-  have hbin0 : 0 < binOrbitRow G p (root i₀) n :=
-    (mul_pos (mul_pos ha hn') hπ).trans_le hbin
-  have hB : a*π*u^k ≤ B := by
-    apply (le_div_iff₀ hn').mpr
-    calc
-      _ = u^k*(a*n*π) := by ring
-      _ ≤ _ := mul_le_mul_of_nonneg_left hbin huk.le
-  have hm := q.amplificationLoad_moment_factor hconn root weight Γ hΓ htr hcov hw hw1
-    c hc hmin i₀ hw₀ hL p k n d hn hdeg a t π ha ht hπ.le hbin hwindow hcode
-  have hmoment : (∫ ω, (q.amplificationLoad (root i₀) hL k n p).row ω (root i₀)^2 ∂law p)/b ≤ B/4 := by
-    apply (div_le_iff₀ hb).mpr
-    change _ ≤ Q*u^k*B at hm
-    calc
-      _ ≤ Q*u^k*B := hm
-      _ = _ := by dsimp [b]; ring
-  have hbad' : b*(law p).real {ω | CappedBadGoodness G ω (root i₀) (2*n+k*M) L'} ≤ B/4 := by
-    have hh := mul_le_mul_of_nonneg_right hbad huk.le
-    dsimp [b]
-    nlinarith only [hh,hB]
-  obtain ⟨q',hq',A',hA'pos,hA'norm,hA'⟩ := q.amplification_truncation_step root weight Γ htr
-    (fun e he => hΓ he) hw i₀ hw₀ hL p k n hn hu hbin0 b L' hb hmoment hbad' A hA
-  refine ⟨q',?_,A',hA'pos,hA'norm,hA'⟩
-  apply le_trans _ hq'
-  change a*π/(8*Q) ≤ B/(2*b)
-  apply (div_le_div_iff₀ (mul_pos (by norm_num) hQ) (mul_pos (by norm_num) hb)).mpr
-  have hh := mul_le_mul_of_nonneg_left hB (show 0 ≤ 8*Q by positivity)
-  dsimp [b]
-  nlinarith only [hh]
-
-end LocalEndpointRule
-end QuantitativeAmplificationStep
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section ConnectionAdjustedRule
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-noncomputable def cappedConnectionMass (G : BondGraph V E) (p : unitInterval)
-    (S : V → V → ℝ) (c : ℝ) (M : ℕ) (L : ℝ) (ω : Set E) (x y : V) : ℝ :=
-  if CappedGoodness G ω x M L ∧ Conn G ω x y then c*S x y/twoPoint G p x y else 0
-
-theorem cappedConnectionMass_nonneg (G : BondGraph V E) (p : unitInterval)
-    (S : V → V → ℝ) (c : ℝ) (hc : 0 ≤ c) (hS : ∀ x y, 0 ≤ S x y)
-    (M : ℕ) (L : ℝ) (ω : Set E) (x y : V) : 0 ≤ cappedConnectionMass G p S c M L ω x y := by
-  unfold cappedConnectionMass
-  split_ifs
-  · exact div_nonneg (mul_nonneg hc (hS x y)) (twoPoint_nonneg G p x y)
-  · rfl
-
-theorem cappedConnectionMass_support (G : BondGraph V E) (p : unitInterval)
-    (S : V → V → ℝ) (c : ℝ) (M : ℕ) (L : ℝ) (ω : Set E) (x y : V)
-    (h : cappedConnectionMass G p S c M L ω x y ≠ 0) :
-    S x y ≠ 0 ∧ CappedGoodness G ω x M L ∧ Conn G ω x y := by
-  unfold cappedConnectionMass at h
-  split_ifs at h with hh
-  · refine ⟨?_,hh⟩
-    intro hz
-    simp only [hz,mul_zero,zero_div] at h
-    exact h rfl
-  · exact (h rfl).elim
-
-theorem cappedConnectionMass_le (G : BondGraph V E) (p : unitInterval)
-    (S : V → V → ℝ) (c : ℝ) (hc : 0 < c) (hS : ∀ x y, 0 ≤ S x y)
-    (hτ : ∀ x y, S x y ≠ 0 → c ≤ twoPoint G p x y)
-    (M : ℕ) (L : ℝ) (ω : Set E) (x y : V) : cappedConnectionMass G p S c M L ω x y ≤ S x y := by
-  by_cases hs : S x y = 0
-  · simp only [cappedConnectionMass,hs,mul_zero,zero_div,ite_self,le_refl]
-  · unfold cappedConnectionMass
-    split_ifs
-    · apply (div_le_iff₀ (hc.trans_le (hτ x y hs))).mpr
-      exact (mul_le_mul_of_nonneg_right (hτ x y hs) (hS x y)).trans_eq (mul_comm _ _)
-    · exact hS x y
-
-theorem cappedConnectionMass_local (G : BondGraph V E) (hloc : LocallyFinite G)
-    (p : unitInterval) (S : V → V → ℝ) (c : ℝ) (M : ℕ) (L : ℝ) (ω : Set E) (x y : V) :
-    let Q := incidentSupport G hloc (finite_reachBall hloc x M).toFinset
-    cappedConnectionMass G p S c M L ω x y = cappedConnectionMass G p S c M L (ω ∩ (Q:Set E)) x y := by
-  let Q := incidentSupport G hloc (finite_reachBall hloc x M).toFinset
-  change cappedConnectionMass G p S c M L ω x y = cappedConnectionMass G p S c M L (ω ∩ (Q:Set E)) x y
-  have hg := cappedGoodness_local G hloc ω x M L
-  change CappedGoodness G ω x M L ↔ CappedGoodness G (ω ∩ (Q:Set E)) x M L at hg
-  unfold cappedConnectionMass
-  apply if_congr _ rfl rfl
-  constructor
-  · rintro ⟨h,hxy⟩
-    have h' := hg.mp h
-    have he := cluster_restrict_of_cap G hloc Q ω x M (Finset.Subset.refl _) h'.1 h'.2.1
-    refine ⟨h',?_⟩
-    change y ∈ cluster G (ω ∩ (Q:Set E)) x
-    rw [he]
-    exact hxy
-  · rintro ⟨h,hxy⟩
-    exact ⟨hg.mpr h,conn_mono G Set.inter_subset_left hxy⟩
-
-theorem cappedConnectionMass_covariant [Countable V] [Countable E]
-    (G : BondGraph V E) (p : unitInterval) (S : V → V → ℝ) (c : ℝ) (M : ℕ) (L : ℝ)
-    (a : V ≃ V) (b : E ≃ E) (hab : IsAut G a b)
-    (hS : ∀ x y, S (a x) (a y) = S x y) (ω : Set E) (x y : V) :
-    cappedConnectionMass G p S c M L (b ⁻¹' ω) x y = cappedConnectionMass G p S c M L ω (a x) (a y) := by
-  simp only [cappedConnectionMass,cappedGoodness_aut_iff G hab,conn_aut_iff G hab,
-    hS,twoPoint_aut G p hab]
-
-noncomputable def connectionAdjustedRule [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (O : Set V) (p : unitInterval)
-    (S : V → V → ℝ) (c : ℝ) (hc : 0 < c) (hS : ∀ x y, 0 ≤ S x y)
-    (hSsum : ∀ x, Summable (S x)) (hrow : ∀ x, ∑' y, S x y ≤ 1)
-    (hτ : ∀ x y, S x y ≠ 0 → c ≤ twoPoint G p x y)
-    (hO : ∀ x y, S x y ≠ 0 → x ∈ O ∧ y ∈ O)
-    (hcov : ∀ a b, IsAut G a b → ∀ x y, S (a x) (a y) = S x y)
-    (M : ℕ) (L : ℝ) : LocalEndpointRule G hloc O M L where
-  mass := cappedConnectionMass G p S c M L
-  radius := M
-  nonneg := cappedConnectionMass_nonneg G p S c hc.le hS M L
-  support := by
-    intro ω x y h
-    obtain ⟨hs,hg,hxy⟩ := cappedConnectionMass_support G p S c M L ω x y h
-    exact ⟨(hO x y hs).1,(hO x y hs).2,hxy,hg⟩
-  locality := fun ω x y => cappedConnectionMass_local G hloc p S c M L ω x y
-  covariance := fun a b hab ω x y => cappedConnectionMass_covariant G p S c M L a b hab (hcov a b hab) ω x y
-  row_le_one := by
-    intro ω x
-    have hs : Summable (cappedConnectionMass G p S c M L ω x) := by
-      apply summable_of_ne_finset_zero (s := (finite_reachBall hloc x M).toFinset)
-      intro y hy
-      by_contra hn
-      obtain ⟨_,hg,hxy⟩ := cappedConnectionMass_support G p S c M L ω x y hn
-      apply hy
-      simp only [Set.Finite.mem_toFinset]
-      exact finite_cluster_subset_reachBall G ω x hg.1 M hg.2.1 hxy
-    exact (hs.tsum_le_tsum (cappedConnectionMass_le G p S c hc hS hτ M L ω x) (hSsum x)).trans (hrow x)
-
-end ConnectionAdjustedRule
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section ConnectionAdjustedKernel
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Countable V] [Countable E]
-
-theorem integral_cappedConnectionMass (G : BondGraph V E) (hloc : LocallyFinite G)
-    (p : unitInterval) (S : V → V → ℝ) (c : ℝ) (M : ℕ) (L : ℝ) (x y : V) :
-    (∫ ω, cappedConnectionMass G p S c M L ω x y ∂law p) =
-      c*S x y/twoPoint G p x y * (law p).real {ω | CappedGoodness G ω x M L ∧ Conn G ω x y} := by
-  have hm : MeasurableSet {ω | CappedGoodness G ω x M L ∧ Conn G ω x y} :=
-    (measurableSet_capped_good G hloc x M L).inter (measurableSet_conn G x y)
-  simpa only [cappedConnectionMass,Set.indicator,Set.mem_ofPred_eq,smul_eq_mul,mul_comm] using
-    integral_indicator_const (μ := law p) (c*S x y/twoPoint G p x y) hm
-
-theorem integral_cappedConnectionMass_bounds (G : BondGraph V E) (hloc : LocallyFinite G)
-    (p : unitInterval) (S : V → V → ℝ) (c : ℝ) (hc : 0 < c) (hS : ∀ x y, 0 ≤ S x y)
-    (hτ : ∀ x y, S x y ≠ 0 → c ≤ twoPoint G p x y)
-    (M : ℕ) (L : ℝ) (x y : V) :
-    let K := ∫ ω, cappedConnectionMass G p S c M L ω x y ∂law p
-    K ≤ c*S x y ∧ c*S x y ≤ K + S x y * (law p).real {ω | ¬ CappedGoodness G ω x M L} := by
-  dsimp only
-  rw [integral_cappedConnectionMass G hloc]
-  by_cases hs : S x y = 0
-  · simp only [hs,mul_zero,zero_div,zero_mul,zero_add,le_refl,and_self]
-  have ht : 0 < twoPoint G p x y := hc.trans_le (hτ x y hs)
-  let a := c*S x y/twoPoint G p x y
-  have ha : 0 ≤ a := div_nonneg (mul_nonneg hc.le (hS x y)) ht.le
-  have hle : a ≤ S x y := by
-    apply (div_le_iff₀ ht).mpr
-    exact (mul_le_mul_of_nonneg_right (hτ x y hs) (hS x y)).trans_eq (mul_comm _ _)
-  have he : a*twoPoint G p x y = c*S x y := div_mul_cancel₀ _ ht.ne'
-  have hsub : {ω : Set E | CappedGoodness G ω x M L ∧ Conn G ω x y} ⊆ {ω | Conn G ω x y} :=
-    fun _ h => h.2
-  have hm : (law p).real {ω | CappedGoodness G ω x M L ∧ Conn G ω x y} ≤ twoPoint G p x y := by
-    exact measureReal_mono hsub (measure_ne_top _ _)
-  have hcover : {ω : Set E | Conn G ω x y} ⊆
-      {ω | CappedGoodness G ω x M L ∧ Conn G ω x y} ∪ {ω | ¬ CappedGoodness G ω x M L} := by
-    intro ω h
-    by_cases hg : CappedGoodness G ω x M L
-    · exact Or.inl ⟨hg,h⟩
-    · exact Or.inr hg
-  have hb : twoPoint G p x y ≤
-      (law p).real {ω | CappedGoodness G ω x M L ∧ Conn G ω x y} +
-        (law p).real {ω | ¬ CappedGoodness G ω x M L} :=
-    (measureReal_mono hcover (measure_ne_top _ _)).trans (measureReal_union_le _ _)
-  constructor
-  · exact (mul_le_mul_of_nonneg_left hm ha).trans_eq he
-  · calc
-      c*S x y = a*twoPoint G p x y := he.symm
-      _ ≤ a*((law p).real {ω | CappedGoodness G ω x M L ∧ Conn G ω x y} +
-          (law p).real {ω | ¬ CappedGoodness G ω x M L}) := mul_le_mul_of_nonneg_left hb ha
-      _ ≤ a*(law p).real {ω | CappedGoodness G ω x M L ∧ Conn G ω x y} +
-          S x y*(law p).real {ω | ¬ CappedGoodness G ω x M L} := by
-        rw [mul_add]
-        exact add_le_add le_rfl (mul_le_mul_of_nonneg_right hle measureReal_nonneg)
-
- 
-
-theorem connectionAdjustedRule_success_lower
-    (G : BondGraph V E) (hloc : LocallyFinite G) (O : Set V) (p : unitInterval)
-    (S : V → V → ℝ) (c : ℝ) (hc : 0 < c) (hS : ∀ x y, 0 ≤ S x y)
-    (hSsum : ∀ x, Summable (S x)) (hrow : ∀ x, ∑' y, S x y ≤ 1)
-    (hτ : ∀ x y, S x y ≠ 0 → c ≤ twoPoint G p x y)
-    (hO : ∀ x y, S x y ≠ 0 → x ∈ O ∧ y ∈ O)
-    (hcov : ∀ a b, IsAut G a b → ∀ x y, S (a x) (a y) = S x y)
-    (M : ℕ) (L : ℝ) (x : V) :
-    c*(∑' y, S x y) ≤
-      (connectionAdjustedRule G hloc O p S c hc hS hSsum hrow hτ hO hcov M L).successProbability p x +
-        (law p).real {ω | ¬ CappedGoodness G ω x M L} := by
-  let q := connectionAdjustedRule G hloc O p S c hc hS hSsum hrow hτ hO hcov M L
-  let ε := (law p).real {ω | ¬ CappedGoodness G ω x M L}
-  have h (y : V) : c*S x y ≤ q.kernel p x y + S x y * ε :=
-    (integral_cappedConnectionMass_bounds G hloc p S c hc hS hτ M L x y).2
-  have hh := ((hSsum x).mul_left c).tsum_le_tsum h
-    ((q.summable_kernel p x).add ((hSsum x).mul_right ε))
-  rw [(q.summable_kernel p x).tsum_add ((hSsum x).mul_right ε),tsum_mul_left,tsum_mul_right] at hh
-  exact hh.trans (add_le_add le_rfl ((mul_le_mul_of_nonneg_right (hrow x) measureReal_nonneg).trans_eq (one_mul ε)))
-
-end ConnectionAdjustedKernel
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section ConnectionAdjustedOperator
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Countable V] [Countable E]
-
- 
-
-theorem connectionAdjustedRule_operator
-    (G : BondGraph V E) (hloc : LocallyFinite G) (o : V) (p : unitInterval)
-    (S : V → V → ℝ) (c : ℝ) (hc : 0 < c) (hS : ∀ x y, 0 ≤ S x y)
-    (hSsum : ∀ x, Summable (S x)) (hrow : ∀ x, ∑' y, S x y ≤ 1)
-    (hτ : ∀ x y, S x y ≠ 0 → c ≤ twoPoint G p x y)
-    (hO : ∀ x y, S x y ≠ 0 → x ∈ vertexOrbit G o ∧ y ∈ vertexOrbit G o)
-    (hcov : ∀ a b, IsAut G a b → ∀ x y, S (a x) (a y) = S x y)
-    (M : ℕ) (L b : ℝ) (hb : 0 < b) (hret : b ≤ ∑' y, S o y)
-    (hbad : (law p).real {ω | ¬ CappedGoodness G ω o M L} ≤ c*b/2)
-    (A : Hilbert V →L[ℂ] Hilbert V)
-    (hA : ∀ x y, matrixCoeff A x y = (S x y : ℂ)) :
-    ∃ q : LocalEndpointRule G hloc (vertexOrbit G o) M L,
-      c*b/2 ≤ q.successProbability p o ∧
-      ∃ T : Hilbert V →L[ℂ] Hilbert V, 0 < ‖T‖ ∧ ‖T‖ ≤ 2*‖A‖/b ∧
-        ∀ x y, matrixCoeff T x y = (q.normalizedKernel o p x y : ℂ) := by
-  let q := connectionAdjustedRule G hloc (vertexOrbit G o) p S c hc hS hSsum hrow hτ hO hcov M L
-  have hlo := connectionAdjustedRule_success_lower G hloc (vertexOrbit G o) p S c hc hS hSsum hrow hτ hO hcov M L o
-  have hs : c*b/2 ≤ q.successProbability p o := by
-    have hmul := mul_le_mul_of_nonneg_left hret hc.le
-    change _ ≤ q.successProbability p o + _ at hlo
-    linarith
-  have hu : 0 < q.successProbability p o := (by positivity : 0 < c*b/2).trans_le hs
-  have hcA (x y : V) : matrixCoeff ((c:ℂ) • A) x y = (c*S x y : ℝ) := by
-    change (c:ℂ)*matrixCoeff A x y = _
-    rw [hA,Complex.ofReal_mul]
-  obtain ⟨B,hBn,hB⟩ := exists_operator_of_matrix_domination
-    (fun x y => c*S x y) (q.kernel p) (fun x y => mul_nonneg hc.le (hS x y))
-    (fun x y => by
-      rw [abs_of_nonneg (q.kernel_nonneg p x y)]
-      exact (integral_cappedConnectionMass_bounds G hloc p S c hc hS hτ M L x y).1)
-    ((c:ℂ) • A) hcA
-  have hBn' : ‖B‖ ≤ c*‖A‖ := by
-    simpa only [norm_smul,Complex.norm_of_nonneg hc.le] using hBn
-  obtain ⟨T,hTn,hT⟩ := q.normalize_mass_operator o p hu B _ hBn' hB
-  refine ⟨q,hs,T,q.normalized_operator_norm_pos o p hu T hT,hTn.trans ?_,hT⟩
-  apply (div_le_iff₀ hu).mpr
-  rw [div_mul_eq_mul_div]
-  apply (le_div_iff₀ hb).mpr
-  have hh := mul_le_mul_of_nonneg_right hs (show 0 ≤ 2*‖A‖ by positivity)
-  nlinarith only [hh]
-
-end ConnectionAdjustedOperator
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter Real
-open scoped Topology
-section CappedGoodnessLimit
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Countable V] [Countable E]
-
-omit [Countable V] in
-theorem clusterLarge_probability_tendsto_zero (G : BondGraph V E) (p : unitInterval) (x : V)
-    (hfin : ∀ᵐ ω ∂law p, (cluster G ω x).Finite) :
-    Tendsto (fun M : ℕ => (law p).real (clusterLarge G x M)) atTop (𝓝 0) := by
-  have hz : law p {ω | (cluster G ω x).Infinite} = 0 := by
-    simpa only [Set.Infinite] using (ae_iff.mp hfin)
-  have ht := (ENNReal.tendsto_toReal (measure_ne_top (law p) (⋂ M, clusterLarge G x M))).comp
-    (tendsto_measure_iInter_atTop (fun M => (measurableSet_clusterLarge G x M).nullMeasurableSet)
-      (clusterLarge_antitone G x) ⟨0,measure_ne_top _ _⟩)
-  rw [← infinite_cluster_eq_iInter,hz,ENNReal.toReal_zero] at ht
-  exact ht
-
-omit [Countable V] [Countable E] in
-theorem cappedGoodness_failure_le (G : BondGraph V E) (p : unitInterval)
-    (x : V) (M : ℕ) (L : ℝ) :
-    (law p).real {ω | ¬ CappedGoodness G ω x M L} ≤
-      (law p).real (clusterLarge G x M) + (law p).real {ω | CappedBadGoodness G ω x M L} := by
-  have hsub : {ω : Set E | ¬ CappedGoodness G ω x M L} ⊆
-      clusterLarge G x M ∪ {ω | CappedBadGoodness G ω x M L} := by
-    intro ω hbad
-    by_cases hf : (cluster G ω x).Finite
-    · by_cases hM : (cluster G ω x).ncard ≤ M
-      · exact Or.inr ⟨hf,hM,fun hg => hbad ⟨hf,hM,hg⟩⟩
-      · apply Or.inl
-        refine ⟨hf.toFinset,?_,fun y hy => hf.mem_toFinset.mp hy⟩
-        rw [← Set.ncard_eq_toFinset_card _ hf]
-        omega
-    · obtain ⟨t,ht,hcard⟩ := (show (cluster G ω x).Infinite from hf).exists_subset_card_eq M
-      exact Or.inl ⟨t,hcard.ge,ht⟩
-  exact (measureReal_mono hsub (measure_ne_top _ _)).trans (measureReal_union_le _ _)
-
-omit [Countable V] in
- 
-
-theorem cappedGoodness_failure_eventually_small [Nonempty V] (G : BondGraph V E)
-    (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : (p:ℝ) < 1) (x : V)
-    (hfin : ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
-    (ε : ℝ) (hε : 0 < ε) (hε1 : ε < 1/2) (δ : ℝ) (hδ : 0 < δ) :
-    ∀ᶠ M : ℕ in atTop, (law p).real {ω | ¬ CappedGoodness G ω x M ((M:ℝ)^(1/2+ε))} ≤ δ := by
-  have htail := (clusterLarge_probability_tendsto_zero G p x hfin).eventually
-    (Iio_mem_nhds (show (0:ℝ) < δ/2 by positivity))
-  have hr : Tendsto (fun M : ℕ => (M:ℝ)^ε) atTop atTop :=
-    (tendsto_rpow_atTop hε).comp tendsto_natCast_atTop_atTop
-  have hexp : Tendsto (fun M : ℕ => Real.exp (-(M:ℝ)^ε)) atTop (𝓝 0) :=
-    Real.tendsto_exp_atBot.comp (tendsto_neg_atTop_atBot.comp hr)
-  have he := hexp.eventually (Iio_mem_nhds (show (0:ℝ) < δ/2 by positivity))
-  filter_upwards [htail,he,bad_goodness_stretched_exp G hloc hqt p hp ε hε hε1] with M ht he hg
-  have hbad := hg x p le_rfl
-  have hle := cappedGoodness_failure_le G p x M ((M:ℝ)^(1/2+ε))
-  linarith
-
-end CappedGoodnessLimit
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section WalkPowerSupport
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem walkKernel_zero_of_not_step (G : BondGraph V E) (hloc : LocallyFinite G)
-    (a : ℝ) (x y : V) (hxy : x ≠ y) (hadj : ¬ Adj G x y) :
-    walkKernel G hloc a x y = 0 := by simp only [walkKernel,ite_eq_right hxy,ite_eq_right hadj,zero_add]
-
-theorem walkKernel_hasSum (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ) (x : V) :
-    HasSum (walkKernel G hloc a x) 1 := by
-  rw [← walkKernel_row_sum G hloc a x]
-  apply hasSum_sum_of_ne_finset_zero
-  intro y hy
-  simp only [Finset.mem_insert,mem_neighbors,not_or] at hy
-  exact walkKernel_zero_of_not_step G hloc a x y (Ne.symm hy.1) hy.2
-
-theorem walkPower_support (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (k : ℕ) (x y : V) (hy : y ∉ reachBall G x k) : (matrixCoeff (P^k) x y).re = 0 := by
-  induction k generalizing y with
-  | zero =>
-    have he : y ≠ x := by simpa only [reachBall,Set.mem_singleton_iff] using hy
-    simp only [pow_zero,matrixCoeff,one_apply_eq_self,lp.single_apply,Pi.single_apply,ite_eq_right (Ne.symm he),Complex.zero_re]
-  | succ k ih =>
-    rw [pow_succ,← ((hpos.pow k).mul_hasSum P x y).tsum_eq]
-    rw [← tsum_zero]
-    apply tsum_congr
-    intro z
-    by_cases hz : z ∈ reachBall G x k
-    · have hzy : z ≠ y := by intro he; subst y; exact hy (Or.inl hz)
-      have hadj : ¬ Adj G z y := by
-        intro h
-        apply hy
-        apply Or.inr
-        simp only [Set.mem_iUnion,Set.mem_ofPred_eq]
-        exact ⟨z,hz,h⟩
-      rw [hP,walkKernel_zero_of_not_step G hloc a z y hzy hadj,Complex.ofReal_zero,Complex.zero_re,mul_zero]
-    · rw [ih z hz,zero_mul]
-
-theorem walkPower_hasSum (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (k : ℕ) (x : V) : HasSum (fun y => (matrixCoeff (P^k) x y).re) 1 := by
-  induction k with
-  | zero =>
-    have h : HasSum (fun y : V => if x=y then (1:ℝ) else 0) 1 := by
-      exact hasSum_ite_eq' x (1:ℝ)
-    simpa only [pow_zero,matrixCoeff,one_apply_eq_self,lp.single_apply,Pi.single_apply,apply_ite,Complex.one_re,Complex.zero_re] using h
-  | succ k ih =>
-    let B := (finite_reachBall hloc x k).toFinset
-    have he (y : V) : (matrixCoeff (P^(k+1)) x y).re =
-        ∑ z ∈ B, (matrixCoeff (P^k) x z).re * walkKernel G hloc a z y := by
-      rw [pow_succ,← ((hpos.pow k).mul_hasSum P x y).tsum_eq]
-      simp only [hP,Complex.ofReal_re]
-      apply tsum_eq_sum
-      intro z hz
-      rw [walkPower_support G hloc a P hpos hP k x z (by simpa only [B,Set.Finite.mem_toFinset] using hz),zero_mul]
-    have hs := hasSum_sum (s := B) (fun z _ => (walkKernel_hasSum G hloc a z).mul_left (matrixCoeff (P^k) x z).re)
-    simp only [mul_one] at hs
-    have hr : (∑ z ∈ B, (matrixCoeff (P^k) x z).re) = 1 := by
-      rw [← ih.tsum_eq]
-      exact (tsum_eq_sum fun z hz => walkPower_support G hloc a P hpos hP k x z
-        (by simpa only [B,Set.Finite.mem_toFinset] using hz)).symm
-    simpa only [← he,hr] using hs
-
-end WalkPowerSupport
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section WalkCovariance
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem neighbors_aut_card (G : BondGraph V E) (hloc : LocallyFinite G)
-    {g : V ≃ V} {b : E ≃ E} (hg : IsAut G g b) (x : V) :
-    (neighbors G hloc (g x)).card = (neighbors G hloc x).card := by
-  have he : neighbors G hloc (g x) = (neighbors G hloc x).map g.toEmbedding := by
-    ext y
-    simp only [mem_neighbors,Finset.mem_map,Equiv.coe_toEmbedding]
-    constructor
-    · intro h
-      refine ⟨g.symm y,?_,by simp⟩
-      simpa only [g.symm_apply_apply] using adj_aut G (isAut_symm G hg) h
-    · rintro ⟨z,hz,rfl⟩
-      exact adj_aut G hg hz
-  rw [he,Finset.card_map]
-
-theorem walkKernel_covariant (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    {g : V ≃ V} {b : E ≃ E} (hg : IsAut G g b) (x y : V) :
-    walkKernel G hloc a (g x) (g y) = walkKernel G hloc a x y := by
-  have hadj : Adj G (g x) (g y) ↔ Adj G x y :=
-    ⟨fun h => by simpa using adj_aut G (isAut_symm G hg) h,adj_aut G hg⟩
-  simp only [walkKernel,g.apply_eq_iff_eq,neighbors_aut_card G hloc hg,hadj]
-
-theorem twoPoint_reachBall_lower [Countable E] (G : BondGraph V E) (p : unitInterval)
-    (x : V) (k : ℕ) (y : V) (hy : y ∈ reachBall G x k) :
-    (p:ℝ)^k ≤ twoPoint G p x y := by
-  induction k generalizing y with
-  | zero =>
-    simp only [reachBall,Set.mem_singleton_iff] at hy
-    subst y
-    simp only [pow_zero,twoPoint_self,le_refl]
-  | succ k ih =>
-    rcases hy with hy | hy
-    · have h : (p:ℝ)^(k+1) ≤ (p:ℝ)^k := by
-        rw [pow_succ]
-        exact mul_le_of_le_one_right (pow_nonneg p.2.1 k) p.2.2
-      exact h.trans (ih y hy)
-    · simp only [Set.mem_iUnion,Set.mem_ofPred_eq] at hy
-      obtain ⟨z,hz,hzy⟩ := hy
-      have h := mul_le_mul_of_nonneg_left (ih z hz) p.2.1
-      rw [←pow_succ'] at h
-      exact h.trans (twoPoint_adj_lower G p x z y hzy)
-
-end WalkCovariance
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section OrbitReturn
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-lemma finiteKernel_summable_mul (K : V → V → ℝ)
-    (hf : ∀ x, (Function.support (K x)).Finite) (x : V) (f : V → ℝ) :
-    Summable (fun y => K x y * f y) := by
-  apply summable_of_ne_finset_zero (s := (hf x).toFinset)
-  intro y hy
-  have hd : K x y = 0 := by simpa only [Set.Finite.mem_toFinset,Function.mem_support,not_not] using hy
-  rw [hd,zero_mul]
-
- 
-
-theorem orbit_two_step_return_lower (G : BondGraph V E)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
-    (i₀ : ι) (hw₀ : 0 < weight i₀)
-    (K : V → V → ℝ)
-    (hfin : ∀ x, (Function.support (K x)).Finite)
-    (hrow : ∀ x, HasSum (K x) 1) (hsym : ∀ x y, K x y = K y x)
-    (hcov : ∀ a b, IsAut G a b → ∀ x y, K (a x) (a y) = K x y) :
-    (∑ i, if root i ∈ vertexOrbit G (root i₀) then weight i else 0) ≤
-      ∑' z, K (root i₀) z * ∑' y, if y ∈ vertexOrbit G (root i₀) then K z y else 0 := by
-  let O := vertexOrbit G (root i₀)
-  let I : V → ℝ := fun x => if x ∈ O then 1 else 0
-  let f : V → ℝ := fun x => ∑' y, K x y * I y
-  let g : V → ℝ := fun x => ∑' y, K x y * f y
-  let σ := ∑ i, weight i * I (root i)
-  have hI (a : V ≃ V) (b : E ≃ E) (hab : IsAut G a b) (x : V) : I (a x) = I x := by
-    simp only [I,O,mem_vertexOrbit_aut_iff G (root i₀) hab]
-  have hf (a : V ≃ V) (b : E ≃ E) (hab : IsAut G a b) (x : V) : f (a x) = f x := by
-    dsimp only [f]
-    rw [←a.tsum_eq (fun y => K (a x) y*I y)]
-    simp only [hcov a b hab,hI a b hab]
-  have hg (a : V ≃ V) (b : E ≃ E) (hab : IsAut G a b) (x : V) : g (a x) = g x := by
-    dsimp only [g]
-    rw [←a.tsum_eq (fun y => K (a x) y*f y)]
-    simp only [hcov a b hab,hf a b hab]
-  have hgO (x : V) (hx : x ∈ O) : g x = g (root i₀) := by
-    obtain ⟨a,b,hab,rfl⟩ := hx
-    exact hg a b hab _
-  have hmean : ∑ i, weight i*f (root i) = σ := by
-    have ht := htr.real root weight Γ (fun x y => K x y*I y)
-      (fun a ha x y => by obtain ⟨b,hab⟩ := hΓ ha; rw [hcov a b hab,hI a b hab])
-      (fun x => finiteKernel_summable_mul K hfin x I)
-      (fun y => by simpa only [hsym, mul_comm] using (hrow y).summable.mul_left (I y))
-    calc
-      _ = ∑ i, weight i * ∑' x, K x (root i)*I (root i) := ht
-      _ = σ := by simp only [tsum_mul_right,show ∀ y, ∑' x, K x y = 1 from fun y => by simpa only [hsym] using (hrow y).tsum_eq,one_mul,σ]
-  have hsecond : σ*g (root i₀) = ∑ i, weight i*f (root i)^2 := by
-    have ht := htr.real root weight Γ (fun x y => I x*K x y*f y)
-      (fun a ha x y => by obtain ⟨b,hab⟩ := hΓ ha; rw [hcov a b hab,hI a b hab,hf a b hab])
-      (fun x => by simpa only [mul_assoc] using (finiteKernel_summable_mul K hfin x f).mul_left (I x))
-      (fun y => by simpa only [hsym,mul_comm,mul_left_comm,mul_assoc] using (finiteKernel_summable_mul K hfin y I).mul_right (f y))
-    have hr (i : ι) : I (root i)*g (root i) = I (root i)*g (root i₀) := by
-      by_cases hi : root i ∈ O
-      · rw [hgO _ hi]
-      · simp only [I,ite_eq_right hi,zero_mul]
-    calc
-      _ = ∑ i, weight i*(I (root i)*g (root i)) := by
-        simp only [hr,σ,Finset.sum_mul,mul_assoc]
-      _ = ∑ i, weight i*∑' x, I x*K x (root i)*f (root i) := by
-        simpa only [mul_assoc,tsum_mul_left,g] using ht
-      _ = ∑ i, weight i*f (root i)^2 := by
-        simp only [tsum_mul_right]
-        congr 1
-        ext i
-        congr 1
-        rw [show (∑' x, I x*K x (root i)) = f (root i) by simp only [f,hsym,mul_comm]]
-        ring
-  have hvariance : 0 ≤ ∑ i, weight i*(f (root i)-σ)^2 :=
-    Finset.sum_nonneg (fun i _ => mul_nonneg (hw i) (sq_nonneg _))
-  have he : (∑ i, weight i*(f (root i)-σ)^2) = (∑ i, weight i*f (root i)^2)-σ^2 := by
-    simp_rw [sub_sq,mul_add,mul_sub]
-    rw [Finset.sum_add_distrib,Finset.sum_sub_distrib]
-    have hc : (∑ i, weight i*(2*f (root i)*σ)) = 2*σ*(∑ i, weight i*f (root i)) := by
-      rw [Finset.mul_sum]; apply Finset.sum_congr rfl; intro i hi; ring
-    rw [hc,hmean,←Finset.sum_mul,hw1,one_mul]
-    ring
-  rw [he,←hsecond] at hvariance
-  have hσ : 0 < σ := by
-    have hx : root i₀ ∈ O := ⟨Equiv.refl _,Equiv.refl _,fun _ => by simp,rfl⟩
-    have hIn (x : V) : 0 ≤ I x := by dsimp [I]; split_ifs <;> norm_num
-    have hb := Finset.single_le_sum (fun i _ => mul_nonneg (hw i) (hIn (root i))) (Finset.mem_univ i₀)
-    have he₀ : weight i₀*I (root i₀) = weight i₀ := by simp only [I,ite_eq_left hx,mul_one]
-    rw [he₀] at hb
-    exact hw₀.trans_le hb
-  have hbound : σ ≤ g (root i₀) := by nlinarith only [hvariance,hσ]
-  simpa only [σ,g,f,I,mul_ite,mul_one,mul_zero,zero_mul] using hbound
-
-end OrbitReturn
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section WalkOrbitReturn
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem matrixCoeff_real_symm (A : Hilbert V →L[ℂ] Hilbert V)
-    (hA : IsSelfAdjoint A) (x y : V) : (matrixCoeff A x y).re = (matrixCoeff A y x).re := by
-  have h := congrArg Complex.re (matrixCoeff_adjoint A x y)
-  simpa only [hA.star_eq,Complex.star_def,Complex.conj_re] using h
-
-theorem finite_positive_square_row (A : Hilbert V →L[ℂ] Hilbert V)
-    (hA : PositiveRealMatrix A) (hfin : ∀ x, (Function.support (fun y => (matrixCoeff A x y).re)).Finite)
-    (f : V → ℝ) (x : V) :
-    (∑' y, (matrixCoeff (A*A) x y).re*f y) =
-      ∑' z, (matrixCoeff A x z).re*∑' y, (matrixCoeff A z y).re*f y := by
-  let K := fun x y => (matrixCoeff A x y).re
-  let B := (hfin x).toFinset
-  have hz (z : V) (hz : z ∉ B) : K x z = 0 := by
-    simpa only [B,Set.Finite.mem_toFinset,Function.mem_support,not_not] using hz
-  have he (y : V) : (matrixCoeff (A*A) x y).re*f y = ∑ z ∈ B, K x z*(K z y*f y) := by
-    rw [←(hA.mul_hasSum A x y).tsum_eq,tsum_eq_sum (s := B) (fun z h => by change K x z*K z y = 0; rw [hz z h,zero_mul]),Finset.sum_mul]
-    apply Finset.sum_congr rfl
-    intro z h
-    exact mul_assoc _ _ _
-  have hs := hasSum_sum (s := B) (fun z _ =>
-    (finiteKernel_summable_mul K hfin z f).hasSum.mul_left (K x z))
-  have hr : (∑' z, K x z*∑' y, K z y*f y) = ∑ z ∈ B, K x z*∑' y, K z y*f y :=
-    tsum_eq_sum (fun z h => by rw [hz z h,zero_mul])
-  simpa only [←he,←hr] using hs.tsum_eq
-
-theorem walkPower_finite_support (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ)) (k : ℕ) (x : V) :
-    (Function.support (fun y => (matrixCoeff (P^k) x y).re)).Finite := by
-  apply (finite_reachBall hloc x k).subset
-  intro y hy
-  by_contra hn
-  exact hy (walkPower_support G hloc a P hpos hP k x y hn)
-
-theorem walkPower_invariant (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (P : Hilbert V →L[ℂ] Hilbert V)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ)) (k : ℕ) :
-    KernelInvariant (vertexAut G) (P^k) := by
-  apply KernelInvariant.pow
-  intro g hg x y
-  obtain ⟨b,hb⟩ := hg
-  simp only [hP,walkKernel_covariant G hloc a hb]
-
- 
-
-theorem walkPower_even_orbit_return (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
-    (i₀ : ι) (hw₀ : 0 < weight i₀)
-    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ)) (k : ℕ) :
-    (∑ i, if root i ∈ vertexOrbit G (root i₀) then weight i else 0) ≤
-      ∑' y, if y ∈ vertexOrbit G (root i₀) then (matrixCoeff (P^(2*k)) (root i₀) y).re else 0 := by
-  have h := orbit_two_step_return_lower G root weight Γ hΓ htr hw hw1 i₀ hw₀
-    (fun x y => (matrixCoeff (P^k) x y).re)
-    (walkPower_finite_support G hloc a P hpos hP k)
-    (walkPower_hasSum G hloc a P hpos hP k)
-    (matrixCoeff_real_symm (P^k) (hsym.pow k))
-    (fun g b hb x y => congrArg Complex.re (walkPower_invariant G hloc a P hP k g ⟨b,hb⟩ x y))
-  have he := finite_positive_square_row (P^k) (hpos.pow k)
-    (walkPower_finite_support G hloc a P hpos hP k)
-    (fun y => if y ∈ vertexOrbit G (root i₀) then 1 else 0) (root i₀)
-  simp only [mul_ite,mul_one,mul_zero] at he
-  rw [←he] at h
-  simpa only [←pow_add,←two_mul] using h
-
-end WalkOrbitReturn
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-open scoped Topology
-section StartingEndpointRule
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Countable V] [Countable E] [Nonempty V]
-
- 
-
-theorem starting_endpoint_rule_of_walk (G : BondGraph V E) (hloc : LocallyFinite G)
-    (hqt : QuasiTransitive G) (p : unitInterval) (hp : 0 < (p:ℝ)) (hp1 : (p:ℝ) < 1)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
-    (i₀ : ι) (hw₀ : 0 < weight i₀)
-    (hfin : ∀ᵐ ω ∂law p, (cluster G ω (root i₀)).Finite)
-    (a : ℝ) (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
-    (hsym : IsSelfAdjoint P) (hnorm : ‖P‖ < 1)
-    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
-    (ε : ℝ) (hε : 0 < ε) (hε1 : ε < 1/2) (η : ℝ) (hη : 0 < η) (M₀ : ℕ) :
-    ∃ M ≥ M₀, ∃ q : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) M ((M:ℝ)^(1/2+ε)),
-      0 < q.successProbability p (root i₀) ∧
-      ∃ T : Hilbert V →L[ℂ] Hilbert V, 0 < ‖T‖ ∧ ‖T‖ < η ∧
-        ∀ x y, matrixCoeff T x y = (q.normalizedKernel (root i₀) p x y : ℂ) := by
-  let O := vertexOrbit G (root i₀)
-  have ho : root i₀ ∈ O := ⟨Equiv.refl _,Equiv.refl _,fun _ => by simp,rfl⟩
-  let b : ℝ := ∑ i, if root i ∈ O then weight i else 0
-  have hb : 0 < b := by
-    have hn (i : ι) : 0 ≤ if root i ∈ O then weight i else 0 := by split_ifs; exact hw i; exact le_rfl
-    have hh := Finset.single_le_sum (fun i _ => hn i) (Finset.mem_univ i₀)
-    simp only [ite_eq_left ho] at hh
-    exact hw₀.trans_le hh
-  have ht := tendsto_pow_atTop_nhds_zero_of_lt_one (norm_nonneg P) hnorm
-  obtain ⟨k,hk⟩ := (ht.eventually (Iio_mem_nhds (show (0:ℝ) < η*b/2 by positivity))).exists
-  have hk1 : ‖P‖^k ≤ 1 := pow_le_one₀ (norm_nonneg P) hnorm.le
-  have hk2 : 2*‖P‖^(2*k)/b < η := by
-    rw [show 2*k = k+k from Nat.two_mul k,pow_add]
-    have hh : ‖P‖^k*‖P‖^k ≤ ‖P‖^k := mul_le_of_le_one_right (pow_nonneg (norm_nonneg P) _) hk1
-    apply (div_lt_iff₀ hb).mpr
-    change ‖P‖^k < η*b/2 at hk
-    nlinarith only [hh,hk]
-  let K : V → V → ℝ := fun x y => (matrixCoeff (P^(2*k)) x y).re
-  let S : V → V → ℝ := fun x y => if x ∈ O ∧ y ∈ O then K x y else 0
-  have hK (x y : V) : 0 ≤ K x y := (hpos.pow (2*k) x y).2
-  have hS (x y : V) : 0 ≤ S x y := by dsimp [S]; split_ifs; exact hK x y; exact le_rfl
-  have hSK (x y : V) : S x y ≤ K x y := by dsimp [S]; split_ifs; exact le_rfl; exact hK x y
-  have hsum (x : V) : HasSum (K x) 1 := walkPower_hasSum G hloc a P hpos hP (2*k) x
-  have hSsum (x : V) : Summable (S x) := Summable.of_nonneg_of_le (hS x) (hSK x) (hsum x).summable
-  have hrow (x : V) : ∑' y, S x y ≤ 1 := by
-    rw [←(hsum x).tsum_eq]
-    exact (hSsum x).tsum_le_tsum (hSK x) (hsum x).summable
-  have hτ (x y : V) (hxy : S x y ≠ 0) : (p:ℝ)^(2*k) ≤ twoPoint G p x y := by
-    apply twoPoint_reachBall_lower G p x (2*k) y
-    by_contra hn
-    have hz := walkPower_support G hloc a P hpos hP (2*k) x y hn
-    apply hxy
-    dsimp [S,K]
-    rw [hz]
-    split_ifs <;> rfl
-  have hO (x y : V) (hxy : S x y ≠ 0) : x ∈ O ∧ y ∈ O := by
-    by_contra ho
-    exact hxy (ite_eq_right ho)
-  have hcov (g : V ≃ V) (d : E ≃ E) (hg : IsAut G g d) (x y : V) : S (g x) (g y) = S x y := by
-    simp only [S,O,mem_vertexOrbit_aut_iff G (root i₀) hg,K,
-      walkPower_invariant G hloc a P hP (2*k) g ⟨d,hg⟩]
-  have hret : b ≤ ∑' y, S (root i₀) y := by
-    simpa only [b,S,K,ho,true_and,O] using
-      walkPower_even_orbit_return G hloc a root weight Γ hΓ htr hw hw1 i₀ hw₀ P hpos hsym hP k
-  obtain ⟨A,hAn,hA⟩ := exists_operator_of_matrix_domination K S hK
-    (fun x y => by rw [abs_of_nonneg (hS x y)]; exact hSK x y) (P^(2*k)) (hpos.pow (2*k)).real
-  have hc : 0 < (p:ℝ)^(2*k) := pow_pos hp _
-  have hbad := cappedGoodness_failure_eventually_small G hloc hqt p hp1 (root i₀) hfin ε hε hε1
-    ((p:ℝ)^(2*k)*b/2) (by positivity)
-  obtain ⟨M,hM,hMb⟩ := ((eventually_ge_atTop M₀).and hbad).exists
-  obtain ⟨q,hq,T,hTpos,hTn,hT⟩ := connectionAdjustedRule_operator G hloc (root i₀) p S
-    ((p:ℝ)^(2*k)) hc hS hSsum hrow hτ hO hcov M ((M:ℝ)^(1/2+ε)) b hb hret hMb A hA
-  refine ⟨M,hM,q,(by positivity : 0 < (p:ℝ)^(2*k)*b/2).trans_le hq,T,hTpos,?_,hT⟩
-  apply lt_of_le_of_lt hTn
-  apply lt_of_le_of_lt _ hk2
-  have hpow (j : ℕ) : ‖P^j‖ ≤ ‖P‖^j := by
-    induction j with
-    | zero => simpa only [pow_zero, ContinuousLinearMap.one_def] using (ContinuousLinearMap.norm_id_le (𝕜 := ℂ) (E := Hilbert V))
-    | succ j ih =>
-      rw [pow_succ,pow_succ]
-      exact (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right ih (norm_nonneg P))
-  exact div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_left (hAn.trans (hpow (2*k))) (by norm_num)) hb.le
-
-end StartingEndpointRule
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section NonamenableStartingMap
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Countable V] [Countable E] [Nonempty V]
-
- 
-
-theorem nonamenable_starting_endpoint_rule (G : BondGraph V E) (hloc : LocallyFinite G)
-    (hqt : QuasiTransitive G) (hh : 0 < hV G)
-    (p : unitInterval) (hp : 0 < (p:ℝ)) (hp1 : (p:ℝ) < 1)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
-    (i₀ : ι) (hw₀ : 0 < weight i₀)
-    (hfin : ∀ᵐ ω ∂law p, (cluster G ω (root i₀)).Finite)
-    (ε : ℝ) (hε : 0 < ε) (hε1 : ε < 1/2) (η : ℝ) (hη : 0 < η) (M₀ : ℕ) :
-    ∃ M ≥ M₀, ∃ q : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) M ((M:ℝ)^(1/2+ε)),
-      0 < q.successProbability p (root i₀) ∧
-      ∃ T : Hilbert V →L[ℂ] Hilbert V, 0 < ‖T‖ ∧ ‖T‖ < η ∧
-        ∀ x y, matrixCoeff T x y = (q.normalizedKernel (root i₀) p x y : ℂ) := by
-  obtain ⟨D,P,hD,hdeg,hPn,hPs,hP⟩ := exists_strict_walk_operator G hloc hqt hh
-  have hDr : (0:ℝ) < D := Nat.cast_pos.mpr hD
-  have ha : 0 ≤ (1:ℝ)/(2*D) := by positivity
-  have haD : (1:ℝ)/(2*D)*D ≤ 1 := by
-    rw [div_mul_eq_mul_div]
-    apply (div_le_iff₀ (by positivity)).mpr
-    nlinarith
-  have hpos : PositiveRealMatrix P := by
-    intro x y
-    change (P (lp.single 2 y (1:ℂ)) x).im = 0 ∧ 0 ≤ (P (lp.single 2 y (1:ℂ)) x).re
-    rw [hP]
-    exact ⟨Complex.ofReal_im _,walkKernel_nonneg G hloc D hdeg _ ha haD x y⟩
-  exact starting_endpoint_rule_of_walk G hloc hqt p hp hp1 root weight Γ hΓ htr hw hw1 i₀ hw₀ hfin
-    _ P hpos hPs.isSelfAdjoint hPn hP ε hε hε1 η hη M₀
-
-end NonamenableStartingMap
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section FiniteOrbitPartition
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem finite_partition_square_bound {α ι : Type*} [Fintype ι]
-    (C : Finset α) (color : α → ι) (q : α → ℝ)
-    (hq : ∀ x ∈ C, (∑ y ∈ C, if color y = color x then (1:ℝ) else 0) ≤ q x) :
-    (C.card:ℝ)^2 ≤ Fintype.card ι * ∑ x ∈ C, q x := by
-  let c : ι → ℝ := fun i => ∑ x ∈ C, if color x=i then 1 else 0
-  have hsum : (∑ i, c i) = C.card := by
-    dsimp only [c]
-    rw [Finset.sum_comm]
-    simp
-  have hsq : (∑ i, c i^2) = ∑ x ∈ C, c (color x) := by
-    have he (i : ι) : c i^2 = ∑ x ∈ C, if color x=i then c i else 0 := by
-      rw [pow_two]
-      conv_lhs => lhs; change ∑ x ∈ C, if color x=i then (1:ℝ) else 0
-      rw [Finset.sum_mul]
-      apply Finset.sum_congr rfl
-      intro x hx
-      split_ifs <;> simp only [one_mul,zero_mul]
-    simp only [he]
-    rw [Finset.sum_comm]
-    apply Finset.sum_congr rfl
-    intro x hx
-    simp
-  have h := Finset.sum_mul_sq_le_sq_mul_sq Finset.univ (fun _ : ι => (1:ℝ)) c
-  simp only [one_mul,one_pow,Finset.sum_const,Finset.card_univ,nsmul_eq_mul,mul_one,hsum,hsq] at h
-  exact h.trans (mul_le_mul_of_nonneg_left (Finset.sum_le_sum hq) (Nat.cast_nonneg _))
-
-theorem vertexOrbit_eq_of_mem (G : BondGraph V E) (o x : V) (hx : x ∈ vertexOrbit G o) :
-    vertexOrbit G x = vertexOrbit G o := by
-  obtain ⟨a,b,hab,rfl⟩ := hx
-  ext y
-  constructor
-  · rintro ⟨c,d,hcd,hc⟩
-    exact ⟨a.trans c,b.trans d,isAut_trans G hab hcd,hc⟩
-  · rintro ⟨c,d,hcd,hc⟩
-    exact ⟨a.symm.trans c,b.symm.trans d,isAut_trans G (isAut_symm G hab) hcd,by simpa using hc⟩
-
- 
-
-theorem finite_sameOrbit_pairs_lower (G : BondGraph V E)
-    {ι : Type*} [Fintype ι] (root : ι → V)
-    (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i)) (C : Finset V) :
-    (C.card:ℝ)^2 ≤ Fintype.card ι *
-      ∑ x ∈ C, ∑ y ∈ C, if y ∈ vertexOrbit G x then (1:ℝ) else 0 := by
-  let color : V → ι := fun x => (hcover x).choose
-  have hc (x : V) : x ∈ vertexOrbit G (root (color x)) := (hcover x).choose_spec
-  apply finite_partition_square_bound C color
-  intro x hx
-  apply Finset.sum_le_sum
-  intro y hy
-  by_cases h : color y = color x
-  · have hm : y ∈ vertexOrbit G x := by
-      rw [vertexOrbit_eq_of_mem G (root (color x)) x (hc x)]
-      simpa only [h] using hc y
-    simp only [ite_eq_left h,ite_eq_left hm,le_refl]
-  · simp only [ite_eq_right h]
-    split_ifs <;> norm_num
-
-end FiniteOrbitPartition
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section BinOrbitRowMass
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def binSizeMass (G : BondGraph V E) (n : ℕ) (ω : Set E) (x : V) : ℝ :=
-  ∑' y, binConnectionMass G n ω x y
-
-noncomputable def binOrbitRowMass (G : BondGraph V E) (n : ℕ) (ω : Set E) (x : V) : ℝ :=
-  ∑' y, if y ∈ vertexOrbit G x then binConnectionMass G n ω x y else 0
-
-theorem binMass_eq_of_window (G : BondGraph V E) (n : ℕ) (ω : Set E) (x y : V)
-    (hb : ClusterWindow G x n (2*n) ω) :
-    binConnectionMass G n ω x y = if y ∈ cluster G ω x then 1 else 0 := by
-  simp only [binConnectionMass,hb.1,hb.2.1,hb.2.2,true_and]
-  rfl
-
-theorem binMass_zero_of_not_window (G : BondGraph V E) (n : ℕ) (ω : Set E) (x y : V)
-    (hb : ¬ ClusterWindow G x n (2*n) ω) : binConnectionMass G n ω x y = 0 := by
-  apply ite_eq_right
-  exact fun h => hb ⟨h.1,h.2.1,h.2.2.1⟩
-
-theorem binSizeMass_eq (G : BondGraph V E) (n : ℕ) (ω : Set E) (x : V)
-    (hb : ClusterWindow G x n (2*n) ω) : binSizeMass G n ω x = (cluster G ω x).ncard := by
-  unfold binSizeMass
-  simp only [binMass_eq_of_window G n ω x _ hb]
-  rw [tsum_eq_sum (s := hb.1.toFinset) (fun y hy => by simp only [Set.Finite.mem_toFinset] at hy; simp only [ite_eq_right hy])]
-  have he : (∑ y ∈ hb.1.toFinset, if y ∈ cluster G ω x then (1:ℝ) else 0) = ∑ _y ∈ hb.1.toFinset, (1:ℝ) := by
-    apply Finset.sum_congr rfl
-    intro y hy
-    exact ite_eq_left (hb.1.mem_toFinset.mp hy)
-  rw [he]
-  simp only [Finset.sum_const,nsmul_eq_mul,mul_one,←Set.ncard_eq_toFinset_card _ hb.1]
-
-theorem binOrbitRowMass_eq (G : BondGraph V E) (n : ℕ) (ω : Set E) (x : V)
-    (hb : ClusterWindow G x n (2*n) ω) :
-    binOrbitRowMass G n ω x = ∑ y ∈ hb.1.toFinset, if y ∈ vertexOrbit G x then (1:ℝ) else 0 := by
-  unfold binOrbitRowMass
-  rw [tsum_eq_sum (s := hb.1.toFinset) (fun y hy => by
-    have hc : y ∉ cluster G ω x := fun h => hy (hb.1.mem_toFinset.mpr h)
-    rw [binMass_eq_of_window G n ω x y hb,ite_eq_right hc,ite_self])]
-  apply Finset.sum_congr rfl
-  intro y hy
-  rw [binMass_eq_of_window G n ω x y hb,ite_eq_left (hb.1.mem_toFinset.mp hy)]
-
-theorem binOrbitRowMass_nonneg (G : BondGraph V E) (n : ℕ) (ω : Set E) (x : V) :
-    0 ≤ binOrbitRowMass G n ω x := by
-  apply tsum_nonneg
-  intro y
-  split_ifs
-  · exact binConnectionMass_nonneg G n ω x y
-  · exact le_rfl
-
-theorem binOrbitRowMass_local (G : BondGraph V E) (hloc : LocallyFinite G)
-    (n : ℕ) (ω : Set E) (x : V) :
-    let Q := incidentSupport G hloc (finite_reachBall hloc x (2*n)).toFinset
-    binOrbitRowMass G n ω x = binOrbitRowMass G n (ω ∩ (Q : Set E)) x := by
-  dsimp only [binOrbitRowMass]
-  apply tsum_congr
-  intro y
-  rw [binConnectionMass_local G hloc n ω x y]
-
-theorem binSizeMass_local (G : BondGraph V E) (hloc : LocallyFinite G)
-    (n : ℕ) (ω : Set E) (x : V) :
-    let Q := incidentSupport G hloc (finite_reachBall hloc x (2*n)).toFinset
-    binSizeMass G n ω x = binSizeMass G n (ω ∩ (Q : Set E)) x := by
-  dsimp only [binSizeMass]
-  apply tsum_congr
-  intro y
-  exact binConnectionMass_local G hloc n ω x y
-
-theorem binOrbitRowMass_covariant (G : BondGraph V E) (n : ℕ) (ω : Set E)
-    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (x : V) :
-    binOrbitRowMass G n (b ⁻¹' ω) x = binOrbitRowMass G n ω (a x) := by
-  unfold binOrbitRowMass
-  rw [←a.tsum_eq (fun y => if y ∈ vertexOrbit G (a x) then binConnectionMass G n ω (a x) y else 0)]
-  have ho : vertexOrbit G (a x) = vertexOrbit G x := vertexOrbit_eq_of_mem G x (a x) ⟨a,b,hab,rfl⟩
-  simp only [ho,mem_vertexOrbit_aut_iff G x hab,binConnectionMass_covariant G hab]
-
-theorem binSizeMass_covariant (G : BondGraph V E) (n : ℕ) (ω : Set E)
-    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (x : V) :
-    binSizeMass G n (b ⁻¹' ω) x = binSizeMass G n ω (a x) := by
-  unfold binSizeMass
-  rw [←a.tsum_eq (fun y => binConnectionMass G n ω (a x) y)]
-  simp only [binConnectionMass_covariant G hab]
-
-end BinOrbitRowMass
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section BinOrbitTransport
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def binOrbitTransportMass (G : BondGraph V E) (n : ℕ) (ω : Set E) (x y : V) : ℝ :=
-  binConnectionMass G n ω x y * binOrbitRowMass G n ω y / binSizeMass G n ω x
-
-theorem binOrbitRowMass_zero (G : BondGraph V E) (n : ℕ) (ω : Set E) (x : V)
-    (hb : ¬ ClusterWindow G x n (2*n) ω) : binOrbitRowMass G n ω x = 0 := by
-  simp only [binOrbitRowMass,binMass_zero_of_not_window G n ω x _ hb,ite_self,tsum_zero]
-
-theorem binOrbitTransportMass_nonneg (G : BondGraph V E) (n : ℕ) (ω : Set E) (x y : V) :
-    0 ≤ binOrbitTransportMass G n ω x y := by
-  exact div_nonneg (mul_nonneg (binConnectionMass_nonneg G n ω x y) (binOrbitRowMass_nonneg G n ω y))
-    (tsum_nonneg (binConnectionMass_nonneg G n ω x))
-
-theorem binOrbitTransportMass_column (G : BondGraph V E) (n : ℕ) (hn : 0 < n)
-    (ω : Set E) (y : V) : (∑' x, binOrbitTransportMass G n ω x y) = binOrbitRowMass G n ω y := by
-  by_cases hb : ClusterWindow G y n (2*n) ω
-  · have he (x : V) : binOrbitTransportMass G n ω x y =
-        binConnectionMass G n ω x y * (binOrbitRowMass G n ω y / binSizeMass G n ω y) := by
-      by_cases hz : binConnectionMass G n ω x y = 0
-      · simp only [binOrbitTransportMass,hz,zero_mul,zero_div]
-      · obtain ⟨hf,hN,hc⟩ := binConnectionMass_support G n ω x y hz
-        have hset := cluster_eq_of_conn G ω hc
-        have hbx : ClusterWindow G x n (2*n) ω := by
-          unfold ClusterWindow at hb ⊢
-          simpa only [hset] using hb
-        rw [binOrbitTransportMass,binSizeMass_eq G n ω x hbx,binSizeMass_eq G n ω y hb,hset]
-        exact mul_div_assoc _ _ _
-    rw [show (∑' x, binOrbitTransportMass G n ω x y) =
-      ∑' x, binConnectionMass G n ω x y*(binOrbitRowMass G n ω y/binSizeMass G n ω y) from tsum_congr he,
-      tsum_mul_right]
-    have hcol : (∑' x, binConnectionMass G n ω x y) = binSizeMass G n ω y := by
-      simp only [binConnectionMass_symm G n ω,binSizeMass]
-    rw [hcol]
-    have hs : binSizeMass G n ω y ≠ 0 := by
-      rw [binSizeMass_eq G n ω y hb]
-      exact Nat.cast_ne_zero.mpr (Nat.ne_of_gt (hn.trans_le hb.2.1))
-    exact mul_div_cancel₀ _ hs
-  · have he : ∀ x, binOrbitTransportMass G n ω x y = 0 := by
-      intro x
-      simp only [binOrbitTransportMass,binOrbitRowMass_zero G n ω y hb,mul_zero,zero_div]
-    simp only [he,tsum_zero,binOrbitRowMass_zero G n ω y hb]
-
-theorem binOrbitTransportMass_row (G : BondGraph V E)
-    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V)
-    (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i)) (n : ℕ) (hn : 0 < n) (ω : Set E) (x : V) :
-    (if ClusterWindow G x n (2*n) ω then (n:ℝ)/Fintype.card ι else 0) ≤
-      ∑' y, binOrbitTransportMass G n ω x y := by
-  by_cases hb : ClusterWindow G x n (2*n) ω
-  · rw [ite_eq_left hb]
-    let C := hb.1.toFinset
-    have hs : (0:ℝ) < (cluster G ω x).ncard := Nat.cast_pos.mpr (hn.trans_le hb.2.1)
-    have ht : (0:ℝ) < Fintype.card ι := Nat.cast_pos.mpr Fintype.card_pos
-    have hqy (y : V) (hy : y ∈ C) : binOrbitRowMass G n ω y =
-        ∑ z ∈ C, if z ∈ vertexOrbit G y then (1:ℝ) else 0 := by
-      have hc : Conn G ω x y := hb.1.mem_toFinset.mp hy
-      have he := cluster_eq_of_conn G ω hc
-      have hby : ClusterWindow G y n (2*n) ω := by
-        unfold ClusterWindow at hb ⊢
-        simpa only [←he] using hb
-      rw [binOrbitRowMass_eq G n ω y hby]
-      have hC : hby.1.toFinset = C := by
-        apply Finset.coe_injective
-        simpa only [C,Set.Finite.coe_toFinset] using he.symm
-      rw [hC]
-    have he : (∑' y, binOrbitTransportMass G n ω x y) =
-        (∑ y ∈ C, ∑ z ∈ C, if z ∈ vertexOrbit G y then (1:ℝ) else 0)/(cluster G ω x).ncard := by
-      rw [tsum_eq_sum (s := C) (fun y hy => by
-        have hc : y ∉ cluster G ω x := fun h => hy (hb.1.mem_toFinset.mpr h)
-        simp only [binOrbitTransportMass,binMass_eq_of_window G n ω x y hb,ite_eq_right hc,zero_mul,zero_div])]
-      rw [Finset.sum_div]
-      apply Finset.sum_congr rfl
-      intro y hy
-      simp only [binOrbitTransportMass,binMass_eq_of_window G n ω x y hb,
-        ite_eq_left (hb.1.mem_toFinset.mp hy),one_mul,binSizeMass_eq G n ω x hb,hqy y hy]
-    rw [he]
-    apply (div_le_div_iff₀ ht hs).mpr
-    have h := finite_sameOrbit_pairs_lower G root hcover C
-    have hC : (C.card:ℝ) = (cluster G ω x).ncard := by
-      exact_mod_cast (Set.ncard_eq_toFinset_card _ hb.1).symm
-    rw [hC] at h
-    have hn' : (n:ℝ) ≤ (cluster G ω x).ncard := by exact_mod_cast hb.2.1
-    nlinarith [mul_le_mul_of_nonneg_right hn' hs.le]
-  · rw [ite_eq_right hb]
-    exact tsum_nonneg (binOrbitTransportMass_nonneg G n ω x)
-
-end BinOrbitTransport
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Kernel.Covariance
+
+namespace OAI
+
+/-! Connections for bond percolation. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section ClusterWindowLaw
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+def ClusterWindow (G : BondGraph V E) (x : V) (lo hi : ℕ) (ω : Set E) : Prop :=
+  (cluster G ω x).Finite ∧ lo ≤ (cluster G ω x).ncard ∧ (cluster G ω x).ncard < hi
+
+theorem clusterWindow_local (G : BondGraph V E) (hloc : LocallyFinite G)
+    (x : V) (lo hi : ℕ) (ω : Set E) :
+    let Q := incidentSupport G hloc (finite_reachBall hloc x hi).toFinset
+    ClusterWindow G x lo hi ω ↔ ClusterWindow G x lo hi (ω ∩ (Q : Set E)) := by
+  let Q := incidentSupport G hloc (finite_reachBall hloc x hi).toFinset
+  change ClusterWindow G x lo hi ω ↔ ClusterWindow G x lo hi (ω ∩ (Q : Set E))
+  constructor
+  · rintro ⟨hf,hl,hh⟩
+    have hs : cluster G ω x ⊆ (finite_reachBall hloc x hi).toFinset := by
+      simpa only [Set.Finite.coe_toFinset] using finite_cluster_subset_reachBall G ω x hf hi hh.le
+    have he := cluster_restrict_incident_of_full G hloc _ ω x hs
+    change cluster G (ω ∩ (Q : Set E)) x = cluster G ω x at he
+    exact ⟨he.symm ▸ hf,he.symm ▸ hl,he.symm ▸ hh⟩
+  · rintro ⟨hf,hl,hh⟩
+    have he := cluster_restrict_of_cap G hloc Q ω x hi (Finset.Subset.refl _) hf hh.le
+    exact ⟨he ▸ hf,he ▸ hl,he ▸ hh⟩
+
+theorem measurableSet_clusterWindow (G : BondGraph V E) (hloc : LocallyFinite G)
+    (x : V) (lo hi : ℕ) : MeasurableSet {ω | ClusterWindow G x lo hi ω} := by
+  let Q := incidentSupport G hloc (finite_reachBall hloc x hi).toFinset
+  have he : {ω | ClusterWindow G x lo hi ω} =
+      {ω | ω ∩ (Q : Set E) ∈ {η | ClusterWindow G x lo hi η}} := by
+    ext ω
+    exact clusterWindow_local G hloc x lo hi ω
+  rw [he]
+  exact measurableSet_finite_restriction Q _
+
+noncomputable def clusterWindowProbability (G : BondGraph V E) (p : unitInterval)
+    (x : V) (lo hi : ℕ) : ℝ := (law p).real {ω | ClusterWindow G x lo hi ω}
+
+theorem finiteClusterWindowMass_eq (G : BondGraph V E) (hloc : LocallyFinite G)
+    (Q : Finset E) (p : unitInterval) (x : V) (lo hi : ℕ)
+    (hQ : incidentSupport G hloc (finite_reachBall hloc x hi).toFinset ⊆ Q) :
+    finiteClusterWindowMass G Q p x lo hi = clusterWindowProbability G p x lo hi := by
+  let S := incidentSupport G hloc (finite_reachBall hloc x hi).toFinset
+  let f : Set E → ℝ := fun ω => if ClusterWindow G x lo hi ω then 1 else 0
+  have hf (ω) : f ω = f (ω ∩ (S : Set E)) :=
+    if_congr (clusterWindow_local G hloc x lo hi ω) rfl rfl
+  have he := integral_finite_local Q p f (finite_local_of_subset S Q hQ f hf)
+  trans ∫ ω, f ω ∂law p
+  · rw [he]
+    unfold finiteClusterWindowMass finiteFieldIntegral
+    apply Finset.sum_congr rfl
+    intro η _
+    congr 1
+    unfold f ClusterWindow
+    dsimp only
+    split_ifs <;> rfl
+  change (∫ ω, ({η | ClusterWindow G x lo hi η}.indicator (1 : Set E → ℝ)) ω ∂law p) = _
+  exact integral_indicator_one (measurableSet_clusterWindow G hloc x lo hi)
+
+theorem clusterWindowProbability_nonneg (G : BondGraph V E) (p : unitInterval)
+    (x : V) (lo hi : ℕ) : 0 ≤ clusterWindowProbability G p x lo hi := ENNReal.toReal_nonneg
+
+theorem clusterWindowProbability_mono (G : BondGraph V E) (p : unitInterval)
+    (x : V) {a b c d : ℕ} (hac : c ≤ a) (hbd : b ≤ d) :
+    clusterWindowProbability G p x a b ≤ clusterWindowProbability G p x c d := by
+  apply measureReal_mono
+  · intro ω h
+    exact ⟨h.1,hac.trans h.2.1,h.2.2.trans_le hbd⟩
+  · exact measure_ne_top _ _
+
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
+
+theorem finiteClusterWindowMass_amplification (o : V)
+    (q : LocalEndpointRule G hloc (vertexOrbit G o) M L) (k n : ℕ) (p : unitInterval) (x : V) :
+    finiteClusterWindowMass G (q.amplificationBonds k n x) p x (n-k*M) (2*n) =
+      clusterWindowProbability G p x (n-k*M) (2*n) := by
+  apply finiteClusterWindowMass_eq G hloc
+  exact incidentSupport_ball_mono G hloc x (by dsimp [amplificationRadius]; omega)
+
+end LocalEndpointRule
+end ClusterWindowLaw
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section QuantitativeAmplificationMoment
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+noncomputable def amplificationMomentFactor (k M d : ℕ) (c a t : ℝ) : ℝ :=
+  4 + (((k*M)+(k*M)^2+d*(k*M)^3:ℕ):ℝ) + (2*c*(k*M:ℕ))^2*t/a
+
+theorem amplificationMomentFactor_pos (k M d : ℕ) (c a t : ℝ) (ha : 0 < a) (ht : 0 ≤ t) :
+    0 < amplificationMomentFactor k M d c a t := by
+  unfold amplificationMomentFactor
+  positivity
+
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
+
+theorem amplificationLoad_moment_factor [Nonempty E] (hconn : Connected G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
+    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (i₀ : ι) (hw₀ : 0 < weight i₀)
+    (q : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) M L) (hL : 1 ≤ L)
+    (p : unitInterval) (k n d : ℕ) (hn : 0 < n) (hdeg : ∀ x, degree G x ≤ d)
+    (a t π : ℝ) (ha : 0 < a) (ht : 0 ≤ t) (hπ : 0 ≤ π)
+    (hbin : a*n*π ≤ binOrbitRow G p (root i₀) n)
+    (hwindow : clusterWindowProbability G p (root i₀) (n-k*M) (2*n) ≤ t*π)
+    (hcode : inverseCodeMass M L^k ≤ n*q.successProbability p (root i₀)^k) :
+    (∫ ω, (q.amplificationLoad (root i₀) hL k n p).row ω (root i₀)^2 ∂law p) ≤
+      amplificationMomentFactor k M d c a t * q.successProbability p (root i₀)^k *
+        (q.successProbability p (root i₀)^k * binOrbitRow G p (root i₀) n / n) := by
+  let u := q.successProbability p (root i₀)
+  let B := u^k * binOrbitRow G p (root i₀) n / n
+  let D : ℝ := (((k*M)+(k*M)^2+d*(k*M)^3:ℕ):ℝ)
+  have hu : 0 ≤ u^k := pow_nonneg (q.successProbability_nonneg p _) _
+  have hn' : (0:ℝ) < n := Nat.cast_pos.mpr hn
+  have hB : a*π*u^k ≤ B := by
+    apply (le_div_iff₀ hn').mpr
+    calc
+      _ = u^k*(a*n*π) := by ring
+      _ ≤ _ := mul_le_mul_of_nonneg_left hbin hu
+  have hB0 : 0 ≤ B := (mul_nonneg (mul_nonneg ha.le hπ) hu).trans hB
+  have hcoef : 4*u^k+D*inverseCodeMass M L^k/n ≤ (4+D)*u^k := by
+    have h := mul_le_mul_of_nonneg_left hcode (show 0 ≤ D by positivity)
+    have h' : D*inverseCodeMass M L^k/n ≤ D*u^k := by
+      apply (div_le_iff₀ hn').mpr
+      nlinarith only [h]
+    nlinarith only [h']
+  have hrem : (2*(c*((k*M:ℕ)*u^k)))^2*(t*π) ≤
+      ((2*c*(k*M:ℕ))^2*t/a)*u^k*B := by
+    have h := mul_le_mul_of_nonneg_left hB
+      (show 0 ≤ ((2*c*(k*M:ℕ))^2*t/a)*u^k from
+        mul_nonneg (div_nonneg (mul_nonneg (sq_nonneg _) ht) ha.le) hu)
+    calc
+      _ = (((2*c*(k*M:ℕ))^2*t/a)*u^k)*(a*π*u^k) := by
+        field_simp
+      _ ≤ _ := h
+  have hself : root i₀ ∈ vertexOrbit G (root i₀) := by
+    refine ⟨Equiv.refl V,Equiv.refl E,?_,rfl⟩
+    intro e
+    simp
+  have he := q.amplificationLoad_mean root weight Γ htr
+    (fun e he => hΓ he) hw i₀ hw₀ hL p k n (root i₀) hself
+  have hm := q.amplificationLoad_second_moment_le hconn root weight Γ hΓ htr hcov hw hw1
+    c hc hmin i₀ hw₀ hL p (root i₀) k n d hn hdeg
+  rw [he,q.finiteClusterWindowMass_amplification] at hm
+  change _ ≤ (4*u^k+D*inverseCodeMass M L^k/n)*B +
+    (2*(c*((k*M:ℕ)*u^k)))^2*clusterWindowProbability G p (root i₀) (n-k*M) (2*n) at hm
+  calc
+    _ ≤ _ := hm
+    _ ≤ (4+D)*u^k*B + (2*(c*((k*M:ℕ)*u^k)))^2*(t*π) :=
+      add_le_add (mul_le_mul_of_nonneg_right hcoef hB0)
+        (mul_le_mul_of_nonneg_left hwindow (sq_nonneg _))
+    _ ≤ (4+D)*u^k*B + ((2*c*(k*M:ℕ))^2*t/a)*u^k*B := add_le_add (le_refl _) hrem
+    _ = _ := by dsimp [amplificationMomentFactor,D,B,u]; ring
+
+end LocalEndpointRule
+end QuantitativeAmplificationMoment
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section QuantitativeAmplificationStep
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
+
+ 
+
+theorem quantitative_amplification_step [Nonempty E] (hconn : Connected G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
+    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (i₀ : ι) (hw₀ : 0 < weight i₀)
+    (q : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) M L) (hL : 1 ≤ L)
+    (p : unitInterval) (k n d : ℕ) (hn : 0 < n) (hdeg : ∀ x, degree G x ≤ d)
+    (hu : 0 < q.successProbability p (root i₀))
+    (a t π L' : ℝ) (ha : 0 < a) (ht : 0 ≤ t) (hπ : 0 < π)
+    (hbin : a*n*π ≤ binOrbitRow G p (root i₀) n)
+    (hwindow : clusterWindowProbability G p (root i₀) (n-k*M) (2*n) ≤ t*π)
+    (hcode : inverseCodeMass M L^k ≤ n*q.successProbability p (root i₀)^k)
+    (hbad : 16*amplificationMomentFactor k M d c a t *
+      (law p).real {ω | CappedBadGoodness G ω (root i₀) (2*n+k*M) L'} ≤ a*π)
+    (A : Hilbert V →L[ℂ] Hilbert V)
+    (hA : ∀ x y, matrixCoeff A x y = (q.normalizedKernel (root i₀) p x y : ℂ)) :
+    ∃ q' : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) (2*n+k*M) L',
+      a*π/(8*amplificationMomentFactor k M d c a t) ≤ q'.successProbability p (root i₀) ∧
+      ∃ A' : Hilbert V →L[ℂ] Hilbert V, 0 < ‖A'‖ ∧ ‖A'‖ ≤ 2*‖A‖^k ∧
+        ∀ x y, matrixCoeff A' x y = (q'.normalizedKernel (root i₀) p x y : ℂ) := by
+  let Q := amplificationMomentFactor k M d c a t
+  let u := q.successProbability p (root i₀)
+  let B := u^k*binOrbitRow G p (root i₀) n / n
+  let b := 4*Q*u^k
+  have hQ : 0 < Q := amplificationMomentFactor_pos k M d c a t ha ht
+  have huk : 0 < u^k := pow_pos hu _
+  have hb : 0 < b := mul_pos (mul_pos (by norm_num) hQ) huk
+  have hn' : (0:ℝ) < n := Nat.cast_pos.mpr hn
+  have hbin0 : 0 < binOrbitRow G p (root i₀) n :=
+    (mul_pos (mul_pos ha hn') hπ).trans_le hbin
+  have hB : a*π*u^k ≤ B := by
+    apply (le_div_iff₀ hn').mpr
+    calc
+      _ = u^k*(a*n*π) := by ring
+      _ ≤ _ := mul_le_mul_of_nonneg_left hbin huk.le
+  have hm := q.amplificationLoad_moment_factor hconn root weight Γ hΓ htr hcov hw hw1
+    c hc hmin i₀ hw₀ hL p k n d hn hdeg a t π ha ht hπ.le hbin hwindow hcode
+  have hmoment : (∫ ω, (q.amplificationLoad (root i₀) hL k n p).row ω (root i₀)^2 ∂law p)/b ≤ B/4 := by
+    apply (div_le_iff₀ hb).mpr
+    change _ ≤ Q*u^k*B at hm
+    calc
+      _ ≤ Q*u^k*B := hm
+      _ = _ := by dsimp [b]; ring
+  have hbad' : b*(law p).real {ω | CappedBadGoodness G ω (root i₀) (2*n+k*M) L'} ≤ B/4 := by
+    have hh := mul_le_mul_of_nonneg_right hbad huk.le
+    dsimp [b]
+    nlinarith only [hh,hB]
+  obtain ⟨q',hq',A',hA'pos,hA'norm,hA'⟩ := q.amplification_truncation_step root weight Γ htr
+    (fun e he => hΓ he) hw i₀ hw₀ hL p k n hn hu hbin0 b L' hb hmoment hbad' A hA
+  refine ⟨q',?_,A',hA'pos,hA'norm,hA'⟩
+  apply le_trans _ hq'
+  change a*π/(8*Q) ≤ B/(2*b)
+  apply (div_le_div_iff₀ (mul_pos (by norm_num) hQ) (mul_pos (by norm_num) hb)).mpr
+  have hh := mul_le_mul_of_nonneg_left hB (show 0 ≤ 8*Q by positivity)
+  dsimp [b]
+  nlinarith only [hh]
+
+end LocalEndpointRule
+end QuantitativeAmplificationStep
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section ConnectionAdjustedRule
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+noncomputable def cappedConnectionMass (G : BondGraph V E) (p : unitInterval)
+    (S : V → V → ℝ) (c : ℝ) (M : ℕ) (L : ℝ) (ω : Set E) (x y : V) : ℝ :=
+  if CappedGoodness G ω x M L ∧ Conn G ω x y then c*S x y/twoPoint G p x y else 0
+
+theorem cappedConnectionMass_nonneg (G : BondGraph V E) (p : unitInterval)
+    (S : V → V → ℝ) (c : ℝ) (hc : 0 ≤ c) (hS : ∀ x y, 0 ≤ S x y)
+    (M : ℕ) (L : ℝ) (ω : Set E) (x y : V) : 0 ≤ cappedConnectionMass G p S c M L ω x y := by
+  unfold cappedConnectionMass
+  split_ifs
+  · exact div_nonneg (mul_nonneg hc (hS x y)) (twoPoint_nonneg G p x y)
+  · rfl
+
+theorem cappedConnectionMass_support (G : BondGraph V E) (p : unitInterval)
+    (S : V → V → ℝ) (c : ℝ) (M : ℕ) (L : ℝ) (ω : Set E) (x y : V)
+    (h : cappedConnectionMass G p S c M L ω x y ≠ 0) :
+    S x y ≠ 0 ∧ CappedGoodness G ω x M L ∧ Conn G ω x y := by
+  unfold cappedConnectionMass at h
+  split_ifs at h with hh
+  · refine ⟨?_,hh⟩
+    intro hz
+    simp only [hz,mul_zero,zero_div] at h
+    exact h rfl
+  · exact (h rfl).elim
+
+theorem cappedConnectionMass_le (G : BondGraph V E) (p : unitInterval)
+    (S : V → V → ℝ) (c : ℝ) (hc : 0 < c) (hS : ∀ x y, 0 ≤ S x y)
+    (hτ : ∀ x y, S x y ≠ 0 → c ≤ twoPoint G p x y)
+    (M : ℕ) (L : ℝ) (ω : Set E) (x y : V) : cappedConnectionMass G p S c M L ω x y ≤ S x y := by
+  by_cases hs : S x y = 0
+  · simp only [cappedConnectionMass,hs,mul_zero,zero_div,ite_self,le_refl]
+  · unfold cappedConnectionMass
+    split_ifs
+    · apply (div_le_iff₀ (hc.trans_le (hτ x y hs))).mpr
+      exact (mul_le_mul_of_nonneg_right (hτ x y hs) (hS x y)).trans_eq (mul_comm _ _)
+    · exact hS x y
+
+theorem cappedConnectionMass_local (G : BondGraph V E) (hloc : LocallyFinite G)
+    (p : unitInterval) (S : V → V → ℝ) (c : ℝ) (M : ℕ) (L : ℝ) (ω : Set E) (x y : V) :
+    let Q := incidentSupport G hloc (finite_reachBall hloc x M).toFinset
+    cappedConnectionMass G p S c M L ω x y = cappedConnectionMass G p S c M L (ω ∩ (Q:Set E)) x y := by
+  let Q := incidentSupport G hloc (finite_reachBall hloc x M).toFinset
+  change cappedConnectionMass G p S c M L ω x y = cappedConnectionMass G p S c M L (ω ∩ (Q:Set E)) x y
+  have hg := cappedGoodness_local G hloc ω x M L
+  change CappedGoodness G ω x M L ↔ CappedGoodness G (ω ∩ (Q:Set E)) x M L at hg
+  unfold cappedConnectionMass
+  apply if_congr _ rfl rfl
+  constructor
+  · rintro ⟨h,hxy⟩
+    have h' := hg.mp h
+    have he := cluster_restrict_of_cap G hloc Q ω x M (Finset.Subset.refl _) h'.1 h'.2.1
+    refine ⟨h',?_⟩
+    change y ∈ cluster G (ω ∩ (Q:Set E)) x
+    rw [he]
+    exact hxy
+  · rintro ⟨h,hxy⟩
+    exact ⟨hg.mpr h,conn_mono G Set.inter_subset_left hxy⟩
+
+theorem cappedConnectionMass_covariant [Countable V] [Countable E]
+    (G : BondGraph V E) (p : unitInterval) (S : V → V → ℝ) (c : ℝ) (M : ℕ) (L : ℝ)
+    (a : V ≃ V) (b : E ≃ E) (hab : IsAut G a b)
+    (hS : ∀ x y, S (a x) (a y) = S x y) (ω : Set E) (x y : V) :
+    cappedConnectionMass G p S c M L (b ⁻¹' ω) x y = cappedConnectionMass G p S c M L ω (a x) (a y) := by
+  simp only [cappedConnectionMass,cappedGoodness_aut_iff G hab,conn_aut_iff G hab,
+    hS,twoPoint_aut G p hab]
+
+noncomputable def connectionAdjustedRule [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (O : Set V) (p : unitInterval)
+    (S : V → V → ℝ) (c : ℝ) (hc : 0 < c) (hS : ∀ x y, 0 ≤ S x y)
+    (hSsum : ∀ x, Summable (S x)) (hrow : ∀ x, ∑' y, S x y ≤ 1)
+    (hτ : ∀ x y, S x y ≠ 0 → c ≤ twoPoint G p x y)
+    (hO : ∀ x y, S x y ≠ 0 → x ∈ O ∧ y ∈ O)
+    (hcov : ∀ a b, IsAut G a b → ∀ x y, S (a x) (a y) = S x y)
+    (M : ℕ) (L : ℝ) : LocalEndpointRule G hloc O M L where
+  mass := cappedConnectionMass G p S c M L
+  radius := M
+  nonneg := cappedConnectionMass_nonneg G p S c hc.le hS M L
+  support := by
+    intro ω x y h
+    obtain ⟨hs,hg,hxy⟩ := cappedConnectionMass_support G p S c M L ω x y h
+    exact ⟨(hO x y hs).1,(hO x y hs).2,hxy,hg⟩
+  locality := fun ω x y => cappedConnectionMass_local G hloc p S c M L ω x y
+  covariance := fun a b hab ω x y => cappedConnectionMass_covariant G p S c M L a b hab (hcov a b hab) ω x y
+  row_le_one := by
+    intro ω x
+    have hs : Summable (cappedConnectionMass G p S c M L ω x) := by
+      apply summable_of_ne_finset_zero (s := (finite_reachBall hloc x M).toFinset)
+      intro y hy
+      by_contra hn
+      obtain ⟨_,hg,hxy⟩ := cappedConnectionMass_support G p S c M L ω x y hn
+      apply hy
+      simp only [Set.Finite.mem_toFinset]
+      exact finite_cluster_subset_reachBall G ω x hg.1 M hg.2.1 hxy
+    exact (hs.tsum_le_tsum (cappedConnectionMass_le G p S c hc hS hτ M L ω x) (hSsum x)).trans (hrow x)
+
+end ConnectionAdjustedRule
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section ConnectionAdjustedKernel
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Countable V] [Countable E]
+
+theorem integral_cappedConnectionMass (G : BondGraph V E) (hloc : LocallyFinite G)
+    (p : unitInterval) (S : V → V → ℝ) (c : ℝ) (M : ℕ) (L : ℝ) (x y : V) :
+    (∫ ω, cappedConnectionMass G p S c M L ω x y ∂law p) =
+      c*S x y/twoPoint G p x y * (law p).real {ω | CappedGoodness G ω x M L ∧ Conn G ω x y} := by
+  have hm : MeasurableSet {ω | CappedGoodness G ω x M L ∧ Conn G ω x y} :=
+    (measurableSet_capped_good G hloc x M L).inter (measurableSet_conn G x y)
+  simpa only [cappedConnectionMass,Set.indicator,Set.mem_ofPred_eq,smul_eq_mul,mul_comm] using
+    integral_indicator_const (μ := law p) (c*S x y/twoPoint G p x y) hm
+
+theorem integral_cappedConnectionMass_bounds (G : BondGraph V E) (hloc : LocallyFinite G)
+    (p : unitInterval) (S : V → V → ℝ) (c : ℝ) (hc : 0 < c) (hS : ∀ x y, 0 ≤ S x y)
+    (hτ : ∀ x y, S x y ≠ 0 → c ≤ twoPoint G p x y)
+    (M : ℕ) (L : ℝ) (x y : V) :
+    let K := ∫ ω, cappedConnectionMass G p S c M L ω x y ∂law p
+    K ≤ c*S x y ∧ c*S x y ≤ K + S x y * (law p).real {ω | ¬ CappedGoodness G ω x M L} := by
+  dsimp only
+  rw [integral_cappedConnectionMass G hloc]
+  by_cases hs : S x y = 0
+  · simp only [hs,mul_zero,zero_div,zero_mul,zero_add,le_refl,and_self]
+  have ht : 0 < twoPoint G p x y := hc.trans_le (hτ x y hs)
+  let a := c*S x y/twoPoint G p x y
+  have ha : 0 ≤ a := div_nonneg (mul_nonneg hc.le (hS x y)) ht.le
+  have hle : a ≤ S x y := by
+    apply (div_le_iff₀ ht).mpr
+    exact (mul_le_mul_of_nonneg_right (hτ x y hs) (hS x y)).trans_eq (mul_comm _ _)
+  have he : a*twoPoint G p x y = c*S x y := div_mul_cancel₀ _ ht.ne'
+  have hsub : {ω : Set E | CappedGoodness G ω x M L ∧ Conn G ω x y} ⊆ {ω | Conn G ω x y} :=
+    fun _ h => h.2
+  have hm : (law p).real {ω | CappedGoodness G ω x M L ∧ Conn G ω x y} ≤ twoPoint G p x y := by
+    exact measureReal_mono hsub (measure_ne_top _ _)
+  have hcover : {ω : Set E | Conn G ω x y} ⊆
+      {ω | CappedGoodness G ω x M L ∧ Conn G ω x y} ∪ {ω | ¬ CappedGoodness G ω x M L} := by
+    intro ω h
+    by_cases hg : CappedGoodness G ω x M L
+    · exact Or.inl ⟨hg,h⟩
+    · exact Or.inr hg
+  have hb : twoPoint G p x y ≤
+      (law p).real {ω | CappedGoodness G ω x M L ∧ Conn G ω x y} +
+        (law p).real {ω | ¬ CappedGoodness G ω x M L} :=
+    (measureReal_mono hcover (measure_ne_top _ _)).trans (measureReal_union_le _ _)
+  constructor
+  · exact (mul_le_mul_of_nonneg_left hm ha).trans_eq he
+  · calc
+      c*S x y = a*twoPoint G p x y := he.symm
+      _ ≤ a*((law p).real {ω | CappedGoodness G ω x M L ∧ Conn G ω x y} +
+          (law p).real {ω | ¬ CappedGoodness G ω x M L}) := mul_le_mul_of_nonneg_left hb ha
+      _ ≤ a*(law p).real {ω | CappedGoodness G ω x M L ∧ Conn G ω x y} +
+          S x y*(law p).real {ω | ¬ CappedGoodness G ω x M L} := by
+        rw [mul_add]
+        exact add_le_add le_rfl (mul_le_mul_of_nonneg_right hle measureReal_nonneg)
+
+ 
+
+theorem connectionAdjustedRule_success_lower
+    (G : BondGraph V E) (hloc : LocallyFinite G) (O : Set V) (p : unitInterval)
+    (S : V → V → ℝ) (c : ℝ) (hc : 0 < c) (hS : ∀ x y, 0 ≤ S x y)
+    (hSsum : ∀ x, Summable (S x)) (hrow : ∀ x, ∑' y, S x y ≤ 1)
+    (hτ : ∀ x y, S x y ≠ 0 → c ≤ twoPoint G p x y)
+    (hO : ∀ x y, S x y ≠ 0 → x ∈ O ∧ y ∈ O)
+    (hcov : ∀ a b, IsAut G a b → ∀ x y, S (a x) (a y) = S x y)
+    (M : ℕ) (L : ℝ) (x : V) :
+    c*(∑' y, S x y) ≤
+      (connectionAdjustedRule G hloc O p S c hc hS hSsum hrow hτ hO hcov M L).successProbability p x +
+        (law p).real {ω | ¬ CappedGoodness G ω x M L} := by
+  let q := connectionAdjustedRule G hloc O p S c hc hS hSsum hrow hτ hO hcov M L
+  let ε := (law p).real {ω | ¬ CappedGoodness G ω x M L}
+  have h (y : V) : c*S x y ≤ q.kernel p x y + S x y * ε :=
+    (integral_cappedConnectionMass_bounds G hloc p S c hc hS hτ M L x y).2
+  have hh := ((hSsum x).mul_left c).tsum_le_tsum h
+    ((q.summable_kernel p x).add ((hSsum x).mul_right ε))
+  rw [(q.summable_kernel p x).tsum_add ((hSsum x).mul_right ε),tsum_mul_left,tsum_mul_right] at hh
+  exact hh.trans (add_le_add le_rfl ((mul_le_mul_of_nonneg_right (hrow x) measureReal_nonneg).trans_eq (one_mul ε)))
+
+end ConnectionAdjustedKernel
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section ConnectionAdjustedOperator
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Countable V] [Countable E]
+
+ 
+
+theorem connectionAdjustedRule_operator
+    (G : BondGraph V E) (hloc : LocallyFinite G) (o : V) (p : unitInterval)
+    (S : V → V → ℝ) (c : ℝ) (hc : 0 < c) (hS : ∀ x y, 0 ≤ S x y)
+    (hSsum : ∀ x, Summable (S x)) (hrow : ∀ x, ∑' y, S x y ≤ 1)
+    (hτ : ∀ x y, S x y ≠ 0 → c ≤ twoPoint G p x y)
+    (hO : ∀ x y, S x y ≠ 0 → x ∈ vertexOrbit G o ∧ y ∈ vertexOrbit G o)
+    (hcov : ∀ a b, IsAut G a b → ∀ x y, S (a x) (a y) = S x y)
+    (M : ℕ) (L b : ℝ) (hb : 0 < b) (hret : b ≤ ∑' y, S o y)
+    (hbad : (law p).real {ω | ¬ CappedGoodness G ω o M L} ≤ c*b/2)
+    (A : Hilbert V →L[ℂ] Hilbert V)
+    (hA : ∀ x y, matrixCoeff A x y = (S x y : ℂ)) :
+    ∃ q : LocalEndpointRule G hloc (vertexOrbit G o) M L,
+      c*b/2 ≤ q.successProbability p o ∧
+      ∃ T : Hilbert V →L[ℂ] Hilbert V, 0 < ‖T‖ ∧ ‖T‖ ≤ 2*‖A‖/b ∧
+        ∀ x y, matrixCoeff T x y = (q.normalizedKernel o p x y : ℂ) := by
+  let q := connectionAdjustedRule G hloc (vertexOrbit G o) p S c hc hS hSsum hrow hτ hO hcov M L
+  have hlo := connectionAdjustedRule_success_lower G hloc (vertexOrbit G o) p S c hc hS hSsum hrow hτ hO hcov M L o
+  have hs : c*b/2 ≤ q.successProbability p o := by
+    have hmul := mul_le_mul_of_nonneg_left hret hc.le
+    change _ ≤ q.successProbability p o + _ at hlo
+    linarith
+  have hu : 0 < q.successProbability p o := (by positivity : 0 < c*b/2).trans_le hs
+  have hcA (x y : V) : matrixCoeff ((c:ℂ) • A) x y = (c*S x y : ℝ) := by
+    change (c:ℂ)*matrixCoeff A x y = _
+    rw [hA,Complex.ofReal_mul]
+  obtain ⟨B,hBn,hB⟩ := exists_operator_of_matrix_domination
+    (fun x y => c*S x y) (q.kernel p) (fun x y => mul_nonneg hc.le (hS x y))
+    (fun x y => by
+      rw [abs_of_nonneg (q.kernel_nonneg p x y)]
+      exact (integral_cappedConnectionMass_bounds G hloc p S c hc hS hτ M L x y).1)
+    ((c:ℂ) • A) hcA
+  have hBn' : ‖B‖ ≤ c*‖A‖ := by
+    simpa only [norm_smul,Complex.norm_of_nonneg hc.le] using hBn
+  obtain ⟨T,hTn,hT⟩ := q.normalize_mass_operator o p hu B _ hBn' hB
+  refine ⟨q,hs,T,q.normalized_operator_norm_pos o p hu T hT,hTn.trans ?_,hT⟩
+  apply (div_le_iff₀ hu).mpr
+  rw [div_mul_eq_mul_div]
+  apply (le_div_iff₀ hb).mpr
+  have hh := mul_le_mul_of_nonneg_right hs (show 0 ≤ 2*‖A‖ by positivity)
+  nlinarith only [hh]
+
+end ConnectionAdjustedOperator
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter Real
+open scoped Topology
+section CappedGoodnessLimit
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Countable V] [Countable E]
+
+omit [Countable V] in
+theorem clusterLarge_probability_tendsto_zero (G : BondGraph V E) (p : unitInterval) (x : V)
+    (hfin : ∀ᵐ ω ∂law p, (cluster G ω x).Finite) :
+    Tendsto (fun M : ℕ => (law p).real (clusterLarge G x M)) atTop (𝓝 0) := by
+  have hz : law p {ω | (cluster G ω x).Infinite} = 0 := by
+    simpa only [Set.Infinite] using (ae_iff.mp hfin)
+  have ht := (ENNReal.tendsto_toReal (measure_ne_top (law p) (⋂ M, clusterLarge G x M))).comp
+    (tendsto_measure_iInter_atTop (fun M => (measurableSet_clusterLarge G x M).nullMeasurableSet)
+      (clusterLarge_antitone G x) ⟨0,measure_ne_top _ _⟩)
+  rw [← infinite_cluster_eq_iInter,hz,ENNReal.toReal_zero] at ht
+  exact ht
+
+omit [Countable V] [Countable E] in
+theorem cappedGoodness_failure_le (G : BondGraph V E) (p : unitInterval)
+    (x : V) (M : ℕ) (L : ℝ) :
+    (law p).real {ω | ¬ CappedGoodness G ω x M L} ≤
+      (law p).real (clusterLarge G x M) + (law p).real {ω | CappedBadGoodness G ω x M L} := by
+  have hsub : {ω : Set E | ¬ CappedGoodness G ω x M L} ⊆
+      clusterLarge G x M ∪ {ω | CappedBadGoodness G ω x M L} := by
+    intro ω hbad
+    by_cases hf : (cluster G ω x).Finite
+    · by_cases hM : (cluster G ω x).ncard ≤ M
+      · exact Or.inr ⟨hf,hM,fun hg => hbad ⟨hf,hM,hg⟩⟩
+      · apply Or.inl
+        refine ⟨hf.toFinset,?_,fun y hy => hf.mem_toFinset.mp hy⟩
+        rw [← Set.ncard_eq_toFinset_card _ hf]
+        omega
+    · obtain ⟨t,ht,hcard⟩ := (show (cluster G ω x).Infinite from hf).exists_subset_card_eq M
+      exact Or.inl ⟨t,hcard.ge,ht⟩
+  exact (measureReal_mono hsub (measure_ne_top _ _)).trans (measureReal_union_le _ _)
+
+omit [Countable V] in
+ 
+
+theorem cappedGoodness_failure_eventually_small [Nonempty V] (G : BondGraph V E)
+    (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : (p:ℝ) < 1) (x : V)
+    (hfin : ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
+    (ε : ℝ) (hε : 0 < ε) (hε1 : ε < 1/2) (δ : ℝ) (hδ : 0 < δ) :
+    ∀ᶠ M : ℕ in atTop, (law p).real {ω | ¬ CappedGoodness G ω x M ((M:ℝ)^(1/2+ε))} ≤ δ := by
+  have htail := (clusterLarge_probability_tendsto_zero G p x hfin).eventually
+    (Iio_mem_nhds (show (0:ℝ) < δ/2 by positivity))
+  have hr : Tendsto (fun M : ℕ => (M:ℝ)^ε) atTop atTop :=
+    (tendsto_rpow_atTop hε).comp tendsto_natCast_atTop_atTop
+  have hexp : Tendsto (fun M : ℕ => Real.exp (-(M:ℝ)^ε)) atTop (𝓝 0) :=
+    Real.tendsto_exp_atBot.comp (tendsto_neg_atTop_atBot.comp hr)
+  have he := hexp.eventually (Iio_mem_nhds (show (0:ℝ) < δ/2 by positivity))
+  filter_upwards [htail,he,bad_goodness_stretched_exp G hloc hqt p hp ε hε hε1] with M ht he hg
+  have hbad := hg x p le_rfl
+  have hle := cappedGoodness_failure_le G p x M ((M:ℝ)^(1/2+ε))
+  linarith
+
+end CappedGoodnessLimit
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section WalkPowerSupport
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem walkKernel_zero_of_not_step (G : BondGraph V E) (hloc : LocallyFinite G)
+    (a : ℝ) (x y : V) (hxy : x ≠ y) (hadj : ¬ Adj G x y) :
+    walkKernel G hloc a x y = 0 := by simp only [walkKernel,ite_eq_right hxy,ite_eq_right hadj,zero_add]
+
+theorem walkKernel_hasSum (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ) (x : V) :
+    HasSum (walkKernel G hloc a x) 1 := by
+  rw [← walkKernel_row_sum G hloc a x]
+  apply hasSum_sum_of_ne_finset_zero
+  intro y hy
+  simp only [Finset.mem_insert,mem_neighbors,not_or] at hy
+  exact walkKernel_zero_of_not_step G hloc a x y (Ne.symm hy.1) hy.2
+
+theorem walkPower_support (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (k : ℕ) (x y : V) (hy : y ∉ reachBall G x k) : (matrixCoeff (P^k) x y).re = 0 := by
+  induction k generalizing y with
+  | zero =>
+    have he : y ≠ x := by simpa only [reachBall,Set.mem_singleton_iff] using hy
+    simp only [pow_zero,matrixCoeff,one_apply_eq_self,lp.single_apply,Pi.single_apply,ite_eq_right (Ne.symm he),Complex.zero_re]
+  | succ k ih =>
+    rw [pow_succ,← ((hpos.pow k).mul_hasSum P x y).tsum_eq]
+    rw [← tsum_zero]
+    apply tsum_congr
+    intro z
+    by_cases hz : z ∈ reachBall G x k
+    · have hzy : z ≠ y := by intro he; subst y; exact hy (Or.inl hz)
+      have hadj : ¬ Adj G z y := by
+        intro h
+        apply hy
+        apply Or.inr
+        simp only [Set.mem_iUnion,Set.mem_ofPred_eq]
+        exact ⟨z,hz,h⟩
+      rw [hP,walkKernel_zero_of_not_step G hloc a z y hzy hadj,Complex.ofReal_zero,Complex.zero_re,mul_zero]
+    · rw [ih z hz,zero_mul]
+
+theorem walkPower_hasSum (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (k : ℕ) (x : V) : HasSum (fun y => (matrixCoeff (P^k) x y).re) 1 := by
+  induction k with
+  | zero =>
+    have h : HasSum (fun y : V => if x=y then (1:ℝ) else 0) 1 := by
+      exact hasSum_ite_eq' x (1:ℝ)
+    simpa only [pow_zero,matrixCoeff,one_apply_eq_self,lp.single_apply,Pi.single_apply,apply_ite,Complex.one_re,Complex.zero_re] using h
+  | succ k ih =>
+    let B := (finite_reachBall hloc x k).toFinset
+    have he (y : V) : (matrixCoeff (P^(k+1)) x y).re =
+        ∑ z ∈ B, (matrixCoeff (P^k) x z).re * walkKernel G hloc a z y := by
+      rw [pow_succ,← ((hpos.pow k).mul_hasSum P x y).tsum_eq]
+      simp only [hP,Complex.ofReal_re]
+      apply tsum_eq_sum
+      intro z hz
+      rw [walkPower_support G hloc a P hpos hP k x z (by simpa only [B,Set.Finite.mem_toFinset] using hz),zero_mul]
+    have hs := hasSum_sum (s := B) (fun z _ => (walkKernel_hasSum G hloc a z).mul_left (matrixCoeff (P^k) x z).re)
+    simp only [mul_one] at hs
+    have hr : (∑ z ∈ B, (matrixCoeff (P^k) x z).re) = 1 := by
+      rw [← ih.tsum_eq]
+      exact (tsum_eq_sum fun z hz => walkPower_support G hloc a P hpos hP k x z
+        (by simpa only [B,Set.Finite.mem_toFinset] using hz)).symm
+    simpa only [← he,hr] using hs
+
+end WalkPowerSupport
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section WalkCovariance
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem neighbors_aut_card (G : BondGraph V E) (hloc : LocallyFinite G)
+    {g : V ≃ V} {b : E ≃ E} (hg : IsAut G g b) (x : V) :
+    (neighbors G hloc (g x)).card = (neighbors G hloc x).card := by
+  have he : neighbors G hloc (g x) = (neighbors G hloc x).map g.toEmbedding := by
+    ext y
+    simp only [mem_neighbors,Finset.mem_map,Equiv.coe_toEmbedding]
+    constructor
+    · intro h
+      refine ⟨g.symm y,?_,by simp⟩
+      simpa only [g.symm_apply_apply] using adj_aut G (isAut_symm G hg) h
+    · rintro ⟨z,hz,rfl⟩
+      exact adj_aut G hg hz
+  rw [he,Finset.card_map]
+
+theorem walkKernel_covariant (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    {g : V ≃ V} {b : E ≃ E} (hg : IsAut G g b) (x y : V) :
+    walkKernel G hloc a (g x) (g y) = walkKernel G hloc a x y := by
+  have hadj : Adj G (g x) (g y) ↔ Adj G x y :=
+    ⟨fun h => by simpa using adj_aut G (isAut_symm G hg) h,adj_aut G hg⟩
+  simp only [walkKernel,g.apply_eq_iff_eq,neighbors_aut_card G hloc hg,hadj]
+
+theorem twoPoint_reachBall_lower [Countable E] (G : BondGraph V E) (p : unitInterval)
+    (x : V) (k : ℕ) (y : V) (hy : y ∈ reachBall G x k) :
+    (p:ℝ)^k ≤ twoPoint G p x y := by
+  induction k generalizing y with
+  | zero =>
+    simp only [reachBall,Set.mem_singleton_iff] at hy
+    subst y
+    simp only [pow_zero,twoPoint_self,le_refl]
+  | succ k ih =>
+    rcases hy with hy | hy
+    · have h : (p:ℝ)^(k+1) ≤ (p:ℝ)^k := by
+        rw [pow_succ]
+        exact mul_le_of_le_one_right (pow_nonneg p.2.1 k) p.2.2
+      exact h.trans (ih y hy)
+    · simp only [Set.mem_iUnion,Set.mem_ofPred_eq] at hy
+      obtain ⟨z,hz,hzy⟩ := hy
+      have h := mul_le_mul_of_nonneg_left (ih z hz) p.2.1
+      rw [←pow_succ'] at h
+      exact h.trans (twoPoint_adj_lower G p x z y hzy)
+
+end WalkCovariance
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section OrbitReturn
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+lemma finiteKernel_summable_mul (K : V → V → ℝ)
+    (hf : ∀ x, (Function.support (K x)).Finite) (x : V) (f : V → ℝ) :
+    Summable (fun y => K x y * f y) := by
+  apply summable_of_ne_finset_zero (s := (hf x).toFinset)
+  intro y hy
+  have hd : K x y = 0 := by simpa only [Set.Finite.mem_toFinset,Function.mem_support,not_not] using hy
+  rw [hd,zero_mul]
+
+ 
+
+theorem orbit_two_step_return_lower (G : BondGraph V E)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
+    (i₀ : ι) (hw₀ : 0 < weight i₀)
+    (K : V → V → ℝ)
+    (hfin : ∀ x, (Function.support (K x)).Finite)
+    (hrow : ∀ x, HasSum (K x) 1) (hsym : ∀ x y, K x y = K y x)
+    (hcov : ∀ a b, IsAut G a b → ∀ x y, K (a x) (a y) = K x y) :
+    (∑ i, if root i ∈ vertexOrbit G (root i₀) then weight i else 0) ≤
+      ∑' z, K (root i₀) z * ∑' y, if y ∈ vertexOrbit G (root i₀) then K z y else 0 := by
+  let O := vertexOrbit G (root i₀)
+  let I : V → ℝ := fun x => if x ∈ O then 1 else 0
+  let f : V → ℝ := fun x => ∑' y, K x y * I y
+  let g : V → ℝ := fun x => ∑' y, K x y * f y
+  let σ := ∑ i, weight i * I (root i)
+  have hI (a : V ≃ V) (b : E ≃ E) (hab : IsAut G a b) (x : V) : I (a x) = I x := by
+    simp only [I,O,mem_vertexOrbit_aut_iff G (root i₀) hab]
+  have hf (a : V ≃ V) (b : E ≃ E) (hab : IsAut G a b) (x : V) : f (a x) = f x := by
+    dsimp only [f]
+    rw [←a.tsum_eq (fun y => K (a x) y*I y)]
+    simp only [hcov a b hab,hI a b hab]
+  have hg (a : V ≃ V) (b : E ≃ E) (hab : IsAut G a b) (x : V) : g (a x) = g x := by
+    dsimp only [g]
+    rw [←a.tsum_eq (fun y => K (a x) y*f y)]
+    simp only [hcov a b hab,hf a b hab]
+  have hgO (x : V) (hx : x ∈ O) : g x = g (root i₀) := by
+    obtain ⟨a,b,hab,rfl⟩ := hx
+    exact hg a b hab _
+  have hmean : ∑ i, weight i*f (root i) = σ := by
+    have ht := htr.real root weight Γ (fun x y => K x y*I y)
+      (fun a ha x y => by obtain ⟨b,hab⟩ := hΓ ha; rw [hcov a b hab,hI a b hab])
+      (fun x => finiteKernel_summable_mul K hfin x I)
+      (fun y => by simpa only [hsym, mul_comm] using (hrow y).summable.mul_left (I y))
+    calc
+      _ = ∑ i, weight i * ∑' x, K x (root i)*I (root i) := ht
+      _ = σ := by simp only [tsum_mul_right,show ∀ y, ∑' x, K x y = 1 from fun y => by simpa only [hsym] using (hrow y).tsum_eq,one_mul,σ]
+  have hsecond : σ*g (root i₀) = ∑ i, weight i*f (root i)^2 := by
+    have ht := htr.real root weight Γ (fun x y => I x*K x y*f y)
+      (fun a ha x y => by obtain ⟨b,hab⟩ := hΓ ha; rw [hcov a b hab,hI a b hab,hf a b hab])
+      (fun x => by simpa only [mul_assoc] using (finiteKernel_summable_mul K hfin x f).mul_left (I x))
+      (fun y => by simpa only [hsym,mul_comm,mul_left_comm,mul_assoc] using (finiteKernel_summable_mul K hfin y I).mul_right (f y))
+    have hr (i : ι) : I (root i)*g (root i) = I (root i)*g (root i₀) := by
+      by_cases hi : root i ∈ O
+      · rw [hgO _ hi]
+      · simp only [I,ite_eq_right hi,zero_mul]
+    calc
+      _ = ∑ i, weight i*(I (root i)*g (root i)) := by
+        simp only [hr,σ,Finset.sum_mul,mul_assoc]
+      _ = ∑ i, weight i*∑' x, I x*K x (root i)*f (root i) := by
+        simpa only [mul_assoc,tsum_mul_left,g] using ht
+      _ = ∑ i, weight i*f (root i)^2 := by
+        simp only [tsum_mul_right]
+        congr 1
+        ext i
+        congr 1
+        rw [show (∑' x, I x*K x (root i)) = f (root i) by simp only [f,hsym,mul_comm]]
+        ring
+  have hvariance : 0 ≤ ∑ i, weight i*(f (root i)-σ)^2 :=
+    Finset.sum_nonneg (fun i _ => mul_nonneg (hw i) (sq_nonneg _))
+  have he : (∑ i, weight i*(f (root i)-σ)^2) = (∑ i, weight i*f (root i)^2)-σ^2 := by
+    simp_rw [sub_sq,mul_add,mul_sub]
+    rw [Finset.sum_add_distrib,Finset.sum_sub_distrib]
+    have hc : (∑ i, weight i*(2*f (root i)*σ)) = 2*σ*(∑ i, weight i*f (root i)) := by
+      rw [Finset.mul_sum]; apply Finset.sum_congr rfl; intro i hi; ring
+    rw [hc,hmean,←Finset.sum_mul,hw1,one_mul]
+    ring
+  rw [he,←hsecond] at hvariance
+  have hσ : 0 < σ := by
+    have hx : root i₀ ∈ O := ⟨Equiv.refl _,Equiv.refl _,fun _ => by simp,rfl⟩
+    have hIn (x : V) : 0 ≤ I x := by dsimp [I]; split_ifs <;> norm_num
+    have hb := Finset.single_le_sum (fun i _ => mul_nonneg (hw i) (hIn (root i))) (Finset.mem_univ i₀)
+    have he₀ : weight i₀*I (root i₀) = weight i₀ := by simp only [I,ite_eq_left hx,mul_one]
+    rw [he₀] at hb
+    exact hw₀.trans_le hb
+  have hbound : σ ≤ g (root i₀) := by nlinarith only [hvariance,hσ]
+  simpa only [σ,g,f,I,mul_ite,mul_one,mul_zero,zero_mul] using hbound
+
+end OrbitReturn
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section WalkOrbitReturn
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem matrixCoeff_real_symm (A : Hilbert V →L[ℂ] Hilbert V)
+    (hA : IsSelfAdjoint A) (x y : V) : (matrixCoeff A x y).re = (matrixCoeff A y x).re := by
+  have h := congrArg Complex.re (matrixCoeff_adjoint A x y)
+  simpa only [hA.star_eq,Complex.star_def,Complex.conj_re] using h
+
+theorem finite_positive_square_row (A : Hilbert V →L[ℂ] Hilbert V)
+    (hA : PositiveRealMatrix A) (hfin : ∀ x, (Function.support (fun y => (matrixCoeff A x y).re)).Finite)
+    (f : V → ℝ) (x : V) :
+    (∑' y, (matrixCoeff (A*A) x y).re*f y) =
+      ∑' z, (matrixCoeff A x z).re*∑' y, (matrixCoeff A z y).re*f y := by
+  let K := fun x y => (matrixCoeff A x y).re
+  let B := (hfin x).toFinset
+  have hz (z : V) (hz : z ∉ B) : K x z = 0 := by
+    simpa only [B,Set.Finite.mem_toFinset,Function.mem_support,not_not] using hz
+  have he (y : V) : (matrixCoeff (A*A) x y).re*f y = ∑ z ∈ B, K x z*(K z y*f y) := by
+    rw [←(hA.mul_hasSum A x y).tsum_eq,tsum_eq_sum (s := B) (fun z h => by change K x z*K z y = 0; rw [hz z h,zero_mul]),Finset.sum_mul]
+    apply Finset.sum_congr rfl
+    intro z h
+    exact mul_assoc _ _ _
+  have hs := hasSum_sum (s := B) (fun z _ =>
+    (finiteKernel_summable_mul K hfin z f).hasSum.mul_left (K x z))
+  have hr : (∑' z, K x z*∑' y, K z y*f y) = ∑ z ∈ B, K x z*∑' y, K z y*f y :=
+    tsum_eq_sum (fun z h => by rw [hz z h,zero_mul])
+  simpa only [←he,←hr] using hs.tsum_eq
+
+theorem walkPower_finite_support (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ)) (k : ℕ) (x : V) :
+    (Function.support (fun y => (matrixCoeff (P^k) x y).re)).Finite := by
+  apply (finite_reachBall hloc x k).subset
+  intro y hy
+  by_contra hn
+  exact hy (walkPower_support G hloc a P hpos hP k x y hn)
+
+theorem walkPower_invariant (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (P : Hilbert V →L[ℂ] Hilbert V)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ)) (k : ℕ) :
+    KernelInvariant (vertexAut G) (P^k) := by
+  apply KernelInvariant.pow
+  intro g hg x y
+  obtain ⟨b,hb⟩ := hg
+  simp only [hP,walkKernel_covariant G hloc a hb]
+
+ 
+
+theorem walkPower_even_orbit_return (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
+    (i₀ : ι) (hw₀ : 0 < weight i₀)
+    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ)) (k : ℕ) :
+    (∑ i, if root i ∈ vertexOrbit G (root i₀) then weight i else 0) ≤
+      ∑' y, if y ∈ vertexOrbit G (root i₀) then (matrixCoeff (P^(2*k)) (root i₀) y).re else 0 := by
+  have h := orbit_two_step_return_lower G root weight Γ hΓ htr hw hw1 i₀ hw₀
+    (fun x y => (matrixCoeff (P^k) x y).re)
+    (walkPower_finite_support G hloc a P hpos hP k)
+    (walkPower_hasSum G hloc a P hpos hP k)
+    (matrixCoeff_real_symm (P^k) (hsym.pow k))
+    (fun g b hb x y => congrArg Complex.re (walkPower_invariant G hloc a P hP k g ⟨b,hb⟩ x y))
+  have he := finite_positive_square_row (P^k) (hpos.pow k)
+    (walkPower_finite_support G hloc a P hpos hP k)
+    (fun y => if y ∈ vertexOrbit G (root i₀) then 1 else 0) (root i₀)
+  simp only [mul_ite,mul_one,mul_zero] at he
+  rw [←he] at h
+  simpa only [←pow_add,←two_mul] using h
+
+end WalkOrbitReturn
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+open scoped Topology
+section StartingEndpointRule
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Countable V] [Countable E] [Nonempty V]
+
+ 
+
+theorem starting_endpoint_rule_of_walk (G : BondGraph V E) (hloc : LocallyFinite G)
+    (hqt : QuasiTransitive G) (p : unitInterval) (hp : 0 < (p:ℝ)) (hp1 : (p:ℝ) < 1)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
+    (i₀ : ι) (hw₀ : 0 < weight i₀)
+    (hfin : ∀ᵐ ω ∂law p, (cluster G ω (root i₀)).Finite)
+    (a : ℝ) (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
+    (hsym : IsSelfAdjoint P) (hnorm : ‖P‖ < 1)
+    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
+    (ε : ℝ) (hε : 0 < ε) (hε1 : ε < 1/2) (η : ℝ) (hη : 0 < η) (M₀ : ℕ) :
+    ∃ M ≥ M₀, ∃ q : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) M ((M:ℝ)^(1/2+ε)),
+      0 < q.successProbability p (root i₀) ∧
+      ∃ T : Hilbert V →L[ℂ] Hilbert V, 0 < ‖T‖ ∧ ‖T‖ < η ∧
+        ∀ x y, matrixCoeff T x y = (q.normalizedKernel (root i₀) p x y : ℂ) := by
+  let O := vertexOrbit G (root i₀)
+  have ho : root i₀ ∈ O := ⟨Equiv.refl _,Equiv.refl _,fun _ => by simp,rfl⟩
+  let b : ℝ := ∑ i, if root i ∈ O then weight i else 0
+  have hb : 0 < b := by
+    have hn (i : ι) : 0 ≤ if root i ∈ O then weight i else 0 := by split_ifs; exact hw i; exact le_rfl
+    have hh := Finset.single_le_sum (fun i _ => hn i) (Finset.mem_univ i₀)
+    simp only [ite_eq_left ho] at hh
+    exact hw₀.trans_le hh
+  have ht := tendsto_pow_atTop_nhds_zero_of_lt_one (norm_nonneg P) hnorm
+  obtain ⟨k,hk⟩ := (ht.eventually (Iio_mem_nhds (show (0:ℝ) < η*b/2 by positivity))).exists
+  have hk1 : ‖P‖^k ≤ 1 := pow_le_one₀ (norm_nonneg P) hnorm.le
+  have hk2 : 2*‖P‖^(2*k)/b < η := by
+    rw [show 2*k = k+k from Nat.two_mul k,pow_add]
+    have hh : ‖P‖^k*‖P‖^k ≤ ‖P‖^k := mul_le_of_le_one_right (pow_nonneg (norm_nonneg P) _) hk1
+    apply (div_lt_iff₀ hb).mpr
+    change ‖P‖^k < η*b/2 at hk
+    nlinarith only [hh,hk]
+  let K : V → V → ℝ := fun x y => (matrixCoeff (P^(2*k)) x y).re
+  let S : V → V → ℝ := fun x y => if x ∈ O ∧ y ∈ O then K x y else 0
+  have hK (x y : V) : 0 ≤ K x y := (hpos.pow (2*k) x y).2
+  have hS (x y : V) : 0 ≤ S x y := by dsimp [S]; split_ifs; exact hK x y; exact le_rfl
+  have hSK (x y : V) : S x y ≤ K x y := by dsimp [S]; split_ifs; exact le_rfl; exact hK x y
+  have hsum (x : V) : HasSum (K x) 1 := walkPower_hasSum G hloc a P hpos hP (2*k) x
+  have hSsum (x : V) : Summable (S x) := Summable.of_nonneg_of_le (hS x) (hSK x) (hsum x).summable
+  have hrow (x : V) : ∑' y, S x y ≤ 1 := by
+    rw [←(hsum x).tsum_eq]
+    exact (hSsum x).tsum_le_tsum (hSK x) (hsum x).summable
+  have hτ (x y : V) (hxy : S x y ≠ 0) : (p:ℝ)^(2*k) ≤ twoPoint G p x y := by
+    apply twoPoint_reachBall_lower G p x (2*k) y
+    by_contra hn
+    have hz := walkPower_support G hloc a P hpos hP (2*k) x y hn
+    apply hxy
+    dsimp [S,K]
+    rw [hz]
+    split_ifs <;> rfl
+  have hO (x y : V) (hxy : S x y ≠ 0) : x ∈ O ∧ y ∈ O := by
+    by_contra ho
+    exact hxy (ite_eq_right ho)
+  have hcov (g : V ≃ V) (d : E ≃ E) (hg : IsAut G g d) (x y : V) : S (g x) (g y) = S x y := by
+    simp only [S,O,mem_vertexOrbit_aut_iff G (root i₀) hg,K,
+      walkPower_invariant G hloc a P hP (2*k) g ⟨d,hg⟩]
+  have hret : b ≤ ∑' y, S (root i₀) y := by
+    simpa only [b,S,K,ho,true_and,O] using
+      walkPower_even_orbit_return G hloc a root weight Γ hΓ htr hw hw1 i₀ hw₀ P hpos hsym hP k
+  obtain ⟨A,hAn,hA⟩ := exists_operator_of_matrix_domination K S hK
+    (fun x y => by rw [abs_of_nonneg (hS x y)]; exact hSK x y) (P^(2*k)) (hpos.pow (2*k)).real
+  have hc : 0 < (p:ℝ)^(2*k) := pow_pos hp _
+  have hbad := cappedGoodness_failure_eventually_small G hloc hqt p hp1 (root i₀) hfin ε hε hε1
+    ((p:ℝ)^(2*k)*b/2) (by positivity)
+  obtain ⟨M,hM,hMb⟩ := ((eventually_ge_atTop M₀).and hbad).exists
+  obtain ⟨q,hq,T,hTpos,hTn,hT⟩ := connectionAdjustedRule_operator G hloc (root i₀) p S
+    ((p:ℝ)^(2*k)) hc hS hSsum hrow hτ hO hcov M ((M:ℝ)^(1/2+ε)) b hb hret hMb A hA
+  refine ⟨M,hM,q,(by positivity : 0 < (p:ℝ)^(2*k)*b/2).trans_le hq,T,hTpos,?_,hT⟩
+  apply lt_of_le_of_lt hTn
+  apply lt_of_le_of_lt _ hk2
+  have hpow (j : ℕ) : ‖P^j‖ ≤ ‖P‖^j := by
+    induction j with
+    | zero => simpa only [pow_zero, ContinuousLinearMap.one_def] using (ContinuousLinearMap.norm_id_le (𝕜 := ℂ) (E := Hilbert V))
+    | succ j ih =>
+      rw [pow_succ,pow_succ]
+      exact (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right ih (norm_nonneg P))
+  exact div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_left (hAn.trans (hpow (2*k))) (by norm_num)) hb.le
+
+end StartingEndpointRule
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section NonamenableStartingMap
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Countable V] [Countable E] [Nonempty V]
+
+ 
+
+theorem nonamenable_starting_endpoint_rule (G : BondGraph V E) (hloc : LocallyFinite G)
+    (hqt : QuasiTransitive G) (hh : 0 < hV G)
+    (p : unitInterval) (hp : 0 < (p:ℝ)) (hp1 : (p:ℝ) < 1)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
+    (i₀ : ι) (hw₀ : 0 < weight i₀)
+    (hfin : ∀ᵐ ω ∂law p, (cluster G ω (root i₀)).Finite)
+    (ε : ℝ) (hε : 0 < ε) (hε1 : ε < 1/2) (η : ℝ) (hη : 0 < η) (M₀ : ℕ) :
+    ∃ M ≥ M₀, ∃ q : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) M ((M:ℝ)^(1/2+ε)),
+      0 < q.successProbability p (root i₀) ∧
+      ∃ T : Hilbert V →L[ℂ] Hilbert V, 0 < ‖T‖ ∧ ‖T‖ < η ∧
+        ∀ x y, matrixCoeff T x y = (q.normalizedKernel (root i₀) p x y : ℂ) := by
+  obtain ⟨D,P,hD,hdeg,hPn,hPs,hP⟩ := exists_strict_walk_operator G hloc hqt hh
+  have hDr : (0:ℝ) < D := Nat.cast_pos.mpr hD
+  have ha : 0 ≤ (1:ℝ)/(2*D) := by positivity
+  have haD : (1:ℝ)/(2*D)*D ≤ 1 := by
+    rw [div_mul_eq_mul_div]
+    apply (div_le_iff₀ (by positivity)).mpr
+    nlinarith
+  have hpos : PositiveRealMatrix P := by
+    intro x y
+    change (P (lp.single 2 y (1:ℂ)) x).im = 0 ∧ 0 ≤ (P (lp.single 2 y (1:ℂ)) x).re
+    rw [hP]
+    exact ⟨Complex.ofReal_im _,walkKernel_nonneg G hloc D hdeg _ ha haD x y⟩
+  exact starting_endpoint_rule_of_walk G hloc hqt p hp hp1 root weight Γ hΓ htr hw hw1 i₀ hw₀ hfin
+    _ P hpos hPs.isSelfAdjoint hPn hP ε hε hε1 η hη M₀
+
+end NonamenableStartingMap
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section FiniteOrbitPartition
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem finite_partition_square_bound {α ι : Type*} [Fintype ι]
+    (C : Finset α) (color : α → ι) (q : α → ℝ)
+    (hq : ∀ x ∈ C, (∑ y ∈ C, if color y = color x then (1:ℝ) else 0) ≤ q x) :
+    (C.card:ℝ)^2 ≤ Fintype.card ι * ∑ x ∈ C, q x := by
+  let c : ι → ℝ := fun i => ∑ x ∈ C, if color x=i then 1 else 0
+  have hsum : (∑ i, c i) = C.card := by
+    dsimp only [c]
+    rw [Finset.sum_comm]
+    simp
+  have hsq : (∑ i, c i^2) = ∑ x ∈ C, c (color x) := by
+    have he (i : ι) : c i^2 = ∑ x ∈ C, if color x=i then c i else 0 := by
+      rw [pow_two]
+      conv_lhs => lhs; change ∑ x ∈ C, if color x=i then (1:ℝ) else 0
+      rw [Finset.sum_mul]
+      apply Finset.sum_congr rfl
+      intro x hx
+      split_ifs <;> simp only [one_mul,zero_mul]
+    simp only [he]
+    rw [Finset.sum_comm]
+    apply Finset.sum_congr rfl
+    intro x hx
+    simp
+  have h := Finset.sum_mul_sq_le_sq_mul_sq Finset.univ (fun _ : ι => (1:ℝ)) c
+  simp only [one_mul,one_pow,Finset.sum_const,Finset.card_univ,nsmul_eq_mul,mul_one,hsum,hsq] at h
+  exact h.trans (mul_le_mul_of_nonneg_left (Finset.sum_le_sum hq) (Nat.cast_nonneg _))
+
+theorem vertexOrbit_eq_of_mem (G : BondGraph V E) (o x : V) (hx : x ∈ vertexOrbit G o) :
+    vertexOrbit G x = vertexOrbit G o := by
+  obtain ⟨a,b,hab,rfl⟩ := hx
+  ext y
+  constructor
+  · rintro ⟨c,d,hcd,hc⟩
+    exact ⟨a.trans c,b.trans d,isAut_trans G hab hcd,hc⟩
+  · rintro ⟨c,d,hcd,hc⟩
+    exact ⟨a.symm.trans c,b.symm.trans d,isAut_trans G (isAut_symm G hab) hcd,by simpa using hc⟩
+
+ 
+
+theorem finite_sameOrbit_pairs_lower (G : BondGraph V E)
+    {ι : Type*} [Fintype ι] (root : ι → V)
+    (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i)) (C : Finset V) :
+    (C.card:ℝ)^2 ≤ Fintype.card ι *
+      ∑ x ∈ C, ∑ y ∈ C, if y ∈ vertexOrbit G x then (1:ℝ) else 0 := by
+  let color : V → ι := fun x => (hcover x).choose
+  have hc (x : V) : x ∈ vertexOrbit G (root (color x)) := (hcover x).choose_spec
+  apply finite_partition_square_bound C color
+  intro x hx
+  apply Finset.sum_le_sum
+  intro y hy
+  by_cases h : color y = color x
+  · have hm : y ∈ vertexOrbit G x := by
+      rw [vertexOrbit_eq_of_mem G (root (color x)) x (hc x)]
+      simpa only [h] using hc y
+    simp only [ite_eq_left h,ite_eq_left hm,le_refl]
+  · simp only [ite_eq_right h]
+    split_ifs <;> norm_num
+
+end FiniteOrbitPartition
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section BinOrbitRowMass
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def binSizeMass (G : BondGraph V E) (n : ℕ) (ω : Set E) (x : V) : ℝ :=
+  ∑' y, binConnectionMass G n ω x y
+
+noncomputable def binOrbitRowMass (G : BondGraph V E) (n : ℕ) (ω : Set E) (x : V) : ℝ :=
+  ∑' y, if y ∈ vertexOrbit G x then binConnectionMass G n ω x y else 0
+
+theorem binMass_eq_of_window (G : BondGraph V E) (n : ℕ) (ω : Set E) (x y : V)
+    (hb : ClusterWindow G x n (2*n) ω) :
+    binConnectionMass G n ω x y = if y ∈ cluster G ω x then 1 else 0 := by
+  simp only [binConnectionMass,hb.1,hb.2.1,hb.2.2,true_and]
+  rfl
+
+theorem binMass_zero_of_not_window (G : BondGraph V E) (n : ℕ) (ω : Set E) (x y : V)
+    (hb : ¬ ClusterWindow G x n (2*n) ω) : binConnectionMass G n ω x y = 0 := by
+  apply ite_eq_right
+  exact fun h => hb ⟨h.1,h.2.1,h.2.2.1⟩
+
+theorem binSizeMass_eq (G : BondGraph V E) (n : ℕ) (ω : Set E) (x : V)
+    (hb : ClusterWindow G x n (2*n) ω) : binSizeMass G n ω x = (cluster G ω x).ncard := by
+  unfold binSizeMass
+  simp only [binMass_eq_of_window G n ω x _ hb]
+  rw [tsum_eq_sum (s := hb.1.toFinset) (fun y hy => by simp only [Set.Finite.mem_toFinset] at hy; simp only [ite_eq_right hy])]
+  have he : (∑ y ∈ hb.1.toFinset, if y ∈ cluster G ω x then (1:ℝ) else 0) = ∑ _y ∈ hb.1.toFinset, (1:ℝ) := by
+    apply Finset.sum_congr rfl
+    intro y hy
+    exact ite_eq_left (hb.1.mem_toFinset.mp hy)
+  rw [he]
+  simp only [Finset.sum_const,nsmul_eq_mul,mul_one,←Set.ncard_eq_toFinset_card _ hb.1]
+
+theorem binOrbitRowMass_eq (G : BondGraph V E) (n : ℕ) (ω : Set E) (x : V)
+    (hb : ClusterWindow G x n (2*n) ω) :
+    binOrbitRowMass G n ω x = ∑ y ∈ hb.1.toFinset, if y ∈ vertexOrbit G x then (1:ℝ) else 0 := by
+  unfold binOrbitRowMass
+  rw [tsum_eq_sum (s := hb.1.toFinset) (fun y hy => by
+    have hc : y ∉ cluster G ω x := fun h => hy (hb.1.mem_toFinset.mpr h)
+    rw [binMass_eq_of_window G n ω x y hb,ite_eq_right hc,ite_self])]
+  apply Finset.sum_congr rfl
+  intro y hy
+  rw [binMass_eq_of_window G n ω x y hb,ite_eq_left (hb.1.mem_toFinset.mp hy)]
+
+theorem binOrbitRowMass_nonneg (G : BondGraph V E) (n : ℕ) (ω : Set E) (x : V) :
+    0 ≤ binOrbitRowMass G n ω x := by
+  apply tsum_nonneg
+  intro y
+  split_ifs
+  · exact binConnectionMass_nonneg G n ω x y
+  · exact le_rfl
+
+theorem binOrbitRowMass_local (G : BondGraph V E) (hloc : LocallyFinite G)
+    (n : ℕ) (ω : Set E) (x : V) :
+    let Q := incidentSupport G hloc (finite_reachBall hloc x (2*n)).toFinset
+    binOrbitRowMass G n ω x = binOrbitRowMass G n (ω ∩ (Q : Set E)) x := by
+  dsimp only [binOrbitRowMass]
+  apply tsum_congr
+  intro y
+  rw [binConnectionMass_local G hloc n ω x y]
+
+theorem binSizeMass_local (G : BondGraph V E) (hloc : LocallyFinite G)
+    (n : ℕ) (ω : Set E) (x : V) :
+    let Q := incidentSupport G hloc (finite_reachBall hloc x (2*n)).toFinset
+    binSizeMass G n ω x = binSizeMass G n (ω ∩ (Q : Set E)) x := by
+  dsimp only [binSizeMass]
+  apply tsum_congr
+  intro y
+  exact binConnectionMass_local G hloc n ω x y
+
+theorem binOrbitRowMass_covariant (G : BondGraph V E) (n : ℕ) (ω : Set E)
+    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (x : V) :
+    binOrbitRowMass G n (b ⁻¹' ω) x = binOrbitRowMass G n ω (a x) := by
+  unfold binOrbitRowMass
+  rw [←a.tsum_eq (fun y => if y ∈ vertexOrbit G (a x) then binConnectionMass G n ω (a x) y else 0)]
+  have ho : vertexOrbit G (a x) = vertexOrbit G x := vertexOrbit_eq_of_mem G x (a x) ⟨a,b,hab,rfl⟩
+  simp only [ho,mem_vertexOrbit_aut_iff G x hab,binConnectionMass_covariant G hab]
+
+theorem binSizeMass_covariant (G : BondGraph V E) (n : ℕ) (ω : Set E)
+    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (x : V) :
+    binSizeMass G n (b ⁻¹' ω) x = binSizeMass G n ω (a x) := by
+  unfold binSizeMass
+  rw [←a.tsum_eq (fun y => binConnectionMass G n ω (a x) y)]
+  simp only [binConnectionMass_covariant G hab]
+
+end BinOrbitRowMass
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section BinOrbitTransport
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def binOrbitTransportMass (G : BondGraph V E) (n : ℕ) (ω : Set E) (x y : V) : ℝ :=
+  binConnectionMass G n ω x y * binOrbitRowMass G n ω y / binSizeMass G n ω x
+
+theorem binOrbitRowMass_zero (G : BondGraph V E) (n : ℕ) (ω : Set E) (x : V)
+    (hb : ¬ ClusterWindow G x n (2*n) ω) : binOrbitRowMass G n ω x = 0 := by
+  simp only [binOrbitRowMass,binMass_zero_of_not_window G n ω x _ hb,ite_self,tsum_zero]
+
+theorem binOrbitTransportMass_nonneg (G : BondGraph V E) (n : ℕ) (ω : Set E) (x y : V) :
+    0 ≤ binOrbitTransportMass G n ω x y := by
+  exact div_nonneg (mul_nonneg (binConnectionMass_nonneg G n ω x y) (binOrbitRowMass_nonneg G n ω y))
+    (tsum_nonneg (binConnectionMass_nonneg G n ω x))
+
+theorem binOrbitTransportMass_column (G : BondGraph V E) (n : ℕ) (hn : 0 < n)
+    (ω : Set E) (y : V) : (∑' x, binOrbitTransportMass G n ω x y) = binOrbitRowMass G n ω y := by
+  by_cases hb : ClusterWindow G y n (2*n) ω
+  · have he (x : V) : binOrbitTransportMass G n ω x y =
+        binConnectionMass G n ω x y * (binOrbitRowMass G n ω y / binSizeMass G n ω y) := by
+      by_cases hz : binConnectionMass G n ω x y = 0
+      · simp only [binOrbitTransportMass,hz,zero_mul,zero_div]
+      · obtain ⟨hf,hN,hc⟩ := binConnectionMass_support G n ω x y hz
+        have hset := cluster_eq_of_conn G ω hc
+        have hbx : ClusterWindow G x n (2*n) ω := by
+          unfold ClusterWindow at hb ⊢
+          simpa only [hset] using hb
+        rw [binOrbitTransportMass,binSizeMass_eq G n ω x hbx,binSizeMass_eq G n ω y hb,hset]
+        exact mul_div_assoc _ _ _
+    rw [show (∑' x, binOrbitTransportMass G n ω x y) =
+      ∑' x, binConnectionMass G n ω x y*(binOrbitRowMass G n ω y/binSizeMass G n ω y) from tsum_congr he,
+      tsum_mul_right]
+    have hcol : (∑' x, binConnectionMass G n ω x y) = binSizeMass G n ω y := by
+      simp only [binConnectionMass_symm G n ω,binSizeMass]
+    rw [hcol]
+    have hs : binSizeMass G n ω y ≠ 0 := by
+      rw [binSizeMass_eq G n ω y hb]
+      exact Nat.cast_ne_zero.mpr (Nat.ne_of_gt (hn.trans_le hb.2.1))
+    exact mul_div_cancel₀ _ hs
+  · have he : ∀ x, binOrbitTransportMass G n ω x y = 0 := by
+      intro x
+      simp only [binOrbitTransportMass,binOrbitRowMass_zero G n ω y hb,mul_zero,zero_div]
+    simp only [he,tsum_zero,binOrbitRowMass_zero G n ω y hb]
+
+theorem binOrbitTransportMass_row (G : BondGraph V E)
+    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V)
+    (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i)) (n : ℕ) (hn : 0 < n) (ω : Set E) (x : V) :
+    (if ClusterWindow G x n (2*n) ω then (n:ℝ)/Fintype.card ι else 0) ≤
+      ∑' y, binOrbitTransportMass G n ω x y := by
+  by_cases hb : ClusterWindow G x n (2*n) ω
+  · rw [ite_eq_left hb]
+    let C := hb.1.toFinset
+    have hs : (0:ℝ) < (cluster G ω x).ncard := Nat.cast_pos.mpr (hn.trans_le hb.2.1)
+    have ht : (0:ℝ) < Fintype.card ι := Nat.cast_pos.mpr Fintype.card_pos
+    have hqy (y : V) (hy : y ∈ C) : binOrbitRowMass G n ω y =
+        ∑ z ∈ C, if z ∈ vertexOrbit G y then (1:ℝ) else 0 := by
+      have hc : Conn G ω x y := hb.1.mem_toFinset.mp hy
+      have he := cluster_eq_of_conn G ω hc
+      have hby : ClusterWindow G y n (2*n) ω := by
+        unfold ClusterWindow at hb ⊢
+        simpa only [←he] using hb
+      rw [binOrbitRowMass_eq G n ω y hby]
+      have hC : hby.1.toFinset = C := by
+        apply Finset.coe_injective
+        simpa only [C,Set.Finite.coe_toFinset] using he.symm
+      rw [hC]
+    have he : (∑' y, binOrbitTransportMass G n ω x y) =
+        (∑ y ∈ C, ∑ z ∈ C, if z ∈ vertexOrbit G y then (1:ℝ) else 0)/(cluster G ω x).ncard := by
+      rw [tsum_eq_sum (s := C) (fun y hy => by
+        have hc : y ∉ cluster G ω x := fun h => hy (hb.1.mem_toFinset.mpr h)
+        simp only [binOrbitTransportMass,binMass_eq_of_window G n ω x y hb,ite_eq_right hc,zero_mul,zero_div])]
+      rw [Finset.sum_div]
+      apply Finset.sum_congr rfl
+      intro y hy
+      simp only [binOrbitTransportMass,binMass_eq_of_window G n ω x y hb,
+        ite_eq_left (hb.1.mem_toFinset.mp hy),one_mul,binSizeMass_eq G n ω x hb,hqy y hy]
+    rw [he]
+    apply (div_le_div_iff₀ ht hs).mpr
+    have h := finite_sameOrbit_pairs_lower G root hcover C
+    have hC : (C.card:ℝ) = (cluster G ω x).ncard := by
+      exact_mod_cast (Set.ncard_eq_toFinset_card _ hb.1).symm
+    rw [hC] at h
+    have hn' : (n:ℝ) ≤ (cluster G ω x).ncard := by exact_mod_cast hb.2.1
+    nlinarith [mul_le_mul_of_nonneg_right hn' hs.le]
+  · rw [ite_eq_right hb]
+    exact tsum_nonneg (binOrbitTransportMass_nonneg G n ω x)
+
+end BinOrbitTransport
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Kernel/ContactTrace.lean b/lean/OAI/Probability/UnimodularPercolation/Kernel/ContactTrace.lean
index a558686dd94a2380f077e07046e38e09da33cbc8..0290a085308329d3e44259e9176f4481b758376a 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Kernel/ContactTrace.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Kernel/ContactTrace.lean
@@ -1,1240 +1,1240 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Clusters.LargeContacts
-
-namespace OAI
-
-/-! Contact Trace for bond percolation. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-open scoped ENNReal
-section IndexedEdgeKernel
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem enn_ncard_eq_tsum {α : Type*} (A : Set α) (hA : A.Finite) :
-    (A.ncard:ℝ≥0∞) = ∑' a, if a ∈ A then (1:ℝ≥0∞) else 0 := by
-  rw [tsum_eq_sum (s := hA.toFinset) (fun a ha => by
-    simp only [Set.Finite.mem_toFinset] at ha
-    exact ite_eq_right ha)]
-  simp only [Set.ncard_eq_toFinset_card A hA]
-  rw [Finset.sum_congr rfl (fun a ha => ite_eq_left ((Set.Finite.mem_toFinset _).mp ha))]
-  simp
-
- 
-
-noncomputable def indexedEdgeKernel (G : BondGraph V E) (x y : V) : ℝ≥0∞ :=
-  ∑' e, if G e = s(x,y) then 1 else 0
-
-theorem indexedEdgeKernel_eq_ncard (G : BondGraph V E) (hloc : LocallyFinite G) (x y : V) :
-    indexedEdgeKernel G x y = ({e | G e = s(x,y)}.ncard:ℝ≥0∞) := by
-  have hf : ({e | G e = s(x,y)} : Set E).Finite := (hloc x).subset (fun e he => by change x ∈ G e; change G e = s(x,y) at he; rw [he]; simp)
-  unfold indexedEdgeKernel
-  convert (enn_ncard_eq_tsum _ hf).symm using 1
-  apply tsum_congr
-  intro e
-  by_cases h : G e = s(x,y) <;> simp only [Set.mem_ofPred_eq,h,ite_true,ite_false]
-
-theorem indexedEdgeKernel_symm (G : BondGraph V E) (x y : V) :
-    indexedEdgeKernel G x y = indexedEdgeKernel G y x := by rw [indexedEdgeKernel,indexedEdgeKernel,Sym2.eq_swap]
-
-theorem indexedEdgeKernel_row (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) :
-    (∑' y, indexedEdgeKernel G x y) = (degree G x:ℝ≥0∞) := by
-  change (∑' y, ∑' e, if G e = s(x,y) then (1:ℝ≥0∞) else 0) = _
-  rw [ENNReal.tsum_comm,←ENNReal.tsum_prod]
-  have hn := (enn_ncard_eq_tsum _ (finite_steps G hloc x)).symm
-  have he : (∑' (p : E × V), if G p.1 = s(x,p.2) then (1:ℝ≥0∞) else 0) =
-      (({ey : E × V | G ey.1 = s(x,ey.2)} : Set (E × V)).ncard:ℝ≥0∞) := by
-    convert hn using 1
-    apply tsum_congr
-    intro ey
-    by_cases h : G ey.1 = s(x,ey.2) <;> simp only [Set.mem_ofPred_eq,h,ite_true,ite_false]
-  rw [he]
-  change (({ey : E × V | G ey.1 = s(x,ey.2)} : Set (E × V)).ncard:ℝ≥0∞) = (Nat.card (Step G x):ℝ≥0∞)
-  rw [←Nat.card_coe_set_eq]; rfl
-
-theorem indexedEdgeKernel_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (x y : V) :
-    indexedEdgeKernel G (a x) (a y) = indexedEdgeKernel G x y := by
-  unfold indexedEdgeKernel
-  rw [←b.tsum_eq (fun e => if G e = s(a x,a y) then (1:ℝ≥0∞) else 0)]
-  apply tsum_congr
-  intro e
-  have he : G (b e) = s(a x,a y) ↔ G e = s(x,y) := by
-    rw [hab e,←Sym2.map_mk]
-    exact (Sym2.map.injective a.injective).eq_iff
-  simp only [he]
-
-end IndexedEdgeKernel
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-open scoped ENNReal
-section ContactKernelCounting
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem contact_endpoints_unique (G : BondGraph V E) (ω : Set E) (x y : V)
-    (hd : ¬Conn G ω x y) {e : E} {a b c d : V}
-    (ha : Conn G ω x a) (_hb : Conn G ω y b)
-    (_hc : Conn G ω x c) (he : Conn G ω y d)
-    (hab : G e = s(a,b)) (hcd : G e = s(c,d)) : a = c ∧ b = d := by
-  rcases Sym2.eq_iff.mp (hab.symm.trans hcd) with h | h
-  · exact h
-  · exfalso
-    apply hd
-    exact conn_trans G ω ha (by simpa only [h.1] using conn_symm G ω he)
-
-theorem closedContactSet_finite (G : BondGraph V E) (hloc : LocallyFinite G)
-    (A B : Set V) (hA : A.Finite) : (closedContactSet G A B).Finite := by
-  have hs := contactSet_subset_incident G hloc hA.toFinset B
-  rw [Set.Finite.coe_toFinset] at hs
-  exact (incidentSupport G hloc hA.toFinset).finite_toSet.subset hs
-
-theorem clusterPairKernel_edge (G : BondGraph V E) (hloc : LocallyFinite G)
-    (ω : Set E) (x y : V) (hd : ¬Conn G ω x y) (hx : (cluster G ω x).Finite) :
-    clusterPairKernel G ω (indexedEdgeKernel G) x y = (clusterContactCount G ω x y:ℝ≥0∞) := by
-  have hed (e : E) :
-      (∑' a, ∑' b, if Conn G ω x a ∧ Conn G ω y b ∧ G e = s(a,b) then (1:ℝ≥0∞) else 0) =
-      if e ∈ closedContactSet G (cluster G ω x) (cluster G ω y) then 1 else 0 := by
-    by_cases h : e ∈ closedContactSet G (cluster G ω x) (cluster G ω y)
-    · obtain ⟨a,ha,b,hb,hab⟩ := h
-      change Conn G ω x a at ha
-      change Conn G ω y b at hb
-      have hh : e ∈ closedContactSet G (cluster G ω x) (cluster G ω y) := ⟨a,ha,b,hb,hab⟩
-      rw [ite_eq_left hh,tsum_eq_single a]
-      · rw [tsum_eq_single b]
-        · exact ite_eq_left ⟨ha,hb,hab⟩
-        · intro d hdb
-          apply ite_eq_right
-          intro hc
-          exact hdb (contact_endpoints_unique G ω x y hd hc.1 hc.2.1 ha hb hc.2.2 hab).2
-      · intro c hca
-        apply ENNReal.tsum_eq_zero.mpr
-        intro d
-        apply ite_eq_right
-        intro hc
-        exact hca (contact_endpoints_unique G ω x y hd hc.1 hc.2.1 ha hb hc.2.2 hab).1
-    · rw [ite_eq_right h]
-      apply ENNReal.tsum_eq_zero.mpr
-      intro a
-      apply ENNReal.tsum_eq_zero.mpr
-      intro b
-      apply ite_eq_right
-      intro hab
-      exact h ⟨a,hab.1,b,hab.2.1,hab.2.2⟩
-  have he (a b : V) : (if Conn G ω x a ∧ Conn G ω y b then indexedEdgeKernel G a b else 0) =
-      ∑' e, if Conn G ω x a ∧ Conn G ω y b ∧ G e = s(a,b) then (1:ℝ≥0∞) else 0 := by
-    by_cases ha : Conn G ω x a <;> by_cases hb : Conn G ω y b <;>
-      simp only [ha,hb,and_self,true_and,false_and,ite_true,ite_false,indexedEdgeKernel,tsum_zero]
-  rw [clusterPairKernel]
-  simp_rw [he]
-  calc
-    _ = ∑' e, ∑' a, ∑' b, if Conn G ω x a ∧ Conn G ω y b ∧ G e = s(a,b) then (1:ℝ≥0∞) else 0 := by
-      calc
-        _ = ∑' a, ∑' e, ∑' b, if Conn G ω x a ∧ Conn G ω y b ∧ G e = s(a,b) then (1:ℝ≥0∞) else 0 := by
-          apply tsum_congr; intro a; exact ENNReal.tsum_comm
-        _ = _ := ENNReal.tsum_comm
-    _ = ∑' e, if e ∈ closedContactSet G (cluster G ω x) (cluster G ω y) then (1:ℝ≥0∞) else 0 := tsum_congr hed
-    _ = _ := (enn_ncard_eq_tsum _ (closedContactSet_finite G hloc _ _ hx)).symm
-
-end ContactKernelCounting
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-open scoped ENNReal
-section ContactCovariance
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem closedContactSet_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (A B : Set V) :
-    b '' closedContactSet G A B = closedContactSet G (a '' A) (a '' B) := by
-  ext e
-  constructor
-  · rintro ⟨f,⟨x,hx,y,hy,hf⟩,rfl⟩
-    exact ⟨a x,Set.mem_image_of_mem a hx,a y,Set.mem_image_of_mem a hy,by rw [hab f,hf,Sym2.map_mk]⟩
-  · rintro ⟨z,⟨x,hx,rfl⟩,w,⟨y,hy,rfl⟩,he⟩
-    refine ⟨b.symm e,⟨x,hx,y,hy,?_⟩,b.apply_symm_apply e⟩
-    apply Sym2.map.injective a.injective
-    rw [←hab,Sym2.map_mk]
-    simpa only [Equiv.apply_symm_apply] using he
-
-theorem clusterContactCount_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (ω : Set E) (x y : V) :
-    clusterContactCount G (b ⁻¹' ω) x y = clusterContactCount G ω (a x) (a y) := by
-  unfold clusterContactCount
-  rw [←cluster_aut G hab ω x,←cluster_aut G hab ω y,←closedContactSet_aut G hab,
-    Set.ncard_image_of_injective _ b.injective]
-
-theorem fullLargeContact_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (ω : Set E) (x y : V) (L : ℕ) :
-    fullLargeContact G x y L (b ⁻¹' ω) = fullLargeContact G (a x) (a y) L ω := by
-  simp only [fullLargeContact,conn_aut_iff G hab,cluster_ncard_covariant G hab,clusterContactCount_aut G hab]
-
-theorem fullLargeContact_cluster_constant (G : BondGraph V E) (ω : Set E)
-    (L : ℕ) (x y a b : V) (hxa : Conn G ω x a) (hyb : Conn G ω y b) :
-    fullLargeContact G x y L ω = fullLargeContact G a b L ω := by
-  have hc : Conn G ω x y ↔ Conn G ω a b := by
-    constructor
-    · intro h; exact conn_trans G ω (conn_symm G ω hxa) (conn_trans G ω h hyb)
-    · intro h; exact conn_trans G ω hxa (conn_trans G ω h (conn_symm G ω hyb))
-  simp only [fullLargeContact,clusterContactCount,cluster_eq_of_conn G ω hxa,
-    cluster_eq_of_conn G ω hyb,hc]
-
-theorem fullLargeContact_distinct (G : BondGraph V E) (ω : Set E) (x y : V) (L : ℕ) :
-    (if ¬Conn G ω x y then ENNReal.ofReal (fullLargeContact G x y L ω) else 0) =
-      ENNReal.ofReal (fullLargeContact G x y L ω) := by
-  by_cases h : Conn G ω x y
-  · simp only [fullLargeContact,h,not_true_eq_false,false_and,ite_false,ENNReal.ofReal_zero]
-  · exact ite_eq_left h
-
-end ContactCovariance
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-open scoped ENNReal
-section ContactSquareTrace
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def contactSquareLarge (G : BondGraph V E) (L : ℕ)
-    (ω : Set E) (x y : V) : ℝ≥0∞ :=
-  if L ≤ min (cluster G ω x).ncard (cluster G ω y).ncard then
-    (clusterContactCount G ω x y:ℝ≥0∞)^2 else 0
-
-theorem contactSquareLarge_reroot [Countable V] [Countable E]
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hw : ∀ i, 0 ≤ weight i) (p : unitInterval)
-    (hfin : ∀ x, ∀ᵐ ω ∂law p, (cluster G ω x).Finite) (L : ℕ) :
-    clusterPairFunctional G p root weight (contactSquareLarge G L) =
-      ∑ i, ENNReal.ofReal (weight i) * ∫⁻ ω, ∑' y,
-        indexedEdgeKernel G (root i) y * ENNReal.ofReal (fullLargeContact G (root i) y L ω) ∂law p := by
-  have hj : ∀ a ∈ Γ, ∀ x y, indexedEdgeKernel G (a x) (a y) = indexedEdgeKernel G x y := by
-    intro a ha x y
-    obtain ⟨b,hab⟩ := hΓ ha
-    exact indexedEdgeKernel_aut G hab x y
-  have ht := clusterPairFunctional_reroot G hconn hloc root weight Γ hΓ htr hw p hfin
-    (indexedEdgeKernel G) hj (fun ω x y => ENNReal.ofReal (fullLargeContact G x y L ω))
-    (fun x y => (measurable_fullLargeContact G hloc x y L).ennreal_ofReal)
-    (fun ω x y a b hxa hyb => congrArg ENNReal.ofReal (fullLargeContact_cluster_constant G ω L x y a b hxa hyb))
-    (fun a _ b hab ω x y => congrArg ENNReal.ofReal (fullLargeContact_aut G hab ω x y L))
-  simp_rw [fullLargeContact_distinct] at ht
-  rw [←ht]
-  unfold clusterPairFunctional
-  apply Finset.sum_congr rfl
-  intro i _
-  congr 1
-  apply lintegral_congr_ae
-  filter_upwards [hfin (root i)] with ω hω
-  unfold clusterPairDensity
-  congr 1
-  apply tsum_congr
-  intro y
-  congr 1
-  by_cases hd : Conn G ω (root i) y
-  · simp only [hd,not_true_eq_false,ite_false]
-  · rw [ite_eq_left hd,ite_eq_left hd]
-    dsimp only
-    rw [clusterPairKernel_edge G hloc ω (root i) y hd hω]
-    unfold contactSquareLarge fullLargeContact
-    by_cases hs : L ≤ min (cluster G ω (root i)).ncard (cluster G ω y).ncard
-    · simp only [hd,hs,not_false_eq_true,true_and,ite_true,ENNReal.ofReal_natCast,pow_two]
-    · simp only [hs,and_false,ite_false,ENNReal.ofReal_zero,mul_zero]
-
-theorem contactSquareLarge_bound [Countable V] [Countable E]
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1) (p : unitInterval)
-    (hfin : ∀ x, ∀ᵐ ω ∂law p, (cluster G ω x).Finite) (L d : ℕ)
-    (hd : ∀ x, degree G x ≤ d) (B : ℝ) (_hB : 0 ≤ B)
-    (hI : ∀ x y, Integrable (fullLargeContact G x y L) (law p))
-    (hb : ∀ x y, (∫ ω, fullLargeContact G x y L ω ∂law p) ≤ B) :
-    clusterPairFunctional G p root weight (contactSquareLarge G L) ≤ ENNReal.ofReal ((d:ℝ)*B) := by
-  have hi (x y : V) : (∫⁻ ω, ENNReal.ofReal (fullLargeContact G x y L ω) ∂law p) ≤ ENNReal.ofReal B := by
-    rw [←ofReal_integral_eq_lintegral_ofReal (hI x y) (Filter.Eventually.of_forall (fullLargeContact_nonneg G x y L))]
-    exact ENNReal.ofReal_le_ofReal (hb x y)
-  have hr (x : V) : (∫⁻ ω, ∑' y, indexedEdgeKernel G x y * ENNReal.ofReal (fullLargeContact G x y L ω) ∂law p) ≤
-      ENNReal.ofReal ((d:ℝ)*B) := by
-    have hm (y : V) : Measurable (fun ω => indexedEdgeKernel G x y * ENNReal.ofReal (fullLargeContact G x y L ω)) :=
-      (measurable_const : Measurable (fun _ : Set E => indexedEdgeKernel G x y)).mul (measurable_fullLargeContact G hloc x y L).ennreal_ofReal
-    rw [lintegral_tsum (fun y => (hm y).aemeasurable)]
-    simp_rw [lintegral_const_mul _ (measurable_fullLargeContact G hloc x _ L).ennreal_ofReal]
-    calc
-      _ ≤ ∑' y, indexedEdgeKernel G x y * ENNReal.ofReal B := ENNReal.tsum_le_tsum (fun y => mul_le_mul_right (hi x y) _)
-      _ = (degree G x:ℝ≥0∞)*ENNReal.ofReal B := by rw [ENNReal.tsum_mul_right,indexedEdgeKernel_row G hloc x]
-      _ ≤ (d:ℝ≥0∞)*ENNReal.ofReal B := mul_le_mul_left (by exact_mod_cast hd x) _
-      _ = _ := by rw [ENNReal.ofReal_mul (by positivity),ENNReal.ofReal_natCast]
-  rw [contactSquareLarge_reroot G hconn hloc root weight Γ hΓ htr hw p hfin L]
-  calc
-    _ ≤ ∑ i, ENNReal.ofReal (weight i)*ENNReal.ofReal ((d:ℝ)*B) :=
-      Finset.sum_le_sum (fun i _ => mul_le_mul_right (hr (root i)) _)
-    _ = _ := by rw [←Finset.sum_mul,←ENNReal.ofReal_sum_of_nonneg (fun i _ => hw i),hw1,ENNReal.ofReal_one,one_mul]
-
-end ContactSquareTrace
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-open scoped ENNReal
-section CriticalContactSquare
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem critical_contact_square_bound_nat
-    [Countable V] [Countable E] [Nonempty V] [Nonempty E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hconn : Connected G)
-    (hqt : QuasiTransitive G) (hh : 0 < hV G)
-    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
-    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
-    (hp₀ : 0 < (pc G:ℝ)) (hp₁ : (pc G:ℝ) < 1)
-    (hfin : ∀ i, ∀ᵐ ω ∂law (pc G), (cluster G ω (root i)).Finite)
-    (α ε : ℝ) (hα : 0 < α) (hα1 : α < 1/2) (hε : 0 < ε)
-    (hr : (1:ℝ)/2+ε-2*α < 0) :
-    ∃ C : ℝ, 0 < C ∧ ∀ (p : unitInterval), (pc G:ℝ)/2 ≤ (p:ℝ) → p ≤ pc G →
-      ∀ L : ℕ, 1 ≤ L → clusterPairFunctional G p root weight (contactSquareLarge G L) ≤
-        ENNReal.ofReal (C*(L:ℝ)^((1:ℝ)/2+ε-2*α)) := by
-  obtain ⟨A,hA,hb⟩ := critical_large_contact_bound G hloc hconn hqt hh root weight Γ hΓ htr
-    hcov hw hw1 hp₀ hp₁ hfin α ε hα hα1 hε hr
-  obtain ⟨d,hd,hdeg⟩ := exists_degree_bound G hqt
-  refine ⟨(d:ℝ)*A,mul_pos (by exact_mod_cast hd) hA,?_⟩
-  intro p hp hpq L hL
-  have hf : ∀ x, ∀ᵐ ω ∂law p, (cluster G ω x).Finite := by
-    intro x
-    obtain ⟨i,a,ha,rfl⟩ := hcov x
-    obtain ⟨b,hab⟩ := hΓ ha
-    exact cluster_finite_ae_mono G (a (root i)) p (pc G) hpq
-      (cluster_finite_ae_covariant G (pc G) hab (root i) (hfin i))
-  have h := contactSquareLarge_bound G hconn hloc root weight Γ hΓ htr (fun i => (hw i).le) hw1 p hf
-    L d hdeg (A*(L:ℝ)^((1:ℝ)/2+ε-2*α)) (mul_nonneg hA.le (Real.rpow_nonneg (by positivity) _))
-    (fun x y => (hb x y p hp hpq L hL).1) (fun x y => (hb x y p hp hpq L hL).2)
-  simpa only [mul_assoc] using h
-
-end CriticalContactSquare
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-open scoped ENNReal
-section DistinctClusterBK
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem distinct_cluster_disjoint_occurrence (G : BondGraph V E) (ω : Set E)
-    (x y a b : V) (hxa : Conn G ω x a) (hyb : Conn G ω y b) (hd : ¬Conn G ω x y) :
-    ω ∈ DisjointOccurrence {η | Conn G η x a} {η | Conn G η y b} := by
-  obtain ⟨s,hs,hsa⟩ := finiteWitness_conn G x a _ (conn_restrict_cluster G ω x (mem_cluster_self G ω x) hxa)
-  obtain ⟨t,ht,htb⟩ := finiteWitness_conn G y b _ (conn_restrict_cluster_compl G ω x hd hyb)
-  refine ⟨s,t,fun e he => (hs he).1,fun e he => (ht he).1,?_,hsa,htb⟩
-  apply Finset.disjoint_left.mpr
-  intro e he ht'
-  obtain ⟨⟨z,w⟩,hr⟩ := Quot.exists_rep (G e)
-  have hz : z ∈ G e := by rw [←hr]; exact Sym2.mem_mk_left z w
-  exact (ht ht').2 z hz ((hs he).2 z hz)
-
-theorem distinct_cluster_two_point_bound [Countable E]
-    (G : BondGraph V E) (p : unitInterval) (x y a b : V) :
-    law p {ω | Conn G ω x a ∧ Conn G ω y b ∧ ¬Conn G ω x y} ≤
-      law p {ω | Conn G ω x a} * law p {ω | Conn G ω y b} := by
-  calc
-    _ ≤ law p (DisjointOccurrence {ω | Conn G ω x a} {ω | Conn G ω y b}) :=
-      measure_mono (fun ω h => distinct_cluster_disjoint_occurrence G ω x y a b h.1 h.2.1 h.2.2)
-    _ ≤ _ := bk_ennreal p _ _ (fun _ _ h hc => conn_mono G h hc)
-      (fun _ _ h hc => conn_mono G h hc) (finiteWitness_conn G x a) (finiteWitness_conn G y b)
-
-theorem measurable_clusterPairKernel [Countable V] [Countable E]
-    (G : BondGraph V E) (J : V → V → ℝ≥0∞) (x y : V) :
-    Measurable (fun ω => clusterPairKernel G ω J x y) := by
-  apply Measurable.tsum
-  intro a
-  apply Measurable.tsum
-  intro b
-  exact measurable_const.ite ((measurableSet_conn G x a).inter (measurableSet_conn G y b)) measurable_const
-
-theorem clusterPairKernel_cluster_constant (G : BondGraph V E) (ω : Set E)
-    (J : V → V → ℝ≥0∞) (x y a b : V) (hxa : Conn G ω x a) (hyb : Conn G ω y b) :
-    clusterPairKernel G ω J x y = clusterPairKernel G ω J a b := by
-  have hx (z : V) : Conn G ω x z ↔ Conn G ω a z :=
-    ⟨fun h => conn_trans G ω (conn_symm G ω hxa) h,fun h => conn_trans G ω hxa h⟩
-  have hy (z : V) : Conn G ω y z ↔ Conn G ω b z :=
-    ⟨fun h => conn_trans G ω (conn_symm G ω hyb) h,fun h => conn_trans G ω hyb h⟩
-  simp only [clusterPairKernel,hx,hy]
-
-theorem clusterPairKernel_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (J : V → V → ℝ≥0∞) (hJ : ∀ x y, J (a x) (a y) = J x y)
-    (ω : Set E) (x y : V) :
-    clusterPairKernel G (b ⁻¹' ω) J x y = clusterPairKernel G ω J (a x) (a y) := by
-  unfold clusterPairKernel
-  rw [←a.tsum_eq (fun x' => ∑' y', if Conn G ω (a x) x' ∧ Conn G ω (a y) y' then J x' y' else 0)]
-  apply tsum_congr
-  intro x'
-  rw [←a.tsum_eq (fun y' => if Conn G ω (a x) (a x') ∧ Conn G ω (a y) y' then J (a x') y' else 0)]
-  simp only [hJ,conn_aut_iff G hab]
-
-end DistinctClusterBK
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-open scoped ENNReal
-section PairSquareDiagram
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem clusterPairKernel_distinct_integral [Countable V] [Countable E]
-    (G : BondGraph V E) (p : unitInterval) (J : V → V → ℝ≥0∞) (x y : V) :
-    (∫⁻ ω, if ¬Conn G ω x y then clusterPairKernel G ω J x y else 0 ∂law p) =
-      ∑' a, ∑' b, J a b * law p {ω | Conn G ω x a ∧ Conn G ω y b ∧ ¬Conn G ω x y} := by
-  have he (ω : Set E) : (if ¬Conn G ω x y then clusterPairKernel G ω J x y else 0) =
-      ∑' a, ∑' b, J a b * (if Conn G ω x a ∧ Conn G ω y b ∧ ¬Conn G ω x y then (1:ℝ≥0∞) else 0) := by
-    by_cases hd : Conn G ω x y
-    · simp only [hd,not_true_eq_false,and_false,ite_false,mul_zero,tsum_zero]
-    · rw [ite_eq_left hd,clusterPairKernel]
-      apply tsum_congr; intro a
-      apply tsum_congr; intro b
-      by_cases h : Conn G ω x a ∧ Conn G ω y b
-      · simp only [h.1,h.2,hd,not_false_eq_true,and_self,ite_true,mul_one]
-      · have hn : ¬ (Conn G ω x a ∧ Conn G ω y b ∧ ¬Conn G ω x y) := fun h' => h ⟨h'.1,h'.2.1⟩
-        simp only [ite_eq_right h,ite_eq_right hn,mul_zero]
-  have hm (a b : V) : Measurable (fun ω => if Conn G ω x a ∧ Conn G ω y b ∧ ¬Conn G ω x y then (1:ℝ≥0∞) else 0) :=
-    measurable_const.ite ((measurableSet_conn G x a).inter ((measurableSet_conn G y b).inter (measurableSet_conn G x y).compl)) measurable_const
-  have hm' (a b : V) : Measurable (fun ω : Set E => J a b * (if Conn G ω x a ∧ Conn G ω y b ∧ ¬Conn G ω x y then (1:ℝ≥0∞) else 0)) :=
-    (measurable_const : Measurable (fun _ : Set E => J a b)).mul (hm a b)
-  simp_rw [he]
-  rw [lintegral_tsum (fun a => (Measurable.tsum (fun b => hm' a b)).aemeasurable)]
-  apply tsum_congr; intro a
-  rw [lintegral_tsum (fun b => (hm' a b).aemeasurable)]
-  apply tsum_congr; intro b
-  rw [lintegral_const_mul _ (hm a b)]
-  congr 1
-  convert
-    (lintegral_indicator_one (μ := law p) ((measurableSet_conn G x a).inter ((measurableSet_conn G y b).inter (measurableSet_conn G x y).compl))) using 1
-  · apply lintegral_congr
-    intro ω
-    simp only [Set.indicator,Set.mem_inter_iff,Set.mem_compl_iff,Set.mem_ofPred_eq,Pi.one_apply]
-  · congr 1
-
-theorem clusterPairKernel_distinct_bound [Countable V] [Countable E]
-    (G : BondGraph V E) (p : unitInterval) (J : V → V → ℝ≥0∞) (x y : V) :
-    (∫⁻ ω, if ¬Conn G ω x y then clusterPairKernel G ω J x y else 0 ∂law p) ≤
-      ∑' a, ∑' b, J a b * (law p {ω | Conn G ω x a} * law p {ω | Conn G ω y b}) := by
-  rw [clusterPairKernel_distinct_integral]
-  exact ENNReal.tsum_le_tsum (fun a => ENNReal.tsum_le_tsum (fun b =>
-    mul_le_mul_right (distinct_cluster_two_point_bound G p x y a b) _))
-
- 
-
-theorem clusterPairFunctional_square_diagram [Countable V] [Countable E]
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hw : ∀ i, 0 ≤ weight i) (p : unitInterval)
-    (hfin : ∀ x, ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
-    (J : V → V → ℝ≥0∞) (hJ : ∀ a ∈ Γ, ∀ x y, J (a x) (a y) = J x y) :
-    clusterPairFunctional G p root weight (fun ω x y => (clusterPairKernel G ω J x y)^2) ≤
-      ∑ i, ENNReal.ofReal (weight i) * ∑' v, ∑' a, ∑' b,
-        J (root i) v * (J a b * (law p {ω | Conn G ω (root i) a} * law p {ω | Conn G ω v b})) := by
-  have ht := clusterPairFunctional_reroot G hconn hloc root weight Γ hΓ htr hw p hfin J hJ
-    (fun ω x y => clusterPairKernel G ω J x y) (measurable_clusterPairKernel G J)
-    (fun ω x y a b => clusterPairKernel_cluster_constant G ω J x y a b)
-    (fun a ha b hab ω x y => clusterPairKernel_aut G hab J (hJ a ha) ω x y)
-  simp only [←pow_two] at ht
-  rw [ht]
-  apply Finset.sum_le_sum
-  intro i _
-  apply mul_le_mul_right
-  have hm (v : V) : Measurable (fun ω : Set E => if ¬Conn G ω (root i) v then clusterPairKernel G ω J (root i) v else 0) :=
-    (measurable_clusterPairKernel G J (root i) v).ite (measurableSet_conn G (root i) v).compl measurable_const
-  have hm' (v : V) : Measurable (fun ω : Set E => J (root i) v * (if ¬Conn G ω (root i) v then clusterPairKernel G ω J (root i) v else 0)) :=
-    (measurable_const : Measurable (fun _ : Set E => J (root i) v)).mul (hm v)
-  rw [lintegral_tsum (fun v => (hm' v).aemeasurable)]
-  apply ENNReal.tsum_le_tsum
-  intro v
-  rw [lintegral_const_mul _ (hm v)]
-  simp_rw [ENNReal.tsum_mul_left]
-  exact mul_le_mul_right (clusterPairKernel_distinct_bound G p J (root i) v) _
-
-end PairSquareDiagram
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-open scoped ENNReal
-section ContactRealCutoff
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def contactSquareLargeReal (G : BondGraph V E) (L : ℝ)
-    (ω : Set E) (x y : V) : ℝ≥0∞ :=
-  if L ≤ min ((cluster G ω x).ncard:ℝ) ((cluster G ω y).ncard:ℝ) then
-    (clusterContactCount G ω x y:ℝ≥0∞)^2 else 0
-
-theorem contactSquareLargeReal_eq_ceil (G : BondGraph V E) (L : ℝ)
-    (ω : Set E) (x y : V) :
-    contactSquareLargeReal G L ω x y = contactSquareLarge G ⌈L⌉₊ ω x y := by
-  have he : L ≤ min ((cluster G ω x).ncard:ℝ) ((cluster G ω y).ncard:ℝ) ↔
-      ⌈L⌉₊ ≤ min (cluster G ω x).ncard (cluster G ω y).ncard := by
-    rw [←Nat.cast_min, Nat.ceil_le]
-  simp only [contactSquareLargeReal,contactSquareLarge,he]
-
- 
-
-theorem critical_contact_square_bound
-    [Countable V] [Countable E] [Nonempty V] [Nonempty E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hconn : Connected G)
-    (hqt : QuasiTransitive G) (hh : 0 < hV G)
-    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
-    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
-    (hp₀ : 0 < (pc G:ℝ)) (hp₁ : (pc G:ℝ) < 1)
-    (hfin : ∀ i, ∀ᵐ ω ∂law (pc G), (cluster G ω (root i)).Finite)
-    (α ε : ℝ) (hα : 0 < α) (hα1 : α < 1/2) (hε : 0 < ε)
-    (hr : (1:ℝ)/2+ε-2*α < 0) :
-    ∃ C : ℝ, 0 < C ∧ ∀ (p : unitInterval), (pc G:ℝ)/2 ≤ (p:ℝ) → p ≤ pc G →
-      ∀ L : ℝ, 1 ≤ L → clusterPairFunctional G p root weight (contactSquareLargeReal G L) ≤
-        ENNReal.ofReal (C*L^((1:ℝ)/2+ε-2*α)) := by
-  obtain ⟨C,hC,hb⟩ := critical_contact_square_bound_nat G hloc hconn hqt hh root weight Γ hΓ htr
-    hcov hw hw1 hp₀ hp₁ hfin α ε hα hα1 hε hr
-  refine ⟨C,hC,?_⟩
-  intro p hp hpq L hL
-  have hLn : 1 ≤ ⌈L⌉₊ := by
-    have hc : (1:ℝ) ≤ (⌈L⌉₊:ℝ) := hL.trans (Nat.le_ceil L)
-    exact_mod_cast hc
-  have hEq : contactSquareLargeReal G L = contactSquareLarge G ⌈L⌉₊ := funext fun ω => funext fun x => funext fun y => contactSquareLargeReal_eq_ceil G L ω x y
-  rw [hEq]
-  exact (hb p hp hpq _ hLn).trans (ENNReal.ofReal_le_ofReal
-    (mul_le_mul_of_nonneg_left (Real.rpow_le_rpow_of_nonpos (by linarith : 0 < L) (Nat.le_ceil L) hr.le) hC.le))
-
-end ContactRealCutoff
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-open scoped ENNReal
-section NonnegativeOperatorDiagram
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def ennMatrixCoeff (A : Hilbert V →L[ℂ] Hilbert V) (x y : V) : ℝ≥0∞ :=
-  ENNReal.ofReal (matrixCoeff A x y).re
-
-theorem ennMatrixCoeff_mul {A B : Hilbert V →L[ℂ] Hilbert V}
-    (hA : PositiveRealMatrix A) (hB : PositiveRealMatrix B) (x y : V) :
-    ennMatrixCoeff (A*B) x y = ∑' z, ennMatrixCoeff A x z * ennMatrixCoeff B z y := by
-  unfold ennMatrixCoeff
-  rw [←(hA.mul_hasSum B x y).tsum_eq, ENNReal.ofReal_tsum_of_nonneg
-    (fun z => mul_nonneg (hA x z).2 (hB z y).2) (hA.mul_hasSum B x y).summable]
-  apply tsum_congr; intro z
-  exact ENNReal.ofReal_mul (hA x z).2
-
-theorem ennMatrixCoeff_symm {A : Hilbert V →L[ℂ] Hilbert V}
-    (hA : IsSelfAdjoint A) (x y : V) : ennMatrixCoeff A x y = ennMatrixCoeff A y x := by
-  unfold ennMatrixCoeff
-  rw [matrixCoeff_real_symm A hA x y]
-
-theorem HasKernel.ennMatrixCoeff [Countable E] (G : BondGraph V E) (p : unitInterval)
-    (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T) (x y : V) :
-    ennMatrixCoeff T x y = law p {ω | Conn G ω x y} := by
-  unfold Percolation.ennMatrixCoeff matrixCoeff
-  rw [hT x y]
-  simp only [Complex.ofReal_re,twoPoint,ENNReal.ofReal_toReal (measure_ne_top _ _)]
-
-theorem HasKernel.positiveRealMatrix (G : BondGraph V E) (p : unitInterval)
-    (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T) : PositiveRealMatrix T := by
-  intro x y
-  change (T (lp.single 2 y (1:ℂ)) x).im = 0 ∧ 0 ≤ (T (lp.single 2 y (1:ℂ)) x).re
-  rw [hT x y]
-  exact ⟨rfl,twoPoint_nonneg G p x y⟩
-
-theorem square_kernel_diagram (J T : Hilbert V →L[ℂ] Hilbert V)
-    (hJ : PositiveRealMatrix J) (hT : PositiveRealMatrix T)
-    (hsJ : IsSelfAdjoint J) (hsT : IsSelfAdjoint T) (x : V) :
-    (∑' v, ∑' a, ∑' b, ennMatrixCoeff J x v * (ennMatrixCoeff J a b *
-      (ennMatrixCoeff T x a * ennMatrixCoeff T v b))) = ennMatrixCoeff (J*T*J*T) x x := by
-  rw [mul_assoc,mul_assoc,ennMatrixCoeff_mul hJ (hT.mul (hJ.mul hT))]
-  apply tsum_congr; intro v
-  rw [ennMatrixCoeff_mul hT (hJ.mul hT)]
-  simp_rw [ennMatrixCoeff_mul hJ hT]
-  simp_rw [←ENNReal.tsum_mul_left]
-  rw [ENNReal.tsum_comm]
-  apply tsum_congr; intro b
-  apply tsum_congr; intro a
-  rw [ennMatrixCoeff_symm hsJ b a,ennMatrixCoeff_symm hsT a x]
-  ac_rfl
-
-theorem enn_rootTrace {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (hw : ∀ i, 0 ≤ weight i) (A : Hilbert V →L[ℂ] Hilbert V) (hA : PositiveRealMatrix A) :
-    (∑ i, ENNReal.ofReal (weight i) * ennMatrixCoeff A (root i) (root i)) =
-      ENNReal.ofReal (rootTrace root weight A).re := by
-  unfold rootTrace ennMatrixCoeff
-  simp only [Complex.re_sum,Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero]
-  rw [ENNReal.ofReal_sum_of_nonneg (fun i _ => mul_nonneg (hw i) (hA _ _).2)]
-  apply Finset.sum_congr rfl; intro i _
-  exact (ENNReal.ofReal_mul (hw i)).symm
-
-end NonnegativeOperatorDiagram
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-open scoped ENNReal ComplexOrder
-section KernelSquareTrace
-attribute [local instance] Classical.propDecidable Classical.decEq
-attribute [local instance] spectralRealOperatorScalarTower spectralRealOperatorSMulCommClass
-local instance : NonUnitalContinuousFunctionalCalculus ℝ
-    (Hilbert V →L[ℂ] Hilbert V) IsSelfAdjoint :=
-  IsSelfAdjoint.instNonUnitalContinuousFunctionalCalculus (A := Hilbert V →L[ℂ] Hilbert V)
-local instance : ContinuousFunctionalCalculus ℝ
-    (Hilbert V →L[ℂ] Hilbert V) IsSelfAdjoint :=
-  IsSelfAdjoint.instContinuousFunctionalCalculus (A := Hilbert V →L[ℂ] Hilbert V)
-
-theorem rootTrace_six_le_cube {ι : Type*} [Fintype ι]
-    (root : ι → V) (weight : ι → ℝ) {Γ : Set (V ≃ V)}
-    (htr : RootTransport root weight Γ) (hw : ∀ i, 0 ≤ weight i)
-    (T : Hilbert V →L[ℂ] Hilbert V) (hT : KernelInvariant Γ T) (hpos : 0 ≤ T) :
-    (rootTrace root weight (T^6)).re ≤ ‖T‖^3*(rootTrace root weight (T^3)).re := by
-  have hp3 : 0 ≤ T^3 := CStarAlgebra.pow_nonneg T 3 hpos
-  have h := rootTrace_positive_mul_le root weight htr hw (T^3) (T^3)
-    (hT.pow 3) (hT.pow 3) hp3 (IsSelfAdjoint.of_nonneg hp3)
-  have he : T^3*T^3 = T^6 := by rw [←pow_add]
-  rw [he] at h
-  have hr := (Complex.le_def.mp h).1
-  simp only [Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero] at hr
-  exact hr.trans (mul_le_mul_of_nonneg_right (norm_pow_le' T (by norm_num : 0 < (3:ℕ)))
-    (Complex.nonneg_iff.mp (rootTrace_nonneg root weight hw (T^3) hp3)).1)
-
- 
-
-theorem clusterPairFunctional_twoPoint_square_bound [Countable V] [Countable E]
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hw : ∀ i, 0 ≤ weight i) (p : unitInterval)
-    (hfin : ∀ x, ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
-    (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T) :
-    clusterPairFunctional G p root weight (fun ω x y =>
-      (clusterPairKernel G ω (ennMatrixCoeff (T^2)) x y)^2) ≤
-      ENNReal.ofReal (‖T‖^3*(rootTrace root weight (T^3)).re) := by
-  have hp := hT.positiveRealMatrix G p T
-  have hpos : 0 ≤ T := ContinuousLinearMap.nonneg_iff_isPositive.mpr (hT.isPositive G p T)
-  have hsa := IsSelfAdjoint.of_nonneg hpos
-  have hI : KernelInvariant Γ T := fun a ha => hT.invariant G p T a (hΓ ha)
-  have hJ : ∀ a ∈ Γ, ∀ x y, ennMatrixCoeff (T^2) (a x) (a y) = ennMatrixCoeff (T^2) x y := by
-    intro a ha x y
-    unfold ennMatrixCoeff
-    rw [(hI.pow 2) a ha]
-  have hb := clusterPairFunctional_square_diagram G hconn hloc root weight Γ hΓ htr hw p hfin
-    (ennMatrixCoeff (T^2)) hJ
-  have hd (x : V) : (∑' v, ∑' a, ∑' b, ennMatrixCoeff (T^2) x v * (ennMatrixCoeff (T^2) a b *
-      (law p {ω | Conn G ω x a} * law p {ω | Conn G ω v b}))) = ennMatrixCoeff (T^6) x x := by
-    simp_rw [←hT.ennMatrixCoeff G p T]
-    rw [square_kernel_diagram (T^2) T (hp.pow 2) hp (hsa.pow 2) hsa]
-    congr 1
-  simp_rw [hd] at hb
-  rw [enn_rootTrace root weight hw (T^6) (hp.pow 6)] at hb
-  exact hb.trans (ENNReal.ofReal_le_ofReal (rootTrace_six_le_cube root weight htr hw T hI hpos))
-
-end KernelSquareTrace
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section FiniteClusterKernel
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def finiteClusterWeight (G : BondGraph V E) (f : V → ℝ)
-    (x : V) (ω : Finset E) : ℝ := ∑ y ∈ finiteRootComponent G ω x, f y
-
-theorem finiteClusterWeight_expand (G : BondGraph V E) (f : V → ℝ)
-    (Q ω : Finset E) (hω : ω ⊆ Q) (x : V) :
-    finiteClusterWeight G f x ω = ∑ y ∈ insert x (searchVertices G Q),
-      f y * (if Conn G (ω:Set E) x y then (1:ℝ) else 0) := by
-  unfold finiteClusterWeight
-  calc
-    _ = ∑ y ∈ finiteRootComponent G ω x, f y * (if Conn G (ω:Set E) x y then (1:ℝ) else 0) := by
-      apply Finset.sum_congr rfl
-      intro y hy
-      have hc : Conn G (ω:Set E) x y := by
-        change y ∈ (finiteRootComponent G ω x:Set V) at hy
-        rwa [coe_finiteRootComponent] at hy
-      rw [ite_eq_left hc,mul_one]
-    _ = _ := Finset.sum_subset (finiteRootComponent_subset G Q _ x ω hω (by rfl)) (by
-      intro y _ hy
-      have hn : ¬Conn G (ω:Set E) x y := by
-        intro h
-        apply hy
-        change y ∈ (finiteRootComponent G ω x:Set V)
-        rwa [coe_finiteRootComponent]
-      rw [ite_eq_right hn,mul_zero])
-
-theorem finiteClusterWeight_expectation_le [Countable E]
-    (G : BondGraph V E) (Q : Finset E) (f : V → ℝ) (hf : ∀ y, 0 ≤ f y)
-    (x : V) (p : unitInterval) (hs : Summable (fun y => f y*twoPoint G p x y)) :
-    bernoulliPolynomial Q (finiteClusterWeight G f x) p ≤ ∑' y, f y*twoPoint G p x y := by
-  have he : bernoulliPolynomial Q (finiteClusterWeight G f x) p =
-      bernoulliPolynomial Q (fun ω => ∑ y ∈ insert x (searchVertices G Q), f y*(if Conn G (ω:Set E) x y then (1:ℝ) else 0)) p := by
-    apply bernoulliPolynomial_congr
-    intro ω hω
-    exact finiteClusterWeight_expand G f Q ω hω x
-  rw [he,bernoulliPolynomial_sum]
-  calc
-    _ ≤ ∑ y ∈ insert x (searchVertices G Q), f y*twoPoint G p x y := by
-      apply Finset.sum_le_sum
-      intro y _
-      have hpQ : bernoulliPolynomial Q (fun ω => if Conn G (ω:Set E) x y then (1:ℝ) else 0) p =
-          (law p).real {ω | Conn G (ω ∩ (Q:Set E)) x y} :=
-        finiteEventPolynomial_restrict Q {ω | Conn G ω x y} p
-      rw [bernoulliPolynomial_const_mul,hpQ]
-      apply mul_le_mul_of_nonneg_left _ (hf y)
-      exact measureReal_mono (fun ω h => conn_mono G Set.inter_subset_left h)
-    _ ≤ _ := hs.sum_le_tsum _ (fun y _ => mul_nonneg (hf y) (twoPoint_nonneg G p x y))
-
-noncomputable def finitePairKernel (G : BondGraph V E) (J : V → V → ℝ)
-    (ω : Finset E) (x y : V) : ℝ :=
-  ∑ a ∈ finiteRootComponent G ω x, finiteClusterWeight G (J a) y ω
-
-theorem finitePairKernel_nonneg (G : BondGraph V E) (J : V → V → ℝ)
-    (hJ : ∀ a b, 0 ≤ J a b) (ω : Finset E) (x y : V) : 0 ≤ finitePairKernel G J ω x y :=
-  Finset.sum_nonneg fun a _ => Finset.sum_nonneg fun b _ => hJ a b
-
-theorem finitePairKernel_exterior (G : BondGraph V E) (J : V → V → ℝ)
-    (ω : Finset E) (x y : V) (hd : ¬Conn G (ω:Set E) x y) :
-    finitePairKernel G J ω x y = ∑ a ∈ finiteRootComponent G ω x,
-      finiteClusterWeight G (J a) y (exteriorBonds G ω (finiteRootComponent G ω x:Set V)) := by
-  unfold finitePairKernel finiteClusterWeight
-  rw [finiteRootComponent_exterior G ω x y hd]
-
-end FiniteClusterKernel
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section SmallFinitePairKernel
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def finiteSmallPairKernel (G : BondGraph V E) (J : V → V → ℝ)
-    (L : ℝ) (x y : V) (ω : Finset E) : ℝ :=
-  if (finiteRootComponent G ω x).card < L ∧ ¬Conn G (ω:Set E) x y then finitePairKernel G J ω x y else 0
-
-noncomputable def finiteSmallClusterSize (G : BondGraph V E) (L : ℝ) (x : V) (ω : Finset E) : ℝ :=
-  if (finiteRootComponent G ω x).card < L then (finiteRootComponent G ω x).card else 0
-
- 
-theorem finiteSmallPairKernel_bound [Countable E]
-    (G : BondGraph V E) (Q : Finset E) (J : V → V → ℝ) (hJ : ∀ a b, 0 ≤ J a b)
-    (x y : V) (p : unitInterval) (L M : ℝ) (hM : 0 ≤ M)
-    (hs : ∀ a, Summable (fun b => J a b*twoPoint G p y b))
-    (hb : ∀ a, (∑' b, J a b*twoPoint G p y b) ≤ M) :
-    bernoulliPolynomial Q (finiteSmallPairKernel G J L x y) p ≤
-      M*bernoulliPolynomial Q (finiteSmallClusterSize G L x) p := by
-  let f (W : Finset V) (η : Finset E) : ℝ :=
-    (if (W.card:ℝ) < L ∧ y ∉ W then ∑ a ∈ W, finiteClusterWeight G (J a) y η else 0) -
-      M*(if (W.card:ℝ) < L then (W.card:ℝ) else 0)
-  have hf (W : Finset V) : bernoulliPolynomial (exteriorBonds G Q (W:Set V)) (f W) p ≤ 0 := by
-    by_cases hW : (W.card:ℝ) < L
-    · by_cases hy : y ∈ W
-      · simp only [f,hW,hy,not_true_eq_false,and_false,ite_false,ite_true,zero_sub,
-          bernoulliPolynomial_const]
-        exact neg_nonpos.mpr (mul_nonneg hM (Nat.cast_nonneg _))
-      · simp only [f,hW,hy,not_false_eq_true,and_self,ite_true,bernoulliPolynomial_sub,
-          bernoulliPolynomial_const,bernoulliPolynomial_sum]
-        apply sub_nonpos.mpr
-        calc
-          _ ≤ ∑ _a ∈ W, M := Finset.sum_le_sum fun a _ =>
-            (finiteClusterWeight_expectation_le G _ (J a) (hJ a) y p (hs a)).trans (hb a)
-          _ = _ := by simp only [Finset.sum_const,nsmul_eq_mul]; ring
-    · simp only [f,hW,false_and,ite_false,mul_zero,sub_zero,bernoulliPolynomial_zero]
-      exact le_rfl
-  have ht := cluster_exterior_expectation_bound G Q x p f 0 hf
-  have he (ω : Finset E) : f (finiteRootComponent G ω x)
-      (exteriorBonds G ω (finiteRootComponent G ω x:Set V)) =
-      finiteSmallPairKernel G J L x y ω - M*finiteSmallClusterSize G L x ω := by
-    have hm : y ∈ finiteRootComponent G ω x ↔ Conn G (ω:Set E) x y := by
-      change y ∈ (finiteRootComponent G ω x:Set V) ↔ _
-      rw [coe_finiteRootComponent]
-      rfl
-    unfold f finiteSmallPairKernel finiteSmallClusterSize
-    simp only [hm]
-    by_cases hc : (finiteRootComponent G ω x).card < L ∧ ¬Conn G (ω:Set E) x y
-    · rw [ite_eq_left hc,ite_eq_left hc,finitePairKernel_exterior G J ω x y hc.2]
-    · rw [ite_eq_right hc,ite_eq_right hc]
-  simp_rw [he] at ht
-  rw [bernoulliPolynomial_sub,bernoulliPolynomial_const_mul] at ht
-  exact sub_nonpos.mp ht
-
-end SmallFinitePairKernel
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section SmallSizeFractionalMoment
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem small_count_le_power (s L α : ℝ) (hs : 0 ≤ s) (hL : 0 ≤ L) (hα : α ≤ 1) :
-    (if s < L then s else 0) ≤ L^(1-α)*s^α := by
-  by_cases h : s < L
-  · rw [ite_eq_left h]
-    rcases eq_or_lt_of_le hs with rfl | hs'
-    · positivity
-    · calc
-        s = s^(1-α)*s^α := by rw [←Real.rpow_add hs']; simp only [sub_add_cancel,Real.rpow_one]
-        _ ≤ _ := mul_le_mul_of_nonneg_right (Real.rpow_le_rpow hs h.le (by linarith)) (Real.rpow_nonneg hs α)
-  · rw [ite_eq_right h]
-    exact mul_nonneg (Real.rpow_nonneg hL _) (Real.rpow_nonneg hs _)
-
-theorem finiteSmallClusterSize_moment_bound [Countable E]
-    (G : BondGraph V E) (Q : Finset E) (x : V) (p : unitInterval)
-    (L α : ℝ) (hL : 0 ≤ L) (hα0 : 0 ≤ α) (hα1 : α ≤ 1)
-    (hfin : ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
-    (hI : Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^α) (law p)) :
-    bernoulliPolynomial Q (finiteSmallClusterSize G L x) p ≤
-      L^(1-α)*(∫ ω, ((cluster G ω x).ncard:ℝ)^α ∂law p) := by
-  let f (ω : Set E) := finiteSmallClusterSize G L x (domainBits Q ω)
-  have hf : Integrable f (law p) := integrable_finite_local Q p f (fun ω => by
-    dsimp [f]; rw [domainBits_local])
-  rw [←integral_domainBits Q p (finiteSmallClusterSize G L x),←integral_const_mul]
-  apply integral_mono_ae hf (hI.const_mul _)
-  filter_upwards [hfin] with ω hω
-  have hc : (finiteRootComponent G (domainBits Q ω) x).card ≤ (cluster G ω x).ncard := by
-    rw [←Set.ncard_coe_finset,coe_finiteRootComponent,coe_domainBits]
-    exact Set.ncard_le_ncard (fun y hy => conn_mono G Set.inter_subset_left hy) hω
-  have hc' : ((finiteRootComponent G (domainBits Q ω) x).card:ℝ) ≤ (cluster G ω x).ncard := by exact_mod_cast hc
-  exact (small_count_le_power _ L α (Nat.cast_nonneg _) hL hα1).trans
-    (mul_le_mul_of_nonneg_left (Real.rpow_le_rpow (Nat.cast_nonneg _) hc' hα0) (Real.rpow_nonneg hL _))
-
-theorem finiteSmallPairKernel_moment_bound [Countable E]
-    (G : BondGraph V E) (Q : Finset E) (J : V → V → ℝ) (hJ : ∀ a b, 0 ≤ J a b)
-    (x y : V) (p : unitInterval) (L M α : ℝ) (hL : 0 ≤ L) (hM : 0 ≤ M)
-    (hα0 : 0 ≤ α) (hα1 : α ≤ 1)
-    (hfin : ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
-    (hI : Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^α) (law p))
-    (hs : ∀ a, Summable (fun b => J a b*twoPoint G p y b))
-    (hb : ∀ a, (∑' b, J a b*twoPoint G p y b) ≤ M) :
-    bernoulliPolynomial Q (finiteSmallPairKernel G J L x y) p ≤
-      M*(L^(1-α)*(∫ ω, ((cluster G ω x).ncard:ℝ)^α ∂law p)) :=
-  (finiteSmallPairKernel_bound G Q J hJ x y p L M hM hs hb).trans
-    (mul_le_mul_of_nonneg_left (finiteSmallClusterSize_moment_bound G Q x p L α hL hα0 hα1 hfin hI) hM)
-
-end SmallSizeFractionalMoment
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-open scoped ENNReal
-section SmallKernelLimit
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem clusterPairKernel_finite (G : BondGraph V E) (J : V → V → ℝ≥0∞)
-    (ω : Finset E) (x y : V) :
-    clusterPairKernel G ω J x y =
-      ∑ a ∈ finiteRootComponent G ω x, ∑ b ∈ finiteRootComponent G ω y, J a b := by
-  have hm (x a : V) : a ∈ finiteRootComponent G ω x ↔ Conn G (ω:Set E) x a := by
-    change a ∈ (finiteRootComponent G ω x:Set V) ↔ _
-    rw [coe_finiteRootComponent]
-    rfl
-  rw [clusterPairKernel,tsum_eq_sum (s := finiteRootComponent G ω x) (fun a ha => by
-    simp only [←hm x a,ha,false_and,ite_false,tsum_zero])]
-  apply Finset.sum_congr rfl
-  intro a ha
-  rw [tsum_eq_sum (s := finiteRootComponent G ω y) (fun b hb => by
-    simp only [←hm y b,hb,and_false,ite_false])]
-  apply Finset.sum_congr rfl
-  intro b hb
-  rw [ite_eq_left ⟨(hm x a).mp ha,(hm y b).mp hb⟩]
-
-theorem ofReal_finitePairKernel (G : BondGraph V E) (J : V → V → ℝ)
-    (hJ : ∀ a b, 0 ≤ J a b) (ω : Finset E) (x y : V) :
-    ENNReal.ofReal (finitePairKernel G J ω x y) =
-      clusterPairKernel G ω (fun a b => ENNReal.ofReal (J a b)) x y := by
-  rw [clusterPairKernel_finite]
-  unfold finitePairKernel finiteClusterWeight
-  rw [ENNReal.ofReal_sum_of_nonneg (fun a _ => Finset.sum_nonneg fun b _ => hJ a b)]
-  apply Finset.sum_congr rfl
-  intro a _
-  exact ENNReal.ofReal_sum_of_nonneg (fun b _ => hJ a b)
-
-noncomputable def smallPairKernel (G : BondGraph V E) (J : V → V → ℝ≥0∞)
-    (L : ℝ) (x y : V) (ω : Set E) : ℝ≥0∞ :=
-  if ((cluster G ω x).ncard:ℝ) < L ∧ ¬Conn G ω x y then clusterPairKernel G ω J x y else 0
-
-theorem smallPairKernel_eq_finite (G : BondGraph V E) (hloc : LocallyFinite G)
-    (J : V → V → ℝ) (hJ : ∀ a b, 0 ≤ J a b) (L : ℝ)
-    (ω : Set E) (x y : V) (hx : (cluster G ω x).Finite) (hy : (cluster G ω y).Finite)
-    (Q : Finset E) (hQx : incidentSupport G hloc hx.toFinset ⊆ Q)
-    (hQy : incidentSupport G hloc hy.toFinset ⊆ Q) :
-    ENNReal.ofReal (finiteSmallPairKernel G J L x y (domainBits Q ω)) =
-      smallPairKernel G (fun a b => ENNReal.ofReal (J a b)) L x y ω := by
-  have hcluster (z : V) (hz : (cluster G ω z).Finite) (hQ : incidentSupport G hloc hz.toFinset ⊆ Q) :
-      cluster G (domainBits Q ω:Set E) z = cluster G ω z := by
-    rw [coe_domainBits]
-    have hh := (cluster_restrict_incident_eq_iff G hloc hz.toFinset Q hQ ω z).mpr
-      (Set.Finite.coe_toFinset hz).symm
-    simpa only [Set.Finite.coe_toFinset] using hh
-  have hxC := hcluster x hx hQx
-  have hyC := hcluster y hy hQy
-  have hxN : (finiteRootComponent G (domainBits Q ω) x).card = (cluster G ω x).ncard := by
-    rw [←Set.ncard_coe_finset,coe_finiteRootComponent,hxC]
-  have hxconn (a : V) : Conn G (domainBits Q ω:Set E) x a ↔ Conn G ω x a := by
-    change a ∈ cluster G (domainBits Q ω:Set E) x ↔ a ∈ cluster G ω x
-    rw [hxC]
-  have hyconn (b : V) : Conn G (domainBits Q ω:Set E) y b ↔ Conn G ω y b := by
-    change b ∈ cluster G (domainBits Q ω:Set E) y ↔ b ∈ cluster G ω y
-    rw [hyC]
-  unfold finiteSmallPairKernel smallPairKernel
-  rw [hxN]
-  simp only [hxconn]
-  split_ifs
-  · rw [ofReal_finitePairKernel G J hJ]
-    simp only [clusterPairKernel,hxconn,hyconn]
-  · exact ENNReal.ofReal_zero
-
-end SmallKernelLimit
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-open scoped ENNReal
-section SmallKernelFatou
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem smallPairKernel_lintegral_le_of_finite [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G)
-    (J : V → V → ℝ) (hJ : ∀ a b, 0 ≤ J a b) (L : ℝ) (x y : V) (p : unitInterval)
-    (hx : ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
-    (hy : ∀ᵐ ω ∂law p, (cluster G ω y).Finite) (B : ℝ)
-    (hB : ∀ Q : Finset E, bernoulliPolynomial Q (finiteSmallPairKernel G J L x y) p ≤ B) :
-    (∫⁻ ω, smallPairKernel G (fun a b => ENNReal.ofReal (J a b)) L x y ω ∂law p) ≤ ENNReal.ofReal B := by
-  let : Encodable E := Encodable.ofCountable E
-  let Q := vertexExhaustion (V := E)
-  let f := fun j ω => finiteSmallPairKernel G J L x y (domainBits (Q j) ω)
-  have hfi (j : ℕ) : Integrable (f j) (law p) := integrable_finite_local (Q j) p (f j) (fun ω => by
-    dsimp [f]; rw [domainBits_local])
-  have hfn (j : ℕ) : 0 ≤ᵐ[law p] f j := Eventually.of_forall (fun ω => by
-    dsimp [f,finiteSmallPairKernel]
-    split_ifs
-    · exact finitePairKernel_nonneg G J hJ _ x y
-    · exact le_rfl)
-  have heq : ∀ᵐ ω ∂law p,
-      liminf (fun j => ENNReal.ofReal (f j ω)) atTop =
-        smallPairKernel G (fun a b => ENNReal.ofReal (J a b)) L x y ω := by
-    filter_upwards [hx,hy] with ω hωx hωy
-    obtain ⟨jx,hjx⟩ := subset_vertexExhaustion (incidentSupport G hloc hωx.toFinset)
-    obtain ⟨jy,hjy⟩ := subset_vertexExhaustion (incidentSupport G hloc hωy.toFinset)
-    have he : (fun j => ENNReal.ofReal (f j ω)) =ᶠ[atTop]
-        (fun _ => smallPairKernel G (fun a b => ENNReal.ofReal (J a b)) L x y ω) := by
-      filter_upwards [eventually_ge_atTop (max jx jy)] with j hj
-      exact smallPairKernel_eq_finite G hloc J hJ L ω x y hωx hωy (Q j)
-        (hjx.trans (vertexExhaustion_mono (le_max_left jx jy |>.trans hj)))
-        (hjy.trans (vertexExhaustion_mono (le_max_right jx jy |>.trans hj)))
-    exact (tendsto_const_nhds.congr' he.symm).liminf_eq
-  calc
-    _ = ∫⁻ ω, liminf (fun j => ENNReal.ofReal (f j ω)) atTop ∂law p := lintegral_congr_ae (heq.mono (fun _ h => h.symm))
-    _ ≤ liminf (fun j => ∫⁻ ω, ENNReal.ofReal (f j ω) ∂law p) atTop :=
-      lintegral_liminf_le' (fun j => (hfi j).aemeasurable.ennreal_ofReal)
-    _ ≤ ENNReal.ofReal B := liminf_le_of_frequently_le' (Eventually.frequently (Eventually.of_forall (fun j => by
-      rw [←ofReal_integral_eq_lintegral_ofReal (hfi j) (hfn j)]
-      apply ENNReal.ofReal_le_ofReal
-      rw [show (∫ ω, f j ω ∂law p) = bernoulliPolynomial (Q j) (finiteSmallPairKernel G J L x y) p from
-        integral_domainBits (Q j) p (finiteSmallPairKernel G J L x y)]
-      exact hB (Q j))))
-
- 
-
-theorem smallPairKernel_moment_bound [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G)
-    (J : V → V → ℝ) (hJ : ∀ a b, 0 ≤ J a b)
-    (x y : V) (p : unitInterval) (L M α : ℝ) (hL : 0 ≤ L) (hM : 0 ≤ M)
-    (hα0 : 0 ≤ α) (hα1 : α ≤ 1)
-    (hx : ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
-    (hy : ∀ᵐ ω ∂law p, (cluster G ω y).Finite)
-    (hI : Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^α) (law p))
-    (hs : ∀ a, Summable (fun b => J a b*twoPoint G p y b))
-    (hb : ∀ a, (∑' b, J a b*twoPoint G p y b) ≤ M) :
-    (∫⁻ ω, smallPairKernel G (fun a b => ENNReal.ofReal (J a b)) L x y ω ∂law p) ≤
-      ENNReal.ofReal (M*(L^(1-α)*(∫ ω, ((cluster G ω x).ncard:ℝ)^α ∂law p))) :=
-  smallPairKernel_lintegral_le_of_finite G hloc J hJ L x y p hx hy _ (fun Q =>
-    finiteSmallPairKernel_moment_bound G Q J hJ x y p L M α hL hM hα0 hα1 hx hI hs hb)
-
-end SmallKernelFatou
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-open scoped ENNReal ComplexOrder
-section SmallTwoPointKernel
-attribute [local instance] Classical.propDecidable Classical.decEq
-attribute [local instance] spectralRealOperatorScalarTower spectralRealOperatorSMulCommClass
-local instance : NonUnitalContinuousFunctionalCalculus ℝ
-    (Hilbert V →L[ℂ] Hilbert V) IsSelfAdjoint :=
-  IsSelfAdjoint.instNonUnitalContinuousFunctionalCalculus (A := Hilbert V →L[ℂ] Hilbert V)
-local instance : ContinuousFunctionalCalculus ℝ
-    (Hilbert V →L[ℂ] Hilbert V) IsSelfAdjoint :=
-  IsSelfAdjoint.instContinuousFunctionalCalculus (A := Hilbert V →L[ℂ] Hilbert V)
-
-theorem matrixCoeff_positive_trace_norm {ι : Type*} [Fintype ι]
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
-    (hw : ∀ i, 0 ≤ weight i) (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (A : Hilbert V →L[ℂ] Hilbert V) (hA : KernelInvariant Γ A) (hpos : 0 ≤ A) (x y : V) :
-    ‖matrixCoeff A x y‖ ≤ c*(rootTrace root weight A).re := by
-  have hd (z : V) : (matrixCoeff A z z).re ≤ c*(rootTrace root weight A).re := by
-    have h := (Complex.le_def.mp (rootTrace_diag_le root weight hcov hw c hc hmin A hA hpos z)).1
-    simpa only [Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero] using h
-  have hn (z : V) : 0 ≤ (matrixCoeff A z z).re := (Complex.nonneg_iff.mp (matrixCoeff_nonneg A hpos z)).1
-  have hC : 0 ≤ c*(rootTrace root weight A).re := mul_nonneg hc
-    (Complex.nonneg_iff.mp (rootTrace_nonneg root weight hw A hpos)).1
-  have hs := matrixCoeff_positive_product_norm_sq_le A 1 hpos x y
-  simp only [one_mul,star_one,mul_one] at hs
-  have hp := mul_le_mul (hd x) (hd y) (hn y) hC
-  nlinarith [norm_nonneg (matrixCoeff A x y)]
-
-theorem smallPairKernel_twoPoint_bound [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G)
-    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
-    (hw : ∀ i, 0 ≤ weight i) (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (x y : V) (p : unitInterval) (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T)
-    (L α : ℝ) (hL : 0 ≤ L) (hα0 : 0 ≤ α) (hα1 : α ≤ 1)
-    (hx : ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
-    (hy : ∀ᵐ ω ∂law p, (cluster G ω y).Finite)
-    (hI : Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^α) (law p)) :
-    (∫⁻ ω, smallPairKernel G (ennMatrixCoeff (T^2)) L x y ω ∂law p) ≤
-      ENNReal.ofReal ((c*(rootTrace root weight (T^3)).re)*
-        (L^(1-α)*(∫ ω, ((cluster G ω x).ncard:ℝ)^α ∂law p))) := by
-  have hp := hT.positiveRealMatrix G p T
-  have hpos : 0 ≤ T := ContinuousLinearMap.nonneg_iff_isPositive.mpr (hT.isPositive G p T)
-  have hp3 := CStarAlgebra.pow_nonneg T 3 hpos
-  have hTI : KernelInvariant Γ T := fun a ha => hT.invariant G p T a (hΓ ha)
-  have hM : 0 ≤ c*(rootTrace root weight (T^3)).re := mul_nonneg hc
-    (Complex.nonneg_iff.mp (rootTrace_nonneg root weight hw _ hp3)).1
-  have he (b : V) : (matrixCoeff T b y).re = twoPoint G p y b := by
-    change (T (lp.single 2 y (1:ℂ)) b).re = _
-    rw [hT b y,Complex.ofReal_re,twoPoint_symm G p b y]
-  have hs (a : V) : HasSum (fun b => (matrixCoeff (T^2) a b).re * twoPoint G p y b)
-      (matrixCoeff (T^3) a y).re := by
-    simpa only [he,←pow_succ] using (hp.pow 2).mul_hasSum T a y
-  apply smallPairKernel_moment_bound G hloc (fun a b => (matrixCoeff (T^2) a b).re)
-    (fun a b => ((hp.pow 2) a b).2) x y p L _ α hL hM hα0 hα1 hx hy hI
-    (fun a => (hs a).summable)
-  intro a
-  rw [(hs a).tsum_eq]
-  exact (Complex.re_le_norm _).trans (matrixCoeff_positive_trace_norm root weight Γ hcov hw c hc hmin _ (hTI.pow 3) hp3 a y)
-
-end SmallTwoPointKernel
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-open scoped ENNReal
-section ContactSmallReroot
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem measurable_smallPairKernel [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (J : V → V → ℝ≥0∞) (L : ℝ) (x y : V) :
-    Measurable (smallPairKernel G J L x y) := by
-  exact (measurable_clusterPairKernel G J x y).ite
-    ((measurableSet_lt ((measurable_of_countable (fun n : ℕ => (n:ℝ))).comp (measurable_cluster_ncard G hloc x))
-      measurable_const).inter (measurableSet_conn G x y).compl) measurable_const
-
-theorem smallPairKernel_cluster_constant (G : BondGraph V E) (J : V → V → ℝ≥0∞)
-    (L : ℝ) (ω : Set E) (x y a b : V) (hxa : Conn G ω x a) (hyb : Conn G ω y b) :
-    smallPairKernel G J L x y ω = smallPairKernel G J L a b ω := by
-  have hc : Conn G ω x y ↔ Conn G ω a b := by
-    constructor
-    · intro h; exact conn_trans G ω (conn_symm G ω hxa) (conn_trans G ω h hyb)
-    · intro h; exact conn_trans G ω hxa (conn_trans G ω h (conn_symm G ω hyb))
-  simp only [smallPairKernel,cluster_eq_of_conn G ω hxa,hc,
-    clusterPairKernel_cluster_constant G ω J x y a b hxa hyb]
-
-theorem smallPairKernel_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (J : V → V → ℝ≥0∞) (hJ : ∀ x y, J (a x) (a y) = J x y)
-    (L : ℝ) (ω : Set E) (x y : V) :
-    smallPairKernel G J L x y (b ⁻¹' ω) = smallPairKernel G J L (a x) (a y) ω := by
-  simp only [smallPairKernel,cluster_ncard_covariant G hab,conn_aut_iff G hab,
-    clusterPairKernel_aut G hab J hJ]
-
-theorem smallPairKernel_distinct (G : BondGraph V E) (J : V → V → ℝ≥0∞)
-    (L : ℝ) (ω : Set E) (x y : V) :
-    (if ¬Conn G ω x y then smallPairKernel G J L x y ω else 0) = smallPairKernel G J L x y ω := by
-  by_cases h : Conn G ω x y
-  · simp only [smallPairKernel,h,not_true_eq_false,and_false,ite_false]
-  · exact ite_eq_left h
-
- 
-theorem contact_smallPairKernel_bound [Countable V] [Countable E]
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1) (p : unitInterval)
-    (hfin : ∀ x, ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
-    (J : V → V → ℝ≥0∞) (hJ : ∀ a ∈ Γ, ∀ x y, J (a x) (a y) = J x y)
-    (L : ℝ) (d : ℕ) (hd : ∀ x, degree G x ≤ d) (B : ℝ)
-    (hb : ∀ x y, (∫⁻ ω, smallPairKernel G J L x y ω ∂law p) ≤ ENNReal.ofReal B) :
-    clusterPairFunctional G p root weight (fun ω x y =>
-      clusterPairKernel G ω (indexedEdgeKernel G) x y * smallPairKernel G J L x y ω) ≤
-      ENNReal.ofReal ((d:ℝ)*B) := by
-  have hj : ∀ a ∈ Γ, ∀ x y, indexedEdgeKernel G (a x) (a y) = indexedEdgeKernel G x y := by
-    intro a ha x y
-    obtain ⟨b,hab⟩ := hΓ ha
-    exact indexedEdgeKernel_aut G hab x y
-  rw [clusterPairFunctional_reroot G hconn hloc root weight Γ hΓ htr hw p hfin
-    (indexedEdgeKernel G) hj (fun ω x y => smallPairKernel G J L x y ω)
-    (measurable_smallPairKernel G hloc J L)
-    (fun ω x y a b hxa hyb => smallPairKernel_cluster_constant G J L ω x y a b hxa hyb)
-    (fun a ha b hab ω x y => smallPairKernel_aut G hab J (hJ a ha) L ω x y)]
-  simp_rw [smallPairKernel_distinct]
-  have hr (x : V) : (∫⁻ ω, ∑' y, indexedEdgeKernel G x y * smallPairKernel G J L x y ω ∂law p) ≤
-      ENNReal.ofReal ((d:ℝ)*B) := by
-    have hm (y : V) : Measurable (fun ω => indexedEdgeKernel G x y * smallPairKernel G J L x y ω) :=
-      (measurable_const : Measurable (fun _ : Set E => indexedEdgeKernel G x y)).mul (measurable_smallPairKernel G hloc J L x y)
-    rw [lintegral_tsum (fun y => (hm y).aemeasurable)]
-    simp_rw [lintegral_const_mul _ (measurable_smallPairKernel G hloc J L x _)]
-    calc
-      _ ≤ ∑' y, indexedEdgeKernel G x y * ENNReal.ofReal B := ENNReal.tsum_le_tsum (fun y => mul_le_mul_right (hb x y) _)
-      _ = (degree G x:ℝ≥0∞)*ENNReal.ofReal B := by rw [ENNReal.tsum_mul_right,indexedEdgeKernel_row G hloc x]
-      _ ≤ (d:ℝ≥0∞)*ENNReal.ofReal B := mul_le_mul_left (by exact_mod_cast hd x) _
-      _ = _ := by rw [ENNReal.ofReal_mul (by positivity),ENNReal.ofReal_natCast]
-  calc
-    _ ≤ ∑ i, ENNReal.ofReal (weight i)*ENNReal.ofReal ((d:ℝ)*B) :=
-      Finset.sum_le_sum (fun i _ => mul_le_mul_right (hr (root i)) _)
-    _ = _ := by rw [←Finset.sum_mul,←ENNReal.ofReal_sum_of_nonneg (fun i _ => hw i),hw1,ENNReal.ofReal_one,one_mul]
-
-end ContactSmallReroot
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-open scoped ENNReal
-section PairMeasure
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def clusterPairMeasure [MeasurableSpace V]
-    {ι : Type*} [MeasurableSpace ι] (G : BondGraph V E) (p : unitInterval)
-    (root : ι → V) (weight : ι → ℝ) : Measure (Set E × (ι × V)) :=
-  ((law p).prod Measure.count).withDensity (fun t => ENNReal.ofReal (weight t.2.1) *
-    clusterPairMass G t.1 (fun _ _ => 1) (root t.2.1) t.2.2)
-
-theorem clusterPairFunctional_eq_lintegral [Countable V] [Countable E]
-    [MeasurableSpace V] [MeasurableSingletonClass V]
-    {ι : Type*} [Fintype ι] [MeasurableSpace ι] [MeasurableSingletonClass ι]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (p : unitInterval)
-    (root : ι → V) (weight : ι → ℝ)
-    (F : Set E → V → V → ℝ≥0∞) (hF : ∀ x y, Measurable (fun ω => F ω x y)) :
-    clusterPairFunctional G p root weight F =
-      ∫⁻ t, F t.1 (root t.2.1) t.2.2 ∂clusterPairMeasure G p root weight := by
-  let d : Set E × (ι × V) → ℝ≥0∞ := fun t => ENNReal.ofReal (weight t.2.1) *
-    clusterPairMass G t.1 (fun _ _ => 1) (root t.2.1) t.2.2
-  let f : Set E × (ι × V) → ℝ≥0∞ := fun t => F t.1 (root t.2.1) t.2.2
-  have hd : Measurable d := by
-    apply measurable_from_prod_countable_left
-    intro q
-    exact (measurable_const : Measurable (fun _ : Set E => ENNReal.ofReal (weight q.1))).mul
-      (measurable_clusterPairMass G hloc (fun _ _ _ => 1) (fun _ _ => measurable_const) (root q.1) q.2)
-  have hf : Measurable f := measurable_from_prod_countable_left (fun q => hF (root q.1) q.2)
-  change _ = ∫⁻ t, f t ∂((law p).prod Measure.count).withDensity d
-  rw [lintegral_withDensity_eq_lintegral_mul _ hd hf,lintegral_prod _ ((hd.mul hf).aemeasurable)]
-  have he (ω : Set E) (i : ι) (y : V) :
-      (d*f) (ω,(i,y)) = ENNReal.ofReal (weight i)*clusterPairMass G ω (F ω) (root i) y := by
-    dsimp [d,f,clusterPairMass]
-    (split_ifs <;> simp only [mul_one,mul_zero,zero_mul]); ring
-  simp_rw [lintegral_count,ENNReal.tsum_prod',he,tsum_fintype]
-  have hm (i : ι) : Measurable (fun ω => ∑' y, ENNReal.ofReal (weight i)*clusterPairMass G ω (F ω) (root i) y) :=
-    Measurable.tsum (fun y => measurable_const.mul (measurable_clusterPairMass G hloc F hF (root i) y))
-  rw [lintegral_finsetSum _ (fun i _ => hm i)]
-  simp_rw [ENNReal.tsum_mul_left]
-  rw [clusterPairFunctional]
-  apply Finset.sum_congr rfl
-  intro i _
-  rw [lintegral_const_mul _ (Measurable.tsum (fun y => measurable_clusterPairMass G hloc F hF (root i) y))]
-  congr 1
-  apply lintegral_congr
-  intro ω
-  exact clusterPairDensity_eq_tsum G ω (F ω) (root i)
-
- 
-
-theorem clusterPairFunctional_cauchy_schwarz [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (p : unitInterval)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (F H : Set E → V → V → ℝ≥0∞)
-    (hF : ∀ x y, Measurable (fun ω => F ω x y))
-    (hH : ∀ x y, Measurable (fun ω => H ω x y)) :
-    clusterPairFunctional G p root weight (fun ω x y => F ω x y*H ω x y) ≤
-      (clusterPairFunctional G p root weight (fun ω x y => (F ω x y)^2))^((1:ℝ)/2) *
-      (clusterPairFunctional G p root weight (fun ω x y => (H ω x y)^2))^((1:ℝ)/2) := by
-  let : MeasurableSpace V := ⊤
-  let : MeasurableSingletonClass V := ⟨fun _ => trivial⟩
-  let : MeasurableSpace ι := ⊤
-  let : MeasurableSingletonClass ι := ⟨fun _ => trivial⟩
-  rw [clusterPairFunctional_eq_lintegral G hloc p root weight (fun ω x y => F ω x y*H ω x y) (fun x y => (hF x y).mul (hH x y)),
-    clusterPairFunctional_eq_lintegral G hloc p root weight (fun ω x y => (F ω x y)^2) (fun x y => (hF x y).pow_const 2),
-    clusterPairFunctional_eq_lintegral G hloc p root weight (fun ω x y => (H ω x y)^2) (fun x y => (hH x y).pow_const 2)]
-  have hf : Measurable (fun t : Set E × (ι × V) => F t.1 (root t.2.1) t.2.2) :=
-    measurable_from_prod_countable_left (fun q => hF (root q.1) q.2)
-  have hh : Measurable (fun t : Set E × (ι × V) => H t.1 (root t.2.1) t.2.2) :=
-    measurable_from_prod_countable_left (fun q => hH (root q.1) q.2)
-  simpa only [Pi.mul_apply,ENNReal.rpow_two] using
-    ENNReal.lintegral_mul_le_Lp_mul_Lq (clusterPairMeasure G p root weight) Real.HolderConjugate.two_two hf.aemeasurable hh.aemeasurable
-
-end PairMeasure
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Clusters.LargeContacts
+
+namespace OAI
+
+/-! Contact Trace for bond percolation. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+open scoped ENNReal
+section IndexedEdgeKernel
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem enn_ncard_eq_tsum {α : Type*} (A : Set α) (hA : A.Finite) :
+    (A.ncard:ℝ≥0∞) = ∑' a, if a ∈ A then (1:ℝ≥0∞) else 0 := by
+  rw [tsum_eq_sum (s := hA.toFinset) (fun a ha => by
+    simp only [Set.Finite.mem_toFinset] at ha
+    exact ite_eq_right ha)]
+  simp only [Set.ncard_eq_toFinset_card A hA]
+  rw [Finset.sum_congr rfl (fun a ha => ite_eq_left ((Set.Finite.mem_toFinset _).mp ha))]
+  simp
+
+ 
+
+noncomputable def indexedEdgeKernel (G : BondGraph V E) (x y : V) : ℝ≥0∞ :=
+  ∑' e, if G e = s(x,y) then 1 else 0
+
+theorem indexedEdgeKernel_eq_ncard (G : BondGraph V E) (hloc : LocallyFinite G) (x y : V) :
+    indexedEdgeKernel G x y = ({e | G e = s(x,y)}.ncard:ℝ≥0∞) := by
+  have hf : ({e | G e = s(x,y)} : Set E).Finite := (hloc x).subset (fun e he => by change x ∈ G e; change G e = s(x,y) at he; rw [he]; simp)
+  unfold indexedEdgeKernel
+  convert (enn_ncard_eq_tsum _ hf).symm using 1
+  apply tsum_congr
+  intro e
+  by_cases h : G e = s(x,y) <;> simp only [Set.mem_ofPred_eq,h,ite_true,ite_false]
+
+theorem indexedEdgeKernel_symm (G : BondGraph V E) (x y : V) :
+    indexedEdgeKernel G x y = indexedEdgeKernel G y x := by rw [indexedEdgeKernel,indexedEdgeKernel,Sym2.eq_swap]
+
+theorem indexedEdgeKernel_row (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) :
+    (∑' y, indexedEdgeKernel G x y) = (degree G x:ℝ≥0∞) := by
+  change (∑' y, ∑' e, if G e = s(x,y) then (1:ℝ≥0∞) else 0) = _
+  rw [ENNReal.tsum_comm,←ENNReal.tsum_prod]
+  have hn := (enn_ncard_eq_tsum _ (finite_steps G hloc x)).symm
+  have he : (∑' (p : E × V), if G p.1 = s(x,p.2) then (1:ℝ≥0∞) else 0) =
+      (({ey : E × V | G ey.1 = s(x,ey.2)} : Set (E × V)).ncard:ℝ≥0∞) := by
+    convert hn using 1
+    apply tsum_congr
+    intro ey
+    by_cases h : G ey.1 = s(x,ey.2) <;> simp only [Set.mem_ofPred_eq,h,ite_true,ite_false]
+  rw [he]
+  change (({ey : E × V | G ey.1 = s(x,ey.2)} : Set (E × V)).ncard:ℝ≥0∞) = (Nat.card (Step G x):ℝ≥0∞)
+  rw [←Nat.card_coe_set_eq]; rfl
+
+theorem indexedEdgeKernel_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (x y : V) :
+    indexedEdgeKernel G (a x) (a y) = indexedEdgeKernel G x y := by
+  unfold indexedEdgeKernel
+  rw [←b.tsum_eq (fun e => if G e = s(a x,a y) then (1:ℝ≥0∞) else 0)]
+  apply tsum_congr
+  intro e
+  have he : G (b e) = s(a x,a y) ↔ G e = s(x,y) := by
+    rw [hab e,←Sym2.map_mk]
+    exact (Sym2.map.injective a.injective).eq_iff
+  simp only [he]
+
+end IndexedEdgeKernel
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+open scoped ENNReal
+section ContactKernelCounting
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem contact_endpoints_unique (G : BondGraph V E) (ω : Set E) (x y : V)
+    (hd : ¬Conn G ω x y) {e : E} {a b c d : V}
+    (ha : Conn G ω x a) (_hb : Conn G ω y b)
+    (_hc : Conn G ω x c) (he : Conn G ω y d)
+    (hab : G e = s(a,b)) (hcd : G e = s(c,d)) : a = c ∧ b = d := by
+  rcases Sym2.eq_iff.mp (hab.symm.trans hcd) with h | h
+  · exact h
+  · exfalso
+    apply hd
+    exact conn_trans G ω ha (by simpa only [h.1] using conn_symm G ω he)
+
+theorem closedContactSet_finite (G : BondGraph V E) (hloc : LocallyFinite G)
+    (A B : Set V) (hA : A.Finite) : (closedContactSet G A B).Finite := by
+  have hs := contactSet_subset_incident G hloc hA.toFinset B
+  rw [Set.Finite.coe_toFinset] at hs
+  exact (incidentSupport G hloc hA.toFinset).finite_toSet.subset hs
+
+theorem clusterPairKernel_edge (G : BondGraph V E) (hloc : LocallyFinite G)
+    (ω : Set E) (x y : V) (hd : ¬Conn G ω x y) (hx : (cluster G ω x).Finite) :
+    clusterPairKernel G ω (indexedEdgeKernel G) x y = (clusterContactCount G ω x y:ℝ≥0∞) := by
+  have hed (e : E) :
+      (∑' a, ∑' b, if Conn G ω x a ∧ Conn G ω y b ∧ G e = s(a,b) then (1:ℝ≥0∞) else 0) =
+      if e ∈ closedContactSet G (cluster G ω x) (cluster G ω y) then 1 else 0 := by
+    by_cases h : e ∈ closedContactSet G (cluster G ω x) (cluster G ω y)
+    · obtain ⟨a,ha,b,hb,hab⟩ := h
+      change Conn G ω x a at ha
+      change Conn G ω y b at hb
+      have hh : e ∈ closedContactSet G (cluster G ω x) (cluster G ω y) := ⟨a,ha,b,hb,hab⟩
+      rw [ite_eq_left hh,tsum_eq_single a]
+      · rw [tsum_eq_single b]
+        · exact ite_eq_left ⟨ha,hb,hab⟩
+        · intro d hdb
+          apply ite_eq_right
+          intro hc
+          exact hdb (contact_endpoints_unique G ω x y hd hc.1 hc.2.1 ha hb hc.2.2 hab).2
+      · intro c hca
+        apply ENNReal.tsum_eq_zero.mpr
+        intro d
+        apply ite_eq_right
+        intro hc
+        exact hca (contact_endpoints_unique G ω x y hd hc.1 hc.2.1 ha hb hc.2.2 hab).1
+    · rw [ite_eq_right h]
+      apply ENNReal.tsum_eq_zero.mpr
+      intro a
+      apply ENNReal.tsum_eq_zero.mpr
+      intro b
+      apply ite_eq_right
+      intro hab
+      exact h ⟨a,hab.1,b,hab.2.1,hab.2.2⟩
+  have he (a b : V) : (if Conn G ω x a ∧ Conn G ω y b then indexedEdgeKernel G a b else 0) =
+      ∑' e, if Conn G ω x a ∧ Conn G ω y b ∧ G e = s(a,b) then (1:ℝ≥0∞) else 0 := by
+    by_cases ha : Conn G ω x a <;> by_cases hb : Conn G ω y b <;>
+      simp only [ha,hb,and_self,true_and,false_and,ite_true,ite_false,indexedEdgeKernel,tsum_zero]
+  rw [clusterPairKernel]
+  simp_rw [he]
+  calc
+    _ = ∑' e, ∑' a, ∑' b, if Conn G ω x a ∧ Conn G ω y b ∧ G e = s(a,b) then (1:ℝ≥0∞) else 0 := by
+      calc
+        _ = ∑' a, ∑' e, ∑' b, if Conn G ω x a ∧ Conn G ω y b ∧ G e = s(a,b) then (1:ℝ≥0∞) else 0 := by
+          apply tsum_congr; intro a; exact ENNReal.tsum_comm
+        _ = _ := ENNReal.tsum_comm
+    _ = ∑' e, if e ∈ closedContactSet G (cluster G ω x) (cluster G ω y) then (1:ℝ≥0∞) else 0 := tsum_congr hed
+    _ = _ := (enn_ncard_eq_tsum _ (closedContactSet_finite G hloc _ _ hx)).symm
+
+end ContactKernelCounting
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+open scoped ENNReal
+section ContactCovariance
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem closedContactSet_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (A B : Set V) :
+    b '' closedContactSet G A B = closedContactSet G (a '' A) (a '' B) := by
+  ext e
+  constructor
+  · rintro ⟨f,⟨x,hx,y,hy,hf⟩,rfl⟩
+    exact ⟨a x,Set.mem_image_of_mem a hx,a y,Set.mem_image_of_mem a hy,by rw [hab f,hf,Sym2.map_mk]⟩
+  · rintro ⟨z,⟨x,hx,rfl⟩,w,⟨y,hy,rfl⟩,he⟩
+    refine ⟨b.symm e,⟨x,hx,y,hy,?_⟩,b.apply_symm_apply e⟩
+    apply Sym2.map.injective a.injective
+    rw [←hab,Sym2.map_mk]
+    simpa only [Equiv.apply_symm_apply] using he
+
+theorem clusterContactCount_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (ω : Set E) (x y : V) :
+    clusterContactCount G (b ⁻¹' ω) x y = clusterContactCount G ω (a x) (a y) := by
+  unfold clusterContactCount
+  rw [←cluster_aut G hab ω x,←cluster_aut G hab ω y,←closedContactSet_aut G hab,
+    Set.ncard_image_of_injective _ b.injective]
+
+theorem fullLargeContact_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (ω : Set E) (x y : V) (L : ℕ) :
+    fullLargeContact G x y L (b ⁻¹' ω) = fullLargeContact G (a x) (a y) L ω := by
+  simp only [fullLargeContact,conn_aut_iff G hab,cluster_ncard_covariant G hab,clusterContactCount_aut G hab]
+
+theorem fullLargeContact_cluster_constant (G : BondGraph V E) (ω : Set E)
+    (L : ℕ) (x y a b : V) (hxa : Conn G ω x a) (hyb : Conn G ω y b) :
+    fullLargeContact G x y L ω = fullLargeContact G a b L ω := by
+  have hc : Conn G ω x y ↔ Conn G ω a b := by
+    constructor
+    · intro h; exact conn_trans G ω (conn_symm G ω hxa) (conn_trans G ω h hyb)
+    · intro h; exact conn_trans G ω hxa (conn_trans G ω h (conn_symm G ω hyb))
+  simp only [fullLargeContact,clusterContactCount,cluster_eq_of_conn G ω hxa,
+    cluster_eq_of_conn G ω hyb,hc]
+
+theorem fullLargeContact_distinct (G : BondGraph V E) (ω : Set E) (x y : V) (L : ℕ) :
+    (if ¬Conn G ω x y then ENNReal.ofReal (fullLargeContact G x y L ω) else 0) =
+      ENNReal.ofReal (fullLargeContact G x y L ω) := by
+  by_cases h : Conn G ω x y
+  · simp only [fullLargeContact,h,not_true_eq_false,false_and,ite_false,ENNReal.ofReal_zero]
+  · exact ite_eq_left h
+
+end ContactCovariance
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+open scoped ENNReal
+section ContactSquareTrace
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def contactSquareLarge (G : BondGraph V E) (L : ℕ)
+    (ω : Set E) (x y : V) : ℝ≥0∞ :=
+  if L ≤ min (cluster G ω x).ncard (cluster G ω y).ncard then
+    (clusterContactCount G ω x y:ℝ≥0∞)^2 else 0
+
+theorem contactSquareLarge_reroot [Countable V] [Countable E]
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hw : ∀ i, 0 ≤ weight i) (p : unitInterval)
+    (hfin : ∀ x, ∀ᵐ ω ∂law p, (cluster G ω x).Finite) (L : ℕ) :
+    clusterPairFunctional G p root weight (contactSquareLarge G L) =
+      ∑ i, ENNReal.ofReal (weight i) * ∫⁻ ω, ∑' y,
+        indexedEdgeKernel G (root i) y * ENNReal.ofReal (fullLargeContact G (root i) y L ω) ∂law p := by
+  have hj : ∀ a ∈ Γ, ∀ x y, indexedEdgeKernel G (a x) (a y) = indexedEdgeKernel G x y := by
+    intro a ha x y
+    obtain ⟨b,hab⟩ := hΓ ha
+    exact indexedEdgeKernel_aut G hab x y
+  have ht := clusterPairFunctional_reroot G hconn hloc root weight Γ hΓ htr hw p hfin
+    (indexedEdgeKernel G) hj (fun ω x y => ENNReal.ofReal (fullLargeContact G x y L ω))
+    (fun x y => (measurable_fullLargeContact G hloc x y L).ennreal_ofReal)
+    (fun ω x y a b hxa hyb => congrArg ENNReal.ofReal (fullLargeContact_cluster_constant G ω L x y a b hxa hyb))
+    (fun a _ b hab ω x y => congrArg ENNReal.ofReal (fullLargeContact_aut G hab ω x y L))
+  simp_rw [fullLargeContact_distinct] at ht
+  rw [←ht]
+  unfold clusterPairFunctional
+  apply Finset.sum_congr rfl
+  intro i _
+  congr 1
+  apply lintegral_congr_ae
+  filter_upwards [hfin (root i)] with ω hω
+  unfold clusterPairDensity
+  congr 1
+  apply tsum_congr
+  intro y
+  congr 1
+  by_cases hd : Conn G ω (root i) y
+  · simp only [hd,not_true_eq_false,ite_false]
+  · rw [ite_eq_left hd,ite_eq_left hd]
+    dsimp only
+    rw [clusterPairKernel_edge G hloc ω (root i) y hd hω]
+    unfold contactSquareLarge fullLargeContact
+    by_cases hs : L ≤ min (cluster G ω (root i)).ncard (cluster G ω y).ncard
+    · simp only [hd,hs,not_false_eq_true,true_and,ite_true,ENNReal.ofReal_natCast,pow_two]
+    · simp only [hs,and_false,ite_false,ENNReal.ofReal_zero,mul_zero]
+
+theorem contactSquareLarge_bound [Countable V] [Countable E]
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1) (p : unitInterval)
+    (hfin : ∀ x, ∀ᵐ ω ∂law p, (cluster G ω x).Finite) (L d : ℕ)
+    (hd : ∀ x, degree G x ≤ d) (B : ℝ) (_hB : 0 ≤ B)
+    (hI : ∀ x y, Integrable (fullLargeContact G x y L) (law p))
+    (hb : ∀ x y, (∫ ω, fullLargeContact G x y L ω ∂law p) ≤ B) :
+    clusterPairFunctional G p root weight (contactSquareLarge G L) ≤ ENNReal.ofReal ((d:ℝ)*B) := by
+  have hi (x y : V) : (∫⁻ ω, ENNReal.ofReal (fullLargeContact G x y L ω) ∂law p) ≤ ENNReal.ofReal B := by
+    rw [←ofReal_integral_eq_lintegral_ofReal (hI x y) (Filter.Eventually.of_forall (fullLargeContact_nonneg G x y L))]
+    exact ENNReal.ofReal_le_ofReal (hb x y)
+  have hr (x : V) : (∫⁻ ω, ∑' y, indexedEdgeKernel G x y * ENNReal.ofReal (fullLargeContact G x y L ω) ∂law p) ≤
+      ENNReal.ofReal ((d:ℝ)*B) := by
+    have hm (y : V) : Measurable (fun ω => indexedEdgeKernel G x y * ENNReal.ofReal (fullLargeContact G x y L ω)) :=
+      (measurable_const : Measurable (fun _ : Set E => indexedEdgeKernel G x y)).mul (measurable_fullLargeContact G hloc x y L).ennreal_ofReal
+    rw [lintegral_tsum (fun y => (hm y).aemeasurable)]
+    simp_rw [lintegral_const_mul _ (measurable_fullLargeContact G hloc x _ L).ennreal_ofReal]
+    calc
+      _ ≤ ∑' y, indexedEdgeKernel G x y * ENNReal.ofReal B := ENNReal.tsum_le_tsum (fun y => mul_le_mul_right (hi x y) _)
+      _ = (degree G x:ℝ≥0∞)*ENNReal.ofReal B := by rw [ENNReal.tsum_mul_right,indexedEdgeKernel_row G hloc x]
+      _ ≤ (d:ℝ≥0∞)*ENNReal.ofReal B := mul_le_mul_left (by exact_mod_cast hd x) _
+      _ = _ := by rw [ENNReal.ofReal_mul (by positivity),ENNReal.ofReal_natCast]
+  rw [contactSquareLarge_reroot G hconn hloc root weight Γ hΓ htr hw p hfin L]
+  calc
+    _ ≤ ∑ i, ENNReal.ofReal (weight i)*ENNReal.ofReal ((d:ℝ)*B) :=
+      Finset.sum_le_sum (fun i _ => mul_le_mul_right (hr (root i)) _)
+    _ = _ := by rw [←Finset.sum_mul,←ENNReal.ofReal_sum_of_nonneg (fun i _ => hw i),hw1,ENNReal.ofReal_one,one_mul]
+
+end ContactSquareTrace
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+open scoped ENNReal
+section CriticalContactSquare
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem critical_contact_square_bound_nat
+    [Countable V] [Countable E] [Nonempty V] [Nonempty E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hconn : Connected G)
+    (hqt : QuasiTransitive G) (hh : 0 < hV G)
+    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
+    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
+    (hp₀ : 0 < (pc G:ℝ)) (hp₁ : (pc G:ℝ) < 1)
+    (hfin : ∀ i, ∀ᵐ ω ∂law (pc G), (cluster G ω (root i)).Finite)
+    (α ε : ℝ) (hα : 0 < α) (hα1 : α < 1/2) (hε : 0 < ε)
+    (hr : (1:ℝ)/2+ε-2*α < 0) :
+    ∃ C : ℝ, 0 < C ∧ ∀ (p : unitInterval), (pc G:ℝ)/2 ≤ (p:ℝ) → p ≤ pc G →
+      ∀ L : ℕ, 1 ≤ L → clusterPairFunctional G p root weight (contactSquareLarge G L) ≤
+        ENNReal.ofReal (C*(L:ℝ)^((1:ℝ)/2+ε-2*α)) := by
+  obtain ⟨A,hA,hb⟩ := critical_large_contact_bound G hloc hconn hqt hh root weight Γ hΓ htr
+    hcov hw hw1 hp₀ hp₁ hfin α ε hα hα1 hε hr
+  obtain ⟨d,hd,hdeg⟩ := exists_degree_bound G hqt
+  refine ⟨(d:ℝ)*A,mul_pos (by exact_mod_cast hd) hA,?_⟩
+  intro p hp hpq L hL
+  have hf : ∀ x, ∀ᵐ ω ∂law p, (cluster G ω x).Finite := by
+    intro x
+    obtain ⟨i,a,ha,rfl⟩ := hcov x
+    obtain ⟨b,hab⟩ := hΓ ha
+    exact cluster_finite_ae_mono G (a (root i)) p (pc G) hpq
+      (cluster_finite_ae_covariant G (pc G) hab (root i) (hfin i))
+  have h := contactSquareLarge_bound G hconn hloc root weight Γ hΓ htr (fun i => (hw i).le) hw1 p hf
+    L d hdeg (A*(L:ℝ)^((1:ℝ)/2+ε-2*α)) (mul_nonneg hA.le (Real.rpow_nonneg (by positivity) _))
+    (fun x y => (hb x y p hp hpq L hL).1) (fun x y => (hb x y p hp hpq L hL).2)
+  simpa only [mul_assoc] using h
+
+end CriticalContactSquare
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+open scoped ENNReal
+section DistinctClusterBK
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem distinct_cluster_disjoint_occurrence (G : BondGraph V E) (ω : Set E)
+    (x y a b : V) (hxa : Conn G ω x a) (hyb : Conn G ω y b) (hd : ¬Conn G ω x y) :
+    ω ∈ DisjointOccurrence {η | Conn G η x a} {η | Conn G η y b} := by
+  obtain ⟨s,hs,hsa⟩ := finiteWitness_conn G x a _ (conn_restrict_cluster G ω x (mem_cluster_self G ω x) hxa)
+  obtain ⟨t,ht,htb⟩ := finiteWitness_conn G y b _ (conn_restrict_cluster_compl G ω x hd hyb)
+  refine ⟨s,t,fun e he => (hs he).1,fun e he => (ht he).1,?_,hsa,htb⟩
+  apply Finset.disjoint_left.mpr
+  intro e he ht'
+  obtain ⟨⟨z,w⟩,hr⟩ := Quot.exists_rep (G e)
+  have hz : z ∈ G e := by rw [←hr]; exact Sym2.mem_mk_left z w
+  exact (ht ht').2 z hz ((hs he).2 z hz)
+
+theorem distinct_cluster_two_point_bound [Countable E]
+    (G : BondGraph V E) (p : unitInterval) (x y a b : V) :
+    law p {ω | Conn G ω x a ∧ Conn G ω y b ∧ ¬Conn G ω x y} ≤
+      law p {ω | Conn G ω x a} * law p {ω | Conn G ω y b} := by
+  calc
+    _ ≤ law p (DisjointOccurrence {ω | Conn G ω x a} {ω | Conn G ω y b}) :=
+      measure_mono (fun ω h => distinct_cluster_disjoint_occurrence G ω x y a b h.1 h.2.1 h.2.2)
+    _ ≤ _ := bk_ennreal p _ _ (fun _ _ h hc => conn_mono G h hc)
+      (fun _ _ h hc => conn_mono G h hc) (finiteWitness_conn G x a) (finiteWitness_conn G y b)
+
+theorem measurable_clusterPairKernel [Countable V] [Countable E]
+    (G : BondGraph V E) (J : V → V → ℝ≥0∞) (x y : V) :
+    Measurable (fun ω => clusterPairKernel G ω J x y) := by
+  apply Measurable.tsum
+  intro a
+  apply Measurable.tsum
+  intro b
+  exact measurable_const.ite ((measurableSet_conn G x a).inter (measurableSet_conn G y b)) measurable_const
+
+theorem clusterPairKernel_cluster_constant (G : BondGraph V E) (ω : Set E)
+    (J : V → V → ℝ≥0∞) (x y a b : V) (hxa : Conn G ω x a) (hyb : Conn G ω y b) :
+    clusterPairKernel G ω J x y = clusterPairKernel G ω J a b := by
+  have hx (z : V) : Conn G ω x z ↔ Conn G ω a z :=
+    ⟨fun h => conn_trans G ω (conn_symm G ω hxa) h,fun h => conn_trans G ω hxa h⟩
+  have hy (z : V) : Conn G ω y z ↔ Conn G ω b z :=
+    ⟨fun h => conn_trans G ω (conn_symm G ω hyb) h,fun h => conn_trans G ω hyb h⟩
+  simp only [clusterPairKernel,hx,hy]
+
+theorem clusterPairKernel_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (J : V → V → ℝ≥0∞) (hJ : ∀ x y, J (a x) (a y) = J x y)
+    (ω : Set E) (x y : V) :
+    clusterPairKernel G (b ⁻¹' ω) J x y = clusterPairKernel G ω J (a x) (a y) := by
+  unfold clusterPairKernel
+  rw [←a.tsum_eq (fun x' => ∑' y', if Conn G ω (a x) x' ∧ Conn G ω (a y) y' then J x' y' else 0)]
+  apply tsum_congr
+  intro x'
+  rw [←a.tsum_eq (fun y' => if Conn G ω (a x) (a x') ∧ Conn G ω (a y) y' then J (a x') y' else 0)]
+  simp only [hJ,conn_aut_iff G hab]
+
+end DistinctClusterBK
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+open scoped ENNReal
+section PairSquareDiagram
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem clusterPairKernel_distinct_integral [Countable V] [Countable E]
+    (G : BondGraph V E) (p : unitInterval) (J : V → V → ℝ≥0∞) (x y : V) :
+    (∫⁻ ω, if ¬Conn G ω x y then clusterPairKernel G ω J x y else 0 ∂law p) =
+      ∑' a, ∑' b, J a b * law p {ω | Conn G ω x a ∧ Conn G ω y b ∧ ¬Conn G ω x y} := by
+  have he (ω : Set E) : (if ¬Conn G ω x y then clusterPairKernel G ω J x y else 0) =
+      ∑' a, ∑' b, J a b * (if Conn G ω x a ∧ Conn G ω y b ∧ ¬Conn G ω x y then (1:ℝ≥0∞) else 0) := by
+    by_cases hd : Conn G ω x y
+    · simp only [hd,not_true_eq_false,and_false,ite_false,mul_zero,tsum_zero]
+    · rw [ite_eq_left hd,clusterPairKernel]
+      apply tsum_congr; intro a
+      apply tsum_congr; intro b
+      by_cases h : Conn G ω x a ∧ Conn G ω y b
+      · simp only [h.1,h.2,hd,not_false_eq_true,and_self,ite_true,mul_one]
+      · have hn : ¬ (Conn G ω x a ∧ Conn G ω y b ∧ ¬Conn G ω x y) := fun h' => h ⟨h'.1,h'.2.1⟩
+        simp only [ite_eq_right h,ite_eq_right hn,mul_zero]
+  have hm (a b : V) : Measurable (fun ω => if Conn G ω x a ∧ Conn G ω y b ∧ ¬Conn G ω x y then (1:ℝ≥0∞) else 0) :=
+    measurable_const.ite ((measurableSet_conn G x a).inter ((measurableSet_conn G y b).inter (measurableSet_conn G x y).compl)) measurable_const
+  have hm' (a b : V) : Measurable (fun ω : Set E => J a b * (if Conn G ω x a ∧ Conn G ω y b ∧ ¬Conn G ω x y then (1:ℝ≥0∞) else 0)) :=
+    (measurable_const : Measurable (fun _ : Set E => J a b)).mul (hm a b)
+  simp_rw [he]
+  rw [lintegral_tsum (fun a => (Measurable.tsum (fun b => hm' a b)).aemeasurable)]
+  apply tsum_congr; intro a
+  rw [lintegral_tsum (fun b => (hm' a b).aemeasurable)]
+  apply tsum_congr; intro b
+  rw [lintegral_const_mul _ (hm a b)]
+  congr 1
+  convert
+    (lintegral_indicator_one (μ := law p) ((measurableSet_conn G x a).inter ((measurableSet_conn G y b).inter (measurableSet_conn G x y).compl))) using 1
+  · apply lintegral_congr
+    intro ω
+    simp only [Set.indicator,Set.mem_inter_iff,Set.mem_compl_iff,Set.mem_ofPred_eq,Pi.one_apply]
+  · congr 1
+
+theorem clusterPairKernel_distinct_bound [Countable V] [Countable E]
+    (G : BondGraph V E) (p : unitInterval) (J : V → V → ℝ≥0∞) (x y : V) :
+    (∫⁻ ω, if ¬Conn G ω x y then clusterPairKernel G ω J x y else 0 ∂law p) ≤
+      ∑' a, ∑' b, J a b * (law p {ω | Conn G ω x a} * law p {ω | Conn G ω y b}) := by
+  rw [clusterPairKernel_distinct_integral]
+  exact ENNReal.tsum_le_tsum (fun a => ENNReal.tsum_le_tsum (fun b =>
+    mul_le_mul_right (distinct_cluster_two_point_bound G p x y a b) _))
+
+ 
+
+theorem clusterPairFunctional_square_diagram [Countable V] [Countable E]
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hw : ∀ i, 0 ≤ weight i) (p : unitInterval)
+    (hfin : ∀ x, ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
+    (J : V → V → ℝ≥0∞) (hJ : ∀ a ∈ Γ, ∀ x y, J (a x) (a y) = J x y) :
+    clusterPairFunctional G p root weight (fun ω x y => (clusterPairKernel G ω J x y)^2) ≤
+      ∑ i, ENNReal.ofReal (weight i) * ∑' v, ∑' a, ∑' b,
+        J (root i) v * (J a b * (law p {ω | Conn G ω (root i) a} * law p {ω | Conn G ω v b})) := by
+  have ht := clusterPairFunctional_reroot G hconn hloc root weight Γ hΓ htr hw p hfin J hJ
+    (fun ω x y => clusterPairKernel G ω J x y) (measurable_clusterPairKernel G J)
+    (fun ω x y a b => clusterPairKernel_cluster_constant G ω J x y a b)
+    (fun a ha b hab ω x y => clusterPairKernel_aut G hab J (hJ a ha) ω x y)
+  simp only [←pow_two] at ht
+  rw [ht]
+  apply Finset.sum_le_sum
+  intro i _
+  apply mul_le_mul_right
+  have hm (v : V) : Measurable (fun ω : Set E => if ¬Conn G ω (root i) v then clusterPairKernel G ω J (root i) v else 0) :=
+    (measurable_clusterPairKernel G J (root i) v).ite (measurableSet_conn G (root i) v).compl measurable_const
+  have hm' (v : V) : Measurable (fun ω : Set E => J (root i) v * (if ¬Conn G ω (root i) v then clusterPairKernel G ω J (root i) v else 0)) :=
+    (measurable_const : Measurable (fun _ : Set E => J (root i) v)).mul (hm v)
+  rw [lintegral_tsum (fun v => (hm' v).aemeasurable)]
+  apply ENNReal.tsum_le_tsum
+  intro v
+  rw [lintegral_const_mul _ (hm v)]
+  simp_rw [ENNReal.tsum_mul_left]
+  exact mul_le_mul_right (clusterPairKernel_distinct_bound G p J (root i) v) _
+
+end PairSquareDiagram
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+open scoped ENNReal
+section ContactRealCutoff
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def contactSquareLargeReal (G : BondGraph V E) (L : ℝ)
+    (ω : Set E) (x y : V) : ℝ≥0∞ :=
+  if L ≤ min ((cluster G ω x).ncard:ℝ) ((cluster G ω y).ncard:ℝ) then
+    (clusterContactCount G ω x y:ℝ≥0∞)^2 else 0
+
+theorem contactSquareLargeReal_eq_ceil (G : BondGraph V E) (L : ℝ)
+    (ω : Set E) (x y : V) :
+    contactSquareLargeReal G L ω x y = contactSquareLarge G ⌈L⌉₊ ω x y := by
+  have he : L ≤ min ((cluster G ω x).ncard:ℝ) ((cluster G ω y).ncard:ℝ) ↔
+      ⌈L⌉₊ ≤ min (cluster G ω x).ncard (cluster G ω y).ncard := by
+    rw [←Nat.cast_min, Nat.ceil_le]
+  simp only [contactSquareLargeReal,contactSquareLarge,he]
+
+ 
+
+theorem critical_contact_square_bound
+    [Countable V] [Countable E] [Nonempty V] [Nonempty E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hconn : Connected G)
+    (hqt : QuasiTransitive G) (hh : 0 < hV G)
+    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
+    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
+    (hp₀ : 0 < (pc G:ℝ)) (hp₁ : (pc G:ℝ) < 1)
+    (hfin : ∀ i, ∀ᵐ ω ∂law (pc G), (cluster G ω (root i)).Finite)
+    (α ε : ℝ) (hα : 0 < α) (hα1 : α < 1/2) (hε : 0 < ε)
+    (hr : (1:ℝ)/2+ε-2*α < 0) :
+    ∃ C : ℝ, 0 < C ∧ ∀ (p : unitInterval), (pc G:ℝ)/2 ≤ (p:ℝ) → p ≤ pc G →
+      ∀ L : ℝ, 1 ≤ L → clusterPairFunctional G p root weight (contactSquareLargeReal G L) ≤
+        ENNReal.ofReal (C*L^((1:ℝ)/2+ε-2*α)) := by
+  obtain ⟨C,hC,hb⟩ := critical_contact_square_bound_nat G hloc hconn hqt hh root weight Γ hΓ htr
+    hcov hw hw1 hp₀ hp₁ hfin α ε hα hα1 hε hr
+  refine ⟨C,hC,?_⟩
+  intro p hp hpq L hL
+  have hLn : 1 ≤ ⌈L⌉₊ := by
+    have hc : (1:ℝ) ≤ (⌈L⌉₊:ℝ) := hL.trans (Nat.le_ceil L)
+    exact_mod_cast hc
+  have hEq : contactSquareLargeReal G L = contactSquareLarge G ⌈L⌉₊ := funext fun ω => funext fun x => funext fun y => contactSquareLargeReal_eq_ceil G L ω x y
+  rw [hEq]
+  exact (hb p hp hpq _ hLn).trans (ENNReal.ofReal_le_ofReal
+    (mul_le_mul_of_nonneg_left (Real.rpow_le_rpow_of_nonpos (by linarith : 0 < L) (Nat.le_ceil L) hr.le) hC.le))
+
+end ContactRealCutoff
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+open scoped ENNReal
+section NonnegativeOperatorDiagram
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def ennMatrixCoeff (A : Hilbert V →L[ℂ] Hilbert V) (x y : V) : ℝ≥0∞ :=
+  ENNReal.ofReal (matrixCoeff A x y).re
+
+theorem ennMatrixCoeff_mul {A B : Hilbert V →L[ℂ] Hilbert V}
+    (hA : PositiveRealMatrix A) (hB : PositiveRealMatrix B) (x y : V) :
+    ennMatrixCoeff (A*B) x y = ∑' z, ennMatrixCoeff A x z * ennMatrixCoeff B z y := by
+  unfold ennMatrixCoeff
+  rw [←(hA.mul_hasSum B x y).tsum_eq, ENNReal.ofReal_tsum_of_nonneg
+    (fun z => mul_nonneg (hA x z).2 (hB z y).2) (hA.mul_hasSum B x y).summable]
+  apply tsum_congr; intro z
+  exact ENNReal.ofReal_mul (hA x z).2
+
+theorem ennMatrixCoeff_symm {A : Hilbert V →L[ℂ] Hilbert V}
+    (hA : IsSelfAdjoint A) (x y : V) : ennMatrixCoeff A x y = ennMatrixCoeff A y x := by
+  unfold ennMatrixCoeff
+  rw [matrixCoeff_real_symm A hA x y]
+
+theorem HasKernel.ennMatrixCoeff [Countable E] (G : BondGraph V E) (p : unitInterval)
+    (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T) (x y : V) :
+    ennMatrixCoeff T x y = law p {ω | Conn G ω x y} := by
+  unfold Percolation.ennMatrixCoeff matrixCoeff
+  rw [hT x y]
+  simp only [Complex.ofReal_re,twoPoint,ENNReal.ofReal_toReal (measure_ne_top _ _)]
+
+theorem HasKernel.positiveRealMatrix (G : BondGraph V E) (p : unitInterval)
+    (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T) : PositiveRealMatrix T := by
+  intro x y
+  change (T (lp.single 2 y (1:ℂ)) x).im = 0 ∧ 0 ≤ (T (lp.single 2 y (1:ℂ)) x).re
+  rw [hT x y]
+  exact ⟨rfl,twoPoint_nonneg G p x y⟩
+
+theorem square_kernel_diagram (J T : Hilbert V →L[ℂ] Hilbert V)
+    (hJ : PositiveRealMatrix J) (hT : PositiveRealMatrix T)
+    (hsJ : IsSelfAdjoint J) (hsT : IsSelfAdjoint T) (x : V) :
+    (∑' v, ∑' a, ∑' b, ennMatrixCoeff J x v * (ennMatrixCoeff J a b *
+      (ennMatrixCoeff T x a * ennMatrixCoeff T v b))) = ennMatrixCoeff (J*T*J*T) x x := by
+  rw [mul_assoc,mul_assoc,ennMatrixCoeff_mul hJ (hT.mul (hJ.mul hT))]
+  apply tsum_congr; intro v
+  rw [ennMatrixCoeff_mul hT (hJ.mul hT)]
+  simp_rw [ennMatrixCoeff_mul hJ hT]
+  simp_rw [←ENNReal.tsum_mul_left]
+  rw [ENNReal.tsum_comm]
+  apply tsum_congr; intro b
+  apply tsum_congr; intro a
+  rw [ennMatrixCoeff_symm hsJ b a,ennMatrixCoeff_symm hsT a x]
+  ac_rfl
+
+theorem enn_rootTrace {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (hw : ∀ i, 0 ≤ weight i) (A : Hilbert V →L[ℂ] Hilbert V) (hA : PositiveRealMatrix A) :
+    (∑ i, ENNReal.ofReal (weight i) * ennMatrixCoeff A (root i) (root i)) =
+      ENNReal.ofReal (rootTrace root weight A).re := by
+  unfold rootTrace ennMatrixCoeff
+  simp only [Complex.re_sum,Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero]
+  rw [ENNReal.ofReal_sum_of_nonneg (fun i _ => mul_nonneg (hw i) (hA _ _).2)]
+  apply Finset.sum_congr rfl; intro i _
+  exact (ENNReal.ofReal_mul (hw i)).symm
+
+end NonnegativeOperatorDiagram
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+open scoped ENNReal ComplexOrder
+section KernelSquareTrace
+attribute [local instance] Classical.propDecidable Classical.decEq
+attribute [local instance] spectralRealOperatorScalarTower spectralRealOperatorSMulCommClass
+local instance : NonUnitalContinuousFunctionalCalculus ℝ
+    (Hilbert V →L[ℂ] Hilbert V) IsSelfAdjoint :=
+  IsSelfAdjoint.instNonUnitalContinuousFunctionalCalculus (A := Hilbert V →L[ℂ] Hilbert V)
+local instance : ContinuousFunctionalCalculus ℝ
+    (Hilbert V →L[ℂ] Hilbert V) IsSelfAdjoint :=
+  IsSelfAdjoint.instContinuousFunctionalCalculus (A := Hilbert V →L[ℂ] Hilbert V)
+
+theorem rootTrace_six_le_cube {ι : Type*} [Fintype ι]
+    (root : ι → V) (weight : ι → ℝ) {Γ : Set (V ≃ V)}
+    (htr : RootTransport root weight Γ) (hw : ∀ i, 0 ≤ weight i)
+    (T : Hilbert V →L[ℂ] Hilbert V) (hT : KernelInvariant Γ T) (hpos : 0 ≤ T) :
+    (rootTrace root weight (T^6)).re ≤ ‖T‖^3*(rootTrace root weight (T^3)).re := by
+  have hp3 : 0 ≤ T^3 := CStarAlgebra.pow_nonneg T 3 hpos
+  have h := rootTrace_positive_mul_le root weight htr hw (T^3) (T^3)
+    (hT.pow 3) (hT.pow 3) hp3 (IsSelfAdjoint.of_nonneg hp3)
+  have he : T^3*T^3 = T^6 := by rw [←pow_add]
+  rw [he] at h
+  have hr := (Complex.le_def.mp h).1
+  simp only [Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero] at hr
+  exact hr.trans (mul_le_mul_of_nonneg_right (norm_pow_le' T (by norm_num : 0 < (3:ℕ)))
+    (Complex.nonneg_iff.mp (rootTrace_nonneg root weight hw (T^3) hp3)).1)
+
+ 
+
+theorem clusterPairFunctional_twoPoint_square_bound [Countable V] [Countable E]
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hw : ∀ i, 0 ≤ weight i) (p : unitInterval)
+    (hfin : ∀ x, ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
+    (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T) :
+    clusterPairFunctional G p root weight (fun ω x y =>
+      (clusterPairKernel G ω (ennMatrixCoeff (T^2)) x y)^2) ≤
+      ENNReal.ofReal (‖T‖^3*(rootTrace root weight (T^3)).re) := by
+  have hp := hT.positiveRealMatrix G p T
+  have hpos : 0 ≤ T := ContinuousLinearMap.nonneg_iff_isPositive.mpr (hT.isPositive G p T)
+  have hsa := IsSelfAdjoint.of_nonneg hpos
+  have hI : KernelInvariant Γ T := fun a ha => hT.invariant G p T a (hΓ ha)
+  have hJ : ∀ a ∈ Γ, ∀ x y, ennMatrixCoeff (T^2) (a x) (a y) = ennMatrixCoeff (T^2) x y := by
+    intro a ha x y
+    unfold ennMatrixCoeff
+    rw [(hI.pow 2) a ha]
+  have hb := clusterPairFunctional_square_diagram G hconn hloc root weight Γ hΓ htr hw p hfin
+    (ennMatrixCoeff (T^2)) hJ
+  have hd (x : V) : (∑' v, ∑' a, ∑' b, ennMatrixCoeff (T^2) x v * (ennMatrixCoeff (T^2) a b *
+      (law p {ω | Conn G ω x a} * law p {ω | Conn G ω v b}))) = ennMatrixCoeff (T^6) x x := by
+    simp_rw [←hT.ennMatrixCoeff G p T]
+    rw [square_kernel_diagram (T^2) T (hp.pow 2) hp (hsa.pow 2) hsa]
+    congr 1
+  simp_rw [hd] at hb
+  rw [enn_rootTrace root weight hw (T^6) (hp.pow 6)] at hb
+  exact hb.trans (ENNReal.ofReal_le_ofReal (rootTrace_six_le_cube root weight htr hw T hI hpos))
+
+end KernelSquareTrace
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section FiniteClusterKernel
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def finiteClusterWeight (G : BondGraph V E) (f : V → ℝ)
+    (x : V) (ω : Finset E) : ℝ := ∑ y ∈ finiteRootComponent G ω x, f y
+
+theorem finiteClusterWeight_expand (G : BondGraph V E) (f : V → ℝ)
+    (Q ω : Finset E) (hω : ω ⊆ Q) (x : V) :
+    finiteClusterWeight G f x ω = ∑ y ∈ insert x (searchVertices G Q),
+      f y * (if Conn G (ω:Set E) x y then (1:ℝ) else 0) := by
+  unfold finiteClusterWeight
+  calc
+    _ = ∑ y ∈ finiteRootComponent G ω x, f y * (if Conn G (ω:Set E) x y then (1:ℝ) else 0) := by
+      apply Finset.sum_congr rfl
+      intro y hy
+      have hc : Conn G (ω:Set E) x y := by
+        change y ∈ (finiteRootComponent G ω x:Set V) at hy
+        rwa [coe_finiteRootComponent] at hy
+      rw [ite_eq_left hc,mul_one]
+    _ = _ := Finset.sum_subset (finiteRootComponent_subset G Q _ x ω hω (by rfl)) (by
+      intro y _ hy
+      have hn : ¬Conn G (ω:Set E) x y := by
+        intro h
+        apply hy
+        change y ∈ (finiteRootComponent G ω x:Set V)
+        rwa [coe_finiteRootComponent]
+      rw [ite_eq_right hn,mul_zero])
+
+theorem finiteClusterWeight_expectation_le [Countable E]
+    (G : BondGraph V E) (Q : Finset E) (f : V → ℝ) (hf : ∀ y, 0 ≤ f y)
+    (x : V) (p : unitInterval) (hs : Summable (fun y => f y*twoPoint G p x y)) :
+    bernoulliPolynomial Q (finiteClusterWeight G f x) p ≤ ∑' y, f y*twoPoint G p x y := by
+  have he : bernoulliPolynomial Q (finiteClusterWeight G f x) p =
+      bernoulliPolynomial Q (fun ω => ∑ y ∈ insert x (searchVertices G Q), f y*(if Conn G (ω:Set E) x y then (1:ℝ) else 0)) p := by
+    apply bernoulliPolynomial_congr
+    intro ω hω
+    exact finiteClusterWeight_expand G f Q ω hω x
+  rw [he,bernoulliPolynomial_sum]
+  calc
+    _ ≤ ∑ y ∈ insert x (searchVertices G Q), f y*twoPoint G p x y := by
+      apply Finset.sum_le_sum
+      intro y _
+      have hpQ : bernoulliPolynomial Q (fun ω => if Conn G (ω:Set E) x y then (1:ℝ) else 0) p =
+          (law p).real {ω | Conn G (ω ∩ (Q:Set E)) x y} :=
+        finiteEventPolynomial_restrict Q {ω | Conn G ω x y} p
+      rw [bernoulliPolynomial_const_mul,hpQ]
+      apply mul_le_mul_of_nonneg_left _ (hf y)
+      exact measureReal_mono (fun ω h => conn_mono G Set.inter_subset_left h)
+    _ ≤ _ := hs.sum_le_tsum _ (fun y _ => mul_nonneg (hf y) (twoPoint_nonneg G p x y))
+
+noncomputable def finitePairKernel (G : BondGraph V E) (J : V → V → ℝ)
+    (ω : Finset E) (x y : V) : ℝ :=
+  ∑ a ∈ finiteRootComponent G ω x, finiteClusterWeight G (J a) y ω
+
+theorem finitePairKernel_nonneg (G : BondGraph V E) (J : V → V → ℝ)
+    (hJ : ∀ a b, 0 ≤ J a b) (ω : Finset E) (x y : V) : 0 ≤ finitePairKernel G J ω x y :=
+  Finset.sum_nonneg fun a _ => Finset.sum_nonneg fun b _ => hJ a b
+
+theorem finitePairKernel_exterior (G : BondGraph V E) (J : V → V → ℝ)
+    (ω : Finset E) (x y : V) (hd : ¬Conn G (ω:Set E) x y) :
+    finitePairKernel G J ω x y = ∑ a ∈ finiteRootComponent G ω x,
+      finiteClusterWeight G (J a) y (exteriorBonds G ω (finiteRootComponent G ω x:Set V)) := by
+  unfold finitePairKernel finiteClusterWeight
+  rw [finiteRootComponent_exterior G ω x y hd]
+
+end FiniteClusterKernel
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section SmallFinitePairKernel
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def finiteSmallPairKernel (G : BondGraph V E) (J : V → V → ℝ)
+    (L : ℝ) (x y : V) (ω : Finset E) : ℝ :=
+  if (finiteRootComponent G ω x).card < L ∧ ¬Conn G (ω:Set E) x y then finitePairKernel G J ω x y else 0
+
+noncomputable def finiteSmallClusterSize (G : BondGraph V E) (L : ℝ) (x : V) (ω : Finset E) : ℝ :=
+  if (finiteRootComponent G ω x).card < L then (finiteRootComponent G ω x).card else 0
+
+ 
+theorem finiteSmallPairKernel_bound [Countable E]
+    (G : BondGraph V E) (Q : Finset E) (J : V → V → ℝ) (hJ : ∀ a b, 0 ≤ J a b)
+    (x y : V) (p : unitInterval) (L M : ℝ) (hM : 0 ≤ M)
+    (hs : ∀ a, Summable (fun b => J a b*twoPoint G p y b))
+    (hb : ∀ a, (∑' b, J a b*twoPoint G p y b) ≤ M) :
+    bernoulliPolynomial Q (finiteSmallPairKernel G J L x y) p ≤
+      M*bernoulliPolynomial Q (finiteSmallClusterSize G L x) p := by
+  let f (W : Finset V) (η : Finset E) : ℝ :=
+    (if (W.card:ℝ) < L ∧ y ∉ W then ∑ a ∈ W, finiteClusterWeight G (J a) y η else 0) -
+      M*(if (W.card:ℝ) < L then (W.card:ℝ) else 0)
+  have hf (W : Finset V) : bernoulliPolynomial (exteriorBonds G Q (W:Set V)) (f W) p ≤ 0 := by
+    by_cases hW : (W.card:ℝ) < L
+    · by_cases hy : y ∈ W
+      · simp only [f,hW,hy,not_true_eq_false,and_false,ite_false,ite_true,zero_sub,
+          bernoulliPolynomial_const]
+        exact neg_nonpos.mpr (mul_nonneg hM (Nat.cast_nonneg _))
+      · simp only [f,hW,hy,not_false_eq_true,and_self,ite_true,bernoulliPolynomial_sub,
+          bernoulliPolynomial_const,bernoulliPolynomial_sum]
+        apply sub_nonpos.mpr
+        calc
+          _ ≤ ∑ _a ∈ W, M := Finset.sum_le_sum fun a _ =>
+            (finiteClusterWeight_expectation_le G _ (J a) (hJ a) y p (hs a)).trans (hb a)
+          _ = _ := by simp only [Finset.sum_const,nsmul_eq_mul]; ring
+    · simp only [f,hW,false_and,ite_false,mul_zero,sub_zero,bernoulliPolynomial_zero]
+      exact le_rfl
+  have ht := cluster_exterior_expectation_bound G Q x p f 0 hf
+  have he (ω : Finset E) : f (finiteRootComponent G ω x)
+      (exteriorBonds G ω (finiteRootComponent G ω x:Set V)) =
+      finiteSmallPairKernel G J L x y ω - M*finiteSmallClusterSize G L x ω := by
+    have hm : y ∈ finiteRootComponent G ω x ↔ Conn G (ω:Set E) x y := by
+      change y ∈ (finiteRootComponent G ω x:Set V) ↔ _
+      rw [coe_finiteRootComponent]
+      rfl
+    unfold f finiteSmallPairKernel finiteSmallClusterSize
+    simp only [hm]
+    by_cases hc : (finiteRootComponent G ω x).card < L ∧ ¬Conn G (ω:Set E) x y
+    · rw [ite_eq_left hc,ite_eq_left hc,finitePairKernel_exterior G J ω x y hc.2]
+    · rw [ite_eq_right hc,ite_eq_right hc]
+  simp_rw [he] at ht
+  rw [bernoulliPolynomial_sub,bernoulliPolynomial_const_mul] at ht
+  exact sub_nonpos.mp ht
+
+end SmallFinitePairKernel
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section SmallSizeFractionalMoment
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem small_count_le_power (s L α : ℝ) (hs : 0 ≤ s) (hL : 0 ≤ L) (hα : α ≤ 1) :
+    (if s < L then s else 0) ≤ L^(1-α)*s^α := by
+  by_cases h : s < L
+  · rw [ite_eq_left h]
+    rcases eq_or_lt_of_le hs with rfl | hs'
+    · positivity
+    · calc
+        s = s^(1-α)*s^α := by rw [←Real.rpow_add hs']; simp only [sub_add_cancel,Real.rpow_one]
+        _ ≤ _ := mul_le_mul_of_nonneg_right (Real.rpow_le_rpow hs h.le (by linarith)) (Real.rpow_nonneg hs α)
+  · rw [ite_eq_right h]
+    exact mul_nonneg (Real.rpow_nonneg hL _) (Real.rpow_nonneg hs _)
+
+theorem finiteSmallClusterSize_moment_bound [Countable E]
+    (G : BondGraph V E) (Q : Finset E) (x : V) (p : unitInterval)
+    (L α : ℝ) (hL : 0 ≤ L) (hα0 : 0 ≤ α) (hα1 : α ≤ 1)
+    (hfin : ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
+    (hI : Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^α) (law p)) :
+    bernoulliPolynomial Q (finiteSmallClusterSize G L x) p ≤
+      L^(1-α)*(∫ ω, ((cluster G ω x).ncard:ℝ)^α ∂law p) := by
+  let f (ω : Set E) := finiteSmallClusterSize G L x (domainBits Q ω)
+  have hf : Integrable f (law p) := integrable_finite_local Q p f (fun ω => by
+    dsimp [f]; rw [domainBits_local])
+  rw [←integral_domainBits Q p (finiteSmallClusterSize G L x),←integral_const_mul]
+  apply integral_mono_ae hf (hI.const_mul _)
+  filter_upwards [hfin] with ω hω
+  have hc : (finiteRootComponent G (domainBits Q ω) x).card ≤ (cluster G ω x).ncard := by
+    rw [←Set.ncard_coe_finset,coe_finiteRootComponent,coe_domainBits]
+    exact Set.ncard_le_ncard (fun y hy => conn_mono G Set.inter_subset_left hy) hω
+  have hc' : ((finiteRootComponent G (domainBits Q ω) x).card:ℝ) ≤ (cluster G ω x).ncard := by exact_mod_cast hc
+  exact (small_count_le_power _ L α (Nat.cast_nonneg _) hL hα1).trans
+    (mul_le_mul_of_nonneg_left (Real.rpow_le_rpow (Nat.cast_nonneg _) hc' hα0) (Real.rpow_nonneg hL _))
+
+theorem finiteSmallPairKernel_moment_bound [Countable E]
+    (G : BondGraph V E) (Q : Finset E) (J : V → V → ℝ) (hJ : ∀ a b, 0 ≤ J a b)
+    (x y : V) (p : unitInterval) (L M α : ℝ) (hL : 0 ≤ L) (hM : 0 ≤ M)
+    (hα0 : 0 ≤ α) (hα1 : α ≤ 1)
+    (hfin : ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
+    (hI : Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^α) (law p))
+    (hs : ∀ a, Summable (fun b => J a b*twoPoint G p y b))
+    (hb : ∀ a, (∑' b, J a b*twoPoint G p y b) ≤ M) :
+    bernoulliPolynomial Q (finiteSmallPairKernel G J L x y) p ≤
+      M*(L^(1-α)*(∫ ω, ((cluster G ω x).ncard:ℝ)^α ∂law p)) :=
+  (finiteSmallPairKernel_bound G Q J hJ x y p L M hM hs hb).trans
+    (mul_le_mul_of_nonneg_left (finiteSmallClusterSize_moment_bound G Q x p L α hL hα0 hα1 hfin hI) hM)
+
+end SmallSizeFractionalMoment
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+open scoped ENNReal
+section SmallKernelLimit
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem clusterPairKernel_finite (G : BondGraph V E) (J : V → V → ℝ≥0∞)
+    (ω : Finset E) (x y : V) :
+    clusterPairKernel G ω J x y =
+      ∑ a ∈ finiteRootComponent G ω x, ∑ b ∈ finiteRootComponent G ω y, J a b := by
+  have hm (x a : V) : a ∈ finiteRootComponent G ω x ↔ Conn G (ω:Set E) x a := by
+    change a ∈ (finiteRootComponent G ω x:Set V) ↔ _
+    rw [coe_finiteRootComponent]
+    rfl
+  rw [clusterPairKernel,tsum_eq_sum (s := finiteRootComponent G ω x) (fun a ha => by
+    simp only [←hm x a,ha,false_and,ite_false,tsum_zero])]
+  apply Finset.sum_congr rfl
+  intro a ha
+  rw [tsum_eq_sum (s := finiteRootComponent G ω y) (fun b hb => by
+    simp only [←hm y b,hb,and_false,ite_false])]
+  apply Finset.sum_congr rfl
+  intro b hb
+  rw [ite_eq_left ⟨(hm x a).mp ha,(hm y b).mp hb⟩]
+
+theorem ofReal_finitePairKernel (G : BondGraph V E) (J : V → V → ℝ)
+    (hJ : ∀ a b, 0 ≤ J a b) (ω : Finset E) (x y : V) :
+    ENNReal.ofReal (finitePairKernel G J ω x y) =
+      clusterPairKernel G ω (fun a b => ENNReal.ofReal (J a b)) x y := by
+  rw [clusterPairKernel_finite]
+  unfold finitePairKernel finiteClusterWeight
+  rw [ENNReal.ofReal_sum_of_nonneg (fun a _ => Finset.sum_nonneg fun b _ => hJ a b)]
+  apply Finset.sum_congr rfl
+  intro a _
+  exact ENNReal.ofReal_sum_of_nonneg (fun b _ => hJ a b)
+
+noncomputable def smallPairKernel (G : BondGraph V E) (J : V → V → ℝ≥0∞)
+    (L : ℝ) (x y : V) (ω : Set E) : ℝ≥0∞ :=
+  if ((cluster G ω x).ncard:ℝ) < L ∧ ¬Conn G ω x y then clusterPairKernel G ω J x y else 0
+
+theorem smallPairKernel_eq_finite (G : BondGraph V E) (hloc : LocallyFinite G)
+    (J : V → V → ℝ) (hJ : ∀ a b, 0 ≤ J a b) (L : ℝ)
+    (ω : Set E) (x y : V) (hx : (cluster G ω x).Finite) (hy : (cluster G ω y).Finite)
+    (Q : Finset E) (hQx : incidentSupport G hloc hx.toFinset ⊆ Q)
+    (hQy : incidentSupport G hloc hy.toFinset ⊆ Q) :
+    ENNReal.ofReal (finiteSmallPairKernel G J L x y (domainBits Q ω)) =
+      smallPairKernel G (fun a b => ENNReal.ofReal (J a b)) L x y ω := by
+  have hcluster (z : V) (hz : (cluster G ω z).Finite) (hQ : incidentSupport G hloc hz.toFinset ⊆ Q) :
+      cluster G (domainBits Q ω:Set E) z = cluster G ω z := by
+    rw [coe_domainBits]
+    have hh := (cluster_restrict_incident_eq_iff G hloc hz.toFinset Q hQ ω z).mpr
+      (Set.Finite.coe_toFinset hz).symm
+    simpa only [Set.Finite.coe_toFinset] using hh
+  have hxC := hcluster x hx hQx
+  have hyC := hcluster y hy hQy
+  have hxN : (finiteRootComponent G (domainBits Q ω) x).card = (cluster G ω x).ncard := by
+    rw [←Set.ncard_coe_finset,coe_finiteRootComponent,hxC]
+  have hxconn (a : V) : Conn G (domainBits Q ω:Set E) x a ↔ Conn G ω x a := by
+    change a ∈ cluster G (domainBits Q ω:Set E) x ↔ a ∈ cluster G ω x
+    rw [hxC]
+  have hyconn (b : V) : Conn G (domainBits Q ω:Set E) y b ↔ Conn G ω y b := by
+    change b ∈ cluster G (domainBits Q ω:Set E) y ↔ b ∈ cluster G ω y
+    rw [hyC]
+  unfold finiteSmallPairKernel smallPairKernel
+  rw [hxN]
+  simp only [hxconn]
+  split_ifs
+  · rw [ofReal_finitePairKernel G J hJ]
+    simp only [clusterPairKernel,hxconn,hyconn]
+  · exact ENNReal.ofReal_zero
+
+end SmallKernelLimit
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+open scoped ENNReal
+section SmallKernelFatou
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem smallPairKernel_lintegral_le_of_finite [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G)
+    (J : V → V → ℝ) (hJ : ∀ a b, 0 ≤ J a b) (L : ℝ) (x y : V) (p : unitInterval)
+    (hx : ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
+    (hy : ∀ᵐ ω ∂law p, (cluster G ω y).Finite) (B : ℝ)
+    (hB : ∀ Q : Finset E, bernoulliPolynomial Q (finiteSmallPairKernel G J L x y) p ≤ B) :
+    (∫⁻ ω, smallPairKernel G (fun a b => ENNReal.ofReal (J a b)) L x y ω ∂law p) ≤ ENNReal.ofReal B := by
+  let : Encodable E := Encodable.ofCountable E
+  let Q := vertexExhaustion (V := E)
+  let f := fun j ω => finiteSmallPairKernel G J L x y (domainBits (Q j) ω)
+  have hfi (j : ℕ) : Integrable (f j) (law p) := integrable_finite_local (Q j) p (f j) (fun ω => by
+    dsimp [f]; rw [domainBits_local])
+  have hfn (j : ℕ) : 0 ≤ᵐ[law p] f j := Eventually.of_forall (fun ω => by
+    dsimp [f,finiteSmallPairKernel]
+    split_ifs
+    · exact finitePairKernel_nonneg G J hJ _ x y
+    · exact le_rfl)
+  have heq : ∀ᵐ ω ∂law p,
+      liminf (fun j => ENNReal.ofReal (f j ω)) atTop =
+        smallPairKernel G (fun a b => ENNReal.ofReal (J a b)) L x y ω := by
+    filter_upwards [hx,hy] with ω hωx hωy
+    obtain ⟨jx,hjx⟩ := subset_vertexExhaustion (incidentSupport G hloc hωx.toFinset)
+    obtain ⟨jy,hjy⟩ := subset_vertexExhaustion (incidentSupport G hloc hωy.toFinset)
+    have he : (fun j => ENNReal.ofReal (f j ω)) =ᶠ[atTop]
+        (fun _ => smallPairKernel G (fun a b => ENNReal.ofReal (J a b)) L x y ω) := by
+      filter_upwards [eventually_ge_atTop (max jx jy)] with j hj
+      exact smallPairKernel_eq_finite G hloc J hJ L ω x y hωx hωy (Q j)
+        (hjx.trans (vertexExhaustion_mono (le_max_left jx jy |>.trans hj)))
+        (hjy.trans (vertexExhaustion_mono (le_max_right jx jy |>.trans hj)))
+    exact (tendsto_const_nhds.congr' he.symm).liminf_eq
+  calc
+    _ = ∫⁻ ω, liminf (fun j => ENNReal.ofReal (f j ω)) atTop ∂law p := lintegral_congr_ae (heq.mono (fun _ h => h.symm))
+    _ ≤ liminf (fun j => ∫⁻ ω, ENNReal.ofReal (f j ω) ∂law p) atTop :=
+      lintegral_liminf_le' (fun j => (hfi j).aemeasurable.ennreal_ofReal)
+    _ ≤ ENNReal.ofReal B := liminf_le_of_frequently_le' (Eventually.frequently (Eventually.of_forall (fun j => by
+      rw [←ofReal_integral_eq_lintegral_ofReal (hfi j) (hfn j)]
+      apply ENNReal.ofReal_le_ofReal
+      rw [show (∫ ω, f j ω ∂law p) = bernoulliPolynomial (Q j) (finiteSmallPairKernel G J L x y) p from
+        integral_domainBits (Q j) p (finiteSmallPairKernel G J L x y)]
+      exact hB (Q j))))
+
+ 
+
+theorem smallPairKernel_moment_bound [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G)
+    (J : V → V → ℝ) (hJ : ∀ a b, 0 ≤ J a b)
+    (x y : V) (p : unitInterval) (L M α : ℝ) (hL : 0 ≤ L) (hM : 0 ≤ M)
+    (hα0 : 0 ≤ α) (hα1 : α ≤ 1)
+    (hx : ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
+    (hy : ∀ᵐ ω ∂law p, (cluster G ω y).Finite)
+    (hI : Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^α) (law p))
+    (hs : ∀ a, Summable (fun b => J a b*twoPoint G p y b))
+    (hb : ∀ a, (∑' b, J a b*twoPoint G p y b) ≤ M) :
+    (∫⁻ ω, smallPairKernel G (fun a b => ENNReal.ofReal (J a b)) L x y ω ∂law p) ≤
+      ENNReal.ofReal (M*(L^(1-α)*(∫ ω, ((cluster G ω x).ncard:ℝ)^α ∂law p))) :=
+  smallPairKernel_lintegral_le_of_finite G hloc J hJ L x y p hx hy _ (fun Q =>
+    finiteSmallPairKernel_moment_bound G Q J hJ x y p L M α hL hM hα0 hα1 hx hI hs hb)
+
+end SmallKernelFatou
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+open scoped ENNReal ComplexOrder
+section SmallTwoPointKernel
+attribute [local instance] Classical.propDecidable Classical.decEq
+attribute [local instance] spectralRealOperatorScalarTower spectralRealOperatorSMulCommClass
+local instance : NonUnitalContinuousFunctionalCalculus ℝ
+    (Hilbert V →L[ℂ] Hilbert V) IsSelfAdjoint :=
+  IsSelfAdjoint.instNonUnitalContinuousFunctionalCalculus (A := Hilbert V →L[ℂ] Hilbert V)
+local instance : ContinuousFunctionalCalculus ℝ
+    (Hilbert V →L[ℂ] Hilbert V) IsSelfAdjoint :=
+  IsSelfAdjoint.instContinuousFunctionalCalculus (A := Hilbert V →L[ℂ] Hilbert V)
+
+theorem matrixCoeff_positive_trace_norm {ι : Type*} [Fintype ι]
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
+    (hw : ∀ i, 0 ≤ weight i) (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (A : Hilbert V →L[ℂ] Hilbert V) (hA : KernelInvariant Γ A) (hpos : 0 ≤ A) (x y : V) :
+    ‖matrixCoeff A x y‖ ≤ c*(rootTrace root weight A).re := by
+  have hd (z : V) : (matrixCoeff A z z).re ≤ c*(rootTrace root weight A).re := by
+    have h := (Complex.le_def.mp (rootTrace_diag_le root weight hcov hw c hc hmin A hA hpos z)).1
+    simpa only [Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero] using h
+  have hn (z : V) : 0 ≤ (matrixCoeff A z z).re := (Complex.nonneg_iff.mp (matrixCoeff_nonneg A hpos z)).1
+  have hC : 0 ≤ c*(rootTrace root weight A).re := mul_nonneg hc
+    (Complex.nonneg_iff.mp (rootTrace_nonneg root weight hw A hpos)).1
+  have hs := matrixCoeff_positive_product_norm_sq_le A 1 hpos x y
+  simp only [one_mul,star_one,mul_one] at hs
+  have hp := mul_le_mul (hd x) (hd y) (hn y) hC
+  nlinarith [norm_nonneg (matrixCoeff A x y)]
+
+theorem smallPairKernel_twoPoint_bound [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G)
+    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
+    (hw : ∀ i, 0 ≤ weight i) (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (x y : V) (p : unitInterval) (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T)
+    (L α : ℝ) (hL : 0 ≤ L) (hα0 : 0 ≤ α) (hα1 : α ≤ 1)
+    (hx : ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
+    (hy : ∀ᵐ ω ∂law p, (cluster G ω y).Finite)
+    (hI : Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^α) (law p)) :
+    (∫⁻ ω, smallPairKernel G (ennMatrixCoeff (T^2)) L x y ω ∂law p) ≤
+      ENNReal.ofReal ((c*(rootTrace root weight (T^3)).re)*
+        (L^(1-α)*(∫ ω, ((cluster G ω x).ncard:ℝ)^α ∂law p))) := by
+  have hp := hT.positiveRealMatrix G p T
+  have hpos : 0 ≤ T := ContinuousLinearMap.nonneg_iff_isPositive.mpr (hT.isPositive G p T)
+  have hp3 := CStarAlgebra.pow_nonneg T 3 hpos
+  have hTI : KernelInvariant Γ T := fun a ha => hT.invariant G p T a (hΓ ha)
+  have hM : 0 ≤ c*(rootTrace root weight (T^3)).re := mul_nonneg hc
+    (Complex.nonneg_iff.mp (rootTrace_nonneg root weight hw _ hp3)).1
+  have he (b : V) : (matrixCoeff T b y).re = twoPoint G p y b := by
+    change (T (lp.single 2 y (1:ℂ)) b).re = _
+    rw [hT b y,Complex.ofReal_re,twoPoint_symm G p b y]
+  have hs (a : V) : HasSum (fun b => (matrixCoeff (T^2) a b).re * twoPoint G p y b)
+      (matrixCoeff (T^3) a y).re := by
+    simpa only [he,←pow_succ] using (hp.pow 2).mul_hasSum T a y
+  apply smallPairKernel_moment_bound G hloc (fun a b => (matrixCoeff (T^2) a b).re)
+    (fun a b => ((hp.pow 2) a b).2) x y p L _ α hL hM hα0 hα1 hx hy hI
+    (fun a => (hs a).summable)
+  intro a
+  rw [(hs a).tsum_eq]
+  exact (Complex.re_le_norm _).trans (matrixCoeff_positive_trace_norm root weight Γ hcov hw c hc hmin _ (hTI.pow 3) hp3 a y)
+
+end SmallTwoPointKernel
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+open scoped ENNReal
+section ContactSmallReroot
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem measurable_smallPairKernel [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (J : V → V → ℝ≥0∞) (L : ℝ) (x y : V) :
+    Measurable (smallPairKernel G J L x y) := by
+  exact (measurable_clusterPairKernel G J x y).ite
+    ((measurableSet_lt ((measurable_of_countable (fun n : ℕ => (n:ℝ))).comp (measurable_cluster_ncard G hloc x))
+      measurable_const).inter (measurableSet_conn G x y).compl) measurable_const
+
+theorem smallPairKernel_cluster_constant (G : BondGraph V E) (J : V → V → ℝ≥0∞)
+    (L : ℝ) (ω : Set E) (x y a b : V) (hxa : Conn G ω x a) (hyb : Conn G ω y b) :
+    smallPairKernel G J L x y ω = smallPairKernel G J L a b ω := by
+  have hc : Conn G ω x y ↔ Conn G ω a b := by
+    constructor
+    · intro h; exact conn_trans G ω (conn_symm G ω hxa) (conn_trans G ω h hyb)
+    · intro h; exact conn_trans G ω hxa (conn_trans G ω h (conn_symm G ω hyb))
+  simp only [smallPairKernel,cluster_eq_of_conn G ω hxa,hc,
+    clusterPairKernel_cluster_constant G ω J x y a b hxa hyb]
+
+theorem smallPairKernel_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (J : V → V → ℝ≥0∞) (hJ : ∀ x y, J (a x) (a y) = J x y)
+    (L : ℝ) (ω : Set E) (x y : V) :
+    smallPairKernel G J L x y (b ⁻¹' ω) = smallPairKernel G J L (a x) (a y) ω := by
+  simp only [smallPairKernel,cluster_ncard_covariant G hab,conn_aut_iff G hab,
+    clusterPairKernel_aut G hab J hJ]
+
+theorem smallPairKernel_distinct (G : BondGraph V E) (J : V → V → ℝ≥0∞)
+    (L : ℝ) (ω : Set E) (x y : V) :
+    (if ¬Conn G ω x y then smallPairKernel G J L x y ω else 0) = smallPairKernel G J L x y ω := by
+  by_cases h : Conn G ω x y
+  · simp only [smallPairKernel,h,not_true_eq_false,and_false,ite_false]
+  · exact ite_eq_left h
+
+ 
+theorem contact_smallPairKernel_bound [Countable V] [Countable E]
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1) (p : unitInterval)
+    (hfin : ∀ x, ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
+    (J : V → V → ℝ≥0∞) (hJ : ∀ a ∈ Γ, ∀ x y, J (a x) (a y) = J x y)
+    (L : ℝ) (d : ℕ) (hd : ∀ x, degree G x ≤ d) (B : ℝ)
+    (hb : ∀ x y, (∫⁻ ω, smallPairKernel G J L x y ω ∂law p) ≤ ENNReal.ofReal B) :
+    clusterPairFunctional G p root weight (fun ω x y =>
+      clusterPairKernel G ω (indexedEdgeKernel G) x y * smallPairKernel G J L x y ω) ≤
+      ENNReal.ofReal ((d:ℝ)*B) := by
+  have hj : ∀ a ∈ Γ, ∀ x y, indexedEdgeKernel G (a x) (a y) = indexedEdgeKernel G x y := by
+    intro a ha x y
+    obtain ⟨b,hab⟩ := hΓ ha
+    exact indexedEdgeKernel_aut G hab x y
+  rw [clusterPairFunctional_reroot G hconn hloc root weight Γ hΓ htr hw p hfin
+    (indexedEdgeKernel G) hj (fun ω x y => smallPairKernel G J L x y ω)
+    (measurable_smallPairKernel G hloc J L)
+    (fun ω x y a b hxa hyb => smallPairKernel_cluster_constant G J L ω x y a b hxa hyb)
+    (fun a ha b hab ω x y => smallPairKernel_aut G hab J (hJ a ha) L ω x y)]
+  simp_rw [smallPairKernel_distinct]
+  have hr (x : V) : (∫⁻ ω, ∑' y, indexedEdgeKernel G x y * smallPairKernel G J L x y ω ∂law p) ≤
+      ENNReal.ofReal ((d:ℝ)*B) := by
+    have hm (y : V) : Measurable (fun ω => indexedEdgeKernel G x y * smallPairKernel G J L x y ω) :=
+      (measurable_const : Measurable (fun _ : Set E => indexedEdgeKernel G x y)).mul (measurable_smallPairKernel G hloc J L x y)
+    rw [lintegral_tsum (fun y => (hm y).aemeasurable)]
+    simp_rw [lintegral_const_mul _ (measurable_smallPairKernel G hloc J L x _)]
+    calc
+      _ ≤ ∑' y, indexedEdgeKernel G x y * ENNReal.ofReal B := ENNReal.tsum_le_tsum (fun y => mul_le_mul_right (hb x y) _)
+      _ = (degree G x:ℝ≥0∞)*ENNReal.ofReal B := by rw [ENNReal.tsum_mul_right,indexedEdgeKernel_row G hloc x]
+      _ ≤ (d:ℝ≥0∞)*ENNReal.ofReal B := mul_le_mul_left (by exact_mod_cast hd x) _
+      _ = _ := by rw [ENNReal.ofReal_mul (by positivity),ENNReal.ofReal_natCast]
+  calc
+    _ ≤ ∑ i, ENNReal.ofReal (weight i)*ENNReal.ofReal ((d:ℝ)*B) :=
+      Finset.sum_le_sum (fun i _ => mul_le_mul_right (hr (root i)) _)
+    _ = _ := by rw [←Finset.sum_mul,←ENNReal.ofReal_sum_of_nonneg (fun i _ => hw i),hw1,ENNReal.ofReal_one,one_mul]
+
+end ContactSmallReroot
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+open scoped ENNReal
+section PairMeasure
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def clusterPairMeasure [MeasurableSpace V]
+    {ι : Type*} [MeasurableSpace ι] (G : BondGraph V E) (p : unitInterval)
+    (root : ι → V) (weight : ι → ℝ) : Measure (Set E × (ι × V)) :=
+  ((law p).prod Measure.count).withDensity (fun t => ENNReal.ofReal (weight t.2.1) *
+    clusterPairMass G t.1 (fun _ _ => 1) (root t.2.1) t.2.2)
+
+theorem clusterPairFunctional_eq_lintegral [Countable V] [Countable E]
+    [MeasurableSpace V] [MeasurableSingletonClass V]
+    {ι : Type*} [Fintype ι] [MeasurableSpace ι] [MeasurableSingletonClass ι]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (p : unitInterval)
+    (root : ι → V) (weight : ι → ℝ)
+    (F : Set E → V → V → ℝ≥0∞) (hF : ∀ x y, Measurable (fun ω => F ω x y)) :
+    clusterPairFunctional G p root weight F =
+      ∫⁻ t, F t.1 (root t.2.1) t.2.2 ∂clusterPairMeasure G p root weight := by
+  let d : Set E × (ι × V) → ℝ≥0∞ := fun t => ENNReal.ofReal (weight t.2.1) *
+    clusterPairMass G t.1 (fun _ _ => 1) (root t.2.1) t.2.2
+  let f : Set E × (ι × V) → ℝ≥0∞ := fun t => F t.1 (root t.2.1) t.2.2
+  have hd : Measurable d := by
+    apply measurable_from_prod_countable_left
+    intro q
+    exact (measurable_const : Measurable (fun _ : Set E => ENNReal.ofReal (weight q.1))).mul
+      (measurable_clusterPairMass G hloc (fun _ _ _ => 1) (fun _ _ => measurable_const) (root q.1) q.2)
+  have hf : Measurable f := measurable_from_prod_countable_left (fun q => hF (root q.1) q.2)
+  change _ = ∫⁻ t, f t ∂((law p).prod Measure.count).withDensity d
+  rw [lintegral_withDensity_eq_lintegral_mul _ hd hf,lintegral_prod _ ((hd.mul hf).aemeasurable)]
+  have he (ω : Set E) (i : ι) (y : V) :
+      (d*f) (ω,(i,y)) = ENNReal.ofReal (weight i)*clusterPairMass G ω (F ω) (root i) y := by
+    dsimp [d,f,clusterPairMass]
+    (split_ifs <;> simp only [mul_one,mul_zero,zero_mul]); ring
+  simp_rw [lintegral_count,ENNReal.tsum_prod',he,tsum_fintype]
+  have hm (i : ι) : Measurable (fun ω => ∑' y, ENNReal.ofReal (weight i)*clusterPairMass G ω (F ω) (root i) y) :=
+    Measurable.tsum (fun y => measurable_const.mul (measurable_clusterPairMass G hloc F hF (root i) y))
+  rw [lintegral_finsetSum _ (fun i _ => hm i)]
+  simp_rw [ENNReal.tsum_mul_left]
+  rw [clusterPairFunctional]
+  apply Finset.sum_congr rfl
+  intro i _
+  rw [lintegral_const_mul _ (Measurable.tsum (fun y => measurable_clusterPairMass G hloc F hF (root i) y))]
+  congr 1
+  apply lintegral_congr
+  intro ω
+  exact clusterPairDensity_eq_tsum G ω (F ω) (root i)
+
+ 
+
+theorem clusterPairFunctional_cauchy_schwarz [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (p : unitInterval)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (F H : Set E → V → V → ℝ≥0∞)
+    (hF : ∀ x y, Measurable (fun ω => F ω x y))
+    (hH : ∀ x y, Measurable (fun ω => H ω x y)) :
+    clusterPairFunctional G p root weight (fun ω x y => F ω x y*H ω x y) ≤
+      (clusterPairFunctional G p root weight (fun ω x y => (F ω x y)^2))^((1:ℝ)/2) *
+      (clusterPairFunctional G p root weight (fun ω x y => (H ω x y)^2))^((1:ℝ)/2) := by
+  let : MeasurableSpace V := ⊤
+  let : MeasurableSingletonClass V := ⟨fun _ => trivial⟩
+  let : MeasurableSpace ι := ⊤
+  let : MeasurableSingletonClass ι := ⟨fun _ => trivial⟩
+  rw [clusterPairFunctional_eq_lintegral G hloc p root weight (fun ω x y => F ω x y*H ω x y) (fun x y => (hF x y).mul (hH x y)),
+    clusterPairFunctional_eq_lintegral G hloc p root weight (fun ω x y => (F ω x y)^2) (fun x y => (hF x y).pow_const 2),
+    clusterPairFunctional_eq_lintegral G hloc p root weight (fun ω x y => (H ω x y)^2) (fun x y => (hH x y).pow_const 2)]
+  have hf : Measurable (fun t : Set E × (ι × V) => F t.1 (root t.2.1) t.2.2) :=
+    measurable_from_prod_countable_left (fun q => hF (root q.1) q.2)
+  have hh : Measurable (fun t : Set E × (ι × V) => H t.1 (root t.2.1) t.2.2) :=
+    measurable_from_prod_countable_left (fun q => hH (root q.1) q.2)
+  simpa only [Pi.mul_apply,ENNReal.rpow_two] using
+    ENNReal.lintegral_mul_le_Lp_mul_Lq (clusterPairMeasure G p root weight) Real.HolderConjugate.two_two hf.aemeasurable hh.aemeasurable
+
+end PairMeasure
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Kernel/Covariance.lean b/lean/OAI/Probability/UnimodularPercolation/Kernel/Covariance.lean
index 698a2f1207762b087b09ea269d993c4d9348adf7..96a53ae5b9012f619663864e74adc36edbcbef71 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Kernel/Covariance.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Kernel/Covariance.lean
@@ -1,1192 +1,1192 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Splicing.EndpointMoments
-
-namespace OAI
-
-/-! Covariance for bond percolation. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-section FiniteTraceCovariance
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-@[simp] theorem edgeFieldEquiv_symm_edgeFieldEquiv (b : E ≃ E) (ω : Set E) :
-    edgeFieldEquiv b.symm (edgeFieldEquiv b ω) = ω := by
-  ext e
-  simp only [edgeFieldEquiv_apply,Set.mem_preimage,Equiv.symm_symm,Equiv.symm_apply_apply]
-
-@[simp] theorem spliceStateEquiv_symm_spliceStateEquiv (b : E ≃ E) (s : SpliceState E) :
-    spliceStateEquiv b.symm (spliceStateEquiv b s) = s := by
-  apply Prod.ext
-  · exact edgeFieldEquiv_symm_edgeFieldEquiv b s.1
-  · funext i
-    exact edgeFieldEquiv_symm_edgeFieldEquiv b (s.2 i)
-
-@[simp] theorem spliceCodeEquiv_symm_spliceCodeEquiv (b : E ≃ E) (c : List (ℕ × E)) :
-    spliceCodeEquiv b.symm (spliceCodeEquiv b c) = c := by
-  induction c with
-  | nil => rfl
-  | cons q c ih =>
-    rcases q with ⟨i,e⟩
-    simpa only [spliceCodeEquiv_apply,List.map_cons,Prod.map_apply,id_eq,Equiv.symm_apply_apply] using
-      congrArg (List.cons (i,e)) ih
-
-theorem spliceTrace_aut_map_iff (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (x : V) (η : ℕ → Set E) (y : ℕ → V) (k j : ℕ)
-    (ω ωf : Set E) (c : List (ℕ × E)) :
-    SpliceTrace G (a x) (fun i => edgeFieldEquiv b (η i)) (fun i => a (y i)) k j
-      (edgeFieldEquiv b ω) (spliceCodeEquiv b c) (edgeFieldEquiv b ωf) ↔
-    SpliceTrace G x η y k j ω c ωf := by
-  constructor
-  · intro ht
-    have hh := ht.aut_map G (isAut_symm G hab) (a x) (fun i => edgeFieldEquiv b (η i))
-      (fun i => a (y i)) k j (edgeFieldEquiv b ω) (edgeFieldEquiv b ωf) (spliceCodeEquiv b c)
-    simpa only [Equiv.symm_apply_apply,edgeFieldEquiv_symm_edgeFieldEquiv,
-      spliceCodeEquiv_symm_spliceCodeEquiv] using hh
-  · exact SpliceTrace.aut_map G hab x η y k j ω ωf c
-
-theorem boundedTraceCodes_aut_map_iff (b : E ≃ E) (Q Q' : Finset E)
-    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (k : ℕ) (c : List (ℕ × E)) :
-    spliceCodeEquiv b c ∈ boundedTraceCodes Q' k ↔ c ∈ boundedTraceCodes Q k := by
-  simp only [mem_boundedTraceCodes,spliceCodeEquiv_apply,List.length_map]
-  apply and_congr_right
-  intro _
-  constructor
-  · intro h q hq
-    have hh := h (q.1,b q.2) (List.mem_map.mpr ⟨q,hq,rfl⟩)
-    exact ⟨hh.1,hh.2.1,(hQ q.2).mpr hh.2.2⟩
-  · intro h r hr
-    obtain ⟨q,hq,rfl⟩ := List.mem_map.mp hr
-    exact ⟨(h q hq).1,(h q hq).2.1,(hQ q.2).mp (h q hq).2.2⟩
-
-theorem finiteTraceCodes_aut_map_iff (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (Q Q' : Finset E) (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q')
-    (x : V) (η : ℕ → Set E) (y : ℕ → V) (k j : ℕ) (ω : Set E) (c : List (ℕ × E)) :
-    spliceCodeEquiv b c ∈ finiteTraceCodes G (a x) (fun i => edgeFieldEquiv b (η i))
-      (fun i => a (y i)) k j (edgeFieldEquiv b ω) Q' ↔
-      c ∈ finiteTraceCodes G x η y k j ω Q := by
-  rw [mem_finiteTraceCodes_iff,mem_finiteTraceCodes_iff,boundedTraceCodes_aut_map_iff b Q Q' hQ,
-    replaySplice_aut_map G hab,spliceTrace_aut_map_iff G hab]
-
- 
-def finiteTraceCodesEquiv (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (Q Q' : Finset E) (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q')
-    (x : V) (η : ℕ → Set E) (y : ℕ → V) (k j : ℕ) (ω : Set E) :
-    ↥(finiteTraceCodes G x η y k j ω Q) ≃
-      ↥(finiteTraceCodes G (a x) (fun i => edgeFieldEquiv b (η i)) (fun i => a (y i)) k j
-        (edgeFieldEquiv b ω) Q') :=
-  (spliceCodeEquiv b).subtypeEquiv (fun c => (finiteTraceCodes_aut_map_iff G hab Q Q' hQ x η y k j ω c).symm)
-
-end FiniteTraceCovariance
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-section FiniteIndexCovariance
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-def finiteTraceIndexEquiv (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (x : V) (k : ℕ) (Q Q' : Finset E) (A A' : Finset (SpliceState E))
-    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (hA : ∀ s, s ∈ A ↔ spliceStateEquiv b s ∈ A')
-    (y y' : SpliceState E → ℕ → V) (hy : ∀ s i, y' (spliceStateEquiv b s) i = a (y s i)) :
-    FiniteTraceIndex G x k Q A y ≃ FiniteTraceIndex G (a x) k Q' A' y' :=
-  Equiv.sigmaCongr ((spliceStateEquiv b).subtypeEquiv hA) (fun s =>
-    (spliceCodeEquiv b).subtypeEquiv (fun c => by
-      have hy' : y' (spliceStateEquiv b s.val) = fun i => a (y s.val i) := funext (hy s.val)
-      change c ∈ finiteTraceCodes G x s.val.2 (y s.val) k 0 s.val.1 Q ↔
-        spliceCodeEquiv b c ∈ finiteTraceCodes G (a x) (fun i => edgeFieldEquiv b (s.val.2 i))
-          (y' (spliceStateEquiv b s.val)) k 0 (edgeFieldEquiv b s.val.1) Q'
-      rw [hy']
-      exact (finiteTraceCodes_aut_map_iff G hab Q Q' hQ x s.val.2 (y s.val) k 0 s.val.1 c).symm))
-
-namespace FiniteTraceIndexEquiv
-variable (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (x : V) (k : ℕ) (Q Q' : Finset E) (A A' : Finset (SpliceState E))
-    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (hA : ∀ s, s ∈ A ↔ spliceStateEquiv b s ∈ A')
-    (y y' : SpliceState E → ℕ → V) (hy : ∀ s i, y' (spliceStateEquiv b s) i = a (y s i))
-
-@[simp] theorem input (t : FiniteTraceIndex G x k Q A y) :
-    traceIndexInput G (a x) k Q' A' y' (finiteTraceIndexEquiv G hab x k Q Q' A A' hQ hA y y' hy t) =
-      spliceStateEquiv b (traceIndexInput G x k Q A y t) := rfl
-
-@[simp] theorem code (t : FiniteTraceIndex G x k Q A y) :
-    traceIndexCode G (a x) k Q' A' y' (finiteTraceIndexEquiv G hab x k Q Q' A A' hQ hA y y' hy t) =
-      spliceCodeEquiv b (traceIndexCode G x k Q A y t) := rfl
-
-@[simp] theorem output (t : FiniteTraceIndex G x k Q A y) :
-    traceIndexOutput G (a x) k Q' A' y' (finiteTraceIndexEquiv G hab x k Q Q' A A' hQ hA y y' hy t) =
-      spliceStateEquiv b (traceIndexOutput G x k Q A y t) := by
-  have hy' : y' (spliceStateEquiv b t.1.val) = fun i => a (y t.1.val i) := funext (hy t.1.val)
-  change runMasks (replayMasks G (a x) (fun i => edgeFieldEquiv b (t.1.val.2 i))
-    (y' (spliceStateEquiv b t.1.val)) (spliceCodeEquiv b t.2.val) (edgeFieldEquiv b t.1.val.1))
-      (spliceStateEquiv b t.1.val) = _
-  rw [hy',replayMasks_aut_map G hab,runMasks_aut_map]
-  rfl
-
-@[simp] theorem mass (t : FiniteTraceIndex G x k Q A y) :
-    traceIndexMass G (a x) k Q' A' y' (finiteTraceIndexEquiv G hab x k Q Q' A A' hQ hA y y' hy t) =
-      traceIndexMass G x k Q A y t := by
-  unfold traceIndexMass
-  rw [output G hab x k Q Q' A A' hQ hA y y' hy t]
-  apply Finset.prod_congr rfl
-  intro i _
-  have hy' : y' (spliceStateEquiv b t.1.val) = fun i => a (y t.1.val i) := funext (hy t.1.val)
-  change (traceContactCount G (fun j => edgeFieldEquiv b (t.1.val.2 j))
-    (y' (spliceStateEquiv b t.1.val)) (edgeFieldEquiv b (traceIndexOutput G x k Q A y t).1)
-    (a x) (spliceCodeEquiv b t.2.val) (i.val+1):ℝ)⁻¹ = _
-  rw [hy',traceContactCount_aut_map G hab]
-
-end FiniteTraceIndexEquiv
-
- 
-
-theorem finiteForwardLoad_aut_map (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (x : V) (k : ℕ) (Q Q' : Finset E) (A A' : Finset (SpliceState E))
-    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (hA : ∀ s, s ∈ A ↔ spliceStateEquiv b s ∈ A')
-    (y y' : SpliceState E → ℕ → V) (hy : ∀ s i, y' (spliceStateEquiv b s) i = a (y s i))
-    (s : SpliceState E) :
-    finiteForwardLoad G (a x) k Q' A' y' (spliceStateEquiv b s) = finiteForwardLoad G x k Q A y s := by
-  unfold finiteForwardLoad finiteInverseLoad
-  rw [← (finiteTraceIndexEquiv G hab x k Q Q' A A' hQ hA y y' hy).sum_comp]
-  apply Finset.sum_congr rfl
-  intro t _
-  rw [FiniteTraceIndexEquiv.output,FiniteTraceIndexEquiv.mass]
-  simp only [(spliceStateEquiv b).injective.eq_iff]
-
-end FiniteIndexCovariance
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-section FiniteFieldCovariance
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem finset_aut_eq_map (b : E ≃ E) (Q Q' : Finset E)
-    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') : Q' = Q.map b.toEmbedding := by
-  ext e
-  simpa only [Finset.mem_map_equiv,b.apply_symm_apply] using (hQ (b.symm e)).symm
-
-noncomputable def finiteFieldAutEquiv (b : E ≃ E) (Q Q' : Finset E)
-    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') : FiniteField Q ≃ FiniteField Q' :=
-  (Equiv.finsetCongr b).subtypeEquiv (fun ω => by
-    simp only [Finset.mem_powerset]
-    change ω ⊆ Q ↔ ω.map b.toEmbedding ⊆ Q'
-    rw [finset_aut_eq_map b Q Q' hQ,Finset.map_subset_map])
-
-@[simp] theorem finiteFieldSet_aut_map (b : E ≃ E) (Q Q' : Finset E)
-    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (ω : FiniteField Q) :
-    finiteFieldSet Q' (finiteFieldAutEquiv b Q Q' hQ ω) = edgeFieldEquiv b (finiteFieldSet Q ω) := by
-  change (ω.val.map b.toEmbedding : Set E) = b.symm ⁻¹' (ω.val : Set E)
-  rw [Finset.coe_map]
-  exact b.image_eq_preimage_symm _
-
-@[simp] theorem finiteFieldWeight_aut_map (b : E ≃ E) (Q Q' : Finset E)
-    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (p : ℝ) (ω : FiniteField Q) :
-    finiteFieldWeight Q' p (finiteFieldAutEquiv b Q Q' hQ ω) = finiteFieldWeight Q p ω := by
-  change p^(ω.val.map b.toEmbedding).card * (1-p)^(Q'.card-(ω.val.map b.toEmbedding).card) = _
-  simp only [Finset.card_map,finset_aut_eq_map b Q Q' hQ]
-  rfl
-
-noncomputable def finiteFamilyEquiv (b : E ≃ E) (Q Q' : Finset E)
-    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (J : Finset ℕ) :
-    (↥J → FiniteField Q) ≃ (↥J → FiniteField Q') :=
-  Equiv.piCongrRight fun _ => finiteFieldAutEquiv b Q Q' hQ
-
-@[simp] theorem finiteFieldFamily_aut_map (b : E ≃ E) (Q Q' : Finset E)
-    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (J : Finset ℕ) (η : ↥J → FiniteField Q) (i : ℕ) :
-    finiteFieldFamily Q' J (finiteFamilyEquiv b Q Q' hQ J η) i =
-      edgeFieldEquiv b (finiteFieldFamily Q J η i) := by
-  unfold finiteFieldFamily
-  split_ifs with hi
-  · exact finiteFieldSet_aut_map b Q Q' hQ (η ⟨i,hi⟩)
-  · rfl
-
-@[simp] theorem finiteFamilyWeight_aut_map (b : E ≃ E) (Q Q' : Finset E)
-    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (J : Finset ℕ) (p : ℝ) (η : ↥J → FiniteField Q) :
-    finiteFamilyWeight Q' J p (finiteFamilyEquiv b Q Q' hQ J η) = finiteFamilyWeight Q J p η := by
-  apply Finset.prod_congr rfl
-  intro i _
-  exact finiteFieldWeight_aut_map b Q Q' hQ p (η i)
-
-@[simp] theorem restrictFiniteField_aut_map (b : E ≃ E) (Q Q' : Finset E)
-    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (ω : Set E) :
-    restrictFiniteField Q' (edgeFieldEquiv b ω) = finiteFieldAutEquiv b Q Q' hQ (restrictFiniteField Q ω) := by
-  apply finiteFieldSet_injective Q'
-  rw [finiteFieldSet_aut_map,finiteFieldSet_restrict,finiteFieldSet_restrict]
-  ext e
-  change (b.symm e ∈ ω ∧ e ∈ Q') ↔ (b.symm e ∈ ω ∧ b.symm e ∈ Q)
-  rw [hQ (b.symm e),b.apply_symm_apply]
-
- 
-
-theorem finiteRootLoad_aut_map (b : E ≃ E) (Q Q' : Finset E)
-    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (J : Finset ℕ) (p : ℝ)
-    (f f' : SpliceState E → ℝ) (hf : ∀ s, f' (spliceStateEquiv b s) = f s)
-    (ω : FiniteField Q) :
-    finiteRootLoad Q' J p f' (finiteFieldAutEquiv b Q Q' hQ ω) = finiteRootLoad Q J p f ω := by
-  unfold finiteRootLoad
-  rw [← (finiteFamilyEquiv b Q Q' hQ J).sum_comp]
-  apply Finset.sum_congr rfl
-  intro η _
-  rw [finiteFamilyWeight_aut_map,finiteFieldSet_aut_map]
-  have he : finiteFieldFamily Q' J (finiteFamilyEquiv b Q Q' hQ J η) =
-      fun i => edgeFieldEquiv b (finiteFieldFamily Q J η i) :=
-    funext (finiteFieldFamily_aut_map b Q Q' hQ J η)
-  rw [he]
-  exact congrArg (fun t => finiteFamilyWeight Q J p η * t)
-    (hf (finiteFieldSet Q ω,finiteFieldFamily Q J η))
-
-end FiniteFieldCovariance
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-section ChoiceIndexCovariance
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {P P' : Type*}
-
-def finiteChoiceTraceIndexEquiv (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (x : V) (k : ℕ) (Q Q' : Finset E)
-    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (e : P ≃ P')
-    (A : P → Finset (SpliceState E)) (A' : P' → Finset (SpliceState E))
-    (hA : ∀ r s, s ∈ A r ↔ spliceStateEquiv b s ∈ A' (e r))
-    (y : P → SpliceState E → ℕ → V) (y' : P' → SpliceState E → ℕ → V)
-    (hy : ∀ r s i, y' (e r) (spliceStateEquiv b s) i = a (y r s i)) :
-    FiniteChoiceTraceIndex G x k Q A y ≃ FiniteChoiceTraceIndex G (a x) k Q' A' y' :=
-  Equiv.sigmaCongr e (fun r => finiteTraceIndexEquiv G hab x k Q Q' (A r) (A' (e r)) hQ (hA r)
-    (y r) (y' (e r)) (hy r))
-
-namespace FiniteChoiceTraceIndexEquiv
-variable (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (x : V) (k : ℕ) (Q Q' : Finset E)
-    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (e : P ≃ P')
-    (A : P → Finset (SpliceState E)) (A' : P' → Finset (SpliceState E))
-    (hA : ∀ r s, s ∈ A r ↔ spliceStateEquiv b s ∈ A' (e r))
-    (y : P → SpliceState E → ℕ → V) (y' : P' → SpliceState E → ℕ → V)
-    (hy : ∀ r s i, y' (e r) (spliceStateEquiv b s) i = a (y r s i))
-
-@[simp] theorem input (t : FiniteChoiceTraceIndex G x k Q A y) :
-    choiceTraceInput G (a x) k Q' A' y' (finiteChoiceTraceIndexEquiv G hab x k Q Q' hQ e A A' hA y y' hy t) =
-      spliceStateEquiv b (choiceTraceInput G x k Q A y t) := rfl
-
-@[simp] theorem output (t : FiniteChoiceTraceIndex G x k Q A y) :
-    choiceTraceOutput G (a x) k Q' A' y' (finiteChoiceTraceIndexEquiv G hab x k Q Q' hQ e A A' hA y y' hy t) =
-      spliceStateEquiv b (choiceTraceOutput G x k Q A y t) :=
-  FiniteTraceIndexEquiv.output G hab x k Q Q' (A t.1) (A' (e t.1)) hQ (hA t.1)
-    (y t.1) (y' (e t.1)) (hy t.1) t.2
-
-@[simp] theorem mass (q : SpliceState E → P → ℝ) (q' : SpliceState E → P' → ℝ)
-    (hq : ∀ s r, q' (spliceStateEquiv b s) (e r) = q s r)
-    (t : FiniteChoiceTraceIndex G x k Q A y) :
-    choiceTraceMass G (a x) k Q' A' y' q'
-      (finiteChoiceTraceIndexEquiv G hab x k Q Q' hQ e A A' hA y y' hy t) =
-      choiceTraceMass G x k Q A y q t := by
-  unfold choiceTraceMass
-  rw [input G hab x k Q Q' hQ e A A' hA y y' hy]
-  change q' (spliceStateEquiv b (choiceTraceInput G x k Q A y t)) (e t.1) * _ = _
-  rw [hq]
-  exact congrArg (fun m => q (choiceTraceInput G x k Q A y t) t.1 * m)
-    (FiniteTraceIndexEquiv.mass G hab x k Q Q' (A t.1) (A' (e t.1)) hQ (hA t.1)
-      (y t.1) (y' (e t.1)) (hy t.1) t.2)
-
-end FiniteChoiceTraceIndexEquiv
-
-theorem finiteChoiceForwardLoad_aut_map [Fintype P] [Fintype P']
-    (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (x : V) (k : ℕ) (Q Q' : Finset E)
-    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (e : P ≃ P')
-    (A : P → Finset (SpliceState E)) (A' : P' → Finset (SpliceState E))
-    (hA : ∀ r s, s ∈ A r ↔ spliceStateEquiv b s ∈ A' (e r))
-    (y : P → SpliceState E → ℕ → V) (y' : P' → SpliceState E → ℕ → V)
-    (hy : ∀ r s i, y' (e r) (spliceStateEquiv b s) i = a (y r s i))
-    (q : SpliceState E → P → ℝ) (q' : SpliceState E → P' → ℝ)
-    (hq : ∀ s r, q' (spliceStateEquiv b s) (e r) = q s r) (s : SpliceState E) :
-    finiteChoiceForwardLoad G (a x) k Q' A' y' q' (spliceStateEquiv b s) =
-      finiteChoiceForwardLoad G x k Q A y q s := by
-  unfold finiteChoiceForwardLoad finiteInverseLoad
-  rw [← (finiteChoiceTraceIndexEquiv G hab x k Q Q' hQ e A A' hA y y' hy).sum_comp]
-  apply Finset.sum_congr rfl
-  intro t _
-  rw [FiniteChoiceTraceIndexEquiv.output,FiniteChoiceTraceIndexEquiv.mass G hab x k Q Q' hQ e A A' hA y y' hy q q' hq]
-  simp only [(spliceStateEquiv b).injective.eq_iff]
-
-end ChoiceIndexCovariance
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-section AmplificationCovarianceInput
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem edgeField_subset_aut_iff (b : E ≃ E) (Q Q' : Finset E)
-    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (ω : Set E) :
-    edgeFieldEquiv b ω ⊆ (Q' : Set E) ↔ ω ⊆ (Q : Set E) := by
-  constructor
-  · intro h e he
-    exact (hQ e).mpr (h (by simpa using he : b e ∈ edgeFieldEquiv b ω))
-  · intro h e he
-    have hh := (hQ (b.symm e)).mp (h he)
-    simpa only [b.apply_symm_apply,Finset.mem_coe] using hh
-
-theorem finiteSpliceDomain_aut_map_iff (b : E ≃ E) (Q Q' : Finset E)
-    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (J : Finset ℕ) (s : SpliceState E) :
-    spliceStateEquiv b s ∈ finiteSpliceDomain Q' J ↔ s ∈ finiteSpliceDomain Q J := by
-  rw [mem_finiteSpliceDomain_iff,mem_finiteSpliceDomain_iff]
-  change (edgeFieldEquiv b s.1 ⊆ (Q':Set E) ∧
-    (∀ i,edgeFieldEquiv b (s.2 i) ⊆ (Q':Set E)) ∧ ∀ i ∉ J,edgeFieldEquiv b (s.2 i)=∅) ↔ _
-  have hz (η : Set E) : edgeFieldEquiv b η = ∅ ↔ η = ∅ := by
-    change edgeFieldEquiv b η = edgeFieldEquiv b ∅ ↔ _
-    exact (edgeFieldEquiv b).injective.eq_iff
-  simp only [edgeField_subset_aut_iff b Q Q' hQ,hz]
-
-theorem oldRunVertices_map {W W' : Type*} (f : W → W') (k : ℕ) (z : W)
-    (r : Fin k → W) (i : ℕ) :
-    oldRunVertices k (f z) (fun j => f (r j)) i = f (oldRunVertices k z r i) := by
-  induction k generalizing z with
-  | zero => rfl
-  | succ k ih =>
-    simp only [oldRunVertices]
-    split_ifs with hi
-    · exact ih (r 0) (Fin.tail r)
-    · rfl
-
-theorem oldRunWeight_map {W W' : Type*} (f : W → W')
-    (K : ℕ → W → W → ℝ) (K' : ℕ → W' → W' → ℝ)
-    (hK : ∀ i a b, K' i (f a) (f b) = K i a b) (k : ℕ) (z : W) (r : Fin k → W) :
-    oldRunWeight K' k (f z) (fun i => f (r i)) = oldRunWeight K k z r := by
-  induction k generalizing z with
-  | zero => rfl
-  | succ k ih =>
-    simp only [oldRunWeight,hK]
-    rw [show Fin.tail (fun i => f (r i)) = (fun i => f (Fin.tail r i)) from rfl,ih]
-
-def finiteVertexEquiv (a : V ≃ V) (B B' : Finset V) (hB : ∀ x, x ∈ B ↔ a x ∈ B') : ↥B ≃ ↥B' :=
-  a.subtypeEquiv hB
-
-def finiteRunEquiv (a : V ≃ V) (B B' : Finset V) (hB : ∀ x, x ∈ B ↔ a x ∈ B') (k : ℕ) :
-    (Fin k → ↥B) ≃ (Fin k → ↥B') :=
-  Equiv.piCongrRight fun _ => finiteVertexEquiv a B B' hB
-
-theorem amplificationVertices_aut_map (a : V ≃ V) (B B' : Finset V)
-    (hB : ∀ x, x ∈ B ↔ a x ∈ B') (k : ℕ) (z : ↥B) (r : Fin k → ↥B) (i : ℕ) :
-    amplificationVertices B' k (finiteVertexEquiv a B B' hB z) (finiteRunEquiv a B B' hB k r) i =
-      a (amplificationVertices B k z r i) :=
-  congrArg Subtype.val (oldRunVertices_map (finiteVertexEquiv a B B' hB) k z r i)
-
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
-
-theorem mass_aut_map (q : LocalEndpointRule G hloc O M L) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (ω : Set E) (x y : V) :
-    q.mass (edgeFieldEquiv b ω) (a x) (a y) = q.mass ω x y := by
-  simpa only [Equiv.symm_apply_apply,edgeFieldEquiv_apply] using
-    q.covariance a.symm b.symm (isAut_symm G hab) ω (a x) (a y)
-
-theorem amplificationWeight_aut_map (q : LocalEndpointRule G hloc O M L) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (B B' : Finset V) (hB : ∀ x, x ∈ B ↔ a x ∈ B')
-    (k : ℕ) (z : ↥B) (s : SpliceState E) (r : Fin k → ↥B) :
-    q.amplificationWeight B' k (finiteVertexEquiv a B B' hB z) (spliceStateEquiv b s)
-      (finiteRunEquiv a B B' hB k r) = q.amplificationWeight B k z s r := by
-  apply oldRunWeight_map (finiteVertexEquiv a B B' hB)
-  intro i v w
-  exact q.mass_aut_map hab (s.2 i) v.val w.val
-
-theorem amplificationInputs_aut_map_iff (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
-    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (Q Q' : Finset E)
-    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (B B' : Finset V) (hB : ∀ x, x ∈ B ↔ a x ∈ B')
-    (x : V) (k n : ℕ) (z : ↥B) (r : Fin k → ↥B) (s : SpliceState E) :
-    spliceStateEquiv b s ∈ q.amplificationInputs Q' B' (a x) k n
-      (finiteVertexEquiv a B B' hB z) (finiteRunEquiv a B B' hB k r) ↔
-      s ∈ q.amplificationInputs Q B x k n z r := by
-  rw [q.mem_amplificationInputs,q.mem_amplificationInputs]
-  have hn : (cluster G (edgeFieldEquiv b s.1) (a x)).ncard = (cluster G s.1 x).ncard := by
-    rw [cluster_edgeFieldEquiv G hab,← a.image_eq_preimage_symm,Set.ncard_image_of_injective _ a.injective]
-  change (spliceStateEquiv b s ∈ finiteSpliceDomain Q' (Finset.Icc 1 k) ∧
-    a x ∈ vertexOrbit G o ∧ a z.val ∈ vertexOrbit G o ∧
-    n ≤ (cluster G (edgeFieldEquiv b s.1) (a x)).ncard ∧
-    (cluster G (edgeFieldEquiv b s.1) (a x)).ncard < 2*n ∧
-    amplificationVertices B' k (finiteVertexEquiv a B B' hB z) (finiteRunEquiv a B B' hB k r) 0 ∈
-      cluster G (edgeFieldEquiv b s.1) (a x) ∧ _) ↔ _
-  rw [finiteSpliceDomain_aut_map_iff b Q Q' hQ,mem_vertexOrbit_aut_iff G o hab,
-    mem_vertexOrbit_aut_iff G o hab,hn,amplificationVertices_aut_map,
-    cluster_edgeFieldEquiv G hab,q.amplificationWeight_aut_map hab B B' hB]
-  simp only [Set.mem_preimage,Equiv.symm_apply_apply]
-
-end LocalEndpointRule
-end AmplificationCovarianceInput
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-section AmplificationCovarianceMarginal
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
-
- 
-
-theorem amplificationMarginal_aut_map (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
-    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (Q Q' : Finset E)
-    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (B B' : Finset V) (hB : ∀ x, x ∈ B ↔ a x ∈ B')
-    (x : V) (k n : ℕ) (z : ↥B) (p : unitInterval) (ω : Set E) :
-    q.amplificationMarginal Q' B' (a x) k n (finiteVertexEquiv a B B' hB z) p (edgeFieldEquiv b ω) =
-      q.amplificationMarginal Q B x k n z p ω := by
-  unfold amplificationMarginal
-  rw [restrictFiniteField_aut_map b Q Q' hQ]
-  apply finiteRootLoad_aut_map
-  intro s
-  apply finiteChoiceForwardLoad_aut_map G hab x k Q Q' hQ (finiteRunEquiv a B B' hB k)
-  · intro r s
-    exact (q.amplificationInputs_aut_map_iff o hab Q Q' hQ B B' hB x k n z r s).symm
-  · intro r s i
-    exact amplificationVertices_aut_map a B B' hB k z r i
-  · intro s r
-    exact q.amplificationWeight_aut_map hab B B' hB k z s r
-
-end LocalEndpointRule
-
-theorem mem_incidentSupport_aut_iff (G : BondGraph V E) (hloc : LocallyFinite G)
-    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (B B' : Finset V)
-    (hB : ∀ x, x ∈ B ↔ a x ∈ B') (e : E) :
-    b e ∈ incidentSupport G hloc B' ↔ e ∈ incidentSupport G hloc B := by
-  rw [finset_aut_eq_map a B B' hB,incidentSupport_aut G hloc hab]
-  simp only [Finset.mem_map_equiv,Equiv.symm_apply_apply]
-
-theorem finiteBall_aut_iff (G : BondGraph V E) (hloc : LocallyFinite G)
-    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (x y : V) (R : ℕ) :
-    y ∈ (finite_reachBall hloc x R).toFinset ↔ a y ∈ (finite_reachBall hloc (a x) R).toFinset := by
-  simpa only [Set.Finite.mem_toFinset] using (mem_reachBall_aut_iff G hab x y R).symm
-
-theorem localBondBall_aut_iff (G : BondGraph V E) (hloc : LocallyFinite G)
-    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (x : V) (R : ℕ) (e : E) :
-    e ∈ incidentSupport G hloc (finite_reachBall hloc x R).toFinset ↔
-      b e ∈ incidentSupport G hloc (finite_reachBall hloc (a x) R).toFinset :=
-  (mem_incidentSupport_aut_iff G hloc hab _ _ (fun y => finiteBall_aut_iff G hloc hab x y R) e).symm
-
-end AmplificationCovarianceMarginal
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-open MeasureTheory
-section CanonicalAmplificationLoad
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
-
-def amplificationRadius (q : LocalEndpointRule G hloc O M L) (k n : ℕ) : ℕ :=
-  2*n+k*M+max q.radius M
-
-noncomputable def amplificationBall (_q : LocalEndpointRule G hloc O M L) (k n : ℕ) (x : V) : Finset V :=
-  (finite_reachBall hloc x (2*n+k*M)).toFinset
-
-noncomputable def amplificationBonds (q : LocalEndpointRule G hloc O M L) (k n : ℕ) (x : V) : Finset E :=
-  incidentSupport G hloc (finite_reachBall hloc x (q.amplificationRadius k n)).toFinset
-
- 
-
-noncomputable def amplifiedMass (q : LocalEndpointRule G hloc O M L) (k n : ℕ) (p : unitInterval)
-    (ω : Set E) (x z : V) : ℝ :=
-  if hz : z ∈ q.amplificationBall k n x then
-    q.amplificationMarginal (q.amplificationBonds k n x) (q.amplificationBall k n x) x k n ⟨z,hz⟩ p ω / n
-  else 0
-
-theorem amplifiedMass_nonneg (q : LocalEndpointRule G hloc O M L) (hL : 1 ≤ L)
-    (k n : ℕ) (p : unitInterval) (ω : Set E) (x z : V) : 0 ≤ q.amplifiedMass k n p ω x z := by
-  unfold amplifiedMass
-  split_ifs
-  · exact div_nonneg (q.amplificationMarginal_bounds hL _ _ _ _ _ _ _ _).1 (Nat.cast_nonneg _)
-  · exact le_rfl
-
-theorem amplifiedMass_support (q : LocalEndpointRule G hloc O M L)
-    (k n : ℕ) (p : unitInterval) (ω : Set E) (x z : V) (h : q.amplifiedMass k n p ω x z ≠ 0) :
-    x ∈ O ∧ z ∈ O ∧ Conn G ω x z ∧ (cluster G ω x).Finite ∧ (cluster G ω x).ncard ≤ 2*n+k*M := by
-  unfold amplifiedMass at h
-  split_ifs at h with hz
-  · apply q.amplificationMarginal_physical_support (q.amplificationBonds k n x) (q.amplificationBall k n x) x k n ⟨z,hz⟩ p ω
-    · exact incidentSupport_ball_mono G hloc x (Nat.le_add_right (2*n+k*M) (max q.radius M))
-    · intro he
-      exact h (by rw [he,zero_div])
-  · exact (h rfl).elim
-
-theorem amplifiedMass_local (q : LocalEndpointRule G hloc O M L)
-    (k n : ℕ) (p : unitInterval) (ω : Set E) (x z : V) :
-    q.amplifiedMass k n p ω x z =
-      q.amplifiedMass k n p (ω ∩ (q.amplificationBonds k n x : Set E)) x z := by
-  unfold amplifiedMass
-  split_ifs
-  · exact congrArg (fun t : ℝ => t / n) (q.amplificationMarginal_local _ _ _ _ _ _ _ _)
-  · rfl
-
-theorem amplifiedMass_aut_map (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
-    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (k n : ℕ) (p : unitInterval)
-    (ω : Set E) (x z : V) :
-    q.amplifiedMass k n p (edgeFieldEquiv b ω) (a x) (a z) = q.amplifiedMass k n p ω x z := by
-  have hB (y : V) : y ∈ q.amplificationBall k n x ↔ a y ∈ q.amplificationBall k n (a x) :=
-    finiteBall_aut_iff G hloc hab x y (2*n+k*M)
-  have hQ (e : E) : e ∈ q.amplificationBonds k n x ↔ b e ∈ q.amplificationBonds k n (a x) :=
-    localBondBall_aut_iff G hloc hab x (q.amplificationRadius k n) e
-  unfold amplifiedMass
-  by_cases hz : z ∈ q.amplificationBall k n x
-  · simp only [dite_eq_left hz,dite_eq_left ((hB z).mp hz)]
-    exact congrArg (fun t : ℝ => t / n)
-      (q.amplificationMarginal_aut_map o hab _ _ hQ _ _ hB x k n ⟨z,hz⟩ p ω)
-  · simp only [dite_eq_right hz,dite_eq_right (mt (hB z).mpr hz)]
-
- 
-
-noncomputable def amplificationLoad (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
-    (hL : 1 ≤ L) (k n : ℕ) (p : unitInterval) :
-    CappedLocalLoad G hloc (vertexOrbit G o) (2*n+k*M) where
-  mass := q.amplifiedMass k n p
-  radius := q.amplificationRadius k n
-  nonneg := q.amplifiedMass_nonneg hL k n p
-  support := q.amplifiedMass_support k n p
-  locality := q.amplifiedMass_local k n p
-  covariance := by
-    intro a b hab ω x z
-    simpa only [Equiv.symm_apply_apply,edgeFieldEquiv_apply,Equiv.symm_symm] using
-      q.amplifiedMass_aut_map o (isAut_symm G hab) k n p ω (a x) (a z)
-
-end LocalEndpointRule
-end CanonicalAmplificationLoad
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-open MeasureTheory
-section CanonicalAmplificationRow
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
-
-theorem amplificationBonds_query {O : Set V} (q : LocalEndpointRule G hloc O M L)
-    (k n : ℕ) (x a : V) (ha : a ∈ q.amplificationBall k n x) :
-    incidentSupport G hloc (finite_reachBall hloc a (max q.radius M)).toFinset ⊆
-      q.amplificationBonds k n x := by
-  apply incidentSupport_mono
-  intro y hy
-  simp only [Set.Finite.mem_toFinset] at hy ⊢
-  exact reachBall_triangle G x a (2*n+k*M) (max q.radius M)
-    (by simpa only [amplificationBall,Set.Finite.mem_toFinset] using ha) hy
-
-theorem amplificationBonds_root {O : Set V} (q : LocalEndpointRule G hloc O M L)
-    (k n : ℕ) (x : V) :
-    incidentSupport G hloc (finite_reachBall hloc x (2*n)).toFinset ⊆
-      q.amplificationBonds k n x := by
-  apply incidentSupport_ball_mono
-  unfold amplificationRadius
-  omega
-
-theorem amplificationLoad_row (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
-    (hL : 1 ≤ L) (k n : ℕ) (p : unitInterval) (ω : Set E) (x : V) :
-    (q.amplificationLoad o hL k n p).row ω x =
-      q.amplificationRow (q.amplificationBonds k n x) (q.amplificationBall k n x) x k n p ω / n := by
-  change (∑' z, q.amplifiedMass k n p ω x z) = _
-  rw [tsum_eq_sum (s := q.amplificationBall k n x) (fun z hz => by
-    simp only [amplifiedMass,dite_eq_right hz])]
-  rw [← Finset.sum_coe_sort]
-  simp only [amplifiedMass,dite_eq_left (Subtype.prop _)]
-  exact (Finset.sum_div _ _ _).symm
-
-theorem amplificationRow_local {O : Set V} (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (p : unitInterval) (ω : Set E) :
-    q.amplificationRow Q B x k n p ω = q.amplificationRow Q B x k n p (ω ∩ (Q : Set E)) := by
-  unfold amplificationRow
-  apply Finset.sum_congr rfl
-  intro z _
-  exact q.amplificationMarginal_local Q B x k n z p ω
-
-theorem amplificationLoad_mean_finite (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
-    (hL : 1 ≤ L) (k n : ℕ) (p : unitInterval) (x : V) :
-    (∫ ω, (q.amplificationLoad o hL k n p).row ω x ∂law p) =
-      finiteFieldIntegral (q.amplificationBonds k n x) p
-        (q.amplificationRow (q.amplificationBonds k n x) (q.amplificationBall k n x) x k n p) / n := by
-  simp_rw [q.amplificationLoad_row o hL k n p]
-  rw [integral_div]
-  congr 1
-  exact integral_finite_local _ p _ (q.amplificationRow_local _ _ x k n p)
-
-theorem amplificationLoad_second_moment_finite (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
-    (hL : 1 ≤ L) (k n : ℕ) (p : unitInterval) (x : V) :
-    (∫ ω, (q.amplificationLoad o hL k n p).row ω x ^ 2 ∂law p) =
-      finiteFieldIntegral (q.amplificationBonds k n x) p (fun ω =>
-        q.amplificationRow (q.amplificationBonds k n x) (q.amplificationBall k n x) x k n p ω ^ 2) / (n:ℝ)^2 := by
-  simp_rw [q.amplificationLoad_row o hL k n p,div_pow]
-  rw [integral_div]
-  congr 1
-  exact integral_finite_local _ p _ (fun ω => congrArg (·^2) (q.amplificationRow_local _ _ x k n p ω))
-
-end LocalEndpointRule
-end CanonicalAmplificationRow
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-open MeasureTheory
-section NormalizedAmplificationMoment
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
-
- 
-
-theorem amplificationLoad_second_moment_le [Nonempty E] (hconn : Connected G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
-    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (i₀ : ι) (hw₀ : 0 < weight i₀)
-    (q : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) M L) (hL : 1 ≤ L)
-    (p : unitInterval) (x : V) (k n d : ℕ) (hn : 0 < n)
-    (hdeg : ∀ a, degree G a ≤ d) :
-    (∫ ω, (q.amplificationLoad (root i₀) hL k n p).row ω x ^ 2 ∂law p) ≤
-      (4 * q.successProbability p (root i₀)^k +
-        (((k*M)+(k*M)^2+d*(k*M)^3:ℕ):ℝ) * inverseCodeMass M L^k / n) *
-          (∫ ω, (q.amplificationLoad (root i₀) hL k n p).row ω x ∂law p) +
-      (2 * (c*((k*M:ℕ)*q.successProbability p (root i₀)^k)))^2 *
-        finiteClusterWindowMass G (q.amplificationBonds k n x) p x (n-k*M) (2*n) := by
-  rw [q.amplificationLoad_second_moment_finite,q.amplificationLoad_mean_finite]
-  have h := q.amplificationRow_second_moment_le hconn root weight Γ hΓ htr hcov hw hw1 c hc hmin
-    i₀ hw₀ hL (q.amplificationBonds k n x) (q.amplificationBall k n x) p x k n d hdeg
-    (fun a ha => q.amplificationBonds_query k n x a ha)
-  have hn' : (n:ℝ) ≠ 0 := ne_of_gt (Nat.cast_pos.mpr hn)
-  apply (div_le_div_of_nonneg_right h (sq_nonneg (n:ℝ))).trans_eq
-  push_cast
-  field_simp
-
-end LocalEndpointRule
-end NormalizedAmplificationMoment
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-section FinitePathExhaustion
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def finitePathTest (B : Finset V) (K : V → V → ℝ) (F : V → ℝ) (k : ℕ) (z : ↥B) : ℝ :=
-  ∑ r : Fin k → ↥B, oldRunWeight (fun _ a b => K a.val b.val) k z r * F (oldRunVertices k z r 0).val
-
-@[simp] theorem finitePathTest_zero (B : Finset V) (K : V → V → ℝ) (F : V → ℝ) (z : ↥B) :
-    finitePathTest B K F 0 z = F z.val := by
-  simp [finitePathTest,oldRunWeight,oldRunVertices]
-
-theorem finitePathTest_succ (B : Finset V) (K : V → V → ℝ) (F : V → ℝ) (k : ℕ) (z : ↥B) :
-    finitePathTest B K F (k+1) z = ∑ a : ↥B, K z.val a.val * finitePathTest B K F k a :=
-  oldRun_test_succ (fun _ (a b : ↥B) => K a.val b.val) (fun (a : ↥B) => F a.val) k z
-
- 
-
-theorem finitePathTest_support (G : BondGraph V E) (B : Finset V) (K : V → V → ℝ) (F : V → ℝ)
-    (O : Set V) (x : V) (R M : ℕ)
-    (hK : ∀ a b, K a b ≠ 0 → a ∈ O ∧ a ∈ reachBall G b M)
-    (hF : ∀ a, F a ≠ 0 → a ∈ O ∧ a ∈ reachBall G x R)
-    (k : ℕ) (z : ↥B) (hz : finitePathTest B K F k z ≠ 0) :
-    z.val ∈ O ∧ z.val ∈ reachBall G x (R+k*M) := by
-  induction k generalizing z with
-  | zero => simpa using hF z.val (by simpa using hz)
-  | succ k ih =>
-    rw [finitePathTest_succ] at hz
-    obtain ⟨a,_,ha⟩ := Finset.exists_ne_zero_of_sum_ne_zero hz
-    have hh := mul_ne_zero_iff.mp ha
-    obtain ⟨haO,haB⟩ := ih a hh.2
-    obtain ⟨hzO,hzB⟩ := hK z.val a.val hh.1
-    refine ⟨hzO,?_⟩
-    have hg := reachBall_triangle G x a.val (R+k*M) M haB hzB
-    simpa only [Nat.add_mul,one_mul,Nat.add_assoc] using hg
-
- 
-
-theorem finitePathTest_total (G : BondGraph V E) (B : Finset V) (K : V → V → ℝ) (F : V → ℝ)
-    (O : Set V) (x : V) (R M N : ℕ) (u : ℝ)
-    (hK : ∀ a b, K a b ≠ 0 → a ∈ O ∧ a ∈ reachBall G b M)
-    (hF : ∀ a, F a ≠ 0 → a ∈ O ∧ a ∈ reachBall G x R)
-    (hcolumn : ∀ j < N, ∀ a : ↥B, a.val ∈ O → a.val ∈ reachBall G x (R+j*M) →
-      ∑ z : ↥B, K z.val a.val = u) :
-    ∀ j ≤ N, (∑ z : ↥B, finitePathTest B K F j z) = u^j * ∑ z : ↥B, F z.val := by
-  intro j hj
-  induction j with
-  | zero => simp
-  | succ j ih =>
-    simp_rw [finitePathTest_succ]
-    rw [Finset.sum_comm]
-    have he : (∑ a : ↥B, ∑ z : ↥B, K z.val a.val * finitePathTest B K F j a) =
-        u * ∑ a : ↥B, finitePathTest B K F j a := by
-      rw [Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro a _
-      rw [← Finset.sum_mul]
-      by_cases ha : finitePathTest B K F j a = 0
-      · simp only [ha,mul_zero]
-      · obtain ⟨haO,haB⟩ := finitePathTest_support G B K F O x R M hK hF j a ha
-        rw [hcolumn j (by omega) a haO haB]
-    rw [he,ih (by omega)]
-    ring
-
-end FinitePathExhaustion
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-open MeasureTheory
-section AmplificationBoundaryExhaustion
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
-
-theorem kernel_range_support {O : Set V} (q : LocalEndpointRule G hloc O M L)
-    (p : unitInterval) (a b : V) (h : q.kernel p a b ≠ 0) : a ∈ O ∧ a ∈ reachBall G b M := by
-  constructor
-  · by_contra ha
-    exact h (q.kernel_zero_of_start_not_mem p a b ha)
-  · by_contra ha
-    exact h (q.kernel_outside_reverse_ball_zero p a b ha)
-
-theorem orbitBinConnection_range_support (o : V) (p : unitInterval) (n : ℕ) (x a : V)
-    (h : orbitBinConnection G p o n x a ≠ 0) :
-    a ∈ vertexOrbit G o ∧ a ∈ reachBall G x (2*n) := by
-  have ha : a ∈ vertexOrbit G o := by
-    by_contra ha
-    exact h (by simp only [orbitBinConnection,ha,and_false,ite_false])
-  refine ⟨ha,?_⟩
-  by_contra hb
-  exact h (by simp only [orbitBinConnection,binConnection_outside G p n x a hb,ite_self])
-
-theorem amplificationBall_column {O : Set V} (q : LocalEndpointRule G hloc O M L)
-    (p : unitInterval) (k n j : ℕ) (hj : j < k) (x a : V)
-    (ha : a ∈ reachBall G x (2*n+j*M)) :
-    (∑ z : ↥(q.amplificationBall k n x), q.kernel p z.val a) = ∑' z, q.kernel p z a := by
-  rw [Finset.sum_coe_sort (q.amplificationBall k n x) (fun z => q.kernel p z a)]
-  symm
-  apply tsum_eq_sum
-  intro z hz
-  apply q.kernel_outside_reverse_ball_zero
-  intro hza
-  have hh := reachBall_triangle G x a (2*n+j*M) M ha hza
-  have hle : (2*n+j*M)+M ≤ 2*n+k*M := by
-    calc
-      _ = 2*n+(j+1)*M := by ring
-      _ ≤ _ := Nat.add_le_add_left (Nat.mul_le_mul_right M (by omega)) _
-  exact hz (by
-    simpa only [amplificationBall,Set.Finite.mem_toFinset] using (reachBall_mono G x hle hh))
-
-theorem amplificationPath_total {ι : Type*} [Fintype ι]
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (htr : RootTransport root weight Γ)
-    (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b) (hw : ∀ i, 0 ≤ weight i)
-    (i₀ : ι) (hw₀ : 0 < weight i₀)
-    (q : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) M L)
-    (p : unitInterval) (k n : ℕ) (x : V) (hx : x ∈ vertexOrbit G (root i₀)) :
-    (∑ z : ↥(q.amplificationBall k n x),
-      finitePathTest (q.amplificationBall k n x) (q.kernel p) (orbitBinConnection G p (root i₀) n x) k z) =
-      q.successProbability p (root i₀)^k * binOrbitRow G p (root i₀) n := by
-  have ht := finitePathTest_total G (q.amplificationBall k n x) (q.kernel p)
-    (orbitBinConnection G p (root i₀) n x) (vertexOrbit G (root i₀)) x (2*n) M k
-    (q.successProbability p (root i₀)) (q.kernel_range_support p)
-    (orbitBinConnection_range_support (root i₀) p n x) (by
-      intro j hj a haO haB
-      rw [q.amplificationBall_column p k n j hj x a.val haB]
-      exact q.kernel_column_eq_success root weight Γ htr hΓ hw i₀ hw₀ p a.val haO) k le_rfl
-  rw [ht]
-  congr 1
-  rw [Finset.sum_coe_sort (q.amplificationBall k n x) (orbitBinConnection G p (root i₀) n x),← orbitBinConnection_row G hloc p (root i₀) n x hx]
-  symm
-  apply tsum_eq_sum
-  intro z hz
-  by_contra h
-  have hb := (orbitBinConnection_range_support (root i₀) p n x z h).2
-  exact hz (by
-    simpa only [amplificationBall,Set.Finite.mem_toFinset] using
-      reachBall_mono G x (Nat.le_add_right (2*n) (k*M)) hb)
-
-end LocalEndpointRule
-end AmplificationBoundaryExhaustion
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-open MeasureTheory
-section AmplificationExactMean
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
-
-theorem amplificationMarginal_mean_pathTest [Nonempty E] (o : V)
-    (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (z : ↥B) (p : unitInterval)
-    (hx : x ∈ vertexOrbit G o)
-    (hroot : incidentSupport G hloc (finite_reachBall hloc x (2*n)).toFinset ⊆ Q)
-    (hquery : ∀ a ∈ B, incidentSupport G hloc (finite_reachBall hloc a q.radius).toFinset ⊆ Q) :
-    (∫ ω, q.amplificationMarginal Q B x k n z p ω ∂law p) =
-      finitePathTest B (q.kernel p) (orbitBinConnection G p o n x) k z := by
-  by_cases hz : z.val ∈ vertexOrbit G o
-  · rw [q.amplificationMarginal_mean_kernel Q B x k n z p hx hz hquery]
-    unfold finitePathTest
-    apply Finset.sum_congr rfl
-    intro r _
-    rw [finiteBinConnection_eq_binConnection G hloc Q p x n _ hroot]
-    by_cases hr : oldRunWeight (fun _ (a b : ↥B) => q.kernel p a.val b.val) k z r = 0
-    · simp only [hr,mul_zero,zero_mul]
-    · have hy : (oldRunVertices k z r 0).val ∈ vertexOrbit G o :=
-        oldRun_endpoint_mem (fun a : ↥B => a.val ∈ vertexOrbit G o)
-          (fun _ (a b : ↥B) => q.kernel p a.val b.val)
-          (fun _ a b hab => by
-            by_contra hb
-            exact hab (q.kernel_zero_of_end_not_mem p a.val b.val hb)) k z hz r hr
-      rw [orbitBinConnection,ite_eq_left ⟨hx,hy⟩]
-      exact mul_comm _ _
-  · have hm (ω : Set E) : q.amplificationMarginal Q B x k n z p ω = 0 := by
-      by_contra h
-      exact hz (q.amplificationMarginal_support Q B x k n z p ω h).2.1
-    simp_rw [hm,integral_zero]
-    symm
-    by_contra h
-    exact hz (finitePathTest_support G B (q.kernel p) (orbitBinConnection G p o n x)
-      (vertexOrbit G o) x (2*n) M (q.kernel_range_support p)
-      (orbitBinConnection_range_support o p n x) k z h).1
-
- 
-
-theorem amplificationLoad_mean [Nonempty E] {ι : Type*} [Fintype ι]
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (htr : RootTransport root weight Γ)
-    (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b) (hw : ∀ i, 0 ≤ weight i)
-    (i₀ : ι) (hw₀ : 0 < weight i₀)
-    (q : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) M L) (hL : 1 ≤ L)
-    (p : unitInterval) (k n : ℕ) (x : V) (hx : x ∈ vertexOrbit G (root i₀)) :
-    (∫ ω, (q.amplificationLoad (root i₀) hL k n p).row ω x ∂law p) =
-      q.successProbability p (root i₀)^k * binOrbitRow G p (root i₀) n / n := by
-  simp_rw [q.amplificationLoad_row]
-  rw [integral_div]
-  congr 1
-  unfold amplificationRow
-  rw [integral_finsetSum _ (fun z _ => integrable_finite_local _ p _
-    (q.amplificationMarginal_local _ _ x k n z p))]
-  simp_rw [q.amplificationMarginal_mean_pathTest (root i₀) _ _ x k n _ p hx
-    (q.amplificationBonds_root k n x) (fun a ha =>
-      (incidentSupport_ball_mono G hloc a (le_max_left _ _)).trans (q.amplificationBonds_query k n x a ha))]
-  exact q.amplificationPath_total root weight Γ htr hΓ hw i₀ hw₀ p k n x hx
-
-end LocalEndpointRule
-end AmplificationExactMean
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-open MeasureTheory
-section AmplificationLoadOperator
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace CappedLocalLoad
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ}
-
-noncomputable def kernel (q : CappedLocalLoad G hloc O M) (p : unitInterval) (x y : V) : ℝ :=
-  ∫ ω, q.mass ω x y ∂law p
-
-theorem kernel_nonneg (q : CappedLocalLoad G hloc O M) (p : unitInterval) (x y : V) :
-    0 ≤ q.kernel p x y := integral_nonneg (q.nonneg · x y)
-
-end CappedLocalLoad
-
- 
-theorem exists_operator_of_scaled_matrix (F : V → V → ℝ) (c C : ℝ) (hc : 0 < c)
-    (A : Hilbert V →L[ℂ] Hilbert V) (hAnorm : ‖A‖ ≤ C)
-    (hA : ∀ x y, matrixCoeff A x y = (c * F x y : ℂ)) :
-    ∃ T : Hilbert V →L[ℂ] Hilbert V, ‖T‖ ≤ C/c ∧
-      ∀ x y, matrixCoeff T x y = (F x y : ℂ) := by
-  refine ⟨((c⁻¹:ℝ):ℂ) • A,?_,?_⟩
-  · rw [norm_smul,Complex.norm_of_nonneg (inv_nonneg.mpr hc.le)]
-    simpa only [div_eq_mul_inv,mul_comm] using
-      mul_le_mul_of_nonneg_left hAnorm (inv_nonneg.mpr hc.le)
-  · intro x y
-    change ((c⁻¹:ℝ):ℂ)*matrixCoeff A x y = _
-    rw [hA]
-    have hc' : (c:ℂ) ≠ 0 := Complex.ofReal_ne_zero.mpr hc.ne'
-    simp only [Complex.ofReal_inv]
-    field_simp
-
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
-
-theorem amplificationLoad_scaled_kernel [Nonempty E] (o : V)
-    (q : LocalEndpointRule G hloc (vertexOrbit G o) M L) (hL : 1 ≤ L)
-    (k n : ℕ) (hn : 0 < n) (p : unitInterval) (x z : V)
-    (hz : z ∈ q.amplificationBall k n x) :
-    (n:ℝ) * (q.amplificationLoad o hL k n p).kernel p x z =
-      ∫ ω, q.amplificationMarginal (q.amplificationBonds k n x) (q.amplificationBall k n x)
-        x k n ⟨z,hz⟩ p ω ∂law p := by
-  change (n:ℝ) * (∫ ω, q.amplifiedMass k n p ω x z ∂law p) = _
-  simp only [amplifiedMass,dite_eq_left hz,integral_div]
-  have hn' : (n:ℝ) ≠ 0 := ne_of_gt (Nat.cast_pos.mpr hn)
-  field_simp
-
- 
-
-theorem amplificationLoad_exists_operator [Nonempty E] (o : V)
-    (q : LocalEndpointRule G hloc (vertexOrbit G o) M L) (hL : 1 ≤ L)
-    (k n : ℕ) (hn : 0 < n) (p : unitInterval)
-    (A : Hilbert V →L[ℂ] Hilbert V)
-    (hA : ∀ a b, matrixCoeff A a b = (q.kernel p a b : ℂ)) :
-    ∃ T : Hilbert V →L[ℂ] Hilbert V, ‖T‖ ≤ binOrbitRow G p o n * ‖A‖^k / n ∧
-      ∀ x z, matrixCoeff T x z = ((q.amplificationLoad o hL k n p).kernel p x z : ℂ) := by
-  let F := (q.amplificationLoad o hL k n p).kernel p
-  have hn' : (0:ℝ) < n := Nat.cast_pos.mpr hn
-  have hposA : PositiveRealMatrix A := fun x z => by
-    rw [hA]
-    exact ⟨rfl,q.kernel_nonneg p x z⟩
-  obtain ⟨S,hSnorm,hS⟩ := exists_operator_of_amplification_domination G hloc p o n k A hposA
-    (fun x z => (n:ℝ)*F x z) (fun x z => mul_nonneg hn'.le ((q.amplificationLoad o hL k n p).kernel_nonneg p x z))
-    (by
-      intro H hH x z
-      have hposH : PositiveRealMatrix H := fun a b => by
-        rw [hH]
-        exact ⟨rfl,orbitBinConnection_nonneg G p o n a b⟩
-      have hpos := (((hposA.pow k).mul hposH) z x).2
-      by_cases hz : z ∈ q.amplificationBall k n x
-      · rw [q.amplificationLoad_scaled_kernel o hL k n hn p x z hz]
-        by_cases ho : x ∈ vertexOrbit G o ∧ z ∈ vertexOrbit G o
-        · exact q.amplificationMarginal_operator_le o _ _ x k n ⟨z,hz⟩ p ho.1 ho.2
-            (q.amplificationBonds_root k n x) (fun a ha =>
-              (incidentSupport_ball_mono G hloc a (le_max_left _ _)).trans (q.amplificationBonds_query k n x a ha)) A H hA hH
-        · have hm (ω : Set E) : q.amplificationMarginal (q.amplificationBonds k n x)
-              (q.amplificationBall k n x) x k n ⟨z,hz⟩ p ω = 0 := by
-            by_contra h
-            have hs := q.amplificationMarginal_support _ _ x k n ⟨z,hz⟩ p ω h
-            exact ho ⟨hs.1,hs.2.1⟩
-          simpa only [hm,integral_zero] using hpos
-      · have he : F x z = 0 := by
-          change (∫ ω, q.amplifiedMass k n p ω x z ∂law p) = 0
-          simp only [amplifiedMass,dite_eq_right hz,integral_zero]
-        simpa only [he,mul_zero] using hpos)
-  exact exists_operator_of_scaled_matrix F n _ hn' S hSnorm (by
-    intro x z
-    simpa only [Complex.ofReal_mul] using hS x z)
-
-end LocalEndpointRule
-end AmplificationLoadOperator
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-open MeasureTheory
-section TruncationOperator
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace CappedLocalLoad
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ}
-
-theorem truncate_kernel_le (q : CappedLocalLoad G hloc O M) (p : unitInterval)
-    (b : ℝ) (hb : 0 < b) (L : ℝ) (x y : V) :
-    (q.truncate b hb L).kernel p x y ≤ b⁻¹ * q.kernel p x y := by
-  rw [kernel,← integral_const_mul]
-  exact integral_mono ((q.truncate b hb L).integrable_mass p x y)
-    ((q.integrable_mass p x y).const_mul b⁻¹) (q.truncate_mass_le b hb L · x y)
-
- 
-
-theorem truncate_exists_operator (q : CappedLocalLoad G hloc O M) (p : unitInterval)
-    (b : ℝ) (hb : 0 < b) (L : ℝ)
-    (A : Hilbert V →L[ℂ] Hilbert V)
-    (hA : ∀ x y, matrixCoeff A x y = (q.kernel p x y : ℂ)) :
-    ∃ T : Hilbert V →L[ℂ] Hilbert V, ‖T‖ ≤ ‖A‖/b ∧
-      ∀ x y, matrixCoeff T x y = ((q.truncate b hb L).kernel p x y : ℂ) := by
-  have hscale (x y : V) : matrixCoeff (((b⁻¹:ℝ):ℂ) • A) x y = (b⁻¹ * q.kernel p x y : ℝ) := by
-    change ((b⁻¹:ℝ):ℂ)*matrixCoeff A x y = _
-    rw [hA,Complex.ofReal_mul]
-  obtain ⟨T,hT,hK⟩ := exists_operator_of_matrix_domination
-    (fun x y => b⁻¹ * q.kernel p x y) ((q.truncate b hb L).kernel p)
-    (fun x y => mul_nonneg (inv_nonneg.mpr hb.le) (q.kernel_nonneg p x y))
-    (fun x y => by
-      rw [abs_of_nonneg ((q.truncate b hb L).kernel_nonneg p x y)]
-      exact q.truncate_kernel_le p b hb L x y)
-    (((b⁻¹:ℝ):ℂ) • A) hscale
-  refine ⟨T,hT.trans_eq ?_,hK⟩
-  rw [norm_smul,Complex.norm_of_nonneg (inv_nonneg.mpr hb.le)]
-  ring
-
-end CappedLocalLoad
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
-
- 
-theorem normalize_mass_operator (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
-    (p : unitInterval) (hu : 0 < q.successProbability p o)
-    (A : Hilbert V →L[ℂ] Hilbert V) (C : ℝ) (hAnorm : ‖A‖ ≤ C)
-    (hA : ∀ x y, matrixCoeff A x y = (q.kernel p x y : ℂ)) :
-    ∃ T : Hilbert V →L[ℂ] Hilbert V, ‖T‖ ≤ C/q.successProbability p o ∧
-      ∀ x y, matrixCoeff T x y = (q.normalizedKernel o p x y : ℂ) := by
-  apply exists_operator_of_scaled_matrix (q.normalizedKernel o p) (q.successProbability p o) C hu A hAnorm
-  intro x y
-  rw [hA]
-  simp only [normalizedKernel,← Complex.ofReal_mul]
-  congr 1
-  rw [← mul_assoc,mul_inv_cancel₀ hu.ne',one_mul]
-
- 
-
-theorem mass_operator_of_normalized (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
-    (p : unitInterval) (hu : 0 < q.successProbability p o)
-    (A : Hilbert V →L[ℂ] Hilbert V)
-    (hA : ∀ x y, matrixCoeff A x y = (q.normalizedKernel o p x y : ℂ)) :
-    ∃ T : Hilbert V →L[ℂ] Hilbert V, ‖T‖ = q.successProbability p o * ‖A‖ ∧
-      ∀ x y, matrixCoeff T x y = (q.kernel p x y : ℂ) := by
-  refine ⟨(q.successProbability p o : ℂ) • A,?_,?_⟩
-  · rw [norm_smul,Complex.norm_of_nonneg hu.le]
-  · intro x y
-    change (q.successProbability p o : ℂ)*matrixCoeff A x y = _
-    rw [hA]
-    simp only [normalizedKernel,← Complex.ofReal_mul]
-    congr 1
-    rw [← mul_assoc,mul_inv_cancel₀ hu.ne',one_mul]
-
-end LocalEndpointRule
-end TruncationOperator
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-open MeasureTheory
-section AmplificationTruncationStep
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
-
-theorem normalized_operator_norm_pos (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
-    (p : unitInterval) (hu : 0 < q.successProbability p o)
-    (A : Hilbert V →L[ℂ] Hilbert V)
-    (hA : ∀ x y, matrixCoeff A x y = (q.normalizedKernel o p x y : ℂ)) : 0 < ‖A‖ := by
-  apply norm_pos_iff.mpr
-  intro ha
-  have hz (y : V) : q.kernel p o y = 0 := by
-    have he := hA o y
-    rw [ha] at he
-    have he' : (q.successProbability p o)⁻¹ * q.kernel p o y = 0 :=
-      Complex.ofReal_eq_zero.mp he.symm
-    exact (mul_eq_zero.mp he').resolve_left (inv_ne_zero hu.ne')
-  have he : q.successProbability p o = 0 := by
-    simp only [successProbability,hz,tsum_zero]
-  exact hu.ne' he
-
- 
-
-theorem amplification_truncation_step [Nonempty E] {ι : Type*} [Fintype ι]
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (htr : RootTransport root weight Γ)
-    (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b) (hw : ∀ i, 0 ≤ weight i)
-    (i₀ : ι) (hw₀ : 0 < weight i₀)
-    (q : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) M L) (hL : 1 ≤ L)
-    (p : unitInterval) (k n : ℕ) (hn : 0 < n)
-    (hu : 0 < q.successProbability p (root i₀))
-    (hbin : 0 < binOrbitRow G p (root i₀) n)
-    (b L' : ℝ) (hb : 0 < b)
-    (hmoment : (∫ ω, (q.amplificationLoad (root i₀) hL k n p).row ω (root i₀)^2 ∂law p) / b ≤
-      (q.successProbability p (root i₀)^k * binOrbitRow G p (root i₀) n / n)/4)
-    (hbad : b * (law p).real {ω | CappedBadGoodness G ω (root i₀) (2*n+k*M) L'} ≤
-      (q.successProbability p (root i₀)^k * binOrbitRow G p (root i₀) n / n)/4)
-    (A : Hilbert V →L[ℂ] Hilbert V)
-    (hA : ∀ x y, matrixCoeff A x y = (q.normalizedKernel (root i₀) p x y : ℂ)) :
-    ∃ q' : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) (2*n+k*M) L',
-      (q.successProbability p (root i₀)^k * binOrbitRow G p (root i₀) n / n)/(2*b) ≤
-        q'.successProbability p (root i₀) ∧
-      ∃ A' : Hilbert V →L[ℂ] Hilbert V, 0 < ‖A'‖ ∧ ‖A'‖ ≤ 2*‖A‖^k ∧
-        ∀ x y, matrixCoeff A' x y = (q'.normalizedKernel (root i₀) p x y : ℂ) := by
-  let load := q.amplificationLoad (root i₀) hL k n p
-  let q' := load.truncate b hb L'
-  have hn' : (0:ℝ) < n := Nat.cast_pos.mpr hn
-  have hb₀ : 0 < q.successProbability p (root i₀)^k * binOrbitRow G p (root i₀) n / n :=
-    div_pos (mul_pos (pow_pos hu _) hbin) hn'
-  have hself : root i₀ ∈ vertexOrbit G (root i₀) := by
-    refine ⟨Equiv.refl V,Equiv.refl E,?_,rfl⟩
-    intro e
-    simp
-  have hmean := q.amplificationLoad_mean root weight Γ htr hΓ hw i₀ hw₀ hL p k n (root i₀) hself
-  have hnew := load.truncate_success_lower p b hb L' (root i₀) _ hmean.ge hmoment hbad
-  have hu' : 0 < q'.successProbability p (root i₀) :=
-    (div_pos hb₀ (mul_pos (by norm_num) hb)).trans_le hnew
-  obtain ⟨B,hBnorm,hB⟩ := q.mass_operator_of_normalized (root i₀) p hu A hA
-  obtain ⟨C,hCnorm,hC⟩ := q.amplificationLoad_exists_operator (root i₀) hL k n hn p B hB
-  obtain ⟨D,hDnorm,hD⟩ := load.truncate_exists_operator p b hb L' C hC
-  have hD' : ‖D‖ ≤ (binOrbitRow G p (root i₀) n * (q.successProbability p (root i₀)*‖A‖)^k/n)/b := by
-    rw [hBnorm] at hCnorm
-    exact hDnorm.trans (div_le_div_of_nonneg_right hCnorm hb.le)
-  obtain ⟨A',hA'norm,hA'⟩ := q'.normalize_mass_operator (root i₀) p hu' D _ hD' hD
-  refine ⟨q',hnew,A',q'.normalized_operator_norm_pos (root i₀) p hu' A' hA',hA'norm.trans ?_,hA'⟩
-  apply (div_le_iff₀ hu').mpr
-  apply (div_le_iff₀ hb).mpr
-  apply (div_le_iff₀ hn').mpr
-  have hret := (div_le_iff₀ (mul_pos (show (0:ℝ)<2 by norm_num) hb)).mp hnew
-  have hret' := (div_le_iff₀ hn').mp hret
-  rw [mul_pow]
-  calc
-    _ = (q.successProbability p (root i₀)^k * binOrbitRow G p (root i₀) n) * ‖A‖^k := by ring
-    _ ≤ (q'.successProbability p (root i₀)*(2*b)*n)*‖A‖^k :=
-      mul_le_mul_of_nonneg_right hret' (pow_nonneg (norm_nonneg _) _)
-    _ = _ := by ring
-
-end LocalEndpointRule
-end AmplificationTruncationStep
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Splicing.EndpointMoments
+
+namespace OAI
+
+/-! Covariance for bond percolation. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+section FiniteTraceCovariance
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+@[simp] theorem edgeFieldEquiv_symm_edgeFieldEquiv (b : E ≃ E) (ω : Set E) :
+    edgeFieldEquiv b.symm (edgeFieldEquiv b ω) = ω := by
+  ext e
+  simp only [edgeFieldEquiv_apply,Set.mem_preimage,Equiv.symm_symm,Equiv.symm_apply_apply]
+
+@[simp] theorem spliceStateEquiv_symm_spliceStateEquiv (b : E ≃ E) (s : SpliceState E) :
+    spliceStateEquiv b.symm (spliceStateEquiv b s) = s := by
+  apply Prod.ext
+  · exact edgeFieldEquiv_symm_edgeFieldEquiv b s.1
+  · funext i
+    exact edgeFieldEquiv_symm_edgeFieldEquiv b (s.2 i)
+
+@[simp] theorem spliceCodeEquiv_symm_spliceCodeEquiv (b : E ≃ E) (c : List (ℕ × E)) :
+    spliceCodeEquiv b.symm (spliceCodeEquiv b c) = c := by
+  induction c with
+  | nil => rfl
+  | cons q c ih =>
+    rcases q with ⟨i,e⟩
+    simpa only [spliceCodeEquiv_apply,List.map_cons,Prod.map_apply,id_eq,Equiv.symm_apply_apply] using
+      congrArg (List.cons (i,e)) ih
+
+theorem spliceTrace_aut_map_iff (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (x : V) (η : ℕ → Set E) (y : ℕ → V) (k j : ℕ)
+    (ω ωf : Set E) (c : List (ℕ × E)) :
+    SpliceTrace G (a x) (fun i => edgeFieldEquiv b (η i)) (fun i => a (y i)) k j
+      (edgeFieldEquiv b ω) (spliceCodeEquiv b c) (edgeFieldEquiv b ωf) ↔
+    SpliceTrace G x η y k j ω c ωf := by
+  constructor
+  · intro ht
+    have hh := ht.aut_map G (isAut_symm G hab) (a x) (fun i => edgeFieldEquiv b (η i))
+      (fun i => a (y i)) k j (edgeFieldEquiv b ω) (edgeFieldEquiv b ωf) (spliceCodeEquiv b c)
+    simpa only [Equiv.symm_apply_apply,edgeFieldEquiv_symm_edgeFieldEquiv,
+      spliceCodeEquiv_symm_spliceCodeEquiv] using hh
+  · exact SpliceTrace.aut_map G hab x η y k j ω ωf c
+
+theorem boundedTraceCodes_aut_map_iff (b : E ≃ E) (Q Q' : Finset E)
+    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (k : ℕ) (c : List (ℕ × E)) :
+    spliceCodeEquiv b c ∈ boundedTraceCodes Q' k ↔ c ∈ boundedTraceCodes Q k := by
+  simp only [mem_boundedTraceCodes,spliceCodeEquiv_apply,List.length_map]
+  apply and_congr_right
+  intro _
+  constructor
+  · intro h q hq
+    have hh := h (q.1,b q.2) (List.mem_map.mpr ⟨q,hq,rfl⟩)
+    exact ⟨hh.1,hh.2.1,(hQ q.2).mpr hh.2.2⟩
+  · intro h r hr
+    obtain ⟨q,hq,rfl⟩ := List.mem_map.mp hr
+    exact ⟨(h q hq).1,(h q hq).2.1,(hQ q.2).mp (h q hq).2.2⟩
+
+theorem finiteTraceCodes_aut_map_iff (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (Q Q' : Finset E) (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q')
+    (x : V) (η : ℕ → Set E) (y : ℕ → V) (k j : ℕ) (ω : Set E) (c : List (ℕ × E)) :
+    spliceCodeEquiv b c ∈ finiteTraceCodes G (a x) (fun i => edgeFieldEquiv b (η i))
+      (fun i => a (y i)) k j (edgeFieldEquiv b ω) Q' ↔
+      c ∈ finiteTraceCodes G x η y k j ω Q := by
+  rw [mem_finiteTraceCodes_iff,mem_finiteTraceCodes_iff,boundedTraceCodes_aut_map_iff b Q Q' hQ,
+    replaySplice_aut_map G hab,spliceTrace_aut_map_iff G hab]
+
+ 
+def finiteTraceCodesEquiv (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (Q Q' : Finset E) (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q')
+    (x : V) (η : ℕ → Set E) (y : ℕ → V) (k j : ℕ) (ω : Set E) :
+    ↥(finiteTraceCodes G x η y k j ω Q) ≃
+      ↥(finiteTraceCodes G (a x) (fun i => edgeFieldEquiv b (η i)) (fun i => a (y i)) k j
+        (edgeFieldEquiv b ω) Q') :=
+  (spliceCodeEquiv b).subtypeEquiv (fun c => (finiteTraceCodes_aut_map_iff G hab Q Q' hQ x η y k j ω c).symm)
+
+end FiniteTraceCovariance
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+section FiniteIndexCovariance
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+def finiteTraceIndexEquiv (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (x : V) (k : ℕ) (Q Q' : Finset E) (A A' : Finset (SpliceState E))
+    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (hA : ∀ s, s ∈ A ↔ spliceStateEquiv b s ∈ A')
+    (y y' : SpliceState E → ℕ → V) (hy : ∀ s i, y' (spliceStateEquiv b s) i = a (y s i)) :
+    FiniteTraceIndex G x k Q A y ≃ FiniteTraceIndex G (a x) k Q' A' y' :=
+  Equiv.sigmaCongr ((spliceStateEquiv b).subtypeEquiv hA) (fun s =>
+    (spliceCodeEquiv b).subtypeEquiv (fun c => by
+      have hy' : y' (spliceStateEquiv b s.val) = fun i => a (y s.val i) := funext (hy s.val)
+      change c ∈ finiteTraceCodes G x s.val.2 (y s.val) k 0 s.val.1 Q ↔
+        spliceCodeEquiv b c ∈ finiteTraceCodes G (a x) (fun i => edgeFieldEquiv b (s.val.2 i))
+          (y' (spliceStateEquiv b s.val)) k 0 (edgeFieldEquiv b s.val.1) Q'
+      rw [hy']
+      exact (finiteTraceCodes_aut_map_iff G hab Q Q' hQ x s.val.2 (y s.val) k 0 s.val.1 c).symm))
+
+namespace FiniteTraceIndexEquiv
+variable (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (x : V) (k : ℕ) (Q Q' : Finset E) (A A' : Finset (SpliceState E))
+    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (hA : ∀ s, s ∈ A ↔ spliceStateEquiv b s ∈ A')
+    (y y' : SpliceState E → ℕ → V) (hy : ∀ s i, y' (spliceStateEquiv b s) i = a (y s i))
+
+@[simp] theorem input (t : FiniteTraceIndex G x k Q A y) :
+    traceIndexInput G (a x) k Q' A' y' (finiteTraceIndexEquiv G hab x k Q Q' A A' hQ hA y y' hy t) =
+      spliceStateEquiv b (traceIndexInput G x k Q A y t) := rfl
+
+@[simp] theorem code (t : FiniteTraceIndex G x k Q A y) :
+    traceIndexCode G (a x) k Q' A' y' (finiteTraceIndexEquiv G hab x k Q Q' A A' hQ hA y y' hy t) =
+      spliceCodeEquiv b (traceIndexCode G x k Q A y t) := rfl
+
+@[simp] theorem output (t : FiniteTraceIndex G x k Q A y) :
+    traceIndexOutput G (a x) k Q' A' y' (finiteTraceIndexEquiv G hab x k Q Q' A A' hQ hA y y' hy t) =
+      spliceStateEquiv b (traceIndexOutput G x k Q A y t) := by
+  have hy' : y' (spliceStateEquiv b t.1.val) = fun i => a (y t.1.val i) := funext (hy t.1.val)
+  change runMasks (replayMasks G (a x) (fun i => edgeFieldEquiv b (t.1.val.2 i))
+    (y' (spliceStateEquiv b t.1.val)) (spliceCodeEquiv b t.2.val) (edgeFieldEquiv b t.1.val.1))
+      (spliceStateEquiv b t.1.val) = _
+  rw [hy',replayMasks_aut_map G hab,runMasks_aut_map]
+  rfl
+
+@[simp] theorem mass (t : FiniteTraceIndex G x k Q A y) :
+    traceIndexMass G (a x) k Q' A' y' (finiteTraceIndexEquiv G hab x k Q Q' A A' hQ hA y y' hy t) =
+      traceIndexMass G x k Q A y t := by
+  unfold traceIndexMass
+  rw [output G hab x k Q Q' A A' hQ hA y y' hy t]
+  apply Finset.prod_congr rfl
+  intro i _
+  have hy' : y' (spliceStateEquiv b t.1.val) = fun i => a (y t.1.val i) := funext (hy t.1.val)
+  change (traceContactCount G (fun j => edgeFieldEquiv b (t.1.val.2 j))
+    (y' (spliceStateEquiv b t.1.val)) (edgeFieldEquiv b (traceIndexOutput G x k Q A y t).1)
+    (a x) (spliceCodeEquiv b t.2.val) (i.val+1):ℝ)⁻¹ = _
+  rw [hy',traceContactCount_aut_map G hab]
+
+end FiniteTraceIndexEquiv
+
+ 
+
+theorem finiteForwardLoad_aut_map (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (x : V) (k : ℕ) (Q Q' : Finset E) (A A' : Finset (SpliceState E))
+    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (hA : ∀ s, s ∈ A ↔ spliceStateEquiv b s ∈ A')
+    (y y' : SpliceState E → ℕ → V) (hy : ∀ s i, y' (spliceStateEquiv b s) i = a (y s i))
+    (s : SpliceState E) :
+    finiteForwardLoad G (a x) k Q' A' y' (spliceStateEquiv b s) = finiteForwardLoad G x k Q A y s := by
+  unfold finiteForwardLoad finiteInverseLoad
+  rw [← (finiteTraceIndexEquiv G hab x k Q Q' A A' hQ hA y y' hy).sum_comp]
+  apply Finset.sum_congr rfl
+  intro t _
+  rw [FiniteTraceIndexEquiv.output,FiniteTraceIndexEquiv.mass]
+  simp only [(spliceStateEquiv b).injective.eq_iff]
+
+end FiniteIndexCovariance
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+section FiniteFieldCovariance
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem finset_aut_eq_map (b : E ≃ E) (Q Q' : Finset E)
+    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') : Q' = Q.map b.toEmbedding := by
+  ext e
+  simpa only [Finset.mem_map_equiv,b.apply_symm_apply] using (hQ (b.symm e)).symm
+
+noncomputable def finiteFieldAutEquiv (b : E ≃ E) (Q Q' : Finset E)
+    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') : FiniteField Q ≃ FiniteField Q' :=
+  (Equiv.finsetCongr b).subtypeEquiv (fun ω => by
+    simp only [Finset.mem_powerset]
+    change ω ⊆ Q ↔ ω.map b.toEmbedding ⊆ Q'
+    rw [finset_aut_eq_map b Q Q' hQ,Finset.map_subset_map])
+
+@[simp] theorem finiteFieldSet_aut_map (b : E ≃ E) (Q Q' : Finset E)
+    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (ω : FiniteField Q) :
+    finiteFieldSet Q' (finiteFieldAutEquiv b Q Q' hQ ω) = edgeFieldEquiv b (finiteFieldSet Q ω) := by
+  change (ω.val.map b.toEmbedding : Set E) = b.symm ⁻¹' (ω.val : Set E)
+  rw [Finset.coe_map]
+  exact b.image_eq_preimage_symm _
+
+@[simp] theorem finiteFieldWeight_aut_map (b : E ≃ E) (Q Q' : Finset E)
+    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (p : ℝ) (ω : FiniteField Q) :
+    finiteFieldWeight Q' p (finiteFieldAutEquiv b Q Q' hQ ω) = finiteFieldWeight Q p ω := by
+  change p^(ω.val.map b.toEmbedding).card * (1-p)^(Q'.card-(ω.val.map b.toEmbedding).card) = _
+  simp only [Finset.card_map,finset_aut_eq_map b Q Q' hQ]
+  rfl
+
+noncomputable def finiteFamilyEquiv (b : E ≃ E) (Q Q' : Finset E)
+    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (J : Finset ℕ) :
+    (↥J → FiniteField Q) ≃ (↥J → FiniteField Q') :=
+  Equiv.piCongrRight fun _ => finiteFieldAutEquiv b Q Q' hQ
+
+@[simp] theorem finiteFieldFamily_aut_map (b : E ≃ E) (Q Q' : Finset E)
+    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (J : Finset ℕ) (η : ↥J → FiniteField Q) (i : ℕ) :
+    finiteFieldFamily Q' J (finiteFamilyEquiv b Q Q' hQ J η) i =
+      edgeFieldEquiv b (finiteFieldFamily Q J η i) := by
+  unfold finiteFieldFamily
+  split_ifs with hi
+  · exact finiteFieldSet_aut_map b Q Q' hQ (η ⟨i,hi⟩)
+  · rfl
+
+@[simp] theorem finiteFamilyWeight_aut_map (b : E ≃ E) (Q Q' : Finset E)
+    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (J : Finset ℕ) (p : ℝ) (η : ↥J → FiniteField Q) :
+    finiteFamilyWeight Q' J p (finiteFamilyEquiv b Q Q' hQ J η) = finiteFamilyWeight Q J p η := by
+  apply Finset.prod_congr rfl
+  intro i _
+  exact finiteFieldWeight_aut_map b Q Q' hQ p (η i)
+
+@[simp] theorem restrictFiniteField_aut_map (b : E ≃ E) (Q Q' : Finset E)
+    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (ω : Set E) :
+    restrictFiniteField Q' (edgeFieldEquiv b ω) = finiteFieldAutEquiv b Q Q' hQ (restrictFiniteField Q ω) := by
+  apply finiteFieldSet_injective Q'
+  rw [finiteFieldSet_aut_map,finiteFieldSet_restrict,finiteFieldSet_restrict]
+  ext e
+  change (b.symm e ∈ ω ∧ e ∈ Q') ↔ (b.symm e ∈ ω ∧ b.symm e ∈ Q)
+  rw [hQ (b.symm e),b.apply_symm_apply]
+
+ 
+
+theorem finiteRootLoad_aut_map (b : E ≃ E) (Q Q' : Finset E)
+    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (J : Finset ℕ) (p : ℝ)
+    (f f' : SpliceState E → ℝ) (hf : ∀ s, f' (spliceStateEquiv b s) = f s)
+    (ω : FiniteField Q) :
+    finiteRootLoad Q' J p f' (finiteFieldAutEquiv b Q Q' hQ ω) = finiteRootLoad Q J p f ω := by
+  unfold finiteRootLoad
+  rw [← (finiteFamilyEquiv b Q Q' hQ J).sum_comp]
+  apply Finset.sum_congr rfl
+  intro η _
+  rw [finiteFamilyWeight_aut_map,finiteFieldSet_aut_map]
+  have he : finiteFieldFamily Q' J (finiteFamilyEquiv b Q Q' hQ J η) =
+      fun i => edgeFieldEquiv b (finiteFieldFamily Q J η i) :=
+    funext (finiteFieldFamily_aut_map b Q Q' hQ J η)
+  rw [he]
+  exact congrArg (fun t => finiteFamilyWeight Q J p η * t)
+    (hf (finiteFieldSet Q ω,finiteFieldFamily Q J η))
+
+end FiniteFieldCovariance
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+section ChoiceIndexCovariance
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {P P' : Type*}
+
+def finiteChoiceTraceIndexEquiv (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (x : V) (k : ℕ) (Q Q' : Finset E)
+    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (e : P ≃ P')
+    (A : P → Finset (SpliceState E)) (A' : P' → Finset (SpliceState E))
+    (hA : ∀ r s, s ∈ A r ↔ spliceStateEquiv b s ∈ A' (e r))
+    (y : P → SpliceState E → ℕ → V) (y' : P' → SpliceState E → ℕ → V)
+    (hy : ∀ r s i, y' (e r) (spliceStateEquiv b s) i = a (y r s i)) :
+    FiniteChoiceTraceIndex G x k Q A y ≃ FiniteChoiceTraceIndex G (a x) k Q' A' y' :=
+  Equiv.sigmaCongr e (fun r => finiteTraceIndexEquiv G hab x k Q Q' (A r) (A' (e r)) hQ (hA r)
+    (y r) (y' (e r)) (hy r))
+
+namespace FiniteChoiceTraceIndexEquiv
+variable (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (x : V) (k : ℕ) (Q Q' : Finset E)
+    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (e : P ≃ P')
+    (A : P → Finset (SpliceState E)) (A' : P' → Finset (SpliceState E))
+    (hA : ∀ r s, s ∈ A r ↔ spliceStateEquiv b s ∈ A' (e r))
+    (y : P → SpliceState E → ℕ → V) (y' : P' → SpliceState E → ℕ → V)
+    (hy : ∀ r s i, y' (e r) (spliceStateEquiv b s) i = a (y r s i))
+
+@[simp] theorem input (t : FiniteChoiceTraceIndex G x k Q A y) :
+    choiceTraceInput G (a x) k Q' A' y' (finiteChoiceTraceIndexEquiv G hab x k Q Q' hQ e A A' hA y y' hy t) =
+      spliceStateEquiv b (choiceTraceInput G x k Q A y t) := rfl
+
+@[simp] theorem output (t : FiniteChoiceTraceIndex G x k Q A y) :
+    choiceTraceOutput G (a x) k Q' A' y' (finiteChoiceTraceIndexEquiv G hab x k Q Q' hQ e A A' hA y y' hy t) =
+      spliceStateEquiv b (choiceTraceOutput G x k Q A y t) :=
+  FiniteTraceIndexEquiv.output G hab x k Q Q' (A t.1) (A' (e t.1)) hQ (hA t.1)
+    (y t.1) (y' (e t.1)) (hy t.1) t.2
+
+@[simp] theorem mass (q : SpliceState E → P → ℝ) (q' : SpliceState E → P' → ℝ)
+    (hq : ∀ s r, q' (spliceStateEquiv b s) (e r) = q s r)
+    (t : FiniteChoiceTraceIndex G x k Q A y) :
+    choiceTraceMass G (a x) k Q' A' y' q'
+      (finiteChoiceTraceIndexEquiv G hab x k Q Q' hQ e A A' hA y y' hy t) =
+      choiceTraceMass G x k Q A y q t := by
+  unfold choiceTraceMass
+  rw [input G hab x k Q Q' hQ e A A' hA y y' hy]
+  change q' (spliceStateEquiv b (choiceTraceInput G x k Q A y t)) (e t.1) * _ = _
+  rw [hq]
+  exact congrArg (fun m => q (choiceTraceInput G x k Q A y t) t.1 * m)
+    (FiniteTraceIndexEquiv.mass G hab x k Q Q' (A t.1) (A' (e t.1)) hQ (hA t.1)
+      (y t.1) (y' (e t.1)) (hy t.1) t.2)
+
+end FiniteChoiceTraceIndexEquiv
+
+theorem finiteChoiceForwardLoad_aut_map [Fintype P] [Fintype P']
+    (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (x : V) (k : ℕ) (Q Q' : Finset E)
+    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (e : P ≃ P')
+    (A : P → Finset (SpliceState E)) (A' : P' → Finset (SpliceState E))
+    (hA : ∀ r s, s ∈ A r ↔ spliceStateEquiv b s ∈ A' (e r))
+    (y : P → SpliceState E → ℕ → V) (y' : P' → SpliceState E → ℕ → V)
+    (hy : ∀ r s i, y' (e r) (spliceStateEquiv b s) i = a (y r s i))
+    (q : SpliceState E → P → ℝ) (q' : SpliceState E → P' → ℝ)
+    (hq : ∀ s r, q' (spliceStateEquiv b s) (e r) = q s r) (s : SpliceState E) :
+    finiteChoiceForwardLoad G (a x) k Q' A' y' q' (spliceStateEquiv b s) =
+      finiteChoiceForwardLoad G x k Q A y q s := by
+  unfold finiteChoiceForwardLoad finiteInverseLoad
+  rw [← (finiteChoiceTraceIndexEquiv G hab x k Q Q' hQ e A A' hA y y' hy).sum_comp]
+  apply Finset.sum_congr rfl
+  intro t _
+  rw [FiniteChoiceTraceIndexEquiv.output,FiniteChoiceTraceIndexEquiv.mass G hab x k Q Q' hQ e A A' hA y y' hy q q' hq]
+  simp only [(spliceStateEquiv b).injective.eq_iff]
+
+end ChoiceIndexCovariance
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+section AmplificationCovarianceInput
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem edgeField_subset_aut_iff (b : E ≃ E) (Q Q' : Finset E)
+    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (ω : Set E) :
+    edgeFieldEquiv b ω ⊆ (Q' : Set E) ↔ ω ⊆ (Q : Set E) := by
+  constructor
+  · intro h e he
+    exact (hQ e).mpr (h (by simpa using he : b e ∈ edgeFieldEquiv b ω))
+  · intro h e he
+    have hh := (hQ (b.symm e)).mp (h he)
+    simpa only [b.apply_symm_apply,Finset.mem_coe] using hh
+
+theorem finiteSpliceDomain_aut_map_iff (b : E ≃ E) (Q Q' : Finset E)
+    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (J : Finset ℕ) (s : SpliceState E) :
+    spliceStateEquiv b s ∈ finiteSpliceDomain Q' J ↔ s ∈ finiteSpliceDomain Q J := by
+  rw [mem_finiteSpliceDomain_iff,mem_finiteSpliceDomain_iff]
+  change (edgeFieldEquiv b s.1 ⊆ (Q':Set E) ∧
+    (∀ i,edgeFieldEquiv b (s.2 i) ⊆ (Q':Set E)) ∧ ∀ i ∉ J,edgeFieldEquiv b (s.2 i)=∅) ↔ _
+  have hz (η : Set E) : edgeFieldEquiv b η = ∅ ↔ η = ∅ := by
+    change edgeFieldEquiv b η = edgeFieldEquiv b ∅ ↔ _
+    exact (edgeFieldEquiv b).injective.eq_iff
+  simp only [edgeField_subset_aut_iff b Q Q' hQ,hz]
+
+theorem oldRunVertices_map {W W' : Type*} (f : W → W') (k : ℕ) (z : W)
+    (r : Fin k → W) (i : ℕ) :
+    oldRunVertices k (f z) (fun j => f (r j)) i = f (oldRunVertices k z r i) := by
+  induction k generalizing z with
+  | zero => rfl
+  | succ k ih =>
+    simp only [oldRunVertices]
+    split_ifs with hi
+    · exact ih (r 0) (Fin.tail r)
+    · rfl
+
+theorem oldRunWeight_map {W W' : Type*} (f : W → W')
+    (K : ℕ → W → W → ℝ) (K' : ℕ → W' → W' → ℝ)
+    (hK : ∀ i a b, K' i (f a) (f b) = K i a b) (k : ℕ) (z : W) (r : Fin k → W) :
+    oldRunWeight K' k (f z) (fun i => f (r i)) = oldRunWeight K k z r := by
+  induction k generalizing z with
+  | zero => rfl
+  | succ k ih =>
+    simp only [oldRunWeight,hK]
+    rw [show Fin.tail (fun i => f (r i)) = (fun i => f (Fin.tail r i)) from rfl,ih]
+
+def finiteVertexEquiv (a : V ≃ V) (B B' : Finset V) (hB : ∀ x, x ∈ B ↔ a x ∈ B') : ↥B ≃ ↥B' :=
+  a.subtypeEquiv hB
+
+def finiteRunEquiv (a : V ≃ V) (B B' : Finset V) (hB : ∀ x, x ∈ B ↔ a x ∈ B') (k : ℕ) :
+    (Fin k → ↥B) ≃ (Fin k → ↥B') :=
+  Equiv.piCongrRight fun _ => finiteVertexEquiv a B B' hB
+
+theorem amplificationVertices_aut_map (a : V ≃ V) (B B' : Finset V)
+    (hB : ∀ x, x ∈ B ↔ a x ∈ B') (k : ℕ) (z : ↥B) (r : Fin k → ↥B) (i : ℕ) :
+    amplificationVertices B' k (finiteVertexEquiv a B B' hB z) (finiteRunEquiv a B B' hB k r) i =
+      a (amplificationVertices B k z r i) :=
+  congrArg Subtype.val (oldRunVertices_map (finiteVertexEquiv a B B' hB) k z r i)
+
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
+
+theorem mass_aut_map (q : LocalEndpointRule G hloc O M L) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (ω : Set E) (x y : V) :
+    q.mass (edgeFieldEquiv b ω) (a x) (a y) = q.mass ω x y := by
+  simpa only [Equiv.symm_apply_apply,edgeFieldEquiv_apply] using
+    q.covariance a.symm b.symm (isAut_symm G hab) ω (a x) (a y)
+
+theorem amplificationWeight_aut_map (q : LocalEndpointRule G hloc O M L) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (B B' : Finset V) (hB : ∀ x, x ∈ B ↔ a x ∈ B')
+    (k : ℕ) (z : ↥B) (s : SpliceState E) (r : Fin k → ↥B) :
+    q.amplificationWeight B' k (finiteVertexEquiv a B B' hB z) (spliceStateEquiv b s)
+      (finiteRunEquiv a B B' hB k r) = q.amplificationWeight B k z s r := by
+  apply oldRunWeight_map (finiteVertexEquiv a B B' hB)
+  intro i v w
+  exact q.mass_aut_map hab (s.2 i) v.val w.val
+
+theorem amplificationInputs_aut_map_iff (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
+    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (Q Q' : Finset E)
+    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (B B' : Finset V) (hB : ∀ x, x ∈ B ↔ a x ∈ B')
+    (x : V) (k n : ℕ) (z : ↥B) (r : Fin k → ↥B) (s : SpliceState E) :
+    spliceStateEquiv b s ∈ q.amplificationInputs Q' B' (a x) k n
+      (finiteVertexEquiv a B B' hB z) (finiteRunEquiv a B B' hB k r) ↔
+      s ∈ q.amplificationInputs Q B x k n z r := by
+  rw [q.mem_amplificationInputs,q.mem_amplificationInputs]
+  have hn : (cluster G (edgeFieldEquiv b s.1) (a x)).ncard = (cluster G s.1 x).ncard := by
+    rw [cluster_edgeFieldEquiv G hab,← a.image_eq_preimage_symm,Set.ncard_image_of_injective _ a.injective]
+  change (spliceStateEquiv b s ∈ finiteSpliceDomain Q' (Finset.Icc 1 k) ∧
+    a x ∈ vertexOrbit G o ∧ a z.val ∈ vertexOrbit G o ∧
+    n ≤ (cluster G (edgeFieldEquiv b s.1) (a x)).ncard ∧
+    (cluster G (edgeFieldEquiv b s.1) (a x)).ncard < 2*n ∧
+    amplificationVertices B' k (finiteVertexEquiv a B B' hB z) (finiteRunEquiv a B B' hB k r) 0 ∈
+      cluster G (edgeFieldEquiv b s.1) (a x) ∧ _) ↔ _
+  rw [finiteSpliceDomain_aut_map_iff b Q Q' hQ,mem_vertexOrbit_aut_iff G o hab,
+    mem_vertexOrbit_aut_iff G o hab,hn,amplificationVertices_aut_map,
+    cluster_edgeFieldEquiv G hab,q.amplificationWeight_aut_map hab B B' hB]
+  simp only [Set.mem_preimage,Equiv.symm_apply_apply]
+
+end LocalEndpointRule
+end AmplificationCovarianceInput
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+section AmplificationCovarianceMarginal
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
+
+ 
+
+theorem amplificationMarginal_aut_map (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
+    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (Q Q' : Finset E)
+    (hQ : ∀ e, e ∈ Q ↔ b e ∈ Q') (B B' : Finset V) (hB : ∀ x, x ∈ B ↔ a x ∈ B')
+    (x : V) (k n : ℕ) (z : ↥B) (p : unitInterval) (ω : Set E) :
+    q.amplificationMarginal Q' B' (a x) k n (finiteVertexEquiv a B B' hB z) p (edgeFieldEquiv b ω) =
+      q.amplificationMarginal Q B x k n z p ω := by
+  unfold amplificationMarginal
+  rw [restrictFiniteField_aut_map b Q Q' hQ]
+  apply finiteRootLoad_aut_map
+  intro s
+  apply finiteChoiceForwardLoad_aut_map G hab x k Q Q' hQ (finiteRunEquiv a B B' hB k)
+  · intro r s
+    exact (q.amplificationInputs_aut_map_iff o hab Q Q' hQ B B' hB x k n z r s).symm
+  · intro r s i
+    exact amplificationVertices_aut_map a B B' hB k z r i
+  · intro s r
+    exact q.amplificationWeight_aut_map hab B B' hB k z s r
+
+end LocalEndpointRule
+
+theorem mem_incidentSupport_aut_iff (G : BondGraph V E) (hloc : LocallyFinite G)
+    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (B B' : Finset V)
+    (hB : ∀ x, x ∈ B ↔ a x ∈ B') (e : E) :
+    b e ∈ incidentSupport G hloc B' ↔ e ∈ incidentSupport G hloc B := by
+  rw [finset_aut_eq_map a B B' hB,incidentSupport_aut G hloc hab]
+  simp only [Finset.mem_map_equiv,Equiv.symm_apply_apply]
+
+theorem finiteBall_aut_iff (G : BondGraph V E) (hloc : LocallyFinite G)
+    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (x y : V) (R : ℕ) :
+    y ∈ (finite_reachBall hloc x R).toFinset ↔ a y ∈ (finite_reachBall hloc (a x) R).toFinset := by
+  simpa only [Set.Finite.mem_toFinset] using (mem_reachBall_aut_iff G hab x y R).symm
+
+theorem localBondBall_aut_iff (G : BondGraph V E) (hloc : LocallyFinite G)
+    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (x : V) (R : ℕ) (e : E) :
+    e ∈ incidentSupport G hloc (finite_reachBall hloc x R).toFinset ↔
+      b e ∈ incidentSupport G hloc (finite_reachBall hloc (a x) R).toFinset :=
+  (mem_incidentSupport_aut_iff G hloc hab _ _ (fun y => finiteBall_aut_iff G hloc hab x y R) e).symm
+
+end AmplificationCovarianceMarginal
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+open MeasureTheory
+section CanonicalAmplificationLoad
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
+
+def amplificationRadius (q : LocalEndpointRule G hloc O M L) (k n : ℕ) : ℕ :=
+  2*n+k*M+max q.radius M
+
+noncomputable def amplificationBall (_q : LocalEndpointRule G hloc O M L) (k n : ℕ) (x : V) : Finset V :=
+  (finite_reachBall hloc x (2*n+k*M)).toFinset
+
+noncomputable def amplificationBonds (q : LocalEndpointRule G hloc O M L) (k n : ℕ) (x : V) : Finset E :=
+  incidentSupport G hloc (finite_reachBall hloc x (q.amplificationRadius k n)).toFinset
+
+ 
+
+noncomputable def amplifiedMass (q : LocalEndpointRule G hloc O M L) (k n : ℕ) (p : unitInterval)
+    (ω : Set E) (x z : V) : ℝ :=
+  if hz : z ∈ q.amplificationBall k n x then
+    q.amplificationMarginal (q.amplificationBonds k n x) (q.amplificationBall k n x) x k n ⟨z,hz⟩ p ω / n
+  else 0
+
+theorem amplifiedMass_nonneg (q : LocalEndpointRule G hloc O M L) (hL : 1 ≤ L)
+    (k n : ℕ) (p : unitInterval) (ω : Set E) (x z : V) : 0 ≤ q.amplifiedMass k n p ω x z := by
+  unfold amplifiedMass
+  split_ifs
+  · exact div_nonneg (q.amplificationMarginal_bounds hL _ _ _ _ _ _ _ _).1 (Nat.cast_nonneg _)
+  · exact le_rfl
+
+theorem amplifiedMass_support (q : LocalEndpointRule G hloc O M L)
+    (k n : ℕ) (p : unitInterval) (ω : Set E) (x z : V) (h : q.amplifiedMass k n p ω x z ≠ 0) :
+    x ∈ O ∧ z ∈ O ∧ Conn G ω x z ∧ (cluster G ω x).Finite ∧ (cluster G ω x).ncard ≤ 2*n+k*M := by
+  unfold amplifiedMass at h
+  split_ifs at h with hz
+  · apply q.amplificationMarginal_physical_support (q.amplificationBonds k n x) (q.amplificationBall k n x) x k n ⟨z,hz⟩ p ω
+    · exact incidentSupport_ball_mono G hloc x (Nat.le_add_right (2*n+k*M) (max q.radius M))
+    · intro he
+      exact h (by rw [he,zero_div])
+  · exact (h rfl).elim
+
+theorem amplifiedMass_local (q : LocalEndpointRule G hloc O M L)
+    (k n : ℕ) (p : unitInterval) (ω : Set E) (x z : V) :
+    q.amplifiedMass k n p ω x z =
+      q.amplifiedMass k n p (ω ∩ (q.amplificationBonds k n x : Set E)) x z := by
+  unfold amplifiedMass
+  split_ifs
+  · exact congrArg (fun t : ℝ => t / n) (q.amplificationMarginal_local _ _ _ _ _ _ _ _)
+  · rfl
+
+theorem amplifiedMass_aut_map (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
+    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (k n : ℕ) (p : unitInterval)
+    (ω : Set E) (x z : V) :
+    q.amplifiedMass k n p (edgeFieldEquiv b ω) (a x) (a z) = q.amplifiedMass k n p ω x z := by
+  have hB (y : V) : y ∈ q.amplificationBall k n x ↔ a y ∈ q.amplificationBall k n (a x) :=
+    finiteBall_aut_iff G hloc hab x y (2*n+k*M)
+  have hQ (e : E) : e ∈ q.amplificationBonds k n x ↔ b e ∈ q.amplificationBonds k n (a x) :=
+    localBondBall_aut_iff G hloc hab x (q.amplificationRadius k n) e
+  unfold amplifiedMass
+  by_cases hz : z ∈ q.amplificationBall k n x
+  · simp only [dite_eq_left hz,dite_eq_left ((hB z).mp hz)]
+    exact congrArg (fun t : ℝ => t / n)
+      (q.amplificationMarginal_aut_map o hab _ _ hQ _ _ hB x k n ⟨z,hz⟩ p ω)
+  · simp only [dite_eq_right hz,dite_eq_right (mt (hB z).mpr hz)]
+
+ 
+
+noncomputable def amplificationLoad (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
+    (hL : 1 ≤ L) (k n : ℕ) (p : unitInterval) :
+    CappedLocalLoad G hloc (vertexOrbit G o) (2*n+k*M) where
+  mass := q.amplifiedMass k n p
+  radius := q.amplificationRadius k n
+  nonneg := q.amplifiedMass_nonneg hL k n p
+  support := q.amplifiedMass_support k n p
+  locality := q.amplifiedMass_local k n p
+  covariance := by
+    intro a b hab ω x z
+    simpa only [Equiv.symm_apply_apply,edgeFieldEquiv_apply,Equiv.symm_symm] using
+      q.amplifiedMass_aut_map o (isAut_symm G hab) k n p ω (a x) (a z)
+
+end LocalEndpointRule
+end CanonicalAmplificationLoad
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+open MeasureTheory
+section CanonicalAmplificationRow
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
+
+theorem amplificationBonds_query {O : Set V} (q : LocalEndpointRule G hloc O M L)
+    (k n : ℕ) (x a : V) (ha : a ∈ q.amplificationBall k n x) :
+    incidentSupport G hloc (finite_reachBall hloc a (max q.radius M)).toFinset ⊆
+      q.amplificationBonds k n x := by
+  apply incidentSupport_mono
+  intro y hy
+  simp only [Set.Finite.mem_toFinset] at hy ⊢
+  exact reachBall_triangle G x a (2*n+k*M) (max q.radius M)
+    (by simpa only [amplificationBall,Set.Finite.mem_toFinset] using ha) hy
+
+theorem amplificationBonds_root {O : Set V} (q : LocalEndpointRule G hloc O M L)
+    (k n : ℕ) (x : V) :
+    incidentSupport G hloc (finite_reachBall hloc x (2*n)).toFinset ⊆
+      q.amplificationBonds k n x := by
+  apply incidentSupport_ball_mono
+  unfold amplificationRadius
+  omega
+
+theorem amplificationLoad_row (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
+    (hL : 1 ≤ L) (k n : ℕ) (p : unitInterval) (ω : Set E) (x : V) :
+    (q.amplificationLoad o hL k n p).row ω x =
+      q.amplificationRow (q.amplificationBonds k n x) (q.amplificationBall k n x) x k n p ω / n := by
+  change (∑' z, q.amplifiedMass k n p ω x z) = _
+  rw [tsum_eq_sum (s := q.amplificationBall k n x) (fun z hz => by
+    simp only [amplifiedMass,dite_eq_right hz])]
+  rw [← Finset.sum_coe_sort]
+  simp only [amplifiedMass,dite_eq_left (Subtype.prop _)]
+  exact (Finset.sum_div _ _ _).symm
+
+theorem amplificationRow_local {O : Set V} (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (p : unitInterval) (ω : Set E) :
+    q.amplificationRow Q B x k n p ω = q.amplificationRow Q B x k n p (ω ∩ (Q : Set E)) := by
+  unfold amplificationRow
+  apply Finset.sum_congr rfl
+  intro z _
+  exact q.amplificationMarginal_local Q B x k n z p ω
+
+theorem amplificationLoad_mean_finite (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
+    (hL : 1 ≤ L) (k n : ℕ) (p : unitInterval) (x : V) :
+    (∫ ω, (q.amplificationLoad o hL k n p).row ω x ∂law p) =
+      finiteFieldIntegral (q.amplificationBonds k n x) p
+        (q.amplificationRow (q.amplificationBonds k n x) (q.amplificationBall k n x) x k n p) / n := by
+  simp_rw [q.amplificationLoad_row o hL k n p]
+  rw [integral_div]
+  congr 1
+  exact integral_finite_local _ p _ (q.amplificationRow_local _ _ x k n p)
+
+theorem amplificationLoad_second_moment_finite (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
+    (hL : 1 ≤ L) (k n : ℕ) (p : unitInterval) (x : V) :
+    (∫ ω, (q.amplificationLoad o hL k n p).row ω x ^ 2 ∂law p) =
+      finiteFieldIntegral (q.amplificationBonds k n x) p (fun ω =>
+        q.amplificationRow (q.amplificationBonds k n x) (q.amplificationBall k n x) x k n p ω ^ 2) / (n:ℝ)^2 := by
+  simp_rw [q.amplificationLoad_row o hL k n p,div_pow]
+  rw [integral_div]
+  congr 1
+  exact integral_finite_local _ p _ (fun ω => congrArg (·^2) (q.amplificationRow_local _ _ x k n p ω))
+
+end LocalEndpointRule
+end CanonicalAmplificationRow
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+open MeasureTheory
+section NormalizedAmplificationMoment
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
+
+ 
+
+theorem amplificationLoad_second_moment_le [Nonempty E] (hconn : Connected G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
+    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (i₀ : ι) (hw₀ : 0 < weight i₀)
+    (q : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) M L) (hL : 1 ≤ L)
+    (p : unitInterval) (x : V) (k n d : ℕ) (hn : 0 < n)
+    (hdeg : ∀ a, degree G a ≤ d) :
+    (∫ ω, (q.amplificationLoad (root i₀) hL k n p).row ω x ^ 2 ∂law p) ≤
+      (4 * q.successProbability p (root i₀)^k +
+        (((k*M)+(k*M)^2+d*(k*M)^3:ℕ):ℝ) * inverseCodeMass M L^k / n) *
+          (∫ ω, (q.amplificationLoad (root i₀) hL k n p).row ω x ∂law p) +
+      (2 * (c*((k*M:ℕ)*q.successProbability p (root i₀)^k)))^2 *
+        finiteClusterWindowMass G (q.amplificationBonds k n x) p x (n-k*M) (2*n) := by
+  rw [q.amplificationLoad_second_moment_finite,q.amplificationLoad_mean_finite]
+  have h := q.amplificationRow_second_moment_le hconn root weight Γ hΓ htr hcov hw hw1 c hc hmin
+    i₀ hw₀ hL (q.amplificationBonds k n x) (q.amplificationBall k n x) p x k n d hdeg
+    (fun a ha => q.amplificationBonds_query k n x a ha)
+  have hn' : (n:ℝ) ≠ 0 := ne_of_gt (Nat.cast_pos.mpr hn)
+  apply (div_le_div_of_nonneg_right h (sq_nonneg (n:ℝ))).trans_eq
+  push_cast
+  field_simp
+
+end LocalEndpointRule
+end NormalizedAmplificationMoment
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+section FinitePathExhaustion
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def finitePathTest (B : Finset V) (K : V → V → ℝ) (F : V → ℝ) (k : ℕ) (z : ↥B) : ℝ :=
+  ∑ r : Fin k → ↥B, oldRunWeight (fun _ a b => K a.val b.val) k z r * F (oldRunVertices k z r 0).val
+
+@[simp] theorem finitePathTest_zero (B : Finset V) (K : V → V → ℝ) (F : V → ℝ) (z : ↥B) :
+    finitePathTest B K F 0 z = F z.val := by
+  simp [finitePathTest,oldRunWeight,oldRunVertices]
+
+theorem finitePathTest_succ (B : Finset V) (K : V → V → ℝ) (F : V → ℝ) (k : ℕ) (z : ↥B) :
+    finitePathTest B K F (k+1) z = ∑ a : ↥B, K z.val a.val * finitePathTest B K F k a :=
+  oldRun_test_succ (fun _ (a b : ↥B) => K a.val b.val) (fun (a : ↥B) => F a.val) k z
+
+ 
+
+theorem finitePathTest_support (G : BondGraph V E) (B : Finset V) (K : V → V → ℝ) (F : V → ℝ)
+    (O : Set V) (x : V) (R M : ℕ)
+    (hK : ∀ a b, K a b ≠ 0 → a ∈ O ∧ a ∈ reachBall G b M)
+    (hF : ∀ a, F a ≠ 0 → a ∈ O ∧ a ∈ reachBall G x R)
+    (k : ℕ) (z : ↥B) (hz : finitePathTest B K F k z ≠ 0) :
+    z.val ∈ O ∧ z.val ∈ reachBall G x (R+k*M) := by
+  induction k generalizing z with
+  | zero => simpa using hF z.val (by simpa using hz)
+  | succ k ih =>
+    rw [finitePathTest_succ] at hz
+    obtain ⟨a,_,ha⟩ := Finset.exists_ne_zero_of_sum_ne_zero hz
+    have hh := mul_ne_zero_iff.mp ha
+    obtain ⟨haO,haB⟩ := ih a hh.2
+    obtain ⟨hzO,hzB⟩ := hK z.val a.val hh.1
+    refine ⟨hzO,?_⟩
+    have hg := reachBall_triangle G x a.val (R+k*M) M haB hzB
+    simpa only [Nat.add_mul,one_mul,Nat.add_assoc] using hg
+
+ 
+
+theorem finitePathTest_total (G : BondGraph V E) (B : Finset V) (K : V → V → ℝ) (F : V → ℝ)
+    (O : Set V) (x : V) (R M N : ℕ) (u : ℝ)
+    (hK : ∀ a b, K a b ≠ 0 → a ∈ O ∧ a ∈ reachBall G b M)
+    (hF : ∀ a, F a ≠ 0 → a ∈ O ∧ a ∈ reachBall G x R)
+    (hcolumn : ∀ j < N, ∀ a : ↥B, a.val ∈ O → a.val ∈ reachBall G x (R+j*M) →
+      ∑ z : ↥B, K z.val a.val = u) :
+    ∀ j ≤ N, (∑ z : ↥B, finitePathTest B K F j z) = u^j * ∑ z : ↥B, F z.val := by
+  intro j hj
+  induction j with
+  | zero => simp
+  | succ j ih =>
+    simp_rw [finitePathTest_succ]
+    rw [Finset.sum_comm]
+    have he : (∑ a : ↥B, ∑ z : ↥B, K z.val a.val * finitePathTest B K F j a) =
+        u * ∑ a : ↥B, finitePathTest B K F j a := by
+      rw [Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro a _
+      rw [← Finset.sum_mul]
+      by_cases ha : finitePathTest B K F j a = 0
+      · simp only [ha,mul_zero]
+      · obtain ⟨haO,haB⟩ := finitePathTest_support G B K F O x R M hK hF j a ha
+        rw [hcolumn j (by omega) a haO haB]
+    rw [he,ih (by omega)]
+    ring
+
+end FinitePathExhaustion
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+open MeasureTheory
+section AmplificationBoundaryExhaustion
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
+
+theorem kernel_range_support {O : Set V} (q : LocalEndpointRule G hloc O M L)
+    (p : unitInterval) (a b : V) (h : q.kernel p a b ≠ 0) : a ∈ O ∧ a ∈ reachBall G b M := by
+  constructor
+  · by_contra ha
+    exact h (q.kernel_zero_of_start_not_mem p a b ha)
+  · by_contra ha
+    exact h (q.kernel_outside_reverse_ball_zero p a b ha)
+
+theorem orbitBinConnection_range_support (o : V) (p : unitInterval) (n : ℕ) (x a : V)
+    (h : orbitBinConnection G p o n x a ≠ 0) :
+    a ∈ vertexOrbit G o ∧ a ∈ reachBall G x (2*n) := by
+  have ha : a ∈ vertexOrbit G o := by
+    by_contra ha
+    exact h (by simp only [orbitBinConnection,ha,and_false,ite_false])
+  refine ⟨ha,?_⟩
+  by_contra hb
+  exact h (by simp only [orbitBinConnection,binConnection_outside G p n x a hb,ite_self])
+
+theorem amplificationBall_column {O : Set V} (q : LocalEndpointRule G hloc O M L)
+    (p : unitInterval) (k n j : ℕ) (hj : j < k) (x a : V)
+    (ha : a ∈ reachBall G x (2*n+j*M)) :
+    (∑ z : ↥(q.amplificationBall k n x), q.kernel p z.val a) = ∑' z, q.kernel p z a := by
+  rw [Finset.sum_coe_sort (q.amplificationBall k n x) (fun z => q.kernel p z a)]
+  symm
+  apply tsum_eq_sum
+  intro z hz
+  apply q.kernel_outside_reverse_ball_zero
+  intro hza
+  have hh := reachBall_triangle G x a (2*n+j*M) M ha hza
+  have hle : (2*n+j*M)+M ≤ 2*n+k*M := by
+    calc
+      _ = 2*n+(j+1)*M := by ring
+      _ ≤ _ := Nat.add_le_add_left (Nat.mul_le_mul_right M (by omega)) _
+  exact hz (by
+    simpa only [amplificationBall,Set.Finite.mem_toFinset] using (reachBall_mono G x hle hh))
+
+theorem amplificationPath_total {ι : Type*} [Fintype ι]
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (htr : RootTransport root weight Γ)
+    (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b) (hw : ∀ i, 0 ≤ weight i)
+    (i₀ : ι) (hw₀ : 0 < weight i₀)
+    (q : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) M L)
+    (p : unitInterval) (k n : ℕ) (x : V) (hx : x ∈ vertexOrbit G (root i₀)) :
+    (∑ z : ↥(q.amplificationBall k n x),
+      finitePathTest (q.amplificationBall k n x) (q.kernel p) (orbitBinConnection G p (root i₀) n x) k z) =
+      q.successProbability p (root i₀)^k * binOrbitRow G p (root i₀) n := by
+  have ht := finitePathTest_total G (q.amplificationBall k n x) (q.kernel p)
+    (orbitBinConnection G p (root i₀) n x) (vertexOrbit G (root i₀)) x (2*n) M k
+    (q.successProbability p (root i₀)) (q.kernel_range_support p)
+    (orbitBinConnection_range_support (root i₀) p n x) (by
+      intro j hj a haO haB
+      rw [q.amplificationBall_column p k n j hj x a.val haB]
+      exact q.kernel_column_eq_success root weight Γ htr hΓ hw i₀ hw₀ p a.val haO) k le_rfl
+  rw [ht]
+  congr 1
+  rw [Finset.sum_coe_sort (q.amplificationBall k n x) (orbitBinConnection G p (root i₀) n x),← orbitBinConnection_row G hloc p (root i₀) n x hx]
+  symm
+  apply tsum_eq_sum
+  intro z hz
+  by_contra h
+  have hb := (orbitBinConnection_range_support (root i₀) p n x z h).2
+  exact hz (by
+    simpa only [amplificationBall,Set.Finite.mem_toFinset] using
+      reachBall_mono G x (Nat.le_add_right (2*n) (k*M)) hb)
+
+end LocalEndpointRule
+end AmplificationBoundaryExhaustion
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+open MeasureTheory
+section AmplificationExactMean
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
+
+theorem amplificationMarginal_mean_pathTest [Nonempty E] (o : V)
+    (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (z : ↥B) (p : unitInterval)
+    (hx : x ∈ vertexOrbit G o)
+    (hroot : incidentSupport G hloc (finite_reachBall hloc x (2*n)).toFinset ⊆ Q)
+    (hquery : ∀ a ∈ B, incidentSupport G hloc (finite_reachBall hloc a q.radius).toFinset ⊆ Q) :
+    (∫ ω, q.amplificationMarginal Q B x k n z p ω ∂law p) =
+      finitePathTest B (q.kernel p) (orbitBinConnection G p o n x) k z := by
+  by_cases hz : z.val ∈ vertexOrbit G o
+  · rw [q.amplificationMarginal_mean_kernel Q B x k n z p hx hz hquery]
+    unfold finitePathTest
+    apply Finset.sum_congr rfl
+    intro r _
+    rw [finiteBinConnection_eq_binConnection G hloc Q p x n _ hroot]
+    by_cases hr : oldRunWeight (fun _ (a b : ↥B) => q.kernel p a.val b.val) k z r = 0
+    · simp only [hr,mul_zero,zero_mul]
+    · have hy : (oldRunVertices k z r 0).val ∈ vertexOrbit G o :=
+        oldRun_endpoint_mem (fun a : ↥B => a.val ∈ vertexOrbit G o)
+          (fun _ (a b : ↥B) => q.kernel p a.val b.val)
+          (fun _ a b hab => by
+            by_contra hb
+            exact hab (q.kernel_zero_of_end_not_mem p a.val b.val hb)) k z hz r hr
+      rw [orbitBinConnection,ite_eq_left ⟨hx,hy⟩]
+      exact mul_comm _ _
+  · have hm (ω : Set E) : q.amplificationMarginal Q B x k n z p ω = 0 := by
+      by_contra h
+      exact hz (q.amplificationMarginal_support Q B x k n z p ω h).2.1
+    simp_rw [hm,integral_zero]
+    symm
+    by_contra h
+    exact hz (finitePathTest_support G B (q.kernel p) (orbitBinConnection G p o n x)
+      (vertexOrbit G o) x (2*n) M (q.kernel_range_support p)
+      (orbitBinConnection_range_support o p n x) k z h).1
+
+ 
+
+theorem amplificationLoad_mean [Nonempty E] {ι : Type*} [Fintype ι]
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (htr : RootTransport root weight Γ)
+    (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b) (hw : ∀ i, 0 ≤ weight i)
+    (i₀ : ι) (hw₀ : 0 < weight i₀)
+    (q : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) M L) (hL : 1 ≤ L)
+    (p : unitInterval) (k n : ℕ) (x : V) (hx : x ∈ vertexOrbit G (root i₀)) :
+    (∫ ω, (q.amplificationLoad (root i₀) hL k n p).row ω x ∂law p) =
+      q.successProbability p (root i₀)^k * binOrbitRow G p (root i₀) n / n := by
+  simp_rw [q.amplificationLoad_row]
+  rw [integral_div]
+  congr 1
+  unfold amplificationRow
+  rw [integral_finsetSum _ (fun z _ => integrable_finite_local _ p _
+    (q.amplificationMarginal_local _ _ x k n z p))]
+  simp_rw [q.amplificationMarginal_mean_pathTest (root i₀) _ _ x k n _ p hx
+    (q.amplificationBonds_root k n x) (fun a ha =>
+      (incidentSupport_ball_mono G hloc a (le_max_left _ _)).trans (q.amplificationBonds_query k n x a ha))]
+  exact q.amplificationPath_total root weight Γ htr hΓ hw i₀ hw₀ p k n x hx
+
+end LocalEndpointRule
+end AmplificationExactMean
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+open MeasureTheory
+section AmplificationLoadOperator
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace CappedLocalLoad
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ}
+
+noncomputable def kernel (q : CappedLocalLoad G hloc O M) (p : unitInterval) (x y : V) : ℝ :=
+  ∫ ω, q.mass ω x y ∂law p
+
+theorem kernel_nonneg (q : CappedLocalLoad G hloc O M) (p : unitInterval) (x y : V) :
+    0 ≤ q.kernel p x y := integral_nonneg (q.nonneg · x y)
+
+end CappedLocalLoad
+
+ 
+theorem exists_operator_of_scaled_matrix (F : V → V → ℝ) (c C : ℝ) (hc : 0 < c)
+    (A : Hilbert V →L[ℂ] Hilbert V) (hAnorm : ‖A‖ ≤ C)
+    (hA : ∀ x y, matrixCoeff A x y = (c * F x y : ℂ)) :
+    ∃ T : Hilbert V →L[ℂ] Hilbert V, ‖T‖ ≤ C/c ∧
+      ∀ x y, matrixCoeff T x y = (F x y : ℂ) := by
+  refine ⟨((c⁻¹:ℝ):ℂ) • A,?_,?_⟩
+  · rw [norm_smul,Complex.norm_of_nonneg (inv_nonneg.mpr hc.le)]
+    simpa only [div_eq_mul_inv,mul_comm] using
+      mul_le_mul_of_nonneg_left hAnorm (inv_nonneg.mpr hc.le)
+  · intro x y
+    change ((c⁻¹:ℝ):ℂ)*matrixCoeff A x y = _
+    rw [hA]
+    have hc' : (c:ℂ) ≠ 0 := Complex.ofReal_ne_zero.mpr hc.ne'
+    simp only [Complex.ofReal_inv]
+    field_simp
+
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
+
+theorem amplificationLoad_scaled_kernel [Nonempty E] (o : V)
+    (q : LocalEndpointRule G hloc (vertexOrbit G o) M L) (hL : 1 ≤ L)
+    (k n : ℕ) (hn : 0 < n) (p : unitInterval) (x z : V)
+    (hz : z ∈ q.amplificationBall k n x) :
+    (n:ℝ) * (q.amplificationLoad o hL k n p).kernel p x z =
+      ∫ ω, q.amplificationMarginal (q.amplificationBonds k n x) (q.amplificationBall k n x)
+        x k n ⟨z,hz⟩ p ω ∂law p := by
+  change (n:ℝ) * (∫ ω, q.amplifiedMass k n p ω x z ∂law p) = _
+  simp only [amplifiedMass,dite_eq_left hz,integral_div]
+  have hn' : (n:ℝ) ≠ 0 := ne_of_gt (Nat.cast_pos.mpr hn)
+  field_simp
+
+ 
+
+theorem amplificationLoad_exists_operator [Nonempty E] (o : V)
+    (q : LocalEndpointRule G hloc (vertexOrbit G o) M L) (hL : 1 ≤ L)
+    (k n : ℕ) (hn : 0 < n) (p : unitInterval)
+    (A : Hilbert V →L[ℂ] Hilbert V)
+    (hA : ∀ a b, matrixCoeff A a b = (q.kernel p a b : ℂ)) :
+    ∃ T : Hilbert V →L[ℂ] Hilbert V, ‖T‖ ≤ binOrbitRow G p o n * ‖A‖^k / n ∧
+      ∀ x z, matrixCoeff T x z = ((q.amplificationLoad o hL k n p).kernel p x z : ℂ) := by
+  let F := (q.amplificationLoad o hL k n p).kernel p
+  have hn' : (0:ℝ) < n := Nat.cast_pos.mpr hn
+  have hposA : PositiveRealMatrix A := fun x z => by
+    rw [hA]
+    exact ⟨rfl,q.kernel_nonneg p x z⟩
+  obtain ⟨S,hSnorm,hS⟩ := exists_operator_of_amplification_domination G hloc p o n k A hposA
+    (fun x z => (n:ℝ)*F x z) (fun x z => mul_nonneg hn'.le ((q.amplificationLoad o hL k n p).kernel_nonneg p x z))
+    (by
+      intro H hH x z
+      have hposH : PositiveRealMatrix H := fun a b => by
+        rw [hH]
+        exact ⟨rfl,orbitBinConnection_nonneg G p o n a b⟩
+      have hpos := (((hposA.pow k).mul hposH) z x).2
+      by_cases hz : z ∈ q.amplificationBall k n x
+      · rw [q.amplificationLoad_scaled_kernel o hL k n hn p x z hz]
+        by_cases ho : x ∈ vertexOrbit G o ∧ z ∈ vertexOrbit G o
+        · exact q.amplificationMarginal_operator_le o _ _ x k n ⟨z,hz⟩ p ho.1 ho.2
+            (q.amplificationBonds_root k n x) (fun a ha =>
+              (incidentSupport_ball_mono G hloc a (le_max_left _ _)).trans (q.amplificationBonds_query k n x a ha)) A H hA hH
+        · have hm (ω : Set E) : q.amplificationMarginal (q.amplificationBonds k n x)
+              (q.amplificationBall k n x) x k n ⟨z,hz⟩ p ω = 0 := by
+            by_contra h
+            have hs := q.amplificationMarginal_support _ _ x k n ⟨z,hz⟩ p ω h
+            exact ho ⟨hs.1,hs.2.1⟩
+          simpa only [hm,integral_zero] using hpos
+      · have he : F x z = 0 := by
+          change (∫ ω, q.amplifiedMass k n p ω x z ∂law p) = 0
+          simp only [amplifiedMass,dite_eq_right hz,integral_zero]
+        simpa only [he,mul_zero] using hpos)
+  exact exists_operator_of_scaled_matrix F n _ hn' S hSnorm (by
+    intro x z
+    simpa only [Complex.ofReal_mul] using hS x z)
+
+end LocalEndpointRule
+end AmplificationLoadOperator
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+open MeasureTheory
+section TruncationOperator
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace CappedLocalLoad
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ}
+
+theorem truncate_kernel_le (q : CappedLocalLoad G hloc O M) (p : unitInterval)
+    (b : ℝ) (hb : 0 < b) (L : ℝ) (x y : V) :
+    (q.truncate b hb L).kernel p x y ≤ b⁻¹ * q.kernel p x y := by
+  rw [kernel,← integral_const_mul]
+  exact integral_mono ((q.truncate b hb L).integrable_mass p x y)
+    ((q.integrable_mass p x y).const_mul b⁻¹) (q.truncate_mass_le b hb L · x y)
+
+ 
+
+theorem truncate_exists_operator (q : CappedLocalLoad G hloc O M) (p : unitInterval)
+    (b : ℝ) (hb : 0 < b) (L : ℝ)
+    (A : Hilbert V →L[ℂ] Hilbert V)
+    (hA : ∀ x y, matrixCoeff A x y = (q.kernel p x y : ℂ)) :
+    ∃ T : Hilbert V →L[ℂ] Hilbert V, ‖T‖ ≤ ‖A‖/b ∧
+      ∀ x y, matrixCoeff T x y = ((q.truncate b hb L).kernel p x y : ℂ) := by
+  have hscale (x y : V) : matrixCoeff (((b⁻¹:ℝ):ℂ) • A) x y = (b⁻¹ * q.kernel p x y : ℝ) := by
+    change ((b⁻¹:ℝ):ℂ)*matrixCoeff A x y = _
+    rw [hA,Complex.ofReal_mul]
+  obtain ⟨T,hT,hK⟩ := exists_operator_of_matrix_domination
+    (fun x y => b⁻¹ * q.kernel p x y) ((q.truncate b hb L).kernel p)
+    (fun x y => mul_nonneg (inv_nonneg.mpr hb.le) (q.kernel_nonneg p x y))
+    (fun x y => by
+      rw [abs_of_nonneg ((q.truncate b hb L).kernel_nonneg p x y)]
+      exact q.truncate_kernel_le p b hb L x y)
+    (((b⁻¹:ℝ):ℂ) • A) hscale
+  refine ⟨T,hT.trans_eq ?_,hK⟩
+  rw [norm_smul,Complex.norm_of_nonneg (inv_nonneg.mpr hb.le)]
+  ring
+
+end CappedLocalLoad
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
+
+ 
+theorem normalize_mass_operator (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
+    (p : unitInterval) (hu : 0 < q.successProbability p o)
+    (A : Hilbert V →L[ℂ] Hilbert V) (C : ℝ) (hAnorm : ‖A‖ ≤ C)
+    (hA : ∀ x y, matrixCoeff A x y = (q.kernel p x y : ℂ)) :
+    ∃ T : Hilbert V →L[ℂ] Hilbert V, ‖T‖ ≤ C/q.successProbability p o ∧
+      ∀ x y, matrixCoeff T x y = (q.normalizedKernel o p x y : ℂ) := by
+  apply exists_operator_of_scaled_matrix (q.normalizedKernel o p) (q.successProbability p o) C hu A hAnorm
+  intro x y
+  rw [hA]
+  simp only [normalizedKernel,← Complex.ofReal_mul]
+  congr 1
+  rw [← mul_assoc,mul_inv_cancel₀ hu.ne',one_mul]
+
+ 
+
+theorem mass_operator_of_normalized (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
+    (p : unitInterval) (hu : 0 < q.successProbability p o)
+    (A : Hilbert V →L[ℂ] Hilbert V)
+    (hA : ∀ x y, matrixCoeff A x y = (q.normalizedKernel o p x y : ℂ)) :
+    ∃ T : Hilbert V →L[ℂ] Hilbert V, ‖T‖ = q.successProbability p o * ‖A‖ ∧
+      ∀ x y, matrixCoeff T x y = (q.kernel p x y : ℂ) := by
+  refine ⟨(q.successProbability p o : ℂ) • A,?_,?_⟩
+  · rw [norm_smul,Complex.norm_of_nonneg hu.le]
+  · intro x y
+    change (q.successProbability p o : ℂ)*matrixCoeff A x y = _
+    rw [hA]
+    simp only [normalizedKernel,← Complex.ofReal_mul]
+    congr 1
+    rw [← mul_assoc,mul_inv_cancel₀ hu.ne',one_mul]
+
+end LocalEndpointRule
+end TruncationOperator
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+open MeasureTheory
+section AmplificationTruncationStep
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
+
+theorem normalized_operator_norm_pos (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
+    (p : unitInterval) (hu : 0 < q.successProbability p o)
+    (A : Hilbert V →L[ℂ] Hilbert V)
+    (hA : ∀ x y, matrixCoeff A x y = (q.normalizedKernel o p x y : ℂ)) : 0 < ‖A‖ := by
+  apply norm_pos_iff.mpr
+  intro ha
+  have hz (y : V) : q.kernel p o y = 0 := by
+    have he := hA o y
+    rw [ha] at he
+    have he' : (q.successProbability p o)⁻¹ * q.kernel p o y = 0 :=
+      Complex.ofReal_eq_zero.mp he.symm
+    exact (mul_eq_zero.mp he').resolve_left (inv_ne_zero hu.ne')
+  have he : q.successProbability p o = 0 := by
+    simp only [successProbability,hz,tsum_zero]
+  exact hu.ne' he
+
+ 
+
+theorem amplification_truncation_step [Nonempty E] {ι : Type*} [Fintype ι]
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (htr : RootTransport root weight Γ)
+    (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b) (hw : ∀ i, 0 ≤ weight i)
+    (i₀ : ι) (hw₀ : 0 < weight i₀)
+    (q : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) M L) (hL : 1 ≤ L)
+    (p : unitInterval) (k n : ℕ) (hn : 0 < n)
+    (hu : 0 < q.successProbability p (root i₀))
+    (hbin : 0 < binOrbitRow G p (root i₀) n)
+    (b L' : ℝ) (hb : 0 < b)
+    (hmoment : (∫ ω, (q.amplificationLoad (root i₀) hL k n p).row ω (root i₀)^2 ∂law p) / b ≤
+      (q.successProbability p (root i₀)^k * binOrbitRow G p (root i₀) n / n)/4)
+    (hbad : b * (law p).real {ω | CappedBadGoodness G ω (root i₀) (2*n+k*M) L'} ≤
+      (q.successProbability p (root i₀)^k * binOrbitRow G p (root i₀) n / n)/4)
+    (A : Hilbert V →L[ℂ] Hilbert V)
+    (hA : ∀ x y, matrixCoeff A x y = (q.normalizedKernel (root i₀) p x y : ℂ)) :
+    ∃ q' : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) (2*n+k*M) L',
+      (q.successProbability p (root i₀)^k * binOrbitRow G p (root i₀) n / n)/(2*b) ≤
+        q'.successProbability p (root i₀) ∧
+      ∃ A' : Hilbert V →L[ℂ] Hilbert V, 0 < ‖A'‖ ∧ ‖A'‖ ≤ 2*‖A‖^k ∧
+        ∀ x y, matrixCoeff A' x y = (q'.normalizedKernel (root i₀) p x y : ℂ) := by
+  let load := q.amplificationLoad (root i₀) hL k n p
+  let q' := load.truncate b hb L'
+  have hn' : (0:ℝ) < n := Nat.cast_pos.mpr hn
+  have hb₀ : 0 < q.successProbability p (root i₀)^k * binOrbitRow G p (root i₀) n / n :=
+    div_pos (mul_pos (pow_pos hu _) hbin) hn'
+  have hself : root i₀ ∈ vertexOrbit G (root i₀) := by
+    refine ⟨Equiv.refl V,Equiv.refl E,?_,rfl⟩
+    intro e
+    simp
+  have hmean := q.amplificationLoad_mean root weight Γ htr hΓ hw i₀ hw₀ hL p k n (root i₀) hself
+  have hnew := load.truncate_success_lower p b hb L' (root i₀) _ hmean.ge hmoment hbad
+  have hu' : 0 < q'.successProbability p (root i₀) :=
+    (div_pos hb₀ (mul_pos (by norm_num) hb)).trans_le hnew
+  obtain ⟨B,hBnorm,hB⟩ := q.mass_operator_of_normalized (root i₀) p hu A hA
+  obtain ⟨C,hCnorm,hC⟩ := q.amplificationLoad_exists_operator (root i₀) hL k n hn p B hB
+  obtain ⟨D,hDnorm,hD⟩ := load.truncate_exists_operator p b hb L' C hC
+  have hD' : ‖D‖ ≤ (binOrbitRow G p (root i₀) n * (q.successProbability p (root i₀)*‖A‖)^k/n)/b := by
+    rw [hBnorm] at hCnorm
+    exact hDnorm.trans (div_le_div_of_nonneg_right hCnorm hb.le)
+  obtain ⟨A',hA'norm,hA'⟩ := q'.normalize_mass_operator (root i₀) p hu' D _ hD' hD
+  refine ⟨q',hnew,A',q'.normalized_operator_norm_pos (root i₀) p hu' A' hA',hA'norm.trans ?_,hA'⟩
+  apply (div_le_iff₀ hu').mpr
+  apply (div_le_iff₀ hb).mpr
+  apply (div_le_iff₀ hn').mpr
+  have hret := (div_le_iff₀ (mul_pos (show (0:ℝ)<2 by norm_num) hb)).mp hnew
+  have hret' := (div_le_iff₀ hn').mp hret
+  rw [mul_pow]
+  calc
+    _ = (q.successProbability p (root i₀)^k * binOrbitRow G p (root i₀) n) * ‖A‖^k := by ring
+    _ ≤ (q'.successProbability p (root i₀)*(2*b)*n)*‖A‖^k :=
+      mul_le_mul_of_nonneg_right hret' (pow_nonneg (norm_nonneg _) _)
+    _ = _ := by ring
+
+end LocalEndpointRule
+end AmplificationTruncationStep
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Kernel/DistinctMoments.lean b/lean/OAI/Probability/UnimodularPercolation/Kernel/DistinctMoments.lean
index fb349efe8071d036f45d2882312f3850fbc62ce4..f3be7c90a8d03e4c43bae6538b31922f9b72601f 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Kernel/DistinctMoments.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Kernel/DistinctMoments.lean
@@ -1,1213 +1,1213 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Clusters.ExteriorMoments
-
-namespace OAI
-
-/-! Distinct Moments for bond percolation. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section ExposedBoundaryPotential
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def exposedVertexPotential (G : BondGraph V E) (p : unitInterval)
-    (x a b : V) (ω : Set E) : ℝ :=
-  if Conn G ω x a then freshExteriorMoment G p a b ω else 0
-
-theorem integrable_exposedVertexPotential [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (x a b : V) :
-    Integrable (exposedVertexPotential G p x a b) (law p) := by
-  have he : exposedVertexPotential G p x a b = {ω | Conn G ω x a}.indicator (freshExteriorMoment G p a b) := by
-    funext ω
-    simp only [exposedVertexPotential,Set.indicator,Set.mem_ofPred_eq]
-  rw [he]
-  exact (integrable_freshExteriorMoment G hloc hqt p hp a b).indicator (measurableSet_conn G x a)
-
-theorem integral_exposedVertexPotential_le [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (x a b : V) :
-    (∫ ω, exposedVertexPotential G p x a b ω ∂law p) ≤ distinctClusterMoment G p a b := by
-  rw [←show (∫ ω, freshExteriorMoment G p a b ω ∂law p) = distinctClusterMoment G p a b from
-    integral_freshExteriorMoment G hloc hqt p hp a b]
-  apply integral_mono (integrable_exposedVertexPotential G hloc hqt p hp x a b)
-    (integrable_freshExteriorMoment G hloc hqt p hp a b)
-  intro ω
-  unfold exposedVertexPotential
-  split_ifs
-  · rfl
-  · exact freshExteriorMoment_nonneg G p a b ω
-
-theorem exposedBoundaryPotential_eq (G : BondGraph V E) (hloc : LocallyFinite G)
-    (p : unitInterval) (x v : V) (M : ℕ) (ω : Set E) :
-    ((cluster G ω x).ncard:ℝ)*deletedBoundaryPotential G hloc (cluster G ω x)ᶜ p v M =
-      ∑ a ∈ (finite_reachBall hloc v M).toFinset,
-        ∑ b ∈ neighbors G hloc a, exposedVertexPotential G p x a b ω := by
-  unfold deletedBoundaryPotential
-  rw [Finset.sum_filter,Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro a _
-  by_cases ha : Conn G ω x a
-  · have hc := cluster_eq_of_conn G ω ha
-    have hn : a ∉ (cluster G ω x)ᶜ := by exact not_not.mpr ha
-    simp only [ite_eq_left hn,Finset.mul_sum]
-    simp only [exposedVertexPotential,ite_eq_left ha,freshExteriorMoment,hc]
-  · have hn : ¬a ∉ (cluster G ω x)ᶜ := by exact not_not.mpr ha
-    simp only [ite_eq_right hn,mul_zero,exposedVertexPotential,ite_eq_right ha,Finset.sum_const_zero]
-
- 
-
-theorem integral_exposedBoundaryPotential_le [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (x v : V) (M : ℕ) :
-    Integrable (fun ω => ((cluster G ω x).ncard:ℝ)*deletedBoundaryPotential G hloc (cluster G ω x)ᶜ p v M) (law p) ∧
-    (∫ ω, ((cluster G ω x).ncard:ℝ)*deletedBoundaryPotential G hloc (cluster G ω x)ᶜ p v M ∂law p) ≤
-      ∑ a ∈ (finite_reachBall hloc v M).toFinset, neighborClusterMoment G hloc p a := by
-  simp_rw [exposedBoundaryPotential_eq G hloc p x v M]
-  have hi (a : V) := integrable_finsetSum (neighbors G hloc a)
-    (fun b _ => integrable_exposedVertexPotential G hloc hqt p hp x a b)
-  refine ⟨integrable_finsetSum _ (fun a _ => hi a),?_⟩
-  rw [integral_finsetSum _ (fun a _ => hi a)]
-  apply Finset.sum_le_sum
-  intro a _
-  rw [integral_finsetSum _ (fun b _ => integrable_exposedVertexPotential G hloc hqt p hp x a b)]
-  exact Finset.sum_le_sum (fun b _ => integral_exposedVertexPotential_le G hloc hqt p hp x a b)
-
-end ExposedBoundaryPotential
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section UniformExposedBoundary
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem reachBall_ncard_covariant (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (x : V) (M : ℕ) :
-    (reachBall G (a x) M).ncard = (reachBall G x M).ncard := by
-  have he : reachBall G (a x) M = a '' reachBall G x M := by
-    ext y
-    constructor
-    · intro h
-      exact ⟨a.symm y,by simpa only [Equiv.symm_apply_apply] using mem_reachBall_aut G (isAut_symm G hab) h,by simp⟩
-    · rintro ⟨z,hz,rfl⟩
-      exact mem_reachBall_aut G hab hz
-  rw [he,Set.ncard_image_of_injective _ a.injective]
-
-theorem reachBall_card_le_root_sum (G : BondGraph V E) (hloc : LocallyFinite G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i))
-    (M : ℕ) (x : V) : (finite_reachBall hloc x M).toFinset.card ≤
-      ∑ i, (reachBall G (root i) M).ncard := by
-  obtain ⟨i,a,b,hab,hax⟩ := hcover x
-  rw [←Set.ncard_eq_toFinset_card (reachBall G x M) (finite_reachBall hloc x M),←hax,reachBall_ncard_covariant G hab]
-  exact Finset.single_le_sum (f := fun i => (reachBall G (root i) M).ncard) (fun _ _ => Nat.zero_le _) (Finset.mem_univ i)
-
-theorem rootNeighborMoment_nonneg (G : BondGraph V E) (hloc : LocallyFinite G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 ≤ weight i)
-    (p : unitInterval) : 0 ≤ rootNeighborMoment G hloc root weight p :=
-  Finset.sum_nonneg (fun i _ => mul_nonneg (hw i) (neighborClusterMoment_nonneg G hloc p _))
-
- 
-
-theorem integral_exposedBoundaryPotential_uniform [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (hw : ∀ i, 0 ≤ weight i) (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i))
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (p : unitInterval) (hp : p < pc G) (x v : V) (M : ℕ) :
-    (∫ ω, ((cluster G ω x).ncard:ℝ)*deletedBoundaryPotential G hloc (cluster G ω x)ᶜ p v M ∂law p) ≤
-      (∑ i, ((reachBall G (root i) M).ncard:ℝ))*c*rootNeighborMoment G hloc root weight p := by
-  let B := (finite_reachBall hloc v M).toFinset
-  calc
-    _ ≤ ∑ a ∈ B, neighborClusterMoment G hloc p a :=
-      (integral_exposedBoundaryPotential_le G hloc hqt p hp x v M).2
-    _ ≤ ∑ _a ∈ B, c*rootNeighborMoment G hloc root weight p :=
-      Finset.sum_le_sum (fun a _ => neighborMoment_le_rootMoment G hloc root weight hw hcover c hc hmin p a)
-    _ = (B.card:ℝ)*(c*rootNeighborMoment G hloc root weight p) := by rw [Finset.sum_const,nsmul_eq_mul]
-    _ ≤ _ := by
-      have hb : (B.card:ℝ) ≤ ∑ i, ((reachBall G (root i) M).ncard:ℝ) := by
-        exact_mod_cast reachBall_card_le_root_sum G hloc root hcover M v
-      simpa only [mul_assoc] using mul_le_mul_of_nonneg_right hb
-        (mul_nonneg hc (rootNeighborMoment_nonneg G hloc root weight hw p))
-
-end UniformExposedBoundary
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section DistinctMomentPotential
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
-
- 
-
-theorem kernel_distinctMoment_integral [Countable V] [Countable E] [Nonempty V]
-    (q : LocalEndpointRule G hloc O M L) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (x v : V) :
-    (∑' y, q.kernel p v y * distinctClusterMoment G p x y) =
-      ∫ ω, (∑' y, q.kernel p v y * freshExteriorMoment G p x y ω) ∂law p := by
-  rw [integral_tsum_of_finite_support (law p) (finite_reachBall hloc v M).toFinset _
-    (fun y => (integrable_freshExteriorMoment G hloc hqt p hp x y).const_mul (q.kernel p v y))
-    (fun ω y hy => by rw [q.kernel_outside_ball_zero p v y (by simpa only [Set.Finite.mem_toFinset] using hy),zero_mul])]
-  apply tsum_congr
-  intro y
-  rw [integral_const_mul,integral_freshExteriorMoment G hloc hqt p hp]
-  rfl
-
-theorem integrable_kernel_freshMoment [Countable V] [Countable E] [Nonempty V]
-    (q : LocalEndpointRule G hloc O M L) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (x v : V) :
-    Integrable (fun ω => ∑' y, q.kernel p v y * freshExteriorMoment G p x y ω) (law p) := by
-  have he (ω) : (∑' y, q.kernel p v y * freshExteriorMoment G p x y ω) =
-      ∑ y ∈ (finite_reachBall hloc v M).toFinset, q.kernel p v y * freshExteriorMoment G p x y ω :=
-    tsum_eq_sum (fun y hy => by rw [q.kernel_outside_ball_zero p v y (by simpa only [Set.Finite.mem_toFinset] using hy),zero_mul])
-  simp_rw [he]
-  exact integrable_finsetSum _ (fun y _ => (integrable_freshExteriorMoment G hloc hqt p hp x y).const_mul _)
-
- 
-theorem kernel_distinctMoment_bound [Countable V] [Countable E] [Nonempty V]
-    (q : LocalEndpointRule G hloc O M L) (hqt : QuasiTransitive G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (hw : ∀ i, 0 ≤ weight i) (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i))
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (p : unitInterval) (hp : 0 < (p:ℝ)) (hpc : p < pc G) (x v : V) :
-    (∑' y, q.kernel p v y * distinctClusterMoment G p x y) ≤
-      (1+(1-(p:ℝ))/p*M)*distinctClusterMoment G p x v +
-        (p:ℝ)^(-(M:ℤ))*(∑ i, ((reachBall G (root i) M).ncard:ℝ))*c*rootNeighborMoment G hloc root weight p := by
-  let C := 1+(1-(p:ℝ))/p*M
-  let R := (p:ℝ)^(-(M:ℤ))
-  let b := fun ω : Set E => ((cluster G ω x).ncard:ℝ)*deletedBoundaryPotential G hloc (cluster G ω x)ᶜ p v M
-  have hi := (integral_exposedBoundaryPotential_le G hloc hqt p hpc x v M).1
-  have hpoint (ω : Set E) : (∑' y, q.kernel p v y * freshExteriorMoment G p x y ω) ≤
-      C*freshExteriorMoment G p x v ω + R*b ω := by
-    have h := mul_le_mul_of_nonneg_left (q.kernel_deleted_potential_bound hqt (cluster G ω x)ᶜ p hp hpc v)
-      (Nat.cast_nonneg (cluster G ω x).ncard : (0:ℝ) ≤ _)
-    rw [←tsum_mul_left] at h
-    calc
-      _ = ∑' y, ((cluster G ω x).ncard:ℝ)*(q.kernel p v y * inducedSusceptibility G (cluster G ω x)ᶜ p y) := by
-        apply tsum_congr
-        intro y
-        simp only [freshExteriorMoment]
-        ring
-      _ ≤ _ := h
-      _ = _ := by dsimp [C,R,b,freshExteriorMoment]; ring
-  rw [q.kernel_distinctMoment_integral hqt p hpc x v]
-  calc
-    _ ≤ ∫ ω, C*freshExteriorMoment G p x v ω + R*b ω ∂law p :=
-      integral_mono (q.integrable_kernel_freshMoment hqt p hpc x v)
-        (((integrable_freshExteriorMoment G hloc hqt p hpc x v).const_mul C).add (hi.const_mul R)) hpoint
-    _ = C*distinctClusterMoment G p x v + R*(∫ ω, b ω ∂law p) := by
-      rw [integral_add ((integrable_freshExteriorMoment G hloc hqt p hpc x v).const_mul C) (hi.const_mul R),
-        integral_const_mul,integral_const_mul,integral_freshExteriorMoment G hloc hqt p hpc]
-      rfl
-    _ ≤ _ := by
-      have h := mul_le_mul_of_nonneg_left
-        (integral_exposedBoundaryPotential_uniform G hloc hqt root weight hw hcover c hc hmin p hpc x v M)
-        (zpow_nonneg p.2.1 (-(M:ℤ)))
-      dsimp [C,R,b] at *
-      nlinarith
-
-end LocalEndpointRule
-end DistinctMomentPotential
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section FinitePositiveOperatorPowers
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem finite_positive_product_support (A B : Hilbert V →L[ℂ] Hilbert V)
-    (hA : PositiveRealMatrix A)
-    (ha : ∀ x, (Function.support (fun y => (matrixCoeff A x y).re)).Finite)
-    (hb : ∀ x, (Function.support (fun y => (matrixCoeff B x y).re)).Finite) (x : V) :
-    (Function.support (fun y => (matrixCoeff (A*B) x y).re)).Finite := by
-  let S := (ha x).toFinset
-  let T := S.biUnion (fun z => (hb z).toFinset)
-  apply T.finite_toSet.subset
-  intro y hy
-  by_contra hn
-  apply hy
-  change (matrixCoeff (A*B) x y).re = 0
-  rw [←(hA.mul_hasSum B x y).tsum_eq]
-  rw [←tsum_zero]
-  apply tsum_congr
-  intro z
-  by_cases hz : z ∈ S
-  · have hzy : (matrixCoeff B z y).re = 0 := by
-      by_contra hh
-      apply hn
-      exact Finset.mem_biUnion.mpr ⟨z,hz,by simpa only [Set.Finite.mem_toFinset,Function.mem_support] using hh⟩
-    rw [hzy,mul_zero]
-  · have hzx : (matrixCoeff A x z).re = 0 := by
-      simpa only [S,Set.Finite.mem_toFinset,Function.mem_support,not_not] using hz
-    rw [hzx,zero_mul]
-
-theorem finite_positive_product_action (A B : Hilbert V →L[ℂ] Hilbert V)
-    (hA : PositiveRealMatrix A)
-    (ha : ∀ x, (Function.support (fun y => (matrixCoeff A x y).re)).Finite)
-    (hb : ∀ x, (Function.support (fun y => (matrixCoeff B x y).re)).Finite)
-    (f : V → ℝ) (x : V) :
-    (∑' y, (matrixCoeff (A*B) x y).re*f y) =
-      ∑' z, (matrixCoeff A x z).re*∑' y, (matrixCoeff B z y).re*f y := by
-  let K := fun x y => (matrixCoeff A x y).re
-  let J := fun x y => (matrixCoeff B x y).re
-  let S := (ha x).toFinset
-  have hz (z : V) (hz : z ∉ S) : K x z = 0 := by
-    simpa only [S,Set.Finite.mem_toFinset,Function.mem_support,not_not] using hz
-  have he (y : V) : (matrixCoeff (A*B) x y).re*f y = ∑ z ∈ S, K x z*(J z y*f y) := by
-    rw [←(hA.mul_hasSum B x y).tsum_eq,tsum_eq_sum (s := S)
-      (fun z h => by change K x z*J z y = 0; rw [hz z h,zero_mul]),Finset.sum_mul]
-    exact Finset.sum_congr rfl (fun z _ => mul_assoc _ _ _)
-  have hs := hasSum_sum (s := S) (fun z _ =>
-    (finiteKernel_summable_mul J hb z f).hasSum.mul_left (K x z))
-  have hr : (∑' z, K x z*∑' y, J z y*f y) = ∑ z ∈ S, K x z*∑' y, J z y*f y :=
-    tsum_eq_sum (fun z h => by rw [hz z h,zero_mul])
-  simpa only [←he,←hr] using hs.tsum_eq
-
-theorem finite_positive_pow_support (A : Hilbert V →L[ℂ] Hilbert V)
-    (hA : PositiveRealMatrix A)
-    (ha : ∀ x, (Function.support (fun y => (matrixCoeff A x y).re)).Finite)
-    (j : ℕ) (x : V) : (Function.support (fun y => (matrixCoeff (A^j) x y).re)).Finite := by
-  induction j generalizing x with
-  | zero =>
-      apply (Set.finite_singleton x).subset
-      intro y hy
-      by_contra hn
-      apply hy
-      have he : x ≠ y := fun h => hn (by simp only [Set.mem_singleton_iff]; exact h.symm)
-      simp only [pow_zero,matrixCoeff,one_apply_eq_self,lp.single_apply,Pi.single_apply,ite_eq_right he,Complex.zero_re]
-  | succ j ih =>
-      rw [pow_succ']
-      exact finite_positive_product_support A (A^j) hA ha ih x
-
-theorem positive_identity_action (f : V → ℝ) (x : V) :
-    (∑' y, (matrixCoeff (1 : Hilbert V →L[ℂ] Hilbert V) x y).re*f y) = f x := by
-  rw [tsum_eq_single x (fun y hy => by
-    simp only [matrixCoeff,one_apply_eq_self,lp.single_apply,Pi.single_apply,ite_eq_right (Ne.symm hy),Complex.zero_re,zero_mul])]
-  simp only [matrixCoeff,one_apply_eq_self,lp.single_apply,Pi.single_apply,ite_true,Complex.one_re,one_mul]
-
-end FinitePositiveOperatorPowers
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section PositivePotentialIteration
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def realAction (A : Hilbert V →L[ℂ] Hilbert V) (f : V → ℝ) (x : V) : ℝ :=
-  ∑' y, (matrixCoeff A x y).re*f y
-
-theorem realAction_mono (A : Hilbert V →L[ℂ] Hilbert V) (hA : PositiveRealMatrix A)
-    (ha : ∀ x, (Function.support (fun y => (matrixCoeff A x y).re)).Finite)
-    (f g : V → ℝ) (hfg : ∀ x, f x ≤ g x) (x : V) : realAction A f x ≤ realAction A g x := by
-  exact (finiteKernel_summable_mul _ ha x f).tsum_le_tsum
-    (fun y => mul_le_mul_of_nonneg_left (hfg y) (hA x y).2) (finiteKernel_summable_mul _ ha x g)
-
-theorem realAction_affine (A : Hilbert V →L[ℂ] Hilbert V)
-    (ha : ∀ x, (Function.support (fun y => (matrixCoeff A x y).re)).Finite)
-    (f : V → ℝ) (a b : ℝ) (x : V) :
-    realAction A (fun y => a*f y+b) x = a*realAction A f x + b*realAction A (fun _ => 1) x := by
-  unfold realAction
-  simp_rw [mul_add]
-  rw [Summable.tsum_add]
-  · have he (y : V) : (matrixCoeff A x y).re*(a*f y) = a*((matrixCoeff A x y).re*f y) := by ring
-    simp_rw [he]
-    simp only [tsum_mul_left,tsum_mul_right,mul_one]
-    ring
-  · exact finiteKernel_summable_mul _ ha x (fun y => a*f y)
-  · exact finiteKernel_summable_mul _ ha x (fun _ => b)
-
-theorem realAction_pow_succ (A : Hilbert V →L[ℂ] Hilbert V) (hA : PositiveRealMatrix A)
-    (ha : ∀ x, (Function.support (fun y => (matrixCoeff A x y).re)).Finite)
-    (j : ℕ) (f : V → ℝ) (x : V) :
-    realAction (A^(j+1)) f x = realAction A (realAction (A^j) f) x := by
-  rw [pow_succ']
-  exact finite_positive_product_action A (A^j) hA ha (finite_positive_pow_support A hA ha j) f x
-
- 
-
-theorem realAction_pow_potential_bound (A : Hilbert V →L[ℂ] Hilbert V) (hA : PositiveRealMatrix A)
-    (ha : ∀ x, (Function.support (fun y => (matrixCoeff A x y).re)).Finite)
-    (hrow : ∀ x, realAction A (fun _ => 1) x ≤ 1)
-    (f : V → ℝ) (Z B : ℝ) (hZ : 2 ≤ Z) (hB : 0 ≤ B)
-    (hf : ∀ x, realAction A f x ≤ Z*f x+B) (j : ℕ) (x : V) :
-    realAction (A^j) f x ≤ Z^j*f x + B*(Z^j-1) := by
-  have hz : 0 ≤ Z := by linarith
-  have hz1 : 1 ≤ Z := by linarith
-  induction j generalizing x with
-  | zero =>
-      simp only [pow_zero,realAction,positive_identity_action,one_mul,sub_self,mul_zero,add_zero,le_refl]
-  | succ j ih =>
-      rw [realAction_pow_succ A hA ha]
-      calc
-        _ ≤ realAction A (fun y => Z^j*f y+B*(Z^j-1)) x := realAction_mono A hA ha _ _ ih x
-        _ = Z^j*realAction A f x + (B*(Z^j-1))*realAction A (fun _ => 1) x := realAction_affine A ha f _ _ x
-        _ ≤ Z^j*(Z*f x+B) + B*(Z^j-1) := add_le_add
-          (mul_le_mul_of_nonneg_left (hf x) (pow_nonneg hz _))
-          (by
-            simpa only [mul_one] using (mul_le_mul_of_nonneg_left (hrow x)
-              (mul_nonneg hB (sub_nonneg.mpr (one_le_pow₀ hz1)))))
-        _ ≤ Z^(j+1)*f x+B*(Z^(j+1)-1) := by
-          have hh := mul_nonneg (mul_nonneg hB (pow_nonneg hz j)) (show 0 ≤ Z-2 by linarith)
-          rw [pow_succ]
-          nlinarith only [hh]
-
-theorem realAction_pow_potential_at_zero (A : Hilbert V →L[ℂ] Hilbert V) (hA : PositiveRealMatrix A)
-    (ha : ∀ x, (Function.support (fun y => (matrixCoeff A x y).re)).Finite)
-    (hrow : ∀ x, realAction A (fun _ => 1) x ≤ 1)
-    (f : V → ℝ) (Z B : ℝ) (hZ : 2 ≤ Z) (hB : 0 ≤ B)
-    (hf : ∀ x, realAction A f x ≤ Z*f x+B) (j : ℕ) (x : V) (hx : f x = 0) :
-    realAction (A^j) f x ≤ B*Z^j := by
-  have h := realAction_pow_potential_bound A hA ha hrow f Z B hZ hB hf j x
-  rw [hx,mul_zero,zero_add] at h
-  nlinarith
-
-end PositivePotentialIteration
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section NormalizedPositiveOperator
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
-
-theorem normalized_operator_positive (o : V)
-    (q : LocalEndpointRule G hloc (vertexOrbit G o) M L) (p : unitInterval)
-    (A : Hilbert V →L[ℂ] Hilbert V)
-    (hA : ∀ x y, matrixCoeff A x y = (q.normalizedKernel o p x y : ℂ)) : PositiveRealMatrix A := by
-  intro x y
-  rw [hA]
-  exact ⟨rfl,mul_nonneg (inv_nonneg.mpr (q.successProbability_nonneg p o)) (q.kernel_nonneg p x y)⟩
-
-theorem normalized_operator_finite (o : V)
-    (q : LocalEndpointRule G hloc (vertexOrbit G o) M L) (p : unitInterval)
-    (A : Hilbert V →L[ℂ] Hilbert V)
-    (hA : ∀ x y, matrixCoeff A x y = (q.normalizedKernel o p x y : ℂ)) (x : V) :
-    (Function.support (fun y => (matrixCoeff A x y).re)).Finite := by
-  apply (finite_reachBall hloc x M).subset
-  intro y hy
-  by_contra hn
-  apply hy
-  change (matrixCoeff A x y).re = 0
-  simp only [hA,normalizedKernel,q.kernel_outside_ball_zero p x y hn,mul_zero,Complex.ofReal_zero,Complex.zero_re]
-
-theorem normalized_operator_action (o : V)
-    (q : LocalEndpointRule G hloc (vertexOrbit G o) M L) (p : unitInterval)
-    (A : Hilbert V →L[ℂ] Hilbert V)
-    (hA : ∀ x y, matrixCoeff A x y = (q.normalizedKernel o p x y : ℂ)) (f : V → ℝ) (x : V) :
-    realAction A f x = (q.successProbability p o)⁻¹ * (∑' y, q.kernel p x y*f y) := by
-  simp only [realAction,hA,Complex.ofReal_re,normalizedKernel,mul_assoc,tsum_mul_left]
-
-theorem normalized_operator_row_le (o : V)
-    (q : LocalEndpointRule G hloc (vertexOrbit G o) M L) (p : unitInterval)
-    (hu : 0 < q.successProbability p o) (A : Hilbert V →L[ℂ] Hilbert V)
-    (hA : ∀ x y, matrixCoeff A x y = (q.normalizedKernel o p x y : ℂ)) (x : V) :
-    realAction A (fun _ => 1) x ≤ 1 := by
-  rw [q.normalized_operator_action o p A hA]
-  simp only [mul_one]
-  calc
-    _ ≤ (q.successProbability p o)⁻¹ * q.successProbability p o :=
-      mul_le_mul_of_nonneg_left (q.kernel_row_le_success o p x) (inv_nonneg.mpr hu.le)
-    _ = 1 := inv_mul_cancel₀ hu.ne'
-
-theorem normalized_operator_row_eq (o : V)
-    (q : LocalEndpointRule G hloc (vertexOrbit G o) M L) (p : unitInterval)
-    (hu : 0 < q.successProbability p o) (A : Hilbert V →L[ℂ] Hilbert V)
-    (hA : ∀ x y, matrixCoeff A x y = (q.normalizedKernel o p x y : ℂ)) (x : V)
-    (hx : x ∈ vertexOrbit G o) : realAction A (fun _ => 1) x = 1 := by
-  rw [q.normalized_operator_action o p A hA]
-  simp only [mul_one]
-  change (q.successProbability p o)⁻¹ * q.successProbability p x = 1
-  rw [q.successProbability_orbit p o x hx,inv_mul_cancel₀ hu.ne']
-
-theorem normalized_operator_invariant (o : V)
-    (q : LocalEndpointRule G hloc (vertexOrbit G o) M L) (p : unitInterval)
-    (A : Hilbert V →L[ℂ] Hilbert V)
-    (hA : ∀ x y, matrixCoeff A x y = (q.normalizedKernel o p x y : ℂ)) :
-    KernelInvariant (vertexAut G) A := by
-  rintro a ⟨b,hab⟩ x y
-  simp only [hA,normalizedKernel,q.kernel_covariant p a b hab]
-
-theorem normalized_operator_end_support (o : V)
-    (q : LocalEndpointRule G hloc (vertexOrbit G o) M L) (p : unitInterval)
-    (A : Hilbert V →L[ℂ] Hilbert V)
-    (hA : ∀ x y, matrixCoeff A x y = (q.normalizedKernel o p x y : ℂ))
-    (x y : V) (hy : y ∉ vertexOrbit G o) : (matrixCoeff A x y).re = 0 := by
-  simp only [hA,normalizedKernel,q.kernel_zero_of_end_not_mem p x y hy,mul_zero,Complex.ofReal_zero,Complex.zero_re]
-
-end LocalEndpointRule
-end NormalizedPositiveOperator
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section UpperDistinctMomentIteration
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem distinctClusterMoment_self (G : BondGraph V E) (p : unitInterval) (x : V) :
-    distinctClusterMoment G p x x = 0 := by
-  have hs (ω : Set E) : Conn G ω x x := .refl
-  simp only [distinctClusterMoment,hs,not_true_eq_false,ite_false,integral_zero]
-
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
-
- 
-
-theorem normalized_power_distinctMoment_upper [Countable V] [Countable E] [Nonempty V]
-    (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L) (hqt : QuasiTransitive G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (hw : ∀ i, 0 ≤ weight i) (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i))
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (p : unitInterval) (hp : 0 < (p:ℝ)) (hpc : p < pc G)
-    (hu : 0 < q.successProbability p o)
-    (A : Hilbert V →L[ℂ] Hilbert V)
-    (hA : ∀ x y, matrixCoeff A x y = (q.normalizedKernel o p x y : ℂ))
-    (Z : ℝ) (hZ : 2 ≤ Z)
-    (hscale : (q.successProbability p o)⁻¹*(1+(1-(p:ℝ))/p*M) ≤ Z)
-    (j : ℕ) (x : V) :
-    realAction (A^j) (distinctClusterMoment G p x) x ≤
-      ((q.successProbability p o)⁻¹*(p:ℝ)^(-(M:ℤ))*(∑ i, ((reachBall G (root i) M).ncard:ℝ))*c)*
-        rootNeighborMoment G hloc root weight p * Z^j := by
-  let B := ((q.successProbability p o)⁻¹*(p:ℝ)^(-(M:ℤ))*(∑ i, ((reachBall G (root i) M).ncard:ℝ))*c)*
-    rootNeighborMoment G hloc root weight p
-  have hB : 0 ≤ B := by
-    exact mul_nonneg (mul_nonneg (mul_nonneg
-      (mul_nonneg (inv_nonneg.mpr hu.le) (zpow_nonneg p.2.1 _))
-      (Finset.sum_nonneg (fun _ _ => Nat.cast_nonneg _))) hc)
-      (rootNeighborMoment_nonneg G hloc root weight hw p)
-  have hf (v : V) : realAction A (distinctClusterMoment G p x) v ≤ Z*distinctClusterMoment G p x v+B := by
-    rw [q.normalized_operator_action o p A hA]
-    calc
-      _ ≤ (q.successProbability p o)⁻¹ *
-          ((1+(1-(p:ℝ))/p*M)*distinctClusterMoment G p x v+
-            (p:ℝ)^(-(M:ℤ))*(∑ i, ((reachBall G (root i) M).ncard:ℝ))*c*rootNeighborMoment G hloc root weight p) :=
-        mul_le_mul_of_nonneg_left
-          (q.kernel_distinctMoment_bound hqt root weight hw hcover c hc hmin p hp hpc x v) (inv_nonneg.mpr hu.le)
-      _ = ((q.successProbability p o)⁻¹*(1+(1-(p:ℝ))/p*M))*distinctClusterMoment G p x v+B := by dsimp [B]; ring
-      _ ≤ _ := by
-        have hh := mul_le_mul_of_nonneg_right hscale (distinctClusterMoment_nonneg G p x v)
-        linarith
-  exact realAction_pow_potential_at_zero A (q.normalized_operator_positive o p A hA)
-    (q.normalized_operator_finite o p A hA) (q.normalized_operator_row_le o p hu A hA)
-    (distinctClusterMoment G p x) Z B hZ hB hf j x (distinctClusterMoment_self G p x)
-
-end LocalEndpointRule
-end UpperDistinctMomentIteration
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section PositivePowerStochasticity
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem realAction_pow_row_le (A : Hilbert V →L[ℂ] Hilbert V) (hA : PositiveRealMatrix A)
-    (ha : ∀ x, (Function.support (fun y => (matrixCoeff A x y).re)).Finite)
-    (hrow : ∀ x, realAction A (fun _ => 1) x ≤ 1) (j : ℕ) (x : V) :
-    realAction (A^j) (fun _ => 1) x ≤ 1 := by
-  induction j generalizing x with
-  | zero => rw [pow_zero]; exact (positive_identity_action (fun _ => 1) x).le
-  | succ j ih =>
-      rw [realAction_pow_succ A hA ha]
-      exact (realAction_mono A hA ha _ (fun _ => 1) ih x).trans (hrow x)
-
-theorem positive_pow_end_support (A : Hilbert V →L[ℂ] Hilbert V) (hA : PositiveRealMatrix A)
-    (O : Set V) (hO : ∀ x y, y ∉ O → (matrixCoeff A x y).re = 0)
-    (j : ℕ) (x y : V) (hx : x ∈ O) (hy : y ∉ O) : (matrixCoeff (A^j) x y).re = 0 := by
-  cases j with
-  | zero =>
-      have hxy : x ≠ y := fun h => hy (h ▸ hx)
-      simp only [pow_zero,matrixCoeff,one_apply_eq_self,lp.single_apply,Pi.single_apply,ite_eq_right hxy,Complex.zero_re]
-  | succ j =>
-      rw [pow_succ,←((hA.pow j).mul_hasSum A x y).tsum_eq]
-      simp only [hO _ y hy,mul_zero,tsum_zero]
-
-theorem realAction_pow_row_eq_on (A : Hilbert V →L[ℂ] Hilbert V) (hA : PositiveRealMatrix A)
-    (ha : ∀ x, (Function.support (fun y => (matrixCoeff A x y).re)).Finite)
-    (O : Set V) (hO : ∀ x y, y ∉ O → (matrixCoeff A x y).re = 0)
-    (hrow : ∀ x ∈ O, realAction A (fun _ => 1) x = 1)
-    (j : ℕ) (x : V) (hx : x ∈ O) : realAction (A^j) (fun _ => 1) x = 1 := by
-  induction j generalizing x with
-  | zero => rw [pow_zero]; exact positive_identity_action (fun _ => 1) x
-  | succ j ih =>
-      rw [realAction_pow_succ A hA ha]
-      calc
-        _ = realAction A (fun _ => 1) x := by
-          apply tsum_congr
-          intro y
-          by_cases hy : y ∈ O
-          · rw [ih y hy]
-          · rw [hO x y hy,zero_mul,zero_mul]
-        _ = 1 := hrow x hx
-
-end PositivePowerStochasticity
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section ConnectedMomentLoss
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def connectedClusterMoment (G : BondGraph V E) (p : unitInterval) (x y : V) : ℝ :=
-  ∫ ω, (if Conn G ω x y then ((cluster G ω x).ncard:ℝ)^2 else 0) ∂law p
-
-theorem integrable_connected_cluster_square [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (x y : V) :
-    Integrable (fun ω => if Conn G ω x y then ((cluster G ω x).ncard:ℝ)^2 else 0) (law p) := by
-  obtain ⟨_,_,hs⟩ := subcritical_second_moment G hloc hqt p hp
-  have he : (fun ω : Set E => if Conn G ω x y then ((cluster G ω x).ncard:ℝ)^2 else 0) =
-      {ω | Conn G ω x y}.indicator (fun ω => ((cluster G ω x).ncard:ℝ)^2) := by
-    funext ω
-    simp only [Set.indicator,Set.mem_ofPred_eq]
-  rw [he]
-  exact (hs x).1.indicator (measurableSet_conn G x y)
-
-theorem distinct_add_connected_cluster_moment [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (x y : V) :
-    distinctClusterMoment G p x y + connectedClusterMoment G p x y =
-      ∫ ω, ((cluster G ω x).ncard:ℝ)*((cluster G ω y).ncard:ℝ) ∂law p := by
-  rw [distinctClusterMoment,connectedClusterMoment,
-    ←integral_add (integrable_distinct_cluster_product G hloc hqt p hp x y)
-      (integrable_connected_cluster_square G hloc hqt p hp x y)]
-  apply integral_congr_ae
-  exact Filter.Eventually.of_forall (fun ω => by
-    by_cases h : Conn G ω x y
-    · simp only [h,not_true_eq_false,ite_false,ite_true,zero_add,cluster_eq_of_conn G ω h,pow_two]
-    · simp only [h,not_false_eq_true,ite_true,ite_false,add_zero])
-
-theorem distinctClusterMoment_lower [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (x y : V) :
-    susceptibility G p x * susceptibility G p y - connectedClusterMoment G p x y ≤
-      distinctClusterMoment G p x y := by
-  have h := (subcritical_cluster_association G hloc hqt p hp x y).2
-  rw [←distinct_add_connected_cluster_moment G hloc hqt p hp] at h
-  linarith
-
- 
-theorem realAction_connectedMoment_eq_diagonalLoss [Countable V] [Countable E]
-    (G : BondGraph V E) (o : V) (A : Hilbert V →L[ℂ] Hilbert V) (hA : PositiveRealMatrix A)
-    (ha : ∀ x, (Function.support (fun y => (matrixCoeff A x y).re)).Finite)
-    (hrow : ∀ x, realAction A (fun _ => 1) x ≤ 1)
-    (x : V) (hend : ∀ y, y ∉ vertexOrbit G o → (matrixCoeff A x y).re = 0)
-    (p : unitInterval) (hs : Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^2) (law p)) :
-    realAction A (connectedClusterMoment G p x) x =
-      ∫ ω, diagonalLossMass G o (fun a b => (matrixCoeff A a b).re) ω x ∂law p := by
-  have hr (z : V) (S : Finset V) : (∑ y ∈ S, (matrixCoeff A z y).re) ≤ 1 := by
-    have hsrow : Summable (fun y => (matrixCoeff A z y).re) := by
-      simpa only [mul_one] using finiteKernel_summable_mul (fun a b => (matrixCoeff A a b).re) ha z (fun _ => 1)
-    exact (hsrow.sum_le_tsum S (fun y _ => (hA z y).2)).trans (by simpa only [realAction,mul_one] using hrow z)
-  rw [integral_diagonalLoss_eq_series G o _ (fun a b => (hA a b).2) 1 hr x p hs]
-  apply tsum_congr
-  intro y
-  by_cases hy : y ∈ vertexOrbit G o
-  · rw [ite_eq_left hy]
-    rfl
-  · simp only [ite_eq_right hy,hend y hy,zero_mul]
-
-end ConnectedMomentLoss
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section LowerDistinctMomentOperator
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem realAction_distinctMoment_lower [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (htr : RootTransport root weight Γ) (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b)
-    (hw : ∀ i, 0 ≤ weight i) (i₀ : ι) (hw₀ : 0 < weight i₀)
-    (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i))
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (p : unitInterval) (hp : p < pc G) (b : ℝ) (hb : 0 ≤ b)
-    (hχ : ∀ x, b ≤ susceptibility G p x)
-    (A : Hilbert V →L[ℂ] Hilbert V) (hA : PositiveRealMatrix A)
-    (ha : ∀ x, (Function.support (fun y => (matrixCoeff A x y).re)).Finite)
-    (hinv : KernelInvariant (vertexAut G) A)
-    (hrow : ∀ x, realAction A (fun _ => 1) x ≤ 1)
-    (heq : realAction A (fun _ => 1) (root i₀) = 1)
-    (hend : ∀ y, y ∉ vertexOrbit G (root i₀) → (matrixCoeff A (root i₀) y).re = 0) :
-    b^2-‖A‖*(c*rootSusceptibility G root weight p)^3 ≤
-      realAction A (distinctClusterMoment G p (root i₀)) (root i₀) := by
-  let x := root i₀
-  let f := connectedClusterMoment G p x
-  have hr (z : V) (S : Finset V) : (∑ y ∈ S, (matrixCoeff A z y).re) ≤ 1 := by
-    have hsrow : Summable (fun y => (matrixCoeff A z y).re) := by
-      simpa only [mul_one] using finiteKernel_summable_mul (fun a b => (matrixCoeff A a b).re) ha z (fun _ => 1)
-    exact (hsrow.sum_le_tsum S (fun y _ => (hA z y).2)).trans (by simpa only [realAction,mul_one] using hrow z)
-  obtain ⟨_,_,hs⟩ := subcritical_second_moment G hloc hqt p hp
-  have hf : realAction A f x ≤ ‖A‖*(c*rootSusceptibility G root weight p)^3 := by
-    rw [realAction_connectedMoment_eq_diagonalLoss G (root i₀) A hA ha hrow x hend p (hs x).1]
-    exact diagonalLoss_le_rootSusceptibility_cube G hloc hqt root weight Γ htr hΓ hw i₀ hw₀ hcover c hc hmin p hp
-      (fun a b => (matrixCoeff A a b).re) (fun a b => (hA a b).2)
-      (fun a b hab z y => congrArg Complex.re (hinv a ⟨b,hab⟩ z y)) 1 hr A (hA.real)
-  have hpnt (y : V) : (-1)*f y+b^2 ≤ distinctClusterMoment G p x y := by
-    have h := distinctClusterMoment_lower G hloc hqt p hp x y
-    have hprod : b*b ≤ susceptibility G p x * susceptibility G p y :=
-      mul_le_mul (hχ x) (hχ y) hb (susceptibility_nonneg G p x)
-    dsimp [f]
-    nlinarith only [h,hprod]
-  have hi := realAction_mono A hA ha (fun y => (-1)*f y+b^2)
-    (distinctClusterMoment G p x) hpnt x
-  rw [realAction_affine A ha f (-1) (b^2) x,heq] at hi
-  nlinarith only [hi,hf]
-
-end LowerDistinctMomentOperator
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section DistinctMomentSandwich
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
-
- 
-
-theorem normalized_power_moment_sandwich [Countable V] [Countable E] [Nonempty V]
-    (hqt : QuasiTransitive G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (htr : RootTransport root weight Γ) (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b)
-    (hw : ∀ i, 0 ≤ weight i) (i₀ : ι) (hw₀ : 0 < weight i₀)
-    (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i))
-    (q : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) M L)
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (p : unitInterval) (hp : 0 < (p:ℝ)) (hpc : p < pc G)
-    (hu : 0 < q.successProbability p (root i₀))
-    (A : Hilbert V →L[ℂ] Hilbert V)
-    (hA : ∀ x y, matrixCoeff A x y = (q.normalizedKernel (root i₀) p x y : ℂ))
-    (a : ℝ) (ha : 0 ≤ a) (hχ : ∀ x, a*rootSusceptibility G root weight p ≤ susceptibility G p x)
-    (Z : ℝ) (hZ : 2 ≤ Z)
-    (hscale : (q.successProbability p (root i₀))⁻¹*(1+(1-(p:ℝ))/p*M) ≤ Z) (j : ℕ) :
-    (a*rootSusceptibility G root weight p)^2 - ‖A‖^j*(c*rootSusceptibility G root weight p)^3 ≤
-      ((q.successProbability p (root i₀))⁻¹*(p:ℝ)^(-(M:ℤ))*(∑ i, ((reachBall G (root i) M).ncard:ℝ))*c)*
-        rootNeighborMoment G hloc root weight p * Z^j := by
-  have hpos := q.normalized_operator_positive (root i₀) p A hA
-  have hfin := q.normalized_operator_finite (root i₀) p A hA
-  have hrow := q.normalized_operator_row_le (root i₀) p hu A hA
-  have hend := q.normalized_operator_end_support (root i₀) p A hA
-  have hx : root i₀ ∈ vertexOrbit G (root i₀) := ⟨Equiv.refl V,Equiv.refl E,(fun _ => by simp),rfl⟩
-  have hlo := realAction_distinctMoment_lower G hloc hqt root weight Γ htr hΓ hw i₀ hw₀ hcover c hc hmin p hpc
-    (a*rootSusceptibility G root weight p) (mul_nonneg ha (rootSusceptibility_nonneg G root weight hw p)) hχ
-    (A^j) (hpos.pow j) (finite_positive_pow_support A hpos hfin j)
-    ((q.normalized_operator_invariant (root i₀) p A hA).pow j)
-    (realAction_pow_row_le A hpos hfin hrow j)
-    (realAction_pow_row_eq_on A hpos hfin _ hend
-      (q.normalized_operator_row_eq (root i₀) p hu A hA) j (root i₀) hx)
-    (positive_pow_end_support A hpos _ hend j (root i₀) · hx)
-  have hhi := q.normalized_power_distinctMoment_upper (root i₀) hqt root weight hw hcover c hc hmin p hp hpc hu A hA Z hZ hscale j (root i₀)
-  have hpow : ‖A^j‖ ≤ ‖A‖^j := by
-    cases j with
-    | zero =>
-      simp only [pow_zero]
-      have he : (1 : Hilbert V →L[ℂ] Hilbert V) = ContinuousLinearMap.id ℂ (Hilbert V) := by ext x; rfl
-      rw [he]
-      exact ContinuousLinearMap.norm_id_le
-    | succ j => exact norm_pow_le' A (Nat.succ_pos j)
-  have hn := mul_le_mul_of_nonneg_right hpow
-    (pow_nonneg (mul_nonneg hc (rootSusceptibility_nonneg G root weight hw p)) 3)
-  linarith
-
-end LocalEndpointRule
-end DistinctMomentSandwich
-end Percolation
-
- 
-namespace Percolation
-section LogarithmicPowerChoice
-
- 
-
-theorem exists_logarithmic_power (r h Z T : ℝ) (hr : 0 ≤ r) (hh : 0 < h)
-    (hrh : r ≤ Real.exp (-h)) (hZ : 1 ≤ Z) (hT : 1 ≤ T) :
-    ∃ j : ℕ, r^j ≤ T⁻¹ ∧ Z^j ≤ Z*T^(Real.log Z/h) := by
-  let t := Real.log T/h
-  have ht : 0 ≤ t := div_nonneg (Real.log_nonneg hT) hh.le
-  let j := Nat.ceil t
-  have hlo : t ≤ (j:ℝ) := Nat.le_ceil t
-  have hhi : (j:ℝ) ≤ t+1 := (Nat.ceil_lt_add_one ht).le
-  have hTp : 0 < T := zero_lt_one.trans_le hT
-  have hZp : 0 < Z := zero_lt_one.trans_le hZ
-  have hlogZ : 0 ≤ Real.log Z := Real.log_nonneg hZ
-  refine ⟨j,?_,?_⟩
-  · calc
-      r^j ≤ (Real.exp (-h))^j := pow_le_pow_left₀ hr hrh j
-      _ = Real.exp ((j:ℝ)*(-h)) := (Real.exp_nat_mul _ _).symm
-      _ ≤ Real.exp (-Real.log T) := by
-        apply Real.exp_le_exp.mpr
-        have hl : Real.log T ≤ (j:ℝ)*h := (div_le_iff₀ hh).mp hlo
-        nlinarith
-      _ = T⁻¹ := by rw [Real.exp_neg,Real.exp_log hTp]
-  · calc
-      Z^j = Real.exp ((j:ℝ)*Real.log Z) := by rw [Real.exp_nat_mul,Real.exp_log hZp]
-      _ ≤ Real.exp ((t+1)*Real.log Z) := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right hhi hlogZ)
-      _ = Z*T^(Real.log Z/h) := by
-        rw [Real.rpow_def_of_pos hTp,←Real.exp_log hZp,←Real.exp_add]
-        simp only [Real.log_exp]
-        congr 1
-        dsimp [t]
-        ring
-
- 
-
-theorem moment_sandwich_power_law (a c C Z h r χ N : ℝ)
-    (ha : 0 < a) (hc : 0 ≤ c) (hC : 0 < C) (hZ : 1 ≤ Z) (hh : 0 < h)
-    (hr : 0 ≤ r) (hrh : r ≤ Real.exp (-h)) (hχ : 1 ≤ χ) (hN : 0 ≤ N)
-    (hs : ∀ j : ℕ, (a*χ)^2-r^j*(c*χ)^3 ≤ C*N*Z^j) :
-    let T := 1+2*c^3/a^2
-    a^2/(2*C*Z*T^(Real.log Z/h))*χ^(2-Real.log Z/h) ≤ N := by
-  let T := 1+2*c^3/a^2
-  have hap : 0 < a^2 := sq_pos_of_pos ha
-  have hT : 1 ≤ T := by
-    have hn : 0 ≤ 2*c^3/a^2 := by positivity
-    dsimp [T]
-    linarith
-  have hTp : 0 < T := zero_lt_one.trans_le hT
-  have hχp : 0 < χ := zero_lt_one.trans_le hχ
-  have hZp : 0 < Z := zero_lt_one.trans_le hZ
-  have hTx : 1 ≤ T*χ := by nlinarith
-  obtain ⟨j,hrj,hZj⟩ := exists_logarithmic_power r h Z (T*χ) hr hh hrh hZ hTx
-  have hbudget : 2*c^3 ≤ a^2*T := by dsimp [T]; field_simp; nlinarith
-  have hsmall : r^j*(c*χ)^3 ≤ a^2/2*χ^2 := by
-    calc
-      _ ≤ (T*χ)⁻¹*(c*χ)^3 := mul_le_mul_of_nonneg_right hrj (pow_nonneg (mul_nonneg hc hχp.le) _)
-      _ = c^3/T*χ^2 := by field_simp
-      _ ≤ a^2/2*χ^2 := mul_le_mul_of_nonneg_right ((div_le_iff₀ hTp).mpr (by nlinarith [hbudget])) (sq_nonneg χ)
-  have hlarge : a^2/2*χ^2 ≤ C*N*(Z*(T*χ)^(Real.log Z/h)) := by
-    have hs' := hs j
-    have hprod := mul_le_mul_of_nonneg_left hZj (mul_nonneg hC.le hN)
-    nlinarith only [hsmall,hs',hprod]
-  have hpow : 0 < χ^(Real.log Z/h) := Real.rpow_pos_of_pos hχp _
-  have hTpow : 0 < T^(Real.log Z/h) := Real.rpow_pos_of_pos hTp _
-  have he : C*N*(Z*(T*χ)^(Real.log Z/h)) =
-      (C*Z*T^(Real.log Z/h))*N*χ^(Real.log Z/h) := by
-    rw [Real.mul_rpow hTp.le hχp.le]
-    ring
-  rw [he] at hlarge
-  have hden : 0 < 2*C*Z*T^(Real.log Z/h) := by positivity
-  change a^2/(2*C*Z*T^(Real.log Z/h))*χ^(2-Real.log Z/h) ≤ N
-  rw [Real.rpow_sub hχp,Real.rpow_two]
-  calc
-    _ = (a^2*χ^2)/(2*C*Z*T^(Real.log Z/h)*χ^(Real.log Z/h)) := by
-      rw [div_mul_div_comm]
-    _ ≤ N := (div_le_iff₀ (mul_pos hden hpow)).mpr (by nlinarith only [hlarge])
-
-end LogarithmicPowerChoice
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open Filter
-section ProfileContinuity
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace EndpointProfile
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {p₀ : unitInterval} {o : V} {s : ℝ}
-
-theorem exp_neg_H (q : EndpointProfile G hloc p₀ o s) : Real.exp (-q.H) = ‖q.op‖ := by
-  rw [H,one_div,Real.log_inv,neg_neg,Real.exp_log q.norm_pos]
-
- 
-
-theorem exists_nearby_operators {ι : Type*} [Fintype ι]
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (htr : RootTransport root weight Γ) (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b)
-    (hw : ∀ i, 0 ≤ weight i) (i₀ : ι) (hw₀ : 0 < weight i₀)
-    (q : EndpointProfile G hloc p₀ (root i₀) s) (hp₀ : 0 < (p₀:ℝ)) :
-    ∃ A : unitInterval → (Hilbert V →L[ℂ] Hilbert V),
-      (∀ p x y, matrixCoeff (A p) x y = (q.rule.normalizedKernel (root i₀) p x y : ℂ)) ∧
-      ∀ᶠ p in nhds p₀, q.rule.successProbability p₀ (root i₀)/2 ≤ q.rule.successProbability p (root i₀) ∧
-        ‖A p‖ ≤ Real.exp (-(q.H-1)) ∧ (p₀:ℝ)/2 ≤ (p:ℝ) := by
-  obtain ⟨A,_,hA,hcont⟩ := q.rule.exists_continuous_normalized_operators root weight Γ htr hΓ hw i₀ hw₀
-  have he : A p₀ = q.op := kernel_extension_unique (hA p₀) q.coeff
-  have hnorm : ContinuousAt (fun p => ‖A p‖) p₀ := (hcont p₀ q.success_pos.ne').norm
-  have hb : ‖A p₀‖ < Real.exp (-(q.H-1)) := by
-    rw [he,←q.exp_neg_H]
-    exact Real.exp_lt_exp.mpr (by linarith)
-  have ha := hnorm.eventually_lt continuousAt_const hb
-  have hu : ∀ᶠ p in nhds p₀, q.rule.successProbability p₀ (root i₀)/2 < q.rule.successProbability p (root i₀) :=
-    continuousAt_const.eventually_lt (q.rule.continuous_successProbability (root i₀)).continuousAt (by linarith [q.success_pos])
-  have hp : ∀ᶠ p : unitInterval in nhds p₀, (p₀:ℝ)/2 < (p:ℝ) :=
-    continuousAt_const.eventually_lt continuous_subtype_val.continuousAt (by linarith)
-  refine ⟨A,hA,?_⟩
-  filter_upwards [ha,hu,hp] with p ha hu hp
-  exact ⟨hu.le,ha.le,hp.le⟩
-
-end EndpointProfile
-end ProfileContinuity
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open Filter
-section MapScaleBudget
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace EndpointProfile
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {p₀ : unitInterval} {o : V} {s : ℝ}
-
-noncomputable def potentialScale (q : EndpointProfile G hloc p₀ o s) (p : unitInterval) : ℝ :=
-  (q.rule.successProbability p o)⁻¹*(1+(1-(p:ℝ))/p*q.cap)
-
-noncomputable def geometricScale (q : EndpointProfile G hloc p₀ o s) : ℝ := 2*q.potentialScale p₀
-
-theorem potentialScale_pos (q : EndpointProfile G hloc p₀ o s) (hp₀ : 0 < (p₀:ℝ)) :
-    0 < q.potentialScale p₀ := by
-  exact mul_pos (inv_pos.mpr q.success_pos) (by
-    have hn : 0 ≤ (1-(p₀:ℝ))/p₀*q.cap := mul_nonneg (div_nonneg (sub_nonneg.mpr p₀.2.2) hp₀.le) (Nat.cast_nonneg _)
-    linarith)
-
-theorem geometricScale_two_le (q : EndpointProfile G hloc p₀ o s) (hp₀ : 0 < (p₀:ℝ)) :
-    2 ≤ q.geometricScale := by
-  have hu : 1 ≤ (q.rule.successProbability p₀ o)⁻¹ :=
-    (one_le_inv₀ q.success_pos).mpr (q.rule.successProbability_le_one p₀ o)
-  have hb : 0 ≤ (1-(p₀:ℝ))/p₀*q.cap :=
-    mul_nonneg (div_nonneg (sub_nonneg.mpr p₀.2.2) hp₀.le) (Nat.cast_nonneg _)
-  dsimp [geometricScale,potentialScale]
-  nlinarith
-
-theorem log_geometricScale_le (q : EndpointProfile G hloc p₀ o s) (hp₀ : 0 < (p₀:ℝ))
-    (hM : 1 ≤ q.cap) :
-    Real.log q.geometricScale ≤ q.U+q.m+Real.log (2*(1+1/(p₀:ℝ))) := by
-  have hM' : (1:ℝ) ≤ q.cap := by exact_mod_cast hM
-  have hMp : (0:ℝ) < q.cap := zero_lt_one.trans_le hM'
-  have hb : 1+(1-(p₀:ℝ))/p₀*q.cap ≤ (1+1/(p₀:ℝ))*q.cap := by
-    apply (mul_le_mul_iff_of_pos_right hp₀).mp
-    field_simp
-    nlinarith
-  have Hz : q.geometricScale ≤ (1/q.rule.successProbability p₀ o)*(q.cap:ℝ)*(2*(1+1/(p₀:ℝ))) := by
-    have h := mul_le_mul_of_nonneg_left hb (mul_nonneg (by norm_num : (0:ℝ) ≤ 2) (inv_nonneg.mpr q.success_pos.le))
-    simpa only [geometricScale,potentialScale,one_div,mul_assoc,mul_comm,mul_left_comm] using h
-  have hc : 0 < 2*(1+1/(p₀:ℝ)) := by positivity
-  have h := Real.log_le_log (lt_of_lt_of_le (by norm_num : (0:ℝ)<2) (q.geometricScale_two_le hp₀)) Hz
-  rw [Real.log_mul (mul_ne_zero (one_div_ne_zero q.success_pos.ne') hMp.ne') hc.ne',
-    Real.log_mul (one_div_ne_zero q.success_pos.ne') hMp.ne'] at h
-  exact h
-
-theorem eventually_potentialScale_le (q : EndpointProfile G hloc p₀ o s) (hp₀ : 0 < (p₀:ℝ)) :
-    ∀ᶠ p in nhds p₀, q.potentialScale p ≤ q.geometricScale := by
-  have hu := (q.rule.continuous_successProbability o).continuousAt.inv₀ q.success_pos.ne'
-  have hv : ContinuousAt (fun p : unitInterval => ((1-(p:ℝ))/p)) p₀ :=
-    (continuousAt_const.sub continuous_subtype_val.continuousAt).div continuous_subtype_val.continuousAt hp₀.ne'
-  have he : ContinuousAt q.potentialScale p₀ := hu.mul (continuousAt_const.add (hv.mul continuousAt_const))
-  have hlt : q.potentialScale p₀ < q.geometricScale := by
-    dsimp only [geometricScale]
-    linarith [q.potentialScale_pos hp₀]
-  filter_upwards [he.eventually_lt continuousAt_const hlt] with p hp
-  exact hp.le
-
-end EndpointProfile
-end MapScaleBudget
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open Filter
-section ProfileNeighborMoment
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace EndpointProfile
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {p₀ : unitInterval} {o : V} {s : ℝ}
-
- 
-
-theorem eventually_neighborMoment_lower [Countable V] [Countable E] [Nonempty V]
-    (hconn : Connected G) (hqt : QuasiTransitive G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (htr : RootTransport root weight Γ) (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b)
-    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
-    (i₀ : ι) (hw₀ : 0 < weight i₀) (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i))
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (q : EndpointProfile G hloc p₀ (root i₀) s) (hp₀ : 0 < (p₀:ℝ)) (hH : 1 < q.H) :
-    ∃ b : ℝ, 0 < b ∧ ∀ᶠ p in nhds p₀, p < pc G →
-      b*(rootSusceptibility G root weight p)^(2-Real.log q.geometricScale/(q.H-1)) ≤
-        rootNeighborMoment G hloc root weight p := by
-  obtain ⟨a,ha,hχ⟩ := exists_rootSusceptibility_lower G hconn hloc hqt root weight hw hw1 hcover
-    ((p₀:ℝ)/2) (by linarith)
-  obtain ⟨A,hA,hnear⟩ := q.exists_nearby_operators root weight Γ htr hΓ hw i₀ hw₀ hp₀
-  let B : unitInterval → ℝ := fun p =>
-    (q.rule.successProbability p (root i₀))⁻¹*(p:ℝ)^(-(q.cap:ℤ))*(∑ i, ((reachBall G (root i) q.cap).ncard:ℝ))*c
-  have hB (p : unitInterval) : 0 ≤ B p := by
-    dsimp [B]
-    exact mul_nonneg (mul_nonneg (mul_nonneg (inv_nonneg.mpr (q.rule.successProbability_nonneg p _))
-      (zpow_nonneg p.2.1 _)) (Finset.sum_nonneg (fun _ _ => Nat.cast_nonneg _))) hc
-  have hBc : ContinuousAt B p₀ := by
-    exact ((((q.rule.continuous_successProbability (root i₀)).continuousAt.inv₀ q.success_pos.ne').mul
-      (continuous_subtype_val.continuousAt.zpow₀ (-(q.cap:ℤ)) (Or.inl hp₀.ne'))).mul continuousAt_const).mul continuousAt_const
-  let C := B p₀+1
-  have hC : 0 < C := by dsimp [C]; linarith [hB p₀]
-  have hCB : ∀ᶠ p in nhds p₀, B p ≤ C := by
-    filter_upwards [hBc.eventually_lt continuousAt_const (show B p₀ < C by dsimp [C]; linarith)] with p hp
-    exact hp.le
-  let Z := q.geometricScale
-  let h := q.H-1
-  let T := 1+2*c^3/a^2
-  let b := a^2/(2*C*Z*T^(Real.log Z/h))
-  have hz : 2 ≤ Z := q.geometricScale_two_le hp₀
-  have ht : 0 < T := by
-    have hh : 0 ≤ 2*c^3/a^2 := by positivity
-    dsimp [T]
-    linarith
-  have hb : 0 < b := div_pos (sq_pos_of_pos ha) (by
-    have hp := Real.rpow_pos_of_pos ht (Real.log Z/h)
-    have hZp : 0 < Z := by linarith
-    positivity)
-  refine ⟨b,hb,?_⟩
-  filter_upwards [hnear,hCB,q.eventually_potentialScale_le hp₀] with p hnear hCB hscale
-  intro hpc
-  obtain ⟨hu,hAn,hpp⟩ := hnear
-  have hp : 0 < (p:ℝ) := lt_of_lt_of_le (by linarith : 0 < (p₀:ℝ)/2) hpp
-  have hup : 0 < q.rule.successProbability p (root i₀) :=
-    lt_of_lt_of_le (by linarith [q.success_pos]) hu
-  have hχ1 : 1 ≤ rootSusceptibility G root weight p := by
-    calc
-      1 = ∑ i, weight i*1 := by simp only [mul_one,hw1]
-      _ ≤ _ := Finset.sum_le_sum (fun i _ => mul_le_mul_of_nonneg_left
-        (susceptibility_one_le G hloc hqt p hpc (root i)) (hw i))
-  apply moment_sandwich_power_law a c C Z h ‖A p‖ (rootSusceptibility G root weight p)
-    (rootNeighborMoment G hloc root weight p) ha hc hC (by linarith) (by dsimp [h]; linarith)
-    (norm_nonneg _) hAn hχ1 (rootNeighborMoment_nonneg G hloc root weight hw p)
-  intro j
-  have hs := q.rule.normalized_power_moment_sandwich hqt root weight Γ htr hΓ hw i₀ hw₀ hcover c hc hmin p hp hpc hup
-    (A p) (hA p) a ha.le (hχ p hpp hpc) Z hz hscale j
-  exact hs.trans (mul_le_mul_of_nonneg_right
-    (mul_le_mul_of_nonneg_right hCB (rootNeighborMoment_nonneg G hloc root weight hw p)) (pow_nonneg (by linarith) j))
-
-end EndpointProfile
-end ProfileNeighborMoment
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section EfficientProfileMomentConsequence
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem neighborMoment_growth_of_divergent_moment
-    [Countable V] [Countable E] [Nonempty V] [Nonempty E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hconn : Connected G)
-    (hqt : QuasiTransitive G) (hh : 0 < hV G)
-    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
-    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
-    (p₀ : unitInterval) (hp₀ : 0 < (p₀:ℝ)) (hp₁ : (p₀:ℝ) < 1)
-    (hfin : ∀ i, ∀ᵐ ω ∂law p₀, (cluster G ω (root i)).Finite)
-    (α β ε : ℝ) (hα : 0 < α) (hαβ : α < β)
-    (hε : 0 < ε) (hε1 : ε < 1/2) (hsmall : β+(1/2+ε) < 1)
-    (hdiv : ∃ i, ¬ Integrable (fun ω => ((cluster G ω (root i)).ncard:ℝ)^α) (law p₀)) :
-    ∀ γ : ℝ, 0 < γ → ∃ b : ℝ, 0 < b ∧ ∀ᶠ p in nhds p₀, p < pc G →
-      b*(rootSusceptibility G root weight p)^(2-γ) ≤ rootNeighborMoment G hloc root weight p := by
-  obtain ⟨i,heff⟩ := efficient_endpoint_profiles_of_divergent_moment G hloc hconn hqt hh root weight Γ hΓ htr
-    hcov hw hw1 p₀ hp₀ hp₁ hfin α β ε hα hαβ hε hε1 hsmall hdiv
-  have hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i) := by
-    intro x
-    obtain ⟨i,a,ha,hax⟩ := hcov x
-    obtain ⟨b,hab⟩ := hΓ ha
-    exact ⟨i,a,b,hab,hax⟩
-  let c : ℝ := ∑ i, 1/weight i
-  have hc : 0 ≤ c := Finset.sum_nonneg (fun i _ => (one_div_pos.mpr (hw i)).le)
-  have hmin (i : ι) : 1 ≤ c*weight i := by
-    apply (div_le_iff₀ (hw i)).mp
-    exact Finset.single_le_sum (fun j _ => (one_div_pos.mpr (hw j)).le) (Finset.mem_univ i)
-  intro γ hγ
-  let C := Real.log (2*(1+1/(p₀:ℝ)))
-  have hC : 0 ≤ C := Real.log_nonneg (by
-    have hinv : 0 ≤ 1/(p₀:ℝ) := (one_div_pos.mpr hp₀).le
-    nlinarith)
-  obtain ⟨q,hM,hH,hR,hcost⟩ := heff (γ/4) (by linarith) (2+4*C/γ)
-  have hH2 : 2 < q.H := by
-    have hn : 0 ≤ 4*C/γ := div_nonneg (mul_nonneg (by norm_num) hC) hγ.le
-    linarith
-  have hCsmall : C < γ*q.H/4 := by
-    have hx : 4*C < (q.H-2)*γ := (div_lt_iff₀ hγ).mp (by linarith : 4*C/γ < q.H-2)
-    nlinarith
-  have hexp : Real.log q.geometricScale/(q.H-1) ≤ γ := by
-    apply (div_le_iff₀ (by linarith : 0 < q.H-1)).mpr
-    have hz := q.log_geometricScale_le hp₀ (by omega)
-    have hm : γ*q.H/2 ≤ γ*(q.H-1) := by nlinarith
-    dsimp [C] at hCsmall
-    nlinarith only [hz,hcost,hCsmall,hm]
-  obtain ⟨b,hb,hnear⟩ := q.eventually_neighborMoment_lower hconn hqt root weight Γ htr
-    (fun a ha => hΓ ha) (fun i => (hw i).le) hw1 i (hw i) hcover c hc hmin hp₀ hH
-  refine ⟨b,hb,?_⟩
-  filter_upwards [hnear] with p hp
-  intro hpc
-  have hχ : 1 ≤ rootSusceptibility G root weight p := by
-    calc
-      1 = ∑ i, weight i*1 := by simp only [mul_one,hw1]
-      _ ≤ _ := Finset.sum_le_sum (fun i _ => mul_le_mul_of_nonneg_left
-        (susceptibility_one_le G hloc hqt p hpc (root i)) (hw i).le)
-  exact (mul_le_mul_of_nonneg_left (Real.rpow_le_rpow_of_exponent_le hχ (by linarith :
-    2-γ ≤ 2-Real.log q.geometricScale/(q.H-1))) hb.le).trans (hp hpc)
-
-end EfficientProfileMomentConsequence
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory ProbabilityTheory Function
-section PivotalSprinklingGain
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem measurable_set_union_pair : Measurable (fun f : Set E × Set E => f.1 ∪ f.2) := by
-  apply measurable_setOfPred.comp
-  apply Measurable.of_eval
-  intro e
-  apply measurableSet_setOfPred.mp
-  change MeasurableSet ({f : Set E × Set E | e ∈ f.1} ∪ {f | e ∈ f.2})
-  apply MeasurableSet.union <;> apply measurableSet_setOfPred.mpr
-  · exact ((measurable_pi_apply e).comp MeasurableEquiv.setOfPred.symm.measurable).comp measurable_fst
-  · exact ((measurable_pi_apply e).comp MeasurableEquiv.setOfPred.symm.measurable).comp measurable_snd
-
-theorem measurable_set_insert (e : E) : Measurable (fun ω : Set E => insert e ω) := by
-  have h : Measurable (fun ω : Set E => ({e}:Set E) ∪ ω) :=
-    measurable_set_union_pair.comp (measurable_const.prodMk measurable_id)
-  simpa only [Set.singleton_union] using (h : Measurable (fun ω : Set E => {e} ∪ ω))
-
-theorem measurableSet_closedPivotal (A : Set (Set E)) (hA : MeasurableSet A) (e : E) :
-    MeasurableSet (closedPivotal A e) := by
-  exact (measurableSet_mem e).compl.inter (hA.compl.inter (hA.preimage (measurable_set_insert e)))
-
-def singleSprinkle (Q : Finset E) (e : (Q : Set E)) : Set (Set E) :=
-  {η | Subtype.val ⁻¹' η = {e}}
-
-theorem measurableSet_singleSprinkle (Q : Finset E) (e : (Q : Set E)) :
-    MeasurableSet (singleSprinkle Q e) :=
-  (measurableSet_singleton _).preimage (measurable_set_restrict (Q:Set E))
-
-theorem singleSprinkle_disjoint (Q : Finset E) :
-    Pairwise (Disjoint on singleSprinkle Q) := by
-  intro a b hab
-  apply Set.disjoint_left.mpr
-  intro η ha hb
-  apply hab
-  have h : ({a} : Set (Q:Set E)) = {b} := ha.symm.trans hb
-  simpa only [Set.singleton_eq_singleton_iff] using h
-
-theorem singleSprinkle_mem (Q : Finset E) (e : (Q:Set E)) {η : Set E}
-    (hη : η ∈ singleSprinkle Q e) : e.val ∈ η := by
-  have h : e ∈ Subtype.val ⁻¹' η := by rw [hη]; exact Set.mem_singleton e
-  exact h
-
-theorem singleSprinkle_probability (Q : Finset E) (e : (Q:Set E)) (a : unitInterval) :
-    (law a).real (singleSprinkle Q e) = (a:ℝ)*(1-(a:ℝ))^(Q.card-1) := by
-  have h := congrArg (fun μ : Measure (Set (Q:Set E)) => μ.real {({e}:Set (Q:Set E))})
-    (law_map_restrict a (Q:Set E))
-  rw [Measure.real,Measure.map_apply (measurable_set_restrict _) (measurableSet_singleton _)] at h
-  change (law a).real (singleSprinkle Q e) = _ at h
-  rw [h,law,setBernoulli_real_singleton a (Set.subset_univ _) Set.finite_univ,
-    Set.ncard_sdiff' (Set.subset_univ _) Set.finite_univ,Set.ncard_singleton]
-  simp only [Set.ncard_univ,Nat.card_eq_fintype_card,pow_one]
-  rw [show Fintype.card (Q:Set E) = Q.card from Fintype.card_coe Q]
-
- 
-
-theorem event_sprinkling_gain (p a : unitInterval) (Q : Finset E)
-    (A : Set (Set E)) (hA : MeasurableSet A) (hup : IsUpperSet A) :
-    (a:ℝ)*(1-(a:ℝ))^(Q.card-1) * (∑ e ∈ Q, (law p).real (closedPivotal A e)) ≤
-      (law (sprinkleParameter p a)).real A - (law p).real A := by
-  let μ : Measure (Set E × Set E) := (law p).prod (law a)
-  let B : Set (Set E × Set E) := {f | f.1 ∉ A ∧ f.1 ∪ f.2 ∈ A}
-  let C (e : (Q:Set E)) := (closedPivotal A e.val) ×ˢ singleSprinkle Q e
-  have hB : MeasurableSet B := (hA.compl.preimage measurable_fst).inter (hA.preimage measurable_set_union_pair)
-  have hC (e : (Q:Set E)) : MeasurableSet (C e) :=
-    (measurableSet_closedPivotal A hA e.val).prod (measurableSet_singleSprinkle Q e)
-  have hd : Pairwise (Disjoint on C) := by
-    intro e f hef
-    apply Set.disjoint_left.mpr
-    intro z hz hz'
-    exact Set.disjoint_left.mp (singleSprinkle_disjoint Q hef) hz.2 hz'.2
-  have hsub : (⋃ e, C e) ⊆ B := by
-    intro z hz
-    obtain ⟨e,he⟩ := Set.mem_iUnion.mp hz
-    refine ⟨he.1.2.1,?_⟩
-    apply hup _ he.1.2.2
-    intro f hf
-    rcases Set.mem_insert_iff.mp hf with rfl | hf
-    · exact Or.inr (singleSprinkle_mem Q e he.2)
-    · exact Or.inl hf
-  have hgain : μ.real B = (law (sprinkleParameter p a)).real A - (law p).real A := by
-    have he : (fun f : Set E × Set E => f.1 ∪ f.2) ⁻¹' A = (A ×ˢ Set.univ) ∪ B := by
-      ext f
-      simp only [Set.mem_preimage,Set.mem_union,Set.mem_prod,Set.mem_univ,and_true,B,Set.mem_ofPred_eq]
-      exact ⟨fun h => by by_cases ha : f.1 ∈ A; exact Or.inl ha; exact Or.inr ⟨ha,h⟩,
-        fun h => h.elim (fun ha => hup Set.subset_union_left ha) And.right⟩
-    have hd : Disjoint (A ×ˢ (Set.univ : Set (Set E))) B := by
-      apply Set.disjoint_left.mpr
-      intro f hf hf'
-      exact hf'.1 hf.1
-    have hm := congrArg (fun ν : Measure (Set E) => ν.real A) (law_map_union p a)
-    rw [Measure.real,Measure.map_apply measurable_set_union_pair hA] at hm
-    change μ.real ((fun f : Set E × Set E => f.1 ∪ f.2) ⁻¹' A) = _ at hm
-    rw [he,measureReal_union hd hB] at hm
-    have hp : μ.real (A ×ˢ Set.univ) = (law p).real A := by
-      rw [measureReal_prod_prod,probReal_univ,mul_one]
-    rw [hp] at hm
-    linarith
-  have hs : μ.real (⋃ e, C e) =
-      (a:ℝ)*(1-(a:ℝ))^(Q.card-1) * (∑ e ∈ Q, (law p).real (closedPivotal A e)) := by
-    rw [measureReal_iUnion_fintype hd hC]
-    simp only [C,μ,measureReal_prod_prod,singleSprinkle_probability]
-    rw [←Finset.sum_mul,mul_comm]
-    congr 1
-    apply Finset.sum_bij (fun e _ => e.val)
-    · intro e _; exact e.property
-    · intro e _ f _ h; exact Subtype.ext h
-    · intro e he; exact ⟨⟨e,he⟩,Finset.mem_univ _,rfl⟩
-    · intro _ _; rfl
-  rw [←hs,←hgain]
-  exact measureReal_mono hsub
-
-end PivotalSprinklingGain
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Clusters.ExteriorMoments
+
+namespace OAI
+
+/-! Distinct Moments for bond percolation. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section ExposedBoundaryPotential
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def exposedVertexPotential (G : BondGraph V E) (p : unitInterval)
+    (x a b : V) (ω : Set E) : ℝ :=
+  if Conn G ω x a then freshExteriorMoment G p a b ω else 0
+
+theorem integrable_exposedVertexPotential [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (x a b : V) :
+    Integrable (exposedVertexPotential G p x a b) (law p) := by
+  have he : exposedVertexPotential G p x a b = {ω | Conn G ω x a}.indicator (freshExteriorMoment G p a b) := by
+    funext ω
+    simp only [exposedVertexPotential,Set.indicator,Set.mem_ofPred_eq]
+  rw [he]
+  exact (integrable_freshExteriorMoment G hloc hqt p hp a b).indicator (measurableSet_conn G x a)
+
+theorem integral_exposedVertexPotential_le [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (x a b : V) :
+    (∫ ω, exposedVertexPotential G p x a b ω ∂law p) ≤ distinctClusterMoment G p a b := by
+  rw [←show (∫ ω, freshExteriorMoment G p a b ω ∂law p) = distinctClusterMoment G p a b from
+    integral_freshExteriorMoment G hloc hqt p hp a b]
+  apply integral_mono (integrable_exposedVertexPotential G hloc hqt p hp x a b)
+    (integrable_freshExteriorMoment G hloc hqt p hp a b)
+  intro ω
+  unfold exposedVertexPotential
+  split_ifs
+  · rfl
+  · exact freshExteriorMoment_nonneg G p a b ω
+
+theorem exposedBoundaryPotential_eq (G : BondGraph V E) (hloc : LocallyFinite G)
+    (p : unitInterval) (x v : V) (M : ℕ) (ω : Set E) :
+    ((cluster G ω x).ncard:ℝ)*deletedBoundaryPotential G hloc (cluster G ω x)ᶜ p v M =
+      ∑ a ∈ (finite_reachBall hloc v M).toFinset,
+        ∑ b ∈ neighbors G hloc a, exposedVertexPotential G p x a b ω := by
+  unfold deletedBoundaryPotential
+  rw [Finset.sum_filter,Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro a _
+  by_cases ha : Conn G ω x a
+  · have hc := cluster_eq_of_conn G ω ha
+    have hn : a ∉ (cluster G ω x)ᶜ := by exact not_not.mpr ha
+    simp only [ite_eq_left hn,Finset.mul_sum]
+    simp only [exposedVertexPotential,ite_eq_left ha,freshExteriorMoment,hc]
+  · have hn : ¬a ∉ (cluster G ω x)ᶜ := by exact not_not.mpr ha
+    simp only [ite_eq_right hn,mul_zero,exposedVertexPotential,ite_eq_right ha,Finset.sum_const_zero]
+
+ 
+
+theorem integral_exposedBoundaryPotential_le [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (x v : V) (M : ℕ) :
+    Integrable (fun ω => ((cluster G ω x).ncard:ℝ)*deletedBoundaryPotential G hloc (cluster G ω x)ᶜ p v M) (law p) ∧
+    (∫ ω, ((cluster G ω x).ncard:ℝ)*deletedBoundaryPotential G hloc (cluster G ω x)ᶜ p v M ∂law p) ≤
+      ∑ a ∈ (finite_reachBall hloc v M).toFinset, neighborClusterMoment G hloc p a := by
+  simp_rw [exposedBoundaryPotential_eq G hloc p x v M]
+  have hi (a : V) := integrable_finsetSum (neighbors G hloc a)
+    (fun b _ => integrable_exposedVertexPotential G hloc hqt p hp x a b)
+  refine ⟨integrable_finsetSum _ (fun a _ => hi a),?_⟩
+  rw [integral_finsetSum _ (fun a _ => hi a)]
+  apply Finset.sum_le_sum
+  intro a _
+  rw [integral_finsetSum _ (fun b _ => integrable_exposedVertexPotential G hloc hqt p hp x a b)]
+  exact Finset.sum_le_sum (fun b _ => integral_exposedVertexPotential_le G hloc hqt p hp x a b)
+
+end ExposedBoundaryPotential
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section UniformExposedBoundary
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem reachBall_ncard_covariant (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (x : V) (M : ℕ) :
+    (reachBall G (a x) M).ncard = (reachBall G x M).ncard := by
+  have he : reachBall G (a x) M = a '' reachBall G x M := by
+    ext y
+    constructor
+    · intro h
+      exact ⟨a.symm y,by simpa only [Equiv.symm_apply_apply] using mem_reachBall_aut G (isAut_symm G hab) h,by simp⟩
+    · rintro ⟨z,hz,rfl⟩
+      exact mem_reachBall_aut G hab hz
+  rw [he,Set.ncard_image_of_injective _ a.injective]
+
+theorem reachBall_card_le_root_sum (G : BondGraph V E) (hloc : LocallyFinite G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i))
+    (M : ℕ) (x : V) : (finite_reachBall hloc x M).toFinset.card ≤
+      ∑ i, (reachBall G (root i) M).ncard := by
+  obtain ⟨i,a,b,hab,hax⟩ := hcover x
+  rw [←Set.ncard_eq_toFinset_card (reachBall G x M) (finite_reachBall hloc x M),←hax,reachBall_ncard_covariant G hab]
+  exact Finset.single_le_sum (f := fun i => (reachBall G (root i) M).ncard) (fun _ _ => Nat.zero_le _) (Finset.mem_univ i)
+
+theorem rootNeighborMoment_nonneg (G : BondGraph V E) (hloc : LocallyFinite G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 ≤ weight i)
+    (p : unitInterval) : 0 ≤ rootNeighborMoment G hloc root weight p :=
+  Finset.sum_nonneg (fun i _ => mul_nonneg (hw i) (neighborClusterMoment_nonneg G hloc p _))
+
+ 
+
+theorem integral_exposedBoundaryPotential_uniform [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (hw : ∀ i, 0 ≤ weight i) (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i))
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (p : unitInterval) (hp : p < pc G) (x v : V) (M : ℕ) :
+    (∫ ω, ((cluster G ω x).ncard:ℝ)*deletedBoundaryPotential G hloc (cluster G ω x)ᶜ p v M ∂law p) ≤
+      (∑ i, ((reachBall G (root i) M).ncard:ℝ))*c*rootNeighborMoment G hloc root weight p := by
+  let B := (finite_reachBall hloc v M).toFinset
+  calc
+    _ ≤ ∑ a ∈ B, neighborClusterMoment G hloc p a :=
+      (integral_exposedBoundaryPotential_le G hloc hqt p hp x v M).2
+    _ ≤ ∑ _a ∈ B, c*rootNeighborMoment G hloc root weight p :=
+      Finset.sum_le_sum (fun a _ => neighborMoment_le_rootMoment G hloc root weight hw hcover c hc hmin p a)
+    _ = (B.card:ℝ)*(c*rootNeighborMoment G hloc root weight p) := by rw [Finset.sum_const,nsmul_eq_mul]
+    _ ≤ _ := by
+      have hb : (B.card:ℝ) ≤ ∑ i, ((reachBall G (root i) M).ncard:ℝ) := by
+        exact_mod_cast reachBall_card_le_root_sum G hloc root hcover M v
+      simpa only [mul_assoc] using mul_le_mul_of_nonneg_right hb
+        (mul_nonneg hc (rootNeighborMoment_nonneg G hloc root weight hw p))
+
+end UniformExposedBoundary
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section DistinctMomentPotential
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
+
+ 
+
+theorem kernel_distinctMoment_integral [Countable V] [Countable E] [Nonempty V]
+    (q : LocalEndpointRule G hloc O M L) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (x v : V) :
+    (∑' y, q.kernel p v y * distinctClusterMoment G p x y) =
+      ∫ ω, (∑' y, q.kernel p v y * freshExteriorMoment G p x y ω) ∂law p := by
+  rw [integral_tsum_of_finite_support (law p) (finite_reachBall hloc v M).toFinset _
+    (fun y => (integrable_freshExteriorMoment G hloc hqt p hp x y).const_mul (q.kernel p v y))
+    (fun ω y hy => by rw [q.kernel_outside_ball_zero p v y (by simpa only [Set.Finite.mem_toFinset] using hy),zero_mul])]
+  apply tsum_congr
+  intro y
+  rw [integral_const_mul,integral_freshExteriorMoment G hloc hqt p hp]
+  rfl
+
+theorem integrable_kernel_freshMoment [Countable V] [Countable E] [Nonempty V]
+    (q : LocalEndpointRule G hloc O M L) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (x v : V) :
+    Integrable (fun ω => ∑' y, q.kernel p v y * freshExteriorMoment G p x y ω) (law p) := by
+  have he (ω) : (∑' y, q.kernel p v y * freshExteriorMoment G p x y ω) =
+      ∑ y ∈ (finite_reachBall hloc v M).toFinset, q.kernel p v y * freshExteriorMoment G p x y ω :=
+    tsum_eq_sum (fun y hy => by rw [q.kernel_outside_ball_zero p v y (by simpa only [Set.Finite.mem_toFinset] using hy),zero_mul])
+  simp_rw [he]
+  exact integrable_finsetSum _ (fun y _ => (integrable_freshExteriorMoment G hloc hqt p hp x y).const_mul _)
+
+ 
+theorem kernel_distinctMoment_bound [Countable V] [Countable E] [Nonempty V]
+    (q : LocalEndpointRule G hloc O M L) (hqt : QuasiTransitive G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (hw : ∀ i, 0 ≤ weight i) (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i))
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (p : unitInterval) (hp : 0 < (p:ℝ)) (hpc : p < pc G) (x v : V) :
+    (∑' y, q.kernel p v y * distinctClusterMoment G p x y) ≤
+      (1+(1-(p:ℝ))/p*M)*distinctClusterMoment G p x v +
+        (p:ℝ)^(-(M:ℤ))*(∑ i, ((reachBall G (root i) M).ncard:ℝ))*c*rootNeighborMoment G hloc root weight p := by
+  let C := 1+(1-(p:ℝ))/p*M
+  let R := (p:ℝ)^(-(M:ℤ))
+  let b := fun ω : Set E => ((cluster G ω x).ncard:ℝ)*deletedBoundaryPotential G hloc (cluster G ω x)ᶜ p v M
+  have hi := (integral_exposedBoundaryPotential_le G hloc hqt p hpc x v M).1
+  have hpoint (ω : Set E) : (∑' y, q.kernel p v y * freshExteriorMoment G p x y ω) ≤
+      C*freshExteriorMoment G p x v ω + R*b ω := by
+    have h := mul_le_mul_of_nonneg_left (q.kernel_deleted_potential_bound hqt (cluster G ω x)ᶜ p hp hpc v)
+      (Nat.cast_nonneg (cluster G ω x).ncard : (0:ℝ) ≤ _)
+    rw [←tsum_mul_left] at h
+    calc
+      _ = ∑' y, ((cluster G ω x).ncard:ℝ)*(q.kernel p v y * inducedSusceptibility G (cluster G ω x)ᶜ p y) := by
+        apply tsum_congr
+        intro y
+        simp only [freshExteriorMoment]
+        ring
+      _ ≤ _ := h
+      _ = _ := by dsimp [C,R,b,freshExteriorMoment]; ring
+  rw [q.kernel_distinctMoment_integral hqt p hpc x v]
+  calc
+    _ ≤ ∫ ω, C*freshExteriorMoment G p x v ω + R*b ω ∂law p :=
+      integral_mono (q.integrable_kernel_freshMoment hqt p hpc x v)
+        (((integrable_freshExteriorMoment G hloc hqt p hpc x v).const_mul C).add (hi.const_mul R)) hpoint
+    _ = C*distinctClusterMoment G p x v + R*(∫ ω, b ω ∂law p) := by
+      rw [integral_add ((integrable_freshExteriorMoment G hloc hqt p hpc x v).const_mul C) (hi.const_mul R),
+        integral_const_mul,integral_const_mul,integral_freshExteriorMoment G hloc hqt p hpc]
+      rfl
+    _ ≤ _ := by
+      have h := mul_le_mul_of_nonneg_left
+        (integral_exposedBoundaryPotential_uniform G hloc hqt root weight hw hcover c hc hmin p hpc x v M)
+        (zpow_nonneg p.2.1 (-(M:ℤ)))
+      dsimp [C,R,b] at *
+      nlinarith
+
+end LocalEndpointRule
+end DistinctMomentPotential
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section FinitePositiveOperatorPowers
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem finite_positive_product_support (A B : Hilbert V →L[ℂ] Hilbert V)
+    (hA : PositiveRealMatrix A)
+    (ha : ∀ x, (Function.support (fun y => (matrixCoeff A x y).re)).Finite)
+    (hb : ∀ x, (Function.support (fun y => (matrixCoeff B x y).re)).Finite) (x : V) :
+    (Function.support (fun y => (matrixCoeff (A*B) x y).re)).Finite := by
+  let S := (ha x).toFinset
+  let T := S.biUnion (fun z => (hb z).toFinset)
+  apply T.finite_toSet.subset
+  intro y hy
+  by_contra hn
+  apply hy
+  change (matrixCoeff (A*B) x y).re = 0
+  rw [←(hA.mul_hasSum B x y).tsum_eq]
+  rw [←tsum_zero]
+  apply tsum_congr
+  intro z
+  by_cases hz : z ∈ S
+  · have hzy : (matrixCoeff B z y).re = 0 := by
+      by_contra hh
+      apply hn
+      exact Finset.mem_biUnion.mpr ⟨z,hz,by simpa only [Set.Finite.mem_toFinset,Function.mem_support] using hh⟩
+    rw [hzy,mul_zero]
+  · have hzx : (matrixCoeff A x z).re = 0 := by
+      simpa only [S,Set.Finite.mem_toFinset,Function.mem_support,not_not] using hz
+    rw [hzx,zero_mul]
+
+theorem finite_positive_product_action (A B : Hilbert V →L[ℂ] Hilbert V)
+    (hA : PositiveRealMatrix A)
+    (ha : ∀ x, (Function.support (fun y => (matrixCoeff A x y).re)).Finite)
+    (hb : ∀ x, (Function.support (fun y => (matrixCoeff B x y).re)).Finite)
+    (f : V → ℝ) (x : V) :
+    (∑' y, (matrixCoeff (A*B) x y).re*f y) =
+      ∑' z, (matrixCoeff A x z).re*∑' y, (matrixCoeff B z y).re*f y := by
+  let K := fun x y => (matrixCoeff A x y).re
+  let J := fun x y => (matrixCoeff B x y).re
+  let S := (ha x).toFinset
+  have hz (z : V) (hz : z ∉ S) : K x z = 0 := by
+    simpa only [S,Set.Finite.mem_toFinset,Function.mem_support,not_not] using hz
+  have he (y : V) : (matrixCoeff (A*B) x y).re*f y = ∑ z ∈ S, K x z*(J z y*f y) := by
+    rw [←(hA.mul_hasSum B x y).tsum_eq,tsum_eq_sum (s := S)
+      (fun z h => by change K x z*J z y = 0; rw [hz z h,zero_mul]),Finset.sum_mul]
+    exact Finset.sum_congr rfl (fun z _ => mul_assoc _ _ _)
+  have hs := hasSum_sum (s := S) (fun z _ =>
+    (finiteKernel_summable_mul J hb z f).hasSum.mul_left (K x z))
+  have hr : (∑' z, K x z*∑' y, J z y*f y) = ∑ z ∈ S, K x z*∑' y, J z y*f y :=
+    tsum_eq_sum (fun z h => by rw [hz z h,zero_mul])
+  simpa only [←he,←hr] using hs.tsum_eq
+
+theorem finite_positive_pow_support (A : Hilbert V →L[ℂ] Hilbert V)
+    (hA : PositiveRealMatrix A)
+    (ha : ∀ x, (Function.support (fun y => (matrixCoeff A x y).re)).Finite)
+    (j : ℕ) (x : V) : (Function.support (fun y => (matrixCoeff (A^j) x y).re)).Finite := by
+  induction j generalizing x with
+  | zero =>
+      apply (Set.finite_singleton x).subset
+      intro y hy
+      by_contra hn
+      apply hy
+      have he : x ≠ y := fun h => hn (by simp only [Set.mem_singleton_iff]; exact h.symm)
+      simp only [pow_zero,matrixCoeff,one_apply_eq_self,lp.single_apply,Pi.single_apply,ite_eq_right he,Complex.zero_re]
+  | succ j ih =>
+      rw [pow_succ']
+      exact finite_positive_product_support A (A^j) hA ha ih x
+
+theorem positive_identity_action (f : V → ℝ) (x : V) :
+    (∑' y, (matrixCoeff (1 : Hilbert V →L[ℂ] Hilbert V) x y).re*f y) = f x := by
+  rw [tsum_eq_single x (fun y hy => by
+    simp only [matrixCoeff,one_apply_eq_self,lp.single_apply,Pi.single_apply,ite_eq_right (Ne.symm hy),Complex.zero_re,zero_mul])]
+  simp only [matrixCoeff,one_apply_eq_self,lp.single_apply,Pi.single_apply,ite_true,Complex.one_re,one_mul]
+
+end FinitePositiveOperatorPowers
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section PositivePotentialIteration
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def realAction (A : Hilbert V →L[ℂ] Hilbert V) (f : V → ℝ) (x : V) : ℝ :=
+  ∑' y, (matrixCoeff A x y).re*f y
+
+theorem realAction_mono (A : Hilbert V →L[ℂ] Hilbert V) (hA : PositiveRealMatrix A)
+    (ha : ∀ x, (Function.support (fun y => (matrixCoeff A x y).re)).Finite)
+    (f g : V → ℝ) (hfg : ∀ x, f x ≤ g x) (x : V) : realAction A f x ≤ realAction A g x := by
+  exact (finiteKernel_summable_mul _ ha x f).tsum_le_tsum
+    (fun y => mul_le_mul_of_nonneg_left (hfg y) (hA x y).2) (finiteKernel_summable_mul _ ha x g)
+
+theorem realAction_affine (A : Hilbert V →L[ℂ] Hilbert V)
+    (ha : ∀ x, (Function.support (fun y => (matrixCoeff A x y).re)).Finite)
+    (f : V → ℝ) (a b : ℝ) (x : V) :
+    realAction A (fun y => a*f y+b) x = a*realAction A f x + b*realAction A (fun _ => 1) x := by
+  unfold realAction
+  simp_rw [mul_add]
+  rw [Summable.tsum_add]
+  · have he (y : V) : (matrixCoeff A x y).re*(a*f y) = a*((matrixCoeff A x y).re*f y) := by ring
+    simp_rw [he]
+    simp only [tsum_mul_left,tsum_mul_right,mul_one]
+    ring
+  · exact finiteKernel_summable_mul _ ha x (fun y => a*f y)
+  · exact finiteKernel_summable_mul _ ha x (fun _ => b)
+
+theorem realAction_pow_succ (A : Hilbert V →L[ℂ] Hilbert V) (hA : PositiveRealMatrix A)
+    (ha : ∀ x, (Function.support (fun y => (matrixCoeff A x y).re)).Finite)
+    (j : ℕ) (f : V → ℝ) (x : V) :
+    realAction (A^(j+1)) f x = realAction A (realAction (A^j) f) x := by
+  rw [pow_succ']
+  exact finite_positive_product_action A (A^j) hA ha (finite_positive_pow_support A hA ha j) f x
+
+ 
+
+theorem realAction_pow_potential_bound (A : Hilbert V →L[ℂ] Hilbert V) (hA : PositiveRealMatrix A)
+    (ha : ∀ x, (Function.support (fun y => (matrixCoeff A x y).re)).Finite)
+    (hrow : ∀ x, realAction A (fun _ => 1) x ≤ 1)
+    (f : V → ℝ) (Z B : ℝ) (hZ : 2 ≤ Z) (hB : 0 ≤ B)
+    (hf : ∀ x, realAction A f x ≤ Z*f x+B) (j : ℕ) (x : V) :
+    realAction (A^j) f x ≤ Z^j*f x + B*(Z^j-1) := by
+  have hz : 0 ≤ Z := by linarith
+  have hz1 : 1 ≤ Z := by linarith
+  induction j generalizing x with
+  | zero =>
+      simp only [pow_zero,realAction,positive_identity_action,one_mul,sub_self,mul_zero,add_zero,le_refl]
+  | succ j ih =>
+      rw [realAction_pow_succ A hA ha]
+      calc
+        _ ≤ realAction A (fun y => Z^j*f y+B*(Z^j-1)) x := realAction_mono A hA ha _ _ ih x
+        _ = Z^j*realAction A f x + (B*(Z^j-1))*realAction A (fun _ => 1) x := realAction_affine A ha f _ _ x
+        _ ≤ Z^j*(Z*f x+B) + B*(Z^j-1) := add_le_add
+          (mul_le_mul_of_nonneg_left (hf x) (pow_nonneg hz _))
+          (by
+            simpa only [mul_one] using (mul_le_mul_of_nonneg_left (hrow x)
+              (mul_nonneg hB (sub_nonneg.mpr (one_le_pow₀ hz1)))))
+        _ ≤ Z^(j+1)*f x+B*(Z^(j+1)-1) := by
+          have hh := mul_nonneg (mul_nonneg hB (pow_nonneg hz j)) (show 0 ≤ Z-2 by linarith)
+          rw [pow_succ]
+          nlinarith only [hh]
+
+theorem realAction_pow_potential_at_zero (A : Hilbert V →L[ℂ] Hilbert V) (hA : PositiveRealMatrix A)
+    (ha : ∀ x, (Function.support (fun y => (matrixCoeff A x y).re)).Finite)
+    (hrow : ∀ x, realAction A (fun _ => 1) x ≤ 1)
+    (f : V → ℝ) (Z B : ℝ) (hZ : 2 ≤ Z) (hB : 0 ≤ B)
+    (hf : ∀ x, realAction A f x ≤ Z*f x+B) (j : ℕ) (x : V) (hx : f x = 0) :
+    realAction (A^j) f x ≤ B*Z^j := by
+  have h := realAction_pow_potential_bound A hA ha hrow f Z B hZ hB hf j x
+  rw [hx,mul_zero,zero_add] at h
+  nlinarith
+
+end PositivePotentialIteration
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section NormalizedPositiveOperator
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
+
+theorem normalized_operator_positive (o : V)
+    (q : LocalEndpointRule G hloc (vertexOrbit G o) M L) (p : unitInterval)
+    (A : Hilbert V →L[ℂ] Hilbert V)
+    (hA : ∀ x y, matrixCoeff A x y = (q.normalizedKernel o p x y : ℂ)) : PositiveRealMatrix A := by
+  intro x y
+  rw [hA]
+  exact ⟨rfl,mul_nonneg (inv_nonneg.mpr (q.successProbability_nonneg p o)) (q.kernel_nonneg p x y)⟩
+
+theorem normalized_operator_finite (o : V)
+    (q : LocalEndpointRule G hloc (vertexOrbit G o) M L) (p : unitInterval)
+    (A : Hilbert V →L[ℂ] Hilbert V)
+    (hA : ∀ x y, matrixCoeff A x y = (q.normalizedKernel o p x y : ℂ)) (x : V) :
+    (Function.support (fun y => (matrixCoeff A x y).re)).Finite := by
+  apply (finite_reachBall hloc x M).subset
+  intro y hy
+  by_contra hn
+  apply hy
+  change (matrixCoeff A x y).re = 0
+  simp only [hA,normalizedKernel,q.kernel_outside_ball_zero p x y hn,mul_zero,Complex.ofReal_zero,Complex.zero_re]
+
+theorem normalized_operator_action (o : V)
+    (q : LocalEndpointRule G hloc (vertexOrbit G o) M L) (p : unitInterval)
+    (A : Hilbert V →L[ℂ] Hilbert V)
+    (hA : ∀ x y, matrixCoeff A x y = (q.normalizedKernel o p x y : ℂ)) (f : V → ℝ) (x : V) :
+    realAction A f x = (q.successProbability p o)⁻¹ * (∑' y, q.kernel p x y*f y) := by
+  simp only [realAction,hA,Complex.ofReal_re,normalizedKernel,mul_assoc,tsum_mul_left]
+
+theorem normalized_operator_row_le (o : V)
+    (q : LocalEndpointRule G hloc (vertexOrbit G o) M L) (p : unitInterval)
+    (hu : 0 < q.successProbability p o) (A : Hilbert V →L[ℂ] Hilbert V)
+    (hA : ∀ x y, matrixCoeff A x y = (q.normalizedKernel o p x y : ℂ)) (x : V) :
+    realAction A (fun _ => 1) x ≤ 1 := by
+  rw [q.normalized_operator_action o p A hA]
+  simp only [mul_one]
+  calc
+    _ ≤ (q.successProbability p o)⁻¹ * q.successProbability p o :=
+      mul_le_mul_of_nonneg_left (q.kernel_row_le_success o p x) (inv_nonneg.mpr hu.le)
+    _ = 1 := inv_mul_cancel₀ hu.ne'
+
+theorem normalized_operator_row_eq (o : V)
+    (q : LocalEndpointRule G hloc (vertexOrbit G o) M L) (p : unitInterval)
+    (hu : 0 < q.successProbability p o) (A : Hilbert V →L[ℂ] Hilbert V)
+    (hA : ∀ x y, matrixCoeff A x y = (q.normalizedKernel o p x y : ℂ)) (x : V)
+    (hx : x ∈ vertexOrbit G o) : realAction A (fun _ => 1) x = 1 := by
+  rw [q.normalized_operator_action o p A hA]
+  simp only [mul_one]
+  change (q.successProbability p o)⁻¹ * q.successProbability p x = 1
+  rw [q.successProbability_orbit p o x hx,inv_mul_cancel₀ hu.ne']
+
+theorem normalized_operator_invariant (o : V)
+    (q : LocalEndpointRule G hloc (vertexOrbit G o) M L) (p : unitInterval)
+    (A : Hilbert V →L[ℂ] Hilbert V)
+    (hA : ∀ x y, matrixCoeff A x y = (q.normalizedKernel o p x y : ℂ)) :
+    KernelInvariant (vertexAut G) A := by
+  rintro a ⟨b,hab⟩ x y
+  simp only [hA,normalizedKernel,q.kernel_covariant p a b hab]
+
+theorem normalized_operator_end_support (o : V)
+    (q : LocalEndpointRule G hloc (vertexOrbit G o) M L) (p : unitInterval)
+    (A : Hilbert V →L[ℂ] Hilbert V)
+    (hA : ∀ x y, matrixCoeff A x y = (q.normalizedKernel o p x y : ℂ))
+    (x y : V) (hy : y ∉ vertexOrbit G o) : (matrixCoeff A x y).re = 0 := by
+  simp only [hA,normalizedKernel,q.kernel_zero_of_end_not_mem p x y hy,mul_zero,Complex.ofReal_zero,Complex.zero_re]
+
+end LocalEndpointRule
+end NormalizedPositiveOperator
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section UpperDistinctMomentIteration
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem distinctClusterMoment_self (G : BondGraph V E) (p : unitInterval) (x : V) :
+    distinctClusterMoment G p x x = 0 := by
+  have hs (ω : Set E) : Conn G ω x x := .refl
+  simp only [distinctClusterMoment,hs,not_true_eq_false,ite_false,integral_zero]
+
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
+
+ 
+
+theorem normalized_power_distinctMoment_upper [Countable V] [Countable E] [Nonempty V]
+    (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L) (hqt : QuasiTransitive G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (hw : ∀ i, 0 ≤ weight i) (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i))
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (p : unitInterval) (hp : 0 < (p:ℝ)) (hpc : p < pc G)
+    (hu : 0 < q.successProbability p o)
+    (A : Hilbert V →L[ℂ] Hilbert V)
+    (hA : ∀ x y, matrixCoeff A x y = (q.normalizedKernel o p x y : ℂ))
+    (Z : ℝ) (hZ : 2 ≤ Z)
+    (hscale : (q.successProbability p o)⁻¹*(1+(1-(p:ℝ))/p*M) ≤ Z)
+    (j : ℕ) (x : V) :
+    realAction (A^j) (distinctClusterMoment G p x) x ≤
+      ((q.successProbability p o)⁻¹*(p:ℝ)^(-(M:ℤ))*(∑ i, ((reachBall G (root i) M).ncard:ℝ))*c)*
+        rootNeighborMoment G hloc root weight p * Z^j := by
+  let B := ((q.successProbability p o)⁻¹*(p:ℝ)^(-(M:ℤ))*(∑ i, ((reachBall G (root i) M).ncard:ℝ))*c)*
+    rootNeighborMoment G hloc root weight p
+  have hB : 0 ≤ B := by
+    exact mul_nonneg (mul_nonneg (mul_nonneg
+      (mul_nonneg (inv_nonneg.mpr hu.le) (zpow_nonneg p.2.1 _))
+      (Finset.sum_nonneg (fun _ _ => Nat.cast_nonneg _))) hc)
+      (rootNeighborMoment_nonneg G hloc root weight hw p)
+  have hf (v : V) : realAction A (distinctClusterMoment G p x) v ≤ Z*distinctClusterMoment G p x v+B := by
+    rw [q.normalized_operator_action o p A hA]
+    calc
+      _ ≤ (q.successProbability p o)⁻¹ *
+          ((1+(1-(p:ℝ))/p*M)*distinctClusterMoment G p x v+
+            (p:ℝ)^(-(M:ℤ))*(∑ i, ((reachBall G (root i) M).ncard:ℝ))*c*rootNeighborMoment G hloc root weight p) :=
+        mul_le_mul_of_nonneg_left
+          (q.kernel_distinctMoment_bound hqt root weight hw hcover c hc hmin p hp hpc x v) (inv_nonneg.mpr hu.le)
+      _ = ((q.successProbability p o)⁻¹*(1+(1-(p:ℝ))/p*M))*distinctClusterMoment G p x v+B := by dsimp [B]; ring
+      _ ≤ _ := by
+        have hh := mul_le_mul_of_nonneg_right hscale (distinctClusterMoment_nonneg G p x v)
+        linarith
+  exact realAction_pow_potential_at_zero A (q.normalized_operator_positive o p A hA)
+    (q.normalized_operator_finite o p A hA) (q.normalized_operator_row_le o p hu A hA)
+    (distinctClusterMoment G p x) Z B hZ hB hf j x (distinctClusterMoment_self G p x)
+
+end LocalEndpointRule
+end UpperDistinctMomentIteration
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section PositivePowerStochasticity
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem realAction_pow_row_le (A : Hilbert V →L[ℂ] Hilbert V) (hA : PositiveRealMatrix A)
+    (ha : ∀ x, (Function.support (fun y => (matrixCoeff A x y).re)).Finite)
+    (hrow : ∀ x, realAction A (fun _ => 1) x ≤ 1) (j : ℕ) (x : V) :
+    realAction (A^j) (fun _ => 1) x ≤ 1 := by
+  induction j generalizing x with
+  | zero => rw [pow_zero]; exact (positive_identity_action (fun _ => 1) x).le
+  | succ j ih =>
+      rw [realAction_pow_succ A hA ha]
+      exact (realAction_mono A hA ha _ (fun _ => 1) ih x).trans (hrow x)
+
+theorem positive_pow_end_support (A : Hilbert V →L[ℂ] Hilbert V) (hA : PositiveRealMatrix A)
+    (O : Set V) (hO : ∀ x y, y ∉ O → (matrixCoeff A x y).re = 0)
+    (j : ℕ) (x y : V) (hx : x ∈ O) (hy : y ∉ O) : (matrixCoeff (A^j) x y).re = 0 := by
+  cases j with
+  | zero =>
+      have hxy : x ≠ y := fun h => hy (h ▸ hx)
+      simp only [pow_zero,matrixCoeff,one_apply_eq_self,lp.single_apply,Pi.single_apply,ite_eq_right hxy,Complex.zero_re]
+  | succ j =>
+      rw [pow_succ,←((hA.pow j).mul_hasSum A x y).tsum_eq]
+      simp only [hO _ y hy,mul_zero,tsum_zero]
+
+theorem realAction_pow_row_eq_on (A : Hilbert V →L[ℂ] Hilbert V) (hA : PositiveRealMatrix A)
+    (ha : ∀ x, (Function.support (fun y => (matrixCoeff A x y).re)).Finite)
+    (O : Set V) (hO : ∀ x y, y ∉ O → (matrixCoeff A x y).re = 0)
+    (hrow : ∀ x ∈ O, realAction A (fun _ => 1) x = 1)
+    (j : ℕ) (x : V) (hx : x ∈ O) : realAction (A^j) (fun _ => 1) x = 1 := by
+  induction j generalizing x with
+  | zero => rw [pow_zero]; exact positive_identity_action (fun _ => 1) x
+  | succ j ih =>
+      rw [realAction_pow_succ A hA ha]
+      calc
+        _ = realAction A (fun _ => 1) x := by
+          apply tsum_congr
+          intro y
+          by_cases hy : y ∈ O
+          · rw [ih y hy]
+          · rw [hO x y hy,zero_mul,zero_mul]
+        _ = 1 := hrow x hx
+
+end PositivePowerStochasticity
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section ConnectedMomentLoss
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def connectedClusterMoment (G : BondGraph V E) (p : unitInterval) (x y : V) : ℝ :=
+  ∫ ω, (if Conn G ω x y then ((cluster G ω x).ncard:ℝ)^2 else 0) ∂law p
+
+theorem integrable_connected_cluster_square [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (x y : V) :
+    Integrable (fun ω => if Conn G ω x y then ((cluster G ω x).ncard:ℝ)^2 else 0) (law p) := by
+  obtain ⟨_,_,hs⟩ := subcritical_second_moment G hloc hqt p hp
+  have he : (fun ω : Set E => if Conn G ω x y then ((cluster G ω x).ncard:ℝ)^2 else 0) =
+      {ω | Conn G ω x y}.indicator (fun ω => ((cluster G ω x).ncard:ℝ)^2) := by
+    funext ω
+    simp only [Set.indicator,Set.mem_ofPred_eq]
+  rw [he]
+  exact (hs x).1.indicator (measurableSet_conn G x y)
+
+theorem distinct_add_connected_cluster_moment [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (x y : V) :
+    distinctClusterMoment G p x y + connectedClusterMoment G p x y =
+      ∫ ω, ((cluster G ω x).ncard:ℝ)*((cluster G ω y).ncard:ℝ) ∂law p := by
+  rw [distinctClusterMoment,connectedClusterMoment,
+    ←integral_add (integrable_distinct_cluster_product G hloc hqt p hp x y)
+      (integrable_connected_cluster_square G hloc hqt p hp x y)]
+  apply integral_congr_ae
+  exact Filter.Eventually.of_forall (fun ω => by
+    by_cases h : Conn G ω x y
+    · simp only [h,not_true_eq_false,ite_false,ite_true,zero_add,cluster_eq_of_conn G ω h,pow_two]
+    · simp only [h,not_false_eq_true,ite_true,ite_false,add_zero])
+
+theorem distinctClusterMoment_lower [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (x y : V) :
+    susceptibility G p x * susceptibility G p y - connectedClusterMoment G p x y ≤
+      distinctClusterMoment G p x y := by
+  have h := (subcritical_cluster_association G hloc hqt p hp x y).2
+  rw [←distinct_add_connected_cluster_moment G hloc hqt p hp] at h
+  linarith
+
+ 
+theorem realAction_connectedMoment_eq_diagonalLoss [Countable V] [Countable E]
+    (G : BondGraph V E) (o : V) (A : Hilbert V →L[ℂ] Hilbert V) (hA : PositiveRealMatrix A)
+    (ha : ∀ x, (Function.support (fun y => (matrixCoeff A x y).re)).Finite)
+    (hrow : ∀ x, realAction A (fun _ => 1) x ≤ 1)
+    (x : V) (hend : ∀ y, y ∉ vertexOrbit G o → (matrixCoeff A x y).re = 0)
+    (p : unitInterval) (hs : Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^2) (law p)) :
+    realAction A (connectedClusterMoment G p x) x =
+      ∫ ω, diagonalLossMass G o (fun a b => (matrixCoeff A a b).re) ω x ∂law p := by
+  have hr (z : V) (S : Finset V) : (∑ y ∈ S, (matrixCoeff A z y).re) ≤ 1 := by
+    have hsrow : Summable (fun y => (matrixCoeff A z y).re) := by
+      simpa only [mul_one] using finiteKernel_summable_mul (fun a b => (matrixCoeff A a b).re) ha z (fun _ => 1)
+    exact (hsrow.sum_le_tsum S (fun y _ => (hA z y).2)).trans (by simpa only [realAction,mul_one] using hrow z)
+  rw [integral_diagonalLoss_eq_series G o _ (fun a b => (hA a b).2) 1 hr x p hs]
+  apply tsum_congr
+  intro y
+  by_cases hy : y ∈ vertexOrbit G o
+  · rw [ite_eq_left hy]
+    rfl
+  · simp only [ite_eq_right hy,hend y hy,zero_mul]
+
+end ConnectedMomentLoss
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section LowerDistinctMomentOperator
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem realAction_distinctMoment_lower [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (htr : RootTransport root weight Γ) (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b)
+    (hw : ∀ i, 0 ≤ weight i) (i₀ : ι) (hw₀ : 0 < weight i₀)
+    (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i))
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (p : unitInterval) (hp : p < pc G) (b : ℝ) (hb : 0 ≤ b)
+    (hχ : ∀ x, b ≤ susceptibility G p x)
+    (A : Hilbert V →L[ℂ] Hilbert V) (hA : PositiveRealMatrix A)
+    (ha : ∀ x, (Function.support (fun y => (matrixCoeff A x y).re)).Finite)
+    (hinv : KernelInvariant (vertexAut G) A)
+    (hrow : ∀ x, realAction A (fun _ => 1) x ≤ 1)
+    (heq : realAction A (fun _ => 1) (root i₀) = 1)
+    (hend : ∀ y, y ∉ vertexOrbit G (root i₀) → (matrixCoeff A (root i₀) y).re = 0) :
+    b^2-‖A‖*(c*rootSusceptibility G root weight p)^3 ≤
+      realAction A (distinctClusterMoment G p (root i₀)) (root i₀) := by
+  let x := root i₀
+  let f := connectedClusterMoment G p x
+  have hr (z : V) (S : Finset V) : (∑ y ∈ S, (matrixCoeff A z y).re) ≤ 1 := by
+    have hsrow : Summable (fun y => (matrixCoeff A z y).re) := by
+      simpa only [mul_one] using finiteKernel_summable_mul (fun a b => (matrixCoeff A a b).re) ha z (fun _ => 1)
+    exact (hsrow.sum_le_tsum S (fun y _ => (hA z y).2)).trans (by simpa only [realAction,mul_one] using hrow z)
+  obtain ⟨_,_,hs⟩ := subcritical_second_moment G hloc hqt p hp
+  have hf : realAction A f x ≤ ‖A‖*(c*rootSusceptibility G root weight p)^3 := by
+    rw [realAction_connectedMoment_eq_diagonalLoss G (root i₀) A hA ha hrow x hend p (hs x).1]
+    exact diagonalLoss_le_rootSusceptibility_cube G hloc hqt root weight Γ htr hΓ hw i₀ hw₀ hcover c hc hmin p hp
+      (fun a b => (matrixCoeff A a b).re) (fun a b => (hA a b).2)
+      (fun a b hab z y => congrArg Complex.re (hinv a ⟨b,hab⟩ z y)) 1 hr A (hA.real)
+  have hpnt (y : V) : (-1)*f y+b^2 ≤ distinctClusterMoment G p x y := by
+    have h := distinctClusterMoment_lower G hloc hqt p hp x y
+    have hprod : b*b ≤ susceptibility G p x * susceptibility G p y :=
+      mul_le_mul (hχ x) (hχ y) hb (susceptibility_nonneg G p x)
+    dsimp [f]
+    nlinarith only [h,hprod]
+  have hi := realAction_mono A hA ha (fun y => (-1)*f y+b^2)
+    (distinctClusterMoment G p x) hpnt x
+  rw [realAction_affine A ha f (-1) (b^2) x,heq] at hi
+  nlinarith only [hi,hf]
+
+end LowerDistinctMomentOperator
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section DistinctMomentSandwich
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
+
+ 
+
+theorem normalized_power_moment_sandwich [Countable V] [Countable E] [Nonempty V]
+    (hqt : QuasiTransitive G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (htr : RootTransport root weight Γ) (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b)
+    (hw : ∀ i, 0 ≤ weight i) (i₀ : ι) (hw₀ : 0 < weight i₀)
+    (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i))
+    (q : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) M L)
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (p : unitInterval) (hp : 0 < (p:ℝ)) (hpc : p < pc G)
+    (hu : 0 < q.successProbability p (root i₀))
+    (A : Hilbert V →L[ℂ] Hilbert V)
+    (hA : ∀ x y, matrixCoeff A x y = (q.normalizedKernel (root i₀) p x y : ℂ))
+    (a : ℝ) (ha : 0 ≤ a) (hχ : ∀ x, a*rootSusceptibility G root weight p ≤ susceptibility G p x)
+    (Z : ℝ) (hZ : 2 ≤ Z)
+    (hscale : (q.successProbability p (root i₀))⁻¹*(1+(1-(p:ℝ))/p*M) ≤ Z) (j : ℕ) :
+    (a*rootSusceptibility G root weight p)^2 - ‖A‖^j*(c*rootSusceptibility G root weight p)^3 ≤
+      ((q.successProbability p (root i₀))⁻¹*(p:ℝ)^(-(M:ℤ))*(∑ i, ((reachBall G (root i) M).ncard:ℝ))*c)*
+        rootNeighborMoment G hloc root weight p * Z^j := by
+  have hpos := q.normalized_operator_positive (root i₀) p A hA
+  have hfin := q.normalized_operator_finite (root i₀) p A hA
+  have hrow := q.normalized_operator_row_le (root i₀) p hu A hA
+  have hend := q.normalized_operator_end_support (root i₀) p A hA
+  have hx : root i₀ ∈ vertexOrbit G (root i₀) := ⟨Equiv.refl V,Equiv.refl E,(fun _ => by simp),rfl⟩
+  have hlo := realAction_distinctMoment_lower G hloc hqt root weight Γ htr hΓ hw i₀ hw₀ hcover c hc hmin p hpc
+    (a*rootSusceptibility G root weight p) (mul_nonneg ha (rootSusceptibility_nonneg G root weight hw p)) hχ
+    (A^j) (hpos.pow j) (finite_positive_pow_support A hpos hfin j)
+    ((q.normalized_operator_invariant (root i₀) p A hA).pow j)
+    (realAction_pow_row_le A hpos hfin hrow j)
+    (realAction_pow_row_eq_on A hpos hfin _ hend
+      (q.normalized_operator_row_eq (root i₀) p hu A hA) j (root i₀) hx)
+    (positive_pow_end_support A hpos _ hend j (root i₀) · hx)
+  have hhi := q.normalized_power_distinctMoment_upper (root i₀) hqt root weight hw hcover c hc hmin p hp hpc hu A hA Z hZ hscale j (root i₀)
+  have hpow : ‖A^j‖ ≤ ‖A‖^j := by
+    cases j with
+    | zero =>
+      simp only [pow_zero]
+      have he : (1 : Hilbert V →L[ℂ] Hilbert V) = ContinuousLinearMap.id ℂ (Hilbert V) := by ext x; rfl
+      rw [he]
+      exact ContinuousLinearMap.norm_id_le
+    | succ j => exact norm_pow_le' A (Nat.succ_pos j)
+  have hn := mul_le_mul_of_nonneg_right hpow
+    (pow_nonneg (mul_nonneg hc (rootSusceptibility_nonneg G root weight hw p)) 3)
+  linarith
+
+end LocalEndpointRule
+end DistinctMomentSandwich
+end Percolation
+
+ 
+namespace Percolation
+section LogarithmicPowerChoice
+
+ 
+
+theorem exists_logarithmic_power (r h Z T : ℝ) (hr : 0 ≤ r) (hh : 0 < h)
+    (hrh : r ≤ Real.exp (-h)) (hZ : 1 ≤ Z) (hT : 1 ≤ T) :
+    ∃ j : ℕ, r^j ≤ T⁻¹ ∧ Z^j ≤ Z*T^(Real.log Z/h) := by
+  let t := Real.log T/h
+  have ht : 0 ≤ t := div_nonneg (Real.log_nonneg hT) hh.le
+  let j := Nat.ceil t
+  have hlo : t ≤ (j:ℝ) := Nat.le_ceil t
+  have hhi : (j:ℝ) ≤ t+1 := (Nat.ceil_lt_add_one ht).le
+  have hTp : 0 < T := zero_lt_one.trans_le hT
+  have hZp : 0 < Z := zero_lt_one.trans_le hZ
+  have hlogZ : 0 ≤ Real.log Z := Real.log_nonneg hZ
+  refine ⟨j,?_,?_⟩
+  · calc
+      r^j ≤ (Real.exp (-h))^j := pow_le_pow_left₀ hr hrh j
+      _ = Real.exp ((j:ℝ)*(-h)) := (Real.exp_nat_mul _ _).symm
+      _ ≤ Real.exp (-Real.log T) := by
+        apply Real.exp_le_exp.mpr
+        have hl : Real.log T ≤ (j:ℝ)*h := (div_le_iff₀ hh).mp hlo
+        nlinarith
+      _ = T⁻¹ := by rw [Real.exp_neg,Real.exp_log hTp]
+  · calc
+      Z^j = Real.exp ((j:ℝ)*Real.log Z) := by rw [Real.exp_nat_mul,Real.exp_log hZp]
+      _ ≤ Real.exp ((t+1)*Real.log Z) := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_right hhi hlogZ)
+      _ = Z*T^(Real.log Z/h) := by
+        rw [Real.rpow_def_of_pos hTp,←Real.exp_log hZp,←Real.exp_add]
+        simp only [Real.log_exp]
+        congr 1
+        dsimp [t]
+        ring
+
+ 
+
+theorem moment_sandwich_power_law (a c C Z h r χ N : ℝ)
+    (ha : 0 < a) (hc : 0 ≤ c) (hC : 0 < C) (hZ : 1 ≤ Z) (hh : 0 < h)
+    (hr : 0 ≤ r) (hrh : r ≤ Real.exp (-h)) (hχ : 1 ≤ χ) (hN : 0 ≤ N)
+    (hs : ∀ j : ℕ, (a*χ)^2-r^j*(c*χ)^3 ≤ C*N*Z^j) :
+    let T := 1+2*c^3/a^2
+    a^2/(2*C*Z*T^(Real.log Z/h))*χ^(2-Real.log Z/h) ≤ N := by
+  let T := 1+2*c^3/a^2
+  have hap : 0 < a^2 := sq_pos_of_pos ha
+  have hT : 1 ≤ T := by
+    have hn : 0 ≤ 2*c^3/a^2 := by positivity
+    dsimp [T]
+    linarith
+  have hTp : 0 < T := zero_lt_one.trans_le hT
+  have hχp : 0 < χ := zero_lt_one.trans_le hχ
+  have hZp : 0 < Z := zero_lt_one.trans_le hZ
+  have hTx : 1 ≤ T*χ := by nlinarith
+  obtain ⟨j,hrj,hZj⟩ := exists_logarithmic_power r h Z (T*χ) hr hh hrh hZ hTx
+  have hbudget : 2*c^3 ≤ a^2*T := by dsimp [T]; field_simp; nlinarith
+  have hsmall : r^j*(c*χ)^3 ≤ a^2/2*χ^2 := by
+    calc
+      _ ≤ (T*χ)⁻¹*(c*χ)^3 := mul_le_mul_of_nonneg_right hrj (pow_nonneg (mul_nonneg hc hχp.le) _)
+      _ = c^3/T*χ^2 := by field_simp
+      _ ≤ a^2/2*χ^2 := mul_le_mul_of_nonneg_right ((div_le_iff₀ hTp).mpr (by nlinarith [hbudget])) (sq_nonneg χ)
+  have hlarge : a^2/2*χ^2 ≤ C*N*(Z*(T*χ)^(Real.log Z/h)) := by
+    have hs' := hs j
+    have hprod := mul_le_mul_of_nonneg_left hZj (mul_nonneg hC.le hN)
+    nlinarith only [hsmall,hs',hprod]
+  have hpow : 0 < χ^(Real.log Z/h) := Real.rpow_pos_of_pos hχp _
+  have hTpow : 0 < T^(Real.log Z/h) := Real.rpow_pos_of_pos hTp _
+  have he : C*N*(Z*(T*χ)^(Real.log Z/h)) =
+      (C*Z*T^(Real.log Z/h))*N*χ^(Real.log Z/h) := by
+    rw [Real.mul_rpow hTp.le hχp.le]
+    ring
+  rw [he] at hlarge
+  have hden : 0 < 2*C*Z*T^(Real.log Z/h) := by positivity
+  change a^2/(2*C*Z*T^(Real.log Z/h))*χ^(2-Real.log Z/h) ≤ N
+  rw [Real.rpow_sub hχp,Real.rpow_two]
+  calc
+    _ = (a^2*χ^2)/(2*C*Z*T^(Real.log Z/h)*χ^(Real.log Z/h)) := by
+      rw [div_mul_div_comm]
+    _ ≤ N := (div_le_iff₀ (mul_pos hden hpow)).mpr (by nlinarith only [hlarge])
+
+end LogarithmicPowerChoice
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open Filter
+section ProfileContinuity
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace EndpointProfile
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {p₀ : unitInterval} {o : V} {s : ℝ}
+
+theorem exp_neg_H (q : EndpointProfile G hloc p₀ o s) : Real.exp (-q.H) = ‖q.op‖ := by
+  rw [H,one_div,Real.log_inv,neg_neg,Real.exp_log q.norm_pos]
+
+ 
+
+theorem exists_nearby_operators {ι : Type*} [Fintype ι]
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (htr : RootTransport root weight Γ) (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b)
+    (hw : ∀ i, 0 ≤ weight i) (i₀ : ι) (hw₀ : 0 < weight i₀)
+    (q : EndpointProfile G hloc p₀ (root i₀) s) (hp₀ : 0 < (p₀:ℝ)) :
+    ∃ A : unitInterval → (Hilbert V →L[ℂ] Hilbert V),
+      (∀ p x y, matrixCoeff (A p) x y = (q.rule.normalizedKernel (root i₀) p x y : ℂ)) ∧
+      ∀ᶠ p in nhds p₀, q.rule.successProbability p₀ (root i₀)/2 ≤ q.rule.successProbability p (root i₀) ∧
+        ‖A p‖ ≤ Real.exp (-(q.H-1)) ∧ (p₀:ℝ)/2 ≤ (p:ℝ) := by
+  obtain ⟨A,_,hA,hcont⟩ := q.rule.exists_continuous_normalized_operators root weight Γ htr hΓ hw i₀ hw₀
+  have he : A p₀ = q.op := kernel_extension_unique (hA p₀) q.coeff
+  have hnorm : ContinuousAt (fun p => ‖A p‖) p₀ := (hcont p₀ q.success_pos.ne').norm
+  have hb : ‖A p₀‖ < Real.exp (-(q.H-1)) := by
+    rw [he,←q.exp_neg_H]
+    exact Real.exp_lt_exp.mpr (by linarith)
+  have ha := hnorm.eventually_lt continuousAt_const hb
+  have hu : ∀ᶠ p in nhds p₀, q.rule.successProbability p₀ (root i₀)/2 < q.rule.successProbability p (root i₀) :=
+    continuousAt_const.eventually_lt (q.rule.continuous_successProbability (root i₀)).continuousAt (by linarith [q.success_pos])
+  have hp : ∀ᶠ p : unitInterval in nhds p₀, (p₀:ℝ)/2 < (p:ℝ) :=
+    continuousAt_const.eventually_lt continuous_subtype_val.continuousAt (by linarith)
+  refine ⟨A,hA,?_⟩
+  filter_upwards [ha,hu,hp] with p ha hu hp
+  exact ⟨hu.le,ha.le,hp.le⟩
+
+end EndpointProfile
+end ProfileContinuity
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open Filter
+section MapScaleBudget
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace EndpointProfile
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {p₀ : unitInterval} {o : V} {s : ℝ}
+
+noncomputable def potentialScale (q : EndpointProfile G hloc p₀ o s) (p : unitInterval) : ℝ :=
+  (q.rule.successProbability p o)⁻¹*(1+(1-(p:ℝ))/p*q.cap)
+
+noncomputable def geometricScale (q : EndpointProfile G hloc p₀ o s) : ℝ := 2*q.potentialScale p₀
+
+theorem potentialScale_pos (q : EndpointProfile G hloc p₀ o s) (hp₀ : 0 < (p₀:ℝ)) :
+    0 < q.potentialScale p₀ := by
+  exact mul_pos (inv_pos.mpr q.success_pos) (by
+    have hn : 0 ≤ (1-(p₀:ℝ))/p₀*q.cap := mul_nonneg (div_nonneg (sub_nonneg.mpr p₀.2.2) hp₀.le) (Nat.cast_nonneg _)
+    linarith)
+
+theorem geometricScale_two_le (q : EndpointProfile G hloc p₀ o s) (hp₀ : 0 < (p₀:ℝ)) :
+    2 ≤ q.geometricScale := by
+  have hu : 1 ≤ (q.rule.successProbability p₀ o)⁻¹ :=
+    (one_le_inv₀ q.success_pos).mpr (q.rule.successProbability_le_one p₀ o)
+  have hb : 0 ≤ (1-(p₀:ℝ))/p₀*q.cap :=
+    mul_nonneg (div_nonneg (sub_nonneg.mpr p₀.2.2) hp₀.le) (Nat.cast_nonneg _)
+  dsimp [geometricScale,potentialScale]
+  nlinarith
+
+theorem log_geometricScale_le (q : EndpointProfile G hloc p₀ o s) (hp₀ : 0 < (p₀:ℝ))
+    (hM : 1 ≤ q.cap) :
+    Real.log q.geometricScale ≤ q.U+q.m+Real.log (2*(1+1/(p₀:ℝ))) := by
+  have hM' : (1:ℝ) ≤ q.cap := by exact_mod_cast hM
+  have hMp : (0:ℝ) < q.cap := zero_lt_one.trans_le hM'
+  have hb : 1+(1-(p₀:ℝ))/p₀*q.cap ≤ (1+1/(p₀:ℝ))*q.cap := by
+    apply (mul_le_mul_iff_of_pos_right hp₀).mp
+    field_simp
+    nlinarith
+  have Hz : q.geometricScale ≤ (1/q.rule.successProbability p₀ o)*(q.cap:ℝ)*(2*(1+1/(p₀:ℝ))) := by
+    have h := mul_le_mul_of_nonneg_left hb (mul_nonneg (by norm_num : (0:ℝ) ≤ 2) (inv_nonneg.mpr q.success_pos.le))
+    simpa only [geometricScale,potentialScale,one_div,mul_assoc,mul_comm,mul_left_comm] using h
+  have hc : 0 < 2*(1+1/(p₀:ℝ)) := by positivity
+  have h := Real.log_le_log (lt_of_lt_of_le (by norm_num : (0:ℝ)<2) (q.geometricScale_two_le hp₀)) Hz
+  rw [Real.log_mul (mul_ne_zero (one_div_ne_zero q.success_pos.ne') hMp.ne') hc.ne',
+    Real.log_mul (one_div_ne_zero q.success_pos.ne') hMp.ne'] at h
+  exact h
+
+theorem eventually_potentialScale_le (q : EndpointProfile G hloc p₀ o s) (hp₀ : 0 < (p₀:ℝ)) :
+    ∀ᶠ p in nhds p₀, q.potentialScale p ≤ q.geometricScale := by
+  have hu := (q.rule.continuous_successProbability o).continuousAt.inv₀ q.success_pos.ne'
+  have hv : ContinuousAt (fun p : unitInterval => ((1-(p:ℝ))/p)) p₀ :=
+    (continuousAt_const.sub continuous_subtype_val.continuousAt).div continuous_subtype_val.continuousAt hp₀.ne'
+  have he : ContinuousAt q.potentialScale p₀ := hu.mul (continuousAt_const.add (hv.mul continuousAt_const))
+  have hlt : q.potentialScale p₀ < q.geometricScale := by
+    dsimp only [geometricScale]
+    linarith [q.potentialScale_pos hp₀]
+  filter_upwards [he.eventually_lt continuousAt_const hlt] with p hp
+  exact hp.le
+
+end EndpointProfile
+end MapScaleBudget
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open Filter
+section ProfileNeighborMoment
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace EndpointProfile
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {p₀ : unitInterval} {o : V} {s : ℝ}
+
+ 
+
+theorem eventually_neighborMoment_lower [Countable V] [Countable E] [Nonempty V]
+    (hconn : Connected G) (hqt : QuasiTransitive G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (htr : RootTransport root weight Γ) (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b)
+    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
+    (i₀ : ι) (hw₀ : 0 < weight i₀) (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i))
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (q : EndpointProfile G hloc p₀ (root i₀) s) (hp₀ : 0 < (p₀:ℝ)) (hH : 1 < q.H) :
+    ∃ b : ℝ, 0 < b ∧ ∀ᶠ p in nhds p₀, p < pc G →
+      b*(rootSusceptibility G root weight p)^(2-Real.log q.geometricScale/(q.H-1)) ≤
+        rootNeighborMoment G hloc root weight p := by
+  obtain ⟨a,ha,hχ⟩ := exists_rootSusceptibility_lower G hconn hloc hqt root weight hw hw1 hcover
+    ((p₀:ℝ)/2) (by linarith)
+  obtain ⟨A,hA,hnear⟩ := q.exists_nearby_operators root weight Γ htr hΓ hw i₀ hw₀ hp₀
+  let B : unitInterval → ℝ := fun p =>
+    (q.rule.successProbability p (root i₀))⁻¹*(p:ℝ)^(-(q.cap:ℤ))*(∑ i, ((reachBall G (root i) q.cap).ncard:ℝ))*c
+  have hB (p : unitInterval) : 0 ≤ B p := by
+    dsimp [B]
+    exact mul_nonneg (mul_nonneg (mul_nonneg (inv_nonneg.mpr (q.rule.successProbability_nonneg p _))
+      (zpow_nonneg p.2.1 _)) (Finset.sum_nonneg (fun _ _ => Nat.cast_nonneg _))) hc
+  have hBc : ContinuousAt B p₀ := by
+    exact ((((q.rule.continuous_successProbability (root i₀)).continuousAt.inv₀ q.success_pos.ne').mul
+      (continuous_subtype_val.continuousAt.zpow₀ (-(q.cap:ℤ)) (Or.inl hp₀.ne'))).mul continuousAt_const).mul continuousAt_const
+  let C := B p₀+1
+  have hC : 0 < C := by dsimp [C]; linarith [hB p₀]
+  have hCB : ∀ᶠ p in nhds p₀, B p ≤ C := by
+    filter_upwards [hBc.eventually_lt continuousAt_const (show B p₀ < C by dsimp [C]; linarith)] with p hp
+    exact hp.le
+  let Z := q.geometricScale
+  let h := q.H-1
+  let T := 1+2*c^3/a^2
+  let b := a^2/(2*C*Z*T^(Real.log Z/h))
+  have hz : 2 ≤ Z := q.geometricScale_two_le hp₀
+  have ht : 0 < T := by
+    have hh : 0 ≤ 2*c^3/a^2 := by positivity
+    dsimp [T]
+    linarith
+  have hb : 0 < b := div_pos (sq_pos_of_pos ha) (by
+    have hp := Real.rpow_pos_of_pos ht (Real.log Z/h)
+    have hZp : 0 < Z := by linarith
+    positivity)
+  refine ⟨b,hb,?_⟩
+  filter_upwards [hnear,hCB,q.eventually_potentialScale_le hp₀] with p hnear hCB hscale
+  intro hpc
+  obtain ⟨hu,hAn,hpp⟩ := hnear
+  have hp : 0 < (p:ℝ) := lt_of_lt_of_le (by linarith : 0 < (p₀:ℝ)/2) hpp
+  have hup : 0 < q.rule.successProbability p (root i₀) :=
+    lt_of_lt_of_le (by linarith [q.success_pos]) hu
+  have hχ1 : 1 ≤ rootSusceptibility G root weight p := by
+    calc
+      1 = ∑ i, weight i*1 := by simp only [mul_one,hw1]
+      _ ≤ _ := Finset.sum_le_sum (fun i _ => mul_le_mul_of_nonneg_left
+        (susceptibility_one_le G hloc hqt p hpc (root i)) (hw i))
+  apply moment_sandwich_power_law a c C Z h ‖A p‖ (rootSusceptibility G root weight p)
+    (rootNeighborMoment G hloc root weight p) ha hc hC (by linarith) (by dsimp [h]; linarith)
+    (norm_nonneg _) hAn hχ1 (rootNeighborMoment_nonneg G hloc root weight hw p)
+  intro j
+  have hs := q.rule.normalized_power_moment_sandwich hqt root weight Γ htr hΓ hw i₀ hw₀ hcover c hc hmin p hp hpc hup
+    (A p) (hA p) a ha.le (hχ p hpp hpc) Z hz hscale j
+  exact hs.trans (mul_le_mul_of_nonneg_right
+    (mul_le_mul_of_nonneg_right hCB (rootNeighborMoment_nonneg G hloc root weight hw p)) (pow_nonneg (by linarith) j))
+
+end EndpointProfile
+end ProfileNeighborMoment
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section EfficientProfileMomentConsequence
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem neighborMoment_growth_of_divergent_moment
+    [Countable V] [Countable E] [Nonempty V] [Nonempty E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hconn : Connected G)
+    (hqt : QuasiTransitive G) (hh : 0 < hV G)
+    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
+    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
+    (p₀ : unitInterval) (hp₀ : 0 < (p₀:ℝ)) (hp₁ : (p₀:ℝ) < 1)
+    (hfin : ∀ i, ∀ᵐ ω ∂law p₀, (cluster G ω (root i)).Finite)
+    (α β ε : ℝ) (hα : 0 < α) (hαβ : α < β)
+    (hε : 0 < ε) (hε1 : ε < 1/2) (hsmall : β+(1/2+ε) < 1)
+    (hdiv : ∃ i, ¬ Integrable (fun ω => ((cluster G ω (root i)).ncard:ℝ)^α) (law p₀)) :
+    ∀ γ : ℝ, 0 < γ → ∃ b : ℝ, 0 < b ∧ ∀ᶠ p in nhds p₀, p < pc G →
+      b*(rootSusceptibility G root weight p)^(2-γ) ≤ rootNeighborMoment G hloc root weight p := by
+  obtain ⟨i,heff⟩ := efficient_endpoint_profiles_of_divergent_moment G hloc hconn hqt hh root weight Γ hΓ htr
+    hcov hw hw1 p₀ hp₀ hp₁ hfin α β ε hα hαβ hε hε1 hsmall hdiv
+  have hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i) := by
+    intro x
+    obtain ⟨i,a,ha,hax⟩ := hcov x
+    obtain ⟨b,hab⟩ := hΓ ha
+    exact ⟨i,a,b,hab,hax⟩
+  let c : ℝ := ∑ i, 1/weight i
+  have hc : 0 ≤ c := Finset.sum_nonneg (fun i _ => (one_div_pos.mpr (hw i)).le)
+  have hmin (i : ι) : 1 ≤ c*weight i := by
+    apply (div_le_iff₀ (hw i)).mp
+    exact Finset.single_le_sum (fun j _ => (one_div_pos.mpr (hw j)).le) (Finset.mem_univ i)
+  intro γ hγ
+  let C := Real.log (2*(1+1/(p₀:ℝ)))
+  have hC : 0 ≤ C := Real.log_nonneg (by
+    have hinv : 0 ≤ 1/(p₀:ℝ) := (one_div_pos.mpr hp₀).le
+    nlinarith)
+  obtain ⟨q,hM,hH,hR,hcost⟩ := heff (γ/4) (by linarith) (2+4*C/γ)
+  have hH2 : 2 < q.H := by
+    have hn : 0 ≤ 4*C/γ := div_nonneg (mul_nonneg (by norm_num) hC) hγ.le
+    linarith
+  have hCsmall : C < γ*q.H/4 := by
+    have hx : 4*C < (q.H-2)*γ := (div_lt_iff₀ hγ).mp (by linarith : 4*C/γ < q.H-2)
+    nlinarith
+  have hexp : Real.log q.geometricScale/(q.H-1) ≤ γ := by
+    apply (div_le_iff₀ (by linarith : 0 < q.H-1)).mpr
+    have hz := q.log_geometricScale_le hp₀ (by omega)
+    have hm : γ*q.H/2 ≤ γ*(q.H-1) := by nlinarith
+    dsimp [C] at hCsmall
+    nlinarith only [hz,hcost,hCsmall,hm]
+  obtain ⟨b,hb,hnear⟩ := q.eventually_neighborMoment_lower hconn hqt root weight Γ htr
+    (fun a ha => hΓ ha) (fun i => (hw i).le) hw1 i (hw i) hcover c hc hmin hp₀ hH
+  refine ⟨b,hb,?_⟩
+  filter_upwards [hnear] with p hp
+  intro hpc
+  have hχ : 1 ≤ rootSusceptibility G root weight p := by
+    calc
+      1 = ∑ i, weight i*1 := by simp only [mul_one,hw1]
+      _ ≤ _ := Finset.sum_le_sum (fun i _ => mul_le_mul_of_nonneg_left
+        (susceptibility_one_le G hloc hqt p hpc (root i)) (hw i).le)
+  exact (mul_le_mul_of_nonneg_left (Real.rpow_le_rpow_of_exponent_le hχ (by linarith :
+    2-γ ≤ 2-Real.log q.geometricScale/(q.H-1))) hb.le).trans (hp hpc)
+
+end EfficientProfileMomentConsequence
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory ProbabilityTheory Function
+section PivotalSprinklingGain
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem measurable_set_union_pair : Measurable (fun f : Set E × Set E => f.1 ∪ f.2) := by
+  apply measurable_setOfPred.comp
+  apply Measurable.of_eval
+  intro e
+  apply measurableSet_setOfPred.mp
+  change MeasurableSet ({f : Set E × Set E | e ∈ f.1} ∪ {f | e ∈ f.2})
+  apply MeasurableSet.union <;> apply measurableSet_setOfPred.mpr
+  · exact ((measurable_pi_apply e).comp MeasurableEquiv.setOfPred.symm.measurable).comp measurable_fst
+  · exact ((measurable_pi_apply e).comp MeasurableEquiv.setOfPred.symm.measurable).comp measurable_snd
+
+theorem measurable_set_insert (e : E) : Measurable (fun ω : Set E => insert e ω) := by
+  have h : Measurable (fun ω : Set E => ({e}:Set E) ∪ ω) :=
+    measurable_set_union_pair.comp (measurable_const.prodMk measurable_id)
+  simpa only [Set.singleton_union] using (h : Measurable (fun ω : Set E => {e} ∪ ω))
+
+theorem measurableSet_closedPivotal (A : Set (Set E)) (hA : MeasurableSet A) (e : E) :
+    MeasurableSet (closedPivotal A e) := by
+  exact (measurableSet_mem e).compl.inter (hA.compl.inter (hA.preimage (measurable_set_insert e)))
+
+def singleSprinkle (Q : Finset E) (e : (Q : Set E)) : Set (Set E) :=
+  {η | Subtype.val ⁻¹' η = {e}}
+
+theorem measurableSet_singleSprinkle (Q : Finset E) (e : (Q : Set E)) :
+    MeasurableSet (singleSprinkle Q e) :=
+  (measurableSet_singleton _).preimage (measurable_set_restrict (Q:Set E))
+
+theorem singleSprinkle_disjoint (Q : Finset E) :
+    Pairwise (Disjoint on singleSprinkle Q) := by
+  intro a b hab
+  apply Set.disjoint_left.mpr
+  intro η ha hb
+  apply hab
+  have h : ({a} : Set (Q:Set E)) = {b} := ha.symm.trans hb
+  simpa only [Set.singleton_eq_singleton_iff] using h
+
+theorem singleSprinkle_mem (Q : Finset E) (e : (Q:Set E)) {η : Set E}
+    (hη : η ∈ singleSprinkle Q e) : e.val ∈ η := by
+  have h : e ∈ Subtype.val ⁻¹' η := by rw [hη]; exact Set.mem_singleton e
+  exact h
+
+theorem singleSprinkle_probability (Q : Finset E) (e : (Q:Set E)) (a : unitInterval) :
+    (law a).real (singleSprinkle Q e) = (a:ℝ)*(1-(a:ℝ))^(Q.card-1) := by
+  have h := congrArg (fun μ : Measure (Set (Q:Set E)) => μ.real {({e}:Set (Q:Set E))})
+    (law_map_restrict a (Q:Set E))
+  rw [Measure.real,Measure.map_apply (measurable_set_restrict _) (measurableSet_singleton _)] at h
+  change (law a).real (singleSprinkle Q e) = _ at h
+  rw [h,law,setBernoulli_real_singleton a (Set.subset_univ _) Set.finite_univ,
+    Set.ncard_sdiff' (Set.subset_univ _) Set.finite_univ,Set.ncard_singleton]
+  simp only [Set.ncard_univ,Nat.card_eq_fintype_card,pow_one]
+  rw [show Fintype.card (Q:Set E) = Q.card from Fintype.card_coe Q]
+
+ 
+
+theorem event_sprinkling_gain (p a : unitInterval) (Q : Finset E)
+    (A : Set (Set E)) (hA : MeasurableSet A) (hup : IsUpperSet A) :
+    (a:ℝ)*(1-(a:ℝ))^(Q.card-1) * (∑ e ∈ Q, (law p).real (closedPivotal A e)) ≤
+      (law (sprinkleParameter p a)).real A - (law p).real A := by
+  let μ : Measure (Set E × Set E) := (law p).prod (law a)
+  let B : Set (Set E × Set E) := {f | f.1 ∉ A ∧ f.1 ∪ f.2 ∈ A}
+  let C (e : (Q:Set E)) := (closedPivotal A e.val) ×ˢ singleSprinkle Q e
+  have hB : MeasurableSet B := (hA.compl.preimage measurable_fst).inter (hA.preimage measurable_set_union_pair)
+  have hC (e : (Q:Set E)) : MeasurableSet (C e) :=
+    (measurableSet_closedPivotal A hA e.val).prod (measurableSet_singleSprinkle Q e)
+  have hd : Pairwise (Disjoint on C) := by
+    intro e f hef
+    apply Set.disjoint_left.mpr
+    intro z hz hz'
+    exact Set.disjoint_left.mp (singleSprinkle_disjoint Q hef) hz.2 hz'.2
+  have hsub : (⋃ e, C e) ⊆ B := by
+    intro z hz
+    obtain ⟨e,he⟩ := Set.mem_iUnion.mp hz
+    refine ⟨he.1.2.1,?_⟩
+    apply hup _ he.1.2.2
+    intro f hf
+    rcases Set.mem_insert_iff.mp hf with rfl | hf
+    · exact Or.inr (singleSprinkle_mem Q e he.2)
+    · exact Or.inl hf
+  have hgain : μ.real B = (law (sprinkleParameter p a)).real A - (law p).real A := by
+    have he : (fun f : Set E × Set E => f.1 ∪ f.2) ⁻¹' A = (A ×ˢ Set.univ) ∪ B := by
+      ext f
+      simp only [Set.mem_preimage,Set.mem_union,Set.mem_prod,Set.mem_univ,and_true,B,Set.mem_ofPred_eq]
+      exact ⟨fun h => by by_cases ha : f.1 ∈ A; exact Or.inl ha; exact Or.inr ⟨ha,h⟩,
+        fun h => h.elim (fun ha => hup Set.subset_union_left ha) And.right⟩
+    have hd : Disjoint (A ×ˢ (Set.univ : Set (Set E))) B := by
+      apply Set.disjoint_left.mpr
+      intro f hf hf'
+      exact hf'.1 hf.1
+    have hm := congrArg (fun ν : Measure (Set E) => ν.real A) (law_map_union p a)
+    rw [Measure.real,Measure.map_apply measurable_set_union_pair hA] at hm
+    change μ.real ((fun f : Set E × Set E => f.1 ∪ f.2) ⁻¹' A) = _ at hm
+    rw [he,measureReal_union hd hB] at hm
+    have hp : μ.real (A ×ˢ Set.univ) = (law p).real A := by
+      rw [measureReal_prod_prod,probReal_univ,mul_one]
+    rw [hp] at hm
+    linarith
+  have hs : μ.real (⋃ e, C e) =
+      (a:ℝ)*(1-(a:ℝ))^(Q.card-1) * (∑ e ∈ Q, (law p).real (closedPivotal A e)) := by
+    rw [measureReal_iUnion_fintype hd hC]
+    simp only [C,μ,measureReal_prod_prod,singleSprinkle_probability]
+    rw [←Finset.sum_mul,mul_comm]
+    congr 1
+    apply Finset.sum_bij (fun e _ => e.val)
+    · intro e _; exact e.property
+    · intro e _ f _ h; exact Subtype.ext h
+    · intro e he; exact ⟨⟨e,he⟩,Finset.mem_univ _,rfl⟩
+    · intro _ _; rfl
+  rw [←hs,←hgain]
+  exact measureReal_mono hsub
+
+end PivotalSprinklingGain
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Kernel/RussoRows.lean b/lean/OAI/Probability/UnimodularPercolation/Kernel/RussoRows.lean
index b88be35a9645bf463ab8a8c96556684e8441539a..f671e759c7bb95efd24fe223cc08b7a8203b0483 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Kernel/RussoRows.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Kernel/RussoRows.lean
@@ -1,1201 +1,1201 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Kernel.ContactTrace
-
-namespace OAI
-
-/-! Russo Rows for bond percolation. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-open scoped ENNReal
-section PairFunctionalAlgebra
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem clusterPairFunctional_mono_distinct (G : BondGraph V E) (p : unitInterval)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (F H : Set E → V → V → ℝ≥0∞)
-    (h : ∀ ω x y, ¬Conn G ω x y → F ω x y ≤ H ω x y) :
-    clusterPairFunctional G p root weight F ≤ clusterPairFunctional G p root weight H := by
-  apply Finset.sum_le_sum
-  intro i _
-  apply mul_le_mul_right
-  apply lintegral_mono
-  intro ω
-  unfold clusterPairDensity
-  apply mul_le_mul_right
-  apply ENNReal.tsum_le_tsum
-  intro y
-  apply mul_le_mul_right
-  by_cases hc : Conn G ω (root i) y
-  · simp only [hc,not_true_eq_false,ite_false,le_refl]
-  · simpa only [ite_eq_left hc] using h ω (root i) y hc
-
-theorem clusterPairFunctional_congr_ae (G : BondGraph V E) (p : unitInterval)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (F H : Set E → V → V → ℝ≥0∞)
-    (h : ∀ᵐ ω ∂law p, ∀ x y, ¬Conn G ω x y → F ω x y = H ω x y) :
-    clusterPairFunctional G p root weight F = clusterPairFunctional G p root weight H := by
-  apply Finset.sum_congr rfl
-  intro i _
-  congr 1
-  apply lintegral_congr_ae
-  filter_upwards [h] with ω hω
-  unfold clusterPairDensity
-  congr 1
-  apply tsum_congr
-  intro y
-  congr 1
-  by_cases hc : Conn G ω (root i) y
-  · simp only [hc,not_true_eq_false,ite_false]
-  · rw [ite_eq_left hc,ite_eq_left hc,hω (root i) y hc]
-
-theorem clusterPairFunctional_add [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (p : unitInterval)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (F H : Set E → V → V → ℝ≥0∞)
-    (hF : ∀ x y, Measurable (fun ω => F ω x y))
-    (hH : ∀ x y, Measurable (fun ω => H ω x y)) :
-    clusterPairFunctional G p root weight (fun ω x y => F ω x y+H ω x y) =
-      clusterPairFunctional G p root weight F + clusterPairFunctional G p root weight H := by
-  let : MeasurableSpace V := ⊤
-  let : MeasurableSingletonClass V := ⟨fun _ => trivial⟩
-  let : MeasurableSpace ι := ⊤
-  let : MeasurableSingletonClass ι := ⟨fun _ => trivial⟩
-  rw [clusterPairFunctional_eq_lintegral G hloc p root weight _ hF,
-    clusterPairFunctional_eq_lintegral G hloc p root weight _ hH,
-    clusterPairFunctional_eq_lintegral G hloc p root weight (fun ω x y => F ω x y+H ω x y) (fun x y => (hF x y).add (hH x y))]
-  exact lintegral_add_left (measurable_from_prod_countable_left (fun q => hF (root q.1) q.2)) _
-
-theorem clusterPairKernel_symm (G : BondGraph V E) (ω : Set E)
-    (J : V → V → ℝ≥0∞) (hJ : ∀ x y, J x y = J y x) (x y : V) :
-    clusterPairKernel G ω J x y = clusterPairKernel G ω J y x := by
-  unfold clusterPairKernel
-  rw [ENNReal.tsum_comm]
-  apply tsum_congr
-  intro a
-  apply tsum_congr
-  intro b
-  simp only [and_comm,hJ b a]
-
-end PairFunctionalAlgebra
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-open scoped ENNReal
-section LargeContactCauchySchwarz
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def largeContactKernel (G : BondGraph V E) (L : ℝ)
-    (ω : Set E) (x y : V) : ℝ≥0∞ :=
-  if L ≤ min ((cluster G ω x).ncard:ℝ) ((cluster G ω y).ncard:ℝ) then
-    clusterPairKernel G ω (indexedEdgeKernel G) x y else 0
-
-theorem measurable_largeContactKernel [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (L : ℝ) (x y : V) :
-    Measurable (fun ω => largeContactKernel G L ω x y) := by
-  have hm (z : V) : Measurable (fun ω : Set E => ((cluster G ω z).ncard:ℝ)) :=
-    (measurable_of_countable (fun n : ℕ => (n:ℝ))).comp (measurable_cluster_ncard G hloc z)
-  exact (measurable_clusterPairKernel G (indexedEdgeKernel G) x y).ite
-    (measurableSet_le measurable_const ((hm x).min (hm y))) measurable_const
-
-theorem clusterPairFunctional_largeContact_sq [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (L : ℝ) (p : unitInterval)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (hfin : ∀ x, ∀ᵐ ω ∂law p, (cluster G ω x).Finite) :
-    clusterPairFunctional G p root weight (fun ω x y => (largeContactKernel G L ω x y)^2) =
-      clusterPairFunctional G p root weight (contactSquareLargeReal G L) := by
-  apply clusterPairFunctional_congr_ae G p root weight
-  filter_upwards [ae_all_iff.mpr hfin] with ω hω
-  intro x y hxy
-  dsimp only [largeContactKernel,contactSquareLargeReal]
-  split_ifs
-  · rw [clusterPairKernel_edge G hloc ω x y hxy (hω x)]
-  · rw [zero_pow (by decide : 2 ≠ 0)]
-
-theorem clusterPairFunctional_largeContact_bound [Countable V] [Countable E]
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hw : ∀ i, 0 ≤ weight i) (p : unitInterval)
-    (hfin : ∀ x, ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
-    (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T) (L : ℝ) (B : ℝ≥0∞)
-    (hB : clusterPairFunctional G p root weight (contactSquareLargeReal G L) ≤ B) :
-    clusterPairFunctional G p root weight (fun ω x y =>
-      largeContactKernel G L ω x y * clusterPairKernel G ω (ennMatrixCoeff (T^2)) x y) ≤
-      B^((1:ℝ)/2)* (ENNReal.ofReal (‖T‖^3*(rootTrace root weight (T^3)).re))^((1:ℝ)/2) := by
-  have hh := clusterPairFunctional_cauchy_schwarz G hloc p root weight
-    (largeContactKernel G L) (fun ω x y => clusterPairKernel G ω (ennMatrixCoeff (T^2)) x y)
-    (measurable_largeContactKernel G hloc L) (measurable_clusterPairKernel G _)
-  rw [clusterPairFunctional_largeContact_sq G hloc L p root weight hfin] at hh
-  exact hh.trans (mul_le_mul
-    (ENNReal.rpow_le_rpow hB (by positivity))
-    (ENNReal.rpow_le_rpow (clusterPairFunctional_twoPoint_square_bound G hconn hloc root weight Γ hΓ htr hw p hfin T hT) (by positivity))
-    bot_le bot_le)
-
-end LargeContactCauchySchwarz
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-open scoped ENNReal ComplexOrder
-section SmallTraceTerm
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem contact_smallTwoPoint_bound [Countable V] [Countable E]
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
-    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (p : unitInterval) (hfin : ∀ x, ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
-    (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T)
-    (L α C : ℝ) (hL : 0 ≤ L) (hα0 : 0 ≤ α) (hα1 : α ≤ 1)
-    (hI : ∀ x, Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^α) (law p))
-    (hC : ∀ x, (∫ ω, ((cluster G ω x).ncard:ℝ)^α ∂law p) ≤ C)
-    (d : ℕ) (hd : ∀ x, degree G x ≤ d) :
-    clusterPairFunctional G p root weight (fun ω x y =>
-      clusterPairKernel G ω (indexedEdgeKernel G) x y * smallPairKernel G (ennMatrixCoeff (T^2)) L x y ω) ≤
-      ENNReal.ofReal ((d:ℝ)*c*C*L^(1-α)*(rootTrace root weight (T^3)).re) := by
-  have hTI : KernelInvariant Γ T := fun a ha => hT.invariant G p T a (hΓ ha)
-  have hJ : ∀ a ∈ Γ, ∀ x y, ennMatrixCoeff (T^2) (a x) (a y) = ennMatrixCoeff (T^2) x y := by
-    intro a ha x y
-    unfold ennMatrixCoeff
-    rw [(hTI.pow 2) a ha]
-  have hpos : 0 ≤ T := ContinuousLinearMap.nonneg_iff_isPositive.mpr (hT.isPositive G p T)
-  have hMt : 0 ≤ c*(rootTrace root weight (T^3)).re := mul_nonneg hc
-    (Complex.nonneg_iff.mp (rootTrace_nonneg root weight hw _ (CStarAlgebra.pow_nonneg T 3 hpos))).1
-  have hh := contact_smallPairKernel_bound G hconn hloc root weight Γ hΓ htr hw hw1 p hfin
-    (ennMatrixCoeff (T^2)) hJ L d hd (c*C*L^(1-α)*(rootTrace root weight (T^3)).re) (fun x y => by
-      refine (smallPairKernel_twoPoint_bound G hloc root weight Γ hΓ hcov hw c hc hmin x y p T hT L α hL hα0 hα1 (hfin x) (hfin y) (hI x)).trans ?_
-      apply ENNReal.ofReal_le_ofReal
-      calc
-        _ ≤ (c*(rootTrace root weight (T^3)).re)*(L^(1-α)*C) :=
-          mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_left (hC x) (Real.rpow_nonneg hL _)) hMt
-        _ = _ := by ring)
-  convert hh using 1; congr 1; ring
-
-end SmallTraceTerm
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-open scoped ENNReal ComplexOrder
-section MixedTraceTerm
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem contact_smallPairKernel_swap [Countable V] [Countable E]
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hw : ∀ i, 0 ≤ weight i) (p : unitInterval)
-    (J : V → V → ℝ≥0∞) (hJ : ∀ a ∈ Γ, ∀ x y, J (a x) (a y) = J x y) (L : ℝ) :
-    clusterPairFunctional G p root weight (fun ω x y =>
-      clusterPairKernel G ω (indexedEdgeKernel G) y x * smallPairKernel G J L y x ω) =
-    clusterPairFunctional G p root weight (fun ω x y =>
-      clusterPairKernel G ω (indexedEdgeKernel G) x y * smallPairKernel G J L x y ω) := by
-  symm
-  apply clusterPairFunctional_swap G hconn hloc root weight Γ hΓ htr hw p
-    (fun ω x y => clusterPairKernel G ω (indexedEdgeKernel G) x y * smallPairKernel G J L x y ω)
-  · intro x y
-    exact (measurable_clusterPairKernel G _ x y).mul (measurable_smallPairKernel G hloc J L x y)
-  · intro a ha b hab ω x y
-    rw [clusterPairKernel_aut G hab _ (indexedEdgeKernel_aut G hab),smallPairKernel_aut G hab J (hJ a ha)]
-
- 
-
-theorem clusterPairFunctional_twoPoint_mixed_bound [Countable V] [Countable E]
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
-    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (p : unitInterval) (hfin : ∀ x, ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
-    (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T)
-    (L α C : ℝ) (hL : 0 ≤ L) (hα0 : 0 ≤ α) (hα1 : α ≤ 1)
-    (hI : ∀ x, Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^α) (law p))
-    (hC : ∀ x, (∫ ω, ((cluster G ω x).ncard:ℝ)^α ∂law p) ≤ C)
-    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (B : ℝ≥0∞)
-    (hB : clusterPairFunctional G p root weight (contactSquareLargeReal G L) ≤ B) :
-    clusterPairFunctional G p root weight (fun ω x y =>
-      clusterPairKernel G ω (indexedEdgeKernel G) x y * clusterPairKernel G ω (ennMatrixCoeff (T^2)) x y) ≤
-    2*ENNReal.ofReal ((d:ℝ)*c*C*L^(1-α)*(rootTrace root weight (T^3)).re) +
-      B^((1:ℝ)/2)* (ENNReal.ofReal (‖T‖^3*(rootTrace root weight (T^3)).re))^((1:ℝ)/2) := by
-  let J := ennMatrixCoeff (T^2)
-  let A := fun ω x y => clusterPairKernel G ω (indexedEdgeKernel G) x y*smallPairKernel G J L x y ω
-  let D := fun ω x y => largeContactKernel G L ω x y*clusterPairKernel G ω J x y
-  have heS (ω : Set E) (x y : V) : clusterPairKernel G ω (indexedEdgeKernel G) y x =
-      clusterPairKernel G ω (indexedEdgeKernel G) x y := clusterPairKernel_symm G ω _ (indexedEdgeKernel_symm G) y x
-  have hpos : 0 ≤ T := ContinuousLinearMap.nonneg_iff_isPositive.mpr (hT.isPositive G p T)
-  have hJS (ω : Set E) (x y : V) : clusterPairKernel G ω J y x = clusterPairKernel G ω J x y :=
-    clusterPairKernel_symm G ω J (ennMatrixCoeff_symm ((IsSelfAdjoint.of_nonneg hpos).pow 2)) y x
-  have hTI : KernelInvariant Γ T := fun a ha => hT.invariant G p T a (hΓ ha)
-  have hJI : ∀ a ∈ Γ, ∀ x y, J (a x) (a y) = J x y := by
-    intro a ha x y
-    dsimp [J,ennMatrixCoeff]
-    rw [(hTI.pow 2) a ha]
-  have hA (x y : V) : Measurable (fun ω => A ω x y) :=
-    (measurable_clusterPairKernel G _ x y).mul (measurable_smallPairKernel G hloc J L x y)
-  have hD (x y : V) : Measurable (fun ω => D ω x y) :=
-    (measurable_largeContactKernel G hloc L x y).mul (measurable_clusterPairKernel G J x y)
-  have hpw (ω : Set E) (x y : V) (hxy : ¬Conn G ω x y) :
-      clusterPairKernel G ω (indexedEdgeKernel G) x y * clusterPairKernel G ω J x y ≤ A ω x y + A ω y x + D ω x y := by
-    have hyx : ¬Conn G ω y x := fun h => hxy (conn_symm G ω h)
-    by_cases hx : ((cluster G ω x).ncard:ℝ) < L
-    · have ha : A ω x y = clusterPairKernel G ω (indexedEdgeKernel G) x y * clusterPairKernel G ω J x y := by
-        dsimp [A,smallPairKernel]; rw [ite_eq_left ⟨hx,hxy⟩]
-      rw [←ha]
-      exact le_add_of_le_left (le_add_right le_rfl)
-    · by_cases hy : ((cluster G ω y).ncard:ℝ) < L
-      · have ha : A ω y x = clusterPairKernel G ω (indexedEdgeKernel G) x y * clusterPairKernel G ω J x y := by
-          dsimp [A,smallPairKernel]; rw [ite_eq_left ⟨hy,hyx⟩,heS ω x y,hJS ω x y]
-        rw [←ha]
-        exact le_add_of_le_left (le_add_left le_rfl)
-      · have hg : L ≤ min ((cluster G ω x).ncard:ℝ) ((cluster G ω y).ncard:ℝ) := le_min (le_of_not_gt hx) (le_of_not_gt hy)
-        have hh : D ω x y = clusterPairKernel G ω (indexedEdgeKernel G) x y * clusterPairKernel G ω J x y := by
-          dsimp [D,largeContactKernel]; rw [ite_eq_left hg]
-        rw [←hh]
-        exact le_add_left le_rfl
-  have ha := contact_smallTwoPoint_bound G hconn hloc root weight Γ hΓ htr hcov hw hw1 c hc hmin p hfin T hT L α C hL hα0 hα1 hI hC d hd
-  have hb := clusterPairFunctional_largeContact_bound G hconn hloc root weight Γ hΓ htr hw p hfin T hT L B hB
-  calc
-    _ ≤ clusterPairFunctional G p root weight (fun ω x y => A ω x y+A ω y x+D ω x y) :=
-      clusterPairFunctional_mono_distinct G p root weight _ _ hpw
-    _ = (clusterPairFunctional G p root weight A + clusterPairFunctional G p root weight A) + clusterPairFunctional G p root weight D := by
-      rw [clusterPairFunctional_add G hloc p root weight (fun ω x y => A ω x y+A ω y x) D (fun x y => (hA x y).add (hA y x)) hD,
-        clusterPairFunctional_add G hloc p root weight A (fun ω x y => A ω y x) hA (fun x y => hA y x)]
-      rw [contact_smallPairKernel_swap G hconn hloc root weight Γ hΓ htr hw p J hJI L]
-    _ ≤ _ := by simpa only [two_mul] using add_le_add (add_le_add ha ha) hb
-
-end MixedTraceTerm
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-open scoped ENNReal
-section UniformMixedTrace
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem critical_mixed_trace_bound
-    [Countable V] [Countable E] [Nonempty V] [Nonempty E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hconn : Connected G)
-    (hqt : QuasiTransitive G) (hh : 0 < hV G)
-    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
-    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
-    (hp₀ : 0 < (pc G:ℝ)) (hp₁ : (pc G:ℝ) < 1)
-    (hfin : ∀ i, ∀ᵐ ω ∂law (pc G), (cluster G ω (root i)).Finite)
-    (α ε : ℝ) (hα : 0 < α) (hα1 : α < 1/2) (hε : 0 < ε)
-    (hr : (1:ℝ)/2+ε-2*α < 0) :
-    ∃ A B : ℝ, 0 < A ∧ 0 < B ∧ ∀ (p : unitInterval),
-      (pc G:ℝ)/2 ≤ (p:ℝ) → p ≤ pc G →
-      ∀ (T : Hilbert V →L[ℂ] Hilbert V), HasKernel G p T →
-      ∀ L : ℝ, 1 ≤ L →
-        clusterPairFunctional G p root weight (fun ω x y =>
-          clusterPairKernel G ω (indexedEdgeKernel G) x y *
-          clusterPairKernel G ω (ennMatrixCoeff (T^2)) x y) ≤
-        2*ENNReal.ofReal (A*L^(1-α)*(rootTrace root weight (T^3)).re) +
-          (ENNReal.ofReal (B*L^((1:ℝ)/2+ε-2*α)))^((1:ℝ)/2) *
-          (ENNReal.ofReal (‖T‖^3*(rootTrace root weight (T^3)).re))^((1:ℝ)/2) := by
-  have hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i) := by
-    intro x
-    obtain ⟨i,a,ha,hax⟩ := hcov x
-    obtain ⟨b,hab⟩ := hΓ ha
-    exact ⟨i,a,b,hab,hax⟩
-  have hfinite : ∀ x, ∀ᵐ ω ∂law (pc G), (cluster G ω x).Finite := by
-    intro x
-    obtain ⟨i,a,b,hab,rfl⟩ := hcover x
-    exact cluster_finite_ae_covariant G (pc G) hab (root i) (hfin i)
-  obtain ⟨C,hC,hCb⟩ := uniform_cluster_power_of_root G hloc (pc G) root hcover α
-    (critical_root_power_integrable G hloc hconn hqt hh root weight Γ hΓ htr hcov hw hw1 hp₀ hp₁ hfin α hα hα1)
-  obtain ⟨B,hB,hBb⟩ := critical_contact_square_bound G hloc hconn hqt hh root weight Γ hΓ htr hcov hw hw1 hp₀ hp₁ hfin α ε hα hα1 hε hr
-  obtain ⟨d,hd,hdg⟩ := exists_degree_bound G hqt
-  let c : ℝ := ∑ i, 1/weight i
-  have hc : 0 < c := Finset.sum_pos (fun i _ => one_div_pos.mpr (hw i)) Finset.univ_nonempty
-  have hmin (i : ι) : 1 ≤ c*weight i := by
-    apply (div_le_iff₀ (hw i)).mp
-    exact Finset.single_le_sum (fun j _ => (one_div_pos.mpr (hw j)).le) (Finset.mem_univ i)
-  refine ⟨(d:ℝ)*c*C,B,mul_pos (mul_pos (by exact_mod_cast hd) hc) hC,hB,?_⟩
-  intro p hp hpq T hT L hL
-  have hm (x : V) := cluster_power_mono_integral G hloc x p (pc G) hpq α hα.le (hfinite x) (hCb x).1
-  exact clusterPairFunctional_twoPoint_mixed_bound G hconn hloc root weight Γ hΓ htr hcov
-    (fun i => (hw i).le) hw1 c hc.le hmin p
-    (fun x => cluster_finite_ae_mono G x p (pc G) hpq (hfinite x))
-    T hT L α C (by linarith) hα.le (by linarith)
-    (fun x => (hm x).1) (fun x => (hm x).2.trans (hCb x).2) d hdg _ (hBb p hp hpq L hL)
-
-end UniformMixedTrace
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section ConnectionRussoContacts
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem closedPivotal_connection_iff (G : BondGraph V E) (ω : Set E) (x y : V) (e : E) :
-    ω ∈ closedPivotal {η | Conn G η x y} e ↔
-      ¬Conn G ω x y ∧ e ∈ closedContactSet G (cluster G ω x) (cluster G ω y) := by
-  constructor
-  · rintro ⟨_,hxy,hins⟩
-    refine ⟨hxy,?_⟩
-    obtain ⟨⟨a,b⟩,hab⟩ := Quot.exists_rep (G e)
-    have he : G e = s(a,b) := hab.symm
-    rcases (conn_insert_iff G ω e a b x y he).mp hins with h | h | h
-    · exact (hxy h).elim
-    · exact ⟨a,h.1,b,conn_symm G ω h.2,he⟩
-    · exact ⟨b,h.1,a,conn_symm G ω h.2,by rw [he,Sym2.eq_swap]⟩
-  · rintro ⟨hxy,a,hxa,b,hyb,he⟩
-    have hclosed : e ∉ ω := by
-      intro heω
-      exact hxy (hxa.trans ((Relation.ReflTransGen.single ⟨e,heω,he⟩).trans (conn_symm G ω hyb)))
-    refine ⟨hclosed,hxy,?_⟩
-    exact (conn_insert_iff G ω e a b x y he).mpr (Or.inr (Or.inl ⟨hxa,conn_symm G ω hyb⟩))
-
-noncomputable def finiteClosedContact (G : BondGraph V E) (Q : Finset E) (x y : V)
-    (ω : Finset E) : ℝ :=
-  if ¬Conn G (ω:Set E) x y then
-    (contactBonds G Q (finiteRootComponent G ω x) (finiteRootComponent G ω y)).card else 0
-
-theorem finiteClosedContact_eq_sum (G : BondGraph V E) (Q : Finset E) (x y : V)
-    (ω : Finset E) :
-    finiteClosedContact G Q x y ω =
-      ∑ e ∈ Q, if (ω:Set E) ∈ closedPivotal {η | Conn G η x y} e then (1:ℝ) else 0 := by
-  have hm (z a : V) : a ∈ finiteRootComponent G ω z ↔ Conn G (ω:Set E) z a := by
-    change a ∈ (finiteRootComponent G ω z:Set V) ↔ _
-    rw [coe_finiteRootComponent]
-    rfl
-  simp only [closedPivotal_connection_iff,closedContactSet,Set.mem_ofPred_eq,finiteClosedContact]
-  by_cases hxy : Conn G (ω:Set E) x y
-  · simp only [hxy,not_true_eq_false,false_and,ite_false,Finset.sum_const_zero]
-  · rw [ite_eq_left hxy]
-    simp only [hxy,not_false_eq_true,true_and,contactBonds,hm,cluster,Set.mem_ofPred_eq]
-    rw [Finset.sum_boole]
-
-theorem finite_connection_russo_contacts (G : BondGraph V E) (Q : Finset E)
-    (x y : V) (p : ℝ) :
-    (1-p)*deriv (finiteEventPolynomial Q {ω | Conn G ω x y}) p =
-      bernoulliPolynomial Q (finiteClosedContact G Q x y) p := by
-  change (1-p)*deriv (bernoulliPolynomial Q (fun ω => if Conn G (ω:Set E) x y then 1 else 0)) p = _
-  rw [closed_difference_russo]
-  have he : finiteClosedContact G Q x y = fun (ω : Finset E) =>
-      ∑ e ∈ Q, if (ω:Set E) ∈ closedPivotal {η | Conn G η x y} e then (1:ℝ) else 0 :=
-    funext (finiteClosedContact_eq_sum G Q x y)
-  rw [he,bernoulliPolynomial_sum]
-  apply Finset.sum_congr rfl
-  intro e _
-  apply bernoulliPolynomial_congr
-  intro ω _
-  exact (closedPivotal_difference {η | Conn G η x y}
-    (fun _ _ h hc => conn_mono G h hc) e ω).symm
-
-end ConnectionRussoContacts
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section UniformContactApproximation
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem finiteClosedContact_eq_full (G : BondGraph V E) (hloc : LocallyFinite G)
-    (ω : Set E) (x y : V) (hx : (cluster G ω x).Finite) (hy : (cluster G ω y).Finite)
-    (Q : Finset E) (hQx : incidentSupport G hloc hx.toFinset ⊆ Q)
-    (hQy : incidentSupport G hloc hy.toFinset ⊆ Q) :
-    finiteClosedContact G Q x y (domainBits Q ω) = fullLargeContact G x y 0 ω := by
-  have hcluster (z : V) (hz : (cluster G ω z).Finite) (hQ : incidentSupport G hloc hz.toFinset ⊆ Q) :
-      cluster G (domainBits Q ω:Set E) z = cluster G ω z := by
-    rw [coe_domainBits]
-    have hh := (cluster_restrict_incident_eq_iff G hloc hz.toFinset Q hQ ω z).mpr
-      (Set.Finite.coe_toFinset hz).symm
-    simpa only [Set.Finite.coe_toFinset] using hh
-  have hcx := hcluster x hx hQx
-  have hcy := hcluster y hy hQy
-  have hFx : finiteRootComponent G (domainBits Q ω) x = hx.toFinset := by
-    apply Finset.coe_injective
-    rw [coe_finiteRootComponent,hcx,Set.Finite.coe_toFinset]
-  have hFy : finiteRootComponent G (domainBits Q ω) y = hy.toFinset := by
-    apply Finset.coe_injective
-    rw [coe_finiteRootComponent,hcy,Set.Finite.coe_toFinset]
-  have hconn : Conn G (domainBits Q ω:Set E) x y ↔ Conn G ω x y := by
-    change y ∈ cluster G (domainBits Q ω:Set E) x ↔ y ∈ cluster G ω x
-    rw [hcx]
-  have hc : (contactBonds G Q hx.toFinset hy.toFinset).card = clusterContactCount G ω x y := by
-    rw [←Set.ncard_coe_finset,coe_contactBonds_eq G hloc Q _ _ hQx]
-    simp only [Set.Finite.coe_toFinset,clusterContactCount]
-  simp only [finiteClosedContact,fullLargeContact,hconn,hFx,hFy,hc,Nat.zero_le,and_true]
-
-theorem finiteClosedContact_nonneg (G : BondGraph V E) (Q : Finset E) (x y : V) (ω : Finset E) :
-    0 ≤ finiteClosedContact G Q x y ω := by
-  unfold finiteClosedContact
-  split_ifs <;> positivity
-
-theorem finiteClosedContact_le_size (G : BondGraph V E) (hloc : LocallyFinite G)
-    (d : ℕ) (hd : ∀ z, degree G z ≤ d) (Q η : Finset E) (ω : Set E)
-    (hη : (η:Set E) ⊆ ω) (x y : V) (hfin : (cluster G ω x).Finite) :
-    finiteClosedContact G Q x y η ≤ (d:ℝ)*(cluster G ω x).ncard := by
-  have hdQ (z : V) : (Q.filter fun e => z ∈ G e).card ≤ d :=
-    (incident_filter_card_le_degree G hloc Q z).trans (hd z)
-  have hs : (finiteRootComponent G η x).card ≤ (cluster G ω x).ncard := by
-    rw [←Set.ncard_coe_finset,coe_finiteRootComponent]
-    exact Set.ncard_le_ncard (fun z hz => conn_mono G hη hz) hfin
-  unfold finiteClosedContact
-  split_ifs
-  · positivity
-  · exact_mod_cast (contactBonds_card_le G Q d hdQ _ _).trans
-      (Nat.mul_le_mul_left d ((min_le_left _ _).trans hs))
-
- 
-
-theorem lower_cluster_restrict_eq (G : BondGraph V E) (hloc : LocallyFinite G)
-    (ω η : Set E) (hη : η ⊆ ω) (x : V) (hx : (cluster G ω x).Finite)
-    (Q : Finset E) (hQ : incidentSupport G hloc hx.toFinset ⊆ Q) :
-    cluster G (η ∩ (Q:Set E)) x = cluster G η x := by
-  have hfin : (cluster G η x).Finite := hx.subset (fun z hz => conn_mono G hη hz)
-  have hs : hfin.toFinset ⊆ hx.toFinset := by
-    intro z hz
-    exact hx.mem_toFinset.mpr (conn_mono G hη (hfin.mem_toFinset.mp hz))
-  have hQ' := (incidentSupport_mono G hloc hs).trans hQ
-  have hh := (cluster_restrict_incident_eq_iff G hloc hfin.toFinset Q hQ' η x).mpr
-    (Set.Finite.coe_toFinset hfin).symm
-  simpa only [Set.Finite.coe_toFinset] using hh
-
-end UniformContactApproximation
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section ContactRegularity
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem fullLargeContact_zero_le_size (G : BondGraph V E) (hloc : LocallyFinite G)
-    (d : ℕ) (hd : ∀ z, degree G z ≤ d) (ω : Set E) (x y : V)
-    (hx : (cluster G ω x).Finite) (hy : (cluster G ω y).Finite) :
-    fullLargeContact G x y 0 ω ≤ (d:ℝ)*(cluster G ω x).ncard := by
-  let Q := incidentSupport G hloc hx.toFinset ∪ incidentSupport G hloc hy.toFinset
-  rw [←finiteClosedContact_eq_full G hloc ω x y hx hy Q Finset.subset_union_left Finset.subset_union_right]
-  exact finiteClosedContact_le_size G hloc d hd Q (domainBits Q ω) ω
-    (by rw [coe_domainBits]; exact Set.inter_subset_left) x y hx
-
-theorem integrable_fullContact_subcritical [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (x y : V) :
-    Integrable (fullLargeContact G x y 0) (law p) := by
-  obtain ⟨d,_,hd⟩ := exists_degree_bound G hqt
-  apply ((susceptibility_eq_integral G hloc hqt p hp x).1.const_mul (d:ℝ)).mono'
-    (measurable_fullLargeContact G hloc x y 0).aestronglyMeasurable
-  filter_upwards [subcritical_cluster_finite_ae G p hp x,subcritical_cluster_finite_ae G p hp y] with ω hx hy
-  rw [Real.norm_of_nonneg (fullLargeContact_nonneg G x y 0 ω)]
-  exact fullLargeContact_zero_le_size G hloc d hd ω x y hx hy
-
-theorem fullContact_label_integral [Countable E] [Countable V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (p : unitInterval) (x y : V) :
-    (∫ labels, fullLargeContact G x y 0 (openEdges labels p) ∂labelLaw) =
-      ∫ ω, fullLargeContact G x y 0 ω ∂law p := by
-  rw [←map_openEdges p]
-  exact (integral_map (measurable_openEdges p).aemeasurable
-    (measurable_fullLargeContact G hloc x y 0).aestronglyMeasurable).symm
-
-theorem fullContact_label_integrable [Countable E] [Countable V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (p : unitInterval) (x y : V) :
-    Integrable (fun labels => fullLargeContact G x y 0 (openEdges labels p)) labelLaw ↔
-      Integrable (fullLargeContact G x y 0) (law p) := by
-  rw [←map_openEdges p]
-  exact (integrable_map_measure (measurable_fullLargeContact G hloc x y 0).aestronglyMeasurable
-    (measurable_openEdges p).aemeasurable).symm
-
-theorem fullContact_eq_of_clusters (G : BondGraph V E) (ω η : Set E) (x y : V)
-    (hx : cluster G ω x = cluster G η x) (hy : cluster G ω y = cluster G η y) :
-    fullLargeContact G x y 0 ω = fullLargeContact G x y 0 η := by
-  have hc : Conn G ω x y ↔ Conn G η x y := by
-    change y ∈ cluster G ω x ↔ y ∈ cluster G η x
-    rw [hx]
-  simp only [fullLargeContact,clusterContactCount,hx,hy,hc]
-
- 
-
-theorem continuousAt_fullContact_expectation [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (x y : V) :
-    ContinuousAt (fun q => ∫ ω, fullLargeContact G x y 0 ω ∂law q) p := by
-  obtain ⟨r,hpr,hr⟩ := exists_between hp
-  obtain ⟨d,_,hd⟩ := exists_degree_bound G hqt
-  let F (q : unitInterval) (labels : E → unitInterval) := fullLargeContact G x y 0 (openEdges labels q)
-  let B (labels : E → unitInterval) : ℝ := (d:ℝ)*(cluster G (openEdges labels r) x).ncard
-  have hm (q : unitInterval) : Measurable (F q) :=
-    (measurable_fullLargeContact G hloc x y 0).comp (measurable_openEdges q)
-  have hc : ContinuousAt (fun q => ∫ labels, F q labels ∂labelLaw) p := by
-    apply continuousAt_of_dominated (bound := B)
-      (Eventually.of_forall (fun q => (hm q).aestronglyMeasurable))
-    · filter_upwards [eventually_lt_nhds hpr] with q hqr
-      filter_upwards [label_cluster_finite_ae_subcritical G r hr x,label_cluster_finite_ae_subcritical G r hr y] with labels hx hy
-      have hsub (z : V) : cluster G (openEdges labels q) z ⊆ cluster G (openEdges labels r) z :=
-        fun w hw => conn_mono G (openEdges_mono labels hqr.le) hw
-      rw [Real.norm_of_nonneg (fullLargeContact_nonneg G x y 0 _)]
-      calc
-        _ ≤ (d:ℝ)*(cluster G (openEdges labels q) x).ncard :=
-          fullLargeContact_zero_le_size G hloc d hd _ x y (hx.subset (hsub x)) (hy.subset (hsub y))
-        _ ≤ B labels := mul_le_mul_of_nonneg_left (by exact_mod_cast Set.ncard_le_ncard (hsub x) hx) (Nat.cast_nonneg _)
-    · exact (susceptibility_eq_label_integral G hloc hqt r hr x).1.const_mul (d:ℝ)
-    · filter_upwards [ae_labels_ne (E := E) p,label_cluster_finite_ae_subcritical G p hp x,
-        label_cluster_finite_ae_subcritical G p hp y] with labels hneq hx hy
-      apply Filter.EventuallyEq.continuousAt
-      filter_upwards [eventually_cluster_eq_at_finite G hloc labels p x hneq hx,
-        eventually_cluster_eq_at_finite G hloc labels p y hneq hy] with q hqx hqy
-      exact fullContact_eq_of_clusters G _ _ x y hqx hqy
-  simpa only [F,fullContact_label_integral G hloc] using hc
-
-end ContactRegularity
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section UniformContactError
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-def ClusterSupportIn (G : BondGraph V E) (Q : Finset E) (ω : Set E) (x : V) : Prop :=
-  ∀ a, Conn G ω x a → ∀ e, a ∈ G e → e ∈ Q
-
-theorem measurableSet_clusterSupportIn [Countable V] [Countable E]
-    (G : BondGraph V E) (Q : Finset E) (x : V) :
-    MeasurableSet {ω : Set E | ClusterSupportIn G Q ω x} := by
-  simp only [ClusterSupportIn,Set.ofPred_forall]
-  apply MeasurableSet.iInter
-  intro a
-  have he : {ω : Set E | Conn G ω x a → ∀ e, a ∈ G e → e ∈ Q} =
-      {ω | ¬Conn G ω x a} ∪ {ω | ∀ e, a ∈ G e → e ∈ Q} := by ext ω; simp only [Set.mem_union,Set.mem_ofPred_eq]; tauto
-  rw [he]
-  apply (measurableSet_conn G x a).compl.union
-  by_cases h : ∀ e, a ∈ G e → e ∈ Q
-  · convert (MeasurableSet.univ : MeasurableSet (Set.univ : Set (Set E))) using 1
-    ext ω
-    simp only [Set.mem_ofPred_eq,Set.mem_univ,iff_true]
-    exact h
-  · convert (MeasurableSet.empty : MeasurableSet (∅ : Set (Set E))) using 1
-    ext ω
-    simp only [Set.mem_ofPred_eq,Set.mem_empty_iff_false,iff_false]
-    exact h
-
-theorem clusterSupportIn_of_incident (G : BondGraph V E) (hloc : LocallyFinite G)
-    (Q : Finset E) (ω : Set E) (x : V) (hx : (cluster G ω x).Finite)
-    (hQ : incidentSupport G hloc hx.toFinset ⊆ Q) : ClusterSupportIn G Q ω x := by
-  intro a ha e he
-  exact hQ ((mem_incidentSupport G hloc hx.toFinset e).mpr ⟨a,hx.mem_toFinset.mpr ha,he⟩)
-
-theorem finiteClosedContact_eq_full_of_support (G : BondGraph V E) (hloc : LocallyFinite G)
-    (Q : Finset E) (ω η : Set E) (hη : η ⊆ ω) (x y : V)
-    (hx : (cluster G ω x).Finite) (hy : (cluster G ω y).Finite)
-    (hQx : ClusterSupportIn G Q ω x) (hQy : ClusterSupportIn G Q ω y) :
-    finiteClosedContact G Q x y (domainBits Q η) = fullLargeContact G x y 0 η := by
-  have hsub (z : V) : cluster G η z ⊆ cluster G ω z := fun a ha => conn_mono G hη ha
-  have hfx := hx.subset (hsub x)
-  have hfy := hy.subset (hsub y)
-  apply finiteClosedContact_eq_full G hloc η x y hfx hfy
-  · intro e he
-    obtain ⟨a,ha,he⟩ := (mem_incidentSupport G hloc hfx.toFinset e).mp he
-    exact hQx a (hsub x (hfx.mem_toFinset.mp ha)) e he
-  · intro e he
-    obtain ⟨a,ha,he⟩ := (mem_incidentSupport G hloc hfy.toFinset e).mp he
-    exact hQy a (hsub y (hfy.mem_toFinset.mp ha)) e he
-
-noncomputable def contactApproxError (G : BondGraph V E) (Q : Finset E) (d : ℕ)
-    (x y : V) (ω : Set E) : ℝ :=
-  if ClusterSupportIn G Q ω x ∧ ClusterSupportIn G Q ω y then 0
-  else (d:ℝ)*(cluster G ω x).ncard
-
-theorem measurable_contactApproxError [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (Q : Finset E) (d : ℕ) (x y : V) :
-    Measurable (contactApproxError G Q d x y) := by
-  exact measurable_const.ite ((measurableSet_clusterSupportIn G Q x).inter
-    (measurableSet_clusterSupportIn G Q y))
-    (measurable_const.mul ((measurable_of_countable (fun n : ℕ => (n:ℝ))).comp (measurable_cluster_ncard G hloc x)))
-
-theorem contactApproxError_nonneg (G : BondGraph V E) (Q : Finset E) (d : ℕ) (x y : V) (ω : Set E) :
-    0 ≤ contactApproxError G Q d x y ω := by unfold contactApproxError; split_ifs <;> positivity
-
-theorem contactApproxError_le (G : BondGraph V E) (Q : Finset E) (d : ℕ) (x y : V) (ω : Set E) :
-    contactApproxError G Q d x y ω ≤ (d:ℝ)*(cluster G ω x).ncard := by
-  unfold contactApproxError; split_ifs <;> first | positivity | exact le_rfl
-
-theorem contact_uniform_error_bound (G : BondGraph V E) (hloc : LocallyFinite G)
-    (d : ℕ) (hd : ∀ z, degree G z ≤ d) (Q : Finset E) (ω η : Set E)
-    (hη : η ⊆ ω) (x y : V) (hx : (cluster G ω x).Finite) (hy : (cluster G ω y).Finite) :
-    ‖finiteClosedContact G Q x y (domainBits Q η) - fullLargeContact G x y 0 η‖ ≤
-      contactApproxError G Q d x y ω := by
-  unfold contactApproxError
-  split_ifs with h
-  · rw [finiteClosedContact_eq_full_of_support G hloc Q ω η hη x y hx hy h.1 h.2,sub_self,norm_zero]
-  · have hsub (z : V) : cluster G η z ⊆ cluster G ω z := fun a ha => conn_mono G hη ha
-    have hf := finiteClosedContact_le_size G hloc d hd Q (domainBits Q η) ω
-      (by rw [coe_domainBits]; exact Set.inter_subset_left.trans hη) x y hx
-    have hg : fullLargeContact G x y 0 η ≤ (d:ℝ)*(cluster G ω x).ncard := by
-      exact (fullLargeContact_zero_le_size G hloc d hd η x y (hx.subset (hsub x)) (hy.subset (hsub y))).trans
-        (mul_le_mul_of_nonneg_left (by exact_mod_cast Set.ncard_le_ncard (hsub x) hx) (Nat.cast_nonneg _))
-    rw [Real.norm_eq_abs,abs_sub_le_iff]
-    have hfn := finiteClosedContact_nonneg G Q x y (domainBits Q η)
-    have hgn := fullLargeContact_nonneg G x y 0 η
-    constructor <;> linarith
-
-end UniformContactError
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section UniformContactExpectation
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem integrable_contactApproxError [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (Q : Finset E) (d : ℕ) (x y : V) :
-    Integrable (contactApproxError G Q d x y) (law p) := by
-  apply ((susceptibility_eq_integral G hloc hqt p hp x).1.const_mul (d:ℝ)).mono'
-    (measurable_contactApproxError G hloc Q d x y).aestronglyMeasurable
-  exact Eventually.of_forall (fun ω => by rw [Real.norm_of_nonneg (contactApproxError_nonneg G Q d x y ω)]; exact contactApproxError_le G Q d x y ω)
-
-theorem contactApproxError_integral_tendsto_zero [Countable V] [Encodable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (d : ℕ) (x y : V) :
-    Tendsto (fun n => ∫ ω, contactApproxError G (vertexExhaustion n) d x y ω ∂law p) atTop (nhds 0) := by
-  have ht : ∀ᵐ ω ∂law p, Tendsto (fun n => contactApproxError G (vertexExhaustion n) d x y ω) atTop (nhds 0) := by
-    filter_upwards [subcritical_cluster_finite_ae G p hp x,subcritical_cluster_finite_ae G p hp y] with ω hx hy
-    obtain ⟨nx,hnx⟩ := subset_vertexExhaustion (incidentSupport G hloc hx.toFinset)
-    obtain ⟨ny,hny⟩ := subset_vertexExhaustion (incidentSupport G hloc hy.toFinset)
-    apply tendsto_const_nhds.congr'
-    filter_upwards [eventually_ge_atTop (max nx ny)] with n hn
-    exact (ite_eq_left ⟨clusterSupportIn_of_incident G hloc _ ω x hx
-      (hnx.trans (vertexExhaustion_mono ((le_max_left nx ny).trans hn))),
-      clusterSupportIn_of_incident G hloc _ ω y hy
-      (hny.trans (vertexExhaustion_mono ((le_max_right nx ny).trans hn)))⟩).symm
-  have hh := tendsto_integral_of_dominated_convergence (fun ω : Set E => (d:ℝ)*(cluster G ω x).ncard)
-    (fun n => (measurable_contactApproxError G hloc (vertexExhaustion n) d x y).aestronglyMeasurable)
-    ((susceptibility_eq_integral G hloc hqt p hp x).1.const_mul (d:ℝ))
-    (fun n => Eventually.of_forall (fun ω => by rw [Real.norm_of_nonneg (contactApproxError_nonneg G _ d x y ω)]; exact contactApproxError_le G _ d x y ω)) ht
-  simpa only [integral_zero] using hh
-
-theorem finiteClosedContact_label_integral [Countable E] (G : BondGraph V E)
-    (Q : Finset E) (p : unitInterval) (x y : V) :
-    (∫ labels, finiteClosedContact G Q x y (domainBits Q (openEdges labels p)) ∂labelLaw) =
-      bernoulliPolynomial Q (finiteClosedContact G Q x y) p := by
-  rw [←integral_domainBits Q p (finiteClosedContact G Q x y),←map_openEdges p]
-  exact (integral_map (measurable_openEdges p).aemeasurable
-    (measurable_finite_local Q (fun ω => finiteClosedContact G Q x y (domainBits Q ω)) (fun ω => by rw [domainBits_local])).aestronglyMeasurable).symm
-
-theorem contactApproxError_label_integral [Countable V] [Countable E] (G : BondGraph V E)
-    (hloc : LocallyFinite G) (Q : Finset E) (p : unitInterval) (d : ℕ) (x y : V) :
-    (∫ labels, contactApproxError G Q d x y (openEdges labels p) ∂labelLaw) =
-      ∫ ω, contactApproxError G Q d x y ω ∂law p := by
-  rw [←map_openEdges p]
-  exact (integral_map (measurable_openEdges p).aemeasurable
-    (measurable_contactApproxError G hloc Q d x y).aestronglyMeasurable).symm
-
-theorem contact_expectation_uniform_error [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (d : ℕ) (hd : ∀ z, degree G z ≤ d) (Q : Finset E) (b q : unitInterval)
-    (hb : b < pc G) (hq : q ≤ b) (x y : V) :
-    ‖bernoulliPolynomial Q (finiteClosedContact G Q x y) q -
-      ∫ ω, fullLargeContact G x y 0 ω ∂law q‖ ≤
-      ∫ ω, contactApproxError G Q d x y ω ∂law b := by
-  have hf : Integrable (fun labels => finiteClosedContact G Q x y (domainBits Q (openEdges labels q))) labelLaw := by
-    have hi := integrable_finite_local Q q (fun ω => finiteClosedContact G Q x y (domainBits Q ω))
-      (fun ω => by rw [domainBits_local])
-    rw [←map_openEdges q] at hi
-    exact hi.comp_measurable (measurable_openEdges q)
-  have hg := (fullContact_label_integrable G hloc q x y).mpr
-    (integrable_fullContact_subcritical G hloc hqt q (hq.trans_lt hb) x y)
-  have hi : Integrable (fun labels => contactApproxError G Q d x y (openEdges labels b)) labelLaw := by
-    have hh := integrable_contactApproxError G hloc hqt b hb Q d x y
-    rw [←map_openEdges b] at hh
-    exact hh.comp_measurable (measurable_openEdges b)
-  rw [←finiteClosedContact_label_integral G Q q x y,←fullContact_label_integral G hloc q x y,
-    ←integral_sub hf hg,←contactApproxError_label_integral G hloc Q b d x y]
-  apply (norm_integral_le_integral_norm _).trans
-  apply integral_mono_ae (hf.sub hg).norm hi
-  filter_upwards [label_cluster_finite_ae_subcritical G b hb x,label_cluster_finite_ae_subcritical G b hb y] with labels hx hy
-  exact contact_uniform_error_bound G hloc d hd Q _ _ (openEdges_mono labels hq) x y hx hy
-
-end UniformContactExpectation
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section PointwiseRussoLimit
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def realContactExpectation (G : BondGraph V E) (x y : V) (t : ℝ) : ℝ :=
-  ∫ ω, fullLargeContact G x y 0 ω ∂law (Set.projIcc 0 1 zero_le_one t)
-
-@[simp] theorem realContactExpectation_coe (G : BondGraph V E) (x y : V) (p : unitInterval) :
-    realContactExpectation G x y (p:ℝ) = ∫ ω, fullLargeContact G x y 0 ω ∂law p := by
-  simp only [realContactExpectation,Set.projIcc_val]
-
-noncomputable def realTwoPoint (G : BondGraph V E) (x y : V) (t : ℝ) : ℝ :=
-  twoPoint G (Set.projIcc 0 1 zero_le_one t) x y
-
-@[simp] theorem realTwoPoint_coe (G : BondGraph V E) (x y : V) (p : unitInterval) :
-    realTwoPoint G x y (p:ℝ) = twoPoint G p x y := by
-  simp only [realTwoPoint,Set.projIcc_val]
-
-theorem finite_connection_deriv_eq (G : BondGraph V E) (Q : Finset E)
-    (x y : V) (t : ℝ) (ht : t ≠ 1) :
-    deriv (finiteEventPolynomial Q {ω | Conn G ω x y}) t =
-      bernoulliPolynomial Q (finiteClosedContact G Q x y) t / (1-t) := by
-  apply (eq_div_iff (sub_ne_zero.mpr (Ne.symm ht))).mpr
-  rw [mul_comm]
-  exact finite_connection_russo_contacts G Q x y t
-
-theorem finite_connection_deriv_tendstoUniformlyOn [Countable V] [Encodable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (b : unitInterval) (hb : b < pc G) (x y : V) :
-    TendstoUniformlyOn (fun n => deriv (finiteEventPolynomial (vertexExhaustion n) {ω | Conn G ω x y}))
-      (fun t => realContactExpectation G x y t / (1-t)) atTop (Set.Icc 0 (b:ℝ)) := by
-  obtain ⟨d,_,hd⟩ := exists_degree_bound G hqt
-  have hb1 : (b:ℝ) < 1 := lt_of_lt_of_le hb (pc G).2.2
-  have hden : 0 < 1-(b:ℝ) := sub_pos.mpr hb1
-  have ht := (contactApproxError_integral_tendsto_zero G hloc hqt b hb d x y).div_const (1-(b:ℝ))
-  rw [zero_div] at ht
-  rw [Metric.tendstoUniformlyOn_iff]
-  intro ε hε
-  filter_upwards [ht.eventually (eventually_lt_nhds hε)] with n hn t ht
-  let q : unitInterval := ⟨t,ht.1,ht.2.trans b.2.2⟩
-  have hqb : q ≤ b := ht.2
-  have hqt1 : t < 1 := ht.2.trans_lt hb1
-  have he := contact_expectation_uniform_error G hloc hqt d hd (vertexExhaustion n) b q hb hqb x y
-  have hn0 : 0 ≤ ∫ ω, contactApproxError G (vertexExhaustion n) d x y ω ∂law b :=
-    integral_nonneg (contactApproxError_nonneg G _ d x y)
-  rw [dist_comm,Real.dist_eq,finite_connection_deriv_eq G _ x y t (ne_of_lt hqt1),←sub_div,abs_div,
-    abs_of_pos (sub_pos.mpr hqt1)]
-  change |bernoulliPolynomial _ _ (q:ℝ) - realContactExpectation G x y (q:ℝ)| / (1-t) < ε
-  rw [realContactExpectation_coe]
-  apply lt_of_le_of_lt _ hn
-  exact div_le_div₀ hn0 (by simpa only [Real.norm_eq_abs] using he) hden (sub_le_sub_left ht.2 1)
-
-theorem finite_connection_polynomial_tendsto_twoPoint [Encodable E]
-    (G : BondGraph V E) (p : unitInterval) (x y : V) :
-    Tendsto (fun n => finiteEventPolynomial (vertexExhaustion n) {ω | Conn G ω x y} p)
-      atTop (nhds (twoPoint G p x y)) := by
-  have ht := finite_connection_polynomial_tendsto G Set.univ (vertexExhaustion (V := E))
-    vertexExhaustion_mono (fun _ => Set.subset_univ _) (fun e _ => by
-      obtain ⟨n,hn⟩ := subset_vertexExhaustion ({e}:Finset E)
-      exact ⟨n,hn (by simp)⟩) p x y
-  simpa only [Set.inter_univ,finiteEventPolynomial,Set.mem_ofPred_eq,twoPoint,Measure.real] using ht
-
- 
-
-theorem hasDerivAt_realTwoPoint_subcritical [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp : p < pc G) (x y : V) :
-    HasDerivAt (realTwoPoint G x y)
-      ((∫ ω, fullLargeContact G x y 0 ω ∂law p)/(1-(p:ℝ))) (p:ℝ) := by
-  let : Encodable E := Encodable.ofCountable E
-  obtain ⟨b,hpb,hb⟩ := exists_between hp
-  have hu := (finite_connection_deriv_tendstoUniformlyOn G hloc hqt b hb x y).mono
-    (show Set.Ioo 0 (b:ℝ) ⊆ Set.Icc 0 (b:ℝ) from Set.Ioo_subset_Icc_self)
-  have hh := hasDerivAt_of_tendstoUniformlyOn (f := fun n => finiteEventPolynomial (vertexExhaustion n) {ω | Conn G ω x y})
-    (g := realTwoPoint G x y) (g' := fun t => realContactExpectation G x y t/(1-t)) isOpen_Ioo hu
-    (Eventually.of_forall (fun n t _ => by
-      change HasDerivAt (bernoulliPolynomial (vertexExhaustion n) (fun (ω : Finset E) => if Conn G (ω:Set E) x y then (1:ℝ) else 0)) _ t
-      exact (hasDerivAt_bernoulliPolynomial (vertexExhaustion n)
-        (fun (ω : Finset E) => if Conn G (ω:Set E) x y then (1:ℝ) else 0) t).differentiableAt.hasDerivAt))
-    (fun t ht => by
-      let q : unitInterval := ⟨t,ht.1.le,ht.2.le.trans b.2.2⟩
-      simpa only [←realTwoPoint_coe G x y q] using finite_connection_polynomial_tendsto_twoPoint G q x y)
-    (show (p:ℝ) ∈ Set.Ioo 0 (b:ℝ) from ⟨hp0,hpb⟩)
-  simpa only [realContactExpectation_coe] using hh
-
-end PointwiseRussoLimit
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-open scoped ENNReal
-section RussoDiagram
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def russoDiagram (G : BondGraph V E) (p : unitInterval) (x y : V) : ℝ≥0∞ :=
-  ∑' a, ∑' b, indexedEdgeKernel G a b *
-    (law p {ω | Conn G ω x a} * law p {ω | Conn G ω y b})
-
-theorem conn_probability_symm (G : BondGraph V E) (p : unitInterval) (x y : V) :
-    law p {ω | Conn G ω x y} = law p {ω | Conn G ω y x} := by
-  congr 1
-  ext ω
-  exact ⟨conn_symm G ω,conn_symm G ω⟩
-
-theorem conn_probability_mono [Countable V] [Countable E] (G : BondGraph V E)
-    (p q : unitInterval) (hpq : p ≤ q) (x y : V) :
-    law p {ω | Conn G ω x y} ≤ law q {ω | Conn G ω x y} := by
-  rw [←ENNReal.ofReal_toReal (measure_ne_top (law p) _),←ENNReal.ofReal_toReal (measure_ne_top (law q) _)]
-  exact ENNReal.ofReal_le_ofReal (twoPoint_mono G x y hpq)
-
-theorem fullContact_lintegral_eq_kernel [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (p : unitInterval) (x y : V)
-    (hx : ∀ᵐ ω ∂law p, (cluster G ω x).Finite) :
-    (∫⁻ ω, ENNReal.ofReal (fullLargeContact G x y 0 ω) ∂law p) =
-      ∫⁻ ω, if ¬Conn G ω x y then clusterPairKernel G ω (indexedEdgeKernel G) x y else 0 ∂law p := by
-  apply lintegral_congr_ae
-  filter_upwards [hx] with ω hω
-  by_cases hxy : Conn G ω x y
-  · simp only [fullLargeContact,hxy,not_true_eq_false,false_and,ite_false,ENNReal.ofReal_zero]
-  · rw [ite_eq_left hxy,clusterPairKernel_edge G hloc ω x y hxy hω]
-    simp only [fullLargeContact,hxy,Nat.zero_le,not_false_eq_true,and_self,ite_true,ENNReal.ofReal_natCast]
-
-theorem fullContact_expectation_le_diagram [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (x y : V) :
-    ENNReal.ofReal (∫ ω, fullLargeContact G x y 0 ω ∂law p) ≤ russoDiagram G p x y := by
-  rw [ofReal_integral_eq_lintegral_ofReal (integrable_fullContact_subcritical G hloc hqt p hp x y)
-    (Eventually.of_forall (fullLargeContact_nonneg G x y 0))]
-  rw [fullContact_lintegral_eq_kernel G hloc p x y (subcritical_cluster_finite_ae G p hp x)]
-  exact clusterPairKernel_distinct_bound G p (indexedEdgeKernel G) x y
-
-theorem russoDiagram_mono [Countable V] [Countable E] (G : BondGraph V E)
-    (p q : unitInterval) (hpq : p ≤ q) (x y : V) : russoDiagram G p x y ≤ russoDiagram G q x y := by
-  exact ENNReal.tsum_le_tsum (fun a => ENNReal.tsum_le_tsum (fun b =>
-    mul_le_mul_right (mul_le_mul' (conn_probability_mono G p q hpq x a)
-      (conn_probability_mono G p q hpq y b)) _))
-
-theorem russoDiagram_symm (G : BondGraph V E) (p : unitInterval) (x y : V) :
-    russoDiagram G p x y = russoDiagram G p y x := by
-  unfold russoDiagram
-  rw [ENNReal.tsum_comm]
-  apply tsum_congr; intro b
-  apply tsum_congr; intro a
-  rw [indexedEdgeKernel_symm G a b,mul_comm (law p {ω | Conn G ω x a})]
-
-theorem russoDiagram_row_bound (G : BondGraph V E) (hloc : LocallyFinite G)
-    (d : ℕ) (hd : ∀ z, degree G z ≤ d) (p : unitInterval) (C : ℝ≥0∞)
-    (hC : ∀ x, ∑' y, law p {ω | Conn G ω x y} ≤ C) (x : V) :
-    (∑' y, russoDiagram G p x y) ≤ (d:ℝ≥0∞)*C^2 := by
-  unfold russoDiagram
-  rw [ENNReal.tsum_comm]
-  calc
-    _ = ∑' a, ∑' b, indexedEdgeKernel G a b * law p {ω | Conn G ω x a} *
-        ∑' y, law p {ω | Conn G ω b y} := by
-      apply tsum_congr; intro a
-      rw [ENNReal.tsum_comm]
-      apply tsum_congr; intro b
-      simp_rw [←mul_assoc,conn_probability_symm G p _ b]
-      rw [ENNReal.tsum_mul_left]
-    _ ≤ ∑' a, ∑' b, indexedEdgeKernel G a b * law p {ω | Conn G ω x a} * C := by
-      exact ENNReal.tsum_le_tsum (fun a => ENNReal.tsum_le_tsum (fun b => mul_le_mul_right (hC b) _))
-    _ = ∑' a, (degree G a:ℝ≥0∞) * law p {ω | Conn G ω x a} * C := by
-      apply tsum_congr; intro a
-      rw [ENNReal.tsum_mul_right,ENNReal.tsum_mul_right,indexedEdgeKernel_row G hloc a]
-    _ ≤ ∑' a, (d:ℝ≥0∞) * law p {ω | Conn G ω x a} * C := by
-      exact ENNReal.tsum_le_tsum (fun a => mul_le_mul_left (mul_le_mul_left (show (degree G a:ℝ≥0∞) ≤ (d:ℝ≥0∞) by exact_mod_cast hd a) _) _)
-    _ = (d:ℝ≥0∞)*(∑' a, law p {ω | Conn G ω x a})*C := by
-      rw [ENNReal.tsum_mul_right,ENNReal.tsum_mul_left]
-    _ ≤ (d:ℝ≥0∞)*C^2 := by
-      rw [pow_two,←mul_assoc]
-      exact mul_le_mul_left (mul_le_mul_right (hC x) _) _
-
-end RussoDiagram
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-open scoped ENNReal
-section RussoMajorant
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def russoSlope (G : BondGraph V E) (x y : V) (t : ℝ) : ℝ :=
-  realContactExpectation G x y t / (1-t)
-
-noncomputable def russoMajorant (G : BondGraph V E) (b : unitInterval) (x y : V) : ℝ :=
-  (russoDiagram G b x y).toReal / (1-(b:ℝ))
-
-theorem russoDiagram_row_ne_top [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (b : unitInterval) (hb : b < pc G) (x : V) : (∑' y, russoDiagram G b x y) ≠ ⊤ := by
-  obtain ⟨d,_,hd⟩ := exists_degree_bound G hqt
-  obtain ⟨C,_,hC⟩ := subcritical_ennreal_row_bound G hloc hqt b hb
-  exact ne_top_of_le_ne_top (by finiteness) (russoDiagram_row_bound G hloc d hd b _ hC x)
-
-theorem russoDiagram_ne_top [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (b : unitInterval) (hb : b < pc G) (x y : V) : russoDiagram G b x y ≠ ⊤ :=
-  ne_top_of_le_ne_top (russoDiagram_row_ne_top G hloc hqt b hb x) (ENNReal.le_tsum y)
-
-theorem summable_russoMajorant [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (b : unitInterval) (hb : b < pc G) (x : V) : Summable (russoMajorant G b x) :=
-  (ENNReal.summable_toReal (russoDiagram_row_ne_top G hloc hqt b hb x)).div_const _
-
-theorem russoSlope_nonneg (G : BondGraph V E) (x y : V) (t : ℝ) (ht : t ≤ 1) :
-    0 ≤ russoSlope G x y t := by
-  exact div_nonneg (integral_nonneg (fullLargeContact_nonneg G x y 0)) (sub_nonneg.mpr ht)
-
-theorem russoMajorant_nonneg (G : BondGraph V E) (b : unitInterval) (x y : V) :
-    0 ≤ russoMajorant G b x y := div_nonneg ENNReal.toReal_nonneg (sub_nonneg.mpr b.2.2)
-
-theorem russoSlope_le_majorant [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (b p : unitInterval) (hb : b < pc G) (hp : p ≤ b) (x y : V) :
-    russoSlope G x y (p:ℝ) ≤ russoMajorant G b x y := by
-  have hp' := hp.trans_lt hb
-  have he := (fullContact_expectation_le_diagram G hloc hqt p hp' x y).trans (russoDiagram_mono G p b hp x y)
-  have hn : 0 ≤ ∫ ω, fullLargeContact G x y 0 ω ∂law p := integral_nonneg (fullLargeContact_nonneg G x y 0)
-  have hc : (∫ ω, fullLargeContact G x y 0 ω ∂law p) ≤ (russoDiagram G b x y).toReal := by
-    simpa only [ENNReal.toReal_ofReal hn] using ENNReal.toReal_mono (russoDiagram_ne_top G hloc hqt b hb x y) he
-  simp only [russoSlope,realContactExpectation_coe,russoMajorant]
-  exact div_le_div₀ ENNReal.toReal_nonneg hc (sub_pos.mpr ((show (b:ℝ) < (pc G:ℝ) from hb).trans_le (pc G).2.2)) (sub_le_sub_left (show (p:ℝ) ≤ (b:ℝ) from hp) 1)
-
-theorem hasDerivAt_realTwoPoint_slope [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (t : ℝ) (ht : t ∈ Set.Ioo 0 (pc G:ℝ)) (x y : V) :
-    HasDerivAt (realTwoPoint G x y) (russoSlope G x y t) t := by
-  let p : unitInterval := ⟨t, ht.1.le,ht.2.le.trans (pc G).2.2⟩
-  change HasDerivAt (realTwoPoint G x y) (russoSlope G x y (p:ℝ)) (p:ℝ)
-  simpa only [russoSlope,realContactExpectation_coe] using hasDerivAt_realTwoPoint_subcritical G hloc hqt p ht.1 ht.2 x y
-
-end RussoMajorant
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-open scoped ENNReal
-section RussoRows
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem russoMajorant_symm (G : BondGraph V E) (b : unitInterval) (x y : V) :
-    russoMajorant G b x y = russoMajorant G b y x := by
-  rw [russoMajorant,russoDiagram_symm,russoMajorant]
-
-theorem russoMajorant_uniform_row_bound [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (b : unitInterval) (hb : b < pc G) :
-    ∃ R : ℝ, 0 ≤ R ∧ ∀ x, ∑' y, russoMajorant G b x y ≤ R := by
-  obtain ⟨d,_,hd⟩ := exists_degree_bound G hqt
-  obtain ⟨C,_,hC⟩ := subcritical_ennreal_row_bound G hloc hqt b hb
-  let R : ℝ≥0∞ := (d:ℝ≥0∞)*(ENNReal.ofReal C)^2
-  have hR : R ≠ ⊤ := by dsimp [R]; finiteness
-  have hden : 0 < 1-(b:ℝ) := sub_pos.mpr ((show (b:ℝ) < (pc G:ℝ) from hb).trans_le (pc G).2.2)
-  refine ⟨R.toReal/(1-(b:ℝ)),div_nonneg ENNReal.toReal_nonneg hden.le,fun x => ?_⟩
-  simp only [russoMajorant,tsum_div_const]
-  rw [←ENNReal.tsum_toReal_eq (russoDiagram_ne_top G hloc hqt b hb x)]
-  exact div_le_div_of_nonneg_right (ENNReal.toReal_mono hR (russoDiagram_row_bound G hloc d hd b _ hC x)) hden.le
-
- 
-
-theorem summable_kernel_pair (A B : V → V → ℝ)
-    (hA : ∀ x y, 0 ≤ A x y) (hB : ∀ x y, 0 ≤ B x y)
-    (hsA : ∀ x, Summable (A x)) (hsB : ∀ x, Summable (B x))
-    (C : ℝ) (hC : ∀ x, ∑' y, B x y ≤ C) (x : V) :
-    Summable (fun ab : V×V => A x ab.1 * B ab.1 ab.2) := by
-  apply (summable_prod_of_nonneg (f := fun ab : V×V => A x ab.1 * B ab.1 ab.2)
-    (fun ab => mul_nonneg (hA x ab.1) (hB ab.1 ab.2))).mpr
-  refine ⟨fun a => (hsB a).mul_left (A x a),?_⟩
-  simp only [tsum_mul_left]
-  exact Summable.of_nonneg_of_le (fun a => mul_nonneg (hA x a) (tsum_nonneg (hB a)))
-    (fun a => mul_le_mul_of_nonneg_left (hC a) (hA x a)) ((hsA x).mul_right C)
-
-theorem summable_twoPoint_pair [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (x : V) :
-    Summable (fun ab : V×V => twoPoint G p x ab.1 * twoPoint G p ab.1 ab.2) := by
-  obtain ⟨C,_,hrow⟩ := subcritical_row_bound G hloc hqt p hp
-  have hs := summable_twoPoint_subcritical G hloc hqt p hp
-  exact summable_kernel_pair (twoPoint G p) (twoPoint G p) (twoPoint_nonneg G p)
-    (twoPoint_nonneg G p) hs hs C (fun x => (hs x).tsum_le_of_sum_le (hrow x)) x
-
-theorem summable_russoMajorant_pair [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (x : V) :
-    Summable (fun ab : V×V => russoMajorant G p x ab.1 * twoPoint G p ab.1 ab.2) ∧
-    Summable (fun ab : V×V => twoPoint G p x ab.1 * russoMajorant G p ab.1 ab.2) := by
-  obtain ⟨C,_,hrow⟩ := subcritical_row_bound G hloc hqt p hp
-  have hs := summable_twoPoint_subcritical G hloc hqt p hp
-  have hr := summable_russoMajorant G hloc hqt p hp
-  obtain ⟨R,_,hR⟩ := russoMajorant_uniform_row_bound G hloc hqt p hp
-  exact ⟨summable_kernel_pair _ _ (russoMajorant_nonneg G p) (twoPoint_nonneg G p) hr hs C
-    (fun x => (hs x).tsum_le_of_sum_le (hrow x)) x,
-    summable_kernel_pair _ _ (twoPoint_nonneg G p) (russoMajorant_nonneg G p) hs hr R hR x⟩
-
-end RussoRows
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section RealTriangle
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def realTriangleTerm (G : BondGraph V E) (x : V) (ab : V×V) (t : ℝ) : ℝ :=
-  realTwoPoint G x ab.1 t * realTwoPoint G ab.1 ab.2 t * realTwoPoint G ab.2 x t
-
-noncomputable def realTriangleSlope (G : BondGraph V E) (x : V) (ab : V×V) (t : ℝ) : ℝ :=
-  russoSlope G x ab.1 t * realTwoPoint G ab.1 ab.2 t * realTwoPoint G ab.2 x t +
-  realTwoPoint G x ab.1 t * russoSlope G ab.1 ab.2 t * realTwoPoint G ab.2 x t +
-  realTwoPoint G x ab.1 t * realTwoPoint G ab.1 ab.2 t * russoSlope G ab.2 x t
-
-noncomputable def triangleMajorant (G : BondGraph V E) (b : unitInterval) (x : V) (ab : V×V) : ℝ :=
-  russoMajorant G b x ab.1 * twoPoint G b ab.1 ab.2 +
-  twoPoint G b x ab.1 * russoMajorant G b ab.1 ab.2 +
-  russoMajorant G b x ab.2 * twoPoint G b ab.2 ab.1
-
-theorem summable_triangleMajorant [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (b : unitInterval) (hb : b < pc G) (x : V) : Summable (triangleMajorant G b x) := by
-  obtain ⟨h₁,h₂⟩ := summable_russoMajorant_pair G hloc hqt b hb x
-  have h₃ : Summable (fun ab : V×V => russoMajorant G b x ab.2 * twoPoint G b ab.2 ab.1) := by
-    exact h₁.comp_injective (i := fun ab : V×V => (ab.2,ab.1)) (Equiv.prodComm V V).injective
-  exact (h₁.add h₂).add h₃
-
-theorem hasDerivAt_realTriangleTerm [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (t : ℝ) (ht : t ∈ Set.Ioo 0 (pc G:ℝ)) (x : V) (ab : V×V) :
-    HasDerivAt (realTriangleTerm G x ab) (realTriangleSlope G x ab t) t := by
-  have hs := hasDerivAt_realTwoPoint_slope G hloc hqt t ht
-  convert! ((hs x ab.1).mul (hs ab.1 ab.2)).mul (hs ab.2 x) using 1
-  simp only [realTriangleSlope,Pi.mul_apply]
-  ring
-
-theorem realTriangleSlope_nonneg (G : BondGraph V E) (x : V) (ab : V×V) (t : ℝ) (ht : t ≤ 1) :
-    0 ≤ realTriangleSlope G x ab t := by
-  have hT (a b : V) : 0 ≤ realTwoPoint G a b t := twoPoint_nonneg G _ a b
-  dsimp [realTriangleSlope]
-  exact add_nonneg (add_nonneg (mul_nonneg (mul_nonneg (russoSlope_nonneg G _ _ t ht) (hT _ _)) (hT _ _))
-    (mul_nonneg (mul_nonneg (hT _ _) (russoSlope_nonneg G _ _ t ht)) (hT _ _)))
-    (mul_nonneg (mul_nonneg (hT _ _) (hT _ _)) (russoSlope_nonneg G _ _ t ht))
-
-theorem realTriangleSlope_le_majorant [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (b p : unitInterval) (hb : b < pc G) (hp : p ≤ b) (x : V) (ab : V×V) :
-    realTriangleSlope G x ab (p:ℝ) ≤ triangleMajorant G b x ab := by
-  have hs := russoSlope_le_majorant G hloc hqt b p hb hp
-  have hn := russoSlope_nonneg (G := G)
-  have hm (a z : V) : twoPoint G p a z ≤ twoPoint G b a z := twoPoint_mono G a z hp
-  simp only [realTriangleSlope,realTwoPoint_coe,triangleMajorant]
-  apply add_le_add
-  · apply add_le_add
-    · calc
-        _ ≤ russoMajorant G b x ab.1 * twoPoint G b ab.1 ab.2 * 1 :=
-          mul_le_mul (mul_le_mul (hs _ _) (hm _ _) (twoPoint_nonneg G p _ _) (russoMajorant_nonneg G b _ _))
-            (twoPoint_le_one G p _ _) (twoPoint_nonneg G p _ _) (mul_nonneg (russoMajorant_nonneg G b _ _) (twoPoint_nonneg G b _ _))
-        _ = _ := mul_one _
-    · calc
-        _ ≤ twoPoint G b x ab.1 * russoMajorant G b ab.1 ab.2 * 1 :=
-          mul_le_mul (mul_le_mul (hm _ _) (hs _ _) (hn _ _ _ p.2.2) (twoPoint_nonneg G b _ _))
-            (twoPoint_le_one G p _ _) (twoPoint_nonneg G p _ _) (mul_nonneg (twoPoint_nonneg G b _ _) (russoMajorant_nonneg G b _ _))
-        _ = _ := mul_one _
-  · calc
-      _ ≤ 1 * twoPoint G b ab.1 ab.2 * russoMajorant G b ab.2 x :=
-        mul_le_mul (mul_le_mul (twoPoint_le_one G p _ _) (hm _ _) (twoPoint_nonneg G p _ _) zero_le_one)
-          (hs _ _) (hn _ _ _ p.2.2) (mul_nonneg zero_le_one (twoPoint_nonneg G b _ _))
-      _ = _ := by rw [one_mul,twoPoint_symm G b,russoMajorant_symm G b]; ring
-
-end RealTriangle
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section TriangleDerivative
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def realTriangle (G : BondGraph V E) (x : V) (t : ℝ) : ℝ :=
-  ∑' ab, realTriangleTerm G x ab t
-
-noncomputable def rootTriangle (G : BondGraph V E) {ι : Type*} [Fintype ι]
-    (root : ι → V) (weight : ι → ℝ) (t : ℝ) : ℝ :=
-  ∑ i, weight i * realTriangle G (root i) t
-
-theorem summable_realTriangleTerm [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (x : V) :
-    Summable (fun ab => realTriangleTerm G x ab (p:ℝ)) := by
-  simp only [realTriangleTerm,realTwoPoint_coe]
-  apply (summable_twoPoint_pair G hloc hqt p hp x).of_nonneg_of_le
-  · intro ab
-    exact mul_nonneg (mul_nonneg (twoPoint_nonneg G p _ _) (twoPoint_nonneg G p _ _)) (twoPoint_nonneg G p _ _)
-  · intro ab
-    exact mul_le_of_le_one_right (mul_nonneg (twoPoint_nonneg G p _ _) (twoPoint_nonneg G p _ _)) (twoPoint_le_one G p _ _)
-
-theorem summable_realTriangleSlope [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (x : V) :
-    Summable (fun ab => realTriangleSlope G x ab (p:ℝ)) := by
-  exact (summable_triangleMajorant G hloc hqt p hp x).of_nonneg_of_le
-    (fun ab => realTriangleSlope_nonneg G x ab p p.2.2)
-    (fun ab => realTriangleSlope_le_majorant G hloc hqt p p hp le_rfl x ab)
-
- 
-
-theorem hasDerivAt_realTriangle_subcritical [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp : p < pc G) (x : V) :
-    HasDerivAt (realTriangle G x) (∑' ab, realTriangleSlope G x ab (p:ℝ)) (p:ℝ) := by
-  obtain ⟨b,hpb,hb⟩ := exists_between hp
-  apply hasDerivAt_tsum_of_isPreconnected (summable_triangleMajorant G hloc hqt b hb x)
-    isOpen_Ioo isPreconnected_Ioo
-    (fun ab t ht => hasDerivAt_realTriangleTerm G hloc hqt t ⟨ht.1,lt_trans ht.2 hb⟩ x ab)
-    (fun ab t ht => ?_) (show (p:ℝ) ∈ Set.Ioo 0 (b:ℝ) from ⟨hp0,hpb⟩)
-    (summable_realTriangleTerm G hloc hqt p hp x) ⟨hp0,hpb⟩
-  let q : unitInterval := ⟨t,ht.1.le,ht.2.le.trans b.2.2⟩
-  have hq : q ≤ b := ht.2.le
-  rw [Real.norm_eq_abs,abs_of_nonneg (realTriangleSlope_nonneg G x ab t q.2.2)]
-  exact realTriangleSlope_le_majorant G hloc hqt b q hb hq x ab
-
-theorem hasDerivAt_rootTriangle_subcritical [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp : p < pc G) :
-    HasDerivAt (rootTriangle G root weight)
-      (∑ i, weight i * ∑' ab, realTriangleSlope G (root i) ab (p:ℝ)) (p:ℝ) := by
-  exact HasDerivAt.fun_sum (fun i _ => (hasDerivAt_realTriangle_subcritical G hloc hqt p hp0 hp (root i)).const_mul (weight i))
-
-end TriangleDerivative
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Kernel.ContactTrace
+
+namespace OAI
+
+/-! Russo Rows for bond percolation. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+open scoped ENNReal
+section PairFunctionalAlgebra
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem clusterPairFunctional_mono_distinct (G : BondGraph V E) (p : unitInterval)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (F H : Set E → V → V → ℝ≥0∞)
+    (h : ∀ ω x y, ¬Conn G ω x y → F ω x y ≤ H ω x y) :
+    clusterPairFunctional G p root weight F ≤ clusterPairFunctional G p root weight H := by
+  apply Finset.sum_le_sum
+  intro i _
+  apply mul_le_mul_right
+  apply lintegral_mono
+  intro ω
+  unfold clusterPairDensity
+  apply mul_le_mul_right
+  apply ENNReal.tsum_le_tsum
+  intro y
+  apply mul_le_mul_right
+  by_cases hc : Conn G ω (root i) y
+  · simp only [hc,not_true_eq_false,ite_false,le_refl]
+  · simpa only [ite_eq_left hc] using h ω (root i) y hc
+
+theorem clusterPairFunctional_congr_ae (G : BondGraph V E) (p : unitInterval)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (F H : Set E → V → V → ℝ≥0∞)
+    (h : ∀ᵐ ω ∂law p, ∀ x y, ¬Conn G ω x y → F ω x y = H ω x y) :
+    clusterPairFunctional G p root weight F = clusterPairFunctional G p root weight H := by
+  apply Finset.sum_congr rfl
+  intro i _
+  congr 1
+  apply lintegral_congr_ae
+  filter_upwards [h] with ω hω
+  unfold clusterPairDensity
+  congr 1
+  apply tsum_congr
+  intro y
+  congr 1
+  by_cases hc : Conn G ω (root i) y
+  · simp only [hc,not_true_eq_false,ite_false]
+  · rw [ite_eq_left hc,ite_eq_left hc,hω (root i) y hc]
+
+theorem clusterPairFunctional_add [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (p : unitInterval)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (F H : Set E → V → V → ℝ≥0∞)
+    (hF : ∀ x y, Measurable (fun ω => F ω x y))
+    (hH : ∀ x y, Measurable (fun ω => H ω x y)) :
+    clusterPairFunctional G p root weight (fun ω x y => F ω x y+H ω x y) =
+      clusterPairFunctional G p root weight F + clusterPairFunctional G p root weight H := by
+  let : MeasurableSpace V := ⊤
+  let : MeasurableSingletonClass V := ⟨fun _ => trivial⟩
+  let : MeasurableSpace ι := ⊤
+  let : MeasurableSingletonClass ι := ⟨fun _ => trivial⟩
+  rw [clusterPairFunctional_eq_lintegral G hloc p root weight _ hF,
+    clusterPairFunctional_eq_lintegral G hloc p root weight _ hH,
+    clusterPairFunctional_eq_lintegral G hloc p root weight (fun ω x y => F ω x y+H ω x y) (fun x y => (hF x y).add (hH x y))]
+  exact lintegral_add_left (measurable_from_prod_countable_left (fun q => hF (root q.1) q.2)) _
+
+theorem clusterPairKernel_symm (G : BondGraph V E) (ω : Set E)
+    (J : V → V → ℝ≥0∞) (hJ : ∀ x y, J x y = J y x) (x y : V) :
+    clusterPairKernel G ω J x y = clusterPairKernel G ω J y x := by
+  unfold clusterPairKernel
+  rw [ENNReal.tsum_comm]
+  apply tsum_congr
+  intro a
+  apply tsum_congr
+  intro b
+  simp only [and_comm,hJ b a]
+
+end PairFunctionalAlgebra
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+open scoped ENNReal
+section LargeContactCauchySchwarz
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def largeContactKernel (G : BondGraph V E) (L : ℝ)
+    (ω : Set E) (x y : V) : ℝ≥0∞ :=
+  if L ≤ min ((cluster G ω x).ncard:ℝ) ((cluster G ω y).ncard:ℝ) then
+    clusterPairKernel G ω (indexedEdgeKernel G) x y else 0
+
+theorem measurable_largeContactKernel [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (L : ℝ) (x y : V) :
+    Measurable (fun ω => largeContactKernel G L ω x y) := by
+  have hm (z : V) : Measurable (fun ω : Set E => ((cluster G ω z).ncard:ℝ)) :=
+    (measurable_of_countable (fun n : ℕ => (n:ℝ))).comp (measurable_cluster_ncard G hloc z)
+  exact (measurable_clusterPairKernel G (indexedEdgeKernel G) x y).ite
+    (measurableSet_le measurable_const ((hm x).min (hm y))) measurable_const
+
+theorem clusterPairFunctional_largeContact_sq [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (L : ℝ) (p : unitInterval)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (hfin : ∀ x, ∀ᵐ ω ∂law p, (cluster G ω x).Finite) :
+    clusterPairFunctional G p root weight (fun ω x y => (largeContactKernel G L ω x y)^2) =
+      clusterPairFunctional G p root weight (contactSquareLargeReal G L) := by
+  apply clusterPairFunctional_congr_ae G p root weight
+  filter_upwards [ae_all_iff.mpr hfin] with ω hω
+  intro x y hxy
+  dsimp only [largeContactKernel,contactSquareLargeReal]
+  split_ifs
+  · rw [clusterPairKernel_edge G hloc ω x y hxy (hω x)]
+  · rw [zero_pow (by decide : 2 ≠ 0)]
+
+theorem clusterPairFunctional_largeContact_bound [Countable V] [Countable E]
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hw : ∀ i, 0 ≤ weight i) (p : unitInterval)
+    (hfin : ∀ x, ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
+    (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T) (L : ℝ) (B : ℝ≥0∞)
+    (hB : clusterPairFunctional G p root weight (contactSquareLargeReal G L) ≤ B) :
+    clusterPairFunctional G p root weight (fun ω x y =>
+      largeContactKernel G L ω x y * clusterPairKernel G ω (ennMatrixCoeff (T^2)) x y) ≤
+      B^((1:ℝ)/2)* (ENNReal.ofReal (‖T‖^3*(rootTrace root weight (T^3)).re))^((1:ℝ)/2) := by
+  have hh := clusterPairFunctional_cauchy_schwarz G hloc p root weight
+    (largeContactKernel G L) (fun ω x y => clusterPairKernel G ω (ennMatrixCoeff (T^2)) x y)
+    (measurable_largeContactKernel G hloc L) (measurable_clusterPairKernel G _)
+  rw [clusterPairFunctional_largeContact_sq G hloc L p root weight hfin] at hh
+  exact hh.trans (mul_le_mul
+    (ENNReal.rpow_le_rpow hB (by positivity))
+    (ENNReal.rpow_le_rpow (clusterPairFunctional_twoPoint_square_bound G hconn hloc root weight Γ hΓ htr hw p hfin T hT) (by positivity))
+    bot_le bot_le)
+
+end LargeContactCauchySchwarz
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+open scoped ENNReal ComplexOrder
+section SmallTraceTerm
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem contact_smallTwoPoint_bound [Countable V] [Countable E]
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
+    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (p : unitInterval) (hfin : ∀ x, ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
+    (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T)
+    (L α C : ℝ) (hL : 0 ≤ L) (hα0 : 0 ≤ α) (hα1 : α ≤ 1)
+    (hI : ∀ x, Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^α) (law p))
+    (hC : ∀ x, (∫ ω, ((cluster G ω x).ncard:ℝ)^α ∂law p) ≤ C)
+    (d : ℕ) (hd : ∀ x, degree G x ≤ d) :
+    clusterPairFunctional G p root weight (fun ω x y =>
+      clusterPairKernel G ω (indexedEdgeKernel G) x y * smallPairKernel G (ennMatrixCoeff (T^2)) L x y ω) ≤
+      ENNReal.ofReal ((d:ℝ)*c*C*L^(1-α)*(rootTrace root weight (T^3)).re) := by
+  have hTI : KernelInvariant Γ T := fun a ha => hT.invariant G p T a (hΓ ha)
+  have hJ : ∀ a ∈ Γ, ∀ x y, ennMatrixCoeff (T^2) (a x) (a y) = ennMatrixCoeff (T^2) x y := by
+    intro a ha x y
+    unfold ennMatrixCoeff
+    rw [(hTI.pow 2) a ha]
+  have hpos : 0 ≤ T := ContinuousLinearMap.nonneg_iff_isPositive.mpr (hT.isPositive G p T)
+  have hMt : 0 ≤ c*(rootTrace root weight (T^3)).re := mul_nonneg hc
+    (Complex.nonneg_iff.mp (rootTrace_nonneg root weight hw _ (CStarAlgebra.pow_nonneg T 3 hpos))).1
+  have hh := contact_smallPairKernel_bound G hconn hloc root weight Γ hΓ htr hw hw1 p hfin
+    (ennMatrixCoeff (T^2)) hJ L d hd (c*C*L^(1-α)*(rootTrace root weight (T^3)).re) (fun x y => by
+      refine (smallPairKernel_twoPoint_bound G hloc root weight Γ hΓ hcov hw c hc hmin x y p T hT L α hL hα0 hα1 (hfin x) (hfin y) (hI x)).trans ?_
+      apply ENNReal.ofReal_le_ofReal
+      calc
+        _ ≤ (c*(rootTrace root weight (T^3)).re)*(L^(1-α)*C) :=
+          mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_left (hC x) (Real.rpow_nonneg hL _)) hMt
+        _ = _ := by ring)
+  convert hh using 1; congr 1; ring
+
+end SmallTraceTerm
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+open scoped ENNReal ComplexOrder
+section MixedTraceTerm
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem contact_smallPairKernel_swap [Countable V] [Countable E]
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hw : ∀ i, 0 ≤ weight i) (p : unitInterval)
+    (J : V → V → ℝ≥0∞) (hJ : ∀ a ∈ Γ, ∀ x y, J (a x) (a y) = J x y) (L : ℝ) :
+    clusterPairFunctional G p root weight (fun ω x y =>
+      clusterPairKernel G ω (indexedEdgeKernel G) y x * smallPairKernel G J L y x ω) =
+    clusterPairFunctional G p root weight (fun ω x y =>
+      clusterPairKernel G ω (indexedEdgeKernel G) x y * smallPairKernel G J L x y ω) := by
+  symm
+  apply clusterPairFunctional_swap G hconn hloc root weight Γ hΓ htr hw p
+    (fun ω x y => clusterPairKernel G ω (indexedEdgeKernel G) x y * smallPairKernel G J L x y ω)
+  · intro x y
+    exact (measurable_clusterPairKernel G _ x y).mul (measurable_smallPairKernel G hloc J L x y)
+  · intro a ha b hab ω x y
+    rw [clusterPairKernel_aut G hab _ (indexedEdgeKernel_aut G hab),smallPairKernel_aut G hab J (hJ a ha)]
+
+ 
+
+theorem clusterPairFunctional_twoPoint_mixed_bound [Countable V] [Countable E]
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
+    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (p : unitInterval) (hfin : ∀ x, ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
+    (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T)
+    (L α C : ℝ) (hL : 0 ≤ L) (hα0 : 0 ≤ α) (hα1 : α ≤ 1)
+    (hI : ∀ x, Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^α) (law p))
+    (hC : ∀ x, (∫ ω, ((cluster G ω x).ncard:ℝ)^α ∂law p) ≤ C)
+    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (B : ℝ≥0∞)
+    (hB : clusterPairFunctional G p root weight (contactSquareLargeReal G L) ≤ B) :
+    clusterPairFunctional G p root weight (fun ω x y =>
+      clusterPairKernel G ω (indexedEdgeKernel G) x y * clusterPairKernel G ω (ennMatrixCoeff (T^2)) x y) ≤
+    2*ENNReal.ofReal ((d:ℝ)*c*C*L^(1-α)*(rootTrace root weight (T^3)).re) +
+      B^((1:ℝ)/2)* (ENNReal.ofReal (‖T‖^3*(rootTrace root weight (T^3)).re))^((1:ℝ)/2) := by
+  let J := ennMatrixCoeff (T^2)
+  let A := fun ω x y => clusterPairKernel G ω (indexedEdgeKernel G) x y*smallPairKernel G J L x y ω
+  let D := fun ω x y => largeContactKernel G L ω x y*clusterPairKernel G ω J x y
+  have heS (ω : Set E) (x y : V) : clusterPairKernel G ω (indexedEdgeKernel G) y x =
+      clusterPairKernel G ω (indexedEdgeKernel G) x y := clusterPairKernel_symm G ω _ (indexedEdgeKernel_symm G) y x
+  have hpos : 0 ≤ T := ContinuousLinearMap.nonneg_iff_isPositive.mpr (hT.isPositive G p T)
+  have hJS (ω : Set E) (x y : V) : clusterPairKernel G ω J y x = clusterPairKernel G ω J x y :=
+    clusterPairKernel_symm G ω J (ennMatrixCoeff_symm ((IsSelfAdjoint.of_nonneg hpos).pow 2)) y x
+  have hTI : KernelInvariant Γ T := fun a ha => hT.invariant G p T a (hΓ ha)
+  have hJI : ∀ a ∈ Γ, ∀ x y, J (a x) (a y) = J x y := by
+    intro a ha x y
+    dsimp [J,ennMatrixCoeff]
+    rw [(hTI.pow 2) a ha]
+  have hA (x y : V) : Measurable (fun ω => A ω x y) :=
+    (measurable_clusterPairKernel G _ x y).mul (measurable_smallPairKernel G hloc J L x y)
+  have hD (x y : V) : Measurable (fun ω => D ω x y) :=
+    (measurable_largeContactKernel G hloc L x y).mul (measurable_clusterPairKernel G J x y)
+  have hpw (ω : Set E) (x y : V) (hxy : ¬Conn G ω x y) :
+      clusterPairKernel G ω (indexedEdgeKernel G) x y * clusterPairKernel G ω J x y ≤ A ω x y + A ω y x + D ω x y := by
+    have hyx : ¬Conn G ω y x := fun h => hxy (conn_symm G ω h)
+    by_cases hx : ((cluster G ω x).ncard:ℝ) < L
+    · have ha : A ω x y = clusterPairKernel G ω (indexedEdgeKernel G) x y * clusterPairKernel G ω J x y := by
+        dsimp [A,smallPairKernel]; rw [ite_eq_left ⟨hx,hxy⟩]
+      rw [←ha]
+      exact le_add_of_le_left (le_add_right le_rfl)
+    · by_cases hy : ((cluster G ω y).ncard:ℝ) < L
+      · have ha : A ω y x = clusterPairKernel G ω (indexedEdgeKernel G) x y * clusterPairKernel G ω J x y := by
+          dsimp [A,smallPairKernel]; rw [ite_eq_left ⟨hy,hyx⟩,heS ω x y,hJS ω x y]
+        rw [←ha]
+        exact le_add_of_le_left (le_add_left le_rfl)
+      · have hg : L ≤ min ((cluster G ω x).ncard:ℝ) ((cluster G ω y).ncard:ℝ) := le_min (le_of_not_gt hx) (le_of_not_gt hy)
+        have hh : D ω x y = clusterPairKernel G ω (indexedEdgeKernel G) x y * clusterPairKernel G ω J x y := by
+          dsimp [D,largeContactKernel]; rw [ite_eq_left hg]
+        rw [←hh]
+        exact le_add_left le_rfl
+  have ha := contact_smallTwoPoint_bound G hconn hloc root weight Γ hΓ htr hcov hw hw1 c hc hmin p hfin T hT L α C hL hα0 hα1 hI hC d hd
+  have hb := clusterPairFunctional_largeContact_bound G hconn hloc root weight Γ hΓ htr hw p hfin T hT L B hB
+  calc
+    _ ≤ clusterPairFunctional G p root weight (fun ω x y => A ω x y+A ω y x+D ω x y) :=
+      clusterPairFunctional_mono_distinct G p root weight _ _ hpw
+    _ = (clusterPairFunctional G p root weight A + clusterPairFunctional G p root weight A) + clusterPairFunctional G p root weight D := by
+      rw [clusterPairFunctional_add G hloc p root weight (fun ω x y => A ω x y+A ω y x) D (fun x y => (hA x y).add (hA y x)) hD,
+        clusterPairFunctional_add G hloc p root weight A (fun ω x y => A ω y x) hA (fun x y => hA y x)]
+      rw [contact_smallPairKernel_swap G hconn hloc root weight Γ hΓ htr hw p J hJI L]
+    _ ≤ _ := by simpa only [two_mul] using add_le_add (add_le_add ha ha) hb
+
+end MixedTraceTerm
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+open scoped ENNReal
+section UniformMixedTrace
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem critical_mixed_trace_bound
+    [Countable V] [Countable E] [Nonempty V] [Nonempty E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hconn : Connected G)
+    (hqt : QuasiTransitive G) (hh : 0 < hV G)
+    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
+    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
+    (hp₀ : 0 < (pc G:ℝ)) (hp₁ : (pc G:ℝ) < 1)
+    (hfin : ∀ i, ∀ᵐ ω ∂law (pc G), (cluster G ω (root i)).Finite)
+    (α ε : ℝ) (hα : 0 < α) (hα1 : α < 1/2) (hε : 0 < ε)
+    (hr : (1:ℝ)/2+ε-2*α < 0) :
+    ∃ A B : ℝ, 0 < A ∧ 0 < B ∧ ∀ (p : unitInterval),
+      (pc G:ℝ)/2 ≤ (p:ℝ) → p ≤ pc G →
+      ∀ (T : Hilbert V →L[ℂ] Hilbert V), HasKernel G p T →
+      ∀ L : ℝ, 1 ≤ L →
+        clusterPairFunctional G p root weight (fun ω x y =>
+          clusterPairKernel G ω (indexedEdgeKernel G) x y *
+          clusterPairKernel G ω (ennMatrixCoeff (T^2)) x y) ≤
+        2*ENNReal.ofReal (A*L^(1-α)*(rootTrace root weight (T^3)).re) +
+          (ENNReal.ofReal (B*L^((1:ℝ)/2+ε-2*α)))^((1:ℝ)/2) *
+          (ENNReal.ofReal (‖T‖^3*(rootTrace root weight (T^3)).re))^((1:ℝ)/2) := by
+  have hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i) := by
+    intro x
+    obtain ⟨i,a,ha,hax⟩ := hcov x
+    obtain ⟨b,hab⟩ := hΓ ha
+    exact ⟨i,a,b,hab,hax⟩
+  have hfinite : ∀ x, ∀ᵐ ω ∂law (pc G), (cluster G ω x).Finite := by
+    intro x
+    obtain ⟨i,a,b,hab,rfl⟩ := hcover x
+    exact cluster_finite_ae_covariant G (pc G) hab (root i) (hfin i)
+  obtain ⟨C,hC,hCb⟩ := uniform_cluster_power_of_root G hloc (pc G) root hcover α
+    (critical_root_power_integrable G hloc hconn hqt hh root weight Γ hΓ htr hcov hw hw1 hp₀ hp₁ hfin α hα hα1)
+  obtain ⟨B,hB,hBb⟩ := critical_contact_square_bound G hloc hconn hqt hh root weight Γ hΓ htr hcov hw hw1 hp₀ hp₁ hfin α ε hα hα1 hε hr
+  obtain ⟨d,hd,hdg⟩ := exists_degree_bound G hqt
+  let c : ℝ := ∑ i, 1/weight i
+  have hc : 0 < c := Finset.sum_pos (fun i _ => one_div_pos.mpr (hw i)) Finset.univ_nonempty
+  have hmin (i : ι) : 1 ≤ c*weight i := by
+    apply (div_le_iff₀ (hw i)).mp
+    exact Finset.single_le_sum (fun j _ => (one_div_pos.mpr (hw j)).le) (Finset.mem_univ i)
+  refine ⟨(d:ℝ)*c*C,B,mul_pos (mul_pos (by exact_mod_cast hd) hc) hC,hB,?_⟩
+  intro p hp hpq T hT L hL
+  have hm (x : V) := cluster_power_mono_integral G hloc x p (pc G) hpq α hα.le (hfinite x) (hCb x).1
+  exact clusterPairFunctional_twoPoint_mixed_bound G hconn hloc root weight Γ hΓ htr hcov
+    (fun i => (hw i).le) hw1 c hc.le hmin p
+    (fun x => cluster_finite_ae_mono G x p (pc G) hpq (hfinite x))
+    T hT L α C (by linarith) hα.le (by linarith)
+    (fun x => (hm x).1) (fun x => (hm x).2.trans (hCb x).2) d hdg _ (hBb p hp hpq L hL)
+
+end UniformMixedTrace
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section ConnectionRussoContacts
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem closedPivotal_connection_iff (G : BondGraph V E) (ω : Set E) (x y : V) (e : E) :
+    ω ∈ closedPivotal {η | Conn G η x y} e ↔
+      ¬Conn G ω x y ∧ e ∈ closedContactSet G (cluster G ω x) (cluster G ω y) := by
+  constructor
+  · rintro ⟨_,hxy,hins⟩
+    refine ⟨hxy,?_⟩
+    obtain ⟨⟨a,b⟩,hab⟩ := Quot.exists_rep (G e)
+    have he : G e = s(a,b) := hab.symm
+    rcases (conn_insert_iff G ω e a b x y he).mp hins with h | h | h
+    · exact (hxy h).elim
+    · exact ⟨a,h.1,b,conn_symm G ω h.2,he⟩
+    · exact ⟨b,h.1,a,conn_symm G ω h.2,by rw [he,Sym2.eq_swap]⟩
+  · rintro ⟨hxy,a,hxa,b,hyb,he⟩
+    have hclosed : e ∉ ω := by
+      intro heω
+      exact hxy (hxa.trans ((Relation.ReflTransGen.single ⟨e,heω,he⟩).trans (conn_symm G ω hyb)))
+    refine ⟨hclosed,hxy,?_⟩
+    exact (conn_insert_iff G ω e a b x y he).mpr (Or.inr (Or.inl ⟨hxa,conn_symm G ω hyb⟩))
+
+noncomputable def finiteClosedContact (G : BondGraph V E) (Q : Finset E) (x y : V)
+    (ω : Finset E) : ℝ :=
+  if ¬Conn G (ω:Set E) x y then
+    (contactBonds G Q (finiteRootComponent G ω x) (finiteRootComponent G ω y)).card else 0
+
+theorem finiteClosedContact_eq_sum (G : BondGraph V E) (Q : Finset E) (x y : V)
+    (ω : Finset E) :
+    finiteClosedContact G Q x y ω =
+      ∑ e ∈ Q, if (ω:Set E) ∈ closedPivotal {η | Conn G η x y} e then (1:ℝ) else 0 := by
+  have hm (z a : V) : a ∈ finiteRootComponent G ω z ↔ Conn G (ω:Set E) z a := by
+    change a ∈ (finiteRootComponent G ω z:Set V) ↔ _
+    rw [coe_finiteRootComponent]
+    rfl
+  simp only [closedPivotal_connection_iff,closedContactSet,Set.mem_ofPred_eq,finiteClosedContact]
+  by_cases hxy : Conn G (ω:Set E) x y
+  · simp only [hxy,not_true_eq_false,false_and,ite_false,Finset.sum_const_zero]
+  · rw [ite_eq_left hxy]
+    simp only [hxy,not_false_eq_true,true_and,contactBonds,hm,cluster,Set.mem_ofPred_eq]
+    rw [Finset.sum_boole]
+
+theorem finite_connection_russo_contacts (G : BondGraph V E) (Q : Finset E)
+    (x y : V) (p : ℝ) :
+    (1-p)*deriv (finiteEventPolynomial Q {ω | Conn G ω x y}) p =
+      bernoulliPolynomial Q (finiteClosedContact G Q x y) p := by
+  change (1-p)*deriv (bernoulliPolynomial Q (fun ω => if Conn G (ω:Set E) x y then 1 else 0)) p = _
+  rw [closed_difference_russo]
+  have he : finiteClosedContact G Q x y = fun (ω : Finset E) =>
+      ∑ e ∈ Q, if (ω:Set E) ∈ closedPivotal {η | Conn G η x y} e then (1:ℝ) else 0 :=
+    funext (finiteClosedContact_eq_sum G Q x y)
+  rw [he,bernoulliPolynomial_sum]
+  apply Finset.sum_congr rfl
+  intro e _
+  apply bernoulliPolynomial_congr
+  intro ω _
+  exact (closedPivotal_difference {η | Conn G η x y}
+    (fun _ _ h hc => conn_mono G h hc) e ω).symm
+
+end ConnectionRussoContacts
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section UniformContactApproximation
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem finiteClosedContact_eq_full (G : BondGraph V E) (hloc : LocallyFinite G)
+    (ω : Set E) (x y : V) (hx : (cluster G ω x).Finite) (hy : (cluster G ω y).Finite)
+    (Q : Finset E) (hQx : incidentSupport G hloc hx.toFinset ⊆ Q)
+    (hQy : incidentSupport G hloc hy.toFinset ⊆ Q) :
+    finiteClosedContact G Q x y (domainBits Q ω) = fullLargeContact G x y 0 ω := by
+  have hcluster (z : V) (hz : (cluster G ω z).Finite) (hQ : incidentSupport G hloc hz.toFinset ⊆ Q) :
+      cluster G (domainBits Q ω:Set E) z = cluster G ω z := by
+    rw [coe_domainBits]
+    have hh := (cluster_restrict_incident_eq_iff G hloc hz.toFinset Q hQ ω z).mpr
+      (Set.Finite.coe_toFinset hz).symm
+    simpa only [Set.Finite.coe_toFinset] using hh
+  have hcx := hcluster x hx hQx
+  have hcy := hcluster y hy hQy
+  have hFx : finiteRootComponent G (domainBits Q ω) x = hx.toFinset := by
+    apply Finset.coe_injective
+    rw [coe_finiteRootComponent,hcx,Set.Finite.coe_toFinset]
+  have hFy : finiteRootComponent G (domainBits Q ω) y = hy.toFinset := by
+    apply Finset.coe_injective
+    rw [coe_finiteRootComponent,hcy,Set.Finite.coe_toFinset]
+  have hconn : Conn G (domainBits Q ω:Set E) x y ↔ Conn G ω x y := by
+    change y ∈ cluster G (domainBits Q ω:Set E) x ↔ y ∈ cluster G ω x
+    rw [hcx]
+  have hc : (contactBonds G Q hx.toFinset hy.toFinset).card = clusterContactCount G ω x y := by
+    rw [←Set.ncard_coe_finset,coe_contactBonds_eq G hloc Q _ _ hQx]
+    simp only [Set.Finite.coe_toFinset,clusterContactCount]
+  simp only [finiteClosedContact,fullLargeContact,hconn,hFx,hFy,hc,Nat.zero_le,and_true]
+
+theorem finiteClosedContact_nonneg (G : BondGraph V E) (Q : Finset E) (x y : V) (ω : Finset E) :
+    0 ≤ finiteClosedContact G Q x y ω := by
+  unfold finiteClosedContact
+  split_ifs <;> positivity
+
+theorem finiteClosedContact_le_size (G : BondGraph V E) (hloc : LocallyFinite G)
+    (d : ℕ) (hd : ∀ z, degree G z ≤ d) (Q η : Finset E) (ω : Set E)
+    (hη : (η:Set E) ⊆ ω) (x y : V) (hfin : (cluster G ω x).Finite) :
+    finiteClosedContact G Q x y η ≤ (d:ℝ)*(cluster G ω x).ncard := by
+  have hdQ (z : V) : (Q.filter fun e => z ∈ G e).card ≤ d :=
+    (incident_filter_card_le_degree G hloc Q z).trans (hd z)
+  have hs : (finiteRootComponent G η x).card ≤ (cluster G ω x).ncard := by
+    rw [←Set.ncard_coe_finset,coe_finiteRootComponent]
+    exact Set.ncard_le_ncard (fun z hz => conn_mono G hη hz) hfin
+  unfold finiteClosedContact
+  split_ifs
+  · positivity
+  · exact_mod_cast (contactBonds_card_le G Q d hdQ _ _).trans
+      (Nat.mul_le_mul_left d ((min_le_left _ _).trans hs))
+
+ 
+
+theorem lower_cluster_restrict_eq (G : BondGraph V E) (hloc : LocallyFinite G)
+    (ω η : Set E) (hη : η ⊆ ω) (x : V) (hx : (cluster G ω x).Finite)
+    (Q : Finset E) (hQ : incidentSupport G hloc hx.toFinset ⊆ Q) :
+    cluster G (η ∩ (Q:Set E)) x = cluster G η x := by
+  have hfin : (cluster G η x).Finite := hx.subset (fun z hz => conn_mono G hη hz)
+  have hs : hfin.toFinset ⊆ hx.toFinset := by
+    intro z hz
+    exact hx.mem_toFinset.mpr (conn_mono G hη (hfin.mem_toFinset.mp hz))
+  have hQ' := (incidentSupport_mono G hloc hs).trans hQ
+  have hh := (cluster_restrict_incident_eq_iff G hloc hfin.toFinset Q hQ' η x).mpr
+    (Set.Finite.coe_toFinset hfin).symm
+  simpa only [Set.Finite.coe_toFinset] using hh
+
+end UniformContactApproximation
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section ContactRegularity
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem fullLargeContact_zero_le_size (G : BondGraph V E) (hloc : LocallyFinite G)
+    (d : ℕ) (hd : ∀ z, degree G z ≤ d) (ω : Set E) (x y : V)
+    (hx : (cluster G ω x).Finite) (hy : (cluster G ω y).Finite) :
+    fullLargeContact G x y 0 ω ≤ (d:ℝ)*(cluster G ω x).ncard := by
+  let Q := incidentSupport G hloc hx.toFinset ∪ incidentSupport G hloc hy.toFinset
+  rw [←finiteClosedContact_eq_full G hloc ω x y hx hy Q Finset.subset_union_left Finset.subset_union_right]
+  exact finiteClosedContact_le_size G hloc d hd Q (domainBits Q ω) ω
+    (by rw [coe_domainBits]; exact Set.inter_subset_left) x y hx
+
+theorem integrable_fullContact_subcritical [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (x y : V) :
+    Integrable (fullLargeContact G x y 0) (law p) := by
+  obtain ⟨d,_,hd⟩ := exists_degree_bound G hqt
+  apply ((susceptibility_eq_integral G hloc hqt p hp x).1.const_mul (d:ℝ)).mono'
+    (measurable_fullLargeContact G hloc x y 0).aestronglyMeasurable
+  filter_upwards [subcritical_cluster_finite_ae G p hp x,subcritical_cluster_finite_ae G p hp y] with ω hx hy
+  rw [Real.norm_of_nonneg (fullLargeContact_nonneg G x y 0 ω)]
+  exact fullLargeContact_zero_le_size G hloc d hd ω x y hx hy
+
+theorem fullContact_label_integral [Countable E] [Countable V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (p : unitInterval) (x y : V) :
+    (∫ labels, fullLargeContact G x y 0 (openEdges labels p) ∂labelLaw) =
+      ∫ ω, fullLargeContact G x y 0 ω ∂law p := by
+  rw [←map_openEdges p]
+  exact (integral_map (measurable_openEdges p).aemeasurable
+    (measurable_fullLargeContact G hloc x y 0).aestronglyMeasurable).symm
+
+theorem fullContact_label_integrable [Countable E] [Countable V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (p : unitInterval) (x y : V) :
+    Integrable (fun labels => fullLargeContact G x y 0 (openEdges labels p)) labelLaw ↔
+      Integrable (fullLargeContact G x y 0) (law p) := by
+  rw [←map_openEdges p]
+  exact (integrable_map_measure (measurable_fullLargeContact G hloc x y 0).aestronglyMeasurable
+    (measurable_openEdges p).aemeasurable).symm
+
+theorem fullContact_eq_of_clusters (G : BondGraph V E) (ω η : Set E) (x y : V)
+    (hx : cluster G ω x = cluster G η x) (hy : cluster G ω y = cluster G η y) :
+    fullLargeContact G x y 0 ω = fullLargeContact G x y 0 η := by
+  have hc : Conn G ω x y ↔ Conn G η x y := by
+    change y ∈ cluster G ω x ↔ y ∈ cluster G η x
+    rw [hx]
+  simp only [fullLargeContact,clusterContactCount,hx,hy,hc]
+
+ 
+
+theorem continuousAt_fullContact_expectation [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (x y : V) :
+    ContinuousAt (fun q => ∫ ω, fullLargeContact G x y 0 ω ∂law q) p := by
+  obtain ⟨r,hpr,hr⟩ := exists_between hp
+  obtain ⟨d,_,hd⟩ := exists_degree_bound G hqt
+  let F (q : unitInterval) (labels : E → unitInterval) := fullLargeContact G x y 0 (openEdges labels q)
+  let B (labels : E → unitInterval) : ℝ := (d:ℝ)*(cluster G (openEdges labels r) x).ncard
+  have hm (q : unitInterval) : Measurable (F q) :=
+    (measurable_fullLargeContact G hloc x y 0).comp (measurable_openEdges q)
+  have hc : ContinuousAt (fun q => ∫ labels, F q labels ∂labelLaw) p := by
+    apply continuousAt_of_dominated (bound := B)
+      (Eventually.of_forall (fun q => (hm q).aestronglyMeasurable))
+    · filter_upwards [eventually_lt_nhds hpr] with q hqr
+      filter_upwards [label_cluster_finite_ae_subcritical G r hr x,label_cluster_finite_ae_subcritical G r hr y] with labels hx hy
+      have hsub (z : V) : cluster G (openEdges labels q) z ⊆ cluster G (openEdges labels r) z :=
+        fun w hw => conn_mono G (openEdges_mono labels hqr.le) hw
+      rw [Real.norm_of_nonneg (fullLargeContact_nonneg G x y 0 _)]
+      calc
+        _ ≤ (d:ℝ)*(cluster G (openEdges labels q) x).ncard :=
+          fullLargeContact_zero_le_size G hloc d hd _ x y (hx.subset (hsub x)) (hy.subset (hsub y))
+        _ ≤ B labels := mul_le_mul_of_nonneg_left (by exact_mod_cast Set.ncard_le_ncard (hsub x) hx) (Nat.cast_nonneg _)
+    · exact (susceptibility_eq_label_integral G hloc hqt r hr x).1.const_mul (d:ℝ)
+    · filter_upwards [ae_labels_ne (E := E) p,label_cluster_finite_ae_subcritical G p hp x,
+        label_cluster_finite_ae_subcritical G p hp y] with labels hneq hx hy
+      apply Filter.EventuallyEq.continuousAt
+      filter_upwards [eventually_cluster_eq_at_finite G hloc labels p x hneq hx,
+        eventually_cluster_eq_at_finite G hloc labels p y hneq hy] with q hqx hqy
+      exact fullContact_eq_of_clusters G _ _ x y hqx hqy
+  simpa only [F,fullContact_label_integral G hloc] using hc
+
+end ContactRegularity
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section UniformContactError
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+def ClusterSupportIn (G : BondGraph V E) (Q : Finset E) (ω : Set E) (x : V) : Prop :=
+  ∀ a, Conn G ω x a → ∀ e, a ∈ G e → e ∈ Q
+
+theorem measurableSet_clusterSupportIn [Countable V] [Countable E]
+    (G : BondGraph V E) (Q : Finset E) (x : V) :
+    MeasurableSet {ω : Set E | ClusterSupportIn G Q ω x} := by
+  simp only [ClusterSupportIn,Set.ofPred_forall]
+  apply MeasurableSet.iInter
+  intro a
+  have he : {ω : Set E | Conn G ω x a → ∀ e, a ∈ G e → e ∈ Q} =
+      {ω | ¬Conn G ω x a} ∪ {ω | ∀ e, a ∈ G e → e ∈ Q} := by ext ω; simp only [Set.mem_union,Set.mem_ofPred_eq]; tauto
+  rw [he]
+  apply (measurableSet_conn G x a).compl.union
+  by_cases h : ∀ e, a ∈ G e → e ∈ Q
+  · convert (MeasurableSet.univ : MeasurableSet (Set.univ : Set (Set E))) using 1
+    ext ω
+    simp only [Set.mem_ofPred_eq,Set.mem_univ,iff_true]
+    exact h
+  · convert (MeasurableSet.empty : MeasurableSet (∅ : Set (Set E))) using 1
+    ext ω
+    simp only [Set.mem_ofPred_eq,Set.mem_empty_iff_false,iff_false]
+    exact h
+
+theorem clusterSupportIn_of_incident (G : BondGraph V E) (hloc : LocallyFinite G)
+    (Q : Finset E) (ω : Set E) (x : V) (hx : (cluster G ω x).Finite)
+    (hQ : incidentSupport G hloc hx.toFinset ⊆ Q) : ClusterSupportIn G Q ω x := by
+  intro a ha e he
+  exact hQ ((mem_incidentSupport G hloc hx.toFinset e).mpr ⟨a,hx.mem_toFinset.mpr ha,he⟩)
+
+theorem finiteClosedContact_eq_full_of_support (G : BondGraph V E) (hloc : LocallyFinite G)
+    (Q : Finset E) (ω η : Set E) (hη : η ⊆ ω) (x y : V)
+    (hx : (cluster G ω x).Finite) (hy : (cluster G ω y).Finite)
+    (hQx : ClusterSupportIn G Q ω x) (hQy : ClusterSupportIn G Q ω y) :
+    finiteClosedContact G Q x y (domainBits Q η) = fullLargeContact G x y 0 η := by
+  have hsub (z : V) : cluster G η z ⊆ cluster G ω z := fun a ha => conn_mono G hη ha
+  have hfx := hx.subset (hsub x)
+  have hfy := hy.subset (hsub y)
+  apply finiteClosedContact_eq_full G hloc η x y hfx hfy
+  · intro e he
+    obtain ⟨a,ha,he⟩ := (mem_incidentSupport G hloc hfx.toFinset e).mp he
+    exact hQx a (hsub x (hfx.mem_toFinset.mp ha)) e he
+  · intro e he
+    obtain ⟨a,ha,he⟩ := (mem_incidentSupport G hloc hfy.toFinset e).mp he
+    exact hQy a (hsub y (hfy.mem_toFinset.mp ha)) e he
+
+noncomputable def contactApproxError (G : BondGraph V E) (Q : Finset E) (d : ℕ)
+    (x y : V) (ω : Set E) : ℝ :=
+  if ClusterSupportIn G Q ω x ∧ ClusterSupportIn G Q ω y then 0
+  else (d:ℝ)*(cluster G ω x).ncard
+
+theorem measurable_contactApproxError [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (Q : Finset E) (d : ℕ) (x y : V) :
+    Measurable (contactApproxError G Q d x y) := by
+  exact measurable_const.ite ((measurableSet_clusterSupportIn G Q x).inter
+    (measurableSet_clusterSupportIn G Q y))
+    (measurable_const.mul ((measurable_of_countable (fun n : ℕ => (n:ℝ))).comp (measurable_cluster_ncard G hloc x)))
+
+theorem contactApproxError_nonneg (G : BondGraph V E) (Q : Finset E) (d : ℕ) (x y : V) (ω : Set E) :
+    0 ≤ contactApproxError G Q d x y ω := by unfold contactApproxError; split_ifs <;> positivity
+
+theorem contactApproxError_le (G : BondGraph V E) (Q : Finset E) (d : ℕ) (x y : V) (ω : Set E) :
+    contactApproxError G Q d x y ω ≤ (d:ℝ)*(cluster G ω x).ncard := by
+  unfold contactApproxError; split_ifs <;> first | positivity | exact le_rfl
+
+theorem contact_uniform_error_bound (G : BondGraph V E) (hloc : LocallyFinite G)
+    (d : ℕ) (hd : ∀ z, degree G z ≤ d) (Q : Finset E) (ω η : Set E)
+    (hη : η ⊆ ω) (x y : V) (hx : (cluster G ω x).Finite) (hy : (cluster G ω y).Finite) :
+    ‖finiteClosedContact G Q x y (domainBits Q η) - fullLargeContact G x y 0 η‖ ≤
+      contactApproxError G Q d x y ω := by
+  unfold contactApproxError
+  split_ifs with h
+  · rw [finiteClosedContact_eq_full_of_support G hloc Q ω η hη x y hx hy h.1 h.2,sub_self,norm_zero]
+  · have hsub (z : V) : cluster G η z ⊆ cluster G ω z := fun a ha => conn_mono G hη ha
+    have hf := finiteClosedContact_le_size G hloc d hd Q (domainBits Q η) ω
+      (by rw [coe_domainBits]; exact Set.inter_subset_left.trans hη) x y hx
+    have hg : fullLargeContact G x y 0 η ≤ (d:ℝ)*(cluster G ω x).ncard := by
+      exact (fullLargeContact_zero_le_size G hloc d hd η x y (hx.subset (hsub x)) (hy.subset (hsub y))).trans
+        (mul_le_mul_of_nonneg_left (by exact_mod_cast Set.ncard_le_ncard (hsub x) hx) (Nat.cast_nonneg _))
+    rw [Real.norm_eq_abs,abs_sub_le_iff]
+    have hfn := finiteClosedContact_nonneg G Q x y (domainBits Q η)
+    have hgn := fullLargeContact_nonneg G x y 0 η
+    constructor <;> linarith
+
+end UniformContactError
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section UniformContactExpectation
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem integrable_contactApproxError [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (Q : Finset E) (d : ℕ) (x y : V) :
+    Integrable (contactApproxError G Q d x y) (law p) := by
+  apply ((susceptibility_eq_integral G hloc hqt p hp x).1.const_mul (d:ℝ)).mono'
+    (measurable_contactApproxError G hloc Q d x y).aestronglyMeasurable
+  exact Eventually.of_forall (fun ω => by rw [Real.norm_of_nonneg (contactApproxError_nonneg G Q d x y ω)]; exact contactApproxError_le G Q d x y ω)
+
+theorem contactApproxError_integral_tendsto_zero [Countable V] [Encodable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (d : ℕ) (x y : V) :
+    Tendsto (fun n => ∫ ω, contactApproxError G (vertexExhaustion n) d x y ω ∂law p) atTop (nhds 0) := by
+  have ht : ∀ᵐ ω ∂law p, Tendsto (fun n => contactApproxError G (vertexExhaustion n) d x y ω) atTop (nhds 0) := by
+    filter_upwards [subcritical_cluster_finite_ae G p hp x,subcritical_cluster_finite_ae G p hp y] with ω hx hy
+    obtain ⟨nx,hnx⟩ := subset_vertexExhaustion (incidentSupport G hloc hx.toFinset)
+    obtain ⟨ny,hny⟩ := subset_vertexExhaustion (incidentSupport G hloc hy.toFinset)
+    apply tendsto_const_nhds.congr'
+    filter_upwards [eventually_ge_atTop (max nx ny)] with n hn
+    exact (ite_eq_left ⟨clusterSupportIn_of_incident G hloc _ ω x hx
+      (hnx.trans (vertexExhaustion_mono ((le_max_left nx ny).trans hn))),
+      clusterSupportIn_of_incident G hloc _ ω y hy
+      (hny.trans (vertexExhaustion_mono ((le_max_right nx ny).trans hn)))⟩).symm
+  have hh := tendsto_integral_of_dominated_convergence (fun ω : Set E => (d:ℝ)*(cluster G ω x).ncard)
+    (fun n => (measurable_contactApproxError G hloc (vertexExhaustion n) d x y).aestronglyMeasurable)
+    ((susceptibility_eq_integral G hloc hqt p hp x).1.const_mul (d:ℝ))
+    (fun n => Eventually.of_forall (fun ω => by rw [Real.norm_of_nonneg (contactApproxError_nonneg G _ d x y ω)]; exact contactApproxError_le G _ d x y ω)) ht
+  simpa only [integral_zero] using hh
+
+theorem finiteClosedContact_label_integral [Countable E] (G : BondGraph V E)
+    (Q : Finset E) (p : unitInterval) (x y : V) :
+    (∫ labels, finiteClosedContact G Q x y (domainBits Q (openEdges labels p)) ∂labelLaw) =
+      bernoulliPolynomial Q (finiteClosedContact G Q x y) p := by
+  rw [←integral_domainBits Q p (finiteClosedContact G Q x y),←map_openEdges p]
+  exact (integral_map (measurable_openEdges p).aemeasurable
+    (measurable_finite_local Q (fun ω => finiteClosedContact G Q x y (domainBits Q ω)) (fun ω => by rw [domainBits_local])).aestronglyMeasurable).symm
+
+theorem contactApproxError_label_integral [Countable V] [Countable E] (G : BondGraph V E)
+    (hloc : LocallyFinite G) (Q : Finset E) (p : unitInterval) (d : ℕ) (x y : V) :
+    (∫ labels, contactApproxError G Q d x y (openEdges labels p) ∂labelLaw) =
+      ∫ ω, contactApproxError G Q d x y ω ∂law p := by
+  rw [←map_openEdges p]
+  exact (integral_map (measurable_openEdges p).aemeasurable
+    (measurable_contactApproxError G hloc Q d x y).aestronglyMeasurable).symm
+
+theorem contact_expectation_uniform_error [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (d : ℕ) (hd : ∀ z, degree G z ≤ d) (Q : Finset E) (b q : unitInterval)
+    (hb : b < pc G) (hq : q ≤ b) (x y : V) :
+    ‖bernoulliPolynomial Q (finiteClosedContact G Q x y) q -
+      ∫ ω, fullLargeContact G x y 0 ω ∂law q‖ ≤
+      ∫ ω, contactApproxError G Q d x y ω ∂law b := by
+  have hf : Integrable (fun labels => finiteClosedContact G Q x y (domainBits Q (openEdges labels q))) labelLaw := by
+    have hi := integrable_finite_local Q q (fun ω => finiteClosedContact G Q x y (domainBits Q ω))
+      (fun ω => by rw [domainBits_local])
+    rw [←map_openEdges q] at hi
+    exact hi.comp_measurable (measurable_openEdges q)
+  have hg := (fullContact_label_integrable G hloc q x y).mpr
+    (integrable_fullContact_subcritical G hloc hqt q (hq.trans_lt hb) x y)
+  have hi : Integrable (fun labels => contactApproxError G Q d x y (openEdges labels b)) labelLaw := by
+    have hh := integrable_contactApproxError G hloc hqt b hb Q d x y
+    rw [←map_openEdges b] at hh
+    exact hh.comp_measurable (measurable_openEdges b)
+  rw [←finiteClosedContact_label_integral G Q q x y,←fullContact_label_integral G hloc q x y,
+    ←integral_sub hf hg,←contactApproxError_label_integral G hloc Q b d x y]
+  apply (norm_integral_le_integral_norm _).trans
+  apply integral_mono_ae (hf.sub hg).norm hi
+  filter_upwards [label_cluster_finite_ae_subcritical G b hb x,label_cluster_finite_ae_subcritical G b hb y] with labels hx hy
+  exact contact_uniform_error_bound G hloc d hd Q _ _ (openEdges_mono labels hq) x y hx hy
+
+end UniformContactExpectation
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section PointwiseRussoLimit
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def realContactExpectation (G : BondGraph V E) (x y : V) (t : ℝ) : ℝ :=
+  ∫ ω, fullLargeContact G x y 0 ω ∂law (Set.projIcc 0 1 zero_le_one t)
+
+@[simp] theorem realContactExpectation_coe (G : BondGraph V E) (x y : V) (p : unitInterval) :
+    realContactExpectation G x y (p:ℝ) = ∫ ω, fullLargeContact G x y 0 ω ∂law p := by
+  simp only [realContactExpectation,Set.projIcc_val]
+
+noncomputable def realTwoPoint (G : BondGraph V E) (x y : V) (t : ℝ) : ℝ :=
+  twoPoint G (Set.projIcc 0 1 zero_le_one t) x y
+
+@[simp] theorem realTwoPoint_coe (G : BondGraph V E) (x y : V) (p : unitInterval) :
+    realTwoPoint G x y (p:ℝ) = twoPoint G p x y := by
+  simp only [realTwoPoint,Set.projIcc_val]
+
+theorem finite_connection_deriv_eq (G : BondGraph V E) (Q : Finset E)
+    (x y : V) (t : ℝ) (ht : t ≠ 1) :
+    deriv (finiteEventPolynomial Q {ω | Conn G ω x y}) t =
+      bernoulliPolynomial Q (finiteClosedContact G Q x y) t / (1-t) := by
+  apply (eq_div_iff (sub_ne_zero.mpr (Ne.symm ht))).mpr
+  rw [mul_comm]
+  exact finite_connection_russo_contacts G Q x y t
+
+theorem finite_connection_deriv_tendstoUniformlyOn [Countable V] [Encodable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (b : unitInterval) (hb : b < pc G) (x y : V) :
+    TendstoUniformlyOn (fun n => deriv (finiteEventPolynomial (vertexExhaustion n) {ω | Conn G ω x y}))
+      (fun t => realContactExpectation G x y t / (1-t)) atTop (Set.Icc 0 (b:ℝ)) := by
+  obtain ⟨d,_,hd⟩ := exists_degree_bound G hqt
+  have hb1 : (b:ℝ) < 1 := lt_of_lt_of_le hb (pc G).2.2
+  have hden : 0 < 1-(b:ℝ) := sub_pos.mpr hb1
+  have ht := (contactApproxError_integral_tendsto_zero G hloc hqt b hb d x y).div_const (1-(b:ℝ))
+  rw [zero_div] at ht
+  rw [Metric.tendstoUniformlyOn_iff]
+  intro ε hε
+  filter_upwards [ht.eventually (eventually_lt_nhds hε)] with n hn t ht
+  let q : unitInterval := ⟨t,ht.1,ht.2.trans b.2.2⟩
+  have hqb : q ≤ b := ht.2
+  have hqt1 : t < 1 := ht.2.trans_lt hb1
+  have he := contact_expectation_uniform_error G hloc hqt d hd (vertexExhaustion n) b q hb hqb x y
+  have hn0 : 0 ≤ ∫ ω, contactApproxError G (vertexExhaustion n) d x y ω ∂law b :=
+    integral_nonneg (contactApproxError_nonneg G _ d x y)
+  rw [dist_comm,Real.dist_eq,finite_connection_deriv_eq G _ x y t (ne_of_lt hqt1),←sub_div,abs_div,
+    abs_of_pos (sub_pos.mpr hqt1)]
+  change |bernoulliPolynomial _ _ (q:ℝ) - realContactExpectation G x y (q:ℝ)| / (1-t) < ε
+  rw [realContactExpectation_coe]
+  apply lt_of_le_of_lt _ hn
+  exact div_le_div₀ hn0 (by simpa only [Real.norm_eq_abs] using he) hden (sub_le_sub_left ht.2 1)
+
+theorem finite_connection_polynomial_tendsto_twoPoint [Encodable E]
+    (G : BondGraph V E) (p : unitInterval) (x y : V) :
+    Tendsto (fun n => finiteEventPolynomial (vertexExhaustion n) {ω | Conn G ω x y} p)
+      atTop (nhds (twoPoint G p x y)) := by
+  have ht := finite_connection_polynomial_tendsto G Set.univ (vertexExhaustion (V := E))
+    vertexExhaustion_mono (fun _ => Set.subset_univ _) (fun e _ => by
+      obtain ⟨n,hn⟩ := subset_vertexExhaustion ({e}:Finset E)
+      exact ⟨n,hn (by simp)⟩) p x y
+  simpa only [Set.inter_univ,finiteEventPolynomial,Set.mem_ofPred_eq,twoPoint,Measure.real] using ht
+
+ 
+
+theorem hasDerivAt_realTwoPoint_subcritical [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp : p < pc G) (x y : V) :
+    HasDerivAt (realTwoPoint G x y)
+      ((∫ ω, fullLargeContact G x y 0 ω ∂law p)/(1-(p:ℝ))) (p:ℝ) := by
+  let : Encodable E := Encodable.ofCountable E
+  obtain ⟨b,hpb,hb⟩ := exists_between hp
+  have hu := (finite_connection_deriv_tendstoUniformlyOn G hloc hqt b hb x y).mono
+    (show Set.Ioo 0 (b:ℝ) ⊆ Set.Icc 0 (b:ℝ) from Set.Ioo_subset_Icc_self)
+  have hh := hasDerivAt_of_tendstoUniformlyOn (f := fun n => finiteEventPolynomial (vertexExhaustion n) {ω | Conn G ω x y})
+    (g := realTwoPoint G x y) (g' := fun t => realContactExpectation G x y t/(1-t)) isOpen_Ioo hu
+    (Eventually.of_forall (fun n t _ => by
+      change HasDerivAt (bernoulliPolynomial (vertexExhaustion n) (fun (ω : Finset E) => if Conn G (ω:Set E) x y then (1:ℝ) else 0)) _ t
+      exact (hasDerivAt_bernoulliPolynomial (vertexExhaustion n)
+        (fun (ω : Finset E) => if Conn G (ω:Set E) x y then (1:ℝ) else 0) t).differentiableAt.hasDerivAt))
+    (fun t ht => by
+      let q : unitInterval := ⟨t,ht.1.le,ht.2.le.trans b.2.2⟩
+      simpa only [←realTwoPoint_coe G x y q] using finite_connection_polynomial_tendsto_twoPoint G q x y)
+    (show (p:ℝ) ∈ Set.Ioo 0 (b:ℝ) from ⟨hp0,hpb⟩)
+  simpa only [realContactExpectation_coe] using hh
+
+end PointwiseRussoLimit
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+open scoped ENNReal
+section RussoDiagram
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def russoDiagram (G : BondGraph V E) (p : unitInterval) (x y : V) : ℝ≥0∞ :=
+  ∑' a, ∑' b, indexedEdgeKernel G a b *
+    (law p {ω | Conn G ω x a} * law p {ω | Conn G ω y b})
+
+theorem conn_probability_symm (G : BondGraph V E) (p : unitInterval) (x y : V) :
+    law p {ω | Conn G ω x y} = law p {ω | Conn G ω y x} := by
+  congr 1
+  ext ω
+  exact ⟨conn_symm G ω,conn_symm G ω⟩
+
+theorem conn_probability_mono [Countable V] [Countable E] (G : BondGraph V E)
+    (p q : unitInterval) (hpq : p ≤ q) (x y : V) :
+    law p {ω | Conn G ω x y} ≤ law q {ω | Conn G ω x y} := by
+  rw [←ENNReal.ofReal_toReal (measure_ne_top (law p) _),←ENNReal.ofReal_toReal (measure_ne_top (law q) _)]
+  exact ENNReal.ofReal_le_ofReal (twoPoint_mono G x y hpq)
+
+theorem fullContact_lintegral_eq_kernel [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (p : unitInterval) (x y : V)
+    (hx : ∀ᵐ ω ∂law p, (cluster G ω x).Finite) :
+    (∫⁻ ω, ENNReal.ofReal (fullLargeContact G x y 0 ω) ∂law p) =
+      ∫⁻ ω, if ¬Conn G ω x y then clusterPairKernel G ω (indexedEdgeKernel G) x y else 0 ∂law p := by
+  apply lintegral_congr_ae
+  filter_upwards [hx] with ω hω
+  by_cases hxy : Conn G ω x y
+  · simp only [fullLargeContact,hxy,not_true_eq_false,false_and,ite_false,ENNReal.ofReal_zero]
+  · rw [ite_eq_left hxy,clusterPairKernel_edge G hloc ω x y hxy hω]
+    simp only [fullLargeContact,hxy,Nat.zero_le,not_false_eq_true,and_self,ite_true,ENNReal.ofReal_natCast]
+
+theorem fullContact_expectation_le_diagram [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (x y : V) :
+    ENNReal.ofReal (∫ ω, fullLargeContact G x y 0 ω ∂law p) ≤ russoDiagram G p x y := by
+  rw [ofReal_integral_eq_lintegral_ofReal (integrable_fullContact_subcritical G hloc hqt p hp x y)
+    (Eventually.of_forall (fullLargeContact_nonneg G x y 0))]
+  rw [fullContact_lintegral_eq_kernel G hloc p x y (subcritical_cluster_finite_ae G p hp x)]
+  exact clusterPairKernel_distinct_bound G p (indexedEdgeKernel G) x y
+
+theorem russoDiagram_mono [Countable V] [Countable E] (G : BondGraph V E)
+    (p q : unitInterval) (hpq : p ≤ q) (x y : V) : russoDiagram G p x y ≤ russoDiagram G q x y := by
+  exact ENNReal.tsum_le_tsum (fun a => ENNReal.tsum_le_tsum (fun b =>
+    mul_le_mul_right (mul_le_mul' (conn_probability_mono G p q hpq x a)
+      (conn_probability_mono G p q hpq y b)) _))
+
+theorem russoDiagram_symm (G : BondGraph V E) (p : unitInterval) (x y : V) :
+    russoDiagram G p x y = russoDiagram G p y x := by
+  unfold russoDiagram
+  rw [ENNReal.tsum_comm]
+  apply tsum_congr; intro b
+  apply tsum_congr; intro a
+  rw [indexedEdgeKernel_symm G a b,mul_comm (law p {ω | Conn G ω x a})]
+
+theorem russoDiagram_row_bound (G : BondGraph V E) (hloc : LocallyFinite G)
+    (d : ℕ) (hd : ∀ z, degree G z ≤ d) (p : unitInterval) (C : ℝ≥0∞)
+    (hC : ∀ x, ∑' y, law p {ω | Conn G ω x y} ≤ C) (x : V) :
+    (∑' y, russoDiagram G p x y) ≤ (d:ℝ≥0∞)*C^2 := by
+  unfold russoDiagram
+  rw [ENNReal.tsum_comm]
+  calc
+    _ = ∑' a, ∑' b, indexedEdgeKernel G a b * law p {ω | Conn G ω x a} *
+        ∑' y, law p {ω | Conn G ω b y} := by
+      apply tsum_congr; intro a
+      rw [ENNReal.tsum_comm]
+      apply tsum_congr; intro b
+      simp_rw [←mul_assoc,conn_probability_symm G p _ b]
+      rw [ENNReal.tsum_mul_left]
+    _ ≤ ∑' a, ∑' b, indexedEdgeKernel G a b * law p {ω | Conn G ω x a} * C := by
+      exact ENNReal.tsum_le_tsum (fun a => ENNReal.tsum_le_tsum (fun b => mul_le_mul_right (hC b) _))
+    _ = ∑' a, (degree G a:ℝ≥0∞) * law p {ω | Conn G ω x a} * C := by
+      apply tsum_congr; intro a
+      rw [ENNReal.tsum_mul_right,ENNReal.tsum_mul_right,indexedEdgeKernel_row G hloc a]
+    _ ≤ ∑' a, (d:ℝ≥0∞) * law p {ω | Conn G ω x a} * C := by
+      exact ENNReal.tsum_le_tsum (fun a => mul_le_mul_left (mul_le_mul_left (show (degree G a:ℝ≥0∞) ≤ (d:ℝ≥0∞) by exact_mod_cast hd a) _) _)
+    _ = (d:ℝ≥0∞)*(∑' a, law p {ω | Conn G ω x a})*C := by
+      rw [ENNReal.tsum_mul_right,ENNReal.tsum_mul_left]
+    _ ≤ (d:ℝ≥0∞)*C^2 := by
+      rw [pow_two,←mul_assoc]
+      exact mul_le_mul_left (mul_le_mul_right (hC x) _) _
+
+end RussoDiagram
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+open scoped ENNReal
+section RussoMajorant
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def russoSlope (G : BondGraph V E) (x y : V) (t : ℝ) : ℝ :=
+  realContactExpectation G x y t / (1-t)
+
+noncomputable def russoMajorant (G : BondGraph V E) (b : unitInterval) (x y : V) : ℝ :=
+  (russoDiagram G b x y).toReal / (1-(b:ℝ))
+
+theorem russoDiagram_row_ne_top [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (b : unitInterval) (hb : b < pc G) (x : V) : (∑' y, russoDiagram G b x y) ≠ ⊤ := by
+  obtain ⟨d,_,hd⟩ := exists_degree_bound G hqt
+  obtain ⟨C,_,hC⟩ := subcritical_ennreal_row_bound G hloc hqt b hb
+  exact ne_top_of_le_ne_top (by finiteness) (russoDiagram_row_bound G hloc d hd b _ hC x)
+
+theorem russoDiagram_ne_top [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (b : unitInterval) (hb : b < pc G) (x y : V) : russoDiagram G b x y ≠ ⊤ :=
+  ne_top_of_le_ne_top (russoDiagram_row_ne_top G hloc hqt b hb x) (ENNReal.le_tsum y)
+
+theorem summable_russoMajorant [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (b : unitInterval) (hb : b < pc G) (x : V) : Summable (russoMajorant G b x) :=
+  (ENNReal.summable_toReal (russoDiagram_row_ne_top G hloc hqt b hb x)).div_const _
+
+theorem russoSlope_nonneg (G : BondGraph V E) (x y : V) (t : ℝ) (ht : t ≤ 1) :
+    0 ≤ russoSlope G x y t := by
+  exact div_nonneg (integral_nonneg (fullLargeContact_nonneg G x y 0)) (sub_nonneg.mpr ht)
+
+theorem russoMajorant_nonneg (G : BondGraph V E) (b : unitInterval) (x y : V) :
+    0 ≤ russoMajorant G b x y := div_nonneg ENNReal.toReal_nonneg (sub_nonneg.mpr b.2.2)
+
+theorem russoSlope_le_majorant [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (b p : unitInterval) (hb : b < pc G) (hp : p ≤ b) (x y : V) :
+    russoSlope G x y (p:ℝ) ≤ russoMajorant G b x y := by
+  have hp' := hp.trans_lt hb
+  have he := (fullContact_expectation_le_diagram G hloc hqt p hp' x y).trans (russoDiagram_mono G p b hp x y)
+  have hn : 0 ≤ ∫ ω, fullLargeContact G x y 0 ω ∂law p := integral_nonneg (fullLargeContact_nonneg G x y 0)
+  have hc : (∫ ω, fullLargeContact G x y 0 ω ∂law p) ≤ (russoDiagram G b x y).toReal := by
+    simpa only [ENNReal.toReal_ofReal hn] using ENNReal.toReal_mono (russoDiagram_ne_top G hloc hqt b hb x y) he
+  simp only [russoSlope,realContactExpectation_coe,russoMajorant]
+  exact div_le_div₀ ENNReal.toReal_nonneg hc (sub_pos.mpr ((show (b:ℝ) < (pc G:ℝ) from hb).trans_le (pc G).2.2)) (sub_le_sub_left (show (p:ℝ) ≤ (b:ℝ) from hp) 1)
+
+theorem hasDerivAt_realTwoPoint_slope [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (t : ℝ) (ht : t ∈ Set.Ioo 0 (pc G:ℝ)) (x y : V) :
+    HasDerivAt (realTwoPoint G x y) (russoSlope G x y t) t := by
+  let p : unitInterval := ⟨t, ht.1.le,ht.2.le.trans (pc G).2.2⟩
+  change HasDerivAt (realTwoPoint G x y) (russoSlope G x y (p:ℝ)) (p:ℝ)
+  simpa only [russoSlope,realContactExpectation_coe] using hasDerivAt_realTwoPoint_subcritical G hloc hqt p ht.1 ht.2 x y
+
+end RussoMajorant
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+open scoped ENNReal
+section RussoRows
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem russoMajorant_symm (G : BondGraph V E) (b : unitInterval) (x y : V) :
+    russoMajorant G b x y = russoMajorant G b y x := by
+  rw [russoMajorant,russoDiagram_symm,russoMajorant]
+
+theorem russoMajorant_uniform_row_bound [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (b : unitInterval) (hb : b < pc G) :
+    ∃ R : ℝ, 0 ≤ R ∧ ∀ x, ∑' y, russoMajorant G b x y ≤ R := by
+  obtain ⟨d,_,hd⟩ := exists_degree_bound G hqt
+  obtain ⟨C,_,hC⟩ := subcritical_ennreal_row_bound G hloc hqt b hb
+  let R : ℝ≥0∞ := (d:ℝ≥0∞)*(ENNReal.ofReal C)^2
+  have hR : R ≠ ⊤ := by dsimp [R]; finiteness
+  have hden : 0 < 1-(b:ℝ) := sub_pos.mpr ((show (b:ℝ) < (pc G:ℝ) from hb).trans_le (pc G).2.2)
+  refine ⟨R.toReal/(1-(b:ℝ)),div_nonneg ENNReal.toReal_nonneg hden.le,fun x => ?_⟩
+  simp only [russoMajorant,tsum_div_const]
+  rw [←ENNReal.tsum_toReal_eq (russoDiagram_ne_top G hloc hqt b hb x)]
+  exact div_le_div_of_nonneg_right (ENNReal.toReal_mono hR (russoDiagram_row_bound G hloc d hd b _ hC x)) hden.le
+
+ 
+
+theorem summable_kernel_pair (A B : V → V → ℝ)
+    (hA : ∀ x y, 0 ≤ A x y) (hB : ∀ x y, 0 ≤ B x y)
+    (hsA : ∀ x, Summable (A x)) (hsB : ∀ x, Summable (B x))
+    (C : ℝ) (hC : ∀ x, ∑' y, B x y ≤ C) (x : V) :
+    Summable (fun ab : V×V => A x ab.1 * B ab.1 ab.2) := by
+  apply (summable_prod_of_nonneg (f := fun ab : V×V => A x ab.1 * B ab.1 ab.2)
+    (fun ab => mul_nonneg (hA x ab.1) (hB ab.1 ab.2))).mpr
+  refine ⟨fun a => (hsB a).mul_left (A x a),?_⟩
+  simp only [tsum_mul_left]
+  exact Summable.of_nonneg_of_le (fun a => mul_nonneg (hA x a) (tsum_nonneg (hB a)))
+    (fun a => mul_le_mul_of_nonneg_left (hC a) (hA x a)) ((hsA x).mul_right C)
+
+theorem summable_twoPoint_pair [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (x : V) :
+    Summable (fun ab : V×V => twoPoint G p x ab.1 * twoPoint G p ab.1 ab.2) := by
+  obtain ⟨C,_,hrow⟩ := subcritical_row_bound G hloc hqt p hp
+  have hs := summable_twoPoint_subcritical G hloc hqt p hp
+  exact summable_kernel_pair (twoPoint G p) (twoPoint G p) (twoPoint_nonneg G p)
+    (twoPoint_nonneg G p) hs hs C (fun x => (hs x).tsum_le_of_sum_le (hrow x)) x
+
+theorem summable_russoMajorant_pair [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (x : V) :
+    Summable (fun ab : V×V => russoMajorant G p x ab.1 * twoPoint G p ab.1 ab.2) ∧
+    Summable (fun ab : V×V => twoPoint G p x ab.1 * russoMajorant G p ab.1 ab.2) := by
+  obtain ⟨C,_,hrow⟩ := subcritical_row_bound G hloc hqt p hp
+  have hs := summable_twoPoint_subcritical G hloc hqt p hp
+  have hr := summable_russoMajorant G hloc hqt p hp
+  obtain ⟨R,_,hR⟩ := russoMajorant_uniform_row_bound G hloc hqt p hp
+  exact ⟨summable_kernel_pair _ _ (russoMajorant_nonneg G p) (twoPoint_nonneg G p) hr hs C
+    (fun x => (hs x).tsum_le_of_sum_le (hrow x)) x,
+    summable_kernel_pair _ _ (twoPoint_nonneg G p) (russoMajorant_nonneg G p) hs hr R hR x⟩
+
+end RussoRows
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section RealTriangle
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def realTriangleTerm (G : BondGraph V E) (x : V) (ab : V×V) (t : ℝ) : ℝ :=
+  realTwoPoint G x ab.1 t * realTwoPoint G ab.1 ab.2 t * realTwoPoint G ab.2 x t
+
+noncomputable def realTriangleSlope (G : BondGraph V E) (x : V) (ab : V×V) (t : ℝ) : ℝ :=
+  russoSlope G x ab.1 t * realTwoPoint G ab.1 ab.2 t * realTwoPoint G ab.2 x t +
+  realTwoPoint G x ab.1 t * russoSlope G ab.1 ab.2 t * realTwoPoint G ab.2 x t +
+  realTwoPoint G x ab.1 t * realTwoPoint G ab.1 ab.2 t * russoSlope G ab.2 x t
+
+noncomputable def triangleMajorant (G : BondGraph V E) (b : unitInterval) (x : V) (ab : V×V) : ℝ :=
+  russoMajorant G b x ab.1 * twoPoint G b ab.1 ab.2 +
+  twoPoint G b x ab.1 * russoMajorant G b ab.1 ab.2 +
+  russoMajorant G b x ab.2 * twoPoint G b ab.2 ab.1
+
+theorem summable_triangleMajorant [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (b : unitInterval) (hb : b < pc G) (x : V) : Summable (triangleMajorant G b x) := by
+  obtain ⟨h₁,h₂⟩ := summable_russoMajorant_pair G hloc hqt b hb x
+  have h₃ : Summable (fun ab : V×V => russoMajorant G b x ab.2 * twoPoint G b ab.2 ab.1) := by
+    exact h₁.comp_injective (i := fun ab : V×V => (ab.2,ab.1)) (Equiv.prodComm V V).injective
+  exact (h₁.add h₂).add h₃
+
+theorem hasDerivAt_realTriangleTerm [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (t : ℝ) (ht : t ∈ Set.Ioo 0 (pc G:ℝ)) (x : V) (ab : V×V) :
+    HasDerivAt (realTriangleTerm G x ab) (realTriangleSlope G x ab t) t := by
+  have hs := hasDerivAt_realTwoPoint_slope G hloc hqt t ht
+  convert! ((hs x ab.1).mul (hs ab.1 ab.2)).mul (hs ab.2 x) using 1
+  simp only [realTriangleSlope,Pi.mul_apply]
+  ring
+
+theorem realTriangleSlope_nonneg (G : BondGraph V E) (x : V) (ab : V×V) (t : ℝ) (ht : t ≤ 1) :
+    0 ≤ realTriangleSlope G x ab t := by
+  have hT (a b : V) : 0 ≤ realTwoPoint G a b t := twoPoint_nonneg G _ a b
+  dsimp [realTriangleSlope]
+  exact add_nonneg (add_nonneg (mul_nonneg (mul_nonneg (russoSlope_nonneg G _ _ t ht) (hT _ _)) (hT _ _))
+    (mul_nonneg (mul_nonneg (hT _ _) (russoSlope_nonneg G _ _ t ht)) (hT _ _)))
+    (mul_nonneg (mul_nonneg (hT _ _) (hT _ _)) (russoSlope_nonneg G _ _ t ht))
+
+theorem realTriangleSlope_le_majorant [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (b p : unitInterval) (hb : b < pc G) (hp : p ≤ b) (x : V) (ab : V×V) :
+    realTriangleSlope G x ab (p:ℝ) ≤ triangleMajorant G b x ab := by
+  have hs := russoSlope_le_majorant G hloc hqt b p hb hp
+  have hn := russoSlope_nonneg (G := G)
+  have hm (a z : V) : twoPoint G p a z ≤ twoPoint G b a z := twoPoint_mono G a z hp
+  simp only [realTriangleSlope,realTwoPoint_coe,triangleMajorant]
+  apply add_le_add
+  · apply add_le_add
+    · calc
+        _ ≤ russoMajorant G b x ab.1 * twoPoint G b ab.1 ab.2 * 1 :=
+          mul_le_mul (mul_le_mul (hs _ _) (hm _ _) (twoPoint_nonneg G p _ _) (russoMajorant_nonneg G b _ _))
+            (twoPoint_le_one G p _ _) (twoPoint_nonneg G p _ _) (mul_nonneg (russoMajorant_nonneg G b _ _) (twoPoint_nonneg G b _ _))
+        _ = _ := mul_one _
+    · calc
+        _ ≤ twoPoint G b x ab.1 * russoMajorant G b ab.1 ab.2 * 1 :=
+          mul_le_mul (mul_le_mul (hm _ _) (hs _ _) (hn _ _ _ p.2.2) (twoPoint_nonneg G b _ _))
+            (twoPoint_le_one G p _ _) (twoPoint_nonneg G p _ _) (mul_nonneg (twoPoint_nonneg G b _ _) (russoMajorant_nonneg G b _ _))
+        _ = _ := mul_one _
+  · calc
+      _ ≤ 1 * twoPoint G b ab.1 ab.2 * russoMajorant G b ab.2 x :=
+        mul_le_mul (mul_le_mul (twoPoint_le_one G p _ _) (hm _ _) (twoPoint_nonneg G p _ _) zero_le_one)
+          (hs _ _) (hn _ _ _ p.2.2) (mul_nonneg zero_le_one (twoPoint_nonneg G b _ _))
+      _ = _ := by rw [one_mul,twoPoint_symm G b,russoMajorant_symm G b]; ring
+
+end RealTriangle
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section TriangleDerivative
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def realTriangle (G : BondGraph V E) (x : V) (t : ℝ) : ℝ :=
+  ∑' ab, realTriangleTerm G x ab t
+
+noncomputable def rootTriangle (G : BondGraph V E) {ι : Type*} [Fintype ι]
+    (root : ι → V) (weight : ι → ℝ) (t : ℝ) : ℝ :=
+  ∑ i, weight i * realTriangle G (root i) t
+
+theorem summable_realTriangleTerm [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (x : V) :
+    Summable (fun ab => realTriangleTerm G x ab (p:ℝ)) := by
+  simp only [realTriangleTerm,realTwoPoint_coe]
+  apply (summable_twoPoint_pair G hloc hqt p hp x).of_nonneg_of_le
+  · intro ab
+    exact mul_nonneg (mul_nonneg (twoPoint_nonneg G p _ _) (twoPoint_nonneg G p _ _)) (twoPoint_nonneg G p _ _)
+  · intro ab
+    exact mul_le_of_le_one_right (mul_nonneg (twoPoint_nonneg G p _ _) (twoPoint_nonneg G p _ _)) (twoPoint_le_one G p _ _)
+
+theorem summable_realTriangleSlope [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (x : V) :
+    Summable (fun ab => realTriangleSlope G x ab (p:ℝ)) := by
+  exact (summable_triangleMajorant G hloc hqt p hp x).of_nonneg_of_le
+    (fun ab => realTriangleSlope_nonneg G x ab p p.2.2)
+    (fun ab => realTriangleSlope_le_majorant G hloc hqt p p hp le_rfl x ab)
+
+ 
+
+theorem hasDerivAt_realTriangle_subcritical [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp : p < pc G) (x : V) :
+    HasDerivAt (realTriangle G x) (∑' ab, realTriangleSlope G x ab (p:ℝ)) (p:ℝ) := by
+  obtain ⟨b,hpb,hb⟩ := exists_between hp
+  apply hasDerivAt_tsum_of_isPreconnected (summable_triangleMajorant G hloc hqt b hb x)
+    isOpen_Ioo isPreconnected_Ioo
+    (fun ab t ht => hasDerivAt_realTriangleTerm G hloc hqt t ⟨ht.1,lt_trans ht.2 hb⟩ x ab)
+    (fun ab t ht => ?_) (show (p:ℝ) ∈ Set.Ioo 0 (b:ℝ) from ⟨hp0,hpb⟩)
+    (summable_realTriangleTerm G hloc hqt p hp x) ⟨hp0,hpb⟩
+  let q : unitInterval := ⟨t,ht.1.le,ht.2.le.trans b.2.2⟩
+  have hq : q ≤ b := ht.2.le
+  rw [Real.norm_eq_abs,abs_of_nonneg (realTriangleSlope_nonneg G x ab t q.2.2)]
+  exact realTriangleSlope_le_majorant G hloc hqt b q hb hq x ab
+
+theorem hasDerivAt_rootTriangle_subcritical [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp : p < pc G) :
+    HasDerivAt (rootTriangle G root weight)
+      (∑ i, weight i * ∑' ab, realTriangleSlope G (root i) ab (p:ℝ)) (p:ℝ) := by
+  exact HasDerivAt.fun_sum (fun i _ => (hasDerivAt_realTriangle_subcritical G hloc hqt p hp0 hp (root i)).const_mul (weight i))
+
+end TriangleDerivative
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Kernel/ScaleBudget.lean b/lean/OAI/Probability/UnimodularPercolation/Kernel/ScaleBudget.lean
index 17d6d36e7320ddd77217a1717d8bd64264c311f8..f80d3839163bcb6ff3d4cef83402663b3750d724 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Kernel/ScaleBudget.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Kernel/ScaleBudget.lean
@@ -1,1214 +1,1214 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Kernel.Connections
-
-namespace OAI
-
-/-! Scale Budget for bond percolation. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section AveragedBinOrbitTransport
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem integral_tsum_of_finite_support {α Ω : Type*} [MeasurableSpace Ω]
-    (μ : Measure Ω) (S : Finset α) (f : Ω → α → ℝ)
-    (hi : ∀ x, Integrable (fun ω => f ω x) μ)
-    (hs : ∀ ω x, x ∉ S → f ω x = 0) :
-    (∫ ω, ∑' x, f ω x ∂μ) = ∑' x, ∫ ω, f ω x ∂μ := by
-  have he (ω) : (∑' x, f ω x) = ∑ x ∈ S, f ω x := tsum_eq_sum (hs ω)
-  rw [show (∫ ω, ∑' x, f ω x ∂μ) = ∫ ω, ∑ x ∈ S, f ω x ∂μ by simp only [he],
-    integral_finsetSum _ (fun x _ => hi x),tsum_eq_sum (s := S) (fun x hx => by simp only [hs _ x hx,integral_zero])]
-
-theorem binOrbitTransportMass_local (G : BondGraph V E) (hloc : LocallyFinite G)
-    (n : ℕ) (ω : Set E) (x y : V) :
-    let Qx := incidentSupport G hloc (finite_reachBall hloc x (2*n)).toFinset
-    let Qy := incidentSupport G hloc (finite_reachBall hloc y (2*n)).toFinset
-    binOrbitTransportMass G n ω x y = binOrbitTransportMass G n (ω ∩ ((Qx∪Qy : Finset E) : Set E)) x y := by
-  let Qx := incidentSupport G hloc (finite_reachBall hloc x (2*n)).toFinset
-  let Qy := incidentSupport G hloc (finite_reachBall hloc y (2*n)).toFinset
-  have hB := finite_local_of_subset Qx (Qx∪Qy) Finset.subset_union_left
-    (fun ω => binConnectionMass G n ω x y) (fun ω => binConnectionMass_local G hloc n ω x y) ω
-  have hq := finite_local_of_subset Qy (Qx∪Qy) Finset.subset_union_right
-    (fun ω => binOrbitRowMass G n ω y) (fun ω => binOrbitRowMass_local G hloc n ω y) ω
-  have hs := finite_local_of_subset Qx (Qx∪Qy) Finset.subset_union_left
-    (fun ω => binSizeMass G n ω x) (fun ω => binSizeMass_local G hloc n ω x) ω
-  exact congrArg₂ (· / ·) (congrArg₂ (· * ·) hB hq) hs
-
-theorem binOrbitTransportMass_integrable (G : BondGraph V E) (hloc : LocallyFinite G)
-    (p : unitInterval) (n : ℕ) (x y : V) : Integrable (fun ω => binOrbitTransportMass G n ω x y) (law p) :=
-  integrable_finite_local _ p _ (fun ω => binOrbitTransportMass_local G hloc n ω x y)
-
-theorem binOrbitTransportMass_outside (G : BondGraph V E) (n : ℕ) (ω : Set E) (x y : V)
-    (hy : y ∉ reachBall G x (2*n)) : binOrbitTransportMass G n ω x y = 0 := by
-  simp only [binOrbitTransportMass,binConnectionMass_outside G n ω x y hy,zero_mul,zero_div]
-
-noncomputable def binOrbitTransport (G : BondGraph V E) (p : unitInterval) (n : ℕ) (x y : V) : ℝ :=
-  ∫ ω, binOrbitTransportMass G n ω x y ∂law p
-
-theorem binOrbitTransport_covariant (G : BondGraph V E) (hloc : LocallyFinite G)
-    (p : unitInterval) {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (n : ℕ) (x y : V) :
-    binOrbitTransport G p n (a x) (a y) = binOrbitTransport G p n x y := by
-  have hm : Measurable (fun ω => binOrbitTransportMass G n ω x y) :=
-    measurable_finite_local _ _ (fun ω => binOrbitTransportMass_local G hloc n ω x y)
-  have h := integral_map (μ := law p) (measurable_set_preimage b).aemeasurable hm.aestronglyMeasurable
-  rw [law_map_preimage_equiv p b] at h
-  simpa only [binOrbitTransport,binOrbitTransportMass,binConnectionMass_covariant G hab,
-    binOrbitRowMass_covariant G n _ hab,binSizeMass_covariant G n _ hab] using h.symm
-
-theorem binOrbitTransport_outside (G : BondGraph V E) (p : unitInterval) (n : ℕ) (x y : V)
-    (hy : y ∉ reachBall G x (2*n)) : binOrbitTransport G p n x y = 0 := by
-  simp only [binOrbitTransport,binOrbitTransportMass_outside G n _ x y hy,integral_zero]
-
-theorem binOrbitTransportMass_outside_column (G : BondGraph V E) (n : ℕ) (ω : Set E) (x y : V)
-    (hx : x ∉ reachBall G y (2*n)) : binOrbitTransportMass G n ω x y = 0 := by
-  have hB : binConnectionMass G n ω x y = 0 := by
-    rw [binConnectionMass_symm]
-    exact binConnectionMass_outside G n ω y x hx
-  simp only [binOrbitTransportMass,hB,zero_mul,zero_div]
-
-theorem binOrbitTransport_row_integral (G : BondGraph V E) (hloc : LocallyFinite G)
-    (p : unitInterval) (n : ℕ) (x : V) :
-    (∑' y, binOrbitTransport G p n x y) = ∫ ω, ∑' y, binOrbitTransportMass G n ω x y ∂law p := by
-  symm
-  apply integral_tsum_of_finite_support (law p) (finite_reachBall hloc x (2*n)).toFinset _
-    (binOrbitTransportMass_integrable G hloc p n x)
-  intro ω y hy
-  exact binOrbitTransportMass_outside G n ω x y (by simpa only [Set.Finite.mem_toFinset] using hy)
-
-theorem integral_binOrbitRowMass (G : BondGraph V E) (hloc : LocallyFinite G)
-    (p : unitInterval) (n : ℕ) (x : V) :
-    (∫ ω, binOrbitRowMass G n ω x ∂law p) = binOrbitRow G p x n := by
-  have hx : x ∈ vertexOrbit G x := ⟨Equiv.refl _,Equiv.refl _,fun _ => by simp,rfl⟩
-  unfold binOrbitRowMass
-  rw [integral_tsum_of_finite_support (law p) (finite_reachBall hloc x (2*n)).toFinset _
-    (fun y => by
-      by_cases hy : y ∈ vertexOrbit G x
-      · simp only [ite_eq_left hy]
-        exact integrable_finite_local _ p _ (fun ω => binConnectionMass_local G hloc n ω x y)
-      · simp only [ite_eq_right hy]; exact integrable_const (0:ℝ))
-    (fun ω y hy => by
-      have hz := binConnectionMass_outside G n ω x y (by simpa only [Set.Finite.mem_toFinset] using hy)
-      simp only [hz,ite_self])]
-  apply tsum_congr
-  intro y
-  by_cases hy : y ∈ vertexOrbit G x
-  · simp only [orbitBinConnection,hx,hy,true_and,ite_true,binConnection]
-  · simp only [orbitBinConnection,hy,and_false,ite_false,integral_zero]
-
-theorem binOrbitTransport_column (G : BondGraph V E) (hloc : LocallyFinite G)
-    (p : unitInterval) (n : ℕ) (hn : 0 < n) (y : V) :
-    (∑' x, binOrbitTransport G p n x y) = binOrbitRow G p y n := by
-  rw [←integral_binOrbitRowMass G hloc p n y,←show (∫ ω, ∑' x, binOrbitTransportMass G n ω x y ∂law p) =
-    ∑' x, binOrbitTransport G p n x y from
-      integral_tsum_of_finite_support (law p) (finite_reachBall hloc y (2*n)).toFinset _
-        (fun x => binOrbitTransportMass_integrable G hloc p n x y)
-        (fun ω x hx => binOrbitTransportMass_outside_column G n ω x y (by simpa only [Set.Finite.mem_toFinset] using hx))]
-  simp only [binOrbitTransportMass_column G n hn]
-
-end AveragedBinOrbitTransport
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section WeightedOrbitBinLower
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem binOrbitTransport_row_lower (G : BondGraph V E) (hloc : LocallyFinite G)
-    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V)
-    (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i))
-    (p : unitInterval) (n : ℕ) (hn : 0 < n) (x : V) :
-    (n:ℝ)/Fintype.card ι * clusterWindowProbability G p x n (2*n) ≤
-      ∑' y, binOrbitTransport G p n x y := by
-  let B := (finite_reachBall hloc x (2*n)).toFinset
-  have he (ω) : (∑' y, binOrbitTransportMass G n ω x y) =
-      ∑ y ∈ B, binOrbitTransportMass G n ω x y := by
-    apply tsum_eq_sum
-    intro y hy
-    exact binOrbitTransportMass_outside G n ω x y (by simpa only [B,Set.Finite.mem_toFinset] using hy)
-  have hi : Integrable (fun ω => ∑' y, binOrbitTransportMass G n ω x y) (law p) := by
-    simp only [he]
-    exact integrable_finsetSum B (fun y _ => binOrbitTransportMass_integrable G hloc p n x y)
-  let W := {ω | ClusterWindow G x n (2*n) ω}
-  have hm : MeasurableSet W := measurableSet_clusterWindow G hloc x n (2*n)
-  have hl : Integrable (W.indicator (fun _ => (n:ℝ)/Fintype.card ι)) (law p) :=
-    (integrable_const _).indicator hm
-  calc
-    _ = ∫ ω, W.indicator (fun _ => (n:ℝ)/Fintype.card ι) ω ∂law p := by
-      rw [integral_indicator_const _ hm]
-      simp only [smul_eq_mul,clusterWindowProbability,W,mul_comm]
-    _ ≤ ∫ ω, ∑' y, binOrbitTransportMass G n ω x y ∂law p := by
-      apply integral_mono hl hi
-      intro ω
-      exact binOrbitTransportMass_row G root hcover n hn ω x
-    _ = _ := (binOrbitTransport_row_integral G hloc p n x).symm
-
- 
-
-theorem weighted_binOrbitRow_lower (G : BondGraph V E) (hloc : LocallyFinite G)
-    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hw : ∀ i, 0 ≤ weight i) (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i))
-    (p : unitInterval) (n : ℕ) (hn : 0 < n) :
-    (n:ℝ)/Fintype.card ι * (∑ i, weight i * clusterWindowProbability G p (root i) n (2*n)) ≤
-      ∑ i, weight i * binOrbitRow G p (root i) n := by
-  have hr (x : V) : Summable (binOrbitTransport G p n x) := by
-    apply summable_of_ne_finset_zero (s := (finite_reachBall hloc x (2*n)).toFinset)
-    intro y hy
-    exact binOrbitTransport_outside G p n x y (by simpa only [Set.Finite.mem_toFinset] using hy)
-  have hc (y : V) : Summable (fun x => binOrbitTransport G p n x y) := by
-    apply summable_of_ne_finset_zero (s := (finite_reachBall hloc y (2*n)).toFinset)
-    intro x hx
-    have hz (ω) := binOrbitTransportMass_outside_column G n ω x y (by simpa only [Set.Finite.mem_toFinset] using hx)
-    simp only [binOrbitTransport,hz,integral_zero]
-  have ht := htr.real root weight Γ (binOrbitTransport G p n)
-    (fun a ha x y => by obtain ⟨b,hab⟩ := hΓ ha; exact binOrbitTransport_covariant G hloc p hab n x y) hr hc
-  calc
-    _ = ∑ i, weight i * ((n:ℝ)/Fintype.card ι * clusterWindowProbability G p (root i) n (2*n)) := by
-      rw [Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro i hi
-      ring
-    _ ≤ ∑ i, weight i * ∑' y, binOrbitTransport G p n (root i) y :=
-      Finset.sum_le_sum (fun i _ => mul_le_mul_of_nonneg_left (binOrbitTransport_row_lower G hloc root hcover p n hn (root i)) (hw i))
-    _ = ∑ i, weight i * ∑' x, binOrbitTransport G p n x (root i) := ht
-    _ = _ := by simp only [binOrbitTransport_column G hloc p n hn]
-
-theorem exists_binOrbitRow_lower (G : BondGraph V E) (hloc : LocallyFinite G)
-    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
-    (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i))
-    (p : unitInterval) (n : ℕ) (hn : 0 < n) :
-    ∃ i, (n:ℝ)/Fintype.card ι * (∑ j, weight j * clusterWindowProbability G p (root j) n (2*n)) ≤
-      binOrbitRow G p (root i) n := by
-  have h := weighted_binOrbitRow_lower G hloc root weight Γ hΓ htr (fun i => (hw i).le) hcover p n hn
-  by_contra! he
-  let c := (n:ℝ)/Fintype.card ι * (∑ j, weight j * clusterWindowProbability G p (root j) n (2*n))
-  have hs : (∑ i, weight i * binOrbitRow G p (root i) n) < ∑ i, weight i*c := by
-    apply Finset.sum_lt_sum (fun i _ => (mul_lt_mul_of_pos_left (he i) (hw i)).le)
-    exact ⟨Classical.arbitrary ι,Finset.mem_univ _,mul_lt_mul_of_pos_left (he _) (hw _)⟩
-  rw [←Finset.sum_mul,hw1,one_mul] at hs
-  exact (not_lt_of_ge h) hs
-
-end WeightedOrbitBinLower
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section DyadicRecords
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem summable_dyadic_of_scaled_bound (π : ℕ → ℝ) (hπ : ∀ j, 0 ≤ π j)
-    (α β C : ℝ) (hαβ : α < β)
-    (hC : ∀ j : ℕ, (2:ℝ)^(β*(j:ℝ))*π j ≤ C) :
-    Summable (fun j : ℕ => (2:ℝ)^(α*(j:ℝ))*π j) := by
-  let r := (2:ℝ)^(α-β)
-  have hr0 : 0 ≤ r := Real.rpow_nonneg (by norm_num) _
-  have hr1 : r < 1 := Real.rpow_lt_one_of_one_lt_of_neg (by norm_num) (sub_neg.mpr hαβ)
-  have hs := (summable_geometric_of_lt_one hr0 hr1).mul_left C
-  apply hs.of_nonneg_of_le (fun j => mul_nonneg (Real.rpow_nonneg (by norm_num) _) (hπ j))
-  intro j
-  have he : (2:ℝ)^(α*(j:ℝ)) = r^j * (2:ℝ)^(β*(j:ℝ)) := by
-    rw [show r^j = (2:ℝ)^((α-β)*(j:ℝ)) by rw [Real.rpow_mul (by norm_num),Real.rpow_natCast],
-      ←Real.rpow_add (by norm_num)]
-    congr 1
-    ring
-  rw [he,mul_assoc]
-  exact (mul_le_mul_of_nonneg_left (hC j) (pow_nonneg hr0 j)).trans_eq (mul_comm _ _)
-
-theorem unbounded_scaled_dyadic (π : ℕ → ℝ) (hπ : ∀ j, 0 ≤ π j)
-    (α β : ℝ) (hαβ : α < β)
-    (hdiv : ¬ Summable (fun j : ℕ => (2:ℝ)^(α*(j:ℝ))*π j)) :
-    ∀ C : ℝ, ∃ j : ℕ, C < (2:ℝ)^(β*(j:ℝ))*π j := by
-  intro C
-  by_contra! hC
-  exact hdiv (summable_dyadic_of_scaled_bound π hπ α β C hαβ hC)
-
- 
-theorem late_record_of_unbounded (f : ℕ → ℝ) (hf : ∀ C : ℝ, ∃ j, C < f j) (N : ℕ) :
-    ∃ j, N < j ∧ 1 ≤ f j ∧ ∀ i ≤ j, f i ≤ f j := by
-  obtain ⟨a,ha,hamax⟩ := (Finset.range (N+1)).exists_max_image f ⟨0,by simp⟩
-  obtain ⟨b,hb⟩ := hf (max 1 (f a))
-  obtain ⟨j,hj,hjmax⟩ := (Finset.range (b+1)).exists_max_image f ⟨0,by simp⟩
-  have hb' : max 1 (f a) < f j := hb.trans_le (hjmax b (by simp))
-  refine ⟨j,?_,(le_max_left _ _).trans hb'.le,?_⟩
-  · by_contra! hjN
-    have he := hamax j (Finset.mem_range.mpr (by omega))
-    exact (not_lt_of_ge he) ((le_max_right _ _).trans_lt hb')
-  · intro i hi
-    exact hjmax i (Finset.mem_range.mpr (by have := Finset.mem_range.mp hj; omega))
-
- 
-
-theorem finite_family_unbounded_choice {ι : Type*} [Fintype ι]
-    (P : ι → ℕ → Prop) (h : ∀ N, ∃ j, N ≤ j ∧ ∃ i, P i j) :
-    ∃ i, ∀ N, ∃ j, N ≤ j ∧ P i j := by
-  by_contra! hnot
-  choose L hL using hnot
-  obtain ⟨j,hj,i,hi⟩ := h (∑ i, L i)
-  have hLi : L i ≤ ∑ t, L t := Finset.single_le_sum (fun t _ => Nat.zero_le (L t)) (Finset.mem_univ i)
-  exact hL i j (hLi.trans hj) hi
-
-theorem record_dyadic_bounds (π : ℕ → ℝ) (β : ℝ) (j : ℕ)
-    (hrec : ∀ i ≤ j+1, (2:ℝ)^(β*(i:ℝ))*π i ≤ (2:ℝ)^(β*((j+1:ℕ):ℝ))*π (j+1))
-    (h1 : 1 ≤ (2:ℝ)^(β*((j+1:ℕ):ℝ))*π (j+1)) :
-    (2:ℝ)^(-β*((j+1:ℕ):ℝ)) ≤ π (j+1) ∧ π j ≤ (2:ℝ)^β*π (j+1) := by
-  have hpos : 0 < (2:ℝ)^(β*((j+1:ℕ):ℝ)) := Real.rpow_pos_of_pos (by norm_num) _
-  have hinv : (2:ℝ)^(-β*((j+1:ℕ):ℝ)) = 1/(2:ℝ)^(β*((j+1:ℕ):ℝ)) := by
-    rw [show -β*((j+1:ℕ):ℝ) = -(β*((j+1:ℕ):ℝ)) by ring,Real.rpow_neg (by norm_num)]
-    simp only [one_div]
-  refine ⟨?_,?_⟩
-  · rw [hinv]
-    exact (div_le_iff₀ hpos).mpr (by simpa only [mul_comm] using h1)
-  · have hr := hrec j (by omega)
-    have he : (2:ℝ)^(β*((j+1:ℕ):ℝ)) = (2:ℝ)^(β*(j:ℝ))*(2:ℝ)^β := by
-      rw [Nat.cast_add,Nat.cast_one,mul_add,mul_one,Real.rpow_add (by norm_num)]
-    rw [he,mul_assoc] at hr
-    exact (mul_le_mul_iff_right₀ (Real.rpow_pos_of_pos (by norm_num : (0:ℝ)<2) _)).mp hr
-
-end DyadicRecords
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section OrbitDyadicScales
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem clusterWindowProbability_split (G : BondGraph V E) (hloc : LocallyFinite G)
-    (p : unitInterval) (x : V) (a b c : ℕ) (hab : a ≤ b) (hbc : b ≤ c) :
-    clusterWindowProbability G p x a c = clusterWindowProbability G p x a b + clusterWindowProbability G p x b c := by
-  have he : {ω | ClusterWindow G x a c ω} =
-      {ω | ClusterWindow G x a b ω} ∪ {ω | ClusterWindow G x b c ω} := by
-    ext ω
-    simp only [Set.mem_ofPred_eq,Set.mem_union,ClusterWindow]
-    constructor
-    · rintro ⟨hf,ha,hc⟩
-      by_cases hb : (cluster G ω x).ncard < b
-      · exact Or.inl ⟨hf,ha,hb⟩
-      · exact Or.inr ⟨hf,by omega,hc⟩
-    · rintro (⟨hf,ha,hb⟩|⟨hf,hb,hc⟩)
-      · exact ⟨hf,ha,hb.trans_le hbc⟩
-      · exact ⟨hf,hab.trans hb,hc⟩
-  have hd : Disjoint {ω | ClusterWindow G x a b ω} {ω | ClusterWindow G x b c ω} := by
-    apply Set.disjoint_left.mpr
-    intro ω h₁ h₂
-    exact (not_lt_of_ge h₂.2.1) h₁.2.2
-  unfold clusterWindowProbability
-  rw [he,measureReal_union hd (measurableSet_clusterWindow G hloc x b c)]
-
-noncomputable def weightedWindowProbability {ι : Type*} [Fintype ι]
-    (G : BondGraph V E) (p : unitInterval) (root : ι → V) (weight : ι → ℝ) (a b : ℕ) : ℝ :=
-  ∑ i, weight i * clusterWindowProbability G p (root i) a b
-
-theorem weightedWindowProbability_nonneg {ι : Type*} [Fintype ι]
-    (G : BondGraph V E) (p : unitInterval) (root : ι → V) (weight : ι → ℝ)
-    (hw : ∀ i, 0 ≤ weight i) (a b : ℕ) :
-    0 ≤ weightedWindowProbability G p root weight a b := by
-  exact Finset.sum_nonneg (fun i _ => mul_nonneg (hw i) (clusterWindowProbability_nonneg G p (root i) a b))
-
-theorem weightedWindowProbability_split {ι : Type*} [Fintype ι]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (p : unitInterval) (root : ι → V) (weight : ι → ℝ)
-    (a b c : ℕ) (hab : a ≤ b) (hbc : b ≤ c) :
-    weightedWindowProbability G p root weight a c =
-      weightedWindowProbability G p root weight a b + weightedWindowProbability G p root weight b c := by
-  simp only [weightedWindowProbability,clusterWindowProbability_split G hloc p _ a b c hab hbc,
-    mul_add,Finset.sum_add_distrib]
-
-noncomputable def dyadicBinProbability {ι : Type*} [Fintype ι]
-    (G : BondGraph V E) (p : unitInterval) (root : ι → V) (weight : ι → ℝ) (j : ℕ) : ℝ :=
-  weightedWindowProbability G p root weight (2^j) (2^(j+1))
-
- 
-
-theorem orbit_dyadic_scales (G : BondGraph V E) (hloc : LocallyFinite G)
-    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
-    (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i)) (p : unitInterval)
-    (α β : ℝ) (hαβ : α < β)
-    (hdiv : ¬ Summable (fun j : ℕ => (2:ℝ)^(α*(j:ℝ))*dyadicBinProbability G p root weight j)) :
-    ∃ i, ∀ N, ∃ j, N ≤ j ∧
-      ((2^(j+1):ℕ):ℝ)^(-β) ≤ dyadicBinProbability G p root weight (j+1) ∧
-      weightedWindowProbability G p root weight (2^j) (2^(j+2)) ≤
-        (1+(2:ℝ)^β)*dyadicBinProbability G p root weight (j+1) ∧
-      (((2^(j+1):ℕ):ℝ)/Fintype.card ι)*dyadicBinProbability G p root weight (j+1) ≤
-        binOrbitRow G p (root i) (2^(j+1)) := by
-  let π := dyadicBinProbability G p root weight
-  let f := fun j : ℕ => (2:ℝ)^(β*(j:ℝ))*π j
-  have hπ : ∀ j, 0 ≤ π j := fun j => weightedWindowProbability_nonneg G p root weight (fun i => (hw i).le) _ _
-  have hf := unbounded_scaled_dyadic π hπ α β hαβ hdiv
-  let P := fun i j => 1 ≤ f (j+1) ∧ (∀ t ≤ j+1, f t ≤ f (j+1)) ∧
-    (((2^(j+1):ℕ):ℝ)/Fintype.card ι)*π (j+1) ≤ binOrbitRow G p (root i) (2^(j+1))
-  have hpick : ∀ N, ∃ j, N ≤ j ∧ ∃ i, P i j := by
-    intro N
-    obtain ⟨t,ht,h1,hr⟩ := late_record_of_unbounded f hf N
-    obtain ⟨j,rfl⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : t ≠ 0)
-    obtain ⟨i,hi⟩ := exists_binOrbitRow_lower G hloc root weight Γ hΓ htr hw hw1 hcover p (2^(j+1)) (by positivity)
-    have he : 2*2^(j+1) = 2^(j+2) := by rw [show j+2 = (j+1)+1 by omega,pow_succ]; omega
-    refine ⟨j,by omega,i,h1,hr,?_⟩
-    simpa only [P,π,dyadicBinProbability,weightedWindowProbability,he] using hi
-  obtain ⟨i,hi⟩ := finite_family_unbounded_choice P hpick
-  refine ⟨i,?_⟩
-  intro N
-  obtain ⟨j,hj,h1,hr,hb⟩ := hi N
-  have hs := record_dyadic_bounds π β j hr h1
-  refine ⟨j,hj,?_,?_,hb⟩
-  · have he : (((2^(j+1):ℕ):ℝ)^(-β)) = (2:ℝ)^(-β*((j+1:ℕ):ℝ)) := by
-      rw [Nat.cast_pow,Nat.cast_ofNat,←Real.rpow_natCast,←Real.rpow_mul (by norm_num)]
-      congr 1
-      ring
-    rw [he]
-    exact hs.1
-  · rw [weightedWindowProbability_split G hloc p root weight (2^j) (2^(j+1)) (2^(j+2))
-      (Nat.pow_le_pow_right (by norm_num) (by omega)) (Nat.pow_le_pow_right (by norm_num) (by omega))]
-    change π j + π (j+1) ≤ (1+(2:ℝ)^β)*π (j+1)
-    nlinarith only [hs.2]
-
-end OrbitDyadicScales
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section ClusterSizeMomentIntegrable
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem measurable_cluster_ncard (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) :
-    Measurable (fun ω : Set E => (cluster G ω x).ncard) := by
-  have hm (k : ℕ) (hk : k ≠ 0) : MeasurableSet {ω : Set E | (cluster G ω x).ncard = k} := by
-    have he : {ω : Set E | (cluster G ω x).ncard = k} = {ω | ClusterWindow G x k (k+1) ω} := by
-      ext ω
-      constructor
-      · intro h
-        have hf := Set.finite_of_ncard_ne_zero (by rw [h]; exact hk)
-        exact ⟨hf,by rw [h],by rw [h]; omega⟩
-      · rintro ⟨hf,hl,hu⟩
-        exact le_antisymm (by omega) hl
-    rw [he]
-    exact measurableSet_clusterWindow G hloc x k (k+1)
-  have hm0 : MeasurableSet {ω : Set E | (cluster G ω x).ncard = 0} := by
-    have he : {ω : Set E | (cluster G ω x).ncard = 0} =
-        (⋃ k : ℕ, {ω : Set E | (cluster G ω x).ncard = k+1})ᶜ := by
-      ext ω
-      simp only [Set.mem_compl_iff,Set.mem_iUnion,Set.mem_ofPred_eq,not_exists]
-      constructor
-      · intro h k
-        omega
-      · intro h
-        by_contra hz
-        obtain ⟨k,hk⟩ := Nat.exists_eq_succ_of_ne_zero hz
-        exact h k hk
-    rw [he]
-    exact (MeasurableSet.iUnion (fun k => hm (k+1) (by omega))).compl
-  apply measurable_to_countable
-  intro ω
-  change MeasurableSet {ω' | (cluster G ω' x).ncard = (cluster G ω x).ncard}
-  by_cases h : (cluster G ω x).ncard = 0
-  · simpa only [h] using hm0
-  · exact hm _ h
-
- 
-
-theorem integrable_cluster_power_of_dyadic (G : BondGraph V E) (hloc : LocallyFinite G)
-    (p : unitInterval) (x : V) (α : ℝ) (hα : 0 < α)
-    (hs : Summable (fun j : ℕ => (2:ℝ)^(α*(j:ℝ))*clusterWindowProbability G p x (2^j) (2^(j+1)))) :
-    Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^α) (law p) := by
-  let f := fun ω : Set E => ((cluster G ω x).ncard:ℝ)^α
-  let F := fun j : ℕ => {ω | ClusterWindow G x (2^j) (2^(j+1)) ω}.indicator
-    (fun _ => (2:ℝ)^(α*((j+1:ℕ):ℝ)))
-  have hFm (j : ℕ) : Measurable (F j) :=
-    measurable_const.indicator (measurableSet_clusterWindow G hloc x _ _)
-  have hFi (j : ℕ) : Integrable (F j) (law p) :=
-    (integrable_const _).indicator (measurableSet_clusterWindow G hloc x _ _)
-  have hFn (j : ℕ) (ω : Set E) : 0 ≤ F j ω := by
-    exact Set.indicator_nonneg (fun _ _ => (Real.rpow_pos_of_pos (by norm_num) _).le) _
-  have hFe (j : ℕ) : (∫ ω, F j ω ∂law p) =
-      (2:ℝ)^α * ((2:ℝ)^(α*(j:ℝ))*clusterWindowProbability G p x (2^j) (2^(j+1))) := by
-    rw [integral_indicator_const _ (measurableSet_clusterWindow G hloc x _ _)]
-    simp only [smul_eq_mul,clusterWindowProbability,Nat.cast_add,Nat.cast_one,mul_add,mul_one,
-      Real.rpow_add (by norm_num : (0:ℝ)<2)]
-    ring
-  have hsum : Summable (fun j => ∫ ω, F j ω ∂law p) := by
-    simp only [hFe]
-    exact hs.mul_left _
-  have hbound (ω : Set E) : ENNReal.ofReal (f ω) ≤ ∑' j, ENNReal.ofReal (F j ω) := by
-    by_cases hz : (cluster G ω x).ncard = 0
-    · simp only [f,hz,Nat.cast_zero,Real.zero_rpow hα.ne',ENNReal.ofReal_zero]
-      exact bot_le
-    · let j := Nat.log 2 (cluster G ω x).ncard
-      have hl := Nat.pow_log_le_self 2 hz
-      have hu := Nat.lt_pow_succ_log_self (by norm_num : 1 < (2:ℕ)) (cluster G ω x).ncard
-      have hb : ClusterWindow G x (2^j) (2^(j+1)) ω :=
-        ⟨Set.finite_of_ncard_ne_zero hz,hl,hu⟩
-      have hle : f ω ≤ (2:ℝ)^(α*((j+1:ℕ):ℝ)) := by
-        have hcast : ((cluster G ω x).ncard:ℝ) ≤ ((2^(j+1):ℕ):ℝ) := by exact_mod_cast hu.le
-        have h := Real.rpow_le_rpow (Nat.cast_nonneg _) hcast hα.le
-        have he : (((2^(j+1):ℕ):ℝ)^α) = (2:ℝ)^(α*((j+1:ℕ):ℝ)) := by
-          rw [Nat.cast_pow,Nat.cast_ofNat,←Real.rpow_natCast,←Real.rpow_mul (by norm_num)]
-          congr 1
-          ring
-        exact h.trans_eq he
-      have he : F j ω = (2:ℝ)^(α*((j+1:ℕ):ℝ)) := Set.indicator_of_mem (show ω ∈ {ω | ClusterWindow G x (2^j) (2^(j+1)) ω} from hb) _
-      exact (ENNReal.ofReal_le_ofReal (hle.trans_eq he.symm)).trans
-        (ENNReal.le_tsum (f := fun scale => ENNReal.ofReal (F scale ω)) j)
-  have hfn (ω : Set E) : 0 ≤ f ω := Real.rpow_nonneg (Nat.cast_nonneg _) _
-  refine ⟨((measurable_of_countable (fun m : ℕ => (m:ℝ)^α)).comp (measurable_cluster_ncard G hloc x)).aestronglyMeasurable,?_⟩
-  apply (hasFiniteIntegral_iff_ofReal (Filter.Eventually.of_forall hfn)).mpr
-  have hlin : (∫⁻ ω, ∑' j, ENNReal.ofReal (F j ω) ∂law p) < ⊤ := by
-    rw [lintegral_tsum (fun j => (hFm j).ennreal_ofReal.aemeasurable)]
-    simp_rw [←ofReal_integral_eq_lintegral_ofReal (hFi _) (Filter.Eventually.of_forall (hFn _))]
-    rw [←ENNReal.ofReal_tsum_of_nonneg (fun j => integral_nonneg (hFn j)) hsum]
-    exact ENNReal.ofReal_lt_top
-  exact (lintegral_mono hbound).trans_lt hlin
-
-end ClusterSizeMomentIntegrable
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section WeightedMomentScales
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem integrable_cluster_power_of_weighted_dyadic (G : BondGraph V E)
-    (hloc : LocallyFinite G) {ι : Type*} [Fintype ι] (p : unitInterval)
-    (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 < weight i)
-    (α : ℝ) (hα : 0 < α)
-    (hs : Summable (fun j : ℕ => (2:ℝ)^(α*(j:ℝ))*dyadicBinProbability G p root weight j))
-    (i : ι) : Integrable (fun ω => ((cluster G ω (root i)).ncard:ℝ)^α) (law p) := by
-  have hle (j : ℕ) : weight i * clusterWindowProbability G p (root i) (2^j) (2^(j+1)) ≤
-      dyadicBinProbability G p root weight j := by
-    exact Finset.single_le_sum
-      (fun t _ => mul_nonneg (hw t).le (clusterWindowProbability_nonneg G p (root t) _ _))
-      (Finset.mem_univ i)
-  have hs' : Summable (fun j : ℕ => weight i *
-      ((2:ℝ)^(α*(j:ℝ))*clusterWindowProbability G p (root i) (2^j) (2^(j+1)))) := by
-    apply hs.of_nonneg_of_le
-    · intro j
-      exact mul_nonneg (hw i).le (mul_nonneg (Real.rpow_nonneg (by norm_num) _)
-        (clusterWindowProbability_nonneg G p (root i) _ _))
-    · intro j
-      have h := mul_le_mul_of_nonneg_left (hle j) (Real.rpow_nonneg (by norm_num : (0:ℝ) ≤ 2) (α*(j:ℝ)))
-      nlinarith only [h]
-  apply integrable_cluster_power_of_dyadic G hloc p (root i) α hα
-  have he : (fun j : ℕ => (2:ℝ)^(α*(j:ℝ))*clusterWindowProbability G p (root i) (2^j) (2^(j+1))) =
-      (fun j : ℕ => (weight i)⁻¹ * (weight i *((2:ℝ)^(α*(j:ℝ))*clusterWindowProbability G p (root i) (2^j) (2^(j+1))))) := by
-    funext j
-    rw [←mul_assoc,inv_mul_cancel₀ (hw i).ne',one_mul]
-  rw [he]
-  exact hs'.mul_left _
-
- 
-
-theorem orbit_scales_of_divergent_moment (G : BondGraph V E) (hloc : LocallyFinite G)
-    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
-    (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i)) (p : unitInterval)
-    (α β : ℝ) (hα : 0 < α) (hαβ : α < β)
-    (hdiv : ∃ i, ¬ Integrable (fun ω => ((cluster G ω (root i)).ncard:ℝ)^α) (law p)) :
-    ∃ i, ∀ N, ∃ j, N ≤ j ∧
-      ((2^(j+1):ℕ):ℝ)^(-β) ≤ dyadicBinProbability G p root weight (j+1) ∧
-      weightedWindowProbability G p root weight (2^j) (2^(j+2)) ≤
-        (1+(2:ℝ)^β)*dyadicBinProbability G p root weight (j+1) ∧
-      (((2^(j+1):ℕ):ℝ)/Fintype.card ι)*dyadicBinProbability G p root weight (j+1) ≤
-        binOrbitRow G p (root i) (2^(j+1)) := by
-  apply orbit_dyadic_scales G hloc root weight Γ hΓ htr hw hw1 hcover p α β hαβ
-  intro hs
-  obtain ⟨i,hi⟩ := hdiv
-  exact hi (integrable_cluster_power_of_weighted_dyadic G hloc p root weight hw α hα hs i)
-
-end WeightedMomentScales
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open Filter
-section LogarithmicChainLength
-
-noncomputable def logarithmicChainLength (t : ℝ) (n : ℕ) : ℕ := ⌊Real.log n / t⌋₊
-
-theorem logarithmicChainLength_bounds (t : ℝ) (ht : 0 < t) (n : ℕ) (hn : 1 ≤ n) :
-    (logarithmicChainLength t n:ℝ)*t ≤ Real.log n ∧
-      Real.log n < ((logarithmicChainLength t n:ℝ)+1)*t := by
-  have hn' : (1:ℝ) ≤ n := by exact_mod_cast hn
-  have hlog := Real.log_nonneg hn'
-  constructor
-  · exact (le_div_iff₀ ht).mp (Nat.floor_le (div_nonneg hlog ht.le))
-  · exact (div_lt_iff₀ ht).mp (Nat.lt_floor_add_one (Real.log n/t))
-
-theorem tendsto_logarithmicChainLength (t : ℝ) (ht : 0 < t) :
-    Tendsto (logarithmicChainLength t) atTop atTop :=
-  tendsto_nat_floor_atTop.comp ((Real.tendsto_log_atTop.comp tendsto_natCast_atTop_atTop).atTop_div_const ht)
-
-theorem tendsto_log_div_chainLength (t : ℝ) (ht : 0 < t) :
-    Tendsto (fun n : ℕ => Real.log n / logarithmicChainLength t n) atTop (nhds t) := by
-  have hk : Tendsto (fun n : ℕ => (logarithmicChainLength t n:ℝ)) atTop atTop :=
-    tendsto_natCast_atTop_atTop.comp (tendsto_logarithmicChainLength t ht)
-  have hlim : Tendsto (fun n : ℕ => t + t/(logarithmicChainLength t n:ℝ)) atTop (nhds t) := by
-    simpa only [add_zero] using tendsto_const_nhds.add (tendsto_const_nhds.div_atTop hk)
-  apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hlim
-  · filter_upwards [eventually_ge_atTop (1:ℕ),hk.eventually_gt_atTop 0] with n hn hk0
-    exact (le_div_iff₀ hk0).mpr (by simpa only [mul_comm] using (logarithmicChainLength_bounds t ht n hn).1)
-  · filter_upwards [eventually_ge_atTop (1:ℕ),hk.eventually_gt_atTop 0] with n hn hk0
-    have h := (logarithmicChainLength_bounds t ht n hn).2.le
-    apply (div_le_iff₀ hk0).mpr
-    calc
-      _ ≤ ((logarithmicChainLength t n:ℝ)+1)*t := h
-      _ = _ := by field_simp
-
-theorem tendsto_chainLength_div_scale (t : ℝ) (ht : 0 < t) :
-    Tendsto (fun n : ℕ => (logarithmicChainLength t n:ℝ)/n) atTop (nhds 0) := by
-  have hlog : Tendsto (fun n : ℕ => (Real.log n/(n:ℝ))/t) atTop (nhds 0) := by
-    simpa only [zero_div,Function.comp_def,id_eq] using
-      (Real.isLittleO_log_id_atTop.tendsto_div_nhds_zero.comp tendsto_natCast_atTop_atTop).div_const t
-  apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hlog
-  · exact Eventually.of_forall (fun n => div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _))
-  · filter_upwards [eventually_ge_atTop (1:ℕ)] with n hn
-    have hn' : (0:ℝ) < n := by exact_mod_cast (show 0 < n by omega)
-    apply (le_div_iff₀ ht).mpr
-    rw [div_mul_eq_mul_div]
-    exact (div_le_div_iff_of_pos_right hn').mpr (logarithmicChainLength_bounds t ht n hn).1
-
-theorem eventually_chain_volume_small (t : ℝ) (ht : 0 < t) (M : ℕ) :
-    ∀ᶠ n : ℕ in atTop, 2*logarithmicChainLength t n*M ≤ n := by
-  have hlim : Tendsto (fun n : ℕ => (2*(logarithmicChainLength t n:ℝ)*M)/(n:ℝ)) atTop (nhds 0) := by
-    have h := (tendsto_chainLength_div_scale t ht).const_mul (2*(M:ℝ))
-    simpa only [mul_zero] using h.congr (fun n => by ring)
-  filter_upwards [(tendsto_order.mp hlim).2 1 (by norm_num),eventually_ge_atTop (1:ℕ)] with n hn hn1
-  have hn' : (0:ℝ) < n := by exact_mod_cast (show 0 < n by omega)
-  have h := (div_lt_iff₀ hn').mp hn
-  norm_num only [one_mul] at h
-  exact_mod_cast h.le
-
-end LogarithmicChainLength
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open Filter
-section AmplificationPolynomial
-
-theorem amplificationMomentFactor_ge_four (k M d : ℕ) (c a t : ℝ) (ha : 0 < a) (ht : 0 ≤ t) :
-    4 ≤ amplificationMomentFactor k M d c a t := by
-  unfold amplificationMomentFactor
-  have h1 : (0:ℝ) ≤ ((k*M)+(k*M)^2+d*(k*M)^3:ℕ) := Nat.cast_nonneg _
-  have h2 : (0:ℝ) ≤ (2*c*(k*M:ℕ))^2*t/a := by positivity
-  linarith
-
-theorem amplificationMomentFactor_cubic (k M d : ℕ) (c a t : ℝ) (ha : 0 < a) (ht : 0 ≤ t) :
-    amplificationMomentFactor k M d c a t ≤ amplificationMomentFactor 1 M d c a t * ((k:ℝ)+1)^3 := by
-  let b := (2*c*(M:ℝ))^2*t/a
-  have hb : 0 ≤ b := by dsimp [b]; positivity
-  have hk : (0:ℝ) ≤ k := Nat.cast_nonneg _
-  have hk1 : (1:ℝ) ≤ (k:ℝ)+1 := by linarith
-  have h13 : (1:ℝ) ≤ ((k:ℝ)+1)^3 := one_le_pow₀ hk1
-  have h1 : (k:ℝ) ≤ ((k:ℝ)+1)^3 := by
-    calc
-      _ ≤ (k:ℝ)+1 := by linarith
-      _ = ((k:ℝ)+1)^1 := (pow_one _).symm
-      _ ≤ _ := pow_le_pow_right₀ hk1 (by omega)
-  have h2 : (k:ℝ)^2 ≤ ((k:ℝ)+1)^3 :=
-    (pow_le_pow_left₀ hk (by linarith : (k:ℝ) ≤ (k:ℝ)+1) 2).trans (pow_le_pow_right₀ hk1 (by omega))
-  have h3 : (k:ℝ)^3 ≤ ((k:ℝ)+1)^3 := pow_le_pow_left₀ hk (by linarith) 3
-  have he (j : ℕ) : amplificationMomentFactor j M d c a t =
-      4 + (M:ℝ)*j + (M:ℝ)^2*(j:ℝ)^2 + ((d:ℝ)*(M:ℝ)^3)*(j:ℝ)^3 + b*(j:ℝ)^2 := by
-    simp only [amplificationMomentFactor,Nat.cast_add,Nat.cast_mul,Nat.cast_pow,b]
-    ring
-  rw [he k,he 1]
-  norm_num only [Nat.cast_one,one_pow,mul_one]
-  have hh0 := mul_le_mul_of_nonneg_left h13 (by norm_num : (0:ℝ) ≤ 4)
-  have hh1 := mul_le_mul_of_nonneg_left h1 (Nat.cast_nonneg M : (0:ℝ) ≤ M)
-  have hh2 := mul_le_mul_of_nonneg_left h2 (sq_nonneg (M:ℝ))
-  have hh3 := mul_le_mul_of_nonneg_left h3 (show 0 ≤ (d:ℝ)*(M:ℝ)^3 by positivity)
-  have hhb := mul_le_mul_of_nonneg_left h2 hb
-  nlinarith only [hh0,hh1,hh2,hh3,hhb]
-
-theorem tendsto_log_amplificationMomentFactor_div (M d : ℕ) (c a t : ℝ) (ha : 0 < a) (ht : 0 ≤ t) :
-    Tendsto (fun k : ℕ => Real.log (amplificationMomentFactor k M d c a t)/(k:ℝ)) atTop (nhds 0) := by
-  let C := amplificationMomentFactor 1 M d c a t
-  have hC : 0 < C := amplificationMomentFactor_pos _ _ _ _ _ _ ha ht
-  have hplus : Tendsto (fun k : ℕ => (k:ℝ)+1) atTop atTop := by
-    simpa only [Function.comp_def,Nat.cast_add,Nat.cast_one] using
-      (tendsto_natCast_atTop_atTop (R := ℝ)).comp (tendsto_add_atTop_nat 1)
-  have hlog : Tendsto (fun k : ℕ => Real.log ((k:ℝ)+1)/(k:ℝ)) atTop (nhds 0) := by
-    simpa only [Function.comp_def,pow_one,one_mul,add_neg_cancel_right] using
-      (Real.tendsto_pow_log_div_mul_add_atTop 1 (-1) 1 one_ne_zero).comp hplus
-  have hupper : Tendsto (fun k : ℕ => (Real.log C + 3*Real.log ((k:ℝ)+1))/(k:ℝ)) atTop (nhds 0) := by
-    have h := ((tendsto_const_nhds (x := Real.log C)).div_atTop (tendsto_natCast_atTop_atTop (R := ℝ))).add (hlog.const_mul 3)
-    simpa only [zero_add,mul_zero,add_div,mul_div_assoc] using h
-  apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hupper
-  · exact Eventually.of_forall (fun k => div_nonneg
-      (Real.log_nonneg (by have := amplificationMomentFactor_ge_four k M d c a t ha ht; linarith)) (Nat.cast_nonneg _))
-  · exact Eventually.of_forall (fun k => by
-      apply div_le_div_of_nonneg_right _ (Nat.cast_nonneg _)
-      have hh := Real.log_le_log (amplificationMomentFactor_pos k M d c a t ha ht)
-        (amplificationMomentFactor_cubic k M d c a t ha ht)
-      have hp : (0:ℝ) < (k:ℝ)+1 := by positivity
-      simpa only [C,Real.log_mul hC.ne' (pow_pos hp 3).ne',Real.log_pow,Nat.cast_ofNat] using hh)
-
-end AmplificationPolynomial
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open Filter
-section AmplificationScalarBudget
-
-theorem eventually_inverseCodeMass_budget (s ξ : ℝ) (hs : 0 ≤ s) (hξ : 0 < ξ) :
-    ∀ᶠ M : ℕ in atTop, 2 ≤ M ∧ 1 ≤ (M:ℝ)^s ∧
-      inverseCodeMass M ((M:ℝ)^s) ≤ (M:ℝ)^(s+ξ) := by
-  have hl := (isLittleO_log_rpow_atTop hξ).bound (by norm_num : (0:ℝ) < 1/8)
-  have hcast : Tendsto (fun M : ℕ => (M:ℝ)) atTop atTop := tendsto_natCast_atTop_atTop
-  filter_upwards [eventually_ge_atTop (2:ℕ),hcast.eventually hl] with M hM hl
-  have hM' : (1:ℝ) ≤ M := by exact_mod_cast (show 1 ≤ M by omega)
-  have hM0 : (0:ℝ) < M := lt_of_lt_of_le zero_lt_one hM'
-  have hL : 1 ≤ (M:ℝ)^s := Real.one_le_rpow hM' hs
-  refine ⟨hM,hL,?_⟩
-  simp only [Real.norm_eq_abs,abs_of_nonneg (Real.log_nonneg hM'),
-    abs_of_nonneg (Real.rpow_nonneg hM0.le ξ)] at hl
-  calc
-    _ ≤ 8*(M:ℝ)^s*Real.log M := inverseCodeMass_le_log M _ hM hL
-    _ ≤ (M:ℝ)^s*(M:ℝ)^ξ := by
-      have hh := mul_le_mul_of_nonneg_left hl (Real.rpow_nonneg hM0.le s)
-      nlinarith only [hh]
-    _ = _ := (Real.rpow_add hM0 _ _).symm
-
-theorem code_bound_of_logarithmic_chain (u W t : ℝ) (hu : 0 < u) (hW : 0 ≤ W)
-    (ht : 0 < t) (hbudget : W ≤ u*Real.exp t) (n : ℕ) (hn : 1 ≤ n) :
-    W^logarithmicChainLength t n ≤ (n:ℝ)*u^logarithmicChainLength t n := by
-  let k := logarithmicChainLength t n
-  have hexp : Real.exp ((k:ℝ)*t) ≤ (n:ℝ) := by
-    have h := Real.exp_le_exp.mpr (logarithmicChainLength_bounds t ht n hn).1
-    have hnp : (0:ℝ) < n := by exact_mod_cast (show 0 < n by omega)
-    simpa only [Real.exp_log hnp] using h
-  calc
-    W^k ≤ (u*Real.exp t)^k := pow_le_pow_left₀ hW hbudget _
-    _ = u^k*Real.exp ((k:ℝ)*t) := by rw [mul_pow,←Real.exp_nat_mul]
-    _ ≤ u^k*n := mul_le_mul_of_nonneg_left hexp (pow_nonneg hu.le _)
-    _ = _ := mul_comm _ _
-
-theorem tendsto_rpow_mul_stretched_exp (γ ε : ℝ) (hε : 0 < ε) :
-    Tendsto (fun x : ℝ => x^γ * Real.exp (-(x^ε))) atTop (nhds 0) := by
-  have h := (tendsto_rpow_mul_exp_neg_mul_atTop_nhds_zero (γ/ε) 1 zero_lt_one).comp (tendsto_rpow_atTop hε)
-  apply h.congr'
-  filter_upwards [eventually_ge_atTop (0:ℝ)] with x hx
-  simp only [Function.comp_def,neg_mul,one_mul,←Real.rpow_mul hx]
-  rw [mul_div_cancel₀ γ hε.ne']
-
-end AmplificationScalarBudget
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open Filter
-section AmplificationGoodnessDecay
-
-theorem eventually_amplification_goodness_decay (T : ℝ) (hT : 0 < T)
-    (M d : ℕ) (c a b β ε : ℝ) (ha : 0 < a) (hb : 0 ≤ b) (hε : 0 < ε) :
-    ∀ᶠ n : ℕ in atTop,
-      16*amplificationMomentFactor (logarithmicChainLength T n) M d c a b *
-        Real.exp (-((2*n+logarithmicChainLength T n*M:ℕ):ℝ)^ε) ≤ a*(n:ℝ)^(-β) := by
-  let C := amplificationMomentFactor 1 M d c a b
-  have hC : 0 < C := amplificationMomentFactor_pos _ _ _ _ _ _ ha hb
-  have hd : Tendsto (fun n : ℕ => 128*C*((n:ℝ)^(β+3)*Real.exp (-(n:ℝ)^ε))) atTop (nhds 0) := by
-    simpa only [mul_zero,Function.comp_def] using
-      ((tendsto_rpow_mul_stretched_exp (β+3) ε hε).comp tendsto_natCast_atTop_atTop).const_mul (128*C)
-  have hk := (tendsto_order.mp (tendsto_chainLength_div_scale T hT)).2 1 (by norm_num)
-  filter_upwards [eventually_ge_atTop (1:ℕ),hk,(tendsto_order.mp hd).2 a ha] with n hn hk hd
-  let k := logarithmicChainLength T n
-  have hn0 : (0:ℝ) < n := by exact_mod_cast (show 0 < n by omega)
-  have hn1 : (1:ℝ) ≤ n := by exact_mod_cast hn
-  have hkn : (k:ℝ) ≤ n := by
-    exact ((div_lt_iff₀ hn0).mp hk).le.trans_eq (one_mul _)
-  have hQ : amplificationMomentFactor k M d c a b ≤ 8*C*(n:ℝ)^3 := by
-    have hp : ((k:ℝ)+1)^3 ≤ (2*(n:ℝ))^3 :=
-      pow_le_pow_left₀ (by positivity) (by linarith) 3
-    calc
-      _ ≤ C*((k:ℝ)+1)^3 := amplificationMomentFactor_cubic k M d c a b ha hb
-      _ ≤ C*(2*(n:ℝ))^3 := mul_le_mul_of_nonneg_left hp hC.le
-      _ = _ := by ring
-  have he : Real.exp (-((2*n+k*M:ℕ):ℝ)^ε) ≤ Real.exp (-(n:ℝ)^ε) := by
-    apply Real.exp_le_exp.mpr
-    apply neg_le_neg
-    apply Real.rpow_le_rpow (Nat.cast_nonneg _) _ hε.le
-    exact_mod_cast (show n ≤ 2*n+k*M by omega)
-  have hpow : 0 < (n:ℝ)^β := Real.rpow_pos_of_pos hn0 _
-  rw [Real.rpow_neg hn0.le,←div_eq_mul_inv]
-  apply (le_div_iff₀ hpow).mpr
-  calc
-    _ ≤ (16*(8*C*(n:ℝ)^3)*Real.exp (-(n:ℝ)^ε))*(n:ℝ)^β := by
-      apply mul_le_mul_of_nonneg_right _ hpow.le
-      exact mul_le_mul (mul_le_mul_of_nonneg_left hQ (by norm_num)) he
-        (Real.exp_pos _).le (by positivity)
-    _ = 128*C*((n:ℝ)^(β+3)*Real.exp (-(n:ℝ)^ε)) := by
-      rw [Real.rpow_add hn0,show (n:ℝ)^(3:ℝ) = (n:ℝ)^3 by norm_cast]
-      ring
-    _ ≤ a := hd.le
-
-end AmplificationGoodnessDecay
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open Filter
-section AmplificationLogEstimates
-
-theorem log_inv_success_le (a Q n β u : ℝ) (ha : 0 < a) (hQ : 0 < Q)
-    (hn : 0 < n) (hbound : a*n^(-β)/(8*Q) ≤ u) :
-    Real.log (1/u) ≤ β*Real.log n + Real.log Q + Real.log (8/a) := by
-  have hl : 0 < a*n^(-β)/(8*Q) := by positivity
-  have hu : 0 < u := hl.trans_le hbound
-  have h := Real.log_le_log hl hbound
-  rw [Real.log_div (mul_pos ha (Real.rpow_pos_of_pos hn _)).ne' (mul_pos (by norm_num) hQ).ne',
-    Real.log_mul ha.ne' (Real.rpow_pos_of_pos hn _).ne',Real.log_rpow hn,
-    Real.log_mul (by norm_num : (8:ℝ) ≠ 0) hQ.ne'] at h
-  rw [one_div,Real.log_inv,Real.log_div (by norm_num : (8:ℝ) ≠ 0) ha.ne']
-  linarith
-
-theorem log_inv_norm_ge (r r' : ℝ) (k : ℕ) (hr : 0 < r) (hr' : 0 < r')
-    (hbound : r' ≤ 2*r^k) :
-    (k:ℝ)*Real.log (1/r) - Real.log 2 ≤ Real.log (1/r') := by
-  have h := Real.log_le_log hr' hbound
-  rw [Real.log_mul (by norm_num : (2:ℝ) ≠ 0) (pow_pos hr _).ne',Real.log_pow] at h
-  simp only [one_div,Real.log_inv]
-  linarith
-
- 
-
-theorem eventually_amplification_log_estimates (T ξ β H : ℝ)
-    (hT : 0 < T) (hξ : 0 < ξ) (hξ1 : ξ < 1) (hH : 0 < H)
-    (M d : ℕ) (c a b : ℝ) (ha : 0 < a) (hb : 0 ≤ b) :
-    ∀ᶠ n : ℕ in atTop,
-      let k := logarithmicChainLength T n
-      β*Real.log n + Real.log (amplificationMomentFactor k M d c a b) + Real.log (8/a) ≤
-        (k:ℝ)*(β+ξ)*T ∧
-      Real.log ((2*n+k*M:ℕ):ℝ) ≤ (k:ℝ)*(1+ξ)*T ∧
-      (k:ℝ)*(1-ξ)*H ≤ (k:ℝ)*H-Real.log 2 ∧
-      2*H ≤ (k:ℝ)*(1-ξ)*H ∧ 1 ≤ k := by
-  let k := logarithmicChainLength T
-  have hk : Tendsto k atTop atTop := tendsto_logarithmicChainLength T hT
-  have hkR : Tendsto (fun n => (k n:ℝ)) atTop atTop := tendsto_natCast_atTop_atTop.comp hk
-  have hU : Tendsto (fun n : ℕ =>
-      (β*Real.log n + Real.log (amplificationMomentFactor (k n) M d c a b) + Real.log (8/a))/(k n:ℝ))
-      atTop (nhds (β*T)) := by
-    have h1 := (tendsto_log_div_chainLength T hT).const_mul β
-    have h2 := (tendsto_log_amplificationMomentFactor_div M d c a b ha hb).comp hk
-    have h3 := (tendsto_const_nhds (x := Real.log (8/a))).div_atTop hkR
-    simpa only [add_zero,add_div,mul_div_assoc,Function.comp_def] using (h1.add h2).add h3
-  have hm : Tendsto (fun n : ℕ => (Real.log n+Real.log 3)/(k n:ℝ)) atTop (nhds T) := by
-    simpa only [add_zero,add_div] using (tendsto_log_div_chainLength T hT).add
-      ((tendsto_const_nhds (x := Real.log 3)).div_atTop hkR)
-  have hHlim : Tendsto (fun n : ℕ => Real.log 2/(k n:ℝ)) atTop (nhds 0) :=
-    tendsto_const_nhds.div_atTop hkR
-  have hU' := (tendsto_order.mp hU).2 ((β+ξ)*T) (by nlinarith)
-  have hm' := (tendsto_order.mp hm).2 ((1+ξ)*T) (by nlinarith)
-  have hH' := (tendsto_order.mp hHlim).2 (ξ*H) (mul_pos hξ hH)
-  filter_upwards [hU',hm',hH',hkR.eventually_ge_atTop (2/(1-ξ)),
-    hk.eventually_ge_atTop 1,eventually_ge_atTop (1:ℕ),eventually_chain_volume_small T hT M]
-    with n hUn hmn hHn hklarge hk1 hn hvol
-  have hkp : (0:ℝ) < k n := by exact_mod_cast (show 0 < k n by omega)
-  have hnp : (0:ℝ) < n := by exact_mod_cast (show 0 < n by omega)
-  have hcap : (2*n+k n*M:ℕ) ≤ 3*n := by
-    dsimp [k] at *
-    rw [Nat.mul_assoc] at hvol
-    omega
-  have hcap0 : (0:ℝ) < (2*n+k n*M:ℕ) := by exact_mod_cast (show 0 < 2*n+k n*M by omega)
-  refine ⟨?_,?_,?_,?_,hk1⟩
-  · have h := (div_lt_iff₀ hkp).mp hUn
-    nlinarith only [h]
-  · calc
-      _ ≤ Real.log (3*(n:ℝ)) := Real.log_le_log hcap0 (by exact_mod_cast hcap)
-      _ = Real.log n + Real.log 3 := by rw [Real.log_mul (by norm_num : (3:ℝ) ≠ 0) hnp.ne']; ring
-      _ ≤ _ := by have h := (div_lt_iff₀ hkp).mp hmn; nlinarith only [h]
-  · have h := (div_lt_iff₀ hkp).mp hHn
-    nlinarith only [h]
-  · have h := (div_le_iff₀ (sub_pos.mpr hξ1)).mp hklarge
-    have hh := mul_le_mul_of_nonneg_right h hH.le
-    nlinarith only [hh]
-
-end AmplificationLogEstimates
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section AsymptoticAmplificationStep
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
-
- 
-
-theorem asymptotic_amplification_step [Nonempty V] [Nonempty E]
-    (hconn : Connected G) (hqt : QuasiTransitive G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
-    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (i₀ : ι) (hw₀ : 0 < weight i₀)
-    (q : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) M L) (hL : 1 ≤ L)
-    (p : unitInterval) (hp : (p:ℝ) < 1) (d : ℕ) (hdeg : ∀ x, degree G x ≤ d)
-    (hu : 0 < q.successProbability p (root i₀))
-    (a b β ε ξ T : ℝ) (ha : 0 < a) (hb : 0 ≤ b)
-    (hε : 0 < ε) (hε1 : ε < 1/2) (hξ : 0 < ξ) (hξ1 : ξ < 1) (hT : 0 < T)
-    (hbudget : inverseCodeMass M L ≤ q.successProbability p (root i₀)*Real.exp T)
-    (A : Hilbert V →L[ℂ] Hilbert V) (hApos : 0 < ‖A‖)
-    (hH : 0 < Real.log (1/‖A‖))
-    (hA : ∀ x y, matrixCoeff A x y = (q.normalizedKernel (root i₀) p x y : ℂ)) :
-    ∀ᶠ n : ℕ in atTop, ∀ π : ℝ,
-      (n:ℝ)^(-β) ≤ π →
-      a*n*π ≤ binOrbitRow G p (root i₀) n →
-      clusterWindowProbability G p (root i₀) (n-logarithmicChainLength T n*M) (2*n) ≤ b*π →
-      ∃ q' : LocalEndpointRule G hloc (vertexOrbit G (root i₀))
-          (2*n+logarithmicChainLength T n*M) (((2*n+logarithmicChainLength T n*M:ℕ):ℝ)^(1/2+ε)),
-        0 < q'.successProbability p (root i₀) ∧
-        Real.log (1/q'.successProbability p (root i₀)) ≤ (logarithmicChainLength T n:ℝ)*(β+ξ)*T ∧
-        Real.log ((2*n+logarithmicChainLength T n*M:ℕ):ℝ) ≤ (logarithmicChainLength T n:ℝ)*(1+ξ)*T ∧
-        ∃ A' : Hilbert V →L[ℂ] Hilbert V, 0 < ‖A'‖ ∧
-          (logarithmicChainLength T n:ℝ)*(1-ξ)*Real.log (1/‖A‖) ≤ Real.log (1/‖A'‖) ∧
-          2*Real.log (1/‖A‖) ≤ Real.log (1/‖A'‖) ∧
-          (∀ x y, matrixCoeff A' x y = (q'.normalizedKernel (root i₀) p x y : ℂ)) ∧
-          1 ≤ logarithmicChainLength T n := by
-  let k := logarithmicChainLength T
-  let cap := fun n => 2*n+k n*M
-  have hcap : Tendsto cap atTop atTop :=
-    tendsto_atTop_mono (fun n => show n ≤ cap n by dsimp [cap]; omega) tendsto_id
-  have hgood := hcap.eventually (bad_goodness_stretched_exp G hloc hqt p hp ε hε hε1)
-  have hdecay := eventually_amplification_goodness_decay T hT M d c a b β ε ha hb hε
-  have hlogs := eventually_amplification_log_estimates T ξ β (Real.log (1/‖A‖))
-    hT hξ hξ1 hH M d c a b ha hb
-  filter_upwards [hgood,hdecay,hlogs,eventually_ge_atTop (1:ℕ)] with n hgood hdecay hlogs hn
-  intro π hπ hbin hwindow
-  have hn0 : 0 < n := by omega
-  have hnR : (0:ℝ) < n := Nat.cast_pos.mpr hn0
-  have hπ0 : 0 < π := (Real.rpow_pos_of_pos hnR (-β)).trans_le hπ
-  have hQ : 0 < amplificationMomentFactor (k n) M d c a b := amplificationMomentFactor_pos _ _ _ _ _ _ ha hb
-  have hcoden : inverseCodeMass M L^(k n) ≤ n*q.successProbability p (root i₀)^(k n) := by
-    apply code_bound_of_logarithmic_chain _ _ _ hu _ hT hbudget n hn
-    unfold inverseCodeMass
-    exact add_nonneg zero_le_one (Finset.sum_nonneg (fun l _ => gapWeight_nonneg L (by linarith) l))
-  have hbad : 16*amplificationMomentFactor (k n) M d c a b *
-      (law p).real {ω | CappedBadGoodness G ω (root i₀) (cap n) ((cap n:ℝ)^(1/2+ε))} ≤ a*π := by
-    calc
-      _ ≤ 16*amplificationMomentFactor (k n) M d c a b * Real.exp (-(cap n:ℝ)^ε) :=
-        mul_le_mul_of_nonneg_left (hgood (root i₀) p le_rfl) (by positivity)
-      _ ≤ a*(n:ℝ)^(-β) := hdecay
-      _ ≤ _ := mul_le_mul_of_nonneg_left hπ ha.le
-  obtain ⟨q',hup,A',hAp,hAr,hA'⟩ := q.quantitative_amplification_step hconn root weight Γ hΓ htr hcov
-    hw hw1 c hc hmin i₀ hw₀ hL p (k n) n d hn0 hdeg hu a b π ((cap n:ℝ)^(1/2+ε))
-    ha hb hπ0 hbin hwindow hcoden hbad A hA
-  have hup0 : 0 < q'.successProbability p (root i₀) := (by positivity : 0 < a*π/(8*amplificationMomentFactor (k n) M d c a b)).trans_le hup
-  have hulo : a*(n:ℝ)^(-β)/(8*amplificationMomentFactor (k n) M d c a b) ≤ q'.successProbability p (root i₀) := by
-    apply le_trans _ hup
-    exact div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_left hπ ha.le) (by positivity)
-  have hulo' := log_inv_success_le a (amplificationMomentFactor (k n) M d c a b) (n:ℝ) β
-    (q'.successProbability p (root i₀)) ha hQ hnR hulo
-  have hH' := log_inv_norm_ge ‖A‖ ‖A'‖ (k n) hApos hAp hAr
-  exact ⟨q',hup0,hulo'.trans hlogs.1,hlogs.2.1,A',hAp,hlogs.2.2.1.trans hH',
-    hlogs.2.2.2.1.trans (hlogs.2.2.1.trans hH'),hA',hlogs.2.2.2.2⟩
-
-end LocalEndpointRule
-end AsymptoticAmplificationStep
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section WeightedScaleWindow
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem weightedWindowProbability_term_le {ι : Type*} [Fintype ι]
-    (G : BondGraph V E) (p : unitInterval) (root : ι → V) (weight : ι → ℝ)
-    (hw : ∀ i, 0 ≤ weight i) (i : ι) (a b : ℕ) :
-    weight i * clusterWindowProbability G p (root i) a b ≤ weightedWindowProbability G p root weight a b :=
-  Finset.single_le_sum (fun j _ => mul_nonneg (hw j) (clusterWindowProbability_nonneg G p (root j) a b)) (Finset.mem_univ i)
-
-theorem orbit_scale_window_bound {ι : Type*} [Fintype ι]
-    (G : BondGraph V E) (p : unitInterval) (root : ι → V) (weight : ι → ℝ)
-    (hw : ∀ i, 0 ≤ weight i) (i : ι) (hi : 0 < weight i)
-    (j k M : ℕ) (hvol : 2*k*M ≤ 2^(j+1)) (β π : ℝ)
-    (hwindow : weightedWindowProbability G p root weight (2^j) (2^(j+2)) ≤ (1+(2:ℝ)^β)*π) :
-    clusterWindowProbability G p (root i) (2^(j+1)-k*M) (2*2^(j+1)) ≤
-      ((1+(2:ℝ)^β)/weight i)*π := by
-  have he : 2^(j+1) = 2*2^j := by rw [pow_succ]; omega
-  have he' : 2*2^(j+1) = 2^(j+2) := by rw [show j+2 = (j+1)+1 by omega,pow_succ]; omega
-  have hl : 2^j ≤ 2^(j+1)-k*M := by
-    rw [he,Nat.mul_assoc] at hvol
-    rw [he]
-    omega
-  have hmono := clusterWindowProbability_mono G p (root i) hl (le_of_eq he')
-  have hterm := weightedWindowProbability_term_le G p root weight hw i (2^j) (2^(j+2))
-  have hb := (mul_le_mul_of_nonneg_left hmono hi.le).trans (hterm.trans hwindow)
-  rw [div_mul_eq_mul_div]
-  exact (le_div_iff₀ hi).mpr (by simpa only [mul_comm] using hb)
-
-theorem tendsto_dyadic_scale : Tendsto (fun j : ℕ => 2^(j+1)) atTop atTop :=
-  tendsto_atTop_mono (fun j => (Nat.le_succ j).trans (j+1).lt_two_pow_self.le) tendsto_id
-
-end WeightedScaleWindow
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section EndpointProfile
-
- 
-
-structure EndpointProfile (G : BondGraph V E) (hloc : LocallyFinite G)
-    (p : unitInterval) (o : V) (s : ℝ) where
-  cap : ℕ
-  rule : LocalEndpointRule G hloc (vertexOrbit G o) cap ((cap:ℝ)^s)
-  success_pos : 0 < rule.successProbability p o
-  op : Hilbert V →L[ℂ] Hilbert V
-  norm_pos : 0 < ‖op‖
-  coeff : ∀ x y, matrixCoeff op x y = (rule.normalizedKernel o p x y : ℂ)
-
-namespace EndpointProfile
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {p : unitInterval} {o : V} {s : ℝ}
-
-noncomputable def U (q : EndpointProfile G hloc p o s) : ℝ := Real.log (1/q.rule.successProbability p o)
-noncomputable def m (q : EndpointProfile G hloc p o s) : ℝ := Real.log q.cap
-noncomputable def H (q : EndpointProfile G hloc p o s) : ℝ := Real.log (1/‖q.op‖)
-noncomputable def cost (q : EndpointProfile G hloc p o s) (ξ : ℝ) : ℝ := q.U+(s+ξ)*q.m
-
-theorem U_nonneg (q : EndpointProfile G hloc p o s) : 0 ≤ q.U := by
-  unfold U
-  rw [one_div,Real.log_inv]
-  have h := Real.log_nonpos q.success_pos.le (q.rule.successProbability_le_one p o)
-  linarith
-
-theorem m_pos (q : EndpointProfile G hloc p o s) (hcap : 2 ≤ q.cap) : 0 < q.m := by
-  exact Real.log_pos (by exact_mod_cast (show 1 < q.cap by omega))
-
-theorem cost_pos (q : EndpointProfile G hloc p o s) (hcap : 2 ≤ q.cap)
-    (ξ : ℝ) (hsξ : 0 < s+ξ) : 0 < q.cost ξ := by
-  exact add_pos_of_nonneg_of_pos q.U_nonneg (mul_pos hsξ (q.m_pos hcap))
-
-theorem exp_cost_identity (q : EndpointProfile G hloc p o s) (hcap : 0 < q.cap) (ξ : ℝ) :
-    q.rule.successProbability p o * Real.exp (q.cost ξ) = (q.cap:ℝ)^(s+ξ) := by
-  have hcp : (0:ℝ) < q.cap := Nat.cast_pos.mpr hcap
-  simp only [cost,U,m,Real.exp_add,Real.exp_log (one_div_pos.mpr q.success_pos)]
-  calc
-    _ = Real.exp ((s+ξ)*Real.log q.cap) := by field_simp [ne_of_gt q.success_pos]
-    _ = _ := by rw [Real.rpow_def_of_pos hcp]; congr 1; ring
-
-end EndpointProfile
-
-variable [Countable V] [Countable E] [Nonempty V]
-
-theorem starting_endpoint_profile (G : BondGraph V E) (hloc : LocallyFinite G)
-    (hqt : QuasiTransitive G) (hh : 0 < hV G)
-    (p : unitInterval) (hp : 0 < (p:ℝ)) (hp1 : (p:ℝ) < 1)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
-    (i₀ : ι) (hw₀ : 0 < weight i₀)
-    (hfin : ∀ᵐ ω ∂law p, (cluster G ω (root i₀)).Finite)
-    (ε : ℝ) (hε : 0 < ε) (hε1 : ε < 1/2) (M₀ : ℕ) :
-    ∃ q : EndpointProfile G hloc p (root i₀) (1/2+ε), M₀ ≤ q.cap ∧ 1 < q.H := by
-  obtain ⟨M,hM,q,hq,A,hAp,hAn,hA⟩ := nonamenable_starting_endpoint_rule G hloc hqt hh p hp hp1
-    root weight Γ hΓ htr hw hw1 i₀ hw₀ hfin ε hε hε1 (Real.exp (-2)) (Real.exp_pos _) M₀
-  refine ⟨⟨M,q,hq,A,hAp,hA⟩,hM,?_⟩
-  have h := Real.log_lt_log hAp hAn
-  rw [Real.log_exp] at h
-  change 1 < Real.log (1/‖A‖)
-  rw [one_div,Real.log_inv]
-  linarith
-
-end EndpointProfile
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section EndpointProfileStep
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-namespace EndpointProfile
-variable {G : BondGraph V E} {hloc : LocallyFinite G}
-
-theorem amplify_at_scales [Nonempty V] [Nonempty E]
-    (hconn : Connected G) (hqt : QuasiTransitive G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
-    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (i : ι) (p : unitInterval) (hp : (p:ℝ) < 1)
-    (d : ℕ) (hdeg : ∀ x, degree G x ≤ d)
-    (ε ξ β : ℝ) (hε : 0 < ε) (hε1 : ε < 1/2) (hξ : 0 < ξ) (hξ1 : ξ < 1)
-    (hscales : ∀ N, ∃ j, N ≤ j ∧
-      ((2^(j+1):ℕ):ℝ)^(-β) ≤ dyadicBinProbability G p root weight (j+1) ∧
-      weightedWindowProbability G p root weight (2^j) (2^(j+2)) ≤
-        (1+(2:ℝ)^β)*dyadicBinProbability G p root weight (j+1) ∧
-      (((2^(j+1):ℕ):ℝ)/Fintype.card ι)*dyadicBinProbability G p root weight (j+1) ≤
-        binOrbitRow G p (root i) (2^(j+1)))
-    (q : EndpointProfile G hloc p (root i) (1/2+ε)) (hM : 2 ≤ q.cap)
-    (hL : 1 ≤ (q.cap:ℝ)^(1/2+ε))
-    (hbudget : inverseCodeMass q.cap ((q.cap:ℝ)^(1/2+ε)) ≤ (q.cap:ℝ)^(1/2+ε+ξ))
-    (hH : 0 < q.H) (N : ℕ) :
-    ∃ (q' : EndpointProfile G hloc p (root i) (1/2+ε)) (k : ℕ),
-      N ≤ q'.cap ∧ 1 ≤ k ∧
-      q'.U ≤ (k:ℝ)*(β+ξ)*q.cost ξ ∧
-      q'.m ≤ (k:ℝ)*(1+ξ)*q.cost ξ ∧
-      (k:ℝ)*(1-ξ)*q.H ≤ q'.H ∧ 2*q.H ≤ q'.H := by
-  let a : ℝ := 1/Fintype.card ι
-  let b : ℝ := (1+(2:ℝ)^β)/weight i
-  let T := q.cost ξ
-  have ht : 0 < T := q.cost_pos hM ξ (by linarith)
-  have hcard : (0:ℝ) < Fintype.card ι := by
-    exact_mod_cast (Fintype.card_pos_iff.mpr ⟨i⟩)
-  have ha : 0 < a := one_div_pos.mpr hcard
-  have hb : 0 ≤ b := div_nonneg (by positivity) (hw i).le
-  have hbudget' : inverseCodeMass q.cap ((q.cap:ℝ)^(1/2+ε)) ≤
-      q.rule.successProbability p (root i)*Real.exp T := by
-    rw [q.exp_cost_identity (by omega) ξ]
-    exact hbudget
-  have hstep := q.rule.asymptotic_amplification_step hconn hqt root weight Γ hΓ htr hcov
-    (fun i => (hw i).le) hw1 c hc hmin i (hw i) hL p hp d hdeg q.success_pos
-    a b β ε ξ T ha hb hε hε1 hξ hξ1 ht hbudget' q.op q.norm_pos hH q.coeff
-  have hvol := eventually_chain_volume_small T ht q.cap
-  have he := tendsto_dyadic_scale.eventually (hstep.and (hvol.and (eventually_ge_atTop N)))
-  obtain ⟨J,hJ⟩ := eventually_atTop.1 he
-  obtain ⟨j,hj,hπ,hwide,hbin⟩ := hscales J
-  obtain ⟨hs,hvol,hN⟩ := hJ j hj
-  have hwin := orbit_scale_window_bound G p root weight (fun i => (hw i).le) i (hw i)
-    j (logarithmicChainLength T (2^(j+1))) q.cap hvol β _ hwide
-  have hbin' : a*(2^(j+1):ℕ)*dyadicBinProbability G p root weight (j+1) ≤
-      binOrbitRow G p (root i) (2^(j+1)) := by
-    dsimp [a]
-    simpa only [div_eq_mul_inv,one_mul,mul_comm (_⁻¹)] using hbin
-  obtain ⟨q',hup,hU,hm,A',hAp,hH',hH'',hA',hk⟩ := hs _ hπ hbin' hwin
-  refine ⟨⟨_,q',hup,A',hAp,hA'⟩,_,?_,hk,hU,hm,hH',hH''⟩
-  change N ≤ 2*2^(j+1)+logarithmicChainLength T (2^(j+1))*q.cap
-  omega
-
-end EndpointProfile
-end EndpointProfileStep
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open Filter
-section AmplificationContraction
-
-noncomputable def amplificationContraction (β s ξ : ℝ) : ℝ :=
-  (β+ξ+(s+ξ)*(1+ξ))/(1-ξ)
-
-theorem exists_amplification_contraction (β s : ℝ) (hβ : 0 ≤ β) (hs : 0 ≤ s)
-    (hsmall : β+s < 1) : ∃ ξ : ℝ, 0 < ξ ∧ ξ < 1 ∧
-      0 ≤ amplificationContraction β s ξ ∧ amplificationContraction β s ξ < 1 := by
-  have hd : 0 < 4+s := by linarith
-  have hδ : 0 < (1-(β+s))/(4+s) := div_pos (by linarith) hd
-  obtain ⟨ξ,hξ,hξupper⟩ := exists_between (lt_min zero_lt_one hδ)
-  have hξ1 := hξupper.trans_le (min_le_left _ _)
-  have hprod : ξ*(4+s) < 1-(β+s) := (lt_div_iff₀ hd).mp (hξupper.trans_le (min_le_right _ _))
-  have hx2 : ξ^2 < ξ := by nlinarith [mul_pos hξ (sub_pos.mpr hξ1)]
-  refine ⟨ξ,hξ,hξ1,?_,?_⟩
-  · unfold amplificationContraction
-    positivity
-  · unfold amplificationContraction
-    apply (div_lt_one (by linarith : 0 < 1-ξ)).mpr
-    nlinarith
-
- 
-
-theorem amplification_cost_contraction (t t' H H' β s ξ : ℝ) (k : ℕ)
-    (ht : 0 ≤ t) (hH : 0 < H) (hξ1 : ξ < 1) (hk : 1 ≤ k)
-    (hC : 0 ≤ amplificationContraction β s ξ)
-    (ht' : t' ≤ (k:ℝ)*(β+ξ+(s+ξ)*(1+ξ))*t)
-    (hH' : (k:ℝ)*(1-ξ)*H ≤ H') :
-    t'/H' ≤ amplificationContraction β s ξ*(t/H) := by
-  have hk0 : (0:ℝ) < k := by exact_mod_cast (show 0 < k by omega)
-  have hd : 0 < (k:ℝ)*(1-ξ)*H := mul_pos (mul_pos hk0 (by linarith)) hH
-  have hH'p := hd.trans_le hH'
-  apply (div_le_iff₀ hH'p).mpr
-  have he : (k:ℝ)*(β+ξ+(s+ξ)*(1+ξ))*t =
-      amplificationContraction β s ξ*(t/H)*((k:ℝ)*(1-ξ)*H) := by
-    unfold amplificationContraction
-    field_simp [hH.ne', (sub_pos.mpr hξ1).ne']
-  rw [he] at ht'
-  exact ht'.trans (mul_le_mul_of_nonneg_left hH' (mul_nonneg hC (div_nonneg ht hH.le)))
-
-namespace EndpointProfile
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {p : unitInterval} {o : V} {s : ℝ}
-
-theorem cost_contraction (q q' : EndpointProfile G hloc p o s) (β ξ : ℝ) (k : ℕ)
-    (hM : 2 ≤ q.cap) (hsξ : 0 < s+ξ) (hH : 0 < q.H) (hξ1 : ξ < 1) (hk : 1 ≤ k)
-    (hC : 0 ≤ amplificationContraction β s ξ)
-    (hU : q'.U ≤ (k:ℝ)*(β+ξ)*q.cost ξ)
-    (hm : q'.m ≤ (k:ℝ)*(1+ξ)*q.cost ξ)
-    (hH' : (k:ℝ)*(1-ξ)*q.H ≤ q'.H) :
-    q'.cost ξ/q'.H ≤ amplificationContraction β s ξ*(q.cost ξ/q.H) := by
-  apply amplification_cost_contraction _ _ _ _ β s ξ k (q.cost_pos hM ξ hsξ).le hH hξ1 hk hC _ hH'
-  unfold cost at *
-  have hh := mul_le_mul_of_nonneg_left hm hsξ.le
-  nlinarith only [hU,hh]
-
-theorem total_cost_le (q : EndpointProfile G hloc p o s) (hM : 2 ≤ q.cap)
-    (ξ : ℝ) (hsξ : 0 < s+ξ) : q.U+q.m ≤ (1+1/(s+ξ))*q.cost ξ := by
-  have hm := (q.m_pos hM).le
-  have hU := q.U_nonneg
-  have hU' : q.U ≤ q.cost ξ := by unfold cost; nlinarith
-  have hm' : q.m ≤ q.cost ξ/(s+ξ) := by
-    apply (le_div_iff₀ hsξ).mpr
-    unfold cost
-    nlinarith
-  calc
-    _ ≤ q.cost ξ+q.cost ξ/(s+ξ) := add_le_add hU' hm'
-    _ = _ := by ring
-
-end EndpointProfile
-end AmplificationContraction
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Kernel.Connections
+
+namespace OAI
+
+/-! Scale Budget for bond percolation. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section AveragedBinOrbitTransport
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem integral_tsum_of_finite_support {α Ω : Type*} [MeasurableSpace Ω]
+    (μ : Measure Ω) (S : Finset α) (f : Ω → α → ℝ)
+    (hi : ∀ x, Integrable (fun ω => f ω x) μ)
+    (hs : ∀ ω x, x ∉ S → f ω x = 0) :
+    (∫ ω, ∑' x, f ω x ∂μ) = ∑' x, ∫ ω, f ω x ∂μ := by
+  have he (ω) : (∑' x, f ω x) = ∑ x ∈ S, f ω x := tsum_eq_sum (hs ω)
+  rw [show (∫ ω, ∑' x, f ω x ∂μ) = ∫ ω, ∑ x ∈ S, f ω x ∂μ by simp only [he],
+    integral_finsetSum _ (fun x _ => hi x),tsum_eq_sum (s := S) (fun x hx => by simp only [hs _ x hx,integral_zero])]
+
+theorem binOrbitTransportMass_local (G : BondGraph V E) (hloc : LocallyFinite G)
+    (n : ℕ) (ω : Set E) (x y : V) :
+    let Qx := incidentSupport G hloc (finite_reachBall hloc x (2*n)).toFinset
+    let Qy := incidentSupport G hloc (finite_reachBall hloc y (2*n)).toFinset
+    binOrbitTransportMass G n ω x y = binOrbitTransportMass G n (ω ∩ ((Qx∪Qy : Finset E) : Set E)) x y := by
+  let Qx := incidentSupport G hloc (finite_reachBall hloc x (2*n)).toFinset
+  let Qy := incidentSupport G hloc (finite_reachBall hloc y (2*n)).toFinset
+  have hB := finite_local_of_subset Qx (Qx∪Qy) Finset.subset_union_left
+    (fun ω => binConnectionMass G n ω x y) (fun ω => binConnectionMass_local G hloc n ω x y) ω
+  have hq := finite_local_of_subset Qy (Qx∪Qy) Finset.subset_union_right
+    (fun ω => binOrbitRowMass G n ω y) (fun ω => binOrbitRowMass_local G hloc n ω y) ω
+  have hs := finite_local_of_subset Qx (Qx∪Qy) Finset.subset_union_left
+    (fun ω => binSizeMass G n ω x) (fun ω => binSizeMass_local G hloc n ω x) ω
+  exact congrArg₂ (· / ·) (congrArg₂ (· * ·) hB hq) hs
+
+theorem binOrbitTransportMass_integrable (G : BondGraph V E) (hloc : LocallyFinite G)
+    (p : unitInterval) (n : ℕ) (x y : V) : Integrable (fun ω => binOrbitTransportMass G n ω x y) (law p) :=
+  integrable_finite_local _ p _ (fun ω => binOrbitTransportMass_local G hloc n ω x y)
+
+theorem binOrbitTransportMass_outside (G : BondGraph V E) (n : ℕ) (ω : Set E) (x y : V)
+    (hy : y ∉ reachBall G x (2*n)) : binOrbitTransportMass G n ω x y = 0 := by
+  simp only [binOrbitTransportMass,binConnectionMass_outside G n ω x y hy,zero_mul,zero_div]
+
+noncomputable def binOrbitTransport (G : BondGraph V E) (p : unitInterval) (n : ℕ) (x y : V) : ℝ :=
+  ∫ ω, binOrbitTransportMass G n ω x y ∂law p
+
+theorem binOrbitTransport_covariant (G : BondGraph V E) (hloc : LocallyFinite G)
+    (p : unitInterval) {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (n : ℕ) (x y : V) :
+    binOrbitTransport G p n (a x) (a y) = binOrbitTransport G p n x y := by
+  have hm : Measurable (fun ω => binOrbitTransportMass G n ω x y) :=
+    measurable_finite_local _ _ (fun ω => binOrbitTransportMass_local G hloc n ω x y)
+  have h := integral_map (μ := law p) (measurable_set_preimage b).aemeasurable hm.aestronglyMeasurable
+  rw [law_map_preimage_equiv p b] at h
+  simpa only [binOrbitTransport,binOrbitTransportMass,binConnectionMass_covariant G hab,
+    binOrbitRowMass_covariant G n _ hab,binSizeMass_covariant G n _ hab] using h.symm
+
+theorem binOrbitTransport_outside (G : BondGraph V E) (p : unitInterval) (n : ℕ) (x y : V)
+    (hy : y ∉ reachBall G x (2*n)) : binOrbitTransport G p n x y = 0 := by
+  simp only [binOrbitTransport,binOrbitTransportMass_outside G n _ x y hy,integral_zero]
+
+theorem binOrbitTransportMass_outside_column (G : BondGraph V E) (n : ℕ) (ω : Set E) (x y : V)
+    (hx : x ∉ reachBall G y (2*n)) : binOrbitTransportMass G n ω x y = 0 := by
+  have hB : binConnectionMass G n ω x y = 0 := by
+    rw [binConnectionMass_symm]
+    exact binConnectionMass_outside G n ω y x hx
+  simp only [binOrbitTransportMass,hB,zero_mul,zero_div]
+
+theorem binOrbitTransport_row_integral (G : BondGraph V E) (hloc : LocallyFinite G)
+    (p : unitInterval) (n : ℕ) (x : V) :
+    (∑' y, binOrbitTransport G p n x y) = ∫ ω, ∑' y, binOrbitTransportMass G n ω x y ∂law p := by
+  symm
+  apply integral_tsum_of_finite_support (law p) (finite_reachBall hloc x (2*n)).toFinset _
+    (binOrbitTransportMass_integrable G hloc p n x)
+  intro ω y hy
+  exact binOrbitTransportMass_outside G n ω x y (by simpa only [Set.Finite.mem_toFinset] using hy)
+
+theorem integral_binOrbitRowMass (G : BondGraph V E) (hloc : LocallyFinite G)
+    (p : unitInterval) (n : ℕ) (x : V) :
+    (∫ ω, binOrbitRowMass G n ω x ∂law p) = binOrbitRow G p x n := by
+  have hx : x ∈ vertexOrbit G x := ⟨Equiv.refl _,Equiv.refl _,fun _ => by simp,rfl⟩
+  unfold binOrbitRowMass
+  rw [integral_tsum_of_finite_support (law p) (finite_reachBall hloc x (2*n)).toFinset _
+    (fun y => by
+      by_cases hy : y ∈ vertexOrbit G x
+      · simp only [ite_eq_left hy]
+        exact integrable_finite_local _ p _ (fun ω => binConnectionMass_local G hloc n ω x y)
+      · simp only [ite_eq_right hy]; exact integrable_const (0:ℝ))
+    (fun ω y hy => by
+      have hz := binConnectionMass_outside G n ω x y (by simpa only [Set.Finite.mem_toFinset] using hy)
+      simp only [hz,ite_self])]
+  apply tsum_congr
+  intro y
+  by_cases hy : y ∈ vertexOrbit G x
+  · simp only [orbitBinConnection,hx,hy,true_and,ite_true,binConnection]
+  · simp only [orbitBinConnection,hy,and_false,ite_false,integral_zero]
+
+theorem binOrbitTransport_column (G : BondGraph V E) (hloc : LocallyFinite G)
+    (p : unitInterval) (n : ℕ) (hn : 0 < n) (y : V) :
+    (∑' x, binOrbitTransport G p n x y) = binOrbitRow G p y n := by
+  rw [←integral_binOrbitRowMass G hloc p n y,←show (∫ ω, ∑' x, binOrbitTransportMass G n ω x y ∂law p) =
+    ∑' x, binOrbitTransport G p n x y from
+      integral_tsum_of_finite_support (law p) (finite_reachBall hloc y (2*n)).toFinset _
+        (fun x => binOrbitTransportMass_integrable G hloc p n x y)
+        (fun ω x hx => binOrbitTransportMass_outside_column G n ω x y (by simpa only [Set.Finite.mem_toFinset] using hx))]
+  simp only [binOrbitTransportMass_column G n hn]
+
+end AveragedBinOrbitTransport
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section WeightedOrbitBinLower
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem binOrbitTransport_row_lower (G : BondGraph V E) (hloc : LocallyFinite G)
+    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V)
+    (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i))
+    (p : unitInterval) (n : ℕ) (hn : 0 < n) (x : V) :
+    (n:ℝ)/Fintype.card ι * clusterWindowProbability G p x n (2*n) ≤
+      ∑' y, binOrbitTransport G p n x y := by
+  let B := (finite_reachBall hloc x (2*n)).toFinset
+  have he (ω) : (∑' y, binOrbitTransportMass G n ω x y) =
+      ∑ y ∈ B, binOrbitTransportMass G n ω x y := by
+    apply tsum_eq_sum
+    intro y hy
+    exact binOrbitTransportMass_outside G n ω x y (by simpa only [B,Set.Finite.mem_toFinset] using hy)
+  have hi : Integrable (fun ω => ∑' y, binOrbitTransportMass G n ω x y) (law p) := by
+    simp only [he]
+    exact integrable_finsetSum B (fun y _ => binOrbitTransportMass_integrable G hloc p n x y)
+  let W := {ω | ClusterWindow G x n (2*n) ω}
+  have hm : MeasurableSet W := measurableSet_clusterWindow G hloc x n (2*n)
+  have hl : Integrable (W.indicator (fun _ => (n:ℝ)/Fintype.card ι)) (law p) :=
+    (integrable_const _).indicator hm
+  calc
+    _ = ∫ ω, W.indicator (fun _ => (n:ℝ)/Fintype.card ι) ω ∂law p := by
+      rw [integral_indicator_const _ hm]
+      simp only [smul_eq_mul,clusterWindowProbability,W,mul_comm]
+    _ ≤ ∫ ω, ∑' y, binOrbitTransportMass G n ω x y ∂law p := by
+      apply integral_mono hl hi
+      intro ω
+      exact binOrbitTransportMass_row G root hcover n hn ω x
+    _ = _ := (binOrbitTransport_row_integral G hloc p n x).symm
+
+ 
+
+theorem weighted_binOrbitRow_lower (G : BondGraph V E) (hloc : LocallyFinite G)
+    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hw : ∀ i, 0 ≤ weight i) (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i))
+    (p : unitInterval) (n : ℕ) (hn : 0 < n) :
+    (n:ℝ)/Fintype.card ι * (∑ i, weight i * clusterWindowProbability G p (root i) n (2*n)) ≤
+      ∑ i, weight i * binOrbitRow G p (root i) n := by
+  have hr (x : V) : Summable (binOrbitTransport G p n x) := by
+    apply summable_of_ne_finset_zero (s := (finite_reachBall hloc x (2*n)).toFinset)
+    intro y hy
+    exact binOrbitTransport_outside G p n x y (by simpa only [Set.Finite.mem_toFinset] using hy)
+  have hc (y : V) : Summable (fun x => binOrbitTransport G p n x y) := by
+    apply summable_of_ne_finset_zero (s := (finite_reachBall hloc y (2*n)).toFinset)
+    intro x hx
+    have hz (ω) := binOrbitTransportMass_outside_column G n ω x y (by simpa only [Set.Finite.mem_toFinset] using hx)
+    simp only [binOrbitTransport,hz,integral_zero]
+  have ht := htr.real root weight Γ (binOrbitTransport G p n)
+    (fun a ha x y => by obtain ⟨b,hab⟩ := hΓ ha; exact binOrbitTransport_covariant G hloc p hab n x y) hr hc
+  calc
+    _ = ∑ i, weight i * ((n:ℝ)/Fintype.card ι * clusterWindowProbability G p (root i) n (2*n)) := by
+      rw [Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro i hi
+      ring
+    _ ≤ ∑ i, weight i * ∑' y, binOrbitTransport G p n (root i) y :=
+      Finset.sum_le_sum (fun i _ => mul_le_mul_of_nonneg_left (binOrbitTransport_row_lower G hloc root hcover p n hn (root i)) (hw i))
+    _ = ∑ i, weight i * ∑' x, binOrbitTransport G p n x (root i) := ht
+    _ = _ := by simp only [binOrbitTransport_column G hloc p n hn]
+
+theorem exists_binOrbitRow_lower (G : BondGraph V E) (hloc : LocallyFinite G)
+    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
+    (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i))
+    (p : unitInterval) (n : ℕ) (hn : 0 < n) :
+    ∃ i, (n:ℝ)/Fintype.card ι * (∑ j, weight j * clusterWindowProbability G p (root j) n (2*n)) ≤
+      binOrbitRow G p (root i) n := by
+  have h := weighted_binOrbitRow_lower G hloc root weight Γ hΓ htr (fun i => (hw i).le) hcover p n hn
+  by_contra! he
+  let c := (n:ℝ)/Fintype.card ι * (∑ j, weight j * clusterWindowProbability G p (root j) n (2*n))
+  have hs : (∑ i, weight i * binOrbitRow G p (root i) n) < ∑ i, weight i*c := by
+    apply Finset.sum_lt_sum (fun i _ => (mul_lt_mul_of_pos_left (he i) (hw i)).le)
+    exact ⟨Classical.arbitrary ι,Finset.mem_univ _,mul_lt_mul_of_pos_left (he _) (hw _)⟩
+  rw [←Finset.sum_mul,hw1,one_mul] at hs
+  exact (not_lt_of_ge h) hs
+
+end WeightedOrbitBinLower
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section DyadicRecords
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem summable_dyadic_of_scaled_bound (π : ℕ → ℝ) (hπ : ∀ j, 0 ≤ π j)
+    (α β C : ℝ) (hαβ : α < β)
+    (hC : ∀ j : ℕ, (2:ℝ)^(β*(j:ℝ))*π j ≤ C) :
+    Summable (fun j : ℕ => (2:ℝ)^(α*(j:ℝ))*π j) := by
+  let r := (2:ℝ)^(α-β)
+  have hr0 : 0 ≤ r := Real.rpow_nonneg (by norm_num) _
+  have hr1 : r < 1 := Real.rpow_lt_one_of_one_lt_of_neg (by norm_num) (sub_neg.mpr hαβ)
+  have hs := (summable_geometric_of_lt_one hr0 hr1).mul_left C
+  apply hs.of_nonneg_of_le (fun j => mul_nonneg (Real.rpow_nonneg (by norm_num) _) (hπ j))
+  intro j
+  have he : (2:ℝ)^(α*(j:ℝ)) = r^j * (2:ℝ)^(β*(j:ℝ)) := by
+    rw [show r^j = (2:ℝ)^((α-β)*(j:ℝ)) by rw [Real.rpow_mul (by norm_num),Real.rpow_natCast],
+      ←Real.rpow_add (by norm_num)]
+    congr 1
+    ring
+  rw [he,mul_assoc]
+  exact (mul_le_mul_of_nonneg_left (hC j) (pow_nonneg hr0 j)).trans_eq (mul_comm _ _)
+
+theorem unbounded_scaled_dyadic (π : ℕ → ℝ) (hπ : ∀ j, 0 ≤ π j)
+    (α β : ℝ) (hαβ : α < β)
+    (hdiv : ¬ Summable (fun j : ℕ => (2:ℝ)^(α*(j:ℝ))*π j)) :
+    ∀ C : ℝ, ∃ j : ℕ, C < (2:ℝ)^(β*(j:ℝ))*π j := by
+  intro C
+  by_contra! hC
+  exact hdiv (summable_dyadic_of_scaled_bound π hπ α β C hαβ hC)
+
+ 
+theorem late_record_of_unbounded (f : ℕ → ℝ) (hf : ∀ C : ℝ, ∃ j, C < f j) (N : ℕ) :
+    ∃ j, N < j ∧ 1 ≤ f j ∧ ∀ i ≤ j, f i ≤ f j := by
+  obtain ⟨a,ha,hamax⟩ := (Finset.range (N+1)).exists_max_image f ⟨0,by simp⟩
+  obtain ⟨b,hb⟩ := hf (max 1 (f a))
+  obtain ⟨j,hj,hjmax⟩ := (Finset.range (b+1)).exists_max_image f ⟨0,by simp⟩
+  have hb' : max 1 (f a) < f j := hb.trans_le (hjmax b (by simp))
+  refine ⟨j,?_,(le_max_left _ _).trans hb'.le,?_⟩
+  · by_contra! hjN
+    have he := hamax j (Finset.mem_range.mpr (by omega))
+    exact (not_lt_of_ge he) ((le_max_right _ _).trans_lt hb')
+  · intro i hi
+    exact hjmax i (Finset.mem_range.mpr (by have := Finset.mem_range.mp hj; omega))
+
+ 
+
+theorem finite_family_unbounded_choice {ι : Type*} [Fintype ι]
+    (P : ι → ℕ → Prop) (h : ∀ N, ∃ j, N ≤ j ∧ ∃ i, P i j) :
+    ∃ i, ∀ N, ∃ j, N ≤ j ∧ P i j := by
+  by_contra! hnot
+  choose L hL using hnot
+  obtain ⟨j,hj,i,hi⟩ := h (∑ i, L i)
+  have hLi : L i ≤ ∑ t, L t := Finset.single_le_sum (fun t _ => Nat.zero_le (L t)) (Finset.mem_univ i)
+  exact hL i j (hLi.trans hj) hi
+
+theorem record_dyadic_bounds (π : ℕ → ℝ) (β : ℝ) (j : ℕ)
+    (hrec : ∀ i ≤ j+1, (2:ℝ)^(β*(i:ℝ))*π i ≤ (2:ℝ)^(β*((j+1:ℕ):ℝ))*π (j+1))
+    (h1 : 1 ≤ (2:ℝ)^(β*((j+1:ℕ):ℝ))*π (j+1)) :
+    (2:ℝ)^(-β*((j+1:ℕ):ℝ)) ≤ π (j+1) ∧ π j ≤ (2:ℝ)^β*π (j+1) := by
+  have hpos : 0 < (2:ℝ)^(β*((j+1:ℕ):ℝ)) := Real.rpow_pos_of_pos (by norm_num) _
+  have hinv : (2:ℝ)^(-β*((j+1:ℕ):ℝ)) = 1/(2:ℝ)^(β*((j+1:ℕ):ℝ)) := by
+    rw [show -β*((j+1:ℕ):ℝ) = -(β*((j+1:ℕ):ℝ)) by ring,Real.rpow_neg (by norm_num)]
+    simp only [one_div]
+  refine ⟨?_,?_⟩
+  · rw [hinv]
+    exact (div_le_iff₀ hpos).mpr (by simpa only [mul_comm] using h1)
+  · have hr := hrec j (by omega)
+    have he : (2:ℝ)^(β*((j+1:ℕ):ℝ)) = (2:ℝ)^(β*(j:ℝ))*(2:ℝ)^β := by
+      rw [Nat.cast_add,Nat.cast_one,mul_add,mul_one,Real.rpow_add (by norm_num)]
+    rw [he,mul_assoc] at hr
+    exact (mul_le_mul_iff_right₀ (Real.rpow_pos_of_pos (by norm_num : (0:ℝ)<2) _)).mp hr
+
+end DyadicRecords
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section OrbitDyadicScales
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem clusterWindowProbability_split (G : BondGraph V E) (hloc : LocallyFinite G)
+    (p : unitInterval) (x : V) (a b c : ℕ) (hab : a ≤ b) (hbc : b ≤ c) :
+    clusterWindowProbability G p x a c = clusterWindowProbability G p x a b + clusterWindowProbability G p x b c := by
+  have he : {ω | ClusterWindow G x a c ω} =
+      {ω | ClusterWindow G x a b ω} ∪ {ω | ClusterWindow G x b c ω} := by
+    ext ω
+    simp only [Set.mem_ofPred_eq,Set.mem_union,ClusterWindow]
+    constructor
+    · rintro ⟨hf,ha,hc⟩
+      by_cases hb : (cluster G ω x).ncard < b
+      · exact Or.inl ⟨hf,ha,hb⟩
+      · exact Or.inr ⟨hf,by omega,hc⟩
+    · rintro (⟨hf,ha,hb⟩|⟨hf,hb,hc⟩)
+      · exact ⟨hf,ha,hb.trans_le hbc⟩
+      · exact ⟨hf,hab.trans hb,hc⟩
+  have hd : Disjoint {ω | ClusterWindow G x a b ω} {ω | ClusterWindow G x b c ω} := by
+    apply Set.disjoint_left.mpr
+    intro ω h₁ h₂
+    exact (not_lt_of_ge h₂.2.1) h₁.2.2
+  unfold clusterWindowProbability
+  rw [he,measureReal_union hd (measurableSet_clusterWindow G hloc x b c)]
+
+noncomputable def weightedWindowProbability {ι : Type*} [Fintype ι]
+    (G : BondGraph V E) (p : unitInterval) (root : ι → V) (weight : ι → ℝ) (a b : ℕ) : ℝ :=
+  ∑ i, weight i * clusterWindowProbability G p (root i) a b
+
+theorem weightedWindowProbability_nonneg {ι : Type*} [Fintype ι]
+    (G : BondGraph V E) (p : unitInterval) (root : ι → V) (weight : ι → ℝ)
+    (hw : ∀ i, 0 ≤ weight i) (a b : ℕ) :
+    0 ≤ weightedWindowProbability G p root weight a b := by
+  exact Finset.sum_nonneg (fun i _ => mul_nonneg (hw i) (clusterWindowProbability_nonneg G p (root i) a b))
+
+theorem weightedWindowProbability_split {ι : Type*} [Fintype ι]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (p : unitInterval) (root : ι → V) (weight : ι → ℝ)
+    (a b c : ℕ) (hab : a ≤ b) (hbc : b ≤ c) :
+    weightedWindowProbability G p root weight a c =
+      weightedWindowProbability G p root weight a b + weightedWindowProbability G p root weight b c := by
+  simp only [weightedWindowProbability,clusterWindowProbability_split G hloc p _ a b c hab hbc,
+    mul_add,Finset.sum_add_distrib]
+
+noncomputable def dyadicBinProbability {ι : Type*} [Fintype ι]
+    (G : BondGraph V E) (p : unitInterval) (root : ι → V) (weight : ι → ℝ) (j : ℕ) : ℝ :=
+  weightedWindowProbability G p root weight (2^j) (2^(j+1))
+
+ 
+
+theorem orbit_dyadic_scales (G : BondGraph V E) (hloc : LocallyFinite G)
+    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
+    (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i)) (p : unitInterval)
+    (α β : ℝ) (hαβ : α < β)
+    (hdiv : ¬ Summable (fun j : ℕ => (2:ℝ)^(α*(j:ℝ))*dyadicBinProbability G p root weight j)) :
+    ∃ i, ∀ N, ∃ j, N ≤ j ∧
+      ((2^(j+1):ℕ):ℝ)^(-β) ≤ dyadicBinProbability G p root weight (j+1) ∧
+      weightedWindowProbability G p root weight (2^j) (2^(j+2)) ≤
+        (1+(2:ℝ)^β)*dyadicBinProbability G p root weight (j+1) ∧
+      (((2^(j+1):ℕ):ℝ)/Fintype.card ι)*dyadicBinProbability G p root weight (j+1) ≤
+        binOrbitRow G p (root i) (2^(j+1)) := by
+  let π := dyadicBinProbability G p root weight
+  let f := fun j : ℕ => (2:ℝ)^(β*(j:ℝ))*π j
+  have hπ : ∀ j, 0 ≤ π j := fun j => weightedWindowProbability_nonneg G p root weight (fun i => (hw i).le) _ _
+  have hf := unbounded_scaled_dyadic π hπ α β hαβ hdiv
+  let P := fun i j => 1 ≤ f (j+1) ∧ (∀ t ≤ j+1, f t ≤ f (j+1)) ∧
+    (((2^(j+1):ℕ):ℝ)/Fintype.card ι)*π (j+1) ≤ binOrbitRow G p (root i) (2^(j+1))
+  have hpick : ∀ N, ∃ j, N ≤ j ∧ ∃ i, P i j := by
+    intro N
+    obtain ⟨t,ht,h1,hr⟩ := late_record_of_unbounded f hf N
+    obtain ⟨j,rfl⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : t ≠ 0)
+    obtain ⟨i,hi⟩ := exists_binOrbitRow_lower G hloc root weight Γ hΓ htr hw hw1 hcover p (2^(j+1)) (by positivity)
+    have he : 2*2^(j+1) = 2^(j+2) := by rw [show j+2 = (j+1)+1 by omega,pow_succ]; omega
+    refine ⟨j,by omega,i,h1,hr,?_⟩
+    simpa only [P,π,dyadicBinProbability,weightedWindowProbability,he] using hi
+  obtain ⟨i,hi⟩ := finite_family_unbounded_choice P hpick
+  refine ⟨i,?_⟩
+  intro N
+  obtain ⟨j,hj,h1,hr,hb⟩ := hi N
+  have hs := record_dyadic_bounds π β j hr h1
+  refine ⟨j,hj,?_,?_,hb⟩
+  · have he : (((2^(j+1):ℕ):ℝ)^(-β)) = (2:ℝ)^(-β*((j+1:ℕ):ℝ)) := by
+      rw [Nat.cast_pow,Nat.cast_ofNat,←Real.rpow_natCast,←Real.rpow_mul (by norm_num)]
+      congr 1
+      ring
+    rw [he]
+    exact hs.1
+  · rw [weightedWindowProbability_split G hloc p root weight (2^j) (2^(j+1)) (2^(j+2))
+      (Nat.pow_le_pow_right (by norm_num) (by omega)) (Nat.pow_le_pow_right (by norm_num) (by omega))]
+    change π j + π (j+1) ≤ (1+(2:ℝ)^β)*π (j+1)
+    nlinarith only [hs.2]
+
+end OrbitDyadicScales
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section ClusterSizeMomentIntegrable
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem measurable_cluster_ncard (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) :
+    Measurable (fun ω : Set E => (cluster G ω x).ncard) := by
+  have hm (k : ℕ) (hk : k ≠ 0) : MeasurableSet {ω : Set E | (cluster G ω x).ncard = k} := by
+    have he : {ω : Set E | (cluster G ω x).ncard = k} = {ω | ClusterWindow G x k (k+1) ω} := by
+      ext ω
+      constructor
+      · intro h
+        have hf := Set.finite_of_ncard_ne_zero (by rw [h]; exact hk)
+        exact ⟨hf,by rw [h],by rw [h]; omega⟩
+      · rintro ⟨hf,hl,hu⟩
+        exact le_antisymm (by omega) hl
+    rw [he]
+    exact measurableSet_clusterWindow G hloc x k (k+1)
+  have hm0 : MeasurableSet {ω : Set E | (cluster G ω x).ncard = 0} := by
+    have he : {ω : Set E | (cluster G ω x).ncard = 0} =
+        (⋃ k : ℕ, {ω : Set E | (cluster G ω x).ncard = k+1})ᶜ := by
+      ext ω
+      simp only [Set.mem_compl_iff,Set.mem_iUnion,Set.mem_ofPred_eq,not_exists]
+      constructor
+      · intro h k
+        omega
+      · intro h
+        by_contra hz
+        obtain ⟨k,hk⟩ := Nat.exists_eq_succ_of_ne_zero hz
+        exact h k hk
+    rw [he]
+    exact (MeasurableSet.iUnion (fun k => hm (k+1) (by omega))).compl
+  apply measurable_to_countable
+  intro ω
+  change MeasurableSet {ω' | (cluster G ω' x).ncard = (cluster G ω x).ncard}
+  by_cases h : (cluster G ω x).ncard = 0
+  · simpa only [h] using hm0
+  · exact hm _ h
+
+ 
+
+theorem integrable_cluster_power_of_dyadic (G : BondGraph V E) (hloc : LocallyFinite G)
+    (p : unitInterval) (x : V) (α : ℝ) (hα : 0 < α)
+    (hs : Summable (fun j : ℕ => (2:ℝ)^(α*(j:ℝ))*clusterWindowProbability G p x (2^j) (2^(j+1)))) :
+    Integrable (fun ω => ((cluster G ω x).ncard:ℝ)^α) (law p) := by
+  let f := fun ω : Set E => ((cluster G ω x).ncard:ℝ)^α
+  let F := fun j : ℕ => {ω | ClusterWindow G x (2^j) (2^(j+1)) ω}.indicator
+    (fun _ => (2:ℝ)^(α*((j+1:ℕ):ℝ)))
+  have hFm (j : ℕ) : Measurable (F j) :=
+    measurable_const.indicator (measurableSet_clusterWindow G hloc x _ _)
+  have hFi (j : ℕ) : Integrable (F j) (law p) :=
+    (integrable_const _).indicator (measurableSet_clusterWindow G hloc x _ _)
+  have hFn (j : ℕ) (ω : Set E) : 0 ≤ F j ω := by
+    exact Set.indicator_nonneg (fun _ _ => (Real.rpow_pos_of_pos (by norm_num) _).le) _
+  have hFe (j : ℕ) : (∫ ω, F j ω ∂law p) =
+      (2:ℝ)^α * ((2:ℝ)^(α*(j:ℝ))*clusterWindowProbability G p x (2^j) (2^(j+1))) := by
+    rw [integral_indicator_const _ (measurableSet_clusterWindow G hloc x _ _)]
+    simp only [smul_eq_mul,clusterWindowProbability,Nat.cast_add,Nat.cast_one,mul_add,mul_one,
+      Real.rpow_add (by norm_num : (0:ℝ)<2)]
+    ring
+  have hsum : Summable (fun j => ∫ ω, F j ω ∂law p) := by
+    simp only [hFe]
+    exact hs.mul_left _
+  have hbound (ω : Set E) : ENNReal.ofReal (f ω) ≤ ∑' j, ENNReal.ofReal (F j ω) := by
+    by_cases hz : (cluster G ω x).ncard = 0
+    · simp only [f,hz,Nat.cast_zero,Real.zero_rpow hα.ne',ENNReal.ofReal_zero]
+      exact bot_le
+    · let j := Nat.log 2 (cluster G ω x).ncard
+      have hl := Nat.pow_log_le_self 2 hz
+      have hu := Nat.lt_pow_succ_log_self (by norm_num : 1 < (2:ℕ)) (cluster G ω x).ncard
+      have hb : ClusterWindow G x (2^j) (2^(j+1)) ω :=
+        ⟨Set.finite_of_ncard_ne_zero hz,hl,hu⟩
+      have hle : f ω ≤ (2:ℝ)^(α*((j+1:ℕ):ℝ)) := by
+        have hcast : ((cluster G ω x).ncard:ℝ) ≤ ((2^(j+1):ℕ):ℝ) := by exact_mod_cast hu.le
+        have h := Real.rpow_le_rpow (Nat.cast_nonneg _) hcast hα.le
+        have he : (((2^(j+1):ℕ):ℝ)^α) = (2:ℝ)^(α*((j+1:ℕ):ℝ)) := by
+          rw [Nat.cast_pow,Nat.cast_ofNat,←Real.rpow_natCast,←Real.rpow_mul (by norm_num)]
+          congr 1
+          ring
+        exact h.trans_eq he
+      have he : F j ω = (2:ℝ)^(α*((j+1:ℕ):ℝ)) := Set.indicator_of_mem (show ω ∈ {ω | ClusterWindow G x (2^j) (2^(j+1)) ω} from hb) _
+      exact (ENNReal.ofReal_le_ofReal (hle.trans_eq he.symm)).trans
+        (ENNReal.le_tsum (f := fun scale => ENNReal.ofReal (F scale ω)) j)
+  have hfn (ω : Set E) : 0 ≤ f ω := Real.rpow_nonneg (Nat.cast_nonneg _) _
+  refine ⟨((measurable_of_countable (fun m : ℕ => (m:ℝ)^α)).comp (measurable_cluster_ncard G hloc x)).aestronglyMeasurable,?_⟩
+  apply (hasFiniteIntegral_iff_ofReal (Filter.Eventually.of_forall hfn)).mpr
+  have hlin : (∫⁻ ω, ∑' j, ENNReal.ofReal (F j ω) ∂law p) < ⊤ := by
+    rw [lintegral_tsum (fun j => (hFm j).ennreal_ofReal.aemeasurable)]
+    simp_rw [←ofReal_integral_eq_lintegral_ofReal (hFi _) (Filter.Eventually.of_forall (hFn _))]
+    rw [←ENNReal.ofReal_tsum_of_nonneg (fun j => integral_nonneg (hFn j)) hsum]
+    exact ENNReal.ofReal_lt_top
+  exact (lintegral_mono hbound).trans_lt hlin
+
+end ClusterSizeMomentIntegrable
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section WeightedMomentScales
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem integrable_cluster_power_of_weighted_dyadic (G : BondGraph V E)
+    (hloc : LocallyFinite G) {ι : Type*} [Fintype ι] (p : unitInterval)
+    (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 < weight i)
+    (α : ℝ) (hα : 0 < α)
+    (hs : Summable (fun j : ℕ => (2:ℝ)^(α*(j:ℝ))*dyadicBinProbability G p root weight j))
+    (i : ι) : Integrable (fun ω => ((cluster G ω (root i)).ncard:ℝ)^α) (law p) := by
+  have hle (j : ℕ) : weight i * clusterWindowProbability G p (root i) (2^j) (2^(j+1)) ≤
+      dyadicBinProbability G p root weight j := by
+    exact Finset.single_le_sum
+      (fun t _ => mul_nonneg (hw t).le (clusterWindowProbability_nonneg G p (root t) _ _))
+      (Finset.mem_univ i)
+  have hs' : Summable (fun j : ℕ => weight i *
+      ((2:ℝ)^(α*(j:ℝ))*clusterWindowProbability G p (root i) (2^j) (2^(j+1)))) := by
+    apply hs.of_nonneg_of_le
+    · intro j
+      exact mul_nonneg (hw i).le (mul_nonneg (Real.rpow_nonneg (by norm_num) _)
+        (clusterWindowProbability_nonneg G p (root i) _ _))
+    · intro j
+      have h := mul_le_mul_of_nonneg_left (hle j) (Real.rpow_nonneg (by norm_num : (0:ℝ) ≤ 2) (α*(j:ℝ)))
+      nlinarith only [h]
+  apply integrable_cluster_power_of_dyadic G hloc p (root i) α hα
+  have he : (fun j : ℕ => (2:ℝ)^(α*(j:ℝ))*clusterWindowProbability G p (root i) (2^j) (2^(j+1))) =
+      (fun j : ℕ => (weight i)⁻¹ * (weight i *((2:ℝ)^(α*(j:ℝ))*clusterWindowProbability G p (root i) (2^j) (2^(j+1))))) := by
+    funext j
+    rw [←mul_assoc,inv_mul_cancel₀ (hw i).ne',one_mul]
+  rw [he]
+  exact hs'.mul_left _
+
+ 
+
+theorem orbit_scales_of_divergent_moment (G : BondGraph V E) (hloc : LocallyFinite G)
+    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
+    (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i)) (p : unitInterval)
+    (α β : ℝ) (hα : 0 < α) (hαβ : α < β)
+    (hdiv : ∃ i, ¬ Integrable (fun ω => ((cluster G ω (root i)).ncard:ℝ)^α) (law p)) :
+    ∃ i, ∀ N, ∃ j, N ≤ j ∧
+      ((2^(j+1):ℕ):ℝ)^(-β) ≤ dyadicBinProbability G p root weight (j+1) ∧
+      weightedWindowProbability G p root weight (2^j) (2^(j+2)) ≤
+        (1+(2:ℝ)^β)*dyadicBinProbability G p root weight (j+1) ∧
+      (((2^(j+1):ℕ):ℝ)/Fintype.card ι)*dyadicBinProbability G p root weight (j+1) ≤
+        binOrbitRow G p (root i) (2^(j+1)) := by
+  apply orbit_dyadic_scales G hloc root weight Γ hΓ htr hw hw1 hcover p α β hαβ
+  intro hs
+  obtain ⟨i,hi⟩ := hdiv
+  exact hi (integrable_cluster_power_of_weighted_dyadic G hloc p root weight hw α hα hs i)
+
+end WeightedMomentScales
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open Filter
+section LogarithmicChainLength
+
+noncomputable def logarithmicChainLength (t : ℝ) (n : ℕ) : ℕ := ⌊Real.log n / t⌋₊
+
+theorem logarithmicChainLength_bounds (t : ℝ) (ht : 0 < t) (n : ℕ) (hn : 1 ≤ n) :
+    (logarithmicChainLength t n:ℝ)*t ≤ Real.log n ∧
+      Real.log n < ((logarithmicChainLength t n:ℝ)+1)*t := by
+  have hn' : (1:ℝ) ≤ n := by exact_mod_cast hn
+  have hlog := Real.log_nonneg hn'
+  constructor
+  · exact (le_div_iff₀ ht).mp (Nat.floor_le (div_nonneg hlog ht.le))
+  · exact (div_lt_iff₀ ht).mp (Nat.lt_floor_add_one (Real.log n/t))
+
+theorem tendsto_logarithmicChainLength (t : ℝ) (ht : 0 < t) :
+    Tendsto (logarithmicChainLength t) atTop atTop :=
+  tendsto_nat_floor_atTop.comp ((Real.tendsto_log_atTop.comp tendsto_natCast_atTop_atTop).atTop_div_const ht)
+
+theorem tendsto_log_div_chainLength (t : ℝ) (ht : 0 < t) :
+    Tendsto (fun n : ℕ => Real.log n / logarithmicChainLength t n) atTop (nhds t) := by
+  have hk : Tendsto (fun n : ℕ => (logarithmicChainLength t n:ℝ)) atTop atTop :=
+    tendsto_natCast_atTop_atTop.comp (tendsto_logarithmicChainLength t ht)
+  have hlim : Tendsto (fun n : ℕ => t + t/(logarithmicChainLength t n:ℝ)) atTop (nhds t) := by
+    simpa only [add_zero] using tendsto_const_nhds.add (tendsto_const_nhds.div_atTop hk)
+  apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hlim
+  · filter_upwards [eventually_ge_atTop (1:ℕ),hk.eventually_gt_atTop 0] with n hn hk0
+    exact (le_div_iff₀ hk0).mpr (by simpa only [mul_comm] using (logarithmicChainLength_bounds t ht n hn).1)
+  · filter_upwards [eventually_ge_atTop (1:ℕ),hk.eventually_gt_atTop 0] with n hn hk0
+    have h := (logarithmicChainLength_bounds t ht n hn).2.le
+    apply (div_le_iff₀ hk0).mpr
+    calc
+      _ ≤ ((logarithmicChainLength t n:ℝ)+1)*t := h
+      _ = _ := by field_simp
+
+theorem tendsto_chainLength_div_scale (t : ℝ) (ht : 0 < t) :
+    Tendsto (fun n : ℕ => (logarithmicChainLength t n:ℝ)/n) atTop (nhds 0) := by
+  have hlog : Tendsto (fun n : ℕ => (Real.log n/(n:ℝ))/t) atTop (nhds 0) := by
+    simpa only [zero_div,Function.comp_def,id_eq] using
+      (Real.isLittleO_log_id_atTop.tendsto_div_nhds_zero.comp tendsto_natCast_atTop_atTop).div_const t
+  apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hlog
+  · exact Eventually.of_forall (fun n => div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _))
+  · filter_upwards [eventually_ge_atTop (1:ℕ)] with n hn
+    have hn' : (0:ℝ) < n := by exact_mod_cast (show 0 < n by omega)
+    apply (le_div_iff₀ ht).mpr
+    rw [div_mul_eq_mul_div]
+    exact (div_le_div_iff_of_pos_right hn').mpr (logarithmicChainLength_bounds t ht n hn).1
+
+theorem eventually_chain_volume_small (t : ℝ) (ht : 0 < t) (M : ℕ) :
+    ∀ᶠ n : ℕ in atTop, 2*logarithmicChainLength t n*M ≤ n := by
+  have hlim : Tendsto (fun n : ℕ => (2*(logarithmicChainLength t n:ℝ)*M)/(n:ℝ)) atTop (nhds 0) := by
+    have h := (tendsto_chainLength_div_scale t ht).const_mul (2*(M:ℝ))
+    simpa only [mul_zero] using h.congr (fun n => by ring)
+  filter_upwards [(tendsto_order.mp hlim).2 1 (by norm_num),eventually_ge_atTop (1:ℕ)] with n hn hn1
+  have hn' : (0:ℝ) < n := by exact_mod_cast (show 0 < n by omega)
+  have h := (div_lt_iff₀ hn').mp hn
+  norm_num only [one_mul] at h
+  exact_mod_cast h.le
+
+end LogarithmicChainLength
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open Filter
+section AmplificationPolynomial
+
+theorem amplificationMomentFactor_ge_four (k M d : ℕ) (c a t : ℝ) (ha : 0 < a) (ht : 0 ≤ t) :
+    4 ≤ amplificationMomentFactor k M d c a t := by
+  unfold amplificationMomentFactor
+  have h1 : (0:ℝ) ≤ ((k*M)+(k*M)^2+d*(k*M)^3:ℕ) := Nat.cast_nonneg _
+  have h2 : (0:ℝ) ≤ (2*c*(k*M:ℕ))^2*t/a := by positivity
+  linarith
+
+theorem amplificationMomentFactor_cubic (k M d : ℕ) (c a t : ℝ) (ha : 0 < a) (ht : 0 ≤ t) :
+    amplificationMomentFactor k M d c a t ≤ amplificationMomentFactor 1 M d c a t * ((k:ℝ)+1)^3 := by
+  let b := (2*c*(M:ℝ))^2*t/a
+  have hb : 0 ≤ b := by dsimp [b]; positivity
+  have hk : (0:ℝ) ≤ k := Nat.cast_nonneg _
+  have hk1 : (1:ℝ) ≤ (k:ℝ)+1 := by linarith
+  have h13 : (1:ℝ) ≤ ((k:ℝ)+1)^3 := one_le_pow₀ hk1
+  have h1 : (k:ℝ) ≤ ((k:ℝ)+1)^3 := by
+    calc
+      _ ≤ (k:ℝ)+1 := by linarith
+      _ = ((k:ℝ)+1)^1 := (pow_one _).symm
+      _ ≤ _ := pow_le_pow_right₀ hk1 (by omega)
+  have h2 : (k:ℝ)^2 ≤ ((k:ℝ)+1)^3 :=
+    (pow_le_pow_left₀ hk (by linarith : (k:ℝ) ≤ (k:ℝ)+1) 2).trans (pow_le_pow_right₀ hk1 (by omega))
+  have h3 : (k:ℝ)^3 ≤ ((k:ℝ)+1)^3 := pow_le_pow_left₀ hk (by linarith) 3
+  have he (j : ℕ) : amplificationMomentFactor j M d c a t =
+      4 + (M:ℝ)*j + (M:ℝ)^2*(j:ℝ)^2 + ((d:ℝ)*(M:ℝ)^3)*(j:ℝ)^3 + b*(j:ℝ)^2 := by
+    simp only [amplificationMomentFactor,Nat.cast_add,Nat.cast_mul,Nat.cast_pow,b]
+    ring
+  rw [he k,he 1]
+  norm_num only [Nat.cast_one,one_pow,mul_one]
+  have hh0 := mul_le_mul_of_nonneg_left h13 (by norm_num : (0:ℝ) ≤ 4)
+  have hh1 := mul_le_mul_of_nonneg_left h1 (Nat.cast_nonneg M : (0:ℝ) ≤ M)
+  have hh2 := mul_le_mul_of_nonneg_left h2 (sq_nonneg (M:ℝ))
+  have hh3 := mul_le_mul_of_nonneg_left h3 (show 0 ≤ (d:ℝ)*(M:ℝ)^3 by positivity)
+  have hhb := mul_le_mul_of_nonneg_left h2 hb
+  nlinarith only [hh0,hh1,hh2,hh3,hhb]
+
+theorem tendsto_log_amplificationMomentFactor_div (M d : ℕ) (c a t : ℝ) (ha : 0 < a) (ht : 0 ≤ t) :
+    Tendsto (fun k : ℕ => Real.log (amplificationMomentFactor k M d c a t)/(k:ℝ)) atTop (nhds 0) := by
+  let C := amplificationMomentFactor 1 M d c a t
+  have hC : 0 < C := amplificationMomentFactor_pos _ _ _ _ _ _ ha ht
+  have hplus : Tendsto (fun k : ℕ => (k:ℝ)+1) atTop atTop := by
+    simpa only [Function.comp_def,Nat.cast_add,Nat.cast_one] using
+      (tendsto_natCast_atTop_atTop (R := ℝ)).comp (tendsto_add_atTop_nat 1)
+  have hlog : Tendsto (fun k : ℕ => Real.log ((k:ℝ)+1)/(k:ℝ)) atTop (nhds 0) := by
+    simpa only [Function.comp_def,pow_one,one_mul,add_neg_cancel_right] using
+      (Real.tendsto_pow_log_div_mul_add_atTop 1 (-1) 1 one_ne_zero).comp hplus
+  have hupper : Tendsto (fun k : ℕ => (Real.log C + 3*Real.log ((k:ℝ)+1))/(k:ℝ)) atTop (nhds 0) := by
+    have h := ((tendsto_const_nhds (x := Real.log C)).div_atTop (tendsto_natCast_atTop_atTop (R := ℝ))).add (hlog.const_mul 3)
+    simpa only [zero_add,mul_zero,add_div,mul_div_assoc] using h
+  apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hupper
+  · exact Eventually.of_forall (fun k => div_nonneg
+      (Real.log_nonneg (by have := amplificationMomentFactor_ge_four k M d c a t ha ht; linarith)) (Nat.cast_nonneg _))
+  · exact Eventually.of_forall (fun k => by
+      apply div_le_div_of_nonneg_right _ (Nat.cast_nonneg _)
+      have hh := Real.log_le_log (amplificationMomentFactor_pos k M d c a t ha ht)
+        (amplificationMomentFactor_cubic k M d c a t ha ht)
+      have hp : (0:ℝ) < (k:ℝ)+1 := by positivity
+      simpa only [C,Real.log_mul hC.ne' (pow_pos hp 3).ne',Real.log_pow,Nat.cast_ofNat] using hh)
+
+end AmplificationPolynomial
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open Filter
+section AmplificationScalarBudget
+
+theorem eventually_inverseCodeMass_budget (s ξ : ℝ) (hs : 0 ≤ s) (hξ : 0 < ξ) :
+    ∀ᶠ M : ℕ in atTop, 2 ≤ M ∧ 1 ≤ (M:ℝ)^s ∧
+      inverseCodeMass M ((M:ℝ)^s) ≤ (M:ℝ)^(s+ξ) := by
+  have hl := (isLittleO_log_rpow_atTop hξ).bound (by norm_num : (0:ℝ) < 1/8)
+  have hcast : Tendsto (fun M : ℕ => (M:ℝ)) atTop atTop := tendsto_natCast_atTop_atTop
+  filter_upwards [eventually_ge_atTop (2:ℕ),hcast.eventually hl] with M hM hl
+  have hM' : (1:ℝ) ≤ M := by exact_mod_cast (show 1 ≤ M by omega)
+  have hM0 : (0:ℝ) < M := lt_of_lt_of_le zero_lt_one hM'
+  have hL : 1 ≤ (M:ℝ)^s := Real.one_le_rpow hM' hs
+  refine ⟨hM,hL,?_⟩
+  simp only [Real.norm_eq_abs,abs_of_nonneg (Real.log_nonneg hM'),
+    abs_of_nonneg (Real.rpow_nonneg hM0.le ξ)] at hl
+  calc
+    _ ≤ 8*(M:ℝ)^s*Real.log M := inverseCodeMass_le_log M _ hM hL
+    _ ≤ (M:ℝ)^s*(M:ℝ)^ξ := by
+      have hh := mul_le_mul_of_nonneg_left hl (Real.rpow_nonneg hM0.le s)
+      nlinarith only [hh]
+    _ = _ := (Real.rpow_add hM0 _ _).symm
+
+theorem code_bound_of_logarithmic_chain (u W t : ℝ) (hu : 0 < u) (hW : 0 ≤ W)
+    (ht : 0 < t) (hbudget : W ≤ u*Real.exp t) (n : ℕ) (hn : 1 ≤ n) :
+    W^logarithmicChainLength t n ≤ (n:ℝ)*u^logarithmicChainLength t n := by
+  let k := logarithmicChainLength t n
+  have hexp : Real.exp ((k:ℝ)*t) ≤ (n:ℝ) := by
+    have h := Real.exp_le_exp.mpr (logarithmicChainLength_bounds t ht n hn).1
+    have hnp : (0:ℝ) < n := by exact_mod_cast (show 0 < n by omega)
+    simpa only [Real.exp_log hnp] using h
+  calc
+    W^k ≤ (u*Real.exp t)^k := pow_le_pow_left₀ hW hbudget _
+    _ = u^k*Real.exp ((k:ℝ)*t) := by rw [mul_pow,←Real.exp_nat_mul]
+    _ ≤ u^k*n := mul_le_mul_of_nonneg_left hexp (pow_nonneg hu.le _)
+    _ = _ := mul_comm _ _
+
+theorem tendsto_rpow_mul_stretched_exp (γ ε : ℝ) (hε : 0 < ε) :
+    Tendsto (fun x : ℝ => x^γ * Real.exp (-(x^ε))) atTop (nhds 0) := by
+  have h := (tendsto_rpow_mul_exp_neg_mul_atTop_nhds_zero (γ/ε) 1 zero_lt_one).comp (tendsto_rpow_atTop hε)
+  apply h.congr'
+  filter_upwards [eventually_ge_atTop (0:ℝ)] with x hx
+  simp only [Function.comp_def,neg_mul,one_mul,←Real.rpow_mul hx]
+  rw [mul_div_cancel₀ γ hε.ne']
+
+end AmplificationScalarBudget
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open Filter
+section AmplificationGoodnessDecay
+
+theorem eventually_amplification_goodness_decay (T : ℝ) (hT : 0 < T)
+    (M d : ℕ) (c a b β ε : ℝ) (ha : 0 < a) (hb : 0 ≤ b) (hε : 0 < ε) :
+    ∀ᶠ n : ℕ in atTop,
+      16*amplificationMomentFactor (logarithmicChainLength T n) M d c a b *
+        Real.exp (-((2*n+logarithmicChainLength T n*M:ℕ):ℝ)^ε) ≤ a*(n:ℝ)^(-β) := by
+  let C := amplificationMomentFactor 1 M d c a b
+  have hC : 0 < C := amplificationMomentFactor_pos _ _ _ _ _ _ ha hb
+  have hd : Tendsto (fun n : ℕ => 128*C*((n:ℝ)^(β+3)*Real.exp (-(n:ℝ)^ε))) atTop (nhds 0) := by
+    simpa only [mul_zero,Function.comp_def] using
+      ((tendsto_rpow_mul_stretched_exp (β+3) ε hε).comp tendsto_natCast_atTop_atTop).const_mul (128*C)
+  have hk := (tendsto_order.mp (tendsto_chainLength_div_scale T hT)).2 1 (by norm_num)
+  filter_upwards [eventually_ge_atTop (1:ℕ),hk,(tendsto_order.mp hd).2 a ha] with n hn hk hd
+  let k := logarithmicChainLength T n
+  have hn0 : (0:ℝ) < n := by exact_mod_cast (show 0 < n by omega)
+  have hn1 : (1:ℝ) ≤ n := by exact_mod_cast hn
+  have hkn : (k:ℝ) ≤ n := by
+    exact ((div_lt_iff₀ hn0).mp hk).le.trans_eq (one_mul _)
+  have hQ : amplificationMomentFactor k M d c a b ≤ 8*C*(n:ℝ)^3 := by
+    have hp : ((k:ℝ)+1)^3 ≤ (2*(n:ℝ))^3 :=
+      pow_le_pow_left₀ (by positivity) (by linarith) 3
+    calc
+      _ ≤ C*((k:ℝ)+1)^3 := amplificationMomentFactor_cubic k M d c a b ha hb
+      _ ≤ C*(2*(n:ℝ))^3 := mul_le_mul_of_nonneg_left hp hC.le
+      _ = _ := by ring
+  have he : Real.exp (-((2*n+k*M:ℕ):ℝ)^ε) ≤ Real.exp (-(n:ℝ)^ε) := by
+    apply Real.exp_le_exp.mpr
+    apply neg_le_neg
+    apply Real.rpow_le_rpow (Nat.cast_nonneg _) _ hε.le
+    exact_mod_cast (show n ≤ 2*n+k*M by omega)
+  have hpow : 0 < (n:ℝ)^β := Real.rpow_pos_of_pos hn0 _
+  rw [Real.rpow_neg hn0.le,←div_eq_mul_inv]
+  apply (le_div_iff₀ hpow).mpr
+  calc
+    _ ≤ (16*(8*C*(n:ℝ)^3)*Real.exp (-(n:ℝ)^ε))*(n:ℝ)^β := by
+      apply mul_le_mul_of_nonneg_right _ hpow.le
+      exact mul_le_mul (mul_le_mul_of_nonneg_left hQ (by norm_num)) he
+        (Real.exp_pos _).le (by positivity)
+    _ = 128*C*((n:ℝ)^(β+3)*Real.exp (-(n:ℝ)^ε)) := by
+      rw [Real.rpow_add hn0,show (n:ℝ)^(3:ℝ) = (n:ℝ)^3 by norm_cast]
+      ring
+    _ ≤ a := hd.le
+
+end AmplificationGoodnessDecay
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open Filter
+section AmplificationLogEstimates
+
+theorem log_inv_success_le (a Q n β u : ℝ) (ha : 0 < a) (hQ : 0 < Q)
+    (hn : 0 < n) (hbound : a*n^(-β)/(8*Q) ≤ u) :
+    Real.log (1/u) ≤ β*Real.log n + Real.log Q + Real.log (8/a) := by
+  have hl : 0 < a*n^(-β)/(8*Q) := by positivity
+  have hu : 0 < u := hl.trans_le hbound
+  have h := Real.log_le_log hl hbound
+  rw [Real.log_div (mul_pos ha (Real.rpow_pos_of_pos hn _)).ne' (mul_pos (by norm_num) hQ).ne',
+    Real.log_mul ha.ne' (Real.rpow_pos_of_pos hn _).ne',Real.log_rpow hn,
+    Real.log_mul (by norm_num : (8:ℝ) ≠ 0) hQ.ne'] at h
+  rw [one_div,Real.log_inv,Real.log_div (by norm_num : (8:ℝ) ≠ 0) ha.ne']
+  linarith
+
+theorem log_inv_norm_ge (r r' : ℝ) (k : ℕ) (hr : 0 < r) (hr' : 0 < r')
+    (hbound : r' ≤ 2*r^k) :
+    (k:ℝ)*Real.log (1/r) - Real.log 2 ≤ Real.log (1/r') := by
+  have h := Real.log_le_log hr' hbound
+  rw [Real.log_mul (by norm_num : (2:ℝ) ≠ 0) (pow_pos hr _).ne',Real.log_pow] at h
+  simp only [one_div,Real.log_inv]
+  linarith
+
+ 
+
+theorem eventually_amplification_log_estimates (T ξ β H : ℝ)
+    (hT : 0 < T) (hξ : 0 < ξ) (hξ1 : ξ < 1) (hH : 0 < H)
+    (M d : ℕ) (c a b : ℝ) (ha : 0 < a) (hb : 0 ≤ b) :
+    ∀ᶠ n : ℕ in atTop,
+      let k := logarithmicChainLength T n
+      β*Real.log n + Real.log (amplificationMomentFactor k M d c a b) + Real.log (8/a) ≤
+        (k:ℝ)*(β+ξ)*T ∧
+      Real.log ((2*n+k*M:ℕ):ℝ) ≤ (k:ℝ)*(1+ξ)*T ∧
+      (k:ℝ)*(1-ξ)*H ≤ (k:ℝ)*H-Real.log 2 ∧
+      2*H ≤ (k:ℝ)*(1-ξ)*H ∧ 1 ≤ k := by
+  let k := logarithmicChainLength T
+  have hk : Tendsto k atTop atTop := tendsto_logarithmicChainLength T hT
+  have hkR : Tendsto (fun n => (k n:ℝ)) atTop atTop := tendsto_natCast_atTop_atTop.comp hk
+  have hU : Tendsto (fun n : ℕ =>
+      (β*Real.log n + Real.log (amplificationMomentFactor (k n) M d c a b) + Real.log (8/a))/(k n:ℝ))
+      atTop (nhds (β*T)) := by
+    have h1 := (tendsto_log_div_chainLength T hT).const_mul β
+    have h2 := (tendsto_log_amplificationMomentFactor_div M d c a b ha hb).comp hk
+    have h3 := (tendsto_const_nhds (x := Real.log (8/a))).div_atTop hkR
+    simpa only [add_zero,add_div,mul_div_assoc,Function.comp_def] using (h1.add h2).add h3
+  have hm : Tendsto (fun n : ℕ => (Real.log n+Real.log 3)/(k n:ℝ)) atTop (nhds T) := by
+    simpa only [add_zero,add_div] using (tendsto_log_div_chainLength T hT).add
+      ((tendsto_const_nhds (x := Real.log 3)).div_atTop hkR)
+  have hHlim : Tendsto (fun n : ℕ => Real.log 2/(k n:ℝ)) atTop (nhds 0) :=
+    tendsto_const_nhds.div_atTop hkR
+  have hU' := (tendsto_order.mp hU).2 ((β+ξ)*T) (by nlinarith)
+  have hm' := (tendsto_order.mp hm).2 ((1+ξ)*T) (by nlinarith)
+  have hH' := (tendsto_order.mp hHlim).2 (ξ*H) (mul_pos hξ hH)
+  filter_upwards [hU',hm',hH',hkR.eventually_ge_atTop (2/(1-ξ)),
+    hk.eventually_ge_atTop 1,eventually_ge_atTop (1:ℕ),eventually_chain_volume_small T hT M]
+    with n hUn hmn hHn hklarge hk1 hn hvol
+  have hkp : (0:ℝ) < k n := by exact_mod_cast (show 0 < k n by omega)
+  have hnp : (0:ℝ) < n := by exact_mod_cast (show 0 < n by omega)
+  have hcap : (2*n+k n*M:ℕ) ≤ 3*n := by
+    dsimp [k] at *
+    rw [Nat.mul_assoc] at hvol
+    omega
+  have hcap0 : (0:ℝ) < (2*n+k n*M:ℕ) := by exact_mod_cast (show 0 < 2*n+k n*M by omega)
+  refine ⟨?_,?_,?_,?_,hk1⟩
+  · have h := (div_lt_iff₀ hkp).mp hUn
+    nlinarith only [h]
+  · calc
+      _ ≤ Real.log (3*(n:ℝ)) := Real.log_le_log hcap0 (by exact_mod_cast hcap)
+      _ = Real.log n + Real.log 3 := by rw [Real.log_mul (by norm_num : (3:ℝ) ≠ 0) hnp.ne']; ring
+      _ ≤ _ := by have h := (div_lt_iff₀ hkp).mp hmn; nlinarith only [h]
+  · have h := (div_lt_iff₀ hkp).mp hHn
+    nlinarith only [h]
+  · have h := (div_le_iff₀ (sub_pos.mpr hξ1)).mp hklarge
+    have hh := mul_le_mul_of_nonneg_right h hH.le
+    nlinarith only [hh]
+
+end AmplificationLogEstimates
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section AsymptoticAmplificationStep
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
+
+ 
+
+theorem asymptotic_amplification_step [Nonempty V] [Nonempty E]
+    (hconn : Connected G) (hqt : QuasiTransitive G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
+    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (i₀ : ι) (hw₀ : 0 < weight i₀)
+    (q : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) M L) (hL : 1 ≤ L)
+    (p : unitInterval) (hp : (p:ℝ) < 1) (d : ℕ) (hdeg : ∀ x, degree G x ≤ d)
+    (hu : 0 < q.successProbability p (root i₀))
+    (a b β ε ξ T : ℝ) (ha : 0 < a) (hb : 0 ≤ b)
+    (hε : 0 < ε) (hε1 : ε < 1/2) (hξ : 0 < ξ) (hξ1 : ξ < 1) (hT : 0 < T)
+    (hbudget : inverseCodeMass M L ≤ q.successProbability p (root i₀)*Real.exp T)
+    (A : Hilbert V →L[ℂ] Hilbert V) (hApos : 0 < ‖A‖)
+    (hH : 0 < Real.log (1/‖A‖))
+    (hA : ∀ x y, matrixCoeff A x y = (q.normalizedKernel (root i₀) p x y : ℂ)) :
+    ∀ᶠ n : ℕ in atTop, ∀ π : ℝ,
+      (n:ℝ)^(-β) ≤ π →
+      a*n*π ≤ binOrbitRow G p (root i₀) n →
+      clusterWindowProbability G p (root i₀) (n-logarithmicChainLength T n*M) (2*n) ≤ b*π →
+      ∃ q' : LocalEndpointRule G hloc (vertexOrbit G (root i₀))
+          (2*n+logarithmicChainLength T n*M) (((2*n+logarithmicChainLength T n*M:ℕ):ℝ)^(1/2+ε)),
+        0 < q'.successProbability p (root i₀) ∧
+        Real.log (1/q'.successProbability p (root i₀)) ≤ (logarithmicChainLength T n:ℝ)*(β+ξ)*T ∧
+        Real.log ((2*n+logarithmicChainLength T n*M:ℕ):ℝ) ≤ (logarithmicChainLength T n:ℝ)*(1+ξ)*T ∧
+        ∃ A' : Hilbert V →L[ℂ] Hilbert V, 0 < ‖A'‖ ∧
+          (logarithmicChainLength T n:ℝ)*(1-ξ)*Real.log (1/‖A‖) ≤ Real.log (1/‖A'‖) ∧
+          2*Real.log (1/‖A‖) ≤ Real.log (1/‖A'‖) ∧
+          (∀ x y, matrixCoeff A' x y = (q'.normalizedKernel (root i₀) p x y : ℂ)) ∧
+          1 ≤ logarithmicChainLength T n := by
+  let k := logarithmicChainLength T
+  let cap := fun n => 2*n+k n*M
+  have hcap : Tendsto cap atTop atTop :=
+    tendsto_atTop_mono (fun n => show n ≤ cap n by dsimp [cap]; omega) tendsto_id
+  have hgood := hcap.eventually (bad_goodness_stretched_exp G hloc hqt p hp ε hε hε1)
+  have hdecay := eventually_amplification_goodness_decay T hT M d c a b β ε ha hb hε
+  have hlogs := eventually_amplification_log_estimates T ξ β (Real.log (1/‖A‖))
+    hT hξ hξ1 hH M d c a b ha hb
+  filter_upwards [hgood,hdecay,hlogs,eventually_ge_atTop (1:ℕ)] with n hgood hdecay hlogs hn
+  intro π hπ hbin hwindow
+  have hn0 : 0 < n := by omega
+  have hnR : (0:ℝ) < n := Nat.cast_pos.mpr hn0
+  have hπ0 : 0 < π := (Real.rpow_pos_of_pos hnR (-β)).trans_le hπ
+  have hQ : 0 < amplificationMomentFactor (k n) M d c a b := amplificationMomentFactor_pos _ _ _ _ _ _ ha hb
+  have hcoden : inverseCodeMass M L^(k n) ≤ n*q.successProbability p (root i₀)^(k n) := by
+    apply code_bound_of_logarithmic_chain _ _ _ hu _ hT hbudget n hn
+    unfold inverseCodeMass
+    exact add_nonneg zero_le_one (Finset.sum_nonneg (fun l _ => gapWeight_nonneg L (by linarith) l))
+  have hbad : 16*amplificationMomentFactor (k n) M d c a b *
+      (law p).real {ω | CappedBadGoodness G ω (root i₀) (cap n) ((cap n:ℝ)^(1/2+ε))} ≤ a*π := by
+    calc
+      _ ≤ 16*amplificationMomentFactor (k n) M d c a b * Real.exp (-(cap n:ℝ)^ε) :=
+        mul_le_mul_of_nonneg_left (hgood (root i₀) p le_rfl) (by positivity)
+      _ ≤ a*(n:ℝ)^(-β) := hdecay
+      _ ≤ _ := mul_le_mul_of_nonneg_left hπ ha.le
+  obtain ⟨q',hup,A',hAp,hAr,hA'⟩ := q.quantitative_amplification_step hconn root weight Γ hΓ htr hcov
+    hw hw1 c hc hmin i₀ hw₀ hL p (k n) n d hn0 hdeg hu a b π ((cap n:ℝ)^(1/2+ε))
+    ha hb hπ0 hbin hwindow hcoden hbad A hA
+  have hup0 : 0 < q'.successProbability p (root i₀) := (by positivity : 0 < a*π/(8*amplificationMomentFactor (k n) M d c a b)).trans_le hup
+  have hulo : a*(n:ℝ)^(-β)/(8*amplificationMomentFactor (k n) M d c a b) ≤ q'.successProbability p (root i₀) := by
+    apply le_trans _ hup
+    exact div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_left hπ ha.le) (by positivity)
+  have hulo' := log_inv_success_le a (amplificationMomentFactor (k n) M d c a b) (n:ℝ) β
+    (q'.successProbability p (root i₀)) ha hQ hnR hulo
+  have hH' := log_inv_norm_ge ‖A‖ ‖A'‖ (k n) hApos hAp hAr
+  exact ⟨q',hup0,hulo'.trans hlogs.1,hlogs.2.1,A',hAp,hlogs.2.2.1.trans hH',
+    hlogs.2.2.2.1.trans (hlogs.2.2.1.trans hH'),hA',hlogs.2.2.2.2⟩
+
+end LocalEndpointRule
+end AsymptoticAmplificationStep
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section WeightedScaleWindow
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem weightedWindowProbability_term_le {ι : Type*} [Fintype ι]
+    (G : BondGraph V E) (p : unitInterval) (root : ι → V) (weight : ι → ℝ)
+    (hw : ∀ i, 0 ≤ weight i) (i : ι) (a b : ℕ) :
+    weight i * clusterWindowProbability G p (root i) a b ≤ weightedWindowProbability G p root weight a b :=
+  Finset.single_le_sum (fun j _ => mul_nonneg (hw j) (clusterWindowProbability_nonneg G p (root j) a b)) (Finset.mem_univ i)
+
+theorem orbit_scale_window_bound {ι : Type*} [Fintype ι]
+    (G : BondGraph V E) (p : unitInterval) (root : ι → V) (weight : ι → ℝ)
+    (hw : ∀ i, 0 ≤ weight i) (i : ι) (hi : 0 < weight i)
+    (j k M : ℕ) (hvol : 2*k*M ≤ 2^(j+1)) (β π : ℝ)
+    (hwindow : weightedWindowProbability G p root weight (2^j) (2^(j+2)) ≤ (1+(2:ℝ)^β)*π) :
+    clusterWindowProbability G p (root i) (2^(j+1)-k*M) (2*2^(j+1)) ≤
+      ((1+(2:ℝ)^β)/weight i)*π := by
+  have he : 2^(j+1) = 2*2^j := by rw [pow_succ]; omega
+  have he' : 2*2^(j+1) = 2^(j+2) := by rw [show j+2 = (j+1)+1 by omega,pow_succ]; omega
+  have hl : 2^j ≤ 2^(j+1)-k*M := by
+    rw [he,Nat.mul_assoc] at hvol
+    rw [he]
+    omega
+  have hmono := clusterWindowProbability_mono G p (root i) hl (le_of_eq he')
+  have hterm := weightedWindowProbability_term_le G p root weight hw i (2^j) (2^(j+2))
+  have hb := (mul_le_mul_of_nonneg_left hmono hi.le).trans (hterm.trans hwindow)
+  rw [div_mul_eq_mul_div]
+  exact (le_div_iff₀ hi).mpr (by simpa only [mul_comm] using hb)
+
+theorem tendsto_dyadic_scale : Tendsto (fun j : ℕ => 2^(j+1)) atTop atTop :=
+  tendsto_atTop_mono (fun j => (Nat.le_succ j).trans (j+1).lt_two_pow_self.le) tendsto_id
+
+end WeightedScaleWindow
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section EndpointProfile
+
+ 
+
+structure EndpointProfile (G : BondGraph V E) (hloc : LocallyFinite G)
+    (p : unitInterval) (o : V) (s : ℝ) where
+  cap : ℕ
+  rule : LocalEndpointRule G hloc (vertexOrbit G o) cap ((cap:ℝ)^s)
+  success_pos : 0 < rule.successProbability p o
+  op : Hilbert V →L[ℂ] Hilbert V
+  norm_pos : 0 < ‖op‖
+  coeff : ∀ x y, matrixCoeff op x y = (rule.normalizedKernel o p x y : ℂ)
+
+namespace EndpointProfile
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {p : unitInterval} {o : V} {s : ℝ}
+
+noncomputable def U (q : EndpointProfile G hloc p o s) : ℝ := Real.log (1/q.rule.successProbability p o)
+noncomputable def m (q : EndpointProfile G hloc p o s) : ℝ := Real.log q.cap
+noncomputable def H (q : EndpointProfile G hloc p o s) : ℝ := Real.log (1/‖q.op‖)
+noncomputable def cost (q : EndpointProfile G hloc p o s) (ξ : ℝ) : ℝ := q.U+(s+ξ)*q.m
+
+theorem U_nonneg (q : EndpointProfile G hloc p o s) : 0 ≤ q.U := by
+  unfold U
+  rw [one_div,Real.log_inv]
+  have h := Real.log_nonpos q.success_pos.le (q.rule.successProbability_le_one p o)
+  linarith
+
+theorem m_pos (q : EndpointProfile G hloc p o s) (hcap : 2 ≤ q.cap) : 0 < q.m := by
+  exact Real.log_pos (by exact_mod_cast (show 1 < q.cap by omega))
+
+theorem cost_pos (q : EndpointProfile G hloc p o s) (hcap : 2 ≤ q.cap)
+    (ξ : ℝ) (hsξ : 0 < s+ξ) : 0 < q.cost ξ := by
+  exact add_pos_of_nonneg_of_pos q.U_nonneg (mul_pos hsξ (q.m_pos hcap))
+
+theorem exp_cost_identity (q : EndpointProfile G hloc p o s) (hcap : 0 < q.cap) (ξ : ℝ) :
+    q.rule.successProbability p o * Real.exp (q.cost ξ) = (q.cap:ℝ)^(s+ξ) := by
+  have hcp : (0:ℝ) < q.cap := Nat.cast_pos.mpr hcap
+  simp only [cost,U,m,Real.exp_add,Real.exp_log (one_div_pos.mpr q.success_pos)]
+  calc
+    _ = Real.exp ((s+ξ)*Real.log q.cap) := by field_simp [ne_of_gt q.success_pos]
+    _ = _ := by rw [Real.rpow_def_of_pos hcp]; congr 1; ring
+
+end EndpointProfile
+
+variable [Countable V] [Countable E] [Nonempty V]
+
+theorem starting_endpoint_profile (G : BondGraph V E) (hloc : LocallyFinite G)
+    (hqt : QuasiTransitive G) (hh : 0 < hV G)
+    (p : unitInterval) (hp : 0 < (p:ℝ)) (hp1 : (p:ℝ) < 1)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
+    (i₀ : ι) (hw₀ : 0 < weight i₀)
+    (hfin : ∀ᵐ ω ∂law p, (cluster G ω (root i₀)).Finite)
+    (ε : ℝ) (hε : 0 < ε) (hε1 : ε < 1/2) (M₀ : ℕ) :
+    ∃ q : EndpointProfile G hloc p (root i₀) (1/2+ε), M₀ ≤ q.cap ∧ 1 < q.H := by
+  obtain ⟨M,hM,q,hq,A,hAp,hAn,hA⟩ := nonamenable_starting_endpoint_rule G hloc hqt hh p hp hp1
+    root weight Γ hΓ htr hw hw1 i₀ hw₀ hfin ε hε hε1 (Real.exp (-2)) (Real.exp_pos _) M₀
+  refine ⟨⟨M,q,hq,A,hAp,hA⟩,hM,?_⟩
+  have h := Real.log_lt_log hAp hAn
+  rw [Real.log_exp] at h
+  change 1 < Real.log (1/‖A‖)
+  rw [one_div,Real.log_inv]
+  linarith
+
+end EndpointProfile
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section EndpointProfileStep
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+namespace EndpointProfile
+variable {G : BondGraph V E} {hloc : LocallyFinite G}
+
+theorem amplify_at_scales [Nonempty V] [Nonempty E]
+    (hconn : Connected G) (hqt : QuasiTransitive G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
+    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (i : ι) (p : unitInterval) (hp : (p:ℝ) < 1)
+    (d : ℕ) (hdeg : ∀ x, degree G x ≤ d)
+    (ε ξ β : ℝ) (hε : 0 < ε) (hε1 : ε < 1/2) (hξ : 0 < ξ) (hξ1 : ξ < 1)
+    (hscales : ∀ N, ∃ j, N ≤ j ∧
+      ((2^(j+1):ℕ):ℝ)^(-β) ≤ dyadicBinProbability G p root weight (j+1) ∧
+      weightedWindowProbability G p root weight (2^j) (2^(j+2)) ≤
+        (1+(2:ℝ)^β)*dyadicBinProbability G p root weight (j+1) ∧
+      (((2^(j+1):ℕ):ℝ)/Fintype.card ι)*dyadicBinProbability G p root weight (j+1) ≤
+        binOrbitRow G p (root i) (2^(j+1)))
+    (q : EndpointProfile G hloc p (root i) (1/2+ε)) (hM : 2 ≤ q.cap)
+    (hL : 1 ≤ (q.cap:ℝ)^(1/2+ε))
+    (hbudget : inverseCodeMass q.cap ((q.cap:ℝ)^(1/2+ε)) ≤ (q.cap:ℝ)^(1/2+ε+ξ))
+    (hH : 0 < q.H) (N : ℕ) :
+    ∃ (q' : EndpointProfile G hloc p (root i) (1/2+ε)) (k : ℕ),
+      N ≤ q'.cap ∧ 1 ≤ k ∧
+      q'.U ≤ (k:ℝ)*(β+ξ)*q.cost ξ ∧
+      q'.m ≤ (k:ℝ)*(1+ξ)*q.cost ξ ∧
+      (k:ℝ)*(1-ξ)*q.H ≤ q'.H ∧ 2*q.H ≤ q'.H := by
+  let a : ℝ := 1/Fintype.card ι
+  let b : ℝ := (1+(2:ℝ)^β)/weight i
+  let T := q.cost ξ
+  have ht : 0 < T := q.cost_pos hM ξ (by linarith)
+  have hcard : (0:ℝ) < Fintype.card ι := by
+    exact_mod_cast (Fintype.card_pos_iff.mpr ⟨i⟩)
+  have ha : 0 < a := one_div_pos.mpr hcard
+  have hb : 0 ≤ b := div_nonneg (by positivity) (hw i).le
+  have hbudget' : inverseCodeMass q.cap ((q.cap:ℝ)^(1/2+ε)) ≤
+      q.rule.successProbability p (root i)*Real.exp T := by
+    rw [q.exp_cost_identity (by omega) ξ]
+    exact hbudget
+  have hstep := q.rule.asymptotic_amplification_step hconn hqt root weight Γ hΓ htr hcov
+    (fun i => (hw i).le) hw1 c hc hmin i (hw i) hL p hp d hdeg q.success_pos
+    a b β ε ξ T ha hb hε hε1 hξ hξ1 ht hbudget' q.op q.norm_pos hH q.coeff
+  have hvol := eventually_chain_volume_small T ht q.cap
+  have he := tendsto_dyadic_scale.eventually (hstep.and (hvol.and (eventually_ge_atTop N)))
+  obtain ⟨J,hJ⟩ := eventually_atTop.1 he
+  obtain ⟨j,hj,hπ,hwide,hbin⟩ := hscales J
+  obtain ⟨hs,hvol,hN⟩ := hJ j hj
+  have hwin := orbit_scale_window_bound G p root weight (fun i => (hw i).le) i (hw i)
+    j (logarithmicChainLength T (2^(j+1))) q.cap hvol β _ hwide
+  have hbin' : a*(2^(j+1):ℕ)*dyadicBinProbability G p root weight (j+1) ≤
+      binOrbitRow G p (root i) (2^(j+1)) := by
+    dsimp [a]
+    simpa only [div_eq_mul_inv,one_mul,mul_comm (_⁻¹)] using hbin
+  obtain ⟨q',hup,hU,hm,A',hAp,hH',hH'',hA',hk⟩ := hs _ hπ hbin' hwin
+  refine ⟨⟨_,q',hup,A',hAp,hA'⟩,_,?_,hk,hU,hm,hH',hH''⟩
+  change N ≤ 2*2^(j+1)+logarithmicChainLength T (2^(j+1))*q.cap
+  omega
+
+end EndpointProfile
+end EndpointProfileStep
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open Filter
+section AmplificationContraction
+
+noncomputable def amplificationContraction (β s ξ : ℝ) : ℝ :=
+  (β+ξ+(s+ξ)*(1+ξ))/(1-ξ)
+
+theorem exists_amplification_contraction (β s : ℝ) (hβ : 0 ≤ β) (hs : 0 ≤ s)
+    (hsmall : β+s < 1) : ∃ ξ : ℝ, 0 < ξ ∧ ξ < 1 ∧
+      0 ≤ amplificationContraction β s ξ ∧ amplificationContraction β s ξ < 1 := by
+  have hd : 0 < 4+s := by linarith
+  have hδ : 0 < (1-(β+s))/(4+s) := div_pos (by linarith) hd
+  obtain ⟨ξ,hξ,hξupper⟩ := exists_between (lt_min zero_lt_one hδ)
+  have hξ1 := hξupper.trans_le (min_le_left _ _)
+  have hprod : ξ*(4+s) < 1-(β+s) := (lt_div_iff₀ hd).mp (hξupper.trans_le (min_le_right _ _))
+  have hx2 : ξ^2 < ξ := by nlinarith [mul_pos hξ (sub_pos.mpr hξ1)]
+  refine ⟨ξ,hξ,hξ1,?_,?_⟩
+  · unfold amplificationContraction
+    positivity
+  · unfold amplificationContraction
+    apply (div_lt_one (by linarith : 0 < 1-ξ)).mpr
+    nlinarith
+
+ 
+
+theorem amplification_cost_contraction (t t' H H' β s ξ : ℝ) (k : ℕ)
+    (ht : 0 ≤ t) (hH : 0 < H) (hξ1 : ξ < 1) (hk : 1 ≤ k)
+    (hC : 0 ≤ amplificationContraction β s ξ)
+    (ht' : t' ≤ (k:ℝ)*(β+ξ+(s+ξ)*(1+ξ))*t)
+    (hH' : (k:ℝ)*(1-ξ)*H ≤ H') :
+    t'/H' ≤ amplificationContraction β s ξ*(t/H) := by
+  have hk0 : (0:ℝ) < k := by exact_mod_cast (show 0 < k by omega)
+  have hd : 0 < (k:ℝ)*(1-ξ)*H := mul_pos (mul_pos hk0 (by linarith)) hH
+  have hH'p := hd.trans_le hH'
+  apply (div_le_iff₀ hH'p).mpr
+  have he : (k:ℝ)*(β+ξ+(s+ξ)*(1+ξ))*t =
+      amplificationContraction β s ξ*(t/H)*((k:ℝ)*(1-ξ)*H) := by
+    unfold amplificationContraction
+    field_simp [hH.ne', (sub_pos.mpr hξ1).ne']
+  rw [he] at ht'
+  exact ht'.trans (mul_le_mul_of_nonneg_left hH' (mul_nonneg hC (div_nonneg ht hH.le)))
+
+namespace EndpointProfile
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {p : unitInterval} {o : V} {s : ℝ}
+
+theorem cost_contraction (q q' : EndpointProfile G hloc p o s) (β ξ : ℝ) (k : ℕ)
+    (hM : 2 ≤ q.cap) (hsξ : 0 < s+ξ) (hH : 0 < q.H) (hξ1 : ξ < 1) (hk : 1 ≤ k)
+    (hC : 0 ≤ amplificationContraction β s ξ)
+    (hU : q'.U ≤ (k:ℝ)*(β+ξ)*q.cost ξ)
+    (hm : q'.m ≤ (k:ℝ)*(1+ξ)*q.cost ξ)
+    (hH' : (k:ℝ)*(1-ξ)*q.H ≤ q'.H) :
+    q'.cost ξ/q'.H ≤ amplificationContraction β s ξ*(q.cost ξ/q.H) := by
+  apply amplification_cost_contraction _ _ _ _ β s ξ k (q.cost_pos hM ξ hsξ).le hH hξ1 hk hC _ hH'
+  unfold cost at *
+  have hh := mul_le_mul_of_nonneg_left hm hsξ.le
+  nlinarith only [hU,hh]
+
+theorem total_cost_le (q : EndpointProfile G hloc p o s) (hM : 2 ≤ q.cap)
+    (ξ : ℝ) (hsξ : 0 < s+ξ) : q.U+q.m ≤ (1+1/(s+ξ))*q.cost ξ := by
+  have hm := (q.m_pos hM).le
+  have hU := q.U_nonneg
+  have hU' : q.U ≤ q.cost ξ := by unfold cost; nlinarith
+  have hm' : q.m ≤ q.cost ξ/(s+ξ) := by
+    apply (le_div_iff₀ hsξ).mpr
+    unfold cost
+    nlinarith
+  calc
+    _ ≤ q.cost ξ+q.cost ξ/(s+ξ) := add_le_add hU' hm'
+    _ = _ := by ring
+
+end EndpointProfile
+end AmplificationContraction
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Kernel/Spectral.lean b/lean/OAI/Probability/UnimodularPercolation/Kernel/Spectral.lean
index 8cbcbe6a45a60473e2be0b3042cf4cd6639e3bb3..273261e40a13079fdc0f7e063735f9cbf525971a 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Kernel/Spectral.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Kernel/Spectral.lean
@@ -1,1166 +1,1166 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Kernel.Sprinkling
-
-namespace OAI
-
-/-! Isoperimetric energy, symmetric walks and invariant operator traces. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-section IsoperimetricEnergy
-attribute [local instance] Classical.propDecidable
-
- 
-noncomputable def levelIndicator (a t : ℝ) : ℝ := if 0 ≤ t ∧ t < a then 1 else 0
-
-theorem levelIndicator_eq (a : ℝ) :
-    levelIndicator a = Set.indicator (Set.Ico 0 a) (fun _ => (1:ℝ)) := by
-  funext t
-  simp [levelIndicator, Set.indicator_apply, Set.mem_Ico]
-
-theorem integrable_levelIndicator (a : ℝ) : Integrable (levelIndicator a) := by
-  rw [levelIndicator_eq]
-  exact (integrableOn_const (isCompact_Icc.measure_lt_top.ne) :
-    IntegrableOn (fun _ : ℝ => (1:ℝ)) (Set.Icc 0 a)).mono_set
-      Set.Ico_subset_Icc_self |>.integrable_indicator measurableSet_Ico
-
-theorem integral_levelIndicator {a : ℝ} (ha : 0 ≤ a) :
-    ∫ t, levelIndicator a t = a := by
-  rw [levelIndicator_eq, integral_indicator measurableSet_Ico]
-  simp [ha]
-
-theorem abs_sub_levelIndicator {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) :
-    (fun t => |levelIndicator a t - levelIndicator b t|) =
-      Set.indicator (Set.Ico (min a b) (max a b)) (fun _ => (1:ℝ)) := by
-  funext t
-  rcases le_total a b with hab | hba
-  · simp only [min_eq_left hab, max_eq_right hab, Set.indicator_apply, Set.mem_Ico,
-      levelIndicator]
-    by_cases ht : 0 ≤ t
-    · by_cases hta : t < a
-      · have htb : t < b := hta.trans_le hab
-        have hna : ¬a ≤ t := not_le.mpr hta
-        simp [ht,hta,htb,hna]
-      · by_cases htb : t < b <;> simp [ht,hta,htb,le_of_not_gt hta]
-    · have hna : ¬a ≤ t := fun h => ht (ha.trans h)
-      simp [ht,hna]
-  · simp only [min_eq_right hba, max_eq_left hba, Set.indicator_apply, Set.mem_Ico,
-      levelIndicator]
-    by_cases ht : 0 ≤ t
-    · by_cases htb : t < b
-      · have hta : t < a := htb.trans_le hba
-        have hnb : ¬b ≤ t := not_le.mpr htb
-        simp [ht,hta,htb,hnb]
-      · by_cases hta : t < a <;> simp [ht,hta,htb,le_of_not_gt htb]
-    · have hnb : ¬b ≤ t := fun h => ht (hb.trans h)
-      simp [ht,hnb]
-
-theorem integrable_abs_sub_levelIndicator (a b : ℝ) :
-    Integrable (fun t => |levelIndicator a t - levelIndicator b t|) :=
-  ((integrable_levelIndicator a).sub (integrable_levelIndicator b)).abs
-
-theorem integral_abs_sub_levelIndicator {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) :
-    ∫ t, |levelIndicator a t - levelIndicator b t| = |a-b| := by
-  rw [abs_sub_levelIndicator ha hb, integral_indicator measurableSet_Ico]
-  simp only [setIntegral_const, smul_eq_mul, mul_one,
-    Real.volume_real_Ico_of_le min_le_max]
-  exact (max_sub_min_eq_abs a b).trans (abs_sub_comm _ _)
-
- 
-theorem finite_coarea_lower_bound {α : Type*} (S : Finset α) (P : Finset (α × α))
-    (f : α → ℝ) (hf : ∀ x, 0 ≤ f x) (hout : ∀ x ∉ S, f x = 0) (h : ℝ)
-    (hcut : ∀ A : Finset α, A ⊆ S →
-      h * A.card ≤ ∑ e ∈ P,
-        |(if e.1 ∈ A then (1:ℝ) else 0) - (if e.2 ∈ A then (1:ℝ) else 0)|) :
-    h * ∑ x ∈ S, f x ≤ ∑ e ∈ P, |f e.1 - f e.2| := by
-  let F : ℝ → ℝ := fun t => h * ∑ x ∈ S, levelIndicator (f x) t
-  let J : ℝ → ℝ := fun t => ∑ e ∈ P,
-    |levelIndicator (f e.1) t - levelIndicator (f e.2) t|
-  have hF : Integrable F :=
-    (integrable_finsetSum S (fun x _ => integrable_levelIndicator (f x))).const_mul h
-  have hJ : Integrable J :=
-    integrable_finsetSum P (fun e _ => integrable_abs_sub_levelIndicator _ _)
-  have hle : ∀ t, F t ≤ J t := by
-    intro t
-    let A := S.filter (fun x => 0 ≤ t ∧ t < f x)
-    have hi (x : α) : levelIndicator (f x) t = if x ∈ A then (1:ℝ) else 0 := by
-      by_cases hx : x ∈ S
-      · simp [A,levelIndicator,hx]
-      · rw [hout x hx]
-        have hz : ¬(0 ≤ t ∧ t < 0) := fun ht => (not_lt_of_ge ht.1) ht.2
-        simp [A,levelIndicator,hx,hz]
-    have hc : ∑ x ∈ S, levelIndicator (f x) t = (A.card : ℝ) := by
-      simp [levelIndicator,A,Finset.sum_boole]
-    dsimp [F,J]
-    rw [hc]
-    simpa only [hi] using hcut A (Finset.filter_subset _ _)
-  have H := integral_mono hF hJ hle
-  dsimp [F,J] at H
-  rw [integral_const_mul, integral_finsetSum _
-    (fun x _ => integrable_levelIndicator (f x)), integral_finsetSum _
-    (fun e _ => integrable_abs_sub_levelIndicator _ _)] at H
-  simpa only [integral_levelIndicator (hf _),
-    integral_abs_sub_levelIndicator (hf _) (hf _)] using H
-
-theorem hV_mul_card_le (G : BondGraph V E) (A : Finset V) :
-    hV G * A.card ≤ (vertexBoundary G A).ncard := by
-  by_cases hA : A.Nonempty
-  · have hb : BddBelow {r : ℝ | ∃ B : Finset V, B.Nonempty ∧
-        r = (vertexBoundary G B).ncard / (B.card : ℝ)} := by
-      refine ⟨0, ?_⟩
-      rintro _ ⟨B,_,rfl⟩
-      positivity
-    have hh : hV G ≤ (vertexBoundary G A).ncard / (A.card : ℝ) :=
-      csInf_le hb ⟨A,hA,rfl⟩
-    exact (le_div_iff₀ (by exact_mod_cast hA.card_pos)).mp hh
-  · simp [Finset.not_nonempty_iff_eq_empty.mp hA]
-
-theorem finite_vertexBoundary (G : BondGraph V E) (hloc : LocallyFinite G)
-    (A : Finset V) : (vertexBoundary G A).Finite := by
-  apply (A.finite_toSet.biUnion (fun x _ => finite_neighbors hloc x)).subset
-  rintro y ⟨_,x,hx,hxy⟩
-  exact Set.mem_iUnion.mpr ⟨x,Set.mem_iUnion.mpr ⟨hx,hxy⟩⟩
-
-noncomputable def neighbors (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) :
-    Finset V := (finite_neighbors hloc x).toFinset
-
-@[simp] theorem mem_neighbors (G : BondGraph V E) (hloc : LocallyFinite G) (x y : V) :
-    y ∈ neighbors G hloc x ↔ Adj G x y := by simp [neighbors]
-
-noncomputable def neighborClosure (G : BondGraph V E) (hloc : LocallyFinite G)
-    (S : Finset V) : Finset V := S ∪ S.biUnion (neighbors G hloc)
-
-theorem subset_neighborClosure (G : BondGraph V E) (hloc : LocallyFinite G)
-    (S : Finset V) : S ⊆ neighborClosure G hloc S := Finset.subset_union_left
-
-theorem neighbor_mem_neighborClosure (G : BondGraph V E) (hloc : LocallyFinite G)
-    (S : Finset V) {x y : V} (hx : x ∈ S) (hxy : Adj G x y) :
-    y ∈ neighborClosure G hloc S := by
-  exact Finset.mem_union_right _ (Finset.mem_biUnion.mpr ⟨x,hx,(mem_neighbors ..).mpr hxy⟩)
-
-noncomputable def finiteAdjPairs (G : BondGraph V E) (K : Finset V) : Finset (V × V) :=
-  (K ×ˢ K).filter (fun e => Adj G e.1 e.2)
-
-@[simp] theorem mem_finiteAdjPairs (G : BondGraph V E) (K : Finset V) (e : V × V) :
-    e ∈ finiteAdjPairs G K ↔ e.1 ∈ K ∧ e.2 ∈ K ∧ Adj G e.1 e.2 := by
-  simp [finiteAdjPairs,and_assoc]
-
-theorem adj_symm (G : BondGraph V E) {x y : V} (h : Adj G x y) : Adj G y x := by
-  obtain ⟨e,he⟩ := h
-  exact ⟨e,he.trans (Sym2.eq_swap)⟩
-
-theorem sum_swap_finiteAdjPairs (G : BondGraph V E) (K : Finset V)
-    (f : V × V → ℝ) :
-    ∑ e ∈ finiteAdjPairs G K, f e.swap = ∑ e ∈ finiteAdjPairs G K, f e := by
-  apply Finset.sum_bij (fun e _ => e.swap)
-  · intro e he
-    rw [mem_finiteAdjPairs] at he ⊢
-    exact ⟨he.2.1,he.1,adj_symm G he.2.2⟩
-  · intro a _ b _ h
-    exact Prod.swap_injective h
-  · intro b hb
-    refine ⟨b.swap,?_,Prod.swap_swap b⟩
-    rw [mem_finiteAdjPairs] at hb ⊢
-    exact ⟨hb.2.1,hb.1,adj_symm G hb.2.2⟩
-  · intros; rfl
-
- 
-theorem finiteAdjPairs_cut_bound (G : BondGraph V E) (hloc : LocallyFinite G)
-    (S A : Finset V) (hAS : A ⊆ S) :
-    (2 * hV G) * A.card ≤ ∑ e ∈ finiteAdjPairs G (neighborClosure G hloc S),
-      |(if e.1 ∈ A then (1:ℝ) else 0) - (if e.2 ∈ A then (1:ℝ) else 0)| := by
-  let K := neighborClosure G hloc S
-  let P := finiteAdjPairs G K
-  let R := P.filter (fun e => e.1 ∈ A ∧ e.2 ∉ A)
-  have hs : (finite_vertexBoundary G hloc A).toFinset ⊆ R.image Prod.snd := by
-    intro y hy
-    have hy' : y ∈ vertexBoundary G A := by simpa using hy
-    obtain ⟨hyA,x,hx,hxy⟩ := hy'
-    apply Finset.mem_image.mpr
-    refine ⟨(x,y),Finset.mem_filter.mpr ⟨?_,hx,hyA⟩,rfl⟩
-    exact (mem_finiteAdjPairs G K (x,y)).mpr
-      ⟨subset_neighborClosure G hloc S (hAS hx),
-        neighbor_mem_neighborClosure G hloc S (hAS hx) hxy,hxy⟩
-  have hc : (vertexBoundary G A).ncard ≤ R.card := by
-    rw [Set.ncard_eq_toFinset_card (vertexBoundary G A) (finite_vertexBoundary G hloc A)]
-    exact (Finset.card_le_card hs).trans (Finset.card_image_le)
-  have hcross : hV G * A.card ≤ ∑ e ∈ P,
-      if e.1 ∈ A ∧ e.2 ∉ A then (1:ℝ) else 0 := by
-    rw [Finset.sum_boole]
-    exact (hV_mul_card_le G A).trans (by exact_mod_cast hc)
-  have hsplit : ∑ e ∈ P,
-      |(if e.1 ∈ A then (1:ℝ) else 0) - (if e.2 ∈ A then (1:ℝ) else 0)| =
-      2 * ∑ e ∈ P, if e.1 ∈ A ∧ e.2 ∉ A then (1:ℝ) else 0 := by
-    have ht (e : V × V) :
-        |(if e.1 ∈ A then (1:ℝ) else 0) - (if e.2 ∈ A then (1:ℝ) else 0)| =
-        (if e.1 ∈ A ∧ e.2 ∉ A then (1:ℝ) else 0) +
-        (if e.2 ∈ A ∧ e.1 ∉ A then (1:ℝ) else 0) := by
-      by_cases h₁ : e.1 ∈ A <;> by_cases h₂ : e.2 ∈ A <;> simp [h₁,h₂]
-    simp_rw [ht]
-    rw [Finset.sum_add_distrib]
-    have he := sum_swap_finiteAdjPairs G K
-      (fun e => if e.1 ∈ A ∧ e.2 ∉ A then (1:ℝ) else 0)
-    change (∑ e ∈ P, if e.2 ∈ A ∧ e.1 ∉ A then (1:ℝ) else 0) = _ at he
-    rw [he]
-    ring
-  change _ ≤ ∑ e ∈ P, _
-  rw [hsplit]
-  nlinarith
-
- 
-theorem bondGraph_coarea (G : BondGraph V E) (hloc : LocallyFinite G)
-    (S : Finset V) (f : V → ℝ) (hf : ∀ x, 0 ≤ f x)
-    (hout : ∀ x ∉ S, f x = 0) :
-    (2 * hV G) * ∑ x ∈ S, f x ≤
-      ∑ e ∈ finiteAdjPairs G (neighborClosure G hloc S), |f e.1 - f e.2| :=
-  finite_coarea_lower_bound S _ f hf hout _
-    (fun A hAS => finiteAdjPairs_cut_bound G hloc S A hAS)
-
-theorem neighbors_card_le_degree (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) :
-    (neighbors G hloc x).card ≤ degree G x := by
-  let : Fintype (Step G x) := (finite_steps G hloc x).fintype
-  let p : Step G x → {y // y ∈ neighbors G hloc x} :=
-    fun d => ⟨d.val.2, (mem_neighbors G hloc x d.val.2).mpr ⟨d.val.1,d.property⟩⟩
-  have hp : Function.Surjective p := by
-    intro y
-    obtain ⟨e,he⟩ := (mem_neighbors G hloc x y.val).mp y.property
-    exact ⟨⟨(e,y.val),he⟩,Subtype.ext rfl⟩
-  simpa only [degree,Nat.card_eq_fintype_card,Fintype.card_coe] using
-    Fintype.card_le_of_surjective p hp
-
-theorem finiteAdjPairs_source_sum_le (G : BondGraph V E) (hloc : LocallyFinite G)
-    (D : ℕ) (hD : ∀ x, degree G x ≤ D) (K : Finset V) (f : V → ℝ)
-    (hf : ∀ x, 0 ≤ f x) :
-    ∑ e ∈ finiteAdjPairs G K, f e.1 ≤ (D:ℝ) * ∑ x ∈ K, f x := by
-  rw [Finset.mul_sum]
-  simp only [finiteAdjPairs,Finset.sum_filter,Finset.sum_product]
-  apply Finset.sum_le_sum
-  intro x _
-  have hc : (K.filter (fun y => Adj G x y)).card ≤ D := by
-    apply (Finset.card_le_card ?_).trans ((neighbors_card_le_degree G hloc x).trans (hD x))
-    intro y hy
-    exact (mem_neighbors G hloc x y).mpr (Finset.mem_filter.mp hy).2
-  calc
-    (∑ y ∈ K, if Adj G x y then f x else 0) =
-        ((K.filter (fun y => Adj G x y)).card:ℝ) * f x := by
-      rw [← Finset.sum_filter]
-      simp
-    _ ≤ (D:ℝ) * f x := mul_le_mul_of_nonneg_right (by exact_mod_cast hc) (hf x)
-
-theorem finiteAdjPairs_square_sum_le (G : BondGraph V E) (hloc : LocallyFinite G)
-    (D : ℕ) (hD : ∀ x, degree G x ≤ D) (S : Finset V) (f : V → ℝ)
-    (hout : ∀ x ∉ S, f x = 0) :
-    ∑ e ∈ finiteAdjPairs G (neighborClosure G hloc S), (f e.1 + f e.2)^2 ≤
-      4 * (D:ℝ) * ∑ x ∈ S, (f x)^2 := by
-  let K := neighborClosure G hloc S
-  let P := finiteAdjPairs G K
-  have hs : ∑ x ∈ K, (f x)^2 = ∑ x ∈ S, (f x)^2 := by
-    symm
-    apply Finset.sum_subset (subset_neighborClosure G hloc S)
-    intro x _ hx
-    simp [hout x hx]
-  have hfirst := finiteAdjPairs_source_sum_le G hloc D hD K (fun x => (f x)^2) (fun _ => sq_nonneg _)
-  rw [hs] at hfirst
-  have hsecond := sum_swap_finiteAdjPairs G K (fun e => (f e.1)^2)
-  change (∑ e ∈ P, (f e.2)^2) = ∑ e ∈ P, (f e.1)^2 at hsecond
-  calc
-    (∑ e ∈ P, (f e.1 + f e.2)^2) ≤ ∑ e ∈ P, (2*(f e.1)^2+2*(f e.2)^2) := by
-      apply Finset.sum_le_sum
-      intro e _
-      nlinarith [sq_nonneg (f e.1 - f e.2)]
-    _ = 4 * ∑ e ∈ P, (f e.1)^2 := by
-      rw [Finset.sum_add_distrib,←Finset.mul_sum,←Finset.mul_sum,hsecond]
-      ring
-    _ ≤ 4 * (D:ℝ) * ∑ x ∈ S, (f x)^2 := by
-      nlinarith
-
- 
-
-theorem bondGraph_cheeger_energy (G : BondGraph V E) (hloc : LocallyFinite G)
-    (hh : 0 < hV G) (D : ℕ) (hD : ∀ x, degree G x ≤ D)
-    (S : Finset V) (f : V → ℝ) (hf : ∀ x, 0 ≤ f x)
-    (hout : ∀ x ∉ S, f x = 0) :
-    (hV G)^2 * ∑ x ∈ S, (f x)^2 ≤
-      (D:ℝ) * ∑ e ∈ finiteAdjPairs G (neighborClosure G hloc S), (f e.1-f e.2)^2 := by
-  let P := finiteAdjPairs G (neighborClosure G hloc S)
-  let N := ∑ x ∈ S, (f x)^2
-  let E := ∑ e ∈ P, (f e.1-f e.2)^2
-  have hN : 0 ≤ N := Finset.sum_nonneg (fun _ _ => sq_nonneg _)
-  have hE : 0 ≤ E := Finset.sum_nonneg (fun _ _ => sq_nonneg _)
-  have hc := bondGraph_coarea G hloc S (fun x => (f x)^2)
-    (fun _ => sq_nonneg _) (fun x hx => by rw [hout x hx]; norm_num)
-  have ht (e : V × V) : |(f e.1)^2-(f e.2)^2| = |f e.1-f e.2| * (f e.1+f e.2) := by
-    rw [show (f e.1)^2-(f e.2)^2 = (f e.1-f e.2)*(f e.1+f e.2) by ring,
-      abs_mul,abs_of_nonneg (add_nonneg (hf _) (hf _))]
-  simp_rw [ht] at hc
-  have hc' : ((2*hV G)*N)^2 ≤
-      (∑ e ∈ P, |f e.1-f e.2| * (f e.1+f e.2))^2 := by
-    exact sq_le_sq₀ (by positivity) (Finset.sum_nonneg (fun e _ => mul_nonneg (abs_nonneg _)
-      (add_nonneg (hf _) (hf _)))) |>.mpr hc
-  have hcs := Finset.sum_mul_sq_le_sq_mul_sq P
-    (fun e => |f e.1-f e.2|) (fun e => f e.1+f e.2)
-  simp only [sq_abs] at hcs
-  have hbound := finiteAdjPairs_square_sum_le G hloc D hD S f hout
-  have hall : ((2*hV G)*N)^2 ≤ E*(4*(D:ℝ)*N) :=
-    (hc'.trans hcs).trans (mul_le_mul_of_nonneg_left hbound hE)
-  change (hV G)^2*N ≤ (D:ℝ)*E
-  rcases hN.eq_or_lt with hzero | hpos
-  · rw [←hzero,mul_zero]
-    positivity
-  · apply (mul_le_mul_iff_left₀ (by positivity : 0 < 4*N)).mp
-    nlinarith [hall]
-
-end IsoperimetricEnergy
-
-section FiniteQuadraticForms
-attribute [local instance] Classical.propDecidable
-
-@[simp] theorem finiteEmbedding_inner_left (f : V →₀ ℂ) (g : Hilbert V) :
-    inner ℂ (finiteEmbedding f) g = ∑ x ∈ f.support, star (f x) * g x := by
-  simp [finiteEmbedding, Finsupp.linearCombination_apply, Finsupp.sum,
-    lp.inner_single_left, mul_comm]
-
-@[simp] theorem finiteEmbedding_inner_right (f : Hilbert V) (g : V →₀ ℂ) :
-    inner ℂ f (finiteEmbedding g) = ∑ x ∈ g.support, star (f x) * g x := by
-  simp [finiteEmbedding, Finsupp.linearCombination_apply, Finsupp.sum,
-    lp.inner_single_right, mul_comm]
-
-@[simp] theorem finiteEmbedding_norm_sq (f : V →₀ ℂ) :
-    ‖finiteEmbedding f‖^2 = ∑ x ∈ f.support, ‖f x‖^2 := by
-  rw [norm_sq_eq_re_inner (𝕜 := ℂ)]
-  rw [finiteEmbedding_inner_left]
-  simp only [map_sum, finiteEmbedding_apply]
-  apply Finset.sum_congr rfl
-  intro x _
-  simp [← Complex.normSq_eq_norm_sq, Complex.normSq_apply]
-
- theorem kernel_finiteEmbedding (A : Hilbert V →L[ℂ] Hilbert V)
-    (U : V → V → ℝ) (hA : ∀ x y, A (lp.single 2 y (1 : ℂ)) x = (U x y : ℂ))
-    (f : V →₀ ℂ) (x : V) :
-    A (finiteEmbedding f) x = ∑ y ∈ f.support, f y * (U x y : ℂ) := by
-  simp only [finiteEmbedding, Finsupp.linearCombination_apply, Finsupp.sum,
-    map_sum, map_smul]
-  change (∑ y ∈ f.support, f y • A (lp.single 2 y (1 : ℂ))) x = _
-  simp only [lp.coeFn_sum, Finset.sum_apply, lp.coeFn_smul, Pi.smul_apply,
-    hA, smul_eq_mul]
-
- theorem symmetric_of_symmetric_kernel (A : Hilbert V →L[ℂ] Hilbert V)
-    (U : V → V → ℝ) (hA : ∀ x y, A (lp.single 2 y (1 : ℂ)) x = (U x y : ℂ))
-    (hU : ∀ x y, U x y = U y x) : A.toLinearMap.IsSymmetric := by
-  apply finiteEmbedding_dense.induction_on₂
-    (p := fun f g => inner ℂ (A f) g = inner ℂ f (A g))
-  · exact isClosed_eq (by fun_prop) (by fun_prop)
-  · intro f g
-    simp only [finiteEmbedding_inner_left, finiteEmbedding_inner_right,
-      kernel_finiteEmbedding A U hA, map_sum, map_mul, RCLike.star_def,
-      Complex.conj_ofReal]
-    simp_rw [Finset.sum_mul, Finset.mul_sum]
-    rw [Finset.sum_comm]
-    apply Finset.sum_congr rfl
-    intro x _
-    apply Finset.sum_congr rfl
-    intro y _
-    rw [hU y x]
-    ring
-
- theorem norm_le_of_finite_quadratic (A : Hilbert V →L[ℂ] Hilbert V)
-    (hA : A.toLinearMap.IsSymmetric) (C : ℝ) (hC : 0 ≤ C)
-    (h : ∀ f : V →₀ ℂ,
-      |(inner ℂ (A (finiteEmbedding f)) (finiteEmbedding f)).re| ≤
-        C * ‖finiteEmbedding f‖^2) : ‖A‖ ≤ C := by
-  have hall (f : Hilbert V) : |(inner ℂ (A f) f).re| ≤ C * ‖f‖^2 := by
-    exact finiteEmbedding_dense.induction_on f
-      (p := fun g => |(inner ℂ (A g) g).re| ≤ C * ‖g‖^2)
-      (isClosed_le (by fun_prop) (by fun_prop)) h
-  rw [A.norm_eq_iSup_rayleighQuotient hA]
-  apply ciSup_le
-  intro f
-  by_cases hf : f = 0
-  · simp [hf, hC]
-  · rw [ContinuousLinearMap.rayleighQuotient, abs_div, abs_sq]
-    exact (div_le_iff₀ (sq_pos_of_pos (norm_pos_iff.mpr hf))).mpr (hall f)
-
- theorem abs_inner_kernel_finite_le (A : Hilbert V →L[ℂ] Hilbert V)
-    (U : V → V → ℝ) (hA : ∀ x y, A (lp.single 2 y (1 : ℂ)) x = (U x y : ℂ))
-    (hU : ∀ x y, 0 ≤ U x y) (f : V →₀ ℂ) :
-    |(inner ℂ (A (finiteEmbedding f)) (finiteEmbedding f)).re| ≤
-      ∑ x ∈ f.support, ∑ y ∈ f.support, ‖f x‖ * ‖f y‖ * U x y := by
-  change |RCLike.re (inner ℂ (A (finiteEmbedding f)) (finiteEmbedding f))| ≤ _
-  rw [inner_re_symm]
-  apply (Complex.abs_re_le_norm _).trans
-  rw [finiteEmbedding_inner_left]
-  simp_rw [kernel_finiteEmbedding A U hA, Finset.mul_sum]
-  apply (norm_sum_le _ _).trans
-  apply Finset.sum_le_sum
-  intro x _
-  apply (norm_sum_le _ _).trans
-  apply Finset.sum_le_sum
-  intro y _
-  simp [abs_of_nonneg (hU x y), mul_assoc]
-
-end FiniteQuadraticForms
-
-section SymmetricWalk
-attribute [local instance] Classical.propDecidable
-
- 
-
-noncomputable def walkKernel (G : BondGraph V E) (hloc : LocallyFinite G)
-    (a : ℝ) (x y : V) : ℝ :=
-  (if x = y then 1 - a * (neighbors G hloc x).card else 0) +
-    (if Adj G x y then a else 0)
-
- theorem walkKernel_symmetric (G : BondGraph V E) (hloc : LocallyFinite G)
-    (a : ℝ) (x y : V) : walkKernel G hloc a x y = walkKernel G hloc a y x := by
-  have ha : Adj G x y ↔ Adj G y x := ⟨adj_symm G,adj_symm G⟩
-  by_cases h : x = y
-  · subst y; rfl
-  · simp [walkKernel,h,Ne.symm h,ha]
-
- theorem walkKernel_nonneg (G : BondGraph V E) (hloc : LocallyFinite G)
-    (D : ℕ) (hD : ∀ x, degree G x ≤ D) (a : ℝ) (ha : 0 ≤ a) (haD : a*D ≤ 1)
-    (x y : V) : 0 ≤ walkKernel G hloc a x y := by
-  have hn : ((neighbors G hloc x).card : ℝ) ≤ D := by
-    exact_mod_cast (neighbors_card_le_degree G hloc x).trans (hD x)
-  have hh : 0 ≤ 1-a*(neighbors G hloc x).card := by nlinarith
-  unfold walkKernel
-  split_ifs <;> positivity
-
- theorem walkKernel_row_sum (G : BondGraph V E) (hloc : LocallyFinite G)
-    (a : ℝ) (x : V) :
-    ∑ y ∈ insert x (neighbors G hloc x), walkKernel G hloc a x y = 1 := by
-  let N := neighbors G hloc x
-  have hf : (insert x N).filter (fun y => Adj G x y) = N := by
-    ext y
-    simp only [Finset.mem_filter, Finset.mem_insert]
-    rw [mem_neighbors]
-    exact ⟨fun h => h.2, fun h => ⟨Or.inr h,h⟩⟩
-  simp only [walkKernel, Finset.sum_add_distrib]
-  rw [Finset.sum_ite_eq]
-  simp only [Finset.mem_insert_self,ite_true]
-  rw [← Finset.sum_filter,hf]
-  simp only [Finset.sum_const, nsmul_eq_mul]
-  dsimp [N]
-  ring
-
- theorem walkKernel_row_bound (G : BondGraph V E) (hloc : LocallyFinite G)
-    (D : ℕ) (hD : ∀ x, degree G x ≤ D) (a : ℝ) (ha : 0 ≤ a) (haD : a*D ≤ 1)
-    (x : V) (S : Finset V) : ∑ y ∈ S, walkKernel G hloc a x y ≤ 1 := by
-  let N := insert x (neighbors G hloc x)
-  calc
-    ∑ y ∈ S, walkKernel G hloc a x y ≤ ∑ y ∈ S ∪ N, walkKernel G hloc a x y :=
-      Finset.sum_le_sum_of_subset_of_nonneg Finset.subset_union_left
-        (fun y _ _ => walkKernel_nonneg G hloc D hD a ha haD x y)
-    _ = ∑ y ∈ N, walkKernel G hloc a x y := by
-      symm
-      apply Finset.sum_subset Finset.subset_union_right
-      intro y _ hy
-      have hxy : x ≠ y := by intro h; subst y; exact hy (by simp [N])
-      have hna : ¬ Adj G x y := by intro h; exact hy (by simp [N,h])
-      simp [walkKernel,hxy,hna]
-    _ = 1 := walkKernel_row_sum G hloc a x
-
- theorem exists_walk_operator (G : BondGraph V E) (hloc : LocallyFinite G)
-    (D : ℕ) (hD : ∀ x, degree G x ≤ D) (a : ℝ) (ha : 0 ≤ a) (haD : a*D ≤ 1) :
-    ∃ P : Hilbert V →L[ℂ] Hilbert V, ‖P‖ ≤ 1 ∧
-      P.toLinearMap.IsSymmetric ∧
-      ∀ x y, P (lp.single 2 y (1 : ℂ)) x = (walkKernel G hloc a x y : ℂ) := by
-  obtain ⟨P,hP,hK⟩ := exists_operator_of_row_column_bound (walkKernel G hloc a) 1
-    zero_le_one (walkKernel_nonneg G hloc D hD a ha haD)
-    (walkKernel_row_bound G hloc D hD a ha haD) (by
-      intro y S
-      simp_rw [walkKernel_symmetric G hloc a]
-      exact walkKernel_row_bound G hloc D hD a ha haD y S)
-  exact ⟨P,hP,symmetric_of_symmetric_kernel P _ hK (walkKernel_symmetric G hloc a),hK⟩
-
- theorem finiteAdjPairs_sum (G : BondGraph V E) (K : Finset V) (f : V × V → ℝ) :
-    (∑ e ∈ finiteAdjPairs G K, f e) =
-      ∑ x ∈ K, ∑ y ∈ K, if Adj G x y then f (x,y) else 0 := by
-  simp [finiteAdjPairs,Finset.sum_filter,Finset.sum_product]
-
- theorem finiteAdjPairs_source_sum (G : BondGraph V E) (hloc : LocallyFinite G)
-    (S : Finset V) (f : V → ℝ) (hout : ∀ x ∉ S, f x = 0) :
-    (∑ e ∈ finiteAdjPairs G (neighborClosure G hloc S), f e.1) =
-      ∑ x ∈ S, ((neighbors G hloc x).card : ℝ) * f x := by
-  let K := neighborClosure G hloc S
-  rw [finiteAdjPairs_sum]
-  have he (x : V) (hx : x ∈ S) :
-      K.filter (fun y => Adj G x y) = neighbors G hloc x := by
-    ext y
-    simp only [Finset.mem_filter,mem_neighbors]
-    exact ⟨fun h => h.2,fun h => ⟨neighbor_mem_neighborClosure G hloc S hx h,h⟩⟩
-  have hout' : ∀ x ∉ S, (∑ y ∈ K, if Adj G x y then f x else 0) = 0 := by
-    intro x hx; simp [hout x hx]
-  change (∑ x ∈ K, ∑ y ∈ K, if Adj G x y then f x else 0) = _
-  rw [← Finset.sum_subset (subset_neighborClosure G hloc S)
-    (fun x _ hx => hout' x hx)]
-  apply Finset.sum_congr rfl
-  intro x hx
-  rw [← Finset.sum_filter,he x hx]
-  simp
-
- theorem finiteAdjPairs_product_sum (G : BondGraph V E) (hloc : LocallyFinite G)
-    (S : Finset V) (f : V → ℝ) (hout : ∀ x ∉ S, f x = 0) :
-    (∑ e ∈ finiteAdjPairs G (neighborClosure G hloc S), f e.1 * f e.2) =
-      ∑ x ∈ S, ∑ y ∈ S, if Adj G x y then f x*f y else 0 := by
-  rw [finiteAdjPairs_sum]
-  have hs := subset_neighborClosure G hloc S
-  calc
-    _ = ∑ x ∈ S, ∑ y ∈ neighborClosure G hloc S,
-        if Adj G x y then f x*f y else 0 := by
-      symm; apply Finset.sum_subset hs
-      intro x _ hx; simp [hout x hx]
-    _ = _ := by
-      apply Finset.sum_congr rfl
-      intro x _
-      symm; apply Finset.sum_subset hs
-      intro y _ hy; simp [hout y hy]
-
- theorem walkKernel_energy (G : BondGraph V E) (hloc : LocallyFinite G)
-    (a : ℝ) (S : Finset V) (f : V → ℝ) (hout : ∀ x ∉ S, f x = 0) :
-    (∑ x ∈ S, ∑ y ∈ S, f x*f y*walkKernel G hloc a x y) =
-      ∑ x ∈ S, (f x)^2 - a/2 *
-        ∑ e ∈ finiteAdjPairs G (neighborClosure G hloc S), (f e.1-f e.2)^2 := by
-  let P := finiteAdjPairs G (neighborClosure G hloc S)
-  have henergy : (∑ e ∈ P, (f e.1-f e.2)^2) =
-      2*(∑ x ∈ S, (neighbors G hloc x).card * (f x)^2) -
-      2*(∑ x ∈ S, ∑ y ∈ S, if Adj G x y then f x*f y else 0) := by
-    have hs := sum_swap_finiteAdjPairs G (neighborClosure G hloc S)
-      (fun e => (f e.1)^2)
-    change (∑ e ∈ P,(f e.2)^2) = (∑ e ∈ P,(f e.1)^2) at hs
-    have hfirst := finiteAdjPairs_source_sum G hloc S (fun x => (f x)^2)
-      (fun x hx => by rw [hout x hx]; norm_num)
-    calc
-      _ = (∑ e ∈ P, (f e.1)^2) + (∑ e ∈ P,(f e.2)^2) -
-          2*(∑ e ∈ P,f e.1*f e.2) := by
-        simp_rw [sub_sq]
-        rw [Finset.sum_add_distrib,Finset.sum_sub_distrib]
-        simp_rw [mul_assoc,← Finset.mul_sum]
-        ring
-      _ = _ := by rw [hs,hfirst,finiteAdjPairs_product_sum G hloc S f hout]; ring
-  rw [henergy]
-  have hpoint (x : V) (hx : x ∈ S) :
-      (∑ y ∈ S, f x*f y*walkKernel G hloc a x y) =
-        (f x)^2 - a*((neighbors G hloc x).card*(f x)^2) +
-         a*(∑ y ∈ S, if Adj G x y then f x*f y else 0) := by
-    simp only [walkKernel,mul_add,Finset.sum_add_distrib]
-    have hi : (∑ y ∈ S, f x*f y*(if x=y then 1-a*(neighbors G hloc x).card else 0)) =
-        (f x)^2*(1-a*(neighbors G hloc x).card) := by
-      simp_rw [mul_ite,mul_zero]
-      rw [Finset.sum_ite_eq]
-      simp only [hx,ite_true]
-      ring
-    rw [hi,Finset.mul_sum]
-    have hj : (∑ y ∈ S, f x*f y*(if Adj G x y then a else 0)) =
-        ∑ y ∈ S, a*(if Adj G x y then f x*f y else 0) := by
-      apply Finset.sum_congr rfl
-      intro y _
-      split_ifs <;> ring
-    rw [hj]
-    ring
-  simp_rw [Finset.sum_congr rfl hpoint,Finset.sum_add_distrib,
-    Finset.sum_sub_distrib,←Finset.mul_sum]
-  ring
-
- 
-
-theorem walkKernel_quadratic_bound (G : BondGraph V E) (hloc : LocallyFinite G)
-    (hh : 0 < hV G) (D : ℕ) (hDp : 0 < D) (hD : ∀ x, degree G x ≤ D)
-    (a : ℝ) (ha : 0 ≤ a) (S : Finset V) (f : V → ℝ)
-    (hf : ∀ x, 0 ≤ f x) (hout : ∀ x ∉ S, f x = 0) :
-    (∑ x ∈ S, ∑ y ∈ S, f x*f y*walkKernel G hloc a x y) ≤
-      (1 - a*(hV G)^2/(2*D)) * ∑ x ∈ S, (f x)^2 := by
-  have hDp' : 0 < (D:ℝ) := by exact_mod_cast hDp
-  have hc := bondGraph_cheeger_energy G hloc hh D hD S f hf hout
-  have he : ((hV G)^2 * ∑ x ∈ S, (f x)^2)/(D:ℝ) ≤
-      ∑ e ∈ finiteAdjPairs G (neighborClosure G hloc S), (f e.1-f e.2)^2 :=
-    (div_le_iff₀ hDp').mpr (by simpa only [mul_comm] using hc)
-  rw [walkKernel_energy G hloc a S f hout]
-  calc
-    _ ≤ (∑ x ∈ S, (f x)^2) - a/2 *
-        (((hV G)^2 * ∑ x ∈ S, (f x)^2)/(D:ℝ)) :=
-      sub_le_sub_left (mul_le_mul_of_nonneg_left he (by positivity)) _
-    _ = _ := by ring
-
- 
-
-theorem walk_operator_norm_lt_one (G : BondGraph V E) (hloc : LocallyFinite G)
-    (hh : 0 < hV G) (D : ℕ) (hDp : 0 < D) (hD : ∀ x, degree G x ≤ D)
-    (a : ℝ) (ha : 0 < a) (haD : a*D ≤ 1)
-    (P : Hilbert V →L[ℂ] Hilbert V)
-    (hP : ∀ x y, P (lp.single 2 y (1:ℂ)) x = (walkKernel G hloc a x y : ℂ)) :
-    ‖P‖ < 1 := by
-  let C := max 0 (1-a*(hV G)^2/(2*D))
-  have hDp' : 0 < (D:ℝ) := by exact_mod_cast hDp
-  have hC : 0 ≤ C := le_max_left _ _
-  have hC1 : C < 1 := by
-    apply max_lt (by norm_num)
-    have : 0 < a*(hV G)^2/(2*D) := by positivity
-    linarith
-  have hn : ‖P‖ ≤ C := by
-    apply norm_le_of_finite_quadratic P
-      (symmetric_of_symmetric_kernel P _ hP (walkKernel_symmetric G hloc a)) C hC
-    intro f
-    calc
-      _ ≤ ∑ x ∈ f.support, ∑ y ∈ f.support, ‖f x‖*‖f y‖*walkKernel G hloc a x y :=
-        abs_inner_kernel_finite_le P _ hP (walkKernel_nonneg G hloc D hD a ha.le haD) f
-      _ ≤ (1-a*(hV G)^2/(2*D)) * ∑ x ∈ f.support, ‖f x‖^2 :=
-        walkKernel_quadratic_bound G hloc hh D hDp hD a ha.le f.support
-          (fun x => ‖f x‖) (fun _ => norm_nonneg _) (by
-            intro x hx; simp [Finsupp.notMem_support_iff.mp hx])
-      _ ≤ C * ∑ x ∈ f.support, ‖f x‖^2 :=
-        mul_le_mul_of_nonneg_right (le_max_right _ _) (Finset.sum_nonneg (fun _ _ => sq_nonneg _))
-      _ = C * ‖finiteEmbedding f‖^2 := by rw [finiteEmbedding_norm_sq]
-  exact hn.trans_lt hC1
-
- 
-
-theorem exists_strict_walk_operator (G : BondGraph V E) (hloc : LocallyFinite G)
-    (hqt : QuasiTransitive G) (hh : 0 < hV G) :
-    ∃ (D : ℕ) (P : Hilbert V →L[ℂ] Hilbert V), 0 < D ∧
-      (∀ x, degree G x ≤ D) ∧ ‖P‖ < 1 ∧ P.toLinearMap.IsSymmetric ∧
-      ∀ x y, P (lp.single 2 y (1:ℂ)) x =
-        (walkKernel G hloc (1/(2*(D:ℝ))) x y : ℂ) := by
-  obtain ⟨D,hDp,hD⟩ := exists_degree_bound G hqt
-  have hDp' : 0 < (D:ℝ) := by exact_mod_cast hDp
-  have ha : 0 < (1:ℝ)/(2*D) := by positivity
-  have haD : (1:ℝ)/(2*D)*D ≤ 1 := by
-    rw [div_mul_eq_mul_div]
-    apply (div_le_iff₀ (by positivity)).mpr
-    nlinarith
-  obtain ⟨P,_,hPs,hP⟩ := exists_walk_operator G hloc D hD _ ha.le haD
-  exact ⟨D,P,hDp,hD,walk_operator_norm_lt_one G hloc hh D hDp hD _ ha haD P hP,hPs,hP⟩
-
-end SymmetricWalk
-
-section PositiveConnectionKernel
-open scoped ComplexOrder Matrix
-attribute [local instance] Classical.propDecidable
-
- 
-
-theorem equivalenceMatrix_posSemidef {α : Type*} [Fintype α] (R : Setoid α) :
-    Matrix.PosSemidef (fun x y => if R.r x y then (1:ℂ) else 0) := by
-  classical
-  let : Fintype (Quotient R) := Fintype.ofFinite (Quotient R)
-  let J : Matrix (Quotient R) α ℂ := fun q x => if q = Quotient.mk R x then 1 else 0
-  have hJ : Jᴴ * J = (fun x y => if R.r x y then (1:ℂ) else 0) := by
-    ext x y
-    change (∑ q : Quotient R, star (J q x) * J q y) = _
-    have hr : R.r y x ↔ R.r x y := ⟨R.symm, R.symm⟩
-    simp [J, Quotient.eq, hr]
-  rw [← hJ]
-  exact Matrix.posSemidef_conjTranspose_mul_self J
-
-theorem connection_quadratic_nonneg (G : BondGraph V E) (ω : Set E)
-    (S : Finset V) (f : V → ℂ) :
-    0 ≤ ∑ x ∈ S, ∑ y ∈ S,
-      star (f x) * (if Conn G ω x y then (1:ℂ) else 0) * f y := by
-  classical
-  let R : Setoid S := ⟨fun x y => Conn G ω x y,
-    ⟨fun x => conn_refl G ω x, fun h => conn_symm G ω h, fun h h' => conn_trans G ω h h'⟩⟩
-  have h := (equivalenceMatrix_posSemidef R).dotProduct_mulVec_nonneg
-    (fun x : S => f x)
-  simp only [dotProduct, Matrix.mulVec, Finset.mul_sum, Pi.star_apply] at h
-  change 0 ≤ ∑ x : S, ∑ y : S,
-    star (f x) * ((if Conn G ω x y then (1:ℂ) else 0) * f y) at h
-  have he : (∑ x : S, ∑ y : S,
-      star (f x) * ((if Conn G ω x y then (1:ℂ) else 0) * f y)) =
-      ∑ x ∈ S, ∑ y ∈ S,
-      star (f x) * ((if Conn G ω x y then (1:ℂ) else 0) * f y) := by
-    rw [Finset.sum_coe_sort S (fun x => ∑ y : S,
-      star (f x) * ((if Conn G ω x y then (1:ℂ) else 0) * f y))]
-    apply Finset.sum_congr rfl
-    intro x _
-    exact Finset.sum_coe_sort S (fun y =>
-      star (f x) * ((if Conn G ω x y then (1:ℂ) else 0) * f y))
-  rw [he] at h
-  simpa only [mul_assoc] using h
-
-variable [Countable V] [Countable E]
-
-theorem conn_indicator_integrable (G : BondGraph V E) (p : unitInterval) (x y : V) :
-    Integrable (fun ω => if Conn G ω x y then (1:ℂ) else 0) (law p) := by
-  change Integrable ({ω | Conn G ω x y}.indicator (fun _ => (1:ℂ))) (law p)
-  exact (integrable_const (1:ℂ)).indicator (measurableSet_conn G x y)
-
-theorem twoPoint_eq_integral (G : BondGraph V E) (p : unitInterval) (x y : V) :
-    (twoPoint G p x y : ℂ) =
-      ∫ ω, (if Conn G ω x y then (1:ℂ) else 0) ∂law p := by
-  simpa only [Set.indicator, Set.mem_ofPred_eq, Measure.real, twoPoint,
-    Complex.real_smul, mul_one] using
-    (integral_indicator_const (μ := law p) (1:ℂ) (measurableSet_conn G x y)).symm
-
-theorem twoPoint_quadratic_nonneg (G : BondGraph V E) (p : unitInterval)
-    (S : Finset V) (f : V → ℂ) :
-    0 ≤ ∑ x ∈ S, ∑ y ∈ S, star (f x) * (twoPoint G p x y : ℂ) * f y := by
-  have hi (x y : V) : Integrable
-      (fun ω => star (f x) * (if Conn G ω x y then (1:ℂ) else 0) * f y) (law p) :=
-    ((conn_indicator_integrable G p x y).const_mul _).mul_const _
-  have hsum : (∑ x ∈ S, ∑ y ∈ S, star (f x) * (twoPoint G p x y : ℂ) * f y) =
-      ∫ ω, ∑ x ∈ S, ∑ y ∈ S,
-        star (f x) * (if Conn G ω x y then (1:ℂ) else 0) * f y ∂law p := by
-    rw [integral_finsetSum S (fun x _ => integrable_finsetSum S (fun y _ => hi x y))]
-    apply Finset.sum_congr rfl
-    intro x _
-    rw [integral_finsetSum S (fun y _ => hi x y)]
-    apply Finset.sum_congr rfl
-    intro y _
-    rw [integral_mul_const, integral_const_mul, twoPoint_eq_integral]
-  rw [hsum]
-  exact integral_nonneg (fun ω => connection_quadratic_nonneg G ω S f)
-
- 
-
-theorem HasKernel.isPositive (G : BondGraph V E) (p : unitInterval)
-    (A : Hilbert V →L[ℂ] Hilbert V) (hA : HasKernel G p A) : A.IsPositive := by
-  have hs : A.toLinearMap.IsSymmetric :=
-    symmetric_of_symmetric_kernel A (twoPoint G p) (fun x y => hA x y) (twoPoint_symm G p)
-  refine ⟨hs, ?_⟩
-  intro f
-  apply finiteEmbedding_dense.induction_on f
-    (p := fun g => 0 ≤ (inner ℂ (A g) g).re)
-  · exact isClosed_le continuous_const (by fun_prop)
-  · intro g
-    rw [show inner ℂ (A (finiteEmbedding g)) (finiteEmbedding g) =
-      inner ℂ (finiteEmbedding g) (A (finiteEmbedding g)) from hs _ _, finiteEmbedding_inner_left]
-    simp only [kernel_finiteEmbedding A (twoPoint G p) (fun x y => hA x y)]
-    simp_rw [Finset.mul_sum]
-    have h := twoPoint_quadratic_nonneg G p g.support g
-    apply Complex.nonneg_iff.mp at h
-    convert h.1 using 1
-    congr 1
-    apply Finset.sum_congr rfl
-    intro x _
-    apply Finset.sum_congr rfl
-    intro y _
-    ring
-
-end PositiveConnectionKernel
-
-section InvariantOperatorTrace
-open scoped ComplexOrder InnerProduct
-attribute [local instance] Classical.propDecidable
-local instance spectralRealOperatorScalarTower :
-    IsScalarTower ℝ (Hilbert V →L[ℂ] Hilbert V) (Hilbert V →L[ℂ] Hilbert V) where
-  smul_assoc scalar first second := by
-    ext vector vertex
-    rfl
-local instance spectralRealOperatorSMulCommClass :
-    SMulCommClass ℝ (Hilbert V →L[ℂ] Hilbert V) (Hilbert V →L[ℂ] Hilbert V) where
-  smul_comm scalar first second := by
-    apply ContinuousLinearMap.ext
-    intro vector
-    exact (first.map_smul_of_tower scalar (second vector)).symm
-local instance spectralNonUnitalContinuousFunctionalCalculus : NonUnitalContinuousFunctionalCalculus ℝ
-    (Hilbert V →L[ℂ] Hilbert V) IsSelfAdjoint :=
-  IsSelfAdjoint.instNonUnitalContinuousFunctionalCalculus (A := Hilbert V →L[ℂ] Hilbert V)
-local instance spectralContinuousFunctionalCalculus : ContinuousFunctionalCalculus ℝ
-    (Hilbert V →L[ℂ] Hilbert V) IsSelfAdjoint :=
-  IsSelfAdjoint.instContinuousFunctionalCalculus (A := Hilbert V →L[ℂ] Hilbert V)
-noncomputable def permuteHilbert (a : V ≃ V) : Hilbert V ≃ₗᵢ[ℂ] Hilbert V where
-  toFun f := ⟨fun x => f (a.symm x), by
-    apply memℓp_gen
-    exact a.symm.summable_iff.mpr (lp.memℓp f |>.summable (by norm_num))⟩
-  invFun f := ⟨fun x => f (a x), by
-    apply memℓp_gen
-    exact a.summable_iff.mpr (lp.memℓp f |>.summable (by norm_num))⟩
-  left_inv f := by ext x; simp
-  right_inv f := by ext x; simp
-  map_add' f g := by ext x; rfl
-  map_smul' c f := by ext x; rfl
-  norm_map' f := by
-    rw [lp.norm_eq_tsum_rpow (by norm_num), lp.norm_eq_tsum_rpow (by norm_num)]
-    congr 1
-    exact a.symm.tsum_eq (fun x => ‖f x‖ ^ (2:ℝ≥0∞).toReal)
-
-@[simp] theorem permuteHilbert_apply (a : V ≃ V) (f : Hilbert V) (x : V) :
-    permuteHilbert a f x = f (a.symm x) := rfl
-
-@[simp] theorem permuteHilbert_single (a : V ≃ V) (x : V) (c : ℂ) :
-    permuteHilbert a (lp.single 2 x c) = lp.single 2 (a x) c := by
-  ext y
-  simp only [permuteHilbert_apply, lp.coeFn_single]
-  simp [Pi.single_apply, eq_comm, Equiv.eq_symm_apply]
-
-noncomputable def matrixCoeff (A : Hilbert V →L[ℂ] Hilbert V) (x y : V) : ℂ :=
-  A (lp.single 2 y (1:ℂ)) x
-
-@[simp] theorem matrixCoeff_eq_inner (A : Hilbert V →L[ℂ] Hilbert V) (x y : V) :
-    matrixCoeff A x y = inner ℂ (lp.single 2 x (1:ℂ)) (A (lp.single 2 y (1:ℂ))) := by
-  simp [matrixCoeff,lp.inner_single_left]
-
-theorem operator_kernel_hasSum (A : Hilbert V →L[ℂ] Hilbert V) (f : Hilbert V) (x : V) :
-    HasSum (fun y => matrixCoeff A x y * f y) (A f x) := by
-  have h := ((lp.evalCLM ℂ (fun _ : V => ℂ) 2 x).comp A).hasSum
-    (lp.hasSum_single (by norm_num : (2:ℝ≥0∞) ≠ ⊤) f)
-  change HasSum (fun y => A (lp.single 2 y (f y)) x) (A f x) at h
-  have heq : (fun y : V => A (lp.single 2 y (f y)) x) =
-      (fun y => matrixCoeff A x y * f y) := by
-    funext y
-    have he : lp.single (E := fun _ : V => ℂ) 2 y (f y) =
-        f y • lp.single 2 y (1:ℂ) := by
-      rw [← lp.single_smul]; simp
-    simp [he, matrixCoeff, mul_comm]
-  rw [heq] at h
-  exact h
-
-@[simp] theorem matrixCoeff_mul (A B : Hilbert V →L[ℂ] Hilbert V) (x y : V) :
-    matrixCoeff (A*B) x y = ∑' z, matrixCoeff A x z * matrixCoeff B z y := by
-  exact (operator_kernel_hasSum A (B (lp.single 2 y (1:ℂ))) x).tsum_eq.symm
-
-@[simp] theorem matrixCoeff_adjoint (A : Hilbert V →L[ℂ] Hilbert V) (x y : V) :
-    matrixCoeff (star A) x y = star (matrixCoeff A y x) := by
-  rw [matrixCoeff_eq_inner, matrixCoeff_eq_inner]
-  change inner ℂ _ ((ContinuousLinearMap.adjoint A) _) = _
-  rw [ContinuousLinearMap.adjoint_inner_right]
-  simpa only [RCLike.star_def] using
-    (inner_conj_symm (A (lp.single 2 x (1:ℂ))) (lp.single 2 y (1:ℂ))).symm
-
- 
-theorem matrixCoeff_invariant_iff (a : V ≃ V) (A : Hilbert V →L[ℂ] Hilbert V) :
-    (∀ x y, matrixCoeff A (a x) (a y) = matrixCoeff A x y) ↔
-      Commute A (permuteHilbert a).toContinuousLinearEquiv.toContinuousLinearMap := by
-  constructor
-  · intro h
-    change A * (permuteHilbert a).toContinuousLinearEquiv.toContinuousLinearMap =
-      (permuteHilbert a).toContinuousLinearEquiv.toContinuousLinearMap * A
-    apply lp.ext_continuousLinearMap (by norm_num : (2:ℝ≥0∞) ≠ ⊤)
-    intro y
-    apply ContinuousLinearMap.ext
-    intro c
-    ext x
-    have hs : lp.single (E := fun _ : V => ℂ) 2 y c =
-        c • lp.single 2 y (1:ℂ) := by rw [← lp.single_smul]; simp
-    have hh := h (a.symm x) y
-    simp only [Equiv.apply_symm_apply] at hh
-    change A (permuteHilbert a (lp.single 2 y c)) x =
-      permuteHilbert a (A (lp.single 2 y c)) x
-    rw [hs]
-    simp only [map_smul, permuteHilbert_single, lp.coeFn_smul, Pi.smul_apply,
-      permuteHilbert_apply]
-    change c * matrixCoeff A x (a y) = c * matrixCoeff A (a.symm x) y
-    rw [hh]
-  · intro h x y
-    have hh := congrArg (fun F : Hilbert V →L[ℂ] Hilbert V =>
-      F (lp.single 2 y (1:ℂ)) (a x)) h.eq
-    simpa only [mul_apply_eq_comp, LinearIsometryEquiv.coe_toContinuousLinearEquiv,
-      ContinuousLinearEquiv.coe_coe, permuteHilbert_single, permuteHilbert_apply,
-      Equiv.symm_apply_apply, matrixCoeff] using hh
-
- theorem matrixCoeff_sqrt_invariant (a : V ≃ V) (A : Hilbert V →L[ℂ] Hilbert V)
-    (h : ∀ x y, matrixCoeff A (a x) (a y) = matrixCoeff A x y) :
-    ∀ x y, matrixCoeff (CFC.sqrt A) (a x) (a y) = matrixCoeff (CFC.sqrt A) x y := by
-  apply (matrixCoeff_invariant_iff a _).mpr
-  rw [CFC.sqrt_eq_cfc]
-  exact ((matrixCoeff_invariant_iff a A).mp h).cfc_nnreal NNReal.sqrt
-
- theorem matrixCoeff_mul_invariant (a : V ≃ V) (A B : Hilbert V →L[ℂ] Hilbert V)
-    (hA : ∀ x y, matrixCoeff A (a x) (a y) = matrixCoeff A x y)
-    (hB : ∀ x y, matrixCoeff B (a x) (a y) = matrixCoeff B x y) :
-    ∀ x y, matrixCoeff (A*B) (a x) (a y) = matrixCoeff (A*B) x y := by
-  intro x y
-  simp only [matrixCoeff_mul]
-  rw [← a.tsum_eq (fun z => matrixCoeff A (a x) z * matrixCoeff B z (a y))]
-  simp only [hA,hB]
-
- theorem matrixCoeff_nonneg (A : Hilbert V →L[ℂ] Hilbert V) (hA : 0 ≤ A) (x : V) :
-    0 ≤ matrixCoeff A x x := by
-  rw [matrixCoeff_eq_inner]
-  exact (ContinuousLinearMap.nonneg_iff_isPositive |>.mp hA).inner_nonneg_right _
-
- theorem matrixCoeff_mono {A B : Hilbert V →L[ℂ] Hilbert V} (h : A ≤ B) (x : V) :
-    matrixCoeff A x x ≤ matrixCoeff B x x := by
-  have hh := matrixCoeff_nonneg (B-A) (sub_nonneg.mpr h) x
-  simpa only [matrixCoeff, sub_apply, lp.coeFn_sub,
-    Pi.sub_apply, sub_nonneg] using hh
-
-section RootTrace
-variable {ι : Type*} [Fintype ι]
-variable (root : ι → V) (weight : ι → ℝ)
-
-noncomputable def rootTrace (A : Hilbert V →L[ℂ] Hilbert V) : ℂ :=
-  ∑ i, (weight i : ℂ) * matrixCoeff A (root i) (root i)
-
-@[simp] theorem rootTrace_add (A B : Hilbert V →L[ℂ] Hilbert V) :
-    rootTrace root weight (A+B) = rootTrace root weight A + rootTrace root weight B := by
-  simp [rootTrace, matrixCoeff, add_apply, mul_add, Finset.sum_add_distrib]
-
-@[simp] theorem rootTrace_smul (c : ℂ) (A : Hilbert V →L[ℂ] Hilbert V) :
-    rootTrace root weight (c • A) = c * rootTrace root weight A := by
-  simp [rootTrace, matrixCoeff, smul_apply, ← Finset.mul_sum, mul_left_comm]
-
-theorem rootTrace_nonneg (hw : ∀ i, 0 ≤ weight i)
-    (A : Hilbert V →L[ℂ] Hilbert V) (hA : 0 ≤ A) : 0 ≤ rootTrace root weight A := by
-  exact Finset.sum_nonneg fun i _ => mul_nonneg (by simpa using hw i)
-    (matrixCoeff_nonneg A hA (root i))
-
-theorem rootTrace_mono (hw : ∀ i, 0 ≤ weight i)
-    {A B : Hilbert V →L[ℂ] Hilbert V} (h : A ≤ B) :
-    rootTrace root weight A ≤ rootTrace root weight B := by
-  apply Finset.sum_le_sum
-  intro i _
-  exact mul_le_mul_of_nonneg_left (matrixCoeff_mono h (root i)) (by simpa using hw i)
-
-@[simp] theorem rootTrace_one (hw : ∑ i, weight i = 1) :
-    rootTrace root weight (1 : Hilbert V →L[ℂ] Hilbert V) = 1 := by
-  simpa [rootTrace, matrixCoeff, lp.coeFn_single, ← Complex.ofReal_sum] using
-    congrArg (fun t : ℝ => (t : ℂ)) hw
-
- 
-def KernelInvariant (Γ : Set (V ≃ V)) (A : Hilbert V →L[ℂ] Hilbert V) : Prop :=
-  ∀ a ∈ Γ, ∀ x y, matrixCoeff A (a x) (a y) = matrixCoeff A x y
-
-theorem KernelInvariant.mul {Γ : Set (V ≃ V)}
-    {A B : Hilbert V →L[ℂ] Hilbert V}
-    (hA : KernelInvariant Γ A) (hB : KernelInvariant Γ B) :
-    KernelInvariant Γ (A*B) := by
-  intro a ha
-  exact matrixCoeff_mul_invariant a A B (hA a ha) (hB a ha)
-
-theorem KernelInvariant.star {Γ : Set (V ≃ V)}
-    {A : Hilbert V →L[ℂ] Hilbert V} (hA : KernelInvariant Γ A) :
-    KernelInvariant Γ (star A) := by
-  intro a ha x y
-  simp only [matrixCoeff_adjoint, hA a ha]
-
-theorem KernelInvariant.sqrt {Γ : Set (V ≃ V)}
-    {A : Hilbert V →L[ℂ] Hilbert V} (hA : KernelInvariant Γ A) :
-    KernelInvariant Γ (CFC.sqrt A) := by
-  intro a ha
-  exact matrixCoeff_sqrt_invariant a A (hA a ha)
-
- 
-
-def RootTransport (Γ : Set (V ≃ V)) : Prop :=
-  ∀ m : V → V → ℂ,
-    (∀ a ∈ Γ, ∀ x y, m (a x) (a y) = m x y) →
-    (∀ x, Summable (m x)) → (∀ y, Summable (fun x => m x y)) →
-    (∑ i, (weight i : ℂ) * ∑' y, m (root i) y) =
-    (∑ i, (weight i : ℂ) * ∑' x, m x (root i))
-
-theorem rootTrace_cyclic {Γ : Set (V ≃ V)}
-    (htr : RootTransport root weight Γ)
-    (A B : Hilbert V →L[ℂ] Hilbert V)
-    (hA : KernelInvariant Γ A) (hB : KernelInvariant Γ B) :
-    rootTrace root weight (A*B) = rootTrace root weight (B*A) := by
-  have ht := htr (fun x y => matrixCoeff A x y * matrixCoeff B y x)
-    (by intro a ha x y; rw [hA a ha, hB a ha])
-    (fun x => (operator_kernel_hasSum A (B (lp.single 2 x (1:ℂ))) x).summable)
-    (fun y => by
-      simpa only [matrixCoeff, mul_comm] using
-        (operator_kernel_hasSum B (A (lp.single 2 y (1:ℂ))) y).summable)
-  simpa only [rootTrace, matrixCoeff_mul, mul_comm] using ht
-
-theorem rootTrace_diag_le {Γ : Set (V ≃ V)}
-    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
-    (hw : ∀ i, 0 ≤ weight i) (c : ℝ) (hc : 0 ≤ c)
-    (hmin : ∀ i, 1 ≤ c * weight i)
-    (A : Hilbert V →L[ℂ] Hilbert V) (hA : KernelInvariant Γ A)
-    (hApos : 0 ≤ A) (x : V) :
-    matrixCoeff A x x ≤ (c : ℂ) * rootTrace root weight A := by
-  obtain ⟨i,a,ha,hax⟩ := hcov x
-  have hi : (weight i : ℂ) * matrixCoeff A (root i) (root i) ≤
-      rootTrace root weight A := by
-    unfold rootTrace
-    exact Finset.single_le_sum (s := Finset.univ)
-      (f := fun j => (weight j : ℂ) * matrixCoeff A (root j) (root j))
-      (fun j _ => mul_nonneg (by simpa using hw j) (matrixCoeff_nonneg A hApos _))
-      (Finset.mem_univ i)
-  calc
-    matrixCoeff A x x = matrixCoeff A (root i) (root i) := by
-      rw [← hax, hA a ha]
-    _ ≤ (c * weight i : ℝ) * matrixCoeff A (root i) (root i) := by
-      simpa only [one_mul] using mul_le_mul_of_nonneg_right
-        (show (1:ℂ) ≤ ((c * weight i : ℝ) : ℂ) by exact_mod_cast hmin i)
-        (matrixCoeff_nonneg A hApos (root i))
-    _ = (c : ℂ) * ((weight i : ℂ) * matrixCoeff A (root i) (root i)) := by
-      push_cast; ring
-    _ ≤ (c : ℂ) * rootTrace root weight A :=
-      mul_le_mul_of_nonneg_left hi (by exact_mod_cast hc)
-
-@[simp] theorem rootTrace_real_smul (c : ℝ) (A : Hilbert V →L[ℂ] Hilbert V) :
-    rootTrace root weight (c • A) = (c : ℂ) * rootTrace root weight A := by
-  simp [rootTrace, matrixCoeff, smul_apply, Complex.real_smul,
-    ← Finset.mul_sum, mul_left_comm]
-
-theorem rootTrace_positive_mul_le {Γ : Set (V ≃ V)}
-    (htr : RootTransport root weight Γ) (hw : ∀ i, 0 ≤ weight i)
-    (A B : Hilbert V →L[ℂ] Hilbert V)
-    (hA : KernelInvariant Γ A) (hB : KernelInvariant Γ B)
-    (hApos : 0 ≤ A) (hBsa : IsSelfAdjoint B) :
-    rootTrace root weight (B*A) ≤ (‖B‖ : ℂ) * rootTrace root weight A := by
-  let S := CFC.sqrt A
-  have hS : S*S = A := CFC.sqrt_mul_sqrt_self A hApos
-  have hSstar : star S = S := (IsSelfAdjoint.of_nonneg (CFC.sqrt_nonneg A)).star_eq
-  have hSI : KernelInvariant Γ S := hA.sqrt
-  have hbound : S*B*S ≤ ‖B‖ • A := by
-    convert (CStarAlgebra.star_left_conjugate_le_norm_smul (a := S) (b := B) hBsa) using 1
-    · rw [hSstar]
-    · rw [hSstar, hS]
-      ext f x
-      rfl
-  calc
-    rootTrace root weight (B*A) = rootTrace root weight (S*(B*S)) := by
-      rw [rootTrace_cyclic root weight htr S (B*S) hSI (hB.mul hSI)]
-      rw [mul_assoc, hS]
-    _ = rootTrace root weight (S*B*S) := by rw [mul_assoc]
-    _ ≤ rootTrace root weight (‖B‖ • A) := rootTrace_mono root weight hw hbound
-    _ = (‖B‖ : ℂ) * rootTrace root weight A := rootTrace_real_smul root weight _ _
-
-theorem rootTrace_conjugate_le {Γ : Set (V ≃ V)}
-    (htr : RootTransport root weight Γ) (hw : ∀ i, 0 ≤ weight i)
-    (A B : Hilbert V →L[ℂ] Hilbert V)
-    (hA : KernelInvariant Γ A) (hB : KernelInvariant Γ B)
-    (hApos : 0 ≤ A) :
-    rootTrace root weight (B*A*star B) ≤ ((‖B‖ ^ 2 : ℝ) : ℂ) * rootTrace root weight A := by
-  calc
-    rootTrace root weight (B*A*star B) = rootTrace root weight (star B*(B*A)) :=
-      rootTrace_cyclic root weight htr (B*A) (star B) (hB.mul hA) hB.star
-    _ = rootTrace root weight ((star B*B)*A) := by rw [mul_assoc]
-    _ ≤ (‖star B*B‖ : ℂ) * rootTrace root weight A :=
-      rootTrace_positive_mul_le root weight htr hw A (star B*B) hA (hB.star.mul hB)
-        hApos (IsSelfAdjoint.of_nonneg (star_mul_self_nonneg B))
-    _ = ((‖B‖ ^ 2 : ℝ) : ℂ) * rootTrace root weight A := by
-      have hnorm := CStarRing.norm_star_mul_self (x := B)
-      have hn : ‖star B * B‖ = ‖B‖ ^ 2 := by
-        convert hnorm using 1; simp only [pow_two]
-      rw [hn]
-
- theorem matrixCoeff_gram (L M : Hilbert V →L[ℂ] Hilbert V) (x y : V) :
-    matrixCoeff (L*star M) x y =
-      inner ℂ (star L (lp.single 2 x (1:ℂ))) (star M (lp.single 2 y (1:ℂ))) := by
-  rw [matrixCoeff_eq_inner]
-  change inner ℂ _ (L ((ContinuousLinearMap.adjoint M) _)) = _
-  rw [← ContinuousLinearMap.adjoint_inner_left]
-  rfl
-
- theorem matrixCoeff_gram_norm_sq_le
-    (L M : Hilbert V →L[ℂ] Hilbert V) (x y : V) :
-    ‖matrixCoeff (L*star M) x y‖ ^ 2 ≤
-      (matrixCoeff (L*star L) x x).re * (matrixCoeff (M*star M) y y).re := by
-  simp only [matrixCoeff_gram]
-  have h := inner_mul_inner_self_le (𝕜 := ℂ)
-      (star L (lp.single 2 x (1:ℂ))) (star M (lp.single 2 y (1:ℂ)))
-  rw [norm_inner_symm (star M (lp.single 2 y (1:ℂ)))] at h
-  simpa only [pow_two, RCLike.re_to_complex] using h
-
- theorem matrixCoeff_positive_product_norm_sq_le
-    (A B : Hilbert V →L[ℂ] Hilbert V) (hA : 0 ≤ A) (x y : V) :
-    ‖matrixCoeff (B*A) x y‖ ^ 2 ≤
-      (matrixCoeff (B*A*star B) x x).re * (matrixCoeff A y y).re := by
-  let S := CFC.sqrt A
-  have hSS : S*S = A := CFC.sqrt_mul_sqrt_self A hA
-  have hSstar : star S = S := (IsSelfAdjoint.of_nonneg (CFC.sqrt_nonneg A)).star_eq
-  convert matrixCoeff_gram_norm_sq_le (B*S) S x y using 1 <;>
-    simp only [star_mul, hSstar, mul_assoc, ← mul_assoc S S, hSS]
-
- 
-
-theorem matrixCoeff_smoothing {Γ : Set (V ≃ V)}
-    (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
-    (hw : ∀ i, 0 ≤ weight i) (hsum : ∑ i, weight i = 1)
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c * weight i)
-    (A B : Hilbert V →L[ℂ] Hilbert V)
-    (hA : KernelInvariant Γ A) (hB : KernelInvariant Γ B)
-    (hApos : 0 ≤ A) (hdiag : ∀ y, matrixCoeff A y y = 1) (x y : V) :
-    ‖matrixCoeff (B*A) x y‖ ≤ Real.sqrt c * ‖B‖ := by
-  have htA : rootTrace root weight A = 1 := by
-    simp only [rootTrace, hdiag, mul_one, ← Complex.ofReal_sum, hsum, Complex.ofReal_one]
-  have hD : 0 ≤ B*A*star B := star_right_conjugate_nonneg hApos B
-  have hi : KernelInvariant Γ (B*A*star B) := (hB.mul hA).mul hB.star
-  have ht := rootTrace_conjugate_le root weight htr hw A B hA hB hApos
-  rw [htA, mul_one] at ht
-  have hd : matrixCoeff (B*A*star B) x x ≤ ((c * ‖B‖^2 : ℝ) : ℂ) := by
-    calc
-      _ ≤ (c : ℂ) * rootTrace root weight (B*A*star B) :=
-        rootTrace_diag_le root weight hcov hw c hc hmin _ hi hD x
-      _ ≤ (c : ℂ) * ((‖B‖^2 : ℝ) : ℂ) :=
-        mul_le_mul_of_nonneg_left ht (by exact_mod_cast hc)
-      _ = _ := by push_cast; rfl
-  have hd' : (matrixCoeff (B*A*star B) x x).re ≤ c * ‖B‖^2 :=
-    (Complex.le_def.mp hd).1
-  have hs : ‖matrixCoeff (B*A) x y‖ ^ 2 ≤ c * ‖B‖ ^ 2 := by
-    have hh := matrixCoeff_positive_product_norm_sq_le A B hApos x y
-    rw [hdiag, Complex.one_re, mul_one] at hh
-    exact hh.trans hd'
-  apply (sq_le_sq₀ (norm_nonneg _) (mul_nonneg (Real.sqrt_nonneg _) (norm_nonneg _))).mp
-  rwa [mul_pow, Real.sq_sqrt hc]
-
-end RootTrace
-
-end InvariantOperatorTrace
-
-section InvariantPercolationOperators
-open scoped ComplexOrder InnerProduct
-attribute [local instance] Classical.propDecidable
-
- 
-def vertexAut (G : BondGraph V E) : Set (V ≃ V) := {a | ∃ b : E ≃ E, IsAut G a b}
-
-theorem twoPoint_aut [Countable V] [Countable E] (G : BondGraph V E) (p : unitInterval)
-    {a : V ≃ V} {b : E ≃ E} (h : IsAut G a b) (x y : V) :
-    twoPoint G p (a x) (a y) = twoPoint G p x y := by
-  have he : {ω | Conn G ω (a x) (a y)} =
-      (fun ω => b ⁻¹' ω) ⁻¹' {ω | Conn G ω x y} := by
-    ext ω
-    exact (conn_aut_iff G h ω x y).symm
-  unfold twoPoint
-  rw [he, ← Measure.map_apply (measurable_set_preimage b)
-    (measurableSet_conn G x y), law_map_preimage_equiv]
-
-theorem HasKernel.invariant [Countable V] [Countable E] (G : BondGraph V E) (p : unitInterval)
-    (A : Hilbert V →L[ℂ] Hilbert V) (hA : HasKernel G p A) :
-    KernelInvariant (vertexAut G) A := by
-  rintro a ⟨b, hab⟩ x y
-  change A (lp.single 2 (a y) (1:ℂ)) (a x) = A (lp.single 2 y (1:ℂ)) x
-  rw [hA, hA, twoPoint_aut G p hab]
-
-theorem KernelInvariant.one (Γ : Set (V ≃ V)) :
-    KernelInvariant Γ (1 : Hilbert V →L[ℂ] Hilbert V) := by
-  intro a _ x y
-  simp only [matrixCoeff, one_apply_eq_self, lp.coeFn_single]
-  simp [Pi.single_apply, a.injective.eq_iff]
-
-theorem KernelInvariant.pow {Γ : Set (V ≃ V)}
-    {A : Hilbert V →L[ℂ] Hilbert V} (hA : KernelInvariant Γ A) (n : ℕ) :
-    KernelInvariant Γ (A^n) := by
-  induction n with
-  | zero => simpa only [pow_zero] using KernelInvariant.one Γ
-  | succ n ih => simpa only [pow_succ] using ih.mul hA
-
- 
-
-theorem twoPoint_operator_smoothing [Countable V] [Countable E] [Nonempty V] {ι : Type*} [Fintype ι]
-    (G : BondGraph V E) (p : unitInterval) (root : ι → V) (weight : ι → ℝ)
-    (htr : RootTransport root weight (vertexAut G))
-    (hcov : ∀ x, ∃ i, ∃ a ∈ vertexAut G, a (root i) = x)
-    (hw : ∀ i, 0 ≤ weight i) (hsum : ∑ i, weight i = 1)
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c * weight i)
-    (A P : Hilbert V →L[ℂ] Hilbert V) (hA : HasKernel G p A)
-    (hP : KernelInvariant (vertexAut G) P) (l : ℕ) (x y : V) :
-    ‖matrixCoeff (P^l*A) x y‖ ≤ Real.sqrt c * ‖P‖^l := by
-  have hdiag (y : V) : matrixCoeff A y y = 1 := by
-    change A (lp.single 2 y (1:ℂ)) y = 1
-    rw [hA y y, twoPoint_self, Complex.ofReal_one]
-  have h := matrixCoeff_smoothing root weight htr hcov hw hsum c hc hmin A (P^l)
-    (hA.invariant G p A) (hP.pow l) (ContinuousLinearMap.nonneg_iff_isPositive.mpr
-      (hA.isPositive G p A)) hdiag x y
-  have hn (n : ℕ) : ‖P^n‖ ≤ ‖P‖^n := by
-    induction n with
-    | zero => simpa only [pow_zero, ContinuousLinearMap.one_def] using (ContinuousLinearMap.norm_id_le (𝕜 := ℂ) (E := Hilbert V))
-    | succ n ih =>
-      rw [pow_succ, pow_succ]
-      exact (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right ih (norm_nonneg P))
-  exact h.trans (mul_le_mul_of_nonneg_left (hn l) (Real.sqrt_nonneg c))
-end InvariantPercolationOperators
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Kernel.Sprinkling
+
+namespace OAI
+
+/-! Isoperimetric energy, symmetric walks and invariant operator traces. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+section IsoperimetricEnergy
+attribute [local instance] Classical.propDecidable
+
+ 
+noncomputable def levelIndicator (a t : ℝ) : ℝ := if 0 ≤ t ∧ t < a then 1 else 0
+
+theorem levelIndicator_eq (a : ℝ) :
+    levelIndicator a = Set.indicator (Set.Ico 0 a) (fun _ => (1:ℝ)) := by
+  funext t
+  simp [levelIndicator, Set.indicator_apply, Set.mem_Ico]
+
+theorem integrable_levelIndicator (a : ℝ) : Integrable (levelIndicator a) := by
+  rw [levelIndicator_eq]
+  exact (integrableOn_const (isCompact_Icc.measure_lt_top.ne) :
+    IntegrableOn (fun _ : ℝ => (1:ℝ)) (Set.Icc 0 a)).mono_set
+      Set.Ico_subset_Icc_self |>.integrable_indicator measurableSet_Ico
+
+theorem integral_levelIndicator {a : ℝ} (ha : 0 ≤ a) :
+    ∫ t, levelIndicator a t = a := by
+  rw [levelIndicator_eq, integral_indicator measurableSet_Ico]
+  simp [ha]
+
+theorem abs_sub_levelIndicator {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) :
+    (fun t => |levelIndicator a t - levelIndicator b t|) =
+      Set.indicator (Set.Ico (min a b) (max a b)) (fun _ => (1:ℝ)) := by
+  funext t
+  rcases le_total a b with hab | hba
+  · simp only [min_eq_left hab, max_eq_right hab, Set.indicator_apply, Set.mem_Ico,
+      levelIndicator]
+    by_cases ht : 0 ≤ t
+    · by_cases hta : t < a
+      · have htb : t < b := hta.trans_le hab
+        have hna : ¬a ≤ t := not_le.mpr hta
+        simp [ht,hta,htb,hna]
+      · by_cases htb : t < b <;> simp [ht,hta,htb,le_of_not_gt hta]
+    · have hna : ¬a ≤ t := fun h => ht (ha.trans h)
+      simp [ht,hna]
+  · simp only [min_eq_right hba, max_eq_left hba, Set.indicator_apply, Set.mem_Ico,
+      levelIndicator]
+    by_cases ht : 0 ≤ t
+    · by_cases htb : t < b
+      · have hta : t < a := htb.trans_le hba
+        have hnb : ¬b ≤ t := not_le.mpr htb
+        simp [ht,hta,htb,hnb]
+      · by_cases hta : t < a <;> simp [ht,hta,htb,le_of_not_gt htb]
+    · have hnb : ¬b ≤ t := fun h => ht (hb.trans h)
+      simp [ht,hnb]
+
+theorem integrable_abs_sub_levelIndicator (a b : ℝ) :
+    Integrable (fun t => |levelIndicator a t - levelIndicator b t|) :=
+  ((integrable_levelIndicator a).sub (integrable_levelIndicator b)).abs
+
+theorem integral_abs_sub_levelIndicator {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) :
+    ∫ t, |levelIndicator a t - levelIndicator b t| = |a-b| := by
+  rw [abs_sub_levelIndicator ha hb, integral_indicator measurableSet_Ico]
+  simp only [setIntegral_const, smul_eq_mul, mul_one,
+    Real.volume_real_Ico_of_le min_le_max]
+  exact (max_sub_min_eq_abs a b).trans (abs_sub_comm _ _)
+
+ 
+theorem finite_coarea_lower_bound {α : Type*} (S : Finset α) (P : Finset (α × α))
+    (f : α → ℝ) (hf : ∀ x, 0 ≤ f x) (hout : ∀ x ∉ S, f x = 0) (h : ℝ)
+    (hcut : ∀ A : Finset α, A ⊆ S →
+      h * A.card ≤ ∑ e ∈ P,
+        |(if e.1 ∈ A then (1:ℝ) else 0) - (if e.2 ∈ A then (1:ℝ) else 0)|) :
+    h * ∑ x ∈ S, f x ≤ ∑ e ∈ P, |f e.1 - f e.2| := by
+  let F : ℝ → ℝ := fun t => h * ∑ x ∈ S, levelIndicator (f x) t
+  let J : ℝ → ℝ := fun t => ∑ e ∈ P,
+    |levelIndicator (f e.1) t - levelIndicator (f e.2) t|
+  have hF : Integrable F :=
+    (integrable_finsetSum S (fun x _ => integrable_levelIndicator (f x))).const_mul h
+  have hJ : Integrable J :=
+    integrable_finsetSum P (fun e _ => integrable_abs_sub_levelIndicator _ _)
+  have hle : ∀ t, F t ≤ J t := by
+    intro t
+    let A := S.filter (fun x => 0 ≤ t ∧ t < f x)
+    have hi (x : α) : levelIndicator (f x) t = if x ∈ A then (1:ℝ) else 0 := by
+      by_cases hx : x ∈ S
+      · simp [A,levelIndicator,hx]
+      · rw [hout x hx]
+        have hz : ¬(0 ≤ t ∧ t < 0) := fun ht => (not_lt_of_ge ht.1) ht.2
+        simp [A,levelIndicator,hx,hz]
+    have hc : ∑ x ∈ S, levelIndicator (f x) t = (A.card : ℝ) := by
+      simp [levelIndicator,A,Finset.sum_boole]
+    dsimp [F,J]
+    rw [hc]
+    simpa only [hi] using hcut A (Finset.filter_subset _ _)
+  have H := integral_mono hF hJ hle
+  dsimp [F,J] at H
+  rw [integral_const_mul, integral_finsetSum _
+    (fun x _ => integrable_levelIndicator (f x)), integral_finsetSum _
+    (fun e _ => integrable_abs_sub_levelIndicator _ _)] at H
+  simpa only [integral_levelIndicator (hf _),
+    integral_abs_sub_levelIndicator (hf _) (hf _)] using H
+
+theorem hV_mul_card_le (G : BondGraph V E) (A : Finset V) :
+    hV G * A.card ≤ (vertexBoundary G A).ncard := by
+  by_cases hA : A.Nonempty
+  · have hb : BddBelow {r : ℝ | ∃ B : Finset V, B.Nonempty ∧
+        r = (vertexBoundary G B).ncard / (B.card : ℝ)} := by
+      refine ⟨0, ?_⟩
+      rintro _ ⟨B,_,rfl⟩
+      positivity
+    have hh : hV G ≤ (vertexBoundary G A).ncard / (A.card : ℝ) :=
+      csInf_le hb ⟨A,hA,rfl⟩
+    exact (le_div_iff₀ (by exact_mod_cast hA.card_pos)).mp hh
+  · simp [Finset.not_nonempty_iff_eq_empty.mp hA]
+
+theorem finite_vertexBoundary (G : BondGraph V E) (hloc : LocallyFinite G)
+    (A : Finset V) : (vertexBoundary G A).Finite := by
+  apply (A.finite_toSet.biUnion (fun x _ => finite_neighbors hloc x)).subset
+  rintro y ⟨_,x,hx,hxy⟩
+  exact Set.mem_iUnion.mpr ⟨x,Set.mem_iUnion.mpr ⟨hx,hxy⟩⟩
+
+noncomputable def neighbors (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) :
+    Finset V := (finite_neighbors hloc x).toFinset
+
+@[simp] theorem mem_neighbors (G : BondGraph V E) (hloc : LocallyFinite G) (x y : V) :
+    y ∈ neighbors G hloc x ↔ Adj G x y := by simp [neighbors]
+
+noncomputable def neighborClosure (G : BondGraph V E) (hloc : LocallyFinite G)
+    (S : Finset V) : Finset V := S ∪ S.biUnion (neighbors G hloc)
+
+theorem subset_neighborClosure (G : BondGraph V E) (hloc : LocallyFinite G)
+    (S : Finset V) : S ⊆ neighborClosure G hloc S := Finset.subset_union_left
+
+theorem neighbor_mem_neighborClosure (G : BondGraph V E) (hloc : LocallyFinite G)
+    (S : Finset V) {x y : V} (hx : x ∈ S) (hxy : Adj G x y) :
+    y ∈ neighborClosure G hloc S := by
+  exact Finset.mem_union_right _ (Finset.mem_biUnion.mpr ⟨x,hx,(mem_neighbors ..).mpr hxy⟩)
+
+noncomputable def finiteAdjPairs (G : BondGraph V E) (K : Finset V) : Finset (V × V) :=
+  (K ×ˢ K).filter (fun e => Adj G e.1 e.2)
+
+@[simp] theorem mem_finiteAdjPairs (G : BondGraph V E) (K : Finset V) (e : V × V) :
+    e ∈ finiteAdjPairs G K ↔ e.1 ∈ K ∧ e.2 ∈ K ∧ Adj G e.1 e.2 := by
+  simp [finiteAdjPairs,and_assoc]
+
+theorem adj_symm (G : BondGraph V E) {x y : V} (h : Adj G x y) : Adj G y x := by
+  obtain ⟨e,he⟩ := h
+  exact ⟨e,he.trans (Sym2.eq_swap)⟩
+
+theorem sum_swap_finiteAdjPairs (G : BondGraph V E) (K : Finset V)
+    (f : V × V → ℝ) :
+    ∑ e ∈ finiteAdjPairs G K, f e.swap = ∑ e ∈ finiteAdjPairs G K, f e := by
+  apply Finset.sum_bij (fun e _ => e.swap)
+  · intro e he
+    rw [mem_finiteAdjPairs] at he ⊢
+    exact ⟨he.2.1,he.1,adj_symm G he.2.2⟩
+  · intro a _ b _ h
+    exact Prod.swap_injective h
+  · intro b hb
+    refine ⟨b.swap,?_,Prod.swap_swap b⟩
+    rw [mem_finiteAdjPairs] at hb ⊢
+    exact ⟨hb.2.1,hb.1,adj_symm G hb.2.2⟩
+  · intros; rfl
+
+ 
+theorem finiteAdjPairs_cut_bound (G : BondGraph V E) (hloc : LocallyFinite G)
+    (S A : Finset V) (hAS : A ⊆ S) :
+    (2 * hV G) * A.card ≤ ∑ e ∈ finiteAdjPairs G (neighborClosure G hloc S),
+      |(if e.1 ∈ A then (1:ℝ) else 0) - (if e.2 ∈ A then (1:ℝ) else 0)| := by
+  let K := neighborClosure G hloc S
+  let P := finiteAdjPairs G K
+  let R := P.filter (fun e => e.1 ∈ A ∧ e.2 ∉ A)
+  have hs : (finite_vertexBoundary G hloc A).toFinset ⊆ R.image Prod.snd := by
+    intro y hy
+    have hy' : y ∈ vertexBoundary G A := by simpa using hy
+    obtain ⟨hyA,x,hx,hxy⟩ := hy'
+    apply Finset.mem_image.mpr
+    refine ⟨(x,y),Finset.mem_filter.mpr ⟨?_,hx,hyA⟩,rfl⟩
+    exact (mem_finiteAdjPairs G K (x,y)).mpr
+      ⟨subset_neighborClosure G hloc S (hAS hx),
+        neighbor_mem_neighborClosure G hloc S (hAS hx) hxy,hxy⟩
+  have hc : (vertexBoundary G A).ncard ≤ R.card := by
+    rw [Set.ncard_eq_toFinset_card (vertexBoundary G A) (finite_vertexBoundary G hloc A)]
+    exact (Finset.card_le_card hs).trans (Finset.card_image_le)
+  have hcross : hV G * A.card ≤ ∑ e ∈ P,
+      if e.1 ∈ A ∧ e.2 ∉ A then (1:ℝ) else 0 := by
+    rw [Finset.sum_boole]
+    exact (hV_mul_card_le G A).trans (by exact_mod_cast hc)
+  have hsplit : ∑ e ∈ P,
+      |(if e.1 ∈ A then (1:ℝ) else 0) - (if e.2 ∈ A then (1:ℝ) else 0)| =
+      2 * ∑ e ∈ P, if e.1 ∈ A ∧ e.2 ∉ A then (1:ℝ) else 0 := by
+    have ht (e : V × V) :
+        |(if e.1 ∈ A then (1:ℝ) else 0) - (if e.2 ∈ A then (1:ℝ) else 0)| =
+        (if e.1 ∈ A ∧ e.2 ∉ A then (1:ℝ) else 0) +
+        (if e.2 ∈ A ∧ e.1 ∉ A then (1:ℝ) else 0) := by
+      by_cases h₁ : e.1 ∈ A <;> by_cases h₂ : e.2 ∈ A <;> simp [h₁,h₂]
+    simp_rw [ht]
+    rw [Finset.sum_add_distrib]
+    have he := sum_swap_finiteAdjPairs G K
+      (fun e => if e.1 ∈ A ∧ e.2 ∉ A then (1:ℝ) else 0)
+    change (∑ e ∈ P, if e.2 ∈ A ∧ e.1 ∉ A then (1:ℝ) else 0) = _ at he
+    rw [he]
+    ring
+  change _ ≤ ∑ e ∈ P, _
+  rw [hsplit]
+  nlinarith
+
+ 
+theorem bondGraph_coarea (G : BondGraph V E) (hloc : LocallyFinite G)
+    (S : Finset V) (f : V → ℝ) (hf : ∀ x, 0 ≤ f x)
+    (hout : ∀ x ∉ S, f x = 0) :
+    (2 * hV G) * ∑ x ∈ S, f x ≤
+      ∑ e ∈ finiteAdjPairs G (neighborClosure G hloc S), |f e.1 - f e.2| :=
+  finite_coarea_lower_bound S _ f hf hout _
+    (fun A hAS => finiteAdjPairs_cut_bound G hloc S A hAS)
+
+theorem neighbors_card_le_degree (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) :
+    (neighbors G hloc x).card ≤ degree G x := by
+  let : Fintype (Step G x) := (finite_steps G hloc x).fintype
+  let p : Step G x → {y // y ∈ neighbors G hloc x} :=
+    fun d => ⟨d.val.2, (mem_neighbors G hloc x d.val.2).mpr ⟨d.val.1,d.property⟩⟩
+  have hp : Function.Surjective p := by
+    intro y
+    obtain ⟨e,he⟩ := (mem_neighbors G hloc x y.val).mp y.property
+    exact ⟨⟨(e,y.val),he⟩,Subtype.ext rfl⟩
+  simpa only [degree,Nat.card_eq_fintype_card,Fintype.card_coe] using
+    Fintype.card_le_of_surjective p hp
+
+theorem finiteAdjPairs_source_sum_le (G : BondGraph V E) (hloc : LocallyFinite G)
+    (D : ℕ) (hD : ∀ x, degree G x ≤ D) (K : Finset V) (f : V → ℝ)
+    (hf : ∀ x, 0 ≤ f x) :
+    ∑ e ∈ finiteAdjPairs G K, f e.1 ≤ (D:ℝ) * ∑ x ∈ K, f x := by
+  rw [Finset.mul_sum]
+  simp only [finiteAdjPairs,Finset.sum_filter,Finset.sum_product]
+  apply Finset.sum_le_sum
+  intro x _
+  have hc : (K.filter (fun y => Adj G x y)).card ≤ D := by
+    apply (Finset.card_le_card ?_).trans ((neighbors_card_le_degree G hloc x).trans (hD x))
+    intro y hy
+    exact (mem_neighbors G hloc x y).mpr (Finset.mem_filter.mp hy).2
+  calc
+    (∑ y ∈ K, if Adj G x y then f x else 0) =
+        ((K.filter (fun y => Adj G x y)).card:ℝ) * f x := by
+      rw [← Finset.sum_filter]
+      simp
+    _ ≤ (D:ℝ) * f x := mul_le_mul_of_nonneg_right (by exact_mod_cast hc) (hf x)
+
+theorem finiteAdjPairs_square_sum_le (G : BondGraph V E) (hloc : LocallyFinite G)
+    (D : ℕ) (hD : ∀ x, degree G x ≤ D) (S : Finset V) (f : V → ℝ)
+    (hout : ∀ x ∉ S, f x = 0) :
+    ∑ e ∈ finiteAdjPairs G (neighborClosure G hloc S), (f e.1 + f e.2)^2 ≤
+      4 * (D:ℝ) * ∑ x ∈ S, (f x)^2 := by
+  let K := neighborClosure G hloc S
+  let P := finiteAdjPairs G K
+  have hs : ∑ x ∈ K, (f x)^2 = ∑ x ∈ S, (f x)^2 := by
+    symm
+    apply Finset.sum_subset (subset_neighborClosure G hloc S)
+    intro x _ hx
+    simp [hout x hx]
+  have hfirst := finiteAdjPairs_source_sum_le G hloc D hD K (fun x => (f x)^2) (fun _ => sq_nonneg _)
+  rw [hs] at hfirst
+  have hsecond := sum_swap_finiteAdjPairs G K (fun e => (f e.1)^2)
+  change (∑ e ∈ P, (f e.2)^2) = ∑ e ∈ P, (f e.1)^2 at hsecond
+  calc
+    (∑ e ∈ P, (f e.1 + f e.2)^2) ≤ ∑ e ∈ P, (2*(f e.1)^2+2*(f e.2)^2) := by
+      apply Finset.sum_le_sum
+      intro e _
+      nlinarith [sq_nonneg (f e.1 - f e.2)]
+    _ = 4 * ∑ e ∈ P, (f e.1)^2 := by
+      rw [Finset.sum_add_distrib,←Finset.mul_sum,←Finset.mul_sum,hsecond]
+      ring
+    _ ≤ 4 * (D:ℝ) * ∑ x ∈ S, (f x)^2 := by
+      nlinarith
+
+ 
+
+theorem bondGraph_cheeger_energy (G : BondGraph V E) (hloc : LocallyFinite G)
+    (hh : 0 < hV G) (D : ℕ) (hD : ∀ x, degree G x ≤ D)
+    (S : Finset V) (f : V → ℝ) (hf : ∀ x, 0 ≤ f x)
+    (hout : ∀ x ∉ S, f x = 0) :
+    (hV G)^2 * ∑ x ∈ S, (f x)^2 ≤
+      (D:ℝ) * ∑ e ∈ finiteAdjPairs G (neighborClosure G hloc S), (f e.1-f e.2)^2 := by
+  let P := finiteAdjPairs G (neighborClosure G hloc S)
+  let N := ∑ x ∈ S, (f x)^2
+  let E := ∑ e ∈ P, (f e.1-f e.2)^2
+  have hN : 0 ≤ N := Finset.sum_nonneg (fun _ _ => sq_nonneg _)
+  have hE : 0 ≤ E := Finset.sum_nonneg (fun _ _ => sq_nonneg _)
+  have hc := bondGraph_coarea G hloc S (fun x => (f x)^2)
+    (fun _ => sq_nonneg _) (fun x hx => by rw [hout x hx]; norm_num)
+  have ht (e : V × V) : |(f e.1)^2-(f e.2)^2| = |f e.1-f e.2| * (f e.1+f e.2) := by
+    rw [show (f e.1)^2-(f e.2)^2 = (f e.1-f e.2)*(f e.1+f e.2) by ring,
+      abs_mul,abs_of_nonneg (add_nonneg (hf _) (hf _))]
+  simp_rw [ht] at hc
+  have hc' : ((2*hV G)*N)^2 ≤
+      (∑ e ∈ P, |f e.1-f e.2| * (f e.1+f e.2))^2 := by
+    exact sq_le_sq₀ (by positivity) (Finset.sum_nonneg (fun e _ => mul_nonneg (abs_nonneg _)
+      (add_nonneg (hf _) (hf _)))) |>.mpr hc
+  have hcs := Finset.sum_mul_sq_le_sq_mul_sq P
+    (fun e => |f e.1-f e.2|) (fun e => f e.1+f e.2)
+  simp only [sq_abs] at hcs
+  have hbound := finiteAdjPairs_square_sum_le G hloc D hD S f hout
+  have hall : ((2*hV G)*N)^2 ≤ E*(4*(D:ℝ)*N) :=
+    (hc'.trans hcs).trans (mul_le_mul_of_nonneg_left hbound hE)
+  change (hV G)^2*N ≤ (D:ℝ)*E
+  rcases hN.eq_or_lt with hzero | hpos
+  · rw [←hzero,mul_zero]
+    positivity
+  · apply (mul_le_mul_iff_left₀ (by positivity : 0 < 4*N)).mp
+    nlinarith [hall]
+
+end IsoperimetricEnergy
+
+section FiniteQuadraticForms
+attribute [local instance] Classical.propDecidable
+
+@[simp] theorem finiteEmbedding_inner_left (f : V →₀ ℂ) (g : Hilbert V) :
+    inner ℂ (finiteEmbedding f) g = ∑ x ∈ f.support, star (f x) * g x := by
+  simp [finiteEmbedding, Finsupp.linearCombination_apply, Finsupp.sum,
+    lp.inner_single_left, mul_comm]
+
+@[simp] theorem finiteEmbedding_inner_right (f : Hilbert V) (g : V →₀ ℂ) :
+    inner ℂ f (finiteEmbedding g) = ∑ x ∈ g.support, star (f x) * g x := by
+  simp [finiteEmbedding, Finsupp.linearCombination_apply, Finsupp.sum,
+    lp.inner_single_right, mul_comm]
+
+@[simp] theorem finiteEmbedding_norm_sq (f : V →₀ ℂ) :
+    ‖finiteEmbedding f‖^2 = ∑ x ∈ f.support, ‖f x‖^2 := by
+  rw [norm_sq_eq_re_inner (𝕜 := ℂ)]
+  rw [finiteEmbedding_inner_left]
+  simp only [map_sum, finiteEmbedding_apply]
+  apply Finset.sum_congr rfl
+  intro x _
+  simp [← Complex.normSq_eq_norm_sq, Complex.normSq_apply]
+
+ theorem kernel_finiteEmbedding (A : Hilbert V →L[ℂ] Hilbert V)
+    (U : V → V → ℝ) (hA : ∀ x y, A (lp.single 2 y (1 : ℂ)) x = (U x y : ℂ))
+    (f : V →₀ ℂ) (x : V) :
+    A (finiteEmbedding f) x = ∑ y ∈ f.support, f y * (U x y : ℂ) := by
+  simp only [finiteEmbedding, Finsupp.linearCombination_apply, Finsupp.sum,
+    map_sum, map_smul]
+  change (∑ y ∈ f.support, f y • A (lp.single 2 y (1 : ℂ))) x = _
+  simp only [lp.coeFn_sum, Finset.sum_apply, lp.coeFn_smul, Pi.smul_apply,
+    hA, smul_eq_mul]
+
+ theorem symmetric_of_symmetric_kernel (A : Hilbert V →L[ℂ] Hilbert V)
+    (U : V → V → ℝ) (hA : ∀ x y, A (lp.single 2 y (1 : ℂ)) x = (U x y : ℂ))
+    (hU : ∀ x y, U x y = U y x) : A.toLinearMap.IsSymmetric := by
+  apply finiteEmbedding_dense.induction_on₂
+    (p := fun f g => inner ℂ (A f) g = inner ℂ f (A g))
+  · exact isClosed_eq (by fun_prop) (by fun_prop)
+  · intro f g
+    simp only [finiteEmbedding_inner_left, finiteEmbedding_inner_right,
+      kernel_finiteEmbedding A U hA, map_sum, map_mul, RCLike.star_def,
+      Complex.conj_ofReal]
+    simp_rw [Finset.sum_mul, Finset.mul_sum]
+    rw [Finset.sum_comm]
+    apply Finset.sum_congr rfl
+    intro x _
+    apply Finset.sum_congr rfl
+    intro y _
+    rw [hU y x]
+    ring
+
+ theorem norm_le_of_finite_quadratic (A : Hilbert V →L[ℂ] Hilbert V)
+    (hA : A.toLinearMap.IsSymmetric) (C : ℝ) (hC : 0 ≤ C)
+    (h : ∀ f : V →₀ ℂ,
+      |(inner ℂ (A (finiteEmbedding f)) (finiteEmbedding f)).re| ≤
+        C * ‖finiteEmbedding f‖^2) : ‖A‖ ≤ C := by
+  have hall (f : Hilbert V) : |(inner ℂ (A f) f).re| ≤ C * ‖f‖^2 := by
+    exact finiteEmbedding_dense.induction_on f
+      (p := fun g => |(inner ℂ (A g) g).re| ≤ C * ‖g‖^2)
+      (isClosed_le (by fun_prop) (by fun_prop)) h
+  rw [A.norm_eq_iSup_rayleighQuotient hA]
+  apply ciSup_le
+  intro f
+  by_cases hf : f = 0
+  · simp [hf, hC]
+  · rw [ContinuousLinearMap.rayleighQuotient, abs_div, abs_sq]
+    exact (div_le_iff₀ (sq_pos_of_pos (norm_pos_iff.mpr hf))).mpr (hall f)
+
+ theorem abs_inner_kernel_finite_le (A : Hilbert V →L[ℂ] Hilbert V)
+    (U : V → V → ℝ) (hA : ∀ x y, A (lp.single 2 y (1 : ℂ)) x = (U x y : ℂ))
+    (hU : ∀ x y, 0 ≤ U x y) (f : V →₀ ℂ) :
+    |(inner ℂ (A (finiteEmbedding f)) (finiteEmbedding f)).re| ≤
+      ∑ x ∈ f.support, ∑ y ∈ f.support, ‖f x‖ * ‖f y‖ * U x y := by
+  change |RCLike.re (inner ℂ (A (finiteEmbedding f)) (finiteEmbedding f))| ≤ _
+  rw [inner_re_symm]
+  apply (Complex.abs_re_le_norm _).trans
+  rw [finiteEmbedding_inner_left]
+  simp_rw [kernel_finiteEmbedding A U hA, Finset.mul_sum]
+  apply (norm_sum_le _ _).trans
+  apply Finset.sum_le_sum
+  intro x _
+  apply (norm_sum_le _ _).trans
+  apply Finset.sum_le_sum
+  intro y _
+  simp [abs_of_nonneg (hU x y), mul_assoc]
+
+end FiniteQuadraticForms
+
+section SymmetricWalk
+attribute [local instance] Classical.propDecidable
+
+ 
+
+noncomputable def walkKernel (G : BondGraph V E) (hloc : LocallyFinite G)
+    (a : ℝ) (x y : V) : ℝ :=
+  (if x = y then 1 - a * (neighbors G hloc x).card else 0) +
+    (if Adj G x y then a else 0)
+
+ theorem walkKernel_symmetric (G : BondGraph V E) (hloc : LocallyFinite G)
+    (a : ℝ) (x y : V) : walkKernel G hloc a x y = walkKernel G hloc a y x := by
+  have ha : Adj G x y ↔ Adj G y x := ⟨adj_symm G,adj_symm G⟩
+  by_cases h : x = y
+  · subst y; rfl
+  · simp [walkKernel,h,Ne.symm h,ha]
+
+ theorem walkKernel_nonneg (G : BondGraph V E) (hloc : LocallyFinite G)
+    (D : ℕ) (hD : ∀ x, degree G x ≤ D) (a : ℝ) (ha : 0 ≤ a) (haD : a*D ≤ 1)
+    (x y : V) : 0 ≤ walkKernel G hloc a x y := by
+  have hn : ((neighbors G hloc x).card : ℝ) ≤ D := by
+    exact_mod_cast (neighbors_card_le_degree G hloc x).trans (hD x)
+  have hh : 0 ≤ 1-a*(neighbors G hloc x).card := by nlinarith
+  unfold walkKernel
+  split_ifs <;> positivity
+
+ theorem walkKernel_row_sum (G : BondGraph V E) (hloc : LocallyFinite G)
+    (a : ℝ) (x : V) :
+    ∑ y ∈ insert x (neighbors G hloc x), walkKernel G hloc a x y = 1 := by
+  let N := neighbors G hloc x
+  have hf : (insert x N).filter (fun y => Adj G x y) = N := by
+    ext y
+    simp only [Finset.mem_filter, Finset.mem_insert]
+    rw [mem_neighbors]
+    exact ⟨fun h => h.2, fun h => ⟨Or.inr h,h⟩⟩
+  simp only [walkKernel, Finset.sum_add_distrib]
+  rw [Finset.sum_ite_eq]
+  simp only [Finset.mem_insert_self,ite_true]
+  rw [← Finset.sum_filter,hf]
+  simp only [Finset.sum_const, nsmul_eq_mul]
+  dsimp [N]
+  ring
+
+ theorem walkKernel_row_bound (G : BondGraph V E) (hloc : LocallyFinite G)
+    (D : ℕ) (hD : ∀ x, degree G x ≤ D) (a : ℝ) (ha : 0 ≤ a) (haD : a*D ≤ 1)
+    (x : V) (S : Finset V) : ∑ y ∈ S, walkKernel G hloc a x y ≤ 1 := by
+  let N := insert x (neighbors G hloc x)
+  calc
+    ∑ y ∈ S, walkKernel G hloc a x y ≤ ∑ y ∈ S ∪ N, walkKernel G hloc a x y :=
+      Finset.sum_le_sum_of_subset_of_nonneg Finset.subset_union_left
+        (fun y _ _ => walkKernel_nonneg G hloc D hD a ha haD x y)
+    _ = ∑ y ∈ N, walkKernel G hloc a x y := by
+      symm
+      apply Finset.sum_subset Finset.subset_union_right
+      intro y _ hy
+      have hxy : x ≠ y := by intro h; subst y; exact hy (by simp [N])
+      have hna : ¬ Adj G x y := by intro h; exact hy (by simp [N,h])
+      simp [walkKernel,hxy,hna]
+    _ = 1 := walkKernel_row_sum G hloc a x
+
+ theorem exists_walk_operator (G : BondGraph V E) (hloc : LocallyFinite G)
+    (D : ℕ) (hD : ∀ x, degree G x ≤ D) (a : ℝ) (ha : 0 ≤ a) (haD : a*D ≤ 1) :
+    ∃ P : Hilbert V →L[ℂ] Hilbert V, ‖P‖ ≤ 1 ∧
+      P.toLinearMap.IsSymmetric ∧
+      ∀ x y, P (lp.single 2 y (1 : ℂ)) x = (walkKernel G hloc a x y : ℂ) := by
+  obtain ⟨P,hP,hK⟩ := exists_operator_of_row_column_bound (walkKernel G hloc a) 1
+    zero_le_one (walkKernel_nonneg G hloc D hD a ha haD)
+    (walkKernel_row_bound G hloc D hD a ha haD) (by
+      intro y S
+      simp_rw [walkKernel_symmetric G hloc a]
+      exact walkKernel_row_bound G hloc D hD a ha haD y S)
+  exact ⟨P,hP,symmetric_of_symmetric_kernel P _ hK (walkKernel_symmetric G hloc a),hK⟩
+
+ theorem finiteAdjPairs_sum (G : BondGraph V E) (K : Finset V) (f : V × V → ℝ) :
+    (∑ e ∈ finiteAdjPairs G K, f e) =
+      ∑ x ∈ K, ∑ y ∈ K, if Adj G x y then f (x,y) else 0 := by
+  simp [finiteAdjPairs,Finset.sum_filter,Finset.sum_product]
+
+ theorem finiteAdjPairs_source_sum (G : BondGraph V E) (hloc : LocallyFinite G)
+    (S : Finset V) (f : V → ℝ) (hout : ∀ x ∉ S, f x = 0) :
+    (∑ e ∈ finiteAdjPairs G (neighborClosure G hloc S), f e.1) =
+      ∑ x ∈ S, ((neighbors G hloc x).card : ℝ) * f x := by
+  let K := neighborClosure G hloc S
+  rw [finiteAdjPairs_sum]
+  have he (x : V) (hx : x ∈ S) :
+      K.filter (fun y => Adj G x y) = neighbors G hloc x := by
+    ext y
+    simp only [Finset.mem_filter,mem_neighbors]
+    exact ⟨fun h => h.2,fun h => ⟨neighbor_mem_neighborClosure G hloc S hx h,h⟩⟩
+  have hout' : ∀ x ∉ S, (∑ y ∈ K, if Adj G x y then f x else 0) = 0 := by
+    intro x hx; simp [hout x hx]
+  change (∑ x ∈ K, ∑ y ∈ K, if Adj G x y then f x else 0) = _
+  rw [← Finset.sum_subset (subset_neighborClosure G hloc S)
+    (fun x _ hx => hout' x hx)]
+  apply Finset.sum_congr rfl
+  intro x hx
+  rw [← Finset.sum_filter,he x hx]
+  simp
+
+ theorem finiteAdjPairs_product_sum (G : BondGraph V E) (hloc : LocallyFinite G)
+    (S : Finset V) (f : V → ℝ) (hout : ∀ x ∉ S, f x = 0) :
+    (∑ e ∈ finiteAdjPairs G (neighborClosure G hloc S), f e.1 * f e.2) =
+      ∑ x ∈ S, ∑ y ∈ S, if Adj G x y then f x*f y else 0 := by
+  rw [finiteAdjPairs_sum]
+  have hs := subset_neighborClosure G hloc S
+  calc
+    _ = ∑ x ∈ S, ∑ y ∈ neighborClosure G hloc S,
+        if Adj G x y then f x*f y else 0 := by
+      symm; apply Finset.sum_subset hs
+      intro x _ hx; simp [hout x hx]
+    _ = _ := by
+      apply Finset.sum_congr rfl
+      intro x _
+      symm; apply Finset.sum_subset hs
+      intro y _ hy; simp [hout y hy]
+
+ theorem walkKernel_energy (G : BondGraph V E) (hloc : LocallyFinite G)
+    (a : ℝ) (S : Finset V) (f : V → ℝ) (hout : ∀ x ∉ S, f x = 0) :
+    (∑ x ∈ S, ∑ y ∈ S, f x*f y*walkKernel G hloc a x y) =
+      ∑ x ∈ S, (f x)^2 - a/2 *
+        ∑ e ∈ finiteAdjPairs G (neighborClosure G hloc S), (f e.1-f e.2)^2 := by
+  let P := finiteAdjPairs G (neighborClosure G hloc S)
+  have henergy : (∑ e ∈ P, (f e.1-f e.2)^2) =
+      2*(∑ x ∈ S, (neighbors G hloc x).card * (f x)^2) -
+      2*(∑ x ∈ S, ∑ y ∈ S, if Adj G x y then f x*f y else 0) := by
+    have hs := sum_swap_finiteAdjPairs G (neighborClosure G hloc S)
+      (fun e => (f e.1)^2)
+    change (∑ e ∈ P,(f e.2)^2) = (∑ e ∈ P,(f e.1)^2) at hs
+    have hfirst := finiteAdjPairs_source_sum G hloc S (fun x => (f x)^2)
+      (fun x hx => by rw [hout x hx]; norm_num)
+    calc
+      _ = (∑ e ∈ P, (f e.1)^2) + (∑ e ∈ P,(f e.2)^2) -
+          2*(∑ e ∈ P,f e.1*f e.2) := by
+        simp_rw [sub_sq]
+        rw [Finset.sum_add_distrib,Finset.sum_sub_distrib]
+        simp_rw [mul_assoc,← Finset.mul_sum]
+        ring
+      _ = _ := by rw [hs,hfirst,finiteAdjPairs_product_sum G hloc S f hout]; ring
+  rw [henergy]
+  have hpoint (x : V) (hx : x ∈ S) :
+      (∑ y ∈ S, f x*f y*walkKernel G hloc a x y) =
+        (f x)^2 - a*((neighbors G hloc x).card*(f x)^2) +
+         a*(∑ y ∈ S, if Adj G x y then f x*f y else 0) := by
+    simp only [walkKernel,mul_add,Finset.sum_add_distrib]
+    have hi : (∑ y ∈ S, f x*f y*(if x=y then 1-a*(neighbors G hloc x).card else 0)) =
+        (f x)^2*(1-a*(neighbors G hloc x).card) := by
+      simp_rw [mul_ite,mul_zero]
+      rw [Finset.sum_ite_eq]
+      simp only [hx,ite_true]
+      ring
+    rw [hi,Finset.mul_sum]
+    have hj : (∑ y ∈ S, f x*f y*(if Adj G x y then a else 0)) =
+        ∑ y ∈ S, a*(if Adj G x y then f x*f y else 0) := by
+      apply Finset.sum_congr rfl
+      intro y _
+      split_ifs <;> ring
+    rw [hj]
+    ring
+  simp_rw [Finset.sum_congr rfl hpoint,Finset.sum_add_distrib,
+    Finset.sum_sub_distrib,←Finset.mul_sum]
+  ring
+
+ 
+
+theorem walkKernel_quadratic_bound (G : BondGraph V E) (hloc : LocallyFinite G)
+    (hh : 0 < hV G) (D : ℕ) (hDp : 0 < D) (hD : ∀ x, degree G x ≤ D)
+    (a : ℝ) (ha : 0 ≤ a) (S : Finset V) (f : V → ℝ)
+    (hf : ∀ x, 0 ≤ f x) (hout : ∀ x ∉ S, f x = 0) :
+    (∑ x ∈ S, ∑ y ∈ S, f x*f y*walkKernel G hloc a x y) ≤
+      (1 - a*(hV G)^2/(2*D)) * ∑ x ∈ S, (f x)^2 := by
+  have hDp' : 0 < (D:ℝ) := by exact_mod_cast hDp
+  have hc := bondGraph_cheeger_energy G hloc hh D hD S f hf hout
+  have he : ((hV G)^2 * ∑ x ∈ S, (f x)^2)/(D:ℝ) ≤
+      ∑ e ∈ finiteAdjPairs G (neighborClosure G hloc S), (f e.1-f e.2)^2 :=
+    (div_le_iff₀ hDp').mpr (by simpa only [mul_comm] using hc)
+  rw [walkKernel_energy G hloc a S f hout]
+  calc
+    _ ≤ (∑ x ∈ S, (f x)^2) - a/2 *
+        (((hV G)^2 * ∑ x ∈ S, (f x)^2)/(D:ℝ)) :=
+      sub_le_sub_left (mul_le_mul_of_nonneg_left he (by positivity)) _
+    _ = _ := by ring
+
+ 
+
+theorem walk_operator_norm_lt_one (G : BondGraph V E) (hloc : LocallyFinite G)
+    (hh : 0 < hV G) (D : ℕ) (hDp : 0 < D) (hD : ∀ x, degree G x ≤ D)
+    (a : ℝ) (ha : 0 < a) (haD : a*D ≤ 1)
+    (P : Hilbert V →L[ℂ] Hilbert V)
+    (hP : ∀ x y, P (lp.single 2 y (1:ℂ)) x = (walkKernel G hloc a x y : ℂ)) :
+    ‖P‖ < 1 := by
+  let C := max 0 (1-a*(hV G)^2/(2*D))
+  have hDp' : 0 < (D:ℝ) := by exact_mod_cast hDp
+  have hC : 0 ≤ C := le_max_left _ _
+  have hC1 : C < 1 := by
+    apply max_lt (by norm_num)
+    have : 0 < a*(hV G)^2/(2*D) := by positivity
+    linarith
+  have hn : ‖P‖ ≤ C := by
+    apply norm_le_of_finite_quadratic P
+      (symmetric_of_symmetric_kernel P _ hP (walkKernel_symmetric G hloc a)) C hC
+    intro f
+    calc
+      _ ≤ ∑ x ∈ f.support, ∑ y ∈ f.support, ‖f x‖*‖f y‖*walkKernel G hloc a x y :=
+        abs_inner_kernel_finite_le P _ hP (walkKernel_nonneg G hloc D hD a ha.le haD) f
+      _ ≤ (1-a*(hV G)^2/(2*D)) * ∑ x ∈ f.support, ‖f x‖^2 :=
+        walkKernel_quadratic_bound G hloc hh D hDp hD a ha.le f.support
+          (fun x => ‖f x‖) (fun _ => norm_nonneg _) (by
+            intro x hx; simp [Finsupp.notMem_support_iff.mp hx])
+      _ ≤ C * ∑ x ∈ f.support, ‖f x‖^2 :=
+        mul_le_mul_of_nonneg_right (le_max_right _ _) (Finset.sum_nonneg (fun _ _ => sq_nonneg _))
+      _ = C * ‖finiteEmbedding f‖^2 := by rw [finiteEmbedding_norm_sq]
+  exact hn.trans_lt hC1
+
+ 
+
+theorem exists_strict_walk_operator (G : BondGraph V E) (hloc : LocallyFinite G)
+    (hqt : QuasiTransitive G) (hh : 0 < hV G) :
+    ∃ (D : ℕ) (P : Hilbert V →L[ℂ] Hilbert V), 0 < D ∧
+      (∀ x, degree G x ≤ D) ∧ ‖P‖ < 1 ∧ P.toLinearMap.IsSymmetric ∧
+      ∀ x y, P (lp.single 2 y (1:ℂ)) x =
+        (walkKernel G hloc (1/(2*(D:ℝ))) x y : ℂ) := by
+  obtain ⟨D,hDp,hD⟩ := exists_degree_bound G hqt
+  have hDp' : 0 < (D:ℝ) := by exact_mod_cast hDp
+  have ha : 0 < (1:ℝ)/(2*D) := by positivity
+  have haD : (1:ℝ)/(2*D)*D ≤ 1 := by
+    rw [div_mul_eq_mul_div]
+    apply (div_le_iff₀ (by positivity)).mpr
+    nlinarith
+  obtain ⟨P,_,hPs,hP⟩ := exists_walk_operator G hloc D hD _ ha.le haD
+  exact ⟨D,P,hDp,hD,walk_operator_norm_lt_one G hloc hh D hDp hD _ ha haD P hP,hPs,hP⟩
+
+end SymmetricWalk
+
+section PositiveConnectionKernel
+open scoped ComplexOrder Matrix
+attribute [local instance] Classical.propDecidable
+
+ 
+
+theorem equivalenceMatrix_posSemidef {α : Type*} [Fintype α] (R : Setoid α) :
+    Matrix.PosSemidef (fun x y => if R.r x y then (1:ℂ) else 0) := by
+  classical
+  let : Fintype (Quotient R) := Fintype.ofFinite (Quotient R)
+  let J : Matrix (Quotient R) α ℂ := fun q x => if q = Quotient.mk R x then 1 else 0
+  have hJ : Jᴴ * J = (fun x y => if R.r x y then (1:ℂ) else 0) := by
+    ext x y
+    change (∑ q : Quotient R, star (J q x) * J q y) = _
+    have hr : R.r y x ↔ R.r x y := ⟨R.symm, R.symm⟩
+    simp [J, Quotient.eq, hr]
+  rw [← hJ]
+  exact Matrix.posSemidef_conjTranspose_mul_self J
+
+theorem connection_quadratic_nonneg (G : BondGraph V E) (ω : Set E)
+    (S : Finset V) (f : V → ℂ) :
+    0 ≤ ∑ x ∈ S, ∑ y ∈ S,
+      star (f x) * (if Conn G ω x y then (1:ℂ) else 0) * f y := by
+  classical
+  let R : Setoid S := ⟨fun x y => Conn G ω x y,
+    ⟨fun x => conn_refl G ω x, fun h => conn_symm G ω h, fun h h' => conn_trans G ω h h'⟩⟩
+  have h := (equivalenceMatrix_posSemidef R).dotProduct_mulVec_nonneg
+    (fun x : S => f x)
+  simp only [dotProduct, Matrix.mulVec, Finset.mul_sum, Pi.star_apply] at h
+  change 0 ≤ ∑ x : S, ∑ y : S,
+    star (f x) * ((if Conn G ω x y then (1:ℂ) else 0) * f y) at h
+  have he : (∑ x : S, ∑ y : S,
+      star (f x) * ((if Conn G ω x y then (1:ℂ) else 0) * f y)) =
+      ∑ x ∈ S, ∑ y ∈ S,
+      star (f x) * ((if Conn G ω x y then (1:ℂ) else 0) * f y) := by
+    rw [Finset.sum_coe_sort S (fun x => ∑ y : S,
+      star (f x) * ((if Conn G ω x y then (1:ℂ) else 0) * f y))]
+    apply Finset.sum_congr rfl
+    intro x _
+    exact Finset.sum_coe_sort S (fun y =>
+      star (f x) * ((if Conn G ω x y then (1:ℂ) else 0) * f y))
+  rw [he] at h
+  simpa only [mul_assoc] using h
+
+variable [Countable V] [Countable E]
+
+theorem conn_indicator_integrable (G : BondGraph V E) (p : unitInterval) (x y : V) :
+    Integrable (fun ω => if Conn G ω x y then (1:ℂ) else 0) (law p) := by
+  change Integrable ({ω | Conn G ω x y}.indicator (fun _ => (1:ℂ))) (law p)
+  exact (integrable_const (1:ℂ)).indicator (measurableSet_conn G x y)
+
+theorem twoPoint_eq_integral (G : BondGraph V E) (p : unitInterval) (x y : V) :
+    (twoPoint G p x y : ℂ) =
+      ∫ ω, (if Conn G ω x y then (1:ℂ) else 0) ∂law p := by
+  simpa only [Set.indicator, Set.mem_ofPred_eq, Measure.real, twoPoint,
+    Complex.real_smul, mul_one] using
+    (integral_indicator_const (μ := law p) (1:ℂ) (measurableSet_conn G x y)).symm
+
+theorem twoPoint_quadratic_nonneg (G : BondGraph V E) (p : unitInterval)
+    (S : Finset V) (f : V → ℂ) :
+    0 ≤ ∑ x ∈ S, ∑ y ∈ S, star (f x) * (twoPoint G p x y : ℂ) * f y := by
+  have hi (x y : V) : Integrable
+      (fun ω => star (f x) * (if Conn G ω x y then (1:ℂ) else 0) * f y) (law p) :=
+    ((conn_indicator_integrable G p x y).const_mul _).mul_const _
+  have hsum : (∑ x ∈ S, ∑ y ∈ S, star (f x) * (twoPoint G p x y : ℂ) * f y) =
+      ∫ ω, ∑ x ∈ S, ∑ y ∈ S,
+        star (f x) * (if Conn G ω x y then (1:ℂ) else 0) * f y ∂law p := by
+    rw [integral_finsetSum S (fun x _ => integrable_finsetSum S (fun y _ => hi x y))]
+    apply Finset.sum_congr rfl
+    intro x _
+    rw [integral_finsetSum S (fun y _ => hi x y)]
+    apply Finset.sum_congr rfl
+    intro y _
+    rw [integral_mul_const, integral_const_mul, twoPoint_eq_integral]
+  rw [hsum]
+  exact integral_nonneg (fun ω => connection_quadratic_nonneg G ω S f)
+
+ 
+
+theorem HasKernel.isPositive (G : BondGraph V E) (p : unitInterval)
+    (A : Hilbert V →L[ℂ] Hilbert V) (hA : HasKernel G p A) : A.IsPositive := by
+  have hs : A.toLinearMap.IsSymmetric :=
+    symmetric_of_symmetric_kernel A (twoPoint G p) (fun x y => hA x y) (twoPoint_symm G p)
+  refine ⟨hs, ?_⟩
+  intro f
+  apply finiteEmbedding_dense.induction_on f
+    (p := fun g => 0 ≤ (inner ℂ (A g) g).re)
+  · exact isClosed_le continuous_const (by fun_prop)
+  · intro g
+    rw [show inner ℂ (A (finiteEmbedding g)) (finiteEmbedding g) =
+      inner ℂ (finiteEmbedding g) (A (finiteEmbedding g)) from hs _ _, finiteEmbedding_inner_left]
+    simp only [kernel_finiteEmbedding A (twoPoint G p) (fun x y => hA x y)]
+    simp_rw [Finset.mul_sum]
+    have h := twoPoint_quadratic_nonneg G p g.support g
+    apply Complex.nonneg_iff.mp at h
+    convert h.1 using 1
+    congr 1
+    apply Finset.sum_congr rfl
+    intro x _
+    apply Finset.sum_congr rfl
+    intro y _
+    ring
+
+end PositiveConnectionKernel
+
+section InvariantOperatorTrace
+open scoped ComplexOrder InnerProduct
+attribute [local instance] Classical.propDecidable
+local instance spectralRealOperatorScalarTower :
+    IsScalarTower ℝ (Hilbert V →L[ℂ] Hilbert V) (Hilbert V →L[ℂ] Hilbert V) where
+  smul_assoc scalar first second := by
+    ext vector vertex
+    rfl
+local instance spectralRealOperatorSMulCommClass :
+    SMulCommClass ℝ (Hilbert V →L[ℂ] Hilbert V) (Hilbert V →L[ℂ] Hilbert V) where
+  smul_comm scalar first second := by
+    apply ContinuousLinearMap.ext
+    intro vector
+    exact (first.map_smul_of_tower scalar (second vector)).symm
+local instance spectralNonUnitalContinuousFunctionalCalculus : NonUnitalContinuousFunctionalCalculus ℝ
+    (Hilbert V →L[ℂ] Hilbert V) IsSelfAdjoint :=
+  IsSelfAdjoint.instNonUnitalContinuousFunctionalCalculus (A := Hilbert V →L[ℂ] Hilbert V)
+local instance spectralContinuousFunctionalCalculus : ContinuousFunctionalCalculus ℝ
+    (Hilbert V →L[ℂ] Hilbert V) IsSelfAdjoint :=
+  IsSelfAdjoint.instContinuousFunctionalCalculus (A := Hilbert V →L[ℂ] Hilbert V)
+noncomputable def permuteHilbert (a : V ≃ V) : Hilbert V ≃ₗᵢ[ℂ] Hilbert V where
+  toFun f := ⟨fun x => f (a.symm x), by
+    apply memℓp_gen
+    exact a.symm.summable_iff.mpr (lp.memℓp f |>.summable (by norm_num))⟩
+  invFun f := ⟨fun x => f (a x), by
+    apply memℓp_gen
+    exact a.summable_iff.mpr (lp.memℓp f |>.summable (by norm_num))⟩
+  left_inv f := by ext x; simp
+  right_inv f := by ext x; simp
+  map_add' f g := by ext x; rfl
+  map_smul' c f := by ext x; rfl
+  norm_map' f := by
+    rw [lp.norm_eq_tsum_rpow (by norm_num), lp.norm_eq_tsum_rpow (by norm_num)]
+    congr 1
+    exact a.symm.tsum_eq (fun x => ‖f x‖ ^ (2:ℝ≥0∞).toReal)
+
+@[simp] theorem permuteHilbert_apply (a : V ≃ V) (f : Hilbert V) (x : V) :
+    permuteHilbert a f x = f (a.symm x) := rfl
+
+@[simp] theorem permuteHilbert_single (a : V ≃ V) (x : V) (c : ℂ) :
+    permuteHilbert a (lp.single 2 x c) = lp.single 2 (a x) c := by
+  ext y
+  simp only [permuteHilbert_apply, lp.coeFn_single]
+  simp [Pi.single_apply, eq_comm, Equiv.eq_symm_apply]
+
+noncomputable def matrixCoeff (A : Hilbert V →L[ℂ] Hilbert V) (x y : V) : ℂ :=
+  A (lp.single 2 y (1:ℂ)) x
+
+@[simp] theorem matrixCoeff_eq_inner (A : Hilbert V →L[ℂ] Hilbert V) (x y : V) :
+    matrixCoeff A x y = inner ℂ (lp.single 2 x (1:ℂ)) (A (lp.single 2 y (1:ℂ))) := by
+  simp [matrixCoeff,lp.inner_single_left]
+
+theorem operator_kernel_hasSum (A : Hilbert V →L[ℂ] Hilbert V) (f : Hilbert V) (x : V) :
+    HasSum (fun y => matrixCoeff A x y * f y) (A f x) := by
+  have h := ((lp.evalCLM ℂ (fun _ : V => ℂ) 2 x).comp A).hasSum
+    (lp.hasSum_single (by norm_num : (2:ℝ≥0∞) ≠ ⊤) f)
+  change HasSum (fun y => A (lp.single 2 y (f y)) x) (A f x) at h
+  have heq : (fun y : V => A (lp.single 2 y (f y)) x) =
+      (fun y => matrixCoeff A x y * f y) := by
+    funext y
+    have he : lp.single (E := fun _ : V => ℂ) 2 y (f y) =
+        f y • lp.single 2 y (1:ℂ) := by
+      rw [← lp.single_smul]; simp
+    simp [he, matrixCoeff, mul_comm]
+  rw [heq] at h
+  exact h
+
+@[simp] theorem matrixCoeff_mul (A B : Hilbert V →L[ℂ] Hilbert V) (x y : V) :
+    matrixCoeff (A*B) x y = ∑' z, matrixCoeff A x z * matrixCoeff B z y := by
+  exact (operator_kernel_hasSum A (B (lp.single 2 y (1:ℂ))) x).tsum_eq.symm
+
+@[simp] theorem matrixCoeff_adjoint (A : Hilbert V →L[ℂ] Hilbert V) (x y : V) :
+    matrixCoeff (star A) x y = star (matrixCoeff A y x) := by
+  rw [matrixCoeff_eq_inner, matrixCoeff_eq_inner]
+  change inner ℂ _ ((ContinuousLinearMap.adjoint A) _) = _
+  rw [ContinuousLinearMap.adjoint_inner_right]
+  simpa only [RCLike.star_def] using
+    (inner_conj_symm (A (lp.single 2 x (1:ℂ))) (lp.single 2 y (1:ℂ))).symm
+
+ 
+theorem matrixCoeff_invariant_iff (a : V ≃ V) (A : Hilbert V →L[ℂ] Hilbert V) :
+    (∀ x y, matrixCoeff A (a x) (a y) = matrixCoeff A x y) ↔
+      Commute A (permuteHilbert a).toContinuousLinearEquiv.toContinuousLinearMap := by
+  constructor
+  · intro h
+    change A * (permuteHilbert a).toContinuousLinearEquiv.toContinuousLinearMap =
+      (permuteHilbert a).toContinuousLinearEquiv.toContinuousLinearMap * A
+    apply lp.ext_continuousLinearMap (by norm_num : (2:ℝ≥0∞) ≠ ⊤)
+    intro y
+    apply ContinuousLinearMap.ext
+    intro c
+    ext x
+    have hs : lp.single (E := fun _ : V => ℂ) 2 y c =
+        c • lp.single 2 y (1:ℂ) := by rw [← lp.single_smul]; simp
+    have hh := h (a.symm x) y
+    simp only [Equiv.apply_symm_apply] at hh
+    change A (permuteHilbert a (lp.single 2 y c)) x =
+      permuteHilbert a (A (lp.single 2 y c)) x
+    rw [hs]
+    simp only [map_smul, permuteHilbert_single, lp.coeFn_smul, Pi.smul_apply,
+      permuteHilbert_apply]
+    change c * matrixCoeff A x (a y) = c * matrixCoeff A (a.symm x) y
+    rw [hh]
+  · intro h x y
+    have hh := congrArg (fun F : Hilbert V →L[ℂ] Hilbert V =>
+      F (lp.single 2 y (1:ℂ)) (a x)) h.eq
+    simpa only [mul_apply_eq_comp, LinearIsometryEquiv.coe_toContinuousLinearEquiv,
+      ContinuousLinearEquiv.coe_coe, permuteHilbert_single, permuteHilbert_apply,
+      Equiv.symm_apply_apply, matrixCoeff] using hh
+
+ theorem matrixCoeff_sqrt_invariant (a : V ≃ V) (A : Hilbert V →L[ℂ] Hilbert V)
+    (h : ∀ x y, matrixCoeff A (a x) (a y) = matrixCoeff A x y) :
+    ∀ x y, matrixCoeff (CFC.sqrt A) (a x) (a y) = matrixCoeff (CFC.sqrt A) x y := by
+  apply (matrixCoeff_invariant_iff a _).mpr
+  rw [CFC.sqrt_eq_cfc]
+  exact ((matrixCoeff_invariant_iff a A).mp h).cfc_nnreal NNReal.sqrt
+
+ theorem matrixCoeff_mul_invariant (a : V ≃ V) (A B : Hilbert V →L[ℂ] Hilbert V)
+    (hA : ∀ x y, matrixCoeff A (a x) (a y) = matrixCoeff A x y)
+    (hB : ∀ x y, matrixCoeff B (a x) (a y) = matrixCoeff B x y) :
+    ∀ x y, matrixCoeff (A*B) (a x) (a y) = matrixCoeff (A*B) x y := by
+  intro x y
+  simp only [matrixCoeff_mul]
+  rw [← a.tsum_eq (fun z => matrixCoeff A (a x) z * matrixCoeff B z (a y))]
+  simp only [hA,hB]
+
+ theorem matrixCoeff_nonneg (A : Hilbert V →L[ℂ] Hilbert V) (hA : 0 ≤ A) (x : V) :
+    0 ≤ matrixCoeff A x x := by
+  rw [matrixCoeff_eq_inner]
+  exact (ContinuousLinearMap.nonneg_iff_isPositive |>.mp hA).inner_nonneg_right _
+
+ theorem matrixCoeff_mono {A B : Hilbert V →L[ℂ] Hilbert V} (h : A ≤ B) (x : V) :
+    matrixCoeff A x x ≤ matrixCoeff B x x := by
+  have hh := matrixCoeff_nonneg (B-A) (sub_nonneg.mpr h) x
+  simpa only [matrixCoeff, sub_apply, lp.coeFn_sub,
+    Pi.sub_apply, sub_nonneg] using hh
+
+section RootTrace
+variable {ι : Type*} [Fintype ι]
+variable (root : ι → V) (weight : ι → ℝ)
+
+noncomputable def rootTrace (A : Hilbert V →L[ℂ] Hilbert V) : ℂ :=
+  ∑ i, (weight i : ℂ) * matrixCoeff A (root i) (root i)
+
+@[simp] theorem rootTrace_add (A B : Hilbert V →L[ℂ] Hilbert V) :
+    rootTrace root weight (A+B) = rootTrace root weight A + rootTrace root weight B := by
+  simp [rootTrace, matrixCoeff, add_apply, mul_add, Finset.sum_add_distrib]
+
+@[simp] theorem rootTrace_smul (c : ℂ) (A : Hilbert V →L[ℂ] Hilbert V) :
+    rootTrace root weight (c • A) = c * rootTrace root weight A := by
+  simp [rootTrace, matrixCoeff, smul_apply, ← Finset.mul_sum, mul_left_comm]
+
+theorem rootTrace_nonneg (hw : ∀ i, 0 ≤ weight i)
+    (A : Hilbert V →L[ℂ] Hilbert V) (hA : 0 ≤ A) : 0 ≤ rootTrace root weight A := by
+  exact Finset.sum_nonneg fun i _ => mul_nonneg (by simpa using hw i)
+    (matrixCoeff_nonneg A hA (root i))
+
+theorem rootTrace_mono (hw : ∀ i, 0 ≤ weight i)
+    {A B : Hilbert V →L[ℂ] Hilbert V} (h : A ≤ B) :
+    rootTrace root weight A ≤ rootTrace root weight B := by
+  apply Finset.sum_le_sum
+  intro i _
+  exact mul_le_mul_of_nonneg_left (matrixCoeff_mono h (root i)) (by simpa using hw i)
+
+@[simp] theorem rootTrace_one (hw : ∑ i, weight i = 1) :
+    rootTrace root weight (1 : Hilbert V →L[ℂ] Hilbert V) = 1 := by
+  simpa [rootTrace, matrixCoeff, lp.coeFn_single, ← Complex.ofReal_sum] using
+    congrArg (fun t : ℝ => (t : ℂ)) hw
+
+ 
+def KernelInvariant (Γ : Set (V ≃ V)) (A : Hilbert V →L[ℂ] Hilbert V) : Prop :=
+  ∀ a ∈ Γ, ∀ x y, matrixCoeff A (a x) (a y) = matrixCoeff A x y
+
+theorem KernelInvariant.mul {Γ : Set (V ≃ V)}
+    {A B : Hilbert V →L[ℂ] Hilbert V}
+    (hA : KernelInvariant Γ A) (hB : KernelInvariant Γ B) :
+    KernelInvariant Γ (A*B) := by
+  intro a ha
+  exact matrixCoeff_mul_invariant a A B (hA a ha) (hB a ha)
+
+theorem KernelInvariant.star {Γ : Set (V ≃ V)}
+    {A : Hilbert V →L[ℂ] Hilbert V} (hA : KernelInvariant Γ A) :
+    KernelInvariant Γ (star A) := by
+  intro a ha x y
+  simp only [matrixCoeff_adjoint, hA a ha]
+
+theorem KernelInvariant.sqrt {Γ : Set (V ≃ V)}
+    {A : Hilbert V →L[ℂ] Hilbert V} (hA : KernelInvariant Γ A) :
+    KernelInvariant Γ (CFC.sqrt A) := by
+  intro a ha
+  exact matrixCoeff_sqrt_invariant a A (hA a ha)
+
+ 
+
+def RootTransport (Γ : Set (V ≃ V)) : Prop :=
+  ∀ m : V → V → ℂ,
+    (∀ a ∈ Γ, ∀ x y, m (a x) (a y) = m x y) →
+    (∀ x, Summable (m x)) → (∀ y, Summable (fun x => m x y)) →
+    (∑ i, (weight i : ℂ) * ∑' y, m (root i) y) =
+    (∑ i, (weight i : ℂ) * ∑' x, m x (root i))
+
+theorem rootTrace_cyclic {Γ : Set (V ≃ V)}
+    (htr : RootTransport root weight Γ)
+    (A B : Hilbert V →L[ℂ] Hilbert V)
+    (hA : KernelInvariant Γ A) (hB : KernelInvariant Γ B) :
+    rootTrace root weight (A*B) = rootTrace root weight (B*A) := by
+  have ht := htr (fun x y => matrixCoeff A x y * matrixCoeff B y x)
+    (by intro a ha x y; rw [hA a ha, hB a ha])
+    (fun x => (operator_kernel_hasSum A (B (lp.single 2 x (1:ℂ))) x).summable)
+    (fun y => by
+      simpa only [matrixCoeff, mul_comm] using
+        (operator_kernel_hasSum B (A (lp.single 2 y (1:ℂ))) y).summable)
+  simpa only [rootTrace, matrixCoeff_mul, mul_comm] using ht
+
+theorem rootTrace_diag_le {Γ : Set (V ≃ V)}
+    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
+    (hw : ∀ i, 0 ≤ weight i) (c : ℝ) (hc : 0 ≤ c)
+    (hmin : ∀ i, 1 ≤ c * weight i)
+    (A : Hilbert V →L[ℂ] Hilbert V) (hA : KernelInvariant Γ A)
+    (hApos : 0 ≤ A) (x : V) :
+    matrixCoeff A x x ≤ (c : ℂ) * rootTrace root weight A := by
+  obtain ⟨i,a,ha,hax⟩ := hcov x
+  have hi : (weight i : ℂ) * matrixCoeff A (root i) (root i) ≤
+      rootTrace root weight A := by
+    unfold rootTrace
+    exact Finset.single_le_sum (s := Finset.univ)
+      (f := fun j => (weight j : ℂ) * matrixCoeff A (root j) (root j))
+      (fun j _ => mul_nonneg (by simpa using hw j) (matrixCoeff_nonneg A hApos _))
+      (Finset.mem_univ i)
+  calc
+    matrixCoeff A x x = matrixCoeff A (root i) (root i) := by
+      rw [← hax, hA a ha]
+    _ ≤ (c * weight i : ℝ) * matrixCoeff A (root i) (root i) := by
+      simpa only [one_mul] using mul_le_mul_of_nonneg_right
+        (show (1:ℂ) ≤ ((c * weight i : ℝ) : ℂ) by exact_mod_cast hmin i)
+        (matrixCoeff_nonneg A hApos (root i))
+    _ = (c : ℂ) * ((weight i : ℂ) * matrixCoeff A (root i) (root i)) := by
+      push_cast; ring
+    _ ≤ (c : ℂ) * rootTrace root weight A :=
+      mul_le_mul_of_nonneg_left hi (by exact_mod_cast hc)
+
+@[simp] theorem rootTrace_real_smul (c : ℝ) (A : Hilbert V →L[ℂ] Hilbert V) :
+    rootTrace root weight (c • A) = (c : ℂ) * rootTrace root weight A := by
+  simp [rootTrace, matrixCoeff, smul_apply, Complex.real_smul,
+    ← Finset.mul_sum, mul_left_comm]
+
+theorem rootTrace_positive_mul_le {Γ : Set (V ≃ V)}
+    (htr : RootTransport root weight Γ) (hw : ∀ i, 0 ≤ weight i)
+    (A B : Hilbert V →L[ℂ] Hilbert V)
+    (hA : KernelInvariant Γ A) (hB : KernelInvariant Γ B)
+    (hApos : 0 ≤ A) (hBsa : IsSelfAdjoint B) :
+    rootTrace root weight (B*A) ≤ (‖B‖ : ℂ) * rootTrace root weight A := by
+  let S := CFC.sqrt A
+  have hS : S*S = A := CFC.sqrt_mul_sqrt_self A hApos
+  have hSstar : star S = S := (IsSelfAdjoint.of_nonneg (CFC.sqrt_nonneg A)).star_eq
+  have hSI : KernelInvariant Γ S := hA.sqrt
+  have hbound : S*B*S ≤ ‖B‖ • A := by
+    convert (CStarAlgebra.star_left_conjugate_le_norm_smul (a := S) (b := B) hBsa) using 1
+    · rw [hSstar]
+    · rw [hSstar, hS]
+      ext f x
+      rfl
+  calc
+    rootTrace root weight (B*A) = rootTrace root weight (S*(B*S)) := by
+      rw [rootTrace_cyclic root weight htr S (B*S) hSI (hB.mul hSI)]
+      rw [mul_assoc, hS]
+    _ = rootTrace root weight (S*B*S) := by rw [mul_assoc]
+    _ ≤ rootTrace root weight (‖B‖ • A) := rootTrace_mono root weight hw hbound
+    _ = (‖B‖ : ℂ) * rootTrace root weight A := rootTrace_real_smul root weight _ _
+
+theorem rootTrace_conjugate_le {Γ : Set (V ≃ V)}
+    (htr : RootTransport root weight Γ) (hw : ∀ i, 0 ≤ weight i)
+    (A B : Hilbert V →L[ℂ] Hilbert V)
+    (hA : KernelInvariant Γ A) (hB : KernelInvariant Γ B)
+    (hApos : 0 ≤ A) :
+    rootTrace root weight (B*A*star B) ≤ ((‖B‖ ^ 2 : ℝ) : ℂ) * rootTrace root weight A := by
+  calc
+    rootTrace root weight (B*A*star B) = rootTrace root weight (star B*(B*A)) :=
+      rootTrace_cyclic root weight htr (B*A) (star B) (hB.mul hA) hB.star
+    _ = rootTrace root weight ((star B*B)*A) := by rw [mul_assoc]
+    _ ≤ (‖star B*B‖ : ℂ) * rootTrace root weight A :=
+      rootTrace_positive_mul_le root weight htr hw A (star B*B) hA (hB.star.mul hB)
+        hApos (IsSelfAdjoint.of_nonneg (star_mul_self_nonneg B))
+    _ = ((‖B‖ ^ 2 : ℝ) : ℂ) * rootTrace root weight A := by
+      have hnorm := CStarRing.norm_star_mul_self (x := B)
+      have hn : ‖star B * B‖ = ‖B‖ ^ 2 := by
+        convert hnorm using 1; simp only [pow_two]
+      rw [hn]
+
+ theorem matrixCoeff_gram (L M : Hilbert V →L[ℂ] Hilbert V) (x y : V) :
+    matrixCoeff (L*star M) x y =
+      inner ℂ (star L (lp.single 2 x (1:ℂ))) (star M (lp.single 2 y (1:ℂ))) := by
+  rw [matrixCoeff_eq_inner]
+  change inner ℂ _ (L ((ContinuousLinearMap.adjoint M) _)) = _
+  rw [← ContinuousLinearMap.adjoint_inner_left]
+  rfl
+
+ theorem matrixCoeff_gram_norm_sq_le
+    (L M : Hilbert V →L[ℂ] Hilbert V) (x y : V) :
+    ‖matrixCoeff (L*star M) x y‖ ^ 2 ≤
+      (matrixCoeff (L*star L) x x).re * (matrixCoeff (M*star M) y y).re := by
+  simp only [matrixCoeff_gram]
+  have h := inner_mul_inner_self_le (𝕜 := ℂ)
+      (star L (lp.single 2 x (1:ℂ))) (star M (lp.single 2 y (1:ℂ)))
+  rw [norm_inner_symm (star M (lp.single 2 y (1:ℂ)))] at h
+  simpa only [pow_two, RCLike.re_to_complex] using h
+
+ theorem matrixCoeff_positive_product_norm_sq_le
+    (A B : Hilbert V →L[ℂ] Hilbert V) (hA : 0 ≤ A) (x y : V) :
+    ‖matrixCoeff (B*A) x y‖ ^ 2 ≤
+      (matrixCoeff (B*A*star B) x x).re * (matrixCoeff A y y).re := by
+  let S := CFC.sqrt A
+  have hSS : S*S = A := CFC.sqrt_mul_sqrt_self A hA
+  have hSstar : star S = S := (IsSelfAdjoint.of_nonneg (CFC.sqrt_nonneg A)).star_eq
+  convert matrixCoeff_gram_norm_sq_le (B*S) S x y using 1 <;>
+    simp only [star_mul, hSstar, mul_assoc, ← mul_assoc S S, hSS]
+
+ 
+
+theorem matrixCoeff_smoothing {Γ : Set (V ≃ V)}
+    (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
+    (hw : ∀ i, 0 ≤ weight i) (hsum : ∑ i, weight i = 1)
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c * weight i)
+    (A B : Hilbert V →L[ℂ] Hilbert V)
+    (hA : KernelInvariant Γ A) (hB : KernelInvariant Γ B)
+    (hApos : 0 ≤ A) (hdiag : ∀ y, matrixCoeff A y y = 1) (x y : V) :
+    ‖matrixCoeff (B*A) x y‖ ≤ Real.sqrt c * ‖B‖ := by
+  have htA : rootTrace root weight A = 1 := by
+    simp only [rootTrace, hdiag, mul_one, ← Complex.ofReal_sum, hsum, Complex.ofReal_one]
+  have hD : 0 ≤ B*A*star B := star_right_conjugate_nonneg hApos B
+  have hi : KernelInvariant Γ (B*A*star B) := (hB.mul hA).mul hB.star
+  have ht := rootTrace_conjugate_le root weight htr hw A B hA hB hApos
+  rw [htA, mul_one] at ht
+  have hd : matrixCoeff (B*A*star B) x x ≤ ((c * ‖B‖^2 : ℝ) : ℂ) := by
+    calc
+      _ ≤ (c : ℂ) * rootTrace root weight (B*A*star B) :=
+        rootTrace_diag_le root weight hcov hw c hc hmin _ hi hD x
+      _ ≤ (c : ℂ) * ((‖B‖^2 : ℝ) : ℂ) :=
+        mul_le_mul_of_nonneg_left ht (by exact_mod_cast hc)
+      _ = _ := by push_cast; rfl
+  have hd' : (matrixCoeff (B*A*star B) x x).re ≤ c * ‖B‖^2 :=
+    (Complex.le_def.mp hd).1
+  have hs : ‖matrixCoeff (B*A) x y‖ ^ 2 ≤ c * ‖B‖ ^ 2 := by
+    have hh := matrixCoeff_positive_product_norm_sq_le A B hApos x y
+    rw [hdiag, Complex.one_re, mul_one] at hh
+    exact hh.trans hd'
+  apply (sq_le_sq₀ (norm_nonneg _) (mul_nonneg (Real.sqrt_nonneg _) (norm_nonneg _))).mp
+  rwa [mul_pow, Real.sq_sqrt hc]
+
+end RootTrace
+
+end InvariantOperatorTrace
+
+section InvariantPercolationOperators
+open scoped ComplexOrder InnerProduct
+attribute [local instance] Classical.propDecidable
+
+ 
+def vertexAut (G : BondGraph V E) : Set (V ≃ V) := {a | ∃ b : E ≃ E, IsAut G a b}
+
+theorem twoPoint_aut [Countable V] [Countable E] (G : BondGraph V E) (p : unitInterval)
+    {a : V ≃ V} {b : E ≃ E} (h : IsAut G a b) (x y : V) :
+    twoPoint G p (a x) (a y) = twoPoint G p x y := by
+  have he : {ω | Conn G ω (a x) (a y)} =
+      (fun ω => b ⁻¹' ω) ⁻¹' {ω | Conn G ω x y} := by
+    ext ω
+    exact (conn_aut_iff G h ω x y).symm
+  unfold twoPoint
+  rw [he, ← Measure.map_apply (measurable_set_preimage b)
+    (measurableSet_conn G x y), law_map_preimage_equiv]
+
+theorem HasKernel.invariant [Countable V] [Countable E] (G : BondGraph V E) (p : unitInterval)
+    (A : Hilbert V →L[ℂ] Hilbert V) (hA : HasKernel G p A) :
+    KernelInvariant (vertexAut G) A := by
+  rintro a ⟨b, hab⟩ x y
+  change A (lp.single 2 (a y) (1:ℂ)) (a x) = A (lp.single 2 y (1:ℂ)) x
+  rw [hA, hA, twoPoint_aut G p hab]
+
+theorem KernelInvariant.one (Γ : Set (V ≃ V)) :
+    KernelInvariant Γ (1 : Hilbert V →L[ℂ] Hilbert V) := by
+  intro a _ x y
+  simp only [matrixCoeff, one_apply_eq_self, lp.coeFn_single]
+  simp [Pi.single_apply, a.injective.eq_iff]
+
+theorem KernelInvariant.pow {Γ : Set (V ≃ V)}
+    {A : Hilbert V →L[ℂ] Hilbert V} (hA : KernelInvariant Γ A) (n : ℕ) :
+    KernelInvariant Γ (A^n) := by
+  induction n with
+  | zero => simpa only [pow_zero] using KernelInvariant.one Γ
+  | succ n ih => simpa only [pow_succ] using ih.mul hA
+
+ 
+
+theorem twoPoint_operator_smoothing [Countable V] [Countable E] [Nonempty V] {ι : Type*} [Fintype ι]
+    (G : BondGraph V E) (p : unitInterval) (root : ι → V) (weight : ι → ℝ)
+    (htr : RootTransport root weight (vertexAut G))
+    (hcov : ∀ x, ∃ i, ∃ a ∈ vertexAut G, a (root i) = x)
+    (hw : ∀ i, 0 ≤ weight i) (hsum : ∑ i, weight i = 1)
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c * weight i)
+    (A P : Hilbert V →L[ℂ] Hilbert V) (hA : HasKernel G p A)
+    (hP : KernelInvariant (vertexAut G) P) (l : ℕ) (x y : V) :
+    ‖matrixCoeff (P^l*A) x y‖ ≤ Real.sqrt c * ‖P‖^l := by
+  have hdiag (y : V) : matrixCoeff A y y = 1 := by
+    change A (lp.single 2 y (1:ℂ)) y = 1
+    rw [hA y y, twoPoint_self, Complex.ofReal_one]
+  have h := matrixCoeff_smoothing root weight htr hcov hw hsum c hc hmin A (P^l)
+    (hA.invariant G p A) (hP.pow l) (ContinuousLinearMap.nonneg_iff_isPositive.mpr
+      (hA.isPositive G p A)) hdiag x y
+  have hn (n : ℕ) : ‖P^n‖ ≤ ‖P‖^n := by
+    induction n with
+    | zero => simpa only [pow_zero, ContinuousLinearMap.one_def] using (ContinuousLinearMap.norm_id_le (𝕜 := ℂ) (E := Hilbert V))
+    | succ n ih =>
+      rw [pow_succ, pow_succ]
+      exact (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right ih (norm_nonneg P))
+  exact h.trans (mul_le_mul_of_nonneg_left (hn l) (Real.sqrt_nonneg c))
+end InvariantPercolationOperators
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Kernel/Sprinkling.lean b/lean/OAI/Probability/UnimodularPercolation/Kernel/Sprinkling.lean
index 753a5c5756a491db8b4f46ac514c01c4c473de29..575402ef5017ea7418e6d807ecbed88b344572eb 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Kernel/Sprinkling.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Kernel/Sprinkling.lean
@@ -1,840 +1,840 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Graph.Paths
-
-namespace OAI
-
-/-! Sprinkling estimates for finite and infinite graph operators. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-section SprinklingLaw
-attribute [local instance] Classical.propDecidable
-
-noncomputable def sprinkleParameter (p a : unitInterval) : unitInterval :=
-  ⟨(p : ℝ) + (1-p)*a, by
-    constructor
-    · exact add_nonneg p.2.1 (mul_nonneg (sub_nonneg.mpr p.2.2) a.2.1)
-    · nlinarith [mul_nonneg (sub_nonneg.mpr p.2.2) (sub_nonneg.mpr a.2.2)]⟩
-
- def pairParameter (p a : unitInterval) (b : Bool) : unitInterval := if b then p else a
-
- theorem map_bool_or (p a : unitInterval) :
-    (Measure.infinitePi (fun b : Bool => bitLaw (pairParameter p a b))).map
-      (fun f : Bool → Prop => ∃ b, f b) = bitLaw (sprinkleParameter p a) := by
-  let μ := (Measure.infinitePi (fun b : Bool => bitLaw (pairParameter p a b))).map
-      (fun f : Bool → Prop => ∃ b, f b)
-  have hm : Measurable (fun f : Bool → Prop => ∃ b, f b) := .of_discrete
-  have : IsProbabilityMeasure μ := inferInstance
-  have hf : μ.real {False} = (1-(p : ℝ)) * (1-(a : ℝ)) := by
-    change ((Measure.infinitePi _).map _).real _ = _
-    rw [Measure.real, Measure.map_apply hm (measurableSet_singleton _)]
-    have he : (fun f : Bool → Prop => ∃ b, f b) ⁻¹' ({False} : Set Prop) =
-        ((Finset.univ : Finset Bool) : Set Bool).pi (fun _ => ({False} : Set Prop)) := by
-      ext f
-      simp [and_comm]
-    rw [he, Measure.infinitePi_pi _ (fun _ _ => measurableSet_singleton _)]
-    simp [pairParameter, bitLaw, ENNReal.toReal_mul]
-  have ht : μ.real {True} = (p : ℝ) + (1-p)*a := by
-    have he : ({True} : Set Prop) = ({False} : Set Prop)ᶜ := by ext b; simp
-    rw [he, measureReal_compl (measurableSet_singleton _), probReal_univ, hf]
-    ring
-  apply Measure.ext_of_singleton
-  intro b
-  apply (ENNReal.toReal_eq_toReal_iff' (measure_ne_top _ _) (measure_ne_top _ _)).mp
-  change μ.real {b} = (bitLaw (sprinkleParameter p a)).real {b}
-  by_cases hb : b
-  · have he : b = True := propext (iff_true_intro hb)
-    subst b
-    rw [ht]
-    simp [bitLaw, Measure.real, sprinkleParameter]
-  · have he : b = False := propext (iff_false_intro hb)
-    subst b
-    rw [hf]
-    simp [bitLaw, Measure.real, sprinkleParameter]
-    ring
-
-noncomputable def fieldLaw (p a : unitInterval) : Measure (Bool → Set E) :=
-  Measure.infinitePi (fun b => law (pairParameter p a b))
-
-def fieldUnion (f : Bool → Set E) : Set E := f true ∪ f false
-
- theorem measurable_fieldUnion : Measurable (fieldUnion : (Bool → Set E) → Set E) := by
-  have hm (e : E) : Measurable (fun f : Bool → Set E => e ∈ fieldUnion f) := by
-    apply measurableSet_setOfPred.mp
-    change MeasurableSet ({f : Bool → Set E | e ∈ f true} ∪ {f | e ∈ f false})
-    apply MeasurableSet.union <;> apply measurableSet_setOfPred.mpr
-    · exact ((measurable_pi_apply e).comp MeasurableEquiv.setOfPred.symm.measurable).comp
-        (measurable_pi_apply true)
-    · exact ((measurable_pi_apply e).comp MeasurableEquiv.setOfPred.symm.measurable).comp
-        (measurable_pi_apply false)
-  exact measurable_setOfPred.comp (Measurable.of_eval hm)
-
- theorem map_fieldUnion (p a : unitInterval) :
-    (fieldLaw p a).map (fieldUnion : (Bool → Set E) → Set E) = law (sprinkleParameter p a) := by
-  let μ := fun b : Bool => bitLaw (pairParameter p a b)
-  let s := fun f : Bool → E → Prop => fun b => {e | f b e}
-  have hs : Measurable s := Measurable.of_eval
-    (fun b => measurable_setOfPred.comp (measurable_pi_apply b))
-  have hstart : (Measure.infinitePi (fun b : Bool => Measure.infinitePi (fun _ : E => μ b))).map s =
-      fieldLaw p a := by
-    rw [Measure.infinitePi_map_pi _ (fun _ => measurable_setOfPred)]
-    congr 1
-    funext b
-    rw [law, setBernoulli_eq_map]
-    rfl
-  rw [← hstart, Measure.map_map measurable_fieldUnion hs]
-  let f := (MeasurableEquiv.curry Bool E Prop).symm
-  let swap := MeasurableEquiv.piCongrLeft (fun _ : E × Bool => Prop) (Equiv.prodComm Bool E)
-  let c := MeasurableEquiv.curry E Bool Prop
-  let b := fun g : E → Bool → Prop => fun e => ∃ j, g e j
-  have hb : Measurable b := Measurable.of_eval
-    (fun e => (Measurable.of_discrete : Measurable (fun f : Bool → Prop => ∃ j, f j)).comp
-      (measurable_pi_apply e))
-  have he : fieldUnion ∘ s = (fun g : E → Prop => {e | g e}) ∘ b ∘ c ∘ swap ∘ f := by
-    funext g
-    ext e
-    simp [fieldUnion, s, b, c, swap, f, Function.comp_def, Bool.exists_bool, or_comm, MeasurableEquiv.coe_piCongrLeft, Equiv.piCongrLeft]
-  rw [he, ← Measure.map_map measurable_setOfPred (hb.comp (c.measurable.comp (swap.measurable.comp f.measurable))),
-    ← Measure.map_map hb (c.measurable.comp (swap.measurable.comp f.measurable))]
-  rw [← Measure.map_map c.measurable (swap.measurable.comp f.measurable),
-    ← Measure.map_map swap.measurable f.measurable]
-  rw [Measure.infinitePi_map_curry_symm]
-  rw [show (Measure.infinitePi (fun x : Bool × E => μ x.1)).map swap =
-      Measure.infinitePi (fun x : E × Bool => μ x.2) from
-    Measure.infinitePi_map_piCongrLeft (fun x : E × Bool => μ x.2) (Equiv.prodComm Bool E)]
-  rw [Measure.infinitePi_map_curry (fun (_ : E) (j : Bool) => μ j)]
-  dsimp only [b]
-  rw [Measure.infinitePi_map_pi _ (fun _ => Measurable.of_discrete)]
-  have hm : (Measure.infinitePi (fun j : Bool => μ j)).map Exists =
-      bitLaw (sprinkleParameter p a) := map_bool_or p a
-  rw [hm, law, setBernoulli_eq_map]
-  rfl
-
- theorem fieldLaw_map_pair (p a : unitInterval) :
-    (fieldLaw p a).map (fun f : Bool → Set E => (f true, f false)) =
-      (law p).prod (law a) := by
-  simpa [fieldLaw, pairParameter] using
-    (Measure.infinitePi_map_eval_prod (P := fun b : Bool => (law (pairParameter p a b) : Measure (Set E)))
-    (by decide : true ≠ false))
-
- theorem law_map_union (p a : unitInterval) :
-    ((law p : Measure (Set E)).prod (law a)).map (fun f => f.1 ∪ f.2) =
-      law (sprinkleParameter p a) := by
-  rw [← fieldLaw_map_pair p a, Measure.map_map]
-  · exact map_fieldUnion p a
-  · have h : Measurable (fun f : Set E × Set E => fun e => e ∈ f.1 ∪ f.2) := by
-      apply Measurable.of_eval
-      intro e
-      apply measurableSet_setOfPred.mp
-      change MeasurableSet ({f : Set E × Set E | e ∈ f.1} ∪ {f | e ∈ f.2})
-      apply MeasurableSet.union <;> apply measurableSet_setOfPred.mpr
-      · exact ((measurable_pi_apply e).comp MeasurableEquiv.setOfPred.symm.measurable).comp measurable_fst
-      · exact ((measurable_pi_apply e).comp MeasurableEquiv.setOfPred.symm.measurable).comp measurable_snd
-    exact measurable_setOfPred.comp h
-  · exact (measurable_pi_apply true).prodMk (measurable_pi_apply false)
-
-end SprinklingLaw
-
-section Schur
-attribute [local instance] Classical.propDecidable
-
- 
-theorem exists_operator_of_row_column_bound (U : V → V → ℝ) (D : ℝ)
-    (hD : 0 ≤ D) (hU : ∀ x y, 0 ≤ U x y)
-    (hrow : ∀ x (s : Finset V), ∑ y ∈ s, U x y ≤ D)
-    (hcol : ∀ y (s : Finset V), ∑ x ∈ s, U x y ≤ D) :
-    ∃ A : Hilbert V →L[ℂ] Hilbert V, ‖A‖ ≤ D ∧
-      ∀ x y, A (lp.single 2 y (1 : ℂ)) x = (U x y : ℂ) := by
-  have hentry (x y : V) : U x y ≤ D := by simpa using hrow x {y}
-  have hcm (y : V) : Memℓp (fun x => (U x y : ℂ)) 2 := by
-    apply (memℓp_gen_iff'' (by norm_num : 0 < (2 : ℝ≥0∞).toReal)).mpr
-    refine ⟨D^2, sq_nonneg _, fun s => ?_⟩
-    simp only [ENNReal.toReal_ofNat, Real.rpow_two, Complex.norm_real,
-      Real.norm_eq_abs, sq_abs]
-    calc
-      ∑ x ∈ s, U x y ^ 2 ≤ ∑ x ∈ s, D * U x y := by
-        apply Finset.sum_le_sum
-        intro x _
-        nlinarith [hentry x y, hU x y]
-      _ = D * ∑ x ∈ s, U x y := by rw [Finset.mul_sum]
-      _ ≤ D * D := mul_le_mul_of_nonneg_left (hcol y s) hD
-      _ = D^2 := by ring
-  let col (y : V) : Hilbert V := ⟨fun x => (U x y : ℂ), hcm y⟩
-  let L : (V →₀ ℂ) →ₗ[ℂ] Hilbert V := Finsupp.linearCombination ℂ col
-  have hn (f : V →₀ ℂ) : ‖L f‖ ≤ D * ‖finiteEmbedding f‖ := by
-    apply lp.norm_le_of_forall_sum_le (by norm_num)
-      (mul_nonneg hD (norm_nonneg _))
-    intro s
-    simp only [ENNReal.toReal_ofNat, Real.rpow_two]
-    have hpoint (x : V) : ‖L f x‖^2 ≤ D * ∑ y ∈ f.support, ‖f y‖^2 * U x y := by
-      have hn0 : 0 ≤ ∑ y ∈ f.support, ‖f y‖ * U x y :=
-        Finset.sum_nonneg fun y _ => mul_nonneg (norm_nonneg _) (hU x y)
-      have hn1 : ‖L f x‖ ≤ ∑ y ∈ f.support, ‖f y‖ * U x y := by
-        simp only [L, Finsupp.linearCombination_apply, Finsupp.sum, lp.coeFn_sum,
-          Finset.sum_apply, lp.coeFn_smul, Pi.smul_apply, smul_eq_mul]
-        apply (norm_sum_le _ _).trans
-        apply Finset.sum_le_sum
-        intro y _
-        simp [col, abs_of_nonneg (hU x y)]
-      have hcs := Finset.sum_sq_le_sum_mul_sum_of_sq_le_mul f.support
-        (f := fun y => U x y) (g := fun y => ‖f y‖^2 * U x y)
-        (r := fun y => ‖f y‖ * U x y)
-        (fun y _ => hU x y)
-        (fun y _ => mul_nonneg (sq_nonneg _) (hU x y))
-        (fun y _ => by nlinarith)
-      exact (sq_le_sq₀ (norm_nonneg _) hn0 |>.mpr hn1).trans
-        (hcs.trans (mul_le_mul_of_nonneg_right (hrow x f.support)
-          (Finset.sum_nonneg fun y _ => mul_nonneg (sq_nonneg _) (hU x y))))
-    calc
-      ∑ x ∈ s, ‖L f x‖^2 ≤ ∑ x ∈ s, D * ∑ y ∈ f.support, ‖f y‖^2 * U x y :=
-        Finset.sum_le_sum fun x _ => hpoint x
-      _ = D * ∑ y ∈ f.support, ‖f y‖^2 * ∑ x ∈ s, U x y := by
-        rw [← Finset.mul_sum, Finset.sum_comm]
-        simp_rw [← Finset.mul_sum]
-      _ ≤ D * ∑ y ∈ f.support, ‖f y‖^2 * D := by
-        apply mul_le_mul_of_nonneg_left _ hD
-        apply Finset.sum_le_sum
-        intro y _
-        exact mul_le_mul_of_nonneg_left (hcol y s) (sq_nonneg _)
-      _ = D^2 * ∑ y ∈ f.support, ‖finiteEmbedding f y‖^2 := by
-        simp only [finiteEmbedding_apply, ← Finset.sum_mul]
-        ring
-      _ ≤ D^2 * ‖finiteEmbedding f‖^2 := by
-        apply mul_le_mul_of_nonneg_left _ (sq_nonneg _)
-        simpa only [ENNReal.toReal_ofNat, Real.rpow_two] using
-          (lp.sum_rpow_le_norm_rpow (by norm_num : 0 < (2 : ℝ≥0∞).toReal)
-            (finiteEmbedding f) f.support)
-      _ = (D * ‖finiteEmbedding f‖)^2 := by ring
-  refine ⟨L.extendOfNorm finiteEmbedding,
-    LinearMap.opNorm_extendOfNorm_le finiteEmbedding_dense hD hn, ?_⟩
-  intro x y
-  have he := LinearMap.extendOfNorm_eq (f := L) finiteEmbedding_dense ⟨D, hn⟩
-    (Finsupp.single y (1 : ℂ))
-  simp only [finiteEmbedding_single] at he
-  rw [he]
-  simp [L, col]
-
-end Schur
-
-section SprinklingPaths
-attribute [local instance] Classical.propDecidable
-
- 
-theorem Along.first_outside {G : BondGraph V E} {x y : V} {l : List E}
-    (h : Along G x l y) (S : Set E) :
-    (∀ e ∈ l, e ∈ S) ∨ ∃ a b e l₁ l₂,
-      l = l₁ ++ e :: l₂ ∧ Along G x l₁ a ∧ G e = s(a,b) ∧
-      Along G b l₂ y ∧ (∀ d ∈ l₁, d ∈ S) ∧ e ∉ S := by
-  induction h with
-  | nil => left; simp
-  | @cons x z y l e he ht ih =>
-    by_cases hes : e ∈ S
-    · rcases ih with hall | ⟨a,b,d,l₁,l₂,hl,h₁,hd,h₂,hopen,hnot⟩
-      · left
-        simpa only [List.mem_cons, forall_eq_or_imp] using And.intro hes hall
-      · right
-        refine ⟨a,b,d,e::l₁,l₂,by simp [hl], .cons e he h₁, hd, h₂, ?_, hnot⟩
-        simpa only [List.mem_cons, forall_eq_or_imp] using And.intro hes hopen
-    · right
-      exact ⟨x,z,e,[],l,rfl,.nil x,he,ht,by simp,hes⟩
-
- 
-theorem conn_first_outside (G : BondGraph V E) (S : Set E) {ω : Set E} {x y : V}
-    (h : Conn G ω x y) :
-    Conn G (ω ∩ S) x y ∨ ∃ a b e, G e = s(a,b) ∧ e ∉ S ∧
-      ω ∈ DisjointOccurrence {η | Conn G (η ∩ S) x a}
-        (DisjointOccurrence {η | e ∈ η} {η | Conn G η b y}) := by
-  obtain ⟨l,hl,hn,ho⟩ := conn_exists_bond_trail G h
-  rcases hl.first_outside S with hall | ⟨a,b,e,l₁,l₂,heq,h₁,he,h₂,hfirst,hnot⟩
-  · exact Or.inl (hl.conn fun e he => ⟨ho e he,hall e he⟩)
-  · subst l
-    have hnod := List.nodup_append.mp hn
-    have hen : e ∉ l₂ := (List.nodup_cons.mp hnod.2.1).1
-    right
-    refine ⟨a,b,e,he,hnot,l₁.toFinset,(e::l₂).toFinset,?_,?_,?_,?_,?_⟩
-    · intro d hd
-      exact ho d (List.mem_append_left _ (List.mem_toFinset.mp hd))
-    · intro d hd
-      exact ho d (List.mem_append_right _ (List.mem_toFinset.mp hd))
-    · apply Finset.disjoint_left.mpr
-      intro d hd hd'
-      exact hnod.2.2 d (List.mem_toFinset.mp hd) d (List.mem_toFinset.mp hd') rfl
-    · exact h₁.conn (fun d hd => ⟨by simpa using hd, hfirst d hd⟩)
-    · refine ⟨{e},l₂.toFinset,?_,?_,?_,by simp,?_⟩
-      · simp
-      · simp
-      · simpa using hen
-      · exact h₂.conn (fun d hd => by simpa using hd)
-
- 
-theorem law_map_preimage_embedding {F : Type*} (p : unitInterval) (f : F ↪ E) :
-    (law p : Measure (Set E)).map (fun ω => f ⁻¹' ω) = law p := by
-  have hm : Measurable (fun ω : Set E => f ⁻¹' ω) :=
-    measurable_setOfPred.comp (Measurable.of_eval
-      (fun e => (measurable_pi_apply (f e)).comp MeasurableEquiv.setOfPred.symm.measurable))
-  unfold law
-  rw [setBernoulli_eq_map, Measure.map_map hm measurable_setOfPred]
-  have he : (fun ω : Set E => f ⁻¹' ω) ∘ (fun g => {e | g e}) =
-      (fun g : F → Prop => {e | g e}) ∘ (fun g : E → Prop => fun e => g (f e)) := rfl
-  rw [he, ← Measure.map_map measurable_setOfPred (by fun_prop)]
-  simp only [Set.mem_univ]
-  have hi := (iIndepFun_infinitePi (P := fun _ : E => bernoulliMeasure True False p)
-    (X := fun (_ : E) (b : Prop) => b) (fun _ => measurable_id)).precomp f.injective
-  have hh := hi.map_fun_eq_infinitePi_map (fun _ => measurable_pi_apply _)
-  simp only [Measure.infinitePi_map_eval] at hh
-  rw [hh, setBernoulli_eq_map]
-  rfl
-
- 
-
-theorem exists_orParameter_small (p : unitInterval) (hp : p < 1)
-    (ε : ℝ) (hε : 0 < ε) (hε1 : ε ≤ 1) :
-    ∃ m : ℕ, ∃ q : unitInterval, (q : ℝ) ≤ ε ∧ orParameter q m = p := by
-  obtain ⟨m, hm⟩ := exists_pow_lt_of_lt_one (show 0 < 1 - (p : ℝ) by exact sub_pos.mpr hp)
-    (show 1-ε < 1 by linarith)
-  let f := fun x : ℝ => 1 - (1-x)^m
-  have hf : Continuous f := by fun_prop
-  have h0 : f 0 = 0 := by simp [f]
-  have hp' : (p : ℝ) ∈ Set.Icc (f 0) (f ε) := by
-    rw [h0]
-    exact ⟨p.2.1, by dsimp [f]; linarith⟩
-  obtain ⟨q,hq,hqp⟩ := intermediate_value_Icc hε.le hf.continuousOn hp'
-  exact ⟨m,⟨q,hq.1,hq.2.trans hε1⟩,hq.2,Subtype.ext hqp⟩
-
-end SprinklingPaths
-
-section ColoredSprinkling
-attribute [local instance] Classical.propDecidable
-
- theorem conn_map_edges {F : Type*} (G : BondGraph V E) (f : F → E)
-    (ω : Set F) (x y : V) : Conn (G ∘ f) ω x y ↔ Conn G (f '' ω) x y := by
-  constructor <;> intro h
-  · induction h with
-    | refl => exact conn_refl _ _ _
-    | @tail a b hab he ih =>
-      rcases he with ⟨e,he,heq⟩
-      exact ih.tail ⟨f e,Set.mem_image_of_mem _ he,heq⟩
-  · induction h with
-    | refl => exact conn_refl _ _ _
-    | @tail a b hab he ih =>
-      rcases he with ⟨e,⟨d,hd,rfl⟩,heq⟩
-      exact ih.tail ⟨d,hd,heq⟩
-
- theorem finiteWitness_inter (A : Set (Set E)) (hA : FiniteWitness A) (S : Set E) :
-    FiniteWitness {ω | ω ∩ S ∈ A} := by
-  intro ω h
-  obtain ⟨s,hs,hsa⟩ := hA (ω ∩ S) h
-  refine ⟨s,hs.trans Set.inter_subset_left,?_⟩
-  have he : (s : Set E) ∩ S = s := Set.inter_eq_left.mpr (hs.trans Set.inter_subset_right)
-  change (s : Set E) ∩ S ∈ A
-  rwa [he]
-
- def colorInclusion (m : ℕ) : E × Fin m ↪ E × Fin (m+1) :=
-  Function.Embedding.prodMap (Function.Embedding.refl E) Fin.castSuccEmb
-
- def baseColors (m : ℕ) : Set (E × Fin (m+1)) := Set.range (colorInclusion (E := E) m)
-
- def baseField (m : ℕ) (ω : Set (E × Fin (m+1))) : Set E :=
-  Prod.fst '' (colorInclusion m ⁻¹' ω)
-
- theorem not_mem_baseColors (m : ℕ) (e : E) (j : Fin (m+1)) :
-    (e,j) ∉ baseColors m ↔ j = Fin.last m := by
-  constructor
-  · intro h
-    apply Fin.eq_last_of_not_lt
-    intro hj
-    apply h
-    exact ⟨(e,⟨j.val,hj⟩),Prod.ext rfl (Fin.ext rfl)⟩
-  · rintro rfl ⟨⟨e',i⟩,hi⟩
-    exact Fin.castSucc_ne_last i (congrArg Prod.snd hi)
-
- theorem fst_inter_baseColors (m : ℕ) (ω : Set (E × Fin (m+1))) :
-    Prod.fst '' (ω ∩ baseColors m) = baseField m ω := by
-  ext e
-  constructor
-  · rintro ⟨⟨e',j⟩,⟨ho,⟨⟨f,i⟩,he⟩⟩,rfl⟩
-    refine ⟨(f,i),?_,?_⟩
-    · change colorInclusion m (f,i) ∈ ω
-      rwa [he]
-    · exact congrArg Prod.fst he
-  · rintro ⟨⟨e',i⟩,ho,rfl⟩
-    exact ⟨colorInclusion m (e',i),⟨ho,Set.mem_range_self _⟩,rfl⟩
-
- theorem measurable_baseField (m : ℕ) : Measurable (baseField (E := E) m) := by
-  apply (measurable_image_fst (Fin m)).comp
-  exact measurable_setOfPred.comp (Measurable.of_eval
-    (fun e => (measurable_pi_apply (colorInclusion m e)).comp
-      MeasurableEquiv.setOfPred.symm.measurable))
-
- theorem law_map_baseField (q : unitInterval) (m : ℕ) :
-    (law q : Measure (Set (E × Fin (m+1)))).map (baseField m) = law (orParameter q m) := by
-  have hm : Measurable (fun ω : Set (E × Fin (m+1)) => colorInclusion m ⁻¹' ω) :=
-    measurable_setOfPred.comp (Measurable.of_eval
-      (fun e => (measurable_pi_apply (colorInclusion m e)).comp
-        MeasurableEquiv.setOfPred.symm.measurable))
-  change (law q).map ((fun ω : Set (E × Fin m) => Prod.fst '' ω) ∘
-    (fun ω => colorInclusion m ⁻¹' ω)) = _
-  rw [← Measure.map_map (measurable_image_fst (Fin m)) hm,
-    law_map_preimage_embedding, law_map_image_fst, Fintype.card_fin]
-
- noncomputable def dartSource (G : BondGraph V E) (d : E × Bool) : V :=
-   if d.2 then (G d.1).out.1 else (G d.1).out.2
- noncomputable def dartTarget (G : BondGraph V E) (d : E × Bool) : V :=
-   if d.2 then (G d.1).out.2 else (G d.1).out.1
-
- theorem exists_dart (G : BondGraph V E) {e : E} {a b : V} (h : G e = s(a,b)) :
-    ∃ j : Bool, dartSource G (e,j) = a ∧ dartTarget G (e,j) = b := by
-  have he : s((G e).out.1, (G e).out.2) = s(a,b) := (G e).out_eq.trans h
-  rcases Sym2.eq_iff.mp he with h | h
-  · exact ⟨true,h⟩
-  · exact ⟨false,h.2,h.1⟩
-
- theorem colored_conn_probability [Countable V] [Countable E] (G : BondGraph V E)
-    (q : unitInterval) (m : ℕ) (x y : V) :
-    (law q).real {ω : Set (E × Fin m) | Conn (G ∘ Prod.fst) ω x y} =
-      twoPoint G (orParameter q m) x y := by
-  have h := law_map_image_fst (E := E) q (Fin m)
-  simp only [Fintype.card_fin] at h
-  rw [twoPoint, ← h, Measure.map_apply (measurable_image_fst (Fin m))
-    (measurableSet_conn G x y)]
-  congr 1
-  ext ω
-  exact conn_map_edges G Prod.fst ω x y
-
- theorem base_conn_probability [Countable V] [Countable E] (G : BondGraph V E)
-    (q : unitInterval) (m : ℕ) (x y : V) :
-    (law q).real {ω : Set (E × Fin (m+1)) |
-      Conn (G ∘ Prod.fst) (ω ∩ baseColors m) x y} =
-      twoPoint G (orParameter q m) x y := by
-  rw [twoPoint, ← law_map_baseField (E := E) q m,
-    Measure.map_apply (measurable_baseField m) (measurableSet_conn G x y)]
-  congr 1
-  ext ω
-  change Conn (G ∘ Prod.fst) (ω ∩ baseColors m) x y ↔ Conn G (baseField m ω) x y
-  rw [conn_map_edges, fst_inter_baseColors]
-
- 
-
- theorem twoPoint_or_succ_le [Countable V] [Fintype E] (G : BondGraph V E) (q : unitInterval)
-    (m : ℕ) (x y : V) :
-    twoPoint G (orParameter q (m+1)) x y ≤ twoPoint G (orParameter q m) x y +
-      (q : ℝ) * ∑ d : E × Bool,
-        twoPoint G (orParameter q m) x (dartSource G d) *
-        twoPoint G (orParameter q (m+1)) (dartTarget G d) y := by
-  let H : BondGraph V (E × Fin (m+1)) := G ∘ Prod.fst
-  let A (a : V) : Set (Set (E × Fin (m+1))) :=
-    {ω | Conn H (ω ∩ baseColors m) x a}
-  let B (b : V) : Set (Set (E × Fin (m+1))) := {ω | Conn H ω b y}
-  let C (d : E × Bool) := DisjointOccurrence (A (dartSource G d))
-    (DisjointOccurrence {ω | (d.1,Fin.last m) ∈ ω} (B (dartTarget G d)))
-  have huA (a : V) : IsUpperSet (A a) :=
-    fun _ _ h hω => conn_mono H (Set.inter_subset_inter_left _ h) hω
-  have hfA (a : V) : FiniteWitness (A a) :=
-    finiteWitness_inter _ (finiteWitness_conn H x a) _
-  have huB (b : V) : IsUpperSet (B b) := fun _ _ h hω => conn_mono H h hω
-  have hfB (b : V) : FiniteWitness (B b) := finiteWitness_conn H b y
-  have hsub : {ω | Conn H ω x y} ⊆ A y ∪ ⋃ d : E × Bool, C d := by
-    intro ω hω
-    rcases conn_first_outside H (baseColors m) hω with ha | ⟨a,b,⟨e,j⟩,he,hj,hc⟩
-    · exact Or.inl ha
-    · have hj' := (not_mem_baseColors m e j).mp hj
-      subst j
-      obtain ⟨k,hk₁,hk₂⟩ := exists_dart G he
-      right
-      refine Set.mem_iUnion.mpr ⟨(e,k),?_⟩
-      simpa only [C,hk₁,hk₂] using hc
-  calc
-    twoPoint G (orParameter q (m+1)) x y = (law q).real {ω | Conn H ω x y} :=
-      (colored_conn_probability G q (m+1) x y).symm
-    _ ≤ (law q).real (A y ∪ ⋃ d : E × Bool, C d) := measureReal_mono hsub
-    _ ≤ (law q).real (A y) + (law q).real (⋃ d : E × Bool, C d) := measureReal_union_le _ _
-    _ ≤ (law q).real (A y) + ∑ d : E × Bool, (law q).real (C d) :=
-      add_le_add_right (measureReal_iUnion_fintype_le _) _
-    _ ≤ twoPoint G (orParameter q m) x y +
-        ∑ d : E × Bool, twoPoint G (orParameter q m) x (dartSource G d) *
-          ((q : ℝ) * twoPoint G (orParameter q (m+1)) (dartTarget G d) y) := by
-      rw [show (law q).real (A y) = twoPoint G (orParameter q m) x y from
-        base_conn_probability G q m x y]
-      apply add_le_add_right
-      apply Finset.sum_le_sum
-      intro d _
-      have hb := bk q {ω | (d.1,Fin.last m) ∈ ω} (B (dartTarget G d))
-        (fun _ _ hh he => hh he) (huB _) (finiteWitness_edge _) (hfB _)
-      have heq : (law q).real {ω : Set (E × Fin (m+1)) | (d.1,Fin.last m) ∈ ω} =
-          (q : ℝ) := by simpa using finite_open_probability q {(d.1,Fin.last m)}
-      rw [heq, show (law q).real (B (dartTarget G d)) =
-        twoPoint G (orParameter q (m+1)) (dartTarget G d) y from
-        colored_conn_probability G q (m+1) _ _] at hb
-      have ha := bk q (A (dartSource G d))
-        (DisjointOccurrence {ω | (d.1,Fin.last m) ∈ ω} (B (dartTarget G d)))
-        (huA _) (disjointOccurrence_upper _ _) (hfA _) (finiteWitness_disjointOccurrence _ _)
-      rw [show (law q).real (A (dartSource G d)) =
-        twoPoint G (orParameter q m) x (dartSource G d) from base_conn_probability G q m _ _] at ha
-      exact ha.trans (mul_le_mul_of_nonneg_left hb (twoPoint_nonneg G _ _ _))
-    _ = _ := by rw [Finset.mul_sum]; congr 1; apply Finset.sum_congr rfl; intro d _; ring
-
-end ColoredSprinkling
-
-section FiniteGraphOperators
-attribute [local instance] Classical.propDecidable
-
- noncomputable def edgeVertices [Fintype E] (G : BondGraph V E) : Finset V :=
-   Finset.univ.biUnion (fun e => (G e).toFinset)
-
- theorem mem_edgeVertices [Fintype E] (G : BondGraph V E) (x : V) :
-    x ∈ edgeVertices G ↔ ∃ e, x ∈ G e := by simp [edgeVertices]
-
- theorem conn_eq_or_mem_edgeVertices [Fintype E] (G : BondGraph V E)
-    {ω : Set E} {x y : V} (h : Conn G ω x y) : x = y ∨ y ∈ edgeVertices G := by
-  cases h with
-  | refl => exact Or.inl rfl
-  | @tail a b hab he =>
-    rcases he with ⟨e,_,he⟩
-    exact Or.inr ((mem_edgeVertices G y).mpr ⟨e,by rw [he]; simp⟩)
-
- theorem operatorBounded_finite_edges [Fintype E] (G : BondGraph V E) (p : unitInterval) :
-    OperatorBounded G p := by
-  have hr (x : V) (s : Finset V) : ∑ y ∈ s, twoPoint G p x y ≤
-      (edgeVertices G).card + (1 : ℝ) := by
-    let F := insert x (edgeVertices G)
-    have bound (y : V) : twoPoint G p x y ≤ if y ∈ F then 1 else 0 := by
-      split_ifs with hy
-      · exact twoPoint_le_one G p x y
-      · have he : {ω | Conn G ω x y} = ∅ := by
-          apply Set.eq_empty_iff_forall_notMem.mpr
-          intro ω h
-          rcases conn_eq_or_mem_edgeVertices G h with rfl | hmem
-          · exact hy (by simp [F])
-          · exact hy (by simp [F,hmem])
-        simp [twoPoint,he]
-    calc
-      ∑ y ∈ s, twoPoint G p x y ≤ ∑ y ∈ s, if y ∈ F then (1:ℝ) else 0 :=
-        Finset.sum_le_sum (fun y _ => bound y)
-      _ = ((s ∩ F).card : ℝ) := by simp
-      _ ≤ (F.card : ℝ) := by
-        exact_mod_cast Finset.card_le_card Finset.inter_subset_right
-      _ ≤ (edgeVertices G).card + (1 : ℝ) := by
-        exact_mod_cast Finset.card_insert_le x (edgeVertices G)
-  obtain ⟨A,_,hA⟩ := exists_operator_of_row_column_bound (twoPoint G p)
-    ((edgeVertices G).card+1) (by positivity) (twoPoint_nonneg G p) hr
-    (fun y s => by simpa only [twoPoint_symm G p] using hr y s)
-  exact ⟨A,hA⟩
-
- theorem dart_edge (G : BondGraph V E) (d : E × Bool) :
-    G d.1 = s(dartSource G d, dartTarget G d) := by
-  rcases d with ⟨e,j⟩
-  cases j
-  · exact (G e).out_eq.symm.trans (Sym2.eq_swap ..)
-  · exact (G e).out_eq.symm
-
- theorem card_darts_at_source [Fintype E] (G : BondGraph V E) (x : V) :
-    Fintype.card {d : E × Bool // dartSource G d = x} ≤ 2 * degree G x := by
-  let : Fintype (Step G x) := (finite_steps G (fun _ => Set.toFinite _) x).fintype
-  let f : {d : E × Bool // dartSource G d = x} → Step G x × Bool := fun d =>
-    (⟨(d.val.1,dartTarget G d.val), by rw [dart_edge G d.val,d.property]⟩,d.val.2)
-  have hf : Function.Injective f := by
-    intro d d' h
-    apply Subtype.ext
-    apply Prod.ext
-    · exact congrArg (fun t : Step G x × Bool => t.1.val.1) h
-    · exact congrArg (fun t : Step G x × Bool => t.2) h
-  have hc := Fintype.card_le_of_injective f hf
-  simpa only [Fintype.card_prod,Fintype.card_bool,degree,Nat.card_eq_fintype_card,Nat.mul_comm] using hc
-
- theorem card_darts_at_target [Fintype E] (G : BondGraph V E) (x : V) :
-    Fintype.card {d : E × Bool // dartTarget G d = x} ≤ 2 * degree G x := by
-  let : Fintype (Step G x) := (finite_steps G (fun _ => Set.toFinite _) x).fintype
-  let f : {d : E × Bool // dartTarget G d = x} → Step G x × Bool := fun d =>
-    (⟨(d.val.1,dartSource G d.val), by
-      rw [dart_edge G d.val,d.property]; exact Sym2.eq_swap ..⟩,d.val.2)
-  have hf : Function.Injective f := by
-    intro d d' h
-    apply Subtype.ext
-    apply Prod.ext
-    · exact congrArg (fun t : Step G x × Bool => t.1.val.1) h
-    · exact congrArg (fun t : Step G x × Bool => t.2) h
-  have hc := Fintype.card_le_of_injective f hf
-  simpa only [Fintype.card_prod,Fintype.card_bool,degree,Nat.card_eq_fintype_card,Nat.mul_comm] using hc
-
- noncomputable def adjacencyOperator [Fintype E] (G : BondGraph V E) :
-    Hilbert V →L[ℂ] Hilbert V :=
-  ∑ d : E × Bool, (lp.singleContinuousLinearMap ℂ (fun _ : V => ℂ) 2 (dartSource G d)).comp
-    (lp.evalCLM ℂ (fun _ : V => ℂ) 2 (dartTarget G d))
-
- theorem adjacencyOperator_apply [Fintype E] (G : BondGraph V E) (f : Hilbert V) :
-    adjacencyOperator G f = ∑ d : E × Bool, lp.single 2 (dartSource G d) (f (dartTarget G d)) := by
-  simp [adjacencyOperator,lp.singleContinuousLinearMap_apply,
-    lp.evalCLM,lp.evalₗ]
-  rfl
-
- theorem adjacencyOperator_norm_le [Fintype E] (G : BondGraph V E)
-    (D : ℕ) (hD : ∀ x, degree G x ≤ D) : ‖adjacencyOperator G‖ ≤ 2 * (D : ℝ) := by
-  let U (x y : V) : ℝ := ∑ d : E × Bool,
-    if dartSource G d = x ∧ dartTarget G d = y then 1 else 0
-  have hU : ∀ x y, 0 ≤ U x y := by intro x y; apply Finset.sum_nonneg; intro d _; positivity
-  have hmat : ∀ x y, adjacencyOperator G (lp.single 2 y (1:ℂ)) x = (U x y : ℂ) := by
-    intro x y
-    rw [adjacencyOperator_apply]
-    simp only [lp.coeFn_sum,Finset.sum_apply,U,Complex.ofReal_sum]
-    apply Finset.sum_congr rfl
-    intro d _
-    simp only [lp.single_apply]
-    by_cases hx : dartSource G d = x <;> by_cases hy : dartTarget G d = y <;>
-      simp [hx,hy]
-  have hrow (x : V) (s : Finset V) : ∑ y ∈ s, U x y ≤ 2 * (D : ℝ) := by
-    calc
-      ∑ y ∈ s, U x y = ∑ d : E × Bool,
-          ∑ y ∈ s, if dartSource G d = x ∧ dartTarget G d = y then (1:ℝ) else 0 :=
-        Finset.sum_comm
-      _ ≤ ∑ d : E × Bool, if dartSource G d = x then (1:ℝ) else 0 := by
-        apply Finset.sum_le_sum
-        intro d _
-        by_cases hx : dartSource G d = x
-        · simp only [hx,true_and,ite_true]
-          rw [Finset.sum_ite_eq]
-          split_ifs <;> norm_num
-        · simp [hx]
-      _ = (Fintype.card {d : E × Bool // dartSource G d = x} : ℝ) := by simp [Fintype.card_subtype]
-      _ ≤ 2 * (D : ℝ) := by exact_mod_cast (card_darts_at_source G x).trans (Nat.mul_le_mul_left 2 (hD x))
-  have hcol (y : V) (s : Finset V) : ∑ x ∈ s, U x y ≤ 2 * (D : ℝ) := by
-    calc
-      ∑ x ∈ s, U x y = ∑ d : E × Bool,
-          ∑ x ∈ s, if dartSource G d = x ∧ dartTarget G d = y then (1:ℝ) else 0 :=
-        Finset.sum_comm
-      _ ≤ ∑ d : E × Bool, if dartTarget G d = y then (1:ℝ) else 0 := by
-        apply Finset.sum_le_sum
-        intro d _
-        by_cases hy : dartTarget G d = y
-        · simp only [hy,and_true,ite_true]
-          rw [Finset.sum_ite_eq]
-          split_ifs <;> norm_num
-        · simp [hy]
-      _ = (Fintype.card {d : E × Bool // dartTarget G d = y} : ℝ) := by simp [Fintype.card_subtype]
-      _ ≤ 2 * (D : ℝ) := by exact_mod_cast (card_darts_at_target G y).trans (Nat.mul_le_mul_left 2 (hD y))
-  obtain ⟨A,hn,hA⟩ := exists_operator_of_row_column_bound U (2*D) (by positivity) hU hrow hcol
-  rwa [kernel_extension_unique hmat hA]
-
-end FiniteGraphOperators
-
-section FiniteSprinklingBound
-attribute [local instance] Classical.propDecidable
-variable [Countable V] [Fintype E]
-
- theorem finite_sprinkling_bound (G : BondGraph V E) (q : unitInterval) (m D : ℕ)
-    (hD : ∀ x, degree G x ≤ D) (A : Hilbert V →L[ℂ] Hilbert V)
-    (hA : HasKernel G (orParameter q m) A) (M : ℝ) (hM : ‖A‖ ≤ M)
-    (hq : (q : ℝ) * (2*D) * M < 1) :
-    ∃ B : Hilbert V →L[ℂ] Hilbert V, HasKernel G (orParameter q (m+1)) B ∧
-      ‖B‖ ≤ M / (1-(q:ℝ)*(2*D)*M) := by
-  obtain ⟨B,hB⟩ := operatorBounded_finite_edges G (orParameter q (m+1))
-  let R := (adjacencyOperator G).comp B
-  let C := A + (q : ℂ) • A.comp R
-  let U (x y : V) := twoPoint G (orParameter q m) x y + (q:ℝ) *
-    ∑ d : E × Bool, twoPoint G (orParameter q m) x (dartSource G d) *
-      twoPoint G (orParameter q (m+1)) (dartTarget G d) y
-  have hU : ∀ x y, 0 ≤ U x y := by
-    intro x y
-    exact add_nonneg (twoPoint_nonneg G _ _ _)
-      (mul_nonneg q.2.1 (Finset.sum_nonneg fun d _ =>
-        mul_nonneg (twoPoint_nonneg G _ _ _) (twoPoint_nonneg G _ _ _)))
-  have hC : ∀ x y, C (lp.single 2 y (1:ℂ)) x = (U x y : ℂ) := by
-    intro x y
-    simp only [C,add_apply,smul_apply,
-      ContinuousLinearMap.comp_apply,R,adjacencyOperator_apply,map_sum,lp.coeFn_add,Pi.add_apply,
-      lp.coeFn_smul,Pi.smul_apply,smul_eq_mul,lp.coeFn_sum,Finset.sum_apply,U,
-      Complex.ofReal_add,Complex.ofReal_mul,Complex.ofReal_sum]
-    rw [hA x y]
-    congr 1
-    congr 1
-    apply Finset.sum_congr rfl
-    intro d _
-    have hs (c : ℂ) : (lp.single 2 (dartSource G d) c : Hilbert V) = c • lp.single 2 (dartSource G d) (1:ℂ) := by
-      rw [← lp.single_smul]; simp
-    rw [hs,map_smul,lp.coeFn_smul,Pi.smul_apply,smul_eq_mul,hA,hB]
-    ring
-  obtain ⟨B',hn,hB'⟩ := exists_operator_of_matrix_domination U
-    (twoPoint G (orParameter q (m+1))) hU
-    (fun x y => by rw [abs_of_nonneg (twoPoint_nonneg G _ _ _)]; exact twoPoint_or_succ_le G q m x y)
-    C hC
-  have he : B' = B := kernel_extension_unique hB' hB
-  rw [he] at hn
-  have hqn : ‖(q : ℂ)‖ = (q : ℝ) := by simp [abs_of_nonneg q.2.1]
-  have hcoef : (q : ℝ) * ‖A‖ * (2*D) * ‖B‖ ≤ (q : ℝ) * M * (2*D) * ‖B‖ := by
-    gcongr
-    exact q.2.1
-  have hbnd : ‖B‖ ≤ M + (q : ℝ) * (2*D) * M * ‖B‖ := by
-    calc
-      ‖B‖ ≤ ‖C‖ := hn
-      _ ≤ ‖A‖ + ‖(q : ℂ) • A.comp R‖ := norm_add_le _ _
-      _ = ‖A‖ + (q : ℝ) * ‖A.comp R‖ := by rw [norm_smul,hqn]
-      _ ≤ ‖A‖ + (q : ℝ) * (‖A‖ * (‖adjacencyOperator G‖ * ‖B‖)) := by
-        apply add_le_add_right
-        apply mul_le_mul_of_nonneg_left _ q.2.1
-        exact (A.opNorm_comp_le R).trans
-          (mul_le_mul_of_nonneg_left ((adjacencyOperator G).opNorm_comp_le B) (norm_nonneg _))
-      _ ≤ ‖A‖ + (q : ℝ) * (‖A‖ * ((2*D) * ‖B‖)) := by
-        gcongr
-        · exact q.2.1
-        · exact adjacencyOperator_norm_le G D hD
-      _ ≤ M + (q : ℝ) * M * (2*D) * ‖B‖ := by
-        rw [show (q : ℝ) * (‖A‖ * ((2*D) * ‖B‖)) = (q : ℝ) * ‖A‖ * (2*D) * ‖B‖ by ring]
-        exact add_le_add hM hcoef
-      _ = _ := by ring
-  refine ⟨B,hB,?_⟩
-  apply (le_div_iff₀ (sub_pos.mpr hq)).mpr
-  nlinarith
-
-end FiniteSprinklingBound
-
-section ExhaustionSprinkling
-attribute [local instance] Classical.propDecidable
-open Filter Topology
-variable [Countable V] [Countable E]
-
- theorem twoPoint_restrict_edges (G : BondGraph V E) (p : unitInterval) (S : Set E) (x y : V) :
-    twoPoint (G ∘ (Subtype.val : S → E)) p x y =
-      (law p).real {ω : Set E | Conn G (ω ∩ S) x y} := by
-  unfold twoPoint
-  rw [← law_map_restrict p S, Measure.map_apply (measurable_set_restrict S)
-    (measurableSet_conn (G ∘ Subtype.val) x y)]
-  apply congrArg ENNReal.toReal
-  congr 1
-  ext ω
-  change Conn (G ∘ Subtype.val) (Subtype.val ⁻¹' ω) x y ↔ _
-  rw [conn_map_edges]
-  have he : (Subtype.val '' (Subtype.val ⁻¹' ω : Set S)) = ω ∩ S := by
-    ext e
-    simp only [Set.mem_image,Set.mem_preimage,Set.mem_inter_iff]
-    constructor
-    · rintro ⟨d,hd,rfl⟩; exact ⟨hd,d.property⟩
-    · rintro ⟨ho,hs⟩; exact ⟨⟨e,hs⟩,ho,rfl⟩
-  rw [he]
-  rfl
-
- theorem twoPoint_restrict_le (G : BondGraph V E) (p : unitInterval) (S : Set E) (x y : V) :
-    twoPoint (G ∘ (Subtype.val : S → E)) p x y ≤ twoPoint G p x y := by
-  rw [twoPoint_restrict_edges]
-  exact measureReal_mono (fun ω h => conn_mono G Set.inter_subset_left h) (measure_ne_top _ _)
-
- omit [Countable V] [Countable E] in
- theorem degree_restrict_le (G : BondGraph V E) (hloc : LocallyFinite G) (S : Set E) (x : V) :
-    degree (G ∘ (Subtype.val : S → E)) x ≤ degree G x := by
-  let : Fintype (Step G x) := (finite_steps G hloc x).fintype
-  let f : Step (G ∘ (Subtype.val : S → E)) x → Step G x :=
-    fun d => ⟨(d.val.1.val,d.val.2),d.property⟩
-  apply Nat.card_le_card_of_injective f
-  intro d d' h
-  apply Subtype.ext
-  apply Prod.ext
-  · apply Subtype.ext
-    exact congrArg (fun t : Step G x => t.val.1) h
-  · exact congrArg (fun t : Step G x => t.val.2) h
-
- theorem sprinkling_bound (G : BondGraph V E) (hloc : LocallyFinite G)
-    (q : unitInterval) (m D : ℕ) (hD : ∀ x, degree G x ≤ D)
-    (A : Hilbert V →L[ℂ] Hilbert V) (hA : HasKernel G (orParameter q m) A)
-    (hq : (q : ℝ) * (2*D) * ‖A‖ < 1) :
-    ∃ B : Hilbert V →L[ℂ] Hilbert V, HasKernel G (orParameter q (m+1)) B ∧
-      ‖B‖ ≤ ‖A‖ / (1-(q:ℝ)*(2*D)*‖A‖) := by
-  let := Encodable.ofCountable E
-  let F (n : ℕ) := edgeExhaustion (E := E) n
-  let H (n : ℕ) : BondGraph V (F n) := G ∘ Subtype.val
-  have hb (n : ℕ) : ∃ B : Hilbert V →L[ℂ] Hilbert V,
-      HasKernel (H n) (orParameter q (m+1)) B ∧
-      ‖B‖ ≤ ‖A‖ / (1-(q:ℝ)*(2*D)*‖A‖) := by
-    obtain ⟨A',hn,hA'⟩ := exists_operator_of_matrix_domination
-      (twoPoint G (orParameter q m)) (twoPoint (H n) (orParameter q m))
-      (twoPoint_nonneg G _) (fun x y => by
-        rw [abs_of_nonneg (twoPoint_nonneg (H n) _ _ _)]
-        exact twoPoint_restrict_le G _ (F n) x y) A hA
-    exact finite_sprinkling_bound (H n) q m D
-      (fun x => (degree_restrict_le G hloc (F n) x).trans (hD x)) A' hA' ‖A‖ hn hq
-  choose B hB hn using hb
-  have hlim (x y : V) : Tendsto (fun n => twoPoint (H n) (orParameter q (m+1)) x y)
-      atTop (𝓝 (twoPoint G (orParameter q (m+1)) x y)) := by
-    have hm : Monotone (fun n => {ω : Set E | Conn G (ω ∩ (F n : Set E)) x y}) := by
-      intro m n h ω hω
-      exact conn_mono G (Set.inter_subset_inter_right _ (edgeExhaustion_mono h)) hω
-    have h := tendsto_measure_iUnion_atTop (μ := law (orParameter q (m+1))) hm
-    have he := witness_eq_union {ω | Conn G ω x y} (fun _ _ h hc => conn_mono G h hc)
-      (finiteWitness_conn G x y)
-    change {ω : Set E | Conn G ω x y} = ⋃ n, {ω | Conn G (ω ∩ (F n : Set E)) x y} at he
-    rw [← he] at h
-    have hr := (ENNReal.tendsto_toReal (measure_ne_top _ _)).comp h
-    apply hr.congr'
-    apply Filter.Eventually.of_forall
-    intro n
-    exact (twoPoint_restrict_edges G (orParameter q (m+1)) (F n) x y).symm
-  obtain ⟨C,hCn,hC⟩ := exists_operator_of_matrix_tendsto B
-    (fun x y => (twoPoint G (orParameter q (m+1)) x y : ℂ))
-    (div_nonneg (norm_nonneg _) (sub_pos.mpr hq).le) hn (fun x y => by
-      have hBm : ∀ n x y, B n (lp.single 2 y (1:ℂ)) x = (twoPoint (H n) (orParameter q (m+1)) x y : ℂ) := hB
-      simp only [hBm]
-      exact Complex.continuous_ofReal.continuousAt.tendsto.comp (hlim x y))
-  exact ⟨C,hC,hCn⟩
-
-end ExhaustionSprinkling
-
-section Openness
-attribute [local instance] Classical.propDecidable
-
- theorem operatorBounded_open_right [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (D : ℕ) (hD : ∀ x, degree G x ≤ D)
-    (p : unitInterval) (hp0 : 0 < p) (hp1 : p < 1) (hb : OperatorBounded G p) :
-    ∃ r : unitInterval, p < r ∧ OperatorBounded G r := by
-  obtain ⟨A,hA⟩ := hb
-  let c : ℝ := (2*D)*‖A‖
-  have hc : 0 ≤ c := by dsimp [c]; positivity
-  obtain ⟨m,q,hq,hqp⟩ := exists_orParameter_small p hp1 (1/(c+1))
-    (by positivity) (by apply (div_le_one (by positivity)).mpr; linarith)
-  have hq0 : (0:ℝ) < q := by
-    by_contra h
-    have hqe : q = 0 := Subtype.ext ((le_of_not_gt h).antisymm q.2.1)
-    have hpe : p = 0 := by rw [← hqp,hqe]; apply Subtype.ext; simp [orParameter]
-    exact (ne_of_gt hp0) hpe
-  have hsmall : (q:ℝ)*(2*D)*‖A‖ < 1 := by
-    have hm := mul_le_mul_of_nonneg_right hq hc
-    have hh : c/(c+1) < 1 := (div_lt_one (by positivity)).mpr (by linarith)
-    calc
-      (q:ℝ)*(2*D)*‖A‖ = (q:ℝ)*c := by dsimp [c]; ring
-      _ ≤ 1/(c+1)*c := hm
-      _ = c/(c+1) := by ring
-      _ < 1 := hh
-  obtain ⟨B,hB,_⟩ := sprinkling_bound G hloc q m D hD A (hqp ▸ hA) hsmall
-  refine ⟨orParameter q (m+1),?_,B,hB⟩
-  change (p:ℝ) < (orParameter q (m+1):ℝ)
-  have he : (orParameter q (m+1):ℝ) = (p:ℝ)+(1-p)*q := by
-    rw [← hqp]
-    simp only [orParameter,pow_succ]
-    ring
-  rw [he]
-  have ht := mul_pos (show (0:ℝ) < 1-p from sub_pos.mpr hp1) hq0
-  linarith
-
- 
-
- theorem mainConclusion_of_critical_bounded [Infinite V] (G : BondGraph V E)
-    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (hc : OperatorBounded G (pc G)) : MainConclusion G := by
-  let : Countable V := countable_vertices hloc hconn
-  let : Countable E := countable_edges hloc
-  have hpc1 : pc G < 1 := by
-    apply lt_of_le_of_ne (show pc G ≤ 1 from (pc G).2.2)
-    intro he
-    exact not_operatorBounded_one G hconn (he ▸ hc)
-  obtain ⟨D,_,hD⟩ := exists_degree_bound G hqt
-  obtain ⟨q,hq,hb⟩ := operatorBounded_open_right G hloc D hD (pc G)
-    (pc_pos G hloc hqt) hpc1 hc
-  exact mainConclusion_of_strict_gap G hconn hloc hqt
-    (hq.trans_le (le_sSup hb))
-
-end Openness
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Graph.Paths
+
+namespace OAI
+
+/-! Sprinkling estimates for finite and infinite graph operators. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+section SprinklingLaw
+attribute [local instance] Classical.propDecidable
+
+noncomputable def sprinkleParameter (p a : unitInterval) : unitInterval :=
+  ⟨(p : ℝ) + (1-p)*a, by
+    constructor
+    · exact add_nonneg p.2.1 (mul_nonneg (sub_nonneg.mpr p.2.2) a.2.1)
+    · nlinarith [mul_nonneg (sub_nonneg.mpr p.2.2) (sub_nonneg.mpr a.2.2)]⟩
+
+ def pairParameter (p a : unitInterval) (b : Bool) : unitInterval := if b then p else a
+
+ theorem map_bool_or (p a : unitInterval) :
+    (Measure.infinitePi (fun b : Bool => bitLaw (pairParameter p a b))).map
+      (fun f : Bool → Prop => ∃ b, f b) = bitLaw (sprinkleParameter p a) := by
+  let μ := (Measure.infinitePi (fun b : Bool => bitLaw (pairParameter p a b))).map
+      (fun f : Bool → Prop => ∃ b, f b)
+  have hm : Measurable (fun f : Bool → Prop => ∃ b, f b) := .of_discrete
+  have : IsProbabilityMeasure μ := inferInstance
+  have hf : μ.real {False} = (1-(p : ℝ)) * (1-(a : ℝ)) := by
+    change ((Measure.infinitePi _).map _).real _ = _
+    rw [Measure.real, Measure.map_apply hm (measurableSet_singleton _)]
+    have he : (fun f : Bool → Prop => ∃ b, f b) ⁻¹' ({False} : Set Prop) =
+        ((Finset.univ : Finset Bool) : Set Bool).pi (fun _ => ({False} : Set Prop)) := by
+      ext f
+      simp [and_comm]
+    rw [he, Measure.infinitePi_pi _ (fun _ _ => measurableSet_singleton _)]
+    simp [pairParameter, bitLaw, ENNReal.toReal_mul]
+  have ht : μ.real {True} = (p : ℝ) + (1-p)*a := by
+    have he : ({True} : Set Prop) = ({False} : Set Prop)ᶜ := by ext b; simp
+    rw [he, measureReal_compl (measurableSet_singleton _), probReal_univ, hf]
+    ring
+  apply Measure.ext_of_singleton
+  intro b
+  apply (ENNReal.toReal_eq_toReal_iff' (measure_ne_top _ _) (measure_ne_top _ _)).mp
+  change μ.real {b} = (bitLaw (sprinkleParameter p a)).real {b}
+  by_cases hb : b
+  · have he : b = True := propext (iff_true_intro hb)
+    subst b
+    rw [ht]
+    simp [bitLaw, Measure.real, sprinkleParameter]
+  · have he : b = False := propext (iff_false_intro hb)
+    subst b
+    rw [hf]
+    simp [bitLaw, Measure.real, sprinkleParameter]
+    ring
+
+noncomputable def fieldLaw (p a : unitInterval) : Measure (Bool → Set E) :=
+  Measure.infinitePi (fun b => law (pairParameter p a b))
+
+def fieldUnion (f : Bool → Set E) : Set E := f true ∪ f false
+
+ theorem measurable_fieldUnion : Measurable (fieldUnion : (Bool → Set E) → Set E) := by
+  have hm (e : E) : Measurable (fun f : Bool → Set E => e ∈ fieldUnion f) := by
+    apply measurableSet_setOfPred.mp
+    change MeasurableSet ({f : Bool → Set E | e ∈ f true} ∪ {f | e ∈ f false})
+    apply MeasurableSet.union <;> apply measurableSet_setOfPred.mpr
+    · exact ((measurable_pi_apply e).comp MeasurableEquiv.setOfPred.symm.measurable).comp
+        (measurable_pi_apply true)
+    · exact ((measurable_pi_apply e).comp MeasurableEquiv.setOfPred.symm.measurable).comp
+        (measurable_pi_apply false)
+  exact measurable_setOfPred.comp (Measurable.of_eval hm)
+
+ theorem map_fieldUnion (p a : unitInterval) :
+    (fieldLaw p a).map (fieldUnion : (Bool → Set E) → Set E) = law (sprinkleParameter p a) := by
+  let μ := fun b : Bool => bitLaw (pairParameter p a b)
+  let s := fun f : Bool → E → Prop => fun b => {e | f b e}
+  have hs : Measurable s := Measurable.of_eval
+    (fun b => measurable_setOfPred.comp (measurable_pi_apply b))
+  have hstart : (Measure.infinitePi (fun b : Bool => Measure.infinitePi (fun _ : E => μ b))).map s =
+      fieldLaw p a := by
+    rw [Measure.infinitePi_map_pi _ (fun _ => measurable_setOfPred)]
+    congr 1
+    funext b
+    rw [law, setBernoulli_eq_map]
+    rfl
+  rw [← hstart, Measure.map_map measurable_fieldUnion hs]
+  let f := (MeasurableEquiv.curry Bool E Prop).symm
+  let swap := MeasurableEquiv.piCongrLeft (fun _ : E × Bool => Prop) (Equiv.prodComm Bool E)
+  let c := MeasurableEquiv.curry E Bool Prop
+  let b := fun g : E → Bool → Prop => fun e => ∃ j, g e j
+  have hb : Measurable b := Measurable.of_eval
+    (fun e => (Measurable.of_discrete : Measurable (fun f : Bool → Prop => ∃ j, f j)).comp
+      (measurable_pi_apply e))
+  have he : fieldUnion ∘ s = (fun g : E → Prop => {e | g e}) ∘ b ∘ c ∘ swap ∘ f := by
+    funext g
+    ext e
+    simp [fieldUnion, s, b, c, swap, f, Function.comp_def, Bool.exists_bool, or_comm, MeasurableEquiv.coe_piCongrLeft, Equiv.piCongrLeft]
+  rw [he, ← Measure.map_map measurable_setOfPred (hb.comp (c.measurable.comp (swap.measurable.comp f.measurable))),
+    ← Measure.map_map hb (c.measurable.comp (swap.measurable.comp f.measurable))]
+  rw [← Measure.map_map c.measurable (swap.measurable.comp f.measurable),
+    ← Measure.map_map swap.measurable f.measurable]
+  rw [Measure.infinitePi_map_curry_symm]
+  rw [show (Measure.infinitePi (fun x : Bool × E => μ x.1)).map swap =
+      Measure.infinitePi (fun x : E × Bool => μ x.2) from
+    Measure.infinitePi_map_piCongrLeft (fun x : E × Bool => μ x.2) (Equiv.prodComm Bool E)]
+  rw [Measure.infinitePi_map_curry (fun (_ : E) (j : Bool) => μ j)]
+  dsimp only [b]
+  rw [Measure.infinitePi_map_pi _ (fun _ => Measurable.of_discrete)]
+  have hm : (Measure.infinitePi (fun j : Bool => μ j)).map Exists =
+      bitLaw (sprinkleParameter p a) := map_bool_or p a
+  rw [hm, law, setBernoulli_eq_map]
+  rfl
+
+ theorem fieldLaw_map_pair (p a : unitInterval) :
+    (fieldLaw p a).map (fun f : Bool → Set E => (f true, f false)) =
+      (law p).prod (law a) := by
+  simpa [fieldLaw, pairParameter] using
+    (Measure.infinitePi_map_eval_prod (P := fun b : Bool => (law (pairParameter p a b) : Measure (Set E)))
+    (by decide : true ≠ false))
+
+ theorem law_map_union (p a : unitInterval) :
+    ((law p : Measure (Set E)).prod (law a)).map (fun f => f.1 ∪ f.2) =
+      law (sprinkleParameter p a) := by
+  rw [← fieldLaw_map_pair p a, Measure.map_map]
+  · exact map_fieldUnion p a
+  · have h : Measurable (fun f : Set E × Set E => fun e => e ∈ f.1 ∪ f.2) := by
+      apply Measurable.of_eval
+      intro e
+      apply measurableSet_setOfPred.mp
+      change MeasurableSet ({f : Set E × Set E | e ∈ f.1} ∪ {f | e ∈ f.2})
+      apply MeasurableSet.union <;> apply measurableSet_setOfPred.mpr
+      · exact ((measurable_pi_apply e).comp MeasurableEquiv.setOfPred.symm.measurable).comp measurable_fst
+      · exact ((measurable_pi_apply e).comp MeasurableEquiv.setOfPred.symm.measurable).comp measurable_snd
+    exact measurable_setOfPred.comp h
+  · exact (measurable_pi_apply true).prodMk (measurable_pi_apply false)
+
+end SprinklingLaw
+
+section Schur
+attribute [local instance] Classical.propDecidable
+
+ 
+theorem exists_operator_of_row_column_bound (U : V → V → ℝ) (D : ℝ)
+    (hD : 0 ≤ D) (hU : ∀ x y, 0 ≤ U x y)
+    (hrow : ∀ x (s : Finset V), ∑ y ∈ s, U x y ≤ D)
+    (hcol : ∀ y (s : Finset V), ∑ x ∈ s, U x y ≤ D) :
+    ∃ A : Hilbert V →L[ℂ] Hilbert V, ‖A‖ ≤ D ∧
+      ∀ x y, A (lp.single 2 y (1 : ℂ)) x = (U x y : ℂ) := by
+  have hentry (x y : V) : U x y ≤ D := by simpa using hrow x {y}
+  have hcm (y : V) : Memℓp (fun x => (U x y : ℂ)) 2 := by
+    apply (memℓp_gen_iff'' (by norm_num : 0 < (2 : ℝ≥0∞).toReal)).mpr
+    refine ⟨D^2, sq_nonneg _, fun s => ?_⟩
+    simp only [ENNReal.toReal_ofNat, Real.rpow_two, Complex.norm_real,
+      Real.norm_eq_abs, sq_abs]
+    calc
+      ∑ x ∈ s, U x y ^ 2 ≤ ∑ x ∈ s, D * U x y := by
+        apply Finset.sum_le_sum
+        intro x _
+        nlinarith [hentry x y, hU x y]
+      _ = D * ∑ x ∈ s, U x y := by rw [Finset.mul_sum]
+      _ ≤ D * D := mul_le_mul_of_nonneg_left (hcol y s) hD
+      _ = D^2 := by ring
+  let col (y : V) : Hilbert V := ⟨fun x => (U x y : ℂ), hcm y⟩
+  let L : (V →₀ ℂ) →ₗ[ℂ] Hilbert V := Finsupp.linearCombination ℂ col
+  have hn (f : V →₀ ℂ) : ‖L f‖ ≤ D * ‖finiteEmbedding f‖ := by
+    apply lp.norm_le_of_forall_sum_le (by norm_num)
+      (mul_nonneg hD (norm_nonneg _))
+    intro s
+    simp only [ENNReal.toReal_ofNat, Real.rpow_two]
+    have hpoint (x : V) : ‖L f x‖^2 ≤ D * ∑ y ∈ f.support, ‖f y‖^2 * U x y := by
+      have hn0 : 0 ≤ ∑ y ∈ f.support, ‖f y‖ * U x y :=
+        Finset.sum_nonneg fun y _ => mul_nonneg (norm_nonneg _) (hU x y)
+      have hn1 : ‖L f x‖ ≤ ∑ y ∈ f.support, ‖f y‖ * U x y := by
+        simp only [L, Finsupp.linearCombination_apply, Finsupp.sum, lp.coeFn_sum,
+          Finset.sum_apply, lp.coeFn_smul, Pi.smul_apply, smul_eq_mul]
+        apply (norm_sum_le _ _).trans
+        apply Finset.sum_le_sum
+        intro y _
+        simp [col, abs_of_nonneg (hU x y)]
+      have hcs := Finset.sum_sq_le_sum_mul_sum_of_sq_le_mul f.support
+        (f := fun y => U x y) (g := fun y => ‖f y‖^2 * U x y)
+        (r := fun y => ‖f y‖ * U x y)
+        (fun y _ => hU x y)
+        (fun y _ => mul_nonneg (sq_nonneg _) (hU x y))
+        (fun y _ => by nlinarith)
+      exact (sq_le_sq₀ (norm_nonneg _) hn0 |>.mpr hn1).trans
+        (hcs.trans (mul_le_mul_of_nonneg_right (hrow x f.support)
+          (Finset.sum_nonneg fun y _ => mul_nonneg (sq_nonneg _) (hU x y))))
+    calc
+      ∑ x ∈ s, ‖L f x‖^2 ≤ ∑ x ∈ s, D * ∑ y ∈ f.support, ‖f y‖^2 * U x y :=
+        Finset.sum_le_sum fun x _ => hpoint x
+      _ = D * ∑ y ∈ f.support, ‖f y‖^2 * ∑ x ∈ s, U x y := by
+        rw [← Finset.mul_sum, Finset.sum_comm]
+        simp_rw [← Finset.mul_sum]
+      _ ≤ D * ∑ y ∈ f.support, ‖f y‖^2 * D := by
+        apply mul_le_mul_of_nonneg_left _ hD
+        apply Finset.sum_le_sum
+        intro y _
+        exact mul_le_mul_of_nonneg_left (hcol y s) (sq_nonneg _)
+      _ = D^2 * ∑ y ∈ f.support, ‖finiteEmbedding f y‖^2 := by
+        simp only [finiteEmbedding_apply, ← Finset.sum_mul]
+        ring
+      _ ≤ D^2 * ‖finiteEmbedding f‖^2 := by
+        apply mul_le_mul_of_nonneg_left _ (sq_nonneg _)
+        simpa only [ENNReal.toReal_ofNat, Real.rpow_two] using
+          (lp.sum_rpow_le_norm_rpow (by norm_num : 0 < (2 : ℝ≥0∞).toReal)
+            (finiteEmbedding f) f.support)
+      _ = (D * ‖finiteEmbedding f‖)^2 := by ring
+  refine ⟨L.extendOfNorm finiteEmbedding,
+    LinearMap.opNorm_extendOfNorm_le finiteEmbedding_dense hD hn, ?_⟩
+  intro x y
+  have he := LinearMap.extendOfNorm_eq (f := L) finiteEmbedding_dense ⟨D, hn⟩
+    (Finsupp.single y (1 : ℂ))
+  simp only [finiteEmbedding_single] at he
+  rw [he]
+  simp [L, col]
+
+end Schur
+
+section SprinklingPaths
+attribute [local instance] Classical.propDecidable
+
+ 
+theorem Along.first_outside {G : BondGraph V E} {x y : V} {l : List E}
+    (h : Along G x l y) (S : Set E) :
+    (∀ e ∈ l, e ∈ S) ∨ ∃ a b e l₁ l₂,
+      l = l₁ ++ e :: l₂ ∧ Along G x l₁ a ∧ G e = s(a,b) ∧
+      Along G b l₂ y ∧ (∀ d ∈ l₁, d ∈ S) ∧ e ∉ S := by
+  induction h with
+  | nil => left; simp
+  | @cons x z y l e he ht ih =>
+    by_cases hes : e ∈ S
+    · rcases ih with hall | ⟨a,b,d,l₁,l₂,hl,h₁,hd,h₂,hopen,hnot⟩
+      · left
+        simpa only [List.mem_cons, forall_eq_or_imp] using And.intro hes hall
+      · right
+        refine ⟨a,b,d,e::l₁,l₂,by simp [hl], .cons e he h₁, hd, h₂, ?_, hnot⟩
+        simpa only [List.mem_cons, forall_eq_or_imp] using And.intro hes hopen
+    · right
+      exact ⟨x,z,e,[],l,rfl,.nil x,he,ht,by simp,hes⟩
+
+ 
+theorem conn_first_outside (G : BondGraph V E) (S : Set E) {ω : Set E} {x y : V}
+    (h : Conn G ω x y) :
+    Conn G (ω ∩ S) x y ∨ ∃ a b e, G e = s(a,b) ∧ e ∉ S ∧
+      ω ∈ DisjointOccurrence {η | Conn G (η ∩ S) x a}
+        (DisjointOccurrence {η | e ∈ η} {η | Conn G η b y}) := by
+  obtain ⟨l,hl,hn,ho⟩ := conn_exists_bond_trail G h
+  rcases hl.first_outside S with hall | ⟨a,b,e,l₁,l₂,heq,h₁,he,h₂,hfirst,hnot⟩
+  · exact Or.inl (hl.conn fun e he => ⟨ho e he,hall e he⟩)
+  · subst l
+    have hnod := List.nodup_append.mp hn
+    have hen : e ∉ l₂ := (List.nodup_cons.mp hnod.2.1).1
+    right
+    refine ⟨a,b,e,he,hnot,l₁.toFinset,(e::l₂).toFinset,?_,?_,?_,?_,?_⟩
+    · intro d hd
+      exact ho d (List.mem_append_left _ (List.mem_toFinset.mp hd))
+    · intro d hd
+      exact ho d (List.mem_append_right _ (List.mem_toFinset.mp hd))
+    · apply Finset.disjoint_left.mpr
+      intro d hd hd'
+      exact hnod.2.2 d (List.mem_toFinset.mp hd) d (List.mem_toFinset.mp hd') rfl
+    · exact h₁.conn (fun d hd => ⟨by simpa using hd, hfirst d hd⟩)
+    · refine ⟨{e},l₂.toFinset,?_,?_,?_,by simp,?_⟩
+      · simp
+      · simp
+      · simpa using hen
+      · exact h₂.conn (fun d hd => by simpa using hd)
+
+ 
+theorem law_map_preimage_embedding {F : Type*} (p : unitInterval) (f : F ↪ E) :
+    (law p : Measure (Set E)).map (fun ω => f ⁻¹' ω) = law p := by
+  have hm : Measurable (fun ω : Set E => f ⁻¹' ω) :=
+    measurable_setOfPred.comp (Measurable.of_eval
+      (fun e => (measurable_pi_apply (f e)).comp MeasurableEquiv.setOfPred.symm.measurable))
+  unfold law
+  rw [setBernoulli_eq_map, Measure.map_map hm measurable_setOfPred]
+  have he : (fun ω : Set E => f ⁻¹' ω) ∘ (fun g => {e | g e}) =
+      (fun g : F → Prop => {e | g e}) ∘ (fun g : E → Prop => fun e => g (f e)) := rfl
+  rw [he, ← Measure.map_map measurable_setOfPred (by fun_prop)]
+  simp only [Set.mem_univ]
+  have hi := (iIndepFun_infinitePi (P := fun _ : E => bernoulliMeasure True False p)
+    (X := fun (_ : E) (b : Prop) => b) (fun _ => measurable_id)).precomp f.injective
+  have hh := hi.map_fun_eq_infinitePi_map (fun _ => measurable_pi_apply _)
+  simp only [Measure.infinitePi_map_eval] at hh
+  rw [hh, setBernoulli_eq_map]
+  rfl
+
+ 
+
+theorem exists_orParameter_small (p : unitInterval) (hp : p < 1)
+    (ε : ℝ) (hε : 0 < ε) (hε1 : ε ≤ 1) :
+    ∃ m : ℕ, ∃ q : unitInterval, (q : ℝ) ≤ ε ∧ orParameter q m = p := by
+  obtain ⟨m, hm⟩ := exists_pow_lt_of_lt_one (show 0 < 1 - (p : ℝ) by exact sub_pos.mpr hp)
+    (show 1-ε < 1 by linarith)
+  let f := fun x : ℝ => 1 - (1-x)^m
+  have hf : Continuous f := by fun_prop
+  have h0 : f 0 = 0 := by simp [f]
+  have hp' : (p : ℝ) ∈ Set.Icc (f 0) (f ε) := by
+    rw [h0]
+    exact ⟨p.2.1, by dsimp [f]; linarith⟩
+  obtain ⟨q,hq,hqp⟩ := intermediate_value_Icc hε.le hf.continuousOn hp'
+  exact ⟨m,⟨q,hq.1,hq.2.trans hε1⟩,hq.2,Subtype.ext hqp⟩
+
+end SprinklingPaths
+
+section ColoredSprinkling
+attribute [local instance] Classical.propDecidable
+
+ theorem conn_map_edges {F : Type*} (G : BondGraph V E) (f : F → E)
+    (ω : Set F) (x y : V) : Conn (G ∘ f) ω x y ↔ Conn G (f '' ω) x y := by
+  constructor <;> intro h
+  · induction h with
+    | refl => exact conn_refl _ _ _
+    | @tail a b hab he ih =>
+      rcases he with ⟨e,he,heq⟩
+      exact ih.tail ⟨f e,Set.mem_image_of_mem _ he,heq⟩
+  · induction h with
+    | refl => exact conn_refl _ _ _
+    | @tail a b hab he ih =>
+      rcases he with ⟨e,⟨d,hd,rfl⟩,heq⟩
+      exact ih.tail ⟨d,hd,heq⟩
+
+ theorem finiteWitness_inter (A : Set (Set E)) (hA : FiniteWitness A) (S : Set E) :
+    FiniteWitness {ω | ω ∩ S ∈ A} := by
+  intro ω h
+  obtain ⟨s,hs,hsa⟩ := hA (ω ∩ S) h
+  refine ⟨s,hs.trans Set.inter_subset_left,?_⟩
+  have he : (s : Set E) ∩ S = s := Set.inter_eq_left.mpr (hs.trans Set.inter_subset_right)
+  change (s : Set E) ∩ S ∈ A
+  rwa [he]
+
+ def colorInclusion (m : ℕ) : E × Fin m ↪ E × Fin (m+1) :=
+  Function.Embedding.prodMap (Function.Embedding.refl E) Fin.castSuccEmb
+
+ def baseColors (m : ℕ) : Set (E × Fin (m+1)) := Set.range (colorInclusion (E := E) m)
+
+ def baseField (m : ℕ) (ω : Set (E × Fin (m+1))) : Set E :=
+  Prod.fst '' (colorInclusion m ⁻¹' ω)
+
+ theorem not_mem_baseColors (m : ℕ) (e : E) (j : Fin (m+1)) :
+    (e,j) ∉ baseColors m ↔ j = Fin.last m := by
+  constructor
+  · intro h
+    apply Fin.eq_last_of_not_lt
+    intro hj
+    apply h
+    exact ⟨(e,⟨j.val,hj⟩),Prod.ext rfl (Fin.ext rfl)⟩
+  · rintro rfl ⟨⟨e',i⟩,hi⟩
+    exact Fin.castSucc_ne_last i (congrArg Prod.snd hi)
+
+ theorem fst_inter_baseColors (m : ℕ) (ω : Set (E × Fin (m+1))) :
+    Prod.fst '' (ω ∩ baseColors m) = baseField m ω := by
+  ext e
+  constructor
+  · rintro ⟨⟨e',j⟩,⟨ho,⟨⟨f,i⟩,he⟩⟩,rfl⟩
+    refine ⟨(f,i),?_,?_⟩
+    · change colorInclusion m (f,i) ∈ ω
+      rwa [he]
+    · exact congrArg Prod.fst he
+  · rintro ⟨⟨e',i⟩,ho,rfl⟩
+    exact ⟨colorInclusion m (e',i),⟨ho,Set.mem_range_self _⟩,rfl⟩
+
+ theorem measurable_baseField (m : ℕ) : Measurable (baseField (E := E) m) := by
+  apply (measurable_image_fst (Fin m)).comp
+  exact measurable_setOfPred.comp (Measurable.of_eval
+    (fun e => (measurable_pi_apply (colorInclusion m e)).comp
+      MeasurableEquiv.setOfPred.symm.measurable))
+
+ theorem law_map_baseField (q : unitInterval) (m : ℕ) :
+    (law q : Measure (Set (E × Fin (m+1)))).map (baseField m) = law (orParameter q m) := by
+  have hm : Measurable (fun ω : Set (E × Fin (m+1)) => colorInclusion m ⁻¹' ω) :=
+    measurable_setOfPred.comp (Measurable.of_eval
+      (fun e => (measurable_pi_apply (colorInclusion m e)).comp
+        MeasurableEquiv.setOfPred.symm.measurable))
+  change (law q).map ((fun ω : Set (E × Fin m) => Prod.fst '' ω) ∘
+    (fun ω => colorInclusion m ⁻¹' ω)) = _
+  rw [← Measure.map_map (measurable_image_fst (Fin m)) hm,
+    law_map_preimage_embedding, law_map_image_fst, Fintype.card_fin]
+
+ noncomputable def dartSource (G : BondGraph V E) (d : E × Bool) : V :=
+   if d.2 then (G d.1).out.1 else (G d.1).out.2
+ noncomputable def dartTarget (G : BondGraph V E) (d : E × Bool) : V :=
+   if d.2 then (G d.1).out.2 else (G d.1).out.1
+
+ theorem exists_dart (G : BondGraph V E) {e : E} {a b : V} (h : G e = s(a,b)) :
+    ∃ j : Bool, dartSource G (e,j) = a ∧ dartTarget G (e,j) = b := by
+  have he : s((G e).out.1, (G e).out.2) = s(a,b) := (G e).out_eq.trans h
+  rcases Sym2.eq_iff.mp he with h | h
+  · exact ⟨true,h⟩
+  · exact ⟨false,h.2,h.1⟩
+
+ theorem colored_conn_probability [Countable V] [Countable E] (G : BondGraph V E)
+    (q : unitInterval) (m : ℕ) (x y : V) :
+    (law q).real {ω : Set (E × Fin m) | Conn (G ∘ Prod.fst) ω x y} =
+      twoPoint G (orParameter q m) x y := by
+  have h := law_map_image_fst (E := E) q (Fin m)
+  simp only [Fintype.card_fin] at h
+  rw [twoPoint, ← h, Measure.map_apply (measurable_image_fst (Fin m))
+    (measurableSet_conn G x y)]
+  congr 1
+  ext ω
+  exact conn_map_edges G Prod.fst ω x y
+
+ theorem base_conn_probability [Countable V] [Countable E] (G : BondGraph V E)
+    (q : unitInterval) (m : ℕ) (x y : V) :
+    (law q).real {ω : Set (E × Fin (m+1)) |
+      Conn (G ∘ Prod.fst) (ω ∩ baseColors m) x y} =
+      twoPoint G (orParameter q m) x y := by
+  rw [twoPoint, ← law_map_baseField (E := E) q m,
+    Measure.map_apply (measurable_baseField m) (measurableSet_conn G x y)]
+  congr 1
+  ext ω
+  change Conn (G ∘ Prod.fst) (ω ∩ baseColors m) x y ↔ Conn G (baseField m ω) x y
+  rw [conn_map_edges, fst_inter_baseColors]
+
+ 
+
+ theorem twoPoint_or_succ_le [Countable V] [Fintype E] (G : BondGraph V E) (q : unitInterval)
+    (m : ℕ) (x y : V) :
+    twoPoint G (orParameter q (m+1)) x y ≤ twoPoint G (orParameter q m) x y +
+      (q : ℝ) * ∑ d : E × Bool,
+        twoPoint G (orParameter q m) x (dartSource G d) *
+        twoPoint G (orParameter q (m+1)) (dartTarget G d) y := by
+  let H : BondGraph V (E × Fin (m+1)) := G ∘ Prod.fst
+  let A (a : V) : Set (Set (E × Fin (m+1))) :=
+    {ω | Conn H (ω ∩ baseColors m) x a}
+  let B (b : V) : Set (Set (E × Fin (m+1))) := {ω | Conn H ω b y}
+  let C (d : E × Bool) := DisjointOccurrence (A (dartSource G d))
+    (DisjointOccurrence {ω | (d.1,Fin.last m) ∈ ω} (B (dartTarget G d)))
+  have huA (a : V) : IsUpperSet (A a) :=
+    fun _ _ h hω => conn_mono H (Set.inter_subset_inter_left _ h) hω
+  have hfA (a : V) : FiniteWitness (A a) :=
+    finiteWitness_inter _ (finiteWitness_conn H x a) _
+  have huB (b : V) : IsUpperSet (B b) := fun _ _ h hω => conn_mono H h hω
+  have hfB (b : V) : FiniteWitness (B b) := finiteWitness_conn H b y
+  have hsub : {ω | Conn H ω x y} ⊆ A y ∪ ⋃ d : E × Bool, C d := by
+    intro ω hω
+    rcases conn_first_outside H (baseColors m) hω with ha | ⟨a,b,⟨e,j⟩,he,hj,hc⟩
+    · exact Or.inl ha
+    · have hj' := (not_mem_baseColors m e j).mp hj
+      subst j
+      obtain ⟨k,hk₁,hk₂⟩ := exists_dart G he
+      right
+      refine Set.mem_iUnion.mpr ⟨(e,k),?_⟩
+      simpa only [C,hk₁,hk₂] using hc
+  calc
+    twoPoint G (orParameter q (m+1)) x y = (law q).real {ω | Conn H ω x y} :=
+      (colored_conn_probability G q (m+1) x y).symm
+    _ ≤ (law q).real (A y ∪ ⋃ d : E × Bool, C d) := measureReal_mono hsub
+    _ ≤ (law q).real (A y) + (law q).real (⋃ d : E × Bool, C d) := measureReal_union_le _ _
+    _ ≤ (law q).real (A y) + ∑ d : E × Bool, (law q).real (C d) :=
+      add_le_add_right (measureReal_iUnion_fintype_le _) _
+    _ ≤ twoPoint G (orParameter q m) x y +
+        ∑ d : E × Bool, twoPoint G (orParameter q m) x (dartSource G d) *
+          ((q : ℝ) * twoPoint G (orParameter q (m+1)) (dartTarget G d) y) := by
+      rw [show (law q).real (A y) = twoPoint G (orParameter q m) x y from
+        base_conn_probability G q m x y]
+      apply add_le_add_right
+      apply Finset.sum_le_sum
+      intro d _
+      have hb := bk q {ω | (d.1,Fin.last m) ∈ ω} (B (dartTarget G d))
+        (fun _ _ hh he => hh he) (huB _) (finiteWitness_edge _) (hfB _)
+      have heq : (law q).real {ω : Set (E × Fin (m+1)) | (d.1,Fin.last m) ∈ ω} =
+          (q : ℝ) := by simpa using finite_open_probability q {(d.1,Fin.last m)}
+      rw [heq, show (law q).real (B (dartTarget G d)) =
+        twoPoint G (orParameter q (m+1)) (dartTarget G d) y from
+        colored_conn_probability G q (m+1) _ _] at hb
+      have ha := bk q (A (dartSource G d))
+        (DisjointOccurrence {ω | (d.1,Fin.last m) ∈ ω} (B (dartTarget G d)))
+        (huA _) (disjointOccurrence_upper _ _) (hfA _) (finiteWitness_disjointOccurrence _ _)
+      rw [show (law q).real (A (dartSource G d)) =
+        twoPoint G (orParameter q m) x (dartSource G d) from base_conn_probability G q m _ _] at ha
+      exact ha.trans (mul_le_mul_of_nonneg_left hb (twoPoint_nonneg G _ _ _))
+    _ = _ := by rw [Finset.mul_sum]; congr 1; apply Finset.sum_congr rfl; intro d _; ring
+
+end ColoredSprinkling
+
+section FiniteGraphOperators
+attribute [local instance] Classical.propDecidable
+
+ noncomputable def edgeVertices [Fintype E] (G : BondGraph V E) : Finset V :=
+   Finset.univ.biUnion (fun e => (G e).toFinset)
+
+ theorem mem_edgeVertices [Fintype E] (G : BondGraph V E) (x : V) :
+    x ∈ edgeVertices G ↔ ∃ e, x ∈ G e := by simp [edgeVertices]
+
+ theorem conn_eq_or_mem_edgeVertices [Fintype E] (G : BondGraph V E)
+    {ω : Set E} {x y : V} (h : Conn G ω x y) : x = y ∨ y ∈ edgeVertices G := by
+  cases h with
+  | refl => exact Or.inl rfl
+  | @tail a b hab he =>
+    rcases he with ⟨e,_,he⟩
+    exact Or.inr ((mem_edgeVertices G y).mpr ⟨e,by rw [he]; simp⟩)
+
+ theorem operatorBounded_finite_edges [Fintype E] (G : BondGraph V E) (p : unitInterval) :
+    OperatorBounded G p := by
+  have hr (x : V) (s : Finset V) : ∑ y ∈ s, twoPoint G p x y ≤
+      (edgeVertices G).card + (1 : ℝ) := by
+    let F := insert x (edgeVertices G)
+    have bound (y : V) : twoPoint G p x y ≤ if y ∈ F then 1 else 0 := by
+      split_ifs with hy
+      · exact twoPoint_le_one G p x y
+      · have he : {ω | Conn G ω x y} = ∅ := by
+          apply Set.eq_empty_iff_forall_notMem.mpr
+          intro ω h
+          rcases conn_eq_or_mem_edgeVertices G h with rfl | hmem
+          · exact hy (by simp [F])
+          · exact hy (by simp [F,hmem])
+        simp [twoPoint,he]
+    calc
+      ∑ y ∈ s, twoPoint G p x y ≤ ∑ y ∈ s, if y ∈ F then (1:ℝ) else 0 :=
+        Finset.sum_le_sum (fun y _ => bound y)
+      _ = ((s ∩ F).card : ℝ) := by simp
+      _ ≤ (F.card : ℝ) := by
+        exact_mod_cast Finset.card_le_card Finset.inter_subset_right
+      _ ≤ (edgeVertices G).card + (1 : ℝ) := by
+        exact_mod_cast Finset.card_insert_le x (edgeVertices G)
+  obtain ⟨A,_,hA⟩ := exists_operator_of_row_column_bound (twoPoint G p)
+    ((edgeVertices G).card+1) (by positivity) (twoPoint_nonneg G p) hr
+    (fun y s => by simpa only [twoPoint_symm G p] using hr y s)
+  exact ⟨A,hA⟩
+
+ theorem dart_edge (G : BondGraph V E) (d : E × Bool) :
+    G d.1 = s(dartSource G d, dartTarget G d) := by
+  rcases d with ⟨e,j⟩
+  cases j
+  · exact (G e).out_eq.symm.trans (Sym2.eq_swap ..)
+  · exact (G e).out_eq.symm
+
+ theorem card_darts_at_source [Fintype E] (G : BondGraph V E) (x : V) :
+    Fintype.card {d : E × Bool // dartSource G d = x} ≤ 2 * degree G x := by
+  let : Fintype (Step G x) := (finite_steps G (fun _ => Set.toFinite _) x).fintype
+  let f : {d : E × Bool // dartSource G d = x} → Step G x × Bool := fun d =>
+    (⟨(d.val.1,dartTarget G d.val), by rw [dart_edge G d.val,d.property]⟩,d.val.2)
+  have hf : Function.Injective f := by
+    intro d d' h
+    apply Subtype.ext
+    apply Prod.ext
+    · exact congrArg (fun t : Step G x × Bool => t.1.val.1) h
+    · exact congrArg (fun t : Step G x × Bool => t.2) h
+  have hc := Fintype.card_le_of_injective f hf
+  simpa only [Fintype.card_prod,Fintype.card_bool,degree,Nat.card_eq_fintype_card,Nat.mul_comm] using hc
+
+ theorem card_darts_at_target [Fintype E] (G : BondGraph V E) (x : V) :
+    Fintype.card {d : E × Bool // dartTarget G d = x} ≤ 2 * degree G x := by
+  let : Fintype (Step G x) := (finite_steps G (fun _ => Set.toFinite _) x).fintype
+  let f : {d : E × Bool // dartTarget G d = x} → Step G x × Bool := fun d =>
+    (⟨(d.val.1,dartSource G d.val), by
+      rw [dart_edge G d.val,d.property]; exact Sym2.eq_swap ..⟩,d.val.2)
+  have hf : Function.Injective f := by
+    intro d d' h
+    apply Subtype.ext
+    apply Prod.ext
+    · exact congrArg (fun t : Step G x × Bool => t.1.val.1) h
+    · exact congrArg (fun t : Step G x × Bool => t.2) h
+  have hc := Fintype.card_le_of_injective f hf
+  simpa only [Fintype.card_prod,Fintype.card_bool,degree,Nat.card_eq_fintype_card,Nat.mul_comm] using hc
+
+ noncomputable def adjacencyOperator [Fintype E] (G : BondGraph V E) :
+    Hilbert V →L[ℂ] Hilbert V :=
+  ∑ d : E × Bool, (lp.singleContinuousLinearMap ℂ (fun _ : V => ℂ) 2 (dartSource G d)).comp
+    (lp.evalCLM ℂ (fun _ : V => ℂ) 2 (dartTarget G d))
+
+ theorem adjacencyOperator_apply [Fintype E] (G : BondGraph V E) (f : Hilbert V) :
+    adjacencyOperator G f = ∑ d : E × Bool, lp.single 2 (dartSource G d) (f (dartTarget G d)) := by
+  simp [adjacencyOperator,lp.singleContinuousLinearMap_apply,
+    lp.evalCLM,lp.evalₗ]
+  rfl
+
+ theorem adjacencyOperator_norm_le [Fintype E] (G : BondGraph V E)
+    (D : ℕ) (hD : ∀ x, degree G x ≤ D) : ‖adjacencyOperator G‖ ≤ 2 * (D : ℝ) := by
+  let U (x y : V) : ℝ := ∑ d : E × Bool,
+    if dartSource G d = x ∧ dartTarget G d = y then 1 else 0
+  have hU : ∀ x y, 0 ≤ U x y := by intro x y; apply Finset.sum_nonneg; intro d _; positivity
+  have hmat : ∀ x y, adjacencyOperator G (lp.single 2 y (1:ℂ)) x = (U x y : ℂ) := by
+    intro x y
+    rw [adjacencyOperator_apply]
+    simp only [lp.coeFn_sum,Finset.sum_apply,U,Complex.ofReal_sum]
+    apply Finset.sum_congr rfl
+    intro d _
+    simp only [lp.single_apply]
+    by_cases hx : dartSource G d = x <;> by_cases hy : dartTarget G d = y <;>
+      simp [hx,hy]
+  have hrow (x : V) (s : Finset V) : ∑ y ∈ s, U x y ≤ 2 * (D : ℝ) := by
+    calc
+      ∑ y ∈ s, U x y = ∑ d : E × Bool,
+          ∑ y ∈ s, if dartSource G d = x ∧ dartTarget G d = y then (1:ℝ) else 0 :=
+        Finset.sum_comm
+      _ ≤ ∑ d : E × Bool, if dartSource G d = x then (1:ℝ) else 0 := by
+        apply Finset.sum_le_sum
+        intro d _
+        by_cases hx : dartSource G d = x
+        · simp only [hx,true_and,ite_true]
+          rw [Finset.sum_ite_eq]
+          split_ifs <;> norm_num
+        · simp [hx]
+      _ = (Fintype.card {d : E × Bool // dartSource G d = x} : ℝ) := by simp [Fintype.card_subtype]
+      _ ≤ 2 * (D : ℝ) := by exact_mod_cast (card_darts_at_source G x).trans (Nat.mul_le_mul_left 2 (hD x))
+  have hcol (y : V) (s : Finset V) : ∑ x ∈ s, U x y ≤ 2 * (D : ℝ) := by
+    calc
+      ∑ x ∈ s, U x y = ∑ d : E × Bool,
+          ∑ x ∈ s, if dartSource G d = x ∧ dartTarget G d = y then (1:ℝ) else 0 :=
+        Finset.sum_comm
+      _ ≤ ∑ d : E × Bool, if dartTarget G d = y then (1:ℝ) else 0 := by
+        apply Finset.sum_le_sum
+        intro d _
+        by_cases hy : dartTarget G d = y
+        · simp only [hy,and_true,ite_true]
+          rw [Finset.sum_ite_eq]
+          split_ifs <;> norm_num
+        · simp [hy]
+      _ = (Fintype.card {d : E × Bool // dartTarget G d = y} : ℝ) := by simp [Fintype.card_subtype]
+      _ ≤ 2 * (D : ℝ) := by exact_mod_cast (card_darts_at_target G y).trans (Nat.mul_le_mul_left 2 (hD y))
+  obtain ⟨A,hn,hA⟩ := exists_operator_of_row_column_bound U (2*D) (by positivity) hU hrow hcol
+  rwa [kernel_extension_unique hmat hA]
+
+end FiniteGraphOperators
+
+section FiniteSprinklingBound
+attribute [local instance] Classical.propDecidable
+variable [Countable V] [Fintype E]
+
+ theorem finite_sprinkling_bound (G : BondGraph V E) (q : unitInterval) (m D : ℕ)
+    (hD : ∀ x, degree G x ≤ D) (A : Hilbert V →L[ℂ] Hilbert V)
+    (hA : HasKernel G (orParameter q m) A) (M : ℝ) (hM : ‖A‖ ≤ M)
+    (hq : (q : ℝ) * (2*D) * M < 1) :
+    ∃ B : Hilbert V →L[ℂ] Hilbert V, HasKernel G (orParameter q (m+1)) B ∧
+      ‖B‖ ≤ M / (1-(q:ℝ)*(2*D)*M) := by
+  obtain ⟨B,hB⟩ := operatorBounded_finite_edges G (orParameter q (m+1))
+  let R := (adjacencyOperator G).comp B
+  let C := A + (q : ℂ) • A.comp R
+  let U (x y : V) := twoPoint G (orParameter q m) x y + (q:ℝ) *
+    ∑ d : E × Bool, twoPoint G (orParameter q m) x (dartSource G d) *
+      twoPoint G (orParameter q (m+1)) (dartTarget G d) y
+  have hU : ∀ x y, 0 ≤ U x y := by
+    intro x y
+    exact add_nonneg (twoPoint_nonneg G _ _ _)
+      (mul_nonneg q.2.1 (Finset.sum_nonneg fun d _ =>
+        mul_nonneg (twoPoint_nonneg G _ _ _) (twoPoint_nonneg G _ _ _)))
+  have hC : ∀ x y, C (lp.single 2 y (1:ℂ)) x = (U x y : ℂ) := by
+    intro x y
+    simp only [C,add_apply,smul_apply,
+      ContinuousLinearMap.comp_apply,R,adjacencyOperator_apply,map_sum,lp.coeFn_add,Pi.add_apply,
+      lp.coeFn_smul,Pi.smul_apply,smul_eq_mul,lp.coeFn_sum,Finset.sum_apply,U,
+      Complex.ofReal_add,Complex.ofReal_mul,Complex.ofReal_sum]
+    rw [hA x y]
+    congr 1
+    congr 1
+    apply Finset.sum_congr rfl
+    intro d _
+    have hs (c : ℂ) : (lp.single 2 (dartSource G d) c : Hilbert V) = c • lp.single 2 (dartSource G d) (1:ℂ) := by
+      rw [← lp.single_smul]; simp
+    rw [hs,map_smul,lp.coeFn_smul,Pi.smul_apply,smul_eq_mul,hA,hB]
+    ring
+  obtain ⟨B',hn,hB'⟩ := exists_operator_of_matrix_domination U
+    (twoPoint G (orParameter q (m+1))) hU
+    (fun x y => by rw [abs_of_nonneg (twoPoint_nonneg G _ _ _)]; exact twoPoint_or_succ_le G q m x y)
+    C hC
+  have he : B' = B := kernel_extension_unique hB' hB
+  rw [he] at hn
+  have hqn : ‖(q : ℂ)‖ = (q : ℝ) := by simp [abs_of_nonneg q.2.1]
+  have hcoef : (q : ℝ) * ‖A‖ * (2*D) * ‖B‖ ≤ (q : ℝ) * M * (2*D) * ‖B‖ := by
+    gcongr
+    exact q.2.1
+  have hbnd : ‖B‖ ≤ M + (q : ℝ) * (2*D) * M * ‖B‖ := by
+    calc
+      ‖B‖ ≤ ‖C‖ := hn
+      _ ≤ ‖A‖ + ‖(q : ℂ) • A.comp R‖ := norm_add_le _ _
+      _ = ‖A‖ + (q : ℝ) * ‖A.comp R‖ := by rw [norm_smul,hqn]
+      _ ≤ ‖A‖ + (q : ℝ) * (‖A‖ * (‖adjacencyOperator G‖ * ‖B‖)) := by
+        apply add_le_add_right
+        apply mul_le_mul_of_nonneg_left _ q.2.1
+        exact (A.opNorm_comp_le R).trans
+          (mul_le_mul_of_nonneg_left ((adjacencyOperator G).opNorm_comp_le B) (norm_nonneg _))
+      _ ≤ ‖A‖ + (q : ℝ) * (‖A‖ * ((2*D) * ‖B‖)) := by
+        gcongr
+        · exact q.2.1
+        · exact adjacencyOperator_norm_le G D hD
+      _ ≤ M + (q : ℝ) * M * (2*D) * ‖B‖ := by
+        rw [show (q : ℝ) * (‖A‖ * ((2*D) * ‖B‖)) = (q : ℝ) * ‖A‖ * (2*D) * ‖B‖ by ring]
+        exact add_le_add hM hcoef
+      _ = _ := by ring
+  refine ⟨B,hB,?_⟩
+  apply (le_div_iff₀ (sub_pos.mpr hq)).mpr
+  nlinarith
+
+end FiniteSprinklingBound
+
+section ExhaustionSprinkling
+attribute [local instance] Classical.propDecidable
+open Filter Topology
+variable [Countable V] [Countable E]
+
+ theorem twoPoint_restrict_edges (G : BondGraph V E) (p : unitInterval) (S : Set E) (x y : V) :
+    twoPoint (G ∘ (Subtype.val : S → E)) p x y =
+      (law p).real {ω : Set E | Conn G (ω ∩ S) x y} := by
+  unfold twoPoint
+  rw [← law_map_restrict p S, Measure.map_apply (measurable_set_restrict S)
+    (measurableSet_conn (G ∘ Subtype.val) x y)]
+  apply congrArg ENNReal.toReal
+  congr 1
+  ext ω
+  change Conn (G ∘ Subtype.val) (Subtype.val ⁻¹' ω) x y ↔ _
+  rw [conn_map_edges]
+  have he : (Subtype.val '' (Subtype.val ⁻¹' ω : Set S)) = ω ∩ S := by
+    ext e
+    simp only [Set.mem_image,Set.mem_preimage,Set.mem_inter_iff]
+    constructor
+    · rintro ⟨d,hd,rfl⟩; exact ⟨hd,d.property⟩
+    · rintro ⟨ho,hs⟩; exact ⟨⟨e,hs⟩,ho,rfl⟩
+  rw [he]
+  rfl
+
+ theorem twoPoint_restrict_le (G : BondGraph V E) (p : unitInterval) (S : Set E) (x y : V) :
+    twoPoint (G ∘ (Subtype.val : S → E)) p x y ≤ twoPoint G p x y := by
+  rw [twoPoint_restrict_edges]
+  exact measureReal_mono (fun ω h => conn_mono G Set.inter_subset_left h) (measure_ne_top _ _)
+
+ omit [Countable V] [Countable E] in
+ theorem degree_restrict_le (G : BondGraph V E) (hloc : LocallyFinite G) (S : Set E) (x : V) :
+    degree (G ∘ (Subtype.val : S → E)) x ≤ degree G x := by
+  let : Fintype (Step G x) := (finite_steps G hloc x).fintype
+  let f : Step (G ∘ (Subtype.val : S → E)) x → Step G x :=
+    fun d => ⟨(d.val.1.val,d.val.2),d.property⟩
+  apply Nat.card_le_card_of_injective f
+  intro d d' h
+  apply Subtype.ext
+  apply Prod.ext
+  · apply Subtype.ext
+    exact congrArg (fun t : Step G x => t.val.1) h
+  · exact congrArg (fun t : Step G x => t.val.2) h
+
+ theorem sprinkling_bound (G : BondGraph V E) (hloc : LocallyFinite G)
+    (q : unitInterval) (m D : ℕ) (hD : ∀ x, degree G x ≤ D)
+    (A : Hilbert V →L[ℂ] Hilbert V) (hA : HasKernel G (orParameter q m) A)
+    (hq : (q : ℝ) * (2*D) * ‖A‖ < 1) :
+    ∃ B : Hilbert V →L[ℂ] Hilbert V, HasKernel G (orParameter q (m+1)) B ∧
+      ‖B‖ ≤ ‖A‖ / (1-(q:ℝ)*(2*D)*‖A‖) := by
+  let := Encodable.ofCountable E
+  let F (n : ℕ) := edgeExhaustion (E := E) n
+  let H (n : ℕ) : BondGraph V (F n) := G ∘ Subtype.val
+  have hb (n : ℕ) : ∃ B : Hilbert V →L[ℂ] Hilbert V,
+      HasKernel (H n) (orParameter q (m+1)) B ∧
+      ‖B‖ ≤ ‖A‖ / (1-(q:ℝ)*(2*D)*‖A‖) := by
+    obtain ⟨A',hn,hA'⟩ := exists_operator_of_matrix_domination
+      (twoPoint G (orParameter q m)) (twoPoint (H n) (orParameter q m))
+      (twoPoint_nonneg G _) (fun x y => by
+        rw [abs_of_nonneg (twoPoint_nonneg (H n) _ _ _)]
+        exact twoPoint_restrict_le G _ (F n) x y) A hA
+    exact finite_sprinkling_bound (H n) q m D
+      (fun x => (degree_restrict_le G hloc (F n) x).trans (hD x)) A' hA' ‖A‖ hn hq
+  choose B hB hn using hb
+  have hlim (x y : V) : Tendsto (fun n => twoPoint (H n) (orParameter q (m+1)) x y)
+      atTop (𝓝 (twoPoint G (orParameter q (m+1)) x y)) := by
+    have hm : Monotone (fun n => {ω : Set E | Conn G (ω ∩ (F n : Set E)) x y}) := by
+      intro m n h ω hω
+      exact conn_mono G (Set.inter_subset_inter_right _ (edgeExhaustion_mono h)) hω
+    have h := tendsto_measure_iUnion_atTop (μ := law (orParameter q (m+1))) hm
+    have he := witness_eq_union {ω | Conn G ω x y} (fun _ _ h hc => conn_mono G h hc)
+      (finiteWitness_conn G x y)
+    change {ω : Set E | Conn G ω x y} = ⋃ n, {ω | Conn G (ω ∩ (F n : Set E)) x y} at he
+    rw [← he] at h
+    have hr := (ENNReal.tendsto_toReal (measure_ne_top _ _)).comp h
+    apply hr.congr'
+    apply Filter.Eventually.of_forall
+    intro n
+    exact (twoPoint_restrict_edges G (orParameter q (m+1)) (F n) x y).symm
+  obtain ⟨C,hCn,hC⟩ := exists_operator_of_matrix_tendsto B
+    (fun x y => (twoPoint G (orParameter q (m+1)) x y : ℂ))
+    (div_nonneg (norm_nonneg _) (sub_pos.mpr hq).le) hn (fun x y => by
+      have hBm : ∀ n x y, B n (lp.single 2 y (1:ℂ)) x = (twoPoint (H n) (orParameter q (m+1)) x y : ℂ) := hB
+      simp only [hBm]
+      exact Complex.continuous_ofReal.continuousAt.tendsto.comp (hlim x y))
+  exact ⟨C,hC,hCn⟩
+
+end ExhaustionSprinkling
+
+section Openness
+attribute [local instance] Classical.propDecidable
+
+ theorem operatorBounded_open_right [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (D : ℕ) (hD : ∀ x, degree G x ≤ D)
+    (p : unitInterval) (hp0 : 0 < p) (hp1 : p < 1) (hb : OperatorBounded G p) :
+    ∃ r : unitInterval, p < r ∧ OperatorBounded G r := by
+  obtain ⟨A,hA⟩ := hb
+  let c : ℝ := (2*D)*‖A‖
+  have hc : 0 ≤ c := by dsimp [c]; positivity
+  obtain ⟨m,q,hq,hqp⟩ := exists_orParameter_small p hp1 (1/(c+1))
+    (by positivity) (by apply (div_le_one (by positivity)).mpr; linarith)
+  have hq0 : (0:ℝ) < q := by
+    by_contra h
+    have hqe : q = 0 := Subtype.ext ((le_of_not_gt h).antisymm q.2.1)
+    have hpe : p = 0 := by rw [← hqp,hqe]; apply Subtype.ext; simp [orParameter]
+    exact (ne_of_gt hp0) hpe
+  have hsmall : (q:ℝ)*(2*D)*‖A‖ < 1 := by
+    have hm := mul_le_mul_of_nonneg_right hq hc
+    have hh : c/(c+1) < 1 := (div_lt_one (by positivity)).mpr (by linarith)
+    calc
+      (q:ℝ)*(2*D)*‖A‖ = (q:ℝ)*c := by dsimp [c]; ring
+      _ ≤ 1/(c+1)*c := hm
+      _ = c/(c+1) := by ring
+      _ < 1 := hh
+  obtain ⟨B,hB,_⟩ := sprinkling_bound G hloc q m D hD A (hqp ▸ hA) hsmall
+  refine ⟨orParameter q (m+1),?_,B,hB⟩
+  change (p:ℝ) < (orParameter q (m+1):ℝ)
+  have he : (orParameter q (m+1):ℝ) = (p:ℝ)+(1-p)*q := by
+    rw [← hqp]
+    simp only [orParameter,pow_succ]
+    ring
+  rw [he]
+  have ht := mul_pos (show (0:ℝ) < 1-p from sub_pos.mpr hp1) hq0
+  linarith
+
+ 
+
+ theorem mainConclusion_of_critical_bounded [Infinite V] (G : BondGraph V E)
+    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (hc : OperatorBounded G (pc G)) : MainConclusion G := by
+  let : Countable V := countable_vertices hloc hconn
+  let : Countable E := countable_edges hloc
+  have hpc1 : pc G < 1 := by
+    apply lt_of_le_of_ne (show pc G ≤ 1 from (pc G).2.2)
+    intro he
+    exact not_operatorBounded_one G hconn (he ▸ hc)
+  obtain ⟨D,_,hD⟩ := exists_degree_bound G hqt
+  obtain ⟨q,hq,hb⟩ := operatorBounded_open_right G hloc D hD (pc G)
+    (pc_pos G hloc hqt) hpc1 hc
+  exact mainConclusion_of_strict_gap G hconn hloc hqt
+    (hq.trans_le (le_sSup hb))
+
+end Openness
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Kernel/TraceBounds.lean b/lean/OAI/Probability/UnimodularPercolation/Kernel/TraceBounds.lean
index c05739004a95ab08a01fd2b7605f724242c78d71..4324e98aa9eec1d5f2652dfb6bc79bfd6dc281c7 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Kernel/TraceBounds.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Kernel/TraceBounds.lean
@@ -1,1270 +1,1270 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Kernel.RussoRows
-
-namespace OAI
-
-/-! Trace Bounds for bond percolation. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section RussoOperator
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem russoSlope_symm [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp : p < pc G) (x y : V) :
-    russoSlope G x y (p:ℝ) = russoSlope G y x (p:ℝ) := by
-  have hh := hasDerivAt_realTwoPoint_slope G hloc hqt p ⟨hp0,hp⟩ x y
-  have he : realTwoPoint G x y = realTwoPoint G y x := funext (fun _ => twoPoint_symm G _ x y)
-  rw [he] at hh
-  exact hh.unique (hasDerivAt_realTwoPoint_slope G hloc hqt p ⟨hp0,hp⟩ y x)
-
-theorem russoSlope_aut [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp : p < pc G)
-    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (x y : V) :
-    russoSlope G (a x) (a y) (p:ℝ) = russoSlope G x y (p:ℝ) := by
-  have hh := hasDerivAt_realTwoPoint_slope G hloc hqt p ⟨hp0,hp⟩ (a x) (a y)
-  have he : realTwoPoint G (a x) (a y) = realTwoPoint G x y := funext (fun _ => twoPoint_aut G _ hab x y)
-  rw [he] at hh
-  exact hh.unique (hasDerivAt_realTwoPoint_slope G hloc hqt p ⟨hp0,hp⟩ x y)
-
- 
-
-theorem exists_russoOperator [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp : p < pc G) :
-    ∃ D : Hilbert V →L[ℂ] Hilbert V,
-      ∀ x y, matrixCoeff D x y = (russoSlope G x y (p:ℝ):ℂ) := by
-  obtain ⟨R,hR,hrow⟩ := russoMajorant_uniform_row_bound G hloc hqt p hp
-  have hb (x : V) (s : Finset V) : ∑ y ∈ s, russoSlope G x y (p:ℝ) ≤ R := by
-    calc
-      _ ≤ ∑ y ∈ s, russoMajorant G p x y := Finset.sum_le_sum (fun y _ => russoSlope_le_majorant G hloc hqt p p hp le_rfl x y)
-      _ ≤ ∑' y, russoMajorant G p x y := Summable.sum_le_tsum s (fun y _ => russoMajorant_nonneg G p x y)
-        (summable_russoMajorant G hloc hqt p hp x)
-      _ ≤ R := hrow x
-  obtain ⟨D,_,hD⟩ := exists_operator_of_row_column_bound (fun x y => russoSlope G x y (p:ℝ)) R hR
-    (fun x y => russoSlope_nonneg G x y p p.2.2) hb
-    (fun y s => by simpa only [russoSlope_symm G hloc hqt p hp0 hp] using hb y s)
-  exact ⟨D,hD⟩
-
-theorem russoOperator_positiveReal (G : BondGraph V E) (p : unitInterval)
-    (D : Hilbert V →L[ℂ] Hilbert V)
-    (hD : ∀ x y, matrixCoeff D x y = (russoSlope G x y (p:ℝ):ℂ)) : PositiveRealMatrix D := by
-  intro x y
-  rw [hD x y]
-  exact ⟨rfl,russoSlope_nonneg G x y p p.2.2⟩
-
-theorem russoOperator_invariant [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp : p < pc G)
-    (D : Hilbert V →L[ℂ] Hilbert V)
-    (hD : ∀ x y, matrixCoeff D x y = (russoSlope G x y (p:ℝ):ℂ))
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) : KernelInvariant Γ D := by
-  intro a ha x y
-  obtain ⟨b,hab⟩ := hΓ ha
-  simp only [hD,russoSlope_aut G hloc hqt p hp0 hp hab]
-
-end RussoOperator
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section PositiveTriangleSeries
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem PositiveRealMatrix.triple_hasSum
-    {A B C : Hilbert V →L[ℂ] Hilbert V}
-    (hA : PositiveRealMatrix A) (hB : PositiveRealMatrix B) (hC : PositiveRealMatrix C)
-    (x y : V) :
-    HasSum (fun ab : V×V => (matrixCoeff A x ab.1).re * (matrixCoeff B ab.1 ab.2).re *
-      (matrixCoeff C ab.2 y).re) (matrixCoeff (A*B*C) x y).re := by
-  have hi (a : V) : HasSum (fun b => (matrixCoeff A x a).re * (matrixCoeff B a b).re * (matrixCoeff C b y).re)
-      ((matrixCoeff A x a).re * (matrixCoeff (B*C) a y).re) := by
-    simpa only [mul_assoc] using (hB.mul_hasSum C a y).mul_left (matrixCoeff A x a).re
-  have ho := hA.mul_hasSum (B*C) x y
-  have hs : Summable (fun ab : V×V => (matrixCoeff A x ab.1).re * (matrixCoeff B ab.1 ab.2).re * (matrixCoeff C ab.2 y).re) := by
-    apply (summable_prod_of_nonneg (f := fun ab : V×V => (matrixCoeff A x ab.1).re * (matrixCoeff B ab.1 ab.2).re * (matrixCoeff C ab.2 y).re)
-      (fun ab => mul_nonneg (mul_nonneg (hA _ _).2 (hB _ _).2) (hC _ _).2)).mpr
-    refine ⟨fun a => (hi a).summable,?_⟩
-    simp only [(hi _).tsum_eq]
-    exact ho.summable
-  have he : (∑' ab : V×V, (matrixCoeff A x ab.1).re * (matrixCoeff B ab.1 ab.2).re * (matrixCoeff C ab.2 y).re) =
-      (matrixCoeff (A*B*C) x y).re := by
-    rw [hs.tsum_prod]
-    simp only [(hi _).tsum_eq]
-    rw [ho.tsum_eq,mul_assoc]
-  exact he ▸ hs.hasSum
-
-theorem HasKernel.matrixCoeff_re (G : BondGraph V E) (p : unitInterval)
-    (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T) (x y : V) :
-    (matrixCoeff T x y).re = twoPoint G p x y := by
-  change (T (lp.single 2 y (1:ℂ)) x).re = _
-  rw [hT x y,Complex.ofReal_re]
-
-theorem realTriangle_eq_matrixCoeff (G : BondGraph V E) (p : unitInterval)
-    (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T) (x : V) :
-    realTriangle G x (p:ℝ) = (matrixCoeff (T^3) x x).re := by
-  have ht := hT.positiveRealMatrix G p T
-  have hs := ht.triple_hasSum ht ht x x
-  rw [show T*T*T = T^3 by simp only [pow_succ,pow_zero,one_mul]] at hs
-  simpa only [HasKernel.matrixCoeff_re G p T hT,realTriangle,realTriangleTerm,realTwoPoint_coe] using hs.tsum_eq
-
-theorem rootTriangle_eq_trace (G : BondGraph V E) (p : unitInterval)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T) :
-    rootTriangle G root weight (p:ℝ) = (rootTrace root weight (T^3)).re := by
-  simp only [rootTriangle,realTriangle_eq_matrixCoeff G p T hT,rootTrace,Complex.re_sum,
-    Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero]
-
-theorem triangleSlope_eq_matrixCoeff (G : BondGraph V E) (p : unitInterval)
-    (T D : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T)
-    (hD : ∀ x y, matrixCoeff D x y = (russoSlope G x y (p:ℝ):ℂ)) (x : V) :
-    (∑' ab, realTriangleSlope G x ab (p:ℝ)) =
-      (matrixCoeff (D*T*T + T*D*T + T*T*D) x x).re := by
-  have ht := hT.positiveRealMatrix G p T
-  have hd := russoOperator_positiveReal G p D hD
-  have h₁ := (hd.triple_hasSum ht ht x x)
-  have h₂ := (ht.triple_hasSum hd ht x x)
-  have h₃ := (ht.triple_hasSum ht hd x x)
-  have h := (h₁.add h₂).add h₃
-  have ha (A B : Hilbert V →L[ℂ] Hilbert V) : matrixCoeff (A+B) x x = matrixCoeff A x x + matrixCoeff B x x := rfl
-  simp only [ha,Complex.add_re]
-  simpa only [hD,Complex.ofReal_re,HasKernel.matrixCoeff_re G p T hT,
-    realTriangleSlope,realTwoPoint_coe] using h.tsum_eq
-
-end PositiveTriangleSeries
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section RootTriangleTraceDerivative
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem rootTrace_triangleSlope_cyclic
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    {Γ : Set (V ≃ V)} (htr : RootTransport root weight Γ)
-    (T D : Hilbert V →L[ℂ] Hilbert V) (hT : KernelInvariant Γ T) (hD : KernelInvariant Γ D) :
-    rootTrace root weight (D*T*T+T*D*T+T*T*D) = 3*rootTrace root weight (T^2*D) := by
-  have h₁ : rootTrace root weight (D*T*T) = rootTrace root weight (T^2*D) := by
-    rw [mul_assoc,rootTrace_cyclic root weight htr D (T*T) hD (hT.mul hT),pow_two]
-  have h₂ : rootTrace root weight (T*D*T) = rootTrace root weight (T^2*D) := by
-    rw [mul_assoc,rootTrace_cyclic root weight htr T (D*T) hT (hD.mul hT),h₁]
-  rw [rootTrace_add,rootTrace_add,h₁,h₂,←pow_two]
-  ring
-
- 
-
-theorem hasDerivAt_rootTriangle_trace [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp : p < pc G)
-    (T D : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T)
-    (hD : ∀ x y, matrixCoeff D x y = (russoSlope G x y (p:ℝ):ℂ)) :
-    HasDerivAt (rootTriangle G root weight) (3*(rootTrace root weight (T^2*D)).re) (p:ℝ) := by
-  have hti : KernelInvariant Γ T := fun a ha => hT.invariant G p T a (hΓ ha)
-  have hdi := russoOperator_invariant G hloc hqt p hp0 hp D hD Γ hΓ
-  have hs := hasDerivAt_rootTriangle_subcritical G hloc hqt root weight p hp0 hp
-  have he : (∑ i, weight i * ∑' ab, realTriangleSlope G (root i) ab (p:ℝ)) =
-      3*(rootTrace root weight (T^2*D)).re := by
-    simp only [triangleSlope_eq_matrixCoeff G p T D hT hD]
-    have hr := congrArg Complex.re (rootTrace_triangleSlope_cyclic root weight htr T D hti hdi)
-    simpa only [rootTrace,Complex.re_sum,Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,
-      Complex.re_ofNat,Complex.im_ofNat,zero_mul,sub_zero] using hr
-  exact he ▸ hs
-
-end RootTriangleTraceDerivative
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-open scoped ENNReal
-section ContactKernelReroot
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem contactKernel_reroot [Countable V] [Countable E]
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hw : ∀ i, 0 ≤ weight i) (p : unitInterval)
-    (hfin : ∀ x, ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
-    (J : V → V → ℝ≥0∞) (hJ : ∀ a ∈ Γ, ∀ x y, J (a x) (a y) = J x y) :
-    clusterPairFunctional G p root weight (fun ω x y =>
-      clusterPairKernel G ω (indexedEdgeKernel G) x y * clusterPairKernel G ω J x y) =
-      ∑ i, ENNReal.ofReal (weight i) * ∫⁻ ω, ∑' y,
-        J (root i) y * ENNReal.ofReal (fullLargeContact G (root i) y 0 ω) ∂law p := by
-  have ht := clusterPairFunctional_reroot G hconn hloc root weight Γ hΓ htr hw p hfin
-    J hJ (fun ω x y => ENNReal.ofReal (fullLargeContact G x y 0 ω))
-    (fun x y => (measurable_fullLargeContact G hloc x y 0).ennreal_ofReal)
-    (fun ω x y a b hxa hyb => congrArg ENNReal.ofReal (fullLargeContact_cluster_constant G ω 0 x y a b hxa hyb))
-    (fun a _ b hab ω x y => congrArg ENNReal.ofReal (fullLargeContact_aut G hab ω x y 0))
-  simp_rw [fullLargeContact_distinct] at ht
-  rw [←ht]
-  unfold clusterPairFunctional
-  apply Finset.sum_congr rfl
-  intro i _
-  congr 1
-  apply lintegral_congr_ae
-  filter_upwards [hfin (root i)] with ω hω
-  unfold clusterPairDensity
-  congr 1
-  apply tsum_congr
-  intro y
-  congr 1
-  by_cases hd : Conn G ω (root i) y
-  · simp only [hd,not_true_eq_false,ite_false]
-  · rw [ite_eq_left hd,ite_eq_left hd]
-    dsimp only
-    rw [clusterPairKernel_edge G hloc ω (root i) y hd hω]
-    simp only [fullLargeContact,hd,not_false_eq_true,Nat.zero_le,and_self,ite_true,ENNReal.ofReal_natCast]
-    exact mul_comm _ _
-
-theorem contact_lintegral_eq_russoOperator [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (D : Hilbert V →L[ℂ] Hilbert V)
-    (hD : ∀ x y, matrixCoeff D x y = (russoSlope G x y (p:ℝ):ℂ)) (x y : V) :
-    (∫⁻ ω, ENNReal.ofReal (fullLargeContact G x y 0 ω) ∂law p) =
-      ENNReal.ofReal (1-(p:ℝ)) * ennMatrixCoeff D x y := by
-  rw [←ofReal_integral_eq_lintegral_ofReal (integrable_fullContact_subcritical G hloc hqt p hp x y)
-    (Eventually.of_forall (fullLargeContact_nonneg G x y 0))]
-  simp only [ennMatrixCoeff,hD,Complex.ofReal_re,russoSlope,realContactExpectation_coe]
-  rw [←ENNReal.ofReal_mul (sub_nonneg.mpr p.2.2)]
-  congr 1
-  have hd : 1-(p:ℝ) ≠ 0 := ne_of_gt (sub_pos.mpr ((show (p:ℝ) < (pc G:ℝ) from hp).trans_le (pc G).2.2))
-  exact (mul_div_cancel₀ _ hd).symm
-
-end ContactKernelReroot
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-open scoped ENNReal
-section MixedRussoTrace
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem mixedFunctional_eq_russoTrace [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hw : ∀ i, 0 ≤ weight i) (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp : p < pc G)
-    (J D : Hilbert V →L[ℂ] Hilbert V) (hJ : PositiveRealMatrix J) (hiJ : KernelInvariant Γ J)
-    (hD : ∀ x y, matrixCoeff D x y = (russoSlope G x y (p:ℝ):ℂ)) :
-    clusterPairFunctional G p root weight (fun ω x y =>
-      clusterPairKernel G ω (indexedEdgeKernel G) x y * clusterPairKernel G ω (ennMatrixCoeff J) x y) =
-      ENNReal.ofReal ((1-(p:ℝ))*(rootTrace root weight (J*D)).re) := by
-  have hd := russoOperator_positiveReal G p D hD
-  have hdi (x y : V) : ennMatrixCoeff D x y = ennMatrixCoeff D y x := by
-    simp only [ennMatrixCoeff,hD,Complex.ofReal_re,russoSlope_symm G hloc hqt p hp0 hp x y]
-  have hji : ∀ a ∈ Γ, ∀ x y, ennMatrixCoeff J (a x) (a y) = ennMatrixCoeff J x y := by
-    intro a ha x y; unfold ennMatrixCoeff; rw [hiJ a ha]
-  rw [contactKernel_reroot G hconn hloc root weight Γ hΓ htr hw p
-    (subcritical_cluster_finite_ae G p hp) (ennMatrixCoeff J) hji]
-  have he (x : V) : (∫⁻ ω, ∑' y, ennMatrixCoeff J x y * ENNReal.ofReal (fullLargeContact G x y 0 ω) ∂law p) =
-      ENNReal.ofReal (1-(p:ℝ)) * ennMatrixCoeff (J*D) x x := by
-    have hm (y : V) : Measurable (fun ω : Set E => ennMatrixCoeff J x y * ENNReal.ofReal (fullLargeContact G x y 0 ω)) :=
-      (measurable_const : Measurable (fun _ : Set E => ennMatrixCoeff J x y)).mul
-        (measurable_fullLargeContact G hloc x y 0).ennreal_ofReal
-    rw [lintegral_tsum (fun y => (hm y).aemeasurable)]
-    simp_rw [lintegral_const_mul _ (measurable_fullLargeContact G hloc x _ 0).ennreal_ofReal,
-      contact_lintegral_eq_russoOperator G hloc hqt p hp D hD]
-    rw [ennMatrixCoeff_mul hJ hd,←ENNReal.tsum_mul_left]
-    apply tsum_congr
-    intro y
-    rw [hdi x y]
-    ac_rfl
-  simp_rw [he]
-  rw [ENNReal.ofReal_mul (sub_nonneg.mpr p.2.2),←enn_rootTrace root weight hw (J*D) (hJ.mul hd),Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro i _
-  ac_rfl
-
-theorem mixedFunctional_toReal_eq_russoTrace [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hw : ∀ i, 0 ≤ weight i) (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp : p < pc G)
-    (J D : Hilbert V →L[ℂ] Hilbert V) (hJ : PositiveRealMatrix J) (hiJ : KernelInvariant Γ J)
-    (hD : ∀ x y, matrixCoeff D x y = (russoSlope G x y (p:ℝ):ℂ)) :
-    (clusterPairFunctional G p root weight (fun ω x y =>
-      clusterPairKernel G ω (indexedEdgeKernel G) x y * clusterPairKernel G ω (ennMatrixCoeff J) x y)).toReal =
-      (1-(p:ℝ))*(rootTrace root weight (J*D)).re := by
-  rw [mixedFunctional_eq_russoTrace G hconn hloc hqt root weight Γ hΓ htr hw p hp0 hp J D hJ hiJ hD]
-  apply ENNReal.toReal_ofReal
-  apply mul_nonneg (sub_nonneg.mpr p.2.2)
-  have hd := hJ.mul (russoOperator_positiveReal G p D hD)
-  simp only [rootTrace,Complex.re_sum,Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero]
-  exact Finset.sum_nonneg (fun i _ => mul_nonneg (hw i) (hd _ _).2)
-
-end MixedRussoTrace
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory Filter
-section RootTriangleRusso
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem rootTriangle_russo [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hw : ∀ i, 0 ≤ weight i) (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp : p < pc G)
-    (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T) :
-    (1-(p:ℝ))*deriv (rootTriangle G root weight) (p:ℝ) =
-      3*(clusterPairFunctional G p root weight (fun ω x y =>
-        clusterPairKernel G ω (indexedEdgeKernel G) x y * clusterPairKernel G ω (ennMatrixCoeff (T^2)) x y)).toReal := by
-  obtain ⟨D,hD⟩ := exists_russoOperator G hloc hqt p hp0 hp
-  have hi : KernelInvariant Γ T := fun a ha => hT.invariant G p T a (hΓ ha)
-  rw [(hasDerivAt_rootTriangle_trace G hloc hqt root weight Γ hΓ htr p hp0 hp T D hT hD).deriv,
-    mixedFunctional_toReal_eq_russoTrace G hconn hloc hqt root weight Γ hΓ htr hw p hp0 hp
-      (T^2) D ((hT.positiveRealMatrix G p T).pow 2) (hi.pow 2) hD]
-  ring
-
-theorem realTriangle_one_le [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (x : V) : 1 ≤ realTriangle G x (p:ℝ) := by
-  have hh := (summable_realTriangleTerm G hloc hqt p hp x).le_tsum (x,x)
-    (fun ab _ => by
-      simp only [realTriangleTerm,realTwoPoint_coe]
-      exact mul_nonneg (mul_nonneg (twoPoint_nonneg G p _ _) (twoPoint_nonneg G p _ _)) (twoPoint_nonneg G p _ _))
-  simpa only [realTriangleTerm,realTwoPoint_coe,twoPoint_self,mul_one,realTriangle] using hh
-
-theorem rootTriangle_one_le [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
-    (p : unitInterval) (hp : p < pc G) : 1 ≤ rootTriangle G root weight (p:ℝ) := by
-  calc
-    1 = ∑ i, weight i * 1 := by simpa using hw1.symm
-    _ ≤ _ := Finset.sum_le_sum (fun i _ => mul_le_mul_of_nonneg_left
-      (realTriangle_one_le G hloc hqt p hp (root i)) (hw i))
-
-end RootTriangleRusso
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-open scoped ENNReal
-section RealMixedTrace
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem critical_mixed_trace_bound_real
-    [Countable V] [Countable E] [Nonempty V] [Nonempty E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hconn : Connected G)
-    (hqt : QuasiTransitive G) (hh : 0 < hV G)
-    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
-    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
-    (hp₀ : 0 < (pc G:ℝ)) (hp₁ : (pc G:ℝ) < 1)
-    (hfin : ∀ i, ∀ᵐ ω ∂law (pc G), (cluster G ω (root i)).Finite)
-    (α ε : ℝ) (hα : 0 < α) (hα1 : α < 1/2) (hε : 0 < ε)
-    (hr : (1:ℝ)/2+ε-2*α < 0) :
-    ∃ A B : ℝ, 0 < A ∧ 0 < B ∧ ∀ (p : unitInterval),
-      (pc G:ℝ)/2 ≤ (p:ℝ) → p < pc G →
-      ∀ (T : Hilbert V →L[ℂ] Hilbert V), HasKernel G p T →
-      ∀ L : ℝ, 1 ≤ L →
-        (clusterPairFunctional G p root weight (fun ω x y =>
-          clusterPairKernel G ω (indexedEdgeKernel G) x y *
-          clusterPairKernel G ω (ennMatrixCoeff (T^2)) x y)).toReal ≤
-        2*(A*L^(1-α)*rootTriangle G root weight (p:ℝ)) +
-          (B*L^((1:ℝ)/2+ε-2*α))^((1:ℝ)/2) *
-          (‖T‖^3*rootTriangle G root weight (p:ℝ))^((1:ℝ)/2) := by
-  obtain ⟨A,B,hA,hB,hb⟩ := critical_mixed_trace_bound G hloc hconn hqt hh root weight Γ hΓ htr hcov hw hw1 hp₀ hp₁ hfin α ε hα hα1 hε hr
-  refine ⟨A,B,hA,hB,?_⟩
-  intro p hp hpq T hT L hL
-  have ht : 0 ≤ rootTriangle G root weight (p:ℝ) := (rootTriangle_one_le G hloc hqt root weight (fun i => (hw i).le) hw1 p hpq).trans' (by norm_num)
-  have ha : 0 ≤ A*L^(1-α)*rootTriangle G root weight (p:ℝ) := mul_nonneg (mul_nonneg hA.le (Real.rpow_nonneg (by linarith) _)) ht
-  have hc : 0 ≤ B*L^((1:ℝ)/2+ε-2*α) := mul_nonneg hB.le (Real.rpow_nonneg (by linarith) _)
-  have hn : 0 ≤ ‖T‖^3*rootTriangle G root weight (p:ℝ) := mul_nonneg (pow_nonneg (norm_nonneg _) _) ht
-  have hb' := hb p hp hpq.le T hT L hL
-  rw [←rootTriangle_eq_trace G p root weight T hT] at hb'
-  have htop : 2*ENNReal.ofReal (A*L^(1-α)*rootTriangle G root weight (p:ℝ)) +
-      (ENNReal.ofReal (B*L^((1:ℝ)/2+ε-2*α)))^((1:ℝ)/2) *
-      (ENNReal.ofReal (‖T‖^3*rootTriangle G root weight (p:ℝ)))^((1:ℝ)/2) ≠ ⊤ := by finiteness
-  have hh := ENNReal.toReal_mono htop hb'
-  rw [ENNReal.toReal_add (by finiteness) (by finiteness)] at hh
-  simpa only [ENNReal.toReal_mul,
-    ENNReal.toReal_ofNat,←ENNReal.toReal_rpow,ENNReal.toReal_ofReal ha,
-    ENNReal.toReal_ofReal hc,ENNReal.toReal_ofReal hn] using hh
-
-end RealMixedTrace
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section TraceNumerics
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem HasKernel.norm_le_susceptibility_bound [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (T : Hilbert V →L[ℂ] Hilbert V)
-    (hT : HasKernel G p T) (C : ℝ) (hC : 0 ≤ C) (hb : ∀ x, susceptibility G p x ≤ C) :
-    ‖T‖ ≤ C := by
-  have hr (x : V) (s : Finset V) : ∑ y ∈ s, twoPoint G p x y ≤ C :=
-    ((summable_twoPoint_subcritical G hloc hqt p hp x).sum_le_tsum s
-      (fun y _ => twoPoint_nonneg G p x y)).trans (hb x)
-  obtain ⟨S,hS,hK⟩ := exists_operator_of_row_column_bound (twoPoint G p) C hC (twoPoint_nonneg G p)
-    hr (fun y s => by simpa only [twoPoint_symm G p] using hr y s)
-  rwa [hT.unique hK]
-
-lemma sqrt_mixed_power (B L m a r : ℝ) (hB : 0 ≤ B) (hL : 0 ≤ L) (hm : 0 ≤ m) (ha : 0 ≤ a) :
-    (B*L^r)^((1:ℝ)/2) * (m^3*a)^((1:ℝ)/2) =
-      Real.sqrt B * L^(r/2) * m^((3:ℝ)/2) * Real.sqrt a := by
-  rw [Real.mul_rpow hB (Real.rpow_nonneg hL _), Real.mul_rpow (pow_nonneg hm _) ha,
-    ←Real.rpow_mul hL,←Real.rpow_natCast m 3,←Real.rpow_mul hm,
-    ←Real.sqrt_eq_rpow,←Real.sqrt_eq_rpow]
-  rw [show r*(1/2:ℝ) = r/2 by ring]
-  norm_num only [Nat.cast_ofNat]
-  ring
-
-lemma sqrt_derivative_estimate (p q A B L m a a' F : ℝ)
-    (hp : p ≤ q) (hq : q < 1) (hA : 0 ≤ A) (hB : 0 ≤ B)
-    (hL : 1 ≤ L) (hm : 0 ≤ m) (ha : 1 ≤ a)
-    (α r : ℝ) (hid : (1-p)*a' = 3*F)
-    (hF : F ≤ 2*(A*L^(1-α)*a) + (B*L^r)^((1:ℝ)/2)*(m^3*a)^((1:ℝ)/2)) :
-    a'/(2*Real.sqrt a) ≤
-      (3*(A+Real.sqrt B)/(1-q))*L^(1-α)*Real.sqrt a +
-      (3*(A+Real.sqrt B)/(1-q))*L^(r/2)*m^((3:ℝ)/2) := by
-  have ha0 : 0 < a := by linarith
-  have hs : 0 < Real.sqrt a := Real.sqrt_pos.mpr ha0
-  have hp1 : 0 < 1-p := by linarith
-  have hq1 : 0 < 1-q := by linarith
-  have he := Real.sq_sqrt ha0.le
-  have hL0 : 0 ≤ L := by linarith
-  rw [sqrt_mixed_power B L m a r hB hL0 hm ha0.le] at hF
-  have hu : 0 ≤ L^(1-α) := Real.rpow_nonneg hL0 _
-  have hv : 0 ≤ L^(r/2)*m^((3:ℝ)/2) := mul_nonneg (Real.rpow_nonneg hL0 _) (Real.rpow_nonneg hm _)
-  have hB0 := Real.sqrt_nonneg B
-  have hcoef : 3*A/(1-p) ≤ 3*(A+Real.sqrt B)/(1-q) := by
-    apply (div_le_div_iff₀ hp1 hq1).mpr
-    nlinarith
-  have hcoef2 : 3*Real.sqrt B/(2*(1-p)) ≤ 3*(A+Real.sqrt B)/(1-q) := by
-    apply (div_le_div_iff₀ (mul_pos (by norm_num) hp1) hq1).mpr
-    nlinarith
-  have hbase : a'/(2*Real.sqrt a) ≤ 3*A/(1-p)*L^(1-α)*Real.sqrt a +
-      (3*Real.sqrt B/(2*(1-p)))*(L^(r/2)*m^((3:ℝ)/2)) := by
-    apply (div_le_iff₀ (mul_pos (by norm_num) hs)).mpr
-    apply (mul_le_mul_iff_right₀ hp1).mp
-    have hex : (3*A/(1-p)*L^(1-α)*Real.sqrt a +
-        (3*Real.sqrt B/(2*(1-p)))*(L^(r/2)*m^((3:ℝ)/2))) * (2*Real.sqrt a) * (1-p) =
-        3*(2*(A*L^(1-α)*a) + Real.sqrt B*L^(r/2)*m^((3:ℝ)/2)*Real.sqrt a) := by
-      field_simp
-      ring_nf
-      rw [he]
-      ring
-    rw [hid,mul_comm (1-p) _,hex]
-    linarith
-  exact hbase.trans (add_le_add
-    (mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hcoef hu) hs.le)
-    (by simpa only [mul_assoc] using mul_le_mul_of_nonneg_right hcoef2 hv))
-
-end TraceNumerics
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section TraceSquareRoot
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def rootTriangleSqrt (G : BondGraph V E) {ι : Type*} [Fintype ι]
-    (root : ι → V) (weight : ι → ℝ) (t : ℝ) : ℝ := Real.sqrt (rootTriangle G root weight t)
-
-theorem hasDerivAt_rootTriangleSqrt [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
-    (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp : p < pc G) :
-    HasDerivAt (rootTriangleSqrt G root weight)
-      (deriv (rootTriangle G root weight) (p:ℝ) / (2*rootTriangleSqrt G root weight (p:ℝ))) (p:ℝ) := by
-  apply (hasDerivAt_rootTriangle_subcritical G hloc hqt root weight p hp0 hp).differentiableAt.hasDerivAt.sqrt
-  have hh := rootTriangle_one_le G hloc hqt root weight hw hw1 p hp
-  linarith
-
- 
-theorem critical_rootTriangleSqrt_derivative
-    [Countable V] [Countable E] [Nonempty V] [Nonempty E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hconn : Connected G)
-    (hqt : QuasiTransitive G) (hh : 0 < hV G)
-    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
-    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
-    (hp₀ : 0 < (pc G:ℝ)) (hp₁ : (pc G:ℝ) < 1)
-    (hfin : ∀ i, ∀ᵐ ω ∂law (pc G), (cluster G ω (root i)).Finite)
-    (α ε : ℝ) (hα : 0 < α) (hα1 : α < 1/2) (hε : 0 < ε)
-    (hr : (1:ℝ)/2+ε-2*α < 0) :
-    ∃ C : ℝ, 0 < C ∧ ∀ (p : unitInterval),
-      (pc G:ℝ)/2 ≤ (p:ℝ) → p < pc G →
-      ∀ (T : Hilbert V →L[ℂ] Hilbert V), HasKernel G p T →
-      ∀ L : ℝ, 1 ≤ L →
-        deriv (rootTriangleSqrt G root weight) (p:ℝ) ≤
-        C*L^(1-α)*rootTriangleSqrt G root weight (p:ℝ) +
-          C*L^(((1:ℝ)/2+ε-2*α)/2)*‖T‖^((3:ℝ)/2) := by
-  obtain ⟨A,B,hA,hB,hb⟩ := critical_mixed_trace_bound_real G hloc hconn hqt hh root weight Γ hΓ htr hcov hw hw1 hp₀ hp₁ hfin α ε hα hα1 hε hr
-  refine ⟨3*(A+Real.sqrt B)/(1-(pc G:ℝ)),div_pos (mul_pos (by norm_num)
-    (add_pos_of_pos_of_nonneg hA (Real.sqrt_nonneg B))) (sub_pos.mpr hp₁),?_⟩
-  intro p hp hpq T hT L hL
-  have hp0 : 0 < (p:ℝ) := lt_of_lt_of_le (half_pos hp₀) hp
-  rw [(hasDerivAt_rootTriangleSqrt G hloc hqt root weight (fun i => (hw i).le) hw1 p hp0 hpq).deriv]
-  exact sqrt_derivative_estimate p (pc G) A B L ‖T‖ (rootTriangle G root weight p)
-    (deriv (rootTriangle G root weight) p) _ hpq.le hp₁ hA.le hB.le hL (norm_nonneg _)
-    (rootTriangle_one_le G hloc hqt root weight (fun i => (hw i).le) hw1 p hpq) α _
-    (rootTriangle_russo G hconn hloc hqt root weight Γ hΓ htr (fun i => (hw i).le) p hp0 hpq T hT)
-    (hb p hp hpq T hT L hL)
-
-end TraceSquareRoot
-end Percolation
-
- 
-namespace Percolation
-section TraceExponents
-
-lemma trace_exponent_choices (η : ℝ) (hη : 0 < η) :
-    ∃ α ε ρ lambda : ℝ, 0 < α ∧ α < 1/2 ∧ 0 < ε ∧ 0 < ρ ∧ 0 < lambda ∧
-      (1:ℝ)/2+ε-2*α < 0 ∧ lambda*(1-α) < 1 ∧
-      lambda*(((1:ℝ)/2+ε-2*α)/2) + (3:ℝ)/2*(1+ρ) < 1+η/2 := by
-  let ζ := min (1/2:ℝ) η
-  let d := min (ζ/16) (η/32)
-  have hz : 0 < ζ := lt_min (by norm_num) hη
-  have hz1 : ζ ≤ 1/2 := min_le_left _ _
-  have hze : ζ ≤ η := min_le_right _ _
-  have hd : 0 < d := lt_min (div_pos hz (by norm_num)) (div_pos hη (by norm_num))
-  have hdz : d ≤ ζ/16 := min_le_left _ _
-  have hde : d ≤ η/32 := min_le_right _ _
-  have hzd : 0 ≤ ζ*d := mul_nonneg hz.le hd.le
-  refine ⟨1/2-d,d,d,2-ζ,?_,?_,hd,hd,?_,?_,?_,?_⟩ <;> nlinarith
-
-lemma trace_scale_large_term (δ D lambda r ρ : ℝ) (hδ : 0 < δ) (hD : 0 ≤ D) :
-    (δ^(-lambda))^(r/2)*(D*δ^(-1-ρ))^((3:ℝ)/2) =
-      D^((3:ℝ)/2)*δ^(-(lambda*(r/2)+(3:ℝ)/2*(1+ρ))) := by
-  rw [Real.mul_rpow hD (Real.rpow_nonneg hδ.le _),←Real.rpow_mul hδ.le,←Real.rpow_mul hδ.le]
-  rw [show δ^(-lambda*(r/2))*(D^((3:ℝ)/2)*δ^((-1-ρ)*((3:ℝ)/2))) =
-    D^((3:ℝ)/2)*(δ^(-lambda*(r/2))*δ^((-1-ρ)*((3:ℝ)/2))) by ring,
-    ←Real.rpow_add hδ]
-  congr 2
-  ring
-
-lemma trace_scale_derivative (δ D C lambda r ρ α b b' m : ℝ)
-    (hδ : 0 < δ) (hD : 0 ≤ D) (hC : 0 ≤ C)
-    (hb : 0 ≤ b) (hm : 0 ≤ m) (hmD : m ≤ D*δ^(-1-ρ))
-    (hd : b' ≤ C*(δ^(-lambda))^(1-α)*b + C*(δ^(-lambda))^(r/2)*m^((3:ℝ)/2)) :
-    b' ≤ (C*(1+D^((3:ℝ)/2)))*δ^(-(lambda*(1-α)))*b +
-      (C*(1+D^((3:ℝ)/2)))*δ^(-(lambda*(r/2)+(3:ℝ)/2*(1+ρ))) := by
-  have hs := Real.rpow_le_rpow hm hmD (by norm_num : (0:ℝ) ≤ 3/2)
-  have he : (δ^(-lambda))^(1-α) = δ^(-(lambda*(1-α))) := by rw [←Real.rpow_mul hδ.le]; congr 1; ring
-  have hc1 : C ≤ C*(1+D^((3:ℝ)/2)) := by nlinarith [Real.rpow_nonneg hD ((3:ℝ)/2)]
-  have hc2 : C*D^((3:ℝ)/2) ≤ C*(1+D^((3:ℝ)/2)) := by nlinarith
-  calc
-    b' ≤ C*(δ^(-lambda))^(1-α)*b + C*(δ^(-lambda))^(r/2)*m^((3:ℝ)/2) := hd
-    _ ≤ C*(δ^(-lambda))^(1-α)*b + C*(δ^(-lambda))^(r/2)*(D*δ^(-1-ρ))^((3:ℝ)/2) :=
-      add_le_add le_rfl (mul_le_mul_of_nonneg_left hs (mul_nonneg hC (Real.rpow_nonneg (Real.rpow_nonneg hδ.le _) _)))
-    _ = C*δ^(-(lambda*(1-α)))*b + (C*D^((3:ℝ)/2))*δ^(-(lambda*(r/2)+(3:ℝ)/2*(1+ρ))) := by
-      rw [he,mul_assoc C ((δ^(-lambda))^(r/2)),trace_scale_large_term δ D lambda r ρ hδ hD]; ring
-    _ ≤ _ := add_le_add (mul_le_mul_of_nonneg_right
-      (mul_le_mul_of_nonneg_right hc1 (Real.rpow_nonneg hδ.le _)) hb)
-      (mul_le_mul_of_nonneg_right hc2 (Real.rpow_nonneg hδ.le _))
-
-end TraceExponents
-end Percolation
-
- 
-namespace Percolation
-section IntegratingFactor
-
-lemma hasDerivAt_gap_rpow (q r t : ℝ) (ht : t < q) :
-    HasDerivAt (fun x : ℝ => (q-x)^r) (-r*(q-t)^(r-1)) t := by
-  convert! ((hasDerivAt_const t q).sub (hasDerivAt_id t)).rpow_const
-    (Or.inl (ne_of_gt (sub_pos.mpr ht))) using 1
-  simp only [Pi.sub_apply, id_eq]
-  ring
-
-noncomputable def traceIntegratingFactor (q C A t : ℝ) : ℝ :=
-  Real.exp (C/(1-A)*(q-t)^(1-A))
-
-lemma hasDerivAt_traceIntegratingFactor (q C A t : ℝ) (hA : A < 1) (ht : t < q) :
-    HasDerivAt (traceIntegratingFactor q C A)
-      (-C*(q-t)^(-A)*traceIntegratingFactor q C A t) t := by
-  have hd := ((hasDerivAt_gap_rpow q (1-A) t ht).const_mul (C/(1-A))).exp
-  convert! hd using 1
-  unfold traceIntegratingFactor
-  rw [show 1-A-1 = -A by ring]
-  field_simp [ne_of_gt (sub_pos.mpr hA)]
-
-lemma traceIntegratingFactor_bounds (q C A t : ℝ) (hC : 0 ≤ C) (hA : A < 1)
-    (ht : t < q) (hd : q-t ≤ 1) :
-    1 ≤ traceIntegratingFactor q C A t ∧
-      traceIntegratingFactor q C A t ≤ Real.exp (C/(1-A)) := by
-  have hk : 0 ≤ C/(1-A) := div_nonneg hC (sub_pos.mpr hA).le
-  have hr : 0 ≤ (q-t)^(1-A) := Real.rpow_nonneg (sub_pos.mpr ht).le _
-  constructor
-  · exact Real.one_le_exp (mul_nonneg hk hr)
-  · apply Real.exp_le_exp.mpr
-    exact mul_le_of_le_one_right hk (Real.rpow_le_one (sub_pos.mpr ht).le hd (sub_pos.mpr hA).le)
-
-lemma hasDerivAt_traceComparison (f : ℝ → ℝ) (q C A κ t : ℝ)
-    (hA : A < 1) (hκ : 0 < κ) (ht : t < q) (hf : DifferentiableAt ℝ f t) :
-    HasDerivAt (fun x => traceIntegratingFactor q C A x*f x -
-      (C*Real.exp (C/(1-A))/κ)*(q-x)^(-κ))
-      (traceIntegratingFactor q C A t*(deriv f t-C*(q-t)^(-A)*f t) -
-        C*Real.exp (C/(1-A))*(q-t)^(-κ-1)) t := by
-  have hh := ((hasDerivAt_traceIntegratingFactor q C A t hA ht).mul hf.hasDerivAt).sub
-    ((hasDerivAt_gap_rpow q (-κ) t ht).const_mul (C*Real.exp (C/(1-A))/κ))
-  convert! hh using 1
-  field_simp [ne_of_gt hκ]
-  ring
-
- 
-
-theorem singular_ode_subpower (f : ℝ → ℝ) (q s C A B κ : ℝ)
-    (hs : s < q) (hqs : q-s ≤ 1) (hC : 0 < C) (hA : A < 1)
-    (hκ : 0 < κ) (hB : B ≤ 1+κ)
-    (hf : ∀ t ∈ Set.Ico s q, DifferentiableAt ℝ f t)
-    (hn : ∀ t ∈ Set.Ico s q, 0 ≤ f t)
-    (hd : ∀ t ∈ Set.Ico s q,
-      deriv f t ≤ C*(q-t)^(-A)*f t + C*(q-t)^(-B)) :
-    ∃ K : ℝ, 0 < K ∧ ∀ t ∈ Set.Ico s q, f t ≤ K*(q-t)^(-κ) := by
-  let M := Real.exp (C/(1-A))
-  let Q := C*M/κ
-  let g := fun t => traceIntegratingFactor q C A t*f t - Q*(q-t)^(-κ)
-  have hM : 0 < M := Real.exp_pos _
-  have hQ : 0 < Q := div_pos (mul_pos hC hM) hκ
-  have hgs (t : ℝ) (ht : t ∈ Set.Ico s q) :
-      HasDerivAt g (traceIntegratingFactor q C A t*(deriv f t-C*(q-t)^(-A)*f t) -
-        C*M*(q-t)^(-κ-1)) t := hasDerivAt_traceComparison f q C A κ t hA hκ ht.2 (hf t ht)
-  have hgn (t : ℝ) (ht : t ∈ Set.Ico s q) : deriv g t ≤ 0 := by
-    have htq : 0 < q-t := sub_pos.mpr ht.2
-    have ht1 : q-t ≤ 1 := by linarith [ht.1]
-    obtain ⟨hw,hwM⟩ := traceIntegratingFactor_bounds q C A t hC.le hA ht.2 ht1
-    have hpow : (q-t)^(-B) ≤ (q-t)^(-κ-1) :=
-      Real.rpow_le_rpow_of_exponent_ge htq ht1 (by linarith)
-    have hmul := mul_le_mul_of_nonneg_left (hd t ht) (show 0 ≤ traceIntegratingFactor q C A t by linarith)
-    have h₂ := mul_le_mul_of_nonneg_right hwM (mul_nonneg hC.le (Real.rpow_nonneg htq.le (-B)))
-    have h₃ := mul_le_mul_of_nonneg_left hpow (mul_nonneg hC.le hM.le)
-    rw [(hgs t ht).deriv]
-    dsimp only [M] at hM ⊢
-    dsimp only [M] at h₃
-    nlinarith
-  have hgmono : AntitoneOn g (Set.Ico s q) := by
-    apply antitoneOn_of_deriv_nonpos (convex_Ico s q)
-    · intro t ht
-      exact (hgs t ht).continuousAt.continuousWithinAt
-    · intro t ht
-      exact (hgs t (interior_subset ht)).differentiableAt.differentiableWithinAt
-    · intro t ht
-      exact hgn t (interior_subset ht)
-  have hsI : s ∈ Set.Ico s q := ⟨le_rfl,hs⟩
-  refine ⟨M*f s+Q,add_pos_of_nonneg_of_pos (mul_nonneg hM.le (hn s hsI)) hQ,?_⟩
-  intro t ht
-  have htq : 0 < q-t := sub_pos.mpr ht.2
-  have ht1 : q-t ≤ 1 := by linarith [ht.1]
-  obtain ⟨hwt,_⟩ := traceIntegratingFactor_bounds q C A t hC.le hA ht.2 ht1
-  obtain ⟨_,hws⟩ := traceIntegratingFactor_bounds q C A s hC.le hA hs hqs
-  have hmono := hgmono hsI ht ht.1
-  have hneg : 0 ≤ Q*(q-s)^(-κ) := mul_nonneg hQ.le (Real.rpow_nonneg (sub_pos.mpr hs).le _)
-  have hft := mul_le_mul_of_nonneg_right hwt (hn t ht)
-  have hfs := mul_le_mul_of_nonneg_right hws (hn s hsI)
-  have hpow : 1 ≤ (q-t)^(-κ) := Real.one_le_rpow_of_pos_of_le_one_of_nonpos htq ht1 (neg_nonpos.mpr hκ.le)
-  have hscale := mul_le_mul_of_nonneg_left hpow (mul_nonneg hM.le (hn s hsI))
-  dsimp only [g] at hmono
-  dsimp only [M] at hscale ⊢
-  nlinarith
-
-end IntegratingFactor
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section SubpowerTrace
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem critical_rootTriangle_subpower
-    [Countable V] [Countable E] [Nonempty V] [Nonempty E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hconn : Connected G)
-    (hqt : QuasiTransitive G) (hh : 0 < hV G)
-    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
-    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
-    (hp₀ : 0 < (pc G:ℝ)) (hp₁ : (pc G:ℝ) < 1)
-    (hfin : ∀ i, ∀ᵐ ω ∂law (pc G), (cluster G ω (root i)).Finite)
-    (η : ℝ) (hη : 0 < η) :
-    ∃ K : ℝ, 0 < K ∧ ∀ (p : unitInterval), (pc G:ℝ)/2 ≤ (p:ℝ) → p < pc G →
-      rootTriangle G root weight (p:ℝ) ≤ K*((pc G:ℝ)-(p:ℝ))^(-η) := by
-  obtain ⟨α,ε,ρ,lambda,hα,hα1,hε,hρ,hlambda,hr,hA,hB⟩ := trace_exponent_choices η hη
-  obtain ⟨C₀,hC₀,hder⟩ := critical_rootTriangleSqrt_derivative G hloc hconn hqt hh root weight Γ hΓ htr hcov hw hw1 hp₀ hp₁ hfin α ε hα hα1 hε hr
-  obtain ⟨D,hD,hχ⟩ := almost_linear_susceptibility G hloc hconn hqt hh root weight Γ hΓ htr hcov hw hw1 hp₀ hp₁ hfin ρ hρ
-  let C := C₀*(1+D^((3:ℝ)/2))
-  let A := lambda*(1-α)
-  let B := lambda*(((1:ℝ)/2+ε-2*α)/2)+(3:ℝ)/2*(1+ρ)
-  have hC : 0 < C := mul_pos hC₀ (by linarith [Real.rpow_nonneg hD.le ((3:ℝ)/2)])
-  have hunit (t : ℝ) (ht : t ∈ Set.Ico ((pc G:ℝ)/2) (pc G:ℝ)) : t ∈ unitInterval :=
-    ⟨(half_pos hp₀).le.trans ht.1,ht.2.le.trans (pc G).2.2⟩
-  have hdf (t : ℝ) (ht : t ∈ Set.Ico ((pc G:ℝ)/2) (pc G:ℝ)) :
-      DifferentiableAt ℝ (rootTriangleSqrt G root weight) t := by
-    exact (hasDerivAt_rootTriangleSqrt G hloc hqt root weight (fun i => (hw i).le) hw1
-      ⟨t,hunit t ht⟩ ((half_pos hp₀).trans_le ht.1) ht.2).differentiableAt
-  have hod (t : ℝ) (ht : t ∈ Set.Ico ((pc G:ℝ)/2) (pc G:ℝ)) :
-      deriv (rootTriangleSqrt G root weight) t ≤ C*((pc G:ℝ)-t)^(-A)*rootTriangleSqrt G root weight t +
-        C*((pc G:ℝ)-t)^(-B) := by
-    let p : unitInterval := ⟨t,hunit t ht⟩
-    have hδ : 0 < (pc G:ℝ)-t := sub_pos.mpr ht.2
-    have hδ1 : (pc G:ℝ)-t ≤ 1 := by linarith [(pc G).2.2,ht.1]
-    obtain ⟨T,hT⟩ := operatorBounded_of_lt_pc G hloc hqt p ht.2
-    have hm := hT.norm_le_susceptibility_bound G hloc hqt p ht.2 T (D*((pc G:ℝ)-t)^(-1-ρ))
-      (mul_nonneg hD.le (Real.rpow_nonneg hδ.le _)) (hχ p ht.2)
-    have hL : 1 ≤ (((pc G:ℝ)-t)^(-lambda)) :=
-      Real.one_le_rpow_of_pos_of_le_one_of_nonpos hδ hδ1 (neg_nonpos.mpr hlambda.le)
-    exact trace_scale_derivative ((pc G:ℝ)-t) D C₀ lambda ((1:ℝ)/2+ε-2*α) ρ α
-      (rootTriangleSqrt G root weight t) (deriv (rootTriangleSqrt G root weight) t) ‖T‖
-      hδ hD.le hC₀.le (Real.sqrt_nonneg _) (norm_nonneg _) hm
-      (hder p ht.1 ht.2 T hT (((pc G:ℝ)-t)^(-lambda)) hL)
-  obtain ⟨K,hK,hb⟩ := singular_ode_subpower (rootTriangleSqrt G root weight) (pc G) ((pc G:ℝ)/2)
-    C A B (η/2) (by linarith) (by linarith [(pc G).2.2]) hC hA (half_pos hη) hB.le
-    hdf (fun _ _ => Real.sqrt_nonneg _) hod
-  refine ⟨K^2,sq_pos_of_pos hK,?_⟩
-  intro p hp hpq
-  have ha : 0 ≤ rootTriangle G root weight (p:ℝ) :=
-    le_trans (by norm_num) (rootTriangle_one_le G hloc hqt root weight (fun i => (hw i).le) hw1 p hpq)
-  have hb' := pow_le_pow_left₀ (Real.sqrt_nonneg (rootTriangle G root weight (p:ℝ))) (hb p ⟨hp,hpq⟩) 2
-  change (Real.sqrt (rootTriangle G root weight (p:ℝ)))^2 ≤ _ at hb'
-  rw [Real.sq_sqrt ha,mul_pow] at hb'
-  have he : (((pc G:ℝ)-(p:ℝ))^(-(η/2)))^2 = ((pc G:ℝ)-(p:ℝ))^(-η) := by
-    rw [←Real.rpow_natCast _ 2,←Real.rpow_mul (sub_pos.mpr (show (p:ℝ)<(pc G:ℝ) from hpq)).le]
-    congr 1
-    norm_num
-  rw [he] at hb'
-  exact hb'
-
-end SubpowerTrace
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section QuantitativeSprinkling
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-lemma riccati_step (a M n x : ℝ) (ha : 0 ≤ a) (hM : 0 ≤ M) (_hn : 0 ≤ n)
-    (_hx : 0 ≤ x) (hb : x ≤ M/(1-a*n*M)) (hh : a*(n+1)*M < 1) :
-    a*x < 1 ∧ x/(1-a*x) ≤ M/(1-a*(n+1)*M) := by
-  have haM : 0 ≤ a*M := mul_nonneg ha hM
-  have hp : 0 < 1-a*n*M := by nlinarith
-  have hb' := (le_div_iff₀ hp).mp hb
-  have hs : a*(M/(1-a*n*M)) < 1 := by
-    rw [←mul_div_assoc,div_lt_one hp]
-    nlinarith
-  have hax : a*x < 1 := (mul_le_mul_of_nonneg_left hb ha).trans_lt hs
-  refine ⟨hax,?_⟩
-  apply (div_le_div_iff₀ (sub_pos.mpr hax) (sub_pos.mpr hh)).mpr
-  nlinarith
-
- 
-
-theorem sprinkling_iterated_bound [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (q : unitInterval) (m D : ℕ)
-    (hD : ∀ x, degree G x ≤ D) (A : Hilbert V →L[ℂ] Hilbert V)
-    (hA : HasKernel G (orParameter q m) A) (n : ℕ)
-    (hn : (q:ℝ)*(2*D)*(n:ℝ)*‖A‖ < 1) :
-    ∃ B : Hilbert V →L[ℂ] Hilbert V, HasKernel G (orParameter q (m+n)) B ∧
-      ‖B‖ ≤ ‖A‖/(1-(q:ℝ)*(2*D)*(n:ℝ)*‖A‖) := by
-  induction n with
-  | zero => exact ⟨A,by simpa using hA,by simp⟩
-  | succ n ih =>
-    have ha : 0 ≤ (q:ℝ)*(2*D) := mul_nonneg q.2.1 (by positivity)
-    have hnext : (q:ℝ)*(2*D)*((n:ℝ)+1)*‖A‖ < 1 := by simpa using hn
-    have hprev : (q:ℝ)*(2*D)*(n:ℝ)*‖A‖ < 1 := by
-      nlinarith [mul_nonneg ha (norm_nonneg A)]
-    obtain ⟨B,hB,hBn⟩ := ih hprev
-    obtain ⟨hs,hbound⟩ := riccati_step ((q:ℝ)*(2*D)) ‖A‖ n ‖B‖ ha (norm_nonneg _) (Nat.cast_nonneg _) (norm_nonneg _) hBn hnext
-    obtain ⟨C,hC,hCn⟩ := sprinkling_bound G hloc q (m+n) D hD B hB hs
-    exact ⟨C,by simpa [Nat.add_assoc] using hC,by simpa using hCn.trans hbound⟩
-
-lemma closed_power_bound (q : unitInterval) (n : ℕ) :
-    (1-(q:ℝ))^n * (1+(n:ℝ)*(q:ℝ)) ≤ 1 := by
-  induction n with
-  | zero => simp
-  | succ n ih =>
-    have hp : 0 ≤ (1-(q:ℝ))^n := pow_nonneg (sub_nonneg.mpr q.2.2) _
-    have hn : 0 ≤ (n:ℝ)*(q:ℝ)^2 := mul_nonneg (Nat.cast_nonneg _) (sq_nonneg _)
-    rw [pow_succ,Nat.cast_add,Nat.cast_one]
-    nlinarith [mul_nonneg hp hn,mul_nonneg hp (sq_nonneg (q:ℝ))]
-
-lemma orParameter_gap_bound (p q : unitInterval) (m n : ℕ) (hm : orParameter q m = p) :
-    (1+(n:ℝ)*(q:ℝ))*(1-(orParameter q (m+n):ℝ)) ≤ 1-(p:ℝ) := by
-  have hh := mul_le_mul_of_nonneg_left (closed_power_bound q n) (sub_nonneg.mpr p.2.2)
-  have he : 1-(p:ℝ) = (1-(q:ℝ))^m := by rw [←hm]; simp [orParameter]
-  rw [he] at hh ⊢
-  change (1+(n:ℝ)*(q:ℝ))*(1-(1-(1-(q:ℝ))^(m+n))) ≤ _
-  rw [sub_sub_cancel,pow_add]
-  nlinarith
-
-end QuantitativeSprinkling
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section QuantitativeOpenness
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem operatorBounded_of_quantitative_gap [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (D : ℕ)
-    (hD : ∀ x, degree G x ≤ D) (p r : unitInterval)
-    (hp : 0 < p) (hpr : p < r) (hr : r < 1)
-    (A : Hilbert V →L[ℂ] Hilbert V) (hA : HasKernel G p A)
-    (hsmall : ((r:ℝ)-(p:ℝ))*(4*D)*‖A‖ < 1-(r:ℝ)) : OperatorBounded G r := by
-  let a := ((r:ℝ)-(p:ℝ))/(1-(r:ℝ))
-  have hden : 0 < 1-(r:ℝ) := sub_pos.mpr hr
-  have ha : 0 < a := div_pos (sub_pos.mpr hpr) hden
-  obtain ⟨m,q,hq,hqp⟩ := exists_orParameter_small p (hpr.trans hr) (min a 1)
-    (lt_min ha zero_lt_one) (min_le_right _ _)
-  have hq0 : (0:ℝ) < q := by
-    by_contra hn
-    have hqe : q = 0 := Subtype.ext ((le_of_not_gt hn).antisymm q.2.1)
-    have hpe : p = 0 := by rw [←hqp,hqe]; apply Subtype.ext; simp [orParameter]
-    exact (ne_of_gt hp) hpe
-  let n := ⌈a/(q:ℝ)⌉₊
-  have hna : a ≤ (n:ℝ)*(q:ℝ) := (div_le_iff₀ hq0).mp (Nat.le_ceil _)
-  have han : (n:ℝ)*(q:ℝ) < a+(q:ℝ) := by
-    have hn := Nat.ceil_lt_add_one (div_nonneg ha.le hq0.le)
-    have hh := mul_lt_mul_of_pos_right hn hq0
-    dsimp only [n] at *
-    field_simp at hh
-    nlinarith
-  have hn2 : (n:ℝ)*(q:ℝ) < 2*a := by linarith [hq.trans (min_le_left _ _)]
-  have hbase : (2*a)*(2*D)*‖A‖ < 1 := by
-    rw [show (2*a)*(2*D)*‖A‖ = (((r:ℝ)-(p:ℝ))*(4*D)*‖A‖)/(1-(r:ℝ)) by dsimp [a]; ring]
-    exact (div_lt_one hden).mpr hsmall
-  have hnsmall : (q:ℝ)*(2*D)*(n:ℝ)*‖A‖ < 1 := by
-    have hs := mul_le_mul_of_nonneg_right hn2.le (show 0 ≤ (2*(D:ℝ))*‖A‖ by positivity)
-    nlinarith
-  obtain ⟨B,hB,_⟩ := sprinkling_iterated_bound G hloc q m D hD A (hqp ▸ hA) n hnsmall
-  apply operatorBounded_of_le G (q := orParameter q (m+n)) ?_ ⟨B,hB⟩
-  have hgap := orParameter_gap_bound p q m n hqp
-  have hae : a*(1-(r:ℝ)) = (r:ℝ)-(p:ℝ) := div_mul_cancel₀ _ (ne_of_gt hden)
-  have hs := mul_le_mul_of_nonneg_right hna hden.le
-  change (r:ℝ) ≤ (orParameter q (m+n):ℝ)
-  nlinarith [mul_nonneg (Nat.cast_nonneg n) q.2.1]
-
- 
-
-theorem norm_gap_lower_of_endpoint_unbounded [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (D : ℕ)
-    (hD : ∀ x, degree G x ≤ D) (p r : unitInterval)
-    (hp : 0 < p) (hpr : p < r) (hr : r < 1)
-    (A : Hilbert V →L[ℂ] Hilbert V) (hA : HasKernel G p A)
-    (hu : ¬OperatorBounded G r) :
-    1-(r:ℝ) ≤ ((r:ℝ)-(p:ℝ))*(4*D)*‖A‖ := by
-  by_contra h
-  exact hu (operatorBounded_of_quantitative_gap G hloc D hD p r hp hpr hr A hA (lt_of_not_ge h))
-
-end QuantitativeOpenness
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section NormRatio
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-noncomputable def maxSusceptibility (G : BondGraph V E) (p : unitInterval) : ℝ :=
-  sSup (Set.range (susceptibility G p))
-
-theorem susceptibility_bddAbove [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) : BddAbove (Set.range (susceptibility G p)) := by
-  obtain ⟨C,_,hC⟩ := subcritical_row_bound G hloc hqt p hp
-  refine ⟨C,?_⟩
-  rintro _ ⟨x,rfl⟩
-  exact (summable_twoPoint_subcritical G hloc hqt p hp x).tsum_le_of_sum_le (hC x)
-
-theorem susceptibility_le_max [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (x : V) :
-    susceptibility G p x ≤ maxSusceptibility G p :=
-  le_csSup (susceptibility_bddAbove G hloc hqt p hp) (Set.mem_range_self x)
-
-theorem maxSusceptibility_le [Nonempty V] (G : BondGraph V E) (p : unitInterval)
-    (C : ℝ) (hC : ∀ x, susceptibility G p x ≤ C) : maxSusceptibility G p ≤ C := by
-  apply csSup_le (Set.range_nonempty _)
-  rintro _ ⟨x,rfl⟩
-  exact hC x
-
-theorem maxSusceptibility_one_le [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) : 1 ≤ maxSusceptibility G p := by
-  let x : V := Classical.choice inferInstance
-  exact (susceptibility_one_le G hloc hqt p hp x).trans (susceptibility_le_max G hloc hqt p hp x)
-
-theorem HasKernel.norm_le_maxSusceptibility [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (p : unitInterval) (hp : p < pc G) (T : Hilbert V →L[ℂ] Hilbert V)
-    (hT : HasKernel G p T) : ‖T‖ ≤ maxSusceptibility G p :=
-  hT.norm_le_susceptibility_bound G hloc hqt p hp T _
-    (le_trans zero_le_one (maxSusceptibility_one_le G hloc hqt p hp))
-    (susceptibility_le_max G hloc hqt p hp)
-
-theorem HasKernel.norm_one_le [Countable E] [Nonempty V]
-    (G : BondGraph V E) (p : unitInterval) (T : Hilbert V →L[ℂ] Hilbert V)
-    (hT : HasKernel G p T) : 1 ≤ ‖T‖ := by
-  let x : V := Classical.choice inferInstance
-  have hd : twoPoint G p x x = 1 := twoPoint_self G p x
-  have hh := (lp.norm_apply_le_norm (by norm_num : (2:ENNReal) ≠ 0) (T (lp.single 2 x (1:ℂ))) x).trans (T.le_opNorm (lp.single 2 x (1:ℂ)))
-  simpa [hT x x,hd,lp.norm_single] using hh
-
-end NormRatio
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section SourceNormRatio
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem critical_norm_inverse_lower [Countable V] [Countable E] [Nonempty V]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (hp₁ : (pc G:ℝ) < 1) (hu : ¬OperatorBounded G (pc G)) :
-    ∃ c : ℝ, 0 < c ∧ ∀ (p : unitInterval), 0 < p → p < pc G →
-      ∀ T : Hilbert V →L[ℂ] Hilbert V, HasKernel G p T →
-        c/((pc G:ℝ)-(p:ℝ)) ≤ ‖T‖ := by
-  obtain ⟨D,hDp,hD⟩ := exists_degree_bound G hqt
-  have hDpos : 0 < (4*(D:ℝ)) := by positivity
-  refine ⟨(1-(pc G:ℝ))/(4*D),div_pos (sub_pos.mpr hp₁) hDpos,?_⟩
-  intro p hp hpq T hT
-  have hh := norm_gap_lower_of_endpoint_unbounded G hloc D hD p (pc G) hp hpq hp₁ T hT hu
-  apply (div_le_iff₀ (sub_pos.mpr (show (p:ℝ)<(pc G:ℝ) from hpq))).mpr
-  apply (div_le_iff₀ hDpos).mpr
-  nlinarith
-
- 
-
-theorem critical_normRatio_subpower
-    [Countable V] [Countable E] [Nonempty V] [Nonempty E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (hconn : Connected G)
-    (hqt : QuasiTransitive G) (hh : 0 < hV G)
-    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
-    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
-    (hp₀ : 0 < (pc G:ℝ)) (hp₁ : (pc G:ℝ) < 1)
-    (hfin : ∀ i, ∀ᵐ ω ∂law (pc G), (cluster G ω (root i)).Finite)
-    (hu : ¬OperatorBounded G (pc G)) (η : ℝ) (hη : 0 < η) :
-    ∃ K : ℝ, 0 < K ∧ ∀ (p : unitInterval), (pc G:ℝ)/2 ≤ (p:ℝ) → p < pc G →
-      ∀ T : Hilbert V →L[ℂ] Hilbert V, HasKernel G p T →
-      1 ≤ maxSusceptibility G p/‖T‖ ∧
-      maxSusceptibility G p/‖T‖ ≤ K*((pc G:ℝ)-(p:ℝ))^(-η) := by
-  obtain ⟨c,hc,hm⟩ := critical_norm_inverse_lower G hloc hqt hp₁ hu
-  obtain ⟨D,hD,hχ⟩ := almost_linear_susceptibility G hloc hconn hqt hh root weight Γ hΓ htr hcov hw hw1 hp₀ hp₁ hfin η hη
-  refine ⟨D/c,div_pos hD hc,?_⟩
-  intro p hp hpq T hT
-  have hδ : 0 < (pc G:ℝ)-(p:ℝ) := sub_pos.mpr hpq
-  have hnorm : 0 < ‖T‖ := zero_lt_one.trans_le (hT.norm_one_le G p T)
-  have hM := hm p ((half_pos hp₀).trans_le hp) hpq T hT
-  have hχb := maxSusceptibility_le G p _ (hχ p hpq)
-  have he : ((pc G:ℝ)-(p:ℝ))^(-1-η)*((pc G:ℝ)-(p:ℝ)) = ((pc G:ℝ)-(p:ℝ))^(-η) := by
-    calc
-      _ = ((pc G:ℝ)-(p:ℝ))^(-1-η)*((pc G:ℝ)-(p:ℝ))^(1:ℝ) := by rw [Real.rpow_one]
-      _ = _ := by rw [←Real.rpow_add hδ]; congr 1; ring
-  refine ⟨(one_le_div hnorm).mpr (hT.norm_le_maxSusceptibility G hloc hqt p hpq T),?_⟩
-  calc
-    maxSusceptibility G p/‖T‖ ≤ (D*((pc G:ℝ)-(p:ℝ))^(-1-η))/(c/((pc G:ℝ)-(p:ℝ))) :=
-      div_le_div₀ (mul_nonneg hD.le (Real.rpow_nonneg hδ.le _)) hχb (div_pos hc hδ) hM
-    _ = (D/c)*((pc G:ℝ)-(p:ℝ))^(-η) := by rw [←he]; field_simp
-
-end SourceNormRatio
-end Percolation
-
- 
-namespace Percolation
-universe u
-section ChronologicalErasure
-open List
-variable {V : Type u}
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-inductive HoldingWalk (H : SimpleGraph V) : ℕ → V → V → Type u
-  | nil (x : V) : HoldingWalk H 0 x x
-  | hold {k : ℕ} {x y : V} (w : HoldingWalk H k x y) : HoldingWalk H (k+1) x y
-  | step {k : ℕ} {x z y : V} (h : H.Adj x z) (w : HoldingWalk H k z y) :
-      HoldingWalk H (k+1) x y
-
-namespace HoldingWalk
-variable {H : SimpleGraph V} {k : ℕ} {x y : V}
-
-def withoutHolds : {k : ℕ} → {x y : V} → HoldingWalk H k x y → H.Walk x y
-  | _,_,_,.nil _ => .nil
-  | _,_,_,.hold w => w.withoutHolds
-  | _,_,_,.step h w => .cons h w.withoutHolds
-
-def vertices : {k : ℕ} → {x y : V} → HoldingWalk H k x y → List V
-  | _,x,_,.nil _ => [x]
-  | _,x,_,.hold w => x :: w.vertices
-  | _,x,_,.step _ w => x :: w.vertices
-
-noncomputable def mass (P : V → V → ℝ) : {k : ℕ} → {x y : V} → HoldingWalk H k x y → ℝ
-  | _,_,_,.nil _ => 1
-  | _,x,_,.hold w => P x x * w.mass P
-  | _,x,_,.step (z := z) _ w => P x z * w.mass P
-
-theorem mass_nonneg (P : V → V → ℝ) (hP : ∀ a b, 0 ≤ P a b) (w : HoldingWalk H k x y) :
-    0 ≤ w.mass P := by
-  induction w with
-  | nil => exact zero_le_one
-  | hold w ih => exact mul_nonneg (hP _ _) ih
-  | step h w ih => exact mul_nonneg (hP _ _) ih
-
-theorem length_vertices (w : HoldingWalk H k x y) : w.vertices.length = k+1 := by
-  induction w with
-  | nil => rfl
-  | hold w ih => simpa only [vertices,List.length_cons] using congrArg (fun n => n+1) ih
-  | step h w ih => simpa only [vertices,List.length_cons] using congrArg (fun n => n+1) ih
-
-theorem withoutHolds_length (w : HoldingWalk H k x y) : w.withoutHolds.length ≤ k := by
-  induction w with
-  | nil => simp [withoutHolds]
-  | hold w ih => exact ih.trans (Nat.le_succ _)
-  | step h w ih => simpa only [withoutHolds,SimpleGraph.Walk.length_cons] using Nat.succ_le_succ ih
-
-theorem withoutHolds_support (w : HoldingWalk H k x y) : w.withoutHolds.support <+ w.vertices := by
-  induction w with
-  | nil => exact List.Sublist.refl _
-  | hold w ih => exact ih.cons _
-  | step h w ih => exact ih.cons_cons _
-
- 
-
-noncomputable def erase (w : HoldingWalk H k x y) : H.Walk x y :=
-  w.withoutHolds.reverse.bypass.reverse
-
-theorem erase_isPath (w : HoldingWalk H k x y) : w.erase.IsPath :=
-  w.withoutHolds.reverse.bypass_isPath.reverse
-
-theorem erase_length (w : HoldingWalk H k x y) : w.erase.length ≤ k := by
-  have hh := w.withoutHolds.reverse.length_bypass_le_length
-  simpa only [erase,SimpleGraph.Walk.length_reverse] using hh.trans (by simpa using w.withoutHolds_length)
-
-theorem erase_support (w : HoldingWalk H k x y) : w.erase.support <+ w.vertices := by
-  have hh := w.withoutHolds.reverse.support_bypass_sublist_support.reverse
-  simp only [SimpleGraph.Walk.support_reverse,List.reverse_reverse] at hh
-  simpa only [erase,SimpleGraph.Walk.support_reverse] using hh.trans w.withoutHolds_support
-
-theorem erase_edges (w : HoldingWalk H k x y) : w.erase.edges <+ w.withoutHolds.edges := by
-  have hh := w.withoutHolds.reverse.edges_bypass_sublist_edges.reverse
-  simpa only [erase,SimpleGraph.Walk.edges_reverse,List.reverse_reverse] using hh
-
-end HoldingWalk
-end ChronologicalErasure
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section CorridorMonotonicity
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-def corridorCounted (G : BondGraph V E) (retained blockers : Set E)
-    (D : Set V) (ω : Set E) (x y : V) : Prop :=
-  Conn G (ω ∩ internalEdges G D) x y ∧
-    (∀ e ∈ retained, e ∈ ω ∩ internalEdges G D ∧
-      ¬Conn G ((ω ∩ internalEdges G D) \ {e}) x y) ∧
-    Disjoint blockers (ω ∩ internalEdges G D)
-
-lemma pivotal_restrict_of_conn (G : BondGraph V E) {S T : Set E} (hST : S ⊆ T)
-    {x y : V} (hc : Conn G S x y) (e : E) (hp : ¬Conn G (T \ {e}) x y) :
-    e ∈ S ∧ ¬Conn G (S \ {e}) x y := by
-  constructor
-  · by_contra he
-    apply hp
-    apply conn_mono G ?_ hc
-    intro d hd
-    exact ⟨hST hd,fun hde => he ((Set.mem_singleton_iff.mp hde) ▸ hd)⟩
-  · intro h
-    exact hp (conn_mono G (Set.sdiff_subset_sdiff_left hST) h)
-
- 
-
-theorem corridorCounted_restrict (G : BondGraph V E) (retained blockers : Set E)
-    {D D' : Set V} {ω ω' : Set E} (hD : D ⊆ D') (hω : ω ⊆ ω') (x y : V)
-    (hc : Conn G (ω ∩ internalEdges G D) x y)
-    (ht : corridorCounted G retained blockers D' ω' x y) :
-    corridorCounted G retained blockers D ω x y := by
-  have hs : ω ∩ internalEdges G D ⊆ ω' ∩ internalEdges G D' :=
-    Set.inter_subset_inter hω (internalEdges_mono G hD)
-  refine ⟨hc,fun e he => pivotal_restrict_of_conn G hs hc e (ht.2.1 e he).2,?_⟩
-  exact ht.2.2.mono_right hs
-
-lemma corridorCounted_symm (G : BondGraph V E) (retained blockers : Set E)
-    (D : Set V) (ω : Set E) (x y : V) :
-    corridorCounted G retained blockers D ω x y ↔ corridorCounted G retained blockers D ω y x := by
-  constructor <;> rintro ⟨hc,hp,hb⟩
-  · exact ⟨conn_symm G _ hc,fun e he => ⟨(hp e he).1,fun hc => (hp e he).2 (conn_symm G _ hc)⟩,hb⟩
-  · exact ⟨conn_symm G _ hc,fun e he => ⟨(hp e he).1,fun hc => (hp e he).2 (conn_symm G _ hc)⟩,hb⟩
-
- 
-
-theorem measurableSet_corridorCounted [Countable E] (G : BondGraph V E)
-    (retained blockers : Set E) (D : Set V) (x y : V) :
-    MeasurableSet {ω | corridorCounted G retained blockers D ω x y} := by
-  have hmem (e : E) : MeasurableSet {ω : Set E | e ∈ ω ∩ internalEdges G D} := by
-    by_cases he : e ∈ internalEdges G D
-    · simpa only [Set.mem_inter_iff,he,and_true] using measurableSet_mem e
-    · simp only [Set.mem_inter_iff,he,and_false,Set.ofPred_false]; exact MeasurableSet.empty
-  have hp (e : E) : MeasurableSet {ω : Set E | e ∈ ω ∩ internalEdges G D ∧
-      ¬Conn G ((ω ∩ internalEdges G D) \ {e}) x y} := by
-    apply (hmem e).inter
-    have heq (ω : Set E) : (ω ∩ internalEdges G D) \ {e} = ω ∩ (internalEdges G D \ {e}) := by
-      ext d; simp only [Set.mem_inter_iff,Set.mem_sdiff]; tauto
-    simp_rw [heq]
-    exact (measurableSet_restrictedConn G _ x y).compl
-  have hb : MeasurableSet {ω : Set E | Disjoint blockers (ω ∩ internalEdges G D)} := by
-    simp only [Set.disjoint_left,Set.ofPred_forall]
-    exact MeasurableSet.iInter (fun e => MeasurableSet.iInter (fun (_ : e ∈ blockers) => (hmem e).compl))
-  have hkeep : MeasurableSet {ω : Set E | ∀ e ∈ retained, e ∈ ω ∩ internalEdges G D ∧
-      ¬Conn G ((ω ∩ internalEdges G D) \ {e}) x y} := by
-    simp only [Set.ofPred_forall]
-    exact MeasurableSet.iInter (fun e => MeasurableSet.iInter (fun (_ : e ∈ retained) => hp e))
-  exact (measurableSet_restrictedConn G _ x y).inter (hkeep.inter hb)
-
-end CorridorMonotonicity
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section LiteralCorridors
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-instance HoldingWalk.countable [Countable V] (H : SimpleGraph V) (k : ℕ) (x y : V) :
-    Countable (HoldingWalk H k x y) := by
-  induction k generalizing x y with
-  | zero =>
-    apply Function.Injective.countable (f := fun _ : HoldingWalk H 0 x y => ())
-    intro a b _
-    cases a
-    cases b
-    rfl
-  | succ k ih =>
-    let (a b : V) : Countable (HoldingWalk H k a b) := ih a b
-    let f : HoldingWalk H (k+1) x y →
-        HoldingWalk H k x y ⊕ (Σ z : {z // H.Adj x z}, HoldingWalk H k z.val y)
-      | .hold w => .inl w
-      | .step (z := z) h w => .inr ⟨⟨z,h⟩,w⟩
-    apply Function.Injective.countable (f := f)
-    intro a b hab
-    cases a with
-    | hold a =>
-      cases b with
-      | hold b => exact congrArg HoldingWalk.hold (Sum.inl.inj hab)
-      | step h b => cases hab
-    | @step _ _ z _ h a =>
-      cases b with
-      | hold b => cases hab
-      | @step _ _ z' _ h' b =>
-        have he := Sum.inr.inj hab
-        have hz := congrArg (fun t => t.1.val) he
-        change z = z' at hz
-        subst z'
-        have hab' : a = b := by simpa only [Sigma.mk.inj_iff,heq_eq_eq,true_and] using he
-        subst b
-        rfl
-
- 
-abbrev RawCorridor (G : BondGraph V E) (k : ℕ) :=
-  Σ x y : V, HoldingWalk (openSimpleGraph G Set.univ) k x y
-
-namespace RawCorridor
-variable {G : BondGraph V E} {k : ℕ}
-
-def base (w : RawCorridor G k) : V := w.1
-def endpoint (w : RawCorridor G k) : V := w.2.1
-noncomputable def path (w : RawCorridor G k) :
-    (openSimpleGraph G Set.univ).Walk w.base w.endpoint := w.2.2.erase
-noncomputable def interior (w : RawCorridor G k) : Finset V :=
-  w.path.support.toFinset \ {w.base,w.endpoint}
-def retained (w : RawCorridor G k) : Set E := {e | G e ∈ w.path.edges}
-noncomputable def blockers (w : RawCorridor G k) : Set E :=
-  {e | ∃ z ∈ w.interior, z ∈ G e} \ w.retained
-noncomputable def probability (P : V → V → ℝ) (w : RawCorridor G k) : ℝ := w.2.2.mass P
-noncomputable def weight (P : V → V → ℝ) (p : ℝ) (w : RawCorridor G k) : ℝ :=
-  w.probability P * (p^w.path.length)⁻¹ * ((1-p)^w.blockers.ncard)⁻¹
-
-theorem path_isPath (w : RawCorridor G k) : w.path.IsPath := w.2.2.erase_isPath
-
-theorem path_length (w : RawCorridor G k) : w.path.length ≤ k := w.2.2.erase_length
-
-theorem retained_finite (hG : Function.Injective G) (w : RawCorridor G k) : w.retained.Finite :=
-  (List.finite_toSet w.path.edges).preimage hG.injOn
-
-theorem blockers_finite (hloc : LocallyFinite G) (w : RawCorridor G k) : w.blockers.Finite := by
-  apply Set.Finite.subset (s := (incidentSupport G hloc w.interior : Set E)) (Finset.finite_toSet _)
-  intro e he
-  exact (mem_incidentSupport G hloc w.interior e).mpr he.1
-
-theorem retained_disjoint_blockers (w : RawCorridor G k) : Disjoint w.retained w.blockers := by
-  exact Set.disjoint_left.mpr (fun e he hb => hb.2 he)
-
-theorem weight_nonneg (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y)
-    (p : unitInterval) (w : RawCorridor G k) : 0 ≤ w.weight P p := by
-  exact mul_nonneg (mul_nonneg (w.2.2.mass_nonneg P hP) (inv_nonneg.mpr (pow_nonneg p.2.1 _)))
-    (inv_nonneg.mpr (pow_nonneg (sub_nonneg.mpr p.2.2) _))
-
-end RawCorridor
-end LiteralCorridors
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Kernel.RussoRows
+
+namespace OAI
+
+/-! Trace Bounds for bond percolation. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section RussoOperator
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem russoSlope_symm [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp : p < pc G) (x y : V) :
+    russoSlope G x y (p:ℝ) = russoSlope G y x (p:ℝ) := by
+  have hh := hasDerivAt_realTwoPoint_slope G hloc hqt p ⟨hp0,hp⟩ x y
+  have he : realTwoPoint G x y = realTwoPoint G y x := funext (fun _ => twoPoint_symm G _ x y)
+  rw [he] at hh
+  exact hh.unique (hasDerivAt_realTwoPoint_slope G hloc hqt p ⟨hp0,hp⟩ y x)
+
+theorem russoSlope_aut [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp : p < pc G)
+    {a : V ≃ V} {b : E ≃ E} (hab : IsAut G a b) (x y : V) :
+    russoSlope G (a x) (a y) (p:ℝ) = russoSlope G x y (p:ℝ) := by
+  have hh := hasDerivAt_realTwoPoint_slope G hloc hqt p ⟨hp0,hp⟩ (a x) (a y)
+  have he : realTwoPoint G (a x) (a y) = realTwoPoint G x y := funext (fun _ => twoPoint_aut G _ hab x y)
+  rw [he] at hh
+  exact hh.unique (hasDerivAt_realTwoPoint_slope G hloc hqt p ⟨hp0,hp⟩ x y)
+
+ 
+
+theorem exists_russoOperator [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp : p < pc G) :
+    ∃ D : Hilbert V →L[ℂ] Hilbert V,
+      ∀ x y, matrixCoeff D x y = (russoSlope G x y (p:ℝ):ℂ) := by
+  obtain ⟨R,hR,hrow⟩ := russoMajorant_uniform_row_bound G hloc hqt p hp
+  have hb (x : V) (s : Finset V) : ∑ y ∈ s, russoSlope G x y (p:ℝ) ≤ R := by
+    calc
+      _ ≤ ∑ y ∈ s, russoMajorant G p x y := Finset.sum_le_sum (fun y _ => russoSlope_le_majorant G hloc hqt p p hp le_rfl x y)
+      _ ≤ ∑' y, russoMajorant G p x y := Summable.sum_le_tsum s (fun y _ => russoMajorant_nonneg G p x y)
+        (summable_russoMajorant G hloc hqt p hp x)
+      _ ≤ R := hrow x
+  obtain ⟨D,_,hD⟩ := exists_operator_of_row_column_bound (fun x y => russoSlope G x y (p:ℝ)) R hR
+    (fun x y => russoSlope_nonneg G x y p p.2.2) hb
+    (fun y s => by simpa only [russoSlope_symm G hloc hqt p hp0 hp] using hb y s)
+  exact ⟨D,hD⟩
+
+theorem russoOperator_positiveReal (G : BondGraph V E) (p : unitInterval)
+    (D : Hilbert V →L[ℂ] Hilbert V)
+    (hD : ∀ x y, matrixCoeff D x y = (russoSlope G x y (p:ℝ):ℂ)) : PositiveRealMatrix D := by
+  intro x y
+  rw [hD x y]
+  exact ⟨rfl,russoSlope_nonneg G x y p p.2.2⟩
+
+theorem russoOperator_invariant [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp : p < pc G)
+    (D : Hilbert V →L[ℂ] Hilbert V)
+    (hD : ∀ x y, matrixCoeff D x y = (russoSlope G x y (p:ℝ):ℂ))
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) : KernelInvariant Γ D := by
+  intro a ha x y
+  obtain ⟨b,hab⟩ := hΓ ha
+  simp only [hD,russoSlope_aut G hloc hqt p hp0 hp hab]
+
+end RussoOperator
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section PositiveTriangleSeries
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem PositiveRealMatrix.triple_hasSum
+    {A B C : Hilbert V →L[ℂ] Hilbert V}
+    (hA : PositiveRealMatrix A) (hB : PositiveRealMatrix B) (hC : PositiveRealMatrix C)
+    (x y : V) :
+    HasSum (fun ab : V×V => (matrixCoeff A x ab.1).re * (matrixCoeff B ab.1 ab.2).re *
+      (matrixCoeff C ab.2 y).re) (matrixCoeff (A*B*C) x y).re := by
+  have hi (a : V) : HasSum (fun b => (matrixCoeff A x a).re * (matrixCoeff B a b).re * (matrixCoeff C b y).re)
+      ((matrixCoeff A x a).re * (matrixCoeff (B*C) a y).re) := by
+    simpa only [mul_assoc] using (hB.mul_hasSum C a y).mul_left (matrixCoeff A x a).re
+  have ho := hA.mul_hasSum (B*C) x y
+  have hs : Summable (fun ab : V×V => (matrixCoeff A x ab.1).re * (matrixCoeff B ab.1 ab.2).re * (matrixCoeff C ab.2 y).re) := by
+    apply (summable_prod_of_nonneg (f := fun ab : V×V => (matrixCoeff A x ab.1).re * (matrixCoeff B ab.1 ab.2).re * (matrixCoeff C ab.2 y).re)
+      (fun ab => mul_nonneg (mul_nonneg (hA _ _).2 (hB _ _).2) (hC _ _).2)).mpr
+    refine ⟨fun a => (hi a).summable,?_⟩
+    simp only [(hi _).tsum_eq]
+    exact ho.summable
+  have he : (∑' ab : V×V, (matrixCoeff A x ab.1).re * (matrixCoeff B ab.1 ab.2).re * (matrixCoeff C ab.2 y).re) =
+      (matrixCoeff (A*B*C) x y).re := by
+    rw [hs.tsum_prod]
+    simp only [(hi _).tsum_eq]
+    rw [ho.tsum_eq,mul_assoc]
+  exact he ▸ hs.hasSum
+
+theorem HasKernel.matrixCoeff_re (G : BondGraph V E) (p : unitInterval)
+    (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T) (x y : V) :
+    (matrixCoeff T x y).re = twoPoint G p x y := by
+  change (T (lp.single 2 y (1:ℂ)) x).re = _
+  rw [hT x y,Complex.ofReal_re]
+
+theorem realTriangle_eq_matrixCoeff (G : BondGraph V E) (p : unitInterval)
+    (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T) (x : V) :
+    realTriangle G x (p:ℝ) = (matrixCoeff (T^3) x x).re := by
+  have ht := hT.positiveRealMatrix G p T
+  have hs := ht.triple_hasSum ht ht x x
+  rw [show T*T*T = T^3 by simp only [pow_succ,pow_zero,one_mul]] at hs
+  simpa only [HasKernel.matrixCoeff_re G p T hT,realTriangle,realTriangleTerm,realTwoPoint_coe] using hs.tsum_eq
+
+theorem rootTriangle_eq_trace (G : BondGraph V E) (p : unitInterval)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T) :
+    rootTriangle G root weight (p:ℝ) = (rootTrace root weight (T^3)).re := by
+  simp only [rootTriangle,realTriangle_eq_matrixCoeff G p T hT,rootTrace,Complex.re_sum,
+    Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero]
+
+theorem triangleSlope_eq_matrixCoeff (G : BondGraph V E) (p : unitInterval)
+    (T D : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T)
+    (hD : ∀ x y, matrixCoeff D x y = (russoSlope G x y (p:ℝ):ℂ)) (x : V) :
+    (∑' ab, realTriangleSlope G x ab (p:ℝ)) =
+      (matrixCoeff (D*T*T + T*D*T + T*T*D) x x).re := by
+  have ht := hT.positiveRealMatrix G p T
+  have hd := russoOperator_positiveReal G p D hD
+  have h₁ := (hd.triple_hasSum ht ht x x)
+  have h₂ := (ht.triple_hasSum hd ht x x)
+  have h₃ := (ht.triple_hasSum ht hd x x)
+  have h := (h₁.add h₂).add h₃
+  have ha (A B : Hilbert V →L[ℂ] Hilbert V) : matrixCoeff (A+B) x x = matrixCoeff A x x + matrixCoeff B x x := rfl
+  simp only [ha,Complex.add_re]
+  simpa only [hD,Complex.ofReal_re,HasKernel.matrixCoeff_re G p T hT,
+    realTriangleSlope,realTwoPoint_coe] using h.tsum_eq
+
+end PositiveTriangleSeries
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section RootTriangleTraceDerivative
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem rootTrace_triangleSlope_cyclic
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    {Γ : Set (V ≃ V)} (htr : RootTransport root weight Γ)
+    (T D : Hilbert V →L[ℂ] Hilbert V) (hT : KernelInvariant Γ T) (hD : KernelInvariant Γ D) :
+    rootTrace root weight (D*T*T+T*D*T+T*T*D) = 3*rootTrace root weight (T^2*D) := by
+  have h₁ : rootTrace root weight (D*T*T) = rootTrace root weight (T^2*D) := by
+    rw [mul_assoc,rootTrace_cyclic root weight htr D (T*T) hD (hT.mul hT),pow_two]
+  have h₂ : rootTrace root weight (T*D*T) = rootTrace root weight (T^2*D) := by
+    rw [mul_assoc,rootTrace_cyclic root weight htr T (D*T) hT (hD.mul hT),h₁]
+  rw [rootTrace_add,rootTrace_add,h₁,h₂,←pow_two]
+  ring
+
+ 
+
+theorem hasDerivAt_rootTriangle_trace [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp : p < pc G)
+    (T D : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T)
+    (hD : ∀ x y, matrixCoeff D x y = (russoSlope G x y (p:ℝ):ℂ)) :
+    HasDerivAt (rootTriangle G root weight) (3*(rootTrace root weight (T^2*D)).re) (p:ℝ) := by
+  have hti : KernelInvariant Γ T := fun a ha => hT.invariant G p T a (hΓ ha)
+  have hdi := russoOperator_invariant G hloc hqt p hp0 hp D hD Γ hΓ
+  have hs := hasDerivAt_rootTriangle_subcritical G hloc hqt root weight p hp0 hp
+  have he : (∑ i, weight i * ∑' ab, realTriangleSlope G (root i) ab (p:ℝ)) =
+      3*(rootTrace root weight (T^2*D)).re := by
+    simp only [triangleSlope_eq_matrixCoeff G p T D hT hD]
+    have hr := congrArg Complex.re (rootTrace_triangleSlope_cyclic root weight htr T D hti hdi)
+    simpa only [rootTrace,Complex.re_sum,Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,
+      Complex.re_ofNat,Complex.im_ofNat,zero_mul,sub_zero] using hr
+  exact he ▸ hs
+
+end RootTriangleTraceDerivative
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+open scoped ENNReal
+section ContactKernelReroot
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem contactKernel_reroot [Countable V] [Countable E]
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hw : ∀ i, 0 ≤ weight i) (p : unitInterval)
+    (hfin : ∀ x, ∀ᵐ ω ∂law p, (cluster G ω x).Finite)
+    (J : V → V → ℝ≥0∞) (hJ : ∀ a ∈ Γ, ∀ x y, J (a x) (a y) = J x y) :
+    clusterPairFunctional G p root weight (fun ω x y =>
+      clusterPairKernel G ω (indexedEdgeKernel G) x y * clusterPairKernel G ω J x y) =
+      ∑ i, ENNReal.ofReal (weight i) * ∫⁻ ω, ∑' y,
+        J (root i) y * ENNReal.ofReal (fullLargeContact G (root i) y 0 ω) ∂law p := by
+  have ht := clusterPairFunctional_reroot G hconn hloc root weight Γ hΓ htr hw p hfin
+    J hJ (fun ω x y => ENNReal.ofReal (fullLargeContact G x y 0 ω))
+    (fun x y => (measurable_fullLargeContact G hloc x y 0).ennreal_ofReal)
+    (fun ω x y a b hxa hyb => congrArg ENNReal.ofReal (fullLargeContact_cluster_constant G ω 0 x y a b hxa hyb))
+    (fun a _ b hab ω x y => congrArg ENNReal.ofReal (fullLargeContact_aut G hab ω x y 0))
+  simp_rw [fullLargeContact_distinct] at ht
+  rw [←ht]
+  unfold clusterPairFunctional
+  apply Finset.sum_congr rfl
+  intro i _
+  congr 1
+  apply lintegral_congr_ae
+  filter_upwards [hfin (root i)] with ω hω
+  unfold clusterPairDensity
+  congr 1
+  apply tsum_congr
+  intro y
+  congr 1
+  by_cases hd : Conn G ω (root i) y
+  · simp only [hd,not_true_eq_false,ite_false]
+  · rw [ite_eq_left hd,ite_eq_left hd]
+    dsimp only
+    rw [clusterPairKernel_edge G hloc ω (root i) y hd hω]
+    simp only [fullLargeContact,hd,not_false_eq_true,Nat.zero_le,and_self,ite_true,ENNReal.ofReal_natCast]
+    exact mul_comm _ _
+
+theorem contact_lintegral_eq_russoOperator [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (D : Hilbert V →L[ℂ] Hilbert V)
+    (hD : ∀ x y, matrixCoeff D x y = (russoSlope G x y (p:ℝ):ℂ)) (x y : V) :
+    (∫⁻ ω, ENNReal.ofReal (fullLargeContact G x y 0 ω) ∂law p) =
+      ENNReal.ofReal (1-(p:ℝ)) * ennMatrixCoeff D x y := by
+  rw [←ofReal_integral_eq_lintegral_ofReal (integrable_fullContact_subcritical G hloc hqt p hp x y)
+    (Eventually.of_forall (fullLargeContact_nonneg G x y 0))]
+  simp only [ennMatrixCoeff,hD,Complex.ofReal_re,russoSlope,realContactExpectation_coe]
+  rw [←ENNReal.ofReal_mul (sub_nonneg.mpr p.2.2)]
+  congr 1
+  have hd : 1-(p:ℝ) ≠ 0 := ne_of_gt (sub_pos.mpr ((show (p:ℝ) < (pc G:ℝ) from hp).trans_le (pc G).2.2))
+  exact (mul_div_cancel₀ _ hd).symm
+
+end ContactKernelReroot
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+open scoped ENNReal
+section MixedRussoTrace
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem mixedFunctional_eq_russoTrace [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hw : ∀ i, 0 ≤ weight i) (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp : p < pc G)
+    (J D : Hilbert V →L[ℂ] Hilbert V) (hJ : PositiveRealMatrix J) (hiJ : KernelInvariant Γ J)
+    (hD : ∀ x y, matrixCoeff D x y = (russoSlope G x y (p:ℝ):ℂ)) :
+    clusterPairFunctional G p root weight (fun ω x y =>
+      clusterPairKernel G ω (indexedEdgeKernel G) x y * clusterPairKernel G ω (ennMatrixCoeff J) x y) =
+      ENNReal.ofReal ((1-(p:ℝ))*(rootTrace root weight (J*D)).re) := by
+  have hd := russoOperator_positiveReal G p D hD
+  have hdi (x y : V) : ennMatrixCoeff D x y = ennMatrixCoeff D y x := by
+    simp only [ennMatrixCoeff,hD,Complex.ofReal_re,russoSlope_symm G hloc hqt p hp0 hp x y]
+  have hji : ∀ a ∈ Γ, ∀ x y, ennMatrixCoeff J (a x) (a y) = ennMatrixCoeff J x y := by
+    intro a ha x y; unfold ennMatrixCoeff; rw [hiJ a ha]
+  rw [contactKernel_reroot G hconn hloc root weight Γ hΓ htr hw p
+    (subcritical_cluster_finite_ae G p hp) (ennMatrixCoeff J) hji]
+  have he (x : V) : (∫⁻ ω, ∑' y, ennMatrixCoeff J x y * ENNReal.ofReal (fullLargeContact G x y 0 ω) ∂law p) =
+      ENNReal.ofReal (1-(p:ℝ)) * ennMatrixCoeff (J*D) x x := by
+    have hm (y : V) : Measurable (fun ω : Set E => ennMatrixCoeff J x y * ENNReal.ofReal (fullLargeContact G x y 0 ω)) :=
+      (measurable_const : Measurable (fun _ : Set E => ennMatrixCoeff J x y)).mul
+        (measurable_fullLargeContact G hloc x y 0).ennreal_ofReal
+    rw [lintegral_tsum (fun y => (hm y).aemeasurable)]
+    simp_rw [lintegral_const_mul _ (measurable_fullLargeContact G hloc x _ 0).ennreal_ofReal,
+      contact_lintegral_eq_russoOperator G hloc hqt p hp D hD]
+    rw [ennMatrixCoeff_mul hJ hd,←ENNReal.tsum_mul_left]
+    apply tsum_congr
+    intro y
+    rw [hdi x y]
+    ac_rfl
+  simp_rw [he]
+  rw [ENNReal.ofReal_mul (sub_nonneg.mpr p.2.2),←enn_rootTrace root weight hw (J*D) (hJ.mul hd),Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro i _
+  ac_rfl
+
+theorem mixedFunctional_toReal_eq_russoTrace [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hw : ∀ i, 0 ≤ weight i) (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp : p < pc G)
+    (J D : Hilbert V →L[ℂ] Hilbert V) (hJ : PositiveRealMatrix J) (hiJ : KernelInvariant Γ J)
+    (hD : ∀ x y, matrixCoeff D x y = (russoSlope G x y (p:ℝ):ℂ)) :
+    (clusterPairFunctional G p root weight (fun ω x y =>
+      clusterPairKernel G ω (indexedEdgeKernel G) x y * clusterPairKernel G ω (ennMatrixCoeff J) x y)).toReal =
+      (1-(p:ℝ))*(rootTrace root weight (J*D)).re := by
+  rw [mixedFunctional_eq_russoTrace G hconn hloc hqt root weight Γ hΓ htr hw p hp0 hp J D hJ hiJ hD]
+  apply ENNReal.toReal_ofReal
+  apply mul_nonneg (sub_nonneg.mpr p.2.2)
+  have hd := hJ.mul (russoOperator_positiveReal G p D hD)
+  simp only [rootTrace,Complex.re_sum,Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero]
+  exact Finset.sum_nonneg (fun i _ => mul_nonneg (hw i) (hd _ _).2)
+
+end MixedRussoTrace
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory Filter
+section RootTriangleRusso
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem rootTriangle_russo [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hw : ∀ i, 0 ≤ weight i) (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp : p < pc G)
+    (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T) :
+    (1-(p:ℝ))*deriv (rootTriangle G root weight) (p:ℝ) =
+      3*(clusterPairFunctional G p root weight (fun ω x y =>
+        clusterPairKernel G ω (indexedEdgeKernel G) x y * clusterPairKernel G ω (ennMatrixCoeff (T^2)) x y)).toReal := by
+  obtain ⟨D,hD⟩ := exists_russoOperator G hloc hqt p hp0 hp
+  have hi : KernelInvariant Γ T := fun a ha => hT.invariant G p T a (hΓ ha)
+  rw [(hasDerivAt_rootTriangle_trace G hloc hqt root weight Γ hΓ htr p hp0 hp T D hT hD).deriv,
+    mixedFunctional_toReal_eq_russoTrace G hconn hloc hqt root weight Γ hΓ htr hw p hp0 hp
+      (T^2) D ((hT.positiveRealMatrix G p T).pow 2) (hi.pow 2) hD]
+  ring
+
+theorem realTriangle_one_le [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (x : V) : 1 ≤ realTriangle G x (p:ℝ) := by
+  have hh := (summable_realTriangleTerm G hloc hqt p hp x).le_tsum (x,x)
+    (fun ab _ => by
+      simp only [realTriangleTerm,realTwoPoint_coe]
+      exact mul_nonneg (mul_nonneg (twoPoint_nonneg G p _ _) (twoPoint_nonneg G p _ _)) (twoPoint_nonneg G p _ _))
+  simpa only [realTriangleTerm,realTwoPoint_coe,twoPoint_self,mul_one,realTriangle] using hh
+
+theorem rootTriangle_one_le [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
+    (p : unitInterval) (hp : p < pc G) : 1 ≤ rootTriangle G root weight (p:ℝ) := by
+  calc
+    1 = ∑ i, weight i * 1 := by simpa using hw1.symm
+    _ ≤ _ := Finset.sum_le_sum (fun i _ => mul_le_mul_of_nonneg_left
+      (realTriangle_one_le G hloc hqt p hp (root i)) (hw i))
+
+end RootTriangleRusso
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+open scoped ENNReal
+section RealMixedTrace
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem critical_mixed_trace_bound_real
+    [Countable V] [Countable E] [Nonempty V] [Nonempty E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hconn : Connected G)
+    (hqt : QuasiTransitive G) (hh : 0 < hV G)
+    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
+    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
+    (hp₀ : 0 < (pc G:ℝ)) (hp₁ : (pc G:ℝ) < 1)
+    (hfin : ∀ i, ∀ᵐ ω ∂law (pc G), (cluster G ω (root i)).Finite)
+    (α ε : ℝ) (hα : 0 < α) (hα1 : α < 1/2) (hε : 0 < ε)
+    (hr : (1:ℝ)/2+ε-2*α < 0) :
+    ∃ A B : ℝ, 0 < A ∧ 0 < B ∧ ∀ (p : unitInterval),
+      (pc G:ℝ)/2 ≤ (p:ℝ) → p < pc G →
+      ∀ (T : Hilbert V →L[ℂ] Hilbert V), HasKernel G p T →
+      ∀ L : ℝ, 1 ≤ L →
+        (clusterPairFunctional G p root weight (fun ω x y =>
+          clusterPairKernel G ω (indexedEdgeKernel G) x y *
+          clusterPairKernel G ω (ennMatrixCoeff (T^2)) x y)).toReal ≤
+        2*(A*L^(1-α)*rootTriangle G root weight (p:ℝ)) +
+          (B*L^((1:ℝ)/2+ε-2*α))^((1:ℝ)/2) *
+          (‖T‖^3*rootTriangle G root weight (p:ℝ))^((1:ℝ)/2) := by
+  obtain ⟨A,B,hA,hB,hb⟩ := critical_mixed_trace_bound G hloc hconn hqt hh root weight Γ hΓ htr hcov hw hw1 hp₀ hp₁ hfin α ε hα hα1 hε hr
+  refine ⟨A,B,hA,hB,?_⟩
+  intro p hp hpq T hT L hL
+  have ht : 0 ≤ rootTriangle G root weight (p:ℝ) := (rootTriangle_one_le G hloc hqt root weight (fun i => (hw i).le) hw1 p hpq).trans' (by norm_num)
+  have ha : 0 ≤ A*L^(1-α)*rootTriangle G root weight (p:ℝ) := mul_nonneg (mul_nonneg hA.le (Real.rpow_nonneg (by linarith) _)) ht
+  have hc : 0 ≤ B*L^((1:ℝ)/2+ε-2*α) := mul_nonneg hB.le (Real.rpow_nonneg (by linarith) _)
+  have hn : 0 ≤ ‖T‖^3*rootTriangle G root weight (p:ℝ) := mul_nonneg (pow_nonneg (norm_nonneg _) _) ht
+  have hb' := hb p hp hpq.le T hT L hL
+  rw [←rootTriangle_eq_trace G p root weight T hT] at hb'
+  have htop : 2*ENNReal.ofReal (A*L^(1-α)*rootTriangle G root weight (p:ℝ)) +
+      (ENNReal.ofReal (B*L^((1:ℝ)/2+ε-2*α)))^((1:ℝ)/2) *
+      (ENNReal.ofReal (‖T‖^3*rootTriangle G root weight (p:ℝ)))^((1:ℝ)/2) ≠ ⊤ := by finiteness
+  have hh := ENNReal.toReal_mono htop hb'
+  rw [ENNReal.toReal_add (by finiteness) (by finiteness)] at hh
+  simpa only [ENNReal.toReal_mul,
+    ENNReal.toReal_ofNat,←ENNReal.toReal_rpow,ENNReal.toReal_ofReal ha,
+    ENNReal.toReal_ofReal hc,ENNReal.toReal_ofReal hn] using hh
+
+end RealMixedTrace
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section TraceNumerics
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem HasKernel.norm_le_susceptibility_bound [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (T : Hilbert V →L[ℂ] Hilbert V)
+    (hT : HasKernel G p T) (C : ℝ) (hC : 0 ≤ C) (hb : ∀ x, susceptibility G p x ≤ C) :
+    ‖T‖ ≤ C := by
+  have hr (x : V) (s : Finset V) : ∑ y ∈ s, twoPoint G p x y ≤ C :=
+    ((summable_twoPoint_subcritical G hloc hqt p hp x).sum_le_tsum s
+      (fun y _ => twoPoint_nonneg G p x y)).trans (hb x)
+  obtain ⟨S,hS,hK⟩ := exists_operator_of_row_column_bound (twoPoint G p) C hC (twoPoint_nonneg G p)
+    hr (fun y s => by simpa only [twoPoint_symm G p] using hr y s)
+  rwa [hT.unique hK]
+
+lemma sqrt_mixed_power (B L m a r : ℝ) (hB : 0 ≤ B) (hL : 0 ≤ L) (hm : 0 ≤ m) (ha : 0 ≤ a) :
+    (B*L^r)^((1:ℝ)/2) * (m^3*a)^((1:ℝ)/2) =
+      Real.sqrt B * L^(r/2) * m^((3:ℝ)/2) * Real.sqrt a := by
+  rw [Real.mul_rpow hB (Real.rpow_nonneg hL _), Real.mul_rpow (pow_nonneg hm _) ha,
+    ←Real.rpow_mul hL,←Real.rpow_natCast m 3,←Real.rpow_mul hm,
+    ←Real.sqrt_eq_rpow,←Real.sqrt_eq_rpow]
+  rw [show r*(1/2:ℝ) = r/2 by ring]
+  norm_num only [Nat.cast_ofNat]
+  ring
+
+lemma sqrt_derivative_estimate (p q A B L m a a' F : ℝ)
+    (hp : p ≤ q) (hq : q < 1) (hA : 0 ≤ A) (hB : 0 ≤ B)
+    (hL : 1 ≤ L) (hm : 0 ≤ m) (ha : 1 ≤ a)
+    (α r : ℝ) (hid : (1-p)*a' = 3*F)
+    (hF : F ≤ 2*(A*L^(1-α)*a) + (B*L^r)^((1:ℝ)/2)*(m^3*a)^((1:ℝ)/2)) :
+    a'/(2*Real.sqrt a) ≤
+      (3*(A+Real.sqrt B)/(1-q))*L^(1-α)*Real.sqrt a +
+      (3*(A+Real.sqrt B)/(1-q))*L^(r/2)*m^((3:ℝ)/2) := by
+  have ha0 : 0 < a := by linarith
+  have hs : 0 < Real.sqrt a := Real.sqrt_pos.mpr ha0
+  have hp1 : 0 < 1-p := by linarith
+  have hq1 : 0 < 1-q := by linarith
+  have he := Real.sq_sqrt ha0.le
+  have hL0 : 0 ≤ L := by linarith
+  rw [sqrt_mixed_power B L m a r hB hL0 hm ha0.le] at hF
+  have hu : 0 ≤ L^(1-α) := Real.rpow_nonneg hL0 _
+  have hv : 0 ≤ L^(r/2)*m^((3:ℝ)/2) := mul_nonneg (Real.rpow_nonneg hL0 _) (Real.rpow_nonneg hm _)
+  have hB0 := Real.sqrt_nonneg B
+  have hcoef : 3*A/(1-p) ≤ 3*(A+Real.sqrt B)/(1-q) := by
+    apply (div_le_div_iff₀ hp1 hq1).mpr
+    nlinarith
+  have hcoef2 : 3*Real.sqrt B/(2*(1-p)) ≤ 3*(A+Real.sqrt B)/(1-q) := by
+    apply (div_le_div_iff₀ (mul_pos (by norm_num) hp1) hq1).mpr
+    nlinarith
+  have hbase : a'/(2*Real.sqrt a) ≤ 3*A/(1-p)*L^(1-α)*Real.sqrt a +
+      (3*Real.sqrt B/(2*(1-p)))*(L^(r/2)*m^((3:ℝ)/2)) := by
+    apply (div_le_iff₀ (mul_pos (by norm_num) hs)).mpr
+    apply (mul_le_mul_iff_right₀ hp1).mp
+    have hex : (3*A/(1-p)*L^(1-α)*Real.sqrt a +
+        (3*Real.sqrt B/(2*(1-p)))*(L^(r/2)*m^((3:ℝ)/2))) * (2*Real.sqrt a) * (1-p) =
+        3*(2*(A*L^(1-α)*a) + Real.sqrt B*L^(r/2)*m^((3:ℝ)/2)*Real.sqrt a) := by
+      field_simp
+      ring_nf
+      rw [he]
+      ring
+    rw [hid,mul_comm (1-p) _,hex]
+    linarith
+  exact hbase.trans (add_le_add
+    (mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hcoef hu) hs.le)
+    (by simpa only [mul_assoc] using mul_le_mul_of_nonneg_right hcoef2 hv))
+
+end TraceNumerics
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section TraceSquareRoot
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def rootTriangleSqrt (G : BondGraph V E) {ι : Type*} [Fintype ι]
+    (root : ι → V) (weight : ι → ℝ) (t : ℝ) : ℝ := Real.sqrt (rootTriangle G root weight t)
+
+theorem hasDerivAt_rootTriangleSqrt [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
+    (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp : p < pc G) :
+    HasDerivAt (rootTriangleSqrt G root weight)
+      (deriv (rootTriangle G root weight) (p:ℝ) / (2*rootTriangleSqrt G root weight (p:ℝ))) (p:ℝ) := by
+  apply (hasDerivAt_rootTriangle_subcritical G hloc hqt root weight p hp0 hp).differentiableAt.hasDerivAt.sqrt
+  have hh := rootTriangle_one_le G hloc hqt root weight hw hw1 p hp
+  linarith
+
+ 
+theorem critical_rootTriangleSqrt_derivative
+    [Countable V] [Countable E] [Nonempty V] [Nonempty E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hconn : Connected G)
+    (hqt : QuasiTransitive G) (hh : 0 < hV G)
+    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
+    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
+    (hp₀ : 0 < (pc G:ℝ)) (hp₁ : (pc G:ℝ) < 1)
+    (hfin : ∀ i, ∀ᵐ ω ∂law (pc G), (cluster G ω (root i)).Finite)
+    (α ε : ℝ) (hα : 0 < α) (hα1 : α < 1/2) (hε : 0 < ε)
+    (hr : (1:ℝ)/2+ε-2*α < 0) :
+    ∃ C : ℝ, 0 < C ∧ ∀ (p : unitInterval),
+      (pc G:ℝ)/2 ≤ (p:ℝ) → p < pc G →
+      ∀ (T : Hilbert V →L[ℂ] Hilbert V), HasKernel G p T →
+      ∀ L : ℝ, 1 ≤ L →
+        deriv (rootTriangleSqrt G root weight) (p:ℝ) ≤
+        C*L^(1-α)*rootTriangleSqrt G root weight (p:ℝ) +
+          C*L^(((1:ℝ)/2+ε-2*α)/2)*‖T‖^((3:ℝ)/2) := by
+  obtain ⟨A,B,hA,hB,hb⟩ := critical_mixed_trace_bound_real G hloc hconn hqt hh root weight Γ hΓ htr hcov hw hw1 hp₀ hp₁ hfin α ε hα hα1 hε hr
+  refine ⟨3*(A+Real.sqrt B)/(1-(pc G:ℝ)),div_pos (mul_pos (by norm_num)
+    (add_pos_of_pos_of_nonneg hA (Real.sqrt_nonneg B))) (sub_pos.mpr hp₁),?_⟩
+  intro p hp hpq T hT L hL
+  have hp0 : 0 < (p:ℝ) := lt_of_lt_of_le (half_pos hp₀) hp
+  rw [(hasDerivAt_rootTriangleSqrt G hloc hqt root weight (fun i => (hw i).le) hw1 p hp0 hpq).deriv]
+  exact sqrt_derivative_estimate p (pc G) A B L ‖T‖ (rootTriangle G root weight p)
+    (deriv (rootTriangle G root weight) p) _ hpq.le hp₁ hA.le hB.le hL (norm_nonneg _)
+    (rootTriangle_one_le G hloc hqt root weight (fun i => (hw i).le) hw1 p hpq) α _
+    (rootTriangle_russo G hconn hloc hqt root weight Γ hΓ htr (fun i => (hw i).le) p hp0 hpq T hT)
+    (hb p hp hpq T hT L hL)
+
+end TraceSquareRoot
+end Percolation
+
+ 
+namespace Percolation
+section TraceExponents
+
+lemma trace_exponent_choices (η : ℝ) (hη : 0 < η) :
+    ∃ α ε ρ lambda : ℝ, 0 < α ∧ α < 1/2 ∧ 0 < ε ∧ 0 < ρ ∧ 0 < lambda ∧
+      (1:ℝ)/2+ε-2*α < 0 ∧ lambda*(1-α) < 1 ∧
+      lambda*(((1:ℝ)/2+ε-2*α)/2) + (3:ℝ)/2*(1+ρ) < 1+η/2 := by
+  let ζ := min (1/2:ℝ) η
+  let d := min (ζ/16) (η/32)
+  have hz : 0 < ζ := lt_min (by norm_num) hη
+  have hz1 : ζ ≤ 1/2 := min_le_left _ _
+  have hze : ζ ≤ η := min_le_right _ _
+  have hd : 0 < d := lt_min (div_pos hz (by norm_num)) (div_pos hη (by norm_num))
+  have hdz : d ≤ ζ/16 := min_le_left _ _
+  have hde : d ≤ η/32 := min_le_right _ _
+  have hzd : 0 ≤ ζ*d := mul_nonneg hz.le hd.le
+  refine ⟨1/2-d,d,d,2-ζ,?_,?_,hd,hd,?_,?_,?_,?_⟩ <;> nlinarith
+
+lemma trace_scale_large_term (δ D lambda r ρ : ℝ) (hδ : 0 < δ) (hD : 0 ≤ D) :
+    (δ^(-lambda))^(r/2)*(D*δ^(-1-ρ))^((3:ℝ)/2) =
+      D^((3:ℝ)/2)*δ^(-(lambda*(r/2)+(3:ℝ)/2*(1+ρ))) := by
+  rw [Real.mul_rpow hD (Real.rpow_nonneg hδ.le _),←Real.rpow_mul hδ.le,←Real.rpow_mul hδ.le]
+  rw [show δ^(-lambda*(r/2))*(D^((3:ℝ)/2)*δ^((-1-ρ)*((3:ℝ)/2))) =
+    D^((3:ℝ)/2)*(δ^(-lambda*(r/2))*δ^((-1-ρ)*((3:ℝ)/2))) by ring,
+    ←Real.rpow_add hδ]
+  congr 2
+  ring
+
+lemma trace_scale_derivative (δ D C lambda r ρ α b b' m : ℝ)
+    (hδ : 0 < δ) (hD : 0 ≤ D) (hC : 0 ≤ C)
+    (hb : 0 ≤ b) (hm : 0 ≤ m) (hmD : m ≤ D*δ^(-1-ρ))
+    (hd : b' ≤ C*(δ^(-lambda))^(1-α)*b + C*(δ^(-lambda))^(r/2)*m^((3:ℝ)/2)) :
+    b' ≤ (C*(1+D^((3:ℝ)/2)))*δ^(-(lambda*(1-α)))*b +
+      (C*(1+D^((3:ℝ)/2)))*δ^(-(lambda*(r/2)+(3:ℝ)/2*(1+ρ))) := by
+  have hs := Real.rpow_le_rpow hm hmD (by norm_num : (0:ℝ) ≤ 3/2)
+  have he : (δ^(-lambda))^(1-α) = δ^(-(lambda*(1-α))) := by rw [←Real.rpow_mul hδ.le]; congr 1; ring
+  have hc1 : C ≤ C*(1+D^((3:ℝ)/2)) := by nlinarith [Real.rpow_nonneg hD ((3:ℝ)/2)]
+  have hc2 : C*D^((3:ℝ)/2) ≤ C*(1+D^((3:ℝ)/2)) := by nlinarith
+  calc
+    b' ≤ C*(δ^(-lambda))^(1-α)*b + C*(δ^(-lambda))^(r/2)*m^((3:ℝ)/2) := hd
+    _ ≤ C*(δ^(-lambda))^(1-α)*b + C*(δ^(-lambda))^(r/2)*(D*δ^(-1-ρ))^((3:ℝ)/2) :=
+      add_le_add le_rfl (mul_le_mul_of_nonneg_left hs (mul_nonneg hC (Real.rpow_nonneg (Real.rpow_nonneg hδ.le _) _)))
+    _ = C*δ^(-(lambda*(1-α)))*b + (C*D^((3:ℝ)/2))*δ^(-(lambda*(r/2)+(3:ℝ)/2*(1+ρ))) := by
+      rw [he,mul_assoc C ((δ^(-lambda))^(r/2)),trace_scale_large_term δ D lambda r ρ hδ hD]; ring
+    _ ≤ _ := add_le_add (mul_le_mul_of_nonneg_right
+      (mul_le_mul_of_nonneg_right hc1 (Real.rpow_nonneg hδ.le _)) hb)
+      (mul_le_mul_of_nonneg_right hc2 (Real.rpow_nonneg hδ.le _))
+
+end TraceExponents
+end Percolation
+
+ 
+namespace Percolation
+section IntegratingFactor
+
+lemma hasDerivAt_gap_rpow (q r t : ℝ) (ht : t < q) :
+    HasDerivAt (fun x : ℝ => (q-x)^r) (-r*(q-t)^(r-1)) t := by
+  convert! ((hasDerivAt_const t q).sub (hasDerivAt_id t)).rpow_const
+    (Or.inl (ne_of_gt (sub_pos.mpr ht))) using 1
+  simp only [Pi.sub_apply, id_eq]
+  ring
+
+noncomputable def traceIntegratingFactor (q C A t : ℝ) : ℝ :=
+  Real.exp (C/(1-A)*(q-t)^(1-A))
+
+lemma hasDerivAt_traceIntegratingFactor (q C A t : ℝ) (hA : A < 1) (ht : t < q) :
+    HasDerivAt (traceIntegratingFactor q C A)
+      (-C*(q-t)^(-A)*traceIntegratingFactor q C A t) t := by
+  have hd := ((hasDerivAt_gap_rpow q (1-A) t ht).const_mul (C/(1-A))).exp
+  convert! hd using 1
+  unfold traceIntegratingFactor
+  rw [show 1-A-1 = -A by ring]
+  field_simp [ne_of_gt (sub_pos.mpr hA)]
+
+lemma traceIntegratingFactor_bounds (q C A t : ℝ) (hC : 0 ≤ C) (hA : A < 1)
+    (ht : t < q) (hd : q-t ≤ 1) :
+    1 ≤ traceIntegratingFactor q C A t ∧
+      traceIntegratingFactor q C A t ≤ Real.exp (C/(1-A)) := by
+  have hk : 0 ≤ C/(1-A) := div_nonneg hC (sub_pos.mpr hA).le
+  have hr : 0 ≤ (q-t)^(1-A) := Real.rpow_nonneg (sub_pos.mpr ht).le _
+  constructor
+  · exact Real.one_le_exp (mul_nonneg hk hr)
+  · apply Real.exp_le_exp.mpr
+    exact mul_le_of_le_one_right hk (Real.rpow_le_one (sub_pos.mpr ht).le hd (sub_pos.mpr hA).le)
+
+lemma hasDerivAt_traceComparison (f : ℝ → ℝ) (q C A κ t : ℝ)
+    (hA : A < 1) (hκ : 0 < κ) (ht : t < q) (hf : DifferentiableAt ℝ f t) :
+    HasDerivAt (fun x => traceIntegratingFactor q C A x*f x -
+      (C*Real.exp (C/(1-A))/κ)*(q-x)^(-κ))
+      (traceIntegratingFactor q C A t*(deriv f t-C*(q-t)^(-A)*f t) -
+        C*Real.exp (C/(1-A))*(q-t)^(-κ-1)) t := by
+  have hh := ((hasDerivAt_traceIntegratingFactor q C A t hA ht).mul hf.hasDerivAt).sub
+    ((hasDerivAt_gap_rpow q (-κ) t ht).const_mul (C*Real.exp (C/(1-A))/κ))
+  convert! hh using 1
+  field_simp [ne_of_gt hκ]
+  ring
+
+ 
+
+theorem singular_ode_subpower (f : ℝ → ℝ) (q s C A B κ : ℝ)
+    (hs : s < q) (hqs : q-s ≤ 1) (hC : 0 < C) (hA : A < 1)
+    (hκ : 0 < κ) (hB : B ≤ 1+κ)
+    (hf : ∀ t ∈ Set.Ico s q, DifferentiableAt ℝ f t)
+    (hn : ∀ t ∈ Set.Ico s q, 0 ≤ f t)
+    (hd : ∀ t ∈ Set.Ico s q,
+      deriv f t ≤ C*(q-t)^(-A)*f t + C*(q-t)^(-B)) :
+    ∃ K : ℝ, 0 < K ∧ ∀ t ∈ Set.Ico s q, f t ≤ K*(q-t)^(-κ) := by
+  let M := Real.exp (C/(1-A))
+  let Q := C*M/κ
+  let g := fun t => traceIntegratingFactor q C A t*f t - Q*(q-t)^(-κ)
+  have hM : 0 < M := Real.exp_pos _
+  have hQ : 0 < Q := div_pos (mul_pos hC hM) hκ
+  have hgs (t : ℝ) (ht : t ∈ Set.Ico s q) :
+      HasDerivAt g (traceIntegratingFactor q C A t*(deriv f t-C*(q-t)^(-A)*f t) -
+        C*M*(q-t)^(-κ-1)) t := hasDerivAt_traceComparison f q C A κ t hA hκ ht.2 (hf t ht)
+  have hgn (t : ℝ) (ht : t ∈ Set.Ico s q) : deriv g t ≤ 0 := by
+    have htq : 0 < q-t := sub_pos.mpr ht.2
+    have ht1 : q-t ≤ 1 := by linarith [ht.1]
+    obtain ⟨hw,hwM⟩ := traceIntegratingFactor_bounds q C A t hC.le hA ht.2 ht1
+    have hpow : (q-t)^(-B) ≤ (q-t)^(-κ-1) :=
+      Real.rpow_le_rpow_of_exponent_ge htq ht1 (by linarith)
+    have hmul := mul_le_mul_of_nonneg_left (hd t ht) (show 0 ≤ traceIntegratingFactor q C A t by linarith)
+    have h₂ := mul_le_mul_of_nonneg_right hwM (mul_nonneg hC.le (Real.rpow_nonneg htq.le (-B)))
+    have h₃ := mul_le_mul_of_nonneg_left hpow (mul_nonneg hC.le hM.le)
+    rw [(hgs t ht).deriv]
+    dsimp only [M] at hM ⊢
+    dsimp only [M] at h₃
+    nlinarith
+  have hgmono : AntitoneOn g (Set.Ico s q) := by
+    apply antitoneOn_of_deriv_nonpos (convex_Ico s q)
+    · intro t ht
+      exact (hgs t ht).continuousAt.continuousWithinAt
+    · intro t ht
+      exact (hgs t (interior_subset ht)).differentiableAt.differentiableWithinAt
+    · intro t ht
+      exact hgn t (interior_subset ht)
+  have hsI : s ∈ Set.Ico s q := ⟨le_rfl,hs⟩
+  refine ⟨M*f s+Q,add_pos_of_nonneg_of_pos (mul_nonneg hM.le (hn s hsI)) hQ,?_⟩
+  intro t ht
+  have htq : 0 < q-t := sub_pos.mpr ht.2
+  have ht1 : q-t ≤ 1 := by linarith [ht.1]
+  obtain ⟨hwt,_⟩ := traceIntegratingFactor_bounds q C A t hC.le hA ht.2 ht1
+  obtain ⟨_,hws⟩ := traceIntegratingFactor_bounds q C A s hC.le hA hs hqs
+  have hmono := hgmono hsI ht ht.1
+  have hneg : 0 ≤ Q*(q-s)^(-κ) := mul_nonneg hQ.le (Real.rpow_nonneg (sub_pos.mpr hs).le _)
+  have hft := mul_le_mul_of_nonneg_right hwt (hn t ht)
+  have hfs := mul_le_mul_of_nonneg_right hws (hn s hsI)
+  have hpow : 1 ≤ (q-t)^(-κ) := Real.one_le_rpow_of_pos_of_le_one_of_nonpos htq ht1 (neg_nonpos.mpr hκ.le)
+  have hscale := mul_le_mul_of_nonneg_left hpow (mul_nonneg hM.le (hn s hsI))
+  dsimp only [g] at hmono
+  dsimp only [M] at hscale ⊢
+  nlinarith
+
+end IntegratingFactor
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section SubpowerTrace
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem critical_rootTriangle_subpower
+    [Countable V] [Countable E] [Nonempty V] [Nonempty E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hconn : Connected G)
+    (hqt : QuasiTransitive G) (hh : 0 < hV G)
+    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
+    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
+    (hp₀ : 0 < (pc G:ℝ)) (hp₁ : (pc G:ℝ) < 1)
+    (hfin : ∀ i, ∀ᵐ ω ∂law (pc G), (cluster G ω (root i)).Finite)
+    (η : ℝ) (hη : 0 < η) :
+    ∃ K : ℝ, 0 < K ∧ ∀ (p : unitInterval), (pc G:ℝ)/2 ≤ (p:ℝ) → p < pc G →
+      rootTriangle G root weight (p:ℝ) ≤ K*((pc G:ℝ)-(p:ℝ))^(-η) := by
+  obtain ⟨α,ε,ρ,lambda,hα,hα1,hε,hρ,hlambda,hr,hA,hB⟩ := trace_exponent_choices η hη
+  obtain ⟨C₀,hC₀,hder⟩ := critical_rootTriangleSqrt_derivative G hloc hconn hqt hh root weight Γ hΓ htr hcov hw hw1 hp₀ hp₁ hfin α ε hα hα1 hε hr
+  obtain ⟨D,hD,hχ⟩ := almost_linear_susceptibility G hloc hconn hqt hh root weight Γ hΓ htr hcov hw hw1 hp₀ hp₁ hfin ρ hρ
+  let C := C₀*(1+D^((3:ℝ)/2))
+  let A := lambda*(1-α)
+  let B := lambda*(((1:ℝ)/2+ε-2*α)/2)+(3:ℝ)/2*(1+ρ)
+  have hC : 0 < C := mul_pos hC₀ (by linarith [Real.rpow_nonneg hD.le ((3:ℝ)/2)])
+  have hunit (t : ℝ) (ht : t ∈ Set.Ico ((pc G:ℝ)/2) (pc G:ℝ)) : t ∈ unitInterval :=
+    ⟨(half_pos hp₀).le.trans ht.1,ht.2.le.trans (pc G).2.2⟩
+  have hdf (t : ℝ) (ht : t ∈ Set.Ico ((pc G:ℝ)/2) (pc G:ℝ)) :
+      DifferentiableAt ℝ (rootTriangleSqrt G root weight) t := by
+    exact (hasDerivAt_rootTriangleSqrt G hloc hqt root weight (fun i => (hw i).le) hw1
+      ⟨t,hunit t ht⟩ ((half_pos hp₀).trans_le ht.1) ht.2).differentiableAt
+  have hod (t : ℝ) (ht : t ∈ Set.Ico ((pc G:ℝ)/2) (pc G:ℝ)) :
+      deriv (rootTriangleSqrt G root weight) t ≤ C*((pc G:ℝ)-t)^(-A)*rootTriangleSqrt G root weight t +
+        C*((pc G:ℝ)-t)^(-B) := by
+    let p : unitInterval := ⟨t,hunit t ht⟩
+    have hδ : 0 < (pc G:ℝ)-t := sub_pos.mpr ht.2
+    have hδ1 : (pc G:ℝ)-t ≤ 1 := by linarith [(pc G).2.2,ht.1]
+    obtain ⟨T,hT⟩ := operatorBounded_of_lt_pc G hloc hqt p ht.2
+    have hm := hT.norm_le_susceptibility_bound G hloc hqt p ht.2 T (D*((pc G:ℝ)-t)^(-1-ρ))
+      (mul_nonneg hD.le (Real.rpow_nonneg hδ.le _)) (hχ p ht.2)
+    have hL : 1 ≤ (((pc G:ℝ)-t)^(-lambda)) :=
+      Real.one_le_rpow_of_pos_of_le_one_of_nonpos hδ hδ1 (neg_nonpos.mpr hlambda.le)
+    exact trace_scale_derivative ((pc G:ℝ)-t) D C₀ lambda ((1:ℝ)/2+ε-2*α) ρ α
+      (rootTriangleSqrt G root weight t) (deriv (rootTriangleSqrt G root weight) t) ‖T‖
+      hδ hD.le hC₀.le (Real.sqrt_nonneg _) (norm_nonneg _) hm
+      (hder p ht.1 ht.2 T hT (((pc G:ℝ)-t)^(-lambda)) hL)
+  obtain ⟨K,hK,hb⟩ := singular_ode_subpower (rootTriangleSqrt G root weight) (pc G) ((pc G:ℝ)/2)
+    C A B (η/2) (by linarith) (by linarith [(pc G).2.2]) hC hA (half_pos hη) hB.le
+    hdf (fun _ _ => Real.sqrt_nonneg _) hod
+  refine ⟨K^2,sq_pos_of_pos hK,?_⟩
+  intro p hp hpq
+  have ha : 0 ≤ rootTriangle G root weight (p:ℝ) :=
+    le_trans (by norm_num) (rootTriangle_one_le G hloc hqt root weight (fun i => (hw i).le) hw1 p hpq)
+  have hb' := pow_le_pow_left₀ (Real.sqrt_nonneg (rootTriangle G root weight (p:ℝ))) (hb p ⟨hp,hpq⟩) 2
+  change (Real.sqrt (rootTriangle G root weight (p:ℝ)))^2 ≤ _ at hb'
+  rw [Real.sq_sqrt ha,mul_pow] at hb'
+  have he : (((pc G:ℝ)-(p:ℝ))^(-(η/2)))^2 = ((pc G:ℝ)-(p:ℝ))^(-η) := by
+    rw [←Real.rpow_natCast _ 2,←Real.rpow_mul (sub_pos.mpr (show (p:ℝ)<(pc G:ℝ) from hpq)).le]
+    congr 1
+    norm_num
+  rw [he] at hb'
+  exact hb'
+
+end SubpowerTrace
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section QuantitativeSprinkling
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+lemma riccati_step (a M n x : ℝ) (ha : 0 ≤ a) (hM : 0 ≤ M) (_hn : 0 ≤ n)
+    (_hx : 0 ≤ x) (hb : x ≤ M/(1-a*n*M)) (hh : a*(n+1)*M < 1) :
+    a*x < 1 ∧ x/(1-a*x) ≤ M/(1-a*(n+1)*M) := by
+  have haM : 0 ≤ a*M := mul_nonneg ha hM
+  have hp : 0 < 1-a*n*M := by nlinarith
+  have hb' := (le_div_iff₀ hp).mp hb
+  have hs : a*(M/(1-a*n*M)) < 1 := by
+    rw [←mul_div_assoc,div_lt_one hp]
+    nlinarith
+  have hax : a*x < 1 := (mul_le_mul_of_nonneg_left hb ha).trans_lt hs
+  refine ⟨hax,?_⟩
+  apply (div_le_div_iff₀ (sub_pos.mpr hax) (sub_pos.mpr hh)).mpr
+  nlinarith
+
+ 
+
+theorem sprinkling_iterated_bound [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (q : unitInterval) (m D : ℕ)
+    (hD : ∀ x, degree G x ≤ D) (A : Hilbert V →L[ℂ] Hilbert V)
+    (hA : HasKernel G (orParameter q m) A) (n : ℕ)
+    (hn : (q:ℝ)*(2*D)*(n:ℝ)*‖A‖ < 1) :
+    ∃ B : Hilbert V →L[ℂ] Hilbert V, HasKernel G (orParameter q (m+n)) B ∧
+      ‖B‖ ≤ ‖A‖/(1-(q:ℝ)*(2*D)*(n:ℝ)*‖A‖) := by
+  induction n with
+  | zero => exact ⟨A,by simpa using hA,by simp⟩
+  | succ n ih =>
+    have ha : 0 ≤ (q:ℝ)*(2*D) := mul_nonneg q.2.1 (by positivity)
+    have hnext : (q:ℝ)*(2*D)*((n:ℝ)+1)*‖A‖ < 1 := by simpa using hn
+    have hprev : (q:ℝ)*(2*D)*(n:ℝ)*‖A‖ < 1 := by
+      nlinarith [mul_nonneg ha (norm_nonneg A)]
+    obtain ⟨B,hB,hBn⟩ := ih hprev
+    obtain ⟨hs,hbound⟩ := riccati_step ((q:ℝ)*(2*D)) ‖A‖ n ‖B‖ ha (norm_nonneg _) (Nat.cast_nonneg _) (norm_nonneg _) hBn hnext
+    obtain ⟨C,hC,hCn⟩ := sprinkling_bound G hloc q (m+n) D hD B hB hs
+    exact ⟨C,by simpa [Nat.add_assoc] using hC,by simpa using hCn.trans hbound⟩
+
+lemma closed_power_bound (q : unitInterval) (n : ℕ) :
+    (1-(q:ℝ))^n * (1+(n:ℝ)*(q:ℝ)) ≤ 1 := by
+  induction n with
+  | zero => simp
+  | succ n ih =>
+    have hp : 0 ≤ (1-(q:ℝ))^n := pow_nonneg (sub_nonneg.mpr q.2.2) _
+    have hn : 0 ≤ (n:ℝ)*(q:ℝ)^2 := mul_nonneg (Nat.cast_nonneg _) (sq_nonneg _)
+    rw [pow_succ,Nat.cast_add,Nat.cast_one]
+    nlinarith [mul_nonneg hp hn,mul_nonneg hp (sq_nonneg (q:ℝ))]
+
+lemma orParameter_gap_bound (p q : unitInterval) (m n : ℕ) (hm : orParameter q m = p) :
+    (1+(n:ℝ)*(q:ℝ))*(1-(orParameter q (m+n):ℝ)) ≤ 1-(p:ℝ) := by
+  have hh := mul_le_mul_of_nonneg_left (closed_power_bound q n) (sub_nonneg.mpr p.2.2)
+  have he : 1-(p:ℝ) = (1-(q:ℝ))^m := by rw [←hm]; simp [orParameter]
+  rw [he] at hh ⊢
+  change (1+(n:ℝ)*(q:ℝ))*(1-(1-(1-(q:ℝ))^(m+n))) ≤ _
+  rw [sub_sub_cancel,pow_add]
+  nlinarith
+
+end QuantitativeSprinkling
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section QuantitativeOpenness
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem operatorBounded_of_quantitative_gap [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (D : ℕ)
+    (hD : ∀ x, degree G x ≤ D) (p r : unitInterval)
+    (hp : 0 < p) (hpr : p < r) (hr : r < 1)
+    (A : Hilbert V →L[ℂ] Hilbert V) (hA : HasKernel G p A)
+    (hsmall : ((r:ℝ)-(p:ℝ))*(4*D)*‖A‖ < 1-(r:ℝ)) : OperatorBounded G r := by
+  let a := ((r:ℝ)-(p:ℝ))/(1-(r:ℝ))
+  have hden : 0 < 1-(r:ℝ) := sub_pos.mpr hr
+  have ha : 0 < a := div_pos (sub_pos.mpr hpr) hden
+  obtain ⟨m,q,hq,hqp⟩ := exists_orParameter_small p (hpr.trans hr) (min a 1)
+    (lt_min ha zero_lt_one) (min_le_right _ _)
+  have hq0 : (0:ℝ) < q := by
+    by_contra hn
+    have hqe : q = 0 := Subtype.ext ((le_of_not_gt hn).antisymm q.2.1)
+    have hpe : p = 0 := by rw [←hqp,hqe]; apply Subtype.ext; simp [orParameter]
+    exact (ne_of_gt hp) hpe
+  let n := ⌈a/(q:ℝ)⌉₊
+  have hna : a ≤ (n:ℝ)*(q:ℝ) := (div_le_iff₀ hq0).mp (Nat.le_ceil _)
+  have han : (n:ℝ)*(q:ℝ) < a+(q:ℝ) := by
+    have hn := Nat.ceil_lt_add_one (div_nonneg ha.le hq0.le)
+    have hh := mul_lt_mul_of_pos_right hn hq0
+    dsimp only [n] at *
+    field_simp at hh
+    nlinarith
+  have hn2 : (n:ℝ)*(q:ℝ) < 2*a := by linarith [hq.trans (min_le_left _ _)]
+  have hbase : (2*a)*(2*D)*‖A‖ < 1 := by
+    rw [show (2*a)*(2*D)*‖A‖ = (((r:ℝ)-(p:ℝ))*(4*D)*‖A‖)/(1-(r:ℝ)) by dsimp [a]; ring]
+    exact (div_lt_one hden).mpr hsmall
+  have hnsmall : (q:ℝ)*(2*D)*(n:ℝ)*‖A‖ < 1 := by
+    have hs := mul_le_mul_of_nonneg_right hn2.le (show 0 ≤ (2*(D:ℝ))*‖A‖ by positivity)
+    nlinarith
+  obtain ⟨B,hB,_⟩ := sprinkling_iterated_bound G hloc q m D hD A (hqp ▸ hA) n hnsmall
+  apply operatorBounded_of_le G (q := orParameter q (m+n)) ?_ ⟨B,hB⟩
+  have hgap := orParameter_gap_bound p q m n hqp
+  have hae : a*(1-(r:ℝ)) = (r:ℝ)-(p:ℝ) := div_mul_cancel₀ _ (ne_of_gt hden)
+  have hs := mul_le_mul_of_nonneg_right hna hden.le
+  change (r:ℝ) ≤ (orParameter q (m+n):ℝ)
+  nlinarith [mul_nonneg (Nat.cast_nonneg n) q.2.1]
+
+ 
+
+theorem norm_gap_lower_of_endpoint_unbounded [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (D : ℕ)
+    (hD : ∀ x, degree G x ≤ D) (p r : unitInterval)
+    (hp : 0 < p) (hpr : p < r) (hr : r < 1)
+    (A : Hilbert V →L[ℂ] Hilbert V) (hA : HasKernel G p A)
+    (hu : ¬OperatorBounded G r) :
+    1-(r:ℝ) ≤ ((r:ℝ)-(p:ℝ))*(4*D)*‖A‖ := by
+  by_contra h
+  exact hu (operatorBounded_of_quantitative_gap G hloc D hD p r hp hpr hr A hA (lt_of_not_ge h))
+
+end QuantitativeOpenness
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section NormRatio
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+noncomputable def maxSusceptibility (G : BondGraph V E) (p : unitInterval) : ℝ :=
+  sSup (Set.range (susceptibility G p))
+
+theorem susceptibility_bddAbove [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) : BddAbove (Set.range (susceptibility G p)) := by
+  obtain ⟨C,_,hC⟩ := subcritical_row_bound G hloc hqt p hp
+  refine ⟨C,?_⟩
+  rintro _ ⟨x,rfl⟩
+  exact (summable_twoPoint_subcritical G hloc hqt p hp x).tsum_le_of_sum_le (hC x)
+
+theorem susceptibility_le_max [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (x : V) :
+    susceptibility G p x ≤ maxSusceptibility G p :=
+  le_csSup (susceptibility_bddAbove G hloc hqt p hp) (Set.mem_range_self x)
+
+theorem maxSusceptibility_le [Nonempty V] (G : BondGraph V E) (p : unitInterval)
+    (C : ℝ) (hC : ∀ x, susceptibility G p x ≤ C) : maxSusceptibility G p ≤ C := by
+  apply csSup_le (Set.range_nonempty _)
+  rintro _ ⟨x,rfl⟩
+  exact hC x
+
+theorem maxSusceptibility_one_le [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) : 1 ≤ maxSusceptibility G p := by
+  let x : V := Classical.choice inferInstance
+  exact (susceptibility_one_le G hloc hqt p hp x).trans (susceptibility_le_max G hloc hqt p hp x)
+
+theorem HasKernel.norm_le_maxSusceptibility [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (p : unitInterval) (hp : p < pc G) (T : Hilbert V →L[ℂ] Hilbert V)
+    (hT : HasKernel G p T) : ‖T‖ ≤ maxSusceptibility G p :=
+  hT.norm_le_susceptibility_bound G hloc hqt p hp T _
+    (le_trans zero_le_one (maxSusceptibility_one_le G hloc hqt p hp))
+    (susceptibility_le_max G hloc hqt p hp)
+
+theorem HasKernel.norm_one_le [Countable E] [Nonempty V]
+    (G : BondGraph V E) (p : unitInterval) (T : Hilbert V →L[ℂ] Hilbert V)
+    (hT : HasKernel G p T) : 1 ≤ ‖T‖ := by
+  let x : V := Classical.choice inferInstance
+  have hd : twoPoint G p x x = 1 := twoPoint_self G p x
+  have hh := (lp.norm_apply_le_norm (by norm_num : (2:ENNReal) ≠ 0) (T (lp.single 2 x (1:ℂ))) x).trans (T.le_opNorm (lp.single 2 x (1:ℂ)))
+  simpa [hT x x,hd,lp.norm_single] using hh
+
+end NormRatio
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section SourceNormRatio
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem critical_norm_inverse_lower [Countable V] [Countable E] [Nonempty V]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (hp₁ : (pc G:ℝ) < 1) (hu : ¬OperatorBounded G (pc G)) :
+    ∃ c : ℝ, 0 < c ∧ ∀ (p : unitInterval), 0 < p → p < pc G →
+      ∀ T : Hilbert V →L[ℂ] Hilbert V, HasKernel G p T →
+        c/((pc G:ℝ)-(p:ℝ)) ≤ ‖T‖ := by
+  obtain ⟨D,hDp,hD⟩ := exists_degree_bound G hqt
+  have hDpos : 0 < (4*(D:ℝ)) := by positivity
+  refine ⟨(1-(pc G:ℝ))/(4*D),div_pos (sub_pos.mpr hp₁) hDpos,?_⟩
+  intro p hp hpq T hT
+  have hh := norm_gap_lower_of_endpoint_unbounded G hloc D hD p (pc G) hp hpq hp₁ T hT hu
+  apply (div_le_iff₀ (sub_pos.mpr (show (p:ℝ)<(pc G:ℝ) from hpq))).mpr
+  apply (div_le_iff₀ hDpos).mpr
+  nlinarith
+
+ 
+
+theorem critical_normRatio_subpower
+    [Countable V] [Countable E] [Nonempty V] [Nonempty E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (hconn : Connected G)
+    (hqt : QuasiTransitive G) (hh : 0 < hV G)
+    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
+    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
+    (hp₀ : 0 < (pc G:ℝ)) (hp₁ : (pc G:ℝ) < 1)
+    (hfin : ∀ i, ∀ᵐ ω ∂law (pc G), (cluster G ω (root i)).Finite)
+    (hu : ¬OperatorBounded G (pc G)) (η : ℝ) (hη : 0 < η) :
+    ∃ K : ℝ, 0 < K ∧ ∀ (p : unitInterval), (pc G:ℝ)/2 ≤ (p:ℝ) → p < pc G →
+      ∀ T : Hilbert V →L[ℂ] Hilbert V, HasKernel G p T →
+      1 ≤ maxSusceptibility G p/‖T‖ ∧
+      maxSusceptibility G p/‖T‖ ≤ K*((pc G:ℝ)-(p:ℝ))^(-η) := by
+  obtain ⟨c,hc,hm⟩ := critical_norm_inverse_lower G hloc hqt hp₁ hu
+  obtain ⟨D,hD,hχ⟩ := almost_linear_susceptibility G hloc hconn hqt hh root weight Γ hΓ htr hcov hw hw1 hp₀ hp₁ hfin η hη
+  refine ⟨D/c,div_pos hD hc,?_⟩
+  intro p hp hpq T hT
+  have hδ : 0 < (pc G:ℝ)-(p:ℝ) := sub_pos.mpr hpq
+  have hnorm : 0 < ‖T‖ := zero_lt_one.trans_le (hT.norm_one_le G p T)
+  have hM := hm p ((half_pos hp₀).trans_le hp) hpq T hT
+  have hχb := maxSusceptibility_le G p _ (hχ p hpq)
+  have he : ((pc G:ℝ)-(p:ℝ))^(-1-η)*((pc G:ℝ)-(p:ℝ)) = ((pc G:ℝ)-(p:ℝ))^(-η) := by
+    calc
+      _ = ((pc G:ℝ)-(p:ℝ))^(-1-η)*((pc G:ℝ)-(p:ℝ))^(1:ℝ) := by rw [Real.rpow_one]
+      _ = _ := by rw [←Real.rpow_add hδ]; congr 1; ring
+  refine ⟨(one_le_div hnorm).mpr (hT.norm_le_maxSusceptibility G hloc hqt p hpq T),?_⟩
+  calc
+    maxSusceptibility G p/‖T‖ ≤ (D*((pc G:ℝ)-(p:ℝ))^(-1-η))/(c/((pc G:ℝ)-(p:ℝ))) :=
+      div_le_div₀ (mul_nonneg hD.le (Real.rpow_nonneg hδ.le _)) hχb (div_pos hc hδ) hM
+    _ = (D/c)*((pc G:ℝ)-(p:ℝ))^(-η) := by rw [←he]; field_simp
+
+end SourceNormRatio
+end Percolation
+
+ 
+namespace Percolation
+universe u
+section ChronologicalErasure
+open List
+variable {V : Type u}
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+inductive HoldingWalk (H : SimpleGraph V) : ℕ → V → V → Type u
+  | nil (x : V) : HoldingWalk H 0 x x
+  | hold {k : ℕ} {x y : V} (w : HoldingWalk H k x y) : HoldingWalk H (k+1) x y
+  | step {k : ℕ} {x z y : V} (h : H.Adj x z) (w : HoldingWalk H k z y) :
+      HoldingWalk H (k+1) x y
+
+namespace HoldingWalk
+variable {H : SimpleGraph V} {k : ℕ} {x y : V}
+
+def withoutHolds : {k : ℕ} → {x y : V} → HoldingWalk H k x y → H.Walk x y
+  | _,_,_,.nil _ => .nil
+  | _,_,_,.hold w => w.withoutHolds
+  | _,_,_,.step h w => .cons h w.withoutHolds
+
+def vertices : {k : ℕ} → {x y : V} → HoldingWalk H k x y → List V
+  | _,x,_,.nil _ => [x]
+  | _,x,_,.hold w => x :: w.vertices
+  | _,x,_,.step _ w => x :: w.vertices
+
+noncomputable def mass (P : V → V → ℝ) : {k : ℕ} → {x y : V} → HoldingWalk H k x y → ℝ
+  | _,_,_,.nil _ => 1
+  | _,x,_,.hold w => P x x * w.mass P
+  | _,x,_,.step (z := z) _ w => P x z * w.mass P
+
+theorem mass_nonneg (P : V → V → ℝ) (hP : ∀ a b, 0 ≤ P a b) (w : HoldingWalk H k x y) :
+    0 ≤ w.mass P := by
+  induction w with
+  | nil => exact zero_le_one
+  | hold w ih => exact mul_nonneg (hP _ _) ih
+  | step h w ih => exact mul_nonneg (hP _ _) ih
+
+theorem length_vertices (w : HoldingWalk H k x y) : w.vertices.length = k+1 := by
+  induction w with
+  | nil => rfl
+  | hold w ih => simpa only [vertices,List.length_cons] using congrArg (fun n => n+1) ih
+  | step h w ih => simpa only [vertices,List.length_cons] using congrArg (fun n => n+1) ih
+
+theorem withoutHolds_length (w : HoldingWalk H k x y) : w.withoutHolds.length ≤ k := by
+  induction w with
+  | nil => simp [withoutHolds]
+  | hold w ih => exact ih.trans (Nat.le_succ _)
+  | step h w ih => simpa only [withoutHolds,SimpleGraph.Walk.length_cons] using Nat.succ_le_succ ih
+
+theorem withoutHolds_support (w : HoldingWalk H k x y) : w.withoutHolds.support <+ w.vertices := by
+  induction w with
+  | nil => exact List.Sublist.refl _
+  | hold w ih => exact ih.cons _
+  | step h w ih => exact ih.cons_cons _
+
+ 
+
+noncomputable def erase (w : HoldingWalk H k x y) : H.Walk x y :=
+  w.withoutHolds.reverse.bypass.reverse
+
+theorem erase_isPath (w : HoldingWalk H k x y) : w.erase.IsPath :=
+  w.withoutHolds.reverse.bypass_isPath.reverse
+
+theorem erase_length (w : HoldingWalk H k x y) : w.erase.length ≤ k := by
+  have hh := w.withoutHolds.reverse.length_bypass_le_length
+  simpa only [erase,SimpleGraph.Walk.length_reverse] using hh.trans (by simpa using w.withoutHolds_length)
+
+theorem erase_support (w : HoldingWalk H k x y) : w.erase.support <+ w.vertices := by
+  have hh := w.withoutHolds.reverse.support_bypass_sublist_support.reverse
+  simp only [SimpleGraph.Walk.support_reverse,List.reverse_reverse] at hh
+  simpa only [erase,SimpleGraph.Walk.support_reverse] using hh.trans w.withoutHolds_support
+
+theorem erase_edges (w : HoldingWalk H k x y) : w.erase.edges <+ w.withoutHolds.edges := by
+  have hh := w.withoutHolds.reverse.edges_bypass_sublist_edges.reverse
+  simpa only [erase,SimpleGraph.Walk.edges_reverse,List.reverse_reverse] using hh
+
+end HoldingWalk
+end ChronologicalErasure
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section CorridorMonotonicity
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+def corridorCounted (G : BondGraph V E) (retained blockers : Set E)
+    (D : Set V) (ω : Set E) (x y : V) : Prop :=
+  Conn G (ω ∩ internalEdges G D) x y ∧
+    (∀ e ∈ retained, e ∈ ω ∩ internalEdges G D ∧
+      ¬Conn G ((ω ∩ internalEdges G D) \ {e}) x y) ∧
+    Disjoint blockers (ω ∩ internalEdges G D)
+
+lemma pivotal_restrict_of_conn (G : BondGraph V E) {S T : Set E} (hST : S ⊆ T)
+    {x y : V} (hc : Conn G S x y) (e : E) (hp : ¬Conn G (T \ {e}) x y) :
+    e ∈ S ∧ ¬Conn G (S \ {e}) x y := by
+  constructor
+  · by_contra he
+    apply hp
+    apply conn_mono G ?_ hc
+    intro d hd
+    exact ⟨hST hd,fun hde => he ((Set.mem_singleton_iff.mp hde) ▸ hd)⟩
+  · intro h
+    exact hp (conn_mono G (Set.sdiff_subset_sdiff_left hST) h)
+
+ 
+
+theorem corridorCounted_restrict (G : BondGraph V E) (retained blockers : Set E)
+    {D D' : Set V} {ω ω' : Set E} (hD : D ⊆ D') (hω : ω ⊆ ω') (x y : V)
+    (hc : Conn G (ω ∩ internalEdges G D) x y)
+    (ht : corridorCounted G retained blockers D' ω' x y) :
+    corridorCounted G retained blockers D ω x y := by
+  have hs : ω ∩ internalEdges G D ⊆ ω' ∩ internalEdges G D' :=
+    Set.inter_subset_inter hω (internalEdges_mono G hD)
+  refine ⟨hc,fun e he => pivotal_restrict_of_conn G hs hc e (ht.2.1 e he).2,?_⟩
+  exact ht.2.2.mono_right hs
+
+lemma corridorCounted_symm (G : BondGraph V E) (retained blockers : Set E)
+    (D : Set V) (ω : Set E) (x y : V) :
+    corridorCounted G retained blockers D ω x y ↔ corridorCounted G retained blockers D ω y x := by
+  constructor <;> rintro ⟨hc,hp,hb⟩
+  · exact ⟨conn_symm G _ hc,fun e he => ⟨(hp e he).1,fun hc => (hp e he).2 (conn_symm G _ hc)⟩,hb⟩
+  · exact ⟨conn_symm G _ hc,fun e he => ⟨(hp e he).1,fun hc => (hp e he).2 (conn_symm G _ hc)⟩,hb⟩
+
+ 
+
+theorem measurableSet_corridorCounted [Countable E] (G : BondGraph V E)
+    (retained blockers : Set E) (D : Set V) (x y : V) :
+    MeasurableSet {ω | corridorCounted G retained blockers D ω x y} := by
+  have hmem (e : E) : MeasurableSet {ω : Set E | e ∈ ω ∩ internalEdges G D} := by
+    by_cases he : e ∈ internalEdges G D
+    · simpa only [Set.mem_inter_iff,he,and_true] using measurableSet_mem e
+    · simp only [Set.mem_inter_iff,he,and_false,Set.ofPred_false]; exact MeasurableSet.empty
+  have hp (e : E) : MeasurableSet {ω : Set E | e ∈ ω ∩ internalEdges G D ∧
+      ¬Conn G ((ω ∩ internalEdges G D) \ {e}) x y} := by
+    apply (hmem e).inter
+    have heq (ω : Set E) : (ω ∩ internalEdges G D) \ {e} = ω ∩ (internalEdges G D \ {e}) := by
+      ext d; simp only [Set.mem_inter_iff,Set.mem_sdiff]; tauto
+    simp_rw [heq]
+    exact (measurableSet_restrictedConn G _ x y).compl
+  have hb : MeasurableSet {ω : Set E | Disjoint blockers (ω ∩ internalEdges G D)} := by
+    simp only [Set.disjoint_left,Set.ofPred_forall]
+    exact MeasurableSet.iInter (fun e => MeasurableSet.iInter (fun (_ : e ∈ blockers) => (hmem e).compl))
+  have hkeep : MeasurableSet {ω : Set E | ∀ e ∈ retained, e ∈ ω ∩ internalEdges G D ∧
+      ¬Conn G ((ω ∩ internalEdges G D) \ {e}) x y} := by
+    simp only [Set.ofPred_forall]
+    exact MeasurableSet.iInter (fun e => MeasurableSet.iInter (fun (_ : e ∈ retained) => hp e))
+  exact (measurableSet_restrictedConn G _ x y).inter (hkeep.inter hb)
+
+end CorridorMonotonicity
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section LiteralCorridors
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+instance HoldingWalk.countable [Countable V] (H : SimpleGraph V) (k : ℕ) (x y : V) :
+    Countable (HoldingWalk H k x y) := by
+  induction k generalizing x y with
+  | zero =>
+    apply Function.Injective.countable (f := fun _ : HoldingWalk H 0 x y => ())
+    intro a b _
+    cases a
+    cases b
+    rfl
+  | succ k ih =>
+    let (a b : V) : Countable (HoldingWalk H k a b) := ih a b
+    let f : HoldingWalk H (k+1) x y →
+        HoldingWalk H k x y ⊕ (Σ z : {z // H.Adj x z}, HoldingWalk H k z.val y)
+      | .hold w => .inl w
+      | .step (z := z) h w => .inr ⟨⟨z,h⟩,w⟩
+    apply Function.Injective.countable (f := f)
+    intro a b hab
+    cases a with
+    | hold a =>
+      cases b with
+      | hold b => exact congrArg HoldingWalk.hold (Sum.inl.inj hab)
+      | step h b => cases hab
+    | @step _ _ z _ h a =>
+      cases b with
+      | hold b => cases hab
+      | @step _ _ z' _ h' b =>
+        have he := Sum.inr.inj hab
+        have hz := congrArg (fun t => t.1.val) he
+        change z = z' at hz
+        subst z'
+        have hab' : a = b := by simpa only [Sigma.mk.inj_iff,heq_eq_eq,true_and] using he
+        subst b
+        rfl
+
+ 
+abbrev RawCorridor (G : BondGraph V E) (k : ℕ) :=
+  Σ x y : V, HoldingWalk (openSimpleGraph G Set.univ) k x y
+
+namespace RawCorridor
+variable {G : BondGraph V E} {k : ℕ}
+
+def base (w : RawCorridor G k) : V := w.1
+def endpoint (w : RawCorridor G k) : V := w.2.1
+noncomputable def path (w : RawCorridor G k) :
+    (openSimpleGraph G Set.univ).Walk w.base w.endpoint := w.2.2.erase
+noncomputable def interior (w : RawCorridor G k) : Finset V :=
+  w.path.support.toFinset \ {w.base,w.endpoint}
+def retained (w : RawCorridor G k) : Set E := {e | G e ∈ w.path.edges}
+noncomputable def blockers (w : RawCorridor G k) : Set E :=
+  {e | ∃ z ∈ w.interior, z ∈ G e} \ w.retained
+noncomputable def probability (P : V → V → ℝ) (w : RawCorridor G k) : ℝ := w.2.2.mass P
+noncomputable def weight (P : V → V → ℝ) (p : ℝ) (w : RawCorridor G k) : ℝ :=
+  w.probability P * (p^w.path.length)⁻¹ * ((1-p)^w.blockers.ncard)⁻¹
+
+theorem path_isPath (w : RawCorridor G k) : w.path.IsPath := w.2.2.erase_isPath
+
+theorem path_length (w : RawCorridor G k) : w.path.length ≤ k := w.2.2.erase_length
+
+theorem retained_finite (hG : Function.Injective G) (w : RawCorridor G k) : w.retained.Finite :=
+  (List.finite_toSet w.path.edges).preimage hG.injOn
+
+theorem blockers_finite (hloc : LocallyFinite G) (w : RawCorridor G k) : w.blockers.Finite := by
+  apply Set.Finite.subset (s := (incidentSupport G hloc w.interior : Set E)) (Finset.finite_toSet _)
+  intro e he
+  exact (mem_incidentSupport G hloc w.interior e).mpr he.1
+
+theorem retained_disjoint_blockers (w : RawCorridor G k) : Disjoint w.retained w.blockers := by
+  exact Set.disjoint_left.mpr (fun e he hb => hb.2 he)
+
+theorem weight_nonneg (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y)
+    (p : unitInterval) (w : RawCorridor G k) : 0 ≤ w.weight P p := by
+  exact mul_nonneg (mul_nonneg (w.2.2.mass_nonneg P hP) (inv_nonneg.mpr (pow_nonneg p.2.1 _)))
+    (inv_nonneg.mpr (pow_nonneg (sub_nonneg.mpr p.2.2) _))
+
+end RawCorridor
+end LiteralCorridors
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Main.lean b/lean/OAI/Probability/UnimodularPercolation/Main.lean
index 3032a0f771039ca88071b90304a8976ec29bccb3..be76339a5626383fefb6e1d54bb220cdabdc56ff 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Main.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Main.lean
@@ -1,2 +1,2 @@
-import OAI.Probability.UnimodularPercolation.Criticality.Operator
-import OAI.Probability.UnimodularPercolation.Subdivision.Operator
+import OAI.Probability.UnimodularPercolation.Criticality.Operator
+import OAI.Probability.UnimodularPercolation.Subdivision.Operator
diff --git a/lean/OAI/Probability/UnimodularPercolation/ProductMeasure/Association.lean b/lean/OAI/Probability/UnimodularPercolation/ProductMeasure/Association.lean
index 650d99c1c6ddcbe0c7ebd7f1e6d95edec7f89fac..2d60b090e30e95ebd16e1ea65fcd2d17da876be9 100644
--- a/lean/OAI/Probability/UnimodularPercolation/ProductMeasure/Association.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/ProductMeasure/Association.lean
@@ -1,1094 +1,1094 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Kernel.Bounds
-
-namespace OAI
-
-/-! Association, disjoint witnesses and the BK inequality. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-section FiniteAssociation
-attribute [local instance] Classical.propDecidable
-variable [Fintype E]
-
-private theorem bernoulli_atom_product (p : unitInterval) (ω : Set E) :
-    (law p).real {ω} = ∏ e : E, if e ∈ ω then (p : ℝ) else 1 - p := by
-  rw [law, setBernoulli_real_singleton p (Set.subset_univ ω) (Set.finite_univ)]
-  simp [Finset.prod_ite, ← Set.ncard_coe_finset, Set.sdiff_eq]
-  exact Or.inl rfl
-
-private theorem bernoulli_atom_modular (p : unitInterval) (a b : Set E) :
-    (law p).real {a} * (law p).real {b} =
-      (law p).real {a ∩ b} * (law p).real {a ∪ b} := by
-  simp only [bernoulli_atom_product, ← Finset.prod_mul_distrib]
-  apply Finset.prod_congr rfl
-  intro e _
-  by_cases ha : e ∈ a <;> by_cases hb : e ∈ b <;> simp [ha, hb, mul_comm]
-
- 
-
-theorem association_finite (p : unitInterval) (f g : Set E → ℝ)
-    (hf : Monotone f) (hg : Monotone g) (hf0 : ∀ ω, 0 ≤ f ω) (hg0 : ∀ ω, 0 ≤ g ω) :
-    (∫ ω, f ω ∂law p) * (∫ ω, g ω ∂law p) ≤ ∫ ω, f ω * g ω ∂law p := by
-  have h := fkg f g (fun ω : Set E => (law p).real {ω})
-    (fun _ => measureReal_nonneg) hf0 hg0 hf hg
-    (fun a b => (bernoulli_atom_modular p a b).le)
-  have hsum : (∑ ω : Set E, (law p).real {ω}) = 1 := by
-    simp
-  rw [hsum, one_mul] at h
-  simpa only [integral_fintype Integrable.of_finite, smul_eq_mul] using h
-
- 
-theorem association_events_finite (p : unitInterval) (A B : Set (Set E))
-    (hA : IsUpperSet A) (hB : IsUpperSet B) :
-    (law p).real A * (law p).real B ≤ (law p).real (A ∩ B) := by
-  have hm (S : Set (Set E)) (hS : IsUpperSet S) :
-      Monotone (S.indicator (fun _ => (1 : ℝ))) := by
-    intro a b hab
-    by_cases ha : a ∈ S
-    · simp [ha, hS hab ha]
-    · simp only [Set.indicator_of_notMem ha]
-      exact Set.indicator_nonneg (fun _ _ => zero_le_one) _
-  have he : (fun ω => A.indicator (fun _ => (1 : ℝ)) ω *
-      B.indicator (fun _ => (1 : ℝ)) ω) = (A ∩ B).indicator (fun _ => 1) := by
-    funext ω
-    by_cases ha : ω ∈ A <;> by_cases hb : ω ∈ B <;> simp [ha, hb]
-  have h := association_finite p _ _ (hm A hA) (hm B hB)
-    (Set.indicator_nonneg (fun _ _ => zero_le_one))
-    (Set.indicator_nonneg (fun _ _ => zero_le_one))
-  rw [he] at h
-  simpa only [integral_indicator_const (1 : ℝ) (.of_discrete : MeasurableSet A),
-    integral_indicator_const (1 : ℝ) (.of_discrete : MeasurableSet B),
-    integral_indicator_const (1 : ℝ) (.of_discrete : MeasurableSet (A ∩ B)),
-    smul_eq_mul, mul_one] using h
-
-end FiniteAssociation
-
-section Restriction
-
-theorem measurable_set_restrict (s : Set E) :
-    Measurable (fun ω : Set E => (Subtype.val ⁻¹' ω : Set s)) :=
-  measurable_setOfPred.comp (Measurable.of_eval
-    (fun e => (measurable_pi_apply e.val).comp MeasurableEquiv.setOfPred.symm.measurable))
-
- 
-theorem law_map_restrict (p : unitInterval) (s : Set E) :
-    (law p : Measure (Set E)).map (fun ω => (Subtype.val ⁻¹' ω : Set s)) = law p := by
-  have hm := measurable_set_restrict s
-  unfold law
-  rw [setBernoulli_eq_map, Measure.map_map hm measurable_setOfPred]
-  have he : (fun ω : Set E => (Subtype.val ⁻¹' ω : Set s)) ∘
-      (fun f : E → Prop => {e | f e}) =
-      (fun f : s → Prop => {e | f e}) ∘ s.domRestrict := rfl
-  rw [he, ← Measure.map_map measurable_setOfPred (by fun_prop),
-    Measure.infinitePi_map_restrict', setBernoulli_eq_map]
-  simp only [Set.mem_univ]
-
-end Restriction
-
-section WitnessAssociation
-attribute [local instance] Classical.propDecidable
-open Filter Topology
-
- 
-def FiniteWitness (A : Set (Set E)) : Prop :=
-  ∀ ω ∈ A, ∃ s : Finset E, (s : Set E) ⊆ ω ∧ (s : Set E) ∈ A
-
-theorem association_restricted (p : unitInterval) (A B : Set (Set E))
-    (hA : IsUpperSet A) (hB : IsUpperSet B) (s : Finset E) :
-    (law p).real {ω | ω ∩ (s : Set E) ∈ A} *
-      (law p).real {ω | ω ∩ (s : Set E) ∈ B} ≤
-      (law p).real {ω | ω ∩ (s : Set E) ∈ A ∩ B} := by
-  let A' : Set (Set (s : Set E)) := {η | Subtype.val '' η ∈ A}
-  let B' : Set (Set (s : Set E)) := {η | Subtype.val '' η ∈ B}
-  have hA' : IsUpperSet A' := fun _ _ h ha => hA (Set.image_mono h) ha
-  have hB' : IsUpperSet B' := fun _ _ h hb => hB (Set.image_mono h) hb
-  have he (ω : Set E) : Subtype.val '' (Subtype.val ⁻¹' ω : Set (s : Set E)) =
-      ω ∩ (s : Set E) := by
-    ext e
-    simp [and_comm]
-  have hl (C : Set (Set E)) :
-      (law p : Measure (Set (s : Set E))).real {η | Subtype.val '' η ∈ C} =
-      (law p).real {ω | ω ∩ (s : Set E) ∈ C} := by
-    rw [← law_map_restrict p (s : Set E), measureReal_def,
-      Measure.map_apply (measurable_set_restrict _) .of_discrete]
-    simp only [Set.preimage_ofPred_eq, he, measureReal_def]
-  have ht := association_events_finite p A' B' hA' hB'
-  change (law p).real {η | Subtype.val '' η ∈ A} *
-    (law p).real {η | Subtype.val '' η ∈ B} ≤
-    (law p).real {η | Subtype.val '' η ∈ A ∩ B} at ht
-  simpa only [hl] using ht
-
- noncomputable def edgeExhaustion [Encodable E] (n : ℕ) : Finset E :=
-  (Finset.range n).preimage Encodable.encode Encodable.encode_injective.injOn
-
- theorem edgeExhaustion_mono [Encodable E] :
-    Monotone (edgeExhaustion : ℕ → Finset E) := by
-  intro m n h e he
-  simp only [edgeExhaustion, Finset.mem_preimage, Finset.mem_range] at he ⊢
-  exact he.trans_le h
-
-private theorem subset_edgeExhaustion [Encodable E] (s : Finset E) :
-    ∃ n, s ⊆ edgeExhaustion n := by
-  refine ⟨s.sup Encodable.encode + 1, ?_⟩
-  intro e he
-  simp only [edgeExhaustion, Finset.mem_preimage, Finset.mem_range]
-  exact Nat.lt_succ_of_le (Finset.le_sup he)
-
- theorem witness_eq_union [Encodable E] (A : Set (Set E))
-    (hA : IsUpperSet A) (hf : FiniteWitness A) :
-    A = ⋃ n, {ω | ω ∩ (edgeExhaustion (E := E) n : Set E) ∈ A} := by
-  ext ω
-  simp only [Set.mem_iUnion, Set.mem_ofPred_eq]
-  constructor
-  · intro hω
-    obtain ⟨s, hs, hsa⟩ := hf ω hω
-    obtain ⟨n, hn⟩ := subset_edgeExhaustion s
-    exact ⟨n, hA (fun _ he => ⟨hs he, hn he⟩) hsa⟩
-  · rintro ⟨n, hn⟩
-    exact hA Set.inter_subset_left hn
-
- 
-
-theorem association_finiteWitness [Countable E] (p : unitInterval)
-    (A B : Set (Set E)) (hA : IsUpperSet A) (hB : IsUpperSet B)
-    (hfA : FiniteWitness A) (hfB : FiniteWitness B) :
-    (law p).real A * (law p).real B ≤ (law p).real (A ∩ B) := by
-  let := Encodable.ofCountable E
-  have hfAB : FiniteWitness (A ∩ B) := by
-    intro ω hω
-    obtain ⟨s, hs, hsa⟩ := hfA ω hω.1
-    obtain ⟨t, ht, htb⟩ := hfB ω hω.2
-    refine ⟨s ∪ t, ?_, hA ?_ hsa, hB ?_ htb⟩
-    · simpa only [Finset.coe_union] using Set.union_subset hs ht
-    · simp only [Finset.coe_union]; exact Set.subset_union_left
-    · simp only [Finset.coe_union]; exact Set.subset_union_right
-  have hlim (C : Set (Set E)) (hC : IsUpperSet C) (hfC : FiniteWitness C) :
-      Tendsto (fun n => (law p).real {ω | ω ∩ (edgeExhaustion (E := E) n : Set E) ∈ C})
-        atTop (𝓝 ((law p).real C)) := by
-    have hm : Monotone (fun n => {ω : Set E | ω ∩ (edgeExhaustion (E := E) n : Set E) ∈ C}) := by
-      intro m n h ω hω
-      exact hC (Set.inter_subset_inter_right _ (edgeExhaustion_mono h)) hω
-    have h := tendsto_measure_iUnion_atTop (μ := law p) hm
-    rw [← witness_eq_union C hC hfC] at h
-    exact (ENNReal.tendsto_toReal (measure_ne_top _ _)).comp h
-  exact le_of_tendsto_of_tendsto
-    ((hlim A hA hfA).mul (hlim B hB hfB)) (hlim (A ∩ B) (hA.inter hB) hfAB)
-    (Filter.Eventually.of_forall fun n => association_restricted p A B hA hB (edgeExhaustion n))
-
-end WitnessAssociation
-
-section ClusterEvents
-attribute [local instance] Classical.propDecidable
-open Filter Topology
-
- 
-theorem measurableSet_finiteWitness [Countable E] (A : Set (Set E))
-    (hA : IsUpperSet A) (hfA : FiniteWitness A) : MeasurableSet A := by
-  let := Encodable.ofCountable E
-  rw [witness_eq_union A hA hfA]
-  apply MeasurableSet.iUnion
-  intro n
-  let s := edgeExhaustion (E := E) n
-  have he : {ω : Set E | ω ∩ (s : Set E) ∈ A} =
-      (fun ω => (Subtype.val ⁻¹' ω : Set (s : Set E))) ⁻¹'
-        {η | Subtype.val '' η ∈ A} := by
-    ext ω
-    change ω ∩ (s : Set E) ∈ A ↔ Subtype.val '' (Subtype.val ⁻¹' ω : Set (s : Set E)) ∈ A
-    have hi : Subtype.val '' (Subtype.val ⁻¹' ω : Set (s : Set E)) = ω ∩ (s : Set E) := by
-      ext e
-      simp [and_comm]
-    rw [hi]
-  change MeasurableSet {ω : Set E | ω ∩ (s : Set E) ∈ A}
-  rw [he]
-  exact MeasurableSet.preimage .of_discrete (measurable_set_restrict _)
-
-theorem finiteWitness_conn (G : BondGraph V E) (x y : V) :
-    FiniteWitness {ω | Conn G ω x y} := by
-  intro ω h
-  induction h with
-  | refl => exact ⟨∅, by simp, by simp⟩
-  | @tail b c hab hbc ih =>
-    obtain ⟨s, hs, hconn⟩ := ih
-    obtain ⟨e, he, hGe⟩ := hbc
-    refine ⟨insert e s, ?_, ?_⟩
-    · simpa only [Finset.coe_insert] using Set.insert_subset he hs
-    · exact (conn_mono G (by simp) hconn).tail ⟨e, by simp, hGe⟩
-
-theorem finiteWitness_conn_finset (G : BondGraph V E) (x : V) (t : Finset V) :
-    FiniteWitness {ω | ∀ y ∈ t, Conn G ω x y} := by
-  intro ω h
-  induction t using Finset.induction_on with
-  | empty => exact ⟨∅, by simp, by simp⟩
-  | @insert a t ha ih =>
-    obtain ⟨s, hs, hsa⟩ := finiteWitness_conn G x a ω (h a (by simp))
-    obtain ⟨r, hr, hrt⟩ := ih (fun y hy => h y (by simp [hy]))
-    refine ⟨s ∪ r, ?_, ?_⟩
-    · simpa only [Finset.coe_union] using Set.union_subset hs hr
-    · intro y hy
-      rcases Finset.mem_insert.mp hy with rfl | hy
-      · exact conn_mono G (by simp) hsa
-      · exact conn_mono G (by simp) (hrt y hy)
-
- 
-def clusterLarge (G : BondGraph V E) (x : V) (n : ℕ) : Set (Set E) :=
-  {ω | ∃ t : Finset V, n ≤ t.card ∧ ∀ y ∈ t, Conn G ω x y}
-
-theorem clusterLarge_upper (G : BondGraph V E) (x : V) (n : ℕ) :
-    IsUpperSet (clusterLarge G x n) := by
-  rintro ω η h ⟨t, ht, hc⟩
-  exact ⟨t, ht, fun y hy => conn_mono G h (hc y hy)⟩
-
-theorem finiteWitness_clusterLarge (G : BondGraph V E) (x : V) (n : ℕ) :
-    FiniteWitness (clusterLarge G x n) := by
-  rintro ω ⟨t, ht, hc⟩
-  obtain ⟨s, hs, hsc⟩ := finiteWitness_conn_finset G x t ω hc
-  exact ⟨s, hs, t, ht, hsc⟩
-
-theorem measurableSet_clusterLarge [Countable E] (G : BondGraph V E) (x : V) (n : ℕ) :
-    MeasurableSet (clusterLarge G x n) :=
-  measurableSet_finiteWitness _ (clusterLarge_upper G x n) (finiteWitness_clusterLarge G x n)
-
-theorem clusterLarge_antitone (G : BondGraph V E) (x : V) :
-    Antitone (clusterLarge G x) := by
-  rintro m n h ω ⟨t, ht, hc⟩
-  exact ⟨t, h.trans ht, hc⟩
-
-theorem infinite_cluster_eq_iInter (G : BondGraph V E) (x : V) :
-    {ω | (cluster G ω x).Infinite} = ⋂ n, clusterLarge G x n := by
-  ext ω
-  simp only [Set.mem_iInter, Set.mem_ofPred_eq]
-  constructor
-  · intro h n
-    obtain ⟨t, ht, hcard⟩ := h.exists_subset_card_eq n
-    exact ⟨t, hcard.ge, ht⟩
-  · intro h hfin
-    obtain ⟨t, ht, hc⟩ := h (hfin.toFinset.card + 1)
-    have hsub : t ⊆ hfin.toFinset := fun y hy => hfin.mem_toFinset.mpr (hc y hy)
-    have := Finset.card_le_card hsub
-    omega
-
-theorem measurableSet_infinite_cluster [Countable E] (G : BondGraph V E) (x : V) :
-    MeasurableSet {ω | (cluster G ω x).Infinite} := by
-  rw [infinite_cluster_eq_iInter]
-  exact MeasurableSet.iInter (measurableSet_clusterLarge G x)
-
- 
-theorem association_decreasing [Countable E] (p : unitInterval)
-    (A B : ℕ → Set (Set E)) (hA : Antitone A) (hB : Antitone B)
-    (huA : ∀ n, IsUpperSet (A n)) (huB : ∀ n, IsUpperSet (B n))
-    (hfA : ∀ n, FiniteWitness (A n)) (hfB : ∀ n, FiniteWitness (B n)) :
-    (law p).real (⋂ n, A n) * (law p).real (⋂ n, B n) ≤
-      (law p).real ((⋂ n, A n) ∩ (⋂ n, B n)) := by
-  have hlim (C : ℕ → Set (Set E)) (hm : Antitone C) (hC : ∀ n, MeasurableSet (C n)) :
-      Tendsto (fun n => (law p).real (C n)) atTop (𝓝 ((law p).real (⋂ n, C n))) :=
-    (ENNReal.tendsto_toReal (measure_ne_top _ _)).comp
-      (tendsto_measure_iInter_atTop (fun n => (hC n).nullMeasurableSet) hm
-        ⟨0, measure_ne_top _ _⟩)
-  have hmA n := measurableSet_finiteWitness (A n) (huA n) (hfA n)
-  have hmB n := measurableSet_finiteWitness (B n) (huB n) (hfB n)
-  have h := le_of_tendsto_of_tendsto ((hlim A hA hmA).mul (hlim B hB hmB))
-    (hlim (fun n => A n ∩ B n) (fun m n h => Set.inter_subset_inter (hA h) (hB h))
-      (fun n => (hmA n).inter (hmB n)))
-    (Filter.Eventually.of_forall fun n =>
-      association_finiteWitness p (A n) (B n) (huA n) (huB n) (hfA n) (hfB n))
-  simpa only [Set.iInter_inter_distrib] using h
-
- 
-
-theorem association_infinite_cluster [Countable E]
-    (G : BondGraph V E) (p : unitInterval) (x y : V) :
-    (law p).real {ω | (cluster G ω x).Infinite} *
-      (law p).real {ω | (cluster G ω y).Infinite} ≤
-      (law p).real {ω | (cluster G ω x).Infinite ∧ (cluster G ω y).Infinite} := by
-  rw [infinite_cluster_eq_iInter, infinite_cluster_eq_iInter]
-  change _ ≤ (law p).real ({ω | (cluster G ω x).Infinite} ∩
-    {ω | (cluster G ω y).Infinite})
-  rw [infinite_cluster_eq_iInter, infinite_cluster_eq_iInter]
-  exact association_decreasing p _ _ (clusterLarge_antitone G x) (clusterLarge_antitone G y)
-    (clusterLarge_upper G x) (clusterLarge_upper G y)
-    (finiteWitness_clusterLarge G x) (finiteWitness_clusterLarge G y)
-
-end ClusterEvents
-
-section Automorphisms
-attribute [local instance] Classical.propDecidable
-
-theorem measurable_set_preimage (b : E ≃ E) :
-    Measurable (fun ω : Set E => b ⁻¹' ω) :=
-  measurable_setOfPred.comp (Measurable.of_eval
-    (fun e => (measurable_pi_apply (b e)).comp MeasurableEquiv.setOfPred.symm.measurable))
-
- 
-theorem law_map_preimage_equiv (p : unitInterval) (b : E ≃ E) :
-    (law p : Measure (Set E)).map (fun ω => b ⁻¹' ω) = law p := by
-  unfold law
-  rw [setBernoulli_eq_map, Measure.map_map (measurable_set_preimage b) measurable_setOfPred]
-  have he : (fun ω : Set E => b ⁻¹' ω) ∘ (fun f : E → Prop => {e | f e}) =
-      (fun f : E → Prop => {e | f e}) ∘
-        MeasurableEquiv.piCongrLeft (fun _ : E => Prop) b.symm := by
-    funext f
-    ext e
-    simp only [Function.comp_apply, Set.mem_preimage, Set.mem_ofPred_eq]
-    have hh := MeasurableEquiv.piCongrLeft_apply_apply (β := fun _ : E => Prop) b.symm f (b e)
-    exact iff_of_eq (by simpa only [Equiv.symm_apply_apply] using hh.symm)
-  rw [he, ← Measure.map_map measurable_setOfPred (MeasurableEquiv.measurable _)]
-  simp only [Set.mem_univ]
-  congr 1
-  exact Measure.infinitePi_map_piCongrLeft
-    (fun _ : E => unitInterval.toNNReal p • Measure.dirac True +
-      unitInterval.toNNReal (unitInterval.symm p) • Measure.dirac False) b.symm
-
-theorem isAut_symm (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (h : IsAut G a b) : IsAut G a.symm b.symm := by
-  intro e
-  have hh := congrArg (Sym2.map a.symm) (h (b.symm e))
-  simpa only [Equiv.apply_symm_apply, Sym2.map_map, Equiv.symm_comp_self,
-    Sym2.map_id', id_eq] using hh.symm
-
-theorem conn_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (h : IsAut G a b) {ω : Set E} {x y : V} (hc : Conn G ω x y) :
-    Conn G (b '' ω) (a x) (a y) := by
-  induction hc with
-  | refl => exact conn_refl G _ _
-  | @tail v w hvw hed ih =>
-    obtain ⟨e, he, hGe⟩ := hed
-    exact ih.tail ⟨b e, Set.mem_image_of_mem _ he, by rw [h e, hGe, Sym2.map_mk]⟩
-
-theorem conn_aut_iff (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (h : IsAut G a b) (ω : Set E) (x y : V) :
-    Conn G (b ⁻¹' ω) x y ↔ Conn G ω (a x) (a y) := by
-  constructor
-  · intro hc
-    simpa only [Set.image_preimage_eq _ b.surjective] using conn_aut G h hc
-  · intro hc
-    have he : b.symm '' ω = b ⁻¹' ω := by
-      ext e
-      simp
-    simpa only [Equiv.symm_apply_apply, he] using
-      conn_aut G (isAut_symm G h) hc
-
-theorem cluster_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (h : IsAut G a b) (ω : Set E) (x : V) :
-    a '' cluster G (b ⁻¹' ω) x = cluster G ω (a x) := by
-  ext y
-  constructor
-  · rintro ⟨z, hz, rfl⟩
-    exact (conn_aut_iff G h ω x z).mp hz
-  · intro hy
-    change Conn G ω (a x) y at hy
-    refine ⟨a.symm y, ?_, by simp⟩
-    exact (conn_aut_iff G h ω x (a.symm y)).mpr (by simpa only [Equiv.apply_symm_apply] using hy)
-
-theorem infinite_cluster_aut [Countable E] (G : BondGraph V E) (p : unitInterval)
-    {a : V ≃ V} {b : E ≃ E} (h : IsAut G a b) (x : V) :
-    (law p).real {ω | (cluster G ω (a x)).Infinite} =
-      (law p).real {ω | (cluster G ω x).Infinite} := by
-  have he : {ω | (cluster G ω (a x)).Infinite} =
-      (fun ω => b ⁻¹' ω) ⁻¹' {ω | (cluster G ω x).Infinite} := by
-    ext ω
-    change (cluster G ω (a x)).Infinite ↔ (cluster G (b ⁻¹' ω) x).Infinite
-    rw [← cluster_aut G h]
-    exact Set.infinite_image_iff (fun _ _ _ _ hab => a.injective hab)
-  unfold Measure.real
-  rw [he, ← Measure.map_apply (measurable_set_preimage b)
-    (measurableSet_infinite_cluster G x), law_map_preimage_equiv]
-
- 
-theorem finite_open_probability (p : unitInterval) (s : Finset E) :
-    (law p).real {ω | (s : Set E) ⊆ ω} = (p : ℝ) ^ s.card := by
-  have he : {ω : Set E | (s : Set E) ⊆ ω} =
-      (fun ω => (Subtype.val ⁻¹' ω : Set (s : Set E))) ⁻¹' {Set.univ} := by
-    ext ω
-    simp only [Set.mem_ofPred_eq, Set.mem_preimage, Set.mem_singleton_iff]
-    exact ⟨fun h => Set.eq_univ_of_forall fun e => h e.property,
-      fun h e he => show (⟨e, he⟩ : (s : Set E)) ∈ (Subtype.val ⁻¹' ω) from
-        h.symm ▸ Set.mem_univ _⟩
-  unfold Measure.real
-  rw [he, ← Measure.map_apply (measurable_set_restrict _) (by measurability),
-    law_map_restrict]
-  change (law p : Measure (Set (s : Set E))).real {Set.univ} = _
-  simp [law, setBernoulli_real_singleton p (Set.Subset.refl _) (Set.toFinite _)]
-
-theorem twoPoint_pos [Countable E] (G : BondGraph V E) (hG : Connected G)
-    (p : unitInterval) (hp : 0 < p) (x y : V) : 0 < twoPoint G p x y := by
-  obtain ⟨s, _, hs⟩ := finiteWitness_conn G x y Set.univ (conn_univ_of_connected G hG x y)
-  have hpos : 0 < (law p).real {ω | (s : Set E) ⊆ ω} := by
-    rw [finite_open_probability]
-    exact pow_pos hp _
-  apply lt_of_lt_of_le hpos
-  exact measureReal_mono (fun _ hω => conn_mono G hω hs)
-
-end Automorphisms
-
-section UniquenessThreshold
-attribute [local instance] Classical.propDecidable
-variable [Countable V] [Countable E]
-
-omit [Countable V] in
- 
-theorem association_infinite_conn (G : BondGraph V E) (p : unitInterval) (x y z : V) :
-    (law p).real {ω | (cluster G ω x).Infinite} * twoPoint G p y z ≤
-      (law p).real {ω | (cluster G ω x).Infinite ∧ Conn G ω y z} := by
-  have h := association_decreasing p (clusterLarge G x) (fun _ => {ω | Conn G ω y z})
-    (clusterLarge_antitone G x) antitone_const (clusterLarge_upper G x)
-    (fun _ _ _ hsub hconn => conn_mono G hsub hconn) (finiteWitness_clusterLarge G x)
-    (fun _ => finiteWitness_conn G y z)
-  simpa only [← infinite_cluster_eq_iInter, Set.iInter_const, twoPoint, Measure.real, Set.inter_def, Set.mem_ofPred_eq] using h
-
-omit [Countable V] in
-theorem survival_pos_of_survival_pos (G : BondGraph V E) (hG : Connected G)
-    (p : unitInterval) (hp : 0 < p) {x : V}
-    (hx : 0 < (law p).real {ω | (cluster G ω x).Infinite}) (y : V) :
-    0 < (law p).real {ω | (cluster G ω y).Infinite} := by
-  apply lt_of_lt_of_le (mul_pos hx (twoPoint_pos G hG p hp x y))
-  refine (association_infinite_conn G p x x y).trans (measureReal_mono ?_)
-  rintro ω ⟨hi, hc⟩
-  change (cluster G ω y).Infinite
-  rwa [← cluster_eq_of_conn G ω hc]
-
-omit [Countable V] [Countable E] in
-theorem exists_infinite_cluster_iff (G : BondGraph V E) (ω : Set E) :
-    (infiniteClusters G ω).Nonempty ↔ ∃ x, (cluster G ω x).Infinite := by
-  constructor
-  · rintro ⟨C, hi, x, rfl⟩
-    exact ⟨x, hi⟩
-  · rintro ⟨x, hx⟩
-    exact ⟨cluster G ω x, hx, x, rfl⟩
-
-omit [Countable V] [Countable E] in
-theorem unique_infinite_cluster_iff (G : BondGraph V E) (ω : Set E) :
-    (∃! C, C ∈ infiniteClusters G ω) ↔
-      ∃ x, (cluster G ω x).Infinite ∧ ∀ y, (cluster G ω y).Infinite → Conn G ω x y := by
-  constructor
-  · rintro ⟨C, ⟨hi, x, rfl⟩, hu⟩
-    refine ⟨x, hi, fun y hy => ?_⟩
-    have he := hu (cluster G ω y) ⟨hy, y, rfl⟩
-    have hm := mem_cluster_self G ω y
-    rwa [he] at hm
-  · rintro ⟨x, hx, hu⟩
-    refine ⟨cluster G ω x, ⟨hx, x, rfl⟩, ?_⟩
-    rintro C ⟨hi, y, rfl⟩
-    exact (cluster_eq_of_conn G ω (hu y hi)).symm
-
-theorem measurableSet_unique_infinite_cluster (G : BondGraph V E) :
-    MeasurableSet {ω | ∃! C, C ∈ infiniteClusters G ω} := by
-  have he : {ω | ∃! C, C ∈ infiniteClusters G ω} =
-      ⋃ x, ({ω | (cluster G ω x).Infinite} ∩
-        ⋂ y, ({ω | (cluster G ω y).Infinite}ᶜ ∪ {ω | Conn G ω x y})) := by
-    ext ω
-    simp only [Set.mem_ofPred_eq, Set.mem_iUnion, Set.mem_inter_iff,
-      Set.mem_iInter, Set.mem_union, Set.mem_compl_iff, unique_infinite_cluster_iff,
-      imp_iff_not_or]
-  rw [he]
-  exact MeasurableSet.iUnion fun x => (measurableSet_infinite_cluster G x).inter
-    (MeasurableSet.iInter fun y =>
-      (measurableSet_infinite_cluster G y).compl.union (measurableSet_conn G x y))
-
-theorem uniform_survival_pos (G : BondGraph V E) (hconn : Connected G)
-    (hqt : QuasiTransitive G) (p : unitInterval) (hp : 0 < p)
-    (hpos : 0 < law p {ω | (infiniteClusters G ω).Nonempty}) :
-    ∃ θ : ℝ, 0 < θ ∧ ∀ x, θ ≤ (law p).real {ω | (cluster G ω x).Infinite} := by
-  have hunion : {ω | (infiniteClusters G ω).Nonempty} =
-      ⋃ x, {ω | (cluster G ω x).Infinite} := by
-    ext ω
-    simp only [Set.mem_ofPred_eq, Set.mem_iUnion, exists_infinite_cluster_iff]
-  rw [hunion] at hpos
-  obtain ⟨x, hx⟩ := exists_measure_pos_of_not_measure_iUnion_null (ne_of_gt hpos)
-  have hxp : 0 < (law p).real {ω | (cluster G ω x).Infinite} :=
-    ENNReal.toReal_pos hx.ne' (measure_ne_top _ _)
-  have hroot := survival_pos_of_survival_pos G hconn p hp hxp
-  obtain ⟨s, hs⟩ := hqt
-  have hsn : s.Nonempty := by obtain ⟨o, ho, _⟩ := hs x; exact ⟨o, ho⟩
-  let vals := s.image (fun o => (law p).real {ω | (cluster G ω o).Infinite})
-  have hv : vals.Nonempty := hsn.image _
-  refine ⟨vals.min' hv, ?_, fun y => ?_⟩
-  · obtain ⟨o, _, he⟩ := Finset.mem_image.mp (Finset.min'_mem vals hv)
-    rw [← he]
-    exact hroot o
-  · obtain ⟨o, ho, a, b, ha, hay⟩ := hs y
-    rw [← hay, infinite_cluster_aut G p ha]
-    exact Finset.min'_le _ _ (Finset.mem_image.mpr ⟨o, ho, rfl⟩)
-
- 
-
-theorem not_operatorBounded_of_unique [Infinite V] (G : BondGraph V E)
-    (hconn : Connected G) (hqt : QuasiTransitive G) (p : unitInterval)
-    (hu : law p {ω | ∃! C, C ∈ infiniteClusters G ω} = 1) : ¬ OperatorBounded G p := by
-  have hp : 0 < p := by
-    by_contra h
-    have hp0 : p = 0 := le_antisymm (not_lt.mp h) (bot_le)
-    subst p
-    simp [law] at hu
-  have hpos : 0 < law p {ω | (infiniteClusters G ω).Nonempty} := by
-    exact lt_of_lt_of_le (show 0 < law p {ω | ∃! C, C ∈ infiniteClusters G ω} by rw [hu]; exact zero_lt_one)
-      (measure_mono fun ω hω => ⟨hω.choose, hω.choose_spec.1⟩)
-  obtain ⟨θ, hθ, hroot⟩ := uniform_survival_pos G hconn hqt p hp hpos
-  have hue : ∀ᵐ ω ∂law p, ∃! C, C ∈ infiniteClusters G ω :=
-    (mem_ae_iff_prob_eq_one (measurableSet_unique_infinite_cluster G)).mpr hu
-  refine not_operatorBounded_of_positive_column G p (Classical.arbitrary V) (mul_pos hθ hθ)
-    (fun x => ?_)
-  refine (mul_le_mul (hroot x) (hroot (Classical.arbitrary V)) hθ.le measureReal_nonneg).trans
-    ((association_infinite_cluster G p x (Classical.arbitrary V)).trans ?_)
-  apply ENNReal.toReal_mono (measure_ne_top _ _)
-  apply measure_mono_ae
-  filter_upwards [hue] with ω hω
-  intro hi
-  obtain ⟨r, _, hconnr⟩ := (unique_infinite_cluster_iff G ω).mp hω
-  exact conn_trans G ω (conn_symm G ω (hconnr x hi.1)) (hconnr _ hi.2)
-
- 
-
-theorem ptwo_le_pu [Infinite V] (G : BondGraph V E)
-    (hconn : Connected G) (hqt : QuasiTransitive G) : ptwo G ≤ pu G := by
-  unfold pu ptwo
-  apply le_sInf
-  intro p hp
-  apply sSup_le
-  intro q hq
-  by_contra hqp
-  have hb := operatorBounded_of_le G (le_of_lt (not_le.mp hqp)) hq
-  exact not_operatorBounded_of_unique G hconn hqt p hp hb
-
-end UniquenessThreshold
-
-section DisjointWitness
-attribute [local instance] Classical.propDecidable
-
- 
-def DisjointOccurrence (A B : Set (Set E)) : Set (Set E) :=
-  {ω | ∃ s t : Finset E, (s : Set E) ⊆ ω ∧ (t : Set E) ⊆ ω ∧
-    Disjoint s t ∧ (s : Set E) ∈ A ∧ (t : Set E) ∈ B}
-
-theorem disjointOccurrence_upper (A B : Set (Set E)) : IsUpperSet (DisjointOccurrence A B) := by
-  rintro ω η h ⟨s, t, hs, ht, hd, ha, hb⟩
-  exact ⟨s, t, hs.trans h, ht.trans h, hd, ha, hb⟩
-
-theorem finiteWitness_disjointOccurrence (A B : Set (Set E)) :
-    FiniteWitness (DisjointOccurrence A B) := by
-  rintro ω ⟨s, t, hs, ht, hd, ha, hb⟩
-  refine ⟨s ∪ t, ?_, s, t, ?_, ?_, hd, ha, hb⟩
-  · simpa only [Finset.coe_union] using Set.union_subset hs ht
-  · simp
-  · simp
-
-theorem disjointOccurrence_mono {A A' B B' : Set (Set E)}
-    (hA : A ⊆ A') (hB : B ⊆ B') : DisjointOccurrence A B ⊆ DisjointOccurrence A' B' := by
-  rintro ω ⟨s, t, hs, ht, hd, ha, hb⟩
-  exact ⟨s, t, hs, ht, hd, hA ha, hB hb⟩
-
- 
-def openSection (A : Set (Set E)) (e : E) : Set (Set E) := {ω | insert e ω ∈ A}
-
-theorem openSection_upper {A : Set (Set E)} (hA : IsUpperSet A) (e : E) :
-    IsUpperSet (openSection A e) :=
-  fun _ _ h hω => hA (Set.insert_subset_insert h) hω
-
-theorem subset_openSection {A : Set (Set E)} (hA : IsUpperSet A) (e : E) :
-    A ⊆ openSection A e := fun ω hω => hA (Set.subset_insert e ω) hω
-
- 
-theorem openSection_disjointOccurrence (A B : Set (Set E)) (_hA : IsUpperSet A)
-    (hB : IsUpperSet B) (e : E) :
-    openSection (DisjointOccurrence A B) e ⊆
-      DisjointOccurrence (openSection A e) B ∪ DisjointOccurrence A (openSection B e) := by
-  rintro ω ⟨s, t, hs, ht, hd, ha, hb⟩
-  by_cases he : e ∈ s
-  · left
-    have het : e ∉ t := fun h => Finset.disjoint_left.mp hd he h
-    refine ⟨s.erase e, t, ?_, ?_, Finset.disjoint_of_subset_left (Finset.erase_subset _ _) hd, ?_, hb⟩
-    · intro z hz
-      exact (hs (Finset.mem_of_mem_erase hz)).resolve_left (Finset.ne_of_mem_erase hz)
-    · intro z hz
-      rcases ht hz with rfl | hz'
-      · exact (het hz).elim
-      · exact hz'
-    · change insert e (↑(s.erase e) : Set E) ∈ A
-      simpa only [← Finset.coe_insert, Finset.insert_erase he] using ha
-  · right
-    refine ⟨s, t.erase e, ?_, ?_, Finset.disjoint_of_subset_right (Finset.erase_subset _ _) hd, ha, ?_⟩
-    · intro z hz
-      rcases hs hz with rfl | hz'
-      · exact (he hz).elim
-      · exact hz'
-    · intro z hz
-      exact (ht (Finset.mem_of_mem_erase hz)).resolve_left (Finset.ne_of_mem_erase hz)
-    · apply hB ?_ hb
-      intro z hz
-      by_cases hze : z = e
-      · exact Or.inl hze
-      · exact Or.inr (Finset.mem_erase.mpr ⟨hze, hz⟩)
-
- 
-
-noncomputable def finiteProbability (p : unitInterval) (s : Finset E) (A : Set (Set E)) : ℝ :=
-  ∑ t ∈ s.powerset, if (t : Set E) ∈ A then
-    (p : ℝ) ^ t.card * (1 - p : ℝ) ^ (s.card - t.card) else 0
-
-theorem finiteProbability_nonneg (p : unitInterval) (s : Finset E) (A : Set (Set E)) :
-    0 ≤ finiteProbability p s A := by
-  apply Finset.sum_nonneg
-  intro t _
-  split_ifs
-  · exact mul_nonneg (pow_nonneg p.2.1 _) (pow_nonneg (sub_nonneg.mpr p.2.2) _)
-  · exact le_rfl
-
-theorem finiteProbability_mono (p : unitInterval) (s : Finset E)
-    {A B : Set (Set E)} (h : A ⊆ B) : finiteProbability p s A ≤ finiteProbability p s B := by
-  apply Finset.sum_le_sum
-  intro t _
-  split_ifs with ha hb hb
-  · exact le_rfl
-  · exact (hb (h ha)).elim
-  · exact mul_nonneg (pow_nonneg p.2.1 _) (pow_nonneg (sub_nonneg.mpr p.2.2) _)
-  · exact le_rfl
-
-theorem finiteProbability_union_add_inter (p : unitInterval) (s : Finset E)
-    (A B : Set (Set E)) :
-    finiteProbability p s (A ∪ B) + finiteProbability p s (A ∩ B) =
-      finiteProbability p s A + finiteProbability p s B := by
-  simp only [finiteProbability, ← Finset.sum_add_distrib]
-  apply Finset.sum_congr rfl
-  intro t _
-  by_cases ha : (t : Set E) ∈ A <;> by_cases hb : (t : Set E) ∈ B <;> simp [ha, hb]
-
-@[simp] theorem finiteProbability_empty (p : unitInterval) (A : Set (Set E)) :
-    finiteProbability p ∅ A = if ∅ ∈ A then 1 else 0 := by
-  simp [finiteProbability]
-
-theorem finiteProbability_insert (p : unitInterval) (s : Finset E) (e : E) (he : e ∉ s)
-    (A : Set (Set E)) :
-    finiteProbability p (insert e s) A =
-      (1 - p : ℝ) * finiteProbability p s A + (p : ℝ) * finiteProbability p s (openSection A e) := by
-  have hd : Disjoint s.powerset (s.powerset.image (insert e)) := by
-    apply Finset.disjoint_left.mpr
-    intro t ht himg
-    obtain ⟨r, hr, rfl⟩ := Finset.mem_image.mp himg
-    exact he ((Finset.mem_powerset.mp ht) (by simp))
-  have hinj : Set.InjOn (insert e) (↑s.powerset : Set (Finset E)) := by
-    intro t ht r hr h
-    have ht' : e ∉ t := fun h => he (Finset.mem_powerset.mp ht h)
-    have hr' : e ∉ r := fun h => he (Finset.mem_powerset.mp hr h)
-    have hh := congrArg (fun t : Finset E => t.erase e) h
-    simpa [ht', hr'] using hh
-  simp only [finiteProbability, Finset.powerset_insert, Finset.sum_union hd,
-    Finset.sum_image hinj, Finset.mul_sum, Finset.card_insert_of_notMem he]
-  congr 1
-  · apply Finset.sum_congr rfl
-    intro t ht
-    have hcard := Finset.card_le_card (Finset.mem_powerset.mp ht)
-    have hexp : s.card + 1 - t.card = (s.card - t.card) + 1 := by omega
-    rw [hexp, pow_succ]
-    split_ifs <;> simp_all [mul_comm, mul_left_comm]
-  · apply Finset.sum_congr rfl
-    intro t ht
-    have het : e ∉ t := fun h => he (Finset.mem_powerset.mp ht h)
-    simp only [Finset.card_insert_of_notMem het, Nat.add_sub_add_right, pow_succ,
-      Finset.coe_insert, openSection, Set.mem_ofPred_eq]
-    split_ifs <;> simp_all [mul_assoc, mul_comm]
-
-end DisjointWitness
-
-section FiniteBK
-attribute [local instance] Classical.propDecidable
-
-@[simp] theorem empty_mem_disjointOccurrence (A B : Set (Set E)) :
-    (∅ : Set E) ∈ DisjointOccurrence A B ↔ ∅ ∈ A ∧ ∅ ∈ B := by
-  constructor
-  · rintro ⟨s, t, hs, ht, _, ha, hb⟩
-    have hse : (s : Set E) = ∅ := Set.subset_empty_iff.mp hs
-    have hte : (t : Set E) = ∅ := Set.subset_empty_iff.mp ht
-    exact ⟨hse ▸ ha, hte ▸ hb⟩
-  · rintro ⟨ha, hb⟩
-    exact ⟨∅, ∅, by simp, by simp, by simp, by simpa using ha, by simpa using hb⟩
-
- 
-
-theorem finiteProbability_disjointOccurrence_le_half (p : unitInterval)
-    (hp : (p : ℝ) ≤ 1 / 2) (s : Finset E) (A B : Set (Set E))
-    (hA : IsUpperSet A) (hB : IsUpperSet B) :
-    finiteProbability p s (DisjointOccurrence A B) ≤
-      finiteProbability p s A * finiteProbability p s B := by
-  induction s using Finset.induction_on generalizing A B with
-  | empty =>
-    simp only [finiteProbability_empty, empty_mem_disjointOccurrence]
-    by_cases ha : (∅ : Set E) ∈ A <;> by_cases hb : (∅ : Set E) ∈ B <;> simp [ha, hb]
-  | @insert e s he ih =>
-    let A1 := openSection A e
-    let B1 := openSection B e
-    let D00 := DisjointOccurrence A B
-    let D10 := DisjointOccurrence A1 B
-    let D01 := DisjointOccurrence A B1
-    have hD : D00 ⊆ D10 ∩ D01 := by
-      intro ω hω
-      exact ⟨disjointOccurrence_mono (subset_openSection hA e) (Set.Subset.refl _) hω,
-        disjointOccurrence_mono (Set.Subset.refl _) (subset_openSection hB e) hω⟩
-    have hsec : finiteProbability p s (openSection D00 e) ≤
-        finiteProbability p s D10 + finiteProbability p s D01 - finiteProbability p s D00 := by
-      have h1 := finiteProbability_mono p s (openSection_disjointOccurrence A B hA hB e)
-      have h2 := finiteProbability_mono p s hD
-      have h3 := finiteProbability_union_add_inter p s D10 D01
-      change finiteProbability p s (openSection D00 e) ≤
-        finiteProbability p s (D10 ∪ D01) at h1
-      linarith
-    have h00 := ih A B hA hB
-    have h10 := ih A1 B (openSection_upper hA e) hB
-    have h01 := ih A B1 hA (openSection_upper hB e)
-    have ha := finiteProbability_mono p s (subset_openSection hA e)
-    have hb := finiteProbability_mono p s (subset_openSection hB e)
-    have hcoef : 0 ≤ 1 - 2 * (p : ℝ) := by linarith
-    have hdif : 0 ≤ ((p : ℝ) ^ 2) *
-        (finiteProbability p s A1 - finiteProbability p s A) *
-        (finiteProbability p s B1 - finiteProbability p s B) :=
-      mul_nonneg (mul_nonneg (sq_nonneg _) (sub_nonneg.mpr ha)) (sub_nonneg.mpr hb)
-    rw [finiteProbability_insert p s e he, finiteProbability_insert p s e he,
-      finiteProbability_insert p s e he]
-    have hm := mul_le_mul_of_nonneg_left hsec p.2.1
-    have hm00 := mul_le_mul_of_nonneg_left h00 hcoef
-    have hm10 := mul_le_mul_of_nonneg_left h10 p.2.1
-    have hm01 := mul_le_mul_of_nonneg_left h01 p.2.1
-    change (1 - (p : ℝ)) * finiteProbability p s D00 +
-      (p : ℝ) * finiteProbability p s (openSection D00 e) ≤
-      ((1 - (p : ℝ)) * finiteProbability p s A + (p : ℝ) * finiteProbability p s A1) *
-      ((1 - (p : ℝ)) * finiteProbability p s B + (p : ℝ) * finiteProbability p s B1)
-    nlinarith
-
-end FiniteBK
-
-section BKMeasure
-attribute [local instance] Classical.propDecidable
-open Filter Topology
-
- 
-theorem finiteProbability_univ [Fintype E] (p : unitInterval) (A : Set (Set E)) :
-    finiteProbability p Finset.univ A = (law p).real A := by
-  have he : (law p).real A = ∑ ω : Set E, if ω ∈ A then (law p).real {ω} else 0 := by
-    have h := integral_fintype (μ := law p) (f := A.indicator (fun _ => (1 : ℝ)))
-      Integrable.of_finite
-    rw [integral_indicator_const (1 : ℝ) (.of_discrete : MeasurableSet A)] at h
-    simpa [smul_eq_mul, mul_ite, Set.indicator_apply] using h
-  rw [he, finiteProbability]
-  apply Finset.sum_bij (fun (t : Finset E) _ => (t : Set E))
-  · simp
-  · intro a _ b _ h
-    exact Finset.coe_injective h
-  · intro ω _
-    exact ⟨ω.toFinset, by simp, by simp⟩
-  · intro t _
-    rw [law, setBernoulli_real_singleton p (Set.subset_univ _) Set.finite_univ]
-    simp [Set.ncard_sdiff' (Set.subset_univ _) Set.finite_univ, Set.ncard_univ,
-      Nat.card_eq_fintype_card]
-
-theorem disjointOccurrence_image {F : Type*} (f : F ↪ E) (A B : Set (Set E)) :
-    (fun ω => f '' ω) ⁻¹' DisjointOccurrence A B =
-      DisjointOccurrence ((fun ω => f '' ω) ⁻¹' A) ((fun ω => f '' ω) ⁻¹' B) := by
-  ext ω
-  constructor
-  · rintro ⟨s, t, hs, ht, hd, ha, hb⟩
-    let s' := s.preimage f f.injective.injOn
-    let t' := t.preimage f f.injective.injOn
-    have hs' : (s' : Set F) = f ⁻¹' (s : Set E) := by simp [s']
-    have ht' : (t' : Set F) = f ⁻¹' (t : Set E) := by simp [t']
-    have hsI : f '' (s' : Set F) = (s : Set E) := by
-      rw [hs', Set.image_preimage_eq_of_subset]
-      exact hs.trans (Set.image_subset_range _ _)
-    have htI : f '' (t' : Set F) = (t : Set E) := by
-      rw [ht', Set.image_preimage_eq_of_subset]
-      exact ht.trans (Set.image_subset_range _ _)
-    refine ⟨s', t', ?_, ?_, ?_, ?_, ?_⟩
-    · rw [hs']
-      exact (Set.preimage_mono hs).trans (by rw [Set.preimage_image_eq _ f.injective])
-    · rw [ht']
-      exact (Set.preimage_mono ht).trans (by rw [Set.preimage_image_eq _ f.injective])
-    · apply Finset.disjoint_left.mpr
-      intro x hx hy
-      exact Finset.disjoint_left.mp hd (by simpa [s'] using hx) (by simpa [t'] using hy)
-    · change f '' (s' : Set F) ∈ A
-      rwa [hsI]
-    · change f '' (t' : Set F) ∈ B
-      rwa [htI]
-  · rintro ⟨s, t, hs, ht, hd, ha, hb⟩
-    refine ⟨s.map f, t.map f, ?_, ?_, ?_, ?_, ?_⟩
-    · simpa only [Finset.coe_map] using Set.image_mono hs
-    · simpa only [Finset.coe_map] using Set.image_mono ht
-    · exact (Finset.disjoint_map f).mpr hd
-    · simpa only [Finset.coe_map, Set.mem_preimage] using ha
-    · simpa only [Finset.coe_map, Set.mem_preimage] using hb
-
-theorem bk_finite_half [Fintype E] (p : unitInterval) (hp : (p : ℝ) ≤ 1 / 2)
-    (A B : Set (Set E)) (hA : IsUpperSet A) (hB : IsUpperSet B) :
-    (law p).real (DisjointOccurrence A B) ≤ (law p).real A * (law p).real B := by
-  simpa only [finiteProbability_univ] using
-    finiteProbability_disjointOccurrence_le_half p hp Finset.univ A B hA hB
-
-theorem bk_restricted_half (p : unitInterval) (hp : (p : ℝ) ≤ 1 / 2)
-    (A B : Set (Set E)) (hA : IsUpperSet A) (hB : IsUpperSet B) (s : Finset E) :
-    (law p).real {ω | ω ∩ (s : Set E) ∈ DisjointOccurrence A B} ≤
-      (law p).real {ω | ω ∩ (s : Set E) ∈ A} *
-      (law p).real {ω | ω ∩ (s : Set E) ∈ B} := by
-  let f : (s : Set E) ↪ E := ⟨Subtype.val, Subtype.val_injective⟩
-  have hreal (C : Set (Set E)) :
-      (law p).real {ω | ω ∩ (s : Set E) ∈ C} =
-        (law p).real {η : Set (s : Set E) | f '' η ∈ C} := by
-    rw [← law_map_restrict p (s : Set E)]
-    simp only [Measure.real]
-    rw [Measure.map_apply (measurable_set_restrict _) MeasurableSet.of_discrete]
-    congr 2
-    ext ω
-    change ω ∩ (s : Set E) ∈ C ↔ f '' (Subtype.val ⁻¹' ω) ∈ C
-    have he : f '' (Subtype.val ⁻¹' ω) = ω ∩ (s : Set E) := by
-      ext e
-      simp [f, and_comm]
-    rw [he]
-  rw [hreal, hreal, hreal]
-  have hh := bk_finite_half p hp ((fun ω => f '' ω) ⁻¹' A)
-    ((fun ω => f '' ω) ⁻¹' B)
-    (fun _ _ h hω => hA (Set.image_mono h) hω)
-    (fun _ _ h hω => hB (Set.image_mono h) hω)
-  rw [← disjointOccurrence_image] at hh
-  exact hh
-
- 
-theorem bk_finiteWitness_half [Countable E] (p : unitInterval) (hp : (p : ℝ) ≤ 1 / 2)
-    (A B : Set (Set E)) (hA : IsUpperSet A) (hB : IsUpperSet B)
-    (hfA : FiniteWitness A) (hfB : FiniteWitness B) :
-    (law p).real (DisjointOccurrence A B) ≤ (law p).real A * (law p).real B := by
-  let := Encodable.ofCountable E
-  have hlim (C : Set (Set E)) (hC : IsUpperSet C) (hfC : FiniteWitness C) :
-      Tendsto (fun n => (law p).real {ω | ω ∩ (edgeExhaustion (E := E) n : Set E) ∈ C})
-        atTop (𝓝 ((law p).real C)) := by
-    have hm : Monotone (fun n => {ω : Set E | ω ∩ (edgeExhaustion (E := E) n : Set E) ∈ C}) := by
-      intro m n h ω hω
-      exact hC (Set.inter_subset_inter_right _ (edgeExhaustion_mono h)) hω
-    have h := tendsto_measure_iUnion_atTop (μ := law p) hm
-    rw [← witness_eq_union C hC hfC] at h
-    exact (ENNReal.tendsto_toReal (measure_ne_top _ _)).comp h
-  apply le_of_tendsto_of_tendsto
-    (hlim _ (disjointOccurrence_upper A B) (finiteWitness_disjointOccurrence A B))
-    ((hlim A hA hfA).mul (hlim B hB hfB))
-  exact Filter.Eventually.of_forall fun n => bk_restricted_half p hp A B hA hB (edgeExhaustion n)
-
-end BKMeasure
-
-section ORLift
-attribute [local instance] Classical.propDecidable
-
-noncomputable def orParameter (q : unitInterval) (m : ℕ) : unitInterval :=
-  ⟨1 - (1 - (q : ℝ)) ^ m,
-    sub_nonneg.mpr (pow_le_one₀ (sub_nonneg.mpr q.2.2) (by linarith [q.2.1])),
-    by linarith [pow_nonneg (sub_nonneg.mpr q.2.2) m]⟩
-
-noncomputable abbrev bitLaw (q : unitInterval) : Measure Prop :=
-  unitInterval.toNNReal q • Measure.dirac True +
-    unitInterval.toNNReal (unitInterval.symm q) • Measure.dirac False
-
-instance bitLaw_isProbabilityMeasure (q : unitInterval) : IsProbabilityMeasure (bitLaw q) := by
-  rw [bitLaw, ← map_threshold q]
-  infer_instance
-
- 
-theorem map_nonempty (q : unitInterval) (F : Type*) [Fintype F] :
-    (law q : Measure (Set F)).map Set.Nonempty = bitLaw (orParameter q (Fintype.card F)) := by
-  let μ := (law q : Measure (Set F)).map Set.Nonempty
-  have hm : Measurable (Set.Nonempty : Set F → Prop) := .of_discrete
-  have : IsProbabilityMeasure μ := inferInstance
-  have hfalse : μ.real {False} = (1 - (q : ℝ)) ^ Fintype.card F := by
-    change ((law q).map Set.Nonempty).real {False} = _
-    simp only [Measure.real, Measure.map_apply hm (measurableSet_singleton _)]
-    have he : Set.Nonempty ⁻¹' ({False} : Set Prop) = ({∅} : Set (Set F)) := by
-      ext ω
-      simp [Set.not_nonempty_iff_eq_empty]
-    rw [he]
-    change (law q).real {∅} = _
-    rw [law, setBernoulli_real_singleton q (Set.empty_subset _) Set.finite_univ]
-    simp [Set.ncard_univ, Nat.card_eq_fintype_card]
-  have htrue : μ.real {True} = 1 - (1 - (q : ℝ)) ^ Fintype.card F := by
-    have he : ({True} : Set Prop) = ({False} : Set Prop)ᶜ := by ext b; simp
-    rw [he, measureReal_compl (measurableSet_singleton _), probReal_univ, hfalse]
-  apply Measure.ext_of_singleton
-  intro b
-  apply (ENNReal.toReal_eq_toReal_iff' (measure_ne_top _ _) (measure_ne_top _ _)).mp
-  change μ.real {b} = (bitLaw (orParameter q (Fintype.card F))).real {b}
-  by_cases hb : b
-  · have he : b = True := propext (iff_true_intro hb)
-    subst b
-    rw [htrue]
-    simp [bitLaw, Measure.real, orParameter]
-  · have he : b = False := propext (iff_false_intro hb)
-    subst b
-    rw [hfalse]
-    simp [bitLaw, Measure.real, orParameter]
-
- 
-theorem finite_image_lift {F : Type*} (f : F → E) {ω : Set F} (s : Finset E)
-    (hs : (s : Set E) ⊆ f '' ω) :
-    ∃ t : Finset F, (t : Set F) ⊆ ω ∧ f '' (t : Set F) = (s : Set E) := by
-  have hex : ∃ t ⊆ f '' ω, t.Finite ∧ t = (s : Set E) :=
-    ⟨s, hs, s.finite_toSet, rfl⟩
-  obtain ⟨t, ht, htf, hti⟩ := Set.exists_subset_image_finite_and.mp hex
-  exact ⟨htf.toFinset, by simpa using ht, by simpa using hti⟩
-
-theorem finiteWitness_image_preimage {F : Type*} (f : F → E) (A : Set (Set E))
-    (hA : FiniteWitness A) : FiniteWitness ((fun ω => f '' ω) ⁻¹' A) := by
-  intro ω hω
-  obtain ⟨s, hs, ha⟩ := hA (f '' ω) hω
-  obtain ⟨t, ht, htI⟩ := finite_image_lift f s hs
-  exact ⟨t, ht, by simpa only [Set.mem_preimage, htI] using ha⟩
-
-theorem image_preimage_disjointOccurrence {F : Type*} (f : F → E) (A B : Set (Set E)) :
-    (fun ω => f '' ω) ⁻¹' DisjointOccurrence A B ⊆
-      DisjointOccurrence ((fun ω => f '' ω) ⁻¹' A) ((fun ω => f '' ω) ⁻¹' B) := by
-  rintro ω ⟨s, t, hs, ht, hd, ha, hb⟩
-  obtain ⟨s', hs', hsI⟩ := finite_image_lift f s hs
-  obtain ⟨t', ht', htI⟩ := finite_image_lift f t ht
-  refine ⟨s', t', hs', ht', ?_, ?_, ?_⟩
-  · apply Finset.disjoint_left.mpr
-    intro x hx hy
-    have hxs : f x ∈ (s : Set E) := hsI ▸ Set.mem_image_of_mem f hx
-    have hxt : f x ∈ (t : Set E) := htI ▸ Set.mem_image_of_mem f hy
-    exact Finset.disjoint_left.mp hd hxs hxt
-  · simpa only [Set.mem_preimage, hsI] using ha
-  · simpa only [Set.mem_preimage, htI] using hb
-
-end ORLift
-
-section ORLaw
-attribute [local instance] Classical.propDecidable
-
-theorem map_exists_bitLaw (q : unitInterval) (F : Type*) [Fintype F] :
-    (Measure.infinitePi (fun _ : F => bitLaw q)).map (fun f : F → Prop => ∃ j, f j) =
-      bitLaw (orParameter q (Fintype.card F)) := by
-  have h := map_nonempty q F
-  unfold law at h
-  rw [setBernoulli_eq_map, Measure.map_map (Measurable.of_discrete) measurable_setOfPred] at h
-  simpa only [Set.mem_univ, Function.comp_def, Set.Nonempty, Set.mem_ofPred_eq,
-    bernoulliMeasure_def, bitLaw] using h
-
-theorem measurable_image_fst (F : Type*) [Fintype F] :
-    Measurable (fun ω : Set (E × F) => Prod.fst '' ω) := by
-  have hm : Measurable (fun ω : Set (E × F) => fun e => ∃ j : F, (e,j) ∈ ω) := by
-    apply Measurable.of_eval
-    intro e
-    apply measurableSet_setOfPred.mp
-    have he : {ω : Set (E × F) | ∃ j : F, (e,j) ∈ ω} =
-        ⋃ j : F, {ω : Set (E × F) | (e,j) ∈ ω} := by ext ω; simp
-    rw [he]
-    exact MeasurableSet.iUnion fun j => measurableSet_setOfPred.mpr
-      ((measurable_pi_apply (e,j)).comp MeasurableEquiv.setOfPred.symm.measurable)
-  have he : (fun ω : Set (E × F) => Prod.fst '' ω) =
-      (fun f : E → Prop => {e | f e}) ∘ (fun ω e => ∃ j : F, (e,j) ∈ ω) := by
-    funext ω
-    ext e
-    simp [Prod.exists]
-  rw [he]
-  exact measurable_setOfPred.comp hm
-
- 
-theorem law_map_image_fst (q : unitInterval) (F : Type*) [Fintype F] :
-    (law q : Measure (Set (E × F))).map (fun ω => Prod.fst '' ω) =
-      law (orParameter q (Fintype.card F)) := by
-  let c := MeasurableEquiv.curry E F Prop
-  let b := fun f : E → F → Prop => fun e => ∃ j, f e j
-  have hb : Measurable b := Measurable.of_eval
-    (fun e => (Measurable.of_discrete : Measurable (fun f : F → Prop => ∃j, f j)).comp
-      (measurable_pi_apply e))
-  unfold law
-  rw [setBernoulli_eq_map, Measure.map_map (measurable_image_fst F) measurable_setOfPred]
-  simp only [Set.mem_univ]
-  have he : (fun ω : Set (E × F) => Prod.fst '' ω) ∘ (fun f => {e | f e}) =
-      (fun f : E → Prop => {e | f e}) ∘ b ∘ c := by
-    funext f
-    ext e
-    simp [b, c, Function.comp_def, Prod.exists]
-  rw [he, ← Measure.map_map measurable_setOfPred (hb.comp c.measurable),
-    ← Measure.map_map hb c.measurable]
-  change ((Measure.infinitePi (fun _ : E × F => bitLaw q)).map c |>.map b |>.map
-    (fun f : E → Prop => {e | f e})) = _
-  rw [Measure.infinitePi_map_curry (fun (_ : E) (_ : F) => bitLaw q)]
-  dsimp only [b]
-  rw [Measure.infinitePi_map_pi _ (fun _ => Measurable.of_discrete)]
-  rw [map_exists_bitLaw q F]
-  rw [setBernoulli_eq_map]
-  rfl
-
- 
-theorem exists_orParameter (p : unitInterval) (hp : p < 1) :
-    ∃ m : ℕ, ∃ q : unitInterval, (q : ℝ) ≤ 1 / 2 ∧ orParameter q m = p := by
-  obtain ⟨m, hm⟩ := exists_pow_lt_of_lt_one (show 0 < 1 - (p : ℝ) by exact sub_pos.mpr hp)
-    (show (1 / 2 : ℝ) < 1 by norm_num)
-  let f := fun x : ℝ => 1 - (1-x)^m
-  have hf : Continuous f := by fun_prop
-  have h0 : f 0 = 0 := by simp [f]
-  have hp' : (p : ℝ) ∈ Set.Icc (f 0) (f (1/2)) := by
-    rw [h0]
-    refine ⟨p.2.1, ?_⟩
-    dsimp [f]
-    norm_num at hm ⊢
-    linarith
-  obtain ⟨q, hq, hqp⟩ := intermediate_value_Icc (by norm_num : (0 : ℝ) ≤ 1/2)
-    hf.continuousOn hp'
-  refine ⟨m, ⟨q, hq.1, hq.2.trans (by norm_num)⟩, hq.2, ?_⟩
-  apply Subtype.ext
-  exact hqp
-
-end ORLaw
-
-section BK
-attribute [local instance] Classical.propDecidable
-
-theorem disjointOccurrence_subset_inter (A B : Set (Set E))
-    (hA : IsUpperSet A) (hB : IsUpperSet B) : DisjointOccurrence A B ⊆ A ∩ B := by
-  rintro ω ⟨s,t,hs,ht,_,ha,hb⟩
-  exact ⟨hA hs ha, hB ht hb⟩
-
- 
-
-theorem bk [Countable E] (p : unitInterval) (A B : Set (Set E))
-    (hA : IsUpperSet A) (hB : IsUpperSet B)
-    (hfA : FiniteWitness A) (hfB : FiniteWitness B) :
-    (law p).real (DisjointOccurrence A B) ≤ (law p).real A * (law p).real B := by
-  by_cases hp : p = 1
-  · subst p
-    by_cases hu : Set.univ ∈ DisjointOccurrence A B
-    · have ha := (disjointOccurrence_subset_inter A B hA hB hu).1
-      have hb := (disjointOccurrence_subset_inter A B hA hB hu).2
-      simp [law, Measure.real, hu, ha, hb]
-    · simp [law, Measure.real, hu]
-      positivity
-  · have hp' : p < 1 := lt_of_le_of_ne p.2.2 hp
-    obtain ⟨m, q, hq, hqp⟩ := exists_orParameter p hp'
-    let f : E × Fin m → E := Prod.fst
-    let A' := (fun ω => f '' ω) ⁻¹' A
-    let B' := (fun ω => f '' ω) ⁻¹' B
-    have hreal (C : Set (Set E)) (hC : MeasurableSet C) :
-        (law p).real C = (law q).real ((fun ω : Set (E × Fin m) => f '' ω) ⁻¹' C) := by
-      have hh := law_map_image_fst (E := E) q (Fin m)
-      simp only [Fintype.card_fin, hqp] at hh
-      rw [← hh]
-      simp only [Measure.real, Measure.map_apply (measurable_image_fst (Fin m)) hC]
-      rfl
-    calc
-      (law p).real (DisjointOccurrence A B) =
-          (law q).real ((fun ω : Set (E × Fin m) => f '' ω) ⁻¹' DisjointOccurrence A B) :=
-        hreal _ (measurableSet_finiteWitness _ (disjointOccurrence_upper A B)
-          (finiteWitness_disjointOccurrence A B))
-      _ ≤ (law q).real (DisjointOccurrence A' B') :=
-        measureReal_mono (image_preimage_disjointOccurrence f A B)
-      _ ≤ (law q).real A' * (law q).real B' := bk_finiteWitness_half q hq A' B'
-        (fun _ _ h hω => hA (Set.image_mono h) hω)
-        (fun _ _ h hω => hB (Set.image_mono h) hω)
-        (finiteWitness_image_preimage f A hfA) (finiteWitness_image_preimage f B hfB)
-      _ = (law p).real A * (law p).real B := by
-        rw [hreal A (measurableSet_finiteWitness A hA hfA),
-          hreal B (measurableSet_finiteWitness B hB hfB)]
-
-end BK
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Kernel.Bounds
+
+namespace OAI
+
+/-! Association, disjoint witnesses and the BK inequality. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+section FiniteAssociation
+attribute [local instance] Classical.propDecidable
+variable [Fintype E]
+
+private theorem bernoulli_atom_product (p : unitInterval) (ω : Set E) :
+    (law p).real {ω} = ∏ e : E, if e ∈ ω then (p : ℝ) else 1 - p := by
+  rw [law, setBernoulli_real_singleton p (Set.subset_univ ω) (Set.finite_univ)]
+  simp [Finset.prod_ite, ← Set.ncard_coe_finset, Set.sdiff_eq]
+  exact Or.inl rfl
+
+private theorem bernoulli_atom_modular (p : unitInterval) (a b : Set E) :
+    (law p).real {a} * (law p).real {b} =
+      (law p).real {a ∩ b} * (law p).real {a ∪ b} := by
+  simp only [bernoulli_atom_product, ← Finset.prod_mul_distrib]
+  apply Finset.prod_congr rfl
+  intro e _
+  by_cases ha : e ∈ a <;> by_cases hb : e ∈ b <;> simp [ha, hb, mul_comm]
+
+ 
+
+theorem association_finite (p : unitInterval) (f g : Set E → ℝ)
+    (hf : Monotone f) (hg : Monotone g) (hf0 : ∀ ω, 0 ≤ f ω) (hg0 : ∀ ω, 0 ≤ g ω) :
+    (∫ ω, f ω ∂law p) * (∫ ω, g ω ∂law p) ≤ ∫ ω, f ω * g ω ∂law p := by
+  have h := fkg f g (fun ω : Set E => (law p).real {ω})
+    (fun _ => measureReal_nonneg) hf0 hg0 hf hg
+    (fun a b => (bernoulli_atom_modular p a b).le)
+  have hsum : (∑ ω : Set E, (law p).real {ω}) = 1 := by
+    simp
+  rw [hsum, one_mul] at h
+  simpa only [integral_fintype Integrable.of_finite, smul_eq_mul] using h
+
+ 
+theorem association_events_finite (p : unitInterval) (A B : Set (Set E))
+    (hA : IsUpperSet A) (hB : IsUpperSet B) :
+    (law p).real A * (law p).real B ≤ (law p).real (A ∩ B) := by
+  have hm (S : Set (Set E)) (hS : IsUpperSet S) :
+      Monotone (S.indicator (fun _ => (1 : ℝ))) := by
+    intro a b hab
+    by_cases ha : a ∈ S
+    · simp [ha, hS hab ha]
+    · simp only [Set.indicator_of_notMem ha]
+      exact Set.indicator_nonneg (fun _ _ => zero_le_one) _
+  have he : (fun ω => A.indicator (fun _ => (1 : ℝ)) ω *
+      B.indicator (fun _ => (1 : ℝ)) ω) = (A ∩ B).indicator (fun _ => 1) := by
+    funext ω
+    by_cases ha : ω ∈ A <;> by_cases hb : ω ∈ B <;> simp [ha, hb]
+  have h := association_finite p _ _ (hm A hA) (hm B hB)
+    (Set.indicator_nonneg (fun _ _ => zero_le_one))
+    (Set.indicator_nonneg (fun _ _ => zero_le_one))
+  rw [he] at h
+  simpa only [integral_indicator_const (1 : ℝ) (.of_discrete : MeasurableSet A),
+    integral_indicator_const (1 : ℝ) (.of_discrete : MeasurableSet B),
+    integral_indicator_const (1 : ℝ) (.of_discrete : MeasurableSet (A ∩ B)),
+    smul_eq_mul, mul_one] using h
+
+end FiniteAssociation
+
+section Restriction
+
+theorem measurable_set_restrict (s : Set E) :
+    Measurable (fun ω : Set E => (Subtype.val ⁻¹' ω : Set s)) :=
+  measurable_setOfPred.comp (Measurable.of_eval
+    (fun e => (measurable_pi_apply e.val).comp MeasurableEquiv.setOfPred.symm.measurable))
+
+ 
+theorem law_map_restrict (p : unitInterval) (s : Set E) :
+    (law p : Measure (Set E)).map (fun ω => (Subtype.val ⁻¹' ω : Set s)) = law p := by
+  have hm := measurable_set_restrict s
+  unfold law
+  rw [setBernoulli_eq_map, Measure.map_map hm measurable_setOfPred]
+  have he : (fun ω : Set E => (Subtype.val ⁻¹' ω : Set s)) ∘
+      (fun f : E → Prop => {e | f e}) =
+      (fun f : s → Prop => {e | f e}) ∘ s.domRestrict := rfl
+  rw [he, ← Measure.map_map measurable_setOfPred (by fun_prop),
+    Measure.infinitePi_map_restrict', setBernoulli_eq_map]
+  simp only [Set.mem_univ]
+
+end Restriction
+
+section WitnessAssociation
+attribute [local instance] Classical.propDecidable
+open Filter Topology
+
+ 
+def FiniteWitness (A : Set (Set E)) : Prop :=
+  ∀ ω ∈ A, ∃ s : Finset E, (s : Set E) ⊆ ω ∧ (s : Set E) ∈ A
+
+theorem association_restricted (p : unitInterval) (A B : Set (Set E))
+    (hA : IsUpperSet A) (hB : IsUpperSet B) (s : Finset E) :
+    (law p).real {ω | ω ∩ (s : Set E) ∈ A} *
+      (law p).real {ω | ω ∩ (s : Set E) ∈ B} ≤
+      (law p).real {ω | ω ∩ (s : Set E) ∈ A ∩ B} := by
+  let A' : Set (Set (s : Set E)) := {η | Subtype.val '' η ∈ A}
+  let B' : Set (Set (s : Set E)) := {η | Subtype.val '' η ∈ B}
+  have hA' : IsUpperSet A' := fun _ _ h ha => hA (Set.image_mono h) ha
+  have hB' : IsUpperSet B' := fun _ _ h hb => hB (Set.image_mono h) hb
+  have he (ω : Set E) : Subtype.val '' (Subtype.val ⁻¹' ω : Set (s : Set E)) =
+      ω ∩ (s : Set E) := by
+    ext e
+    simp [and_comm]
+  have hl (C : Set (Set E)) :
+      (law p : Measure (Set (s : Set E))).real {η | Subtype.val '' η ∈ C} =
+      (law p).real {ω | ω ∩ (s : Set E) ∈ C} := by
+    rw [← law_map_restrict p (s : Set E), measureReal_def,
+      Measure.map_apply (measurable_set_restrict _) .of_discrete]
+    simp only [Set.preimage_ofPred_eq, he, measureReal_def]
+  have ht := association_events_finite p A' B' hA' hB'
+  change (law p).real {η | Subtype.val '' η ∈ A} *
+    (law p).real {η | Subtype.val '' η ∈ B} ≤
+    (law p).real {η | Subtype.val '' η ∈ A ∩ B} at ht
+  simpa only [hl] using ht
+
+ noncomputable def edgeExhaustion [Encodable E] (n : ℕ) : Finset E :=
+  (Finset.range n).preimage Encodable.encode Encodable.encode_injective.injOn
+
+ theorem edgeExhaustion_mono [Encodable E] :
+    Monotone (edgeExhaustion : ℕ → Finset E) := by
+  intro m n h e he
+  simp only [edgeExhaustion, Finset.mem_preimage, Finset.mem_range] at he ⊢
+  exact he.trans_le h
+
+private theorem subset_edgeExhaustion [Encodable E] (s : Finset E) :
+    ∃ n, s ⊆ edgeExhaustion n := by
+  refine ⟨s.sup Encodable.encode + 1, ?_⟩
+  intro e he
+  simp only [edgeExhaustion, Finset.mem_preimage, Finset.mem_range]
+  exact Nat.lt_succ_of_le (Finset.le_sup he)
+
+ theorem witness_eq_union [Encodable E] (A : Set (Set E))
+    (hA : IsUpperSet A) (hf : FiniteWitness A) :
+    A = ⋃ n, {ω | ω ∩ (edgeExhaustion (E := E) n : Set E) ∈ A} := by
+  ext ω
+  simp only [Set.mem_iUnion, Set.mem_ofPred_eq]
+  constructor
+  · intro hω
+    obtain ⟨s, hs, hsa⟩ := hf ω hω
+    obtain ⟨n, hn⟩ := subset_edgeExhaustion s
+    exact ⟨n, hA (fun _ he => ⟨hs he, hn he⟩) hsa⟩
+  · rintro ⟨n, hn⟩
+    exact hA Set.inter_subset_left hn
+
+ 
+
+theorem association_finiteWitness [Countable E] (p : unitInterval)
+    (A B : Set (Set E)) (hA : IsUpperSet A) (hB : IsUpperSet B)
+    (hfA : FiniteWitness A) (hfB : FiniteWitness B) :
+    (law p).real A * (law p).real B ≤ (law p).real (A ∩ B) := by
+  let := Encodable.ofCountable E
+  have hfAB : FiniteWitness (A ∩ B) := by
+    intro ω hω
+    obtain ⟨s, hs, hsa⟩ := hfA ω hω.1
+    obtain ⟨t, ht, htb⟩ := hfB ω hω.2
+    refine ⟨s ∪ t, ?_, hA ?_ hsa, hB ?_ htb⟩
+    · simpa only [Finset.coe_union] using Set.union_subset hs ht
+    · simp only [Finset.coe_union]; exact Set.subset_union_left
+    · simp only [Finset.coe_union]; exact Set.subset_union_right
+  have hlim (C : Set (Set E)) (hC : IsUpperSet C) (hfC : FiniteWitness C) :
+      Tendsto (fun n => (law p).real {ω | ω ∩ (edgeExhaustion (E := E) n : Set E) ∈ C})
+        atTop (𝓝 ((law p).real C)) := by
+    have hm : Monotone (fun n => {ω : Set E | ω ∩ (edgeExhaustion (E := E) n : Set E) ∈ C}) := by
+      intro m n h ω hω
+      exact hC (Set.inter_subset_inter_right _ (edgeExhaustion_mono h)) hω
+    have h := tendsto_measure_iUnion_atTop (μ := law p) hm
+    rw [← witness_eq_union C hC hfC] at h
+    exact (ENNReal.tendsto_toReal (measure_ne_top _ _)).comp h
+  exact le_of_tendsto_of_tendsto
+    ((hlim A hA hfA).mul (hlim B hB hfB)) (hlim (A ∩ B) (hA.inter hB) hfAB)
+    (Filter.Eventually.of_forall fun n => association_restricted p A B hA hB (edgeExhaustion n))
+
+end WitnessAssociation
+
+section ClusterEvents
+attribute [local instance] Classical.propDecidable
+open Filter Topology
+
+ 
+theorem measurableSet_finiteWitness [Countable E] (A : Set (Set E))
+    (hA : IsUpperSet A) (hfA : FiniteWitness A) : MeasurableSet A := by
+  let := Encodable.ofCountable E
+  rw [witness_eq_union A hA hfA]
+  apply MeasurableSet.iUnion
+  intro n
+  let s := edgeExhaustion (E := E) n
+  have he : {ω : Set E | ω ∩ (s : Set E) ∈ A} =
+      (fun ω => (Subtype.val ⁻¹' ω : Set (s : Set E))) ⁻¹'
+        {η | Subtype.val '' η ∈ A} := by
+    ext ω
+    change ω ∩ (s : Set E) ∈ A ↔ Subtype.val '' (Subtype.val ⁻¹' ω : Set (s : Set E)) ∈ A
+    have hi : Subtype.val '' (Subtype.val ⁻¹' ω : Set (s : Set E)) = ω ∩ (s : Set E) := by
+      ext e
+      simp [and_comm]
+    rw [hi]
+  change MeasurableSet {ω : Set E | ω ∩ (s : Set E) ∈ A}
+  rw [he]
+  exact MeasurableSet.preimage .of_discrete (measurable_set_restrict _)
+
+theorem finiteWitness_conn (G : BondGraph V E) (x y : V) :
+    FiniteWitness {ω | Conn G ω x y} := by
+  intro ω h
+  induction h with
+  | refl => exact ⟨∅, by simp, by simp⟩
+  | @tail b c hab hbc ih =>
+    obtain ⟨s, hs, hconn⟩ := ih
+    obtain ⟨e, he, hGe⟩ := hbc
+    refine ⟨insert e s, ?_, ?_⟩
+    · simpa only [Finset.coe_insert] using Set.insert_subset he hs
+    · exact (conn_mono G (by simp) hconn).tail ⟨e, by simp, hGe⟩
+
+theorem finiteWitness_conn_finset (G : BondGraph V E) (x : V) (t : Finset V) :
+    FiniteWitness {ω | ∀ y ∈ t, Conn G ω x y} := by
+  intro ω h
+  induction t using Finset.induction_on with
+  | empty => exact ⟨∅, by simp, by simp⟩
+  | @insert a t ha ih =>
+    obtain ⟨s, hs, hsa⟩ := finiteWitness_conn G x a ω (h a (by simp))
+    obtain ⟨r, hr, hrt⟩ := ih (fun y hy => h y (by simp [hy]))
+    refine ⟨s ∪ r, ?_, ?_⟩
+    · simpa only [Finset.coe_union] using Set.union_subset hs hr
+    · intro y hy
+      rcases Finset.mem_insert.mp hy with rfl | hy
+      · exact conn_mono G (by simp) hsa
+      · exact conn_mono G (by simp) (hrt y hy)
+
+ 
+def clusterLarge (G : BondGraph V E) (x : V) (n : ℕ) : Set (Set E) :=
+  {ω | ∃ t : Finset V, n ≤ t.card ∧ ∀ y ∈ t, Conn G ω x y}
+
+theorem clusterLarge_upper (G : BondGraph V E) (x : V) (n : ℕ) :
+    IsUpperSet (clusterLarge G x n) := by
+  rintro ω η h ⟨t, ht, hc⟩
+  exact ⟨t, ht, fun y hy => conn_mono G h (hc y hy)⟩
+
+theorem finiteWitness_clusterLarge (G : BondGraph V E) (x : V) (n : ℕ) :
+    FiniteWitness (clusterLarge G x n) := by
+  rintro ω ⟨t, ht, hc⟩
+  obtain ⟨s, hs, hsc⟩ := finiteWitness_conn_finset G x t ω hc
+  exact ⟨s, hs, t, ht, hsc⟩
+
+theorem measurableSet_clusterLarge [Countable E] (G : BondGraph V E) (x : V) (n : ℕ) :
+    MeasurableSet (clusterLarge G x n) :=
+  measurableSet_finiteWitness _ (clusterLarge_upper G x n) (finiteWitness_clusterLarge G x n)
+
+theorem clusterLarge_antitone (G : BondGraph V E) (x : V) :
+    Antitone (clusterLarge G x) := by
+  rintro m n h ω ⟨t, ht, hc⟩
+  exact ⟨t, h.trans ht, hc⟩
+
+theorem infinite_cluster_eq_iInter (G : BondGraph V E) (x : V) :
+    {ω | (cluster G ω x).Infinite} = ⋂ n, clusterLarge G x n := by
+  ext ω
+  simp only [Set.mem_iInter, Set.mem_ofPred_eq]
+  constructor
+  · intro h n
+    obtain ⟨t, ht, hcard⟩ := h.exists_subset_card_eq n
+    exact ⟨t, hcard.ge, ht⟩
+  · intro h hfin
+    obtain ⟨t, ht, hc⟩ := h (hfin.toFinset.card + 1)
+    have hsub : t ⊆ hfin.toFinset := fun y hy => hfin.mem_toFinset.mpr (hc y hy)
+    have := Finset.card_le_card hsub
+    omega
+
+theorem measurableSet_infinite_cluster [Countable E] (G : BondGraph V E) (x : V) :
+    MeasurableSet {ω | (cluster G ω x).Infinite} := by
+  rw [infinite_cluster_eq_iInter]
+  exact MeasurableSet.iInter (measurableSet_clusterLarge G x)
+
+ 
+theorem association_decreasing [Countable E] (p : unitInterval)
+    (A B : ℕ → Set (Set E)) (hA : Antitone A) (hB : Antitone B)
+    (huA : ∀ n, IsUpperSet (A n)) (huB : ∀ n, IsUpperSet (B n))
+    (hfA : ∀ n, FiniteWitness (A n)) (hfB : ∀ n, FiniteWitness (B n)) :
+    (law p).real (⋂ n, A n) * (law p).real (⋂ n, B n) ≤
+      (law p).real ((⋂ n, A n) ∩ (⋂ n, B n)) := by
+  have hlim (C : ℕ → Set (Set E)) (hm : Antitone C) (hC : ∀ n, MeasurableSet (C n)) :
+      Tendsto (fun n => (law p).real (C n)) atTop (𝓝 ((law p).real (⋂ n, C n))) :=
+    (ENNReal.tendsto_toReal (measure_ne_top _ _)).comp
+      (tendsto_measure_iInter_atTop (fun n => (hC n).nullMeasurableSet) hm
+        ⟨0, measure_ne_top _ _⟩)
+  have hmA n := measurableSet_finiteWitness (A n) (huA n) (hfA n)
+  have hmB n := measurableSet_finiteWitness (B n) (huB n) (hfB n)
+  have h := le_of_tendsto_of_tendsto ((hlim A hA hmA).mul (hlim B hB hmB))
+    (hlim (fun n => A n ∩ B n) (fun m n h => Set.inter_subset_inter (hA h) (hB h))
+      (fun n => (hmA n).inter (hmB n)))
+    (Filter.Eventually.of_forall fun n =>
+      association_finiteWitness p (A n) (B n) (huA n) (huB n) (hfA n) (hfB n))
+  simpa only [Set.iInter_inter_distrib] using h
+
+ 
+
+theorem association_infinite_cluster [Countable E]
+    (G : BondGraph V E) (p : unitInterval) (x y : V) :
+    (law p).real {ω | (cluster G ω x).Infinite} *
+      (law p).real {ω | (cluster G ω y).Infinite} ≤
+      (law p).real {ω | (cluster G ω x).Infinite ∧ (cluster G ω y).Infinite} := by
+  rw [infinite_cluster_eq_iInter, infinite_cluster_eq_iInter]
+  change _ ≤ (law p).real ({ω | (cluster G ω x).Infinite} ∩
+    {ω | (cluster G ω y).Infinite})
+  rw [infinite_cluster_eq_iInter, infinite_cluster_eq_iInter]
+  exact association_decreasing p _ _ (clusterLarge_antitone G x) (clusterLarge_antitone G y)
+    (clusterLarge_upper G x) (clusterLarge_upper G y)
+    (finiteWitness_clusterLarge G x) (finiteWitness_clusterLarge G y)
+
+end ClusterEvents
+
+section Automorphisms
+attribute [local instance] Classical.propDecidable
+
+theorem measurable_set_preimage (b : E ≃ E) :
+    Measurable (fun ω : Set E => b ⁻¹' ω) :=
+  measurable_setOfPred.comp (Measurable.of_eval
+    (fun e => (measurable_pi_apply (b e)).comp MeasurableEquiv.setOfPred.symm.measurable))
+
+ 
+theorem law_map_preimage_equiv (p : unitInterval) (b : E ≃ E) :
+    (law p : Measure (Set E)).map (fun ω => b ⁻¹' ω) = law p := by
+  unfold law
+  rw [setBernoulli_eq_map, Measure.map_map (measurable_set_preimage b) measurable_setOfPred]
+  have he : (fun ω : Set E => b ⁻¹' ω) ∘ (fun f : E → Prop => {e | f e}) =
+      (fun f : E → Prop => {e | f e}) ∘
+        MeasurableEquiv.piCongrLeft (fun _ : E => Prop) b.symm := by
+    funext f
+    ext e
+    simp only [Function.comp_apply, Set.mem_preimage, Set.mem_ofPred_eq]
+    have hh := MeasurableEquiv.piCongrLeft_apply_apply (β := fun _ : E => Prop) b.symm f (b e)
+    exact iff_of_eq (by simpa only [Equiv.symm_apply_apply] using hh.symm)
+  rw [he, ← Measure.map_map measurable_setOfPred (MeasurableEquiv.measurable _)]
+  simp only [Set.mem_univ]
+  congr 1
+  exact Measure.infinitePi_map_piCongrLeft
+    (fun _ : E => unitInterval.toNNReal p • Measure.dirac True +
+      unitInterval.toNNReal (unitInterval.symm p) • Measure.dirac False) b.symm
+
+theorem isAut_symm (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (h : IsAut G a b) : IsAut G a.symm b.symm := by
+  intro e
+  have hh := congrArg (Sym2.map a.symm) (h (b.symm e))
+  simpa only [Equiv.apply_symm_apply, Sym2.map_map, Equiv.symm_comp_self,
+    Sym2.map_id', id_eq] using hh.symm
+
+theorem conn_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (h : IsAut G a b) {ω : Set E} {x y : V} (hc : Conn G ω x y) :
+    Conn G (b '' ω) (a x) (a y) := by
+  induction hc with
+  | refl => exact conn_refl G _ _
+  | @tail v w hvw hed ih =>
+    obtain ⟨e, he, hGe⟩ := hed
+    exact ih.tail ⟨b e, Set.mem_image_of_mem _ he, by rw [h e, hGe, Sym2.map_mk]⟩
+
+theorem conn_aut_iff (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (h : IsAut G a b) (ω : Set E) (x y : V) :
+    Conn G (b ⁻¹' ω) x y ↔ Conn G ω (a x) (a y) := by
+  constructor
+  · intro hc
+    simpa only [Set.image_preimage_eq _ b.surjective] using conn_aut G h hc
+  · intro hc
+    have he : b.symm '' ω = b ⁻¹' ω := by
+      ext e
+      simp
+    simpa only [Equiv.symm_apply_apply, he] using
+      conn_aut G (isAut_symm G h) hc
+
+theorem cluster_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (h : IsAut G a b) (ω : Set E) (x : V) :
+    a '' cluster G (b ⁻¹' ω) x = cluster G ω (a x) := by
+  ext y
+  constructor
+  · rintro ⟨z, hz, rfl⟩
+    exact (conn_aut_iff G h ω x z).mp hz
+  · intro hy
+    change Conn G ω (a x) y at hy
+    refine ⟨a.symm y, ?_, by simp⟩
+    exact (conn_aut_iff G h ω x (a.symm y)).mpr (by simpa only [Equiv.apply_symm_apply] using hy)
+
+theorem infinite_cluster_aut [Countable E] (G : BondGraph V E) (p : unitInterval)
+    {a : V ≃ V} {b : E ≃ E} (h : IsAut G a b) (x : V) :
+    (law p).real {ω | (cluster G ω (a x)).Infinite} =
+      (law p).real {ω | (cluster G ω x).Infinite} := by
+  have he : {ω | (cluster G ω (a x)).Infinite} =
+      (fun ω => b ⁻¹' ω) ⁻¹' {ω | (cluster G ω x).Infinite} := by
+    ext ω
+    change (cluster G ω (a x)).Infinite ↔ (cluster G (b ⁻¹' ω) x).Infinite
+    rw [← cluster_aut G h]
+    exact Set.infinite_image_iff (fun _ _ _ _ hab => a.injective hab)
+  unfold Measure.real
+  rw [he, ← Measure.map_apply (measurable_set_preimage b)
+    (measurableSet_infinite_cluster G x), law_map_preimage_equiv]
+
+ 
+theorem finite_open_probability (p : unitInterval) (s : Finset E) :
+    (law p).real {ω | (s : Set E) ⊆ ω} = (p : ℝ) ^ s.card := by
+  have he : {ω : Set E | (s : Set E) ⊆ ω} =
+      (fun ω => (Subtype.val ⁻¹' ω : Set (s : Set E))) ⁻¹' {Set.univ} := by
+    ext ω
+    simp only [Set.mem_ofPred_eq, Set.mem_preimage, Set.mem_singleton_iff]
+    exact ⟨fun h => Set.eq_univ_of_forall fun e => h e.property,
+      fun h e he => show (⟨e, he⟩ : (s : Set E)) ∈ (Subtype.val ⁻¹' ω) from
+        h.symm ▸ Set.mem_univ _⟩
+  unfold Measure.real
+  rw [he, ← Measure.map_apply (measurable_set_restrict _) (by measurability),
+    law_map_restrict]
+  change (law p : Measure (Set (s : Set E))).real {Set.univ} = _
+  simp [law, setBernoulli_real_singleton p (Set.Subset.refl _) (Set.toFinite _)]
+
+theorem twoPoint_pos [Countable E] (G : BondGraph V E) (hG : Connected G)
+    (p : unitInterval) (hp : 0 < p) (x y : V) : 0 < twoPoint G p x y := by
+  obtain ⟨s, _, hs⟩ := finiteWitness_conn G x y Set.univ (conn_univ_of_connected G hG x y)
+  have hpos : 0 < (law p).real {ω | (s : Set E) ⊆ ω} := by
+    rw [finite_open_probability]
+    exact pow_pos hp _
+  apply lt_of_lt_of_le hpos
+  exact measureReal_mono (fun _ hω => conn_mono G hω hs)
+
+end Automorphisms
+
+section UniquenessThreshold
+attribute [local instance] Classical.propDecidable
+variable [Countable V] [Countable E]
+
+omit [Countable V] in
+ 
+theorem association_infinite_conn (G : BondGraph V E) (p : unitInterval) (x y z : V) :
+    (law p).real {ω | (cluster G ω x).Infinite} * twoPoint G p y z ≤
+      (law p).real {ω | (cluster G ω x).Infinite ∧ Conn G ω y z} := by
+  have h := association_decreasing p (clusterLarge G x) (fun _ => {ω | Conn G ω y z})
+    (clusterLarge_antitone G x) antitone_const (clusterLarge_upper G x)
+    (fun _ _ _ hsub hconn => conn_mono G hsub hconn) (finiteWitness_clusterLarge G x)
+    (fun _ => finiteWitness_conn G y z)
+  simpa only [← infinite_cluster_eq_iInter, Set.iInter_const, twoPoint, Measure.real, Set.inter_def, Set.mem_ofPred_eq] using h
+
+omit [Countable V] in
+theorem survival_pos_of_survival_pos (G : BondGraph V E) (hG : Connected G)
+    (p : unitInterval) (hp : 0 < p) {x : V}
+    (hx : 0 < (law p).real {ω | (cluster G ω x).Infinite}) (y : V) :
+    0 < (law p).real {ω | (cluster G ω y).Infinite} := by
+  apply lt_of_lt_of_le (mul_pos hx (twoPoint_pos G hG p hp x y))
+  refine (association_infinite_conn G p x x y).trans (measureReal_mono ?_)
+  rintro ω ⟨hi, hc⟩
+  change (cluster G ω y).Infinite
+  rwa [← cluster_eq_of_conn G ω hc]
+
+omit [Countable V] [Countable E] in
+theorem exists_infinite_cluster_iff (G : BondGraph V E) (ω : Set E) :
+    (infiniteClusters G ω).Nonempty ↔ ∃ x, (cluster G ω x).Infinite := by
+  constructor
+  · rintro ⟨C, hi, x, rfl⟩
+    exact ⟨x, hi⟩
+  · rintro ⟨x, hx⟩
+    exact ⟨cluster G ω x, hx, x, rfl⟩
+
+omit [Countable V] [Countable E] in
+theorem unique_infinite_cluster_iff (G : BondGraph V E) (ω : Set E) :
+    (∃! C, C ∈ infiniteClusters G ω) ↔
+      ∃ x, (cluster G ω x).Infinite ∧ ∀ y, (cluster G ω y).Infinite → Conn G ω x y := by
+  constructor
+  · rintro ⟨C, ⟨hi, x, rfl⟩, hu⟩
+    refine ⟨x, hi, fun y hy => ?_⟩
+    have he := hu (cluster G ω y) ⟨hy, y, rfl⟩
+    have hm := mem_cluster_self G ω y
+    rwa [he] at hm
+  · rintro ⟨x, hx, hu⟩
+    refine ⟨cluster G ω x, ⟨hx, x, rfl⟩, ?_⟩
+    rintro C ⟨hi, y, rfl⟩
+    exact (cluster_eq_of_conn G ω (hu y hi)).symm
+
+theorem measurableSet_unique_infinite_cluster (G : BondGraph V E) :
+    MeasurableSet {ω | ∃! C, C ∈ infiniteClusters G ω} := by
+  have he : {ω | ∃! C, C ∈ infiniteClusters G ω} =
+      ⋃ x, ({ω | (cluster G ω x).Infinite} ∩
+        ⋂ y, ({ω | (cluster G ω y).Infinite}ᶜ ∪ {ω | Conn G ω x y})) := by
+    ext ω
+    simp only [Set.mem_ofPred_eq, Set.mem_iUnion, Set.mem_inter_iff,
+      Set.mem_iInter, Set.mem_union, Set.mem_compl_iff, unique_infinite_cluster_iff,
+      imp_iff_not_or]
+  rw [he]
+  exact MeasurableSet.iUnion fun x => (measurableSet_infinite_cluster G x).inter
+    (MeasurableSet.iInter fun y =>
+      (measurableSet_infinite_cluster G y).compl.union (measurableSet_conn G x y))
+
+theorem uniform_survival_pos (G : BondGraph V E) (hconn : Connected G)
+    (hqt : QuasiTransitive G) (p : unitInterval) (hp : 0 < p)
+    (hpos : 0 < law p {ω | (infiniteClusters G ω).Nonempty}) :
+    ∃ θ : ℝ, 0 < θ ∧ ∀ x, θ ≤ (law p).real {ω | (cluster G ω x).Infinite} := by
+  have hunion : {ω | (infiniteClusters G ω).Nonempty} =
+      ⋃ x, {ω | (cluster G ω x).Infinite} := by
+    ext ω
+    simp only [Set.mem_ofPred_eq, Set.mem_iUnion, exists_infinite_cluster_iff]
+  rw [hunion] at hpos
+  obtain ⟨x, hx⟩ := exists_measure_pos_of_not_measure_iUnion_null (ne_of_gt hpos)
+  have hxp : 0 < (law p).real {ω | (cluster G ω x).Infinite} :=
+    ENNReal.toReal_pos hx.ne' (measure_ne_top _ _)
+  have hroot := survival_pos_of_survival_pos G hconn p hp hxp
+  obtain ⟨s, hs⟩ := hqt
+  have hsn : s.Nonempty := by obtain ⟨o, ho, _⟩ := hs x; exact ⟨o, ho⟩
+  let vals := s.image (fun o => (law p).real {ω | (cluster G ω o).Infinite})
+  have hv : vals.Nonempty := hsn.image _
+  refine ⟨vals.min' hv, ?_, fun y => ?_⟩
+  · obtain ⟨o, _, he⟩ := Finset.mem_image.mp (Finset.min'_mem vals hv)
+    rw [← he]
+    exact hroot o
+  · obtain ⟨o, ho, a, b, ha, hay⟩ := hs y
+    rw [← hay, infinite_cluster_aut G p ha]
+    exact Finset.min'_le _ _ (Finset.mem_image.mpr ⟨o, ho, rfl⟩)
+
+ 
+
+theorem not_operatorBounded_of_unique [Infinite V] (G : BondGraph V E)
+    (hconn : Connected G) (hqt : QuasiTransitive G) (p : unitInterval)
+    (hu : law p {ω | ∃! C, C ∈ infiniteClusters G ω} = 1) : ¬ OperatorBounded G p := by
+  have hp : 0 < p := by
+    by_contra h
+    have hp0 : p = 0 := le_antisymm (not_lt.mp h) (bot_le)
+    subst p
+    simp [law] at hu
+  have hpos : 0 < law p {ω | (infiniteClusters G ω).Nonempty} := by
+    exact lt_of_lt_of_le (show 0 < law p {ω | ∃! C, C ∈ infiniteClusters G ω} by rw [hu]; exact zero_lt_one)
+      (measure_mono fun ω hω => ⟨hω.choose, hω.choose_spec.1⟩)
+  obtain ⟨θ, hθ, hroot⟩ := uniform_survival_pos G hconn hqt p hp hpos
+  have hue : ∀ᵐ ω ∂law p, ∃! C, C ∈ infiniteClusters G ω :=
+    (mem_ae_iff_prob_eq_one (measurableSet_unique_infinite_cluster G)).mpr hu
+  refine not_operatorBounded_of_positive_column G p (Classical.arbitrary V) (mul_pos hθ hθ)
+    (fun x => ?_)
+  refine (mul_le_mul (hroot x) (hroot (Classical.arbitrary V)) hθ.le measureReal_nonneg).trans
+    ((association_infinite_cluster G p x (Classical.arbitrary V)).trans ?_)
+  apply ENNReal.toReal_mono (measure_ne_top _ _)
+  apply measure_mono_ae
+  filter_upwards [hue] with ω hω
+  intro hi
+  obtain ⟨r, _, hconnr⟩ := (unique_infinite_cluster_iff G ω).mp hω
+  exact conn_trans G ω (conn_symm G ω (hconnr x hi.1)) (hconnr _ hi.2)
+
+ 
+
+theorem ptwo_le_pu [Infinite V] (G : BondGraph V E)
+    (hconn : Connected G) (hqt : QuasiTransitive G) : ptwo G ≤ pu G := by
+  unfold pu ptwo
+  apply le_sInf
+  intro p hp
+  apply sSup_le
+  intro q hq
+  by_contra hqp
+  have hb := operatorBounded_of_le G (le_of_lt (not_le.mp hqp)) hq
+  exact not_operatorBounded_of_unique G hconn hqt p hp hb
+
+end UniquenessThreshold
+
+section DisjointWitness
+attribute [local instance] Classical.propDecidable
+
+ 
+def DisjointOccurrence (A B : Set (Set E)) : Set (Set E) :=
+  {ω | ∃ s t : Finset E, (s : Set E) ⊆ ω ∧ (t : Set E) ⊆ ω ∧
+    Disjoint s t ∧ (s : Set E) ∈ A ∧ (t : Set E) ∈ B}
+
+theorem disjointOccurrence_upper (A B : Set (Set E)) : IsUpperSet (DisjointOccurrence A B) := by
+  rintro ω η h ⟨s, t, hs, ht, hd, ha, hb⟩
+  exact ⟨s, t, hs.trans h, ht.trans h, hd, ha, hb⟩
+
+theorem finiteWitness_disjointOccurrence (A B : Set (Set E)) :
+    FiniteWitness (DisjointOccurrence A B) := by
+  rintro ω ⟨s, t, hs, ht, hd, ha, hb⟩
+  refine ⟨s ∪ t, ?_, s, t, ?_, ?_, hd, ha, hb⟩
+  · simpa only [Finset.coe_union] using Set.union_subset hs ht
+  · simp
+  · simp
+
+theorem disjointOccurrence_mono {A A' B B' : Set (Set E)}
+    (hA : A ⊆ A') (hB : B ⊆ B') : DisjointOccurrence A B ⊆ DisjointOccurrence A' B' := by
+  rintro ω ⟨s, t, hs, ht, hd, ha, hb⟩
+  exact ⟨s, t, hs, ht, hd, hA ha, hB hb⟩
+
+ 
+def openSection (A : Set (Set E)) (e : E) : Set (Set E) := {ω | insert e ω ∈ A}
+
+theorem openSection_upper {A : Set (Set E)} (hA : IsUpperSet A) (e : E) :
+    IsUpperSet (openSection A e) :=
+  fun _ _ h hω => hA (Set.insert_subset_insert h) hω
+
+theorem subset_openSection {A : Set (Set E)} (hA : IsUpperSet A) (e : E) :
+    A ⊆ openSection A e := fun ω hω => hA (Set.subset_insert e ω) hω
+
+ 
+theorem openSection_disjointOccurrence (A B : Set (Set E)) (_hA : IsUpperSet A)
+    (hB : IsUpperSet B) (e : E) :
+    openSection (DisjointOccurrence A B) e ⊆
+      DisjointOccurrence (openSection A e) B ∪ DisjointOccurrence A (openSection B e) := by
+  rintro ω ⟨s, t, hs, ht, hd, ha, hb⟩
+  by_cases he : e ∈ s
+  · left
+    have het : e ∉ t := fun h => Finset.disjoint_left.mp hd he h
+    refine ⟨s.erase e, t, ?_, ?_, Finset.disjoint_of_subset_left (Finset.erase_subset _ _) hd, ?_, hb⟩
+    · intro z hz
+      exact (hs (Finset.mem_of_mem_erase hz)).resolve_left (Finset.ne_of_mem_erase hz)
+    · intro z hz
+      rcases ht hz with rfl | hz'
+      · exact (het hz).elim
+      · exact hz'
+    · change insert e (↑(s.erase e) : Set E) ∈ A
+      simpa only [← Finset.coe_insert, Finset.insert_erase he] using ha
+  · right
+    refine ⟨s, t.erase e, ?_, ?_, Finset.disjoint_of_subset_right (Finset.erase_subset _ _) hd, ha, ?_⟩
+    · intro z hz
+      rcases hs hz with rfl | hz'
+      · exact (he hz).elim
+      · exact hz'
+    · intro z hz
+      exact (ht (Finset.mem_of_mem_erase hz)).resolve_left (Finset.ne_of_mem_erase hz)
+    · apply hB ?_ hb
+      intro z hz
+      by_cases hze : z = e
+      · exact Or.inl hze
+      · exact Or.inr (Finset.mem_erase.mpr ⟨hze, hz⟩)
+
+ 
+
+noncomputable def finiteProbability (p : unitInterval) (s : Finset E) (A : Set (Set E)) : ℝ :=
+  ∑ t ∈ s.powerset, if (t : Set E) ∈ A then
+    (p : ℝ) ^ t.card * (1 - p : ℝ) ^ (s.card - t.card) else 0
+
+theorem finiteProbability_nonneg (p : unitInterval) (s : Finset E) (A : Set (Set E)) :
+    0 ≤ finiteProbability p s A := by
+  apply Finset.sum_nonneg
+  intro t _
+  split_ifs
+  · exact mul_nonneg (pow_nonneg p.2.1 _) (pow_nonneg (sub_nonneg.mpr p.2.2) _)
+  · exact le_rfl
+
+theorem finiteProbability_mono (p : unitInterval) (s : Finset E)
+    {A B : Set (Set E)} (h : A ⊆ B) : finiteProbability p s A ≤ finiteProbability p s B := by
+  apply Finset.sum_le_sum
+  intro t _
+  split_ifs with ha hb hb
+  · exact le_rfl
+  · exact (hb (h ha)).elim
+  · exact mul_nonneg (pow_nonneg p.2.1 _) (pow_nonneg (sub_nonneg.mpr p.2.2) _)
+  · exact le_rfl
+
+theorem finiteProbability_union_add_inter (p : unitInterval) (s : Finset E)
+    (A B : Set (Set E)) :
+    finiteProbability p s (A ∪ B) + finiteProbability p s (A ∩ B) =
+      finiteProbability p s A + finiteProbability p s B := by
+  simp only [finiteProbability, ← Finset.sum_add_distrib]
+  apply Finset.sum_congr rfl
+  intro t _
+  by_cases ha : (t : Set E) ∈ A <;> by_cases hb : (t : Set E) ∈ B <;> simp [ha, hb]
+
+@[simp] theorem finiteProbability_empty (p : unitInterval) (A : Set (Set E)) :
+    finiteProbability p ∅ A = if ∅ ∈ A then 1 else 0 := by
+  simp [finiteProbability]
+
+theorem finiteProbability_insert (p : unitInterval) (s : Finset E) (e : E) (he : e ∉ s)
+    (A : Set (Set E)) :
+    finiteProbability p (insert e s) A =
+      (1 - p : ℝ) * finiteProbability p s A + (p : ℝ) * finiteProbability p s (openSection A e) := by
+  have hd : Disjoint s.powerset (s.powerset.image (insert e)) := by
+    apply Finset.disjoint_left.mpr
+    intro t ht himg
+    obtain ⟨r, hr, rfl⟩ := Finset.mem_image.mp himg
+    exact he ((Finset.mem_powerset.mp ht) (by simp))
+  have hinj : Set.InjOn (insert e) (↑s.powerset : Set (Finset E)) := by
+    intro t ht r hr h
+    have ht' : e ∉ t := fun h => he (Finset.mem_powerset.mp ht h)
+    have hr' : e ∉ r := fun h => he (Finset.mem_powerset.mp hr h)
+    have hh := congrArg (fun t : Finset E => t.erase e) h
+    simpa [ht', hr'] using hh
+  simp only [finiteProbability, Finset.powerset_insert, Finset.sum_union hd,
+    Finset.sum_image hinj, Finset.mul_sum, Finset.card_insert_of_notMem he]
+  congr 1
+  · apply Finset.sum_congr rfl
+    intro t ht
+    have hcard := Finset.card_le_card (Finset.mem_powerset.mp ht)
+    have hexp : s.card + 1 - t.card = (s.card - t.card) + 1 := by omega
+    rw [hexp, pow_succ]
+    split_ifs <;> simp_all [mul_comm, mul_left_comm]
+  · apply Finset.sum_congr rfl
+    intro t ht
+    have het : e ∉ t := fun h => he (Finset.mem_powerset.mp ht h)
+    simp only [Finset.card_insert_of_notMem het, Nat.add_sub_add_right, pow_succ,
+      Finset.coe_insert, openSection, Set.mem_ofPred_eq]
+    split_ifs <;> simp_all [mul_assoc, mul_comm]
+
+end DisjointWitness
+
+section FiniteBK
+attribute [local instance] Classical.propDecidable
+
+@[simp] theorem empty_mem_disjointOccurrence (A B : Set (Set E)) :
+    (∅ : Set E) ∈ DisjointOccurrence A B ↔ ∅ ∈ A ∧ ∅ ∈ B := by
+  constructor
+  · rintro ⟨s, t, hs, ht, _, ha, hb⟩
+    have hse : (s : Set E) = ∅ := Set.subset_empty_iff.mp hs
+    have hte : (t : Set E) = ∅ := Set.subset_empty_iff.mp ht
+    exact ⟨hse ▸ ha, hte ▸ hb⟩
+  · rintro ⟨ha, hb⟩
+    exact ⟨∅, ∅, by simp, by simp, by simp, by simpa using ha, by simpa using hb⟩
+
+ 
+
+theorem finiteProbability_disjointOccurrence_le_half (p : unitInterval)
+    (hp : (p : ℝ) ≤ 1 / 2) (s : Finset E) (A B : Set (Set E))
+    (hA : IsUpperSet A) (hB : IsUpperSet B) :
+    finiteProbability p s (DisjointOccurrence A B) ≤
+      finiteProbability p s A * finiteProbability p s B := by
+  induction s using Finset.induction_on generalizing A B with
+  | empty =>
+    simp only [finiteProbability_empty, empty_mem_disjointOccurrence]
+    by_cases ha : (∅ : Set E) ∈ A <;> by_cases hb : (∅ : Set E) ∈ B <;> simp [ha, hb]
+  | @insert e s he ih =>
+    let A1 := openSection A e
+    let B1 := openSection B e
+    let D00 := DisjointOccurrence A B
+    let D10 := DisjointOccurrence A1 B
+    let D01 := DisjointOccurrence A B1
+    have hD : D00 ⊆ D10 ∩ D01 := by
+      intro ω hω
+      exact ⟨disjointOccurrence_mono (subset_openSection hA e) (Set.Subset.refl _) hω,
+        disjointOccurrence_mono (Set.Subset.refl _) (subset_openSection hB e) hω⟩
+    have hsec : finiteProbability p s (openSection D00 e) ≤
+        finiteProbability p s D10 + finiteProbability p s D01 - finiteProbability p s D00 := by
+      have h1 := finiteProbability_mono p s (openSection_disjointOccurrence A B hA hB e)
+      have h2 := finiteProbability_mono p s hD
+      have h3 := finiteProbability_union_add_inter p s D10 D01
+      change finiteProbability p s (openSection D00 e) ≤
+        finiteProbability p s (D10 ∪ D01) at h1
+      linarith
+    have h00 := ih A B hA hB
+    have h10 := ih A1 B (openSection_upper hA e) hB
+    have h01 := ih A B1 hA (openSection_upper hB e)
+    have ha := finiteProbability_mono p s (subset_openSection hA e)
+    have hb := finiteProbability_mono p s (subset_openSection hB e)
+    have hcoef : 0 ≤ 1 - 2 * (p : ℝ) := by linarith
+    have hdif : 0 ≤ ((p : ℝ) ^ 2) *
+        (finiteProbability p s A1 - finiteProbability p s A) *
+        (finiteProbability p s B1 - finiteProbability p s B) :=
+      mul_nonneg (mul_nonneg (sq_nonneg _) (sub_nonneg.mpr ha)) (sub_nonneg.mpr hb)
+    rw [finiteProbability_insert p s e he, finiteProbability_insert p s e he,
+      finiteProbability_insert p s e he]
+    have hm := mul_le_mul_of_nonneg_left hsec p.2.1
+    have hm00 := mul_le_mul_of_nonneg_left h00 hcoef
+    have hm10 := mul_le_mul_of_nonneg_left h10 p.2.1
+    have hm01 := mul_le_mul_of_nonneg_left h01 p.2.1
+    change (1 - (p : ℝ)) * finiteProbability p s D00 +
+      (p : ℝ) * finiteProbability p s (openSection D00 e) ≤
+      ((1 - (p : ℝ)) * finiteProbability p s A + (p : ℝ) * finiteProbability p s A1) *
+      ((1 - (p : ℝ)) * finiteProbability p s B + (p : ℝ) * finiteProbability p s B1)
+    nlinarith
+
+end FiniteBK
+
+section BKMeasure
+attribute [local instance] Classical.propDecidable
+open Filter Topology
+
+ 
+theorem finiteProbability_univ [Fintype E] (p : unitInterval) (A : Set (Set E)) :
+    finiteProbability p Finset.univ A = (law p).real A := by
+  have he : (law p).real A = ∑ ω : Set E, if ω ∈ A then (law p).real {ω} else 0 := by
+    have h := integral_fintype (μ := law p) (f := A.indicator (fun _ => (1 : ℝ)))
+      Integrable.of_finite
+    rw [integral_indicator_const (1 : ℝ) (.of_discrete : MeasurableSet A)] at h
+    simpa [smul_eq_mul, mul_ite, Set.indicator_apply] using h
+  rw [he, finiteProbability]
+  apply Finset.sum_bij (fun (t : Finset E) _ => (t : Set E))
+  · simp
+  · intro a _ b _ h
+    exact Finset.coe_injective h
+  · intro ω _
+    exact ⟨ω.toFinset, by simp, by simp⟩
+  · intro t _
+    rw [law, setBernoulli_real_singleton p (Set.subset_univ _) Set.finite_univ]
+    simp [Set.ncard_sdiff' (Set.subset_univ _) Set.finite_univ, Set.ncard_univ,
+      Nat.card_eq_fintype_card]
+
+theorem disjointOccurrence_image {F : Type*} (f : F ↪ E) (A B : Set (Set E)) :
+    (fun ω => f '' ω) ⁻¹' DisjointOccurrence A B =
+      DisjointOccurrence ((fun ω => f '' ω) ⁻¹' A) ((fun ω => f '' ω) ⁻¹' B) := by
+  ext ω
+  constructor
+  · rintro ⟨s, t, hs, ht, hd, ha, hb⟩
+    let s' := s.preimage f f.injective.injOn
+    let t' := t.preimage f f.injective.injOn
+    have hs' : (s' : Set F) = f ⁻¹' (s : Set E) := by simp [s']
+    have ht' : (t' : Set F) = f ⁻¹' (t : Set E) := by simp [t']
+    have hsI : f '' (s' : Set F) = (s : Set E) := by
+      rw [hs', Set.image_preimage_eq_of_subset]
+      exact hs.trans (Set.image_subset_range _ _)
+    have htI : f '' (t' : Set F) = (t : Set E) := by
+      rw [ht', Set.image_preimage_eq_of_subset]
+      exact ht.trans (Set.image_subset_range _ _)
+    refine ⟨s', t', ?_, ?_, ?_, ?_, ?_⟩
+    · rw [hs']
+      exact (Set.preimage_mono hs).trans (by rw [Set.preimage_image_eq _ f.injective])
+    · rw [ht']
+      exact (Set.preimage_mono ht).trans (by rw [Set.preimage_image_eq _ f.injective])
+    · apply Finset.disjoint_left.mpr
+      intro x hx hy
+      exact Finset.disjoint_left.mp hd (by simpa [s'] using hx) (by simpa [t'] using hy)
+    · change f '' (s' : Set F) ∈ A
+      rwa [hsI]
+    · change f '' (t' : Set F) ∈ B
+      rwa [htI]
+  · rintro ⟨s, t, hs, ht, hd, ha, hb⟩
+    refine ⟨s.map f, t.map f, ?_, ?_, ?_, ?_, ?_⟩
+    · simpa only [Finset.coe_map] using Set.image_mono hs
+    · simpa only [Finset.coe_map] using Set.image_mono ht
+    · exact (Finset.disjoint_map f).mpr hd
+    · simpa only [Finset.coe_map, Set.mem_preimage] using ha
+    · simpa only [Finset.coe_map, Set.mem_preimage] using hb
+
+theorem bk_finite_half [Fintype E] (p : unitInterval) (hp : (p : ℝ) ≤ 1 / 2)
+    (A B : Set (Set E)) (hA : IsUpperSet A) (hB : IsUpperSet B) :
+    (law p).real (DisjointOccurrence A B) ≤ (law p).real A * (law p).real B := by
+  simpa only [finiteProbability_univ] using
+    finiteProbability_disjointOccurrence_le_half p hp Finset.univ A B hA hB
+
+theorem bk_restricted_half (p : unitInterval) (hp : (p : ℝ) ≤ 1 / 2)
+    (A B : Set (Set E)) (hA : IsUpperSet A) (hB : IsUpperSet B) (s : Finset E) :
+    (law p).real {ω | ω ∩ (s : Set E) ∈ DisjointOccurrence A B} ≤
+      (law p).real {ω | ω ∩ (s : Set E) ∈ A} *
+      (law p).real {ω | ω ∩ (s : Set E) ∈ B} := by
+  let f : (s : Set E) ↪ E := ⟨Subtype.val, Subtype.val_injective⟩
+  have hreal (C : Set (Set E)) :
+      (law p).real {ω | ω ∩ (s : Set E) ∈ C} =
+        (law p).real {η : Set (s : Set E) | f '' η ∈ C} := by
+    rw [← law_map_restrict p (s : Set E)]
+    simp only [Measure.real]
+    rw [Measure.map_apply (measurable_set_restrict _) MeasurableSet.of_discrete]
+    congr 2
+    ext ω
+    change ω ∩ (s : Set E) ∈ C ↔ f '' (Subtype.val ⁻¹' ω) ∈ C
+    have he : f '' (Subtype.val ⁻¹' ω) = ω ∩ (s : Set E) := by
+      ext e
+      simp [f, and_comm]
+    rw [he]
+  rw [hreal, hreal, hreal]
+  have hh := bk_finite_half p hp ((fun ω => f '' ω) ⁻¹' A)
+    ((fun ω => f '' ω) ⁻¹' B)
+    (fun _ _ h hω => hA (Set.image_mono h) hω)
+    (fun _ _ h hω => hB (Set.image_mono h) hω)
+  rw [← disjointOccurrence_image] at hh
+  exact hh
+
+ 
+theorem bk_finiteWitness_half [Countable E] (p : unitInterval) (hp : (p : ℝ) ≤ 1 / 2)
+    (A B : Set (Set E)) (hA : IsUpperSet A) (hB : IsUpperSet B)
+    (hfA : FiniteWitness A) (hfB : FiniteWitness B) :
+    (law p).real (DisjointOccurrence A B) ≤ (law p).real A * (law p).real B := by
+  let := Encodable.ofCountable E
+  have hlim (C : Set (Set E)) (hC : IsUpperSet C) (hfC : FiniteWitness C) :
+      Tendsto (fun n => (law p).real {ω | ω ∩ (edgeExhaustion (E := E) n : Set E) ∈ C})
+        atTop (𝓝 ((law p).real C)) := by
+    have hm : Monotone (fun n => {ω : Set E | ω ∩ (edgeExhaustion (E := E) n : Set E) ∈ C}) := by
+      intro m n h ω hω
+      exact hC (Set.inter_subset_inter_right _ (edgeExhaustion_mono h)) hω
+    have h := tendsto_measure_iUnion_atTop (μ := law p) hm
+    rw [← witness_eq_union C hC hfC] at h
+    exact (ENNReal.tendsto_toReal (measure_ne_top _ _)).comp h
+  apply le_of_tendsto_of_tendsto
+    (hlim _ (disjointOccurrence_upper A B) (finiteWitness_disjointOccurrence A B))
+    ((hlim A hA hfA).mul (hlim B hB hfB))
+  exact Filter.Eventually.of_forall fun n => bk_restricted_half p hp A B hA hB (edgeExhaustion n)
+
+end BKMeasure
+
+section ORLift
+attribute [local instance] Classical.propDecidable
+
+noncomputable def orParameter (q : unitInterval) (m : ℕ) : unitInterval :=
+  ⟨1 - (1 - (q : ℝ)) ^ m,
+    sub_nonneg.mpr (pow_le_one₀ (sub_nonneg.mpr q.2.2) (by linarith [q.2.1])),
+    by linarith [pow_nonneg (sub_nonneg.mpr q.2.2) m]⟩
+
+noncomputable abbrev bitLaw (q : unitInterval) : Measure Prop :=
+  unitInterval.toNNReal q • Measure.dirac True +
+    unitInterval.toNNReal (unitInterval.symm q) • Measure.dirac False
+
+instance bitLaw_isProbabilityMeasure (q : unitInterval) : IsProbabilityMeasure (bitLaw q) := by
+  rw [bitLaw, ← map_threshold q]
+  infer_instance
+
+ 
+theorem map_nonempty (q : unitInterval) (F : Type*) [Fintype F] :
+    (law q : Measure (Set F)).map Set.Nonempty = bitLaw (orParameter q (Fintype.card F)) := by
+  let μ := (law q : Measure (Set F)).map Set.Nonempty
+  have hm : Measurable (Set.Nonempty : Set F → Prop) := .of_discrete
+  have : IsProbabilityMeasure μ := inferInstance
+  have hfalse : μ.real {False} = (1 - (q : ℝ)) ^ Fintype.card F := by
+    change ((law q).map Set.Nonempty).real {False} = _
+    simp only [Measure.real, Measure.map_apply hm (measurableSet_singleton _)]
+    have he : Set.Nonempty ⁻¹' ({False} : Set Prop) = ({∅} : Set (Set F)) := by
+      ext ω
+      simp [Set.not_nonempty_iff_eq_empty]
+    rw [he]
+    change (law q).real {∅} = _
+    rw [law, setBernoulli_real_singleton q (Set.empty_subset _) Set.finite_univ]
+    simp [Set.ncard_univ, Nat.card_eq_fintype_card]
+  have htrue : μ.real {True} = 1 - (1 - (q : ℝ)) ^ Fintype.card F := by
+    have he : ({True} : Set Prop) = ({False} : Set Prop)ᶜ := by ext b; simp
+    rw [he, measureReal_compl (measurableSet_singleton _), probReal_univ, hfalse]
+  apply Measure.ext_of_singleton
+  intro b
+  apply (ENNReal.toReal_eq_toReal_iff' (measure_ne_top _ _) (measure_ne_top _ _)).mp
+  change μ.real {b} = (bitLaw (orParameter q (Fintype.card F))).real {b}
+  by_cases hb : b
+  · have he : b = True := propext (iff_true_intro hb)
+    subst b
+    rw [htrue]
+    simp [bitLaw, Measure.real, orParameter]
+  · have he : b = False := propext (iff_false_intro hb)
+    subst b
+    rw [hfalse]
+    simp [bitLaw, Measure.real, orParameter]
+
+ 
+theorem finite_image_lift {F : Type*} (f : F → E) {ω : Set F} (s : Finset E)
+    (hs : (s : Set E) ⊆ f '' ω) :
+    ∃ t : Finset F, (t : Set F) ⊆ ω ∧ f '' (t : Set F) = (s : Set E) := by
+  have hex : ∃ t ⊆ f '' ω, t.Finite ∧ t = (s : Set E) :=
+    ⟨s, hs, s.finite_toSet, rfl⟩
+  obtain ⟨t, ht, htf, hti⟩ := Set.exists_subset_image_finite_and.mp hex
+  exact ⟨htf.toFinset, by simpa using ht, by simpa using hti⟩
+
+theorem finiteWitness_image_preimage {F : Type*} (f : F → E) (A : Set (Set E))
+    (hA : FiniteWitness A) : FiniteWitness ((fun ω => f '' ω) ⁻¹' A) := by
+  intro ω hω
+  obtain ⟨s, hs, ha⟩ := hA (f '' ω) hω
+  obtain ⟨t, ht, htI⟩ := finite_image_lift f s hs
+  exact ⟨t, ht, by simpa only [Set.mem_preimage, htI] using ha⟩
+
+theorem image_preimage_disjointOccurrence {F : Type*} (f : F → E) (A B : Set (Set E)) :
+    (fun ω => f '' ω) ⁻¹' DisjointOccurrence A B ⊆
+      DisjointOccurrence ((fun ω => f '' ω) ⁻¹' A) ((fun ω => f '' ω) ⁻¹' B) := by
+  rintro ω ⟨s, t, hs, ht, hd, ha, hb⟩
+  obtain ⟨s', hs', hsI⟩ := finite_image_lift f s hs
+  obtain ⟨t', ht', htI⟩ := finite_image_lift f t ht
+  refine ⟨s', t', hs', ht', ?_, ?_, ?_⟩
+  · apply Finset.disjoint_left.mpr
+    intro x hx hy
+    have hxs : f x ∈ (s : Set E) := hsI ▸ Set.mem_image_of_mem f hx
+    have hxt : f x ∈ (t : Set E) := htI ▸ Set.mem_image_of_mem f hy
+    exact Finset.disjoint_left.mp hd hxs hxt
+  · simpa only [Set.mem_preimage, hsI] using ha
+  · simpa only [Set.mem_preimage, htI] using hb
+
+end ORLift
+
+section ORLaw
+attribute [local instance] Classical.propDecidable
+
+theorem map_exists_bitLaw (q : unitInterval) (F : Type*) [Fintype F] :
+    (Measure.infinitePi (fun _ : F => bitLaw q)).map (fun f : F → Prop => ∃ j, f j) =
+      bitLaw (orParameter q (Fintype.card F)) := by
+  have h := map_nonempty q F
+  unfold law at h
+  rw [setBernoulli_eq_map, Measure.map_map (Measurable.of_discrete) measurable_setOfPred] at h
+  simpa only [Set.mem_univ, Function.comp_def, Set.Nonempty, Set.mem_ofPred_eq,
+    bernoulliMeasure_def, bitLaw] using h
+
+theorem measurable_image_fst (F : Type*) [Fintype F] :
+    Measurable (fun ω : Set (E × F) => Prod.fst '' ω) := by
+  have hm : Measurable (fun ω : Set (E × F) => fun e => ∃ j : F, (e,j) ∈ ω) := by
+    apply Measurable.of_eval
+    intro e
+    apply measurableSet_setOfPred.mp
+    have he : {ω : Set (E × F) | ∃ j : F, (e,j) ∈ ω} =
+        ⋃ j : F, {ω : Set (E × F) | (e,j) ∈ ω} := by ext ω; simp
+    rw [he]
+    exact MeasurableSet.iUnion fun j => measurableSet_setOfPred.mpr
+      ((measurable_pi_apply (e,j)).comp MeasurableEquiv.setOfPred.symm.measurable)
+  have he : (fun ω : Set (E × F) => Prod.fst '' ω) =
+      (fun f : E → Prop => {e | f e}) ∘ (fun ω e => ∃ j : F, (e,j) ∈ ω) := by
+    funext ω
+    ext e
+    simp [Prod.exists]
+  rw [he]
+  exact measurable_setOfPred.comp hm
+
+ 
+theorem law_map_image_fst (q : unitInterval) (F : Type*) [Fintype F] :
+    (law q : Measure (Set (E × F))).map (fun ω => Prod.fst '' ω) =
+      law (orParameter q (Fintype.card F)) := by
+  let c := MeasurableEquiv.curry E F Prop
+  let b := fun f : E → F → Prop => fun e => ∃ j, f e j
+  have hb : Measurable b := Measurable.of_eval
+    (fun e => (Measurable.of_discrete : Measurable (fun f : F → Prop => ∃j, f j)).comp
+      (measurable_pi_apply e))
+  unfold law
+  rw [setBernoulli_eq_map, Measure.map_map (measurable_image_fst F) measurable_setOfPred]
+  simp only [Set.mem_univ]
+  have he : (fun ω : Set (E × F) => Prod.fst '' ω) ∘ (fun f => {e | f e}) =
+      (fun f : E → Prop => {e | f e}) ∘ b ∘ c := by
+    funext f
+    ext e
+    simp [b, c, Function.comp_def, Prod.exists]
+  rw [he, ← Measure.map_map measurable_setOfPred (hb.comp c.measurable),
+    ← Measure.map_map hb c.measurable]
+  change ((Measure.infinitePi (fun _ : E × F => bitLaw q)).map c |>.map b |>.map
+    (fun f : E → Prop => {e | f e})) = _
+  rw [Measure.infinitePi_map_curry (fun (_ : E) (_ : F) => bitLaw q)]
+  dsimp only [b]
+  rw [Measure.infinitePi_map_pi _ (fun _ => Measurable.of_discrete)]
+  rw [map_exists_bitLaw q F]
+  rw [setBernoulli_eq_map]
+  rfl
+
+ 
+theorem exists_orParameter (p : unitInterval) (hp : p < 1) :
+    ∃ m : ℕ, ∃ q : unitInterval, (q : ℝ) ≤ 1 / 2 ∧ orParameter q m = p := by
+  obtain ⟨m, hm⟩ := exists_pow_lt_of_lt_one (show 0 < 1 - (p : ℝ) by exact sub_pos.mpr hp)
+    (show (1 / 2 : ℝ) < 1 by norm_num)
+  let f := fun x : ℝ => 1 - (1-x)^m
+  have hf : Continuous f := by fun_prop
+  have h0 : f 0 = 0 := by simp [f]
+  have hp' : (p : ℝ) ∈ Set.Icc (f 0) (f (1/2)) := by
+    rw [h0]
+    refine ⟨p.2.1, ?_⟩
+    dsimp [f]
+    norm_num at hm ⊢
+    linarith
+  obtain ⟨q, hq, hqp⟩ := intermediate_value_Icc (by norm_num : (0 : ℝ) ≤ 1/2)
+    hf.continuousOn hp'
+  refine ⟨m, ⟨q, hq.1, hq.2.trans (by norm_num)⟩, hq.2, ?_⟩
+  apply Subtype.ext
+  exact hqp
+
+end ORLaw
+
+section BK
+attribute [local instance] Classical.propDecidable
+
+theorem disjointOccurrence_subset_inter (A B : Set (Set E))
+    (hA : IsUpperSet A) (hB : IsUpperSet B) : DisjointOccurrence A B ⊆ A ∩ B := by
+  rintro ω ⟨s,t,hs,ht,_,ha,hb⟩
+  exact ⟨hA hs ha, hB ht hb⟩
+
+ 
+
+theorem bk [Countable E] (p : unitInterval) (A B : Set (Set E))
+    (hA : IsUpperSet A) (hB : IsUpperSet B)
+    (hfA : FiniteWitness A) (hfB : FiniteWitness B) :
+    (law p).real (DisjointOccurrence A B) ≤ (law p).real A * (law p).real B := by
+  by_cases hp : p = 1
+  · subst p
+    by_cases hu : Set.univ ∈ DisjointOccurrence A B
+    · have ha := (disjointOccurrence_subset_inter A B hA hB hu).1
+      have hb := (disjointOccurrence_subset_inter A B hA hB hu).2
+      simp [law, Measure.real, hu, ha, hb]
+    · simp [law, Measure.real, hu]
+      positivity
+  · have hp' : p < 1 := lt_of_le_of_ne p.2.2 hp
+    obtain ⟨m, q, hq, hqp⟩ := exists_orParameter p hp'
+    let f : E × Fin m → E := Prod.fst
+    let A' := (fun ω => f '' ω) ⁻¹' A
+    let B' := (fun ω => f '' ω) ⁻¹' B
+    have hreal (C : Set (Set E)) (hC : MeasurableSet C) :
+        (law p).real C = (law q).real ((fun ω : Set (E × Fin m) => f '' ω) ⁻¹' C) := by
+      have hh := law_map_image_fst (E := E) q (Fin m)
+      simp only [Fintype.card_fin, hqp] at hh
+      rw [← hh]
+      simp only [Measure.real, Measure.map_apply (measurable_image_fst (Fin m)) hC]
+      rfl
+    calc
+      (law p).real (DisjointOccurrence A B) =
+          (law q).real ((fun ω : Set (E × Fin m) => f '' ω) ⁻¹' DisjointOccurrence A B) :=
+        hreal _ (measurableSet_finiteWitness _ (disjointOccurrence_upper A B)
+          (finiteWitness_disjointOccurrence A B))
+      _ ≤ (law q).real (DisjointOccurrence A' B') :=
+        measureReal_mono (image_preimage_disjointOccurrence f A B)
+      _ ≤ (law q).real A' * (law q).real B' := bk_finiteWitness_half q hq A' B'
+        (fun _ _ h hω => hA (Set.image_mono h) hω)
+        (fun _ _ h hω => hB (Set.image_mono h) hω)
+        (finiteWitness_image_preimage f A hfA) (finiteWitness_image_preimage f B hfB)
+      _ = (law p).real A * (law p).real B := by
+        rw [hreal A (measurableSet_finiteWitness A hA hfA),
+          hreal B (measurableSet_finiteWitness B hB hfB)]
+
+end BK
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Splicing/EndpointMoments.lean b/lean/OAI/Probability/UnimodularPercolation/Splicing/EndpointMoments.lean
index 5e3d48e180c603ef781b2ed8839b934314728916..99cf8ccbfaf811fed15070fde1cd355326bb5b22 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Splicing/EndpointMoments.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Splicing/EndpointMoments.lean
@@ -1,1272 +1,1272 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Kernel.Amplification
-
-namespace OAI
-
-/-! Endpoint Moments for bond percolation. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section TrajectoryHitTransport
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
-variable {ι : Type*} [Fintype ι]
-
- 
-
-theorem visitEnvelope_column_bound (hconn : Connected G)
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
-    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
-    (p : unitInterval) (k : ℕ) (a : V) :
-    Summable (fun z => q.visitEnvelope o p k z a) ∧
-      (∑' z, q.visitEnvelope o p k z a) ≤ c*((k*M:ℕ)*q.successProbability p o ^ k) := by
-  apply transported_column_of_row_bound G hconn hloc root weight Γ hΓ htr hcov hw hw1 c hc hmin
-    (q.visitEnvelope o p k) (q.visitEnvelope_nonneg o p k) _
-    (fun z => (q.visitEnvelope_row_bound o p k z).1) _
-    (fun z => (q.visitEnvelope_row_bound o p k z).2) a
-  intro b hb x y
-  obtain ⟨d,hd⟩ := hΓ hb
-  exact q.visitEnvelope_covariant o p k b d hd x y
-
- 
-
-theorem finiteTrajectoryHit_sum_le (hconn : Connected G)
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
-    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
-    (Q : Finset E) (B : Finset V) (p : unitInterval) (k : ℕ) (A : Finset V)
-    (hQ : ∀ x ∈ B, incidentSupport G hloc (finite_reachBall hloc x (max q.radius M)).toFinset ⊆ Q) :
-    (∑ z : ↥B, finiteTrajectoryHit G Subtype.val Q p k
-      (fun ω (x y : ↥B) => q.mass ω x.val y.val) z A) ≤
-      (A.card:ℝ) * (c*((k*M:ℕ)*q.successProbability p o ^ k)) := by
-  have hcol := fun a => q.visitEnvelope_column_bound hconn root weight Γ hΓ htr hcov hw hw1 c hc hmin o p k a
-  calc
-    _ ≤ ∑ z : ↥B, ∑ a ∈ A, q.visitEnvelope o p k z.val a := by
-      apply Finset.sum_le_sum
-      intro z _
-      exact (finiteTrajectoryHit_le_sum G Subtype.val Q p k _ (fun ω x y => q.nonneg ω x.val y.val) z A).trans
-        (Finset.sum_le_sum fun a _ => q.finiteTrajectoryMass_le_envelope o Q B p k z a hQ)
-    _ = ∑ a ∈ A, ∑ z : ↥B, q.visitEnvelope o p k z.val a := Finset.sum_comm
-    _ ≤ ∑ _a ∈ A, c*((k*M:ℕ)*q.successProbability p o ^ k) := by
-      apply Finset.sum_le_sum
-      intro a _
-      rw [Finset.sum_coe_sort B (fun z => q.visitEnvelope o p k z a)]
-      exact ((hcol a).1.sum_le_tsum B (fun z _ => q.visitEnvelope_nonneg o p k z a)).trans (hcol a).2
-    _ = _ := by simp only [Finset.sum_const,nsmul_eq_mul]
-
-end LocalEndpointRule
-end TrajectoryHitTransport
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section AllEndpointHit
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem finiteChoiceHit_allRuns {W : Type*} [Fintype W]
-    (G : BondGraph V E) (e : W → V) (Q : Finset E) (p : ℝ) (k : ℕ)
-    (q : Set E → W → W → ℝ) (A : Finset V) :
-    finiteChoiceHit G k Q p
-      (fun (r : W × (Fin k → W)) _ => e ∘ oldRunVertices k r.1 r.2)
-      (fun r η => oldRunWeight (fun i => q (η i)) k r.1 r.2) (A : Set V) =
-      ∑ z : W, finiteTrajectoryHit G e Q p k q z A := by
-  unfold finiteChoiceHit finiteTrajectoryHit
-  rw [Fintype.sum_prod_type]
-  simp only [finiteFamilyIntegral_sum]
-
-noncomputable def allAmplificationVertices (B : Finset V) (k : ℕ)
-    (r : ↥B × (Fin k → ↥B)) (_η : ℕ → Set E) : ℕ → V :=
-  amplificationVertices B k r.1 r.2
-
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
-
-noncomputable def allAmplificationWeight (q : LocalEndpointRule G hloc O M L)
-    (B : Finset V) (k : ℕ) (r : ↥B × (Fin k → ↥B)) (η : ℕ → Set E) : ℝ :=
-  oldRunWeight (fun i (a b : ↥B) => q.mass (η i) a.val b.val) k r.1 r.2
-
-noncomputable def allAmplificationInputs (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ)
-    (r : ↥B × (Fin k → ↥B)) : Finset (SpliceState E) :=
-  q.amplificationInputs Q B x k n r.1 r.2
-
-theorem allAmplificationWeight_nonneg (q : LocalEndpointRule G hloc O M L)
-    (B : Finset V) (k : ℕ) (r : ↥B × (Fin k → ↥B)) (η : ℕ → Set E) :
-    0 ≤ q.allAmplificationWeight B k r η :=
-  oldRunWeight_nonneg (fun i (a b : ↥B) => q.mass (η i) a.val b.val) (fun i a b => q.nonneg (η i) a.val b.val) k r.1 r.2
-
-variable {ι : Type*} [Fintype ι]
-
-theorem allAmplificationHit_bound (hconn : Connected G)
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
-    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
-    (Q : Finset E) (B : Finset V) (p : unitInterval) (k : ℕ) (A : Set V) (hf : A.Finite)
-    (hQ : ∀ x ∈ B, incidentSupport G hloc (finite_reachBall hloc x (max q.radius M)).toFinset ⊆ Q) :
-    finiteChoiceHit G k Q p (allAmplificationVertices B k) (q.allAmplificationWeight B k) A ≤
-      (A.ncard:ℝ) * (c*((k*M:ℕ)*q.successProbability p o ^ k)) := by
-  have he : finiteChoiceHit G k Q p (allAmplificationVertices B k) (q.allAmplificationWeight B k) A =
-      ∑ z : ↥B, finiteTrajectoryHit G Subtype.val Q p k
-        (fun ω (a b : ↥B) => q.mass ω a.val b.val) z hf.toFinset := by
-    rw [← finiteChoiceHit_allRuns]
-    rw [hf.coe_toFinset]
-    rfl
-  rw [he,Set.ncard_eq_toFinset_card A hf]
-  exact q.finiteTrajectoryHit_sum_le hconn root weight Γ hΓ htr hcov hw hw1 c hc hmin o Q B p k hf.toFinset hQ
-
-end LocalEndpointRule
-end AllEndpointHit
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section AllEndpointSeparated
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def finiteClusterWindowMass (G : BondGraph V E) (Q : Finset E) (p : ℝ)
-    (x : V) (lo hi : ℕ) : ℝ :=
-  finiteFieldIntegral Q p (fun ω => if (cluster G ω x).Finite ∧
-    lo ≤ (cluster G ω x).ncard ∧ (cluster G ω x).ncard < hi then 1 else 0)
-
-theorem finiteChoiceHit_nonneg {P : Type*} [Fintype P]
-    (G : BondGraph V E) (k : ℕ) (Q : Finset E) (p : unitInterval)
-    (y : P → (ℕ → Set E) → ℕ → V) (m : P → (ℕ → Set E) → ℝ)
-    (hm : ∀ r η, 0 ≤ m r η) (A : Set V) : 0 ≤ finiteChoiceHit G k Q p y m A := by
-  apply Finset.sum_nonneg
-  intro r _
-  apply finiteFamilyIntegral_nonneg
-  intro η
-  split_ifs
-  · exact hm r η
-  · exact le_refl _
-
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
-variable {ι : Type*} [Fintype ι]
-
- 
-
-theorem amplificationSeparatedSecondMoment_le [Nonempty E] (hconn : Connected G)
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
-    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
-    (Q : Finset E) (B : Finset V) (p : unitInterval) (x : V) (k n : ℕ)
-    (hQ : ∀ a ∈ B, incidentSupport G hloc (finite_reachBall hloc a (max q.radius M)).toFinset ⊆ Q) :
-    finiteChoiceSeparatedPairMass G x k Q p
-      (q.allAmplificationInputs Q B x k n) (q.allAmplificationInputs Q B x k n)
-      (allAmplificationVertices B k) (allAmplificationVertices B k)
-      (q.allAmplificationWeight B k) (q.allAmplificationWeight B k) ≤
-      (((2*n:ℕ):ℝ) * (c*((k*M:ℕ)*q.successProbability p o ^ k)))^2 *
-        finiteClusterWindowMass G Q p x (n-k*M) (2*n) := by
-  let A := q.allAmplificationInputs Q B x k n
-  let y := allAmplificationVertices (E := E) B k
-  let m := q.allAmplificationWeight B k
-  have hA (r) : A r ⊆ finiteSpliceDomain Q (Finset.Icc 1 k) :=
-    q.amplificationInputs_subset Q B x k n r.1 r.2
-  have hbin r s (hs : s ∈ A r) : (cluster G s.1 x).Finite ∧
-      n ≤ (cluster G s.1 x).ncard ∧ (cluster G s.1 x).ncard < 2*n := by
-    have hd := (mem_finiteSpliceDomain_iff Q (Finset.Icc 1 k) s).mp (hA r hs)
-    have hb := (q.mem_amplificationInputs Q B x k n r.1 r.2 s).mp hs
-    exact ⟨cluster_finite_of_finite_edges G (Q.finite_toSet.subset hd.1) x,hb.2.2.2.1,hb.2.2.2.2.1⟩
-  have hcap r s (hs : s ∈ A r) (i : ℕ) (hi : 1 ≤ i) (hik : i ≤ k) :
-      (cluster G (s.2 i) (y r s.2 i)).Finite ∧ (cluster G (s.2 i) (y r s.2 i)).ncard ≤ M := by
-    have hd := q.amplificationInputs_chain Q B x k n r.1 r.2 s hs i hi hik
-    exact ⟨hd.2.1,hd.2.2.1⟩
-  have hm := q.allAmplificationWeight_nonneg B k
-  have hb := finiteChoiceSeparatedPairMass_le G hloc x k n M Q p A A y y m m hm hm hA hA hbin
-    (fun r s hs i hi hik => (hcap r s hs i hi hik).1) hcap
-  let d : ℝ := c*((k*M:ℕ)*q.successProbability p o ^ k)
-  have hd : 0 ≤ d := mul_nonneg hc (mul_nonneg (Nat.cast_nonneg _) (pow_nonneg (q.successProbability_nonneg p o) _))
-  let t : ℝ := ((2*n:ℕ):ℝ)*d
-  apply hb.trans
-  unfold finiteClusterWindowMass finiteFieldIntegral
-  rw [Finset.mul_sum]
-  apply Finset.sum_le_sum
-  intro ω _
-  have he : t^2 * (finiteFieldWeight Q p ω * (if (cluster G (finiteFieldSet Q ω) x).Finite ∧
-      n-k*M ≤ (cluster G (finiteFieldSet Q ω) x).ncard ∧
-      (cluster G (finiteFieldSet Q ω) x).ncard < 2*n then 1 else 0)) =
-      finiteFieldWeight Q p ω * (if (cluster G (finiteFieldSet Q ω) x).Finite ∧
-      n-k*M ≤ (cluster G (finiteFieldSet Q ω) x).ncard ∧
-      (cluster G (finiteFieldSet Q ω) x).ncard < 2*n then t^2 else 0) := by
-    split_ifs <;> ring
-  change _ ≤ t^2 * _
-  rw [he]
-  apply mul_le_mul_of_nonneg_left _ (finiteFieldWeight_nonneg Q p ω)
-  by_cases h : (cluster G (finiteFieldSet Q ω) x).Finite ∧
-      n-k*M ≤ (cluster G (finiteFieldSet Q ω) x).ncard ∧
-      (cluster G (finiteFieldSet Q ω) x).ncard < 2*n
-  · rw [ite_eq_left h,ite_eq_left h]
-    have hhit := q.allAmplificationHit_bound hconn root weight Γ hΓ htr hcov hw hw1 c hc hmin
-      o Q B p k (cluster G (finiteFieldSet Q ω) x) h.1 hQ
-    have hc' : ((cluster G (finiteFieldSet Q ω) x).ncard:ℝ) ≤ ((2*n:ℕ):ℝ) := by exact_mod_cast h.2.2.le
-    have hhit' : finiteChoiceHit G k Q p y m (cluster G (finiteFieldSet Q ω) x) ≤ t :=
-      hhit.trans (mul_le_mul_of_nonneg_right hc' hd)
-    simpa only [pow_two] using mul_self_le_mul_self
-      (finiteChoiceHit_nonneg G k Q p y m hm _) hhit'
-  · rw [ite_eq_right h,ite_eq_right h]
-
-end LocalEndpointRule
-end AllEndpointSeparated
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section AllEndpointMarginal
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem finiteRootLoad_sum {P : Type*} (Q : Finset E) (J : Finset ℕ) (p : ℝ)
-    (S : Finset P) (f : P → SpliceState E → ℝ) (ω : FiniteField Q) :
-    finiteRootLoad Q J p (fun s => ∑ z ∈ S, f z s) ω =
-      ∑ z ∈ S, finiteRootLoad Q J p (f z) ω := by
-  simp only [finiteRootLoad,Finset.mul_sum]
-  exact Finset.sum_comm
-
- 
-
-theorem finiteChoiceForwardLoad_prod {P R : Type*} [Fintype P] [Fintype R]
-    (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
-    (A : P × R → Finset (SpliceState E)) (y : P × R → SpliceState E → ℕ → V)
-    (m : SpliceState E → P × R → ℝ) (t : SpliceState E) :
-    finiteChoiceForwardLoad G x k Q A y m t = ∑ z : P,
-      finiteChoiceForwardLoad G x k Q (fun r => A (z,r)) (fun r => y (z,r))
-        (fun s r => m s (z,r)) t := by
-  unfold finiteChoiceForwardLoad finiteInverseLoad
-  rw [Fintype.sum_sigma,Fintype.sum_prod_type]
-  simp only [Fintype.sum_sigma,choiceTraceOutput,choiceTraceMass,choiceTraceInput]
-  rfl
-
- 
-theorem endpointRootMass_choice_prod {P R : Type*} [Fintype P] [Fintype R]
-    (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E) (p : ℝ)
-    (A : P × R → Finset (SpliceState E)) (y : P × R → SpliceState E → ℕ → V)
-    (m : SpliceState E → P × R → ℝ) (e : P → V) (he : Function.Injective e)
-    (z : P) (ω : Set E) :
-    endpointRootMass Q (Finset.Icc 1 k) p (choiceTraceOutput G x k Q A y)
-      (fun a => e a.1.1) (choiceTraceMass G x k Q A y m) ω (e z) =
-      ∑ a : FiniteChoiceTraceIndex G x k Q (fun r => A (z,r)) (fun r => y (z,r)),
-        if (choiceTraceOutput G x k Q (fun r => A (z,r)) (fun r => y (z,r)) a).1 = ω then
-          auxiliaryTupleWeight Q (Finset.Icc 1 k) p
-            (choiceTraceOutput G x k Q (fun r => A (z,r)) (fun r => y (z,r)) a) *
-          choiceTraceMass G x k Q (fun r => A (z,r)) (fun r => y (z,r)) (fun s r => m s (z,r)) a else 0 := by
-  unfold endpointRootMass
-  rw [Fintype.sum_sigma,Fintype.sum_prod_type]
-  rw [Finset.sum_eq_single z]
-  · rw [Fintype.sum_sigma]
-    simp only [choiceTraceOutput,choiceTraceMass,choiceTraceInput,and_true]
-    rfl
-  · intro z' _ hz'
-    have hh : e z' ≠ e z := fun h => hz' (he h)
-    simp only [hh,and_false,ite_false,Finset.sum_const_zero]
-  · intro hz
-    exact (hz (Finset.mem_univ z)).elim
-
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
-
-abbrev AmplificationIndex (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) :=
-  FiniteChoiceTraceIndex G x k Q (q.allAmplificationInputs Q B x k n)
-    (fun r s => allAmplificationVertices B k r s.2)
-
-noncomputable def amplificationOutput (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) : q.AmplificationIndex Q B x k n → SpliceState E :=
-  choiceTraceOutput G x k Q (q.allAmplificationInputs Q B x k n)
-    (fun r s => allAmplificationVertices B k r s.2)
-
-noncomputable def amplificationInput (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) : q.AmplificationIndex Q B x k n → SpliceState E :=
-  choiceTraceInput G x k Q (q.allAmplificationInputs Q B x k n)
-    (fun r s => allAmplificationVertices B k r s.2)
-
-noncomputable def amplificationCode (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) : q.AmplificationIndex Q B x k n → List (ℕ × E) :=
-  choiceTraceCode G x k Q (q.allAmplificationInputs Q B x k n)
-    (fun r s => allAmplificationVertices B k r s.2)
-
-noncomputable def amplificationMass (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) : q.AmplificationIndex Q B x k n → ℝ :=
-  choiceTraceMass G x k Q (q.allAmplificationInputs Q B x k n)
-    (fun r s => allAmplificationVertices B k r s.2) (fun s r => q.allAmplificationWeight B k r s.2)
-
-noncomputable def amplificationRow (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (p : unitInterval) (ω : Set E) : ℝ :=
-  ∑ z : ↥B, q.amplificationMarginal Q B x k n z p ω
-
-theorem amplificationRow_eq_load (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (p : unitInterval) (ω : Set E) :
-    q.amplificationRow Q B x k n p ω = finiteRootLoad Q (Finset.Icc 1 k) p
-      (finiteInverseLoad (q.amplificationOutput Q B x k n) (q.amplificationMass Q B x k n))
-      (restrictFiniteField Q ω) := by
-  change _ = finiteRootLoad Q (Finset.Icc 1 k) p
-    (finiteChoiceForwardLoad G x k Q (q.allAmplificationInputs Q B x k n)
-      (fun r s => allAmplificationVertices B k r s.2) (fun s r => q.allAmplificationWeight B k r s.2)) _
-  have hf : (finiteChoiceForwardLoad G x k Q (q.allAmplificationInputs Q B x k n)
-      (fun r s => allAmplificationVertices B k r s.2) (fun s r => q.allAmplificationWeight B k r s.2)) =
-      (fun t => ∑ z : ↥B, finiteChoiceForwardLoad G x k Q (q.amplificationInputs Q B x k n z)
-        (fun r _ => amplificationVertices B k z r) (q.amplificationWeight B k z) t) := by
-    funext t
-    exact finiteChoiceForwardLoad_prod G x k Q (q.allAmplificationInputs Q B x k n)
-      (fun r s => allAmplificationVertices B k r s.2) (fun s r => q.allAmplificationWeight B k r s.2) t
-  rw [hf,finiteRootLoad_sum]
-  rfl
-
-theorem amplificationEndpointMass_eq (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (p : unitInterval) (ω : FiniteField Q) (z : ↥B) :
-    endpointRootMass Q (Finset.Icc 1 k) p (q.amplificationOutput Q B x k n)
-      (fun a => a.1.1.val) (q.amplificationMass Q B x k n) (finiteFieldSet Q ω) z.val =
-      q.amplificationMarginal Q B x k n z p (finiteFieldSet Q ω) := by
-  change endpointRootMass Q (Finset.Icc 1 k) p (choiceTraceOutput G x k Q _ _)
-    _ (choiceTraceMass G x k Q _ _ _) _ _ = _
-  rw [endpointRootMass_choice_prod G x k Q p _ _ _ Subtype.val Subtype.val_injective z]
-  unfold amplificationMarginal finiteChoiceForwardLoad
-  rw [restrictFiniteField_fieldSet,finiteRootLoad_eq_sum]
-  · rfl
-  · intro a
-    exact traceIndexOutput_mem G x k Q _ _ (q.amplificationInputs_subset Q B x k n z a.1) a.2
-
-end LocalEndpointRule
-end AllEndpointMarginal
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section AmplificationIndexFacts
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
-
-theorem amplificationOutput_mem (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (a : q.AmplificationIndex Q B x k n) :
-    q.amplificationOutput Q B x k n a ∈ finiteSpliceDomain Q (Finset.Icc 1 k) :=
-  traceIndexOutput_mem G x k Q _ _ (q.amplificationInputs_subset Q B x k n a.1.1 a.1.2) a.2
-
-theorem amplificationMass_nonneg (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (a : q.AmplificationIndex Q B x k n) :
-    0 ≤ q.amplificationMass Q B x k n a :=
-  mul_nonneg (q.allAmplificationWeight_nonneg B k a.1 (q.amplificationInput Q B x k n a).2)
-    (Finset.prod_nonneg fun _ _ => inv_nonneg.mpr (Nat.cast_nonneg _))
-
-theorem amplificationIndex_trace (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (a : q.AmplificationIndex Q B x k n) :
-    SpliceTrace G x (q.amplificationInput Q B x k n a).2
-      (amplificationVertices B k a.1.1 a.1.2) k 0 (q.amplificationInput Q B x k n a).1
-      (q.amplificationCode Q B x k n a) (q.amplificationOutput Q B x k n a).1 :=
-  (traceIndex_valid G x k Q _ _ a.2).1
-
-theorem amplificationIndex_execution (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (a : q.AmplificationIndex Q B x k n) :
-    SpliceExecution G x (q.amplificationInput Q B x k n a)
-      (q.amplificationCode Q B x k n a) (q.amplificationOutput Q B x k n a) :=
-  (traceIndex_valid G x k Q _ _ a.2).2
-
-theorem amplificationIndex_support (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (a : q.AmplificationIndex Q B x k n) :
-    x ∈ O ∧ a.1.1.val ∈ O ∧ Conn G (q.amplificationOutput Q B x k n a).1 x a.1.1.val ∧
-      (cluster G (q.amplificationOutput Q B x k n a).1 x).Finite ∧
-      (cluster G (q.amplificationOutput Q B x k n a).1 x).ncard ≤ 2*n+k*M :=
-  q.amplificationTrace_support Q B x k n a.1.1 a.1.2 a.2.1.val a.2.1.property _ _
-    (q.amplificationIndex_trace Q B x k n a)
-
-theorem amplificationIndex_caps (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (a : q.AmplificationIndex Q B x k n)
-    (i : ℕ) (hi : 1 ≤ i) (hik : i ≤ k) :
-    (cluster G ((q.amplificationInput Q B x k n a).2 i) (amplificationVertices B k a.1.1 a.1.2 i)).Finite ∧
-    (cluster G ((q.amplificationInput Q B x k n a).2 i) (amplificationVertices B k a.1.1 a.1.2 i)).ncard ≤ M := by
-  have hc := q.amplificationInputs_chain Q B x k n a.1.1 a.1.2 a.2.1.val a.2.1.property i hi hik
-  exact ⟨hc.2.1,hc.2.2.1⟩
-
-theorem amplificationEndpointMass_outside (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (p : unitInterval) (ω : Set E)
-    (z : V) (hz : z ∉ B) :
-    endpointRootMass Q (Finset.Icc 1 k) p (q.amplificationOutput Q B x k n)
-      (fun a => a.1.1.val) (q.amplificationMass Q B x k n) ω z = 0 := by
-  apply Finset.sum_eq_zero
-  intro a _
-  split_ifs with h
-  · exact (hz (h.2 ▸ a.1.1.property)).elim
-  · rfl
-
- 
-
-theorem amplificationEndpointMass_bound (q : LocalEndpointRule G hloc O M L) (hL : 1 ≤ L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (p : unitInterval) (ω : Set E)
-    (hω : ω ⊆ (Q : Set E)) (z : V) :
-    endpointRootMass Q (Finset.Icc 1 k) p (q.amplificationOutput Q B x k n)
-      (fun a => a.1.1.val) (q.amplificationMass Q B x k n) ω z ≤ inverseCodeMass M L ^ k := by
-  by_cases hz : z ∈ B
-  · let z' : ↥B := ⟨z,hz⟩
-    have hfield : finiteFieldSet Q (restrictFiniteField Q ω) = ω := by
-      rw [finiteFieldSet_restrict,Set.inter_eq_left.mpr hω]
-    rw [← hfield,q.amplificationEndpointMass_eq Q B x k n p _ z']
-    exact (q.amplificationMarginal_bounds hL Q B x k n z' p _).2
-  · rw [q.amplificationEndpointMass_outside Q B x k n p ω z hz]
-    apply pow_nonneg
-    exact add_nonneg zero_le_one (Finset.sum_nonneg fun l _ => gapWeight_nonneg L (by linarith) l)
-
-end LocalEndpointRule
-end AmplificationIndexFacts
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section NoSwitchDensity
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-@[simp] theorem SpliceExecution.nil_iff (G : BondGraph V E) (x : V) (s t : SpliceState E) :
-    SpliceExecution G x s [] t ↔ s = t := by
-  constructor
-  · intro h
-    cases h
-    rfl
-  · rintro rfl
-    exact SpliceExecution.nil s
-
-theorem traceIndexMass_nil (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
-    (A : Finset (SpliceState E)) (y : SpliceState E → ℕ → V)
-    (a : FiniteTraceIndex G x k Q A y) (hc : traceIndexCode G x k Q A y a = []) :
-    traceIndexMass G x k Q A y a = 1 := by
-  change a.2.val = [] at hc
-  simp [traceIndexMass,hc,traceContactCount]
-
- 
-
-theorem finiteChoiceNilLoad_le {P : Type*} [Fintype P]
-    (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
-    (A : P → Finset (SpliceState E)) (y : P → SpliceState E → ℕ → V)
-    (m : SpliceState E → P → ℝ) (hm : ∀ s r, 0 ≤ m s r) (t : SpliceState E) :
-    finiteInverseLoad (choiceTraceOutput G x k Q A y)
-      (fun a => if choiceTraceCode G x k Q A y a = [] then choiceTraceMass G x k Q A y m a else 0) t ≤
-      ∑ r, if t ∈ A r then m t r else 0 := by
-  let I := FiniteChoiceTraceIndex G x k Q A y
-  let S : I → Prop := fun a => choiceTraceOutput G x k Q A y a = t ∧ choiceTraceCode G x k Q A y a = []
-  let J := {a : I // S a}
-  have hin (a : J) : choiceTraceInput G x k Q A y a.val = t := by
-    have he := (traceIndex_valid G x k Q (A a.val.1) (y a.val.1) a.val.2).2
-    change SpliceExecution G x (choiceTraceInput G x k Q A y a.val)
-      (choiceTraceCode G x k Q A y a.val) (choiceTraceOutput G x k Q A y a.val) at he
-    rw [a.property.1,a.property.2,SpliceExecution.nil_iff] at he
-    exact he
-  have hi : Function.Injective (fun a : J => a.val.1) := by
-    intro a b hab
-    apply Subtype.ext
-    apply choiceTrace_injective G x k Q A y
-    exact Prod.ext ((hin a).trans (hin b).symm)
-      (Prod.ext hab (a.property.2.trans b.property.2.symm))
-  have hm' (a : J) : choiceTraceMass G x k Q A y m a.val = m t a.val.1 := by
-    unfold choiceTraceMass
-    rw [hin a,traceIndexMass_nil G x k Q (A a.val.1) (y a.val.1) a.val.2 a.property.2,mul_one]
-  have hA (a : J) : t ∈ A a.val.1 := (hin a) ▸ a.val.2.1.property
-  have he : finiteInverseLoad (choiceTraceOutput G x k Q A y)
-      (fun a => if choiceTraceCode G x k Q A y a = [] then choiceTraceMass G x k Q A y m a else 0) t =
-      ∑ a : J, m t a.val.1 := by
-    unfold finiteInverseLoad
-    simp only [← ite_and]
-    change (∑ a : I, if S a then choiceTraceMass G x k Q A y m a else 0) = _
-    calc
-      _ = ∑ a : J, choiceTraceMass G x k Q A y m a.val := by
-        rw [← Finset.sum_filter]
-        exact Finset.sum_subtype _ (by intro a; simp) _
-      _ = _ := Finset.sum_congr rfl (fun a _ => hm' a)
-  rw [he]
-  have hb := sum_comp_le_sum_of_injective (fun a : J => a.val.1) hi
-    (fun r => if t ∈ A r then m t r else 0) (by intro r; split_ifs; exact hm t r; exact le_refl _)
-  simpa only [ite_eq_left (hA _)] using hb
-
-end NoSwitchDensity
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section AllEndpointNil
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem oldRunArrival_integral {W : Type*} [Fintype W]
-    (Q : Finset E) (p : ℝ) (k : ℕ) (q : Set E → W → W → ℝ) (z : W) (A : Finset W) :
-    finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η => oldRunArrival (fun i => q (η i)) k z A) =
-      oldRunArrival (fun _ => finiteOldKernel Q p q) k z A := by
-  unfold oldRunArrival
-  rw [finiteFamilyIntegral_sum]
-  apply Finset.sum_congr rfl
-  intro r _
-  by_cases h : oldRunVertices k z r 0 ∈ A
-  · simp only [ite_eq_left h]
-    exact oldRunWeight_integral Q p k q z r
-  · simp [h,finiteFamilyIntegral]
-
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
-
-noncomputable def amplificationNilRow (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (p : unitInterval) (ω : FiniteField Q) : ℝ :=
-  finiteRootLoad Q (Finset.Icc 1 k) p
-    (finiteInverseLoad (q.amplificationOutput Q B x k n)
-      (fun a => if q.amplificationCode Q B x k n a = [] then q.amplificationMass Q B x k n a else 0)) ω
-
- 
-
-theorem amplificationNilRow_arrival_le (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (p : unitInterval) (ω : FiniteField Q) :
-    q.amplificationNilRow Q B x k n p ω ≤
-      if n ≤ (cluster G (finiteFieldSet Q ω) x).ncard ∧ (cluster G (finiteFieldSet Q ω) x).ncard < 2*n then
-        ∑ z : ↥B, oldRunArrival (fun _ => finiteOldKernel Q p (fun η (a b : ↥B) => q.mass η a.val b.val)) k z
-          (Finset.univ.filter fun a : ↥B => a.val ∈ cluster G (finiteFieldSet Q ω) x) else 0 := by
-  let A : Finset ↥B := Finset.univ.filter fun a : ↥B => a.val ∈ cluster G (finiteFieldSet Q ω) x
-  let H := n ≤ (cluster G (finiteFieldSet Q ω) x).ncard ∧ (cluster G (finiteFieldSet Q ω) x).ncard < 2*n
-  have hb (η : ℕ → Set E) :
-      finiteInverseLoad (q.amplificationOutput Q B x k n)
-        (fun a => if q.amplificationCode Q B x k n a = [] then q.amplificationMass Q B x k n a else 0)
-        (finiteFieldSet Q ω,η) ≤
-      if H then ∑ z : ↥B, oldRunArrival (fun i (a b : ↥B) => q.mass (η i) a.val b.val) k z A else 0 := by
-    apply (finiteChoiceNilLoad_le G x k Q (q.allAmplificationInputs Q B x k n)
-      (fun r s => allAmplificationVertices B k r s.2)
-      (fun s r => q.allAmplificationWeight B k r s.2)
-      (fun s r => q.allAmplificationWeight_nonneg B k r s.2) (finiteFieldSet Q ω,η)).trans
-    rw [Fintype.sum_prod_type]
-    by_cases h : H
-    · rw [ite_eq_left h]
-      apply Finset.sum_le_sum
-      intro z _
-      apply Finset.sum_le_sum
-      intro r _
-      by_cases hs : (finiteFieldSet Q ω,η) ∈ q.allAmplificationInputs Q B x k n (z,r)
-      · rw [ite_eq_left hs]
-        have hi := (q.mem_amplificationInputs Q B x k n z r _).mp hs
-        have ha : oldRunVertices k z r 0 ∈ A := by
-          simpa only [A,Finset.mem_filter,Finset.mem_univ,true_and,amplificationVertices] using hi.2.2.2.2.2.1
-        rw [ite_eq_left ha]
-        exact le_rfl
-      · rw [ite_eq_right hs]
-        split_ifs
-        · exact oldRunWeight_nonneg _ (fun i a b => q.nonneg (η i) a.val b.val) k z r
-        · exact le_refl _
-    · rw [ite_eq_right h]
-      apply le_of_eq
-      apply Finset.sum_eq_zero
-      intro z _
-      apply Finset.sum_eq_zero
-      intro r _
-      rw [ite_eq_right (by
-        intro hs
-        have hi := (q.mem_amplificationInputs Q B x k n z r _).mp hs
-        exact h ⟨hi.2.2.2.1,hi.2.2.2.2.1⟩)]
-  change _ ≤ if H then ∑ z : ↥B,
-    oldRunArrival (fun _ => finiteOldKernel Q p (fun η (a b : ↥B) => q.mass η a.val b.val)) k z A else 0
-  calc
-    _ ≤ finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η => if H then
-        ∑ z : ↥B, oldRunArrival (fun i (a b : ↥B) => q.mass (η i) a.val b.val) k z A else 0) := by
-      apply Finset.sum_le_sum
-      intro η _
-      exact mul_le_mul_of_nonneg_left (hb _) (finiteFamilyWeight_nonneg Q (Finset.Icc 1 k) p η)
-    _ = _ := by
-      by_cases h : H
-      · simp only [ite_eq_left h,finiteFamilyIntegral_sum]
-        apply Finset.sum_congr rfl
-        intro z _
-        exact oldRunArrival_integral Q p k (fun η (a b : ↥B) => q.mass η a.val b.val) z A
-      · simp only [ite_eq_right h,finiteFamilyIntegral,mul_zero,Finset.sum_const_zero]
-
-end LocalEndpointRule
-end AllEndpointNil
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-section AllEndpointNilBound
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
-
-theorem amplificationNilRow_nonneg {O : Set V} (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (p : unitInterval) (ω : FiniteField Q) :
-    0 ≤ q.amplificationNilRow Q B x k n p ω := by
-  apply finiteRootLoad_nonneg
-  intro s _
-  apply finiteInverseLoad_nonneg
-  intro a
-  split_ifs
-  · exact q.amplificationMass_nonneg Q B x k n a
-  all_goals exact le_refl _
-
- 
-
-theorem amplificationNilRow_bound {ι : Type*} [Fintype ι]
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (htr : RootTransport root weight Γ) (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b)
-    (hw : ∀ i, 0 ≤ weight i) (i₀ : ι) (hw₀ : 0 < weight i₀)
-    (q : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (p : unitInterval) (ω : FiniteField Q)
-    (hQ : ∀ a ∈ B, incidentSupport G hloc (finite_reachBall hloc a q.radius).toFinset ⊆ Q) :
-    q.amplificationNilRow Q B x k n p ω ≤ ((2*n:ℕ):ℝ) * q.successProbability p (root i₀)^k := by
-  apply (q.amplificationNilRow_arrival_le Q B x k n p ω).trans
-  split_ifs with h
-  · let A : Finset ↥B := Finset.univ.filter fun a : ↥B => a.val ∈ cluster G (finiteFieldSet Q ω) x
-    let K := fun (_ : ℕ) => finiteOldKernel Q p (fun η (a b : ↥B) => q.mass η a.val b.val)
-    have hK (i : ℕ) (a b : ↥B) : K i a b = q.kernel p a.val b.val :=
-      q.finiteOldKernel_eq_kernel Q p a.val b.val (hQ a.val a.property)
-    have hnK (i : ℕ) (a b : ↥B) : 0 ≤ K i a b := by rw [hK i a b]; exact q.kernel_nonneg p a.val b.val
-    have hc (i : ℕ) (b : ↥B) : (∑ a, K i a b) ≤ q.successProbability p (root i₀) := by
-      simp only [hK]
-      rw [Finset.sum_coe_sort B (fun a => q.kernel p a b.val)]
-      exact ((q.summable_kernel_column p b.val).sum_le_tsum B (fun a _ => q.kernel_nonneg p a b.val)).trans
-        (q.kernel_column_le_success root weight Γ htr hΓ hw i₀ hw₀ p b.val)
-    have hf : (cluster G (finiteFieldSet Q ω) x).Finite := cluster_finite_of_finite_edges G
-      ω.val.finite_toSet x
-    have hcard : A.card ≤ 2*n := by
-      have hh : A.card ≤ hf.toFinset.card := Finset.card_le_card_of_injOn Subtype.val
-        (by intro a ha; exact hf.mem_toFinset.mpr (Finset.mem_filter.mp ha).2)
-        (fun _ _ _ _ he => Subtype.val_injective he)
-      rw [← Set.ncard_eq_toFinset_card (cluster G (finiteFieldSet Q ω) x) hf] at hh
-      exact hh.trans h.2.le
-    calc
-      _ ≤ (A.card:ℝ) * q.successProbability p (root i₀)^k :=
-        sum_oldRunArrival_le K hnK _ (q.successProbability_nonneg p (root i₀)) hc k A
-      _ ≤ ((2*n:ℕ):ℝ) * q.successProbability p (root i₀)^k :=
-        mul_le_mul_of_nonneg_right (by exact_mod_cast hcard) (pow_nonneg (q.successProbability_nonneg p (root i₀)) _)
-  · exact mul_nonneg (Nat.cast_nonneg _) (pow_nonneg (q.successProbability_nonneg p (root i₀)) _)
-
-end LocalEndpointRule
-end AllEndpointNilBound
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-section AllEndpointNear
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
-
-noncomputable def amplificationAdded (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (a : q.AmplificationIndex Q B x k n) : Set V :=
-  cluster G (q.amplificationOutput Q B x k n a).1 x \ cluster G (q.amplificationInput Q B x k n a).1 x
-
-noncomputable def amplificationNearPairMass (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (p : unitInterval) : ℝ :=
-  ∑ a : q.AmplificationIndex Q B x k n, ∑ b : q.AmplificationIndex Q B x k n,
-    if (q.amplificationOutput Q B x k n a).1 = (q.amplificationOutput Q B x k n b).1 ∧
-      q.amplificationCode Q B x k n a ≠ [] ∧ q.amplificationCode Q B x k n b ≠ [] ∧
-      ¬GraphSeparated G (q.amplificationAdded Q B x k n a) (q.amplificationAdded Q B x k n b)
-    then jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p
-      (pairSpliceOutput (q.amplificationOutput Q B x k n a) (q.amplificationOutput Q B x k n b)) *
-        q.amplificationMass Q B x k n a * q.amplificationMass Q B x k n b else 0
-
- 
-
-theorem amplificationNearPairMass_bound [Nonempty E]
-    (q : LocalEndpointRule G hloc O M L) (hL : 1 ≤ L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n d : ℕ) (p : unitInterval)
-    (hdeg : ∀ a, degree G a ≤ d) :
-    q.amplificationNearPairMass Q B x k n p ≤
-      (((k*M)+(k*M)^2+d*(k*M)^3:ℕ):ℝ) * inverseCodeMass M L ^ k *
-        ∑ a : q.AmplificationIndex Q B x k n,
-          spliceTupleWeight Q (Finset.Icc 1 k) p (q.amplificationOutput Q B x k n a) *
-            q.amplificationMass Q B x k n a := by
-  let out := q.amplificationOutput Q B x k n
-  let code := q.amplificationCode Q B x k n
-  let added := q.amplificationAdded Q B x k n
-  let m := q.amplificationMass Q B x k n
-  let e (a : q.AmplificationIndex Q B x k n) := (code a).headD (0,Classical.choice ‹Nonempty E›) |>.2
-  have hfin (a : q.AmplificationIndex Q B x k n) : (cluster G (out a).1 x).Finite :=
-    (q.amplificationIndex_support Q B x k n a).2.2.2.1
-  have hhead (a : q.AmplificationIndex Q B x k n) (hc : code a ≠ []) :
-      OpenMarkedPivotal G (out a).1 {a.1.1.val} x (e a) ∧
-      hangingSide G (out a).1 x (e a) = added a ∧
-      (hangingSide G (out a).1 x (e a)).ncard ≤ k*M := by
-    cases hh : code a with
-    | nil => exact (hc hh).elim
-    | cons w rest =>
-      have ht := q.amplificationIndex_trace Q B x k n a
-      change SpliceTrace G x (q.amplificationInput Q B x k n a).2
-        (amplificationVertices B k a.1.1 a.1.2) k 0 (q.amplificationInput Q B x k n a).1
-        (code a) (out a).1 at ht
-      rw [hh] at ht
-      have he : e a = w.2 := by simp only [e,hh,List.headD_cons]
-      rw [he]
-      refine ⟨?_,ht.head_hanging G x _ _ k 0 w.1 _ _ w.2 rest,
-        ht.head_hanging_cap G x _ _ k 0 w.1 M _ _ w.2 rest (q.amplificationIndex_caps Q B x k n a)⟩
-      simpa only [amplificationVertices,oldRunVertices_last] using
-        ht.seams_pivotal G x _ _ k 0 _ _ (w::rest) w (List.mem_cons_self)
-  let S (a : q.AmplificationIndex Q B x k n) : Finset V := if code a = [] then ∅ else
-    nearHangingEndpoints G hloc (out a).1 x a.1.1.val (e a) (k*M) (hfin a)
-  let R (a b : q.AmplificationIndex Q B x k n) := code a ≠ [] ∧ code b ≠ [] ∧
-    ¬GraphSeparated G (added a) (added b)
-  have hS (a b : q.AmplificationIndex Q B x k n) (he : (out a).1 = (out b).1) (h : R a b) :
-      b.1.1.val ∈ S a := by
-    obtain ⟨hpa,hea,_⟩ := hhead a h.1
-    obtain ⟨hpb,heb,hcb⟩ := hhead b h.2.1
-    have hn := h.2.2
-    rw [← hea,← heb,← he] at hn
-    rw [← he] at hpb hcb
-    simp only [S,ite_eq_right h.1]
-    exact mem_nearHangingEndpoints_of_not_separated G hloc (out a).1 x a.1.1.val b.1.1.val
-      (e a) (e b) (k*M) (hfin a) hpa hpb hcb hn
-  have hN (a : q.AmplificationIndex Q B x k n) : (S a).card ≤ (k*M)+(k*M)^2+d*(k*M)^3 := by
-    by_cases h : code a = []
-    · simp only [S,ite_eq_left h,Finset.card_empty,Nat.zero_le]
-    · simp only [S,ite_eq_right h]
-      exact nearHangingEndpoints_card_le G hloc (out a).1 x a.1.1.val (e a) (k*M) d
-        (hfin a) (hhead a h).2.2 hdeg
-  have hC : 0 ≤ inverseCodeMass M L ^ k := pow_nonneg
-    (add_nonneg zero_le_one (Finset.sum_nonneg fun l _ => gapWeight_nonneg L (by linarith) l)) _
-  simpa only [amplificationNearPairMass,out,m,R,added] using finite_near_pair_mass_le Q (Finset.Icc 1 k) (Finset.Icc 1 k) p out out m m
-    (q.amplificationMass_nonneg Q B x k n) (q.amplificationMass_nonneg Q B x k n)
-    (fun a => a.1.1.val) R S hS ((k*M)+(k*M)^2+d*(k*M)^3) hN _ hC
-    (fun a z _ => q.amplificationEndpointMass_bound hL Q B x k n p (out a).1
-      ((mem_finiteSpliceDomain_iff Q (Finset.Icc 1 k) (out a)).mp (q.amplificationOutput_mem Q B x k n a)).1 z)
-
-end LocalEndpointRule
-end AllEndpointNear
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-section FiniteThreeClass
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def commonRootPairMass {I : Type*} [Fintype I]
-    (Q : Finset E) (J : Finset ℕ) (p : ℝ) (out : I → SpliceState E)
-    (m : I → ℝ) (P : I → I → Prop) : ℝ :=
-  ∑ a, ∑ b, if (out a).1 = (out b).1 ∧ P a b then
-    jointTupleWeight Q J J p (pairSpliceOutput (out a) (out b)) * m a * m b else 0
-
- 
-
-theorem finiteRootLoad_three_classes_le {I : Type*} [Fintype I]
-    (Q : Finset E) (J : Finset ℕ) (p : unitInterval) (out : I → SpliceState E)
-    (m : I → ℝ) (hm : ∀ a, 0 ≤ m a) (hs : ∀ a, out a ∈ finiteSpliceDomain Q J)
-    (C : I → Prop) (R : I → I → Prop) :
-    (∑ ω : FiniteField Q, finiteFieldWeight Q p ω *
-      finiteRootLoad Q J p (finiteInverseLoad out m) ω ^ 2) ≤
-      2 * (∑ ω : FiniteField Q, finiteFieldWeight Q p ω *
-        finiteRootLoad Q J p (finiteInverseLoad out (fun a => if C a then m a else 0)) ω *
-        finiteRootLoad Q J p (finiteInverseLoad out m) ω) +
-      commonRootPairMass Q J p out m (fun a b => ¬C a ∧ ¬C b ∧ R a b) +
-      commonRootPairMass Q J p out m (fun a b => ¬C a ∧ ¬C b ∧ ¬R a b) := by
-  let m₀ := fun a => if C a then m a else 0
-  let row := finiteRootLoad Q J p (finiteInverseLoad out m)
-  let nilrow := finiteRootLoad Q J p (finiteInverseLoad out m₀)
-  have hcross : (∑ ω : FiniteField Q, finiteFieldWeight Q p ω * row ω * nilrow ω) =
-      ∑ ω : FiniteField Q, finiteFieldWeight Q p ω * nilrow ω * row ω := by
-    apply Finset.sum_congr rfl
-    intro ω _
-    ring
-  have heq : 2 * (∑ ω : FiniteField Q, finiteFieldWeight Q p ω * nilrow ω * row ω) =
-      (∑ ω : FiniteField Q, finiteFieldWeight Q p ω * nilrow ω * row ω) +
-      (∑ ω : FiniteField Q, finiteFieldWeight Q p ω * row ω * nilrow ω) := by
-    rw [hcross]; ring
-  change _ ≤ 2 * (∑ ω : FiniteField Q, finiteFieldWeight Q p ω * nilrow ω * row ω) + _ + _
-  rw [heq]
-  dsimp only [nilrow,row]
-  simp only [pow_two,← mul_assoc]
-  rw [finiteRootLoad_pair_expansion Q J J p out out m m hs hs,
-    finiteRootLoad_pair_expansion Q J J p out out m₀ m hs hs,
-    finiteRootLoad_pair_expansion Q J J p out out m m₀ hs hs]
-  unfold commonRootPairMass
-  simp only [← Finset.sum_add_distrib]
-  apply Finset.sum_le_sum
-  intro a _
-  apply Finset.sum_le_sum
-  intro b _
-  have hw : 0 ≤ jointTupleWeight Q J J p (pairSpliceOutput (out a) (out b)) :=
-    jointTupleWeight_nonneg Q J J p _
-  have ht : 0 ≤ jointTupleWeight Q J J p (pairSpliceOutput (out a) (out b)) * m a * m b :=
-    mul_nonneg (mul_nonneg hw (hm a)) (hm b)
-  dsimp only [m₀]
-  by_cases hab : (out a).1 = (out b).1
-  · simp only [hab,true_and,ite_true]
-    by_cases ha : C a <;> by_cases hb : C b <;> by_cases hr : R a b <;>
-      simp only [ha,hb,hr,not_true_eq_false,not_false_eq_true,and_true,and_false,
-        ite_true,ite_false,mul_zero,zero_mul] <;> linarith
-  · simp only [hab,false_and,ite_false,zero_add,le_refl]
-
-end FiniteThreeClass
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-section AllEndpointMomentIdentities
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem finiteRootLoad_inverse_mean {I : Type*} [Fintype I]
-    (Q : Finset E) (J : Finset ℕ) (p : ℝ) (out : I → SpliceState E) (m : I → ℝ)
-    (hs : ∀ a, out a ∈ finiteSpliceDomain Q J) :
-    (∑ ω : FiniteField Q, finiteFieldWeight Q p ω * finiteRootLoad Q J p (finiteInverseLoad out m) ω) =
-      ∑ a, spliceTupleWeight Q J p (out a) * m a := by
-  rw [finiteRootLoad_mean]
-  simp only [finiteInverseLoad,Finset.mul_sum,mul_ite,mul_zero]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro a _
-  rw [Finset.sum_ite_eq,ite_eq_left (hs a)]
-
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
-
-theorem amplificationRow_nonneg (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (p : unitInterval) (ω : Set E) :
-    0 ≤ q.amplificationRow Q B x k n p ω := by
-  rw [q.amplificationRow_eq_load]
-  apply finiteRootLoad_nonneg
-  intro s _
-  exact finiteInverseLoad_nonneg _ _ (q.amplificationMass_nonneg Q B x k n) s
-
-theorem amplificationRow_mean_eq (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (p : unitInterval) :
-    finiteFieldIntegral Q p (q.amplificationRow Q B x k n p) =
-      ∑ a : q.AmplificationIndex Q B x k n,
-        spliceTupleWeight Q (Finset.Icc 1 k) p (q.amplificationOutput Q B x k n a) *
-          q.amplificationMass Q B x k n a := by
-  simp only [finiteFieldIntegral,q.amplificationRow_eq_load,restrictFiniteField_fieldSet]
-  exact finiteRootLoad_inverse_mean Q (Finset.Icc 1 k) p _ _ (q.amplificationOutput_mem Q B x k n)
-
- 
-
-theorem amplificationSeparated_class_eq (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (p : unitInterval) :
-    commonRootPairMass Q (Finset.Icc 1 k) p (q.amplificationOutput Q B x k n)
-      (q.amplificationMass Q B x k n) (fun a b =>
-        q.amplificationCode Q B x k n a ≠ [] ∧ q.amplificationCode Q B x k n b ≠ [] ∧
-        GraphSeparated G (q.amplificationAdded Q B x k n a) (q.amplificationAdded Q B x k n b)) =
-    finiteChoiceSeparatedPairMass G x k Q p
-      (q.allAmplificationInputs Q B x k n) (q.allAmplificationInputs Q B x k n)
-      (allAmplificationVertices B k) (allAmplificationVertices B k)
-      (q.allAmplificationWeight B k) (q.allAmplificationWeight B k) := by
-  unfold commonRootPairMass finiteChoiceSeparatedPairMass
-  apply Finset.sum_congr rfl
-  intro a _
-  apply Finset.sum_congr rfl
-  intro b _
-  split_ifs with ha hb hb
-  · rfl
-  · exfalso
-    apply hb
-    dsimp only [amplificationAdded,amplificationOutput,amplificationCode,amplificationInput] at ha
-    rcases ha with ⟨he,hca,hcb,hsep⟩
-    refine ⟨he,hca,hcb,?_⟩
-    simpa only [he] using hsep
-  · exfalso
-    apply ha
-    dsimp only [amplificationAdded,amplificationOutput,amplificationCode,amplificationInput]
-    rcases hb with ⟨he,hca,hcb,hsep⟩
-    refine ⟨he,hca,hcb,?_⟩
-    simpa only [he] using hsep
-  · rfl
-
-end LocalEndpointRule
-end AllEndpointMomentIdentities
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-section AllEndpointSecondMoment
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
-
- 
-
-theorem amplificationRow_second_moment_le [Nonempty E] (hconn : Connected G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
-    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (i₀ : ι) (hw₀ : 0 < weight i₀)
-    (q : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) M L) (hL : 1 ≤ L)
-    (Q : Finset E) (B : Finset V) (p : unitInterval) (x : V) (k n d : ℕ)
-    (hdeg : ∀ a, degree G a ≤ d)
-    (hQ : ∀ a ∈ B, incidentSupport G hloc (finite_reachBall hloc a (max q.radius M)).toFinset ⊆ Q) :
-    finiteFieldIntegral Q p (fun ω => q.amplificationRow Q B x k n p ω ^ 2) ≤
-      (((4*n:ℕ):ℝ) * q.successProbability p (root i₀)^k +
-        (((k*M)+(k*M)^2+d*(k*M)^3:ℕ):ℝ) * inverseCodeMass M L ^ k) *
-          finiteFieldIntegral Q p (q.amplificationRow Q B x k n p) +
-      (((2*n:ℕ):ℝ) * (c*((k*M:ℕ)*q.successProbability p (root i₀)^k)))^2 *
-        finiteClusterWindowMass G Q p x (n-k*M) (2*n) := by
-  let out := q.amplificationOutput Q B x k n
-  let m := q.amplificationMass Q B x k n
-  let C := fun a => q.amplificationCode Q B x k n a = []
-  let R := fun a b => ¬GraphSeparated G (q.amplificationAdded Q B x k n a) (q.amplificationAdded Q B x k n b)
-  let row := finiteRootLoad Q (Finset.Icc 1 k) p (finiteInverseLoad out m)
-  let nilrow := q.amplificationNilRow Q B x k n p
-  let b₀ := ∑ ω : FiniteField Q, finiteFieldWeight Q p ω * row ω
-  let bnil := ∑ ω : FiniteField Q, finiteFieldWeight Q p ω * nilrow ω * row ω
-  let S := finiteChoiceSeparatedPairMass G x k Q p
-    (q.allAmplificationInputs Q B x k n) (q.allAmplificationInputs Q B x k n)
-    (allAmplificationVertices B k) (allAmplificationVertices B k)
-    (q.allAmplificationWeight B k) (q.allAmplificationWeight B k)
-  have hrow (ω : FiniteField Q) : q.amplificationRow Q B x k n p (finiteFieldSet Q ω) = row ω := by
-    rw [q.amplificationRow_eq_load,restrictFiniteField_fieldSet]
-  have hmean : finiteFieldIntegral Q p (q.amplificationRow Q B x k n p) = b₀ := by
-    simp only [finiteFieldIntegral,hrow,b₀]
-  have hpart := finiteRootLoad_three_classes_le Q (Finset.Icc 1 k) p out m
-    (q.amplificationMass_nonneg Q B x k n) (q.amplificationOutput_mem Q B x k n) C R
-  have hnear_eq : commonRootPairMass Q (Finset.Icc 1 k) p out m
-      (fun a b => ¬C a ∧ ¬C b ∧ R a b) = q.amplificationNearPairMass Q B x k n p := by
-    simp only [commonRootPairMass,amplificationNearPairMass,C,R,out,m]
-  have hsep_eq : commonRootPairMass Q (Finset.Icc 1 k) p out m
-      (fun a b => ¬C a ∧ ¬C b ∧ ¬R a b) = S := by
-    simpa only [C,R,not_not] using q.amplificationSeparated_class_eq Q B x k n p
-  rw [hnear_eq,hsep_eq] at hpart
-  have hnilfun : finiteInverseLoad out (fun a => @ite ℝ (C a) (Classical.propDecidable _) (m a) 0) =
-      finiteInverseLoad (q.amplificationOutput Q B x k n)
-        (fun a => if q.amplificationCode Q B x k n a = [] then q.amplificationMass Q B x k n a else 0) := by
-    apply congrArg (finiteInverseLoad out)
-    funext a
-    dsimp only [C,m]
-    split_ifs <;> rfl
-  rw [hnilfun] at hpart
-  change (∑ ω : FiniteField Q, finiteFieldWeight Q p ω * row ω ^ 2) ≤
-    2*bnil + q.amplificationNearPairMass Q B x k n p + S at hpart
-  have hnil : bnil ≤ ((2*n:ℕ):ℝ) * q.successProbability p (root i₀)^k * b₀ := by
-    dsimp only [bnil,b₀]
-    rw [Finset.mul_sum]
-    apply Finset.sum_le_sum
-    intro ω _
-    have hr : 0 ≤ row ω := by rw [← hrow]; exact q.amplificationRow_nonneg Q B x k n p _
-    have hn := q.amplificationNilRow_bound root weight Γ htr (fun a ha => hΓ ha) hw i₀ hw₀ Q B x k n p ω
-      (fun a ha => (incidentSupport_ball_mono G hloc a (le_max_left _ _)).trans (hQ a ha))
-    calc
-      _ ≤ finiteFieldWeight Q p ω * (((2*n:ℕ):ℝ) * q.successProbability p (root i₀)^k) * row ω :=
-        mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left hn (finiteFieldWeight_nonneg Q p ω)) hr
-      _ = _ := by ring
-  have hnear := q.amplificationNearPairMass_bound hL Q B x k n d p hdeg
-  rw [← q.amplificationRow_mean_eq Q B x k n p,hmean] at hnear
-  have hsep := q.amplificationSeparatedSecondMoment_le hconn root weight Γ hΓ htr hcov hw hw1 c hc hmin
-    (root i₀) Q B p x k n hQ
-  have hfinal := add_le_add (add_le_add (mul_le_mul_of_nonneg_left hnil (show (0:ℝ) ≤ 2 by norm_num)) hnear) hsep
-  rw [hmean]
-  simp only [finiteFieldIntegral,hrow]
-  apply hpart.trans
-  convert hfinal using 1
-  push_cast
-  ring
-
-end LocalEndpointRule
-end AllEndpointSecondMoment
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-section SpliceCovarianceGeometry
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem cluster_aut_preimage (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (ω : Set E) (x : V) :
-    cluster G (b ⁻¹' ω) x = a ⁻¹' cluster G ω (a x) := by
-  ext y
-  exact conn_aut_iff G hab ω x y
-
-theorem exchangeOn_preimage (b : E ≃ E) (mask ω η : Set E) :
-    exchangeOn (b ⁻¹' mask) (b ⁻¹' ω) (b ⁻¹' η) = b ⁻¹' exchangeOn mask ω η := by
-  rfl
-
-theorem spliceMask_aut_preimage (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (S B : Set V) (e : E) :
-    spliceMask G (a ⁻¹' S) (a ⁻¹' B) e = b ⁻¹' spliceMask G S B (b e) := by
-  ext f
-  simp only [spliceMask,Set.mem_insert_iff,Set.mem_ofPred_eq,Set.mem_preimage,b.injective.eq_iff]
-  apply or_congr_right
-  have hi : internalEdges G (a ⁻¹' S)ᶜ = b ⁻¹' internalEdges G Sᶜ :=
-    internalEdges_aut_preimage G hab Sᶜ
-  rw [hi]
-  apply and_congr_left
-  intro _
-  constructor
-  · rintro ⟨v,hv,hvf⟩
-    refine ⟨a v,hv,?_⟩
-    rw [hab f]
-    exact Sym2.mem_map.mpr ⟨v,hvf,rfl⟩
-  · rintro ⟨v,hv,hvf⟩
-    rw [hab f] at hvf
-    obtain ⟨w,hw,rfl⟩ := Sym2.mem_map.mp hvf
-    exact ⟨w,hv,hw⟩
-
-theorem spliceField_aut_preimage (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (S B : Set V) (e : E) (ω η : Set E) :
-    spliceField G (a ⁻¹' S) (a ⁻¹' B) e (b ⁻¹' ω) (b ⁻¹' η) =
-      b ⁻¹' spliceField G S B (b e) ω η := by
-  rw [spliceField,spliceMask_aut_preimage G hab,exchangeOn_preimage]
-  rfl
-
-theorem seamContactCount_aut_preimage (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (η : ℕ → Set E) (y : ℕ → V) (ωf : Set E) (x : V) (i : ℕ) (e : E) :
-    seamContactCount G (fun j => b ⁻¹' η j) y (b ⁻¹' ωf) x i e =
-      seamContactCount G η (fun j => a (y j)) ωf (a x) i (b e) := by
-  have he : (b ⁻¹' ωf) \ {e} = b ⁻¹' (ωf \ {b e}) := by
-    ext f
-    simp only [Set.mem_sdiff,Set.mem_preimage,Set.mem_singleton_iff,b.injective.eq_iff]
-  unfold seamContactCount
-  rw [he,cluster_aut_preimage G hab]
-  rw [show (a ⁻¹' cluster G (ωf \ {b e}) (a x))ᶜ =
-    a ⁻¹' (cluster G (ωf \ {b e}) (a x))ᶜ from rfl]
-  rw [internalEdges_aut_preimage G hab,← Set.preimage_inter,cluster_aut_preimage G hab,
-    contactSet_aut_preimage G hab,Set.ncard_preimage_of_injective_subset_range b.injective]
-  exact fun f _ => b.surjective f
-
-theorem traceContactCount_aut_preimage (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (η : ℕ → Set E) (y : ℕ → V) (ωf : Set E) (x : V)
-    (c : List (ℕ × E)) (i : ℕ) :
-    traceContactCount G (fun j => b ⁻¹' η j) y (b ⁻¹' ωf) x c i =
-      traceContactCount G η (fun j => a (y j)) ωf (a x) (c.map (Prod.map id b)) i := by
-  induction c with
-  | nil => rfl
-  | cons q c ih =>
-    rcases q with ⟨j,e⟩
-    simp only [traceContactCount,List.map_cons,Prod.map_apply,id_eq]
-    split_ifs
-    · exact seamContactCount_aut_preimage G hab η y ωf x j e
-    · exact ih
-
-end SpliceCovarianceGeometry
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-section SpliceCovarianceTrace
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-def edgeFieldEquiv (b : E ≃ E) : Set E ≃ Set E where
-  toFun ω := b.symm ⁻¹' ω
-  invFun ω := b ⁻¹' ω
-  left_inv ω := by ext e; simp
-  right_inv ω := by ext e; simp
-
-def spliceStateEquiv (b : E ≃ E) : SpliceState E ≃ SpliceState E :=
-  (edgeFieldEquiv b).prodCongr (Equiv.piCongrRight fun _ => edgeFieldEquiv b)
-
-def spliceCodeEquiv (b : E ≃ E) : List (ℕ × E) ≃ List (ℕ × E) :=
-  Equiv.listEquivOfEquiv ((Equiv.refl ℕ).prodCongr b)
-
-@[simp] theorem edgeFieldEquiv_apply (b : E ≃ E) (ω : Set E) :
-    edgeFieldEquiv b ω = b.symm ⁻¹' ω := rfl
-
-@[simp] theorem spliceCodeEquiv_apply (b : E ≃ E) (c : List (ℕ × E)) :
-    spliceCodeEquiv b c = c.map (Prod.map id b) := rfl
-
-theorem cluster_edgeFieldEquiv (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (ω : Set E) (x : V) :
-    cluster G (edgeFieldEquiv b ω) (a x) = a.symm ⁻¹' cluster G ω x := by
-  simpa only [edgeFieldEquiv_apply,Equiv.symm_apply_apply] using
-    cluster_aut_preimage G (isAut_symm G hab) ω (a x)
-
-theorem slitCluster_edgeFieldEquiv (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (ω η : Set E) (x y : V) :
-    cluster G (edgeFieldEquiv b η ∩ internalEdges G (cluster G (edgeFieldEquiv b ω) (a x))ᶜ) (a y) =
-      a.symm ⁻¹' cluster G (η ∩ internalEdges G (cluster G ω x)ᶜ) y := by
-  rw [cluster_edgeFieldEquiv G hab]
-  rw [show (a.symm ⁻¹' cluster G ω x)ᶜ = a.symm ⁻¹' (cluster G ω x)ᶜ from rfl]
-  rw [internalEdges_aut_preimage G (isAut_symm G hab),edgeFieldEquiv_apply,← Set.preimage_inter,
-    cluster_aut_preimage G (isAut_symm G hab),Equiv.symm_apply_apply]
-
-theorem spliceStep_edgeFieldEquiv (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (ω η : Set E) (x y : V) (e : E) :
-    spliceField G (cluster G (edgeFieldEquiv b ω) (a x))
-      (cluster G (edgeFieldEquiv b η ∩ internalEdges G (cluster G (edgeFieldEquiv b ω) (a x))ᶜ) (a y))
-      (b e) (edgeFieldEquiv b ω) (edgeFieldEquiv b η) =
-    edgeFieldEquiv b (spliceField G (cluster G ω x)
-      (cluster G (η ∩ internalEdges G (cluster G ω x)ᶜ) y) e ω η) := by
-  rw [slitCluster_edgeFieldEquiv G hab,cluster_edgeFieldEquiv G hab]
-  rw [edgeFieldEquiv_apply,edgeFieldEquiv_apply,
-    spliceField_aut_preimage G (isAut_symm G hab),Equiv.symm_apply_apply]
-  rfl
-
- 
-
-theorem SpliceTrace.aut_map (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (x : V) (η : ℕ → Set E) (y : ℕ → V) (k j : ℕ)
-    (ω ωf : Set E) (c : List (ℕ × E)) (ht : SpliceTrace G x η y k j ω c ωf) :
-    SpliceTrace G (a x) (fun i => edgeFieldEquiv b (η i)) (fun i => a (y i)) k j
-      (edgeFieldEquiv b ω) (spliceCodeEquiv b c) (edgeFieldEquiv b ωf) := by
-  induction ht with
-  | stop j ω hs =>
-    apply SpliceTrace.stop
-    simpa only [cluster_edgeFieldEquiv G hab,Set.mem_preimage,Equiv.symm_apply_apply] using hs
-  | step j i ω e v w rest ωf hn hji hik hmax hy he hev hv hw ht ih =>
-    change SpliceTrace G (a x) _ _ k j _ ((i,b e)::spliceCodeEquiv b rest) _
-    refine SpliceTrace.step j i _ (b e) (a v) (a w) _ _ ?_ hji hik ?_ ?_ ?_ ?_ ?_ ?_ ?_
-    · simpa only [cluster_edgeFieldEquiv G hab,Set.mem_preimage,Equiv.symm_apply_apply] using hn
-    · intro l hil hlk
-      rw [cluster_edgeFieldEquiv G hab,cluster_edgeFieldEquiv G hab,
-        Set.disjoint_preimage_iff a.symm.surjective]
-      exact hmax l hil hlk
-    · simpa only [cluster_edgeFieldEquiv G hab,Set.mem_preimage,Equiv.symm_apply_apply] using hy
-    · simpa only [edgeFieldEquiv_apply,Set.mem_preimage,Equiv.symm_apply_apply] using he
-    · rw [hab e,hev,Sym2.map_mk]
-    · simpa only [cluster_edgeFieldEquiv G hab,Set.mem_preimage,Equiv.symm_apply_apply] using hv
-    · simpa only [slitCluster_edgeFieldEquiv G hab,Set.mem_preimage,Equiv.symm_apply_apply] using hw
-    · rw [spliceStep_edgeFieldEquiv G hab]
-      exact ih
-
-end SpliceCovarianceTrace
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-section SpliceCovarianceReplay
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem spliceStepMask_edgeFieldEquiv (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (ω η : Set E) (x y : V) (e : E) :
-    spliceMask G (cluster G (edgeFieldEquiv b ω) (a x))
-      (cluster G (edgeFieldEquiv b η ∩ internalEdges G (cluster G (edgeFieldEquiv b ω) (a x))ᶜ) (a y))
-      (b e) =
-    edgeFieldEquiv b (spliceMask G (cluster G ω x)
-      (cluster G (η ∩ internalEdges G (cluster G ω x)ᶜ) y) e) := by
-  rw [slitCluster_edgeFieldEquiv G hab,cluster_edgeFieldEquiv G hab,
-    spliceMask_aut_preimage G (isAut_symm G hab),Equiv.symm_apply_apply]
-  rfl
-
-theorem exchangeState_aut_map (b : E ≃ E) (mask : Set E) (i : ℕ) (s : SpliceState E) :
-    exchangeState (edgeFieldEquiv b mask) i (spliceStateEquiv b s) =
-      spliceStateEquiv b (exchangeState mask i s) := by
-  apply Prod.ext
-  · rfl
-  · funext j
-    by_cases hji : j = i
-    · subst j
-      rw [exchangeState_at]
-      change _ = edgeFieldEquiv b ((exchangeState mask i s).2 i)
-      rw [exchangeState_at]
-      rfl
-    · rw [exchangeState_away _ _ _ _ hji]
-      change edgeFieldEquiv b (s.2 j) = edgeFieldEquiv b ((exchangeState mask i s).2 j)
-      rw [exchangeState_away _ _ _ _ hji]
-
-theorem replaySplice_aut_map (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (x : V) (η : ℕ → Set E) (y : ℕ → V)
-    (c : List (ℕ × E)) (ω : Set E) :
-    replaySplice G (a x) (fun i => edgeFieldEquiv b (η i)) (fun i => a (y i))
-      (spliceCodeEquiv b c) (edgeFieldEquiv b ω) =
-    edgeFieldEquiv b (replaySplice G x η y c ω) := by
-  induction c generalizing ω with
-  | nil => rfl
-  | cons q c ih =>
-    rcases q with ⟨i,e⟩
-    change replaySplice G (a x) _ _ (spliceCodeEquiv b c) _ = _
-    rw [spliceStep_edgeFieldEquiv G hab,ih]
-    rfl
-
-theorem replayMasks_aut_map (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (x : V) (η : ℕ → Set E) (y : ℕ → V)
-    (c : List (ℕ × E)) (ω : Set E) :
-    replayMasks G (a x) (fun i => edgeFieldEquiv b (η i)) (fun i => a (y i))
-      (spliceCodeEquiv b c) (edgeFieldEquiv b ω) =
-    (replayMasks G x η y c ω).map (Prod.map id (edgeFieldEquiv b)) := by
-  induction c generalizing ω with
-  | nil => rfl
-  | cons q c ih =>
-    rcases q with ⟨i,e⟩
-    simp only [spliceCodeEquiv_apply,List.map_cons,Prod.map_apply,id_eq,replayMasks]
-    rw [spliceStepMask_edgeFieldEquiv G hab,spliceStep_edgeFieldEquiv G hab]
-    exact congrArg (List.cons _) (ih _)
-
-theorem runMasks_aut_map (b : E ≃ E) (c : List (ℕ × Set E)) (s : SpliceState E) :
-    runMasks (c.map (Prod.map id (edgeFieldEquiv b))) (spliceStateEquiv b s) =
-      spliceStateEquiv b (runMasks c s) := by
-  induction c generalizing s with
-  | nil => rfl
-  | cons q c ih =>
-    rcases q with ⟨i,mask⟩
-    simp only [List.map_cons,Prod.map_apply,id_eq,runMasks]
-    rw [exchangeState_aut_map,ih]
-
-theorem seamContactCount_aut_map (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (η : ℕ → Set E) (y : ℕ → V) (ωf : Set E) (x : V) (i : ℕ) (e : E) :
-    seamContactCount G (fun j => edgeFieldEquiv b (η j)) (fun j => a (y j))
-      (edgeFieldEquiv b ωf) (a x) i (b e) = seamContactCount G η y ωf x i e := by
-  simpa only [edgeFieldEquiv_apply,Equiv.symm_apply_apply] using
-    seamContactCount_aut_preimage G (isAut_symm G hab) η (fun j => a (y j)) ωf (a x) i (b e)
-
-theorem traceContactCount_aut_map (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (η : ℕ → Set E) (y : ℕ → V) (ωf : Set E) (x : V)
-    (c : List (ℕ × E)) (i : ℕ) :
-    traceContactCount G (fun j => edgeFieldEquiv b (η j)) (fun j => a (y j))
-      (edgeFieldEquiv b ωf) (a x) (spliceCodeEquiv b c) i =
-      traceContactCount G η y ωf x c i := by
-  induction c with
-  | nil => rfl
-  | cons q c ih =>
-    rcases q with ⟨j,e⟩
-    simp only [spliceCodeEquiv_apply,List.map_cons,Prod.map_apply,id_eq,traceContactCount]
-    split_ifs
-    · exact seamContactCount_aut_map G hab η y ωf x j e
-    · exact ih
-
-end SpliceCovarianceReplay
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Kernel.Amplification
+
+namespace OAI
+
+/-! Endpoint Moments for bond percolation. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section TrajectoryHitTransport
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
+variable {ι : Type*} [Fintype ι]
+
+ 
+
+theorem visitEnvelope_column_bound (hconn : Connected G)
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
+    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
+    (p : unitInterval) (k : ℕ) (a : V) :
+    Summable (fun z => q.visitEnvelope o p k z a) ∧
+      (∑' z, q.visitEnvelope o p k z a) ≤ c*((k*M:ℕ)*q.successProbability p o ^ k) := by
+  apply transported_column_of_row_bound G hconn hloc root weight Γ hΓ htr hcov hw hw1 c hc hmin
+    (q.visitEnvelope o p k) (q.visitEnvelope_nonneg o p k) _
+    (fun z => (q.visitEnvelope_row_bound o p k z).1) _
+    (fun z => (q.visitEnvelope_row_bound o p k z).2) a
+  intro b hb x y
+  obtain ⟨d,hd⟩ := hΓ hb
+  exact q.visitEnvelope_covariant o p k b d hd x y
+
+ 
+
+theorem finiteTrajectoryHit_sum_le (hconn : Connected G)
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
+    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
+    (Q : Finset E) (B : Finset V) (p : unitInterval) (k : ℕ) (A : Finset V)
+    (hQ : ∀ x ∈ B, incidentSupport G hloc (finite_reachBall hloc x (max q.radius M)).toFinset ⊆ Q) :
+    (∑ z : ↥B, finiteTrajectoryHit G Subtype.val Q p k
+      (fun ω (x y : ↥B) => q.mass ω x.val y.val) z A) ≤
+      (A.card:ℝ) * (c*((k*M:ℕ)*q.successProbability p o ^ k)) := by
+  have hcol := fun a => q.visitEnvelope_column_bound hconn root weight Γ hΓ htr hcov hw hw1 c hc hmin o p k a
+  calc
+    _ ≤ ∑ z : ↥B, ∑ a ∈ A, q.visitEnvelope o p k z.val a := by
+      apply Finset.sum_le_sum
+      intro z _
+      exact (finiteTrajectoryHit_le_sum G Subtype.val Q p k _ (fun ω x y => q.nonneg ω x.val y.val) z A).trans
+        (Finset.sum_le_sum fun a _ => q.finiteTrajectoryMass_le_envelope o Q B p k z a hQ)
+    _ = ∑ a ∈ A, ∑ z : ↥B, q.visitEnvelope o p k z.val a := Finset.sum_comm
+    _ ≤ ∑ _a ∈ A, c*((k*M:ℕ)*q.successProbability p o ^ k) := by
+      apply Finset.sum_le_sum
+      intro a _
+      rw [Finset.sum_coe_sort B (fun z => q.visitEnvelope o p k z a)]
+      exact ((hcol a).1.sum_le_tsum B (fun z _ => q.visitEnvelope_nonneg o p k z a)).trans (hcol a).2
+    _ = _ := by simp only [Finset.sum_const,nsmul_eq_mul]
+
+end LocalEndpointRule
+end TrajectoryHitTransport
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section AllEndpointHit
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem finiteChoiceHit_allRuns {W : Type*} [Fintype W]
+    (G : BondGraph V E) (e : W → V) (Q : Finset E) (p : ℝ) (k : ℕ)
+    (q : Set E → W → W → ℝ) (A : Finset V) :
+    finiteChoiceHit G k Q p
+      (fun (r : W × (Fin k → W)) _ => e ∘ oldRunVertices k r.1 r.2)
+      (fun r η => oldRunWeight (fun i => q (η i)) k r.1 r.2) (A : Set V) =
+      ∑ z : W, finiteTrajectoryHit G e Q p k q z A := by
+  unfold finiteChoiceHit finiteTrajectoryHit
+  rw [Fintype.sum_prod_type]
+  simp only [finiteFamilyIntegral_sum]
+
+noncomputable def allAmplificationVertices (B : Finset V) (k : ℕ)
+    (r : ↥B × (Fin k → ↥B)) (_η : ℕ → Set E) : ℕ → V :=
+  amplificationVertices B k r.1 r.2
+
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
+
+noncomputable def allAmplificationWeight (q : LocalEndpointRule G hloc O M L)
+    (B : Finset V) (k : ℕ) (r : ↥B × (Fin k → ↥B)) (η : ℕ → Set E) : ℝ :=
+  oldRunWeight (fun i (a b : ↥B) => q.mass (η i) a.val b.val) k r.1 r.2
+
+noncomputable def allAmplificationInputs (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ)
+    (r : ↥B × (Fin k → ↥B)) : Finset (SpliceState E) :=
+  q.amplificationInputs Q B x k n r.1 r.2
+
+theorem allAmplificationWeight_nonneg (q : LocalEndpointRule G hloc O M L)
+    (B : Finset V) (k : ℕ) (r : ↥B × (Fin k → ↥B)) (η : ℕ → Set E) :
+    0 ≤ q.allAmplificationWeight B k r η :=
+  oldRunWeight_nonneg (fun i (a b : ↥B) => q.mass (η i) a.val b.val) (fun i a b => q.nonneg (η i) a.val b.val) k r.1 r.2
+
+variable {ι : Type*} [Fintype ι]
+
+theorem allAmplificationHit_bound (hconn : Connected G)
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
+    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
+    (Q : Finset E) (B : Finset V) (p : unitInterval) (k : ℕ) (A : Set V) (hf : A.Finite)
+    (hQ : ∀ x ∈ B, incidentSupport G hloc (finite_reachBall hloc x (max q.radius M)).toFinset ⊆ Q) :
+    finiteChoiceHit G k Q p (allAmplificationVertices B k) (q.allAmplificationWeight B k) A ≤
+      (A.ncard:ℝ) * (c*((k*M:ℕ)*q.successProbability p o ^ k)) := by
+  have he : finiteChoiceHit G k Q p (allAmplificationVertices B k) (q.allAmplificationWeight B k) A =
+      ∑ z : ↥B, finiteTrajectoryHit G Subtype.val Q p k
+        (fun ω (a b : ↥B) => q.mass ω a.val b.val) z hf.toFinset := by
+    rw [← finiteChoiceHit_allRuns]
+    rw [hf.coe_toFinset]
+    rfl
+  rw [he,Set.ncard_eq_toFinset_card A hf]
+  exact q.finiteTrajectoryHit_sum_le hconn root weight Γ hΓ htr hcov hw hw1 c hc hmin o Q B p k hf.toFinset hQ
+
+end LocalEndpointRule
+end AllEndpointHit
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section AllEndpointSeparated
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def finiteClusterWindowMass (G : BondGraph V E) (Q : Finset E) (p : ℝ)
+    (x : V) (lo hi : ℕ) : ℝ :=
+  finiteFieldIntegral Q p (fun ω => if (cluster G ω x).Finite ∧
+    lo ≤ (cluster G ω x).ncard ∧ (cluster G ω x).ncard < hi then 1 else 0)
+
+theorem finiteChoiceHit_nonneg {P : Type*} [Fintype P]
+    (G : BondGraph V E) (k : ℕ) (Q : Finset E) (p : unitInterval)
+    (y : P → (ℕ → Set E) → ℕ → V) (m : P → (ℕ → Set E) → ℝ)
+    (hm : ∀ r η, 0 ≤ m r η) (A : Set V) : 0 ≤ finiteChoiceHit G k Q p y m A := by
+  apply Finset.sum_nonneg
+  intro r _
+  apply finiteFamilyIntegral_nonneg
+  intro η
+  split_ifs
+  · exact hm r η
+  · exact le_refl _
+
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
+variable {ι : Type*} [Fintype ι]
+
+ 
+
+theorem amplificationSeparatedSecondMoment_le [Nonempty E] (hconn : Connected G)
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
+    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
+    (Q : Finset E) (B : Finset V) (p : unitInterval) (x : V) (k n : ℕ)
+    (hQ : ∀ a ∈ B, incidentSupport G hloc (finite_reachBall hloc a (max q.radius M)).toFinset ⊆ Q) :
+    finiteChoiceSeparatedPairMass G x k Q p
+      (q.allAmplificationInputs Q B x k n) (q.allAmplificationInputs Q B x k n)
+      (allAmplificationVertices B k) (allAmplificationVertices B k)
+      (q.allAmplificationWeight B k) (q.allAmplificationWeight B k) ≤
+      (((2*n:ℕ):ℝ) * (c*((k*M:ℕ)*q.successProbability p o ^ k)))^2 *
+        finiteClusterWindowMass G Q p x (n-k*M) (2*n) := by
+  let A := q.allAmplificationInputs Q B x k n
+  let y := allAmplificationVertices (E := E) B k
+  let m := q.allAmplificationWeight B k
+  have hA (r) : A r ⊆ finiteSpliceDomain Q (Finset.Icc 1 k) :=
+    q.amplificationInputs_subset Q B x k n r.1 r.2
+  have hbin r s (hs : s ∈ A r) : (cluster G s.1 x).Finite ∧
+      n ≤ (cluster G s.1 x).ncard ∧ (cluster G s.1 x).ncard < 2*n := by
+    have hd := (mem_finiteSpliceDomain_iff Q (Finset.Icc 1 k) s).mp (hA r hs)
+    have hb := (q.mem_amplificationInputs Q B x k n r.1 r.2 s).mp hs
+    exact ⟨cluster_finite_of_finite_edges G (Q.finite_toSet.subset hd.1) x,hb.2.2.2.1,hb.2.2.2.2.1⟩
+  have hcap r s (hs : s ∈ A r) (i : ℕ) (hi : 1 ≤ i) (hik : i ≤ k) :
+      (cluster G (s.2 i) (y r s.2 i)).Finite ∧ (cluster G (s.2 i) (y r s.2 i)).ncard ≤ M := by
+    have hd := q.amplificationInputs_chain Q B x k n r.1 r.2 s hs i hi hik
+    exact ⟨hd.2.1,hd.2.2.1⟩
+  have hm := q.allAmplificationWeight_nonneg B k
+  have hb := finiteChoiceSeparatedPairMass_le G hloc x k n M Q p A A y y m m hm hm hA hA hbin
+    (fun r s hs i hi hik => (hcap r s hs i hi hik).1) hcap
+  let d : ℝ := c*((k*M:ℕ)*q.successProbability p o ^ k)
+  have hd : 0 ≤ d := mul_nonneg hc (mul_nonneg (Nat.cast_nonneg _) (pow_nonneg (q.successProbability_nonneg p o) _))
+  let t : ℝ := ((2*n:ℕ):ℝ)*d
+  apply hb.trans
+  unfold finiteClusterWindowMass finiteFieldIntegral
+  rw [Finset.mul_sum]
+  apply Finset.sum_le_sum
+  intro ω _
+  have he : t^2 * (finiteFieldWeight Q p ω * (if (cluster G (finiteFieldSet Q ω) x).Finite ∧
+      n-k*M ≤ (cluster G (finiteFieldSet Q ω) x).ncard ∧
+      (cluster G (finiteFieldSet Q ω) x).ncard < 2*n then 1 else 0)) =
+      finiteFieldWeight Q p ω * (if (cluster G (finiteFieldSet Q ω) x).Finite ∧
+      n-k*M ≤ (cluster G (finiteFieldSet Q ω) x).ncard ∧
+      (cluster G (finiteFieldSet Q ω) x).ncard < 2*n then t^2 else 0) := by
+    split_ifs <;> ring
+  change _ ≤ t^2 * _
+  rw [he]
+  apply mul_le_mul_of_nonneg_left _ (finiteFieldWeight_nonneg Q p ω)
+  by_cases h : (cluster G (finiteFieldSet Q ω) x).Finite ∧
+      n-k*M ≤ (cluster G (finiteFieldSet Q ω) x).ncard ∧
+      (cluster G (finiteFieldSet Q ω) x).ncard < 2*n
+  · rw [ite_eq_left h,ite_eq_left h]
+    have hhit := q.allAmplificationHit_bound hconn root weight Γ hΓ htr hcov hw hw1 c hc hmin
+      o Q B p k (cluster G (finiteFieldSet Q ω) x) h.1 hQ
+    have hc' : ((cluster G (finiteFieldSet Q ω) x).ncard:ℝ) ≤ ((2*n:ℕ):ℝ) := by exact_mod_cast h.2.2.le
+    have hhit' : finiteChoiceHit G k Q p y m (cluster G (finiteFieldSet Q ω) x) ≤ t :=
+      hhit.trans (mul_le_mul_of_nonneg_right hc' hd)
+    simpa only [pow_two] using mul_self_le_mul_self
+      (finiteChoiceHit_nonneg G k Q p y m hm _) hhit'
+  · rw [ite_eq_right h,ite_eq_right h]
+
+end LocalEndpointRule
+end AllEndpointSeparated
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section AllEndpointMarginal
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem finiteRootLoad_sum {P : Type*} (Q : Finset E) (J : Finset ℕ) (p : ℝ)
+    (S : Finset P) (f : P → SpliceState E → ℝ) (ω : FiniteField Q) :
+    finiteRootLoad Q J p (fun s => ∑ z ∈ S, f z s) ω =
+      ∑ z ∈ S, finiteRootLoad Q J p (f z) ω := by
+  simp only [finiteRootLoad,Finset.mul_sum]
+  exact Finset.sum_comm
+
+ 
+
+theorem finiteChoiceForwardLoad_prod {P R : Type*} [Fintype P] [Fintype R]
+    (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
+    (A : P × R → Finset (SpliceState E)) (y : P × R → SpliceState E → ℕ → V)
+    (m : SpliceState E → P × R → ℝ) (t : SpliceState E) :
+    finiteChoiceForwardLoad G x k Q A y m t = ∑ z : P,
+      finiteChoiceForwardLoad G x k Q (fun r => A (z,r)) (fun r => y (z,r))
+        (fun s r => m s (z,r)) t := by
+  unfold finiteChoiceForwardLoad finiteInverseLoad
+  rw [Fintype.sum_sigma,Fintype.sum_prod_type]
+  simp only [Fintype.sum_sigma,choiceTraceOutput,choiceTraceMass,choiceTraceInput]
+  rfl
+
+ 
+theorem endpointRootMass_choice_prod {P R : Type*} [Fintype P] [Fintype R]
+    (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E) (p : ℝ)
+    (A : P × R → Finset (SpliceState E)) (y : P × R → SpliceState E → ℕ → V)
+    (m : SpliceState E → P × R → ℝ) (e : P → V) (he : Function.Injective e)
+    (z : P) (ω : Set E) :
+    endpointRootMass Q (Finset.Icc 1 k) p (choiceTraceOutput G x k Q A y)
+      (fun a => e a.1.1) (choiceTraceMass G x k Q A y m) ω (e z) =
+      ∑ a : FiniteChoiceTraceIndex G x k Q (fun r => A (z,r)) (fun r => y (z,r)),
+        if (choiceTraceOutput G x k Q (fun r => A (z,r)) (fun r => y (z,r)) a).1 = ω then
+          auxiliaryTupleWeight Q (Finset.Icc 1 k) p
+            (choiceTraceOutput G x k Q (fun r => A (z,r)) (fun r => y (z,r)) a) *
+          choiceTraceMass G x k Q (fun r => A (z,r)) (fun r => y (z,r)) (fun s r => m s (z,r)) a else 0 := by
+  unfold endpointRootMass
+  rw [Fintype.sum_sigma,Fintype.sum_prod_type]
+  rw [Finset.sum_eq_single z]
+  · rw [Fintype.sum_sigma]
+    simp only [choiceTraceOutput,choiceTraceMass,choiceTraceInput,and_true]
+    rfl
+  · intro z' _ hz'
+    have hh : e z' ≠ e z := fun h => hz' (he h)
+    simp only [hh,and_false,ite_false,Finset.sum_const_zero]
+  · intro hz
+    exact (hz (Finset.mem_univ z)).elim
+
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
+
+abbrev AmplificationIndex (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) :=
+  FiniteChoiceTraceIndex G x k Q (q.allAmplificationInputs Q B x k n)
+    (fun r s => allAmplificationVertices B k r s.2)
+
+noncomputable def amplificationOutput (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) : q.AmplificationIndex Q B x k n → SpliceState E :=
+  choiceTraceOutput G x k Q (q.allAmplificationInputs Q B x k n)
+    (fun r s => allAmplificationVertices B k r s.2)
+
+noncomputable def amplificationInput (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) : q.AmplificationIndex Q B x k n → SpliceState E :=
+  choiceTraceInput G x k Q (q.allAmplificationInputs Q B x k n)
+    (fun r s => allAmplificationVertices B k r s.2)
+
+noncomputable def amplificationCode (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) : q.AmplificationIndex Q B x k n → List (ℕ × E) :=
+  choiceTraceCode G x k Q (q.allAmplificationInputs Q B x k n)
+    (fun r s => allAmplificationVertices B k r s.2)
+
+noncomputable def amplificationMass (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) : q.AmplificationIndex Q B x k n → ℝ :=
+  choiceTraceMass G x k Q (q.allAmplificationInputs Q B x k n)
+    (fun r s => allAmplificationVertices B k r s.2) (fun s r => q.allAmplificationWeight B k r s.2)
+
+noncomputable def amplificationRow (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (p : unitInterval) (ω : Set E) : ℝ :=
+  ∑ z : ↥B, q.amplificationMarginal Q B x k n z p ω
+
+theorem amplificationRow_eq_load (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (p : unitInterval) (ω : Set E) :
+    q.amplificationRow Q B x k n p ω = finiteRootLoad Q (Finset.Icc 1 k) p
+      (finiteInverseLoad (q.amplificationOutput Q B x k n) (q.amplificationMass Q B x k n))
+      (restrictFiniteField Q ω) := by
+  change _ = finiteRootLoad Q (Finset.Icc 1 k) p
+    (finiteChoiceForwardLoad G x k Q (q.allAmplificationInputs Q B x k n)
+      (fun r s => allAmplificationVertices B k r s.2) (fun s r => q.allAmplificationWeight B k r s.2)) _
+  have hf : (finiteChoiceForwardLoad G x k Q (q.allAmplificationInputs Q B x k n)
+      (fun r s => allAmplificationVertices B k r s.2) (fun s r => q.allAmplificationWeight B k r s.2)) =
+      (fun t => ∑ z : ↥B, finiteChoiceForwardLoad G x k Q (q.amplificationInputs Q B x k n z)
+        (fun r _ => amplificationVertices B k z r) (q.amplificationWeight B k z) t) := by
+    funext t
+    exact finiteChoiceForwardLoad_prod G x k Q (q.allAmplificationInputs Q B x k n)
+      (fun r s => allAmplificationVertices B k r s.2) (fun s r => q.allAmplificationWeight B k r s.2) t
+  rw [hf,finiteRootLoad_sum]
+  rfl
+
+theorem amplificationEndpointMass_eq (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (p : unitInterval) (ω : FiniteField Q) (z : ↥B) :
+    endpointRootMass Q (Finset.Icc 1 k) p (q.amplificationOutput Q B x k n)
+      (fun a => a.1.1.val) (q.amplificationMass Q B x k n) (finiteFieldSet Q ω) z.val =
+      q.amplificationMarginal Q B x k n z p (finiteFieldSet Q ω) := by
+  change endpointRootMass Q (Finset.Icc 1 k) p (choiceTraceOutput G x k Q _ _)
+    _ (choiceTraceMass G x k Q _ _ _) _ _ = _
+  rw [endpointRootMass_choice_prod G x k Q p _ _ _ Subtype.val Subtype.val_injective z]
+  unfold amplificationMarginal finiteChoiceForwardLoad
+  rw [restrictFiniteField_fieldSet,finiteRootLoad_eq_sum]
+  · rfl
+  · intro a
+    exact traceIndexOutput_mem G x k Q _ _ (q.amplificationInputs_subset Q B x k n z a.1) a.2
+
+end LocalEndpointRule
+end AllEndpointMarginal
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section AmplificationIndexFacts
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
+
+theorem amplificationOutput_mem (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (a : q.AmplificationIndex Q B x k n) :
+    q.amplificationOutput Q B x k n a ∈ finiteSpliceDomain Q (Finset.Icc 1 k) :=
+  traceIndexOutput_mem G x k Q _ _ (q.amplificationInputs_subset Q B x k n a.1.1 a.1.2) a.2
+
+theorem amplificationMass_nonneg (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (a : q.AmplificationIndex Q B x k n) :
+    0 ≤ q.amplificationMass Q B x k n a :=
+  mul_nonneg (q.allAmplificationWeight_nonneg B k a.1 (q.amplificationInput Q B x k n a).2)
+    (Finset.prod_nonneg fun _ _ => inv_nonneg.mpr (Nat.cast_nonneg _))
+
+theorem amplificationIndex_trace (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (a : q.AmplificationIndex Q B x k n) :
+    SpliceTrace G x (q.amplificationInput Q B x k n a).2
+      (amplificationVertices B k a.1.1 a.1.2) k 0 (q.amplificationInput Q B x k n a).1
+      (q.amplificationCode Q B x k n a) (q.amplificationOutput Q B x k n a).1 :=
+  (traceIndex_valid G x k Q _ _ a.2).1
+
+theorem amplificationIndex_execution (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (a : q.AmplificationIndex Q B x k n) :
+    SpliceExecution G x (q.amplificationInput Q B x k n a)
+      (q.amplificationCode Q B x k n a) (q.amplificationOutput Q B x k n a) :=
+  (traceIndex_valid G x k Q _ _ a.2).2
+
+theorem amplificationIndex_support (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (a : q.AmplificationIndex Q B x k n) :
+    x ∈ O ∧ a.1.1.val ∈ O ∧ Conn G (q.amplificationOutput Q B x k n a).1 x a.1.1.val ∧
+      (cluster G (q.amplificationOutput Q B x k n a).1 x).Finite ∧
+      (cluster G (q.amplificationOutput Q B x k n a).1 x).ncard ≤ 2*n+k*M :=
+  q.amplificationTrace_support Q B x k n a.1.1 a.1.2 a.2.1.val a.2.1.property _ _
+    (q.amplificationIndex_trace Q B x k n a)
+
+theorem amplificationIndex_caps (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (a : q.AmplificationIndex Q B x k n)
+    (i : ℕ) (hi : 1 ≤ i) (hik : i ≤ k) :
+    (cluster G ((q.amplificationInput Q B x k n a).2 i) (amplificationVertices B k a.1.1 a.1.2 i)).Finite ∧
+    (cluster G ((q.amplificationInput Q B x k n a).2 i) (amplificationVertices B k a.1.1 a.1.2 i)).ncard ≤ M := by
+  have hc := q.amplificationInputs_chain Q B x k n a.1.1 a.1.2 a.2.1.val a.2.1.property i hi hik
+  exact ⟨hc.2.1,hc.2.2.1⟩
+
+theorem amplificationEndpointMass_outside (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (p : unitInterval) (ω : Set E)
+    (z : V) (hz : z ∉ B) :
+    endpointRootMass Q (Finset.Icc 1 k) p (q.amplificationOutput Q B x k n)
+      (fun a => a.1.1.val) (q.amplificationMass Q B x k n) ω z = 0 := by
+  apply Finset.sum_eq_zero
+  intro a _
+  split_ifs with h
+  · exact (hz (h.2 ▸ a.1.1.property)).elim
+  · rfl
+
+ 
+
+theorem amplificationEndpointMass_bound (q : LocalEndpointRule G hloc O M L) (hL : 1 ≤ L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (p : unitInterval) (ω : Set E)
+    (hω : ω ⊆ (Q : Set E)) (z : V) :
+    endpointRootMass Q (Finset.Icc 1 k) p (q.amplificationOutput Q B x k n)
+      (fun a => a.1.1.val) (q.amplificationMass Q B x k n) ω z ≤ inverseCodeMass M L ^ k := by
+  by_cases hz : z ∈ B
+  · let z' : ↥B := ⟨z,hz⟩
+    have hfield : finiteFieldSet Q (restrictFiniteField Q ω) = ω := by
+      rw [finiteFieldSet_restrict,Set.inter_eq_left.mpr hω]
+    rw [← hfield,q.amplificationEndpointMass_eq Q B x k n p _ z']
+    exact (q.amplificationMarginal_bounds hL Q B x k n z' p _).2
+  · rw [q.amplificationEndpointMass_outside Q B x k n p ω z hz]
+    apply pow_nonneg
+    exact add_nonneg zero_le_one (Finset.sum_nonneg fun l _ => gapWeight_nonneg L (by linarith) l)
+
+end LocalEndpointRule
+end AmplificationIndexFacts
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section NoSwitchDensity
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+@[simp] theorem SpliceExecution.nil_iff (G : BondGraph V E) (x : V) (s t : SpliceState E) :
+    SpliceExecution G x s [] t ↔ s = t := by
+  constructor
+  · intro h
+    cases h
+    rfl
+  · rintro rfl
+    exact SpliceExecution.nil s
+
+theorem traceIndexMass_nil (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
+    (A : Finset (SpliceState E)) (y : SpliceState E → ℕ → V)
+    (a : FiniteTraceIndex G x k Q A y) (hc : traceIndexCode G x k Q A y a = []) :
+    traceIndexMass G x k Q A y a = 1 := by
+  change a.2.val = [] at hc
+  simp [traceIndexMass,hc,traceContactCount]
+
+ 
+
+theorem finiteChoiceNilLoad_le {P : Type*} [Fintype P]
+    (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
+    (A : P → Finset (SpliceState E)) (y : P → SpliceState E → ℕ → V)
+    (m : SpliceState E → P → ℝ) (hm : ∀ s r, 0 ≤ m s r) (t : SpliceState E) :
+    finiteInverseLoad (choiceTraceOutput G x k Q A y)
+      (fun a => if choiceTraceCode G x k Q A y a = [] then choiceTraceMass G x k Q A y m a else 0) t ≤
+      ∑ r, if t ∈ A r then m t r else 0 := by
+  let I := FiniteChoiceTraceIndex G x k Q A y
+  let S : I → Prop := fun a => choiceTraceOutput G x k Q A y a = t ∧ choiceTraceCode G x k Q A y a = []
+  let J := {a : I // S a}
+  have hin (a : J) : choiceTraceInput G x k Q A y a.val = t := by
+    have he := (traceIndex_valid G x k Q (A a.val.1) (y a.val.1) a.val.2).2
+    change SpliceExecution G x (choiceTraceInput G x k Q A y a.val)
+      (choiceTraceCode G x k Q A y a.val) (choiceTraceOutput G x k Q A y a.val) at he
+    rw [a.property.1,a.property.2,SpliceExecution.nil_iff] at he
+    exact he
+  have hi : Function.Injective (fun a : J => a.val.1) := by
+    intro a b hab
+    apply Subtype.ext
+    apply choiceTrace_injective G x k Q A y
+    exact Prod.ext ((hin a).trans (hin b).symm)
+      (Prod.ext hab (a.property.2.trans b.property.2.symm))
+  have hm' (a : J) : choiceTraceMass G x k Q A y m a.val = m t a.val.1 := by
+    unfold choiceTraceMass
+    rw [hin a,traceIndexMass_nil G x k Q (A a.val.1) (y a.val.1) a.val.2 a.property.2,mul_one]
+  have hA (a : J) : t ∈ A a.val.1 := (hin a) ▸ a.val.2.1.property
+  have he : finiteInverseLoad (choiceTraceOutput G x k Q A y)
+      (fun a => if choiceTraceCode G x k Q A y a = [] then choiceTraceMass G x k Q A y m a else 0) t =
+      ∑ a : J, m t a.val.1 := by
+    unfold finiteInverseLoad
+    simp only [← ite_and]
+    change (∑ a : I, if S a then choiceTraceMass G x k Q A y m a else 0) = _
+    calc
+      _ = ∑ a : J, choiceTraceMass G x k Q A y m a.val := by
+        rw [← Finset.sum_filter]
+        exact Finset.sum_subtype _ (by intro a; simp) _
+      _ = _ := Finset.sum_congr rfl (fun a _ => hm' a)
+  rw [he]
+  have hb := sum_comp_le_sum_of_injective (fun a : J => a.val.1) hi
+    (fun r => if t ∈ A r then m t r else 0) (by intro r; split_ifs; exact hm t r; exact le_refl _)
+  simpa only [ite_eq_left (hA _)] using hb
+
+end NoSwitchDensity
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section AllEndpointNil
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem oldRunArrival_integral {W : Type*} [Fintype W]
+    (Q : Finset E) (p : ℝ) (k : ℕ) (q : Set E → W → W → ℝ) (z : W) (A : Finset W) :
+    finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η => oldRunArrival (fun i => q (η i)) k z A) =
+      oldRunArrival (fun _ => finiteOldKernel Q p q) k z A := by
+  unfold oldRunArrival
+  rw [finiteFamilyIntegral_sum]
+  apply Finset.sum_congr rfl
+  intro r _
+  by_cases h : oldRunVertices k z r 0 ∈ A
+  · simp only [ite_eq_left h]
+    exact oldRunWeight_integral Q p k q z r
+  · simp [h,finiteFamilyIntegral]
+
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
+
+noncomputable def amplificationNilRow (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (p : unitInterval) (ω : FiniteField Q) : ℝ :=
+  finiteRootLoad Q (Finset.Icc 1 k) p
+    (finiteInverseLoad (q.amplificationOutput Q B x k n)
+      (fun a => if q.amplificationCode Q B x k n a = [] then q.amplificationMass Q B x k n a else 0)) ω
+
+ 
+
+theorem amplificationNilRow_arrival_le (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (p : unitInterval) (ω : FiniteField Q) :
+    q.amplificationNilRow Q B x k n p ω ≤
+      if n ≤ (cluster G (finiteFieldSet Q ω) x).ncard ∧ (cluster G (finiteFieldSet Q ω) x).ncard < 2*n then
+        ∑ z : ↥B, oldRunArrival (fun _ => finiteOldKernel Q p (fun η (a b : ↥B) => q.mass η a.val b.val)) k z
+          (Finset.univ.filter fun a : ↥B => a.val ∈ cluster G (finiteFieldSet Q ω) x) else 0 := by
+  let A : Finset ↥B := Finset.univ.filter fun a : ↥B => a.val ∈ cluster G (finiteFieldSet Q ω) x
+  let H := n ≤ (cluster G (finiteFieldSet Q ω) x).ncard ∧ (cluster G (finiteFieldSet Q ω) x).ncard < 2*n
+  have hb (η : ℕ → Set E) :
+      finiteInverseLoad (q.amplificationOutput Q B x k n)
+        (fun a => if q.amplificationCode Q B x k n a = [] then q.amplificationMass Q B x k n a else 0)
+        (finiteFieldSet Q ω,η) ≤
+      if H then ∑ z : ↥B, oldRunArrival (fun i (a b : ↥B) => q.mass (η i) a.val b.val) k z A else 0 := by
+    apply (finiteChoiceNilLoad_le G x k Q (q.allAmplificationInputs Q B x k n)
+      (fun r s => allAmplificationVertices B k r s.2)
+      (fun s r => q.allAmplificationWeight B k r s.2)
+      (fun s r => q.allAmplificationWeight_nonneg B k r s.2) (finiteFieldSet Q ω,η)).trans
+    rw [Fintype.sum_prod_type]
+    by_cases h : H
+    · rw [ite_eq_left h]
+      apply Finset.sum_le_sum
+      intro z _
+      apply Finset.sum_le_sum
+      intro r _
+      by_cases hs : (finiteFieldSet Q ω,η) ∈ q.allAmplificationInputs Q B x k n (z,r)
+      · rw [ite_eq_left hs]
+        have hi := (q.mem_amplificationInputs Q B x k n z r _).mp hs
+        have ha : oldRunVertices k z r 0 ∈ A := by
+          simpa only [A,Finset.mem_filter,Finset.mem_univ,true_and,amplificationVertices] using hi.2.2.2.2.2.1
+        rw [ite_eq_left ha]
+        exact le_rfl
+      · rw [ite_eq_right hs]
+        split_ifs
+        · exact oldRunWeight_nonneg _ (fun i a b => q.nonneg (η i) a.val b.val) k z r
+        · exact le_refl _
+    · rw [ite_eq_right h]
+      apply le_of_eq
+      apply Finset.sum_eq_zero
+      intro z _
+      apply Finset.sum_eq_zero
+      intro r _
+      rw [ite_eq_right (by
+        intro hs
+        have hi := (q.mem_amplificationInputs Q B x k n z r _).mp hs
+        exact h ⟨hi.2.2.2.1,hi.2.2.2.2.1⟩)]
+  change _ ≤ if H then ∑ z : ↥B,
+    oldRunArrival (fun _ => finiteOldKernel Q p (fun η (a b : ↥B) => q.mass η a.val b.val)) k z A else 0
+  calc
+    _ ≤ finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η => if H then
+        ∑ z : ↥B, oldRunArrival (fun i (a b : ↥B) => q.mass (η i) a.val b.val) k z A else 0) := by
+      apply Finset.sum_le_sum
+      intro η _
+      exact mul_le_mul_of_nonneg_left (hb _) (finiteFamilyWeight_nonneg Q (Finset.Icc 1 k) p η)
+    _ = _ := by
+      by_cases h : H
+      · simp only [ite_eq_left h,finiteFamilyIntegral_sum]
+        apply Finset.sum_congr rfl
+        intro z _
+        exact oldRunArrival_integral Q p k (fun η (a b : ↥B) => q.mass η a.val b.val) z A
+      · simp only [ite_eq_right h,finiteFamilyIntegral,mul_zero,Finset.sum_const_zero]
+
+end LocalEndpointRule
+end AllEndpointNil
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+section AllEndpointNilBound
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
+
+theorem amplificationNilRow_nonneg {O : Set V} (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (p : unitInterval) (ω : FiniteField Q) :
+    0 ≤ q.amplificationNilRow Q B x k n p ω := by
+  apply finiteRootLoad_nonneg
+  intro s _
+  apply finiteInverseLoad_nonneg
+  intro a
+  split_ifs
+  · exact q.amplificationMass_nonneg Q B x k n a
+  all_goals exact le_refl _
+
+ 
+
+theorem amplificationNilRow_bound {ι : Type*} [Fintype ι]
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (htr : RootTransport root weight Γ) (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b)
+    (hw : ∀ i, 0 ≤ weight i) (i₀ : ι) (hw₀ : 0 < weight i₀)
+    (q : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (p : unitInterval) (ω : FiniteField Q)
+    (hQ : ∀ a ∈ B, incidentSupport G hloc (finite_reachBall hloc a q.radius).toFinset ⊆ Q) :
+    q.amplificationNilRow Q B x k n p ω ≤ ((2*n:ℕ):ℝ) * q.successProbability p (root i₀)^k := by
+  apply (q.amplificationNilRow_arrival_le Q B x k n p ω).trans
+  split_ifs with h
+  · let A : Finset ↥B := Finset.univ.filter fun a : ↥B => a.val ∈ cluster G (finiteFieldSet Q ω) x
+    let K := fun (_ : ℕ) => finiteOldKernel Q p (fun η (a b : ↥B) => q.mass η a.val b.val)
+    have hK (i : ℕ) (a b : ↥B) : K i a b = q.kernel p a.val b.val :=
+      q.finiteOldKernel_eq_kernel Q p a.val b.val (hQ a.val a.property)
+    have hnK (i : ℕ) (a b : ↥B) : 0 ≤ K i a b := by rw [hK i a b]; exact q.kernel_nonneg p a.val b.val
+    have hc (i : ℕ) (b : ↥B) : (∑ a, K i a b) ≤ q.successProbability p (root i₀) := by
+      simp only [hK]
+      rw [Finset.sum_coe_sort B (fun a => q.kernel p a b.val)]
+      exact ((q.summable_kernel_column p b.val).sum_le_tsum B (fun a _ => q.kernel_nonneg p a b.val)).trans
+        (q.kernel_column_le_success root weight Γ htr hΓ hw i₀ hw₀ p b.val)
+    have hf : (cluster G (finiteFieldSet Q ω) x).Finite := cluster_finite_of_finite_edges G
+      ω.val.finite_toSet x
+    have hcard : A.card ≤ 2*n := by
+      have hh : A.card ≤ hf.toFinset.card := Finset.card_le_card_of_injOn Subtype.val
+        (by intro a ha; exact hf.mem_toFinset.mpr (Finset.mem_filter.mp ha).2)
+        (fun _ _ _ _ he => Subtype.val_injective he)
+      rw [← Set.ncard_eq_toFinset_card (cluster G (finiteFieldSet Q ω) x) hf] at hh
+      exact hh.trans h.2.le
+    calc
+      _ ≤ (A.card:ℝ) * q.successProbability p (root i₀)^k :=
+        sum_oldRunArrival_le K hnK _ (q.successProbability_nonneg p (root i₀)) hc k A
+      _ ≤ ((2*n:ℕ):ℝ) * q.successProbability p (root i₀)^k :=
+        mul_le_mul_of_nonneg_right (by exact_mod_cast hcard) (pow_nonneg (q.successProbability_nonneg p (root i₀)) _)
+  · exact mul_nonneg (Nat.cast_nonneg _) (pow_nonneg (q.successProbability_nonneg p (root i₀)) _)
+
+end LocalEndpointRule
+end AllEndpointNilBound
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+section AllEndpointNear
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
+
+noncomputable def amplificationAdded (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (a : q.AmplificationIndex Q B x k n) : Set V :=
+  cluster G (q.amplificationOutput Q B x k n a).1 x \ cluster G (q.amplificationInput Q B x k n a).1 x
+
+noncomputable def amplificationNearPairMass (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (p : unitInterval) : ℝ :=
+  ∑ a : q.AmplificationIndex Q B x k n, ∑ b : q.AmplificationIndex Q B x k n,
+    if (q.amplificationOutput Q B x k n a).1 = (q.amplificationOutput Q B x k n b).1 ∧
+      q.amplificationCode Q B x k n a ≠ [] ∧ q.amplificationCode Q B x k n b ≠ [] ∧
+      ¬GraphSeparated G (q.amplificationAdded Q B x k n a) (q.amplificationAdded Q B x k n b)
+    then jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p
+      (pairSpliceOutput (q.amplificationOutput Q B x k n a) (q.amplificationOutput Q B x k n b)) *
+        q.amplificationMass Q B x k n a * q.amplificationMass Q B x k n b else 0
+
+ 
+
+theorem amplificationNearPairMass_bound [Nonempty E]
+    (q : LocalEndpointRule G hloc O M L) (hL : 1 ≤ L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n d : ℕ) (p : unitInterval)
+    (hdeg : ∀ a, degree G a ≤ d) :
+    q.amplificationNearPairMass Q B x k n p ≤
+      (((k*M)+(k*M)^2+d*(k*M)^3:ℕ):ℝ) * inverseCodeMass M L ^ k *
+        ∑ a : q.AmplificationIndex Q B x k n,
+          spliceTupleWeight Q (Finset.Icc 1 k) p (q.amplificationOutput Q B x k n a) *
+            q.amplificationMass Q B x k n a := by
+  let out := q.amplificationOutput Q B x k n
+  let code := q.amplificationCode Q B x k n
+  let added := q.amplificationAdded Q B x k n
+  let m := q.amplificationMass Q B x k n
+  let e (a : q.AmplificationIndex Q B x k n) := (code a).headD (0,Classical.choice ‹Nonempty E›) |>.2
+  have hfin (a : q.AmplificationIndex Q B x k n) : (cluster G (out a).1 x).Finite :=
+    (q.amplificationIndex_support Q B x k n a).2.2.2.1
+  have hhead (a : q.AmplificationIndex Q B x k n) (hc : code a ≠ []) :
+      OpenMarkedPivotal G (out a).1 {a.1.1.val} x (e a) ∧
+      hangingSide G (out a).1 x (e a) = added a ∧
+      (hangingSide G (out a).1 x (e a)).ncard ≤ k*M := by
+    cases hh : code a with
+    | nil => exact (hc hh).elim
+    | cons w rest =>
+      have ht := q.amplificationIndex_trace Q B x k n a
+      change SpliceTrace G x (q.amplificationInput Q B x k n a).2
+        (amplificationVertices B k a.1.1 a.1.2) k 0 (q.amplificationInput Q B x k n a).1
+        (code a) (out a).1 at ht
+      rw [hh] at ht
+      have he : e a = w.2 := by simp only [e,hh,List.headD_cons]
+      rw [he]
+      refine ⟨?_,ht.head_hanging G x _ _ k 0 w.1 _ _ w.2 rest,
+        ht.head_hanging_cap G x _ _ k 0 w.1 M _ _ w.2 rest (q.amplificationIndex_caps Q B x k n a)⟩
+      simpa only [amplificationVertices,oldRunVertices_last] using
+        ht.seams_pivotal G x _ _ k 0 _ _ (w::rest) w (List.mem_cons_self)
+  let S (a : q.AmplificationIndex Q B x k n) : Finset V := if code a = [] then ∅ else
+    nearHangingEndpoints G hloc (out a).1 x a.1.1.val (e a) (k*M) (hfin a)
+  let R (a b : q.AmplificationIndex Q B x k n) := code a ≠ [] ∧ code b ≠ [] ∧
+    ¬GraphSeparated G (added a) (added b)
+  have hS (a b : q.AmplificationIndex Q B x k n) (he : (out a).1 = (out b).1) (h : R a b) :
+      b.1.1.val ∈ S a := by
+    obtain ⟨hpa,hea,_⟩ := hhead a h.1
+    obtain ⟨hpb,heb,hcb⟩ := hhead b h.2.1
+    have hn := h.2.2
+    rw [← hea,← heb,← he] at hn
+    rw [← he] at hpb hcb
+    simp only [S,ite_eq_right h.1]
+    exact mem_nearHangingEndpoints_of_not_separated G hloc (out a).1 x a.1.1.val b.1.1.val
+      (e a) (e b) (k*M) (hfin a) hpa hpb hcb hn
+  have hN (a : q.AmplificationIndex Q B x k n) : (S a).card ≤ (k*M)+(k*M)^2+d*(k*M)^3 := by
+    by_cases h : code a = []
+    · simp only [S,ite_eq_left h,Finset.card_empty,Nat.zero_le]
+    · simp only [S,ite_eq_right h]
+      exact nearHangingEndpoints_card_le G hloc (out a).1 x a.1.1.val (e a) (k*M) d
+        (hfin a) (hhead a h).2.2 hdeg
+  have hC : 0 ≤ inverseCodeMass M L ^ k := pow_nonneg
+    (add_nonneg zero_le_one (Finset.sum_nonneg fun l _ => gapWeight_nonneg L (by linarith) l)) _
+  simpa only [amplificationNearPairMass,out,m,R,added] using finite_near_pair_mass_le Q (Finset.Icc 1 k) (Finset.Icc 1 k) p out out m m
+    (q.amplificationMass_nonneg Q B x k n) (q.amplificationMass_nonneg Q B x k n)
+    (fun a => a.1.1.val) R S hS ((k*M)+(k*M)^2+d*(k*M)^3) hN _ hC
+    (fun a z _ => q.amplificationEndpointMass_bound hL Q B x k n p (out a).1
+      ((mem_finiteSpliceDomain_iff Q (Finset.Icc 1 k) (out a)).mp (q.amplificationOutput_mem Q B x k n a)).1 z)
+
+end LocalEndpointRule
+end AllEndpointNear
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+section FiniteThreeClass
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def commonRootPairMass {I : Type*} [Fintype I]
+    (Q : Finset E) (J : Finset ℕ) (p : ℝ) (out : I → SpliceState E)
+    (m : I → ℝ) (P : I → I → Prop) : ℝ :=
+  ∑ a, ∑ b, if (out a).1 = (out b).1 ∧ P a b then
+    jointTupleWeight Q J J p (pairSpliceOutput (out a) (out b)) * m a * m b else 0
+
+ 
+
+theorem finiteRootLoad_three_classes_le {I : Type*} [Fintype I]
+    (Q : Finset E) (J : Finset ℕ) (p : unitInterval) (out : I → SpliceState E)
+    (m : I → ℝ) (hm : ∀ a, 0 ≤ m a) (hs : ∀ a, out a ∈ finiteSpliceDomain Q J)
+    (C : I → Prop) (R : I → I → Prop) :
+    (∑ ω : FiniteField Q, finiteFieldWeight Q p ω *
+      finiteRootLoad Q J p (finiteInverseLoad out m) ω ^ 2) ≤
+      2 * (∑ ω : FiniteField Q, finiteFieldWeight Q p ω *
+        finiteRootLoad Q J p (finiteInverseLoad out (fun a => if C a then m a else 0)) ω *
+        finiteRootLoad Q J p (finiteInverseLoad out m) ω) +
+      commonRootPairMass Q J p out m (fun a b => ¬C a ∧ ¬C b ∧ R a b) +
+      commonRootPairMass Q J p out m (fun a b => ¬C a ∧ ¬C b ∧ ¬R a b) := by
+  let m₀ := fun a => if C a then m a else 0
+  let row := finiteRootLoad Q J p (finiteInverseLoad out m)
+  let nilrow := finiteRootLoad Q J p (finiteInverseLoad out m₀)
+  have hcross : (∑ ω : FiniteField Q, finiteFieldWeight Q p ω * row ω * nilrow ω) =
+      ∑ ω : FiniteField Q, finiteFieldWeight Q p ω * nilrow ω * row ω := by
+    apply Finset.sum_congr rfl
+    intro ω _
+    ring
+  have heq : 2 * (∑ ω : FiniteField Q, finiteFieldWeight Q p ω * nilrow ω * row ω) =
+      (∑ ω : FiniteField Q, finiteFieldWeight Q p ω * nilrow ω * row ω) +
+      (∑ ω : FiniteField Q, finiteFieldWeight Q p ω * row ω * nilrow ω) := by
+    rw [hcross]; ring
+  change _ ≤ 2 * (∑ ω : FiniteField Q, finiteFieldWeight Q p ω * nilrow ω * row ω) + _ + _
+  rw [heq]
+  dsimp only [nilrow,row]
+  simp only [pow_two,← mul_assoc]
+  rw [finiteRootLoad_pair_expansion Q J J p out out m m hs hs,
+    finiteRootLoad_pair_expansion Q J J p out out m₀ m hs hs,
+    finiteRootLoad_pair_expansion Q J J p out out m m₀ hs hs]
+  unfold commonRootPairMass
+  simp only [← Finset.sum_add_distrib]
+  apply Finset.sum_le_sum
+  intro a _
+  apply Finset.sum_le_sum
+  intro b _
+  have hw : 0 ≤ jointTupleWeight Q J J p (pairSpliceOutput (out a) (out b)) :=
+    jointTupleWeight_nonneg Q J J p _
+  have ht : 0 ≤ jointTupleWeight Q J J p (pairSpliceOutput (out a) (out b)) * m a * m b :=
+    mul_nonneg (mul_nonneg hw (hm a)) (hm b)
+  dsimp only [m₀]
+  by_cases hab : (out a).1 = (out b).1
+  · simp only [hab,true_and,ite_true]
+    by_cases ha : C a <;> by_cases hb : C b <;> by_cases hr : R a b <;>
+      simp only [ha,hb,hr,not_true_eq_false,not_false_eq_true,and_true,and_false,
+        ite_true,ite_false,mul_zero,zero_mul] <;> linarith
+  · simp only [hab,false_and,ite_false,zero_add,le_refl]
+
+end FiniteThreeClass
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+section AllEndpointMomentIdentities
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem finiteRootLoad_inverse_mean {I : Type*} [Fintype I]
+    (Q : Finset E) (J : Finset ℕ) (p : ℝ) (out : I → SpliceState E) (m : I → ℝ)
+    (hs : ∀ a, out a ∈ finiteSpliceDomain Q J) :
+    (∑ ω : FiniteField Q, finiteFieldWeight Q p ω * finiteRootLoad Q J p (finiteInverseLoad out m) ω) =
+      ∑ a, spliceTupleWeight Q J p (out a) * m a := by
+  rw [finiteRootLoad_mean]
+  simp only [finiteInverseLoad,Finset.mul_sum,mul_ite,mul_zero]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro a _
+  rw [Finset.sum_ite_eq,ite_eq_left (hs a)]
+
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
+
+theorem amplificationRow_nonneg (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (p : unitInterval) (ω : Set E) :
+    0 ≤ q.amplificationRow Q B x k n p ω := by
+  rw [q.amplificationRow_eq_load]
+  apply finiteRootLoad_nonneg
+  intro s _
+  exact finiteInverseLoad_nonneg _ _ (q.amplificationMass_nonneg Q B x k n) s
+
+theorem amplificationRow_mean_eq (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (p : unitInterval) :
+    finiteFieldIntegral Q p (q.amplificationRow Q B x k n p) =
+      ∑ a : q.AmplificationIndex Q B x k n,
+        spliceTupleWeight Q (Finset.Icc 1 k) p (q.amplificationOutput Q B x k n a) *
+          q.amplificationMass Q B x k n a := by
+  simp only [finiteFieldIntegral,q.amplificationRow_eq_load,restrictFiniteField_fieldSet]
+  exact finiteRootLoad_inverse_mean Q (Finset.Icc 1 k) p _ _ (q.amplificationOutput_mem Q B x k n)
+
+ 
+
+theorem amplificationSeparated_class_eq (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (p : unitInterval) :
+    commonRootPairMass Q (Finset.Icc 1 k) p (q.amplificationOutput Q B x k n)
+      (q.amplificationMass Q B x k n) (fun a b =>
+        q.amplificationCode Q B x k n a ≠ [] ∧ q.amplificationCode Q B x k n b ≠ [] ∧
+        GraphSeparated G (q.amplificationAdded Q B x k n a) (q.amplificationAdded Q B x k n b)) =
+    finiteChoiceSeparatedPairMass G x k Q p
+      (q.allAmplificationInputs Q B x k n) (q.allAmplificationInputs Q B x k n)
+      (allAmplificationVertices B k) (allAmplificationVertices B k)
+      (q.allAmplificationWeight B k) (q.allAmplificationWeight B k) := by
+  unfold commonRootPairMass finiteChoiceSeparatedPairMass
+  apply Finset.sum_congr rfl
+  intro a _
+  apply Finset.sum_congr rfl
+  intro b _
+  split_ifs with ha hb hb
+  · rfl
+  · exfalso
+    apply hb
+    dsimp only [amplificationAdded,amplificationOutput,amplificationCode,amplificationInput] at ha
+    rcases ha with ⟨he,hca,hcb,hsep⟩
+    refine ⟨he,hca,hcb,?_⟩
+    simpa only [he] using hsep
+  · exfalso
+    apply ha
+    dsimp only [amplificationAdded,amplificationOutput,amplificationCode,amplificationInput]
+    rcases hb with ⟨he,hca,hcb,hsep⟩
+    refine ⟨he,hca,hcb,?_⟩
+    simpa only [he] using hsep
+  · rfl
+
+end LocalEndpointRule
+end AllEndpointMomentIdentities
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+section AllEndpointSecondMoment
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
+
+ 
+
+theorem amplificationRow_second_moment_le [Nonempty E] (hconn : Connected G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
+    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (i₀ : ι) (hw₀ : 0 < weight i₀)
+    (q : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) M L) (hL : 1 ≤ L)
+    (Q : Finset E) (B : Finset V) (p : unitInterval) (x : V) (k n d : ℕ)
+    (hdeg : ∀ a, degree G a ≤ d)
+    (hQ : ∀ a ∈ B, incidentSupport G hloc (finite_reachBall hloc a (max q.radius M)).toFinset ⊆ Q) :
+    finiteFieldIntegral Q p (fun ω => q.amplificationRow Q B x k n p ω ^ 2) ≤
+      (((4*n:ℕ):ℝ) * q.successProbability p (root i₀)^k +
+        (((k*M)+(k*M)^2+d*(k*M)^3:ℕ):ℝ) * inverseCodeMass M L ^ k) *
+          finiteFieldIntegral Q p (q.amplificationRow Q B x k n p) +
+      (((2*n:ℕ):ℝ) * (c*((k*M:ℕ)*q.successProbability p (root i₀)^k)))^2 *
+        finiteClusterWindowMass G Q p x (n-k*M) (2*n) := by
+  let out := q.amplificationOutput Q B x k n
+  let m := q.amplificationMass Q B x k n
+  let C := fun a => q.amplificationCode Q B x k n a = []
+  let R := fun a b => ¬GraphSeparated G (q.amplificationAdded Q B x k n a) (q.amplificationAdded Q B x k n b)
+  let row := finiteRootLoad Q (Finset.Icc 1 k) p (finiteInverseLoad out m)
+  let nilrow := q.amplificationNilRow Q B x k n p
+  let b₀ := ∑ ω : FiniteField Q, finiteFieldWeight Q p ω * row ω
+  let bnil := ∑ ω : FiniteField Q, finiteFieldWeight Q p ω * nilrow ω * row ω
+  let S := finiteChoiceSeparatedPairMass G x k Q p
+    (q.allAmplificationInputs Q B x k n) (q.allAmplificationInputs Q B x k n)
+    (allAmplificationVertices B k) (allAmplificationVertices B k)
+    (q.allAmplificationWeight B k) (q.allAmplificationWeight B k)
+  have hrow (ω : FiniteField Q) : q.amplificationRow Q B x k n p (finiteFieldSet Q ω) = row ω := by
+    rw [q.amplificationRow_eq_load,restrictFiniteField_fieldSet]
+  have hmean : finiteFieldIntegral Q p (q.amplificationRow Q B x k n p) = b₀ := by
+    simp only [finiteFieldIntegral,hrow,b₀]
+  have hpart := finiteRootLoad_three_classes_le Q (Finset.Icc 1 k) p out m
+    (q.amplificationMass_nonneg Q B x k n) (q.amplificationOutput_mem Q B x k n) C R
+  have hnear_eq : commonRootPairMass Q (Finset.Icc 1 k) p out m
+      (fun a b => ¬C a ∧ ¬C b ∧ R a b) = q.amplificationNearPairMass Q B x k n p := by
+    simp only [commonRootPairMass,amplificationNearPairMass,C,R,out,m]
+  have hsep_eq : commonRootPairMass Q (Finset.Icc 1 k) p out m
+      (fun a b => ¬C a ∧ ¬C b ∧ ¬R a b) = S := by
+    simpa only [C,R,not_not] using q.amplificationSeparated_class_eq Q B x k n p
+  rw [hnear_eq,hsep_eq] at hpart
+  have hnilfun : finiteInverseLoad out (fun a => @ite ℝ (C a) (Classical.propDecidable _) (m a) 0) =
+      finiteInverseLoad (q.amplificationOutput Q B x k n)
+        (fun a => if q.amplificationCode Q B x k n a = [] then q.amplificationMass Q B x k n a else 0) := by
+    apply congrArg (finiteInverseLoad out)
+    funext a
+    dsimp only [C,m]
+    split_ifs <;> rfl
+  rw [hnilfun] at hpart
+  change (∑ ω : FiniteField Q, finiteFieldWeight Q p ω * row ω ^ 2) ≤
+    2*bnil + q.amplificationNearPairMass Q B x k n p + S at hpart
+  have hnil : bnil ≤ ((2*n:ℕ):ℝ) * q.successProbability p (root i₀)^k * b₀ := by
+    dsimp only [bnil,b₀]
+    rw [Finset.mul_sum]
+    apply Finset.sum_le_sum
+    intro ω _
+    have hr : 0 ≤ row ω := by rw [← hrow]; exact q.amplificationRow_nonneg Q B x k n p _
+    have hn := q.amplificationNilRow_bound root weight Γ htr (fun a ha => hΓ ha) hw i₀ hw₀ Q B x k n p ω
+      (fun a ha => (incidentSupport_ball_mono G hloc a (le_max_left _ _)).trans (hQ a ha))
+    calc
+      _ ≤ finiteFieldWeight Q p ω * (((2*n:ℕ):ℝ) * q.successProbability p (root i₀)^k) * row ω :=
+        mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left hn (finiteFieldWeight_nonneg Q p ω)) hr
+      _ = _ := by ring
+  have hnear := q.amplificationNearPairMass_bound hL Q B x k n d p hdeg
+  rw [← q.amplificationRow_mean_eq Q B x k n p,hmean] at hnear
+  have hsep := q.amplificationSeparatedSecondMoment_le hconn root weight Γ hΓ htr hcov hw hw1 c hc hmin
+    (root i₀) Q B p x k n hQ
+  have hfinal := add_le_add (add_le_add (mul_le_mul_of_nonneg_left hnil (show (0:ℝ) ≤ 2 by norm_num)) hnear) hsep
+  rw [hmean]
+  simp only [finiteFieldIntegral,hrow]
+  apply hpart.trans
+  convert hfinal using 1
+  push_cast
+  ring
+
+end LocalEndpointRule
+end AllEndpointSecondMoment
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+section SpliceCovarianceGeometry
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem cluster_aut_preimage (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (ω : Set E) (x : V) :
+    cluster G (b ⁻¹' ω) x = a ⁻¹' cluster G ω (a x) := by
+  ext y
+  exact conn_aut_iff G hab ω x y
+
+theorem exchangeOn_preimage (b : E ≃ E) (mask ω η : Set E) :
+    exchangeOn (b ⁻¹' mask) (b ⁻¹' ω) (b ⁻¹' η) = b ⁻¹' exchangeOn mask ω η := by
+  rfl
+
+theorem spliceMask_aut_preimage (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (S B : Set V) (e : E) :
+    spliceMask G (a ⁻¹' S) (a ⁻¹' B) e = b ⁻¹' spliceMask G S B (b e) := by
+  ext f
+  simp only [spliceMask,Set.mem_insert_iff,Set.mem_ofPred_eq,Set.mem_preimage,b.injective.eq_iff]
+  apply or_congr_right
+  have hi : internalEdges G (a ⁻¹' S)ᶜ = b ⁻¹' internalEdges G Sᶜ :=
+    internalEdges_aut_preimage G hab Sᶜ
+  rw [hi]
+  apply and_congr_left
+  intro _
+  constructor
+  · rintro ⟨v,hv,hvf⟩
+    refine ⟨a v,hv,?_⟩
+    rw [hab f]
+    exact Sym2.mem_map.mpr ⟨v,hvf,rfl⟩
+  · rintro ⟨v,hv,hvf⟩
+    rw [hab f] at hvf
+    obtain ⟨w,hw,rfl⟩ := Sym2.mem_map.mp hvf
+    exact ⟨w,hv,hw⟩
+
+theorem spliceField_aut_preimage (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (S B : Set V) (e : E) (ω η : Set E) :
+    spliceField G (a ⁻¹' S) (a ⁻¹' B) e (b ⁻¹' ω) (b ⁻¹' η) =
+      b ⁻¹' spliceField G S B (b e) ω η := by
+  rw [spliceField,spliceMask_aut_preimage G hab,exchangeOn_preimage]
+  rfl
+
+theorem seamContactCount_aut_preimage (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (η : ℕ → Set E) (y : ℕ → V) (ωf : Set E) (x : V) (i : ℕ) (e : E) :
+    seamContactCount G (fun j => b ⁻¹' η j) y (b ⁻¹' ωf) x i e =
+      seamContactCount G η (fun j => a (y j)) ωf (a x) i (b e) := by
+  have he : (b ⁻¹' ωf) \ {e} = b ⁻¹' (ωf \ {b e}) := by
+    ext f
+    simp only [Set.mem_sdiff,Set.mem_preimage,Set.mem_singleton_iff,b.injective.eq_iff]
+  unfold seamContactCount
+  rw [he,cluster_aut_preimage G hab]
+  rw [show (a ⁻¹' cluster G (ωf \ {b e}) (a x))ᶜ =
+    a ⁻¹' (cluster G (ωf \ {b e}) (a x))ᶜ from rfl]
+  rw [internalEdges_aut_preimage G hab,← Set.preimage_inter,cluster_aut_preimage G hab,
+    contactSet_aut_preimage G hab,Set.ncard_preimage_of_injective_subset_range b.injective]
+  exact fun f _ => b.surjective f
+
+theorem traceContactCount_aut_preimage (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (η : ℕ → Set E) (y : ℕ → V) (ωf : Set E) (x : V)
+    (c : List (ℕ × E)) (i : ℕ) :
+    traceContactCount G (fun j => b ⁻¹' η j) y (b ⁻¹' ωf) x c i =
+      traceContactCount G η (fun j => a (y j)) ωf (a x) (c.map (Prod.map id b)) i := by
+  induction c with
+  | nil => rfl
+  | cons q c ih =>
+    rcases q with ⟨j,e⟩
+    simp only [traceContactCount,List.map_cons,Prod.map_apply,id_eq]
+    split_ifs
+    · exact seamContactCount_aut_preimage G hab η y ωf x j e
+    · exact ih
+
+end SpliceCovarianceGeometry
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+section SpliceCovarianceTrace
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+def edgeFieldEquiv (b : E ≃ E) : Set E ≃ Set E where
+  toFun ω := b.symm ⁻¹' ω
+  invFun ω := b ⁻¹' ω
+  left_inv ω := by ext e; simp
+  right_inv ω := by ext e; simp
+
+def spliceStateEquiv (b : E ≃ E) : SpliceState E ≃ SpliceState E :=
+  (edgeFieldEquiv b).prodCongr (Equiv.piCongrRight fun _ => edgeFieldEquiv b)
+
+def spliceCodeEquiv (b : E ≃ E) : List (ℕ × E) ≃ List (ℕ × E) :=
+  Equiv.listEquivOfEquiv ((Equiv.refl ℕ).prodCongr b)
+
+@[simp] theorem edgeFieldEquiv_apply (b : E ≃ E) (ω : Set E) :
+    edgeFieldEquiv b ω = b.symm ⁻¹' ω := rfl
+
+@[simp] theorem spliceCodeEquiv_apply (b : E ≃ E) (c : List (ℕ × E)) :
+    spliceCodeEquiv b c = c.map (Prod.map id b) := rfl
+
+theorem cluster_edgeFieldEquiv (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (ω : Set E) (x : V) :
+    cluster G (edgeFieldEquiv b ω) (a x) = a.symm ⁻¹' cluster G ω x := by
+  simpa only [edgeFieldEquiv_apply,Equiv.symm_apply_apply] using
+    cluster_aut_preimage G (isAut_symm G hab) ω (a x)
+
+theorem slitCluster_edgeFieldEquiv (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (ω η : Set E) (x y : V) :
+    cluster G (edgeFieldEquiv b η ∩ internalEdges G (cluster G (edgeFieldEquiv b ω) (a x))ᶜ) (a y) =
+      a.symm ⁻¹' cluster G (η ∩ internalEdges G (cluster G ω x)ᶜ) y := by
+  rw [cluster_edgeFieldEquiv G hab]
+  rw [show (a.symm ⁻¹' cluster G ω x)ᶜ = a.symm ⁻¹' (cluster G ω x)ᶜ from rfl]
+  rw [internalEdges_aut_preimage G (isAut_symm G hab),edgeFieldEquiv_apply,← Set.preimage_inter,
+    cluster_aut_preimage G (isAut_symm G hab),Equiv.symm_apply_apply]
+
+theorem spliceStep_edgeFieldEquiv (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (ω η : Set E) (x y : V) (e : E) :
+    spliceField G (cluster G (edgeFieldEquiv b ω) (a x))
+      (cluster G (edgeFieldEquiv b η ∩ internalEdges G (cluster G (edgeFieldEquiv b ω) (a x))ᶜ) (a y))
+      (b e) (edgeFieldEquiv b ω) (edgeFieldEquiv b η) =
+    edgeFieldEquiv b (spliceField G (cluster G ω x)
+      (cluster G (η ∩ internalEdges G (cluster G ω x)ᶜ) y) e ω η) := by
+  rw [slitCluster_edgeFieldEquiv G hab,cluster_edgeFieldEquiv G hab]
+  rw [edgeFieldEquiv_apply,edgeFieldEquiv_apply,
+    spliceField_aut_preimage G (isAut_symm G hab),Equiv.symm_apply_apply]
+  rfl
+
+ 
+
+theorem SpliceTrace.aut_map (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (x : V) (η : ℕ → Set E) (y : ℕ → V) (k j : ℕ)
+    (ω ωf : Set E) (c : List (ℕ × E)) (ht : SpliceTrace G x η y k j ω c ωf) :
+    SpliceTrace G (a x) (fun i => edgeFieldEquiv b (η i)) (fun i => a (y i)) k j
+      (edgeFieldEquiv b ω) (spliceCodeEquiv b c) (edgeFieldEquiv b ωf) := by
+  induction ht with
+  | stop j ω hs =>
+    apply SpliceTrace.stop
+    simpa only [cluster_edgeFieldEquiv G hab,Set.mem_preimage,Equiv.symm_apply_apply] using hs
+  | step j i ω e v w rest ωf hn hji hik hmax hy he hev hv hw ht ih =>
+    change SpliceTrace G (a x) _ _ k j _ ((i,b e)::spliceCodeEquiv b rest) _
+    refine SpliceTrace.step j i _ (b e) (a v) (a w) _ _ ?_ hji hik ?_ ?_ ?_ ?_ ?_ ?_ ?_
+    · simpa only [cluster_edgeFieldEquiv G hab,Set.mem_preimage,Equiv.symm_apply_apply] using hn
+    · intro l hil hlk
+      rw [cluster_edgeFieldEquiv G hab,cluster_edgeFieldEquiv G hab,
+        Set.disjoint_preimage_iff a.symm.surjective]
+      exact hmax l hil hlk
+    · simpa only [cluster_edgeFieldEquiv G hab,Set.mem_preimage,Equiv.symm_apply_apply] using hy
+    · simpa only [edgeFieldEquiv_apply,Set.mem_preimage,Equiv.symm_apply_apply] using he
+    · rw [hab e,hev,Sym2.map_mk]
+    · simpa only [cluster_edgeFieldEquiv G hab,Set.mem_preimage,Equiv.symm_apply_apply] using hv
+    · simpa only [slitCluster_edgeFieldEquiv G hab,Set.mem_preimage,Equiv.symm_apply_apply] using hw
+    · rw [spliceStep_edgeFieldEquiv G hab]
+      exact ih
+
+end SpliceCovarianceTrace
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+section SpliceCovarianceReplay
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem spliceStepMask_edgeFieldEquiv (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (ω η : Set E) (x y : V) (e : E) :
+    spliceMask G (cluster G (edgeFieldEquiv b ω) (a x))
+      (cluster G (edgeFieldEquiv b η ∩ internalEdges G (cluster G (edgeFieldEquiv b ω) (a x))ᶜ) (a y))
+      (b e) =
+    edgeFieldEquiv b (spliceMask G (cluster G ω x)
+      (cluster G (η ∩ internalEdges G (cluster G ω x)ᶜ) y) e) := by
+  rw [slitCluster_edgeFieldEquiv G hab,cluster_edgeFieldEquiv G hab,
+    spliceMask_aut_preimage G (isAut_symm G hab),Equiv.symm_apply_apply]
+  rfl
+
+theorem exchangeState_aut_map (b : E ≃ E) (mask : Set E) (i : ℕ) (s : SpliceState E) :
+    exchangeState (edgeFieldEquiv b mask) i (spliceStateEquiv b s) =
+      spliceStateEquiv b (exchangeState mask i s) := by
+  apply Prod.ext
+  · rfl
+  · funext j
+    by_cases hji : j = i
+    · subst j
+      rw [exchangeState_at]
+      change _ = edgeFieldEquiv b ((exchangeState mask i s).2 i)
+      rw [exchangeState_at]
+      rfl
+    · rw [exchangeState_away _ _ _ _ hji]
+      change edgeFieldEquiv b (s.2 j) = edgeFieldEquiv b ((exchangeState mask i s).2 j)
+      rw [exchangeState_away _ _ _ _ hji]
+
+theorem replaySplice_aut_map (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (x : V) (η : ℕ → Set E) (y : ℕ → V)
+    (c : List (ℕ × E)) (ω : Set E) :
+    replaySplice G (a x) (fun i => edgeFieldEquiv b (η i)) (fun i => a (y i))
+      (spliceCodeEquiv b c) (edgeFieldEquiv b ω) =
+    edgeFieldEquiv b (replaySplice G x η y c ω) := by
+  induction c generalizing ω with
+  | nil => rfl
+  | cons q c ih =>
+    rcases q with ⟨i,e⟩
+    change replaySplice G (a x) _ _ (spliceCodeEquiv b c) _ = _
+    rw [spliceStep_edgeFieldEquiv G hab,ih]
+    rfl
+
+theorem replayMasks_aut_map (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (x : V) (η : ℕ → Set E) (y : ℕ → V)
+    (c : List (ℕ × E)) (ω : Set E) :
+    replayMasks G (a x) (fun i => edgeFieldEquiv b (η i)) (fun i => a (y i))
+      (spliceCodeEquiv b c) (edgeFieldEquiv b ω) =
+    (replayMasks G x η y c ω).map (Prod.map id (edgeFieldEquiv b)) := by
+  induction c generalizing ω with
+  | nil => rfl
+  | cons q c ih =>
+    rcases q with ⟨i,e⟩
+    simp only [spliceCodeEquiv_apply,List.map_cons,Prod.map_apply,id_eq,replayMasks]
+    rw [spliceStepMask_edgeFieldEquiv G hab,spliceStep_edgeFieldEquiv G hab]
+    exact congrArg (List.cons _) (ih _)
+
+theorem runMasks_aut_map (b : E ≃ E) (c : List (ℕ × Set E)) (s : SpliceState E) :
+    runMasks (c.map (Prod.map id (edgeFieldEquiv b))) (spliceStateEquiv b s) =
+      spliceStateEquiv b (runMasks c s) := by
+  induction c generalizing s with
+  | nil => rfl
+  | cons q c ih =>
+    rcases q with ⟨i,mask⟩
+    simp only [List.map_cons,Prod.map_apply,id_eq,runMasks]
+    rw [exchangeState_aut_map,ih]
+
+theorem seamContactCount_aut_map (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (η : ℕ → Set E) (y : ℕ → V) (ωf : Set E) (x : V) (i : ℕ) (e : E) :
+    seamContactCount G (fun j => edgeFieldEquiv b (η j)) (fun j => a (y j))
+      (edgeFieldEquiv b ωf) (a x) i (b e) = seamContactCount G η y ωf x i e := by
+  simpa only [edgeFieldEquiv_apply,Equiv.symm_apply_apply] using
+    seamContactCount_aut_preimage G (isAut_symm G hab) η (fun j => a (y j)) ωf (a x) i (b e)
+
+theorem traceContactCount_aut_map (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (η : ℕ → Set E) (y : ℕ → V) (ωf : Set E) (x : V)
+    (c : List (ℕ × E)) (i : ℕ) :
+    traceContactCount G (fun j => edgeFieldEquiv b (η j)) (fun j => a (y j))
+      (edgeFieldEquiv b ωf) (a x) (spliceCodeEquiv b c) i =
+      traceContactCount G η y ωf x c i := by
+  induction c with
+  | nil => rfl
+  | cons q c ih =>
+    rcases q with ⟨j,e⟩
+    simp only [spliceCodeEquiv_apply,List.map_cons,Prod.map_apply,id_eq,traceContactCount]
+    split_ifs
+    · exact seamContactCount_aut_map G hab η y ωf x j e
+    · exact ih
+
+end SpliceCovarianceReplay
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Splicing/GapCodes.lean b/lean/OAI/Probability/UnimodularPercolation/Splicing/GapCodes.lean
index 5609233f7dadc8e5f07960c527d684016473b1f8..0543c7dfb92998575df033f7bb3b13e7f557cbb7 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Splicing/GapCodes.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Splicing/GapCodes.lean
@@ -1,1327 +1,1327 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Splicing.Geometry
-
-namespace OAI
-
-/-! Gap Codes for bond percolation. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section SpliceExecution
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-abbrev SpliceState (E : Type v) := Set E × (ℕ → Set E)
-
- 
-
-def exchangeState (mask : Set E) (i : ℕ) (s : SpliceState E) : SpliceState E :=
-  (exchangeOn mask s.1 (s.2 i), Function.update s.2 i (exchangeOn mask (s.2 i) s.1))
-
-@[simp] theorem exchangeState_first (mask : Set E) (i : ℕ) (s : SpliceState E) :
-    (exchangeState mask i s).1 = exchangeOn mask s.1 (s.2 i) := rfl
-
-@[simp] theorem exchangeState_at (mask : Set E) (i : ℕ) (s : SpliceState E) :
-    (exchangeState mask i s).2 i = exchangeOn mask (s.2 i) s.1 := by
-  simp [exchangeState]
-
-theorem exchangeState_away (mask : Set E) (i l : ℕ) (s : SpliceState E) (hli : l ≠ i) :
-    (exchangeState mask i s).2 l = s.2 l := by
-  simp [exchangeState, Function.update_of_ne hli]
-
-theorem exchangeState_involutive (mask : Set E) (i : ℕ) :
-    Function.Involutive (exchangeState mask i (E := E)) := by
-  intro s
-  apply Prod.ext
-  · simp only [exchangeState_first, exchangeState_at]
-    exact exchangeOn_involutive mask s.1 (s.2 i)
-  · funext l
-    by_cases hli : l = i
-    · subst l
-      simp only [exchangeState_at, exchangeState_first]
-      exact exchangeOn_involutive mask (s.2 i) s.1
-    · rw [exchangeState_away _ _ _ _ hli, exchangeState_away _ _ _ _ hli]
-
- 
-
-def undoState (G : BondGraph V E) (x : V) (q : ℕ × E) (s : SpliceState E) :
-    SpliceState E := exchangeState (recoveredMask G s.1 x q.2) q.1 s
-
- 
-
-def ValidSpliceStep (G : BondGraph V E) (x : V) (s : SpliceState E)
-    (q : ℕ × E) (s' : SpliceState E) : Prop :=
-  ∃ y a b,
-    y ∉ cluster G s.1 x ∧ q.2 ∈ s.2 q.1 ∧ G q.2 = s(a,b) ∧
-    a ∈ cluster G s.1 x ∧
-    b ∈ cluster G (s.2 q.1 ∩ internalEdges G (cluster G s.1 x)ᶜ) y ∧
-    s' = exchangeState
-      (spliceMask G (cluster G s.1 x)
-        (cluster G (s.2 q.1 ∩ internalEdges G (cluster G s.1 x)ᶜ) y) q.2) q.1 s
-
-theorem undoState_of_valid (G : BondGraph V E) (x : V) (s s' : SpliceState E)
-    (q : ℕ × E) (h : ValidSpliceStep G x s q s') : undoState G x q s' = s := by
-  obtain ⟨y,a,b,hy,he,hab,ha,hb,rfl⟩ := h
-  unfold undoState
-  rw [show recoveredMask G
-      (exchangeState (spliceMask G (cluster G s.1 x)
-        (cluster G (s.2 q.1 ∩ internalEdges G (cluster G s.1 x)ᶜ) y) q.2) q.1 s).1 x q.2 =
-      spliceMask G (cluster G s.1 x)
-        (cluster G (s.2 q.1 ∩ internalEdges G (cluster G s.1 x)ᶜ) y) q.2 from
-    recoveredMask_splice G s.1 (s.2 q.1) x y hy q.2 a b hab he ha hb]
-  exact exchangeState_involutive _ _ s
-
- 
-inductive SpliceExecution (G : BondGraph V E) (x : V) :
-    SpliceState E → List (ℕ × E) → SpliceState E → Prop
-  | nil (s) : SpliceExecution G x s [] s
-  | cons (s s' s'') (q qs)
-      (hstep : ValidSpliceStep G x s q s')
-      (hrest : SpliceExecution G x s' qs s'') : SpliceExecution G x s (q::qs) s''
-
- 
-def undoExecution (G : BondGraph V E) (x : V) :
-    List (ℕ × E) → SpliceState E → SpliceState E
-  | [], s => s
-  | q::qs, s => undoState G x q (undoExecution G x qs s)
-
-theorem SpliceExecution.undo (G : BondGraph V E) (x : V)
-    (s s' : SpliceState E) (code : List (ℕ × E))
-    (h : SpliceExecution G x s code s') : undoExecution G x code s' = s := by
-  induction h with
-  | nil => rfl
-  | cons s s' s'' q qs hstep hrest ih =>
-    change undoState G x q (undoExecution G x qs s'') = s
-    rw [ih]
-    exact undoState_of_valid G x s s' q hstep
-
- 
-
-theorem SpliceExecution.input_unique (G : BondGraph V E) (x : V)
-    (s₁ s₂ out : SpliceState E) (code : List (ℕ × E))
-    (h₁ : SpliceExecution G x s₁ code out) (h₂ : SpliceExecution G x s₂ code out) :
-    s₁ = s₂ := by
-  exact (h₁.undo G x s₁ out code).symm.trans (h₂.undo G x s₂ out code)
-
- 
-
-theorem SpliceTrace.realize (G : BondGraph V E) (x : V) (η : ℕ → Set E)
-    (y : ℕ → V) (k j : ℕ) (ω ωf : Set E) (code : List (ℕ × E))
-    (h : SpliceTrace G x η y k j ω code ωf) :
-    ∀ θ : ℕ → Set E, (∀ l, j < l → l ≤ k → θ l = η l) →
-    ∃ θf, SpliceExecution G x (ω,θ) code (ωf,θf) := by
-  induction h with
-  | stop j ω hz =>
-    intro θ _
-    exact ⟨θ, SpliceExecution.nil _⟩
-  | @step j i ω e a b rest ωf _ hji hik _ hy he hab ha hb ht ih =>
-    intro θ hθ
-    have hθi := hθ i hji hik
-    let S := cluster G ω x
-    let B := cluster G (η i ∩ internalEdges G Sᶜ) (y i)
-    let mask := spliceMask G S B e
-    let next := exchangeState mask i (ω,θ)
-    have hnext : next.1 = spliceField G S B e ω (η i) := by
-      change exchangeOn mask ω (θ i) = exchangeOn mask ω (η i)
-      rw [hθi]
-    obtain ⟨θf,hexec⟩ := ih next.2 (by
-      intro l hil hlk
-      rw [exchangeState_away _ _ _ _ (by omega)]
-      exact hθ l (hji.trans hil) hlk)
-    refine ⟨θf, SpliceExecution.cons (ω,θ) next (ωf,θf) (i,e) rest ?_ ?_⟩
-    · refine ⟨y i,a,b,hy,?_,hab,ha,?_,?_⟩
-      · simpa only [hθi] using he
-      · simpa only [hθi] using hb
-      · dsimp only [next, mask, B, S]
-        rw [hθi]
-    · change SpliceExecution G x (next.1,next.2) rest (ωf,θf)
-      rw [hnext]
-      exact hexec
-
-end SpliceExecution
-
-end Percolation
-
- 
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section BridgeCode
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem pivotal_mem_trail (G : BondGraph V E) (ω : Set E) (x z : V)
-    (l : List E) (hl : Along G x l z) (hopen : ∀ e ∈ l, e ∈ ω)
-    {e : E} (he : OpenMarkedPivotal G ω {z} x e) : e ∈ l := by
-  by_contra hnot
-  apply he.2.2
-  refine ⟨z,rfl,hl.conn ?_⟩
-  intro f hf
-  exact ⟨hopen f hf,fun hfe => hnot ((Set.mem_singleton_iff.mp hfe) ▸ hf)⟩
-
- 
-theorem finite_separatingBonds (G : BondGraph V E) (ω : Set E) (x z : V) :
-    {e | OpenMarkedPivotal G ω {z} x e}.Finite := by
-  by_cases hc : Conn G ω x z
-  · obtain ⟨l,hl,_,ho⟩ := conn_exists_bond_trail G hc
-    exact l.toFinset.finite_toSet.subset fun e he =>
-      List.mem_toFinset.mpr (pivotal_mem_trail G ω x z l hl ho he)
-  · have hempty : {e | OpenMarkedPivotal G ω {z} x e} = ∅ := by
-      ext e
-      simp only [Set.mem_ofPred_eq,Set.mem_empty_iff_false,iff_false]
-      rintro ⟨_,⟨y,hy,hxy⟩,_⟩
-      exact hc ((Set.mem_singleton_iff.mp hy) ▸ hxy)
-    rw [hempty]
-    exact Set.finite_empty
-
-noncomputable def separatingBonds (G : BondGraph V E) (ω : Set E) (x z : V) : Finset E :=
-  (finite_separatingBonds G ω x z).toFinset
-
-@[simp] theorem mem_separatingBonds (G : BondGraph V E) (ω : Set E) (x z : V) (e : E) :
-    e ∈ separatingBonds G ω x z ↔ OpenMarkedPivotal G ω {z} x e := by
-  exact (finite_separatingBonds G ω x z).mem_toFinset
-
- 
-theorem pivotalRank_injective_of_cluster_finite (G : BondGraph V E) (ω : Set E)
-    (m : Set V) (x : V) (hC : (cluster G ω x).Finite) :
-    Set.InjOn (pivotalRank G ω x) {e | OpenMarkedPivotal G ω m x e} := by
-  have hcut (f : E) : (cluster G (ω \ {f}) x).Finite :=
-    hC.subset (fun _ h => conn_mono G Set.sdiff_subset h)
-  intro e he f hf heq
-  by_contra hne
-  rcases pivotal_rootSides_nested G ω m x he hf hne with h | h
-  · exact (Nat.ne_of_lt (Set.ncard_lt_ncard h (hcut f))) heq
-  · exact (Nat.ne_of_lt (Set.ncard_lt_ncard h (hcut e))) heq.symm
-
- 
-
-noncomputable def bridgeIndex (G : BondGraph V E) (ω : Set E) (x z : V) (e : E) : ℕ :=
-  ((separatingBonds G ω x z).filter fun f => pivotalRank G ω x f ≤ pivotalRank G ω x e).card
-
-theorem bridgeIndex_pos (G : BondGraph V E) (ω : Set E) (x z : V) {e : E}
-    (he : OpenMarkedPivotal G ω {z} x e) : 0 < bridgeIndex G ω x z e := by
-  apply Finset.card_pos.mpr
-  exact ⟨e,Finset.mem_filter.mpr ⟨(mem_separatingBonds G ω x z e).mpr he,le_refl _⟩⟩
-
-theorem bridgeIndex_lt_of_rank_lt (G : BondGraph V E) (ω : Set E) (x z : V) {e f : E}
-    (hf : OpenMarkedPivotal G ω {z} x f)
-    (hlt : pivotalRank G ω x e < pivotalRank G ω x f) :
-    bridgeIndex G ω x z e < bridgeIndex G ω x z f := by
-  apply Finset.card_lt_card
-  apply Finset.ssubset_iff_subset_ne.mpr
-  refine ⟨?_,?_⟩
-  · intro d hd
-    exact Finset.mem_filter.mpr ⟨(Finset.mem_filter.mp hd).1,
-      (Finset.mem_filter.mp hd).2.trans (Nat.le_of_lt hlt)⟩
-  · intro heq
-    have hmem : f ∈ (separatingBonds G ω x z).filter
-        (fun d => pivotalRank G ω x d ≤ pivotalRank G ω x f) :=
-      Finset.mem_filter.mpr ⟨(mem_separatingBonds G ω x z f).mpr hf,le_refl _⟩
-    rw [← heq] at hmem
-    exact (Nat.not_le_of_lt hlt) (Finset.mem_filter.mp hmem).2
-
-theorem bridgeIndex_injective (G : BondGraph V E) (ω : Set E) (x z : V)
-    (hC : (cluster G ω x).Finite) :
-    Set.InjOn (bridgeIndex G ω x z) {e | OpenMarkedPivotal G ω {z} x e} := by
-  intro e he f hf heq
-  apply pivotalRank_injective_of_cluster_finite G ω {z} x hC he hf
-  rcases lt_trichotomy (pivotalRank G ω x e) (pivotalRank G ω x f) with h | h | h
-  · exact ((Nat.ne_of_lt (bridgeIndex_lt_of_rank_lt G ω x z hf h)) heq).elim
-  · exact h
-  · exact ((Nat.ne_of_lt (bridgeIndex_lt_of_rank_lt G ω x z he h)) heq.symm).elim
-
- 
-noncomputable def ordinalSpliceCode (G : BondGraph V E) (ω : Set E) (x z : V)
-    (code : List (ℕ × E)) : List (ℕ × ℕ) :=
-  code.map fun q => (q.1,bridgeIndex G ω x z q.2)
-
-theorem ordinalSpliceCode_injective (G : BondGraph V E) (ω : Set E) (x z : V)
-    (hC : (cluster G ω x).Finite) (c₁ c₂ : List (ℕ × E))
-    (h₁ : ∀ q ∈ c₁, OpenMarkedPivotal G ω {z} x q.2)
-    (h₂ : ∀ q ∈ c₂, OpenMarkedPivotal G ω {z} x q.2)
-    (heq : ordinalSpliceCode G ω x z c₁ = ordinalSpliceCode G ω x z c₂) : c₁ = c₂ := by
-  induction c₁ generalizing c₂ with
-  | nil =>
-    cases c₂ with
-    | nil => rfl
-    | cons q qs => simp [ordinalSpliceCode] at heq
-  | cons q qs ih =>
-    cases c₂ with
-    | nil => simp [ordinalSpliceCode] at heq
-    | cons r rs =>
-      have heq' := List.cons.inj heq
-      have hqr : q = r := by
-        apply Prod.ext
-        · exact congrArg (fun t : ℕ × ℕ => t.1) heq'.1
-        · exact bridgeIndex_injective G ω x z hC (h₁ q (by simp)) (h₂ r (by simp))
-            (congrArg Prod.snd heq'.1)
-      exact congrArg₂ List.cons hqr (ih rs
-        (fun a ha => h₁ a (List.mem_cons_of_mem _ ha))
-        (fun a ha => h₂ a (List.mem_cons_of_mem _ ha)) heq'.2)
-
- 
-
-theorem SpliceTrace.input_unique_of_ordinalCode (G : BondGraph V E) (x : V)
-    (η₁ η₂ : ℕ → Set E) (y₁ y₂ : ℕ → V) (k j : ℕ) (ω₁ ω₂ ωf : Set E)
-    (c₁ c₂ : List (ℕ × E))
-    (ht₁ : SpliceTrace G x η₁ y₁ k j ω₁ c₁ ωf)
-    (ht₂ : SpliceTrace G x η₂ y₂ k j ω₂ c₂ ωf)
-    (hz : y₁ k = y₂ k) (hC : (cluster G ωf x).Finite)
-    (heq : ordinalSpliceCode G ωf x (y₁ k) c₁ = ordinalSpliceCode G ωf x (y₁ k) c₂)
-    (θf : ℕ → Set E)
-    (he₁ : SpliceExecution G x (ω₁,η₁) c₁ (ωf,θf))
-    (he₂ : SpliceExecution G x (ω₂,η₂) c₂ (ωf,θf)) :
-    (ω₁,η₁) = (ω₂,η₂) := by
-  have hc : c₁ = c₂ := ordinalSpliceCode_injective G ωf x (y₁ k) hC c₁ c₂
-    (ht₁.seams_pivotal G x η₁ y₁ k j ω₁ ωf c₁)
-    (by simpa only [hz] using ht₂.seams_pivotal G x η₂ y₂ k j ω₂ ωf c₂) heq
-  subst c₂
-  exact he₁.input_unique G x (ω₁,η₁) (ω₂,η₂) (ωf,θf) c₁ he₂
-
-end BridgeCode
-
-end Percolation
-
- 
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section SpliceWeight
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-noncomputable def domainBits (Q : Finset E) (ω : Set E) : Finset E := Q.filter (· ∈ ω)
-
-@[simp] theorem mem_domainBits (Q : Finset E) (ω : Set E) (e : E) :
-    e ∈ domainBits Q ω ↔ e ∈ Q ∧ e ∈ ω := Finset.mem_filter
-
-theorem domainBits_subset (Q : Finset E) (ω : Set E) : domainBits Q ω ⊆ Q :=
-  Finset.filter_subset _ _
-
-theorem domainBits_exchange (Q : Finset E) (M ω η : Set E) :
-    domainBits Q (exchangeOn M ω η) =
-      finiteExchange (domainBits Q M) (domainBits Q ω) (domainBits Q η) := by
-  ext e
-  simp only [mem_domainBits,exchangeOn,Set.mem_union,Set.mem_sdiff,Set.mem_inter_iff,
-    finiteExchange,Finset.mem_union,Finset.mem_sdiff,Finset.mem_inter]
-  tauto
-
- 
-
-noncomputable def spliceTupleWeight (Q : Finset E) (J : Finset ℕ) (p : ℝ)
-    (s : SpliceState E) : ℝ :=
-  bernoulliWeight Q (domainBits Q s.1) p *
-    ∏ i ∈ J, bernoulliWeight Q (domainBits Q (s.2 i)) p
-
-theorem spliceTupleWeight_nonneg (Q : Finset E) (J : Finset ℕ) (p : unitInterval)
-    (s : SpliceState E) : 0 ≤ spliceTupleWeight Q J p s := by
-  exact mul_nonneg (bernoulliWeight_nonneg _ _ p)
-    (Finset.prod_nonneg fun _ _ => bernoulliWeight_nonneg _ _ p)
-
- 
-theorem spliceTupleWeight_exchange (Q : Finset E) (J : Finset ℕ) (p : ℝ)
-    (s : SpliceState E) (M : Set E) (i : ℕ) (hi : i ∈ J) :
-    spliceTupleWeight Q J p (exchangeState M i s) = spliceTupleWeight Q J p s := by
-  unfold spliceTupleWeight
-  rw [← Finset.mul_prod_erase J (fun l =>
-    bernoulliWeight Q (domainBits Q ((exchangeState M i s).2 l)) p) hi,
-    ← Finset.mul_prod_erase J (fun l => bernoulliWeight Q (domainBits Q (s.2 l)) p) hi]
-  have herase : (∏ l ∈ J.erase i,
-      bernoulliWeight Q (domainBits Q ((exchangeState M i s).2 l)) p) =
-      ∏ l ∈ J.erase i, bernoulliWeight Q (domainBits Q (s.2 l)) p := by
-    apply Finset.prod_congr rfl
-    intro l hl
-    rw [exchangeState_away _ _ _ _ (Finset.mem_erase.mp hl).1]
-  rw [herase,exchangeState_first,exchangeState_at,← mul_assoc,← mul_assoc,
-    domainBits_exchange,domainBits_exchange,
-    bernoulliWeight_exchange _ _ _ _ (domainBits_subset Q s.1) (domainBits_subset Q (s.2 i)) p]
-
- 
-theorem SpliceExecution.weight (G : BondGraph V E) (x : V)
-    (s s' : SpliceState E) (code : List (ℕ × E))
-    (h : SpliceExecution G x s code s') (Q : Finset E) (J : Finset ℕ)
-    (hJ : ∀ q ∈ code, q.1 ∈ J) (p : ℝ) :
-    spliceTupleWeight Q J p s' = spliceTupleWeight Q J p s := by
-  induction h with
-  | nil => rfl
-  | cons s s' s'' q code hstep hrest ih =>
-    rw [ih (fun r hr => hJ r (List.mem_cons_of_mem _ hr))]
-    obtain ⟨y,a,b,_,_,_,_,_,rfl⟩ := hstep
-    exact spliceTupleWeight_exchange Q J p s _ q.1 (hJ q (List.mem_cons_self ..))
-
- 
-
-theorem splice_finite_weight_comparison (G : BondGraph V E) (x : V)
-    (Q : Finset E) (J : Finset ℕ) (p : unitInterval) (code : List (ℕ × E))
-    (hJ : ∀ q ∈ code, q.1 ∈ J) (A B : Finset (SpliceState E))
-    (F : SpliceState E → SpliceState E)
-    (hex : ∀ s ∈ A, SpliceExecution G x s code (F s))
-    (hB : ∀ s ∈ A, F s ∈ B) :
-    (∑ s ∈ A, spliceTupleWeight Q J p s) ≤ ∑ s ∈ B, spliceTupleWeight Q J p s := by
-  have hinj : Set.InjOn F (A : Set (SpliceState E)) := by
-    intro s hs t ht heq
-    exact (hex s hs).input_unique G x s t (F s) code (heq.symm ▸ hex t ht)
-  have hweight : (∑ s ∈ A, spliceTupleWeight Q J p s) =
-      ∑ s ∈ A.image F, spliceTupleWeight Q J p s := by
-    rw [Finset.sum_image hinj]
-    apply Finset.sum_congr rfl
-    intro s hs
-    exact ((hex s hs).weight G x s (F s) code Q J hJ p).symm
-  rw [hweight]
-  apply Finset.sum_le_sum_of_subset_of_nonneg
-  · intro t ht
-    obtain ⟨s,hs,rfl⟩ := Finset.mem_image.mp ht
-    exact hB s hs
-  · intro s _ _
-    exact spliceTupleWeight_nonneg Q J p s
-
-end SpliceWeight
-
-end Percolation
-
- 
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section BridgeGapGeometry
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem pivotalRank_lt_iff_of_cluster_finite (G : BondGraph V E) (ω : Set E)
-    (m : Set V) (x : V) (hC : (cluster G ω x).Finite) {e f : E}
-    (he : OpenMarkedPivotal G ω m x e) (hf : OpenMarkedPivotal G ω m x f) :
-    pivotalRank G ω x e < pivotalRank G ω x f ↔
-      f ∈ internalEdges G (cluster G (ω \ {e}) x)ᶜ := by
-  have hcut (q : E) : (cluster G (ω \ {q}) x).Finite :=
-    hC.subset (fun _ h => conn_mono G Set.sdiff_subset h)
-  have hexterior (hout : f ∈ internalEdges G (cluster G (ω \ {e}) x)ᶜ) : e ≠ f := by
-    obtain ⟨a,b,hab,hxa,_,_⟩ := openMarkedPivotal_oriented G ω m x he
-    intro h; subst f
-    exact ((mem_internalEdges_of_ends G _ e a b hab).mp hout).1 hxa
-  constructor
-  · intro hlt
-    have hne : e ≠ f := fun h => (Nat.ne_of_lt hlt) (congrArg (pivotalRank G ω x) h)
-    rcases pivotal_other_sides G ω m x he hf hne.symm with hinside | hout
-    · obtain ⟨a,b,hab,_,hxb,hbm⟩ := openMarkedPivotal_oriented G ω m x he
-      obtain ⟨y,hy,hby⟩ := hbm
-      have hby' : Conn G (ω \ {f}) b y := conn_mono G
-        (ω₁ := (ω \ {e}) \ {f}) (ω₂ := ω \ {f}) (fun _ h => ⟨h.1.1,h.2⟩)
-        (conn_delete_internal_of_outside G (ω \ {e}) x hinside hxb hby)
-      have hb : b ∉ cluster G (ω \ {f}) x := fun h => hf.2.2 ⟨y,hy,h.trans hby'⟩
-      have ha : a ∉ cluster G (ω \ {f}) x := fun h =>
-        hb ((edge_cluster_sides G (ω \ {f}) x ⟨he.1,hne⟩ hab).mp h)
-      have hs := pivotal_rootSide_ssubset G ω m x hf hne.symm
-        ((mem_internalEdges_of_ends G _ e a b hab).mpr ⟨ha,hb⟩)
-      have hrev := Set.ncard_lt_ncard hs (hcut e)
-      exact (Nat.not_lt_of_ge (Nat.le_of_lt hrev) hlt).elim
-    · exact hout
-  · intro hout
-    exact Set.ncard_lt_ncard (pivotal_rootSide_ssubset G ω m x he (hexterior hout) hout) (hcut f)
-
-theorem pivotal_internal_of_rank_lt (G : BondGraph V E) (ω : Set E)
-    (m : Set V) (x : V) (hC : (cluster G ω x).Finite) {e f : E}
-    (he : OpenMarkedPivotal G ω m x e) (hf : OpenMarkedPivotal G ω m x f)
-    (hlt : pivotalRank G ω x e < pivotalRank G ω x f) :
-    e ∈ internalEdges G (cluster G (ω \ {f}) x) := by
-  have hne : e ≠ f := fun h => (Nat.ne_of_lt hlt) (congrArg (pivotalRank G ω x) h)
-  rcases pivotal_other_sides G ω m x hf he hne with h | h
-  · exact h
-  · have hr := (pivotalRank_lt_iff_of_cluster_finite G ω m x hC hf he).mpr h
-    omega
-
- 
-theorem conn_far_end_avoids_later_pivotal (G : BondGraph V E) (ω : Set E)
-    (x z : V) (hC : (cluster G ω x).Finite) {e g : E}
-    (he : OpenMarkedPivotal G ω {z} x e) (hg : OpenMarkedPivotal G ω {z} x g)
-    (hlt : pivotalRank G ω x e < pivotalRank G ω x g)
-    (a b : V) (hab : G e = s(a,b)) (ha : Conn G (ω \ {e}) x a) :
-    Conn G (ω \ {g}) x b := by
-  have hout := (pivotalRank_lt_iff_of_cluster_finite G ω {z} x hC he hg).mp hlt
-  have hne : e ≠ g := fun h => (Nat.ne_of_lt hlt) (congrArg (pivotalRank G ω x) h)
-  have hc : Conn G (ω \ {g}) x a :=
-    conn_mono G (ω₁ := (ω \ {e}) \ {g}) (ω₂ := ω \ {g}) (fun _ h => ⟨h.1.1,h.2⟩)
-      (conn_delete_exterior_of_inside G (ω \ {e}) x hout (conn_refl G _ x) ha)
-  exact hc.tail ⟨e,⟨he.1,hne⟩,hab⟩
-
- 
-theorem conn_near_end_avoids_earlier_pivotal (G : BondGraph V E) (ω : Set E)
-    (x z : V) (hC : (cluster G ω x).Finite) {f g : E}
-    (hf : OpenMarkedPivotal G ω {z} x f) (hg : OpenMarkedPivotal G ω {z} x g)
-    (hlt : pivotalRank G ω x g < pivotalRank G ω x f)
-    (c d : V) (hcd : G f = s(c,d)) (hd : ¬Conn G (ω \ {f}) x d)
-    (hdz : Conn G (ω \ {f}) d z) : Conn G (ω \ {g}) c z := by
-  have hin := pivotal_internal_of_rank_lt G ω {z} x hC hg hf hlt
-  have hne : f ≠ g := fun h => (Nat.ne_of_lt hlt) (congrArg (pivotalRank G ω x) h.symm)
-  have hc : Conn G (ω \ {g}) d z :=
-    conn_mono G (ω₁ := (ω \ {f}) \ {g}) (ω₂ := ω \ {g}) (fun _ h => ⟨h.1.1,h.2⟩)
-      (conn_delete_internal_of_outside G (ω \ {f}) x hin hd hdz)
-  exact (Relation.ReflTransGen.single ⟨f,⟨hf.1,hne⟩,hcd⟩).trans hc
-
- 
-
-theorem pivotal_between_is_internal (G : BondGraph V E) (ω : Set E)
-    (x z : V) (hC : (cluster G ω x).Finite) {e f g : E}
-    (he : OpenMarkedPivotal G ω {z} x e) (hf : OpenMarkedPivotal G ω {z} x f)
-    (hg : OpenMarkedPivotal G ω {z} x g)
-    (heg : pivotalRank G ω x e < pivotalRank G ω x g)
-    (hgf : pivotalRank G ω x g < pivotalRank G ω x f)
-    (a b c d : V) (hab : G e = s(a,b)) (hcd : G f = s(c,d))
-    (ha : Conn G (ω \ {e}) x a) (hd : ¬Conn G (ω \ {f}) x d)
-    (hdz : Conn G (ω \ {f}) d z)
-    (hbc : Conn G (ω ∩ internalEdges G
-      (cluster G (ω \ {f}) x \ cluster G (ω \ {e}) x)) b c) :
-    OpenMarkedPivotal G (ω ∩ internalEdges G
-      (cluster G (ω \ {f}) x \ cluster G (ω \ {e}) x)) {c} b g := by
-  have hout := (pivotalRank_lt_iff_of_cluster_finite G ω {z} x hC he hg).mp heg
-  have hin := pivotal_internal_of_rank_lt G ω {z} x hC hg hf hgf
-  refine ⟨⟨hg.1,fun v hv => ⟨hin v hv,hout v hv⟩⟩,⟨c,rfl,hbc⟩,?_⟩
-  rintro ⟨w,hw,hbw⟩
-  have hbc' : Conn G (ω \ {g}) b c := by
-    have hw' : w = c := Set.mem_singleton_iff.mp hw
-    subst w
-    apply conn_mono G ?_ hbw
-    intro q hq
-    exact ⟨hq.1.1,hq.2⟩
-  apply hg.2.2
-  exact ⟨z,rfl,(conn_far_end_avoids_later_pivotal G ω x z hC he hg heg a b hab ha).trans
-    (hbc'.trans (conn_near_end_avoids_earlier_pivotal G ω x z hC hf hg hgf c d hcd hd hdz))⟩
-
- 
-theorem pivotal_after_is_internal (G : BondGraph V E) (ω : Set E)
-    (x z : V) (hC : (cluster G ω x).Finite) {e g : E}
-    (he : OpenMarkedPivotal G ω {z} x e) (hg : OpenMarkedPivotal G ω {z} x g)
-    (heg : pivotalRank G ω x e < pivotalRank G ω x g)
-    (a b : V) (hab : G e = s(a,b)) (ha : Conn G (ω \ {e}) x a)
-    (hbz : Conn G (ω ∩ internalEdges G
-      (cluster G ω x \ cluster G (ω \ {e}) x)) b z) :
-    OpenMarkedPivotal G (ω ∩ internalEdges G
-      (cluster G ω x \ cluster G (ω \ {e}) x)) {z} b g := by
-  have hout := (pivotalRank_lt_iff_of_cluster_finite G ω {z} x hC he hg).mp heg
-  have hin : g ∈ internalEdges G (cluster G ω x) := by
-    obtain ⟨u,v,huv,hxu,_,_⟩ := openMarkedPivotal_oriented G ω {z} x hg
-    have hu : u ∈ cluster G ω x := conn_mono G Set.sdiff_subset hxu
-    exact (mem_internalEdges_of_ends G _ g u v huv).mpr
-      ⟨hu,(edge_cluster_sides G ω x hg.1 huv).mp hu⟩
-  refine ⟨⟨hg.1,fun v hv => ⟨hin v hv,hout v hv⟩⟩,⟨z,rfl,hbz⟩,?_⟩
-  rintro ⟨w,hw,hbw⟩
-  have hbz' : Conn G (ω \ {g}) b z := by
-    have hw' : w = z := Set.mem_singleton_iff.mp hw
-    subst w
-    apply conn_mono G ?_ hbw
-    intro q hq
-    exact ⟨hq.1.1,hq.2⟩
-  exact hg.2.2 ⟨z,rfl,
-    (conn_far_end_avoids_later_pivotal G ω x z hC he hg heg a b hab ha).trans hbz'⟩
-
-end BridgeGapGeometry
-
-end Percolation
-
- 
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section BridgeGapCount
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-@[simp] theorem separatingBonds_card (G : BondGraph V E) (ω : Set E) (x z : V) :
-    (separatingBonds G ω x z).card = pivotalDistance G ω x z := by
-  exact Set.ncard_coe_finset (separatingBonds G ω x z) |>.symm.trans (by
-    congr 1
-    ext e
-    exact mem_separatingBonds G ω x z e)
-
- 
-noncomputable def bridgeInterval (G : BondGraph V E) (ω : Set E) (x z : V) (e f : E) : Finset E :=
-  (separatingBonds G ω x z).filter fun g =>
-    pivotalRank G ω x e < pivotalRank G ω x g ∧ pivotalRank G ω x g < pivotalRank G ω x f
-
-theorem bridgeIndex_interval_identity (G : BondGraph V E) (ω : Set E) (x z : V)
-    (hC : (cluster G ω x).Finite) {e f : E}
-    (hf : OpenMarkedPivotal G ω {z} x f)
-    (hlt : pivotalRank G ω x e < pivotalRank G ω x f) :
-    bridgeIndex G ω x z f = bridgeIndex G ω x z e + (bridgeInterval G ω x z e f).card + 1 := by
-  let P := separatingBonds G ω x z
-  let A := P.filter fun g => pivotalRank G ω x g ≤ pivotalRank G ω x e
-  let B := bridgeInterval G ω x z e f
-  have hdis : Disjoint A B := by
-    apply Finset.disjoint_left.mpr
-    intro g hgA hgB
-    have h₁ := (Finset.mem_filter.mp hgA).2
-    have h₂ := (Finset.mem_filter.mp hgB).2.1
-    omega
-  have hnot : f ∉ A ∪ B := by
-    intro hh
-    rcases Finset.mem_union.mp hh with h | h
-    · have hr := (Finset.mem_filter.mp h).2
-      omega
-    · exact (Nat.lt_irrefl _) (Finset.mem_filter.mp h).2.2
-  have hset : P.filter (fun g => pivotalRank G ω x g ≤ pivotalRank G ω x f) = insert f (A ∪ B) := by
-    ext g
-    constructor
-    · intro hg
-      obtain ⟨hgP,hgf⟩ := Finset.mem_filter.mp hg
-      by_cases hge : g = f
-      · exact Finset.mem_insert.mpr (Or.inl hge)
-      · apply Finset.mem_insert.mpr
-        apply Or.inr
-        by_cases hga : pivotalRank G ω x g ≤ pivotalRank G ω x e
-        · exact Finset.mem_union_left _ (Finset.mem_filter.mpr ⟨hgP,hga⟩)
-        · apply Finset.mem_union_right
-          apply Finset.mem_filter.mpr
-          refine ⟨hgP,Nat.lt_of_not_ge hga,?_⟩
-          apply lt_of_le_of_ne hgf
-          intro hr
-          exact hge (pivotalRank_injective_of_cluster_finite G ω {z} x hC
-            ((mem_separatingBonds G ω x z g).mp hgP) hf hr)
-    · intro hg
-      rcases Finset.mem_insert.mp hg with heq | hg
-      · subst g
-        exact Finset.mem_filter.mpr ⟨(mem_separatingBonds G ω x z f).mpr hf,le_refl _⟩
-      · rcases Finset.mem_union.mp hg with hg | hg
-        · exact Finset.mem_filter.mpr ⟨(Finset.mem_filter.mp hg).1,
-            (Finset.mem_filter.mp hg).2.trans (Nat.le_of_lt hlt)⟩
-        · exact Finset.mem_filter.mpr ⟨(Finset.mem_filter.mp hg).1,
-            Nat.le_of_lt (Finset.mem_filter.mp hg).2.2⟩
-  change (P.filter _).card = A.card+B.card+1
-  rw [hset,Finset.card_insert_of_notMem hnot,Finset.card_union_of_disjoint hdis]
-
-theorem bridgeInterval_card_le (G : BondGraph V E) (ω : Set E)
-    (x z : V) (hC : (cluster G ω x).Finite) {e f : E}
-    (he : OpenMarkedPivotal G ω {z} x e) (hf : OpenMarkedPivotal G ω {z} x f)
-    (a b c d : V) (hab : G e = s(a,b)) (hcd : G f = s(c,d))
-    (ha : Conn G (ω \ {e}) x a) (hd : ¬Conn G (ω \ {f}) x d)
-    (hdz : Conn G (ω \ {f}) d z)
-    (hbc : Conn G (ω ∩ internalEdges G
-      (cluster G (ω \ {f}) x \ cluster G (ω \ {e}) x)) b c) :
-    (bridgeInterval G ω x z e f).card ≤ pivotalDistance G (ω ∩ internalEdges G
-      (cluster G (ω \ {f}) x \ cluster G (ω \ {e}) x)) b c := by
-  rw [← separatingBonds_card]
-  apply Finset.card_le_card
-  intro g hg
-  obtain ⟨hgP,heg,hgf⟩ := Finset.mem_filter.mp hg
-  apply (mem_separatingBonds _ _ _ _ _).mpr
-  exact pivotal_between_is_internal G ω x z hC he hf
-    ((mem_separatingBonds G ω x z g).mp hgP) heg hgf a b c d hab hcd ha hd hdz hbc
-
- 
-theorem bridge_gap_le_internal_distance (G : BondGraph V E) (ω : Set E)
-    (x z : V) (hC : (cluster G ω x).Finite) {e f : E}
-    (he : OpenMarkedPivotal G ω {z} x e) (hf : OpenMarkedPivotal G ω {z} x f)
-    (hlt : pivotalRank G ω x e < pivotalRank G ω x f)
-    (a b c d : V) (hab : G e = s(a,b)) (hcd : G f = s(c,d))
-    (ha : Conn G (ω \ {e}) x a) (hd : ¬Conn G (ω \ {f}) x d)
-    (hdz : Conn G (ω \ {f}) d z)
-    (hbc : Conn G (ω ∩ internalEdges G
-      (cluster G (ω \ {f}) x \ cluster G (ω \ {e}) x)) b c) :
-    bridgeIndex G ω x z f - bridgeIndex G ω x z e ≤
-      1 + pivotalDistance G (ω ∩ internalEdges G
-        (cluster G (ω \ {f}) x \ cluster G (ω \ {e}) x)) b c := by
-  have hid := bridgeIndex_interval_identity G ω x z hC hf hlt
-  have hle := bridgeInterval_card_le G ω x z hC he hf a b c d hab hcd ha hd hdz hbc
-  omega
-
- 
-theorem bridge_terminal_gap_le_internal_distance (G : BondGraph V E) (ω : Set E)
-    (x z : V) (hC : (cluster G ω x).Finite) {e : E}
-    (he : OpenMarkedPivotal G ω {z} x e)
-    (a b : V) (hab : G e = s(a,b)) (ha : Conn G (ω \ {e}) x a)
-    (hbz : Conn G (ω ∩ internalEdges G
-      (cluster G ω x \ cluster G (ω \ {e}) x)) b z) :
-    (separatingBonds G ω x z).card + 1 - bridgeIndex G ω x z e ≤
-      1 + pivotalDistance G (ω ∩ internalEdges G
-        (cluster G ω x \ cluster G (ω \ {e}) x)) b z := by
-  let P := separatingBonds G ω x z
-  let A := P.filter fun g => pivotalRank G ω x g ≤ pivotalRank G ω x e
-  let B := P.filter fun g => pivotalRank G ω x e < pivotalRank G ω x g
-  have hid : A.card + B.card = P.card := by
-    simpa only [A,B,not_le] using Finset.card_filter_add_card_filter_not
-      (s := P) (fun g => pivotalRank G ω x g ≤ pivotalRank G ω x e)
-  have hle : B.card ≤ pivotalDistance G (ω ∩ internalEdges G
-      (cluster G ω x \ cluster G (ω \ {e}) x)) b z := by
-    rw [← separatingBonds_card]
-    apply Finset.card_le_card
-    intro g hg
-    obtain ⟨hgP,heg⟩ := Finset.mem_filter.mp hg
-    apply (mem_separatingBonds _ _ _ _ _).mpr
-    exact pivotal_after_is_internal G ω x z hC he
-      ((mem_separatingBonds G ω x z g).mp hgP) heg a b hab ha hbz
-  change P.card+1-A.card ≤ _
-  omega
-
- 
-theorem pivotalDistance_le_cluster_card (G : BondGraph V E) (ω : Set E) (x z : V)
-    (hC : (cluster G ω x).Finite) :
-    pivotalDistance G ω x z ≤ (cluster G ω x).ncard - 1 := by
-  rw [← separatingBonds_card]
-  let N := (cluster G ω x).ncard
-  have hm : Set.MapsTo (pivotalRank G ω x) (separatingBonds G ω x z : Set E)
-      (Finset.Ico 1 N : Set ℕ) := by
-    intro e he
-    have he' := (mem_separatingBonds G ω x z e).mp he
-    have hfin : (cluster G (ω \ {e}) x).Finite :=
-      hC.subset (fun _ h => conn_mono G Set.sdiff_subset h)
-    have hpos : 0 < pivotalRank G ω x e :=
-      (Set.ncard_pos hfin).mpr ⟨x,conn_refl G _ x⟩
-    have hlt : pivotalRank G ω x e < N := by
-      apply Set.ncard_lt_ncard _ hC
-      refine Set.ssubset_iff_subset_ne.mpr ⟨fun v hv => conn_mono G Set.sdiff_subset hv,?_⟩
-      intro heq
-      obtain ⟨v,hv,hxv⟩ := he'.2.1
-      apply he'.2.2
-      refine ⟨v,hv,?_⟩
-      change v ∈ cluster G (ω \ {e}) x
-      rw [heq]
-      exact hxv
-    exact Finset.mem_Ico.mpr ⟨hpos,hlt⟩
-  have hi : Set.InjOn (pivotalRank G ω x) (separatingBonds G ω x z : Set E) := by
-    intro e he f hf h
-    exact pivotalRank_injective_of_cluster_finite G ω {z} x hC
-      ((mem_separatingBonds G ω x z e).mp he) ((mem_separatingBonds G ω x z f).mp hf) h
-  simpa only [Nat.card_Ico,N] using Finset.card_le_card_of_injOn (pivotalRank G ω x) hm hi
-
- 
-
-theorem inverse_contact_weight_le (K l : ℕ) (L : ℝ) (hK : 1 ≤ K) (hl : 1 ≤ l)
-    (hL : 1 ≤ L) (hgap : (l:ℝ) ≤ 1 + K*L) :
-    (K:ℝ)⁻¹ ≤ min 1 (2*L/l) := by
-  have hKr : (1:ℝ) ≤ K := by exact_mod_cast hK
-  have hlr : (1:ℝ) ≤ l := by exact_mod_cast hl
-  have hKpos : (0:ℝ) < K := lt_of_lt_of_le zero_lt_one hKr
-  have hlpos : (0:ℝ) < l := lt_of_lt_of_le zero_lt_one hlr
-  apply le_min
-  · exact inv_le_one_of_one_le₀ hKr
-  · apply (le_div_iff₀ hlpos).mpr
-    apply (inv_mul_le_iff₀ hKpos).mpr
-    by_cases h1 : l = 1
-    · simp only [h1,Nat.cast_one]
-      nlinarith
-    · have h2 : (2:ℝ) ≤ l := by exact_mod_cast (show 2 ≤ l by omega)
-      nlinarith
-
-end BridgeGapCount
-
-end Percolation
-
- 
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section BridgePiecePath
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem conn_restrict_cluster_of_subset (G : BondGraph V E) (ω : Set E) (x : V)
-    {η : Set E} (hη : η ⊆ ω) {a b : V} (ha : a ∈ cluster G ω x)
-    (hab : Conn G η a b) : Conn G (η ∩ internalEdges G (cluster G ω x)) a b := by
-  induction hab with
-  | refl => exact conn_refl G _ a
-  | @tail u v hau huv ih =>
-    obtain ⟨e,he,heq⟩ := huv
-    have hu : u ∈ cluster G ω x := ha.trans (conn_mono G hη hau)
-    have hv : v ∈ cluster G ω x := (edge_cluster_sides G ω x (hη he) heq).mp hu
-    exact ih.tail ⟨e,⟨he,(mem_internalEdges_of_ends G _ e u v heq).mpr ⟨hu,hv⟩⟩,heq⟩
-
- 
-
-theorem conn_between_pivotals_internal (G : BondGraph V E) (ω : Set E)
-    (x z : V) (hC : (cluster G ω x).Finite) {e f : E}
-    (he : OpenMarkedPivotal G ω {z} x e) (hf : OpenMarkedPivotal G ω {z} x f)
-    (hlt : pivotalRank G ω x e < pivotalRank G ω x f)
-    (a b c d : V) (hab : G e = s(a,b)) (hcd : G f = s(c,d))
-    (ha : Conn G (ω \ {e}) x a) (hb : ¬Conn G (ω \ {e}) x b)
-    (hc : Conn G (ω \ {f}) x c) :
-    Conn G (ω ∩ internalEdges G
-      (cluster G (ω \ {f}) x \ cluster G (ω \ {e}) x)) b c := by
-  have hne : e ≠ f := fun h => (Nat.ne_of_lt hlt) (congrArg (pivotalRank G ω x) h)
-  have hout := (pivotalRank_lt_iff_of_cluster_finite G ω {z} x hC he hf).mp hlt
-  have hcnot : c ∉ cluster G (ω \ {e}) x :=
-    ((mem_internalEdges_of_ends G _ f c d hcd).mp hout).1
-  have hins : insert e ((ω \ {e}) \ {f}) = ω \ {f} := by
-    ext q
-    simp only [Set.mem_insert_iff,Set.mem_sdiff,Set.mem_singleton_iff]
-    constructor
-    · rintro (rfl | ⟨⟨hq,hqe⟩,hqf⟩)
-      · exact ⟨he.1,hne⟩
-      · exact ⟨hq,hqf⟩
-    · rintro ⟨hq,hqf⟩
-      by_cases hqe : q = e
-      · exact Or.inl hqe
-      · exact Or.inr ⟨⟨hq,hqe⟩,hqf⟩
-  have hbc : Conn G ((ω \ {e}) \ {f}) b c := by
-    rw [← hins,conn_insert_iff G _ e a b x c hab] at hc
-    rcases hc with h | ⟨_,h⟩ | ⟨h,_⟩
-    · exact (hcnot (conn_mono G Set.sdiff_subset h)).elim
-    · exact h
-    · exact (hb (conn_mono G Set.sdiff_subset h)).elim
-  have hbo : b ∈ cluster G (ω \ {f}) x :=
-    conn_far_end_avoids_later_pivotal G ω x z hC he hf hlt a b hab ha
-  have hp := conn_restrict_cluster_compl_of_subset G (ω \ {e}) x Set.sdiff_subset hb hbc
-  have hp' := conn_restrict_cluster_of_subset G (ω \ {f}) x
-    (η := ((ω \ {e}) \ {f}) ∩ internalEdges G (cluster G (ω \ {e}) x)ᶜ)
-    (fun _ h => ⟨h.1.1.1,h.1.2⟩) hbo hp
-  apply conn_mono G ?_ hp'
-  intro q hq
-  exact ⟨hq.1.1.1.1,fun v hv => ⟨hq.2 v hv,hq.1.2 v hv⟩⟩
-
- 
-theorem conn_after_pivotal_internal (G : BondGraph V E) (ω : Set E)
-    (x z : V) {e : E} (he : OpenMarkedPivotal G ω {z} x e)
-    (a b : V) (hab : G e = s(a,b)) (ha : Conn G (ω \ {e}) x a)
-    (hb : ¬Conn G (ω \ {e}) x b) (hbz : Conn G (ω \ {e}) b z) :
-    Conn G (ω ∩ internalEdges G (cluster G ω x \ cluster G (ω \ {e}) x)) b z := by
-  have hbo : b ∈ cluster G ω x :=
-    (conn_mono G Set.sdiff_subset ha).tail ⟨e,he.1,hab⟩
-  have hp := conn_restrict_cluster_compl G (ω \ {e}) x hb hbz
-  have hp' := conn_restrict_cluster_of_subset G ω x
-    (η := (ω \ {e}) ∩ internalEdges G (cluster G (ω \ {e}) x)ᶜ)
-    (fun _ h => h.1.1) hbo hp
-  apply conn_mono G ?_ hp'
-  intro q hq
-  exact ⟨hq.1.1.1,fun v hv => ⟨hq.2 v hv,hq.1.2 v hv⟩⟩
-
- 
-theorem pivotalDistance_le_induced_card (G : BondGraph V E) (ω : Set E)
-    (B : Set V) (hB : B.Finite) {a b : V} (ha : a ∈ B) :
-    pivotalDistance G (ω ∩ internalEdges G B) a b ≤ B.ncard-1 := by
-  have hsub : cluster G (ω ∩ internalEdges G B) a ⊆ B :=
-    fun _ h => conn_internal_endpoint G B ha h
-  exact (pivotalDistance_le_cluster_card G (ω ∩ internalEdges G B) a b (hB.subset hsub)).trans
-    (Nat.sub_le_sub_right (Set.ncard_le_ncard hsub hB) 1)
-
-end BridgePiecePath
-
-end Percolation
-
- 
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section SplicePieceGeometry
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem conn_after_pivotal_of_orientation (G : BondGraph V E) (ω : Set E)
-    (x z : V) {e : E} (he : OpenMarkedPivotal G ω {z} x e)
-    (a b : V) (hab : G e = s(a,b)) (hb : ¬Conn G (ω \ {e}) x b) :
-    Conn G (ω \ {e}) b z := by
-  have hins : insert e (ω \ {e}) = ω := by
-    ext q
-    simp only [Set.mem_insert_iff,Set.mem_sdiff,Set.mem_singleton_iff]
-    by_cases hqe : q = e <;> simp_all [he.1]
-  obtain ⟨w,hw,hxw⟩ := he.2.1
-  have hw' : w = z := Set.mem_singleton_iff.mp hw
-  subst w
-  rw [← hins,conn_insert_iff G _ e a b x z hab] at hxw
-  rcases hxw with h | ⟨_,h⟩ | ⟨h,_⟩
-  · exact (he.2.2 ⟨z,rfl,h⟩).elim
-  · exact h
-  · exact (hb h).elim
-
- 
-theorem SpliceTrace.internal_agreement (G : BondGraph V E) (x : V) (η : ℕ → Set E)
-    (y : ℕ → V) (k j : ℕ) (ω ωf : Set E) (steps : List (ℕ × E))
-    (h : SpliceTrace G x η y k j ω steps ωf) :
-    ∀ q ∈ internalEdges G (cluster G ω x), q ∈ ωf ↔ q ∈ ω := by
-  induction h with
-  | stop => exact fun _ _ => Iff.rfl
-  | @step j i ω e a b rest ωf _ _ _ _ hy he hab ha hb ht ih =>
-    intro q hq
-    let S := cluster G ω x
-    let B := cluster G (η i ∩ internalEdges G Sᶜ) (y i)
-    let ω' := spliceField G S B e ω (η i)
-    have hB : B ⊆ Sᶜ := slit_cluster_subset G S (η i) (y i) hy
-    have hbS : b ∉ S := hB hb
-    have hqq : q ∈ internalEdges G (cluster G ω' x) := by
-      rw [splice_cluster G ω (η i) x (y i) hy e a b hab he ha hb]
-      exact fun v hv => Or.inl (hq v hv)
-    rw [ih q hqq]
-    change q ∈ internalEdges G S at hq
-    have hh := (splice_internal_fields G S B e ω (η i) a b hab hbS hB).1
-    exact Iff.of_eq (show (q ∈ ω') = (q ∈ ω) from by
-      have := congrArg (fun A : Set E => q ∈ A) hh
-      simpa only [Set.mem_inter_iff,hq,and_true] using this)
-
- 
-theorem SpliceTrace.head_cut (G : BondGraph V E) (x : V) (η : ℕ → Set E)
-    (y : ℕ → V) (k j i : ℕ) (ω ωf : Set E) (e : E) (rest : List (ℕ × E))
-    (h : SpliceTrace G x η y k j ω ((i,e)::rest) ωf) :
-    cluster G (ωf \ {e}) x = cluster G ω x := by
-  cases h with
-  | step j i ω e a b rest ωf _ _ _ hmax hy he hab ha hb ht =>
-    have hc := splice_close_seam_cluster G ω (η i) x (y i) hy e a b hab he ha hb
-    rw [SpliceTrace.preserve_cut G x η y k i _ ωf rest ht e]
-    · exact hc
-    · intro l hil hlk
-      rw [hc]
-      exact hmax l hil hlk
-
- 
-theorem contactSet_finite (G : BondGraph V E) (hloc : LocallyFinite G)
-    (ω : Set E) (B : Set V) (hB : B.Finite) : (contactSet G ω B).Finite := by
-  let W := hB.toFinset
-  have hWB : (W : Set V) = B := hB.coe_toFinset
-  apply (incidentSupport G hloc W).finite_toSet.subset
-  rintro e ⟨_,a,b,hab,ha,_⟩
-  apply (mem_incidentSupport G hloc W e).mpr
-  refine ⟨a,?_,?_⟩
-  · change a ∈ (W : Set V)
-    rw [hWB]
-    exact ha
-  · simp [hab]
-
- 
-
-theorem goodForScale_piece (G : BondGraph V E) (ω : Set E) (x : V) (L : ℝ)
-    (hg : GoodForScale G ω x L) (B : Set V) (hB : B.Finite)
-    (hBC : B ⊆ cluster G ω x)
-    (hconn : ∀ a ∈ B, ∀ b ∈ B, Conn G (ω ∩ internalEdges G B) a b)
-    (hK : 0 < (contactSet G ω B).ncard) {a b : V} (ha : a ∈ B) (hb : b ∈ B) :
-    (pivotalDistance G (ω ∩ internalEdges G B) a b : ℝ) ≤
-      (contactSet G ω B).ncard*L := by
-  let W := hB.toFinset
-  have hWB : (W : Set V) = B := hB.coe_toFinset
-  have hh := hg W (by simpa only [hWB] using hBC)
-    (by simpa only [← Finset.mem_coe,hWB] using hconn)
-    (by simpa only [hWB] using hK) a (by simpa only [← Finset.mem_coe,hWB] using ha)
-    b (by simpa only [← Finset.mem_coe,hWB] using hb)
-  simpa only [hWB] using hh
-
- 
-theorem slit_piece_connected (G : BondGraph V E) (η : Set E) (S : Set V) (y : V)
-    (_hy : y ∉ S) :
-    let B := cluster G (η ∩ internalEdges G Sᶜ) y
-    ∀ a ∈ B, ∀ b ∈ B, Conn G (η ∩ internalEdges G B) a b := by
-  intro B a ha b hb
-  have hc := conn_restrict_cluster G (η ∩ internalEdges G Sᶜ) y ha
-    (conn_trans G _ (conn_symm G _ ha) hb)
-  apply conn_mono G ?_ hc
-  exact fun _ h => ⟨h.1.1,h.2⟩
-
-end SplicePieceGeometry
-
-end Percolation
-
- 
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section SpliceGapBound
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-noncomputable def traceHeadGap (G : BondGraph V E) (ωf : Set E) (x z : V)
-    (e : E) (rest : List (ℕ × E)) : ℕ :=
-  (match rest with
-    | [] => (separatingBonds G ωf x z).card + 1
-    | (_,f)::_ => bridgeIndex G ωf x z f) - bridgeIndex G ωf x z e
-
- 
-
-theorem SpliceTrace.head_gap_bounds (G : BondGraph V E) (hloc : LocallyFinite G)
-    (x : V) (η : ℕ → Set E) (y : ℕ → V) (k j i M : ℕ)
-    (ω ωf : Set E) (e : E) (rest : List (ℕ × E))
-    (h : SpliceTrace G x η y k j ω ((i,e)::rest) ωf)
-    (hCf : (cluster G ωf x).Finite)
-    (hCi : (cluster G (η i) (y i)).Finite)
-    (hM : (cluster G (η i) (y i)).ncard ≤ M)
-    (L : ℝ) (hgood : GoodForScale G (η i) (y i) L) :
-    let B := cluster G (η i ∩ internalEdges G (cluster G ω x)ᶜ) (y i)
-    let K := (contactSet G (η i) B).ncard
-    let l := traceHeadGap G ωf x (y k) e rest
-    1 ≤ K ∧ 1 ≤ l ∧ l ≤ M ∧ (l:ℝ) ≤ 1 + K*L := by
-  have hfull := h
-  cases h with
-  | step j i ω e a b rest ωf hn _ _ _ hy he hab ha hb ht =>
-    let S := cluster G ω x
-    let B := cluster G (η i ∩ internalEdges G Sᶜ) (y i)
-    let ω' := spliceField G S B e ω (η i)
-    let K := (contactSet G (η i) B).ncard
-    change 1 ≤ K ∧ 1 ≤ traceHeadGap G ωf x (y k) e rest ∧
-      traceHeadGap G ωf x (y k) e rest ≤ M ∧
-      (traceHeadGap G ωf x (y k) e rest:ℝ) ≤ 1+K*L
-    have hBo : B ⊆ Sᶜ := slit_cluster_subset G S (η i) (y i) hy
-    have hBC : B ⊆ cluster G (η i) (y i) := fun _ h => conn_mono G Set.inter_subset_left h
-    have hB : B.Finite := hCi.subset hBC
-    have hKB : (contactSet G (η i) B).Finite := contactSet_finite G hloc (η i) B hB
-    have heK : e ∈ contactSet G (η i) B :=
-      ⟨he,b,a,hab.trans (Sym2.eq_swap ..),hb,fun h => hBo h ha⟩
-    have hK : 0 < K := (Set.ncard_pos hKB).mpr ⟨e,heK⟩
-    have hBpos : 0 < B.ncard := (Set.ncard_pos hB).mpr ⟨b,hb⟩
-    have hBM : B.ncard ≤ M := (Set.ncard_le_ncard hBC hCi).trans hM
-    have hcluster : cluster G ω' x = S ∪ B :=
-      splice_cluster G ω (η i) x (y i) hy e a b hab he ha hb
-    have hcut : cluster G (ωf \ {e}) x = S :=
-      SpliceTrace.head_cut G x η y k j i ω ωf e rest hfull
-    have hpe : OpenMarkedPivotal G ωf {y k} x e :=
-      SpliceTrace.seams_pivotal G x η y k j ω ωf ((i,e)::rest) hfull (i,e) (by simp)
-    have hxa : Conn G (ωf \ {e}) x a := by
-      change a ∈ cluster G (ωf \ {e}) x
-      rw [hcut]
-      exact ha
-    have hxb : ¬Conn G (ωf \ {e}) x b := by
-      change b ∉ cluster G (ωf \ {e}) x
-      rw [hcut]
-      exact hBo hb
-    have hbits : ωf ∩ internalEdges G B = η i ∩ internalEdges G B := by
-      have hii := (splice_internal_fields G S B e ω (η i) a b hab (hBo hb) hBo).2
-      ext q
-      by_cases hq : q ∈ internalEdges G B
-      · have hq' : q ∈ internalEdges G (cluster G ω' x) := by
-          rw [hcluster]
-          exact fun v hv => Or.inr (hq v hv)
-        have hiff := SpliceTrace.internal_agreement G x η y k i ω' ωf rest ht q hq'
-        have hbit := congrArg (fun A : Set E => q ∈ A) hii
-        simpa only [Set.mem_inter_iff,hq,and_true,hiff] using Iff.of_eq hbit
-      · simp only [Set.mem_inter_iff,hq,and_false]
-    have hconn := slit_piece_connected G (η i) S (y i) hy
-    have finish (l : ℕ) (c : V) (hc : c ∈ B) (hl : 1 ≤ l)
-        (hg : l ≤ 1 + pivotalDistance G (ωf ∩ internalEdges G B) b c) :
-        1 ≤ l ∧ l ≤ M ∧ (l:ℝ) ≤ 1+K*L := by
-      rw [hbits] at hg
-      have hp := pivotalDistance_le_induced_card G (η i) B hB hb (b := c)
-      have hg' : (l:ℝ) ≤ 1+(pivotalDistance G (η i ∩ internalEdges G B) b c:ℝ) := by
-        exact_mod_cast hg
-      have hgood' := goodForScale_piece G (η i) (y i) L hgood B hB hBC hconn hK hb hc
-      refine ⟨hl,by omega,?_⟩
-      change (l:ℝ) ≤ 1+(contactSet G (η i) B).ncard*L
-      linarith
-    refine ⟨hK,?_⟩
-    cases rest with
-    | nil =>
-      have hfω : ωf = ω' := by cases ht; rfl
-      have hdiff : cluster G ωf x \ cluster G (ωf \ {e}) x = B := by
-        rw [hcut,hfω,hcluster]
-        ext v
-        constructor
-        · rintro ⟨hv,hnv⟩
-          exact hv.resolve_left hnv
-        · intro hv
-          exact ⟨Or.inr hv,hBo hv⟩
-      have hyk : y k ∈ B := by
-        have hz := SpliceTrace.endpoint G x η y k j ω ωf [(i,e)] hfull
-        rw [hfω,hcluster] at hz
-        exact hz.resolve_left hn
-      have hconnz := conn_after_pivotal_of_orientation G ωf x (y k) hpe a b hab hxb
-      have hp := conn_after_pivotal_internal G ωf x (y k) hpe a b hab hxa hxb hconnz
-      have hg := bridge_terminal_gap_le_internal_distance G ωf x (y k) hCf hpe a b hab hxa hp
-      rw [hdiff] at hg
-      apply finish _ (y k) hyk _ hg
-      have hcard : bridgeIndex G ωf x (y k) e ≤ (separatingBonds G ωf x (y k)).card :=
-        Finset.card_filter_le _ _
-      omega
-    | cons q more =>
-      obtain ⟨l,f⟩ := q
-      have hcutf : cluster G (ωf \ {f}) x = S ∪ B := by
-        rw [SpliceTrace.head_cut G x η y k i l ω' ωf f more ht,hcluster]
-      have hss : cluster G (ωf \ {e}) x ⊂ cluster G (ωf \ {f}) x := by
-        rw [hcut,hcutf]
-        exact Set.ssubset_iff_subset_ne.mpr ⟨Set.subset_union_left,fun heq =>
-          hBo hb (heq.symm ▸ (show b ∈ S ∪ B from Or.inr hb))⟩
-      have hfFin : (cluster G (ωf \ {f}) x).Finite :=
-        hCf.subset (fun _ h => conn_mono G Set.sdiff_subset h)
-      have hr : pivotalRank G ωf x e < pivotalRank G ωf x f := Set.ncard_lt_ncard hss hfFin
-      have hpf : OpenMarkedPivotal G ωf {y k} x f :=
-        SpliceTrace.seams_pivotal G x η y k j ω ωf ((i,e)::(l,f)::more) hfull (l,f) (by simp)
-      obtain ⟨c,d,hcd,hxc,hxd,hdyk⟩ := openMarkedPivotal_oriented G ωf {y k} x hpf
-      have hdy : Conn G (ωf \ {f}) d (y k) := by
-        simpa only [MarkedEvent,Set.mem_singleton_iff,exists_eq_left] using hdyk
-      have hdiff : cluster G (ωf \ {f}) x \ cluster G (ωf \ {e}) x = B := by
-        rw [hcut,hcutf]
-        ext v
-        constructor
-        · rintro ⟨hv,hnv⟩
-          exact hv.resolve_left hnv
-        · intro hv
-          exact ⟨Or.inr hv,hBo hv⟩
-      have hc : c ∈ B := by
-        have hfout := (pivotalRank_lt_iff_of_cluster_finite G ωf {y k} x hCf hpe hpf).mp hr
-        have hcnot := ((mem_internalEdges_of_ends G _ f c d hcd).mp hfout).1
-        rw [hcut] at hcnot
-        change c ∈ cluster G (ωf \ {f}) x at hxc
-        rw [hcutf] at hxc
-        exact hxc.resolve_left hcnot
-      have hp := conn_between_pivotals_internal G ωf x (y k) hCf hpe hpf hr
-        a b c d hab hcd hxa hxb (by change c ∈ cluster G (ωf \ {f}) x; rw [hcutf]; exact Or.inr hc)
-      have hg := bridge_gap_le_internal_distance G ωf x (y k) hCf hpe hpf hr
-        a b c d hab hcd hxa hxd hdy hp
-      rw [hdiff] at hg
-      apply finish _ c hc _ hg
-      have hind := bridgeIndex_interval_identity G ωf x (y k) hCf hpf hr
-      omega
-
-end SpliceGapBound
-
-end Percolation
-
- 
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section GapCode
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem SpliceTrace.seam_data (G : BondGraph V E) (x : V) (η : ℕ → Set E)
-    (y : ℕ → V) (k j : ℕ) (ω ωf : Set E) (steps : List (ℕ × E))
-    (h : SpliceTrace G x η y k j ω steps ωf) :
-    ∀ q ∈ steps, j < q.1 ∧ q.1 ≤ k ∧ q.2 ∈ η q.1 ∧
-      q.2 ∈ internalEdges G (cluster G (η q.1) (y q.1)) := by
-  induction h with
-  | stop => simp
-  | @step j i ω e a b rest ωf _ hji hik _ _ he hab _ hb ht ih =>
-    intro q hq
-    rcases List.mem_cons.mp hq with rfl | hq
-    · have hb' : b ∈ cluster G (η i) (y i) := conn_mono G Set.inter_subset_left hb
-      exact ⟨hji,hik,he,(mem_internalEdges_of_ends G _ e a b hab).mpr
-        ⟨(edge_cluster_sides G (η i) (y i) he hab).mpr hb',hb'⟩⟩
-    · obtain ⟨hqi,hqk,hqe,hqC⟩ := ih q hq
-      exact ⟨hji.trans hqi,hqk,hqe,hqC⟩
-
- 
-theorem SpliceTrace.head_index_lt (G : BondGraph V E) (x : V) (η : ℕ → Set E)
-    (y : ℕ → V) (k j i : ℕ) (ω ωf : Set E) (e : E) (rest : List (ℕ × E))
-    (h : SpliceTrace G x η y k j ω ((i,e)::rest) ωf)
-    (hC : (cluster G ωf x).Finite) :
-    ∀ q ∈ rest, bridgeIndex G ωf x (y k) e < bridgeIndex G ωf x (y k) q.2 := by
-  have hfull := h
-  have hc := SpliceTrace.head_cut G x η y k j i ω ωf e rest h
-  have hseams := SpliceTrace.seams_pivotal G x η y k j ω ωf ((i,e)::rest) h
-  cases h with
-  | step j i ω e a b rest ωf _ _ _ hmax _ _ _ _ _ ht =>
-    intro q hq
-    obtain ⟨hqi,hqk,_,hqC⟩ := SpliceTrace.seam_data G x η y k i _ ωf rest ht q hq
-    have hout : q.2 ∈ internalEdges G (cluster G (ωf \ {e}) x)ᶜ := by
-      rw [hc]
-      intro v hv hvS
-      exact Set.disjoint_left.mp (hmax q.1 hqi hqk) (hqC v hv) hvS
-    have hpe := hseams (i,e) (by simp)
-    have hpq := hseams q (List.mem_cons_of_mem _ hq)
-    exact bridgeIndex_lt_of_rank_lt G ωf x (y k) hpq
-      ((pivotalRank_lt_iff_of_cluster_finite G ωf {y k} x hC hpe hpq).mpr hout)
-
- 
-noncomputable def gapSpliceCode (G : BondGraph V E) (ωf : Set E) (x z : V) :
-    List (ℕ × E) → List (ℕ × ℕ)
-  | [] => []
-  | (i,e)::rest => (i,traceHeadGap G ωf x z e rest)::gapSpliceCode G ωf x z rest
-
- 
-def decodeGapOrdinals (N : ℕ) : List (ℕ × ℕ) → List (ℕ × ℕ)
-  | [] => []
-  | (i,l)::rest =>
-    let decoded := decodeGapOrdinals N rest
-    (i, (match decoded with | [] => N+1 | (_,r)::_ => r)-l)::decoded
-
- 
-theorem SpliceTrace.decode_gap_code (G : BondGraph V E) (x : V) (η : ℕ → Set E)
-    (y : ℕ → V) (k j : ℕ) (ω ωf : Set E) (steps : List (ℕ × E))
-    (h : SpliceTrace G x η y k j ω steps ωf) (hC : (cluster G ωf x).Finite) :
-    decodeGapOrdinals (separatingBonds G ωf x (y k)).card
-      (gapSpliceCode G ωf x (y k) steps) = ordinalSpliceCode G ωf x (y k) steps := by
-  induction steps generalizing j ω with
-  | nil => rfl
-  | cons q rest ih =>
-    obtain ⟨i,e⟩ := q
-    have hlater := SpliceTrace.head_index_lt G x η y k j i ω ωf e rest h hC
-    cases h with
-    | step j i ω e a b rest ωf _ _ _ _ _ _ _ _ _ ht =>
-      have ih' := ih i _ ht
-      simp only [gapSpliceCode,decodeGapOrdinals,ih',ordinalSpliceCode,List.map_cons]
-      congr 2
-      cases rest with
-      | nil =>
-        simp only [List.map_nil,traceHeadGap]
-        have hle : bridgeIndex G ωf x (y k) e ≤ (separatingBonds G ωf x (y k)).card :=
-          Finset.card_filter_le _ _
-        omega
-      | cons q qs =>
-        obtain ⟨l,f⟩ := q
-        have hle := hlater (l,f) (by simp)
-        change bridgeIndex G ωf x (y k) e < bridgeIndex G ωf x (y k) f at hle
-        simp only [List.map_cons,traceHeadGap]
-        omega
-
- 
-theorem SpliceTrace.input_unique_of_gapCode (G : BondGraph V E) (x : V)
-    (η₁ η₂ : ℕ → Set E) (y₁ y₂ : ℕ → V) (k j : ℕ) (ω₁ ω₂ ωf : Set E)
-    (c₁ c₂ : List (ℕ × E))
-    (ht₁ : SpliceTrace G x η₁ y₁ k j ω₁ c₁ ωf)
-    (ht₂ : SpliceTrace G x η₂ y₂ k j ω₂ c₂ ωf)
-    (hz : y₁ k = y₂ k) (hC : (cluster G ωf x).Finite)
-    (heq : gapSpliceCode G ωf x (y₁ k) c₁ = gapSpliceCode G ωf x (y₁ k) c₂)
-    (θf : ℕ → Set E)
-    (he₁ : SpliceExecution G x (ω₁,η₁) c₁ (ωf,θf))
-    (he₂ : SpliceExecution G x (ω₂,η₂) c₂ (ωf,θf)) :
-    (ω₁,η₁) = (ω₂,η₂) := by
-  apply SpliceTrace.input_unique_of_ordinalCode G x η₁ η₂ y₁ y₂ k j ω₁ ω₂ ωf
-    c₁ c₂ ht₁ ht₂ hz hC _ θf he₁ he₂
-  rw [← SpliceTrace.decode_gap_code G x η₁ y₁ k j ω₁ ωf c₁ ht₁ hC,
-    heq,hz,SpliceTrace.decode_gap_code G x η₂ y₂ k j ω₂ ωf c₂ ht₂ hC]
-
-end GapCode
-
-end Percolation
-
- 
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section InverseCodeMass
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-noncomputable def gapWeight (L : ℝ) (l : ℕ) : ℝ := min 1 (2*L/l)
-
- 
-noncomputable def inverseCodeMass (M : ℕ) (L : ℝ) : ℝ :=
-  1 + ∑ l ∈ Finset.Icc 1 M, gapWeight L l
-
- 
-noncomputable def gapLetterWeight (L : ℝ) (l : ℕ) : ℝ := if l = 0 then 1 else gapWeight L l
-
-theorem gapWeight_nonneg (L : ℝ) (hL : 0 ≤ L) (l : ℕ) : 0 ≤ gapWeight L l := by
-  exact le_min zero_le_one (by positivity)
-
-theorem gapLetterWeight_nonneg (L : ℝ) (hL : 0 ≤ L) (l : ℕ) : 0 ≤ gapLetterWeight L l := by
-  unfold gapLetterWeight
-  split_ifs
-  · exact zero_le_one
-  · exact gapWeight_nonneg L hL l
-
-theorem sum_fin_succ_eq_Icc (M : ℕ) (f : ℕ → ℝ) :
-    ∑ l : Fin M, f (l.val+1) = ∑ l ∈ Finset.Icc 1 M, f l := by
-  rw [Fin.sum_univ_eq_sum_range (fun l => f (l+1)) M]
-  apply Finset.sum_bij (fun l _ => l+1)
-  · intro l hl
-    exact Finset.mem_Icc.mpr ⟨by omega,by have := Finset.mem_range.mp hl; omega⟩
-  · intro l hl m hm h
-    omega
-  · intro l hl
-    obtain ⟨h1,hM⟩ := Finset.mem_Icc.mp hl
-    exact ⟨l-1,Finset.mem_range.mpr (by omega),by omega⟩
-  · intros
-    rfl
-
-theorem sum_gapLetterWeight (M : ℕ) (L : ℝ) :
-    ∑ l : Fin (M+1), gapLetterWeight L l.val = inverseCodeMass M L := by
-  rw [Fin.sum_univ_succ]
-  simp only [Fin.val_zero,gapLetterWeight,Fin.val_succ,Nat.add_eq_zero_iff,
-    Nat.one_ne_zero,and_false,ite_false]
-  exact congrArg (1+·) (sum_fin_succ_eq_Icc M (gapWeight L))
-
- 
-
-theorem total_gap_code_mass (k M : ℕ) (L : ℝ) :
-    ∑ c : Fin k → Fin (M+1), ∏ i : Fin k, gapLetterWeight L (c i).val =
-      inverseCodeMass M L ^ k := by
-  rw [← Fintype.prod_sum (fun (_ : Fin k) (l : Fin (M+1)) => gapLetterWeight L l.val)]
-  simp only [sum_gapLetterWeight,Finset.prod_const,Finset.card_univ,Fintype.card_fin]
-
- 
-theorem valid_gap_code_mass_le (k M : ℕ) (L : ℝ) (hL : 0 ≤ L)
-    (valid : Finset (Fin k → Fin (M+1))) :
-    ∑ c ∈ valid, ∏ i : Fin k, gapLetterWeight L (c i).val ≤ inverseCodeMass M L ^ k := by
-  rw [← total_gap_code_mass]
-  exact Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _)
-    (fun c _ _ => Finset.prod_nonneg (fun i _ => gapLetterWeight_nonneg L hL (c i).val))
-
-theorem inverseCodeMass_le_harmonic (M : ℕ) (L : ℝ) (hL : 0 ≤ L) :
-    inverseCodeMass M L ≤ 1+2*L*(1+Real.log M) := by
-  have hsum : ∑ l ∈ Finset.Icc 1 M, gapWeight L l ≤
-      2*L*∑ l ∈ Finset.Icc 1 M, (l:ℝ)⁻¹ := by
-    rw [Finset.mul_sum]
-    apply Finset.sum_le_sum
-    intro l _
-    simpa only [gapWeight,div_eq_mul_inv] using min_le_right (1:ℝ) (2*L/l)
-  have hh : ∑ l ∈ Finset.Icc 1 M, (l:ℝ)⁻¹ ≤ 1+Real.log M := by
-    simpa only [harmonic_eq_sum_Icc,Rat.cast_sum,Rat.cast_inv,Rat.cast_natCast] using
-      harmonic_le_one_add_log M
-  unfold inverseCodeMass
-  have h := hsum.trans (mul_le_mul_of_nonneg_left hh (by positivity))
-  linarith
-
- 
-theorem inverseCodeMass_le_log (M : ℕ) (L : ℝ) (hM : 2 ≤ M) (hL : 1 ≤ L) :
-    inverseCodeMass M L ≤ 8*L*Real.log M := by
-  have hlog2 : (1:ℝ)/2 ≤ Real.log 2 := by
-    have hh := Real.one_sub_inv_le_log_of_pos (show (0:ℝ) < 2 by norm_num)
-    norm_num at hh ⊢
-    exact hh
-  have hlog : (1:ℝ)/2 ≤ Real.log M := hlog2.trans
-    (Real.log_le_log (by norm_num) (by exact_mod_cast hM))
-  have hLl : L/2 ≤ L*Real.log M := by nlinarith
-  have hbound := inverseCodeMass_le_harmonic M L (by linarith)
-  nlinarith
-
- 
-def contactInterval (K a : ℕ) : Set unitInterval :=
-  {u | (a:ℝ)/K ≤ (u:ℝ) ∧ (u:ℝ) < (a+1:ℕ)/K}
-
-theorem volume_contactInterval (K a : ℕ) (ha : a < K) :
-    volume (contactInterval K a) = ENNReal.ofReal ((K:ℝ)⁻¹) := by
-  have hK : (0:ℝ) < K := by exact_mod_cast (show 0 < K by omega)
-  have hleft : (a:ℝ)/K ∈ Set.Icc (0:ℝ) 1 := by
-    refine ⟨by positivity,(div_le_one hK).mpr ?_⟩
-    exact_mod_cast (Nat.le_of_lt ha)
-  have hright : ((a+1:ℕ):ℝ)/K ∈ Set.Icc (0:ℝ) 1 := by
-    refine ⟨by positivity,(div_le_one hK).mpr ?_⟩
-    exact_mod_cast (show a+1 ≤ K by omega)
-  let lo : unitInterval := ⟨_,hleft⟩
-  let hi : unitInterval := ⟨_,hright⟩
-  have heq : contactInterval K a = Set.Ico lo hi := rfl
-  rw [heq,unitInterval.volume_Ico]
-  congr 1
-  dsimp only [lo,hi]
-  push_cast
-  field_simp
-  ring
-
- 
-
-theorem volume_contact_cube {I : Type*} [Fintype I] (K a : I → ℕ)
-    (ha : ∀ i, a i < K i) :
-    (volume : Measure (I → unitInterval)) (Set.pi Set.univ (fun i => contactInterval (K i) (a i))) =
-      ∏ i, ENNReal.ofReal ((K i:ℝ)⁻¹) := by
-  rw [MeasureTheory.volume_pi,Measure.pi_pi]
-  exact Finset.prod_congr rfl (fun i _ => volume_contactInterval (K i) (a i) (ha i))
-
-theorem real_volume_contact_cube {I : Type*} [Fintype I] (K a : I → ℕ)
-    (ha : ∀ i, a i < K i) :
-    ((volume : Measure (I → unitInterval))
-      (Set.pi Set.univ (fun i => contactInterval (K i) (a i)))).toReal =
-      ∏ i, (K i:ℝ)⁻¹ := by
-  rw [volume_contact_cube K a ha,ENNReal.toReal_prod]
-  exact Finset.prod_congr rfl (fun i _ => ENNReal.toReal_ofReal (by positivity))
-
-end InverseCodeMass
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Splicing.Geometry
+
+namespace OAI
+
+/-! Gap Codes for bond percolation. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section SpliceExecution
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+abbrev SpliceState (E : Type v) := Set E × (ℕ → Set E)
+
+ 
+
+def exchangeState (mask : Set E) (i : ℕ) (s : SpliceState E) : SpliceState E :=
+  (exchangeOn mask s.1 (s.2 i), Function.update s.2 i (exchangeOn mask (s.2 i) s.1))
+
+@[simp] theorem exchangeState_first (mask : Set E) (i : ℕ) (s : SpliceState E) :
+    (exchangeState mask i s).1 = exchangeOn mask s.1 (s.2 i) := rfl
+
+@[simp] theorem exchangeState_at (mask : Set E) (i : ℕ) (s : SpliceState E) :
+    (exchangeState mask i s).2 i = exchangeOn mask (s.2 i) s.1 := by
+  simp [exchangeState]
+
+theorem exchangeState_away (mask : Set E) (i l : ℕ) (s : SpliceState E) (hli : l ≠ i) :
+    (exchangeState mask i s).2 l = s.2 l := by
+  simp [exchangeState, Function.update_of_ne hli]
+
+theorem exchangeState_involutive (mask : Set E) (i : ℕ) :
+    Function.Involutive (exchangeState mask i (E := E)) := by
+  intro s
+  apply Prod.ext
+  · simp only [exchangeState_first, exchangeState_at]
+    exact exchangeOn_involutive mask s.1 (s.2 i)
+  · funext l
+    by_cases hli : l = i
+    · subst l
+      simp only [exchangeState_at, exchangeState_first]
+      exact exchangeOn_involutive mask (s.2 i) s.1
+    · rw [exchangeState_away _ _ _ _ hli, exchangeState_away _ _ _ _ hli]
+
+ 
+
+def undoState (G : BondGraph V E) (x : V) (q : ℕ × E) (s : SpliceState E) :
+    SpliceState E := exchangeState (recoveredMask G s.1 x q.2) q.1 s
+
+ 
+
+def ValidSpliceStep (G : BondGraph V E) (x : V) (s : SpliceState E)
+    (q : ℕ × E) (s' : SpliceState E) : Prop :=
+  ∃ y a b,
+    y ∉ cluster G s.1 x ∧ q.2 ∈ s.2 q.1 ∧ G q.2 = s(a,b) ∧
+    a ∈ cluster G s.1 x ∧
+    b ∈ cluster G (s.2 q.1 ∩ internalEdges G (cluster G s.1 x)ᶜ) y ∧
+    s' = exchangeState
+      (spliceMask G (cluster G s.1 x)
+        (cluster G (s.2 q.1 ∩ internalEdges G (cluster G s.1 x)ᶜ) y) q.2) q.1 s
+
+theorem undoState_of_valid (G : BondGraph V E) (x : V) (s s' : SpliceState E)
+    (q : ℕ × E) (h : ValidSpliceStep G x s q s') : undoState G x q s' = s := by
+  obtain ⟨y,a,b,hy,he,hab,ha,hb,rfl⟩ := h
+  unfold undoState
+  rw [show recoveredMask G
+      (exchangeState (spliceMask G (cluster G s.1 x)
+        (cluster G (s.2 q.1 ∩ internalEdges G (cluster G s.1 x)ᶜ) y) q.2) q.1 s).1 x q.2 =
+      spliceMask G (cluster G s.1 x)
+        (cluster G (s.2 q.1 ∩ internalEdges G (cluster G s.1 x)ᶜ) y) q.2 from
+    recoveredMask_splice G s.1 (s.2 q.1) x y hy q.2 a b hab he ha hb]
+  exact exchangeState_involutive _ _ s
+
+ 
+inductive SpliceExecution (G : BondGraph V E) (x : V) :
+    SpliceState E → List (ℕ × E) → SpliceState E → Prop
+  | nil (s) : SpliceExecution G x s [] s
+  | cons (s s' s'') (q qs)
+      (hstep : ValidSpliceStep G x s q s')
+      (hrest : SpliceExecution G x s' qs s'') : SpliceExecution G x s (q::qs) s''
+
+ 
+def undoExecution (G : BondGraph V E) (x : V) :
+    List (ℕ × E) → SpliceState E → SpliceState E
+  | [], s => s
+  | q::qs, s => undoState G x q (undoExecution G x qs s)
+
+theorem SpliceExecution.undo (G : BondGraph V E) (x : V)
+    (s s' : SpliceState E) (code : List (ℕ × E))
+    (h : SpliceExecution G x s code s') : undoExecution G x code s' = s := by
+  induction h with
+  | nil => rfl
+  | cons s s' s'' q qs hstep hrest ih =>
+    change undoState G x q (undoExecution G x qs s'') = s
+    rw [ih]
+    exact undoState_of_valid G x s s' q hstep
+
+ 
+
+theorem SpliceExecution.input_unique (G : BondGraph V E) (x : V)
+    (s₁ s₂ out : SpliceState E) (code : List (ℕ × E))
+    (h₁ : SpliceExecution G x s₁ code out) (h₂ : SpliceExecution G x s₂ code out) :
+    s₁ = s₂ := by
+  exact (h₁.undo G x s₁ out code).symm.trans (h₂.undo G x s₂ out code)
+
+ 
+
+theorem SpliceTrace.realize (G : BondGraph V E) (x : V) (η : ℕ → Set E)
+    (y : ℕ → V) (k j : ℕ) (ω ωf : Set E) (code : List (ℕ × E))
+    (h : SpliceTrace G x η y k j ω code ωf) :
+    ∀ θ : ℕ → Set E, (∀ l, j < l → l ≤ k → θ l = η l) →
+    ∃ θf, SpliceExecution G x (ω,θ) code (ωf,θf) := by
+  induction h with
+  | stop j ω hz =>
+    intro θ _
+    exact ⟨θ, SpliceExecution.nil _⟩
+  | @step j i ω e a b rest ωf _ hji hik _ hy he hab ha hb ht ih =>
+    intro θ hθ
+    have hθi := hθ i hji hik
+    let S := cluster G ω x
+    let B := cluster G (η i ∩ internalEdges G Sᶜ) (y i)
+    let mask := spliceMask G S B e
+    let next := exchangeState mask i (ω,θ)
+    have hnext : next.1 = spliceField G S B e ω (η i) := by
+      change exchangeOn mask ω (θ i) = exchangeOn mask ω (η i)
+      rw [hθi]
+    obtain ⟨θf,hexec⟩ := ih next.2 (by
+      intro l hil hlk
+      rw [exchangeState_away _ _ _ _ (by omega)]
+      exact hθ l (hji.trans hil) hlk)
+    refine ⟨θf, SpliceExecution.cons (ω,θ) next (ωf,θf) (i,e) rest ?_ ?_⟩
+    · refine ⟨y i,a,b,hy,?_,hab,ha,?_,?_⟩
+      · simpa only [hθi] using he
+      · simpa only [hθi] using hb
+      · dsimp only [next, mask, B, S]
+        rw [hθi]
+    · change SpliceExecution G x (next.1,next.2) rest (ωf,θf)
+      rw [hnext]
+      exact hexec
+
+end SpliceExecution
+
+end Percolation
+
+ 
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section BridgeCode
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem pivotal_mem_trail (G : BondGraph V E) (ω : Set E) (x z : V)
+    (l : List E) (hl : Along G x l z) (hopen : ∀ e ∈ l, e ∈ ω)
+    {e : E} (he : OpenMarkedPivotal G ω {z} x e) : e ∈ l := by
+  by_contra hnot
+  apply he.2.2
+  refine ⟨z,rfl,hl.conn ?_⟩
+  intro f hf
+  exact ⟨hopen f hf,fun hfe => hnot ((Set.mem_singleton_iff.mp hfe) ▸ hf)⟩
+
+ 
+theorem finite_separatingBonds (G : BondGraph V E) (ω : Set E) (x z : V) :
+    {e | OpenMarkedPivotal G ω {z} x e}.Finite := by
+  by_cases hc : Conn G ω x z
+  · obtain ⟨l,hl,_,ho⟩ := conn_exists_bond_trail G hc
+    exact l.toFinset.finite_toSet.subset fun e he =>
+      List.mem_toFinset.mpr (pivotal_mem_trail G ω x z l hl ho he)
+  · have hempty : {e | OpenMarkedPivotal G ω {z} x e} = ∅ := by
+      ext e
+      simp only [Set.mem_ofPred_eq,Set.mem_empty_iff_false,iff_false]
+      rintro ⟨_,⟨y,hy,hxy⟩,_⟩
+      exact hc ((Set.mem_singleton_iff.mp hy) ▸ hxy)
+    rw [hempty]
+    exact Set.finite_empty
+
+noncomputable def separatingBonds (G : BondGraph V E) (ω : Set E) (x z : V) : Finset E :=
+  (finite_separatingBonds G ω x z).toFinset
+
+@[simp] theorem mem_separatingBonds (G : BondGraph V E) (ω : Set E) (x z : V) (e : E) :
+    e ∈ separatingBonds G ω x z ↔ OpenMarkedPivotal G ω {z} x e := by
+  exact (finite_separatingBonds G ω x z).mem_toFinset
+
+ 
+theorem pivotalRank_injective_of_cluster_finite (G : BondGraph V E) (ω : Set E)
+    (m : Set V) (x : V) (hC : (cluster G ω x).Finite) :
+    Set.InjOn (pivotalRank G ω x) {e | OpenMarkedPivotal G ω m x e} := by
+  have hcut (f : E) : (cluster G (ω \ {f}) x).Finite :=
+    hC.subset (fun _ h => conn_mono G Set.sdiff_subset h)
+  intro e he f hf heq
+  by_contra hne
+  rcases pivotal_rootSides_nested G ω m x he hf hne with h | h
+  · exact (Nat.ne_of_lt (Set.ncard_lt_ncard h (hcut f))) heq
+  · exact (Nat.ne_of_lt (Set.ncard_lt_ncard h (hcut e))) heq.symm
+
+ 
+
+noncomputable def bridgeIndex (G : BondGraph V E) (ω : Set E) (x z : V) (e : E) : ℕ :=
+  ((separatingBonds G ω x z).filter fun f => pivotalRank G ω x f ≤ pivotalRank G ω x e).card
+
+theorem bridgeIndex_pos (G : BondGraph V E) (ω : Set E) (x z : V) {e : E}
+    (he : OpenMarkedPivotal G ω {z} x e) : 0 < bridgeIndex G ω x z e := by
+  apply Finset.card_pos.mpr
+  exact ⟨e,Finset.mem_filter.mpr ⟨(mem_separatingBonds G ω x z e).mpr he,le_refl _⟩⟩
+
+theorem bridgeIndex_lt_of_rank_lt (G : BondGraph V E) (ω : Set E) (x z : V) {e f : E}
+    (hf : OpenMarkedPivotal G ω {z} x f)
+    (hlt : pivotalRank G ω x e < pivotalRank G ω x f) :
+    bridgeIndex G ω x z e < bridgeIndex G ω x z f := by
+  apply Finset.card_lt_card
+  apply Finset.ssubset_iff_subset_ne.mpr
+  refine ⟨?_,?_⟩
+  · intro d hd
+    exact Finset.mem_filter.mpr ⟨(Finset.mem_filter.mp hd).1,
+      (Finset.mem_filter.mp hd).2.trans (Nat.le_of_lt hlt)⟩
+  · intro heq
+    have hmem : f ∈ (separatingBonds G ω x z).filter
+        (fun d => pivotalRank G ω x d ≤ pivotalRank G ω x f) :=
+      Finset.mem_filter.mpr ⟨(mem_separatingBonds G ω x z f).mpr hf,le_refl _⟩
+    rw [← heq] at hmem
+    exact (Nat.not_le_of_lt hlt) (Finset.mem_filter.mp hmem).2
+
+theorem bridgeIndex_injective (G : BondGraph V E) (ω : Set E) (x z : V)
+    (hC : (cluster G ω x).Finite) :
+    Set.InjOn (bridgeIndex G ω x z) {e | OpenMarkedPivotal G ω {z} x e} := by
+  intro e he f hf heq
+  apply pivotalRank_injective_of_cluster_finite G ω {z} x hC he hf
+  rcases lt_trichotomy (pivotalRank G ω x e) (pivotalRank G ω x f) with h | h | h
+  · exact ((Nat.ne_of_lt (bridgeIndex_lt_of_rank_lt G ω x z hf h)) heq).elim
+  · exact h
+  · exact ((Nat.ne_of_lt (bridgeIndex_lt_of_rank_lt G ω x z he h)) heq.symm).elim
+
+ 
+noncomputable def ordinalSpliceCode (G : BondGraph V E) (ω : Set E) (x z : V)
+    (code : List (ℕ × E)) : List (ℕ × ℕ) :=
+  code.map fun q => (q.1,bridgeIndex G ω x z q.2)
+
+theorem ordinalSpliceCode_injective (G : BondGraph V E) (ω : Set E) (x z : V)
+    (hC : (cluster G ω x).Finite) (c₁ c₂ : List (ℕ × E))
+    (h₁ : ∀ q ∈ c₁, OpenMarkedPivotal G ω {z} x q.2)
+    (h₂ : ∀ q ∈ c₂, OpenMarkedPivotal G ω {z} x q.2)
+    (heq : ordinalSpliceCode G ω x z c₁ = ordinalSpliceCode G ω x z c₂) : c₁ = c₂ := by
+  induction c₁ generalizing c₂ with
+  | nil =>
+    cases c₂ with
+    | nil => rfl
+    | cons q qs => simp [ordinalSpliceCode] at heq
+  | cons q qs ih =>
+    cases c₂ with
+    | nil => simp [ordinalSpliceCode] at heq
+    | cons r rs =>
+      have heq' := List.cons.inj heq
+      have hqr : q = r := by
+        apply Prod.ext
+        · exact congrArg (fun t : ℕ × ℕ => t.1) heq'.1
+        · exact bridgeIndex_injective G ω x z hC (h₁ q (by simp)) (h₂ r (by simp))
+            (congrArg Prod.snd heq'.1)
+      exact congrArg₂ List.cons hqr (ih rs
+        (fun a ha => h₁ a (List.mem_cons_of_mem _ ha))
+        (fun a ha => h₂ a (List.mem_cons_of_mem _ ha)) heq'.2)
+
+ 
+
+theorem SpliceTrace.input_unique_of_ordinalCode (G : BondGraph V E) (x : V)
+    (η₁ η₂ : ℕ → Set E) (y₁ y₂ : ℕ → V) (k j : ℕ) (ω₁ ω₂ ωf : Set E)
+    (c₁ c₂ : List (ℕ × E))
+    (ht₁ : SpliceTrace G x η₁ y₁ k j ω₁ c₁ ωf)
+    (ht₂ : SpliceTrace G x η₂ y₂ k j ω₂ c₂ ωf)
+    (hz : y₁ k = y₂ k) (hC : (cluster G ωf x).Finite)
+    (heq : ordinalSpliceCode G ωf x (y₁ k) c₁ = ordinalSpliceCode G ωf x (y₁ k) c₂)
+    (θf : ℕ → Set E)
+    (he₁ : SpliceExecution G x (ω₁,η₁) c₁ (ωf,θf))
+    (he₂ : SpliceExecution G x (ω₂,η₂) c₂ (ωf,θf)) :
+    (ω₁,η₁) = (ω₂,η₂) := by
+  have hc : c₁ = c₂ := ordinalSpliceCode_injective G ωf x (y₁ k) hC c₁ c₂
+    (ht₁.seams_pivotal G x η₁ y₁ k j ω₁ ωf c₁)
+    (by simpa only [hz] using ht₂.seams_pivotal G x η₂ y₂ k j ω₂ ωf c₂) heq
+  subst c₂
+  exact he₁.input_unique G x (ω₁,η₁) (ω₂,η₂) (ωf,θf) c₁ he₂
+
+end BridgeCode
+
+end Percolation
+
+ 
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section SpliceWeight
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+noncomputable def domainBits (Q : Finset E) (ω : Set E) : Finset E := Q.filter (· ∈ ω)
+
+@[simp] theorem mem_domainBits (Q : Finset E) (ω : Set E) (e : E) :
+    e ∈ domainBits Q ω ↔ e ∈ Q ∧ e ∈ ω := Finset.mem_filter
+
+theorem domainBits_subset (Q : Finset E) (ω : Set E) : domainBits Q ω ⊆ Q :=
+  Finset.filter_subset _ _
+
+theorem domainBits_exchange (Q : Finset E) (M ω η : Set E) :
+    domainBits Q (exchangeOn M ω η) =
+      finiteExchange (domainBits Q M) (domainBits Q ω) (domainBits Q η) := by
+  ext e
+  simp only [mem_domainBits,exchangeOn,Set.mem_union,Set.mem_sdiff,Set.mem_inter_iff,
+    finiteExchange,Finset.mem_union,Finset.mem_sdiff,Finset.mem_inter]
+  tauto
+
+ 
+
+noncomputable def spliceTupleWeight (Q : Finset E) (J : Finset ℕ) (p : ℝ)
+    (s : SpliceState E) : ℝ :=
+  bernoulliWeight Q (domainBits Q s.1) p *
+    ∏ i ∈ J, bernoulliWeight Q (domainBits Q (s.2 i)) p
+
+theorem spliceTupleWeight_nonneg (Q : Finset E) (J : Finset ℕ) (p : unitInterval)
+    (s : SpliceState E) : 0 ≤ spliceTupleWeight Q J p s := by
+  exact mul_nonneg (bernoulliWeight_nonneg _ _ p)
+    (Finset.prod_nonneg fun _ _ => bernoulliWeight_nonneg _ _ p)
+
+ 
+theorem spliceTupleWeight_exchange (Q : Finset E) (J : Finset ℕ) (p : ℝ)
+    (s : SpliceState E) (M : Set E) (i : ℕ) (hi : i ∈ J) :
+    spliceTupleWeight Q J p (exchangeState M i s) = spliceTupleWeight Q J p s := by
+  unfold spliceTupleWeight
+  rw [← Finset.mul_prod_erase J (fun l =>
+    bernoulliWeight Q (domainBits Q ((exchangeState M i s).2 l)) p) hi,
+    ← Finset.mul_prod_erase J (fun l => bernoulliWeight Q (domainBits Q (s.2 l)) p) hi]
+  have herase : (∏ l ∈ J.erase i,
+      bernoulliWeight Q (domainBits Q ((exchangeState M i s).2 l)) p) =
+      ∏ l ∈ J.erase i, bernoulliWeight Q (domainBits Q (s.2 l)) p := by
+    apply Finset.prod_congr rfl
+    intro l hl
+    rw [exchangeState_away _ _ _ _ (Finset.mem_erase.mp hl).1]
+  rw [herase,exchangeState_first,exchangeState_at,← mul_assoc,← mul_assoc,
+    domainBits_exchange,domainBits_exchange,
+    bernoulliWeight_exchange _ _ _ _ (domainBits_subset Q s.1) (domainBits_subset Q (s.2 i)) p]
+
+ 
+theorem SpliceExecution.weight (G : BondGraph V E) (x : V)
+    (s s' : SpliceState E) (code : List (ℕ × E))
+    (h : SpliceExecution G x s code s') (Q : Finset E) (J : Finset ℕ)
+    (hJ : ∀ q ∈ code, q.1 ∈ J) (p : ℝ) :
+    spliceTupleWeight Q J p s' = spliceTupleWeight Q J p s := by
+  induction h with
+  | nil => rfl
+  | cons s s' s'' q code hstep hrest ih =>
+    rw [ih (fun r hr => hJ r (List.mem_cons_of_mem _ hr))]
+    obtain ⟨y,a,b,_,_,_,_,_,rfl⟩ := hstep
+    exact spliceTupleWeight_exchange Q J p s _ q.1 (hJ q (List.mem_cons_self ..))
+
+ 
+
+theorem splice_finite_weight_comparison (G : BondGraph V E) (x : V)
+    (Q : Finset E) (J : Finset ℕ) (p : unitInterval) (code : List (ℕ × E))
+    (hJ : ∀ q ∈ code, q.1 ∈ J) (A B : Finset (SpliceState E))
+    (F : SpliceState E → SpliceState E)
+    (hex : ∀ s ∈ A, SpliceExecution G x s code (F s))
+    (hB : ∀ s ∈ A, F s ∈ B) :
+    (∑ s ∈ A, spliceTupleWeight Q J p s) ≤ ∑ s ∈ B, spliceTupleWeight Q J p s := by
+  have hinj : Set.InjOn F (A : Set (SpliceState E)) := by
+    intro s hs t ht heq
+    exact (hex s hs).input_unique G x s t (F s) code (heq.symm ▸ hex t ht)
+  have hweight : (∑ s ∈ A, spliceTupleWeight Q J p s) =
+      ∑ s ∈ A.image F, spliceTupleWeight Q J p s := by
+    rw [Finset.sum_image hinj]
+    apply Finset.sum_congr rfl
+    intro s hs
+    exact ((hex s hs).weight G x s (F s) code Q J hJ p).symm
+  rw [hweight]
+  apply Finset.sum_le_sum_of_subset_of_nonneg
+  · intro t ht
+    obtain ⟨s,hs,rfl⟩ := Finset.mem_image.mp ht
+    exact hB s hs
+  · intro s _ _
+    exact spliceTupleWeight_nonneg Q J p s
+
+end SpliceWeight
+
+end Percolation
+
+ 
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section BridgeGapGeometry
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem pivotalRank_lt_iff_of_cluster_finite (G : BondGraph V E) (ω : Set E)
+    (m : Set V) (x : V) (hC : (cluster G ω x).Finite) {e f : E}
+    (he : OpenMarkedPivotal G ω m x e) (hf : OpenMarkedPivotal G ω m x f) :
+    pivotalRank G ω x e < pivotalRank G ω x f ↔
+      f ∈ internalEdges G (cluster G (ω \ {e}) x)ᶜ := by
+  have hcut (q : E) : (cluster G (ω \ {q}) x).Finite :=
+    hC.subset (fun _ h => conn_mono G Set.sdiff_subset h)
+  have hexterior (hout : f ∈ internalEdges G (cluster G (ω \ {e}) x)ᶜ) : e ≠ f := by
+    obtain ⟨a,b,hab,hxa,_,_⟩ := openMarkedPivotal_oriented G ω m x he
+    intro h; subst f
+    exact ((mem_internalEdges_of_ends G _ e a b hab).mp hout).1 hxa
+  constructor
+  · intro hlt
+    have hne : e ≠ f := fun h => (Nat.ne_of_lt hlt) (congrArg (pivotalRank G ω x) h)
+    rcases pivotal_other_sides G ω m x he hf hne.symm with hinside | hout
+    · obtain ⟨a,b,hab,_,hxb,hbm⟩ := openMarkedPivotal_oriented G ω m x he
+      obtain ⟨y,hy,hby⟩ := hbm
+      have hby' : Conn G (ω \ {f}) b y := conn_mono G
+        (ω₁ := (ω \ {e}) \ {f}) (ω₂ := ω \ {f}) (fun _ h => ⟨h.1.1,h.2⟩)
+        (conn_delete_internal_of_outside G (ω \ {e}) x hinside hxb hby)
+      have hb : b ∉ cluster G (ω \ {f}) x := fun h => hf.2.2 ⟨y,hy,h.trans hby'⟩
+      have ha : a ∉ cluster G (ω \ {f}) x := fun h =>
+        hb ((edge_cluster_sides G (ω \ {f}) x ⟨he.1,hne⟩ hab).mp h)
+      have hs := pivotal_rootSide_ssubset G ω m x hf hne.symm
+        ((mem_internalEdges_of_ends G _ e a b hab).mpr ⟨ha,hb⟩)
+      have hrev := Set.ncard_lt_ncard hs (hcut e)
+      exact (Nat.not_lt_of_ge (Nat.le_of_lt hrev) hlt).elim
+    · exact hout
+  · intro hout
+    exact Set.ncard_lt_ncard (pivotal_rootSide_ssubset G ω m x he (hexterior hout) hout) (hcut f)
+
+theorem pivotal_internal_of_rank_lt (G : BondGraph V E) (ω : Set E)
+    (m : Set V) (x : V) (hC : (cluster G ω x).Finite) {e f : E}
+    (he : OpenMarkedPivotal G ω m x e) (hf : OpenMarkedPivotal G ω m x f)
+    (hlt : pivotalRank G ω x e < pivotalRank G ω x f) :
+    e ∈ internalEdges G (cluster G (ω \ {f}) x) := by
+  have hne : e ≠ f := fun h => (Nat.ne_of_lt hlt) (congrArg (pivotalRank G ω x) h)
+  rcases pivotal_other_sides G ω m x hf he hne with h | h
+  · exact h
+  · have hr := (pivotalRank_lt_iff_of_cluster_finite G ω m x hC hf he).mpr h
+    omega
+
+ 
+theorem conn_far_end_avoids_later_pivotal (G : BondGraph V E) (ω : Set E)
+    (x z : V) (hC : (cluster G ω x).Finite) {e g : E}
+    (he : OpenMarkedPivotal G ω {z} x e) (hg : OpenMarkedPivotal G ω {z} x g)
+    (hlt : pivotalRank G ω x e < pivotalRank G ω x g)
+    (a b : V) (hab : G e = s(a,b)) (ha : Conn G (ω \ {e}) x a) :
+    Conn G (ω \ {g}) x b := by
+  have hout := (pivotalRank_lt_iff_of_cluster_finite G ω {z} x hC he hg).mp hlt
+  have hne : e ≠ g := fun h => (Nat.ne_of_lt hlt) (congrArg (pivotalRank G ω x) h)
+  have hc : Conn G (ω \ {g}) x a :=
+    conn_mono G (ω₁ := (ω \ {e}) \ {g}) (ω₂ := ω \ {g}) (fun _ h => ⟨h.1.1,h.2⟩)
+      (conn_delete_exterior_of_inside G (ω \ {e}) x hout (conn_refl G _ x) ha)
+  exact hc.tail ⟨e,⟨he.1,hne⟩,hab⟩
+
+ 
+theorem conn_near_end_avoids_earlier_pivotal (G : BondGraph V E) (ω : Set E)
+    (x z : V) (hC : (cluster G ω x).Finite) {f g : E}
+    (hf : OpenMarkedPivotal G ω {z} x f) (hg : OpenMarkedPivotal G ω {z} x g)
+    (hlt : pivotalRank G ω x g < pivotalRank G ω x f)
+    (c d : V) (hcd : G f = s(c,d)) (hd : ¬Conn G (ω \ {f}) x d)
+    (hdz : Conn G (ω \ {f}) d z) : Conn G (ω \ {g}) c z := by
+  have hin := pivotal_internal_of_rank_lt G ω {z} x hC hg hf hlt
+  have hne : f ≠ g := fun h => (Nat.ne_of_lt hlt) (congrArg (pivotalRank G ω x) h.symm)
+  have hc : Conn G (ω \ {g}) d z :=
+    conn_mono G (ω₁ := (ω \ {f}) \ {g}) (ω₂ := ω \ {g}) (fun _ h => ⟨h.1.1,h.2⟩)
+      (conn_delete_internal_of_outside G (ω \ {f}) x hin hd hdz)
+  exact (Relation.ReflTransGen.single ⟨f,⟨hf.1,hne⟩,hcd⟩).trans hc
+
+ 
+
+theorem pivotal_between_is_internal (G : BondGraph V E) (ω : Set E)
+    (x z : V) (hC : (cluster G ω x).Finite) {e f g : E}
+    (he : OpenMarkedPivotal G ω {z} x e) (hf : OpenMarkedPivotal G ω {z} x f)
+    (hg : OpenMarkedPivotal G ω {z} x g)
+    (heg : pivotalRank G ω x e < pivotalRank G ω x g)
+    (hgf : pivotalRank G ω x g < pivotalRank G ω x f)
+    (a b c d : V) (hab : G e = s(a,b)) (hcd : G f = s(c,d))
+    (ha : Conn G (ω \ {e}) x a) (hd : ¬Conn G (ω \ {f}) x d)
+    (hdz : Conn G (ω \ {f}) d z)
+    (hbc : Conn G (ω ∩ internalEdges G
+      (cluster G (ω \ {f}) x \ cluster G (ω \ {e}) x)) b c) :
+    OpenMarkedPivotal G (ω ∩ internalEdges G
+      (cluster G (ω \ {f}) x \ cluster G (ω \ {e}) x)) {c} b g := by
+  have hout := (pivotalRank_lt_iff_of_cluster_finite G ω {z} x hC he hg).mp heg
+  have hin := pivotal_internal_of_rank_lt G ω {z} x hC hg hf hgf
+  refine ⟨⟨hg.1,fun v hv => ⟨hin v hv,hout v hv⟩⟩,⟨c,rfl,hbc⟩,?_⟩
+  rintro ⟨w,hw,hbw⟩
+  have hbc' : Conn G (ω \ {g}) b c := by
+    have hw' : w = c := Set.mem_singleton_iff.mp hw
+    subst w
+    apply conn_mono G ?_ hbw
+    intro q hq
+    exact ⟨hq.1.1,hq.2⟩
+  apply hg.2.2
+  exact ⟨z,rfl,(conn_far_end_avoids_later_pivotal G ω x z hC he hg heg a b hab ha).trans
+    (hbc'.trans (conn_near_end_avoids_earlier_pivotal G ω x z hC hf hg hgf c d hcd hd hdz))⟩
+
+ 
+theorem pivotal_after_is_internal (G : BondGraph V E) (ω : Set E)
+    (x z : V) (hC : (cluster G ω x).Finite) {e g : E}
+    (he : OpenMarkedPivotal G ω {z} x e) (hg : OpenMarkedPivotal G ω {z} x g)
+    (heg : pivotalRank G ω x e < pivotalRank G ω x g)
+    (a b : V) (hab : G e = s(a,b)) (ha : Conn G (ω \ {e}) x a)
+    (hbz : Conn G (ω ∩ internalEdges G
+      (cluster G ω x \ cluster G (ω \ {e}) x)) b z) :
+    OpenMarkedPivotal G (ω ∩ internalEdges G
+      (cluster G ω x \ cluster G (ω \ {e}) x)) {z} b g := by
+  have hout := (pivotalRank_lt_iff_of_cluster_finite G ω {z} x hC he hg).mp heg
+  have hin : g ∈ internalEdges G (cluster G ω x) := by
+    obtain ⟨u,v,huv,hxu,_,_⟩ := openMarkedPivotal_oriented G ω {z} x hg
+    have hu : u ∈ cluster G ω x := conn_mono G Set.sdiff_subset hxu
+    exact (mem_internalEdges_of_ends G _ g u v huv).mpr
+      ⟨hu,(edge_cluster_sides G ω x hg.1 huv).mp hu⟩
+  refine ⟨⟨hg.1,fun v hv => ⟨hin v hv,hout v hv⟩⟩,⟨z,rfl,hbz⟩,?_⟩
+  rintro ⟨w,hw,hbw⟩
+  have hbz' : Conn G (ω \ {g}) b z := by
+    have hw' : w = z := Set.mem_singleton_iff.mp hw
+    subst w
+    apply conn_mono G ?_ hbw
+    intro q hq
+    exact ⟨hq.1.1,hq.2⟩
+  exact hg.2.2 ⟨z,rfl,
+    (conn_far_end_avoids_later_pivotal G ω x z hC he hg heg a b hab ha).trans hbz'⟩
+
+end BridgeGapGeometry
+
+end Percolation
+
+ 
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section BridgeGapCount
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+@[simp] theorem separatingBonds_card (G : BondGraph V E) (ω : Set E) (x z : V) :
+    (separatingBonds G ω x z).card = pivotalDistance G ω x z := by
+  exact Set.ncard_coe_finset (separatingBonds G ω x z) |>.symm.trans (by
+    congr 1
+    ext e
+    exact mem_separatingBonds G ω x z e)
+
+ 
+noncomputable def bridgeInterval (G : BondGraph V E) (ω : Set E) (x z : V) (e f : E) : Finset E :=
+  (separatingBonds G ω x z).filter fun g =>
+    pivotalRank G ω x e < pivotalRank G ω x g ∧ pivotalRank G ω x g < pivotalRank G ω x f
+
+theorem bridgeIndex_interval_identity (G : BondGraph V E) (ω : Set E) (x z : V)
+    (hC : (cluster G ω x).Finite) {e f : E}
+    (hf : OpenMarkedPivotal G ω {z} x f)
+    (hlt : pivotalRank G ω x e < pivotalRank G ω x f) :
+    bridgeIndex G ω x z f = bridgeIndex G ω x z e + (bridgeInterval G ω x z e f).card + 1 := by
+  let P := separatingBonds G ω x z
+  let A := P.filter fun g => pivotalRank G ω x g ≤ pivotalRank G ω x e
+  let B := bridgeInterval G ω x z e f
+  have hdis : Disjoint A B := by
+    apply Finset.disjoint_left.mpr
+    intro g hgA hgB
+    have h₁ := (Finset.mem_filter.mp hgA).2
+    have h₂ := (Finset.mem_filter.mp hgB).2.1
+    omega
+  have hnot : f ∉ A ∪ B := by
+    intro hh
+    rcases Finset.mem_union.mp hh with h | h
+    · have hr := (Finset.mem_filter.mp h).2
+      omega
+    · exact (Nat.lt_irrefl _) (Finset.mem_filter.mp h).2.2
+  have hset : P.filter (fun g => pivotalRank G ω x g ≤ pivotalRank G ω x f) = insert f (A ∪ B) := by
+    ext g
+    constructor
+    · intro hg
+      obtain ⟨hgP,hgf⟩ := Finset.mem_filter.mp hg
+      by_cases hge : g = f
+      · exact Finset.mem_insert.mpr (Or.inl hge)
+      · apply Finset.mem_insert.mpr
+        apply Or.inr
+        by_cases hga : pivotalRank G ω x g ≤ pivotalRank G ω x e
+        · exact Finset.mem_union_left _ (Finset.mem_filter.mpr ⟨hgP,hga⟩)
+        · apply Finset.mem_union_right
+          apply Finset.mem_filter.mpr
+          refine ⟨hgP,Nat.lt_of_not_ge hga,?_⟩
+          apply lt_of_le_of_ne hgf
+          intro hr
+          exact hge (pivotalRank_injective_of_cluster_finite G ω {z} x hC
+            ((mem_separatingBonds G ω x z g).mp hgP) hf hr)
+    · intro hg
+      rcases Finset.mem_insert.mp hg with heq | hg
+      · subst g
+        exact Finset.mem_filter.mpr ⟨(mem_separatingBonds G ω x z f).mpr hf,le_refl _⟩
+      · rcases Finset.mem_union.mp hg with hg | hg
+        · exact Finset.mem_filter.mpr ⟨(Finset.mem_filter.mp hg).1,
+            (Finset.mem_filter.mp hg).2.trans (Nat.le_of_lt hlt)⟩
+        · exact Finset.mem_filter.mpr ⟨(Finset.mem_filter.mp hg).1,
+            Nat.le_of_lt (Finset.mem_filter.mp hg).2.2⟩
+  change (P.filter _).card = A.card+B.card+1
+  rw [hset,Finset.card_insert_of_notMem hnot,Finset.card_union_of_disjoint hdis]
+
+theorem bridgeInterval_card_le (G : BondGraph V E) (ω : Set E)
+    (x z : V) (hC : (cluster G ω x).Finite) {e f : E}
+    (he : OpenMarkedPivotal G ω {z} x e) (hf : OpenMarkedPivotal G ω {z} x f)
+    (a b c d : V) (hab : G e = s(a,b)) (hcd : G f = s(c,d))
+    (ha : Conn G (ω \ {e}) x a) (hd : ¬Conn G (ω \ {f}) x d)
+    (hdz : Conn G (ω \ {f}) d z)
+    (hbc : Conn G (ω ∩ internalEdges G
+      (cluster G (ω \ {f}) x \ cluster G (ω \ {e}) x)) b c) :
+    (bridgeInterval G ω x z e f).card ≤ pivotalDistance G (ω ∩ internalEdges G
+      (cluster G (ω \ {f}) x \ cluster G (ω \ {e}) x)) b c := by
+  rw [← separatingBonds_card]
+  apply Finset.card_le_card
+  intro g hg
+  obtain ⟨hgP,heg,hgf⟩ := Finset.mem_filter.mp hg
+  apply (mem_separatingBonds _ _ _ _ _).mpr
+  exact pivotal_between_is_internal G ω x z hC he hf
+    ((mem_separatingBonds G ω x z g).mp hgP) heg hgf a b c d hab hcd ha hd hdz hbc
+
+ 
+theorem bridge_gap_le_internal_distance (G : BondGraph V E) (ω : Set E)
+    (x z : V) (hC : (cluster G ω x).Finite) {e f : E}
+    (he : OpenMarkedPivotal G ω {z} x e) (hf : OpenMarkedPivotal G ω {z} x f)
+    (hlt : pivotalRank G ω x e < pivotalRank G ω x f)
+    (a b c d : V) (hab : G e = s(a,b)) (hcd : G f = s(c,d))
+    (ha : Conn G (ω \ {e}) x a) (hd : ¬Conn G (ω \ {f}) x d)
+    (hdz : Conn G (ω \ {f}) d z)
+    (hbc : Conn G (ω ∩ internalEdges G
+      (cluster G (ω \ {f}) x \ cluster G (ω \ {e}) x)) b c) :
+    bridgeIndex G ω x z f - bridgeIndex G ω x z e ≤
+      1 + pivotalDistance G (ω ∩ internalEdges G
+        (cluster G (ω \ {f}) x \ cluster G (ω \ {e}) x)) b c := by
+  have hid := bridgeIndex_interval_identity G ω x z hC hf hlt
+  have hle := bridgeInterval_card_le G ω x z hC he hf a b c d hab hcd ha hd hdz hbc
+  omega
+
+ 
+theorem bridge_terminal_gap_le_internal_distance (G : BondGraph V E) (ω : Set E)
+    (x z : V) (hC : (cluster G ω x).Finite) {e : E}
+    (he : OpenMarkedPivotal G ω {z} x e)
+    (a b : V) (hab : G e = s(a,b)) (ha : Conn G (ω \ {e}) x a)
+    (hbz : Conn G (ω ∩ internalEdges G
+      (cluster G ω x \ cluster G (ω \ {e}) x)) b z) :
+    (separatingBonds G ω x z).card + 1 - bridgeIndex G ω x z e ≤
+      1 + pivotalDistance G (ω ∩ internalEdges G
+        (cluster G ω x \ cluster G (ω \ {e}) x)) b z := by
+  let P := separatingBonds G ω x z
+  let A := P.filter fun g => pivotalRank G ω x g ≤ pivotalRank G ω x e
+  let B := P.filter fun g => pivotalRank G ω x e < pivotalRank G ω x g
+  have hid : A.card + B.card = P.card := by
+    simpa only [A,B,not_le] using Finset.card_filter_add_card_filter_not
+      (s := P) (fun g => pivotalRank G ω x g ≤ pivotalRank G ω x e)
+  have hle : B.card ≤ pivotalDistance G (ω ∩ internalEdges G
+      (cluster G ω x \ cluster G (ω \ {e}) x)) b z := by
+    rw [← separatingBonds_card]
+    apply Finset.card_le_card
+    intro g hg
+    obtain ⟨hgP,heg⟩ := Finset.mem_filter.mp hg
+    apply (mem_separatingBonds _ _ _ _ _).mpr
+    exact pivotal_after_is_internal G ω x z hC he
+      ((mem_separatingBonds G ω x z g).mp hgP) heg a b hab ha hbz
+  change P.card+1-A.card ≤ _
+  omega
+
+ 
+theorem pivotalDistance_le_cluster_card (G : BondGraph V E) (ω : Set E) (x z : V)
+    (hC : (cluster G ω x).Finite) :
+    pivotalDistance G ω x z ≤ (cluster G ω x).ncard - 1 := by
+  rw [← separatingBonds_card]
+  let N := (cluster G ω x).ncard
+  have hm : Set.MapsTo (pivotalRank G ω x) (separatingBonds G ω x z : Set E)
+      (Finset.Ico 1 N : Set ℕ) := by
+    intro e he
+    have he' := (mem_separatingBonds G ω x z e).mp he
+    have hfin : (cluster G (ω \ {e}) x).Finite :=
+      hC.subset (fun _ h => conn_mono G Set.sdiff_subset h)
+    have hpos : 0 < pivotalRank G ω x e :=
+      (Set.ncard_pos hfin).mpr ⟨x,conn_refl G _ x⟩
+    have hlt : pivotalRank G ω x e < N := by
+      apply Set.ncard_lt_ncard _ hC
+      refine Set.ssubset_iff_subset_ne.mpr ⟨fun v hv => conn_mono G Set.sdiff_subset hv,?_⟩
+      intro heq
+      obtain ⟨v,hv,hxv⟩ := he'.2.1
+      apply he'.2.2
+      refine ⟨v,hv,?_⟩
+      change v ∈ cluster G (ω \ {e}) x
+      rw [heq]
+      exact hxv
+    exact Finset.mem_Ico.mpr ⟨hpos,hlt⟩
+  have hi : Set.InjOn (pivotalRank G ω x) (separatingBonds G ω x z : Set E) := by
+    intro e he f hf h
+    exact pivotalRank_injective_of_cluster_finite G ω {z} x hC
+      ((mem_separatingBonds G ω x z e).mp he) ((mem_separatingBonds G ω x z f).mp hf) h
+  simpa only [Nat.card_Ico,N] using Finset.card_le_card_of_injOn (pivotalRank G ω x) hm hi
+
+ 
+
+theorem inverse_contact_weight_le (K l : ℕ) (L : ℝ) (hK : 1 ≤ K) (hl : 1 ≤ l)
+    (hL : 1 ≤ L) (hgap : (l:ℝ) ≤ 1 + K*L) :
+    (K:ℝ)⁻¹ ≤ min 1 (2*L/l) := by
+  have hKr : (1:ℝ) ≤ K := by exact_mod_cast hK
+  have hlr : (1:ℝ) ≤ l := by exact_mod_cast hl
+  have hKpos : (0:ℝ) < K := lt_of_lt_of_le zero_lt_one hKr
+  have hlpos : (0:ℝ) < l := lt_of_lt_of_le zero_lt_one hlr
+  apply le_min
+  · exact inv_le_one_of_one_le₀ hKr
+  · apply (le_div_iff₀ hlpos).mpr
+    apply (inv_mul_le_iff₀ hKpos).mpr
+    by_cases h1 : l = 1
+    · simp only [h1,Nat.cast_one]
+      nlinarith
+    · have h2 : (2:ℝ) ≤ l := by exact_mod_cast (show 2 ≤ l by omega)
+      nlinarith
+
+end BridgeGapCount
+
+end Percolation
+
+ 
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section BridgePiecePath
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem conn_restrict_cluster_of_subset (G : BondGraph V E) (ω : Set E) (x : V)
+    {η : Set E} (hη : η ⊆ ω) {a b : V} (ha : a ∈ cluster G ω x)
+    (hab : Conn G η a b) : Conn G (η ∩ internalEdges G (cluster G ω x)) a b := by
+  induction hab with
+  | refl => exact conn_refl G _ a
+  | @tail u v hau huv ih =>
+    obtain ⟨e,he,heq⟩ := huv
+    have hu : u ∈ cluster G ω x := ha.trans (conn_mono G hη hau)
+    have hv : v ∈ cluster G ω x := (edge_cluster_sides G ω x (hη he) heq).mp hu
+    exact ih.tail ⟨e,⟨he,(mem_internalEdges_of_ends G _ e u v heq).mpr ⟨hu,hv⟩⟩,heq⟩
+
+ 
+
+theorem conn_between_pivotals_internal (G : BondGraph V E) (ω : Set E)
+    (x z : V) (hC : (cluster G ω x).Finite) {e f : E}
+    (he : OpenMarkedPivotal G ω {z} x e) (hf : OpenMarkedPivotal G ω {z} x f)
+    (hlt : pivotalRank G ω x e < pivotalRank G ω x f)
+    (a b c d : V) (hab : G e = s(a,b)) (hcd : G f = s(c,d))
+    (ha : Conn G (ω \ {e}) x a) (hb : ¬Conn G (ω \ {e}) x b)
+    (hc : Conn G (ω \ {f}) x c) :
+    Conn G (ω ∩ internalEdges G
+      (cluster G (ω \ {f}) x \ cluster G (ω \ {e}) x)) b c := by
+  have hne : e ≠ f := fun h => (Nat.ne_of_lt hlt) (congrArg (pivotalRank G ω x) h)
+  have hout := (pivotalRank_lt_iff_of_cluster_finite G ω {z} x hC he hf).mp hlt
+  have hcnot : c ∉ cluster G (ω \ {e}) x :=
+    ((mem_internalEdges_of_ends G _ f c d hcd).mp hout).1
+  have hins : insert e ((ω \ {e}) \ {f}) = ω \ {f} := by
+    ext q
+    simp only [Set.mem_insert_iff,Set.mem_sdiff,Set.mem_singleton_iff]
+    constructor
+    · rintro (rfl | ⟨⟨hq,hqe⟩,hqf⟩)
+      · exact ⟨he.1,hne⟩
+      · exact ⟨hq,hqf⟩
+    · rintro ⟨hq,hqf⟩
+      by_cases hqe : q = e
+      · exact Or.inl hqe
+      · exact Or.inr ⟨⟨hq,hqe⟩,hqf⟩
+  have hbc : Conn G ((ω \ {e}) \ {f}) b c := by
+    rw [← hins,conn_insert_iff G _ e a b x c hab] at hc
+    rcases hc with h | ⟨_,h⟩ | ⟨h,_⟩
+    · exact (hcnot (conn_mono G Set.sdiff_subset h)).elim
+    · exact h
+    · exact (hb (conn_mono G Set.sdiff_subset h)).elim
+  have hbo : b ∈ cluster G (ω \ {f}) x :=
+    conn_far_end_avoids_later_pivotal G ω x z hC he hf hlt a b hab ha
+  have hp := conn_restrict_cluster_compl_of_subset G (ω \ {e}) x Set.sdiff_subset hb hbc
+  have hp' := conn_restrict_cluster_of_subset G (ω \ {f}) x
+    (η := ((ω \ {e}) \ {f}) ∩ internalEdges G (cluster G (ω \ {e}) x)ᶜ)
+    (fun _ h => ⟨h.1.1.1,h.1.2⟩) hbo hp
+  apply conn_mono G ?_ hp'
+  intro q hq
+  exact ⟨hq.1.1.1.1,fun v hv => ⟨hq.2 v hv,hq.1.2 v hv⟩⟩
+
+ 
+theorem conn_after_pivotal_internal (G : BondGraph V E) (ω : Set E)
+    (x z : V) {e : E} (he : OpenMarkedPivotal G ω {z} x e)
+    (a b : V) (hab : G e = s(a,b)) (ha : Conn G (ω \ {e}) x a)
+    (hb : ¬Conn G (ω \ {e}) x b) (hbz : Conn G (ω \ {e}) b z) :
+    Conn G (ω ∩ internalEdges G (cluster G ω x \ cluster G (ω \ {e}) x)) b z := by
+  have hbo : b ∈ cluster G ω x :=
+    (conn_mono G Set.sdiff_subset ha).tail ⟨e,he.1,hab⟩
+  have hp := conn_restrict_cluster_compl G (ω \ {e}) x hb hbz
+  have hp' := conn_restrict_cluster_of_subset G ω x
+    (η := (ω \ {e}) ∩ internalEdges G (cluster G (ω \ {e}) x)ᶜ)
+    (fun _ h => h.1.1) hbo hp
+  apply conn_mono G ?_ hp'
+  intro q hq
+  exact ⟨hq.1.1.1,fun v hv => ⟨hq.2 v hv,hq.1.2 v hv⟩⟩
+
+ 
+theorem pivotalDistance_le_induced_card (G : BondGraph V E) (ω : Set E)
+    (B : Set V) (hB : B.Finite) {a b : V} (ha : a ∈ B) :
+    pivotalDistance G (ω ∩ internalEdges G B) a b ≤ B.ncard-1 := by
+  have hsub : cluster G (ω ∩ internalEdges G B) a ⊆ B :=
+    fun _ h => conn_internal_endpoint G B ha h
+  exact (pivotalDistance_le_cluster_card G (ω ∩ internalEdges G B) a b (hB.subset hsub)).trans
+    (Nat.sub_le_sub_right (Set.ncard_le_ncard hsub hB) 1)
+
+end BridgePiecePath
+
+end Percolation
+
+ 
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section SplicePieceGeometry
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem conn_after_pivotal_of_orientation (G : BondGraph V E) (ω : Set E)
+    (x z : V) {e : E} (he : OpenMarkedPivotal G ω {z} x e)
+    (a b : V) (hab : G e = s(a,b)) (hb : ¬Conn G (ω \ {e}) x b) :
+    Conn G (ω \ {e}) b z := by
+  have hins : insert e (ω \ {e}) = ω := by
+    ext q
+    simp only [Set.mem_insert_iff,Set.mem_sdiff,Set.mem_singleton_iff]
+    by_cases hqe : q = e <;> simp_all [he.1]
+  obtain ⟨w,hw,hxw⟩ := he.2.1
+  have hw' : w = z := Set.mem_singleton_iff.mp hw
+  subst w
+  rw [← hins,conn_insert_iff G _ e a b x z hab] at hxw
+  rcases hxw with h | ⟨_,h⟩ | ⟨h,_⟩
+  · exact (he.2.2 ⟨z,rfl,h⟩).elim
+  · exact h
+  · exact (hb h).elim
+
+ 
+theorem SpliceTrace.internal_agreement (G : BondGraph V E) (x : V) (η : ℕ → Set E)
+    (y : ℕ → V) (k j : ℕ) (ω ωf : Set E) (steps : List (ℕ × E))
+    (h : SpliceTrace G x η y k j ω steps ωf) :
+    ∀ q ∈ internalEdges G (cluster G ω x), q ∈ ωf ↔ q ∈ ω := by
+  induction h with
+  | stop => exact fun _ _ => Iff.rfl
+  | @step j i ω e a b rest ωf _ _ _ _ hy he hab ha hb ht ih =>
+    intro q hq
+    let S := cluster G ω x
+    let B := cluster G (η i ∩ internalEdges G Sᶜ) (y i)
+    let ω' := spliceField G S B e ω (η i)
+    have hB : B ⊆ Sᶜ := slit_cluster_subset G S (η i) (y i) hy
+    have hbS : b ∉ S := hB hb
+    have hqq : q ∈ internalEdges G (cluster G ω' x) := by
+      rw [splice_cluster G ω (η i) x (y i) hy e a b hab he ha hb]
+      exact fun v hv => Or.inl (hq v hv)
+    rw [ih q hqq]
+    change q ∈ internalEdges G S at hq
+    have hh := (splice_internal_fields G S B e ω (η i) a b hab hbS hB).1
+    exact Iff.of_eq (show (q ∈ ω') = (q ∈ ω) from by
+      have := congrArg (fun A : Set E => q ∈ A) hh
+      simpa only [Set.mem_inter_iff,hq,and_true] using this)
+
+ 
+theorem SpliceTrace.head_cut (G : BondGraph V E) (x : V) (η : ℕ → Set E)
+    (y : ℕ → V) (k j i : ℕ) (ω ωf : Set E) (e : E) (rest : List (ℕ × E))
+    (h : SpliceTrace G x η y k j ω ((i,e)::rest) ωf) :
+    cluster G (ωf \ {e}) x = cluster G ω x := by
+  cases h with
+  | step j i ω e a b rest ωf _ _ _ hmax hy he hab ha hb ht =>
+    have hc := splice_close_seam_cluster G ω (η i) x (y i) hy e a b hab he ha hb
+    rw [SpliceTrace.preserve_cut G x η y k i _ ωf rest ht e]
+    · exact hc
+    · intro l hil hlk
+      rw [hc]
+      exact hmax l hil hlk
+
+ 
+theorem contactSet_finite (G : BondGraph V E) (hloc : LocallyFinite G)
+    (ω : Set E) (B : Set V) (hB : B.Finite) : (contactSet G ω B).Finite := by
+  let W := hB.toFinset
+  have hWB : (W : Set V) = B := hB.coe_toFinset
+  apply (incidentSupport G hloc W).finite_toSet.subset
+  rintro e ⟨_,a,b,hab,ha,_⟩
+  apply (mem_incidentSupport G hloc W e).mpr
+  refine ⟨a,?_,?_⟩
+  · change a ∈ (W : Set V)
+    rw [hWB]
+    exact ha
+  · simp [hab]
+
+ 
+
+theorem goodForScale_piece (G : BondGraph V E) (ω : Set E) (x : V) (L : ℝ)
+    (hg : GoodForScale G ω x L) (B : Set V) (hB : B.Finite)
+    (hBC : B ⊆ cluster G ω x)
+    (hconn : ∀ a ∈ B, ∀ b ∈ B, Conn G (ω ∩ internalEdges G B) a b)
+    (hK : 0 < (contactSet G ω B).ncard) {a b : V} (ha : a ∈ B) (hb : b ∈ B) :
+    (pivotalDistance G (ω ∩ internalEdges G B) a b : ℝ) ≤
+      (contactSet G ω B).ncard*L := by
+  let W := hB.toFinset
+  have hWB : (W : Set V) = B := hB.coe_toFinset
+  have hh := hg W (by simpa only [hWB] using hBC)
+    (by simpa only [← Finset.mem_coe,hWB] using hconn)
+    (by simpa only [hWB] using hK) a (by simpa only [← Finset.mem_coe,hWB] using ha)
+    b (by simpa only [← Finset.mem_coe,hWB] using hb)
+  simpa only [hWB] using hh
+
+ 
+theorem slit_piece_connected (G : BondGraph V E) (η : Set E) (S : Set V) (y : V)
+    (_hy : y ∉ S) :
+    let B := cluster G (η ∩ internalEdges G Sᶜ) y
+    ∀ a ∈ B, ∀ b ∈ B, Conn G (η ∩ internalEdges G B) a b := by
+  intro B a ha b hb
+  have hc := conn_restrict_cluster G (η ∩ internalEdges G Sᶜ) y ha
+    (conn_trans G _ (conn_symm G _ ha) hb)
+  apply conn_mono G ?_ hc
+  exact fun _ h => ⟨h.1.1,h.2⟩
+
+end SplicePieceGeometry
+
+end Percolation
+
+ 
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section SpliceGapBound
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+noncomputable def traceHeadGap (G : BondGraph V E) (ωf : Set E) (x z : V)
+    (e : E) (rest : List (ℕ × E)) : ℕ :=
+  (match rest with
+    | [] => (separatingBonds G ωf x z).card + 1
+    | (_,f)::_ => bridgeIndex G ωf x z f) - bridgeIndex G ωf x z e
+
+ 
+
+theorem SpliceTrace.head_gap_bounds (G : BondGraph V E) (hloc : LocallyFinite G)
+    (x : V) (η : ℕ → Set E) (y : ℕ → V) (k j i M : ℕ)
+    (ω ωf : Set E) (e : E) (rest : List (ℕ × E))
+    (h : SpliceTrace G x η y k j ω ((i,e)::rest) ωf)
+    (hCf : (cluster G ωf x).Finite)
+    (hCi : (cluster G (η i) (y i)).Finite)
+    (hM : (cluster G (η i) (y i)).ncard ≤ M)
+    (L : ℝ) (hgood : GoodForScale G (η i) (y i) L) :
+    let B := cluster G (η i ∩ internalEdges G (cluster G ω x)ᶜ) (y i)
+    let K := (contactSet G (η i) B).ncard
+    let l := traceHeadGap G ωf x (y k) e rest
+    1 ≤ K ∧ 1 ≤ l ∧ l ≤ M ∧ (l:ℝ) ≤ 1 + K*L := by
+  have hfull := h
+  cases h with
+  | step j i ω e a b rest ωf hn _ _ _ hy he hab ha hb ht =>
+    let S := cluster G ω x
+    let B := cluster G (η i ∩ internalEdges G Sᶜ) (y i)
+    let ω' := spliceField G S B e ω (η i)
+    let K := (contactSet G (η i) B).ncard
+    change 1 ≤ K ∧ 1 ≤ traceHeadGap G ωf x (y k) e rest ∧
+      traceHeadGap G ωf x (y k) e rest ≤ M ∧
+      (traceHeadGap G ωf x (y k) e rest:ℝ) ≤ 1+K*L
+    have hBo : B ⊆ Sᶜ := slit_cluster_subset G S (η i) (y i) hy
+    have hBC : B ⊆ cluster G (η i) (y i) := fun _ h => conn_mono G Set.inter_subset_left h
+    have hB : B.Finite := hCi.subset hBC
+    have hKB : (contactSet G (η i) B).Finite := contactSet_finite G hloc (η i) B hB
+    have heK : e ∈ contactSet G (η i) B :=
+      ⟨he,b,a,hab.trans (Sym2.eq_swap ..),hb,fun h => hBo h ha⟩
+    have hK : 0 < K := (Set.ncard_pos hKB).mpr ⟨e,heK⟩
+    have hBpos : 0 < B.ncard := (Set.ncard_pos hB).mpr ⟨b,hb⟩
+    have hBM : B.ncard ≤ M := (Set.ncard_le_ncard hBC hCi).trans hM
+    have hcluster : cluster G ω' x = S ∪ B :=
+      splice_cluster G ω (η i) x (y i) hy e a b hab he ha hb
+    have hcut : cluster G (ωf \ {e}) x = S :=
+      SpliceTrace.head_cut G x η y k j i ω ωf e rest hfull
+    have hpe : OpenMarkedPivotal G ωf {y k} x e :=
+      SpliceTrace.seams_pivotal G x η y k j ω ωf ((i,e)::rest) hfull (i,e) (by simp)
+    have hxa : Conn G (ωf \ {e}) x a := by
+      change a ∈ cluster G (ωf \ {e}) x
+      rw [hcut]
+      exact ha
+    have hxb : ¬Conn G (ωf \ {e}) x b := by
+      change b ∉ cluster G (ωf \ {e}) x
+      rw [hcut]
+      exact hBo hb
+    have hbits : ωf ∩ internalEdges G B = η i ∩ internalEdges G B := by
+      have hii := (splice_internal_fields G S B e ω (η i) a b hab (hBo hb) hBo).2
+      ext q
+      by_cases hq : q ∈ internalEdges G B
+      · have hq' : q ∈ internalEdges G (cluster G ω' x) := by
+          rw [hcluster]
+          exact fun v hv => Or.inr (hq v hv)
+        have hiff := SpliceTrace.internal_agreement G x η y k i ω' ωf rest ht q hq'
+        have hbit := congrArg (fun A : Set E => q ∈ A) hii
+        simpa only [Set.mem_inter_iff,hq,and_true,hiff] using Iff.of_eq hbit
+      · simp only [Set.mem_inter_iff,hq,and_false]
+    have hconn := slit_piece_connected G (η i) S (y i) hy
+    have finish (l : ℕ) (c : V) (hc : c ∈ B) (hl : 1 ≤ l)
+        (hg : l ≤ 1 + pivotalDistance G (ωf ∩ internalEdges G B) b c) :
+        1 ≤ l ∧ l ≤ M ∧ (l:ℝ) ≤ 1+K*L := by
+      rw [hbits] at hg
+      have hp := pivotalDistance_le_induced_card G (η i) B hB hb (b := c)
+      have hg' : (l:ℝ) ≤ 1+(pivotalDistance G (η i ∩ internalEdges G B) b c:ℝ) := by
+        exact_mod_cast hg
+      have hgood' := goodForScale_piece G (η i) (y i) L hgood B hB hBC hconn hK hb hc
+      refine ⟨hl,by omega,?_⟩
+      change (l:ℝ) ≤ 1+(contactSet G (η i) B).ncard*L
+      linarith
+    refine ⟨hK,?_⟩
+    cases rest with
+    | nil =>
+      have hfω : ωf = ω' := by cases ht; rfl
+      have hdiff : cluster G ωf x \ cluster G (ωf \ {e}) x = B := by
+        rw [hcut,hfω,hcluster]
+        ext v
+        constructor
+        · rintro ⟨hv,hnv⟩
+          exact hv.resolve_left hnv
+        · intro hv
+          exact ⟨Or.inr hv,hBo hv⟩
+      have hyk : y k ∈ B := by
+        have hz := SpliceTrace.endpoint G x η y k j ω ωf [(i,e)] hfull
+        rw [hfω,hcluster] at hz
+        exact hz.resolve_left hn
+      have hconnz := conn_after_pivotal_of_orientation G ωf x (y k) hpe a b hab hxb
+      have hp := conn_after_pivotal_internal G ωf x (y k) hpe a b hab hxa hxb hconnz
+      have hg := bridge_terminal_gap_le_internal_distance G ωf x (y k) hCf hpe a b hab hxa hp
+      rw [hdiff] at hg
+      apply finish _ (y k) hyk _ hg
+      have hcard : bridgeIndex G ωf x (y k) e ≤ (separatingBonds G ωf x (y k)).card :=
+        Finset.card_filter_le _ _
+      omega
+    | cons q more =>
+      obtain ⟨l,f⟩ := q
+      have hcutf : cluster G (ωf \ {f}) x = S ∪ B := by
+        rw [SpliceTrace.head_cut G x η y k i l ω' ωf f more ht,hcluster]
+      have hss : cluster G (ωf \ {e}) x ⊂ cluster G (ωf \ {f}) x := by
+        rw [hcut,hcutf]
+        exact Set.ssubset_iff_subset_ne.mpr ⟨Set.subset_union_left,fun heq =>
+          hBo hb (heq.symm ▸ (show b ∈ S ∪ B from Or.inr hb))⟩
+      have hfFin : (cluster G (ωf \ {f}) x).Finite :=
+        hCf.subset (fun _ h => conn_mono G Set.sdiff_subset h)
+      have hr : pivotalRank G ωf x e < pivotalRank G ωf x f := Set.ncard_lt_ncard hss hfFin
+      have hpf : OpenMarkedPivotal G ωf {y k} x f :=
+        SpliceTrace.seams_pivotal G x η y k j ω ωf ((i,e)::(l,f)::more) hfull (l,f) (by simp)
+      obtain ⟨c,d,hcd,hxc,hxd,hdyk⟩ := openMarkedPivotal_oriented G ωf {y k} x hpf
+      have hdy : Conn G (ωf \ {f}) d (y k) := by
+        simpa only [MarkedEvent,Set.mem_singleton_iff,exists_eq_left] using hdyk
+      have hdiff : cluster G (ωf \ {f}) x \ cluster G (ωf \ {e}) x = B := by
+        rw [hcut,hcutf]
+        ext v
+        constructor
+        · rintro ⟨hv,hnv⟩
+          exact hv.resolve_left hnv
+        · intro hv
+          exact ⟨Or.inr hv,hBo hv⟩
+      have hc : c ∈ B := by
+        have hfout := (pivotalRank_lt_iff_of_cluster_finite G ωf {y k} x hCf hpe hpf).mp hr
+        have hcnot := ((mem_internalEdges_of_ends G _ f c d hcd).mp hfout).1
+        rw [hcut] at hcnot
+        change c ∈ cluster G (ωf \ {f}) x at hxc
+        rw [hcutf] at hxc
+        exact hxc.resolve_left hcnot
+      have hp := conn_between_pivotals_internal G ωf x (y k) hCf hpe hpf hr
+        a b c d hab hcd hxa hxb (by change c ∈ cluster G (ωf \ {f}) x; rw [hcutf]; exact Or.inr hc)
+      have hg := bridge_gap_le_internal_distance G ωf x (y k) hCf hpe hpf hr
+        a b c d hab hcd hxa hxd hdy hp
+      rw [hdiff] at hg
+      apply finish _ c hc _ hg
+      have hind := bridgeIndex_interval_identity G ωf x (y k) hCf hpf hr
+      omega
+
+end SpliceGapBound
+
+end Percolation
+
+ 
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section GapCode
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem SpliceTrace.seam_data (G : BondGraph V E) (x : V) (η : ℕ → Set E)
+    (y : ℕ → V) (k j : ℕ) (ω ωf : Set E) (steps : List (ℕ × E))
+    (h : SpliceTrace G x η y k j ω steps ωf) :
+    ∀ q ∈ steps, j < q.1 ∧ q.1 ≤ k ∧ q.2 ∈ η q.1 ∧
+      q.2 ∈ internalEdges G (cluster G (η q.1) (y q.1)) := by
+  induction h with
+  | stop => simp
+  | @step j i ω e a b rest ωf _ hji hik _ _ he hab _ hb ht ih =>
+    intro q hq
+    rcases List.mem_cons.mp hq with rfl | hq
+    · have hb' : b ∈ cluster G (η i) (y i) := conn_mono G Set.inter_subset_left hb
+      exact ⟨hji,hik,he,(mem_internalEdges_of_ends G _ e a b hab).mpr
+        ⟨(edge_cluster_sides G (η i) (y i) he hab).mpr hb',hb'⟩⟩
+    · obtain ⟨hqi,hqk,hqe,hqC⟩ := ih q hq
+      exact ⟨hji.trans hqi,hqk,hqe,hqC⟩
+
+ 
+theorem SpliceTrace.head_index_lt (G : BondGraph V E) (x : V) (η : ℕ → Set E)
+    (y : ℕ → V) (k j i : ℕ) (ω ωf : Set E) (e : E) (rest : List (ℕ × E))
+    (h : SpliceTrace G x η y k j ω ((i,e)::rest) ωf)
+    (hC : (cluster G ωf x).Finite) :
+    ∀ q ∈ rest, bridgeIndex G ωf x (y k) e < bridgeIndex G ωf x (y k) q.2 := by
+  have hfull := h
+  have hc := SpliceTrace.head_cut G x η y k j i ω ωf e rest h
+  have hseams := SpliceTrace.seams_pivotal G x η y k j ω ωf ((i,e)::rest) h
+  cases h with
+  | step j i ω e a b rest ωf _ _ _ hmax _ _ _ _ _ ht =>
+    intro q hq
+    obtain ⟨hqi,hqk,_,hqC⟩ := SpliceTrace.seam_data G x η y k i _ ωf rest ht q hq
+    have hout : q.2 ∈ internalEdges G (cluster G (ωf \ {e}) x)ᶜ := by
+      rw [hc]
+      intro v hv hvS
+      exact Set.disjoint_left.mp (hmax q.1 hqi hqk) (hqC v hv) hvS
+    have hpe := hseams (i,e) (by simp)
+    have hpq := hseams q (List.mem_cons_of_mem _ hq)
+    exact bridgeIndex_lt_of_rank_lt G ωf x (y k) hpq
+      ((pivotalRank_lt_iff_of_cluster_finite G ωf {y k} x hC hpe hpq).mpr hout)
+
+ 
+noncomputable def gapSpliceCode (G : BondGraph V E) (ωf : Set E) (x z : V) :
+    List (ℕ × E) → List (ℕ × ℕ)
+  | [] => []
+  | (i,e)::rest => (i,traceHeadGap G ωf x z e rest)::gapSpliceCode G ωf x z rest
+
+ 
+def decodeGapOrdinals (N : ℕ) : List (ℕ × ℕ) → List (ℕ × ℕ)
+  | [] => []
+  | (i,l)::rest =>
+    let decoded := decodeGapOrdinals N rest
+    (i, (match decoded with | [] => N+1 | (_,r)::_ => r)-l)::decoded
+
+ 
+theorem SpliceTrace.decode_gap_code (G : BondGraph V E) (x : V) (η : ℕ → Set E)
+    (y : ℕ → V) (k j : ℕ) (ω ωf : Set E) (steps : List (ℕ × E))
+    (h : SpliceTrace G x η y k j ω steps ωf) (hC : (cluster G ωf x).Finite) :
+    decodeGapOrdinals (separatingBonds G ωf x (y k)).card
+      (gapSpliceCode G ωf x (y k) steps) = ordinalSpliceCode G ωf x (y k) steps := by
+  induction steps generalizing j ω with
+  | nil => rfl
+  | cons q rest ih =>
+    obtain ⟨i,e⟩ := q
+    have hlater := SpliceTrace.head_index_lt G x η y k j i ω ωf e rest h hC
+    cases h with
+    | step j i ω e a b rest ωf _ _ _ _ _ _ _ _ _ ht =>
+      have ih' := ih i _ ht
+      simp only [gapSpliceCode,decodeGapOrdinals,ih',ordinalSpliceCode,List.map_cons]
+      congr 2
+      cases rest with
+      | nil =>
+        simp only [List.map_nil,traceHeadGap]
+        have hle : bridgeIndex G ωf x (y k) e ≤ (separatingBonds G ωf x (y k)).card :=
+          Finset.card_filter_le _ _
+        omega
+      | cons q qs =>
+        obtain ⟨l,f⟩ := q
+        have hle := hlater (l,f) (by simp)
+        change bridgeIndex G ωf x (y k) e < bridgeIndex G ωf x (y k) f at hle
+        simp only [List.map_cons,traceHeadGap]
+        omega
+
+ 
+theorem SpliceTrace.input_unique_of_gapCode (G : BondGraph V E) (x : V)
+    (η₁ η₂ : ℕ → Set E) (y₁ y₂ : ℕ → V) (k j : ℕ) (ω₁ ω₂ ωf : Set E)
+    (c₁ c₂ : List (ℕ × E))
+    (ht₁ : SpliceTrace G x η₁ y₁ k j ω₁ c₁ ωf)
+    (ht₂ : SpliceTrace G x η₂ y₂ k j ω₂ c₂ ωf)
+    (hz : y₁ k = y₂ k) (hC : (cluster G ωf x).Finite)
+    (heq : gapSpliceCode G ωf x (y₁ k) c₁ = gapSpliceCode G ωf x (y₁ k) c₂)
+    (θf : ℕ → Set E)
+    (he₁ : SpliceExecution G x (ω₁,η₁) c₁ (ωf,θf))
+    (he₂ : SpliceExecution G x (ω₂,η₂) c₂ (ωf,θf)) :
+    (ω₁,η₁) = (ω₂,η₂) := by
+  apply SpliceTrace.input_unique_of_ordinalCode G x η₁ η₂ y₁ y₂ k j ω₁ ω₂ ωf
+    c₁ c₂ ht₁ ht₂ hz hC _ θf he₁ he₂
+  rw [← SpliceTrace.decode_gap_code G x η₁ y₁ k j ω₁ ωf c₁ ht₁ hC,
+    heq,hz,SpliceTrace.decode_gap_code G x η₂ y₂ k j ω₂ ωf c₂ ht₂ hC]
+
+end GapCode
+
+end Percolation
+
+ 
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section InverseCodeMass
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+noncomputable def gapWeight (L : ℝ) (l : ℕ) : ℝ := min 1 (2*L/l)
+
+ 
+noncomputable def inverseCodeMass (M : ℕ) (L : ℝ) : ℝ :=
+  1 + ∑ l ∈ Finset.Icc 1 M, gapWeight L l
+
+ 
+noncomputable def gapLetterWeight (L : ℝ) (l : ℕ) : ℝ := if l = 0 then 1 else gapWeight L l
+
+theorem gapWeight_nonneg (L : ℝ) (hL : 0 ≤ L) (l : ℕ) : 0 ≤ gapWeight L l := by
+  exact le_min zero_le_one (by positivity)
+
+theorem gapLetterWeight_nonneg (L : ℝ) (hL : 0 ≤ L) (l : ℕ) : 0 ≤ gapLetterWeight L l := by
+  unfold gapLetterWeight
+  split_ifs
+  · exact zero_le_one
+  · exact gapWeight_nonneg L hL l
+
+theorem sum_fin_succ_eq_Icc (M : ℕ) (f : ℕ → ℝ) :
+    ∑ l : Fin M, f (l.val+1) = ∑ l ∈ Finset.Icc 1 M, f l := by
+  rw [Fin.sum_univ_eq_sum_range (fun l => f (l+1)) M]
+  apply Finset.sum_bij (fun l _ => l+1)
+  · intro l hl
+    exact Finset.mem_Icc.mpr ⟨by omega,by have := Finset.mem_range.mp hl; omega⟩
+  · intro l hl m hm h
+    omega
+  · intro l hl
+    obtain ⟨h1,hM⟩ := Finset.mem_Icc.mp hl
+    exact ⟨l-1,Finset.mem_range.mpr (by omega),by omega⟩
+  · intros
+    rfl
+
+theorem sum_gapLetterWeight (M : ℕ) (L : ℝ) :
+    ∑ l : Fin (M+1), gapLetterWeight L l.val = inverseCodeMass M L := by
+  rw [Fin.sum_univ_succ]
+  simp only [Fin.val_zero,gapLetterWeight,Fin.val_succ,Nat.add_eq_zero_iff,
+    Nat.one_ne_zero,and_false,ite_false]
+  exact congrArg (1+·) (sum_fin_succ_eq_Icc M (gapWeight L))
+
+ 
+
+theorem total_gap_code_mass (k M : ℕ) (L : ℝ) :
+    ∑ c : Fin k → Fin (M+1), ∏ i : Fin k, gapLetterWeight L (c i).val =
+      inverseCodeMass M L ^ k := by
+  rw [← Fintype.prod_sum (fun (_ : Fin k) (l : Fin (M+1)) => gapLetterWeight L l.val)]
+  simp only [sum_gapLetterWeight,Finset.prod_const,Finset.card_univ,Fintype.card_fin]
+
+ 
+theorem valid_gap_code_mass_le (k M : ℕ) (L : ℝ) (hL : 0 ≤ L)
+    (valid : Finset (Fin k → Fin (M+1))) :
+    ∑ c ∈ valid, ∏ i : Fin k, gapLetterWeight L (c i).val ≤ inverseCodeMass M L ^ k := by
+  rw [← total_gap_code_mass]
+  exact Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _)
+    (fun c _ _ => Finset.prod_nonneg (fun i _ => gapLetterWeight_nonneg L hL (c i).val))
+
+theorem inverseCodeMass_le_harmonic (M : ℕ) (L : ℝ) (hL : 0 ≤ L) :
+    inverseCodeMass M L ≤ 1+2*L*(1+Real.log M) := by
+  have hsum : ∑ l ∈ Finset.Icc 1 M, gapWeight L l ≤
+      2*L*∑ l ∈ Finset.Icc 1 M, (l:ℝ)⁻¹ := by
+    rw [Finset.mul_sum]
+    apply Finset.sum_le_sum
+    intro l _
+    simpa only [gapWeight,div_eq_mul_inv] using min_le_right (1:ℝ) (2*L/l)
+  have hh : ∑ l ∈ Finset.Icc 1 M, (l:ℝ)⁻¹ ≤ 1+Real.log M := by
+    simpa only [harmonic_eq_sum_Icc,Rat.cast_sum,Rat.cast_inv,Rat.cast_natCast] using
+      harmonic_le_one_add_log M
+  unfold inverseCodeMass
+  have h := hsum.trans (mul_le_mul_of_nonneg_left hh (by positivity))
+  linarith
+
+ 
+theorem inverseCodeMass_le_log (M : ℕ) (L : ℝ) (hM : 2 ≤ M) (hL : 1 ≤ L) :
+    inverseCodeMass M L ≤ 8*L*Real.log M := by
+  have hlog2 : (1:ℝ)/2 ≤ Real.log 2 := by
+    have hh := Real.one_sub_inv_le_log_of_pos (show (0:ℝ) < 2 by norm_num)
+    norm_num at hh ⊢
+    exact hh
+  have hlog : (1:ℝ)/2 ≤ Real.log M := hlog2.trans
+    (Real.log_le_log (by norm_num) (by exact_mod_cast hM))
+  have hLl : L/2 ≤ L*Real.log M := by nlinarith
+  have hbound := inverseCodeMass_le_harmonic M L (by linarith)
+  nlinarith
+
+ 
+def contactInterval (K a : ℕ) : Set unitInterval :=
+  {u | (a:ℝ)/K ≤ (u:ℝ) ∧ (u:ℝ) < (a+1:ℕ)/K}
+
+theorem volume_contactInterval (K a : ℕ) (ha : a < K) :
+    volume (contactInterval K a) = ENNReal.ofReal ((K:ℝ)⁻¹) := by
+  have hK : (0:ℝ) < K := by exact_mod_cast (show 0 < K by omega)
+  have hleft : (a:ℝ)/K ∈ Set.Icc (0:ℝ) 1 := by
+    refine ⟨by positivity,(div_le_one hK).mpr ?_⟩
+    exact_mod_cast (Nat.le_of_lt ha)
+  have hright : ((a+1:ℕ):ℝ)/K ∈ Set.Icc (0:ℝ) 1 := by
+    refine ⟨by positivity,(div_le_one hK).mpr ?_⟩
+    exact_mod_cast (show a+1 ≤ K by omega)
+  let lo : unitInterval := ⟨_,hleft⟩
+  let hi : unitInterval := ⟨_,hright⟩
+  have heq : contactInterval K a = Set.Ico lo hi := rfl
+  rw [heq,unitInterval.volume_Ico]
+  congr 1
+  dsimp only [lo,hi]
+  push_cast
+  field_simp
+  ring
+
+ 
+
+theorem volume_contact_cube {I : Type*} [Fintype I] (K a : I → ℕ)
+    (ha : ∀ i, a i < K i) :
+    (volume : Measure (I → unitInterval)) (Set.pi Set.univ (fun i => contactInterval (K i) (a i))) =
+      ∏ i, ENNReal.ofReal ((K i:ℝ)⁻¹) := by
+  rw [MeasureTheory.volume_pi,Measure.pi_pi]
+  exact Finset.prod_congr rfl (fun i _ => volume_contactInterval (K i) (a i) (ha i))
+
+theorem real_volume_contact_cube {I : Type*} [Fintype I] (K a : I → ℕ)
+    (ha : ∀ i, a i < K i) :
+    ((volume : Measure (I → unitInterval))
+      (Set.pi Set.univ (fun i => contactInterval (K i) (a i)))).toReal =
+      ∏ i, (K i:ℝ)⁻¹ := by
+  rw [volume_contact_cube K a ha,ENNReal.toReal_prod]
+  exact Finset.prod_congr rfl (fun i _ => ENNReal.toReal_ofReal (by positivity))
+
+end InverseCodeMass
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Splicing/Geometry.lean b/lean/OAI/Probability/UnimodularPercolation/Splicing/Geometry.lean
index 67647ec52b40d78b4e1773121e70ceeb3f88c9a0..3d11bb8b0aa4dfa0d47ea0ef7d87c0af3d8e34ca 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Splicing/Geometry.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Splicing/Geometry.lean
@@ -1,836 +1,836 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Exploration.Goodness
-import OAI.Probability.UnimodularPercolation.Kernel.Spectral
-
-namespace OAI
-
-/-! Critical parameters and the geometry, selection and traces of splices. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-
-section CriticalParameter
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem twoPoint_adj_lower [Countable E] (G : BondGraph V E) (p : unitInterval)
-    (z x y : V) (hxy : Adj G x y) : (p:ℝ)*twoPoint G p z x ≤ twoPoint G p z y := by
-  obtain ⟨e,he⟩ := hxy
-  have h := association_finiteWitness p {ω : Set E | e ∈ ω} {ω | Conn G ω z x}
-    (fun _ _ hω h => hω h) (fun _ _ hω h => conn_mono G hω h)
-    (finiteWitness_edge e) (finiteWitness_conn G z x)
-  have hp : (law p).real {ω : Set E | e ∈ ω} = (p:ℝ) := by
-    simpa using finite_open_probability p {e}
-  rw [hp] at h
-  exact h.trans (measureReal_mono (fun ω hω =>
-    conn_trans G ω hω.2 (Relation.ReflTransGen.single ⟨e,hω.1,he⟩)))
-
- 
-theorem walk_operator_apply (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
-    (P : Hilbert V →L[ℂ] Hilbert V)
-    (hP : ∀ x y, P (lp.single 2 y (1:ℂ)) x = (walkKernel G hloc a x y : ℂ))
-    (f : Hilbert V) (x : V) :
-    P f x = ∑ y ∈ insert x (neighbors G hloc x), (walkKernel G hloc a x y : ℂ)*f y := by
-  have hz (y : V) (hy : y ∉ insert x (neighbors G hloc x)) : matrixCoeff P x y*f y = 0 := by
-    have hxy : x ≠ y := by intro h; subst y; exact hy (Finset.mem_insert_self _ _)
-    have hAdj : ¬Adj G x y := fun h => hy (Finset.mem_insert_of_mem ((mem_neighbors ..).mpr h))
-    simp only [matrixCoeff,hP,walkKernel,ite_eq_right hxy,ite_eq_right hAdj,add_zero,Complex.ofReal_zero,zero_mul]
-  have hsum := (operator_kernel_hasSum P f x).unique (hasSum_sum_of_ne_finset_zero hz)
-  simpa only [matrixCoeff,hP] using hsum
-
- 
-
-theorem parameter_le_walk_norm [Countable E] [Nonempty V] (G : BondGraph V E)
-    (hloc : LocallyFinite G) (D : ℕ) (hD : ∀ x, degree G x ≤ D)
-    (a : ℝ) (ha : 0 ≤ a) (haD : a*D ≤ 1)
-    (P : Hilbert V →L[ℂ] Hilbert V)
-    (hP : ∀ x y, P (lp.single 2 y (1:ℂ)) x = (walkKernel G hloc a x y : ℂ))
-    (p : unitInterval) (hp : OperatorBounded G p) : (p:ℝ) ≤ ‖P‖ := by
-  obtain ⟨T,hT⟩ := hp
-  let z := Classical.arbitrary V
-  let f := T (lp.single 2 z (1:ℂ))
-  have hf (x : V) : f x = (twoPoint G p x z : ℂ) := hT x z
-  have hf0 : 0 < ‖f‖ := by
-    apply norm_pos_iff.mpr
-    intro h
-    have hval := congrArg (fun g : Hilbert V => g z) h
-    simp [hf] at hval
-  have hrow (x : V) : (p:ℝ)*twoPoint G p x z ≤
-      ∑ y ∈ insert x (neighbors G hloc x), walkKernel G hloc a x y*twoPoint G p y z := by
-    calc
-      _ = ∑ y ∈ insert x (neighbors G hloc x), walkKernel G hloc a x y*((p:ℝ)*twoPoint G p x z) := by
-        rw [← Finset.sum_mul,walkKernel_row_sum,one_mul]
-      _ ≤ _ := by
-        apply Finset.sum_le_sum
-        intro y hy
-        apply mul_le_mul_of_nonneg_left _ (walkKernel_nonneg G hloc D hD a ha haD x y)
-        rcases Finset.mem_insert.mp hy with rfl | hy
-        · exact mul_le_of_le_one_left (twoPoint_nonneg G p y z) p.2.2
-        · simpa only [twoPoint_symm G p] using twoPoint_adj_lower G p z x y ((mem_neighbors ..).mp hy)
-  have hpoint (x : V) : ‖((p:ℂ) • f) x‖ ≤ ‖P f x‖ := by
-    rw [lp.coeFn_smul,Pi.smul_apply,smul_eq_mul,hf,← Complex.ofReal_mul,
-      Complex.norm_of_nonneg (mul_nonneg p.2.1 (twoPoint_nonneg G p x z)),
-      walk_operator_apply G hloc a P hP]
-    simp only [hf,← Complex.ofReal_mul,← Complex.ofReal_sum]
-    rw [Complex.norm_of_nonneg (le_trans (mul_nonneg p.2.1 (twoPoint_nonneg G p x z)) (hrow x))]
-    exact hrow x
-  have hn := (lp.norm_mono (by norm_num : (2:ℝ≥0∞) ≠ 0) hpoint).trans (P.le_opNorm f)
-  rw [norm_smul,Complex.norm_of_nonneg p.2.1] at hn
-  exact (mul_le_mul_iff_left₀ hf0).mp hn
-
- 
-theorem pc_lt_one_of_nonamenable [Nonempty V] (G : BondGraph V E)
-    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
-    (hh : 0 < hV G) : pc G < 1 := by
-  let : Countable V := countable_vertices hloc hconn
-  let : Countable E := countable_edges hloc
-  obtain ⟨D,P,hD0,hD,hPn,_,hP⟩ := exists_strict_walk_operator G hloc hqt hh
-  have hDr : 0 < (D:ℝ) := by exact_mod_cast hD0
-  let p : unitInterval := ⟨(‖P‖+1)/2,by constructor <;> linarith [norm_nonneg P]⟩
-  have hp1 : p < 1 := by change (‖P‖+1)/2 < 1; linarith
-  have ha : 0 ≤ (1:ℝ)/(2*D) := by positivity
-  have haD : (1:ℝ)/(2*D)*D ≤ 1 := by
-    rw [div_mul_eq_mul_div]
-    apply (div_le_iff₀ (by positivity)).mpr
-    linarith
-  apply lt_of_le_of_ne (show pc G ≤ 1 from (pc G).2.2)
-  intro he
-  have hb := operatorBounded_of_lt_pc G hloc hqt p (he ▸ hp1)
-  have hle := parameter_le_walk_norm G hloc D hD _ ha haD P hP p hb
-  change (‖P‖+1)/2 ≤ ‖P‖ at hle
-  linarith
-
-end CriticalParameter
-
-section SpliceGeometry
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-def exchangeOn (mask ω η : Set E) : Set E := (ω \ mask) ∪ (η ∩ mask)
-
-@[simp] theorem mem_exchangeOn {mask ω η : Set E} {e : E} :
-    e ∈ exchangeOn mask ω η ↔ if e ∈ mask then e ∈ η else e ∈ ω := by
-  by_cases h : e ∈ mask <;> simp [exchangeOn,h]
-
-theorem exchangeOn_involutive (mask ω η : Set E) :
-    exchangeOn mask (exchangeOn mask ω η) (exchangeOn mask η ω) = ω := by
-  ext e
-  by_cases h : e ∈ mask <;> simp [h]
-
- 
-
-def spliceMask (G : BondGraph V E) (S B : Set V) (e : E) : Set E :=
-  insert e {f | (∃ b ∈ B, b ∈ G f) ∧ f ∈ internalEdges G Sᶜ}
-
-def spliceField (G : BondGraph V E) (S B : Set V) (e : E) (ω η : Set E) : Set E :=
-  exchangeOn (spliceMask G S B e) ω η
-
-theorem slit_cluster_subset (G : BondGraph V E) (S : Set V) (η : Set E)
-    (y : V) (hy : y ∉ S) : cluster G (η ∩ internalEdges G Sᶜ) y ⊆ Sᶜ := by
-  exact fun z hz => conn_internal_endpoint G Sᶜ hy hz
-
-theorem spliceMask_not_internal (G : BondGraph V E) (S B : Set V) (e : E)
-    (a b : V) (he : G e = s(a,b)) (hb : b ∉ S) {f : E}
-    (hf : f ∈ internalEdges G S) : f ∉ spliceMask G S B e := by
-  rintro (rfl | ⟨_,hout⟩)
-  · exact hb (((mem_internalEdges_of_ends G S f a b he).mp hf).2)
-  · exact (hout _ (Sym2.out_fst_mem _)) (hf _ (Sym2.out_fst_mem _))
-
-theorem spliceMask_of_internal_piece (G : BondGraph V E) (S B : Set V) (e : E)
-    (hB : B ⊆ Sᶜ) {f : E} (hf : f ∈ internalEdges G B) : f ∈ spliceMask G S B e := by
-  right
-  exact ⟨⟨(G f).out.1,hf _ (Sym2.out_fst_mem _),Sym2.out_fst_mem _⟩,
-    fun v hv => hB (hf v hv)⟩
-
- 
-theorem splice_internal_fields (G : BondGraph V E) (S B : Set V) (e : E)
-    (ω η : Set E) (a b : V) (he : G e = s(a,b)) (hb : b ∉ S) (hB : B ⊆ Sᶜ) :
-    spliceField G S B e ω η ∩ internalEdges G S = ω ∩ internalEdges G S ∧
-    spliceField G S B e ω η ∩ internalEdges G B = η ∩ internalEdges G B := by
-  constructor
-  · ext f
-    by_cases hf : f ∈ internalEdges G S
-    · simp only [Set.mem_inter_iff,hf,and_true,spliceField,mem_exchangeOn,
-        ite_eq_right (spliceMask_not_internal G S B e a b he hb hf)]
-    · simp only [Set.mem_inter_iff,hf,and_false]
-  · ext f
-    by_cases hf : f ∈ internalEdges G B
-    · simp only [Set.mem_inter_iff,hf,and_true,spliceField,mem_exchangeOn,
-        ite_eq_left (spliceMask_of_internal_piece G S B e hB hf)]
-    · simp only [Set.mem_inter_iff,hf,and_false]
-
- 
-theorem splice_crossing_unique (G : BondGraph V E) (ω η : Set E) (x : V)
-    (B : Set V) (e : E) {f : E} {a b : V}
-    (hf : f ∈ spliceField G (cluster G ω x) B e ω η)
-    (hab : G f = s(a,b)) (ha : a ∈ cluster G ω x) (hb : b ∉ cluster G ω x) : f = e := by
-  have hno : f ∉ ω := fun h => hb ((edge_cluster_sides G ω x h hab).mp ha)
-  have hm : f ∈ spliceMask G (cluster G ω x) B e := by
-    by_contra h
-    have hh := hf
-    simp only [spliceField,mem_exchangeOn,ite_eq_right h] at hh
-    exact hno hh
-  rcases hm with rfl | ⟨_,hout⟩
-  · rfl
-  · exact ((((mem_internalEdges_of_ends G _ f a b hab).mp hout).1) ha).elim
-
- 
-
-theorem splice_cluster (G : BondGraph V E) (ω η : Set E) (x y : V)
-    (hy : y ∉ cluster G ω x) (e : E) (a b : V)
-    (he : G e = s(a,b)) (heη : e ∈ η) (ha : a ∈ cluster G ω x)
-    (hb : b ∈ cluster G (η ∩ internalEdges G (cluster G ω x)ᶜ) y) :
-    let S := cluster G ω x
-    let B := cluster G (η ∩ internalEdges G Sᶜ) y
-    cluster G (spliceField G S B e ω η) x = S ∪ B := by
-  let S := cluster G ω x
-  let ζ := η ∩ internalEdges G Sᶜ
-  let B := cluster G ζ y
-  let ω' := spliceField G S B e ω η
-  change cluster G ω' x = S ∪ B
-  have hB : B ⊆ Sᶜ := slit_cluster_subset G S η y hy
-  have hbS : b ∉ S := hB hb
-  have hfields := splice_internal_fields G S B e ω η a b he hbS hB
-  have he' : e ∈ ω' := by
-    change e ∈ exchangeOn (spliceMask G S B e) ω η
-    rw [mem_exchangeOn,ite_eq_left (show e ∈ spliceMask G S B e from Set.mem_insert _ _)]
-    exact heη
-  have hSconn (z : V) (hz : z ∈ S) : Conn G ω' x z := by
-    have h := conn_restrict_cluster G ω x (mem_cluster_self G ω x) hz
-    apply conn_mono G _ h
-    intro f hf
-    exact (hfields.1.symm ▸ hf).1
-  have hBconn (u v : V) (hu : u ∈ B) (hv : v ∈ B) : Conn G ω' u v := by
-    have h := conn_restrict_cluster G ζ y hu (conn_trans G ζ (conn_symm G ζ hu) hv)
-    apply conn_mono G _ h
-    intro f hf
-    exact (hfields.2.symm ▸ (show f ∈ η ∩ internalEdges G B from ⟨hf.1.1,hf.2⟩)).1
-  apply Set.Subset.antisymm
-  · intro z hz
-    induction hz with
-    | refl => exact Or.inl (mem_cluster_self G ω x)
-    | @tail u v hxu huv ih =>
-      obtain ⟨f,hf,hfv⟩ := huv
-      by_cases hvS : v ∈ S
-      · exact Or.inl hvS
-      · rcases ih with huS | huB
-        · have hfe := splice_crossing_unique G ω η x B e hf hfv huS hvS
-          subst f
-          have hends := Sym2.eq_iff.mp (he.symm.trans hfv)
-          rcases hends with ⟨hau,hbv⟩ | ⟨hav,hbu⟩
-          · exact Or.inr (hbv ▸ hb)
-          · exact (hvS (hav ▸ ha)).elim
-        · have huS : u ∉ S := hB huB
-          have hm : f ∈ spliceMask G S B e := by
-            right
-            exact ⟨⟨u,huB,by simp [hfv]⟩,(mem_internalEdges_of_ends G _ f u v hfv).mpr ⟨huS,hvS⟩⟩
-          have hfη : f ∈ η := by
-            change f ∈ exchangeOn (spliceMask G S B e) ω η at hf
-            rwa [mem_exchangeOn,ite_eq_left hm] at hf
-          exact Or.inr ((edge_cluster_sides G ζ y
-            ⟨hfη,(mem_internalEdges_of_ends G _ f u v hfv).mpr ⟨huS,hvS⟩⟩ hfv).mp huB)
-  · intro z hz
-    rcases hz with hz | hz
-    · exact hSconn z hz
-    · exact ((hSconn a ha).tail ⟨e,he',he⟩).trans (hBconn b z hb hz)
-
- 
-theorem splice_seam_pivotal (G : BondGraph V E) (ω η : Set E) (x y : V)
-    (hy : y ∉ cluster G ω x) (e : E) (a b : V)
-    (he : G e = s(a,b)) (heη : e ∈ η) (ha : a ∈ cluster G ω x)
-    (hb : b ∈ cluster G (η ∩ internalEdges G (cluster G ω x)ᶜ) y)
-    (z : V) (hz : z ∈ cluster G (η ∩ internalEdges G (cluster G ω x)ᶜ) y) :
-    let S := cluster G ω x
-    let B := cluster G (η ∩ internalEdges G Sᶜ) y
-    OpenMarkedPivotal G (spliceField G S B e ω η) {z} x e := by
-  let S := cluster G ω x
-  let B := cluster G (η ∩ internalEdges G Sᶜ) y
-  let ω' := spliceField G S B e ω η
-  have hB := slit_cluster_subset G S η y hy
-  have hC := splice_cluster G ω η x y hy e a b he heη ha hb
-  change cluster G ω' x = S ∪ B at hC
-  refine ⟨?_,⟨z,Set.mem_singleton z,by change z ∈ cluster G ω' x; rw [hC]; exact Or.inr hz⟩,?_⟩
-  · change e ∈ exchangeOn (spliceMask G S B e) ω η
-    rw [mem_exchangeOn,ite_eq_left (show e ∈ spliceMask G S B e from Set.mem_insert _ _)]
-    exact heη
-  · rintro ⟨v,hv,hxv⟩
-    have hvz : v = z := Set.mem_singleton_iff.mp hv
-    subst v
-    have hsub : cluster G (ω' \ {e}) x ⊆ S := by
-      intro v hxv
-      induction hxv with
-      | refl => exact mem_cluster_self G ω x
-      | @tail u v hxu huv ih =>
-        obtain ⟨f,hf,hfv⟩ := huv
-        by_contra hvS
-        have hfe := splice_crossing_unique G ω η x B e hf.1 hfv ih hvS
-        exact hf.2 (by simpa only [Set.mem_singleton_iff] using hfe)
-    exact hB hz (hsub hxv)
-
-end SpliceGeometry
-
-section SpliceInverse
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem splice_close_seam_cluster (G : BondGraph V E) (ω η : Set E) (x y : V)
-    (hy : y ∉ cluster G ω x) (e : E) (a b : V)
-    (he : G e = s(a,b)) (_heη : e ∈ η) (_ha : a ∈ cluster G ω x)
-    (hb : b ∈ cluster G (η ∩ internalEdges G (cluster G ω x)ᶜ) y) :
-    let S := cluster G ω x
-    let B := cluster G (η ∩ internalEdges G Sᶜ) y
-    cluster G (spliceField G S B e ω η \ {e}) x = S := by
-  let S := cluster G ω x
-  let B := cluster G (η ∩ internalEdges G Sᶜ) y
-  let ω' := spliceField G S B e ω η
-  have hB := slit_cluster_subset G S η y hy
-  have hf := (splice_internal_fields G S B e ω η a b he (hB hb) hB).1
-  apply Set.Subset.antisymm
-  · intro v hxv
-    induction hxv with
-    | refl => exact mem_cluster_self G ω x
-    | @tail u v _ huv ih =>
-      obtain ⟨f,hf,hfv⟩ := huv
-      by_contra hvS
-      have hfe := splice_crossing_unique G ω η x B e hf.1 hfv ih hvS
-      exact hf.2 (Set.mem_singleton_iff.mpr hfe)
-  · intro z hz
-    have hc := conn_restrict_cluster G ω x (mem_cluster_self G ω x) hz
-    apply conn_mono G _ hc
-    intro f hfin
-    refine ⟨(hf.symm ▸ hfin).1,?_⟩
-    intro hfe
-    have hfe' := Set.mem_singleton_iff.mp hfe
-    subst f
-    exact hB hb (((mem_internalEdges_of_ends G S e a b he).mp hfin.2).2)
-
- 
-theorem splice_recover_piece (G : BondGraph V E) (ω η : Set E) (x y : V)
-    (hy : y ∉ cluster G ω x) (e : E) (a b : V)
-    (he : G e = s(a,b)) (heη : e ∈ η) (ha : a ∈ cluster G ω x)
-    (hb : b ∈ cluster G (η ∩ internalEdges G (cluster G ω x)ᶜ) y) :
-    let S := cluster G ω x
-    let B := cluster G (η ∩ internalEdges G Sᶜ) y
-    let ω' := spliceField G S B e ω η
-    cluster G ω' x \ cluster G (ω' \ {e}) x = B := by
-  dsimp only
-  rw [splice_cluster G ω η x y hy e a b he heη ha hb,
-    splice_close_seam_cluster G ω η x y hy e a b he heη ha hb]
-  have hB := slit_cluster_subset G (cluster G ω x) η y hy
-  ext z
-  constructor
-  · rintro ⟨hz,hzS⟩
-    exact hz.resolve_left hzS
-  · intro hz
-    exact ⟨Or.inr hz,hB hz⟩
-
- 
-def recoveredMask (G : BondGraph V E) (ω' : Set E) (x : V) (e : E) : Set E :=
-  spliceMask G (cluster G (ω' \ {e}) x)
-    (cluster G ω' x \ cluster G (ω' \ {e}) x) e
-
-theorem recoveredMask_splice (G : BondGraph V E) (ω η : Set E) (x y : V)
-    (hy : y ∉ cluster G ω x) (e : E) (a b : V)
-    (he : G e = s(a,b)) (heη : e ∈ η) (ha : a ∈ cluster G ω x)
-    (hb : b ∈ cluster G (η ∩ internalEdges G (cluster G ω x)ᶜ) y) :
-    let S := cluster G ω x
-    let B := cluster G (η ∩ internalEdges G Sᶜ) y
-    recoveredMask G (spliceField G S B e ω η) x e = spliceMask G S B e := by
-  unfold recoveredMask
-  dsimp only
-  rw [splice_recover_piece G ω η x y hy e a b he heη ha hb,
-    splice_close_seam_cluster G ω η x y hy e a b he heη ha hb]
-
- 
-def undoSplice (G : BondGraph V E) (x : V) (e : E) (ω' η' : Set E) : Set E × Set E :=
-  let M := recoveredMask G ω' x e
-  (exchangeOn M ω' η',exchangeOn M η' ω')
-
- 
-theorem undoSplice_of_valid (G : BondGraph V E) (ω η : Set E) (x y : V)
-    (hy : y ∉ cluster G ω x) (e : E) (a b : V)
-    (he : G e = s(a,b)) (heη : e ∈ η) (ha : a ∈ cluster G ω x)
-    (hb : b ∈ cluster G (η ∩ internalEdges G (cluster G ω x)ᶜ) y) :
-    let S := cluster G ω x
-    let B := cluster G (η ∩ internalEdges G Sᶜ) y
-    let M := spliceMask G S B e
-    undoSplice G x e (exchangeOn M ω η) (exchangeOn M η ω) = (ω,η) := by
-  dsimp only [undoSplice]
-  rw [show recoveredMask G (exchangeOn
-      (spliceMask G (cluster G ω x) (cluster G (η ∩ internalEdges G (cluster G ω x)ᶜ) y) e) ω η) x e =
-      spliceMask G (cluster G ω x) (cluster G (η ∩ internalEdges G (cluster G ω x)ᶜ) y) e from
-    recoveredMask_splice G ω η x y hy e a b he heη ha hb]
-  rw [exchangeOn_involutive,exchangeOn_involutive]
-
- 
-theorem spliceMask_finite (G : BondGraph V E) (hloc : LocallyFinite G)
-    (S B : Set V) (hB : B.Finite) (e : E) : (spliceMask G S B e).Finite := by
-  apply Set.Finite.insert
-  apply (hB.biUnion (fun b _ => hloc b)).subset
-  rintro f ⟨⟨b,hb,hbf⟩,_⟩
-  exact Set.mem_iUnion₂.mpr ⟨b,hb,hbf⟩
-
-end SpliceInverse
-
-section FiniteExchange
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def finiteExchange (M ω η : Finset E) : Finset E := (ω \ M) ∪ (η ∩ M)
-
-@[simp] theorem coe_finiteExchange (M ω η : Finset E) :
-    (finiteExchange M ω η : Set E) = exchangeOn (M:Set E) (ω:Set E) (η:Set E) := by
-  simp only [finiteExchange,exchangeOn,Finset.coe_union,Finset.coe_sdiff,Finset.coe_inter]
-
-theorem finiteExchange_card (M ω η : Finset E) :
-    (finiteExchange M ω η).card = (ω \ M).card+(η ∩ M).card := by
-  apply Finset.card_union_of_disjoint
-  exact Finset.disjoint_left.mpr fun e he he' => (Finset.mem_sdiff.mp he).2 (Finset.mem_inter.mp he').2
-
-theorem finiteExchange_card_sum (M ω η : Finset E) :
-    (finiteExchange M ω η).card+(finiteExchange M η ω).card = ω.card+η.card := by
-  rw [finiteExchange_card,finiteExchange_card]
-  have hω := Finset.card_sdiff_add_card_inter ω M
-  have hη := Finset.card_sdiff_add_card_inter η M
-  omega
-
-theorem finiteExchange_subset (M Q ω η : Finset E) (hω : ω ⊆ Q) (hη : η ⊆ Q) :
-    finiteExchange M ω η ⊆ Q := by
-  intro e he
-  rcases Finset.mem_union.mp he with he | he
-  · exact hω (Finset.mem_sdiff.mp he).1
-  · exact hη (Finset.mem_inter.mp he).1
-
- 
-theorem bernoulliWeight_exchange (M Q ω η : Finset E) (hω : ω ⊆ Q) (hη : η ⊆ Q)
-    (p : ℝ) :
-    bernoulliWeight Q (finiteExchange M ω η) p * bernoulliWeight Q (finiteExchange M η ω) p =
-      bernoulliWeight Q ω p * bernoulliWeight Q η p := by
-  have hc := finiteExchange_card_sum M ω η
-  have hcω := Finset.card_le_card hω
-  have hcη := Finset.card_le_card hη
-  have hcω' := Finset.card_le_card (finiteExchange_subset M Q ω η hω hη)
-  have hcη' := Finset.card_le_card (finiteExchange_subset M Q η ω hη hω)
-  have hd : (Q.card-(finiteExchange M ω η).card)+(Q.card-(finiteExchange M η ω).card) =
-      (Q.card-ω.card)+(Q.card-η.card) := by omega
-  unfold bernoulliWeight
-  calc
-    _ = p^((finiteExchange M ω η).card+(finiteExchange M η ω).card) *
-      (1-p)^((Q.card-(finiteExchange M ω η).card)+(Q.card-(finiteExchange M η ω).card)) := by
-        rw [pow_add,pow_add]; ring
-    _ = p^(ω.card+η.card)*(1-p)^((Q.card-ω.card)+(Q.card-η.card)) := by rw [hc,hd]
-    _ = _ := by rw [pow_add,pow_add]; ring
-
-end FiniteExchange
-
-section DeletionGoodness
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-def deletionContacts (G : BondGraph V E) (ω : Set E) (A B : Set V) : Set E :=
-  {e | e ∈ ω ∧ ∃ a ∈ B, ∃ b ∈ A, G e = s(a,b)}
-
- 
-
-def DeletionGoodForScale (G : BondGraph V E) (ω : Set E) (x : V) (L : ℝ) : Prop :=
-  ∀ A ⊆ cluster G ω x, ∀ y ∈ cluster G ω x \ A,
-    let B := cluster G (ω ∩ internalEdges G Aᶜ) y
-    0 < (deletionContacts G ω A B).ncard →
-    ∀ a ∈ B, ∀ b ∈ B,
-      (pivotalDistance G (ω ∩ internalEdges G B) a b : ℝ) ≤
-        (deletionContacts G ω A B).ncard*L
-
-theorem deleted_cluster_subset (G : BondGraph V E) (ω : Set E) (x y : V)
-    (A : Set V) (hy : y ∈ cluster G ω x \ A) :
-    cluster G (ω ∩ internalEdges G Aᶜ) y ⊆ cluster G ω x \ A := by
-  intro z hz
-  exact ⟨conn_trans G ω hy.1 (conn_mono G Set.inter_subset_left hz),
-    conn_internal_endpoint G Aᶜ hy.2 hz⟩
-
-theorem contactSet_deleted_cluster (G : BondGraph V E) (ω : Set E) (A : Set V)
-    (y : V) (hy : y ∉ A) :
-    let B := cluster G (ω ∩ internalEdges G Aᶜ) y
-    contactSet G ω B = deletionContacts G ω A B := by
-  let ζ := ω ∩ internalEdges G Aᶜ
-  let B := cluster G ζ y
-  have hB : B ⊆ Aᶜ := slit_cluster_subset G A ω y hy
-  ext e
-  constructor
-  · rintro ⟨he,a,b,hab,ha,hb⟩
-    refine ⟨he,a,ha,b,?_,hab⟩
-    by_contra hbA
-    have heζ : e ∈ ζ := ⟨he,(mem_internalEdges_of_ends G Aᶜ e a b hab).mpr ⟨hB ha,hbA⟩⟩
-    exact hb ((edge_cluster_sides G ζ y heζ hab).mp ha)
-  · rintro ⟨he,a,ha,b,hb,hab⟩
-    exact ⟨he,a,b,hab,ha,fun hh => hB hh hb⟩
-
- 
-
-theorem goodForScale_iff_deletion (G : BondGraph V E) (ω : Set E) (x : V) (L : ℝ)
-    (hC : (cluster G ω x).Finite) :
-    GoodForScale G ω x L ↔ DeletionGoodForScale G ω x L := by
-  constructor
-  · intro h A _ y hy
-    let ζ := ω ∩ internalEdges G Aᶜ
-    let B := cluster G ζ y
-    have hBC := deleted_cluster_subset G ω x y A hy
-    have hfin : B.Finite := hC.subset (fun _ hz => (hBC hz).1)
-    let W := hfin.toFinset
-    have hW : (W : Set V) = B := hfin.coe_toFinset
-    have hsub : (W : Set V) ⊆ cluster G ω x := by rw [hW]; exact fun _ hz => (hBC hz).1
-    have hi (a : V) (ha : a ∈ W) (b : V) (hb : b ∈ W) :
-        Conn G (ω ∩ internalEdges G (W : Set V)) a b := by
-      have ha' : a ∈ B := hW ▸ ha
-      have hb' : b ∈ B := hW ▸ hb
-      have hc := conn_restrict_cluster G ζ y ha' (conn_trans G ζ (conn_symm G ζ ha') hb')
-      rw [hW]
-      apply conn_mono G ?_ hc
-      intro e he
-      exact ⟨he.1.1,he.2⟩
-    have hh := h W hsub hi
-    rw [hW,contactSet_deleted_cluster G ω A y hy.2] at hh
-    dsimp only
-    intro hK a ha b hb
-    exact hh hK a (by change a ∈ (W : Set V); rw [hW]; exact ha) b
-      (by change b ∈ (W : Set V); rw [hW]; exact hb)
-  · intro h W hW hconn hK a ha b hb
-    let C := cluster G ω x
-    let A := C \ (W : Set V)
-    have haA : a ∈ C \ A := ⟨hW ha,fun hh => hh.2 ha⟩
-    have hWA : (W : Set V) ⊆ Aᶜ := fun _ hz hh => hh.2 hz
-    have heq : cluster G (ω ∩ internalEdges G Aᶜ) a = (W : Set V) := by
-      apply Set.Subset.antisymm
-      · intro z hz
-        have hzC := deleted_cluster_subset G ω x a A haA hz
-        by_contra hzW
-        exact hzC.2 ⟨hzC.1,hzW⟩
-      · intro z hz
-        apply conn_mono G ?_ (hconn a ha z hz)
-        intro e he
-        exact ⟨he.1,fun v hv => hWA (he.2 v hv)⟩
-    have hh := h A Set.sdiff_subset a haA
-    dsimp only at hh
-    have hc := contactSet_deleted_cluster G ω A a haA.2
-    dsimp only at hc
-    rw [heq] at hh hc
-    rw [← hc] at hh
-    exact hh hK a ha b hb
-
- 
-theorem capped_good_local (G : BondGraph V E) (hloc : LocallyFinite G)
-    (x : V) (M : ℕ) (L : ℝ) (ω : Set E) :
-    let Q := incidentSupport G hloc (finite_reachBall hloc x M).toFinset
-    ((cluster G ω x).Finite ∧ (cluster G ω x).ncard ≤ M ∧ GoodForScale G ω x L) ↔
-      ((cluster G (ω ∩ (Q : Set E)) x).Finite ∧
-        (cluster G (ω ∩ (Q : Set E)) x).ncard ≤ M ∧
-          GoodForScale G (ω ∩ (Q : Set E)) x L) := by
-  have hc := capped_cluster_local G hloc x M ω
-  have hb := cappedBadGoodness_local G hloc x M L ω
-  unfold CappedBadGoodness at hb
-  dsimp only at hc hb ⊢
-  constructor
-  · rintro ⟨hf,hM,hg⟩
-    obtain ⟨hf',hM'⟩ := hc.mp ⟨hf,hM⟩
-    refine ⟨hf',hM',?_⟩
-    by_contra hng
-    exact (hb.mpr ⟨hf',hM',hng⟩).2.2 hg
-  · rintro ⟨hf,hM,hg⟩
-    obtain ⟨hf',hM'⟩ := hc.mpr ⟨hf,hM⟩
-    refine ⟨hf',hM',?_⟩
-    by_contra hng
-    exact (hb.mp ⟨hf',hM',hng⟩).2.2 hg
-
-theorem measurableSet_capped_good (G : BondGraph V E) (hloc : LocallyFinite G)
-    (x : V) (M : ℕ) (L : ℝ) :
-    MeasurableSet {ω : Set E | (cluster G ω x).Finite ∧
-      (cluster G ω x).ncard ≤ M ∧ GoodForScale G ω x L} := by
-  let Q := incidentSupport G hloc (finite_reachBall hloc x M).toFinset
-  have he : {ω : Set E | (cluster G ω x).Finite ∧
-      (cluster G ω x).ncard ≤ M ∧ GoodForScale G ω x L} =
-    {ω : Set E | (cluster G (ω ∩ (Q : Set E)) x).Finite ∧
-      (cluster G (ω ∩ (Q : Set E)) x).ncard ≤ M ∧ GoodForScale G (ω ∩ (Q : Set E)) x L} := by
-    ext ω
-    exact capped_good_local G hloc x M L ω
-  rw [he]
-  exact measurableSet_finite_restriction Q {η : Set E | (cluster G η x).Finite ∧
-      (cluster G η x).ncard ≤ M ∧ GoodForScale G η x L}
-
-end DeletionGoodness
-
-section SpliceSelection
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem exists_maximal_splice_index (G : BondGraph V E) (η : ℕ → Set E) (y : ℕ → V)
-    (k j : ℕ) (hj : j ≤ k)
-    (hchain : ∀ i ∈ Finset.Icc 1 k, Conn G (η i) (y i) (y (i-1)))
-    (S : Set V) (hyj : y j ∈ S) (hyk : y k ∉ S) :
-    ∃ i ∈ Finset.Icc (j+1) k,
-      ((cluster G (η i) (y i)) ∩ S).Nonempty ∧ y i ∉ S ∧
-      ∀ l, i < l → l ≤ k → Disjoint (cluster G (η l) (y l)) S := by
-  have hjk : j < k := by
-    by_contra h
-    have hEq : j = k := by omega
-    exact hyk (hEq ▸ hyj)
-  let T := (Finset.Icc (j+1) k).filter fun i => ((cluster G (η i) (y i)) ∩ S).Nonempty
-  have hT : T.Nonempty := by
-    refine ⟨j+1,Finset.mem_filter.mpr ⟨Finset.mem_Icc.mpr ⟨le_rfl,by omega⟩,?_⟩⟩
-    refine ⟨y j,?_,hyj⟩
-    simpa only [Nat.add_sub_cancel,cluster,Set.mem_ofPred_eq] using hchain (j+1) (Finset.mem_Icc.mpr ⟨by omega,by omega⟩)
-  let i := T.max' hT
-  have hi : i ∈ T := Finset.max'_mem T hT
-  obtain ⟨hiI,hiS⟩ := Finset.mem_filter.mp hi
-  have hiU := (Finset.mem_Icc.mp hiI).2
-  have hiL := (Finset.mem_Icc.mp hiI).1
-  have hl (l : ℕ) (hil : i < l) (hlk : l ≤ k) : Disjoint (cluster G (η l) (y l)) S := by
-    apply Set.disjoint_left.mpr
-    intro z hz hzS
-    have hlT : l ∈ T := Finset.mem_filter.mpr
-      ⟨Finset.mem_Icc.mpr ⟨by omega,hlk⟩,⟨z,hz,hzS⟩⟩
-    have hli : l ≤ i := Finset.le_max' T l hlT
-    omega
-  refine ⟨i,hiI,hiS,?_,hl⟩
-  intro hyi
-  by_cases hik : i = k
-  · exact hyk (hik ▸ hyi)
-  · have hic : y i ∈ cluster G (η (i+1)) (y (i+1)) := by
-      simpa only [Nat.add_sub_cancel,cluster,Set.mem_ofPred_eq] using hchain (i+1) (Finset.mem_Icc.mpr ⟨by omega,by omega⟩)
-    exact Set.disjoint_left.mp (hl (i+1) (by omega) (by omega)) hic hyi
-
- 
-
-theorem exists_splice_contact (G : BondGraph V E) (η : Set E) (S : Set V) (y : V)
-    (hy : y ∉ S) (hmeet : (cluster G η y ∩ S).Nonempty) :
-    ∃ e a b, e ∈ η ∧ G e = s(a,b) ∧ a ∈ S ∧
-      b ∈ cluster G (η ∩ internalEdges G Sᶜ) y := by
-  obtain ⟨v,hv,hvS⟩ := hmeet
-  obtain ⟨b,a,e,_,ha,he,hab,hb⟩ := conn_first_entry G S η hy hvS hv
-  exact ⟨e,a,b,he,hab.trans (Sym2.eq_swap),ha,hb⟩
-
- 
-theorem splice_cluster_cap (G : BondGraph V E) (ω η : Set E) (x y : V)
-    (hy : y ∉ cluster G ω x) (e : E) (a b : V)
-    (he : G e = s(a,b)) (heη : e ∈ η) (ha : a ∈ cluster G ω x)
-    (hb : b ∈ cluster G (η ∩ internalEdges G (cluster G ω x)ᶜ) y)
-    (N M : ℕ) (hS : (cluster G ω x).Finite) (hN : (cluster G ω x).ncard ≤ N)
-    (hC : (cluster G η y).Finite) (hM : (cluster G η y).ncard ≤ M) :
-    let S := cluster G ω x
-    let B := cluster G (η ∩ internalEdges G Sᶜ) y
-    (cluster G (spliceField G S B e ω η) x).Finite ∧
-      (cluster G (spliceField G S B e ω η) x).ncard ≤ N+M := by
-  let S := cluster G ω x
-  let B := cluster G (η ∩ internalEdges G Sᶜ) y
-  have hBC : B ⊆ cluster G η y := fun _ hz => conn_mono G Set.inter_subset_left hz
-  have hB : B.Finite := hC.subset hBC
-  dsimp only
-  rw [splice_cluster G ω η x y hy e a b he heη ha hb]
-  refine ⟨hS.union hB,?_⟩
-  exact (Set.ncard_union_le _ _).trans (Nat.add_le_add hN ((Set.ncard_le_ncard hBC hC).trans hM))
-
- 
-
-theorem cluster_eq_of_incident_agreement (G : BondGraph V E) (ω η : Set E) (x : V)
-    (he : ∀ e, (∃ v ∈ cluster G ω x, v ∈ G e) → (e ∈ ω ↔ e ∈ η)) :
-    cluster G η x = cluster G ω x := by
-  have hforward : cluster G ω x ⊆ cluster G η x := by
-    intro y hy
-    induction hy with
-    | refl => exact conn_refl G _ _
-    | @tail a b hxa hab ih =>
-      obtain ⟨e,heω,hab⟩ := hab
-      exact ih.tail ⟨e,(he e ⟨a,hxa,by simp [hab]⟩).mp heω,hab⟩
-  apply Set.Subset.antisymm _ hforward
-  intro y hy
-  induction hy with
-  | refl => exact conn_refl G _ _
-  | @tail a b _ hab ih =>
-    obtain ⟨e,heη,hab⟩ := hab
-    exact ih.tail ⟨e,(he e ⟨a,ih,by simp [hab]⟩).mpr heη,hab⟩
-
- 
-
-theorem spliceMask_avoids_prefix (G : BondGraph V E) (S B R : Set V) (hRS : R ⊆ S)
-    (e : E) (a b : V) (hab : G e = s(a,b)) (ha : a ∉ R) (hb : b ∉ S)
-    {f : E} (hf : ∃ v ∈ R, v ∈ G f) : f ∉ spliceMask G S B e := by
-  rintro (rfl | ⟨_,hfo⟩)
-  · obtain ⟨v,hv,hvf⟩ := hf
-    rw [hab] at hvf
-    rcases Sym2.mem_iff.mp hvf with rfl | rfl
-    · exact ha hv
-    · exact hb (hRS hv)
-  · obtain ⟨v,hv,hvf⟩ := hf
-    exact hfo v hvf (hRS hv)
-
- 
-
-theorem splice_preserves_prefix_cut (G : BondGraph V E) (ω η : Set E) (x : V)
-    (B : Set V) (e f : E) (a b : V) (hab : G e = s(a,b))
-    (ha : a ∉ cluster G (ω \ {f}) x) (hb : b ∉ cluster G ω x) :
-    cluster G ((spliceField G (cluster G ω x) B e ω η) \ {f}) x =
-      cluster G (ω \ {f}) x := by
-  apply cluster_eq_of_incident_agreement
-  intro g hg
-  have hRS : cluster G (ω \ {f}) x ⊆ cluster G ω x := fun _ h => conn_mono G Set.sdiff_subset h
-  have hno := spliceMask_avoids_prefix G _ B _ hRS e a b hab ha hb hg
-  simp only [Set.mem_sdiff,spliceField,mem_exchangeOn,ite_eq_right hno]
-
-end SpliceSelection
-
-section SpliceTrace
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-inductive SpliceTrace (G : BondGraph V E) (x : V) (η : ℕ → Set E) (y : ℕ → V) (k : ℕ) :
-    ℕ → Set E → List (ℕ × E) → Set E → Prop
-  | stop (j : ℕ) (ω : Set E) (hstop : y k ∈ cluster G ω x) :
-      SpliceTrace G x η y k j ω [] ω
-  | step (j i : ℕ) (ω : Set E) (e : E) (a b : V) (rest : List (ℕ × E)) (ωf : Set E)
-      (hnot : y k ∉ cluster G ω x) (hji : j < i) (hik : i ≤ k)
-      (hmax : ∀ l, i < l → l ≤ k → Disjoint (cluster G (η l) (y l)) (cluster G ω x))
-      (hy : y i ∉ cluster G ω x) (he : e ∈ η i) (hab : G e = s(a,b))
-      (ha : a ∈ cluster G ω x)
-      (hb : b ∈ cluster G (η i ∩ internalEdges G (cluster G ω x)ᶜ) (y i))
-      (htail : SpliceTrace G x η y k i
-        (spliceField G (cluster G ω x)
-          (cluster G (η i ∩ internalEdges G (cluster G ω x)ᶜ) (y i)) e ω (η i)) rest ωf) :
-      SpliceTrace G x η y k j ω ((i,e)::rest) ωf
-
- 
-
-theorem exists_spliceTrace (G : BondGraph V E) (x : V) (η : ℕ → Set E) (y : ℕ → V)
-    (k M : ℕ)
-    (hchain : ∀ i ∈ Finset.Icc 1 k, Conn G (η i) (y i) (y (i-1)))
-    (hcaps : ∀ i ∈ Finset.Icc 1 k, (cluster G (η i) (y i)).Finite ∧
-      (cluster G (η i) (y i)).ncard ≤ M)
-    (j : ℕ) (hj : j ≤ k) (ω : Set E) (hyj : y j ∈ cluster G ω x)
-    (N : ℕ) (hfin : (cluster G ω x).Finite) (hN : (cluster G ω x).ncard ≤ N) :
-    ∃ steps ωf, SpliceTrace G x η y k j ω steps ωf ∧ steps.length ≤ k-j ∧
-      y k ∈ cluster G ωf x ∧ (cluster G ωf x).Finite ∧
-        (cluster G ωf x).ncard ≤ N+(k-j)*M := by
-  generalize hn : k-j = n
-  induction n using Nat.strong_induction_on generalizing j ω N with
-  | h n ih =>
-    by_cases hz : y k ∈ cluster G ω x
-    · refine ⟨[],ω,SpliceTrace.stop j ω hz,by simp,hz,hfin,?_⟩
-      exact hN.trans (Nat.le_add_right ..)
-    · obtain ⟨i,hiI,hmeet,hyi,hmax⟩ :=
-        exists_maximal_splice_index G η y k j hj hchain (cluster G ω x) hyj hz
-      obtain ⟨hji,hik⟩ := Finset.mem_Icc.mp hiI
-      obtain ⟨e,a,b,he,hab,ha,hb⟩ := exists_splice_contact G (η i) (cluster G ω x) (y i) hyi hmeet
-      let B := cluster G (η i ∩ internalEdges G (cluster G ω x)ᶜ) (y i)
-      let ω' := spliceField G (cluster G ω x) B e ω (η i)
-      have hi1 : i ∈ Finset.Icc 1 k := Finset.mem_Icc.mpr ⟨by omega,hik⟩
-      have hcap := splice_cluster_cap G ω (η i) x (y i) hyi e a b hab he ha hb N M hfin hN
-        (hcaps i hi1).1 (hcaps i hi1).2
-      have hyt : y i ∈ cluster G ω' x := by
-        change y i ∈ cluster G (spliceField G (cluster G ω x) B e ω (η i)) x
-        rw [splice_cluster G ω (η i) x (y i) hyi e a b hab he ha hb]
-        exact Or.inr (mem_cluster_self G _ _)
-      have hlt : k-i < n := by omega
-      obtain ⟨steps,ωf,htrace,hlen,hconn,hf,hs⟩ :=
-        ih (k-i) hlt i hik ω' hyt (N+M) hcap.1 hcap.2 rfl
-      refine ⟨(i,e)::steps,ωf,
-        SpliceTrace.step j i ω e a b steps ωf hz (by omega) hik hmax hyi he hab ha hb htrace,
-        ?_,hconn,hf,?_⟩
-      · simp only [List.length_cons]
-        omega
-      · have hm : (k-i+1)*M ≤ (k-j)*M := Nat.mul_le_mul_right M (by omega)
-        rw [Nat.add_mul,one_mul] at hm
-        calc
-          _ ≤ (N+M)+(k-i)*M := hs
-          _ ≤ N+(k-j)*M := by omega
-          _ = N+n*M := by rw [hn]
-
- 
-theorem SpliceTrace.cluster_subset (G : BondGraph V E) (x : V) (η : ℕ → Set E)
-    (y : ℕ → V) (k j : ℕ) (ω ωf : Set E) (steps : List (ℕ × E))
-    (h : SpliceTrace G x η y k j ω steps ωf) : cluster G ω x ⊆ cluster G ωf x := by
-  induction h with
-  | stop => exact Set.Subset.rfl
-  | @step j i ω e a b rest ωf _ _ _ _ hy he hab ha hb ht ih =>
-    apply Set.Subset.trans _ ih
-    rw [splice_cluster G ω (η i) x (y i) hy e a b hab he ha hb]
-    exact Set.subset_union_left
-
- 
-
-theorem SpliceTrace.preserve_cut (G : BondGraph V E) (x : V) (η : ℕ → Set E)
-    (y : ℕ → V) (k j : ℕ) (ω ωf : Set E) (steps : List (ℕ × E))
-    (h : SpliceTrace G x η y k j ω steps ωf) (f : E)
-    (havoid : ∀ l, j < l → l ≤ k →
-      Disjoint (cluster G (η l) (y l)) (cluster G (ω \ {f}) x)) :
-    cluster G (ωf \ {f}) x = cluster G (ω \ {f}) x := by
-  induction h with
-  | stop => rfl
-  | @step j i ω e a b rest ωf hn hji hik hmax hy he hab ha hb ht ih =>
-    have hBC : b ∈ cluster G (η i) (y i) := conn_mono G Set.inter_subset_left hb
-    have haC : a ∈ cluster G (η i) (y i) := (edge_cluster_sides G (η i) (y i) he hab).mpr hBC
-    have haR : a ∉ cluster G (ω \ {f}) x := fun hh => Set.disjoint_left.mp (havoid i hji hik) haC hh
-    have hbS : b ∉ cluster G ω x := slit_cluster_subset G _ (η i) (y i) hy hb
-    have heq := splice_preserves_prefix_cut G ω (η i) x
-      (cluster G (η i ∩ internalEdges G (cluster G ω x)ᶜ) (y i)) e f a b hab haR hbS
-    rw [ih]
-    · exact heq
-    · intro l hil hlk
-      rw [heq]
-      exact havoid l (hji.trans hil) hlk
-
- 
-theorem SpliceTrace.endpoint (G : BondGraph V E) (x : V) (η : ℕ → Set E)
-    (y : ℕ → V) (k j : ℕ) (ω ωf : Set E) (steps : List (ℕ × E))
-    (h : SpliceTrace G x η y k j ω steps ωf) : y k ∈ cluster G ωf x := by
-  induction h with
-  | stop _ _ hs => exact hs
-  | step => assumption
-
- 
-
-theorem SpliceTrace.seams_pivotal (G : BondGraph V E) (x : V) (η : ℕ → Set E)
-    (y : ℕ → V) (k j : ℕ) (ω ωf : Set E) (steps : List (ℕ × E))
-    (h : SpliceTrace G x η y k j ω steps ωf) :
-    ∀ q ∈ steps, OpenMarkedPivotal G ωf {y k} x q.2 := by
-  induction h with
-  | stop => simp
-  | @step j i ω e a b rest ωf hn hji hik hmax hy he hab ha hb ht ih =>
-    intro q hq
-    rcases List.mem_cons.mp hq with rfl | hq
-    · let S := cluster G ω x
-      let B := cluster G (η i ∩ internalEdges G Sᶜ) (y i)
-      let ω' := spliceField G S B e ω (η i)
-      have hcut : cluster G (ω' \ {e}) x = S :=
-        splice_close_seam_cluster G ω (η i) x (y i) hy e a b hab he ha hb
-      have hfinal : cluster G (ωf \ {e}) x = S := by
-        rw [SpliceTrace.preserve_cut G x η y k i ω' ωf rest ht e]
-        · exact hcut
-        · intro l hil hlk
-          rw [hcut]
-          exact hmax l hil hlk
-      have hconn := SpliceTrace.endpoint G x η y k i ω' ωf rest ht
-      have hno : ¬Conn G (ωf \ {e}) x (y k) := by
-        intro hc
-        have : y k ∈ cluster G (ωf \ {e}) x := hc
-        rw [hfinal] at this
-        exact hn this
-      have heω : e ∈ ωf := by
-        by_contra heω
-        have heq : ωf \ {e} = ωf := by
-          ext f
-          constructor
-          · exact fun hh => hh.1
-          · intro hh
-            exact ⟨hh,fun hfe => heω ((Set.mem_singleton_iff.mp hfe) ▸ hh)⟩
-        exact hno (heq.symm ▸ hconn)
-      exact ⟨heω,⟨y k,rfl,hconn⟩,by simpa only [MarkedEvent,Set.mem_singleton_iff,exists_eq_left] using hno⟩
-    · exact ih q hq
-
-end SpliceTrace
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Exploration.Goodness
+import OAI.Probability.UnimodularPercolation.Kernel.Spectral
+
+namespace OAI
+
+/-! Critical parameters and the geometry, selection and traces of splices. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+
+section CriticalParameter
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem twoPoint_adj_lower [Countable E] (G : BondGraph V E) (p : unitInterval)
+    (z x y : V) (hxy : Adj G x y) : (p:ℝ)*twoPoint G p z x ≤ twoPoint G p z y := by
+  obtain ⟨e,he⟩ := hxy
+  have h := association_finiteWitness p {ω : Set E | e ∈ ω} {ω | Conn G ω z x}
+    (fun _ _ hω h => hω h) (fun _ _ hω h => conn_mono G hω h)
+    (finiteWitness_edge e) (finiteWitness_conn G z x)
+  have hp : (law p).real {ω : Set E | e ∈ ω} = (p:ℝ) := by
+    simpa using finite_open_probability p {e}
+  rw [hp] at h
+  exact h.trans (measureReal_mono (fun ω hω =>
+    conn_trans G ω hω.2 (Relation.ReflTransGen.single ⟨e,hω.1,he⟩)))
+
+ 
+theorem walk_operator_apply (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
+    (P : Hilbert V →L[ℂ] Hilbert V)
+    (hP : ∀ x y, P (lp.single 2 y (1:ℂ)) x = (walkKernel G hloc a x y : ℂ))
+    (f : Hilbert V) (x : V) :
+    P f x = ∑ y ∈ insert x (neighbors G hloc x), (walkKernel G hloc a x y : ℂ)*f y := by
+  have hz (y : V) (hy : y ∉ insert x (neighbors G hloc x)) : matrixCoeff P x y*f y = 0 := by
+    have hxy : x ≠ y := by intro h; subst y; exact hy (Finset.mem_insert_self _ _)
+    have hAdj : ¬Adj G x y := fun h => hy (Finset.mem_insert_of_mem ((mem_neighbors ..).mpr h))
+    simp only [matrixCoeff,hP,walkKernel,ite_eq_right hxy,ite_eq_right hAdj,add_zero,Complex.ofReal_zero,zero_mul]
+  have hsum := (operator_kernel_hasSum P f x).unique (hasSum_sum_of_ne_finset_zero hz)
+  simpa only [matrixCoeff,hP] using hsum
+
+ 
+
+theorem parameter_le_walk_norm [Countable E] [Nonempty V] (G : BondGraph V E)
+    (hloc : LocallyFinite G) (D : ℕ) (hD : ∀ x, degree G x ≤ D)
+    (a : ℝ) (ha : 0 ≤ a) (haD : a*D ≤ 1)
+    (P : Hilbert V →L[ℂ] Hilbert V)
+    (hP : ∀ x y, P (lp.single 2 y (1:ℂ)) x = (walkKernel G hloc a x y : ℂ))
+    (p : unitInterval) (hp : OperatorBounded G p) : (p:ℝ) ≤ ‖P‖ := by
+  obtain ⟨T,hT⟩ := hp
+  let z := Classical.arbitrary V
+  let f := T (lp.single 2 z (1:ℂ))
+  have hf (x : V) : f x = (twoPoint G p x z : ℂ) := hT x z
+  have hf0 : 0 < ‖f‖ := by
+    apply norm_pos_iff.mpr
+    intro h
+    have hval := congrArg (fun g : Hilbert V => g z) h
+    simp [hf] at hval
+  have hrow (x : V) : (p:ℝ)*twoPoint G p x z ≤
+      ∑ y ∈ insert x (neighbors G hloc x), walkKernel G hloc a x y*twoPoint G p y z := by
+    calc
+      _ = ∑ y ∈ insert x (neighbors G hloc x), walkKernel G hloc a x y*((p:ℝ)*twoPoint G p x z) := by
+        rw [← Finset.sum_mul,walkKernel_row_sum,one_mul]
+      _ ≤ _ := by
+        apply Finset.sum_le_sum
+        intro y hy
+        apply mul_le_mul_of_nonneg_left _ (walkKernel_nonneg G hloc D hD a ha haD x y)
+        rcases Finset.mem_insert.mp hy with rfl | hy
+        · exact mul_le_of_le_one_left (twoPoint_nonneg G p y z) p.2.2
+        · simpa only [twoPoint_symm G p] using twoPoint_adj_lower G p z x y ((mem_neighbors ..).mp hy)
+  have hpoint (x : V) : ‖((p:ℂ) • f) x‖ ≤ ‖P f x‖ := by
+    rw [lp.coeFn_smul,Pi.smul_apply,smul_eq_mul,hf,← Complex.ofReal_mul,
+      Complex.norm_of_nonneg (mul_nonneg p.2.1 (twoPoint_nonneg G p x z)),
+      walk_operator_apply G hloc a P hP]
+    simp only [hf,← Complex.ofReal_mul,← Complex.ofReal_sum]
+    rw [Complex.norm_of_nonneg (le_trans (mul_nonneg p.2.1 (twoPoint_nonneg G p x z)) (hrow x))]
+    exact hrow x
+  have hn := (lp.norm_mono (by norm_num : (2:ℝ≥0∞) ≠ 0) hpoint).trans (P.le_opNorm f)
+  rw [norm_smul,Complex.norm_of_nonneg p.2.1] at hn
+  exact (mul_le_mul_iff_left₀ hf0).mp hn
+
+ 
+theorem pc_lt_one_of_nonamenable [Nonempty V] (G : BondGraph V E)
+    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
+    (hh : 0 < hV G) : pc G < 1 := by
+  let : Countable V := countable_vertices hloc hconn
+  let : Countable E := countable_edges hloc
+  obtain ⟨D,P,hD0,hD,hPn,_,hP⟩ := exists_strict_walk_operator G hloc hqt hh
+  have hDr : 0 < (D:ℝ) := by exact_mod_cast hD0
+  let p : unitInterval := ⟨(‖P‖+1)/2,by constructor <;> linarith [norm_nonneg P]⟩
+  have hp1 : p < 1 := by change (‖P‖+1)/2 < 1; linarith
+  have ha : 0 ≤ (1:ℝ)/(2*D) := by positivity
+  have haD : (1:ℝ)/(2*D)*D ≤ 1 := by
+    rw [div_mul_eq_mul_div]
+    apply (div_le_iff₀ (by positivity)).mpr
+    linarith
+  apply lt_of_le_of_ne (show pc G ≤ 1 from (pc G).2.2)
+  intro he
+  have hb := operatorBounded_of_lt_pc G hloc hqt p (he ▸ hp1)
+  have hle := parameter_le_walk_norm G hloc D hD _ ha haD P hP p hb
+  change (‖P‖+1)/2 ≤ ‖P‖ at hle
+  linarith
+
+end CriticalParameter
+
+section SpliceGeometry
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+def exchangeOn (mask ω η : Set E) : Set E := (ω \ mask) ∪ (η ∩ mask)
+
+@[simp] theorem mem_exchangeOn {mask ω η : Set E} {e : E} :
+    e ∈ exchangeOn mask ω η ↔ if e ∈ mask then e ∈ η else e ∈ ω := by
+  by_cases h : e ∈ mask <;> simp [exchangeOn,h]
+
+theorem exchangeOn_involutive (mask ω η : Set E) :
+    exchangeOn mask (exchangeOn mask ω η) (exchangeOn mask η ω) = ω := by
+  ext e
+  by_cases h : e ∈ mask <;> simp [h]
+
+ 
+
+def spliceMask (G : BondGraph V E) (S B : Set V) (e : E) : Set E :=
+  insert e {f | (∃ b ∈ B, b ∈ G f) ∧ f ∈ internalEdges G Sᶜ}
+
+def spliceField (G : BondGraph V E) (S B : Set V) (e : E) (ω η : Set E) : Set E :=
+  exchangeOn (spliceMask G S B e) ω η
+
+theorem slit_cluster_subset (G : BondGraph V E) (S : Set V) (η : Set E)
+    (y : V) (hy : y ∉ S) : cluster G (η ∩ internalEdges G Sᶜ) y ⊆ Sᶜ := by
+  exact fun z hz => conn_internal_endpoint G Sᶜ hy hz
+
+theorem spliceMask_not_internal (G : BondGraph V E) (S B : Set V) (e : E)
+    (a b : V) (he : G e = s(a,b)) (hb : b ∉ S) {f : E}
+    (hf : f ∈ internalEdges G S) : f ∉ spliceMask G S B e := by
+  rintro (rfl | ⟨_,hout⟩)
+  · exact hb (((mem_internalEdges_of_ends G S f a b he).mp hf).2)
+  · exact (hout _ (Sym2.out_fst_mem _)) (hf _ (Sym2.out_fst_mem _))
+
+theorem spliceMask_of_internal_piece (G : BondGraph V E) (S B : Set V) (e : E)
+    (hB : B ⊆ Sᶜ) {f : E} (hf : f ∈ internalEdges G B) : f ∈ spliceMask G S B e := by
+  right
+  exact ⟨⟨(G f).out.1,hf _ (Sym2.out_fst_mem _),Sym2.out_fst_mem _⟩,
+    fun v hv => hB (hf v hv)⟩
+
+ 
+theorem splice_internal_fields (G : BondGraph V E) (S B : Set V) (e : E)
+    (ω η : Set E) (a b : V) (he : G e = s(a,b)) (hb : b ∉ S) (hB : B ⊆ Sᶜ) :
+    spliceField G S B e ω η ∩ internalEdges G S = ω ∩ internalEdges G S ∧
+    spliceField G S B e ω η ∩ internalEdges G B = η ∩ internalEdges G B := by
+  constructor
+  · ext f
+    by_cases hf : f ∈ internalEdges G S
+    · simp only [Set.mem_inter_iff,hf,and_true,spliceField,mem_exchangeOn,
+        ite_eq_right (spliceMask_not_internal G S B e a b he hb hf)]
+    · simp only [Set.mem_inter_iff,hf,and_false]
+  · ext f
+    by_cases hf : f ∈ internalEdges G B
+    · simp only [Set.mem_inter_iff,hf,and_true,spliceField,mem_exchangeOn,
+        ite_eq_left (spliceMask_of_internal_piece G S B e hB hf)]
+    · simp only [Set.mem_inter_iff,hf,and_false]
+
+ 
+theorem splice_crossing_unique (G : BondGraph V E) (ω η : Set E) (x : V)
+    (B : Set V) (e : E) {f : E} {a b : V}
+    (hf : f ∈ spliceField G (cluster G ω x) B e ω η)
+    (hab : G f = s(a,b)) (ha : a ∈ cluster G ω x) (hb : b ∉ cluster G ω x) : f = e := by
+  have hno : f ∉ ω := fun h => hb ((edge_cluster_sides G ω x h hab).mp ha)
+  have hm : f ∈ spliceMask G (cluster G ω x) B e := by
+    by_contra h
+    have hh := hf
+    simp only [spliceField,mem_exchangeOn,ite_eq_right h] at hh
+    exact hno hh
+  rcases hm with rfl | ⟨_,hout⟩
+  · rfl
+  · exact ((((mem_internalEdges_of_ends G _ f a b hab).mp hout).1) ha).elim
+
+ 
+
+theorem splice_cluster (G : BondGraph V E) (ω η : Set E) (x y : V)
+    (hy : y ∉ cluster G ω x) (e : E) (a b : V)
+    (he : G e = s(a,b)) (heη : e ∈ η) (ha : a ∈ cluster G ω x)
+    (hb : b ∈ cluster G (η ∩ internalEdges G (cluster G ω x)ᶜ) y) :
+    let S := cluster G ω x
+    let B := cluster G (η ∩ internalEdges G Sᶜ) y
+    cluster G (spliceField G S B e ω η) x = S ∪ B := by
+  let S := cluster G ω x
+  let ζ := η ∩ internalEdges G Sᶜ
+  let B := cluster G ζ y
+  let ω' := spliceField G S B e ω η
+  change cluster G ω' x = S ∪ B
+  have hB : B ⊆ Sᶜ := slit_cluster_subset G S η y hy
+  have hbS : b ∉ S := hB hb
+  have hfields := splice_internal_fields G S B e ω η a b he hbS hB
+  have he' : e ∈ ω' := by
+    change e ∈ exchangeOn (spliceMask G S B e) ω η
+    rw [mem_exchangeOn,ite_eq_left (show e ∈ spliceMask G S B e from Set.mem_insert _ _)]
+    exact heη
+  have hSconn (z : V) (hz : z ∈ S) : Conn G ω' x z := by
+    have h := conn_restrict_cluster G ω x (mem_cluster_self G ω x) hz
+    apply conn_mono G _ h
+    intro f hf
+    exact (hfields.1.symm ▸ hf).1
+  have hBconn (u v : V) (hu : u ∈ B) (hv : v ∈ B) : Conn G ω' u v := by
+    have h := conn_restrict_cluster G ζ y hu (conn_trans G ζ (conn_symm G ζ hu) hv)
+    apply conn_mono G _ h
+    intro f hf
+    exact (hfields.2.symm ▸ (show f ∈ η ∩ internalEdges G B from ⟨hf.1.1,hf.2⟩)).1
+  apply Set.Subset.antisymm
+  · intro z hz
+    induction hz with
+    | refl => exact Or.inl (mem_cluster_self G ω x)
+    | @tail u v hxu huv ih =>
+      obtain ⟨f,hf,hfv⟩ := huv
+      by_cases hvS : v ∈ S
+      · exact Or.inl hvS
+      · rcases ih with huS | huB
+        · have hfe := splice_crossing_unique G ω η x B e hf hfv huS hvS
+          subst f
+          have hends := Sym2.eq_iff.mp (he.symm.trans hfv)
+          rcases hends with ⟨hau,hbv⟩ | ⟨hav,hbu⟩
+          · exact Or.inr (hbv ▸ hb)
+          · exact (hvS (hav ▸ ha)).elim
+        · have huS : u ∉ S := hB huB
+          have hm : f ∈ spliceMask G S B e := by
+            right
+            exact ⟨⟨u,huB,by simp [hfv]⟩,(mem_internalEdges_of_ends G _ f u v hfv).mpr ⟨huS,hvS⟩⟩
+          have hfη : f ∈ η := by
+            change f ∈ exchangeOn (spliceMask G S B e) ω η at hf
+            rwa [mem_exchangeOn,ite_eq_left hm] at hf
+          exact Or.inr ((edge_cluster_sides G ζ y
+            ⟨hfη,(mem_internalEdges_of_ends G _ f u v hfv).mpr ⟨huS,hvS⟩⟩ hfv).mp huB)
+  · intro z hz
+    rcases hz with hz | hz
+    · exact hSconn z hz
+    · exact ((hSconn a ha).tail ⟨e,he',he⟩).trans (hBconn b z hb hz)
+
+ 
+theorem splice_seam_pivotal (G : BondGraph V E) (ω η : Set E) (x y : V)
+    (hy : y ∉ cluster G ω x) (e : E) (a b : V)
+    (he : G e = s(a,b)) (heη : e ∈ η) (ha : a ∈ cluster G ω x)
+    (hb : b ∈ cluster G (η ∩ internalEdges G (cluster G ω x)ᶜ) y)
+    (z : V) (hz : z ∈ cluster G (η ∩ internalEdges G (cluster G ω x)ᶜ) y) :
+    let S := cluster G ω x
+    let B := cluster G (η ∩ internalEdges G Sᶜ) y
+    OpenMarkedPivotal G (spliceField G S B e ω η) {z} x e := by
+  let S := cluster G ω x
+  let B := cluster G (η ∩ internalEdges G Sᶜ) y
+  let ω' := spliceField G S B e ω η
+  have hB := slit_cluster_subset G S η y hy
+  have hC := splice_cluster G ω η x y hy e a b he heη ha hb
+  change cluster G ω' x = S ∪ B at hC
+  refine ⟨?_,⟨z,Set.mem_singleton z,by change z ∈ cluster G ω' x; rw [hC]; exact Or.inr hz⟩,?_⟩
+  · change e ∈ exchangeOn (spliceMask G S B e) ω η
+    rw [mem_exchangeOn,ite_eq_left (show e ∈ spliceMask G S B e from Set.mem_insert _ _)]
+    exact heη
+  · rintro ⟨v,hv,hxv⟩
+    have hvz : v = z := Set.mem_singleton_iff.mp hv
+    subst v
+    have hsub : cluster G (ω' \ {e}) x ⊆ S := by
+      intro v hxv
+      induction hxv with
+      | refl => exact mem_cluster_self G ω x
+      | @tail u v hxu huv ih =>
+        obtain ⟨f,hf,hfv⟩ := huv
+        by_contra hvS
+        have hfe := splice_crossing_unique G ω η x B e hf.1 hfv ih hvS
+        exact hf.2 (by simpa only [Set.mem_singleton_iff] using hfe)
+    exact hB hz (hsub hxv)
+
+end SpliceGeometry
+
+section SpliceInverse
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem splice_close_seam_cluster (G : BondGraph V E) (ω η : Set E) (x y : V)
+    (hy : y ∉ cluster G ω x) (e : E) (a b : V)
+    (he : G e = s(a,b)) (_heη : e ∈ η) (_ha : a ∈ cluster G ω x)
+    (hb : b ∈ cluster G (η ∩ internalEdges G (cluster G ω x)ᶜ) y) :
+    let S := cluster G ω x
+    let B := cluster G (η ∩ internalEdges G Sᶜ) y
+    cluster G (spliceField G S B e ω η \ {e}) x = S := by
+  let S := cluster G ω x
+  let B := cluster G (η ∩ internalEdges G Sᶜ) y
+  let ω' := spliceField G S B e ω η
+  have hB := slit_cluster_subset G S η y hy
+  have hf := (splice_internal_fields G S B e ω η a b he (hB hb) hB).1
+  apply Set.Subset.antisymm
+  · intro v hxv
+    induction hxv with
+    | refl => exact mem_cluster_self G ω x
+    | @tail u v _ huv ih =>
+      obtain ⟨f,hf,hfv⟩ := huv
+      by_contra hvS
+      have hfe := splice_crossing_unique G ω η x B e hf.1 hfv ih hvS
+      exact hf.2 (Set.mem_singleton_iff.mpr hfe)
+  · intro z hz
+    have hc := conn_restrict_cluster G ω x (mem_cluster_self G ω x) hz
+    apply conn_mono G _ hc
+    intro f hfin
+    refine ⟨(hf.symm ▸ hfin).1,?_⟩
+    intro hfe
+    have hfe' := Set.mem_singleton_iff.mp hfe
+    subst f
+    exact hB hb (((mem_internalEdges_of_ends G S e a b he).mp hfin.2).2)
+
+ 
+theorem splice_recover_piece (G : BondGraph V E) (ω η : Set E) (x y : V)
+    (hy : y ∉ cluster G ω x) (e : E) (a b : V)
+    (he : G e = s(a,b)) (heη : e ∈ η) (ha : a ∈ cluster G ω x)
+    (hb : b ∈ cluster G (η ∩ internalEdges G (cluster G ω x)ᶜ) y) :
+    let S := cluster G ω x
+    let B := cluster G (η ∩ internalEdges G Sᶜ) y
+    let ω' := spliceField G S B e ω η
+    cluster G ω' x \ cluster G (ω' \ {e}) x = B := by
+  dsimp only
+  rw [splice_cluster G ω η x y hy e a b he heη ha hb,
+    splice_close_seam_cluster G ω η x y hy e a b he heη ha hb]
+  have hB := slit_cluster_subset G (cluster G ω x) η y hy
+  ext z
+  constructor
+  · rintro ⟨hz,hzS⟩
+    exact hz.resolve_left hzS
+  · intro hz
+    exact ⟨Or.inr hz,hB hz⟩
+
+ 
+def recoveredMask (G : BondGraph V E) (ω' : Set E) (x : V) (e : E) : Set E :=
+  spliceMask G (cluster G (ω' \ {e}) x)
+    (cluster G ω' x \ cluster G (ω' \ {e}) x) e
+
+theorem recoveredMask_splice (G : BondGraph V E) (ω η : Set E) (x y : V)
+    (hy : y ∉ cluster G ω x) (e : E) (a b : V)
+    (he : G e = s(a,b)) (heη : e ∈ η) (ha : a ∈ cluster G ω x)
+    (hb : b ∈ cluster G (η ∩ internalEdges G (cluster G ω x)ᶜ) y) :
+    let S := cluster G ω x
+    let B := cluster G (η ∩ internalEdges G Sᶜ) y
+    recoveredMask G (spliceField G S B e ω η) x e = spliceMask G S B e := by
+  unfold recoveredMask
+  dsimp only
+  rw [splice_recover_piece G ω η x y hy e a b he heη ha hb,
+    splice_close_seam_cluster G ω η x y hy e a b he heη ha hb]
+
+ 
+def undoSplice (G : BondGraph V E) (x : V) (e : E) (ω' η' : Set E) : Set E × Set E :=
+  let M := recoveredMask G ω' x e
+  (exchangeOn M ω' η',exchangeOn M η' ω')
+
+ 
+theorem undoSplice_of_valid (G : BondGraph V E) (ω η : Set E) (x y : V)
+    (hy : y ∉ cluster G ω x) (e : E) (a b : V)
+    (he : G e = s(a,b)) (heη : e ∈ η) (ha : a ∈ cluster G ω x)
+    (hb : b ∈ cluster G (η ∩ internalEdges G (cluster G ω x)ᶜ) y) :
+    let S := cluster G ω x
+    let B := cluster G (η ∩ internalEdges G Sᶜ) y
+    let M := spliceMask G S B e
+    undoSplice G x e (exchangeOn M ω η) (exchangeOn M η ω) = (ω,η) := by
+  dsimp only [undoSplice]
+  rw [show recoveredMask G (exchangeOn
+      (spliceMask G (cluster G ω x) (cluster G (η ∩ internalEdges G (cluster G ω x)ᶜ) y) e) ω η) x e =
+      spliceMask G (cluster G ω x) (cluster G (η ∩ internalEdges G (cluster G ω x)ᶜ) y) e from
+    recoveredMask_splice G ω η x y hy e a b he heη ha hb]
+  rw [exchangeOn_involutive,exchangeOn_involutive]
+
+ 
+theorem spliceMask_finite (G : BondGraph V E) (hloc : LocallyFinite G)
+    (S B : Set V) (hB : B.Finite) (e : E) : (spliceMask G S B e).Finite := by
+  apply Set.Finite.insert
+  apply (hB.biUnion (fun b _ => hloc b)).subset
+  rintro f ⟨⟨b,hb,hbf⟩,_⟩
+  exact Set.mem_iUnion₂.mpr ⟨b,hb,hbf⟩
+
+end SpliceInverse
+
+section FiniteExchange
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def finiteExchange (M ω η : Finset E) : Finset E := (ω \ M) ∪ (η ∩ M)
+
+@[simp] theorem coe_finiteExchange (M ω η : Finset E) :
+    (finiteExchange M ω η : Set E) = exchangeOn (M:Set E) (ω:Set E) (η:Set E) := by
+  simp only [finiteExchange,exchangeOn,Finset.coe_union,Finset.coe_sdiff,Finset.coe_inter]
+
+theorem finiteExchange_card (M ω η : Finset E) :
+    (finiteExchange M ω η).card = (ω \ M).card+(η ∩ M).card := by
+  apply Finset.card_union_of_disjoint
+  exact Finset.disjoint_left.mpr fun e he he' => (Finset.mem_sdiff.mp he).2 (Finset.mem_inter.mp he').2
+
+theorem finiteExchange_card_sum (M ω η : Finset E) :
+    (finiteExchange M ω η).card+(finiteExchange M η ω).card = ω.card+η.card := by
+  rw [finiteExchange_card,finiteExchange_card]
+  have hω := Finset.card_sdiff_add_card_inter ω M
+  have hη := Finset.card_sdiff_add_card_inter η M
+  omega
+
+theorem finiteExchange_subset (M Q ω η : Finset E) (hω : ω ⊆ Q) (hη : η ⊆ Q) :
+    finiteExchange M ω η ⊆ Q := by
+  intro e he
+  rcases Finset.mem_union.mp he with he | he
+  · exact hω (Finset.mem_sdiff.mp he).1
+  · exact hη (Finset.mem_inter.mp he).1
+
+ 
+theorem bernoulliWeight_exchange (M Q ω η : Finset E) (hω : ω ⊆ Q) (hη : η ⊆ Q)
+    (p : ℝ) :
+    bernoulliWeight Q (finiteExchange M ω η) p * bernoulliWeight Q (finiteExchange M η ω) p =
+      bernoulliWeight Q ω p * bernoulliWeight Q η p := by
+  have hc := finiteExchange_card_sum M ω η
+  have hcω := Finset.card_le_card hω
+  have hcη := Finset.card_le_card hη
+  have hcω' := Finset.card_le_card (finiteExchange_subset M Q ω η hω hη)
+  have hcη' := Finset.card_le_card (finiteExchange_subset M Q η ω hη hω)
+  have hd : (Q.card-(finiteExchange M ω η).card)+(Q.card-(finiteExchange M η ω).card) =
+      (Q.card-ω.card)+(Q.card-η.card) := by omega
+  unfold bernoulliWeight
+  calc
+    _ = p^((finiteExchange M ω η).card+(finiteExchange M η ω).card) *
+      (1-p)^((Q.card-(finiteExchange M ω η).card)+(Q.card-(finiteExchange M η ω).card)) := by
+        rw [pow_add,pow_add]; ring
+    _ = p^(ω.card+η.card)*(1-p)^((Q.card-ω.card)+(Q.card-η.card)) := by rw [hc,hd]
+    _ = _ := by rw [pow_add,pow_add]; ring
+
+end FiniteExchange
+
+section DeletionGoodness
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+def deletionContacts (G : BondGraph V E) (ω : Set E) (A B : Set V) : Set E :=
+  {e | e ∈ ω ∧ ∃ a ∈ B, ∃ b ∈ A, G e = s(a,b)}
+
+ 
+
+def DeletionGoodForScale (G : BondGraph V E) (ω : Set E) (x : V) (L : ℝ) : Prop :=
+  ∀ A ⊆ cluster G ω x, ∀ y ∈ cluster G ω x \ A,
+    let B := cluster G (ω ∩ internalEdges G Aᶜ) y
+    0 < (deletionContacts G ω A B).ncard →
+    ∀ a ∈ B, ∀ b ∈ B,
+      (pivotalDistance G (ω ∩ internalEdges G B) a b : ℝ) ≤
+        (deletionContacts G ω A B).ncard*L
+
+theorem deleted_cluster_subset (G : BondGraph V E) (ω : Set E) (x y : V)
+    (A : Set V) (hy : y ∈ cluster G ω x \ A) :
+    cluster G (ω ∩ internalEdges G Aᶜ) y ⊆ cluster G ω x \ A := by
+  intro z hz
+  exact ⟨conn_trans G ω hy.1 (conn_mono G Set.inter_subset_left hz),
+    conn_internal_endpoint G Aᶜ hy.2 hz⟩
+
+theorem contactSet_deleted_cluster (G : BondGraph V E) (ω : Set E) (A : Set V)
+    (y : V) (hy : y ∉ A) :
+    let B := cluster G (ω ∩ internalEdges G Aᶜ) y
+    contactSet G ω B = deletionContacts G ω A B := by
+  let ζ := ω ∩ internalEdges G Aᶜ
+  let B := cluster G ζ y
+  have hB : B ⊆ Aᶜ := slit_cluster_subset G A ω y hy
+  ext e
+  constructor
+  · rintro ⟨he,a,b,hab,ha,hb⟩
+    refine ⟨he,a,ha,b,?_,hab⟩
+    by_contra hbA
+    have heζ : e ∈ ζ := ⟨he,(mem_internalEdges_of_ends G Aᶜ e a b hab).mpr ⟨hB ha,hbA⟩⟩
+    exact hb ((edge_cluster_sides G ζ y heζ hab).mp ha)
+  · rintro ⟨he,a,ha,b,hb,hab⟩
+    exact ⟨he,a,b,hab,ha,fun hh => hB hh hb⟩
+
+ 
+
+theorem goodForScale_iff_deletion (G : BondGraph V E) (ω : Set E) (x : V) (L : ℝ)
+    (hC : (cluster G ω x).Finite) :
+    GoodForScale G ω x L ↔ DeletionGoodForScale G ω x L := by
+  constructor
+  · intro h A _ y hy
+    let ζ := ω ∩ internalEdges G Aᶜ
+    let B := cluster G ζ y
+    have hBC := deleted_cluster_subset G ω x y A hy
+    have hfin : B.Finite := hC.subset (fun _ hz => (hBC hz).1)
+    let W := hfin.toFinset
+    have hW : (W : Set V) = B := hfin.coe_toFinset
+    have hsub : (W : Set V) ⊆ cluster G ω x := by rw [hW]; exact fun _ hz => (hBC hz).1
+    have hi (a : V) (ha : a ∈ W) (b : V) (hb : b ∈ W) :
+        Conn G (ω ∩ internalEdges G (W : Set V)) a b := by
+      have ha' : a ∈ B := hW ▸ ha
+      have hb' : b ∈ B := hW ▸ hb
+      have hc := conn_restrict_cluster G ζ y ha' (conn_trans G ζ (conn_symm G ζ ha') hb')
+      rw [hW]
+      apply conn_mono G ?_ hc
+      intro e he
+      exact ⟨he.1.1,he.2⟩
+    have hh := h W hsub hi
+    rw [hW,contactSet_deleted_cluster G ω A y hy.2] at hh
+    dsimp only
+    intro hK a ha b hb
+    exact hh hK a (by change a ∈ (W : Set V); rw [hW]; exact ha) b
+      (by change b ∈ (W : Set V); rw [hW]; exact hb)
+  · intro h W hW hconn hK a ha b hb
+    let C := cluster G ω x
+    let A := C \ (W : Set V)
+    have haA : a ∈ C \ A := ⟨hW ha,fun hh => hh.2 ha⟩
+    have hWA : (W : Set V) ⊆ Aᶜ := fun _ hz hh => hh.2 hz
+    have heq : cluster G (ω ∩ internalEdges G Aᶜ) a = (W : Set V) := by
+      apply Set.Subset.antisymm
+      · intro z hz
+        have hzC := deleted_cluster_subset G ω x a A haA hz
+        by_contra hzW
+        exact hzC.2 ⟨hzC.1,hzW⟩
+      · intro z hz
+        apply conn_mono G ?_ (hconn a ha z hz)
+        intro e he
+        exact ⟨he.1,fun v hv => hWA (he.2 v hv)⟩
+    have hh := h A Set.sdiff_subset a haA
+    dsimp only at hh
+    have hc := contactSet_deleted_cluster G ω A a haA.2
+    dsimp only at hc
+    rw [heq] at hh hc
+    rw [← hc] at hh
+    exact hh hK a ha b hb
+
+ 
+theorem capped_good_local (G : BondGraph V E) (hloc : LocallyFinite G)
+    (x : V) (M : ℕ) (L : ℝ) (ω : Set E) :
+    let Q := incidentSupport G hloc (finite_reachBall hloc x M).toFinset
+    ((cluster G ω x).Finite ∧ (cluster G ω x).ncard ≤ M ∧ GoodForScale G ω x L) ↔
+      ((cluster G (ω ∩ (Q : Set E)) x).Finite ∧
+        (cluster G (ω ∩ (Q : Set E)) x).ncard ≤ M ∧
+          GoodForScale G (ω ∩ (Q : Set E)) x L) := by
+  have hc := capped_cluster_local G hloc x M ω
+  have hb := cappedBadGoodness_local G hloc x M L ω
+  unfold CappedBadGoodness at hb
+  dsimp only at hc hb ⊢
+  constructor
+  · rintro ⟨hf,hM,hg⟩
+    obtain ⟨hf',hM'⟩ := hc.mp ⟨hf,hM⟩
+    refine ⟨hf',hM',?_⟩
+    by_contra hng
+    exact (hb.mpr ⟨hf',hM',hng⟩).2.2 hg
+  · rintro ⟨hf,hM,hg⟩
+    obtain ⟨hf',hM'⟩ := hc.mpr ⟨hf,hM⟩
+    refine ⟨hf',hM',?_⟩
+    by_contra hng
+    exact (hb.mp ⟨hf',hM',hng⟩).2.2 hg
+
+theorem measurableSet_capped_good (G : BondGraph V E) (hloc : LocallyFinite G)
+    (x : V) (M : ℕ) (L : ℝ) :
+    MeasurableSet {ω : Set E | (cluster G ω x).Finite ∧
+      (cluster G ω x).ncard ≤ M ∧ GoodForScale G ω x L} := by
+  let Q := incidentSupport G hloc (finite_reachBall hloc x M).toFinset
+  have he : {ω : Set E | (cluster G ω x).Finite ∧
+      (cluster G ω x).ncard ≤ M ∧ GoodForScale G ω x L} =
+    {ω : Set E | (cluster G (ω ∩ (Q : Set E)) x).Finite ∧
+      (cluster G (ω ∩ (Q : Set E)) x).ncard ≤ M ∧ GoodForScale G (ω ∩ (Q : Set E)) x L} := by
+    ext ω
+    exact capped_good_local G hloc x M L ω
+  rw [he]
+  exact measurableSet_finite_restriction Q {η : Set E | (cluster G η x).Finite ∧
+      (cluster G η x).ncard ≤ M ∧ GoodForScale G η x L}
+
+end DeletionGoodness
+
+section SpliceSelection
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem exists_maximal_splice_index (G : BondGraph V E) (η : ℕ → Set E) (y : ℕ → V)
+    (k j : ℕ) (hj : j ≤ k)
+    (hchain : ∀ i ∈ Finset.Icc 1 k, Conn G (η i) (y i) (y (i-1)))
+    (S : Set V) (hyj : y j ∈ S) (hyk : y k ∉ S) :
+    ∃ i ∈ Finset.Icc (j+1) k,
+      ((cluster G (η i) (y i)) ∩ S).Nonempty ∧ y i ∉ S ∧
+      ∀ l, i < l → l ≤ k → Disjoint (cluster G (η l) (y l)) S := by
+  have hjk : j < k := by
+    by_contra h
+    have hEq : j = k := by omega
+    exact hyk (hEq ▸ hyj)
+  let T := (Finset.Icc (j+1) k).filter fun i => ((cluster G (η i) (y i)) ∩ S).Nonempty
+  have hT : T.Nonempty := by
+    refine ⟨j+1,Finset.mem_filter.mpr ⟨Finset.mem_Icc.mpr ⟨le_rfl,by omega⟩,?_⟩⟩
+    refine ⟨y j,?_,hyj⟩
+    simpa only [Nat.add_sub_cancel,cluster,Set.mem_ofPred_eq] using hchain (j+1) (Finset.mem_Icc.mpr ⟨by omega,by omega⟩)
+  let i := T.max' hT
+  have hi : i ∈ T := Finset.max'_mem T hT
+  obtain ⟨hiI,hiS⟩ := Finset.mem_filter.mp hi
+  have hiU := (Finset.mem_Icc.mp hiI).2
+  have hiL := (Finset.mem_Icc.mp hiI).1
+  have hl (l : ℕ) (hil : i < l) (hlk : l ≤ k) : Disjoint (cluster G (η l) (y l)) S := by
+    apply Set.disjoint_left.mpr
+    intro z hz hzS
+    have hlT : l ∈ T := Finset.mem_filter.mpr
+      ⟨Finset.mem_Icc.mpr ⟨by omega,hlk⟩,⟨z,hz,hzS⟩⟩
+    have hli : l ≤ i := Finset.le_max' T l hlT
+    omega
+  refine ⟨i,hiI,hiS,?_,hl⟩
+  intro hyi
+  by_cases hik : i = k
+  · exact hyk (hik ▸ hyi)
+  · have hic : y i ∈ cluster G (η (i+1)) (y (i+1)) := by
+      simpa only [Nat.add_sub_cancel,cluster,Set.mem_ofPred_eq] using hchain (i+1) (Finset.mem_Icc.mpr ⟨by omega,by omega⟩)
+    exact Set.disjoint_left.mp (hl (i+1) (by omega) (by omega)) hic hyi
+
+ 
+
+theorem exists_splice_contact (G : BondGraph V E) (η : Set E) (S : Set V) (y : V)
+    (hy : y ∉ S) (hmeet : (cluster G η y ∩ S).Nonempty) :
+    ∃ e a b, e ∈ η ∧ G e = s(a,b) ∧ a ∈ S ∧
+      b ∈ cluster G (η ∩ internalEdges G Sᶜ) y := by
+  obtain ⟨v,hv,hvS⟩ := hmeet
+  obtain ⟨b,a,e,_,ha,he,hab,hb⟩ := conn_first_entry G S η hy hvS hv
+  exact ⟨e,a,b,he,hab.trans (Sym2.eq_swap),ha,hb⟩
+
+ 
+theorem splice_cluster_cap (G : BondGraph V E) (ω η : Set E) (x y : V)
+    (hy : y ∉ cluster G ω x) (e : E) (a b : V)
+    (he : G e = s(a,b)) (heη : e ∈ η) (ha : a ∈ cluster G ω x)
+    (hb : b ∈ cluster G (η ∩ internalEdges G (cluster G ω x)ᶜ) y)
+    (N M : ℕ) (hS : (cluster G ω x).Finite) (hN : (cluster G ω x).ncard ≤ N)
+    (hC : (cluster G η y).Finite) (hM : (cluster G η y).ncard ≤ M) :
+    let S := cluster G ω x
+    let B := cluster G (η ∩ internalEdges G Sᶜ) y
+    (cluster G (spliceField G S B e ω η) x).Finite ∧
+      (cluster G (spliceField G S B e ω η) x).ncard ≤ N+M := by
+  let S := cluster G ω x
+  let B := cluster G (η ∩ internalEdges G Sᶜ) y
+  have hBC : B ⊆ cluster G η y := fun _ hz => conn_mono G Set.inter_subset_left hz
+  have hB : B.Finite := hC.subset hBC
+  dsimp only
+  rw [splice_cluster G ω η x y hy e a b he heη ha hb]
+  refine ⟨hS.union hB,?_⟩
+  exact (Set.ncard_union_le _ _).trans (Nat.add_le_add hN ((Set.ncard_le_ncard hBC hC).trans hM))
+
+ 
+
+theorem cluster_eq_of_incident_agreement (G : BondGraph V E) (ω η : Set E) (x : V)
+    (he : ∀ e, (∃ v ∈ cluster G ω x, v ∈ G e) → (e ∈ ω ↔ e ∈ η)) :
+    cluster G η x = cluster G ω x := by
+  have hforward : cluster G ω x ⊆ cluster G η x := by
+    intro y hy
+    induction hy with
+    | refl => exact conn_refl G _ _
+    | @tail a b hxa hab ih =>
+      obtain ⟨e,heω,hab⟩ := hab
+      exact ih.tail ⟨e,(he e ⟨a,hxa,by simp [hab]⟩).mp heω,hab⟩
+  apply Set.Subset.antisymm _ hforward
+  intro y hy
+  induction hy with
+  | refl => exact conn_refl G _ _
+  | @tail a b _ hab ih =>
+    obtain ⟨e,heη,hab⟩ := hab
+    exact ih.tail ⟨e,(he e ⟨a,ih,by simp [hab]⟩).mpr heη,hab⟩
+
+ 
+
+theorem spliceMask_avoids_prefix (G : BondGraph V E) (S B R : Set V) (hRS : R ⊆ S)
+    (e : E) (a b : V) (hab : G e = s(a,b)) (ha : a ∉ R) (hb : b ∉ S)
+    {f : E} (hf : ∃ v ∈ R, v ∈ G f) : f ∉ spliceMask G S B e := by
+  rintro (rfl | ⟨_,hfo⟩)
+  · obtain ⟨v,hv,hvf⟩ := hf
+    rw [hab] at hvf
+    rcases Sym2.mem_iff.mp hvf with rfl | rfl
+    · exact ha hv
+    · exact hb (hRS hv)
+  · obtain ⟨v,hv,hvf⟩ := hf
+    exact hfo v hvf (hRS hv)
+
+ 
+
+theorem splice_preserves_prefix_cut (G : BondGraph V E) (ω η : Set E) (x : V)
+    (B : Set V) (e f : E) (a b : V) (hab : G e = s(a,b))
+    (ha : a ∉ cluster G (ω \ {f}) x) (hb : b ∉ cluster G ω x) :
+    cluster G ((spliceField G (cluster G ω x) B e ω η) \ {f}) x =
+      cluster G (ω \ {f}) x := by
+  apply cluster_eq_of_incident_agreement
+  intro g hg
+  have hRS : cluster G (ω \ {f}) x ⊆ cluster G ω x := fun _ h => conn_mono G Set.sdiff_subset h
+  have hno := spliceMask_avoids_prefix G _ B _ hRS e a b hab ha hb hg
+  simp only [Set.mem_sdiff,spliceField,mem_exchangeOn,ite_eq_right hno]
+
+end SpliceSelection
+
+section SpliceTrace
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+inductive SpliceTrace (G : BondGraph V E) (x : V) (η : ℕ → Set E) (y : ℕ → V) (k : ℕ) :
+    ℕ → Set E → List (ℕ × E) → Set E → Prop
+  | stop (j : ℕ) (ω : Set E) (hstop : y k ∈ cluster G ω x) :
+      SpliceTrace G x η y k j ω [] ω
+  | step (j i : ℕ) (ω : Set E) (e : E) (a b : V) (rest : List (ℕ × E)) (ωf : Set E)
+      (hnot : y k ∉ cluster G ω x) (hji : j < i) (hik : i ≤ k)
+      (hmax : ∀ l, i < l → l ≤ k → Disjoint (cluster G (η l) (y l)) (cluster G ω x))
+      (hy : y i ∉ cluster G ω x) (he : e ∈ η i) (hab : G e = s(a,b))
+      (ha : a ∈ cluster G ω x)
+      (hb : b ∈ cluster G (η i ∩ internalEdges G (cluster G ω x)ᶜ) (y i))
+      (htail : SpliceTrace G x η y k i
+        (spliceField G (cluster G ω x)
+          (cluster G (η i ∩ internalEdges G (cluster G ω x)ᶜ) (y i)) e ω (η i)) rest ωf) :
+      SpliceTrace G x η y k j ω ((i,e)::rest) ωf
+
+ 
+
+theorem exists_spliceTrace (G : BondGraph V E) (x : V) (η : ℕ → Set E) (y : ℕ → V)
+    (k M : ℕ)
+    (hchain : ∀ i ∈ Finset.Icc 1 k, Conn G (η i) (y i) (y (i-1)))
+    (hcaps : ∀ i ∈ Finset.Icc 1 k, (cluster G (η i) (y i)).Finite ∧
+      (cluster G (η i) (y i)).ncard ≤ M)
+    (j : ℕ) (hj : j ≤ k) (ω : Set E) (hyj : y j ∈ cluster G ω x)
+    (N : ℕ) (hfin : (cluster G ω x).Finite) (hN : (cluster G ω x).ncard ≤ N) :
+    ∃ steps ωf, SpliceTrace G x η y k j ω steps ωf ∧ steps.length ≤ k-j ∧
+      y k ∈ cluster G ωf x ∧ (cluster G ωf x).Finite ∧
+        (cluster G ωf x).ncard ≤ N+(k-j)*M := by
+  generalize hn : k-j = n
+  induction n using Nat.strong_induction_on generalizing j ω N with
+  | h n ih =>
+    by_cases hz : y k ∈ cluster G ω x
+    · refine ⟨[],ω,SpliceTrace.stop j ω hz,by simp,hz,hfin,?_⟩
+      exact hN.trans (Nat.le_add_right ..)
+    · obtain ⟨i,hiI,hmeet,hyi,hmax⟩ :=
+        exists_maximal_splice_index G η y k j hj hchain (cluster G ω x) hyj hz
+      obtain ⟨hji,hik⟩ := Finset.mem_Icc.mp hiI
+      obtain ⟨e,a,b,he,hab,ha,hb⟩ := exists_splice_contact G (η i) (cluster G ω x) (y i) hyi hmeet
+      let B := cluster G (η i ∩ internalEdges G (cluster G ω x)ᶜ) (y i)
+      let ω' := spliceField G (cluster G ω x) B e ω (η i)
+      have hi1 : i ∈ Finset.Icc 1 k := Finset.mem_Icc.mpr ⟨by omega,hik⟩
+      have hcap := splice_cluster_cap G ω (η i) x (y i) hyi e a b hab he ha hb N M hfin hN
+        (hcaps i hi1).1 (hcaps i hi1).2
+      have hyt : y i ∈ cluster G ω' x := by
+        change y i ∈ cluster G (spliceField G (cluster G ω x) B e ω (η i)) x
+        rw [splice_cluster G ω (η i) x (y i) hyi e a b hab he ha hb]
+        exact Or.inr (mem_cluster_self G _ _)
+      have hlt : k-i < n := by omega
+      obtain ⟨steps,ωf,htrace,hlen,hconn,hf,hs⟩ :=
+        ih (k-i) hlt i hik ω' hyt (N+M) hcap.1 hcap.2 rfl
+      refine ⟨(i,e)::steps,ωf,
+        SpliceTrace.step j i ω e a b steps ωf hz (by omega) hik hmax hyi he hab ha hb htrace,
+        ?_,hconn,hf,?_⟩
+      · simp only [List.length_cons]
+        omega
+      · have hm : (k-i+1)*M ≤ (k-j)*M := Nat.mul_le_mul_right M (by omega)
+        rw [Nat.add_mul,one_mul] at hm
+        calc
+          _ ≤ (N+M)+(k-i)*M := hs
+          _ ≤ N+(k-j)*M := by omega
+          _ = N+n*M := by rw [hn]
+
+ 
+theorem SpliceTrace.cluster_subset (G : BondGraph V E) (x : V) (η : ℕ → Set E)
+    (y : ℕ → V) (k j : ℕ) (ω ωf : Set E) (steps : List (ℕ × E))
+    (h : SpliceTrace G x η y k j ω steps ωf) : cluster G ω x ⊆ cluster G ωf x := by
+  induction h with
+  | stop => exact Set.Subset.rfl
+  | @step j i ω e a b rest ωf _ _ _ _ hy he hab ha hb ht ih =>
+    apply Set.Subset.trans _ ih
+    rw [splice_cluster G ω (η i) x (y i) hy e a b hab he ha hb]
+    exact Set.subset_union_left
+
+ 
+
+theorem SpliceTrace.preserve_cut (G : BondGraph V E) (x : V) (η : ℕ → Set E)
+    (y : ℕ → V) (k j : ℕ) (ω ωf : Set E) (steps : List (ℕ × E))
+    (h : SpliceTrace G x η y k j ω steps ωf) (f : E)
+    (havoid : ∀ l, j < l → l ≤ k →
+      Disjoint (cluster G (η l) (y l)) (cluster G (ω \ {f}) x)) :
+    cluster G (ωf \ {f}) x = cluster G (ω \ {f}) x := by
+  induction h with
+  | stop => rfl
+  | @step j i ω e a b rest ωf hn hji hik hmax hy he hab ha hb ht ih =>
+    have hBC : b ∈ cluster G (η i) (y i) := conn_mono G Set.inter_subset_left hb
+    have haC : a ∈ cluster G (η i) (y i) := (edge_cluster_sides G (η i) (y i) he hab).mpr hBC
+    have haR : a ∉ cluster G (ω \ {f}) x := fun hh => Set.disjoint_left.mp (havoid i hji hik) haC hh
+    have hbS : b ∉ cluster G ω x := slit_cluster_subset G _ (η i) (y i) hy hb
+    have heq := splice_preserves_prefix_cut G ω (η i) x
+      (cluster G (η i ∩ internalEdges G (cluster G ω x)ᶜ) (y i)) e f a b hab haR hbS
+    rw [ih]
+    · exact heq
+    · intro l hil hlk
+      rw [heq]
+      exact havoid l (hji.trans hil) hlk
+
+ 
+theorem SpliceTrace.endpoint (G : BondGraph V E) (x : V) (η : ℕ → Set E)
+    (y : ℕ → V) (k j : ℕ) (ω ωf : Set E) (steps : List (ℕ × E))
+    (h : SpliceTrace G x η y k j ω steps ωf) : y k ∈ cluster G ωf x := by
+  induction h with
+  | stop _ _ hs => exact hs
+  | step => assumption
+
+ 
+
+theorem SpliceTrace.seams_pivotal (G : BondGraph V E) (x : V) (η : ℕ → Set E)
+    (y : ℕ → V) (k j : ℕ) (ω ωf : Set E) (steps : List (ℕ × E))
+    (h : SpliceTrace G x η y k j ω steps ωf) :
+    ∀ q ∈ steps, OpenMarkedPivotal G ωf {y k} x q.2 := by
+  induction h with
+  | stop => simp
+  | @step j i ω e a b rest ωf hn hji hik hmax hy he hab ha hb ht ih =>
+    intro q hq
+    rcases List.mem_cons.mp hq with rfl | hq
+    · let S := cluster G ω x
+      let B := cluster G (η i ∩ internalEdges G Sᶜ) (y i)
+      let ω' := spliceField G S B e ω (η i)
+      have hcut : cluster G (ω' \ {e}) x = S :=
+        splice_close_seam_cluster G ω (η i) x (y i) hy e a b hab he ha hb
+      have hfinal : cluster G (ωf \ {e}) x = S := by
+        rw [SpliceTrace.preserve_cut G x η y k i ω' ωf rest ht e]
+        · exact hcut
+        · intro l hil hlk
+          rw [hcut]
+          exact hmax l hil hlk
+      have hconn := SpliceTrace.endpoint G x η y k i ω' ωf rest ht
+      have hno : ¬Conn G (ωf \ {e}) x (y k) := by
+        intro hc
+        have : y k ∈ cluster G (ωf \ {e}) x := hc
+        rw [hfinal] at this
+        exact hn this
+      have heω : e ∈ ωf := by
+        by_contra heω
+        have heq : ωf \ {e} = ωf := by
+          ext f
+          constructor
+          · exact fun hh => hh.1
+          · intro hh
+            exact ⟨hh,fun hfe => heω ((Set.mem_singleton_iff.mp hfe) ▸ hh)⟩
+        exact hno (heq.symm ▸ hconn)
+      exact ⟨heω,⟨y k,rfl,hconn⟩,by simpa only [MarkedEvent,Set.mem_singleton_iff,exists_eq_left] using hno⟩
+    · exact ih q hq
+
+end SpliceTrace
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Splicing/MomentBounds.lean b/lean/OAI/Probability/UnimodularPercolation/Splicing/MomentBounds.lean
index c2464e15ebf686ee4896a34179ecd66c2bc8dd0e..de5eebc5b27b99a6aaed39bc36f3a56cfd6e0d5e 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Splicing/MomentBounds.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Splicing/MomentBounds.lean
@@ -1,1252 +1,1252 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Splicing.ReferenceLaws
-
-namespace OAI
-
-/-! Moment Bounds for bond percolation. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section SeededTraceMeasurable
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem uniformRank_fiber_eq (K : ℕ) (hK : 0 < K) (a : Fin K) :
-    {t : unitInterval | uniformRank K hK t = a} = contactInterval K a.val ∪
-      (if uniformRank K hK 1 = a then {1} else ∅) := by
-  ext t
-  by_cases ht : t = 1
-  · subst t
-    have hnot : (1 : unitInterval) ∉ contactInterval K a.val := by
-      intro h
-      have hp : (0:ℝ) < K := by exact_mod_cast hK
-      have hlt := (lt_div_iff₀ hp).mp h.2
-      have ha : (a.val:ℝ)+1 ≤ K := by exact_mod_cast (show a.val+1 ≤ K by omega)
-      push_cast at hlt
-      norm_num at hlt
-      linarith
-    simp [hnot]
-  · have hlt : (t:ℝ) < 1 := lt_of_le_of_ne t.property.2 (fun he => ht (Subtype.ext he))
-    have he := uniformRank_eq_iff K hK t hlt a
-    simp only [Set.mem_union,Set.mem_ofPred_eq]
-    split_ifs <;> simpa [ht] using he
-
-theorem measurableSet_contactInterval (K a : ℕ) : MeasurableSet (contactInterval K a) :=
-  (measurableSet_le measurable_const measurable_subtype_coe).inter
-    (measurableSet_lt measurable_subtype_coe measurable_const)
-
-theorem measurableSet_uniformRank_fiber (K : ℕ) (hK : 0 < K) (a : Fin K) :
-    MeasurableSet {t : unitInterval | uniformRank K hK t = a} := by
-  rw [uniformRank_fiber_eq]
-  apply (measurableSet_contactInterval K a.val).union
-  split_ifs
-  · exact measurableSet_singleton _
-  · exact MeasurableSet.empty
-
-theorem measurableSet_uniformContact_fiber (K : Finset E) (hK : K.Nonempty) (e : E) :
-    MeasurableSet {t : unitInterval | uniformContact K hK t = e} := by
-  by_cases he : e ∈ K
-  · simp_rw [uniformContact_eq_iff K hK _ e he]
-    exact measurableSet_uniformRank_fiber _ _ _
-  · have hz : {t : unitInterval | uniformContact K hK t = e} = ∅ := by
-      ext t
-      simp only [Set.mem_ofPred_eq,Set.mem_empty_iff_false,iff_false]
-      exact fun h => he (h ▸ uniformContact_mem K hK t)
-    rw [hz]
-    exact MeasurableSet.empty
-
-theorem measurableSet_chooseContact_fiber [Nonempty E] (i : ℕ) (K : Set E) (e : E) :
-    MeasurableSet {t : unitInterval | chooseContact (fun _ => t) i K = e} := by
-  unfold chooseContact
-  split_ifs
-  · exact measurableSet_uniformContact_fiber _ _ _
-  · by_cases he : Classical.arbitrary E = e <;> simp [he]
-  · by_cases he : Classical.arbitrary E = e <;> simp [he]
-
-theorem measurableSet_replaySeedFiber [Nonempty E] (G : BondGraph V E) (x : V)
-    (η : ℕ → Set E) (y : ℕ → V) (c : List (ℕ × E)) (ω : Set E) (i : ℕ) :
-    MeasurableSet (replaySeedFiber G x η y c ω i) := by
-  induction c generalizing ω with
-  | nil => exact MeasurableSet.univ
-  | cons q c ih =>
-    obtain ⟨j,e⟩ := q
-    simp only [replaySeedFiber]
-    split_ifs
-    · exact measurableSet_chooseContact_fiber _ _ _
-    · exact ih _
-
-theorem SpliceTrace.measurableSet_seed_event [Nonempty E] (G : BondGraph V E) (x : V)
-    (η : ℕ → Set E) (y : ℕ → V) (k j : ℕ) (ω ωf : Set E)
-    (c : List (ℕ × E)) (h : SpliceTrace G x η y k j ω c ωf) :
-    MeasurableSet {u : Fin k → unitInterval | FollowsContactRule G x η y
-      (chooseContact (finiteContactSeeds k u)) c ω} := by
-  rw [h.seed_event_eq_pi G x η y k j ω ωf c]
-  exact MeasurableSet.univ_pi (fun i => measurableSet_replaySeedFiber G x η y c ω (i.val+1))
-
-end SeededTraceMeasurable
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section FiniteTraceDistribution
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem finite_bounded_lists {A : Type*} (Q : Finset A) (k : ℕ) :
-    {c : List A | c.length ≤ k ∧ ∀ q ∈ c, q ∈ Q}.Finite := by
-  have hf := (List.finite_length_le ↥Q k).image (List.map Subtype.val)
-  apply hf.subset
-  rintro c ⟨hc,hQ⟩
-  let d : List ↥Q := c.attach.map (fun q => ⟨q.val,hQ q.val q.property⟩)
-  refine ⟨d,?_,?_⟩
-  · simpa only [d,List.length_map,List.length_attach,Set.mem_ofPred_eq] using hc
-  · simp [d,List.map_map]
-
-noncomputable def boundedTraceCodes (Q : Finset E) (k : ℕ) : Finset (List (ℕ × E)) :=
-  (finite_bounded_lists ((Finset.Icc 1 k) ×ˢ Q) k).toFinset
-
-@[simp] theorem mem_boundedTraceCodes (Q : Finset E) (k : ℕ) (c : List (ℕ × E)) :
-    c ∈ boundedTraceCodes Q k ↔ c.length ≤ k ∧ ∀ q ∈ c, 1 ≤ q.1 ∧ q.1 ≤ k ∧ q.2 ∈ Q := by
-  simp [boundedTraceCodes,Finset.mem_product,and_assoc]
-
-theorem SpliceTrace.length_le (G : BondGraph V E) (x : V) (η : ℕ → Set E)
-    (y : ℕ → V) (k j : ℕ) (ω ωf : Set E) (c : List (ℕ × E))
-    (h : SpliceTrace G x η y k j ω c ωf) : c.length ≤ k-j := by
-  induction h with
-  | stop => simp
-  | step => simp only [List.length_cons]; omega
-
-theorem SpliceTrace.mem_boundedTraceCodes (G : BondGraph V E) (x : V) (η : ℕ → Set E)
-    (y : ℕ → V) (k j : ℕ) (ω ωf : Set E) (c : List (ℕ × E))
-    (h : SpliceTrace G x η y k j ω c ωf) (Q : Finset E)
-    (hQ : ∀ i, 1 ≤ i → i ≤ k → η i ⊆ (Q : Set E)) : c ∈ boundedTraceCodes Q k := by
-  rw [Percolation.mem_boundedTraceCodes]
-  refine ⟨(h.length_le G x η y k j ω ωf c).trans (Nat.sub_le ..),?_⟩
-  intro q hq
-  obtain ⟨hj,hk,he,_⟩ := h.seam_data G x η y k j ω ωf c q hq
-  exact ⟨by omega,hk,hQ q.1 (by omega) hk he⟩
-
- 
-
-noncomputable def finiteTraceCodes (G : BondGraph V E) (x : V) (η : ℕ → Set E)
-    (y : ℕ → V) (k j : ℕ) (ω : Set E) (Q : Finset E) : Finset (List (ℕ × E)) :=
-  (boundedTraceCodes Q k).filter (fun c =>
-    SpliceTrace G x η y k j ω c (replaySplice G x η y c ω))
-
-theorem mem_finiteTraceCodes_iff (G : BondGraph V E) (x : V) (η : ℕ → Set E)
-    (y : ℕ → V) (k j : ℕ) (ω : Set E) (Q : Finset E) (c : List (ℕ × E)) :
-    c ∈ finiteTraceCodes G x η y k j ω Q ↔ c ∈ boundedTraceCodes Q k ∧
-      SpliceTrace G x η y k j ω c (replaySplice G x η y c ω) := by
-  simp [finiteTraceCodes]
-
- 
-
-theorem finite_trace_seed_partition [Nonempty E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) (η : ℕ → Set E)
-    (y : ℕ → V) (k M : ℕ) (Q : Finset E)
-    (hQ : ∀ i, 1 ≤ i → i ≤ k → η i ⊆ (Q : Set E))
-    (hchain : ∀ i ∈ Finset.Icc 1 k, Conn G (η i) (y i) (y (i-1)))
-    (hcaps : ∀ i ∈ Finset.Icc 1 k, (cluster G (η i) (y i)).Finite ∧
-      (cluster G (η i) (y i)).ncard ≤ M)
-    (j : ℕ) (hj : j ≤ k) (ω : Set E) (hyj : y j ∈ cluster G ω x)
-    (hfin : (cluster G ω x).Finite) :
-    let S := finiteTraceCodes G x η y k j ω Q
-    let A := fun c => {u : Fin k → unitInterval | FollowsContactRule G x η y
-      (chooseContact (finiteContactSeeds k u)) c ω}
-    (⋃ c ∈ S, A c) = Set.univ ∧ Set.PairwiseDisjoint (↑S) A := by
-  intro S A
-  constructor
-  · apply Set.eq_univ_of_forall
-    intro u
-    obtain ⟨c,ωf,ht,hr,_⟩ := exists_seeded_spliceTrace G hloc x η y k M hchain hcaps
-      (chooseContact (finiteContactSeeds k u)) (chooseContact_mem _) j hj ω hyj _ hfin (le_refl _)
-    have hc : c ∈ S := (mem_finiteTraceCodes_iff ..).mpr
-      ⟨ht.mem_boundedTraceCodes G x η y k j ω ωf c Q hQ,
-        (ht.replay_eq G x η y k j ω ωf c).symm ▸ ht⟩
-    exact Set.mem_iUnion.mpr ⟨c,Set.mem_iUnion.mpr ⟨hc,hr⟩⟩
-  · intro c hc d hd hcd
-    apply Set.disjoint_left.mpr
-    intro u huc hud
-    have ht := (mem_finiteTraceCodes_iff G x η y k j ω Q c).mp hc
-    have hs := (mem_finiteTraceCodes_iff G x η y k j ω Q d).mp hd
-    exact hcd (ht.2.seeded_unique G x η y k j ω _ _ c d hs.2
-      (chooseContact (finiteContactSeeds k u)) huc hud).1
-
- 
-theorem sum_finite_trace_probabilities [Nonempty E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) (η : ℕ → Set E)
-    (y : ℕ → V) (k M : ℕ) (Q : Finset E)
-    (hQ : ∀ i, 1 ≤ i → i ≤ k → η i ⊆ (Q : Set E))
-    (hchain : ∀ i ∈ Finset.Icc 1 k, Conn G (η i) (y i) (y (i-1)))
-    (hcaps : ∀ i ∈ Finset.Icc 1 k, (cluster G (η i) (y i)).Finite ∧
-      (cluster G (η i) (y i)).ncard ≤ M)
-    (j : ℕ) (hj : j ≤ k) (ω : Set E) (hyj : y j ∈ cluster G ω x)
-    (hfin : (cluster G ω x).Finite) :
-    (∑ c ∈ finiteTraceCodes G x η y k j ω Q,
-      ∏ i : Fin k, (traceContactCount G η y (replaySplice G x η y c ω) x c (i.val+1):ℝ)⁻¹) = 1 := by
-  let S := finiteTraceCodes G x η y k j ω Q
-  let A := fun c => {u : Fin k → unitInterval | FollowsContactRule G x η y
-    (chooseContact (finiteContactSeeds k u)) c ω}
-  have hpart := finite_trace_seed_partition G hloc x η y k M Q hQ hchain hcaps j hj ω hyj hfin
-  have hm : ∀ c ∈ S, MeasurableSet (A c) := by
-    intro c hc
-    exact ((mem_finiteTraceCodes_iff G x η y k j ω Q c).mp hc).2.measurableSet_seed_event
-      G x η y k j ω _ c
-  have hsum : ∑ c ∈ S, volume (A c) = 1 := by
-    rw [← measure_biUnion_finset hpart.2 hm,hpart.1,measure_univ]
-  have hr := congrArg ENNReal.toReal hsum
-  rw [ENNReal.toReal_sum (fun c _ => measure_ne_top volume (A c)),ENNReal.toReal_one] at hr
-  rw [← hr]
-  apply Finset.sum_congr rfl
-  intro c hc
-  exact (((mem_finiteTraceCodes_iff G x η y k j ω Q c).mp hc).2.seed_probability
-    G hloc x η y k j ω _ c (fun i hi hik => (hcaps i (Finset.mem_Icc.mpr ⟨hi,hik⟩)).1)).symm
-
-end FiniteTraceDistribution
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section FiniteSpliceState
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem SpliceTrace.runMasks_execution (G : BondGraph V E) (x : V)
-    (η : ℕ → Set E) (y : ℕ → V) (k j : ℕ) (ω ωf : Set E) (c : List (ℕ × E))
-    (h : SpliceTrace G x η y k j ω c ωf) :
-    ∀ θ : ℕ → Set E, (∀ l, j < l → l ≤ k → θ l = η l) →
-      SpliceExecution G x (ω,θ) c (runMasks (replayMasks G x η y c ω) (ω,θ)) := by
-  induction h with
-  | stop => intro θ _; exact SpliceExecution.nil _
-  | @step j i ω e a b c ωf hn hji hik hm hy he hab ha hb ht ih =>
-    intro θ hθ
-    let S := cluster G ω x
-    let B := cluster G (η i ∩ internalEdges G Sᶜ) (y i)
-    let M := spliceMask G S B e
-    let ω' := spliceField G S B e ω (η i)
-    let θ' := Function.update θ i (exchangeOn M (θ i) ω)
-    have hθi := hθ i hji hik
-    have hstep : exchangeState M i (ω,θ) = (ω',θ') := by
-      apply Prod.ext
-      · change exchangeOn M ω (θ i) = exchangeOn M ω (η i)
-        rw [hθi]
-      · rfl
-    change SpliceExecution G x (ω,θ) ((i,e)::c)
-      (runMasks (replayMasks G x η y c ω') (exchangeState M i (ω,θ)))
-    rw [hstep]
-    refine SpliceExecution.cons (ω,θ) (ω',θ') _ (i,e) c ?_ ?_
-    · refine ⟨y i,a,b,hy,?_,hab,ha,?_,?_⟩
-      · simpa only [hθi] using he
-      · simpa only [hθi] using hb
-      · simpa only [hθi] using hstep.symm
-    · apply ih θ'
-      intro l hil hlk
-      rw [show θ' l = θ l from Function.update_of_ne (by omega) _ _]
-      exact hθ l (hji.trans hil) hlk
-
-abbrev FiniteSpliceData (Q : Finset E) (J : Finset ℕ) :=
-  FiniteField Q × (↥J → FiniteField Q)
-
-def encodeFiniteSplice (Q : Finset E) (J : Finset ℕ)
-    (s : FiniteSpliceData Q J) : SpliceState E :=
-  (finiteFieldSet Q s.1,finiteFieldFamily Q J s.2)
-
-theorem encodeFiniteSplice_injective (Q : Finset E) (J : Finset ℕ) :
-    Function.Injective (encodeFiniteSplice Q J) := by
-  intro a b he
-  exact Prod.ext (finiteFieldSet_injective Q (congrArg Prod.fst he))
-    (finiteFieldFamily_injective Q J (congrArg Prod.snd he))
-
-noncomputable def finiteSpliceDomain (Q : Finset E) (J : Finset ℕ) : Finset (SpliceState E) :=
-  Finset.univ.image (encodeFiniteSplice Q J)
-
-theorem spliceTupleWeight_encode (Q : Finset E) (J : Finset ℕ) (p : ℝ)
-    (s : FiniteSpliceData Q J) : spliceTupleWeight Q J p (encodeFiniteSplice Q J s) =
-      finiteFieldWeight Q p s.1 * finiteFamilyWeight Q J p s.2 := by
-  simp only [spliceTupleWeight,encodeFiniteSplice,domainBits_finiteFieldSet,
-    prod_fieldFamilyWeight,finiteFieldWeight]
-
-theorem sum_finiteSpliceDomain (Q : Finset E) (J : Finset ℕ) (p : ℝ) :
-    ∑ s ∈ finiteSpliceDomain Q J, spliceTupleWeight Q J p s = 1 := by
-  rw [finiteSpliceDomain,Finset.sum_image (encodeFiniteSplice_injective Q J).injOn]
-  simp only [spliceTupleWeight_encode,Fintype.sum_prod_type,← Finset.mul_sum,
-    sum_finiteFamilyWeight,mul_one,sum_finiteFieldWeight]
-
- 
-theorem mem_finiteSpliceDomain_iff (Q : Finset E) (J : Finset ℕ) (s : SpliceState E) :
-    s ∈ finiteSpliceDomain Q J ↔ s.1 ⊆ (Q : Set E) ∧
-      (∀ i, s.2 i ⊆ (Q : Set E)) ∧ ∀ i ∉ J, s.2 i = ∅ := by
-  constructor
-  · intro hs
-    obtain ⟨a,_,rfl⟩ := Finset.mem_image.mp hs
-    refine ⟨?_,?_,?_⟩
-    · exact Finset.coe_subset.mpr (Finset.mem_powerset.mp a.1.property)
-    · intro i
-      by_cases hi : i ∈ J
-      · simpa only [encodeFiniteSplice,finiteFieldFamily,dite_eq_left hi,finiteFieldSet] using
-          Finset.coe_subset.mpr (Finset.mem_powerset.mp (a.2 ⟨i,hi⟩).property)
-      · simp [encodeFiniteSplice,finiteFieldFamily,hi]
-    · intro i hi
-      simp [encodeFiniteSplice,finiteFieldFamily,hi]
-  · rintro ⟨hω,hη,hz⟩
-    let desc (ω : Set E) : FiniteField Q := ⟨domainBits Q ω,Finset.mem_powerset.mpr (domainBits_subset Q ω)⟩
-    have he (ω : Set E) (h : ω ⊆ (Q : Set E)) : finiteFieldSet Q (desc ω) = ω := by
-      ext e
-      simp only [finiteFieldSet,desc,Finset.mem_coe,mem_domainBits]
-      exact ⟨fun hh => hh.2,fun hh => ⟨h hh,hh⟩⟩
-    refine Finset.mem_image.mpr ⟨(desc s.1,fun i => desc (s.2 i)),Finset.mem_univ _,?_⟩
-    apply Prod.ext (he s.1 hω)
-    funext i
-    by_cases hi : i ∈ J
-    · simp only [encodeFiniteSplice,finiteFieldFamily,dite_eq_left hi,he _ (hη i)]
-    · simp only [encodeFiniteSplice,finiteFieldFamily,dite_eq_right hi,hz i hi]
-
-theorem exchangeState_mem_finiteSpliceDomain (Q : Finset E) (J : Finset ℕ)
-    (s : SpliceState E) (hs : s ∈ finiteSpliceDomain Q J) (i : ℕ) (hi : i ∈ J) (M : Set E) :
-    exchangeState M i s ∈ finiteSpliceDomain Q J := by
-  obtain ⟨hω,hη,hz⟩ := (mem_finiteSpliceDomain_iff Q J s).mp hs
-  apply (mem_finiteSpliceDomain_iff Q J _).mpr
-  refine ⟨exchangeOn_subset M _ _ _ hω (hη i),?_,?_⟩
-  · intro l
-    by_cases hli : l = i
-    · simpa only [hli,exchangeState_at] using exchangeOn_subset M _ _ _ (hη i) hω
-    · rw [exchangeState_away _ _ _ _ hli]
-      exact hη l
-  · intro l hl
-    have hli : l ≠ i := by intro he; subst l; exact hl hi
-    rw [exchangeState_away _ _ _ _ hli]
-    exact hz l hl
-
-theorem runMasks_mem_finiteSpliceDomain (Q : Finset E) (J : Finset ℕ)
-    (s : SpliceState E) (hs : s ∈ finiteSpliceDomain Q J) (ms : List (ℕ × Set E))
-    (hJ : ∀ q ∈ ms, q.1 ∈ J) : runMasks ms s ∈ finiteSpliceDomain Q J := by
-  induction ms generalizing s with
-  | nil => exact hs
-  | cons q ms ih =>
-    apply ih _ (exchangeState_mem_finiteSpliceDomain Q J s hs q.1 (hJ q (by simp)) q.2)
-    exact fun q hq => hJ q (by simp [hq])
-
-end FiniteSpliceState
-
-end Percolation
-
- 
-open MeasureTheory
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section FiniteSwitchDensity
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-noncomputable def finiteInverseLoad {I : Type*} [Fintype I]
-    (out : I → SpliceState E) (mass : I → ℝ) (t : SpliceState E) : ℝ :=
-  ∑ a, if out a = t then mass a else 0
-
-theorem finiteInverseLoad_nonneg {I : Type*} [Fintype I]
-    (out : I → SpliceState E) (mass : I → ℝ) (hm : ∀ a, 0 ≤ mass a) (t : SpliceState E) :
-    0 ≤ finiteInverseLoad out mass t := by
-  apply Finset.sum_nonneg
-  intro a _
-  split_ifs
-  · exact hm a
-  · exact le_refl 0
-
- 
-
-theorem finiteInverseLoad_le {I : Type*} [Fintype I]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (x z : V) (k M : ℕ)
-    (L : ℝ) (hL : 1 ≤ L)
-    (inp out : I → SpliceState E) (steps : I → List (ℕ × E)) (y : I → ℕ → V)
-    (hir : Function.Injective (fun a => (inp a,steps a)))
-    (htrace : ∀ a, SpliceTrace G x (inp a).2 (y a) k 0 (inp a).1 (steps a) (out a).1)
-    (hex : ∀ a, SpliceExecution G x (inp a) (steps a) (out a))
-    (hz : ∀ a, y a k = z) (hfin : ∀ a, (cluster G (out a).1 x).Finite)
-    (hgood : ∀ a i, 1 ≤ i → i ≤ k → (cluster G ((inp a).2 i) (y a i)).Finite ∧
-      (cluster G ((inp a).2 i) (y a i)).ncard ≤ M ∧ GoodForScale G ((inp a).2 i) (y a i) L)
-    (t : SpliceState E) :
-    finiteInverseLoad out (fun a => ∏ i : Fin k,
-      (traceContactCount G (inp a).2 (y a) (out a).1 x (steps a) (i.val+1):ℝ)⁻¹) t ≤
-      inverseCodeMass M L ^ k := by
-  let J := {a : I // out a = t}
-  have hr (a : J) (i : Fin k) : 0 < traceContactCount G (inp a.val).2 (y a.val)
-      (out a.val).1 x (steps a.val) (i.val+1) := by
-    have hh := ((htrace a.val).contact_weight_bound G hloc x (inp a.val).2 (y a.val)
-      k 0 M (inp a.val).1 (out a.val).1 (steps a.val) (hfin a.val) L hL
-      (hgood a.val) (i.val+1)).1
-    omega
-  have hb := finite_inverse_contact_mass_le G hloc x z k M L hL (0 : unitInterval) ∅
-    (fun a : J => inp a.val) (fun a : J => out a.val) (fun a : J => steps a.val)
-    (fun a : J => y a.val)
-    (fun a b he => Subtype.ext (hir he)) (fun a => htrace a.val) (fun a => hex a.val)
-    (fun a => hz a.val) (fun a => hfin a.val) (fun a => hgood a.val)
-    (fun _ _ => 0) hr {t} (fun a => by simpa only [Finset.mem_singleton] using a.property)
-  simp only [real_volume_contact_choice_cube _ _ (hr _),spliceTupleWeight,domainBits,
-    Finset.filter_empty,bernoulliWeight,Finset.card_empty,Nat.sub_self,pow_zero,
-    Finset.prod_const_one,mul_one,one_mul,Finset.sum_singleton] at hb
-  rw [finiteInverseLoad,← Finset.sum_filter]
-  rw [Finset.sum_subtype (p := fun a : I => out a = t) _ (by intro a; simp) _]
-  exact hb
-
- 
-
-theorem finiteInverseLoad_density {I : Type*} [Fintype I]
-    (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E) (p : ℝ)
-    (inp out : I → SpliceState E) (steps : I → List (ℕ × E)) (y : I → ℕ → V)
-    (htrace : ∀ a, SpliceTrace G x (inp a).2 (y a) k 0 (inp a).1 (steps a) (out a).1)
-    (hex : ∀ a, SpliceExecution G x (inp a) (steps a) (out a))
-    (mass : I → ℝ) (B : Finset (SpliceState E)) (hB : ∀ a, out a ∈ B)
-    (f : SpliceState E → ℝ) :
-    (∑ a, spliceTupleWeight Q (Finset.Icc 1 k) p (inp a) * mass a * f (out a)) =
-      ∑ t ∈ B, spliceTupleWeight Q (Finset.Icc 1 k) p t * finiteInverseLoad out mass t * f t := by
-  have hw (a : I) : spliceTupleWeight Q (Finset.Icc 1 k) p (inp a) =
-      spliceTupleWeight Q (Finset.Icc 1 k) p (out a) := by
-    symm
-    apply (hex a).weight G x (inp a) (out a) (steps a) Q (Finset.Icc 1 k) _ p
-    intro q hq
-    have h := (htrace a).seam_data G x (inp a).2 (y a) k 0 (inp a).1 (out a).1 (steps a) q hq
-    exact Finset.mem_Icc.mpr ⟨h.1,h.2.1⟩
-  simp only [finiteInverseLoad,Finset.mul_sum,Finset.sum_mul]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro a _
-  simp only [mul_ite,mul_zero,ite_mul,zero_mul]
-  rw [Finset.sum_ite_eq,ite_eq_left (hB a),hw a]
-
-end FiniteSwitchDensity
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section FiniteForwardDensity
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-abbrev FiniteTraceIndex (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
-    (A : Finset (SpliceState E)) (y : SpliceState E → ℕ → V) :=
-  (s : ↥A) × ↥(finiteTraceCodes G x s.val.2 (y s.val) k 0 s.val.1 Q)
-
-def traceIndexInput (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
-    (A : Finset (SpliceState E)) (y : SpliceState E → ℕ → V)
-    (a : FiniteTraceIndex G x k Q A y) : SpliceState E := a.1.val
-
-def traceIndexCode (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
-    (A : Finset (SpliceState E)) (y : SpliceState E → ℕ → V)
-    (a : FiniteTraceIndex G x k Q A y) : List (ℕ × E) := a.2.val
-
-def traceIndexOutput (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
-    (A : Finset (SpliceState E)) (y : SpliceState E → ℕ → V)
-    (a : FiniteTraceIndex G x k Q A y) : SpliceState E :=
-  runMasks (replayMasks G x a.1.val.2 (y a.1.val) a.2.val a.1.val.1) a.1.val
-
-theorem traceIndex_valid (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
-    (A : Finset (SpliceState E)) (y : SpliceState E → ℕ → V)
-    (a : FiniteTraceIndex G x k Q A y) :
-    SpliceTrace G x a.1.val.2 (y a.1.val) k 0 a.1.val.1 a.2.val
-      (traceIndexOutput G x k Q A y a).1 ∧
-    SpliceExecution G x a.1.val a.2.val (traceIndexOutput G x k Q A y a) := by
-  have ht := ((mem_finiteTraceCodes_iff G x a.1.val.2 (y a.1.val) k 0 a.1.val.1 Q a.2.val).mp
-    a.2.property).2
-  have hf := ht.runMasks_first G x a.1.val.2 (y a.1.val) k 0 a.1.val.1 _ a.2.val
-    a.1.val.2 (by intros; rfl)
-  refine ⟨?_,?_⟩
-  · simpa only [traceIndexOutput,hf] using ht
-  · exact ht.runMasks_execution G x a.1.val.2 (y a.1.val) k 0 a.1.val.1 _ a.2.val
-      a.1.val.2 (by intros; rfl)
-
-theorem traceIndex_injective (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
-    (A : Finset (SpliceState E)) (y : SpliceState E → ℕ → V) :
-    Function.Injective (fun a : FiniteTraceIndex G x k Q A y =>
-      (traceIndexInput G x k Q A y a,traceIndexCode G x k Q A y a)) := by
-  rintro ⟨a,c⟩ ⟨b,d⟩ he
-  have hah : a = b := Subtype.ext (congrArg Prod.fst he)
-  subst b
-  have hcd : c = d := Subtype.ext (congrArg Prod.snd he)
-  subst d
-  rfl
-
-theorem traceIndexOutput_mem (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
-    (A : Finset (SpliceState E)) (y : SpliceState E → ℕ → V)
-    (hA : A ⊆ finiteSpliceDomain Q (Finset.Icc 1 k))
-    (a : FiniteTraceIndex G x k Q A y) :
-    traceIndexOutput G x k Q A y a ∈ finiteSpliceDomain Q (Finset.Icc 1 k) := by
-  apply runMasks_mem_finiteSpliceDomain Q (Finset.Icc 1 k) a.1.val (hA a.1.property)
-  intro q hq
-  have hmap : q.1 ∈ (replayMasks G x a.1.val.2 (y a.1.val) a.2.val a.1.val.1).map Prod.fst :=
-    List.mem_map.mpr ⟨q,hq,rfl⟩
-  rw [replayMasks_indices] at hmap
-  obtain ⟨r,hr,her⟩ := List.mem_map.mp hmap
-  have h := (traceIndex_valid G x k Q A y a).1.seam_data G x a.1.val.2 (y a.1.val)
-    k 0 a.1.val.1 _ a.2.val r hr
-  exact her ▸ Finset.mem_Icc.mpr ⟨h.1,h.2.1⟩
-
- 
-
-noncomputable def traceIndexMass (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
-    (A : Finset (SpliceState E)) (y : SpliceState E → ℕ → V)
-    (a : FiniteTraceIndex G x k Q A y) : ℝ :=
-  ∏ i : Fin k, (traceContactCount G a.1.val.2 (y a.1.val)
-    (traceIndexOutput G x k Q A y a).1 x a.2.val (i.val+1):ℝ)⁻¹
-
-noncomputable def finiteForwardLoad (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
-    (A : Finset (SpliceState E)) (y : SpliceState E → ℕ → V) (t : SpliceState E) : ℝ :=
-  finiteInverseLoad (traceIndexOutput G x k Q A y) (traceIndexMass G x k Q A y) t
-
- 
-
-theorem finiteForwardLoad_bounds (G : BondGraph V E) (hloc : LocallyFinite G)
-    (x z : V) (k M : ℕ) (L : ℝ) (hL : 1 ≤ L) (Q : Finset E)
-    (A : Finset (SpliceState E)) (y : SpliceState E → ℕ → V)
-    (hA : A ⊆ finiteSpliceDomain Q (Finset.Icc 1 k))
-    (hz : ∀ s ∈ A, y s k = z)
-    (hgood : ∀ s ∈ A, ∀ i, 1 ≤ i → i ≤ k → (cluster G (s.2 i) (y s i)).Finite ∧
-      (cluster G (s.2 i) (y s i)).ncard ≤ M ∧ GoodForScale G (s.2 i) (y s i) L)
-    (t : SpliceState E) :
-    0 ≤ finiteForwardLoad G x k Q A y t ∧ finiteForwardLoad G x k Q A y t ≤ inverseCodeMass M L ^ k := by
-  constructor
-  · exact finiteInverseLoad_nonneg _ _ (by intro a; unfold traceIndexMass; positivity) t
-  · apply finiteInverseLoad_le G hloc x z k M L hL
-      (traceIndexInput G x k Q A y) (traceIndexOutput G x k Q A y) (traceIndexCode G x k Q A y)
-      (fun a => y a.1.val) (traceIndex_injective G x k Q A y)
-      (fun a => (traceIndex_valid G x k Q A y a).1)
-      (fun a => (traceIndex_valid G x k Q A y a).2) (fun a => hz a.1.val a.1.property)
-      _ (fun a => hgood a.1.val a.1.property) t
-    intro a
-    have hs := (mem_finiteSpliceDomain_iff Q (Finset.Icc 1 k) _).mp
-      (traceIndexOutput_mem G x k Q A y hA a)
-    exact cluster_finite_of_finite_edges G (Q.finite_toSet.subset hs.1) x
-
- 
-
-theorem finiteForwardLoad_mean [Nonempty E] (G : BondGraph V E) (hloc : LocallyFinite G)
-    (x : V) (k M : ℕ) (Q : Finset E) (p : ℝ)
-    (A : Finset (SpliceState E)) (y : SpliceState E → ℕ → V)
-    (hA : A ⊆ finiteSpliceDomain Q (Finset.Icc 1 k))
-    (hy : ∀ s ∈ A, y s 0 ∈ cluster G s.1 x)
-    (hchain : ∀ s ∈ A, ∀ i ∈ Finset.Icc 1 k, Conn G (s.2 i) (y s i) (y s (i-1)))
-    (hcaps : ∀ s ∈ A, ∀ i ∈ Finset.Icc 1 k, (cluster G (s.2 i) (y s i)).Finite ∧
-      (cluster G (s.2 i) (y s i)).ncard ≤ M) :
-    (∑ t ∈ finiteSpliceDomain Q (Finset.Icc 1 k),
-      spliceTupleWeight Q (Finset.Icc 1 k) p t * finiteForwardLoad G x k Q A y t) =
-      ∑ s ∈ A, spliceTupleWeight Q (Finset.Icc 1 k) p s := by
-  have hd := finiteInverseLoad_density G x k Q p
-    (traceIndexInput G x k Q A y) (traceIndexOutput G x k Q A y) (traceIndexCode G x k Q A y)
-    (fun a => y a.1.val) (fun a => (traceIndex_valid G x k Q A y a).1)
-    (fun a => (traceIndex_valid G x k Q A y a).2)
-    (traceIndexMass G x k Q A y) (finiteSpliceDomain Q (Finset.Icc 1 k))
-    (traceIndexOutput_mem G x k Q A y hA) (fun _ => 1)
-  simp only [mul_one] at hd
-  unfold finiteForwardLoad
-  rw [← hd,Fintype.sum_sigma]
-  rw [← Finset.sum_coe_sort A (fun s => spliceTupleWeight Q (Finset.Icc 1 k) p s)]
-  apply Finset.sum_congr rfl
-  intro s _
-  change (∑ c : ↥(finiteTraceCodes G x s.val.2 (y s.val) k 0 s.val.1 Q),
-    spliceTupleWeight Q (Finset.Icc 1 k) p s.val * traceIndexMass G x k Q A y ⟨s,c⟩) = _
-  rw [← Finset.mul_sum]
-  suffices hm : (∑ c : ↥(finiteTraceCodes G x s.val.2 (y s.val) k 0 s.val.1 Q),
-      traceIndexMass G x k Q A y ⟨s,c⟩) = 1 by rw [hm,mul_one]
-  have hs := (mem_finiteSpliceDomain_iff Q (Finset.Icc 1 k) s.val).mp (hA s.property)
-  have hn := sum_finite_trace_probabilities G hloc x s.val.2 (y s.val) k M Q
-    (fun i _ _ => hs.2.1 i) (hchain s.val s.property) (hcaps s.val s.property)
-    0 (Nat.zero_le _) s.val.1 (hy s.val s.property)
-    (cluster_finite_of_finite_edges G (Q.finite_toSet.subset hs.1) x)
-  rw [← Finset.sum_coe_sort (finiteTraceCodes G x s.val.2 (y s.val) k 0 s.val.1 Q)] at hn
-  rw [← hn]
-  apply Finset.sum_congr rfl
-  intro c _
-  have ht := ((mem_finiteTraceCodes_iff G x s.val.2 (y s.val) k 0 s.val.1 Q c.val).mp c.property).2
-  have hf := ht.runMasks_first G x s.val.2 (y s.val) k 0 s.val.1 _ c.val s.val.2 (by intros; rfl)
-  simp only [traceIndexMass,traceIndexOutput,hf]
-
-end FiniteForwardDensity
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section ContactIntegratedSecondMoment
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem finite_event_mass_le {Ω I : Type*} [MeasurableSpace Ω] [Fintype I]
-    (μ : Measure Ω) [IsProbabilityMeasure μ] (P : I → Set Ω)
-    (hm : ∀ a, MeasurableSet (P a)) (w : I → ℝ) (B : ℝ)
-    (hpoint : ∀ u, (∑ a, if u ∈ P a then w a else 0) ≤ B) :
-    (∑ a, w a * μ.real (P a)) ≤ B := by
-  let f (a : I) : Ω → ℝ := (P a).indicator (fun _ => w a)
-  have hf : ∀ a, Integrable (f a) μ := fun a => (integrable_const _).indicator (hm a)
-  have hi := integral_mono (integrable_finsetSum _ (fun a _ => hf a))
-    (integrable_const B) (fun u => hpoint u)
-  change (∫ u, ∑ a, f a u ∂μ) ≤ (∫ _ : Ω, B ∂μ) at hi
-  rw [integral_finsetSum _ (fun a _ => hf a)] at hi
-  simpa only [f,integral_indicator_const _ (hm _),smul_eq_mul,mul_comm,
-    integral_const,probReal_univ,one_smul,mul_one] using hi
-
- 
-
-theorem finite_separated_contact_second_moment_le {I : Type*} [Fintype I] [Nonempty E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) (k n M : ℕ) (p : unitInterval) (Q : Finset E)
-    (η₁ η₂ : I → ℕ → Set E) (y₁ y₂ : (ℕ → Set E) → ℕ → V)
-    (success₁ success₂ : (ℕ → Set E) → Prop)
-    (ω₁ ω₂ ωf : I → Set E) (c₁ c₂ : I → List (ℕ × E))
-    (hir : Function.Injective (fun a =>
-      (jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a),
-        c₁ a,c₂ a)))
-    (h₁ : ∀ a, SpliceTrace G x (η₁ a) (y₁ (η₁ a)) k 0 (ω₁ a) (c₁ a) (ωf a))
-    (h₂ : ∀ a, SpliceTrace G x (η₂ a) (y₂ (η₂ a)) k 0 (ω₂ a) (c₂ a) (ωf a))
-    (hc₁ : ∀ a, c₁ a ≠ []) (hc₂ : ∀ a, c₂ a ≠ [])
-    (hsep : ∀ a, GraphSeparated G (cluster G (ωf a) x \ cluster G (ω₁ a) x)
-      (cluster G (ωf a) x \ cluster G (ω₂ a) x))
-    (hbin : ∀ a, (cluster G (ω₁ a) x).Finite ∧ n ≤ (cluster G (ω₁ a) x).ncard ∧
-      (cluster G (ω₁ a) x).ncard < 2*n)
-    (hcap₁ : ∀ a i, 1 ≤ i → i ≤ k → (cluster G (η₁ a i) (y₁ (η₁ a) i)).Finite)
-    (hcap₂ : ∀ a i, 1 ≤ i → i ≤ k → (cluster G (η₂ a i) (y₂ (η₂ a) i)).Finite ∧
-      (cluster G (η₂ a i) (y₂ (η₂ a) i)).ncard ≤ M)
-    (hsc₁ : ∀ a, success₁ (η₁ a)) (hsc₂ : ∀ a, success₂ (η₂ a))
-    (hsω₁ : ∀ a, ω₁ a ⊆ (Q : Set E)) (hsω₂ : ∀ a, ω₂ a ⊆ (Q : Set E))
-    (hsη₁ : ∀ a i, η₁ a i ⊆ (Q : Set E)) (hsη₂ : ∀ a i, η₂ a i ⊆ (Q : Set E))
-    (hz₁ : ∀ a i, i ∉ Finset.Icc 1 k → η₁ a i = ∅)
-    (hz₂ : ∀ a i, i ∉ Finset.Icc 1 k → η₂ a i = ∅) :
-    (∑ a, jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p
-      (jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a)) *
-      (∏ i : Fin k, (traceContactCount G (η₁ a) (y₁ (η₁ a)) (ωf a) x (c₁ a) (i.val+1):ℝ)⁻¹) *
-      (∏ i : Fin k, (traceContactCount G (η₂ a) (y₂ (η₂ a)) (ωf a) x (c₂ a) (i.val+1):ℝ)⁻¹)) ≤
-    ∑ ω : FiniteField Q, finiteFieldWeight Q p ω *
-      (if (cluster G (finiteFieldSet Q ω) x).Finite ∧
-        n-k*M ≤ (cluster G (finiteFieldSet Q ω) x).ncard ∧
-        (cluster G (finiteFieldSet Q ω) x).ncard < 2*n then
-        finiteFamilyEvent Q (Finset.Icc 1 k) p (fun η => success₁ η ∧
-          (cluster G (finiteFieldSet Q ω) x ∩ trajectoryUnion G η (y₁ η) k).Nonempty) *
-        finiteFamilyEvent Q (Finset.Icc 1 k) p (fun η => success₂ η ∧
-          (cluster G (finiteFieldSet Q ω) x ∩ trajectoryUnion G η (y₂ η) k).Nonempty) else 0) := by
-  let S := Fin k → unitInterval
-  let μ : Measure (S × S) := (volume : Measure S).prod volume
-  let P₁ (a : I) : Set S := {u | FollowsContactRule G x (η₁ a) (y₁ (η₁ a))
-    (chooseContact (finiteContactSeeds k u)) (c₁ a) (ω₁ a)}
-  let P₂ (a : I) : Set S := {u | FollowsContactRule G x (η₂ a) (y₂ (η₂ a))
-    (chooseContact (finiteContactSeeds k u)) (c₂ a) (ω₂ a)}
-  let w (a : I) := jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p
-    (jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a))
-  have hm₁ (a : I) : MeasurableSet (P₁ a) :=
-    (h₁ a).measurableSet_seed_event G x (η₁ a) (y₁ (η₁ a)) k 0 (ω₁ a) (ωf a) (c₁ a)
-  have hm₂ (a : I) : MeasurableSet (P₂ a) :=
-    (h₂ a).measurableSet_seed_event G x (η₂ a) (y₂ (η₂ a)) k 0 (ω₂ a) (ωf a) (c₂ a)
-  have hp (a : I) : μ.real (P₁ a ×ˢ P₂ a) =
-      (∏ i : Fin k, (traceContactCount G (η₁ a) (y₁ (η₁ a)) (ωf a) x (c₁ a) (i.val+1):ℝ)⁻¹) *
-      (∏ i : Fin k, (traceContactCount G (η₂ a) (y₂ (η₂ a)) (ωf a) x (c₂ a) (i.val+1):ℝ)⁻¹) := by
-    rw [measureReal_def,Measure.prod_prod,ENNReal.toReal_mul,
-      (h₁ a).seed_probability G hloc x (η₁ a) (y₁ (η₁ a)) k 0 (ω₁ a) (ωf a) (c₁ a) (hcap₁ a),
-      (h₂ a).seed_probability G hloc x (η₂ a) (y₂ (η₂ a)) k 0 (ω₂ a) (ωf a) (c₂ a) (fun i hi hik => (hcap₂ a i hi hik).1)]
-  conv_lhs => arg 2; ext a; rw [mul_assoc,← hp a]
-  apply finite_event_mass_le μ (fun a => P₁ a ×ˢ P₂ a) (fun a => (hm₁ a).prod (hm₂ a)) w
-  intro u
-  let J := {a : I // u ∈ P₁ a ×ˢ P₂ a}
-  have hb := finite_separated_second_moment_le G x k n M p Q
-    (fun a : J => η₁ a.val) (fun a : J => η₂ a.val) y₁ y₂ success₁ success₂
-    (chooseContact (finiteContactSeeds k u.1)) (chooseContact (finiteContactSeeds k u.2))
-    (fun a : J => ω₁ a.val) (fun a : J => ω₂ a.val) (fun a : J => ωf a.val)
-    (fun a : J => c₁ a.val) (fun a : J => c₂ a.val)
-    (fun _ _ he => Subtype.ext (hir he)) (fun a => h₁ a.val) (fun a => h₂ a.val)
-    (fun a => hc₁ a.val) (fun a => hc₂ a.val) (fun a => hsep a.val)
-    (fun a => a.property.1) (fun a => a.property.2)
-    (fun a => hbin a.val) (fun a => hcap₂ a.val) (fun a => hsc₁ a.val) (fun a => hsc₂ a.val)
-    (fun a => hsω₁ a.val) (fun a => hsω₂ a.val) (fun a => hsη₁ a.val) (fun a => hsη₂ a.val)
-    (fun a => hz₁ a.val) (fun a => hz₂ a.val)
-  convert hb using 1
-  have he := Finset.sum_subtype (p := fun a => u ∈ P₁ a ×ˢ P₂ a) (F := inferInstance) (Finset.univ.filter (fun a => u ∈ P₁ a ×ˢ P₂ a))
-    (by intro a; simp) w
-  simpa [Finset.sum_filter,w] using he
-
-end ContactIntegratedSecondMoment
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section FiniteRootLoad
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-noncomputable def finiteRootLoad (Q : Finset E) (J : Finset ℕ) (p : ℝ)
-    (f : SpliceState E → ℝ) (ω : FiniteField Q) : ℝ :=
-  ∑ η : ↥J → FiniteField Q, finiteFamilyWeight Q J p η *
-    f (finiteFieldSet Q ω,finiteFieldFamily Q J η)
-
-theorem finiteRootLoad_nonneg (Q : Finset E) (J : Finset ℕ) (p : unitInterval)
-    (f : SpliceState E → ℝ) (hf : ∀ s ∈ finiteSpliceDomain Q J, 0 ≤ f s)
-    (ω : FiniteField Q) : 0 ≤ finiteRootLoad Q J p f ω := by
-  apply Finset.sum_nonneg
-  intro η _
-  apply mul_nonneg (finiteFamilyWeight_nonneg Q J p η)
-  exact hf _ (Finset.mem_image.mpr ⟨(ω,η),Finset.mem_univ _,rfl⟩)
-
-theorem finiteRootLoad_le (Q : Finset E) (J : Finset ℕ) (p : unitInterval)
-    (f : SpliceState E → ℝ) (C : ℝ)
-    (hf : ∀ s ∈ finiteSpliceDomain Q J, f s ≤ C)
-    (ω : FiniteField Q) : finiteRootLoad Q J p f ω ≤ C := by
-  calc
-    _ ≤ ∑ η : ↥J → FiniteField Q, finiteFamilyWeight Q J p η * C := by
-      apply Finset.sum_le_sum
-      intro η _
-      apply mul_le_mul_of_nonneg_left _ (finiteFamilyWeight_nonneg Q J p η)
-      exact hf _ (Finset.mem_image.mpr ⟨(ω,η),Finset.mem_univ _,rfl⟩)
-    _ = C := by rw [← Finset.sum_mul,sum_finiteFamilyWeight,one_mul]
-
- 
-theorem finiteRootLoad_mean (Q : Finset E) (J : Finset ℕ) (p : ℝ)
-    (f : SpliceState E → ℝ) :
-    (∑ ω : FiniteField Q, finiteFieldWeight Q p ω * finiteRootLoad Q J p f ω) =
-      ∑ s ∈ finiteSpliceDomain Q J, spliceTupleWeight Q J p s * f s := by
-  rw [finiteSpliceDomain,Finset.sum_image (encodeFiniteSplice_injective Q J).injOn,
-    Fintype.sum_prod_type]
-  simp only [spliceTupleWeight,finiteRootLoad,Finset.mul_sum,encodeFiniteSplice,
-    domainBits_finiteFieldSet,prod_fieldFamilyWeight,finiteFieldWeight,mul_assoc]
-
- 
-
-theorem finiteRootLoad_second_moment (Q : Finset E) (J₁ J₂ : Finset ℕ) (p : ℝ)
-    (f₁ f₂ : SpliceState E → ℝ) :
-    (∑ ω : FiniteField Q, finiteFieldWeight Q p ω *
-      finiteRootLoad Q J₁ p f₁ ω * finiteRootLoad Q J₂ p f₂ ω) =
-    ∑ s : FiniteJointData Q J₁ J₂, jointTupleWeight Q J₁ J₂ p (encodeFiniteJoint Q J₁ J₂ s) *
-      f₁ (finiteFieldSet Q s.1,finiteFieldFamily Q J₁ s.2.1) *
-      f₂ (finiteFieldSet Q s.1,finiteFieldFamily Q J₂ s.2.2) := by
-  rw [Fintype.sum_prod_type]
-  apply Finset.sum_congr rfl
-  intro ω _
-  rw [Fintype.sum_prod_type]
-  simp only [finiteRootLoad,Finset.mul_sum,Finset.sum_mul,jointTupleWeight_encode]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro η₁ _
-  apply Finset.sum_congr rfl
-  intro η₂ _
-  ring
-
- 
-theorem finiteForwardRootLoad_bounds (G : BondGraph V E) (hloc : LocallyFinite G)
-    (x z : V) (k M : ℕ) (L : ℝ) (hL : 1 ≤ L) (Q : Finset E) (p : unitInterval)
-    (A : Finset (SpliceState E)) (y : SpliceState E → ℕ → V)
-    (hA : A ⊆ finiteSpliceDomain Q (Finset.Icc 1 k))
-    (hz : ∀ s ∈ A, y s k = z)
-    (hgood : ∀ s ∈ A, ∀ i, 1 ≤ i → i ≤ k → (cluster G (s.2 i) (y s i)).Finite ∧
-      (cluster G (s.2 i) (y s i)).ncard ≤ M ∧ GoodForScale G (s.2 i) (y s i) L)
-    (ω : FiniteField Q) :
-    0 ≤ finiteRootLoad Q (Finset.Icc 1 k) p (finiteForwardLoad G x k Q A y) ω ∧
-    finiteRootLoad Q (Finset.Icc 1 k) p (finiteForwardLoad G x k Q A y) ω ≤ inverseCodeMass M L ^ k := by
-  have hb := finiteForwardLoad_bounds G hloc x z k M L hL Q A y hA hz hgood
-  exact ⟨finiteRootLoad_nonneg _ _ p _ (fun s _ => (hb s).1) _,
-    finiteRootLoad_le _ _ p _ _ (fun s _ => (hb s).2) _⟩
-
- 
-
-theorem finiteForwardRootLoad_mean [Nonempty E] (G : BondGraph V E) (hloc : LocallyFinite G)
-    (x : V) (k M : ℕ) (Q : Finset E) (p : ℝ)
-    (A : Finset (SpliceState E)) (y : SpliceState E → ℕ → V)
-    (hA : A ⊆ finiteSpliceDomain Q (Finset.Icc 1 k))
-    (hy : ∀ s ∈ A, y s 0 ∈ cluster G s.1 x)
-    (hchain : ∀ s ∈ A, ∀ i ∈ Finset.Icc 1 k, Conn G (s.2 i) (y s i) (y s (i-1)))
-    (hcaps : ∀ s ∈ A, ∀ i ∈ Finset.Icc 1 k, (cluster G (s.2 i) (y s i)).Finite ∧
-      (cluster G (s.2 i) (y s i)).ncard ≤ M) :
-    (∑ ω : FiniteField Q, finiteFieldWeight Q p ω *
-      finiteRootLoad Q (Finset.Icc 1 k) p (finiteForwardLoad G x k Q A y) ω) =
-      ∑ s ∈ A, spliceTupleWeight Q (Finset.Icc 1 k) p s := by
-  rw [finiteRootLoad_mean]
-  exact finiteForwardLoad_mean G hloc x k M Q p A y hA hy hchain hcaps
-
-end FiniteRootLoad
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section FinitePairLoad
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-def pairSpliceOutput (s t : SpliceState E) : JointSpliceState E := (s.1,s.2,t.2)
-
-theorem pairSpliceOutput_mem (Q : Finset E) (J₁ J₂ : Finset ℕ)
-    (s t : SpliceState E) (hs : s ∈ finiteSpliceDomain Q J₁)
-    (ht : t ∈ finiteSpliceDomain Q J₂) :
-    pairSpliceOutput s t ∈ finiteJointDomain Q J₁ J₂ := by
-  have h₁ := (mem_finiteSpliceDomain_iff Q J₁ s).mp hs
-  have h₂ := (mem_finiteSpliceDomain_iff Q J₂ t).mp ht
-  exact mem_finiteJointDomain_of_support Q J₁ J₂ _ h₁.1 h₁.2.1 h₂.2.1 h₁.2.2 h₂.2.2
-
- 
-theorem finiteInverseLoad_mul {I₁ I₂ : Type*} [Fintype I₁] [Fintype I₂]
-    (out₁ : I₁ → SpliceState E) (out₂ : I₂ → SpliceState E)
-    (m₁ : I₁ → ℝ) (m₂ : I₂ → ℝ) (t : JointSpliceState E) :
-    finiteInverseLoad out₁ m₁ (t.1,t.2.1) * finiteInverseLoad out₂ m₂ (t.1,t.2.2) =
-    ∑ a, ∑ b, if (out₁ a).1 = (out₂ b).1 ∧ pairSpliceOutput (out₁ a) (out₂ b) = t
-      then m₁ a * m₂ b else 0 := by
-  simp only [finiteInverseLoad,Finset.sum_mul,Finset.mul_sum]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro a _
-  apply Finset.sum_congr rfl
-  intro b _
-  have he : (out₁ a = (t.1,t.2.1) ∧ out₂ b = (t.1,t.2.2)) ↔
-      ((out₁ a).1 = (out₂ b).1 ∧ pairSpliceOutput (out₁ a) (out₂ b) = t) := by
-    cases h₁ : out₁ a with | mk r₁ u₁ =>
-      cases h₂ : out₂ b with | mk r₂ u₂ =>
-        rcases t with ⟨r,u,v⟩
-        simpa [pairSpliceOutput,Prod.mk.injEq] using
-          (show (r₁ = r ∧ u₁ = u) ∧ (r₂ = r ∧ u₂ = v) ↔
-            r₁ = r₂ ∧ r₁ = r ∧ u₁ = u ∧ u₂ = v from by aesop)
-  simp only [← he]
-  split_ifs <;> simp_all
-
- 
-
-theorem finiteRootLoad_pair_expansion {I₁ I₂ : Type*} [Fintype I₁] [Fintype I₂]
-    (Q : Finset E) (J₁ J₂ : Finset ℕ) (p : ℝ)
-    (out₁ : I₁ → SpliceState E) (out₂ : I₂ → SpliceState E)
-    (m₁ : I₁ → ℝ) (m₂ : I₂ → ℝ)
-    (hs₁ : ∀ a, out₁ a ∈ finiteSpliceDomain Q J₁)
-    (hs₂ : ∀ b, out₂ b ∈ finiteSpliceDomain Q J₂) :
-    (∑ ω : FiniteField Q, finiteFieldWeight Q p ω *
-      finiteRootLoad Q J₁ p (finiteInverseLoad out₁ m₁) ω *
-      finiteRootLoad Q J₂ p (finiteInverseLoad out₂ m₂) ω) =
-    ∑ a, ∑ b, if (out₁ a).1 = (out₂ b).1 then
-      jointTupleWeight Q J₁ J₂ p (pairSpliceOutput (out₁ a) (out₂ b)) * m₁ a * m₂ b else 0 := by
-  rw [finiteRootLoad_second_moment]
-  have hdom : (∑ s : FiniteJointData Q J₁ J₂,
-      jointTupleWeight Q J₁ J₂ p (encodeFiniteJoint Q J₁ J₂ s) *
-        finiteInverseLoad out₁ m₁ ((encodeFiniteJoint Q J₁ J₂ s).1,(encodeFiniteJoint Q J₁ J₂ s).2.1) *
-        finiteInverseLoad out₂ m₂ ((encodeFiniteJoint Q J₁ J₂ s).1,(encodeFiniteJoint Q J₁ J₂ s).2.2)) =
-      ∑ t ∈ finiteJointDomain Q J₁ J₂, jointTupleWeight Q J₁ J₂ p t *
-        finiteInverseLoad out₁ m₁ (t.1,t.2.1) * finiteInverseLoad out₂ m₂ (t.1,t.2.2) := by
-    rw [finiteJointDomain,Finset.sum_image (encodeFiniteJoint_injective Q J₁ J₂).injOn]
-  apply hdom.trans
-  simp only [mul_assoc,finiteInverseLoad_mul,Finset.mul_sum,mul_ite,mul_zero]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro a _
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro b _
-  by_cases he : (out₁ a).1 = (out₂ b).1
-  · simp only [he,true_and,ite_true]
-    rw [Finset.sum_ite_eq,ite_eq_left (pairSpliceOutput_mem Q J₁ J₂ _ _ (hs₁ a) (hs₂ b))]
-  · simp [he]
-
-theorem traceIndex_output_code_injective (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
-    (A : Finset (SpliceState E)) (y : SpliceState E → ℕ → V) :
-    Function.Injective (fun a : FiniteTraceIndex G x k Q A y =>
-      (traceIndexOutput G x k Q A y a,traceIndexCode G x k Q A y a)) := by
-  intro a b he
-  have hc : traceIndexCode G x k Q A y a = traceIndexCode G x k Q A y b := congrArg Prod.snd he
-  have ho : traceIndexOutput G x k Q A y a = traceIndexOutput G x k Q A y b := congrArg Prod.fst he
-  apply traceIndex_injective G x k Q A y
-  apply Prod.ext
-  · apply SpliceExecution.input_unique G x _ _ _ (traceIndexCode G x k Q A y a)
-      (traceIndex_valid G x k Q A y a).2
-    change SpliceExecution G x b.1.val (traceIndexCode G x k Q A y a) (traceIndexOutput G x k Q A y a)
-    rw [hc,ho]
-    exact (traceIndex_valid G x k Q A y b).2
-  · exact hc
-
-end FinitePairLoad
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section WeightedJointMass
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem jointRestored_injective {I : Type*}
-    (G : BondGraph V E) (x : V) (k : ℕ)
-    (η₁ η₂ : I → ℕ → Set E) (y₁ y₂ : (ℕ → Set E) → ℕ → V)
-    (choose₁ choose₂ : ℕ → Set E → E)
-    (ω₁ ω₂ ωf : I → Set E) (c₁ c₂ : I → List (ℕ × E))
-    (hir : Function.Injective (fun a =>
-      (jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a),
-        c₁ a,c₂ a)))
-    (h₁ : ∀ a, SpliceTrace G x (η₁ a) (y₁ (η₁ a)) k 0 (ω₁ a) (c₁ a) (ωf a))
-    (h₂ : ∀ a, SpliceTrace G x (η₂ a) (y₂ (η₂ a)) k 0 (ω₂ a) (c₂ a) (ωf a))
-    (hc₁ : ∀ a, c₁ a ≠ []) (hc₂ : ∀ a, c₂ a ≠ [])
-    (hsep : ∀ a, GraphSeparated G (cluster G (ωf a) x \ cluster G (ω₁ a) x)
-      (cluster G (ωf a) x \ cluster G (ω₂ a) x))
-    (hr₁ : ∀ a, FollowsContactRule G x (η₁ a) (y₁ (η₁ a)) choose₁ (c₁ a) (ω₁ a))
-    (hr₂ : ∀ a, FollowsContactRule G x (η₂ a) (y₂ (η₂ a)) choose₂ (c₂ a) (ω₂ a))
-    : Function.Injective (fun a => jointRestored G x (η₁ a) (η₂ a)
-      (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (c₂ a)) := by
-  let out (a : I) := jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a))
-    (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a)
-  let rest (a : I) := jointRestored G x (η₁ a) (η₂ a) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (c₂ a)
-  change Function.Injective rest
-  intro a b he
-  have hη₁ : η₁ a = η₁ b := congrArg (fun s : JointSpliceState E => s.2.1) he
-  have hη₂ : η₂ a = η₂ b := congrArg (fun s : JointSpliceState E => s.2.2) he
-  have gb₁ : SpliceTrace G x (η₁ a) (y₁ (η₁ a)) k 0 (ω₁ b) (c₁ b) (ωf b) := by
-    rw [hη₁]; exact h₁ b
-  have gb₂ : SpliceTrace G x (η₂ a) (y₂ (η₂ a)) k 0 (ω₂ b) (c₂ b) (ωf b) := by
-    rw [hη₂]; exact h₂ b
-  have grb₁ : FollowsContactRule G x (η₁ a) (y₁ (η₁ a)) choose₁ (c₁ b) (ω₁ b) := by
-    rw [hη₁]; exact hr₁ b
-  have grb₂ : FollowsContactRule G x (η₂ a) (y₂ (η₂ a)) choose₂ (c₂ b) (ω₂ b) := by
-    rw [hη₂]; exact hr₂ b
-  have hfield : exchangeOn (maskUnion (replayMasks G x (η₂ a) (y₂ (η₂ a)) (c₂ a) (ω₂ a))) (ω₁ a) (ω₂ a) =
-      exchangeOn (maskUnion (replayMasks G x (η₂ a) (y₂ (η₂ a)) (c₂ b) (ω₂ b))) (ω₁ b) (ω₂ b) := by
-    have hh := congrArg Prod.fst he
-    simpa only [rest,jointRestored,← hη₂] using hh
-  obtain ⟨hcode₁,hcode₂,hfieldf,haux₁,haux₂⟩ := joint_restore_seeded_unique G x
-    (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) choose₁ choose₂ k 0 0
-    (ω₁ a) (ω₂ a) (ωf a) (ω₁ b) (ω₂ b) (ωf b)
-    (c₁ a) (c₂ a) (c₁ b) (c₂ b) (h₁ a) (h₂ a) gb₁ gb₂
-    (hc₁ a) (hc₂ a) (hc₁ b) (hc₂ b) (hsep a) (hsep b)
-    (hr₁ a) (hr₂ a) grb₁ grb₂ hfield
-  have hout : out a = out b := by
-    apply Prod.ext hfieldf
-    apply Prod.ext
-    · simpa only [out,jointOutput,hη₁] using haux₁
-    · simpa only [out,jointOutput,hη₂] using haux₂
-  exact hir (Prod.ext hout (Prod.ext hcode₁ hcode₂))
-
- 
-
-theorem finite_joint_weighted_mass_le {I : Type*} [Fintype I]
-    (G : BondGraph V E) (x : V) (k : ℕ) (p : unitInterval) (Q : Finset E)
-    (η₁ η₂ : I → ℕ → Set E) (y₁ y₂ : (ℕ → Set E) → ℕ → V)
-    (choose₁ choose₂ : ℕ → Set E → E)
-    (ω₁ ω₂ ωf : I → Set E) (c₁ c₂ : I → List (ℕ × E))
-    (hir : Function.Injective (fun a =>
-      (jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a),
-        c₁ a,c₂ a)))
-    (h₁ : ∀ a, SpliceTrace G x (η₁ a) (y₁ (η₁ a)) k 0 (ω₁ a) (c₁ a) (ωf a))
-    (h₂ : ∀ a, SpliceTrace G x (η₂ a) (y₂ (η₂ a)) k 0 (ω₂ a) (c₂ a) (ωf a))
-    (hc₁ : ∀ a, c₁ a ≠ []) (hc₂ : ∀ a, c₂ a ≠ [])
-    (hsep : ∀ a, GraphSeparated G (cluster G (ωf a) x \ cluster G (ω₁ a) x)
-      (cluster G (ωf a) x \ cluster G (ω₂ a) x))
-    (hr₁ : ∀ a, FollowsContactRule G x (η₁ a) (y₁ (η₁ a)) choose₁ (c₁ a) (ω₁ a))
-    (hr₂ : ∀ a, FollowsContactRule G x (η₂ a) (y₂ (η₂ a)) choose₂ (c₂ a) (ω₂ a))
-    (m₁ m₂ : (ℕ → Set E) → ℝ) (hm₁ : ∀ η, 0 ≤ m₁ η) (hm₂ : ∀ η, 0 ≤ m₂ η)
-    (B : Finset (JointSpliceState E))
-    (hB : ∀ a, jointRestored G x (η₁ a) (η₂ a) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (c₂ a) ∈ B) :
-    (∑ a, jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p
-      (jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a)) * m₁ (η₁ a) * m₂ (η₂ a)) ≤
-      ∑ s ∈ B, jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p s * m₁ s.2.1 * m₂ s.2.2 := by
-  let out (a : I) := jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a))
-    (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a)
-  let rest (a : I) := jointRestored G x (η₁ a) (η₂ a) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (c₂ a)
-  have hinj : Function.Injective rest := jointRestored_injective G x k η₁ η₂ y₁ y₂ choose₁ choose₂
-    ω₁ ω₂ ωf c₁ c₂ hir h₁ h₂ hc₁ hc₂ hsep hr₁ hr₂
-  have hw : ∀ a, jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p (out a) =
-      jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p (rest a) := by
-    intro a
-    apply (h₁ a).joint_weight G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) k 0 0
-      (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a) (h₂ a) (hsep a) Q (Finset.Icc 1 k) (Finset.Icc 1 k) p
-    · intro q hq
-      have hr := (h₁ a).seam_data G x (η₁ a) (y₁ (η₁ a)) k 0 (ω₁ a) (ωf a) (c₁ a) q hq
-      exact Finset.mem_Icc.mpr ⟨hr.1,hr.2.1⟩
-    · intro q hq
-      have hr := (h₂ a).seam_data G x (η₂ a) (y₂ (η₂ a)) k 0 (ω₂ a) (ωf a) (c₂ a) q hq
-      exact Finset.mem_Icc.mpr ⟨hr.1,hr.2.1⟩
-  change (∑ a, jointTupleWeight Q _ _ p (out a) * m₁ (η₁ a) * m₂ (η₂ a)) ≤ _
-  simp_rw [hw]
-  change (∑ a, jointTupleWeight Q _ _ p (rest a) * m₁ (rest a).2.1 * m₂ (rest a).2.2) ≤ _
-  rw [← Finset.sum_image (f := fun s : JointSpliceState E =>
-    jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p s * m₁ s.2.1 * m₂ s.2.2) hinj.injOn]
-  apply Finset.sum_le_sum_of_subset_of_nonneg
-  · intro s hs
-    obtain ⟨a,_,rfl⟩ := Finset.mem_image.mp hs
-    exact hB a
-  · intro s _ _
-    exact mul_nonneg (mul_nonneg (jointTupleWeight_nonneg Q _ _ p s) (hm₁ _)) (hm₂ _)
-
-end WeightedJointMass
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section FiniteFamilyIntegral
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-noncomputable def finiteFamilyIntegral (Q : Finset E) (J : Finset ℕ) (p : ℝ)
-    (f : (ℕ → Set E) → ℝ) : ℝ :=
-  ∑ η : ↥J → FiniteField Q, finiteFamilyWeight Q J p η * f (finiteFieldFamily Q J η)
-
-theorem finiteFamilyIntegral_nonneg (Q : Finset E) (J : Finset ℕ) (p : unitInterval)
-    (f : (ℕ → Set E) → ℝ) (hf : ∀ η, 0 ≤ f η) : 0 ≤ finiteFamilyIntegral Q J p f := by
-  exact Finset.sum_nonneg fun η _ => mul_nonneg (finiteFamilyWeight_nonneg Q J p η) (hf _)
-
-theorem finiteFamilyIntegral_mono (Q : Finset E) (J : Finset ℕ) (p : unitInterval)
-    (f g : (ℕ → Set E) → ℝ) (hfg : ∀ η, f η ≤ g η) :
-    finiteFamilyIntegral Q J p f ≤ finiteFamilyIntegral Q J p g := by
-  exact Finset.sum_le_sum fun η _ => mul_le_mul_of_nonneg_left (hfg _) (finiteFamilyWeight_nonneg Q J p η)
-
-theorem finiteFamilyIntegral_const (Q : Finset E) (J : Finset ℕ) (p c : ℝ) :
-    finiteFamilyIntegral Q J p (fun _ => c) = c := by
-  rw [finiteFamilyIntegral,← Finset.sum_mul,sum_finiteFamilyWeight,one_mul]
-
-theorem finiteFamilyIntegral_sum {I : Type*} (Q : Finset E) (J : Finset ℕ) (p : ℝ)
-    (S : Finset I) (f : I → (ℕ → Set E) → ℝ) :
-    finiteFamilyIntegral Q J p (fun η => ∑ i ∈ S, f i η) =
-      ∑ i ∈ S, finiteFamilyIntegral Q J p (f i) := by
-  simp only [finiteFamilyIntegral,Finset.mul_sum]
-  exact Finset.sum_comm
-
- 
-theorem finiteJointDomain_integral_factor (Q : Finset E) (J₁ J₂ : Finset ℕ) (p : ℝ)
-    (f₁ f₂ : Set E → (ℕ → Set E) → ℝ) :
-    (∑ s ∈ finiteJointDomain Q J₁ J₂, jointTupleWeight Q J₁ J₂ p s * f₁ s.1 s.2.1 * f₂ s.1 s.2.2) =
-    ∑ ω : FiniteField Q, finiteFieldWeight Q p ω *
-      finiteFamilyIntegral Q J₁ p (f₁ (finiteFieldSet Q ω)) *
-      finiteFamilyIntegral Q J₂ p (f₂ (finiteFieldSet Q ω)) := by
-  rw [finiteJointDomain,Finset.sum_image (encodeFiniteJoint_injective Q J₁ J₂).injOn]
-  simp only [jointTupleWeight_encode]
-  rw [Fintype.sum_prod_type]
-  apply Finset.sum_congr rfl
-  intro ω _
-  rw [Fintype.sum_prod_type]
-  simp only [finiteFamilyIntegral,Finset.mul_sum,Finset.sum_mul,encodeFiniteJoint]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro η₂ _
-  apply Finset.sum_congr rfl
-  intro η₁ _
-  ring
-
-theorem finiteJointDomain_weighted_event_factor (Q : Finset E) (J₁ J₂ : Finset ℕ) (p : ℝ)
-    (P : Set E → Prop) (H₁ H₂ : Set E → (ℕ → Set E) → Prop)
-    (m₁ m₂ : (ℕ → Set E) → ℝ) :
-    (∑ s ∈ (finiteJointDomain Q J₁ J₂).filter (fun s => P s.1 ∧ H₁ s.1 s.2.1 ∧ H₂ s.1 s.2.2),
-      jointTupleWeight Q J₁ J₂ p s * m₁ s.2.1 * m₂ s.2.2) =
-    ∑ ω : FiniteField Q, finiteFieldWeight Q p ω *
-      (if P (finiteFieldSet Q ω) then
-        finiteFamilyIntegral Q J₁ p (fun η => if H₁ (finiteFieldSet Q ω) η then m₁ η else 0) *
-        finiteFamilyIntegral Q J₂ p (fun η => if H₂ (finiteFieldSet Q ω) η then m₂ η else 0) else 0) := by
-  rw [Finset.sum_filter]
-  have he : (∑ s ∈ finiteJointDomain Q J₁ J₂,
-      if P s.1 ∧ H₁ s.1 s.2.1 ∧ H₂ s.1 s.2.2 then
-        jointTupleWeight Q J₁ J₂ p s * m₁ s.2.1 * m₂ s.2.2 else 0) =
-    ∑ s ∈ finiteJointDomain Q J₁ J₂, jointTupleWeight Q J₁ J₂ p s *
-      (if P s.1 ∧ H₁ s.1 s.2.1 then m₁ s.2.1 else 0) *
-      (if H₂ s.1 s.2.2 then m₂ s.2.2 else 0) := by
-    apply Finset.sum_congr rfl
-    intro s _
-    split_ifs <;> simp_all
-  rw [he,finiteJointDomain_integral_factor Q J₁ J₂ p
-    (fun ω η => if P ω ∧ H₁ ω η then m₁ η else 0)
-    (fun ω η => if H₂ ω η then m₂ η else 0)]
-  apply Finset.sum_congr rfl
-  intro ω _
-  by_cases hP : P (finiteFieldSet Q ω)
-  · simp only [hP,true_and,ite_true,mul_assoc]
-  · simp [hP,finiteFamilyIntegral]
-
-end FiniteFamilyIntegral
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section WeightedSeparatedSecondMoment
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem finite_separated_weighted_second_moment_le {I : Type*} [Fintype I]
-    (G : BondGraph V E) (x : V) (k n M : ℕ) (p : unitInterval) (Q : Finset E)
-    (η₁ η₂ : I → ℕ → Set E) (y₁ y₂ : (ℕ → Set E) → ℕ → V)
-    (success₁ success₂ : (ℕ → Set E) → Prop)
-    (m₁ m₂ : (ℕ → Set E) → ℝ) (hm₁ : ∀ η, 0 ≤ m₁ η) (hm₂ : ∀ η, 0 ≤ m₂ η)
-    (choose₁ choose₂ : ℕ → Set E → E)
-    (ω₁ ω₂ ωf : I → Set E) (c₁ c₂ : I → List (ℕ × E))
-    (hir : Function.Injective (fun a =>
-      (jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a),
-        c₁ a,c₂ a)))
-    (h₁ : ∀ a, SpliceTrace G x (η₁ a) (y₁ (η₁ a)) k 0 (ω₁ a) (c₁ a) (ωf a))
-    (h₂ : ∀ a, SpliceTrace G x (η₂ a) (y₂ (η₂ a)) k 0 (ω₂ a) (c₂ a) (ωf a))
-    (hc₁ : ∀ a, c₁ a ≠ []) (hc₂ : ∀ a, c₂ a ≠ [])
-    (hsep : ∀ a, GraphSeparated G (cluster G (ωf a) x \ cluster G (ω₁ a) x)
-      (cluster G (ωf a) x \ cluster G (ω₂ a) x))
-    (hr₁ : ∀ a, FollowsContactRule G x (η₁ a) (y₁ (η₁ a)) choose₁ (c₁ a) (ω₁ a))
-    (hr₂ : ∀ a, FollowsContactRule G x (η₂ a) (y₂ (η₂ a)) choose₂ (c₂ a) (ω₂ a))
-    (hbin : ∀ a, (cluster G (ω₁ a) x).Finite ∧ n ≤ (cluster G (ω₁ a) x).ncard ∧
-      (cluster G (ω₁ a) x).ncard < 2*n)
-    (hcap₂ : ∀ a i, 1 ≤ i → i ≤ k → (cluster G (η₂ a i) (y₂ (η₂ a) i)).Finite ∧
-      (cluster G (η₂ a i) (y₂ (η₂ a) i)).ncard ≤ M)
-    (hsc₁ : ∀ a, success₁ (η₁ a)) (hsc₂ : ∀ a, success₂ (η₂ a))
-    (hsω₁ : ∀ a, ω₁ a ⊆ (Q : Set E)) (hsω₂ : ∀ a, ω₂ a ⊆ (Q : Set E))
-    (hsη₁ : ∀ a i, η₁ a i ⊆ (Q : Set E)) (hsη₂ : ∀ a i, η₂ a i ⊆ (Q : Set E))
-    (hz₁ : ∀ a i, i ∉ Finset.Icc 1 k → η₁ a i = ∅)
-    (hz₂ : ∀ a i, i ∉ Finset.Icc 1 k → η₂ a i = ∅) :
-    (∑ a, jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p
-      (jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a)) * m₁ (η₁ a) * m₂ (η₂ a)) ≤
-    ∑ ω : FiniteField Q, finiteFieldWeight Q p ω *
-      (if (cluster G (finiteFieldSet Q ω) x).Finite ∧
-        n-k*M ≤ (cluster G (finiteFieldSet Q ω) x).ncard ∧
-        (cluster G (finiteFieldSet Q ω) x).ncard < 2*n then
-        finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η => if success₁ η ∧
-          (cluster G (finiteFieldSet Q ω) x ∩ trajectoryUnion G η (y₁ η) k).Nonempty then m₁ η else 0) *
-        finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η => if success₂ η ∧
-          (cluster G (finiteFieldSet Q ω) x ∩ trajectoryUnion G η (y₂ η) k).Nonempty then m₂ η else 0) else 0) := by
-  let J := Finset.Icc 1 k
-  let P (ω : Set E) := (cluster G ω x).Finite ∧ n-k*M ≤ (cluster G ω x).ncard ∧
-    (cluster G ω x).ncard < 2*n
-  let H₁ (ω : Set E) (η : ℕ → Set E) := success₁ η ∧
-    (cluster G ω x ∩ trajectoryUnion G η (y₁ η) k).Nonempty
-  let H₂ (ω : Set E) (η : ℕ → Set E) := success₂ η ∧
-    (cluster G ω x ∩ trajectoryUnion G η (y₂ η) k).Nonempty
-  let B := (finiteJointDomain Q J J).filter fun s => P s.1 ∧ H₁ s.1 s.2.1 ∧ H₂ s.1 s.2.2
-  have hB (a : I) : jointRestored G x (η₁ a) (η₂ a) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (c₂ a) ∈ B := by
-    apply Finset.mem_filter.mpr
-    refine ⟨mem_finiteJointDomain_of_support Q J J _
-      (exchangeOn_subset _ _ _ _ (hsω₁ a) (hsω₂ a)) (hsη₁ a) (hsη₂ a) (hz₁ a) (hz₂ a),?_⟩
-    have hroot := (h₁ a).joint_restored_size G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a))
-      k 0 0 n M (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a) (h₂ a) (hc₁ a) (hc₂ a) (hsep a)
-      (hbin a).1 (hbin a).2.1 (hbin a).2.2 (hcap₂ a)
-    have hhits := (h₁ a).joint_trajectory_hits G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a))
-      k 0 0 (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a) (h₂ a) (hc₁ a) (hc₂ a) (hsep a)
-    exact ⟨hroot,⟨hsc₁ a,hhits.1⟩,hsc₂ a,hhits.2⟩
-  have h := finite_joint_weighted_mass_le G x k p Q η₁ η₂ y₁ y₂ choose₁ choose₂ ω₁ ω₂ ωf c₁ c₂
-    hir h₁ h₂ hc₁ hc₂ hsep hr₁ hr₂ m₁ m₂ hm₁ hm₂ B hB
-  apply h.trans_eq
-  convert finiteJointDomain_weighted_event_factor Q J J p P H₁ H₂ m₁ m₂ using 1 <;>
-    simp only [B, J, P, H₁, H₂]
-  apply Finset.sum_congr rfl
-  intro ω _
-  apply congrArg (fun z : ℝ => finiteFieldWeight Q p ω * z)
-  split_ifs
-  · apply congrArg₂ (· * ·)
-    · apply congrArg (finiteFamilyIntegral Q (Finset.Icc 1 k) p)
-      funext η
-      split_ifs <;> rfl
-    · apply congrArg (finiteFamilyIntegral Q (Finset.Icc 1 k) p)
-      funext η
-      split_ifs <;> rfl
-  · rfl
-
-end WeightedSeparatedSecondMoment
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section WeightedContactSecondMoment
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem finite_separated_weighted_contact_second_moment_le {I : Type*} [Fintype I] [Nonempty E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) (k n M : ℕ) (p : unitInterval) (Q : Finset E)
-    (η₁ η₂ : I → ℕ → Set E) (y₁ y₂ : (ℕ → Set E) → ℕ → V)
-    (success₁ success₂ : (ℕ → Set E) → Prop)
-    (m₁ m₂ : (ℕ → Set E) → ℝ) (hm₁ : ∀ η, 0 ≤ m₁ η) (hm₂ : ∀ η, 0 ≤ m₂ η)
-    (ω₁ ω₂ ωf : I → Set E) (c₁ c₂ : I → List (ℕ × E))
-    (hir : Function.Injective (fun a =>
-      (jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a),
-        c₁ a,c₂ a)))
-    (h₁ : ∀ a, SpliceTrace G x (η₁ a) (y₁ (η₁ a)) k 0 (ω₁ a) (c₁ a) (ωf a))
-    (h₂ : ∀ a, SpliceTrace G x (η₂ a) (y₂ (η₂ a)) k 0 (ω₂ a) (c₂ a) (ωf a))
-    (hc₁ : ∀ a, c₁ a ≠ []) (hc₂ : ∀ a, c₂ a ≠ [])
-    (hsep : ∀ a, GraphSeparated G (cluster G (ωf a) x \ cluster G (ω₁ a) x)
-      (cluster G (ωf a) x \ cluster G (ω₂ a) x))
-    (hbin : ∀ a, (cluster G (ω₁ a) x).Finite ∧ n ≤ (cluster G (ω₁ a) x).ncard ∧
-      (cluster G (ω₁ a) x).ncard < 2*n)
-    (hcap₁ : ∀ a i, 1 ≤ i → i ≤ k → (cluster G (η₁ a i) (y₁ (η₁ a) i)).Finite)
-    (hcap₂ : ∀ a i, 1 ≤ i → i ≤ k → (cluster G (η₂ a i) (y₂ (η₂ a) i)).Finite ∧
-      (cluster G (η₂ a i) (y₂ (η₂ a) i)).ncard ≤ M)
-    (hsc₁ : ∀ a, success₁ (η₁ a)) (hsc₂ : ∀ a, success₂ (η₂ a))
-    (hsω₁ : ∀ a, ω₁ a ⊆ (Q : Set E)) (hsω₂ : ∀ a, ω₂ a ⊆ (Q : Set E))
-    (hsη₁ : ∀ a i, η₁ a i ⊆ (Q : Set E)) (hsη₂ : ∀ a i, η₂ a i ⊆ (Q : Set E))
-    (hz₁ : ∀ a i, i ∉ Finset.Icc 1 k → η₁ a i = ∅)
-    (hz₂ : ∀ a i, i ∉ Finset.Icc 1 k → η₂ a i = ∅) :
-    (∑ a, jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p
-      (jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a)) * m₁ (η₁ a) * m₂ (η₂ a) *
-      (∏ i : Fin k, (traceContactCount G (η₁ a) (y₁ (η₁ a)) (ωf a) x (c₁ a) (i.val+1):ℝ)⁻¹) *
-      (∏ i : Fin k, (traceContactCount G (η₂ a) (y₂ (η₂ a)) (ωf a) x (c₂ a) (i.val+1):ℝ)⁻¹)) ≤
-    ∑ ω : FiniteField Q, finiteFieldWeight Q p ω *
-      (if (cluster G (finiteFieldSet Q ω) x).Finite ∧
-        n-k*M ≤ (cluster G (finiteFieldSet Q ω) x).ncard ∧
-        (cluster G (finiteFieldSet Q ω) x).ncard < 2*n then
-        finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η => if success₁ η ∧
-          (cluster G (finiteFieldSet Q ω) x ∩ trajectoryUnion G η (y₁ η) k).Nonempty then m₁ η else 0) *
-        finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η => if success₂ η ∧
-          (cluster G (finiteFieldSet Q ω) x ∩ trajectoryUnion G η (y₂ η) k).Nonempty then m₂ η else 0) else 0) := by
-  let S := Fin k → unitInterval
-  let μ : Measure (S × S) := (volume : Measure S).prod volume
-  let P₁ (a : I) : Set S := {u | FollowsContactRule G x (η₁ a) (y₁ (η₁ a))
-    (chooseContact (finiteContactSeeds k u)) (c₁ a) (ω₁ a)}
-  let P₂ (a : I) : Set S := {u | FollowsContactRule G x (η₂ a) (y₂ (η₂ a))
-    (chooseContact (finiteContactSeeds k u)) (c₂ a) (ω₂ a)}
-  let w (a : I) := jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p
-    (jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a)) * m₁ (η₁ a) * m₂ (η₂ a)
-  have hmeas₁ (a : I) : MeasurableSet (P₁ a) :=
-    (h₁ a).measurableSet_seed_event G x (η₁ a) (y₁ (η₁ a)) k 0 (ω₁ a) (ωf a) (c₁ a)
-  have hmeas₂ (a : I) : MeasurableSet (P₂ a) :=
-    (h₂ a).measurableSet_seed_event G x (η₂ a) (y₂ (η₂ a)) k 0 (ω₂ a) (ωf a) (c₂ a)
-  have hp (a : I) : μ.real (P₁ a ×ˢ P₂ a) =
-      (∏ i : Fin k, (traceContactCount G (η₁ a) (y₁ (η₁ a)) (ωf a) x (c₁ a) (i.val+1):ℝ)⁻¹) *
-      (∏ i : Fin k, (traceContactCount G (η₂ a) (y₂ (η₂ a)) (ωf a) x (c₂ a) (i.val+1):ℝ)⁻¹) := by
-    rw [measureReal_def,Measure.prod_prod,ENNReal.toReal_mul,
-      (h₁ a).seed_probability G hloc x (η₁ a) (y₁ (η₁ a)) k 0 (ω₁ a) (ωf a) (c₁ a) (hcap₁ a),
-      (h₂ a).seed_probability G hloc x (η₂ a) (y₂ (η₂ a)) k 0 (ω₂ a) (ωf a) (c₂ a) (fun i hi hik => (hcap₂ a i hi hik).1)]
-  conv_lhs => arg 2; ext a; rw [mul_assoc,← hp a]
-  apply finite_event_mass_le μ (fun a => P₁ a ×ˢ P₂ a) (fun a => (hmeas₁ a).prod (hmeas₂ a)) w
-  intro u
-  let J := {a : I // u ∈ P₁ a ×ˢ P₂ a}
-  have hb := finite_separated_weighted_second_moment_le G x k n M p Q
-    (fun a : J => η₁ a.val) (fun a : J => η₂ a.val) y₁ y₂ success₁ success₂ m₁ m₂ hm₁ hm₂
-    (chooseContact (finiteContactSeeds k u.1)) (chooseContact (finiteContactSeeds k u.2))
-    (fun a : J => ω₁ a.val) (fun a : J => ω₂ a.val) (fun a : J => ωf a.val)
-    (fun a : J => c₁ a.val) (fun a : J => c₂ a.val)
-    (fun _ _ he => Subtype.ext (hir he)) (fun a => h₁ a.val) (fun a => h₂ a.val)
-    (fun a => hc₁ a.val) (fun a => hc₂ a.val) (fun a => hsep a.val)
-    (fun a => a.property.1) (fun a => a.property.2)
-    (fun a => hbin a.val) (fun a => hcap₂ a.val) (fun a => hsc₁ a.val) (fun a => hsc₂ a.val)
-    (fun a => hsω₁ a.val) (fun a => hsω₂ a.val) (fun a => hsη₁ a.val) (fun a => hsη₂ a.val)
-    (fun a => hz₁ a.val) (fun a => hz₂ a.val)
-  convert hb using 1
-  have he := Finset.sum_subtype (p := fun a => u ∈ P₁ a ×ˢ P₂ a) (F := inferInstance) (Finset.univ.filter (fun a => u ∈ P₁ a ×ˢ P₂ a))
-    (by intro a; simp) w
-  simpa [Finset.sum_filter,w] using he
-
-end WeightedContactSecondMoment
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Splicing.ReferenceLaws
+
+namespace OAI
+
+/-! Moment Bounds for bond percolation. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section SeededTraceMeasurable
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem uniformRank_fiber_eq (K : ℕ) (hK : 0 < K) (a : Fin K) :
+    {t : unitInterval | uniformRank K hK t = a} = contactInterval K a.val ∪
+      (if uniformRank K hK 1 = a then {1} else ∅) := by
+  ext t
+  by_cases ht : t = 1
+  · subst t
+    have hnot : (1 : unitInterval) ∉ contactInterval K a.val := by
+      intro h
+      have hp : (0:ℝ) < K := by exact_mod_cast hK
+      have hlt := (lt_div_iff₀ hp).mp h.2
+      have ha : (a.val:ℝ)+1 ≤ K := by exact_mod_cast (show a.val+1 ≤ K by omega)
+      push_cast at hlt
+      norm_num at hlt
+      linarith
+    simp [hnot]
+  · have hlt : (t:ℝ) < 1 := lt_of_le_of_ne t.property.2 (fun he => ht (Subtype.ext he))
+    have he := uniformRank_eq_iff K hK t hlt a
+    simp only [Set.mem_union,Set.mem_ofPred_eq]
+    split_ifs <;> simpa [ht] using he
+
+theorem measurableSet_contactInterval (K a : ℕ) : MeasurableSet (contactInterval K a) :=
+  (measurableSet_le measurable_const measurable_subtype_coe).inter
+    (measurableSet_lt measurable_subtype_coe measurable_const)
+
+theorem measurableSet_uniformRank_fiber (K : ℕ) (hK : 0 < K) (a : Fin K) :
+    MeasurableSet {t : unitInterval | uniformRank K hK t = a} := by
+  rw [uniformRank_fiber_eq]
+  apply (measurableSet_contactInterval K a.val).union
+  split_ifs
+  · exact measurableSet_singleton _
+  · exact MeasurableSet.empty
+
+theorem measurableSet_uniformContact_fiber (K : Finset E) (hK : K.Nonempty) (e : E) :
+    MeasurableSet {t : unitInterval | uniformContact K hK t = e} := by
+  by_cases he : e ∈ K
+  · simp_rw [uniformContact_eq_iff K hK _ e he]
+    exact measurableSet_uniformRank_fiber _ _ _
+  · have hz : {t : unitInterval | uniformContact K hK t = e} = ∅ := by
+      ext t
+      simp only [Set.mem_ofPred_eq,Set.mem_empty_iff_false,iff_false]
+      exact fun h => he (h ▸ uniformContact_mem K hK t)
+    rw [hz]
+    exact MeasurableSet.empty
+
+theorem measurableSet_chooseContact_fiber [Nonempty E] (i : ℕ) (K : Set E) (e : E) :
+    MeasurableSet {t : unitInterval | chooseContact (fun _ => t) i K = e} := by
+  unfold chooseContact
+  split_ifs
+  · exact measurableSet_uniformContact_fiber _ _ _
+  · by_cases he : Classical.arbitrary E = e <;> simp [he]
+  · by_cases he : Classical.arbitrary E = e <;> simp [he]
+
+theorem measurableSet_replaySeedFiber [Nonempty E] (G : BondGraph V E) (x : V)
+    (η : ℕ → Set E) (y : ℕ → V) (c : List (ℕ × E)) (ω : Set E) (i : ℕ) :
+    MeasurableSet (replaySeedFiber G x η y c ω i) := by
+  induction c generalizing ω with
+  | nil => exact MeasurableSet.univ
+  | cons q c ih =>
+    obtain ⟨j,e⟩ := q
+    simp only [replaySeedFiber]
+    split_ifs
+    · exact measurableSet_chooseContact_fiber _ _ _
+    · exact ih _
+
+theorem SpliceTrace.measurableSet_seed_event [Nonempty E] (G : BondGraph V E) (x : V)
+    (η : ℕ → Set E) (y : ℕ → V) (k j : ℕ) (ω ωf : Set E)
+    (c : List (ℕ × E)) (h : SpliceTrace G x η y k j ω c ωf) :
+    MeasurableSet {u : Fin k → unitInterval | FollowsContactRule G x η y
+      (chooseContact (finiteContactSeeds k u)) c ω} := by
+  rw [h.seed_event_eq_pi G x η y k j ω ωf c]
+  exact MeasurableSet.univ_pi (fun i => measurableSet_replaySeedFiber G x η y c ω (i.val+1))
+
+end SeededTraceMeasurable
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section FiniteTraceDistribution
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem finite_bounded_lists {A : Type*} (Q : Finset A) (k : ℕ) :
+    {c : List A | c.length ≤ k ∧ ∀ q ∈ c, q ∈ Q}.Finite := by
+  have hf := (List.finite_length_le ↥Q k).image (List.map Subtype.val)
+  apply hf.subset
+  rintro c ⟨hc,hQ⟩
+  let d : List ↥Q := c.attach.map (fun q => ⟨q.val,hQ q.val q.property⟩)
+  refine ⟨d,?_,?_⟩
+  · simpa only [d,List.length_map,List.length_attach,Set.mem_ofPred_eq] using hc
+  · simp [d,List.map_map]
+
+noncomputable def boundedTraceCodes (Q : Finset E) (k : ℕ) : Finset (List (ℕ × E)) :=
+  (finite_bounded_lists ((Finset.Icc 1 k) ×ˢ Q) k).toFinset
+
+@[simp] theorem mem_boundedTraceCodes (Q : Finset E) (k : ℕ) (c : List (ℕ × E)) :
+    c ∈ boundedTraceCodes Q k ↔ c.length ≤ k ∧ ∀ q ∈ c, 1 ≤ q.1 ∧ q.1 ≤ k ∧ q.2 ∈ Q := by
+  simp [boundedTraceCodes,Finset.mem_product,and_assoc]
+
+theorem SpliceTrace.length_le (G : BondGraph V E) (x : V) (η : ℕ → Set E)
+    (y : ℕ → V) (k j : ℕ) (ω ωf : Set E) (c : List (ℕ × E))
+    (h : SpliceTrace G x η y k j ω c ωf) : c.length ≤ k-j := by
+  induction h with
+  | stop => simp
+  | step => simp only [List.length_cons]; omega
+
+theorem SpliceTrace.mem_boundedTraceCodes (G : BondGraph V E) (x : V) (η : ℕ → Set E)
+    (y : ℕ → V) (k j : ℕ) (ω ωf : Set E) (c : List (ℕ × E))
+    (h : SpliceTrace G x η y k j ω c ωf) (Q : Finset E)
+    (hQ : ∀ i, 1 ≤ i → i ≤ k → η i ⊆ (Q : Set E)) : c ∈ boundedTraceCodes Q k := by
+  rw [Percolation.mem_boundedTraceCodes]
+  refine ⟨(h.length_le G x η y k j ω ωf c).trans (Nat.sub_le ..),?_⟩
+  intro q hq
+  obtain ⟨hj,hk,he,_⟩ := h.seam_data G x η y k j ω ωf c q hq
+  exact ⟨by omega,hk,hQ q.1 (by omega) hk he⟩
+
+ 
+
+noncomputable def finiteTraceCodes (G : BondGraph V E) (x : V) (η : ℕ → Set E)
+    (y : ℕ → V) (k j : ℕ) (ω : Set E) (Q : Finset E) : Finset (List (ℕ × E)) :=
+  (boundedTraceCodes Q k).filter (fun c =>
+    SpliceTrace G x η y k j ω c (replaySplice G x η y c ω))
+
+theorem mem_finiteTraceCodes_iff (G : BondGraph V E) (x : V) (η : ℕ → Set E)
+    (y : ℕ → V) (k j : ℕ) (ω : Set E) (Q : Finset E) (c : List (ℕ × E)) :
+    c ∈ finiteTraceCodes G x η y k j ω Q ↔ c ∈ boundedTraceCodes Q k ∧
+      SpliceTrace G x η y k j ω c (replaySplice G x η y c ω) := by
+  simp [finiteTraceCodes]
+
+ 
+
+theorem finite_trace_seed_partition [Nonempty E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) (η : ℕ → Set E)
+    (y : ℕ → V) (k M : ℕ) (Q : Finset E)
+    (hQ : ∀ i, 1 ≤ i → i ≤ k → η i ⊆ (Q : Set E))
+    (hchain : ∀ i ∈ Finset.Icc 1 k, Conn G (η i) (y i) (y (i-1)))
+    (hcaps : ∀ i ∈ Finset.Icc 1 k, (cluster G (η i) (y i)).Finite ∧
+      (cluster G (η i) (y i)).ncard ≤ M)
+    (j : ℕ) (hj : j ≤ k) (ω : Set E) (hyj : y j ∈ cluster G ω x)
+    (hfin : (cluster G ω x).Finite) :
+    let S := finiteTraceCodes G x η y k j ω Q
+    let A := fun c => {u : Fin k → unitInterval | FollowsContactRule G x η y
+      (chooseContact (finiteContactSeeds k u)) c ω}
+    (⋃ c ∈ S, A c) = Set.univ ∧ Set.PairwiseDisjoint (↑S) A := by
+  intro S A
+  constructor
+  · apply Set.eq_univ_of_forall
+    intro u
+    obtain ⟨c,ωf,ht,hr,_⟩ := exists_seeded_spliceTrace G hloc x η y k M hchain hcaps
+      (chooseContact (finiteContactSeeds k u)) (chooseContact_mem _) j hj ω hyj _ hfin (le_refl _)
+    have hc : c ∈ S := (mem_finiteTraceCodes_iff ..).mpr
+      ⟨ht.mem_boundedTraceCodes G x η y k j ω ωf c Q hQ,
+        (ht.replay_eq G x η y k j ω ωf c).symm ▸ ht⟩
+    exact Set.mem_iUnion.mpr ⟨c,Set.mem_iUnion.mpr ⟨hc,hr⟩⟩
+  · intro c hc d hd hcd
+    apply Set.disjoint_left.mpr
+    intro u huc hud
+    have ht := (mem_finiteTraceCodes_iff G x η y k j ω Q c).mp hc
+    have hs := (mem_finiteTraceCodes_iff G x η y k j ω Q d).mp hd
+    exact hcd (ht.2.seeded_unique G x η y k j ω _ _ c d hs.2
+      (chooseContact (finiteContactSeeds k u)) huc hud).1
+
+ 
+theorem sum_finite_trace_probabilities [Nonempty E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) (η : ℕ → Set E)
+    (y : ℕ → V) (k M : ℕ) (Q : Finset E)
+    (hQ : ∀ i, 1 ≤ i → i ≤ k → η i ⊆ (Q : Set E))
+    (hchain : ∀ i ∈ Finset.Icc 1 k, Conn G (η i) (y i) (y (i-1)))
+    (hcaps : ∀ i ∈ Finset.Icc 1 k, (cluster G (η i) (y i)).Finite ∧
+      (cluster G (η i) (y i)).ncard ≤ M)
+    (j : ℕ) (hj : j ≤ k) (ω : Set E) (hyj : y j ∈ cluster G ω x)
+    (hfin : (cluster G ω x).Finite) :
+    (∑ c ∈ finiteTraceCodes G x η y k j ω Q,
+      ∏ i : Fin k, (traceContactCount G η y (replaySplice G x η y c ω) x c (i.val+1):ℝ)⁻¹) = 1 := by
+  let S := finiteTraceCodes G x η y k j ω Q
+  let A := fun c => {u : Fin k → unitInterval | FollowsContactRule G x η y
+    (chooseContact (finiteContactSeeds k u)) c ω}
+  have hpart := finite_trace_seed_partition G hloc x η y k M Q hQ hchain hcaps j hj ω hyj hfin
+  have hm : ∀ c ∈ S, MeasurableSet (A c) := by
+    intro c hc
+    exact ((mem_finiteTraceCodes_iff G x η y k j ω Q c).mp hc).2.measurableSet_seed_event
+      G x η y k j ω _ c
+  have hsum : ∑ c ∈ S, volume (A c) = 1 := by
+    rw [← measure_biUnion_finset hpart.2 hm,hpart.1,measure_univ]
+  have hr := congrArg ENNReal.toReal hsum
+  rw [ENNReal.toReal_sum (fun c _ => measure_ne_top volume (A c)),ENNReal.toReal_one] at hr
+  rw [← hr]
+  apply Finset.sum_congr rfl
+  intro c hc
+  exact (((mem_finiteTraceCodes_iff G x η y k j ω Q c).mp hc).2.seed_probability
+    G hloc x η y k j ω _ c (fun i hi hik => (hcaps i (Finset.mem_Icc.mpr ⟨hi,hik⟩)).1)).symm
+
+end FiniteTraceDistribution
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section FiniteSpliceState
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem SpliceTrace.runMasks_execution (G : BondGraph V E) (x : V)
+    (η : ℕ → Set E) (y : ℕ → V) (k j : ℕ) (ω ωf : Set E) (c : List (ℕ × E))
+    (h : SpliceTrace G x η y k j ω c ωf) :
+    ∀ θ : ℕ → Set E, (∀ l, j < l → l ≤ k → θ l = η l) →
+      SpliceExecution G x (ω,θ) c (runMasks (replayMasks G x η y c ω) (ω,θ)) := by
+  induction h with
+  | stop => intro θ _; exact SpliceExecution.nil _
+  | @step j i ω e a b c ωf hn hji hik hm hy he hab ha hb ht ih =>
+    intro θ hθ
+    let S := cluster G ω x
+    let B := cluster G (η i ∩ internalEdges G Sᶜ) (y i)
+    let M := spliceMask G S B e
+    let ω' := spliceField G S B e ω (η i)
+    let θ' := Function.update θ i (exchangeOn M (θ i) ω)
+    have hθi := hθ i hji hik
+    have hstep : exchangeState M i (ω,θ) = (ω',θ') := by
+      apply Prod.ext
+      · change exchangeOn M ω (θ i) = exchangeOn M ω (η i)
+        rw [hθi]
+      · rfl
+    change SpliceExecution G x (ω,θ) ((i,e)::c)
+      (runMasks (replayMasks G x η y c ω') (exchangeState M i (ω,θ)))
+    rw [hstep]
+    refine SpliceExecution.cons (ω,θ) (ω',θ') _ (i,e) c ?_ ?_
+    · refine ⟨y i,a,b,hy,?_,hab,ha,?_,?_⟩
+      · simpa only [hθi] using he
+      · simpa only [hθi] using hb
+      · simpa only [hθi] using hstep.symm
+    · apply ih θ'
+      intro l hil hlk
+      rw [show θ' l = θ l from Function.update_of_ne (by omega) _ _]
+      exact hθ l (hji.trans hil) hlk
+
+abbrev FiniteSpliceData (Q : Finset E) (J : Finset ℕ) :=
+  FiniteField Q × (↥J → FiniteField Q)
+
+def encodeFiniteSplice (Q : Finset E) (J : Finset ℕ)
+    (s : FiniteSpliceData Q J) : SpliceState E :=
+  (finiteFieldSet Q s.1,finiteFieldFamily Q J s.2)
+
+theorem encodeFiniteSplice_injective (Q : Finset E) (J : Finset ℕ) :
+    Function.Injective (encodeFiniteSplice Q J) := by
+  intro a b he
+  exact Prod.ext (finiteFieldSet_injective Q (congrArg Prod.fst he))
+    (finiteFieldFamily_injective Q J (congrArg Prod.snd he))
+
+noncomputable def finiteSpliceDomain (Q : Finset E) (J : Finset ℕ) : Finset (SpliceState E) :=
+  Finset.univ.image (encodeFiniteSplice Q J)
+
+theorem spliceTupleWeight_encode (Q : Finset E) (J : Finset ℕ) (p : ℝ)
+    (s : FiniteSpliceData Q J) : spliceTupleWeight Q J p (encodeFiniteSplice Q J s) =
+      finiteFieldWeight Q p s.1 * finiteFamilyWeight Q J p s.2 := by
+  simp only [spliceTupleWeight,encodeFiniteSplice,domainBits_finiteFieldSet,
+    prod_fieldFamilyWeight,finiteFieldWeight]
+
+theorem sum_finiteSpliceDomain (Q : Finset E) (J : Finset ℕ) (p : ℝ) :
+    ∑ s ∈ finiteSpliceDomain Q J, spliceTupleWeight Q J p s = 1 := by
+  rw [finiteSpliceDomain,Finset.sum_image (encodeFiniteSplice_injective Q J).injOn]
+  simp only [spliceTupleWeight_encode,Fintype.sum_prod_type,← Finset.mul_sum,
+    sum_finiteFamilyWeight,mul_one,sum_finiteFieldWeight]
+
+ 
+theorem mem_finiteSpliceDomain_iff (Q : Finset E) (J : Finset ℕ) (s : SpliceState E) :
+    s ∈ finiteSpliceDomain Q J ↔ s.1 ⊆ (Q : Set E) ∧
+      (∀ i, s.2 i ⊆ (Q : Set E)) ∧ ∀ i ∉ J, s.2 i = ∅ := by
+  constructor
+  · intro hs
+    obtain ⟨a,_,rfl⟩ := Finset.mem_image.mp hs
+    refine ⟨?_,?_,?_⟩
+    · exact Finset.coe_subset.mpr (Finset.mem_powerset.mp a.1.property)
+    · intro i
+      by_cases hi : i ∈ J
+      · simpa only [encodeFiniteSplice,finiteFieldFamily,dite_eq_left hi,finiteFieldSet] using
+          Finset.coe_subset.mpr (Finset.mem_powerset.mp (a.2 ⟨i,hi⟩).property)
+      · simp [encodeFiniteSplice,finiteFieldFamily,hi]
+    · intro i hi
+      simp [encodeFiniteSplice,finiteFieldFamily,hi]
+  · rintro ⟨hω,hη,hz⟩
+    let desc (ω : Set E) : FiniteField Q := ⟨domainBits Q ω,Finset.mem_powerset.mpr (domainBits_subset Q ω)⟩
+    have he (ω : Set E) (h : ω ⊆ (Q : Set E)) : finiteFieldSet Q (desc ω) = ω := by
+      ext e
+      simp only [finiteFieldSet,desc,Finset.mem_coe,mem_domainBits]
+      exact ⟨fun hh => hh.2,fun hh => ⟨h hh,hh⟩⟩
+    refine Finset.mem_image.mpr ⟨(desc s.1,fun i => desc (s.2 i)),Finset.mem_univ _,?_⟩
+    apply Prod.ext (he s.1 hω)
+    funext i
+    by_cases hi : i ∈ J
+    · simp only [encodeFiniteSplice,finiteFieldFamily,dite_eq_left hi,he _ (hη i)]
+    · simp only [encodeFiniteSplice,finiteFieldFamily,dite_eq_right hi,hz i hi]
+
+theorem exchangeState_mem_finiteSpliceDomain (Q : Finset E) (J : Finset ℕ)
+    (s : SpliceState E) (hs : s ∈ finiteSpliceDomain Q J) (i : ℕ) (hi : i ∈ J) (M : Set E) :
+    exchangeState M i s ∈ finiteSpliceDomain Q J := by
+  obtain ⟨hω,hη,hz⟩ := (mem_finiteSpliceDomain_iff Q J s).mp hs
+  apply (mem_finiteSpliceDomain_iff Q J _).mpr
+  refine ⟨exchangeOn_subset M _ _ _ hω (hη i),?_,?_⟩
+  · intro l
+    by_cases hli : l = i
+    · simpa only [hli,exchangeState_at] using exchangeOn_subset M _ _ _ (hη i) hω
+    · rw [exchangeState_away _ _ _ _ hli]
+      exact hη l
+  · intro l hl
+    have hli : l ≠ i := by intro he; subst l; exact hl hi
+    rw [exchangeState_away _ _ _ _ hli]
+    exact hz l hl
+
+theorem runMasks_mem_finiteSpliceDomain (Q : Finset E) (J : Finset ℕ)
+    (s : SpliceState E) (hs : s ∈ finiteSpliceDomain Q J) (ms : List (ℕ × Set E))
+    (hJ : ∀ q ∈ ms, q.1 ∈ J) : runMasks ms s ∈ finiteSpliceDomain Q J := by
+  induction ms generalizing s with
+  | nil => exact hs
+  | cons q ms ih =>
+    apply ih _ (exchangeState_mem_finiteSpliceDomain Q J s hs q.1 (hJ q (by simp)) q.2)
+    exact fun q hq => hJ q (by simp [hq])
+
+end FiniteSpliceState
+
+end Percolation
+
+ 
+open MeasureTheory
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section FiniteSwitchDensity
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+noncomputable def finiteInverseLoad {I : Type*} [Fintype I]
+    (out : I → SpliceState E) (mass : I → ℝ) (t : SpliceState E) : ℝ :=
+  ∑ a, if out a = t then mass a else 0
+
+theorem finiteInverseLoad_nonneg {I : Type*} [Fintype I]
+    (out : I → SpliceState E) (mass : I → ℝ) (hm : ∀ a, 0 ≤ mass a) (t : SpliceState E) :
+    0 ≤ finiteInverseLoad out mass t := by
+  apply Finset.sum_nonneg
+  intro a _
+  split_ifs
+  · exact hm a
+  · exact le_refl 0
+
+ 
+
+theorem finiteInverseLoad_le {I : Type*} [Fintype I]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (x z : V) (k M : ℕ)
+    (L : ℝ) (hL : 1 ≤ L)
+    (inp out : I → SpliceState E) (steps : I → List (ℕ × E)) (y : I → ℕ → V)
+    (hir : Function.Injective (fun a => (inp a,steps a)))
+    (htrace : ∀ a, SpliceTrace G x (inp a).2 (y a) k 0 (inp a).1 (steps a) (out a).1)
+    (hex : ∀ a, SpliceExecution G x (inp a) (steps a) (out a))
+    (hz : ∀ a, y a k = z) (hfin : ∀ a, (cluster G (out a).1 x).Finite)
+    (hgood : ∀ a i, 1 ≤ i → i ≤ k → (cluster G ((inp a).2 i) (y a i)).Finite ∧
+      (cluster G ((inp a).2 i) (y a i)).ncard ≤ M ∧ GoodForScale G ((inp a).2 i) (y a i) L)
+    (t : SpliceState E) :
+    finiteInverseLoad out (fun a => ∏ i : Fin k,
+      (traceContactCount G (inp a).2 (y a) (out a).1 x (steps a) (i.val+1):ℝ)⁻¹) t ≤
+      inverseCodeMass M L ^ k := by
+  let J := {a : I // out a = t}
+  have hr (a : J) (i : Fin k) : 0 < traceContactCount G (inp a.val).2 (y a.val)
+      (out a.val).1 x (steps a.val) (i.val+1) := by
+    have hh := ((htrace a.val).contact_weight_bound G hloc x (inp a.val).2 (y a.val)
+      k 0 M (inp a.val).1 (out a.val).1 (steps a.val) (hfin a.val) L hL
+      (hgood a.val) (i.val+1)).1
+    omega
+  have hb := finite_inverse_contact_mass_le G hloc x z k M L hL (0 : unitInterval) ∅
+    (fun a : J => inp a.val) (fun a : J => out a.val) (fun a : J => steps a.val)
+    (fun a : J => y a.val)
+    (fun a b he => Subtype.ext (hir he)) (fun a => htrace a.val) (fun a => hex a.val)
+    (fun a => hz a.val) (fun a => hfin a.val) (fun a => hgood a.val)
+    (fun _ _ => 0) hr {t} (fun a => by simpa only [Finset.mem_singleton] using a.property)
+  simp only [real_volume_contact_choice_cube _ _ (hr _),spliceTupleWeight,domainBits,
+    Finset.filter_empty,bernoulliWeight,Finset.card_empty,Nat.sub_self,pow_zero,
+    Finset.prod_const_one,mul_one,one_mul,Finset.sum_singleton] at hb
+  rw [finiteInverseLoad,← Finset.sum_filter]
+  rw [Finset.sum_subtype (p := fun a : I => out a = t) _ (by intro a; simp) _]
+  exact hb
+
+ 
+
+theorem finiteInverseLoad_density {I : Type*} [Fintype I]
+    (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E) (p : ℝ)
+    (inp out : I → SpliceState E) (steps : I → List (ℕ × E)) (y : I → ℕ → V)
+    (htrace : ∀ a, SpliceTrace G x (inp a).2 (y a) k 0 (inp a).1 (steps a) (out a).1)
+    (hex : ∀ a, SpliceExecution G x (inp a) (steps a) (out a))
+    (mass : I → ℝ) (B : Finset (SpliceState E)) (hB : ∀ a, out a ∈ B)
+    (f : SpliceState E → ℝ) :
+    (∑ a, spliceTupleWeight Q (Finset.Icc 1 k) p (inp a) * mass a * f (out a)) =
+      ∑ t ∈ B, spliceTupleWeight Q (Finset.Icc 1 k) p t * finiteInverseLoad out mass t * f t := by
+  have hw (a : I) : spliceTupleWeight Q (Finset.Icc 1 k) p (inp a) =
+      spliceTupleWeight Q (Finset.Icc 1 k) p (out a) := by
+    symm
+    apply (hex a).weight G x (inp a) (out a) (steps a) Q (Finset.Icc 1 k) _ p
+    intro q hq
+    have h := (htrace a).seam_data G x (inp a).2 (y a) k 0 (inp a).1 (out a).1 (steps a) q hq
+    exact Finset.mem_Icc.mpr ⟨h.1,h.2.1⟩
+  simp only [finiteInverseLoad,Finset.mul_sum,Finset.sum_mul]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro a _
+  simp only [mul_ite,mul_zero,ite_mul,zero_mul]
+  rw [Finset.sum_ite_eq,ite_eq_left (hB a),hw a]
+
+end FiniteSwitchDensity
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section FiniteForwardDensity
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+abbrev FiniteTraceIndex (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
+    (A : Finset (SpliceState E)) (y : SpliceState E → ℕ → V) :=
+  (s : ↥A) × ↥(finiteTraceCodes G x s.val.2 (y s.val) k 0 s.val.1 Q)
+
+def traceIndexInput (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
+    (A : Finset (SpliceState E)) (y : SpliceState E → ℕ → V)
+    (a : FiniteTraceIndex G x k Q A y) : SpliceState E := a.1.val
+
+def traceIndexCode (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
+    (A : Finset (SpliceState E)) (y : SpliceState E → ℕ → V)
+    (a : FiniteTraceIndex G x k Q A y) : List (ℕ × E) := a.2.val
+
+def traceIndexOutput (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
+    (A : Finset (SpliceState E)) (y : SpliceState E → ℕ → V)
+    (a : FiniteTraceIndex G x k Q A y) : SpliceState E :=
+  runMasks (replayMasks G x a.1.val.2 (y a.1.val) a.2.val a.1.val.1) a.1.val
+
+theorem traceIndex_valid (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
+    (A : Finset (SpliceState E)) (y : SpliceState E → ℕ → V)
+    (a : FiniteTraceIndex G x k Q A y) :
+    SpliceTrace G x a.1.val.2 (y a.1.val) k 0 a.1.val.1 a.2.val
+      (traceIndexOutput G x k Q A y a).1 ∧
+    SpliceExecution G x a.1.val a.2.val (traceIndexOutput G x k Q A y a) := by
+  have ht := ((mem_finiteTraceCodes_iff G x a.1.val.2 (y a.1.val) k 0 a.1.val.1 Q a.2.val).mp
+    a.2.property).2
+  have hf := ht.runMasks_first G x a.1.val.2 (y a.1.val) k 0 a.1.val.1 _ a.2.val
+    a.1.val.2 (by intros; rfl)
+  refine ⟨?_,?_⟩
+  · simpa only [traceIndexOutput,hf] using ht
+  · exact ht.runMasks_execution G x a.1.val.2 (y a.1.val) k 0 a.1.val.1 _ a.2.val
+      a.1.val.2 (by intros; rfl)
+
+theorem traceIndex_injective (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
+    (A : Finset (SpliceState E)) (y : SpliceState E → ℕ → V) :
+    Function.Injective (fun a : FiniteTraceIndex G x k Q A y =>
+      (traceIndexInput G x k Q A y a,traceIndexCode G x k Q A y a)) := by
+  rintro ⟨a,c⟩ ⟨b,d⟩ he
+  have hah : a = b := Subtype.ext (congrArg Prod.fst he)
+  subst b
+  have hcd : c = d := Subtype.ext (congrArg Prod.snd he)
+  subst d
+  rfl
+
+theorem traceIndexOutput_mem (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
+    (A : Finset (SpliceState E)) (y : SpliceState E → ℕ → V)
+    (hA : A ⊆ finiteSpliceDomain Q (Finset.Icc 1 k))
+    (a : FiniteTraceIndex G x k Q A y) :
+    traceIndexOutput G x k Q A y a ∈ finiteSpliceDomain Q (Finset.Icc 1 k) := by
+  apply runMasks_mem_finiteSpliceDomain Q (Finset.Icc 1 k) a.1.val (hA a.1.property)
+  intro q hq
+  have hmap : q.1 ∈ (replayMasks G x a.1.val.2 (y a.1.val) a.2.val a.1.val.1).map Prod.fst :=
+    List.mem_map.mpr ⟨q,hq,rfl⟩
+  rw [replayMasks_indices] at hmap
+  obtain ⟨r,hr,her⟩ := List.mem_map.mp hmap
+  have h := (traceIndex_valid G x k Q A y a).1.seam_data G x a.1.val.2 (y a.1.val)
+    k 0 a.1.val.1 _ a.2.val r hr
+  exact her ▸ Finset.mem_Icc.mpr ⟨h.1,h.2.1⟩
+
+ 
+
+noncomputable def traceIndexMass (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
+    (A : Finset (SpliceState E)) (y : SpliceState E → ℕ → V)
+    (a : FiniteTraceIndex G x k Q A y) : ℝ :=
+  ∏ i : Fin k, (traceContactCount G a.1.val.2 (y a.1.val)
+    (traceIndexOutput G x k Q A y a).1 x a.2.val (i.val+1):ℝ)⁻¹
+
+noncomputable def finiteForwardLoad (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
+    (A : Finset (SpliceState E)) (y : SpliceState E → ℕ → V) (t : SpliceState E) : ℝ :=
+  finiteInverseLoad (traceIndexOutput G x k Q A y) (traceIndexMass G x k Q A y) t
+
+ 
+
+theorem finiteForwardLoad_bounds (G : BondGraph V E) (hloc : LocallyFinite G)
+    (x z : V) (k M : ℕ) (L : ℝ) (hL : 1 ≤ L) (Q : Finset E)
+    (A : Finset (SpliceState E)) (y : SpliceState E → ℕ → V)
+    (hA : A ⊆ finiteSpliceDomain Q (Finset.Icc 1 k))
+    (hz : ∀ s ∈ A, y s k = z)
+    (hgood : ∀ s ∈ A, ∀ i, 1 ≤ i → i ≤ k → (cluster G (s.2 i) (y s i)).Finite ∧
+      (cluster G (s.2 i) (y s i)).ncard ≤ M ∧ GoodForScale G (s.2 i) (y s i) L)
+    (t : SpliceState E) :
+    0 ≤ finiteForwardLoad G x k Q A y t ∧ finiteForwardLoad G x k Q A y t ≤ inverseCodeMass M L ^ k := by
+  constructor
+  · exact finiteInverseLoad_nonneg _ _ (by intro a; unfold traceIndexMass; positivity) t
+  · apply finiteInverseLoad_le G hloc x z k M L hL
+      (traceIndexInput G x k Q A y) (traceIndexOutput G x k Q A y) (traceIndexCode G x k Q A y)
+      (fun a => y a.1.val) (traceIndex_injective G x k Q A y)
+      (fun a => (traceIndex_valid G x k Q A y a).1)
+      (fun a => (traceIndex_valid G x k Q A y a).2) (fun a => hz a.1.val a.1.property)
+      _ (fun a => hgood a.1.val a.1.property) t
+    intro a
+    have hs := (mem_finiteSpliceDomain_iff Q (Finset.Icc 1 k) _).mp
+      (traceIndexOutput_mem G x k Q A y hA a)
+    exact cluster_finite_of_finite_edges G (Q.finite_toSet.subset hs.1) x
+
+ 
+
+theorem finiteForwardLoad_mean [Nonempty E] (G : BondGraph V E) (hloc : LocallyFinite G)
+    (x : V) (k M : ℕ) (Q : Finset E) (p : ℝ)
+    (A : Finset (SpliceState E)) (y : SpliceState E → ℕ → V)
+    (hA : A ⊆ finiteSpliceDomain Q (Finset.Icc 1 k))
+    (hy : ∀ s ∈ A, y s 0 ∈ cluster G s.1 x)
+    (hchain : ∀ s ∈ A, ∀ i ∈ Finset.Icc 1 k, Conn G (s.2 i) (y s i) (y s (i-1)))
+    (hcaps : ∀ s ∈ A, ∀ i ∈ Finset.Icc 1 k, (cluster G (s.2 i) (y s i)).Finite ∧
+      (cluster G (s.2 i) (y s i)).ncard ≤ M) :
+    (∑ t ∈ finiteSpliceDomain Q (Finset.Icc 1 k),
+      spliceTupleWeight Q (Finset.Icc 1 k) p t * finiteForwardLoad G x k Q A y t) =
+      ∑ s ∈ A, spliceTupleWeight Q (Finset.Icc 1 k) p s := by
+  have hd := finiteInverseLoad_density G x k Q p
+    (traceIndexInput G x k Q A y) (traceIndexOutput G x k Q A y) (traceIndexCode G x k Q A y)
+    (fun a => y a.1.val) (fun a => (traceIndex_valid G x k Q A y a).1)
+    (fun a => (traceIndex_valid G x k Q A y a).2)
+    (traceIndexMass G x k Q A y) (finiteSpliceDomain Q (Finset.Icc 1 k))
+    (traceIndexOutput_mem G x k Q A y hA) (fun _ => 1)
+  simp only [mul_one] at hd
+  unfold finiteForwardLoad
+  rw [← hd,Fintype.sum_sigma]
+  rw [← Finset.sum_coe_sort A (fun s => spliceTupleWeight Q (Finset.Icc 1 k) p s)]
+  apply Finset.sum_congr rfl
+  intro s _
+  change (∑ c : ↥(finiteTraceCodes G x s.val.2 (y s.val) k 0 s.val.1 Q),
+    spliceTupleWeight Q (Finset.Icc 1 k) p s.val * traceIndexMass G x k Q A y ⟨s,c⟩) = _
+  rw [← Finset.mul_sum]
+  suffices hm : (∑ c : ↥(finiteTraceCodes G x s.val.2 (y s.val) k 0 s.val.1 Q),
+      traceIndexMass G x k Q A y ⟨s,c⟩) = 1 by rw [hm,mul_one]
+  have hs := (mem_finiteSpliceDomain_iff Q (Finset.Icc 1 k) s.val).mp (hA s.property)
+  have hn := sum_finite_trace_probabilities G hloc x s.val.2 (y s.val) k M Q
+    (fun i _ _ => hs.2.1 i) (hchain s.val s.property) (hcaps s.val s.property)
+    0 (Nat.zero_le _) s.val.1 (hy s.val s.property)
+    (cluster_finite_of_finite_edges G (Q.finite_toSet.subset hs.1) x)
+  rw [← Finset.sum_coe_sort (finiteTraceCodes G x s.val.2 (y s.val) k 0 s.val.1 Q)] at hn
+  rw [← hn]
+  apply Finset.sum_congr rfl
+  intro c _
+  have ht := ((mem_finiteTraceCodes_iff G x s.val.2 (y s.val) k 0 s.val.1 Q c.val).mp c.property).2
+  have hf := ht.runMasks_first G x s.val.2 (y s.val) k 0 s.val.1 _ c.val s.val.2 (by intros; rfl)
+  simp only [traceIndexMass,traceIndexOutput,hf]
+
+end FiniteForwardDensity
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section ContactIntegratedSecondMoment
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem finite_event_mass_le {Ω I : Type*} [MeasurableSpace Ω] [Fintype I]
+    (μ : Measure Ω) [IsProbabilityMeasure μ] (P : I → Set Ω)
+    (hm : ∀ a, MeasurableSet (P a)) (w : I → ℝ) (B : ℝ)
+    (hpoint : ∀ u, (∑ a, if u ∈ P a then w a else 0) ≤ B) :
+    (∑ a, w a * μ.real (P a)) ≤ B := by
+  let f (a : I) : Ω → ℝ := (P a).indicator (fun _ => w a)
+  have hf : ∀ a, Integrable (f a) μ := fun a => (integrable_const _).indicator (hm a)
+  have hi := integral_mono (integrable_finsetSum _ (fun a _ => hf a))
+    (integrable_const B) (fun u => hpoint u)
+  change (∫ u, ∑ a, f a u ∂μ) ≤ (∫ _ : Ω, B ∂μ) at hi
+  rw [integral_finsetSum _ (fun a _ => hf a)] at hi
+  simpa only [f,integral_indicator_const _ (hm _),smul_eq_mul,mul_comm,
+    integral_const,probReal_univ,one_smul,mul_one] using hi
+
+ 
+
+theorem finite_separated_contact_second_moment_le {I : Type*} [Fintype I] [Nonempty E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) (k n M : ℕ) (p : unitInterval) (Q : Finset E)
+    (η₁ η₂ : I → ℕ → Set E) (y₁ y₂ : (ℕ → Set E) → ℕ → V)
+    (success₁ success₂ : (ℕ → Set E) → Prop)
+    (ω₁ ω₂ ωf : I → Set E) (c₁ c₂ : I → List (ℕ × E))
+    (hir : Function.Injective (fun a =>
+      (jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a),
+        c₁ a,c₂ a)))
+    (h₁ : ∀ a, SpliceTrace G x (η₁ a) (y₁ (η₁ a)) k 0 (ω₁ a) (c₁ a) (ωf a))
+    (h₂ : ∀ a, SpliceTrace G x (η₂ a) (y₂ (η₂ a)) k 0 (ω₂ a) (c₂ a) (ωf a))
+    (hc₁ : ∀ a, c₁ a ≠ []) (hc₂ : ∀ a, c₂ a ≠ [])
+    (hsep : ∀ a, GraphSeparated G (cluster G (ωf a) x \ cluster G (ω₁ a) x)
+      (cluster G (ωf a) x \ cluster G (ω₂ a) x))
+    (hbin : ∀ a, (cluster G (ω₁ a) x).Finite ∧ n ≤ (cluster G (ω₁ a) x).ncard ∧
+      (cluster G (ω₁ a) x).ncard < 2*n)
+    (hcap₁ : ∀ a i, 1 ≤ i → i ≤ k → (cluster G (η₁ a i) (y₁ (η₁ a) i)).Finite)
+    (hcap₂ : ∀ a i, 1 ≤ i → i ≤ k → (cluster G (η₂ a i) (y₂ (η₂ a) i)).Finite ∧
+      (cluster G (η₂ a i) (y₂ (η₂ a) i)).ncard ≤ M)
+    (hsc₁ : ∀ a, success₁ (η₁ a)) (hsc₂ : ∀ a, success₂ (η₂ a))
+    (hsω₁ : ∀ a, ω₁ a ⊆ (Q : Set E)) (hsω₂ : ∀ a, ω₂ a ⊆ (Q : Set E))
+    (hsη₁ : ∀ a i, η₁ a i ⊆ (Q : Set E)) (hsη₂ : ∀ a i, η₂ a i ⊆ (Q : Set E))
+    (hz₁ : ∀ a i, i ∉ Finset.Icc 1 k → η₁ a i = ∅)
+    (hz₂ : ∀ a i, i ∉ Finset.Icc 1 k → η₂ a i = ∅) :
+    (∑ a, jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p
+      (jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a)) *
+      (∏ i : Fin k, (traceContactCount G (η₁ a) (y₁ (η₁ a)) (ωf a) x (c₁ a) (i.val+1):ℝ)⁻¹) *
+      (∏ i : Fin k, (traceContactCount G (η₂ a) (y₂ (η₂ a)) (ωf a) x (c₂ a) (i.val+1):ℝ)⁻¹)) ≤
+    ∑ ω : FiniteField Q, finiteFieldWeight Q p ω *
+      (if (cluster G (finiteFieldSet Q ω) x).Finite ∧
+        n-k*M ≤ (cluster G (finiteFieldSet Q ω) x).ncard ∧
+        (cluster G (finiteFieldSet Q ω) x).ncard < 2*n then
+        finiteFamilyEvent Q (Finset.Icc 1 k) p (fun η => success₁ η ∧
+          (cluster G (finiteFieldSet Q ω) x ∩ trajectoryUnion G η (y₁ η) k).Nonempty) *
+        finiteFamilyEvent Q (Finset.Icc 1 k) p (fun η => success₂ η ∧
+          (cluster G (finiteFieldSet Q ω) x ∩ trajectoryUnion G η (y₂ η) k).Nonempty) else 0) := by
+  let S := Fin k → unitInterval
+  let μ : Measure (S × S) := (volume : Measure S).prod volume
+  let P₁ (a : I) : Set S := {u | FollowsContactRule G x (η₁ a) (y₁ (η₁ a))
+    (chooseContact (finiteContactSeeds k u)) (c₁ a) (ω₁ a)}
+  let P₂ (a : I) : Set S := {u | FollowsContactRule G x (η₂ a) (y₂ (η₂ a))
+    (chooseContact (finiteContactSeeds k u)) (c₂ a) (ω₂ a)}
+  let w (a : I) := jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p
+    (jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a))
+  have hm₁ (a : I) : MeasurableSet (P₁ a) :=
+    (h₁ a).measurableSet_seed_event G x (η₁ a) (y₁ (η₁ a)) k 0 (ω₁ a) (ωf a) (c₁ a)
+  have hm₂ (a : I) : MeasurableSet (P₂ a) :=
+    (h₂ a).measurableSet_seed_event G x (η₂ a) (y₂ (η₂ a)) k 0 (ω₂ a) (ωf a) (c₂ a)
+  have hp (a : I) : μ.real (P₁ a ×ˢ P₂ a) =
+      (∏ i : Fin k, (traceContactCount G (η₁ a) (y₁ (η₁ a)) (ωf a) x (c₁ a) (i.val+1):ℝ)⁻¹) *
+      (∏ i : Fin k, (traceContactCount G (η₂ a) (y₂ (η₂ a)) (ωf a) x (c₂ a) (i.val+1):ℝ)⁻¹) := by
+    rw [measureReal_def,Measure.prod_prod,ENNReal.toReal_mul,
+      (h₁ a).seed_probability G hloc x (η₁ a) (y₁ (η₁ a)) k 0 (ω₁ a) (ωf a) (c₁ a) (hcap₁ a),
+      (h₂ a).seed_probability G hloc x (η₂ a) (y₂ (η₂ a)) k 0 (ω₂ a) (ωf a) (c₂ a) (fun i hi hik => (hcap₂ a i hi hik).1)]
+  conv_lhs => arg 2; ext a; rw [mul_assoc,← hp a]
+  apply finite_event_mass_le μ (fun a => P₁ a ×ˢ P₂ a) (fun a => (hm₁ a).prod (hm₂ a)) w
+  intro u
+  let J := {a : I // u ∈ P₁ a ×ˢ P₂ a}
+  have hb := finite_separated_second_moment_le G x k n M p Q
+    (fun a : J => η₁ a.val) (fun a : J => η₂ a.val) y₁ y₂ success₁ success₂
+    (chooseContact (finiteContactSeeds k u.1)) (chooseContact (finiteContactSeeds k u.2))
+    (fun a : J => ω₁ a.val) (fun a : J => ω₂ a.val) (fun a : J => ωf a.val)
+    (fun a : J => c₁ a.val) (fun a : J => c₂ a.val)
+    (fun _ _ he => Subtype.ext (hir he)) (fun a => h₁ a.val) (fun a => h₂ a.val)
+    (fun a => hc₁ a.val) (fun a => hc₂ a.val) (fun a => hsep a.val)
+    (fun a => a.property.1) (fun a => a.property.2)
+    (fun a => hbin a.val) (fun a => hcap₂ a.val) (fun a => hsc₁ a.val) (fun a => hsc₂ a.val)
+    (fun a => hsω₁ a.val) (fun a => hsω₂ a.val) (fun a => hsη₁ a.val) (fun a => hsη₂ a.val)
+    (fun a => hz₁ a.val) (fun a => hz₂ a.val)
+  convert hb using 1
+  have he := Finset.sum_subtype (p := fun a => u ∈ P₁ a ×ˢ P₂ a) (F := inferInstance) (Finset.univ.filter (fun a => u ∈ P₁ a ×ˢ P₂ a))
+    (by intro a; simp) w
+  simpa [Finset.sum_filter,w] using he
+
+end ContactIntegratedSecondMoment
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section FiniteRootLoad
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+noncomputable def finiteRootLoad (Q : Finset E) (J : Finset ℕ) (p : ℝ)
+    (f : SpliceState E → ℝ) (ω : FiniteField Q) : ℝ :=
+  ∑ η : ↥J → FiniteField Q, finiteFamilyWeight Q J p η *
+    f (finiteFieldSet Q ω,finiteFieldFamily Q J η)
+
+theorem finiteRootLoad_nonneg (Q : Finset E) (J : Finset ℕ) (p : unitInterval)
+    (f : SpliceState E → ℝ) (hf : ∀ s ∈ finiteSpliceDomain Q J, 0 ≤ f s)
+    (ω : FiniteField Q) : 0 ≤ finiteRootLoad Q J p f ω := by
+  apply Finset.sum_nonneg
+  intro η _
+  apply mul_nonneg (finiteFamilyWeight_nonneg Q J p η)
+  exact hf _ (Finset.mem_image.mpr ⟨(ω,η),Finset.mem_univ _,rfl⟩)
+
+theorem finiteRootLoad_le (Q : Finset E) (J : Finset ℕ) (p : unitInterval)
+    (f : SpliceState E → ℝ) (C : ℝ)
+    (hf : ∀ s ∈ finiteSpliceDomain Q J, f s ≤ C)
+    (ω : FiniteField Q) : finiteRootLoad Q J p f ω ≤ C := by
+  calc
+    _ ≤ ∑ η : ↥J → FiniteField Q, finiteFamilyWeight Q J p η * C := by
+      apply Finset.sum_le_sum
+      intro η _
+      apply mul_le_mul_of_nonneg_left _ (finiteFamilyWeight_nonneg Q J p η)
+      exact hf _ (Finset.mem_image.mpr ⟨(ω,η),Finset.mem_univ _,rfl⟩)
+    _ = C := by rw [← Finset.sum_mul,sum_finiteFamilyWeight,one_mul]
+
+ 
+theorem finiteRootLoad_mean (Q : Finset E) (J : Finset ℕ) (p : ℝ)
+    (f : SpliceState E → ℝ) :
+    (∑ ω : FiniteField Q, finiteFieldWeight Q p ω * finiteRootLoad Q J p f ω) =
+      ∑ s ∈ finiteSpliceDomain Q J, spliceTupleWeight Q J p s * f s := by
+  rw [finiteSpliceDomain,Finset.sum_image (encodeFiniteSplice_injective Q J).injOn,
+    Fintype.sum_prod_type]
+  simp only [spliceTupleWeight,finiteRootLoad,Finset.mul_sum,encodeFiniteSplice,
+    domainBits_finiteFieldSet,prod_fieldFamilyWeight,finiteFieldWeight,mul_assoc]
+
+ 
+
+theorem finiteRootLoad_second_moment (Q : Finset E) (J₁ J₂ : Finset ℕ) (p : ℝ)
+    (f₁ f₂ : SpliceState E → ℝ) :
+    (∑ ω : FiniteField Q, finiteFieldWeight Q p ω *
+      finiteRootLoad Q J₁ p f₁ ω * finiteRootLoad Q J₂ p f₂ ω) =
+    ∑ s : FiniteJointData Q J₁ J₂, jointTupleWeight Q J₁ J₂ p (encodeFiniteJoint Q J₁ J₂ s) *
+      f₁ (finiteFieldSet Q s.1,finiteFieldFamily Q J₁ s.2.1) *
+      f₂ (finiteFieldSet Q s.1,finiteFieldFamily Q J₂ s.2.2) := by
+  rw [Fintype.sum_prod_type]
+  apply Finset.sum_congr rfl
+  intro ω _
+  rw [Fintype.sum_prod_type]
+  simp only [finiteRootLoad,Finset.mul_sum,Finset.sum_mul,jointTupleWeight_encode]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro η₁ _
+  apply Finset.sum_congr rfl
+  intro η₂ _
+  ring
+
+ 
+theorem finiteForwardRootLoad_bounds (G : BondGraph V E) (hloc : LocallyFinite G)
+    (x z : V) (k M : ℕ) (L : ℝ) (hL : 1 ≤ L) (Q : Finset E) (p : unitInterval)
+    (A : Finset (SpliceState E)) (y : SpliceState E → ℕ → V)
+    (hA : A ⊆ finiteSpliceDomain Q (Finset.Icc 1 k))
+    (hz : ∀ s ∈ A, y s k = z)
+    (hgood : ∀ s ∈ A, ∀ i, 1 ≤ i → i ≤ k → (cluster G (s.2 i) (y s i)).Finite ∧
+      (cluster G (s.2 i) (y s i)).ncard ≤ M ∧ GoodForScale G (s.2 i) (y s i) L)
+    (ω : FiniteField Q) :
+    0 ≤ finiteRootLoad Q (Finset.Icc 1 k) p (finiteForwardLoad G x k Q A y) ω ∧
+    finiteRootLoad Q (Finset.Icc 1 k) p (finiteForwardLoad G x k Q A y) ω ≤ inverseCodeMass M L ^ k := by
+  have hb := finiteForwardLoad_bounds G hloc x z k M L hL Q A y hA hz hgood
+  exact ⟨finiteRootLoad_nonneg _ _ p _ (fun s _ => (hb s).1) _,
+    finiteRootLoad_le _ _ p _ _ (fun s _ => (hb s).2) _⟩
+
+ 
+
+theorem finiteForwardRootLoad_mean [Nonempty E] (G : BondGraph V E) (hloc : LocallyFinite G)
+    (x : V) (k M : ℕ) (Q : Finset E) (p : ℝ)
+    (A : Finset (SpliceState E)) (y : SpliceState E → ℕ → V)
+    (hA : A ⊆ finiteSpliceDomain Q (Finset.Icc 1 k))
+    (hy : ∀ s ∈ A, y s 0 ∈ cluster G s.1 x)
+    (hchain : ∀ s ∈ A, ∀ i ∈ Finset.Icc 1 k, Conn G (s.2 i) (y s i) (y s (i-1)))
+    (hcaps : ∀ s ∈ A, ∀ i ∈ Finset.Icc 1 k, (cluster G (s.2 i) (y s i)).Finite ∧
+      (cluster G (s.2 i) (y s i)).ncard ≤ M) :
+    (∑ ω : FiniteField Q, finiteFieldWeight Q p ω *
+      finiteRootLoad Q (Finset.Icc 1 k) p (finiteForwardLoad G x k Q A y) ω) =
+      ∑ s ∈ A, spliceTupleWeight Q (Finset.Icc 1 k) p s := by
+  rw [finiteRootLoad_mean]
+  exact finiteForwardLoad_mean G hloc x k M Q p A y hA hy hchain hcaps
+
+end FiniteRootLoad
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section FinitePairLoad
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+def pairSpliceOutput (s t : SpliceState E) : JointSpliceState E := (s.1,s.2,t.2)
+
+theorem pairSpliceOutput_mem (Q : Finset E) (J₁ J₂ : Finset ℕ)
+    (s t : SpliceState E) (hs : s ∈ finiteSpliceDomain Q J₁)
+    (ht : t ∈ finiteSpliceDomain Q J₂) :
+    pairSpliceOutput s t ∈ finiteJointDomain Q J₁ J₂ := by
+  have h₁ := (mem_finiteSpliceDomain_iff Q J₁ s).mp hs
+  have h₂ := (mem_finiteSpliceDomain_iff Q J₂ t).mp ht
+  exact mem_finiteJointDomain_of_support Q J₁ J₂ _ h₁.1 h₁.2.1 h₂.2.1 h₁.2.2 h₂.2.2
+
+ 
+theorem finiteInverseLoad_mul {I₁ I₂ : Type*} [Fintype I₁] [Fintype I₂]
+    (out₁ : I₁ → SpliceState E) (out₂ : I₂ → SpliceState E)
+    (m₁ : I₁ → ℝ) (m₂ : I₂ → ℝ) (t : JointSpliceState E) :
+    finiteInverseLoad out₁ m₁ (t.1,t.2.1) * finiteInverseLoad out₂ m₂ (t.1,t.2.2) =
+    ∑ a, ∑ b, if (out₁ a).1 = (out₂ b).1 ∧ pairSpliceOutput (out₁ a) (out₂ b) = t
+      then m₁ a * m₂ b else 0 := by
+  simp only [finiteInverseLoad,Finset.sum_mul,Finset.mul_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro a _
+  apply Finset.sum_congr rfl
+  intro b _
+  have he : (out₁ a = (t.1,t.2.1) ∧ out₂ b = (t.1,t.2.2)) ↔
+      ((out₁ a).1 = (out₂ b).1 ∧ pairSpliceOutput (out₁ a) (out₂ b) = t) := by
+    cases h₁ : out₁ a with | mk r₁ u₁ =>
+      cases h₂ : out₂ b with | mk r₂ u₂ =>
+        rcases t with ⟨r,u,v⟩
+        simpa [pairSpliceOutput,Prod.mk.injEq] using
+          (show (r₁ = r ∧ u₁ = u) ∧ (r₂ = r ∧ u₂ = v) ↔
+            r₁ = r₂ ∧ r₁ = r ∧ u₁ = u ∧ u₂ = v from by aesop)
+  simp only [← he]
+  split_ifs <;> simp_all
+
+ 
+
+theorem finiteRootLoad_pair_expansion {I₁ I₂ : Type*} [Fintype I₁] [Fintype I₂]
+    (Q : Finset E) (J₁ J₂ : Finset ℕ) (p : ℝ)
+    (out₁ : I₁ → SpliceState E) (out₂ : I₂ → SpliceState E)
+    (m₁ : I₁ → ℝ) (m₂ : I₂ → ℝ)
+    (hs₁ : ∀ a, out₁ a ∈ finiteSpliceDomain Q J₁)
+    (hs₂ : ∀ b, out₂ b ∈ finiteSpliceDomain Q J₂) :
+    (∑ ω : FiniteField Q, finiteFieldWeight Q p ω *
+      finiteRootLoad Q J₁ p (finiteInverseLoad out₁ m₁) ω *
+      finiteRootLoad Q J₂ p (finiteInverseLoad out₂ m₂) ω) =
+    ∑ a, ∑ b, if (out₁ a).1 = (out₂ b).1 then
+      jointTupleWeight Q J₁ J₂ p (pairSpliceOutput (out₁ a) (out₂ b)) * m₁ a * m₂ b else 0 := by
+  rw [finiteRootLoad_second_moment]
+  have hdom : (∑ s : FiniteJointData Q J₁ J₂,
+      jointTupleWeight Q J₁ J₂ p (encodeFiniteJoint Q J₁ J₂ s) *
+        finiteInverseLoad out₁ m₁ ((encodeFiniteJoint Q J₁ J₂ s).1,(encodeFiniteJoint Q J₁ J₂ s).2.1) *
+        finiteInverseLoad out₂ m₂ ((encodeFiniteJoint Q J₁ J₂ s).1,(encodeFiniteJoint Q J₁ J₂ s).2.2)) =
+      ∑ t ∈ finiteJointDomain Q J₁ J₂, jointTupleWeight Q J₁ J₂ p t *
+        finiteInverseLoad out₁ m₁ (t.1,t.2.1) * finiteInverseLoad out₂ m₂ (t.1,t.2.2) := by
+    rw [finiteJointDomain,Finset.sum_image (encodeFiniteJoint_injective Q J₁ J₂).injOn]
+  apply hdom.trans
+  simp only [mul_assoc,finiteInverseLoad_mul,Finset.mul_sum,mul_ite,mul_zero]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro a _
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro b _
+  by_cases he : (out₁ a).1 = (out₂ b).1
+  · simp only [he,true_and,ite_true]
+    rw [Finset.sum_ite_eq,ite_eq_left (pairSpliceOutput_mem Q J₁ J₂ _ _ (hs₁ a) (hs₂ b))]
+  · simp [he]
+
+theorem traceIndex_output_code_injective (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
+    (A : Finset (SpliceState E)) (y : SpliceState E → ℕ → V) :
+    Function.Injective (fun a : FiniteTraceIndex G x k Q A y =>
+      (traceIndexOutput G x k Q A y a,traceIndexCode G x k Q A y a)) := by
+  intro a b he
+  have hc : traceIndexCode G x k Q A y a = traceIndexCode G x k Q A y b := congrArg Prod.snd he
+  have ho : traceIndexOutput G x k Q A y a = traceIndexOutput G x k Q A y b := congrArg Prod.fst he
+  apply traceIndex_injective G x k Q A y
+  apply Prod.ext
+  · apply SpliceExecution.input_unique G x _ _ _ (traceIndexCode G x k Q A y a)
+      (traceIndex_valid G x k Q A y a).2
+    change SpliceExecution G x b.1.val (traceIndexCode G x k Q A y a) (traceIndexOutput G x k Q A y a)
+    rw [hc,ho]
+    exact (traceIndex_valid G x k Q A y b).2
+  · exact hc
+
+end FinitePairLoad
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section WeightedJointMass
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem jointRestored_injective {I : Type*}
+    (G : BondGraph V E) (x : V) (k : ℕ)
+    (η₁ η₂ : I → ℕ → Set E) (y₁ y₂ : (ℕ → Set E) → ℕ → V)
+    (choose₁ choose₂ : ℕ → Set E → E)
+    (ω₁ ω₂ ωf : I → Set E) (c₁ c₂ : I → List (ℕ × E))
+    (hir : Function.Injective (fun a =>
+      (jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a),
+        c₁ a,c₂ a)))
+    (h₁ : ∀ a, SpliceTrace G x (η₁ a) (y₁ (η₁ a)) k 0 (ω₁ a) (c₁ a) (ωf a))
+    (h₂ : ∀ a, SpliceTrace G x (η₂ a) (y₂ (η₂ a)) k 0 (ω₂ a) (c₂ a) (ωf a))
+    (hc₁ : ∀ a, c₁ a ≠ []) (hc₂ : ∀ a, c₂ a ≠ [])
+    (hsep : ∀ a, GraphSeparated G (cluster G (ωf a) x \ cluster G (ω₁ a) x)
+      (cluster G (ωf a) x \ cluster G (ω₂ a) x))
+    (hr₁ : ∀ a, FollowsContactRule G x (η₁ a) (y₁ (η₁ a)) choose₁ (c₁ a) (ω₁ a))
+    (hr₂ : ∀ a, FollowsContactRule G x (η₂ a) (y₂ (η₂ a)) choose₂ (c₂ a) (ω₂ a))
+    : Function.Injective (fun a => jointRestored G x (η₁ a) (η₂ a)
+      (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (c₂ a)) := by
+  let out (a : I) := jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a))
+    (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a)
+  let rest (a : I) := jointRestored G x (η₁ a) (η₂ a) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (c₂ a)
+  change Function.Injective rest
+  intro a b he
+  have hη₁ : η₁ a = η₁ b := congrArg (fun s : JointSpliceState E => s.2.1) he
+  have hη₂ : η₂ a = η₂ b := congrArg (fun s : JointSpliceState E => s.2.2) he
+  have gb₁ : SpliceTrace G x (η₁ a) (y₁ (η₁ a)) k 0 (ω₁ b) (c₁ b) (ωf b) := by
+    rw [hη₁]; exact h₁ b
+  have gb₂ : SpliceTrace G x (η₂ a) (y₂ (η₂ a)) k 0 (ω₂ b) (c₂ b) (ωf b) := by
+    rw [hη₂]; exact h₂ b
+  have grb₁ : FollowsContactRule G x (η₁ a) (y₁ (η₁ a)) choose₁ (c₁ b) (ω₁ b) := by
+    rw [hη₁]; exact hr₁ b
+  have grb₂ : FollowsContactRule G x (η₂ a) (y₂ (η₂ a)) choose₂ (c₂ b) (ω₂ b) := by
+    rw [hη₂]; exact hr₂ b
+  have hfield : exchangeOn (maskUnion (replayMasks G x (η₂ a) (y₂ (η₂ a)) (c₂ a) (ω₂ a))) (ω₁ a) (ω₂ a) =
+      exchangeOn (maskUnion (replayMasks G x (η₂ a) (y₂ (η₂ a)) (c₂ b) (ω₂ b))) (ω₁ b) (ω₂ b) := by
+    have hh := congrArg Prod.fst he
+    simpa only [rest,jointRestored,← hη₂] using hh
+  obtain ⟨hcode₁,hcode₂,hfieldf,haux₁,haux₂⟩ := joint_restore_seeded_unique G x
+    (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) choose₁ choose₂ k 0 0
+    (ω₁ a) (ω₂ a) (ωf a) (ω₁ b) (ω₂ b) (ωf b)
+    (c₁ a) (c₂ a) (c₁ b) (c₂ b) (h₁ a) (h₂ a) gb₁ gb₂
+    (hc₁ a) (hc₂ a) (hc₁ b) (hc₂ b) (hsep a) (hsep b)
+    (hr₁ a) (hr₂ a) grb₁ grb₂ hfield
+  have hout : out a = out b := by
+    apply Prod.ext hfieldf
+    apply Prod.ext
+    · simpa only [out,jointOutput,hη₁] using haux₁
+    · simpa only [out,jointOutput,hη₂] using haux₂
+  exact hir (Prod.ext hout (Prod.ext hcode₁ hcode₂))
+
+ 
+
+theorem finite_joint_weighted_mass_le {I : Type*} [Fintype I]
+    (G : BondGraph V E) (x : V) (k : ℕ) (p : unitInterval) (Q : Finset E)
+    (η₁ η₂ : I → ℕ → Set E) (y₁ y₂ : (ℕ → Set E) → ℕ → V)
+    (choose₁ choose₂ : ℕ → Set E → E)
+    (ω₁ ω₂ ωf : I → Set E) (c₁ c₂ : I → List (ℕ × E))
+    (hir : Function.Injective (fun a =>
+      (jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a),
+        c₁ a,c₂ a)))
+    (h₁ : ∀ a, SpliceTrace G x (η₁ a) (y₁ (η₁ a)) k 0 (ω₁ a) (c₁ a) (ωf a))
+    (h₂ : ∀ a, SpliceTrace G x (η₂ a) (y₂ (η₂ a)) k 0 (ω₂ a) (c₂ a) (ωf a))
+    (hc₁ : ∀ a, c₁ a ≠ []) (hc₂ : ∀ a, c₂ a ≠ [])
+    (hsep : ∀ a, GraphSeparated G (cluster G (ωf a) x \ cluster G (ω₁ a) x)
+      (cluster G (ωf a) x \ cluster G (ω₂ a) x))
+    (hr₁ : ∀ a, FollowsContactRule G x (η₁ a) (y₁ (η₁ a)) choose₁ (c₁ a) (ω₁ a))
+    (hr₂ : ∀ a, FollowsContactRule G x (η₂ a) (y₂ (η₂ a)) choose₂ (c₂ a) (ω₂ a))
+    (m₁ m₂ : (ℕ → Set E) → ℝ) (hm₁ : ∀ η, 0 ≤ m₁ η) (hm₂ : ∀ η, 0 ≤ m₂ η)
+    (B : Finset (JointSpliceState E))
+    (hB : ∀ a, jointRestored G x (η₁ a) (η₂ a) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (c₂ a) ∈ B) :
+    (∑ a, jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p
+      (jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a)) * m₁ (η₁ a) * m₂ (η₂ a)) ≤
+      ∑ s ∈ B, jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p s * m₁ s.2.1 * m₂ s.2.2 := by
+  let out (a : I) := jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a))
+    (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a)
+  let rest (a : I) := jointRestored G x (η₁ a) (η₂ a) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (c₂ a)
+  have hinj : Function.Injective rest := jointRestored_injective G x k η₁ η₂ y₁ y₂ choose₁ choose₂
+    ω₁ ω₂ ωf c₁ c₂ hir h₁ h₂ hc₁ hc₂ hsep hr₁ hr₂
+  have hw : ∀ a, jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p (out a) =
+      jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p (rest a) := by
+    intro a
+    apply (h₁ a).joint_weight G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) k 0 0
+      (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a) (h₂ a) (hsep a) Q (Finset.Icc 1 k) (Finset.Icc 1 k) p
+    · intro q hq
+      have hr := (h₁ a).seam_data G x (η₁ a) (y₁ (η₁ a)) k 0 (ω₁ a) (ωf a) (c₁ a) q hq
+      exact Finset.mem_Icc.mpr ⟨hr.1,hr.2.1⟩
+    · intro q hq
+      have hr := (h₂ a).seam_data G x (η₂ a) (y₂ (η₂ a)) k 0 (ω₂ a) (ωf a) (c₂ a) q hq
+      exact Finset.mem_Icc.mpr ⟨hr.1,hr.2.1⟩
+  change (∑ a, jointTupleWeight Q _ _ p (out a) * m₁ (η₁ a) * m₂ (η₂ a)) ≤ _
+  simp_rw [hw]
+  change (∑ a, jointTupleWeight Q _ _ p (rest a) * m₁ (rest a).2.1 * m₂ (rest a).2.2) ≤ _
+  rw [← Finset.sum_image (f := fun s : JointSpliceState E =>
+    jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p s * m₁ s.2.1 * m₂ s.2.2) hinj.injOn]
+  apply Finset.sum_le_sum_of_subset_of_nonneg
+  · intro s hs
+    obtain ⟨a,_,rfl⟩ := Finset.mem_image.mp hs
+    exact hB a
+  · intro s _ _
+    exact mul_nonneg (mul_nonneg (jointTupleWeight_nonneg Q _ _ p s) (hm₁ _)) (hm₂ _)
+
+end WeightedJointMass
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section FiniteFamilyIntegral
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+noncomputable def finiteFamilyIntegral (Q : Finset E) (J : Finset ℕ) (p : ℝ)
+    (f : (ℕ → Set E) → ℝ) : ℝ :=
+  ∑ η : ↥J → FiniteField Q, finiteFamilyWeight Q J p η * f (finiteFieldFamily Q J η)
+
+theorem finiteFamilyIntegral_nonneg (Q : Finset E) (J : Finset ℕ) (p : unitInterval)
+    (f : (ℕ → Set E) → ℝ) (hf : ∀ η, 0 ≤ f η) : 0 ≤ finiteFamilyIntegral Q J p f := by
+  exact Finset.sum_nonneg fun η _ => mul_nonneg (finiteFamilyWeight_nonneg Q J p η) (hf _)
+
+theorem finiteFamilyIntegral_mono (Q : Finset E) (J : Finset ℕ) (p : unitInterval)
+    (f g : (ℕ → Set E) → ℝ) (hfg : ∀ η, f η ≤ g η) :
+    finiteFamilyIntegral Q J p f ≤ finiteFamilyIntegral Q J p g := by
+  exact Finset.sum_le_sum fun η _ => mul_le_mul_of_nonneg_left (hfg _) (finiteFamilyWeight_nonneg Q J p η)
+
+theorem finiteFamilyIntegral_const (Q : Finset E) (J : Finset ℕ) (p c : ℝ) :
+    finiteFamilyIntegral Q J p (fun _ => c) = c := by
+  rw [finiteFamilyIntegral,← Finset.sum_mul,sum_finiteFamilyWeight,one_mul]
+
+theorem finiteFamilyIntegral_sum {I : Type*} (Q : Finset E) (J : Finset ℕ) (p : ℝ)
+    (S : Finset I) (f : I → (ℕ → Set E) → ℝ) :
+    finiteFamilyIntegral Q J p (fun η => ∑ i ∈ S, f i η) =
+      ∑ i ∈ S, finiteFamilyIntegral Q J p (f i) := by
+  simp only [finiteFamilyIntegral,Finset.mul_sum]
+  exact Finset.sum_comm
+
+ 
+theorem finiteJointDomain_integral_factor (Q : Finset E) (J₁ J₂ : Finset ℕ) (p : ℝ)
+    (f₁ f₂ : Set E → (ℕ → Set E) → ℝ) :
+    (∑ s ∈ finiteJointDomain Q J₁ J₂, jointTupleWeight Q J₁ J₂ p s * f₁ s.1 s.2.1 * f₂ s.1 s.2.2) =
+    ∑ ω : FiniteField Q, finiteFieldWeight Q p ω *
+      finiteFamilyIntegral Q J₁ p (f₁ (finiteFieldSet Q ω)) *
+      finiteFamilyIntegral Q J₂ p (f₂ (finiteFieldSet Q ω)) := by
+  rw [finiteJointDomain,Finset.sum_image (encodeFiniteJoint_injective Q J₁ J₂).injOn]
+  simp only [jointTupleWeight_encode]
+  rw [Fintype.sum_prod_type]
+  apply Finset.sum_congr rfl
+  intro ω _
+  rw [Fintype.sum_prod_type]
+  simp only [finiteFamilyIntegral,Finset.mul_sum,Finset.sum_mul,encodeFiniteJoint]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro η₂ _
+  apply Finset.sum_congr rfl
+  intro η₁ _
+  ring
+
+theorem finiteJointDomain_weighted_event_factor (Q : Finset E) (J₁ J₂ : Finset ℕ) (p : ℝ)
+    (P : Set E → Prop) (H₁ H₂ : Set E → (ℕ → Set E) → Prop)
+    (m₁ m₂ : (ℕ → Set E) → ℝ) :
+    (∑ s ∈ (finiteJointDomain Q J₁ J₂).filter (fun s => P s.1 ∧ H₁ s.1 s.2.1 ∧ H₂ s.1 s.2.2),
+      jointTupleWeight Q J₁ J₂ p s * m₁ s.2.1 * m₂ s.2.2) =
+    ∑ ω : FiniteField Q, finiteFieldWeight Q p ω *
+      (if P (finiteFieldSet Q ω) then
+        finiteFamilyIntegral Q J₁ p (fun η => if H₁ (finiteFieldSet Q ω) η then m₁ η else 0) *
+        finiteFamilyIntegral Q J₂ p (fun η => if H₂ (finiteFieldSet Q ω) η then m₂ η else 0) else 0) := by
+  rw [Finset.sum_filter]
+  have he : (∑ s ∈ finiteJointDomain Q J₁ J₂,
+      if P s.1 ∧ H₁ s.1 s.2.1 ∧ H₂ s.1 s.2.2 then
+        jointTupleWeight Q J₁ J₂ p s * m₁ s.2.1 * m₂ s.2.2 else 0) =
+    ∑ s ∈ finiteJointDomain Q J₁ J₂, jointTupleWeight Q J₁ J₂ p s *
+      (if P s.1 ∧ H₁ s.1 s.2.1 then m₁ s.2.1 else 0) *
+      (if H₂ s.1 s.2.2 then m₂ s.2.2 else 0) := by
+    apply Finset.sum_congr rfl
+    intro s _
+    split_ifs <;> simp_all
+  rw [he,finiteJointDomain_integral_factor Q J₁ J₂ p
+    (fun ω η => if P ω ∧ H₁ ω η then m₁ η else 0)
+    (fun ω η => if H₂ ω η then m₂ η else 0)]
+  apply Finset.sum_congr rfl
+  intro ω _
+  by_cases hP : P (finiteFieldSet Q ω)
+  · simp only [hP,true_and,ite_true,mul_assoc]
+  · simp [hP,finiteFamilyIntegral]
+
+end FiniteFamilyIntegral
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section WeightedSeparatedSecondMoment
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem finite_separated_weighted_second_moment_le {I : Type*} [Fintype I]
+    (G : BondGraph V E) (x : V) (k n M : ℕ) (p : unitInterval) (Q : Finset E)
+    (η₁ η₂ : I → ℕ → Set E) (y₁ y₂ : (ℕ → Set E) → ℕ → V)
+    (success₁ success₂ : (ℕ → Set E) → Prop)
+    (m₁ m₂ : (ℕ → Set E) → ℝ) (hm₁ : ∀ η, 0 ≤ m₁ η) (hm₂ : ∀ η, 0 ≤ m₂ η)
+    (choose₁ choose₂ : ℕ → Set E → E)
+    (ω₁ ω₂ ωf : I → Set E) (c₁ c₂ : I → List (ℕ × E))
+    (hir : Function.Injective (fun a =>
+      (jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a),
+        c₁ a,c₂ a)))
+    (h₁ : ∀ a, SpliceTrace G x (η₁ a) (y₁ (η₁ a)) k 0 (ω₁ a) (c₁ a) (ωf a))
+    (h₂ : ∀ a, SpliceTrace G x (η₂ a) (y₂ (η₂ a)) k 0 (ω₂ a) (c₂ a) (ωf a))
+    (hc₁ : ∀ a, c₁ a ≠ []) (hc₂ : ∀ a, c₂ a ≠ [])
+    (hsep : ∀ a, GraphSeparated G (cluster G (ωf a) x \ cluster G (ω₁ a) x)
+      (cluster G (ωf a) x \ cluster G (ω₂ a) x))
+    (hr₁ : ∀ a, FollowsContactRule G x (η₁ a) (y₁ (η₁ a)) choose₁ (c₁ a) (ω₁ a))
+    (hr₂ : ∀ a, FollowsContactRule G x (η₂ a) (y₂ (η₂ a)) choose₂ (c₂ a) (ω₂ a))
+    (hbin : ∀ a, (cluster G (ω₁ a) x).Finite ∧ n ≤ (cluster G (ω₁ a) x).ncard ∧
+      (cluster G (ω₁ a) x).ncard < 2*n)
+    (hcap₂ : ∀ a i, 1 ≤ i → i ≤ k → (cluster G (η₂ a i) (y₂ (η₂ a) i)).Finite ∧
+      (cluster G (η₂ a i) (y₂ (η₂ a) i)).ncard ≤ M)
+    (hsc₁ : ∀ a, success₁ (η₁ a)) (hsc₂ : ∀ a, success₂ (η₂ a))
+    (hsω₁ : ∀ a, ω₁ a ⊆ (Q : Set E)) (hsω₂ : ∀ a, ω₂ a ⊆ (Q : Set E))
+    (hsη₁ : ∀ a i, η₁ a i ⊆ (Q : Set E)) (hsη₂ : ∀ a i, η₂ a i ⊆ (Q : Set E))
+    (hz₁ : ∀ a i, i ∉ Finset.Icc 1 k → η₁ a i = ∅)
+    (hz₂ : ∀ a i, i ∉ Finset.Icc 1 k → η₂ a i = ∅) :
+    (∑ a, jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p
+      (jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a)) * m₁ (η₁ a) * m₂ (η₂ a)) ≤
+    ∑ ω : FiniteField Q, finiteFieldWeight Q p ω *
+      (if (cluster G (finiteFieldSet Q ω) x).Finite ∧
+        n-k*M ≤ (cluster G (finiteFieldSet Q ω) x).ncard ∧
+        (cluster G (finiteFieldSet Q ω) x).ncard < 2*n then
+        finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η => if success₁ η ∧
+          (cluster G (finiteFieldSet Q ω) x ∩ trajectoryUnion G η (y₁ η) k).Nonempty then m₁ η else 0) *
+        finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η => if success₂ η ∧
+          (cluster G (finiteFieldSet Q ω) x ∩ trajectoryUnion G η (y₂ η) k).Nonempty then m₂ η else 0) else 0) := by
+  let J := Finset.Icc 1 k
+  let P (ω : Set E) := (cluster G ω x).Finite ∧ n-k*M ≤ (cluster G ω x).ncard ∧
+    (cluster G ω x).ncard < 2*n
+  let H₁ (ω : Set E) (η : ℕ → Set E) := success₁ η ∧
+    (cluster G ω x ∩ trajectoryUnion G η (y₁ η) k).Nonempty
+  let H₂ (ω : Set E) (η : ℕ → Set E) := success₂ η ∧
+    (cluster G ω x ∩ trajectoryUnion G η (y₂ η) k).Nonempty
+  let B := (finiteJointDomain Q J J).filter fun s => P s.1 ∧ H₁ s.1 s.2.1 ∧ H₂ s.1 s.2.2
+  have hB (a : I) : jointRestored G x (η₁ a) (η₂ a) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (c₂ a) ∈ B := by
+    apply Finset.mem_filter.mpr
+    refine ⟨mem_finiteJointDomain_of_support Q J J _
+      (exchangeOn_subset _ _ _ _ (hsω₁ a) (hsω₂ a)) (hsη₁ a) (hsη₂ a) (hz₁ a) (hz₂ a),?_⟩
+    have hroot := (h₁ a).joint_restored_size G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a))
+      k 0 0 n M (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a) (h₂ a) (hc₁ a) (hc₂ a) (hsep a)
+      (hbin a).1 (hbin a).2.1 (hbin a).2.2 (hcap₂ a)
+    have hhits := (h₁ a).joint_trajectory_hits G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a))
+      k 0 0 (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a) (h₂ a) (hc₁ a) (hc₂ a) (hsep a)
+    exact ⟨hroot,⟨hsc₁ a,hhits.1⟩,hsc₂ a,hhits.2⟩
+  have h := finite_joint_weighted_mass_le G x k p Q η₁ η₂ y₁ y₂ choose₁ choose₂ ω₁ ω₂ ωf c₁ c₂
+    hir h₁ h₂ hc₁ hc₂ hsep hr₁ hr₂ m₁ m₂ hm₁ hm₂ B hB
+  apply h.trans_eq
+  convert finiteJointDomain_weighted_event_factor Q J J p P H₁ H₂ m₁ m₂ using 1 <;>
+    simp only [B, J, P, H₁, H₂]
+  apply Finset.sum_congr rfl
+  intro ω _
+  apply congrArg (fun z : ℝ => finiteFieldWeight Q p ω * z)
+  split_ifs
+  · apply congrArg₂ (· * ·)
+    · apply congrArg (finiteFamilyIntegral Q (Finset.Icc 1 k) p)
+      funext η
+      split_ifs <;> rfl
+    · apply congrArg (finiteFamilyIntegral Q (Finset.Icc 1 k) p)
+      funext η
+      split_ifs <;> rfl
+  · rfl
+
+end WeightedSeparatedSecondMoment
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section WeightedContactSecondMoment
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem finite_separated_weighted_contact_second_moment_le {I : Type*} [Fintype I] [Nonempty E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) (k n M : ℕ) (p : unitInterval) (Q : Finset E)
+    (η₁ η₂ : I → ℕ → Set E) (y₁ y₂ : (ℕ → Set E) → ℕ → V)
+    (success₁ success₂ : (ℕ → Set E) → Prop)
+    (m₁ m₂ : (ℕ → Set E) → ℝ) (hm₁ : ∀ η, 0 ≤ m₁ η) (hm₂ : ∀ η, 0 ≤ m₂ η)
+    (ω₁ ω₂ ωf : I → Set E) (c₁ c₂ : I → List (ℕ × E))
+    (hir : Function.Injective (fun a =>
+      (jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a),
+        c₁ a,c₂ a)))
+    (h₁ : ∀ a, SpliceTrace G x (η₁ a) (y₁ (η₁ a)) k 0 (ω₁ a) (c₁ a) (ωf a))
+    (h₂ : ∀ a, SpliceTrace G x (η₂ a) (y₂ (η₂ a)) k 0 (ω₂ a) (c₂ a) (ωf a))
+    (hc₁ : ∀ a, c₁ a ≠ []) (hc₂ : ∀ a, c₂ a ≠ [])
+    (hsep : ∀ a, GraphSeparated G (cluster G (ωf a) x \ cluster G (ω₁ a) x)
+      (cluster G (ωf a) x \ cluster G (ω₂ a) x))
+    (hbin : ∀ a, (cluster G (ω₁ a) x).Finite ∧ n ≤ (cluster G (ω₁ a) x).ncard ∧
+      (cluster G (ω₁ a) x).ncard < 2*n)
+    (hcap₁ : ∀ a i, 1 ≤ i → i ≤ k → (cluster G (η₁ a i) (y₁ (η₁ a) i)).Finite)
+    (hcap₂ : ∀ a i, 1 ≤ i → i ≤ k → (cluster G (η₂ a i) (y₂ (η₂ a) i)).Finite ∧
+      (cluster G (η₂ a i) (y₂ (η₂ a) i)).ncard ≤ M)
+    (hsc₁ : ∀ a, success₁ (η₁ a)) (hsc₂ : ∀ a, success₂ (η₂ a))
+    (hsω₁ : ∀ a, ω₁ a ⊆ (Q : Set E)) (hsω₂ : ∀ a, ω₂ a ⊆ (Q : Set E))
+    (hsη₁ : ∀ a i, η₁ a i ⊆ (Q : Set E)) (hsη₂ : ∀ a i, η₂ a i ⊆ (Q : Set E))
+    (hz₁ : ∀ a i, i ∉ Finset.Icc 1 k → η₁ a i = ∅)
+    (hz₂ : ∀ a i, i ∉ Finset.Icc 1 k → η₂ a i = ∅) :
+    (∑ a, jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p
+      (jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a)) * m₁ (η₁ a) * m₂ (η₂ a) *
+      (∏ i : Fin k, (traceContactCount G (η₁ a) (y₁ (η₁ a)) (ωf a) x (c₁ a) (i.val+1):ℝ)⁻¹) *
+      (∏ i : Fin k, (traceContactCount G (η₂ a) (y₂ (η₂ a)) (ωf a) x (c₂ a) (i.val+1):ℝ)⁻¹)) ≤
+    ∑ ω : FiniteField Q, finiteFieldWeight Q p ω *
+      (if (cluster G (finiteFieldSet Q ω) x).Finite ∧
+        n-k*M ≤ (cluster G (finiteFieldSet Q ω) x).ncard ∧
+        (cluster G (finiteFieldSet Q ω) x).ncard < 2*n then
+        finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η => if success₁ η ∧
+          (cluster G (finiteFieldSet Q ω) x ∩ trajectoryUnion G η (y₁ η) k).Nonempty then m₁ η else 0) *
+        finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η => if success₂ η ∧
+          (cluster G (finiteFieldSet Q ω) x ∩ trajectoryUnion G η (y₂ η) k).Nonempty then m₂ η else 0) else 0) := by
+  let S := Fin k → unitInterval
+  let μ : Measure (S × S) := (volume : Measure S).prod volume
+  let P₁ (a : I) : Set S := {u | FollowsContactRule G x (η₁ a) (y₁ (η₁ a))
+    (chooseContact (finiteContactSeeds k u)) (c₁ a) (ω₁ a)}
+  let P₂ (a : I) : Set S := {u | FollowsContactRule G x (η₂ a) (y₂ (η₂ a))
+    (chooseContact (finiteContactSeeds k u)) (c₂ a) (ω₂ a)}
+  let w (a : I) := jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p
+    (jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a)) * m₁ (η₁ a) * m₂ (η₂ a)
+  have hmeas₁ (a : I) : MeasurableSet (P₁ a) :=
+    (h₁ a).measurableSet_seed_event G x (η₁ a) (y₁ (η₁ a)) k 0 (ω₁ a) (ωf a) (c₁ a)
+  have hmeas₂ (a : I) : MeasurableSet (P₂ a) :=
+    (h₂ a).measurableSet_seed_event G x (η₂ a) (y₂ (η₂ a)) k 0 (ω₂ a) (ωf a) (c₂ a)
+  have hp (a : I) : μ.real (P₁ a ×ˢ P₂ a) =
+      (∏ i : Fin k, (traceContactCount G (η₁ a) (y₁ (η₁ a)) (ωf a) x (c₁ a) (i.val+1):ℝ)⁻¹) *
+      (∏ i : Fin k, (traceContactCount G (η₂ a) (y₂ (η₂ a)) (ωf a) x (c₂ a) (i.val+1):ℝ)⁻¹) := by
+    rw [measureReal_def,Measure.prod_prod,ENNReal.toReal_mul,
+      (h₁ a).seed_probability G hloc x (η₁ a) (y₁ (η₁ a)) k 0 (ω₁ a) (ωf a) (c₁ a) (hcap₁ a),
+      (h₂ a).seed_probability G hloc x (η₂ a) (y₂ (η₂ a)) k 0 (ω₂ a) (ωf a) (c₂ a) (fun i hi hik => (hcap₂ a i hi hik).1)]
+  conv_lhs => arg 2; ext a; rw [mul_assoc,← hp a]
+  apply finite_event_mass_le μ (fun a => P₁ a ×ˢ P₂ a) (fun a => (hmeas₁ a).prod (hmeas₂ a)) w
+  intro u
+  let J := {a : I // u ∈ P₁ a ×ˢ P₂ a}
+  have hb := finite_separated_weighted_second_moment_le G x k n M p Q
+    (fun a : J => η₁ a.val) (fun a : J => η₂ a.val) y₁ y₂ success₁ success₂ m₁ m₂ hm₁ hm₂
+    (chooseContact (finiteContactSeeds k u.1)) (chooseContact (finiteContactSeeds k u.2))
+    (fun a : J => ω₁ a.val) (fun a : J => ω₂ a.val) (fun a : J => ωf a.val)
+    (fun a : J => c₁ a.val) (fun a : J => c₂ a.val)
+    (fun _ _ he => Subtype.ext (hir he)) (fun a => h₁ a.val) (fun a => h₂ a.val)
+    (fun a => hc₁ a.val) (fun a => hc₂ a.val) (fun a => hsep a.val)
+    (fun a => a.property.1) (fun a => a.property.2)
+    (fun a => hbin a.val) (fun a => hcap₂ a.val) (fun a => hsc₁ a.val) (fun a => hsc₂ a.val)
+    (fun a => hsω₁ a.val) (fun a => hsω₂ a.val) (fun a => hsη₁ a.val) (fun a => hsη₂ a.val)
+    (fun a => hz₁ a.val) (fun a => hz₂ a.val)
+  convert hb using 1
+  have he := Finset.sum_subtype (p := fun a => u ∈ P₁ a ×ˢ P₂ a) (F := inferInstance) (Finset.univ.filter (fun a => u ∈ P₁ a ×ˢ P₂ a))
+    (by intro a; simp) w
+  simpa [Finset.sum_filter,w] using he
+
+end WeightedContactSecondMoment
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Splicing/ReferenceLaws.lean b/lean/OAI/Probability/UnimodularPercolation/Splicing/ReferenceLaws.lean
index b714cd9a0606301b8d8089e2aadc88c7ca316518..4f24a3671f96dee3c1f403aaad8fdf0548a99a22 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Splicing/ReferenceLaws.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Splicing/ReferenceLaws.lean
@@ -1,1200 +1,1200 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Splicing.Replay
-
-namespace OAI
-
-/-! Reference Laws for bond percolation. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section FiniteJointMass
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-def jointOutput (G : BondGraph V E) (x : V) (η₁ η₂ : ℕ → Set E)
-    (y₁ y₂ : ℕ → V) (ω₁ ω₂ ωf : Set E) (c₁ c₂ : List (ℕ × E)) : JointSpliceState E :=
-  (ωf,(runMasks (replayMasks G x η₁ y₁ c₁ ω₁) (ω₁,η₁)).2,
-    (runMasks (replayMasks G x η₂ y₂ c₂ ω₂) (ω₂,η₂)).2)
-
-def jointRestored (G : BondGraph V E) (x : V) (η₁ η₂ : ℕ → Set E)
-    (y₂ : ℕ → V) (ω₁ ω₂ : Set E) (c₂ : List (ℕ × E)) : JointSpliceState E :=
-  (exchangeOn (maskUnion (replayMasks G x η₂ y₂ c₂ ω₂)) ω₁ ω₂,η₁,η₂)
-
- 
-
-theorem finite_joint_mass_le {I : Type*} [Fintype I]
-    (G : BondGraph V E) (x : V) (k : ℕ) (p : unitInterval) (Q : Finset E)
-    (η₁ η₂ : I → ℕ → Set E) (y₁ y₂ : (ℕ → Set E) → ℕ → V)
-    (choose₁ choose₂ : ℕ → Set E → E)
-    (ω₁ ω₂ ωf : I → Set E) (c₁ c₂ : I → List (ℕ × E))
-    (hir : Function.Injective (fun a =>
-      (jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a),
-        c₁ a,c₂ a)))
-    (h₁ : ∀ a, SpliceTrace G x (η₁ a) (y₁ (η₁ a)) k 0 (ω₁ a) (c₁ a) (ωf a))
-    (h₂ : ∀ a, SpliceTrace G x (η₂ a) (y₂ (η₂ a)) k 0 (ω₂ a) (c₂ a) (ωf a))
-    (hc₁ : ∀ a, c₁ a ≠ []) (hc₂ : ∀ a, c₂ a ≠ [])
-    (hsep : ∀ a, GraphSeparated G (cluster G (ωf a) x \ cluster G (ω₁ a) x)
-      (cluster G (ωf a) x \ cluster G (ω₂ a) x))
-    (hr₁ : ∀ a, FollowsContactRule G x (η₁ a) (y₁ (η₁ a)) choose₁ (c₁ a) (ω₁ a))
-    (hr₂ : ∀ a, FollowsContactRule G x (η₂ a) (y₂ (η₂ a)) choose₂ (c₂ a) (ω₂ a))
-    (B : Finset (JointSpliceState E))
-    (hB : ∀ a, jointRestored G x (η₁ a) (η₂ a) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (c₂ a) ∈ B) :
-    (∑ a, jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p
-      (jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a))) ≤
-      ∑ s ∈ B, jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p s := by
-  let out (a : I) := jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a))
-    (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a)
-  let rest (a : I) := jointRestored G x (η₁ a) (η₂ a) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (c₂ a)
-  have hinj : Function.Injective rest := by
-    intro a b he
-    have hη₁ : η₁ a = η₁ b := congrArg (fun s : JointSpliceState E => s.2.1) he
-    have hη₂ : η₂ a = η₂ b := congrArg (fun s : JointSpliceState E => s.2.2) he
-    have gb₁ : SpliceTrace G x (η₁ a) (y₁ (η₁ a)) k 0 (ω₁ b) (c₁ b) (ωf b) := by
-      rw [hη₁]; exact h₁ b
-    have gb₂ : SpliceTrace G x (η₂ a) (y₂ (η₂ a)) k 0 (ω₂ b) (c₂ b) (ωf b) := by
-      rw [hη₂]; exact h₂ b
-    have grb₁ : FollowsContactRule G x (η₁ a) (y₁ (η₁ a)) choose₁ (c₁ b) (ω₁ b) := by
-      rw [hη₁]; exact hr₁ b
-    have grb₂ : FollowsContactRule G x (η₂ a) (y₂ (η₂ a)) choose₂ (c₂ b) (ω₂ b) := by
-      rw [hη₂]; exact hr₂ b
-    have hfield : exchangeOn (maskUnion (replayMasks G x (η₂ a) (y₂ (η₂ a)) (c₂ a) (ω₂ a))) (ω₁ a) (ω₂ a) =
-        exchangeOn (maskUnion (replayMasks G x (η₂ a) (y₂ (η₂ a)) (c₂ b) (ω₂ b))) (ω₁ b) (ω₂ b) := by
-      have hh := congrArg Prod.fst he
-      simpa only [rest,jointRestored,← hη₂] using hh
-    obtain ⟨hcode₁,hcode₂,hfieldf,haux₁,haux₂⟩ := joint_restore_seeded_unique G x
-      (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) choose₁ choose₂ k 0 0
-      (ω₁ a) (ω₂ a) (ωf a) (ω₁ b) (ω₂ b) (ωf b)
-      (c₁ a) (c₂ a) (c₁ b) (c₂ b) (h₁ a) (h₂ a) gb₁ gb₂
-      (hc₁ a) (hc₂ a) (hc₁ b) (hc₂ b) (hsep a) (hsep b)
-      (hr₁ a) (hr₂ a) grb₁ grb₂ hfield
-    have hout : out a = out b := by
-      apply Prod.ext hfieldf
-      apply Prod.ext
-      · simpa only [out,jointOutput,hη₁] using haux₁
-      · simpa only [out,jointOutput,hη₂] using haux₂
-    exact hir (Prod.ext hout (Prod.ext hcode₁ hcode₂))
-  have hw : ∀ a, jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p (out a) =
-      jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p (rest a) := by
-    intro a
-    apply (h₁ a).joint_weight G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) k 0 0
-      (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a) (h₂ a) (hsep a) Q (Finset.Icc 1 k) (Finset.Icc 1 k) p
-    · intro q hq
-      have hr := (h₁ a).seam_data G x (η₁ a) (y₁ (η₁ a)) k 0 (ω₁ a) (ωf a) (c₁ a) q hq
-      exact Finset.mem_Icc.mpr ⟨hr.1,hr.2.1⟩
-    · intro q hq
-      have hr := (h₂ a).seam_data G x (η₂ a) (y₂ (η₂ a)) k 0 (ω₂ a) (ωf a) (c₂ a) q hq
-      exact Finset.mem_Icc.mpr ⟨hr.1,hr.2.1⟩
-  change (∑ a, jointTupleWeight Q _ _ p (out a)) ≤ _
-  simp_rw [hw]
-  rw [← Finset.sum_image hinj.injOn]
-  apply Finset.sum_le_sum_of_subset_of_nonneg
-  · intro s hs
-    obtain ⟨a,_,rfl⟩ := Finset.mem_image.mp hs
-    exact hB a
-  · intro s _ _
-    exact jointTupleWeight_nonneg Q _ _ p s
-
-end FiniteJointMass
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section TrajectorySupport
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-def trajectoryUnion (G : BondGraph V E) (η : ℕ → Set E) (y : ℕ → V) : ℕ → Set V
-  | 0 => ∅
-  | k+1 => trajectoryUnion G η y k ∪ cluster G (η (k+1)) (y (k+1))
-
-theorem mem_trajectoryUnion (G : BondGraph V E) (η : ℕ → Set E) (y : ℕ → V)
-    (k : ℕ) (v : V) : v ∈ trajectoryUnion G η y k ↔
-      ∃ i, 1 ≤ i ∧ i ≤ k ∧ v ∈ cluster G (η i) (y i) := by
-  induction k with
-  | zero => simp [trajectoryUnion]
-  | succ k ih =>
-    simp only [trajectoryUnion,Set.mem_union,ih]
-    constructor
-    · rintro (⟨i,hi,hik,hv⟩ | hv)
-      · exact ⟨i,hi,by omega,hv⟩
-      · exact ⟨k+1,by omega,by omega,hv⟩
-    · rintro ⟨i,hi,hik,hv⟩
-      by_cases he : i = k+1
-      · exact Or.inr (he ▸ hv)
-      · exact Or.inl ⟨i,hi,by omega,hv⟩
-
-theorem trajectoryUnion_finite_cap (G : BondGraph V E) (η : ℕ → Set E) (y : ℕ → V)
-    (k M : ℕ) (hcaps : ∀ i, 1 ≤ i → i ≤ k →
-      (cluster G (η i) (y i)).Finite ∧ (cluster G (η i) (y i)).ncard ≤ M) :
-    (trajectoryUnion G η y k).Finite ∧ (trajectoryUnion G η y k).ncard ≤ k*M := by
-  induction k with
-  | zero => simp [trajectoryUnion]
-  | succ k ih =>
-    obtain ⟨hf,hc⟩ := ih (fun i hi hik => hcaps i hi (by omega))
-    obtain ⟨hf',hc'⟩ := hcaps (k+1) (by omega) (by omega)
-    refine ⟨hf.union hf',?_⟩
-    calc
-      _ ≤ (trajectoryUnion G η y k).ncard + (cluster G (η (k+1)) (y (k+1))).ncard := Set.ncard_union_le ..
-      _ ≤ k*M+M := Nat.add_le_add hc hc'
-      _ = (k+1)*M := by ring
-
- 
-
-theorem SpliceTrace.added_subset_trajectory (G : BondGraph V E) (x : V)
-    (η : ℕ → Set E) (y : ℕ → V) (k j : ℕ) (ω ωf : Set E) (c : List (ℕ × E))
-    (h : SpliceTrace G x η y k j ω c ωf) :
-    cluster G ωf x \ cluster G ω x ⊆ trajectoryUnion G η y k := by
-  induction h with
-  | stop => exact fun _ hv => (hv.2 hv.1).elim
-  | @step j i ω e a b rest ωf hn hji hik hmax hy he hab ha hb ht ih =>
-    let S := cluster G ω x
-    let B := cluster G (η i ∩ internalEdges G Sᶜ) (y i)
-    let ω' := spliceField G S B e ω (η i)
-    have hC : cluster G ω' x = S ∪ B := splice_cluster G ω (η i) x (y i) hy e a b hab he ha hb
-    intro v hv
-    by_cases hv' : v ∈ cluster G ω' x
-    · rw [hC] at hv'
-      have hvB := hv'.resolve_left hv.2
-      exact (mem_trajectoryUnion G η y k v).mpr
-        ⟨i,by omega,hik,conn_mono G Set.inter_subset_left hvB⟩
-    · exact ih ⟨hv.1,hv'⟩
-
- 
-
-theorem SpliceTrace.trajectory_hits (G : BondGraph V E) (x : V)
-    (η : ℕ → Set E) (y : ℕ → V) (k j : ℕ) (ω ωf : Set E) (c : List (ℕ × E))
-    (h : SpliceTrace G x η y k j ω c ωf) (hc : c ≠ []) :
-    (cluster G ω x ∩ trajectoryUnion G η y k).Nonempty := by
-  cases h with
-  | stop => exact (hc rfl).elim
-  | step j i ω e a b rest ωf hn hji hik hmax hy he hab ha hb ht =>
-    exact ⟨a,ha,(mem_trajectoryUnion G η y k a).mpr ⟨i,by omega,hik,
-      (edge_cluster_sides G (η i) (y i) he hab).mpr (conn_mono G Set.inter_subset_left hb)⟩⟩
-
- 
-
-theorem SpliceTrace.added_finite_cap (G : BondGraph V E) (x : V)
-    (η : ℕ → Set E) (y : ℕ → V) (k j M : ℕ) (ω ωf : Set E) (c : List (ℕ × E))
-    (h : SpliceTrace G x η y k j ω c ωf)
-    (hcaps : ∀ i, 1 ≤ i → i ≤ k →
-      (cluster G (η i) (y i)).Finite ∧ (cluster G (η i) (y i)).ncard ≤ M) :
-    (cluster G ωf x \ cluster G ω x).Finite ∧ (cluster G ωf x \ cluster G ω x).ncard ≤ k*M := by
-  obtain ⟨hf,hc⟩ := trajectoryUnion_finite_cap G η y k M hcaps
-  have hs := h.added_subset_trajectory G x η y k j ω ωf c
-  exact ⟨hf.subset hs,(Set.ncard_le_ncard hs hf).trans hc⟩
-
-end TrajectorySupport
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section JointReferenceEvent
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem SpliceTrace.joint_restored_field_symm (G : BondGraph V E) (x : V)
-    (η₁ η₂ : ℕ → Set E) (y₁ y₂ : ℕ → V) (k j₁ j₂ : ℕ)
-    (ω₁ ω₂ ωf : Set E) (c₁ c₂ : List (ℕ × E))
-    (h₁ : SpliceTrace G x η₁ y₁ k j₁ ω₁ c₁ ωf)
-    (h₂ : SpliceTrace G x η₂ y₂ k j₂ ω₂ c₂ ωf)
-    (hsep : GraphSeparated G (cluster G ωf x \ cluster G ω₁ x)
-      (cluster G ωf x \ cluster G ω₂ x)) :
-    (jointRestored G x η₁ η₂ y₂ ω₁ ω₂ c₂).1 =
-      (jointRestored G x η₂ η₁ y₁ ω₂ ω₁ c₁).1 := by
-  apply exchange_patch_symm _ _ ω₁ ω₂ ωf
-    (h₁.masks_disjoint G x η₁ η₂ y₁ y₂ k j₁ j₂ ω₁ ω₂ ωf c₁ c₂ h₂ hsep)
-  · intro e he
-    rw [← h₁.replay_eq G x η₁ y₁ k j₁ ω₁ ωf c₁]
-    exact (replaySplice_off_masks G x η₁ y₁ c₁ ω₁ e he).symm
-  · intro e he
-    rw [← h₂.replay_eq G x η₂ y₂ k j₂ ω₂ ωf c₂]
-    exact (replaySplice_off_masks G x η₂ y₂ c₂ ω₂ e he).symm
-
-theorem SpliceTrace.joint_restored_root (G : BondGraph V E) (x : V)
-    (η₁ η₂ : ℕ → Set E) (y₁ y₂ : ℕ → V) (k j₁ j₂ : ℕ)
-    (ω₁ ω₂ ωf : Set E) (c₁ c₂ : List (ℕ × E))
-    (h₁ : SpliceTrace G x η₁ y₁ k j₁ ω₁ c₁ ωf)
-    (h₂ : SpliceTrace G x η₂ y₂ k j₂ ω₂ c₂ ωf)
-    (hc₁ : c₁ ≠ []) (hc₂ : c₂ ≠ [])
-    (hsep : GraphSeparated G (cluster G ωf x \ cluster G ω₁ x)
-      (cluster G ωf x \ cluster G ω₂ x)) :
-    cluster G (jointRestored G x η₁ η₂ y₂ ω₁ ω₂ c₂).1 x =
-      cluster G ω₁ x ∩ cluster G ω₂ x := by
-  cases c₁ with
-  | nil => exact (hc₁ rfl).elim
-  | cons q r₁ =>
-    obtain ⟨i₁,e₁⟩ := q
-    cases c₂ with
-    | nil => exact (hc₂ rfl).elim
-    | cons q r₂ =>
-      obtain ⟨i₂,e₂⟩ := q
-      exact h₁.joint_root G x η₁ η₂ y₁ y₂ k j₁ j₂ i₁ i₂ ω₁ ω₂ ωf e₁ e₂ r₁ r₂ h₂ hsep
-
- 
-
-theorem SpliceTrace.joint_trajectory_hits (G : BondGraph V E) (x : V)
-    (η₁ η₂ : ℕ → Set E) (y₁ y₂ : ℕ → V) (k j₁ j₂ : ℕ)
-    (ω₁ ω₂ ωf : Set E) (c₁ c₂ : List (ℕ × E))
-    (h₁ : SpliceTrace G x η₁ y₁ k j₁ ω₁ c₁ ωf)
-    (h₂ : SpliceTrace G x η₂ y₂ k j₂ ω₂ c₂ ωf)
-    (hc₁ : c₁ ≠ []) (hc₂ : c₂ ≠ [])
-    (hsep : GraphSeparated G (cluster G ωf x \ cluster G ω₁ x)
-      (cluster G ωf x \ cluster G ω₂ x)) :
-    let A := cluster G (jointRestored G x η₁ η₂ y₂ ω₁ ω₂ c₂).1 x
-    (A ∩ trajectoryUnion G η₁ y₁ k).Nonempty ∧ (A ∩ trajectoryUnion G η₂ y₂ k).Nonempty := by
-  intro A
-  have hv₁ := (h₁.joint_virtual G x η₁ η₂ y₁ y₂ k j₁ j₂ ω₁ ω₂ ωf c₁ c₂ h₂ hc₁ hc₂ hsep).1
-  have hv₂ := (h₂.joint_virtual G x η₂ η₁ y₂ y₁ k j₂ j₁ ω₂ ω₁ ωf c₂ c₁ h₁ hc₂ hc₁
-    (hsep.symm G _ _)).1
-  refine ⟨hv₁.trajectory_hits G x η₁ y₁ k j₁ _ ω₂ c₁ hc₁,?_⟩
-  change (cluster G (jointRestored G x η₁ η₂ y₂ ω₁ ω₂ c₂).1 x ∩ _).Nonempty
-  rw [h₁.joint_restored_field_symm G x η₁ η₂ y₁ y₂ k j₁ j₂ ω₁ ω₂ ωf c₁ c₂ h₂ hsep]
-  exact hv₂.trajectory_hits G x η₂ y₂ k j₂ _ ω₁ c₂ hc₂
-
- 
-
-theorem SpliceTrace.joint_restored_size (G : BondGraph V E) (x : V)
-    (η₁ η₂ : ℕ → Set E) (y₁ y₂ : ℕ → V) (k j₁ j₂ n M : ℕ)
-    (ω₁ ω₂ ωf : Set E) (c₁ c₂ : List (ℕ × E))
-    (h₁ : SpliceTrace G x η₁ y₁ k j₁ ω₁ c₁ ωf)
-    (h₂ : SpliceTrace G x η₂ y₂ k j₂ ω₂ c₂ ωf)
-    (hc₁ : c₁ ≠ []) (hc₂ : c₂ ≠ [])
-    (hsep : GraphSeparated G (cluster G ωf x \ cluster G ω₁ x)
-      (cluster G ωf x \ cluster G ω₂ x))
-    (hf : (cluster G ω₁ x).Finite) (hn : n ≤ (cluster G ω₁ x).ncard)
-    (h2n : (cluster G ω₁ x).ncard < 2*n)
-    (hcap₂ : ∀ i, 1 ≤ i → i ≤ k →
-      (cluster G (η₂ i) (y₂ i)).Finite ∧ (cluster G (η₂ i) (y₂ i)).ncard ≤ M) :
-    let A := cluster G (jointRestored G x η₁ η₂ y₂ ω₁ ω₂ c₂).1 x
-    A.Finite ∧ n-k*M ≤ A.ncard ∧ A.ncard < 2*n := by
-  intro A
-  have hA : A = cluster G ω₁ x ∩ cluster G ω₂ x :=
-    h₁.joint_restored_root G x η₁ η₂ y₁ y₂ k j₁ j₂ ω₁ ω₂ ωf c₁ c₂ h₂ hc₁ hc₂ hsep
-  have hdiff : A = cluster G ω₁ x \ (cluster G ωf x \ cluster G ω₂ x) := by
-    rw [hA]
-    ext v
-    constructor
-    · rintro ⟨h1,h2⟩
-      exact ⟨h1,fun hh => hh.2 h2⟩
-    · rintro ⟨h1,h2⟩
-      refine ⟨h1,?_⟩
-      by_contra hn
-      exact h2 ⟨h₁.cluster_subset G x η₁ y₁ k j₁ ω₁ ωf c₁ h1,hn⟩
-  obtain ⟨hBfin,hBcap⟩ := h₂.added_finite_cap G x η₂ y₂ k j₂ M ω₂ ωf c₂ hcap₂
-  have hlower := Set.ncard_le_ncard_sdiff_add_ncard (cluster G ω₁ x)
-    (cluster G ωf x \ cluster G ω₂ x) hBfin
-  rw [← hdiff] at hlower
-  refine ⟨hA ▸ hf.inter_of_left _,by omega,?_⟩
-  rw [hA]
-  exact (Set.ncard_inter_le_ncard_left _ _ hf).trans_lt h2n
-
-end JointReferenceEvent
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section SmallHanging
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-def hangingSide (G : BondGraph V E) (ω : Set E) (x : V) (e : E) : Set V :=
-  cluster G ω x \ cluster G (ω \ {e}) x
-
-theorem hangingSide_finite (G : BondGraph V E) (ω : Set E) (x : V) (e : E)
-    (hC : (cluster G ω x).Finite) : (hangingSide G ω x e).Finite := hC.subset Set.sdiff_subset
-
-theorem mem_hangingSide_iff_pivotal (G : BondGraph V E) (ω : Set E) (x z : V)
-    (e : E) (he : e ∈ ω) : z ∈ hangingSide G ω x e ↔ OpenMarkedPivotal G ω {z} x e := by
-  constructor
-  · rintro ⟨hz,hn⟩
-    exact ⟨he,⟨z,rfl,hz⟩,fun ⟨y,hy,hxy⟩ => hn ((Set.mem_singleton_iff.mp hy) ▸ hxy)⟩
-  · rintro ⟨_,⟨y,hy,hxy⟩,hn⟩
-    exact ⟨(Set.mem_singleton_iff.mp hy) ▸ hxy,fun hz => hn ⟨z,rfl,hz⟩⟩
-
-theorem hangingSide_ncard_add_rank (G : BondGraph V E) (ω : Set E) (x : V)
-    (e : E) (hC : (cluster G ω x).Finite) :
-    (hangingSide G ω x e).ncard + pivotalRank G ω x e = (cluster G ω x).ncard := by
-  have hs : cluster G (ω \ {e}) x ⊆ cluster G ω x := fun _ h => conn_mono G Set.sdiff_subset h
-  rw [hangingSide,Set.ncard_sdiff hs (hC.subset hs)]
-  exact Nat.sub_add_cancel (Set.ncard_le_ncard hs hC)
-
- 
-
-noncomputable def smallSeparatingBonds (G : BondGraph V E) (ω : Set E) (x z : V)
-    (v : ℕ) : Finset E :=
-  (separatingBonds G ω x z).filter fun e => (hangingSide G ω x e).ncard ≤ v
-
-theorem smallSeparatingBonds_card_le (G : BondGraph V E) (ω : Set E) (x z : V)
-    (v : ℕ) (hC : (cluster G ω x).Finite) : (smallSeparatingBonds G ω x z v).card ≤ v := by
-  have hh := Finset.card_le_card_of_injOn (s := smallSeparatingBonds G ω x z v)
-    (t := Finset.Icc 1 v) (fun e => (hangingSide G ω x e).ncard)
-  have hmap : Set.MapsTo (fun e => (hangingSide G ω x e).ncard)
-      (↑(smallSeparatingBonds G ω x z v)) (↑(Finset.Icc 1 v)) := by
-    intro e he
-    obtain ⟨he,hcap⟩ := Finset.mem_filter.mp he
-    have hp := (mem_separatingBonds G ω x z e).mp he
-    exact Finset.mem_Icc.mpr ⟨(Set.ncard_pos (hangingSide_finite G ω x e hC)).mpr
-      ⟨z,(mem_hangingSide_iff_pivotal G ω x z e hp.1).mpr hp⟩,hcap⟩
-  have hinj : Set.InjOn (fun e => (hangingSide G ω x e).ncard)
-      (↑(smallSeparatingBonds G ω x z v)) := by
-    intro e he f hf heq
-    apply pivotalRank_injective_of_cluster_finite G ω {z} x hC
-      ((mem_separatingBonds G ω x z e).mp (Finset.mem_filter.mp he).1)
-      ((mem_separatingBonds G ω x z f).mp (Finset.mem_filter.mp hf).1)
-    change (hangingSide G ω x e).ncard = (hangingSide G ω x f).ncard at heq
-    have he' := hangingSide_ncard_add_rank G ω x e hC
-    have hf' := hangingSide_ncard_add_rank G ω x f hC
-    omega
-  simpa only [Nat.card_Icc, Nat.add_sub_cancel] using hh hmap hinj
-
-noncomputable def smallHangingThrough (G : BondGraph V E) (ω : Set E) (x z : V)
-    (v : ℕ) (hC : (cluster G ω x).Finite) : Finset V :=
-  (smallSeparatingBonds G ω x z v).biUnion fun e => (hangingSide_finite G ω x e hC).toFinset
-
-theorem mem_smallHangingThrough (G : BondGraph V E) (ω : Set E) (x z w : V)
-    (v : ℕ) (hC : (cluster G ω x).Finite) : w ∈ smallHangingThrough G ω x z v hC ↔
-      ∃ e, OpenMarkedPivotal G ω {z} x e ∧ (hangingSide G ω x e).ncard ≤ v ∧
-        w ∈ hangingSide G ω x e := by
-  simp [smallHangingThrough, smallSeparatingBonds, and_assoc]
-
-theorem smallHangingThrough_card_le (G : BondGraph V E) (ω : Set E) (x z : V)
-    (v : ℕ) (hC : (cluster G ω x).Finite) :
-    (smallHangingThrough G ω x z v hC).card ≤ v*v := by
-  calc
-    _ ≤ ∑ e ∈ smallSeparatingBonds G ω x z v, (hangingSide_finite G ω x e hC).toFinset.card :=
-      Finset.card_biUnion_le
-    _ ≤ ∑ _e ∈ smallSeparatingBonds G ω x z v, v := by
-      apply Finset.sum_le_sum
-      intro e he
-      rw [← Set.ncard_eq_toFinset_card (hangingSide G ω x e) (hangingSide_finite G ω x e hC)]
-      exact (Finset.mem_filter.mp he).2
-    _ = (smallSeparatingBonds G ω x z v).card*v := by simp
-    _ ≤ v*v := Nat.mul_le_mul_right v (smallSeparatingBonds_card_le G ω x z v hC)
-
-end SmallHanging
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section NearHanging
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem hangingSide_laminar (G : BondGraph V E) (ω : Set E) (x : V)
-    (e f : E) (he : e ∈ ω) (hf : f ∈ ω) :
-    Disjoint (hangingSide G ω x e) (hangingSide G ω x f) ∨
-      hangingSide G ω x e ⊆ hangingSide G ω x f ∨
-      hangingSide G ω x f ⊆ hangingSide G ω x e := by
-  by_cases hd : Disjoint (hangingSide G ω x e) (hangingSide G ω x f)
-  · exact Or.inl hd
-  obtain ⟨z,hze,hzf⟩ := Set.not_disjoint_iff.mp hd
-  by_cases hef : e = f
-  · exact Or.inr (Or.inl (hef ▸ Set.Subset.refl _))
-  rcases pivotal_rootSides_nested G ω {z} x
-    ((mem_hangingSide_iff_pivotal G ω x z e he).mp hze)
-    ((mem_hangingSide_iff_pivotal G ω x z f hf).mp hzf) hef with h | h
-  · exact Or.inr (Or.inr (Set.sdiff_subset_sdiff_right h.1))
-  · exact Or.inr (Or.inl (Set.sdiff_subset_sdiff_right h.1))
-
-noncomputable def graphNeighbors (G : BondGraph V E) (hloc : LocallyFinite G)
-    (A : Finset V) : Finset V := A.biUnion (neighbors G hloc)
-
-theorem graphNeighbors_card_le (G : BondGraph V E) (hloc : LocallyFinite G)
-    (A : Finset V) (d : ℕ) (hdeg : ∀ a, degree G a ≤ d) :
-    (graphNeighbors G hloc A).card ≤ A.card*d := by
-  calc
-    _ ≤ ∑ a ∈ A, (neighbors G hloc a).card := Finset.card_biUnion_le
-    _ ≤ ∑ _a ∈ A, d := Finset.sum_le_sum fun a _ =>
-      (neighbors_card_le_degree G hloc a).trans (hdeg a)
-    _ = _ := by simp
-
- 
-
-noncomputable def nearHangingEndpoints (G : BondGraph V E) (hloc : LocallyFinite G)
-    (ω : Set E) (x z : V) (e : E) (v : ℕ) (hC : (cluster G ω x).Finite) : Finset V :=
-  ((hangingSide_finite G ω x e hC).toFinset ∪ smallHangingThrough G ω x z v hC) ∪
-    (graphNeighbors G hloc (hangingSide_finite G ω x e hC).toFinset).biUnion
-      (fun a => smallHangingThrough G ω x a v hC)
-
-theorem nearHangingEndpoints_card_le (G : BondGraph V E) (hloc : LocallyFinite G)
-    (ω : Set E) (x z : V) (e : E) (v d : ℕ) (hC : (cluster G ω x).Finite)
-    (hcap : (hangingSide G ω x e).ncard ≤ v) (hdeg : ∀ a, degree G a ≤ d) :
-    (nearHangingEndpoints G hloc ω x z e v hC).card ≤ v+v^2+d*v^3 := by
-  let B := (hangingSide_finite G ω x e hC).toFinset
-  let N := graphNeighbors G hloc B
-  have hB : B.card ≤ v := by
-    simpa only [B,← Set.ncard_eq_toFinset_card (hangingSide G ω x e)
-      (hangingSide_finite G ω x e hC)] using hcap
-  have hN : N.card ≤ v*d := (graphNeighbors_card_le G hloc B d hdeg).trans
-    (Nat.mul_le_mul_right d hB)
-  have hU : (N.biUnion fun a => smallHangingThrough G ω x a v hC).card ≤ d*v^3 := by
-    calc
-      _ ≤ ∑ a ∈ N, (smallHangingThrough G ω x a v hC).card := Finset.card_biUnion_le
-      _ ≤ ∑ _a ∈ N, v*v := Finset.sum_le_sum fun a _ => smallHangingThrough_card_le G ω x a v hC
-      _ = N.card*(v*v) := by simp
-      _ ≤ (v*d)*(v*v) := Nat.mul_le_mul_right (v*v) hN
-      _ = d*v^3 := by ring
-  calc
-    _ ≤ (B ∪ smallHangingThrough G ω x z v hC).card +
-        (N.biUnion fun a => smallHangingThrough G ω x a v hC).card := Finset.card_union_le ..
-    _ ≤ (B.card + (smallHangingThrough G ω x z v hC).card) + d*v^3 :=
-      Nat.add_le_add (Finset.card_union_le ..) hU
-    _ ≤ (v+v*v)+d*v^3 := Nat.add_le_add_right
-      (Nat.add_le_add hB (smallHangingThrough_card_le G ω x z v hC)) _
-    _ = v+v^2+d*v^3 := by ring
-
-theorem mem_nearHangingEndpoints_of_not_separated (G : BondGraph V E)
-    (hloc : LocallyFinite G) (ω : Set E) (x z w : V) (e f : E) (v : ℕ)
-    (hC : (cluster G ω x).Finite)
-    (he : OpenMarkedPivotal G ω {z} x e) (hf : OpenMarkedPivotal G ω {w} x f)
-    (hcap : (hangingSide G ω x f).ncard ≤ v)
-    (hn : ¬GraphSeparated G (hangingSide G ω x e) (hangingSide G ω x f)) :
-    w ∈ nearHangingEndpoints G hloc ω x z e v hC := by
-  have hz := (mem_hangingSide_iff_pivotal G ω x z e he.1).mpr he
-  have hw := (mem_hangingSide_iff_pivotal G ω x w f hf.1).mpr hf
-  rcases hangingSide_laminar G ω x e f he.1 hf.1 with hd | hsub | hsub
-  · have hadj : ∃ a ∈ hangingSide G ω x e, ∃ b ∈ hangingSide G ω x f, Adj G a b := by
-      by_contra h
-      apply hn
-      refine ⟨hd,?_⟩
-      intro a ha b hb hab
-      exact h ⟨a,ha,b,hb,hab⟩
-    obtain ⟨a,ha,b,hb,hab⟩ := hadj
-    apply Finset.mem_union_right
-    apply Finset.mem_biUnion.mpr
-    refine ⟨b,Finset.mem_biUnion.mpr ⟨a,?_,(mem_neighbors G hloc a b).mpr hab⟩,?_⟩
-    · exact (hangingSide_finite G ω x e hC).mem_toFinset.mpr ha
-    · exact (mem_smallHangingThrough G ω x b w v hC).mpr
-        ⟨f,(mem_hangingSide_iff_pivotal G ω x b f hf.1).mp hb,hcap,hw⟩
-  · apply Finset.mem_union_left
-    apply Finset.mem_union_right
-    exact (mem_smallHangingThrough G ω x z w v hC).mpr
-      ⟨f,(mem_hangingSide_iff_pivotal G ω x z f hf.1).mp (hsub hz),hcap,hw⟩
-  · apply Finset.mem_union_left
-    apply Finset.mem_union_left
-    exact (hangingSide_finite G ω x e hC).mem_toFinset.mpr (hsub hw)
-
-end NearHanging
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section FiniteReferenceLaw
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-abbrev FiniteField (Q : Finset E) := ↥Q.powerset
-
-def finiteFieldSet (Q : Finset E) (ω : FiniteField Q) : Set E := (ω.val : Set E)
-
-noncomputable def finiteFieldWeight (Q : Finset E) (p : ℝ) (ω : FiniteField Q) : ℝ :=
-  bernoulliWeight Q ω.val p
-
-@[simp] theorem domainBits_finiteFieldSet (Q : Finset E) (ω : FiniteField Q) :
-    domainBits Q (finiteFieldSet Q ω) = ω.val := by
-  ext e
-  have hs := Finset.mem_powerset.mp ω.property
-  simp only [mem_domainBits,finiteFieldSet,Finset.mem_coe]
-  exact ⟨fun h => h.2,fun h => ⟨hs h,h⟩⟩
-
-theorem finiteFieldWeight_nonneg (Q : Finset E) (p : unitInterval) (ω : FiniteField Q) :
-    0 ≤ finiteFieldWeight Q p ω := bernoulliWeight_nonneg _ _ p
-
-theorem sum_finiteFieldWeight (Q : Finset E) (p : ℝ) :
-    ∑ ω : FiniteField Q, finiteFieldWeight Q p ω = 1 := by
-  rw [show (fun ω : FiniteField Q => finiteFieldWeight Q p ω) =
-    (fun ω : FiniteField Q => bernoulliWeight Q ω.val p) from rfl,Finset.sum_coe_sort Q.powerset (fun ω => bernoulliWeight Q ω p)]
-  simpa only [bernoulliPolynomial,bernoulliWeight,mul_one,one_mul] using bernoulliPolynomial_const Q 1 p
-
- 
-
-def finiteFieldFamily (Q : Finset E) (J : Finset ℕ) (η : ↥J → FiniteField Q) : ℕ → Set E :=
-  fun i => if hi : i ∈ J then finiteFieldSet Q (η ⟨i,hi⟩) else ∅
-
-@[simp] theorem finiteFieldFamily_at (Q : Finset E) (J : Finset ℕ)
-    (η : ↥J → FiniteField Q) (i : ↥J) :
-    finiteFieldFamily Q J η i = finiteFieldSet Q (η i) := by
-  simp [finiteFieldFamily,i.property]
-
-noncomputable def finiteFamilyWeight (Q : Finset E) (J : Finset ℕ) (p : ℝ)
-    (η : ↥J → FiniteField Q) : ℝ := ∏ i, finiteFieldWeight Q p (η i)
-
-theorem finiteFamilyWeight_nonneg (Q : Finset E) (J : Finset ℕ) (p : unitInterval)
-    (η : ↥J → FiniteField Q) : 0 ≤ finiteFamilyWeight Q J p η :=
-  Finset.prod_nonneg fun _ _ => finiteFieldWeight_nonneg Q p _
-
-theorem sum_finiteFamilyWeight (Q : Finset E) (J : Finset ℕ) (p : ℝ) :
-    ∑ η : ↥J → FiniteField Q, finiteFamilyWeight Q J p η = 1 := by
-  unfold finiteFamilyWeight
-  rw [← Fintype.prod_sum]
-  simp only [sum_finiteFieldWeight,Finset.prod_const_one]
-
-theorem prod_fieldFamilyWeight (Q : Finset E) (J : Finset ℕ) (p : ℝ)
-    (η : ↥J → FiniteField Q) :
-    (∏ i ∈ J, bernoulliWeight Q (domainBits Q (finiteFieldFamily Q J η i)) p) =
-      finiteFamilyWeight Q J p η := by
-  rw [← Finset.prod_coe_sort J]
-  simp only [finiteFieldFamily_at,domainBits_finiteFieldSet,finiteFamilyWeight,finiteFieldWeight]
-
-abbrev FiniteJointData (Q : Finset E) (J₁ J₂ : Finset ℕ) :=
-  FiniteField Q × (↥J₁ → FiniteField Q) × (↥J₂ → FiniteField Q)
-
-def encodeFiniteJoint (Q : Finset E) (J₁ J₂ : Finset ℕ)
-    (s : FiniteJointData Q J₁ J₂) : JointSpliceState E :=
-  (finiteFieldSet Q s.1,finiteFieldFamily Q J₁ s.2.1,finiteFieldFamily Q J₂ s.2.2)
-
-theorem finiteFieldSet_injective (Q : Finset E) : Function.Injective (finiteFieldSet Q) := by
-  intro a b he
-  exact Subtype.ext (Finset.coe_injective he)
-
-theorem finiteFieldFamily_injective (Q : Finset E) (J : Finset ℕ) :
-    Function.Injective (finiteFieldFamily Q J) := by
-  intro a b he
-  funext i
-  apply finiteFieldSet_injective Q
-  simpa only [finiteFieldFamily_at] using congrFun he i
-
-theorem encodeFiniteJoint_injective (Q : Finset E) (J₁ J₂ : Finset ℕ) :
-    Function.Injective (encodeFiniteJoint Q J₁ J₂) := by
-  intro a b he
-  apply Prod.ext (finiteFieldSet_injective Q (congrArg Prod.fst he))
-  exact Prod.ext (finiteFieldFamily_injective Q J₁ (congrArg (fun s => s.2.1) he))
-    (finiteFieldFamily_injective Q J₂ (congrArg (fun s => s.2.2) he))
-
-theorem jointTupleWeight_encode (Q : Finset E) (J₁ J₂ : Finset ℕ) (p : ℝ)
-    (s : FiniteJointData Q J₁ J₂) :
-    jointTupleWeight Q J₁ J₂ p (encodeFiniteJoint Q J₁ J₂ s) =
-      finiteFieldWeight Q p s.1 * finiteFamilyWeight Q J₁ p s.2.1 *
-        finiteFamilyWeight Q J₂ p s.2.2 := by
-  simp only [jointTupleWeight,encodeFiniteJoint,domainBits_finiteFieldSet,
-    prod_fieldFamilyWeight,finiteFieldWeight]
-
- 
-
-theorem sum_jointTupleWeight_encode (Q : Finset E) (J₁ J₂ : Finset ℕ) (p : ℝ) :
-    ∑ s : FiniteJointData Q J₁ J₂, jointTupleWeight Q J₁ J₂ p (encodeFiniteJoint Q J₁ J₂ s) = 1 := by
-  simp_rw [jointTupleWeight_encode]
-  rw [Fintype.sum_prod_type]
-  simp_rw [Fintype.sum_prod_type,← Finset.mul_sum,sum_finiteFamilyWeight,mul_one,
-    ← Finset.mul_sum,sum_finiteFamilyWeight,mul_one]
-  exact sum_finiteFieldWeight Q p
-
-end FiniteReferenceLaw
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section FiniteReferenceFactor
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-noncomputable def finiteFamilyEvent (Q : Finset E) (J : Finset ℕ) (p : ℝ)
-    (H : (ℕ → Set E) → Prop) : ℝ :=
-  ∑ η : ↥J → FiniteField Q, finiteFamilyWeight Q J p η *
-    if H (finiteFieldFamily Q J η) then 1 else 0
-
-theorem finiteFamilyEvent_nonneg (Q : Finset E) (J : Finset ℕ) (p : unitInterval)
-    (H : (ℕ → Set E) → Prop) : 0 ≤ finiteFamilyEvent Q J p H := by
-  apply Finset.sum_nonneg
-  intro η _
-  exact mul_nonneg (finiteFamilyWeight_nonneg Q J p η) (by split <;> norm_num)
-
-theorem finiteFamilyEvent_le_one (Q : Finset E) (J : Finset ℕ) (p : unitInterval)
-    (H : (ℕ → Set E) → Prop) : finiteFamilyEvent Q J p H ≤ 1 := by
-  rw [← sum_finiteFamilyWeight Q J p]
-  apply Finset.sum_le_sum
-  intro η _
-  exact mul_le_of_le_one_right (finiteFamilyWeight_nonneg Q J p η) (by split <;> norm_num)
-
-noncomputable def finiteJointDomain (Q : Finset E) (J₁ J₂ : Finset ℕ) :
-    Finset (JointSpliceState E) := Finset.univ.image (encodeFiniteJoint Q J₁ J₂)
-
-theorem finiteJointDomain_event_factor (Q : Finset E) (J₁ J₂ : Finset ℕ) (p : ℝ)
-    (P : Set E → Prop) (H₁ H₂ : Set E → (ℕ → Set E) → Prop) :
-    (∑ s ∈ (finiteJointDomain Q J₁ J₂).filter (fun s => P s.1 ∧ H₁ s.1 s.2.1 ∧ H₂ s.1 s.2.2),
-      jointTupleWeight Q J₁ J₂ p s) =
-    ∑ ω : FiniteField Q, finiteFieldWeight Q p ω *
-      (if P (finiteFieldSet Q ω) then
-        finiteFamilyEvent Q J₁ p (H₁ (finiteFieldSet Q ω)) *
-        finiteFamilyEvent Q J₂ p (H₂ (finiteFieldSet Q ω)) else 0) := by
-  rw [Finset.sum_filter,finiteJointDomain,Finset.sum_image (encodeFiniteJoint_injective Q J₁ J₂).injOn,
-    Fintype.sum_prod_type]
-  apply Finset.sum_congr rfl
-  intro ω _
-  rw [Fintype.sum_prod_type]
-  by_cases hP : P (finiteFieldSet Q ω)
-  · simp only [encodeFiniteJoint, hP, true_and,ite_true]
-    have heq (η₁ : ↥J₁ → FiniteField Q) (η₂ : ↥J₂ → FiniteField Q) :
-      (if H₁ (finiteFieldSet Q ω) (finiteFieldFamily Q J₁ η₁) ∧
-        H₂ (finiteFieldSet Q ω) (finiteFieldFamily Q J₂ η₂) then
-        jointTupleWeight Q J₁ J₂ p
-          (finiteFieldSet Q ω,finiteFieldFamily Q J₁ η₁,finiteFieldFamily Q J₂ η₂) else 0) =
-      finiteFieldWeight Q p ω *
-        (finiteFamilyWeight Q J₁ p η₁ * if H₁ (finiteFieldSet Q ω) (finiteFieldFamily Q J₁ η₁) then 1 else 0) *
-        (finiteFamilyWeight Q J₂ p η₂ * if H₂ (finiteFieldSet Q ω) (finiteFieldFamily Q J₂ η₂) then 1 else 0) := by
-      have hw := jointTupleWeight_encode Q J₁ J₂ p (ω,η₁,η₂)
-      change jointTupleWeight Q J₁ J₂ p (finiteFieldSet Q ω,finiteFieldFamily Q J₁ η₁,finiteFieldFamily Q J₂ η₂) = _ at hw
-      rw [hw]
-      split_ifs <;> simp_all
-    simp_rw [heq,← Finset.mul_sum,← Finset.sum_mul]
-    rw [← Finset.mul_sum,mul_assoc]
-    rfl
-  · simp [encodeFiniteJoint,hP]
-
- 
-
-theorem mem_finiteJointDomain_of_support (Q : Finset E) (J₁ J₂ : Finset ℕ)
-    (s : JointSpliceState E) (hω : s.1 ⊆ (Q : Set E))
-    (hη₁ : ∀ i, s.2.1 i ⊆ (Q : Set E)) (hη₂ : ∀ i, s.2.2 i ⊆ (Q : Set E))
-    (hz₁ : ∀ i ∉ J₁, s.2.1 i = ∅) (hz₂ : ∀ i ∉ J₂, s.2.2 i = ∅) :
-    s ∈ finiteJointDomain Q J₁ J₂ := by
-  let desc (ω : Set E) : FiniteField Q := ⟨domainBits Q ω,Finset.mem_powerset.mpr (domainBits_subset Q ω)⟩
-  have hdesc (ω : Set E) (h : ω ⊆ (Q : Set E)) : finiteFieldSet Q (desc ω) = ω := by
-    ext e
-    simp only [finiteFieldSet,desc,Finset.mem_coe,mem_domainBits]
-    exact ⟨fun hh => hh.2,fun hh => ⟨h hh,hh⟩⟩
-  let a : FiniteJointData Q J₁ J₂ := (desc s.1,fun i => desc (s.2.1 i),fun i => desc (s.2.2 i))
-  apply Finset.mem_image.mpr
-  refine ⟨a,Finset.mem_univ _,?_⟩
-  apply Prod.ext (hdesc s.1 hω)
-  apply Prod.ext
-  · funext i
-    change (if hi : i ∈ J₁ then finiteFieldSet Q (desc (s.2.1 i)) else ∅) = s.2.1 i
-    by_cases hi : i ∈ J₁
-    · simp only [hi,dite_true,hdesc _ (hη₁ i)]
-    · simp only [hi,dite_false,hz₁ i hi]
-  · funext i
-    change (if hi : i ∈ J₂ then finiteFieldSet Q (desc (s.2.2 i)) else ∅) = s.2.2 i
-    by_cases hi : i ∈ J₂
-    · simp only [hi,dite_true,hdesc _ (hη₂ i)]
-    · simp only [hi,dite_false,hz₂ i hi]
-
-theorem exchangeOn_subset (M ω η Q : Set E) (hω : ω ⊆ Q) (hη : η ⊆ Q) :
-    exchangeOn M ω η ⊆ Q := by
-  rintro e (⟨he,_⟩ | ⟨he,_⟩)
-  · exact hω he
-  · exact hη he
-
-end FiniteReferenceFactor
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section FiniteSeparatedSecondMoment
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem finite_separated_second_moment_le {I : Type*} [Fintype I]
-    (G : BondGraph V E) (x : V) (k n M : ℕ) (p : unitInterval) (Q : Finset E)
-    (η₁ η₂ : I → ℕ → Set E) (y₁ y₂ : (ℕ → Set E) → ℕ → V)
-    (success₁ success₂ : (ℕ → Set E) → Prop)
-    (choose₁ choose₂ : ℕ → Set E → E)
-    (ω₁ ω₂ ωf : I → Set E) (c₁ c₂ : I → List (ℕ × E))
-    (hir : Function.Injective (fun a =>
-      (jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a),
-        c₁ a,c₂ a)))
-    (h₁ : ∀ a, SpliceTrace G x (η₁ a) (y₁ (η₁ a)) k 0 (ω₁ a) (c₁ a) (ωf a))
-    (h₂ : ∀ a, SpliceTrace G x (η₂ a) (y₂ (η₂ a)) k 0 (ω₂ a) (c₂ a) (ωf a))
-    (hc₁ : ∀ a, c₁ a ≠ []) (hc₂ : ∀ a, c₂ a ≠ [])
-    (hsep : ∀ a, GraphSeparated G (cluster G (ωf a) x \ cluster G (ω₁ a) x)
-      (cluster G (ωf a) x \ cluster G (ω₂ a) x))
-    (hr₁ : ∀ a, FollowsContactRule G x (η₁ a) (y₁ (η₁ a)) choose₁ (c₁ a) (ω₁ a))
-    (hr₂ : ∀ a, FollowsContactRule G x (η₂ a) (y₂ (η₂ a)) choose₂ (c₂ a) (ω₂ a))
-    (hbin : ∀ a, (cluster G (ω₁ a) x).Finite ∧ n ≤ (cluster G (ω₁ a) x).ncard ∧
-      (cluster G (ω₁ a) x).ncard < 2*n)
-    (hcap₂ : ∀ a i, 1 ≤ i → i ≤ k → (cluster G (η₂ a i) (y₂ (η₂ a) i)).Finite ∧
-      (cluster G (η₂ a i) (y₂ (η₂ a) i)).ncard ≤ M)
-    (hsc₁ : ∀ a, success₁ (η₁ a)) (hsc₂ : ∀ a, success₂ (η₂ a))
-    (hsω₁ : ∀ a, ω₁ a ⊆ (Q : Set E)) (hsω₂ : ∀ a, ω₂ a ⊆ (Q : Set E))
-    (hsη₁ : ∀ a i, η₁ a i ⊆ (Q : Set E)) (hsη₂ : ∀ a i, η₂ a i ⊆ (Q : Set E))
-    (hz₁ : ∀ a i, i ∉ Finset.Icc 1 k → η₁ a i = ∅)
-    (hz₂ : ∀ a i, i ∉ Finset.Icc 1 k → η₂ a i = ∅) :
-    (∑ a, jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p
-      (jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a))) ≤
-    ∑ ω : FiniteField Q, finiteFieldWeight Q p ω *
-      (if (cluster G (finiteFieldSet Q ω) x).Finite ∧
-        n-k*M ≤ (cluster G (finiteFieldSet Q ω) x).ncard ∧
-        (cluster G (finiteFieldSet Q ω) x).ncard < 2*n then
-        finiteFamilyEvent Q (Finset.Icc 1 k) p (fun η => success₁ η ∧
-          (cluster G (finiteFieldSet Q ω) x ∩ trajectoryUnion G η (y₁ η) k).Nonempty) *
-        finiteFamilyEvent Q (Finset.Icc 1 k) p (fun η => success₂ η ∧
-          (cluster G (finiteFieldSet Q ω) x ∩ trajectoryUnion G η (y₂ η) k).Nonempty) else 0) := by
-  let J := Finset.Icc 1 k
-  let P (ω : Set E) := (cluster G ω x).Finite ∧ n-k*M ≤ (cluster G ω x).ncard ∧
-    (cluster G ω x).ncard < 2*n
-  let H₁ (ω : Set E) (η : ℕ → Set E) := success₁ η ∧
-    (cluster G ω x ∩ trajectoryUnion G η (y₁ η) k).Nonempty
-  let H₂ (ω : Set E) (η : ℕ → Set E) := success₂ η ∧
-    (cluster G ω x ∩ trajectoryUnion G η (y₂ η) k).Nonempty
-  let B := (finiteJointDomain Q J J).filter fun s => P s.1 ∧ H₁ s.1 s.2.1 ∧ H₂ s.1 s.2.2
-  have hB (a : I) : jointRestored G x (η₁ a) (η₂ a) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (c₂ a) ∈ B := by
-    apply Finset.mem_filter.mpr
-    refine ⟨mem_finiteJointDomain_of_support Q J J _
-      (exchangeOn_subset _ _ _ _ (hsω₁ a) (hsω₂ a)) (hsη₁ a) (hsη₂ a) (hz₁ a) (hz₂ a),?_⟩
-    have hroot := (h₁ a).joint_restored_size G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a))
-      k 0 0 n M (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a) (h₂ a) (hc₁ a) (hc₂ a) (hsep a)
-      (hbin a).1 (hbin a).2.1 (hbin a).2.2 (hcap₂ a)
-    have hhits := (h₁ a).joint_trajectory_hits G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a))
-      k 0 0 (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a) (h₂ a) (hc₁ a) (hc₂ a) (hsep a)
-    exact ⟨hroot,⟨hsc₁ a,hhits.1⟩,hsc₂ a,hhits.2⟩
-  have h := finite_joint_mass_le G x k p Q η₁ η₂ y₁ y₂ choose₁ choose₂ ω₁ ω₂ ωf c₁ c₂
-    hir h₁ h₂ hc₁ hc₂ hsep hr₁ hr₂ B hB
-  apply h.trans_eq
-  convert finiteJointDomain_event_factor Q J J p P H₁ H₂ using 1 <;> simp [B,J,P,H₁,H₂]
-
-end FiniteSeparatedSecondMoment
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section SeededForward
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem exists_seeded_spliceTrace (G : BondGraph V E) (hloc : LocallyFinite G)
-    (x : V) (η : ℕ → Set E) (y : ℕ → V) (k M : ℕ)
-    (hchain : ∀ i ∈ Finset.Icc 1 k, Conn G (η i) (y i) (y (i-1)))
-    (hcaps : ∀ i ∈ Finset.Icc 1 k, (cluster G (η i) (y i)).Finite ∧
-      (cluster G (η i) (y i)).ncard ≤ M)
-    (choose : ℕ → Set E → E)
-    (hchoose : ∀ i K, K.Finite → K.Nonempty → choose i K ∈ K)
-    (j : ℕ) (hj : j ≤ k) (ω : Set E) (hyj : y j ∈ cluster G ω x)
-    (N : ℕ) (hfin : (cluster G ω x).Finite) (hN : (cluster G ω x).ncard ≤ N) :
-    ∃ steps ωf, SpliceTrace G x η y k j ω steps ωf ∧
-      FollowsContactRule G x η y choose steps ω ∧ steps.length ≤ k-j ∧
-      y k ∈ cluster G ωf x ∧ (cluster G ωf x).Finite ∧
-      (cluster G ωf x).ncard ≤ N+(k-j)*M := by
-  generalize hn : k-j = n
-  induction n using Nat.strong_induction_on generalizing j ω N with
-  | h n ih =>
-    by_cases hz : y k ∈ cluster G ω x
-    · refine ⟨[],ω,SpliceTrace.stop j ω hz,?_,by simp,hz,hfin,?_⟩
-      · exact List.Forall₂.nil
-      · exact hN.trans (Nat.le_add_right ..)
-    · obtain ⟨i,hiI,hmeet,hyi,hmax⟩ :=
-        exists_maximal_splice_index G η y k j hj hchain (cluster G ω x) hyj hz
-      obtain ⟨hji,hik⟩ := Finset.mem_Icc.mp hiI
-      have hi1 : i ∈ Finset.Icc 1 k := Finset.mem_Icc.mpr ⟨by omega,hik⟩
-      let B := cluster G (η i ∩ internalEdges G (cluster G ω x)ᶜ) (y i)
-      let K := contactSet G (η i) B
-      have hBC : B ⊆ cluster G (η i) (y i) := fun _ h => conn_mono G Set.inter_subset_left h
-      have hBK : K.Finite := contactSet_finite G hloc (η i) B ((hcaps i hi1).1.subset hBC)
-      have hK : K.Nonempty := by
-        obtain ⟨e,a,b,he,hab,ha,hb⟩ := exists_splice_contact G (η i) (cluster G ω x) (y i) hyi hmeet
-        exact ⟨e,he,b,a,hab.trans (Sym2.eq_swap ..),hb,
-          fun h => slit_cluster_subset G (cluster G ω x) (η i) (y i) hyi h ha⟩
-      let e := choose i K
-      have heK : e ∈ contactSet G (η i) B := hchoose i K hBK hK
-      change e ∈ contactSet G (η i)
-        (cluster G (η i ∩ internalEdges G (cluster G ω x)ᶜ) (y i)) at heK
-      rw [contactSet_deleted_cluster G (η i) (cluster G ω x) (y i) hyi] at heK
-      obtain ⟨he,b,hb,a,ha,hba⟩ := heK
-      have hab : G e = s(a,b) := hba.trans (Sym2.eq_swap ..)
-      let ω' := spliceField G (cluster G ω x) B e ω (η i)
-      have hcap := splice_cluster_cap G ω (η i) x (y i) hyi e a b hab he ha hb N M hfin hN
-        (hcaps i hi1).1 (hcaps i hi1).2
-      have hyt : y i ∈ cluster G ω' x := by
-        change y i ∈ cluster G (spliceField G (cluster G ω x) B e ω (η i)) x
-        rw [splice_cluster G ω (η i) x (y i) hyi e a b hab he ha hb]
-        exact Or.inr (mem_cluster_self G _ _)
-      obtain ⟨steps,ωf,htrace,hrule,hlen,hconn,hf,hs⟩ :=
-        ih (k-i) (by omega) i hik ω' hyt (N+M) hcap.1 hcap.2 rfl
-      refine ⟨(i,e)::steps,ωf,
-        SpliceTrace.step j i ω e a b steps ωf hz (by omega) hik hmax hyi he hab ha hb htrace,
-        ?_,?_,hconn,hf,?_⟩
-      · exact (followsContactRule_cons G x η y choose i e steps ω).mpr ⟨rfl,hrule⟩
-      · simp only [List.length_cons]
-        omega
-      · have hm : (k-i+1)*M ≤ (k-j)*M := Nat.mul_le_mul_right M (by omega)
-        rw [Nat.add_mul,one_mul] at hm
-        calc
-          _ ≤ (N+M)+(k-i)*M := hs
-          _ ≤ N+(k-j)*M := by omega
-          _ = N+n*M := by rw [hn]
-
- 
-
-theorem existsUnique_seeded_spliceTrace (G : BondGraph V E) (hloc : LocallyFinite G)
-    (x : V) (η : ℕ → Set E) (y : ℕ → V) (k M : ℕ)
-    (hchain : ∀ i ∈ Finset.Icc 1 k, Conn G (η i) (y i) (y (i-1)))
-    (hcaps : ∀ i ∈ Finset.Icc 1 k, (cluster G (η i) (y i)).Finite ∧
-      (cluster G (η i) (y i)).ncard ≤ M)
-    (choose : ℕ → Set E → E)
-    (hchoose : ∀ i K, K.Finite → K.Nonempty → choose i K ∈ K)
-    (j : ℕ) (hj : j ≤ k) (ω : Set E) (hyj : y j ∈ cluster G ω x)
-    (hfin : (cluster G ω x).Finite) :
-    ∃! out : List (ℕ × E) × Set E,
-      SpliceTrace G x η y k j ω out.1 out.2 ∧
-      FollowsContactRule G x η y choose out.1 ω := by
-  obtain ⟨c,ωf,ht,hr,_⟩ := exists_seeded_spliceTrace G hloc x η y k M hchain hcaps
-    choose hchoose j hj ω hyj _ hfin (le_refl _)
-  refine ⟨(c,ωf),⟨ht,hr⟩,?_⟩
-  rintro ⟨d,σf⟩ ⟨hs,hd⟩
-  obtain ⟨hc,hf⟩ := hs.seeded_unique G x η y k j ω σf ωf d c ht choose hd hr
-  exact Prod.ext hc hf
-
-end SeededForward
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section SpliceLocalSupport
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem reachBall_triangle (G : BondGraph V E) (x y : V) (n m : ℕ)
-    (hy : y ∈ reachBall G x n) : reachBall G y m ⊆ reachBall G x (n+m) := by
-  induction m with
-  | zero => simpa only [reachBall,Nat.add_zero,Set.singleton_subset_iff] using hy
-  | succ m ih =>
-    intro z hz
-    rcases hz with hz | hz
-    · exact Or.inl (ih hz)
-    · apply Or.inr
-      simp only [Set.mem_iUnion,Set.mem_ofPred_eq] at hz ⊢
-      obtain ⟨a,ha,haz⟩ := hz
-      exact ⟨a,ih ha,haz⟩
-
- 
-theorem capped_cluster_ball_of_mem (G : BondGraph V E) (ω : Set E) (x y : V)
-    (hfin : (cluster G ω x).Finite) (M : ℕ) (hM : (cluster G ω x).ncard ≤ M)
-    (hy : y ∈ cluster G ω x) : cluster G ω x ⊆ reachBall G y M := by
-  have he := cluster_eq_of_conn G ω hy
-  rw [he] at hfin hM ⊢
-  exact finite_cluster_subset_reachBall G ω y hfin M hM
-
- 
-
-theorem oldChain_local_support (G : BondGraph V E) (x : V)
-    (η : ℕ → Set E) (y : ℕ → V) (k M N : ℕ)
-    (hstart : y 0 ∈ reachBall G x N)
-    (hchain : ∀ i, 1 ≤ i → i ≤ k → Conn G (η i) (y i) (y (i-1)))
-    (hcaps : ∀ i, 1 ≤ i → i ≤ k →
-      (cluster G (η i) (y i)).Finite ∧ (cluster G (η i) (y i)).ncard ≤ M) :
-    (∀ i, i ≤ k → y i ∈ reachBall G x (N+i*M)) ∧
-      trajectoryUnion G η y k ⊆ reachBall G x (N+k*M) := by
-  have hvertex : ∀ i, i ≤ k → y i ∈ reachBall G x (N+i*M) := by
-    intro i
-    induction i with
-    | zero => simpa using hstart
-    | succ i ih =>
-      intro hik
-      obtain ⟨hf,hM⟩ := hcaps (i+1) (by omega) hik
-      have hc := capped_cluster_ball_of_mem G (η (i+1)) (y (i+1)) (y i) hf M hM
-        (by simpa only [Nat.add_sub_cancel,cluster,Set.mem_ofPred_eq] using hchain (i+1) (by omega) hik)
-      have hh := reachBall_triangle G x (y i) (N+i*M) M (ih (by omega))
-        (hc (mem_cluster_self G _ _))
-      convert hh using 1
-      congr 1
-      ring
-  refine ⟨hvertex,?_⟩
-  intro v hv
-  obtain ⟨i,hi,hik,hv⟩ := (mem_trajectoryUnion G η y k v).mp hv
-  obtain ⟨hf,hM⟩ := hcaps i hi hik
-  have hc := capped_cluster_ball_of_mem G (η i) (y i) (y (i-1)) hf M hM (hchain i hi hik)
-  have hh := reachBall_triangle G x (y (i-1)) (N+(i-1)*M) M
-    (hvertex (i-1) (by omega)) (hc hv)
-  apply reachBall_mono_local G x _ hh
-  have hmul : i*M ≤ k*M := Nat.mul_le_mul_right M hik
-  have he : (i-1)*M+M = i*M := by
-    calc _ = (i-1+1)*M := by ring
-         _ = i*M := by rw [Nat.sub_add_cancel hi]
-  omega
-
- 
-
-theorem SpliceTrace.masks_in_deterministic_ball (G : BondGraph V E)
-    (hloc : LocallyFinite G) (x : V) (η : ℕ → Set E) (y : ℕ → V)
-    (k j M N : ℕ) (ω ωf : Set E) (c : List (ℕ × E))
-    (h : SpliceTrace G x η y k j ω c ωf)
-    (hstart : y 0 ∈ reachBall G x N)
-    (hchain : ∀ i, 1 ≤ i → i ≤ k → Conn G (η i) (y i) (y (i-1)))
-    (hcaps : ∀ i, 1 ≤ i → i ≤ k →
-      (cluster G (η i) (y i)).Finite ∧ (cluster G (η i) (y i)).ncard ≤ M) :
-    maskUnion (replayMasks G x η y c ω) ⊆
-      (incidentSupport G hloc (finite_reachBall hloc x (N+k*M)).toFinset : Set E) := by
-  intro e he
-  obtain ⟨b,hb,hbe⟩ := h.masks_incident G x η y k j ω ωf c e he
-  have hbt := h.added_subset_trajectory G x η y k j ω ωf c hb
-  have hball := (oldChain_local_support G x η y k M N hstart hchain hcaps).2 hbt
-  exact (mem_incidentSupport G hloc _ e).mpr ⟨b,by simpa using hball,hbe⟩
-
- 
-
-theorem oldChain_query_support (G : BondGraph V E) (x : V)
-    (η : ℕ → Set E) (y : ℕ → V) (k M N R : ℕ)
-    (hstart : y 0 ∈ reachBall G x N)
-    (hchain : ∀ i, 1 ≤ i → i ≤ k → Conn G (η i) (y i) (y (i-1)))
-    (hcaps : ∀ i, 1 ≤ i → i ≤ k →
-      (cluster G (η i) (y i)).Finite ∧ (cluster G (η i) (y i)).ncard ≤ M)
-    (i : ℕ) (hi : i ≤ k) :
-    reachBall G (y i) R ⊆ reachBall G x (N+k*M+R) := by
-  have hv := (oldChain_local_support G x η y k M N hstart hchain hcaps).1 i hi
-  intro z hz
-  have hh := reachBall_triangle G x (y i) (N+i*M) R hv hz
-  exact reachBall_mono_local G x (by nlinarith) hh
-
-end SpliceLocalSupport
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section UniformContactSelector
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-noncomputable def uniformRank (K : ℕ) (hK : 0 < K) (t : unitInterval) : Fin K :=
-  ⟨min ⌊(K:ℝ)*(t:ℝ)⌋₊ (K-1), lt_of_le_of_lt (min_le_right ..) (by omega)⟩
-
-theorem uniformRank_val_of_lt_one (K : ℕ) (hK : 0 < K) (t : unitInterval)
-    (ht : (t:ℝ) < 1) : (uniformRank K hK t).val = ⌊(K:ℝ)*(t:ℝ)⌋₊ := by
-  have hp : (0:ℝ) < K := by exact_mod_cast hK
-  have hf : ⌊(K:ℝ)*(t:ℝ)⌋₊ < K := (Nat.floor_lt (mul_nonneg (Nat.cast_nonneg _) t.property.1)).mpr (by nlinarith)
-  exact min_eq_left (by omega)
-
-theorem uniformRank_eq_iff (K : ℕ) (hK : 0 < K) (t : unitInterval)
-    (ht : (t:ℝ) < 1) (a : Fin K) :
-    uniformRank K hK t = a ↔ t ∈ contactInterval K a.val := by
-  rw [Fin.ext_iff,uniformRank_val_of_lt_one K hK t ht,
-    Nat.floor_eq_iff (mul_nonneg (Nat.cast_nonneg _) t.property.1)]
-  have hp : (0:ℝ) < K := by exact_mod_cast hK
-  simp only [contactInterval,Set.mem_ofPred_eq,Nat.cast_add,Nat.cast_one]
-  rw [div_le_iff₀ hp,lt_div_iff₀ hp]
-  simp only [mul_comm]
-
-theorem volume_uniformRank_fiber (K : ℕ) (hK : 0 < K) (a : Fin K) :
-    volume {t : unitInterval | uniformRank K hK t = a} =
-      ENNReal.ofReal ((K:ℝ)⁻¹) := by
-  rw [← volume_contactInterval K a.val a.isLt]
-  apply measure_congr
-  filter_upwards [(volume : Measure unitInterval).ae_ne 1] with t ht
-  exact propext (uniformRank_eq_iff K hK t (lt_of_le_of_ne t.property.2
-    (fun he => ht (Subtype.ext he))) a)
-
- 
-
-noncomputable def uniformContact (K : Finset E) (hK : K.Nonempty)
-    (t : unitInterval) : E :=
-  ((K.equivFin).symm (uniformRank K.card (Finset.card_pos.mpr hK) t)).val
-
-theorem uniformContact_mem (K : Finset E) (hK : K.Nonempty) (t : unitInterval) :
-    uniformContact K hK t ∈ K :=
-  ((K.equivFin).symm (uniformRank K.card (Finset.card_pos.mpr hK) t)).property
-
-theorem uniformContact_eq_iff (K : Finset E) (hK : K.Nonempty) (t : unitInterval)
-    (e : E) (he : e ∈ K) :
-    uniformContact K hK t = e ↔
-      uniformRank K.card (Finset.card_pos.mpr hK) t = K.equivFin ⟨e,he⟩ := by
-  unfold uniformContact
-  change ((K.equivFin).symm (uniformRank K.card (Finset.card_pos.mpr hK) t)).val =
-    (⟨e,he⟩ : ↥K).val ↔ _
-  rw [← Subtype.ext_iff]
-  exact K.equivFin.symm_apply_eq
-
-theorem volume_uniformContact_fiber (K : Finset E) (hK : K.Nonempty)
-    (e : E) (he : e ∈ K) : volume {t : unitInterval | uniformContact K hK t = e} =
-      ENNReal.ofReal ((K.card:ℝ)⁻¹) := by
-  simp_rw [uniformContact_eq_iff K hK _ e he]
-  exact volume_uniformRank_fiber K.card (Finset.card_pos.mpr hK) _
-
- 
-
-noncomputable def chooseContact [Nonempty E] (seed : ℕ → unitInterval)
-    (i : ℕ) (K : Set E) : E :=
-  if hf : K.Finite then if hn : K.Nonempty then
-    uniformContact hf.toFinset (by simpa using hn) (seed i)
-  else Classical.arbitrary E else Classical.arbitrary E
-
-theorem chooseContact_mem [Nonempty E] (seed : ℕ → unitInterval)
-    (i : ℕ) (K : Set E) (hf : K.Finite) (hn : K.Nonempty) :
-    chooseContact seed i K ∈ K := by
-  simp only [chooseContact,dite_eq_left hf,dite_eq_left hn]
-  exact hf.mem_toFinset.mp (uniformContact_mem _ _ _)
-
-end UniformContactSelector
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section SeededTraceLaw
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable [Nonempty E]
-
- 
-theorem chooseContact_congr (u v : ℕ → unitInterval) (i : ℕ) (K : Set E)
-    (h : u i = v i) : chooseContact u i K = chooseContact v i K := by
-  unfold chooseContact
-  split_ifs <;> simp only [h]
-
- 
-noncomputable def replaySeedFiber (G : BondGraph V E) (x : V) (η : ℕ → Set E)
-    (y : ℕ → V) : List (ℕ × E) → Set E → ℕ → Set unitInterval
-  | [], _, _ => Set.univ
-  | (j,e)::c, ω, i =>
-    let S := cluster G ω x
-    let B := cluster G (η j ∩ internalEdges G Sᶜ) (y j)
-    if i = j then {t | chooseContact (fun _ => t) j (contactSet G (η j) B) = e}
-    else replaySeedFiber G x η y c (spliceField G S B e ω (η j)) i
-
-theorem replaySeedFiber_univ (G : BondGraph V E) (x : V) (η : ℕ → Set E)
-    (y : ℕ → V) (c : List (ℕ × E)) (ω : Set E) (i : ℕ)
-    (hi : ∀ q ∈ c, i ≠ q.1) : replaySeedFiber G x η y c ω i = Set.univ := by
-  induction c generalizing ω with
-  | nil => rfl
-  | cons q c ih =>
-    obtain ⟨j,e⟩ := q
-    simp only [replaySeedFiber,ite_eq_right (hi (j,e) (by simp))]
-    exact ih _ (fun q hq => hi q (by simp [hq]))
-
-theorem SpliceTrace.follows_iff_seedFiber (G : BondGraph V E) (x : V)
-    (η : ℕ → Set E) (y : ℕ → V) (k j : ℕ) (ω ωf : Set E)
-    (c : List (ℕ × E)) (h : SpliceTrace G x η y k j ω c ωf)
-    (seed : ℕ → unitInterval) :
-    FollowsContactRule G x η y (chooseContact seed) c ω ↔
-      ∀ i, 1 ≤ i → i ≤ k → seed i ∈ replaySeedFiber G x η y c ω i := by
-  induction h with
-  | stop j ω hz => simp [FollowsContactRule,replayContacts,replaySeedFiber]
-  | @step j l ω e a b c ωf hn hjl hlk hm hy he hab ha hb ht ih =>
-    rw [followsContactRule_cons,ih]
-    have htail : replaySeedFiber G x η y c
-        (spliceField G (cluster G ω x)
-          (cluster G (η l ∩ internalEdges G (cluster G ω x)ᶜ) (y l)) e ω (η l)) l = Set.univ := by
-      apply replaySeedFiber_univ
-      intro q hq
-      have hq' := ht.seam_data G x η y k l _ ωf c q hq
-      omega
-    constructor
-    · rintro ⟨hc,hr⟩ i hi hik
-      by_cases hil : i = l
-      · subst i
-        simp only [replaySeedFiber,ite_true,Set.mem_ofPred_eq]
-        exact (chooseContact_congr _ _ l _ rfl).trans hc
-      · simpa only [replaySeedFiber,ite_eq_right hil] using hr i hi hik
-    · intro hr
-      constructor
-      · have hh := hr l (by omega) hlk
-        simp only [replaySeedFiber,ite_true,Set.mem_ofPred_eq] at hh
-        exact (chooseContact_congr _ _ l _ rfl).trans hh
-      · intro i hi hik
-        by_cases hil : i = l
-        · subst i
-          rw [htail]
-          trivial
-        · simpa only [replaySeedFiber,ite_eq_right hil] using hr i hi hik
-
- 
-
-def finiteContactSeeds (k : ℕ) (u : Fin k → unitInterval) (i : ℕ) : unitInterval :=
-  if hi : 1 ≤ i ∧ i ≤ k then u ⟨i-1,by omega⟩ else 0
-
-@[simp] theorem finiteContactSeeds_at (k : ℕ) (u : Fin k → unitInterval) (i : Fin k) :
-    finiteContactSeeds k u (i.val+1) = u i := by
-  have hi : 1 ≤ i.val+1 ∧ i.val+1 ≤ k := ⟨by omega,by omega⟩
-  simp only [finiteContactSeeds,dite_eq_left hi,Nat.add_sub_cancel]
-
-theorem SpliceTrace.seed_event_eq_pi (G : BondGraph V E) (x : V)
-    (η : ℕ → Set E) (y : ℕ → V) (k j : ℕ) (ω ωf : Set E)
-    (c : List (ℕ × E)) (h : SpliceTrace G x η y k j ω c ωf) :
-    {u : Fin k → unitInterval | FollowsContactRule G x η y
-      (chooseContact (finiteContactSeeds k u)) c ω} =
-      Set.pi Set.univ (fun i => replaySeedFiber G x η y c ω (i.val+1)) := by
-  ext u
-  rw [Set.mem_ofPred_eq,h.follows_iff_seedFiber G x η y k j ω ωf c]
-  simp only [Set.mem_pi,Set.mem_univ,true_implies]
-  constructor
-  · intro hu i
-    simpa only [finiteContactSeeds_at] using hu (i.val+1) (by omega) (by omega)
-  · intro hu i hi hik
-    have hh := hu ⟨i-1,by omega⟩
-    have hii : 1 ≤ i ∧ i ≤ k := ⟨hi,hik⟩
-    simpa only [finiteContactSeeds,dite_eq_left hii,Nat.sub_add_cancel hi] using hh
-
- 
-
-theorem SpliceTrace.volume_seedFiber (G : BondGraph V E) (hloc : LocallyFinite G)
-    (x : V) (η : ℕ → Set E) (y : ℕ → V) (k j : ℕ) (ω ωf : Set E)
-    (c : List (ℕ × E)) (h : SpliceTrace G x η y k j ω c ωf)
-    (hcaps : ∀ i, 1 ≤ i → i ≤ k → (cluster G (η i) (y i)).Finite) :
-    ∀ i, volume (replaySeedFiber G x η y c ω i) =
-      ENNReal.ofReal ((traceContactCount G η y ωf x c i : ℝ)⁻¹) := by
-  induction c generalizing j ω with
-  | nil => simp [replaySeedFiber,traceContactCount]
-  | cons q c ih =>
-    obtain ⟨l,e⟩ := q
-    have hcut := h.head_cut G x η y k j l ω ωf e c
-    cases h with
-    | step j l ω e a b c ωf hn hjl hlk hm hy he hab ha hb ht =>
-      intro i
-      by_cases hil : i = l
-      · subst i
-        simp only [replaySeedFiber,ite_true,traceContactCount,seamContactCount,hcut]
-        let B := cluster G (η l ∩ internalEdges G (cluster G ω x)ᶜ) (y l)
-        let K := contactSet G (η l) B
-        have hB : B.Finite := (hcaps l (by omega) hlk).subset (fun _ h => conn_mono G Set.inter_subset_left h)
-        have hK : K.Finite := contactSet_finite G hloc (η l) B hB
-        have heK : e ∈ K := ⟨he,b,a,hab.trans (Sym2.eq_swap ..),hb,
-          fun haB => slit_cluster_subset G (cluster G ω x) (η l) (y l) hy haB ha⟩
-        have hnK : K.Nonempty := ⟨e,heK⟩
-        change volume {t : unitInterval | chooseContact (fun _ => t) l K = e} = _
-        simp only [chooseContact,dite_eq_left hK,dite_eq_left hnK]
-        rw [volume_uniformContact_fiber _ _ e (hK.mem_toFinset.mpr heK)]
-        rw [← Set.ncard_eq_toFinset_card K hK]
-      · simpa only [replaySeedFiber,traceContactCount,ite_eq_right hil] using ih l _ ht i
-
- 
-theorem SpliceTrace.seed_probability (G : BondGraph V E) (hloc : LocallyFinite G)
-    (x : V) (η : ℕ → Set E) (y : ℕ → V) (k j : ℕ) (ω ωf : Set E)
-    (c : List (ℕ × E)) (h : SpliceTrace G x η y k j ω c ωf)
-    (hcaps : ∀ i, 1 ≤ i → i ≤ k → (cluster G (η i) (y i)).Finite) :
-    ((volume : Measure (Fin k → unitInterval))
-      {u | FollowsContactRule G x η y (chooseContact (finiteContactSeeds k u)) c ω}).toReal =
-      ∏ i : Fin k, (traceContactCount G η y ωf x c (i.val+1):ℝ)⁻¹ := by
-  rw [h.seed_event_eq_pi G x η y k j ω ωf c,MeasureTheory.volume_pi,Measure.pi_pi,ENNReal.toReal_prod]
-  apply Finset.prod_congr rfl
-  intro i _
-  rw [h.volume_seedFiber G hloc x η y k j ω ωf c hcaps,ENNReal.toReal_ofReal (by positivity)]
-
-end SeededTraceLaw
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Splicing.Replay
+
+namespace OAI
+
+/-! Reference Laws for bond percolation. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section FiniteJointMass
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+def jointOutput (G : BondGraph V E) (x : V) (η₁ η₂ : ℕ → Set E)
+    (y₁ y₂ : ℕ → V) (ω₁ ω₂ ωf : Set E) (c₁ c₂ : List (ℕ × E)) : JointSpliceState E :=
+  (ωf,(runMasks (replayMasks G x η₁ y₁ c₁ ω₁) (ω₁,η₁)).2,
+    (runMasks (replayMasks G x η₂ y₂ c₂ ω₂) (ω₂,η₂)).2)
+
+def jointRestored (G : BondGraph V E) (x : V) (η₁ η₂ : ℕ → Set E)
+    (y₂ : ℕ → V) (ω₁ ω₂ : Set E) (c₂ : List (ℕ × E)) : JointSpliceState E :=
+  (exchangeOn (maskUnion (replayMasks G x η₂ y₂ c₂ ω₂)) ω₁ ω₂,η₁,η₂)
+
+ 
+
+theorem finite_joint_mass_le {I : Type*} [Fintype I]
+    (G : BondGraph V E) (x : V) (k : ℕ) (p : unitInterval) (Q : Finset E)
+    (η₁ η₂ : I → ℕ → Set E) (y₁ y₂ : (ℕ → Set E) → ℕ → V)
+    (choose₁ choose₂ : ℕ → Set E → E)
+    (ω₁ ω₂ ωf : I → Set E) (c₁ c₂ : I → List (ℕ × E))
+    (hir : Function.Injective (fun a =>
+      (jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a),
+        c₁ a,c₂ a)))
+    (h₁ : ∀ a, SpliceTrace G x (η₁ a) (y₁ (η₁ a)) k 0 (ω₁ a) (c₁ a) (ωf a))
+    (h₂ : ∀ a, SpliceTrace G x (η₂ a) (y₂ (η₂ a)) k 0 (ω₂ a) (c₂ a) (ωf a))
+    (hc₁ : ∀ a, c₁ a ≠ []) (hc₂ : ∀ a, c₂ a ≠ [])
+    (hsep : ∀ a, GraphSeparated G (cluster G (ωf a) x \ cluster G (ω₁ a) x)
+      (cluster G (ωf a) x \ cluster G (ω₂ a) x))
+    (hr₁ : ∀ a, FollowsContactRule G x (η₁ a) (y₁ (η₁ a)) choose₁ (c₁ a) (ω₁ a))
+    (hr₂ : ∀ a, FollowsContactRule G x (η₂ a) (y₂ (η₂ a)) choose₂ (c₂ a) (ω₂ a))
+    (B : Finset (JointSpliceState E))
+    (hB : ∀ a, jointRestored G x (η₁ a) (η₂ a) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (c₂ a) ∈ B) :
+    (∑ a, jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p
+      (jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a))) ≤
+      ∑ s ∈ B, jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p s := by
+  let out (a : I) := jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a))
+    (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a)
+  let rest (a : I) := jointRestored G x (η₁ a) (η₂ a) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (c₂ a)
+  have hinj : Function.Injective rest := by
+    intro a b he
+    have hη₁ : η₁ a = η₁ b := congrArg (fun s : JointSpliceState E => s.2.1) he
+    have hη₂ : η₂ a = η₂ b := congrArg (fun s : JointSpliceState E => s.2.2) he
+    have gb₁ : SpliceTrace G x (η₁ a) (y₁ (η₁ a)) k 0 (ω₁ b) (c₁ b) (ωf b) := by
+      rw [hη₁]; exact h₁ b
+    have gb₂ : SpliceTrace G x (η₂ a) (y₂ (η₂ a)) k 0 (ω₂ b) (c₂ b) (ωf b) := by
+      rw [hη₂]; exact h₂ b
+    have grb₁ : FollowsContactRule G x (η₁ a) (y₁ (η₁ a)) choose₁ (c₁ b) (ω₁ b) := by
+      rw [hη₁]; exact hr₁ b
+    have grb₂ : FollowsContactRule G x (η₂ a) (y₂ (η₂ a)) choose₂ (c₂ b) (ω₂ b) := by
+      rw [hη₂]; exact hr₂ b
+    have hfield : exchangeOn (maskUnion (replayMasks G x (η₂ a) (y₂ (η₂ a)) (c₂ a) (ω₂ a))) (ω₁ a) (ω₂ a) =
+        exchangeOn (maskUnion (replayMasks G x (η₂ a) (y₂ (η₂ a)) (c₂ b) (ω₂ b))) (ω₁ b) (ω₂ b) := by
+      have hh := congrArg Prod.fst he
+      simpa only [rest,jointRestored,← hη₂] using hh
+    obtain ⟨hcode₁,hcode₂,hfieldf,haux₁,haux₂⟩ := joint_restore_seeded_unique G x
+      (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) choose₁ choose₂ k 0 0
+      (ω₁ a) (ω₂ a) (ωf a) (ω₁ b) (ω₂ b) (ωf b)
+      (c₁ a) (c₂ a) (c₁ b) (c₂ b) (h₁ a) (h₂ a) gb₁ gb₂
+      (hc₁ a) (hc₂ a) (hc₁ b) (hc₂ b) (hsep a) (hsep b)
+      (hr₁ a) (hr₂ a) grb₁ grb₂ hfield
+    have hout : out a = out b := by
+      apply Prod.ext hfieldf
+      apply Prod.ext
+      · simpa only [out,jointOutput,hη₁] using haux₁
+      · simpa only [out,jointOutput,hη₂] using haux₂
+    exact hir (Prod.ext hout (Prod.ext hcode₁ hcode₂))
+  have hw : ∀ a, jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p (out a) =
+      jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p (rest a) := by
+    intro a
+    apply (h₁ a).joint_weight G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) k 0 0
+      (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a) (h₂ a) (hsep a) Q (Finset.Icc 1 k) (Finset.Icc 1 k) p
+    · intro q hq
+      have hr := (h₁ a).seam_data G x (η₁ a) (y₁ (η₁ a)) k 0 (ω₁ a) (ωf a) (c₁ a) q hq
+      exact Finset.mem_Icc.mpr ⟨hr.1,hr.2.1⟩
+    · intro q hq
+      have hr := (h₂ a).seam_data G x (η₂ a) (y₂ (η₂ a)) k 0 (ω₂ a) (ωf a) (c₂ a) q hq
+      exact Finset.mem_Icc.mpr ⟨hr.1,hr.2.1⟩
+  change (∑ a, jointTupleWeight Q _ _ p (out a)) ≤ _
+  simp_rw [hw]
+  rw [← Finset.sum_image hinj.injOn]
+  apply Finset.sum_le_sum_of_subset_of_nonneg
+  · intro s hs
+    obtain ⟨a,_,rfl⟩ := Finset.mem_image.mp hs
+    exact hB a
+  · intro s _ _
+    exact jointTupleWeight_nonneg Q _ _ p s
+
+end FiniteJointMass
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section TrajectorySupport
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+def trajectoryUnion (G : BondGraph V E) (η : ℕ → Set E) (y : ℕ → V) : ℕ → Set V
+  | 0 => ∅
+  | k+1 => trajectoryUnion G η y k ∪ cluster G (η (k+1)) (y (k+1))
+
+theorem mem_trajectoryUnion (G : BondGraph V E) (η : ℕ → Set E) (y : ℕ → V)
+    (k : ℕ) (v : V) : v ∈ trajectoryUnion G η y k ↔
+      ∃ i, 1 ≤ i ∧ i ≤ k ∧ v ∈ cluster G (η i) (y i) := by
+  induction k with
+  | zero => simp [trajectoryUnion]
+  | succ k ih =>
+    simp only [trajectoryUnion,Set.mem_union,ih]
+    constructor
+    · rintro (⟨i,hi,hik,hv⟩ | hv)
+      · exact ⟨i,hi,by omega,hv⟩
+      · exact ⟨k+1,by omega,by omega,hv⟩
+    · rintro ⟨i,hi,hik,hv⟩
+      by_cases he : i = k+1
+      · exact Or.inr (he ▸ hv)
+      · exact Or.inl ⟨i,hi,by omega,hv⟩
+
+theorem trajectoryUnion_finite_cap (G : BondGraph V E) (η : ℕ → Set E) (y : ℕ → V)
+    (k M : ℕ) (hcaps : ∀ i, 1 ≤ i → i ≤ k →
+      (cluster G (η i) (y i)).Finite ∧ (cluster G (η i) (y i)).ncard ≤ M) :
+    (trajectoryUnion G η y k).Finite ∧ (trajectoryUnion G η y k).ncard ≤ k*M := by
+  induction k with
+  | zero => simp [trajectoryUnion]
+  | succ k ih =>
+    obtain ⟨hf,hc⟩ := ih (fun i hi hik => hcaps i hi (by omega))
+    obtain ⟨hf',hc'⟩ := hcaps (k+1) (by omega) (by omega)
+    refine ⟨hf.union hf',?_⟩
+    calc
+      _ ≤ (trajectoryUnion G η y k).ncard + (cluster G (η (k+1)) (y (k+1))).ncard := Set.ncard_union_le ..
+      _ ≤ k*M+M := Nat.add_le_add hc hc'
+      _ = (k+1)*M := by ring
+
+ 
+
+theorem SpliceTrace.added_subset_trajectory (G : BondGraph V E) (x : V)
+    (η : ℕ → Set E) (y : ℕ → V) (k j : ℕ) (ω ωf : Set E) (c : List (ℕ × E))
+    (h : SpliceTrace G x η y k j ω c ωf) :
+    cluster G ωf x \ cluster G ω x ⊆ trajectoryUnion G η y k := by
+  induction h with
+  | stop => exact fun _ hv => (hv.2 hv.1).elim
+  | @step j i ω e a b rest ωf hn hji hik hmax hy he hab ha hb ht ih =>
+    let S := cluster G ω x
+    let B := cluster G (η i ∩ internalEdges G Sᶜ) (y i)
+    let ω' := spliceField G S B e ω (η i)
+    have hC : cluster G ω' x = S ∪ B := splice_cluster G ω (η i) x (y i) hy e a b hab he ha hb
+    intro v hv
+    by_cases hv' : v ∈ cluster G ω' x
+    · rw [hC] at hv'
+      have hvB := hv'.resolve_left hv.2
+      exact (mem_trajectoryUnion G η y k v).mpr
+        ⟨i,by omega,hik,conn_mono G Set.inter_subset_left hvB⟩
+    · exact ih ⟨hv.1,hv'⟩
+
+ 
+
+theorem SpliceTrace.trajectory_hits (G : BondGraph V E) (x : V)
+    (η : ℕ → Set E) (y : ℕ → V) (k j : ℕ) (ω ωf : Set E) (c : List (ℕ × E))
+    (h : SpliceTrace G x η y k j ω c ωf) (hc : c ≠ []) :
+    (cluster G ω x ∩ trajectoryUnion G η y k).Nonempty := by
+  cases h with
+  | stop => exact (hc rfl).elim
+  | step j i ω e a b rest ωf hn hji hik hmax hy he hab ha hb ht =>
+    exact ⟨a,ha,(mem_trajectoryUnion G η y k a).mpr ⟨i,by omega,hik,
+      (edge_cluster_sides G (η i) (y i) he hab).mpr (conn_mono G Set.inter_subset_left hb)⟩⟩
+
+ 
+
+theorem SpliceTrace.added_finite_cap (G : BondGraph V E) (x : V)
+    (η : ℕ → Set E) (y : ℕ → V) (k j M : ℕ) (ω ωf : Set E) (c : List (ℕ × E))
+    (h : SpliceTrace G x η y k j ω c ωf)
+    (hcaps : ∀ i, 1 ≤ i → i ≤ k →
+      (cluster G (η i) (y i)).Finite ∧ (cluster G (η i) (y i)).ncard ≤ M) :
+    (cluster G ωf x \ cluster G ω x).Finite ∧ (cluster G ωf x \ cluster G ω x).ncard ≤ k*M := by
+  obtain ⟨hf,hc⟩ := trajectoryUnion_finite_cap G η y k M hcaps
+  have hs := h.added_subset_trajectory G x η y k j ω ωf c
+  exact ⟨hf.subset hs,(Set.ncard_le_ncard hs hf).trans hc⟩
+
+end TrajectorySupport
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section JointReferenceEvent
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem SpliceTrace.joint_restored_field_symm (G : BondGraph V E) (x : V)
+    (η₁ η₂ : ℕ → Set E) (y₁ y₂ : ℕ → V) (k j₁ j₂ : ℕ)
+    (ω₁ ω₂ ωf : Set E) (c₁ c₂ : List (ℕ × E))
+    (h₁ : SpliceTrace G x η₁ y₁ k j₁ ω₁ c₁ ωf)
+    (h₂ : SpliceTrace G x η₂ y₂ k j₂ ω₂ c₂ ωf)
+    (hsep : GraphSeparated G (cluster G ωf x \ cluster G ω₁ x)
+      (cluster G ωf x \ cluster G ω₂ x)) :
+    (jointRestored G x η₁ η₂ y₂ ω₁ ω₂ c₂).1 =
+      (jointRestored G x η₂ η₁ y₁ ω₂ ω₁ c₁).1 := by
+  apply exchange_patch_symm _ _ ω₁ ω₂ ωf
+    (h₁.masks_disjoint G x η₁ η₂ y₁ y₂ k j₁ j₂ ω₁ ω₂ ωf c₁ c₂ h₂ hsep)
+  · intro e he
+    rw [← h₁.replay_eq G x η₁ y₁ k j₁ ω₁ ωf c₁]
+    exact (replaySplice_off_masks G x η₁ y₁ c₁ ω₁ e he).symm
+  · intro e he
+    rw [← h₂.replay_eq G x η₂ y₂ k j₂ ω₂ ωf c₂]
+    exact (replaySplice_off_masks G x η₂ y₂ c₂ ω₂ e he).symm
+
+theorem SpliceTrace.joint_restored_root (G : BondGraph V E) (x : V)
+    (η₁ η₂ : ℕ → Set E) (y₁ y₂ : ℕ → V) (k j₁ j₂ : ℕ)
+    (ω₁ ω₂ ωf : Set E) (c₁ c₂ : List (ℕ × E))
+    (h₁ : SpliceTrace G x η₁ y₁ k j₁ ω₁ c₁ ωf)
+    (h₂ : SpliceTrace G x η₂ y₂ k j₂ ω₂ c₂ ωf)
+    (hc₁ : c₁ ≠ []) (hc₂ : c₂ ≠ [])
+    (hsep : GraphSeparated G (cluster G ωf x \ cluster G ω₁ x)
+      (cluster G ωf x \ cluster G ω₂ x)) :
+    cluster G (jointRestored G x η₁ η₂ y₂ ω₁ ω₂ c₂).1 x =
+      cluster G ω₁ x ∩ cluster G ω₂ x := by
+  cases c₁ with
+  | nil => exact (hc₁ rfl).elim
+  | cons q r₁ =>
+    obtain ⟨i₁,e₁⟩ := q
+    cases c₂ with
+    | nil => exact (hc₂ rfl).elim
+    | cons q r₂ =>
+      obtain ⟨i₂,e₂⟩ := q
+      exact h₁.joint_root G x η₁ η₂ y₁ y₂ k j₁ j₂ i₁ i₂ ω₁ ω₂ ωf e₁ e₂ r₁ r₂ h₂ hsep
+
+ 
+
+theorem SpliceTrace.joint_trajectory_hits (G : BondGraph V E) (x : V)
+    (η₁ η₂ : ℕ → Set E) (y₁ y₂ : ℕ → V) (k j₁ j₂ : ℕ)
+    (ω₁ ω₂ ωf : Set E) (c₁ c₂ : List (ℕ × E))
+    (h₁ : SpliceTrace G x η₁ y₁ k j₁ ω₁ c₁ ωf)
+    (h₂ : SpliceTrace G x η₂ y₂ k j₂ ω₂ c₂ ωf)
+    (hc₁ : c₁ ≠ []) (hc₂ : c₂ ≠ [])
+    (hsep : GraphSeparated G (cluster G ωf x \ cluster G ω₁ x)
+      (cluster G ωf x \ cluster G ω₂ x)) :
+    let A := cluster G (jointRestored G x η₁ η₂ y₂ ω₁ ω₂ c₂).1 x
+    (A ∩ trajectoryUnion G η₁ y₁ k).Nonempty ∧ (A ∩ trajectoryUnion G η₂ y₂ k).Nonempty := by
+  intro A
+  have hv₁ := (h₁.joint_virtual G x η₁ η₂ y₁ y₂ k j₁ j₂ ω₁ ω₂ ωf c₁ c₂ h₂ hc₁ hc₂ hsep).1
+  have hv₂ := (h₂.joint_virtual G x η₂ η₁ y₂ y₁ k j₂ j₁ ω₂ ω₁ ωf c₂ c₁ h₁ hc₂ hc₁
+    (hsep.symm G _ _)).1
+  refine ⟨hv₁.trajectory_hits G x η₁ y₁ k j₁ _ ω₂ c₁ hc₁,?_⟩
+  change (cluster G (jointRestored G x η₁ η₂ y₂ ω₁ ω₂ c₂).1 x ∩ _).Nonempty
+  rw [h₁.joint_restored_field_symm G x η₁ η₂ y₁ y₂ k j₁ j₂ ω₁ ω₂ ωf c₁ c₂ h₂ hsep]
+  exact hv₂.trajectory_hits G x η₂ y₂ k j₂ _ ω₁ c₂ hc₂
+
+ 
+
+theorem SpliceTrace.joint_restored_size (G : BondGraph V E) (x : V)
+    (η₁ η₂ : ℕ → Set E) (y₁ y₂ : ℕ → V) (k j₁ j₂ n M : ℕ)
+    (ω₁ ω₂ ωf : Set E) (c₁ c₂ : List (ℕ × E))
+    (h₁ : SpliceTrace G x η₁ y₁ k j₁ ω₁ c₁ ωf)
+    (h₂ : SpliceTrace G x η₂ y₂ k j₂ ω₂ c₂ ωf)
+    (hc₁ : c₁ ≠ []) (hc₂ : c₂ ≠ [])
+    (hsep : GraphSeparated G (cluster G ωf x \ cluster G ω₁ x)
+      (cluster G ωf x \ cluster G ω₂ x))
+    (hf : (cluster G ω₁ x).Finite) (hn : n ≤ (cluster G ω₁ x).ncard)
+    (h2n : (cluster G ω₁ x).ncard < 2*n)
+    (hcap₂ : ∀ i, 1 ≤ i → i ≤ k →
+      (cluster G (η₂ i) (y₂ i)).Finite ∧ (cluster G (η₂ i) (y₂ i)).ncard ≤ M) :
+    let A := cluster G (jointRestored G x η₁ η₂ y₂ ω₁ ω₂ c₂).1 x
+    A.Finite ∧ n-k*M ≤ A.ncard ∧ A.ncard < 2*n := by
+  intro A
+  have hA : A = cluster G ω₁ x ∩ cluster G ω₂ x :=
+    h₁.joint_restored_root G x η₁ η₂ y₁ y₂ k j₁ j₂ ω₁ ω₂ ωf c₁ c₂ h₂ hc₁ hc₂ hsep
+  have hdiff : A = cluster G ω₁ x \ (cluster G ωf x \ cluster G ω₂ x) := by
+    rw [hA]
+    ext v
+    constructor
+    · rintro ⟨h1,h2⟩
+      exact ⟨h1,fun hh => hh.2 h2⟩
+    · rintro ⟨h1,h2⟩
+      refine ⟨h1,?_⟩
+      by_contra hn
+      exact h2 ⟨h₁.cluster_subset G x η₁ y₁ k j₁ ω₁ ωf c₁ h1,hn⟩
+  obtain ⟨hBfin,hBcap⟩ := h₂.added_finite_cap G x η₂ y₂ k j₂ M ω₂ ωf c₂ hcap₂
+  have hlower := Set.ncard_le_ncard_sdiff_add_ncard (cluster G ω₁ x)
+    (cluster G ωf x \ cluster G ω₂ x) hBfin
+  rw [← hdiff] at hlower
+  refine ⟨hA ▸ hf.inter_of_left _,by omega,?_⟩
+  rw [hA]
+  exact (Set.ncard_inter_le_ncard_left _ _ hf).trans_lt h2n
+
+end JointReferenceEvent
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section SmallHanging
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+def hangingSide (G : BondGraph V E) (ω : Set E) (x : V) (e : E) : Set V :=
+  cluster G ω x \ cluster G (ω \ {e}) x
+
+theorem hangingSide_finite (G : BondGraph V E) (ω : Set E) (x : V) (e : E)
+    (hC : (cluster G ω x).Finite) : (hangingSide G ω x e).Finite := hC.subset Set.sdiff_subset
+
+theorem mem_hangingSide_iff_pivotal (G : BondGraph V E) (ω : Set E) (x z : V)
+    (e : E) (he : e ∈ ω) : z ∈ hangingSide G ω x e ↔ OpenMarkedPivotal G ω {z} x e := by
+  constructor
+  · rintro ⟨hz,hn⟩
+    exact ⟨he,⟨z,rfl,hz⟩,fun ⟨y,hy,hxy⟩ => hn ((Set.mem_singleton_iff.mp hy) ▸ hxy)⟩
+  · rintro ⟨_,⟨y,hy,hxy⟩,hn⟩
+    exact ⟨(Set.mem_singleton_iff.mp hy) ▸ hxy,fun hz => hn ⟨z,rfl,hz⟩⟩
+
+theorem hangingSide_ncard_add_rank (G : BondGraph V E) (ω : Set E) (x : V)
+    (e : E) (hC : (cluster G ω x).Finite) :
+    (hangingSide G ω x e).ncard + pivotalRank G ω x e = (cluster G ω x).ncard := by
+  have hs : cluster G (ω \ {e}) x ⊆ cluster G ω x := fun _ h => conn_mono G Set.sdiff_subset h
+  rw [hangingSide,Set.ncard_sdiff hs (hC.subset hs)]
+  exact Nat.sub_add_cancel (Set.ncard_le_ncard hs hC)
+
+ 
+
+noncomputable def smallSeparatingBonds (G : BondGraph V E) (ω : Set E) (x z : V)
+    (v : ℕ) : Finset E :=
+  (separatingBonds G ω x z).filter fun e => (hangingSide G ω x e).ncard ≤ v
+
+theorem smallSeparatingBonds_card_le (G : BondGraph V E) (ω : Set E) (x z : V)
+    (v : ℕ) (hC : (cluster G ω x).Finite) : (smallSeparatingBonds G ω x z v).card ≤ v := by
+  have hh := Finset.card_le_card_of_injOn (s := smallSeparatingBonds G ω x z v)
+    (t := Finset.Icc 1 v) (fun e => (hangingSide G ω x e).ncard)
+  have hmap : Set.MapsTo (fun e => (hangingSide G ω x e).ncard)
+      (↑(smallSeparatingBonds G ω x z v)) (↑(Finset.Icc 1 v)) := by
+    intro e he
+    obtain ⟨he,hcap⟩ := Finset.mem_filter.mp he
+    have hp := (mem_separatingBonds G ω x z e).mp he
+    exact Finset.mem_Icc.mpr ⟨(Set.ncard_pos (hangingSide_finite G ω x e hC)).mpr
+      ⟨z,(mem_hangingSide_iff_pivotal G ω x z e hp.1).mpr hp⟩,hcap⟩
+  have hinj : Set.InjOn (fun e => (hangingSide G ω x e).ncard)
+      (↑(smallSeparatingBonds G ω x z v)) := by
+    intro e he f hf heq
+    apply pivotalRank_injective_of_cluster_finite G ω {z} x hC
+      ((mem_separatingBonds G ω x z e).mp (Finset.mem_filter.mp he).1)
+      ((mem_separatingBonds G ω x z f).mp (Finset.mem_filter.mp hf).1)
+    change (hangingSide G ω x e).ncard = (hangingSide G ω x f).ncard at heq
+    have he' := hangingSide_ncard_add_rank G ω x e hC
+    have hf' := hangingSide_ncard_add_rank G ω x f hC
+    omega
+  simpa only [Nat.card_Icc, Nat.add_sub_cancel] using hh hmap hinj
+
+noncomputable def smallHangingThrough (G : BondGraph V E) (ω : Set E) (x z : V)
+    (v : ℕ) (hC : (cluster G ω x).Finite) : Finset V :=
+  (smallSeparatingBonds G ω x z v).biUnion fun e => (hangingSide_finite G ω x e hC).toFinset
+
+theorem mem_smallHangingThrough (G : BondGraph V E) (ω : Set E) (x z w : V)
+    (v : ℕ) (hC : (cluster G ω x).Finite) : w ∈ smallHangingThrough G ω x z v hC ↔
+      ∃ e, OpenMarkedPivotal G ω {z} x e ∧ (hangingSide G ω x e).ncard ≤ v ∧
+        w ∈ hangingSide G ω x e := by
+  simp [smallHangingThrough, smallSeparatingBonds, and_assoc]
+
+theorem smallHangingThrough_card_le (G : BondGraph V E) (ω : Set E) (x z : V)
+    (v : ℕ) (hC : (cluster G ω x).Finite) :
+    (smallHangingThrough G ω x z v hC).card ≤ v*v := by
+  calc
+    _ ≤ ∑ e ∈ smallSeparatingBonds G ω x z v, (hangingSide_finite G ω x e hC).toFinset.card :=
+      Finset.card_biUnion_le
+    _ ≤ ∑ _e ∈ smallSeparatingBonds G ω x z v, v := by
+      apply Finset.sum_le_sum
+      intro e he
+      rw [← Set.ncard_eq_toFinset_card (hangingSide G ω x e) (hangingSide_finite G ω x e hC)]
+      exact (Finset.mem_filter.mp he).2
+    _ = (smallSeparatingBonds G ω x z v).card*v := by simp
+    _ ≤ v*v := Nat.mul_le_mul_right v (smallSeparatingBonds_card_le G ω x z v hC)
+
+end SmallHanging
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section NearHanging
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem hangingSide_laminar (G : BondGraph V E) (ω : Set E) (x : V)
+    (e f : E) (he : e ∈ ω) (hf : f ∈ ω) :
+    Disjoint (hangingSide G ω x e) (hangingSide G ω x f) ∨
+      hangingSide G ω x e ⊆ hangingSide G ω x f ∨
+      hangingSide G ω x f ⊆ hangingSide G ω x e := by
+  by_cases hd : Disjoint (hangingSide G ω x e) (hangingSide G ω x f)
+  · exact Or.inl hd
+  obtain ⟨z,hze,hzf⟩ := Set.not_disjoint_iff.mp hd
+  by_cases hef : e = f
+  · exact Or.inr (Or.inl (hef ▸ Set.Subset.refl _))
+  rcases pivotal_rootSides_nested G ω {z} x
+    ((mem_hangingSide_iff_pivotal G ω x z e he).mp hze)
+    ((mem_hangingSide_iff_pivotal G ω x z f hf).mp hzf) hef with h | h
+  · exact Or.inr (Or.inr (Set.sdiff_subset_sdiff_right h.1))
+  · exact Or.inr (Or.inl (Set.sdiff_subset_sdiff_right h.1))
+
+noncomputable def graphNeighbors (G : BondGraph V E) (hloc : LocallyFinite G)
+    (A : Finset V) : Finset V := A.biUnion (neighbors G hloc)
+
+theorem graphNeighbors_card_le (G : BondGraph V E) (hloc : LocallyFinite G)
+    (A : Finset V) (d : ℕ) (hdeg : ∀ a, degree G a ≤ d) :
+    (graphNeighbors G hloc A).card ≤ A.card*d := by
+  calc
+    _ ≤ ∑ a ∈ A, (neighbors G hloc a).card := Finset.card_biUnion_le
+    _ ≤ ∑ _a ∈ A, d := Finset.sum_le_sum fun a _ =>
+      (neighbors_card_le_degree G hloc a).trans (hdeg a)
+    _ = _ := by simp
+
+ 
+
+noncomputable def nearHangingEndpoints (G : BondGraph V E) (hloc : LocallyFinite G)
+    (ω : Set E) (x z : V) (e : E) (v : ℕ) (hC : (cluster G ω x).Finite) : Finset V :=
+  ((hangingSide_finite G ω x e hC).toFinset ∪ smallHangingThrough G ω x z v hC) ∪
+    (graphNeighbors G hloc (hangingSide_finite G ω x e hC).toFinset).biUnion
+      (fun a => smallHangingThrough G ω x a v hC)
+
+theorem nearHangingEndpoints_card_le (G : BondGraph V E) (hloc : LocallyFinite G)
+    (ω : Set E) (x z : V) (e : E) (v d : ℕ) (hC : (cluster G ω x).Finite)
+    (hcap : (hangingSide G ω x e).ncard ≤ v) (hdeg : ∀ a, degree G a ≤ d) :
+    (nearHangingEndpoints G hloc ω x z e v hC).card ≤ v+v^2+d*v^3 := by
+  let B := (hangingSide_finite G ω x e hC).toFinset
+  let N := graphNeighbors G hloc B
+  have hB : B.card ≤ v := by
+    simpa only [B,← Set.ncard_eq_toFinset_card (hangingSide G ω x e)
+      (hangingSide_finite G ω x e hC)] using hcap
+  have hN : N.card ≤ v*d := (graphNeighbors_card_le G hloc B d hdeg).trans
+    (Nat.mul_le_mul_right d hB)
+  have hU : (N.biUnion fun a => smallHangingThrough G ω x a v hC).card ≤ d*v^3 := by
+    calc
+      _ ≤ ∑ a ∈ N, (smallHangingThrough G ω x a v hC).card := Finset.card_biUnion_le
+      _ ≤ ∑ _a ∈ N, v*v := Finset.sum_le_sum fun a _ => smallHangingThrough_card_le G ω x a v hC
+      _ = N.card*(v*v) := by simp
+      _ ≤ (v*d)*(v*v) := Nat.mul_le_mul_right (v*v) hN
+      _ = d*v^3 := by ring
+  calc
+    _ ≤ (B ∪ smallHangingThrough G ω x z v hC).card +
+        (N.biUnion fun a => smallHangingThrough G ω x a v hC).card := Finset.card_union_le ..
+    _ ≤ (B.card + (smallHangingThrough G ω x z v hC).card) + d*v^3 :=
+      Nat.add_le_add (Finset.card_union_le ..) hU
+    _ ≤ (v+v*v)+d*v^3 := Nat.add_le_add_right
+      (Nat.add_le_add hB (smallHangingThrough_card_le G ω x z v hC)) _
+    _ = v+v^2+d*v^3 := by ring
+
+theorem mem_nearHangingEndpoints_of_not_separated (G : BondGraph V E)
+    (hloc : LocallyFinite G) (ω : Set E) (x z w : V) (e f : E) (v : ℕ)
+    (hC : (cluster G ω x).Finite)
+    (he : OpenMarkedPivotal G ω {z} x e) (hf : OpenMarkedPivotal G ω {w} x f)
+    (hcap : (hangingSide G ω x f).ncard ≤ v)
+    (hn : ¬GraphSeparated G (hangingSide G ω x e) (hangingSide G ω x f)) :
+    w ∈ nearHangingEndpoints G hloc ω x z e v hC := by
+  have hz := (mem_hangingSide_iff_pivotal G ω x z e he.1).mpr he
+  have hw := (mem_hangingSide_iff_pivotal G ω x w f hf.1).mpr hf
+  rcases hangingSide_laminar G ω x e f he.1 hf.1 with hd | hsub | hsub
+  · have hadj : ∃ a ∈ hangingSide G ω x e, ∃ b ∈ hangingSide G ω x f, Adj G a b := by
+      by_contra h
+      apply hn
+      refine ⟨hd,?_⟩
+      intro a ha b hb hab
+      exact h ⟨a,ha,b,hb,hab⟩
+    obtain ⟨a,ha,b,hb,hab⟩ := hadj
+    apply Finset.mem_union_right
+    apply Finset.mem_biUnion.mpr
+    refine ⟨b,Finset.mem_biUnion.mpr ⟨a,?_,(mem_neighbors G hloc a b).mpr hab⟩,?_⟩
+    · exact (hangingSide_finite G ω x e hC).mem_toFinset.mpr ha
+    · exact (mem_smallHangingThrough G ω x b w v hC).mpr
+        ⟨f,(mem_hangingSide_iff_pivotal G ω x b f hf.1).mp hb,hcap,hw⟩
+  · apply Finset.mem_union_left
+    apply Finset.mem_union_right
+    exact (mem_smallHangingThrough G ω x z w v hC).mpr
+      ⟨f,(mem_hangingSide_iff_pivotal G ω x z f hf.1).mp (hsub hz),hcap,hw⟩
+  · apply Finset.mem_union_left
+    apply Finset.mem_union_left
+    exact (hangingSide_finite G ω x e hC).mem_toFinset.mpr (hsub hw)
+
+end NearHanging
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section FiniteReferenceLaw
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+abbrev FiniteField (Q : Finset E) := ↥Q.powerset
+
+def finiteFieldSet (Q : Finset E) (ω : FiniteField Q) : Set E := (ω.val : Set E)
+
+noncomputable def finiteFieldWeight (Q : Finset E) (p : ℝ) (ω : FiniteField Q) : ℝ :=
+  bernoulliWeight Q ω.val p
+
+@[simp] theorem domainBits_finiteFieldSet (Q : Finset E) (ω : FiniteField Q) :
+    domainBits Q (finiteFieldSet Q ω) = ω.val := by
+  ext e
+  have hs := Finset.mem_powerset.mp ω.property
+  simp only [mem_domainBits,finiteFieldSet,Finset.mem_coe]
+  exact ⟨fun h => h.2,fun h => ⟨hs h,h⟩⟩
+
+theorem finiteFieldWeight_nonneg (Q : Finset E) (p : unitInterval) (ω : FiniteField Q) :
+    0 ≤ finiteFieldWeight Q p ω := bernoulliWeight_nonneg _ _ p
+
+theorem sum_finiteFieldWeight (Q : Finset E) (p : ℝ) :
+    ∑ ω : FiniteField Q, finiteFieldWeight Q p ω = 1 := by
+  rw [show (fun ω : FiniteField Q => finiteFieldWeight Q p ω) =
+    (fun ω : FiniteField Q => bernoulliWeight Q ω.val p) from rfl,Finset.sum_coe_sort Q.powerset (fun ω => bernoulliWeight Q ω p)]
+  simpa only [bernoulliPolynomial,bernoulliWeight,mul_one,one_mul] using bernoulliPolynomial_const Q 1 p
+
+ 
+
+def finiteFieldFamily (Q : Finset E) (J : Finset ℕ) (η : ↥J → FiniteField Q) : ℕ → Set E :=
+  fun i => if hi : i ∈ J then finiteFieldSet Q (η ⟨i,hi⟩) else ∅
+
+@[simp] theorem finiteFieldFamily_at (Q : Finset E) (J : Finset ℕ)
+    (η : ↥J → FiniteField Q) (i : ↥J) :
+    finiteFieldFamily Q J η i = finiteFieldSet Q (η i) := by
+  simp [finiteFieldFamily,i.property]
+
+noncomputable def finiteFamilyWeight (Q : Finset E) (J : Finset ℕ) (p : ℝ)
+    (η : ↥J → FiniteField Q) : ℝ := ∏ i, finiteFieldWeight Q p (η i)
+
+theorem finiteFamilyWeight_nonneg (Q : Finset E) (J : Finset ℕ) (p : unitInterval)
+    (η : ↥J → FiniteField Q) : 0 ≤ finiteFamilyWeight Q J p η :=
+  Finset.prod_nonneg fun _ _ => finiteFieldWeight_nonneg Q p _
+
+theorem sum_finiteFamilyWeight (Q : Finset E) (J : Finset ℕ) (p : ℝ) :
+    ∑ η : ↥J → FiniteField Q, finiteFamilyWeight Q J p η = 1 := by
+  unfold finiteFamilyWeight
+  rw [← Fintype.prod_sum]
+  simp only [sum_finiteFieldWeight,Finset.prod_const_one]
+
+theorem prod_fieldFamilyWeight (Q : Finset E) (J : Finset ℕ) (p : ℝ)
+    (η : ↥J → FiniteField Q) :
+    (∏ i ∈ J, bernoulliWeight Q (domainBits Q (finiteFieldFamily Q J η i)) p) =
+      finiteFamilyWeight Q J p η := by
+  rw [← Finset.prod_coe_sort J]
+  simp only [finiteFieldFamily_at,domainBits_finiteFieldSet,finiteFamilyWeight,finiteFieldWeight]
+
+abbrev FiniteJointData (Q : Finset E) (J₁ J₂ : Finset ℕ) :=
+  FiniteField Q × (↥J₁ → FiniteField Q) × (↥J₂ → FiniteField Q)
+
+def encodeFiniteJoint (Q : Finset E) (J₁ J₂ : Finset ℕ)
+    (s : FiniteJointData Q J₁ J₂) : JointSpliceState E :=
+  (finiteFieldSet Q s.1,finiteFieldFamily Q J₁ s.2.1,finiteFieldFamily Q J₂ s.2.2)
+
+theorem finiteFieldSet_injective (Q : Finset E) : Function.Injective (finiteFieldSet Q) := by
+  intro a b he
+  exact Subtype.ext (Finset.coe_injective he)
+
+theorem finiteFieldFamily_injective (Q : Finset E) (J : Finset ℕ) :
+    Function.Injective (finiteFieldFamily Q J) := by
+  intro a b he
+  funext i
+  apply finiteFieldSet_injective Q
+  simpa only [finiteFieldFamily_at] using congrFun he i
+
+theorem encodeFiniteJoint_injective (Q : Finset E) (J₁ J₂ : Finset ℕ) :
+    Function.Injective (encodeFiniteJoint Q J₁ J₂) := by
+  intro a b he
+  apply Prod.ext (finiteFieldSet_injective Q (congrArg Prod.fst he))
+  exact Prod.ext (finiteFieldFamily_injective Q J₁ (congrArg (fun s => s.2.1) he))
+    (finiteFieldFamily_injective Q J₂ (congrArg (fun s => s.2.2) he))
+
+theorem jointTupleWeight_encode (Q : Finset E) (J₁ J₂ : Finset ℕ) (p : ℝ)
+    (s : FiniteJointData Q J₁ J₂) :
+    jointTupleWeight Q J₁ J₂ p (encodeFiniteJoint Q J₁ J₂ s) =
+      finiteFieldWeight Q p s.1 * finiteFamilyWeight Q J₁ p s.2.1 *
+        finiteFamilyWeight Q J₂ p s.2.2 := by
+  simp only [jointTupleWeight,encodeFiniteJoint,domainBits_finiteFieldSet,
+    prod_fieldFamilyWeight,finiteFieldWeight]
+
+ 
+
+theorem sum_jointTupleWeight_encode (Q : Finset E) (J₁ J₂ : Finset ℕ) (p : ℝ) :
+    ∑ s : FiniteJointData Q J₁ J₂, jointTupleWeight Q J₁ J₂ p (encodeFiniteJoint Q J₁ J₂ s) = 1 := by
+  simp_rw [jointTupleWeight_encode]
+  rw [Fintype.sum_prod_type]
+  simp_rw [Fintype.sum_prod_type,← Finset.mul_sum,sum_finiteFamilyWeight,mul_one,
+    ← Finset.mul_sum,sum_finiteFamilyWeight,mul_one]
+  exact sum_finiteFieldWeight Q p
+
+end FiniteReferenceLaw
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section FiniteReferenceFactor
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+noncomputable def finiteFamilyEvent (Q : Finset E) (J : Finset ℕ) (p : ℝ)
+    (H : (ℕ → Set E) → Prop) : ℝ :=
+  ∑ η : ↥J → FiniteField Q, finiteFamilyWeight Q J p η *
+    if H (finiteFieldFamily Q J η) then 1 else 0
+
+theorem finiteFamilyEvent_nonneg (Q : Finset E) (J : Finset ℕ) (p : unitInterval)
+    (H : (ℕ → Set E) → Prop) : 0 ≤ finiteFamilyEvent Q J p H := by
+  apply Finset.sum_nonneg
+  intro η _
+  exact mul_nonneg (finiteFamilyWeight_nonneg Q J p η) (by split <;> norm_num)
+
+theorem finiteFamilyEvent_le_one (Q : Finset E) (J : Finset ℕ) (p : unitInterval)
+    (H : (ℕ → Set E) → Prop) : finiteFamilyEvent Q J p H ≤ 1 := by
+  rw [← sum_finiteFamilyWeight Q J p]
+  apply Finset.sum_le_sum
+  intro η _
+  exact mul_le_of_le_one_right (finiteFamilyWeight_nonneg Q J p η) (by split <;> norm_num)
+
+noncomputable def finiteJointDomain (Q : Finset E) (J₁ J₂ : Finset ℕ) :
+    Finset (JointSpliceState E) := Finset.univ.image (encodeFiniteJoint Q J₁ J₂)
+
+theorem finiteJointDomain_event_factor (Q : Finset E) (J₁ J₂ : Finset ℕ) (p : ℝ)
+    (P : Set E → Prop) (H₁ H₂ : Set E → (ℕ → Set E) → Prop) :
+    (∑ s ∈ (finiteJointDomain Q J₁ J₂).filter (fun s => P s.1 ∧ H₁ s.1 s.2.1 ∧ H₂ s.1 s.2.2),
+      jointTupleWeight Q J₁ J₂ p s) =
+    ∑ ω : FiniteField Q, finiteFieldWeight Q p ω *
+      (if P (finiteFieldSet Q ω) then
+        finiteFamilyEvent Q J₁ p (H₁ (finiteFieldSet Q ω)) *
+        finiteFamilyEvent Q J₂ p (H₂ (finiteFieldSet Q ω)) else 0) := by
+  rw [Finset.sum_filter,finiteJointDomain,Finset.sum_image (encodeFiniteJoint_injective Q J₁ J₂).injOn,
+    Fintype.sum_prod_type]
+  apply Finset.sum_congr rfl
+  intro ω _
+  rw [Fintype.sum_prod_type]
+  by_cases hP : P (finiteFieldSet Q ω)
+  · simp only [encodeFiniteJoint, hP, true_and,ite_true]
+    have heq (η₁ : ↥J₁ → FiniteField Q) (η₂ : ↥J₂ → FiniteField Q) :
+      (if H₁ (finiteFieldSet Q ω) (finiteFieldFamily Q J₁ η₁) ∧
+        H₂ (finiteFieldSet Q ω) (finiteFieldFamily Q J₂ η₂) then
+        jointTupleWeight Q J₁ J₂ p
+          (finiteFieldSet Q ω,finiteFieldFamily Q J₁ η₁,finiteFieldFamily Q J₂ η₂) else 0) =
+      finiteFieldWeight Q p ω *
+        (finiteFamilyWeight Q J₁ p η₁ * if H₁ (finiteFieldSet Q ω) (finiteFieldFamily Q J₁ η₁) then 1 else 0) *
+        (finiteFamilyWeight Q J₂ p η₂ * if H₂ (finiteFieldSet Q ω) (finiteFieldFamily Q J₂ η₂) then 1 else 0) := by
+      have hw := jointTupleWeight_encode Q J₁ J₂ p (ω,η₁,η₂)
+      change jointTupleWeight Q J₁ J₂ p (finiteFieldSet Q ω,finiteFieldFamily Q J₁ η₁,finiteFieldFamily Q J₂ η₂) = _ at hw
+      rw [hw]
+      split_ifs <;> simp_all
+    simp_rw [heq,← Finset.mul_sum,← Finset.sum_mul]
+    rw [← Finset.mul_sum,mul_assoc]
+    rfl
+  · simp [encodeFiniteJoint,hP]
+
+ 
+
+theorem mem_finiteJointDomain_of_support (Q : Finset E) (J₁ J₂ : Finset ℕ)
+    (s : JointSpliceState E) (hω : s.1 ⊆ (Q : Set E))
+    (hη₁ : ∀ i, s.2.1 i ⊆ (Q : Set E)) (hη₂ : ∀ i, s.2.2 i ⊆ (Q : Set E))
+    (hz₁ : ∀ i ∉ J₁, s.2.1 i = ∅) (hz₂ : ∀ i ∉ J₂, s.2.2 i = ∅) :
+    s ∈ finiteJointDomain Q J₁ J₂ := by
+  let desc (ω : Set E) : FiniteField Q := ⟨domainBits Q ω,Finset.mem_powerset.mpr (domainBits_subset Q ω)⟩
+  have hdesc (ω : Set E) (h : ω ⊆ (Q : Set E)) : finiteFieldSet Q (desc ω) = ω := by
+    ext e
+    simp only [finiteFieldSet,desc,Finset.mem_coe,mem_domainBits]
+    exact ⟨fun hh => hh.2,fun hh => ⟨h hh,hh⟩⟩
+  let a : FiniteJointData Q J₁ J₂ := (desc s.1,fun i => desc (s.2.1 i),fun i => desc (s.2.2 i))
+  apply Finset.mem_image.mpr
+  refine ⟨a,Finset.mem_univ _,?_⟩
+  apply Prod.ext (hdesc s.1 hω)
+  apply Prod.ext
+  · funext i
+    change (if hi : i ∈ J₁ then finiteFieldSet Q (desc (s.2.1 i)) else ∅) = s.2.1 i
+    by_cases hi : i ∈ J₁
+    · simp only [hi,dite_true,hdesc _ (hη₁ i)]
+    · simp only [hi,dite_false,hz₁ i hi]
+  · funext i
+    change (if hi : i ∈ J₂ then finiteFieldSet Q (desc (s.2.2 i)) else ∅) = s.2.2 i
+    by_cases hi : i ∈ J₂
+    · simp only [hi,dite_true,hdesc _ (hη₂ i)]
+    · simp only [hi,dite_false,hz₂ i hi]
+
+theorem exchangeOn_subset (M ω η Q : Set E) (hω : ω ⊆ Q) (hη : η ⊆ Q) :
+    exchangeOn M ω η ⊆ Q := by
+  rintro e (⟨he,_⟩ | ⟨he,_⟩)
+  · exact hω he
+  · exact hη he
+
+end FiniteReferenceFactor
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section FiniteSeparatedSecondMoment
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem finite_separated_second_moment_le {I : Type*} [Fintype I]
+    (G : BondGraph V E) (x : V) (k n M : ℕ) (p : unitInterval) (Q : Finset E)
+    (η₁ η₂ : I → ℕ → Set E) (y₁ y₂ : (ℕ → Set E) → ℕ → V)
+    (success₁ success₂ : (ℕ → Set E) → Prop)
+    (choose₁ choose₂ : ℕ → Set E → E)
+    (ω₁ ω₂ ωf : I → Set E) (c₁ c₂ : I → List (ℕ × E))
+    (hir : Function.Injective (fun a =>
+      (jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a),
+        c₁ a,c₂ a)))
+    (h₁ : ∀ a, SpliceTrace G x (η₁ a) (y₁ (η₁ a)) k 0 (ω₁ a) (c₁ a) (ωf a))
+    (h₂ : ∀ a, SpliceTrace G x (η₂ a) (y₂ (η₂ a)) k 0 (ω₂ a) (c₂ a) (ωf a))
+    (hc₁ : ∀ a, c₁ a ≠ []) (hc₂ : ∀ a, c₂ a ≠ [])
+    (hsep : ∀ a, GraphSeparated G (cluster G (ωf a) x \ cluster G (ω₁ a) x)
+      (cluster G (ωf a) x \ cluster G (ω₂ a) x))
+    (hr₁ : ∀ a, FollowsContactRule G x (η₁ a) (y₁ (η₁ a)) choose₁ (c₁ a) (ω₁ a))
+    (hr₂ : ∀ a, FollowsContactRule G x (η₂ a) (y₂ (η₂ a)) choose₂ (c₂ a) (ω₂ a))
+    (hbin : ∀ a, (cluster G (ω₁ a) x).Finite ∧ n ≤ (cluster G (ω₁ a) x).ncard ∧
+      (cluster G (ω₁ a) x).ncard < 2*n)
+    (hcap₂ : ∀ a i, 1 ≤ i → i ≤ k → (cluster G (η₂ a i) (y₂ (η₂ a) i)).Finite ∧
+      (cluster G (η₂ a i) (y₂ (η₂ a) i)).ncard ≤ M)
+    (hsc₁ : ∀ a, success₁ (η₁ a)) (hsc₂ : ∀ a, success₂ (η₂ a))
+    (hsω₁ : ∀ a, ω₁ a ⊆ (Q : Set E)) (hsω₂ : ∀ a, ω₂ a ⊆ (Q : Set E))
+    (hsη₁ : ∀ a i, η₁ a i ⊆ (Q : Set E)) (hsη₂ : ∀ a i, η₂ a i ⊆ (Q : Set E))
+    (hz₁ : ∀ a i, i ∉ Finset.Icc 1 k → η₁ a i = ∅)
+    (hz₂ : ∀ a i, i ∉ Finset.Icc 1 k → η₂ a i = ∅) :
+    (∑ a, jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p
+      (jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a))) ≤
+    ∑ ω : FiniteField Q, finiteFieldWeight Q p ω *
+      (if (cluster G (finiteFieldSet Q ω) x).Finite ∧
+        n-k*M ≤ (cluster G (finiteFieldSet Q ω) x).ncard ∧
+        (cluster G (finiteFieldSet Q ω) x).ncard < 2*n then
+        finiteFamilyEvent Q (Finset.Icc 1 k) p (fun η => success₁ η ∧
+          (cluster G (finiteFieldSet Q ω) x ∩ trajectoryUnion G η (y₁ η) k).Nonempty) *
+        finiteFamilyEvent Q (Finset.Icc 1 k) p (fun η => success₂ η ∧
+          (cluster G (finiteFieldSet Q ω) x ∩ trajectoryUnion G η (y₂ η) k).Nonempty) else 0) := by
+  let J := Finset.Icc 1 k
+  let P (ω : Set E) := (cluster G ω x).Finite ∧ n-k*M ≤ (cluster G ω x).ncard ∧
+    (cluster G ω x).ncard < 2*n
+  let H₁ (ω : Set E) (η : ℕ → Set E) := success₁ η ∧
+    (cluster G ω x ∩ trajectoryUnion G η (y₁ η) k).Nonempty
+  let H₂ (ω : Set E) (η : ℕ → Set E) := success₂ η ∧
+    (cluster G ω x ∩ trajectoryUnion G η (y₂ η) k).Nonempty
+  let B := (finiteJointDomain Q J J).filter fun s => P s.1 ∧ H₁ s.1 s.2.1 ∧ H₂ s.1 s.2.2
+  have hB (a : I) : jointRestored G x (η₁ a) (η₂ a) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (c₂ a) ∈ B := by
+    apply Finset.mem_filter.mpr
+    refine ⟨mem_finiteJointDomain_of_support Q J J _
+      (exchangeOn_subset _ _ _ _ (hsω₁ a) (hsω₂ a)) (hsη₁ a) (hsη₂ a) (hz₁ a) (hz₂ a),?_⟩
+    have hroot := (h₁ a).joint_restored_size G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a))
+      k 0 0 n M (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a) (h₂ a) (hc₁ a) (hc₂ a) (hsep a)
+      (hbin a).1 (hbin a).2.1 (hbin a).2.2 (hcap₂ a)
+    have hhits := (h₁ a).joint_trajectory_hits G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a))
+      k 0 0 (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a) (h₂ a) (hc₁ a) (hc₂ a) (hsep a)
+    exact ⟨hroot,⟨hsc₁ a,hhits.1⟩,hsc₂ a,hhits.2⟩
+  have h := finite_joint_mass_le G x k p Q η₁ η₂ y₁ y₂ choose₁ choose₂ ω₁ ω₂ ωf c₁ c₂
+    hir h₁ h₂ hc₁ hc₂ hsep hr₁ hr₂ B hB
+  apply h.trans_eq
+  convert finiteJointDomain_event_factor Q J J p P H₁ H₂ using 1 <;> simp [B,J,P,H₁,H₂]
+
+end FiniteSeparatedSecondMoment
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section SeededForward
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem exists_seeded_spliceTrace (G : BondGraph V E) (hloc : LocallyFinite G)
+    (x : V) (η : ℕ → Set E) (y : ℕ → V) (k M : ℕ)
+    (hchain : ∀ i ∈ Finset.Icc 1 k, Conn G (η i) (y i) (y (i-1)))
+    (hcaps : ∀ i ∈ Finset.Icc 1 k, (cluster G (η i) (y i)).Finite ∧
+      (cluster G (η i) (y i)).ncard ≤ M)
+    (choose : ℕ → Set E → E)
+    (hchoose : ∀ i K, K.Finite → K.Nonempty → choose i K ∈ K)
+    (j : ℕ) (hj : j ≤ k) (ω : Set E) (hyj : y j ∈ cluster G ω x)
+    (N : ℕ) (hfin : (cluster G ω x).Finite) (hN : (cluster G ω x).ncard ≤ N) :
+    ∃ steps ωf, SpliceTrace G x η y k j ω steps ωf ∧
+      FollowsContactRule G x η y choose steps ω ∧ steps.length ≤ k-j ∧
+      y k ∈ cluster G ωf x ∧ (cluster G ωf x).Finite ∧
+      (cluster G ωf x).ncard ≤ N+(k-j)*M := by
+  generalize hn : k-j = n
+  induction n using Nat.strong_induction_on generalizing j ω N with
+  | h n ih =>
+    by_cases hz : y k ∈ cluster G ω x
+    · refine ⟨[],ω,SpliceTrace.stop j ω hz,?_,by simp,hz,hfin,?_⟩
+      · exact List.Forall₂.nil
+      · exact hN.trans (Nat.le_add_right ..)
+    · obtain ⟨i,hiI,hmeet,hyi,hmax⟩ :=
+        exists_maximal_splice_index G η y k j hj hchain (cluster G ω x) hyj hz
+      obtain ⟨hji,hik⟩ := Finset.mem_Icc.mp hiI
+      have hi1 : i ∈ Finset.Icc 1 k := Finset.mem_Icc.mpr ⟨by omega,hik⟩
+      let B := cluster G (η i ∩ internalEdges G (cluster G ω x)ᶜ) (y i)
+      let K := contactSet G (η i) B
+      have hBC : B ⊆ cluster G (η i) (y i) := fun _ h => conn_mono G Set.inter_subset_left h
+      have hBK : K.Finite := contactSet_finite G hloc (η i) B ((hcaps i hi1).1.subset hBC)
+      have hK : K.Nonempty := by
+        obtain ⟨e,a,b,he,hab,ha,hb⟩ := exists_splice_contact G (η i) (cluster G ω x) (y i) hyi hmeet
+        exact ⟨e,he,b,a,hab.trans (Sym2.eq_swap ..),hb,
+          fun h => slit_cluster_subset G (cluster G ω x) (η i) (y i) hyi h ha⟩
+      let e := choose i K
+      have heK : e ∈ contactSet G (η i) B := hchoose i K hBK hK
+      change e ∈ contactSet G (η i)
+        (cluster G (η i ∩ internalEdges G (cluster G ω x)ᶜ) (y i)) at heK
+      rw [contactSet_deleted_cluster G (η i) (cluster G ω x) (y i) hyi] at heK
+      obtain ⟨he,b,hb,a,ha,hba⟩ := heK
+      have hab : G e = s(a,b) := hba.trans (Sym2.eq_swap ..)
+      let ω' := spliceField G (cluster G ω x) B e ω (η i)
+      have hcap := splice_cluster_cap G ω (η i) x (y i) hyi e a b hab he ha hb N M hfin hN
+        (hcaps i hi1).1 (hcaps i hi1).2
+      have hyt : y i ∈ cluster G ω' x := by
+        change y i ∈ cluster G (spliceField G (cluster G ω x) B e ω (η i)) x
+        rw [splice_cluster G ω (η i) x (y i) hyi e a b hab he ha hb]
+        exact Or.inr (mem_cluster_self G _ _)
+      obtain ⟨steps,ωf,htrace,hrule,hlen,hconn,hf,hs⟩ :=
+        ih (k-i) (by omega) i hik ω' hyt (N+M) hcap.1 hcap.2 rfl
+      refine ⟨(i,e)::steps,ωf,
+        SpliceTrace.step j i ω e a b steps ωf hz (by omega) hik hmax hyi he hab ha hb htrace,
+        ?_,?_,hconn,hf,?_⟩
+      · exact (followsContactRule_cons G x η y choose i e steps ω).mpr ⟨rfl,hrule⟩
+      · simp only [List.length_cons]
+        omega
+      · have hm : (k-i+1)*M ≤ (k-j)*M := Nat.mul_le_mul_right M (by omega)
+        rw [Nat.add_mul,one_mul] at hm
+        calc
+          _ ≤ (N+M)+(k-i)*M := hs
+          _ ≤ N+(k-j)*M := by omega
+          _ = N+n*M := by rw [hn]
+
+ 
+
+theorem existsUnique_seeded_spliceTrace (G : BondGraph V E) (hloc : LocallyFinite G)
+    (x : V) (η : ℕ → Set E) (y : ℕ → V) (k M : ℕ)
+    (hchain : ∀ i ∈ Finset.Icc 1 k, Conn G (η i) (y i) (y (i-1)))
+    (hcaps : ∀ i ∈ Finset.Icc 1 k, (cluster G (η i) (y i)).Finite ∧
+      (cluster G (η i) (y i)).ncard ≤ M)
+    (choose : ℕ → Set E → E)
+    (hchoose : ∀ i K, K.Finite → K.Nonempty → choose i K ∈ K)
+    (j : ℕ) (hj : j ≤ k) (ω : Set E) (hyj : y j ∈ cluster G ω x)
+    (hfin : (cluster G ω x).Finite) :
+    ∃! out : List (ℕ × E) × Set E,
+      SpliceTrace G x η y k j ω out.1 out.2 ∧
+      FollowsContactRule G x η y choose out.1 ω := by
+  obtain ⟨c,ωf,ht,hr,_⟩ := exists_seeded_spliceTrace G hloc x η y k M hchain hcaps
+    choose hchoose j hj ω hyj _ hfin (le_refl _)
+  refine ⟨(c,ωf),⟨ht,hr⟩,?_⟩
+  rintro ⟨d,σf⟩ ⟨hs,hd⟩
+  obtain ⟨hc,hf⟩ := hs.seeded_unique G x η y k j ω σf ωf d c ht choose hd hr
+  exact Prod.ext hc hf
+
+end SeededForward
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section SpliceLocalSupport
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem reachBall_triangle (G : BondGraph V E) (x y : V) (n m : ℕ)
+    (hy : y ∈ reachBall G x n) : reachBall G y m ⊆ reachBall G x (n+m) := by
+  induction m with
+  | zero => simpa only [reachBall,Nat.add_zero,Set.singleton_subset_iff] using hy
+  | succ m ih =>
+    intro z hz
+    rcases hz with hz | hz
+    · exact Or.inl (ih hz)
+    · apply Or.inr
+      simp only [Set.mem_iUnion,Set.mem_ofPred_eq] at hz ⊢
+      obtain ⟨a,ha,haz⟩ := hz
+      exact ⟨a,ih ha,haz⟩
+
+ 
+theorem capped_cluster_ball_of_mem (G : BondGraph V E) (ω : Set E) (x y : V)
+    (hfin : (cluster G ω x).Finite) (M : ℕ) (hM : (cluster G ω x).ncard ≤ M)
+    (hy : y ∈ cluster G ω x) : cluster G ω x ⊆ reachBall G y M := by
+  have he := cluster_eq_of_conn G ω hy
+  rw [he] at hfin hM ⊢
+  exact finite_cluster_subset_reachBall G ω y hfin M hM
+
+ 
+
+theorem oldChain_local_support (G : BondGraph V E) (x : V)
+    (η : ℕ → Set E) (y : ℕ → V) (k M N : ℕ)
+    (hstart : y 0 ∈ reachBall G x N)
+    (hchain : ∀ i, 1 ≤ i → i ≤ k → Conn G (η i) (y i) (y (i-1)))
+    (hcaps : ∀ i, 1 ≤ i → i ≤ k →
+      (cluster G (η i) (y i)).Finite ∧ (cluster G (η i) (y i)).ncard ≤ M) :
+    (∀ i, i ≤ k → y i ∈ reachBall G x (N+i*M)) ∧
+      trajectoryUnion G η y k ⊆ reachBall G x (N+k*M) := by
+  have hvertex : ∀ i, i ≤ k → y i ∈ reachBall G x (N+i*M) := by
+    intro i
+    induction i with
+    | zero => simpa using hstart
+    | succ i ih =>
+      intro hik
+      obtain ⟨hf,hM⟩ := hcaps (i+1) (by omega) hik
+      have hc := capped_cluster_ball_of_mem G (η (i+1)) (y (i+1)) (y i) hf M hM
+        (by simpa only [Nat.add_sub_cancel,cluster,Set.mem_ofPred_eq] using hchain (i+1) (by omega) hik)
+      have hh := reachBall_triangle G x (y i) (N+i*M) M (ih (by omega))
+        (hc (mem_cluster_self G _ _))
+      convert hh using 1
+      congr 1
+      ring
+  refine ⟨hvertex,?_⟩
+  intro v hv
+  obtain ⟨i,hi,hik,hv⟩ := (mem_trajectoryUnion G η y k v).mp hv
+  obtain ⟨hf,hM⟩ := hcaps i hi hik
+  have hc := capped_cluster_ball_of_mem G (η i) (y i) (y (i-1)) hf M hM (hchain i hi hik)
+  have hh := reachBall_triangle G x (y (i-1)) (N+(i-1)*M) M
+    (hvertex (i-1) (by omega)) (hc hv)
+  apply reachBall_mono_local G x _ hh
+  have hmul : i*M ≤ k*M := Nat.mul_le_mul_right M hik
+  have he : (i-1)*M+M = i*M := by
+    calc _ = (i-1+1)*M := by ring
+         _ = i*M := by rw [Nat.sub_add_cancel hi]
+  omega
+
+ 
+
+theorem SpliceTrace.masks_in_deterministic_ball (G : BondGraph V E)
+    (hloc : LocallyFinite G) (x : V) (η : ℕ → Set E) (y : ℕ → V)
+    (k j M N : ℕ) (ω ωf : Set E) (c : List (ℕ × E))
+    (h : SpliceTrace G x η y k j ω c ωf)
+    (hstart : y 0 ∈ reachBall G x N)
+    (hchain : ∀ i, 1 ≤ i → i ≤ k → Conn G (η i) (y i) (y (i-1)))
+    (hcaps : ∀ i, 1 ≤ i → i ≤ k →
+      (cluster G (η i) (y i)).Finite ∧ (cluster G (η i) (y i)).ncard ≤ M) :
+    maskUnion (replayMasks G x η y c ω) ⊆
+      (incidentSupport G hloc (finite_reachBall hloc x (N+k*M)).toFinset : Set E) := by
+  intro e he
+  obtain ⟨b,hb,hbe⟩ := h.masks_incident G x η y k j ω ωf c e he
+  have hbt := h.added_subset_trajectory G x η y k j ω ωf c hb
+  have hball := (oldChain_local_support G x η y k M N hstart hchain hcaps).2 hbt
+  exact (mem_incidentSupport G hloc _ e).mpr ⟨b,by simpa using hball,hbe⟩
+
+ 
+
+theorem oldChain_query_support (G : BondGraph V E) (x : V)
+    (η : ℕ → Set E) (y : ℕ → V) (k M N R : ℕ)
+    (hstart : y 0 ∈ reachBall G x N)
+    (hchain : ∀ i, 1 ≤ i → i ≤ k → Conn G (η i) (y i) (y (i-1)))
+    (hcaps : ∀ i, 1 ≤ i → i ≤ k →
+      (cluster G (η i) (y i)).Finite ∧ (cluster G (η i) (y i)).ncard ≤ M)
+    (i : ℕ) (hi : i ≤ k) :
+    reachBall G (y i) R ⊆ reachBall G x (N+k*M+R) := by
+  have hv := (oldChain_local_support G x η y k M N hstart hchain hcaps).1 i hi
+  intro z hz
+  have hh := reachBall_triangle G x (y i) (N+i*M) R hv hz
+  exact reachBall_mono_local G x (by nlinarith) hh
+
+end SpliceLocalSupport
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section UniformContactSelector
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+noncomputable def uniformRank (K : ℕ) (hK : 0 < K) (t : unitInterval) : Fin K :=
+  ⟨min ⌊(K:ℝ)*(t:ℝ)⌋₊ (K-1), lt_of_le_of_lt (min_le_right ..) (by omega)⟩
+
+theorem uniformRank_val_of_lt_one (K : ℕ) (hK : 0 < K) (t : unitInterval)
+    (ht : (t:ℝ) < 1) : (uniformRank K hK t).val = ⌊(K:ℝ)*(t:ℝ)⌋₊ := by
+  have hp : (0:ℝ) < K := by exact_mod_cast hK
+  have hf : ⌊(K:ℝ)*(t:ℝ)⌋₊ < K := (Nat.floor_lt (mul_nonneg (Nat.cast_nonneg _) t.property.1)).mpr (by nlinarith)
+  exact min_eq_left (by omega)
+
+theorem uniformRank_eq_iff (K : ℕ) (hK : 0 < K) (t : unitInterval)
+    (ht : (t:ℝ) < 1) (a : Fin K) :
+    uniformRank K hK t = a ↔ t ∈ contactInterval K a.val := by
+  rw [Fin.ext_iff,uniformRank_val_of_lt_one K hK t ht,
+    Nat.floor_eq_iff (mul_nonneg (Nat.cast_nonneg _) t.property.1)]
+  have hp : (0:ℝ) < K := by exact_mod_cast hK
+  simp only [contactInterval,Set.mem_ofPred_eq,Nat.cast_add,Nat.cast_one]
+  rw [div_le_iff₀ hp,lt_div_iff₀ hp]
+  simp only [mul_comm]
+
+theorem volume_uniformRank_fiber (K : ℕ) (hK : 0 < K) (a : Fin K) :
+    volume {t : unitInterval | uniformRank K hK t = a} =
+      ENNReal.ofReal ((K:ℝ)⁻¹) := by
+  rw [← volume_contactInterval K a.val a.isLt]
+  apply measure_congr
+  filter_upwards [(volume : Measure unitInterval).ae_ne 1] with t ht
+  exact propext (uniformRank_eq_iff K hK t (lt_of_le_of_ne t.property.2
+    (fun he => ht (Subtype.ext he))) a)
+
+ 
+
+noncomputable def uniformContact (K : Finset E) (hK : K.Nonempty)
+    (t : unitInterval) : E :=
+  ((K.equivFin).symm (uniformRank K.card (Finset.card_pos.mpr hK) t)).val
+
+theorem uniformContact_mem (K : Finset E) (hK : K.Nonempty) (t : unitInterval) :
+    uniformContact K hK t ∈ K :=
+  ((K.equivFin).symm (uniformRank K.card (Finset.card_pos.mpr hK) t)).property
+
+theorem uniformContact_eq_iff (K : Finset E) (hK : K.Nonempty) (t : unitInterval)
+    (e : E) (he : e ∈ K) :
+    uniformContact K hK t = e ↔
+      uniformRank K.card (Finset.card_pos.mpr hK) t = K.equivFin ⟨e,he⟩ := by
+  unfold uniformContact
+  change ((K.equivFin).symm (uniformRank K.card (Finset.card_pos.mpr hK) t)).val =
+    (⟨e,he⟩ : ↥K).val ↔ _
+  rw [← Subtype.ext_iff]
+  exact K.equivFin.symm_apply_eq
+
+theorem volume_uniformContact_fiber (K : Finset E) (hK : K.Nonempty)
+    (e : E) (he : e ∈ K) : volume {t : unitInterval | uniformContact K hK t = e} =
+      ENNReal.ofReal ((K.card:ℝ)⁻¹) := by
+  simp_rw [uniformContact_eq_iff K hK _ e he]
+  exact volume_uniformRank_fiber K.card (Finset.card_pos.mpr hK) _
+
+ 
+
+noncomputable def chooseContact [Nonempty E] (seed : ℕ → unitInterval)
+    (i : ℕ) (K : Set E) : E :=
+  if hf : K.Finite then if hn : K.Nonempty then
+    uniformContact hf.toFinset (by simpa using hn) (seed i)
+  else Classical.arbitrary E else Classical.arbitrary E
+
+theorem chooseContact_mem [Nonempty E] (seed : ℕ → unitInterval)
+    (i : ℕ) (K : Set E) (hf : K.Finite) (hn : K.Nonempty) :
+    chooseContact seed i K ∈ K := by
+  simp only [chooseContact,dite_eq_left hf,dite_eq_left hn]
+  exact hf.mem_toFinset.mp (uniformContact_mem _ _ _)
+
+end UniformContactSelector
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section SeededTraceLaw
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable [Nonempty E]
+
+ 
+theorem chooseContact_congr (u v : ℕ → unitInterval) (i : ℕ) (K : Set E)
+    (h : u i = v i) : chooseContact u i K = chooseContact v i K := by
+  unfold chooseContact
+  split_ifs <;> simp only [h]
+
+ 
+noncomputable def replaySeedFiber (G : BondGraph V E) (x : V) (η : ℕ → Set E)
+    (y : ℕ → V) : List (ℕ × E) → Set E → ℕ → Set unitInterval
+  | [], _, _ => Set.univ
+  | (j,e)::c, ω, i =>
+    let S := cluster G ω x
+    let B := cluster G (η j ∩ internalEdges G Sᶜ) (y j)
+    if i = j then {t | chooseContact (fun _ => t) j (contactSet G (η j) B) = e}
+    else replaySeedFiber G x η y c (spliceField G S B e ω (η j)) i
+
+theorem replaySeedFiber_univ (G : BondGraph V E) (x : V) (η : ℕ → Set E)
+    (y : ℕ → V) (c : List (ℕ × E)) (ω : Set E) (i : ℕ)
+    (hi : ∀ q ∈ c, i ≠ q.1) : replaySeedFiber G x η y c ω i = Set.univ := by
+  induction c generalizing ω with
+  | nil => rfl
+  | cons q c ih =>
+    obtain ⟨j,e⟩ := q
+    simp only [replaySeedFiber,ite_eq_right (hi (j,e) (by simp))]
+    exact ih _ (fun q hq => hi q (by simp [hq]))
+
+theorem SpliceTrace.follows_iff_seedFiber (G : BondGraph V E) (x : V)
+    (η : ℕ → Set E) (y : ℕ → V) (k j : ℕ) (ω ωf : Set E)
+    (c : List (ℕ × E)) (h : SpliceTrace G x η y k j ω c ωf)
+    (seed : ℕ → unitInterval) :
+    FollowsContactRule G x η y (chooseContact seed) c ω ↔
+      ∀ i, 1 ≤ i → i ≤ k → seed i ∈ replaySeedFiber G x η y c ω i := by
+  induction h with
+  | stop j ω hz => simp [FollowsContactRule,replayContacts,replaySeedFiber]
+  | @step j l ω e a b c ωf hn hjl hlk hm hy he hab ha hb ht ih =>
+    rw [followsContactRule_cons,ih]
+    have htail : replaySeedFiber G x η y c
+        (spliceField G (cluster G ω x)
+          (cluster G (η l ∩ internalEdges G (cluster G ω x)ᶜ) (y l)) e ω (η l)) l = Set.univ := by
+      apply replaySeedFiber_univ
+      intro q hq
+      have hq' := ht.seam_data G x η y k l _ ωf c q hq
+      omega
+    constructor
+    · rintro ⟨hc,hr⟩ i hi hik
+      by_cases hil : i = l
+      · subst i
+        simp only [replaySeedFiber,ite_true,Set.mem_ofPred_eq]
+        exact (chooseContact_congr _ _ l _ rfl).trans hc
+      · simpa only [replaySeedFiber,ite_eq_right hil] using hr i hi hik
+    · intro hr
+      constructor
+      · have hh := hr l (by omega) hlk
+        simp only [replaySeedFiber,ite_true,Set.mem_ofPred_eq] at hh
+        exact (chooseContact_congr _ _ l _ rfl).trans hh
+      · intro i hi hik
+        by_cases hil : i = l
+        · subst i
+          rw [htail]
+          trivial
+        · simpa only [replaySeedFiber,ite_eq_right hil] using hr i hi hik
+
+ 
+
+def finiteContactSeeds (k : ℕ) (u : Fin k → unitInterval) (i : ℕ) : unitInterval :=
+  if hi : 1 ≤ i ∧ i ≤ k then u ⟨i-1,by omega⟩ else 0
+
+@[simp] theorem finiteContactSeeds_at (k : ℕ) (u : Fin k → unitInterval) (i : Fin k) :
+    finiteContactSeeds k u (i.val+1) = u i := by
+  have hi : 1 ≤ i.val+1 ∧ i.val+1 ≤ k := ⟨by omega,by omega⟩
+  simp only [finiteContactSeeds,dite_eq_left hi,Nat.add_sub_cancel]
+
+theorem SpliceTrace.seed_event_eq_pi (G : BondGraph V E) (x : V)
+    (η : ℕ → Set E) (y : ℕ → V) (k j : ℕ) (ω ωf : Set E)
+    (c : List (ℕ × E)) (h : SpliceTrace G x η y k j ω c ωf) :
+    {u : Fin k → unitInterval | FollowsContactRule G x η y
+      (chooseContact (finiteContactSeeds k u)) c ω} =
+      Set.pi Set.univ (fun i => replaySeedFiber G x η y c ω (i.val+1)) := by
+  ext u
+  rw [Set.mem_ofPred_eq,h.follows_iff_seedFiber G x η y k j ω ωf c]
+  simp only [Set.mem_pi,Set.mem_univ,true_implies]
+  constructor
+  · intro hu i
+    simpa only [finiteContactSeeds_at] using hu (i.val+1) (by omega) (by omega)
+  · intro hu i hi hik
+    have hh := hu ⟨i-1,by omega⟩
+    have hii : 1 ≤ i ∧ i ≤ k := ⟨hi,hik⟩
+    simpa only [finiteContactSeeds,dite_eq_left hii,Nat.sub_add_cancel hi] using hh
+
+ 
+
+theorem SpliceTrace.volume_seedFiber (G : BondGraph V E) (hloc : LocallyFinite G)
+    (x : V) (η : ℕ → Set E) (y : ℕ → V) (k j : ℕ) (ω ωf : Set E)
+    (c : List (ℕ × E)) (h : SpliceTrace G x η y k j ω c ωf)
+    (hcaps : ∀ i, 1 ≤ i → i ≤ k → (cluster G (η i) (y i)).Finite) :
+    ∀ i, volume (replaySeedFiber G x η y c ω i) =
+      ENNReal.ofReal ((traceContactCount G η y ωf x c i : ℝ)⁻¹) := by
+  induction c generalizing j ω with
+  | nil => simp [replaySeedFiber,traceContactCount]
+  | cons q c ih =>
+    obtain ⟨l,e⟩ := q
+    have hcut := h.head_cut G x η y k j l ω ωf e c
+    cases h with
+    | step j l ω e a b c ωf hn hjl hlk hm hy he hab ha hb ht =>
+      intro i
+      by_cases hil : i = l
+      · subst i
+        simp only [replaySeedFiber,ite_true,traceContactCount,seamContactCount,hcut]
+        let B := cluster G (η l ∩ internalEdges G (cluster G ω x)ᶜ) (y l)
+        let K := contactSet G (η l) B
+        have hB : B.Finite := (hcaps l (by omega) hlk).subset (fun _ h => conn_mono G Set.inter_subset_left h)
+        have hK : K.Finite := contactSet_finite G hloc (η l) B hB
+        have heK : e ∈ K := ⟨he,b,a,hab.trans (Sym2.eq_swap ..),hb,
+          fun haB => slit_cluster_subset G (cluster G ω x) (η l) (y l) hy haB ha⟩
+        have hnK : K.Nonempty := ⟨e,heK⟩
+        change volume {t : unitInterval | chooseContact (fun _ => t) l K = e} = _
+        simp only [chooseContact,dite_eq_left hK,dite_eq_left hnK]
+        rw [volume_uniformContact_fiber _ _ e (hK.mem_toFinset.mpr heK)]
+        rw [← Set.ncard_eq_toFinset_card K hK]
+      · simpa only [replaySeedFiber,traceContactCount,ite_eq_right hil] using ih l _ ht i
+
+ 
+theorem SpliceTrace.seed_probability (G : BondGraph V E) (hloc : LocallyFinite G)
+    (x : V) (η : ℕ → Set E) (y : ℕ → V) (k j : ℕ) (ω ωf : Set E)
+    (c : List (ℕ × E)) (h : SpliceTrace G x η y k j ω c ωf)
+    (hcaps : ∀ i, 1 ≤ i → i ≤ k → (cluster G (η i) (y i)).Finite) :
+    ((volume : Measure (Fin k → unitInterval))
+      {u | FollowsContactRule G x η y (chooseContact (finiteContactSeeds k u)) c ω}).toReal =
+      ∏ i : Fin k, (traceContactCount G η y ωf x c (i.val+1):ℝ)⁻¹ := by
+  rw [h.seed_event_eq_pi G x η y k j ω ωf c,MeasureTheory.volume_pi,Measure.pi_pi,ENNReal.toReal_prod]
+  apply Finset.prod_congr rfl
+  intro i _
+  rw [h.volume_seedFiber G hloc x η y k j ω ωf c hcaps,ENNReal.toReal_ofReal (by positivity)]
+
+end SeededTraceLaw
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Splicing/Replay.lean b/lean/OAI/Probability/UnimodularPercolation/Splicing/Replay.lean
index 3a05cef2328613303a355e0a6d44ff021f08cde4..a206dd03529abf8e920845b276fda0b0f48c9177 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Splicing/Replay.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Splicing/Replay.lean
@@ -1,1245 +1,1245 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Splicing.GapCodes
-
-namespace OAI
-
-/-! Replay for bond percolation. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section SparseGapCode
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-def ValidSparseGapCode (k M : ℕ) (c : List (ℕ × ℕ)) : Prop :=
-  c.Pairwise (fun q r => q.1 < r.1) ∧
-    ∀ q ∈ c, 1 ≤ q.1 ∧ q.1 ≤ k ∧ 1 ≤ q.2 ∧ q.2 ≤ M
-
-def sparseGapValue : List (ℕ × ℕ) → ℕ → ℕ
-  | [], _ => 0
-  | (j,l)::c, i => if i = j then l else sparseGapValue c i
-
-theorem sparseGapValue_zero (c : List (ℕ × ℕ)) (i : ℕ)
-    (h : ∀ q ∈ c, i ≠ q.1) : sparseGapValue c i = 0 := by
-  induction c with
-  | nil => rfl
-  | cons q c ih =>
-    obtain ⟨j,l⟩ := q
-    rw [sparseGapValue,ite_eq_right (h (j,l) (by simp))]
-    exact ih (fun q hq => h q (List.mem_cons_of_mem _ hq))
-
-theorem sparseGapValue_bound (k M : ℕ) (c : List (ℕ × ℕ))
-    (h : ValidSparseGapCode k M c) (i : ℕ) : sparseGapValue c i ≤ M := by
-  induction c with
-  | nil => exact Nat.zero_le _
-  | cons q c ih =>
-    obtain ⟨j,l⟩ := q
-    rw [sparseGapValue]
-    split_ifs
-    · exact (h.2 (j,l) (by simp)).2.2.2
-    · exact ih ⟨(List.pairwise_cons.mp h.1).2,fun q hq => h.2 q (List.mem_cons_of_mem _ hq)⟩
-
- 
-theorem sparseGapValue_injective (c d : List (ℕ × ℕ))
-    (hc : c.Pairwise (fun q r => q.1 < r.1))
-    (hd : d.Pairwise (fun q r => q.1 < r.1))
-    (hpc : ∀ q ∈ c, 0 < q.2) (hpd : ∀ q ∈ d, 0 < q.2)
-    (heq : ∀ i, sparseGapValue c i = sparseGapValue d i) : c = d := by
-  induction c generalizing d with
-  | nil =>
-    cases d with
-    | nil => rfl
-    | cons q d =>
-      obtain ⟨j,l⟩ := q
-      have hh := heq j
-      simp [sparseGapValue] at hh
-      have := hpd (j,l) (by simp)
-      omega
-  | cons q c ih =>
-    obtain ⟨i,l⟩ := q
-    cases d with
-    | nil =>
-      have hh := heq i
-      simp [sparseGapValue] at hh
-      have := hpc (i,l) (by simp)
-      omega
-    | cons q d =>
-      obtain ⟨j,m⟩ := q
-      obtain ⟨hc0,hc1⟩ := List.pairwise_cons.mp hc
-      obtain ⟨hd0,hd1⟩ := List.pairwise_cons.mp hd
-      have hij : i = j := by
-        by_contra hne
-        rcases Nat.lt_or_gt_of_ne hne with hij | hji
-        · have hz : sparseGapValue ((j,m)::d) i = 0 := by
-            apply sparseGapValue_zero
-            intro q hq
-            rcases List.mem_cons.mp hq with rfl | hq
-            · exact hne
-            · have := hd0 q hq
-              change j < q.1 at this
-              omega
-          have hh := heq i
-          rw [hz] at hh
-          simp [sparseGapValue] at hh
-          have := hpc (i,l) (by simp)
-          omega
-        · have hz : sparseGapValue ((i,l)::c) j = 0 := by
-            apply sparseGapValue_zero
-            intro q hq
-            rcases List.mem_cons.mp hq with rfl | hq
-            · exact Ne.symm hne
-            · have := hc0 q hq
-              change i < q.1 at this
-              omega
-          have hh := heq j
-          rw [hz] at hh
-          simp [sparseGapValue] at hh
-          have := hpd (j,m) (by simp)
-          omega
-      subst j
-      have hlm : l = m := by simpa [sparseGapValue] using heq i
-      subst m
-      congr 1
-      apply ih d hc1 hd1
-        (fun q hq => hpc q (List.mem_cons_of_mem _ hq))
-        (fun q hq => hpd q (List.mem_cons_of_mem _ hq))
-      intro r
-      by_cases hr : r = i
-      · subst r
-        rw [sparseGapValue_zero c i (fun q hq => (hc0 q hq).ne),
-          sparseGapValue_zero d i (fun q hq => (hd0 q hq).ne)]
-      · simpa only [sparseGapValue,ite_eq_right hr] using heq r
-
- 
-def sparseGapArray (k M : ℕ) (c : List (ℕ × ℕ)) (h : ValidSparseGapCode k M c) :
-    Fin k → Fin (M+1) :=
-  fun i => ⟨sparseGapValue c (i.val+1),Nat.lt_succ_of_le (sparseGapValue_bound k M c h _)⟩
-
-theorem sparseGapArray_injective (k M : ℕ) (c d : List (ℕ × ℕ))
-    (hc : ValidSparseGapCode k M c) (hd : ValidSparseGapCode k M d)
-    (heq : sparseGapArray k M c hc = sparseGapArray k M d hd) : c = d := by
-  apply sparseGapValue_injective c d hc.1 hd.1
-    (fun q hq => (hc.2 q hq).2.2.1) (fun q hq => (hd.2 q hq).2.2.1)
-  intro i
-  by_cases hi : 1 ≤ i ∧ i ≤ k
-  · have he := congrArg (fun f => (f ⟨i-1,by omega⟩).val) heq
-    simpa only [sparseGapArray,show i-1+1 = i by omega] using he
-  · rw [sparseGapValue_zero c i, sparseGapValue_zero d i]
-    · intro q hq he
-      have h := hd.2 q hq
-      omega
-    · intro q hq he
-      have h := hc.2 q hq
-      omega
-
- 
-
-theorem SpliceTrace.valid_gap_code (G : BondGraph V E) (hloc : LocallyFinite G)
-    (x : V) (η : ℕ → Set E) (y : ℕ → V) (k j M : ℕ) (ω ωf : Set E)
-    (steps : List (ℕ × E)) (h : SpliceTrace G x η y k j ω steps ωf)
-    (hCf : (cluster G ωf x).Finite) (L : ℝ)
-    (hgood : ∀ i, 1 ≤ i → i ≤ k → (cluster G (η i) (y i)).Finite ∧
-      (cluster G (η i) (y i)).ncard ≤ M ∧ GoodForScale G (η i) (y i) L) :
-    ValidSparseGapCode k M (gapSpliceCode G ωf x (y k) steps) ∧
-      ∀ q ∈ gapSpliceCode G ωf x (y k) steps, j < q.1 := by
-  induction steps generalizing j ω with
-  | nil => simp [ValidSparseGapCode,gapSpliceCode]
-  | cons q rest ih =>
-    obtain ⟨i,e⟩ := q
-    have hs := h.seam_data G x η y k j ω ωf ((i,e)::rest) (i,e) (by simp)
-    have hi : 1 ≤ i := by omega
-    obtain ⟨hCi,hMi,hGi⟩ := hgood i hi hs.2.1
-    have hg := h.head_gap_bounds G hloc x η y k j i M ω ωf e rest hCf hCi hMi L hGi
-    change 1 ≤ _ ∧ 1 ≤ traceHeadGap G ωf x (y k) e rest ∧
-      traceHeadGap G ωf x (y k) e rest ≤ M ∧ _ at hg
-    cases h with
-    | step j i ω e a b rest ωf _ _ _ _ _ _ _ _ _ ht =>
-      obtain ⟨⟨hpair,hbounds⟩,hlater⟩ := ih i _ ht
-      simp only [gapSpliceCode]
-      refine ⟨⟨List.pairwise_cons.mpr ⟨?_,hpair⟩,?_⟩,?_⟩
-      · exact hlater
-      · intro q hq
-        rcases List.mem_cons.mp hq with rfl | hq
-        · exact ⟨hi,hs.2.1,hg.2.1,hg.2.2.1⟩
-        · exact hbounds q hq
-      · intro q hq
-        rcases List.mem_cons.mp hq with rfl | hq
-        · exact hs.1
-        · exact hs.1.trans (hlater q hq)
-
-end SparseGapCode
-
-end Percolation
-
- 
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section TraceContactMass
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-noncomputable def seamContactCount (G : BondGraph V E) (η : ℕ → Set E)
-    (y : ℕ → V) (ωf : Set E) (x : V) (i : ℕ) (e : E) : ℕ :=
-  (contactSet G (η i)
-    (cluster G (η i ∩ internalEdges G (cluster G (ωf \ {e}) x)ᶜ) (y i))).ncard
-
- 
-noncomputable def traceContactCount (G : BondGraph V E) (η : ℕ → Set E)
-    (y : ℕ → V) (ωf : Set E) (x : V) : List (ℕ × E) → ℕ → ℕ
-  | [], _ => 1
-  | (j,e)::c, i => if i = j then seamContactCount G η y ωf x j e
-      else traceContactCount G η y ωf x c i
-
- 
-theorem SpliceTrace.contact_weight_bound (G : BondGraph V E) (hloc : LocallyFinite G)
-    (x : V) (η : ℕ → Set E) (y : ℕ → V) (k j M : ℕ) (ω ωf : Set E)
-    (steps : List (ℕ × E)) (h : SpliceTrace G x η y k j ω steps ωf)
-    (hCf : (cluster G ωf x).Finite) (L : ℝ) (hL : 1 ≤ L)
-    (hgood : ∀ i, 1 ≤ i → i ≤ k → (cluster G (η i) (y i)).Finite ∧
-      (cluster G (η i) (y i)).ncard ≤ M ∧ GoodForScale G (η i) (y i) L) :
-    ∀ i, 1 ≤ traceContactCount G η y ωf x steps i ∧
-      (traceContactCount G η y ωf x steps i : ℝ)⁻¹ ≤
-        gapLetterWeight L (sparseGapValue (gapSpliceCode G ωf x (y k) steps) i) := by
-  induction steps generalizing j ω with
-  | nil => simp [traceContactCount,gapSpliceCode,sparseGapValue,gapLetterWeight]
-  | cons q rest ih =>
-    obtain ⟨i,e⟩ := q
-    have hs := h.seam_data G x η y k j ω ωf ((i,e)::rest) (i,e) (by simp)
-    have hi : 1 ≤ i := by omega
-    obtain ⟨hCi,hMi,hGi⟩ := hgood i hi hs.2.1
-    have hg := h.head_gap_bounds G hloc x η y k j i M ω ωf e rest hCf hCi hMi L hGi
-    have hcut := h.head_cut G x η y k j i ω ωf e rest
-    change 1 ≤ _ ∧ 1 ≤ traceHeadGap G ωf x (y k) e rest ∧
-      traceHeadGap G ωf x (y k) e rest ≤ M ∧ _ at hg
-    have hk : seamContactCount G η y ωf x i e =
-      (contactSet G (η i) (cluster G (η i ∩ internalEdges G (cluster G ω x)ᶜ) (y i))).ncard := by
-      unfold seamContactCount
-      rw [hcut]
-    rw [← hk] at hg
-    cases h with
-    | step j i ω e a b rest ωf _ _ _ _ _ _ _ _ _ ht =>
-      intro r
-      by_cases hr : r = i
-      · subst r
-        simp only [traceContactCount,gapSpliceCode,sparseGapValue,ite_true]
-        refine ⟨hg.1,?_⟩
-        rw [gapLetterWeight,ite_eq_right (by omega)]
-        exact inverse_contact_weight_le _ _ L hg.1 hg.2.1 hL hg.2.2.2
-      · simpa only [traceContactCount,gapSpliceCode,sparseGapValue,ite_eq_right hr] using ih i _ ht r
-
- 
-
-def contactChoiceSet (K a : ℕ) : Set unitInterval :=
-  if K = 1 then Set.univ else contactInterval K a
-
-theorem volume_contactChoiceSet (K a : ℕ) (ha : a < K) :
-    volume (contactChoiceSet K a) = ENNReal.ofReal ((K:ℝ)⁻¹) := by
-  unfold contactChoiceSet
-  split_ifs with h
-  · simp [h]
-  · exact volume_contactInterval K a ha
-
-theorem real_volume_contact_choice_cube {I : Type*} [Fintype I] (K a : I → ℕ)
-    (ha : ∀ i, a i < K i) :
-    ((volume : Measure (I → unitInterval))
-      (Set.pi Set.univ (fun i => contactChoiceSet (K i) (a i)))).toReal =
-      ∏ i, (K i:ℝ)⁻¹ := by
-  rw [MeasureTheory.volume_pi,Measure.pi_pi,ENNReal.toReal_prod]
-  apply Finset.prod_congr rfl
-  intro i _
-  rw [volume_contactChoiceSet _ _ (ha i),ENNReal.toReal_ofReal (by positivity)]
-
- 
-
-theorem SpliceTrace.contact_cube_le_code_weight (G : BondGraph V E) (hloc : LocallyFinite G)
-    (x : V) (η : ℕ → Set E) (y : ℕ → V) (k j M : ℕ) (ω ωf : Set E)
-    (steps : List (ℕ × E)) (h : SpliceTrace G x η y k j ω steps ωf)
-    (hCf : (cluster G ωf x).Finite) (L : ℝ) (hL : 1 ≤ L)
-    (hgood : ∀ i, 1 ≤ i → i ≤ k → (cluster G (η i) (y i)).Finite ∧
-      (cluster G (η i) (y i)).ncard ≤ M ∧ GoodForScale G (η i) (y i) L)
-    (a : Fin k → ℕ) (ha : ∀ i, a i < traceContactCount G η y ωf x steps (i.val+1)) :
-    ((volume : Measure (Fin k → unitInterval))
-      (Set.pi Set.univ (fun i => contactChoiceSet
-        (traceContactCount G η y ωf x steps (i.val+1)) (a i)))).toReal ≤
-    ∏ i : Fin k, gapLetterWeight L (sparseGapValue (gapSpliceCode G ωf x (y k) steps) (i.val+1)) := by
-  rw [real_volume_contact_choice_cube _ _ ha]
-  apply Finset.prod_le_prod₀
-  · intros
-    positivity
-  · intro i _
-    exact (h.contact_weight_bound G hloc x η y k j M ω ωf steps hCf L hL hgood (i.val+1)).2
-
- 
-theorem SpliceTrace.code_unique_of_gapCode (G : BondGraph V E) (x : V)
-    (η₁ η₂ : ℕ → Set E) (y₁ y₂ : ℕ → V) (k j : ℕ) (ω₁ ω₂ ωf : Set E)
-    (c₁ c₂ : List (ℕ × E))
-    (ht₁ : SpliceTrace G x η₁ y₁ k j ω₁ c₁ ωf)
-    (ht₂ : SpliceTrace G x η₂ y₂ k j ω₂ c₂ ωf)
-    (hz : y₁ k = y₂ k) (hC : (cluster G ωf x).Finite)
-    (heq : gapSpliceCode G ωf x (y₁ k) c₁ = gapSpliceCode G ωf x (y₁ k) c₂) : c₁ = c₂ := by
-  apply ordinalSpliceCode_injective G ωf x (y₁ k) hC c₁ c₂
-    (ht₁.seams_pivotal G x η₁ y₁ k j ω₁ ωf c₁)
-    (by simpa only [hz] using ht₂.seams_pivotal G x η₂ y₂ k j ω₂ ωf c₂)
-  rw [← ht₁.decode_gap_code G x η₁ y₁ k j ω₁ ωf c₁ hC,heq,hz,
-    ht₂.decode_gap_code G x η₂ y₂ k j ω₂ ωf c₂ hC]
-
-end TraceContactMass
-
-end Percolation
-
- 
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section FiniteInverseMass
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem finite_inverse_contact_mass_le {I : Type*} [Fintype I]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (x z : V) (k M : ℕ)
-    (L : ℝ) (hL : 1 ≤ L) (p : unitInterval) (Q : Finset E)
-    (inp out : I → SpliceState E) (steps : I → List (ℕ × E)) (y : I → ℕ → V)
-    (hir : Function.Injective (fun a => (inp a,steps a)))
-    (htrace : ∀ a, SpliceTrace G x (inp a).2 (y a) k 0 (inp a).1 (steps a) (out a).1)
-    (hex : ∀ a, SpliceExecution G x (inp a) (steps a) (out a))
-    (hz : ∀ a, y a k = z) (hfin : ∀ a, (cluster G (out a).1 x).Finite)
-    (hgood : ∀ a i, 1 ≤ i → i ≤ k → (cluster G ((inp a).2 i) (y a i)).Finite ∧
-      (cluster G ((inp a).2 i) (y a i)).ncard ≤ M ∧ GoodForScale G ((inp a).2 i) (y a i) L)
-    (rank : I → Fin k → ℕ)
-    (hrank : ∀ a i, rank a i < traceContactCount G (inp a).2 (y a) (out a).1 x (steps a) (i.val+1))
-    (B : Finset (SpliceState E)) (hB : ∀ a, out a ∈ B) :
-    (∑ a, spliceTupleWeight Q (Finset.Icc 1 k) p (inp a) *
-      ((volume : Measure (Fin k → unitInterval))
-        (Set.pi Set.univ (fun i => contactChoiceSet
-          (traceContactCount G (inp a).2 (y a) (out a).1 x (steps a) (i.val+1))
-          (rank a i)))).toReal) ≤
-      (∑ t ∈ B, spliceTupleWeight Q (Finset.Icc 1 k) p t) * inverseCodeMass M L ^ k := by
-  let c : I → List (ℕ × ℕ) := fun a => gapSpliceCode G (out a).1 x z (steps a)
-  have hvalid : ∀ a, ValidSparseGapCode k M (c a) := by
-    intro a
-    have hv := (htrace a).valid_gap_code G hloc x (inp a).2 (y a) k 0 M
-      (inp a).1 (out a).1 (steps a) (hfin a) L (hgood a)
-    simpa only [c,hz a] using hv.1
-  let arr (a : I) : Fin k → Fin (M+1) := sparseGapArray k M (c a) (hvalid a)
-  let f (a : I) : SpliceState E × (Fin k → Fin (M+1)) := (out a,arr a)
-  have hinj : Function.Injective f := by
-    intro a b he
-    have ho : out a = out b := congrArg Prod.fst he
-    have hc : c a = c b := sparseGapArray_injective k M (c a) (c b)
-      (hvalid a) (hvalid b) (congrArg Prod.snd he)
-    have hze : y a k = y b k := (hz a).trans (hz b).symm
-    have htB : SpliceTrace G x (inp b).2 (y b) k 0 (inp b).1 (steps b) (out a).1 := by
-      rw [ho]
-      exact htrace b
-    have heB : SpliceExecution G x (inp b) (steps b) (out a) := ho.symm ▸ hex b
-    have hg : gapSpliceCode G (out a).1 x (y a k) (steps a) =
-        gapSpliceCode G (out a).1 x (y a k) (steps b) := by
-      simpa only [c,hz a,ho] using hc
-    have hs := (htrace a).code_unique_of_gapCode G x (inp a).2 (inp b).2 (y a) (y b)
-      k 0 (inp a).1 (inp b).1 (out a).1 (steps a) (steps b) htB hze (hfin a) hg
-    have hi := (htrace a).input_unique_of_gapCode G x (inp a).2 (inp b).2 (y a) (y b)
-      k 0 (inp a).1 (inp b).1 (out a).1 (steps a) (steps b) htB hze (hfin a) hg
-      (out a).2 (hex a) heB
-    exact hir (Prod.ext hi hs)
-  let w (t : SpliceState E × (Fin k → Fin (M+1))) : ℝ :=
-    spliceTupleWeight Q (Finset.Icc 1 k) p t.1 * ∏ i, gapLetterWeight L (t.2 i).val
-  have hweight : ∀ a, spliceTupleWeight Q (Finset.Icc 1 k) p (inp a) =
-      spliceTupleWeight Q (Finset.Icc 1 k) p (out a) := by
-    intro a
-    apply ((hex a).weight G x (inp a) (out a) (steps a) Q (Finset.Icc 1 k) ?_ p).symm
-    intro q hq
-    have hq' := (htrace a).seam_data G x (inp a).2 (y a) k 0 (inp a).1 (out a).1 (steps a) q hq
-    exact Finset.mem_Icc.mpr ⟨hq'.1,hq'.2.1⟩
-  calc
-    _ ≤ ∑ a, w (f a) := by
-      apply Finset.sum_le_sum
-      intro a _
-      dsimp only [w,f,arr,sparseGapArray]
-      rw [← hweight a]
-      apply mul_le_mul_of_nonneg_left _ (spliceTupleWeight_nonneg _ _ p _)
-      have hh := (htrace a).contact_cube_le_code_weight G hloc x (inp a).2 (y a)
-        k 0 M (inp a).1 (out a).1 (steps a) (hfin a) L hL (hgood a) (rank a) (hrank a)
-      simpa only [c,hz a] using hh
-    _ = ∑ t ∈ Finset.univ.image f, w t := (Finset.sum_image hinj.injOn).symm
-    _ ≤ ∑ t ∈ B ×ˢ (Finset.univ : Finset (Fin k → Fin (M+1))), w t := by
-      apply Finset.sum_le_sum_of_subset_of_nonneg
-      · intro t ht
-        obtain ⟨a,_,rfl⟩ := Finset.mem_image.mp ht
-        exact Finset.mem_product.mpr ⟨hB a,Finset.mem_univ _⟩
-      · intro t _ _
-        exact mul_nonneg (spliceTupleWeight_nonneg _ _ p _)
-          (Finset.prod_nonneg (fun i _ => gapLetterWeight_nonneg L (by linarith) _))
-    _ = _ := by
-      rw [Finset.sum_product]
-      simp only [w,← Finset.mul_sum,total_gap_code_mass,← Finset.sum_mul]
-
-end FiniteInverseMass
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section RemoteSpliceGeometry
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-def GraphSeparated (G : BondGraph V E) (A B : Set V) : Prop :=
-  Disjoint A B ∧ ∀ a ∈ A, ∀ b ∈ B, ¬Adj G a b
-
-theorem GraphSeparated.symm (G : BondGraph V E) (A B : Set V)
-    (h : GraphSeparated G A B) : GraphSeparated G B A := by
-  refine ⟨h.1.symm,?_⟩
-  intro b hb a ha hab
-  obtain ⟨e,he⟩ := hab
-  exact h.2 a ha b hb ⟨e,he.trans (Sym2.eq_swap ..)⟩
-
-theorem GraphSeparated.mono (G : BondGraph V E) {A B C D : Set V}
-    (h : GraphSeparated G A B) (hC : C ⊆ A) (hD : D ⊆ B) : GraphSeparated G C D := by
-  exact ⟨h.1.mono hC hD,fun c hc d hd => h.2 c (hC hc) d (hD hd)⟩
-
-theorem GraphSeparated.no_shared_incidence (G : BondGraph V E) (A B : Set V)
-    (h : GraphSeparated G A B) (e : E) (a b : V) (ha : a ∈ A) (hb : b ∈ B)
-    (hea : a ∈ G e) (heb : b ∈ G e) : False := by
-  have hne : a ≠ b := by
-    rintro rfl
-    exact Set.disjoint_left.mp h.1 ha hb
-  exact h.2 a ha b hb ⟨e,(Sym2.mem_and_mem_iff hne).mp ⟨hea,heb⟩⟩
-
- 
-
-theorem slit_cluster_remove_remote (G : BondGraph V E) (η : Set E) (S H : Set V)
-    (y : V) (hy : y ∉ S)
-    (hsep : GraphSeparated G (cluster G (η ∩ internalEdges G Sᶜ) y) H) :
-    cluster G (η ∩ internalEdges G (S \ H)ᶜ) y = cluster G (η ∩ internalEdges G Sᶜ) y := by
-  apply Set.Subset.antisymm
-  · intro v hv
-    induction hv with
-    | refl => exact conn_refl G _ y
-    | @tail a b hya hab ih =>
-      obtain ⟨e,he,hab⟩ := hab
-      have hbH : b ∉ H := fun hb => hsep.2 a ih b hb ⟨e,hab⟩
-      have hbS : b ∉ S := by
-        have hb := ((mem_internalEdges_of_ends G _ e a b hab).mp he.2).2
-        exact fun h => hb ⟨h,hbH⟩
-      have haS := slit_cluster_subset G S η y hy ih
-      exact ih.tail ⟨e,⟨he.1,(mem_internalEdges_of_ends G _ e a b hab).mpr ⟨haS,hbS⟩⟩,hab⟩
-  · exact fun _ h => conn_mono G
-      (Set.inter_subset_inter_right _ (internalEdges_mono G (by
-        intro v hv hvS
-        exact hv hvS.1))) h
-
- 
-
-theorem spliceMask_remove_remote (G : BondGraph V E) (S B H : Set V) (e : E)
-    (hsep : GraphSeparated G B H) : spliceMask G (S \ H) B e = spliceMask G S B e := by
-  ext f
-  simp only [spliceMask,Set.mem_insert_iff,Set.mem_ofPred_eq]
-  apply or_congr Iff.rfl
-  constructor
-  · rintro ⟨⟨b,hb,hbf⟩,hi⟩
-    refine ⟨⟨b,hb,hbf⟩,?_⟩
-    intro v hv hvS
-    have hvH : v ∉ H := fun hh => hsep.no_shared_incidence G B H f b v hb hh hbf hv
-    exact hi v hv ⟨hvS,hvH⟩
-  · rintro ⟨hb,hi⟩
-    exact ⟨hb,fun v hv h => hi v hv h.1⟩
-
- 
-theorem spliceMask_incident (G : BondGraph V E) (S B : Set V) (e : E)
-    (b : V) (hb : b ∈ B) (heb : b ∈ G e) :
-    ∀ f ∈ spliceMask G S B e, ∃ v ∈ B, v ∈ G f := by
-  intro f hf
-  rcases Set.mem_insert_iff.mp hf with rfl | hf
-  · exact ⟨b,hb,heb⟩
-  · exact hf.1
-
- 
-
-theorem disjoint_spliceMasks (G : BondGraph V E) (S₁ S₂ B₁ B₂ : Set V) (e₁ e₂ : E)
-    (b₁ b₂ : V) (hb₁ : b₁ ∈ B₁) (hb₂ : b₂ ∈ B₂) (he₁ : b₁ ∈ G e₁) (he₂ : b₂ ∈ G e₂)
-    (hsep : GraphSeparated G B₁ B₂) :
-    Disjoint (spliceMask G S₁ B₁ e₁) (spliceMask G S₂ B₂ e₂) := by
-  apply Set.disjoint_left.mpr
-  intro f hf₁ hf₂
-  obtain ⟨a,ha,haf⟩ := spliceMask_incident G S₁ B₁ e₁ b₁ hb₁ he₁ f hf₁
-  obtain ⟨b,hb,hbf⟩ := spliceMask_incident G S₂ B₂ e₂ b₂ hb₂ he₂ f hf₂
-  exact hsep.no_shared_incidence G B₁ B₂ f a b ha hb haf hbf
-
-end RemoteSpliceGeometry
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section RemoteReplay
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-def replaySplice (G : BondGraph V E) (x : V) (η : ℕ → Set E) (y : ℕ → V) :
-    List (ℕ × E) → Set E → Set E
-  | [], ω => ω
-  | (i,e)::rest, ω =>
-    replaySplice G x η y rest (spliceField G (cluster G ω x)
-      (cluster G (η i ∩ internalEdges G (cluster G ω x)ᶜ) (y i)) e ω (η i))
-
- 
-def replayMasks (G : BondGraph V E) (x : V) (η : ℕ → Set E) (y : ℕ → V) :
-    List (ℕ × E) → Set E → List (ℕ × Set E)
-  | [], _ => []
-  | (i,e)::rest, ω =>
-    let S := cluster G ω x
-    let B := cluster G (η i ∩ internalEdges G Sᶜ) (y i)
-    (i,spliceMask G S B e)::replayMasks G x η y rest (spliceField G S B e ω (η i))
-
- 
-def replayContacts (G : BondGraph V E) (x : V) (η : ℕ → Set E) (y : ℕ → V) :
-    List (ℕ × E) → Set E → List (Set E)
-  | [], _ => []
-  | (i,e)::rest, ω =>
-    let S := cluster G ω x
-    let B := cluster G (η i ∩ internalEdges G Sᶜ) (y i)
-    contactSet G (η i) B::replayContacts G x η y rest (spliceField G S B e ω (η i))
-
-theorem SpliceTrace.replay_eq (G : BondGraph V E) (x : V) (η : ℕ → Set E) (y : ℕ → V)
-    (k j : ℕ) (ω ωf : Set E) (steps : List (ℕ × E))
-    (h : SpliceTrace G x η y k j ω steps ωf) : replaySplice G x η y steps ω = ωf := by
-  induction h with
-  | stop => rfl
-  | step => assumption
-
- 
-
-theorem SpliceTrace.remote_replay (G : BondGraph V E) (x : V) (η : ℕ → Set E) (y : ℕ → V)
-    (k j : ℕ) (ω ωf : Set E) (steps : List (ℕ × E))
-    (h : SpliceTrace G x η y k j ω steps ωf) (H : Set V) (ω₀ : Set E)
-    (hroot : cluster G ω₀ x = cluster G ω x \ H)
-    (hsep : GraphSeparated G (cluster G ωf x \ cluster G ω x) H)
-    (hzH : y k ∉ H) :
-    SpliceTrace G x η y k j ω₀ steps (replaySplice G x η y steps ω₀) ∧
-      cluster G (replaySplice G x η y steps ω₀) x = cluster G ωf x \ H ∧
-      replayMasks G x η y steps ω₀ = replayMasks G x η y steps ω ∧
-      replayContacts G x η y steps ω₀ = replayContacts G x η y steps ω := by
-  induction h generalizing ω₀ with
-  | stop j ω hs =>
-    refine ⟨SpliceTrace.stop j ω₀ ?_,hroot,rfl,rfl⟩
-    rw [hroot]
-    exact ⟨hs,hzH⟩
-  | @step j i ω e a b rest ωf hn hji hik hmax hy he hab ha hb ht ih =>
-    let S := cluster G ω x
-    let B := cluster G (η i ∩ internalEdges G Sᶜ) (y i)
-    let ω' := spliceField G S B e ω (η i)
-    let S₀ := cluster G ω₀ x
-    let B₀ := cluster G (η i ∩ internalEdges G S₀ᶜ) (y i)
-    let ω₀' := spliceField G S₀ B₀ e ω₀ (η i)
-    have hC : cluster G ω' x = S ∪ B := splice_cluster G ω (η i) x (y i) hy e a b hab he ha hb
-    have hBo : B ⊆ Sᶜ := slit_cluster_subset G S (η i) (y i) hy
-    have hretain : cluster G ω' x ⊆ cluster G ωf x := ht.cluster_subset G x η y k i ω' ωf rest
-    have hBT : B ⊆ cluster G ωf x \ S := by
-      intro v hv
-      refine ⟨hretain ?_,hBo hv⟩
-      rw [hC]
-      exact Or.inr hv
-    have hsepB : GraphSeparated G B H := hsep.mono G hBT Set.Subset.rfl
-    have hBB : B₀ = B := by
-      dsimp only [B₀,S₀]
-      rw [hroot]
-      exact slit_cluster_remove_remote G (η i) S H (y i) hy hsepB
-    have haH : a ∉ H := fun hh => hsepB.2 b hb a hh ⟨e,hab.trans (Sym2.eq_swap ..)⟩
-    have ha₀ : a ∈ S₀ := by rw [show S₀ = S \ H from hroot]; exact ⟨ha,haH⟩
-    have hy₀ : y i ∉ S₀ := by rw [show S₀ = S \ H from hroot]; exact fun h => hy h.1
-    have hb₀ : b ∈ B₀ := hBB.symm ▸ hb
-    have hC₀ : cluster G ω₀' x = S₀ ∪ B₀ :=
-      splice_cluster G ω₀ (η i) x (y i) hy₀ e a b hab he ha₀ hb₀
-    have hroot' : cluster G ω₀' x = cluster G ω' x \ H := by
-      rw [hC₀,hC,hBB,show S₀ = S \ H from hroot]
-      ext v
-      constructor
-      · rintro (⟨hv,hvH⟩ | hv)
-        · exact ⟨Or.inl hv,hvH⟩
-        · exact ⟨Or.inr hv,fun hvH => Set.disjoint_left.mp hsepB.1 hv hvH⟩
-      · rintro ⟨hv,hvH⟩
-        exact hv.elim (fun h => Or.inl ⟨h,hvH⟩) Or.inr
-    have hsep' : GraphSeparated G (cluster G ωf x \ cluster G ω' x) H := by
-      apply hsep.mono G _ Set.Subset.rfl
-      intro v hv
-      refine ⟨hv.1,fun h => hv.2 ?_⟩
-      rw [hC]
-      exact Or.inl h
-    obtain ⟨ht₀,hfinal,hmasks,hcontacts⟩ := ih ω₀' hroot' hsep'
-    have hn₀ : y k ∉ S₀ := by rw [show S₀ = S \ H from hroot]; exact fun h => hn h.1
-    have hmax₀ : ∀ l, i < l → l ≤ k → Disjoint (cluster G (η l) (y l)) S₀ := by
-      intro l hil hlk
-      apply (hmax l hil hlk).mono Set.Subset.rfl
-      rw [show S₀ = S \ H from hroot]
-      exact Set.sdiff_subset
-    have hmask : spliceMask G S₀ B₀ e = spliceMask G S B e := by
-      rw [hBB,show S₀ = S \ H from hroot,spliceMask_remove_remote G S B H e hsepB]
-    refine ⟨SpliceTrace.step j i ω₀ e a b rest _ hn₀ hji hik hmax₀ hy₀ he hab ha₀ hb₀ ht₀,
-      hfinal,?_,?_⟩
-    · change (i,spliceMask G S₀ B₀ e)::replayMasks G x η y rest ω₀' =
-      (i,spliceMask G S B e)::replayMasks G x η y rest ω'
-      rw [hmask,hmasks]
-    · change contactSet G (η i) B₀::replayContacts G x η y rest ω₀' =
-        contactSet G (η i) B::replayContacts G x η y rest ω'
-      rw [hBB,hcontacts]
-
-end RemoteReplay
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section DeterministicReplay
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-def FollowsContactRule (G : BondGraph V E) (x : V) (η : ℕ → Set E)
-    (y : ℕ → V) (choose : ℕ → Set E → E) (steps : List (ℕ × E)) (ω : Set E) : Prop :=
-  List.Forall₂ (fun q K => choose q.1 K = q.2) steps (replayContacts G x η y steps ω)
-
-theorem followsContactRule_cons (G : BondGraph V E) (x : V) (η : ℕ → Set E)
-    (y : ℕ → V) (choose : ℕ → Set E → E) (i : ℕ) (e : E)
-    (rest : List (ℕ × E)) (ω : Set E) :
-    FollowsContactRule G x η y choose ((i,e)::rest) ω ↔
-      choose i (contactSet G (η i)
-        (cluster G (η i ∩ internalEdges G (cluster G ω x)ᶜ) (y i))) = e ∧
-      FollowsContactRule G x η y choose rest
-        (spliceField G (cluster G ω x)
-          (cluster G (η i ∩ internalEdges G (cluster G ω x)ᶜ) (y i)) e ω (η i)) := by
-  simp only [FollowsContactRule,replayContacts,List.forall₂_cons]
-
- 
-
-theorem SpliceTrace.remote_follows_contact_rule (G : BondGraph V E) (x : V)
-    (η : ℕ → Set E) (y : ℕ → V) (k j : ℕ) (ω ωf : Set E) (steps : List (ℕ × E))
-    (h : SpliceTrace G x η y k j ω steps ωf) (H : Set V) (ω₀ : Set E)
-    (hroot : cluster G ω₀ x = cluster G ω x \ H)
-    (hsep : GraphSeparated G (cluster G ωf x \ cluster G ω x) H) (hzH : y k ∉ H)
-    (choose : ℕ → Set E → E) (hc : FollowsContactRule G x η y choose steps ω) :
-    FollowsContactRule G x η y choose steps ω₀ := by
-  unfold FollowsContactRule at hc ⊢
-  rw [(h.remote_replay G x η y k j ω ωf steps H ω₀ hroot hsep hzH).2.2.2]
-  exact hc
-
- 
-
-theorem SpliceTrace.seeded_unique (G : BondGraph V E) (x : V) (η : ℕ → Set E)
-    (y : ℕ → V) (k j : ℕ) (ω ω₁ ω₂ : Set E) (c₁ c₂ : List (ℕ × E))
-    (h₁ : SpliceTrace G x η y k j ω c₁ ω₁)
-    (h₂ : SpliceTrace G x η y k j ω c₂ ω₂) (choose : ℕ → Set E → E)
-    (hc₁ : FollowsContactRule G x η y choose c₁ ω)
-    (hc₂ : FollowsContactRule G x η y choose c₂ ω) : c₁ = c₂ ∧ ω₁ = ω₂ := by
-  induction h₁ generalizing c₂ ω₂ with
-  | stop j ω hs =>
-    cases h₂ with
-    | stop => exact ⟨rfl,rfl⟩
-    | step _ _ _ _ _ _ _ _ hn => exact (hn hs).elim
-  | @step j i ω e a b rest ω₁ hn hji hik hmax hy he hab ha hb ht ih =>
-    cases h₂ with
-    | stop _ _ hs => exact (hn hs).elim
-    | step _ l _ f c d tail ω₂ hn₂ hjl hlk hmax₂ hy₂ hf hcd hc hd ht₂ =>
-      have haC : a ∈ cluster G (η i) (y i) :=
-        (edge_cluster_sides G (η i) (y i) he hab).mpr (conn_mono G Set.inter_subset_left hb)
-      have hcC : c ∈ cluster G (η l) (y l) :=
-        (edge_cluster_sides G (η l) (y l) hf hcd).mpr (conn_mono G Set.inter_subset_left hd)
-      have hil : i = l := by
-        rcases lt_trichotomy i l with hil | hil | hli
-        · exact (Set.disjoint_left.mp (hmax l hil hlk) hcC hc).elim
-        · exact hil
-        · exact (Set.disjoint_left.mp (hmax₂ i hli hik) haC ha).elim
-      subst l
-      obtain ⟨hec,hrest⟩ := (followsContactRule_cons G x η y choose i e rest ω).mp hc₁
-      obtain ⟨hfc,htail⟩ := (followsContactRule_cons G x η y choose i f tail ω).mp hc₂
-      have hef : e = f := hec.symm.trans hfc
-      rw [← hef] at ht₂ htail ⊢
-      obtain ⟨hcodes,houts⟩ := ih ω₂ tail ht₂ hrest htail
-      exact ⟨congrArg (List.cons (i,e)) hcodes,houts⟩
-
-end DeterministicReplay
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section TraceMaskSupport
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-def maskUnion (masks : List (ℕ × Set E)) : Set E :=
-  {e | ∃ q ∈ masks, e ∈ q.2}
-
-@[simp] theorem maskUnion_nil : maskUnion ([] : List (ℕ × Set E)) = ∅ := by
-  ext e
-  simp [maskUnion]
-
-@[simp] theorem maskUnion_cons (i : ℕ) (M : Set E) (ms : List (ℕ × Set E)) :
-    maskUnion ((i,M)::ms) = M ∪ maskUnion ms := by
-  ext e
-  simp [maskUnion,or_and_right,exists_or]
-
- 
-theorem SpliceTrace.masks_incident (G : BondGraph V E) (x : V) (η : ℕ → Set E)
-    (y : ℕ → V) (k j : ℕ) (ω ωf : Set E) (steps : List (ℕ × E))
-    (h : SpliceTrace G x η y k j ω steps ωf) :
-    ∀ f ∈ maskUnion (replayMasks G x η y steps ω),
-      ∃ b ∈ cluster G ωf x \ cluster G ω x, b ∈ G f := by
-  induction h with
-  | stop => simp [replayMasks]
-  | @step j i ω e a b rest ωf hn hji hik hmax hy he hab ha hb ht ih =>
-    let S := cluster G ω x
-    let B := cluster G (η i ∩ internalEdges G Sᶜ) (y i)
-    let ω' := spliceField G S B e ω (η i)
-    have hC : cluster G ω' x = S ∪ B := splice_cluster G ω (η i) x (y i) hy e a b hab he ha hb
-    have hB : B ⊆ Sᶜ := slit_cluster_subset G S (η i) (y i) hy
-    have hretain := ht.cluster_subset G x η y k i ω' ωf rest
-    intro f hf
-    change f ∈ maskUnion ((i,spliceMask G S B e)::replayMasks G x η y rest ω') at hf
-    rw [maskUnion_cons] at hf
-    rcases hf with hf | hf
-    · obtain ⟨v,hv,hvf⟩ := spliceMask_incident G S B e b hb (by simp [hab]) f hf
-      refine ⟨v,⟨hretain ?_,hB hv⟩,hvf⟩
-      rw [hC]
-      exact Or.inr hv
-    · obtain ⟨v,hv,hvf⟩ := ih f hf
-      refine ⟨v,⟨hv.1,fun h => hv.2 ?_⟩,hvf⟩
-      rw [hC]
-      exact Or.inl h
-
- 
-theorem replaySplice_off_masks (G : BondGraph V E) (x : V) (η : ℕ → Set E)
-    (y : ℕ → V) (steps : List (ℕ × E)) (ω : Set E) (f : E)
-    (hf : f ∉ maskUnion (replayMasks G x η y steps ω)) :
-    f ∈ replaySplice G x η y steps ω ↔ f ∈ ω := by
-  induction steps generalizing ω with
-  | nil => rfl
-  | cons q rest ih =>
-    obtain ⟨i,e⟩ := q
-    simp only [replayMasks,maskUnion_cons,Set.mem_union,not_or] at hf
-    change f ∈ replaySplice G x η y rest _ ↔ f ∈ ω
-    rw [ih _ hf.2]
-    simp only [spliceField,mem_exchangeOn,ite_eq_right hf.1]
-
- 
-theorem SpliceTrace.masks_disjoint (G : BondGraph V E) (x : V)
-    (η₁ η₂ : ℕ → Set E) (y₁ y₂ : ℕ → V) (k j₁ j₂ : ℕ)
-    (ω₁ ω₂ ωf : Set E) (c₁ c₂ : List (ℕ × E))
-    (h₁ : SpliceTrace G x η₁ y₁ k j₁ ω₁ c₁ ωf)
-    (h₂ : SpliceTrace G x η₂ y₂ k j₂ ω₂ c₂ ωf)
-    (hsep : GraphSeparated G (cluster G ωf x \ cluster G ω₁ x)
-      (cluster G ωf x \ cluster G ω₂ x)) :
-    Disjoint (maskUnion (replayMasks G x η₁ y₁ c₁ ω₁))
-      (maskUnion (replayMasks G x η₂ y₂ c₂ ω₂)) := by
-  apply Set.disjoint_left.mpr
-  intro f hf₁ hf₂
-  obtain ⟨a,ha,hea⟩ := h₁.masks_incident G x η₁ y₁ k j₁ ω₁ ωf c₁ f hf₁
-  obtain ⟨b,hb,heb⟩ := h₂.masks_incident G x η₂ y₂ k j₂ ω₂ ωf c₂ f hf₂
-  exact hsep.no_shared_incidence G _ _ f a b ha hb hea heb
-
- 
-
-theorem conn_delete_rootBridge (G : BondGraph V E) (ω η : Set E) (x z : V)
-    (hη : η ⊆ ω) (e : E) (a b : V) (hab : G e = s(a,b))
-    (hb : b ∉ cluster G (ω \ {e}) x) (hz : z ∈ cluster G (ω \ {e}) x)
-    (hxz : Conn G η x z) : Conn G (η \ {e}) x z := by
-  have hc : Conn G (insert e (η \ {e})) x z := conn_mono G (by
-    intro f hf
-    by_cases hfe : f = e
-    · exact Or.inl hfe
-    · exact Or.inr ⟨hf,hfe⟩) hxz
-  have hs : η \ {e} ⊆ ω \ {e} := Set.sdiff_subset_sdiff_left hη
-  rw [conn_insert_iff G _ e a b x z hab] at hc
-  rcases hc with hc | ⟨_,hc⟩ | ⟨hc,_⟩
-  · exact hc
-  · exact (hb (hz.trans (conn_symm G _ (conn_mono G hs hc)))).elim
-  · exact (hb (conn_mono G hs hc)).elim
-
- 
-theorem cluster_double_cut (G : BondGraph V E) (ω : Set E) (x : V) (e f : E)
-    (a b : V) (hab : G f = s(a,b)) (hb : b ∉ cluster G (ω \ {f}) x) :
-    cluster G ((ω \ {e}) \ {f}) x = cluster G (ω \ {e}) x ∩ cluster G (ω \ {f}) x := by
-  apply Set.Subset.antisymm
-  · intro v hv
-    exact ⟨conn_mono G Set.sdiff_subset hv,
-      conn_mono G (ω₁ := (ω \ {e}) \ {f}) (ω₂ := ω \ {f}) (fun _ h => ⟨h.1.1,h.2⟩) hv⟩
-  · intro v hv
-    exact conn_delete_rootBridge G ω (ω \ {e}) x v Set.sdiff_subset f a b hab hb hv.2 hv.1
-
-end TraceMaskSupport
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section JointRoot
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem mask_not_crossing_remote (G : BondGraph V E) (U : Set E) (A B : Set V)
-    (hU : ∀ e ∈ U, ∃ v ∈ B, v ∈ G e) (hBA : B ⊆ A)
-    (e : E) (a b : V) (hab : G e = s(a,b)) (haB : a ∉ B) (hbA : b ∉ A) : e ∉ U := by
-  intro he
-  obtain ⟨v,hv,hve⟩ := hU e he
-  have hvab : v = a ∨ v = b := by simpa only [hab,Sym2.mem_iff] using hve
-  rcases hvab with rfl | rfl
-  · exact haB hv
-  · exact hbA (hBA hv)
-
- 
-
-theorem exchange_patch_symm (U₁ U₂ ω₁ ω₂ ωf : Set E)
-    (hdis : Disjoint U₁ U₂)
-    (h₁ : ∀ e ∉ U₁, e ∈ ω₁ ↔ e ∈ ωf)
-    (h₂ : ∀ e ∉ U₂, e ∈ ω₂ ↔ e ∈ ωf) :
-    exchangeOn U₂ ω₁ ω₂ = exchangeOn U₁ ω₂ ω₁ := by
-  ext e
-  by_cases he₁ : e ∈ U₁
-  · have he₂ : e ∉ U₂ := fun h => Set.disjoint_left.mp hdis he₁ h
-    simp only [mem_exchangeOn,ite_eq_right he₂,ite_eq_left he₁]
-  · by_cases he₂ : e ∈ U₂
-    · simp only [mem_exchangeOn,ite_eq_left he₂,ite_eq_right he₁]
-    · simp only [mem_exchangeOn,ite_eq_right he₁,ite_eq_right he₂]
-      exact (h₁ e he₁).trans (h₂ e he₂).symm
-
- 
-
-theorem SpliceTrace.joint_root (G : BondGraph V E) (x : V)
-    (η₁ η₂ : ℕ → Set E) (y₁ y₂ : ℕ → V) (k j₁ j₂ i₁ i₂ : ℕ)
-    (ω₁ ω₂ ωf : Set E) (e₁ e₂ : E) (r₁ r₂ : List (ℕ × E))
-    (h₁ : SpliceTrace G x η₁ y₁ k j₁ ω₁ ((i₁,e₁)::r₁) ωf)
-    (h₂ : SpliceTrace G x η₂ y₂ k j₂ ω₂ ((i₂,e₂)::r₂) ωf)
-    (hsep : GraphSeparated G (cluster G ωf x \ cluster G ω₁ x)
-      (cluster G ωf x \ cluster G ω₂ x)) :
-    cluster G (exchangeOn (maskUnion (replayMasks G x η₂ y₂ ((i₂,e₂)::r₂) ω₂)) ω₁ ω₂) x =
-      cluster G ω₁ x ∩ cluster G ω₂ x := by
-  let A₁ := cluster G ω₁ x
-  let A₂ := cluster G ω₂ x
-  let B₁ := cluster G ωf x \ A₁
-  let B₂ := cluster G ωf x \ A₂
-  let U₁ := maskUnion (replayMasks G x η₁ y₁ ((i₁,e₁)::r₁) ω₁)
-  let U₂ := maskUnion (replayMasks G x η₂ y₂ ((i₂,e₂)::r₂) ω₂)
-  let ω₀ := exchangeOn U₂ ω₁ ω₂
-  have hsub₁ := h₁.cluster_subset G x η₁ y₁ k j₁ ω₁ ωf ((i₁,e₁)::r₁)
-  have hsub₂ := h₂.cluster_subset G x η₂ y₂ k j₂ ω₂ ωf ((i₂,e₂)::r₂)
-  have hBA₁ : B₁ ⊆ A₂ := by
-    intro v hv
-    by_contra hn
-    exact Set.disjoint_left.mp hsep.1 hv ⟨hv.1,hn⟩
-  have hBA₂ : B₂ ⊆ A₁ := by
-    intro v hv
-    by_contra hn
-    exact Set.disjoint_left.mp hsep.1 ⟨hv.1,hn⟩ hv
-  have hU₁ : ∀ e ∈ U₁, ∃ v ∈ B₁, v ∈ G e :=
-    h₁.masks_incident G x η₁ y₁ k j₁ ω₁ ωf ((i₁,e₁)::r₁)
-  have hU₂ : ∀ e ∈ U₂, ∃ v ∈ B₂, v ∈ G e :=
-    h₂.masks_incident G x η₂ y₂ k j₂ ω₂ ωf ((i₂,e₂)::r₂)
-  have hdis : Disjoint U₁ U₂ := h₁.masks_disjoint G x η₁ η₂ y₁ y₂ k j₁ j₂ ω₁ ω₂ ωf
-    ((i₁,e₁)::r₁) ((i₂,e₂)::r₂) h₂ hsep
-  have hoff₁ : ∀ e ∉ U₁, e ∈ ω₁ ↔ e ∈ ωf := by
-    intro e he
-    rw [← h₁.replay_eq G x η₁ y₁ k j₁ ω₁ ωf ((i₁,e₁)::r₁)]
-    exact (replaySplice_off_masks G x η₁ y₁ ((i₁,e₁)::r₁) ω₁ e he).symm
-  have hoff₂ : ∀ e ∉ U₂, e ∈ ω₂ ↔ e ∈ ωf := by
-    intro e he
-    rw [← h₂.replay_eq G x η₂ y₂ k j₂ ω₂ ωf ((i₂,e₂)::r₂)]
-    exact (replaySplice_off_masks G x η₂ y₂ ((i₂,e₂)::r₂) ω₂ e he).symm
-  have hsym : ω₀ = exchangeOn U₁ ω₂ ω₁ := exchange_patch_symm U₁ U₂ ω₁ ω₂ ωf hdis hoff₁ hoff₂
-  have hcut₁ := h₁.head_cut G x η₁ y₁ k j₁ i₁ ω₁ ωf e₁ r₁
-  have hcut₂ := h₂.head_cut G x η₂ y₂ k j₂ i₂ ω₂ ωf e₂ r₂
-  obtain ⟨a,b,hab,_,hb,_⟩ := openMarkedPivotal_oriented G ωf {y₂ k} x
-    (h₂.seams_pivotal G x η₂ y₂ k j₂ ω₂ ωf ((i₂,e₂)::r₂) (i₂,e₂) (by simp))
-  have hdouble : cluster G ((ωf \ {e₁}) \ {e₂}) x = A₁ ∩ A₂ := by
-    rw [cluster_double_cut G ωf x e₁ e₂ a b hab hb,hcut₁,hcut₂]
-  change cluster G ω₀ x = A₁ ∩ A₂
-  apply Set.Subset.antisymm
-  · intro v hv
-    induction hv with
-    | refl => exact ⟨conn_refl G _ x,conn_refl G _ x⟩
-    | @tail a b hxa hab ih =>
-      obtain ⟨e,he,hab⟩ := hab
-      constructor
-      · by_contra hb
-        have hno := mask_not_crossing_remote G U₂ A₁ B₂ hU₂ hBA₂ e a b hab
-          (fun hh => hh.2 ih.2) hb
-        have he₁ : e ∈ ω₁ := by simpa only [ω₀,mem_exchangeOn,ite_eq_right hno] using he
-        exact hb ((edge_cluster_sides G ω₁ x he₁ hab).mp ih.1)
-      · by_contra hb
-        have hno := mask_not_crossing_remote G U₁ A₂ B₁ hU₁ hBA₁ e a b hab
-          (fun hh => hh.2 ih.1) hb
-        have he₂ : e ∈ ω₂ := by simpa only [hsym,mem_exchangeOn,ite_eq_right hno] using he
-        exact hb ((edge_cluster_sides G ω₂ x he₂ hab).mp ih.2)
-  · intro v hv
-    have hc : Conn G ((ωf \ {e₁}) \ {e₂}) x v := by
-      change v ∈ cluster G ((ωf \ {e₁}) \ {e₂}) x
-      rw [hdouble]
-      exact hv
-    have hr := conn_restrict_cluster G ((ωf \ {e₁}) \ {e₂}) x (mem_cluster_self G _ x) hc
-    apply conn_mono G _ hr
-    intro e he
-    have hi : e ∈ internalEdges G (A₁ ∩ A₂) := by
-      simpa only [hdouble] using he.2
-    have hno : e ∉ U₂ := by
-      intro hu
-      obtain ⟨b,hb,heb⟩ := hU₂ e hu
-      exact hb.2 (hi b heb).2
-    have he₁ : e ∈ ω₁ := (h₁.internal_agreement G x η₁ y₁ k j₁ ω₁ ωf
-      ((i₁,e₁)::r₁) e (fun a ha => (hi a ha).1)).mp he.1.1.1
-    simpa only [ω₀,mem_exchangeOn,ite_eq_right hno] using he₁
-
-end JointRoot
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section MaskExecution
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-def runMasks : List (ℕ × Set E) → SpliceState E → SpliceState E
-  | [], s => s
-  | (i,M)::ms, s => runMasks ms (exchangeState M i s)
-
- 
-def patchFirst (U τ : Set E) (s : SpliceState E) : SpliceState E :=
-  (exchangeOn U s.1 τ,s.2)
-
-theorem exchangeState_patchFirst (U τ M : Set E) (i : ℕ) (s : SpliceState E)
-    (hdis : Disjoint M U) :
-    exchangeState M i (patchFirst U τ s) = patchFirst U τ (exchangeState M i s) := by
-  apply Prod.ext
-  · ext e
-    by_cases heM : e ∈ M
-    · have heU : e ∉ U := fun he => Set.disjoint_left.mp hdis heM he
-      simp only [exchangeState,patchFirst,mem_exchangeOn,ite_eq_left heM,ite_eq_right heU]
-    · by_cases heU : e ∈ U <;>
-      simp only [exchangeState,patchFirst,mem_exchangeOn,ite_eq_right heM,heU,ite_true,ite_false]
-  · funext l
-    by_cases hli : l = i
-    · subst l
-      ext e
-      by_cases heM : e ∈ M
-      · have heU : e ∉ U := fun he => Set.disjoint_left.mp hdis heM he
-        simp only [exchangeState,patchFirst,Function.update_self,mem_exchangeOn,ite_eq_left heM,ite_eq_right heU]
-      · simp only [exchangeState,patchFirst,Function.update_self,mem_exchangeOn,ite_eq_right heM]
-    · simp only [exchangeState,patchFirst,Function.update_of_ne hli]
-
- 
-
-theorem runMasks_patchFirst (ms : List (ℕ × Set E)) (U τ : Set E) (s : SpliceState E)
-    (hdis : Disjoint (maskUnion ms) U) :
-    runMasks ms (patchFirst U τ s) = patchFirst U τ (runMasks ms s) := by
-  induction ms generalizing s with
-  | nil => rfl
-  | cons q ms ih =>
-    obtain ⟨i,M⟩ := q
-    rw [maskUnion_cons,Set.disjoint_union_left] at hdis
-    simp only [runMasks]
-    rw [exchangeState_patchFirst U τ M i s hdis.1,ih _ hdis.2]
-
- 
-
-theorem SpliceTrace.runMasks_first (G : BondGraph V E) (x : V) (η : ℕ → Set E)
-    (y : ℕ → V) (k j : ℕ) (ω ωf : Set E) (steps : List (ℕ × E))
-    (h : SpliceTrace G x η y k j ω steps ωf) :
-    ∀ θ : ℕ → Set E, (∀ l, j < l → l ≤ k → θ l = η l) →
-    (runMasks (replayMasks G x η y steps ω) (ω,θ)).1 = ωf := by
-  induction h with
-  | stop => exact fun _ _ => rfl
-  | @step j i ω e a b rest ωf hn hji hik hmax hy he hab ha hb ht ih =>
-    intro θ hθ
-    have hθi := hθ i hji hik
-    let S := cluster G ω x
-    let B := cluster G (η i ∩ internalEdges G Sᶜ) (y i)
-    let M := spliceMask G S B e
-    let ω' := spliceField G S B e ω (η i)
-    let θ' := Function.update θ i (exchangeOn M (θ i) ω)
-    have hstep : exchangeState M i (ω,θ) = (ω',θ') := by
-      apply Prod.ext
-      · change exchangeOn M ω (θ i) = exchangeOn M ω (η i)
-        rw [hθi]
-      · rfl
-    change (runMasks (replayMasks G x η y rest ω') (exchangeState M i (ω,θ))).1 = ωf
-    rw [hstep]
-    apply ih θ'
-    intro l hil hlk
-    rw [show θ' l = θ l from Function.update_of_ne (by omega) _ _]
-    exact hθ l (hji.trans hil) hlk
-
- 
-theorem runMasks_weight (ms : List (ℕ × Set E)) (s : SpliceState E)
-    (Q : Finset E) (J : Finset ℕ) (p : ℝ) (hJ : ∀ q ∈ ms, q.1 ∈ J) :
-    spliceTupleWeight Q J p (runMasks ms s) = spliceTupleWeight Q J p s := by
-  induction ms generalizing s with
-  | nil => rfl
-  | cons q ms ih =>
-    obtain ⟨i,M⟩ := q
-    rw [runMasks,ih _ (fun q hq => hJ q (List.mem_cons_of_mem _ hq))]
-    exact spliceTupleWeight_exchange Q J p s M i (hJ (i,M) (List.mem_cons_self ..))
-
- 
-
-theorem SpliceTrace.joint_replay_state (G : BondGraph V E) (x : V)
-    (η₁ η₂ : ℕ → Set E) (y₁ y₂ : ℕ → V) (k j₁ j₂ : ℕ)
-    (ω₁ ω₂ ωf : Set E) (c₁ c₂ : List (ℕ × E))
-    (h₁ : SpliceTrace G x η₁ y₁ k j₁ ω₁ c₁ ωf)
-    (h₂ : SpliceTrace G x η₂ y₂ k j₂ ω₂ c₂ ωf)
-    (hsep : GraphSeparated G (cluster G ωf x \ cluster G ω₁ x)
-      (cluster G ωf x \ cluster G ω₂ x)) :
-    let m₁ := replayMasks G x η₁ y₁ c₁ ω₁
-    let m₂ := replayMasks G x η₂ y₂ c₂ ω₂
-    let ω₀ := exchangeOn (maskUnion m₂) ω₁ ω₂
-    runMasks m₁ (ω₀,η₁) = (ω₂,(runMasks m₁ (ω₁,η₁)).2) := by
-  intro m₁ m₂ ω₀
-  have hdis := h₁.masks_disjoint G x η₁ η₂ y₁ y₂ k j₁ j₂ ω₁ ω₂ ωf c₁ c₂ h₂ hsep
-  have hroot := h₁.runMasks_first G x η₁ y₁ k j₁ ω₁ ωf c₁ η₁ (by intros; rfl)
-  have hpatch : exchangeOn (maskUnion m₂) ωf ω₂ = ω₂ := by
-    ext e
-    by_cases he : e ∈ maskUnion m₂
-    · simp only [mem_exchangeOn,ite_eq_left he]
-    · simp only [mem_exchangeOn,ite_eq_right he]
-      rw [← h₂.replay_eq G x η₂ y₂ k j₂ ω₂ ωf c₂]
-      exact replaySplice_off_masks G x η₂ y₂ c₂ ω₂ e he
-  change runMasks m₁ (patchFirst (maskUnion m₂) ω₂ (ω₁,η₁)) = _
-  rw [runMasks_patchFirst m₁ _ ω₂ _ hdis]
-  change (exchangeOn (maskUnion m₂) (runMasks m₁ (ω₁,η₁)).1 ω₂,_) = _
-  rw [hroot,hpatch]
-
-end MaskExecution
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section JointReplay
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem SpliceTrace.endpoint_not_mem_of_ne_nil (G : BondGraph V E) (x : V)
-    (η : ℕ → Set E) (y : ℕ → V) (k j : ℕ) (ω ωf : Set E) (c : List (ℕ × E))
-    (h : SpliceTrace G x η y k j ω c ωf) (hc : c ≠ []) : y k ∉ cluster G ω x := by
-  cases h with
-  | stop => exact (hc rfl).elim
-  | step _ _ _ _ _ _ _ _ hn => exact hn
-
- 
-
-theorem SpliceTrace.joint_virtual (G : BondGraph V E) (x : V)
-    (η₁ η₂ : ℕ → Set E) (y₁ y₂ : ℕ → V) (k j₁ j₂ : ℕ)
-    (ω₁ ω₂ ωf : Set E) (c₁ c₂ : List (ℕ × E))
-    (h₁ : SpliceTrace G x η₁ y₁ k j₁ ω₁ c₁ ωf)
-    (h₂ : SpliceTrace G x η₂ y₂ k j₂ ω₂ c₂ ωf)
-    (hc₁ : c₁ ≠ []) (hc₂ : c₂ ≠ [])
-    (hsep : GraphSeparated G (cluster G ωf x \ cluster G ω₁ x)
-      (cluster G ωf x \ cluster G ω₂ x)) :
-    let ω₀ := exchangeOn (maskUnion (replayMasks G x η₂ y₂ c₂ ω₂)) ω₁ ω₂
-    SpliceTrace G x η₁ y₁ k j₁ ω₀ c₁ ω₂ ∧
-      replayMasks G x η₁ y₁ c₁ ω₀ = replayMasks G x η₁ y₁ c₁ ω₁ ∧
-      replayContacts G x η₁ y₁ c₁ ω₀ = replayContacts G x η₁ y₁ c₁ ω₁ := by
-  intro ω₀
-  have hroot' : cluster G ω₀ x = cluster G ω₁ x ∩ cluster G ω₂ x := by
-    cases c₁ with
-    | nil => exact (hc₁ rfl).elim
-    | cons q r₁ =>
-      obtain ⟨i₁,e₁⟩ := q
-      cases c₂ with
-      | nil => exact (hc₂ rfl).elim
-      | cons q r₂ =>
-        obtain ⟨i₂,e₂⟩ := q
-        exact h₁.joint_root G x η₁ η₂ y₁ y₂ k j₁ j₂ i₁ i₂ ω₁ ω₂ ωf e₁ e₂ r₁ r₂ h₂ hsep
-  let H := cluster G ωf x \ cluster G ω₂ x
-  have hroot : cluster G ω₀ x = cluster G ω₁ x \ H := by
-    rw [hroot']
-    ext v
-    constructor
-    · rintro ⟨h1,h2⟩
-      exact ⟨h1,fun hh => hh.2 h2⟩
-    · rintro ⟨h1,h2⟩
-      refine ⟨h1,?_⟩
-      by_contra hn
-      exact h2 ⟨h₁.cluster_subset G x η₁ y₁ k j₁ ω₁ ωf c₁ h1,hn⟩
-  have hz : y₁ k ∉ H := fun hz => Set.disjoint_left.mp hsep.1
-    ⟨h₁.endpoint G x η₁ y₁ k j₁ ω₁ ωf c₁,
-      h₁.endpoint_not_mem_of_ne_nil G x η₁ y₁ k j₁ ω₁ ωf c₁ hc₁⟩ hz
-  obtain ⟨ht,hout,hm,hcontacts⟩ := h₁.remote_replay G x η₁ y₁ k j₁ ω₁ ωf c₁ H ω₀ hroot hsep hz
-  have hf := ht.runMasks_first G x η₁ y₁ k j₁ ω₀ (replaySplice G x η₁ y₁ c₁ ω₀) c₁ η₁
-    (by intros; rfl)
-  rw [hm] at hf
-  have hs := h₁.joint_replay_state G x η₁ η₂ y₁ y₂ k j₁ j₂ ω₁ ω₂ ωf c₁ c₂ h₂ hsep
-  have heq : replaySplice G x η₁ y₁ c₁ ω₀ = ω₂ := hf.symm.trans (congrArg Prod.fst hs)
-  rw [heq] at ht
-  exact ⟨ht,hm,hcontacts⟩
-
- 
-
-theorem joint_restore_seeded_unique (G : BondGraph V E) (x : V)
-    (η₁ η₂ : ℕ → Set E) (y₁ y₂ : ℕ → V) (choose₁ choose₂ : ℕ → Set E → E)
-    (k j₁ j₂ : ℕ) (ω₁ ω₂ ωf σ₁ σ₂ σf : Set E)
-    (c₁ c₂ d₁ d₂ : List (ℕ × E))
-    (h₁ : SpliceTrace G x η₁ y₁ k j₁ ω₁ c₁ ωf)
-    (h₂ : SpliceTrace G x η₂ y₂ k j₂ ω₂ c₂ ωf)
-    (g₁ : SpliceTrace G x η₁ y₁ k j₁ σ₁ d₁ σf)
-    (g₂ : SpliceTrace G x η₂ y₂ k j₂ σ₂ d₂ σf)
-    (hc₁ : c₁ ≠ []) (hc₂ : c₂ ≠ []) (hd₁ : d₁ ≠ []) (hd₂ : d₂ ≠ [])
-    (hsep : GraphSeparated G (cluster G ωf x \ cluster G ω₁ x)
-      (cluster G ωf x \ cluster G ω₂ x))
-    (gsep : GraphSeparated G (cluster G σf x \ cluster G σ₁ x)
-      (cluster G σf x \ cluster G σ₂ x))
-    (hr₁ : FollowsContactRule G x η₁ y₁ choose₁ c₁ ω₁)
-    (hr₂ : FollowsContactRule G x η₂ y₂ choose₂ c₂ ω₂)
-    (gr₁ : FollowsContactRule G x η₁ y₁ choose₁ d₁ σ₁)
-    (gr₂ : FollowsContactRule G x η₂ y₂ choose₂ d₂ σ₂)
-    (hrestore : exchangeOn (maskUnion (replayMasks G x η₂ y₂ c₂ ω₂)) ω₁ ω₂ =
-      exchangeOn (maskUnion (replayMasks G x η₂ y₂ d₂ σ₂)) σ₁ σ₂) :
-    c₁ = d₁ ∧ c₂ = d₂ ∧ ωf = σf ∧
-      (runMasks (replayMasks G x η₁ y₁ c₁ ω₁) (ω₁,η₁)).2 =
-        (runMasks (replayMasks G x η₁ y₁ d₁ σ₁) (σ₁,η₁)).2 ∧
-      (runMasks (replayMasks G x η₂ y₂ c₂ ω₂) (ω₂,η₂)).2 =
-        (runMasks (replayMasks G x η₂ y₂ d₂ σ₂) (σ₂,η₂)).2 := by
-  let ω₀ := exchangeOn (maskUnion (replayMasks G x η₂ y₂ c₂ ω₂)) ω₁ ω₂
-  let σ₀ := exchangeOn (maskUnion (replayMasks G x η₂ y₂ d₂ σ₂)) σ₁ σ₂
-  obtain ⟨ht,hm,hcontact⟩ := h₁.joint_virtual G x η₁ η₂ y₁ y₂ k j₁ j₂ ω₁ ω₂ ωf c₁ c₂ h₂ hc₁ hc₂ hsep
-  obtain ⟨gt,gm,gcontact⟩ := g₁.joint_virtual G x η₁ η₂ y₁ y₂ k j₁ j₂ σ₁ σ₂ σf d₁ d₂ g₂ hd₁ hd₂ gsep
-  have hrc : FollowsContactRule G x η₁ y₁ choose₁ c₁ ω₀ := by
-    unfold FollowsContactRule at hr₁ ⊢
-    rw [hcontact]
-    exact hr₁
-  have grc : FollowsContactRule G x η₁ y₁ choose₁ d₁ σ₀ := by
-    unfold FollowsContactRule at gr₁ ⊢
-    rw [gcontact]
-    exact gr₁
-  change SpliceTrace G x η₁ y₁ k j₁ σ₀ d₁ σ₂ at gt
-  change replayMasks G x η₁ y₁ c₁ ω₀ = _ at hm
-  change replayMasks G x η₁ y₁ d₁ σ₀ = _ at gm
-  have h00 : ω₀ = σ₀ := hrestore
-  rw [← h00] at gt grc
-  obtain ⟨hcode₁,hω₂⟩ := ht.seeded_unique G x η₁ y₁ k j₁ ω₀ ω₂ σ₂ c₁ d₁ gt choose₁ hrc grc
-  rw [← hω₂] at g₂ gr₂
-  obtain ⟨hcode₂,hout⟩ := h₂.seeded_unique G x η₂ y₂ k j₂ ω₂ ωf σf c₂ d₂ g₂ choose₂ hr₂ gr₂
-  refine ⟨hcode₁,hcode₂,hout,?_,?_⟩
-  · have hs := congrArg Prod.snd (h₁.joint_replay_state G x η₁ η₂ y₁ y₂ k j₁ j₂ ω₁ ω₂ ωf c₁ c₂ h₂ hsep)
-    have gs := congrArg Prod.snd (g₁.joint_replay_state G x η₁ η₂ y₁ y₂ k j₁ j₂ σ₁ σ₂ σf d₁ d₂
-      (hω₂ ▸ g₂) gsep)
-    change (runMasks _ (ω₀,η₁)).2 = _ at hs
-    change (runMasks _ (σ₀,η₁)).2 = _ at gs
-    rw [← hs,← gs,← hm,← gm,h00,hcode₁]
-  · rw [hcode₂,hω₂]
-
-end JointReplay
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section JointWeight
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-abbrev JointSpliceState (E : Type v) := Set E × (ℕ → Set E) × (ℕ → Set E)
-
-noncomputable def jointTupleWeight (Q : Finset E) (J₁ J₂ : Finset ℕ) (p : ℝ)
-    (s : JointSpliceState E) : ℝ :=
-  bernoulliWeight Q (domainBits Q s.1) p *
-    (∏ i ∈ J₁, bernoulliWeight Q (domainBits Q (s.2.1 i)) p) *
-    ∏ i ∈ J₂, bernoulliWeight Q (domainBits Q (s.2.2 i)) p
-
-theorem jointTupleWeight_nonneg (Q : Finset E) (J₁ J₂ : Finset ℕ)
-    (p : unitInterval) (s : JointSpliceState E) : 0 ≤ jointTupleWeight Q J₁ J₂ p s := by
-  exact mul_nonneg (spliceTupleWeight_nonneg Q J₁ p (s.1,s.2.1))
-    (Finset.prod_nonneg fun _ _ => bernoulliWeight_nonneg _ _ p)
-
-theorem replayMasks_indices (G : BondGraph V E) (x : V) (η : ℕ → Set E)
-    (y : ℕ → V) (c : List (ℕ × E)) (ω : Set E) :
-    (replayMasks G x η y c ω).map Prod.fst = c.map Prod.fst := by
-  induction c generalizing ω with
-  | nil => rfl
-  | cons q c ih =>
-    obtain ⟨i,e⟩ := q
-    simp only [replayMasks,List.map_cons,ih]
-
-theorem replayMasks_indices_mem (G : BondGraph V E) (x : V) (η : ℕ → Set E)
-    (y : ℕ → V) (c : List (ℕ × E)) (ω : Set E) (J : Finset ℕ)
-    (hJ : ∀ q ∈ c, q.1 ∈ J) : ∀ q ∈ replayMasks G x η y c ω, q.1 ∈ J := by
-  intro q hq
-  have hq' : q.1 ∈ (replayMasks G x η y c ω).map Prod.fst := List.mem_map_of_mem hq
-  rw [replayMasks_indices G x η y c ω] at hq'
-  obtain ⟨r,hr,he⟩ := List.mem_map.mp hq'
-  exact he ▸ hJ r hr
-
- 
-
-theorem SpliceTrace.joint_weight (G : BondGraph V E) (x : V)
-    (η₁ η₂ : ℕ → Set E) (y₁ y₂ : ℕ → V) (k j₁ j₂ : ℕ)
-    (ω₁ ω₂ ωf : Set E) (c₁ c₂ : List (ℕ × E))
-    (h₁ : SpliceTrace G x η₁ y₁ k j₁ ω₁ c₁ ωf)
-    (h₂ : SpliceTrace G x η₂ y₂ k j₂ ω₂ c₂ ωf)
-    (hsep : GraphSeparated G (cluster G ωf x \ cluster G ω₁ x)
-      (cluster G ωf x \ cluster G ω₂ x))
-    (Q : Finset E) (J₁ J₂ : Finset ℕ) (p : ℝ)
-    (hJ₁ : ∀ q ∈ c₁, q.1 ∈ J₁) (hJ₂ : ∀ q ∈ c₂, q.1 ∈ J₂) :
-    let m₁ := replayMasks G x η₁ y₁ c₁ ω₁
-    let m₂ := replayMasks G x η₂ y₂ c₂ ω₂
-    let ω₀ := exchangeOn (maskUnion m₂) ω₁ ω₂
-    jointTupleWeight Q J₁ J₂ p (ωf,(runMasks m₁ (ω₁,η₁)).2,(runMasks m₂ (ω₂,η₂)).2) =
-      jointTupleWeight Q J₁ J₂ p (ω₀,η₁,η₂) := by
-  intro m₁ m₂ ω₀
-  have hw₁ := runMasks_weight m₁ (ω₀,η₁) Q J₁ p
-    (replayMasks_indices_mem G x η₁ y₁ c₁ ω₁ J₁ hJ₁)
-  rw [h₁.joint_replay_state G x η₁ η₂ y₁ y₂ k j₁ j₂ ω₁ ω₂ ωf c₁ c₂ h₂ hsep] at hw₁
-  have hw₂ := runMasks_weight m₂ (ω₂,η₂) Q J₂ p
-    (replayMasks_indices_mem G x η₂ y₂ c₂ ω₂ J₂ hJ₂)
-  unfold spliceTupleWeight at hw₁ hw₂
-  rw [h₂.runMasks_first G x η₂ y₂ k j₂ ω₂ ωf c₂ η₂ (by intros; rfl)] at hw₂
-  unfold jointTupleWeight
-  dsimp only at hw₁ hw₂ ⊢
-  calc
-    _ = (bernoulliWeight Q (domainBits Q ωf) p *
-        ∏ i ∈ J₂, bernoulliWeight Q (domainBits Q ((runMasks m₂ (ω₂,η₂)).2 i)) p) *
-        ∏ i ∈ J₁, bernoulliWeight Q (domainBits Q ((runMasks m₁ (ω₁,η₁)).2 i)) p := by ring
-    _ = (bernoulliWeight Q (domainBits Q ω₂) p *
-        ∏ i ∈ J₂, bernoulliWeight Q (domainBits Q (η₂ i)) p) *
-        ∏ i ∈ J₁, bernoulliWeight Q (domainBits Q ((runMasks m₁ (ω₁,η₁)).2 i)) p := by rw [hw₂]
-    _ = (bernoulliWeight Q (domainBits Q ω₂) p *
-        ∏ i ∈ J₁, bernoulliWeight Q (domainBits Q ((runMasks m₁ (ω₁,η₁)).2 i)) p) *
-        ∏ i ∈ J₂, bernoulliWeight Q (domainBits Q (η₂ i)) p := by ring
-    _ = _ := by rw [hw₁]
-
-end JointWeight
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Splicing.GapCodes
+
+namespace OAI
+
+/-! Replay for bond percolation. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section SparseGapCode
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+def ValidSparseGapCode (k M : ℕ) (c : List (ℕ × ℕ)) : Prop :=
+  c.Pairwise (fun q r => q.1 < r.1) ∧
+    ∀ q ∈ c, 1 ≤ q.1 ∧ q.1 ≤ k ∧ 1 ≤ q.2 ∧ q.2 ≤ M
+
+def sparseGapValue : List (ℕ × ℕ) → ℕ → ℕ
+  | [], _ => 0
+  | (j,l)::c, i => if i = j then l else sparseGapValue c i
+
+theorem sparseGapValue_zero (c : List (ℕ × ℕ)) (i : ℕ)
+    (h : ∀ q ∈ c, i ≠ q.1) : sparseGapValue c i = 0 := by
+  induction c with
+  | nil => rfl
+  | cons q c ih =>
+    obtain ⟨j,l⟩ := q
+    rw [sparseGapValue,ite_eq_right (h (j,l) (by simp))]
+    exact ih (fun q hq => h q (List.mem_cons_of_mem _ hq))
+
+theorem sparseGapValue_bound (k M : ℕ) (c : List (ℕ × ℕ))
+    (h : ValidSparseGapCode k M c) (i : ℕ) : sparseGapValue c i ≤ M := by
+  induction c with
+  | nil => exact Nat.zero_le _
+  | cons q c ih =>
+    obtain ⟨j,l⟩ := q
+    rw [sparseGapValue]
+    split_ifs
+    · exact (h.2 (j,l) (by simp)).2.2.2
+    · exact ih ⟨(List.pairwise_cons.mp h.1).2,fun q hq => h.2 q (List.mem_cons_of_mem _ hq)⟩
+
+ 
+theorem sparseGapValue_injective (c d : List (ℕ × ℕ))
+    (hc : c.Pairwise (fun q r => q.1 < r.1))
+    (hd : d.Pairwise (fun q r => q.1 < r.1))
+    (hpc : ∀ q ∈ c, 0 < q.2) (hpd : ∀ q ∈ d, 0 < q.2)
+    (heq : ∀ i, sparseGapValue c i = sparseGapValue d i) : c = d := by
+  induction c generalizing d with
+  | nil =>
+    cases d with
+    | nil => rfl
+    | cons q d =>
+      obtain ⟨j,l⟩ := q
+      have hh := heq j
+      simp [sparseGapValue] at hh
+      have := hpd (j,l) (by simp)
+      omega
+  | cons q c ih =>
+    obtain ⟨i,l⟩ := q
+    cases d with
+    | nil =>
+      have hh := heq i
+      simp [sparseGapValue] at hh
+      have := hpc (i,l) (by simp)
+      omega
+    | cons q d =>
+      obtain ⟨j,m⟩ := q
+      obtain ⟨hc0,hc1⟩ := List.pairwise_cons.mp hc
+      obtain ⟨hd0,hd1⟩ := List.pairwise_cons.mp hd
+      have hij : i = j := by
+        by_contra hne
+        rcases Nat.lt_or_gt_of_ne hne with hij | hji
+        · have hz : sparseGapValue ((j,m)::d) i = 0 := by
+            apply sparseGapValue_zero
+            intro q hq
+            rcases List.mem_cons.mp hq with rfl | hq
+            · exact hne
+            · have := hd0 q hq
+              change j < q.1 at this
+              omega
+          have hh := heq i
+          rw [hz] at hh
+          simp [sparseGapValue] at hh
+          have := hpc (i,l) (by simp)
+          omega
+        · have hz : sparseGapValue ((i,l)::c) j = 0 := by
+            apply sparseGapValue_zero
+            intro q hq
+            rcases List.mem_cons.mp hq with rfl | hq
+            · exact Ne.symm hne
+            · have := hc0 q hq
+              change i < q.1 at this
+              omega
+          have hh := heq j
+          rw [hz] at hh
+          simp [sparseGapValue] at hh
+          have := hpd (j,m) (by simp)
+          omega
+      subst j
+      have hlm : l = m := by simpa [sparseGapValue] using heq i
+      subst m
+      congr 1
+      apply ih d hc1 hd1
+        (fun q hq => hpc q (List.mem_cons_of_mem _ hq))
+        (fun q hq => hpd q (List.mem_cons_of_mem _ hq))
+      intro r
+      by_cases hr : r = i
+      · subst r
+        rw [sparseGapValue_zero c i (fun q hq => (hc0 q hq).ne),
+          sparseGapValue_zero d i (fun q hq => (hd0 q hq).ne)]
+      · simpa only [sparseGapValue,ite_eq_right hr] using heq r
+
+ 
+def sparseGapArray (k M : ℕ) (c : List (ℕ × ℕ)) (h : ValidSparseGapCode k M c) :
+    Fin k → Fin (M+1) :=
+  fun i => ⟨sparseGapValue c (i.val+1),Nat.lt_succ_of_le (sparseGapValue_bound k M c h _)⟩
+
+theorem sparseGapArray_injective (k M : ℕ) (c d : List (ℕ × ℕ))
+    (hc : ValidSparseGapCode k M c) (hd : ValidSparseGapCode k M d)
+    (heq : sparseGapArray k M c hc = sparseGapArray k M d hd) : c = d := by
+  apply sparseGapValue_injective c d hc.1 hd.1
+    (fun q hq => (hc.2 q hq).2.2.1) (fun q hq => (hd.2 q hq).2.2.1)
+  intro i
+  by_cases hi : 1 ≤ i ∧ i ≤ k
+  · have he := congrArg (fun f => (f ⟨i-1,by omega⟩).val) heq
+    simpa only [sparseGapArray,show i-1+1 = i by omega] using he
+  · rw [sparseGapValue_zero c i, sparseGapValue_zero d i]
+    · intro q hq he
+      have h := hd.2 q hq
+      omega
+    · intro q hq he
+      have h := hc.2 q hq
+      omega
+
+ 
+
+theorem SpliceTrace.valid_gap_code (G : BondGraph V E) (hloc : LocallyFinite G)
+    (x : V) (η : ℕ → Set E) (y : ℕ → V) (k j M : ℕ) (ω ωf : Set E)
+    (steps : List (ℕ × E)) (h : SpliceTrace G x η y k j ω steps ωf)
+    (hCf : (cluster G ωf x).Finite) (L : ℝ)
+    (hgood : ∀ i, 1 ≤ i → i ≤ k → (cluster G (η i) (y i)).Finite ∧
+      (cluster G (η i) (y i)).ncard ≤ M ∧ GoodForScale G (η i) (y i) L) :
+    ValidSparseGapCode k M (gapSpliceCode G ωf x (y k) steps) ∧
+      ∀ q ∈ gapSpliceCode G ωf x (y k) steps, j < q.1 := by
+  induction steps generalizing j ω with
+  | nil => simp [ValidSparseGapCode,gapSpliceCode]
+  | cons q rest ih =>
+    obtain ⟨i,e⟩ := q
+    have hs := h.seam_data G x η y k j ω ωf ((i,e)::rest) (i,e) (by simp)
+    have hi : 1 ≤ i := by omega
+    obtain ⟨hCi,hMi,hGi⟩ := hgood i hi hs.2.1
+    have hg := h.head_gap_bounds G hloc x η y k j i M ω ωf e rest hCf hCi hMi L hGi
+    change 1 ≤ _ ∧ 1 ≤ traceHeadGap G ωf x (y k) e rest ∧
+      traceHeadGap G ωf x (y k) e rest ≤ M ∧ _ at hg
+    cases h with
+    | step j i ω e a b rest ωf _ _ _ _ _ _ _ _ _ ht =>
+      obtain ⟨⟨hpair,hbounds⟩,hlater⟩ := ih i _ ht
+      simp only [gapSpliceCode]
+      refine ⟨⟨List.pairwise_cons.mpr ⟨?_,hpair⟩,?_⟩,?_⟩
+      · exact hlater
+      · intro q hq
+        rcases List.mem_cons.mp hq with rfl | hq
+        · exact ⟨hi,hs.2.1,hg.2.1,hg.2.2.1⟩
+        · exact hbounds q hq
+      · intro q hq
+        rcases List.mem_cons.mp hq with rfl | hq
+        · exact hs.1
+        · exact hs.1.trans (hlater q hq)
+
+end SparseGapCode
+
+end Percolation
+
+ 
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section TraceContactMass
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+noncomputable def seamContactCount (G : BondGraph V E) (η : ℕ → Set E)
+    (y : ℕ → V) (ωf : Set E) (x : V) (i : ℕ) (e : E) : ℕ :=
+  (contactSet G (η i)
+    (cluster G (η i ∩ internalEdges G (cluster G (ωf \ {e}) x)ᶜ) (y i))).ncard
+
+ 
+noncomputable def traceContactCount (G : BondGraph V E) (η : ℕ → Set E)
+    (y : ℕ → V) (ωf : Set E) (x : V) : List (ℕ × E) → ℕ → ℕ
+  | [], _ => 1
+  | (j,e)::c, i => if i = j then seamContactCount G η y ωf x j e
+      else traceContactCount G η y ωf x c i
+
+ 
+theorem SpliceTrace.contact_weight_bound (G : BondGraph V E) (hloc : LocallyFinite G)
+    (x : V) (η : ℕ → Set E) (y : ℕ → V) (k j M : ℕ) (ω ωf : Set E)
+    (steps : List (ℕ × E)) (h : SpliceTrace G x η y k j ω steps ωf)
+    (hCf : (cluster G ωf x).Finite) (L : ℝ) (hL : 1 ≤ L)
+    (hgood : ∀ i, 1 ≤ i → i ≤ k → (cluster G (η i) (y i)).Finite ∧
+      (cluster G (η i) (y i)).ncard ≤ M ∧ GoodForScale G (η i) (y i) L) :
+    ∀ i, 1 ≤ traceContactCount G η y ωf x steps i ∧
+      (traceContactCount G η y ωf x steps i : ℝ)⁻¹ ≤
+        gapLetterWeight L (sparseGapValue (gapSpliceCode G ωf x (y k) steps) i) := by
+  induction steps generalizing j ω with
+  | nil => simp [traceContactCount,gapSpliceCode,sparseGapValue,gapLetterWeight]
+  | cons q rest ih =>
+    obtain ⟨i,e⟩ := q
+    have hs := h.seam_data G x η y k j ω ωf ((i,e)::rest) (i,e) (by simp)
+    have hi : 1 ≤ i := by omega
+    obtain ⟨hCi,hMi,hGi⟩ := hgood i hi hs.2.1
+    have hg := h.head_gap_bounds G hloc x η y k j i M ω ωf e rest hCf hCi hMi L hGi
+    have hcut := h.head_cut G x η y k j i ω ωf e rest
+    change 1 ≤ _ ∧ 1 ≤ traceHeadGap G ωf x (y k) e rest ∧
+      traceHeadGap G ωf x (y k) e rest ≤ M ∧ _ at hg
+    have hk : seamContactCount G η y ωf x i e =
+      (contactSet G (η i) (cluster G (η i ∩ internalEdges G (cluster G ω x)ᶜ) (y i))).ncard := by
+      unfold seamContactCount
+      rw [hcut]
+    rw [← hk] at hg
+    cases h with
+    | step j i ω e a b rest ωf _ _ _ _ _ _ _ _ _ ht =>
+      intro r
+      by_cases hr : r = i
+      · subst r
+        simp only [traceContactCount,gapSpliceCode,sparseGapValue,ite_true]
+        refine ⟨hg.1,?_⟩
+        rw [gapLetterWeight,ite_eq_right (by omega)]
+        exact inverse_contact_weight_le _ _ L hg.1 hg.2.1 hL hg.2.2.2
+      · simpa only [traceContactCount,gapSpliceCode,sparseGapValue,ite_eq_right hr] using ih i _ ht r
+
+ 
+
+def contactChoiceSet (K a : ℕ) : Set unitInterval :=
+  if K = 1 then Set.univ else contactInterval K a
+
+theorem volume_contactChoiceSet (K a : ℕ) (ha : a < K) :
+    volume (contactChoiceSet K a) = ENNReal.ofReal ((K:ℝ)⁻¹) := by
+  unfold contactChoiceSet
+  split_ifs with h
+  · simp [h]
+  · exact volume_contactInterval K a ha
+
+theorem real_volume_contact_choice_cube {I : Type*} [Fintype I] (K a : I → ℕ)
+    (ha : ∀ i, a i < K i) :
+    ((volume : Measure (I → unitInterval))
+      (Set.pi Set.univ (fun i => contactChoiceSet (K i) (a i)))).toReal =
+      ∏ i, (K i:ℝ)⁻¹ := by
+  rw [MeasureTheory.volume_pi,Measure.pi_pi,ENNReal.toReal_prod]
+  apply Finset.prod_congr rfl
+  intro i _
+  rw [volume_contactChoiceSet _ _ (ha i),ENNReal.toReal_ofReal (by positivity)]
+
+ 
+
+theorem SpliceTrace.contact_cube_le_code_weight (G : BondGraph V E) (hloc : LocallyFinite G)
+    (x : V) (η : ℕ → Set E) (y : ℕ → V) (k j M : ℕ) (ω ωf : Set E)
+    (steps : List (ℕ × E)) (h : SpliceTrace G x η y k j ω steps ωf)
+    (hCf : (cluster G ωf x).Finite) (L : ℝ) (hL : 1 ≤ L)
+    (hgood : ∀ i, 1 ≤ i → i ≤ k → (cluster G (η i) (y i)).Finite ∧
+      (cluster G (η i) (y i)).ncard ≤ M ∧ GoodForScale G (η i) (y i) L)
+    (a : Fin k → ℕ) (ha : ∀ i, a i < traceContactCount G η y ωf x steps (i.val+1)) :
+    ((volume : Measure (Fin k → unitInterval))
+      (Set.pi Set.univ (fun i => contactChoiceSet
+        (traceContactCount G η y ωf x steps (i.val+1)) (a i)))).toReal ≤
+    ∏ i : Fin k, gapLetterWeight L (sparseGapValue (gapSpliceCode G ωf x (y k) steps) (i.val+1)) := by
+  rw [real_volume_contact_choice_cube _ _ ha]
+  apply Finset.prod_le_prod₀
+  · intros
+    positivity
+  · intro i _
+    exact (h.contact_weight_bound G hloc x η y k j M ω ωf steps hCf L hL hgood (i.val+1)).2
+
+ 
+theorem SpliceTrace.code_unique_of_gapCode (G : BondGraph V E) (x : V)
+    (η₁ η₂ : ℕ → Set E) (y₁ y₂ : ℕ → V) (k j : ℕ) (ω₁ ω₂ ωf : Set E)
+    (c₁ c₂ : List (ℕ × E))
+    (ht₁ : SpliceTrace G x η₁ y₁ k j ω₁ c₁ ωf)
+    (ht₂ : SpliceTrace G x η₂ y₂ k j ω₂ c₂ ωf)
+    (hz : y₁ k = y₂ k) (hC : (cluster G ωf x).Finite)
+    (heq : gapSpliceCode G ωf x (y₁ k) c₁ = gapSpliceCode G ωf x (y₁ k) c₂) : c₁ = c₂ := by
+  apply ordinalSpliceCode_injective G ωf x (y₁ k) hC c₁ c₂
+    (ht₁.seams_pivotal G x η₁ y₁ k j ω₁ ωf c₁)
+    (by simpa only [hz] using ht₂.seams_pivotal G x η₂ y₂ k j ω₂ ωf c₂)
+  rw [← ht₁.decode_gap_code G x η₁ y₁ k j ω₁ ωf c₁ hC,heq,hz,
+    ht₂.decode_gap_code G x η₂ y₂ k j ω₂ ωf c₂ hC]
+
+end TraceContactMass
+
+end Percolation
+
+ 
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section FiniteInverseMass
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem finite_inverse_contact_mass_le {I : Type*} [Fintype I]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (x z : V) (k M : ℕ)
+    (L : ℝ) (hL : 1 ≤ L) (p : unitInterval) (Q : Finset E)
+    (inp out : I → SpliceState E) (steps : I → List (ℕ × E)) (y : I → ℕ → V)
+    (hir : Function.Injective (fun a => (inp a,steps a)))
+    (htrace : ∀ a, SpliceTrace G x (inp a).2 (y a) k 0 (inp a).1 (steps a) (out a).1)
+    (hex : ∀ a, SpliceExecution G x (inp a) (steps a) (out a))
+    (hz : ∀ a, y a k = z) (hfin : ∀ a, (cluster G (out a).1 x).Finite)
+    (hgood : ∀ a i, 1 ≤ i → i ≤ k → (cluster G ((inp a).2 i) (y a i)).Finite ∧
+      (cluster G ((inp a).2 i) (y a i)).ncard ≤ M ∧ GoodForScale G ((inp a).2 i) (y a i) L)
+    (rank : I → Fin k → ℕ)
+    (hrank : ∀ a i, rank a i < traceContactCount G (inp a).2 (y a) (out a).1 x (steps a) (i.val+1))
+    (B : Finset (SpliceState E)) (hB : ∀ a, out a ∈ B) :
+    (∑ a, spliceTupleWeight Q (Finset.Icc 1 k) p (inp a) *
+      ((volume : Measure (Fin k → unitInterval))
+        (Set.pi Set.univ (fun i => contactChoiceSet
+          (traceContactCount G (inp a).2 (y a) (out a).1 x (steps a) (i.val+1))
+          (rank a i)))).toReal) ≤
+      (∑ t ∈ B, spliceTupleWeight Q (Finset.Icc 1 k) p t) * inverseCodeMass M L ^ k := by
+  let c : I → List (ℕ × ℕ) := fun a => gapSpliceCode G (out a).1 x z (steps a)
+  have hvalid : ∀ a, ValidSparseGapCode k M (c a) := by
+    intro a
+    have hv := (htrace a).valid_gap_code G hloc x (inp a).2 (y a) k 0 M
+      (inp a).1 (out a).1 (steps a) (hfin a) L (hgood a)
+    simpa only [c,hz a] using hv.1
+  let arr (a : I) : Fin k → Fin (M+1) := sparseGapArray k M (c a) (hvalid a)
+  let f (a : I) : SpliceState E × (Fin k → Fin (M+1)) := (out a,arr a)
+  have hinj : Function.Injective f := by
+    intro a b he
+    have ho : out a = out b := congrArg Prod.fst he
+    have hc : c a = c b := sparseGapArray_injective k M (c a) (c b)
+      (hvalid a) (hvalid b) (congrArg Prod.snd he)
+    have hze : y a k = y b k := (hz a).trans (hz b).symm
+    have htB : SpliceTrace G x (inp b).2 (y b) k 0 (inp b).1 (steps b) (out a).1 := by
+      rw [ho]
+      exact htrace b
+    have heB : SpliceExecution G x (inp b) (steps b) (out a) := ho.symm ▸ hex b
+    have hg : gapSpliceCode G (out a).1 x (y a k) (steps a) =
+        gapSpliceCode G (out a).1 x (y a k) (steps b) := by
+      simpa only [c,hz a,ho] using hc
+    have hs := (htrace a).code_unique_of_gapCode G x (inp a).2 (inp b).2 (y a) (y b)
+      k 0 (inp a).1 (inp b).1 (out a).1 (steps a) (steps b) htB hze (hfin a) hg
+    have hi := (htrace a).input_unique_of_gapCode G x (inp a).2 (inp b).2 (y a) (y b)
+      k 0 (inp a).1 (inp b).1 (out a).1 (steps a) (steps b) htB hze (hfin a) hg
+      (out a).2 (hex a) heB
+    exact hir (Prod.ext hi hs)
+  let w (t : SpliceState E × (Fin k → Fin (M+1))) : ℝ :=
+    spliceTupleWeight Q (Finset.Icc 1 k) p t.1 * ∏ i, gapLetterWeight L (t.2 i).val
+  have hweight : ∀ a, spliceTupleWeight Q (Finset.Icc 1 k) p (inp a) =
+      spliceTupleWeight Q (Finset.Icc 1 k) p (out a) := by
+    intro a
+    apply ((hex a).weight G x (inp a) (out a) (steps a) Q (Finset.Icc 1 k) ?_ p).symm
+    intro q hq
+    have hq' := (htrace a).seam_data G x (inp a).2 (y a) k 0 (inp a).1 (out a).1 (steps a) q hq
+    exact Finset.mem_Icc.mpr ⟨hq'.1,hq'.2.1⟩
+  calc
+    _ ≤ ∑ a, w (f a) := by
+      apply Finset.sum_le_sum
+      intro a _
+      dsimp only [w,f,arr,sparseGapArray]
+      rw [← hweight a]
+      apply mul_le_mul_of_nonneg_left _ (spliceTupleWeight_nonneg _ _ p _)
+      have hh := (htrace a).contact_cube_le_code_weight G hloc x (inp a).2 (y a)
+        k 0 M (inp a).1 (out a).1 (steps a) (hfin a) L hL (hgood a) (rank a) (hrank a)
+      simpa only [c,hz a] using hh
+    _ = ∑ t ∈ Finset.univ.image f, w t := (Finset.sum_image hinj.injOn).symm
+    _ ≤ ∑ t ∈ B ×ˢ (Finset.univ : Finset (Fin k → Fin (M+1))), w t := by
+      apply Finset.sum_le_sum_of_subset_of_nonneg
+      · intro t ht
+        obtain ⟨a,_,rfl⟩ := Finset.mem_image.mp ht
+        exact Finset.mem_product.mpr ⟨hB a,Finset.mem_univ _⟩
+      · intro t _ _
+        exact mul_nonneg (spliceTupleWeight_nonneg _ _ p _)
+          (Finset.prod_nonneg (fun i _ => gapLetterWeight_nonneg L (by linarith) _))
+    _ = _ := by
+      rw [Finset.sum_product]
+      simp only [w,← Finset.mul_sum,total_gap_code_mass,← Finset.sum_mul]
+
+end FiniteInverseMass
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section RemoteSpliceGeometry
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+def GraphSeparated (G : BondGraph V E) (A B : Set V) : Prop :=
+  Disjoint A B ∧ ∀ a ∈ A, ∀ b ∈ B, ¬Adj G a b
+
+theorem GraphSeparated.symm (G : BondGraph V E) (A B : Set V)
+    (h : GraphSeparated G A B) : GraphSeparated G B A := by
+  refine ⟨h.1.symm,?_⟩
+  intro b hb a ha hab
+  obtain ⟨e,he⟩ := hab
+  exact h.2 a ha b hb ⟨e,he.trans (Sym2.eq_swap ..)⟩
+
+theorem GraphSeparated.mono (G : BondGraph V E) {A B C D : Set V}
+    (h : GraphSeparated G A B) (hC : C ⊆ A) (hD : D ⊆ B) : GraphSeparated G C D := by
+  exact ⟨h.1.mono hC hD,fun c hc d hd => h.2 c (hC hc) d (hD hd)⟩
+
+theorem GraphSeparated.no_shared_incidence (G : BondGraph V E) (A B : Set V)
+    (h : GraphSeparated G A B) (e : E) (a b : V) (ha : a ∈ A) (hb : b ∈ B)
+    (hea : a ∈ G e) (heb : b ∈ G e) : False := by
+  have hne : a ≠ b := by
+    rintro rfl
+    exact Set.disjoint_left.mp h.1 ha hb
+  exact h.2 a ha b hb ⟨e,(Sym2.mem_and_mem_iff hne).mp ⟨hea,heb⟩⟩
+
+ 
+
+theorem slit_cluster_remove_remote (G : BondGraph V E) (η : Set E) (S H : Set V)
+    (y : V) (hy : y ∉ S)
+    (hsep : GraphSeparated G (cluster G (η ∩ internalEdges G Sᶜ) y) H) :
+    cluster G (η ∩ internalEdges G (S \ H)ᶜ) y = cluster G (η ∩ internalEdges G Sᶜ) y := by
+  apply Set.Subset.antisymm
+  · intro v hv
+    induction hv with
+    | refl => exact conn_refl G _ y
+    | @tail a b hya hab ih =>
+      obtain ⟨e,he,hab⟩ := hab
+      have hbH : b ∉ H := fun hb => hsep.2 a ih b hb ⟨e,hab⟩
+      have hbS : b ∉ S := by
+        have hb := ((mem_internalEdges_of_ends G _ e a b hab).mp he.2).2
+        exact fun h => hb ⟨h,hbH⟩
+      have haS := slit_cluster_subset G S η y hy ih
+      exact ih.tail ⟨e,⟨he.1,(mem_internalEdges_of_ends G _ e a b hab).mpr ⟨haS,hbS⟩⟩,hab⟩
+  · exact fun _ h => conn_mono G
+      (Set.inter_subset_inter_right _ (internalEdges_mono G (by
+        intro v hv hvS
+        exact hv hvS.1))) h
+
+ 
+
+theorem spliceMask_remove_remote (G : BondGraph V E) (S B H : Set V) (e : E)
+    (hsep : GraphSeparated G B H) : spliceMask G (S \ H) B e = spliceMask G S B e := by
+  ext f
+  simp only [spliceMask,Set.mem_insert_iff,Set.mem_ofPred_eq]
+  apply or_congr Iff.rfl
+  constructor
+  · rintro ⟨⟨b,hb,hbf⟩,hi⟩
+    refine ⟨⟨b,hb,hbf⟩,?_⟩
+    intro v hv hvS
+    have hvH : v ∉ H := fun hh => hsep.no_shared_incidence G B H f b v hb hh hbf hv
+    exact hi v hv ⟨hvS,hvH⟩
+  · rintro ⟨hb,hi⟩
+    exact ⟨hb,fun v hv h => hi v hv h.1⟩
+
+ 
+theorem spliceMask_incident (G : BondGraph V E) (S B : Set V) (e : E)
+    (b : V) (hb : b ∈ B) (heb : b ∈ G e) :
+    ∀ f ∈ spliceMask G S B e, ∃ v ∈ B, v ∈ G f := by
+  intro f hf
+  rcases Set.mem_insert_iff.mp hf with rfl | hf
+  · exact ⟨b,hb,heb⟩
+  · exact hf.1
+
+ 
+
+theorem disjoint_spliceMasks (G : BondGraph V E) (S₁ S₂ B₁ B₂ : Set V) (e₁ e₂ : E)
+    (b₁ b₂ : V) (hb₁ : b₁ ∈ B₁) (hb₂ : b₂ ∈ B₂) (he₁ : b₁ ∈ G e₁) (he₂ : b₂ ∈ G e₂)
+    (hsep : GraphSeparated G B₁ B₂) :
+    Disjoint (spliceMask G S₁ B₁ e₁) (spliceMask G S₂ B₂ e₂) := by
+  apply Set.disjoint_left.mpr
+  intro f hf₁ hf₂
+  obtain ⟨a,ha,haf⟩ := spliceMask_incident G S₁ B₁ e₁ b₁ hb₁ he₁ f hf₁
+  obtain ⟨b,hb,hbf⟩ := spliceMask_incident G S₂ B₂ e₂ b₂ hb₂ he₂ f hf₂
+  exact hsep.no_shared_incidence G B₁ B₂ f a b ha hb haf hbf
+
+end RemoteSpliceGeometry
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section RemoteReplay
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+def replaySplice (G : BondGraph V E) (x : V) (η : ℕ → Set E) (y : ℕ → V) :
+    List (ℕ × E) → Set E → Set E
+  | [], ω => ω
+  | (i,e)::rest, ω =>
+    replaySplice G x η y rest (spliceField G (cluster G ω x)
+      (cluster G (η i ∩ internalEdges G (cluster G ω x)ᶜ) (y i)) e ω (η i))
+
+ 
+def replayMasks (G : BondGraph V E) (x : V) (η : ℕ → Set E) (y : ℕ → V) :
+    List (ℕ × E) → Set E → List (ℕ × Set E)
+  | [], _ => []
+  | (i,e)::rest, ω =>
+    let S := cluster G ω x
+    let B := cluster G (η i ∩ internalEdges G Sᶜ) (y i)
+    (i,spliceMask G S B e)::replayMasks G x η y rest (spliceField G S B e ω (η i))
+
+ 
+def replayContacts (G : BondGraph V E) (x : V) (η : ℕ → Set E) (y : ℕ → V) :
+    List (ℕ × E) → Set E → List (Set E)
+  | [], _ => []
+  | (i,e)::rest, ω =>
+    let S := cluster G ω x
+    let B := cluster G (η i ∩ internalEdges G Sᶜ) (y i)
+    contactSet G (η i) B::replayContacts G x η y rest (spliceField G S B e ω (η i))
+
+theorem SpliceTrace.replay_eq (G : BondGraph V E) (x : V) (η : ℕ → Set E) (y : ℕ → V)
+    (k j : ℕ) (ω ωf : Set E) (steps : List (ℕ × E))
+    (h : SpliceTrace G x η y k j ω steps ωf) : replaySplice G x η y steps ω = ωf := by
+  induction h with
+  | stop => rfl
+  | step => assumption
+
+ 
+
+theorem SpliceTrace.remote_replay (G : BondGraph V E) (x : V) (η : ℕ → Set E) (y : ℕ → V)
+    (k j : ℕ) (ω ωf : Set E) (steps : List (ℕ × E))
+    (h : SpliceTrace G x η y k j ω steps ωf) (H : Set V) (ω₀ : Set E)
+    (hroot : cluster G ω₀ x = cluster G ω x \ H)
+    (hsep : GraphSeparated G (cluster G ωf x \ cluster G ω x) H)
+    (hzH : y k ∉ H) :
+    SpliceTrace G x η y k j ω₀ steps (replaySplice G x η y steps ω₀) ∧
+      cluster G (replaySplice G x η y steps ω₀) x = cluster G ωf x \ H ∧
+      replayMasks G x η y steps ω₀ = replayMasks G x η y steps ω ∧
+      replayContacts G x η y steps ω₀ = replayContacts G x η y steps ω := by
+  induction h generalizing ω₀ with
+  | stop j ω hs =>
+    refine ⟨SpliceTrace.stop j ω₀ ?_,hroot,rfl,rfl⟩
+    rw [hroot]
+    exact ⟨hs,hzH⟩
+  | @step j i ω e a b rest ωf hn hji hik hmax hy he hab ha hb ht ih =>
+    let S := cluster G ω x
+    let B := cluster G (η i ∩ internalEdges G Sᶜ) (y i)
+    let ω' := spliceField G S B e ω (η i)
+    let S₀ := cluster G ω₀ x
+    let B₀ := cluster G (η i ∩ internalEdges G S₀ᶜ) (y i)
+    let ω₀' := spliceField G S₀ B₀ e ω₀ (η i)
+    have hC : cluster G ω' x = S ∪ B := splice_cluster G ω (η i) x (y i) hy e a b hab he ha hb
+    have hBo : B ⊆ Sᶜ := slit_cluster_subset G S (η i) (y i) hy
+    have hretain : cluster G ω' x ⊆ cluster G ωf x := ht.cluster_subset G x η y k i ω' ωf rest
+    have hBT : B ⊆ cluster G ωf x \ S := by
+      intro v hv
+      refine ⟨hretain ?_,hBo hv⟩
+      rw [hC]
+      exact Or.inr hv
+    have hsepB : GraphSeparated G B H := hsep.mono G hBT Set.Subset.rfl
+    have hBB : B₀ = B := by
+      dsimp only [B₀,S₀]
+      rw [hroot]
+      exact slit_cluster_remove_remote G (η i) S H (y i) hy hsepB
+    have haH : a ∉ H := fun hh => hsepB.2 b hb a hh ⟨e,hab.trans (Sym2.eq_swap ..)⟩
+    have ha₀ : a ∈ S₀ := by rw [show S₀ = S \ H from hroot]; exact ⟨ha,haH⟩
+    have hy₀ : y i ∉ S₀ := by rw [show S₀ = S \ H from hroot]; exact fun h => hy h.1
+    have hb₀ : b ∈ B₀ := hBB.symm ▸ hb
+    have hC₀ : cluster G ω₀' x = S₀ ∪ B₀ :=
+      splice_cluster G ω₀ (η i) x (y i) hy₀ e a b hab he ha₀ hb₀
+    have hroot' : cluster G ω₀' x = cluster G ω' x \ H := by
+      rw [hC₀,hC,hBB,show S₀ = S \ H from hroot]
+      ext v
+      constructor
+      · rintro (⟨hv,hvH⟩ | hv)
+        · exact ⟨Or.inl hv,hvH⟩
+        · exact ⟨Or.inr hv,fun hvH => Set.disjoint_left.mp hsepB.1 hv hvH⟩
+      · rintro ⟨hv,hvH⟩
+        exact hv.elim (fun h => Or.inl ⟨h,hvH⟩) Or.inr
+    have hsep' : GraphSeparated G (cluster G ωf x \ cluster G ω' x) H := by
+      apply hsep.mono G _ Set.Subset.rfl
+      intro v hv
+      refine ⟨hv.1,fun h => hv.2 ?_⟩
+      rw [hC]
+      exact Or.inl h
+    obtain ⟨ht₀,hfinal,hmasks,hcontacts⟩ := ih ω₀' hroot' hsep'
+    have hn₀ : y k ∉ S₀ := by rw [show S₀ = S \ H from hroot]; exact fun h => hn h.1
+    have hmax₀ : ∀ l, i < l → l ≤ k → Disjoint (cluster G (η l) (y l)) S₀ := by
+      intro l hil hlk
+      apply (hmax l hil hlk).mono Set.Subset.rfl
+      rw [show S₀ = S \ H from hroot]
+      exact Set.sdiff_subset
+    have hmask : spliceMask G S₀ B₀ e = spliceMask G S B e := by
+      rw [hBB,show S₀ = S \ H from hroot,spliceMask_remove_remote G S B H e hsepB]
+    refine ⟨SpliceTrace.step j i ω₀ e a b rest _ hn₀ hji hik hmax₀ hy₀ he hab ha₀ hb₀ ht₀,
+      hfinal,?_,?_⟩
+    · change (i,spliceMask G S₀ B₀ e)::replayMasks G x η y rest ω₀' =
+      (i,spliceMask G S B e)::replayMasks G x η y rest ω'
+      rw [hmask,hmasks]
+    · change contactSet G (η i) B₀::replayContacts G x η y rest ω₀' =
+        contactSet G (η i) B::replayContacts G x η y rest ω'
+      rw [hBB,hcontacts]
+
+end RemoteReplay
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section DeterministicReplay
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+def FollowsContactRule (G : BondGraph V E) (x : V) (η : ℕ → Set E)
+    (y : ℕ → V) (choose : ℕ → Set E → E) (steps : List (ℕ × E)) (ω : Set E) : Prop :=
+  List.Forall₂ (fun q K => choose q.1 K = q.2) steps (replayContacts G x η y steps ω)
+
+theorem followsContactRule_cons (G : BondGraph V E) (x : V) (η : ℕ → Set E)
+    (y : ℕ → V) (choose : ℕ → Set E → E) (i : ℕ) (e : E)
+    (rest : List (ℕ × E)) (ω : Set E) :
+    FollowsContactRule G x η y choose ((i,e)::rest) ω ↔
+      choose i (contactSet G (η i)
+        (cluster G (η i ∩ internalEdges G (cluster G ω x)ᶜ) (y i))) = e ∧
+      FollowsContactRule G x η y choose rest
+        (spliceField G (cluster G ω x)
+          (cluster G (η i ∩ internalEdges G (cluster G ω x)ᶜ) (y i)) e ω (η i)) := by
+  simp only [FollowsContactRule,replayContacts,List.forall₂_cons]
+
+ 
+
+theorem SpliceTrace.remote_follows_contact_rule (G : BondGraph V E) (x : V)
+    (η : ℕ → Set E) (y : ℕ → V) (k j : ℕ) (ω ωf : Set E) (steps : List (ℕ × E))
+    (h : SpliceTrace G x η y k j ω steps ωf) (H : Set V) (ω₀ : Set E)
+    (hroot : cluster G ω₀ x = cluster G ω x \ H)
+    (hsep : GraphSeparated G (cluster G ωf x \ cluster G ω x) H) (hzH : y k ∉ H)
+    (choose : ℕ → Set E → E) (hc : FollowsContactRule G x η y choose steps ω) :
+    FollowsContactRule G x η y choose steps ω₀ := by
+  unfold FollowsContactRule at hc ⊢
+  rw [(h.remote_replay G x η y k j ω ωf steps H ω₀ hroot hsep hzH).2.2.2]
+  exact hc
+
+ 
+
+theorem SpliceTrace.seeded_unique (G : BondGraph V E) (x : V) (η : ℕ → Set E)
+    (y : ℕ → V) (k j : ℕ) (ω ω₁ ω₂ : Set E) (c₁ c₂ : List (ℕ × E))
+    (h₁ : SpliceTrace G x η y k j ω c₁ ω₁)
+    (h₂ : SpliceTrace G x η y k j ω c₂ ω₂) (choose : ℕ → Set E → E)
+    (hc₁ : FollowsContactRule G x η y choose c₁ ω)
+    (hc₂ : FollowsContactRule G x η y choose c₂ ω) : c₁ = c₂ ∧ ω₁ = ω₂ := by
+  induction h₁ generalizing c₂ ω₂ with
+  | stop j ω hs =>
+    cases h₂ with
+    | stop => exact ⟨rfl,rfl⟩
+    | step _ _ _ _ _ _ _ _ hn => exact (hn hs).elim
+  | @step j i ω e a b rest ω₁ hn hji hik hmax hy he hab ha hb ht ih =>
+    cases h₂ with
+    | stop _ _ hs => exact (hn hs).elim
+    | step _ l _ f c d tail ω₂ hn₂ hjl hlk hmax₂ hy₂ hf hcd hc hd ht₂ =>
+      have haC : a ∈ cluster G (η i) (y i) :=
+        (edge_cluster_sides G (η i) (y i) he hab).mpr (conn_mono G Set.inter_subset_left hb)
+      have hcC : c ∈ cluster G (η l) (y l) :=
+        (edge_cluster_sides G (η l) (y l) hf hcd).mpr (conn_mono G Set.inter_subset_left hd)
+      have hil : i = l := by
+        rcases lt_trichotomy i l with hil | hil | hli
+        · exact (Set.disjoint_left.mp (hmax l hil hlk) hcC hc).elim
+        · exact hil
+        · exact (Set.disjoint_left.mp (hmax₂ i hli hik) haC ha).elim
+      subst l
+      obtain ⟨hec,hrest⟩ := (followsContactRule_cons G x η y choose i e rest ω).mp hc₁
+      obtain ⟨hfc,htail⟩ := (followsContactRule_cons G x η y choose i f tail ω).mp hc₂
+      have hef : e = f := hec.symm.trans hfc
+      rw [← hef] at ht₂ htail ⊢
+      obtain ⟨hcodes,houts⟩ := ih ω₂ tail ht₂ hrest htail
+      exact ⟨congrArg (List.cons (i,e)) hcodes,houts⟩
+
+end DeterministicReplay
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section TraceMaskSupport
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+def maskUnion (masks : List (ℕ × Set E)) : Set E :=
+  {e | ∃ q ∈ masks, e ∈ q.2}
+
+@[simp] theorem maskUnion_nil : maskUnion ([] : List (ℕ × Set E)) = ∅ := by
+  ext e
+  simp [maskUnion]
+
+@[simp] theorem maskUnion_cons (i : ℕ) (M : Set E) (ms : List (ℕ × Set E)) :
+    maskUnion ((i,M)::ms) = M ∪ maskUnion ms := by
+  ext e
+  simp [maskUnion,or_and_right,exists_or]
+
+ 
+theorem SpliceTrace.masks_incident (G : BondGraph V E) (x : V) (η : ℕ → Set E)
+    (y : ℕ → V) (k j : ℕ) (ω ωf : Set E) (steps : List (ℕ × E))
+    (h : SpliceTrace G x η y k j ω steps ωf) :
+    ∀ f ∈ maskUnion (replayMasks G x η y steps ω),
+      ∃ b ∈ cluster G ωf x \ cluster G ω x, b ∈ G f := by
+  induction h with
+  | stop => simp [replayMasks]
+  | @step j i ω e a b rest ωf hn hji hik hmax hy he hab ha hb ht ih =>
+    let S := cluster G ω x
+    let B := cluster G (η i ∩ internalEdges G Sᶜ) (y i)
+    let ω' := spliceField G S B e ω (η i)
+    have hC : cluster G ω' x = S ∪ B := splice_cluster G ω (η i) x (y i) hy e a b hab he ha hb
+    have hB : B ⊆ Sᶜ := slit_cluster_subset G S (η i) (y i) hy
+    have hretain := ht.cluster_subset G x η y k i ω' ωf rest
+    intro f hf
+    change f ∈ maskUnion ((i,spliceMask G S B e)::replayMasks G x η y rest ω') at hf
+    rw [maskUnion_cons] at hf
+    rcases hf with hf | hf
+    · obtain ⟨v,hv,hvf⟩ := spliceMask_incident G S B e b hb (by simp [hab]) f hf
+      refine ⟨v,⟨hretain ?_,hB hv⟩,hvf⟩
+      rw [hC]
+      exact Or.inr hv
+    · obtain ⟨v,hv,hvf⟩ := ih f hf
+      refine ⟨v,⟨hv.1,fun h => hv.2 ?_⟩,hvf⟩
+      rw [hC]
+      exact Or.inl h
+
+ 
+theorem replaySplice_off_masks (G : BondGraph V E) (x : V) (η : ℕ → Set E)
+    (y : ℕ → V) (steps : List (ℕ × E)) (ω : Set E) (f : E)
+    (hf : f ∉ maskUnion (replayMasks G x η y steps ω)) :
+    f ∈ replaySplice G x η y steps ω ↔ f ∈ ω := by
+  induction steps generalizing ω with
+  | nil => rfl
+  | cons q rest ih =>
+    obtain ⟨i,e⟩ := q
+    simp only [replayMasks,maskUnion_cons,Set.mem_union,not_or] at hf
+    change f ∈ replaySplice G x η y rest _ ↔ f ∈ ω
+    rw [ih _ hf.2]
+    simp only [spliceField,mem_exchangeOn,ite_eq_right hf.1]
+
+ 
+theorem SpliceTrace.masks_disjoint (G : BondGraph V E) (x : V)
+    (η₁ η₂ : ℕ → Set E) (y₁ y₂ : ℕ → V) (k j₁ j₂ : ℕ)
+    (ω₁ ω₂ ωf : Set E) (c₁ c₂ : List (ℕ × E))
+    (h₁ : SpliceTrace G x η₁ y₁ k j₁ ω₁ c₁ ωf)
+    (h₂ : SpliceTrace G x η₂ y₂ k j₂ ω₂ c₂ ωf)
+    (hsep : GraphSeparated G (cluster G ωf x \ cluster G ω₁ x)
+      (cluster G ωf x \ cluster G ω₂ x)) :
+    Disjoint (maskUnion (replayMasks G x η₁ y₁ c₁ ω₁))
+      (maskUnion (replayMasks G x η₂ y₂ c₂ ω₂)) := by
+  apply Set.disjoint_left.mpr
+  intro f hf₁ hf₂
+  obtain ⟨a,ha,hea⟩ := h₁.masks_incident G x η₁ y₁ k j₁ ω₁ ωf c₁ f hf₁
+  obtain ⟨b,hb,heb⟩ := h₂.masks_incident G x η₂ y₂ k j₂ ω₂ ωf c₂ f hf₂
+  exact hsep.no_shared_incidence G _ _ f a b ha hb hea heb
+
+ 
+
+theorem conn_delete_rootBridge (G : BondGraph V E) (ω η : Set E) (x z : V)
+    (hη : η ⊆ ω) (e : E) (a b : V) (hab : G e = s(a,b))
+    (hb : b ∉ cluster G (ω \ {e}) x) (hz : z ∈ cluster G (ω \ {e}) x)
+    (hxz : Conn G η x z) : Conn G (η \ {e}) x z := by
+  have hc : Conn G (insert e (η \ {e})) x z := conn_mono G (by
+    intro f hf
+    by_cases hfe : f = e
+    · exact Or.inl hfe
+    · exact Or.inr ⟨hf,hfe⟩) hxz
+  have hs : η \ {e} ⊆ ω \ {e} := Set.sdiff_subset_sdiff_left hη
+  rw [conn_insert_iff G _ e a b x z hab] at hc
+  rcases hc with hc | ⟨_,hc⟩ | ⟨hc,_⟩
+  · exact hc
+  · exact (hb (hz.trans (conn_symm G _ (conn_mono G hs hc)))).elim
+  · exact (hb (conn_mono G hs hc)).elim
+
+ 
+theorem cluster_double_cut (G : BondGraph V E) (ω : Set E) (x : V) (e f : E)
+    (a b : V) (hab : G f = s(a,b)) (hb : b ∉ cluster G (ω \ {f}) x) :
+    cluster G ((ω \ {e}) \ {f}) x = cluster G (ω \ {e}) x ∩ cluster G (ω \ {f}) x := by
+  apply Set.Subset.antisymm
+  · intro v hv
+    exact ⟨conn_mono G Set.sdiff_subset hv,
+      conn_mono G (ω₁ := (ω \ {e}) \ {f}) (ω₂ := ω \ {f}) (fun _ h => ⟨h.1.1,h.2⟩) hv⟩
+  · intro v hv
+    exact conn_delete_rootBridge G ω (ω \ {e}) x v Set.sdiff_subset f a b hab hb hv.2 hv.1
+
+end TraceMaskSupport
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section JointRoot
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem mask_not_crossing_remote (G : BondGraph V E) (U : Set E) (A B : Set V)
+    (hU : ∀ e ∈ U, ∃ v ∈ B, v ∈ G e) (hBA : B ⊆ A)
+    (e : E) (a b : V) (hab : G e = s(a,b)) (haB : a ∉ B) (hbA : b ∉ A) : e ∉ U := by
+  intro he
+  obtain ⟨v,hv,hve⟩ := hU e he
+  have hvab : v = a ∨ v = b := by simpa only [hab,Sym2.mem_iff] using hve
+  rcases hvab with rfl | rfl
+  · exact haB hv
+  · exact hbA (hBA hv)
+
+ 
+
+theorem exchange_patch_symm (U₁ U₂ ω₁ ω₂ ωf : Set E)
+    (hdis : Disjoint U₁ U₂)
+    (h₁ : ∀ e ∉ U₁, e ∈ ω₁ ↔ e ∈ ωf)
+    (h₂ : ∀ e ∉ U₂, e ∈ ω₂ ↔ e ∈ ωf) :
+    exchangeOn U₂ ω₁ ω₂ = exchangeOn U₁ ω₂ ω₁ := by
+  ext e
+  by_cases he₁ : e ∈ U₁
+  · have he₂ : e ∉ U₂ := fun h => Set.disjoint_left.mp hdis he₁ h
+    simp only [mem_exchangeOn,ite_eq_right he₂,ite_eq_left he₁]
+  · by_cases he₂ : e ∈ U₂
+    · simp only [mem_exchangeOn,ite_eq_left he₂,ite_eq_right he₁]
+    · simp only [mem_exchangeOn,ite_eq_right he₁,ite_eq_right he₂]
+      exact (h₁ e he₁).trans (h₂ e he₂).symm
+
+ 
+
+theorem SpliceTrace.joint_root (G : BondGraph V E) (x : V)
+    (η₁ η₂ : ℕ → Set E) (y₁ y₂ : ℕ → V) (k j₁ j₂ i₁ i₂ : ℕ)
+    (ω₁ ω₂ ωf : Set E) (e₁ e₂ : E) (r₁ r₂ : List (ℕ × E))
+    (h₁ : SpliceTrace G x η₁ y₁ k j₁ ω₁ ((i₁,e₁)::r₁) ωf)
+    (h₂ : SpliceTrace G x η₂ y₂ k j₂ ω₂ ((i₂,e₂)::r₂) ωf)
+    (hsep : GraphSeparated G (cluster G ωf x \ cluster G ω₁ x)
+      (cluster G ωf x \ cluster G ω₂ x)) :
+    cluster G (exchangeOn (maskUnion (replayMasks G x η₂ y₂ ((i₂,e₂)::r₂) ω₂)) ω₁ ω₂) x =
+      cluster G ω₁ x ∩ cluster G ω₂ x := by
+  let A₁ := cluster G ω₁ x
+  let A₂ := cluster G ω₂ x
+  let B₁ := cluster G ωf x \ A₁
+  let B₂ := cluster G ωf x \ A₂
+  let U₁ := maskUnion (replayMasks G x η₁ y₁ ((i₁,e₁)::r₁) ω₁)
+  let U₂ := maskUnion (replayMasks G x η₂ y₂ ((i₂,e₂)::r₂) ω₂)
+  let ω₀ := exchangeOn U₂ ω₁ ω₂
+  have hsub₁ := h₁.cluster_subset G x η₁ y₁ k j₁ ω₁ ωf ((i₁,e₁)::r₁)
+  have hsub₂ := h₂.cluster_subset G x η₂ y₂ k j₂ ω₂ ωf ((i₂,e₂)::r₂)
+  have hBA₁ : B₁ ⊆ A₂ := by
+    intro v hv
+    by_contra hn
+    exact Set.disjoint_left.mp hsep.1 hv ⟨hv.1,hn⟩
+  have hBA₂ : B₂ ⊆ A₁ := by
+    intro v hv
+    by_contra hn
+    exact Set.disjoint_left.mp hsep.1 ⟨hv.1,hn⟩ hv
+  have hU₁ : ∀ e ∈ U₁, ∃ v ∈ B₁, v ∈ G e :=
+    h₁.masks_incident G x η₁ y₁ k j₁ ω₁ ωf ((i₁,e₁)::r₁)
+  have hU₂ : ∀ e ∈ U₂, ∃ v ∈ B₂, v ∈ G e :=
+    h₂.masks_incident G x η₂ y₂ k j₂ ω₂ ωf ((i₂,e₂)::r₂)
+  have hdis : Disjoint U₁ U₂ := h₁.masks_disjoint G x η₁ η₂ y₁ y₂ k j₁ j₂ ω₁ ω₂ ωf
+    ((i₁,e₁)::r₁) ((i₂,e₂)::r₂) h₂ hsep
+  have hoff₁ : ∀ e ∉ U₁, e ∈ ω₁ ↔ e ∈ ωf := by
+    intro e he
+    rw [← h₁.replay_eq G x η₁ y₁ k j₁ ω₁ ωf ((i₁,e₁)::r₁)]
+    exact (replaySplice_off_masks G x η₁ y₁ ((i₁,e₁)::r₁) ω₁ e he).symm
+  have hoff₂ : ∀ e ∉ U₂, e ∈ ω₂ ↔ e ∈ ωf := by
+    intro e he
+    rw [← h₂.replay_eq G x η₂ y₂ k j₂ ω₂ ωf ((i₂,e₂)::r₂)]
+    exact (replaySplice_off_masks G x η₂ y₂ ((i₂,e₂)::r₂) ω₂ e he).symm
+  have hsym : ω₀ = exchangeOn U₁ ω₂ ω₁ := exchange_patch_symm U₁ U₂ ω₁ ω₂ ωf hdis hoff₁ hoff₂
+  have hcut₁ := h₁.head_cut G x η₁ y₁ k j₁ i₁ ω₁ ωf e₁ r₁
+  have hcut₂ := h₂.head_cut G x η₂ y₂ k j₂ i₂ ω₂ ωf e₂ r₂
+  obtain ⟨a,b,hab,_,hb,_⟩ := openMarkedPivotal_oriented G ωf {y₂ k} x
+    (h₂.seams_pivotal G x η₂ y₂ k j₂ ω₂ ωf ((i₂,e₂)::r₂) (i₂,e₂) (by simp))
+  have hdouble : cluster G ((ωf \ {e₁}) \ {e₂}) x = A₁ ∩ A₂ := by
+    rw [cluster_double_cut G ωf x e₁ e₂ a b hab hb,hcut₁,hcut₂]
+  change cluster G ω₀ x = A₁ ∩ A₂
+  apply Set.Subset.antisymm
+  · intro v hv
+    induction hv with
+    | refl => exact ⟨conn_refl G _ x,conn_refl G _ x⟩
+    | @tail a b hxa hab ih =>
+      obtain ⟨e,he,hab⟩ := hab
+      constructor
+      · by_contra hb
+        have hno := mask_not_crossing_remote G U₂ A₁ B₂ hU₂ hBA₂ e a b hab
+          (fun hh => hh.2 ih.2) hb
+        have he₁ : e ∈ ω₁ := by simpa only [ω₀,mem_exchangeOn,ite_eq_right hno] using he
+        exact hb ((edge_cluster_sides G ω₁ x he₁ hab).mp ih.1)
+      · by_contra hb
+        have hno := mask_not_crossing_remote G U₁ A₂ B₁ hU₁ hBA₁ e a b hab
+          (fun hh => hh.2 ih.1) hb
+        have he₂ : e ∈ ω₂ := by simpa only [hsym,mem_exchangeOn,ite_eq_right hno] using he
+        exact hb ((edge_cluster_sides G ω₂ x he₂ hab).mp ih.2)
+  · intro v hv
+    have hc : Conn G ((ωf \ {e₁}) \ {e₂}) x v := by
+      change v ∈ cluster G ((ωf \ {e₁}) \ {e₂}) x
+      rw [hdouble]
+      exact hv
+    have hr := conn_restrict_cluster G ((ωf \ {e₁}) \ {e₂}) x (mem_cluster_self G _ x) hc
+    apply conn_mono G _ hr
+    intro e he
+    have hi : e ∈ internalEdges G (A₁ ∩ A₂) := by
+      simpa only [hdouble] using he.2
+    have hno : e ∉ U₂ := by
+      intro hu
+      obtain ⟨b,hb,heb⟩ := hU₂ e hu
+      exact hb.2 (hi b heb).2
+    have he₁ : e ∈ ω₁ := (h₁.internal_agreement G x η₁ y₁ k j₁ ω₁ ωf
+      ((i₁,e₁)::r₁) e (fun a ha => (hi a ha).1)).mp he.1.1.1
+    simpa only [ω₀,mem_exchangeOn,ite_eq_right hno] using he₁
+
+end JointRoot
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section MaskExecution
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+def runMasks : List (ℕ × Set E) → SpliceState E → SpliceState E
+  | [], s => s
+  | (i,M)::ms, s => runMasks ms (exchangeState M i s)
+
+ 
+def patchFirst (U τ : Set E) (s : SpliceState E) : SpliceState E :=
+  (exchangeOn U s.1 τ,s.2)
+
+theorem exchangeState_patchFirst (U τ M : Set E) (i : ℕ) (s : SpliceState E)
+    (hdis : Disjoint M U) :
+    exchangeState M i (patchFirst U τ s) = patchFirst U τ (exchangeState M i s) := by
+  apply Prod.ext
+  · ext e
+    by_cases heM : e ∈ M
+    · have heU : e ∉ U := fun he => Set.disjoint_left.mp hdis heM he
+      simp only [exchangeState,patchFirst,mem_exchangeOn,ite_eq_left heM,ite_eq_right heU]
+    · by_cases heU : e ∈ U <;>
+      simp only [exchangeState,patchFirst,mem_exchangeOn,ite_eq_right heM,heU,ite_true,ite_false]
+  · funext l
+    by_cases hli : l = i
+    · subst l
+      ext e
+      by_cases heM : e ∈ M
+      · have heU : e ∉ U := fun he => Set.disjoint_left.mp hdis heM he
+        simp only [exchangeState,patchFirst,Function.update_self,mem_exchangeOn,ite_eq_left heM,ite_eq_right heU]
+      · simp only [exchangeState,patchFirst,Function.update_self,mem_exchangeOn,ite_eq_right heM]
+    · simp only [exchangeState,patchFirst,Function.update_of_ne hli]
+
+ 
+
+theorem runMasks_patchFirst (ms : List (ℕ × Set E)) (U τ : Set E) (s : SpliceState E)
+    (hdis : Disjoint (maskUnion ms) U) :
+    runMasks ms (patchFirst U τ s) = patchFirst U τ (runMasks ms s) := by
+  induction ms generalizing s with
+  | nil => rfl
+  | cons q ms ih =>
+    obtain ⟨i,M⟩ := q
+    rw [maskUnion_cons,Set.disjoint_union_left] at hdis
+    simp only [runMasks]
+    rw [exchangeState_patchFirst U τ M i s hdis.1,ih _ hdis.2]
+
+ 
+
+theorem SpliceTrace.runMasks_first (G : BondGraph V E) (x : V) (η : ℕ → Set E)
+    (y : ℕ → V) (k j : ℕ) (ω ωf : Set E) (steps : List (ℕ × E))
+    (h : SpliceTrace G x η y k j ω steps ωf) :
+    ∀ θ : ℕ → Set E, (∀ l, j < l → l ≤ k → θ l = η l) →
+    (runMasks (replayMasks G x η y steps ω) (ω,θ)).1 = ωf := by
+  induction h with
+  | stop => exact fun _ _ => rfl
+  | @step j i ω e a b rest ωf hn hji hik hmax hy he hab ha hb ht ih =>
+    intro θ hθ
+    have hθi := hθ i hji hik
+    let S := cluster G ω x
+    let B := cluster G (η i ∩ internalEdges G Sᶜ) (y i)
+    let M := spliceMask G S B e
+    let ω' := spliceField G S B e ω (η i)
+    let θ' := Function.update θ i (exchangeOn M (θ i) ω)
+    have hstep : exchangeState M i (ω,θ) = (ω',θ') := by
+      apply Prod.ext
+      · change exchangeOn M ω (θ i) = exchangeOn M ω (η i)
+        rw [hθi]
+      · rfl
+    change (runMasks (replayMasks G x η y rest ω') (exchangeState M i (ω,θ))).1 = ωf
+    rw [hstep]
+    apply ih θ'
+    intro l hil hlk
+    rw [show θ' l = θ l from Function.update_of_ne (by omega) _ _]
+    exact hθ l (hji.trans hil) hlk
+
+ 
+theorem runMasks_weight (ms : List (ℕ × Set E)) (s : SpliceState E)
+    (Q : Finset E) (J : Finset ℕ) (p : ℝ) (hJ : ∀ q ∈ ms, q.1 ∈ J) :
+    spliceTupleWeight Q J p (runMasks ms s) = spliceTupleWeight Q J p s := by
+  induction ms generalizing s with
+  | nil => rfl
+  | cons q ms ih =>
+    obtain ⟨i,M⟩ := q
+    rw [runMasks,ih _ (fun q hq => hJ q (List.mem_cons_of_mem _ hq))]
+    exact spliceTupleWeight_exchange Q J p s M i (hJ (i,M) (List.mem_cons_self ..))
+
+ 
+
+theorem SpliceTrace.joint_replay_state (G : BondGraph V E) (x : V)
+    (η₁ η₂ : ℕ → Set E) (y₁ y₂ : ℕ → V) (k j₁ j₂ : ℕ)
+    (ω₁ ω₂ ωf : Set E) (c₁ c₂ : List (ℕ × E))
+    (h₁ : SpliceTrace G x η₁ y₁ k j₁ ω₁ c₁ ωf)
+    (h₂ : SpliceTrace G x η₂ y₂ k j₂ ω₂ c₂ ωf)
+    (hsep : GraphSeparated G (cluster G ωf x \ cluster G ω₁ x)
+      (cluster G ωf x \ cluster G ω₂ x)) :
+    let m₁ := replayMasks G x η₁ y₁ c₁ ω₁
+    let m₂ := replayMasks G x η₂ y₂ c₂ ω₂
+    let ω₀ := exchangeOn (maskUnion m₂) ω₁ ω₂
+    runMasks m₁ (ω₀,η₁) = (ω₂,(runMasks m₁ (ω₁,η₁)).2) := by
+  intro m₁ m₂ ω₀
+  have hdis := h₁.masks_disjoint G x η₁ η₂ y₁ y₂ k j₁ j₂ ω₁ ω₂ ωf c₁ c₂ h₂ hsep
+  have hroot := h₁.runMasks_first G x η₁ y₁ k j₁ ω₁ ωf c₁ η₁ (by intros; rfl)
+  have hpatch : exchangeOn (maskUnion m₂) ωf ω₂ = ω₂ := by
+    ext e
+    by_cases he : e ∈ maskUnion m₂
+    · simp only [mem_exchangeOn,ite_eq_left he]
+    · simp only [mem_exchangeOn,ite_eq_right he]
+      rw [← h₂.replay_eq G x η₂ y₂ k j₂ ω₂ ωf c₂]
+      exact replaySplice_off_masks G x η₂ y₂ c₂ ω₂ e he
+  change runMasks m₁ (patchFirst (maskUnion m₂) ω₂ (ω₁,η₁)) = _
+  rw [runMasks_patchFirst m₁ _ ω₂ _ hdis]
+  change (exchangeOn (maskUnion m₂) (runMasks m₁ (ω₁,η₁)).1 ω₂,_) = _
+  rw [hroot,hpatch]
+
+end MaskExecution
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section JointReplay
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem SpliceTrace.endpoint_not_mem_of_ne_nil (G : BondGraph V E) (x : V)
+    (η : ℕ → Set E) (y : ℕ → V) (k j : ℕ) (ω ωf : Set E) (c : List (ℕ × E))
+    (h : SpliceTrace G x η y k j ω c ωf) (hc : c ≠ []) : y k ∉ cluster G ω x := by
+  cases h with
+  | stop => exact (hc rfl).elim
+  | step _ _ _ _ _ _ _ _ hn => exact hn
+
+ 
+
+theorem SpliceTrace.joint_virtual (G : BondGraph V E) (x : V)
+    (η₁ η₂ : ℕ → Set E) (y₁ y₂ : ℕ → V) (k j₁ j₂ : ℕ)
+    (ω₁ ω₂ ωf : Set E) (c₁ c₂ : List (ℕ × E))
+    (h₁ : SpliceTrace G x η₁ y₁ k j₁ ω₁ c₁ ωf)
+    (h₂ : SpliceTrace G x η₂ y₂ k j₂ ω₂ c₂ ωf)
+    (hc₁ : c₁ ≠ []) (hc₂ : c₂ ≠ [])
+    (hsep : GraphSeparated G (cluster G ωf x \ cluster G ω₁ x)
+      (cluster G ωf x \ cluster G ω₂ x)) :
+    let ω₀ := exchangeOn (maskUnion (replayMasks G x η₂ y₂ c₂ ω₂)) ω₁ ω₂
+    SpliceTrace G x η₁ y₁ k j₁ ω₀ c₁ ω₂ ∧
+      replayMasks G x η₁ y₁ c₁ ω₀ = replayMasks G x η₁ y₁ c₁ ω₁ ∧
+      replayContacts G x η₁ y₁ c₁ ω₀ = replayContacts G x η₁ y₁ c₁ ω₁ := by
+  intro ω₀
+  have hroot' : cluster G ω₀ x = cluster G ω₁ x ∩ cluster G ω₂ x := by
+    cases c₁ with
+    | nil => exact (hc₁ rfl).elim
+    | cons q r₁ =>
+      obtain ⟨i₁,e₁⟩ := q
+      cases c₂ with
+      | nil => exact (hc₂ rfl).elim
+      | cons q r₂ =>
+        obtain ⟨i₂,e₂⟩ := q
+        exact h₁.joint_root G x η₁ η₂ y₁ y₂ k j₁ j₂ i₁ i₂ ω₁ ω₂ ωf e₁ e₂ r₁ r₂ h₂ hsep
+  let H := cluster G ωf x \ cluster G ω₂ x
+  have hroot : cluster G ω₀ x = cluster G ω₁ x \ H := by
+    rw [hroot']
+    ext v
+    constructor
+    · rintro ⟨h1,h2⟩
+      exact ⟨h1,fun hh => hh.2 h2⟩
+    · rintro ⟨h1,h2⟩
+      refine ⟨h1,?_⟩
+      by_contra hn
+      exact h2 ⟨h₁.cluster_subset G x η₁ y₁ k j₁ ω₁ ωf c₁ h1,hn⟩
+  have hz : y₁ k ∉ H := fun hz => Set.disjoint_left.mp hsep.1
+    ⟨h₁.endpoint G x η₁ y₁ k j₁ ω₁ ωf c₁,
+      h₁.endpoint_not_mem_of_ne_nil G x η₁ y₁ k j₁ ω₁ ωf c₁ hc₁⟩ hz
+  obtain ⟨ht,hout,hm,hcontacts⟩ := h₁.remote_replay G x η₁ y₁ k j₁ ω₁ ωf c₁ H ω₀ hroot hsep hz
+  have hf := ht.runMasks_first G x η₁ y₁ k j₁ ω₀ (replaySplice G x η₁ y₁ c₁ ω₀) c₁ η₁
+    (by intros; rfl)
+  rw [hm] at hf
+  have hs := h₁.joint_replay_state G x η₁ η₂ y₁ y₂ k j₁ j₂ ω₁ ω₂ ωf c₁ c₂ h₂ hsep
+  have heq : replaySplice G x η₁ y₁ c₁ ω₀ = ω₂ := hf.symm.trans (congrArg Prod.fst hs)
+  rw [heq] at ht
+  exact ⟨ht,hm,hcontacts⟩
+
+ 
+
+theorem joint_restore_seeded_unique (G : BondGraph V E) (x : V)
+    (η₁ η₂ : ℕ → Set E) (y₁ y₂ : ℕ → V) (choose₁ choose₂ : ℕ → Set E → E)
+    (k j₁ j₂ : ℕ) (ω₁ ω₂ ωf σ₁ σ₂ σf : Set E)
+    (c₁ c₂ d₁ d₂ : List (ℕ × E))
+    (h₁ : SpliceTrace G x η₁ y₁ k j₁ ω₁ c₁ ωf)
+    (h₂ : SpliceTrace G x η₂ y₂ k j₂ ω₂ c₂ ωf)
+    (g₁ : SpliceTrace G x η₁ y₁ k j₁ σ₁ d₁ σf)
+    (g₂ : SpliceTrace G x η₂ y₂ k j₂ σ₂ d₂ σf)
+    (hc₁ : c₁ ≠ []) (hc₂ : c₂ ≠ []) (hd₁ : d₁ ≠ []) (hd₂ : d₂ ≠ [])
+    (hsep : GraphSeparated G (cluster G ωf x \ cluster G ω₁ x)
+      (cluster G ωf x \ cluster G ω₂ x))
+    (gsep : GraphSeparated G (cluster G σf x \ cluster G σ₁ x)
+      (cluster G σf x \ cluster G σ₂ x))
+    (hr₁ : FollowsContactRule G x η₁ y₁ choose₁ c₁ ω₁)
+    (hr₂ : FollowsContactRule G x η₂ y₂ choose₂ c₂ ω₂)
+    (gr₁ : FollowsContactRule G x η₁ y₁ choose₁ d₁ σ₁)
+    (gr₂ : FollowsContactRule G x η₂ y₂ choose₂ d₂ σ₂)
+    (hrestore : exchangeOn (maskUnion (replayMasks G x η₂ y₂ c₂ ω₂)) ω₁ ω₂ =
+      exchangeOn (maskUnion (replayMasks G x η₂ y₂ d₂ σ₂)) σ₁ σ₂) :
+    c₁ = d₁ ∧ c₂ = d₂ ∧ ωf = σf ∧
+      (runMasks (replayMasks G x η₁ y₁ c₁ ω₁) (ω₁,η₁)).2 =
+        (runMasks (replayMasks G x η₁ y₁ d₁ σ₁) (σ₁,η₁)).2 ∧
+      (runMasks (replayMasks G x η₂ y₂ c₂ ω₂) (ω₂,η₂)).2 =
+        (runMasks (replayMasks G x η₂ y₂ d₂ σ₂) (σ₂,η₂)).2 := by
+  let ω₀ := exchangeOn (maskUnion (replayMasks G x η₂ y₂ c₂ ω₂)) ω₁ ω₂
+  let σ₀ := exchangeOn (maskUnion (replayMasks G x η₂ y₂ d₂ σ₂)) σ₁ σ₂
+  obtain ⟨ht,hm,hcontact⟩ := h₁.joint_virtual G x η₁ η₂ y₁ y₂ k j₁ j₂ ω₁ ω₂ ωf c₁ c₂ h₂ hc₁ hc₂ hsep
+  obtain ⟨gt,gm,gcontact⟩ := g₁.joint_virtual G x η₁ η₂ y₁ y₂ k j₁ j₂ σ₁ σ₂ σf d₁ d₂ g₂ hd₁ hd₂ gsep
+  have hrc : FollowsContactRule G x η₁ y₁ choose₁ c₁ ω₀ := by
+    unfold FollowsContactRule at hr₁ ⊢
+    rw [hcontact]
+    exact hr₁
+  have grc : FollowsContactRule G x η₁ y₁ choose₁ d₁ σ₀ := by
+    unfold FollowsContactRule at gr₁ ⊢
+    rw [gcontact]
+    exact gr₁
+  change SpliceTrace G x η₁ y₁ k j₁ σ₀ d₁ σ₂ at gt
+  change replayMasks G x η₁ y₁ c₁ ω₀ = _ at hm
+  change replayMasks G x η₁ y₁ d₁ σ₀ = _ at gm
+  have h00 : ω₀ = σ₀ := hrestore
+  rw [← h00] at gt grc
+  obtain ⟨hcode₁,hω₂⟩ := ht.seeded_unique G x η₁ y₁ k j₁ ω₀ ω₂ σ₂ c₁ d₁ gt choose₁ hrc grc
+  rw [← hω₂] at g₂ gr₂
+  obtain ⟨hcode₂,hout⟩ := h₂.seeded_unique G x η₂ y₂ k j₂ ω₂ ωf σf c₂ d₂ g₂ choose₂ hr₂ gr₂
+  refine ⟨hcode₁,hcode₂,hout,?_,?_⟩
+  · have hs := congrArg Prod.snd (h₁.joint_replay_state G x η₁ η₂ y₁ y₂ k j₁ j₂ ω₁ ω₂ ωf c₁ c₂ h₂ hsep)
+    have gs := congrArg Prod.snd (g₁.joint_replay_state G x η₁ η₂ y₁ y₂ k j₁ j₂ σ₁ σ₂ σf d₁ d₂
+      (hω₂ ▸ g₂) gsep)
+    change (runMasks _ (ω₀,η₁)).2 = _ at hs
+    change (runMasks _ (σ₀,η₁)).2 = _ at gs
+    rw [← hs,← gs,← hm,← gm,h00,hcode₁]
+  · rw [hcode₂,hω₂]
+
+end JointReplay
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section JointWeight
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+abbrev JointSpliceState (E : Type v) := Set E × (ℕ → Set E) × (ℕ → Set E)
+
+noncomputable def jointTupleWeight (Q : Finset E) (J₁ J₂ : Finset ℕ) (p : ℝ)
+    (s : JointSpliceState E) : ℝ :=
+  bernoulliWeight Q (domainBits Q s.1) p *
+    (∏ i ∈ J₁, bernoulliWeight Q (domainBits Q (s.2.1 i)) p) *
+    ∏ i ∈ J₂, bernoulliWeight Q (domainBits Q (s.2.2 i)) p
+
+theorem jointTupleWeight_nonneg (Q : Finset E) (J₁ J₂ : Finset ℕ)
+    (p : unitInterval) (s : JointSpliceState E) : 0 ≤ jointTupleWeight Q J₁ J₂ p s := by
+  exact mul_nonneg (spliceTupleWeight_nonneg Q J₁ p (s.1,s.2.1))
+    (Finset.prod_nonneg fun _ _ => bernoulliWeight_nonneg _ _ p)
+
+theorem replayMasks_indices (G : BondGraph V E) (x : V) (η : ℕ → Set E)
+    (y : ℕ → V) (c : List (ℕ × E)) (ω : Set E) :
+    (replayMasks G x η y c ω).map Prod.fst = c.map Prod.fst := by
+  induction c generalizing ω with
+  | nil => rfl
+  | cons q c ih =>
+    obtain ⟨i,e⟩ := q
+    simp only [replayMasks,List.map_cons,ih]
+
+theorem replayMasks_indices_mem (G : BondGraph V E) (x : V) (η : ℕ → Set E)
+    (y : ℕ → V) (c : List (ℕ × E)) (ω : Set E) (J : Finset ℕ)
+    (hJ : ∀ q ∈ c, q.1 ∈ J) : ∀ q ∈ replayMasks G x η y c ω, q.1 ∈ J := by
+  intro q hq
+  have hq' : q.1 ∈ (replayMasks G x η y c ω).map Prod.fst := List.mem_map_of_mem hq
+  rw [replayMasks_indices G x η y c ω] at hq'
+  obtain ⟨r,hr,he⟩ := List.mem_map.mp hq'
+  exact he ▸ hJ r hr
+
+ 
+
+theorem SpliceTrace.joint_weight (G : BondGraph V E) (x : V)
+    (η₁ η₂ : ℕ → Set E) (y₁ y₂ : ℕ → V) (k j₁ j₂ : ℕ)
+    (ω₁ ω₂ ωf : Set E) (c₁ c₂ : List (ℕ × E))
+    (h₁ : SpliceTrace G x η₁ y₁ k j₁ ω₁ c₁ ωf)
+    (h₂ : SpliceTrace G x η₂ y₂ k j₂ ω₂ c₂ ωf)
+    (hsep : GraphSeparated G (cluster G ωf x \ cluster G ω₁ x)
+      (cluster G ωf x \ cluster G ω₂ x))
+    (Q : Finset E) (J₁ J₂ : Finset ℕ) (p : ℝ)
+    (hJ₁ : ∀ q ∈ c₁, q.1 ∈ J₁) (hJ₂ : ∀ q ∈ c₂, q.1 ∈ J₂) :
+    let m₁ := replayMasks G x η₁ y₁ c₁ ω₁
+    let m₂ := replayMasks G x η₂ y₂ c₂ ω₂
+    let ω₀ := exchangeOn (maskUnion m₂) ω₁ ω₂
+    jointTupleWeight Q J₁ J₂ p (ωf,(runMasks m₁ (ω₁,η₁)).2,(runMasks m₂ (ω₂,η₂)).2) =
+      jointTupleWeight Q J₁ J₂ p (ω₀,η₁,η₂) := by
+  intro m₁ m₂ ω₀
+  have hw₁ := runMasks_weight m₁ (ω₀,η₁) Q J₁ p
+    (replayMasks_indices_mem G x η₁ y₁ c₁ ω₁ J₁ hJ₁)
+  rw [h₁.joint_replay_state G x η₁ η₂ y₁ y₂ k j₁ j₂ ω₁ ω₂ ωf c₁ c₂ h₂ hsep] at hw₁
+  have hw₂ := runMasks_weight m₂ (ω₂,η₂) Q J₂ p
+    (replayMasks_indices_mem G x η₂ y₂ c₂ ω₂ J₂ hJ₂)
+  unfold spliceTupleWeight at hw₁ hw₂
+  rw [h₂.runMasks_first G x η₂ y₂ k j₂ ω₂ ωf c₂ η₂ (by intros; rfl)] at hw₂
+  unfold jointTupleWeight
+  dsimp only at hw₁ hw₂ ⊢
+  calc
+    _ = (bernoulliWeight Q (domainBits Q ωf) p *
+        ∏ i ∈ J₂, bernoulliWeight Q (domainBits Q ((runMasks m₂ (ω₂,η₂)).2 i)) p) *
+        ∏ i ∈ J₁, bernoulliWeight Q (domainBits Q ((runMasks m₁ (ω₁,η₁)).2 i)) p := by ring
+    _ = (bernoulliWeight Q (domainBits Q ω₂) p *
+        ∏ i ∈ J₂, bernoulliWeight Q (domainBits Q (η₂ i)) p) *
+        ∏ i ∈ J₁, bernoulliWeight Q (domainBits Q ((runMasks m₁ (ω₁,η₁)).2 i)) p := by rw [hw₂]
+    _ = (bernoulliWeight Q (domainBits Q ω₂) p *
+        ∏ i ∈ J₁, bernoulliWeight Q (domainBits Q ((runMasks m₁ (ω₁,η₁)).2 i)) p) *
+        ∏ i ∈ J₂, bernoulliWeight Q (domainBits Q (η₂ i)) p := by ring
+    _ = _ := by rw [hw₁]
+
+end JointWeight
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Subdivision/Coupling.lean b/lean/OAI/Probability/UnimodularPercolation/Subdivision/Coupling.lean
index f3a1d06702e7df1fd5fa6e852c128bcd3fdc6199..95c3d79c138c0a18a039987ee6320fd43a52c12e 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Subdivision/Coupling.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Subdivision/Coupling.lean
@@ -1,347 +1,347 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Subdivision.Graph
-
-namespace OAI
-
-/-! Coupling percolation on a graph with percolation on its subdivision. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-
- 
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal
-
-noncomputable def halfBit (G : BondGraph V E) : SubdivisionEdge G ↪ E × Bool := by
-  classical
-  refine ⟨fun ev => (ev.val.1,if ev.val.2 = (G ev.val.1).out.1 then false else true),?_⟩
-  rintro ⟨⟨e,x⟩,hx⟩ ⟨⟨f,y⟩,hy⟩ h
-  have hef : e = f := congrArg Prod.fst h
-  subst f
-  have hb := congrArg Prod.snd h
-  dsimp only at hb
-  have he : G e = s((G e).out.1,(G e).out.2) := (G e).out_eq.symm
-  have hx' : x = (G e).out.1 ∨ x = (G e).out.2 := by
-    apply Sym2.mem_iff.mp
-    rw [←he]
-    exact hx
-  have hy' : y = (G e).out.1 ∨ y = (G e).out.2 := by
-    apply Sym2.mem_iff.mp
-    rw [←he]
-    exact hy
-  have hxy : x = y := by
-    by_cases h₁ : x = (G e).out.1 <;> by_cases h₂ : y = (G e).out.1
-    · exact h₁.trans h₂.symm
-    · simp only [ite_eq_left h₁,ite_eq_right h₂,Bool.false_eq_true] at hb
-    · simp only [ite_eq_right h₁,ite_eq_left h₂,Bool.true_eq_false] at hb
-    · exact (hx'.resolve_left h₁).trans (hy'.resolve_left h₂).symm
-  subst y
-  rfl
-
-def seriesOpen (ν : Set (E × Bool)) : Set E := {e | ∀ b, (e,b) ∈ ν}
-
-noncomputable def subdivisionOpen (G : BondGraph V E) (ν : Set (E × Bool)) :
-    Set (SubdivisionEdge G) := halfBit G ⁻¹' ν
-
-@[simp] theorem halfBit_fst (G : BondGraph V E) (e : SubdivisionEdge G) :
-    (halfBit G e).1 = e.val.1 := rfl
-
- 
-theorem seriesOpen_of_two_incident (G : BondGraph V E) (ν : Set (E × Bool))
-    (e : E) (x y : V) (hx : x ∈ G e) (hy : y ∈ G e) (hne : x ≠ y)
-    (hxo : (⟨(e,x),hx⟩ : SubdivisionEdge G) ∈ subdivisionOpen G ν)
-    (hyo : (⟨(e,y),hy⟩ : SubdivisionEdge G) ∈ subdivisionOpen G ν) :
-    e ∈ seriesOpen ν := by
-  have hbits : (halfBit G ⟨(e,x),hx⟩).2 ≠ (halfBit G ⟨(e,y),hy⟩).2 := by
-    intro h
-    have hh : halfBit G ⟨(e,x),hx⟩ = halfBit G ⟨(e,y),hy⟩ := Prod.ext rfl h
-    have he := (halfBit G).injective hh
-    exact hne (congrArg (fun z : SubdivisionEdge G => z.val.2) he)
-  intro b
-  have hb : b = (halfBit G ⟨(e,x),hx⟩).2 ∨ b = (halfBit G ⟨(e,y),hy⟩).2 := by
-    generalize (halfBit G ⟨(e,x),hx⟩).2 = c at *
-    generalize (halfBit G ⟨(e,y),hy⟩).2 = d at *
-    cases b <;> cases c <;> cases d <;> simp_all
-  rcases hb with rfl | rfl
-  · exact hxo
-  · exact hyo
-
- 
-theorem subdivision_open_adj_iff (G : BondGraph V E) (ν : Set (E × Bool)) (x : V) (e : E) :
-    (∃ d ∈ subdivisionOpen G ν, subdivision G d = s(Sum.inl x, Sum.inr e)) ↔
-      ∃ hx : x ∈ G e, (⟨(e,x),hx⟩ : SubdivisionEdge G) ∈ subdivisionOpen G ν := by
-  constructor
-  · rintro ⟨⟨⟨f,y⟩,hy⟩,ho,he⟩
-    simp only [subdivision,Sym2.eq_iff] at he
-    rcases he with ⟨hyx,hfe⟩ | ⟨hbad,_⟩
-    · have : y = x := Sum.inl.inj hyx
-      have : f = e := Sum.inr.inj hfe
-      subst y; subst f
-      exact ⟨hy,ho⟩
-    · cases hbad
-  · rintro ⟨hx,ho⟩
-    exact ⟨⟨(e,x),hx⟩,ho,rfl⟩
-
- 
-theorem conn_subdivision_iff (G : BondGraph V E) (ν : Set (E × Bool)) (x y : V) :
-    Conn (subdivision G) (subdivisionOpen G ν) (.inl x) (.inl y) ↔
-      Conn G (seriesOpen ν) x y := by
-  constructor
-  · intro hc
-    have reach (z : V ⊕ E) (h : Conn (subdivision G) (subdivisionOpen G ν) (.inl x) z) :
-        match z with
-        | .inl w => Conn G (seriesOpen ν) x w
-        | .inr e => ∃ w : V, ∃ hw : w ∈ G e,
-            (⟨(e,w),hw⟩ : SubdivisionEdge G) ∈ subdivisionOpen G ν ∧ Conn G (seriesOpen ν) x w := by
-      induction h with
-      | refl => exact conn_refl G _ x
-      | @tail u v huv hed ih =>
-        cases u with
-        | inl a =>
-          cases v with
-          | inl b =>
-            obtain ⟨d,_,he⟩ := hed
-            exact (subdivision_adj_inl_inl G a b ⟨d,he⟩).elim
-          | inr e =>
-            obtain ⟨ha,ho⟩ := (subdivision_open_adj_iff G ν a e).mp hed
-            exact ⟨a,ha,ho,ih⟩
-        | inr e =>
-          cases v with
-          | inl b =>
-            have hed' : ∃ d ∈ subdivisionOpen G ν, subdivision G d = s(Sum.inl b, Sum.inr e) := by
-              obtain ⟨d,hd,he⟩ := hed
-              exact ⟨d,hd,he.trans (Sym2.eq_swap)⟩
-            obtain ⟨hb,hbo⟩ := (subdivision_open_adj_iff G ν b e).mp hed'
-            obtain ⟨a,ha,hao,hxa⟩ := ih
-            by_cases hab : a = b
-            · simpa only [hab] using hxa
-            · exact hxa.tail ⟨e,seriesOpen_of_two_incident G ν e a b ha hb hab hao hbo,
-                (Sym2.mem_and_mem_iff hab).mp ⟨ha,hb⟩⟩
-          | inr f =>
-            obtain ⟨d,_,he⟩ := hed
-            exact (subdivision_adj_inr_inr G e f ⟨d,he⟩).elim
-    exact reach (.inl y) hc
-  · intro hc
-    induction hc with
-    | refl => exact conn_refl _ _ _
-    | @tail a b hab hed ih =>
-      obtain ⟨e,he,heq⟩ := hed
-      have ha : a ∈ G e := by simp [heq]
-      have hb : b ∈ G e := by simp [heq]
-      have hao : (⟨(e,a),ha⟩ : SubdivisionEdge G) ∈ subdivisionOpen G ν := he _
-      have hbo : (⟨(e,b),hb⟩ : SubdivisionEdge G) ∈ subdivisionOpen G ν := he _
-      exact (ih.tail ⟨⟨(e,a),ha⟩,hao,rfl⟩).tail ⟨⟨(e,b),hb⟩,hbo,Sym2.eq_swap⟩
-
- 
-theorem map_bool_and (q : unitInterval) :
-    (Measure.infinitePi (fun _ : Bool => bitLaw q)).map
-      (fun f : Bool → Prop => ∀ b, f b) = bitLaw (q^2) := by
-  let μ := (Measure.infinitePi (fun _ : Bool => bitLaw q)).map
-      (fun f : Bool → Prop => ∀ b, f b)
-  have hm : Measurable (fun f : Bool → Prop => ∀ b, f b) := .of_discrete
-  have : IsProbabilityMeasure μ := inferInstance
-  have ht : μ.real {True} = (q:ℝ)^2 := by
-    change ((Measure.infinitePi _).map _).real _ = _
-    rw [Measure.real,Measure.map_apply hm (measurableSet_singleton _)]
-    have he : (fun f : Bool → Prop => ∀ b, f b) ⁻¹' ({True} : Set Prop) =
-        ((Finset.univ : Finset Bool) : Set Bool).pi (fun _ => ({True} : Set Prop)) := by
-      ext f
-      simp [and_comm]
-    rw [he,Measure.infinitePi_pi _ (fun _ _ => measurableSet_singleton _)]
-    simp [bitLaw,ENNReal.toReal_mul,pow_two]
-  have hf : μ.real {False} = 1-(q:ℝ)^2 := by
-    have he : ({False} : Set Prop) = ({True} : Set Prop)ᶜ := by ext b; simp
-    rw [he,measureReal_compl (measurableSet_singleton _),probReal_univ,ht]
-  apply Measure.ext_of_singleton
-  intro b
-  apply (ENNReal.toReal_eq_toReal_iff' (measure_ne_top _ _) (measure_ne_top _ _)).mp
-  change μ.real {b} = (bitLaw (q^2)).real {b}
-  by_cases hb : b
-  · have he : b = True := propext (iff_true_intro hb)
-    subst b
-    rw [ht]
-    simp [bitLaw,Measure.real]
-  · have he : b = False := propext (iff_false_intro hb)
-    subst b
-    rw [hf]
-    simp [bitLaw,Measure.real]
-
-theorem measurable_seriesOpen : Measurable (seriesOpen : Set (E × Bool) → Set E) := by
-  apply measurable_setOfPred.comp
-  apply Measurable.of_eval
-  intro e
-  apply measurableSet_setOfPred.mp
-  have he : {ν : Set (E × Bool) | ∀ b, (e,b) ∈ ν} = ⋂ b : Bool, {ν | (e,b) ∈ ν} := by
-    ext ν; simp
-  rw [he]
-  exact MeasurableSet.iInter fun b => measurableSet_setOfPred.mpr
-    ((measurable_pi_apply (e,b)).comp MeasurableEquiv.setOfPred.symm.measurable)
-
- 
-theorem law_map_seriesOpen (q : unitInterval) :
-    (law q : Measure (Set (E × Bool))).map seriesOpen = law (q^2) := by
-  let c := MeasurableEquiv.curry E Bool Prop
-  let b := fun f : E → Bool → Prop => fun e => ∀ j, f e j
-  have hb : Measurable b := Measurable.of_eval
-    (fun e => (Measurable.of_discrete : Measurable (fun f : Bool → Prop => ∀ j, f j)).comp
-      (measurable_pi_apply e))
-  unfold law
-  rw [setBernoulli_eq_map,Measure.map_map measurable_seriesOpen measurable_setOfPred]
-  simp only [Set.mem_univ]
-  have he : seriesOpen ∘ (fun f : E × Bool → Prop => {e | f e}) =
-      (fun f : E → Prop => {e | f e}) ∘ b ∘ c := rfl
-  rw [he,←Measure.map_map measurable_setOfPred (hb.comp c.measurable),
-    ←Measure.map_map hb c.measurable]
-  change ((Measure.infinitePi (fun _ : E × Bool => bitLaw q)).map c |>.map b |>.map
-    (fun f : E → Prop => {e | f e})) = _
-  rw [Measure.infinitePi_map_curry (fun (_ : E) (_ : Bool) => bitLaw q)]
-  dsimp only [b]
-  rw [Measure.infinitePi_map_pi _ (f := fun _ (g : Bool → Prop) => ∀ j, g j)
-    (fun _ => Measurable.of_discrete)]
-  rw [map_bool_and q,setBernoulli_eq_map]
-  rfl
-
-theorem measurable_subdivisionOpen (G : BondGraph V E) : Measurable (subdivisionOpen G) :=
-  measurable_setOfPred.comp (Measurable.of_eval (fun e =>
-    (measurable_pi_apply (halfBit G e)).comp MeasurableEquiv.setOfPred.symm.measurable))
-
-theorem law_map_subdivisionOpen (G : BondGraph V E) (q : unitInterval) :
-    (law q : Measure (Set (E × Bool))).map (subdivisionOpen G) = law q :=
-  law_map_preimage_embedding q (halfBit G)
-
- 
-
-theorem twoPoint_subdivision [Countable V] [Countable E]
-    (G : BondGraph V E) (q : unitInterval) (x y : V) :
-    twoPoint (subdivision G) q (.inl x) (.inl y) = twoPoint G (q^2) x y := by
-  unfold twoPoint
-  congr 1
-  rw [←law_map_subdivisionOpen G q,Measure.map_apply (measurable_subdivisionOpen G)
-    (measurableSet_conn (subdivision G) (.inl x) (.inl y))]
-  rw [←law_map_seriesOpen q,Measure.map_apply measurable_seriesOpen (measurableSet_conn G x y)]
-  congr 1
-  ext ν
-  exact conn_subdivision_iff G ν x y
-
- 
-theorem conn_subdivision_inr (G : BondGraph V E) (ν : Set (E × Bool)) (x : V) (e : E)
-    (hc : Conn (subdivision G) (subdivisionOpen G ν) (.inl x) (.inr e)) :
-    ∃ y, Conn G (seriesOpen ν) x y ∧ y ∈ G e := by
-  rcases hc.cases_tail with h | ⟨z,hxz,hze⟩
-  · cases h
-  · cases z with
-    | inl y =>
-      obtain ⟨hy,_⟩ := (subdivision_open_adj_iff G ν y e).mp hze
-      exact ⟨y,(conn_subdivision_iff G ν x y).mp hxz,hy⟩
-    | inr f =>
-      obtain ⟨d,_,hd⟩ := hze
-      exact (subdivision_adj_inr_inr G f e ⟨d,hd⟩).elim
-
- 
-
-theorem finite_cluster_subdivision_iff (G : BondGraph V E) (hloc : LocallyFinite G)
-    (ν : Set (E × Bool)) (x : V) :
-    (cluster (subdivision G) (subdivisionOpen G ν) (.inl x)).Finite ↔
-      (cluster G (seriesOpen ν) x).Finite := by
-  constructor
-  · intro hf
-    apply Set.Finite.of_finite_image (f := (Sum.inl : V → V ⊕ E))
-    · apply hf.subset
-      rintro z ⟨y,hy,rfl⟩
-      exact (conn_subdivision_iff G ν x y).mpr hy
-    · exact Sum.inl_injective.injOn
-  · intro hf
-    have he : (⋃ y ∈ cluster G (seriesOpen ν) x, {e | y ∈ G e}).Finite :=
-      hf.biUnion (fun y _ => hloc y)
-    apply ((hf.image (Sum.inl : V → V ⊕ E)).union
-      (he.image (Sum.inr : E → V ⊕ E))).subset
-    intro z hz
-    cases z with
-    | inl y => exact Or.inl ⟨y,(conn_subdivision_iff G ν x y).mp hz,rfl⟩
-    | inr e =>
-      obtain ⟨y,hxy,hy⟩ := conn_subdivision_inr G ν x e hz
-      exact Or.inr ⟨e,Set.mem_iUnion.mpr ⟨y,Set.mem_iUnion.mpr ⟨hxy,hy⟩⟩,rfl⟩
-
- 
-
-theorem exists_infinite_cluster_subdivision_iff (G : BondGraph V E)
-    (hloc : LocallyFinite G) (ν : Set (E × Bool)) :
-    (∃ z, (cluster (subdivision G) (subdivisionOpen G ν) z).Infinite) ↔
-      ∃ x, (cluster G (seriesOpen ν) x).Infinite := by
-  have hi (x : V) : (cluster (subdivision G) (subdivisionOpen G ν) (.inl x)).Infinite ↔
-      (cluster G (seriesOpen ν) x).Infinite := by
-    exact not_congr (finite_cluster_subdivision_iff G hloc ν x)
-  constructor
-  · rintro ⟨z,hz⟩
-    cases z with
-    | inl x => exact ⟨x,(hi x).mp hz⟩
-    | inr e =>
-      obtain ⟨w,hw,hne⟩ := hz.exists_notMem_finset {Sum.inr e}
-      have hwne : w ≠ Sum.inr e := by simpa using hne
-      rcases (show Conn (subdivision G) (subdivisionOpen G ν) (.inr e) w from hw).cases_head
-        with h | ⟨z,hez,hzw⟩
-      · exact (hwne h.symm).elim
-      · cases z with
-        | inl y =>
-          have hconn : Conn (subdivision G) (subdivisionOpen G ν) (.inr e) (.inl y) :=
-            .single hez
-          have heq := cluster_eq_of_conn (subdivision G) (subdivisionOpen G ν) hconn
-          exact ⟨y,(hi y).mp (heq ▸ hz)⟩
-        | inr f =>
-          obtain ⟨d,_,hd⟩ := hez
-          exact (subdivision_adj_inr_inr G e f ⟨d,hd⟩).elim
-  · rintro ⟨x,hx⟩
-    exact ⟨.inl x,(hi x).mpr hx⟩
-
-theorem infinite_cluster_probability_subdivision [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (q : unitInterval) :
-    law q {ω | (infiniteClusters (subdivision G) ω).Nonempty} =
-      law (q^2) {ω | (infiniteClusters G ω).Nonempty} := by
-  simp only [infiniteClusters_nonempty_iff]
-  rw [←law_map_subdivisionOpen G q,Measure.map_apply (measurable_subdivisionOpen G)
-    (measurableSet_exists_infinite_cluster _)]
-  rw [←law_map_seriesOpen q,Measure.map_apply measurable_seriesOpen
-    (measurableSet_exists_infinite_cluster _)]
-  congr 1
-  ext ν
-  exact exists_infinite_cluster_subdivision_iff G hloc ν
-
- 
-noncomputable def squareParameter : unitInterval ≃o unitInterval where
-  toFun q := q^2
-  invFun p := ⟨Real.sqrt p,Real.sqrt_nonneg _,Real.sqrt_le_one.mpr p.2.2⟩
-  left_inv q := by
-    apply Subtype.ext
-    change Real.sqrt ((q:ℝ)^2) = (q:ℝ)
-    exact Real.sqrt_sq q.2.1
-  right_inv p := by
-    apply Subtype.ext
-    change Real.sqrt (p:ℝ)^2 = (p:ℝ)
-    exact Real.sq_sqrt p.2.1
-  map_rel_iff' := by
-    intro p q
-    change (p:ℝ)^2 ≤ (q:ℝ)^2 ↔ (p:ℝ) ≤ (q:ℝ)
-    exact sq_le_sq₀ p.2.1 q.2.1
-
- 
-
-theorem pc_subdivision_sq [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) :
-    pc (subdivision G)^2 = pc G := by
-  change squareParameter (sInf {q | 0 < law q {ω | (infiniteClusters (subdivision G) ω).Nonempty}}) = _
-  rw [squareParameter.map_sInf]
-  change (⨅ q, ⨅ (_ : 0 < law q {ω | (infiniteClusters (subdivision G) ω).Nonempty}), q^2) = _
-  simp only [infinite_cluster_probability_subdivision G hloc]
-  change (⨅ q, ⨅ (_ : 0 < law (squareParameter q) {ω | (infiniteClusters G ω).Nonempty}),
-    squareParameter q) = _
-  simpa only [pc,sInf_eq_iInf,Set.mem_ofPred_eq] using squareParameter.surjective.iInf_comp
-    (g := fun p => ⨅ (_ : 0 < law p {ω | (infiniteClusters G ω).Nonempty}), p)
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Subdivision.Graph
+
+namespace OAI
+
+/-! Coupling percolation on a graph with percolation on its subdivision. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+
+ 
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal
+
+noncomputable def halfBit (G : BondGraph V E) : SubdivisionEdge G ↪ E × Bool := by
+  classical
+  refine ⟨fun ev => (ev.val.1,if ev.val.2 = (G ev.val.1).out.1 then false else true),?_⟩
+  rintro ⟨⟨e,x⟩,hx⟩ ⟨⟨f,y⟩,hy⟩ h
+  have hef : e = f := congrArg Prod.fst h
+  subst f
+  have hb := congrArg Prod.snd h
+  dsimp only at hb
+  have he : G e = s((G e).out.1,(G e).out.2) := (G e).out_eq.symm
+  have hx' : x = (G e).out.1 ∨ x = (G e).out.2 := by
+    apply Sym2.mem_iff.mp
+    rw [←he]
+    exact hx
+  have hy' : y = (G e).out.1 ∨ y = (G e).out.2 := by
+    apply Sym2.mem_iff.mp
+    rw [←he]
+    exact hy
+  have hxy : x = y := by
+    by_cases h₁ : x = (G e).out.1 <;> by_cases h₂ : y = (G e).out.1
+    · exact h₁.trans h₂.symm
+    · simp only [ite_eq_left h₁,ite_eq_right h₂,Bool.false_eq_true] at hb
+    · simp only [ite_eq_right h₁,ite_eq_left h₂,Bool.true_eq_false] at hb
+    · exact (hx'.resolve_left h₁).trans (hy'.resolve_left h₂).symm
+  subst y
+  rfl
+
+def seriesOpen (ν : Set (E × Bool)) : Set E := {e | ∀ b, (e,b) ∈ ν}
+
+noncomputable def subdivisionOpen (G : BondGraph V E) (ν : Set (E × Bool)) :
+    Set (SubdivisionEdge G) := halfBit G ⁻¹' ν
+
+@[simp] theorem halfBit_fst (G : BondGraph V E) (e : SubdivisionEdge G) :
+    (halfBit G e).1 = e.val.1 := rfl
+
+ 
+theorem seriesOpen_of_two_incident (G : BondGraph V E) (ν : Set (E × Bool))
+    (e : E) (x y : V) (hx : x ∈ G e) (hy : y ∈ G e) (hne : x ≠ y)
+    (hxo : (⟨(e,x),hx⟩ : SubdivisionEdge G) ∈ subdivisionOpen G ν)
+    (hyo : (⟨(e,y),hy⟩ : SubdivisionEdge G) ∈ subdivisionOpen G ν) :
+    e ∈ seriesOpen ν := by
+  have hbits : (halfBit G ⟨(e,x),hx⟩).2 ≠ (halfBit G ⟨(e,y),hy⟩).2 := by
+    intro h
+    have hh : halfBit G ⟨(e,x),hx⟩ = halfBit G ⟨(e,y),hy⟩ := Prod.ext rfl h
+    have he := (halfBit G).injective hh
+    exact hne (congrArg (fun z : SubdivisionEdge G => z.val.2) he)
+  intro b
+  have hb : b = (halfBit G ⟨(e,x),hx⟩).2 ∨ b = (halfBit G ⟨(e,y),hy⟩).2 := by
+    generalize (halfBit G ⟨(e,x),hx⟩).2 = c at *
+    generalize (halfBit G ⟨(e,y),hy⟩).2 = d at *
+    cases b <;> cases c <;> cases d <;> simp_all
+  rcases hb with rfl | rfl
+  · exact hxo
+  · exact hyo
+
+ 
+theorem subdivision_open_adj_iff (G : BondGraph V E) (ν : Set (E × Bool)) (x : V) (e : E) :
+    (∃ d ∈ subdivisionOpen G ν, subdivision G d = s(Sum.inl x, Sum.inr e)) ↔
+      ∃ hx : x ∈ G e, (⟨(e,x),hx⟩ : SubdivisionEdge G) ∈ subdivisionOpen G ν := by
+  constructor
+  · rintro ⟨⟨⟨f,y⟩,hy⟩,ho,he⟩
+    simp only [subdivision,Sym2.eq_iff] at he
+    rcases he with ⟨hyx,hfe⟩ | ⟨hbad,_⟩
+    · have : y = x := Sum.inl.inj hyx
+      have : f = e := Sum.inr.inj hfe
+      subst y; subst f
+      exact ⟨hy,ho⟩
+    · cases hbad
+  · rintro ⟨hx,ho⟩
+    exact ⟨⟨(e,x),hx⟩,ho,rfl⟩
+
+ 
+theorem conn_subdivision_iff (G : BondGraph V E) (ν : Set (E × Bool)) (x y : V) :
+    Conn (subdivision G) (subdivisionOpen G ν) (.inl x) (.inl y) ↔
+      Conn G (seriesOpen ν) x y := by
+  constructor
+  · intro hc
+    have reach (z : V ⊕ E) (h : Conn (subdivision G) (subdivisionOpen G ν) (.inl x) z) :
+        match z with
+        | .inl w => Conn G (seriesOpen ν) x w
+        | .inr e => ∃ w : V, ∃ hw : w ∈ G e,
+            (⟨(e,w),hw⟩ : SubdivisionEdge G) ∈ subdivisionOpen G ν ∧ Conn G (seriesOpen ν) x w := by
+      induction h with
+      | refl => exact conn_refl G _ x
+      | @tail u v huv hed ih =>
+        cases u with
+        | inl a =>
+          cases v with
+          | inl b =>
+            obtain ⟨d,_,he⟩ := hed
+            exact (subdivision_adj_inl_inl G a b ⟨d,he⟩).elim
+          | inr e =>
+            obtain ⟨ha,ho⟩ := (subdivision_open_adj_iff G ν a e).mp hed
+            exact ⟨a,ha,ho,ih⟩
+        | inr e =>
+          cases v with
+          | inl b =>
+            have hed' : ∃ d ∈ subdivisionOpen G ν, subdivision G d = s(Sum.inl b, Sum.inr e) := by
+              obtain ⟨d,hd,he⟩ := hed
+              exact ⟨d,hd,he.trans (Sym2.eq_swap)⟩
+            obtain ⟨hb,hbo⟩ := (subdivision_open_adj_iff G ν b e).mp hed'
+            obtain ⟨a,ha,hao,hxa⟩ := ih
+            by_cases hab : a = b
+            · simpa only [hab] using hxa
+            · exact hxa.tail ⟨e,seriesOpen_of_two_incident G ν e a b ha hb hab hao hbo,
+                (Sym2.mem_and_mem_iff hab).mp ⟨ha,hb⟩⟩
+          | inr f =>
+            obtain ⟨d,_,he⟩ := hed
+            exact (subdivision_adj_inr_inr G e f ⟨d,he⟩).elim
+    exact reach (.inl y) hc
+  · intro hc
+    induction hc with
+    | refl => exact conn_refl _ _ _
+    | @tail a b hab hed ih =>
+      obtain ⟨e,he,heq⟩ := hed
+      have ha : a ∈ G e := by simp [heq]
+      have hb : b ∈ G e := by simp [heq]
+      have hao : (⟨(e,a),ha⟩ : SubdivisionEdge G) ∈ subdivisionOpen G ν := he _
+      have hbo : (⟨(e,b),hb⟩ : SubdivisionEdge G) ∈ subdivisionOpen G ν := he _
+      exact (ih.tail ⟨⟨(e,a),ha⟩,hao,rfl⟩).tail ⟨⟨(e,b),hb⟩,hbo,Sym2.eq_swap⟩
+
+ 
+theorem map_bool_and (q : unitInterval) :
+    (Measure.infinitePi (fun _ : Bool => bitLaw q)).map
+      (fun f : Bool → Prop => ∀ b, f b) = bitLaw (q^2) := by
+  let μ := (Measure.infinitePi (fun _ : Bool => bitLaw q)).map
+      (fun f : Bool → Prop => ∀ b, f b)
+  have hm : Measurable (fun f : Bool → Prop => ∀ b, f b) := .of_discrete
+  have : IsProbabilityMeasure μ := inferInstance
+  have ht : μ.real {True} = (q:ℝ)^2 := by
+    change ((Measure.infinitePi _).map _).real _ = _
+    rw [Measure.real,Measure.map_apply hm (measurableSet_singleton _)]
+    have he : (fun f : Bool → Prop => ∀ b, f b) ⁻¹' ({True} : Set Prop) =
+        ((Finset.univ : Finset Bool) : Set Bool).pi (fun _ => ({True} : Set Prop)) := by
+      ext f
+      simp [and_comm]
+    rw [he,Measure.infinitePi_pi _ (fun _ _ => measurableSet_singleton _)]
+    simp [bitLaw,ENNReal.toReal_mul,pow_two]
+  have hf : μ.real {False} = 1-(q:ℝ)^2 := by
+    have he : ({False} : Set Prop) = ({True} : Set Prop)ᶜ := by ext b; simp
+    rw [he,measureReal_compl (measurableSet_singleton _),probReal_univ,ht]
+  apply Measure.ext_of_singleton
+  intro b
+  apply (ENNReal.toReal_eq_toReal_iff' (measure_ne_top _ _) (measure_ne_top _ _)).mp
+  change μ.real {b} = (bitLaw (q^2)).real {b}
+  by_cases hb : b
+  · have he : b = True := propext (iff_true_intro hb)
+    subst b
+    rw [ht]
+    simp [bitLaw,Measure.real]
+  · have he : b = False := propext (iff_false_intro hb)
+    subst b
+    rw [hf]
+    simp [bitLaw,Measure.real]
+
+theorem measurable_seriesOpen : Measurable (seriesOpen : Set (E × Bool) → Set E) := by
+  apply measurable_setOfPred.comp
+  apply Measurable.of_eval
+  intro e
+  apply measurableSet_setOfPred.mp
+  have he : {ν : Set (E × Bool) | ∀ b, (e,b) ∈ ν} = ⋂ b : Bool, {ν | (e,b) ∈ ν} := by
+    ext ν; simp
+  rw [he]
+  exact MeasurableSet.iInter fun b => measurableSet_setOfPred.mpr
+    ((measurable_pi_apply (e,b)).comp MeasurableEquiv.setOfPred.symm.measurable)
+
+ 
+theorem law_map_seriesOpen (q : unitInterval) :
+    (law q : Measure (Set (E × Bool))).map seriesOpen = law (q^2) := by
+  let c := MeasurableEquiv.curry E Bool Prop
+  let b := fun f : E → Bool → Prop => fun e => ∀ j, f e j
+  have hb : Measurable b := Measurable.of_eval
+    (fun e => (Measurable.of_discrete : Measurable (fun f : Bool → Prop => ∀ j, f j)).comp
+      (measurable_pi_apply e))
+  unfold law
+  rw [setBernoulli_eq_map,Measure.map_map measurable_seriesOpen measurable_setOfPred]
+  simp only [Set.mem_univ]
+  have he : seriesOpen ∘ (fun f : E × Bool → Prop => {e | f e}) =
+      (fun f : E → Prop => {e | f e}) ∘ b ∘ c := rfl
+  rw [he,←Measure.map_map measurable_setOfPred (hb.comp c.measurable),
+    ←Measure.map_map hb c.measurable]
+  change ((Measure.infinitePi (fun _ : E × Bool => bitLaw q)).map c |>.map b |>.map
+    (fun f : E → Prop => {e | f e})) = _
+  rw [Measure.infinitePi_map_curry (fun (_ : E) (_ : Bool) => bitLaw q)]
+  dsimp only [b]
+  rw [Measure.infinitePi_map_pi _ (f := fun _ (g : Bool → Prop) => ∀ j, g j)
+    (fun _ => Measurable.of_discrete)]
+  rw [map_bool_and q,setBernoulli_eq_map]
+  rfl
+
+theorem measurable_subdivisionOpen (G : BondGraph V E) : Measurable (subdivisionOpen G) :=
+  measurable_setOfPred.comp (Measurable.of_eval (fun e =>
+    (measurable_pi_apply (halfBit G e)).comp MeasurableEquiv.setOfPred.symm.measurable))
+
+theorem law_map_subdivisionOpen (G : BondGraph V E) (q : unitInterval) :
+    (law q : Measure (Set (E × Bool))).map (subdivisionOpen G) = law q :=
+  law_map_preimage_embedding q (halfBit G)
+
+ 
+
+theorem twoPoint_subdivision [Countable V] [Countable E]
+    (G : BondGraph V E) (q : unitInterval) (x y : V) :
+    twoPoint (subdivision G) q (.inl x) (.inl y) = twoPoint G (q^2) x y := by
+  unfold twoPoint
+  congr 1
+  rw [←law_map_subdivisionOpen G q,Measure.map_apply (measurable_subdivisionOpen G)
+    (measurableSet_conn (subdivision G) (.inl x) (.inl y))]
+  rw [←law_map_seriesOpen q,Measure.map_apply measurable_seriesOpen (measurableSet_conn G x y)]
+  congr 1
+  ext ν
+  exact conn_subdivision_iff G ν x y
+
+ 
+theorem conn_subdivision_inr (G : BondGraph V E) (ν : Set (E × Bool)) (x : V) (e : E)
+    (hc : Conn (subdivision G) (subdivisionOpen G ν) (.inl x) (.inr e)) :
+    ∃ y, Conn G (seriesOpen ν) x y ∧ y ∈ G e := by
+  rcases hc.cases_tail with h | ⟨z,hxz,hze⟩
+  · cases h
+  · cases z with
+    | inl y =>
+      obtain ⟨hy,_⟩ := (subdivision_open_adj_iff G ν y e).mp hze
+      exact ⟨y,(conn_subdivision_iff G ν x y).mp hxz,hy⟩
+    | inr f =>
+      obtain ⟨d,_,hd⟩ := hze
+      exact (subdivision_adj_inr_inr G f e ⟨d,hd⟩).elim
+
+ 
+
+theorem finite_cluster_subdivision_iff (G : BondGraph V E) (hloc : LocallyFinite G)
+    (ν : Set (E × Bool)) (x : V) :
+    (cluster (subdivision G) (subdivisionOpen G ν) (.inl x)).Finite ↔
+      (cluster G (seriesOpen ν) x).Finite := by
+  constructor
+  · intro hf
+    apply Set.Finite.of_finite_image (f := (Sum.inl : V → V ⊕ E))
+    · apply hf.subset
+      rintro z ⟨y,hy,rfl⟩
+      exact (conn_subdivision_iff G ν x y).mpr hy
+    · exact Sum.inl_injective.injOn
+  · intro hf
+    have he : (⋃ y ∈ cluster G (seriesOpen ν) x, {e | y ∈ G e}).Finite :=
+      hf.biUnion (fun y _ => hloc y)
+    apply ((hf.image (Sum.inl : V → V ⊕ E)).union
+      (he.image (Sum.inr : E → V ⊕ E))).subset
+    intro z hz
+    cases z with
+    | inl y => exact Or.inl ⟨y,(conn_subdivision_iff G ν x y).mp hz,rfl⟩
+    | inr e =>
+      obtain ⟨y,hxy,hy⟩ := conn_subdivision_inr G ν x e hz
+      exact Or.inr ⟨e,Set.mem_iUnion.mpr ⟨y,Set.mem_iUnion.mpr ⟨hxy,hy⟩⟩,rfl⟩
+
+ 
+
+theorem exists_infinite_cluster_subdivision_iff (G : BondGraph V E)
+    (hloc : LocallyFinite G) (ν : Set (E × Bool)) :
+    (∃ z, (cluster (subdivision G) (subdivisionOpen G ν) z).Infinite) ↔
+      ∃ x, (cluster G (seriesOpen ν) x).Infinite := by
+  have hi (x : V) : (cluster (subdivision G) (subdivisionOpen G ν) (.inl x)).Infinite ↔
+      (cluster G (seriesOpen ν) x).Infinite := by
+    exact not_congr (finite_cluster_subdivision_iff G hloc ν x)
+  constructor
+  · rintro ⟨z,hz⟩
+    cases z with
+    | inl x => exact ⟨x,(hi x).mp hz⟩
+    | inr e =>
+      obtain ⟨w,hw,hne⟩ := hz.exists_notMem_finset {Sum.inr e}
+      have hwne : w ≠ Sum.inr e := by simpa using hne
+      rcases (show Conn (subdivision G) (subdivisionOpen G ν) (.inr e) w from hw).cases_head
+        with h | ⟨z,hez,hzw⟩
+      · exact (hwne h.symm).elim
+      · cases z with
+        | inl y =>
+          have hconn : Conn (subdivision G) (subdivisionOpen G ν) (.inr e) (.inl y) :=
+            .single hez
+          have heq := cluster_eq_of_conn (subdivision G) (subdivisionOpen G ν) hconn
+          exact ⟨y,(hi y).mp (heq ▸ hz)⟩
+        | inr f =>
+          obtain ⟨d,_,hd⟩ := hez
+          exact (subdivision_adj_inr_inr G e f ⟨d,hd⟩).elim
+  · rintro ⟨x,hx⟩
+    exact ⟨.inl x,(hi x).mpr hx⟩
+
+theorem infinite_cluster_probability_subdivision [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (q : unitInterval) :
+    law q {ω | (infiniteClusters (subdivision G) ω).Nonempty} =
+      law (q^2) {ω | (infiniteClusters G ω).Nonempty} := by
+  simp only [infiniteClusters_nonempty_iff]
+  rw [←law_map_subdivisionOpen G q,Measure.map_apply (measurable_subdivisionOpen G)
+    (measurableSet_exists_infinite_cluster _)]
+  rw [←law_map_seriesOpen q,Measure.map_apply measurable_seriesOpen
+    (measurableSet_exists_infinite_cluster _)]
+  congr 1
+  ext ν
+  exact exists_infinite_cluster_subdivision_iff G hloc ν
+
+ 
+noncomputable def squareParameter : unitInterval ≃o unitInterval where
+  toFun q := q^2
+  invFun p := ⟨Real.sqrt p,Real.sqrt_nonneg _,Real.sqrt_le_one.mpr p.2.2⟩
+  left_inv q := by
+    apply Subtype.ext
+    change Real.sqrt ((q:ℝ)^2) = (q:ℝ)
+    exact Real.sqrt_sq q.2.1
+  right_inv p := by
+    apply Subtype.ext
+    change Real.sqrt (p:ℝ)^2 = (p:ℝ)
+    exact Real.sq_sqrt p.2.1
+  map_rel_iff' := by
+    intro p q
+    change (p:ℝ)^2 ≤ (q:ℝ)^2 ↔ (p:ℝ) ≤ (q:ℝ)
+    exact sq_le_sq₀ p.2.1 q.2.1
+
+ 
+
+theorem pc_subdivision_sq [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) :
+    pc (subdivision G)^2 = pc G := by
+  change squareParameter (sInf {q | 0 < law q {ω | (infiniteClusters (subdivision G) ω).Nonempty}}) = _
+  rw [squareParameter.map_sInf]
+  change (⨅ q, ⨅ (_ : 0 < law q {ω | (infiniteClusters (subdivision G) ω).Nonempty}), q^2) = _
+  simp only [infinite_cluster_probability_subdivision G hloc]
+  change (⨅ q, ⨅ (_ : 0 < law (squareParameter q) {ω | (infiniteClusters G ω).Nonempty}),
+    squareParameter q) = _
+  simpa only [pc,sInf_eq_iInf,Set.mem_ofPred_eq] using squareParameter.surjective.iInf_comp
+    (g := fun p => ⨅ (_ : 0 < law p {ω | (infiniteClusters G ω).Nonempty}), p)
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Subdivision/Graph.lean b/lean/OAI/Probability/UnimodularPercolation/Subdivision/Graph.lean
index 07865eca6891c32fe58837869fa9a72ef347f95c..4156665e13a3a1f938bbe4dc77677e9bcc26c75e 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Subdivision/Graph.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Subdivision/Graph.lean
@@ -1,284 +1,284 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Clusters.LargeContacts
-
-namespace OAI
-
-/-! Subdivision of indexed bond graphs and its basic geometry. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-
- 
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-
-abbrev SubdivisionEdge (G : BondGraph V E) := {ev : E × V // ev.2 ∈ G ev.1}
-
-def subdivision (G : BondGraph V E) : BondGraph (V ⊕ E) (SubdivisionEdge G) :=
-  fun ev => s(Sum.inl ev.val.2, Sum.inr ev.val.1)
-
-theorem subdivision_injective (G : BondGraph V E) : Function.Injective (subdivision G) := by
-  intro ⟨⟨e,x⟩,hx⟩ ⟨⟨f,y⟩,hy⟩ h
-  simp only [subdivision,Sym2.eq_iff] at h
-  rcases h with ⟨hxy,hef⟩ | ⟨hbad,_⟩
-  · have h₁ : x = y := Sum.inl.inj hxy
-    have h₂ : e = f := Sum.inr.inj hef
-    subst y; subst f; rfl
-  · cases hbad
-
-@[simp] theorem subdivision_adj_inl_inr (G : BondGraph V E) (x : V) (e : E) :
-    Adj (subdivision G) (.inl x) (.inr e) ↔ x ∈ G e := by
-  constructor
-  · rintro ⟨⟨⟨f,y⟩,hy⟩,h⟩
-    simp only [subdivision,Sym2.eq_iff] at h
-    rcases h with ⟨hyx,hfe⟩ | ⟨hbad,_⟩
-    · have : y = x := Sum.inl.inj hyx
-      have : f = e := Sum.inr.inj hfe
-      subst y; subst f; exact hy
-    · cases hbad
-  · intro hx
-    exact ⟨⟨(e,x),hx⟩,rfl⟩
-
-@[simp] theorem subdivision_adj_inl_inl (G : BondGraph V E) (x y : V) :
-    ¬ Adj (subdivision G) (.inl x) (.inl y) := by
-  rintro ⟨⟨⟨e,z⟩,hz⟩,h⟩
-  simp [subdivision] at h
-
-@[simp] theorem subdivision_adj_inr_inr (G : BondGraph V E) (e f : E) :
-    ¬ Adj (subdivision G) (.inr e) (.inr f) := by
-  rintro ⟨⟨⟨d,z⟩,hz⟩,h⟩
-  simp [subdivision] at h
-
-@[simp] theorem subdivision_adj_inr_inl (G : BondGraph V E) (e : E) (x : V) :
-    Adj (subdivision G) (.inr e) (.inl x) ↔ x ∈ G e := by
-  exact ⟨fun h => (subdivision_adj_inl_inr G x e).mp (adj_symm _ h),
-    fun h => adj_symm _ ((subdivision_adj_inl_inr G x e).mpr h)⟩
-
-theorem subdivision_loopless (G : BondGraph V E) (x : V ⊕ E) :
-    ¬ Adj (subdivision G) x x := by cases x <;> simp
-
-theorem subdivision_locallyFinite (G : BondGraph V E) (hloc : LocallyFinite G) :
-    LocallyFinite (subdivision G) := by
-  classical
-  intro z
-  cases z with
-  | inl x =>
-    apply Set.Finite.of_finite_image (f := fun e : SubdivisionEdge G => e.val.1)
-    · apply (hloc x).subset
-      rintro e ⟨⟨⟨f,y⟩,hy⟩,hz,rfl⟩
-      have he : x = y := by simpa [subdivision] using hz
-      simpa [he] using hy
-    · intro e he f hf hef
-      apply Subtype.ext
-      apply Prod.ext hef
-      have hx : x = e.val.2 := by simpa [subdivision] using he
-      have hy : x = f.val.2 := by simpa [subdivision] using hf
-      exact hx.symm.trans hy
-  | inr e =>
-    apply Set.Finite.of_finite_image (f := fun f : SubdivisionEdge G => f.val.2)
-    · apply (Sym2.toFinset (G e)).finite_toSet.subset
-      rintro x ⟨⟨⟨f,y⟩,hy⟩,hz,rfl⟩
-      have he : e = f := by simpa [subdivision] using hz
-      simpa [he] using hy
-    · intro f hf g hg hfg
-      apply Subtype.ext
-      apply Prod.ext _ hfg
-      have h₁ : e = f.val.1 := by simpa [subdivision] using hf
-      have h₂ : e = g.val.1 := by simpa [subdivision] using hg
-      exact h₁.symm.trans h₂
-
-theorem subdivision_connected (G : BondGraph V E) (hconn : Connected G) :
-    Connected (subdivision G) := by
-  have hold (x y : V) : Relation.ReflTransGen (Adj (subdivision G)) (.inl x) (.inl y) := by
-    induction hconn x y with
-    | refl => exact .refl
-    | @tail y z hxy hyz ih =>
-      obtain ⟨e,he⟩ := hyz
-      exact (ih.tail ((subdivision_adj_inl_inr G y e).mpr (by simp [he]))).tail
-        ((subdivision_adj_inr_inl G e z).mpr (by simp [he]))
-  have attach (z : V ⊕ E) : ∃ x : V, Relation.ReflTransGen (Adj (subdivision G)) z (.inl x) := by
-    cases z with
-    | inl x => exact ⟨x,.refl⟩
-    | inr e =>
-      exact ⟨(G e).out.1,Relation.ReflTransGen.single
-        ((subdivision_adj_inr_inl G e _).mpr (G e).out_fst_mem)⟩
-  have rev {a b : V ⊕ E} (h : Relation.ReflTransGen (Adj (subdivision G)) a b) :
-      Relation.ReflTransGen (Adj (subdivision G)) b a := by
-    induction h with
-    | refl => exact .refl
-    | tail _ hab ih => exact (Relation.ReflTransGen.single (adj_symm _ hab)).trans ih
-  intro z w
-  obtain ⟨x,hx⟩ := attach z
-  obtain ⟨y,hy⟩ := attach w
-  exact hx.trans ((hold x y).trans (rev hy))
-
- 
-theorem incidence_aut_iff (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (e : E) (x : V) : a x ∈ G (b e) ↔ x ∈ G e := by
-  rw [hab e]
-  simp only [Sym2.mem_map,a.injective.eq_iff,exists_eq_right]
-
- 
-noncomputable def subdivisionAutEdges (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) : SubdivisionEdge G ≃ SubdivisionEdge G :=
-  (b.prodCongr a).subtypeEquiv (fun p => (incidence_aut_iff G hab p.1 p.2).symm)
-
-theorem subdivision_isAut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) :
-    IsAut (subdivision G) (a.sumCongr b) (subdivisionAutEdges G hab) := by
-  intro e
-  rfl
-
- 
-
-theorem finite_edge_roots (G : BondGraph V E) (hloc : LocallyFinite G)
-    (hqt : QuasiTransitive G) :
-    ∃ R : Finset E, ∀ e, ∃ f ∈ R, ∃ a : V ≃ V, ∃ b : E ≃ E,
-      IsAut G a b ∧ b f = e := by
-  classical
-  obtain ⟨S,hS⟩ := hqt
-  let R := S.biUnion (fun x => (hloc x).toFinset)
-  refine ⟨R,fun e => ?_⟩
-  obtain ⟨x,hx,a,b,hab,hax⟩ := hS (G e).out.1
-  refine ⟨b.symm e,Finset.mem_biUnion.mpr ⟨x,hx,?_⟩,a,b,hab,b.apply_symm_apply e⟩
-  apply (hloc x).mem_toFinset.mpr
-  apply (incidence_aut_iff G hab _ _).mp
-  simpa only [Equiv.apply_symm_apply,hax] using (G e).out_fst_mem
-
-theorem subdivision_quasiTransitive (G : BondGraph V E) (hloc : LocallyFinite G)
-    (hqt : QuasiTransitive G) : QuasiTransitive (subdivision G) := by
-  classical
-  obtain ⟨S,hS⟩ := hqt
-  obtain ⟨R,hR⟩ := finite_edge_roots G hloc ⟨S,hS⟩
-  refine ⟨S.image Sum.inl ∪ R.image Sum.inr,?_⟩
-  intro z
-  cases z with
-  | inl x =>
-    obtain ⟨o,ho,a,b,hab,hax⟩ := hS x
-    refine ⟨.inl o,Finset.mem_union_left _ (Finset.mem_image.mpr ⟨o,ho,rfl⟩),
-      a.sumCongr b,subdivisionAutEdges G hab,subdivision_isAut G hab,?_⟩
-    exact congrArg Sum.inl hax
-  | inr e =>
-    obtain ⟨f,hf,a,b,hab,hbe⟩ := hR e
-    refine ⟨.inr f,Finset.mem_union_right _ (Finset.mem_image.mpr ⟨f,hf,rfl⟩),
-      a.sumCongr b,subdivisionAutEdges G hab,subdivision_isAut G hab,?_⟩
-    exact congrArg Sum.inr hbe
-
- 
-
-theorem subdivision_old_boundary_bound (G : BondGraph V E) (hloc : LocallyFinite G)
-    (A : Finset (V ⊕ E)) :
-    hV G * (A.toLeft.card : ℝ) ≤
-      2 * ((vertexBoundary (subdivision G) A).ncard : ℝ) := by
-  classical
-  let B := (finite_vertexBoundary (subdivision G) (subdivision_locallyFinite G hloc) A).toFinset
-  let C := (finite_vertexBoundary G hloc A.toLeft).toFinset
-  have hsub : C ⊆ B.toLeft ∪ B.toRight.biUnion (fun e => (G e).toFinset) := by
-    intro y hy
-    obtain ⟨hy,x,hx,e,he⟩ := (finite_vertexBoundary G hloc A.toLeft).mem_toFinset.mp hy
-    by_cases heA : Sum.inr e ∈ A
-    · apply Finset.mem_union_left
-      apply Finset.mem_toLeft.mpr
-      apply (finite_vertexBoundary (subdivision G) (subdivision_locallyFinite G hloc) A).mem_toFinset.mpr
-      exact ⟨fun h => hy (Finset.mem_toLeft.mpr h),.inr e,heA,
-        (subdivision_adj_inr_inl G e y).mpr (by simp [he])⟩
-    · apply Finset.mem_union_right
-      apply Finset.mem_biUnion.mpr
-      refine ⟨e,?_,Sym2.mem_toFinset.mpr (by simp [he])⟩
-      apply Finset.mem_toRight.mpr
-      apply (finite_vertexBoundary (subdivision G) (subdivision_locallyFinite G hloc) A).mem_toFinset.mpr
-      exact ⟨heA,.inl x,Finset.mem_toLeft.mp hx,
-        (subdivision_adj_inl_inr G x e).mpr (by simp [he])⟩
-  have hcard : C.card ≤ B.toLeft.card + 2*B.toRight.card := by
-    calc
-      C.card ≤ (B.toLeft ∪ B.toRight.biUnion (fun e => (G e).toFinset)).card := Finset.card_le_card hsub
-      _ ≤ B.toLeft.card + (B.toRight.biUnion (fun e => (G e).toFinset)).card := Finset.card_union_le _ _
-      _ ≤ B.toLeft.card + ∑ e ∈ B.toRight, (G e).toFinset.card :=
-        Nat.add_le_add_left (Finset.card_biUnion_le) _
-      _ ≤ B.toLeft.card + ∑ _e ∈ B.toRight, 2 := by
-        apply Nat.add_le_add_left
-        apply Finset.sum_le_sum
-        intro e _
-        rw [Sym2.card_toFinset]
-        split_ifs <;> omega
-      _ = _ := by simp [Nat.mul_comm]
-  have hcard' : C.card ≤ 2*B.card := by
-    have hsum := B.card_toLeft_add_card_toRight
-    omega
-  have hh := hV_mul_card_le G A.toLeft
-  have hc : C.card = (vertexBoundary G A.toLeft).ncard := (Set.ncard_eq_toFinset_card _ _).symm
-  have hb : B.card = (vertexBoundary (subdivision G) A).ncard := (Set.ncard_eq_toFinset_card _ _).symm
-  rw [←hc] at hh
-  exact hh.trans (by exact_mod_cast (hcard'.trans_eq (by rw [hb])))
-
- 
-
-theorem subdivision_new_card_bound (G : BondGraph V E) (hloc : LocallyFinite G)
-    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (A : Finset (V ⊕ E)) :
-    A.toRight.card ≤ d*(A.toLeft.card + (vertexBoundary (subdivision G) A).ncard) := by
-  classical
-  let B := (finite_vertexBoundary (subdivision G) (subdivision_locallyFinite G hloc) A).toFinset
-  let C := A.toLeft ∪ B.toLeft
-  have hsub : A.toRight ⊆ C.biUnion (fun x => (hloc x).toFinset) := by
-    intro e he
-    let x := (G e).out.1
-    have hx : x ∈ G e := (G e).out_fst_mem
-    refine Finset.mem_biUnion.mpr ⟨x,?_,(hloc x).mem_toFinset.mpr hx⟩
-    by_cases hxA : x ∈ A.toLeft
-    · exact Finset.mem_union_left _ hxA
-    · apply Finset.mem_union_right
-      apply Finset.mem_toLeft.mpr
-      apply (finite_vertexBoundary (subdivision G) (subdivision_locallyFinite G hloc) A).mem_toFinset.mpr
-      exact ⟨fun h => hxA (Finset.mem_toLeft.mpr h),.inr e,Finset.mem_toRight.mp he,
-        (subdivision_adj_inr_inl G e x).mpr hx⟩
-  have hc : C.card ≤ A.toLeft.card+B.card :=
-    (Finset.card_union_le _ _).trans (Nat.add_le_add_left B.card_toLeft_le _)
-  have hb : B.card = (vertexBoundary (subdivision G) A).ncard := (Set.ncard_eq_toFinset_card _ _).symm
-  calc
-    A.toRight.card ≤ (C.biUnion (fun x => (hloc x).toFinset)).card := Finset.card_le_card hsub
-    _ ≤ ∑ x ∈ C, (hloc x).toFinset.card := Finset.card_biUnion_le
-    _ ≤ ∑ _x ∈ C, d := by
-      apply Finset.sum_le_sum
-      intro x _
-      rw [←Set.ncard_eq_toFinset_card _ (hloc x)]
-      exact (incident_ncard_le_degree G hloc x).trans (hd x)
-    _ = d*C.card := by simp [Nat.mul_comm]
-    _ ≤ d*(A.toLeft.card+B.card) := Nat.mul_le_mul_left d hc
-    _ = _ := by rw [hb]
-
- 
-
-theorem subdivision_hV_pos [Nonempty V] (G : BondGraph V E) (hloc : LocallyFinite G)
-    (hqt : QuasiTransitive G) (hh : 0 < hV G) : 0 < hV (subdivision G) := by
-  classical
-  obtain ⟨d,hd,hdg⟩ := exists_degree_bound G hqt
-  let c := hV G / (2*((d:ℝ)+1)+(d:ℝ)*hV G)
-  have hden : 0 < 2*((d:ℝ)+1)+(d:ℝ)*hV G := by positivity
-  have hc : 0 < c := div_pos hh hden
-  apply hc.trans_le
-  apply le_csInf
-  · exact ⟨_,{Sum.inl (Classical.arbitrary V)},Finset.singleton_nonempty _,rfl⟩
-  · rintro r ⟨A,hA,rfl⟩
-    have ho := subdivision_old_boundary_bound G hloc A
-    have hn : (A.toRight.card:ℝ) ≤ (d:ℝ)*((A.toLeft.card:ℝ)+
-        ((vertexBoundary (subdivision G) A).ncard:ℝ)) := by
-      exact_mod_cast subdivision_new_card_bound G hloc d hdg A
-    have hs : (A.toLeft.card:ℝ)+(A.toRight.card:ℝ) = A.card := by
-      exact_mod_cast A.card_toLeft_add_card_toRight
-    have hcard : (0:ℝ) < A.card := Nat.cast_pos.mpr hA.card_pos
-    apply (le_div_iff₀ hcard).mpr
-    dsimp only [c]
-    rw [div_mul_eq_mul_div]
-    apply (div_le_iff₀ hden).mpr
-    have h₁ := mul_le_mul_of_nonneg_left ho (show 0 ≤ (d:ℝ)+1 by positivity)
-    have h₂ := mul_le_mul_of_nonneg_left hn hh.le
-    rw [←hs]
-    nlinarith only [h₁,h₂]
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Clusters.LargeContacts
+
+namespace OAI
+
+/-! Subdivision of indexed bond graphs and its basic geometry. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+
+ 
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+
+abbrev SubdivisionEdge (G : BondGraph V E) := {ev : E × V // ev.2 ∈ G ev.1}
+
+def subdivision (G : BondGraph V E) : BondGraph (V ⊕ E) (SubdivisionEdge G) :=
+  fun ev => s(Sum.inl ev.val.2, Sum.inr ev.val.1)
+
+theorem subdivision_injective (G : BondGraph V E) : Function.Injective (subdivision G) := by
+  intro ⟨⟨e,x⟩,hx⟩ ⟨⟨f,y⟩,hy⟩ h
+  simp only [subdivision,Sym2.eq_iff] at h
+  rcases h with ⟨hxy,hef⟩ | ⟨hbad,_⟩
+  · have h₁ : x = y := Sum.inl.inj hxy
+    have h₂ : e = f := Sum.inr.inj hef
+    subst y; subst f; rfl
+  · cases hbad
+
+@[simp] theorem subdivision_adj_inl_inr (G : BondGraph V E) (x : V) (e : E) :
+    Adj (subdivision G) (.inl x) (.inr e) ↔ x ∈ G e := by
+  constructor
+  · rintro ⟨⟨⟨f,y⟩,hy⟩,h⟩
+    simp only [subdivision,Sym2.eq_iff] at h
+    rcases h with ⟨hyx,hfe⟩ | ⟨hbad,_⟩
+    · have : y = x := Sum.inl.inj hyx
+      have : f = e := Sum.inr.inj hfe
+      subst y; subst f; exact hy
+    · cases hbad
+  · intro hx
+    exact ⟨⟨(e,x),hx⟩,rfl⟩
+
+@[simp] theorem subdivision_adj_inl_inl (G : BondGraph V E) (x y : V) :
+    ¬ Adj (subdivision G) (.inl x) (.inl y) := by
+  rintro ⟨⟨⟨e,z⟩,hz⟩,h⟩
+  simp [subdivision] at h
+
+@[simp] theorem subdivision_adj_inr_inr (G : BondGraph V E) (e f : E) :
+    ¬ Adj (subdivision G) (.inr e) (.inr f) := by
+  rintro ⟨⟨⟨d,z⟩,hz⟩,h⟩
+  simp [subdivision] at h
+
+@[simp] theorem subdivision_adj_inr_inl (G : BondGraph V E) (e : E) (x : V) :
+    Adj (subdivision G) (.inr e) (.inl x) ↔ x ∈ G e := by
+  exact ⟨fun h => (subdivision_adj_inl_inr G x e).mp (adj_symm _ h),
+    fun h => adj_symm _ ((subdivision_adj_inl_inr G x e).mpr h)⟩
+
+theorem subdivision_loopless (G : BondGraph V E) (x : V ⊕ E) :
+    ¬ Adj (subdivision G) x x := by cases x <;> simp
+
+theorem subdivision_locallyFinite (G : BondGraph V E) (hloc : LocallyFinite G) :
+    LocallyFinite (subdivision G) := by
+  classical
+  intro z
+  cases z with
+  | inl x =>
+    apply Set.Finite.of_finite_image (f := fun e : SubdivisionEdge G => e.val.1)
+    · apply (hloc x).subset
+      rintro e ⟨⟨⟨f,y⟩,hy⟩,hz,rfl⟩
+      have he : x = y := by simpa [subdivision] using hz
+      simpa [he] using hy
+    · intro e he f hf hef
+      apply Subtype.ext
+      apply Prod.ext hef
+      have hx : x = e.val.2 := by simpa [subdivision] using he
+      have hy : x = f.val.2 := by simpa [subdivision] using hf
+      exact hx.symm.trans hy
+  | inr e =>
+    apply Set.Finite.of_finite_image (f := fun f : SubdivisionEdge G => f.val.2)
+    · apply (Sym2.toFinset (G e)).finite_toSet.subset
+      rintro x ⟨⟨⟨f,y⟩,hy⟩,hz,rfl⟩
+      have he : e = f := by simpa [subdivision] using hz
+      simpa [he] using hy
+    · intro f hf g hg hfg
+      apply Subtype.ext
+      apply Prod.ext _ hfg
+      have h₁ : e = f.val.1 := by simpa [subdivision] using hf
+      have h₂ : e = g.val.1 := by simpa [subdivision] using hg
+      exact h₁.symm.trans h₂
+
+theorem subdivision_connected (G : BondGraph V E) (hconn : Connected G) :
+    Connected (subdivision G) := by
+  have hold (x y : V) : Relation.ReflTransGen (Adj (subdivision G)) (.inl x) (.inl y) := by
+    induction hconn x y with
+    | refl => exact .refl
+    | @tail y z hxy hyz ih =>
+      obtain ⟨e,he⟩ := hyz
+      exact (ih.tail ((subdivision_adj_inl_inr G y e).mpr (by simp [he]))).tail
+        ((subdivision_adj_inr_inl G e z).mpr (by simp [he]))
+  have attach (z : V ⊕ E) : ∃ x : V, Relation.ReflTransGen (Adj (subdivision G)) z (.inl x) := by
+    cases z with
+    | inl x => exact ⟨x,.refl⟩
+    | inr e =>
+      exact ⟨(G e).out.1,Relation.ReflTransGen.single
+        ((subdivision_adj_inr_inl G e _).mpr (G e).out_fst_mem)⟩
+  have rev {a b : V ⊕ E} (h : Relation.ReflTransGen (Adj (subdivision G)) a b) :
+      Relation.ReflTransGen (Adj (subdivision G)) b a := by
+    induction h with
+    | refl => exact .refl
+    | tail _ hab ih => exact (Relation.ReflTransGen.single (adj_symm _ hab)).trans ih
+  intro z w
+  obtain ⟨x,hx⟩ := attach z
+  obtain ⟨y,hy⟩ := attach w
+  exact hx.trans ((hold x y).trans (rev hy))
+
+ 
+theorem incidence_aut_iff (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (e : E) (x : V) : a x ∈ G (b e) ↔ x ∈ G e := by
+  rw [hab e]
+  simp only [Sym2.mem_map,a.injective.eq_iff,exists_eq_right]
+
+ 
+noncomputable def subdivisionAutEdges (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) : SubdivisionEdge G ≃ SubdivisionEdge G :=
+  (b.prodCongr a).subtypeEquiv (fun p => (incidence_aut_iff G hab p.1 p.2).symm)
+
+theorem subdivision_isAut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) :
+    IsAut (subdivision G) (a.sumCongr b) (subdivisionAutEdges G hab) := by
+  intro e
+  rfl
+
+ 
+
+theorem finite_edge_roots (G : BondGraph V E) (hloc : LocallyFinite G)
+    (hqt : QuasiTransitive G) :
+    ∃ R : Finset E, ∀ e, ∃ f ∈ R, ∃ a : V ≃ V, ∃ b : E ≃ E,
+      IsAut G a b ∧ b f = e := by
+  classical
+  obtain ⟨S,hS⟩ := hqt
+  let R := S.biUnion (fun x => (hloc x).toFinset)
+  refine ⟨R,fun e => ?_⟩
+  obtain ⟨x,hx,a,b,hab,hax⟩ := hS (G e).out.1
+  refine ⟨b.symm e,Finset.mem_biUnion.mpr ⟨x,hx,?_⟩,a,b,hab,b.apply_symm_apply e⟩
+  apply (hloc x).mem_toFinset.mpr
+  apply (incidence_aut_iff G hab _ _).mp
+  simpa only [Equiv.apply_symm_apply,hax] using (G e).out_fst_mem
+
+theorem subdivision_quasiTransitive (G : BondGraph V E) (hloc : LocallyFinite G)
+    (hqt : QuasiTransitive G) : QuasiTransitive (subdivision G) := by
+  classical
+  obtain ⟨S,hS⟩ := hqt
+  obtain ⟨R,hR⟩ := finite_edge_roots G hloc ⟨S,hS⟩
+  refine ⟨S.image Sum.inl ∪ R.image Sum.inr,?_⟩
+  intro z
+  cases z with
+  | inl x =>
+    obtain ⟨o,ho,a,b,hab,hax⟩ := hS x
+    refine ⟨.inl o,Finset.mem_union_left _ (Finset.mem_image.mpr ⟨o,ho,rfl⟩),
+      a.sumCongr b,subdivisionAutEdges G hab,subdivision_isAut G hab,?_⟩
+    exact congrArg Sum.inl hax
+  | inr e =>
+    obtain ⟨f,hf,a,b,hab,hbe⟩ := hR e
+    refine ⟨.inr f,Finset.mem_union_right _ (Finset.mem_image.mpr ⟨f,hf,rfl⟩),
+      a.sumCongr b,subdivisionAutEdges G hab,subdivision_isAut G hab,?_⟩
+    exact congrArg Sum.inr hbe
+
+ 
+
+theorem subdivision_old_boundary_bound (G : BondGraph V E) (hloc : LocallyFinite G)
+    (A : Finset (V ⊕ E)) :
+    hV G * (A.toLeft.card : ℝ) ≤
+      2 * ((vertexBoundary (subdivision G) A).ncard : ℝ) := by
+  classical
+  let B := (finite_vertexBoundary (subdivision G) (subdivision_locallyFinite G hloc) A).toFinset
+  let C := (finite_vertexBoundary G hloc A.toLeft).toFinset
+  have hsub : C ⊆ B.toLeft ∪ B.toRight.biUnion (fun e => (G e).toFinset) := by
+    intro y hy
+    obtain ⟨hy,x,hx,e,he⟩ := (finite_vertexBoundary G hloc A.toLeft).mem_toFinset.mp hy
+    by_cases heA : Sum.inr e ∈ A
+    · apply Finset.mem_union_left
+      apply Finset.mem_toLeft.mpr
+      apply (finite_vertexBoundary (subdivision G) (subdivision_locallyFinite G hloc) A).mem_toFinset.mpr
+      exact ⟨fun h => hy (Finset.mem_toLeft.mpr h),.inr e,heA,
+        (subdivision_adj_inr_inl G e y).mpr (by simp [he])⟩
+    · apply Finset.mem_union_right
+      apply Finset.mem_biUnion.mpr
+      refine ⟨e,?_,Sym2.mem_toFinset.mpr (by simp [he])⟩
+      apply Finset.mem_toRight.mpr
+      apply (finite_vertexBoundary (subdivision G) (subdivision_locallyFinite G hloc) A).mem_toFinset.mpr
+      exact ⟨heA,.inl x,Finset.mem_toLeft.mp hx,
+        (subdivision_adj_inl_inr G x e).mpr (by simp [he])⟩
+  have hcard : C.card ≤ B.toLeft.card + 2*B.toRight.card := by
+    calc
+      C.card ≤ (B.toLeft ∪ B.toRight.biUnion (fun e => (G e).toFinset)).card := Finset.card_le_card hsub
+      _ ≤ B.toLeft.card + (B.toRight.biUnion (fun e => (G e).toFinset)).card := Finset.card_union_le _ _
+      _ ≤ B.toLeft.card + ∑ e ∈ B.toRight, (G e).toFinset.card :=
+        Nat.add_le_add_left (Finset.card_biUnion_le) _
+      _ ≤ B.toLeft.card + ∑ _e ∈ B.toRight, 2 := by
+        apply Nat.add_le_add_left
+        apply Finset.sum_le_sum
+        intro e _
+        rw [Sym2.card_toFinset]
+        split_ifs <;> omega
+      _ = _ := by simp [Nat.mul_comm]
+  have hcard' : C.card ≤ 2*B.card := by
+    have hsum := B.card_toLeft_add_card_toRight
+    omega
+  have hh := hV_mul_card_le G A.toLeft
+  have hc : C.card = (vertexBoundary G A.toLeft).ncard := (Set.ncard_eq_toFinset_card _ _).symm
+  have hb : B.card = (vertexBoundary (subdivision G) A).ncard := (Set.ncard_eq_toFinset_card _ _).symm
+  rw [←hc] at hh
+  exact hh.trans (by exact_mod_cast (hcard'.trans_eq (by rw [hb])))
+
+ 
+
+theorem subdivision_new_card_bound (G : BondGraph V E) (hloc : LocallyFinite G)
+    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (A : Finset (V ⊕ E)) :
+    A.toRight.card ≤ d*(A.toLeft.card + (vertexBoundary (subdivision G) A).ncard) := by
+  classical
+  let B := (finite_vertexBoundary (subdivision G) (subdivision_locallyFinite G hloc) A).toFinset
+  let C := A.toLeft ∪ B.toLeft
+  have hsub : A.toRight ⊆ C.biUnion (fun x => (hloc x).toFinset) := by
+    intro e he
+    let x := (G e).out.1
+    have hx : x ∈ G e := (G e).out_fst_mem
+    refine Finset.mem_biUnion.mpr ⟨x,?_,(hloc x).mem_toFinset.mpr hx⟩
+    by_cases hxA : x ∈ A.toLeft
+    · exact Finset.mem_union_left _ hxA
+    · apply Finset.mem_union_right
+      apply Finset.mem_toLeft.mpr
+      apply (finite_vertexBoundary (subdivision G) (subdivision_locallyFinite G hloc) A).mem_toFinset.mpr
+      exact ⟨fun h => hxA (Finset.mem_toLeft.mpr h),.inr e,Finset.mem_toRight.mp he,
+        (subdivision_adj_inr_inl G e x).mpr hx⟩
+  have hc : C.card ≤ A.toLeft.card+B.card :=
+    (Finset.card_union_le _ _).trans (Nat.add_le_add_left B.card_toLeft_le _)
+  have hb : B.card = (vertexBoundary (subdivision G) A).ncard := (Set.ncard_eq_toFinset_card _ _).symm
+  calc
+    A.toRight.card ≤ (C.biUnion (fun x => (hloc x).toFinset)).card := Finset.card_le_card hsub
+    _ ≤ ∑ x ∈ C, (hloc x).toFinset.card := Finset.card_biUnion_le
+    _ ≤ ∑ _x ∈ C, d := by
+      apply Finset.sum_le_sum
+      intro x _
+      rw [←Set.ncard_eq_toFinset_card _ (hloc x)]
+      exact (incident_ncard_le_degree G hloc x).trans (hd x)
+    _ = d*C.card := by simp [Nat.mul_comm]
+    _ ≤ d*(A.toLeft.card+B.card) := Nat.mul_le_mul_left d hc
+    _ = _ := by rw [hb]
+
+ 
+
+theorem subdivision_hV_pos [Nonempty V] (G : BondGraph V E) (hloc : LocallyFinite G)
+    (hqt : QuasiTransitive G) (hh : 0 < hV G) : 0 < hV (subdivision G) := by
+  classical
+  obtain ⟨d,hd,hdg⟩ := exists_degree_bound G hqt
+  let c := hV G / (2*((d:ℝ)+1)+(d:ℝ)*hV G)
+  have hden : 0 < 2*((d:ℝ)+1)+(d:ℝ)*hV G := by positivity
+  have hc : 0 < c := div_pos hh hden
+  apply hc.trans_le
+  apply le_csInf
+  · exact ⟨_,{Sum.inl (Classical.arbitrary V)},Finset.singleton_nonempty _,rfl⟩
+  · rintro r ⟨A,hA,rfl⟩
+    have ho := subdivision_old_boundary_bound G hloc A
+    have hn : (A.toRight.card:ℝ) ≤ (d:ℝ)*((A.toLeft.card:ℝ)+
+        ((vertexBoundary (subdivision G) A).ncard:ℝ)) := by
+      exact_mod_cast subdivision_new_card_bound G hloc d hdg A
+    have hs : (A.toLeft.card:ℝ)+(A.toRight.card:ℝ) = A.card := by
+      exact_mod_cast A.card_toLeft_add_card_toRight
+    have hcard : (0:ℝ) < A.card := Nat.cast_pos.mpr hA.card_pos
+    apply (le_div_iff₀ hcard).mpr
+    dsimp only [c]
+    rw [div_mul_eq_mul_div]
+    apply (div_le_iff₀ hden).mpr
+    have h₁ := mul_le_mul_of_nonneg_left ho (show 0 ≤ (d:ℝ)+1 by positivity)
+    have h₂ := mul_le_mul_of_nonneg_left hn hh.le
+    rw [←hs]
+    nlinarith only [h₁,h₂]
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Subdivision/Operator.lean b/lean/OAI/Probability/UnimodularPercolation/Subdivision/Operator.lean
index 345380eb45a604cbf47658ec12d8e50ce717df12..45229bb3362607e977b02c331d9814158c53e7a1 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Subdivision/Operator.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Subdivision/Operator.lean
@@ -1,115 +1,115 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Subdivision.Coupling
-
-namespace OAI
-
-/-! Operator bounds transferred through graph subdivision. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-
- 
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-open scoped ENNReal
-
-private theorem sum_inl_extend_le (f : Hilbert V) (s : Finset (V ⊕ E)) :
-    ∑ z ∈ s, ‖Sum.elim (fun x => f x) (fun _ : E => (0:ℂ)) z‖ ^ (2:ℝ≥0∞).toReal ≤
-      ‖f‖ ^ (2:ℝ≥0∞).toReal := by
-  classical
-  rw [Finset.sum_sum_eq_sum_toLeft_add_sum_toRight]
-  simp only [Sum.elim_inl,Sum.elim_inr,norm_zero,ENNReal.toReal_ofNat,Real.zero_rpow (by norm_num : (2:ℝ) ≠ 0),
-    Finset.sum_const_zero,add_zero]
-  exact lp.sum_rpow_le_norm_rpow (by norm_num : 0 < (2:ℝ≥0∞).toReal) f s.toLeft
-
-noncomputable def hilbertInlValue (f : Hilbert V) : Hilbert (V ⊕ E) :=
-  ⟨Sum.elim (fun x => f x) (fun _ => 0),memℓp_gen' (sum_inl_extend_le f)⟩
-
-theorem norm_hilbertInlValue_le (f : Hilbert V) : ‖hilbertInlValue (E:=E) f‖ ≤ ‖f‖ :=
-  lp.norm_le_of_forall_sum_le (by norm_num) (norm_nonneg f) (sum_inl_extend_le f)
-
-noncomputable def hilbertInl : Hilbert V →L[ℂ] Hilbert (V ⊕ E) :=
-  LinearMap.mkContinuous
-    { toFun := hilbertInlValue
-      map_add' := by
-        intro f g
-        apply lp.ext
-        funext z
-        cases z with
-        | inl x => rfl
-        | inr e => change (0:ℂ) = 0+0; simp
-      map_smul' := by
-        intro c f
-        apply lp.ext
-        funext z
-        cases z with
-        | inl x => rfl
-        | inr e => change (0:ℂ) = c • 0; simp }
-    1 (fun f => by simpa using norm_hilbertInlValue_le (E:=E) f)
-
-@[simp] theorem hilbertInl_apply_inl (f : Hilbert V) (x : V) :
-    hilbertInl (E:=E) f (.inl x) = f x := rfl
-@[simp] theorem hilbertInl_apply_inr (f : Hilbert V) (e : E) :
-    hilbertInl (E:=E) f (.inr e) = 0 := rfl
-
-private theorem sum_inl_restrict_le (f : Hilbert (V ⊕ E)) (s : Finset V) :
-    ∑ x ∈ s, ‖f (.inl x)‖ ^ (2:ℝ≥0∞).toReal ≤ ‖f‖ ^ (2:ℝ≥0∞).toReal := by
-  classical
-  rw [←Finset.sum_image (f := fun z => ‖f z‖ ^ (2:ℝ≥0∞).toReal) Sum.inl_injective.injOn]
-  exact lp.sum_rpow_le_norm_rpow (by norm_num) f _
-
-noncomputable def hilbertRestrictInlValue (f : Hilbert (V ⊕ E)) : Hilbert V :=
-  ⟨fun x => f (.inl x),memℓp_gen' (sum_inl_restrict_le f)⟩
-
-theorem norm_hilbertRestrictInlValue_le (f : Hilbert (V ⊕ E)) :
-    ‖hilbertRestrictInlValue f‖ ≤ ‖f‖ :=
-  lp.norm_le_of_forall_sum_le (by norm_num) (norm_nonneg f) (sum_inl_restrict_le f)
-
-noncomputable def hilbertRestrictInl : Hilbert (V ⊕ E) →L[ℂ] Hilbert V :=
-  LinearMap.mkContinuous
-    { toFun := hilbertRestrictInlValue
-      map_add' := by intro f g; apply lp.ext; rfl
-      map_smul' := by intro c f; apply lp.ext; rfl }
-    1 (fun f => by simpa using norm_hilbertRestrictInlValue_le f)
-
-@[simp] theorem hilbertRestrictInl_apply (f : Hilbert (V ⊕ E)) (x : V) :
-    hilbertRestrictInl f x = f (.inl x) := rfl
-
-@[simp] theorem hilbertInl_single [DecidableEq V] [DecidableEq E] (x : V) (c : ℂ) :
-    hilbertInl (E:=E) (lp.single 2 x c) = lp.single 2 (.inl x) c := by
-  apply lp.ext
-  funext z
-  cases z <;> simp [lp.single_apply,Pi.single_apply]
-
- 
-theorem operatorBounded_of_subdivision [Countable V] [Countable E]
-    (G : BondGraph V E) (q : unitInterval) (h : OperatorBounded (subdivision G) q) :
-    OperatorBounded G (q^2) := by
-  classical
-  obtain ⟨A,hA⟩ := h
-  refine ⟨hilbertRestrictInl ∘L A ∘L hilbertInl,?_⟩
-  intro x y
-  simp only [ContinuousLinearMap.comp_apply,hilbertInl_single,hilbertRestrictInl_apply]
-  convert (hA (.inl x) (.inl y)).trans (congrArg (fun r : ℝ => (r:ℂ))
-    (twoPoint_subdivision G q x y)) using 1
-  apply congrArg (fun f : Hilbert (V ⊕ E) => A f (.inl x))
-  apply lp.ext
-  funext z
-  simp [lp.single_apply,Pi.single_apply]
-
- 
-
-theorem critical_bounded_of_subdivision [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G)
-    (h : OperatorBounded (subdivision G) (pc (subdivision G))) :
-    OperatorBounded G (pc G) := by
-  simpa only [pc_subdivision_sq G hloc] using
-    operatorBounded_of_subdivision G (pc (subdivision G)) h
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Subdivision.Coupling
+
+namespace OAI
+
+/-! Operator bounds transferred through graph subdivision. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+
+ 
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+open scoped ENNReal
+
+private theorem sum_inl_extend_le (f : Hilbert V) (s : Finset (V ⊕ E)) :
+    ∑ z ∈ s, ‖Sum.elim (fun x => f x) (fun _ : E => (0:ℂ)) z‖ ^ (2:ℝ≥0∞).toReal ≤
+      ‖f‖ ^ (2:ℝ≥0∞).toReal := by
+  classical
+  rw [Finset.sum_sum_eq_sum_toLeft_add_sum_toRight]
+  simp only [Sum.elim_inl,Sum.elim_inr,norm_zero,ENNReal.toReal_ofNat,Real.zero_rpow (by norm_num : (2:ℝ) ≠ 0),
+    Finset.sum_const_zero,add_zero]
+  exact lp.sum_rpow_le_norm_rpow (by norm_num : 0 < (2:ℝ≥0∞).toReal) f s.toLeft
+
+noncomputable def hilbertInlValue (f : Hilbert V) : Hilbert (V ⊕ E) :=
+  ⟨Sum.elim (fun x => f x) (fun _ => 0),memℓp_gen' (sum_inl_extend_le f)⟩
+
+theorem norm_hilbertInlValue_le (f : Hilbert V) : ‖hilbertInlValue (E:=E) f‖ ≤ ‖f‖ :=
+  lp.norm_le_of_forall_sum_le (by norm_num) (norm_nonneg f) (sum_inl_extend_le f)
+
+noncomputable def hilbertInl : Hilbert V →L[ℂ] Hilbert (V ⊕ E) :=
+  LinearMap.mkContinuous
+    { toFun := hilbertInlValue
+      map_add' := by
+        intro f g
+        apply lp.ext
+        funext z
+        cases z with
+        | inl x => rfl
+        | inr e => change (0:ℂ) = 0+0; simp
+      map_smul' := by
+        intro c f
+        apply lp.ext
+        funext z
+        cases z with
+        | inl x => rfl
+        | inr e => change (0:ℂ) = c • 0; simp }
+    1 (fun f => by simpa using norm_hilbertInlValue_le (E:=E) f)
+
+@[simp] theorem hilbertInl_apply_inl (f : Hilbert V) (x : V) :
+    hilbertInl (E:=E) f (.inl x) = f x := rfl
+@[simp] theorem hilbertInl_apply_inr (f : Hilbert V) (e : E) :
+    hilbertInl (E:=E) f (.inr e) = 0 := rfl
+
+private theorem sum_inl_restrict_le (f : Hilbert (V ⊕ E)) (s : Finset V) :
+    ∑ x ∈ s, ‖f (.inl x)‖ ^ (2:ℝ≥0∞).toReal ≤ ‖f‖ ^ (2:ℝ≥0∞).toReal := by
+  classical
+  rw [←Finset.sum_image (f := fun z => ‖f z‖ ^ (2:ℝ≥0∞).toReal) Sum.inl_injective.injOn]
+  exact lp.sum_rpow_le_norm_rpow (by norm_num) f _
+
+noncomputable def hilbertRestrictInlValue (f : Hilbert (V ⊕ E)) : Hilbert V :=
+  ⟨fun x => f (.inl x),memℓp_gen' (sum_inl_restrict_le f)⟩
+
+theorem norm_hilbertRestrictInlValue_le (f : Hilbert (V ⊕ E)) :
+    ‖hilbertRestrictInlValue f‖ ≤ ‖f‖ :=
+  lp.norm_le_of_forall_sum_le (by norm_num) (norm_nonneg f) (sum_inl_restrict_le f)
+
+noncomputable def hilbertRestrictInl : Hilbert (V ⊕ E) →L[ℂ] Hilbert V :=
+  LinearMap.mkContinuous
+    { toFun := hilbertRestrictInlValue
+      map_add' := by intro f g; apply lp.ext; rfl
+      map_smul' := by intro c f; apply lp.ext; rfl }
+    1 (fun f => by simpa using norm_hilbertRestrictInlValue_le f)
+
+@[simp] theorem hilbertRestrictInl_apply (f : Hilbert (V ⊕ E)) (x : V) :
+    hilbertRestrictInl f x = f (.inl x) := rfl
+
+@[simp] theorem hilbertInl_single [DecidableEq V] [DecidableEq E] (x : V) (c : ℂ) :
+    hilbertInl (E:=E) (lp.single 2 x c) = lp.single 2 (.inl x) c := by
+  apply lp.ext
+  funext z
+  cases z <;> simp [lp.single_apply,Pi.single_apply]
+
+ 
+theorem operatorBounded_of_subdivision [Countable V] [Countable E]
+    (G : BondGraph V E) (q : unitInterval) (h : OperatorBounded (subdivision G) q) :
+    OperatorBounded G (q^2) := by
+  classical
+  obtain ⟨A,hA⟩ := h
+  refine ⟨hilbertRestrictInl ∘L A ∘L hilbertInl,?_⟩
+  intro x y
+  simp only [ContinuousLinearMap.comp_apply,hilbertInl_single,hilbertRestrictInl_apply]
+  convert (hA (.inl x) (.inl y)).trans (congrArg (fun r : ℝ => (r:ℂ))
+    (twoPoint_subdivision G q x y)) using 1
+  apply congrArg (fun f : Hilbert (V ⊕ E) => A f (.inl x))
+  apply lp.ext
+  funext z
+  simp [lp.single_apply,Pi.single_apply]
+
+ 
+
+theorem critical_bounded_of_subdivision [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G)
+    (h : OperatorBounded (subdivision G) (pc (subdivision G))) :
+    OperatorBounded G (pc G) := by
+  simpa only [pc_subdivision_sq G hloc] using
+    operatorBounded_of_subdivision G (pc (subdivision G)) h
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Transport/EndpointKernels.lean b/lean/OAI/Probability/UnimodularPercolation/Transport/EndpointKernels.lean
index ccd7cd64d3b5956e8e8f4ab959db8d15fed8966f..f322c2a670e2459cfb386ffef61107c8ac74f45a 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Transport/EndpointKernels.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Transport/EndpointKernels.lean
@@ -1,1226 +1,1226 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Transport.TrajectoryMass
-
-namespace OAI
-
-/-! Endpoint Kernels for bond percolation. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section SeparatedTracePair
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem finite_trace_pair_separated_le [Nonempty E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) (k n M : ℕ)
-    (Q : Finset E) (p : unitInterval)
-    (A₁ A₂ : Finset (SpliceState E)) (y₁ y₂ : (ℕ → Set E) → ℕ → V)
-    (m₁ m₂ : (ℕ → Set E) → ℝ) (hm₁ : ∀ η, 0 ≤ m₁ η) (hm₂ : ∀ η, 0 ≤ m₂ η)
-    (hA₁ : A₁ ⊆ finiteSpliceDomain Q (Finset.Icc 1 k))
-    (hA₂ : A₂ ⊆ finiteSpliceDomain Q (Finset.Icc 1 k))
-    (hbin : ∀ s ∈ A₁, (cluster G s.1 x).Finite ∧ n ≤ (cluster G s.1 x).ncard ∧
-      (cluster G s.1 x).ncard < 2*n)
-    (hcap₁ : ∀ s ∈ A₁, ∀ i, 1 ≤ i → i ≤ k → (cluster G (s.2 i) (y₁ s.2 i)).Finite)
-    (hcap₂ : ∀ s ∈ A₂, ∀ i, 1 ≤ i → i ≤ k → (cluster G (s.2 i) (y₂ s.2 i)).Finite ∧
-      (cluster G (s.2 i) (y₂ s.2 i)).ncard ≤ M) :
-    (∑ a : FiniteTraceIndex G x k Q A₁ (fun s => y₁ s.2),
-      ∑ b : FiniteTraceIndex G x k Q A₂ (fun s => y₂ s.2),
-      if (traceIndexOutput G x k Q A₁ (fun s => y₁ s.2) a).1 =
-          (traceIndexOutput G x k Q A₂ (fun s => y₂ s.2) b).1 ∧
-        a.2.val ≠ [] ∧ b.2.val ≠ [] ∧
-        GraphSeparated G
-          (cluster G (traceIndexOutput G x k Q A₁ (fun s => y₁ s.2) a).1 x \ cluster G a.1.val.1 x)
-          (cluster G (traceIndexOutput G x k Q A₁ (fun s => y₁ s.2) a).1 x \ cluster G b.1.val.1 x)
-      then jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p
-        (pairSpliceOutput (traceIndexOutput G x k Q A₁ (fun s => y₁ s.2) a)
-          (traceIndexOutput G x k Q A₂ (fun s => y₂ s.2) b)) *
-        (m₁ a.1.val.2 * traceIndexMass G x k Q A₁ (fun s => y₁ s.2) a) *
-        (m₂ b.1.val.2 * traceIndexMass G x k Q A₂ (fun s => y₂ s.2) b) else 0) ≤
-    ∑ ω : FiniteField Q, finiteFieldWeight Q p ω *
-      (if (cluster G (finiteFieldSet Q ω) x).Finite ∧
-        n-k*M ≤ (cluster G (finiteFieldSet Q ω) x).ncard ∧
-        (cluster G (finiteFieldSet Q ω) x).ncard < 2*n then
-        finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η => if
-          (cluster G (finiteFieldSet Q ω) x ∩ trajectoryUnion G η (y₁ η) k).Nonempty then m₁ η else 0) *
-        finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η => if
-          (cluster G (finiteFieldSet Q ω) x ∩ trajectoryUnion G η (y₂ η) k).Nonempty then m₂ η else 0) else 0) := by
-  let T₁ := FiniteTraceIndex G x k Q A₁ (fun s => y₁ s.2)
-  let T₂ := FiniteTraceIndex G x k Q A₂ (fun s => y₂ s.2)
-  let out₁ := traceIndexOutput G x k Q A₁ (fun s => y₁ s.2)
-  let out₂ := traceIndexOutput G x k Q A₂ (fun s => y₂ s.2)
-  let P (a : T₁ × T₂) := (out₁ a.1).1 = (out₂ a.2).1 ∧ a.1.2.val ≠ [] ∧ a.2.2.val ≠ [] ∧
-    GraphSeparated G (cluster G (out₁ a.1).1 x \ cluster G a.1.1.val.1 x)
-      (cluster G (out₁ a.1).1 x \ cluster G a.2.1.val.1 x)
-  let I := {a : T₁ × T₂ // P a}
-  let η₁ (a : I) := a.val.1.1.val.2
-  let η₂ (a : I) := a.val.2.1.val.2
-  let ω₁ (a : I) := a.val.1.1.val.1
-  let ω₂ (a : I) := a.val.2.1.val.1
-  let ωf (a : I) := (out₁ a.val.1).1
-  let c₁ (a : I) := a.val.1.2.val
-  let c₂ (a : I) := a.val.2.2.val
-  have hout (a : I) : jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a))
-      (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a) = pairSpliceOutput (out₁ a.val.1) (out₂ a.val.2) := rfl
-  have hinj : Function.Injective (fun a : I =>
-      (jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a),c₁ a,c₂ a)) := by
-    intro a b he
-    dsimp only at he
-    rw [hout a,hout b] at he
-    have ho₁ : out₁ a.val.1 = out₁ b.val.1 := Prod.ext
-      (congrArg (fun z : JointSpliceState E × List (ℕ × E) × List (ℕ × E) => z.1.1) he)
-      (congrArg (fun z : JointSpliceState E × List (ℕ × E) × List (ℕ × E) => z.1.2.1) he)
-    have ho₂ : out₂ a.val.2 = out₂ b.val.2 := Prod.ext
-      (a.property.1.symm.trans ((congrArg Prod.fst ho₁).trans b.property.1))
-      (congrArg (fun z : JointSpliceState E × List (ℕ × E) × List (ℕ × E) => z.1.2.2) he)
-    apply Subtype.ext
-    exact Prod.ext
-      (traceIndex_output_code_injective G x k Q A₁ (fun s => y₁ s.2)
-        (Prod.ext ho₁ (congrArg (fun z => z.2.1) he)))
-      (traceIndex_output_code_injective G x k Q A₂ (fun s => y₂ s.2)
-        (Prod.ext ho₂ (congrArg (fun z => z.2.2) he)))
-  have htr₁ (a : I) : SpliceTrace G x (η₁ a) (y₁ (η₁ a)) k 0 (ω₁ a) (c₁ a) (ωf a) :=
-    (traceIndex_valid G x k Q A₁ (fun s => y₁ s.2) a.val.1).1
-  have htr₂ (a : I) : SpliceTrace G x (η₂ a) (y₂ (η₂ a)) k 0 (ω₂ a) (c₂ a) (ωf a) := by
-    change SpliceTrace G x _ _ _ _ _ _ (out₁ a.val.1).1
-    rw [a.property.1]
-    exact (traceIndex_valid G x k Q A₂ (fun s => y₂ s.2) a.val.2).1
-  have hs₁ (a : I) := (mem_finiteSpliceDomain_iff Q (Finset.Icc 1 k) _).mp (hA₁ a.val.1.1.property)
-  have hs₂ (a : I) := (mem_finiteSpliceDomain_iff Q (Finset.Icc 1 k) _).mp (hA₂ a.val.2.1.property)
-  have hb := finite_separated_weighted_contact_second_moment_le G hloc x k n M p Q η₁ η₂ y₁ y₂
-    (fun _ => True) (fun _ => True) m₁ m₂ hm₁ hm₂ ω₁ ω₂ ωf c₁ c₂ hinj htr₁ htr₂
-    (fun a => a.property.2.1) (fun a => a.property.2.2.1) (fun a => a.property.2.2.2)
-    (fun a => hbin _ a.val.1.1.property) (fun a => hcap₁ _ a.val.1.1.property)
-    (fun a => hcap₂ _ a.val.2.1.property) (fun _ => trivial) (fun _ => trivial)
-    (fun a => (hs₁ a).1) (fun a => (hs₂ a).1) (fun a => (hs₁ a).2.1)
-    (fun a => (hs₂ a).2.1) (fun a => (hs₁ a).2.2) (fun a => (hs₂ a).2.2)
-  simp only [true_and] at hb
-  apply le_trans _ hb
-  let w (a : T₁ × T₂) := jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p
-    (pairSpliceOutput (out₁ a.1) (out₂ a.2)) *
-    (m₁ a.1.1.val.2 * traceIndexMass G x k Q A₁ (fun s => y₁ s.2) a.1) *
-    (m₂ a.2.1.val.2 * traceIndexMass G x k Q A₂ (fun s => y₂ s.2) a.2)
-  have he := Finset.sum_subtype (p := P) (F := inferInstance) (Finset.univ.filter P)
-    (by intro a; simp) w
-  rw [Finset.sum_filter,Fintype.sum_prod_type] at he
-  change (∑ a : T₁, ∑ b : T₂, if P (a,b) then w (a,b) else 0) ≤ _
-  rw [he]
-  apply Finset.sum_le_sum
-  intro a _
-  rw [hout a]
-  change w a.val ≤ _
-  simp only [w,traceIndexMass,η₁,η₂,ωf,c₁,c₂,out₁,out₂]
-  rw [← a.property.1]
-  exact le_of_eq (by ring)
-
-end SeparatedTracePair
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section ChoiceSeparatedMass
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {P₁ P₂ : Type*} [Fintype P₁] [Fintype P₂]
-
- 
-noncomputable def finiteChoiceSeparatedPairMass
-    (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E) (p : ℝ)
-    (A₁ : P₁ → Finset (SpliceState E)) (A₂ : P₂ → Finset (SpliceState E))
-    (y₁ : P₁ → (ℕ → Set E) → ℕ → V) (y₂ : P₂ → (ℕ → Set E) → ℕ → V)
-    (m₁ : P₁ → (ℕ → Set E) → ℝ) (m₂ : P₂ → (ℕ → Set E) → ℝ) : ℝ :=
-  ∑ a : FiniteChoiceTraceIndex G x k Q A₁ (fun r s => y₁ r s.2),
-    ∑ b : FiniteChoiceTraceIndex G x k Q A₂ (fun r s => y₂ r s.2),
-    if (choiceTraceOutput G x k Q A₁ (fun r s => y₁ r s.2) a).1 =
-        (choiceTraceOutput G x k Q A₂ (fun r s => y₂ r s.2) b).1 ∧
-      choiceTraceCode G x k Q A₁ (fun r s => y₁ r s.2) a ≠ [] ∧
-      choiceTraceCode G x k Q A₂ (fun r s => y₂ r s.2) b ≠ [] ∧
-      GraphSeparated G
-        (cluster G (choiceTraceOutput G x k Q A₁ (fun r s => y₁ r s.2) a).1 x \
-          cluster G (choiceTraceInput G x k Q A₁ (fun r s => y₁ r s.2) a).1 x)
-        (cluster G (choiceTraceOutput G x k Q A₁ (fun r s => y₁ r s.2) a).1 x \
-          cluster G (choiceTraceInput G x k Q A₂ (fun r s => y₂ r s.2) b).1 x)
-    then jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p
-      (pairSpliceOutput (choiceTraceOutput G x k Q A₁ (fun r s => y₁ r s.2) a)
-        (choiceTraceOutput G x k Q A₂ (fun r s => y₂ r s.2) b)) *
-      choiceTraceMass G x k Q A₁ (fun r s => y₁ r s.2) (fun s r => m₁ r s.2) a *
-      choiceTraceMass G x k Q A₂ (fun r s => y₂ r s.2) (fun s r => m₂ r s.2) b else 0
-
- 
-noncomputable def finiteChoiceHit (G : BondGraph V E) (k : ℕ) (Q : Finset E) (p : ℝ)
-    (y : P₁ → (ℕ → Set E) → ℕ → V) (m : P₁ → (ℕ → Set E) → ℝ) (B : Set V) : ℝ :=
-  ∑ r, finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η =>
-    if (B ∩ trajectoryUnion G η (y r η) k).Nonempty then m r η else 0)
-
- 
-
-theorem finiteChoiceSeparatedPairMass_le [Nonempty E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) (k n M : ℕ)
-    (Q : Finset E) (p : unitInterval)
-    (A₁ : P₁ → Finset (SpliceState E)) (A₂ : P₂ → Finset (SpliceState E))
-    (y₁ : P₁ → (ℕ → Set E) → ℕ → V) (y₂ : P₂ → (ℕ → Set E) → ℕ → V)
-    (m₁ : P₁ → (ℕ → Set E) → ℝ) (m₂ : P₂ → (ℕ → Set E) → ℝ)
-    (hm₁ : ∀ r η, 0 ≤ m₁ r η) (hm₂ : ∀ r η, 0 ≤ m₂ r η)
-    (hA₁ : ∀ r, A₁ r ⊆ finiteSpliceDomain Q (Finset.Icc 1 k))
-    (hA₂ : ∀ r, A₂ r ⊆ finiteSpliceDomain Q (Finset.Icc 1 k))
-    (hbin : ∀ r s, s ∈ A₁ r → (cluster G s.1 x).Finite ∧ n ≤ (cluster G s.1 x).ncard ∧
-      (cluster G s.1 x).ncard < 2*n)
-    (hcap₁ : ∀ r s, s ∈ A₁ r → ∀ i, 1 ≤ i → i ≤ k →
-      (cluster G (s.2 i) (y₁ r s.2 i)).Finite)
-    (hcap₂ : ∀ r s, s ∈ A₂ r → ∀ i, 1 ≤ i → i ≤ k →
-      (cluster G (s.2 i) (y₂ r s.2 i)).Finite ∧ (cluster G (s.2 i) (y₂ r s.2 i)).ncard ≤ M) :
-    finiteChoiceSeparatedPairMass G x k Q p A₁ A₂ y₁ y₂ m₁ m₂ ≤
-    ∑ ω : FiniteField Q, finiteFieldWeight Q p ω *
-      (if (cluster G (finiteFieldSet Q ω) x).Finite ∧
-        n-k*M ≤ (cluster G (finiteFieldSet Q ω) x).ncard ∧
-        (cluster G (finiteFieldSet Q ω) x).ncard < 2*n then
-        finiteChoiceHit G k Q p y₁ m₁ (cluster G (finiteFieldSet Q ω) x) *
-        finiteChoiceHit G k Q p y₂ m₂ (cluster G (finiteFieldSet Q ω) x) else 0) := by
-  let R (ω : FiniteField Q) := (cluster G (finiteFieldSet Q ω) x).Finite ∧
-    n-k*M ≤ (cluster G (finiteFieldSet Q ω) x).ncard ∧
-    (cluster G (finiteFieldSet Q ω) x).ncard < 2*n
-  let H₁ (r : P₁) (ω : FiniteField Q) := finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η =>
-    if (cluster G (finiteFieldSet Q ω) x ∩ trajectoryUnion G η (y₁ r η) k).Nonempty then m₁ r η else 0)
-  let H₂ (r : P₂) (ω : FiniteField Q) := finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η =>
-    if (cluster G (finiteFieldSet Q ω) x ∩ trajectoryUnion G η (y₂ r η) k).Nonempty then m₂ r η else 0)
-  have hb : finiteChoiceSeparatedPairMass G x k Q p A₁ A₂ y₁ y₂ m₁ m₂ ≤
-      ∑ r : P₁, ∑ s : P₂, ∑ ω : FiniteField Q, finiteFieldWeight Q p ω *
-        (if R ω then H₁ r ω * H₂ s ω else 0) := by
-    unfold finiteChoiceSeparatedPairMass
-    rw [Fintype.sum_sigma]
-    apply Finset.sum_le_sum
-    intro r _
-    conv_lhs => arg 2; ext a; rw [Fintype.sum_sigma]
-    conv_lhs => rw [Finset.sum_comm]
-    simp only [choiceTraceOutput,choiceTraceInput,choiceTraceCode,choiceTraceMass,
-      traceIndexInput,traceIndexCode]
-    apply Finset.sum_le_sum
-    intro s _
-    exact finite_trace_pair_separated_le G hloc x k n M Q p (A₁ r) (A₂ s)
-      (y₁ r) (y₂ s) (m₁ r) (m₂ s) (hm₁ r) (hm₂ s) (hA₁ r) (hA₂ s)
-      (hbin r) (hcap₁ r) (hcap₂ s)
-  apply hb.trans_eq
-  rw [Finset.sum_comm]
-  conv_lhs => arg 2; ext s; rw [Finset.sum_comm]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro ω _
-  change (∑ s : P₂, ∑ r : P₁, finiteFieldWeight Q p ω * (if R ω then H₁ r ω * H₂ s ω else 0)) =
-    finiteFieldWeight Q p ω * (if R ω then (∑ r : P₁, H₁ r ω) * (∑ s : P₂, H₂ s ω) else 0)
-  by_cases h : R ω
-  · simp only [ite_eq_left h,← Finset.mul_sum,← Finset.sum_mul]
-  · simp only [ite_eq_right h,mul_zero,Finset.sum_const_zero]
-
-end ChoiceSeparatedMass
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section LocalEndpointRule
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-structure LocalEndpointRule (G : BondGraph V E) (hloc : LocallyFinite G)
-    (O : Set V) (M : ℕ) (L : ℝ) where
-  mass : Set E → V → V → ℝ
-  radius : ℕ
-  nonneg : ∀ ω x y, 0 ≤ mass ω x y
-  row_le_one : ∀ ω x, (∑' y, mass ω x y) ≤ 1
-  support : ∀ ω x y, mass ω x y ≠ 0 → x ∈ O ∧ y ∈ O ∧ Conn G ω x y ∧
-    (cluster G ω x).Finite ∧ (cluster G ω x).ncard ≤ M ∧ GoodForScale G ω x L
-  locality : ∀ ω x y, mass ω x y =
-    mass (ω ∩ (incidentSupport G hloc (finite_reachBall hloc x radius).toFinset : Set E)) x y
-  covariance : ∀ (a : V ≃ V) (b : E ≃ E), IsAut G a b → ∀ ω x y,
-    mass (b ⁻¹' ω) x y = mass ω (a x) (a y)
-
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
-
- 
-
-theorem outside_ball_zero (q : LocalEndpointRule G hloc O M L) (ω : Set E) (x y : V)
-    (hy : y ∉ reachBall G x M) : q.mass ω x y = 0 := by
-  by_contra h
-  obtain ⟨_,_,hc,hfin,hcap,_⟩ := q.support ω x y h
-  exact hy (finite_cluster_subset_reachBall G ω x hfin M hcap hc)
-
-theorem summable_mass (q : LocalEndpointRule G hloc O M L) (ω : Set E) (x : V) :
-    Summable (q.mass ω x) := by
-  apply summable_of_ne_finset_zero (s := (finite_reachBall hloc x M).toFinset)
-  intro y hy
-  apply q.outside_ball_zero ω x y
-  simpa only [Set.Finite.mem_toFinset] using hy
-
-theorem mass_le_one (q : LocalEndpointRule G hloc O M L) (ω : Set E) (x y : V) :
-    q.mass ω x y ≤ 1 := by
-  have h := (q.summable_mass ω x).sum_le_tsum ({y} : Finset V) (fun z _ => q.nonneg ω x z)
-  simpa only [Finset.sum_singleton] using h.trans (q.row_le_one ω x)
-
- 
-theorem measurable_mass (q : LocalEndpointRule G hloc O M L) (x y : V) :
-    Measurable (fun ω => q.mass ω x y) := by
-  let Q := incidentSupport G hloc (finite_reachBall hloc x q.radius).toFinset
-  let f : Set (Q : Set E) → ℝ := fun η => q.mass (Subtype.val '' η) x y
-  have hm : Measurable f := measurable_of_finite f
-  have he : (fun ω => q.mass ω x y) = f ∘ (fun ω : Set E => Subtype.val ⁻¹' ω : Set E → Set (Q : Set E)) := by
-    funext ω
-    rw [q.locality ω x y]
-    congr 1
-    ext e
-    simp [Q,and_comm]
-  rw [he]
-  exact hm.comp (measurable_set_restrict (Q : Set E))
-
-theorem integrable_mass (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x y : V) :
-    Integrable (fun ω => q.mass ω x y) (law p) := by
-  apply Integrable.of_bound (q.measurable_mass x y).aestronglyMeasurable 1
-  exact ae_of_all _ fun ω => by simpa only [Real.norm_eq_abs,abs_of_nonneg (q.nonneg ω x y)] using q.mass_le_one ω x y
-
- 
-
-noncomputable def kernel (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x y : V) : ℝ :=
-  ∫ ω, q.mass ω x y ∂law p
-
-theorem kernel_nonneg (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x y : V) :
-    0 ≤ q.kernel p x y := integral_nonneg (fun ω => q.nonneg ω x y)
-
-theorem kernel_outside_ball_zero (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x y : V)
-    (hy : y ∉ reachBall G x M) : q.kernel p x y = 0 := by
-  simp only [kernel,q.outside_ball_zero _ x y hy,integral_zero]
-
-theorem summable_kernel (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x : V) :
-    Summable (q.kernel p x) := by
-  apply summable_of_ne_finset_zero (s := (finite_reachBall hloc x M).toFinset)
-  intro y hy
-  apply q.kernel_outside_ball_zero p x y
-  simpa only [Set.Finite.mem_toFinset] using hy
-
- 
-
-theorem kernel_row_integral (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x : V) :
-    (∑' y, q.kernel p x y) = ∫ ω, (∑' y, q.mass ω x y) ∂law p := by
-  let B := (finite_reachBall hloc x M).toFinset
-  have h₁ : (∑' y, q.kernel p x y) = ∑ y ∈ B, q.kernel p x y :=
-    tsum_eq_sum fun y hy => q.kernel_outside_ball_zero p x y (by simpa only [B,Set.Finite.mem_toFinset] using hy)
-  have h₂ (ω : Set E) : (∑' y, q.mass ω x y) = ∑ y ∈ B, q.mass ω x y :=
-    tsum_eq_sum fun y hy => q.outside_ball_zero ω x y (by simpa only [B,Set.Finite.mem_toFinset] using hy)
-  rw [h₁]
-  simp_rw [h₂]
-  exact (integral_finsetSum B (fun y _ => q.integrable_mass p x y)).symm
-
-theorem kernel_row_le_one (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x : V) :
-    (∑' y, q.kernel p x y) ≤ 1 := by
-  rw [q.kernel_row_integral p x]
-  have hi : Integrable (fun ω => ∑' y, q.mass ω x y) (law p) := by
-    let B := (finite_reachBall hloc x M).toFinset
-    have he (ω : Set E) : (∑' y, q.mass ω x y) = ∑ y ∈ B, q.mass ω x y :=
-      tsum_eq_sum fun y hy => q.outside_ball_zero ω x y (by simpa only [B,Set.Finite.mem_toFinset] using hy)
-    simp_rw [he]
-    exact integrable_finsetSum B (fun y _ => q.integrable_mass p x y)
-  calc
-    _ ≤ ∫ _, (1:ℝ) ∂law p := integral_mono hi (integrable_const _) (q.row_le_one · x)
-    _ = 1 := by simp
-
-end LocalEndpointRule
-end LocalEndpointRule
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section LocalEndpointKernel
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
-
- 
-
-theorem outside_reverse_ball_zero (q : LocalEndpointRule G hloc O M L) (ω : Set E) (x y : V)
-    (hx : x ∉ reachBall G y M) : q.mass ω x y = 0 := by
-  by_contra h
-  obtain ⟨_,_,hc,hfin,hcap,_⟩ := q.support ω x y h
-  have he := cluster_eq_of_conn G ω hc
-  rw [he] at hfin hcap
-  exact hx (finite_cluster_subset_reachBall G ω y hfin M hcap (conn_symm G ω hc))
-
-theorem kernel_outside_reverse_ball_zero (q : LocalEndpointRule G hloc O M L)
-    (p : unitInterval) (x y : V) (hx : x ∉ reachBall G y M) : q.kernel p x y = 0 := by
-  simp only [kernel,q.outside_reverse_ball_zero _ x y hx,integral_zero]
-
-theorem summable_kernel_column (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (y : V) :
-    Summable (fun x => q.kernel p x y) := by
-  apply summable_of_ne_finset_zero (s := (finite_reachBall hloc y M).toFinset)
-  intro x hx
-  apply q.kernel_outside_reverse_ball_zero p x y
-  simpa only [Set.Finite.mem_toFinset] using hx
-
-theorem mass_zero_of_start_not_mem (q : LocalEndpointRule G hloc O M L) (ω : Set E)
-    (x y : V) (hx : x ∉ O) : q.mass ω x y = 0 := by
-  by_contra h
-  exact hx (q.support ω x y h).1
-
-theorem mass_zero_of_end_not_mem (q : LocalEndpointRule G hloc O M L) (ω : Set E)
-    (x y : V) (hy : y ∉ O) : q.mass ω x y = 0 := by
-  by_contra h
-  exact hy (q.support ω x y h).2.1
-
-theorem kernel_zero_of_start_not_mem (q : LocalEndpointRule G hloc O M L) (p : unitInterval)
-    (x y : V) (hx : x ∉ O) : q.kernel p x y = 0 := by
-  simp only [kernel,q.mass_zero_of_start_not_mem _ x y hx,integral_zero]
-
-theorem kernel_zero_of_end_not_mem (q : LocalEndpointRule G hloc O M L) (p : unitInterval)
-    (x y : V) (hy : y ∉ O) : q.kernel p x y = 0 := by
-  simp only [kernel,q.mass_zero_of_end_not_mem _ x y hy,integral_zero]
-
- 
-
-theorem kernel_covariant (q : LocalEndpointRule G hloc O M L) (p : unitInterval)
-    (a : V ≃ V) (b : E ≃ E) (ha : IsAut G a b) (x y : V) :
-    q.kernel p (a x) (a y) = q.kernel p x y := by
-  have h := integral_map (μ := law p) (measurable_set_preimage b).aemeasurable
-    (q.measurable_mass x y).aestronglyMeasurable
-  rw [law_map_preimage_equiv p b] at h
-  simpa only [kernel,q.covariance a b ha] using h.symm
-
-noncomputable def successProbability (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x : V) : ℝ :=
-  ∑' y, q.kernel p x y
-
-theorem successProbability_nonneg (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x : V) :
-    0 ≤ q.successProbability p x := tsum_nonneg (q.kernel_nonneg p x)
-
-theorem successProbability_le_one (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x : V) :
-    q.successProbability p x ≤ 1 := q.kernel_row_le_one p x
-
-theorem successProbability_covariant (q : LocalEndpointRule G hloc O M L) (p : unitInterval)
-    (a : V ≃ V) (b : E ≃ E) (ha : IsAut G a b) (x : V) :
-    q.successProbability p (a x) = q.successProbability p x := by
-  unfold successProbability
-  rw [← a.tsum_eq (fun y => q.kernel p (a x) y)]
-  exact tsum_congr (q.kernel_covariant p a b ha x)
-
-theorem successProbability_orbit (q : LocalEndpointRule G hloc O M L) (p : unitInterval)
-    (o x : V) (hx : x ∈ vertexOrbit G o) : q.successProbability p x = q.successProbability p o := by
-  obtain ⟨a,b,ha,rfl⟩ := hx
-  exact q.successProbability_covariant p a b ha o
-
-theorem kernel_column_covariant (q : LocalEndpointRule G hloc O M L) (p : unitInterval)
-    (a : V ≃ V) (b : E ≃ E) (ha : IsAut G a b) (y : V) :
-    (∑' x, q.kernel p x (a y)) = ∑' x, q.kernel p x y := by
-  rw [← a.tsum_eq (fun x => q.kernel p x (a y))]
-  exact tsum_congr (fun x => q.kernel_covariant p a b ha x y)
-
-end LocalEndpointRule
-end LocalEndpointKernel
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section EndpointTransport
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
-
- 
-
-theorem kernel_column_orbit (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
-    (p : unitInterval) (y : V) (hy : y ∈ vertexOrbit G o) :
-    (∑' x, q.kernel p x y) = ∑' x, q.kernel p x o := by
-  obtain ⟨a,b,ha,rfl⟩ := hy
-  exact q.kernel_column_covariant p a b ha o
-
- 
-
-theorem kernel_column_eq_success {ι : Type*} [Fintype ι]
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (htr : RootTransport root weight Γ)
-    (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b) (hw : ∀ i, 0 ≤ weight i)
-    (i₀ : ι) (hw₀ : 0 < weight i₀)
-    (q : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) M L)
-    (p : unitInterval) (y : V) (hy : y ∈ vertexOrbit G (root i₀)) :
-    (∑' x, q.kernel p x y) = q.successProbability p (root i₀) := by
-  let O := vertexOrbit G (root i₀)
-  let w : ι → ℝ := fun i => if root i ∈ O then weight i else 0
-  have ho : root i₀ ∈ O := by
-    refine ⟨Equiv.refl V,Equiv.refl E,?_,rfl⟩
-    intro e
-    simp
-  have hw' (i : ι) : 0 ≤ w i := by
-    dsimp [w]
-    split_ifs
-    · exact hw i
-    · exact le_rfl
-  have hpos : 0 < ∑ i, w i := by
-    have hi : w i₀ ≤ ∑ i, w i := Finset.single_le_sum (fun i _ => hw' i) (Finset.mem_univ i₀)
-    have he : w i₀ = weight i₀ := ite_eq_left ho
-    rw [he] at hi
-    exact hw₀.trans_le hi
-  have ht := htr.real root weight Γ (q.kernel p)
-    (by intro a ha x z; obtain ⟨b,hb⟩ := hΓ a ha; exact q.kernel_covariant p a b hb x z)
-    (q.summable_kernel p) (q.summable_kernel_column p)
-  have hr (i : ι) : weight i * (∑' z, q.kernel p (root i) z) =
-      w i * q.successProbability p (root i₀) := by
-    by_cases hi : root i ∈ O
-    · have he := q.successProbability_orbit p (root i₀) (root i) hi
-      change weight i * q.successProbability p (root i) = _
-      rw [he]
-      simp only [w,ite_eq_left hi]
-    · simp only [w,ite_eq_right hi,zero_mul]
-      have he : (∑' z, q.kernel p (root i) z) = 0 := by
-        simp only [q.kernel_zero_of_start_not_mem p _ _ hi,tsum_zero]
-      rw [he,mul_zero]
-  have hc (i : ι) : weight i * (∑' x, q.kernel p x (root i)) =
-      w i * (∑' x, q.kernel p x (root i₀)) := by
-    by_cases hi : root i ∈ O
-    · rw [q.kernel_column_orbit (root i₀) p (root i) hi]
-      simp only [w,ite_eq_left hi]
-    · simp only [w,ite_eq_right hi,zero_mul]
-      have he : (∑' x, q.kernel p x (root i)) = 0 := by
-        simp only [q.kernel_zero_of_end_not_mem p _ _ hi,tsum_zero]
-      rw [he,mul_zero]
-  simp_rw [hr,hc] at ht
-  rw [← Finset.sum_mul,← Finset.sum_mul] at ht
-  rw [q.kernel_column_orbit (root i₀) p y hy]
-  exact (mul_left_cancel₀ (ne_of_gt hpos) ht).symm
-
- 
-theorem kernel_column_le_success {ι : Type*} [Fintype ι]
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (htr : RootTransport root weight Γ)
-    (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b) (hw : ∀ i, 0 ≤ weight i)
-    (i₀ : ι) (hw₀ : 0 < weight i₀)
-    (q : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) M L)
-    (p : unitInterval) (y : V) :
-    (∑' x, q.kernel p x y) ≤ q.successProbability p (root i₀) := by
-  by_cases hy : y ∈ vertexOrbit G (root i₀)
-  · exact (q.kernel_column_eq_success root weight Γ htr hΓ hw i₀ hw₀ p y hy).le
-  · simpa only [q.kernel_zero_of_end_not_mem p _ _ hy,tsum_zero] using
-      q.successProbability_nonneg p (root i₀)
-
-end LocalEndpointRule
-end EndpointTransport
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section EndpointSchur
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
-
-theorem kernel_row_le_success (o : V)
-    (q : LocalEndpointRule G hloc (vertexOrbit G o) M L) (p : unitInterval) (x : V) :
-    (∑' y, q.kernel p x y) ≤ q.successProbability p o := by
-  by_cases hx : x ∈ vertexOrbit G o
-  · exact (q.successProbability_orbit p o x hx).le
-  · simpa only [q.kernel_zero_of_start_not_mem p _ _ hx,tsum_zero] using
-      q.successProbability_nonneg p o
-
- 
-noncomputable def normalizedKernel (o : V)
-    (q : LocalEndpointRule G hloc (vertexOrbit G o) M L) (p : unitInterval) (x y : V) : ℝ :=
-  (q.successProbability p o)⁻¹ * q.kernel p x y
-
- 
-
-theorem exists_normalized_operator {ι : Type*} [Fintype ι]
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (htr : RootTransport root weight Γ)
-    (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b) (hw : ∀ i, 0 ≤ weight i)
-    (i₀ : ι) (hw₀ : 0 < weight i₀)
-    (q : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) M L)
-    (p : unitInterval) (hu : 0 < q.successProbability p (root i₀)) :
-    ∃ A : Hilbert V →L[ℂ] Hilbert V, 0 < ‖A‖ ∧ ‖A‖ ≤ 1 ∧
-      ∀ x y, A (lp.single 2 y (1:ℂ)) x = (q.normalizedKernel (root i₀) p x y : ℂ) := by
-  let u := q.successProbability p (root i₀)
-  have hu₀ : u ≠ 0 := ne_of_gt hu
-  have hn (x y : V) : 0 ≤ q.normalizedKernel (root i₀) p x y :=
-    mul_nonneg (inv_nonneg.mpr hu.le) (q.kernel_nonneg p x y)
-  have hr (x : V) (S : Finset V) : (∑ y ∈ S, q.normalizedKernel (root i₀) p x y) ≤ 1 := by
-    change (∑ y ∈ S, u⁻¹ * q.kernel p x y) ≤ 1
-    rw [← Finset.mul_sum]
-    calc
-      _ ≤ u⁻¹ * (∑' y, q.kernel p x y) := mul_le_mul_of_nonneg_left
-        ((q.summable_kernel p x).sum_le_tsum S (fun y _ => q.kernel_nonneg p x y)) (inv_nonneg.mpr hu.le)
-      _ ≤ u⁻¹ * u := mul_le_mul_of_nonneg_left
-        (q.kernel_row_le_success (root i₀) p x) (inv_nonneg.mpr hu.le)
-      _ = 1 := inv_mul_cancel₀ hu₀
-  have hc (y : V) (S : Finset V) : (∑ x ∈ S, q.normalizedKernel (root i₀) p x y) ≤ 1 := by
-    change (∑ x ∈ S, u⁻¹ * q.kernel p x y) ≤ 1
-    rw [← Finset.mul_sum]
-    calc
-      _ ≤ u⁻¹ * (∑' x, q.kernel p x y) := mul_le_mul_of_nonneg_left
-        ((q.summable_kernel_column p y).sum_le_tsum S (fun x _ => q.kernel_nonneg p x y)) (inv_nonneg.mpr hu.le)
-      _ ≤ u⁻¹ * u := mul_le_mul_of_nonneg_left
-        (q.kernel_column_le_success root weight Γ htr hΓ hw i₀ hw₀ p y) (inv_nonneg.mpr hu.le)
-      _ = 1 := inv_mul_cancel₀ hu₀
-  obtain ⟨A,hA,hK⟩ := exists_operator_of_row_column_bound (q.normalizedKernel (root i₀) p) 1 (by norm_num) hn hr hc
-  refine ⟨A,?_,hA,hK⟩
-  apply norm_pos_iff.mpr
-  intro ha
-  have hz (y : V) : q.kernel p (root i₀) y = 0 := by
-    have h := hK (root i₀) y
-    rw [ha] at h
-    have he : u⁻¹ * q.kernel p (root i₀) y = 0 := Complex.ofReal_eq_zero.mp h.symm
-    exact (mul_eq_zero.mp he).resolve_left (inv_ne_zero hu₀)
-  have he : u = 0 := by simp only [u,successProbability,hz,tsum_zero]
-  exact hu₀ he
-
-end LocalEndpointRule
-end EndpointSchur
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory ProbabilityTheory
-section FiniteLocalIntegral
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-noncomputable def finiteFieldEquiv (Q : Finset E) : FiniteField Q ≃ Set (Q : Set E) where
-  toFun ω := Subtype.val ⁻¹' finiteFieldSet Q ω
-  invFun η := ⟨η.toFinset.map ⟨Subtype.val,Subtype.val_injective⟩, by
-    apply Finset.mem_powerset.mpr
-    intro e he
-    obtain ⟨a,_,rfl⟩ := Finset.mem_map.mp he
-    exact a.property⟩
-  left_inv ω := by
-    apply Subtype.ext
-    apply Finset.coe_injective
-    ext e
-    simp only [Finset.coe_map,Set.mem_image,finiteFieldSet,Function.Embedding.coeFn_mk]
-    constructor
-    · rintro ⟨a,ha,rfl⟩
-      have ha' : a ∈ (Subtype.val ⁻¹' (ω.val : Set E) : Set (Q : Set E)) := Set.mem_toFinset.mp ha
-      exact ha'
-    · intro he
-      exact ⟨⟨e,Finset.mem_powerset.mp ω.property he⟩,Set.mem_toFinset.mpr he,rfl⟩
-  right_inv η := by
-    ext e
-    simp only [Set.mem_preimage,finiteFieldSet,Finset.coe_map,Set.coe_toFinset,
-      Set.mem_image,Function.Embedding.coeFn_mk,Subtype.val_injective.eq_iff,exists_eq_right]
-
-@[simp] theorem finiteFieldEquiv_apply (Q : Finset E) (ω : FiniteField Q) :
-    finiteFieldEquiv Q ω = Subtype.val ⁻¹' finiteFieldSet Q ω := rfl
-
-@[simp] theorem finiteFieldEquiv_image (Q : Finset E) (ω : FiniteField Q) :
-    Subtype.val '' finiteFieldEquiv Q ω = finiteFieldSet Q ω := by
-  rw [finiteFieldEquiv_apply]
-  apply Set.image_preimage_eq_of_subset
-  intro e he
-  exact ⟨⟨e,Finset.mem_powerset.mp ω.property he⟩,rfl⟩
-
- 
-
-theorem finiteFieldWeight_eq_atom (Q : Finset E) (p : unitInterval) (ω : FiniteField Q) :
-    finiteFieldWeight Q p ω = (law p : Measure (Set (Q : Set E))).real {finiteFieldEquiv Q ω} := by
-  have hn : (finiteFieldEquiv Q ω).ncard = ω.val.card := by
-    rw [finiteFieldEquiv_apply,Set.ncard_preimage_of_injective_subset_range Subtype.val_injective]
-    · exact Set.ncard_coe_finset _
-    · intro e he
-      exact ⟨⟨e,Finset.mem_powerset.mp ω.property he⟩,rfl⟩
-  rw [law,setBernoulli_real_singleton p (Set.subset_univ _) Set.finite_univ,
-    Set.ncard_sdiff' (Set.subset_univ _) Set.finite_univ,hn]
-  have hc : Fintype.card (Q : Set E) = Q.card := Fintype.card_coe Q
-  simp only [finiteFieldWeight,bernoulliWeight,Set.ncard_univ,Nat.card_eq_fintype_card,hc]
-
- 
-
-theorem integral_finite_local (Q : Finset E) (p : unitInterval) (f : Set E → ℝ)
-    (hf : ∀ ω, f ω = f (ω ∩ (Q : Set E))) :
-    (∫ ω, f ω ∂law p) = ∑ η : FiniteField Q, finiteFieldWeight Q p η * f (finiteFieldSet Q η) := by
-  let g : Set (Q : Set E) → ℝ := fun η => f (Subtype.val '' η)
-  have hmap := integral_map (μ := law p) (measurable_set_restrict (Q : Set E)).aemeasurable
-    (measurable_of_finite g).aestronglyMeasurable
-  rw [law_map_restrict p (Q : Set E)] at hmap
-  have he (ω : Set E) : g (Subtype.val ⁻¹' ω) = f ω := by
-    rw [hf ω]
-    dsimp [g]
-    congr 1
-    ext e
-    simp [and_comm]
-  simp_rw [he] at hmap
-  rw [← hmap,integral_fintype Integrable.of_finite]
-  rw [← (finiteFieldEquiv Q).sum_comp]
-  apply Finset.sum_congr rfl
-  intro η _
-  rw [← finiteFieldWeight_eq_atom]
-  simp only [g,finiteFieldEquiv_image,smul_eq_mul]
-
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
-
- 
-
-theorem kernel_finite_sum (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x y : V) :
-    let Q := incidentSupport G hloc (finite_reachBall hloc x q.radius).toFinset
-    q.kernel p x y = ∑ η : FiniteField Q, finiteFieldWeight Q p η * q.mass (finiteFieldSet Q η) x y :=
-  integral_finite_local _ p (fun ω => q.mass ω x y) (fun ω => q.locality ω x y)
-
-theorem continuous_kernel (q : LocalEndpointRule G hloc O M L) (x y : V) :
-    Continuous (fun p : unitInterval => q.kernel p x y) := by
-  simp_rw [q.kernel_finite_sum]
-  apply continuous_finsetSum
-  intro η _
-  unfold finiteFieldWeight bernoulliWeight
-  fun_prop
-
-theorem continuous_successProbability (q : LocalEndpointRule G hloc O M L) (x : V) :
-    Continuous (fun p : unitInterval => q.successProbability p x) := by
-  have he (p : unitInterval) : q.successProbability p x =
-      ∑ y ∈ (finite_reachBall hloc x M).toFinset, q.kernel p x y := by
-    apply tsum_eq_sum
-    intro y hy
-    exact q.kernel_outside_ball_zero p x y (by simpa only [Set.Finite.mem_toFinset] using hy)
-  simp_rw [he]
-  exact continuous_finsetSum _ (fun y _ => q.continuous_kernel x y)
-
-end LocalEndpointRule
-end FiniteLocalIntegral
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section GoodnessCovariance
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem internalEdges_aut_preimage (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (W : Set V) :
-    internalEdges G (a ⁻¹' W) = b ⁻¹' internalEdges G W := by
-  ext e
-  simp only [internalEdges,Set.mem_ofPred_eq,Set.mem_preimage,hab e,Sym2.mem_map]
-  constructor
-  · intro h y hy
-    obtain ⟨z,hz,rfl⟩ := hy
-    exact h z hz
-  · intro h z hz
-    exact h (a z) ⟨z,hz,rfl⟩
-
-theorem cutExists_aut_preimage (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (W : Set V) (e : E) :
-    CutExists G (a ⁻¹' W) e ↔ CutExists G W (b e) := by
-  constructor
-  · rintro ⟨u,v,he,hu,hv⟩
-    exact ⟨a u,a v,by rw [hab e,he,Sym2.map_mk],hu,hv⟩
-  · rintro ⟨u,v,he,hu,hv⟩
-    refine ⟨a.symm u,a.symm v,?_,by simpa using hu,by simpa using hv⟩
-    have h := congrArg (Sym2.map a.symm) he
-    simpa only [hab e,Sym2.map_map,Equiv.symm_comp_self,id_eq,Sym2.map_id',Sym2.map_mk] using h
-
-theorem contactSet_aut_preimage (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (ω : Set E) (W : Set V) :
-    contactSet G (b ⁻¹' ω) (a ⁻¹' W) = b ⁻¹' contactSet G ω W := by
-  ext e
-  simp only [contactSet,Set.mem_ofPred_eq,Set.mem_preimage,cutExists_aut_preimage G hab]
-
-theorem openMarkedPivotal_pair_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (ω : Set E) (x y : V) (e : E) :
-    OpenMarkedPivotal G (b ⁻¹' ω) {y} x e ↔
-      OpenMarkedPivotal G ω {a y} (a x) (b e) := by
-  have he : (b ⁻¹' ω) \ {e} = b ⁻¹' (ω \ {b e}) := by
-    ext f
-    simp only [Set.mem_sdiff,Set.mem_preimage,Set.mem_singleton_iff,b.injective.eq_iff]
-  simp only [OpenMarkedPivotal,MarkedEvent,Set.mem_singleton_iff,exists_eq_left,
-    Set.mem_preimage,he,conn_aut_iff G hab]
-
-theorem pivotalDistance_aut_preimage (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (ω : Set E) (x y : V) :
-    pivotalDistance G (b ⁻¹' ω) x y = pivotalDistance G ω (a x) (a y) := by
-  have he : {e | OpenMarkedPivotal G (b ⁻¹' ω) {y} x e} =
-      b ⁻¹' {e | OpenMarkedPivotal G ω {a y} (a x) e} := by
-    ext e
-    exact openMarkedPivotal_pair_aut G hab ω x y e
-  rw [pivotalDistance,he,Set.ncard_preimage_of_injective_subset_range b.injective]
-  · rfl
-  · exact fun e _ => b.surjective e
-
-theorem goodForScale_aut_pullback (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (ω : Set E) (x : V) (L : ℝ)
-    (hg : GoodForScale G ω (a x) L) : GoodForScale G (b ⁻¹' ω) x L := by
-  intro W hW hconn hpos u hu v hv
-  let W' := W.map a.toEmbedding
-  have hpre : a ⁻¹' (W' : Set V) = (W : Set V) := by
-    ext z
-    simp [W']
-  have hcfg : b ⁻¹' (ω ∩ internalEdges G (W' : Set V)) =
-      (b ⁻¹' ω) ∩ internalEdges G (W : Set V) := by
-    rw [Set.preimage_inter,← internalEdges_aut_preimage G hab,hpre]
-  have hcontact : (contactSet G (b ⁻¹' ω) (W : Set V)).ncard =
-      (contactSet G ω (W' : Set V)).ncard := by
-    rw [← hpre,contactSet_aut_preimage G hab,
-      Set.ncard_preimage_of_injective_subset_range b.injective]
-    exact fun e _ => b.surjective e
-  have hW' : (W' : Set V) ⊆ cluster G ω (a x) := by
-    intro z hz
-    obtain ⟨w,hw,rfl⟩ := Finset.mem_map.mp hz
-    exact (conn_aut_iff G hab ω x w).mp (hW hw)
-  have hc' : ∀ w ∈ W', ∀ z ∈ W', Conn G (ω ∩ internalEdges G (W' : Set V)) w z := by
-    intro w hw z hz
-    obtain ⟨u,hu,rfl⟩ := Finset.mem_map.mp hw
-    obtain ⟨v,hv,rfl⟩ := Finset.mem_map.mp hz
-    apply (conn_aut_iff G hab _ u v).mp
-    rw [hcfg]
-    exact hconn u hu v hv
-  have H := hg W' hW' hc' (by rwa [← hcontact]) (a u)
-    (Finset.mem_map.mpr ⟨u,hu,rfl⟩) (a v) (Finset.mem_map.mpr ⟨v,hv,rfl⟩)
-  rw [← pivotalDistance_aut_preimage G hab,hcfg,← hcontact] at H
-  exact H
-
-theorem goodForScale_aut_iff (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (ω : Set E) (x : V) (L : ℝ) :
-    GoodForScale G (b ⁻¹' ω) x L ↔ GoodForScale G ω (a x) L := by
-  constructor
-  · intro h
-    have H := goodForScale_aut_pullback G (isAut_symm G hab) (b ⁻¹' ω) (a x) L
-      (by simpa only [Equiv.symm_apply_apply] using h)
-    simpa only [← Set.preimage_comp,Equiv.self_comp_symm,Set.preimage_id_eq,id_eq] using H
-  · exact goodForScale_aut_pullback G hab ω x L
-
-end GoodnessCovariance
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section CappedGoodLocality
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem measurable_finite_local (Q : Finset E) (f : Set E → ℝ)
-    (hf : ∀ ω, f ω = f (ω ∩ (Q : Set E))) : Measurable f := by
-  let g : Set (Q : Set E) → ℝ := fun η => f (Subtype.val '' η)
-  have he : f = g ∘ (fun ω : Set E => (Subtype.val ⁻¹' ω : Set (Q : Set E))) := by
-    funext ω
-    rw [hf ω]
-    congr 1
-    ext e
-    simp [and_comm]
-  rw [he]
-  exact (measurable_of_finite g).comp (measurable_set_restrict (Q : Set E))
-
-theorem integrable_finite_local (Q : Finset E) (p : unitInterval) (f : Set E → ℝ)
-    (hf : ∀ ω, f ω = f (ω ∩ (Q : Set E))) : Integrable f (law p) := by
-  let g : Set (Q : Set E) → ℝ := fun η => f (Subtype.val '' η)
-  have he : f = g ∘ (fun ω : Set E => (Subtype.val ⁻¹' ω : Set (Q : Set E))) := by
-    funext ω
-    rw [hf ω]
-    congr 1
-    ext e
-    simp [and_comm]
-  have hi : Integrable g (Measure.map (fun ω : Set E => (Subtype.val ⁻¹' ω : Set (Q : Set E))) (law p)) := by
-    rw [law_map_restrict]
-    exact Integrable.of_finite
-  rw [he]
-  exact hi.comp_measurable (measurable_set_restrict (Q : Set E))
-
-def CappedGoodness (G : BondGraph V E) (ω : Set E) (x : V) (M : ℕ) (L : ℝ) : Prop :=
-  (cluster G ω x).Finite ∧ (cluster G ω x).ncard ≤ M ∧ GoodForScale G ω x L
-
-theorem cappedGoodness_local (G : BondGraph V E) (hloc : LocallyFinite G)
-    (ω : Set E) (x : V) (M : ℕ) (L : ℝ) :
-    let Q := incidentSupport G hloc (finite_reachBall hloc x M).toFinset
-    CappedGoodness G ω x M L ↔ CappedGoodness G (ω ∩ (Q : Set E)) x M L := by
-  have he (η : Set E) : CappedGoodness G η x M L ↔
-      ((cluster G η x).Finite ∧ (cluster G η x).ncard ≤ M) ∧ ¬CappedBadGoodness G η x M L := by
-    constructor
-    · rintro ⟨hf,hM,hg⟩
-      exact ⟨⟨hf,hM⟩,fun h => h.2.2 hg⟩
-    · rintro ⟨⟨hf,hM⟩,hbad⟩
-      refine ⟨hf,hM,?_⟩
-      by_contra hn
-      exact hbad ⟨hf,hM,hn⟩
-  dsimp only
-  have hbad := cappedBadGoodness_local G hloc x M L ω
-  have hcap := capped_cluster_local G hloc x M ω
-  dsimp only at hbad hcap
-  rw [he,he,hbad,hcap]
-
-theorem cappedGoodness_aut_iff (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (ω : Set E) (x : V) (M : ℕ) (L : ℝ) :
-    CappedGoodness G (b ⁻¹' ω) x M L ↔ CappedGoodness G ω (a x) M L := by
-  have hf : (cluster G (b ⁻¹' ω) x).Finite ↔ (cluster G ω (a x)).Finite := by
-    rw [← cluster_aut G hab ω x]
-    exact (Set.finite_image_iff a.injective.injOn).symm
-  have hn : (cluster G (b ⁻¹' ω) x).ncard = (cluster G ω (a x)).ncard := by
-    rw [← cluster_aut G hab ω x,Set.ncard_image_of_injective _ a.injective]
-  simp only [CappedGoodness,hf,hn,goodForScale_aut_iff G hab]
-
-theorem incidentSupport_ball_mono (G : BondGraph V E) (hloc : LocallyFinite G)
-    (x : V) {r R : ℕ} (h : r ≤ R) :
-    incidentSupport G hloc (finite_reachBall hloc x r).toFinset ⊆
-      incidentSupport G hloc (finite_reachBall hloc x R).toFinset := by
-  apply incidentSupport_mono
-  intro z hz
-  simp only [Set.Finite.mem_toFinset] at hz ⊢
-  exact reachBall_mono_local G x h hz
-
-theorem finite_local_of_subset (Q S : Finset E) (hQS : Q ⊆ S) (f : Set E → ℝ)
-    (hf : ∀ ω, f ω = f (ω ∩ (Q : Set E))) (ω : Set E) :
-    f ω = f (ω ∩ (S : Set E)) := by
-  rw [hf ω,hf (ω ∩ (S : Set E))]
-  congr 1
-  exact (Set.inter_assoc _ _ _).trans (congrArg (ω ∩ ·) (Set.inter_eq_right.mpr hQS)) |>.symm
-
-end CappedGoodLocality
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section CappedLocalLoad
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem finite_prop_local_of_subset (Q S : Finset E) (hQS : Q ⊆ S) (P : Set E → Prop)
-    (hf : ∀ ω, P ω ↔ P (ω ∩ (Q : Set E))) (ω : Set E) :
-    P ω ↔ P (ω ∩ (S : Set E)) := by
-  have he : (ω ∩ (S : Set E)) ∩ (Q : Set E) = ω ∩ (Q : Set E) :=
-    (Set.inter_assoc _ _ _).trans (congrArg (ω ∩ ·) (Set.inter_eq_right.mpr hQS))
-  exact (hf ω).trans ((he ▸ hf (ω ∩ (S : Set E))).symm)
-
- 
-
-structure CappedLocalLoad (G : BondGraph V E) (hloc : LocallyFinite G) (O : Set V) (M : ℕ) where
-  mass : Set E → V → V → ℝ
-  radius : ℕ
-  nonneg : ∀ ω x y, 0 ≤ mass ω x y
-  support : ∀ ω x y, mass ω x y ≠ 0 → x ∈ O ∧ y ∈ O ∧ Conn G ω x y ∧
-    (cluster G ω x).Finite ∧ (cluster G ω x).ncard ≤ M
-  locality : ∀ ω x y, mass ω x y =
-    mass (ω ∩ (incidentSupport G hloc (finite_reachBall hloc x radius).toFinset : Set E)) x y
-  covariance : ∀ (a : V ≃ V) (b : E ≃ E), IsAut G a b → ∀ ω x y,
-    mass (b ⁻¹' ω) x y = mass ω (a x) (a y)
-
-namespace CappedLocalLoad
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ}
-
-noncomputable def row (q : CappedLocalLoad G hloc O M) (ω : Set E) (x : V) : ℝ :=
-  ∑' y, q.mass ω x y
-
-theorem row_nonneg (q : CappedLocalLoad G hloc O M) (ω : Set E) (x : V) : 0 ≤ q.row ω x :=
-  tsum_nonneg (q.nonneg ω x)
-
-theorem outside_ball_zero (q : CappedLocalLoad G hloc O M) (ω : Set E) (x y : V)
-    (hy : y ∉ reachBall G x M) : q.mass ω x y = 0 := by
-  by_contra h
-  obtain ⟨_,_,hc,hfin,hcap⟩ := q.support ω x y h
-  exact hy (finite_cluster_subset_reachBall G ω x hfin M hcap hc)
-
-theorem summable_mass (q : CappedLocalLoad G hloc O M) (ω : Set E) (x : V) :
-    Summable (q.mass ω x) := by
-  apply summable_of_ne_finset_zero (s := (finite_reachBall hloc x M).toFinset)
-  intro y hy
-  exact q.outside_ball_zero ω x y (by simpa only [Set.Finite.mem_toFinset] using hy)
-
-theorem mass_le_row (q : CappedLocalLoad G hloc O M) (ω : Set E) (x y : V) :
-    q.mass ω x y ≤ q.row ω x := by
-  simpa only [Finset.sum_singleton,row] using
-    (q.summable_mass ω x).sum_le_tsum {y} (fun z _ => q.nonneg ω x z)
-
-theorem row_locality (q : CappedLocalLoad G hloc O M) (ω : Set E) (x : V) :
-    q.row ω x = q.row
-      (ω ∩ (incidentSupport G hloc (finite_reachBall hloc x q.radius).toFinset : Set E)) x :=
-  tsum_congr (q.locality ω x)
-
-theorem row_covariance (q : CappedLocalLoad G hloc O M) (a : V ≃ V) (b : E ≃ E)
-    (hab : IsAut G a b) (ω : Set E) (x : V) : q.row (b ⁻¹' ω) x = q.row ω (a x) := by
-  unfold row
-  simp_rw [q.covariance a b hab]
-  exact a.tsum_eq _
-
-theorem integrable_mass (q : CappedLocalLoad G hloc O M) (p : unitInterval) (x y : V) :
-    Integrable (fun ω => q.mass ω x y) (law p) :=
-  integrable_finite_local _ p _ (q.locality · x y)
-
-theorem integrable_row (q : CappedLocalLoad G hloc O M) (p : unitInterval) (x : V) :
-    Integrable (fun ω => q.row ω x) (law p) :=
-  integrable_finite_local _ p _ (q.row_locality · x)
-
-theorem integrable_sq_row (q : CappedLocalLoad G hloc O M) (p : unitInterval) (x : V) :
-    Integrable (fun ω => q.row ω x ^ 2) (law p) :=
-  integrable_finite_local _ p _ (fun ω => congrArg (· ^ 2) (q.row_locality ω x))
-
- 
-
-noncomputable def truncate (q : CappedLocalLoad G hloc O M) (b : ℝ) (hb : 0 < b) (L : ℝ) :
-    LocalEndpointRule G hloc O M L where
-  mass ω x y := if q.row ω x ≤ b ∧ CappedGoodness G ω x M L then b⁻¹ * q.mass ω x y else 0
-  radius := max q.radius M
-  nonneg ω x y := by split_ifs; exact mul_nonneg (inv_nonneg.mpr hb.le) (q.nonneg ω x y); rfl
-  row_le_one ω x := by
-    by_cases h : q.row ω x ≤ b ∧ CappedGoodness G ω x M L
-    · simp only [h,and_self,ite_true,tsum_mul_left]
-      exact (mul_le_mul_of_nonneg_left h.1 (inv_nonneg.mpr hb.le)).trans_eq (inv_mul_cancel₀ hb.ne')
-    · simp only [h,ite_false,tsum_zero,zero_le_one]
-  support ω x y h := by
-    split_ifs at h with hkeep
-    · have hm : q.mass ω x y ≠ 0 := fun he => h (by rw [he,mul_zero])
-      obtain ⟨hx,hy,hc,_,_⟩ := q.support ω x y hm
-      exact ⟨hx,hy,hc,hkeep.2⟩
-    · exact (h rfl).elim
-  locality ω x y := by
-    let S := incidentSupport G hloc (finite_reachBall hloc x (max q.radius M)).toFinset
-    have hm := finite_local_of_subset _ S
-      (incidentSupport_ball_mono G hloc x (le_max_left q.radius M))
-      (fun η => q.mass η x y) (q.locality · x y) ω
-    have hr := finite_local_of_subset _ S
-      (incidentSupport_ball_mono G hloc x (le_max_left q.radius M))
-      (fun η => q.row η x) (q.row_locality · x) ω
-    have hg := finite_prop_local_of_subset _ S
-      (incidentSupport_ball_mono G hloc x (le_max_right q.radius M))
-      (fun η => CappedGoodness G η x M L) (fun η => cappedGoodness_local G hloc η x M L) ω
-    simp only [hm,hr,hg]
-    rfl
-  covariance a c hac ω x y := by
-    simp only [q.row_covariance a c hac,q.covariance a c hac,cappedGoodness_aut_iff G hac]
-
-end CappedLocalLoad
-end CappedLocalLoad
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-open MeasureTheory
-section TruncationLoss
-attribute [local instance] Classical.propDecidable Classical.decEq
-namespace CappedLocalLoad
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ}
-
-theorem cap_of_row_ne_zero (q : CappedLocalLoad G hloc O M) (ω : Set E) (x : V)
-    (hr : q.row ω x ≠ 0) : (cluster G ω x).Finite ∧ (cluster G ω x).ncard ≤ M := by
-  obtain ⟨y,hy⟩ : ∃ y, q.mass ω x y ≠ 0 := by
-    by_contra hn
-    push Not at hn
-    exact hr (by simp only [row,hn,tsum_zero])
-  exact (q.support ω x y hy).2.2.2
-
-theorem truncate_scaled_row (q : CappedLocalLoad G hloc O M) (b : ℝ) (hb : 0 < b)
-    (L : ℝ) (ω : Set E) (x : V) :
-    b * (∑' y, (q.truncate b hb L).mass ω x y) =
-      if q.row ω x ≤ b ∧ CappedGoodness G ω x M L then q.row ω x else 0 := by
-  by_cases h : q.row ω x ≤ b ∧ CappedGoodness G ω x M L
-  · simp only [truncate,ite_eq_left h,tsum_mul_left,← mul_assoc,mul_inv_cancel₀ hb.ne',one_mul]
-    rfl
-  · simp only [truncate,ite_eq_right h,tsum_zero,mul_zero]
-
-theorem truncate_mass_le (q : CappedLocalLoad G hloc O M) (b : ℝ) (hb : 0 < b)
-    (L : ℝ) (ω : Set E) (x y : V) :
-    (q.truncate b hb L).mass ω x y ≤ b⁻¹ * q.mass ω x y := by
-  change (if _ then _ else _) ≤ _
-  split_ifs
-  · rfl
-  · exact mul_nonneg (inv_nonneg.mpr hb.le) (q.nonneg ω x y)
-
- 
-
-theorem truncate_pointwise_loss (q : CappedLocalLoad G hloc O M) (b : ℝ) (hb : 0 < b)
-    (L : ℝ) (ω : Set E) (x : V) :
-    q.row ω x ≤ b * (∑' y, (q.truncate b hb L).mass ω x y) + q.row ω x ^ 2 / b +
-      b * ({η | CappedBadGoodness G η x M L}.indicator (1 : Set E → ℝ) ω) := by
-  rw [q.truncate_scaled_row b hb L]
-  have hs : 0 ≤ q.row ω x ^ 2 / b := div_nonneg (sq_nonneg _) hb.le
-  have hi : 0 ≤ ({η | CappedBadGoodness G η x M L}.indicator (1 : Set E → ℝ) ω) := by
-    apply Set.indicator_nonneg
-    intro η _
-    norm_num
-  by_cases hr : q.row ω x ≤ b
-  · by_cases hg : CappedGoodness G ω x M L
-    · simp only [hr,hg,and_self,ite_true]
-      nlinarith
-    · simp only [hg,and_false,ite_false,zero_add]
-      by_cases hz : q.row ω x = 0
-      · rw [hz]
-        positivity
-      · have hc := q.cap_of_row_ne_zero ω x hz
-        have hbad : CappedBadGoodness G ω x M L := ⟨hc.1,hc.2,fun h => hg ⟨hc.1,hc.2,h⟩⟩
-        rw [Set.indicator_of_mem (show ω ∈ {η | CappedBadGoodness G η x M L} from hbad)]
-        change q.row ω x ≤ q.row ω x ^ 2 / b + b * 1
-        linarith
-  · simp only [hr,false_and,ite_false,zero_add]
-    have htail : q.row ω x ≤ q.row ω x ^ 2 / b := by
-      apply (le_div_iff₀ hb).mpr
-      have ht := mul_le_mul_of_nonneg_left (le_of_lt (lt_of_not_ge hr)) (q.row_nonneg ω x)
-      nlinarith
-    exact htail.trans (le_add_of_nonneg_right (mul_nonneg hb.le hi))
-
- 
-
-theorem truncate_mean_loss (q : CappedLocalLoad G hloc O M) (p : unitInterval)
-    (b : ℝ) (hb : 0 < b) (L : ℝ) (x : V) :
-    (∫ ω, q.row ω x ∂law p) ≤ b * (q.truncate b hb L).successProbability p x +
-      (∫ ω, q.row ω x ^ 2 ∂law p) / b + b * (law p).real {ω | CappedBadGoodness G ω x M L} := by
-  let t := q.truncate b hb L
-  have ht : Integrable (fun ω => ∑' y, t.mass ω x y) (law p) := by
-    apply integrable_finite_local
-      (incidentSupport G hloc (finite_reachBall hloc x t.radius).toFinset) p
-    intro ω
-    exact tsum_congr (t.locality ω x)
-  have hbad := measurableSet_cappedBadGoodness G hloc x M L
-  have hi : Integrable ({ω | CappedBadGoodness G ω x M L}.indicator (1 : Set E → ℝ)) (law p) :=
-    (integrable_const (1:ℝ)).indicator hbad
-  have h := integral_mono (q.integrable_row p x)
-    (((ht.const_mul b).add ((q.integrable_sq_row p x).div_const b)).add (hi.const_mul b))
-    (q.truncate_pointwise_loss b hb L · x)
-  dsimp only [Pi.add_apply] at h
-  have he₁ := integral_add ((ht.const_mul b).add ((q.integrable_sq_row p x).div_const b)) (hi.const_mul b)
-  have he₂ := integral_add (ht.const_mul b) ((q.integrable_sq_row p x).div_const b)
-  dsimp only [Pi.add_apply] at he₁ he₂
-  rw [he₁,he₂,integral_const_mul,integral_div,integral_const_mul,
-    integral_indicator_one hbad] at h
-  rw [← t.kernel_row_integral p x] at h
-  exact h
-
-theorem truncate_success_lower (q : CappedLocalLoad G hloc O M) (p : unitInterval)
-    (b : ℝ) (hb : 0 < b) (L : ℝ) (x : V) (a : ℝ)
-    (hmean : a ≤ ∫ ω, q.row ω x ∂law p)
-    (hmoment : (∫ ω, q.row ω x ^ 2 ∂law p) / b ≤ a/4)
-    (hbad : b * (law p).real {ω | CappedBadGoodness G ω x M L} ≤ a/4) :
-    a / (2*b) ≤ (q.truncate b hb L).successProbability p x := by
-  have hh := q.truncate_mean_loss p b hb L x
-  apply (div_le_iff₀ (mul_pos (by norm_num) hb)).mpr
-  nlinarith
-
-end CappedLocalLoad
-end TruncationLoss
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section CategoricalAmplificationInput
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-def amplificationVertices (B : Finset V) (k : ℕ) (z : ↥B) (r : Fin k → ↥B) : ℕ → V :=
-  fun i => (oldRunVertices k z r i).val
-
-namespace LocalEndpointRule
-variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
-
-noncomputable def amplificationWeight (q : LocalEndpointRule G hloc O M L)
-    (B : Finset V) (k : ℕ) (z : ↥B) (s : SpliceState E) (r : Fin k → ↥B) : ℝ :=
-  oldRunWeight (fun i a b => q.mass (s.2 i) a.val b.val) k z r
-
-theorem amplificationWeight_eq (q : LocalEndpointRule G hloc O M L)
-    (B : Finset V) (k : ℕ) (z : ↥B) (s : SpliceState E) (r : Fin k → ↥B) :
-    q.amplificationWeight B k z s r = oldPathWeight k q.mass (amplificationVertices B k z r) s.2 := by
-  exact (oldPathWeight_oldRunVertices k (fun ω a b => q.mass ω a.val b.val) s.2 z r).symm
-
-theorem amplificationWeight_nonneg (q : LocalEndpointRule G hloc O M L)
-    (B : Finset V) (k : ℕ) (z : ↥B) (s : SpliceState E) (r : Fin k → ↥B) :
-    0 ≤ q.amplificationWeight B k z s r :=
-  oldRunWeight_nonneg (fun i (a b : ↥B) => q.mass (s.2 i) a.val b.val)
-    (fun i a b => q.nonneg (s.2 i) a.val b.val) k z r
-
-theorem amplificationWeight_row (q : LocalEndpointRule G hloc O M L)
-    (B : Finset V) (k : ℕ) (z : ↥B) (s : SpliceState E) :
-    (∑ r, q.amplificationWeight B k z s r) ≤ 1 := by
-  have hr (i : ℕ) (a : ↥B) : (∑ b : ↥B, q.mass (s.2 i) a.val b.val) ≤ 1 := by
-    rw [Finset.sum_coe_sort B]
-    exact ((q.summable_mass _ _).sum_le_tsum B (fun y _ => q.nonneg _ _ y)).trans (q.row_le_one _ _)
-  simpa only [one_pow,amplificationWeight] using sum_oldRunWeight_le
-    (fun i (a b : ↥B) => q.mass (s.2 i) a.val b.val)
-    (fun _ a b => q.nonneg _ a.val b.val) 1 zero_le_one hr k z
-
- 
-
-noncomputable def amplificationInputs (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (z : ↥B) (r : Fin k → ↥B) : Finset (SpliceState E) :=
-  (finiteSpliceDomain Q (Finset.Icc 1 k)).filter fun s =>
-    x ∈ O ∧ z.val ∈ O ∧ n ≤ (cluster G s.1 x).ncard ∧ (cluster G s.1 x).ncard < 2*n ∧
-      amplificationVertices B k z r 0 ∈ cluster G s.1 x ∧ q.amplificationWeight B k z s r ≠ 0
-
-theorem mem_amplificationInputs (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (z : ↥B) (r : Fin k → ↥B) (s : SpliceState E) :
-    s ∈ q.amplificationInputs Q B x k n z r ↔
-      s ∈ finiteSpliceDomain Q (Finset.Icc 1 k) ∧
-      x ∈ O ∧ z.val ∈ O ∧ n ≤ (cluster G s.1 x).ncard ∧ (cluster G s.1 x).ncard < 2*n ∧
-      amplificationVertices B k z r 0 ∈ cluster G s.1 x ∧ q.amplificationWeight B k z s r ≠ 0 :=
-  Finset.mem_filter
-
-theorem amplificationInputs_subset (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (z : ↥B) (r : Fin k → ↥B) :
-    q.amplificationInputs Q B x k n z r ⊆ finiteSpliceDomain Q (Finset.Icc 1 k) :=
-  Finset.filter_subset _ _
-
-theorem amplificationInputs_chain (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (z : ↥B) (r : Fin k → ↥B) (s : SpliceState E)
-    (hs : s ∈ q.amplificationInputs Q B x k n z r) :
-    ∀ i, 1 ≤ i → i ≤ k →
-      Conn G (s.2 i) (amplificationVertices B k z r i) (amplificationVertices B k z r (i-1)) ∧
-      (cluster G (s.2 i) (amplificationVertices B k z r i)).Finite ∧
-      (cluster G (s.2 i) (amplificationVertices B k z r i)).ncard ≤ M ∧
-      GoodForScale G (s.2 i) (amplificationVertices B k z r i) L := by
-  apply oldPathWeight_support G k M L q.mass
-  · intro ω a b h
-    exact (q.support ω a b h).2.2
-  · exact (q.amplificationWeight_eq B k z s r) ▸ ((q.mem_amplificationInputs Q B x k n z r s).mp hs).2.2.2.2.2.2
-
- 
-
-theorem amplificationTrace_support (q : LocalEndpointRule G hloc O M L)
-    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (z : ↥B) (r : Fin k → ↥B) (s : SpliceState E)
-    (hs : s ∈ q.amplificationInputs Q B x k n z r) (c : List (ℕ × E)) (ωf : Set E)
-    (ht : SpliceTrace G x s.2 (amplificationVertices B k z r) k 0 s.1 c ωf) :
-    x ∈ O ∧ z.val ∈ O ∧ Conn G ωf x z.val ∧ (cluster G ωf x).Finite ∧
-      (cluster G ωf x).ncard ≤ 2*n+k*M := by
-  obtain ⟨hdom,hx,hz,_,hbin,_,_⟩ := (q.mem_amplificationInputs Q B x k n z r s).mp hs
-  have hc := q.amplificationInputs_chain Q B x k n z r s hs
-  have hroot : (cluster G s.1 x).Finite :=
-    cluster_finite_of_finite_edges G (Q.finite_toSet.subset
-      ((mem_finiteSpliceDomain_iff Q (Finset.Icc 1 k) s).mp hdom).1) x
-  obtain ⟨hf,hcap⟩ := trajectoryUnion_finite_cap G s.2 (amplificationVertices B k z r) k M
-    (fun i hi hik => ⟨(hc i hi hik).2.1,(hc i hi hik).2.2.1⟩)
-  have hsub : cluster G ωf x ⊆ cluster G s.1 x ∪ trajectoryUnion G s.2 (amplificationVertices B k z r) k := by
-    intro v hv
-    by_cases hv' : v ∈ cluster G s.1 x
-    · exact Or.inl hv'
-    · exact Or.inr (ht.added_subset_trajectory G x s.2 (amplificationVertices B k z r) k 0 s.1 ωf c ⟨hv,hv'⟩)
-  refine ⟨hx,hz,?_,(hroot.union hf).subset hsub,?_⟩
-  · simpa only [amplificationVertices,oldRunVertices_last,cluster,Set.mem_ofPred_eq] using
-      ht.endpoint G x s.2 (amplificationVertices B k z r) k 0 s.1 ωf c
-  · exact (Set.ncard_le_ncard hsub (hroot.union hf)).trans
-      ((Set.ncard_union_le ..).trans (Nat.add_le_add (by omega) hcap))
-
-end LocalEndpointRule
-end CategoricalAmplificationInput
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Transport.TrajectoryMass
+
+namespace OAI
+
+/-! Endpoint Kernels for bond percolation. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section SeparatedTracePair
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem finite_trace_pair_separated_le [Nonempty E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) (k n M : ℕ)
+    (Q : Finset E) (p : unitInterval)
+    (A₁ A₂ : Finset (SpliceState E)) (y₁ y₂ : (ℕ → Set E) → ℕ → V)
+    (m₁ m₂ : (ℕ → Set E) → ℝ) (hm₁ : ∀ η, 0 ≤ m₁ η) (hm₂ : ∀ η, 0 ≤ m₂ η)
+    (hA₁ : A₁ ⊆ finiteSpliceDomain Q (Finset.Icc 1 k))
+    (hA₂ : A₂ ⊆ finiteSpliceDomain Q (Finset.Icc 1 k))
+    (hbin : ∀ s ∈ A₁, (cluster G s.1 x).Finite ∧ n ≤ (cluster G s.1 x).ncard ∧
+      (cluster G s.1 x).ncard < 2*n)
+    (hcap₁ : ∀ s ∈ A₁, ∀ i, 1 ≤ i → i ≤ k → (cluster G (s.2 i) (y₁ s.2 i)).Finite)
+    (hcap₂ : ∀ s ∈ A₂, ∀ i, 1 ≤ i → i ≤ k → (cluster G (s.2 i) (y₂ s.2 i)).Finite ∧
+      (cluster G (s.2 i) (y₂ s.2 i)).ncard ≤ M) :
+    (∑ a : FiniteTraceIndex G x k Q A₁ (fun s => y₁ s.2),
+      ∑ b : FiniteTraceIndex G x k Q A₂ (fun s => y₂ s.2),
+      if (traceIndexOutput G x k Q A₁ (fun s => y₁ s.2) a).1 =
+          (traceIndexOutput G x k Q A₂ (fun s => y₂ s.2) b).1 ∧
+        a.2.val ≠ [] ∧ b.2.val ≠ [] ∧
+        GraphSeparated G
+          (cluster G (traceIndexOutput G x k Q A₁ (fun s => y₁ s.2) a).1 x \ cluster G a.1.val.1 x)
+          (cluster G (traceIndexOutput G x k Q A₁ (fun s => y₁ s.2) a).1 x \ cluster G b.1.val.1 x)
+      then jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p
+        (pairSpliceOutput (traceIndexOutput G x k Q A₁ (fun s => y₁ s.2) a)
+          (traceIndexOutput G x k Q A₂ (fun s => y₂ s.2) b)) *
+        (m₁ a.1.val.2 * traceIndexMass G x k Q A₁ (fun s => y₁ s.2) a) *
+        (m₂ b.1.val.2 * traceIndexMass G x k Q A₂ (fun s => y₂ s.2) b) else 0) ≤
+    ∑ ω : FiniteField Q, finiteFieldWeight Q p ω *
+      (if (cluster G (finiteFieldSet Q ω) x).Finite ∧
+        n-k*M ≤ (cluster G (finiteFieldSet Q ω) x).ncard ∧
+        (cluster G (finiteFieldSet Q ω) x).ncard < 2*n then
+        finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η => if
+          (cluster G (finiteFieldSet Q ω) x ∩ trajectoryUnion G η (y₁ η) k).Nonempty then m₁ η else 0) *
+        finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η => if
+          (cluster G (finiteFieldSet Q ω) x ∩ trajectoryUnion G η (y₂ η) k).Nonempty then m₂ η else 0) else 0) := by
+  let T₁ := FiniteTraceIndex G x k Q A₁ (fun s => y₁ s.2)
+  let T₂ := FiniteTraceIndex G x k Q A₂ (fun s => y₂ s.2)
+  let out₁ := traceIndexOutput G x k Q A₁ (fun s => y₁ s.2)
+  let out₂ := traceIndexOutput G x k Q A₂ (fun s => y₂ s.2)
+  let P (a : T₁ × T₂) := (out₁ a.1).1 = (out₂ a.2).1 ∧ a.1.2.val ≠ [] ∧ a.2.2.val ≠ [] ∧
+    GraphSeparated G (cluster G (out₁ a.1).1 x \ cluster G a.1.1.val.1 x)
+      (cluster G (out₁ a.1).1 x \ cluster G a.2.1.val.1 x)
+  let I := {a : T₁ × T₂ // P a}
+  let η₁ (a : I) := a.val.1.1.val.2
+  let η₂ (a : I) := a.val.2.1.val.2
+  let ω₁ (a : I) := a.val.1.1.val.1
+  let ω₂ (a : I) := a.val.2.1.val.1
+  let ωf (a : I) := (out₁ a.val.1).1
+  let c₁ (a : I) := a.val.1.2.val
+  let c₂ (a : I) := a.val.2.2.val
+  have hout (a : I) : jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a))
+      (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a) = pairSpliceOutput (out₁ a.val.1) (out₂ a.val.2) := rfl
+  have hinj : Function.Injective (fun a : I =>
+      (jointOutput G x (η₁ a) (η₂ a) (y₁ (η₁ a)) (y₂ (η₂ a)) (ω₁ a) (ω₂ a) (ωf a) (c₁ a) (c₂ a),c₁ a,c₂ a)) := by
+    intro a b he
+    dsimp only at he
+    rw [hout a,hout b] at he
+    have ho₁ : out₁ a.val.1 = out₁ b.val.1 := Prod.ext
+      (congrArg (fun z : JointSpliceState E × List (ℕ × E) × List (ℕ × E) => z.1.1) he)
+      (congrArg (fun z : JointSpliceState E × List (ℕ × E) × List (ℕ × E) => z.1.2.1) he)
+    have ho₂ : out₂ a.val.2 = out₂ b.val.2 := Prod.ext
+      (a.property.1.symm.trans ((congrArg Prod.fst ho₁).trans b.property.1))
+      (congrArg (fun z : JointSpliceState E × List (ℕ × E) × List (ℕ × E) => z.1.2.2) he)
+    apply Subtype.ext
+    exact Prod.ext
+      (traceIndex_output_code_injective G x k Q A₁ (fun s => y₁ s.2)
+        (Prod.ext ho₁ (congrArg (fun z => z.2.1) he)))
+      (traceIndex_output_code_injective G x k Q A₂ (fun s => y₂ s.2)
+        (Prod.ext ho₂ (congrArg (fun z => z.2.2) he)))
+  have htr₁ (a : I) : SpliceTrace G x (η₁ a) (y₁ (η₁ a)) k 0 (ω₁ a) (c₁ a) (ωf a) :=
+    (traceIndex_valid G x k Q A₁ (fun s => y₁ s.2) a.val.1).1
+  have htr₂ (a : I) : SpliceTrace G x (η₂ a) (y₂ (η₂ a)) k 0 (ω₂ a) (c₂ a) (ωf a) := by
+    change SpliceTrace G x _ _ _ _ _ _ (out₁ a.val.1).1
+    rw [a.property.1]
+    exact (traceIndex_valid G x k Q A₂ (fun s => y₂ s.2) a.val.2).1
+  have hs₁ (a : I) := (mem_finiteSpliceDomain_iff Q (Finset.Icc 1 k) _).mp (hA₁ a.val.1.1.property)
+  have hs₂ (a : I) := (mem_finiteSpliceDomain_iff Q (Finset.Icc 1 k) _).mp (hA₂ a.val.2.1.property)
+  have hb := finite_separated_weighted_contact_second_moment_le G hloc x k n M p Q η₁ η₂ y₁ y₂
+    (fun _ => True) (fun _ => True) m₁ m₂ hm₁ hm₂ ω₁ ω₂ ωf c₁ c₂ hinj htr₁ htr₂
+    (fun a => a.property.2.1) (fun a => a.property.2.2.1) (fun a => a.property.2.2.2)
+    (fun a => hbin _ a.val.1.1.property) (fun a => hcap₁ _ a.val.1.1.property)
+    (fun a => hcap₂ _ a.val.2.1.property) (fun _ => trivial) (fun _ => trivial)
+    (fun a => (hs₁ a).1) (fun a => (hs₂ a).1) (fun a => (hs₁ a).2.1)
+    (fun a => (hs₂ a).2.1) (fun a => (hs₁ a).2.2) (fun a => (hs₂ a).2.2)
+  simp only [true_and] at hb
+  apply le_trans _ hb
+  let w (a : T₁ × T₂) := jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p
+    (pairSpliceOutput (out₁ a.1) (out₂ a.2)) *
+    (m₁ a.1.1.val.2 * traceIndexMass G x k Q A₁ (fun s => y₁ s.2) a.1) *
+    (m₂ a.2.1.val.2 * traceIndexMass G x k Q A₂ (fun s => y₂ s.2) a.2)
+  have he := Finset.sum_subtype (p := P) (F := inferInstance) (Finset.univ.filter P)
+    (by intro a; simp) w
+  rw [Finset.sum_filter,Fintype.sum_prod_type] at he
+  change (∑ a : T₁, ∑ b : T₂, if P (a,b) then w (a,b) else 0) ≤ _
+  rw [he]
+  apply Finset.sum_le_sum
+  intro a _
+  rw [hout a]
+  change w a.val ≤ _
+  simp only [w,traceIndexMass,η₁,η₂,ωf,c₁,c₂,out₁,out₂]
+  rw [← a.property.1]
+  exact le_of_eq (by ring)
+
+end SeparatedTracePair
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section ChoiceSeparatedMass
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {P₁ P₂ : Type*} [Fintype P₁] [Fintype P₂]
+
+ 
+noncomputable def finiteChoiceSeparatedPairMass
+    (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E) (p : ℝ)
+    (A₁ : P₁ → Finset (SpliceState E)) (A₂ : P₂ → Finset (SpliceState E))
+    (y₁ : P₁ → (ℕ → Set E) → ℕ → V) (y₂ : P₂ → (ℕ → Set E) → ℕ → V)
+    (m₁ : P₁ → (ℕ → Set E) → ℝ) (m₂ : P₂ → (ℕ → Set E) → ℝ) : ℝ :=
+  ∑ a : FiniteChoiceTraceIndex G x k Q A₁ (fun r s => y₁ r s.2),
+    ∑ b : FiniteChoiceTraceIndex G x k Q A₂ (fun r s => y₂ r s.2),
+    if (choiceTraceOutput G x k Q A₁ (fun r s => y₁ r s.2) a).1 =
+        (choiceTraceOutput G x k Q A₂ (fun r s => y₂ r s.2) b).1 ∧
+      choiceTraceCode G x k Q A₁ (fun r s => y₁ r s.2) a ≠ [] ∧
+      choiceTraceCode G x k Q A₂ (fun r s => y₂ r s.2) b ≠ [] ∧
+      GraphSeparated G
+        (cluster G (choiceTraceOutput G x k Q A₁ (fun r s => y₁ r s.2) a).1 x \
+          cluster G (choiceTraceInput G x k Q A₁ (fun r s => y₁ r s.2) a).1 x)
+        (cluster G (choiceTraceOutput G x k Q A₁ (fun r s => y₁ r s.2) a).1 x \
+          cluster G (choiceTraceInput G x k Q A₂ (fun r s => y₂ r s.2) b).1 x)
+    then jointTupleWeight Q (Finset.Icc 1 k) (Finset.Icc 1 k) p
+      (pairSpliceOutput (choiceTraceOutput G x k Q A₁ (fun r s => y₁ r s.2) a)
+        (choiceTraceOutput G x k Q A₂ (fun r s => y₂ r s.2) b)) *
+      choiceTraceMass G x k Q A₁ (fun r s => y₁ r s.2) (fun s r => m₁ r s.2) a *
+      choiceTraceMass G x k Q A₂ (fun r s => y₂ r s.2) (fun s r => m₂ r s.2) b else 0
+
+ 
+noncomputable def finiteChoiceHit (G : BondGraph V E) (k : ℕ) (Q : Finset E) (p : ℝ)
+    (y : P₁ → (ℕ → Set E) → ℕ → V) (m : P₁ → (ℕ → Set E) → ℝ) (B : Set V) : ℝ :=
+  ∑ r, finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η =>
+    if (B ∩ trajectoryUnion G η (y r η) k).Nonempty then m r η else 0)
+
+ 
+
+theorem finiteChoiceSeparatedPairMass_le [Nonempty E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) (k n M : ℕ)
+    (Q : Finset E) (p : unitInterval)
+    (A₁ : P₁ → Finset (SpliceState E)) (A₂ : P₂ → Finset (SpliceState E))
+    (y₁ : P₁ → (ℕ → Set E) → ℕ → V) (y₂ : P₂ → (ℕ → Set E) → ℕ → V)
+    (m₁ : P₁ → (ℕ → Set E) → ℝ) (m₂ : P₂ → (ℕ → Set E) → ℝ)
+    (hm₁ : ∀ r η, 0 ≤ m₁ r η) (hm₂ : ∀ r η, 0 ≤ m₂ r η)
+    (hA₁ : ∀ r, A₁ r ⊆ finiteSpliceDomain Q (Finset.Icc 1 k))
+    (hA₂ : ∀ r, A₂ r ⊆ finiteSpliceDomain Q (Finset.Icc 1 k))
+    (hbin : ∀ r s, s ∈ A₁ r → (cluster G s.1 x).Finite ∧ n ≤ (cluster G s.1 x).ncard ∧
+      (cluster G s.1 x).ncard < 2*n)
+    (hcap₁ : ∀ r s, s ∈ A₁ r → ∀ i, 1 ≤ i → i ≤ k →
+      (cluster G (s.2 i) (y₁ r s.2 i)).Finite)
+    (hcap₂ : ∀ r s, s ∈ A₂ r → ∀ i, 1 ≤ i → i ≤ k →
+      (cluster G (s.2 i) (y₂ r s.2 i)).Finite ∧ (cluster G (s.2 i) (y₂ r s.2 i)).ncard ≤ M) :
+    finiteChoiceSeparatedPairMass G x k Q p A₁ A₂ y₁ y₂ m₁ m₂ ≤
+    ∑ ω : FiniteField Q, finiteFieldWeight Q p ω *
+      (if (cluster G (finiteFieldSet Q ω) x).Finite ∧
+        n-k*M ≤ (cluster G (finiteFieldSet Q ω) x).ncard ∧
+        (cluster G (finiteFieldSet Q ω) x).ncard < 2*n then
+        finiteChoiceHit G k Q p y₁ m₁ (cluster G (finiteFieldSet Q ω) x) *
+        finiteChoiceHit G k Q p y₂ m₂ (cluster G (finiteFieldSet Q ω) x) else 0) := by
+  let R (ω : FiniteField Q) := (cluster G (finiteFieldSet Q ω) x).Finite ∧
+    n-k*M ≤ (cluster G (finiteFieldSet Q ω) x).ncard ∧
+    (cluster G (finiteFieldSet Q ω) x).ncard < 2*n
+  let H₁ (r : P₁) (ω : FiniteField Q) := finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η =>
+    if (cluster G (finiteFieldSet Q ω) x ∩ trajectoryUnion G η (y₁ r η) k).Nonempty then m₁ r η else 0)
+  let H₂ (r : P₂) (ω : FiniteField Q) := finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η =>
+    if (cluster G (finiteFieldSet Q ω) x ∩ trajectoryUnion G η (y₂ r η) k).Nonempty then m₂ r η else 0)
+  have hb : finiteChoiceSeparatedPairMass G x k Q p A₁ A₂ y₁ y₂ m₁ m₂ ≤
+      ∑ r : P₁, ∑ s : P₂, ∑ ω : FiniteField Q, finiteFieldWeight Q p ω *
+        (if R ω then H₁ r ω * H₂ s ω else 0) := by
+    unfold finiteChoiceSeparatedPairMass
+    rw [Fintype.sum_sigma]
+    apply Finset.sum_le_sum
+    intro r _
+    conv_lhs => arg 2; ext a; rw [Fintype.sum_sigma]
+    conv_lhs => rw [Finset.sum_comm]
+    simp only [choiceTraceOutput,choiceTraceInput,choiceTraceCode,choiceTraceMass,
+      traceIndexInput,traceIndexCode]
+    apply Finset.sum_le_sum
+    intro s _
+    exact finite_trace_pair_separated_le G hloc x k n M Q p (A₁ r) (A₂ s)
+      (y₁ r) (y₂ s) (m₁ r) (m₂ s) (hm₁ r) (hm₂ s) (hA₁ r) (hA₂ s)
+      (hbin r) (hcap₁ r) (hcap₂ s)
+  apply hb.trans_eq
+  rw [Finset.sum_comm]
+  conv_lhs => arg 2; ext s; rw [Finset.sum_comm]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro ω _
+  change (∑ s : P₂, ∑ r : P₁, finiteFieldWeight Q p ω * (if R ω then H₁ r ω * H₂ s ω else 0)) =
+    finiteFieldWeight Q p ω * (if R ω then (∑ r : P₁, H₁ r ω) * (∑ s : P₂, H₂ s ω) else 0)
+  by_cases h : R ω
+  · simp only [ite_eq_left h,← Finset.mul_sum,← Finset.sum_mul]
+  · simp only [ite_eq_right h,mul_zero,Finset.sum_const_zero]
+
+end ChoiceSeparatedMass
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section LocalEndpointRule
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+structure LocalEndpointRule (G : BondGraph V E) (hloc : LocallyFinite G)
+    (O : Set V) (M : ℕ) (L : ℝ) where
+  mass : Set E → V → V → ℝ
+  radius : ℕ
+  nonneg : ∀ ω x y, 0 ≤ mass ω x y
+  row_le_one : ∀ ω x, (∑' y, mass ω x y) ≤ 1
+  support : ∀ ω x y, mass ω x y ≠ 0 → x ∈ O ∧ y ∈ O ∧ Conn G ω x y ∧
+    (cluster G ω x).Finite ∧ (cluster G ω x).ncard ≤ M ∧ GoodForScale G ω x L
+  locality : ∀ ω x y, mass ω x y =
+    mass (ω ∩ (incidentSupport G hloc (finite_reachBall hloc x radius).toFinset : Set E)) x y
+  covariance : ∀ (a : V ≃ V) (b : E ≃ E), IsAut G a b → ∀ ω x y,
+    mass (b ⁻¹' ω) x y = mass ω (a x) (a y)
+
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
+
+ 
+
+theorem outside_ball_zero (q : LocalEndpointRule G hloc O M L) (ω : Set E) (x y : V)
+    (hy : y ∉ reachBall G x M) : q.mass ω x y = 0 := by
+  by_contra h
+  obtain ⟨_,_,hc,hfin,hcap,_⟩ := q.support ω x y h
+  exact hy (finite_cluster_subset_reachBall G ω x hfin M hcap hc)
+
+theorem summable_mass (q : LocalEndpointRule G hloc O M L) (ω : Set E) (x : V) :
+    Summable (q.mass ω x) := by
+  apply summable_of_ne_finset_zero (s := (finite_reachBall hloc x M).toFinset)
+  intro y hy
+  apply q.outside_ball_zero ω x y
+  simpa only [Set.Finite.mem_toFinset] using hy
+
+theorem mass_le_one (q : LocalEndpointRule G hloc O M L) (ω : Set E) (x y : V) :
+    q.mass ω x y ≤ 1 := by
+  have h := (q.summable_mass ω x).sum_le_tsum ({y} : Finset V) (fun z _ => q.nonneg ω x z)
+  simpa only [Finset.sum_singleton] using h.trans (q.row_le_one ω x)
+
+ 
+theorem measurable_mass (q : LocalEndpointRule G hloc O M L) (x y : V) :
+    Measurable (fun ω => q.mass ω x y) := by
+  let Q := incidentSupport G hloc (finite_reachBall hloc x q.radius).toFinset
+  let f : Set (Q : Set E) → ℝ := fun η => q.mass (Subtype.val '' η) x y
+  have hm : Measurable f := measurable_of_finite f
+  have he : (fun ω => q.mass ω x y) = f ∘ (fun ω : Set E => Subtype.val ⁻¹' ω : Set E → Set (Q : Set E)) := by
+    funext ω
+    rw [q.locality ω x y]
+    congr 1
+    ext e
+    simp [Q,and_comm]
+  rw [he]
+  exact hm.comp (measurable_set_restrict (Q : Set E))
+
+theorem integrable_mass (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x y : V) :
+    Integrable (fun ω => q.mass ω x y) (law p) := by
+  apply Integrable.of_bound (q.measurable_mass x y).aestronglyMeasurable 1
+  exact ae_of_all _ fun ω => by simpa only [Real.norm_eq_abs,abs_of_nonneg (q.nonneg ω x y)] using q.mass_le_one ω x y
+
+ 
+
+noncomputable def kernel (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x y : V) : ℝ :=
+  ∫ ω, q.mass ω x y ∂law p
+
+theorem kernel_nonneg (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x y : V) :
+    0 ≤ q.kernel p x y := integral_nonneg (fun ω => q.nonneg ω x y)
+
+theorem kernel_outside_ball_zero (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x y : V)
+    (hy : y ∉ reachBall G x M) : q.kernel p x y = 0 := by
+  simp only [kernel,q.outside_ball_zero _ x y hy,integral_zero]
+
+theorem summable_kernel (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x : V) :
+    Summable (q.kernel p x) := by
+  apply summable_of_ne_finset_zero (s := (finite_reachBall hloc x M).toFinset)
+  intro y hy
+  apply q.kernel_outside_ball_zero p x y
+  simpa only [Set.Finite.mem_toFinset] using hy
+
+ 
+
+theorem kernel_row_integral (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x : V) :
+    (∑' y, q.kernel p x y) = ∫ ω, (∑' y, q.mass ω x y) ∂law p := by
+  let B := (finite_reachBall hloc x M).toFinset
+  have h₁ : (∑' y, q.kernel p x y) = ∑ y ∈ B, q.kernel p x y :=
+    tsum_eq_sum fun y hy => q.kernel_outside_ball_zero p x y (by simpa only [B,Set.Finite.mem_toFinset] using hy)
+  have h₂ (ω : Set E) : (∑' y, q.mass ω x y) = ∑ y ∈ B, q.mass ω x y :=
+    tsum_eq_sum fun y hy => q.outside_ball_zero ω x y (by simpa only [B,Set.Finite.mem_toFinset] using hy)
+  rw [h₁]
+  simp_rw [h₂]
+  exact (integral_finsetSum B (fun y _ => q.integrable_mass p x y)).symm
+
+theorem kernel_row_le_one (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x : V) :
+    (∑' y, q.kernel p x y) ≤ 1 := by
+  rw [q.kernel_row_integral p x]
+  have hi : Integrable (fun ω => ∑' y, q.mass ω x y) (law p) := by
+    let B := (finite_reachBall hloc x M).toFinset
+    have he (ω : Set E) : (∑' y, q.mass ω x y) = ∑ y ∈ B, q.mass ω x y :=
+      tsum_eq_sum fun y hy => q.outside_ball_zero ω x y (by simpa only [B,Set.Finite.mem_toFinset] using hy)
+    simp_rw [he]
+    exact integrable_finsetSum B (fun y _ => q.integrable_mass p x y)
+  calc
+    _ ≤ ∫ _, (1:ℝ) ∂law p := integral_mono hi (integrable_const _) (q.row_le_one · x)
+    _ = 1 := by simp
+
+end LocalEndpointRule
+end LocalEndpointRule
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section LocalEndpointKernel
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
+
+ 
+
+theorem outside_reverse_ball_zero (q : LocalEndpointRule G hloc O M L) (ω : Set E) (x y : V)
+    (hx : x ∉ reachBall G y M) : q.mass ω x y = 0 := by
+  by_contra h
+  obtain ⟨_,_,hc,hfin,hcap,_⟩ := q.support ω x y h
+  have he := cluster_eq_of_conn G ω hc
+  rw [he] at hfin hcap
+  exact hx (finite_cluster_subset_reachBall G ω y hfin M hcap (conn_symm G ω hc))
+
+theorem kernel_outside_reverse_ball_zero (q : LocalEndpointRule G hloc O M L)
+    (p : unitInterval) (x y : V) (hx : x ∉ reachBall G y M) : q.kernel p x y = 0 := by
+  simp only [kernel,q.outside_reverse_ball_zero _ x y hx,integral_zero]
+
+theorem summable_kernel_column (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (y : V) :
+    Summable (fun x => q.kernel p x y) := by
+  apply summable_of_ne_finset_zero (s := (finite_reachBall hloc y M).toFinset)
+  intro x hx
+  apply q.kernel_outside_reverse_ball_zero p x y
+  simpa only [Set.Finite.mem_toFinset] using hx
+
+theorem mass_zero_of_start_not_mem (q : LocalEndpointRule G hloc O M L) (ω : Set E)
+    (x y : V) (hx : x ∉ O) : q.mass ω x y = 0 := by
+  by_contra h
+  exact hx (q.support ω x y h).1
+
+theorem mass_zero_of_end_not_mem (q : LocalEndpointRule G hloc O M L) (ω : Set E)
+    (x y : V) (hy : y ∉ O) : q.mass ω x y = 0 := by
+  by_contra h
+  exact hy (q.support ω x y h).2.1
+
+theorem kernel_zero_of_start_not_mem (q : LocalEndpointRule G hloc O M L) (p : unitInterval)
+    (x y : V) (hx : x ∉ O) : q.kernel p x y = 0 := by
+  simp only [kernel,q.mass_zero_of_start_not_mem _ x y hx,integral_zero]
+
+theorem kernel_zero_of_end_not_mem (q : LocalEndpointRule G hloc O M L) (p : unitInterval)
+    (x y : V) (hy : y ∉ O) : q.kernel p x y = 0 := by
+  simp only [kernel,q.mass_zero_of_end_not_mem _ x y hy,integral_zero]
+
+ 
+
+theorem kernel_covariant (q : LocalEndpointRule G hloc O M L) (p : unitInterval)
+    (a : V ≃ V) (b : E ≃ E) (ha : IsAut G a b) (x y : V) :
+    q.kernel p (a x) (a y) = q.kernel p x y := by
+  have h := integral_map (μ := law p) (measurable_set_preimage b).aemeasurable
+    (q.measurable_mass x y).aestronglyMeasurable
+  rw [law_map_preimage_equiv p b] at h
+  simpa only [kernel,q.covariance a b ha] using h.symm
+
+noncomputable def successProbability (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x : V) : ℝ :=
+  ∑' y, q.kernel p x y
+
+theorem successProbability_nonneg (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x : V) :
+    0 ≤ q.successProbability p x := tsum_nonneg (q.kernel_nonneg p x)
+
+theorem successProbability_le_one (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x : V) :
+    q.successProbability p x ≤ 1 := q.kernel_row_le_one p x
+
+theorem successProbability_covariant (q : LocalEndpointRule G hloc O M L) (p : unitInterval)
+    (a : V ≃ V) (b : E ≃ E) (ha : IsAut G a b) (x : V) :
+    q.successProbability p (a x) = q.successProbability p x := by
+  unfold successProbability
+  rw [← a.tsum_eq (fun y => q.kernel p (a x) y)]
+  exact tsum_congr (q.kernel_covariant p a b ha x)
+
+theorem successProbability_orbit (q : LocalEndpointRule G hloc O M L) (p : unitInterval)
+    (o x : V) (hx : x ∈ vertexOrbit G o) : q.successProbability p x = q.successProbability p o := by
+  obtain ⟨a,b,ha,rfl⟩ := hx
+  exact q.successProbability_covariant p a b ha o
+
+theorem kernel_column_covariant (q : LocalEndpointRule G hloc O M L) (p : unitInterval)
+    (a : V ≃ V) (b : E ≃ E) (ha : IsAut G a b) (y : V) :
+    (∑' x, q.kernel p x (a y)) = ∑' x, q.kernel p x y := by
+  rw [← a.tsum_eq (fun x => q.kernel p x (a y))]
+  exact tsum_congr (fun x => q.kernel_covariant p a b ha x y)
+
+end LocalEndpointRule
+end LocalEndpointKernel
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section EndpointTransport
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
+
+ 
+
+theorem kernel_column_orbit (o : V) (q : LocalEndpointRule G hloc (vertexOrbit G o) M L)
+    (p : unitInterval) (y : V) (hy : y ∈ vertexOrbit G o) :
+    (∑' x, q.kernel p x y) = ∑' x, q.kernel p x o := by
+  obtain ⟨a,b,ha,rfl⟩ := hy
+  exact q.kernel_column_covariant p a b ha o
+
+ 
+
+theorem kernel_column_eq_success {ι : Type*} [Fintype ι]
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (htr : RootTransport root weight Γ)
+    (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b) (hw : ∀ i, 0 ≤ weight i)
+    (i₀ : ι) (hw₀ : 0 < weight i₀)
+    (q : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) M L)
+    (p : unitInterval) (y : V) (hy : y ∈ vertexOrbit G (root i₀)) :
+    (∑' x, q.kernel p x y) = q.successProbability p (root i₀) := by
+  let O := vertexOrbit G (root i₀)
+  let w : ι → ℝ := fun i => if root i ∈ O then weight i else 0
+  have ho : root i₀ ∈ O := by
+    refine ⟨Equiv.refl V,Equiv.refl E,?_,rfl⟩
+    intro e
+    simp
+  have hw' (i : ι) : 0 ≤ w i := by
+    dsimp [w]
+    split_ifs
+    · exact hw i
+    · exact le_rfl
+  have hpos : 0 < ∑ i, w i := by
+    have hi : w i₀ ≤ ∑ i, w i := Finset.single_le_sum (fun i _ => hw' i) (Finset.mem_univ i₀)
+    have he : w i₀ = weight i₀ := ite_eq_left ho
+    rw [he] at hi
+    exact hw₀.trans_le hi
+  have ht := htr.real root weight Γ (q.kernel p)
+    (by intro a ha x z; obtain ⟨b,hb⟩ := hΓ a ha; exact q.kernel_covariant p a b hb x z)
+    (q.summable_kernel p) (q.summable_kernel_column p)
+  have hr (i : ι) : weight i * (∑' z, q.kernel p (root i) z) =
+      w i * q.successProbability p (root i₀) := by
+    by_cases hi : root i ∈ O
+    · have he := q.successProbability_orbit p (root i₀) (root i) hi
+      change weight i * q.successProbability p (root i) = _
+      rw [he]
+      simp only [w,ite_eq_left hi]
+    · simp only [w,ite_eq_right hi,zero_mul]
+      have he : (∑' z, q.kernel p (root i) z) = 0 := by
+        simp only [q.kernel_zero_of_start_not_mem p _ _ hi,tsum_zero]
+      rw [he,mul_zero]
+  have hc (i : ι) : weight i * (∑' x, q.kernel p x (root i)) =
+      w i * (∑' x, q.kernel p x (root i₀)) := by
+    by_cases hi : root i ∈ O
+    · rw [q.kernel_column_orbit (root i₀) p (root i) hi]
+      simp only [w,ite_eq_left hi]
+    · simp only [w,ite_eq_right hi,zero_mul]
+      have he : (∑' x, q.kernel p x (root i)) = 0 := by
+        simp only [q.kernel_zero_of_end_not_mem p _ _ hi,tsum_zero]
+      rw [he,mul_zero]
+  simp_rw [hr,hc] at ht
+  rw [← Finset.sum_mul,← Finset.sum_mul] at ht
+  rw [q.kernel_column_orbit (root i₀) p y hy]
+  exact (mul_left_cancel₀ (ne_of_gt hpos) ht).symm
+
+ 
+theorem kernel_column_le_success {ι : Type*} [Fintype ι]
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (htr : RootTransport root weight Γ)
+    (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b) (hw : ∀ i, 0 ≤ weight i)
+    (i₀ : ι) (hw₀ : 0 < weight i₀)
+    (q : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) M L)
+    (p : unitInterval) (y : V) :
+    (∑' x, q.kernel p x y) ≤ q.successProbability p (root i₀) := by
+  by_cases hy : y ∈ vertexOrbit G (root i₀)
+  · exact (q.kernel_column_eq_success root weight Γ htr hΓ hw i₀ hw₀ p y hy).le
+  · simpa only [q.kernel_zero_of_end_not_mem p _ _ hy,tsum_zero] using
+      q.successProbability_nonneg p (root i₀)
+
+end LocalEndpointRule
+end EndpointTransport
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section EndpointSchur
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {M : ℕ} {L : ℝ}
+
+theorem kernel_row_le_success (o : V)
+    (q : LocalEndpointRule G hloc (vertexOrbit G o) M L) (p : unitInterval) (x : V) :
+    (∑' y, q.kernel p x y) ≤ q.successProbability p o := by
+  by_cases hx : x ∈ vertexOrbit G o
+  · exact (q.successProbability_orbit p o x hx).le
+  · simpa only [q.kernel_zero_of_start_not_mem p _ _ hx,tsum_zero] using
+      q.successProbability_nonneg p o
+
+ 
+noncomputable def normalizedKernel (o : V)
+    (q : LocalEndpointRule G hloc (vertexOrbit G o) M L) (p : unitInterval) (x y : V) : ℝ :=
+  (q.successProbability p o)⁻¹ * q.kernel p x y
+
+ 
+
+theorem exists_normalized_operator {ι : Type*} [Fintype ι]
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (htr : RootTransport root weight Γ)
+    (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b) (hw : ∀ i, 0 ≤ weight i)
+    (i₀ : ι) (hw₀ : 0 < weight i₀)
+    (q : LocalEndpointRule G hloc (vertexOrbit G (root i₀)) M L)
+    (p : unitInterval) (hu : 0 < q.successProbability p (root i₀)) :
+    ∃ A : Hilbert V →L[ℂ] Hilbert V, 0 < ‖A‖ ∧ ‖A‖ ≤ 1 ∧
+      ∀ x y, A (lp.single 2 y (1:ℂ)) x = (q.normalizedKernel (root i₀) p x y : ℂ) := by
+  let u := q.successProbability p (root i₀)
+  have hu₀ : u ≠ 0 := ne_of_gt hu
+  have hn (x y : V) : 0 ≤ q.normalizedKernel (root i₀) p x y :=
+    mul_nonneg (inv_nonneg.mpr hu.le) (q.kernel_nonneg p x y)
+  have hr (x : V) (S : Finset V) : (∑ y ∈ S, q.normalizedKernel (root i₀) p x y) ≤ 1 := by
+    change (∑ y ∈ S, u⁻¹ * q.kernel p x y) ≤ 1
+    rw [← Finset.mul_sum]
+    calc
+      _ ≤ u⁻¹ * (∑' y, q.kernel p x y) := mul_le_mul_of_nonneg_left
+        ((q.summable_kernel p x).sum_le_tsum S (fun y _ => q.kernel_nonneg p x y)) (inv_nonneg.mpr hu.le)
+      _ ≤ u⁻¹ * u := mul_le_mul_of_nonneg_left
+        (q.kernel_row_le_success (root i₀) p x) (inv_nonneg.mpr hu.le)
+      _ = 1 := inv_mul_cancel₀ hu₀
+  have hc (y : V) (S : Finset V) : (∑ x ∈ S, q.normalizedKernel (root i₀) p x y) ≤ 1 := by
+    change (∑ x ∈ S, u⁻¹ * q.kernel p x y) ≤ 1
+    rw [← Finset.mul_sum]
+    calc
+      _ ≤ u⁻¹ * (∑' x, q.kernel p x y) := mul_le_mul_of_nonneg_left
+        ((q.summable_kernel_column p y).sum_le_tsum S (fun x _ => q.kernel_nonneg p x y)) (inv_nonneg.mpr hu.le)
+      _ ≤ u⁻¹ * u := mul_le_mul_of_nonneg_left
+        (q.kernel_column_le_success root weight Γ htr hΓ hw i₀ hw₀ p y) (inv_nonneg.mpr hu.le)
+      _ = 1 := inv_mul_cancel₀ hu₀
+  obtain ⟨A,hA,hK⟩ := exists_operator_of_row_column_bound (q.normalizedKernel (root i₀) p) 1 (by norm_num) hn hr hc
+  refine ⟨A,?_,hA,hK⟩
+  apply norm_pos_iff.mpr
+  intro ha
+  have hz (y : V) : q.kernel p (root i₀) y = 0 := by
+    have h := hK (root i₀) y
+    rw [ha] at h
+    have he : u⁻¹ * q.kernel p (root i₀) y = 0 := Complex.ofReal_eq_zero.mp h.symm
+    exact (mul_eq_zero.mp he).resolve_left (inv_ne_zero hu₀)
+  have he : u = 0 := by simp only [u,successProbability,hz,tsum_zero]
+  exact hu₀ he
+
+end LocalEndpointRule
+end EndpointSchur
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory ProbabilityTheory
+section FiniteLocalIntegral
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+noncomputable def finiteFieldEquiv (Q : Finset E) : FiniteField Q ≃ Set (Q : Set E) where
+  toFun ω := Subtype.val ⁻¹' finiteFieldSet Q ω
+  invFun η := ⟨η.toFinset.map ⟨Subtype.val,Subtype.val_injective⟩, by
+    apply Finset.mem_powerset.mpr
+    intro e he
+    obtain ⟨a,_,rfl⟩ := Finset.mem_map.mp he
+    exact a.property⟩
+  left_inv ω := by
+    apply Subtype.ext
+    apply Finset.coe_injective
+    ext e
+    simp only [Finset.coe_map,Set.mem_image,finiteFieldSet,Function.Embedding.coeFn_mk]
+    constructor
+    · rintro ⟨a,ha,rfl⟩
+      have ha' : a ∈ (Subtype.val ⁻¹' (ω.val : Set E) : Set (Q : Set E)) := Set.mem_toFinset.mp ha
+      exact ha'
+    · intro he
+      exact ⟨⟨e,Finset.mem_powerset.mp ω.property he⟩,Set.mem_toFinset.mpr he,rfl⟩
+  right_inv η := by
+    ext e
+    simp only [Set.mem_preimage,finiteFieldSet,Finset.coe_map,Set.coe_toFinset,
+      Set.mem_image,Function.Embedding.coeFn_mk,Subtype.val_injective.eq_iff,exists_eq_right]
+
+@[simp] theorem finiteFieldEquiv_apply (Q : Finset E) (ω : FiniteField Q) :
+    finiteFieldEquiv Q ω = Subtype.val ⁻¹' finiteFieldSet Q ω := rfl
+
+@[simp] theorem finiteFieldEquiv_image (Q : Finset E) (ω : FiniteField Q) :
+    Subtype.val '' finiteFieldEquiv Q ω = finiteFieldSet Q ω := by
+  rw [finiteFieldEquiv_apply]
+  apply Set.image_preimage_eq_of_subset
+  intro e he
+  exact ⟨⟨e,Finset.mem_powerset.mp ω.property he⟩,rfl⟩
+
+ 
+
+theorem finiteFieldWeight_eq_atom (Q : Finset E) (p : unitInterval) (ω : FiniteField Q) :
+    finiteFieldWeight Q p ω = (law p : Measure (Set (Q : Set E))).real {finiteFieldEquiv Q ω} := by
+  have hn : (finiteFieldEquiv Q ω).ncard = ω.val.card := by
+    rw [finiteFieldEquiv_apply,Set.ncard_preimage_of_injective_subset_range Subtype.val_injective]
+    · exact Set.ncard_coe_finset _
+    · intro e he
+      exact ⟨⟨e,Finset.mem_powerset.mp ω.property he⟩,rfl⟩
+  rw [law,setBernoulli_real_singleton p (Set.subset_univ _) Set.finite_univ,
+    Set.ncard_sdiff' (Set.subset_univ _) Set.finite_univ,hn]
+  have hc : Fintype.card (Q : Set E) = Q.card := Fintype.card_coe Q
+  simp only [finiteFieldWeight,bernoulliWeight,Set.ncard_univ,Nat.card_eq_fintype_card,hc]
+
+ 
+
+theorem integral_finite_local (Q : Finset E) (p : unitInterval) (f : Set E → ℝ)
+    (hf : ∀ ω, f ω = f (ω ∩ (Q : Set E))) :
+    (∫ ω, f ω ∂law p) = ∑ η : FiniteField Q, finiteFieldWeight Q p η * f (finiteFieldSet Q η) := by
+  let g : Set (Q : Set E) → ℝ := fun η => f (Subtype.val '' η)
+  have hmap := integral_map (μ := law p) (measurable_set_restrict (Q : Set E)).aemeasurable
+    (measurable_of_finite g).aestronglyMeasurable
+  rw [law_map_restrict p (Q : Set E)] at hmap
+  have he (ω : Set E) : g (Subtype.val ⁻¹' ω) = f ω := by
+    rw [hf ω]
+    dsimp [g]
+    congr 1
+    ext e
+    simp [and_comm]
+  simp_rw [he] at hmap
+  rw [← hmap,integral_fintype Integrable.of_finite]
+  rw [← (finiteFieldEquiv Q).sum_comp]
+  apply Finset.sum_congr rfl
+  intro η _
+  rw [← finiteFieldWeight_eq_atom]
+  simp only [g,finiteFieldEquiv_image,smul_eq_mul]
+
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
+
+ 
+
+theorem kernel_finite_sum (q : LocalEndpointRule G hloc O M L) (p : unitInterval) (x y : V) :
+    let Q := incidentSupport G hloc (finite_reachBall hloc x q.radius).toFinset
+    q.kernel p x y = ∑ η : FiniteField Q, finiteFieldWeight Q p η * q.mass (finiteFieldSet Q η) x y :=
+  integral_finite_local _ p (fun ω => q.mass ω x y) (fun ω => q.locality ω x y)
+
+theorem continuous_kernel (q : LocalEndpointRule G hloc O M L) (x y : V) :
+    Continuous (fun p : unitInterval => q.kernel p x y) := by
+  simp_rw [q.kernel_finite_sum]
+  apply continuous_finsetSum
+  intro η _
+  unfold finiteFieldWeight bernoulliWeight
+  fun_prop
+
+theorem continuous_successProbability (q : LocalEndpointRule G hloc O M L) (x : V) :
+    Continuous (fun p : unitInterval => q.successProbability p x) := by
+  have he (p : unitInterval) : q.successProbability p x =
+      ∑ y ∈ (finite_reachBall hloc x M).toFinset, q.kernel p x y := by
+    apply tsum_eq_sum
+    intro y hy
+    exact q.kernel_outside_ball_zero p x y (by simpa only [Set.Finite.mem_toFinset] using hy)
+  simp_rw [he]
+  exact continuous_finsetSum _ (fun y _ => q.continuous_kernel x y)
+
+end LocalEndpointRule
+end FiniteLocalIntegral
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section GoodnessCovariance
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem internalEdges_aut_preimage (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (W : Set V) :
+    internalEdges G (a ⁻¹' W) = b ⁻¹' internalEdges G W := by
+  ext e
+  simp only [internalEdges,Set.mem_ofPred_eq,Set.mem_preimage,hab e,Sym2.mem_map]
+  constructor
+  · intro h y hy
+    obtain ⟨z,hz,rfl⟩ := hy
+    exact h z hz
+  · intro h z hz
+    exact h (a z) ⟨z,hz,rfl⟩
+
+theorem cutExists_aut_preimage (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (W : Set V) (e : E) :
+    CutExists G (a ⁻¹' W) e ↔ CutExists G W (b e) := by
+  constructor
+  · rintro ⟨u,v,he,hu,hv⟩
+    exact ⟨a u,a v,by rw [hab e,he,Sym2.map_mk],hu,hv⟩
+  · rintro ⟨u,v,he,hu,hv⟩
+    refine ⟨a.symm u,a.symm v,?_,by simpa using hu,by simpa using hv⟩
+    have h := congrArg (Sym2.map a.symm) he
+    simpa only [hab e,Sym2.map_map,Equiv.symm_comp_self,id_eq,Sym2.map_id',Sym2.map_mk] using h
+
+theorem contactSet_aut_preimage (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (ω : Set E) (W : Set V) :
+    contactSet G (b ⁻¹' ω) (a ⁻¹' W) = b ⁻¹' contactSet G ω W := by
+  ext e
+  simp only [contactSet,Set.mem_ofPred_eq,Set.mem_preimage,cutExists_aut_preimage G hab]
+
+theorem openMarkedPivotal_pair_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (ω : Set E) (x y : V) (e : E) :
+    OpenMarkedPivotal G (b ⁻¹' ω) {y} x e ↔
+      OpenMarkedPivotal G ω {a y} (a x) (b e) := by
+  have he : (b ⁻¹' ω) \ {e} = b ⁻¹' (ω \ {b e}) := by
+    ext f
+    simp only [Set.mem_sdiff,Set.mem_preimage,Set.mem_singleton_iff,b.injective.eq_iff]
+  simp only [OpenMarkedPivotal,MarkedEvent,Set.mem_singleton_iff,exists_eq_left,
+    Set.mem_preimage,he,conn_aut_iff G hab]
+
+theorem pivotalDistance_aut_preimage (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (ω : Set E) (x y : V) :
+    pivotalDistance G (b ⁻¹' ω) x y = pivotalDistance G ω (a x) (a y) := by
+  have he : {e | OpenMarkedPivotal G (b ⁻¹' ω) {y} x e} =
+      b ⁻¹' {e | OpenMarkedPivotal G ω {a y} (a x) e} := by
+    ext e
+    exact openMarkedPivotal_pair_aut G hab ω x y e
+  rw [pivotalDistance,he,Set.ncard_preimage_of_injective_subset_range b.injective]
+  · rfl
+  · exact fun e _ => b.surjective e
+
+theorem goodForScale_aut_pullback (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (ω : Set E) (x : V) (L : ℝ)
+    (hg : GoodForScale G ω (a x) L) : GoodForScale G (b ⁻¹' ω) x L := by
+  intro W hW hconn hpos u hu v hv
+  let W' := W.map a.toEmbedding
+  have hpre : a ⁻¹' (W' : Set V) = (W : Set V) := by
+    ext z
+    simp [W']
+  have hcfg : b ⁻¹' (ω ∩ internalEdges G (W' : Set V)) =
+      (b ⁻¹' ω) ∩ internalEdges G (W : Set V) := by
+    rw [Set.preimage_inter,← internalEdges_aut_preimage G hab,hpre]
+  have hcontact : (contactSet G (b ⁻¹' ω) (W : Set V)).ncard =
+      (contactSet G ω (W' : Set V)).ncard := by
+    rw [← hpre,contactSet_aut_preimage G hab,
+      Set.ncard_preimage_of_injective_subset_range b.injective]
+    exact fun e _ => b.surjective e
+  have hW' : (W' : Set V) ⊆ cluster G ω (a x) := by
+    intro z hz
+    obtain ⟨w,hw,rfl⟩ := Finset.mem_map.mp hz
+    exact (conn_aut_iff G hab ω x w).mp (hW hw)
+  have hc' : ∀ w ∈ W', ∀ z ∈ W', Conn G (ω ∩ internalEdges G (W' : Set V)) w z := by
+    intro w hw z hz
+    obtain ⟨u,hu,rfl⟩ := Finset.mem_map.mp hw
+    obtain ⟨v,hv,rfl⟩ := Finset.mem_map.mp hz
+    apply (conn_aut_iff G hab _ u v).mp
+    rw [hcfg]
+    exact hconn u hu v hv
+  have H := hg W' hW' hc' (by rwa [← hcontact]) (a u)
+    (Finset.mem_map.mpr ⟨u,hu,rfl⟩) (a v) (Finset.mem_map.mpr ⟨v,hv,rfl⟩)
+  rw [← pivotalDistance_aut_preimage G hab,hcfg,← hcontact] at H
+  exact H
+
+theorem goodForScale_aut_iff (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (ω : Set E) (x : V) (L : ℝ) :
+    GoodForScale G (b ⁻¹' ω) x L ↔ GoodForScale G ω (a x) L := by
+  constructor
+  · intro h
+    have H := goodForScale_aut_pullback G (isAut_symm G hab) (b ⁻¹' ω) (a x) L
+      (by simpa only [Equiv.symm_apply_apply] using h)
+    simpa only [← Set.preimage_comp,Equiv.self_comp_symm,Set.preimage_id_eq,id_eq] using H
+  · exact goodForScale_aut_pullback G hab ω x L
+
+end GoodnessCovariance
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section CappedGoodLocality
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem measurable_finite_local (Q : Finset E) (f : Set E → ℝ)
+    (hf : ∀ ω, f ω = f (ω ∩ (Q : Set E))) : Measurable f := by
+  let g : Set (Q : Set E) → ℝ := fun η => f (Subtype.val '' η)
+  have he : f = g ∘ (fun ω : Set E => (Subtype.val ⁻¹' ω : Set (Q : Set E))) := by
+    funext ω
+    rw [hf ω]
+    congr 1
+    ext e
+    simp [and_comm]
+  rw [he]
+  exact (measurable_of_finite g).comp (measurable_set_restrict (Q : Set E))
+
+theorem integrable_finite_local (Q : Finset E) (p : unitInterval) (f : Set E → ℝ)
+    (hf : ∀ ω, f ω = f (ω ∩ (Q : Set E))) : Integrable f (law p) := by
+  let g : Set (Q : Set E) → ℝ := fun η => f (Subtype.val '' η)
+  have he : f = g ∘ (fun ω : Set E => (Subtype.val ⁻¹' ω : Set (Q : Set E))) := by
+    funext ω
+    rw [hf ω]
+    congr 1
+    ext e
+    simp [and_comm]
+  have hi : Integrable g (Measure.map (fun ω : Set E => (Subtype.val ⁻¹' ω : Set (Q : Set E))) (law p)) := by
+    rw [law_map_restrict]
+    exact Integrable.of_finite
+  rw [he]
+  exact hi.comp_measurable (measurable_set_restrict (Q : Set E))
+
+def CappedGoodness (G : BondGraph V E) (ω : Set E) (x : V) (M : ℕ) (L : ℝ) : Prop :=
+  (cluster G ω x).Finite ∧ (cluster G ω x).ncard ≤ M ∧ GoodForScale G ω x L
+
+theorem cappedGoodness_local (G : BondGraph V E) (hloc : LocallyFinite G)
+    (ω : Set E) (x : V) (M : ℕ) (L : ℝ) :
+    let Q := incidentSupport G hloc (finite_reachBall hloc x M).toFinset
+    CappedGoodness G ω x M L ↔ CappedGoodness G (ω ∩ (Q : Set E)) x M L := by
+  have he (η : Set E) : CappedGoodness G η x M L ↔
+      ((cluster G η x).Finite ∧ (cluster G η x).ncard ≤ M) ∧ ¬CappedBadGoodness G η x M L := by
+    constructor
+    · rintro ⟨hf,hM,hg⟩
+      exact ⟨⟨hf,hM⟩,fun h => h.2.2 hg⟩
+    · rintro ⟨⟨hf,hM⟩,hbad⟩
+      refine ⟨hf,hM,?_⟩
+      by_contra hn
+      exact hbad ⟨hf,hM,hn⟩
+  dsimp only
+  have hbad := cappedBadGoodness_local G hloc x M L ω
+  have hcap := capped_cluster_local G hloc x M ω
+  dsimp only at hbad hcap
+  rw [he,he,hbad,hcap]
+
+theorem cappedGoodness_aut_iff (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (ω : Set E) (x : V) (M : ℕ) (L : ℝ) :
+    CappedGoodness G (b ⁻¹' ω) x M L ↔ CappedGoodness G ω (a x) M L := by
+  have hf : (cluster G (b ⁻¹' ω) x).Finite ↔ (cluster G ω (a x)).Finite := by
+    rw [← cluster_aut G hab ω x]
+    exact (Set.finite_image_iff a.injective.injOn).symm
+  have hn : (cluster G (b ⁻¹' ω) x).ncard = (cluster G ω (a x)).ncard := by
+    rw [← cluster_aut G hab ω x,Set.ncard_image_of_injective _ a.injective]
+  simp only [CappedGoodness,hf,hn,goodForScale_aut_iff G hab]
+
+theorem incidentSupport_ball_mono (G : BondGraph V E) (hloc : LocallyFinite G)
+    (x : V) {r R : ℕ} (h : r ≤ R) :
+    incidentSupport G hloc (finite_reachBall hloc x r).toFinset ⊆
+      incidentSupport G hloc (finite_reachBall hloc x R).toFinset := by
+  apply incidentSupport_mono
+  intro z hz
+  simp only [Set.Finite.mem_toFinset] at hz ⊢
+  exact reachBall_mono_local G x h hz
+
+theorem finite_local_of_subset (Q S : Finset E) (hQS : Q ⊆ S) (f : Set E → ℝ)
+    (hf : ∀ ω, f ω = f (ω ∩ (Q : Set E))) (ω : Set E) :
+    f ω = f (ω ∩ (S : Set E)) := by
+  rw [hf ω,hf (ω ∩ (S : Set E))]
+  congr 1
+  exact (Set.inter_assoc _ _ _).trans (congrArg (ω ∩ ·) (Set.inter_eq_right.mpr hQS)) |>.symm
+
+end CappedGoodLocality
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section CappedLocalLoad
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem finite_prop_local_of_subset (Q S : Finset E) (hQS : Q ⊆ S) (P : Set E → Prop)
+    (hf : ∀ ω, P ω ↔ P (ω ∩ (Q : Set E))) (ω : Set E) :
+    P ω ↔ P (ω ∩ (S : Set E)) := by
+  have he : (ω ∩ (S : Set E)) ∩ (Q : Set E) = ω ∩ (Q : Set E) :=
+    (Set.inter_assoc _ _ _).trans (congrArg (ω ∩ ·) (Set.inter_eq_right.mpr hQS))
+  exact (hf ω).trans ((he ▸ hf (ω ∩ (S : Set E))).symm)
+
+ 
+
+structure CappedLocalLoad (G : BondGraph V E) (hloc : LocallyFinite G) (O : Set V) (M : ℕ) where
+  mass : Set E → V → V → ℝ
+  radius : ℕ
+  nonneg : ∀ ω x y, 0 ≤ mass ω x y
+  support : ∀ ω x y, mass ω x y ≠ 0 → x ∈ O ∧ y ∈ O ∧ Conn G ω x y ∧
+    (cluster G ω x).Finite ∧ (cluster G ω x).ncard ≤ M
+  locality : ∀ ω x y, mass ω x y =
+    mass (ω ∩ (incidentSupport G hloc (finite_reachBall hloc x radius).toFinset : Set E)) x y
+  covariance : ∀ (a : V ≃ V) (b : E ≃ E), IsAut G a b → ∀ ω x y,
+    mass (b ⁻¹' ω) x y = mass ω (a x) (a y)
+
+namespace CappedLocalLoad
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ}
+
+noncomputable def row (q : CappedLocalLoad G hloc O M) (ω : Set E) (x : V) : ℝ :=
+  ∑' y, q.mass ω x y
+
+theorem row_nonneg (q : CappedLocalLoad G hloc O M) (ω : Set E) (x : V) : 0 ≤ q.row ω x :=
+  tsum_nonneg (q.nonneg ω x)
+
+theorem outside_ball_zero (q : CappedLocalLoad G hloc O M) (ω : Set E) (x y : V)
+    (hy : y ∉ reachBall G x M) : q.mass ω x y = 0 := by
+  by_contra h
+  obtain ⟨_,_,hc,hfin,hcap⟩ := q.support ω x y h
+  exact hy (finite_cluster_subset_reachBall G ω x hfin M hcap hc)
+
+theorem summable_mass (q : CappedLocalLoad G hloc O M) (ω : Set E) (x : V) :
+    Summable (q.mass ω x) := by
+  apply summable_of_ne_finset_zero (s := (finite_reachBall hloc x M).toFinset)
+  intro y hy
+  exact q.outside_ball_zero ω x y (by simpa only [Set.Finite.mem_toFinset] using hy)
+
+theorem mass_le_row (q : CappedLocalLoad G hloc O M) (ω : Set E) (x y : V) :
+    q.mass ω x y ≤ q.row ω x := by
+  simpa only [Finset.sum_singleton,row] using
+    (q.summable_mass ω x).sum_le_tsum {y} (fun z _ => q.nonneg ω x z)
+
+theorem row_locality (q : CappedLocalLoad G hloc O M) (ω : Set E) (x : V) :
+    q.row ω x = q.row
+      (ω ∩ (incidentSupport G hloc (finite_reachBall hloc x q.radius).toFinset : Set E)) x :=
+  tsum_congr (q.locality ω x)
+
+theorem row_covariance (q : CappedLocalLoad G hloc O M) (a : V ≃ V) (b : E ≃ E)
+    (hab : IsAut G a b) (ω : Set E) (x : V) : q.row (b ⁻¹' ω) x = q.row ω (a x) := by
+  unfold row
+  simp_rw [q.covariance a b hab]
+  exact a.tsum_eq _
+
+theorem integrable_mass (q : CappedLocalLoad G hloc O M) (p : unitInterval) (x y : V) :
+    Integrable (fun ω => q.mass ω x y) (law p) :=
+  integrable_finite_local _ p _ (q.locality · x y)
+
+theorem integrable_row (q : CappedLocalLoad G hloc O M) (p : unitInterval) (x : V) :
+    Integrable (fun ω => q.row ω x) (law p) :=
+  integrable_finite_local _ p _ (q.row_locality · x)
+
+theorem integrable_sq_row (q : CappedLocalLoad G hloc O M) (p : unitInterval) (x : V) :
+    Integrable (fun ω => q.row ω x ^ 2) (law p) :=
+  integrable_finite_local _ p _ (fun ω => congrArg (· ^ 2) (q.row_locality ω x))
+
+ 
+
+noncomputable def truncate (q : CappedLocalLoad G hloc O M) (b : ℝ) (hb : 0 < b) (L : ℝ) :
+    LocalEndpointRule G hloc O M L where
+  mass ω x y := if q.row ω x ≤ b ∧ CappedGoodness G ω x M L then b⁻¹ * q.mass ω x y else 0
+  radius := max q.radius M
+  nonneg ω x y := by split_ifs; exact mul_nonneg (inv_nonneg.mpr hb.le) (q.nonneg ω x y); rfl
+  row_le_one ω x := by
+    by_cases h : q.row ω x ≤ b ∧ CappedGoodness G ω x M L
+    · simp only [h,and_self,ite_true,tsum_mul_left]
+      exact (mul_le_mul_of_nonneg_left h.1 (inv_nonneg.mpr hb.le)).trans_eq (inv_mul_cancel₀ hb.ne')
+    · simp only [h,ite_false,tsum_zero,zero_le_one]
+  support ω x y h := by
+    split_ifs at h with hkeep
+    · have hm : q.mass ω x y ≠ 0 := fun he => h (by rw [he,mul_zero])
+      obtain ⟨hx,hy,hc,_,_⟩ := q.support ω x y hm
+      exact ⟨hx,hy,hc,hkeep.2⟩
+    · exact (h rfl).elim
+  locality ω x y := by
+    let S := incidentSupport G hloc (finite_reachBall hloc x (max q.radius M)).toFinset
+    have hm := finite_local_of_subset _ S
+      (incidentSupport_ball_mono G hloc x (le_max_left q.radius M))
+      (fun η => q.mass η x y) (q.locality · x y) ω
+    have hr := finite_local_of_subset _ S
+      (incidentSupport_ball_mono G hloc x (le_max_left q.radius M))
+      (fun η => q.row η x) (q.row_locality · x) ω
+    have hg := finite_prop_local_of_subset _ S
+      (incidentSupport_ball_mono G hloc x (le_max_right q.radius M))
+      (fun η => CappedGoodness G η x M L) (fun η => cappedGoodness_local G hloc η x M L) ω
+    simp only [hm,hr,hg]
+    rfl
+  covariance a c hac ω x y := by
+    simp only [q.row_covariance a c hac,q.covariance a c hac,cappedGoodness_aut_iff G hac]
+
+end CappedLocalLoad
+end CappedLocalLoad
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+open MeasureTheory
+section TruncationLoss
+attribute [local instance] Classical.propDecidable Classical.decEq
+namespace CappedLocalLoad
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ}
+
+theorem cap_of_row_ne_zero (q : CappedLocalLoad G hloc O M) (ω : Set E) (x : V)
+    (hr : q.row ω x ≠ 0) : (cluster G ω x).Finite ∧ (cluster G ω x).ncard ≤ M := by
+  obtain ⟨y,hy⟩ : ∃ y, q.mass ω x y ≠ 0 := by
+    by_contra hn
+    push Not at hn
+    exact hr (by simp only [row,hn,tsum_zero])
+  exact (q.support ω x y hy).2.2.2
+
+theorem truncate_scaled_row (q : CappedLocalLoad G hloc O M) (b : ℝ) (hb : 0 < b)
+    (L : ℝ) (ω : Set E) (x : V) :
+    b * (∑' y, (q.truncate b hb L).mass ω x y) =
+      if q.row ω x ≤ b ∧ CappedGoodness G ω x M L then q.row ω x else 0 := by
+  by_cases h : q.row ω x ≤ b ∧ CappedGoodness G ω x M L
+  · simp only [truncate,ite_eq_left h,tsum_mul_left,← mul_assoc,mul_inv_cancel₀ hb.ne',one_mul]
+    rfl
+  · simp only [truncate,ite_eq_right h,tsum_zero,mul_zero]
+
+theorem truncate_mass_le (q : CappedLocalLoad G hloc O M) (b : ℝ) (hb : 0 < b)
+    (L : ℝ) (ω : Set E) (x y : V) :
+    (q.truncate b hb L).mass ω x y ≤ b⁻¹ * q.mass ω x y := by
+  change (if _ then _ else _) ≤ _
+  split_ifs
+  · rfl
+  · exact mul_nonneg (inv_nonneg.mpr hb.le) (q.nonneg ω x y)
+
+ 
+
+theorem truncate_pointwise_loss (q : CappedLocalLoad G hloc O M) (b : ℝ) (hb : 0 < b)
+    (L : ℝ) (ω : Set E) (x : V) :
+    q.row ω x ≤ b * (∑' y, (q.truncate b hb L).mass ω x y) + q.row ω x ^ 2 / b +
+      b * ({η | CappedBadGoodness G η x M L}.indicator (1 : Set E → ℝ) ω) := by
+  rw [q.truncate_scaled_row b hb L]
+  have hs : 0 ≤ q.row ω x ^ 2 / b := div_nonneg (sq_nonneg _) hb.le
+  have hi : 0 ≤ ({η | CappedBadGoodness G η x M L}.indicator (1 : Set E → ℝ) ω) := by
+    apply Set.indicator_nonneg
+    intro η _
+    norm_num
+  by_cases hr : q.row ω x ≤ b
+  · by_cases hg : CappedGoodness G ω x M L
+    · simp only [hr,hg,and_self,ite_true]
+      nlinarith
+    · simp only [hg,and_false,ite_false,zero_add]
+      by_cases hz : q.row ω x = 0
+      · rw [hz]
+        positivity
+      · have hc := q.cap_of_row_ne_zero ω x hz
+        have hbad : CappedBadGoodness G ω x M L := ⟨hc.1,hc.2,fun h => hg ⟨hc.1,hc.2,h⟩⟩
+        rw [Set.indicator_of_mem (show ω ∈ {η | CappedBadGoodness G η x M L} from hbad)]
+        change q.row ω x ≤ q.row ω x ^ 2 / b + b * 1
+        linarith
+  · simp only [hr,false_and,ite_false,zero_add]
+    have htail : q.row ω x ≤ q.row ω x ^ 2 / b := by
+      apply (le_div_iff₀ hb).mpr
+      have ht := mul_le_mul_of_nonneg_left (le_of_lt (lt_of_not_ge hr)) (q.row_nonneg ω x)
+      nlinarith
+    exact htail.trans (le_add_of_nonneg_right (mul_nonneg hb.le hi))
+
+ 
+
+theorem truncate_mean_loss (q : CappedLocalLoad G hloc O M) (p : unitInterval)
+    (b : ℝ) (hb : 0 < b) (L : ℝ) (x : V) :
+    (∫ ω, q.row ω x ∂law p) ≤ b * (q.truncate b hb L).successProbability p x +
+      (∫ ω, q.row ω x ^ 2 ∂law p) / b + b * (law p).real {ω | CappedBadGoodness G ω x M L} := by
+  let t := q.truncate b hb L
+  have ht : Integrable (fun ω => ∑' y, t.mass ω x y) (law p) := by
+    apply integrable_finite_local
+      (incidentSupport G hloc (finite_reachBall hloc x t.radius).toFinset) p
+    intro ω
+    exact tsum_congr (t.locality ω x)
+  have hbad := measurableSet_cappedBadGoodness G hloc x M L
+  have hi : Integrable ({ω | CappedBadGoodness G ω x M L}.indicator (1 : Set E → ℝ)) (law p) :=
+    (integrable_const (1:ℝ)).indicator hbad
+  have h := integral_mono (q.integrable_row p x)
+    (((ht.const_mul b).add ((q.integrable_sq_row p x).div_const b)).add (hi.const_mul b))
+    (q.truncate_pointwise_loss b hb L · x)
+  dsimp only [Pi.add_apply] at h
+  have he₁ := integral_add ((ht.const_mul b).add ((q.integrable_sq_row p x).div_const b)) (hi.const_mul b)
+  have he₂ := integral_add (ht.const_mul b) ((q.integrable_sq_row p x).div_const b)
+  dsimp only [Pi.add_apply] at he₁ he₂
+  rw [he₁,he₂,integral_const_mul,integral_div,integral_const_mul,
+    integral_indicator_one hbad] at h
+  rw [← t.kernel_row_integral p x] at h
+  exact h
+
+theorem truncate_success_lower (q : CappedLocalLoad G hloc O M) (p : unitInterval)
+    (b : ℝ) (hb : 0 < b) (L : ℝ) (x : V) (a : ℝ)
+    (hmean : a ≤ ∫ ω, q.row ω x ∂law p)
+    (hmoment : (∫ ω, q.row ω x ^ 2 ∂law p) / b ≤ a/4)
+    (hbad : b * (law p).real {ω | CappedBadGoodness G ω x M L} ≤ a/4) :
+    a / (2*b) ≤ (q.truncate b hb L).successProbability p x := by
+  have hh := q.truncate_mean_loss p b hb L x
+  apply (div_le_iff₀ (mul_pos (by norm_num) hb)).mpr
+  nlinarith
+
+end CappedLocalLoad
+end TruncationLoss
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section CategoricalAmplificationInput
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+def amplificationVertices (B : Finset V) (k : ℕ) (z : ↥B) (r : Fin k → ↥B) : ℕ → V :=
+  fun i => (oldRunVertices k z r i).val
+
+namespace LocalEndpointRule
+variable {G : BondGraph V E} {hloc : LocallyFinite G} {O : Set V} {M : ℕ} {L : ℝ}
+
+noncomputable def amplificationWeight (q : LocalEndpointRule G hloc O M L)
+    (B : Finset V) (k : ℕ) (z : ↥B) (s : SpliceState E) (r : Fin k → ↥B) : ℝ :=
+  oldRunWeight (fun i a b => q.mass (s.2 i) a.val b.val) k z r
+
+theorem amplificationWeight_eq (q : LocalEndpointRule G hloc O M L)
+    (B : Finset V) (k : ℕ) (z : ↥B) (s : SpliceState E) (r : Fin k → ↥B) :
+    q.amplificationWeight B k z s r = oldPathWeight k q.mass (amplificationVertices B k z r) s.2 := by
+  exact (oldPathWeight_oldRunVertices k (fun ω a b => q.mass ω a.val b.val) s.2 z r).symm
+
+theorem amplificationWeight_nonneg (q : LocalEndpointRule G hloc O M L)
+    (B : Finset V) (k : ℕ) (z : ↥B) (s : SpliceState E) (r : Fin k → ↥B) :
+    0 ≤ q.amplificationWeight B k z s r :=
+  oldRunWeight_nonneg (fun i (a b : ↥B) => q.mass (s.2 i) a.val b.val)
+    (fun i a b => q.nonneg (s.2 i) a.val b.val) k z r
+
+theorem amplificationWeight_row (q : LocalEndpointRule G hloc O M L)
+    (B : Finset V) (k : ℕ) (z : ↥B) (s : SpliceState E) :
+    (∑ r, q.amplificationWeight B k z s r) ≤ 1 := by
+  have hr (i : ℕ) (a : ↥B) : (∑ b : ↥B, q.mass (s.2 i) a.val b.val) ≤ 1 := by
+    rw [Finset.sum_coe_sort B]
+    exact ((q.summable_mass _ _).sum_le_tsum B (fun y _ => q.nonneg _ _ y)).trans (q.row_le_one _ _)
+  simpa only [one_pow,amplificationWeight] using sum_oldRunWeight_le
+    (fun i (a b : ↥B) => q.mass (s.2 i) a.val b.val)
+    (fun _ a b => q.nonneg _ a.val b.val) 1 zero_le_one hr k z
+
+ 
+
+noncomputable def amplificationInputs (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (z : ↥B) (r : Fin k → ↥B) : Finset (SpliceState E) :=
+  (finiteSpliceDomain Q (Finset.Icc 1 k)).filter fun s =>
+    x ∈ O ∧ z.val ∈ O ∧ n ≤ (cluster G s.1 x).ncard ∧ (cluster G s.1 x).ncard < 2*n ∧
+      amplificationVertices B k z r 0 ∈ cluster G s.1 x ∧ q.amplificationWeight B k z s r ≠ 0
+
+theorem mem_amplificationInputs (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (z : ↥B) (r : Fin k → ↥B) (s : SpliceState E) :
+    s ∈ q.amplificationInputs Q B x k n z r ↔
+      s ∈ finiteSpliceDomain Q (Finset.Icc 1 k) ∧
+      x ∈ O ∧ z.val ∈ O ∧ n ≤ (cluster G s.1 x).ncard ∧ (cluster G s.1 x).ncard < 2*n ∧
+      amplificationVertices B k z r 0 ∈ cluster G s.1 x ∧ q.amplificationWeight B k z s r ≠ 0 :=
+  Finset.mem_filter
+
+theorem amplificationInputs_subset (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (z : ↥B) (r : Fin k → ↥B) :
+    q.amplificationInputs Q B x k n z r ⊆ finiteSpliceDomain Q (Finset.Icc 1 k) :=
+  Finset.filter_subset _ _
+
+theorem amplificationInputs_chain (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (z : ↥B) (r : Fin k → ↥B) (s : SpliceState E)
+    (hs : s ∈ q.amplificationInputs Q B x k n z r) :
+    ∀ i, 1 ≤ i → i ≤ k →
+      Conn G (s.2 i) (amplificationVertices B k z r i) (amplificationVertices B k z r (i-1)) ∧
+      (cluster G (s.2 i) (amplificationVertices B k z r i)).Finite ∧
+      (cluster G (s.2 i) (amplificationVertices B k z r i)).ncard ≤ M ∧
+      GoodForScale G (s.2 i) (amplificationVertices B k z r i) L := by
+  apply oldPathWeight_support G k M L q.mass
+  · intro ω a b h
+    exact (q.support ω a b h).2.2
+  · exact (q.amplificationWeight_eq B k z s r) ▸ ((q.mem_amplificationInputs Q B x k n z r s).mp hs).2.2.2.2.2.2
+
+ 
+
+theorem amplificationTrace_support (q : LocalEndpointRule G hloc O M L)
+    (Q : Finset E) (B : Finset V) (x : V) (k n : ℕ) (z : ↥B) (r : Fin k → ↥B) (s : SpliceState E)
+    (hs : s ∈ q.amplificationInputs Q B x k n z r) (c : List (ℕ × E)) (ωf : Set E)
+    (ht : SpliceTrace G x s.2 (amplificationVertices B k z r) k 0 s.1 c ωf) :
+    x ∈ O ∧ z.val ∈ O ∧ Conn G ωf x z.val ∧ (cluster G ωf x).Finite ∧
+      (cluster G ωf x).ncard ≤ 2*n+k*M := by
+  obtain ⟨hdom,hx,hz,_,hbin,_,_⟩ := (q.mem_amplificationInputs Q B x k n z r s).mp hs
+  have hc := q.amplificationInputs_chain Q B x k n z r s hs
+  have hroot : (cluster G s.1 x).Finite :=
+    cluster_finite_of_finite_edges G (Q.finite_toSet.subset
+      ((mem_finiteSpliceDomain_iff Q (Finset.Icc 1 k) s).mp hdom).1) x
+  obtain ⟨hf,hcap⟩ := trajectoryUnion_finite_cap G s.2 (amplificationVertices B k z r) k M
+    (fun i hi hik => ⟨(hc i hi hik).2.1,(hc i hi hik).2.2.1⟩)
+  have hsub : cluster G ωf x ⊆ cluster G s.1 x ∪ trajectoryUnion G s.2 (amplificationVertices B k z r) k := by
+    intro v hv
+    by_cases hv' : v ∈ cluster G s.1 x
+    · exact Or.inl hv'
+    · exact Or.inr (ht.added_subset_trajectory G x s.2 (amplificationVertices B k z r) k 0 s.1 ωf c ⟨hv,hv'⟩)
+  refine ⟨hx,hz,?_,(hroot.union hf).subset hsub,?_⟩
+  · simpa only [amplificationVertices,oldRunVertices_last,cluster,Set.mem_ofPred_eq] using
+      ht.endpoint G x s.2 (amplificationVertices B k z r) k 0 s.1 ωf c
+  · exact (Set.ncard_le_ncard hsub (hroot.union hf)).trans
+      ((Set.ncard_union_le ..).trans (Nat.add_le_add (by omega) hcap))
+
+end LocalEndpointRule
+end CategoricalAmplificationInput
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Transport/Haar.lean b/lean/OAI/Probability/UnimodularPercolation/Transport/Haar.lean
index 31e7afe76157a2f77d09d21b1a4aa70ddb6ff961..1b642122b7b5e7ebb276b34eedbc572e92fd7629 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Transport/Haar.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Transport/Haar.lean
@@ -1,229 +1,229 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Kernel.Spectral
-
-namespace OAI
-
-/-! Haar-measure transport for graph automorphism actions. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-open MeasureTheory
-open scoped ENNReal
-universe u v w
-
- 
-
-theorem haar_invInvariant_of_rightInvariant
-    {H : Type w} [Group H] [TopologicalSpace H] [IsTopologicalGroup H]
-    [MeasurableSpace H] [BorelSpace H] [LocallyCompactSpace H]
-    (μ : Measure H) [μ.IsHaarMeasure] [μ.Regular] [μ.IsMulRightInvariant] :
-    μ.IsInvInvariant := by
-  constructor
-  let c : ℝ≥0∞ := Measure.haarScalarFactor μ.inv μ
-  have hc : μ.inv = c • μ := Measure.isMulLeftInvariant_eq_smul_of_regular μ.inv μ
-  have he : Measure.map Inv.inv (Measure.map Inv.inv μ) = c ^ 2 • μ := by
-    rw [← Measure.inv_def μ, hc, Measure.map_smul c continuous_inv.measurable.aemeasurable,
-      ← Measure.inv_def μ, hc,
-      smul_smul, pow_two]
-  have μeq : μ = c ^ 2 • μ := by
-    rw [Measure.map_map continuous_inv.measurable continuous_inv.measurable] at he
-    simpa only [inv_involutive, Function.Involutive.comp_self, Measure.map_id] using he
-  let K : TopologicalSpace.PositiveCompacts H := Classical.arbitrary _
-  have hpow : c ^ 2 * μ K = 1 ^ 2 * μ K := by
-    conv_rhs => rw [μeq]
-    simp
-  have hpow' : c ^ 2 = 1 ^ 2 :=
-    (ENNReal.mul_left_inj
-      (Measure.measure_pos_of_nonempty_interior μ K.interior_nonempty).ne'
-      K.isCompact.measure_lt_top.ne).mp hpow
-  have hc1 : c = 1 := (ENNReal.pow_right_strictMono two_ne_zero).injective hpow'
-  rw [hc, hc1, one_smul]
-
-section HaarAction
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {H : Type w} [Group H] [MeasurableSpace H] [MeasurableMul₂ H] [MeasurableInv H]
-variable {V : Type u} [Countable V] [MeasurableSpace V] [MeasurableSingletonClass V]
-variable [MulAction H V]
-
- 
-def actionStabilizer (x : V) : Set H := {g | g • x = x}
-
- 
-def actionOrbit (x : V) : Set V := {y | ∃ g : H, g • x = y}
-
-omit [MeasurableSpace H] [MeasurableMul₂ H] [MeasurableInv H] [Countable V]
-  [MeasurableSpace V] [MeasurableSingletonClass V] in
-theorem actionOrbit_self (x : V) : x ∈ actionOrbit (H := H) x := ⟨1,one_smul _ _⟩
-
-omit [MeasurableSpace H] [MeasurableMul₂ H] [MeasurableInv H] [Countable V]
-  [MeasurableSpace V] [MeasurableSingletonClass V] in
-theorem actionOrbit_eq_of_mem {x y : V} (h : y ∈ actionOrbit (H := H) x) :
-    actionOrbit (H := H) y = actionOrbit (H := H) x := by
-  obtain ⟨g,rfl⟩ := h
-  ext z
-  constructor
-  · rintro ⟨k,hk⟩
-    exact ⟨k*g,by simpa only [mul_smul] using hk⟩
-  · rintro ⟨k,hk⟩
-    exact ⟨k*g⁻¹,by simpa only [mul_smul, inv_smul_smul] using hk⟩
-
- 
-
-omit [MeasurableInv H] [Countable V] [MeasurableSingletonClass V] in
-theorem orbit_fiber_measure (μ : Measure H) [μ.IsMulLeftInvariant]
-    (_hmeas : ∀ x : V, Measurable (fun g : H => g • x)) (x y : V) :
-    μ {g : H | g • x = y} = if y ∈ actionOrbit (H := H) x then μ (actionStabilizer (H := H) x) else 0 := by
-  classical
-  split_ifs with h
-  · obtain ⟨g,rfl⟩ := h
-    have he : {k : H | k • x = g • x} = (fun k : H => g⁻¹*k) ⁻¹' actionStabilizer (H := H) x := by
-      ext k
-      simp only [Set.mem_ofPred_eq,Set.mem_preimage,actionStabilizer,mul_smul]
-      exact ⟨fun hk => by rw [hk,inv_smul_smul],fun hk => by simpa using congrArg (fun z => g • z) hk⟩
-    rw [he]
-    exact measure_preimage_mul μ g⁻¹ _
-  · have he : {g : H | g • x = y} = ∅ := by
-      apply Set.eq_empty_iff_forall_notMem.mpr
-      intro g hg
-      exact h ⟨g,hg⟩
-    simp only [he,measure_empty]
-
- 
-
-omit [MeasurableInv H] in
-theorem orbit_map_measure (μ : Measure H) [μ.IsMulLeftInvariant]
-    (hmeas : ∀ x : V, Measurable (fun g : H => g • x)) (x : V) :
-    Measure.map (fun g : H => g • x) μ =
-      Measure.sum (fun y : V => (if y ∈ actionOrbit (H := H) x then μ (actionStabilizer (H := H) x) else 0) • Measure.dirac y) := by
-  classical
-  apply Measure.ext_of_singleton
-  intro y
-  rw [Measure.map_apply (hmeas x) (measurableSet_singleton y),Measure.sum_apply _ (measurableSet_singleton y)]
-  rw [tsum_eq_single y (fun z hz => by split_ifs <;> simp [Measure.smul_apply,Measure.dirac_apply',hz])]
-  simp only [Measure.smul_apply,Measure.dirac_apply_of_mem (Set.mem_singleton y),smul_eq_mul,mul_one]
-  exact orbit_fiber_measure μ hmeas x y
-
- 
-
-omit [MeasurableInv H] in
-theorem integral_orbit_sum (μ : Measure H) [μ.IsMulLeftInvariant]
-    (hmeas : ∀ x : V, Measurable (fun g : H => g • x)) (x : V)
-    (hx : μ (actionStabilizer (H := H) x) ≠ ∞) (f : V → ℂ) :
-    (∫ g : H, f (g • x) ∂μ) =
-      (μ.real (actionStabilizer (H := H) x) : ℂ) *
-        ∑' y : V, if y ∈ actionOrbit (H := H) x then f y else 0 := by
-  classical
-  rw [←integral_map_of_stronglyMeasurable (hmeas x) (measurable_of_countable f).stronglyMeasurable,
-    orbit_map_measure μ hmeas x,integral_sum_dirac (fun y => by split_ifs <;> simp_all)]
-  rw [←tsum_mul_left]
-  apply tsum_congr
-  intro y
-  split_ifs <;> simp [Measure.real,Complex.real_smul]
-
- 
-
-theorem orbit_mass_balance (μ : Measure H) [μ.IsMulLeftInvariant] [μ.IsInvInvariant]
-    (hmeas : ∀ x : V, Measurable (fun g : H => g • x)) (x y : V)
-    (hx : μ (actionStabilizer (H := H) x) ≠ ∞) (hy : μ (actionStabilizer (H := H) y) ≠ ∞)
-    (m : V → V → ℂ) (hinv : ∀ g : H, ∀ x y : V, m (g • x) (g • y) = m x y) :
-    (μ.real (actionStabilizer (H := H) y) : ℂ) * (∑' z, if z ∈ actionOrbit (H := H) y then m x z else 0) =
-      (μ.real (actionStabilizer (H := H) x) : ℂ) * (∑' z, if z ∈ actionOrbit (H := H) x then m z y else 0) := by
-  rw [←integral_orbit_sum μ hmeas y hy (m x),←integral_orbit_sum μ hmeas x hx (fun z => m z y)]
-  calc
-    _ = ∫ g : H, m (g⁻¹ • x) y ∂μ := by
-      apply integral_congr_ae
-      filter_upwards [] with g
-      have h := hinv g⁻¹ x (g • y)
-      simpa only [inv_smul_smul] using h.symm
-    _ = _ := integral_inv_eq_self (fun g : H => m (g • x) y) μ
-
-variable {ι : Type*} [Fintype ι]
-
- 
-
-noncomputable def inverseStabilizerWeight (μ : Measure H) (root : ι → V) (i : ι) : ℝ :=
-  (μ.real (actionStabilizer (H := H) (root i)))⁻¹ /
-    ∑ j, (μ.real (actionStabilizer (H := H) (root j)))⁻¹
-
-omit [MeasurableMul₂ H] [MeasurableInv H] [Countable V] [MeasurableSpace V]
-  [MeasurableSingletonClass V] in
-theorem inverseStabilizerWeight_pos [Nonempty ι] (μ : Measure H) (root : ι → V)
-    (hvol : ∀ i, 0 < μ.real (actionStabilizer (H := H) (root i))) (i : ι) :
-    0 < inverseStabilizerWeight μ root i := by
-  apply div_pos (inv_pos.mpr (hvol i))
-  exact Finset.sum_pos (fun j _ => inv_pos.mpr (hvol j)) Finset.univ_nonempty
-
-omit [MeasurableMul₂ H] [MeasurableInv H] [Countable V] [MeasurableSpace V]
-  [MeasurableSingletonClass V] in
-theorem inverseStabilizerWeight_sum [Nonempty ι] (μ : Measure H) (root : ι → V)
-    (hvol : ∀ i, 0 < μ.real (actionStabilizer (H := H) (root i))) :
-    ∑ i, inverseStabilizerWeight μ root i = 1 := by
-  unfold inverseStabilizerWeight
-  rw [←Finset.sum_div,div_self]
-  exact ne_of_gt (Finset.sum_pos (fun j _ => inv_pos.mpr (hvol j)) Finset.univ_nonempty)
-
- 
-
-omit [MeasurableSpace H] [MeasurableMul₂ H] [MeasurableInv H] [Countable V]
-  [MeasurableSpace V] [MeasurableSingletonClass V] in
-theorem tsum_eq_sum_orbits (root : ι → V)
-    (hcov : ∀ x, ∃ i, x ∈ actionOrbit (H := H) (root i))
-    (hsep : ∀ i j, root j ∈ actionOrbit (H := H) (root i) → i = j)
-    (f : V → ℂ) (hf : Summable f) :
-    ∑' x, f x = ∑ i, ∑' x, if x ∈ actionOrbit (H := H) (root i) then f x else 0 := by
-  have hi (i : ι) : Summable (fun x => if x ∈ actionOrbit (H := H) (root i) then f x else 0) := hf.indicator _
-  rw [←Summable.tsum_finsetSum (s := Finset.univ) (fun i _ => hi i)]
-  apply tsum_congr
-  intro x
-  obtain ⟨i,hi⟩ := hcov x
-  symm
-  rw [Finset.sum_eq_single i]
-  · exact ite_eq_left hi
-  · intro j _ hji
-    have hj : x ∉ actionOrbit (H := H) (root j) := by
-      intro hj
-      apply hji
-      apply hsep j i
-      rw [←actionOrbit_eq_of_mem hj,actionOrbit_eq_of_mem hi]
-      exact actionOrbit_self _
-    exact ite_eq_right hj
-  · simp
-
- 
-
-theorem rootTransport_of_invariant_measure [Nonempty ι]
-    (μ : Measure H) [μ.IsMulLeftInvariant] [μ.IsInvInvariant]
-    (hmeas : ∀ x : V, Measurable (fun g : H => g • x))
-    (root : ι → V) (hvol : ∀ i, 0 < μ.real (actionStabilizer (H := H) (root i)))
-    (hfinite : ∀ i, μ (actionStabilizer (H := H) (root i)) ≠ ∞)
-    (hcov : ∀ x, ∃ i, x ∈ actionOrbit (H := H) (root i))
-    (hsep : ∀ i j, root j ∈ actionOrbit (H := H) (root i) → i = j)
-    (Γ : Set (V ≃ V)) (hΓ : ∀ g : H, MulAction.toPerm g ∈ Γ) :
-    RootTransport root (inverseStabilizerWeight μ root) Γ := by
-  intro m hinv hrow hcol
-  have hm : ∀ g : H, ∀ x y : V, m (g • x) (g • y) = m x y := fun g => hinv _ (hΓ g)
-  have hp (i j : ι) :
-      (inverseStabilizerWeight μ root i : ℂ) * (∑' y, if y ∈ actionOrbit (H := H) (root j) then m (root i) y else 0) =
-      (inverseStabilizerWeight μ root j : ℂ) * (∑' x, if x ∈ actionOrbit (H := H) (root i) then m x (root j) else 0) := by
-    have hb := orbit_mass_balance μ hmeas (root i) (root j) (hfinite i) (hfinite j) m hm
-    have hi : (μ.real (actionStabilizer (H := H) (root i)) : ℂ) ≠ 0 := Complex.ofReal_ne_zero.mpr (hvol i).ne'
-    have hj : (μ.real (actionStabilizer (H := H) (root j)) : ℂ) ≠ 0 := Complex.ofReal_ne_zero.mpr (hvol j).ne'
-    have hZ : ((∑ k, (μ.real (actionStabilizer (H := H) (root k)))⁻¹ : ℝ) : ℂ) ≠ 0 :=
-      Complex.ofReal_ne_zero.mpr (ne_of_gt (Finset.sum_pos (fun k _ => inv_pos.mpr (hvol k)) Finset.univ_nonempty))
-    unfold inverseStabilizerWeight
-    push_cast
-    field_simp
-    simpa only [mul_comm,mul_left_comm,mul_assoc] using congrArg (fun z : ℂ => z / ∑ k, (1 : ℂ) / (μ.real (actionStabilizer (H := H) (root k)) : ℂ)) hb
-  conv_lhs => arg 2; ext i; rw [tsum_eq_sum_orbits root hcov hsep (m (root i)) (hrow (root i)),Finset.mul_sum]
-  simp_rw [hp]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro j _
-  rw [←Finset.mul_sum,←tsum_eq_sum_orbits root hcov hsep (fun x => m x (root j)) (hcol (root j))]
-
-end HaarAction
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Kernel.Spectral
+
+namespace OAI
+
+/-! Haar-measure transport for graph automorphism actions. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+open MeasureTheory
+open scoped ENNReal
+universe u v w
+
+ 
+
+theorem haar_invInvariant_of_rightInvariant
+    {H : Type w} [Group H] [TopologicalSpace H] [IsTopologicalGroup H]
+    [MeasurableSpace H] [BorelSpace H] [LocallyCompactSpace H]
+    (μ : Measure H) [μ.IsHaarMeasure] [μ.Regular] [μ.IsMulRightInvariant] :
+    μ.IsInvInvariant := by
+  constructor
+  let c : ℝ≥0∞ := Measure.haarScalarFactor μ.inv μ
+  have hc : μ.inv = c • μ := Measure.isMulLeftInvariant_eq_smul_of_regular μ.inv μ
+  have he : Measure.map Inv.inv (Measure.map Inv.inv μ) = c ^ 2 • μ := by
+    rw [← Measure.inv_def μ, hc, Measure.map_smul c continuous_inv.measurable.aemeasurable,
+      ← Measure.inv_def μ, hc,
+      smul_smul, pow_two]
+  have μeq : μ = c ^ 2 • μ := by
+    rw [Measure.map_map continuous_inv.measurable continuous_inv.measurable] at he
+    simpa only [inv_involutive, Function.Involutive.comp_self, Measure.map_id] using he
+  let K : TopologicalSpace.PositiveCompacts H := Classical.arbitrary _
+  have hpow : c ^ 2 * μ K = 1 ^ 2 * μ K := by
+    conv_rhs => rw [μeq]
+    simp
+  have hpow' : c ^ 2 = 1 ^ 2 :=
+    (ENNReal.mul_left_inj
+      (Measure.measure_pos_of_nonempty_interior μ K.interior_nonempty).ne'
+      K.isCompact.measure_lt_top.ne).mp hpow
+  have hc1 : c = 1 := (ENNReal.pow_right_strictMono two_ne_zero).injective hpow'
+  rw [hc, hc1, one_smul]
+
+section HaarAction
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {H : Type w} [Group H] [MeasurableSpace H] [MeasurableMul₂ H] [MeasurableInv H]
+variable {V : Type u} [Countable V] [MeasurableSpace V] [MeasurableSingletonClass V]
+variable [MulAction H V]
+
+ 
+def actionStabilizer (x : V) : Set H := {g | g • x = x}
+
+ 
+def actionOrbit (x : V) : Set V := {y | ∃ g : H, g • x = y}
+
+omit [MeasurableSpace H] [MeasurableMul₂ H] [MeasurableInv H] [Countable V]
+  [MeasurableSpace V] [MeasurableSingletonClass V] in
+theorem actionOrbit_self (x : V) : x ∈ actionOrbit (H := H) x := ⟨1,one_smul _ _⟩
+
+omit [MeasurableSpace H] [MeasurableMul₂ H] [MeasurableInv H] [Countable V]
+  [MeasurableSpace V] [MeasurableSingletonClass V] in
+theorem actionOrbit_eq_of_mem {x y : V} (h : y ∈ actionOrbit (H := H) x) :
+    actionOrbit (H := H) y = actionOrbit (H := H) x := by
+  obtain ⟨g,rfl⟩ := h
+  ext z
+  constructor
+  · rintro ⟨k,hk⟩
+    exact ⟨k*g,by simpa only [mul_smul] using hk⟩
+  · rintro ⟨k,hk⟩
+    exact ⟨k*g⁻¹,by simpa only [mul_smul, inv_smul_smul] using hk⟩
+
+ 
+
+omit [MeasurableInv H] [Countable V] [MeasurableSingletonClass V] in
+theorem orbit_fiber_measure (μ : Measure H) [μ.IsMulLeftInvariant]
+    (_hmeas : ∀ x : V, Measurable (fun g : H => g • x)) (x y : V) :
+    μ {g : H | g • x = y} = if y ∈ actionOrbit (H := H) x then μ (actionStabilizer (H := H) x) else 0 := by
+  classical
+  split_ifs with h
+  · obtain ⟨g,rfl⟩ := h
+    have he : {k : H | k • x = g • x} = (fun k : H => g⁻¹*k) ⁻¹' actionStabilizer (H := H) x := by
+      ext k
+      simp only [Set.mem_ofPred_eq,Set.mem_preimage,actionStabilizer,mul_smul]
+      exact ⟨fun hk => by rw [hk,inv_smul_smul],fun hk => by simpa using congrArg (fun z => g • z) hk⟩
+    rw [he]
+    exact measure_preimage_mul μ g⁻¹ _
+  · have he : {g : H | g • x = y} = ∅ := by
+      apply Set.eq_empty_iff_forall_notMem.mpr
+      intro g hg
+      exact h ⟨g,hg⟩
+    simp only [he,measure_empty]
+
+ 
+
+omit [MeasurableInv H] in
+theorem orbit_map_measure (μ : Measure H) [μ.IsMulLeftInvariant]
+    (hmeas : ∀ x : V, Measurable (fun g : H => g • x)) (x : V) :
+    Measure.map (fun g : H => g • x) μ =
+      Measure.sum (fun y : V => (if y ∈ actionOrbit (H := H) x then μ (actionStabilizer (H := H) x) else 0) • Measure.dirac y) := by
+  classical
+  apply Measure.ext_of_singleton
+  intro y
+  rw [Measure.map_apply (hmeas x) (measurableSet_singleton y),Measure.sum_apply _ (measurableSet_singleton y)]
+  rw [tsum_eq_single y (fun z hz => by split_ifs <;> simp [Measure.smul_apply,Measure.dirac_apply',hz])]
+  simp only [Measure.smul_apply,Measure.dirac_apply_of_mem (Set.mem_singleton y),smul_eq_mul,mul_one]
+  exact orbit_fiber_measure μ hmeas x y
+
+ 
+
+omit [MeasurableInv H] in
+theorem integral_orbit_sum (μ : Measure H) [μ.IsMulLeftInvariant]
+    (hmeas : ∀ x : V, Measurable (fun g : H => g • x)) (x : V)
+    (hx : μ (actionStabilizer (H := H) x) ≠ ∞) (f : V → ℂ) :
+    (∫ g : H, f (g • x) ∂μ) =
+      (μ.real (actionStabilizer (H := H) x) : ℂ) *
+        ∑' y : V, if y ∈ actionOrbit (H := H) x then f y else 0 := by
+  classical
+  rw [←integral_map_of_stronglyMeasurable (hmeas x) (measurable_of_countable f).stronglyMeasurable,
+    orbit_map_measure μ hmeas x,integral_sum_dirac (fun y => by split_ifs <;> simp_all)]
+  rw [←tsum_mul_left]
+  apply tsum_congr
+  intro y
+  split_ifs <;> simp [Measure.real,Complex.real_smul]
+
+ 
+
+theorem orbit_mass_balance (μ : Measure H) [μ.IsMulLeftInvariant] [μ.IsInvInvariant]
+    (hmeas : ∀ x : V, Measurable (fun g : H => g • x)) (x y : V)
+    (hx : μ (actionStabilizer (H := H) x) ≠ ∞) (hy : μ (actionStabilizer (H := H) y) ≠ ∞)
+    (m : V → V → ℂ) (hinv : ∀ g : H, ∀ x y : V, m (g • x) (g • y) = m x y) :
+    (μ.real (actionStabilizer (H := H) y) : ℂ) * (∑' z, if z ∈ actionOrbit (H := H) y then m x z else 0) =
+      (μ.real (actionStabilizer (H := H) x) : ℂ) * (∑' z, if z ∈ actionOrbit (H := H) x then m z y else 0) := by
+  rw [←integral_orbit_sum μ hmeas y hy (m x),←integral_orbit_sum μ hmeas x hx (fun z => m z y)]
+  calc
+    _ = ∫ g : H, m (g⁻¹ • x) y ∂μ := by
+      apply integral_congr_ae
+      filter_upwards [] with g
+      have h := hinv g⁻¹ x (g • y)
+      simpa only [inv_smul_smul] using h.symm
+    _ = _ := integral_inv_eq_self (fun g : H => m (g • x) y) μ
+
+variable {ι : Type*} [Fintype ι]
+
+ 
+
+noncomputable def inverseStabilizerWeight (μ : Measure H) (root : ι → V) (i : ι) : ℝ :=
+  (μ.real (actionStabilizer (H := H) (root i)))⁻¹ /
+    ∑ j, (μ.real (actionStabilizer (H := H) (root j)))⁻¹
+
+omit [MeasurableMul₂ H] [MeasurableInv H] [Countable V] [MeasurableSpace V]
+  [MeasurableSingletonClass V] in
+theorem inverseStabilizerWeight_pos [Nonempty ι] (μ : Measure H) (root : ι → V)
+    (hvol : ∀ i, 0 < μ.real (actionStabilizer (H := H) (root i))) (i : ι) :
+    0 < inverseStabilizerWeight μ root i := by
+  apply div_pos (inv_pos.mpr (hvol i))
+  exact Finset.sum_pos (fun j _ => inv_pos.mpr (hvol j)) Finset.univ_nonempty
+
+omit [MeasurableMul₂ H] [MeasurableInv H] [Countable V] [MeasurableSpace V]
+  [MeasurableSingletonClass V] in
+theorem inverseStabilizerWeight_sum [Nonempty ι] (μ : Measure H) (root : ι → V)
+    (hvol : ∀ i, 0 < μ.real (actionStabilizer (H := H) (root i))) :
+    ∑ i, inverseStabilizerWeight μ root i = 1 := by
+  unfold inverseStabilizerWeight
+  rw [←Finset.sum_div,div_self]
+  exact ne_of_gt (Finset.sum_pos (fun j _ => inv_pos.mpr (hvol j)) Finset.univ_nonempty)
+
+ 
+
+omit [MeasurableSpace H] [MeasurableMul₂ H] [MeasurableInv H] [Countable V]
+  [MeasurableSpace V] [MeasurableSingletonClass V] in
+theorem tsum_eq_sum_orbits (root : ι → V)
+    (hcov : ∀ x, ∃ i, x ∈ actionOrbit (H := H) (root i))
+    (hsep : ∀ i j, root j ∈ actionOrbit (H := H) (root i) → i = j)
+    (f : V → ℂ) (hf : Summable f) :
+    ∑' x, f x = ∑ i, ∑' x, if x ∈ actionOrbit (H := H) (root i) then f x else 0 := by
+  have hi (i : ι) : Summable (fun x => if x ∈ actionOrbit (H := H) (root i) then f x else 0) := hf.indicator _
+  rw [←Summable.tsum_finsetSum (s := Finset.univ) (fun i _ => hi i)]
+  apply tsum_congr
+  intro x
+  obtain ⟨i,hi⟩ := hcov x
+  symm
+  rw [Finset.sum_eq_single i]
+  · exact ite_eq_left hi
+  · intro j _ hji
+    have hj : x ∉ actionOrbit (H := H) (root j) := by
+      intro hj
+      apply hji
+      apply hsep j i
+      rw [←actionOrbit_eq_of_mem hj,actionOrbit_eq_of_mem hi]
+      exact actionOrbit_self _
+    exact ite_eq_right hj
+  · simp
+
+ 
+
+theorem rootTransport_of_invariant_measure [Nonempty ι]
+    (μ : Measure H) [μ.IsMulLeftInvariant] [μ.IsInvInvariant]
+    (hmeas : ∀ x : V, Measurable (fun g : H => g • x))
+    (root : ι → V) (hvol : ∀ i, 0 < μ.real (actionStabilizer (H := H) (root i)))
+    (hfinite : ∀ i, μ (actionStabilizer (H := H) (root i)) ≠ ∞)
+    (hcov : ∀ x, ∃ i, x ∈ actionOrbit (H := H) (root i))
+    (hsep : ∀ i j, root j ∈ actionOrbit (H := H) (root i) → i = j)
+    (Γ : Set (V ≃ V)) (hΓ : ∀ g : H, MulAction.toPerm g ∈ Γ) :
+    RootTransport root (inverseStabilizerWeight μ root) Γ := by
+  intro m hinv hrow hcol
+  have hm : ∀ g : H, ∀ x y : V, m (g • x) (g • y) = m x y := fun g => hinv _ (hΓ g)
+  have hp (i j : ι) :
+      (inverseStabilizerWeight μ root i : ℂ) * (∑' y, if y ∈ actionOrbit (H := H) (root j) then m (root i) y else 0) =
+      (inverseStabilizerWeight μ root j : ℂ) * (∑' x, if x ∈ actionOrbit (H := H) (root i) then m x (root j) else 0) := by
+    have hb := orbit_mass_balance μ hmeas (root i) (root j) (hfinite i) (hfinite j) m hm
+    have hi : (μ.real (actionStabilizer (H := H) (root i)) : ℂ) ≠ 0 := Complex.ofReal_ne_zero.mpr (hvol i).ne'
+    have hj : (μ.real (actionStabilizer (H := H) (root j)) : ℂ) ≠ 0 := Complex.ofReal_ne_zero.mpr (hvol j).ne'
+    have hZ : ((∑ k, (μ.real (actionStabilizer (H := H) (root k)))⁻¹ : ℝ) : ℂ) ≠ 0 :=
+      Complex.ofReal_ne_zero.mpr (ne_of_gt (Finset.sum_pos (fun k _ => inv_pos.mpr (hvol k)) Finset.univ_nonempty))
+    unfold inverseStabilizerWeight
+    push_cast
+    field_simp
+    simpa only [mul_comm,mul_left_comm,mul_assoc] using congrArg (fun z : ℂ => z / ∑ k, (1 : ℂ) / (μ.real (actionStabilizer (H := H) (root k)) : ℂ)) hb
+  conv_lhs => arg 2; ext i; rw [tsum_eq_sum_orbits root hcov hsep (m (root i)) (hrow (root i)),Finset.mul_sum]
+  simp_rw [hp]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro j _
+  rw [←Finset.mul_sum,←tsum_eq_sum_orbits root hcov hsep (fun x => m x (root j)) (hcol (root j))]
+
+end HaarAction
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Transport/InvariantRoot.lean b/lean/OAI/Probability/UnimodularPercolation/Transport/InvariantRoot.lean
index ecf3f246506e3d32f0243d015490ccb4e7cd55c3..a79472af9ed8cf6577df03dddbb4824ef2f64247 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Transport/InvariantRoot.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Transport/InvariantRoot.lean
@@ -1,126 +1,126 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Main
-import OAI.Probability.UnimodularPercolation.Graph.FullTopology
-
-namespace OAI
-
-/-! Haar measures and root transport for the full automorphism group. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-open MeasureTheory Topology
-open scoped ENNReal
-universe u v
-variable {V : Type u} {E : Type v}
-variable [TopologicalSpace V] [DiscreteTopology V]
-
-noncomputable instance fullGraphAutMeasurableSpace (G : BondGraph V E) :
-    MeasurableSpace (GraphAut G) := borel (GraphAut G)
-
-instance graphAutBorel (G : BondGraph V E) : BorelSpace (GraphAut G) := ⟨rfl⟩
-
-noncomputable def graphAutHaar (G : BondGraph V E) (hinj : Function.Injective G)
-    (hloc : LocallyFinite G) (hconn : Connected G) (x : V) : Measure (GraphAut G) := by
-  letI := locallyCompact_graphAut G hinj hloc hconn x
-  exact Measure.haar
-
-instance graphAutHaar_isHaar (G : BondGraph V E) (hinj : Function.Injective G)
-    (hloc : LocallyFinite G) (hconn : Connected G) (x : V) :
-    (graphAutHaar G hinj hloc hconn x).IsHaarMeasure := by
-  let := locallyCompact_graphAut G hinj hloc hconn x
-  exact inferInstanceAs (Measure.IsHaarMeasure Measure.haar)
-
-omit [TopologicalSpace V] [DiscreteTopology V] in
-theorem graphStabilizer_eq_action (G : BondGraph V E) (x : V) :
-    graphStabilizer G x = actionStabilizer (H := GraphAut G) x := rfl
-
-omit [TopologicalSpace V] [DiscreteTopology V] in
-theorem graphStabilizer_inv (G : BondGraph V E) (x : V) :
-    (graphStabilizer G x)⁻¹ = graphStabilizer G x := by
-  ext g
-  change (graphAutPerm G g).symm x = x ↔ graphAutPerm G g x = x
-  rw [Equiv.symm_apply_eq,eq_comm]
-
-theorem graphAutHaar_stabilizer_pos (G : BondGraph V E)
-    (hinj : Function.Injective G) (hloc : LocallyFinite G) (hconn : Connected G) (x y : V) :
-    0 < (graphAutHaar G hinj hloc hconn x).real (actionStabilizer (H := GraphAut G) y) := by
-  apply ENNReal.toReal_pos
-  · exact ((isOpen_graphStabilizer G y).measure_pos _ ⟨1,rfl⟩).ne'
-  · exact (isCompact_graphStabilizer G hinj hloc hconn y).measure_lt_top.ne
-
-theorem graphAutHaar_stabilizer_finite (G : BondGraph V E)
-    (hinj : Function.Injective G) (hloc : LocallyFinite G) (hconn : Connected G) (x y : V) :
-    graphAutHaar G hinj hloc hconn x (actionStabilizer (H := GraphAut G) y) ≠ ∞ :=
-  (isCompact_graphStabilizer G hinj hloc hconn y).measure_lt_top.ne
-
-def GraphUnimodular (G : BondGraph V E) (hinj : Function.Injective G)
-    (hloc : LocallyFinite G) (hconn : Connected G) (x : V) : Prop :=
-  (graphAutHaar G hinj hloc hconn x).IsMulRightInvariant
-
-theorem graphAutHaar_invInvariant [Countable V] (G : BondGraph V E)
-    (hinj : Function.Injective G) (hloc : LocallyFinite G) (hconn : Connected G) (x : V)
-    (hu : GraphUnimodular G hinj hloc hconn x) :
-    (graphAutHaar G hinj hloc hconn x).IsInvInvariant := by
-  let := locallyCompact_graphAut G hinj hloc hconn x
-  let μ := graphAutHaar G hinj hloc hconn x
-  let : μ.IsMulRightInvariant := hu
-  constructor
-  have hc := Measure.isMulLeftInvariant_eq_smul_of_regular μ.inv μ
-  have hf : μ (graphStabilizer G x) ≠ ∞ :=
-    (isCompact_graphStabilizer G hinj hloc hconn x).measure_lt_top.ne
-  have hp : μ (graphStabilizer G x) ≠ 0 :=
-    ((isOpen_graphStabilizer G x).measure_pos μ ⟨1,rfl⟩).ne'
-  have he := congrArg (fun ν : Measure (GraphAut G) => ν (graphStabilizer G x)) hc
-  rw [Measure.inv_apply,graphStabilizer_inv,Measure.smul_apply] at he
-  have hz : (μ.inv.haarScalarFactor μ : ℝ≥0∞) = 1 := by
-    apply (ENNReal.mul_left_inj hp hf).mp
-    simpa only [ENNReal.smul_def,smul_eq_mul,one_mul] using he.symm
-  rw [hc,show μ.inv.haarScalarFactor μ = 1 from ENNReal.coe_eq_one.mp hz,one_smul]
-
-theorem exists_rootTransport_of_unimodular [Nonempty V]
-    (G : BondGraph V E) (hinj : Function.Injective G)
-    (hloc : LocallyFinite G) (hconn : Connected G) (hqt : QuasiTransitive G) (x₀ : V)
-    (hu : GraphUnimodular G hinj hloc hconn x₀) :
-    ∃ (n : ℕ) (root : Fin n → V) (weight : Fin n → ℝ),
-      weight = inverseStabilizerWeight (graphAutHaar G hinj hloc hconn x₀) root ∧
-      (∀ i, 0 < weight i) ∧ (∑ i, weight i = 1) ∧
-      RootTransport root weight (vertexAut G) ∧
-      (∀ x, ∃ i, ∃ g ∈ vertexAut G, g (root i) = x) := by
-  classical
-  let : Countable V := countable_vertices hloc hconn
-  let : MeasurableSpace V := borel V
-  let : BorelSpace V := ⟨rfl⟩
-  let : Finite (GraphOrbit G) := finite_graphOrbit G hqt
-  let : Fintype (GraphOrbit G) := Fintype.ofFinite _
-  let n := Fintype.card (GraphOrbit G)
-  let q : Fin n ≃ GraphOrbit G := (Fintype.equivFin _).symm
-  let root : Fin n → V := fun i => fullGraphOrbitRoot G (q i)
-  let μ := graphAutHaar G hinj hloc hconn x₀
-  let : μ.IsInvInvariant := graphAutHaar_invInvariant G hinj hloc hconn x₀ hu
-  have hcov (x : V) : ∃ i : Fin n, x ∈ actionOrbit (H := GraphAut G) (root i) := by
-    obtain ⟨i,g,hg⟩ := graphOrbitRoot_covers G x
-    refine ⟨q.symm i,g,?_⟩
-    simpa only [root,Equiv.apply_symm_apply] using hg
-  let : Nonempty (Fin n) := ⟨(hcov x₀).choose⟩
-  have hsep (i j : Fin n) (h : root j ∈ actionOrbit (H := GraphAut G) (root i)) : i = j :=
-    q.injective (graphOrbitRoot_separates G (q i) (q j) h)
-  have hvol (i : Fin n) : 0 < μ.real (actionStabilizer (H := GraphAut G) (root i)) :=
-    graphAutHaar_stabilizer_pos G hinj hloc hconn x₀ (root i)
-  refine ⟨n,root,inverseStabilizerWeight μ root,rfl,
-    inverseStabilizerWeight_pos μ root hvol,
-    inverseStabilizerWeight_sum μ root hvol,?_,?_⟩
-  · apply rootTransport_of_invariant_measure μ
-      (fun x => (continuous_aut_apply G x).measurable) root hvol
-      (fun i => graphAutHaar_stabilizer_finite G hinj hloc hconn x₀ (root i))
-      hcov hsep (vertexAut G)
-    intro g
-    exact graphAutPerm_mem G g
-  · intro x
-    obtain ⟨i,g,hg⟩ := hcov x
-    exact ⟨i,graphAutPerm G g,graphAutPerm_mem G g,hg⟩
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Main
+import OAI.Probability.UnimodularPercolation.Graph.FullTopology
+
+namespace OAI
+
+/-! Haar measures and root transport for the full automorphism group. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+open MeasureTheory Topology
+open scoped ENNReal
+universe u v
+variable {V : Type u} {E : Type v}
+variable [TopologicalSpace V] [DiscreteTopology V]
+
+noncomputable instance fullGraphAutMeasurableSpace (G : BondGraph V E) :
+    MeasurableSpace (GraphAut G) := borel (GraphAut G)
+
+instance graphAutBorel (G : BondGraph V E) : BorelSpace (GraphAut G) := ⟨rfl⟩
+
+noncomputable def graphAutHaar (G : BondGraph V E) (hinj : Function.Injective G)
+    (hloc : LocallyFinite G) (hconn : Connected G) (x : V) : Measure (GraphAut G) := by
+  letI := locallyCompact_graphAut G hinj hloc hconn x
+  exact Measure.haar
+
+instance graphAutHaar_isHaar (G : BondGraph V E) (hinj : Function.Injective G)
+    (hloc : LocallyFinite G) (hconn : Connected G) (x : V) :
+    (graphAutHaar G hinj hloc hconn x).IsHaarMeasure := by
+  let := locallyCompact_graphAut G hinj hloc hconn x
+  exact inferInstanceAs (Measure.IsHaarMeasure Measure.haar)
+
+omit [TopologicalSpace V] [DiscreteTopology V] in
+theorem graphStabilizer_eq_action (G : BondGraph V E) (x : V) :
+    graphStabilizer G x = actionStabilizer (H := GraphAut G) x := rfl
+
+omit [TopologicalSpace V] [DiscreteTopology V] in
+theorem graphStabilizer_inv (G : BondGraph V E) (x : V) :
+    (graphStabilizer G x)⁻¹ = graphStabilizer G x := by
+  ext g
+  change (graphAutPerm G g).symm x = x ↔ graphAutPerm G g x = x
+  rw [Equiv.symm_apply_eq,eq_comm]
+
+theorem graphAutHaar_stabilizer_pos (G : BondGraph V E)
+    (hinj : Function.Injective G) (hloc : LocallyFinite G) (hconn : Connected G) (x y : V) :
+    0 < (graphAutHaar G hinj hloc hconn x).real (actionStabilizer (H := GraphAut G) y) := by
+  apply ENNReal.toReal_pos
+  · exact ((isOpen_graphStabilizer G y).measure_pos _ ⟨1,rfl⟩).ne'
+  · exact (isCompact_graphStabilizer G hinj hloc hconn y).measure_lt_top.ne
+
+theorem graphAutHaar_stabilizer_finite (G : BondGraph V E)
+    (hinj : Function.Injective G) (hloc : LocallyFinite G) (hconn : Connected G) (x y : V) :
+    graphAutHaar G hinj hloc hconn x (actionStabilizer (H := GraphAut G) y) ≠ ∞ :=
+  (isCompact_graphStabilizer G hinj hloc hconn y).measure_lt_top.ne
+
+def GraphUnimodular (G : BondGraph V E) (hinj : Function.Injective G)
+    (hloc : LocallyFinite G) (hconn : Connected G) (x : V) : Prop :=
+  (graphAutHaar G hinj hloc hconn x).IsMulRightInvariant
+
+theorem graphAutHaar_invInvariant [Countable V] (G : BondGraph V E)
+    (hinj : Function.Injective G) (hloc : LocallyFinite G) (hconn : Connected G) (x : V)
+    (hu : GraphUnimodular G hinj hloc hconn x) :
+    (graphAutHaar G hinj hloc hconn x).IsInvInvariant := by
+  let := locallyCompact_graphAut G hinj hloc hconn x
+  let μ := graphAutHaar G hinj hloc hconn x
+  let : μ.IsMulRightInvariant := hu
+  constructor
+  have hc := Measure.isMulLeftInvariant_eq_smul_of_regular μ.inv μ
+  have hf : μ (graphStabilizer G x) ≠ ∞ :=
+    (isCompact_graphStabilizer G hinj hloc hconn x).measure_lt_top.ne
+  have hp : μ (graphStabilizer G x) ≠ 0 :=
+    ((isOpen_graphStabilizer G x).measure_pos μ ⟨1,rfl⟩).ne'
+  have he := congrArg (fun ν : Measure (GraphAut G) => ν (graphStabilizer G x)) hc
+  rw [Measure.inv_apply,graphStabilizer_inv,Measure.smul_apply] at he
+  have hz : (μ.inv.haarScalarFactor μ : ℝ≥0∞) = 1 := by
+    apply (ENNReal.mul_left_inj hp hf).mp
+    simpa only [ENNReal.smul_def,smul_eq_mul,one_mul] using he.symm
+  rw [hc,show μ.inv.haarScalarFactor μ = 1 from ENNReal.coe_eq_one.mp hz,one_smul]
+
+theorem exists_rootTransport_of_unimodular [Nonempty V]
+    (G : BondGraph V E) (hinj : Function.Injective G)
+    (hloc : LocallyFinite G) (hconn : Connected G) (hqt : QuasiTransitive G) (x₀ : V)
+    (hu : GraphUnimodular G hinj hloc hconn x₀) :
+    ∃ (n : ℕ) (root : Fin n → V) (weight : Fin n → ℝ),
+      weight = inverseStabilizerWeight (graphAutHaar G hinj hloc hconn x₀) root ∧
+      (∀ i, 0 < weight i) ∧ (∑ i, weight i = 1) ∧
+      RootTransport root weight (vertexAut G) ∧
+      (∀ x, ∃ i, ∃ g ∈ vertexAut G, g (root i) = x) := by
+  classical
+  let : Countable V := countable_vertices hloc hconn
+  let : MeasurableSpace V := borel V
+  let : BorelSpace V := ⟨rfl⟩
+  let : Finite (GraphOrbit G) := finite_graphOrbit G hqt
+  let : Fintype (GraphOrbit G) := Fintype.ofFinite _
+  let n := Fintype.card (GraphOrbit G)
+  let q : Fin n ≃ GraphOrbit G := (Fintype.equivFin _).symm
+  let root : Fin n → V := fun i => fullGraphOrbitRoot G (q i)
+  let μ := graphAutHaar G hinj hloc hconn x₀
+  let : μ.IsInvInvariant := graphAutHaar_invInvariant G hinj hloc hconn x₀ hu
+  have hcov (x : V) : ∃ i : Fin n, x ∈ actionOrbit (H := GraphAut G) (root i) := by
+    obtain ⟨i,g,hg⟩ := graphOrbitRoot_covers G x
+    refine ⟨q.symm i,g,?_⟩
+    simpa only [root,Equiv.apply_symm_apply] using hg
+  let : Nonempty (Fin n) := ⟨(hcov x₀).choose⟩
+  have hsep (i j : Fin n) (h : root j ∈ actionOrbit (H := GraphAut G) (root i)) : i = j :=
+    q.injective (graphOrbitRoot_separates G (q i) (q j) h)
+  have hvol (i : Fin n) : 0 < μ.real (actionStabilizer (H := GraphAut G) (root i)) :=
+    graphAutHaar_stabilizer_pos G hinj hloc hconn x₀ (root i)
+  refine ⟨n,root,inverseStabilizerWeight μ root,rfl,
+    inverseStabilizerWeight_pos μ root hvol,
+    inverseStabilizerWeight_sum μ root hvol,?_,?_⟩
+  · apply rootTransport_of_invariant_measure μ
+      (fun x => (continuous_aut_apply G x).measurable) root hvol
+      (fun i => graphAutHaar_stabilizer_finite G hinj hloc hconn x₀ (root i))
+      hcov hsep (vertexAut G)
+    intro g
+    exact graphAutPerm_mem G g
+  · intro x
+    obtain ⟨i,g,hg⟩ := hcov x
+    exact ⟨i,graphAutPerm G g,graphAutPerm_mem G g,hg⟩
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/Transport/TrajectoryMass.lean b/lean/OAI/Probability/UnimodularPercolation/Transport/TrajectoryMass.lean
index 31734f0426dcc0a262469a0fb2ad9de7507f869b..12cf13436bb83ed040c30f915f3db097742cb93f 100644
--- a/lean/OAI/Probability/UnimodularPercolation/Transport/TrajectoryMass.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/Transport/TrajectoryMass.lean
@@ -1,1323 +1,1323 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Splicing.MomentBounds
-
-namespace OAI
-
-/-! Trajectory Mass for bond percolation. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section OldPathWeight
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-noncomputable def finiteFieldIntegral (Q : Finset E) (p : ℝ) (f : Set E → ℝ) : ℝ :=
-  ∑ ω : FiniteField Q, finiteFieldWeight Q p ω * f (finiteFieldSet Q ω)
-
-theorem finiteFieldIntegral_nonneg (Q : Finset E) (p : unitInterval) (f : Set E → ℝ)
-    (hf : ∀ ω, 0 ≤ f ω) : 0 ≤ finiteFieldIntegral Q p f :=
-  Finset.sum_nonneg fun ω _ => mul_nonneg (finiteFieldWeight_nonneg Q p ω) (hf _)
-
-theorem finiteFieldIntegral_sum {I : Type*} (Q : Finset E) (p : ℝ) (S : Finset I)
-    (f : I → Set E → ℝ) :
-    finiteFieldIntegral Q p (fun ω => ∑ i ∈ S, f i ω) =
-      ∑ i ∈ S, finiteFieldIntegral Q p (f i) := by
-  simp only [finiteFieldIntegral, Finset.mul_sum]
-  exact Finset.sum_comm
-
- 
-
-theorem finiteFamilyIntegral_prod (Q : Finset E) (J : Finset ℕ) (p : ℝ)
-    (f : ↥J → Set E → ℝ) :
-    finiteFamilyIntegral Q J p (fun η => ∏ i : ↥J, f i (η i)) =
-      ∏ i : ↥J, finiteFieldIntegral Q p (f i) := by
-  simp only [finiteFamilyIntegral, finiteFamilyWeight, finiteFieldFamily_at,
-    ← Finset.prod_mul_distrib, finiteFieldIntegral]
-  exact (Fintype.prod_sum (fun (i : ↥J) (ω : FiniteField Q) =>
-    finiteFieldWeight Q p ω * f i (finiteFieldSet Q ω))).symm
-
- 
-
-noncomputable def oldPathWeight (k : ℕ) (q : Set E → V → V → ℝ)
-    (y : ℕ → V) (η : ℕ → Set E) : ℝ :=
-  ∏ i : ↥(Finset.Icc 1 k), q (η i) (y i) (y (i.val-1))
-
-theorem oldPathWeight_nonneg (k : ℕ) (q : Set E → V → V → ℝ)
-    (hq : ∀ ω a b, 0 ≤ q ω a b) (y : ℕ → V) (η : ℕ → Set E) :
-    0 ≤ oldPathWeight k q y η :=
-  Finset.prod_nonneg fun _ _ => hq _ _ _
-
-theorem oldPathWeight_integral (Q : Finset E) (k : ℕ) (p : ℝ)
-    (q : Set E → V → V → ℝ) (y : ℕ → V) :
-    finiteFamilyIntegral Q (Finset.Icc 1 k) p (oldPathWeight k q y) =
-      ∏ i : ↥(Finset.Icc 1 k), finiteFieldIntegral Q p (fun ω => q ω (y i) (y (i.val-1))) := by
-  unfold oldPathWeight
-  exact finiteFamilyIntegral_prod Q (Finset.Icc 1 k) p
-    (fun i ω => q ω (y i) (y (i.val-1)))
-
- 
-
-theorem oldPathWeight_support (G : BondGraph V E) (k M : ℕ) (L : ℝ)
-    (q : Set E → V → V → ℝ)
-    (hs : ∀ ω a b, q ω a b ≠ 0 → b ∈ cluster G ω a ∧
-      (cluster G ω a).Finite ∧ (cluster G ω a).ncard ≤ M ∧ GoodForScale G ω a L)
-    (y : ℕ → V) (η : ℕ → Set E) (hy : oldPathWeight k q y η ≠ 0) :
-    ∀ i, 1 ≤ i → i ≤ k → y (i-1) ∈ cluster G (η i) (y i) ∧
-      (cluster G (η i) (y i)).Finite ∧ (cluster G (η i) (y i)).ncard ≤ M ∧
-      GoodForScale G (η i) (y i) L := by
-  intro i hi hik
-  apply hs
-  have h := (Finset.prod_ne_zero_iff.mp hy) ⟨i, Finset.mem_Icc.mpr ⟨hi,hik⟩⟩ (Finset.mem_univ _)
-  exact h
-
-end OldPathWeight
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section FiniteOldRun
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {W : Type*}
-
- 
-def oldRunConsEquiv (k : ℕ) : (Fin (k+1) → W) ≃ W × (Fin k → W) where
-  toFun y := (y 0, Fin.tail y)
-  invFun y := Fin.cons y.1 y.2
-  left_inv := Fin.cons_self_tail
-  right_inv y := by simp
-
- 
-
-noncomputable def oldRunWeight (K : ℕ → W → W → ℝ) : (k : ℕ) → W → (Fin k → W) → ℝ
-  | 0, _, _ => 1
-  | k+1, z, y => K (k+1) z (y 0) * oldRunWeight K k (y 0) (Fin.tail y)
-
-theorem oldRunWeight_nonneg (K : ℕ → W → W → ℝ) (hK : ∀ i a b, 0 ≤ K i a b)
-    (k : ℕ) (z : W) (y : Fin k → W) : 0 ≤ oldRunWeight K k z y := by
-  induction k generalizing z with
-  | zero => exact zero_le_one
-  | succ k ih => exact mul_nonneg (hK _ _ _) (ih _ _)
-
-theorem sum_oldRunWeight_succ [Fintype W] (K : ℕ → W → W → ℝ) (k : ℕ) (z : W) :
-    (∑ y : Fin (k+1) → W, oldRunWeight K (k+1) z y) =
-      ∑ a : W, K (k+1) z a * ∑ y : Fin k → W, oldRunWeight K k a y := by
-  rw [Fintype.sum_equiv (oldRunConsEquiv k) _
-    (fun y => K (k+1) z y.1 * oldRunWeight K k y.1 y.2) (by intro y; rfl),
-    Fintype.sum_prod_type]
-  simp only [Finset.mul_sum]
-
- 
-
-theorem sum_oldRunWeight_le [Fintype W] (K : ℕ → W → W → ℝ)
-    (hK : ∀ i a b, 0 ≤ K i a b) (u : ℝ) (hu : 0 ≤ u)
-    (hrow : ∀ i a, ∑ b, K i a b ≤ u) (k : ℕ) (z : W) :
-    (∑ y : Fin k → W, oldRunWeight K k z y) ≤ u^k := by
-  induction k generalizing z with
-  | zero => simp [oldRunWeight]
-  | succ k ih =>
-    rw [sum_oldRunWeight_succ]
-    calc
-      _ ≤ ∑ a : W, K (k+1) z a * u^k := Finset.sum_le_sum fun a _ =>
-        mul_le_mul_of_nonneg_left (ih a) (hK _ _ _)
-      _ = (∑ a : W, K (k+1) z a) * u^k := (Finset.sum_mul _ _ _).symm
-      _ ≤ u * u^k := mul_le_mul_of_nonneg_right (hrow _ _) (pow_nonneg hu _)
-      _ = u^(k+1) := by ring
-
-theorem sum_oldRunWeight_eq [Fintype W] (K : ℕ → W → W → ℝ)
-    (u : ℝ) (hrow : ∀ i a, ∑ b, K i a b = u) (k : ℕ) (z : W) :
-    (∑ y : Fin k → W, oldRunWeight K k z y) = u^k := by
-  induction k generalizing z with
-  | zero => simp [oldRunWeight]
-  | succ k ih =>
-    simp only [sum_oldRunWeight_succ, ih, ← Finset.sum_mul, hrow]
-    ring
-
- 
-
-def oldRunVertices : (k : ℕ) → W → (Fin k → W) → ℕ → W
-  | 0, z, _ => fun _ => z
-  | k+1, z, y => fun i => if i ≤ k then oldRunVertices k (y 0) (Fin.tail y) i else z
-
-@[simp] theorem oldRunVertices_last (k : ℕ) (z : W) (y : Fin k → W) :
-    oldRunVertices k z y k = z := by
-  cases k <;> simp [oldRunVertices]
-
-theorem oldRunVertices_succ_of_le (k : ℕ) (z : W) (y : Fin (k+1) → W)
-    (i : ℕ) (hi : i ≤ k) :
-    oldRunVertices (k+1) z y i = oldRunVertices k (y 0) (Fin.tail y) i := by
-  simp only [oldRunVertices, ite_eq_left hi]
-
- 
-
-theorem oldPathWeight_oldRunVertices (k : ℕ) (q : Set E → W → W → ℝ)
-    (η : ℕ → Set E) (z : W) (y : Fin k → W) :
-    oldPathWeight k q (oldRunVertices k z y) η =
-      oldRunWeight (fun i => q (η i)) k z y := by
-  induction k generalizing z with
-  | zero =>
-    unfold oldPathWeight
-    rw [Finset.prod_coe_sort (Finset.Icc 1 0)
-      (fun i => q (η i) (oldRunVertices 0 z y i) (oldRunVertices 0 z y (i-1)))]
-    simp [oldRunWeight]
-  | succ k ih =>
-    unfold oldPathWeight
-    rw [Finset.prod_coe_sort (Finset.Icc 1 (k+1))
-      (fun i => q (η i) (oldRunVertices (k+1) z y i) (oldRunVertices (k+1) z y (i-1))),
-      Finset.prod_Icc_succ_top (by omega)]
-    have he : (∏ i ∈ Finset.Icc 1 k,
-        q (η i) (oldRunVertices (k+1) z y i) (oldRunVertices (k+1) z y (i-1))) =
-        oldPathWeight k q (oldRunVertices k (y 0) (Fin.tail y)) η := by
-      unfold oldPathWeight
-      rw [Finset.prod_coe_sort (Finset.Icc 1 k)
-        (fun i => q (η i) (oldRunVertices k (y 0) (Fin.tail y) i)
-          (oldRunVertices k (y 0) (Fin.tail y) (i-1)))]
-      apply Finset.prod_congr rfl
-      intro i hi
-      have hi' := Finset.mem_Icc.mp hi
-      rw [oldRunVertices_succ_of_le _ _ _ _ hi'.2,
-        oldRunVertices_succ_of_le _ _ _ _ (by omega)]
-    rw [he, ih, oldRunVertices_last, Nat.add_sub_cancel,
-      oldRunVertices_succ_of_le _ _ _ _ (le_refl _), oldRunVertices_last]
-    exact mul_comm _ _
-
-end FiniteOldRun
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section IndependentOldRun
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {W : Type*}
-
- 
-noncomputable def finiteOldKernel (Q : Finset E) (p : ℝ)
-    (q : Set E → W → W → ℝ) (a b : W) : ℝ :=
-  finiteFieldIntegral Q p (fun ω => q ω a b)
-
-theorem finiteOldKernel_nonneg (Q : Finset E) (p : unitInterval)
-    (q : Set E → W → W → ℝ) (hq : ∀ ω a b, 0 ≤ q ω a b) (a b : W) :
-    0 ≤ finiteOldKernel Q p q a b :=
-  finiteFieldIntegral_nonneg Q p _ (fun _ => hq _ _ _)
-
- 
-
-theorem oldRunWeight_integral (Q : Finset E) (p : ℝ) (k : ℕ)
-    (q : Set E → W → W → ℝ) (z : W) (y : Fin k → W) :
-    finiteFamilyIntegral Q (Finset.Icc 1 k) p
-      (fun η => oldRunWeight (fun i => q (η i)) k z y) =
-    oldRunWeight (fun _ => finiteOldKernel Q p q) k z y := by
-  calc
-    _ = finiteFamilyIntegral Q (Finset.Icc 1 k) p
-        (fun η => oldPathWeight k q (oldRunVertices k z y) η) := by
-      apply congrArg (finiteFamilyIntegral Q (Finset.Icc 1 k) p)
-      funext η
-      exact (oldPathWeight_oldRunVertices k q η z y).symm
-    _ = ∏ i : ↥(Finset.Icc 1 k), finiteOldKernel Q p q
-        (oldRunVertices k z y i) (oldRunVertices k z y (i.val-1)) :=
-      oldPathWeight_integral Q k p q _
-    _ = oldRunWeight (fun _ => finiteOldKernel Q p q) k z y :=
-      oldPathWeight_oldRunVertices k (fun (_ : Set E) => finiteOldKernel Q p q)
-        (fun _ => ∅) z y
-
- 
-
-theorem independent_oldRun_success [Fintype W] (Q : Finset E) (p : ℝ) (k : ℕ)
-    (q : Set E → W → W → ℝ) (u : ℝ)
-    (hrow : ∀ a, ∑ b, finiteOldKernel Q p q a b = u) (z : W) :
-    finiteFamilyIntegral Q (Finset.Icc 1 k) p
-      (fun η => ∑ y : Fin k → W, oldRunWeight (fun i => q (η i)) k z y) = u^k := by
-  rw [finiteFamilyIntegral_sum]
-  simp only [oldRunWeight_integral]
-  exact sum_oldRunWeight_eq _ u (fun _ => hrow) k z
-
- 
-
-theorem independent_oldRun_success_le [Fintype W] (Q : Finset E) (p : unitInterval) (k : ℕ)
-    (q : Set E → W → W → ℝ) (hq : ∀ ω a b, 0 ≤ q ω a b) (u : ℝ) (hu : 0 ≤ u)
-    (hrow : ∀ a, ∑ b, finiteOldKernel Q p q a b ≤ u) (z : W) :
-    finiteFamilyIntegral Q (Finset.Icc 1 k) p
-      (fun η => ∑ y : Fin k → W, oldRunWeight (fun i => q (η i)) k z y) ≤ u^k := by
-  rw [finiteFamilyIntegral_sum]
-  simp only [oldRunWeight_integral]
-  exact sum_oldRunWeight_le _ (fun _ => finiteOldKernel_nonneg Q p q hq) u hu (fun _ => hrow) k z
-
-end IndependentOldRun
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section WeightedCodeFiber
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem sum_comp_le_sum_of_injective {A B : Type*} [Fintype A] [Fintype B]
-    (f : A → B) (hf : Function.Injective f) (w : B → ℝ) (hw : ∀ b, 0 ≤ w b) :
-    (∑ a, w (f a)) ≤ ∑ b, w b := by
-  rw [← Finset.sum_image hf.injOn]
-  exact Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _)
-    (fun b _ _ => hw b)
-
- 
-theorem finite_sum_group_weight {A B : Type*} [Fintype A] [Fintype B]
-    (f : A → B) (m : A → ℝ) (w : B → ℝ) :
-    (∑ a, m a * w (f a)) = ∑ b, w b * ∑ a, if f a = b then m a else 0 := by
-  simp only [Finset.mul_sum]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro a _
-  simp only [mul_ite,mul_zero,Finset.sum_ite_eq,Finset.mem_univ,ite_true]
-  exact mul_comm _ _
-
- 
-
-theorem finite_choice_fiber_le {A B C P : Type*} [Fintype A] [Fintype P]
-    (f : A → B) (ctx : A → C) (choice : A → P)
-    (hir : Function.Injective (fun a => (ctx a,choice a,f a)))
-    (hctx : ∀ a b, f a = f b → ctx a = ctx b)
-    (q : C → P → ℝ) (hq : ∀ c p, 0 ≤ q c p)
-    (hrow : ∀ c, ∑ p, q c p ≤ 1) (b : B) :
-    (∑ a, if f a = b then q (ctx a) (choice a) else 0) ≤ 1 := by
-  let J := {a : A // f a = b}
-  by_cases hne : Nonempty J
-  · let a₀ : J := Classical.choice hne
-    have hc (a : J) : ctx a.val = ctx a₀.val := hctx _ _ (a.property.trans a₀.property.symm)
-    have hi : Function.Injective (fun a : J => choice a.val) := by
-      intro a a' he
-      apply Subtype.ext
-      apply hir
-      exact Prod.ext (hctx _ _ (a.property.trans a'.property.symm))
-        (Prod.ext he (a.property.trans a'.property.symm))
-    have hb := sum_comp_le_sum_of_injective (fun a : J => choice a.val) hi
-      (q (ctx a₀.val)) (hq _)
-    have he : (∑ a, if f a = b then q (ctx a) (choice a) else 0) =
-        ∑ a : J, q (ctx a) (choice a) := by
-      rw [← Finset.sum_filter]
-      exact Finset.sum_subtype _ (by intro a; simp) _
-    rw [he]
-    simpa only [hc] using hb.trans (hrow (ctx a₀.val))
-  · have hn (a : A) : f a ≠ b := fun h => hne ⟨⟨a,h⟩⟩
-    simp only [ite_eq_right (hn _),Finset.sum_const_zero,zero_le_one]
-
- 
-
-theorem finite_weighted_code_mass_le {A B C P : Type*} [Fintype A] [Fintype B] [Fintype P]
-    (f : A → B) (ctx : A → C) (choice : A → P)
-    (hir : Function.Injective (fun a => (ctx a,choice a,f a)))
-    (hctx : ∀ a b, f a = f b → ctx a = ctx b)
-    (q : C → P → ℝ) (hq : ∀ c p, 0 ≤ q c p) (hrow : ∀ c, ∑ p, q c p ≤ 1)
-    (w : B → ℝ) (hw : ∀ b, 0 ≤ w b) :
-    (∑ a, q (ctx a) (choice a) * w (f a)) ≤ ∑ b, w b := by
-  rw [finite_sum_group_weight]
-  apply Finset.sum_le_sum
-  intro b _
-  calc
-    _ ≤ w b * 1 := mul_le_mul_of_nonneg_left
-      (finite_choice_fiber_le f ctx choice hir hctx q hq hrow b) (hw b)
-    _ = w b := mul_one _
-
-end WeightedCodeFiber
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section WeightedInverseLoad
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem finiteInverseLoad_weighted_le {I P : Type*} [Fintype I] [Fintype P]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (x z : V) (k M : ℕ)
-    (L : ℝ) (hL : 1 ≤ L)
-    (inp out : I → SpliceState E) (steps : I → List (ℕ × E)) (y : I → ℕ → V)
-    (choice : I → P) (hir : Function.Injective (fun a => (inp a,choice a,steps a)))
-    (q : SpliceState E → P → ℝ) (hq : ∀ s b, 0 ≤ q s b)
-    (hrow : ∀ s, ∑ b, q s b ≤ 1)
-    (htrace : ∀ a, SpliceTrace G x (inp a).2 (y a) k 0 (inp a).1 (steps a) (out a).1)
-    (hex : ∀ a, SpliceExecution G x (inp a) (steps a) (out a))
-    (hz : ∀ a, y a k = z) (hfin : ∀ a, (cluster G (out a).1 x).Finite)
-    (hgood : ∀ a i, 1 ≤ i → i ≤ k → (cluster G ((inp a).2 i) (y a i)).Finite ∧
-      (cluster G ((inp a).2 i) (y a i)).ncard ≤ M ∧ GoodForScale G ((inp a).2 i) (y a i) L)
-    (t : SpliceState E) :
-    finiteInverseLoad out (fun a => q (inp a) (choice a) * ∏ i : Fin k,
-      (traceContactCount G (inp a).2 (y a) (out a).1 x (steps a) (i.val+1):ℝ)⁻¹) t ≤
-      inverseCodeMass M L ^ k := by
-  let J := {a : I // out a = t}
-  let c (a : J) := gapSpliceCode G (out a.val).1 x z (steps a.val)
-  have hvalid (a : J) : ValidSparseGapCode k M (c a) := by
-    have hv := (htrace a.val).valid_gap_code G hloc x (inp a.val).2 (y a.val) k 0 M
-      (inp a.val).1 (out a.val).1 (steps a.val) (hfin a.val) L (hgood a.val)
-    simpa only [c,hz a.val] using hv.1
-  let arr (a : J) := sparseGapArray k M (c a) (hvalid a)
-  let ctx (a : J) := (inp a.val,steps a.val)
-  have hc (a b : J) (he : arr a = arr b) : ctx a = ctx b := by
-    have ho : out a.val = out b.val := a.property.trans b.property.symm
-    have hcode : c a = c b := sparseGapArray_injective k M (c a) (c b)
-      (hvalid a) (hvalid b) he
-    have htB : SpliceTrace G x (inp b.val).2 (y b.val) k 0
-        (inp b.val).1 (steps b.val) (out a.val).1 := by
-      rw [ho]
-      exact htrace b.val
-    have heB : SpliceExecution G x (inp b.val) (steps b.val) (out a.val) := ho.symm ▸ hex b.val
-    have hg : gapSpliceCode G (out a.val).1 x (y a.val k) (steps a.val) =
-        gapSpliceCode G (out a.val).1 x (y a.val k) (steps b.val) := by
-      simpa only [c,hz a.val,ho] using hcode
-    have hze : y a.val k = y b.val k := (hz a.val).trans (hz b.val).symm
-    have hs := (htrace a.val).code_unique_of_gapCode G x (inp a.val).2 (inp b.val).2
-      (y a.val) (y b.val) k 0 (inp a.val).1 (inp b.val).1 (out a.val).1
-      (steps a.val) (steps b.val) htB hze (hfin a.val) hg
-    have hi := (htrace a.val).input_unique_of_gapCode G x (inp a.val).2 (inp b.val).2
-      (y a.val) (y b.val) k 0 (inp a.val).1 (inp b.val).1 (out a.val).1
-      (steps a.val) (steps b.val) htB hze (hfin a.val) hg (out a.val).2 (hex a.val) heB
-    exact Prod.ext hi hs
-  have hi : Function.Injective (fun a : J => (ctx a,choice a.val,arr a)) := by
-    intro a b he
-    apply Subtype.ext
-    apply hir
-    have hec := congrArg Prod.fst he
-    have hin : inp a.val = inp b.val := congrArg (fun s : SpliceState E × List (ℕ × E) => s.1) hec
-    have hst : steps a.val = steps b.val := congrArg (fun s : SpliceState E × List (ℕ × E) => s.2) hec
-    exact Prod.ext hin (Prod.ext (congrArg (fun z => z.2.1) he) hst)
-  let w (b : Fin k → Fin (M+1)) := ∏ i, gapLetterWeight L (b i).val
-  have hb := finite_weighted_code_mass_le arr ctx (fun a : J => choice a.val) hi hc
-    (fun s b => q s.1 b) (fun s => hq s.1) (fun s => hrow s.1) w
-    (fun b => Finset.prod_nonneg (fun i _ => gapLetterWeight_nonneg L (by linarith) _))
-  change (∑ a : J, q (inp a.val) (choice a.val) * w (arr a)) ≤ ∑ b, w b at hb
-  have he : finiteInverseLoad out (fun a => q (inp a) (choice a) * ∏ i : Fin k,
-      (traceContactCount G (inp a).2 (y a) (out a).1 x (steps a) (i.val+1):ℝ)⁻¹) t =
-      ∑ a : J, q (inp a.val) (choice a.val) * ∏ i : Fin k,
-      (traceContactCount G (inp a.val).2 (y a.val) (out a.val).1 x (steps a.val) (i.val+1):ℝ)⁻¹ := by
-    rw [finiteInverseLoad,← Finset.sum_filter]
-    exact Finset.sum_subtype _ (by intro a; simp) _
-  rw [he]
-  calc
-    _ ≤ ∑ a : J, q (inp a.val) (choice a.val) * w (arr a) := by
-      apply Finset.sum_le_sum
-      intro a _
-      apply mul_le_mul_of_nonneg_left _ (hq _ _)
-      apply Finset.prod_le_prod₀ (fun _ _ => by positivity)
-      intro i _
-      have h := ((htrace a.val).contact_weight_bound G hloc x (inp a.val).2 (y a.val)
-        k 0 M (inp a.val).1 (out a.val).1 (steps a.val) (hfin a.val) L hL
-        (hgood a.val) (i.val+1)).2
-      simpa only [w,arr,sparseGapArray,c,hz a.val] using h
-    _ ≤ ∑ b, w b := hb
-    _ = _ := total_gap_code_mass k M L
-
-end WeightedInverseLoad
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section OldRunInverseLoad
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem finiteInverseLoad_oldRun_le {I W : Type*} [Fintype I] [Fintype W]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (x z : V) (k M : ℕ)
-    (L : ℝ) (hL : 1 ≤ L)
-    (inp out : I → SpliceState E) (steps : I → List (ℕ × E)) (y : I → ℕ → V)
-    (start : W) (path : I → Fin k → W)
-    (hir : Function.Injective (fun a => (inp a,path a,steps a)))
-    (q : Set E → W → W → ℝ) (hq : ∀ ω a b, 0 ≤ q ω a b)
-    (hrow : ∀ ω a, ∑ b, q ω a b ≤ 1)
-    (htrace : ∀ a, SpliceTrace G x (inp a).2 (y a) k 0 (inp a).1 (steps a) (out a).1)
-    (hex : ∀ a, SpliceExecution G x (inp a) (steps a) (out a))
-    (hz : ∀ a, y a k = z) (hfin : ∀ a, (cluster G (out a).1 x).Finite)
-    (hgood : ∀ a i, 1 ≤ i → i ≤ k → (cluster G ((inp a).2 i) (y a i)).Finite ∧
-      (cluster G ((inp a).2 i) (y a i)).ncard ≤ M ∧ GoodForScale G ((inp a).2 i) (y a i) L)
-    (t : SpliceState E) :
-    finiteInverseLoad out (fun a => oldRunWeight (fun i => q ((inp a).2 i)) k start (path a) *
-      ∏ i : Fin k, (traceContactCount G (inp a).2 (y a) (out a).1 x (steps a) (i.val+1):ℝ)⁻¹) t ≤
-      inverseCodeMass M L ^ k := by
-  apply finiteInverseLoad_weighted_le G hloc x z k M L hL inp out steps y path hir
-    (fun s b => oldRunWeight (fun i => q (s.2 i)) k start b)
-    (fun s b => oldRunWeight_nonneg _ (fun _ => hq _) k start b) _ htrace hex hz hfin hgood t
-  intro s
-  simpa only [one_pow] using sum_oldRunWeight_le (fun i => q (s.2 i))
-    (fun _ => hq _) 1 zero_le_one (fun _ => hrow _) k start
-
-end OldRunInverseLoad
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section ChoiceForwardDensity
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {P : Type*}
-
- 
-
-abbrev FiniteChoiceTraceIndex (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
-    (A : P → Finset (SpliceState E)) (y : P → SpliceState E → ℕ → V) :=
-  (r : P) × FiniteTraceIndex G x k Q (A r) (y r)
-
-def choiceTraceInput (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
-    (A : P → Finset (SpliceState E)) (y : P → SpliceState E → ℕ → V)
-    (a : FiniteChoiceTraceIndex G x k Q A y) : SpliceState E :=
-  traceIndexInput G x k Q (A a.1) (y a.1) a.2
-
-def choiceTraceOutput (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
-    (A : P → Finset (SpliceState E)) (y : P → SpliceState E → ℕ → V)
-    (a : FiniteChoiceTraceIndex G x k Q A y) : SpliceState E :=
-  traceIndexOutput G x k Q (A a.1) (y a.1) a.2
-
-def choiceTraceCode (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
-    (A : P → Finset (SpliceState E)) (y : P → SpliceState E → ℕ → V)
-    (a : FiniteChoiceTraceIndex G x k Q A y) : List (ℕ × E) :=
-  traceIndexCode G x k Q (A a.1) (y a.1) a.2
-
-noncomputable def choiceTraceMass (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
-    (A : P → Finset (SpliceState E)) (y : P → SpliceState E → ℕ → V)
-    (q : SpliceState E → P → ℝ) (a : FiniteChoiceTraceIndex G x k Q A y) : ℝ :=
-  q (choiceTraceInput G x k Q A y a) a.1 *
-    traceIndexMass G x k Q (A a.1) (y a.1) a.2
-
-theorem choiceTrace_injective (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
-    (A : P → Finset (SpliceState E)) (y : P → SpliceState E → ℕ → V) :
-    Function.Injective (fun a : FiniteChoiceTraceIndex G x k Q A y =>
-      (choiceTraceInput G x k Q A y a,a.1,choiceTraceCode G x k Q A y a)) := by
-  rintro ⟨r,a⟩ ⟨r',b⟩ he
-  have hr : r = r' := congrArg (fun z => z.2.1) he
-  subst r'
-  have hin : traceIndexInput G x k Q (A r) (y r) a =
-      traceIndexInput G x k Q (A r) (y r) b :=
-    congrArg (fun z : SpliceState E × P × List (ℕ × E) => z.1) he
-  have hcode : traceIndexCode G x k Q (A r) (y r) a =
-      traceIndexCode G x k Q (A r) (y r) b :=
-    congrArg (fun z : SpliceState E × P × List (ℕ × E) => z.2.2) he
-  have hab : a = b := traceIndex_injective G x k Q (A r) (y r) (Prod.ext hin hcode)
-  subst b
-  rfl
-
-noncomputable def finiteChoiceForwardLoad [Fintype P]
-    (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
-    (A : P → Finset (SpliceState E)) (y : P → SpliceState E → ℕ → V)
-    (q : SpliceState E → P → ℝ) : SpliceState E → ℝ :=
-  finiteInverseLoad (choiceTraceOutput G x k Q A y) (choiceTraceMass G x k Q A y q)
-
- 
-theorem finiteChoiceForwardLoad_bounds [Fintype P]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (x z : V) (k M : ℕ)
-    (L : ℝ) (hL : 1 ≤ L) (Q : Finset E)
-    (A : P → Finset (SpliceState E)) (y : P → SpliceState E → ℕ → V)
-    (q : SpliceState E → P → ℝ) (hq : ∀ s r, 0 ≤ q s r)
-    (hrow : ∀ s, ∑ r, q s r ≤ 1)
-    (hA : ∀ r, A r ⊆ finiteSpliceDomain Q (Finset.Icc 1 k))
-    (hz : ∀ r s, s ∈ A r → y r s k = z)
-    (hgood : ∀ r s, s ∈ A r → ∀ i, 1 ≤ i → i ≤ k →
-      (cluster G (s.2 i) (y r s i)).Finite ∧
-      (cluster G (s.2 i) (y r s i)).ncard ≤ M ∧ GoodForScale G (s.2 i) (y r s i) L)
-    (t : SpliceState E) :
-    0 ≤ finiteChoiceForwardLoad G x k Q A y q t ∧
-      finiteChoiceForwardLoad G x k Q A y q t ≤ inverseCodeMass M L ^ k := by
-  constructor
-  · apply finiteInverseLoad_nonneg
-    intro a
-    exact mul_nonneg (hq _ _) (Finset.prod_nonneg (fun _ _ => by positivity))
-  · apply finiteInverseLoad_weighted_le G hloc x z k M L hL
-      (choiceTraceInput G x k Q A y) (choiceTraceOutput G x k Q A y)
-      (choiceTraceCode G x k Q A y) (fun a => y a.1 a.2.1.val) Sigma.fst
-      (choiceTrace_injective G x k Q A y) q hq hrow
-      (fun a => (traceIndex_valid G x k Q (A a.1) (y a.1) a.2).1)
-      (fun a => (traceIndex_valid G x k Q (A a.1) (y a.1) a.2).2)
-      (fun a => hz a.1 _ a.2.1.property) _
-      (fun a => hgood a.1 _ a.2.1.property) t
-    intro a
-    have hs := (mem_finiteSpliceDomain_iff Q (Finset.Icc 1 k) _).mp
-      (traceIndexOutput_mem G x k Q (A a.1) (y a.1) (hA a.1) a.2)
-    exact cluster_finite_of_finite_edges G (Q.finite_toSet.subset hs.1) x
-
- 
-
-theorem sum_traceIndexMass [Nonempty E] (G : BondGraph V E) (hloc : LocallyFinite G)
-    (x : V) (k M : ℕ) (Q : Finset E) (A : Finset (SpliceState E))
-    (y : SpliceState E → ℕ → V) (s : ↥A)
-    (hs : s.val ∈ finiteSpliceDomain Q (Finset.Icc 1 k))
-    (hy : y s.val 0 ∈ cluster G s.val.1 x)
-    (hchain : ∀ i ∈ Finset.Icc 1 k, Conn G (s.val.2 i) (y s.val i) (y s.val (i-1)))
-    (hcaps : ∀ i ∈ Finset.Icc 1 k, (cluster G (s.val.2 i) (y s.val i)).Finite ∧
-      (cluster G (s.val.2 i) (y s.val i)).ncard ≤ M) :
-    (∑ c : ↥(finiteTraceCodes G x s.val.2 (y s.val) k 0 s.val.1 Q),
-      traceIndexMass G x k Q A y ⟨s,c⟩) = 1 := by
-  have hs' := (mem_finiteSpliceDomain_iff Q (Finset.Icc 1 k) s.val).mp hs
-  have hn := sum_finite_trace_probabilities G hloc x s.val.2 (y s.val) k M Q
-    (fun i _ _ => hs'.2.1 i) hchain hcaps 0 (Nat.zero_le _) s.val.1 hy
-    (cluster_finite_of_finite_edges G (Q.finite_toSet.subset hs'.1) x)
-  rw [← Finset.sum_coe_sort (finiteTraceCodes G x s.val.2 (y s.val) k 0 s.val.1 Q)] at hn
-  rw [← hn]
-  apply Finset.sum_congr rfl
-  intro c _
-  have ht := ((mem_finiteTraceCodes_iff G x s.val.2 (y s.val) k 0 s.val.1 Q c.val).mp c.property).2
-  have hf := ht.runMasks_first G x s.val.2 (y s.val) k 0 s.val.1 _ c.val s.val.2 (by intros; rfl)
-  simp only [traceIndexMass,traceIndexOutput,hf]
-
- 
-theorem finiteChoiceForwardLoad_mean [Fintype P] [Nonempty E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) (k M : ℕ) (Q : Finset E) (p : ℝ)
-    (A : P → Finset (SpliceState E)) (y : P → SpliceState E → ℕ → V)
-    (q : SpliceState E → P → ℝ)
-    (hA : ∀ r, A r ⊆ finiteSpliceDomain Q (Finset.Icc 1 k))
-    (hy : ∀ r s, s ∈ A r → y r s 0 ∈ cluster G s.1 x)
-    (hchain : ∀ r s, s ∈ A r → ∀ i ∈ Finset.Icc 1 k, Conn G (s.2 i) (y r s i) (y r s (i-1)))
-    (hcaps : ∀ r s, s ∈ A r → ∀ i ∈ Finset.Icc 1 k,
-      (cluster G (s.2 i) (y r s i)).Finite ∧ (cluster G (s.2 i) (y r s i)).ncard ≤ M) :
-    (∑ t ∈ finiteSpliceDomain Q (Finset.Icc 1 k), spliceTupleWeight Q (Finset.Icc 1 k) p t *
-      finiteChoiceForwardLoad G x k Q A y q t) =
-      ∑ r, ∑ s ∈ A r, spliceTupleWeight Q (Finset.Icc 1 k) p s * q s r := by
-  have hd := finiteInverseLoad_density G x k Q p
-    (choiceTraceInput G x k Q A y) (choiceTraceOutput G x k Q A y)
-    (choiceTraceCode G x k Q A y) (fun a => y a.1 a.2.1.val)
-    (fun a => (traceIndex_valid G x k Q (A a.1) (y a.1) a.2).1)
-    (fun a => (traceIndex_valid G x k Q (A a.1) (y a.1) a.2).2)
-    (choiceTraceMass G x k Q A y q) (finiteSpliceDomain Q (Finset.Icc 1 k))
-    (fun a => traceIndexOutput_mem G x k Q (A a.1) (y a.1) (hA a.1) a.2) (fun _ => 1)
-  simp only [mul_one] at hd
-  unfold finiteChoiceForwardLoad
-  rw [← hd,Fintype.sum_sigma]
-  apply Finset.sum_congr rfl
-  intro r _
-  rw [Fintype.sum_sigma,← Finset.sum_coe_sort (A r)
-    (fun s => spliceTupleWeight Q (Finset.Icc 1 k) p s * q s r)]
-  apply Finset.sum_congr rfl
-  intro s _
-  change (∑ c : ↥(finiteTraceCodes G x s.val.2 (y r s.val) k 0 s.val.1 Q),
-    spliceTupleWeight Q (Finset.Icc 1 k) p s.val *
-      (q s.val r * traceIndexMass G x k Q (A r) (y r) ⟨s,c⟩)) = _
-  simp only [← mul_assoc,← Finset.mul_sum]
-  rw [sum_traceIndexMass G hloc x k M Q (A r) (y r) s (hA r s.property)
-    (hy r _ s.property) (hchain r _ s.property) (hcaps r _ s.property),mul_one]
-
-end ChoiceForwardDensity
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section EndpointRootMass
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-noncomputable def auxiliaryTupleWeight (Q : Finset E) (J : Finset ℕ) (p : ℝ)
-    (s : SpliceState E) : ℝ := ∏ i ∈ J, bernoulliWeight Q (domainBits Q (s.2 i)) p
-
-theorem auxiliaryTupleWeight_nonneg (Q : Finset E) (J : Finset ℕ) (p : unitInterval)
-    (s : SpliceState E) : 0 ≤ auxiliaryTupleWeight Q J p s :=
-  Finset.prod_nonneg fun _ _ => bernoulliWeight_nonneg _ _ p
-
- 
-
-noncomputable def endpointRootMass {I : Type*} [Fintype I]
-    (Q : Finset E) (J : Finset ℕ) (p : ℝ) (out : I → SpliceState E)
-    (endpoint : I → V) (m : I → ℝ) (ω : Set E) (z : V) : ℝ :=
-  ∑ a, if (out a).1 = ω ∧ endpoint a = z then auxiliaryTupleWeight Q J p (out a) * m a else 0
-
-theorem endpointRootMass_nonneg {I : Type*} [Fintype I]
-    (Q : Finset E) (J : Finset ℕ) (p : unitInterval) (out : I → SpliceState E)
-    (endpoint : I → V) (m : I → ℝ) (hm : ∀ a, 0 ≤ m a) (ω : Set E) (z : V) :
-    0 ≤ endpointRootMass Q J p out endpoint m ω z := by
-  apply Finset.sum_nonneg
-  intro a _
-  split_ifs
-  · exact mul_nonneg (auxiliaryTupleWeight_nonneg Q J p _) (hm a)
-  · exact le_refl 0
-
- 
-theorem finiteRootLoad_eq_sum {I : Type*} [Fintype I]
-    (Q : Finset E) (J : Finset ℕ) (p : ℝ) (out : I → SpliceState E) (m : I → ℝ)
-    (hs : ∀ a, out a ∈ finiteSpliceDomain Q J) (ω : FiniteField Q) :
-    finiteRootLoad Q J p (finiteInverseLoad out m) ω =
-      ∑ a, if (out a).1 = finiteFieldSet Q ω then auxiliaryTupleWeight Q J p (out a) * m a else 0 := by
-  simp only [finiteRootLoad,finiteInverseLoad,Finset.mul_sum,mul_ite,mul_zero]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro a _
-  obtain ⟨⟨r,η⟩,_,ha⟩ := Finset.mem_image.mp (hs a)
-  change (finiteFieldSet Q r,finiteFieldFamily Q J η) = out a at ha
-  rw [← ha]
-  have he (θ : ↥J → FiniteField Q) :
-      (finiteFieldSet Q r,finiteFieldFamily Q J η) = (finiteFieldSet Q ω,finiteFieldFamily Q J θ) ↔
-      r = ω ∧ η = θ := by
-    constructor
-    · intro h
-      exact ⟨finiteFieldSet_injective Q (congrArg Prod.fst h),
-        finiteFieldFamily_injective Q J (congrArg Prod.snd h)⟩
-    · rintro ⟨rfl,rfl⟩; rfl
-  simp only [he]
-  by_cases hr : r = ω
-  · subst r
-    simp only [true_and,ite_true,Finset.sum_ite_eq,Finset.mem_univ]
-    rw [auxiliaryTupleWeight,prod_fieldFamilyWeight]
-  · have hr' : finiteFieldSet Q r ≠ finiteFieldSet Q ω := fun h => hr (finiteFieldSet_injective Q h)
-    simp [hr,hr']
-
- 
-
-theorem endpointRootMass_le {I : Type*} [Fintype I]
-    (Q : Finset E) (J : Finset ℕ) (p : unitInterval) (out : I → SpliceState E)
-    (endpoint : I → V) (m : I → ℝ) (hs : ∀ a, out a ∈ finiteSpliceDomain Q J)
-    (z : V) (C : ℝ)
-    (hC : ∀ t ∈ finiteSpliceDomain Q J, finiteInverseLoad out
-      (fun a => if endpoint a = z then m a else 0) t ≤ C) (ω : FiniteField Q) :
-    endpointRootMass Q J p out endpoint m (finiteFieldSet Q ω) z ≤ C := by
-  have hb := finiteRootLoad_le Q J p _ C hC ω
-  rw [finiteRootLoad_eq_sum Q J p out _ hs ω] at hb
-  convert hb using 1
-  unfold endpointRootMass
-  apply Finset.sum_congr rfl
-  intro a _
-  split_ifs <;> simp_all
-
- 
-
-theorem sum_endpointRootMass {I : Type*} [Fintype I]
-    (Q : Finset E) (J : Finset ℕ) (p : ℝ) (out : I → SpliceState E)
-    (endpoint : I → V) (m : I → ℝ) (ω : Set E) (S : Finset V) :
-    (∑ z ∈ S, endpointRootMass Q J p out endpoint m ω z) =
-      ∑ a, if (out a).1 = ω ∧ endpoint a ∈ S then auxiliaryTupleWeight Q J p (out a) * m a else 0 := by
-  simp only [endpointRootMass]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro a _
-  by_cases hr : (out a).1 = ω
-  · simp only [hr,true_and,Finset.sum_ite_eq]
-  · simp [hr]
-
-end EndpointRootMass
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section NearPairMass
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-theorem finite_near_pair_mass_le_cover {I₁ I₂ : Type*} [Fintype I₁] [Fintype I₂]
-    (Q : Finset E) (J₁ J₂ : Finset ℕ) (p : unitInterval)
-    (out₁ : I₁ → SpliceState E) (out₂ : I₂ → SpliceState E)
-    (m₁ : I₁ → ℝ) (m₂ : I₂ → ℝ) (hm₁ : ∀ a, 0 ≤ m₁ a) (hm₂ : ∀ b, 0 ≤ m₂ b)
-    (endpoint : I₂ → V) (R : I₁ → I₂ → Prop) (S : I₁ → Finset V)
-    (hS : ∀ a b, (out₁ a).1 = (out₂ b).1 → R a b → endpoint b ∈ S a) :
-    (∑ a, ∑ b, if (out₁ a).1 = (out₂ b).1 ∧ R a b then
-      jointTupleWeight Q J₁ J₂ p (pairSpliceOutput (out₁ a) (out₂ b)) * m₁ a * m₂ b else 0) ≤
-    ∑ a, spliceTupleWeight Q J₁ p (out₁ a) * m₁ a *
-      ∑ z ∈ S a, endpointRootMass Q J₂ p out₂ endpoint m₂ (out₁ a).1 z := by
-  apply Finset.sum_le_sum
-  intro a _
-  rw [sum_endpointRootMass,Finset.mul_sum]
-  apply Finset.sum_le_sum
-  intro b _
-  by_cases h : (out₁ a).1 = (out₂ b).1 ∧ R a b
-  · have hh : (out₂ b).1 = (out₁ a).1 ∧ endpoint b ∈ S a :=
-      ⟨h.1.symm,hS a b h.1 h.2⟩
-    rw [ite_eq_left h,ite_eq_left hh]
-    change (spliceTupleWeight Q J₁ p (out₁ a) * auxiliaryTupleWeight Q J₂ p (out₂ b)) *
-      m₁ a * m₂ b ≤ _
-    ring_nf
-    exact le_refl _
-  · rw [ite_eq_right h]
-    apply mul_nonneg (mul_nonneg (spliceTupleWeight_nonneg Q J₁ p _) (hm₁ a))
-    split_ifs
-    · exact mul_nonneg (auxiliaryTupleWeight_nonneg Q J₂ p _) (hm₂ b)
-    · exact le_refl 0
-
- 
-
-theorem finite_near_pair_mass_le {I₁ I₂ : Type*} [Fintype I₁] [Fintype I₂]
-    (Q : Finset E) (J₁ J₂ : Finset ℕ) (p : unitInterval)
-    (out₁ : I₁ → SpliceState E) (out₂ : I₂ → SpliceState E)
-    (m₁ : I₁ → ℝ) (m₂ : I₂ → ℝ) (hm₁ : ∀ a, 0 ≤ m₁ a) (hm₂ : ∀ b, 0 ≤ m₂ b)
-    (endpoint : I₂ → V) (R : I₁ → I₂ → Prop) (S : I₁ → Finset V)
-    (hS : ∀ a b, (out₁ a).1 = (out₂ b).1 → R a b → endpoint b ∈ S a)
-    (N : ℕ) (hN : ∀ a, (S a).card ≤ N) (C : ℝ) (hC : 0 ≤ C)
-    (hb : ∀ a z, z ∈ S a → endpointRootMass Q J₂ p out₂ endpoint m₂ (out₁ a).1 z ≤ C) :
-    (∑ a, ∑ b, if (out₁ a).1 = (out₂ b).1 ∧ R a b then
-      jointTupleWeight Q J₁ J₂ p (pairSpliceOutput (out₁ a) (out₂ b)) * m₁ a * m₂ b else 0) ≤
-    (N:ℝ) * C * ∑ a, spliceTupleWeight Q J₁ p (out₁ a) * m₁ a := by
-  apply (finite_near_pair_mass_le_cover Q J₁ J₂ p out₁ out₂ m₁ m₂ hm₁ hm₂ endpoint R S hS).trans
-  calc
-    _ ≤ ∑ a, spliceTupleWeight Q J₁ p (out₁ a) * m₁ a * ((N:ℝ)*C) := by
-      apply Finset.sum_le_sum
-      intro a _
-      apply mul_le_mul_of_nonneg_left _
-        (mul_nonneg (spliceTupleWeight_nonneg Q J₁ p _) (hm₁ a))
-      calc
-        _ ≤ ∑ _z ∈ S a, C := Finset.sum_le_sum (fun z hz => hb a z hz)
-        _ = ((S a).card:ℝ) * C := by simp
-        _ ≤ (N:ℝ)*C := mul_le_mul_of_nonneg_right (by exact_mod_cast hN a) hC
-    _ = _ := by rw [← Finset.sum_mul]; ring
-
-end NearPairMass
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section EndpointInverseBound
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem finiteInverseLoad_subtype {I : Type*} [Fintype I]
-    (out : I → SpliceState E) (m : I → ℝ) (P : I → Prop) (t : SpliceState E) :
-    finiteInverseLoad out (fun a => if P a then m a else 0) t =
-      finiteInverseLoad (fun a : {a : I // P a} => out a.val) (fun a => m a.val) t := by
-  unfold finiteInverseLoad
-  have he : (∑ a, if out a = t then if P a then m a else 0 else 0) =
-      ∑ a, if P a then if out a = t then m a else 0 else 0 := by
-    apply Finset.sum_congr rfl
-    intro a _
-    split_ifs <;> rfl
-  rw [he,← Finset.sum_filter]
-  exact Finset.sum_subtype _ (by intro a; simp) _
-
- 
-
-theorem endpointInverseLoad_le {I P : Type*} [Fintype I] [Fintype P]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) (k M : ℕ) (L : ℝ) (hL : 1 ≤ L)
-    (inp out : I → SpliceState E) (steps : I → List (ℕ × E)) (y : I → ℕ → V)
-    (endpoint : I → V) (choice : I → P)
-    (hir : Function.Injective (fun a => (inp a,endpoint a,choice a,steps a)))
-    (q : V → SpliceState E → P → ℝ) (hq : ∀ z s b, 0 ≤ q z s b)
-    (hrow : ∀ z s, ∑ b, q z s b ≤ 1)
-    (htrace : ∀ a, SpliceTrace G x (inp a).2 (y a) k 0 (inp a).1 (steps a) (out a).1)
-    (hex : ∀ a, SpliceExecution G x (inp a) (steps a) (out a))
-    (hz : ∀ a, y a k = endpoint a) (hfin : ∀ a, (cluster G (out a).1 x).Finite)
-    (hgood : ∀ a i, 1 ≤ i → i ≤ k → (cluster G ((inp a).2 i) (y a i)).Finite ∧
-      (cluster G ((inp a).2 i) (y a i)).ncard ≤ M ∧ GoodForScale G ((inp a).2 i) (y a i) L)
-    (z : V) (t : SpliceState E) :
-    finiteInverseLoad out (fun a => if endpoint a = z then
-      q (endpoint a) (inp a) (choice a) * ∏ i : Fin k,
-        (traceContactCount G (inp a).2 (y a) (out a).1 x (steps a) (i.val+1):ℝ)⁻¹ else 0) t ≤
-      inverseCodeMass M L ^ k := by
-  rw [finiteInverseLoad_subtype]
-  let J := {a : I // endpoint a = z}
-  have hi : Function.Injective (fun a : J => (inp a.val,choice a.val,steps a.val)) := by
-    intro a b he
-    apply Subtype.ext
-    apply hir
-    have hin : inp a.val = inp b.val := congrArg (fun s : SpliceState E × P × List (ℕ × E) => s.1) he
-    exact Prod.ext hin (Prod.ext (a.property.trans b.property.symm) (congrArg (fun s : SpliceState E × P × List (ℕ × E) => s.2) he))
-  have hb := finiteInverseLoad_weighted_le G hloc x z k M L hL
-    (fun a : J => inp a.val) (fun a : J => out a.val) (fun a : J => steps a.val)
-    (fun a : J => y a.val) (fun a : J => choice a.val) hi (q z) (hq z) (hrow z)
-    (fun a => htrace a.val) (fun a => hex a.val) (fun a => (hz a.val).trans a.property)
-    (fun a => hfin a.val) (fun a => hgood a.val) t
-  convert hb using 1
-  apply congrArg (fun f : J → ℝ => finiteInverseLoad (fun a : J => out a.val) f t)
-  funext a
-  rw [a.property]
-
-end EndpointInverseBound
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section NearHangingMass
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem SpliceTrace.head_hanging (G : BondGraph V E) (x : V) (η : ℕ → Set E)
-    (y : ℕ → V) (k j i : ℕ) (ω ωf : Set E) (e : E) (rest : List (ℕ × E))
-    (h : SpliceTrace G x η y k j ω ((i,e)::rest) ωf) :
-    hangingSide G ωf x e = cluster G ωf x \ cluster G ω x := by
-  rw [hangingSide,h.head_cut G x η y k j i ω ωf e rest]
-
-theorem SpliceTrace.head_hanging_cap (G : BondGraph V E) (x : V) (η : ℕ → Set E)
-    (y : ℕ → V) (k j i M : ℕ) (ω ωf : Set E) (e : E) (rest : List (ℕ × E))
-    (h : SpliceTrace G x η y k j ω ((i,e)::rest) ωf)
-    (hcaps : ∀ l, 1 ≤ l → l ≤ k →
-      (cluster G (η l) (y l)).Finite ∧ (cluster G (η l) (y l)).ncard ≤ M) :
-    (hangingSide G ωf x e).ncard ≤ k*M := by
-  rw [h.head_hanging G x η y k j i ω ωf e rest]
-  exact (h.added_finite_cap G x η y k j M ω ωf ((i,e)::rest) hcaps).2
-
- 
-
-theorem finite_near_hanging_mass_le {I₁ I₂ : Type*} [Fintype I₁] [Fintype I₂]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) (v d : ℕ)
-    (hdeg : ∀ a, degree G a ≤ d)
-    (Q : Finset E) (J₁ J₂ : Finset ℕ) (p : unitInterval)
-    (out₁ : I₁ → SpliceState E) (out₂ : I₂ → SpliceState E)
-    (m₁ : I₁ → ℝ) (m₂ : I₂ → ℝ) (hm₁ : ∀ a, 0 ≤ m₁ a) (hm₂ : ∀ b, 0 ≤ m₂ b)
-    (z₁ : I₁ → V) (z₂ : I₂ → V) (e₁ : I₁ → E) (e₂ : I₂ → E)
-    (hfin : ∀ a, (cluster G (out₁ a).1 x).Finite)
-    (hp₁ : ∀ a, OpenMarkedPivotal G (out₁ a).1 {z₁ a} x (e₁ a))
-    (hp₂ : ∀ b, OpenMarkedPivotal G (out₂ b).1 {z₂ b} x (e₂ b))
-    (hcap₁ : ∀ a, (hangingSide G (out₁ a).1 x (e₁ a)).ncard ≤ v)
-    (hcap₂ : ∀ b, (hangingSide G (out₂ b).1 x (e₂ b)).ncard ≤ v)
-    (C : ℝ) (hC : 0 ≤ C)
-    (hb : ∀ a z, endpointRootMass Q J₂ p out₂ z₂ m₂ (out₁ a).1 z ≤ C) :
-    (∑ a, ∑ b, if (out₁ a).1 = (out₂ b).1 ∧
-      ¬GraphSeparated G (hangingSide G (out₁ a).1 x (e₁ a))
-        (hangingSide G (out₂ b).1 x (e₂ b)) then
-      jointTupleWeight Q J₁ J₂ p (pairSpliceOutput (out₁ a) (out₂ b)) * m₁ a * m₂ b else 0) ≤
-    ((v+v^2+d*v^3:ℕ):ℝ) * C * ∑ a, spliceTupleWeight Q J₁ p (out₁ a) * m₁ a := by
-  let R : I₁ → I₂ → Prop := fun a b =>
-    ¬GraphSeparated G (hangingSide G (out₁ a).1 x (e₁ a)) (hangingSide G (out₂ b).1 x (e₂ b))
-  let S := fun a => nearHangingEndpoints G hloc (out₁ a).1 x (z₁ a) (e₁ a) v (hfin a)
-  have hS : ∀ a b, (out₁ a).1 = (out₂ b).1 → R a b → z₂ b ∈ S a := by
-    intro a b he hn
-    have hp := hp₂ b
-    have hc := hcap₂ b
-    change ¬GraphSeparated G _ _ at hn
-    rw [← he] at hp hc hn
-    exact mem_nearHangingEndpoints_of_not_separated G hloc (out₁ a).1 x (z₁ a) (z₂ b)
-      (e₁ a) (e₂ b) v (hfin a) (hp₁ a) hp hc hn
-  have hN : ∀ a, (S a).card ≤ v+v^2+d*v^3 := fun a =>
-    nearHangingEndpoints_card_le G hloc (out₁ a).1 x (z₁ a) (e₁ a) v d
-      (hfin a) (hcap₁ a) hdeg
-  have ht := finite_near_pair_mass_le Q J₁ J₂ p out₁ out₂ m₁ m₂ hm₁ hm₂ z₂ R S hS
-    (v+v^2+d*v^3) hN C hC (fun a z _ => hb a z)
-  convert ht using 1
-  apply Finset.sum_congr rfl
-  intro a _
-  apply Finset.sum_congr rfl
-  intro b _
-  by_cases h : (out₁ a).1 = (out₂ b).1 ∧ R a b
-  · rw [ite_eq_left h,ite_eq_left h]
-  · rw [ite_eq_right h,ite_eq_right h]
-
-end NearHangingMass
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section OldRunArrival
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {W : Type*} [Fintype W]
-
-noncomputable def oldRunArrival (K : ℕ → W → W → ℝ) (k : ℕ) (z : W) (A : Finset W) : ℝ :=
-  ∑ y : Fin k → W, if oldRunVertices k z y 0 ∈ A then oldRunWeight K k z y else 0
-
-theorem oldRunArrival_nonneg (K : ℕ → W → W → ℝ) (hK : ∀ i a b, 0 ≤ K i a b)
-    (k : ℕ) (z : W) (A : Finset W) : 0 ≤ oldRunArrival K k z A := by
-  apply Finset.sum_nonneg
-  intro y _
-  split_ifs
-  · exact oldRunWeight_nonneg K hK k z y
-  · exact le_refl 0
-
-theorem oldRunArrival_zero (K : ℕ → W → W → ℝ) (z : W) (A : Finset W) :
-    oldRunArrival K 0 z A = if z ∈ A then 1 else 0 := by
-  by_cases h : z ∈ A
-  · simp [oldRunArrival,oldRunVertices,oldRunWeight,h]
-  · simp [oldRunArrival,oldRunVertices,h]
-
-theorem oldRunArrival_succ (K : ℕ → W → W → ℝ) (k : ℕ) (z : W) (A : Finset W) :
-    oldRunArrival K (k+1) z A = ∑ a, K (k+1) z a * oldRunArrival K k a A := by
-  unfold oldRunArrival
-  rw [Fintype.sum_equiv (oldRunConsEquiv k) _
-    (fun y => if oldRunVertices k y.1 y.2 0 ∈ A then
-      K (k+1) z y.1 * oldRunWeight K k y.1 y.2 else 0)
-    (by
-      intro y
-      by_cases h : oldRunVertices k (y 0) (Fin.tail y) 0 ∈ A
-      · simp [oldRunVertices_succ_of_le k z y 0 (Nat.zero_le k),oldRunConsEquiv,oldRunWeight,h]
-      · simp [oldRunVertices_succ_of_le k z y 0 (Nat.zero_le k),oldRunConsEquiv,h]),
-    Fintype.sum_prod_type]
-  apply Finset.sum_congr rfl
-  intro a _
-  simp only [Finset.mul_sum,mul_ite,mul_zero]
-
- 
-
-theorem sum_oldRunArrival_le (K : ℕ → W → W → ℝ) (hK : ∀ i a b, 0 ≤ K i a b)
-    (u : ℝ) (hu : 0 ≤ u) (hcol : ∀ i b, ∑ a, K i a b ≤ u) (k : ℕ) (A : Finset W) :
-    (∑ z, oldRunArrival K k z A) ≤ (A.card:ℝ) * u^k := by
-  induction k with
-  | zero => simp [oldRunArrival_zero]
-  | succ k ih =>
-    simp only [oldRunArrival_succ]
-    rw [Finset.sum_comm]
-    calc
-      _ = ∑ a, (∑ z, K (k+1) z a) * oldRunArrival K k a A := by
-        simp only [Finset.sum_mul]
-      _ ≤ ∑ a, u * oldRunArrival K k a A := Finset.sum_le_sum fun a _ =>
-        mul_le_mul_of_nonneg_right (hcol _ _) (oldRunArrival_nonneg K hK k a A)
-      _ = u * ∑ a, oldRunArrival K k a A := (Finset.mul_sum _ _ _).symm
-      _ ≤ u*((A.card:ℝ)*u^k) := mul_le_mul_of_nonneg_left ih hu
-      _ = (A.card:ℝ)*u^(k+1) := by ring
-
- 
-theorem sum_oldRunArrival_on_le (K : ℕ → W → W → ℝ) (hK : ∀ i a b, 0 ≤ K i a b)
-    (u : ℝ) (hu : 0 ≤ u) (hcol : ∀ i b, ∑ a, K i a b ≤ u)
-    (k : ℕ) (A Z : Finset W) :
-    (∑ z ∈ Z, oldRunArrival K k z A) ≤ (A.card:ℝ) * u^k := by
-  apply (Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ Z)
-    (fun z _ _ => oldRunArrival_nonneg K hK k z A)).trans
-  exact sum_oldRunArrival_le K hK u hu hcol k A
-
- 
-theorem noSwitch_arrival_le (K : ℕ → W → W → ℝ) (hK : ∀ i a b, 0 ≤ K i a b)
-    (u : ℝ) (hu : 0 ≤ u) (hcol : ∀ i b, ∑ a, K i a b ≤ u)
-    (k n : ℕ) (hn : 0 < n) (A Z : Finset W) (hA : A.card ≤ 2*n) :
-    (n:ℝ)⁻¹ * ∑ z ∈ Z, oldRunArrival K k z A ≤ 2*u^k := by
-  have hn' : (0:ℝ) < n := by exact_mod_cast hn
-  calc
-    _ ≤ (n:ℝ)⁻¹ * ((A.card:ℝ)*u^k) :=
-      mul_le_mul_of_nonneg_left (sum_oldRunArrival_on_le K hK u hu hcol k A Z)
-        (inv_nonneg.mpr hn'.le)
-    _ ≤ (n:ℝ)⁻¹ * ((2*(n:ℝ))*u^k) := mul_le_mul_of_nonneg_left
-      (mul_le_mul_of_nonneg_right (by exact_mod_cast hA) (pow_nonneg hu _)) (inv_nonneg.mpr hn'.le)
-    _ = 2*u^k := by field_simp
-
-end OldRunArrival
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section TrajectoryTransport
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {ι : Type*} [Fintype ι]
-
- 
-
-theorem RootTransport.real (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (htr : RootTransport root weight Γ) (m : V → V → ℝ)
-    (hi : ∀ a ∈ Γ, ∀ x y, m (a x) (a y) = m x y)
-    (hr : ∀ x, Summable (m x)) (hc : ∀ y, Summable (fun x => m x y)) :
-    (∑ i, weight i * ∑' y, m (root i) y) =
-      (∑ i, weight i * ∑' x, m x (root i)) := by
-  have ht := htr (fun x y => (m x y:ℂ))
-    (fun a ha x y => congrArg Complex.ofReal (hi a ha x y))
-    (fun x => Complex.summable_ofReal.mpr (hr x))
-    (fun y => Complex.summable_ofReal.mpr (hc y))
-  apply Complex.ofReal_injective
-  simpa only [Complex.ofReal_sum,Complex.ofReal_mul,Complex.ofReal_tsum] using ht
-
- 
-
-theorem transported_column_le (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
-    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (m : V → V → ℝ) (hm : ∀ x y, 0 ≤ m x y)
-    (hi : ∀ a ∈ Γ, ∀ x y, m (a x) (a y) = m x y)
-    (hr : ∀ x, Summable (m x)) (hcol : ∀ y, Summable (fun x => m x y))
-    (B : ℝ) (hB : ∀ x, ∑' y, m x y ≤ B) (y : V) :
-    (∑' x, m x y) ≤ c*B := by
-  have ht := htr.real root weight Γ m hi hr hcol
-  have hb : (∑ i, weight i * ∑' x, m x (root i)) ≤ B := by
-    rw [← ht]
-    calc
-      _ ≤ ∑ i, weight i * B := Finset.sum_le_sum fun i _ =>
-        mul_le_mul_of_nonneg_left (hB _) (hw i)
-      _ = B := by rw [← Finset.sum_mul,hw1,one_mul]
-  obtain ⟨i,a,ha,hay⟩ := hcov y
-  have he : (∑' x, m x y) = ∑' x, m x (root i) := by
-    rw [← hay,← a.tsum_eq (fun x => m x (a (root i)))]
-    exact tsum_congr (fun x => hi a ha x (root i))
-  have hn : 0 ≤ ∑' x, m x (root i) := tsum_nonneg (fun x => hm x _)
-  have hs : weight i * ∑' x, m x (root i) ≤ ∑ j, weight j * ∑' x, m x (root j) :=
-    Finset.single_le_sum (fun j _ => mul_nonneg (hw j) (tsum_nonneg (fun x => hm x (root j))))
-      (Finset.mem_univ i)
-  rw [he]
-  calc
-    _ ≤ (c*weight i) * ∑' x, m x (root i) := by
-      simpa only [one_mul] using mul_le_mul_of_nonneg_right (hmin i) hn
-    _ = c * (weight i * ∑' x, m x (root i)) := by ring
-    _ ≤ c * (∑ j, weight j * ∑' x, m x (root j)) := mul_le_mul_of_nonneg_left hs hc
-    _ ≤ c*B := mul_le_mul_of_nonneg_left hb hc
-
- 
-
-theorem finite_trajectory_hit_le (m : V → V → ℝ) (hm : ∀ x y, 0 ≤ m x y)
-    (hcol : ∀ y, Summable (fun x => m x y)) (B : ℝ)
-    (hb : ∀ y, ∑' x, m x y ≤ B) (h : V → ℝ) (A Z : Finset V)
-    (hh : ∀ x, h x ≤ ∑ y ∈ A, m x y) :
-    (∑ x ∈ Z, h x) ≤ (A.card:ℝ)*B := by
-  calc
-    _ ≤ ∑ x ∈ Z, ∑ y ∈ A, m x y := Finset.sum_le_sum fun x _ => hh x
-    _ = ∑ y ∈ A, ∑ x ∈ Z, m x y := Finset.sum_comm
-    _ ≤ ∑ y ∈ A, ∑' x, m x y := Finset.sum_le_sum fun y _ =>
-      (hcol y).sum_le_tsum Z (fun x _ => hm x y)
-    _ ≤ ∑ _y ∈ A, B := Finset.sum_le_sum fun y _ => hb y
-    _ = (A.card:ℝ)*B := by simp
-
-end TrajectoryTransport
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section NonnegativeTransport
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {ι : Type*} [Fintype ι]
-
-theorem mem_reachBall_aut_iff (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (ha : IsAut G a b) (x y : V) (n : ℕ) :
-    a y ∈ reachBall G (a x) n ↔ y ∈ reachBall G x n := by
-  constructor
-  · intro h
-    simpa only [Equiv.symm_apply_apply] using mem_reachBall_aut G (isAut_symm G ha) h
-  · exact mem_reachBall_aut G ha
-
- 
-def transportCut (G : BondGraph V E) (n : ℕ) (x y : V) : Prop :=
-  y ∈ reachBall G x n ∧ x ∈ reachBall G y n
-
-theorem transportCut_mono (G : BondGraph V E) {n m : ℕ} (h : n ≤ m)
-    {x y : V} (hxy : transportCut G n x y) : transportCut G m x y :=
-  ⟨reachBall_mono G x h hxy.1, reachBall_mono G y h hxy.2⟩
-
-theorem transportCut_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
-    (ha : IsAut G a b) (n : ℕ) (x y : V) :
-    transportCut G n (a x) (a y) ↔ transportCut G n x y := by
-  simp only [transportCut,mem_reachBall_aut_iff G ha]
-
-theorem exists_transportCut (G : BondGraph V E) (hconn : Connected G)
-    (y : V) (Z : Finset V) : ∃ n, ∀ x ∈ Z, transportCut G n x y := by
-  induction Z using Finset.induction_on with
-  | empty => exact ⟨0, by simp⟩
-  | @insert x Z _ ih =>
-    obtain ⟨n,hn⟩ := ih
-    obtain ⟨r,hr⟩ := exists_mem_reachBall (hconn x y)
-    obtain ⟨s,hs⟩ := exists_mem_reachBall (hconn y x)
-    refine ⟨max n (max r s), ?_⟩
-    intro z hz
-    rcases Finset.mem_insert.mp hz with rfl | hz
-    · exact ⟨reachBall_mono G z (le_trans (le_max_left r s) (le_max_right n _)) hr,
-        reachBall_mono G y (le_trans (le_max_right r s) (le_max_right n _)) hs⟩
-    · exact transportCut_mono G (le_max_left n _) (hn z hz)
-
- 
-
-theorem transported_column_of_row_bound (G : BondGraph V E)
-    (hconn : Connected G) (hloc : LocallyFinite G)
-    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
-    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
-    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (m : V → V → ℝ) (hm : ∀ x y, 0 ≤ m x y)
-    (hi : ∀ a ∈ Γ, ∀ x y, m (a x) (a y) = m x y)
-    (hr : ∀ x, Summable (m x)) (B : ℝ) (hB : ∀ x, ∑' y, m x y ≤ B) :
-    ∀ y, Summable (fun x => m x y) ∧ (∑' x, m x y) ≤ c*B := by
-  let cut (n : ℕ) (x y : V) : ℝ := if transportCut G n x y then m x y else 0
-  have hn (n : ℕ) (x y : V) : 0 ≤ cut n x y := by
-    dsimp [cut]; split_ifs; exact hm x y; exact le_refl 0
-  have hcRow (n : ℕ) (x : V) : Summable (cut n x) := by
-    apply summable_of_ne_finset_zero (s := (finite_reachBall hloc x n).toFinset)
-    intro y hy
-    simp only [Set.Finite.mem_toFinset] at hy
-    exact ite_eq_right (fun h => hy h.1)
-  have hcCol (n : ℕ) (y : V) : Summable (fun x => cut n x y) := by
-    apply summable_of_ne_finset_zero (s := (finite_reachBall hloc y n).toFinset)
-    intro x hx
-    simp only [Set.Finite.mem_toFinset] at hx
-    exact ite_eq_right (fun h => hx h.2)
-  have hcInv (n : ℕ) : ∀ a ∈ Γ, ∀ x y, cut n (a x) (a y) = cut n x y := by
-    intro a ha x y
-    obtain ⟨b,hb⟩ := hΓ ha
-    simp only [cut,transportCut_aut G hb,hi a ha]
-  have hcBound (n : ℕ) (x : V) : (∑' y, cut n x y) ≤ B := by
-    apply le_trans ((hcRow n x).tsum_le_tsum (fun y => ?_) (hr x)) (hB x)
-    dsimp [cut]; split_ifs; exact le_refl _; exact hm x y
-  have hbound (y : V) (Z : Finset V) : (∑ x ∈ Z, m x y) ≤ c*B := by
-    obtain ⟨n,hnZ⟩ := exists_transportCut G hconn y Z
-    calc
-      _ = ∑ x ∈ Z, cut n x y := by
-        apply Finset.sum_congr rfl
-        intro x hx
-        exact (ite_eq_left (hnZ x hx)).symm
-      _ ≤ ∑' x, cut n x y := (hcCol n y).sum_le_tsum Z (fun x _ => hn n x y)
-      _ ≤ c*B := transported_column_le root weight Γ htr hcov hw hw1 c hc hmin
-        (cut n) (hn n) (hcInv n) (hcRow n) (hcCol n) B (hcBound n) y
-  intro y
-  exact ⟨summable_of_sum_le (fun x => hm x y) (hbound y),
-    Real.tsum_le_of_sum_le (fun x => hm x y) (hbound y)⟩
-
- 
-
-theorem trajectory_hit_tsum_le (m : V → V → ℝ) (hm : ∀ x y, 0 ≤ m x y)
-    (hcol : ∀ y, Summable (fun x => m x y)) (B : ℝ)
-    (hb : ∀ y, ∑' x, m x y ≤ B) (h : V → ℝ) (hh0 : ∀ x, 0 ≤ h x)
-    (A : Finset V) (hh : ∀ x, h x ≤ ∑ y ∈ A, m x y) :
-    Summable h ∧ (∑' x, h x) ≤ (A.card:ℝ)*B := by
-  have hs := fun Z => finite_trajectory_hit_le m hm hcol B hb h A Z hh
-  exact ⟨summable_of_sum_le hh0 hs, Real.tsum_le_of_sum_le hh0 hs⟩
-
-end NonnegativeTransport
-
-end Percolation
-
- 
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-section FiniteTrajectoryMass
-attribute [local instance] Classical.propDecidable Classical.decEq
-variable {W : Type*} [Fintype W]
-
-theorem finiteFamilyIntegral_mul_const (Q : Finset E) (J : Finset ℕ) (p c : ℝ)
-    (f : (ℕ → Set E) → ℝ) :
-    finiteFamilyIntegral Q J p (fun η => f η * c) = finiteFamilyIntegral Q J p f * c := by
-  simp only [finiteFamilyIntegral,Finset.sum_mul,mul_assoc]
-
-omit [Fintype W] in
-theorem oldRunWeight_factor_ne_zero (q : Set E → W → W → ℝ) (k : ℕ)
-    (η : ℕ → Set E) (z : W) (y : Fin k → W)
-    (hy : oldRunWeight (fun i => q (η i)) k z y ≠ 0) (i : ℕ) (hi : 1 ≤ i) (hik : i ≤ k) :
-    q (η i) (oldRunVertices k z y i) (oldRunVertices k z y (i-1)) ≠ 0 := by
-  rw [← oldPathWeight_oldRunVertices] at hy
-  exact (Finset.prod_ne_zero_iff.mp hy) ⟨i,Finset.mem_Icc.mpr ⟨hi,hik⟩⟩ (Finset.mem_univ _)
-
- 
-
-noncomputable def finiteTrajectoryMass (G : BondGraph V E) (e : W → V)
-    (Q : Finset E) (p : ℝ) (k : ℕ) (q : Set E → W → W → ℝ) (z : W) (a : V) : ℝ :=
-  finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η =>
-    ∑ y : Fin k → W, if a ∈ trajectoryUnion G η (e ∘ oldRunVertices k z y) k then
-      oldRunWeight (fun i => q (η i)) k z y else 0)
-
-noncomputable def finiteTrajectoryHit (G : BondGraph V E) (e : W → V)
-    (Q : Finset E) (p : ℝ) (k : ℕ) (q : Set E → W → W → ℝ) (z : W) (A : Finset V) : ℝ :=
-  finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η =>
-    ∑ y : Fin k → W, if ((A:Set V) ∩ trajectoryUnion G η (e ∘ oldRunVertices k z y) k).Nonempty then
-      oldRunWeight (fun i => q (η i)) k z y else 0)
-
-theorem finiteTrajectoryMass_nonneg (G : BondGraph V E) (e : W → V)
-    (Q : Finset E) (p : unitInterval) (k : ℕ) (q : Set E → W → W → ℝ)
-    (hq : ∀ ω a b, 0 ≤ q ω a b) (z : W) (a : V) :
-    0 ≤ finiteTrajectoryMass G e Q p k q z a := by
-  apply finiteFamilyIntegral_nonneg
-  intro η
-  apply Finset.sum_nonneg
-  intro y _
-  split_ifs
-  · exact oldRunWeight_nonneg _ (fun _ => hq _) k z y
-  · exact le_refl 0
-
-theorem finiteTrajectoryHit_nonneg (G : BondGraph V E) (e : W → V)
-    (Q : Finset E) (p : unitInterval) (k : ℕ) (q : Set E → W → W → ℝ)
-    (hq : ∀ ω a b, 0 ≤ q ω a b) (z : W) (A : Finset V) :
-    0 ≤ finiteTrajectoryHit G e Q p k q z A := by
-  apply finiteFamilyIntegral_nonneg
-  intro η
-  apply Finset.sum_nonneg
-  intro y _
-  split_ifs
-  · exact oldRunWeight_nonneg _ (fun _ => hq _) k z y
-  · exact le_refl 0
-
-theorem finiteTrajectoryHit_le_sum (G : BondGraph V E) (e : W → V)
-    (Q : Finset E) (p : unitInterval) (k : ℕ) (q : Set E → W → W → ℝ)
-    (hq : ∀ ω a b, 0 ≤ q ω a b) (z : W) (A : Finset V) :
-    finiteTrajectoryHit G e Q p k q z A ≤ ∑ a ∈ A, finiteTrajectoryMass G e Q p k q z a := by
-  unfold finiteTrajectoryHit finiteTrajectoryMass
-  rw [← finiteFamilyIntegral_sum]
-  apply finiteFamilyIntegral_mono
-  intro η
-  rw [Finset.sum_comm]
-  apply Finset.sum_le_sum
-  intro y _
-  have hn := oldRunWeight_nonneg (fun i => q (η i)) (fun _ => hq _) k z y
-  by_cases h : ((A:Set V) ∩ trajectoryUnion G η (e ∘ oldRunVertices k z y) k).Nonempty
-  · rw [ite_eq_left h]
-    obtain ⟨a,ha,hat⟩ := h
-    have hs := Finset.single_le_sum (s := A)
-      (f := fun a => if a ∈ trajectoryUnion G η (e ∘ oldRunVertices k z y) k then
-        oldRunWeight (fun i => q (η i)) k z y else 0)
-      (fun a _ => by split_ifs; exact hn; exact le_refl 0) ha
-    simpa only [ite_eq_left hat] using hs
-  · rw [ite_eq_right h]
-    exact Finset.sum_nonneg fun a _ => by split_ifs; exact hn; exact le_refl 0
-
- 
-
-theorem finiteTrajectoryMass_finite_sum_le (G : BondGraph V E) (e : W → V)
-    (Q : Finset E) (p : unitInterval) (k M : ℕ) (q : Set E → W → W → ℝ)
-    (hq : ∀ ω a b, 0 ≤ q ω a b)
-    (hcap : ∀ ω a b, q ω a b ≠ 0 →
-      (cluster G ω (e a)).Finite ∧ (cluster G ω (e a)).ncard ≤ M)
-    (u : ℝ) (hu : 0 ≤ u) (hrow : ∀ a, ∑ b, finiteOldKernel Q p q a b ≤ u)
-    (z : W) (A : Finset V) :
-    (∑ a ∈ A, finiteTrajectoryMass G e Q p k q z a) ≤ (k*M:ℕ)*u^k := by
-  unfold finiteTrajectoryMass
-  rw [← finiteFamilyIntegral_sum]
-  have hb : ∀ η, (∑ a ∈ A, ∑ y : Fin k → W,
-      if a ∈ trajectoryUnion G η (e ∘ oldRunVertices k z y) k then
-        oldRunWeight (fun i => q (η i)) k z y else 0) ≤
-      (∑ y : Fin k → W, oldRunWeight (fun i => q (η i)) k z y) * (k*M:ℕ) := by
-    intro η
-    rw [Finset.sum_comm,Finset.sum_mul]
-    apply Finset.sum_le_sum
-    intro y _
-    by_cases hy : oldRunWeight (fun i => q (η i)) k z y = 0
-    · simp [hy]
-    obtain ⟨hf,hc⟩ := trajectoryUnion_finite_cap G η (e ∘ oldRunVertices k z y) k M
-      (fun i hi hik => hcap _ _ _ (oldRunWeight_factor_ne_zero q k η z y hy i hi hik))
-    have hsub : A.filter (fun a => a ∈ trajectoryUnion G η (e ∘ oldRunVertices k z y) k) ⊆ hf.toFinset := by
-      intro a ha
-      exact hf.mem_toFinset.mpr (Finset.mem_filter.mp ha).2
-    have hcard : ((A.filter (fun a => a ∈ trajectoryUnion G η (e ∘ oldRunVertices k z y) k)).card:ℝ) ≤ (k*M:ℕ) := by
-      exact_mod_cast (Finset.card_le_card hsub).trans (by simpa only [← Set.ncard_eq_toFinset_card _ hf] using hc)
-    rw [← Finset.sum_filter]
-    simp only [Finset.sum_const, nsmul_eq_mul]
-    simpa only [mul_comm] using mul_le_mul_of_nonneg_left hcard
-      (oldRunWeight_nonneg _ (fun _ => hq _) k z y)
-  calc
-    _ ≤ finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η =>
-        (∑ y : Fin k → W, oldRunWeight (fun i => q (η i)) k z y) * (k*M:ℕ)) :=
-      finiteFamilyIntegral_mono Q (Finset.Icc 1 k) p _ _ hb
-    _ = finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η =>
-        ∑ y : Fin k → W, oldRunWeight (fun i => q (η i)) k z y) * (k*M:ℕ) :=
-      finiteFamilyIntegral_mul_const Q (Finset.Icc 1 k) p _ _
-    _ ≤ u^k * (k*M:ℕ) := mul_le_mul_of_nonneg_right
-      (independent_oldRun_success_le Q p k q hq u hu hrow z) (Nat.cast_nonneg _)
-    _ = (k*M:ℕ)*u^k := mul_comm _ _
-
- 
-
-theorem finiteTrajectoryMass_row_bound (G : BondGraph V E) (e : W → V)
-    (Q : Finset E) (p : unitInterval) (k M : ℕ) (q : Set E → W → W → ℝ)
-    (hq : ∀ ω a b, 0 ≤ q ω a b)
-    (hcap : ∀ ω a b, q ω a b ≠ 0 →
-      (cluster G ω (e a)).Finite ∧ (cluster G ω (e a)).ncard ≤ M)
-    (u : ℝ) (hu : 0 ≤ u) (hrow : ∀ a, ∑ b, finiteOldKernel Q p q a b ≤ u) (z : W) :
-    Summable (finiteTrajectoryMass G e Q p k q z) ∧
-      (∑' a, finiteTrajectoryMass G e Q p k q z a) ≤ (k*M:ℕ)*u^k := by
-  have hn := finiteTrajectoryMass_nonneg G e Q p k q hq z
-  have hb := finiteTrajectoryMass_finite_sum_le G e Q p k M q hq hcap u hu hrow z
-  exact ⟨summable_of_sum_le hn hb,Real.tsum_le_of_sum_le hn hb⟩
-
-end FiniteTrajectoryMass
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Splicing.MomentBounds
+
+namespace OAI
+
+/-! Trajectory Mass for bond percolation. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section OldPathWeight
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+noncomputable def finiteFieldIntegral (Q : Finset E) (p : ℝ) (f : Set E → ℝ) : ℝ :=
+  ∑ ω : FiniteField Q, finiteFieldWeight Q p ω * f (finiteFieldSet Q ω)
+
+theorem finiteFieldIntegral_nonneg (Q : Finset E) (p : unitInterval) (f : Set E → ℝ)
+    (hf : ∀ ω, 0 ≤ f ω) : 0 ≤ finiteFieldIntegral Q p f :=
+  Finset.sum_nonneg fun ω _ => mul_nonneg (finiteFieldWeight_nonneg Q p ω) (hf _)
+
+theorem finiteFieldIntegral_sum {I : Type*} (Q : Finset E) (p : ℝ) (S : Finset I)
+    (f : I → Set E → ℝ) :
+    finiteFieldIntegral Q p (fun ω => ∑ i ∈ S, f i ω) =
+      ∑ i ∈ S, finiteFieldIntegral Q p (f i) := by
+  simp only [finiteFieldIntegral, Finset.mul_sum]
+  exact Finset.sum_comm
+
+ 
+
+theorem finiteFamilyIntegral_prod (Q : Finset E) (J : Finset ℕ) (p : ℝ)
+    (f : ↥J → Set E → ℝ) :
+    finiteFamilyIntegral Q J p (fun η => ∏ i : ↥J, f i (η i)) =
+      ∏ i : ↥J, finiteFieldIntegral Q p (f i) := by
+  simp only [finiteFamilyIntegral, finiteFamilyWeight, finiteFieldFamily_at,
+    ← Finset.prod_mul_distrib, finiteFieldIntegral]
+  exact (Fintype.prod_sum (fun (i : ↥J) (ω : FiniteField Q) =>
+    finiteFieldWeight Q p ω * f i (finiteFieldSet Q ω))).symm
+
+ 
+
+noncomputable def oldPathWeight (k : ℕ) (q : Set E → V → V → ℝ)
+    (y : ℕ → V) (η : ℕ → Set E) : ℝ :=
+  ∏ i : ↥(Finset.Icc 1 k), q (η i) (y i) (y (i.val-1))
+
+theorem oldPathWeight_nonneg (k : ℕ) (q : Set E → V → V → ℝ)
+    (hq : ∀ ω a b, 0 ≤ q ω a b) (y : ℕ → V) (η : ℕ → Set E) :
+    0 ≤ oldPathWeight k q y η :=
+  Finset.prod_nonneg fun _ _ => hq _ _ _
+
+theorem oldPathWeight_integral (Q : Finset E) (k : ℕ) (p : ℝ)
+    (q : Set E → V → V → ℝ) (y : ℕ → V) :
+    finiteFamilyIntegral Q (Finset.Icc 1 k) p (oldPathWeight k q y) =
+      ∏ i : ↥(Finset.Icc 1 k), finiteFieldIntegral Q p (fun ω => q ω (y i) (y (i.val-1))) := by
+  unfold oldPathWeight
+  exact finiteFamilyIntegral_prod Q (Finset.Icc 1 k) p
+    (fun i ω => q ω (y i) (y (i.val-1)))
+
+ 
+
+theorem oldPathWeight_support (G : BondGraph V E) (k M : ℕ) (L : ℝ)
+    (q : Set E → V → V → ℝ)
+    (hs : ∀ ω a b, q ω a b ≠ 0 → b ∈ cluster G ω a ∧
+      (cluster G ω a).Finite ∧ (cluster G ω a).ncard ≤ M ∧ GoodForScale G ω a L)
+    (y : ℕ → V) (η : ℕ → Set E) (hy : oldPathWeight k q y η ≠ 0) :
+    ∀ i, 1 ≤ i → i ≤ k → y (i-1) ∈ cluster G (η i) (y i) ∧
+      (cluster G (η i) (y i)).Finite ∧ (cluster G (η i) (y i)).ncard ≤ M ∧
+      GoodForScale G (η i) (y i) L := by
+  intro i hi hik
+  apply hs
+  have h := (Finset.prod_ne_zero_iff.mp hy) ⟨i, Finset.mem_Icc.mpr ⟨hi,hik⟩⟩ (Finset.mem_univ _)
+  exact h
+
+end OldPathWeight
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section FiniteOldRun
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {W : Type*}
+
+ 
+def oldRunConsEquiv (k : ℕ) : (Fin (k+1) → W) ≃ W × (Fin k → W) where
+  toFun y := (y 0, Fin.tail y)
+  invFun y := Fin.cons y.1 y.2
+  left_inv := Fin.cons_self_tail
+  right_inv y := by simp
+
+ 
+
+noncomputable def oldRunWeight (K : ℕ → W → W → ℝ) : (k : ℕ) → W → (Fin k → W) → ℝ
+  | 0, _, _ => 1
+  | k+1, z, y => K (k+1) z (y 0) * oldRunWeight K k (y 0) (Fin.tail y)
+
+theorem oldRunWeight_nonneg (K : ℕ → W → W → ℝ) (hK : ∀ i a b, 0 ≤ K i a b)
+    (k : ℕ) (z : W) (y : Fin k → W) : 0 ≤ oldRunWeight K k z y := by
+  induction k generalizing z with
+  | zero => exact zero_le_one
+  | succ k ih => exact mul_nonneg (hK _ _ _) (ih _ _)
+
+theorem sum_oldRunWeight_succ [Fintype W] (K : ℕ → W → W → ℝ) (k : ℕ) (z : W) :
+    (∑ y : Fin (k+1) → W, oldRunWeight K (k+1) z y) =
+      ∑ a : W, K (k+1) z a * ∑ y : Fin k → W, oldRunWeight K k a y := by
+  rw [Fintype.sum_equiv (oldRunConsEquiv k) _
+    (fun y => K (k+1) z y.1 * oldRunWeight K k y.1 y.2) (by intro y; rfl),
+    Fintype.sum_prod_type]
+  simp only [Finset.mul_sum]
+
+ 
+
+theorem sum_oldRunWeight_le [Fintype W] (K : ℕ → W → W → ℝ)
+    (hK : ∀ i a b, 0 ≤ K i a b) (u : ℝ) (hu : 0 ≤ u)
+    (hrow : ∀ i a, ∑ b, K i a b ≤ u) (k : ℕ) (z : W) :
+    (∑ y : Fin k → W, oldRunWeight K k z y) ≤ u^k := by
+  induction k generalizing z with
+  | zero => simp [oldRunWeight]
+  | succ k ih =>
+    rw [sum_oldRunWeight_succ]
+    calc
+      _ ≤ ∑ a : W, K (k+1) z a * u^k := Finset.sum_le_sum fun a _ =>
+        mul_le_mul_of_nonneg_left (ih a) (hK _ _ _)
+      _ = (∑ a : W, K (k+1) z a) * u^k := (Finset.sum_mul _ _ _).symm
+      _ ≤ u * u^k := mul_le_mul_of_nonneg_right (hrow _ _) (pow_nonneg hu _)
+      _ = u^(k+1) := by ring
+
+theorem sum_oldRunWeight_eq [Fintype W] (K : ℕ → W → W → ℝ)
+    (u : ℝ) (hrow : ∀ i a, ∑ b, K i a b = u) (k : ℕ) (z : W) :
+    (∑ y : Fin k → W, oldRunWeight K k z y) = u^k := by
+  induction k generalizing z with
+  | zero => simp [oldRunWeight]
+  | succ k ih =>
+    simp only [sum_oldRunWeight_succ, ih, ← Finset.sum_mul, hrow]
+    ring
+
+ 
+
+def oldRunVertices : (k : ℕ) → W → (Fin k → W) → ℕ → W
+  | 0, z, _ => fun _ => z
+  | k+1, z, y => fun i => if i ≤ k then oldRunVertices k (y 0) (Fin.tail y) i else z
+
+@[simp] theorem oldRunVertices_last (k : ℕ) (z : W) (y : Fin k → W) :
+    oldRunVertices k z y k = z := by
+  cases k <;> simp [oldRunVertices]
+
+theorem oldRunVertices_succ_of_le (k : ℕ) (z : W) (y : Fin (k+1) → W)
+    (i : ℕ) (hi : i ≤ k) :
+    oldRunVertices (k+1) z y i = oldRunVertices k (y 0) (Fin.tail y) i := by
+  simp only [oldRunVertices, ite_eq_left hi]
+
+ 
+
+theorem oldPathWeight_oldRunVertices (k : ℕ) (q : Set E → W → W → ℝ)
+    (η : ℕ → Set E) (z : W) (y : Fin k → W) :
+    oldPathWeight k q (oldRunVertices k z y) η =
+      oldRunWeight (fun i => q (η i)) k z y := by
+  induction k generalizing z with
+  | zero =>
+    unfold oldPathWeight
+    rw [Finset.prod_coe_sort (Finset.Icc 1 0)
+      (fun i => q (η i) (oldRunVertices 0 z y i) (oldRunVertices 0 z y (i-1)))]
+    simp [oldRunWeight]
+  | succ k ih =>
+    unfold oldPathWeight
+    rw [Finset.prod_coe_sort (Finset.Icc 1 (k+1))
+      (fun i => q (η i) (oldRunVertices (k+1) z y i) (oldRunVertices (k+1) z y (i-1))),
+      Finset.prod_Icc_succ_top (by omega)]
+    have he : (∏ i ∈ Finset.Icc 1 k,
+        q (η i) (oldRunVertices (k+1) z y i) (oldRunVertices (k+1) z y (i-1))) =
+        oldPathWeight k q (oldRunVertices k (y 0) (Fin.tail y)) η := by
+      unfold oldPathWeight
+      rw [Finset.prod_coe_sort (Finset.Icc 1 k)
+        (fun i => q (η i) (oldRunVertices k (y 0) (Fin.tail y) i)
+          (oldRunVertices k (y 0) (Fin.tail y) (i-1)))]
+      apply Finset.prod_congr rfl
+      intro i hi
+      have hi' := Finset.mem_Icc.mp hi
+      rw [oldRunVertices_succ_of_le _ _ _ _ hi'.2,
+        oldRunVertices_succ_of_le _ _ _ _ (by omega)]
+    rw [he, ih, oldRunVertices_last, Nat.add_sub_cancel,
+      oldRunVertices_succ_of_le _ _ _ _ (le_refl _), oldRunVertices_last]
+    exact mul_comm _ _
+
+end FiniteOldRun
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section IndependentOldRun
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {W : Type*}
+
+ 
+noncomputable def finiteOldKernel (Q : Finset E) (p : ℝ)
+    (q : Set E → W → W → ℝ) (a b : W) : ℝ :=
+  finiteFieldIntegral Q p (fun ω => q ω a b)
+
+theorem finiteOldKernel_nonneg (Q : Finset E) (p : unitInterval)
+    (q : Set E → W → W → ℝ) (hq : ∀ ω a b, 0 ≤ q ω a b) (a b : W) :
+    0 ≤ finiteOldKernel Q p q a b :=
+  finiteFieldIntegral_nonneg Q p _ (fun _ => hq _ _ _)
+
+ 
+
+theorem oldRunWeight_integral (Q : Finset E) (p : ℝ) (k : ℕ)
+    (q : Set E → W → W → ℝ) (z : W) (y : Fin k → W) :
+    finiteFamilyIntegral Q (Finset.Icc 1 k) p
+      (fun η => oldRunWeight (fun i => q (η i)) k z y) =
+    oldRunWeight (fun _ => finiteOldKernel Q p q) k z y := by
+  calc
+    _ = finiteFamilyIntegral Q (Finset.Icc 1 k) p
+        (fun η => oldPathWeight k q (oldRunVertices k z y) η) := by
+      apply congrArg (finiteFamilyIntegral Q (Finset.Icc 1 k) p)
+      funext η
+      exact (oldPathWeight_oldRunVertices k q η z y).symm
+    _ = ∏ i : ↥(Finset.Icc 1 k), finiteOldKernel Q p q
+        (oldRunVertices k z y i) (oldRunVertices k z y (i.val-1)) :=
+      oldPathWeight_integral Q k p q _
+    _ = oldRunWeight (fun _ => finiteOldKernel Q p q) k z y :=
+      oldPathWeight_oldRunVertices k (fun (_ : Set E) => finiteOldKernel Q p q)
+        (fun _ => ∅) z y
+
+ 
+
+theorem independent_oldRun_success [Fintype W] (Q : Finset E) (p : ℝ) (k : ℕ)
+    (q : Set E → W → W → ℝ) (u : ℝ)
+    (hrow : ∀ a, ∑ b, finiteOldKernel Q p q a b = u) (z : W) :
+    finiteFamilyIntegral Q (Finset.Icc 1 k) p
+      (fun η => ∑ y : Fin k → W, oldRunWeight (fun i => q (η i)) k z y) = u^k := by
+  rw [finiteFamilyIntegral_sum]
+  simp only [oldRunWeight_integral]
+  exact sum_oldRunWeight_eq _ u (fun _ => hrow) k z
+
+ 
+
+theorem independent_oldRun_success_le [Fintype W] (Q : Finset E) (p : unitInterval) (k : ℕ)
+    (q : Set E → W → W → ℝ) (hq : ∀ ω a b, 0 ≤ q ω a b) (u : ℝ) (hu : 0 ≤ u)
+    (hrow : ∀ a, ∑ b, finiteOldKernel Q p q a b ≤ u) (z : W) :
+    finiteFamilyIntegral Q (Finset.Icc 1 k) p
+      (fun η => ∑ y : Fin k → W, oldRunWeight (fun i => q (η i)) k z y) ≤ u^k := by
+  rw [finiteFamilyIntegral_sum]
+  simp only [oldRunWeight_integral]
+  exact sum_oldRunWeight_le _ (fun _ => finiteOldKernel_nonneg Q p q hq) u hu (fun _ => hrow) k z
+
+end IndependentOldRun
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section WeightedCodeFiber
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem sum_comp_le_sum_of_injective {A B : Type*} [Fintype A] [Fintype B]
+    (f : A → B) (hf : Function.Injective f) (w : B → ℝ) (hw : ∀ b, 0 ≤ w b) :
+    (∑ a, w (f a)) ≤ ∑ b, w b := by
+  rw [← Finset.sum_image hf.injOn]
+  exact Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _)
+    (fun b _ _ => hw b)
+
+ 
+theorem finite_sum_group_weight {A B : Type*} [Fintype A] [Fintype B]
+    (f : A → B) (m : A → ℝ) (w : B → ℝ) :
+    (∑ a, m a * w (f a)) = ∑ b, w b * ∑ a, if f a = b then m a else 0 := by
+  simp only [Finset.mul_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro a _
+  simp only [mul_ite,mul_zero,Finset.sum_ite_eq,Finset.mem_univ,ite_true]
+  exact mul_comm _ _
+
+ 
+
+theorem finite_choice_fiber_le {A B C P : Type*} [Fintype A] [Fintype P]
+    (f : A → B) (ctx : A → C) (choice : A → P)
+    (hir : Function.Injective (fun a => (ctx a,choice a,f a)))
+    (hctx : ∀ a b, f a = f b → ctx a = ctx b)
+    (q : C → P → ℝ) (hq : ∀ c p, 0 ≤ q c p)
+    (hrow : ∀ c, ∑ p, q c p ≤ 1) (b : B) :
+    (∑ a, if f a = b then q (ctx a) (choice a) else 0) ≤ 1 := by
+  let J := {a : A // f a = b}
+  by_cases hne : Nonempty J
+  · let a₀ : J := Classical.choice hne
+    have hc (a : J) : ctx a.val = ctx a₀.val := hctx _ _ (a.property.trans a₀.property.symm)
+    have hi : Function.Injective (fun a : J => choice a.val) := by
+      intro a a' he
+      apply Subtype.ext
+      apply hir
+      exact Prod.ext (hctx _ _ (a.property.trans a'.property.symm))
+        (Prod.ext he (a.property.trans a'.property.symm))
+    have hb := sum_comp_le_sum_of_injective (fun a : J => choice a.val) hi
+      (q (ctx a₀.val)) (hq _)
+    have he : (∑ a, if f a = b then q (ctx a) (choice a) else 0) =
+        ∑ a : J, q (ctx a) (choice a) := by
+      rw [← Finset.sum_filter]
+      exact Finset.sum_subtype _ (by intro a; simp) _
+    rw [he]
+    simpa only [hc] using hb.trans (hrow (ctx a₀.val))
+  · have hn (a : A) : f a ≠ b := fun h => hne ⟨⟨a,h⟩⟩
+    simp only [ite_eq_right (hn _),Finset.sum_const_zero,zero_le_one]
+
+ 
+
+theorem finite_weighted_code_mass_le {A B C P : Type*} [Fintype A] [Fintype B] [Fintype P]
+    (f : A → B) (ctx : A → C) (choice : A → P)
+    (hir : Function.Injective (fun a => (ctx a,choice a,f a)))
+    (hctx : ∀ a b, f a = f b → ctx a = ctx b)
+    (q : C → P → ℝ) (hq : ∀ c p, 0 ≤ q c p) (hrow : ∀ c, ∑ p, q c p ≤ 1)
+    (w : B → ℝ) (hw : ∀ b, 0 ≤ w b) :
+    (∑ a, q (ctx a) (choice a) * w (f a)) ≤ ∑ b, w b := by
+  rw [finite_sum_group_weight]
+  apply Finset.sum_le_sum
+  intro b _
+  calc
+    _ ≤ w b * 1 := mul_le_mul_of_nonneg_left
+      (finite_choice_fiber_le f ctx choice hir hctx q hq hrow b) (hw b)
+    _ = w b := mul_one _
+
+end WeightedCodeFiber
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section WeightedInverseLoad
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem finiteInverseLoad_weighted_le {I P : Type*} [Fintype I] [Fintype P]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (x z : V) (k M : ℕ)
+    (L : ℝ) (hL : 1 ≤ L)
+    (inp out : I → SpliceState E) (steps : I → List (ℕ × E)) (y : I → ℕ → V)
+    (choice : I → P) (hir : Function.Injective (fun a => (inp a,choice a,steps a)))
+    (q : SpliceState E → P → ℝ) (hq : ∀ s b, 0 ≤ q s b)
+    (hrow : ∀ s, ∑ b, q s b ≤ 1)
+    (htrace : ∀ a, SpliceTrace G x (inp a).2 (y a) k 0 (inp a).1 (steps a) (out a).1)
+    (hex : ∀ a, SpliceExecution G x (inp a) (steps a) (out a))
+    (hz : ∀ a, y a k = z) (hfin : ∀ a, (cluster G (out a).1 x).Finite)
+    (hgood : ∀ a i, 1 ≤ i → i ≤ k → (cluster G ((inp a).2 i) (y a i)).Finite ∧
+      (cluster G ((inp a).2 i) (y a i)).ncard ≤ M ∧ GoodForScale G ((inp a).2 i) (y a i) L)
+    (t : SpliceState E) :
+    finiteInverseLoad out (fun a => q (inp a) (choice a) * ∏ i : Fin k,
+      (traceContactCount G (inp a).2 (y a) (out a).1 x (steps a) (i.val+1):ℝ)⁻¹) t ≤
+      inverseCodeMass M L ^ k := by
+  let J := {a : I // out a = t}
+  let c (a : J) := gapSpliceCode G (out a.val).1 x z (steps a.val)
+  have hvalid (a : J) : ValidSparseGapCode k M (c a) := by
+    have hv := (htrace a.val).valid_gap_code G hloc x (inp a.val).2 (y a.val) k 0 M
+      (inp a.val).1 (out a.val).1 (steps a.val) (hfin a.val) L (hgood a.val)
+    simpa only [c,hz a.val] using hv.1
+  let arr (a : J) := sparseGapArray k M (c a) (hvalid a)
+  let ctx (a : J) := (inp a.val,steps a.val)
+  have hc (a b : J) (he : arr a = arr b) : ctx a = ctx b := by
+    have ho : out a.val = out b.val := a.property.trans b.property.symm
+    have hcode : c a = c b := sparseGapArray_injective k M (c a) (c b)
+      (hvalid a) (hvalid b) he
+    have htB : SpliceTrace G x (inp b.val).2 (y b.val) k 0
+        (inp b.val).1 (steps b.val) (out a.val).1 := by
+      rw [ho]
+      exact htrace b.val
+    have heB : SpliceExecution G x (inp b.val) (steps b.val) (out a.val) := ho.symm ▸ hex b.val
+    have hg : gapSpliceCode G (out a.val).1 x (y a.val k) (steps a.val) =
+        gapSpliceCode G (out a.val).1 x (y a.val k) (steps b.val) := by
+      simpa only [c,hz a.val,ho] using hcode
+    have hze : y a.val k = y b.val k := (hz a.val).trans (hz b.val).symm
+    have hs := (htrace a.val).code_unique_of_gapCode G x (inp a.val).2 (inp b.val).2
+      (y a.val) (y b.val) k 0 (inp a.val).1 (inp b.val).1 (out a.val).1
+      (steps a.val) (steps b.val) htB hze (hfin a.val) hg
+    have hi := (htrace a.val).input_unique_of_gapCode G x (inp a.val).2 (inp b.val).2
+      (y a.val) (y b.val) k 0 (inp a.val).1 (inp b.val).1 (out a.val).1
+      (steps a.val) (steps b.val) htB hze (hfin a.val) hg (out a.val).2 (hex a.val) heB
+    exact Prod.ext hi hs
+  have hi : Function.Injective (fun a : J => (ctx a,choice a.val,arr a)) := by
+    intro a b he
+    apply Subtype.ext
+    apply hir
+    have hec := congrArg Prod.fst he
+    have hin : inp a.val = inp b.val := congrArg (fun s : SpliceState E × List (ℕ × E) => s.1) hec
+    have hst : steps a.val = steps b.val := congrArg (fun s : SpliceState E × List (ℕ × E) => s.2) hec
+    exact Prod.ext hin (Prod.ext (congrArg (fun z => z.2.1) he) hst)
+  let w (b : Fin k → Fin (M+1)) := ∏ i, gapLetterWeight L (b i).val
+  have hb := finite_weighted_code_mass_le arr ctx (fun a : J => choice a.val) hi hc
+    (fun s b => q s.1 b) (fun s => hq s.1) (fun s => hrow s.1) w
+    (fun b => Finset.prod_nonneg (fun i _ => gapLetterWeight_nonneg L (by linarith) _))
+  change (∑ a : J, q (inp a.val) (choice a.val) * w (arr a)) ≤ ∑ b, w b at hb
+  have he : finiteInverseLoad out (fun a => q (inp a) (choice a) * ∏ i : Fin k,
+      (traceContactCount G (inp a).2 (y a) (out a).1 x (steps a) (i.val+1):ℝ)⁻¹) t =
+      ∑ a : J, q (inp a.val) (choice a.val) * ∏ i : Fin k,
+      (traceContactCount G (inp a.val).2 (y a.val) (out a.val).1 x (steps a.val) (i.val+1):ℝ)⁻¹ := by
+    rw [finiteInverseLoad,← Finset.sum_filter]
+    exact Finset.sum_subtype _ (by intro a; simp) _
+  rw [he]
+  calc
+    _ ≤ ∑ a : J, q (inp a.val) (choice a.val) * w (arr a) := by
+      apply Finset.sum_le_sum
+      intro a _
+      apply mul_le_mul_of_nonneg_left _ (hq _ _)
+      apply Finset.prod_le_prod₀ (fun _ _ => by positivity)
+      intro i _
+      have h := ((htrace a.val).contact_weight_bound G hloc x (inp a.val).2 (y a.val)
+        k 0 M (inp a.val).1 (out a.val).1 (steps a.val) (hfin a.val) L hL
+        (hgood a.val) (i.val+1)).2
+      simpa only [w,arr,sparseGapArray,c,hz a.val] using h
+    _ ≤ ∑ b, w b := hb
+    _ = _ := total_gap_code_mass k M L
+
+end WeightedInverseLoad
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section OldRunInverseLoad
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem finiteInverseLoad_oldRun_le {I W : Type*} [Fintype I] [Fintype W]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (x z : V) (k M : ℕ)
+    (L : ℝ) (hL : 1 ≤ L)
+    (inp out : I → SpliceState E) (steps : I → List (ℕ × E)) (y : I → ℕ → V)
+    (start : W) (path : I → Fin k → W)
+    (hir : Function.Injective (fun a => (inp a,path a,steps a)))
+    (q : Set E → W → W → ℝ) (hq : ∀ ω a b, 0 ≤ q ω a b)
+    (hrow : ∀ ω a, ∑ b, q ω a b ≤ 1)
+    (htrace : ∀ a, SpliceTrace G x (inp a).2 (y a) k 0 (inp a).1 (steps a) (out a).1)
+    (hex : ∀ a, SpliceExecution G x (inp a) (steps a) (out a))
+    (hz : ∀ a, y a k = z) (hfin : ∀ a, (cluster G (out a).1 x).Finite)
+    (hgood : ∀ a i, 1 ≤ i → i ≤ k → (cluster G ((inp a).2 i) (y a i)).Finite ∧
+      (cluster G ((inp a).2 i) (y a i)).ncard ≤ M ∧ GoodForScale G ((inp a).2 i) (y a i) L)
+    (t : SpliceState E) :
+    finiteInverseLoad out (fun a => oldRunWeight (fun i => q ((inp a).2 i)) k start (path a) *
+      ∏ i : Fin k, (traceContactCount G (inp a).2 (y a) (out a).1 x (steps a) (i.val+1):ℝ)⁻¹) t ≤
+      inverseCodeMass M L ^ k := by
+  apply finiteInverseLoad_weighted_le G hloc x z k M L hL inp out steps y path hir
+    (fun s b => oldRunWeight (fun i => q (s.2 i)) k start b)
+    (fun s b => oldRunWeight_nonneg _ (fun _ => hq _) k start b) _ htrace hex hz hfin hgood t
+  intro s
+  simpa only [one_pow] using sum_oldRunWeight_le (fun i => q (s.2 i))
+    (fun _ => hq _) 1 zero_le_one (fun _ => hrow _) k start
+
+end OldRunInverseLoad
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section ChoiceForwardDensity
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {P : Type*}
+
+ 
+
+abbrev FiniteChoiceTraceIndex (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
+    (A : P → Finset (SpliceState E)) (y : P → SpliceState E → ℕ → V) :=
+  (r : P) × FiniteTraceIndex G x k Q (A r) (y r)
+
+def choiceTraceInput (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
+    (A : P → Finset (SpliceState E)) (y : P → SpliceState E → ℕ → V)
+    (a : FiniteChoiceTraceIndex G x k Q A y) : SpliceState E :=
+  traceIndexInput G x k Q (A a.1) (y a.1) a.2
+
+def choiceTraceOutput (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
+    (A : P → Finset (SpliceState E)) (y : P → SpliceState E → ℕ → V)
+    (a : FiniteChoiceTraceIndex G x k Q A y) : SpliceState E :=
+  traceIndexOutput G x k Q (A a.1) (y a.1) a.2
+
+def choiceTraceCode (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
+    (A : P → Finset (SpliceState E)) (y : P → SpliceState E → ℕ → V)
+    (a : FiniteChoiceTraceIndex G x k Q A y) : List (ℕ × E) :=
+  traceIndexCode G x k Q (A a.1) (y a.1) a.2
+
+noncomputable def choiceTraceMass (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
+    (A : P → Finset (SpliceState E)) (y : P → SpliceState E → ℕ → V)
+    (q : SpliceState E → P → ℝ) (a : FiniteChoiceTraceIndex G x k Q A y) : ℝ :=
+  q (choiceTraceInput G x k Q A y a) a.1 *
+    traceIndexMass G x k Q (A a.1) (y a.1) a.2
+
+theorem choiceTrace_injective (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
+    (A : P → Finset (SpliceState E)) (y : P → SpliceState E → ℕ → V) :
+    Function.Injective (fun a : FiniteChoiceTraceIndex G x k Q A y =>
+      (choiceTraceInput G x k Q A y a,a.1,choiceTraceCode G x k Q A y a)) := by
+  rintro ⟨r,a⟩ ⟨r',b⟩ he
+  have hr : r = r' := congrArg (fun z => z.2.1) he
+  subst r'
+  have hin : traceIndexInput G x k Q (A r) (y r) a =
+      traceIndexInput G x k Q (A r) (y r) b :=
+    congrArg (fun z : SpliceState E × P × List (ℕ × E) => z.1) he
+  have hcode : traceIndexCode G x k Q (A r) (y r) a =
+      traceIndexCode G x k Q (A r) (y r) b :=
+    congrArg (fun z : SpliceState E × P × List (ℕ × E) => z.2.2) he
+  have hab : a = b := traceIndex_injective G x k Q (A r) (y r) (Prod.ext hin hcode)
+  subst b
+  rfl
+
+noncomputable def finiteChoiceForwardLoad [Fintype P]
+    (G : BondGraph V E) (x : V) (k : ℕ) (Q : Finset E)
+    (A : P → Finset (SpliceState E)) (y : P → SpliceState E → ℕ → V)
+    (q : SpliceState E → P → ℝ) : SpliceState E → ℝ :=
+  finiteInverseLoad (choiceTraceOutput G x k Q A y) (choiceTraceMass G x k Q A y q)
+
+ 
+theorem finiteChoiceForwardLoad_bounds [Fintype P]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (x z : V) (k M : ℕ)
+    (L : ℝ) (hL : 1 ≤ L) (Q : Finset E)
+    (A : P → Finset (SpliceState E)) (y : P → SpliceState E → ℕ → V)
+    (q : SpliceState E → P → ℝ) (hq : ∀ s r, 0 ≤ q s r)
+    (hrow : ∀ s, ∑ r, q s r ≤ 1)
+    (hA : ∀ r, A r ⊆ finiteSpliceDomain Q (Finset.Icc 1 k))
+    (hz : ∀ r s, s ∈ A r → y r s k = z)
+    (hgood : ∀ r s, s ∈ A r → ∀ i, 1 ≤ i → i ≤ k →
+      (cluster G (s.2 i) (y r s i)).Finite ∧
+      (cluster G (s.2 i) (y r s i)).ncard ≤ M ∧ GoodForScale G (s.2 i) (y r s i) L)
+    (t : SpliceState E) :
+    0 ≤ finiteChoiceForwardLoad G x k Q A y q t ∧
+      finiteChoiceForwardLoad G x k Q A y q t ≤ inverseCodeMass M L ^ k := by
+  constructor
+  · apply finiteInverseLoad_nonneg
+    intro a
+    exact mul_nonneg (hq _ _) (Finset.prod_nonneg (fun _ _ => by positivity))
+  · apply finiteInverseLoad_weighted_le G hloc x z k M L hL
+      (choiceTraceInput G x k Q A y) (choiceTraceOutput G x k Q A y)
+      (choiceTraceCode G x k Q A y) (fun a => y a.1 a.2.1.val) Sigma.fst
+      (choiceTrace_injective G x k Q A y) q hq hrow
+      (fun a => (traceIndex_valid G x k Q (A a.1) (y a.1) a.2).1)
+      (fun a => (traceIndex_valid G x k Q (A a.1) (y a.1) a.2).2)
+      (fun a => hz a.1 _ a.2.1.property) _
+      (fun a => hgood a.1 _ a.2.1.property) t
+    intro a
+    have hs := (mem_finiteSpliceDomain_iff Q (Finset.Icc 1 k) _).mp
+      (traceIndexOutput_mem G x k Q (A a.1) (y a.1) (hA a.1) a.2)
+    exact cluster_finite_of_finite_edges G (Q.finite_toSet.subset hs.1) x
+
+ 
+
+theorem sum_traceIndexMass [Nonempty E] (G : BondGraph V E) (hloc : LocallyFinite G)
+    (x : V) (k M : ℕ) (Q : Finset E) (A : Finset (SpliceState E))
+    (y : SpliceState E → ℕ → V) (s : ↥A)
+    (hs : s.val ∈ finiteSpliceDomain Q (Finset.Icc 1 k))
+    (hy : y s.val 0 ∈ cluster G s.val.1 x)
+    (hchain : ∀ i ∈ Finset.Icc 1 k, Conn G (s.val.2 i) (y s.val i) (y s.val (i-1)))
+    (hcaps : ∀ i ∈ Finset.Icc 1 k, (cluster G (s.val.2 i) (y s.val i)).Finite ∧
+      (cluster G (s.val.2 i) (y s.val i)).ncard ≤ M) :
+    (∑ c : ↥(finiteTraceCodes G x s.val.2 (y s.val) k 0 s.val.1 Q),
+      traceIndexMass G x k Q A y ⟨s,c⟩) = 1 := by
+  have hs' := (mem_finiteSpliceDomain_iff Q (Finset.Icc 1 k) s.val).mp hs
+  have hn := sum_finite_trace_probabilities G hloc x s.val.2 (y s.val) k M Q
+    (fun i _ _ => hs'.2.1 i) hchain hcaps 0 (Nat.zero_le _) s.val.1 hy
+    (cluster_finite_of_finite_edges G (Q.finite_toSet.subset hs'.1) x)
+  rw [← Finset.sum_coe_sort (finiteTraceCodes G x s.val.2 (y s.val) k 0 s.val.1 Q)] at hn
+  rw [← hn]
+  apply Finset.sum_congr rfl
+  intro c _
+  have ht := ((mem_finiteTraceCodes_iff G x s.val.2 (y s.val) k 0 s.val.1 Q c.val).mp c.property).2
+  have hf := ht.runMasks_first G x s.val.2 (y s.val) k 0 s.val.1 _ c.val s.val.2 (by intros; rfl)
+  simp only [traceIndexMass,traceIndexOutput,hf]
+
+ 
+theorem finiteChoiceForwardLoad_mean [Fintype P] [Nonempty E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) (k M : ℕ) (Q : Finset E) (p : ℝ)
+    (A : P → Finset (SpliceState E)) (y : P → SpliceState E → ℕ → V)
+    (q : SpliceState E → P → ℝ)
+    (hA : ∀ r, A r ⊆ finiteSpliceDomain Q (Finset.Icc 1 k))
+    (hy : ∀ r s, s ∈ A r → y r s 0 ∈ cluster G s.1 x)
+    (hchain : ∀ r s, s ∈ A r → ∀ i ∈ Finset.Icc 1 k, Conn G (s.2 i) (y r s i) (y r s (i-1)))
+    (hcaps : ∀ r s, s ∈ A r → ∀ i ∈ Finset.Icc 1 k,
+      (cluster G (s.2 i) (y r s i)).Finite ∧ (cluster G (s.2 i) (y r s i)).ncard ≤ M) :
+    (∑ t ∈ finiteSpliceDomain Q (Finset.Icc 1 k), spliceTupleWeight Q (Finset.Icc 1 k) p t *
+      finiteChoiceForwardLoad G x k Q A y q t) =
+      ∑ r, ∑ s ∈ A r, spliceTupleWeight Q (Finset.Icc 1 k) p s * q s r := by
+  have hd := finiteInverseLoad_density G x k Q p
+    (choiceTraceInput G x k Q A y) (choiceTraceOutput G x k Q A y)
+    (choiceTraceCode G x k Q A y) (fun a => y a.1 a.2.1.val)
+    (fun a => (traceIndex_valid G x k Q (A a.1) (y a.1) a.2).1)
+    (fun a => (traceIndex_valid G x k Q (A a.1) (y a.1) a.2).2)
+    (choiceTraceMass G x k Q A y q) (finiteSpliceDomain Q (Finset.Icc 1 k))
+    (fun a => traceIndexOutput_mem G x k Q (A a.1) (y a.1) (hA a.1) a.2) (fun _ => 1)
+  simp only [mul_one] at hd
+  unfold finiteChoiceForwardLoad
+  rw [← hd,Fintype.sum_sigma]
+  apply Finset.sum_congr rfl
+  intro r _
+  rw [Fintype.sum_sigma,← Finset.sum_coe_sort (A r)
+    (fun s => spliceTupleWeight Q (Finset.Icc 1 k) p s * q s r)]
+  apply Finset.sum_congr rfl
+  intro s _
+  change (∑ c : ↥(finiteTraceCodes G x s.val.2 (y r s.val) k 0 s.val.1 Q),
+    spliceTupleWeight Q (Finset.Icc 1 k) p s.val *
+      (q s.val r * traceIndexMass G x k Q (A r) (y r) ⟨s,c⟩)) = _
+  simp only [← mul_assoc,← Finset.mul_sum]
+  rw [sum_traceIndexMass G hloc x k M Q (A r) (y r) s (hA r s.property)
+    (hy r _ s.property) (hchain r _ s.property) (hcaps r _ s.property),mul_one]
+
+end ChoiceForwardDensity
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section EndpointRootMass
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+noncomputable def auxiliaryTupleWeight (Q : Finset E) (J : Finset ℕ) (p : ℝ)
+    (s : SpliceState E) : ℝ := ∏ i ∈ J, bernoulliWeight Q (domainBits Q (s.2 i)) p
+
+theorem auxiliaryTupleWeight_nonneg (Q : Finset E) (J : Finset ℕ) (p : unitInterval)
+    (s : SpliceState E) : 0 ≤ auxiliaryTupleWeight Q J p s :=
+  Finset.prod_nonneg fun _ _ => bernoulliWeight_nonneg _ _ p
+
+ 
+
+noncomputable def endpointRootMass {I : Type*} [Fintype I]
+    (Q : Finset E) (J : Finset ℕ) (p : ℝ) (out : I → SpliceState E)
+    (endpoint : I → V) (m : I → ℝ) (ω : Set E) (z : V) : ℝ :=
+  ∑ a, if (out a).1 = ω ∧ endpoint a = z then auxiliaryTupleWeight Q J p (out a) * m a else 0
+
+theorem endpointRootMass_nonneg {I : Type*} [Fintype I]
+    (Q : Finset E) (J : Finset ℕ) (p : unitInterval) (out : I → SpliceState E)
+    (endpoint : I → V) (m : I → ℝ) (hm : ∀ a, 0 ≤ m a) (ω : Set E) (z : V) :
+    0 ≤ endpointRootMass Q J p out endpoint m ω z := by
+  apply Finset.sum_nonneg
+  intro a _
+  split_ifs
+  · exact mul_nonneg (auxiliaryTupleWeight_nonneg Q J p _) (hm a)
+  · exact le_refl 0
+
+ 
+theorem finiteRootLoad_eq_sum {I : Type*} [Fintype I]
+    (Q : Finset E) (J : Finset ℕ) (p : ℝ) (out : I → SpliceState E) (m : I → ℝ)
+    (hs : ∀ a, out a ∈ finiteSpliceDomain Q J) (ω : FiniteField Q) :
+    finiteRootLoad Q J p (finiteInverseLoad out m) ω =
+      ∑ a, if (out a).1 = finiteFieldSet Q ω then auxiliaryTupleWeight Q J p (out a) * m a else 0 := by
+  simp only [finiteRootLoad,finiteInverseLoad,Finset.mul_sum,mul_ite,mul_zero]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro a _
+  obtain ⟨⟨r,η⟩,_,ha⟩ := Finset.mem_image.mp (hs a)
+  change (finiteFieldSet Q r,finiteFieldFamily Q J η) = out a at ha
+  rw [← ha]
+  have he (θ : ↥J → FiniteField Q) :
+      (finiteFieldSet Q r,finiteFieldFamily Q J η) = (finiteFieldSet Q ω,finiteFieldFamily Q J θ) ↔
+      r = ω ∧ η = θ := by
+    constructor
+    · intro h
+      exact ⟨finiteFieldSet_injective Q (congrArg Prod.fst h),
+        finiteFieldFamily_injective Q J (congrArg Prod.snd h)⟩
+    · rintro ⟨rfl,rfl⟩; rfl
+  simp only [he]
+  by_cases hr : r = ω
+  · subst r
+    simp only [true_and,ite_true,Finset.sum_ite_eq,Finset.mem_univ]
+    rw [auxiliaryTupleWeight,prod_fieldFamilyWeight]
+  · have hr' : finiteFieldSet Q r ≠ finiteFieldSet Q ω := fun h => hr (finiteFieldSet_injective Q h)
+    simp [hr,hr']
+
+ 
+
+theorem endpointRootMass_le {I : Type*} [Fintype I]
+    (Q : Finset E) (J : Finset ℕ) (p : unitInterval) (out : I → SpliceState E)
+    (endpoint : I → V) (m : I → ℝ) (hs : ∀ a, out a ∈ finiteSpliceDomain Q J)
+    (z : V) (C : ℝ)
+    (hC : ∀ t ∈ finiteSpliceDomain Q J, finiteInverseLoad out
+      (fun a => if endpoint a = z then m a else 0) t ≤ C) (ω : FiniteField Q) :
+    endpointRootMass Q J p out endpoint m (finiteFieldSet Q ω) z ≤ C := by
+  have hb := finiteRootLoad_le Q J p _ C hC ω
+  rw [finiteRootLoad_eq_sum Q J p out _ hs ω] at hb
+  convert hb using 1
+  unfold endpointRootMass
+  apply Finset.sum_congr rfl
+  intro a _
+  split_ifs <;> simp_all
+
+ 
+
+theorem sum_endpointRootMass {I : Type*} [Fintype I]
+    (Q : Finset E) (J : Finset ℕ) (p : ℝ) (out : I → SpliceState E)
+    (endpoint : I → V) (m : I → ℝ) (ω : Set E) (S : Finset V) :
+    (∑ z ∈ S, endpointRootMass Q J p out endpoint m ω z) =
+      ∑ a, if (out a).1 = ω ∧ endpoint a ∈ S then auxiliaryTupleWeight Q J p (out a) * m a else 0 := by
+  simp only [endpointRootMass]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro a _
+  by_cases hr : (out a).1 = ω
+  · simp only [hr,true_and,Finset.sum_ite_eq]
+  · simp [hr]
+
+end EndpointRootMass
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section NearPairMass
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+theorem finite_near_pair_mass_le_cover {I₁ I₂ : Type*} [Fintype I₁] [Fintype I₂]
+    (Q : Finset E) (J₁ J₂ : Finset ℕ) (p : unitInterval)
+    (out₁ : I₁ → SpliceState E) (out₂ : I₂ → SpliceState E)
+    (m₁ : I₁ → ℝ) (m₂ : I₂ → ℝ) (hm₁ : ∀ a, 0 ≤ m₁ a) (hm₂ : ∀ b, 0 ≤ m₂ b)
+    (endpoint : I₂ → V) (R : I₁ → I₂ → Prop) (S : I₁ → Finset V)
+    (hS : ∀ a b, (out₁ a).1 = (out₂ b).1 → R a b → endpoint b ∈ S a) :
+    (∑ a, ∑ b, if (out₁ a).1 = (out₂ b).1 ∧ R a b then
+      jointTupleWeight Q J₁ J₂ p (pairSpliceOutput (out₁ a) (out₂ b)) * m₁ a * m₂ b else 0) ≤
+    ∑ a, spliceTupleWeight Q J₁ p (out₁ a) * m₁ a *
+      ∑ z ∈ S a, endpointRootMass Q J₂ p out₂ endpoint m₂ (out₁ a).1 z := by
+  apply Finset.sum_le_sum
+  intro a _
+  rw [sum_endpointRootMass,Finset.mul_sum]
+  apply Finset.sum_le_sum
+  intro b _
+  by_cases h : (out₁ a).1 = (out₂ b).1 ∧ R a b
+  · have hh : (out₂ b).1 = (out₁ a).1 ∧ endpoint b ∈ S a :=
+      ⟨h.1.symm,hS a b h.1 h.2⟩
+    rw [ite_eq_left h,ite_eq_left hh]
+    change (spliceTupleWeight Q J₁ p (out₁ a) * auxiliaryTupleWeight Q J₂ p (out₂ b)) *
+      m₁ a * m₂ b ≤ _
+    ring_nf
+    exact le_refl _
+  · rw [ite_eq_right h]
+    apply mul_nonneg (mul_nonneg (spliceTupleWeight_nonneg Q J₁ p _) (hm₁ a))
+    split_ifs
+    · exact mul_nonneg (auxiliaryTupleWeight_nonneg Q J₂ p _) (hm₂ b)
+    · exact le_refl 0
+
+ 
+
+theorem finite_near_pair_mass_le {I₁ I₂ : Type*} [Fintype I₁] [Fintype I₂]
+    (Q : Finset E) (J₁ J₂ : Finset ℕ) (p : unitInterval)
+    (out₁ : I₁ → SpliceState E) (out₂ : I₂ → SpliceState E)
+    (m₁ : I₁ → ℝ) (m₂ : I₂ → ℝ) (hm₁ : ∀ a, 0 ≤ m₁ a) (hm₂ : ∀ b, 0 ≤ m₂ b)
+    (endpoint : I₂ → V) (R : I₁ → I₂ → Prop) (S : I₁ → Finset V)
+    (hS : ∀ a b, (out₁ a).1 = (out₂ b).1 → R a b → endpoint b ∈ S a)
+    (N : ℕ) (hN : ∀ a, (S a).card ≤ N) (C : ℝ) (hC : 0 ≤ C)
+    (hb : ∀ a z, z ∈ S a → endpointRootMass Q J₂ p out₂ endpoint m₂ (out₁ a).1 z ≤ C) :
+    (∑ a, ∑ b, if (out₁ a).1 = (out₂ b).1 ∧ R a b then
+      jointTupleWeight Q J₁ J₂ p (pairSpliceOutput (out₁ a) (out₂ b)) * m₁ a * m₂ b else 0) ≤
+    (N:ℝ) * C * ∑ a, spliceTupleWeight Q J₁ p (out₁ a) * m₁ a := by
+  apply (finite_near_pair_mass_le_cover Q J₁ J₂ p out₁ out₂ m₁ m₂ hm₁ hm₂ endpoint R S hS).trans
+  calc
+    _ ≤ ∑ a, spliceTupleWeight Q J₁ p (out₁ a) * m₁ a * ((N:ℝ)*C) := by
+      apply Finset.sum_le_sum
+      intro a _
+      apply mul_le_mul_of_nonneg_left _
+        (mul_nonneg (spliceTupleWeight_nonneg Q J₁ p _) (hm₁ a))
+      calc
+        _ ≤ ∑ _z ∈ S a, C := Finset.sum_le_sum (fun z hz => hb a z hz)
+        _ = ((S a).card:ℝ) * C := by simp
+        _ ≤ (N:ℝ)*C := mul_le_mul_of_nonneg_right (by exact_mod_cast hN a) hC
+    _ = _ := by rw [← Finset.sum_mul]; ring
+
+end NearPairMass
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section EndpointInverseBound
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem finiteInverseLoad_subtype {I : Type*} [Fintype I]
+    (out : I → SpliceState E) (m : I → ℝ) (P : I → Prop) (t : SpliceState E) :
+    finiteInverseLoad out (fun a => if P a then m a else 0) t =
+      finiteInverseLoad (fun a : {a : I // P a} => out a.val) (fun a => m a.val) t := by
+  unfold finiteInverseLoad
+  have he : (∑ a, if out a = t then if P a then m a else 0 else 0) =
+      ∑ a, if P a then if out a = t then m a else 0 else 0 := by
+    apply Finset.sum_congr rfl
+    intro a _
+    split_ifs <;> rfl
+  rw [he,← Finset.sum_filter]
+  exact Finset.sum_subtype _ (by intro a; simp) _
+
+ 
+
+theorem endpointInverseLoad_le {I P : Type*} [Fintype I] [Fintype P]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) (k M : ℕ) (L : ℝ) (hL : 1 ≤ L)
+    (inp out : I → SpliceState E) (steps : I → List (ℕ × E)) (y : I → ℕ → V)
+    (endpoint : I → V) (choice : I → P)
+    (hir : Function.Injective (fun a => (inp a,endpoint a,choice a,steps a)))
+    (q : V → SpliceState E → P → ℝ) (hq : ∀ z s b, 0 ≤ q z s b)
+    (hrow : ∀ z s, ∑ b, q z s b ≤ 1)
+    (htrace : ∀ a, SpliceTrace G x (inp a).2 (y a) k 0 (inp a).1 (steps a) (out a).1)
+    (hex : ∀ a, SpliceExecution G x (inp a) (steps a) (out a))
+    (hz : ∀ a, y a k = endpoint a) (hfin : ∀ a, (cluster G (out a).1 x).Finite)
+    (hgood : ∀ a i, 1 ≤ i → i ≤ k → (cluster G ((inp a).2 i) (y a i)).Finite ∧
+      (cluster G ((inp a).2 i) (y a i)).ncard ≤ M ∧ GoodForScale G ((inp a).2 i) (y a i) L)
+    (z : V) (t : SpliceState E) :
+    finiteInverseLoad out (fun a => if endpoint a = z then
+      q (endpoint a) (inp a) (choice a) * ∏ i : Fin k,
+        (traceContactCount G (inp a).2 (y a) (out a).1 x (steps a) (i.val+1):ℝ)⁻¹ else 0) t ≤
+      inverseCodeMass M L ^ k := by
+  rw [finiteInverseLoad_subtype]
+  let J := {a : I // endpoint a = z}
+  have hi : Function.Injective (fun a : J => (inp a.val,choice a.val,steps a.val)) := by
+    intro a b he
+    apply Subtype.ext
+    apply hir
+    have hin : inp a.val = inp b.val := congrArg (fun s : SpliceState E × P × List (ℕ × E) => s.1) he
+    exact Prod.ext hin (Prod.ext (a.property.trans b.property.symm) (congrArg (fun s : SpliceState E × P × List (ℕ × E) => s.2) he))
+  have hb := finiteInverseLoad_weighted_le G hloc x z k M L hL
+    (fun a : J => inp a.val) (fun a : J => out a.val) (fun a : J => steps a.val)
+    (fun a : J => y a.val) (fun a : J => choice a.val) hi (q z) (hq z) (hrow z)
+    (fun a => htrace a.val) (fun a => hex a.val) (fun a => (hz a.val).trans a.property)
+    (fun a => hfin a.val) (fun a => hgood a.val) t
+  convert hb using 1
+  apply congrArg (fun f : J → ℝ => finiteInverseLoad (fun a : J => out a.val) f t)
+  funext a
+  rw [a.property]
+
+end EndpointInverseBound
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section NearHangingMass
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem SpliceTrace.head_hanging (G : BondGraph V E) (x : V) (η : ℕ → Set E)
+    (y : ℕ → V) (k j i : ℕ) (ω ωf : Set E) (e : E) (rest : List (ℕ × E))
+    (h : SpliceTrace G x η y k j ω ((i,e)::rest) ωf) :
+    hangingSide G ωf x e = cluster G ωf x \ cluster G ω x := by
+  rw [hangingSide,h.head_cut G x η y k j i ω ωf e rest]
+
+theorem SpliceTrace.head_hanging_cap (G : BondGraph V E) (x : V) (η : ℕ → Set E)
+    (y : ℕ → V) (k j i M : ℕ) (ω ωf : Set E) (e : E) (rest : List (ℕ × E))
+    (h : SpliceTrace G x η y k j ω ((i,e)::rest) ωf)
+    (hcaps : ∀ l, 1 ≤ l → l ≤ k →
+      (cluster G (η l) (y l)).Finite ∧ (cluster G (η l) (y l)).ncard ≤ M) :
+    (hangingSide G ωf x e).ncard ≤ k*M := by
+  rw [h.head_hanging G x η y k j i ω ωf e rest]
+  exact (h.added_finite_cap G x η y k j M ω ωf ((i,e)::rest) hcaps).2
+
+ 
+
+theorem finite_near_hanging_mass_le {I₁ I₂ : Type*} [Fintype I₁] [Fintype I₂]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (x : V) (v d : ℕ)
+    (hdeg : ∀ a, degree G a ≤ d)
+    (Q : Finset E) (J₁ J₂ : Finset ℕ) (p : unitInterval)
+    (out₁ : I₁ → SpliceState E) (out₂ : I₂ → SpliceState E)
+    (m₁ : I₁ → ℝ) (m₂ : I₂ → ℝ) (hm₁ : ∀ a, 0 ≤ m₁ a) (hm₂ : ∀ b, 0 ≤ m₂ b)
+    (z₁ : I₁ → V) (z₂ : I₂ → V) (e₁ : I₁ → E) (e₂ : I₂ → E)
+    (hfin : ∀ a, (cluster G (out₁ a).1 x).Finite)
+    (hp₁ : ∀ a, OpenMarkedPivotal G (out₁ a).1 {z₁ a} x (e₁ a))
+    (hp₂ : ∀ b, OpenMarkedPivotal G (out₂ b).1 {z₂ b} x (e₂ b))
+    (hcap₁ : ∀ a, (hangingSide G (out₁ a).1 x (e₁ a)).ncard ≤ v)
+    (hcap₂ : ∀ b, (hangingSide G (out₂ b).1 x (e₂ b)).ncard ≤ v)
+    (C : ℝ) (hC : 0 ≤ C)
+    (hb : ∀ a z, endpointRootMass Q J₂ p out₂ z₂ m₂ (out₁ a).1 z ≤ C) :
+    (∑ a, ∑ b, if (out₁ a).1 = (out₂ b).1 ∧
+      ¬GraphSeparated G (hangingSide G (out₁ a).1 x (e₁ a))
+        (hangingSide G (out₂ b).1 x (e₂ b)) then
+      jointTupleWeight Q J₁ J₂ p (pairSpliceOutput (out₁ a) (out₂ b)) * m₁ a * m₂ b else 0) ≤
+    ((v+v^2+d*v^3:ℕ):ℝ) * C * ∑ a, spliceTupleWeight Q J₁ p (out₁ a) * m₁ a := by
+  let R : I₁ → I₂ → Prop := fun a b =>
+    ¬GraphSeparated G (hangingSide G (out₁ a).1 x (e₁ a)) (hangingSide G (out₂ b).1 x (e₂ b))
+  let S := fun a => nearHangingEndpoints G hloc (out₁ a).1 x (z₁ a) (e₁ a) v (hfin a)
+  have hS : ∀ a b, (out₁ a).1 = (out₂ b).1 → R a b → z₂ b ∈ S a := by
+    intro a b he hn
+    have hp := hp₂ b
+    have hc := hcap₂ b
+    change ¬GraphSeparated G _ _ at hn
+    rw [← he] at hp hc hn
+    exact mem_nearHangingEndpoints_of_not_separated G hloc (out₁ a).1 x (z₁ a) (z₂ b)
+      (e₁ a) (e₂ b) v (hfin a) (hp₁ a) hp hc hn
+  have hN : ∀ a, (S a).card ≤ v+v^2+d*v^3 := fun a =>
+    nearHangingEndpoints_card_le G hloc (out₁ a).1 x (z₁ a) (e₁ a) v d
+      (hfin a) (hcap₁ a) hdeg
+  have ht := finite_near_pair_mass_le Q J₁ J₂ p out₁ out₂ m₁ m₂ hm₁ hm₂ z₂ R S hS
+    (v+v^2+d*v^3) hN C hC (fun a z _ => hb a z)
+  convert ht using 1
+  apply Finset.sum_congr rfl
+  intro a _
+  apply Finset.sum_congr rfl
+  intro b _
+  by_cases h : (out₁ a).1 = (out₂ b).1 ∧ R a b
+  · rw [ite_eq_left h,ite_eq_left h]
+  · rw [ite_eq_right h,ite_eq_right h]
+
+end NearHangingMass
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section OldRunArrival
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {W : Type*} [Fintype W]
+
+noncomputable def oldRunArrival (K : ℕ → W → W → ℝ) (k : ℕ) (z : W) (A : Finset W) : ℝ :=
+  ∑ y : Fin k → W, if oldRunVertices k z y 0 ∈ A then oldRunWeight K k z y else 0
+
+theorem oldRunArrival_nonneg (K : ℕ → W → W → ℝ) (hK : ∀ i a b, 0 ≤ K i a b)
+    (k : ℕ) (z : W) (A : Finset W) : 0 ≤ oldRunArrival K k z A := by
+  apply Finset.sum_nonneg
+  intro y _
+  split_ifs
+  · exact oldRunWeight_nonneg K hK k z y
+  · exact le_refl 0
+
+theorem oldRunArrival_zero (K : ℕ → W → W → ℝ) (z : W) (A : Finset W) :
+    oldRunArrival K 0 z A = if z ∈ A then 1 else 0 := by
+  by_cases h : z ∈ A
+  · simp [oldRunArrival,oldRunVertices,oldRunWeight,h]
+  · simp [oldRunArrival,oldRunVertices,h]
+
+theorem oldRunArrival_succ (K : ℕ → W → W → ℝ) (k : ℕ) (z : W) (A : Finset W) :
+    oldRunArrival K (k+1) z A = ∑ a, K (k+1) z a * oldRunArrival K k a A := by
+  unfold oldRunArrival
+  rw [Fintype.sum_equiv (oldRunConsEquiv k) _
+    (fun y => if oldRunVertices k y.1 y.2 0 ∈ A then
+      K (k+1) z y.1 * oldRunWeight K k y.1 y.2 else 0)
+    (by
+      intro y
+      by_cases h : oldRunVertices k (y 0) (Fin.tail y) 0 ∈ A
+      · simp [oldRunVertices_succ_of_le k z y 0 (Nat.zero_le k),oldRunConsEquiv,oldRunWeight,h]
+      · simp [oldRunVertices_succ_of_le k z y 0 (Nat.zero_le k),oldRunConsEquiv,h]),
+    Fintype.sum_prod_type]
+  apply Finset.sum_congr rfl
+  intro a _
+  simp only [Finset.mul_sum,mul_ite,mul_zero]
+
+ 
+
+theorem sum_oldRunArrival_le (K : ℕ → W → W → ℝ) (hK : ∀ i a b, 0 ≤ K i a b)
+    (u : ℝ) (hu : 0 ≤ u) (hcol : ∀ i b, ∑ a, K i a b ≤ u) (k : ℕ) (A : Finset W) :
+    (∑ z, oldRunArrival K k z A) ≤ (A.card:ℝ) * u^k := by
+  induction k with
+  | zero => simp [oldRunArrival_zero]
+  | succ k ih =>
+    simp only [oldRunArrival_succ]
+    rw [Finset.sum_comm]
+    calc
+      _ = ∑ a, (∑ z, K (k+1) z a) * oldRunArrival K k a A := by
+        simp only [Finset.sum_mul]
+      _ ≤ ∑ a, u * oldRunArrival K k a A := Finset.sum_le_sum fun a _ =>
+        mul_le_mul_of_nonneg_right (hcol _ _) (oldRunArrival_nonneg K hK k a A)
+      _ = u * ∑ a, oldRunArrival K k a A := (Finset.mul_sum _ _ _).symm
+      _ ≤ u*((A.card:ℝ)*u^k) := mul_le_mul_of_nonneg_left ih hu
+      _ = (A.card:ℝ)*u^(k+1) := by ring
+
+ 
+theorem sum_oldRunArrival_on_le (K : ℕ → W → W → ℝ) (hK : ∀ i a b, 0 ≤ K i a b)
+    (u : ℝ) (hu : 0 ≤ u) (hcol : ∀ i b, ∑ a, K i a b ≤ u)
+    (k : ℕ) (A Z : Finset W) :
+    (∑ z ∈ Z, oldRunArrival K k z A) ≤ (A.card:ℝ) * u^k := by
+  apply (Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ Z)
+    (fun z _ _ => oldRunArrival_nonneg K hK k z A)).trans
+  exact sum_oldRunArrival_le K hK u hu hcol k A
+
+ 
+theorem noSwitch_arrival_le (K : ℕ → W → W → ℝ) (hK : ∀ i a b, 0 ≤ K i a b)
+    (u : ℝ) (hu : 0 ≤ u) (hcol : ∀ i b, ∑ a, K i a b ≤ u)
+    (k n : ℕ) (hn : 0 < n) (A Z : Finset W) (hA : A.card ≤ 2*n) :
+    (n:ℝ)⁻¹ * ∑ z ∈ Z, oldRunArrival K k z A ≤ 2*u^k := by
+  have hn' : (0:ℝ) < n := by exact_mod_cast hn
+  calc
+    _ ≤ (n:ℝ)⁻¹ * ((A.card:ℝ)*u^k) :=
+      mul_le_mul_of_nonneg_left (sum_oldRunArrival_on_le K hK u hu hcol k A Z)
+        (inv_nonneg.mpr hn'.le)
+    _ ≤ (n:ℝ)⁻¹ * ((2*(n:ℝ))*u^k) := mul_le_mul_of_nonneg_left
+      (mul_le_mul_of_nonneg_right (by exact_mod_cast hA) (pow_nonneg hu _)) (inv_nonneg.mpr hn'.le)
+    _ = 2*u^k := by field_simp
+
+end OldRunArrival
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section TrajectoryTransport
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {ι : Type*} [Fintype ι]
+
+ 
+
+theorem RootTransport.real (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (htr : RootTransport root weight Γ) (m : V → V → ℝ)
+    (hi : ∀ a ∈ Γ, ∀ x y, m (a x) (a y) = m x y)
+    (hr : ∀ x, Summable (m x)) (hc : ∀ y, Summable (fun x => m x y)) :
+    (∑ i, weight i * ∑' y, m (root i) y) =
+      (∑ i, weight i * ∑' x, m x (root i)) := by
+  have ht := htr (fun x y => (m x y:ℂ))
+    (fun a ha x y => congrArg Complex.ofReal (hi a ha x y))
+    (fun x => Complex.summable_ofReal.mpr (hr x))
+    (fun y => Complex.summable_ofReal.mpr (hc y))
+  apply Complex.ofReal_injective
+  simpa only [Complex.ofReal_sum,Complex.ofReal_mul,Complex.ofReal_tsum] using ht
+
+ 
+
+theorem transported_column_le (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
+    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (m : V → V → ℝ) (hm : ∀ x y, 0 ≤ m x y)
+    (hi : ∀ a ∈ Γ, ∀ x y, m (a x) (a y) = m x y)
+    (hr : ∀ x, Summable (m x)) (hcol : ∀ y, Summable (fun x => m x y))
+    (B : ℝ) (hB : ∀ x, ∑' y, m x y ≤ B) (y : V) :
+    (∑' x, m x y) ≤ c*B := by
+  have ht := htr.real root weight Γ m hi hr hcol
+  have hb : (∑ i, weight i * ∑' x, m x (root i)) ≤ B := by
+    rw [← ht]
+    calc
+      _ ≤ ∑ i, weight i * B := Finset.sum_le_sum fun i _ =>
+        mul_le_mul_of_nonneg_left (hB _) (hw i)
+      _ = B := by rw [← Finset.sum_mul,hw1,one_mul]
+  obtain ⟨i,a,ha,hay⟩ := hcov y
+  have he : (∑' x, m x y) = ∑' x, m x (root i) := by
+    rw [← hay,← a.tsum_eq (fun x => m x (a (root i)))]
+    exact tsum_congr (fun x => hi a ha x (root i))
+  have hn : 0 ≤ ∑' x, m x (root i) := tsum_nonneg (fun x => hm x _)
+  have hs : weight i * ∑' x, m x (root i) ≤ ∑ j, weight j * ∑' x, m x (root j) :=
+    Finset.single_le_sum (fun j _ => mul_nonneg (hw j) (tsum_nonneg (fun x => hm x (root j))))
+      (Finset.mem_univ i)
+  rw [he]
+  calc
+    _ ≤ (c*weight i) * ∑' x, m x (root i) := by
+      simpa only [one_mul] using mul_le_mul_of_nonneg_right (hmin i) hn
+    _ = c * (weight i * ∑' x, m x (root i)) := by ring
+    _ ≤ c * (∑ j, weight j * ∑' x, m x (root j)) := mul_le_mul_of_nonneg_left hs hc
+    _ ≤ c*B := mul_le_mul_of_nonneg_left hb hc
+
+ 
+
+theorem finite_trajectory_hit_le (m : V → V → ℝ) (hm : ∀ x y, 0 ≤ m x y)
+    (hcol : ∀ y, Summable (fun x => m x y)) (B : ℝ)
+    (hb : ∀ y, ∑' x, m x y ≤ B) (h : V → ℝ) (A Z : Finset V)
+    (hh : ∀ x, h x ≤ ∑ y ∈ A, m x y) :
+    (∑ x ∈ Z, h x) ≤ (A.card:ℝ)*B := by
+  calc
+    _ ≤ ∑ x ∈ Z, ∑ y ∈ A, m x y := Finset.sum_le_sum fun x _ => hh x
+    _ = ∑ y ∈ A, ∑ x ∈ Z, m x y := Finset.sum_comm
+    _ ≤ ∑ y ∈ A, ∑' x, m x y := Finset.sum_le_sum fun y _ =>
+      (hcol y).sum_le_tsum Z (fun x _ => hm x y)
+    _ ≤ ∑ _y ∈ A, B := Finset.sum_le_sum fun y _ => hb y
+    _ = (A.card:ℝ)*B := by simp
+
+end TrajectoryTransport
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section NonnegativeTransport
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {ι : Type*} [Fintype ι]
+
+theorem mem_reachBall_aut_iff (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (ha : IsAut G a b) (x y : V) (n : ℕ) :
+    a y ∈ reachBall G (a x) n ↔ y ∈ reachBall G x n := by
+  constructor
+  · intro h
+    simpa only [Equiv.symm_apply_apply] using mem_reachBall_aut G (isAut_symm G ha) h
+  · exact mem_reachBall_aut G ha
+
+ 
+def transportCut (G : BondGraph V E) (n : ℕ) (x y : V) : Prop :=
+  y ∈ reachBall G x n ∧ x ∈ reachBall G y n
+
+theorem transportCut_mono (G : BondGraph V E) {n m : ℕ} (h : n ≤ m)
+    {x y : V} (hxy : transportCut G n x y) : transportCut G m x y :=
+  ⟨reachBall_mono G x h hxy.1, reachBall_mono G y h hxy.2⟩
+
+theorem transportCut_aut (G : BondGraph V E) {a : V ≃ V} {b : E ≃ E}
+    (ha : IsAut G a b) (n : ℕ) (x y : V) :
+    transportCut G n (a x) (a y) ↔ transportCut G n x y := by
+  simp only [transportCut,mem_reachBall_aut_iff G ha]
+
+theorem exists_transportCut (G : BondGraph V E) (hconn : Connected G)
+    (y : V) (Z : Finset V) : ∃ n, ∀ x ∈ Z, transportCut G n x y := by
+  induction Z using Finset.induction_on with
+  | empty => exact ⟨0, by simp⟩
+  | @insert x Z _ ih =>
+    obtain ⟨n,hn⟩ := ih
+    obtain ⟨r,hr⟩ := exists_mem_reachBall (hconn x y)
+    obtain ⟨s,hs⟩ := exists_mem_reachBall (hconn y x)
+    refine ⟨max n (max r s), ?_⟩
+    intro z hz
+    rcases Finset.mem_insert.mp hz with rfl | hz
+    · exact ⟨reachBall_mono G z (le_trans (le_max_left r s) (le_max_right n _)) hr,
+        reachBall_mono G y (le_trans (le_max_right r s) (le_max_right n _)) hs⟩
+    · exact transportCut_mono G (le_max_left n _) (hn z hz)
+
+ 
+
+theorem transported_column_of_row_bound (G : BondGraph V E)
+    (hconn : Connected G) (hloc : LocallyFinite G)
+    (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
+    (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
+    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (m : V → V → ℝ) (hm : ∀ x y, 0 ≤ m x y)
+    (hi : ∀ a ∈ Γ, ∀ x y, m (a x) (a y) = m x y)
+    (hr : ∀ x, Summable (m x)) (B : ℝ) (hB : ∀ x, ∑' y, m x y ≤ B) :
+    ∀ y, Summable (fun x => m x y) ∧ (∑' x, m x y) ≤ c*B := by
+  let cut (n : ℕ) (x y : V) : ℝ := if transportCut G n x y then m x y else 0
+  have hn (n : ℕ) (x y : V) : 0 ≤ cut n x y := by
+    dsimp [cut]; split_ifs; exact hm x y; exact le_refl 0
+  have hcRow (n : ℕ) (x : V) : Summable (cut n x) := by
+    apply summable_of_ne_finset_zero (s := (finite_reachBall hloc x n).toFinset)
+    intro y hy
+    simp only [Set.Finite.mem_toFinset] at hy
+    exact ite_eq_right (fun h => hy h.1)
+  have hcCol (n : ℕ) (y : V) : Summable (fun x => cut n x y) := by
+    apply summable_of_ne_finset_zero (s := (finite_reachBall hloc y n).toFinset)
+    intro x hx
+    simp only [Set.Finite.mem_toFinset] at hx
+    exact ite_eq_right (fun h => hx h.2)
+  have hcInv (n : ℕ) : ∀ a ∈ Γ, ∀ x y, cut n (a x) (a y) = cut n x y := by
+    intro a ha x y
+    obtain ⟨b,hb⟩ := hΓ ha
+    simp only [cut,transportCut_aut G hb,hi a ha]
+  have hcBound (n : ℕ) (x : V) : (∑' y, cut n x y) ≤ B := by
+    apply le_trans ((hcRow n x).tsum_le_tsum (fun y => ?_) (hr x)) (hB x)
+    dsimp [cut]; split_ifs; exact le_refl _; exact hm x y
+  have hbound (y : V) (Z : Finset V) : (∑ x ∈ Z, m x y) ≤ c*B := by
+    obtain ⟨n,hnZ⟩ := exists_transportCut G hconn y Z
+    calc
+      _ = ∑ x ∈ Z, cut n x y := by
+        apply Finset.sum_congr rfl
+        intro x hx
+        exact (ite_eq_left (hnZ x hx)).symm
+      _ ≤ ∑' x, cut n x y := (hcCol n y).sum_le_tsum Z (fun x _ => hn n x y)
+      _ ≤ c*B := transported_column_le root weight Γ htr hcov hw hw1 c hc hmin
+        (cut n) (hn n) (hcInv n) (hcRow n) (hcCol n) B (hcBound n) y
+  intro y
+  exact ⟨summable_of_sum_le (fun x => hm x y) (hbound y),
+    Real.tsum_le_of_sum_le (fun x => hm x y) (hbound y)⟩
+
+ 
+
+theorem trajectory_hit_tsum_le (m : V → V → ℝ) (hm : ∀ x y, 0 ≤ m x y)
+    (hcol : ∀ y, Summable (fun x => m x y)) (B : ℝ)
+    (hb : ∀ y, ∑' x, m x y ≤ B) (h : V → ℝ) (hh0 : ∀ x, 0 ≤ h x)
+    (A : Finset V) (hh : ∀ x, h x ≤ ∑ y ∈ A, m x y) :
+    Summable h ∧ (∑' x, h x) ≤ (A.card:ℝ)*B := by
+  have hs := fun Z => finite_trajectory_hit_le m hm hcol B hb h A Z hh
+  exact ⟨summable_of_sum_le hh0 hs, Real.tsum_le_of_sum_le hh0 hs⟩
+
+end NonnegativeTransport
+
+end Percolation
+
+ 
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+section FiniteTrajectoryMass
+attribute [local instance] Classical.propDecidable Classical.decEq
+variable {W : Type*} [Fintype W]
+
+theorem finiteFamilyIntegral_mul_const (Q : Finset E) (J : Finset ℕ) (p c : ℝ)
+    (f : (ℕ → Set E) → ℝ) :
+    finiteFamilyIntegral Q J p (fun η => f η * c) = finiteFamilyIntegral Q J p f * c := by
+  simp only [finiteFamilyIntegral,Finset.sum_mul,mul_assoc]
+
+omit [Fintype W] in
+theorem oldRunWeight_factor_ne_zero (q : Set E → W → W → ℝ) (k : ℕ)
+    (η : ℕ → Set E) (z : W) (y : Fin k → W)
+    (hy : oldRunWeight (fun i => q (η i)) k z y ≠ 0) (i : ℕ) (hi : 1 ≤ i) (hik : i ≤ k) :
+    q (η i) (oldRunVertices k z y i) (oldRunVertices k z y (i-1)) ≠ 0 := by
+  rw [← oldPathWeight_oldRunVertices] at hy
+  exact (Finset.prod_ne_zero_iff.mp hy) ⟨i,Finset.mem_Icc.mpr ⟨hi,hik⟩⟩ (Finset.mem_univ _)
+
+ 
+
+noncomputable def finiteTrajectoryMass (G : BondGraph V E) (e : W → V)
+    (Q : Finset E) (p : ℝ) (k : ℕ) (q : Set E → W → W → ℝ) (z : W) (a : V) : ℝ :=
+  finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η =>
+    ∑ y : Fin k → W, if a ∈ trajectoryUnion G η (e ∘ oldRunVertices k z y) k then
+      oldRunWeight (fun i => q (η i)) k z y else 0)
+
+noncomputable def finiteTrajectoryHit (G : BondGraph V E) (e : W → V)
+    (Q : Finset E) (p : ℝ) (k : ℕ) (q : Set E → W → W → ℝ) (z : W) (A : Finset V) : ℝ :=
+  finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η =>
+    ∑ y : Fin k → W, if ((A:Set V) ∩ trajectoryUnion G η (e ∘ oldRunVertices k z y) k).Nonempty then
+      oldRunWeight (fun i => q (η i)) k z y else 0)
+
+theorem finiteTrajectoryMass_nonneg (G : BondGraph V E) (e : W → V)
+    (Q : Finset E) (p : unitInterval) (k : ℕ) (q : Set E → W → W → ℝ)
+    (hq : ∀ ω a b, 0 ≤ q ω a b) (z : W) (a : V) :
+    0 ≤ finiteTrajectoryMass G e Q p k q z a := by
+  apply finiteFamilyIntegral_nonneg
+  intro η
+  apply Finset.sum_nonneg
+  intro y _
+  split_ifs
+  · exact oldRunWeight_nonneg _ (fun _ => hq _) k z y
+  · exact le_refl 0
+
+theorem finiteTrajectoryHit_nonneg (G : BondGraph V E) (e : W → V)
+    (Q : Finset E) (p : unitInterval) (k : ℕ) (q : Set E → W → W → ℝ)
+    (hq : ∀ ω a b, 0 ≤ q ω a b) (z : W) (A : Finset V) :
+    0 ≤ finiteTrajectoryHit G e Q p k q z A := by
+  apply finiteFamilyIntegral_nonneg
+  intro η
+  apply Finset.sum_nonneg
+  intro y _
+  split_ifs
+  · exact oldRunWeight_nonneg _ (fun _ => hq _) k z y
+  · exact le_refl 0
+
+theorem finiteTrajectoryHit_le_sum (G : BondGraph V E) (e : W → V)
+    (Q : Finset E) (p : unitInterval) (k : ℕ) (q : Set E → W → W → ℝ)
+    (hq : ∀ ω a b, 0 ≤ q ω a b) (z : W) (A : Finset V) :
+    finiteTrajectoryHit G e Q p k q z A ≤ ∑ a ∈ A, finiteTrajectoryMass G e Q p k q z a := by
+  unfold finiteTrajectoryHit finiteTrajectoryMass
+  rw [← finiteFamilyIntegral_sum]
+  apply finiteFamilyIntegral_mono
+  intro η
+  rw [Finset.sum_comm]
+  apply Finset.sum_le_sum
+  intro y _
+  have hn := oldRunWeight_nonneg (fun i => q (η i)) (fun _ => hq _) k z y
+  by_cases h : ((A:Set V) ∩ trajectoryUnion G η (e ∘ oldRunVertices k z y) k).Nonempty
+  · rw [ite_eq_left h]
+    obtain ⟨a,ha,hat⟩ := h
+    have hs := Finset.single_le_sum (s := A)
+      (f := fun a => if a ∈ trajectoryUnion G η (e ∘ oldRunVertices k z y) k then
+        oldRunWeight (fun i => q (η i)) k z y else 0)
+      (fun a _ => by split_ifs; exact hn; exact le_refl 0) ha
+    simpa only [ite_eq_left hat] using hs
+  · rw [ite_eq_right h]
+    exact Finset.sum_nonneg fun a _ => by split_ifs; exact hn; exact le_refl 0
+
+ 
+
+theorem finiteTrajectoryMass_finite_sum_le (G : BondGraph V E) (e : W → V)
+    (Q : Finset E) (p : unitInterval) (k M : ℕ) (q : Set E → W → W → ℝ)
+    (hq : ∀ ω a b, 0 ≤ q ω a b)
+    (hcap : ∀ ω a b, q ω a b ≠ 0 →
+      (cluster G ω (e a)).Finite ∧ (cluster G ω (e a)).ncard ≤ M)
+    (u : ℝ) (hu : 0 ≤ u) (hrow : ∀ a, ∑ b, finiteOldKernel Q p q a b ≤ u)
+    (z : W) (A : Finset V) :
+    (∑ a ∈ A, finiteTrajectoryMass G e Q p k q z a) ≤ (k*M:ℕ)*u^k := by
+  unfold finiteTrajectoryMass
+  rw [← finiteFamilyIntegral_sum]
+  have hb : ∀ η, (∑ a ∈ A, ∑ y : Fin k → W,
+      if a ∈ trajectoryUnion G η (e ∘ oldRunVertices k z y) k then
+        oldRunWeight (fun i => q (η i)) k z y else 0) ≤
+      (∑ y : Fin k → W, oldRunWeight (fun i => q (η i)) k z y) * (k*M:ℕ) := by
+    intro η
+    rw [Finset.sum_comm,Finset.sum_mul]
+    apply Finset.sum_le_sum
+    intro y _
+    by_cases hy : oldRunWeight (fun i => q (η i)) k z y = 0
+    · simp [hy]
+    obtain ⟨hf,hc⟩ := trajectoryUnion_finite_cap G η (e ∘ oldRunVertices k z y) k M
+      (fun i hi hik => hcap _ _ _ (oldRunWeight_factor_ne_zero q k η z y hy i hi hik))
+    have hsub : A.filter (fun a => a ∈ trajectoryUnion G η (e ∘ oldRunVertices k z y) k) ⊆ hf.toFinset := by
+      intro a ha
+      exact hf.mem_toFinset.mpr (Finset.mem_filter.mp ha).2
+    have hcard : ((A.filter (fun a => a ∈ trajectoryUnion G η (e ∘ oldRunVertices k z y) k)).card:ℝ) ≤ (k*M:ℕ) := by
+      exact_mod_cast (Finset.card_le_card hsub).trans (by simpa only [← Set.ncard_eq_toFinset_card _ hf] using hc)
+    rw [← Finset.sum_filter]
+    simp only [Finset.sum_const, nsmul_eq_mul]
+    simpa only [mul_comm] using mul_le_mul_of_nonneg_left hcard
+      (oldRunWeight_nonneg _ (fun _ => hq _) k z y)
+  calc
+    _ ≤ finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η =>
+        (∑ y : Fin k → W, oldRunWeight (fun i => q (η i)) k z y) * (k*M:ℕ)) :=
+      finiteFamilyIntegral_mono Q (Finset.Icc 1 k) p _ _ hb
+    _ = finiteFamilyIntegral Q (Finset.Icc 1 k) p (fun η =>
+        ∑ y : Fin k → W, oldRunWeight (fun i => q (η i)) k z y) * (k*M:ℕ) :=
+      finiteFamilyIntegral_mul_const Q (Finset.Icc 1 k) p _ _
+    _ ≤ u^k * (k*M:ℕ) := mul_le_mul_of_nonneg_right
+      (independent_oldRun_success_le Q p k q hq u hu hrow z) (Nat.cast_nonneg _)
+    _ = (k*M:ℕ)*u^k := mul_comm _ _
+
+ 
+
+theorem finiteTrajectoryMass_row_bound (G : BondGraph V E) (e : W → V)
+    (Q : Finset E) (p : unitInterval) (k M : ℕ) (q : Set E → W → W → ℝ)
+    (hq : ∀ ω a b, 0 ≤ q ω a b)
+    (hcap : ∀ ω a b, q ω a b ≠ 0 →
+      (cluster G ω (e a)).Finite ∧ (cluster G ω (e a)).ncard ≤ M)
+    (u : ℝ) (hu : 0 ≤ u) (hrow : ∀ a, ∑ b, finiteOldKernel Q p q a b ≤ u) (z : W) :
+    Summable (finiteTrajectoryMass G e Q p k q z) ∧
+      (∑' a, finiteTrajectoryMass G e Q p k q z a) ≤ (k*M:ℕ)*u^k := by
+  have hn := finiteTrajectoryMass_nonneg G e Q p k q hq z
+  have hb := finiteTrajectoryMass_finite_sum_le G e Q p k M q hq hcap u hu hrow z
+  exact ⟨summable_of_sum_le hn hb,Real.tsum_le_of_sum_le hn hb⟩
+
+end FiniteTrajectoryMass
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/TwoCluster/Bounds.lean b/lean/OAI/Probability/UnimodularPercolation/TwoCluster/Bounds.lean
index d33643a2f717efdb14caddc4db5c11fa3afd95a1..11ab0b9652c297b15f4c33cfe2e453c37e5c3439 100644
--- a/lean/OAI/Probability/UnimodularPercolation/TwoCluster/Bounds.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/TwoCluster/Bounds.lean
@@ -1,258 +1,258 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.TwoCluster.Transport
-
-namespace OAI
-
-/-! Quantitative two-cluster probability bounds. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-open MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem localDensityLoss_integral_bound [Countable V] [Countable E]
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
-    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (d : ℕ) (hd : ∀ y, {e | y ∈ G e}.ncard ≤ d) (n : ℕ) (hn : 0 < n)
-    (p : unitInterval) (hp : 0 < (p:ℝ))
-    (hfin : ∀ y, ∀ᵐ ω ∂law p, (cluster G ω y).Finite) (y : V) :
-    (∫⁻ ω, localDensityLoss G n ω y ∂law p) ≤
-      ENNReal.ofReal (10*c*Real.sqrt (d*n:ℕ)/(p:ℝ)) := by
-  obtain ⟨i,a,ha,hai⟩ := hcov y
-  obtain ⟨b,hab⟩ := hΓ ha
-  have he : (∫⁻ ω, localDensityLoss G n ω (root i) ∂law p) =
-      ∫⁻ ω, localDensityLoss G n ω y ∂law p := by
-    have ht := lintegral_map (μ := law p) (measurable_localDensityLoss G hloc n (root i))
-      (measurable_set_preimage b)
-    rw [law_map_preimage_equiv] at ht
-    rw [ht]
-    apply lintegral_congr
-    intro ω
-    rw [localDensityLoss_aut G n hab,hai]
-  have hwi : (1:ℝ≥0∞) ≤ ENNReal.ofReal c * ENNReal.ofReal (weight i) := by
-    rw [←ENNReal.ofReal_mul hc]
-    exact_mod_cast ENNReal.ofReal_le_ofReal (hmin i)
-  have hsingle : ENNReal.ofReal (weight i) * (∫⁻ ω, localDensityLoss G n ω (root i) ∂law p) ≤
-      ∑ j, ENNReal.ofReal (weight j) * ∫⁻ ω, localDensityLoss G n ω (root j) ∂law p :=
-    Finset.single_le_sum (f := fun j => ENNReal.ofReal (weight j) *
-      ∫⁻ ω, localDensityLoss G n ω (root j) ∂law p) (fun j _ => zero_le) (Finset.mem_univ i)
-  calc
-    _ = 1 * (∫⁻ ω, localDensityLoss G n ω (root i) ∂law p) := by rw [he,one_mul]
-    _ ≤ ENNReal.ofReal c * (ENNReal.ofReal (weight i) *
-      ∫⁻ ω, localDensityLoss G n ω (root i) ∂law p) := by
-      rw [←mul_assoc]
-      exact mul_le_mul_left hwi _
-    _ ≤ ENNReal.ofReal c * (∑ j, ENNReal.ofReal (weight j) *
-      ∫⁻ ω, localDensityLoss G n ω (root j) ∂law p) := mul_le_mul_right hsingle _
-    _ ≤ ENNReal.ofReal c * (2 * ENNReal.ofReal (5*Real.sqrt (d*n:ℕ)/(p:ℝ))) :=
-      mul_le_mul_right (weighted_localDensityLoss_bound G hconn hloc root weight Γ hΓ htr hw hw1
-        d hd n hn p hp hfin) _
-    _ = _ := by
-      rw [←ENNReal.ofReal_ofNat 2,←ENNReal.ofReal_mul (by norm_num),←ENNReal.ofReal_mul hc]
-      congr 1
-      ring
-
- 
-theorem cluster_insert_left_eq_union (G : BondGraph V E) (ω : Set E)
-    (e : E) (a b : V) (he : G e = s(a,b)) :
-    cluster G (insert e ω) a = cluster G ω a ∪ cluster G ω b := by
-  ext y
-  change Conn G (insert e ω) a y ↔ Conn G ω a y ∨ Conn G ω b y
-  rw [conn_insert_iff G ω e a b a y he]
-  constructor
-  · rintro (h | ⟨_,h⟩ | ⟨_,h⟩)
-    · exact Or.inl h
-    · exact Or.inr h
-    · exact Or.inl h
-  · rintro (h | h)
-    · exact Or.inl h
-    · exact Or.inr (Or.inl ⟨conn_refl G ω a,h⟩)
-
-theorem endpoint_density_defect (G : BondGraph V E) (ω : Set E)
-    (e : E) (a b : V) (he : G e = s(a,b)) (hab : ¬Conn G ω a b)
-    (ha : (cluster G ω a).Finite) (hb : (cluster G ω b).Finite)
-    (n : ℕ) (hna : n ≤ (cluster G ω a).ncard) (hnb : n ≤ (cluster G ω b).ncard) :
-    ((cluster G ω a).ncard:ℝ)*densityInfluence G n a e ω +
-      ((cluster G ω b).ncard:ℝ)*densityInfluence G n b e ω = n := by
-  have hc : e ∉ ω := fun h => hab (.single ⟨e,h,he⟩)
-  have hdis : Disjoint (cluster G ω a) (cluster G ω b) := by
-    apply Set.disjoint_left.mpr
-    intro z hz hz'
-    exact hab (conn_trans G ω hz (conn_symm G ω hz'))
-  have hins : Conn G (insert e ω) a b := .single ⟨e,Set.mem_insert e ω,he⟩
-  have hs : (cluster G (insert e ω) a).ncard = (cluster G ω a).ncard+(cluster G ω b).ncard := by
-    rw [cluster_insert_left_eq_union G ω e a b he,Set.ncard_union_eq hdis ha hb]
-  have hsa : 0 < ((cluster G ω a).ncard:ℝ) := by exact_mod_cast cluster_ncard_pos G ω a ha
-  have hsb : 0 < ((cluster G ω b).ncard:ℝ) := by exact_mod_cast cluster_ncard_pos G ω b hb
-  have hna' : (n:ℝ) ≤ (cluster G ω a).ncard := by exact_mod_cast hna
-  have hnb' : (n:ℝ) ≤ (cluster G ω b).ncard := by exact_mod_cast hnb
-  unfold densityInfluence clusterDensity cappedDensity
-  rw [ite_eq_right hc,ite_eq_right hc,←cluster_eq_of_conn G (insert e ω) hins,hs,Nat.cast_add,
-    min_eq_left hna',min_eq_left hnb',min_eq_left (by linarith : (n:ℝ) ≤
-      (cluster G ω a).ncard+(cluster G ω b).ncard)]
-  field_simp
-  ring
-
-theorem localDensityLoss_ge_edge (G : BondGraph V E) (n : ℕ) (ω : Set E)
-    (e : E) (a : V) (ha : a ∈ G e) :
-    ((cluster G ω a).ncard:ℝ≥0∞)*ENNReal.ofReal (densityInfluence G n a e ω) ≤
-      localDensityLoss G n ω a := by
-  apply mul_le_mul_right
-  exact (le_of_eq (ite_eq_left ha).symm).trans
-    (ENNReal.le_tsum (f := fun edge =>
-      if a ∈ G edge then ENNReal.ofReal (densityInfluence G n a edge ω) else 0) e)
-
-theorem two_large_endpoint_loss (G : BondGraph V E) (ω : Set E)
-    (e : E) (a b : V) (he : G e = s(a,b)) (ha : (cluster G ω a).Finite)
-    (hb : (cluster G ω b).Finite) (n : ℕ)
-    (h : ¬Conn G ω a b ∧ ω ∈ clusterLarge G a n ∧ ω ∈ clusterLarge G b n) :
-    (n:ℝ≥0∞) ≤ localDensityLoss G n ω a + localDensityLoss G n ω b := by
-  have hsz (v : V) (hv : (cluster G ω v).Finite) (hh : ω ∈ clusterLarge G v n) :
-      n ≤ (cluster G ω v).ncard := by
-    obtain ⟨t,ht,hct⟩ := hh
-    exact ht.trans (by rw [←Set.ncard_coe_finset]; exact Set.ncard_le_ncard hct hv)
-  have hid := endpoint_density_defect G ω e a b he h.1 ha hb n (hsz a ha h.2.1) (hsz b hb h.2.2)
-  have hid' := congrArg ENNReal.ofReal hid
-  rw [ENNReal.ofReal_add (mul_nonneg (Nat.cast_nonneg _) (densityInfluence_nonneg ..))
-    (mul_nonneg (Nat.cast_nonneg _) (densityInfluence_nonneg ..)),
-    ENNReal.ofReal_mul (Nat.cast_nonneg _),ENNReal.ofReal_mul (Nat.cast_nonneg _)] at hid'
-  simp only [ENNReal.ofReal_natCast] at hid'
-  rw [←hid']
-  exact add_le_add (localDensityLoss_ge_edge G n ω e a (by rw [he]; simp))
-    (localDensityLoss_ge_edge G n ω e b (by rw [he]; simp))
-
-end Percolation
-
- 
-
-open MeasureTheory ProbabilityTheory Filter Topology
-open scoped ENNReal
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-
-def twoLargeEvent (G : BondGraph V E) (a b : V) (n : ℕ) : Set (Set E) :=
-  {ω | ¬Conn G ω a b ∧ ω ∈ clusterLarge G a n ∧ ω ∈ clusterLarge G b n}
-
-theorem measurableSet_twoLargeEvent [Countable V] [Countable E] (G : BondGraph V E) (a b : V) (n : ℕ) :
-    MeasurableSet (twoLargeEvent G a b n) :=
-  (measurableSet_conn G a b).compl.inter
-    ((measurableSet_clusterLarge G a n).inter (measurableSet_clusterLarge G b n))
-
-theorem twoLarge_probability_bound [Countable V] [Countable E]
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
-    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
-    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
-    (d : ℕ) (hd : ∀ y, {e | y ∈ G e}.ncard ≤ d) (n : ℕ) (hn : 0 < n)
-    (p : unitInterval) (hp : 0 < (p:ℝ))
-    (hfin : ∀ y, ∀ᵐ ω ∂law p, (cluster G ω y).Finite)
-    (e : E) (a b : V) (he : G e = s(a,b)) :
-    (n:ℝ) * (law p).real (twoLargeEvent G a b n) ≤
-      20*c*Real.sqrt (d*n:ℕ)/(p:ℝ) := by
-  have hpoint : ∀ᵐ ω ∂law p,
-      (twoLargeEvent G a b n).indicator (fun _ => (n:ℝ≥0∞)) ω ≤
-        localDensityLoss G n ω a + localDensityLoss G n ω b := by
-    filter_upwards [hfin a,hfin b] with ω ha hb
-    by_cases h : ω ∈ twoLargeEvent G a b n
-    · rw [Set.indicator_of_mem h]
-      exact two_large_endpoint_loss G ω e a b he ha hb n h
-    · rw [Set.indicator_of_notMem h]
-      exact zero_le
-  have hi := lintegral_mono_ae hpoint
-  rw [lintegral_indicator (measurableSet_twoLargeEvent G a b n),lintegral_const,
-    lintegral_add_left (measurable_localDensityLoss G hloc n a)] at hi
-  have hbnd (y : V) := localDensityLoss_integral_bound G hconn hloc root weight Γ hΓ htr
-    hcov hw hw1 c hc hmin d hd n hn p hp hfin y
-  have hi' := hi.trans (add_le_add (hbnd a) (hbnd b))
-  rw [←ENNReal.ofReal_add (by positivity : 0 ≤ 10*c*Real.sqrt (d*n:ℕ)/(p:ℝ))
-    (by positivity : 0 ≤ 10*c*Real.sqrt (d*n:ℕ)/(p:ℝ))] at hi'
-  have ht := ENNReal.toReal_mono ENNReal.ofReal_ne_top hi'
-  rw [ENNReal.toReal_mul,ENNReal.toReal_natCast,ENNReal.toReal_ofReal (by positivity)] at ht
-  simp only [Measure.restrict_apply_univ] at ht
-  dsimp [Measure.real] at ht ⊢
-  convert ht using 1
-  ring
-
- 
-
-theorem finiteWitness_eventually_iff (A : Set (Set E)) (hu : IsUpperSet A)
-    (hf : FiniteWitness A) (ω : ℕ → Set E) (hm : Monotone ω) :
-    ∀ᶠ j in atTop, ω j ∈ A ↔ (⋃ k, ω k) ∈ A := by
-  by_cases h : (⋃ k, ω k) ∈ A
-  · obtain ⟨t,ht,htA⟩ := hf _ h
-    have ht' : ∀ e ∈ t, ∃ k, e ∈ ω k := by
-      intro e he
-      exact Set.mem_iUnion.mp (ht he)
-    choose k hk using fun e : t => ht' e e.property
-    let N := t.attach.sup k
-    have hN : (t : Set E) ⊆ ω N := by
-      intro e he
-      exact hm (Finset.le_sup (f := k) (Finset.mem_attach t ⟨e,he⟩)) (hk ⟨e,he⟩)
-    filter_upwards [eventually_ge_atTop N] with j hj
-    exact iff_of_true (hu (hN.trans (hm hj)) htA) h
-  · exact Eventually.of_forall (fun j => iff_of_false
-      (fun hj => h (hu (Set.subset_iUnion ω j) hj)) h)
-
-theorem twoLarge_probability_tendsto [Countable V] [Countable E] (G : BondGraph V E)
-    (p : ℕ → unitInterval) (hp : Monotone p) (q : unitInterval)
-    (hle : ∀ j, p j ≤ q) (hlim : Tendsto p atTop (𝓝 q)) (a b : V) (n : ℕ) :
-    Tendsto (fun j => (law (p j)).real (twoLargeEvent G a b n)) atTop
-      (𝓝 ((law q).real (twoLargeEvent G a b n))) := by
-  let A := twoLargeEvent G a b n
-  let f := fun r labels => A.indicator (fun _ => (1:ℝ)) (openEdges labels r)
-  have hfi (r : unitInterval) : Integrable (f r) labelLaw := by
-    have hi : Integrable (A.indicator (fun _ => (1:ℝ))) (law r) :=
-      (integrable_const _).indicator (measurableSet_twoLargeEvent G a b n)
-    rw [←map_openEdges r] at hi
-    exact hi.comp_measurable (measurable_openEdges r)
-  have ht : ∀ᵐ labels ∂(labelLaw : Measure (E → unitInterval)),
-      Tendsto (fun j => f (p j) labels) atTop (𝓝 (f q labels)) := by
-    filter_upwards [ae_labels_ne (E := E) q] with labels hlabels
-    have hm : Monotone (fun j => openEdges labels (p j)) :=
-      fun j k hjk => openEdges_mono labels (hp hjk)
-    have he := openEdges_eq_iUnion_of_tendsto labels q hlabels p hle hlim
-    have hC := finiteWitness_eventually_iff {ω | Conn G ω a b}
-      (fun _ _ hω h => conn_mono G hω h) (finiteWitness_conn G a b) _ hm
-    have ha := finiteWitness_eventually_iff (clusterLarge G a n)
-      (clusterLarge_upper G a n) (finiteWitness_clusterLarge G a n) _ hm
-    have hb := finiteWitness_eventually_iff (clusterLarge G b n)
-      (clusterLarge_upper G b n) (finiteWitness_clusterLarge G b n) _ hm
-    rw [←he] at hC ha hb
-    apply tendsto_const_nhds.congr'
-    filter_upwards [hC,ha,hb] with j hC ha hb
-    have hi : openEdges labels (p j) ∈ A ↔ openEdges labels q ∈ A := by
-      exact and_congr (not_congr hC) (and_congr ha hb)
-    simp only [f,Set.indicator_apply]
-    rw [hi]
-  have ht' := tendsto_integral_of_dominated_convergence (fun _ : E → unitInterval => (1:ℝ))
-    (fun j => (hfi (p j)).aestronglyMeasurable) (integrable_const _)
-    (fun j => Eventually.of_forall (fun labels => by
-      simp only [f,Set.indicator_apply]
-      split <;> norm_num)) ht
-  have heq (r : unitInterval) : (∫ labels, f r labels ∂labelLaw) = (law r).real A := by
-    calc
-      _ = ∫ ω, A.indicator (fun _ => (1:ℝ)) ω ∂law r := by
-        rw [←map_openEdges r]
-        exact (integral_map (measurable_openEdges r).aemeasurable
-          ((measurable_const.indicator (measurableSet_twoLargeEvent G a b n)).aestronglyMeasurable)).symm
-      _ = _ := by rw [integral_indicator_const _ (measurableSet_twoLargeEvent G a b n)]; simp [A]
-  simpa only [heq] using ht'
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.TwoCluster.Transport
+
+namespace OAI
+
+/-! Quantitative two-cluster probability bounds. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+open MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem localDensityLoss_integral_bound [Countable V] [Countable E]
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
+    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (d : ℕ) (hd : ∀ y, {e | y ∈ G e}.ncard ≤ d) (n : ℕ) (hn : 0 < n)
+    (p : unitInterval) (hp : 0 < (p:ℝ))
+    (hfin : ∀ y, ∀ᵐ ω ∂law p, (cluster G ω y).Finite) (y : V) :
+    (∫⁻ ω, localDensityLoss G n ω y ∂law p) ≤
+      ENNReal.ofReal (10*c*Real.sqrt (d*n:ℕ)/(p:ℝ)) := by
+  obtain ⟨i,a,ha,hai⟩ := hcov y
+  obtain ⟨b,hab⟩ := hΓ ha
+  have he : (∫⁻ ω, localDensityLoss G n ω (root i) ∂law p) =
+      ∫⁻ ω, localDensityLoss G n ω y ∂law p := by
+    have ht := lintegral_map (μ := law p) (measurable_localDensityLoss G hloc n (root i))
+      (measurable_set_preimage b)
+    rw [law_map_preimage_equiv] at ht
+    rw [ht]
+    apply lintegral_congr
+    intro ω
+    rw [localDensityLoss_aut G n hab,hai]
+  have hwi : (1:ℝ≥0∞) ≤ ENNReal.ofReal c * ENNReal.ofReal (weight i) := by
+    rw [←ENNReal.ofReal_mul hc]
+    exact_mod_cast ENNReal.ofReal_le_ofReal (hmin i)
+  have hsingle : ENNReal.ofReal (weight i) * (∫⁻ ω, localDensityLoss G n ω (root i) ∂law p) ≤
+      ∑ j, ENNReal.ofReal (weight j) * ∫⁻ ω, localDensityLoss G n ω (root j) ∂law p :=
+    Finset.single_le_sum (f := fun j => ENNReal.ofReal (weight j) *
+      ∫⁻ ω, localDensityLoss G n ω (root j) ∂law p) (fun j _ => zero_le) (Finset.mem_univ i)
+  calc
+    _ = 1 * (∫⁻ ω, localDensityLoss G n ω (root i) ∂law p) := by rw [he,one_mul]
+    _ ≤ ENNReal.ofReal c * (ENNReal.ofReal (weight i) *
+      ∫⁻ ω, localDensityLoss G n ω (root i) ∂law p) := by
+      rw [←mul_assoc]
+      exact mul_le_mul_left hwi _
+    _ ≤ ENNReal.ofReal c * (∑ j, ENNReal.ofReal (weight j) *
+      ∫⁻ ω, localDensityLoss G n ω (root j) ∂law p) := mul_le_mul_right hsingle _
+    _ ≤ ENNReal.ofReal c * (2 * ENNReal.ofReal (5*Real.sqrt (d*n:ℕ)/(p:ℝ))) :=
+      mul_le_mul_right (weighted_localDensityLoss_bound G hconn hloc root weight Γ hΓ htr hw hw1
+        d hd n hn p hp hfin) _
+    _ = _ := by
+      rw [←ENNReal.ofReal_ofNat 2,←ENNReal.ofReal_mul (by norm_num),←ENNReal.ofReal_mul hc]
+      congr 1
+      ring
+
+ 
+theorem cluster_insert_left_eq_union (G : BondGraph V E) (ω : Set E)
+    (e : E) (a b : V) (he : G e = s(a,b)) :
+    cluster G (insert e ω) a = cluster G ω a ∪ cluster G ω b := by
+  ext y
+  change Conn G (insert e ω) a y ↔ Conn G ω a y ∨ Conn G ω b y
+  rw [conn_insert_iff G ω e a b a y he]
+  constructor
+  · rintro (h | ⟨_,h⟩ | ⟨_,h⟩)
+    · exact Or.inl h
+    · exact Or.inr h
+    · exact Or.inl h
+  · rintro (h | h)
+    · exact Or.inl h
+    · exact Or.inr (Or.inl ⟨conn_refl G ω a,h⟩)
+
+theorem endpoint_density_defect (G : BondGraph V E) (ω : Set E)
+    (e : E) (a b : V) (he : G e = s(a,b)) (hab : ¬Conn G ω a b)
+    (ha : (cluster G ω a).Finite) (hb : (cluster G ω b).Finite)
+    (n : ℕ) (hna : n ≤ (cluster G ω a).ncard) (hnb : n ≤ (cluster G ω b).ncard) :
+    ((cluster G ω a).ncard:ℝ)*densityInfluence G n a e ω +
+      ((cluster G ω b).ncard:ℝ)*densityInfluence G n b e ω = n := by
+  have hc : e ∉ ω := fun h => hab (.single ⟨e,h,he⟩)
+  have hdis : Disjoint (cluster G ω a) (cluster G ω b) := by
+    apply Set.disjoint_left.mpr
+    intro z hz hz'
+    exact hab (conn_trans G ω hz (conn_symm G ω hz'))
+  have hins : Conn G (insert e ω) a b := .single ⟨e,Set.mem_insert e ω,he⟩
+  have hs : (cluster G (insert e ω) a).ncard = (cluster G ω a).ncard+(cluster G ω b).ncard := by
+    rw [cluster_insert_left_eq_union G ω e a b he,Set.ncard_union_eq hdis ha hb]
+  have hsa : 0 < ((cluster G ω a).ncard:ℝ) := by exact_mod_cast cluster_ncard_pos G ω a ha
+  have hsb : 0 < ((cluster G ω b).ncard:ℝ) := by exact_mod_cast cluster_ncard_pos G ω b hb
+  have hna' : (n:ℝ) ≤ (cluster G ω a).ncard := by exact_mod_cast hna
+  have hnb' : (n:ℝ) ≤ (cluster G ω b).ncard := by exact_mod_cast hnb
+  unfold densityInfluence clusterDensity cappedDensity
+  rw [ite_eq_right hc,ite_eq_right hc,←cluster_eq_of_conn G (insert e ω) hins,hs,Nat.cast_add,
+    min_eq_left hna',min_eq_left hnb',min_eq_left (by linarith : (n:ℝ) ≤
+      (cluster G ω a).ncard+(cluster G ω b).ncard)]
+  field_simp
+  ring
+
+theorem localDensityLoss_ge_edge (G : BondGraph V E) (n : ℕ) (ω : Set E)
+    (e : E) (a : V) (ha : a ∈ G e) :
+    ((cluster G ω a).ncard:ℝ≥0∞)*ENNReal.ofReal (densityInfluence G n a e ω) ≤
+      localDensityLoss G n ω a := by
+  apply mul_le_mul_right
+  exact (le_of_eq (ite_eq_left ha).symm).trans
+    (ENNReal.le_tsum (f := fun edge =>
+      if a ∈ G edge then ENNReal.ofReal (densityInfluence G n a edge ω) else 0) e)
+
+theorem two_large_endpoint_loss (G : BondGraph V E) (ω : Set E)
+    (e : E) (a b : V) (he : G e = s(a,b)) (ha : (cluster G ω a).Finite)
+    (hb : (cluster G ω b).Finite) (n : ℕ)
+    (h : ¬Conn G ω a b ∧ ω ∈ clusterLarge G a n ∧ ω ∈ clusterLarge G b n) :
+    (n:ℝ≥0∞) ≤ localDensityLoss G n ω a + localDensityLoss G n ω b := by
+  have hsz (v : V) (hv : (cluster G ω v).Finite) (hh : ω ∈ clusterLarge G v n) :
+      n ≤ (cluster G ω v).ncard := by
+    obtain ⟨t,ht,hct⟩ := hh
+    exact ht.trans (by rw [←Set.ncard_coe_finset]; exact Set.ncard_le_ncard hct hv)
+  have hid := endpoint_density_defect G ω e a b he h.1 ha hb n (hsz a ha h.2.1) (hsz b hb h.2.2)
+  have hid' := congrArg ENNReal.ofReal hid
+  rw [ENNReal.ofReal_add (mul_nonneg (Nat.cast_nonneg _) (densityInfluence_nonneg ..))
+    (mul_nonneg (Nat.cast_nonneg _) (densityInfluence_nonneg ..)),
+    ENNReal.ofReal_mul (Nat.cast_nonneg _),ENNReal.ofReal_mul (Nat.cast_nonneg _)] at hid'
+  simp only [ENNReal.ofReal_natCast] at hid'
+  rw [←hid']
+  exact add_le_add (localDensityLoss_ge_edge G n ω e a (by rw [he]; simp))
+    (localDensityLoss_ge_edge G n ω e b (by rw [he]; simp))
+
+end Percolation
+
+ 
+
+open MeasureTheory ProbabilityTheory Filter Topology
+open scoped ENNReal
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+
+def twoLargeEvent (G : BondGraph V E) (a b : V) (n : ℕ) : Set (Set E) :=
+  {ω | ¬Conn G ω a b ∧ ω ∈ clusterLarge G a n ∧ ω ∈ clusterLarge G b n}
+
+theorem measurableSet_twoLargeEvent [Countable V] [Countable E] (G : BondGraph V E) (a b : V) (n : ℕ) :
+    MeasurableSet (twoLargeEvent G a b n) :=
+  (measurableSet_conn G a b).compl.inter
+    ((measurableSet_clusterLarge G a n).inter (measurableSet_clusterLarge G b n))
+
+theorem twoLarge_probability_bound [Countable V] [Countable E]
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hcov : ∀ x, ∃ i, ∃ a ∈ Γ, a (root i) = x)
+    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
+    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
+    (d : ℕ) (hd : ∀ y, {e | y ∈ G e}.ncard ≤ d) (n : ℕ) (hn : 0 < n)
+    (p : unitInterval) (hp : 0 < (p:ℝ))
+    (hfin : ∀ y, ∀ᵐ ω ∂law p, (cluster G ω y).Finite)
+    (e : E) (a b : V) (he : G e = s(a,b)) :
+    (n:ℝ) * (law p).real (twoLargeEvent G a b n) ≤
+      20*c*Real.sqrt (d*n:ℕ)/(p:ℝ) := by
+  have hpoint : ∀ᵐ ω ∂law p,
+      (twoLargeEvent G a b n).indicator (fun _ => (n:ℝ≥0∞)) ω ≤
+        localDensityLoss G n ω a + localDensityLoss G n ω b := by
+    filter_upwards [hfin a,hfin b] with ω ha hb
+    by_cases h : ω ∈ twoLargeEvent G a b n
+    · rw [Set.indicator_of_mem h]
+      exact two_large_endpoint_loss G ω e a b he ha hb n h
+    · rw [Set.indicator_of_notMem h]
+      exact zero_le
+  have hi := lintegral_mono_ae hpoint
+  rw [lintegral_indicator (measurableSet_twoLargeEvent G a b n),lintegral_const,
+    lintegral_add_left (measurable_localDensityLoss G hloc n a)] at hi
+  have hbnd (y : V) := localDensityLoss_integral_bound G hconn hloc root weight Γ hΓ htr
+    hcov hw hw1 c hc hmin d hd n hn p hp hfin y
+  have hi' := hi.trans (add_le_add (hbnd a) (hbnd b))
+  rw [←ENNReal.ofReal_add (by positivity : 0 ≤ 10*c*Real.sqrt (d*n:ℕ)/(p:ℝ))
+    (by positivity : 0 ≤ 10*c*Real.sqrt (d*n:ℕ)/(p:ℝ))] at hi'
+  have ht := ENNReal.toReal_mono ENNReal.ofReal_ne_top hi'
+  rw [ENNReal.toReal_mul,ENNReal.toReal_natCast,ENNReal.toReal_ofReal (by positivity)] at ht
+  simp only [Measure.restrict_apply_univ] at ht
+  dsimp [Measure.real] at ht ⊢
+  convert ht using 1
+  ring
+
+ 
+
+theorem finiteWitness_eventually_iff (A : Set (Set E)) (hu : IsUpperSet A)
+    (hf : FiniteWitness A) (ω : ℕ → Set E) (hm : Monotone ω) :
+    ∀ᶠ j in atTop, ω j ∈ A ↔ (⋃ k, ω k) ∈ A := by
+  by_cases h : (⋃ k, ω k) ∈ A
+  · obtain ⟨t,ht,htA⟩ := hf _ h
+    have ht' : ∀ e ∈ t, ∃ k, e ∈ ω k := by
+      intro e he
+      exact Set.mem_iUnion.mp (ht he)
+    choose k hk using fun e : t => ht' e e.property
+    let N := t.attach.sup k
+    have hN : (t : Set E) ⊆ ω N := by
+      intro e he
+      exact hm (Finset.le_sup (f := k) (Finset.mem_attach t ⟨e,he⟩)) (hk ⟨e,he⟩)
+    filter_upwards [eventually_ge_atTop N] with j hj
+    exact iff_of_true (hu (hN.trans (hm hj)) htA) h
+  · exact Eventually.of_forall (fun j => iff_of_false
+      (fun hj => h (hu (Set.subset_iUnion ω j) hj)) h)
+
+theorem twoLarge_probability_tendsto [Countable V] [Countable E] (G : BondGraph V E)
+    (p : ℕ → unitInterval) (hp : Monotone p) (q : unitInterval)
+    (hle : ∀ j, p j ≤ q) (hlim : Tendsto p atTop (𝓝 q)) (a b : V) (n : ℕ) :
+    Tendsto (fun j => (law (p j)).real (twoLargeEvent G a b n)) atTop
+      (𝓝 ((law q).real (twoLargeEvent G a b n))) := by
+  let A := twoLargeEvent G a b n
+  let f := fun r labels => A.indicator (fun _ => (1:ℝ)) (openEdges labels r)
+  have hfi (r : unitInterval) : Integrable (f r) labelLaw := by
+    have hi : Integrable (A.indicator (fun _ => (1:ℝ))) (law r) :=
+      (integrable_const _).indicator (measurableSet_twoLargeEvent G a b n)
+    rw [←map_openEdges r] at hi
+    exact hi.comp_measurable (measurable_openEdges r)
+  have ht : ∀ᵐ labels ∂(labelLaw : Measure (E → unitInterval)),
+      Tendsto (fun j => f (p j) labels) atTop (𝓝 (f q labels)) := by
+    filter_upwards [ae_labels_ne (E := E) q] with labels hlabels
+    have hm : Monotone (fun j => openEdges labels (p j)) :=
+      fun j k hjk => openEdges_mono labels (hp hjk)
+    have he := openEdges_eq_iUnion_of_tendsto labels q hlabels p hle hlim
+    have hC := finiteWitness_eventually_iff {ω | Conn G ω a b}
+      (fun _ _ hω h => conn_mono G hω h) (finiteWitness_conn G a b) _ hm
+    have ha := finiteWitness_eventually_iff (clusterLarge G a n)
+      (clusterLarge_upper G a n) (finiteWitness_clusterLarge G a n) _ hm
+    have hb := finiteWitness_eventually_iff (clusterLarge G b n)
+      (clusterLarge_upper G b n) (finiteWitness_clusterLarge G b n) _ hm
+    rw [←he] at hC ha hb
+    apply tendsto_const_nhds.congr'
+    filter_upwards [hC,ha,hb] with j hC ha hb
+    have hi : openEdges labels (p j) ∈ A ↔ openEdges labels q ∈ A := by
+      exact and_congr (not_congr hC) (and_congr ha hb)
+    simp only [f,Set.indicator_apply]
+    rw [hi]
+  have ht' := tendsto_integral_of_dominated_convergence (fun _ : E → unitInterval => (1:ℝ))
+    (fun j => (hfi (p j)).aestronglyMeasurable) (integrable_const _)
+    (fun j => Eventually.of_forall (fun labels => by
+      simp only [f,Set.indicator_apply]
+      split <;> norm_num)) ht
+  have heq (r : unitInterval) : (∫ labels, f r labels ∂labelLaw) = (law r).real A := by
+    calc
+      _ = ∫ ω, A.indicator (fun _ => (1:ℝ)) ω ∂law r := by
+        rw [←map_openEdges r]
+        exact (integral_map (measurable_openEdges r).aemeasurable
+          ((measurable_const.indicator (measurableSet_twoLargeEvent G a b n)).aestronglyMeasurable)).symm
+      _ = _ := by rw [integral_indicator_const _ (measurableSet_twoLargeEvent G a b n)]; simp [A]
+  simpa only [heq] using ht'
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/TwoCluster/Score.lean b/lean/OAI/Probability/UnimodularPercolation/TwoCluster/Score.lean
index d470235d8bd37fbefe9aa7ed558fceb67d13686d..0cee04e07d8842bee5ce02abbae9b242534baf7c 100644
--- a/lean/OAI/Probability/UnimodularPercolation/TwoCluster/Score.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/TwoCluster/Score.lean
@@ -1,465 +1,465 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Exploration.TailBounds
-
-namespace OAI
-
-/-! Decision-tree scores and influences for two-cluster events. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-namespace ProbeTree
-
- 
-theorem score_mean_zero (p h : ℝ) (t : ProbeTree) :
-    ∑ l, weight p h t l * score p t l = 0 := by
-  induction t with
-  | done b => simp [weight,score]
-  | edge t₀ t₁ ih₀ ih₁ =>
-    change (∑ l : t₀.Leaf ⊕ t₁.Leaf, _) = 0
-    simp only [Fintype.sum_sum_type,weight,score]
-    have h₀ (l : t₀.Leaf) : (1-p)*weight p h t₀ l*(score p t₀ l-p) =
-        (1-p)*(weight p h t₀ l*score p t₀ l)-p*(1-p)*weight p h t₀ l := by ring
-    have h₁ (l : t₁.Leaf) : p*weight p h t₁ l*(score p t₁ l+(1-p)) =
-        p*(weight p h t₁ l*score p t₁ l)+p*(1-p)*weight p h t₁ l := by ring
-    simp_rw [h₀,h₁]
-    simp only [Finset.sum_add_distrib,Finset.sum_sub_distrib,←Finset.mul_sum,
-      sum_weight,ih₀,ih₁]
-    ring
-  | mark t₀ t₁ ih₀ ih₁ =>
-    change (∑ l : t₀.Leaf ⊕ t₁.Leaf, _) = 0
-    simp only [Fintype.sum_sum_type,weight,score,mul_assoc,←Finset.mul_sum,ih₀,ih₁,mul_zero,add_zero]
-
- 
-
-theorem score_second_moment (p h : ℝ) (t : ProbeTree) :
-    ∑ l, weight p h t l * score p t l ^ 2 =
-      p*(1-p) * ∑ l, weight p h t l * (queries t l : ℝ) := by
-  induction t with
-  | done b => simp [weight,score,queries]
-  | edge t₀ t₁ ih₀ ih₁ =>
-    change (∑ l : t₀.Leaf ⊕ t₁.Leaf, _) = p*(1-p)*(∑ l : t₀.Leaf ⊕ t₁.Leaf, _)
-    rw [Fintype.sum_sum_type,Fintype.sum_sum_type]
-    have h₀ : ∀ l : t₀.Leaf, weight p h (.edge t₀ t₁) (.inl l) * score p (.edge t₀ t₁) (.inl l)^2 =
-        (1-p)*(weight p h t₀ l*score p t₀ l^2) -
-        (2*p*(1-p))*(weight p h t₀ l*score p t₀ l) + p^2*(1-p)*weight p h t₀ l := by
-      intro l; simp only [weight,score]; ring
-    have h₁ : ∀ l : t₁.Leaf, weight p h (.edge t₀ t₁) (.inr l) * score p (.edge t₀ t₁) (.inr l)^2 =
-        p*(weight p h t₁ l*score p t₁ l^2) +
-        (2*p*(1-p))*(weight p h t₁ l*score p t₁ l) + p*(1-p)^2*weight p h t₁ l := by
-      intro l; simp only [weight,score]; ring
-    simp_rw [h₀,h₁]
-    simp only [Finset.sum_add_distrib,Finset.sum_sub_distrib,←Finset.mul_sum,
-      score_mean_zero,sum_weight,mul_zero,mul_one,sub_zero,add_zero,ih₀,ih₁,
-      weight,queries,Nat.cast_add,Nat.cast_one]
-    simp only [mul_add,mul_one,Finset.sum_add_distrib,mul_assoc,←Finset.mul_sum,sum_weight]
-    ring
-  | mark t₀ t₁ ih₀ ih₁ =>
-    change (∑ l : t₀.Leaf ⊕ t₁.Leaf, _) = p*(1-p)*(∑ l : t₀.Leaf ⊕ t₁.Leaf, _)
-    simp only [Fintype.sum_sum_type,weight,score,queries,mul_assoc,←Finset.mul_sum,ih₀,ih₁]
-    ring
-
- 
-theorem score_abs_mean_le_sqrt (p h : unitInterval) (t : ProbeTree)
-    (N : ℝ) (hN0 : 0 ≤ N)
-    (hN : ∑ l, weight p h t l*(queries t l:ℝ) ≤ N) :
-    ∑ l, weight p h t l * |score p t l| ≤ Real.sqrt N := by
-  have hw := weight_nonneg p h t
-  have hc := Finset.sum_mul_sq_le_sq_mul_sq Finset.univ
-    (fun l => Real.sqrt (weight p h t l))
-    (fun l => Real.sqrt (weight p h t l)*|score p t l|)
-  have he : (∑ l, weight p h t l * |score p t l|)^2 ≤
-      ∑ l, weight p h t l * score p t l ^2 := by
-    simpa only [mul_pow,Real.sq_sqrt (hw _),sq_abs,
-      ←mul_assoc,Real.mul_self_sqrt (hw _),sum_weight,one_mul] using hc
-  have hp : 0 ≤ (p:ℝ)*(1-p) := mul_nonneg p.2.1 (sub_nonneg.mpr p.2.2)
-  have hp1 : (p:ℝ)*(1-p) ≤ 1 := by nlinarith [p.2.1,p.2.2]
-  rw [score_second_moment] at he
-  have hnon : 0 ≤ ∑ l, weight p h t l*|score p t l| :=
-    Finset.sum_nonneg fun l _ => mul_nonneg (hw l) (abs_nonneg _)
-  apply (Real.le_sqrt hnon hN0).mpr
-  exact he.trans ((mul_le_mul_of_nonneg_left hN hp).trans (by nlinarith [hN0]))
-
-end ProbeTree
-namespace EdgeQueryTree
-attribute [local instance] Classical.propDecidable Classical.decEq
-
- 
-theorem capped_score_abs_mean_le (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
-    (p : unitInterval) (N : ℕ) (hN : ∀ ω ⊆ Q, t.queryCount ω ≤ N) :
-    bernoulliPolynomial Q (fun ω => |t.score p ω|) p ≤ Real.sqrt (N:ℝ) := by
-  have hq : ∑ l, ProbeTree.weight p 0 t.forget l*(ProbeTree.queries t.forget l:ℝ) ≤ N := by
-    rw [←t.leaf_expectation ht p (fun l => (ProbeTree.queries t.forget l:ℝ))]
-    calc
-      _ ≤ bernoulliPolynomial Q (fun _ => (N:ℝ)) p := by
-        apply bernoulliPolynomial_mono _ _ _ p p.2
-        intro ω hω
-        exact_mod_cast hN ω hω
-      _ = N := bernoulliPolynomial_const Q (N:ℝ) p
-  change bernoulliPolynomial Q (fun ω => |ProbeTree.score p t.forget (t.leaf ω)|) p ≤ _
-  rw [t.leaf_expectation ht p (fun l => |ProbeTree.score p t.forget l|)]
-  exact ProbeTree.score_abs_mean_le_sqrt p 0 t.forget N (Nat.cast_nonneg N) hq
-
-end EdgeQueryTree
-
-theorem incidenceSearch_abs_mean_le (G : BondGraph V E) (N : ℕ) (Q : Finset E)
-    (R : Finset V) (p : unitInterval) :
-    bernoulliPolynomial Q (fun ω => |(incidenceSearch G N Q R).score p ω|) p ≤
-      Real.sqrt (N:ℝ) := by
-  classical
-  exact EdgeQueryTree.capped_score_abs_mean_le _ (incidenceSearch_valid ..) p N
-    (fun ω _ => incidenceSearch_cap G N Q R ω)
-
-end Percolation
-
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def finiteClusterScore (G : BondGraph V E) (Q : Finset E)
-    (x : V) (p : ℝ) (ω : Finset E) : ℝ :=
-  ∑ e ∈ incidentBonds G Q (finiteRootComponent G ω x), ((if e ∈ ω then 1 else 0)-p)
-
- 
-noncomputable def cappedDensity (n s : ℕ) : ℝ := min (n:ℝ) s / s
-
-theorem cappedDensity_nonneg (n s : ℕ) : 0 ≤ cappedDensity n s := by
-  unfold cappedDensity
-  positivity
-
-theorem cappedDensity_le_one (n s : ℕ) : cappedDensity n s ≤ 1 := by
-  unfold cappedDensity
-  by_cases hs : s = 0
-  · simp [hs]
-  · exact (div_le_one (Nat.cast_pos.mpr (Nat.pos_of_ne_zero hs))).mpr (min_le_right ..)
-
-private theorem exists_four_scale (n s K : ℕ) (hs : n < s) (hK : s ≤ n*4^K) :
-    ∃ j < K, n*4^j < s ∧ s ≤ n*4^(j+1) := by
-  induction K with
-  | zero => simp only [pow_zero,mul_one] at hK; omega
-  | succ K ih =>
-    by_cases h : s ≤ n*4^K
-    · obtain ⟨j,hj,hj₁,hj₂⟩ := ih h
-      exact ⟨j,by omega,hj₁,hj₂⟩
-    · exact ⟨K,by omega,by omega,hK⟩
-
-private theorem nat_le_four_pow (k : ℕ) : k ≤ 4^k := by
-  induction k with
-  | zero => simp
-  | succ k ih => rw [pow_succ]; have := Nat.one_le_pow k 4 (by omega); omega
-
-private theorem cappedDensity_le_four_scale (n s j : ℕ) (hn : 0 < n)
-    (hs : n*4^j < s) : cappedDensity n s ≤ (1/4:ℝ)^j := by
-  have hpow : 1 ≤ 4^j := Nat.one_le_pow j 4 (by omega)
-  have hns : n ≤ s := le_trans (by nlinarith : n ≤ n*4^j) hs.le
-  have hs0 : 0 < (s:ℝ) := by exact_mod_cast (lt_of_lt_of_le hn hns)
-  have hden : 0 < (n:ℝ)*4^j := mul_pos (Nat.cast_pos.mpr hn) (pow_pos (by norm_num) _)
-  unfold cappedDensity
-  rw [min_eq_left (by exact_mod_cast hns)]
-  calc
-    (n:ℝ)/s ≤ n/(n*4^j) := div_le_div_of_nonneg_left (Nat.cast_nonneg _) hden (by exact_mod_cast hs.le)
-    _ = (1/4:ℝ)^j := by rw [div_pow,one_pow]; field_simp
-
- 
-
-theorem cappedDensity_score_le_stopped (G : BondGraph V E) (Q : Finset E)
-    (x : V) (p : ℝ) (d n K : ℕ) (hn : 0 < n)
-    (hdeg : ∀ v, (Q.filter fun e => v ∈ G e).card ≤ d)
-    (ω : Finset E) (hω : ω ⊆ Q)
-    (hK : (finiteRootComponent G ω x).card ≤ n*4^K) :
-    cappedDensity n (finiteRootComponent G ω x).card * |finiteClusterScore G Q x p ω| ≤
-      |(incidenceSearch G (d*n) Q {x}).score p ω| +
-        ∑ j ∈ Finset.range K, (1/4:ℝ)^j *
-          |(incidenceSearch G (d*(n*4^(j+1))) Q {x}).score p ω| := by
-  let s := (finiteRootComponent G ω x).card
-  have hcomplete (N : ℕ) (hN : s ≤ N) :
-      (incidenceSearch G (d*N) Q {x}).score p ω = finiteClusterScore G Q x p ω := by
-    exact incidenceSearch_completed_score G (d*N) Q ω hω x p
-      ((incidentBonds_card_le G Q _ d hdeg).trans (Nat.mul_le_mul_left d hN))
-  have hterms : ∀ j, 0 ≤ (1/4:ℝ)^j *
-      |(incidenceSearch G (d*(n*4^(j+1))) Q {x}).score p ω| := fun j => by positivity
-  by_cases hs : s ≤ n
-  · have he := hcomplete n hs
-    rw [he]
-    exact (mul_le_of_le_one_left (abs_nonneg _) (cappedDensity_le_one n s)).trans
-      (le_add_of_nonneg_right (Finset.sum_nonneg fun j _ => hterms j))
-  · obtain ⟨j,hj,hlo,hhi⟩ := exists_four_scale n s K (by omega) hK
-    have he := hcomplete (n*4^(j+1)) hhi
-    calc
-      _ ≤ (1/4:ℝ)^j * |finiteClusterScore G Q x p ω| :=
-        mul_le_mul_of_nonneg_right (cappedDensity_le_four_scale n s j hn hlo) (abs_nonneg _)
-      _ = (1/4:ℝ)^j * |(incidenceSearch G (d*(n*4^(j+1))) Q {x}).score p ω| := by rw [he]
-      _ ≤ ∑ j ∈ Finset.range K, (1/4:ℝ)^j *
-          |(incidenceSearch G (d*(n*4^(j+1))) Q {x}).score p ω| :=
-        Finset.single_le_sum (fun j _ => hterms j) (Finset.mem_range.mpr hj)
-      _ ≤ _ := le_add_of_nonneg_left (abs_nonneg _)
-
-private theorem sqrt_four_cap (d n j : ℕ) :
-    Real.sqrt (d*(n*4^(j+1)):ℕ) = (2:ℝ)^(j+1)*Real.sqrt (d*n:ℕ) := by
-  have he : (d*(n*4^(j+1)):ℕ) = (d*n)*4^(j+1) := by ring
-  rw [he,Nat.cast_mul,Real.sqrt_mul (Nat.cast_nonneg _)]
-  rw [Nat.cast_pow,Nat.cast_ofNat,show (4:ℝ) = 2^2 by norm_num,←pow_mul,pow_mul',Real.sqrt_sq_eq_abs,
-    abs_of_nonneg (pow_nonneg (by norm_num : (0:ℝ) ≤ 2) _)]
-  ring
-
-theorem cappedDensity_score_expectation_bound (G : BondGraph V E) (Q : Finset E)
-    (x : V) (p : unitInterval) (d n : ℕ) (hn : 0 < n)
-    (hdeg : ∀ v, (Q.filter fun e => v ∈ G e).card ≤ d) :
-    bernoulliPolynomial Q (fun ω => cappedDensity n (finiteRootComponent G ω x).card *
-      |finiteClusterScore G Q x p ω|) p ≤ 5*Real.sqrt (d*n:ℕ) := by
-  let K := (insert x (searchVertices G Q)).card
-  have hK (ω : Finset E) (hω : ω ⊆ Q) : (finiteRootComponent G ω x).card ≤ n*4^K := by
-    have hs : (finiteRootComponent G ω x).card ≤ K :=
-      Finset.card_le_card (finiteRootComponent_subset G Q _ x ω hω le_rfl)
-    exact hs.trans ((nat_le_four_pow K).trans (by nlinarith [Nat.one_le_pow K 4 (by omega)]))
-  have hp := bernoulliPolynomial_mono Q _ _ p p.2 (fun ω hω =>
-    cappedDensity_score_le_stopped G Q x p d n K hn hdeg ω hω (hK ω hω))
-  rw [bernoulliPolynomial_add,bernoulliPolynomial_sum] at hp
-  simp_rw [bernoulliPolynomial_const_mul] at hp
-  have hh := hp.trans (add_le_add (incidenceSearch_abs_mean_le G (d*n) Q {x} p)
-    (Finset.sum_le_sum fun j _ => mul_le_mul_of_nonneg_left
-      (incidenceSearch_abs_mean_le G (d*(n*4^(j+1))) Q {x} p) (by positivity)))
-  simp_rw [sqrt_four_cap] at hh
-  have he (j : ℕ) : (1/4:ℝ)^j*((2:ℝ)^(j+1)*Real.sqrt (d*n:ℕ)) =
-      2*(1/2:ℝ)^j*Real.sqrt (d*n:ℕ) := by
-    rw [pow_succ]
-    calc
-      _ = 2*((1/4:ℝ)^j*2^j)*Real.sqrt (d*n:ℕ) := by ring
-      _ = _ := by rw [←mul_pow]; norm_num
-  simp_rw [he] at hh
-  rw [←Finset.sum_mul,←Finset.mul_sum] at hh
-  have hg : ∑ j ∈ Finset.range K, (1/2:ℝ)^j ≤ 2 := by
-    have hid : ∑ j ∈ Finset.range K, (1/2:ℝ)^j = 2-2*(1/2:ℝ)^K := by
-      induction K with
-      | zero => simp
-      | succ K ih => rw [Finset.sum_range_succ,ih,pow_succ]; ring
-    rw [hid]
-    linarith [pow_nonneg (by norm_num : (0:ℝ) ≤ 1/2) K]
-  exact hh.trans (by nlinarith [Real.sqrt_nonneg (d*n:ℕ)])
-
-end Percolation
-
- 
-
- 
-
-open MeasureTheory ProbabilityTheory
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-theorem incident_cluster_insert_iff (G : BondGraph V E) (ω : Set E) (x : V) (e : E) :
-    (∃ a, Conn G (insert e ω) x a ∧ a ∈ G e) ↔
-      ∃ a, Conn G ω x a ∧ a ∈ G e := by
-  obtain ⟨a,b,hab⟩ : ∃ a b, G e = s(a,b) := by
-    obtain ⟨⟨a,b⟩,h⟩ := Quot.exists_rep (G e)
-    exact ⟨a,b,h.symm⟩
-  constructor
-  · rintro ⟨z,hz,hze⟩
-    rcases (conn_insert_iff G ω e a b x z hab).mp hz with hz | ⟨hxa,_⟩ | ⟨hxb,_⟩
-    · exact ⟨z,hz,hze⟩
-    · exact ⟨a,hxa,hab ▸ by simp⟩
-    · exact ⟨b,hxb,hab ▸ by simp⟩
-  · rintro ⟨z,hz,hze⟩
-    exact ⟨z,conn_mono G (Set.subset_insert e ω) hz,hze⟩
-
-theorem cluster_insert_eq_of_not_incident (G : BondGraph V E) (ω : Set E) (x : V) (e : E)
-    (hn : ¬∃ a, Conn G ω x a ∧ a ∈ G e) :
-    cluster G (insert e ω) x = cluster G ω x := by
-  obtain ⟨a,b,hab⟩ : ∃ a b, G e = s(a,b) := by
-    obtain ⟨⟨a,b⟩,h⟩ := Quot.exists_rep (G e)
-    exact ⟨a,b,h.symm⟩
-  ext z
-  constructor
-  · intro hz
-    rcases (conn_insert_iff G ω e a b x z hab).mp hz with hz | ⟨hxa,_⟩ | ⟨hxb,_⟩
-    · exact hz
-    · exact (hn ⟨a,hxa,hab ▸ by simp⟩).elim
-    · exact (hn ⟨b,hxb,hab ▸ by simp⟩).elim
-  · exact conn_mono G (Set.subset_insert e ω)
-
-theorem finite_incident_insert_iff (G : BondGraph V E) (Q ω : Finset E) (x : V) (e : E) :
-    e ∈ incidentBonds G Q (finiteRootComponent G (insert e ω) x) ↔
-      e ∈ incidentBonds G Q (finiteRootComponent G ω x) := by
-  simp only [mem_incidentBonds]
-  have hc (η : Finset E) (a : V) : a ∈ finiteRootComponent G η x ↔ Conn G (η:Set E) x a := by
-    change a ∈ (finiteRootComponent G η x:Set V) ↔ _
-    rw [coe_finiteRootComponent]
-    rfl
-  simp only [hc,Finset.coe_insert,incident_cluster_insert_iff]
-
-theorem finiteRootComponent_insert_eq_of_not_incident (G : BondGraph V E)
-    (Q ω : Finset E) (x : V) (e : E) (he : e ∈ Q)
-    (hn : e ∉ incidentBonds G Q (finiteRootComponent G ω x)) :
-    finiteRootComponent G (insert e ω) x = finiteRootComponent G ω x := by
-  apply Finset.coe_injective
-  rw [coe_finiteRootComponent,coe_finiteRootComponent,Finset.coe_insert]
-  apply cluster_insert_eq_of_not_incident
-  rintro ⟨a,ha,hae⟩
-  apply hn
-  apply (mem_incidentBonds G Q _ e).mpr
-  refine ⟨he,a,?_,hae⟩
-  change a ∈ (finiteRootComponent G ω x:Set V)
-  rw [coe_finiteRootComponent]
-  exact ha
-
-theorem finiteRootComponent_card_pos (G : BondGraph V E) (ω : Finset E) (x : V) :
-    0 < (finiteRootComponent G ω x).card := by
-  apply Finset.card_pos.mpr
-  refine ⟨x,?_⟩
-  change x ∈ (finiteRootComponent G ω x:Set V)
-  rw [coe_finiteRootComponent]
-  exact mem_cluster_self G _ x
-
-theorem cappedDensity_antitone (n : ℕ) {s t : ℕ} (hs : 0 < s) (hst : s ≤ t) :
-    cappedDensity n t ≤ cappedDensity n s := by
-  by_cases hn : s ≤ n
-  · have he : cappedDensity n s = 1 := by
-      unfold cappedDensity
-      rw [min_eq_right (by exact_mod_cast hn),div_self (by exact_mod_cast Nat.ne_of_gt hs)]
-    rw [he]
-    exact cappedDensity_le_one n t
-  · have hs' : (n:ℝ) ≤ s := by exact_mod_cast (le_of_lt (Nat.lt_of_not_ge hn))
-    have ht' : (n:ℝ) ≤ t := hs'.trans (by exact_mod_cast hst)
-    unfold cappedDensity
-    rw [min_eq_left ht',min_eq_left hs']
-    exact div_le_div_of_nonneg_left (Nat.cast_nonneg n) (Nat.cast_pos.mpr hs) (by exact_mod_cast hst)
-
-noncomputable def finiteDensity (G : BondGraph V E) (n : ℕ) (x : V) (ω : Finset E) : ℝ :=
-  cappedDensity n (finiteRootComponent G ω x).card
-
-noncomputable def finiteDensityInfluence (G : BondGraph V E) (n : ℕ) (x : V)
-    (e : E) (ω : Finset E) : ℝ :=
-  if e ∈ ω then 0 else finiteDensity G n x ω - finiteDensity G n x (insert e ω)
-
-theorem finiteDensityInfluence_nonneg (G : BondGraph V E) (n : ℕ) (x : V)
-    (e : E) (ω : Finset E) : 0 ≤ finiteDensityInfluence G n x e ω := by
-  unfold finiteDensityInfluence
-  split_ifs
-  · rfl
-  · apply sub_nonneg.mpr
-    apply cappedDensity_antitone n (finiteRootComponent_card_pos G ω x)
-    apply Finset.card_le_card
-    intro y hy
-    have hm (η : Finset E) : y ∈ finiteRootComponent G η x ↔ Conn G (η:Set E) x y := by
-      change y ∈ (finiteRootComponent G η x:Set V) ↔ _
-      rw [coe_finiteRootComponent]
-      rfl
-    rw [hm] at hy ⊢
-    exact conn_mono G (by simp) hy
-
-theorem finiteDensityInfluence_le_one (G : BondGraph V E) (n : ℕ) (x : V)
-    (e : E) (ω : Finset E) : finiteDensityInfluence G n x e ω ≤ 1 := by
-  unfold finiteDensityInfluence finiteDensity
-  split_ifs
-  · norm_num
-  · exact (sub_le_self _ (cappedDensity_nonneg ..)).trans (cappedDensity_le_one ..)
-
- 
-theorem finiteDensity_score_one_bit (G : BondGraph V E) (Q : Finset E) (n : ℕ)
-    (x : V) (p : ℝ) (e : E) (he : e ∈ Q) :
-    bernoulliPolynomial Q (fun ω => if e ∈ incidentBonds G Q (finiteRootComponent G ω x)
-      then ((if e ∈ ω then 1 else 0)-p)*finiteDensity G n x ω else 0) p =
-      -p*bernoulliPolynomial Q (finiteDensityInfluence G n x e) p := by
-  let f := finiteDensity G n x
-  let I := fun ω : Finset E => e ∈ incidentBonds G Q (finiteRootComponent G ω x)
-  let R := Q.erase e
-  have heR : e ∉ R := Finset.notMem_erase e Q
-  have hQ : insert e R = Q := Finset.insert_erase he
-  rw [←hQ,bernoulliPolynomial_insert R e heR,bernoulliPolynomial_insert R e heR]
-  rw [hQ]
-  have hclosed (ω : Finset E) (hω : ω ⊆ R) : e ∉ ω := fun h => heR (hω h)
-  have hc : bernoulliPolynomial R (fun ω => if I ω then ((if e ∈ ω then 1 else 0)-p)*f ω else 0) p =
-      -p*bernoulliPolynomial R (fun ω => if I ω then f ω else 0) p := by
-    rw [←bernoulliPolynomial_const_mul]
-    apply bernoulliPolynomial_congr
-    intro ω hω
-    simp only [ite_eq_right (hclosed ω hω),zero_sub]
-    split_ifs <;> ring
-  have ho : bernoulliPolynomial R (fun ω => if I (insert e ω) then
-      ((if e ∈ insert e ω then 1 else 0)-p)*f (insert e ω) else 0) p =
-      (1-p)*bernoulliPolynomial R (fun ω => if I ω then f (insert e ω) else 0) p := by
-    rw [←bernoulliPolynomial_const_mul]
-    apply bernoulliPolynomial_congr
-    intro ω _
-    simp only [I,finite_incident_insert_iff,Finset.mem_insert_self,ite_true]
-    split_ifs <;> ring
-  have hi : bernoulliPolynomial R (finiteDensityInfluence G n x e) p =
-      bernoulliPolynomial R (fun ω => if I ω then f ω else 0) p -
-        bernoulliPolynomial R (fun ω => if I ω then f (insert e ω) else 0) p := by
-    rw [←bernoulliPolynomial_sub]
-    apply bernoulliPolynomial_congr
-    intro ω hω
-    unfold finiteDensityInfluence
-    rw [ite_eq_right (hclosed ω hω)]
-    by_cases hi : I ω
-    · simp only [ite_eq_left hi,f]
-    · have heq : f (insert e ω) = f ω := by
-        unfold f finiteDensity
-        rw [finiteRootComponent_insert_eq_of_not_incident G Q ω x e he hi]
-      simp only [ite_eq_right hi,heq,sub_self,f]
-  have hz : bernoulliPolynomial R (fun ω => finiteDensityInfluence G n x e (insert e ω)) p = 0 := by
-    simp only [finiteDensityInfluence,Finset.mem_insert_self,ite_true,bernoulliPolynomial_zero]
-  change _ = -p*((1-p)*bernoulliPolynomial R (finiteDensityInfluence G n x e) p+
-    p*bernoulliPolynomial R (fun ω => finiteDensityInfluence G n x e (insert e ω)) p)
-  rw [hc,ho,hi,hz]
-  ring
-
- 
-theorem finiteDensity_score_identity (G : BondGraph V E) (Q : Finset E) (n : ℕ)
-    (x : V) (p : ℝ) :
-    bernoulliPolynomial Q (fun ω => finiteDensity G n x ω * finiteClusterScore G Q x p ω) p =
-      -p * ∑ e ∈ Q, bernoulliPolynomial Q (finiteDensityInfluence G n x e) p := by
-  have hf (ω : Finset E) : finiteDensity G n x ω * finiteClusterScore G Q x p ω =
-      ∑ e ∈ Q, if e ∈ incidentBonds G Q (finiteRootComponent G ω x)
-        then ((if e ∈ ω then 1 else 0)-p)*finiteDensity G n x ω else 0 := by
-    unfold finiteClusterScore
-    rw [Finset.mul_sum]
-    symm
-    rw [←Finset.sum_filter]
-    have hs : Q.filter (fun e => e ∈ incidentBonds G Q (finiteRootComponent G ω x)) =
-        incidentBonds G Q (finiteRootComponent G ω x) := by
-      ext e
-      simp only [Finset.mem_filter,mem_incidentBonds]
-      tauto
-    rw [hs]
-    apply Finset.sum_congr rfl
-    intro e _
-    ring
-  simp_rw [hf]
-  rw [bernoulliPolynomial_sum,Finset.mul_sum]
-  exact Finset.sum_congr rfl fun e he => finiteDensity_score_one_bit G Q n x p e he
-
- 
-
-theorem finiteDensityInfluence_sum_bound (G : BondGraph V E) (Q : Finset E)
-    (x : V) (p : unitInterval) (d n : ℕ) (hn : 0 < n)
-    (hdeg : ∀ v, (Q.filter fun e => v ∈ G e).card ≤ d) :
-    (p:ℝ) * ∑ e ∈ Q, bernoulliPolynomial Q (finiteDensityInfluence G n x e) p ≤
-      5*Real.sqrt (d*n:ℕ) := by
-  have hid := finiteDensity_score_identity G Q n x p
-  have hle := bernoulliPolynomial_mono Q
-    (fun ω => -(finiteDensity G n x ω * finiteClusterScore G Q x p ω))
-    (fun ω => cappedDensity n (finiteRootComponent G ω x).card * |finiteClusterScore G Q x p ω|)
-    p p.2 (fun ω _ => by
-      rw [←mul_neg]
-      exact mul_le_mul_of_nonneg_left (neg_le_abs _) (cappedDensity_nonneg ..))
-  have hneg : bernoulliPolynomial Q
-      (fun ω => -(finiteDensity G n x ω * finiteClusterScore G Q x p ω)) p =
-      -bernoulliPolynomial Q (fun ω => finiteDensity G n x ω * finiteClusterScore G Q x p ω) p := by
-    simpa using bernoulliPolynomial_const_mul Q
-      (fun ω => finiteDensity G n x ω * finiteClusterScore G Q x p ω) (-1) (p:ℝ)
-  rw [hneg,hid] at hle
-  have hb := cappedDensity_score_expectation_bound G Q x p d n hn hdeg
-  linarith
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Exploration.TailBounds
+
+namespace OAI
+
+/-! Decision-tree scores and influences for two-cluster events. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+namespace ProbeTree
+
+ 
+theorem score_mean_zero (p h : ℝ) (t : ProbeTree) :
+    ∑ l, weight p h t l * score p t l = 0 := by
+  induction t with
+  | done b => simp [weight,score]
+  | edge t₀ t₁ ih₀ ih₁ =>
+    change (∑ l : t₀.Leaf ⊕ t₁.Leaf, _) = 0
+    simp only [Fintype.sum_sum_type,weight,score]
+    have h₀ (l : t₀.Leaf) : (1-p)*weight p h t₀ l*(score p t₀ l-p) =
+        (1-p)*(weight p h t₀ l*score p t₀ l)-p*(1-p)*weight p h t₀ l := by ring
+    have h₁ (l : t₁.Leaf) : p*weight p h t₁ l*(score p t₁ l+(1-p)) =
+        p*(weight p h t₁ l*score p t₁ l)+p*(1-p)*weight p h t₁ l := by ring
+    simp_rw [h₀,h₁]
+    simp only [Finset.sum_add_distrib,Finset.sum_sub_distrib,←Finset.mul_sum,
+      sum_weight,ih₀,ih₁]
+    ring
+  | mark t₀ t₁ ih₀ ih₁ =>
+    change (∑ l : t₀.Leaf ⊕ t₁.Leaf, _) = 0
+    simp only [Fintype.sum_sum_type,weight,score,mul_assoc,←Finset.mul_sum,ih₀,ih₁,mul_zero,add_zero]
+
+ 
+
+theorem score_second_moment (p h : ℝ) (t : ProbeTree) :
+    ∑ l, weight p h t l * score p t l ^ 2 =
+      p*(1-p) * ∑ l, weight p h t l * (queries t l : ℝ) := by
+  induction t with
+  | done b => simp [weight,score,queries]
+  | edge t₀ t₁ ih₀ ih₁ =>
+    change (∑ l : t₀.Leaf ⊕ t₁.Leaf, _) = p*(1-p)*(∑ l : t₀.Leaf ⊕ t₁.Leaf, _)
+    rw [Fintype.sum_sum_type,Fintype.sum_sum_type]
+    have h₀ : ∀ l : t₀.Leaf, weight p h (.edge t₀ t₁) (.inl l) * score p (.edge t₀ t₁) (.inl l)^2 =
+        (1-p)*(weight p h t₀ l*score p t₀ l^2) -
+        (2*p*(1-p))*(weight p h t₀ l*score p t₀ l) + p^2*(1-p)*weight p h t₀ l := by
+      intro l; simp only [weight,score]; ring
+    have h₁ : ∀ l : t₁.Leaf, weight p h (.edge t₀ t₁) (.inr l) * score p (.edge t₀ t₁) (.inr l)^2 =
+        p*(weight p h t₁ l*score p t₁ l^2) +
+        (2*p*(1-p))*(weight p h t₁ l*score p t₁ l) + p*(1-p)^2*weight p h t₁ l := by
+      intro l; simp only [weight,score]; ring
+    simp_rw [h₀,h₁]
+    simp only [Finset.sum_add_distrib,Finset.sum_sub_distrib,←Finset.mul_sum,
+      score_mean_zero,sum_weight,mul_zero,mul_one,sub_zero,add_zero,ih₀,ih₁,
+      weight,queries,Nat.cast_add,Nat.cast_one]
+    simp only [mul_add,mul_one,Finset.sum_add_distrib,mul_assoc,←Finset.mul_sum,sum_weight]
+    ring
+  | mark t₀ t₁ ih₀ ih₁ =>
+    change (∑ l : t₀.Leaf ⊕ t₁.Leaf, _) = p*(1-p)*(∑ l : t₀.Leaf ⊕ t₁.Leaf, _)
+    simp only [Fintype.sum_sum_type,weight,score,queries,mul_assoc,←Finset.mul_sum,ih₀,ih₁]
+    ring
+
+ 
+theorem score_abs_mean_le_sqrt (p h : unitInterval) (t : ProbeTree)
+    (N : ℝ) (hN0 : 0 ≤ N)
+    (hN : ∑ l, weight p h t l*(queries t l:ℝ) ≤ N) :
+    ∑ l, weight p h t l * |score p t l| ≤ Real.sqrt N := by
+  have hw := weight_nonneg p h t
+  have hc := Finset.sum_mul_sq_le_sq_mul_sq Finset.univ
+    (fun l => Real.sqrt (weight p h t l))
+    (fun l => Real.sqrt (weight p h t l)*|score p t l|)
+  have he : (∑ l, weight p h t l * |score p t l|)^2 ≤
+      ∑ l, weight p h t l * score p t l ^2 := by
+    simpa only [mul_pow,Real.sq_sqrt (hw _),sq_abs,
+      ←mul_assoc,Real.mul_self_sqrt (hw _),sum_weight,one_mul] using hc
+  have hp : 0 ≤ (p:ℝ)*(1-p) := mul_nonneg p.2.1 (sub_nonneg.mpr p.2.2)
+  have hp1 : (p:ℝ)*(1-p) ≤ 1 := by nlinarith [p.2.1,p.2.2]
+  rw [score_second_moment] at he
+  have hnon : 0 ≤ ∑ l, weight p h t l*|score p t l| :=
+    Finset.sum_nonneg fun l _ => mul_nonneg (hw l) (abs_nonneg _)
+  apply (Real.le_sqrt hnon hN0).mpr
+  exact he.trans ((mul_le_mul_of_nonneg_left hN hp).trans (by nlinarith [hN0]))
+
+end ProbeTree
+namespace EdgeQueryTree
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+ 
+theorem capped_score_abs_mean_le (t : EdgeQueryTree E) {Q : Finset E} (ht : t.Valid Q)
+    (p : unitInterval) (N : ℕ) (hN : ∀ ω ⊆ Q, t.queryCount ω ≤ N) :
+    bernoulliPolynomial Q (fun ω => |t.score p ω|) p ≤ Real.sqrt (N:ℝ) := by
+  have hq : ∑ l, ProbeTree.weight p 0 t.forget l*(ProbeTree.queries t.forget l:ℝ) ≤ N := by
+    rw [←t.leaf_expectation ht p (fun l => (ProbeTree.queries t.forget l:ℝ))]
+    calc
+      _ ≤ bernoulliPolynomial Q (fun _ => (N:ℝ)) p := by
+        apply bernoulliPolynomial_mono _ _ _ p p.2
+        intro ω hω
+        exact_mod_cast hN ω hω
+      _ = N := bernoulliPolynomial_const Q (N:ℝ) p
+  change bernoulliPolynomial Q (fun ω => |ProbeTree.score p t.forget (t.leaf ω)|) p ≤ _
+  rw [t.leaf_expectation ht p (fun l => |ProbeTree.score p t.forget l|)]
+  exact ProbeTree.score_abs_mean_le_sqrt p 0 t.forget N (Nat.cast_nonneg N) hq
+
+end EdgeQueryTree
+
+theorem incidenceSearch_abs_mean_le (G : BondGraph V E) (N : ℕ) (Q : Finset E)
+    (R : Finset V) (p : unitInterval) :
+    bernoulliPolynomial Q (fun ω => |(incidenceSearch G N Q R).score p ω|) p ≤
+      Real.sqrt (N:ℝ) := by
+  classical
+  exact EdgeQueryTree.capped_score_abs_mean_le _ (incidenceSearch_valid ..) p N
+    (fun ω _ => incidenceSearch_cap G N Q R ω)
+
+end Percolation
+
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def finiteClusterScore (G : BondGraph V E) (Q : Finset E)
+    (x : V) (p : ℝ) (ω : Finset E) : ℝ :=
+  ∑ e ∈ incidentBonds G Q (finiteRootComponent G ω x), ((if e ∈ ω then 1 else 0)-p)
+
+ 
+noncomputable def cappedDensity (n s : ℕ) : ℝ := min (n:ℝ) s / s
+
+theorem cappedDensity_nonneg (n s : ℕ) : 0 ≤ cappedDensity n s := by
+  unfold cappedDensity
+  positivity
+
+theorem cappedDensity_le_one (n s : ℕ) : cappedDensity n s ≤ 1 := by
+  unfold cappedDensity
+  by_cases hs : s = 0
+  · simp [hs]
+  · exact (div_le_one (Nat.cast_pos.mpr (Nat.pos_of_ne_zero hs))).mpr (min_le_right ..)
+
+private theorem exists_four_scale (n s K : ℕ) (hs : n < s) (hK : s ≤ n*4^K) :
+    ∃ j < K, n*4^j < s ∧ s ≤ n*4^(j+1) := by
+  induction K with
+  | zero => simp only [pow_zero,mul_one] at hK; omega
+  | succ K ih =>
+    by_cases h : s ≤ n*4^K
+    · obtain ⟨j,hj,hj₁,hj₂⟩ := ih h
+      exact ⟨j,by omega,hj₁,hj₂⟩
+    · exact ⟨K,by omega,by omega,hK⟩
+
+private theorem nat_le_four_pow (k : ℕ) : k ≤ 4^k := by
+  induction k with
+  | zero => simp
+  | succ k ih => rw [pow_succ]; have := Nat.one_le_pow k 4 (by omega); omega
+
+private theorem cappedDensity_le_four_scale (n s j : ℕ) (hn : 0 < n)
+    (hs : n*4^j < s) : cappedDensity n s ≤ (1/4:ℝ)^j := by
+  have hpow : 1 ≤ 4^j := Nat.one_le_pow j 4 (by omega)
+  have hns : n ≤ s := le_trans (by nlinarith : n ≤ n*4^j) hs.le
+  have hs0 : 0 < (s:ℝ) := by exact_mod_cast (lt_of_lt_of_le hn hns)
+  have hden : 0 < (n:ℝ)*4^j := mul_pos (Nat.cast_pos.mpr hn) (pow_pos (by norm_num) _)
+  unfold cappedDensity
+  rw [min_eq_left (by exact_mod_cast hns)]
+  calc
+    (n:ℝ)/s ≤ n/(n*4^j) := div_le_div_of_nonneg_left (Nat.cast_nonneg _) hden (by exact_mod_cast hs.le)
+    _ = (1/4:ℝ)^j := by rw [div_pow,one_pow]; field_simp
+
+ 
+
+theorem cappedDensity_score_le_stopped (G : BondGraph V E) (Q : Finset E)
+    (x : V) (p : ℝ) (d n K : ℕ) (hn : 0 < n)
+    (hdeg : ∀ v, (Q.filter fun e => v ∈ G e).card ≤ d)
+    (ω : Finset E) (hω : ω ⊆ Q)
+    (hK : (finiteRootComponent G ω x).card ≤ n*4^K) :
+    cappedDensity n (finiteRootComponent G ω x).card * |finiteClusterScore G Q x p ω| ≤
+      |(incidenceSearch G (d*n) Q {x}).score p ω| +
+        ∑ j ∈ Finset.range K, (1/4:ℝ)^j *
+          |(incidenceSearch G (d*(n*4^(j+1))) Q {x}).score p ω| := by
+  let s := (finiteRootComponent G ω x).card
+  have hcomplete (N : ℕ) (hN : s ≤ N) :
+      (incidenceSearch G (d*N) Q {x}).score p ω = finiteClusterScore G Q x p ω := by
+    exact incidenceSearch_completed_score G (d*N) Q ω hω x p
+      ((incidentBonds_card_le G Q _ d hdeg).trans (Nat.mul_le_mul_left d hN))
+  have hterms : ∀ j, 0 ≤ (1/4:ℝ)^j *
+      |(incidenceSearch G (d*(n*4^(j+1))) Q {x}).score p ω| := fun j => by positivity
+  by_cases hs : s ≤ n
+  · have he := hcomplete n hs
+    rw [he]
+    exact (mul_le_of_le_one_left (abs_nonneg _) (cappedDensity_le_one n s)).trans
+      (le_add_of_nonneg_right (Finset.sum_nonneg fun j _ => hterms j))
+  · obtain ⟨j,hj,hlo,hhi⟩ := exists_four_scale n s K (by omega) hK
+    have he := hcomplete (n*4^(j+1)) hhi
+    calc
+      _ ≤ (1/4:ℝ)^j * |finiteClusterScore G Q x p ω| :=
+        mul_le_mul_of_nonneg_right (cappedDensity_le_four_scale n s j hn hlo) (abs_nonneg _)
+      _ = (1/4:ℝ)^j * |(incidenceSearch G (d*(n*4^(j+1))) Q {x}).score p ω| := by rw [he]
+      _ ≤ ∑ j ∈ Finset.range K, (1/4:ℝ)^j *
+          |(incidenceSearch G (d*(n*4^(j+1))) Q {x}).score p ω| :=
+        Finset.single_le_sum (fun j _ => hterms j) (Finset.mem_range.mpr hj)
+      _ ≤ _ := le_add_of_nonneg_left (abs_nonneg _)
+
+private theorem sqrt_four_cap (d n j : ℕ) :
+    Real.sqrt (d*(n*4^(j+1)):ℕ) = (2:ℝ)^(j+1)*Real.sqrt (d*n:ℕ) := by
+  have he : (d*(n*4^(j+1)):ℕ) = (d*n)*4^(j+1) := by ring
+  rw [he,Nat.cast_mul,Real.sqrt_mul (Nat.cast_nonneg _)]
+  rw [Nat.cast_pow,Nat.cast_ofNat,show (4:ℝ) = 2^2 by norm_num,←pow_mul,pow_mul',Real.sqrt_sq_eq_abs,
+    abs_of_nonneg (pow_nonneg (by norm_num : (0:ℝ) ≤ 2) _)]
+  ring
+
+theorem cappedDensity_score_expectation_bound (G : BondGraph V E) (Q : Finset E)
+    (x : V) (p : unitInterval) (d n : ℕ) (hn : 0 < n)
+    (hdeg : ∀ v, (Q.filter fun e => v ∈ G e).card ≤ d) :
+    bernoulliPolynomial Q (fun ω => cappedDensity n (finiteRootComponent G ω x).card *
+      |finiteClusterScore G Q x p ω|) p ≤ 5*Real.sqrt (d*n:ℕ) := by
+  let K := (insert x (searchVertices G Q)).card
+  have hK (ω : Finset E) (hω : ω ⊆ Q) : (finiteRootComponent G ω x).card ≤ n*4^K := by
+    have hs : (finiteRootComponent G ω x).card ≤ K :=
+      Finset.card_le_card (finiteRootComponent_subset G Q _ x ω hω le_rfl)
+    exact hs.trans ((nat_le_four_pow K).trans (by nlinarith [Nat.one_le_pow K 4 (by omega)]))
+  have hp := bernoulliPolynomial_mono Q _ _ p p.2 (fun ω hω =>
+    cappedDensity_score_le_stopped G Q x p d n K hn hdeg ω hω (hK ω hω))
+  rw [bernoulliPolynomial_add,bernoulliPolynomial_sum] at hp
+  simp_rw [bernoulliPolynomial_const_mul] at hp
+  have hh := hp.trans (add_le_add (incidenceSearch_abs_mean_le G (d*n) Q {x} p)
+    (Finset.sum_le_sum fun j _ => mul_le_mul_of_nonneg_left
+      (incidenceSearch_abs_mean_le G (d*(n*4^(j+1))) Q {x} p) (by positivity)))
+  simp_rw [sqrt_four_cap] at hh
+  have he (j : ℕ) : (1/4:ℝ)^j*((2:ℝ)^(j+1)*Real.sqrt (d*n:ℕ)) =
+      2*(1/2:ℝ)^j*Real.sqrt (d*n:ℕ) := by
+    rw [pow_succ]
+    calc
+      _ = 2*((1/4:ℝ)^j*2^j)*Real.sqrt (d*n:ℕ) := by ring
+      _ = _ := by rw [←mul_pow]; norm_num
+  simp_rw [he] at hh
+  rw [←Finset.sum_mul,←Finset.mul_sum] at hh
+  have hg : ∑ j ∈ Finset.range K, (1/2:ℝ)^j ≤ 2 := by
+    have hid : ∑ j ∈ Finset.range K, (1/2:ℝ)^j = 2-2*(1/2:ℝ)^K := by
+      induction K with
+      | zero => simp
+      | succ K ih => rw [Finset.sum_range_succ,ih,pow_succ]; ring
+    rw [hid]
+    linarith [pow_nonneg (by norm_num : (0:ℝ) ≤ 1/2) K]
+  exact hh.trans (by nlinarith [Real.sqrt_nonneg (d*n:ℕ)])
+
+end Percolation
+
+ 
+
+ 
+
+open MeasureTheory ProbabilityTheory
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+theorem incident_cluster_insert_iff (G : BondGraph V E) (ω : Set E) (x : V) (e : E) :
+    (∃ a, Conn G (insert e ω) x a ∧ a ∈ G e) ↔
+      ∃ a, Conn G ω x a ∧ a ∈ G e := by
+  obtain ⟨a,b,hab⟩ : ∃ a b, G e = s(a,b) := by
+    obtain ⟨⟨a,b⟩,h⟩ := Quot.exists_rep (G e)
+    exact ⟨a,b,h.symm⟩
+  constructor
+  · rintro ⟨z,hz,hze⟩
+    rcases (conn_insert_iff G ω e a b x z hab).mp hz with hz | ⟨hxa,_⟩ | ⟨hxb,_⟩
+    · exact ⟨z,hz,hze⟩
+    · exact ⟨a,hxa,hab ▸ by simp⟩
+    · exact ⟨b,hxb,hab ▸ by simp⟩
+  · rintro ⟨z,hz,hze⟩
+    exact ⟨z,conn_mono G (Set.subset_insert e ω) hz,hze⟩
+
+theorem cluster_insert_eq_of_not_incident (G : BondGraph V E) (ω : Set E) (x : V) (e : E)
+    (hn : ¬∃ a, Conn G ω x a ∧ a ∈ G e) :
+    cluster G (insert e ω) x = cluster G ω x := by
+  obtain ⟨a,b,hab⟩ : ∃ a b, G e = s(a,b) := by
+    obtain ⟨⟨a,b⟩,h⟩ := Quot.exists_rep (G e)
+    exact ⟨a,b,h.symm⟩
+  ext z
+  constructor
+  · intro hz
+    rcases (conn_insert_iff G ω e a b x z hab).mp hz with hz | ⟨hxa,_⟩ | ⟨hxb,_⟩
+    · exact hz
+    · exact (hn ⟨a,hxa,hab ▸ by simp⟩).elim
+    · exact (hn ⟨b,hxb,hab ▸ by simp⟩).elim
+  · exact conn_mono G (Set.subset_insert e ω)
+
+theorem finite_incident_insert_iff (G : BondGraph V E) (Q ω : Finset E) (x : V) (e : E) :
+    e ∈ incidentBonds G Q (finiteRootComponent G (insert e ω) x) ↔
+      e ∈ incidentBonds G Q (finiteRootComponent G ω x) := by
+  simp only [mem_incidentBonds]
+  have hc (η : Finset E) (a : V) : a ∈ finiteRootComponent G η x ↔ Conn G (η:Set E) x a := by
+    change a ∈ (finiteRootComponent G η x:Set V) ↔ _
+    rw [coe_finiteRootComponent]
+    rfl
+  simp only [hc,Finset.coe_insert,incident_cluster_insert_iff]
+
+theorem finiteRootComponent_insert_eq_of_not_incident (G : BondGraph V E)
+    (Q ω : Finset E) (x : V) (e : E) (he : e ∈ Q)
+    (hn : e ∉ incidentBonds G Q (finiteRootComponent G ω x)) :
+    finiteRootComponent G (insert e ω) x = finiteRootComponent G ω x := by
+  apply Finset.coe_injective
+  rw [coe_finiteRootComponent,coe_finiteRootComponent,Finset.coe_insert]
+  apply cluster_insert_eq_of_not_incident
+  rintro ⟨a,ha,hae⟩
+  apply hn
+  apply (mem_incidentBonds G Q _ e).mpr
+  refine ⟨he,a,?_,hae⟩
+  change a ∈ (finiteRootComponent G ω x:Set V)
+  rw [coe_finiteRootComponent]
+  exact ha
+
+theorem finiteRootComponent_card_pos (G : BondGraph V E) (ω : Finset E) (x : V) :
+    0 < (finiteRootComponent G ω x).card := by
+  apply Finset.card_pos.mpr
+  refine ⟨x,?_⟩
+  change x ∈ (finiteRootComponent G ω x:Set V)
+  rw [coe_finiteRootComponent]
+  exact mem_cluster_self G _ x
+
+theorem cappedDensity_antitone (n : ℕ) {s t : ℕ} (hs : 0 < s) (hst : s ≤ t) :
+    cappedDensity n t ≤ cappedDensity n s := by
+  by_cases hn : s ≤ n
+  · have he : cappedDensity n s = 1 := by
+      unfold cappedDensity
+      rw [min_eq_right (by exact_mod_cast hn),div_self (by exact_mod_cast Nat.ne_of_gt hs)]
+    rw [he]
+    exact cappedDensity_le_one n t
+  · have hs' : (n:ℝ) ≤ s := by exact_mod_cast (le_of_lt (Nat.lt_of_not_ge hn))
+    have ht' : (n:ℝ) ≤ t := hs'.trans (by exact_mod_cast hst)
+    unfold cappedDensity
+    rw [min_eq_left ht',min_eq_left hs']
+    exact div_le_div_of_nonneg_left (Nat.cast_nonneg n) (Nat.cast_pos.mpr hs) (by exact_mod_cast hst)
+
+noncomputable def finiteDensity (G : BondGraph V E) (n : ℕ) (x : V) (ω : Finset E) : ℝ :=
+  cappedDensity n (finiteRootComponent G ω x).card
+
+noncomputable def finiteDensityInfluence (G : BondGraph V E) (n : ℕ) (x : V)
+    (e : E) (ω : Finset E) : ℝ :=
+  if e ∈ ω then 0 else finiteDensity G n x ω - finiteDensity G n x (insert e ω)
+
+theorem finiteDensityInfluence_nonneg (G : BondGraph V E) (n : ℕ) (x : V)
+    (e : E) (ω : Finset E) : 0 ≤ finiteDensityInfluence G n x e ω := by
+  unfold finiteDensityInfluence
+  split_ifs
+  · rfl
+  · apply sub_nonneg.mpr
+    apply cappedDensity_antitone n (finiteRootComponent_card_pos G ω x)
+    apply Finset.card_le_card
+    intro y hy
+    have hm (η : Finset E) : y ∈ finiteRootComponent G η x ↔ Conn G (η:Set E) x y := by
+      change y ∈ (finiteRootComponent G η x:Set V) ↔ _
+      rw [coe_finiteRootComponent]
+      rfl
+    rw [hm] at hy ⊢
+    exact conn_mono G (by simp) hy
+
+theorem finiteDensityInfluence_le_one (G : BondGraph V E) (n : ℕ) (x : V)
+    (e : E) (ω : Finset E) : finiteDensityInfluence G n x e ω ≤ 1 := by
+  unfold finiteDensityInfluence finiteDensity
+  split_ifs
+  · norm_num
+  · exact (sub_le_self _ (cappedDensity_nonneg ..)).trans (cappedDensity_le_one ..)
+
+ 
+theorem finiteDensity_score_one_bit (G : BondGraph V E) (Q : Finset E) (n : ℕ)
+    (x : V) (p : ℝ) (e : E) (he : e ∈ Q) :
+    bernoulliPolynomial Q (fun ω => if e ∈ incidentBonds G Q (finiteRootComponent G ω x)
+      then ((if e ∈ ω then 1 else 0)-p)*finiteDensity G n x ω else 0) p =
+      -p*bernoulliPolynomial Q (finiteDensityInfluence G n x e) p := by
+  let f := finiteDensity G n x
+  let I := fun ω : Finset E => e ∈ incidentBonds G Q (finiteRootComponent G ω x)
+  let R := Q.erase e
+  have heR : e ∉ R := Finset.notMem_erase e Q
+  have hQ : insert e R = Q := Finset.insert_erase he
+  rw [←hQ,bernoulliPolynomial_insert R e heR,bernoulliPolynomial_insert R e heR]
+  rw [hQ]
+  have hclosed (ω : Finset E) (hω : ω ⊆ R) : e ∉ ω := fun h => heR (hω h)
+  have hc : bernoulliPolynomial R (fun ω => if I ω then ((if e ∈ ω then 1 else 0)-p)*f ω else 0) p =
+      -p*bernoulliPolynomial R (fun ω => if I ω then f ω else 0) p := by
+    rw [←bernoulliPolynomial_const_mul]
+    apply bernoulliPolynomial_congr
+    intro ω hω
+    simp only [ite_eq_right (hclosed ω hω),zero_sub]
+    split_ifs <;> ring
+  have ho : bernoulliPolynomial R (fun ω => if I (insert e ω) then
+      ((if e ∈ insert e ω then 1 else 0)-p)*f (insert e ω) else 0) p =
+      (1-p)*bernoulliPolynomial R (fun ω => if I ω then f (insert e ω) else 0) p := by
+    rw [←bernoulliPolynomial_const_mul]
+    apply bernoulliPolynomial_congr
+    intro ω _
+    simp only [I,finite_incident_insert_iff,Finset.mem_insert_self,ite_true]
+    split_ifs <;> ring
+  have hi : bernoulliPolynomial R (finiteDensityInfluence G n x e) p =
+      bernoulliPolynomial R (fun ω => if I ω then f ω else 0) p -
+        bernoulliPolynomial R (fun ω => if I ω then f (insert e ω) else 0) p := by
+    rw [←bernoulliPolynomial_sub]
+    apply bernoulliPolynomial_congr
+    intro ω hω
+    unfold finiteDensityInfluence
+    rw [ite_eq_right (hclosed ω hω)]
+    by_cases hi : I ω
+    · simp only [ite_eq_left hi,f]
+    · have heq : f (insert e ω) = f ω := by
+        unfold f finiteDensity
+        rw [finiteRootComponent_insert_eq_of_not_incident G Q ω x e he hi]
+      simp only [ite_eq_right hi,heq,sub_self,f]
+  have hz : bernoulliPolynomial R (fun ω => finiteDensityInfluence G n x e (insert e ω)) p = 0 := by
+    simp only [finiteDensityInfluence,Finset.mem_insert_self,ite_true,bernoulliPolynomial_zero]
+  change _ = -p*((1-p)*bernoulliPolynomial R (finiteDensityInfluence G n x e) p+
+    p*bernoulliPolynomial R (fun ω => finiteDensityInfluence G n x e (insert e ω)) p)
+  rw [hc,ho,hi,hz]
+  ring
+
+ 
+theorem finiteDensity_score_identity (G : BondGraph V E) (Q : Finset E) (n : ℕ)
+    (x : V) (p : ℝ) :
+    bernoulliPolynomial Q (fun ω => finiteDensity G n x ω * finiteClusterScore G Q x p ω) p =
+      -p * ∑ e ∈ Q, bernoulliPolynomial Q (finiteDensityInfluence G n x e) p := by
+  have hf (ω : Finset E) : finiteDensity G n x ω * finiteClusterScore G Q x p ω =
+      ∑ e ∈ Q, if e ∈ incidentBonds G Q (finiteRootComponent G ω x)
+        then ((if e ∈ ω then 1 else 0)-p)*finiteDensity G n x ω else 0 := by
+    unfold finiteClusterScore
+    rw [Finset.mul_sum]
+    symm
+    rw [←Finset.sum_filter]
+    have hs : Q.filter (fun e => e ∈ incidentBonds G Q (finiteRootComponent G ω x)) =
+        incidentBonds G Q (finiteRootComponent G ω x) := by
+      ext e
+      simp only [Finset.mem_filter,mem_incidentBonds]
+      tauto
+    rw [hs]
+    apply Finset.sum_congr rfl
+    intro e _
+    ring
+  simp_rw [hf]
+  rw [bernoulliPolynomial_sum,Finset.mul_sum]
+  exact Finset.sum_congr rfl fun e he => finiteDensity_score_one_bit G Q n x p e he
+
+ 
+
+theorem finiteDensityInfluence_sum_bound (G : BondGraph V E) (Q : Finset E)
+    (x : V) (p : unitInterval) (d n : ℕ) (hn : 0 < n)
+    (hdeg : ∀ v, (Q.filter fun e => v ∈ G e).card ≤ d) :
+    (p:ℝ) * ∑ e ∈ Q, bernoulliPolynomial Q (finiteDensityInfluence G n x e) p ≤
+      5*Real.sqrt (d*n:ℕ) := by
+  have hid := finiteDensity_score_identity G Q n x p
+  have hle := bernoulliPolynomial_mono Q
+    (fun ω => -(finiteDensity G n x ω * finiteClusterScore G Q x p ω))
+    (fun ω => cappedDensity n (finiteRootComponent G ω x).card * |finiteClusterScore G Q x p ω|)
+    p p.2 (fun ω _ => by
+      rw [←mul_neg]
+      exact mul_le_mul_of_nonneg_left (neg_le_abs _) (cappedDensity_nonneg ..))
+  have hneg : bernoulliPolynomial Q
+      (fun ω => -(finiteDensity G n x ω * finiteClusterScore G Q x p ω)) p =
+      -bernoulliPolynomial Q (fun ω => finiteDensity G n x ω * finiteClusterScore G Q x p ω) p := by
+    simpa using bernoulliPolynomial_const_mul Q
+      (fun ω => finiteDensity G n x ω * finiteClusterScore G Q x p ω) (-1) (p:ℝ)
+  rw [hneg,hid] at hle
+  have hb := cappedDensity_score_expectation_bound G Q x p d n hn hdeg
+  linarith
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/TwoCluster/Transport.lean b/lean/OAI/Probability/UnimodularPercolation/TwoCluster/Transport.lean
index 13af820ed9b8c6a278ff74d853a5d1d4a6afb6eb..80b36e24d29bc43f3c6a5a020684ee6870bcd5eb 100644
--- a/lean/OAI/Probability/UnimodularPercolation/TwoCluster/Transport.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/TwoCluster/Transport.lean
@@ -1,357 +1,357 @@
-import Mathlib
-import OAI.Probability.UnimodularPercolation.Clusters.LargeContacts
-import OAI.Probability.UnimodularPercolation.TwoCluster.Score
-
-namespace OAI
-
-/-! Infinite-volume two-cluster limits and transport. -/
-
-open MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-
-open MeasureTheory ProbabilityTheory Filter
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def clusterDensity (G : BondGraph V E) (n : ℕ) (x : V) (ω : Set E) : ℝ :=
-  cappedDensity n (cluster G ω x).ncard
-
-noncomputable def densityInfluence (G : BondGraph V E) (n : ℕ) (x : V)
-    (e : E) (ω : Set E) : ℝ :=
-  if e ∈ ω then 0 else clusterDensity G n x ω - clusterDensity G n x (insert e ω)
-
-theorem clusterDensity_nonneg (G : BondGraph V E) (n : ℕ) (x : V) (ω : Set E) :
-    0 ≤ clusterDensity G n x ω := cappedDensity_nonneg ..
-
-theorem clusterDensity_le_one (G : BondGraph V E) (n : ℕ) (x : V) (ω : Set E) :
-    clusterDensity G n x ω ≤ 1 := cappedDensity_le_one ..
-
-theorem clusterDensity_antitone (G : BondGraph V E) (n : ℕ) (x : V)
-    {ω η : Set E} (h : ω ⊆ η) : clusterDensity G n x η ≤ clusterDensity G n x ω := by
-  by_cases hη : (cluster G η x).Finite
-  · have hω : (cluster G ω x).Finite := hη.subset (fun y hy => conn_mono G h hy)
-    exact cappedDensity_antitone n ((Set.ncard_pos hω).mpr ⟨x,mem_cluster_self G ω x⟩)
-      (Set.ncard_le_ncard (fun y hy => conn_mono G h hy) hη)
-  · rw [clusterDensity,Set.Infinite.ncard hη]
-    simpa [cappedDensity] using clusterDensity_nonneg G n x ω
-
-theorem densityInfluence_nonneg (G : BondGraph V E) (n : ℕ) (x : V) (e : E) (ω : Set E) :
-    0 ≤ densityInfluence G n x e ω := by
-  unfold densityInfluence
-  split_ifs
-  · exact le_rfl
-  · exact sub_nonneg.mpr (clusterDensity_antitone G n x (Set.subset_insert e ω))
-
-theorem densityInfluence_le_one (G : BondGraph V E) (n : ℕ) (x : V) (e : E) (ω : Set E) :
-    densityInfluence G n x e ω ≤ 1 := by
-  unfold densityInfluence
-  split_ifs
-  · norm_num
-  · exact (sub_le_self _ (clusterDensity_nonneg ..)).trans (clusterDensity_le_one ..)
-
-theorem measurable_clusterDensity (G : BondGraph V E) (hloc : LocallyFinite G) (n : ℕ) (x : V) :
-    Measurable (clusterDensity G n x) :=
-  (measurable_of_countable (cappedDensity n)).comp (measurable_cluster_ncard G hloc x)
-
-theorem measurable_densityInfluence (G : BondGraph V E) (hloc : LocallyFinite G)
-    (n : ℕ) (x : V) (e : E) : Measurable (densityInfluence G n x e) := by
-  exact measurable_const.ite (measurableSet_mem e)
-    ((measurable_clusterDensity G hloc n x).sub
-      ((measurable_clusterDensity G hloc n x).comp (measurable_set_insert e)))
-
-theorem integrable_densityInfluence (G : BondGraph V E) (hloc : LocallyFinite G)
-    (n : ℕ) (x : V) (e : E) (p : unitInterval) :
-    Integrable (densityInfluence G n x e) (law p) := by
-  apply Integrable.mono' (integrable_const (1:ℝ)) (measurable_densityInfluence G hloc n x e).aestronglyMeasurable
-  exact Eventually.of_forall fun ω => by
-    rw [Real.norm_eq_abs,abs_of_nonneg (densityInfluence_nonneg ..)]
-    exact densityInfluence_le_one ..
-
-theorem cluster_insert_finite (G : BondGraph V E) (ω : Set E) (e : E) (x a b : V)
-    (he : G e = s(a,b)) (hx : (cluster G ω x).Finite)
-    (ha : (cluster G ω a).Finite) (hb : (cluster G ω b).Finite) :
-    (cluster G (insert e ω) x).Finite := by
-  apply (hx.union (ha.union hb)).subset
-  intro y hy
-  rcases (conn_insert_iff G ω e a b x y he).mp hy with hy | ⟨_,hy⟩ | ⟨_,hy⟩
-  · exact Or.inl hy
-  · exact Or.inr (Or.inr hy)
-  · exact Or.inr (Or.inl hy)
-
-theorem cluster_insert_finite_ae (G : BondGraph V E) (p : unitInterval)
-    (hfin : ∀ y, ∀ᵐ ω ∂law p, (cluster G ω y).Finite) (x : V) (e : E) :
-    ∀ᵐ ω ∂law p, (cluster G (insert e ω) x).Finite := by
-  obtain ⟨a,b,he⟩ : ∃ a b, G e = s(a,b) := by
-    obtain ⟨⟨a,b⟩,h⟩ := Quot.exists_rep (G e)
-    exact ⟨a,b,h.symm⟩
-  filter_upwards [hfin x,hfin a,hfin b] with ω hx ha hb
-  exact cluster_insert_finite G ω e x a b he hx ha hb
-
-theorem finiteDensity_eq_clusterDensity (G : BondGraph V E) (hloc : LocallyFinite G)
-    (n : ℕ) (x : V) (ω : Set E) (hx : (cluster G ω x).Finite)
-    (Q : Finset E) (hQ : incidentSupport G hloc hx.toFinset ⊆ Q) :
-    finiteDensity G n x (domainBits Q ω) = clusterDensity G n x ω := by
-  have hc : cluster G (domainBits Q ω:Set E) x = cluster G ω x := by
-    rw [coe_domainBits]
-    have hh := (cluster_restrict_incident_eq_iff G hloc hx.toFinset Q hQ ω x).mpr
-      (Set.Finite.coe_toFinset hx).symm
-    simpa only [Set.Finite.coe_toFinset] using hh
-  unfold finiteDensity clusterDensity
-  rw [←Set.ncard_coe_finset,coe_finiteRootComponent,hc]
-
-theorem domainBits_insert (Q : Finset E) (e : E) (he : e ∈ Q) (ω : Set E) :
-    domainBits Q (insert e ω) = insert e (domainBits Q ω) := by
-  ext f
-  simp only [mem_domainBits,Set.mem_insert_iff,Finset.mem_insert]
-  aesop
-
-theorem densityInfluence_tendsto_finite [Encodable E] (G : BondGraph V E) (hloc : LocallyFinite G)
-    (n : ℕ) (x : V) (e : E) (ω : Set E) (hx : (cluster G ω x).Finite)
-    (he : (cluster G (insert e ω) x).Finite) (F : Finset E) :
-    Tendsto (fun j => finiteDensityInfluence G n x e
-      (domainBits (F ∪ vertexExhaustion (V := E) j) ω)) atTop
-      (nhds (densityInfluence G n x e ω)) := by
-  obtain ⟨jx,hjx⟩ := subset_vertexExhaustion (incidentSupport G hloc hx.toFinset)
-  obtain ⟨je,hje⟩ := subset_vertexExhaustion (incidentSupport G hloc he.toFinset)
-  obtain ⟨j₀,hj₀'⟩ := subset_vertexExhaustion ({e}:Finset E)
-  have hj₀ : e ∈ vertexExhaustion (V := E) j₀ := hj₀' (by simp)
-  have hev : (fun j => finiteDensityInfluence G n x e
-      (domainBits (F ∪ vertexExhaustion (V := E) j) ω)) =ᶠ[atTop]
-      (fun _ => densityInfluence G n x e ω) := by
-    filter_upwards [eventually_ge_atTop (max jx (max je j₀))] with j hj
-    let Q := F ∪ vertexExhaustion (V := E) j
-    have hxQ : incidentSupport G hloc hx.toFinset ⊆ Q := fun z hz =>
-      Finset.mem_union_right _ (vertexExhaustion_mono (by omega) (hjx hz))
-    have heQ : incidentSupport G hloc he.toFinset ⊆ Q := fun z hz =>
-      Finset.mem_union_right _ (vertexExhaustion_mono (by omega) (hje hz))
-    have heQ' : e ∈ Q := Finset.mem_union_right _ (vertexExhaustion_mono (by omega) hj₀)
-    change finiteDensityInfluence G n x e (domainBits Q ω) = _
-    unfold finiteDensityInfluence densityInfluence
-    simp only [mem_domainBits,heQ',true_and]
-    split_ifs
-    · rfl
-    · rw [finiteDensity_eq_clusterDensity G hloc n x ω hx Q hxQ,
-        ←domainBits_insert Q e heQ' ω,finiteDensity_eq_clusterDensity G hloc n x (insert e ω) he Q heQ]
-  exact tendsto_const_nhds.congr' hev.symm
-
-theorem densityInfluence_integral_tendsto [Encodable E] (G : BondGraph V E)
-    (hloc : LocallyFinite G) (n : ℕ) (x : V) (e : E) (p : unitInterval)
-    (hfin : ∀ y, ∀ᵐ ω ∂law p, (cluster G ω y).Finite) (F : Finset E) :
-    Tendsto (fun j => bernoulliPolynomial (F ∪ vertexExhaustion (V := E) j)
-      (finiteDensityInfluence G n x e) p) atTop
-      (nhds (∫ ω, densityInfluence G n x e ω ∂law p)) := by
-  let Q := fun j => F ∪ vertexExhaustion (V := E) j
-  let f := fun j ω => finiteDensityInfluence G n x e (domainBits (Q j) ω)
-  have hfi (j : ℕ) : Integrable (f j) (law p) :=
-    integrable_finite_local (Q j) p (f j) (fun ω => by dsimp [f]; rw [domainBits_local])
-  have ht : ∀ᵐ ω ∂law p, Tendsto (fun j => f j ω) atTop (nhds (densityInfluence G n x e ω)) := by
-    filter_upwards [hfin x,cluster_insert_finite_ae G p hfin x e] with ω hx he
-    exact densityInfluence_tendsto_finite G hloc n x e ω hx he F
-  have hd := tendsto_integral_of_dominated_convergence (fun _ : Set E => (1:ℝ))
-    (fun j => (hfi j).aestronglyMeasurable) (integrable_const _)
-    (fun j => Eventually.of_forall (fun ω => by
-      rw [Real.norm_eq_abs,abs_of_nonneg (finiteDensityInfluence_nonneg ..)]
-      exact finiteDensityInfluence_le_one ..)) ht
-  simpa only [f,integral_domainBits,Q] using hd
-
- 
-theorem densityInfluence_finite_sum_bound [Countable E] (G : BondGraph V E)
-    (hloc : LocallyFinite G) (d : ℕ) (hd : ∀ y, {e | y ∈ G e}.ncard ≤ d)
-    (n : ℕ) (hn : 0 < n) (x : V) (p : unitInterval)
-    (hfin : ∀ y, ∀ᵐ ω ∂law p, (cluster G ω y).Finite) (F : Finset E) :
-    (p:ℝ) * ∑ e ∈ F, (∫ ω, densityInfluence G n x e ω ∂law p) ≤
-      5*Real.sqrt (d*n:ℕ) := by
-  let : Encodable E := Encodable.ofCountable E
-  let Q := fun j => F ∪ vertexExhaustion (V := E) j
-  have ht := (tendsto_const_nhds (x := (p:ℝ)) (f := atTop)).mul (tendsto_finsetSum F
-    (fun e _ => densityInfluence_integral_tendsto G hloc n x e p hfin F))
-  apply le_of_tendsto ht
-  apply Eventually.of_forall
-  intro j
-  have hdeg (y : V) : ((Q j).filter fun e => y ∈ G e).card ≤ d := by
-    calc
-      _ = (((Q j).filter fun e => y ∈ G e):Set E).ncard := (Set.ncard_coe_finset _).symm
-      _ ≤ {e | y ∈ G e}.ncard := Set.ncard_le_ncard (fun e he => (Finset.mem_filter.mp he).2) (hloc y)
-      _ ≤ d := hd y
-  have hs : (∑ e ∈ F, bernoulliPolynomial (Q j) (finiteDensityInfluence G n x e) p) ≤
-      ∑ e ∈ Q j, bernoulliPolynomial (Q j) (finiteDensityInfluence G n x e) p := by
-    exact Finset.sum_le_sum_of_subset_of_nonneg Finset.subset_union_left
-      (fun e _ _ => bernoulliPolynomial_nonneg (Q j) _ p p.2 (fun ω _ => finiteDensityInfluence_nonneg ..))
-  exact (mul_le_mul_of_nonneg_left hs p.2.1).trans
-    (finiteDensityInfluence_sum_bound G (Q j) x p d n hn hdeg)
-
- 
-theorem densityInfluence_lintegral_sum_bound [Countable E] (G : BondGraph V E)
-    (hloc : LocallyFinite G) (d : ℕ) (hd : ∀ y, {e | y ∈ G e}.ncard ≤ d)
-    (n : ℕ) (hn : 0 < n) (x : V) (p : unitInterval) (hp : 0 < (p:ℝ))
-    (hfin : ∀ y, ∀ᵐ ω ∂law p, (cluster G ω y).Finite) :
-    (∑' e, ∫⁻ ω, ENNReal.ofReal (densityInfluence G n x e ω) ∂law p) ≤
-      ENNReal.ofReal (5*Real.sqrt (d*n:ℕ)/(p:ℝ)) := by
-  rw [ENNReal.tsum_eq_iSup_sum]
-  apply iSup_le
-  intro F
-  simp_rw [←ofReal_integral_eq_lintegral_ofReal (integrable_densityInfluence G hloc n x _ p)
-    (Eventually.of_forall (densityInfluence_nonneg G n x _))]
-  rw [←ENNReal.ofReal_sum_of_nonneg (fun e _ => integral_nonneg (densityInfluence_nonneg G n x e))]
-  apply ENNReal.ofReal_le_ofReal
-  apply (le_div_iff₀ hp).mpr
-  simpa only [mul_comm] using densityInfluence_finite_sum_bound G hloc d hd n hn x p hfin F
-
-end Percolation
-
- 
-
- 
-
-open MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal
-namespace Percolation
-universe u v
-variable {V : Type u} {E : Type v}
-attribute [local instance] Classical.propDecidable Classical.decEq
-
-noncomputable def boundaryDensityInfluence (G : BondGraph V E) (n : ℕ)
-    (ω : Set E) (y : V) : ℝ≥0∞ :=
-  ∑' e, if y ∈ G e then ENNReal.ofReal (densityInfluence G n y e ω) else 0
-
-noncomputable def localDensityLoss (G : BondGraph V E) (n : ℕ)
-    (ω : Set E) (y : V) : ℝ≥0∞ :=
-  (cluster G ω y).ncard * boundaryDensityInfluence G n ω y
-
-theorem measurable_boundaryDensityInfluence [Countable E] (G : BondGraph V E)
-    (hloc : LocallyFinite G) (n : ℕ) (y : V) :
-    Measurable (fun ω => boundaryDensityInfluence G n ω y) := by
-  apply Measurable.tsum
-  intro e
-  by_cases h : y ∈ G e
-  · simpa only [ite_eq_left h] using (measurable_densityInfluence G hloc n y e).ennreal_ofReal
-  · simp only [ite_eq_right h]; exact measurable_const
-
-theorem measurable_localDensityLoss [Countable E] (G : BondGraph V E)
-    (hloc : LocallyFinite G) (n : ℕ) (y : V) :
-    Measurable (fun ω => localDensityLoss G n ω y) := by
-  exact ((measurable_of_countable (fun k : ℕ => (k:ℝ≥0∞))).comp
-    (measurable_cluster_ncard G hloc y)).mul (measurable_boundaryDensityInfluence G hloc n y)
-
-theorem densityInfluence_eq_of_conn (G : BondGraph V E) (n : ℕ) (ω : Set E)
-    (x y : V) (hxy : Conn G ω x y) (e : E) :
-    densityInfluence G n x e ω = densityInfluence G n y e ω := by
-  unfold densityInfluence clusterDensity
-  rw [cluster_eq_of_conn G ω hxy,
-    cluster_eq_of_conn G (insert e ω) (conn_mono G (Set.subset_insert e ω) hxy)]
-
-theorem densityInfluence_aut (G : BondGraph V E) (n : ℕ) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (ω : Set E) (x : V) (e : E) :
-    densityInfluence G n x e (b ⁻¹' ω) = densityInfluence G n (a x) (b e) ω := by
-  have hins : insert e (b ⁻¹' ω) = b ⁻¹' (insert (b e) ω) := by
-    ext f
-    simp only [Set.mem_insert_iff,Set.mem_preimage,b.injective.eq_iff]
-  simp only [densityInfluence,Set.mem_preimage,clusterDensity,hins,
-    cluster_ncard_covariant G hab]
-
-theorem localDensityLoss_aut (G : BondGraph V E) (n : ℕ) {a : V ≃ V} {b : E ≃ E}
-    (hab : IsAut G a b) (ω : Set E) (x : V) :
-    localDensityLoss G n (b ⁻¹' ω) x = localDensityLoss G n ω (a x) := by
-  unfold localDensityLoss boundaryDensityInfluence
-  rw [cluster_ncard_covariant G hab,←b.tsum_eq (fun e =>
-    if a x ∈ G e then ENNReal.ofReal (densityInfluence G n (a x) e ω) else 0)]
-  congr 1
-  apply tsum_congr
-  intro e
-  have hm : a x ∈ G (b e) ↔ x ∈ G e := by
-    rw [hab e,Sym2.mem_map]
-    constructor
-    · rintro ⟨z,hz,hzx⟩
-      exact a.injective hzx ▸ hz
-    · intro h; exact ⟨x,h,rfl⟩
-  simp only [hm,densityInfluence_aut G n hab]
-
- 
-theorem tsum_bond_endpoints_le_two (G : BondGraph V E) (e : E) (c : ℝ≥0∞) :
-    (∑' y, if y ∈ G e then c else 0) ≤ 2*c := by
-  obtain ⟨a,b,he⟩ : ∃ a b, G e = s(a,b) := by
-    obtain ⟨⟨a,b⟩,h⟩ := Quot.exists_rep (G e)
-    exact ⟨a,b,h.symm⟩
-  calc
-    _ ≤ ∑' y, ((if y = a then c else 0)+(if y = b then c else 0)) := by
-      apply ENNReal.tsum_le_tsum
-      intro y
-      by_cases h : y ∈ G e
-      · rw [ite_eq_left h]
-        rw [he,Sym2.mem_iff] at h
-        rcases h with rfl | rfl <;> simp
-      · rw [ite_eq_right h]; exact zero_le
-    _ = 2*c := by rw [ENNReal.tsum_add]; simp; ring
-
- 
-
-theorem clusterAveraged_localDensityLoss_le (G : BondGraph V E) (n : ℕ)
-    (ω : Set E) (x : V) (hx : (cluster G ω x).Finite) :
-    (∑' y, clusterAveragingKernel G ω x y * localDensityLoss G n ω y) ≤
-      2 * ∑' e, ENNReal.ofReal (densityInfluence G n x e ω) := by
-  have hf (y : V) : clusterAveragingKernel G ω x y * localDensityLoss G n ω y ≤
-      ∑' e, if y ∈ G e then ENNReal.ofReal (densityInfluence G n x e ω) else 0 := by
-    by_cases hxy : Conn G ω x y
-    · have hc := cluster_eq_of_conn G ω hxy
-      have hpos : ((cluster G ω x).ncard:ℝ≥0∞) ≠ 0 := by
-        exact_mod_cast (cluster_ncard_pos G ω x hx).ne'
-      unfold clusterAveragingKernel localDensityLoss
-      rw [ite_eq_left hxy,←hc,←mul_assoc,ENNReal.inv_mul_cancel hpos (ENNReal.natCast_ne_top _),one_mul]
-      unfold boundaryDensityInfluence
-      simp_rw [densityInfluence_eq_of_conn G n ω x y hxy]
-      exact le_rfl
-    · simp only [clusterAveragingKernel,ite_eq_right hxy,zero_mul]; exact zero_le
-  calc
-    _ ≤ ∑' y, ∑' e, if y ∈ G e then ENNReal.ofReal (densityInfluence G n x e ω) else 0 :=
-      ENNReal.tsum_le_tsum hf
-    _ = ∑' e, ∑' y, if y ∈ G e then ENNReal.ofReal (densityInfluence G n x e ω) else 0 :=
-      ENNReal.tsum_comm
-    _ ≤ ∑' e, 2*ENNReal.ofReal (densityInfluence G n x e ω) :=
-      ENNReal.tsum_le_tsum (fun e => tsum_bond_endpoints_le_two G e _)
-    _ = _ := ENNReal.tsum_mul_left
-
-theorem clusterAveraged_localDensityLoss_integral_bound [Countable V] [Countable E]
-    (G : BondGraph V E) (hloc : LocallyFinite G) (d : ℕ)
-    (hd : ∀ y, {e | y ∈ G e}.ncard ≤ d) (n : ℕ) (hn : 0 < n)
-    (p : unitInterval) (hp : 0 < (p:ℝ))
-    (hfin : ∀ y, ∀ᵐ ω ∂law p, (cluster G ω y).Finite) (x : V) :
-    (∫⁻ ω, ∑' y, clusterAveragingKernel G ω x y * localDensityLoss G n ω y ∂law p) ≤
-      2 * ENNReal.ofReal (5*Real.sqrt (d*n:ℕ)/(p:ℝ)) := by
-  calc
-    _ ≤ ∫⁻ ω, 2 * ∑' e, ENNReal.ofReal (densityInfluence G n x e ω) ∂law p := by
-      apply lintegral_mono_ae
-      filter_upwards [hfin x] with ω hω
-      exact clusterAveraged_localDensityLoss_le G n ω x hω
-    _ = 2 * ∑' e, ∫⁻ ω, ENNReal.ofReal (densityInfluence G n x e ω) ∂law p := by
-      rw [lintegral_const_mul _ (Measurable.tsum (fun e =>
-        (measurable_densityInfluence G hloc n x e).ennreal_ofReal)),lintegral_tsum]
-      exact fun e => (measurable_densityInfluence G hloc n x e).ennreal_ofReal.aemeasurable
-    _ ≤ _ := mul_le_mul_right (densityInfluence_lintegral_sum_bound G hloc d hd n hn x p hp hfin) 2
-
- 
-theorem weighted_localDensityLoss_bound [Countable V] [Countable E]
-    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
-    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
-    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
-    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
-    (d : ℕ) (hd : ∀ y, {e | y ∈ G e}.ncard ≤ d) (n : ℕ) (hn : 0 < n)
-    (p : unitInterval) (hp : 0 < (p:ℝ))
-    (hfin : ∀ y, ∀ᵐ ω ∂law p, (cluster G ω y).Finite) :
-    (∑ i, ENNReal.ofReal (weight i) * ∫⁻ ω, localDensityLoss G n ω (root i) ∂law p) ≤
-      2 * ENNReal.ofReal (5*Real.sqrt (d*n:ℕ)/(p:ℝ)) := by
-  rw [←root_cluster_average G hconn hloc root weight Γ hΓ htr hw p hfin
-    (localDensityLoss G n) (measurable_localDensityLoss G hloc n)
-    (fun a ha b hab ω x => localDensityLoss_aut G n hab ω x)]
-  calc
-    _ ≤ ∑ i, ENNReal.ofReal (weight i) * (2 * ENNReal.ofReal (5*Real.sqrt (d*n:ℕ)/(p:ℝ))) :=
-      Finset.sum_le_sum fun i _ => mul_le_mul_right
-        (clusterAveraged_localDensityLoss_integral_bound G hloc d hd n hn p hp hfin (root i)) _
-    _ = _ := by
-      rw [←Finset.sum_mul,←ENNReal.ofReal_sum_of_nonneg (fun i _ => hw i),hw1]
-      simp
-
-end Percolation
-
-end OAI
+import Mathlib
+import OAI.Probability.UnimodularPercolation.Clusters.LargeContacts
+import OAI.Probability.UnimodularPercolation.TwoCluster.Score
+
+namespace OAI
+
+/-! Infinite-volume two-cluster limits and transport. -/
+
+open MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+
+open MeasureTheory ProbabilityTheory Filter
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def clusterDensity (G : BondGraph V E) (n : ℕ) (x : V) (ω : Set E) : ℝ :=
+  cappedDensity n (cluster G ω x).ncard
+
+noncomputable def densityInfluence (G : BondGraph V E) (n : ℕ) (x : V)
+    (e : E) (ω : Set E) : ℝ :=
+  if e ∈ ω then 0 else clusterDensity G n x ω - clusterDensity G n x (insert e ω)
+
+theorem clusterDensity_nonneg (G : BondGraph V E) (n : ℕ) (x : V) (ω : Set E) :
+    0 ≤ clusterDensity G n x ω := cappedDensity_nonneg ..
+
+theorem clusterDensity_le_one (G : BondGraph V E) (n : ℕ) (x : V) (ω : Set E) :
+    clusterDensity G n x ω ≤ 1 := cappedDensity_le_one ..
+
+theorem clusterDensity_antitone (G : BondGraph V E) (n : ℕ) (x : V)
+    {ω η : Set E} (h : ω ⊆ η) : clusterDensity G n x η ≤ clusterDensity G n x ω := by
+  by_cases hη : (cluster G η x).Finite
+  · have hω : (cluster G ω x).Finite := hη.subset (fun y hy => conn_mono G h hy)
+    exact cappedDensity_antitone n ((Set.ncard_pos hω).mpr ⟨x,mem_cluster_self G ω x⟩)
+      (Set.ncard_le_ncard (fun y hy => conn_mono G h hy) hη)
+  · rw [clusterDensity,Set.Infinite.ncard hη]
+    simpa [cappedDensity] using clusterDensity_nonneg G n x ω
+
+theorem densityInfluence_nonneg (G : BondGraph V E) (n : ℕ) (x : V) (e : E) (ω : Set E) :
+    0 ≤ densityInfluence G n x e ω := by
+  unfold densityInfluence
+  split_ifs
+  · exact le_rfl
+  · exact sub_nonneg.mpr (clusterDensity_antitone G n x (Set.subset_insert e ω))
+
+theorem densityInfluence_le_one (G : BondGraph V E) (n : ℕ) (x : V) (e : E) (ω : Set E) :
+    densityInfluence G n x e ω ≤ 1 := by
+  unfold densityInfluence
+  split_ifs
+  · norm_num
+  · exact (sub_le_self _ (clusterDensity_nonneg ..)).trans (clusterDensity_le_one ..)
+
+theorem measurable_clusterDensity (G : BondGraph V E) (hloc : LocallyFinite G) (n : ℕ) (x : V) :
+    Measurable (clusterDensity G n x) :=
+  (measurable_of_countable (cappedDensity n)).comp (measurable_cluster_ncard G hloc x)
+
+theorem measurable_densityInfluence (G : BondGraph V E) (hloc : LocallyFinite G)
+    (n : ℕ) (x : V) (e : E) : Measurable (densityInfluence G n x e) := by
+  exact measurable_const.ite (measurableSet_mem e)
+    ((measurable_clusterDensity G hloc n x).sub
+      ((measurable_clusterDensity G hloc n x).comp (measurable_set_insert e)))
+
+theorem integrable_densityInfluence (G : BondGraph V E) (hloc : LocallyFinite G)
+    (n : ℕ) (x : V) (e : E) (p : unitInterval) :
+    Integrable (densityInfluence G n x e) (law p) := by
+  apply Integrable.mono' (integrable_const (1:ℝ)) (measurable_densityInfluence G hloc n x e).aestronglyMeasurable
+  exact Eventually.of_forall fun ω => by
+    rw [Real.norm_eq_abs,abs_of_nonneg (densityInfluence_nonneg ..)]
+    exact densityInfluence_le_one ..
+
+theorem cluster_insert_finite (G : BondGraph V E) (ω : Set E) (e : E) (x a b : V)
+    (he : G e = s(a,b)) (hx : (cluster G ω x).Finite)
+    (ha : (cluster G ω a).Finite) (hb : (cluster G ω b).Finite) :
+    (cluster G (insert e ω) x).Finite := by
+  apply (hx.union (ha.union hb)).subset
+  intro y hy
+  rcases (conn_insert_iff G ω e a b x y he).mp hy with hy | ⟨_,hy⟩ | ⟨_,hy⟩
+  · exact Or.inl hy
+  · exact Or.inr (Or.inr hy)
+  · exact Or.inr (Or.inl hy)
+
+theorem cluster_insert_finite_ae (G : BondGraph V E) (p : unitInterval)
+    (hfin : ∀ y, ∀ᵐ ω ∂law p, (cluster G ω y).Finite) (x : V) (e : E) :
+    ∀ᵐ ω ∂law p, (cluster G (insert e ω) x).Finite := by
+  obtain ⟨a,b,he⟩ : ∃ a b, G e = s(a,b) := by
+    obtain ⟨⟨a,b⟩,h⟩ := Quot.exists_rep (G e)
+    exact ⟨a,b,h.symm⟩
+  filter_upwards [hfin x,hfin a,hfin b] with ω hx ha hb
+  exact cluster_insert_finite G ω e x a b he hx ha hb
+
+theorem finiteDensity_eq_clusterDensity (G : BondGraph V E) (hloc : LocallyFinite G)
+    (n : ℕ) (x : V) (ω : Set E) (hx : (cluster G ω x).Finite)
+    (Q : Finset E) (hQ : incidentSupport G hloc hx.toFinset ⊆ Q) :
+    finiteDensity G n x (domainBits Q ω) = clusterDensity G n x ω := by
+  have hc : cluster G (domainBits Q ω:Set E) x = cluster G ω x := by
+    rw [coe_domainBits]
+    have hh := (cluster_restrict_incident_eq_iff G hloc hx.toFinset Q hQ ω x).mpr
+      (Set.Finite.coe_toFinset hx).symm
+    simpa only [Set.Finite.coe_toFinset] using hh
+  unfold finiteDensity clusterDensity
+  rw [←Set.ncard_coe_finset,coe_finiteRootComponent,hc]
+
+theorem domainBits_insert (Q : Finset E) (e : E) (he : e ∈ Q) (ω : Set E) :
+    domainBits Q (insert e ω) = insert e (domainBits Q ω) := by
+  ext f
+  simp only [mem_domainBits,Set.mem_insert_iff,Finset.mem_insert]
+  aesop
+
+theorem densityInfluence_tendsto_finite [Encodable E] (G : BondGraph V E) (hloc : LocallyFinite G)
+    (n : ℕ) (x : V) (e : E) (ω : Set E) (hx : (cluster G ω x).Finite)
+    (he : (cluster G (insert e ω) x).Finite) (F : Finset E) :
+    Tendsto (fun j => finiteDensityInfluence G n x e
+      (domainBits (F ∪ vertexExhaustion (V := E) j) ω)) atTop
+      (nhds (densityInfluence G n x e ω)) := by
+  obtain ⟨jx,hjx⟩ := subset_vertexExhaustion (incidentSupport G hloc hx.toFinset)
+  obtain ⟨je,hje⟩ := subset_vertexExhaustion (incidentSupport G hloc he.toFinset)
+  obtain ⟨j₀,hj₀'⟩ := subset_vertexExhaustion ({e}:Finset E)
+  have hj₀ : e ∈ vertexExhaustion (V := E) j₀ := hj₀' (by simp)
+  have hev : (fun j => finiteDensityInfluence G n x e
+      (domainBits (F ∪ vertexExhaustion (V := E) j) ω)) =ᶠ[atTop]
+      (fun _ => densityInfluence G n x e ω) := by
+    filter_upwards [eventually_ge_atTop (max jx (max je j₀))] with j hj
+    let Q := F ∪ vertexExhaustion (V := E) j
+    have hxQ : incidentSupport G hloc hx.toFinset ⊆ Q := fun z hz =>
+      Finset.mem_union_right _ (vertexExhaustion_mono (by omega) (hjx hz))
+    have heQ : incidentSupport G hloc he.toFinset ⊆ Q := fun z hz =>
+      Finset.mem_union_right _ (vertexExhaustion_mono (by omega) (hje hz))
+    have heQ' : e ∈ Q := Finset.mem_union_right _ (vertexExhaustion_mono (by omega) hj₀)
+    change finiteDensityInfluence G n x e (domainBits Q ω) = _
+    unfold finiteDensityInfluence densityInfluence
+    simp only [mem_domainBits,heQ',true_and]
+    split_ifs
+    · rfl
+    · rw [finiteDensity_eq_clusterDensity G hloc n x ω hx Q hxQ,
+        ←domainBits_insert Q e heQ' ω,finiteDensity_eq_clusterDensity G hloc n x (insert e ω) he Q heQ]
+  exact tendsto_const_nhds.congr' hev.symm
+
+theorem densityInfluence_integral_tendsto [Encodable E] (G : BondGraph V E)
+    (hloc : LocallyFinite G) (n : ℕ) (x : V) (e : E) (p : unitInterval)
+    (hfin : ∀ y, ∀ᵐ ω ∂law p, (cluster G ω y).Finite) (F : Finset E) :
+    Tendsto (fun j => bernoulliPolynomial (F ∪ vertexExhaustion (V := E) j)
+      (finiteDensityInfluence G n x e) p) atTop
+      (nhds (∫ ω, densityInfluence G n x e ω ∂law p)) := by
+  let Q := fun j => F ∪ vertexExhaustion (V := E) j
+  let f := fun j ω => finiteDensityInfluence G n x e (domainBits (Q j) ω)
+  have hfi (j : ℕ) : Integrable (f j) (law p) :=
+    integrable_finite_local (Q j) p (f j) (fun ω => by dsimp [f]; rw [domainBits_local])
+  have ht : ∀ᵐ ω ∂law p, Tendsto (fun j => f j ω) atTop (nhds (densityInfluence G n x e ω)) := by
+    filter_upwards [hfin x,cluster_insert_finite_ae G p hfin x e] with ω hx he
+    exact densityInfluence_tendsto_finite G hloc n x e ω hx he F
+  have hd := tendsto_integral_of_dominated_convergence (fun _ : Set E => (1:ℝ))
+    (fun j => (hfi j).aestronglyMeasurable) (integrable_const _)
+    (fun j => Eventually.of_forall (fun ω => by
+      rw [Real.norm_eq_abs,abs_of_nonneg (finiteDensityInfluence_nonneg ..)]
+      exact finiteDensityInfluence_le_one ..)) ht
+  simpa only [f,integral_domainBits,Q] using hd
+
+ 
+theorem densityInfluence_finite_sum_bound [Countable E] (G : BondGraph V E)
+    (hloc : LocallyFinite G) (d : ℕ) (hd : ∀ y, {e | y ∈ G e}.ncard ≤ d)
+    (n : ℕ) (hn : 0 < n) (x : V) (p : unitInterval)
+    (hfin : ∀ y, ∀ᵐ ω ∂law p, (cluster G ω y).Finite) (F : Finset E) :
+    (p:ℝ) * ∑ e ∈ F, (∫ ω, densityInfluence G n x e ω ∂law p) ≤
+      5*Real.sqrt (d*n:ℕ) := by
+  let : Encodable E := Encodable.ofCountable E
+  let Q := fun j => F ∪ vertexExhaustion (V := E) j
+  have ht := (tendsto_const_nhds (x := (p:ℝ)) (f := atTop)).mul (tendsto_finsetSum F
+    (fun e _ => densityInfluence_integral_tendsto G hloc n x e p hfin F))
+  apply le_of_tendsto ht
+  apply Eventually.of_forall
+  intro j
+  have hdeg (y : V) : ((Q j).filter fun e => y ∈ G e).card ≤ d := by
+    calc
+      _ = (((Q j).filter fun e => y ∈ G e):Set E).ncard := (Set.ncard_coe_finset _).symm
+      _ ≤ {e | y ∈ G e}.ncard := Set.ncard_le_ncard (fun e he => (Finset.mem_filter.mp he).2) (hloc y)
+      _ ≤ d := hd y
+  have hs : (∑ e ∈ F, bernoulliPolynomial (Q j) (finiteDensityInfluence G n x e) p) ≤
+      ∑ e ∈ Q j, bernoulliPolynomial (Q j) (finiteDensityInfluence G n x e) p := by
+    exact Finset.sum_le_sum_of_subset_of_nonneg Finset.subset_union_left
+      (fun e _ _ => bernoulliPolynomial_nonneg (Q j) _ p p.2 (fun ω _ => finiteDensityInfluence_nonneg ..))
+  exact (mul_le_mul_of_nonneg_left hs p.2.1).trans
+    (finiteDensityInfluence_sum_bound G (Q j) x p d n hn hdeg)
+
+ 
+theorem densityInfluence_lintegral_sum_bound [Countable E] (G : BondGraph V E)
+    (hloc : LocallyFinite G) (d : ℕ) (hd : ∀ y, {e | y ∈ G e}.ncard ≤ d)
+    (n : ℕ) (hn : 0 < n) (x : V) (p : unitInterval) (hp : 0 < (p:ℝ))
+    (hfin : ∀ y, ∀ᵐ ω ∂law p, (cluster G ω y).Finite) :
+    (∑' e, ∫⁻ ω, ENNReal.ofReal (densityInfluence G n x e ω) ∂law p) ≤
+      ENNReal.ofReal (5*Real.sqrt (d*n:ℕ)/(p:ℝ)) := by
+  rw [ENNReal.tsum_eq_iSup_sum]
+  apply iSup_le
+  intro F
+  simp_rw [←ofReal_integral_eq_lintegral_ofReal (integrable_densityInfluence G hloc n x _ p)
+    (Eventually.of_forall (densityInfluence_nonneg G n x _))]
+  rw [←ENNReal.ofReal_sum_of_nonneg (fun e _ => integral_nonneg (densityInfluence_nonneg G n x e))]
+  apply ENNReal.ofReal_le_ofReal
+  apply (le_div_iff₀ hp).mpr
+  simpa only [mul_comm] using densityInfluence_finite_sum_bound G hloc d hd n hn x p hfin F
+
+end Percolation
+
+ 
+
+ 
+
+open MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal
+namespace Percolation
+universe u v
+variable {V : Type u} {E : Type v}
+attribute [local instance] Classical.propDecidable Classical.decEq
+
+noncomputable def boundaryDensityInfluence (G : BondGraph V E) (n : ℕ)
+    (ω : Set E) (y : V) : ℝ≥0∞ :=
+  ∑' e, if y ∈ G e then ENNReal.ofReal (densityInfluence G n y e ω) else 0
+
+noncomputable def localDensityLoss (G : BondGraph V E) (n : ℕ)
+    (ω : Set E) (y : V) : ℝ≥0∞ :=
+  (cluster G ω y).ncard * boundaryDensityInfluence G n ω y
+
+theorem measurable_boundaryDensityInfluence [Countable E] (G : BondGraph V E)
+    (hloc : LocallyFinite G) (n : ℕ) (y : V) :
+    Measurable (fun ω => boundaryDensityInfluence G n ω y) := by
+  apply Measurable.tsum
+  intro e
+  by_cases h : y ∈ G e
+  · simpa only [ite_eq_left h] using (measurable_densityInfluence G hloc n y e).ennreal_ofReal
+  · simp only [ite_eq_right h]; exact measurable_const
+
+theorem measurable_localDensityLoss [Countable E] (G : BondGraph V E)
+    (hloc : LocallyFinite G) (n : ℕ) (y : V) :
+    Measurable (fun ω => localDensityLoss G n ω y) := by
+  exact ((measurable_of_countable (fun k : ℕ => (k:ℝ≥0∞))).comp
+    (measurable_cluster_ncard G hloc y)).mul (measurable_boundaryDensityInfluence G hloc n y)
+
+theorem densityInfluence_eq_of_conn (G : BondGraph V E) (n : ℕ) (ω : Set E)
+    (x y : V) (hxy : Conn G ω x y) (e : E) :
+    densityInfluence G n x e ω = densityInfluence G n y e ω := by
+  unfold densityInfluence clusterDensity
+  rw [cluster_eq_of_conn G ω hxy,
+    cluster_eq_of_conn G (insert e ω) (conn_mono G (Set.subset_insert e ω) hxy)]
+
+theorem densityInfluence_aut (G : BondGraph V E) (n : ℕ) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (ω : Set E) (x : V) (e : E) :
+    densityInfluence G n x e (b ⁻¹' ω) = densityInfluence G n (a x) (b e) ω := by
+  have hins : insert e (b ⁻¹' ω) = b ⁻¹' (insert (b e) ω) := by
+    ext f
+    simp only [Set.mem_insert_iff,Set.mem_preimage,b.injective.eq_iff]
+  simp only [densityInfluence,Set.mem_preimage,clusterDensity,hins,
+    cluster_ncard_covariant G hab]
+
+theorem localDensityLoss_aut (G : BondGraph V E) (n : ℕ) {a : V ≃ V} {b : E ≃ E}
+    (hab : IsAut G a b) (ω : Set E) (x : V) :
+    localDensityLoss G n (b ⁻¹' ω) x = localDensityLoss G n ω (a x) := by
+  unfold localDensityLoss boundaryDensityInfluence
+  rw [cluster_ncard_covariant G hab,←b.tsum_eq (fun e =>
+    if a x ∈ G e then ENNReal.ofReal (densityInfluence G n (a x) e ω) else 0)]
+  congr 1
+  apply tsum_congr
+  intro e
+  have hm : a x ∈ G (b e) ↔ x ∈ G e := by
+    rw [hab e,Sym2.mem_map]
+    constructor
+    · rintro ⟨z,hz,hzx⟩
+      exact a.injective hzx ▸ hz
+    · intro h; exact ⟨x,h,rfl⟩
+  simp only [hm,densityInfluence_aut G n hab]
+
+ 
+theorem tsum_bond_endpoints_le_two (G : BondGraph V E) (e : E) (c : ℝ≥0∞) :
+    (∑' y, if y ∈ G e then c else 0) ≤ 2*c := by
+  obtain ⟨a,b,he⟩ : ∃ a b, G e = s(a,b) := by
+    obtain ⟨⟨a,b⟩,h⟩ := Quot.exists_rep (G e)
+    exact ⟨a,b,h.symm⟩
+  calc
+    _ ≤ ∑' y, ((if y = a then c else 0)+(if y = b then c else 0)) := by
+      apply ENNReal.tsum_le_tsum
+      intro y
+      by_cases h : y ∈ G e
+      · rw [ite_eq_left h]
+        rw [he,Sym2.mem_iff] at h
+        rcases h with rfl | rfl <;> simp
+      · rw [ite_eq_right h]; exact zero_le
+    _ = 2*c := by rw [ENNReal.tsum_add]; simp; ring
+
+ 
+
+theorem clusterAveraged_localDensityLoss_le (G : BondGraph V E) (n : ℕ)
+    (ω : Set E) (x : V) (hx : (cluster G ω x).Finite) :
+    (∑' y, clusterAveragingKernel G ω x y * localDensityLoss G n ω y) ≤
+      2 * ∑' e, ENNReal.ofReal (densityInfluence G n x e ω) := by
+  have hf (y : V) : clusterAveragingKernel G ω x y * localDensityLoss G n ω y ≤
+      ∑' e, if y ∈ G e then ENNReal.ofReal (densityInfluence G n x e ω) else 0 := by
+    by_cases hxy : Conn G ω x y
+    · have hc := cluster_eq_of_conn G ω hxy
+      have hpos : ((cluster G ω x).ncard:ℝ≥0∞) ≠ 0 := by
+        exact_mod_cast (cluster_ncard_pos G ω x hx).ne'
+      unfold clusterAveragingKernel localDensityLoss
+      rw [ite_eq_left hxy,←hc,←mul_assoc,ENNReal.inv_mul_cancel hpos (ENNReal.natCast_ne_top _),one_mul]
+      unfold boundaryDensityInfluence
+      simp_rw [densityInfluence_eq_of_conn G n ω x y hxy]
+      exact le_rfl
+    · simp only [clusterAveragingKernel,ite_eq_right hxy,zero_mul]; exact zero_le
+  calc
+    _ ≤ ∑' y, ∑' e, if y ∈ G e then ENNReal.ofReal (densityInfluence G n x e ω) else 0 :=
+      ENNReal.tsum_le_tsum hf
+    _ = ∑' e, ∑' y, if y ∈ G e then ENNReal.ofReal (densityInfluence G n x e ω) else 0 :=
+      ENNReal.tsum_comm
+    _ ≤ ∑' e, 2*ENNReal.ofReal (densityInfluence G n x e ω) :=
+      ENNReal.tsum_le_tsum (fun e => tsum_bond_endpoints_le_two G e _)
+    _ = _ := ENNReal.tsum_mul_left
+
+theorem clusterAveraged_localDensityLoss_integral_bound [Countable V] [Countable E]
+    (G : BondGraph V E) (hloc : LocallyFinite G) (d : ℕ)
+    (hd : ∀ y, {e | y ∈ G e}.ncard ≤ d) (n : ℕ) (hn : 0 < n)
+    (p : unitInterval) (hp : 0 < (p:ℝ))
+    (hfin : ∀ y, ∀ᵐ ω ∂law p, (cluster G ω y).Finite) (x : V) :
+    (∫⁻ ω, ∑' y, clusterAveragingKernel G ω x y * localDensityLoss G n ω y ∂law p) ≤
+      2 * ENNReal.ofReal (5*Real.sqrt (d*n:ℕ)/(p:ℝ)) := by
+  calc
+    _ ≤ ∫⁻ ω, 2 * ∑' e, ENNReal.ofReal (densityInfluence G n x e ω) ∂law p := by
+      apply lintegral_mono_ae
+      filter_upwards [hfin x] with ω hω
+      exact clusterAveraged_localDensityLoss_le G n ω x hω
+    _ = 2 * ∑' e, ∫⁻ ω, ENNReal.ofReal (densityInfluence G n x e ω) ∂law p := by
+      rw [lintegral_const_mul _ (Measurable.tsum (fun e =>
+        (measurable_densityInfluence G hloc n x e).ennreal_ofReal)),lintegral_tsum]
+      exact fun e => (measurable_densityInfluence G hloc n x e).ennreal_ofReal.aemeasurable
+    _ ≤ _ := mul_le_mul_right (densityInfluence_lintegral_sum_bound G hloc d hd n hn x p hp hfin) 2
+
+ 
+theorem weighted_localDensityLoss_bound [Countable V] [Countable E]
+    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
+    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
+    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
+    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
+    (d : ℕ) (hd : ∀ y, {e | y ∈ G e}.ncard ≤ d) (n : ℕ) (hn : 0 < n)
+    (p : unitInterval) (hp : 0 < (p:ℝ))
+    (hfin : ∀ y, ∀ᵐ ω ∂law p, (cluster G ω y).Finite) :
+    (∑ i, ENNReal.ofReal (weight i) * ∫⁻ ω, localDensityLoss G n ω (root i) ∂law p) ≤
+      2 * ENNReal.ofReal (5*Real.sqrt (d*n:ℕ)/(p:ℝ)) := by
+  rw [←root_cluster_average G hconn hloc root weight Γ hΓ htr hw p hfin
+    (localDensityLoss G n) (measurable_localDensityLoss G hloc n)
+    (fun a ha b hab ω x => localDensityLoss_aut G n hab ω x)]
+  calc
+    _ ≤ ∑ i, ENNReal.ofReal (weight i) * (2 * ENNReal.ofReal (5*Real.sqrt (d*n:ℕ)/(p:ℝ))) :=
+      Finset.sum_le_sum fun i _ => mul_le_mul_right
+        (clusterAveraged_localDensityLoss_integral_bound G hloc d hd n hn p hp hfin (root i)) _
+    _ = _ := by
+      rw [←Finset.sum_mul,←ENNReal.ofReal_sum_of_nonneg (fun i _ => hw i),hw1]
+      simp
+
+end Percolation
+
+end OAI
diff --git a/lean/OAI/Probability/UnimodularPercolation/UnimodularMain.lean b/lean/OAI/Probability/UnimodularPercolation/UnimodularMain.lean
index a9d8170afe5ad281031a241dfd2d9b351fdbc6cf..47ba1946bd06b925c89c27d93524330063b8d248 100644
--- a/lean/OAI/Probability/UnimodularPercolation/UnimodularMain.lean
+++ b/lean/OAI/Probability/UnimodularPercolation/UnimodularMain.lean
@@ -1 +1 @@
-import OAI.Probability.UnimodularPercolation.Criticality.UnimodularMain
+import OAI.Probability.UnimodularPercolation.Criticality.UnimodularMain
diff --git a/lean/OAI/Probability/Voronoi/CardyFormula.lean b/lean/OAI/Probability/Voronoi/CardyFormula.lean
index 22974dabb6449bbf46a5e888b19ca286593a6b6c..19178be1487c36a0c186d2a37a698e4f463791f3 100644
--- a/lean/OAI/Probability/Voronoi/CardyFormula.lean
+++ b/lean/OAI/Probability/Voronoi/CardyFormula.lean
@@ -1,530 +1,530 @@
-import Mathlib
-
-namespace OAI
-
-/-! Cardy integrals and hypergeometric formulas with Poisson-Voronoi crossing definitions. -/
-
-noncomputable section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace Cardy
-
-def PositivelyOriented (D : Set ℂ) (γ : ℝ → ℂ) : Prop :=
-  ∃ p ∈ D, ∃ H : ℝ × ℝ → ℂ,
-    ContinuousOn H (Icc 0 1 ×ˢ Icc 0 1) ∧
-    (∀ t ∈ Icc (0 : ℝ) 1, H (0, t) = γ t) ∧
-    (∀ t ∈ Icc (0 : ℝ) 1,
-      H (1, t) = p + Complex.exp ((2 * Real.pi * t : ℝ) * Complex.I)) ∧
-    (∀ s ∈ Icc (0 : ℝ) 1, H (s, 0) = H (s, 1)) ∧
-    (∀ s ∈ Icc (0 : ℝ) 1, ∀ t ∈ Icc (0 : ℝ) 1, H (s, t) ≠ p)
-
-structure JordanQuadrilateral where
-  domain : Set ℂ
-  domain_open : IsOpen domain
-  domain_connected : IsConnected domain
-  domain_bounded : Bornology.IsBounded domain
-  boundary : ℝ → ℂ
-  boundary_continuous : ContinuousOn boundary (Icc 0 1)
-  boundary_closed : boundary 0 = boundary 1
-  boundary_injective : InjOn boundary (Ico 0 1)
-  boundary_image : boundary '' Icc 0 1 = frontier domain
-  boundary_positive : PositivelyOriented domain boundary
-  tb : ℝ
-  tc : ℝ
-  td : ℝ
-  zero_lt_tb : 0 < tb
-  tb_lt_tc : tb < tc
-  tc_lt_td : tc < td
-  td_lt_one : td < 1
-
-namespace JordanQuadrilateral
-
-def a (Q : JordanQuadrilateral) : ℂ := Q.boundary 0
-
-def b (Q : JordanQuadrilateral) : ℂ := Q.boundary Q.tb
-
-def c (Q : JordanQuadrilateral) : ℂ := Q.boundary Q.tc
-
-def d (Q : JordanQuadrilateral) : ℂ := Q.boundary Q.td
-
-def arcAB (Q : JordanQuadrilateral) : Set ℂ := Q.boundary '' Icc 0 Q.tb
-
-def arcCD (Q : JordanQuadrilateral) : Set ℂ := Q.boundary '' Icc Q.tc Q.td
-
-end JordanQuadrilateral
-
-def upperHalfPlane : Set ℂ := {z | 0 < z.im}
-
-def IsConformalParameter (Q : JordanQuadrilateral) (x : ℝ) : Prop :=
-  x ∈ Ioo 0 1 ∧ ∃ f g : ℂ → ℂ,
-    DifferentiableOn ℂ f Q.domain ∧
-    DifferentiableOn ℂ g upperHalfPlane ∧
-    MapsTo f Q.domain upperHalfPlane ∧
-    MapsTo g upperHalfPlane Q.domain ∧
-    (∀ z ∈ Q.domain, g (f z) = z) ∧
-    (∀ w ∈ upperHalfPlane, f (g w) = w) ∧
-    Tendsto f (𝓝[Q.domain] Q.a) (𝓝 0) ∧
-    Tendsto f (𝓝[Q.domain] Q.b) (𝓝 (x : ℂ)) ∧
-    Tendsto f (𝓝[Q.domain] Q.c) (𝓝 1) ∧
-    Tendsto (fun z => ‖f z‖) (𝓝[Q.domain] Q.d) atTop
-
-abbrev Box := ℤ × ℤ
-abbrev Slot := Box × ℕ
-abbrev GeometrySample := (Box → ℕ) × (Slot → ℝ × ℝ)
-abbrev Sample := GeometrySample × (Slot → Bool)
-
-def unitUniform : Measure ℝ := volume.restrict (Ico 0 1)
-
-instance : IsProbabilityMeasure unitUniform := by
-  constructor
-  simp [unitUniform]
-
-def fairColor : Measure Bool := (PMF.uniformOfFintype Bool).toMeasure
-
-instance : IsProbabilityMeasure fairColor := by
-  unfold fairColor
-  infer_instance
-
-def rate (ε : ℝ) : ℝ≥0 := Real.toNNReal ((ε⁻¹) ^ 2)
-
-def countLaw (ε : ℝ) : Measure (Box → ℕ) :=
-  Measure.infinitePi (fun _ => poissonMeasure (rate ε))
-
-def locationLaw : Measure (Slot → ℝ × ℝ) :=
-  Measure.infinitePi (fun _ => unitUniform.prod unitUniform)
-
-def colorLaw : Measure (Slot → Bool) :=
-  Measure.infinitePi (fun _ => fairColor)
-
-instance (ε : ℝ) : IsProbabilityMeasure (countLaw ε) := by
-  unfold countLaw
-  infer_instance
-
-instance : IsProbabilityMeasure locationLaw := by
-  unfold locationLaw
-  infer_instance
-
-instance : IsProbabilityMeasure colorLaw := by
-  unfold colorLaw
-  infer_instance
-
-def geometryLaw (ε : ℝ) : Measure GeometrySample := (countLaw ε).prod locationLaw
-
-instance (ε : ℝ) : IsProbabilityMeasure (geometryLaw ε) := by
-  unfold geometryLaw
-  infer_instance
-
-def sampleLaw (ε : ℝ) : Measure Sample := (geometryLaw ε).prod colorLaw
-
-instance (ε : ℝ) : IsProbabilityMeasure (sampleLaw ε) := by
-  unfold sampleLaw
-  infer_instance
-
-def Active (ω : Sample) (i : Slot) : Prop := i.2 < ω.1.1 i.1
-
-def position (ω : Sample) (i : Slot) : ℂ :=
-  (((i.1.1 : ℝ) + (ω.1.2 i).1 : ℝ) : ℂ) +
-    (((i.1.2 : ℝ) + (ω.1.2 i).2 : ℝ) : ℂ) * Complex.I
-
-def sites (ω : Sample) : Set ℂ := {v | ∃ i, Active ω i ∧ position ω i = v}
-
-def blackSites (ω : Sample) : Set ℂ :=
-  {v | ∃ i, Active ω i ∧ ω.2 i = true ∧ position ω i = v}
-
-def voronoiCell (η : Set ℂ) (v : ℂ) : Set ℂ :=
-  {z | ∀ u ∈ η, dist z v ≤ dist z u}
-
-def blackRegion (ω : Sample) : Set ℂ :=
-  ⋃ v ∈ blackSites ω, voronoiCell (sites ω) v
-
-def Crosses (Q : JordanQuadrilateral) (B : Set ℂ) : Prop :=
-  ∃ K : Set ℂ, IsConnected K ∧ K ⊆ B ∩ closure Q.domain ∧
-    (K ∩ Q.arcAB).Nonempty ∧ (K ∩ Q.arcCD).Nonempty
-
-def crossingEvent (Q : JordanQuadrilateral) : Set Sample :=
-  {ω | Crosses Q (blackRegion ω)}
-
-def crossingProbability (ε : ℝ) (Q : JordanQuadrilateral) : ℝ :=
-  (sampleLaw ε (crossingEvent Q)).toReal
-
-def cardyWeight (u : ℝ) : ℝ := (u * (1 - u)) ^ (-(2 : ℝ) / 3)
-
-def cardyIntegral (x : ℝ) : ℝ :=
-  (∫ u in 0..x, cardyWeight u) / (∫ u in 0..1, cardyWeight u)
-
-def cardyHypergeometric (x : ℝ) : ℝ :=
-  (Real.Gamma (2 / 3) * x ^ ((1 : ℝ) / 3)) /
-    (Real.Gamma (1 / 3) * Real.Gamma (4 / 3)) *
-      ordinaryHypergeometric (1 / 3 : ℝ) (2 / 3) (4 / 3) x
-
-namespace JordanQuadrilateral
-
-lemma boundary_ne_of_lt (Q : JordanQuadrilateral) {s t : ℝ}
-    (hs : 0 ≤ s) (hst : s < t) (ht : t < 1) : Q.boundary s ≠ Q.boundary t := by
-  intro heq
-  exact hst.ne (Q.boundary_injective ⟨hs, hst.trans ht⟩ ⟨hs.trans hst.le, ht⟩ heq)
-
-theorem marks_distinct (Q : JordanQuadrilateral) :
-    Q.a ≠ Q.b ∧ Q.a ≠ Q.c ∧ Q.a ≠ Q.d ∧ Q.b ≠ Q.c ∧ Q.b ≠ Q.d ∧ Q.c ≠ Q.d := by
-  have h0b := Q.zero_lt_tb
-  have hbc := Q.tb_lt_tc
-  have hcd := Q.tc_lt_td
-  have hd1 := Q.td_lt_one
-  exact ⟨Q.boundary_ne_of_lt le_rfl h0b (by linarith),
-    Q.boundary_ne_of_lt le_rfl (by linarith) (by linarith),
-    Q.boundary_ne_of_lt le_rfl (by linarith) hd1,
-    Q.boundary_ne_of_lt h0b.le hbc (by linarith),
-    Q.boundary_ne_of_lt h0b.le (by linarith) hd1,
-    Q.boundary_ne_of_lt (by linarith) hcd hd1⟩
-
-theorem arcAB_subset_frontier (Q : JordanQuadrilateral) : Q.arcAB ⊆ frontier Q.domain := by
-  rw [← Q.boundary_image]
-  apply image_mono
-  exact Icc_subset_Icc le_rfl (Q.tb_lt_tc.trans (Q.tc_lt_td.trans Q.td_lt_one)).le
-
-theorem arcCD_subset_frontier (Q : JordanQuadrilateral) : Q.arcCD ⊆ frontier Q.domain := by
-  rw [← Q.boundary_image]
-  apply image_mono
-  exact Icc_subset_Icc (Q.zero_lt_tb.trans Q.tb_lt_tc).le Q.td_lt_one.le
-
-theorem endpoints_mem_arcs (Q : JordanQuadrilateral) :
-    Q.a ∈ Q.arcAB ∧ Q.b ∈ Q.arcAB ∧ Q.c ∈ Q.arcCD ∧ Q.d ∈ Q.arcCD := by
-  exact ⟨mem_image_of_mem _ ⟨le_rfl, Q.zero_lt_tb.le⟩,
-    mem_image_of_mem _ ⟨Q.zero_lt_tb.le, le_rfl⟩,
-    mem_image_of_mem _ ⟨le_rfl, Q.tc_lt_td.le⟩,
-    mem_image_of_mem _ ⟨Q.tc_lt_td.le, le_rfl⟩⟩
-
-end JordanQuadrilateral
-
-theorem voronoiCell_closed (η : Set ℂ) (v : ℂ) : IsClosed (voronoiCell η v) := by
-  have h : voronoiCell η v = ⋂ u ∈ η, {z : ℂ | dist z v ≤ dist z u} := by
-    ext z
-    simp [voronoiCell]
-  rw [h]
-  exact isClosed_biInter fun u _ => isClosed_le (by fun_prop) (by fun_prop)
-
-theorem site_mem_voronoiCell (η : Set ℂ) (v : ℂ) : v ∈ voronoiCell η v := by
-  intro u _
-  simp
-
-theorem blackSites_subset_sites (ω : Sample) : blackSites ω ⊆ sites ω := by
-  rintro v ⟨i, hi, _, hv⟩
-  exact ⟨i, hi, hv⟩
-
-theorem blackSites_subset_blackRegion (ω : Sample) : blackSites ω ⊆ blackRegion ω := by
-  intro v hv
-  exact mem_iUnion₂.mpr ⟨v, hv, site_mem_voronoiCell _ _⟩
-
-theorem crosses_mono {Q : JordanQuadrilateral} {B B' : Set ℂ}
-    (hBB' : B ⊆ B') (h : Crosses Q B) : Crosses Q B' := by
-  rcases h with ⟨K, hK, hKB, ha, hc⟩
-  exact ⟨K, hK, fun z hz => ⟨hBB' (hKB hz).1, (hKB hz).2⟩, ha, hc⟩
-
-theorem crossingProbability_nonneg (ε : ℝ) (Q : JordanQuadrilateral) :
-    0 ≤ crossingProbability ε Q := ENNReal.toReal_nonneg
-
-theorem crossingProbability_le_one (ε : ℝ) (Q : JordanQuadrilateral) :
-    crossingProbability ε Q ≤ 1 := by
-  exact measureReal_le_one
-
-theorem countLaw_marginal (ε : ℝ) (j : Box) :
-    (countLaw ε).map (fun N => N j) = poissonMeasure (rate ε) := by
-  exact Measure.infinitePi_map_eval _ j
-
-theorem colorLaw_marginal (i : Slot) :
-    colorLaw.map (fun χ => χ i) = fairColor := by
-  exact Measure.infinitePi_map_eval _ i
-
-theorem rate_coe (ε : ℝ) : (rate ε : ℝ) = (ε⁻¹) ^ 2 := by
-  exact Real.coe_toNNReal _ (sq_nonneg _)
-
-theorem geometry_colors_independent (ε : ℝ) :
-    IndepFun (fun ω : Sample => ω.1) (fun ω : Sample => ω.2) (sampleLaw ε) := by
-  exact indepFun_prod (X := id) (Y := id) measurable_id measurable_id
-
-theorem sampleLaw_color_marginal (ε : ℝ) :
-    (sampleLaw ε).map (fun ω : Sample => ω.2) = colorLaw := by
-  exact measurePreserving_snd.map_eq
-
-theorem sampleLaw_geometry_marginal (ε : ℝ) :
-    (sampleLaw ε).map (fun ω : Sample => ω.1) = geometryLaw ε := by
-  exact measurePreserving_fst.map_eq
-
-theorem counts_independent (ε : ℝ) :
-    iIndepFun (fun j : Box => fun N : Box → ℕ => N j) (countLaw ε) := by
-  exact iIndepFun_infinitePi (fun _ => measurable_id)
-
-theorem colors_independent :
-    iIndepFun (fun i : Slot => fun χ : Slot → Bool => χ i) colorLaw := by
-  exact iIndepFun_infinitePi (fun _ => measurable_id)
-
-theorem fairColor_singleton (b : Bool) : fairColor {b} = (1 / 2 : ℝ≥0∞) := by
-  rw [fairColor, PMF.toMeasure_apply_singleton _ _ (measurableSet_singleton _),
-    PMF.uniformOfFintype_apply]
-  norm_num
-
-theorem colorLaw_singleton (i : Slot) (b : Bool) :
-    colorLaw {χ | χ i = b} = (1 / 2 : ℝ≥0∞) := by
-  rw [← fairColor_singleton b, ← colorLaw_marginal i]
-  exact (Measure.map_apply (by fun_prop) (measurableSet_singleton b)).symm
-
-lemma cardyWeight_complex {u : ℝ} (hu : u ∈ Icc 0 1) :
-    (cardyWeight u : ℂ) =
-      (u : ℂ) ^ ((1 / 3 : ℂ) - 1) * (1 - (u : ℂ)) ^ ((1 / 3 : ℂ) - 1) := by
-  have h₁ : (1 / 3 : ℂ) - 1 = ((-(2 : ℝ) / 3 : ℝ) : ℂ) := by
-    push_cast
-    ring
-  rw [cardyWeight, Real.mul_rpow hu.1 (sub_nonneg.mpr hu.2),
-    Complex.ofReal_mul, Complex.ofReal_cpow hu.1,
-    Complex.ofReal_cpow (sub_nonneg.mpr hu.2), h₁]
-  push_cast
-  rfl
-
-theorem cardyDenominator_eq_beta :
-    ((∫ u in 0..1, cardyWeight u : ℝ) : ℂ) =
-      Complex.betaIntegral (1 / 3) (1 / 3) := by
-  rw [Complex.betaIntegral, ← intervalIntegral.integral_ofReal]
-  apply intervalIntegral.integral_congr
-  intro u hu
-  exact cardyWeight_complex (by simpa only [uIcc_of_le zero_le_one] using hu)
-
-theorem cardyDenominator_eq_gamma :
-    (∫ u in 0..1, cardyWeight u) =
-      Real.Gamma (1 / 3) ^ 2 / Real.Gamma (2 / 3) := by
-  apply Complex.ofReal_injective
-  rw [cardyDenominator_eq_beta,
-    Complex.betaIntegral_eq_Gamma_mul_div _ _ (by norm_num) (by norm_num)]
-  have h : (1 / 3 : ℂ) + 1 / 3 = 2 / 3 := by norm_num
-  rw [h]
-  rw [show (1 / 3 : ℂ) = ((1 / 3 : ℝ) : ℂ) by norm_num,
-    show (2 / 3 : ℂ) = ((2 / 3 : ℝ) : ℂ) by norm_num,
-    Complex.Gamma_ofReal, Complex.Gamma_ofReal]
-  push_cast
-  ring
-
-theorem cardyDenominator_pos : 0 < ∫ u in 0..1, cardyWeight u := by
-  rw [cardyDenominator_eq_gamma]
-  exact div_pos (pow_pos (Real.Gamma_pos_of_pos (by norm_num)) _)
-    (Real.Gamma_pos_of_pos (by norm_num))
-
-@[simp] theorem cardyIntegral_zero : cardyIntegral 0 = 0 := by
-  simp [cardyIntegral]
-
-@[simp] theorem cardyIntegral_one : cardyIntegral 1 = 1 := by
-  exact div_self (ne_of_gt cardyDenominator_pos)
-
-theorem cardyWeight_integrable : IntervalIntegrable cardyWeight volume 0 1 := by
-  have h := Complex.betaIntegral_convergent
-    (u := (1 / 3 : ℂ)) (v := (1 / 3 : ℂ)) (by norm_num) (by norm_num)
-  have h' : IntervalIntegrable (fun u : ℝ =>
-      ((u : ℂ) ^ ((1 / 3 : ℂ) - 1) *
-        (1 - (u : ℂ)) ^ ((1 / 3 : ℂ) - 1)).re) volume 0 1 := ⟨h.1.re, h.2.re⟩
-  refine IntervalIntegrable.congr ?_ h'
-  intro u hu
-  have hu' : u ∈ Icc (0 : ℝ) 1 := by
-    exact Ioc_subset_Icc_self (by simpa only [uIoc_of_le zero_le_one] using hu)
-  simpa only [Complex.ofReal_re] using congrArg Complex.re (cardyWeight_complex hu').symm
-
-theorem cardyWeight_integrable_subinterval {s t : ℝ} (hs : 0 ≤ s)
-    (ht : t ≤ 1) (hst : s ≤ t) : IntervalIntegrable cardyWeight volume s t := by
-  apply cardyWeight_integrable.mono_set
-  rw [uIcc_of_le hst, uIcc_of_le zero_le_one]
-  exact Icc_subset_Icc hs ht
-
-theorem cardyWeight_nonneg {u : ℝ} (hu : u ∈ Icc 0 1) : 0 ≤ cardyWeight u := by
-  exact Real.rpow_nonneg (mul_nonneg hu.1 (sub_nonneg.mpr hu.2)) _
-
-theorem cardyWeight_pos {u : ℝ} (hu : u ∈ Ioo 0 1) : 0 < cardyWeight u := by
-  exact Real.rpow_pos_of_pos (mul_pos hu.1 (sub_pos.mpr hu.2)) _
-
-lemma cardyWeight_symm (u : ℝ) : cardyWeight (1 - u) = cardyWeight u := by
-  unfold cardyWeight
-  congr 1
-  ring
-
-theorem cardyIntegral_continuousOn : ContinuousOn cardyIntegral (Icc 0 1) := by
-  have h := intervalIntegral.continuousOn_primitive_interval
-    ((intervalIntegrable_iff' (by finiteness)).mp cardyWeight_integrable)
-  rw [uIcc_of_le zero_le_one] at h
-  exact h.div_const _
-
-theorem cardyIntegral_strictMonoOn : StrictMonoOn cardyIntegral (Icc 0 1) := by
-  intro a ha b hb hab
-  have h0a := cardyWeight_integrable_subinterval (s := 0) le_rfl ha.2 ha.1
-  have habi := cardyWeight_integrable_subinterval ha.1 hb.2 hab.le
-  have hp : 0 < ∫ u in a..b, cardyWeight u := by
-    apply intervalIntegral.intervalIntegral_pos_of_pos_on habi _ hab
-    intro u hu
-    exact cardyWeight_pos ⟨ha.1.trans_lt hu.1, hu.2.trans_le hb.2⟩
-  have hadd := intervalIntegral.integral_add_adjacent_intervals h0a habi
-  apply (div_lt_div_iff_of_pos_right cardyDenominator_pos).mpr
-  linarith
-
-theorem cardyIntegral_mem_Icc {x : ℝ} (hx : x ∈ Icc 0 1) :
-    cardyIntegral x ∈ Icc 0 1 := by
-  have h := cardyIntegral_strictMonoOn.monotoneOn
-  constructor
-  · simpa using h (show (0 : ℝ) ∈ Icc 0 1 by simp) hx hx.1
-  · simpa using h hx (show (1 : ℝ) ∈ Icc 0 1 by simp) hx.2
-
-theorem cardyIntegral_mem_Ioo {x : ℝ} (hx : x ∈ Ioo 0 1) :
-    cardyIntegral x ∈ Ioo 0 1 := by
-  constructor
-  · simpa using cardyIntegral_strictMonoOn (show (0 : ℝ) ∈ Icc 0 1 by simp)
-      ⟨hx.1.le, hx.2.le⟩ hx.1
-  · simpa using cardyIntegral_strictMonoOn ⟨hx.1.le, hx.2.le⟩
-      (show (1 : ℝ) ∈ Icc 0 1 by simp) hx.2
-
-theorem cardyIntegral_reflection {x : ℝ} (hx : x ∈ Icc 0 1) :
-    cardyIntegral (1 - x) = 1 - cardyIntegral x := by
-  have hx' : 1 - x ∈ Icc (0 : ℝ) 1 := ⟨by linarith [hx.2], by linarith [hx.1]⟩
-  have h0 := cardyWeight_integrable_subinterval (s := 0) le_rfl hx'.2 hx'.1
-  have h1 := cardyWeight_integrable_subinterval hx'.1 le_rfl hx'.2
-  have hsub : (∫ u in 0..x, cardyWeight u) = ∫ u in (1 - x)..1, cardyWeight u := by
-    simpa only [cardyWeight_symm, sub_zero] using
-      intervalIntegral.integral_comp_sub_left (a := 0) (b := x) cardyWeight 1
-  apply (eq_sub_iff_add_eq).mpr
-  unfold cardyIntegral
-  rw [← add_div, hsub, intervalIntegral.integral_add_adjacent_intervals h0 h1]
-  exact div_self (ne_of_gt cardyDenominator_pos)
-
-@[simp] theorem cardyIntegral_half : cardyIntegral (1 / 2) = 1 / 2 := by
-  have h := cardyIntegral_reflection (x := 1 / 2) (by norm_num)
-  norm_num only [show (1 : ℝ) - 1 / 2 = 1 / 2 by norm_num] at h
-  linarith
-
-def cardyBinomialCoefficient (n : ℕ) : ℝ :=
-  (n.factorial : ℝ)⁻¹ * (ascPochhammer ℝ n).eval (2 / 3)
-
-lemma cardyBinomialCoefficient_pos (n : ℕ) : 0 < cardyBinomialCoefficient n := by
-  exact mul_pos (inv_pos.mpr (by positivity)) (ascPochhammer_pos n _ (by norm_num))
-
-lemma cardyBinomialCoefficient_eq_choose (n : ℕ) :
-    cardyBinomialCoefficient n = Ring.choose ((2 / 3 : ℝ) + n - 1) n := by
-  rw [← Ring.multichoose_eq]
-  have h := Ring.factorial_nsmul_multichoose_eq_ascPochhammer (2 / 3 : ℝ) n
-  rw [nsmul_eq_mul, Polynomial.ascPochhammer_smeval_eq_eval] at h
-  rw [cardyBinomialCoefficient, ← h, ← mul_assoc, inv_mul_cancel₀, one_mul]
-  exact_mod_cast n.factorial_ne_zero
-
-lemma cardy_binomial_hasSum {u : ℝ} (hu : u ∈ Ioo 0 1) :
-    HasSum (fun n : ℕ => cardyBinomialCoefficient n * u ^ n)
-      ((1 - u) ^ (-(2 : ℝ) / 3)) := by
-  have h := (Real.one_div_one_sub_rpow_hasFPowerSeriesOnBall_zero (2 / 3)).hasSum
-    (y := u) (by
-      rw [← ENNReal.ofReal_one, Metric.eball_ofReal]
-      simpa [Real.dist_eq, abs_of_pos hu.1] using hu.2)
-  simpa only [FormalMultilinearSeries.ofScalars_apply_eq, smul_eq_mul,
-    ← cardyBinomialCoefficient_eq_choose, zero_add, one_div, neg_div,
-    Real.rpow_neg (sub_pos.mpr hu.2).le]
-    using h
-
-lemma cardyWeight_hasSum {u : ℝ} (hu : u ∈ Ioo 0 1) :
-    HasSum (fun n : ℕ => cardyBinomialCoefficient n * u ^ ((n : ℝ) - 2 / 3))
-      (cardyWeight u) := by
-  have h := (cardy_binomial_hasSum hu).mul_left (u ^ (-(2 : ℝ) / 3))
-  have heq (n : ℕ) : cardyBinomialCoefficient n * u ^ ((n : ℝ) - 2 / 3) =
-      u ^ (-(2 : ℝ) / 3) * (cardyBinomialCoefficient n * u ^ n) := by
-    rw [show (n : ℝ) - 2 / 3 = -(2 : ℝ) / 3 + n by ring,
-      Real.rpow_add hu.1, Real.rpow_natCast]
-    ring
-  simpa only [heq, cardyWeight, Real.mul_rpow hu.1.le (sub_pos.mpr hu.2).le] using h
-
-theorem cardyNumerator_hasSum {x : ℝ} (hx : x ∈ Ioo 0 1) :
-    HasSum (fun n : ℕ => ∫ u in 0..x,
-      cardyBinomialCoefficient n * u ^ ((n : ℝ) - 2 / 3))
-      (∫ u in 0..x, cardyWeight u) := by
-  let S : ℕ → ℝ → ℝ := fun n u =>
-    cardyBinomialCoefficient n * u ^ ((n : ℝ) - 2 / 3)
-  have hpoint {u : ℝ} (hu : u ∈ uIoc 0 x) : HasSum (fun n => S n u) (cardyWeight u) := by
-    rw [uIoc_of_le hx.1.le] at hu
-    exact cardyWeight_hasSum ⟨hu.1, hu.2.trans_lt hx.2⟩
-  have hi (n : ℕ) : IntervalIntegrable (S n) volume 0 x := by
-    exact (intervalIntegral.intervalIntegrable_rpow'
-      (r := (n : ℝ) - 2 / 3) (by linarith [Nat.cast_nonneg (α := ℝ) n])).const_mul _
-  have htotal : IntervalIntegrable (fun u => ∑' n, S n u) volume 0 x := by
-    apply IntervalIntegrable.congr (fun u hu => (hpoint hu).tsum_eq.symm)
-    exact cardyWeight_integrable_subinterval le_rfl hx.2.le hx.1.le
-  apply intervalIntegral.hasSum_integral_of_dominated_convergence S
-    (fun n => ((intervalIntegrable_iff).mp (hi n)).1) ?_ ?_ htotal ?_
-  · intro n
-    refine Filter.Eventually.of_forall fun u hu => ?_
-    rw [uIoc_of_le hx.1.le] at hu
-    have hpos : 0 ≤ S n u :=
-      mul_nonneg (cardyBinomialCoefficient_pos n).le (Real.rpow_nonneg hu.1.le _)
-    exact (Real.norm_of_nonneg hpos).le
-  · exact Filter.Eventually.of_forall fun u hu => (hpoint hu).summable
-  · exact Filter.Eventually.of_forall fun u hu => hpoint hu
-
-lemma cardyBinomialTerm_integral (x : ℝ) (n : ℕ) :
-    (∫ u in 0..x, cardyBinomialCoefficient n * u ^ ((n : ℝ) - 2 / 3)) =
-      cardyBinomialCoefficient n * x ^ ((n : ℝ) + 1 / 3) / ((n : ℝ) + 1 / 3) := by
-  rw [intervalIntegral.integral_const_mul, integral_rpow
-    (Or.inl (by linarith [Nat.cast_nonneg (α := ℝ) n]))]
-  rw [show (n : ℝ) - 2 / 3 + 1 = (n : ℝ) + 1 / 3 by ring,
-    Real.zero_rpow (by linarith [Nat.cast_nonneg (α := ℝ) n]), sub_zero]
-  ring
-
-lemma cardyHypergeometricCoefficient_eq (n : ℕ) :
-    cardyBinomialCoefficient n / ((n : ℝ) + 1 / 3) =
-      3 * ordinaryHypergeometricCoefficient (1 / 3 : ℝ) (2 / 3) (4 / 3) n := by
-  have hc : (ascPochhammer ℝ n).eval (4 / 3) ≠ 0 :=
-    ne_of_gt (ascPochhammer_pos n _ (by norm_num))
-  have hn : (n : ℝ) + 1 / 3 ≠ 0 := by linarith [Nat.cast_nonneg (α := ℝ) n]
-  have hshift : (ascPochhammer ℝ n).eval (1 / 3) * ((1 / 3 : ℝ) + n) =
-      (1 / 3 : ℝ) * (ascPochhammer ℝ n).eval (4 / 3) := by
-    rw [← ascPochhammer_succ_eval, ascPochhammer_succ_left]
-    simp only [Polynomial.eval_mul, Polynomial.eval_X, Polynomial.eval_comp,
-      Polynomial.eval_add, Polynomial.eval_one]
-    norm_num
-  have hratio : ((ascPochhammer ℝ n).eval (1 / 3) /
-      (ascPochhammer ℝ n).eval (4 / 3)) * ((n : ℝ) + 1 / 3) = 1 / 3 := by
-    calc
-      _ = ((ascPochhammer ℝ n).eval (1 / 3) * ((1 / 3 : ℝ) + n)) /
-          (ascPochhammer ℝ n).eval (4 / 3) := by ring
-      _ = ((1 / 3 : ℝ) * (ascPochhammer ℝ n).eval (4 / 3)) /
-          (ascPochhammer ℝ n).eval (4 / 3) := by rw [hshift]
-      _ = 1 / 3 := mul_div_cancel_right₀ _ hc
-  symm
-  apply (eq_div_iff hn).mpr
-  calc
-    _ = (3 * (((ascPochhammer ℝ n).eval (1 / 3) /
-          (ascPochhammer ℝ n).eval (4 / 3)) * ((n : ℝ) + 1 / 3))) *
-        cardyBinomialCoefficient n := by
-      unfold ordinaryHypergeometricCoefficient cardyBinomialCoefficient
-      ring
-    _ = cardyBinomialCoefficient n := by rw [hratio]; ring
-
-theorem cardyNumerator_eq_hypergeometric {x : ℝ} (hx : x ∈ Ioo 0 1) :
-    (∫ u in 0..x, cardyWeight u) =
-      3 * x ^ ((1 : ℝ) / 3) *
-        ordinaryHypergeometric (1 / 3 : ℝ) (2 / 3) (4 / 3) x := by
-  rw [← (cardyNumerator_hasSum hx).tsum_eq, ordinaryHypergeometric_eq_tsum]
-  simp only [smul_eq_mul]
-  rw [← tsum_mul_left]
-  apply tsum_congr
-  intro n
-  rw [cardyBinomialTerm_integral, Real.rpow_add hx.1, Real.rpow_natCast]
-  calc
-    _ = (cardyBinomialCoefficient n / ((n : ℝ) + 1 / 3)) *
-        (x ^ n * x ^ ((1 : ℝ) / 3)) := by ring
-    _ = _ := by rw [cardyHypergeometricCoefficient_eq]; ring
-
-theorem cardyIntegral_eq_hypergeometric {x : ℝ} (hx : x ∈ Ioo 0 1) :
-    cardyIntegral x = cardyHypergeometric x := by
-  have hγ : Real.Gamma (4 / 3) = (1 / 3 : ℝ) * Real.Gamma (1 / 3) := by
-    convert Real.Gamma_add_one (s := 1 / 3) (by norm_num) using 1
-    norm_num
-  have h₁ : Real.Gamma (1 / 3) ≠ 0 := ne_of_gt (Real.Gamma_pos_of_pos (by norm_num))
-  have h₂ : Real.Gamma (2 / 3) ≠ 0 := ne_of_gt (Real.Gamma_pos_of_pos (by norm_num))
-  rw [cardyIntegral, cardyNumerator_eq_hypergeometric hx,
-    cardyDenominator_eq_gamma, cardyHypergeometric, hγ]
-  field_simp
-
-end Cardy
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-! Cardy integrals and hypergeometric formulas with Poisson-Voronoi crossing definitions. -/
+
+noncomputable section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace Cardy
+
+def PositivelyOriented (D : Set ℂ) (γ : ℝ → ℂ) : Prop :=
+  ∃ p ∈ D, ∃ H : ℝ × ℝ → ℂ,
+    ContinuousOn H (Icc 0 1 ×ˢ Icc 0 1) ∧
+    (∀ t ∈ Icc (0 : ℝ) 1, H (0, t) = γ t) ∧
+    (∀ t ∈ Icc (0 : ℝ) 1,
+      H (1, t) = p + Complex.exp ((2 * Real.pi * t : ℝ) * Complex.I)) ∧
+    (∀ s ∈ Icc (0 : ℝ) 1, H (s, 0) = H (s, 1)) ∧
+    (∀ s ∈ Icc (0 : ℝ) 1, ∀ t ∈ Icc (0 : ℝ) 1, H (s, t) ≠ p)
+
+structure JordanQuadrilateral where
+  domain : Set ℂ
+  domain_open : IsOpen domain
+  domain_connected : IsConnected domain
+  domain_bounded : Bornology.IsBounded domain
+  boundary : ℝ → ℂ
+  boundary_continuous : ContinuousOn boundary (Icc 0 1)
+  boundary_closed : boundary 0 = boundary 1
+  boundary_injective : InjOn boundary (Ico 0 1)
+  boundary_image : boundary '' Icc 0 1 = frontier domain
+  boundary_positive : PositivelyOriented domain boundary
+  tb : ℝ
+  tc : ℝ
+  td : ℝ
+  zero_lt_tb : 0 < tb
+  tb_lt_tc : tb < tc
+  tc_lt_td : tc < td
+  td_lt_one : td < 1
+
+namespace JordanQuadrilateral
+
+def a (Q : JordanQuadrilateral) : ℂ := Q.boundary 0
+
+def b (Q : JordanQuadrilateral) : ℂ := Q.boundary Q.tb
+
+def c (Q : JordanQuadrilateral) : ℂ := Q.boundary Q.tc
+
+def d (Q : JordanQuadrilateral) : ℂ := Q.boundary Q.td
+
+def arcAB (Q : JordanQuadrilateral) : Set ℂ := Q.boundary '' Icc 0 Q.tb
+
+def arcCD (Q : JordanQuadrilateral) : Set ℂ := Q.boundary '' Icc Q.tc Q.td
+
+end JordanQuadrilateral
+
+def upperHalfPlane : Set ℂ := {z | 0 < z.im}
+
+def IsConformalParameter (Q : JordanQuadrilateral) (x : ℝ) : Prop :=
+  x ∈ Ioo 0 1 ∧ ∃ f g : ℂ → ℂ,
+    DifferentiableOn ℂ f Q.domain ∧
+    DifferentiableOn ℂ g upperHalfPlane ∧
+    MapsTo f Q.domain upperHalfPlane ∧
+    MapsTo g upperHalfPlane Q.domain ∧
+    (∀ z ∈ Q.domain, g (f z) = z) ∧
+    (∀ w ∈ upperHalfPlane, f (g w) = w) ∧
+    Tendsto f (𝓝[Q.domain] Q.a) (𝓝 0) ∧
+    Tendsto f (𝓝[Q.domain] Q.b) (𝓝 (x : ℂ)) ∧
+    Tendsto f (𝓝[Q.domain] Q.c) (𝓝 1) ∧
+    Tendsto (fun z => ‖f z‖) (𝓝[Q.domain] Q.d) atTop
+
+abbrev Box := ℤ × ℤ
+abbrev Slot := Box × ℕ
+abbrev GeometrySample := (Box → ℕ) × (Slot → ℝ × ℝ)
+abbrev Sample := GeometrySample × (Slot → Bool)
+
+def unitUniform : Measure ℝ := volume.restrict (Ico 0 1)
+
+instance : IsProbabilityMeasure unitUniform := by
+  constructor
+  simp [unitUniform]
+
+def fairColor : Measure Bool := (PMF.uniformOfFintype Bool).toMeasure
+
+instance : IsProbabilityMeasure fairColor := by
+  unfold fairColor
+  infer_instance
+
+def rate (ε : ℝ) : ℝ≥0 := Real.toNNReal ((ε⁻¹) ^ 2)
+
+def countLaw (ε : ℝ) : Measure (Box → ℕ) :=
+  Measure.infinitePi (fun _ => poissonMeasure (rate ε))
+
+def locationLaw : Measure (Slot → ℝ × ℝ) :=
+  Measure.infinitePi (fun _ => unitUniform.prod unitUniform)
+
+def colorLaw : Measure (Slot → Bool) :=
+  Measure.infinitePi (fun _ => fairColor)
+
+instance (ε : ℝ) : IsProbabilityMeasure (countLaw ε) := by
+  unfold countLaw
+  infer_instance
+
+instance : IsProbabilityMeasure locationLaw := by
+  unfold locationLaw
+  infer_instance
+
+instance : IsProbabilityMeasure colorLaw := by
+  unfold colorLaw
+  infer_instance
+
+def geometryLaw (ε : ℝ) : Measure GeometrySample := (countLaw ε).prod locationLaw
+
+instance (ε : ℝ) : IsProbabilityMeasure (geometryLaw ε) := by
+  unfold geometryLaw
+  infer_instance
+
+def sampleLaw (ε : ℝ) : Measure Sample := (geometryLaw ε).prod colorLaw
+
+instance (ε : ℝ) : IsProbabilityMeasure (sampleLaw ε) := by
+  unfold sampleLaw
+  infer_instance
+
+def Active (ω : Sample) (i : Slot) : Prop := i.2 < ω.1.1 i.1
+
+def position (ω : Sample) (i : Slot) : ℂ :=
+  (((i.1.1 : ℝ) + (ω.1.2 i).1 : ℝ) : ℂ) +
+    (((i.1.2 : ℝ) + (ω.1.2 i).2 : ℝ) : ℂ) * Complex.I
+
+def sites (ω : Sample) : Set ℂ := {v | ∃ i, Active ω i ∧ position ω i = v}
+
+def blackSites (ω : Sample) : Set ℂ :=
+  {v | ∃ i, Active ω i ∧ ω.2 i = true ∧ position ω i = v}
+
+def voronoiCell (η : Set ℂ) (v : ℂ) : Set ℂ :=
+  {z | ∀ u ∈ η, dist z v ≤ dist z u}
+
+def blackRegion (ω : Sample) : Set ℂ :=
+  ⋃ v ∈ blackSites ω, voronoiCell (sites ω) v
+
+def Crosses (Q : JordanQuadrilateral) (B : Set ℂ) : Prop :=
+  ∃ K : Set ℂ, IsConnected K ∧ K ⊆ B ∩ closure Q.domain ∧
+    (K ∩ Q.arcAB).Nonempty ∧ (K ∩ Q.arcCD).Nonempty
+
+def crossingEvent (Q : JordanQuadrilateral) : Set Sample :=
+  {ω | Crosses Q (blackRegion ω)}
+
+def crossingProbability (ε : ℝ) (Q : JordanQuadrilateral) : ℝ :=
+  (sampleLaw ε (crossingEvent Q)).toReal
+
+def cardyWeight (u : ℝ) : ℝ := (u * (1 - u)) ^ (-(2 : ℝ) / 3)
+
+def cardyIntegral (x : ℝ) : ℝ :=
+  (∫ u in 0..x, cardyWeight u) / (∫ u in 0..1, cardyWeight u)
+
+def cardyHypergeometric (x : ℝ) : ℝ :=
+  (Real.Gamma (2 / 3) * x ^ ((1 : ℝ) / 3)) /
+    (Real.Gamma (1 / 3) * Real.Gamma (4 / 3)) *
+      ordinaryHypergeometric (1 / 3 : ℝ) (2 / 3) (4 / 3) x
+
+namespace JordanQuadrilateral
+
+lemma boundary_ne_of_lt (Q : JordanQuadrilateral) {s t : ℝ}
+    (hs : 0 ≤ s) (hst : s < t) (ht : t < 1) : Q.boundary s ≠ Q.boundary t := by
+  intro heq
+  exact hst.ne (Q.boundary_injective ⟨hs, hst.trans ht⟩ ⟨hs.trans hst.le, ht⟩ heq)
+
+theorem marks_distinct (Q : JordanQuadrilateral) :
+    Q.a ≠ Q.b ∧ Q.a ≠ Q.c ∧ Q.a ≠ Q.d ∧ Q.b ≠ Q.c ∧ Q.b ≠ Q.d ∧ Q.c ≠ Q.d := by
+  have h0b := Q.zero_lt_tb
+  have hbc := Q.tb_lt_tc
+  have hcd := Q.tc_lt_td
+  have hd1 := Q.td_lt_one
+  exact ⟨Q.boundary_ne_of_lt le_rfl h0b (by linarith),
+    Q.boundary_ne_of_lt le_rfl (by linarith) (by linarith),
+    Q.boundary_ne_of_lt le_rfl (by linarith) hd1,
+    Q.boundary_ne_of_lt h0b.le hbc (by linarith),
+    Q.boundary_ne_of_lt h0b.le (by linarith) hd1,
+    Q.boundary_ne_of_lt (by linarith) hcd hd1⟩
+
+theorem arcAB_subset_frontier (Q : JordanQuadrilateral) : Q.arcAB ⊆ frontier Q.domain := by
+  rw [← Q.boundary_image]
+  apply image_mono
+  exact Icc_subset_Icc le_rfl (Q.tb_lt_tc.trans (Q.tc_lt_td.trans Q.td_lt_one)).le
+
+theorem arcCD_subset_frontier (Q : JordanQuadrilateral) : Q.arcCD ⊆ frontier Q.domain := by
+  rw [← Q.boundary_image]
+  apply image_mono
+  exact Icc_subset_Icc (Q.zero_lt_tb.trans Q.tb_lt_tc).le Q.td_lt_one.le
+
+theorem endpoints_mem_arcs (Q : JordanQuadrilateral) :
+    Q.a ∈ Q.arcAB ∧ Q.b ∈ Q.arcAB ∧ Q.c ∈ Q.arcCD ∧ Q.d ∈ Q.arcCD := by
+  exact ⟨mem_image_of_mem _ ⟨le_rfl, Q.zero_lt_tb.le⟩,
+    mem_image_of_mem _ ⟨Q.zero_lt_tb.le, le_rfl⟩,
+    mem_image_of_mem _ ⟨le_rfl, Q.tc_lt_td.le⟩,
+    mem_image_of_mem _ ⟨Q.tc_lt_td.le, le_rfl⟩⟩
+
+end JordanQuadrilateral
+
+theorem voronoiCell_closed (η : Set ℂ) (v : ℂ) : IsClosed (voronoiCell η v) := by
+  have h : voronoiCell η v = ⋂ u ∈ η, {z : ℂ | dist z v ≤ dist z u} := by
+    ext z
+    simp [voronoiCell]
+  rw [h]
+  exact isClosed_biInter fun u _ => isClosed_le (by fun_prop) (by fun_prop)
+
+theorem site_mem_voronoiCell (η : Set ℂ) (v : ℂ) : v ∈ voronoiCell η v := by
+  intro u _
+  simp
+
+theorem blackSites_subset_sites (ω : Sample) : blackSites ω ⊆ sites ω := by
+  rintro v ⟨i, hi, _, hv⟩
+  exact ⟨i, hi, hv⟩
+
+theorem blackSites_subset_blackRegion (ω : Sample) : blackSites ω ⊆ blackRegion ω := by
+  intro v hv
+  exact mem_iUnion₂.mpr ⟨v, hv, site_mem_voronoiCell _ _⟩
+
+theorem crosses_mono {Q : JordanQuadrilateral} {B B' : Set ℂ}
+    (hBB' : B ⊆ B') (h : Crosses Q B) : Crosses Q B' := by
+  rcases h with ⟨K, hK, hKB, ha, hc⟩
+  exact ⟨K, hK, fun z hz => ⟨hBB' (hKB hz).1, (hKB hz).2⟩, ha, hc⟩
+
+theorem crossingProbability_nonneg (ε : ℝ) (Q : JordanQuadrilateral) :
+    0 ≤ crossingProbability ε Q := ENNReal.toReal_nonneg
+
+theorem crossingProbability_le_one (ε : ℝ) (Q : JordanQuadrilateral) :
+    crossingProbability ε Q ≤ 1 := by
+  exact measureReal_le_one
+
+theorem countLaw_marginal (ε : ℝ) (j : Box) :
+    (countLaw ε).map (fun N => N j) = poissonMeasure (rate ε) := by
+  exact Measure.infinitePi_map_eval _ j
+
+theorem colorLaw_marginal (i : Slot) :
+    colorLaw.map (fun χ => χ i) = fairColor := by
+  exact Measure.infinitePi_map_eval _ i
+
+theorem rate_coe (ε : ℝ) : (rate ε : ℝ) = (ε⁻¹) ^ 2 := by
+  exact Real.coe_toNNReal _ (sq_nonneg _)
+
+theorem geometry_colors_independent (ε : ℝ) :
+    IndepFun (fun ω : Sample => ω.1) (fun ω : Sample => ω.2) (sampleLaw ε) := by
+  exact indepFun_prod (X := id) (Y := id) measurable_id measurable_id
+
+theorem sampleLaw_color_marginal (ε : ℝ) :
+    (sampleLaw ε).map (fun ω : Sample => ω.2) = colorLaw := by
+  exact measurePreserving_snd.map_eq
+
+theorem sampleLaw_geometry_marginal (ε : ℝ) :
+    (sampleLaw ε).map (fun ω : Sample => ω.1) = geometryLaw ε := by
+  exact measurePreserving_fst.map_eq
+
+theorem counts_independent (ε : ℝ) :
+    iIndepFun (fun j : Box => fun N : Box → ℕ => N j) (countLaw ε) := by
+  exact iIndepFun_infinitePi (fun _ => measurable_id)
+
+theorem colors_independent :
+    iIndepFun (fun i : Slot => fun χ : Slot → Bool => χ i) colorLaw := by
+  exact iIndepFun_infinitePi (fun _ => measurable_id)
+
+theorem fairColor_singleton (b : Bool) : fairColor {b} = (1 / 2 : ℝ≥0∞) := by
+  rw [fairColor, PMF.toMeasure_apply_singleton _ _ (measurableSet_singleton _),
+    PMF.uniformOfFintype_apply]
+  norm_num
+
+theorem colorLaw_singleton (i : Slot) (b : Bool) :
+    colorLaw {χ | χ i = b} = (1 / 2 : ℝ≥0∞) := by
+  rw [← fairColor_singleton b, ← colorLaw_marginal i]
+  exact (Measure.map_apply (by fun_prop) (measurableSet_singleton b)).symm
+
+lemma cardyWeight_complex {u : ℝ} (hu : u ∈ Icc 0 1) :
+    (cardyWeight u : ℂ) =
+      (u : ℂ) ^ ((1 / 3 : ℂ) - 1) * (1 - (u : ℂ)) ^ ((1 / 3 : ℂ) - 1) := by
+  have h₁ : (1 / 3 : ℂ) - 1 = ((-(2 : ℝ) / 3 : ℝ) : ℂ) := by
+    push_cast
+    ring
+  rw [cardyWeight, Real.mul_rpow hu.1 (sub_nonneg.mpr hu.2),
+    Complex.ofReal_mul, Complex.ofReal_cpow hu.1,
+    Complex.ofReal_cpow (sub_nonneg.mpr hu.2), h₁]
+  push_cast
+  rfl
+
+theorem cardyDenominator_eq_beta :
+    ((∫ u in 0..1, cardyWeight u : ℝ) : ℂ) =
+      Complex.betaIntegral (1 / 3) (1 / 3) := by
+  rw [Complex.betaIntegral, ← intervalIntegral.integral_ofReal]
+  apply intervalIntegral.integral_congr
+  intro u hu
+  exact cardyWeight_complex (by simpa only [uIcc_of_le zero_le_one] using hu)
+
+theorem cardyDenominator_eq_gamma :
+    (∫ u in 0..1, cardyWeight u) =
+      Real.Gamma (1 / 3) ^ 2 / Real.Gamma (2 / 3) := by
+  apply Complex.ofReal_injective
+  rw [cardyDenominator_eq_beta,
+    Complex.betaIntegral_eq_Gamma_mul_div _ _ (by norm_num) (by norm_num)]
+  have h : (1 / 3 : ℂ) + 1 / 3 = 2 / 3 := by norm_num
+  rw [h]
+  rw [show (1 / 3 : ℂ) = ((1 / 3 : ℝ) : ℂ) by norm_num,
+    show (2 / 3 : ℂ) = ((2 / 3 : ℝ) : ℂ) by norm_num,
+    Complex.Gamma_ofReal, Complex.Gamma_ofReal]
+  push_cast
+  ring
+
+theorem cardyDenominator_pos : 0 < ∫ u in 0..1, cardyWeight u := by
+  rw [cardyDenominator_eq_gamma]
+  exact div_pos (pow_pos (Real.Gamma_pos_of_pos (by norm_num)) _)
+    (Real.Gamma_pos_of_pos (by norm_num))
+
+@[simp] theorem cardyIntegral_zero : cardyIntegral 0 = 0 := by
+  simp [cardyIntegral]
+
+@[simp] theorem cardyIntegral_one : cardyIntegral 1 = 1 := by
+  exact div_self (ne_of_gt cardyDenominator_pos)
+
+theorem cardyWeight_integrable : IntervalIntegrable cardyWeight volume 0 1 := by
+  have h := Complex.betaIntegral_convergent
+    (u := (1 / 3 : ℂ)) (v := (1 / 3 : ℂ)) (by norm_num) (by norm_num)
+  have h' : IntervalIntegrable (fun u : ℝ =>
+      ((u : ℂ) ^ ((1 / 3 : ℂ) - 1) *
+        (1 - (u : ℂ)) ^ ((1 / 3 : ℂ) - 1)).re) volume 0 1 := ⟨h.1.re, h.2.re⟩
+  refine IntervalIntegrable.congr ?_ h'
+  intro u hu
+  have hu' : u ∈ Icc (0 : ℝ) 1 := by
+    exact Ioc_subset_Icc_self (by simpa only [uIoc_of_le zero_le_one] using hu)
+  simpa only [Complex.ofReal_re] using congrArg Complex.re (cardyWeight_complex hu').symm
+
+theorem cardyWeight_integrable_subinterval {s t : ℝ} (hs : 0 ≤ s)
+    (ht : t ≤ 1) (hst : s ≤ t) : IntervalIntegrable cardyWeight volume s t := by
+  apply cardyWeight_integrable.mono_set
+  rw [uIcc_of_le hst, uIcc_of_le zero_le_one]
+  exact Icc_subset_Icc hs ht
+
+theorem cardyWeight_nonneg {u : ℝ} (hu : u ∈ Icc 0 1) : 0 ≤ cardyWeight u := by
+  exact Real.rpow_nonneg (mul_nonneg hu.1 (sub_nonneg.mpr hu.2)) _
+
+theorem cardyWeight_pos {u : ℝ} (hu : u ∈ Ioo 0 1) : 0 < cardyWeight u := by
+  exact Real.rpow_pos_of_pos (mul_pos hu.1 (sub_pos.mpr hu.2)) _
+
+lemma cardyWeight_symm (u : ℝ) : cardyWeight (1 - u) = cardyWeight u := by
+  unfold cardyWeight
+  congr 1
+  ring
+
+theorem cardyIntegral_continuousOn : ContinuousOn cardyIntegral (Icc 0 1) := by
+  have h := intervalIntegral.continuousOn_primitive_interval
+    ((intervalIntegrable_iff' (by finiteness)).mp cardyWeight_integrable)
+  rw [uIcc_of_le zero_le_one] at h
+  exact h.div_const _
+
+theorem cardyIntegral_strictMonoOn : StrictMonoOn cardyIntegral (Icc 0 1) := by
+  intro a ha b hb hab
+  have h0a := cardyWeight_integrable_subinterval (s := 0) le_rfl ha.2 ha.1
+  have habi := cardyWeight_integrable_subinterval ha.1 hb.2 hab.le
+  have hp : 0 < ∫ u in a..b, cardyWeight u := by
+    apply intervalIntegral.intervalIntegral_pos_of_pos_on habi _ hab
+    intro u hu
+    exact cardyWeight_pos ⟨ha.1.trans_lt hu.1, hu.2.trans_le hb.2⟩
+  have hadd := intervalIntegral.integral_add_adjacent_intervals h0a habi
+  apply (div_lt_div_iff_of_pos_right cardyDenominator_pos).mpr
+  linarith
+
+theorem cardyIntegral_mem_Icc {x : ℝ} (hx : x ∈ Icc 0 1) :
+    cardyIntegral x ∈ Icc 0 1 := by
+  have h := cardyIntegral_strictMonoOn.monotoneOn
+  constructor
+  · simpa using h (show (0 : ℝ) ∈ Icc 0 1 by simp) hx hx.1
+  · simpa using h hx (show (1 : ℝ) ∈ Icc 0 1 by simp) hx.2
+
+theorem cardyIntegral_mem_Ioo {x : ℝ} (hx : x ∈ Ioo 0 1) :
+    cardyIntegral x ∈ Ioo 0 1 := by
+  constructor
+  · simpa using cardyIntegral_strictMonoOn (show (0 : ℝ) ∈ Icc 0 1 by simp)
+      ⟨hx.1.le, hx.2.le⟩ hx.1
+  · simpa using cardyIntegral_strictMonoOn ⟨hx.1.le, hx.2.le⟩
+      (show (1 : ℝ) ∈ Icc 0 1 by simp) hx.2
+
+theorem cardyIntegral_reflection {x : ℝ} (hx : x ∈ Icc 0 1) :
+    cardyIntegral (1 - x) = 1 - cardyIntegral x := by
+  have hx' : 1 - x ∈ Icc (0 : ℝ) 1 := ⟨by linarith [hx.2], by linarith [hx.1]⟩
+  have h0 := cardyWeight_integrable_subinterval (s := 0) le_rfl hx'.2 hx'.1
+  have h1 := cardyWeight_integrable_subinterval hx'.1 le_rfl hx'.2
+  have hsub : (∫ u in 0..x, cardyWeight u) = ∫ u in (1 - x)..1, cardyWeight u := by
+    simpa only [cardyWeight_symm, sub_zero] using
+      intervalIntegral.integral_comp_sub_left (a := 0) (b := x) cardyWeight 1
+  apply (eq_sub_iff_add_eq).mpr
+  unfold cardyIntegral
+  rw [← add_div, hsub, intervalIntegral.integral_add_adjacent_intervals h0 h1]
+  exact div_self (ne_of_gt cardyDenominator_pos)
+
+@[simp] theorem cardyIntegral_half : cardyIntegral (1 / 2) = 1 / 2 := by
+  have h := cardyIntegral_reflection (x := 1 / 2) (by norm_num)
+  norm_num only [show (1 : ℝ) - 1 / 2 = 1 / 2 by norm_num] at h
+  linarith
+
+def cardyBinomialCoefficient (n : ℕ) : ℝ :=
+  (n.factorial : ℝ)⁻¹ * (ascPochhammer ℝ n).eval (2 / 3)
+
+lemma cardyBinomialCoefficient_pos (n : ℕ) : 0 < cardyBinomialCoefficient n := by
+  exact mul_pos (inv_pos.mpr (by positivity)) (ascPochhammer_pos n _ (by norm_num))
+
+lemma cardyBinomialCoefficient_eq_choose (n : ℕ) :
+    cardyBinomialCoefficient n = Ring.choose ((2 / 3 : ℝ) + n - 1) n := by
+  rw [← Ring.multichoose_eq]
+  have h := Ring.factorial_nsmul_multichoose_eq_ascPochhammer (2 / 3 : ℝ) n
+  rw [nsmul_eq_mul, Polynomial.ascPochhammer_smeval_eq_eval] at h
+  rw [cardyBinomialCoefficient, ← h, ← mul_assoc, inv_mul_cancel₀, one_mul]
+  exact_mod_cast n.factorial_ne_zero
+
+lemma cardy_binomial_hasSum {u : ℝ} (hu : u ∈ Ioo 0 1) :
+    HasSum (fun n : ℕ => cardyBinomialCoefficient n * u ^ n)
+      ((1 - u) ^ (-(2 : ℝ) / 3)) := by
+  have h := (Real.one_div_one_sub_rpow_hasFPowerSeriesOnBall_zero (2 / 3)).hasSum
+    (y := u) (by
+      rw [← ENNReal.ofReal_one, Metric.eball_ofReal]
+      simpa [Real.dist_eq, abs_of_pos hu.1] using hu.2)
+  simpa only [FormalMultilinearSeries.ofScalars_apply_eq, smul_eq_mul,
+    ← cardyBinomialCoefficient_eq_choose, zero_add, one_div, neg_div,
+    Real.rpow_neg (sub_pos.mpr hu.2).le]
+    using h
+
+lemma cardyWeight_hasSum {u : ℝ} (hu : u ∈ Ioo 0 1) :
+    HasSum (fun n : ℕ => cardyBinomialCoefficient n * u ^ ((n : ℝ) - 2 / 3))
+      (cardyWeight u) := by
+  have h := (cardy_binomial_hasSum hu).mul_left (u ^ (-(2 : ℝ) / 3))
+  have heq (n : ℕ) : cardyBinomialCoefficient n * u ^ ((n : ℝ) - 2 / 3) =
+      u ^ (-(2 : ℝ) / 3) * (cardyBinomialCoefficient n * u ^ n) := by
+    rw [show (n : ℝ) - 2 / 3 = -(2 : ℝ) / 3 + n by ring,
+      Real.rpow_add hu.1, Real.rpow_natCast]
+    ring
+  simpa only [heq, cardyWeight, Real.mul_rpow hu.1.le (sub_pos.mpr hu.2).le] using h
+
+theorem cardyNumerator_hasSum {x : ℝ} (hx : x ∈ Ioo 0 1) :
+    HasSum (fun n : ℕ => ∫ u in 0..x,
+      cardyBinomialCoefficient n * u ^ ((n : ℝ) - 2 / 3))
+      (∫ u in 0..x, cardyWeight u) := by
+  let S : ℕ → ℝ → ℝ := fun n u =>
+    cardyBinomialCoefficient n * u ^ ((n : ℝ) - 2 / 3)
+  have hpoint {u : ℝ} (hu : u ∈ uIoc 0 x) : HasSum (fun n => S n u) (cardyWeight u) := by
+    rw [uIoc_of_le hx.1.le] at hu
+    exact cardyWeight_hasSum ⟨hu.1, hu.2.trans_lt hx.2⟩
+  have hi (n : ℕ) : IntervalIntegrable (S n) volume 0 x := by
+    exact (intervalIntegral.intervalIntegrable_rpow'
+      (r := (n : ℝ) - 2 / 3) (by linarith [Nat.cast_nonneg (α := ℝ) n])).const_mul _
+  have htotal : IntervalIntegrable (fun u => ∑' n, S n u) volume 0 x := by
+    apply IntervalIntegrable.congr (fun u hu => (hpoint hu).tsum_eq.symm)
+    exact cardyWeight_integrable_subinterval le_rfl hx.2.le hx.1.le
+  apply intervalIntegral.hasSum_integral_of_dominated_convergence S
+    (fun n => ((intervalIntegrable_iff).mp (hi n)).1) ?_ ?_ htotal ?_
+  · intro n
+    refine Filter.Eventually.of_forall fun u hu => ?_
+    rw [uIoc_of_le hx.1.le] at hu
+    have hpos : 0 ≤ S n u :=
+      mul_nonneg (cardyBinomialCoefficient_pos n).le (Real.rpow_nonneg hu.1.le _)
+    exact (Real.norm_of_nonneg hpos).le
+  · exact Filter.Eventually.of_forall fun u hu => (hpoint hu).summable
+  · exact Filter.Eventually.of_forall fun u hu => hpoint hu
+
+lemma cardyBinomialTerm_integral (x : ℝ) (n : ℕ) :
+    (∫ u in 0..x, cardyBinomialCoefficient n * u ^ ((n : ℝ) - 2 / 3)) =
+      cardyBinomialCoefficient n * x ^ ((n : ℝ) + 1 / 3) / ((n : ℝ) + 1 / 3) := by
+  rw [intervalIntegral.integral_const_mul, integral_rpow
+    (Or.inl (by linarith [Nat.cast_nonneg (α := ℝ) n]))]
+  rw [show (n : ℝ) - 2 / 3 + 1 = (n : ℝ) + 1 / 3 by ring,
+    Real.zero_rpow (by linarith [Nat.cast_nonneg (α := ℝ) n]), sub_zero]
+  ring
+
+lemma cardyHypergeometricCoefficient_eq (n : ℕ) :
+    cardyBinomialCoefficient n / ((n : ℝ) + 1 / 3) =
+      3 * ordinaryHypergeometricCoefficient (1 / 3 : ℝ) (2 / 3) (4 / 3) n := by
+  have hc : (ascPochhammer ℝ n).eval (4 / 3) ≠ 0 :=
+    ne_of_gt (ascPochhammer_pos n _ (by norm_num))
+  have hn : (n : ℝ) + 1 / 3 ≠ 0 := by linarith [Nat.cast_nonneg (α := ℝ) n]
+  have hshift : (ascPochhammer ℝ n).eval (1 / 3) * ((1 / 3 : ℝ) + n) =
+      (1 / 3 : ℝ) * (ascPochhammer ℝ n).eval (4 / 3) := by
+    rw [← ascPochhammer_succ_eval, ascPochhammer_succ_left]
+    simp only [Polynomial.eval_mul, Polynomial.eval_X, Polynomial.eval_comp,
+      Polynomial.eval_add, Polynomial.eval_one]
+    norm_num
+  have hratio : ((ascPochhammer ℝ n).eval (1 / 3) /
+      (ascPochhammer ℝ n).eval (4 / 3)) * ((n : ℝ) + 1 / 3) = 1 / 3 := by
+    calc
+      _ = ((ascPochhammer ℝ n).eval (1 / 3) * ((1 / 3 : ℝ) + n)) /
+          (ascPochhammer ℝ n).eval (4 / 3) := by ring
+      _ = ((1 / 3 : ℝ) * (ascPochhammer ℝ n).eval (4 / 3)) /
+          (ascPochhammer ℝ n).eval (4 / 3) := by rw [hshift]
+      _ = 1 / 3 := mul_div_cancel_right₀ _ hc
+  symm
+  apply (eq_div_iff hn).mpr
+  calc
+    _ = (3 * (((ascPochhammer ℝ n).eval (1 / 3) /
+          (ascPochhammer ℝ n).eval (4 / 3)) * ((n : ℝ) + 1 / 3))) *
+        cardyBinomialCoefficient n := by
+      unfold ordinaryHypergeometricCoefficient cardyBinomialCoefficient
+      ring
+    _ = cardyBinomialCoefficient n := by rw [hratio]; ring
+
+theorem cardyNumerator_eq_hypergeometric {x : ℝ} (hx : x ∈ Ioo 0 1) :
+    (∫ u in 0..x, cardyWeight u) =
+      3 * x ^ ((1 : ℝ) / 3) *
+        ordinaryHypergeometric (1 / 3 : ℝ) (2 / 3) (4 / 3) x := by
+  rw [← (cardyNumerator_hasSum hx).tsum_eq, ordinaryHypergeometric_eq_tsum]
+  simp only [smul_eq_mul]
+  rw [← tsum_mul_left]
+  apply tsum_congr
+  intro n
+  rw [cardyBinomialTerm_integral, Real.rpow_add hx.1, Real.rpow_natCast]
+  calc
+    _ = (cardyBinomialCoefficient n / ((n : ℝ) + 1 / 3)) *
+        (x ^ n * x ^ ((1 : ℝ) / 3)) := by ring
+    _ = _ := by rw [cardyHypergeometricCoefficient_eq]; ring
+
+theorem cardyIntegral_eq_hypergeometric {x : ℝ} (hx : x ∈ Ioo 0 1) :
+    cardyIntegral x = cardyHypergeometric x := by
+  have hγ : Real.Gamma (4 / 3) = (1 / 3 : ℝ) * Real.Gamma (1 / 3) := by
+    convert Real.Gamma_add_one (s := 1 / 3) (by norm_num) using 1
+    norm_num
+  have h₁ : Real.Gamma (1 / 3) ≠ 0 := ne_of_gt (Real.Gamma_pos_of_pos (by norm_num))
+  have h₂ : Real.Gamma (2 / 3) ≠ 0 := ne_of_gt (Real.Gamma_pos_of_pos (by norm_num))
+  rw [cardyIntegral, cardyNumerator_eq_hypergeometric hx,
+    cardyDenominator_eq_gamma, cardyHypergeometric, hγ]
+  field_simp
+
+end Cardy
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/AlperinWeights/Brauer.lean b/lean/OAI/RepresentationTheory/AlperinWeights/Brauer.lean
index 67bc7a9dab8934b5db602829da65dfa3302f300a..12c99a8e3af19ccd9a9aeddf6c0698f09b684d16 100644
--- a/lean/OAI/RepresentationTheory/AlperinWeights/Brauer.lean
+++ b/lean/OAI/RepresentationTheory/AlperinWeights/Brauer.lean
@@ -1,560 +1,560 @@
-import Mathlib
-
-namespace OAI
-
-/-! Brauer projection and foundations for blockwise Alperin weights. -/
-
-noncomputable section
-namespace BlockwiseAW
-abbrev AlgebraicIntegers := integralClosure ℤ ℂ
-
-lemma exists_integer_reduction (p : ℕ) [Fact p.Prime]
-    (k : Type*) [Field k] [IsAlgClosed k] [CharP k p] :
-    Nonempty (AlgebraicIntegers →+* k) := by
-  let P : Ideal ℤ := RingHom.ker (Int.castRingHom (ZMod p))
-  have hsur : Function.Surjective (Int.castRingHom (ZMod p)) := by
-    intro x
-    exact ⟨x.val, by simp⟩
-  have : P.IsMaximal := RingHom.ker_isMaximal_of_surjective _ hsur
-  have hker : RingHom.ker (algebraMap ℤ AlgebraicIntegers) ≤ P := by
-    rw [(RingHom.injective_iff_ker_eq_bot _).mp
-      (FaithfulSMul.algebraMap_injective ℤ AlgebraicIntegers)]
-    exact bot_le
-  obtain ⟨Q, hQ, hQP⟩ :=
-    Ideal.exists_ideal_over_maximal_of_isIntegral (S := AlgebraicIntegers) P hker
-  have : Q.IsMaximal := hQ
-  let P' := Q.comap (algebraMap ℤ AlgebraicIntegers)
-  have : P'.IsMaximal := by
-    change (Q.under ℤ).IsMaximal
-    rw [hQP]
-    infer_instance
-  let R := ℤ ⧸ P'
-  let S := AlgebraicIntegers ⧸ Q
-  let : Field R := Ideal.Quotient.field P'
-  let : Field S := Ideal.Quotient.field Q
-  let : Algebra R S := Ideal.quotientAlgebra
-  have : Algebra.IsIntegral R S := Algebra.IsIntegral.quotient
-  have : Algebra.IsAlgebraic R S := Algebra.IsIntegral.isAlgebraic
-  let f : R →+* k := Ideal.Quotient.lift P' (Int.castRingHom k) (by
-    intro x hx
-    change (x : k) = 0
-    rw [CharP.intCast_eq_zero_iff k p]
-    have hx' : (x : ZMod p) = 0 := by
-      change x ∈ P
-      rw [← hQP]
-      exact hx
-    exact (CharP.intCast_eq_zero_iff (ZMod p) p x).mp hx')
-  let : Algebra R k := f.toAlgebra
-  let lift : S →ₐ[R] k := IsAlgClosed.lift
-  exact ⟨lift.toRingHom.comp (Ideal.Quotient.mk Q)⟩
-
-def integerReduction (p : ℕ) [Fact p.Prime]
-    (k : Type*) [Field k] [IsAlgClosed k] [CharP k p] :
-    AlgebraicIntegers →+* k :=
-  Classical.choice (exists_integer_reduction p k)
-
-section OrbitCancellation
-open scoped BigOperators
-open MulAction
-
-theorem invariant_sum_eq_fixedPoint_sum
-    {p : ℕ} [Fact p.Prime] {P α R : Type*} [Group P] [MulAction P α]
-    [Fintype α] [Fintype (fixedPoints P α)] [CommSemiring R] [CharP R p] (hP : IsPGroup p P)
-    (f : α → R) (hf : ∀ (g : P) (x : α), f (g • x) = f x) :
-    (∑ x : α, f x) = ∑ x : fixedPoints P α, f x.val := by
-  classical
-  have key (x : α) :
-      (∑ y : { y // (Quotient.mk'' y : Quotient (orbitRel P α)) = Quotient.mk'' x }, f y.val) =
-        (Fintype.card (orbit P x) : R) * f x := by
-    have hval (y : { y // (Quotient.mk'' y : Quotient (orbitRel P α)) = Quotient.mk'' x }) :
-        f y.val = f x := by
-      obtain ⟨g, hg⟩ := mem_orbit_iff.mp (Quotient.exact' y.property)
-      rw [← hg]
-      exact hf g x
-    simp_rw [hval]
-    have hcard : Fintype.card { y // (Quotient.mk'' y : Quotient (orbitRel P α)) = Quotient.mk'' x } =
-        Fintype.card (orbit P x) := by
-      simp only [Quotient.eq'']; congr
-    rw [Finset.sum_const, Finset.card_univ, nsmul_eq_mul, hcard]
-  calc
-    (∑ x : α, f x) = ∑ a : Quotient (orbitRel P α),
-        ∑ y : {y // Quotient.mk'' y = a}, f y.val := by
-      rw [← Fintype.sum_sigma' (fun (a : Quotient (orbitRel P α)) (y : {y // Quotient.mk'' y = a}) => f y.val)]
-      exact (Fintype.sum_equiv (Equiv.sigmaFiberEquiv (@Quotient.mk'' _ (orbitRel P α))) _ _
-        (fun _ => rfl)).symm
-    _ = ∑ x : fixedPoints P α, f x.val := by
-      symm
-      refine Finset.sum_bij_ne_zero (fun a _ _ => Quotient.mk'' a.val)
-        (fun _ _ _ => Finset.mem_univ _)
-        (fun a₁ _ _ a₂ _ _ h =>
-          Subtype.ext (mem_fixedPoints'.mp a₂.property a₁.val (Quotient.exact' h)))
-        (fun b => Quotient.inductionOn' b fun b _ hb => by
-        obtain ⟨n, hn⟩ := hP.card_orbit b
-        rw [Nat.card_eq_fintype_card] at hn
-        have hn0 : n = 0 := by
-          by_contra hne
-          have hp : (p : R) = 0 := CharP.cast_eq_zero R p
-          apply hb
-          rw [key, hn, Nat.cast_pow, hp, zero_pow hne, zero_mul]
-        have hbfix : b ∈ fixedPoints P α :=
-          mem_fixedPoints_iff_card_orbit_eq_one.mpr (by rw [hn, hn0, pow_zero])
-        refine ⟨⟨b, hbfix⟩, Finset.mem_univ _, ?_, rfl⟩
-        simpa [key, hn, hn0] using hb
-      ) (fun a _ _ => by
-        rw [key, mem_fixedPoints_iff_card_orbit_eq_one.mp a.property]
-        simp)
-
-end OrbitCancellation
-
-section GroupAlgebra
-variable {R G : Type*} [CommSemiring R] [Group G]
-
-abbrev Center (R G : Type*) [CommSemiring R] [Group G] :=
-  Subalgebra.center R (MonoidAlgebra R G)
-
-theorem mem_center_iff_coeff_conj (a : MonoidAlgebra R G) :
-    a ∈ Center R G ↔ ∀ g x : G, a.coeff (g * x * g⁻¹) = a.coeff x := by
-  rw [Subalgebra.mem_center_iff]
-  constructor
-  · intro h g x
-    have eq := congrArg (fun b : MonoidAlgebra R G => b.coeff (g * x))
-      (h (MonoidAlgebra.single g 1))
-    simpa using eq.symm
-  · intro h b
-    induction b using MonoidAlgebra.induction_on with
-    | of g =>
-      ext x
-      simpa [MonoidAlgebra.of_apply, mul_assoc] using (h g (g⁻¹ * x)).symm
-    | add b c hb hc => simp only [add_mul, mul_add, hb, hc]
-    | smul r b hb => simp only [smul_mul_assoc, mul_smul_comm, hb]
-
-def coefficientRestriction (H : Subgroup G) :
-    MonoidAlgebra R G →ₗ[R] MonoidAlgebra R H where
-  toFun a := .ofCoeff (a.coeff.comapDomain Subtype.val Subtype.val_injective.injOn)
-  map_add' a b := by ext; rfl
-  map_smul' r a := by ext; rfl
-
-@[simp] theorem coefficientRestriction_coeff (H : Subgroup G)
-    (a : MonoidAlgebra R G) (h : H) :
-    (coefficientRestriction H a).coeff h = a.coeff h := rfl
-
-theorem coefficientRestriction_mem_center (H : Subgroup G)
-    (a : MonoidAlgebra R G) (ha : a ∈ Center R G) :
-    coefficientRestriction H a ∈ Center R H := by
-  rw [mem_center_iff_coeff_conj] at ha ⊢
-  intro g x
-  exact ha g x
-
-def centerRestriction (H : Subgroup G) : Center R G →ₗ[R] Center R H where
-  toFun z := ⟨coefficientRestriction H z.val,
-    coefficientRestriction_mem_center H z.val z.property⟩
-  map_add' _ _ := Subtype.ext (map_add (coefficientRestriction H) _ _)
-  map_smul' _ _ := Subtype.ext (map_smul (coefficientRestriction H) _ _)
-
-def classSum [Finite G] (g : G) : Center R G := by
-  classical
-  refine ⟨.ofCoeff (Finsupp.equivFunOnFinite.symm (fun x => if IsConj g x then 1 else 0)), ?_⟩
-  rw [mem_center_iff_coeff_conj]
-  intro h x
-  change (if IsConj g (h * x * h⁻¹) then (1 : R) else 0) =
-    (if IsConj g x then (1 : R) else 0)
-  have hc : IsConj x (h * x * h⁻¹) := isConj_iff.mpr ⟨h, rfl⟩
-  have hi : IsConj g (h * x * h⁻¹) ↔ IsConj g x :=
-    ⟨fun t => t.trans hc.symm, fun t => t.trans hc⟩
-  simp only [hi]
-
-end GroupAlgebra
-
-section BrauerProjection
-open scoped BigOperators
-variable {R G : Type*} [CommSemiring R] [Group G]
-
-def conjugationInvariants (R : Type*) [CommSemiring R] (Q : Subgroup G) :
-    Subalgebra R (MonoidAlgebra R G) :=
-  Subalgebra.centralizer R (Set.range (fun q : Q => MonoidAlgebra.single (q : G) (1 : R)))
-
-theorem mem_conjugationInvariants_iff (Q : Subgroup G) (a : MonoidAlgebra R G) :
-    a ∈ conjugationInvariants R Q ↔
-      ∀ (q : Q) (x : G), a.coeff ((q : G) * x * (q : G)⁻¹) = a.coeff x := by
-  rw [conjugationInvariants, Subalgebra.mem_centralizer_iff]
-  constructor
-  · intro h q x
-    have eq := congrArg (fun b : MonoidAlgebra R G => b.coeff ((q : G) * x))
-      (h (MonoidAlgebra.single (q : G) 1) ⟨q, rfl⟩)
-    simpa using eq.symm
-  · rintro h _ ⟨q, rfl⟩
-    ext x
-    simpa [mul_assoc] using (h q ((q : G)⁻¹ * x)).symm
-
-theorem coeff_mul_sum [Fintype G] (a b : MonoidAlgebra R G) (g : G) :
-    (a * b).coeff g = ∑ x : G, a.coeff x * b.coeff (x⁻¹ * g) := by
-  rw [MonoidAlgebra.coeff_mul_apply_left, Finsupp.sum_fintype]
-  intro x
-  exact zero_mul _
-
-@[simp] theorem coefficientRestriction_one (H : Subgroup G) :
-    coefficientRestriction (R := R) H 1 = 1 := by
-  classical
-  ext x
-  simp [MonoidAlgebra.one_def, Finsupp.single_apply, eq_comm]
-
-variable [Finite G]
-
-theorem coefficientRestriction_mul_of_invariant
-    {p : ℕ} [Fact p.Prime] [CharP R p] (Q : Subgroup G) (hQ : IsPGroup p Q)
-    (a b : MonoidAlgebra R G) (ha : a ∈ conjugationInvariants R Q)
-    (hb : b ∈ conjugationInvariants R Q) :
-    coefficientRestriction (Subgroup.centralizer (Q : Set G)) (a * b) =
-      coefficientRestriction (Subgroup.centralizer (Q : Set G)) a *
-        coefficientRestriction (Subgroup.centralizer (Q : Set G)) b := by
-  classical
-  let : Fintype G := Fintype.ofFinite G
-  let : MulAction Q G := MulAction.compHom G (ConjAct.toConjAct.toMonoidHom.comp Q.subtype)
-  have hfixed : MulAction.fixedPoints Q G = (Subgroup.centralizer (Q : Set G) : Set G) := by
-    ext g
-    change (g ∈ MulAction.fixedPoints Q G) ↔ g ∈ Subgroup.centralizer (Q : Set G)
-    rw [MulAction.mem_fixedPoints, Subgroup.mem_centralizer_iff]
-    constructor
-    · intro h q hq
-      have hq' := h ⟨q, hq⟩
-      change q * g * q⁻¹ = g at hq'
-      exact (mul_inv_eq_iff_eq_mul).mp hq'
-    · intro h q
-      change (q : G) * g * (q : G)⁻¹ = g
-      exact (mul_inv_eq_iff_eq_mul).mpr (h q q.property)
-  rw [mem_conjugationInvariants_iff] at ha hb
-  ext g
-  rw [coefficientRestriction_coeff, coeff_mul_sum, coeff_mul_sum]
-  simp only [coefficientRestriction_coeff]
-  let f : G → R := fun x => a.coeff x * b.coeff (x⁻¹ * (g : G))
-  have hf : ∀ (q : Q) (x : G), f (q • x) = f x := by
-    intro q x
-    have hg : (q : G) * (g : G) * (q : G)⁻¹ = g :=
-      (mul_inv_eq_iff_eq_mul).mpr (Subgroup.mem_centralizer_iff.mp g.property q q.property)
-    have he : ((q : G) * x * (q : G)⁻¹)⁻¹ * (g : G) =
-        (q : G) * (x⁻¹ * (g : G)) * (q : G)⁻¹ := by
-      calc
-        _ = ((q : G) * x * (q : G)⁻¹)⁻¹ * ((q : G) * (g : G) * (q : G)⁻¹) := by rw [hg]
-        _ = _ := by group
-    change a.coeff ((q : G) * x * (q : G)⁻¹) *
-      b.coeff (((q : G) * x * (q : G)⁻¹)⁻¹ * (g : G)) = f x
-    rw [ha q x, he, hb q (x⁻¹ * (g : G))]
-  have hs := invariant_sum_eq_fixedPoint_sum hQ f hf
-  change (∑ x : G, f x) = ∑ x : Subgroup.centralizer (Q : Set G), f x.val
-  rw [hs]
-  exact (Fintype.sum_equiv (Set.equivOfEq hfixed) _ _ (fun _ => rfl))
-
-def brauerHom {p : ℕ} [Fact p.Prime] [CharP R p]
-    (Q : Subgroup G) (hQ : IsPGroup p Q) :
-    conjugationInvariants R Q →ₐ[R] MonoidAlgebra R (Subgroup.centralizer (Q : Set G)) where
-  toFun a := coefficientRestriction (Subgroup.centralizer (Q : Set G)) a.val
-  map_zero' := map_zero _
-  map_one' := coefficientRestriction_one _
-  map_add' a b := map_add _ a.val b.val
-  map_mul' a b := coefficientRestriction_mul_of_invariant Q hQ a.val b.val a.property b.property
-  commutes' r := by
-    change coefficientRestriction _ (algebraMap R (MonoidAlgebra R G) r) = _
-    simp only [Algebra.algebraMap_eq_smul_one, map_smul, coefficientRestriction_one]
-
-end BrauerProjection
-
-theorem finite_simple_quotients {R M ι : Type*} [Ring R]
-    [AddCommGroup M] [Module R M]
-    (S : ι → Type*) [∀ i, AddCommGroup (S i)] [∀ i, Module R (S i)]
-    [∀ i, IsSimpleModule R (S i)]
-    (hS : ∀ i j, Nonempty (S i ≃ₗ[R] S j) → i = j)
-    (hM : IsFiniteLength R M) :
-    Set.Finite {i | ∃ f : M →ₗ[R] S i, f ≠ 0} := by
-  classical
-  induction hM with
-  | of_subsingleton =>
-    convert Set.finite_empty (α := ι) using 1
-    ext i
-    simp only [Set.mem_ofPred_eq, Set.mem_empty_iff_false, iff_false, not_exists, not_not]
-    intro f
-    ext x
-    rw [Subsingleton.elim x 0, map_zero, LinearMap.zero_apply]
-  | @of_simple_quotient M _ _ N _ hN ih =>
-    have hsimple : Set.Subsingleton {i | Nonempty ((M ⧸ N) ≃ₗ[R] S i)} := by
-      rintro i ⟨e⟩ j ⟨f⟩
-      exact hS i j ⟨e.symm.trans f⟩
-    refine (ih.union hsimple.finite).subset ?_
-    rintro i ⟨f, hf⟩
-    by_cases hn : f.comp N.subtype = 0
-    · right
-      have hker : N ≤ LinearMap.ker f := by
-        intro x hx
-        have hh := congrArg (fun g : N →ₗ[R] S i => g ⟨x, hx⟩) hn
-        exact hh
-      let g : (M ⧸ N) →ₗ[R] S i := N.liftQ f hker
-      have hg : g ≠ 0 := by
-        intro hg0
-        apply hf
-        ext x
-        exact congrArg (fun t : (M ⧸ N) →ₗ[R] S i => t (N.mkQ x)) hg0
-      exact ⟨LinearEquiv.ofBijective g ((LinearMap.bijective_or_eq_zero g).resolve_right hg)⟩
-    · exact Or.inl ⟨f.comp N.subtype, hn⟩
-
-section Blocks
-variable (k G : Type*) [Field k] [Group G]
-
-structure Block where
-  idempotent : Center k G
-  isIdempotent : idempotent * idempotent = idempotent
-  nonzero : idempotent ≠ 0
-  primitive : ∀ c : Center k G, c * c = c → c * idempotent = c →
-    c = 0 ∨ c = idempotent
-
-structure SimpleRepresentation where
-  dim : ℕ
-  representation : Representation k G (Fin dim → k)
-  irreducible : representation.IsIrreducible
-
-variable {k G}
-
-theorem irreducible_moduleFinite {k G V : Type*} [Field k] [Group G] [Finite G]
-    [AddCommGroup V] [Module k V] (ρ : Representation k G V) [ρ.IsIrreducible] :
-    Module.Finite k V := by
-  have : Module.Finite (MonoidAlgebra k G) ρ.asModule := inferInstance
-  have : Module.Finite k ρ.asModule := Module.Finite.trans (MonoidAlgebra k G) ρ.asModule
-  exact Module.Finite.equiv ρ.asModuleEquiv
-
-theorem exists_coordinate_simple {k G V : Type*} [Field k] [Group G] [Finite G]
-    [AddCommGroup V] [Module k V] (ρ : Representation k G V) [ρ.IsIrreducible] :
-    ∃ S : SimpleRepresentation k G, Nonempty (Representation.Equiv ρ S.representation) := by
-  let : Module.Finite k V := irreducible_moduleFinite ρ
-  let n := Module.finrank k V
-  let e : V ≃ₗ[k] (Fin n → k) := (Module.finBasis k V).equivFun
-  let σ : Representation k G (Fin n → k) := e.conjRingEquiv.toMonoidHom.comp ρ
-  let E : Representation.Equiv ρ σ := Representation.Equiv.mk e (by
-    intro g
-    ext x
-    simp [σ, LinearEquiv.conjRingEquiv])
-  let f : ρ.asModule →ₗ[MonoidAlgebra k G] σ.asModule :=
-    Representation.IntertwiningMap.equivLinearMapAsModule ρ σ E.toIntertwiningMap
-  have hσ : σ.IsIrreducible := by
-    rw [Representation.irreducible_iff_isSimpleModule_asModule]
-    exact (f.isSimpleModule_iff_of_bijective e.bijective).mp inferInstance
-  exact ⟨⟨n, σ, hσ⟩, ⟨E⟩⟩
-
-def SimpleRepresentation.BelongsTo (V : SimpleRepresentation k G) (B : Block k G) : Prop :=
-  V.representation.asAlgebraHom B.idempotent.val = 1
-
-def simpleModuleSetoid (B : Block k G) :
-    Setoid {V : SimpleRepresentation k G // V.BelongsTo B} where
-  r V W := Nonempty (Representation.Equiv V.val.representation W.val.representation)
-  iseqv := by
-    constructor
-    · intro V; exact ⟨Representation.Equiv.refl _⟩
-    · rintro V W ⟨e⟩; exact ⟨e.symm⟩
-    · rintro U V W ⟨e⟩ ⟨f⟩; exact ⟨e.trans f⟩
-
-def SimpleModuleClasses (B : Block k G) := Quotient (simpleModuleSetoid B)
-
-instance finiteSimpleModuleClasses {k G : Type*} [Field k] [Group G] [Finite G]
-    (B : Block k G) : Finite (SimpleModuleClasses B) := by
-  classical
-  let R := MonoidAlgebra k G
-  let V (q : SimpleModuleClasses B) := (Quotient.out q).val
-  let S (q : SimpleModuleClasses B) := (V q).representation.asModule
-  let (q : SimpleModuleClasses B) : (V q).representation.IsIrreducible := (V q).irreducible
-  let (q : SimpleModuleClasses B) : IsSimpleModule R (S q) := inferInstance
-  have hS (q t : SimpleModuleClasses B) (h : Nonempty (S q ≃ₗ[R] S t)) : q = t := by
-    obtain ⟨e⟩ := h
-    let f := (Representation.IntertwiningMap.equivLinearMapAsModule
-      (V q).representation (V t).representation).symm e.toLinearMap
-    have hf : Function.Bijective f := e.bijective
-    have hrel : (simpleModuleSetoid B).r (Quotient.out q) (Quotient.out t) :=
-      ⟨f.ofBijective hf⟩
-    exact (Quotient.out_eq q).symm.trans ((Quotient.sound hrel).trans (Quotient.out_eq t))
-  have hNoeth : IsNoetherian R R := isNoetherian_of_tower k (inferInstance : IsNoetherian k R)
-  have hArtin : IsArtinian R R := isArtinian_of_tower k (inferInstance : IsArtinian k R)
-  have hfinite := finite_simple_quotients S hS
-    (isFiniteLength_iff_isNoetherian_isArtinian.mpr ⟨hNoeth, hArtin⟩)
-  apply Finite.of_finite_univ
-  apply hfinite.subset
-  intro q _
-  have : Nontrivial (S q) := IsSimpleModule.nontrivial R (S q)
-  obtain ⟨x, hx⟩ := exists_ne (0 : S q)
-  refine ⟨LinearMap.toSpanSingleton R (S q) x, ?_⟩
-  intro hzero
-  apply hx
-  simpa using congrArg (fun f : R →ₗ[R] S q => f 1) hzero
-
-def l (B : Block k G) : ℕ := Nat.card (SimpleModuleClasses B)
-
-def IsCentralCharacter (B : Block k G) (lam : Center k G →ₐ[k] k) : Prop :=
-  ∀ V : SimpleRepresentation k G, V.BelongsTo B → ∀ z : Center k G,
-    V.representation.asAlgebraHom z.val =
-      algebraMap k (Module.End k (Fin V.dim → k)) (lam z)
-
-end Blocks
-
-def OrdinaryCharacter (G : Type*) [Group G] :=
-  {χ : G → ℂ // ∃ n : ℕ, ∃ ρ : Representation ℂ G (Fin n → ℂ),
-    ρ.IsIrreducible ∧ ρ.character = χ}
-
-namespace OrdinaryCharacter
-variable {G : Type*} [Group G]
-
-def degree (χ : OrdinaryCharacter G) : ℕ := χ.property.choose
-
-def affording (χ : OrdinaryCharacter G) : Representation ℂ G (Fin χ.degree → ℂ) :=
-  χ.property.choose_spec.choose
-
-instance affording_irreducible (χ : OrdinaryCharacter G) : χ.affording.IsIrreducible :=
-  χ.property.choose_spec.choose_spec.left
-
-@[simp] theorem affording_character (χ : OrdinaryCharacter G) : χ.affording.character = χ.val :=
-  χ.property.choose_spec.choose_spec.right
-
-theorem apply_conj (χ : OrdinaryCharacter G) (g x : G) :
-    χ.val (g * x * g⁻¹) = χ.val x := by
-  rw [← χ.affording_character]
-  exact Representation.char_conj χ.affording x g
-
-theorem degree_spec (χ : OrdinaryCharacter G) : χ.val 1 = (χ.degree : ℂ) := by
-  obtain ⟨ρ, -, hρ⟩ := χ.property.choose_spec
-  rw [← hρ, Representation.char_one]
-  simp [degree]
-
-theorem degree_pos (χ : OrdinaryCharacter G) : 0 < χ.degree := by
-  have : Nontrivial χ.affording.asModule := IsSimpleModule.nontrivial (MonoidAlgebra ℂ G) _
-  have : Nontrivial (Fin χ.degree → ℂ) := χ.affording.asModuleEquiv.symm.toEquiv.nontrivial
-  simpa using Module.finrank_pos (R := ℂ) (M := Fin χ.degree → ℂ)
-
-open scoped Classical in
-
-theorem orthogonality [Fintype G] (χ ψ : OrdinaryCharacter G) :
-    (Nat.card G : ℂ)⁻¹ * ∑ g : G, χ.val g * ψ.val g⁻¹ = if χ = ψ then 1 else 0 := by
-  classical
-  let : Invertible (Nat.card G : ℂ) := invertibleOfNonzero (by exact_mod_cast Nat.card_pos.ne')
-  have he : Nonempty (Representation.Equiv ψ.affording χ.affording) ↔ χ = ψ := by
-    constructor
-    · rintro ⟨e⟩
-      apply Subtype.ext
-      simpa using (Representation.char_iso e).symm
-    · rintro rfl
-      exact ⟨Representation.Equiv.refl _⟩
-  simpa [he] using Representation.char_orthonormal χ.affording ψ.affording
-
-theorem linearIndependent [Finite G] :
-    LinearIndependent ℂ (fun χ : OrdinaryCharacter G => χ.val) := by
-  classical
-  let : Fintype G := Fintype.ofFinite G
-  let pair (ψ : OrdinaryCharacter G) : (G → ℂ) →ₗ[ℂ] ℂ :=
-    { toFun := fun f => (Nat.card G : ℂ)⁻¹ * ∑ g : G, f g * ψ.val g⁻¹
-      map_add' := fun f g => by simp [add_mul, Finset.sum_add_distrib, mul_add]
-      map_smul' := fun c f => by
-        simp [Finset.mul_sum, mul_left_comm, mul_assoc] }
-  have hpair (χ ψ : OrdinaryCharacter G) : pair ψ χ.val = if χ = ψ then 1 else 0 :=
-    orthogonality χ ψ
-  refine linearIndependent_iff'.mpr ?_
-  intro s c hs χ hχ
-  have h := congrArg (pair χ) hs
-  simpa [map_sum, map_smul, hpair, ← Finset.mul_sum, hχ] using h
-
-instance finite [Finite G] : Finite (OrdinaryCharacter G) := linearIndependent.finite
-
-end OrdinaryCharacter
-
-def pPart (p n : ℕ) : ℕ := p ^ n.factorization p
-
-abbrev Normalizer {G : Type*} [Group G] (Q : Subgroup G) : Subgroup G :=
-  Subgroup.normalizer (Q : Set G)
-
-abbrev NormalizerQuotient {G : Type*} [Group G] (Q : Subgroup G) :=
-  (Normalizer Q) ⧸ Q.subgroupOf (Normalizer Q)
-
-structure Weight (p : ℕ) (G : Type*) [Group G] where
-  subgroup : Subgroup G
-  isPSubgroup : IsPGroup p subgroup
-  character : OrdinaryCharacter (NormalizerQuotient subgroup)
-  defectZero : pPart p character.degree = pPart p (Nat.card (NormalizerQuotient subgroup))
-
-namespace Weight
-variable {p : ℕ} {G : Type*} [Group G]
-
-def inflatedCharacter (w : Weight p G) (h : (Normalizer w.subgroup)) : ℂ :=
-  w.character.val (QuotientGroup.mk' (w.subgroup.subgroupOf (Normalizer w.subgroup)) h)
-
-def extendedCharacter (w : Weight p G) (g : G) : ℂ := by
-  classical
-  exact if h : g ∈ (Normalizer w.subgroup) then w.inflatedCharacter ⟨g, h⟩ else 0
-
-def IsConjugate (w v : Weight p G) : Prop :=
-  ∃ g : G,
-    (∀ x : G, x ∈ w.subgroup ↔ g * x * g⁻¹ ∈ v.subgroup) ∧
-    (∀ x : G, w.extendedCharacter x = v.extendedCharacter (g * x * g⁻¹))
-
-theorem isConjugate_refl (w : Weight p G) : w.IsConjugate w := by
-  refine ⟨1, ?_, ?_⟩ <;> simp
-
-theorem isConjugate_symm {w v : Weight p G} (hwv : w.IsConjugate v) :
-    v.IsConjugate w := by
-  obtain ⟨g, hQ, hχ⟩ := hwv
-  refine ⟨g⁻¹, ?_, ?_⟩
-  · intro x
-    simpa [mul_assoc] using (hQ (g⁻¹ * x * g)).symm
-  · intro x
-    simpa [mul_assoc] using (hχ (g⁻¹ * x * g)).symm
-
-theorem isConjugate_trans {w v u : Weight p G}
-    (hwv : w.IsConjugate v) (hvu : v.IsConjugate u) : w.IsConjugate u := by
-  obtain ⟨g, hQ, hχ⟩ := hwv
-  obtain ⟨h, hR, hψ⟩ := hvu
-  refine ⟨h * g, ?_, ?_⟩
-  · intro x
-    simpa [mul_assoc] using (hQ x).trans (hR (g * x * g⁻¹))
-  · intro x
-    simpa [mul_assoc] using (hχ x).trans (hψ (g * x * g⁻¹))
-
-instance finite [Finite G] : Finite (Weight p G) := by
-  let f : Weight p G → Σ Q : Subgroup G, OrdinaryCharacter (NormalizerQuotient Q) :=
-    fun w => ⟨w.subgroup, w.character⟩
-  apply Finite.of_injective f
-  rintro ⟨Q, hQ, χ, hχ⟩ ⟨Q', hQ', χ', hχ'⟩ h
-  obtain ⟨rfl, he⟩ := Sigma.mk.inj_iff.mp h
-  obtain rfl := eq_of_heq he
-  rfl
-
-def conjugacySetoid (p : ℕ) (G : Type*) [Group G] : Setoid (Weight p G) where
-  r := IsConjugate
-  iseqv := ⟨isConjugate_refl, isConjugate_symm, isConjugate_trans⟩
-
-end Weight
-
-section BlockAssignment
-variable {k G : Type*} [Field k] [Group G] [Finite G]
-
-def IsOrdinaryReduction (r : AlgebraicIntegers →+* k) (χ : G → ℂ)
-    (lam : Center k G →ₐ[k] k) : Prop :=
-  ∀ g : G, ∃ a : AlgebraicIntegers,
-    (a : ℂ) = (Nat.card {x : G // IsConj g x} : ℂ) * χ g / χ 1 ∧
-    lam (classSum g) = r a
-
-def AssignedTo (r : AlgebraicIntegers →+* k) {p : ℕ}
-    (w : Weight p G) (B : Block k G) : Prop :=
-  ∃ b : Block k (Normalizer w.subgroup),
-    ∃ lam_b : Center k (Normalizer w.subgroup) →ₐ[k] k,
-    ∃ lam_B : Center k G →ₐ[k] k,
-      IsCentralCharacter b lam_b ∧ IsCentralCharacter B lam_B ∧
-      IsOrdinaryReduction r w.inflatedCharacter lam_b ∧
-      (∀ z : Center k G, lam_b (centerRestriction (Normalizer w.subgroup) z) = lam_B z)
-
-def W (p : ℕ) [Fact p.Prime] [IsAlgClosed k] [CharP k p] (B : Block k G) :=
-  Quotient ((Weight.conjugacySetoid p G).comap
-    (fun w : {w : Weight p G // AssignedTo (integerReduction p k) w B} => w.val))
-
-instance finiteW (p : ℕ) [Fact p.Prime] [IsAlgClosed k] [CharP k p] (B : Block k G) :
-    Finite (W p B) := by
-  unfold W
-  infer_instance
-
-end BlockAssignment
-
-end BlockwiseAW
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-! Brauer projection and foundations for blockwise Alperin weights. -/
+
+noncomputable section
+namespace BlockwiseAW
+abbrev AlgebraicIntegers := integralClosure ℤ ℂ
+
+lemma exists_integer_reduction (p : ℕ) [Fact p.Prime]
+    (k : Type*) [Field k] [IsAlgClosed k] [CharP k p] :
+    Nonempty (AlgebraicIntegers →+* k) := by
+  let P : Ideal ℤ := RingHom.ker (Int.castRingHom (ZMod p))
+  have hsur : Function.Surjective (Int.castRingHom (ZMod p)) := by
+    intro x
+    exact ⟨x.val, by simp⟩
+  have : P.IsMaximal := RingHom.ker_isMaximal_of_surjective _ hsur
+  have hker : RingHom.ker (algebraMap ℤ AlgebraicIntegers) ≤ P := by
+    rw [(RingHom.injective_iff_ker_eq_bot _).mp
+      (FaithfulSMul.algebraMap_injective ℤ AlgebraicIntegers)]
+    exact bot_le
+  obtain ⟨Q, hQ, hQP⟩ :=
+    Ideal.exists_ideal_over_maximal_of_isIntegral (S := AlgebraicIntegers) P hker
+  have : Q.IsMaximal := hQ
+  let P' := Q.comap (algebraMap ℤ AlgebraicIntegers)
+  have : P'.IsMaximal := by
+    change (Q.under ℤ).IsMaximal
+    rw [hQP]
+    infer_instance
+  let R := ℤ ⧸ P'
+  let S := AlgebraicIntegers ⧸ Q
+  let : Field R := Ideal.Quotient.field P'
+  let : Field S := Ideal.Quotient.field Q
+  let : Algebra R S := Ideal.quotientAlgebra
+  have : Algebra.IsIntegral R S := Algebra.IsIntegral.quotient
+  have : Algebra.IsAlgebraic R S := Algebra.IsIntegral.isAlgebraic
+  let f : R →+* k := Ideal.Quotient.lift P' (Int.castRingHom k) (by
+    intro x hx
+    change (x : k) = 0
+    rw [CharP.intCast_eq_zero_iff k p]
+    have hx' : (x : ZMod p) = 0 := by
+      change x ∈ P
+      rw [← hQP]
+      exact hx
+    exact (CharP.intCast_eq_zero_iff (ZMod p) p x).mp hx')
+  let : Algebra R k := f.toAlgebra
+  let lift : S →ₐ[R] k := IsAlgClosed.lift
+  exact ⟨lift.toRingHom.comp (Ideal.Quotient.mk Q)⟩
+
+def integerReduction (p : ℕ) [Fact p.Prime]
+    (k : Type*) [Field k] [IsAlgClosed k] [CharP k p] :
+    AlgebraicIntegers →+* k :=
+  Classical.choice (exists_integer_reduction p k)
+
+section OrbitCancellation
+open scoped BigOperators
+open MulAction
+
+theorem invariant_sum_eq_fixedPoint_sum
+    {p : ℕ} [Fact p.Prime] {P α R : Type*} [Group P] [MulAction P α]
+    [Fintype α] [Fintype (fixedPoints P α)] [CommSemiring R] [CharP R p] (hP : IsPGroup p P)
+    (f : α → R) (hf : ∀ (g : P) (x : α), f (g • x) = f x) :
+    (∑ x : α, f x) = ∑ x : fixedPoints P α, f x.val := by
+  classical
+  have key (x : α) :
+      (∑ y : { y // (Quotient.mk'' y : Quotient (orbitRel P α)) = Quotient.mk'' x }, f y.val) =
+        (Fintype.card (orbit P x) : R) * f x := by
+    have hval (y : { y // (Quotient.mk'' y : Quotient (orbitRel P α)) = Quotient.mk'' x }) :
+        f y.val = f x := by
+      obtain ⟨g, hg⟩ := mem_orbit_iff.mp (Quotient.exact' y.property)
+      rw [← hg]
+      exact hf g x
+    simp_rw [hval]
+    have hcard : Fintype.card { y // (Quotient.mk'' y : Quotient (orbitRel P α)) = Quotient.mk'' x } =
+        Fintype.card (orbit P x) := by
+      simp only [Quotient.eq'']; congr
+    rw [Finset.sum_const, Finset.card_univ, nsmul_eq_mul, hcard]
+  calc
+    (∑ x : α, f x) = ∑ a : Quotient (orbitRel P α),
+        ∑ y : {y // Quotient.mk'' y = a}, f y.val := by
+      rw [← Fintype.sum_sigma' (fun (a : Quotient (orbitRel P α)) (y : {y // Quotient.mk'' y = a}) => f y.val)]
+      exact (Fintype.sum_equiv (Equiv.sigmaFiberEquiv (@Quotient.mk'' _ (orbitRel P α))) _ _
+        (fun _ => rfl)).symm
+    _ = ∑ x : fixedPoints P α, f x.val := by
+      symm
+      refine Finset.sum_bij_ne_zero (fun a _ _ => Quotient.mk'' a.val)
+        (fun _ _ _ => Finset.mem_univ _)
+        (fun a₁ _ _ a₂ _ _ h =>
+          Subtype.ext (mem_fixedPoints'.mp a₂.property a₁.val (Quotient.exact' h)))
+        (fun b => Quotient.inductionOn' b fun b _ hb => by
+        obtain ⟨n, hn⟩ := hP.card_orbit b
+        rw [Nat.card_eq_fintype_card] at hn
+        have hn0 : n = 0 := by
+          by_contra hne
+          have hp : (p : R) = 0 := CharP.cast_eq_zero R p
+          apply hb
+          rw [key, hn, Nat.cast_pow, hp, zero_pow hne, zero_mul]
+        have hbfix : b ∈ fixedPoints P α :=
+          mem_fixedPoints_iff_card_orbit_eq_one.mpr (by rw [hn, hn0, pow_zero])
+        refine ⟨⟨b, hbfix⟩, Finset.mem_univ _, ?_, rfl⟩
+        simpa [key, hn, hn0] using hb
+      ) (fun a _ _ => by
+        rw [key, mem_fixedPoints_iff_card_orbit_eq_one.mp a.property]
+        simp)
+
+end OrbitCancellation
+
+section GroupAlgebra
+variable {R G : Type*} [CommSemiring R] [Group G]
+
+abbrev Center (R G : Type*) [CommSemiring R] [Group G] :=
+  Subalgebra.center R (MonoidAlgebra R G)
+
+theorem mem_center_iff_coeff_conj (a : MonoidAlgebra R G) :
+    a ∈ Center R G ↔ ∀ g x : G, a.coeff (g * x * g⁻¹) = a.coeff x := by
+  rw [Subalgebra.mem_center_iff]
+  constructor
+  · intro h g x
+    have eq := congrArg (fun b : MonoidAlgebra R G => b.coeff (g * x))
+      (h (MonoidAlgebra.single g 1))
+    simpa using eq.symm
+  · intro h b
+    induction b using MonoidAlgebra.induction_on with
+    | of g =>
+      ext x
+      simpa [MonoidAlgebra.of_apply, mul_assoc] using (h g (g⁻¹ * x)).symm
+    | add b c hb hc => simp only [add_mul, mul_add, hb, hc]
+    | smul r b hb => simp only [smul_mul_assoc, mul_smul_comm, hb]
+
+def coefficientRestriction (H : Subgroup G) :
+    MonoidAlgebra R G →ₗ[R] MonoidAlgebra R H where
+  toFun a := .ofCoeff (a.coeff.comapDomain Subtype.val Subtype.val_injective.injOn)
+  map_add' a b := by ext; rfl
+  map_smul' r a := by ext; rfl
+
+@[simp] theorem coefficientRestriction_coeff (H : Subgroup G)
+    (a : MonoidAlgebra R G) (h : H) :
+    (coefficientRestriction H a).coeff h = a.coeff h := rfl
+
+theorem coefficientRestriction_mem_center (H : Subgroup G)
+    (a : MonoidAlgebra R G) (ha : a ∈ Center R G) :
+    coefficientRestriction H a ∈ Center R H := by
+  rw [mem_center_iff_coeff_conj] at ha ⊢
+  intro g x
+  exact ha g x
+
+def centerRestriction (H : Subgroup G) : Center R G →ₗ[R] Center R H where
+  toFun z := ⟨coefficientRestriction H z.val,
+    coefficientRestriction_mem_center H z.val z.property⟩
+  map_add' _ _ := Subtype.ext (map_add (coefficientRestriction H) _ _)
+  map_smul' _ _ := Subtype.ext (map_smul (coefficientRestriction H) _ _)
+
+def classSum [Finite G] (g : G) : Center R G := by
+  classical
+  refine ⟨.ofCoeff (Finsupp.equivFunOnFinite.symm (fun x => if IsConj g x then 1 else 0)), ?_⟩
+  rw [mem_center_iff_coeff_conj]
+  intro h x
+  change (if IsConj g (h * x * h⁻¹) then (1 : R) else 0) =
+    (if IsConj g x then (1 : R) else 0)
+  have hc : IsConj x (h * x * h⁻¹) := isConj_iff.mpr ⟨h, rfl⟩
+  have hi : IsConj g (h * x * h⁻¹) ↔ IsConj g x :=
+    ⟨fun t => t.trans hc.symm, fun t => t.trans hc⟩
+  simp only [hi]
+
+end GroupAlgebra
+
+section BrauerProjection
+open scoped BigOperators
+variable {R G : Type*} [CommSemiring R] [Group G]
+
+def conjugationInvariants (R : Type*) [CommSemiring R] (Q : Subgroup G) :
+    Subalgebra R (MonoidAlgebra R G) :=
+  Subalgebra.centralizer R (Set.range (fun q : Q => MonoidAlgebra.single (q : G) (1 : R)))
+
+theorem mem_conjugationInvariants_iff (Q : Subgroup G) (a : MonoidAlgebra R G) :
+    a ∈ conjugationInvariants R Q ↔
+      ∀ (q : Q) (x : G), a.coeff ((q : G) * x * (q : G)⁻¹) = a.coeff x := by
+  rw [conjugationInvariants, Subalgebra.mem_centralizer_iff]
+  constructor
+  · intro h q x
+    have eq := congrArg (fun b : MonoidAlgebra R G => b.coeff ((q : G) * x))
+      (h (MonoidAlgebra.single (q : G) 1) ⟨q, rfl⟩)
+    simpa using eq.symm
+  · rintro h _ ⟨q, rfl⟩
+    ext x
+    simpa [mul_assoc] using (h q ((q : G)⁻¹ * x)).symm
+
+theorem coeff_mul_sum [Fintype G] (a b : MonoidAlgebra R G) (g : G) :
+    (a * b).coeff g = ∑ x : G, a.coeff x * b.coeff (x⁻¹ * g) := by
+  rw [MonoidAlgebra.coeff_mul_apply_left, Finsupp.sum_fintype]
+  intro x
+  exact zero_mul _
+
+@[simp] theorem coefficientRestriction_one (H : Subgroup G) :
+    coefficientRestriction (R := R) H 1 = 1 := by
+  classical
+  ext x
+  simp [MonoidAlgebra.one_def, Finsupp.single_apply, eq_comm]
+
+variable [Finite G]
+
+theorem coefficientRestriction_mul_of_invariant
+    {p : ℕ} [Fact p.Prime] [CharP R p] (Q : Subgroup G) (hQ : IsPGroup p Q)
+    (a b : MonoidAlgebra R G) (ha : a ∈ conjugationInvariants R Q)
+    (hb : b ∈ conjugationInvariants R Q) :
+    coefficientRestriction (Subgroup.centralizer (Q : Set G)) (a * b) =
+      coefficientRestriction (Subgroup.centralizer (Q : Set G)) a *
+        coefficientRestriction (Subgroup.centralizer (Q : Set G)) b := by
+  classical
+  let : Fintype G := Fintype.ofFinite G
+  let : MulAction Q G := MulAction.compHom G (ConjAct.toConjAct.toMonoidHom.comp Q.subtype)
+  have hfixed : MulAction.fixedPoints Q G = (Subgroup.centralizer (Q : Set G) : Set G) := by
+    ext g
+    change (g ∈ MulAction.fixedPoints Q G) ↔ g ∈ Subgroup.centralizer (Q : Set G)
+    rw [MulAction.mem_fixedPoints, Subgroup.mem_centralizer_iff]
+    constructor
+    · intro h q hq
+      have hq' := h ⟨q, hq⟩
+      change q * g * q⁻¹ = g at hq'
+      exact (mul_inv_eq_iff_eq_mul).mp hq'
+    · intro h q
+      change (q : G) * g * (q : G)⁻¹ = g
+      exact (mul_inv_eq_iff_eq_mul).mpr (h q q.property)
+  rw [mem_conjugationInvariants_iff] at ha hb
+  ext g
+  rw [coefficientRestriction_coeff, coeff_mul_sum, coeff_mul_sum]
+  simp only [coefficientRestriction_coeff]
+  let f : G → R := fun x => a.coeff x * b.coeff (x⁻¹ * (g : G))
+  have hf : ∀ (q : Q) (x : G), f (q • x) = f x := by
+    intro q x
+    have hg : (q : G) * (g : G) * (q : G)⁻¹ = g :=
+      (mul_inv_eq_iff_eq_mul).mpr (Subgroup.mem_centralizer_iff.mp g.property q q.property)
+    have he : ((q : G) * x * (q : G)⁻¹)⁻¹ * (g : G) =
+        (q : G) * (x⁻¹ * (g : G)) * (q : G)⁻¹ := by
+      calc
+        _ = ((q : G) * x * (q : G)⁻¹)⁻¹ * ((q : G) * (g : G) * (q : G)⁻¹) := by rw [hg]
+        _ = _ := by group
+    change a.coeff ((q : G) * x * (q : G)⁻¹) *
+      b.coeff (((q : G) * x * (q : G)⁻¹)⁻¹ * (g : G)) = f x
+    rw [ha q x, he, hb q (x⁻¹ * (g : G))]
+  have hs := invariant_sum_eq_fixedPoint_sum hQ f hf
+  change (∑ x : G, f x) = ∑ x : Subgroup.centralizer (Q : Set G), f x.val
+  rw [hs]
+  exact (Fintype.sum_equiv (Set.equivOfEq hfixed) _ _ (fun _ => rfl))
+
+def brauerHom {p : ℕ} [Fact p.Prime] [CharP R p]
+    (Q : Subgroup G) (hQ : IsPGroup p Q) :
+    conjugationInvariants R Q →ₐ[R] MonoidAlgebra R (Subgroup.centralizer (Q : Set G)) where
+  toFun a := coefficientRestriction (Subgroup.centralizer (Q : Set G)) a.val
+  map_zero' := map_zero _
+  map_one' := coefficientRestriction_one _
+  map_add' a b := map_add _ a.val b.val
+  map_mul' a b := coefficientRestriction_mul_of_invariant Q hQ a.val b.val a.property b.property
+  commutes' r := by
+    change coefficientRestriction _ (algebraMap R (MonoidAlgebra R G) r) = _
+    simp only [Algebra.algebraMap_eq_smul_one, map_smul, coefficientRestriction_one]
+
+end BrauerProjection
+
+theorem finite_simple_quotients {R M ι : Type*} [Ring R]
+    [AddCommGroup M] [Module R M]
+    (S : ι → Type*) [∀ i, AddCommGroup (S i)] [∀ i, Module R (S i)]
+    [∀ i, IsSimpleModule R (S i)]
+    (hS : ∀ i j, Nonempty (S i ≃ₗ[R] S j) → i = j)
+    (hM : IsFiniteLength R M) :
+    Set.Finite {i | ∃ f : M →ₗ[R] S i, f ≠ 0} := by
+  classical
+  induction hM with
+  | of_subsingleton =>
+    convert Set.finite_empty (α := ι) using 1
+    ext i
+    simp only [Set.mem_ofPred_eq, Set.mem_empty_iff_false, iff_false, not_exists, not_not]
+    intro f
+    ext x
+    rw [Subsingleton.elim x 0, map_zero, LinearMap.zero_apply]
+  | @of_simple_quotient M _ _ N _ hN ih =>
+    have hsimple : Set.Subsingleton {i | Nonempty ((M ⧸ N) ≃ₗ[R] S i)} := by
+      rintro i ⟨e⟩ j ⟨f⟩
+      exact hS i j ⟨e.symm.trans f⟩
+    refine (ih.union hsimple.finite).subset ?_
+    rintro i ⟨f, hf⟩
+    by_cases hn : f.comp N.subtype = 0
+    · right
+      have hker : N ≤ LinearMap.ker f := by
+        intro x hx
+        have hh := congrArg (fun g : N →ₗ[R] S i => g ⟨x, hx⟩) hn
+        exact hh
+      let g : (M ⧸ N) →ₗ[R] S i := N.liftQ f hker
+      have hg : g ≠ 0 := by
+        intro hg0
+        apply hf
+        ext x
+        exact congrArg (fun t : (M ⧸ N) →ₗ[R] S i => t (N.mkQ x)) hg0
+      exact ⟨LinearEquiv.ofBijective g ((LinearMap.bijective_or_eq_zero g).resolve_right hg)⟩
+    · exact Or.inl ⟨f.comp N.subtype, hn⟩
+
+section Blocks
+variable (k G : Type*) [Field k] [Group G]
+
+structure Block where
+  idempotent : Center k G
+  isIdempotent : idempotent * idempotent = idempotent
+  nonzero : idempotent ≠ 0
+  primitive : ∀ c : Center k G, c * c = c → c * idempotent = c →
+    c = 0 ∨ c = idempotent
+
+structure SimpleRepresentation where
+  dim : ℕ
+  representation : Representation k G (Fin dim → k)
+  irreducible : representation.IsIrreducible
+
+variable {k G}
+
+theorem irreducible_moduleFinite {k G V : Type*} [Field k] [Group G] [Finite G]
+    [AddCommGroup V] [Module k V] (ρ : Representation k G V) [ρ.IsIrreducible] :
+    Module.Finite k V := by
+  have : Module.Finite (MonoidAlgebra k G) ρ.asModule := inferInstance
+  have : Module.Finite k ρ.asModule := Module.Finite.trans (MonoidAlgebra k G) ρ.asModule
+  exact Module.Finite.equiv ρ.asModuleEquiv
+
+theorem exists_coordinate_simple {k G V : Type*} [Field k] [Group G] [Finite G]
+    [AddCommGroup V] [Module k V] (ρ : Representation k G V) [ρ.IsIrreducible] :
+    ∃ S : SimpleRepresentation k G, Nonempty (Representation.Equiv ρ S.representation) := by
+  let : Module.Finite k V := irreducible_moduleFinite ρ
+  let n := Module.finrank k V
+  let e : V ≃ₗ[k] (Fin n → k) := (Module.finBasis k V).equivFun
+  let σ : Representation k G (Fin n → k) := e.conjRingEquiv.toMonoidHom.comp ρ
+  let E : Representation.Equiv ρ σ := Representation.Equiv.mk e (by
+    intro g
+    ext x
+    simp [σ, LinearEquiv.conjRingEquiv])
+  let f : ρ.asModule →ₗ[MonoidAlgebra k G] σ.asModule :=
+    Representation.IntertwiningMap.equivLinearMapAsModule ρ σ E.toIntertwiningMap
+  have hσ : σ.IsIrreducible := by
+    rw [Representation.irreducible_iff_isSimpleModule_asModule]
+    exact (f.isSimpleModule_iff_of_bijective e.bijective).mp inferInstance
+  exact ⟨⟨n, σ, hσ⟩, ⟨E⟩⟩
+
+def SimpleRepresentation.BelongsTo (V : SimpleRepresentation k G) (B : Block k G) : Prop :=
+  V.representation.asAlgebraHom B.idempotent.val = 1
+
+def simpleModuleSetoid (B : Block k G) :
+    Setoid {V : SimpleRepresentation k G // V.BelongsTo B} where
+  r V W := Nonempty (Representation.Equiv V.val.representation W.val.representation)
+  iseqv := by
+    constructor
+    · intro V; exact ⟨Representation.Equiv.refl _⟩
+    · rintro V W ⟨e⟩; exact ⟨e.symm⟩
+    · rintro U V W ⟨e⟩ ⟨f⟩; exact ⟨e.trans f⟩
+
+def SimpleModuleClasses (B : Block k G) := Quotient (simpleModuleSetoid B)
+
+instance finiteSimpleModuleClasses {k G : Type*} [Field k] [Group G] [Finite G]
+    (B : Block k G) : Finite (SimpleModuleClasses B) := by
+  classical
+  let R := MonoidAlgebra k G
+  let V (q : SimpleModuleClasses B) := (Quotient.out q).val
+  let S (q : SimpleModuleClasses B) := (V q).representation.asModule
+  let (q : SimpleModuleClasses B) : (V q).representation.IsIrreducible := (V q).irreducible
+  let (q : SimpleModuleClasses B) : IsSimpleModule R (S q) := inferInstance
+  have hS (q t : SimpleModuleClasses B) (h : Nonempty (S q ≃ₗ[R] S t)) : q = t := by
+    obtain ⟨e⟩ := h
+    let f := (Representation.IntertwiningMap.equivLinearMapAsModule
+      (V q).representation (V t).representation).symm e.toLinearMap
+    have hf : Function.Bijective f := e.bijective
+    have hrel : (simpleModuleSetoid B).r (Quotient.out q) (Quotient.out t) :=
+      ⟨f.ofBijective hf⟩
+    exact (Quotient.out_eq q).symm.trans ((Quotient.sound hrel).trans (Quotient.out_eq t))
+  have hNoeth : IsNoetherian R R := isNoetherian_of_tower k (inferInstance : IsNoetherian k R)
+  have hArtin : IsArtinian R R := isArtinian_of_tower k (inferInstance : IsArtinian k R)
+  have hfinite := finite_simple_quotients S hS
+    (isFiniteLength_iff_isNoetherian_isArtinian.mpr ⟨hNoeth, hArtin⟩)
+  apply Finite.of_finite_univ
+  apply hfinite.subset
+  intro q _
+  have : Nontrivial (S q) := IsSimpleModule.nontrivial R (S q)
+  obtain ⟨x, hx⟩ := exists_ne (0 : S q)
+  refine ⟨LinearMap.toSpanSingleton R (S q) x, ?_⟩
+  intro hzero
+  apply hx
+  simpa using congrArg (fun f : R →ₗ[R] S q => f 1) hzero
+
+def l (B : Block k G) : ℕ := Nat.card (SimpleModuleClasses B)
+
+def IsCentralCharacter (B : Block k G) (lam : Center k G →ₐ[k] k) : Prop :=
+  ∀ V : SimpleRepresentation k G, V.BelongsTo B → ∀ z : Center k G,
+    V.representation.asAlgebraHom z.val =
+      algebraMap k (Module.End k (Fin V.dim → k)) (lam z)
+
+end Blocks
+
+def OrdinaryCharacter (G : Type*) [Group G] :=
+  {χ : G → ℂ // ∃ n : ℕ, ∃ ρ : Representation ℂ G (Fin n → ℂ),
+    ρ.IsIrreducible ∧ ρ.character = χ}
+
+namespace OrdinaryCharacter
+variable {G : Type*} [Group G]
+
+def degree (χ : OrdinaryCharacter G) : ℕ := χ.property.choose
+
+def affording (χ : OrdinaryCharacter G) : Representation ℂ G (Fin χ.degree → ℂ) :=
+  χ.property.choose_spec.choose
+
+instance affording_irreducible (χ : OrdinaryCharacter G) : χ.affording.IsIrreducible :=
+  χ.property.choose_spec.choose_spec.left
+
+@[simp] theorem affording_character (χ : OrdinaryCharacter G) : χ.affording.character = χ.val :=
+  χ.property.choose_spec.choose_spec.right
+
+theorem apply_conj (χ : OrdinaryCharacter G) (g x : G) :
+    χ.val (g * x * g⁻¹) = χ.val x := by
+  rw [← χ.affording_character]
+  exact Representation.char_conj χ.affording x g
+
+theorem degree_spec (χ : OrdinaryCharacter G) : χ.val 1 = (χ.degree : ℂ) := by
+  obtain ⟨ρ, -, hρ⟩ := χ.property.choose_spec
+  rw [← hρ, Representation.char_one]
+  simp [degree]
+
+theorem degree_pos (χ : OrdinaryCharacter G) : 0 < χ.degree := by
+  have : Nontrivial χ.affording.asModule := IsSimpleModule.nontrivial (MonoidAlgebra ℂ G) _
+  have : Nontrivial (Fin χ.degree → ℂ) := χ.affording.asModuleEquiv.symm.toEquiv.nontrivial
+  simpa using Module.finrank_pos (R := ℂ) (M := Fin χ.degree → ℂ)
+
+open scoped Classical in
+
+theorem orthogonality [Fintype G] (χ ψ : OrdinaryCharacter G) :
+    (Nat.card G : ℂ)⁻¹ * ∑ g : G, χ.val g * ψ.val g⁻¹ = if χ = ψ then 1 else 0 := by
+  classical
+  let : Invertible (Nat.card G : ℂ) := invertibleOfNonzero (by exact_mod_cast Nat.card_pos.ne')
+  have he : Nonempty (Representation.Equiv ψ.affording χ.affording) ↔ χ = ψ := by
+    constructor
+    · rintro ⟨e⟩
+      apply Subtype.ext
+      simpa using (Representation.char_iso e).symm
+    · rintro rfl
+      exact ⟨Representation.Equiv.refl _⟩
+  simpa [he] using Representation.char_orthonormal χ.affording ψ.affording
+
+theorem linearIndependent [Finite G] :
+    LinearIndependent ℂ (fun χ : OrdinaryCharacter G => χ.val) := by
+  classical
+  let : Fintype G := Fintype.ofFinite G
+  let pair (ψ : OrdinaryCharacter G) : (G → ℂ) →ₗ[ℂ] ℂ :=
+    { toFun := fun f => (Nat.card G : ℂ)⁻¹ * ∑ g : G, f g * ψ.val g⁻¹
+      map_add' := fun f g => by simp [add_mul, Finset.sum_add_distrib, mul_add]
+      map_smul' := fun c f => by
+        simp [Finset.mul_sum, mul_left_comm, mul_assoc] }
+  have hpair (χ ψ : OrdinaryCharacter G) : pair ψ χ.val = if χ = ψ then 1 else 0 :=
+    orthogonality χ ψ
+  refine linearIndependent_iff'.mpr ?_
+  intro s c hs χ hχ
+  have h := congrArg (pair χ) hs
+  simpa [map_sum, map_smul, hpair, ← Finset.mul_sum, hχ] using h
+
+instance finite [Finite G] : Finite (OrdinaryCharacter G) := linearIndependent.finite
+
+end OrdinaryCharacter
+
+def pPart (p n : ℕ) : ℕ := p ^ n.factorization p
+
+abbrev Normalizer {G : Type*} [Group G] (Q : Subgroup G) : Subgroup G :=
+  Subgroup.normalizer (Q : Set G)
+
+abbrev NormalizerQuotient {G : Type*} [Group G] (Q : Subgroup G) :=
+  (Normalizer Q) ⧸ Q.subgroupOf (Normalizer Q)
+
+structure Weight (p : ℕ) (G : Type*) [Group G] where
+  subgroup : Subgroup G
+  isPSubgroup : IsPGroup p subgroup
+  character : OrdinaryCharacter (NormalizerQuotient subgroup)
+  defectZero : pPart p character.degree = pPart p (Nat.card (NormalizerQuotient subgroup))
+
+namespace Weight
+variable {p : ℕ} {G : Type*} [Group G]
+
+def inflatedCharacter (w : Weight p G) (h : (Normalizer w.subgroup)) : ℂ :=
+  w.character.val (QuotientGroup.mk' (w.subgroup.subgroupOf (Normalizer w.subgroup)) h)
+
+def extendedCharacter (w : Weight p G) (g : G) : ℂ := by
+  classical
+  exact if h : g ∈ (Normalizer w.subgroup) then w.inflatedCharacter ⟨g, h⟩ else 0
+
+def IsConjugate (w v : Weight p G) : Prop :=
+  ∃ g : G,
+    (∀ x : G, x ∈ w.subgroup ↔ g * x * g⁻¹ ∈ v.subgroup) ∧
+    (∀ x : G, w.extendedCharacter x = v.extendedCharacter (g * x * g⁻¹))
+
+theorem isConjugate_refl (w : Weight p G) : w.IsConjugate w := by
+  refine ⟨1, ?_, ?_⟩ <;> simp
+
+theorem isConjugate_symm {w v : Weight p G} (hwv : w.IsConjugate v) :
+    v.IsConjugate w := by
+  obtain ⟨g, hQ, hχ⟩ := hwv
+  refine ⟨g⁻¹, ?_, ?_⟩
+  · intro x
+    simpa [mul_assoc] using (hQ (g⁻¹ * x * g)).symm
+  · intro x
+    simpa [mul_assoc] using (hχ (g⁻¹ * x * g)).symm
+
+theorem isConjugate_trans {w v u : Weight p G}
+    (hwv : w.IsConjugate v) (hvu : v.IsConjugate u) : w.IsConjugate u := by
+  obtain ⟨g, hQ, hχ⟩ := hwv
+  obtain ⟨h, hR, hψ⟩ := hvu
+  refine ⟨h * g, ?_, ?_⟩
+  · intro x
+    simpa [mul_assoc] using (hQ x).trans (hR (g * x * g⁻¹))
+  · intro x
+    simpa [mul_assoc] using (hχ x).trans (hψ (g * x * g⁻¹))
+
+instance finite [Finite G] : Finite (Weight p G) := by
+  let f : Weight p G → Σ Q : Subgroup G, OrdinaryCharacter (NormalizerQuotient Q) :=
+    fun w => ⟨w.subgroup, w.character⟩
+  apply Finite.of_injective f
+  rintro ⟨Q, hQ, χ, hχ⟩ ⟨Q', hQ', χ', hχ'⟩ h
+  obtain ⟨rfl, he⟩ := Sigma.mk.inj_iff.mp h
+  obtain rfl := eq_of_heq he
+  rfl
+
+def conjugacySetoid (p : ℕ) (G : Type*) [Group G] : Setoid (Weight p G) where
+  r := IsConjugate
+  iseqv := ⟨isConjugate_refl, isConjugate_symm, isConjugate_trans⟩
+
+end Weight
+
+section BlockAssignment
+variable {k G : Type*} [Field k] [Group G] [Finite G]
+
+def IsOrdinaryReduction (r : AlgebraicIntegers →+* k) (χ : G → ℂ)
+    (lam : Center k G →ₐ[k] k) : Prop :=
+  ∀ g : G, ∃ a : AlgebraicIntegers,
+    (a : ℂ) = (Nat.card {x : G // IsConj g x} : ℂ) * χ g / χ 1 ∧
+    lam (classSum g) = r a
+
+def AssignedTo (r : AlgebraicIntegers →+* k) {p : ℕ}
+    (w : Weight p G) (B : Block k G) : Prop :=
+  ∃ b : Block k (Normalizer w.subgroup),
+    ∃ lam_b : Center k (Normalizer w.subgroup) →ₐ[k] k,
+    ∃ lam_B : Center k G →ₐ[k] k,
+      IsCentralCharacter b lam_b ∧ IsCentralCharacter B lam_B ∧
+      IsOrdinaryReduction r w.inflatedCharacter lam_b ∧
+      (∀ z : Center k G, lam_b (centerRestriction (Normalizer w.subgroup) z) = lam_B z)
+
+def W (p : ℕ) [Fact p.Prime] [IsAlgClosed k] [CharP k p] (B : Block k G) :=
+  Quotient ((Weight.conjugacySetoid p G).comap
+    (fun w : {w : Weight p G // AssignedTo (integerReduction p k) w B} => w.val))
+
+instance finiteW (p : ℕ) [Fact p.Prime] [IsAlgClosed k] [CharP k p] (B : Block k G) :
+    Finite (W p B) := by
+  unfold W
+  infer_instance
+
+end BlockAssignment
+
+end BlockwiseAW
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/Block/Defect.lean b/lean/OAI/RepresentationTheory/Block/Defect.lean
index c0f7d79791056507d41f3bc5e73b70b7a38bb11e..3c7a42b98f55bc50ab623cb333af799abfe70f03 100644
--- a/lean/OAI/RepresentationTheory/Block/Defect.lean
+++ b/lean/OAI/RepresentationTheory/Block/Defect.lean
@@ -1,408 +1,408 @@
-import Mathlib
-
-namespace OAI
-
-/-!
-# Blocks, defect groups, and the bounded-order formulation of Donovan's conjecture
-
-This file develops supporting results, not a proof of Donovan's conjecture.
-The algebra of a block is the actual corner `b (KG) b = (KG) b`, with identity `b`.
-Morita equivalence is Mathlib's equivalence of module categories by a linear functor.
-
-We prove finiteness of blocks for a fixed finite-dimensional algebra, existence of
-defect groups, the conditional reduction from bounded-order to fixed-defect Morita
-finiteness, and equality of defect subgroups for blocks corresponding under a
-coefficient-field embedding.
-
-Uniform bounded-defect-order Morita finiteness across varying finite groups remains
-unproved. Block-idempotent descent and bijectivity, preservation of primitivity, and
-scalar extension of Morita equivalences are also not established here.
--/
-noncomputable section
-
-universe u v w
-
-namespace Donovan
-
-section Blocks
-
-variable {A : Type u} [Ring A]
-
-/-- A block is a nonzero primitive idempotent in the center, not necessarily
-a primitive idempotent of the whole ring. -/
-def IsBlockIdempotent (b : A) : Prop :=
-  b ≠ 0 ∧ IsIdempotentElem b ∧ IsMulCentral b ∧
-    ∀ e : A, IsIdempotentElem e → IsMulCentral e → e * b = e → e = 0 ∨ e = b
-
-/-- The block idempotents in a ring. -/
-def BlockIdempotent (A : Type u) [Ring A] := {b : A // IsBlockIdempotent b}
-
-namespace BlockIdempotent
-
-instance : Coe (BlockIdempotent A) A := ⟨Subtype.val⟩
-
-@[ext] theorem ext {b c : BlockIdempotent A} (h : (b : A) = c) : b = c :=
-  Subtype.ext h
-
-theorem ne_zero (b : BlockIdempotent A) : (b : A) ≠ 0 := b.property.1
-
-theorem idempotent (b : BlockIdempotent A) : IsIdempotentElem (b : A) :=
-  b.property.2.1
-
-theorem central (b : BlockIdempotent A) : IsMulCentral (b : A) :=
-  b.property.2.2.1
-
-theorem primitive (b : BlockIdempotent A) (e : A)
-    (he : IsIdempotentElem e) (hc : IsMulCentral e) (heb : e * b = e) :
-    e = 0 ∨ e = b :=
-  b.property.2.2.2 e he hc heb
-
-/-- The ring of the block: the corner has identity `b`, rather than the
-identity of the ambient ring. -/
-abbrev Algebra (b : BlockIdempotent A) := b.idempotent.Corner
-
-/-- For a central idempotent the corner is exactly the summand `A b`. -/
-theorem mem_corner_iff (b : BlockIdempotent A) (x : A) :
-    x ∈ Subsemigroup.corner (b : A) ↔ ∃ a : A, a * b = x :=
-  Subsemigroup.mem_corner_iff_mem_range_mul_right b.idempotent b.central
-
-variable (K : Type v) [CommRing K] [_root_.Algebra K A]
-
-/-- Scalars in the block are `k b`. In particular the identity is `b`. -/
-def scalarHom (b : BlockIdempotent A) : K →+* b.Algebra where
-  toFun k := ⟨k • (b : A), by
-    rw [Subsemigroup.mem_corner_iff b.idempotent]
-    constructor
-    · rw [mul_smul_comm, b.idempotent.eq]
-    · rw [smul_mul_assoc, b.idempotent.eq]⟩
-  map_zero' := Subtype.ext (zero_smul K (b : A))
-  map_one' := Subtype.ext (one_smul K (b : A))
-  map_add' k l := Subtype.ext (add_smul k l (b : A))
-  map_mul' k l := Subtype.ext (by
-    change (k * l) • (b : A) = (k • (b : A)) * (l • (b : A))
-    rw [smul_mul_smul_comm, b.idempotent.eq])
-
-instance algebra (b : BlockIdempotent A) : _root_.Algebra K b.Algebra :=
-  (scalarHom K b).toAlgebra' fun k x => Subtype.ext (by
-    change (k • (b : A)) * (x.val : A) = (x.val : A) * (k • (b : A))
-    rw [smul_mul_assoc, mul_smul_comm,
-      ((Subsemigroup.mem_corner_iff b.idempotent).mp x.property).1,
-      ((Subsemigroup.mem_corner_iff b.idempotent).mp x.property).2])
-
-/-- Inclusion of the block into the ambient algebra is linear. It is not
-generally a unital homomorphism of rings, since the identities differ. -/
-def inclusion (b : BlockIdempotent A) : b.Algebra →ₗ[K] A where
-  toFun x := x.val
-  map_add' _ _ := rfl
-  map_smul' k x := by
-    change (k • (b : A)) * x.val = k • x.val
-    rw [smul_mul_assoc, ((Subsemigroup.mem_corner_iff b.idempotent).mp x.property).1]
-
-theorem inclusion_injective (b : BlockIdempotent A) :
-    Function.Injective (inclusion K b) := Subtype.val_injective
-
-end BlockIdempotent
-end Blocks
-
-section FinitelyManyBlocksOfOneAlgebra
-
-variable {A : Type u} [Ring A]
-
-namespace BlockIdempotent
-
-/-- Distinct primitive central idempotents are orthogonal. -/
-theorem mul_eq_zero {b c : BlockIdempotent A} (h : b ≠ c) : (b : A) * c = 0 := by
-  have he : IsIdempotentElem ((b : A) * c) := by
-    change ((b : A) * c) * ((b : A) * c) = (b : A) * c
-    calc
-      ((b : A) * c) * ((b : A) * c) = (b : A) * ((c : A) * b) * c :=
-        by simp only [mul_assoc]
-      _ = (b : A) * ((b : A) * c) * c := by rw [(c.central.comm (b : A)).eq]
-      _ = ((b : A) * b) * ((c : A) * c) := by simp only [mul_assoc]
-      _ = (b : A) * c := by rw [b.idempotent.eq, c.idempotent.eq]
-  have hc : IsMulCentral ((b : A) * c) := Set.mul_mem_center b.central c.central
-  have hb : ((b : A) * c) * b = (b : A) * c := by
-    rw [mul_assoc, (c.central.comm (b : A)).eq, ← mul_assoc, b.idempotent.eq]
-  have hc' : ((b : A) * c) * c = (b : A) * c := by
-    rw [mul_assoc, c.idempotent.eq]
-  obtain hz | hb' := b.primitive _ he hc hb
-  · exact hz
-  obtain hz | hc'' := c.primitive _ he hc hc'
-  · exact hz
-  exact (h (ext (hb'.symm.trans hc''))).elim
-
-variable (K : Type v) [Field K] [_root_.Algebra K A]
-
-/-- Primitive central idempotents form a linearly independent family. -/
-theorem linearIndependent : LinearIndependent K (fun b : BlockIdempotent A => (b : A)) := by
-  classical
-  rw [linearIndependent_iff']
-  intro s f hs i hi
-  have h := congrArg (fun x : A => x * (i : A)) hs
-  rw [Finset.sum_mul, zero_mul, Finset.sum_eq_single_of_mem i hi] at h
-  · rw [smul_mul_assoc, i.idempotent.eq] at h
-    exact (smul_eq_zero.mp h).resolve_right i.ne_zero
-  · intro j _ hji
-    rw [smul_mul_assoc, mul_eq_zero hji, smul_zero]
-
-/-- A single finite-dimensional algebra has only finitely many blocks.
-This does not bound the number of Morita types as the algebra varies. -/
-theorem finite [Module.Finite K A] : Finite (BlockIdempotent A) :=
-  (linearIndependent (A := A) K).finite
-
-/-- A block of a finite-dimensional algebra is itself finite-dimensional,
-for its natural scalar structure. -/
-theorem finiteDimensional_corner [Module.Finite K A] (b : BlockIdempotent A) :
-    FiniteDimensional K b.Algebra :=
-  FiniteDimensional.of_injective (inclusion K b) (inclusion_injective K b)
-
-end BlockIdempotent
-end FinitelyManyBlocksOfOneAlgebra
-
-section DefectGroups
-
-variable {K : Type v} [Field K] {G : Type u} [Group G]
-
-/-- A block idempotent is fixed under conjugation by every group element.
-Thus it belongs to the domain of each genuine Brauer map. -/
-theorem BlockIdempotent.conjugation_fixed (b : BlockIdempotent (MonoidAlgebra K G))
-    (g : G) :
-    MonoidAlgebra.single g 1 * (b : MonoidAlgebra K G) *
-      MonoidAlgebra.single g⁻¹ 1 = (b : MonoidAlgebra K G) := by
-  rw [← (b.central.comm (MonoidAlgebra.single g 1)).eq, mul_assoc,
-    MonoidAlgebra.single_mul_single, mul_inv_cancel, one_mul,
-    ← MonoidAlgebra.one_def, mul_one]
-
-/-- Delete coefficients outside the centralizer of `D`. On the `D`-fixed group algebra
-in characteristic `p`, for a `p`-subgroup `D`, this is the Brauer map. We only use it on
-central idempotents; no multiplicativity on the entire group algebra is asserted. -/
-def brauerProjection (D : Subgroup G) (x : MonoidAlgebra K G) :
-    MonoidAlgebra K (Subgroup.centralizer (D : Set G)) :=
-  MonoidAlgebra.ofCoeff (x.coeff.subtypeDomain
-    (fun g => g ∈ Subgroup.centralizer (D : Set G)))
-
-@[simp] theorem brauerProjection_coeff (D : Subgroup G) (x : MonoidAlgebra K G)
-    (g : Subgroup.centralizer (D : Set G)) :
-    (brauerProjection D x).coeff g = x.coeff g.val := rfl
-
-@[simp] theorem brauerProjection_zero (D : Subgroup G) :
-    brauerProjection D (0 : MonoidAlgebra K G) = 0 := rfl
-
-/-- Defect groups are maximal, under inclusion, among the `p`-subgroups
-with nonzero Brauer image. This does not impose a commutativity condition. -/
-def IsDefectGroup (p : ℕ) (b : BlockIdempotent (MonoidAlgebra K G))
-    (D : Subgroup G) : Prop :=
-  Maximal (fun Q : Subgroup G => IsPGroup p Q ∧ brauerProjection Q (b : MonoidAlgebra K G) ≠ 0) D
-
-@[simp] theorem brauerProjection_bot_eq_zero_iff (x : MonoidAlgebra K G) :
-    brauerProjection (⊥ : Subgroup G) x = 0 ↔ x = 0 := by
-  change MonoidAlgebra.ofCoeff _ = 0 ↔ x = 0
-  rw [MonoidAlgebra.ofCoeff_eq_zero, Finsupp.subtypeDomain_eq_zero_iff]
-  · exact MonoidAlgebra.coeff_eq_zero
-  · intro g _
-    rw [Subgroup.mem_centralizer_iff]
-    intro h hh
-    have : h = 1 := Subgroup.mem_bot.mp hh
-    simp [this]
-
-/-- Any `p`-subgroup with nonzero Brauer image is contained in a defect
-group. This uses finiteness of the subgroup poset. -/
-theorem exists_defectGroup_ge [Finite G] (p : ℕ)
-    (b : BlockIdempotent (MonoidAlgebra K G)) (Q : Subgroup G)
-    (hQ : IsPGroup p Q) (hbQ : brauerProjection Q (b : MonoidAlgebra K G) ≠ 0) :
-    ∃ D : Subgroup G, Q ≤ D ∧ IsDefectGroup p b D :=
-  ({D : Subgroup G | IsPGroup p D ∧
-    brauerProjection D (b : MonoidAlgebra K G) ≠ 0}).toFinite.exists_le_maximal ⟨hQ, hbQ⟩
-
-/-- A finite-group block has at least one defect group as defined above. -/
-theorem exists_defectGroup [Finite G] (p : ℕ)
-    (b : BlockIdempotent (MonoidAlgebra K G)) :
-    ∃ D : Subgroup G, IsDefectGroup p b D := by
-  let s : Set (Subgroup G) :=
-    {D | IsPGroup p D ∧ brauerProjection D (b : MonoidAlgebra K G) ≠ 0}
-  have hs : s.Nonempty := by
-    refine ⟨⊥, IsPGroup.of_bot, ?_⟩
-    exact mt (brauerProjection_bot_eq_zero_iff (b : MonoidAlgebra K G)).mp b.ne_zero
-  exact s.toFinite.exists_maximal hs
-
-end DefectGroups
-
-section MoritaClasses
-
-/-- A block of a finite group algebra over `K`. The group is allowed to
-vary over all finite group types in the specified universe. -/
-structure FiniteGroupBlock (K : Type v) [Field K] where
-  G : Type u
-  [group : Group G]
-  [finite : Finite G]
-  block : BlockIdempotent (MonoidAlgebra K G)
-
-attribute [instance] FiniteGroupBlock.group FiniteGroupBlock.finite
-
-namespace FiniteGroupBlock
-
-variable {K : Type v} [Field K]
-
-/-- The actual block algebra `KGb`. -/
-abbrev Algebra (B : FiniteGroupBlock.{u, v} K) := B.block.Algebra
-
-/-- The equivalence relation is linear Morita equivalence, not ring
-isomorphism or equivalence of the underlying un-enriched categories. -/
-def moritaSetoid (K : Type v) [Field K] : Setoid (FiniteGroupBlock.{u, v} K) where
-  r B C := IsMoritaEquivalent K B.Algebra C.Algebra
-  iseqv := ⟨fun B => IsMoritaEquivalent.refl K B.Algebra,
-    fun h => IsMoritaEquivalent.symm K h, fun h h' => IsMoritaEquivalent.trans K h h'⟩
-
-end FiniteGroupBlock
-
-/-- Morita classes of all finite-group blocks over the fixed field `K`. -/
-def MoritaClass (K : Type v) [Field K] :=
-  Quotient (FiniteGroupBlock.moritaSetoid.{u, v} K)
-
-/-- The Morita class represented by a block. -/
-def moritaClass {K : Type v} [Field K] (B : FiniteGroupBlock.{u, v} K) :
-    MoritaClass.{u, v} K := Quotient.mk _ B
-
-/-- The block has a defect group whose order is at most `M`. -/
-def HasDefectAtMost {K : Type v} [Field K] (p M : ℕ) (B : FiniteGroupBlock.{u, v} K) :
-    Prop :=
-  ∃ D : Subgroup B.G, IsDefectGroup p B.block D ∧ Nat.card D ≤ M
-
-/-- The block has a defect group isomorphic to the given group `P`. -/
-def HasDefectIsomorphic {K : Type v} [Field K] (p : ℕ) (P : Type w) [Group P]
-    (B : FiniteGroupBlock.{u, v} K) : Prop :=
-  ∃ D : Subgroup B.G, IsDefectGroup p B.block D ∧ Nonempty (D ≃* P)
-
-/-- Morita classes represented by blocks with a defect group of order at most `M`. -/
-def boundedClasses (p : ℕ) (K : Type v) [Field K] (M : ℕ) :
-    Set (MoritaClass.{u, v} K) :=
-  moritaClass '' {B | HasDefectAtMost p M B}
-
-/-- Morita classes represented by blocks with a defect group isomorphic to `P`. -/
-def fixedDefectClasses (p : ℕ) (K : Type v) [Field K] (P : Type w) [Group P] :
-    Set (MoritaClass.{u, v} K) :=
-  moritaClass '' {B | HasDefectIsomorphic p P B}
-
-variable {p : ℕ} {K : Type v} [Field K]
-
-/-- Equality of classes means linear Morita equivalence of the actual block algebras. -/
-theorem moritaClass_eq_iff (B C : FiniteGroupBlock.{u, v} K) :
-    moritaClass B = moritaClass C ↔ IsMoritaEquivalent K B.Algebra C.Algebra :=
-  Quotient.eq
-
-/-- The order of a defect group is positive. -/
-theorem HasDefectAtMost.pos {M : ℕ} {B : FiniteGroupBlock.{u, v} K}
-    (h : HasDefectAtMost p M B) : 0 < M := by
-  obtain ⟨D, _, hD⟩ := h
-  exact lt_of_lt_of_le (Nat.card_pos (α := D)) hD
-
-/-- Every block is covered by some positive bound; the bounded-order classes
-are not made empty by the definition of a defect group. -/
-theorem exists_positive_defect_bound (p : ℕ) (B : FiniteGroupBlock.{u, v} K) :
-    ∃ M : ℕ, 0 < M ∧ HasDefectAtMost p M B := by
-  obtain ⟨D, hD⟩ := exists_defectGroup p B.block
-  exact ⟨Nat.card D, Nat.card_pos, D, hD, le_rfl⟩
-
-/-- An isomorphism of defect groups gives equality of their orders. -/
-theorem HasDefectIsomorphic.hasDefectAtMost {P : Type w} [Group P]
-    {B : FiniteGroupBlock.{u, v} K} (h : HasDefectIsomorphic p P B) :
-    HasDefectAtMost p (Nat.card P) B := by
-  obtain ⟨D, hD, ⟨e⟩⟩ := h
-  exact ⟨D, hD, (Nat.card_congr e.toEquiv).le⟩
-
-/-- The fixed-defect classes are a subset of the bounded-order classes. -/
-theorem fixedDefectClasses_subset (p : ℕ) (K : Type v) [Field K]
-    (P : Type w) [Group P] :
-    fixedDefectClasses.{u, v, w} p K P ⊆ boundedClasses p K (Nat.card P) := by
-  rintro _ ⟨B, hB, rfl⟩
-  exact ⟨B, hB.hasDefectAtMost, rfl⟩
-
-/-- The elementary fixed-defect reduction. Uniform bounded-order finiteness is
-an assumption, not a result proved here. There is no abelian hypothesis on `P`. -/
-theorem finite_fixedDefectClasses_of_bounded
-    (h : ∀ M : ℕ, 0 < M → (boundedClasses.{u, v} p K M).Finite)
-    (P : Type w) [Group P] [Finite P] :
-    (fixedDefectClasses.{u, v, w} p K P).Finite :=
-  (h (Nat.card P) Nat.card_pos).subset (fixedDefectClasses_subset p K P)
-
-namespace FiniteGroupBlock
-
-variable {G : Type u} [Group G] [Finite G]
-
-/-- Regard a block for one fixed ambient group as a member of the class of
-all finite-group blocks. -/
-def ofBlock (b : BlockIdempotent (MonoidAlgebra K G)) : FiniteGroupBlock.{u, v} K where
-  G := G
-  block := b
-
-end FiniteGroupBlock
-
-/-- For a fixed finite group, its blocks give a finite collection of Morita
-classes. This is only the fixed-ambient-group claim, not Donovan's theorem. -/
-theorem finite_classes_for_one_group (K : Type v) [Field K]
-    (G : Type u) [Group G] [Finite G] :
-    (Set.range (fun b : BlockIdempotent (MonoidAlgebra K G) =>
-      moritaClass (FiniteGroupBlock.ofBlock b))).Finite := by
-  have : Finite (BlockIdempotent (MonoidAlgebra K G)) := BlockIdempotent.finite K
-  exact Set.finite_range _
-
-end MoritaClasses
-
-section CoefficientExtension
-
-variable {K : Type v} [Field K] {L : Type w} [Field L]
-variable {G : Type u} [Group G]
-
-/-- Applying a field embedding to group-ring coefficients is injective. -/
-theorem coefficientExtension_injective (f : K →+* L) :
-    Function.Injective (MonoidAlgebra.mapRingHom G f) := by
-  intro x y h
-  apply MonoidAlgebra.ext
-  ext g
-  apply f.injective
-  simpa only [MonoidAlgebra.coeff_mapRingHom] using
-    congrArg (fun z : MonoidAlgebra L G => z.coeff g) h
-
-/-- Deleting coefficients outside the centralizer commutes with applying a
-field embedding. This is the coefficient calculation for the Brauer map. -/
-theorem brauerProjection_coefficientExtension (f : K →+* L) (D : Subgroup G)
-    (x : MonoidAlgebra K G) :
-    brauerProjection D (MonoidAlgebra.mapRingHom G f x) =
-      MonoidAlgebra.mapRingHom (Subgroup.centralizer (D : Set G)) f
-        (brauerProjection D x) := by
-  apply MonoidAlgebra.ext
-  ext g
-  simp only [brauerProjection_coeff, MonoidAlgebra.coeff_mapRingHom]
-
-/-- The vanishing criterion used to compute defects is unchanged by a
-coefficient-field embedding. -/
-theorem brauerProjection_coefficientExtension_eq_zero_iff (f : K →+* L)
-    (D : Subgroup G) (x : MonoidAlgebra K G) :
-    brauerProjection D (MonoidAlgebra.mapRingHom G f x) = 0 ↔
-      brauerProjection D x = 0 := by
-  rw [brauerProjection_coefficientExtension]
-  constructor
-  · intro h
-    apply coefficientExtension_injective f
-    simpa only [map_zero] using h
-  · intro h
-    simp only [h, map_zero]
-
-/-- If two actual block idempotents correspond by coefficient extension,
-then their defect subgroups are identical. This lemma does not assert
-that every block descends, or that primitivity is preserved. -/
-theorem isDefectGroup_coefficientExtension_iff (f : K →+* L) (p : ℕ)
-    (b : BlockIdempotent (MonoidAlgebra K G))
-    (c : BlockIdempotent (MonoidAlgebra L G))
-    (h : MonoidAlgebra.mapRingHom G f (b : MonoidAlgebra K G) = (c : MonoidAlgebra L G))
-    (D : Subgroup G) : IsDefectGroup p c D ↔ IsDefectGroup p b D := by
-  unfold IsDefectGroup Maximal
-  simp only [← h, ne_eq, brauerProjection_coefficientExtension_eq_zero_iff]
-
-end CoefficientExtension
-
-end Donovan
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-!
+# Blocks, defect groups, and the bounded-order formulation of Donovan's conjecture
+
+This file develops supporting results, not a proof of Donovan's conjecture.
+The algebra of a block is the actual corner `b (KG) b = (KG) b`, with identity `b`.
+Morita equivalence is Mathlib's equivalence of module categories by a linear functor.
+
+We prove finiteness of blocks for a fixed finite-dimensional algebra, existence of
+defect groups, the conditional reduction from bounded-order to fixed-defect Morita
+finiteness, and equality of defect subgroups for blocks corresponding under a
+coefficient-field embedding.
+
+Uniform bounded-defect-order Morita finiteness across varying finite groups remains
+unproved. Block-idempotent descent and bijectivity, preservation of primitivity, and
+scalar extension of Morita equivalences are also not established here.
+-/
+noncomputable section
+
+universe u v w
+
+namespace Donovan
+
+section Blocks
+
+variable {A : Type u} [Ring A]
+
+/-- A block is a nonzero primitive idempotent in the center, not necessarily
+a primitive idempotent of the whole ring. -/
+def IsBlockIdempotent (b : A) : Prop :=
+  b ≠ 0 ∧ IsIdempotentElem b ∧ IsMulCentral b ∧
+    ∀ e : A, IsIdempotentElem e → IsMulCentral e → e * b = e → e = 0 ∨ e = b
+
+/-- The block idempotents in a ring. -/
+def BlockIdempotent (A : Type u) [Ring A] := {b : A // IsBlockIdempotent b}
+
+namespace BlockIdempotent
+
+instance : Coe (BlockIdempotent A) A := ⟨Subtype.val⟩
+
+@[ext] theorem ext {b c : BlockIdempotent A} (h : (b : A) = c) : b = c :=
+  Subtype.ext h
+
+theorem ne_zero (b : BlockIdempotent A) : (b : A) ≠ 0 := b.property.1
+
+theorem idempotent (b : BlockIdempotent A) : IsIdempotentElem (b : A) :=
+  b.property.2.1
+
+theorem central (b : BlockIdempotent A) : IsMulCentral (b : A) :=
+  b.property.2.2.1
+
+theorem primitive (b : BlockIdempotent A) (e : A)
+    (he : IsIdempotentElem e) (hc : IsMulCentral e) (heb : e * b = e) :
+    e = 0 ∨ e = b :=
+  b.property.2.2.2 e he hc heb
+
+/-- The ring of the block: the corner has identity `b`, rather than the
+identity of the ambient ring. -/
+abbrev Algebra (b : BlockIdempotent A) := b.idempotent.Corner
+
+/-- For a central idempotent the corner is exactly the summand `A b`. -/
+theorem mem_corner_iff (b : BlockIdempotent A) (x : A) :
+    x ∈ Subsemigroup.corner (b : A) ↔ ∃ a : A, a * b = x :=
+  Subsemigroup.mem_corner_iff_mem_range_mul_right b.idempotent b.central
+
+variable (K : Type v) [CommRing K] [_root_.Algebra K A]
+
+/-- Scalars in the block are `k b`. In particular the identity is `b`. -/
+def scalarHom (b : BlockIdempotent A) : K →+* b.Algebra where
+  toFun k := ⟨k • (b : A), by
+    rw [Subsemigroup.mem_corner_iff b.idempotent]
+    constructor
+    · rw [mul_smul_comm, b.idempotent.eq]
+    · rw [smul_mul_assoc, b.idempotent.eq]⟩
+  map_zero' := Subtype.ext (zero_smul K (b : A))
+  map_one' := Subtype.ext (one_smul K (b : A))
+  map_add' k l := Subtype.ext (add_smul k l (b : A))
+  map_mul' k l := Subtype.ext (by
+    change (k * l) • (b : A) = (k • (b : A)) * (l • (b : A))
+    rw [smul_mul_smul_comm, b.idempotent.eq])
+
+instance algebra (b : BlockIdempotent A) : _root_.Algebra K b.Algebra :=
+  (scalarHom K b).toAlgebra' fun k x => Subtype.ext (by
+    change (k • (b : A)) * (x.val : A) = (x.val : A) * (k • (b : A))
+    rw [smul_mul_assoc, mul_smul_comm,
+      ((Subsemigroup.mem_corner_iff b.idempotent).mp x.property).1,
+      ((Subsemigroup.mem_corner_iff b.idempotent).mp x.property).2])
+
+/-- Inclusion of the block into the ambient algebra is linear. It is not
+generally a unital homomorphism of rings, since the identities differ. -/
+def inclusion (b : BlockIdempotent A) : b.Algebra →ₗ[K] A where
+  toFun x := x.val
+  map_add' _ _ := rfl
+  map_smul' k x := by
+    change (k • (b : A)) * x.val = k • x.val
+    rw [smul_mul_assoc, ((Subsemigroup.mem_corner_iff b.idempotent).mp x.property).1]
+
+theorem inclusion_injective (b : BlockIdempotent A) :
+    Function.Injective (inclusion K b) := Subtype.val_injective
+
+end BlockIdempotent
+end Blocks
+
+section FinitelyManyBlocksOfOneAlgebra
+
+variable {A : Type u} [Ring A]
+
+namespace BlockIdempotent
+
+/-- Distinct primitive central idempotents are orthogonal. -/
+theorem mul_eq_zero {b c : BlockIdempotent A} (h : b ≠ c) : (b : A) * c = 0 := by
+  have he : IsIdempotentElem ((b : A) * c) := by
+    change ((b : A) * c) * ((b : A) * c) = (b : A) * c
+    calc
+      ((b : A) * c) * ((b : A) * c) = (b : A) * ((c : A) * b) * c :=
+        by simp only [mul_assoc]
+      _ = (b : A) * ((b : A) * c) * c := by rw [(c.central.comm (b : A)).eq]
+      _ = ((b : A) * b) * ((c : A) * c) := by simp only [mul_assoc]
+      _ = (b : A) * c := by rw [b.idempotent.eq, c.idempotent.eq]
+  have hc : IsMulCentral ((b : A) * c) := Set.mul_mem_center b.central c.central
+  have hb : ((b : A) * c) * b = (b : A) * c := by
+    rw [mul_assoc, (c.central.comm (b : A)).eq, ← mul_assoc, b.idempotent.eq]
+  have hc' : ((b : A) * c) * c = (b : A) * c := by
+    rw [mul_assoc, c.idempotent.eq]
+  obtain hz | hb' := b.primitive _ he hc hb
+  · exact hz
+  obtain hz | hc'' := c.primitive _ he hc hc'
+  · exact hz
+  exact (h (ext (hb'.symm.trans hc''))).elim
+
+variable (K : Type v) [Field K] [_root_.Algebra K A]
+
+/-- Primitive central idempotents form a linearly independent family. -/
+theorem linearIndependent : LinearIndependent K (fun b : BlockIdempotent A => (b : A)) := by
+  classical
+  rw [linearIndependent_iff']
+  intro s f hs i hi
+  have h := congrArg (fun x : A => x * (i : A)) hs
+  rw [Finset.sum_mul, zero_mul, Finset.sum_eq_single_of_mem i hi] at h
+  · rw [smul_mul_assoc, i.idempotent.eq] at h
+    exact (smul_eq_zero.mp h).resolve_right i.ne_zero
+  · intro j _ hji
+    rw [smul_mul_assoc, mul_eq_zero hji, smul_zero]
+
+/-- A single finite-dimensional algebra has only finitely many blocks.
+This does not bound the number of Morita types as the algebra varies. -/
+theorem finite [Module.Finite K A] : Finite (BlockIdempotent A) :=
+  (linearIndependent (A := A) K).finite
+
+/-- A block of a finite-dimensional algebra is itself finite-dimensional,
+for its natural scalar structure. -/
+theorem finiteDimensional_corner [Module.Finite K A] (b : BlockIdempotent A) :
+    FiniteDimensional K b.Algebra :=
+  FiniteDimensional.of_injective (inclusion K b) (inclusion_injective K b)
+
+end BlockIdempotent
+end FinitelyManyBlocksOfOneAlgebra
+
+section DefectGroups
+
+variable {K : Type v} [Field K] {G : Type u} [Group G]
+
+/-- A block idempotent is fixed under conjugation by every group element.
+Thus it belongs to the domain of each genuine Brauer map. -/
+theorem BlockIdempotent.conjugation_fixed (b : BlockIdempotent (MonoidAlgebra K G))
+    (g : G) :
+    MonoidAlgebra.single g 1 * (b : MonoidAlgebra K G) *
+      MonoidAlgebra.single g⁻¹ 1 = (b : MonoidAlgebra K G) := by
+  rw [← (b.central.comm (MonoidAlgebra.single g 1)).eq, mul_assoc,
+    MonoidAlgebra.single_mul_single, mul_inv_cancel, one_mul,
+    ← MonoidAlgebra.one_def, mul_one]
+
+/-- Delete coefficients outside the centralizer of `D`. On the `D`-fixed group algebra
+in characteristic `p`, for a `p`-subgroup `D`, this is the Brauer map. We only use it on
+central idempotents; no multiplicativity on the entire group algebra is asserted. -/
+def brauerProjection (D : Subgroup G) (x : MonoidAlgebra K G) :
+    MonoidAlgebra K (Subgroup.centralizer (D : Set G)) :=
+  MonoidAlgebra.ofCoeff (x.coeff.subtypeDomain
+    (fun g => g ∈ Subgroup.centralizer (D : Set G)))
+
+@[simp] theorem brauerProjection_coeff (D : Subgroup G) (x : MonoidAlgebra K G)
+    (g : Subgroup.centralizer (D : Set G)) :
+    (brauerProjection D x).coeff g = x.coeff g.val := rfl
+
+@[simp] theorem brauerProjection_zero (D : Subgroup G) :
+    brauerProjection D (0 : MonoidAlgebra K G) = 0 := rfl
+
+/-- Defect groups are maximal, under inclusion, among the `p`-subgroups
+with nonzero Brauer image. This does not impose a commutativity condition. -/
+def IsDefectGroup (p : ℕ) (b : BlockIdempotent (MonoidAlgebra K G))
+    (D : Subgroup G) : Prop :=
+  Maximal (fun Q : Subgroup G => IsPGroup p Q ∧ brauerProjection Q (b : MonoidAlgebra K G) ≠ 0) D
+
+@[simp] theorem brauerProjection_bot_eq_zero_iff (x : MonoidAlgebra K G) :
+    brauerProjection (⊥ : Subgroup G) x = 0 ↔ x = 0 := by
+  change MonoidAlgebra.ofCoeff _ = 0 ↔ x = 0
+  rw [MonoidAlgebra.ofCoeff_eq_zero, Finsupp.subtypeDomain_eq_zero_iff]
+  · exact MonoidAlgebra.coeff_eq_zero
+  · intro g _
+    rw [Subgroup.mem_centralizer_iff]
+    intro h hh
+    have : h = 1 := Subgroup.mem_bot.mp hh
+    simp [this]
+
+/-- Any `p`-subgroup with nonzero Brauer image is contained in a defect
+group. This uses finiteness of the subgroup poset. -/
+theorem exists_defectGroup_ge [Finite G] (p : ℕ)
+    (b : BlockIdempotent (MonoidAlgebra K G)) (Q : Subgroup G)
+    (hQ : IsPGroup p Q) (hbQ : brauerProjection Q (b : MonoidAlgebra K G) ≠ 0) :
+    ∃ D : Subgroup G, Q ≤ D ∧ IsDefectGroup p b D :=
+  ({D : Subgroup G | IsPGroup p D ∧
+    brauerProjection D (b : MonoidAlgebra K G) ≠ 0}).toFinite.exists_le_maximal ⟨hQ, hbQ⟩
+
+/-- A finite-group block has at least one defect group as defined above. -/
+theorem exists_defectGroup [Finite G] (p : ℕ)
+    (b : BlockIdempotent (MonoidAlgebra K G)) :
+    ∃ D : Subgroup G, IsDefectGroup p b D := by
+  let s : Set (Subgroup G) :=
+    {D | IsPGroup p D ∧ brauerProjection D (b : MonoidAlgebra K G) ≠ 0}
+  have hs : s.Nonempty := by
+    refine ⟨⊥, IsPGroup.of_bot, ?_⟩
+    exact mt (brauerProjection_bot_eq_zero_iff (b : MonoidAlgebra K G)).mp b.ne_zero
+  exact s.toFinite.exists_maximal hs
+
+end DefectGroups
+
+section MoritaClasses
+
+/-- A block of a finite group algebra over `K`. The group is allowed to
+vary over all finite group types in the specified universe. -/
+structure FiniteGroupBlock (K : Type v) [Field K] where
+  G : Type u
+  [group : Group G]
+  [finite : Finite G]
+  block : BlockIdempotent (MonoidAlgebra K G)
+
+attribute [instance] FiniteGroupBlock.group FiniteGroupBlock.finite
+
+namespace FiniteGroupBlock
+
+variable {K : Type v} [Field K]
+
+/-- The actual block algebra `KGb`. -/
+abbrev Algebra (B : FiniteGroupBlock.{u, v} K) := B.block.Algebra
+
+/-- The equivalence relation is linear Morita equivalence, not ring
+isomorphism or equivalence of the underlying un-enriched categories. -/
+def moritaSetoid (K : Type v) [Field K] : Setoid (FiniteGroupBlock.{u, v} K) where
+  r B C := IsMoritaEquivalent K B.Algebra C.Algebra
+  iseqv := ⟨fun B => IsMoritaEquivalent.refl K B.Algebra,
+    fun h => IsMoritaEquivalent.symm K h, fun h h' => IsMoritaEquivalent.trans K h h'⟩
+
+end FiniteGroupBlock
+
+/-- Morita classes of all finite-group blocks over the fixed field `K`. -/
+def MoritaClass (K : Type v) [Field K] :=
+  Quotient (FiniteGroupBlock.moritaSetoid.{u, v} K)
+
+/-- The Morita class represented by a block. -/
+def moritaClass {K : Type v} [Field K] (B : FiniteGroupBlock.{u, v} K) :
+    MoritaClass.{u, v} K := Quotient.mk _ B
+
+/-- The block has a defect group whose order is at most `M`. -/
+def HasDefectAtMost {K : Type v} [Field K] (p M : ℕ) (B : FiniteGroupBlock.{u, v} K) :
+    Prop :=
+  ∃ D : Subgroup B.G, IsDefectGroup p B.block D ∧ Nat.card D ≤ M
+
+/-- The block has a defect group isomorphic to the given group `P`. -/
+def HasDefectIsomorphic {K : Type v} [Field K] (p : ℕ) (P : Type w) [Group P]
+    (B : FiniteGroupBlock.{u, v} K) : Prop :=
+  ∃ D : Subgroup B.G, IsDefectGroup p B.block D ∧ Nonempty (D ≃* P)
+
+/-- Morita classes represented by blocks with a defect group of order at most `M`. -/
+def boundedClasses (p : ℕ) (K : Type v) [Field K] (M : ℕ) :
+    Set (MoritaClass.{u, v} K) :=
+  moritaClass '' {B | HasDefectAtMost p M B}
+
+/-- Morita classes represented by blocks with a defect group isomorphic to `P`. -/
+def fixedDefectClasses (p : ℕ) (K : Type v) [Field K] (P : Type w) [Group P] :
+    Set (MoritaClass.{u, v} K) :=
+  moritaClass '' {B | HasDefectIsomorphic p P B}
+
+variable {p : ℕ} {K : Type v} [Field K]
+
+/-- Equality of classes means linear Morita equivalence of the actual block algebras. -/
+theorem moritaClass_eq_iff (B C : FiniteGroupBlock.{u, v} K) :
+    moritaClass B = moritaClass C ↔ IsMoritaEquivalent K B.Algebra C.Algebra :=
+  Quotient.eq
+
+/-- The order of a defect group is positive. -/
+theorem HasDefectAtMost.pos {M : ℕ} {B : FiniteGroupBlock.{u, v} K}
+    (h : HasDefectAtMost p M B) : 0 < M := by
+  obtain ⟨D, _, hD⟩ := h
+  exact lt_of_lt_of_le (Nat.card_pos (α := D)) hD
+
+/-- Every block is covered by some positive bound; the bounded-order classes
+are not made empty by the definition of a defect group. -/
+theorem exists_positive_defect_bound (p : ℕ) (B : FiniteGroupBlock.{u, v} K) :
+    ∃ M : ℕ, 0 < M ∧ HasDefectAtMost p M B := by
+  obtain ⟨D, hD⟩ := exists_defectGroup p B.block
+  exact ⟨Nat.card D, Nat.card_pos, D, hD, le_rfl⟩
+
+/-- An isomorphism of defect groups gives equality of their orders. -/
+theorem HasDefectIsomorphic.hasDefectAtMost {P : Type w} [Group P]
+    {B : FiniteGroupBlock.{u, v} K} (h : HasDefectIsomorphic p P B) :
+    HasDefectAtMost p (Nat.card P) B := by
+  obtain ⟨D, hD, ⟨e⟩⟩ := h
+  exact ⟨D, hD, (Nat.card_congr e.toEquiv).le⟩
+
+/-- The fixed-defect classes are a subset of the bounded-order classes. -/
+theorem fixedDefectClasses_subset (p : ℕ) (K : Type v) [Field K]
+    (P : Type w) [Group P] :
+    fixedDefectClasses.{u, v, w} p K P ⊆ boundedClasses p K (Nat.card P) := by
+  rintro _ ⟨B, hB, rfl⟩
+  exact ⟨B, hB.hasDefectAtMost, rfl⟩
+
+/-- The elementary fixed-defect reduction. Uniform bounded-order finiteness is
+an assumption, not a result proved here. There is no abelian hypothesis on `P`. -/
+theorem finite_fixedDefectClasses_of_bounded
+    (h : ∀ M : ℕ, 0 < M → (boundedClasses.{u, v} p K M).Finite)
+    (P : Type w) [Group P] [Finite P] :
+    (fixedDefectClasses.{u, v, w} p K P).Finite :=
+  (h (Nat.card P) Nat.card_pos).subset (fixedDefectClasses_subset p K P)
+
+namespace FiniteGroupBlock
+
+variable {G : Type u} [Group G] [Finite G]
+
+/-- Regard a block for one fixed ambient group as a member of the class of
+all finite-group blocks. -/
+def ofBlock (b : BlockIdempotent (MonoidAlgebra K G)) : FiniteGroupBlock.{u, v} K where
+  G := G
+  block := b
+
+end FiniteGroupBlock
+
+/-- For a fixed finite group, its blocks give a finite collection of Morita
+classes. This is only the fixed-ambient-group claim, not Donovan's theorem. -/
+theorem finite_classes_for_one_group (K : Type v) [Field K]
+    (G : Type u) [Group G] [Finite G] :
+    (Set.range (fun b : BlockIdempotent (MonoidAlgebra K G) =>
+      moritaClass (FiniteGroupBlock.ofBlock b))).Finite := by
+  have : Finite (BlockIdempotent (MonoidAlgebra K G)) := BlockIdempotent.finite K
+  exact Set.finite_range _
+
+end MoritaClasses
+
+section CoefficientExtension
+
+variable {K : Type v} [Field K] {L : Type w} [Field L]
+variable {G : Type u} [Group G]
+
+/-- Applying a field embedding to group-ring coefficients is injective. -/
+theorem coefficientExtension_injective (f : K →+* L) :
+    Function.Injective (MonoidAlgebra.mapRingHom G f) := by
+  intro x y h
+  apply MonoidAlgebra.ext
+  ext g
+  apply f.injective
+  simpa only [MonoidAlgebra.coeff_mapRingHom] using
+    congrArg (fun z : MonoidAlgebra L G => z.coeff g) h
+
+/-- Deleting coefficients outside the centralizer commutes with applying a
+field embedding. This is the coefficient calculation for the Brauer map. -/
+theorem brauerProjection_coefficientExtension (f : K →+* L) (D : Subgroup G)
+    (x : MonoidAlgebra K G) :
+    brauerProjection D (MonoidAlgebra.mapRingHom G f x) =
+      MonoidAlgebra.mapRingHom (Subgroup.centralizer (D : Set G)) f
+        (brauerProjection D x) := by
+  apply MonoidAlgebra.ext
+  ext g
+  simp only [brauerProjection_coeff, MonoidAlgebra.coeff_mapRingHom]
+
+/-- The vanishing criterion used to compute defects is unchanged by a
+coefficient-field embedding. -/
+theorem brauerProjection_coefficientExtension_eq_zero_iff (f : K →+* L)
+    (D : Subgroup G) (x : MonoidAlgebra K G) :
+    brauerProjection D (MonoidAlgebra.mapRingHom G f x) = 0 ↔
+      brauerProjection D x = 0 := by
+  rw [brauerProjection_coefficientExtension]
+  constructor
+  · intro h
+    apply coefficientExtension_injective f
+    simpa only [map_zero] using h
+  · intro h
+    simp only [h, map_zero]
+
+/-- If two actual block idempotents correspond by coefficient extension,
+then their defect subgroups are identical. This lemma does not assert
+that every block descends, or that primitivity is preserved. -/
+theorem isDefectGroup_coefficientExtension_iff (f : K →+* L) (p : ℕ)
+    (b : BlockIdempotent (MonoidAlgebra K G))
+    (c : BlockIdempotent (MonoidAlgebra L G))
+    (h : MonoidAlgebra.mapRingHom G f (b : MonoidAlgebra K G) = (c : MonoidAlgebra L G))
+    (D : Subgroup G) : IsDefectGroup p c D ↔ IsDefectGroup p b D := by
+  unfold IsDefectGroup Maximal
+  simp only [← h, ne_eq, brauerProjection_coefficientExtension_eq_zero_iff]
+
+end CoefficientExtension
+
+end Donovan
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FiniteUnitary/Action.lean b/lean/OAI/RepresentationTheory/FiniteUnitary/Action.lean
index 901d035538555429eb1719d0c2358696af740e0a..473ddc216d436541c24f9cd5b719b0731e67c638 100644
--- a/lean/OAI/RepresentationTheory/FiniteUnitary/Action.lean
+++ b/lean/OAI/RepresentationTheory/FiniteUnitary/Action.lean
@@ -1,57 +1,57 @@
-import OAI.RepresentationTheory.FiniteUnitary.Coefficients
-
-namespace OAI
-
-namespace CubeShuffle.UnitaryFinite
-open scoped BigOperators Classical
-
-variable {α ι : Type*} [Fintype α] [Fintype ι]
-
-def tupleAction : Equiv.Perm α →* Equiv.Perm (ι ↪ α) where
-  toFun p :=
-    { toFun := fun e => e.trans p.toEmbedding
-      invFun := fun e => e.trans p.symm.toEmbedding
-      left_inv := fun e => by ext i; exact p.symm_apply_apply (e i)
-      right_inv := fun e => by ext i; exact p.apply_symm_apply (e i) }
-  map_one' := by ext e i; rfl
-  map_mul' p q := by ext e i; rfl
-
-omit [Fintype α] [Fintype ι] in
-@[simp] lemma tupleAction_apply (p : Equiv.Perm α) (e : ι ↪ α) (i : ι) :
-    tupleAction p e i=p (e i) := rfl
-
-omit [Fintype α] in
-lemma tupleAction_transitive (e f : ι ↪ α) : ∃ p, tupleAction p e=f := by
-  obtain ⟨p,hp⟩ := Equiv.Perm.exists_extending_pair e f e.injective f.injective
-  exact ⟨p,Function.Embedding.ext hp⟩
-
-omit [Fintype α] in
-lemma tupleAction_fixed_iff [DecidableEq α] (e : ι ↪ α) (p : Equiv.Perm α) :
-    tupleAction p e=e ↔ ∀ x ∈ Finset.univ.image e, p x=x := by
-  constructor
-  · intro h x hx
-    obtain ⟨i,_,rfl⟩ := Finset.mem_image.mp hx
-    exact congrArg (fun f : ι ↪ α => f i) h
-  · intro h
-    ext i
-    exact h (e i) (Finset.mem_image.mpr ⟨i,Finset.mem_univ _,rfl⟩)
-
-end CubeShuffle.UnitaryFinite
-namespace CubeShuffle.UnitaryFinite
-open scoped BigOperators Classical
-
-lemma irreducible_of_equiv {G V W : Type*} [Group G]
-    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
-    (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
-    (e : Representation.Equiv ρ σ) [Representation.IsIrreducible ρ] :
-    Representation.IsIrreducible σ := by
-  rw [Representation.irreducible_iff_isSimpleModule_asModule]
-  have hi := (Representation.irreducible_iff_isSimpleModule_asModule ρ).mp inferInstance
-  let f : ρ.asModule →ₗ[MonoidAlgebra ℂ G] σ.asModule :=
-    Representation.IntertwiningMap.equivLinearMapAsModule ρ σ e.toIntertwiningMap
-  have hf : Function.Bijective f := e.toLinearEquiv.bijective
-  exact (f.isSimpleModule_iff_of_bijective hf).mp hi
-
-end CubeShuffle.UnitaryFinite
-
-end OAI
+import OAI.RepresentationTheory.FiniteUnitary.Coefficients
+
+namespace OAI
+
+namespace CubeShuffle.UnitaryFinite
+open scoped BigOperators Classical
+
+variable {α ι : Type*} [Fintype α] [Fintype ι]
+
+def tupleAction : Equiv.Perm α →* Equiv.Perm (ι ↪ α) where
+  toFun p :=
+    { toFun := fun e => e.trans p.toEmbedding
+      invFun := fun e => e.trans p.symm.toEmbedding
+      left_inv := fun e => by ext i; exact p.symm_apply_apply (e i)
+      right_inv := fun e => by ext i; exact p.apply_symm_apply (e i) }
+  map_one' := by ext e i; rfl
+  map_mul' p q := by ext e i; rfl
+
+omit [Fintype α] [Fintype ι] in
+@[simp] lemma tupleAction_apply (p : Equiv.Perm α) (e : ι ↪ α) (i : ι) :
+    tupleAction p e i=p (e i) := rfl
+
+omit [Fintype α] in
+lemma tupleAction_transitive (e f : ι ↪ α) : ∃ p, tupleAction p e=f := by
+  obtain ⟨p,hp⟩ := Equiv.Perm.exists_extending_pair e f e.injective f.injective
+  exact ⟨p,Function.Embedding.ext hp⟩
+
+omit [Fintype α] in
+lemma tupleAction_fixed_iff [DecidableEq α] (e : ι ↪ α) (p : Equiv.Perm α) :
+    tupleAction p e=e ↔ ∀ x ∈ Finset.univ.image e, p x=x := by
+  constructor
+  · intro h x hx
+    obtain ⟨i,_,rfl⟩ := Finset.mem_image.mp hx
+    exact congrArg (fun f : ι ↪ α => f i) h
+  · intro h
+    ext i
+    exact h (e i) (Finset.mem_image.mpr ⟨i,Finset.mem_univ _,rfl⟩)
+
+end CubeShuffle.UnitaryFinite
+namespace CubeShuffle.UnitaryFinite
+open scoped BigOperators Classical
+
+lemma irreducible_of_equiv {G V W : Type*} [Group G]
+    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
+    (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
+    (e : Representation.Equiv ρ σ) [Representation.IsIrreducible ρ] :
+    Representation.IsIrreducible σ := by
+  rw [Representation.irreducible_iff_isSimpleModule_asModule]
+  have hi := (Representation.irreducible_iff_isSimpleModule_asModule ρ).mp inferInstance
+  let f : ρ.asModule →ₗ[MonoidAlgebra ℂ G] σ.asModule :=
+    Representation.IntertwiningMap.equivLinearMapAsModule ρ σ e.toIntertwiningMap
+  have hf : Function.Bijective f := e.toLinearEquiv.bijective
+  exact (f.isSimpleModule_iff_of_bijective hf).mp hi
+
+end CubeShuffle.UnitaryFinite
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FiniteUnitary/Averaging.lean b/lean/OAI/RepresentationTheory/FiniteUnitary/Averaging.lean
index d225ec656d1e4846ff1b83c7805941496c834db3..f2e3fd7415cb2f099a5d640a6f892c2a5a7b41f8 100644
--- a/lean/OAI/RepresentationTheory/FiniteUnitary/Averaging.lean
+++ b/lean/OAI/RepresentationTheory/FiniteUnitary/Averaging.lean
@@ -1,370 +1,370 @@
-import OAI.RepresentationTheory.FiniteUnitary.Coefficients
-
-namespace OAI
-
-namespace CubeShuffle.FiniteConvolution
-open scoped BigOperators Classical
-variable {G : Type*} [Group G] [Fintype G]
-
-noncomputable def conv (p q : G → ℝ) (g : G) : ℝ := ∑ h, p h*q (h⁻¹*g)
-noncomputable def power (p : G → ℝ) : ℕ → G → ℝ
-  | 0 => fun g => if g=1 then 1 else 0
-  | n+1 => conv p (power p n)
-
-lemma conv_nonneg {p q : G → ℝ} (hp : ∀ g, 0 ≤ p g) (hq : ∀ g, 0 ≤ q g) :
-    ∀ g, 0 ≤ conv p q g := fun _ => Finset.sum_nonneg (fun h _ => mul_nonneg (hp h) (hq _))
-
-lemma conv_pos_of {p q : G → ℝ} (hp : ∀ g, 0 ≤ p g) (hq : ∀ g, 0 ≤ q g)
-    {a b : G} (ha : 0 < p a) (hb : 0 < q b) : 0 < conv p q (a*b) := by
-  apply lt_of_lt_of_le (mul_pos ha hb)
-  have h := Finset.single_le_sum (fun g _ => mul_nonneg (hp g) (hq (g⁻¹*(a*b))))
-    (Finset.mem_univ a)
-  simpa [conv] using h
-
-lemma sum_conv (p q : G → ℝ) : ∑ g, conv p q g=(∑ g,p g)*(∑ g,q g) := by
-  unfold conv
-  rw [Finset.sum_comm,Finset.sum_mul]
-  apply Finset.sum_congr rfl
-  intro h _
-  rw [←Finset.mul_sum]
-  congr 1
-  exact Fintype.sum_equiv (Equiv.mulLeft h⁻¹) _ _ (fun _ => rfl)
-
-lemma power_nonneg {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) : ∀ n g, 0 ≤ power p n g := by
-  intro n
-  induction n with
-  | zero => intro g; simp only [power]; split_ifs  <;> norm_num
-  | succ n ih => exact conv_nonneg hp ih
-
-lemma sum_power {p : G → ℝ} (hp : ∑ g,p g=1) : ∀ n, ∑ g,power p n g=1 := by
-  intro n
-  induction n with
-  | zero => simp [power]
-  | succ n ih => rw [power,sum_conv,hp,ih,one_mul]
-
-lemma power_list_pos {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) (l : List G)
-    (hl : ∀ g ∈ l, 0 < p g) : 0 < power p l.length l.prod := by
-  induction l with
-  | nil => simp [power]
-  | cons a l ih =>
-    simpa only [List.length_cons,List.prod_cons,power] using
-      conv_pos_of hp (power_nonneg hp l.length) (hl a (by simp))
-        (ih (fun g hg => hl g (by simp [hg])))
-
-lemma power_pos_mono {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) (h1 : 0 < p 1)
-    {n m : ℕ} (hnm : n ≤ m) {g : G} (hg : 0 < power p n g) : 0 < power p m g := by
-  induction m, hnm using Nat.le_induction with
-  | base => exact hg
-  | succ m _ ih =>
-    simpa only [one_mul,power] using conv_pos_of hp (power_nonneg hp m) h1 ih
-
-/-- Finite generation and a positive holding probability produce a genuine
-finite-time positive minorant, rather than a spectral-gap assumption. -/
-theorem exists_positive_power {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) (h1 : 0 < p 1)
-    (hgen : Subgroup.closure {g | 0 < p g}=⊤) :
-    ∃ n : ℕ, ∀ g, 0 < power p (2^n) g := by
-  have hx (g : G) : g ∈ Submonoid.closure {g | 0 < p g} := by
-    rw [←Subgroup.closure_toSubmonoid_of_finite,hgen]
-    trivial
-  choose l hl hprod using fun g => Submonoid.exists_list_of_mem_closure (hx g)
-  let n := Finset.univ.sup (fun g => (l g).length)
-  refine ⟨n,fun g => ?_⟩
-  have hlen : (l g).length ≤ n := Finset.le_sup (f := fun g => (l g).length) (Finset.mem_univ g)
-  have hpow : n ≤ 2^n := Nat.le_of_lt (Nat.lt_two_pow_self)
-  have hpos := power_list_pos hp (l g) (hl g)
-  rw [hprod g] at hpos
-  exact power_pos_mono hp h1 (hlen.trans hpow) hpos
-
-structure Minorant (p : G → ℝ) where
-  exponent : ℕ
-  mass : ℝ
-  mass_pos : 0 < mass
-  mass_le_one : mass ≤ 1
-  lower : ∀ g, mass/(Fintype.card G:ℝ) ≤ power p (2^exponent) g
-
-noncomputable def minorant {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) (hsum : ∑ g,p g=1)
-    (h1 : 0 < p 1) (hgen : Subgroup.closure {g | 0 < p g}=⊤) : Minorant p := by
-  let n := Classical.choose (exists_positive_power hp h1 hgen)
-  have hn := Classical.choose_spec (exists_positive_power hp h1 hgen)
-  let a := Finset.univ.inf' Finset.univ_nonempty (power p (2^n))
-  have ha : 0 < a := (Finset.lt_inf'_iff _).mpr (fun g _ => hn g)
-  have hag (g : G) : a ≤ power p (2^n) g := Finset.inf'_le _ (Finset.mem_univ g)
-  have hN : (0:ℝ) < Fintype.card G := Nat.cast_pos.mpr Fintype.card_pos
-  have hsum' := Finset.sum_le_sum (s := Finset.univ) (fun g _ => hag g)
-  simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,sum_power hsum] at hsum'
-  refine ⟨n,(Fintype.card G:ℝ)*a,mul_pos hN ha,hsum',fun g => ?_⟩
-  have he : (Fintype.card G:ℝ)*a/(Fintype.card G:ℝ)=a := by field_simp
-  rw [he]
-  exact hag g
-
-end CubeShuffle.FiniteConvolution
-namespace CubeShuffle.UnitaryFinite
-open scoped BigOperators ComplexConjugate Classical
-
-variable {G : Type*} [Group G] [Fintype G]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-noncomputable def continuousRepresentation (ρ : Representation ℂ G V) : G →* (V →L[ℂ] V) where
-  toFun g := (ρ g).toContinuousLinearMap
-  map_one' := by ext v; simp
-  map_mul' g h := by ext v; simp
-
-omit [Fintype G] in
-@[simp] lemma continuousRepresentation_apply (ρ : Representation ℂ G V) (g : G) (v : V) :
-    continuousRepresentation ρ g v=ρ g v := rfl
-
-omit [Fintype G] [FiniteDimensional ℂ V] in
-lemma unitary_norm (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (g : G) (v : V) :
-    ‖ρ g v‖=‖v‖ := by
-  rw [norm_eq_sqrt_re_inner (𝕜 := ℂ),hρ,norm_eq_sqrt_re_inner (𝕜 := ℂ)]
-
-omit [Fintype G] in
-lemma continuousRepresentation_norm_le (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (g : G) :
-    ‖continuousRepresentation ρ g‖≤1 := by
-  apply ContinuousLinearMap.opNorm_le_bound _ zero_le_one
-  intro v
-  simp only [continuousRepresentation_apply,unitary_norm ρ hρ,one_mul,le_refl]
-
-omit [Fintype G] in
-lemma continuousRepresentation_adjoint (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (g : G) :
-    (continuousRepresentation ρ g).adjoint=continuousRepresentation ρ g⁻¹ := by
-  ext w
-  apply ext_inner_left ℂ
-  intro v
-  rw [ContinuousLinearMap.adjoint_inner_right]
-  exact (unitary_inner_inv ρ hρ g v w).symm
-
-noncomputable def weightedOperator (ρ : Representation ℂ G V) (p : G → ℝ) : V →L[ℂ] V :=
-  ∑ g, (p g : ℂ) • continuousRepresentation ρ g
-
-lemma weightedOperator_norm_le (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
-    (p : G → ℝ) (hp : ∀ g, 0≤p g) : ‖weightedOperator ρ p‖ ≤ ∑ g, p g := by
-  apply (norm_sum_le _ _).trans
-  apply Finset.sum_le_sum
-  intro g _
-  rw [norm_smul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (hp g)]
-  exact mul_le_of_le_one_right (hp g) (continuousRepresentation_norm_le ρ hρ g)
-
-lemma weightedOperator_symmetric (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
-    (p : G → ℝ) (hp : ∀ g, p g⁻¹=p g) : IsSelfAdjoint (weightedOperator ρ p) := by
-  change (weightedOperator ρ p).adjoint=weightedOperator ρ p
-  unfold weightedOperator
-  rw [map_sum]
-  simp only [map_smulₛₗ,Complex.conj_ofReal,continuousRepresentation_adjoint ρ hρ]
-  apply Fintype.sum_equiv (Equiv.inv G)
-  intro g
-  simp [hp]
-
-noncomputable def uniformOperator (ρ : Representation ℂ G V) : V →L[ℂ] V :=
-  weightedOperator ρ (fun _ => (Fintype.card G:ℝ)⁻¹)
-
-lemma uniformOperator_symmetric (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) :
-    IsSelfAdjoint (uniformOperator ρ) := weightedOperator_symmetric ρ hρ _ (fun _ => rfl)
-
-lemma uniformOperator_left (ρ : Representation ℂ G V) (g : G) :
-    continuousRepresentation ρ g * uniformOperator ρ=uniformOperator ρ := by
-  unfold uniformOperator weightedOperator
-  rw [Finset.mul_sum]
-  simp only [mul_smul_comm,←map_mul]
-  exact Fintype.sum_equiv (Equiv.mulLeft g) _ _ (fun _ => rfl)
-
-lemma uniformOperator_right (ρ : Representation ℂ G V) (g : G) :
-    uniformOperator ρ * continuousRepresentation ρ g=uniformOperator ρ := by
-  unfold uniformOperator weightedOperator
-  rw [Finset.sum_mul]
-  simp only [smul_mul_assoc,←map_mul]
-  exact Fintype.sum_equiv (Equiv.mulRight g) _ _ (fun _ => rfl)
-
-lemma uniformOperator_idempotent (ρ : Representation ℂ G V) :
-    IsIdempotentElem (uniformOperator ρ) := by
-  change uniformOperator ρ * uniformOperator ρ=uniformOperator ρ
-  nth_rw 1 [uniformOperator,weightedOperator]
-  rw [Finset.sum_mul]
-  simp only [smul_mul_assoc,uniformOperator_left,Finset.sum_const,Finset.card_univ,
-    ←Nat.cast_smul_eq_nsmul ℂ,smul_smul,Complex.ofReal_inv,Complex.ofReal_natCast]
-  rw [mul_inv_cancel₀ (by exact_mod_cast Fintype.card_ne_zero),one_smul]
-
-lemma uniformOperator_projection (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) :
-    IsStarProjection (uniformOperator ρ) :=
-  ⟨uniformOperator_idempotent ρ,uniformOperator_symmetric ρ hρ⟩
-
-lemma weighted_uniform_left (ρ : Representation ℂ G V) (p : G → ℝ) :
-    weightedOperator ρ p * uniformOperator ρ=(∑ g,p g : ℝ) • uniformOperator ρ := by
-  unfold weightedOperator
-  rw [Finset.sum_mul]
-  simp only [smul_mul_assoc,uniformOperator_left,←Finset.sum_smul,←Complex.ofReal_sum]
-  rfl
-
-lemma weighted_uniform_right (ρ : Representation ℂ G V) (p : G → ℝ) :
-    uniformOperator ρ * weightedOperator ρ p=(∑ g,p g : ℝ) • uniformOperator ρ := by
-  unfold weightedOperator
-  rw [Finset.mul_sum]
-  simp only [mul_smul_comm,uniformOperator_right,←Finset.sum_smul,←Complex.ofReal_sum]
-  rfl
-
-/-- A finite Doeblin minorant gives a dimension-free contraction off invariants;
-no irreducible decomposition or unproved spectral-gap hypothesis is used. -/
-theorem doeblin_off_invariants (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
-    (p : G → ℝ) (hp : ∑ g,p g=1) (ε : ℝ)
-    (hminor : ∀ g, ε/(Fintype.card G:ℝ)≤p g) :
-    ‖weightedOperator ρ p * (1-uniformOperator ρ)‖≤1-ε := by
-  let q : G → ℝ := fun g => p g-ε/(Fintype.card G:ℝ)
-  have hq : ∀ g, 0≤q g := fun g => sub_nonneg.mpr (hminor g)
-  have hcard : (Fintype.card G:ℝ)≠0 := by exact_mod_cast Fintype.card_ne_zero
-  have hsum : ∑ g,q g=1-ε := by
-    simp only [q,Finset.sum_sub_distrib,hp,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
-    field_simp
-  have he : weightedOperator ρ p=weightedOperator ρ q+(ε:ℂ) • uniformOperator ρ := by
-    simp only [uniformOperator,weightedOperator]
-    rw [Finset.smul_sum,←Finset.sum_add_distrib]
-    apply Finset.sum_congr rfl
-    intro g _
-    simp only [q,Complex.ofReal_sub,Complex.ofReal_div,smul_smul,←add_smul,
-      Complex.ofReal_inv,Complex.ofReal_natCast]
-    congr 1
-    field_simp
-    ring
-  have hz : uniformOperator ρ*(1-uniformOperator ρ)=0 := by
-    rw [mul_sub,mul_one,(uniformOperator_idempotent ρ).eq,sub_self]
-  rw [he,add_mul,smul_mul_assoc,hz,smul_zero,add_zero]
-  calc
-    _ ≤ ‖weightedOperator ρ q‖*‖1-uniformOperator ρ‖ := norm_mul_le _ _
-    _ ≤ (1-ε)*1 := mul_le_mul (by rw [←hsum]; exact weightedOperator_norm_le ρ hρ q hq)
-      ((uniformOperator_projection ρ hρ).one_sub.norm_le _) (norm_nonneg _)
-      (by rw [←hsum]; exact Finset.sum_nonneg (fun g _ => hq g))
-    _ = _ := mul_one _
-
-end CubeShuffle.UnitaryFinite
-namespace CubeShuffle.UnitaryFinite
-open scoped BigOperators Classical
-variable {G : Type*} [Group G] [Fintype G]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-lemma weightedOperator_conv (ρ : Representation ℂ G V) (p q : G → ℝ) :
-    weightedOperator ρ (FiniteConvolution.conv p q)=weightedOperator ρ p*weightedOperator ρ q := by
-  unfold weightedOperator FiniteConvolution.conv
-  simp only [Complex.ofReal_sum,Finset.sum_smul]
-  rw [Finset.sum_comm,Finset.sum_mul]
-  apply Finset.sum_congr rfl
-  intro h _
-  rw [Finset.mul_sum]
-  simp only [smul_mul_assoc,mul_smul_comm,smul_smul,←map_mul]
-  apply Fintype.sum_equiv (Equiv.mulLeft h⁻¹)
-  intro g
-  simp [Complex.ofReal_mul,mul_comm]
-
-lemma weightedOperator_power (ρ : Representation ℂ G V) (p : G → ℝ) (n : ℕ) :
-    weightedOperator ρ (FiniteConvolution.power p n)=(weightedOperator ρ p)^n := by
-  induction n with
-  | zero => simp [FiniteConvolution.power,weightedOperator]
-  | succ n ih => rw [FiniteConvolution.power,weightedOperator_conv,ih,pow_succ']
-
-lemma off_invariants_pow (ρ : Representation ℂ G V) (p : G → ℝ)
-    (hp : ∑ g,p g=1) (n : ℕ) (hn : n≠0) :
-    (weightedOperator ρ p-uniformOperator ρ)^n=
-      (weightedOperator ρ p)^n*(1-uniformOperator ρ) := by
-  have hAP := weighted_uniform_left ρ p
-  have hPA := weighted_uniform_right ρ p
-  rw [hp,one_smul] at hAP hPA
-  have hc : Commute (weightedOperator ρ p) (1-uniformOperator ρ) := by
-    show _*_= _*_
-    rw [mul_sub,sub_mul,mul_one,one_mul,hAP,hPA]
-  have he : weightedOperator ρ p-uniformOperator ρ=
-      weightedOperator ρ p*(1-uniformOperator ρ) := by rw [mul_sub,mul_one,hAP]
-  rw [he,hc.mul_pow,((uniformOperator_idempotent ρ).one_sub).pow_eq hn]
-
-/-- Explicit gap from the finite-time minorant; its value depends only on the
-finite group law, not on the dimension of any represented space. -/
-noncomputable def minorantGap {p : G → ℝ} (c : FiniteConvolution.Minorant p) : ℝ :=
-  1-c.mass/(2*(2:ℝ)^c.exponent)
-
-omit [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V] in
-lemma minorantGap_pos {p : G → ℝ} (c : FiniteConvolution.Minorant p) : 0 < minorantGap c := by
-  have hn : (1:ℝ) ≤ 2^c.exponent := one_le_pow₀ (by norm_num)
-  have hq : c.mass/(2*(2:ℝ)^c.exponent) ≤ 1/2 := by
-    apply (div_le_iff₀ (by positivity)).mpr
-    nlinarith [c.mass_le_one]
-  unfold minorantGap
-  linarith
-
-omit [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V] in
-lemma minorantGap_lt_one {p : G → ℝ} (c : FiniteConvolution.Minorant p) : minorantGap c < 1 := by
-  unfold minorantGap
-  exact sub_lt_self _ (div_pos c.mass_pos (by positivity))
-
-/-- The one-block strict contraction follows from generation and the genuine
-holding probability via Doeblin; this is uniform over ALL unitary representations. -/
-theorem norm_off_invariants_le_gap (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
-    (p : G → ℝ) (hp : ∑ g,p g=1) (hsym : ∀ g,p g⁻¹=p g)
-    (c : FiniteConvolution.Minorant p) :
-    ‖weightedOperator ρ p-uniformOperator ρ‖ ≤ minorantGap c := by
-  have hself := (weightedOperator_symmetric ρ hρ p hsym).sub (uniformOperator_symmetric ρ hρ)
-  have hbound := doeblin_off_invariants ρ hρ (FiniteConvolution.power p (2^c.exponent))
-    (FiniteConvolution.sum_power hp _) c.mass c.lower
-  rw [weightedOperator_power,←off_invariants_pow ρ p hp _ (by positivity),
-    hself.norm_pow_two_pow] at hbound
-  have hn : (1:ℝ) ≤ 2^c.exponent := one_le_pow₀ (by norm_num)
-  have hq : c.mass/(2*(2:ℝ)^c.exponent) ≤ 1/2 := by
-    apply (div_le_iff₀ (by positivity)).mpr
-    nlinarith [c.mass_le_one]
-  have hbern := one_add_mul_le_pow
-    (a := -(c.mass/(2*(2:ℝ)^c.exponent))) (by linarith : (-2:ℝ) ≤ -(c.mass/(2*(2:ℝ)^c.exponent)))
-    (2^c.exponent)
-  have he : 1+(2^c.exponent:ℕ)*(-(c.mass/(2*(2:ℝ)^c.exponent)))=1-c.mass/2 := by
-    push_cast
-    field_simp
-    ring
-  rw [he,show 1+ -(c.mass/(2*(2:ℝ)^c.exponent))=minorantGap c by rfl] at hbern
-  by_contra h
-  have hlt : minorantGap c < ‖weightedOperator ρ p-uniformOperator ρ‖ := lt_of_not_ge h
-  have hpows := pow_lt_pow_left₀ hlt (minorantGap_pos c).le (by positivity : (2^c.exponent:ℕ)≠0)
-  linarith [c.mass_pos]
-
-end CubeShuffle.UnitaryFinite
-
-namespace CubeShuffle.FiniteAverage
-open scoped BigOperators Classical
-variable {I Ω A : Type*} [Fintype I] [DecidableEq I] [Fintype Ω]
-    [Ring A] [Algebra ℂ A]
-
-noncomputable def orderedProduct : MultilinearMap ℂ (fun _ : I => A) A :=
-  MultilinearMap.domDomCongr (Fintype.equivFin I).symm
-    (MultilinearMap.mkPiAlgebraFin ℂ (Fintype.card I) A)
-
-lemma orderedProduct_eq (f : I → A) (hc : Pairwise fun i j => Commute (f i) (f j)) :
-    orderedProduct f=Finset.univ.noncommProd f (fun _ _ _ _ hij => hc hij) := by
-  unfold orderedProduct
-  rw [MultilinearMap.domDomCongr_apply,MultilinearMap.mkPiAlgebraFin_apply]
-  let l := List.ofFn (Fintype.equivFin I).symm
-  have hl : l.Nodup := List.nodup_ofFn.mpr (Fintype.equivFin I).symm.injective
-  have hu : l.toFinset=Finset.univ := by
-    ext i
-    simp only [List.mem_toFinset,List.mem_ofFn,Finset.mem_univ,iff_true,l]
-    exact (Fintype.equivFin I).symm.surjective i
-  have he := Finset.noncommProd_toFinset l f (by intro i hi j hj hij; exact hc hij) hl
-  rw [hu] at he
-  rw [he]
-  congr 1
-  exact List.map_ofFn.symm
-
-lemma multilinear_average (F : MultilinearMap ℂ (fun _ : I => A) A) (x : I → Ω → A) :
-    F (fun i => (Fintype.card Ω:ℂ)⁻¹ • ∑ ω,x i ω)=
-      (Fintype.card (I → Ω):ℂ)⁻¹ • ∑ ω : I → Ω,F (fun i => x i (ω i)) := by
-  rw [F.map_smul_univ,F.map_sum]
-  simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fun,Nat.cast_pow,inv_pow]
-
-/-- Exact independence factorization for commuting finite random factors. -/
-lemma ordered_average (x : I → Ω → A)
-    (hc : ∀ ω, Pairwise fun i j => Commute (x i (ω i)) (x j (ω j)))
-    (havg : Pairwise fun i j => Commute ((Fintype.card Ω:ℂ)⁻¹ • ∑ ω,x i ω)
-      ((Fintype.card Ω:ℂ)⁻¹ • ∑ ω,x j ω)) :
-    (Finset.univ.noncommProd (fun i => (Fintype.card Ω:ℂ)⁻¹ • ∑ ω,x i ω)
-      (fun _ _ _ _ hij => havg hij))=
-    (Fintype.card (I → Ω):ℂ)⁻¹ • ∑ ω : I → Ω,
-      Finset.univ.noncommProd (fun i => x i (ω i)) (fun _ _ _ _ hij => hc ω hij) := by
-  rw [←orderedProduct_eq _ havg,multilinear_average]
-  simp only [orderedProduct_eq _ (hc _)]
-
-end CubeShuffle.FiniteAverage
-
-end OAI
+import OAI.RepresentationTheory.FiniteUnitary.Coefficients
+
+namespace OAI
+
+namespace CubeShuffle.FiniteConvolution
+open scoped BigOperators Classical
+variable {G : Type*} [Group G] [Fintype G]
+
+noncomputable def conv (p q : G → ℝ) (g : G) : ℝ := ∑ h, p h*q (h⁻¹*g)
+noncomputable def power (p : G → ℝ) : ℕ → G → ℝ
+  | 0 => fun g => if g=1 then 1 else 0
+  | n+1 => conv p (power p n)
+
+lemma conv_nonneg {p q : G → ℝ} (hp : ∀ g, 0 ≤ p g) (hq : ∀ g, 0 ≤ q g) :
+    ∀ g, 0 ≤ conv p q g := fun _ => Finset.sum_nonneg (fun h _ => mul_nonneg (hp h) (hq _))
+
+lemma conv_pos_of {p q : G → ℝ} (hp : ∀ g, 0 ≤ p g) (hq : ∀ g, 0 ≤ q g)
+    {a b : G} (ha : 0 < p a) (hb : 0 < q b) : 0 < conv p q (a*b) := by
+  apply lt_of_lt_of_le (mul_pos ha hb)
+  have h := Finset.single_le_sum (fun g _ => mul_nonneg (hp g) (hq (g⁻¹*(a*b))))
+    (Finset.mem_univ a)
+  simpa [conv] using h
+
+lemma sum_conv (p q : G → ℝ) : ∑ g, conv p q g=(∑ g,p g)*(∑ g,q g) := by
+  unfold conv
+  rw [Finset.sum_comm,Finset.sum_mul]
+  apply Finset.sum_congr rfl
+  intro h _
+  rw [←Finset.mul_sum]
+  congr 1
+  exact Fintype.sum_equiv (Equiv.mulLeft h⁻¹) _ _ (fun _ => rfl)
+
+lemma power_nonneg {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) : ∀ n g, 0 ≤ power p n g := by
+  intro n
+  induction n with
+  | zero => intro g; simp only [power]; split_ifs  <;> norm_num
+  | succ n ih => exact conv_nonneg hp ih
+
+lemma sum_power {p : G → ℝ} (hp : ∑ g,p g=1) : ∀ n, ∑ g,power p n g=1 := by
+  intro n
+  induction n with
+  | zero => simp [power]
+  | succ n ih => rw [power,sum_conv,hp,ih,one_mul]
+
+lemma power_list_pos {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) (l : List G)
+    (hl : ∀ g ∈ l, 0 < p g) : 0 < power p l.length l.prod := by
+  induction l with
+  | nil => simp [power]
+  | cons a l ih =>
+    simpa only [List.length_cons,List.prod_cons,power] using
+      conv_pos_of hp (power_nonneg hp l.length) (hl a (by simp))
+        (ih (fun g hg => hl g (by simp [hg])))
+
+lemma power_pos_mono {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) (h1 : 0 < p 1)
+    {n m : ℕ} (hnm : n ≤ m) {g : G} (hg : 0 < power p n g) : 0 < power p m g := by
+  induction m, hnm using Nat.le_induction with
+  | base => exact hg
+  | succ m _ ih =>
+    simpa only [one_mul,power] using conv_pos_of hp (power_nonneg hp m) h1 ih
+
+/-- Finite generation and a positive holding probability produce a genuine
+finite-time positive minorant, rather than a spectral-gap assumption. -/
+theorem exists_positive_power {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) (h1 : 0 < p 1)
+    (hgen : Subgroup.closure {g | 0 < p g}=⊤) :
+    ∃ n : ℕ, ∀ g, 0 < power p (2^n) g := by
+  have hx (g : G) : g ∈ Submonoid.closure {g | 0 < p g} := by
+    rw [←Subgroup.closure_toSubmonoid_of_finite,hgen]
+    trivial
+  choose l hl hprod using fun g => Submonoid.exists_list_of_mem_closure (hx g)
+  let n := Finset.univ.sup (fun g => (l g).length)
+  refine ⟨n,fun g => ?_⟩
+  have hlen : (l g).length ≤ n := Finset.le_sup (f := fun g => (l g).length) (Finset.mem_univ g)
+  have hpow : n ≤ 2^n := Nat.le_of_lt (Nat.lt_two_pow_self)
+  have hpos := power_list_pos hp (l g) (hl g)
+  rw [hprod g] at hpos
+  exact power_pos_mono hp h1 (hlen.trans hpow) hpos
+
+structure Minorant (p : G → ℝ) where
+  exponent : ℕ
+  mass : ℝ
+  mass_pos : 0 < mass
+  mass_le_one : mass ≤ 1
+  lower : ∀ g, mass/(Fintype.card G:ℝ) ≤ power p (2^exponent) g
+
+noncomputable def minorant {p : G → ℝ} (hp : ∀ g, 0 ≤ p g) (hsum : ∑ g,p g=1)
+    (h1 : 0 < p 1) (hgen : Subgroup.closure {g | 0 < p g}=⊤) : Minorant p := by
+  let n := Classical.choose (exists_positive_power hp h1 hgen)
+  have hn := Classical.choose_spec (exists_positive_power hp h1 hgen)
+  let a := Finset.univ.inf' Finset.univ_nonempty (power p (2^n))
+  have ha : 0 < a := (Finset.lt_inf'_iff _).mpr (fun g _ => hn g)
+  have hag (g : G) : a ≤ power p (2^n) g := Finset.inf'_le _ (Finset.mem_univ g)
+  have hN : (0:ℝ) < Fintype.card G := Nat.cast_pos.mpr Fintype.card_pos
+  have hsum' := Finset.sum_le_sum (s := Finset.univ) (fun g _ => hag g)
+  simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,sum_power hsum] at hsum'
+  refine ⟨n,(Fintype.card G:ℝ)*a,mul_pos hN ha,hsum',fun g => ?_⟩
+  have he : (Fintype.card G:ℝ)*a/(Fintype.card G:ℝ)=a := by field_simp
+  rw [he]
+  exact hag g
+
+end CubeShuffle.FiniteConvolution
+namespace CubeShuffle.UnitaryFinite
+open scoped BigOperators ComplexConjugate Classical
+
+variable {G : Type*} [Group G] [Fintype G]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+noncomputable def continuousRepresentation (ρ : Representation ℂ G V) : G →* (V →L[ℂ] V) where
+  toFun g := (ρ g).toContinuousLinearMap
+  map_one' := by ext v; simp
+  map_mul' g h := by ext v; simp
+
+omit [Fintype G] in
+@[simp] lemma continuousRepresentation_apply (ρ : Representation ℂ G V) (g : G) (v : V) :
+    continuousRepresentation ρ g v=ρ g v := rfl
+
+omit [Fintype G] [FiniteDimensional ℂ V] in
+lemma unitary_norm (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (g : G) (v : V) :
+    ‖ρ g v‖=‖v‖ := by
+  rw [norm_eq_sqrt_re_inner (𝕜 := ℂ),hρ,norm_eq_sqrt_re_inner (𝕜 := ℂ)]
+
+omit [Fintype G] in
+lemma continuousRepresentation_norm_le (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (g : G) :
+    ‖continuousRepresentation ρ g‖≤1 := by
+  apply ContinuousLinearMap.opNorm_le_bound _ zero_le_one
+  intro v
+  simp only [continuousRepresentation_apply,unitary_norm ρ hρ,one_mul,le_refl]
+
+omit [Fintype G] in
+lemma continuousRepresentation_adjoint (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (g : G) :
+    (continuousRepresentation ρ g).adjoint=continuousRepresentation ρ g⁻¹ := by
+  ext w
+  apply ext_inner_left ℂ
+  intro v
+  rw [ContinuousLinearMap.adjoint_inner_right]
+  exact (unitary_inner_inv ρ hρ g v w).symm
+
+noncomputable def weightedOperator (ρ : Representation ℂ G V) (p : G → ℝ) : V →L[ℂ] V :=
+  ∑ g, (p g : ℂ) • continuousRepresentation ρ g
+
+lemma weightedOperator_norm_le (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
+    (p : G → ℝ) (hp : ∀ g, 0≤p g) : ‖weightedOperator ρ p‖ ≤ ∑ g, p g := by
+  apply (norm_sum_le _ _).trans
+  apply Finset.sum_le_sum
+  intro g _
+  rw [norm_smul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (hp g)]
+  exact mul_le_of_le_one_right (hp g) (continuousRepresentation_norm_le ρ hρ g)
+
+lemma weightedOperator_symmetric (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
+    (p : G → ℝ) (hp : ∀ g, p g⁻¹=p g) : IsSelfAdjoint (weightedOperator ρ p) := by
+  change (weightedOperator ρ p).adjoint=weightedOperator ρ p
+  unfold weightedOperator
+  rw [map_sum]
+  simp only [map_smulₛₗ,Complex.conj_ofReal,continuousRepresentation_adjoint ρ hρ]
+  apply Fintype.sum_equiv (Equiv.inv G)
+  intro g
+  simp [hp]
+
+noncomputable def uniformOperator (ρ : Representation ℂ G V) : V →L[ℂ] V :=
+  weightedOperator ρ (fun _ => (Fintype.card G:ℝ)⁻¹)
+
+lemma uniformOperator_symmetric (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) :
+    IsSelfAdjoint (uniformOperator ρ) := weightedOperator_symmetric ρ hρ _ (fun _ => rfl)
+
+lemma uniformOperator_left (ρ : Representation ℂ G V) (g : G) :
+    continuousRepresentation ρ g * uniformOperator ρ=uniformOperator ρ := by
+  unfold uniformOperator weightedOperator
+  rw [Finset.mul_sum]
+  simp only [mul_smul_comm,←map_mul]
+  exact Fintype.sum_equiv (Equiv.mulLeft g) _ _ (fun _ => rfl)
+
+lemma uniformOperator_right (ρ : Representation ℂ G V) (g : G) :
+    uniformOperator ρ * continuousRepresentation ρ g=uniformOperator ρ := by
+  unfold uniformOperator weightedOperator
+  rw [Finset.sum_mul]
+  simp only [smul_mul_assoc,←map_mul]
+  exact Fintype.sum_equiv (Equiv.mulRight g) _ _ (fun _ => rfl)
+
+lemma uniformOperator_idempotent (ρ : Representation ℂ G V) :
+    IsIdempotentElem (uniformOperator ρ) := by
+  change uniformOperator ρ * uniformOperator ρ=uniformOperator ρ
+  nth_rw 1 [uniformOperator,weightedOperator]
+  rw [Finset.sum_mul]
+  simp only [smul_mul_assoc,uniformOperator_left,Finset.sum_const,Finset.card_univ,
+    ←Nat.cast_smul_eq_nsmul ℂ,smul_smul,Complex.ofReal_inv,Complex.ofReal_natCast]
+  rw [mul_inv_cancel₀ (by exact_mod_cast Fintype.card_ne_zero),one_smul]
+
+lemma uniformOperator_projection (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) :
+    IsStarProjection (uniformOperator ρ) :=
+  ⟨uniformOperator_idempotent ρ,uniformOperator_symmetric ρ hρ⟩
+
+lemma weighted_uniform_left (ρ : Representation ℂ G V) (p : G → ℝ) :
+    weightedOperator ρ p * uniformOperator ρ=(∑ g,p g : ℝ) • uniformOperator ρ := by
+  unfold weightedOperator
+  rw [Finset.sum_mul]
+  simp only [smul_mul_assoc,uniformOperator_left,←Finset.sum_smul,←Complex.ofReal_sum]
+  rfl
+
+lemma weighted_uniform_right (ρ : Representation ℂ G V) (p : G → ℝ) :
+    uniformOperator ρ * weightedOperator ρ p=(∑ g,p g : ℝ) • uniformOperator ρ := by
+  unfold weightedOperator
+  rw [Finset.mul_sum]
+  simp only [mul_smul_comm,uniformOperator_right,←Finset.sum_smul,←Complex.ofReal_sum]
+  rfl
+
+/-- A finite Doeblin minorant gives a dimension-free contraction off invariants;
+no irreducible decomposition or unproved spectral-gap hypothesis is used. -/
+theorem doeblin_off_invariants (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
+    (p : G → ℝ) (hp : ∑ g,p g=1) (ε : ℝ)
+    (hminor : ∀ g, ε/(Fintype.card G:ℝ)≤p g) :
+    ‖weightedOperator ρ p * (1-uniformOperator ρ)‖≤1-ε := by
+  let q : G → ℝ := fun g => p g-ε/(Fintype.card G:ℝ)
+  have hq : ∀ g, 0≤q g := fun g => sub_nonneg.mpr (hminor g)
+  have hcard : (Fintype.card G:ℝ)≠0 := by exact_mod_cast Fintype.card_ne_zero
+  have hsum : ∑ g,q g=1-ε := by
+    simp only [q,Finset.sum_sub_distrib,hp,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
+    field_simp
+  have he : weightedOperator ρ p=weightedOperator ρ q+(ε:ℂ) • uniformOperator ρ := by
+    simp only [uniformOperator,weightedOperator]
+    rw [Finset.smul_sum,←Finset.sum_add_distrib]
+    apply Finset.sum_congr rfl
+    intro g _
+    simp only [q,Complex.ofReal_sub,Complex.ofReal_div,smul_smul,←add_smul,
+      Complex.ofReal_inv,Complex.ofReal_natCast]
+    congr 1
+    field_simp
+    ring
+  have hz : uniformOperator ρ*(1-uniformOperator ρ)=0 := by
+    rw [mul_sub,mul_one,(uniformOperator_idempotent ρ).eq,sub_self]
+  rw [he,add_mul,smul_mul_assoc,hz,smul_zero,add_zero]
+  calc
+    _ ≤ ‖weightedOperator ρ q‖*‖1-uniformOperator ρ‖ := norm_mul_le _ _
+    _ ≤ (1-ε)*1 := mul_le_mul (by rw [←hsum]; exact weightedOperator_norm_le ρ hρ q hq)
+      ((uniformOperator_projection ρ hρ).one_sub.norm_le _) (norm_nonneg _)
+      (by rw [←hsum]; exact Finset.sum_nonneg (fun g _ => hq g))
+    _ = _ := mul_one _
+
+end CubeShuffle.UnitaryFinite
+namespace CubeShuffle.UnitaryFinite
+open scoped BigOperators Classical
+variable {G : Type*} [Group G] [Fintype G]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+lemma weightedOperator_conv (ρ : Representation ℂ G V) (p q : G → ℝ) :
+    weightedOperator ρ (FiniteConvolution.conv p q)=weightedOperator ρ p*weightedOperator ρ q := by
+  unfold weightedOperator FiniteConvolution.conv
+  simp only [Complex.ofReal_sum,Finset.sum_smul]
+  rw [Finset.sum_comm,Finset.sum_mul]
+  apply Finset.sum_congr rfl
+  intro h _
+  rw [Finset.mul_sum]
+  simp only [smul_mul_assoc,mul_smul_comm,smul_smul,←map_mul]
+  apply Fintype.sum_equiv (Equiv.mulLeft h⁻¹)
+  intro g
+  simp [Complex.ofReal_mul,mul_comm]
+
+lemma weightedOperator_power (ρ : Representation ℂ G V) (p : G → ℝ) (n : ℕ) :
+    weightedOperator ρ (FiniteConvolution.power p n)=(weightedOperator ρ p)^n := by
+  induction n with
+  | zero => simp [FiniteConvolution.power,weightedOperator]
+  | succ n ih => rw [FiniteConvolution.power,weightedOperator_conv,ih,pow_succ']
+
+lemma off_invariants_pow (ρ : Representation ℂ G V) (p : G → ℝ)
+    (hp : ∑ g,p g=1) (n : ℕ) (hn : n≠0) :
+    (weightedOperator ρ p-uniformOperator ρ)^n=
+      (weightedOperator ρ p)^n*(1-uniformOperator ρ) := by
+  have hAP := weighted_uniform_left ρ p
+  have hPA := weighted_uniform_right ρ p
+  rw [hp,one_smul] at hAP hPA
+  have hc : Commute (weightedOperator ρ p) (1-uniformOperator ρ) := by
+    show _*_= _*_
+    rw [mul_sub,sub_mul,mul_one,one_mul,hAP,hPA]
+  have he : weightedOperator ρ p-uniformOperator ρ=
+      weightedOperator ρ p*(1-uniformOperator ρ) := by rw [mul_sub,mul_one,hAP]
+  rw [he,hc.mul_pow,((uniformOperator_idempotent ρ).one_sub).pow_eq hn]
+
+/-- Explicit gap from the finite-time minorant; its value depends only on the
+finite group law, not on the dimension of any represented space. -/
+noncomputable def minorantGap {p : G → ℝ} (c : FiniteConvolution.Minorant p) : ℝ :=
+  1-c.mass/(2*(2:ℝ)^c.exponent)
+
+omit [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V] in
+lemma minorantGap_pos {p : G → ℝ} (c : FiniteConvolution.Minorant p) : 0 < minorantGap c := by
+  have hn : (1:ℝ) ≤ 2^c.exponent := one_le_pow₀ (by norm_num)
+  have hq : c.mass/(2*(2:ℝ)^c.exponent) ≤ 1/2 := by
+    apply (div_le_iff₀ (by positivity)).mpr
+    nlinarith [c.mass_le_one]
+  unfold minorantGap
+  linarith
+
+omit [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V] in
+lemma minorantGap_lt_one {p : G → ℝ} (c : FiniteConvolution.Minorant p) : minorantGap c < 1 := by
+  unfold minorantGap
+  exact sub_lt_self _ (div_pos c.mass_pos (by positivity))
+
+/-- The one-block strict contraction follows from generation and the genuine
+holding probability via Doeblin; this is uniform over ALL unitary representations. -/
+theorem norm_off_invariants_le_gap (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
+    (p : G → ℝ) (hp : ∑ g,p g=1) (hsym : ∀ g,p g⁻¹=p g)
+    (c : FiniteConvolution.Minorant p) :
+    ‖weightedOperator ρ p-uniformOperator ρ‖ ≤ minorantGap c := by
+  have hself := (weightedOperator_symmetric ρ hρ p hsym).sub (uniformOperator_symmetric ρ hρ)
+  have hbound := doeblin_off_invariants ρ hρ (FiniteConvolution.power p (2^c.exponent))
+    (FiniteConvolution.sum_power hp _) c.mass c.lower
+  rw [weightedOperator_power,←off_invariants_pow ρ p hp _ (by positivity),
+    hself.norm_pow_two_pow] at hbound
+  have hn : (1:ℝ) ≤ 2^c.exponent := one_le_pow₀ (by norm_num)
+  have hq : c.mass/(2*(2:ℝ)^c.exponent) ≤ 1/2 := by
+    apply (div_le_iff₀ (by positivity)).mpr
+    nlinarith [c.mass_le_one]
+  have hbern := one_add_mul_le_pow
+    (a := -(c.mass/(2*(2:ℝ)^c.exponent))) (by linarith : (-2:ℝ) ≤ -(c.mass/(2*(2:ℝ)^c.exponent)))
+    (2^c.exponent)
+  have he : 1+(2^c.exponent:ℕ)*(-(c.mass/(2*(2:ℝ)^c.exponent)))=1-c.mass/2 := by
+    push_cast
+    field_simp
+    ring
+  rw [he,show 1+ -(c.mass/(2*(2:ℝ)^c.exponent))=minorantGap c by rfl] at hbern
+  by_contra h
+  have hlt : minorantGap c < ‖weightedOperator ρ p-uniformOperator ρ‖ := lt_of_not_ge h
+  have hpows := pow_lt_pow_left₀ hlt (minorantGap_pos c).le (by positivity : (2^c.exponent:ℕ)≠0)
+  linarith [c.mass_pos]
+
+end CubeShuffle.UnitaryFinite
+
+namespace CubeShuffle.FiniteAverage
+open scoped BigOperators Classical
+variable {I Ω A : Type*} [Fintype I] [DecidableEq I] [Fintype Ω]
+    [Ring A] [Algebra ℂ A]
+
+noncomputable def orderedProduct : MultilinearMap ℂ (fun _ : I => A) A :=
+  MultilinearMap.domDomCongr (Fintype.equivFin I).symm
+    (MultilinearMap.mkPiAlgebraFin ℂ (Fintype.card I) A)
+
+lemma orderedProduct_eq (f : I → A) (hc : Pairwise fun i j => Commute (f i) (f j)) :
+    orderedProduct f=Finset.univ.noncommProd f (fun _ _ _ _ hij => hc hij) := by
+  unfold orderedProduct
+  rw [MultilinearMap.domDomCongr_apply,MultilinearMap.mkPiAlgebraFin_apply]
+  let l := List.ofFn (Fintype.equivFin I).symm
+  have hl : l.Nodup := List.nodup_ofFn.mpr (Fintype.equivFin I).symm.injective
+  have hu : l.toFinset=Finset.univ := by
+    ext i
+    simp only [List.mem_toFinset,List.mem_ofFn,Finset.mem_univ,iff_true,l]
+    exact (Fintype.equivFin I).symm.surjective i
+  have he := Finset.noncommProd_toFinset l f (by intro i hi j hj hij; exact hc hij) hl
+  rw [hu] at he
+  rw [he]
+  congr 1
+  exact List.map_ofFn.symm
+
+lemma multilinear_average (F : MultilinearMap ℂ (fun _ : I => A) A) (x : I → Ω → A) :
+    F (fun i => (Fintype.card Ω:ℂ)⁻¹ • ∑ ω,x i ω)=
+      (Fintype.card (I → Ω):ℂ)⁻¹ • ∑ ω : I → Ω,F (fun i => x i (ω i)) := by
+  rw [F.map_smul_univ,F.map_sum]
+  simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fun,Nat.cast_pow,inv_pow]
+
+/-- Exact independence factorization for commuting finite random factors. -/
+lemma ordered_average (x : I → Ω → A)
+    (hc : ∀ ω, Pairwise fun i j => Commute (x i (ω i)) (x j (ω j)))
+    (havg : Pairwise fun i j => Commute ((Fintype.card Ω:ℂ)⁻¹ • ∑ ω,x i ω)
+      ((Fintype.card Ω:ℂ)⁻¹ • ∑ ω,x j ω)) :
+    (Finset.univ.noncommProd (fun i => (Fintype.card Ω:ℂ)⁻¹ • ∑ ω,x i ω)
+      (fun _ _ _ _ hij => havg hij))=
+    (Fintype.card (I → Ω):ℂ)⁻¹ • ∑ ω : I → Ω,
+      Finset.univ.noncommProd (fun i => x i (ω i)) (fun _ _ _ _ hij => hc ω hij) := by
+  rw [←orderedProduct_eq _ havg,multilinear_average]
+  simp only [orderedProduct_eq _ (hc _)]
+
+end CubeShuffle.FiniteAverage
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FiniteUnitary/Characters.lean b/lean/OAI/RepresentationTheory/FiniteUnitary/Characters.lean
index 06a0aa30cccc8e4db7fdc12154256714255a6806..6b1f9ed68f9446ed4cccdba2dde3807b59317cfb 100644
--- a/lean/OAI/RepresentationTheory/FiniteUnitary/Characters.lean
+++ b/lean/OAI/RepresentationTheory/FiniteUnitary/Characters.lean
@@ -1,166 +1,166 @@
-import Mathlib
-
-namespace OAI
-
-namespace CubeShuffle.FiniteCharacters
-open scoped BigOperators Classical
-
-section Group
-variable {G : Type*} [Group G] [Fintype G]
-
-noncomputable def classRep (c : ConjClasses G) : G := (ConjClasses.exists_rep c).choose
-omit [Fintype G] in
-lemma classRep_spec (c : ConjClasses G) : ConjClasses.mk (classRep c) = c :=
-  (ConjClasses.exists_rep c).choose_spec
-
-variable {V : Type*} [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
-
-noncomputable def characterClass (ρ : Representation ℂ G V) : ConjClasses G → ℂ :=
-  fun c => ρ.character (classRep c)
-
-omit [Fintype G] [FiniteDimensional ℂ V] in
-lemma characterClass_mk (ρ : Representation ℂ G V) (g : G) :
-    characterClass ρ (ConjClasses.mk g) = ρ.character g := by
-  have h := ConjClasses.mk_eq_mk_iff_isConj.mp (classRep_spec (ConjClasses.mk g))
-  obtain ⟨p,hp⟩ := isConj_iff.mp h
-  change ρ.character (classRep (ConjClasses.mk g)) = ρ.character g
-  exact (Representation.char_conj ρ _ p).symm.trans (congrArg ρ.character hp)
-
-noncomputable def characterDual (ρ : Representation ℂ G V) :
-    (ConjClasses G → ℂ) →ₗ[ℂ] ℂ where
-  toFun f := (Nat.card G : ℂ)⁻¹ * ∑ g, f (ConjClasses.mk g) * ρ.character g⁻¹
-  map_add' f h := by simp [Finset.sum_add_distrib,add_mul,mul_add]
-  map_smul' a f := by
-    simp only [Pi.smul_apply,smul_eq_mul,RingHom.id_apply]
-    simp_rw [mul_assoc]
-    rw [←Finset.mul_sum]
-    ring
-
-lemma characterDual_characterClass (ρ : Representation ℂ G V)
-    {W : Type*} [AddCommGroup W] [Module ℂ W] [FiniteDimensional ℂ W]
-    (σ : Representation ℂ G W) [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ] :
-    characterDual ρ (characterClass σ) = if Nonempty (Representation.Equiv ρ σ) then 1 else 0 := by
-  let : Invertible (Nat.card G : ℂ) := invertibleOfNonzero (by
-    exact_mod_cast (Nat.card_pos (α := G)).ne')
-  change (Nat.card G : ℂ)⁻¹ * ∑ g, characterClass σ (ConjClasses.mk g) * ρ.character g⁻¹ = _
-  simp only [characterClass_mk]
-  exact Representation.char_orthonormal σ ρ
-
-end Group
-
-/-- A finite family of pairwise inequivalent irreducibles whose size is at
-least the number of conjugacy classes is complete. This avoids assuming any
-symmetric-group classification theorem. -/
-theorem complete_of_card_conjugacy_le
-    {G ι : Type*} [Group G] [Fintype G] [Fintype ι]
-    (V : ι → Type*) [∀ i, AddCommGroup (V i)] [∀ i, Module ℂ (V i)]
-    [∀ i, FiniteDimensional ℂ (V i)]
-    (ρ : ∀ i, Representation ℂ G (V i)) [∀ i, Representation.IsIrreducible (ρ i)]
-    (hneq : ∀ i j, i ≠ j → ¬Nonempty (Representation.Equiv (ρ i) (ρ j)))
-    (hcard : Nat.card (ConjClasses G) ≤ Fintype.card ι)
-    {W : Type*} [AddCommGroup W] [Module ℂ W] [FiniteDimensional ℂ W]
-    (σ : Representation ℂ G W) [Representation.IsIrreducible σ] :
-    ∃ i, Nonempty (Representation.Equiv (ρ i) σ) := by
-  let : Fintype (ConjClasses G) := Fintype.ofFinite _
-  have horth (i j : ι) : characterDual (ρ i) (characterClass (ρ j)) = if i=j then 1 else 0 := by
-    rw [characterDual_characterClass]
-    by_cases hij : i=j
-    · subst j
-      simp only [ite_true]
-      exact ite_eq_left ⟨Representation.Equiv.refl _⟩
-    · rw [ite_eq_right hij,ite_eq_right (hneq i j hij)]
-  have hLI : LinearIndependent ℂ (fun i => characterClass (ρ i)) := by
-    apply LinearIndependent.of_pairwise_dual_eq_zero_one _ (fun i => characterDual (ρ i))
-    · intro i j hij
-      exact (horth i j).trans (ite_eq_right hij)
-    · intro i
-      simpa using horth i i
-  have heq : Fintype.card ι = Module.finrank ℂ (ConjClasses G → ℂ) := by
-    apply Nat.le_antisymm hLI.fintype_card_le_finrank
-    simpa only [Module.finrank_pi,Module.finrank_self,Finset.sum_const,Finset.card_univ,
-      smul_eq_mul,mul_one,Nat.card_eq_fintype_card] using hcard
-  have hspan := hLI.span_eq_top_of_card_eq_finrank' heq
-  by_contra h
-  push Not at h
-  have hker : Submodule.span ℂ (Set.range (fun i => characterClass (ρ i))) ≤
-      (characterDual σ).ker := by
-    apply Submodule.span_le.mpr
-    rintro f ⟨i,rfl⟩
-    change characterDual σ (characterClass (ρ i)) = 0
-    rw [characterDual_characterClass,ite_eq_right]
-    rintro ⟨T⟩
-    exact (h i).false T.symm
-  rw [hspan] at hker
-  have hz : characterDual σ (characterClass σ) = 0 := hker (show characterClass σ ∈ ⊤ from trivial)
-  rw [characterDual_characterClass,ite_eq_left ⟨Representation.Equiv.refl σ⟩] at hz
-  exact one_ne_zero hz
-
-end CubeShuffle.FiniteCharacters
-
-namespace CubeShuffle.FiniteCharacters
-open Module
-open scoped BigOperators Classical
-
-variable {G ι : Type*} [Group G] [Fintype G] [Fintype ι]
-variable (V : ι → Type*) [∀ i, AddCommGroup (V i)] [∀ i, Module ℂ (V i)]
-  [∀ i, FiniteDimensional ℂ (V i)]
-variable (ρ : ∀ i, Representation ℂ G (V i)) [∀ i, Representation.IsIrreducible (ρ i)]
-variable (hneq : ∀ i j, i≠j → ¬Nonempty (Representation.Equiv (ρ i) (ρ j)))
-  (hcard : Nat.card (ConjClasses G)≤Fintype.card ι)
-include hneq hcard
-
-lemma character_expansion (f : ConjClasses G → ℂ) :
-    f=∑ i, characterDual (ρ i) f • characterClass (ρ i) := by
-  let : Fintype (ConjClasses G) := Fintype.ofFinite _
-  have horth (i j : ι) : characterDual (ρ i) (characterClass (ρ j))=if i=j then 1 else 0 := by
-    rw [characterDual_characterClass]
-    by_cases h : i=j
-    · subst j; rw [ite_eq_left ⟨Representation.Equiv.refl _⟩,ite_eq_left rfl]
-    · rw [ite_eq_right h,ite_eq_right (hneq i j h)]
-  have hLI : LinearIndependent ℂ (fun i => characterClass (ρ i)) := by
-    apply LinearIndependent.of_pairwise_dual_eq_zero_one _ (fun i => characterDual (ρ i))
-    · intro i j hij; exact (horth i j).trans (ite_eq_right hij)
-    · intro i; simpa using horth i i
-  have heq : Fintype.card ι=Module.finrank ℂ (ConjClasses G → ℂ) := by
-    apply Nat.le_antisymm hLI.fintype_card_le_finrank
-    simpa only [Module.finrank_pi,Module.finrank_self,Finset.sum_const,Finset.card_univ,
-      smul_eq_mul,mul_one,Nat.card_eq_fintype_card] using hcard
-  let b := Basis.mk hLI (by rw [hLI.span_eq_top_of_card_eq_finrank' heq])
-  have hb (i : ι) : b i=characterClass (ρ i) := Basis.mk_apply _ _ _
-  have hd (i : ι) : b.coord i=characterDual (ρ i) := by
-    apply b.ext
-    intro j
-    rw [hb,horth]
-    simp only [←hb,Basis.coord_apply,Basis.repr_self,Finsupp.single_apply]
-    split_ifs <;> simp_all
-  have h := b.sum_repr f
-  simpa only [hb,show ∀ i,b.repr f i=characterDual (ρ i) f from fun i => congrArg (fun l => l f) (hd i)] using h.symm
-
-/-- The sum of squared dimensions is derived from the actual character basis,
-without a regular-module decomposition assumption. -/
-theorem sum_dimension_sq : ∑ i, (Module.finrank ℂ (V i))^2=Fintype.card G := by
-  let f : ConjClasses G → ℂ := fun c => if c=ConjClasses.mk (1:G) then 1 else 0
-  have hf (g : G) : f (ConjClasses.mk g)=if g=1 then 1 else 0 := by
-    simp only [f,ConjClasses.mk_eq_mk_iff_isConj,isConj_one_left]
-  have hd (i : ι) : characterDual (ρ i) f=(Fintype.card G:ℂ)⁻¹*(Module.finrank ℂ (V i):ℂ) := by
-    change (Nat.card G:ℂ)⁻¹ * ∑ g, f (ConjClasses.mk g)*(ρ i).character g⁻¹=_
-    simp only [hf,ite_mul,one_mul,zero_mul,Finset.sum_ite_eq',Finset.mem_univ,↓reduceIte,inv_one,
-      Representation.char_one,Nat.card_eq_fintype_card]
-  have he := congrFun (character_expansion V ρ hneq hcard f) (ConjClasses.mk (1:G))
-  rw [hf,ite_eq_left rfl] at he
-  simp only [Finset.sum_apply,Pi.smul_apply,smul_eq_mul,hd,characterClass_mk,Representation.char_one] at he
-  have hN : (Fintype.card G:ℂ)≠0 := by exact_mod_cast Fintype.card_ne_zero
-  have hs : (∑ i,(Module.finrank ℂ (V i):ℂ)^2)=(Fintype.card G:ℂ) := by
-    calc
-      _ = (Fintype.card G:ℂ) * ∑ i,(Fintype.card G:ℂ)⁻¹ *
-          (Module.finrank ℂ (V i):ℂ)*(Module.finrank ℂ (V i):ℂ) := by
-        rw [Finset.mul_sum]
-        apply Finset.sum_congr rfl
-        intro i _
-        field_simp
-      _ = _ := by rw [←he,mul_one]
-  exact_mod_cast hs
-
-end CubeShuffle.FiniteCharacters
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+namespace CubeShuffle.FiniteCharacters
+open scoped BigOperators Classical
+
+section Group
+variable {G : Type*} [Group G] [Fintype G]
+
+noncomputable def classRep (c : ConjClasses G) : G := (ConjClasses.exists_rep c).choose
+omit [Fintype G] in
+lemma classRep_spec (c : ConjClasses G) : ConjClasses.mk (classRep c) = c :=
+  (ConjClasses.exists_rep c).choose_spec
+
+variable {V : Type*} [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
+
+noncomputable def characterClass (ρ : Representation ℂ G V) : ConjClasses G → ℂ :=
+  fun c => ρ.character (classRep c)
+
+omit [Fintype G] [FiniteDimensional ℂ V] in
+lemma characterClass_mk (ρ : Representation ℂ G V) (g : G) :
+    characterClass ρ (ConjClasses.mk g) = ρ.character g := by
+  have h := ConjClasses.mk_eq_mk_iff_isConj.mp (classRep_spec (ConjClasses.mk g))
+  obtain ⟨p,hp⟩ := isConj_iff.mp h
+  change ρ.character (classRep (ConjClasses.mk g)) = ρ.character g
+  exact (Representation.char_conj ρ _ p).symm.trans (congrArg ρ.character hp)
+
+noncomputable def characterDual (ρ : Representation ℂ G V) :
+    (ConjClasses G → ℂ) →ₗ[ℂ] ℂ where
+  toFun f := (Nat.card G : ℂ)⁻¹ * ∑ g, f (ConjClasses.mk g) * ρ.character g⁻¹
+  map_add' f h := by simp [Finset.sum_add_distrib,add_mul,mul_add]
+  map_smul' a f := by
+    simp only [Pi.smul_apply,smul_eq_mul,RingHom.id_apply]
+    simp_rw [mul_assoc]
+    rw [←Finset.mul_sum]
+    ring
+
+lemma characterDual_characterClass (ρ : Representation ℂ G V)
+    {W : Type*} [AddCommGroup W] [Module ℂ W] [FiniteDimensional ℂ W]
+    (σ : Representation ℂ G W) [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ] :
+    characterDual ρ (characterClass σ) = if Nonempty (Representation.Equiv ρ σ) then 1 else 0 := by
+  let : Invertible (Nat.card G : ℂ) := invertibleOfNonzero (by
+    exact_mod_cast (Nat.card_pos (α := G)).ne')
+  change (Nat.card G : ℂ)⁻¹ * ∑ g, characterClass σ (ConjClasses.mk g) * ρ.character g⁻¹ = _
+  simp only [characterClass_mk]
+  exact Representation.char_orthonormal σ ρ
+
+end Group
+
+/-- A finite family of pairwise inequivalent irreducibles whose size is at
+least the number of conjugacy classes is complete. This avoids assuming any
+symmetric-group classification theorem. -/
+theorem complete_of_card_conjugacy_le
+    {G ι : Type*} [Group G] [Fintype G] [Fintype ι]
+    (V : ι → Type*) [∀ i, AddCommGroup (V i)] [∀ i, Module ℂ (V i)]
+    [∀ i, FiniteDimensional ℂ (V i)]
+    (ρ : ∀ i, Representation ℂ G (V i)) [∀ i, Representation.IsIrreducible (ρ i)]
+    (hneq : ∀ i j, i ≠ j → ¬Nonempty (Representation.Equiv (ρ i) (ρ j)))
+    (hcard : Nat.card (ConjClasses G) ≤ Fintype.card ι)
+    {W : Type*} [AddCommGroup W] [Module ℂ W] [FiniteDimensional ℂ W]
+    (σ : Representation ℂ G W) [Representation.IsIrreducible σ] :
+    ∃ i, Nonempty (Representation.Equiv (ρ i) σ) := by
+  let : Fintype (ConjClasses G) := Fintype.ofFinite _
+  have horth (i j : ι) : characterDual (ρ i) (characterClass (ρ j)) = if i=j then 1 else 0 := by
+    rw [characterDual_characterClass]
+    by_cases hij : i=j
+    · subst j
+      simp only [ite_true]
+      exact ite_eq_left ⟨Representation.Equiv.refl _⟩
+    · rw [ite_eq_right hij,ite_eq_right (hneq i j hij)]
+  have hLI : LinearIndependent ℂ (fun i => characterClass (ρ i)) := by
+    apply LinearIndependent.of_pairwise_dual_eq_zero_one _ (fun i => characterDual (ρ i))
+    · intro i j hij
+      exact (horth i j).trans (ite_eq_right hij)
+    · intro i
+      simpa using horth i i
+  have heq : Fintype.card ι = Module.finrank ℂ (ConjClasses G → ℂ) := by
+    apply Nat.le_antisymm hLI.fintype_card_le_finrank
+    simpa only [Module.finrank_pi,Module.finrank_self,Finset.sum_const,Finset.card_univ,
+      smul_eq_mul,mul_one,Nat.card_eq_fintype_card] using hcard
+  have hspan := hLI.span_eq_top_of_card_eq_finrank' heq
+  by_contra h
+  push Not at h
+  have hker : Submodule.span ℂ (Set.range (fun i => characterClass (ρ i))) ≤
+      (characterDual σ).ker := by
+    apply Submodule.span_le.mpr
+    rintro f ⟨i,rfl⟩
+    change characterDual σ (characterClass (ρ i)) = 0
+    rw [characterDual_characterClass,ite_eq_right]
+    rintro ⟨T⟩
+    exact (h i).false T.symm
+  rw [hspan] at hker
+  have hz : characterDual σ (characterClass σ) = 0 := hker (show characterClass σ ∈ ⊤ from trivial)
+  rw [characterDual_characterClass,ite_eq_left ⟨Representation.Equiv.refl σ⟩] at hz
+  exact one_ne_zero hz
+
+end CubeShuffle.FiniteCharacters
+
+namespace CubeShuffle.FiniteCharacters
+open Module
+open scoped BigOperators Classical
+
+variable {G ι : Type*} [Group G] [Fintype G] [Fintype ι]
+variable (V : ι → Type*) [∀ i, AddCommGroup (V i)] [∀ i, Module ℂ (V i)]
+  [∀ i, FiniteDimensional ℂ (V i)]
+variable (ρ : ∀ i, Representation ℂ G (V i)) [∀ i, Representation.IsIrreducible (ρ i)]
+variable (hneq : ∀ i j, i≠j → ¬Nonempty (Representation.Equiv (ρ i) (ρ j)))
+  (hcard : Nat.card (ConjClasses G)≤Fintype.card ι)
+include hneq hcard
+
+lemma character_expansion (f : ConjClasses G → ℂ) :
+    f=∑ i, characterDual (ρ i) f • characterClass (ρ i) := by
+  let : Fintype (ConjClasses G) := Fintype.ofFinite _
+  have horth (i j : ι) : characterDual (ρ i) (characterClass (ρ j))=if i=j then 1 else 0 := by
+    rw [characterDual_characterClass]
+    by_cases h : i=j
+    · subst j; rw [ite_eq_left ⟨Representation.Equiv.refl _⟩,ite_eq_left rfl]
+    · rw [ite_eq_right h,ite_eq_right (hneq i j h)]
+  have hLI : LinearIndependent ℂ (fun i => characterClass (ρ i)) := by
+    apply LinearIndependent.of_pairwise_dual_eq_zero_one _ (fun i => characterDual (ρ i))
+    · intro i j hij; exact (horth i j).trans (ite_eq_right hij)
+    · intro i; simpa using horth i i
+  have heq : Fintype.card ι=Module.finrank ℂ (ConjClasses G → ℂ) := by
+    apply Nat.le_antisymm hLI.fintype_card_le_finrank
+    simpa only [Module.finrank_pi,Module.finrank_self,Finset.sum_const,Finset.card_univ,
+      smul_eq_mul,mul_one,Nat.card_eq_fintype_card] using hcard
+  let b := Basis.mk hLI (by rw [hLI.span_eq_top_of_card_eq_finrank' heq])
+  have hb (i : ι) : b i=characterClass (ρ i) := Basis.mk_apply _ _ _
+  have hd (i : ι) : b.coord i=characterDual (ρ i) := by
+    apply b.ext
+    intro j
+    rw [hb,horth]
+    simp only [←hb,Basis.coord_apply,Basis.repr_self,Finsupp.single_apply]
+    split_ifs <;> simp_all
+  have h := b.sum_repr f
+  simpa only [hb,show ∀ i,b.repr f i=characterDual (ρ i) f from fun i => congrArg (fun l => l f) (hd i)] using h.symm
+
+/-- The sum of squared dimensions is derived from the actual character basis,
+without a regular-module decomposition assumption. -/
+theorem sum_dimension_sq : ∑ i, (Module.finrank ℂ (V i))^2=Fintype.card G := by
+  let f : ConjClasses G → ℂ := fun c => if c=ConjClasses.mk (1:G) then 1 else 0
+  have hf (g : G) : f (ConjClasses.mk g)=if g=1 then 1 else 0 := by
+    simp only [f,ConjClasses.mk_eq_mk_iff_isConj,isConj_one_left]
+  have hd (i : ι) : characterDual (ρ i) f=(Fintype.card G:ℂ)⁻¹*(Module.finrank ℂ (V i):ℂ) := by
+    change (Nat.card G:ℂ)⁻¹ * ∑ g, f (ConjClasses.mk g)*(ρ i).character g⁻¹=_
+    simp only [hf,ite_mul,one_mul,zero_mul,Finset.sum_ite_eq',Finset.mem_univ,↓reduceIte,inv_one,
+      Representation.char_one,Nat.card_eq_fintype_card]
+  have he := congrFun (character_expansion V ρ hneq hcard f) (ConjClasses.mk (1:G))
+  rw [hf,ite_eq_left rfl] at he
+  simp only [Finset.sum_apply,Pi.smul_apply,smul_eq_mul,hd,characterClass_mk,Representation.char_one] at he
+  have hN : (Fintype.card G:ℂ)≠0 := by exact_mod_cast Fintype.card_ne_zero
+  have hs : (∑ i,(Module.finrank ℂ (V i):ℂ)^2)=(Fintype.card G:ℂ) := by
+    calc
+      _ = (Fintype.card G:ℂ) * ∑ i,(Fintype.card G:ℂ)⁻¹ *
+          (Module.finrank ℂ (V i):ℂ)*(Module.finrank ℂ (V i):ℂ) := by
+        rw [Finset.mul_sum]
+        apply Finset.sum_congr rfl
+        intro i _
+        field_simp
+      _ = _ := by rw [←he,mul_one]
+  exact_mod_cast hs
+
+end CubeShuffle.FiniteCharacters
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FiniteUnitary/Coefficients.lean b/lean/OAI/RepresentationTheory/FiniteUnitary/Coefficients.lean
index b2dc97472063274d0f6040488579aa0be6b27424..638056494edd6bb5d10608fce3c4717a648d402f 100644
--- a/lean/OAI/RepresentationTheory/FiniteUnitary/Coefficients.lean
+++ b/lean/OAI/RepresentationTheory/FiniteUnitary/Coefficients.lean
@@ -1,216 +1,216 @@
-import Mathlib
-
-namespace OAI
-
-namespace CubeShuffle.UnitaryFinite
-open scoped BigOperators ComplexConjugate Classical
-
-variable {G : Type*} [Group G] [Fintype G]
-variable {V W : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-  [NormedAddCommGroup W] [InnerProductSpace ℂ W] [FiniteDimensional ℂ W]
-
-def IsUnitary (ρ : Representation ℂ G V) : Prop :=
-  ∀ g v w, inner ℂ (ρ g v) (ρ g w) = inner ℂ v w
-
-omit [Fintype G] [FiniteDimensional ℂ V] in
-lemma unitary_inner_inv (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (g : G) (v w : V) :
-    inner ℂ v (ρ g⁻¹ w) = inner ℂ (ρ g v) w := by
-  have h := hρ g v (ρ g⁻¹ w)
-  have he : ρ g (ρ g⁻¹ w) = w := by
-    change (ρ g * ρ g⁻¹) w = w
-    rw [←map_mul,mul_inv_cancel,map_one]
-    rfl
-  rw [he] at h
-  exact h.symm
-
-noncomputable def averageMap (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
-    (A : V →ₗ[ℂ] W) : Representation.IntertwiningMap ρ σ where
-  toLinearMap := ∑ g, (σ g).comp (A.comp (ρ g⁻¹))
-  isIntertwining' h := by
-    ext v
-    simp only [LinearMap.comp_apply,LinearMap.sum_apply,map_sum]
-    apply Fintype.sum_equiv (Equiv.mulLeft h⁻¹)
-    intro g
-    simp only [Equiv.coe_mulLeft,mul_inv_rev,inv_inv,map_mul,Module.End.mul_apply]
-    have hh (w : W) : σ h (σ h⁻¹ w)=w := by
-      change (σ h*σ h⁻¹) w=w
-      rw [←map_mul,mul_inv_cancel,map_one]
-      rfl
-    rw [hh]
-
-omit [FiniteDimensional ℂ V] in
-lemma averageMap_self_trace (ρ : Representation ℂ G V) (A : Module.End ℂ V) :
-    LinearMap.trace ℂ V (averageMap ρ ρ A).toLinearMap =
-      (Fintype.card G:ℂ) * LinearMap.trace ℂ V A := by
-  change LinearMap.trace ℂ V (∑ g, ρ g * A * ρ g⁻¹) = _
-  rw [map_sum]
-  have hj (g : G) : LinearMap.trace ℂ V (ρ g*A*ρ g⁻¹) = LinearMap.trace ℂ V A := by
-    rw [LinearMap.trace_mul_cycle,←map_mul,inv_mul_cancel,map_one,one_mul]
-  simp only [hj,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
-
-lemma averageMap_scalar (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
-    (A : Module.End ℂ V) :
-    ∃ c : ℂ, (averageMap ρ ρ A).toLinearMap = c • LinearMap.id ∧
-      c * (Module.finrank ℂ V:ℂ) = (Fintype.card G:ℂ)*LinearMap.trace ℂ V A := by
-  obtain ⟨c,hc⟩ := (Representation.IsIrreducible.algebraMap_intertwiningMap_bijective_of_isAlgClosed (ρ := ρ)).surjective (averageMap ρ ρ A)
-  have he : (averageMap ρ ρ A).toLinearMap = c • LinearMap.id := by
-    rw [←hc]
-    rfl
-  refine ⟨c,he,?_⟩
-  rw [←averageMap_self_trace ρ A,he,map_smul,LinearMap.trace_id,smul_eq_mul]
-
-/-- Schur orthogonality in an actual unitary irreducible, obtained by
-averaging a rank-one operator and applying the proved algebraic Schur lemma. -/
-theorem coefficient_orthogonality (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
-    (hρ : IsUnitary ρ) (u v w z : V) :
-    (Module.finrank ℂ V:ℂ) * ∑ g, inner ℂ u (ρ g v) * inner ℂ (ρ g w) z =
-      (Fintype.card G:ℂ) * inner ℂ w v * inner ℂ u z := by
-  obtain ⟨c,hc,htrace⟩ := averageMap_scalar ρ (InnerProductSpace.rankOne ℂ v w).toLinearMap
-  rw [InnerProductSpace.trace_rankOne] at htrace
-  have happ := congrArg (fun A : Module.End ℂ V => inner ℂ u (A z)) hc
-  change inner ℂ u ((∑ g, ρ g * (InnerProductSpace.rankOne ℂ v w).toLinearMap * ρ g⁻¹) z) = _ at happ
-  simp only [LinearMap.sum_apply,Module.End.mul_apply,ContinuousLinearMap.coe_coe,
-    InnerProductSpace.rankOne_apply,map_smul,unitary_inner_inv ρ hρ,inner_sum,
-    inner_smul_right,LinearMap.smul_apply,LinearMap.id_apply] at happ
-  have hsum : ∑ g, inner ℂ u (ρ g v) * inner ℂ (ρ g w) z = c * inner ℂ u z := by
-    simpa only [mul_comm] using happ
-  rw [hsum]
-  calc
-    _ = (c*(Module.finrank ℂ V:ℂ))*inner ℂ u z := by ring
-    _ = _ := by rw [htrace]
-
-end CubeShuffle.UnitaryFinite
-namespace CubeShuffle.UnitaryFinite
-open scoped BigOperators ComplexConjugate Classical
-
-section Action
-variable {G X : Type*} [Group G] [Fintype G] [Fintype X]
-  (a : G →* Equiv.Perm X) (x₀ : X) (ha : ∀ x, ∃ g, a g x₀=x)
-
-noncomputable def representative (x : X) : G := Classical.choose (ha x)
-
-omit [Fintype G] [Fintype X] in
-lemma representative_apply (x : X) : a (representative a x₀ ha x) x₀=x :=
-  Classical.choose_spec (ha x)
-
-include ha in
-lemma orbit_sum (F : X → ℂ) :
-    (Fintype.card X:ℂ)*∑ g, F (a g x₀) = (Fintype.card G:ℂ)*∑ x, F x := by
-  have ht (x : X) : ∑ g, F (a g x)=∑ g, F (a g x₀) := by
-    obtain ⟨t,rfl⟩ := ha x
-    symm
-    apply Fintype.sum_equiv (Equiv.mulRight t⁻¹)
-    intro g
-    simp
-  calc
-    _ = ∑ x : X, ∑ g : G, F (a g x) := by simp only [ht,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
-    _ = ∑ g : G, ∑ x : X, F (a g x) := Finset.sum_comm
-    _ = ∑ g : G, ∑ x : X, F x := by
-      apply Finset.sum_congr rfl
-      intro g _
-      exact Equiv.sum_comp (a g) F
-    _ = _ := by simp
-
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V)
-
-def IsFixed (w : V) : Prop := ∀ g, a g x₀=x₀ → ρ g w=w
-
-noncomputable def coefficient (w v : V) : EuclideanSpace ℂ X :=
-  WithLp.toLp 2 (fun x => inner ℂ (ρ (representative a x₀ ha x) w) v)
-
-omit [Fintype G] [Fintype X] [FiniteDimensional ℂ V] in
-lemma coefficient_at (w : V) (hw : IsFixed a x₀ ρ w) (v : V) (g : G) :
-    coefficient a x₀ ha ρ w v (a g x₀) = inner ℂ (ρ g w) v := by
-  let r := representative a x₀ ha (a g x₀)
-  have hr : a r x₀=a g x₀ := representative_apply a x₀ ha _
-  have hs : a (g⁻¹*r) x₀=x₀ := by
-    simp only [map_mul,Equiv.Perm.mul_apply]
-    rw [hr,←Equiv.Perm.mul_apply,←map_mul,inv_mul_cancel,map_one]
-    rfl
-  have he := congrArg (ρ g) (hw (g⁻¹*r) hs)
-  have hh : ρ r w=ρ g w := by
-    simpa only [map_mul,Module.End.mul_apply,←Module.End.mul_apply,←map_mul,
-      mul_inv_cancel_left] using he
-  change inner ℂ (ρ r w) v= _
-  rw [hh]
-
-lemma coefficient_inner [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
-    (w z : V) (hw : IsFixed a x₀ ρ w) (hz : IsFixed a x₀ ρ z) (v t : V) :
-    (Module.finrank ℂ V:ℂ)*inner ℂ (coefficient a x₀ ha ρ w v) (coefficient a x₀ ha ρ z t) =
-      (Fintype.card X:ℂ)*inner ℂ z w*inner ℂ v t := by
-  have hs := orbit_sum a x₀ ha (fun x => star (coefficient a x₀ ha ρ w v x)*coefficient a x₀ ha ρ z t x)
-  simp only [coefficient_at a x₀ ha ρ w hw,coefficient_at a x₀ ha ρ z hz] at hs
-  simp only [←starRingEnd_apply,inner_conj_symm] at hs
-  simp only [starRingEnd_apply] at hs
-  have ho := coefficient_orthogonality ρ hρ v w z t
-  have hcard : (Fintype.card G:ℂ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
-  apply mul_left_cancel₀ hcard
-  rw [PiLp.inner_apply]
-  simp only [RCLike.inner_apply,starRingEnd_apply]
-  have hsum : ∑ x, coefficient a x₀ ha ρ z t x * star (coefficient a x₀ ha ρ w v x) =
-      ∑ x, star (coefficient a x₀ ha ρ w v x)*coefficient a x₀ ha ρ z t x := by
-    apply Finset.sum_congr rfl
-    intro x _
-    ring
-  rw [hsum]
-  calc
-    _ = (Module.finrank ℂ V:ℂ)*((Fintype.card G:ℂ)*∑ x,
-        star (coefficient a x₀ ha ρ w v x)*coefficient a x₀ ha ρ z t x) := by ring
-    _ = (Fintype.card X:ℂ)*((Module.finrank ℂ V:ℂ)*∑ g,
-        inner ℂ v (ρ g w)*inner ℂ (ρ g z) t) := by rw [←hs]; ring
-    _ = _ := by rw [ho]; ring
-
-omit [Fintype G] [Fintype X] [FiniteDimensional ℂ V] in
-lemma coefficient_action (hρ : IsUnitary ρ) (w : V) (hw : IsFixed a x₀ ρ w)
-    (v : V) (g : G) (x : X) :
-    coefficient a x₀ ha ρ w v (a g x) = coefficient a x₀ ha ρ w (ρ g⁻¹ v) x := by
-  obtain ⟨r,rfl⟩ := ha x
-  rw [←Equiv.Perm.mul_apply,←map_mul,coefficient_at a x₀ ha ρ w hw,
-    coefficient_at a x₀ ha ρ w hw]
-  simp only [map_mul,Module.End.mul_apply,unitary_inner_inv ρ hρ]
-
-noncomputable def normalizedCoefficient (w v : V) : EuclideanSpace ℂ X :=
-  (Real.sqrt ((Module.finrank ℂ V:ℝ)/(Fintype.card X:ℝ)) : ℂ) • coefficient a x₀ ha ρ w v
-
-lemma normalizedCoefficient_inner [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
-    (w z : V) (hw : IsFixed a x₀ ρ w) (hz : IsFixed a x₀ ρ z) (v t : V) :
-    inner ℂ (normalizedCoefficient a x₀ ha ρ w v) (normalizedCoefficient a x₀ ha ρ z t) =
-      inner ℂ z w*inner ℂ v t := by
-  let : Nonempty X := ⟨x₀⟩
-  have hX : (Fintype.card X:ℂ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
-  have hs : (Real.sqrt ((Module.finrank ℂ V:ℝ)/(Fintype.card X:ℝ)) : ℂ)^2 =
-      (Module.finrank ℂ V:ℂ)/(Fintype.card X:ℂ) := by
-    rw [←Complex.ofReal_pow,Real.sq_sqrt (by positivity)]
-    simp
-  simp only [normalizedCoefficient,inner_smul_left,inner_smul_right]
-  simp only [Complex.conj_ofReal]
-  rw [←mul_assoc,←pow_two,hs]
-  have hi := coefficient_inner a x₀ ha ρ hρ w z hw hz v t
-  field_simp
-  simpa only [mul_assoc] using hi
-
-lemma normalizedCoefficient_orthonormal [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
-    {I : Type*} (w : I → V) (hw : Orthonormal ℂ w)
-    (hfix : ∀ i, IsFixed a x₀ ρ (w i)) (v : V) (hv : ‖v‖=1) :
-    Orthonormal ℂ (fun i => normalizedCoefficient a x₀ ha ρ (w i) v) := by
-  rw [orthonormal_iff_ite] at hw ⊢
-  intro i j
-  rw [normalizedCoefficient_inner a x₀ ha ρ hρ _ _ (hfix i) (hfix j),hw,
-    inner_self_eq_norm_sq_to_K,hv]
-  simp [eq_comm]
-
-lemma normalizedCoefficient_norm [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
-    (w : V) (hw : IsFixed a x₀ ρ w) (hn : ‖w‖=1) (v : V) :
-    ‖normalizedCoefficient a x₀ ha ρ w v‖=‖v‖ := by
-  have he := normalizedCoefficient_inner a x₀ ha ρ hρ w w hw hw v v
-  rw [inner_self_eq_norm_sq_to_K,inner_self_eq_norm_sq_to_K,hn] at he
-  simp only [inner_self_eq_norm_sq_to_K] at he
-  norm_num at he
-  have hs : ‖normalizedCoefficient a x₀ ha ρ w v‖^2=‖v‖^2 := by exact_mod_cast he
-  nlinarith [norm_nonneg (normalizedCoefficient a x₀ ha ρ w v),norm_nonneg v]
-
-end Action
-end CubeShuffle.UnitaryFinite
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+namespace CubeShuffle.UnitaryFinite
+open scoped BigOperators ComplexConjugate Classical
+
+variable {G : Type*} [Group G] [Fintype G]
+variable {V W : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+  [NormedAddCommGroup W] [InnerProductSpace ℂ W] [FiniteDimensional ℂ W]
+
+def IsUnitary (ρ : Representation ℂ G V) : Prop :=
+  ∀ g v w, inner ℂ (ρ g v) (ρ g w) = inner ℂ v w
+
+omit [Fintype G] [FiniteDimensional ℂ V] in
+lemma unitary_inner_inv (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (g : G) (v w : V) :
+    inner ℂ v (ρ g⁻¹ w) = inner ℂ (ρ g v) w := by
+  have h := hρ g v (ρ g⁻¹ w)
+  have he : ρ g (ρ g⁻¹ w) = w := by
+    change (ρ g * ρ g⁻¹) w = w
+    rw [←map_mul,mul_inv_cancel,map_one]
+    rfl
+  rw [he] at h
+  exact h.symm
+
+noncomputable def averageMap (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
+    (A : V →ₗ[ℂ] W) : Representation.IntertwiningMap ρ σ where
+  toLinearMap := ∑ g, (σ g).comp (A.comp (ρ g⁻¹))
+  isIntertwining' h := by
+    ext v
+    simp only [LinearMap.comp_apply,LinearMap.sum_apply,map_sum]
+    apply Fintype.sum_equiv (Equiv.mulLeft h⁻¹)
+    intro g
+    simp only [Equiv.coe_mulLeft,mul_inv_rev,inv_inv,map_mul,Module.End.mul_apply]
+    have hh (w : W) : σ h (σ h⁻¹ w)=w := by
+      change (σ h*σ h⁻¹) w=w
+      rw [←map_mul,mul_inv_cancel,map_one]
+      rfl
+    rw [hh]
+
+omit [FiniteDimensional ℂ V] in
+lemma averageMap_self_trace (ρ : Representation ℂ G V) (A : Module.End ℂ V) :
+    LinearMap.trace ℂ V (averageMap ρ ρ A).toLinearMap =
+      (Fintype.card G:ℂ) * LinearMap.trace ℂ V A := by
+  change LinearMap.trace ℂ V (∑ g, ρ g * A * ρ g⁻¹) = _
+  rw [map_sum]
+  have hj (g : G) : LinearMap.trace ℂ V (ρ g*A*ρ g⁻¹) = LinearMap.trace ℂ V A := by
+    rw [LinearMap.trace_mul_cycle,←map_mul,inv_mul_cancel,map_one,one_mul]
+  simp only [hj,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
+
+lemma averageMap_scalar (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
+    (A : Module.End ℂ V) :
+    ∃ c : ℂ, (averageMap ρ ρ A).toLinearMap = c • LinearMap.id ∧
+      c * (Module.finrank ℂ V:ℂ) = (Fintype.card G:ℂ)*LinearMap.trace ℂ V A := by
+  obtain ⟨c,hc⟩ := (Representation.IsIrreducible.algebraMap_intertwiningMap_bijective_of_isAlgClosed (ρ := ρ)).surjective (averageMap ρ ρ A)
+  have he : (averageMap ρ ρ A).toLinearMap = c • LinearMap.id := by
+    rw [←hc]
+    rfl
+  refine ⟨c,he,?_⟩
+  rw [←averageMap_self_trace ρ A,he,map_smul,LinearMap.trace_id,smul_eq_mul]
+
+/-- Schur orthogonality in an actual unitary irreducible, obtained by
+averaging a rank-one operator and applying the proved algebraic Schur lemma. -/
+theorem coefficient_orthogonality (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
+    (hρ : IsUnitary ρ) (u v w z : V) :
+    (Module.finrank ℂ V:ℂ) * ∑ g, inner ℂ u (ρ g v) * inner ℂ (ρ g w) z =
+      (Fintype.card G:ℂ) * inner ℂ w v * inner ℂ u z := by
+  obtain ⟨c,hc,htrace⟩ := averageMap_scalar ρ (InnerProductSpace.rankOne ℂ v w).toLinearMap
+  rw [InnerProductSpace.trace_rankOne] at htrace
+  have happ := congrArg (fun A : Module.End ℂ V => inner ℂ u (A z)) hc
+  change inner ℂ u ((∑ g, ρ g * (InnerProductSpace.rankOne ℂ v w).toLinearMap * ρ g⁻¹) z) = _ at happ
+  simp only [LinearMap.sum_apply,Module.End.mul_apply,ContinuousLinearMap.coe_coe,
+    InnerProductSpace.rankOne_apply,map_smul,unitary_inner_inv ρ hρ,inner_sum,
+    inner_smul_right,LinearMap.smul_apply,LinearMap.id_apply] at happ
+  have hsum : ∑ g, inner ℂ u (ρ g v) * inner ℂ (ρ g w) z = c * inner ℂ u z := by
+    simpa only [mul_comm] using happ
+  rw [hsum]
+  calc
+    _ = (c*(Module.finrank ℂ V:ℂ))*inner ℂ u z := by ring
+    _ = _ := by rw [htrace]
+
+end CubeShuffle.UnitaryFinite
+namespace CubeShuffle.UnitaryFinite
+open scoped BigOperators ComplexConjugate Classical
+
+section Action
+variable {G X : Type*} [Group G] [Fintype G] [Fintype X]
+  (a : G →* Equiv.Perm X) (x₀ : X) (ha : ∀ x, ∃ g, a g x₀=x)
+
+noncomputable def representative (x : X) : G := Classical.choose (ha x)
+
+omit [Fintype G] [Fintype X] in
+lemma representative_apply (x : X) : a (representative a x₀ ha x) x₀=x :=
+  Classical.choose_spec (ha x)
+
+include ha in
+lemma orbit_sum (F : X → ℂ) :
+    (Fintype.card X:ℂ)*∑ g, F (a g x₀) = (Fintype.card G:ℂ)*∑ x, F x := by
+  have ht (x : X) : ∑ g, F (a g x)=∑ g, F (a g x₀) := by
+    obtain ⟨t,rfl⟩ := ha x
+    symm
+    apply Fintype.sum_equiv (Equiv.mulRight t⁻¹)
+    intro g
+    simp
+  calc
+    _ = ∑ x : X, ∑ g : G, F (a g x) := by simp only [ht,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
+    _ = ∑ g : G, ∑ x : X, F (a g x) := Finset.sum_comm
+    _ = ∑ g : G, ∑ x : X, F x := by
+      apply Finset.sum_congr rfl
+      intro g _
+      exact Equiv.sum_comp (a g) F
+    _ = _ := by simp
+
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V)
+
+def IsFixed (w : V) : Prop := ∀ g, a g x₀=x₀ → ρ g w=w
+
+noncomputable def coefficient (w v : V) : EuclideanSpace ℂ X :=
+  WithLp.toLp 2 (fun x => inner ℂ (ρ (representative a x₀ ha x) w) v)
+
+omit [Fintype G] [Fintype X] [FiniteDimensional ℂ V] in
+lemma coefficient_at (w : V) (hw : IsFixed a x₀ ρ w) (v : V) (g : G) :
+    coefficient a x₀ ha ρ w v (a g x₀) = inner ℂ (ρ g w) v := by
+  let r := representative a x₀ ha (a g x₀)
+  have hr : a r x₀=a g x₀ := representative_apply a x₀ ha _
+  have hs : a (g⁻¹*r) x₀=x₀ := by
+    simp only [map_mul,Equiv.Perm.mul_apply]
+    rw [hr,←Equiv.Perm.mul_apply,←map_mul,inv_mul_cancel,map_one]
+    rfl
+  have he := congrArg (ρ g) (hw (g⁻¹*r) hs)
+  have hh : ρ r w=ρ g w := by
+    simpa only [map_mul,Module.End.mul_apply,←Module.End.mul_apply,←map_mul,
+      mul_inv_cancel_left] using he
+  change inner ℂ (ρ r w) v= _
+  rw [hh]
+
+lemma coefficient_inner [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
+    (w z : V) (hw : IsFixed a x₀ ρ w) (hz : IsFixed a x₀ ρ z) (v t : V) :
+    (Module.finrank ℂ V:ℂ)*inner ℂ (coefficient a x₀ ha ρ w v) (coefficient a x₀ ha ρ z t) =
+      (Fintype.card X:ℂ)*inner ℂ z w*inner ℂ v t := by
+  have hs := orbit_sum a x₀ ha (fun x => star (coefficient a x₀ ha ρ w v x)*coefficient a x₀ ha ρ z t x)
+  simp only [coefficient_at a x₀ ha ρ w hw,coefficient_at a x₀ ha ρ z hz] at hs
+  simp only [←starRingEnd_apply,inner_conj_symm] at hs
+  simp only [starRingEnd_apply] at hs
+  have ho := coefficient_orthogonality ρ hρ v w z t
+  have hcard : (Fintype.card G:ℂ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
+  apply mul_left_cancel₀ hcard
+  rw [PiLp.inner_apply]
+  simp only [RCLike.inner_apply,starRingEnd_apply]
+  have hsum : ∑ x, coefficient a x₀ ha ρ z t x * star (coefficient a x₀ ha ρ w v x) =
+      ∑ x, star (coefficient a x₀ ha ρ w v x)*coefficient a x₀ ha ρ z t x := by
+    apply Finset.sum_congr rfl
+    intro x _
+    ring
+  rw [hsum]
+  calc
+    _ = (Module.finrank ℂ V:ℂ)*((Fintype.card G:ℂ)*∑ x,
+        star (coefficient a x₀ ha ρ w v x)*coefficient a x₀ ha ρ z t x) := by ring
+    _ = (Fintype.card X:ℂ)*((Module.finrank ℂ V:ℂ)*∑ g,
+        inner ℂ v (ρ g w)*inner ℂ (ρ g z) t) := by rw [←hs]; ring
+    _ = _ := by rw [ho]; ring
+
+omit [Fintype G] [Fintype X] [FiniteDimensional ℂ V] in
+lemma coefficient_action (hρ : IsUnitary ρ) (w : V) (hw : IsFixed a x₀ ρ w)
+    (v : V) (g : G) (x : X) :
+    coefficient a x₀ ha ρ w v (a g x) = coefficient a x₀ ha ρ w (ρ g⁻¹ v) x := by
+  obtain ⟨r,rfl⟩ := ha x
+  rw [←Equiv.Perm.mul_apply,←map_mul,coefficient_at a x₀ ha ρ w hw,
+    coefficient_at a x₀ ha ρ w hw]
+  simp only [map_mul,Module.End.mul_apply,unitary_inner_inv ρ hρ]
+
+noncomputable def normalizedCoefficient (w v : V) : EuclideanSpace ℂ X :=
+  (Real.sqrt ((Module.finrank ℂ V:ℝ)/(Fintype.card X:ℝ)) : ℂ) • coefficient a x₀ ha ρ w v
+
+lemma normalizedCoefficient_inner [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
+    (w z : V) (hw : IsFixed a x₀ ρ w) (hz : IsFixed a x₀ ρ z) (v t : V) :
+    inner ℂ (normalizedCoefficient a x₀ ha ρ w v) (normalizedCoefficient a x₀ ha ρ z t) =
+      inner ℂ z w*inner ℂ v t := by
+  let : Nonempty X := ⟨x₀⟩
+  have hX : (Fintype.card X:ℂ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
+  have hs : (Real.sqrt ((Module.finrank ℂ V:ℝ)/(Fintype.card X:ℝ)) : ℂ)^2 =
+      (Module.finrank ℂ V:ℂ)/(Fintype.card X:ℂ) := by
+    rw [←Complex.ofReal_pow,Real.sq_sqrt (by positivity)]
+    simp
+  simp only [normalizedCoefficient,inner_smul_left,inner_smul_right]
+  simp only [Complex.conj_ofReal]
+  rw [←mul_assoc,←pow_two,hs]
+  have hi := coefficient_inner a x₀ ha ρ hρ w z hw hz v t
+  field_simp
+  simpa only [mul_assoc] using hi
+
+lemma normalizedCoefficient_orthonormal [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
+    {I : Type*} (w : I → V) (hw : Orthonormal ℂ w)
+    (hfix : ∀ i, IsFixed a x₀ ρ (w i)) (v : V) (hv : ‖v‖=1) :
+    Orthonormal ℂ (fun i => normalizedCoefficient a x₀ ha ρ (w i) v) := by
+  rw [orthonormal_iff_ite] at hw ⊢
+  intro i j
+  rw [normalizedCoefficient_inner a x₀ ha ρ hρ _ _ (hfix i) (hfix j),hw,
+    inner_self_eq_norm_sq_to_K,hv]
+  simp [eq_comm]
+
+lemma normalizedCoefficient_norm [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
+    (w : V) (hw : IsFixed a x₀ ρ w) (hn : ‖w‖=1) (v : V) :
+    ‖normalizedCoefficient a x₀ ha ρ w v‖=‖v‖ := by
+  have he := normalizedCoefficient_inner a x₀ ha ρ hρ w w hw hw v v
+  rw [inner_self_eq_norm_sq_to_K,inner_self_eq_norm_sq_to_K,hn] at he
+  simp only [inner_self_eq_norm_sq_to_K] at he
+  norm_num at he
+  have hs : ‖normalizedCoefficient a x₀ ha ρ w v‖^2=‖v‖^2 := by exact_mod_cast he
+  nlinarith [norm_nonneg (normalizedCoefficient a x₀ ha ρ w v),norm_nonneg v]
+
+end Action
+end CubeShuffle.UnitaryFinite
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FiniteUnitary/CommutingBlocks.lean b/lean/OAI/RepresentationTheory/FiniteUnitary/CommutingBlocks.lean
index 1c011df242a68cf70e45895c2216e0bba271b0f5..166b4afc3f46cc0728a4a051b5f1c4cf382b09f3 100644
--- a/lean/OAI/RepresentationTheory/FiniteUnitary/CommutingBlocks.lean
+++ b/lean/OAI/RepresentationTheory/FiniteUnitary/CommutingBlocks.lean
@@ -1,404 +1,404 @@
-import Mathlib
-
-namespace OAI
-
-namespace CubeShuffle.CommutingBlocks
-open scoped BigOperators Classical
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
-variable {ι : Type*} [Fintype ι]
-
-omit [CompleteSpace V] in
-lemma orthogonal_sum_norm_sq (v : ι → V) (hv : ∀ i j, i≠j → inner ℂ (v i) (v j)=0) :
-    ‖∑ i,v i‖^2=∑ i,‖v i‖^2 := by
-  rw [norm_sq_eq_re_inner (𝕜 := ℂ),sum_inner]
-  simp only [inner_sum,map_sum]
-  apply Finset.sum_congr rfl
-  intro i _
-  rw [Finset.sum_eq_single i]
-  · exact (norm_sq_eq_re_inner (𝕜 := ℂ) (v i)).symm
-  · intro j _ hji
-    rw [hv i j hji.symm]
-    rfl
-  · simp
-
-lemma projection_inner_zero (P Q : V →L[ℂ] V) (hP : IsStarProjection P)
-    (hPQ : P*Q=0) (v w : V) : inner ℂ (P v) (Q w)=0 := by
-  rw [←ContinuousLinearMap.adjoint_inner_right]
-  change inner ℂ v ((star P*Q) w)=0
-  rw [hP.isSelfAdjoint.star_eq,hPQ]
-  simp
-
-lemma projection_sandwich_norm (P B : V →L[ℂ] V) (hP : IsStarProjection P) :
-    ‖B.adjoint*P*B‖=‖P*B‖^2 := by
-  have he : star (P*B)*(P*B)=B.adjoint*P*B := by
-    rw [star_mul,hP.isSelfAdjoint.star_eq]
-    change B.adjoint*P*(P*B)=_
-    rw [←mul_assoc,mul_assoc B.adjoint P P,hP.isIdempotentElem.eq]
-  rw [←he,CStarRing.norm_star_mul_self]
-  ring
-
-/-- Orthogonal sector norm bound, without tensor-product classification. -/
-theorem resolution_norm_bound (P : ι → V →L[ℂ] V)
-    (hP : ∀ i, IsStarProjection (P i))
-    (horth : ∀ i j, i≠j → P i*P j=0) (hsum : ∑ i,P i=1)
-    (A : V →L[ℂ] V) (hcomm : ∀ i, Commute A (P i))
-    (c : ℝ) (hc : 0≤c) (hbound : ∀ i, ‖A*P i‖≤c) : ‖A‖≤c := by
-  apply ContinuousLinearMap.opNorm_le_bound _ hc
-  intro v
-  have hsplit (w : V) : ∑ i,P i w=w := by
-    rw [←sum_apply,hsum]
-    rfl
-  have hp := orthogonal_sum_norm_sq (fun i => P i v)
-    (fun i j hij => projection_inner_zero _ _ (hP i) (horth i j hij) v v)
-  rw [hsplit] at hp
-  have ha := orthogonal_sum_norm_sq (fun i => P i (A v))
-    (fun i j hij => projection_inner_zero _ _ (hP i) (horth i j hij) (A v) (A v))
-  rw [hsplit] at ha
-  have he (i : ι) : P i (A v)=(A*P i) (P i v) := by
-    change (P i*A) v=(A*P i*P i) v
-    rw [mul_assoc,(hP i).isIdempotentElem.eq,(hcomm i).eq]
-  have hh (i : ι) : ‖P i (A v)‖≤c*‖P i v‖ := by
-    rw [he]
-    exact ((A*P i).le_opNorm _).trans (mul_le_mul_of_nonneg_right (hbound i) (norm_nonneg _))
-  have hsq := Finset.sum_le_sum (s := Finset.univ) (fun i _ =>
-    pow_le_pow_left₀ (norm_nonneg _) (hh i) 2)
-  simp_rw [mul_pow] at hsq
-  rw [←Finset.mul_sum,←hp,←ha,←mul_pow] at hsq
-  exact (sq_le_sq₀ (norm_nonneg _) (mul_nonneg hc (norm_nonneg _))).mp hsq
-
-/-- Sandwich comparison for one orthogonal sector subordinate to a coarser
-invariant projection. This proves the needed PSD comparison at the norm level. -/
-theorem sector_sandwich_le (T P A B : V →L[ℂ] V)
-    (hT : IsStarProjection T) (hP : IsStarProjection P)
-    (hTP : T*P=T) (hcomm : Commute A T) (hA : ‖A‖≤1) :
-    ‖B.adjoint*A*T*B‖≤‖B.adjoint*P*B‖ := by
-  have he : B.adjoint*A*T*B=(T*B).adjoint*A*(T*B) := by
-    change _=star (T*B)*A*(T*B)
-    rw [star_mul,hT.isSelfAdjoint.star_eq]
-    change _=B.adjoint*T*A*(T*B)
-    rw [mul_assoc B.adjoint T A,←hcomm.eq]
-    simp only [mul_assoc]
-    rw [←mul_assoc T T B,hT.isIdempotentElem.eq]
-  have hTB : ‖T*B‖≤‖P*B‖ := by
-    have he : T*B=T*(P*B) := by rw [←mul_assoc,hTP]
-    rw [he]
-    exact (norm_mul_le _ _).trans ((mul_le_mul_of_nonneg_right (hT.norm_le _) (norm_nonneg _)).trans_eq (one_mul _))
-  rw [he,projection_sandwich_norm P B hP]
-  calc
-    _ ≤ ‖(T*B).adjoint‖*‖A‖*‖T*B‖ := (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right (norm_mul_le _ _) (norm_nonneg _))
-    _ ≤ ‖T*B‖^2 := by rw [ContinuousLinearMap.adjoint.norm_map]; nlinarith [norm_nonneg (T*B)]
-    _ ≤ _ := pow_le_pow_left₀ (norm_nonneg _) hTB 2
-
-omit [CompleteSpace V] [Fintype ι] in
-lemma resolution_commute (P : ι → V →L[ℂ] V)
-    (horth : ∀ i j, i≠j → P i*P j=0) (i j : ι) : Commute (P i) (P j) := by
-  by_cases hij : i=j
-  · subst j; exact Commute.refl _
-  · exact (horth i j hij).trans (horth j i (Ne.symm hij)).symm
-
-omit [Fintype ι] in
-lemma resolution_sum_mul (P : ι → V →L[ℂ] V) (hP : ∀ i, IsStarProjection (P i))
-    (horth : ∀ i j, i≠j → P i*P j=0) (S : Finset ι) (j : ι) :
-    (∑ i∈S,P i)*P j=if j∈S then P j else 0 := by
-  rw [Finset.sum_mul]
-  have he (i : ι) : P i*P j=if i=j then P j else 0 := by
-    split_ifs with h
-    · subst i; exact (hP j).isIdempotentElem.eq
-    · exact horth i j h
-  simp only [he,Finset.sum_ite_eq']
-
-lemma resolution_truncated_norm (P : ι → V →L[ℂ] V)
-    (hP : ∀ i, IsStarProjection (P i))
-    (horth : ∀ i j, i≠j → P i*P j=0) (hsum : ∑ i,P i=1)
-    (A : V →L[ℂ] V) (hcomm : ∀ i, Commute A (P i))
-    (S : Finset ι) (c : ℝ) (hc : 0≤c) (hbound : ∀ i∈S, ‖A*P i‖≤c) :
-    ‖A*(∑ i∈S,P i)‖≤c := by
-  apply resolution_norm_bound P hP horth hsum _ _ c hc
-  · intro i
-    rw [mul_assoc,resolution_sum_mul P hP horth]
-    split_ifs with hi
-    · exact hbound i hi
-    · simp [hc]
-  · intro i
-    exact (hcomm i).mul_left (Commute.sum_left S P (P i) fun j _ => resolution_commute P horth j i)
-
-/-- Exceptional-sector comparison, faithful to FAC's positive sandwich
-argument but not requiring tensor-product representation machinery. -/
-theorem resolution_exception_bound (T Q : ι → V →L[ℂ] V)
-    (hT : ∀ i, IsStarProjection (T i))
-    (horth : ∀ i j, i≠j → T i*T j=0) (hsum : ∑ i,T i=1)
-    (hQ : ∀ i, IsStarProjection (Q i)) (hTQ : ∀ i,T i*Q i=T i)
-    (A B : V →L[ℂ] V) (hcomm : ∀ i, Commute A (T i))
-    (hA : ‖A‖≤1) (hB : ‖B‖≤1)
-    (S : Finset ι) (c : ℝ) (hc : 0≤c) (hbound : ∀ i∉S, ‖A*T i‖≤c) :
-    ‖B.adjoint*A*B‖≤c+∑ i∈S,‖B.adjoint*Q i*B‖ := by
-  have hhigh : ‖A*(∑ i∈Sᶜ,T i)‖≤c := resolution_truncated_norm T hT horth hsum A hcomm
-    Sᶜ c hc (fun i hi => hbound i (Finset.mem_compl.mp hi))
-  have hsplit : (∑ i∈Sᶜ,T i)+(∑ i∈S,T i)=1 := by
-    rw [Finset.sum_compl_add_sum,hsum]
-  have he : B.adjoint*A*B=B.adjoint*(A*(∑ i∈Sᶜ,T i))*B+
-      ∑ i∈S,B.adjoint*A*T i*B := by
-    rw [←Finset.sum_mul,←Finset.mul_sum]
-    rw [←add_mul]
-    simp only [mul_assoc]
-    rw [←mul_add,←mul_add,hsplit]
-    simp only [mul_one,mul_assoc]
-  rw [he]
-  apply (norm_add_le _ _).trans
-  apply add_le_add
-  · calc
-      _ ≤ ‖B.adjoint‖*‖A*(∑ i∈Sᶜ,T i)‖*‖B‖ :=
-        (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right (norm_mul_le _ _) (norm_nonneg _))
-      _ ≤ 1*c*1 := mul_le_mul (mul_le_mul (by simpa only [ContinuousLinearMap.adjoint.norm_map] using hB)
-        hhigh (norm_nonneg _) zero_le_one) hB (norm_nonneg _) (by simpa using hc)
-      _ = c := by ring
-  · apply (norm_sum_le _ _).trans
-    exact Finset.sum_le_sum fun i _ => sector_sandwich_le _ _ _ _ (hT i) (hQ i) (hTQ i) (hcomm i) hA
-
-section Sectors
-variable [DecidableEq ι]
-variable (P : ι → V →L[ℂ] V) (hcomm : ∀ i j, Commute (P i) (P j))
-
-noncomputable def choice (E : Finset ι) (i : ι) : V →L[ℂ] V :=
-  if i∈E then 1-P i else P i
-
-include hcomm
-omit [CompleteSpace V] [Fintype ι] in
-lemma choice_commute (E F : Finset ι) (i j : ι) :
-    Commute (choice P E i) (choice P F j) := by
-  unfold choice
-  split_ifs
-  · exact (Commute.one_left _).sub_left ((Commute.one_right _).sub_right (hcomm i j))
-  · exact (Commute.one_left _).sub_left (hcomm i j)
-  · exact (Commute.one_right _).sub_right (hcomm i j)
-  · exact hcomm i j
-
-noncomputable def sector (s E : Finset ι) : V →L[ℂ] V :=
-  s.noncommProd (choice P E) (fun i _ j _ _ => choice_commute P hcomm E E i j)
-
-omit [CompleteSpace V] [Fintype ι] in
-lemma sector_insert (s E : Finset ι) (a : ι) (ha : a∉s) :
-    sector P hcomm (insert a s) E=choice P E a*sector P hcomm s E := by
-  exact Finset.noncommProd_insert_of_notMem _ _ _ _ ha
-
-omit [CompleteSpace V] [Fintype ι] in
-lemma sector_congr (s E F : Finset ι) (h : ∀ i∈s, (i∈E ↔ i∈F)) :
-    sector P hcomm s E=sector P hcomm s F := by
-  apply Finset.noncommProd_congr rfl
-  intro i hi
-  simp only [choice,h i hi]
-
-omit [CompleteSpace V] [Fintype ι] in
-lemma sector_factor_commute (s E F : Finset ι) (i : ι) :
-    Commute (choice P E i) (sector P hcomm s F) :=
-  Finset.noncommProd_commute _ _ _ _ (fun j _ => choice_commute P hcomm E F i j)
-
-omit [Fintype ι] in
-lemma sector_projection (hP : ∀ i, IsStarProjection (P i)) (s E : Finset ι) :
-    IsStarProjection (sector P hcomm s E) := by
-  induction s using Finset.induction_on with
-  | empty => exact IsStarProjection.one _
-  | @insert a s ha ih =>
-    rw [sector_insert P hcomm s E a ha]
-    have hc : IsStarProjection (choice P E a) := by
-      unfold choice
-      split_ifs
-      · exact (hP a).one_sub
-      · exact hP a
-    exact hc.mul ih (sector_factor_commute P hcomm s E E a)
-
-omit [CompleteSpace V] [Fintype ι] in
-lemma sector_sum (s : Finset ι) : ∑ E∈s.powerset, sector P hcomm s E=1 := by
-  induction s using Finset.induction_on with
-  | empty =>
-    simp only [Finset.powerset_empty,Finset.sum_singleton]
-    exact Finset.noncommProd_empty _ _
-  | @insert a s ha ih =>
-    rw [Finset.sum_powerset_insert ha]
-    have h₁ (E : Finset ι) (hE : E∈s.powerset) :
-        sector P hcomm (insert a s) E=P a*sector P hcomm s E := by
-      rw [sector_insert P hcomm s E a ha]
-      congr 1
-      exact ite_eq_right (fun h => ha ((Finset.mem_powerset.mp hE) h))
-    have h₂ (E : Finset ι) (_hE : E∈s.powerset) :
-        sector P hcomm (insert a s) (insert a E)=(1-P a)*sector P hcomm s E := by
-      rw [sector_insert P hcomm s (insert a E) a ha]
-      simp only [choice,Finset.mem_insert_self,↓reduceIte]
-      congr 1
-      apply sector_congr
-      intro i hi
-      simp [ne_of_mem_of_not_mem hi ha]
-    rw [Finset.sum_congr rfl h₁,Finset.sum_congr rfl h₂,
-      ←Finset.mul_sum,←Finset.mul_sum,ih]
-    simp
-
-omit [Fintype ι] in
-lemma sector_absorb_factor (s E : Finset ι) (i : ι) (hi : i∈s)
-    (hP : ∀ i, IsStarProjection (P i)) :
-    choice P E i*sector P hcomm s E=sector P hcomm s E := by
-  have hc : IsIdempotentElem (choice P E i) := by
-    unfold choice
-    split_ifs
-    · exact (hP i).one_sub.isIdempotentElem
-    · exact (hP i).isIdempotentElem
-  have hsplit := Finset.mul_noncommProd_erase s hi (choice P E)
-    (fun i _ j _ _ => choice_commute P hcomm E E i j)
-  change choice P E i*sector P hcomm s E=sector P hcomm s E
-  change _=sector P hcomm s E at hsplit
-  nth_rw 1 [←hsplit]
-  rw [←mul_assoc,hc.eq,hsplit]
-
-omit [Fintype ι] in
-lemma sector_orthogonal (s E F : Finset ι) (hE : E⊆s) (hF : F⊆s) (hEF : E≠F)
-    (hP : ∀ i, IsStarProjection (P i)) :
-    sector P hcomm s E*sector P hcomm s F=0 := by
-  have hdiff : ∃ i∈s, ¬(i∈E ↔ i∈F) := by
-    by_contra h
-    push Not at h
-    exact hEF (Finset.ext fun i => by
-      by_cases hi : i∈s
-      · exact h i hi
-      · exact iff_of_false (fun h => hi (hE h)) (fun h => hi (hF h)))
-  obtain ⟨i,hi,hdiff⟩ := hdiff
-  have hz : choice P E i*choice P F i=0 := by
-    by_cases he : i∈E <;> by_cases hf : i∈F
-    · exact (hdiff (iff_of_true he hf)).elim
-    · simpa only [choice,ite_eq_left he,ite_eq_right hf] using (hP i).one_sub_mul_self
-    · simpa only [choice,ite_eq_right he,ite_eq_left hf] using (hP i).mul_one_sub_self
-    · exact (hdiff (iff_of_false he hf)).elim
-  have hEi := sector_absorb_factor P hcomm s E i hi hP
-  have hFi := sector_absorb_factor P hcomm s F i hi hP
-  calc
-    _ = sector P hcomm s E*(choice P E i*choice P F i)*sector P hcomm s F := by
-      simp only [mul_assoc]
-      rw [hFi]
-      rw [←mul_assoc,(sector_factor_commute P hcomm s E E i).eq.symm,hEi]
-    _ = 0 := by rw [hz,mul_zero,zero_mul]
-
-noncomputable def blockProduct (A : ι → V →L[ℂ] V)
-    (hA : ∀ i j, Commute (A i) (A j)) (s : Finset ι) : V →L[ℂ] V :=
-  s.noncommProd A (fun i _ j _ _ => hA i j)
-
-omit hcomm [Fintype ι] in
-lemma norm_noncommProd_le (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
-    (s : Finset ι) : ‖blockProduct A hA s‖≤∏ i∈s,‖A i‖ := by
-  induction s using Finset.induction_on with
-  | empty => exact ContinuousLinearMap.norm_id_le
-  | @insert a s ha ih =>
-    have he : blockProduct A hA (insert a s)=A a*blockProduct A hA s :=
-      Finset.noncommProd_insert_of_notMem _ _ _ _ ha
-    rw [he,Finset.prod_insert ha]
-    exact (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_left ih (norm_nonneg _))
-
-noncomputable def blocks (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j)) : V →L[ℂ] V :=
-  blockProduct A hA Finset.univ
-
-omit hcomm in
-lemma blocks_norm_le_one (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
-    (hn : ∀ i,‖A i‖≤1) : ‖blocks A hA‖≤1 :=
-  (norm_noncommProd_le A hA Finset.univ).trans (Finset.prod_le_one₀ (fun _ _ => norm_nonneg _) (fun i _ => hn i))
-
-omit [CompleteSpace V] in
-lemma blocks_sector_commute (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
-    (hAP : ∀ i j, Commute (A i) (P j)) (E : Finset ι) :
-    Commute (blocks A hA) (sector P hcomm Finset.univ E) := by
-  unfold blocks blockProduct
-  apply (Finset.noncommProd_commute _ _ (fun i _ j _ _ => hA i j) _ _).symm
-  intro i _
-  unfold sector
-  apply (Finset.noncommProd_commute _ _ (fun i _ j _ _ => choice_commute P hcomm E E i j) _ _).symm
-  intro j _
-  unfold choice
-  split_ifs
-  · exact (Commute.one_right _).sub_right (hAP i j)
-  · exact hAP i j
-
-lemma blocks_sector_norm (hP : ∀ i, IsStarProjection (P i))
-    (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
-    (hAP : ∀ i j, Commute (A i) (P j)) (hn : ∀ i,‖A i‖≤1)
-    (θ : ℝ) (_hθ : 0≤θ) (hgap : ∀ i,‖A i*(1-P i)‖≤θ) (E : Finset ι) :
-    ‖blocks A hA*sector P hcomm Finset.univ E‖≤θ^E.card := by
-  have hAC (i j : ι) : Commute (A i) (choice P E j) := by
-    unfold choice
-    split_ifs
-    · exact (Commute.one_right _).sub_right (hAP i j)
-    · exact hAP i j
-  have hprod := Finset.noncommProd_mul_distrib (s := Finset.univ) A (choice P E)
-    (fun i _ j _ _ => hA i j) (fun i _ j _ _ => choice_commute P hcomm E E i j)
-    (fun i _ j _ _ => (hAC j i).symm)
-  change _=blocks A hA*sector P hcomm Finset.univ E at hprod
-  rw [←hprod]
-  have hc (i j : ι) : Commute ((A*choice P E) i) ((A*choice P E) j) :=
-    ((hA i j).mul_right (hAC i j)).mul_left ((hAC j i).symm.mul_right (choice_commute P hcomm E E i j))
-  apply (norm_noncommProd_le _ hc Finset.univ).trans
-  calc
-    _ ≤ ∏ i : ι, if i∈E then θ else 1 := by
-      apply Finset.prod_le_prod₀ (fun i _ => norm_nonneg _)
-      intro i _
-      dsimp only [Pi.mul_apply]
-      unfold choice
-      split_ifs with hi
-      · exact hgap i
-      · exact (norm_mul_le _ _).trans (by nlinarith [hn i,(hP i).norm_le (P i),norm_nonneg (A i),norm_nonneg (P i)])
-    _ = _ := by rw [Fintype.prod_ite_mem,Finset.prod_const]
-
-noncomputable def coarse (E : Finset ι) : V →L[ℂ] V :=
-  Eᶜ.noncommProd P (fun i _ j _ _ => hcomm i j)
-
-lemma coarse_projection (hP : ∀ i, IsStarProjection (P i)) (E : Finset ι) :
-    IsStarProjection (coarse P hcomm E) := by
-  have he : coarse P hcomm E=sector P hcomm Eᶜ ∅ := by
-    apply Finset.noncommProd_congr rfl
-    intro i _
-    simp [choice]
-  rw [he]
-  exact sector_projection P hcomm hP _ _
-
-lemma sector_coarse (hP : ∀ i, IsStarProjection (P i)) (E : Finset ι) :
-    sector P hcomm Finset.univ E*coarse P hcomm E=sector P hcomm Finset.univ E := by
-  have hpi (i : ι) (hi : i∈Eᶜ) : sector P hcomm Finset.univ E*P i=sector P hcomm Finset.univ E := by
-    have h := sector_absorb_factor P hcomm Finset.univ E i (Finset.mem_univ i) hP
-    rw [(sector_factor_commute P hcomm Finset.univ E E i).eq] at h
-    simpa only [choice,ite_eq_right (Finset.mem_compl.mp hi)] using h
-  have hp (s : Finset ι) (hs : ∀ i∈s,
-      sector P hcomm Finset.univ E*P i=sector P hcomm Finset.univ E) :
-      sector P hcomm Finset.univ E*blockProduct P hcomm s=sector P hcomm Finset.univ E := by
-    induction s using Finset.induction_on with
-    | empty => exact mul_one _
-    | @insert a s ha ih =>
-      have he : blockProduct P hcomm (insert a s)=P a*blockProduct P hcomm s :=
-        Finset.noncommProd_insert_of_notMem _ _ _ _ ha
-      rw [he,←mul_assoc,hs a (Finset.mem_insert_self _ _)]
-      exact ih (fun i hi => hs i (Finset.mem_insert_of_mem hi))
-  exact hp Eᶜ hpi
-
-/-- Actual commuting-block exceptional estimate, with no tensor irreducible
-hypothesis. The local gaps are for each represented finite block. -/
-theorem blocks_exception_bound (hP : ∀ i, IsStarProjection (P i))
-    (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
-    (hAP : ∀ i j, Commute (A i) (P j)) (hn : ∀ i,‖A i‖≤1)
-    (θ : ℝ) (hθ : 0≤θ) (hθ1 : θ≤1) (hgap : ∀ i,‖A i*(1-P i)‖≤θ)
-    (B : V →L[ℂ] V) (hB : ‖B‖≤1) (n : ℕ) :
-    ‖B.adjoint*blocks A hA*B‖≤θ^n+
-      ∑ E : Finset ι, if E.card<n then ‖B.adjoint*coarse P hcomm E*B‖ else 0 := by
-  let T : Finset ι → V →L[ℂ] V := sector P hcomm Finset.univ
-  have hsum : ∑ E,T E=1 := by
-    simpa only [Finset.powerset_univ] using sector_sum P hcomm Finset.univ
-  have hbound := resolution_exception_bound T (coarse P hcomm)
-    (fun E => sector_projection P hcomm hP _ E)
-    (fun E F hEF => sector_orthogonal P hcomm _ E F (Finset.subset_univ _) (Finset.subset_univ _) hEF hP)
-    hsum (coarse_projection P hcomm hP) (sector_coarse P hcomm hP)
-    (blocks A hA) B (blocks_sector_commute P hcomm A hA hAP)
-    (blocks_norm_le_one A hA hn) hB (Finset.univ.filter fun E : Finset ι => E.card<n)
-    (θ^n) (pow_nonneg hθ _) (by
-      intro E hE
-      have hEn : n≤E.card := by simpa using hE
-      exact (blocks_sector_norm P hcomm hP A hA hAP hn θ hθ hgap E).trans
-        (pow_le_pow_of_le_one hθ hθ1 hEn))
-  simpa only [Finset.sum_filter] using hbound
-
-end Sectors
-
-end CubeShuffle.CommutingBlocks
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+namespace CubeShuffle.CommutingBlocks
+open scoped BigOperators Classical
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
+variable {ι : Type*} [Fintype ι]
+
+omit [CompleteSpace V] in
+lemma orthogonal_sum_norm_sq (v : ι → V) (hv : ∀ i j, i≠j → inner ℂ (v i) (v j)=0) :
+    ‖∑ i,v i‖^2=∑ i,‖v i‖^2 := by
+  rw [norm_sq_eq_re_inner (𝕜 := ℂ),sum_inner]
+  simp only [inner_sum,map_sum]
+  apply Finset.sum_congr rfl
+  intro i _
+  rw [Finset.sum_eq_single i]
+  · exact (norm_sq_eq_re_inner (𝕜 := ℂ) (v i)).symm
+  · intro j _ hji
+    rw [hv i j hji.symm]
+    rfl
+  · simp
+
+lemma projection_inner_zero (P Q : V →L[ℂ] V) (hP : IsStarProjection P)
+    (hPQ : P*Q=0) (v w : V) : inner ℂ (P v) (Q w)=0 := by
+  rw [←ContinuousLinearMap.adjoint_inner_right]
+  change inner ℂ v ((star P*Q) w)=0
+  rw [hP.isSelfAdjoint.star_eq,hPQ]
+  simp
+
+lemma projection_sandwich_norm (P B : V →L[ℂ] V) (hP : IsStarProjection P) :
+    ‖B.adjoint*P*B‖=‖P*B‖^2 := by
+  have he : star (P*B)*(P*B)=B.adjoint*P*B := by
+    rw [star_mul,hP.isSelfAdjoint.star_eq]
+    change B.adjoint*P*(P*B)=_
+    rw [←mul_assoc,mul_assoc B.adjoint P P,hP.isIdempotentElem.eq]
+  rw [←he,CStarRing.norm_star_mul_self]
+  ring
+
+/-- Orthogonal sector norm bound, without tensor-product classification. -/
+theorem resolution_norm_bound (P : ι → V →L[ℂ] V)
+    (hP : ∀ i, IsStarProjection (P i))
+    (horth : ∀ i j, i≠j → P i*P j=0) (hsum : ∑ i,P i=1)
+    (A : V →L[ℂ] V) (hcomm : ∀ i, Commute A (P i))
+    (c : ℝ) (hc : 0≤c) (hbound : ∀ i, ‖A*P i‖≤c) : ‖A‖≤c := by
+  apply ContinuousLinearMap.opNorm_le_bound _ hc
+  intro v
+  have hsplit (w : V) : ∑ i,P i w=w := by
+    rw [←sum_apply,hsum]
+    rfl
+  have hp := orthogonal_sum_norm_sq (fun i => P i v)
+    (fun i j hij => projection_inner_zero _ _ (hP i) (horth i j hij) v v)
+  rw [hsplit] at hp
+  have ha := orthogonal_sum_norm_sq (fun i => P i (A v))
+    (fun i j hij => projection_inner_zero _ _ (hP i) (horth i j hij) (A v) (A v))
+  rw [hsplit] at ha
+  have he (i : ι) : P i (A v)=(A*P i) (P i v) := by
+    change (P i*A) v=(A*P i*P i) v
+    rw [mul_assoc,(hP i).isIdempotentElem.eq,(hcomm i).eq]
+  have hh (i : ι) : ‖P i (A v)‖≤c*‖P i v‖ := by
+    rw [he]
+    exact ((A*P i).le_opNorm _).trans (mul_le_mul_of_nonneg_right (hbound i) (norm_nonneg _))
+  have hsq := Finset.sum_le_sum (s := Finset.univ) (fun i _ =>
+    pow_le_pow_left₀ (norm_nonneg _) (hh i) 2)
+  simp_rw [mul_pow] at hsq
+  rw [←Finset.mul_sum,←hp,←ha,←mul_pow] at hsq
+  exact (sq_le_sq₀ (norm_nonneg _) (mul_nonneg hc (norm_nonneg _))).mp hsq
+
+/-- Sandwich comparison for one orthogonal sector subordinate to a coarser
+invariant projection. This proves the needed PSD comparison at the norm level. -/
+theorem sector_sandwich_le (T P A B : V →L[ℂ] V)
+    (hT : IsStarProjection T) (hP : IsStarProjection P)
+    (hTP : T*P=T) (hcomm : Commute A T) (hA : ‖A‖≤1) :
+    ‖B.adjoint*A*T*B‖≤‖B.adjoint*P*B‖ := by
+  have he : B.adjoint*A*T*B=(T*B).adjoint*A*(T*B) := by
+    change _=star (T*B)*A*(T*B)
+    rw [star_mul,hT.isSelfAdjoint.star_eq]
+    change _=B.adjoint*T*A*(T*B)
+    rw [mul_assoc B.adjoint T A,←hcomm.eq]
+    simp only [mul_assoc]
+    rw [←mul_assoc T T B,hT.isIdempotentElem.eq]
+  have hTB : ‖T*B‖≤‖P*B‖ := by
+    have he : T*B=T*(P*B) := by rw [←mul_assoc,hTP]
+    rw [he]
+    exact (norm_mul_le _ _).trans ((mul_le_mul_of_nonneg_right (hT.norm_le _) (norm_nonneg _)).trans_eq (one_mul _))
+  rw [he,projection_sandwich_norm P B hP]
+  calc
+    _ ≤ ‖(T*B).adjoint‖*‖A‖*‖T*B‖ := (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right (norm_mul_le _ _) (norm_nonneg _))
+    _ ≤ ‖T*B‖^2 := by rw [ContinuousLinearMap.adjoint.norm_map]; nlinarith [norm_nonneg (T*B)]
+    _ ≤ _ := pow_le_pow_left₀ (norm_nonneg _) hTB 2
+
+omit [CompleteSpace V] [Fintype ι] in
+lemma resolution_commute (P : ι → V →L[ℂ] V)
+    (horth : ∀ i j, i≠j → P i*P j=0) (i j : ι) : Commute (P i) (P j) := by
+  by_cases hij : i=j
+  · subst j; exact Commute.refl _
+  · exact (horth i j hij).trans (horth j i (Ne.symm hij)).symm
+
+omit [Fintype ι] in
+lemma resolution_sum_mul (P : ι → V →L[ℂ] V) (hP : ∀ i, IsStarProjection (P i))
+    (horth : ∀ i j, i≠j → P i*P j=0) (S : Finset ι) (j : ι) :
+    (∑ i∈S,P i)*P j=if j∈S then P j else 0 := by
+  rw [Finset.sum_mul]
+  have he (i : ι) : P i*P j=if i=j then P j else 0 := by
+    split_ifs with h
+    · subst i; exact (hP j).isIdempotentElem.eq
+    · exact horth i j h
+  simp only [he,Finset.sum_ite_eq']
+
+lemma resolution_truncated_norm (P : ι → V →L[ℂ] V)
+    (hP : ∀ i, IsStarProjection (P i))
+    (horth : ∀ i j, i≠j → P i*P j=0) (hsum : ∑ i,P i=1)
+    (A : V →L[ℂ] V) (hcomm : ∀ i, Commute A (P i))
+    (S : Finset ι) (c : ℝ) (hc : 0≤c) (hbound : ∀ i∈S, ‖A*P i‖≤c) :
+    ‖A*(∑ i∈S,P i)‖≤c := by
+  apply resolution_norm_bound P hP horth hsum _ _ c hc
+  · intro i
+    rw [mul_assoc,resolution_sum_mul P hP horth]
+    split_ifs with hi
+    · exact hbound i hi
+    · simp [hc]
+  · intro i
+    exact (hcomm i).mul_left (Commute.sum_left S P (P i) fun j _ => resolution_commute P horth j i)
+
+/-- Exceptional-sector comparison, faithful to FAC's positive sandwich
+argument but not requiring tensor-product representation machinery. -/
+theorem resolution_exception_bound (T Q : ι → V →L[ℂ] V)
+    (hT : ∀ i, IsStarProjection (T i))
+    (horth : ∀ i j, i≠j → T i*T j=0) (hsum : ∑ i,T i=1)
+    (hQ : ∀ i, IsStarProjection (Q i)) (hTQ : ∀ i,T i*Q i=T i)
+    (A B : V →L[ℂ] V) (hcomm : ∀ i, Commute A (T i))
+    (hA : ‖A‖≤1) (hB : ‖B‖≤1)
+    (S : Finset ι) (c : ℝ) (hc : 0≤c) (hbound : ∀ i∉S, ‖A*T i‖≤c) :
+    ‖B.adjoint*A*B‖≤c+∑ i∈S,‖B.adjoint*Q i*B‖ := by
+  have hhigh : ‖A*(∑ i∈Sᶜ,T i)‖≤c := resolution_truncated_norm T hT horth hsum A hcomm
+    Sᶜ c hc (fun i hi => hbound i (Finset.mem_compl.mp hi))
+  have hsplit : (∑ i∈Sᶜ,T i)+(∑ i∈S,T i)=1 := by
+    rw [Finset.sum_compl_add_sum,hsum]
+  have he : B.adjoint*A*B=B.adjoint*(A*(∑ i∈Sᶜ,T i))*B+
+      ∑ i∈S,B.adjoint*A*T i*B := by
+    rw [←Finset.sum_mul,←Finset.mul_sum]
+    rw [←add_mul]
+    simp only [mul_assoc]
+    rw [←mul_add,←mul_add,hsplit]
+    simp only [mul_one,mul_assoc]
+  rw [he]
+  apply (norm_add_le _ _).trans
+  apply add_le_add
+  · calc
+      _ ≤ ‖B.adjoint‖*‖A*(∑ i∈Sᶜ,T i)‖*‖B‖ :=
+        (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right (norm_mul_le _ _) (norm_nonneg _))
+      _ ≤ 1*c*1 := mul_le_mul (mul_le_mul (by simpa only [ContinuousLinearMap.adjoint.norm_map] using hB)
+        hhigh (norm_nonneg _) zero_le_one) hB (norm_nonneg _) (by simpa using hc)
+      _ = c := by ring
+  · apply (norm_sum_le _ _).trans
+    exact Finset.sum_le_sum fun i _ => sector_sandwich_le _ _ _ _ (hT i) (hQ i) (hTQ i) (hcomm i) hA
+
+section Sectors
+variable [DecidableEq ι]
+variable (P : ι → V →L[ℂ] V) (hcomm : ∀ i j, Commute (P i) (P j))
+
+noncomputable def choice (E : Finset ι) (i : ι) : V →L[ℂ] V :=
+  if i∈E then 1-P i else P i
+
+include hcomm
+omit [CompleteSpace V] [Fintype ι] in
+lemma choice_commute (E F : Finset ι) (i j : ι) :
+    Commute (choice P E i) (choice P F j) := by
+  unfold choice
+  split_ifs
+  · exact (Commute.one_left _).sub_left ((Commute.one_right _).sub_right (hcomm i j))
+  · exact (Commute.one_left _).sub_left (hcomm i j)
+  · exact (Commute.one_right _).sub_right (hcomm i j)
+  · exact hcomm i j
+
+noncomputable def sector (s E : Finset ι) : V →L[ℂ] V :=
+  s.noncommProd (choice P E) (fun i _ j _ _ => choice_commute P hcomm E E i j)
+
+omit [CompleteSpace V] [Fintype ι] in
+lemma sector_insert (s E : Finset ι) (a : ι) (ha : a∉s) :
+    sector P hcomm (insert a s) E=choice P E a*sector P hcomm s E := by
+  exact Finset.noncommProd_insert_of_notMem _ _ _ _ ha
+
+omit [CompleteSpace V] [Fintype ι] in
+lemma sector_congr (s E F : Finset ι) (h : ∀ i∈s, (i∈E ↔ i∈F)) :
+    sector P hcomm s E=sector P hcomm s F := by
+  apply Finset.noncommProd_congr rfl
+  intro i hi
+  simp only [choice,h i hi]
+
+omit [CompleteSpace V] [Fintype ι] in
+lemma sector_factor_commute (s E F : Finset ι) (i : ι) :
+    Commute (choice P E i) (sector P hcomm s F) :=
+  Finset.noncommProd_commute _ _ _ _ (fun j _ => choice_commute P hcomm E F i j)
+
+omit [Fintype ι] in
+lemma sector_projection (hP : ∀ i, IsStarProjection (P i)) (s E : Finset ι) :
+    IsStarProjection (sector P hcomm s E) := by
+  induction s using Finset.induction_on with
+  | empty => exact IsStarProjection.one _
+  | @insert a s ha ih =>
+    rw [sector_insert P hcomm s E a ha]
+    have hc : IsStarProjection (choice P E a) := by
+      unfold choice
+      split_ifs
+      · exact (hP a).one_sub
+      · exact hP a
+    exact hc.mul ih (sector_factor_commute P hcomm s E E a)
+
+omit [CompleteSpace V] [Fintype ι] in
+lemma sector_sum (s : Finset ι) : ∑ E∈s.powerset, sector P hcomm s E=1 := by
+  induction s using Finset.induction_on with
+  | empty =>
+    simp only [Finset.powerset_empty,Finset.sum_singleton]
+    exact Finset.noncommProd_empty _ _
+  | @insert a s ha ih =>
+    rw [Finset.sum_powerset_insert ha]
+    have h₁ (E : Finset ι) (hE : E∈s.powerset) :
+        sector P hcomm (insert a s) E=P a*sector P hcomm s E := by
+      rw [sector_insert P hcomm s E a ha]
+      congr 1
+      exact ite_eq_right (fun h => ha ((Finset.mem_powerset.mp hE) h))
+    have h₂ (E : Finset ι) (_hE : E∈s.powerset) :
+        sector P hcomm (insert a s) (insert a E)=(1-P a)*sector P hcomm s E := by
+      rw [sector_insert P hcomm s (insert a E) a ha]
+      simp only [choice,Finset.mem_insert_self,↓reduceIte]
+      congr 1
+      apply sector_congr
+      intro i hi
+      simp [ne_of_mem_of_not_mem hi ha]
+    rw [Finset.sum_congr rfl h₁,Finset.sum_congr rfl h₂,
+      ←Finset.mul_sum,←Finset.mul_sum,ih]
+    simp
+
+omit [Fintype ι] in
+lemma sector_absorb_factor (s E : Finset ι) (i : ι) (hi : i∈s)
+    (hP : ∀ i, IsStarProjection (P i)) :
+    choice P E i*sector P hcomm s E=sector P hcomm s E := by
+  have hc : IsIdempotentElem (choice P E i) := by
+    unfold choice
+    split_ifs
+    · exact (hP i).one_sub.isIdempotentElem
+    · exact (hP i).isIdempotentElem
+  have hsplit := Finset.mul_noncommProd_erase s hi (choice P E)
+    (fun i _ j _ _ => choice_commute P hcomm E E i j)
+  change choice P E i*sector P hcomm s E=sector P hcomm s E
+  change _=sector P hcomm s E at hsplit
+  nth_rw 1 [←hsplit]
+  rw [←mul_assoc,hc.eq,hsplit]
+
+omit [Fintype ι] in
+lemma sector_orthogonal (s E F : Finset ι) (hE : E⊆s) (hF : F⊆s) (hEF : E≠F)
+    (hP : ∀ i, IsStarProjection (P i)) :
+    sector P hcomm s E*sector P hcomm s F=0 := by
+  have hdiff : ∃ i∈s, ¬(i∈E ↔ i∈F) := by
+    by_contra h
+    push Not at h
+    exact hEF (Finset.ext fun i => by
+      by_cases hi : i∈s
+      · exact h i hi
+      · exact iff_of_false (fun h => hi (hE h)) (fun h => hi (hF h)))
+  obtain ⟨i,hi,hdiff⟩ := hdiff
+  have hz : choice P E i*choice P F i=0 := by
+    by_cases he : i∈E <;> by_cases hf : i∈F
+    · exact (hdiff (iff_of_true he hf)).elim
+    · simpa only [choice,ite_eq_left he,ite_eq_right hf] using (hP i).one_sub_mul_self
+    · simpa only [choice,ite_eq_right he,ite_eq_left hf] using (hP i).mul_one_sub_self
+    · exact (hdiff (iff_of_false he hf)).elim
+  have hEi := sector_absorb_factor P hcomm s E i hi hP
+  have hFi := sector_absorb_factor P hcomm s F i hi hP
+  calc
+    _ = sector P hcomm s E*(choice P E i*choice P F i)*sector P hcomm s F := by
+      simp only [mul_assoc]
+      rw [hFi]
+      rw [←mul_assoc,(sector_factor_commute P hcomm s E E i).eq.symm,hEi]
+    _ = 0 := by rw [hz,mul_zero,zero_mul]
+
+noncomputable def blockProduct (A : ι → V →L[ℂ] V)
+    (hA : ∀ i j, Commute (A i) (A j)) (s : Finset ι) : V →L[ℂ] V :=
+  s.noncommProd A (fun i _ j _ _ => hA i j)
+
+omit hcomm [Fintype ι] in
+lemma norm_noncommProd_le (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
+    (s : Finset ι) : ‖blockProduct A hA s‖≤∏ i∈s,‖A i‖ := by
+  induction s using Finset.induction_on with
+  | empty => exact ContinuousLinearMap.norm_id_le
+  | @insert a s ha ih =>
+    have he : blockProduct A hA (insert a s)=A a*blockProduct A hA s :=
+      Finset.noncommProd_insert_of_notMem _ _ _ _ ha
+    rw [he,Finset.prod_insert ha]
+    exact (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_left ih (norm_nonneg _))
+
+noncomputable def blocks (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j)) : V →L[ℂ] V :=
+  blockProduct A hA Finset.univ
+
+omit hcomm in
+lemma blocks_norm_le_one (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
+    (hn : ∀ i,‖A i‖≤1) : ‖blocks A hA‖≤1 :=
+  (norm_noncommProd_le A hA Finset.univ).trans (Finset.prod_le_one₀ (fun _ _ => norm_nonneg _) (fun i _ => hn i))
+
+omit [CompleteSpace V] in
+lemma blocks_sector_commute (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
+    (hAP : ∀ i j, Commute (A i) (P j)) (E : Finset ι) :
+    Commute (blocks A hA) (sector P hcomm Finset.univ E) := by
+  unfold blocks blockProduct
+  apply (Finset.noncommProd_commute _ _ (fun i _ j _ _ => hA i j) _ _).symm
+  intro i _
+  unfold sector
+  apply (Finset.noncommProd_commute _ _ (fun i _ j _ _ => choice_commute P hcomm E E i j) _ _).symm
+  intro j _
+  unfold choice
+  split_ifs
+  · exact (Commute.one_right _).sub_right (hAP i j)
+  · exact hAP i j
+
+lemma blocks_sector_norm (hP : ∀ i, IsStarProjection (P i))
+    (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
+    (hAP : ∀ i j, Commute (A i) (P j)) (hn : ∀ i,‖A i‖≤1)
+    (θ : ℝ) (_hθ : 0≤θ) (hgap : ∀ i,‖A i*(1-P i)‖≤θ) (E : Finset ι) :
+    ‖blocks A hA*sector P hcomm Finset.univ E‖≤θ^E.card := by
+  have hAC (i j : ι) : Commute (A i) (choice P E j) := by
+    unfold choice
+    split_ifs
+    · exact (Commute.one_right _).sub_right (hAP i j)
+    · exact hAP i j
+  have hprod := Finset.noncommProd_mul_distrib (s := Finset.univ) A (choice P E)
+    (fun i _ j _ _ => hA i j) (fun i _ j _ _ => choice_commute P hcomm E E i j)
+    (fun i _ j _ _ => (hAC j i).symm)
+  change _=blocks A hA*sector P hcomm Finset.univ E at hprod
+  rw [←hprod]
+  have hc (i j : ι) : Commute ((A*choice P E) i) ((A*choice P E) j) :=
+    ((hA i j).mul_right (hAC i j)).mul_left ((hAC j i).symm.mul_right (choice_commute P hcomm E E i j))
+  apply (norm_noncommProd_le _ hc Finset.univ).trans
+  calc
+    _ ≤ ∏ i : ι, if i∈E then θ else 1 := by
+      apply Finset.prod_le_prod₀ (fun i _ => norm_nonneg _)
+      intro i _
+      dsimp only [Pi.mul_apply]
+      unfold choice
+      split_ifs with hi
+      · exact hgap i
+      · exact (norm_mul_le _ _).trans (by nlinarith [hn i,(hP i).norm_le (P i),norm_nonneg (A i),norm_nonneg (P i)])
+    _ = _ := by rw [Fintype.prod_ite_mem,Finset.prod_const]
+
+noncomputable def coarse (E : Finset ι) : V →L[ℂ] V :=
+  Eᶜ.noncommProd P (fun i _ j _ _ => hcomm i j)
+
+lemma coarse_projection (hP : ∀ i, IsStarProjection (P i)) (E : Finset ι) :
+    IsStarProjection (coarse P hcomm E) := by
+  have he : coarse P hcomm E=sector P hcomm Eᶜ ∅ := by
+    apply Finset.noncommProd_congr rfl
+    intro i _
+    simp [choice]
+  rw [he]
+  exact sector_projection P hcomm hP _ _
+
+lemma sector_coarse (hP : ∀ i, IsStarProjection (P i)) (E : Finset ι) :
+    sector P hcomm Finset.univ E*coarse P hcomm E=sector P hcomm Finset.univ E := by
+  have hpi (i : ι) (hi : i∈Eᶜ) : sector P hcomm Finset.univ E*P i=sector P hcomm Finset.univ E := by
+    have h := sector_absorb_factor P hcomm Finset.univ E i (Finset.mem_univ i) hP
+    rw [(sector_factor_commute P hcomm Finset.univ E E i).eq] at h
+    simpa only [choice,ite_eq_right (Finset.mem_compl.mp hi)] using h
+  have hp (s : Finset ι) (hs : ∀ i∈s,
+      sector P hcomm Finset.univ E*P i=sector P hcomm Finset.univ E) :
+      sector P hcomm Finset.univ E*blockProduct P hcomm s=sector P hcomm Finset.univ E := by
+    induction s using Finset.induction_on with
+    | empty => exact mul_one _
+    | @insert a s ha ih =>
+      have he : blockProduct P hcomm (insert a s)=P a*blockProduct P hcomm s :=
+        Finset.noncommProd_insert_of_notMem _ _ _ _ ha
+      rw [he,←mul_assoc,hs a (Finset.mem_insert_self _ _)]
+      exact ih (fun i hi => hs i (Finset.mem_insert_of_mem hi))
+  exact hp Eᶜ hpi
+
+/-- Actual commuting-block exceptional estimate, with no tensor irreducible
+hypothesis. The local gaps are for each represented finite block. -/
+theorem blocks_exception_bound (hP : ∀ i, IsStarProjection (P i))
+    (A : ι → V →L[ℂ] V) (hA : ∀ i j, Commute (A i) (A j))
+    (hAP : ∀ i j, Commute (A i) (P j)) (hn : ∀ i,‖A i‖≤1)
+    (θ : ℝ) (hθ : 0≤θ) (hθ1 : θ≤1) (hgap : ∀ i,‖A i*(1-P i)‖≤θ)
+    (B : V →L[ℂ] V) (hB : ‖B‖≤1) (n : ℕ) :
+    ‖B.adjoint*blocks A hA*B‖≤θ^n+
+      ∑ E : Finset ι, if E.card<n then ‖B.adjoint*coarse P hcomm E*B‖ else 0 := by
+  let T : Finset ι → V →L[ℂ] V := sector P hcomm Finset.univ
+  have hsum : ∑ E,T E=1 := by
+    simpa only [Finset.powerset_univ] using sector_sum P hcomm Finset.univ
+  have hbound := resolution_exception_bound T (coarse P hcomm)
+    (fun E => sector_projection P hcomm hP _ E)
+    (fun E F hEF => sector_orthogonal P hcomm _ E F (Finset.subset_univ _) (Finset.subset_univ _) hEF hP)
+    hsum (coarse_projection P hcomm hP) (sector_coarse P hcomm hP)
+    (blocks A hA) B (blocks_sector_commute P hcomm A hA hAP)
+    (blocks_norm_le_one A hA hn) hB (Finset.univ.filter fun E : Finset ι => E.card<n)
+    (θ^n) (pow_nonneg hθ _) (by
+      intro E hE
+      have hEn : n≤E.card := by simpa using hE
+      exact (blocks_sector_norm P hcomm hP A hA hAP hn θ hθ hgap E).trans
+        (pow_le_pow_of_le_one hθ hθ1 hEn))
+  simpa only [Finset.sum_filter] using hbound
+
+end Sectors
+
+end CubeShuffle.CommutingBlocks
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FiniteUnitary/Fourier.lean b/lean/OAI/RepresentationTheory/FiniteUnitary/Fourier.lean
index 1ea0a6a4c025f26bc3fb1450de78e3f7345e2f13..db9bd570831b049ed20617c1d59793aa6b155af5 100644
--- a/lean/OAI/RepresentationTheory/FiniteUnitary/Fourier.lean
+++ b/lean/OAI/RepresentationTheory/FiniteUnitary/Fourier.lean
@@ -1,149 +1,149 @@
-import OAI.RepresentationTheory.FiniteUnitary.Characters
-import OAI.RepresentationTheory.FiniteUnitary.Sampling
-
-namespace OAI
-
-namespace CubeShuffle.FiniteFourier
-open scoped BigOperators ComplexConjugate Classical
-open UnitaryFinite
-
-variable {G ι : Type*} [Group G] [Fintype G] [Fintype ι]
-variable (V : ι → Type*) [∀ i, NormedAddCommGroup (V i)] [∀ i, InnerProductSpace ℂ (V i)]
-  [∀ i, FiniteDimensional ℂ (V i)]
-variable (ρ : ∀ i, Representation ℂ G (V i)) [∀ i, Representation.IsIrreducible (ρ i)]
-variable (hρ : ∀ i, IsUnitary (ρ i))
-variable (hneq : ∀ i j, i≠j → ¬Nonempty (Representation.Equiv (ρ i) (ρ j)))
-  (hcard : Nat.card (ConjClasses G)≤Fintype.card ι)
-
-abbrev CoefficientIndex := Σ i,Fin (Module.finrank ℂ (V i)) × Fin (Module.finrank ℂ (V i))
-
-noncomputable def coefficientFamily (a : CoefficientIndex V) : EuclideanSpace ℂ G :=
-  regularCoefficient (ρ a.1) (stdOrthonormalBasis ℂ (V a.1) a.2.1) (stdOrthonormalBasis ℂ (V a.1) a.2.2)
-
-omit [Fintype ι] in
-include hρ hneq in
-lemma coefficientFamily_orthonormal : Orthonormal ℂ (coefficientFamily V ρ) := by
-  rw [orthonormal_iff_ite]
-  rintro ⟨i,a,b⟩ ⟨j,c,d⟩
-  dsimp only [coefficientFamily]
-  by_cases hij : i=j
-  · subst j
-    rw [regularCoefficient_inner _ (hρ i)]
-    simp only [(stdOrthonormalBasis ℂ (V i)).inner_eq_ite]
-    by_cases hac : a=c <;> by_cases hbd : b=d <;> simp [hac,hbd,eq_comm]
-  · rw [regularCoefficient_cross _ _ (hρ j) (hneq j i (Ne.symm hij))]
-    rw [ite_eq_right (by intro h; exact hij (congrArg Sigma.fst h))]
-
-include ρ hneq hcard in
-lemma coefficientIndex_card : Fintype.card (CoefficientIndex V)=Fintype.card G := by
-  simp only [CoefficientIndex,Fintype.card_sigma,Fintype.card_prod,Fintype.card_fin,←pow_two]
-  exact FiniteCharacters.sum_dimension_sq V ρ hneq hcard
-
-noncomputable def coefficientBasis : OrthonormalBasis (CoefficientIndex V) ℂ (EuclideanSpace ℂ G) :=
-  OrthonormalBasis.mk (coefficientFamily_orthonormal V ρ hρ hneq)
-    (by rw [(coefficientFamily_orthonormal V ρ hρ hneq).linearIndependent.span_eq_top_of_card_eq_finrank'
-          (by rw [coefficientIndex_card V ρ hneq hcard,finrank_euclideanSpace])])
-
-lemma coefficientBasis_apply (a : CoefficientIndex V) :
-    coefficientBasis V ρ hρ hneq hcard a=coefficientFamily V ρ a := congrFun (OrthonormalBasis.coe_mk _ _) a
-
-include hρ hneq hcard in
-/-- Finite-group Plancherel, produced from the actual complete irreducible
-coefficient basis rather than postulated as a Fourier hypothesis. -/
-theorem parseval (f : EuclideanSpace ℂ G) :
-    ∑ a : CoefficientIndex V, ‖inner ℂ (coefficientFamily V ρ a) f‖^2=‖f‖^2 := by
-  simpa only [coefficientBasis_apply] using
-    (coefficientBasis V ρ hρ hneq hcard).sum_sq_norm_inner_right f
-
-end CubeShuffle.FiniteFourier
-namespace CubeShuffle.FiniteFourier
-open scoped BigOperators ComplexConjugate Classical
-open UnitaryFinite
-
-variable {G ι : Type*} [Group G] [Fintype G] [Fintype ι]
-variable (V : ι → Type*) [∀ i, NormedAddCommGroup (V i)] [∀ i, InnerProductSpace ℂ (V i)]
-  [∀ i, FiniteDimensional ℂ (V i)]
-variable (ρ : ∀ i, Representation ℂ G (V i)) [∀ i, Representation.IsIrreducible (ρ i)]
-variable (hρ : ∀ i, IsUnitary (ρ i))
-variable (hneq : ∀ i j, i≠j → ¬Nonempty (Representation.Equiv (ρ i) (ρ j)))
-  (hcard : Nat.card (ConjClasses G)≤Fintype.card ι)
-
-noncomputable def realVector (p : G → ℝ) : EuclideanSpace ℂ G := WithLp.toLp 2 (fun g => (p g:ℂ))
-
-omit [Fintype ι] [∀ i, Representation.IsIrreducible (ρ i)] in
-lemma coefficient_realVector (p : G → ℝ) (a : CoefficientIndex V) :
-    inner ℂ (coefficientFamily V ρ a) (realVector p)=
-      (Real.sqrt ((Module.finrank ℂ (V a.1):ℝ)/(Fintype.card G:ℝ)):ℂ)*
-      inner ℂ (stdOrthonormalBasis ℂ (V a.1) a.2.2)
-        (weightedOperator (ρ a.1) p (stdOrthonormalBasis ℂ (V a.1) a.2.1)) := by
-  simp only [PiLp.inner_apply,coefficientFamily,regularCoefficient_apply,realVector,
-    RCLike.inner_apply,map_mul,Complex.conj_ofReal,inner_conj_symm,
-    weightedOperator,sum_apply,smul_apply,
-    continuousRepresentation_apply,inner_sum,inner_smul_right,Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro g _
-  ring
-
-omit [Fintype ι] [∀ i, Representation.IsIrreducible (ρ i)] in
-lemma coefficient_realVector_sq (p : G → ℝ) (a : CoefficientIndex V) :
-    ‖inner ℂ (coefficientFamily V ρ a) (realVector p)‖^2=
-      ((Module.finrank ℂ (V a.1):ℝ)/(Fintype.card G:ℝ))*
-      ‖inner ℂ (stdOrthonormalBasis ℂ (V a.1) a.2.2)
-        (weightedOperator (ρ a.1) p (stdOrthonormalBasis ℂ (V a.1) a.2.1))‖^2 := by
-  rw [coefficient_realVector,norm_mul,Complex.norm_real,Real.norm_eq_abs,
-    abs_of_nonneg (Real.sqrt_nonneg _),mul_pow,Real.sq_sqrt (div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _))]
-
-include hρ hneq hcard in
-lemma parseval_operator (p : G → ℝ) :
-    (Fintype.card G:ℝ)*‖realVector p‖^2=
-      ∑ i,(Module.finrank ℂ (V i):ℝ)*∑ a : Fin (Module.finrank ℂ (V i)),
-        ‖weightedOperator (ρ i) p (stdOrthonormalBasis ℂ (V i) a)‖^2 := by
-  rw [←parseval V ρ hρ hneq hcard (realVector p)]
-  simp only [Fintype.sum_sigma,Fintype.sum_prod_type,coefficient_realVector_sq]
-  have hN : (Fintype.card G:ℝ)≠0 := by exact_mod_cast Fintype.card_ne_zero
-  rw [Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro i _
-  have he (a : Fin (Module.finrank ℂ (V i))) :
-      (∑ b : Fin (Module.finrank ℂ (V i)),
-        (Module.finrank ℂ (V i):ℝ)/(Fintype.card G:ℝ)*
-          ‖inner ℂ (stdOrthonormalBasis ℂ (V i) b)
-            (weightedOperator (ρ i) p (stdOrthonormalBasis ℂ (V i) a))‖^2)=
-        (Module.finrank ℂ (V i):ℝ)/(Fintype.card G:ℝ)*
-          ‖weightedOperator (ρ i) p (stdOrthonormalBasis ℂ (V i) a)‖^2 := by
-    rw [←Finset.mul_sum,(stdOrthonormalBasis ℂ (V i)).sum_sq_norm_inner_right]
-  simp_rw [he]
-  rw [←Finset.mul_sum]
-  field_simp
-
-include hρ hneq hcard in
-lemma realVector_bound (p : G → ℝ) :
-    (Fintype.card G:ℝ)*‖realVector p‖^2≤
-      ∑ i,(Module.finrank ℂ (V i):ℝ)^2*‖weightedOperator (ρ i) p‖^2 := by
-  rw [parseval_operator V ρ hρ hneq hcard]
-  apply Finset.sum_le_sum
-  intro i _
-  calc
-    _ ≤ (Module.finrank ℂ (V i):ℝ)*∑ _a : Fin (Module.finrank ℂ (V i)),‖weightedOperator (ρ i) p‖^2 := by
-      apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
-      apply Finset.sum_le_sum
-      intro a _
-      apply pow_le_pow_left₀ (norm_nonneg _)
-      simpa only [(stdOrthonormalBasis ℂ (V i)).norm_eq_one,mul_one] using
-        (weightedOperator (ρ i) p).le_opNorm (stdOrthonormalBasis ℂ (V i) a)
-    _ = _ := by simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]; ring
-
-include hρ hneq hcard in
-/-- The exact full-group Fourier upper bound, with no marginal replacement. -/
-theorem l1_fourier_bound (p : G → ℝ) :
-    (∑ g,|p g|)^2≤∑ i,(Module.finrank ℂ (V i):ℝ)^2*‖weightedOperator (ρ i) p‖^2 := by
-  apply le_trans _ (realVector_bound V ρ hρ hneq hcard p)
-  have he : ‖realVector p‖^2=∑ g,|p g|^2 := by
-    rw [EuclideanSpace.norm_sq_eq]
-    simp only [realVector,PiLp.toLp_apply,Complex.norm_real,Real.norm_eq_abs]
-  rw [he]
-  simpa only [Finset.card_univ] using sq_sum_le_card_mul_sum_sq (s := Finset.univ) (f := fun g => |p g|)
-
-end CubeShuffle.FiniteFourier
-
-end OAI
+import OAI.RepresentationTheory.FiniteUnitary.Characters
+import OAI.RepresentationTheory.FiniteUnitary.Sampling
+
+namespace OAI
+
+namespace CubeShuffle.FiniteFourier
+open scoped BigOperators ComplexConjugate Classical
+open UnitaryFinite
+
+variable {G ι : Type*} [Group G] [Fintype G] [Fintype ι]
+variable (V : ι → Type*) [∀ i, NormedAddCommGroup (V i)] [∀ i, InnerProductSpace ℂ (V i)]
+  [∀ i, FiniteDimensional ℂ (V i)]
+variable (ρ : ∀ i, Representation ℂ G (V i)) [∀ i, Representation.IsIrreducible (ρ i)]
+variable (hρ : ∀ i, IsUnitary (ρ i))
+variable (hneq : ∀ i j, i≠j → ¬Nonempty (Representation.Equiv (ρ i) (ρ j)))
+  (hcard : Nat.card (ConjClasses G)≤Fintype.card ι)
+
+abbrev CoefficientIndex := Σ i,Fin (Module.finrank ℂ (V i)) × Fin (Module.finrank ℂ (V i))
+
+noncomputable def coefficientFamily (a : CoefficientIndex V) : EuclideanSpace ℂ G :=
+  regularCoefficient (ρ a.1) (stdOrthonormalBasis ℂ (V a.1) a.2.1) (stdOrthonormalBasis ℂ (V a.1) a.2.2)
+
+omit [Fintype ι] in
+include hρ hneq in
+lemma coefficientFamily_orthonormal : Orthonormal ℂ (coefficientFamily V ρ) := by
+  rw [orthonormal_iff_ite]
+  rintro ⟨i,a,b⟩ ⟨j,c,d⟩
+  dsimp only [coefficientFamily]
+  by_cases hij : i=j
+  · subst j
+    rw [regularCoefficient_inner _ (hρ i)]
+    simp only [(stdOrthonormalBasis ℂ (V i)).inner_eq_ite]
+    by_cases hac : a=c <;> by_cases hbd : b=d <;> simp [hac,hbd,eq_comm]
+  · rw [regularCoefficient_cross _ _ (hρ j) (hneq j i (Ne.symm hij))]
+    rw [ite_eq_right (by intro h; exact hij (congrArg Sigma.fst h))]
+
+include ρ hneq hcard in
+lemma coefficientIndex_card : Fintype.card (CoefficientIndex V)=Fintype.card G := by
+  simp only [CoefficientIndex,Fintype.card_sigma,Fintype.card_prod,Fintype.card_fin,←pow_two]
+  exact FiniteCharacters.sum_dimension_sq V ρ hneq hcard
+
+noncomputable def coefficientBasis : OrthonormalBasis (CoefficientIndex V) ℂ (EuclideanSpace ℂ G) :=
+  OrthonormalBasis.mk (coefficientFamily_orthonormal V ρ hρ hneq)
+    (by rw [(coefficientFamily_orthonormal V ρ hρ hneq).linearIndependent.span_eq_top_of_card_eq_finrank'
+          (by rw [coefficientIndex_card V ρ hneq hcard,finrank_euclideanSpace])])
+
+lemma coefficientBasis_apply (a : CoefficientIndex V) :
+    coefficientBasis V ρ hρ hneq hcard a=coefficientFamily V ρ a := congrFun (OrthonormalBasis.coe_mk _ _) a
+
+include hρ hneq hcard in
+/-- Finite-group Plancherel, produced from the actual complete irreducible
+coefficient basis rather than postulated as a Fourier hypothesis. -/
+theorem parseval (f : EuclideanSpace ℂ G) :
+    ∑ a : CoefficientIndex V, ‖inner ℂ (coefficientFamily V ρ a) f‖^2=‖f‖^2 := by
+  simpa only [coefficientBasis_apply] using
+    (coefficientBasis V ρ hρ hneq hcard).sum_sq_norm_inner_right f
+
+end CubeShuffle.FiniteFourier
+namespace CubeShuffle.FiniteFourier
+open scoped BigOperators ComplexConjugate Classical
+open UnitaryFinite
+
+variable {G ι : Type*} [Group G] [Fintype G] [Fintype ι]
+variable (V : ι → Type*) [∀ i, NormedAddCommGroup (V i)] [∀ i, InnerProductSpace ℂ (V i)]
+  [∀ i, FiniteDimensional ℂ (V i)]
+variable (ρ : ∀ i, Representation ℂ G (V i)) [∀ i, Representation.IsIrreducible (ρ i)]
+variable (hρ : ∀ i, IsUnitary (ρ i))
+variable (hneq : ∀ i j, i≠j → ¬Nonempty (Representation.Equiv (ρ i) (ρ j)))
+  (hcard : Nat.card (ConjClasses G)≤Fintype.card ι)
+
+noncomputable def realVector (p : G → ℝ) : EuclideanSpace ℂ G := WithLp.toLp 2 (fun g => (p g:ℂ))
+
+omit [Fintype ι] [∀ i, Representation.IsIrreducible (ρ i)] in
+lemma coefficient_realVector (p : G → ℝ) (a : CoefficientIndex V) :
+    inner ℂ (coefficientFamily V ρ a) (realVector p)=
+      (Real.sqrt ((Module.finrank ℂ (V a.1):ℝ)/(Fintype.card G:ℝ)):ℂ)*
+      inner ℂ (stdOrthonormalBasis ℂ (V a.1) a.2.2)
+        (weightedOperator (ρ a.1) p (stdOrthonormalBasis ℂ (V a.1) a.2.1)) := by
+  simp only [PiLp.inner_apply,coefficientFamily,regularCoefficient_apply,realVector,
+    RCLike.inner_apply,map_mul,Complex.conj_ofReal,inner_conj_symm,
+    weightedOperator,sum_apply,smul_apply,
+    continuousRepresentation_apply,inner_sum,inner_smul_right,Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro g _
+  ring
+
+omit [Fintype ι] [∀ i, Representation.IsIrreducible (ρ i)] in
+lemma coefficient_realVector_sq (p : G → ℝ) (a : CoefficientIndex V) :
+    ‖inner ℂ (coefficientFamily V ρ a) (realVector p)‖^2=
+      ((Module.finrank ℂ (V a.1):ℝ)/(Fintype.card G:ℝ))*
+      ‖inner ℂ (stdOrthonormalBasis ℂ (V a.1) a.2.2)
+        (weightedOperator (ρ a.1) p (stdOrthonormalBasis ℂ (V a.1) a.2.1))‖^2 := by
+  rw [coefficient_realVector,norm_mul,Complex.norm_real,Real.norm_eq_abs,
+    abs_of_nonneg (Real.sqrt_nonneg _),mul_pow,Real.sq_sqrt (div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _))]
+
+include hρ hneq hcard in
+lemma parseval_operator (p : G → ℝ) :
+    (Fintype.card G:ℝ)*‖realVector p‖^2=
+      ∑ i,(Module.finrank ℂ (V i):ℝ)*∑ a : Fin (Module.finrank ℂ (V i)),
+        ‖weightedOperator (ρ i) p (stdOrthonormalBasis ℂ (V i) a)‖^2 := by
+  rw [←parseval V ρ hρ hneq hcard (realVector p)]
+  simp only [Fintype.sum_sigma,Fintype.sum_prod_type,coefficient_realVector_sq]
+  have hN : (Fintype.card G:ℝ)≠0 := by exact_mod_cast Fintype.card_ne_zero
+  rw [Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro i _
+  have he (a : Fin (Module.finrank ℂ (V i))) :
+      (∑ b : Fin (Module.finrank ℂ (V i)),
+        (Module.finrank ℂ (V i):ℝ)/(Fintype.card G:ℝ)*
+          ‖inner ℂ (stdOrthonormalBasis ℂ (V i) b)
+            (weightedOperator (ρ i) p (stdOrthonormalBasis ℂ (V i) a))‖^2)=
+        (Module.finrank ℂ (V i):ℝ)/(Fintype.card G:ℝ)*
+          ‖weightedOperator (ρ i) p (stdOrthonormalBasis ℂ (V i) a)‖^2 := by
+    rw [←Finset.mul_sum,(stdOrthonormalBasis ℂ (V i)).sum_sq_norm_inner_right]
+  simp_rw [he]
+  rw [←Finset.mul_sum]
+  field_simp
+
+include hρ hneq hcard in
+lemma realVector_bound (p : G → ℝ) :
+    (Fintype.card G:ℝ)*‖realVector p‖^2≤
+      ∑ i,(Module.finrank ℂ (V i):ℝ)^2*‖weightedOperator (ρ i) p‖^2 := by
+  rw [parseval_operator V ρ hρ hneq hcard]
+  apply Finset.sum_le_sum
+  intro i _
+  calc
+    _ ≤ (Module.finrank ℂ (V i):ℝ)*∑ _a : Fin (Module.finrank ℂ (V i)),‖weightedOperator (ρ i) p‖^2 := by
+      apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
+      apply Finset.sum_le_sum
+      intro a _
+      apply pow_le_pow_left₀ (norm_nonneg _)
+      simpa only [(stdOrthonormalBasis ℂ (V i)).norm_eq_one,mul_one] using
+        (weightedOperator (ρ i) p).le_opNorm (stdOrthonormalBasis ℂ (V i) a)
+    _ = _ := by simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]; ring
+
+include hρ hneq hcard in
+/-- The exact full-group Fourier upper bound, with no marginal replacement. -/
+theorem l1_fourier_bound (p : G → ℝ) :
+    (∑ g,|p g|)^2≤∑ i,(Module.finrank ℂ (V i):ℝ)^2*‖weightedOperator (ρ i) p‖^2 := by
+  apply le_trans _ (realVector_bound V ρ hρ hneq hcard p)
+  have he : ‖realVector p‖^2=∑ g,|p g|^2 := by
+    rw [EuclideanSpace.norm_sq_eq]
+    simp only [realVector,PiLp.toLp_apply,Complex.norm_real,Real.norm_eq_abs]
+  rw [he]
+  simpa only [Finset.card_univ] using sq_sum_le_card_mul_sum_sq (s := Finset.univ) (f := fun g => |p g|)
+
+end CubeShuffle.FiniteFourier
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FiniteUnitary/Frames.lean b/lean/OAI/RepresentationTheory/FiniteUnitary/Frames.lean
index 66356b3ff0ea2fe4f34282397dac83e329496c37..703de26296cddf79aa970cfdb6a85f18905f93e1 100644
--- a/lean/OAI/RepresentationTheory/FiniteUnitary/Frames.lean
+++ b/lean/OAI/RepresentationTheory/FiniteUnitary/Frames.lean
@@ -1,560 +1,560 @@
-import Mathlib
-
-namespace OAI
-
-section
-noncomputable section
-open scoped BigOperators ComplexConjugate
-namespace RowColumn.HilbertSchmidt
-variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
-  [FiniteDimensional ℂ E]
-
-/-- The algebraic Hilbert--Schmidt scalar product, irrespective of the operator
-norm on the carrier. A core is used to avoid changing any source norm. -/
-def hsInner (T U : E →ₗ[ℂ] E) : ℂ :=
-  LinearMap.trace ℂ E (LinearMap.adjoint T ∘ₗ U)
-
-lemma hsInner_basis (T U : E →ₗ[ℂ] E) :
-    hsInner T U = ∑ i, inner ℂ (T (stdOrthonormalBasis ℂ E i))
-      (U (stdOrthonormalBasis ℂ E i)) := by
-  rw [hsInner, LinearMap.trace_eq_sum_inner _ (stdOrthonormalBasis ℂ E)]
-  simp only [LinearMap.comp_apply, LinearMap.adjoint_inner_right]
-
-@[instance_reducible]
-def endCore : InnerProductSpace.Core ℂ (E →ₗ[ℂ] E) where
-  inner := hsInner
-  conj_inner_symm T U := by
-    simp only [hsInner_basis, map_sum, inner_conj_symm]
-  re_inner_nonneg T := by
-    rw [hsInner_basis, map_sum]
-    exact Finset.sum_nonneg fun _ _ => inner_self_nonneg
-  add_left T U V := by
-    simp only [hsInner_basis, LinearMap.add_apply, inner_add_left, Finset.sum_add_distrib]
-  smul_left T U z := by
-    simp only [hsInner_basis, LinearMap.smul_apply, inner_smul_left, Finset.mul_sum]
-  definite T h := by
-    have hh : ∑ i, ‖T (stdOrthonormalBasis ℂ E i)‖^2 = 0 := by
-      have he := congrArg (RCLike.re : ℂ → ℝ) h
-      simpa only [hsInner_basis, map_sum, map_zero, inner_self_eq_norm_sq] using he
-    have hz (i : Fin (Module.finrank ℂ E)) : T (stdOrthonormalBasis ℂ E i) = 0 := by
-      have hi := (Finset.sum_eq_zero_iff_of_nonneg (fun j (_ : j ∈ Finset.univ) =>
-        sq_nonneg ‖T (stdOrthonormalBasis ℂ E j)‖)).mp hh i (Finset.mem_univ i)
-      exact norm_eq_zero.mp (sq_eq_zero_iff.mp hi)
-    apply (stdOrthonormalBasis ℂ E).toBasis.ext
-    intro i
-    simpa using hz i
-
-lemma adjoint_isometry (u : E ≃ₗᵢ[ℂ] E) :
-    LinearMap.adjoint u.toLinearEquiv.toLinearMap = u.symm.toLinearEquiv.toLinearMap := by
-  apply LinearMap.ext
-  intro x
-  apply ext_inner_right ℂ
-  intro y
-  rw [LinearMap.adjoint_inner_left]
-  change inner ℂ x (u y) = inner ℂ (u.symm x) y
-  simpa only [u.apply_symm_apply] using u.inner_map_map (u.symm x) y
-
-lemma hsInner_conj (u : E ≃ₗᵢ[ℂ] E) (T V : E →ₗ[ℂ] E) :
-    hsInner (u.toLinearEquiv.conj T) (u.toLinearEquiv.conj V) = hsInner T V := by
-  rw [hsInner_basis]
-  conv_rhs => rw [hsInner, LinearMap.trace_eq_sum_inner _
-    ((stdOrthonormalBasis ℂ E).map u.symm)]
-  apply Finset.sum_congr rfl
-  intro i _
-  simp only [LinearMap.comp_apply, LinearMap.adjoint_inner_right,
-    OrthonormalBasis.map_apply, LinearEquiv.conj_apply]
-  change inner ℂ (u (T (u.symm (stdOrthonormalBasis ℂ E i))))
-      (u (V (u.symm (stdOrthonormalBasis ℂ E i)))) = _
-  exact u.inner_map_map _ _
-
-end RowColumn.HilbertSchmidt
-
-noncomputable section
-open scoped RealInnerProductSpace
-namespace RowColumn.UnitaryFrame
-variable {E ι : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]
-
-omit [CompleteSpace E] in
-lemma norm_minimizer_unique [CompleteSpace E] (K : Set E) (hK : Convex ℝ K)
-    {x y : E} (hx : x ∈ K) (hy : y ∈ K)
-    (hxm : ∀ z ∈ K, ‖x‖ ≤ ‖z‖) (hym : ∀ z ∈ K, ‖y‖ ≤ ‖z‖) : x = y := by
-  have he : ‖x‖ = ‖y‖ := le_antisymm (hxm y hy) (hym x hx)
-  have hm : (1/2 : ℝ) • x + (1/2 : ℝ) • y ∈ K :=
-    hK hx hy (by norm_num) (by norm_num) (by norm_num)
-  have hl := hxm _ hm
-  have hn : ‖(1/2 : ℝ) • x + (1/2 : ℝ) • y‖ = (1/2 : ℝ)*‖x+y‖ := by
-    rw [← smul_add, norm_smul, Real.norm_eq_abs, abs_of_nonneg (by norm_num)]
-  rw [hn] at hl
-  have hsq := mul_self_le_mul_self (norm_nonneg x) hl
-  have hp := parallelogram_law_with_norm ℝ x y
-  rw [← he] at hp
-  have hd := sq_nonneg ‖x-y‖
-  have hz : ‖x-y‖ = 0 := by
-    nlinarith [norm_nonneg (x-y)]
-  exact sub_eq_zero.mp (norm_eq_zero.mp hz)
-
-/-- Simultaneous fixed point obtained from the unique minimum-norm point of a
-closed convex invariant set. Applied to rank-one unitary frames, this replaces
-Haar integration by finite-dimensional convex averaging. -/
-theorem exists_fixed_minimum (K : Set E) (hne : K.Nonempty) (hclosed : IsClosed K)
-    (hconv : Convex ℝ K) (L : ι → E →ₗᵢ[ℝ] E)
-    (hL : ∀ i x, x ∈ K → L i x ∈ K) :
-    ∃ x ∈ K, ∀ i, L i x = x := by
-  obtain ⟨x,hx,hm⟩ := exists_norm_eq_iInf_of_complete_convex hne hclosed.isComplete hconv (0 : E)
-  have hmin (z : E) (hz : z ∈ K) : ‖x‖ ≤ ‖z‖ := by
-    have hh : (⨅ w : K, ‖(0 : E)-w‖) ≤ ‖(0 : E)-z‖ :=
-      ciInf_le (f := fun w : K => ‖(0 : E) - (w : E)‖)
-        ⟨0, Set.forall_mem_range.mpr (fun _ => norm_nonneg _)⟩ (⟨z,hz⟩ : K)
-    rw [← hm] at hh
-    simpa using hh
-  refine ⟨x,hx,fun i => ?_⟩
-  exact norm_minimizer_unique K hconv (hL i x hx) hx
-    (fun z hz => by rw [(L i).norm_map]; exact hmin z hz) hmin
-
-end RowColumn.UnitaryFrame
-
-noncomputable section
-open scoped BigOperators ComplexConjugate
-namespace RowColumn.UnitaryFrame
-variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
-  [FiniteDimensional ℂ E]
-
-def quadraticForm (T : E →ₗ[ℂ] E) (x : E) : ℝ := (inner ℂ x (T x)).re
-
-omit [FiniteDimensional ℂ E] in
-lemma quadraticForm_real_smul [FiniteDimensional ℂ E] (T : E →ₗ[ℂ] E) (c : ℝ) (x : E) :
-    quadraticForm T (c • x) = c^2 * quadraticForm T x := by
-  simp only [quadraticForm, ← Complex.coe_smul, map_smul, inner_smul_left,
-    inner_smul_right, Complex.conj_ofReal]
-  simp [Complex.mul_re, pow_two, mul_assoc]
-
-lemma quadraticForm_continuous (T : E →ₗ[ℂ] E) : Continuous (quadraticForm T) :=
-  Complex.continuous_re.comp (continuous_id.inner T.toContinuousLinearMap.continuous)
-
-/-- Strict positivity of a finite-dimensional frame is uniformly coercive. -/
-theorem positive_coercive (T : E →ₗ[ℂ] E)
-    (hT : ∀ x : E, x ≠ 0 → 0 < quadraticForm T x) :
-    ∃ ε : ℝ, 0 < ε ∧ ∀ x, ε * ‖x‖^2 ≤ quadraticForm T x := by
-  cases subsingleton_or_nontrivial E with
-  | inl h =>
-    let := h
-    refine ⟨1, by norm_num, fun x => ?_⟩
-    have he : x = 0 := Subsingleton.elim _ _
-    simp [he, quadraticForm]
-  | inr h =>
-    let := h
-    let : ProperSpace E := FiniteDimensional.proper ℂ E
-    have hn : (Metric.sphere (0 : E) 1).Nonempty := NormedSpace.sphere_nonempty.mpr (by norm_num)
-    obtain ⟨y,hy,hmin⟩ := (isCompact_sphere (0 : E) 1).exists_isMinOn hn
-      (quadraticForm_continuous T).continuousOn
-    have hny : ‖y‖ = 1 := by simpa using hy
-    have hpy : 0 < quadraticForm T y := hT y (by intro he; simp [he] at hny)
-    refine ⟨quadraticForm T y, hpy, fun x => ?_⟩
-    by_cases hx : x = 0
-    · simp [hx, quadraticForm]
-    · have hp : 0 < ‖x‖ := norm_pos_iff.mpr hx
-      have hu : ‖x‖⁻¹ • x ∈ Metric.sphere (0 : E) 1 := by
-        simp [norm_smul, ne_of_gt hp]
-      have hm : quadraticForm T y ≤ quadraticForm T (‖x‖⁻¹ • x) := hmin hu
-      rw [quadraticForm_real_smul] at hm
-      have hh := mul_le_mul_of_nonneg_right hm (sq_nonneg ‖x‖)
-      calc
-        _ ≤ (‖x‖⁻¹ ^ 2 * quadraticForm T x) * ‖x‖^2 := hh
-        _ = _ := by field_simp [ne_of_gt hp]
-
-end RowColumn.UnitaryFrame
-
-namespace RowColumn.UnitaryFrame
-open scoped BigOperators ComplexOrder ComplexConjugate
-variable {E G : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
-  [FiniteDimensional ℂ E] [Group G]
-
-local instance : NormedAddCommGroup (E →ₗ[ℂ] E) := HilbertSchmidt.endCore.toNormedAddCommGroup
-local instance : InnerProductSpace ℂ (E →ₗ[ℂ] E) :=
-  InnerProductSpace.ofCore { __ := HilbertSchmidt.endCore (E := E) }
-local instance : InnerProductSpace ℝ (E →ₗ[ℂ] E) := InnerProductSpace.complexToReal
-
-def conjugationIsometry (u : E ≃ₗᵢ[ℂ] E) :
-    (E →ₗ[ℂ] E) →ₗᵢ[ℂ] (E →ₗ[ℂ] E) :=
-  u.toLinearEquiv.conj.toLinearMap.isometryOfInner (HilbertSchmidt.hsInner_conj u)
-
-def conjugationReal (u : E ≃ₗᵢ[ℂ] E) :
-    (E →ₗ[ℂ] E) →ₗᵢ[ℝ] (E →ₗ[ℂ] E) where
-  toLinearMap := (conjugationIsometry u).toLinearMap.restrictScalars ℝ
-  norm_map' := (conjugationIsometry u).norm_map
-
-lemma conjugation_apply (u : E ≃ₗᵢ[ℂ] E) (T : E →ₗ[ℂ] E) (x : E) :
-    conjugationReal u T x = u (T (u.symm x)) := rfl
-
-lemma conjugation_rankOne (u : E ≃ₗᵢ[ℂ] E) (x : E) :
-    conjugationReal u (InnerProductSpace.rankOne ℂ x x).toLinearMap =
-      (InnerProductSpace.rankOne ℂ (u x) (u x)).toLinearMap := by
-  ext y
-  change u (inner ℂ x (u.symm y) • x) = inner ℂ (u x) y • u x
-  rw [map_smul]
-  congr 1
-  simpa only [u.apply_symm_apply] using (u.inner_map_map x (u.symm y)).symm
-
-def matrixEntry (x y : E) : (E →ₗ[ℂ] E) →ₗ[ℂ] ℂ where
-  toFun T := inner ℂ x (T y)
-  map_add' T V := by simp only [LinearMap.add_apply, inner_add_right]
-  map_smul' c T := by simp only [LinearMap.smul_apply, inner_smul_right, smul_eq_mul, RingHom.id_apply]
-
-def quadraticEval (x : E) : (E →ₗ[ℂ] E) →ₗ[ℝ] ℝ where
-  toFun T := quadraticForm T x
-  map_add' T V := by simp [quadraticForm]
-  map_smul' c T := by
-    simp only [quadraticForm, LinearMap.smul_apply, ← Complex.coe_smul, inner_smul_right]
-    simp [Complex.mul_re]
-
-def coerciveSet (ε : ℝ) : Set (E →ₗ[ℂ] E) :=
-  {T | ∀ x, ε * ‖x‖^2 ≤ quadraticForm T x}
-
-lemma coerciveSet_closed (ε : ℝ) : IsClosed (coerciveSet (E := E) ε) := by
-  rw [coerciveSet, Set.ofPred_forall]
-  apply isClosed_iInter
-  intro x
-  exact isClosed_le continuous_const
-    (show Continuous (fun T : E →ₗ[ℂ] E => quadraticForm T x) from
-      (quadraticEval x).toContinuousLinearMap.continuous)
-
-omit [FiniteDimensional ℂ E] in
-lemma coerciveSet_convex [FiniteDimensional ℂ E] (ε : ℝ) :
-    Convex ℝ (coerciveSet (E := E) ε) := by
-  intro T hT V hV a b ha hb hab x
-  have h₁ := mul_le_mul_of_nonneg_left (hT x) ha
-  have h₂ := mul_le_mul_of_nonneg_left (hV x) hb
-  change ε * ‖x‖^2 ≤ quadraticEval x (a • T + b • V)
-  rw [map_add, map_smul, map_smul]
-  change ε * ‖x‖^2 ≤ a * quadraticForm T x + b * quadraticForm V x
-  calc
-    ε * ‖x‖^2 = (a+b) * (ε * ‖x‖^2) := by rw [hab, one_mul]
-    _ = a * (ε * ‖x‖^2) + b * (ε * ‖x‖^2) := add_mul _ _ _
-    _ ≤ _ := add_le_add h₁ h₂
-
-lemma positiveSet_closed : IsClosed {T : E →ₗ[ℂ] E | T.IsPositive} := by
-  have hs : IsClosed {T : E →ₗ[ℂ] E | T.IsSymmetric} := by
-    have he : {T : E →ₗ[ℂ] E | T.IsSymmetric} =
-        ⋂ x : E, ⋂ y : E,
-        {T | starRingEnd ℂ (matrixEntry y x T) = matrixEntry x y T} := by
-      ext T
-      simp only [Set.mem_ofPred_eq, Set.mem_iInter, matrixEntry, LinearMap.coe_mk,
-        AddHom.coe_mk, inner_conj_symm, LinearMap.IsSymmetric]
-    rw [he]
-    exact isClosed_iInter fun x => isClosed_iInter fun y => isClosed_eq
-      (Complex.continuous_conj.comp (matrixEntry y x).toContinuousLinearMap.continuous)
-      (matrixEntry x y).toContinuousLinearMap.continuous
-  have hn : IsClosed {T : E →ₗ[ℂ] E | ∀ x, 0 ≤ quadraticForm T x} := by
-    rw [Set.ofPred_forall]
-    apply isClosed_iInter
-    intro x
-    exact isClosed_le continuous_const
-      (show Continuous (fun T : E →ₗ[ℂ] E => quadraticForm T x) from
-        (quadraticEval x).toContinuousLinearMap.continuous)
-  have he : {T : E →ₗ[ℂ] E | T.IsPositive} =
-      {T | T.IsSymmetric} ∩ {T | ∀ x, 0 ≤ quadraticForm T x} := by
-    ext T
-    constructor
-    · intro h; exact ⟨h.isSymmetric, fun x => h.re_inner_nonneg_right x⟩
-    · rintro ⟨hs,hn⟩
-      refine ⟨hs, fun x => ?_⟩
-      rw [inner_re_symm]
-      exact hn x
-  rw [he]
-  exact hs.inter hn
-
-omit [FiniteDimensional ℂ E] in
-lemma positiveSet_convex [FiniteDimensional ℂ E] : Convex ℝ {T : E →ₗ[ℂ] E | T.IsPositive} := by
-  intro T hT V hV a b ha hb hab
-  change (a • T + b • V).IsPositive
-  rw [← Complex.coe_smul a T, ← Complex.coe_smul b V]
-  exact (hT.smul_of_nonneg (by exact_mod_cast ha)).add
-    (hV.smul_of_nonneg (by exact_mod_cast hb))
-
-def frameHull (π : G →* E ≃ₗᵢ[ℂ] E) (x₀ : E) : Set (E →ₗ[ℂ] E) :=
-  closure (convexHull ℝ (Set.range fun g : G =>
-    (InnerProductSpace.rankOne ℂ (π g x₀) (π g x₀)).toLinearMap))
-
-lemma frameHull_invariant (π : G →* E ≃ₗᵢ[ℂ] E) (x₀ : E) (g : G) :
-    Set.MapsTo (conjugationReal (π g)) (frameHull π x₀) (frameHull π x₀) := by
-  apply Set.MapsTo.closure _ (conjugationReal (π g)).continuous
-  rw [Set.mapsTo_iff_image_subset]
-  rw [← LinearIsometry.coe_toLinearMap, (conjugationReal (π g)).toLinearMap.image_convexHull]
-  apply convexHull_mono
-  rintro _ ⟨_, ⟨h,rfl⟩, rfl⟩
-  refine ⟨g*h, ?_⟩
-  change (InnerProductSpace.rankOne ℂ (π (g*h) x₀) (π (g*h) x₀)).toLinearMap =
-    conjugationReal (π g) (InnerProductSpace.rankOne ℂ (π h x₀) (π h x₀)).toLinearMap
-  rw [conjugation_rankOne, map_mul]
-  rfl
-
-lemma quadraticForm_conj (u : E ≃ₗᵢ[ℂ] E) (T : E →ₗ[ℂ] E) (x : E) :
-    quadraticForm (conjugationReal u T) x = quadraticForm T (u.symm x) := by
-  unfold quadraticForm
-  rw [conjugation_apply]
-  have hi := u.inner_map_map (u.symm x) (T (u.symm x))
-  rw [u.apply_symm_apply] at hi
-  exact congrArg Complex.re hi
-
-/-- A genuine invariant strictly positive frame is obtained by convex averaging;
-no Haar measure or compact-group orthogonality is taken as a hypothesis. -/
-theorem exists_invariant_frame_of_coercive
-    (π : G →* E ≃ₗᵢ[ℂ] E) (x₀ : E)
-    (T₀ : E →ₗ[ℂ] E) (hT₀ : T₀ ∈ frameHull π x₀) (hp₀ : T₀.IsPositive)
-    (ε : ℝ) (_ : 0 < ε) (hε₀ : T₀ ∈ coerciveSet ε) :
-    ∃ T : E →ₗ[ℂ] E, T ∈ frameHull π x₀ ∧ T.IsPositive ∧
-      T ∈ coerciveSet ε ∧ ∀ g, conjugationReal (π g) T = T := by
-  let K := frameHull π x₀ ∩ ({T | T.IsPositive} ∩ coerciveSet ε)
-  have hc : IsClosed K := isClosed_closure.inter (positiveSet_closed.inter (coerciveSet_closed ε))
-  have hv : Convex ℝ K := (convex_convexHull ℝ _).closure.inter
-    (positiveSet_convex.inter (coerciveSet_convex ε))
-  have hL (g : G) (V : E →ₗ[ℂ] E) (hV : V ∈ K) :
-      conjugationReal (π g) V ∈ K := by
-    refine ⟨frameHull_invariant π x₀ g hV.1, ?_, ?_⟩
-    · exact (LinearMap.isPositive_linearIsometryEquiv_conj_iff (π g)).mpr hV.2.1
-    · intro x
-      rw [quadraticForm_conj]
-      simpa only [(π g).symm.norm_map] using hV.2.2 ((π g).symm x)
-  obtain ⟨T,hT,hfix⟩ := exists_fixed_minimum K ⟨T₀,hT₀,hp₀,hε₀⟩ hc hv
-    (fun g => conjugationReal (π g)) hL
-  exact ⟨T,hT.1,hT.2.1,hT.2.2,hfix⟩
-
-end RowColumn.UnitaryFrame
-namespace RowColumn.UnitaryFrame
-open scoped BigOperators ComplexOrder ComplexConjugate
-variable {E G : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
-  [FiniteDimensional ℂ E] [Group G]
-
-omit [FiniteDimensional ℂ E] in
-lemma rankOne_quadratic [FiniteDimensional ℂ E] (v x : E) :
-    quadraticForm (InnerProductSpace.rankOne ℂ v v).toLinearMap x =
-      Complex.normSq (inner ℂ v x) := by
-  change (inner ℂ x (inner ℂ v x • v)).re = _
-  rw [inner_smul_right, ← inner_conj_symm x v, Complex.mul_conj, Complex.ofReal_re]
-
-omit [FiniteDimensional ℂ E] in
-lemma finite_frame_strict [FiniteDimensional ℂ E] (s : Finset E)
-    (hs : Submodule.span ℂ (s : Set E) = ⊤)
-    {x : E} (hx : x ≠ 0) : 0 < ∑ v ∈ s, Complex.normSq (inner ℂ v x) := by
-  have hn : 0 ≤ ∑ v ∈ s, Complex.normSq (inner ℂ v x) :=
-    Finset.sum_nonneg fun _ _ => Complex.normSq_nonneg _
-  by_contra h
-  have hz := le_antisymm (le_of_not_gt h) hn
-  have hi (v : E) (hv : v ∈ s) : inner ℂ v x = 0 :=
-    Complex.normSq_eq_zero.mp ((Finset.sum_eq_zero_iff_of_nonneg
-      (fun _ _ => Complex.normSq_nonneg _)).mp hz v hv)
-  have hall (v : E) (hv : v ∈ Submodule.span ℂ (s : Set E)) : inner ℂ v x = 0 := by
-    induction hv using Submodule.span_induction with
-    | mem v hv => exact hi v hv
-    | zero => simp
-    | add u v hu hv heu hev => simp [heu, hev]
-    | smul c v hv hv' => simp [hv']
-  exact hx ((inner_self_eq_zero (𝕜 := ℂ)).mp (hall x (by rw [hs]; trivial)))
-
-def finiteFrame (s : Finset E) : E →ₗ[ℂ] E :=
-  s.centerMass (fun _ => (1 : ℝ))
-    (fun v => (InnerProductSpace.rankOne ℂ v v).toLinearMap)
-
-lemma finiteFrame_quadratic (s : Finset E) (x : E) :
-    quadraticForm (finiteFrame s) x = (s.card : ℝ)⁻¹ *
-      ∑ v ∈ s, Complex.normSq (inner ℂ v x) := by
-  change quadraticEval x (finiteFrame s) = _
-  simp only [finiteFrame, Finset.centerMass, one_smul, Finset.sum_const,
-    nsmul_eq_mul, mul_one, map_smul, map_sum, smul_eq_mul]
-  congr 1
-  apply Finset.sum_congr rfl
-  intro v _
-  exact rankOne_quadratic v x
-
-local instance : NormedAddCommGroup (E →ₗ[ℂ] E) := HilbertSchmidt.endCore.toNormedAddCommGroup
-local instance : InnerProductSpace ℂ (E →ₗ[ℂ] E) :=
-  InnerProductSpace.ofCore { __ := HilbertSchmidt.endCore (E := E) }
-local instance : InnerProductSpace ℝ (E →ₗ[ℂ] E) := InnerProductSpace.complexToReal
-
-lemma finiteFrame_positive (s : Finset E) (hs : s.Nonempty) : (finiteFrame s).IsPositive := by
-  change s.centerMass (fun _ => (1 : ℝ))
-    (fun v => (InnerProductSpace.rankOne ℂ v v).toLinearMap) ∈
-      {T : E →ₗ[ℂ] E | T.IsPositive}
-  apply (positiveSet_convex (E := E)).centerMass_mem (fun _ _ => by norm_num)
-  · simpa only [Finset.sum_const, nsmul_eq_mul, mul_one] using
-      (show (0 : ℝ) < s.card from by exact_mod_cast hs.card_pos)
-  · intro v _
-    exact (InnerProductSpace.isPositive_rankOne_self v).toLinearMap
-
-/-- Spanning unitary orbits have an invertible positive invariant frame, with
-its approximation by finite probability mixtures of the original orbit. -/
-theorem exists_invariant_frame (π : G →* E ≃ₗᵢ[ℂ] E) (x₀ : E)
-    (hspan : Submodule.span ℂ (Set.range fun g : G => π g x₀) = ⊤) :
-    ∃ T : E →ₗ[ℂ] E, T ∈ frameHull π x₀ ∧ T.IsPositive ∧
-      Function.Bijective T ∧ ∀ g, conjugationReal (π g) T = T := by
-  classical
-  obtain ⟨s,hs,hcard,hss,hindep⟩ := Submodule.exists_finset_span_eq_linearIndepOn ℂ
-    (Set.range fun g : G => π g x₀)
-  let t := insert x₀ s
-  have htn : t.Nonempty := Finset.insert_nonempty _ _
-  have hts : (t : Set E) ⊆ Set.range fun g : G => π g x₀ := by
-    intro x hx
-    rcases Finset.mem_insert.mp hx with rfl | hx
-    · exact ⟨1, by simp⟩
-    · exact hs hx
-  have htspan : Submodule.span ℂ (t : Set E) = ⊤ := by
-    apply top_unique
-    rw [← hspan, ← hss]
-    exact Submodule.span_mono (by intro x hx; exact Finset.mem_insert_of_mem hx)
-  have hmem : finiteFrame t ∈ frameHull π x₀ := by
-    apply subset_closure
-    apply t.centerMass_mem_convexHull (fun _ _ => by norm_num)
-    · simpa only [Finset.sum_const, nsmul_eq_mul, mul_one] using
-        (show (0 : ℝ) < t.card from by exact_mod_cast htn.card_pos)
-    · intro v hv
-      obtain ⟨g,rfl⟩ := hts hv
-      exact ⟨g,rfl⟩
-  have hstrict (x : E) (hx : x ≠ 0) : 0 < quadraticForm (finiteFrame t) x := by
-    rw [finiteFrame_quadratic]
-    exact mul_pos (inv_pos.mpr (by exact_mod_cast htn.card_pos)) (finite_frame_strict t htspan hx)
-  obtain ⟨ε,hε,hε₀⟩ := positive_coercive (finiteFrame t) hstrict
-  obtain ⟨T,hT,hp,hc,hfix⟩ := exists_invariant_frame_of_coercive π x₀ (finiteFrame t)
-    hmem (finiteFrame_positive t htn) ε hε hε₀
-  have hinj : Function.Injective T := by
-    apply LinearMap.ker_eq_bot.mp
-    apply LinearMap.ker_eq_bot'.mpr
-    intro x hx
-    have hh := hc x
-    change ε * ‖x‖^2 ≤ (inner ℂ x (T x)).re at hh
-    rw [hx, inner_zero_right, Complex.zero_re] at hh
-    have hn : ‖x‖^2 = 0 := le_antisymm (nonpos_of_mul_nonpos_right hh hε) (sq_nonneg _)
-    exact norm_eq_zero.mp (sq_eq_zero_iff.mp hn)
-  exact ⟨T,hT,hp,⟨hinj, (LinearMap.injective_iff_surjective).mp hinj⟩,hfix⟩
-
-end RowColumn.UnitaryFrame
-
-end
-end
-end
-end
-
-
-section
-noncomputable section
-open scoped BigOperators ComplexConjugate ComplexOrder
-namespace RowColumn.UnitaryFrame
-
-variable {A G : Type*} [Ring A] [Algebra ℂ A] [StarRing A]
-  [FiniteDimensional ℂ A] [Group G]
-
-/-- Algebraic and convex reconstruction from a spanning unitary group. The
-trace is the physical trace, not the trace of the algebra's regular action. -/
-theorem exists_density (τ : A →ₗ[ℂ] ℂ) (core : InnerProductSpace.Core ℂ A)
-    (hinner : ∀ x y, core.inner x y = τ (star x * y))
-    (u : G →* unitary A)
-    (hspan : Submodule.span ℂ (Set.range fun g : G => (u g : A)) = ⊤) :
-    ∃ D : A, τ D = (Module.finrank ℂ A : ℂ) ∧
-      (∀ x : A, 0 ≤ τ (star x * x * D)) ∧
-      (∀ (B : A) (ℓ : A →ₗ[ℂ] ℂ) (R : ℝ),
-        (∀ g, ‖τ (star (u g : A) * (B*D)) * ℓ (u g : A)‖ ≤ R) → ‖ℓ B‖ ≤ R) := by
-  let : NormedAddCommGroup A := core.toNormedAddCommGroup
-  let : InnerProductSpace ℂ A := InnerProductSpace.ofCore { __ := core }
-  change ∀ x y, inner ℂ x y = τ (star x*y) at hinner
-  let π : G →* A ≃ₗᵢ[ℂ] A :=
-    { toFun := fun g =>
-        (Units.mulLeftLinearEquiv ℂ A (Unitary.toUnits (u g))).isometryOfInner (by
-          intro x y
-          change inner ℂ ((u g : A)*x) ((u g : A)*y) = inner ℂ x y
-          simp only [hinner, star_mul, mul_assoc]
-          rw [← mul_assoc (star (u g : A)), Unitary.coe_star_mul_self, one_mul])
-      map_one' := by ext x; simp
-      map_mul' := by intros g h; ext x; simp }
-  have hπ (g : G) (x : A) : π g x = (u g : A)*x := rfl
-  obtain ⟨T,hT,hpos,hbij,hfix⟩ := exists_invariant_frame π 1 (by simpa only [hπ, mul_one] using hspan)
-  have heq (g : G) (x : A) : T ((u g : A)*x) = (u g : A)*T x := by
-    have he := LinearMap.congr_fun (hfix g) (π g x)
-    change π g (T ((π g).symm (π g x))) = T (π g x) at he
-    rw [(π g).symm_apply_apply] at he
-    exact he.symm
-  let C := T 1
-  have hTm (x : A) : T x = x*C := by
-    have hx : x ∈ Submodule.span ℂ (Set.range fun g : G => (u g : A)) := by rw [hspan]; trivial
-    induction hx using Submodule.span_induction with
-    | mem x hx => obtain ⟨g,rfl⟩ := hx; simpa only [mul_one] using heq g 1
-    | zero => simp
-    | add x y hx hy hxt hyt => simp only [map_add, add_mul, hxt, hyt]
-    | smul c x hx hxt => simpa only [map_smul, smul_mul_assoc] using congrArg (c • ·) hxt
-  let e : A ≃ₗ[ℂ] A := LinearEquiv.ofBijective T hbij
-  let D : A := e.symm 1
-  have hTD : T D = 1 := e.apply_symm_apply 1
-  have hDC : D*C = 1 := (hTm D).symm.trans hTD
-  have hinv (x : A) : T (x*D) = x := by rw [hTm, mul_assoc, hDC, mul_one]
-  have heinv (x : A) : e.symm x = x*D := by
-    apply hbij.injective
-    exact (e.apply_symm_apply x).trans (hinv x).symm
-  have hip : e.symm.toLinearMap.IsPositive := hpos.toLinearMap_symm
-  have hDpos (x : A) : 0 ≤ τ (star x*x*D) := by
-    have hh := hip.inner_nonneg_right x
-    simpa only [LinearEquiv.coe_coe, heinv, hinner, mul_assoc] using hh
-  let f : (A →ₗ[ℂ] A) →ₗ[ℂ] ℂ :=
-    { toFun S := LinearMap.trace ℂ A (S ∘ₗ LinearMap.mulRight ℂ D)
-      map_add' S V := by simp only [LinearMap.add_comp, map_add]
-      map_smul' c S := by simp only [LinearMap.smul_comp, map_smul, RingHom.id_apply] }
-  have hf (g : G) : f (InnerProductSpace.rankOne ℂ (π g 1) (π g 1)).toLinearMap = τ D := by
-    change LinearMap.trace ℂ A (_ ∘ₗ _) = _
-    rw [LinearMap.trace_comp_comm']
-    have he : LinearMap.mulRight ℂ D ∘ₗ
-        (InnerProductSpace.rankOne ℂ (π g 1) (π g 1)).toLinearMap =
-        (InnerProductSpace.rankOne ℂ ((π g 1)*D) (π g 1)).toLinearMap := by
-      ext x
-      simp only [LinearMap.comp_apply, LinearMap.mulRight_apply,
-        ContinuousLinearMap.coe_coe, InnerProductSpace.rankOne_apply, smul_mul_assoc]
-    rw [he]
-    change (InnerProductSpace.rankOne ℂ ((π g 1)*D) (π g 1)).trace ℂ A = _
-    rw [InnerProductSpace.trace_rankOne, hinner, hπ, mul_one,
-      ← mul_assoc, Unitary.coe_star_mul_self, one_mul]
-  let : NormedAddCommGroup (A →ₗ[ℂ] A) := HilbertSchmidt.endCore.toNormedAddCommGroup
-  let : InnerProductSpace ℂ (A →ₗ[ℂ] A) :=
-    InnerProductSpace.ofCore { __ := HilbertSchmidt.endCore (E := A) }
-  let : InnerProductSpace ℝ (A →ₗ[ℂ] A) := InnerProductSpace.complexToReal
-  have hfc : ∀ S ∈ frameHull π 1, f S = τ D := by
-    change closure (convexHull ℝ _) ⊆ {S | f S = τ D}
-    apply closure_minimal
-    · apply convexHull_min
-      · rintro _ ⟨g,rfl⟩; exact hf g
-      · intro S hS V hV a b ha hb hab
-        change f (a • S + b • V) = τ D
-        rw [map_add, ← Complex.coe_smul a S, ← Complex.coe_smul b V, map_smul, map_smul, hS, hV]
-        rw [← add_smul, ← Complex.ofReal_add, hab, Complex.ofReal_one, one_smul]
-    · exact isClosed_eq f.toContinuousLinearMap.continuous continuous_const
-  have hτD : τ D = (Module.finrank ℂ A : ℂ) := by
-    rw [← hfc T hT]
-    have hcomp : T ∘ₗ LinearMap.mulRight ℂ D = LinearMap.id := by ext x; exact hinv x
-    change LinearMap.trace ℂ A (_ ∘ₗ _) = _
-    rw [hcomp, LinearMap.trace_id]
-  refine ⟨D,hτD,hDpos,?_⟩
-  intro B ℓ R hR
-  let l : (A →ₗ[ℂ] A) →ₗ[ℂ] ℂ :=
-    { toFun S := ℓ (S (B*D))
-      map_add' S V := by simp only [LinearMap.add_apply, map_add]
-      map_smul' c S := by simp only [LinearMap.smul_apply, map_smul, RingHom.id_apply] }
-  have hl : ∀ S ∈ frameHull π 1, ‖l S‖ ≤ R := by
-    change closure (convexHull ℝ _) ⊆ {S | ‖l S‖ ≤ R}
-    apply closure_minimal
-    · apply convexHull_min
-      · rintro _ ⟨g,rfl⟩
-        change ‖ℓ ((InnerProductSpace.rankOne ℂ (π g 1) (π g 1)) (B*D))‖ ≤ R
-        simpa only [l, LinearMap.coe_mk, AddHom.coe_mk, ContinuousLinearMap.coe_coe,
-          InnerProductSpace.rankOne_apply, map_smul, hinner, hπ, mul_one, smul_eq_mul] using hR g
-      · intro S hS V hV a b ha hb hab
-        change ‖l (a • S + b • V)‖ ≤ R
-        rw [map_add, ← Complex.coe_smul a S, ← Complex.coe_smul b V, map_smul, map_smul]
-        calc
-          _ ≤ ‖(a : ℂ) • l S‖ + ‖(b : ℂ) • l V‖ := norm_add_le _ _
-          _ = a * ‖l S‖ + b * ‖l V‖ := by simp only [norm_smul, Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg ha, abs_of_nonneg hb]
-          _ ≤ a*R+b*R := add_le_add (mul_le_mul_of_nonneg_left hS ha) (mul_le_mul_of_nonneg_left hV hb)
-          _ = R := by rw [← add_mul, hab, one_mul]
-    · exact isClosed_le l.toContinuousLinearMap.continuous.norm continuous_const
-  simpa only [l, LinearMap.coe_mk, AddHom.coe_mk, hinv] using hl T hT
-
-end RowColumn.UnitaryFrame
-
-end
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+section
+noncomputable section
+open scoped BigOperators ComplexConjugate
+namespace RowColumn.HilbertSchmidt
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
+  [FiniteDimensional ℂ E]
+
+/-- The algebraic Hilbert--Schmidt scalar product, irrespective of the operator
+norm on the carrier. A core is used to avoid changing any source norm. -/
+def hsInner (T U : E →ₗ[ℂ] E) : ℂ :=
+  LinearMap.trace ℂ E (LinearMap.adjoint T ∘ₗ U)
+
+lemma hsInner_basis (T U : E →ₗ[ℂ] E) :
+    hsInner T U = ∑ i, inner ℂ (T (stdOrthonormalBasis ℂ E i))
+      (U (stdOrthonormalBasis ℂ E i)) := by
+  rw [hsInner, LinearMap.trace_eq_sum_inner _ (stdOrthonormalBasis ℂ E)]
+  simp only [LinearMap.comp_apply, LinearMap.adjoint_inner_right]
+
+@[instance_reducible]
+def endCore : InnerProductSpace.Core ℂ (E →ₗ[ℂ] E) where
+  inner := hsInner
+  conj_inner_symm T U := by
+    simp only [hsInner_basis, map_sum, inner_conj_symm]
+  re_inner_nonneg T := by
+    rw [hsInner_basis, map_sum]
+    exact Finset.sum_nonneg fun _ _ => inner_self_nonneg
+  add_left T U V := by
+    simp only [hsInner_basis, LinearMap.add_apply, inner_add_left, Finset.sum_add_distrib]
+  smul_left T U z := by
+    simp only [hsInner_basis, LinearMap.smul_apply, inner_smul_left, Finset.mul_sum]
+  definite T h := by
+    have hh : ∑ i, ‖T (stdOrthonormalBasis ℂ E i)‖^2 = 0 := by
+      have he := congrArg (RCLike.re : ℂ → ℝ) h
+      simpa only [hsInner_basis, map_sum, map_zero, inner_self_eq_norm_sq] using he
+    have hz (i : Fin (Module.finrank ℂ E)) : T (stdOrthonormalBasis ℂ E i) = 0 := by
+      have hi := (Finset.sum_eq_zero_iff_of_nonneg (fun j (_ : j ∈ Finset.univ) =>
+        sq_nonneg ‖T (stdOrthonormalBasis ℂ E j)‖)).mp hh i (Finset.mem_univ i)
+      exact norm_eq_zero.mp (sq_eq_zero_iff.mp hi)
+    apply (stdOrthonormalBasis ℂ E).toBasis.ext
+    intro i
+    simpa using hz i
+
+lemma adjoint_isometry (u : E ≃ₗᵢ[ℂ] E) :
+    LinearMap.adjoint u.toLinearEquiv.toLinearMap = u.symm.toLinearEquiv.toLinearMap := by
+  apply LinearMap.ext
+  intro x
+  apply ext_inner_right ℂ
+  intro y
+  rw [LinearMap.adjoint_inner_left]
+  change inner ℂ x (u y) = inner ℂ (u.symm x) y
+  simpa only [u.apply_symm_apply] using u.inner_map_map (u.symm x) y
+
+lemma hsInner_conj (u : E ≃ₗᵢ[ℂ] E) (T V : E →ₗ[ℂ] E) :
+    hsInner (u.toLinearEquiv.conj T) (u.toLinearEquiv.conj V) = hsInner T V := by
+  rw [hsInner_basis]
+  conv_rhs => rw [hsInner, LinearMap.trace_eq_sum_inner _
+    ((stdOrthonormalBasis ℂ E).map u.symm)]
+  apply Finset.sum_congr rfl
+  intro i _
+  simp only [LinearMap.comp_apply, LinearMap.adjoint_inner_right,
+    OrthonormalBasis.map_apply, LinearEquiv.conj_apply]
+  change inner ℂ (u (T (u.symm (stdOrthonormalBasis ℂ E i))))
+      (u (V (u.symm (stdOrthonormalBasis ℂ E i)))) = _
+  exact u.inner_map_map _ _
+
+end RowColumn.HilbertSchmidt
+
+noncomputable section
+open scoped RealInnerProductSpace
+namespace RowColumn.UnitaryFrame
+variable {E ι : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]
+
+omit [CompleteSpace E] in
+lemma norm_minimizer_unique [CompleteSpace E] (K : Set E) (hK : Convex ℝ K)
+    {x y : E} (hx : x ∈ K) (hy : y ∈ K)
+    (hxm : ∀ z ∈ K, ‖x‖ ≤ ‖z‖) (hym : ∀ z ∈ K, ‖y‖ ≤ ‖z‖) : x = y := by
+  have he : ‖x‖ = ‖y‖ := le_antisymm (hxm y hy) (hym x hx)
+  have hm : (1/2 : ℝ) • x + (1/2 : ℝ) • y ∈ K :=
+    hK hx hy (by norm_num) (by norm_num) (by norm_num)
+  have hl := hxm _ hm
+  have hn : ‖(1/2 : ℝ) • x + (1/2 : ℝ) • y‖ = (1/2 : ℝ)*‖x+y‖ := by
+    rw [← smul_add, norm_smul, Real.norm_eq_abs, abs_of_nonneg (by norm_num)]
+  rw [hn] at hl
+  have hsq := mul_self_le_mul_self (norm_nonneg x) hl
+  have hp := parallelogram_law_with_norm ℝ x y
+  rw [← he] at hp
+  have hd := sq_nonneg ‖x-y‖
+  have hz : ‖x-y‖ = 0 := by
+    nlinarith [norm_nonneg (x-y)]
+  exact sub_eq_zero.mp (norm_eq_zero.mp hz)
+
+/-- Simultaneous fixed point obtained from the unique minimum-norm point of a
+closed convex invariant set. Applied to rank-one unitary frames, this replaces
+Haar integration by finite-dimensional convex averaging. -/
+theorem exists_fixed_minimum (K : Set E) (hne : K.Nonempty) (hclosed : IsClosed K)
+    (hconv : Convex ℝ K) (L : ι → E →ₗᵢ[ℝ] E)
+    (hL : ∀ i x, x ∈ K → L i x ∈ K) :
+    ∃ x ∈ K, ∀ i, L i x = x := by
+  obtain ⟨x,hx,hm⟩ := exists_norm_eq_iInf_of_complete_convex hne hclosed.isComplete hconv (0 : E)
+  have hmin (z : E) (hz : z ∈ K) : ‖x‖ ≤ ‖z‖ := by
+    have hh : (⨅ w : K, ‖(0 : E)-w‖) ≤ ‖(0 : E)-z‖ :=
+      ciInf_le (f := fun w : K => ‖(0 : E) - (w : E)‖)
+        ⟨0, Set.forall_mem_range.mpr (fun _ => norm_nonneg _)⟩ (⟨z,hz⟩ : K)
+    rw [← hm] at hh
+    simpa using hh
+  refine ⟨x,hx,fun i => ?_⟩
+  exact norm_minimizer_unique K hconv (hL i x hx) hx
+    (fun z hz => by rw [(L i).norm_map]; exact hmin z hz) hmin
+
+end RowColumn.UnitaryFrame
+
+noncomputable section
+open scoped BigOperators ComplexConjugate
+namespace RowColumn.UnitaryFrame
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
+  [FiniteDimensional ℂ E]
+
+def quadraticForm (T : E →ₗ[ℂ] E) (x : E) : ℝ := (inner ℂ x (T x)).re
+
+omit [FiniteDimensional ℂ E] in
+lemma quadraticForm_real_smul [FiniteDimensional ℂ E] (T : E →ₗ[ℂ] E) (c : ℝ) (x : E) :
+    quadraticForm T (c • x) = c^2 * quadraticForm T x := by
+  simp only [quadraticForm, ← Complex.coe_smul, map_smul, inner_smul_left,
+    inner_smul_right, Complex.conj_ofReal]
+  simp [Complex.mul_re, pow_two, mul_assoc]
+
+lemma quadraticForm_continuous (T : E →ₗ[ℂ] E) : Continuous (quadraticForm T) :=
+  Complex.continuous_re.comp (continuous_id.inner T.toContinuousLinearMap.continuous)
+
+/-- Strict positivity of a finite-dimensional frame is uniformly coercive. -/
+theorem positive_coercive (T : E →ₗ[ℂ] E)
+    (hT : ∀ x : E, x ≠ 0 → 0 < quadraticForm T x) :
+    ∃ ε : ℝ, 0 < ε ∧ ∀ x, ε * ‖x‖^2 ≤ quadraticForm T x := by
+  cases subsingleton_or_nontrivial E with
+  | inl h =>
+    let := h
+    refine ⟨1, by norm_num, fun x => ?_⟩
+    have he : x = 0 := Subsingleton.elim _ _
+    simp [he, quadraticForm]
+  | inr h =>
+    let := h
+    let : ProperSpace E := FiniteDimensional.proper ℂ E
+    have hn : (Metric.sphere (0 : E) 1).Nonempty := NormedSpace.sphere_nonempty.mpr (by norm_num)
+    obtain ⟨y,hy,hmin⟩ := (isCompact_sphere (0 : E) 1).exists_isMinOn hn
+      (quadraticForm_continuous T).continuousOn
+    have hny : ‖y‖ = 1 := by simpa using hy
+    have hpy : 0 < quadraticForm T y := hT y (by intro he; simp [he] at hny)
+    refine ⟨quadraticForm T y, hpy, fun x => ?_⟩
+    by_cases hx : x = 0
+    · simp [hx, quadraticForm]
+    · have hp : 0 < ‖x‖ := norm_pos_iff.mpr hx
+      have hu : ‖x‖⁻¹ • x ∈ Metric.sphere (0 : E) 1 := by
+        simp [norm_smul, ne_of_gt hp]
+      have hm : quadraticForm T y ≤ quadraticForm T (‖x‖⁻¹ • x) := hmin hu
+      rw [quadraticForm_real_smul] at hm
+      have hh := mul_le_mul_of_nonneg_right hm (sq_nonneg ‖x‖)
+      calc
+        _ ≤ (‖x‖⁻¹ ^ 2 * quadraticForm T x) * ‖x‖^2 := hh
+        _ = _ := by field_simp [ne_of_gt hp]
+
+end RowColumn.UnitaryFrame
+
+namespace RowColumn.UnitaryFrame
+open scoped BigOperators ComplexOrder ComplexConjugate
+variable {E G : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
+  [FiniteDimensional ℂ E] [Group G]
+
+local instance : NormedAddCommGroup (E →ₗ[ℂ] E) := HilbertSchmidt.endCore.toNormedAddCommGroup
+local instance : InnerProductSpace ℂ (E →ₗ[ℂ] E) :=
+  InnerProductSpace.ofCore { __ := HilbertSchmidt.endCore (E := E) }
+local instance : InnerProductSpace ℝ (E →ₗ[ℂ] E) := InnerProductSpace.complexToReal
+
+def conjugationIsometry (u : E ≃ₗᵢ[ℂ] E) :
+    (E →ₗ[ℂ] E) →ₗᵢ[ℂ] (E →ₗ[ℂ] E) :=
+  u.toLinearEquiv.conj.toLinearMap.isometryOfInner (HilbertSchmidt.hsInner_conj u)
+
+def conjugationReal (u : E ≃ₗᵢ[ℂ] E) :
+    (E →ₗ[ℂ] E) →ₗᵢ[ℝ] (E →ₗ[ℂ] E) where
+  toLinearMap := (conjugationIsometry u).toLinearMap.restrictScalars ℝ
+  norm_map' := (conjugationIsometry u).norm_map
+
+lemma conjugation_apply (u : E ≃ₗᵢ[ℂ] E) (T : E →ₗ[ℂ] E) (x : E) :
+    conjugationReal u T x = u (T (u.symm x)) := rfl
+
+lemma conjugation_rankOne (u : E ≃ₗᵢ[ℂ] E) (x : E) :
+    conjugationReal u (InnerProductSpace.rankOne ℂ x x).toLinearMap =
+      (InnerProductSpace.rankOne ℂ (u x) (u x)).toLinearMap := by
+  ext y
+  change u (inner ℂ x (u.symm y) • x) = inner ℂ (u x) y • u x
+  rw [map_smul]
+  congr 1
+  simpa only [u.apply_symm_apply] using (u.inner_map_map x (u.symm y)).symm
+
+def matrixEntry (x y : E) : (E →ₗ[ℂ] E) →ₗ[ℂ] ℂ where
+  toFun T := inner ℂ x (T y)
+  map_add' T V := by simp only [LinearMap.add_apply, inner_add_right]
+  map_smul' c T := by simp only [LinearMap.smul_apply, inner_smul_right, smul_eq_mul, RingHom.id_apply]
+
+def quadraticEval (x : E) : (E →ₗ[ℂ] E) →ₗ[ℝ] ℝ where
+  toFun T := quadraticForm T x
+  map_add' T V := by simp [quadraticForm]
+  map_smul' c T := by
+    simp only [quadraticForm, LinearMap.smul_apply, ← Complex.coe_smul, inner_smul_right]
+    simp [Complex.mul_re]
+
+def coerciveSet (ε : ℝ) : Set (E →ₗ[ℂ] E) :=
+  {T | ∀ x, ε * ‖x‖^2 ≤ quadraticForm T x}
+
+lemma coerciveSet_closed (ε : ℝ) : IsClosed (coerciveSet (E := E) ε) := by
+  rw [coerciveSet, Set.ofPred_forall]
+  apply isClosed_iInter
+  intro x
+  exact isClosed_le continuous_const
+    (show Continuous (fun T : E →ₗ[ℂ] E => quadraticForm T x) from
+      (quadraticEval x).toContinuousLinearMap.continuous)
+
+omit [FiniteDimensional ℂ E] in
+lemma coerciveSet_convex [FiniteDimensional ℂ E] (ε : ℝ) :
+    Convex ℝ (coerciveSet (E := E) ε) := by
+  intro T hT V hV a b ha hb hab x
+  have h₁ := mul_le_mul_of_nonneg_left (hT x) ha
+  have h₂ := mul_le_mul_of_nonneg_left (hV x) hb
+  change ε * ‖x‖^2 ≤ quadraticEval x (a • T + b • V)
+  rw [map_add, map_smul, map_smul]
+  change ε * ‖x‖^2 ≤ a * quadraticForm T x + b * quadraticForm V x
+  calc
+    ε * ‖x‖^2 = (a+b) * (ε * ‖x‖^2) := by rw [hab, one_mul]
+    _ = a * (ε * ‖x‖^2) + b * (ε * ‖x‖^2) := add_mul _ _ _
+    _ ≤ _ := add_le_add h₁ h₂
+
+lemma positiveSet_closed : IsClosed {T : E →ₗ[ℂ] E | T.IsPositive} := by
+  have hs : IsClosed {T : E →ₗ[ℂ] E | T.IsSymmetric} := by
+    have he : {T : E →ₗ[ℂ] E | T.IsSymmetric} =
+        ⋂ x : E, ⋂ y : E,
+        {T | starRingEnd ℂ (matrixEntry y x T) = matrixEntry x y T} := by
+      ext T
+      simp only [Set.mem_ofPred_eq, Set.mem_iInter, matrixEntry, LinearMap.coe_mk,
+        AddHom.coe_mk, inner_conj_symm, LinearMap.IsSymmetric]
+    rw [he]
+    exact isClosed_iInter fun x => isClosed_iInter fun y => isClosed_eq
+      (Complex.continuous_conj.comp (matrixEntry y x).toContinuousLinearMap.continuous)
+      (matrixEntry x y).toContinuousLinearMap.continuous
+  have hn : IsClosed {T : E →ₗ[ℂ] E | ∀ x, 0 ≤ quadraticForm T x} := by
+    rw [Set.ofPred_forall]
+    apply isClosed_iInter
+    intro x
+    exact isClosed_le continuous_const
+      (show Continuous (fun T : E →ₗ[ℂ] E => quadraticForm T x) from
+        (quadraticEval x).toContinuousLinearMap.continuous)
+  have he : {T : E →ₗ[ℂ] E | T.IsPositive} =
+      {T | T.IsSymmetric} ∩ {T | ∀ x, 0 ≤ quadraticForm T x} := by
+    ext T
+    constructor
+    · intro h; exact ⟨h.isSymmetric, fun x => h.re_inner_nonneg_right x⟩
+    · rintro ⟨hs,hn⟩
+      refine ⟨hs, fun x => ?_⟩
+      rw [inner_re_symm]
+      exact hn x
+  rw [he]
+  exact hs.inter hn
+
+omit [FiniteDimensional ℂ E] in
+lemma positiveSet_convex [FiniteDimensional ℂ E] : Convex ℝ {T : E →ₗ[ℂ] E | T.IsPositive} := by
+  intro T hT V hV a b ha hb hab
+  change (a • T + b • V).IsPositive
+  rw [← Complex.coe_smul a T, ← Complex.coe_smul b V]
+  exact (hT.smul_of_nonneg (by exact_mod_cast ha)).add
+    (hV.smul_of_nonneg (by exact_mod_cast hb))
+
+def frameHull (π : G →* E ≃ₗᵢ[ℂ] E) (x₀ : E) : Set (E →ₗ[ℂ] E) :=
+  closure (convexHull ℝ (Set.range fun g : G =>
+    (InnerProductSpace.rankOne ℂ (π g x₀) (π g x₀)).toLinearMap))
+
+lemma frameHull_invariant (π : G →* E ≃ₗᵢ[ℂ] E) (x₀ : E) (g : G) :
+    Set.MapsTo (conjugationReal (π g)) (frameHull π x₀) (frameHull π x₀) := by
+  apply Set.MapsTo.closure _ (conjugationReal (π g)).continuous
+  rw [Set.mapsTo_iff_image_subset]
+  rw [← LinearIsometry.coe_toLinearMap, (conjugationReal (π g)).toLinearMap.image_convexHull]
+  apply convexHull_mono
+  rintro _ ⟨_, ⟨h,rfl⟩, rfl⟩
+  refine ⟨g*h, ?_⟩
+  change (InnerProductSpace.rankOne ℂ (π (g*h) x₀) (π (g*h) x₀)).toLinearMap =
+    conjugationReal (π g) (InnerProductSpace.rankOne ℂ (π h x₀) (π h x₀)).toLinearMap
+  rw [conjugation_rankOne, map_mul]
+  rfl
+
+lemma quadraticForm_conj (u : E ≃ₗᵢ[ℂ] E) (T : E →ₗ[ℂ] E) (x : E) :
+    quadraticForm (conjugationReal u T) x = quadraticForm T (u.symm x) := by
+  unfold quadraticForm
+  rw [conjugation_apply]
+  have hi := u.inner_map_map (u.symm x) (T (u.symm x))
+  rw [u.apply_symm_apply] at hi
+  exact congrArg Complex.re hi
+
+/-- A genuine invariant strictly positive frame is obtained by convex averaging;
+no Haar measure or compact-group orthogonality is taken as a hypothesis. -/
+theorem exists_invariant_frame_of_coercive
+    (π : G →* E ≃ₗᵢ[ℂ] E) (x₀ : E)
+    (T₀ : E →ₗ[ℂ] E) (hT₀ : T₀ ∈ frameHull π x₀) (hp₀ : T₀.IsPositive)
+    (ε : ℝ) (_ : 0 < ε) (hε₀ : T₀ ∈ coerciveSet ε) :
+    ∃ T : E →ₗ[ℂ] E, T ∈ frameHull π x₀ ∧ T.IsPositive ∧
+      T ∈ coerciveSet ε ∧ ∀ g, conjugationReal (π g) T = T := by
+  let K := frameHull π x₀ ∩ ({T | T.IsPositive} ∩ coerciveSet ε)
+  have hc : IsClosed K := isClosed_closure.inter (positiveSet_closed.inter (coerciveSet_closed ε))
+  have hv : Convex ℝ K := (convex_convexHull ℝ _).closure.inter
+    (positiveSet_convex.inter (coerciveSet_convex ε))
+  have hL (g : G) (V : E →ₗ[ℂ] E) (hV : V ∈ K) :
+      conjugationReal (π g) V ∈ K := by
+    refine ⟨frameHull_invariant π x₀ g hV.1, ?_, ?_⟩
+    · exact (LinearMap.isPositive_linearIsometryEquiv_conj_iff (π g)).mpr hV.2.1
+    · intro x
+      rw [quadraticForm_conj]
+      simpa only [(π g).symm.norm_map] using hV.2.2 ((π g).symm x)
+  obtain ⟨T,hT,hfix⟩ := exists_fixed_minimum K ⟨T₀,hT₀,hp₀,hε₀⟩ hc hv
+    (fun g => conjugationReal (π g)) hL
+  exact ⟨T,hT.1,hT.2.1,hT.2.2,hfix⟩
+
+end RowColumn.UnitaryFrame
+namespace RowColumn.UnitaryFrame
+open scoped BigOperators ComplexOrder ComplexConjugate
+variable {E G : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
+  [FiniteDimensional ℂ E] [Group G]
+
+omit [FiniteDimensional ℂ E] in
+lemma rankOne_quadratic [FiniteDimensional ℂ E] (v x : E) :
+    quadraticForm (InnerProductSpace.rankOne ℂ v v).toLinearMap x =
+      Complex.normSq (inner ℂ v x) := by
+  change (inner ℂ x (inner ℂ v x • v)).re = _
+  rw [inner_smul_right, ← inner_conj_symm x v, Complex.mul_conj, Complex.ofReal_re]
+
+omit [FiniteDimensional ℂ E] in
+lemma finite_frame_strict [FiniteDimensional ℂ E] (s : Finset E)
+    (hs : Submodule.span ℂ (s : Set E) = ⊤)
+    {x : E} (hx : x ≠ 0) : 0 < ∑ v ∈ s, Complex.normSq (inner ℂ v x) := by
+  have hn : 0 ≤ ∑ v ∈ s, Complex.normSq (inner ℂ v x) :=
+    Finset.sum_nonneg fun _ _ => Complex.normSq_nonneg _
+  by_contra h
+  have hz := le_antisymm (le_of_not_gt h) hn
+  have hi (v : E) (hv : v ∈ s) : inner ℂ v x = 0 :=
+    Complex.normSq_eq_zero.mp ((Finset.sum_eq_zero_iff_of_nonneg
+      (fun _ _ => Complex.normSq_nonneg _)).mp hz v hv)
+  have hall (v : E) (hv : v ∈ Submodule.span ℂ (s : Set E)) : inner ℂ v x = 0 := by
+    induction hv using Submodule.span_induction with
+    | mem v hv => exact hi v hv
+    | zero => simp
+    | add u v hu hv heu hev => simp [heu, hev]
+    | smul c v hv hv' => simp [hv']
+  exact hx ((inner_self_eq_zero (𝕜 := ℂ)).mp (hall x (by rw [hs]; trivial)))
+
+def finiteFrame (s : Finset E) : E →ₗ[ℂ] E :=
+  s.centerMass (fun _ => (1 : ℝ))
+    (fun v => (InnerProductSpace.rankOne ℂ v v).toLinearMap)
+
+lemma finiteFrame_quadratic (s : Finset E) (x : E) :
+    quadraticForm (finiteFrame s) x = (s.card : ℝ)⁻¹ *
+      ∑ v ∈ s, Complex.normSq (inner ℂ v x) := by
+  change quadraticEval x (finiteFrame s) = _
+  simp only [finiteFrame, Finset.centerMass, one_smul, Finset.sum_const,
+    nsmul_eq_mul, mul_one, map_smul, map_sum, smul_eq_mul]
+  congr 1
+  apply Finset.sum_congr rfl
+  intro v _
+  exact rankOne_quadratic v x
+
+local instance : NormedAddCommGroup (E →ₗ[ℂ] E) := HilbertSchmidt.endCore.toNormedAddCommGroup
+local instance : InnerProductSpace ℂ (E →ₗ[ℂ] E) :=
+  InnerProductSpace.ofCore { __ := HilbertSchmidt.endCore (E := E) }
+local instance : InnerProductSpace ℝ (E →ₗ[ℂ] E) := InnerProductSpace.complexToReal
+
+lemma finiteFrame_positive (s : Finset E) (hs : s.Nonempty) : (finiteFrame s).IsPositive := by
+  change s.centerMass (fun _ => (1 : ℝ))
+    (fun v => (InnerProductSpace.rankOne ℂ v v).toLinearMap) ∈
+      {T : E →ₗ[ℂ] E | T.IsPositive}
+  apply (positiveSet_convex (E := E)).centerMass_mem (fun _ _ => by norm_num)
+  · simpa only [Finset.sum_const, nsmul_eq_mul, mul_one] using
+      (show (0 : ℝ) < s.card from by exact_mod_cast hs.card_pos)
+  · intro v _
+    exact (InnerProductSpace.isPositive_rankOne_self v).toLinearMap
+
+/-- Spanning unitary orbits have an invertible positive invariant frame, with
+its approximation by finite probability mixtures of the original orbit. -/
+theorem exists_invariant_frame (π : G →* E ≃ₗᵢ[ℂ] E) (x₀ : E)
+    (hspan : Submodule.span ℂ (Set.range fun g : G => π g x₀) = ⊤) :
+    ∃ T : E →ₗ[ℂ] E, T ∈ frameHull π x₀ ∧ T.IsPositive ∧
+      Function.Bijective T ∧ ∀ g, conjugationReal (π g) T = T := by
+  classical
+  obtain ⟨s,hs,hcard,hss,hindep⟩ := Submodule.exists_finset_span_eq_linearIndepOn ℂ
+    (Set.range fun g : G => π g x₀)
+  let t := insert x₀ s
+  have htn : t.Nonempty := Finset.insert_nonempty _ _
+  have hts : (t : Set E) ⊆ Set.range fun g : G => π g x₀ := by
+    intro x hx
+    rcases Finset.mem_insert.mp hx with rfl | hx
+    · exact ⟨1, by simp⟩
+    · exact hs hx
+  have htspan : Submodule.span ℂ (t : Set E) = ⊤ := by
+    apply top_unique
+    rw [← hspan, ← hss]
+    exact Submodule.span_mono (by intro x hx; exact Finset.mem_insert_of_mem hx)
+  have hmem : finiteFrame t ∈ frameHull π x₀ := by
+    apply subset_closure
+    apply t.centerMass_mem_convexHull (fun _ _ => by norm_num)
+    · simpa only [Finset.sum_const, nsmul_eq_mul, mul_one] using
+        (show (0 : ℝ) < t.card from by exact_mod_cast htn.card_pos)
+    · intro v hv
+      obtain ⟨g,rfl⟩ := hts hv
+      exact ⟨g,rfl⟩
+  have hstrict (x : E) (hx : x ≠ 0) : 0 < quadraticForm (finiteFrame t) x := by
+    rw [finiteFrame_quadratic]
+    exact mul_pos (inv_pos.mpr (by exact_mod_cast htn.card_pos)) (finite_frame_strict t htspan hx)
+  obtain ⟨ε,hε,hε₀⟩ := positive_coercive (finiteFrame t) hstrict
+  obtain ⟨T,hT,hp,hc,hfix⟩ := exists_invariant_frame_of_coercive π x₀ (finiteFrame t)
+    hmem (finiteFrame_positive t htn) ε hε hε₀
+  have hinj : Function.Injective T := by
+    apply LinearMap.ker_eq_bot.mp
+    apply LinearMap.ker_eq_bot'.mpr
+    intro x hx
+    have hh := hc x
+    change ε * ‖x‖^2 ≤ (inner ℂ x (T x)).re at hh
+    rw [hx, inner_zero_right, Complex.zero_re] at hh
+    have hn : ‖x‖^2 = 0 := le_antisymm (nonpos_of_mul_nonpos_right hh hε) (sq_nonneg _)
+    exact norm_eq_zero.mp (sq_eq_zero_iff.mp hn)
+  exact ⟨T,hT,hp,⟨hinj, (LinearMap.injective_iff_surjective).mp hinj⟩,hfix⟩
+
+end RowColumn.UnitaryFrame
+
+end
+end
+end
+end
+
+
+section
+noncomputable section
+open scoped BigOperators ComplexConjugate ComplexOrder
+namespace RowColumn.UnitaryFrame
+
+variable {A G : Type*} [Ring A] [Algebra ℂ A] [StarRing A]
+  [FiniteDimensional ℂ A] [Group G]
+
+/-- Algebraic and convex reconstruction from a spanning unitary group. The
+trace is the physical trace, not the trace of the algebra's regular action. -/
+theorem exists_density (τ : A →ₗ[ℂ] ℂ) (core : InnerProductSpace.Core ℂ A)
+    (hinner : ∀ x y, core.inner x y = τ (star x * y))
+    (u : G →* unitary A)
+    (hspan : Submodule.span ℂ (Set.range fun g : G => (u g : A)) = ⊤) :
+    ∃ D : A, τ D = (Module.finrank ℂ A : ℂ) ∧
+      (∀ x : A, 0 ≤ τ (star x * x * D)) ∧
+      (∀ (B : A) (ℓ : A →ₗ[ℂ] ℂ) (R : ℝ),
+        (∀ g, ‖τ (star (u g : A) * (B*D)) * ℓ (u g : A)‖ ≤ R) → ‖ℓ B‖ ≤ R) := by
+  let : NormedAddCommGroup A := core.toNormedAddCommGroup
+  let : InnerProductSpace ℂ A := InnerProductSpace.ofCore { __ := core }
+  change ∀ x y, inner ℂ x y = τ (star x*y) at hinner
+  let π : G →* A ≃ₗᵢ[ℂ] A :=
+    { toFun := fun g =>
+        (Units.mulLeftLinearEquiv ℂ A (Unitary.toUnits (u g))).isometryOfInner (by
+          intro x y
+          change inner ℂ ((u g : A)*x) ((u g : A)*y) = inner ℂ x y
+          simp only [hinner, star_mul, mul_assoc]
+          rw [← mul_assoc (star (u g : A)), Unitary.coe_star_mul_self, one_mul])
+      map_one' := by ext x; simp
+      map_mul' := by intros g h; ext x; simp }
+  have hπ (g : G) (x : A) : π g x = (u g : A)*x := rfl
+  obtain ⟨T,hT,hpos,hbij,hfix⟩ := exists_invariant_frame π 1 (by simpa only [hπ, mul_one] using hspan)
+  have heq (g : G) (x : A) : T ((u g : A)*x) = (u g : A)*T x := by
+    have he := LinearMap.congr_fun (hfix g) (π g x)
+    change π g (T ((π g).symm (π g x))) = T (π g x) at he
+    rw [(π g).symm_apply_apply] at he
+    exact he.symm
+  let C := T 1
+  have hTm (x : A) : T x = x*C := by
+    have hx : x ∈ Submodule.span ℂ (Set.range fun g : G => (u g : A)) := by rw [hspan]; trivial
+    induction hx using Submodule.span_induction with
+    | mem x hx => obtain ⟨g,rfl⟩ := hx; simpa only [mul_one] using heq g 1
+    | zero => simp
+    | add x y hx hy hxt hyt => simp only [map_add, add_mul, hxt, hyt]
+    | smul c x hx hxt => simpa only [map_smul, smul_mul_assoc] using congrArg (c • ·) hxt
+  let e : A ≃ₗ[ℂ] A := LinearEquiv.ofBijective T hbij
+  let D : A := e.symm 1
+  have hTD : T D = 1 := e.apply_symm_apply 1
+  have hDC : D*C = 1 := (hTm D).symm.trans hTD
+  have hinv (x : A) : T (x*D) = x := by rw [hTm, mul_assoc, hDC, mul_one]
+  have heinv (x : A) : e.symm x = x*D := by
+    apply hbij.injective
+    exact (e.apply_symm_apply x).trans (hinv x).symm
+  have hip : e.symm.toLinearMap.IsPositive := hpos.toLinearMap_symm
+  have hDpos (x : A) : 0 ≤ τ (star x*x*D) := by
+    have hh := hip.inner_nonneg_right x
+    simpa only [LinearEquiv.coe_coe, heinv, hinner, mul_assoc] using hh
+  let f : (A →ₗ[ℂ] A) →ₗ[ℂ] ℂ :=
+    { toFun S := LinearMap.trace ℂ A (S ∘ₗ LinearMap.mulRight ℂ D)
+      map_add' S V := by simp only [LinearMap.add_comp, map_add]
+      map_smul' c S := by simp only [LinearMap.smul_comp, map_smul, RingHom.id_apply] }
+  have hf (g : G) : f (InnerProductSpace.rankOne ℂ (π g 1) (π g 1)).toLinearMap = τ D := by
+    change LinearMap.trace ℂ A (_ ∘ₗ _) = _
+    rw [LinearMap.trace_comp_comm']
+    have he : LinearMap.mulRight ℂ D ∘ₗ
+        (InnerProductSpace.rankOne ℂ (π g 1) (π g 1)).toLinearMap =
+        (InnerProductSpace.rankOne ℂ ((π g 1)*D) (π g 1)).toLinearMap := by
+      ext x
+      simp only [LinearMap.comp_apply, LinearMap.mulRight_apply,
+        ContinuousLinearMap.coe_coe, InnerProductSpace.rankOne_apply, smul_mul_assoc]
+    rw [he]
+    change (InnerProductSpace.rankOne ℂ ((π g 1)*D) (π g 1)).trace ℂ A = _
+    rw [InnerProductSpace.trace_rankOne, hinner, hπ, mul_one,
+      ← mul_assoc, Unitary.coe_star_mul_self, one_mul]
+  let : NormedAddCommGroup (A →ₗ[ℂ] A) := HilbertSchmidt.endCore.toNormedAddCommGroup
+  let : InnerProductSpace ℂ (A →ₗ[ℂ] A) :=
+    InnerProductSpace.ofCore { __ := HilbertSchmidt.endCore (E := A) }
+  let : InnerProductSpace ℝ (A →ₗ[ℂ] A) := InnerProductSpace.complexToReal
+  have hfc : ∀ S ∈ frameHull π 1, f S = τ D := by
+    change closure (convexHull ℝ _) ⊆ {S | f S = τ D}
+    apply closure_minimal
+    · apply convexHull_min
+      · rintro _ ⟨g,rfl⟩; exact hf g
+      · intro S hS V hV a b ha hb hab
+        change f (a • S + b • V) = τ D
+        rw [map_add, ← Complex.coe_smul a S, ← Complex.coe_smul b V, map_smul, map_smul, hS, hV]
+        rw [← add_smul, ← Complex.ofReal_add, hab, Complex.ofReal_one, one_smul]
+    · exact isClosed_eq f.toContinuousLinearMap.continuous continuous_const
+  have hτD : τ D = (Module.finrank ℂ A : ℂ) := by
+    rw [← hfc T hT]
+    have hcomp : T ∘ₗ LinearMap.mulRight ℂ D = LinearMap.id := by ext x; exact hinv x
+    change LinearMap.trace ℂ A (_ ∘ₗ _) = _
+    rw [hcomp, LinearMap.trace_id]
+  refine ⟨D,hτD,hDpos,?_⟩
+  intro B ℓ R hR
+  let l : (A →ₗ[ℂ] A) →ₗ[ℂ] ℂ :=
+    { toFun S := ℓ (S (B*D))
+      map_add' S V := by simp only [LinearMap.add_apply, map_add]
+      map_smul' c S := by simp only [LinearMap.smul_apply, map_smul, RingHom.id_apply] }
+  have hl : ∀ S ∈ frameHull π 1, ‖l S‖ ≤ R := by
+    change closure (convexHull ℝ _) ⊆ {S | ‖l S‖ ≤ R}
+    apply closure_minimal
+    · apply convexHull_min
+      · rintro _ ⟨g,rfl⟩
+        change ‖ℓ ((InnerProductSpace.rankOne ℂ (π g 1) (π g 1)) (B*D))‖ ≤ R
+        simpa only [l, LinearMap.coe_mk, AddHom.coe_mk, ContinuousLinearMap.coe_coe,
+          InnerProductSpace.rankOne_apply, map_smul, hinner, hπ, mul_one, smul_eq_mul] using hR g
+      · intro S hS V hV a b ha hb hab
+        change ‖l (a • S + b • V)‖ ≤ R
+        rw [map_add, ← Complex.coe_smul a S, ← Complex.coe_smul b V, map_smul, map_smul]
+        calc
+          _ ≤ ‖(a : ℂ) • l S‖ + ‖(b : ℂ) • l V‖ := norm_add_le _ _
+          _ = a * ‖l S‖ + b * ‖l V‖ := by simp only [norm_smul, Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg ha, abs_of_nonneg hb]
+          _ ≤ a*R+b*R := add_le_add (mul_le_mul_of_nonneg_left hS ha) (mul_le_mul_of_nonneg_left hV hb)
+          _ = R := by rw [← add_mul, hab, one_mul]
+    · exact isClosed_le l.toContinuousLinearMap.continuous.norm continuous_const
+  simpa only [l, LinearMap.coe_mk, AddHom.coe_mk, hinv] using hl T hT
+
+end RowColumn.UnitaryFrame
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FiniteUnitary/KernelBounds.lean b/lean/OAI/RepresentationTheory/FiniteUnitary/KernelBounds.lean
index 6eedd88aab3b70d295a8e8594c70ff2d49bcb8d8..313fd809a15475ec72735757d577d62f4eff0998 100644
--- a/lean/OAI/RepresentationTheory/FiniteUnitary/KernelBounds.lean
+++ b/lean/OAI/RepresentationTheory/FiniteUnitary/KernelBounds.lean
@@ -1,145 +1,145 @@
-import OAI.Probability.CubeShuffle.FiniteKernel
-import OAI.RepresentationTheory.FiniteUnitary.Sampling
-
-namespace OAI
-
-namespace CubeShuffle.UnitaryFinite
-open scoped BigOperators ComplexConjugate Classical
-
-variable {G X Ω : Type*} [Group G] [Fintype G] [Fintype X] [Fintype Ω]
-
-noncomputable def actionKernel (a : G →* Equiv.Perm X) (P : Ω → G) (x y : X) : ℝ :=
-  finiteMean (fun ω => if a (P ω) x=y then 1 else 0)
-
-omit [Fintype G] in
-lemma actionKernel_apply (a : G →* Equiv.Perm X) (P : Ω → G)
-    (v : EuclideanSpace ℂ X) (x : X) :
-    DensityTransfer.applyKernel (actionKernel a P) v x =
-      (Fintype.card Ω:ℂ)⁻¹ * ∑ ω, v (a (P ω) x) := by
-  simp only [DensityTransfer.applyKernel_apply,actionKernel,finiteMean,
-    Complex.ofReal_sum,div_eq_mul_inv,Finset.sum_mul]
-  rw [Finset.sum_comm]
-  calc
-    _ = ∑ ω, (Fintype.card Ω:ℂ)⁻¹ * ∑ y : X, if a (P ω) x=y then v y else 0 := by
-      apply Finset.sum_congr rfl
-      intro ω _
-      rw [Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro y _
-      split_ifs <;> simp_all
-    _ = _ := by simp [Finset.mul_sum]
-
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-noncomputable def randomOperator (ρ : Representation ℂ G V) (P : Ω → G) : Module.End ℂ V :=
-  (Fintype.card Ω:ℂ)⁻¹ • ∑ ω, ρ (P ω)⁻¹
-
-omit [Fintype G] [FiniteDimensional ℂ V] in
-lemma coefficient_random (a : G →* Equiv.Perm X) (x₀ : X)
-    (ha : ∀ x, ∃ g, a g x₀=x) (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
-    (P : Ω → G) (w : V) (hw : IsFixed a x₀ ρ w) (v : V) :
-    DensityTransfer.applyKernel (actionKernel a P) (normalizedCoefficient a x₀ ha ρ w v) =
-      normalizedCoefficient a x₀ ha ρ w (randomOperator ρ P v) := by
-  ext x
-  rw [actionKernel_apply]
-  simp only [normalizedCoefficient,PiLp.smul_apply,smul_eq_mul,Finset.mul_sum,
-    coefficient,randomOperator,LinearMap.smul_apply,LinearMap.sum_apply,
-    inner_smul_right,inner_sum]
-  apply Finset.sum_congr rfl
-  intro ω _
-  have h := coefficient_action a x₀ ha ρ hρ w hw v (P ω) x
-  change inner ℂ (ρ (representative a x₀ ha (a (P ω) x)) w) v =
-    inner ℂ (ρ (representative a x₀ ha x) w) (ρ (P ω)⁻¹ v) at h
-  rw [h]
-  ring
-
-noncomputable def fixedSpace (a : G →* Equiv.Perm X) (x₀ : X) (ρ : Representation ℂ G V) :
-    Submodule ℂ V where
-  carrier := {w | IsFixed a x₀ ρ w}
-  zero_mem' := fun g hg => map_zero _
-  add_mem' := fun hw hz g hg => by rw [map_add,hw g hg,hz g hg]
-  smul_mem' := fun c w hw g hg => by rw [map_smul,hw g hg]
-
-/-- Genuine multiplicity amplification: the family is produced from the
-stabilizer-fixed space by unitary Schur orthogonality, not assumed. -/
-theorem operator_amplification (a : G →* Equiv.Perm X) (x₀ : X)
-    (ha : ∀ x, ∃ g, a g x₀=x) (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
-    (hρ : IsUnitary ρ) (P : Ω → G)
-    (hsym : ∀ x y, actionKernel a P x y=actionKernel a P y x)
-    (hrow : ∀ x, ∑ y, actionKernel a P x y=1)
-    (b z M : ℝ) (hb : 0≤b) (hz : 0<z) (hM0 : 0≤M)
-    (hM : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*actionKernel a P x y)^(1+b)) ≤ M)
-    (hm : 0 < Module.finrank ℂ (fixedSpace a x₀ ρ)) (v : V) (hv : ‖v‖=1) :
-    ‖randomOperator ρ P v‖ ≤ M*z^(-b)+Real.sqrt (z/(Module.finrank ℂ (fixedSpace a x₀ ρ):ℝ)) := by
-  let K := fixedSpace a x₀ ρ
-  let m := Module.finrank ℂ K
-  let : Nonempty X := ⟨x₀⟩
-  let : Nonempty (Fin m) := ⟨⟨0,hm⟩⟩
-  let B := stdOrthonormalBasis ℂ K
-  let w : Fin m → V := fun i => (B i : V)
-  have hfix (i : Fin m) : IsFixed a x₀ ρ (w i) := (B i).2
-  have hw : Orthonormal ℂ w := B.orthonormal.comp_linearIsometry K.subtypeₗᵢ
-  have hc := normalizedCoefficient_orthonormal a x₀ ha ρ hρ w hw hfix v hv
-  have hn (i : Fin m) : ‖DensityTransfer.applyKernel (actionKernel a P)
-      (normalizedCoefficient a x₀ ha ρ (w i) v)‖ = ‖randomOperator ρ P v‖ := by
-    rw [coefficient_random a x₀ ha ρ hρ P (w i) (hfix i),
-      normalizedCoefficient_norm a x₀ ha ρ hρ (w i) (hfix i) (hw.norm_eq_one i)]
-  have h := DensityTransfer.multiplicity_amplification (actionKernel a P)
-    (fun x y => sampleLaw_nonneg (fun ω => a (P ω) x) y) hsym hrow
-    b z M hb hz hM0 hM _ hc (‖randomOperator ρ P v‖) (fun i => (hn i).ge)
-  simpa only [Fintype.card_fin] using h
-
-end CubeShuffle.UnitaryFinite
-
-namespace CubeShuffle.UnitaryFinite
-open scoped BigOperators Classical
-
-variable {G H X Ω : Type*} [Group G] [Group H] [Fintype G] [Fintype X] [Fintype Ω]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-omit [Fintype G] in
-lemma sampleOperator_inv_apply (ρ : Representation ℂ G V) (P : Ω → G) (v : V) :
-    sampleOperator ρ (fun ω => (P ω)⁻¹) v=randomOperator ρ P v := by
-  simp only [sampleOperator,randomOperator,smul_apply,
-    sum_apply,continuousRepresentation_apply,LinearMap.smul_apply,LinearMap.sum_apply]
-
-/-- FAC.7 as a genuine operator-norm bound for sample averages in an irrep. -/
-theorem sampleOperator_amplification (a : G →* Equiv.Perm X) (x₀ : X)
-    (ha : ∀ x, ∃ g, a g x₀=x) (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
-    (hρ : IsUnitary ρ) (P : Ω → G)
-    (hsym : ∀ x y, actionKernel a P x y=actionKernel a P y x)
-    (hrow : ∀ x, ∑ y, actionKernel a P x y=1)
-    (b z M : ℝ) (hb : 0≤b) (hz : 0<z) (hM0 : 0≤M)
-    (hM : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*actionKernel a P x y)^(1+b)) ≤ M)
-    (hm : 0 < Module.finrank ℂ (fixedSpace a x₀ ρ)) :
-    ‖sampleOperator ρ P‖ ≤ M*z^(-b)+Real.sqrt (z/(Module.finrank ℂ (fixedSpace a x₀ ρ):ℝ)) := by
-  rw [←ContinuousLinearMap.adjoint.norm_map (sampleOperator ρ P),←sampleOperator_inv ρ hρ]
-  apply ContinuousLinearMap.opNorm_le_of_unit_norm (by positivity)
-  intro v hv
-  rw [sampleOperator_inv_apply]
-  exact operator_amplification a x₀ ha ρ hρ P hsym hrow b z M hb hz hM0 hM hm v hv
-
-omit [Fintype G] [Fintype X] [Fintype Ω] [FiniteDimensional ℂ V] in
-noncomputable def subrepresentationEquiv (ρ : Representation ℂ G V) (e : H ≃* G) :
-    Subrepresentation (ρ.comp e.toMonoidHom) ≃o Subrepresentation ρ where
-  toFun U :=
-    { toSubmodule := U.toSubmodule
-      apply_mem_toSubmodule := fun g v hv => by
-        have h := U.apply_mem_toSubmodule (e.symm g) hv
-        change ρ (e (e.symm g)) v∈U.toSubmodule at h
-        simpa only [e.apply_symm_apply] using h }
-  invFun U :=
-    { toSubmodule := U.toSubmodule
-      apply_mem_toSubmodule := fun h _ hv => U.apply_mem_toSubmodule (e h) hv }
-  left_inv U := by apply Subrepresentation.toSubmodule_injective; rfl
-  right_inv U := by apply Subrepresentation.toSubmodule_injective; rfl
-  map_rel_iff' := Iff.rfl
-
-omit [Fintype G] [Fintype X] [Fintype Ω] [FiniteDimensional ℂ V] in
-lemma irreducible_comp_equiv (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
-    (e : H ≃* G) : Representation.IsIrreducible (ρ.comp e.toMonoidHom) := by
-  exact (subrepresentationEquiv ρ e).isSimpleOrder_iff.mpr inferInstance
-
-end CubeShuffle.UnitaryFinite
-
-end OAI
+import OAI.Probability.CubeShuffle.FiniteKernel
+import OAI.RepresentationTheory.FiniteUnitary.Sampling
+
+namespace OAI
+
+namespace CubeShuffle.UnitaryFinite
+open scoped BigOperators ComplexConjugate Classical
+
+variable {G X Ω : Type*} [Group G] [Fintype G] [Fintype X] [Fintype Ω]
+
+noncomputable def actionKernel (a : G →* Equiv.Perm X) (P : Ω → G) (x y : X) : ℝ :=
+  finiteMean (fun ω => if a (P ω) x=y then 1 else 0)
+
+omit [Fintype G] in
+lemma actionKernel_apply (a : G →* Equiv.Perm X) (P : Ω → G)
+    (v : EuclideanSpace ℂ X) (x : X) :
+    DensityTransfer.applyKernel (actionKernel a P) v x =
+      (Fintype.card Ω:ℂ)⁻¹ * ∑ ω, v (a (P ω) x) := by
+  simp only [DensityTransfer.applyKernel_apply,actionKernel,finiteMean,
+    Complex.ofReal_sum,div_eq_mul_inv,Finset.sum_mul]
+  rw [Finset.sum_comm]
+  calc
+    _ = ∑ ω, (Fintype.card Ω:ℂ)⁻¹ * ∑ y : X, if a (P ω) x=y then v y else 0 := by
+      apply Finset.sum_congr rfl
+      intro ω _
+      rw [Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro y _
+      split_ifs <;> simp_all
+    _ = _ := by simp [Finset.mul_sum]
+
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+noncomputable def randomOperator (ρ : Representation ℂ G V) (P : Ω → G) : Module.End ℂ V :=
+  (Fintype.card Ω:ℂ)⁻¹ • ∑ ω, ρ (P ω)⁻¹
+
+omit [Fintype G] [FiniteDimensional ℂ V] in
+lemma coefficient_random (a : G →* Equiv.Perm X) (x₀ : X)
+    (ha : ∀ x, ∃ g, a g x₀=x) (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
+    (P : Ω → G) (w : V) (hw : IsFixed a x₀ ρ w) (v : V) :
+    DensityTransfer.applyKernel (actionKernel a P) (normalizedCoefficient a x₀ ha ρ w v) =
+      normalizedCoefficient a x₀ ha ρ w (randomOperator ρ P v) := by
+  ext x
+  rw [actionKernel_apply]
+  simp only [normalizedCoefficient,PiLp.smul_apply,smul_eq_mul,Finset.mul_sum,
+    coefficient,randomOperator,LinearMap.smul_apply,LinearMap.sum_apply,
+    inner_smul_right,inner_sum]
+  apply Finset.sum_congr rfl
+  intro ω _
+  have h := coefficient_action a x₀ ha ρ hρ w hw v (P ω) x
+  change inner ℂ (ρ (representative a x₀ ha (a (P ω) x)) w) v =
+    inner ℂ (ρ (representative a x₀ ha x) w) (ρ (P ω)⁻¹ v) at h
+  rw [h]
+  ring
+
+noncomputable def fixedSpace (a : G →* Equiv.Perm X) (x₀ : X) (ρ : Representation ℂ G V) :
+    Submodule ℂ V where
+  carrier := {w | IsFixed a x₀ ρ w}
+  zero_mem' := fun g hg => map_zero _
+  add_mem' := fun hw hz g hg => by rw [map_add,hw g hg,hz g hg]
+  smul_mem' := fun c w hw g hg => by rw [map_smul,hw g hg]
+
+/-- Genuine multiplicity amplification: the family is produced from the
+stabilizer-fixed space by unitary Schur orthogonality, not assumed. -/
+theorem operator_amplification (a : G →* Equiv.Perm X) (x₀ : X)
+    (ha : ∀ x, ∃ g, a g x₀=x) (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
+    (hρ : IsUnitary ρ) (P : Ω → G)
+    (hsym : ∀ x y, actionKernel a P x y=actionKernel a P y x)
+    (hrow : ∀ x, ∑ y, actionKernel a P x y=1)
+    (b z M : ℝ) (hb : 0≤b) (hz : 0<z) (hM0 : 0≤M)
+    (hM : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*actionKernel a P x y)^(1+b)) ≤ M)
+    (hm : 0 < Module.finrank ℂ (fixedSpace a x₀ ρ)) (v : V) (hv : ‖v‖=1) :
+    ‖randomOperator ρ P v‖ ≤ M*z^(-b)+Real.sqrt (z/(Module.finrank ℂ (fixedSpace a x₀ ρ):ℝ)) := by
+  let K := fixedSpace a x₀ ρ
+  let m := Module.finrank ℂ K
+  let : Nonempty X := ⟨x₀⟩
+  let : Nonempty (Fin m) := ⟨⟨0,hm⟩⟩
+  let B := stdOrthonormalBasis ℂ K
+  let w : Fin m → V := fun i => (B i : V)
+  have hfix (i : Fin m) : IsFixed a x₀ ρ (w i) := (B i).2
+  have hw : Orthonormal ℂ w := B.orthonormal.comp_linearIsometry K.subtypeₗᵢ
+  have hc := normalizedCoefficient_orthonormal a x₀ ha ρ hρ w hw hfix v hv
+  have hn (i : Fin m) : ‖DensityTransfer.applyKernel (actionKernel a P)
+      (normalizedCoefficient a x₀ ha ρ (w i) v)‖ = ‖randomOperator ρ P v‖ := by
+    rw [coefficient_random a x₀ ha ρ hρ P (w i) (hfix i),
+      normalizedCoefficient_norm a x₀ ha ρ hρ (w i) (hfix i) (hw.norm_eq_one i)]
+  have h := DensityTransfer.multiplicity_amplification (actionKernel a P)
+    (fun x y => sampleLaw_nonneg (fun ω => a (P ω) x) y) hsym hrow
+    b z M hb hz hM0 hM _ hc (‖randomOperator ρ P v‖) (fun i => (hn i).ge)
+  simpa only [Fintype.card_fin] using h
+
+end CubeShuffle.UnitaryFinite
+
+namespace CubeShuffle.UnitaryFinite
+open scoped BigOperators Classical
+
+variable {G H X Ω : Type*} [Group G] [Group H] [Fintype G] [Fintype X] [Fintype Ω]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+omit [Fintype G] in
+lemma sampleOperator_inv_apply (ρ : Representation ℂ G V) (P : Ω → G) (v : V) :
+    sampleOperator ρ (fun ω => (P ω)⁻¹) v=randomOperator ρ P v := by
+  simp only [sampleOperator,randomOperator,smul_apply,
+    sum_apply,continuousRepresentation_apply,LinearMap.smul_apply,LinearMap.sum_apply]
+
+/-- FAC.7 as a genuine operator-norm bound for sample averages in an irrep. -/
+theorem sampleOperator_amplification (a : G →* Equiv.Perm X) (x₀ : X)
+    (ha : ∀ x, ∃ g, a g x₀=x) (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
+    (hρ : IsUnitary ρ) (P : Ω → G)
+    (hsym : ∀ x y, actionKernel a P x y=actionKernel a P y x)
+    (hrow : ∀ x, ∑ y, actionKernel a P x y=1)
+    (b z M : ℝ) (hb : 0≤b) (hz : 0<z) (hM0 : 0≤M)
+    (hM : ∀ x, finiteMean (fun y => ((Fintype.card X:ℝ)*actionKernel a P x y)^(1+b)) ≤ M)
+    (hm : 0 < Module.finrank ℂ (fixedSpace a x₀ ρ)) :
+    ‖sampleOperator ρ P‖ ≤ M*z^(-b)+Real.sqrt (z/(Module.finrank ℂ (fixedSpace a x₀ ρ):ℝ)) := by
+  rw [←ContinuousLinearMap.adjoint.norm_map (sampleOperator ρ P),←sampleOperator_inv ρ hρ]
+  apply ContinuousLinearMap.opNorm_le_of_unit_norm (by positivity)
+  intro v hv
+  rw [sampleOperator_inv_apply]
+  exact operator_amplification a x₀ ha ρ hρ P hsym hrow b z M hb hz hM0 hM hm v hv
+
+omit [Fintype G] [Fintype X] [Fintype Ω] [FiniteDimensional ℂ V] in
+noncomputable def subrepresentationEquiv (ρ : Representation ℂ G V) (e : H ≃* G) :
+    Subrepresentation (ρ.comp e.toMonoidHom) ≃o Subrepresentation ρ where
+  toFun U :=
+    { toSubmodule := U.toSubmodule
+      apply_mem_toSubmodule := fun g v hv => by
+        have h := U.apply_mem_toSubmodule (e.symm g) hv
+        change ρ (e (e.symm g)) v∈U.toSubmodule at h
+        simpa only [e.apply_symm_apply] using h }
+  invFun U :=
+    { toSubmodule := U.toSubmodule
+      apply_mem_toSubmodule := fun h _ hv => U.apply_mem_toSubmodule (e h) hv }
+  left_inv U := by apply Subrepresentation.toSubmodule_injective; rfl
+  right_inv U := by apply Subrepresentation.toSubmodule_injective; rfl
+  map_rel_iff' := Iff.rfl
+
+omit [Fintype G] [Fintype X] [Fintype Ω] [FiniteDimensional ℂ V] in
+lemma irreducible_comp_equiv (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
+    (e : H ≃* G) : Representation.IsIrreducible (ρ.comp e.toMonoidHom) := by
+  exact (subrepresentationEquiv ρ e).isSimpleOrder_iff.mpr inferInstance
+
+end CubeShuffle.UnitaryFinite
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FiniteUnitary/Reconstruction.lean b/lean/OAI/RepresentationTheory/FiniteUnitary/Reconstruction.lean
index abd98b7251270e6238a74a4f05cbb455f967ab87..d7c21a4bbf897836b3487fa26c00654dae7d9654 100644
--- a/lean/OAI/RepresentationTheory/FiniteUnitary/Reconstruction.lean
+++ b/lean/OAI/RepresentationTheory/FiniteUnitary/Reconstruction.lean
@@ -1,699 +1,699 @@
-import OAI.RepresentationTheory.FiniteUnitary.Frames
-import OAI.RepresentationTheory.RowColumn.Basic
-
-namespace OAI
-
-section
-noncomputable section
-open scoped BigOperators ComplexOrder
-namespace RowColumn.UnitaryFrame
-variable {A G : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A]
-  [FiniteDimensional ℂ A] [Group G]
-
-omit [FiniteDimensional ℂ A] in
-lemma positive_functional_bound (f : A →ₚ[ℂ] ℂ) (x : A) :
-    ‖f x‖ ≤ 4 * ‖f 1‖ * ‖x‖ := by
-  obtain ⟨y, hy_nonneg, hy_norm, hy⟩ := CStarAlgebra.exists_sum_four_nonneg x
-  conv_lhs => rw [hy]
-  simp only [map_sum, map_smul]
-  apply (norm_sum_le _ _).trans
-  simp only [norm_smul, norm_pow, Complex.norm_I, one_pow, one_mul]
-  apply (Finset.sum_le_sum (g := fun _ => ‖f 1‖ * ‖x‖) (fun i _ => ?_)).trans
-  · simp [mul_assoc]
-  · exact (f.norm_apply_le_of_nonneg _ (hy_nonneg i)).trans
-      (mul_le_mul_of_nonneg_left (hy_norm i) (norm_nonneg _))
-
-/-- Finite-dimensional unitary reconstruction with polynomial Fourier cost.
-Only the actual operator norm of B occurs; the auxiliary Hilbert norm is used
-solely to construct the frame and does not alter that norm. -/
-theorem reconstruction_bound (τ : A →ₗ[ℂ] ℂ) (core : InnerProductSpace.Core ℂ A)
-    (hinner : ∀ x y, core.inner x y = τ (star x*y))
-    (u : G →* unitary A)
-    (hspan : Submodule.span ℂ (Set.range fun g : G => (u g : A)) = ⊤)
-    (B : A) (ℓ : A →ₗ[ℂ] ℂ) (R : ℝ) (_ : 0 ≤ R)
-    (hU : ∀ g, ‖ℓ (u g : A)‖ ≤ R) :
-    ‖ℓ B‖ ≤ 4 * (Module.finrank ℂ A : ℝ) * ‖B‖ * R := by
-  obtain ⟨D,hD,hpos,hrec⟩ := exists_density τ core hinner u hspan
-  let φ : A →ₗ[ℂ] ℂ := τ.comp (LinearMap.mulRight ℂ D)
-  have hφpos (a : A) (ha : 0 ≤ a) : 0 ≤ φ a := by
-    have hh := hpos (CFC.sqrt a)
-    have hs : star (CFC.sqrt a) * CFC.sqrt a = a := by
-      rw [(IsSelfAdjoint.of_nonneg (CFC.sqrt_nonneg a)).star_eq]
-      exact CFC.sqrt_mul_sqrt_self a ha
-    change 0 ≤ τ (a*D)
-    simpa only [hs] using hh
-  let f : A →ₚ[ℂ] ℂ := PositiveLinearMap.mk₀ φ hφpos
-  have hf1 : ‖f 1‖ = (Module.finrank ℂ A : ℝ) := by
-    change ‖τ (1*D)‖ = _
-    rw [one_mul, hD]
-    simp
-  apply hrec B ℓ
-  intro g
-  rw [← mul_assoc]
-  change ‖φ (star (u g : A)*B) * ℓ (u g : A)‖ ≤ _
-  rw [norm_mul]
-  have hb : ‖φ (star (u g : A)*B)‖ ≤ 4 * (Module.finrank ℂ A : ℝ) * ‖B‖ := by
-    have hh := positive_functional_bound f (star (u g : A)*B)
-    rw [hf1] at hh
-    change ‖f (star (u g : A)*B)‖ ≤ _
-    simpa only [← Unitary.coe_star, CStarRing.norm_coe_unitary_mul] using hh
-  exact mul_le_mul hb (hU g) (norm_nonneg _) (by positivity)
-
-end RowColumn.UnitaryFrame
-noncomputable section
-open scoped ComplexConjugate
-namespace RowColumn.FiniteStarAlgebra
-variable {A : Type*} [Ring A] [Module ℂ A] [StarRing A] [FiniteDimensional ℂ A]
-
-/-- A finite-dimensional two-sided star ideal has a central orthogonal unit.
-The two inner-product identities hold for the actual Hilbert–Schmidt trace.
-This constructs the central support used in the occupied-board filter. -/
-theorem complementary_central_projection (core : InnerProductSpace.Core ℂ A) (K : Submodule ℂ A)
-    (hL : ∀ a x, x ∈ K → a*x ∈ K) (hR : ∀ a x, x ∈ K → x*a ∈ K)
-    (hstar : ∀ x, x ∈ K → star x ∈ K)
-    (hadjL : ∀ a x y : A, core.inner (a*x) y = core.inner x (star a*y))
-    (hadjR : ∀ a x y : A, core.inner (x*a) y = core.inner x (y*star a)) :
-    ∃ q : A, star q = q ∧ q*q = q ∧ (∀ a, q*a = a*q) ∧
-      1-q ∈ K ∧ (∀ x, x ∈ K → q*x = 0 ∧ x*q = 0) := by
-  let : NormedAddCommGroup A := core.toNormedAddCommGroup
-  let : InnerProductSpace ℂ A := InnerProductSpace.ofCore { __ := core }
-  change ∀ a x y : A, inner ℂ (a*x) y = inner ℂ x (star a*y) at hadjL
-  change ∀ a x y : A, inner ℂ (x*a) y = inner ℂ x (y*star a) at hadjR
-  have hoL (a x : A) (hx : x ∈ Kᗮ) : a*x ∈ Kᗮ := by
-    rw [Submodule.mem_orthogonal] at hx ⊢
-    intro y hy
-    have hh := hx (star a*y) (hL (star a) y hy)
-    simpa only [hadjL, star_star] using hh
-  have hoR (a x : A) (hx : x ∈ Kᗮ) : x*a ∈ Kᗮ := by
-    rw [Submodule.mem_orthogonal] at hx ⊢
-    intro y hy
-    have hh := hx (y*star a) (hR (star a) y hy)
-    simpa only [hadjR, star_star] using hh
-  have hz (x : A) (hk : x ∈ K) (ho : x ∈ Kᗮ) : x = 0 :=
-    (inner_self_eq_zero (𝕜 := ℂ)).mp ((K.mem_orthogonal x).mp ho x hk)
-  let k : A := K.starProjection 1
-  let q : A := 1-k
-  have hk : k ∈ K := (K.orthogonalProjectionOnto 1).property
-  have hq : q ∈ Kᗮ := K.sub_starProjection_mem_orthogonal 1
-  have hqa (x : A) (hx : x ∈ K) : q*x = 0 :=
-    hz _ (hL q x hx) (hoR x q hq)
-  have haq (x : A) (hx : x ∈ K) : x*q = 0 :=
-    hz _ (hR q x hx) (hoL x q hq)
-  have hsk : star k = k := by
-    have e₁ : star k = k*star k := by
-      have he := hqa (star k) (hstar k hk)
-      change (1-k)*star k = 0 at he
-      simpa only [sub_mul, one_mul, sub_eq_zero] using he
-    have e₂ := congrArg star e₁
-    simp only [star_star, star_mul] at e₂
-    exact e₁.trans e₂.symm
-  refine ⟨q, ?_, ?_, ?_, ?_, fun x hx => ⟨hqa x hx, haq x hx⟩⟩
-  · simp only [q, star_sub, star_one, hsk]
-  · change q*(1-k) = q
-    rw [mul_sub, mul_one, hqa k hk, sub_zero]
-  · intro a
-    have hka : k*a = k*a*k := by
-      have he := haq (k*a) (hR a k hk)
-      change (k*a)*(1-k) = 0 at he
-      simpa only [mul_sub, mul_one, sub_eq_zero] using he
-    have hak : a*k = k*a*k := by
-      have he := hqa (a*k) (hL a k hk)
-      change (1-k)*(a*k) = 0 at he
-      simpa only [sub_mul, one_mul, sub_eq_zero, ← mul_assoc] using he
-    change (1-k)*a = a*(1-k)
-    rw [sub_mul, one_mul, mul_sub, mul_one]
-    exact congrArg (fun z => a-z) (hka.trans hak.symm)
-  · simpa only [q, sub_sub_cancel] using hk
-
-end RowColumn.FiniteStarAlgebra
-
-namespace RowColumn.FiniteStarAlgebra
-open scoped ComplexOrder
-variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
-  [FiniteDimensional ℂ E]
-
-abbrev Ambient := E →L[ℂ] E
-
-def algebraToEnd (A : StarSubalgebra ℂ (Ambient (E := E))) : A →ₗ[ℂ] (E →ₗ[ℂ] E) :=
-  { toFun a := (a : Ambient (E := E)).toLinearMap
-    map_add' _ _ := rfl
-    map_smul' _ _ := rfl }
-
-lemma algebraToEnd_injective (A : StarSubalgebra ℂ (Ambient (E := E))) :
-    Function.Injective (algebraToEnd A) := by
-  intro a b h
-  apply Subtype.ext
-  apply ContinuousLinearMap.ext
-  intro x
-  exact LinearMap.congr_fun h x
-
-@[instance_reducible]
-def algebraCore (A : StarSubalgebra ℂ (Ambient (E := E))) : InnerProductSpace.Core ℂ A where
-  inner a b := HilbertSchmidt.endCore.inner (algebraToEnd A a) (algebraToEnd A b)
-  conj_inner_symm a b := HilbertSchmidt.endCore.conj_inner_symm _ _
-  re_inner_nonneg a := HilbertSchmidt.endCore.re_inner_nonneg _
-  add_left a b c := by rw [map_add]; exact HilbertSchmidt.endCore.add_left _ _ _
-  smul_left a b c := by rw [map_smul]; exact HilbertSchmidt.endCore.smul_left _ _ _
-  definite a h := (algebraToEnd_injective A) (by simpa only [map_zero] using HilbertSchmidt.endCore.definite _ h)
-
-def physicalTrace (A : StarSubalgebra ℂ (Ambient (E := E))) : A →ₗ[ℂ] ℂ :=
-  (LinearMap.trace ℂ E).comp (algebraToEnd A)
-
-lemma core_inner_trace (A : StarSubalgebra ℂ (Ambient (E := E))) (a b : A) :
-    (algebraCore A).inner a b = physicalTrace A (star a*b) := by
-  change LinearMap.trace ℂ E (LinearMap.adjoint (a : Ambient (E := E)).toLinearMap ∘ₗ
-    (b : Ambient (E := E)).toLinearMap) = LinearMap.trace ℂ E ((star (a : Ambient (E := E)) * (b : Ambient (E := E))).toLinearMap)
-  rw [ContinuousLinearMap.star_eq_adjoint, ContinuousLinearMap.adjoint_toLinearMap]
-  rfl
-
-lemma physicalTrace_mul_comm (A : StarSubalgebra ℂ (Ambient (E := E))) (a b : A) :
-    physicalTrace A (a*b) = physicalTrace A (b*a) :=
-  LinearMap.trace_comp_comm' (b : Ambient (E := E)).toLinearMap (a : Ambient (E := E)).toLinearMap
-
-/-- Restriction to a subspace preserved by the whole star algebra is a genuine
-star homomorphism (both the subspace and its orthogonal complement are reducing). -/
-def restrictHom (A : StarSubalgebra ℂ (Ambient (E := E))) (F : Submodule ℂ E)
-    (hF : ∀ a : A, ∀ x ∈ F, (a : Ambient (E := E)) x ∈ F) : A →⋆ₐ[ℂ] (F →L[ℂ] F) where
-  toFun a := (a : Ambient (E := E)).restrict (hF a)
-  map_zero' := by ext x; rfl
-  map_add' a b := by ext x; rfl
-  map_one' := by ext x; rfl
-  map_mul' a b := by ext x; rfl
-  commutes' c := by ext x; rfl
-  map_star' a := by
-    apply ContinuousLinearMap.ext
-    intro x
-    apply ext_inner_right ℂ
-    intro y
-    rw [ContinuousLinearMap.star_eq_adjoint, ContinuousLinearMap.adjoint_inner_left]
-    change inner ℂ ((star (a : Ambient (E := E))) (x : E)) (y : E) =
-      inner ℂ (x : E) ((a : Ambient (E := E)) (y : E))
-    rw [ContinuousLinearMap.star_eq_adjoint, ContinuousLinearMap.adjoint_inner_left]
-
-/-- Central support of a reducing physical subspace. Its algebraic kernel is
-removed, so no other unsupported sector can inflate the Fourier norm. -/
-theorem exists_central_support (A : StarSubalgebra ℂ (Ambient (E := E))) (F : Submodule ℂ E)
-    (hF : ∀ a : A, ∀ x ∈ F, (a : Ambient (E := E)) x ∈ F) :
-    ∃ q : A, star q = q ∧ q*q=q ∧ (∀ a, q*a=a*q) ∧
-      restrictHom A F hF q = 1 ∧
-      (∀ a : A, restrictHom A F hF a = 0 → q*a=0) := by
-  let : FiniteDimensional ℂ A := FiniteDimensional.of_injective
-    (algebraToEnd A) (algebraToEnd_injective A)
-  let φ := restrictHom A F hF
-  let K := LinearMap.ker φ.toLinearMap
-  have hk (a : A) : a ∈ K ↔ φ a = 0 := Iff.rfl
-  obtain ⟨q,hs,hqq,hc,hkq,hkill⟩ := complementary_central_projection (algebraCore A) K
-    (fun a x hx => by rw [hk, map_mul, (hk x).mp hx, mul_zero])
-    (fun a x hx => by rw [hk, map_mul, (hk x).mp hx, zero_mul])
-    (fun x hx => by rw [hk, map_star, (hk x).mp hx]; exact star_zero (F →L[ℂ] F))
-    (fun a x y => by simp only [core_inner_trace, star_mul, mul_assoc])
-    (fun a x y => by
-      simp only [core_inner_trace, star_mul]
-      rw [mul_assoc, physicalTrace_mul_comm A (star a) _, mul_assoc])
-  refine ⟨q,hs,hqq,hc,?_,fun a ha => (hkill a ((hk a).mpr ha)).1⟩
-  have hh := (hk (1-q)).mp hkq
-  have he : 1-φ q=0 := by simpa only [map_sub, map_one] using hh
-  exact (sub_eq_zero.mp he).symm
-
-end RowColumn.FiniteStarAlgebra
-noncomputable section
-namespace RowColumn.FiniteStarAlgebra
-variable {A B : Type*} [CStarAlgebra A] [CStarAlgebra B]
-
-def supportedAlgebra (q : A) (hs : star q = q) (hc : ∀ a, q*a=a*q) :
-    NonUnitalStarSubalgebra ℂ A where
-  carrier := {a | q*a=a}
-  zero_mem' := mul_zero q
-  add_mem' := by intro a b ha hb; change q*(a+b)=a+b; rw [mul_add, ha, hb]
-  mul_mem' := by intro a b ha hb; change q*(a*b)=a*b; rw [← mul_assoc, ha]
-  smul_mem' := by intro c a ha; change q*(c • a)=c • a; rw [mul_smul_comm, ha]
-  star_mem' := by
-    intro a ha
-    change q*star a=star a
-    rw [hc]
-    simpa only [star_mul, hs] using congrArg star ha
-
-lemma supportedAlgebra_closed (q : A) (hs : star q = q) (hc : ∀ a, q*a=a*q) :
-    IsClosed (supportedAlgebra q hs hc : Set A) :=
-  isClosed_eq (continuous_const.mul continuous_id) continuous_id
-
-/-- Removing the kernel by its central support makes restriction isometric.
-This avoids choosing irreducible compact-group carriers or using unproved
-Schur--Weyl decompositions. -/
-theorem norm_supported_eq (φ : A →⋆ₐ[ℂ] B) (q : A) (hs : star q = q)
-    (hqq : q*q=q) (hc : ∀ a, q*a=a*q) (hq : φ q=1)
-    (hkill : ∀ a, φ a=0 → q*a=0) (a : A) : ‖q*a‖ = ‖φ a‖ := by
-  let J := supportedAlgebra q hs hc
-  let : IsClosed (J : Set A) := supportedAlgebra_closed q hs hc
-  let ψ : J →⋆ₙₐ[ℂ] B := φ.toNonUnitalStarAlgHom.comp (NonUnitalStarSubalgebraClass.subtype J)
-  have hinj : Function.Injective ψ := by
-    intro x y h
-    apply Subtype.ext
-    have he : φ ((x : A)-(y : A))=0 := by
-      rw [map_sub]
-      exact sub_eq_zero.mpr h
-    have hz := hkill _ he
-    change q*((x : A)-(y : A))=0 at hz
-    rw [mul_sub, x.property, y.property] at hz
-    exact sub_eq_zero.mp hz
-  let x : J := ⟨q*a, by change q*(q*a)=q*a; rw [← mul_assoc, hqq]⟩
-  have hh := NonUnitalStarAlgHom.norm_map ψ hinj x
-  change ‖φ (q*a)‖ = ‖q*a‖ at hh
-  rw [map_mul, hq, one_mul] at hh
-  exact hh.symm
-
-end RowColumn.FiniteStarAlgebra
-noncomputable section
-open scoped BigOperators ComplexConjugate
-namespace RowColumn.Isotypic
-variable {G V E : Type*} [Group G]
-  [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
-  [NormedAddCommGroup E] [InnerProductSpace ℂ E] [FiniteDimensional ℂ E]
-  (ρ : Representation ℂ G V) (τ : Representation ℂ G E)
-
-def carrier : Submodule ℂ E := Submodule.span ℂ
-  (Set.range fun p : Representation.IntertwiningMap ρ τ × V => p.1 p.2)
-
-omit [FiniteDimensional ℂ V] [FiniteDimensional ℂ E] in
-lemma mem_carrier (f : Representation.IntertwiningMap ρ τ) (v : V) : f v ∈ carrier ρ τ :=
-  Submodule.subset_span ⟨(f,v),rfl⟩
-
-lemma projection_intertwines (F : Submodule ℂ E)
-    (hF : ∀ g x, x ∈ F → τ g x ∈ F)
-    (hτ : ∀ g x y, inner ℂ (τ g x) (τ g y) = inner ℂ x y) :
-    ∀ g x, F.starProjection (τ g x) = τ g (F.starProjection x) := by
-  intro g x
-  let e : E →ₗᵢ[ℂ] E := (τ g).isometryOfInner (hτ g)
-  have hmap : F.map e.toLinearMap = F := by
-    apply le_antisymm
-    · rintro _ ⟨y,hy,rfl⟩
-      exact hF g y hy
-    · intro y hy
-      refine ⟨τ g⁻¹ y, hF g⁻¹ y hy, ?_⟩
-      change (τ g * τ g⁻¹) y = y
-      rw [← map_mul, mul_inv_cancel, map_one]
-      rfl
-  have he := e.map_starProjection F x
-  simpa only [hmap, e, LinearMap.coe_isometryOfInner] using he.symm
-
-omit [FiniteDimensional ℂ V] in
-/-- On an isotypic unitary representation, every eigenspace of an operator in
-the group commutant has dimension at least that of the irreducible type. -/
-theorem eigenspace_finrank [Representation.IsIrreducible ρ]
-    (hspan : carrier ρ τ = ⊤)
-    (hτ : ∀ g x y, inner ℂ (τ g x) (τ g y) = inner ℂ x y)
-    (T : E →ₗ[ℂ] E) (hT : ∀ g x, T (τ g x) = τ g (T x))
-    (μ : ℂ) (hμ : Module.End.HasEigenvalue T μ) :
-    Module.finrank ℂ V ≤ Module.finrank ℂ (Module.End.eigenspace T μ) := by
-  classical
-  let F := Module.End.eigenspace T μ
-  have hf (g : G) (x : E) (hx : x ∈ F) : τ g x ∈ F := by
-    rw [Module.End.mem_eigenspace_iff] at hx ⊢
-    rw [hT, hx, map_smul]
-  let P : Representation.IntertwiningMap τ τ :=
-    { toLinearMap := F.starProjection.toLinearMap
-      isIntertwining' g := by ext x; exact projection_intertwines τ F hf hτ g x }
-  have hex : ∃ f : Representation.IntertwiningMap ρ τ, P.comp f ≠ 0 := by
-    by_contra! h
-    have hz (x : E) (hx : x ∈ carrier ρ τ) : P x = 0 := by
-      induction hx using Submodule.span_induction with
-      | mem x hx =>
-        obtain ⟨⟨f,v⟩,rfl⟩ := hx
-        exact congrArg (fun L : Representation.IntertwiningMap ρ τ => L v) (h f)
-      | zero => exact map_zero P
-      | add x y hx hy hpx hpy => simp only [map_add, hpx, hpy, add_zero]
-      | smul c x hx hpx => simp only [map_smul, hpx, smul_zero]
-    obtain ⟨x,hx,hxn⟩ := hμ.exists_hasEigenvector
-    have he := hz x (by rw [hspan]; trivial)
-    change F.starProjection x = 0 at he
-    rw [F.starProjection_eq_self_iff.mpr hx] at he
-    exact hxn he
-  obtain ⟨f,hfne⟩ := hex
-  have hi : Function.Injective (P.comp f) :=
-    (Representation.IsIrreducible.injective_or_eq_zero (P.comp f)).resolve_right hfne
-  let L : V →ₗ[ℂ] F := (P.comp f).toLinearMap.codRestrict F (fun v =>
-    (F.orthogonalProjectionOnto (f v)).property)
-  apply LinearMap.finrank_le_finrank_of_injective (f := L)
-  intro x y h
-  exact hi (congrArg Subtype.val h)
-
-end RowColumn.Isotypic
-namespace RowColumn.Isotypic
-variable {G V E : Type*} [Group G]
-  [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
-  [NormedAddCommGroup E] [InnerProductSpace ℂ E] [FiniteDimensional ℂ E]
-  (ρ : Representation ℂ G V) (τ : Representation ℂ G E)
-
-omit [FiniteDimensional ℂ V] [FiniteDimensional ℂ E] in
-lemma carrier_invariant (g : G) (x : E) (hx : x ∈ carrier ρ τ) : τ g x ∈ carrier ρ τ := by
-  induction hx using Submodule.span_induction with
-  | mem x hx =>
-    obtain ⟨⟨f,v⟩,rfl⟩ := hx
-    rw [← Representation.IntertwiningMap.isIntertwining ρ τ f]
-    exact mem_carrier ρ τ f _
-  | zero => simpa only [map_zero] using (carrier ρ τ).zero_mem
-  | add x y hx hy hx' hy' => simpa only [map_add] using (carrier ρ τ).add_mem hx' hy'
-  | smul c x hx hx' => simpa only [map_smul] using (carrier ρ τ).smul_mem c hx'
-
-omit [FiniteDimensional ℂ V] [FiniteDimensional ℂ E] in
-lemma carrier_commutant (T : E →ₗ[ℂ] E) (hT : ∀ g x, T (τ g x) = τ g (T x))
-    (x : E) (hx : x ∈ carrier ρ τ) : T x ∈ carrier ρ τ := by
-  let t : Representation.IntertwiningMap τ τ :=
-    { toLinearMap := T
-      isIntertwining' g := by ext x; exact hT g x }
-  induction hx using Submodule.span_induction with
-  | mem x hx =>
-    obtain ⟨⟨f,v⟩,rfl⟩ := hx
-    exact mem_carrier ρ τ (t.comp f) v
-  | zero => simpa only [map_zero] using (carrier ρ τ).zero_mem
-  | add x y hx hy hx' hy' => simpa only [map_add] using (carrier ρ τ).add_mem hx' hy'
-  | smul c x hx hx' => simpa only [map_smul] using (carrier ρ τ).smul_mem c hx'
-
-def restricted : Representation ℂ G (carrier ρ τ) where
-  toFun g := (τ g).restrict (carrier_invariant ρ τ g)
-  map_one' := by ext x; exact LinearMap.congr_fun (map_one τ) x
-  map_mul' g h := by ext x; exact LinearMap.congr_fun (map_mul τ g h) x
-
-def restrictIntertwiner (f : Representation.IntertwiningMap ρ τ) :
-    Representation.IntertwiningMap ρ (restricted ρ τ) where
-  toLinearMap := f.toLinearMap.codRestrict (carrier ρ τ) (mem_carrier ρ τ f)
-  isIntertwining' g := by
-    ext v
-    exact Representation.IntertwiningMap.isIntertwining ρ τ f g v
-
-omit [FiniteDimensional ℂ V] [FiniteDimensional ℂ E] in
-lemma restricted_unitary (hτ : ∀ g x y, inner ℂ (τ g x) (τ g y) = inner ℂ x y) :
-    ∀ g (x y : carrier ρ τ), inner ℂ (restricted ρ τ g x) (restricted ρ τ g y) = inner ℂ x y := by
-  intro g x y
-  exact hτ g x y
-
-omit [FiniteDimensional ℂ V] [FiniteDimensional ℂ E] in
-lemma restricted_isotypic : carrier ρ (restricted ρ τ) = ⊤ := by
-  let F := carrier ρ τ
-  let K := (carrier ρ (restricted ρ τ)).map F.subtype
-  have hFK : F ≤ K := by
-    apply Submodule.span_le.mpr
-    rintro _ ⟨⟨f,v⟩,rfl⟩
-    exact ⟨restrictIntertwiner ρ τ f v, mem_carrier ρ (restricted ρ τ)
-      (restrictIntertwiner ρ τ f) v, rfl⟩
-  apply top_unique
-  intro x _
-  obtain ⟨y,hy,hxy⟩ := hFK x.property
-  have he : y=x := Subtype.ext hxy
-  exact he ▸ hy
-
-end RowColumn.Isotypic
-noncomputable section
-open scoped BigOperators ComplexOrder
-namespace RowColumn.SpectralMultiplicity
-variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
-  [FiniteDimensional ℂ E]
-
-lemma positive_norm_le_of_eigenvalues (T : E →L[ℂ] E) (hT : T.IsPositive)
-    (c : ℝ) (hc : 0 ≤ c)
-    (hval : ∀ i, hT.toLinearMap.isSymmetric.eigenvalues rfl i ≤ c) : ‖T‖ ≤ c := by
-  let b := hT.toLinearMap.isSymmetric.eigenvectorBasis rfl
-  apply T.opNorm_le_bound hc
-  intro x
-  apply le_of_sq_le_sq _ (mul_nonneg hc (norm_nonneg _))
-  rw [← b.repr.norm_map (T x), ← b.repr.norm_map x, mul_pow]
-  rw [EuclideanSpace.norm_sq_eq, EuclideanSpace.norm_sq_eq, Finset.mul_sum]
-  apply Finset.sum_le_sum
-  intro i _
-  have hi := hT.toLinearMap.nonneg_eigenvalues rfl i
-  have he := hT.toLinearMap.isSymmetric.eigenvectorBasis_apply_self_apply rfl x i
-  change (b.repr (T x)) i = _ at he
-  change ‖(b.repr (T x)) i‖^2 ≤ _
-  rw [he, norm_mul, RCLike.norm_ofReal, abs_of_nonneg hi, mul_pow]
-  exact mul_le_mul_of_nonneg_right (sq_le_sq₀ hi hc |>.mpr (hval i)) (sq_nonneg _)
-
-/-- An eigenvalue supported on at least d dimensions pays a trace factor d.
-This is the precise spectral normalization needed for the global filter. -/
-theorem norm_le_trace_div (T : E →L[ℂ] E) (hT : T.IsPositive) (d : ℕ) (hd : 0 < d)
-    (hmult : ∀ μ : ℂ, Module.End.HasEigenvalue T.toLinearMap μ →
-      d ≤ Module.finrank ℂ (Module.End.eigenspace T.toLinearMap μ)) :
-    ‖T‖ ≤ (LinearMap.trace ℂ E T.toLinearMap).re / d := by
-  have htrace : 0 ≤ (LinearMap.trace ℂ E T.toLinearMap).re :=
-    (Complex.nonneg_iff.mp hT.toLinearMap.trace_nonneg).1
-  apply positive_norm_le_of_eigenvalues T hT _ (div_nonneg htrace (by positivity))
-  intro i
-  let μ := hT.toLinearMap.isSymmetric.eigenvalues rfl i
-  have hμ : 0 ≤ μ := hT.toLinearMap.nonneg_eigenvalues rfl i
-  have hcount := hmult (μ : ℂ) (hT.toLinearMap.isSymmetric.hasEigenvalue_eigenvalues rfl i)
-  have hcard := hT.toLinearMap.isSymmetric.card_filter_eigenvalues_eq rfl (μ : ℂ)
-  rw [← hcard] at hcount
-  simp only [RCLike.ofReal_eq_complex_ofReal, Complex.ofReal_inj] at hcount
-  apply (le_div_iff₀ (by exact_mod_cast hd)).mpr
-  calc
-    μ * (d : ℝ) ≤ μ * ((Finset.univ.filter fun j =>
-      hT.toLinearMap.isSymmetric.eigenvalues rfl j = μ).card : ℕ) :=
-        mul_le_mul_of_nonneg_left (by exact_mod_cast hcount) hμ
-    _ = ∑ j ∈ Finset.univ.filter (fun j =>
-      hT.toLinearMap.isSymmetric.eigenvalues rfl j = μ),
-        hT.toLinearMap.isSymmetric.eigenvalues rfl j := by
-      symm
-      calc
-        _ = ∑ j ∈ Finset.univ.filter (fun j =>
-          hT.toLinearMap.isSymmetric.eigenvalues rfl j = μ), μ :=
-            Finset.sum_congr rfl (fun j hj => (Finset.mem_filter.mp hj).2)
-        _ = _ := by simp [nsmul_eq_mul, mul_comm]
-    _ ≤ ∑ j, hT.toLinearMap.isSymmetric.eigenvalues rfl j :=
-      Finset.sum_le_sum_of_subset_of_nonneg (Finset.filter_subset _ _)
-        (fun j _ _ => hT.toLinearMap.nonneg_eigenvalues rfl j)
-    _ = _ := (hT.toLinearMap.isSymmetric.re_trace_eq_sum_eigenvalues rfl).symm
-
-end RowColumn.SpectralMultiplicity
-noncomputable section
-open scoped ComplexOrder
-namespace RowColumn.SpectralMultiplicity
-variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
-  [FiniteDimensional ℂ E]
-
-lemma trace_compression (T : E →L[ℂ] E) (F : Submodule ℂ E) :
-    LinearMap.trace ℂ F (F.orthogonalProjectionOnto ∘L T ∘L F.subtypeL).toLinearMap =
-      LinearMap.trace ℂ E (T ∘L F.starProjection).toLinearMap := by
-  have he := LinearMap.trace_comp_comm' (T.toLinearMap ∘ₗ F.subtype)
-    F.orthogonalProjectionOnto.toLinearMap
-  exact he
-
-lemma trace_compression_le (T : E →L[ℂ] E) (hT : T.IsPositive)
-    (F : Submodule ℂ E) :
-    (LinearMap.trace ℂ F (F.orthogonalProjectionOnto ∘L T ∘L F.subtypeL).toLinearMap).re ≤
-      (LinearMap.trace ℂ E T.toLinearMap).re := by
-  have hp := (hT.orthogonalProjectionOnto_comp Fᗮ).toLinearMap.trace_nonneg
-  have hn := (Complex.nonneg_iff.mp hp).1
-  rw [trace_compression] at hn ⊢
-  have he : F.starProjection + Fᗮ.starProjection = (1 : E →L[ℂ] E) := by
-    ext x
-    exact F.starProjection_add_starProjection_orthogonal x
-  have ht : T ∘L F.starProjection + T ∘L Fᗮ.starProjection = T := by
-    rw [← ContinuousLinearMap.comp_add, he]
-    rfl
-  have htr := congrArg (fun S : E →L[ℂ] E => (LinearMap.trace ℂ E S.toLinearMap).re) ht
-  simp only [ContinuousLinearMap.toLinearMap_add, map_add, Complex.add_re] at htr
-  linarith
-
-lemma restrict_positive (T : E →L[ℂ] E) (hT : T.IsPositive)
-    (F : Submodule ℂ E) (hF : ∀ x ∈ F, T x ∈ F) : (T.restrict hF).IsPositive := by
-  have he : T.restrict hF = F.orthogonalProjectionOnto ∘L T ∘L F.subtypeL := by
-    ext x
-    change T x = F.starProjection (T x)
-    exact (F.starProjection_eq_self_iff.mpr (hF x x.property)).symm
-  rw [he]
-  exact hT.orthogonalProjectionOnto_comp F
-
-lemma trace_restrict_le (T : E →L[ℂ] E) (hT : T.IsPositive)
-    (F : Submodule ℂ E) (hF : ∀ x ∈ F, T x ∈ F) :
-    (LinearMap.trace ℂ F (T.restrict hF).toLinearMap).re ≤
-      (LinearMap.trace ℂ E T.toLinearMap).re := by
-  have he : T.restrict hF = F.orthogonalProjectionOnto ∘L T ∘L F.subtypeL := by
-    ext x
-    change T x = F.starProjection (T x)
-    exact (F.starProjection_eq_self_iff.mpr (hF x x.property)).symm
-  rw [he]
-  exact trace_compression_le T hT F
-
-end RowColumn.SpectralMultiplicity
-
-end
-end
-end
-end
-end
-end
-end
-
-
-section
-noncomputable section
-open scoped ComplexOrder
-namespace RowColumn.FiniteStarAlgebra
-variable {G V E : Type*} [Group G]
-  [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
-  [NormedAddCommGroup E] [InnerProductSpace ℂ E] [FiniteDimensional ℂ E]
-
-instance algebraFiniteDimensional (A : StarSubalgebra ℂ (E →L[ℂ] E)) : FiniteDimensional ℂ A :=
-  FiniteDimensional.of_injective (algebraToEnd A) (algebraToEnd_injective A)
-
-instance algebraIsClosed (A : StarSubalgebra ℂ (E →L[ℂ] E)) : IsClosed (A : Set (E →L[ℂ] E)) :=
-  A.toSubalgebra.toSubmodule.closed_of_finiteDimensional
-
-lemma supported_restriction_norm (A : StarSubalgebra ℂ (E →L[ℂ] E)) (F : Submodule ℂ E)
-    (φ : A →⋆ₐ[ℂ] (F →L[ℂ] F))
-    (q : A) (hs : star q=q) (hqq : q*q=q) (hc : ∀ a, q*a=a*q)
-    (hq : φ q=1) (hkill : ∀ a : A, φ a=0 → q*a=0) (a : A) :
-    ‖q*a‖ = ‖φ a‖ := by
-  let J : NonUnitalStarSubalgebra ℂ A :=
-    { carrier := {x | q*x=x}
-      zero_mem' := mul_zero q
-      add_mem' := by intro x y hx hy; change q*(x+y)=x+y; rw [mul_add,hx,hy]
-      mul_mem' := by intro x y hx hy; change q*(x*y)=x*y; rw [← mul_assoc,hx]
-      smul_mem' := by intro c x hx; change q*(c • x)=c • x; rw [mul_smul_comm,hx]
-      star_mem' := by
-        intro x hx
-        change q*star x=star x
-        rw [hc]
-        simpa only [star_mul,hs] using congrArg star hx }
-  let : IsClosed (J : Set A) := isClosed_eq (continuous_const.mul continuous_id) continuous_id
-  let ψ : J →⋆ₙₐ[ℂ] (F →L[ℂ] F) := φ.toNonUnitalStarAlgHom.comp (NonUnitalStarSubalgebraClass.subtype J)
-  have hinj : Function.Injective ψ := by
-    intro x y h
-    apply Subtype.ext
-    have he : φ ((x : A)-(y : A))=0 := by
-      rw [map_sub]
-      exact sub_eq_zero.mpr h
-    have hz := hkill _ he
-    change q*((x : A)-(y : A))=0 at hz
-    rw [mul_sub,x.property,y.property] at hz
-    exact sub_eq_zero.mp hz
-  let x : J := ⟨q*a,by change q*(q*a)=q*a; rw [← mul_assoc,hqq]⟩
-  have hh := NonUnitalStarAlgHom.norm_map ψ hinj x
-  change ‖φ (q*a)‖ = ‖q*a‖ at hh
-  rw [map_mul,hq,one_mul] at hh
-  exact hh.symm
-
-/-- The exact dimension normalization of the occupied-board global filter.
-No highest-weight entropy or Schur--Weyl hypothesis is required: restriction
-of the supported algebra is faithful, and every eigenspace contains the
-irreducible carrier. -/
-theorem exists_isotypic_filter (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
-    (τ : Representation ℂ G E)
-    (hτ : ∀ g x y, inner ℂ (τ g x) (τ g y) = inner ℂ x y)
-    (A : StarSubalgebra ℂ (E →L[ℂ] E))
-    (hA : ∀ a : A, ∀ g x, (a : E →L[ℂ] E) (τ g x) = τ g ((a : E →L[ℂ] E) x)) :
-    ∃ q : A, star q = q ∧ q*q=q ∧ (∀ a, q*a=a*q) ∧
-      (∀ x ∈ Isotypic.carrier ρ τ, (q : E →L[ℂ] E) x = x) ∧
-      ∀ a : A, (a : E →L[ℂ] E).IsPositive →
-        ‖q*a‖ ≤ (physicalTrace A a).re / Module.finrank ℂ V := by
-  let : Nontrivial V := by
-    let := IsSimpleModule.nontrivial (MonoidAlgebra ℂ G) ρ.asModule
-    exact ρ.asModuleEquiv.symm.toEquiv.nontrivial
-  let F := Isotypic.carrier ρ τ
-  have hF : ∀ a : A, ∀ x ∈ F, (a : E →L[ℂ] E) x ∈ F := fun a =>
-    Isotypic.carrier_commutant ρ τ (a : E →L[ℂ] E).toLinearMap (hA a)
-  let φ := restrictHom A F hF
-  obtain ⟨q,hqs,hqq,hqc,hq,hkill⟩ := exists_central_support A F hF
-  refine ⟨q,hqs,hqq,hqc,?_,?_⟩
-  · intro x hx
-    exact congrArg Subtype.val (congrArg (fun T : F →L[ℂ] F => T ⟨x,hx⟩) hq)
-  · intro a ha
-    rw [supported_restriction_norm A F φ q hqs hqq hqc hq hkill a]
-    have hp : (φ a).IsPositive := SpectralMultiplicity.restrict_positive
-      (a : E →L[ℂ] E) ha F (hF a)
-    apply (SpectralMultiplicity.norm_le_trace_div (φ a) hp (Module.finrank ℂ V)
-      (Module.finrank_pos) ?_).trans
-    · apply div_le_div_of_nonneg_right _ (Nat.cast_nonneg _)
-      exact SpectralMultiplicity.trace_restrict_le (a : E →L[ℂ] E) ha F (hF a)
-    · intro μ hμ
-      apply Isotypic.eigenspace_finrank ρ (Isotypic.restricted ρ τ)
-        (Isotypic.restricted_isotypic ρ τ) (Isotypic.restricted_unitary ρ τ hτ)
-        (φ a).toLinearMap _ μ hμ
-      intro g x
-      apply Subtype.ext
-      exact hA a g x
-
-end RowColumn.FiniteStarAlgebra
-noncomputable section
-open scoped BigOperators ComplexOrder
-namespace RowColumn.UnitaryFrame
-variable {A G E : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A]
-  [FiniteDimensional ℂ A] [Group G]
-  [NormedAddCommGroup E] [InnerProductSpace ℂ E]
-
-/-- The finite Fourier reconstruction bound for arbitrary Hilbert-valued
-linear maps, in particular weighted Hilbert–Schmidt sandwich maps. -/
-theorem reconstruction_hilbert_bound (τ : A →ₗ[ℂ] ℂ) (core : InnerProductSpace.Core ℂ A)
-    (hinner : ∀ x y, core.inner x y = τ (star x*y))
-    (u : G →* unitary A)
-    (hspan : Submodule.span ℂ (Set.range fun g : G => (u g : A)) = ⊤)
-    (B : A) (F : A →ₗ[ℂ] E) (R : ℝ) (hR : 0 ≤ R)
-    (hU : ∀ g, ‖F (u g : A)‖ ≤ R) :
-    ‖F B‖ ≤ 4 * (Module.finrank ℂ A : ℝ) * ‖B‖ * R := by
-  let ℓ := (innerSL ℂ (F B)).toLinearMap.comp F
-  have hu (g : G) : ‖ℓ (u g : A)‖ ≤ ‖F B‖ * R :=
-    (norm_inner_le_norm (F B) (F (u g : A))).trans
-      (mul_le_mul_of_nonneg_left (hU g) (norm_nonneg _))
-  have hh := reconstruction_bound τ core hinner u hspan B ℓ (‖F B‖ * R)
-    (mul_nonneg (norm_nonneg _) hR) hu
-  change ‖inner ℂ (F B) (F B)‖ ≤ _ at hh
-  rw [inner_self_eq_norm_sq_to_K, norm_pow, RCLike.norm_ofReal,
-    abs_of_nonneg (norm_nonneg _)] at hh
-  by_cases hz : ‖F B‖=0
-  · rw [hz]
-    positivity
-  · have hp : 0 < ‖F B‖ := lt_of_le_of_ne (norm_nonneg _) (Ne.symm hz)
-    apply (mul_le_mul_iff_right₀ hp).mp
-    nlinarith [hh]
-
-end RowColumn.UnitaryFrame
-noncomputable section
-namespace RowColumn
-variable {A : Type*} [CStarAlgebra A]
-
-lemma supported_fourth (q F : A) (hq : q*q=q) (hc : Commute q F) :
-    (q*F)^4 = q*(F^4) := by
-  rw [hc.mul_pow]
-  have hq4 : q^4=q := by calc
-    q^4=(q*q)*(q*q) := by noncomm_ring
-    _=q := by rw [hq,hq]
-  rw [hq4]
-
-/-- The quarter-root estimate at the level of the genuine operator norm.
-The two fourth-moment bounds imply the exact squared sandwich bound. -/
-theorem filterNorm_sq (q F G : A) (hsq : IsSelfAdjoint q) (hqq : q*q=q)
-    (hF : IsSelfAdjoint F) (hG : IsSelfAdjoint G) (hcF : Commute q F) (hcG : Commute q G)
-    (d : ℝ) (_ : 0 < d) (hFb : ‖q * F^4‖ ≤ 1/d) (hGb : ‖q * G^4‖ ≤ 1/d) :
-    ‖F*q*G‖^2 ≤ 1/d := by
-  have hsF : IsSelfAdjoint (q*F) := by
-    change star (q*F)=q*F
-    rw [star_mul,hsq.star_eq,hF.star_eq,hcF.eq]
-  have hsG : IsSelfAdjoint (q*G) := by
-    change star (q*G)=q*G
-    rw [star_mul,hsq.star_eq,hG.star_eq,hcG.eq]
-  have hnF : ‖q*F‖^4 ≤ 1/d := by
-    have hn : ‖(q*F)^4‖=‖q*F‖^4 := by simpa using hsF.norm_pow_two_pow 2
-    rw [← hn,supported_fourth q F hqq hcF]
-    exact hFb
-  have hnG : ‖q*G‖^4 ≤ 1/d := by
-    have hn : ‖(q*G)^4‖=‖q*G‖^4 := by simpa using hsG.norm_pow_two_pow 2
-    rw [← hn,supported_fourth q G hqq hcG]
-    exact hGb
-  have he : F*q*G=(q*F)*(q*G) := by
-    calc
-      _ = F*(q*q)*G := by rw [hqq]
-      _ = (F*q)*(q*G) := by noncomm_ring
-      _ = _ := by rw [hcF.eq]
-  have hn := norm_mul_le (q*F) (q*G)
-  rw [← he] at hn
-  have hp : (‖q*F‖*‖q*G‖)^2 ≤ 1/d := by
-    have ha := sq_nonneg (‖q*F‖^2-‖q*G‖^2)
-    nlinarith
-  exact (pow_le_pow_left₀ (norm_nonneg _) hn 2).trans hp
-end RowColumn
-
-end
-end
-end
-end
-
-end OAI
+import OAI.RepresentationTheory.FiniteUnitary.Frames
+import OAI.RepresentationTheory.RowColumn.Basic
+
+namespace OAI
+
+section
+noncomputable section
+open scoped BigOperators ComplexOrder
+namespace RowColumn.UnitaryFrame
+variable {A G : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A]
+  [FiniteDimensional ℂ A] [Group G]
+
+omit [FiniteDimensional ℂ A] in
+lemma positive_functional_bound (f : A →ₚ[ℂ] ℂ) (x : A) :
+    ‖f x‖ ≤ 4 * ‖f 1‖ * ‖x‖ := by
+  obtain ⟨y, hy_nonneg, hy_norm, hy⟩ := CStarAlgebra.exists_sum_four_nonneg x
+  conv_lhs => rw [hy]
+  simp only [map_sum, map_smul]
+  apply (norm_sum_le _ _).trans
+  simp only [norm_smul, norm_pow, Complex.norm_I, one_pow, one_mul]
+  apply (Finset.sum_le_sum (g := fun _ => ‖f 1‖ * ‖x‖) (fun i _ => ?_)).trans
+  · simp [mul_assoc]
+  · exact (f.norm_apply_le_of_nonneg _ (hy_nonneg i)).trans
+      (mul_le_mul_of_nonneg_left (hy_norm i) (norm_nonneg _))
+
+/-- Finite-dimensional unitary reconstruction with polynomial Fourier cost.
+Only the actual operator norm of B occurs; the auxiliary Hilbert norm is used
+solely to construct the frame and does not alter that norm. -/
+theorem reconstruction_bound (τ : A →ₗ[ℂ] ℂ) (core : InnerProductSpace.Core ℂ A)
+    (hinner : ∀ x y, core.inner x y = τ (star x*y))
+    (u : G →* unitary A)
+    (hspan : Submodule.span ℂ (Set.range fun g : G => (u g : A)) = ⊤)
+    (B : A) (ℓ : A →ₗ[ℂ] ℂ) (R : ℝ) (_ : 0 ≤ R)
+    (hU : ∀ g, ‖ℓ (u g : A)‖ ≤ R) :
+    ‖ℓ B‖ ≤ 4 * (Module.finrank ℂ A : ℝ) * ‖B‖ * R := by
+  obtain ⟨D,hD,hpos,hrec⟩ := exists_density τ core hinner u hspan
+  let φ : A →ₗ[ℂ] ℂ := τ.comp (LinearMap.mulRight ℂ D)
+  have hφpos (a : A) (ha : 0 ≤ a) : 0 ≤ φ a := by
+    have hh := hpos (CFC.sqrt a)
+    have hs : star (CFC.sqrt a) * CFC.sqrt a = a := by
+      rw [(IsSelfAdjoint.of_nonneg (CFC.sqrt_nonneg a)).star_eq]
+      exact CFC.sqrt_mul_sqrt_self a ha
+    change 0 ≤ τ (a*D)
+    simpa only [hs] using hh
+  let f : A →ₚ[ℂ] ℂ := PositiveLinearMap.mk₀ φ hφpos
+  have hf1 : ‖f 1‖ = (Module.finrank ℂ A : ℝ) := by
+    change ‖τ (1*D)‖ = _
+    rw [one_mul, hD]
+    simp
+  apply hrec B ℓ
+  intro g
+  rw [← mul_assoc]
+  change ‖φ (star (u g : A)*B) * ℓ (u g : A)‖ ≤ _
+  rw [norm_mul]
+  have hb : ‖φ (star (u g : A)*B)‖ ≤ 4 * (Module.finrank ℂ A : ℝ) * ‖B‖ := by
+    have hh := positive_functional_bound f (star (u g : A)*B)
+    rw [hf1] at hh
+    change ‖f (star (u g : A)*B)‖ ≤ _
+    simpa only [← Unitary.coe_star, CStarRing.norm_coe_unitary_mul] using hh
+  exact mul_le_mul hb (hU g) (norm_nonneg _) (by positivity)
+
+end RowColumn.UnitaryFrame
+noncomputable section
+open scoped ComplexConjugate
+namespace RowColumn.FiniteStarAlgebra
+variable {A : Type*} [Ring A] [Module ℂ A] [StarRing A] [FiniteDimensional ℂ A]
+
+/-- A finite-dimensional two-sided star ideal has a central orthogonal unit.
+The two inner-product identities hold for the actual Hilbert–Schmidt trace.
+This constructs the central support used in the occupied-board filter. -/
+theorem complementary_central_projection (core : InnerProductSpace.Core ℂ A) (K : Submodule ℂ A)
+    (hL : ∀ a x, x ∈ K → a*x ∈ K) (hR : ∀ a x, x ∈ K → x*a ∈ K)
+    (hstar : ∀ x, x ∈ K → star x ∈ K)
+    (hadjL : ∀ a x y : A, core.inner (a*x) y = core.inner x (star a*y))
+    (hadjR : ∀ a x y : A, core.inner (x*a) y = core.inner x (y*star a)) :
+    ∃ q : A, star q = q ∧ q*q = q ∧ (∀ a, q*a = a*q) ∧
+      1-q ∈ K ∧ (∀ x, x ∈ K → q*x = 0 ∧ x*q = 0) := by
+  let : NormedAddCommGroup A := core.toNormedAddCommGroup
+  let : InnerProductSpace ℂ A := InnerProductSpace.ofCore { __ := core }
+  change ∀ a x y : A, inner ℂ (a*x) y = inner ℂ x (star a*y) at hadjL
+  change ∀ a x y : A, inner ℂ (x*a) y = inner ℂ x (y*star a) at hadjR
+  have hoL (a x : A) (hx : x ∈ Kᗮ) : a*x ∈ Kᗮ := by
+    rw [Submodule.mem_orthogonal] at hx ⊢
+    intro y hy
+    have hh := hx (star a*y) (hL (star a) y hy)
+    simpa only [hadjL, star_star] using hh
+  have hoR (a x : A) (hx : x ∈ Kᗮ) : x*a ∈ Kᗮ := by
+    rw [Submodule.mem_orthogonal] at hx ⊢
+    intro y hy
+    have hh := hx (y*star a) (hR (star a) y hy)
+    simpa only [hadjR, star_star] using hh
+  have hz (x : A) (hk : x ∈ K) (ho : x ∈ Kᗮ) : x = 0 :=
+    (inner_self_eq_zero (𝕜 := ℂ)).mp ((K.mem_orthogonal x).mp ho x hk)
+  let k : A := K.starProjection 1
+  let q : A := 1-k
+  have hk : k ∈ K := (K.orthogonalProjectionOnto 1).property
+  have hq : q ∈ Kᗮ := K.sub_starProjection_mem_orthogonal 1
+  have hqa (x : A) (hx : x ∈ K) : q*x = 0 :=
+    hz _ (hL q x hx) (hoR x q hq)
+  have haq (x : A) (hx : x ∈ K) : x*q = 0 :=
+    hz _ (hR q x hx) (hoL x q hq)
+  have hsk : star k = k := by
+    have e₁ : star k = k*star k := by
+      have he := hqa (star k) (hstar k hk)
+      change (1-k)*star k = 0 at he
+      simpa only [sub_mul, one_mul, sub_eq_zero] using he
+    have e₂ := congrArg star e₁
+    simp only [star_star, star_mul] at e₂
+    exact e₁.trans e₂.symm
+  refine ⟨q, ?_, ?_, ?_, ?_, fun x hx => ⟨hqa x hx, haq x hx⟩⟩
+  · simp only [q, star_sub, star_one, hsk]
+  · change q*(1-k) = q
+    rw [mul_sub, mul_one, hqa k hk, sub_zero]
+  · intro a
+    have hka : k*a = k*a*k := by
+      have he := haq (k*a) (hR a k hk)
+      change (k*a)*(1-k) = 0 at he
+      simpa only [mul_sub, mul_one, sub_eq_zero] using he
+    have hak : a*k = k*a*k := by
+      have he := hqa (a*k) (hL a k hk)
+      change (1-k)*(a*k) = 0 at he
+      simpa only [sub_mul, one_mul, sub_eq_zero, ← mul_assoc] using he
+    change (1-k)*a = a*(1-k)
+    rw [sub_mul, one_mul, mul_sub, mul_one]
+    exact congrArg (fun z => a-z) (hka.trans hak.symm)
+  · simpa only [q, sub_sub_cancel] using hk
+
+end RowColumn.FiniteStarAlgebra
+
+namespace RowColumn.FiniteStarAlgebra
+open scoped ComplexOrder
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
+  [FiniteDimensional ℂ E]
+
+abbrev Ambient := E →L[ℂ] E
+
+def algebraToEnd (A : StarSubalgebra ℂ (Ambient (E := E))) : A →ₗ[ℂ] (E →ₗ[ℂ] E) :=
+  { toFun a := (a : Ambient (E := E)).toLinearMap
+    map_add' _ _ := rfl
+    map_smul' _ _ := rfl }
+
+lemma algebraToEnd_injective (A : StarSubalgebra ℂ (Ambient (E := E))) :
+    Function.Injective (algebraToEnd A) := by
+  intro a b h
+  apply Subtype.ext
+  apply ContinuousLinearMap.ext
+  intro x
+  exact LinearMap.congr_fun h x
+
+@[instance_reducible]
+def algebraCore (A : StarSubalgebra ℂ (Ambient (E := E))) : InnerProductSpace.Core ℂ A where
+  inner a b := HilbertSchmidt.endCore.inner (algebraToEnd A a) (algebraToEnd A b)
+  conj_inner_symm a b := HilbertSchmidt.endCore.conj_inner_symm _ _
+  re_inner_nonneg a := HilbertSchmidt.endCore.re_inner_nonneg _
+  add_left a b c := by rw [map_add]; exact HilbertSchmidt.endCore.add_left _ _ _
+  smul_left a b c := by rw [map_smul]; exact HilbertSchmidt.endCore.smul_left _ _ _
+  definite a h := (algebraToEnd_injective A) (by simpa only [map_zero] using HilbertSchmidt.endCore.definite _ h)
+
+def physicalTrace (A : StarSubalgebra ℂ (Ambient (E := E))) : A →ₗ[ℂ] ℂ :=
+  (LinearMap.trace ℂ E).comp (algebraToEnd A)
+
+lemma core_inner_trace (A : StarSubalgebra ℂ (Ambient (E := E))) (a b : A) :
+    (algebraCore A).inner a b = physicalTrace A (star a*b) := by
+  change LinearMap.trace ℂ E (LinearMap.adjoint (a : Ambient (E := E)).toLinearMap ∘ₗ
+    (b : Ambient (E := E)).toLinearMap) = LinearMap.trace ℂ E ((star (a : Ambient (E := E)) * (b : Ambient (E := E))).toLinearMap)
+  rw [ContinuousLinearMap.star_eq_adjoint, ContinuousLinearMap.adjoint_toLinearMap]
+  rfl
+
+lemma physicalTrace_mul_comm (A : StarSubalgebra ℂ (Ambient (E := E))) (a b : A) :
+    physicalTrace A (a*b) = physicalTrace A (b*a) :=
+  LinearMap.trace_comp_comm' (b : Ambient (E := E)).toLinearMap (a : Ambient (E := E)).toLinearMap
+
+/-- Restriction to a subspace preserved by the whole star algebra is a genuine
+star homomorphism (both the subspace and its orthogonal complement are reducing). -/
+def restrictHom (A : StarSubalgebra ℂ (Ambient (E := E))) (F : Submodule ℂ E)
+    (hF : ∀ a : A, ∀ x ∈ F, (a : Ambient (E := E)) x ∈ F) : A →⋆ₐ[ℂ] (F →L[ℂ] F) where
+  toFun a := (a : Ambient (E := E)).restrict (hF a)
+  map_zero' := by ext x; rfl
+  map_add' a b := by ext x; rfl
+  map_one' := by ext x; rfl
+  map_mul' a b := by ext x; rfl
+  commutes' c := by ext x; rfl
+  map_star' a := by
+    apply ContinuousLinearMap.ext
+    intro x
+    apply ext_inner_right ℂ
+    intro y
+    rw [ContinuousLinearMap.star_eq_adjoint, ContinuousLinearMap.adjoint_inner_left]
+    change inner ℂ ((star (a : Ambient (E := E))) (x : E)) (y : E) =
+      inner ℂ (x : E) ((a : Ambient (E := E)) (y : E))
+    rw [ContinuousLinearMap.star_eq_adjoint, ContinuousLinearMap.adjoint_inner_left]
+
+/-- Central support of a reducing physical subspace. Its algebraic kernel is
+removed, so no other unsupported sector can inflate the Fourier norm. -/
+theorem exists_central_support (A : StarSubalgebra ℂ (Ambient (E := E))) (F : Submodule ℂ E)
+    (hF : ∀ a : A, ∀ x ∈ F, (a : Ambient (E := E)) x ∈ F) :
+    ∃ q : A, star q = q ∧ q*q=q ∧ (∀ a, q*a=a*q) ∧
+      restrictHom A F hF q = 1 ∧
+      (∀ a : A, restrictHom A F hF a = 0 → q*a=0) := by
+  let : FiniteDimensional ℂ A := FiniteDimensional.of_injective
+    (algebraToEnd A) (algebraToEnd_injective A)
+  let φ := restrictHom A F hF
+  let K := LinearMap.ker φ.toLinearMap
+  have hk (a : A) : a ∈ K ↔ φ a = 0 := Iff.rfl
+  obtain ⟨q,hs,hqq,hc,hkq,hkill⟩ := complementary_central_projection (algebraCore A) K
+    (fun a x hx => by rw [hk, map_mul, (hk x).mp hx, mul_zero])
+    (fun a x hx => by rw [hk, map_mul, (hk x).mp hx, zero_mul])
+    (fun x hx => by rw [hk, map_star, (hk x).mp hx]; exact star_zero (F →L[ℂ] F))
+    (fun a x y => by simp only [core_inner_trace, star_mul, mul_assoc])
+    (fun a x y => by
+      simp only [core_inner_trace, star_mul]
+      rw [mul_assoc, physicalTrace_mul_comm A (star a) _, mul_assoc])
+  refine ⟨q,hs,hqq,hc,?_,fun a ha => (hkill a ((hk a).mpr ha)).1⟩
+  have hh := (hk (1-q)).mp hkq
+  have he : 1-φ q=0 := by simpa only [map_sub, map_one] using hh
+  exact (sub_eq_zero.mp he).symm
+
+end RowColumn.FiniteStarAlgebra
+noncomputable section
+namespace RowColumn.FiniteStarAlgebra
+variable {A B : Type*} [CStarAlgebra A] [CStarAlgebra B]
+
+def supportedAlgebra (q : A) (hs : star q = q) (hc : ∀ a, q*a=a*q) :
+    NonUnitalStarSubalgebra ℂ A where
+  carrier := {a | q*a=a}
+  zero_mem' := mul_zero q
+  add_mem' := by intro a b ha hb; change q*(a+b)=a+b; rw [mul_add, ha, hb]
+  mul_mem' := by intro a b ha hb; change q*(a*b)=a*b; rw [← mul_assoc, ha]
+  smul_mem' := by intro c a ha; change q*(c • a)=c • a; rw [mul_smul_comm, ha]
+  star_mem' := by
+    intro a ha
+    change q*star a=star a
+    rw [hc]
+    simpa only [star_mul, hs] using congrArg star ha
+
+lemma supportedAlgebra_closed (q : A) (hs : star q = q) (hc : ∀ a, q*a=a*q) :
+    IsClosed (supportedAlgebra q hs hc : Set A) :=
+  isClosed_eq (continuous_const.mul continuous_id) continuous_id
+
+/-- Removing the kernel by its central support makes restriction isometric.
+This avoids choosing irreducible compact-group carriers or using unproved
+Schur--Weyl decompositions. -/
+theorem norm_supported_eq (φ : A →⋆ₐ[ℂ] B) (q : A) (hs : star q = q)
+    (hqq : q*q=q) (hc : ∀ a, q*a=a*q) (hq : φ q=1)
+    (hkill : ∀ a, φ a=0 → q*a=0) (a : A) : ‖q*a‖ = ‖φ a‖ := by
+  let J := supportedAlgebra q hs hc
+  let : IsClosed (J : Set A) := supportedAlgebra_closed q hs hc
+  let ψ : J →⋆ₙₐ[ℂ] B := φ.toNonUnitalStarAlgHom.comp (NonUnitalStarSubalgebraClass.subtype J)
+  have hinj : Function.Injective ψ := by
+    intro x y h
+    apply Subtype.ext
+    have he : φ ((x : A)-(y : A))=0 := by
+      rw [map_sub]
+      exact sub_eq_zero.mpr h
+    have hz := hkill _ he
+    change q*((x : A)-(y : A))=0 at hz
+    rw [mul_sub, x.property, y.property] at hz
+    exact sub_eq_zero.mp hz
+  let x : J := ⟨q*a, by change q*(q*a)=q*a; rw [← mul_assoc, hqq]⟩
+  have hh := NonUnitalStarAlgHom.norm_map ψ hinj x
+  change ‖φ (q*a)‖ = ‖q*a‖ at hh
+  rw [map_mul, hq, one_mul] at hh
+  exact hh.symm
+
+end RowColumn.FiniteStarAlgebra
+noncomputable section
+open scoped BigOperators ComplexConjugate
+namespace RowColumn.Isotypic
+variable {G V E : Type*} [Group G]
+  [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
+  [NormedAddCommGroup E] [InnerProductSpace ℂ E] [FiniteDimensional ℂ E]
+  (ρ : Representation ℂ G V) (τ : Representation ℂ G E)
+
+def carrier : Submodule ℂ E := Submodule.span ℂ
+  (Set.range fun p : Representation.IntertwiningMap ρ τ × V => p.1 p.2)
+
+omit [FiniteDimensional ℂ V] [FiniteDimensional ℂ E] in
+lemma mem_carrier (f : Representation.IntertwiningMap ρ τ) (v : V) : f v ∈ carrier ρ τ :=
+  Submodule.subset_span ⟨(f,v),rfl⟩
+
+lemma projection_intertwines (F : Submodule ℂ E)
+    (hF : ∀ g x, x ∈ F → τ g x ∈ F)
+    (hτ : ∀ g x y, inner ℂ (τ g x) (τ g y) = inner ℂ x y) :
+    ∀ g x, F.starProjection (τ g x) = τ g (F.starProjection x) := by
+  intro g x
+  let e : E →ₗᵢ[ℂ] E := (τ g).isometryOfInner (hτ g)
+  have hmap : F.map e.toLinearMap = F := by
+    apply le_antisymm
+    · rintro _ ⟨y,hy,rfl⟩
+      exact hF g y hy
+    · intro y hy
+      refine ⟨τ g⁻¹ y, hF g⁻¹ y hy, ?_⟩
+      change (τ g * τ g⁻¹) y = y
+      rw [← map_mul, mul_inv_cancel, map_one]
+      rfl
+  have he := e.map_starProjection F x
+  simpa only [hmap, e, LinearMap.coe_isometryOfInner] using he.symm
+
+omit [FiniteDimensional ℂ V] in
+/-- On an isotypic unitary representation, every eigenspace of an operator in
+the group commutant has dimension at least that of the irreducible type. -/
+theorem eigenspace_finrank [Representation.IsIrreducible ρ]
+    (hspan : carrier ρ τ = ⊤)
+    (hτ : ∀ g x y, inner ℂ (τ g x) (τ g y) = inner ℂ x y)
+    (T : E →ₗ[ℂ] E) (hT : ∀ g x, T (τ g x) = τ g (T x))
+    (μ : ℂ) (hμ : Module.End.HasEigenvalue T μ) :
+    Module.finrank ℂ V ≤ Module.finrank ℂ (Module.End.eigenspace T μ) := by
+  classical
+  let F := Module.End.eigenspace T μ
+  have hf (g : G) (x : E) (hx : x ∈ F) : τ g x ∈ F := by
+    rw [Module.End.mem_eigenspace_iff] at hx ⊢
+    rw [hT, hx, map_smul]
+  let P : Representation.IntertwiningMap τ τ :=
+    { toLinearMap := F.starProjection.toLinearMap
+      isIntertwining' g := by ext x; exact projection_intertwines τ F hf hτ g x }
+  have hex : ∃ f : Representation.IntertwiningMap ρ τ, P.comp f ≠ 0 := by
+    by_contra! h
+    have hz (x : E) (hx : x ∈ carrier ρ τ) : P x = 0 := by
+      induction hx using Submodule.span_induction with
+      | mem x hx =>
+        obtain ⟨⟨f,v⟩,rfl⟩ := hx
+        exact congrArg (fun L : Representation.IntertwiningMap ρ τ => L v) (h f)
+      | zero => exact map_zero P
+      | add x y hx hy hpx hpy => simp only [map_add, hpx, hpy, add_zero]
+      | smul c x hx hpx => simp only [map_smul, hpx, smul_zero]
+    obtain ⟨x,hx,hxn⟩ := hμ.exists_hasEigenvector
+    have he := hz x (by rw [hspan]; trivial)
+    change F.starProjection x = 0 at he
+    rw [F.starProjection_eq_self_iff.mpr hx] at he
+    exact hxn he
+  obtain ⟨f,hfne⟩ := hex
+  have hi : Function.Injective (P.comp f) :=
+    (Representation.IsIrreducible.injective_or_eq_zero (P.comp f)).resolve_right hfne
+  let L : V →ₗ[ℂ] F := (P.comp f).toLinearMap.codRestrict F (fun v =>
+    (F.orthogonalProjectionOnto (f v)).property)
+  apply LinearMap.finrank_le_finrank_of_injective (f := L)
+  intro x y h
+  exact hi (congrArg Subtype.val h)
+
+end RowColumn.Isotypic
+namespace RowColumn.Isotypic
+variable {G V E : Type*} [Group G]
+  [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
+  [NormedAddCommGroup E] [InnerProductSpace ℂ E] [FiniteDimensional ℂ E]
+  (ρ : Representation ℂ G V) (τ : Representation ℂ G E)
+
+omit [FiniteDimensional ℂ V] [FiniteDimensional ℂ E] in
+lemma carrier_invariant (g : G) (x : E) (hx : x ∈ carrier ρ τ) : τ g x ∈ carrier ρ τ := by
+  induction hx using Submodule.span_induction with
+  | mem x hx =>
+    obtain ⟨⟨f,v⟩,rfl⟩ := hx
+    rw [← Representation.IntertwiningMap.isIntertwining ρ τ f]
+    exact mem_carrier ρ τ f _
+  | zero => simpa only [map_zero] using (carrier ρ τ).zero_mem
+  | add x y hx hy hx' hy' => simpa only [map_add] using (carrier ρ τ).add_mem hx' hy'
+  | smul c x hx hx' => simpa only [map_smul] using (carrier ρ τ).smul_mem c hx'
+
+omit [FiniteDimensional ℂ V] [FiniteDimensional ℂ E] in
+lemma carrier_commutant (T : E →ₗ[ℂ] E) (hT : ∀ g x, T (τ g x) = τ g (T x))
+    (x : E) (hx : x ∈ carrier ρ τ) : T x ∈ carrier ρ τ := by
+  let t : Representation.IntertwiningMap τ τ :=
+    { toLinearMap := T
+      isIntertwining' g := by ext x; exact hT g x }
+  induction hx using Submodule.span_induction with
+  | mem x hx =>
+    obtain ⟨⟨f,v⟩,rfl⟩ := hx
+    exact mem_carrier ρ τ (t.comp f) v
+  | zero => simpa only [map_zero] using (carrier ρ τ).zero_mem
+  | add x y hx hy hx' hy' => simpa only [map_add] using (carrier ρ τ).add_mem hx' hy'
+  | smul c x hx hx' => simpa only [map_smul] using (carrier ρ τ).smul_mem c hx'
+
+def restricted : Representation ℂ G (carrier ρ τ) where
+  toFun g := (τ g).restrict (carrier_invariant ρ τ g)
+  map_one' := by ext x; exact LinearMap.congr_fun (map_one τ) x
+  map_mul' g h := by ext x; exact LinearMap.congr_fun (map_mul τ g h) x
+
+def restrictIntertwiner (f : Representation.IntertwiningMap ρ τ) :
+    Representation.IntertwiningMap ρ (restricted ρ τ) where
+  toLinearMap := f.toLinearMap.codRestrict (carrier ρ τ) (mem_carrier ρ τ f)
+  isIntertwining' g := by
+    ext v
+    exact Representation.IntertwiningMap.isIntertwining ρ τ f g v
+
+omit [FiniteDimensional ℂ V] [FiniteDimensional ℂ E] in
+lemma restricted_unitary (hτ : ∀ g x y, inner ℂ (τ g x) (τ g y) = inner ℂ x y) :
+    ∀ g (x y : carrier ρ τ), inner ℂ (restricted ρ τ g x) (restricted ρ τ g y) = inner ℂ x y := by
+  intro g x y
+  exact hτ g x y
+
+omit [FiniteDimensional ℂ V] [FiniteDimensional ℂ E] in
+lemma restricted_isotypic : carrier ρ (restricted ρ τ) = ⊤ := by
+  let F := carrier ρ τ
+  let K := (carrier ρ (restricted ρ τ)).map F.subtype
+  have hFK : F ≤ K := by
+    apply Submodule.span_le.mpr
+    rintro _ ⟨⟨f,v⟩,rfl⟩
+    exact ⟨restrictIntertwiner ρ τ f v, mem_carrier ρ (restricted ρ τ)
+      (restrictIntertwiner ρ τ f) v, rfl⟩
+  apply top_unique
+  intro x _
+  obtain ⟨y,hy,hxy⟩ := hFK x.property
+  have he : y=x := Subtype.ext hxy
+  exact he ▸ hy
+
+end RowColumn.Isotypic
+noncomputable section
+open scoped BigOperators ComplexOrder
+namespace RowColumn.SpectralMultiplicity
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
+  [FiniteDimensional ℂ E]
+
+lemma positive_norm_le_of_eigenvalues (T : E →L[ℂ] E) (hT : T.IsPositive)
+    (c : ℝ) (hc : 0 ≤ c)
+    (hval : ∀ i, hT.toLinearMap.isSymmetric.eigenvalues rfl i ≤ c) : ‖T‖ ≤ c := by
+  let b := hT.toLinearMap.isSymmetric.eigenvectorBasis rfl
+  apply T.opNorm_le_bound hc
+  intro x
+  apply le_of_sq_le_sq _ (mul_nonneg hc (norm_nonneg _))
+  rw [← b.repr.norm_map (T x), ← b.repr.norm_map x, mul_pow]
+  rw [EuclideanSpace.norm_sq_eq, EuclideanSpace.norm_sq_eq, Finset.mul_sum]
+  apply Finset.sum_le_sum
+  intro i _
+  have hi := hT.toLinearMap.nonneg_eigenvalues rfl i
+  have he := hT.toLinearMap.isSymmetric.eigenvectorBasis_apply_self_apply rfl x i
+  change (b.repr (T x)) i = _ at he
+  change ‖(b.repr (T x)) i‖^2 ≤ _
+  rw [he, norm_mul, RCLike.norm_ofReal, abs_of_nonneg hi, mul_pow]
+  exact mul_le_mul_of_nonneg_right (sq_le_sq₀ hi hc |>.mpr (hval i)) (sq_nonneg _)
+
+/-- An eigenvalue supported on at least d dimensions pays a trace factor d.
+This is the precise spectral normalization needed for the global filter. -/
+theorem norm_le_trace_div (T : E →L[ℂ] E) (hT : T.IsPositive) (d : ℕ) (hd : 0 < d)
+    (hmult : ∀ μ : ℂ, Module.End.HasEigenvalue T.toLinearMap μ →
+      d ≤ Module.finrank ℂ (Module.End.eigenspace T.toLinearMap μ)) :
+    ‖T‖ ≤ (LinearMap.trace ℂ E T.toLinearMap).re / d := by
+  have htrace : 0 ≤ (LinearMap.trace ℂ E T.toLinearMap).re :=
+    (Complex.nonneg_iff.mp hT.toLinearMap.trace_nonneg).1
+  apply positive_norm_le_of_eigenvalues T hT _ (div_nonneg htrace (by positivity))
+  intro i
+  let μ := hT.toLinearMap.isSymmetric.eigenvalues rfl i
+  have hμ : 0 ≤ μ := hT.toLinearMap.nonneg_eigenvalues rfl i
+  have hcount := hmult (μ : ℂ) (hT.toLinearMap.isSymmetric.hasEigenvalue_eigenvalues rfl i)
+  have hcard := hT.toLinearMap.isSymmetric.card_filter_eigenvalues_eq rfl (μ : ℂ)
+  rw [← hcard] at hcount
+  simp only [RCLike.ofReal_eq_complex_ofReal, Complex.ofReal_inj] at hcount
+  apply (le_div_iff₀ (by exact_mod_cast hd)).mpr
+  calc
+    μ * (d : ℝ) ≤ μ * ((Finset.univ.filter fun j =>
+      hT.toLinearMap.isSymmetric.eigenvalues rfl j = μ).card : ℕ) :=
+        mul_le_mul_of_nonneg_left (by exact_mod_cast hcount) hμ
+    _ = ∑ j ∈ Finset.univ.filter (fun j =>
+      hT.toLinearMap.isSymmetric.eigenvalues rfl j = μ),
+        hT.toLinearMap.isSymmetric.eigenvalues rfl j := by
+      symm
+      calc
+        _ = ∑ j ∈ Finset.univ.filter (fun j =>
+          hT.toLinearMap.isSymmetric.eigenvalues rfl j = μ), μ :=
+            Finset.sum_congr rfl (fun j hj => (Finset.mem_filter.mp hj).2)
+        _ = _ := by simp [nsmul_eq_mul, mul_comm]
+    _ ≤ ∑ j, hT.toLinearMap.isSymmetric.eigenvalues rfl j :=
+      Finset.sum_le_sum_of_subset_of_nonneg (Finset.filter_subset _ _)
+        (fun j _ _ => hT.toLinearMap.nonneg_eigenvalues rfl j)
+    _ = _ := (hT.toLinearMap.isSymmetric.re_trace_eq_sum_eigenvalues rfl).symm
+
+end RowColumn.SpectralMultiplicity
+noncomputable section
+open scoped ComplexOrder
+namespace RowColumn.SpectralMultiplicity
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
+  [FiniteDimensional ℂ E]
+
+lemma trace_compression (T : E →L[ℂ] E) (F : Submodule ℂ E) :
+    LinearMap.trace ℂ F (F.orthogonalProjectionOnto ∘L T ∘L F.subtypeL).toLinearMap =
+      LinearMap.trace ℂ E (T ∘L F.starProjection).toLinearMap := by
+  have he := LinearMap.trace_comp_comm' (T.toLinearMap ∘ₗ F.subtype)
+    F.orthogonalProjectionOnto.toLinearMap
+  exact he
+
+lemma trace_compression_le (T : E →L[ℂ] E) (hT : T.IsPositive)
+    (F : Submodule ℂ E) :
+    (LinearMap.trace ℂ F (F.orthogonalProjectionOnto ∘L T ∘L F.subtypeL).toLinearMap).re ≤
+      (LinearMap.trace ℂ E T.toLinearMap).re := by
+  have hp := (hT.orthogonalProjectionOnto_comp Fᗮ).toLinearMap.trace_nonneg
+  have hn := (Complex.nonneg_iff.mp hp).1
+  rw [trace_compression] at hn ⊢
+  have he : F.starProjection + Fᗮ.starProjection = (1 : E →L[ℂ] E) := by
+    ext x
+    exact F.starProjection_add_starProjection_orthogonal x
+  have ht : T ∘L F.starProjection + T ∘L Fᗮ.starProjection = T := by
+    rw [← ContinuousLinearMap.comp_add, he]
+    rfl
+  have htr := congrArg (fun S : E →L[ℂ] E => (LinearMap.trace ℂ E S.toLinearMap).re) ht
+  simp only [ContinuousLinearMap.toLinearMap_add, map_add, Complex.add_re] at htr
+  linarith
+
+lemma restrict_positive (T : E →L[ℂ] E) (hT : T.IsPositive)
+    (F : Submodule ℂ E) (hF : ∀ x ∈ F, T x ∈ F) : (T.restrict hF).IsPositive := by
+  have he : T.restrict hF = F.orthogonalProjectionOnto ∘L T ∘L F.subtypeL := by
+    ext x
+    change T x = F.starProjection (T x)
+    exact (F.starProjection_eq_self_iff.mpr (hF x x.property)).symm
+  rw [he]
+  exact hT.orthogonalProjectionOnto_comp F
+
+lemma trace_restrict_le (T : E →L[ℂ] E) (hT : T.IsPositive)
+    (F : Submodule ℂ E) (hF : ∀ x ∈ F, T x ∈ F) :
+    (LinearMap.trace ℂ F (T.restrict hF).toLinearMap).re ≤
+      (LinearMap.trace ℂ E T.toLinearMap).re := by
+  have he : T.restrict hF = F.orthogonalProjectionOnto ∘L T ∘L F.subtypeL := by
+    ext x
+    change T x = F.starProjection (T x)
+    exact (F.starProjection_eq_self_iff.mpr (hF x x.property)).symm
+  rw [he]
+  exact trace_compression_le T hT F
+
+end RowColumn.SpectralMultiplicity
+
+end
+end
+end
+end
+end
+end
+end
+
+
+section
+noncomputable section
+open scoped ComplexOrder
+namespace RowColumn.FiniteStarAlgebra
+variable {G V E : Type*} [Group G]
+  [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
+  [NormedAddCommGroup E] [InnerProductSpace ℂ E] [FiniteDimensional ℂ E]
+
+instance algebraFiniteDimensional (A : StarSubalgebra ℂ (E →L[ℂ] E)) : FiniteDimensional ℂ A :=
+  FiniteDimensional.of_injective (algebraToEnd A) (algebraToEnd_injective A)
+
+instance algebraIsClosed (A : StarSubalgebra ℂ (E →L[ℂ] E)) : IsClosed (A : Set (E →L[ℂ] E)) :=
+  A.toSubalgebra.toSubmodule.closed_of_finiteDimensional
+
+lemma supported_restriction_norm (A : StarSubalgebra ℂ (E →L[ℂ] E)) (F : Submodule ℂ E)
+    (φ : A →⋆ₐ[ℂ] (F →L[ℂ] F))
+    (q : A) (hs : star q=q) (hqq : q*q=q) (hc : ∀ a, q*a=a*q)
+    (hq : φ q=1) (hkill : ∀ a : A, φ a=0 → q*a=0) (a : A) :
+    ‖q*a‖ = ‖φ a‖ := by
+  let J : NonUnitalStarSubalgebra ℂ A :=
+    { carrier := {x | q*x=x}
+      zero_mem' := mul_zero q
+      add_mem' := by intro x y hx hy; change q*(x+y)=x+y; rw [mul_add,hx,hy]
+      mul_mem' := by intro x y hx hy; change q*(x*y)=x*y; rw [← mul_assoc,hx]
+      smul_mem' := by intro c x hx; change q*(c • x)=c • x; rw [mul_smul_comm,hx]
+      star_mem' := by
+        intro x hx
+        change q*star x=star x
+        rw [hc]
+        simpa only [star_mul,hs] using congrArg star hx }
+  let : IsClosed (J : Set A) := isClosed_eq (continuous_const.mul continuous_id) continuous_id
+  let ψ : J →⋆ₙₐ[ℂ] (F →L[ℂ] F) := φ.toNonUnitalStarAlgHom.comp (NonUnitalStarSubalgebraClass.subtype J)
+  have hinj : Function.Injective ψ := by
+    intro x y h
+    apply Subtype.ext
+    have he : φ ((x : A)-(y : A))=0 := by
+      rw [map_sub]
+      exact sub_eq_zero.mpr h
+    have hz := hkill _ he
+    change q*((x : A)-(y : A))=0 at hz
+    rw [mul_sub,x.property,y.property] at hz
+    exact sub_eq_zero.mp hz
+  let x : J := ⟨q*a,by change q*(q*a)=q*a; rw [← mul_assoc,hqq]⟩
+  have hh := NonUnitalStarAlgHom.norm_map ψ hinj x
+  change ‖φ (q*a)‖ = ‖q*a‖ at hh
+  rw [map_mul,hq,one_mul] at hh
+  exact hh.symm
+
+/-- The exact dimension normalization of the occupied-board global filter.
+No highest-weight entropy or Schur--Weyl hypothesis is required: restriction
+of the supported algebra is faithful, and every eigenspace contains the
+irreducible carrier. -/
+theorem exists_isotypic_filter (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
+    (τ : Representation ℂ G E)
+    (hτ : ∀ g x y, inner ℂ (τ g x) (τ g y) = inner ℂ x y)
+    (A : StarSubalgebra ℂ (E →L[ℂ] E))
+    (hA : ∀ a : A, ∀ g x, (a : E →L[ℂ] E) (τ g x) = τ g ((a : E →L[ℂ] E) x)) :
+    ∃ q : A, star q = q ∧ q*q=q ∧ (∀ a, q*a=a*q) ∧
+      (∀ x ∈ Isotypic.carrier ρ τ, (q : E →L[ℂ] E) x = x) ∧
+      ∀ a : A, (a : E →L[ℂ] E).IsPositive →
+        ‖q*a‖ ≤ (physicalTrace A a).re / Module.finrank ℂ V := by
+  let : Nontrivial V := by
+    let := IsSimpleModule.nontrivial (MonoidAlgebra ℂ G) ρ.asModule
+    exact ρ.asModuleEquiv.symm.toEquiv.nontrivial
+  let F := Isotypic.carrier ρ τ
+  have hF : ∀ a : A, ∀ x ∈ F, (a : E →L[ℂ] E) x ∈ F := fun a =>
+    Isotypic.carrier_commutant ρ τ (a : E →L[ℂ] E).toLinearMap (hA a)
+  let φ := restrictHom A F hF
+  obtain ⟨q,hqs,hqq,hqc,hq,hkill⟩ := exists_central_support A F hF
+  refine ⟨q,hqs,hqq,hqc,?_,?_⟩
+  · intro x hx
+    exact congrArg Subtype.val (congrArg (fun T : F →L[ℂ] F => T ⟨x,hx⟩) hq)
+  · intro a ha
+    rw [supported_restriction_norm A F φ q hqs hqq hqc hq hkill a]
+    have hp : (φ a).IsPositive := SpectralMultiplicity.restrict_positive
+      (a : E →L[ℂ] E) ha F (hF a)
+    apply (SpectralMultiplicity.norm_le_trace_div (φ a) hp (Module.finrank ℂ V)
+      (Module.finrank_pos) ?_).trans
+    · apply div_le_div_of_nonneg_right _ (Nat.cast_nonneg _)
+      exact SpectralMultiplicity.trace_restrict_le (a : E →L[ℂ] E) ha F (hF a)
+    · intro μ hμ
+      apply Isotypic.eigenspace_finrank ρ (Isotypic.restricted ρ τ)
+        (Isotypic.restricted_isotypic ρ τ) (Isotypic.restricted_unitary ρ τ hτ)
+        (φ a).toLinearMap _ μ hμ
+      intro g x
+      apply Subtype.ext
+      exact hA a g x
+
+end RowColumn.FiniteStarAlgebra
+noncomputable section
+open scoped BigOperators ComplexOrder
+namespace RowColumn.UnitaryFrame
+variable {A G E : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A]
+  [FiniteDimensional ℂ A] [Group G]
+  [NormedAddCommGroup E] [InnerProductSpace ℂ E]
+
+/-- The finite Fourier reconstruction bound for arbitrary Hilbert-valued
+linear maps, in particular weighted Hilbert–Schmidt sandwich maps. -/
+theorem reconstruction_hilbert_bound (τ : A →ₗ[ℂ] ℂ) (core : InnerProductSpace.Core ℂ A)
+    (hinner : ∀ x y, core.inner x y = τ (star x*y))
+    (u : G →* unitary A)
+    (hspan : Submodule.span ℂ (Set.range fun g : G => (u g : A)) = ⊤)
+    (B : A) (F : A →ₗ[ℂ] E) (R : ℝ) (hR : 0 ≤ R)
+    (hU : ∀ g, ‖F (u g : A)‖ ≤ R) :
+    ‖F B‖ ≤ 4 * (Module.finrank ℂ A : ℝ) * ‖B‖ * R := by
+  let ℓ := (innerSL ℂ (F B)).toLinearMap.comp F
+  have hu (g : G) : ‖ℓ (u g : A)‖ ≤ ‖F B‖ * R :=
+    (norm_inner_le_norm (F B) (F (u g : A))).trans
+      (mul_le_mul_of_nonneg_left (hU g) (norm_nonneg _))
+  have hh := reconstruction_bound τ core hinner u hspan B ℓ (‖F B‖ * R)
+    (mul_nonneg (norm_nonneg _) hR) hu
+  change ‖inner ℂ (F B) (F B)‖ ≤ _ at hh
+  rw [inner_self_eq_norm_sq_to_K, norm_pow, RCLike.norm_ofReal,
+    abs_of_nonneg (norm_nonneg _)] at hh
+  by_cases hz : ‖F B‖=0
+  · rw [hz]
+    positivity
+  · have hp : 0 < ‖F B‖ := lt_of_le_of_ne (norm_nonneg _) (Ne.symm hz)
+    apply (mul_le_mul_iff_right₀ hp).mp
+    nlinarith [hh]
+
+end RowColumn.UnitaryFrame
+noncomputable section
+namespace RowColumn
+variable {A : Type*} [CStarAlgebra A]
+
+lemma supported_fourth (q F : A) (hq : q*q=q) (hc : Commute q F) :
+    (q*F)^4 = q*(F^4) := by
+  rw [hc.mul_pow]
+  have hq4 : q^4=q := by calc
+    q^4=(q*q)*(q*q) := by noncomm_ring
+    _=q := by rw [hq,hq]
+  rw [hq4]
+
+/-- The quarter-root estimate at the level of the genuine operator norm.
+The two fourth-moment bounds imply the exact squared sandwich bound. -/
+theorem filterNorm_sq (q F G : A) (hsq : IsSelfAdjoint q) (hqq : q*q=q)
+    (hF : IsSelfAdjoint F) (hG : IsSelfAdjoint G) (hcF : Commute q F) (hcG : Commute q G)
+    (d : ℝ) (_ : 0 < d) (hFb : ‖q * F^4‖ ≤ 1/d) (hGb : ‖q * G^4‖ ≤ 1/d) :
+    ‖F*q*G‖^2 ≤ 1/d := by
+  have hsF : IsSelfAdjoint (q*F) := by
+    change star (q*F)=q*F
+    rw [star_mul,hsq.star_eq,hF.star_eq,hcF.eq]
+  have hsG : IsSelfAdjoint (q*G) := by
+    change star (q*G)=q*G
+    rw [star_mul,hsq.star_eq,hG.star_eq,hcG.eq]
+  have hnF : ‖q*F‖^4 ≤ 1/d := by
+    have hn : ‖(q*F)^4‖=‖q*F‖^4 := by simpa using hsF.norm_pow_two_pow 2
+    rw [← hn,supported_fourth q F hqq hcF]
+    exact hFb
+  have hnG : ‖q*G‖^4 ≤ 1/d := by
+    have hn : ‖(q*G)^4‖=‖q*G‖^4 := by simpa using hsG.norm_pow_two_pow 2
+    rw [← hn,supported_fourth q G hqq hcG]
+    exact hGb
+  have he : F*q*G=(q*F)*(q*G) := by
+    calc
+      _ = F*(q*q)*G := by rw [hqq]
+      _ = (F*q)*(q*G) := by noncomm_ring
+      _ = _ := by rw [hcF.eq]
+  have hn := norm_mul_le (q*F) (q*G)
+  rw [← he] at hn
+  have hp : (‖q*F‖*‖q*G‖)^2 ≤ 1/d := by
+    have ha := sq_nonneg (‖q*F‖^2-‖q*G‖^2)
+    nlinarith
+  exact (pow_le_pow_left₀ (norm_nonneg _) hn 2).trans hp
+end RowColumn
+
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FiniteUnitary/Sampling.lean b/lean/OAI/RepresentationTheory/FiniteUnitary/Sampling.lean
index 717e623dc9c3d292b7e354f70b7888d4fdb8db3c..3639c8ade4b115537808118b03eeaa17e55f48a1 100644
--- a/lean/OAI/RepresentationTheory/FiniteUnitary/Sampling.lean
+++ b/lean/OAI/RepresentationTheory/FiniteUnitary/Sampling.lean
@@ -1,256 +1,256 @@
-import OAI.Probability.CubeShuffle.Basic
-import OAI.RepresentationTheory.FiniteUnitary.Averaging
-
-namespace OAI
-
-namespace CubeShuffle
-open scoped BigOperators Classical
-variable {Ω G : Type*} [Fintype Ω] [Fintype G]
-
-noncomputable def sampleLaw (P : Ω → G) (g : G) : ℝ :=
-  finiteMean (fun ω => if P ω=g then 1 else 0)
-
-omit [Fintype G] in
-lemma sampleLaw_nonneg (P : Ω → G) (g : G) : 0 ≤ sampleLaw P g :=
-  div_nonneg (Finset.sum_nonneg (fun _ _ => by split_ifs <;> norm_num)) (Nat.cast_nonneg _)
-
-lemma sampleLaw_sum [Nonempty Ω] (P : Ω → G) : ∑ g,sampleLaw P g=1 := by
-  unfold sampleLaw
-  rw [←finiteMean_sum]
-  simp [finiteMean, Fintype.card_ne_zero]
-
-omit [Fintype G] in
-lemma sampleLaw_pos_iff [Nonempty Ω] (P : Ω → G) (g : G) :
-    0<sampleLaw P g ↔ ∃ ω,P ω=g := by
-  unfold sampleLaw finiteMean
-  rw [div_pos_iff_of_pos_right (Nat.cast_pos.mpr Fintype.card_pos)]
-  rw [Finset.sum_pos_iff_of_nonneg (fun _ _ => by split_ifs <;> norm_num)]
-  simp only [Finset.mem_univ,true_and]
-  apply exists_congr
-  intro ω
-  split_ifs <;> simp_all
-
-omit [Fintype G] in
-lemma sampleLaw_equiv {Ω' : Type*} [Fintype Ω'] (P : Ω → G) (e : Ω' ≃ Ω) :
-    sampleLaw (P ∘ e)=sampleLaw P := by
-  funext g
-  simpa only [sampleLaw,Function.comp_apply,Equiv.apply_symm_apply] using
-    finiteMean_equiv e (fun ω => if P (e ω)=g then (1:ℝ) else 0)
-
-variable [Group G]
-omit [Fintype G] in
-lemma sampleLaw_inverse (P : Ω → G) (g : G) :
-    sampleLaw (fun ω => (P ω)⁻¹) g=sampleLaw P g⁻¹ := by
-  apply finiteMean_congr
-  intro ω
-  rw [inv_eq_iff_eq_inv]
-
-omit [Fintype G] in
-lemma sampleLaw_symmetric (P : Ω → G) (e : Equiv.Perm Ω)
-    (he : ∀ ω,P (e ω)=(P ω)⁻¹) (g : G) : sampleLaw P g⁻¹=sampleLaw P g := by
-  rw [←sampleLaw_inverse]
-  have hh : (fun ω => (P ω)⁻¹)=P ∘ e := funext (fun ω => (he ω).symm)
-  rw [hh,sampleLaw_equiv]
-
-end CubeShuffle
-
-namespace CubeShuffle.UnitaryFinite
-open scoped BigOperators Classical
-
-variable {G : Type*} [Group G] [Fintype G]
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-variable {Ω Ξ : Type*} [Fintype Ω] [Fintype Ξ]
-
-noncomputable def sampleOperator (ρ : Representation ℂ G V) (P : Ω → G) : V →L[ℂ] V :=
-  (Fintype.card Ω:ℂ)⁻¹ • ∑ ω,continuousRepresentation ρ (P ω)
-
-omit [Fintype G] in
-lemma continuousRepresentation_comp {H : Type*} [Group H]
-    (ρ : Representation ℂ G V) (φ : H →* G) (h : H) :
-    continuousRepresentation (ρ.comp φ) h=continuousRepresentation ρ (φ h) := rfl
-
-omit [Fintype G] in
-lemma sampleOperator_comp {H : Type*} [Group H]
-    (ρ : Representation ℂ G V) (φ : H →* G) (P : Ω → H) :
-    sampleOperator (ρ.comp φ) P=sampleOperator ρ (φ ∘ P) := rfl
-
-omit [Fintype G] [FiniteDimensional ℂ V] in
-lemma unitary_comp {H : Type*} [Group H]
-    (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (φ : H →* G) :
-    IsUnitary (ρ.comp φ) := fun h v w => hρ (φ h) v w
-
-omit [Fintype G] in
-lemma sampleOperator_commute (ρ : Representation ℂ G V) (P : Ω → G) (Q : Ξ → G)
-    (h : ∀ ω ξ, Commute (P ω) (Q ξ)) :
-    Commute (sampleOperator ρ P) (sampleOperator ρ Q) := by
-  unfold sampleOperator
-  apply Commute.smul_left
-  apply Commute.smul_right
-  apply Commute.sum_left
-  intro ω _
-  apply Commute.sum_right
-  intro ξ _
-  exact (h ω ξ).map (continuousRepresentation ρ)
-
-lemma sampleOperator_weighted (ρ : Representation ℂ G V) (P : Ω → G) :
-    sampleOperator ρ P=weightedOperator ρ (sampleLaw P) := by
-  unfold sampleOperator weightedOperator sampleLaw finiteMean
-  simp only [Complex.ofReal_sum,div_eq_mul_inv,Finset.sum_mul,Finset.sum_smul]
-  rw [Finset.sum_comm]
-  simp only [Finset.smul_sum]
-  apply Finset.sum_congr rfl
-  intro ω _
-  calc
-    _ = ∑ g, if P ω=g then (Fintype.card Ω:ℂ)⁻¹ • continuousRepresentation ρ g else 0 := by
-      symm
-      simp only [Finset.sum_ite_eq,Finset.mem_univ,↓reduceIte]
-    _ = _ := by
-      apply Finset.sum_congr rfl
-      intro g _
-      split_ifs <;> simp_all
-
-lemma sampleOperator_norm_le [Nonempty Ω] (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (P : Ω → G) :
-    ‖sampleOperator ρ P‖≤1 := by
-  rw [sampleOperator_weighted]
-  exact (weightedOperator_norm_le ρ hρ _ (sampleLaw_nonneg _)).trans_eq (sampleLaw_sum P)
-
-omit [Fintype G] in
-lemma sampleOperator_equiv (ρ : Representation ℂ G V) (P : Ω → G) (e : Ξ ≃ Ω) :
-    sampleOperator ρ (P ∘ e)=sampleOperator ρ P := by
-  simp only [sampleOperator,Fintype.card_congr e,Function.comp_apply]
-  rw [Equiv.sum_comp e (fun ω => continuousRepresentation ρ (P ω))]
-
-omit [Fintype G] in
-lemma sampleOperator_prod (ρ : Representation ℂ G V) (P : Ω → G) (Q : Ξ → G) :
-    sampleOperator ρ (fun ω : Ω × Ξ => P ω.1*Q ω.2)=sampleOperator ρ P*sampleOperator ρ Q := by
-  simp only [sampleOperator,Fintype.card_prod,Nat.cast_mul,mul_inv_rev,map_mul,Fintype.sum_prod_type,
-    Finset.sum_mul,Finset.mul_sum,smul_mul_smul,Finset.smul_sum]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro ω _
-  apply Finset.sum_congr rfl
-  intro ξ _
-  rw [mul_comm]
-
-omit [Fintype G] in
-lemma sampleOperator_inv (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (P : Ω → G) :
-    sampleOperator ρ (fun ω => (P ω)⁻¹)=(sampleOperator ρ P).adjoint := by
-  simp only [sampleOperator,map_smulₛₗ,map_sum,map_inv₀,map_natCast,
-    continuousRepresentation_adjoint ρ hρ]
-
-omit [Fintype G] in
-lemma sampleOperator_one [Nonempty Ω] (ρ : Representation ℂ G V) :
-    sampleOperator ρ (fun _ : Ω => 1)=1 := by
-  simp only [sampleOperator,map_one,Finset.sum_const,Finset.card_univ,←Nat.cast_smul_eq_nsmul ℂ,
-    smul_smul]
-  rw [inv_mul_cancel₀ (by exact_mod_cast Fintype.card_ne_zero),one_smul]
-
-lemma sampleOperator_uniform (ρ : Representation ℂ G V) : sampleOperator ρ (id : G → G)=uniformOperator ρ := by
-  simp only [sampleOperator,uniformOperator,weightedOperator,Complex.ofReal_inv,Complex.ofReal_natCast,
-    Finset.smul_sum,id_eq]
-
-lemma sampleOperator_uniform_mul [Nonempty Ω] (ρ : Representation ℂ G V) (P : Ω → G) :
-    sampleOperator ρ (fun ω : Ω × G => P ω.1*ω.2⁻¹)=uniformOperator ρ := by
-  rw [sampleOperator_prod]
-  have he : sampleOperator ρ (fun g : G => g⁻¹)=uniformOperator ρ := by
-    rw [←sampleOperator_uniform]
-    exact sampleOperator_equiv ρ id (Equiv.inv G)
-  rw [he,sampleOperator_weighted,weighted_uniform_left,sampleLaw_sum,one_smul]
-
-end CubeShuffle.UnitaryFinite
-
-namespace CubeShuffle.UnitaryFinite
-open scoped BigOperators ComplexConjugate Classical
-
-variable {G : Type*} [Group G] [Fintype G]
-variable {V W : Type*} [NormedAddCommGroup V] [NormedAddCommGroup W]
-  [InnerProductSpace ℂ V] [InnerProductSpace ℂ W]
-  [FiniteDimensional ℂ V] [FiniteDimensional ℂ W]
-
-omit [FiniteDimensional ℂ V] [FiniteDimensional ℂ W] in
-lemma averageMap_zero (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
-    [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ]
-    (hne : ¬Nonempty (Representation.Equiv ρ σ)) (A : V →ₗ[ℂ] W) :
-    averageMap ρ σ A=0 := by
-  rcases Representation.IsIrreducible.bijective_or_eq_zero (averageMap ρ σ A) with h|h
-  · exact (hne ⟨Representation.IntertwiningMap.ofBijective _ h⟩).elim
-  · exact h
-
-omit [FiniteDimensional ℂ V] [FiniteDimensional ℂ W] in
-lemma coefficient_cross_orthogonality (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
-    [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ]
-    (hρ : IsUnitary ρ) (hne : ¬Nonempty (Representation.Equiv ρ σ))
-    (u z : W) (v w : V) :
-    ∑ g, inner ℂ u (σ g z)*inner ℂ (ρ g w) v=0 := by
-  have he := averageMap_zero ρ σ hne (InnerProductSpace.rankOne ℂ z w).toLinearMap
-  have hv := congrArg (fun T : Representation.IntertwiningMap ρ σ => inner ℂ u (T v)) he
-  change inner ℂ u ((∑ g, (σ g).comp ((InnerProductSpace.rankOne ℂ z w).toLinearMap.comp (ρ g⁻¹))) v)=_ at hv
-  simpa only [LinearMap.sum_apply,LinearMap.comp_apply,ContinuousLinearMap.coe_coe,
-    InnerProductSpace.rankOne_apply,map_smul,unitary_inner_inv ρ hρ,inner_sum,
-    inner_smul_right,Representation.IntertwiningMap.coe_zero,Pi.zero_apply,inner_zero_right,mul_comm] using hv
-
-end CubeShuffle.UnitaryFinite
-namespace CubeShuffle.UnitaryFinite
-open scoped BigOperators ComplexConjugate Classical
-
-variable {G : Type*} [Group G] [Fintype G]
-
-def regularAction : G →* Equiv.Perm G where
-  toFun := Equiv.mulLeft
-  map_one' := by ext; simp
-  map_mul' _ _ := by ext; simp
-
-omit [Fintype G] in
-lemma regular_transitive (x : G) : ∃ g,regularAction g (1:G)=x := ⟨x,mul_one x⟩
-
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-
-omit [Fintype G] [FiniteDimensional ℂ V] in
-lemma regular_fixed (ρ : Representation ℂ G V) (w : V) : IsFixed regularAction (1:G) ρ w := by
-  intro g hg
-  have he : g=1 := by simpa only [regularAction,MonoidHom.coe_mk,OneHom.coe_mk,Equiv.coe_mulLeft,mul_one] using hg
-  subst g
-  simp
-
-noncomputable def regularCoefficient (ρ : Representation ℂ G V) (w v : V) : EuclideanSpace ℂ G :=
-  normalizedCoefficient regularAction 1 regular_transitive ρ w v
-
-omit [FiniteDimensional ℂ V] in
-lemma regularCoefficient_apply (ρ : Representation ℂ G V) (w v : V) (g : G) :
-    regularCoefficient ρ w v g = (Real.sqrt ((Module.finrank ℂ V:ℝ)/(Fintype.card G:ℝ)):ℂ)*inner ℂ (ρ g w) v := by
-  unfold regularCoefficient normalizedCoefficient
-  simp only [PiLp.smul_apply,smul_eq_mul]
-  congr 1
-  have he := coefficient_at regularAction 1 regular_transitive ρ w (regular_fixed ρ w) v g
-  simpa only [regularAction,MonoidHom.coe_mk,OneHom.coe_mk,Equiv.coe_mulLeft,mul_one] using he
-
-lemma regularCoefficient_inner (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
-    (hρ : IsUnitary ρ) (w z v t : V) :
-    inner ℂ (regularCoefficient ρ w v) (regularCoefficient ρ z t)=inner ℂ z w*inner ℂ v t :=
-  normalizedCoefficient_inner regularAction 1 regular_transitive ρ hρ w z
-    (regular_fixed ρ w) (regular_fixed ρ z) v t
-
-variable {W : Type*} [NormedAddCommGroup W] [InnerProductSpace ℂ W] [FiniteDimensional ℂ W]
-
-omit [FiniteDimensional ℂ V] [FiniteDimensional ℂ W] in
-lemma regularCoefficient_cross (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
-    [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ]
-    (hσ : IsUnitary σ) (hne : ¬Nonempty (Representation.Equiv σ ρ))
-    (w v : V) (z t : W) : inner ℂ (regularCoefficient ρ w v) (regularCoefficient σ z t)=0 := by
-  rw [PiLp.inner_apply]
-  simp only [RCLike.inner_apply,regularCoefficient_apply,map_mul,Complex.conj_ofReal,
-    inner_conj_symm]
-  have he := coefficient_cross_orthogonality σ ρ hσ hne v w t z
-  calc
-    _ = ((Real.sqrt ((Module.finrank ℂ V:ℝ)/(Fintype.card G:ℝ)):ℂ)*
-        (Real.sqrt ((Module.finrank ℂ W:ℝ)/(Fintype.card G:ℝ)):ℂ)) *
-        ∑ g,inner ℂ v (ρ g w)*inner ℂ (σ g z) t := by
-      rw [Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro g _
-      ring
-    _ = 0 := by rw [he,mul_zero]
-
-end CubeShuffle.UnitaryFinite
-
-end OAI
+import OAI.Probability.CubeShuffle.Basic
+import OAI.RepresentationTheory.FiniteUnitary.Averaging
+
+namespace OAI
+
+namespace CubeShuffle
+open scoped BigOperators Classical
+variable {Ω G : Type*} [Fintype Ω] [Fintype G]
+
+noncomputable def sampleLaw (P : Ω → G) (g : G) : ℝ :=
+  finiteMean (fun ω => if P ω=g then 1 else 0)
+
+omit [Fintype G] in
+lemma sampleLaw_nonneg (P : Ω → G) (g : G) : 0 ≤ sampleLaw P g :=
+  div_nonneg (Finset.sum_nonneg (fun _ _ => by split_ifs <;> norm_num)) (Nat.cast_nonneg _)
+
+lemma sampleLaw_sum [Nonempty Ω] (P : Ω → G) : ∑ g,sampleLaw P g=1 := by
+  unfold sampleLaw
+  rw [←finiteMean_sum]
+  simp [finiteMean, Fintype.card_ne_zero]
+
+omit [Fintype G] in
+lemma sampleLaw_pos_iff [Nonempty Ω] (P : Ω → G) (g : G) :
+    0<sampleLaw P g ↔ ∃ ω,P ω=g := by
+  unfold sampleLaw finiteMean
+  rw [div_pos_iff_of_pos_right (Nat.cast_pos.mpr Fintype.card_pos)]
+  rw [Finset.sum_pos_iff_of_nonneg (fun _ _ => by split_ifs <;> norm_num)]
+  simp only [Finset.mem_univ,true_and]
+  apply exists_congr
+  intro ω
+  split_ifs <;> simp_all
+
+omit [Fintype G] in
+lemma sampleLaw_equiv {Ω' : Type*} [Fintype Ω'] (P : Ω → G) (e : Ω' ≃ Ω) :
+    sampleLaw (P ∘ e)=sampleLaw P := by
+  funext g
+  simpa only [sampleLaw,Function.comp_apply,Equiv.apply_symm_apply] using
+    finiteMean_equiv e (fun ω => if P (e ω)=g then (1:ℝ) else 0)
+
+variable [Group G]
+omit [Fintype G] in
+lemma sampleLaw_inverse (P : Ω → G) (g : G) :
+    sampleLaw (fun ω => (P ω)⁻¹) g=sampleLaw P g⁻¹ := by
+  apply finiteMean_congr
+  intro ω
+  rw [inv_eq_iff_eq_inv]
+
+omit [Fintype G] in
+lemma sampleLaw_symmetric (P : Ω → G) (e : Equiv.Perm Ω)
+    (he : ∀ ω,P (e ω)=(P ω)⁻¹) (g : G) : sampleLaw P g⁻¹=sampleLaw P g := by
+  rw [←sampleLaw_inverse]
+  have hh : (fun ω => (P ω)⁻¹)=P ∘ e := funext (fun ω => (he ω).symm)
+  rw [hh,sampleLaw_equiv]
+
+end CubeShuffle
+
+namespace CubeShuffle.UnitaryFinite
+open scoped BigOperators Classical
+
+variable {G : Type*} [Group G] [Fintype G]
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+variable {Ω Ξ : Type*} [Fintype Ω] [Fintype Ξ]
+
+noncomputable def sampleOperator (ρ : Representation ℂ G V) (P : Ω → G) : V →L[ℂ] V :=
+  (Fintype.card Ω:ℂ)⁻¹ • ∑ ω,continuousRepresentation ρ (P ω)
+
+omit [Fintype G] in
+lemma continuousRepresentation_comp {H : Type*} [Group H]
+    (ρ : Representation ℂ G V) (φ : H →* G) (h : H) :
+    continuousRepresentation (ρ.comp φ) h=continuousRepresentation ρ (φ h) := rfl
+
+omit [Fintype G] in
+lemma sampleOperator_comp {H : Type*} [Group H]
+    (ρ : Representation ℂ G V) (φ : H →* G) (P : Ω → H) :
+    sampleOperator (ρ.comp φ) P=sampleOperator ρ (φ ∘ P) := rfl
+
+omit [Fintype G] [FiniteDimensional ℂ V] in
+lemma unitary_comp {H : Type*} [Group H]
+    (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (φ : H →* G) :
+    IsUnitary (ρ.comp φ) := fun h v w => hρ (φ h) v w
+
+omit [Fintype G] in
+lemma sampleOperator_commute (ρ : Representation ℂ G V) (P : Ω → G) (Q : Ξ → G)
+    (h : ∀ ω ξ, Commute (P ω) (Q ξ)) :
+    Commute (sampleOperator ρ P) (sampleOperator ρ Q) := by
+  unfold sampleOperator
+  apply Commute.smul_left
+  apply Commute.smul_right
+  apply Commute.sum_left
+  intro ω _
+  apply Commute.sum_right
+  intro ξ _
+  exact (h ω ξ).map (continuousRepresentation ρ)
+
+lemma sampleOperator_weighted (ρ : Representation ℂ G V) (P : Ω → G) :
+    sampleOperator ρ P=weightedOperator ρ (sampleLaw P) := by
+  unfold sampleOperator weightedOperator sampleLaw finiteMean
+  simp only [Complex.ofReal_sum,div_eq_mul_inv,Finset.sum_mul,Finset.sum_smul]
+  rw [Finset.sum_comm]
+  simp only [Finset.smul_sum]
+  apply Finset.sum_congr rfl
+  intro ω _
+  calc
+    _ = ∑ g, if P ω=g then (Fintype.card Ω:ℂ)⁻¹ • continuousRepresentation ρ g else 0 := by
+      symm
+      simp only [Finset.sum_ite_eq,Finset.mem_univ,↓reduceIte]
+    _ = _ := by
+      apply Finset.sum_congr rfl
+      intro g _
+      split_ifs <;> simp_all
+
+lemma sampleOperator_norm_le [Nonempty Ω] (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (P : Ω → G) :
+    ‖sampleOperator ρ P‖≤1 := by
+  rw [sampleOperator_weighted]
+  exact (weightedOperator_norm_le ρ hρ _ (sampleLaw_nonneg _)).trans_eq (sampleLaw_sum P)
+
+omit [Fintype G] in
+lemma sampleOperator_equiv (ρ : Representation ℂ G V) (P : Ω → G) (e : Ξ ≃ Ω) :
+    sampleOperator ρ (P ∘ e)=sampleOperator ρ P := by
+  simp only [sampleOperator,Fintype.card_congr e,Function.comp_apply]
+  rw [Equiv.sum_comp e (fun ω => continuousRepresentation ρ (P ω))]
+
+omit [Fintype G] in
+lemma sampleOperator_prod (ρ : Representation ℂ G V) (P : Ω → G) (Q : Ξ → G) :
+    sampleOperator ρ (fun ω : Ω × Ξ => P ω.1*Q ω.2)=sampleOperator ρ P*sampleOperator ρ Q := by
+  simp only [sampleOperator,Fintype.card_prod,Nat.cast_mul,mul_inv_rev,map_mul,Fintype.sum_prod_type,
+    Finset.sum_mul,Finset.mul_sum,smul_mul_smul,Finset.smul_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro ω _
+  apply Finset.sum_congr rfl
+  intro ξ _
+  rw [mul_comm]
+
+omit [Fintype G] in
+lemma sampleOperator_inv (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (P : Ω → G) :
+    sampleOperator ρ (fun ω => (P ω)⁻¹)=(sampleOperator ρ P).adjoint := by
+  simp only [sampleOperator,map_smulₛₗ,map_sum,map_inv₀,map_natCast,
+    continuousRepresentation_adjoint ρ hρ]
+
+omit [Fintype G] in
+lemma sampleOperator_one [Nonempty Ω] (ρ : Representation ℂ G V) :
+    sampleOperator ρ (fun _ : Ω => 1)=1 := by
+  simp only [sampleOperator,map_one,Finset.sum_const,Finset.card_univ,←Nat.cast_smul_eq_nsmul ℂ,
+    smul_smul]
+  rw [inv_mul_cancel₀ (by exact_mod_cast Fintype.card_ne_zero),one_smul]
+
+lemma sampleOperator_uniform (ρ : Representation ℂ G V) : sampleOperator ρ (id : G → G)=uniformOperator ρ := by
+  simp only [sampleOperator,uniformOperator,weightedOperator,Complex.ofReal_inv,Complex.ofReal_natCast,
+    Finset.smul_sum,id_eq]
+
+lemma sampleOperator_uniform_mul [Nonempty Ω] (ρ : Representation ℂ G V) (P : Ω → G) :
+    sampleOperator ρ (fun ω : Ω × G => P ω.1*ω.2⁻¹)=uniformOperator ρ := by
+  rw [sampleOperator_prod]
+  have he : sampleOperator ρ (fun g : G => g⁻¹)=uniformOperator ρ := by
+    rw [←sampleOperator_uniform]
+    exact sampleOperator_equiv ρ id (Equiv.inv G)
+  rw [he,sampleOperator_weighted,weighted_uniform_left,sampleLaw_sum,one_smul]
+
+end CubeShuffle.UnitaryFinite
+
+namespace CubeShuffle.UnitaryFinite
+open scoped BigOperators ComplexConjugate Classical
+
+variable {G : Type*} [Group G] [Fintype G]
+variable {V W : Type*} [NormedAddCommGroup V] [NormedAddCommGroup W]
+  [InnerProductSpace ℂ V] [InnerProductSpace ℂ W]
+  [FiniteDimensional ℂ V] [FiniteDimensional ℂ W]
+
+omit [FiniteDimensional ℂ V] [FiniteDimensional ℂ W] in
+lemma averageMap_zero (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
+    [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ]
+    (hne : ¬Nonempty (Representation.Equiv ρ σ)) (A : V →ₗ[ℂ] W) :
+    averageMap ρ σ A=0 := by
+  rcases Representation.IsIrreducible.bijective_or_eq_zero (averageMap ρ σ A) with h|h
+  · exact (hne ⟨Representation.IntertwiningMap.ofBijective _ h⟩).elim
+  · exact h
+
+omit [FiniteDimensional ℂ V] [FiniteDimensional ℂ W] in
+lemma coefficient_cross_orthogonality (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
+    [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ]
+    (hρ : IsUnitary ρ) (hne : ¬Nonempty (Representation.Equiv ρ σ))
+    (u z : W) (v w : V) :
+    ∑ g, inner ℂ u (σ g z)*inner ℂ (ρ g w) v=0 := by
+  have he := averageMap_zero ρ σ hne (InnerProductSpace.rankOne ℂ z w).toLinearMap
+  have hv := congrArg (fun T : Representation.IntertwiningMap ρ σ => inner ℂ u (T v)) he
+  change inner ℂ u ((∑ g, (σ g).comp ((InnerProductSpace.rankOne ℂ z w).toLinearMap.comp (ρ g⁻¹))) v)=_ at hv
+  simpa only [LinearMap.sum_apply,LinearMap.comp_apply,ContinuousLinearMap.coe_coe,
+    InnerProductSpace.rankOne_apply,map_smul,unitary_inner_inv ρ hρ,inner_sum,
+    inner_smul_right,Representation.IntertwiningMap.coe_zero,Pi.zero_apply,inner_zero_right,mul_comm] using hv
+
+end CubeShuffle.UnitaryFinite
+namespace CubeShuffle.UnitaryFinite
+open scoped BigOperators ComplexConjugate Classical
+
+variable {G : Type*} [Group G] [Fintype G]
+
+def regularAction : G →* Equiv.Perm G where
+  toFun := Equiv.mulLeft
+  map_one' := by ext; simp
+  map_mul' _ _ := by ext; simp
+
+omit [Fintype G] in
+lemma regular_transitive (x : G) : ∃ g,regularAction g (1:G)=x := ⟨x,mul_one x⟩
+
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+
+omit [Fintype G] [FiniteDimensional ℂ V] in
+lemma regular_fixed (ρ : Representation ℂ G V) (w : V) : IsFixed regularAction (1:G) ρ w := by
+  intro g hg
+  have he : g=1 := by simpa only [regularAction,MonoidHom.coe_mk,OneHom.coe_mk,Equiv.coe_mulLeft,mul_one] using hg
+  subst g
+  simp
+
+noncomputable def regularCoefficient (ρ : Representation ℂ G V) (w v : V) : EuclideanSpace ℂ G :=
+  normalizedCoefficient regularAction 1 regular_transitive ρ w v
+
+omit [FiniteDimensional ℂ V] in
+lemma regularCoefficient_apply (ρ : Representation ℂ G V) (w v : V) (g : G) :
+    regularCoefficient ρ w v g = (Real.sqrt ((Module.finrank ℂ V:ℝ)/(Fintype.card G:ℝ)):ℂ)*inner ℂ (ρ g w) v := by
+  unfold regularCoefficient normalizedCoefficient
+  simp only [PiLp.smul_apply,smul_eq_mul]
+  congr 1
+  have he := coefficient_at regularAction 1 regular_transitive ρ w (regular_fixed ρ w) v g
+  simpa only [regularAction,MonoidHom.coe_mk,OneHom.coe_mk,Equiv.coe_mulLeft,mul_one] using he
+
+lemma regularCoefficient_inner (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
+    (hρ : IsUnitary ρ) (w z v t : V) :
+    inner ℂ (regularCoefficient ρ w v) (regularCoefficient ρ z t)=inner ℂ z w*inner ℂ v t :=
+  normalizedCoefficient_inner regularAction 1 regular_transitive ρ hρ w z
+    (regular_fixed ρ w) (regular_fixed ρ z) v t
+
+variable {W : Type*} [NormedAddCommGroup W] [InnerProductSpace ℂ W] [FiniteDimensional ℂ W]
+
+omit [FiniteDimensional ℂ V] [FiniteDimensional ℂ W] in
+lemma regularCoefficient_cross (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
+    [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ]
+    (hσ : IsUnitary σ) (hne : ¬Nonempty (Representation.Equiv σ ρ))
+    (w v : V) (z t : W) : inner ℂ (regularCoefficient ρ w v) (regularCoefficient σ z t)=0 := by
+  rw [PiLp.inner_apply]
+  simp only [RCLike.inner_apply,regularCoefficient_apply,map_mul,Complex.conj_ofReal,
+    inner_conj_symm]
+  have he := coefficient_cross_orthogonality σ ρ hσ hne v w t z
+  calc
+    _ = ((Real.sqrt ((Module.finrank ℂ V:ℝ)/(Fintype.card G:ℝ)):ℂ)*
+        (Real.sqrt ((Module.finrank ℂ W:ℝ)/(Fintype.card G:ℝ)):ℂ)) *
+        ∑ g,inner ℂ v (ρ g w)*inner ℂ (σ g z) t := by
+      rw [Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro g _
+      ring
+    _ = 0 := by rw [he,mul_zero]
+
+end CubeShuffle.UnitaryFinite
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FiniteUnitary/Uncertainty.lean b/lean/OAI/RepresentationTheory/FiniteUnitary/Uncertainty.lean
index 37d6fa90a58dfcebf2b983deafb89e5973f13c59..0ec94a5b2df201e4f687f2a23008cba9e6d467ff 100644
--- a/lean/OAI/RepresentationTheory/FiniteUnitary/Uncertainty.lean
+++ b/lean/OAI/RepresentationTheory/FiniteUnitary/Uncertainty.lean
@@ -1,139 +1,139 @@
-import Mathlib
-
-namespace OAI
-
-namespace CubeShuffle.PermutationHilbert
-open scoped BigOperators ComplexConjugate Classical
-open Complex
-
-variable {X : Type*} [Fintype X]
-abbrev H (X : Type*) [Fintype X] := EuclideanSpace ℂ X
-
-noncomputable def act (p : Equiv.Perm X) : H X ≃ₗᵢ[ℂ] H X :=
-  LinearIsometryEquiv.piLpCongrLeft 2 ℂ ℂ p
-
-@[simp] lemma act_apply (p : Equiv.Perm X) (v : H X) (x : X) :
-    act p v x = v (p⁻¹ x) := rfl
-
-lemma act_single [DecidableEq X] (p : Equiv.Perm X) (x : X) :
-    act p (EuclideanSpace.single x (1:ℂ)) = EuclideanSpace.single (p x) 1 := by
-  exact EuclideanSpace.piLpCongrLeft_single p x 1
-
-lemma trace_projection (W : Submodule ℂ (H X)) :
-    ∑ x, (W.starProjection (EuclideanSpace.single x (1:ℂ))) x = Module.finrank ℂ W := by
-  classical
-  have hp : LinearMap.IsProj W W.starProjection.toLinearMap := by
-    constructor
-    · intro x; exact (W.orthogonalProjectionOnto x).2
-    · intro x hx; exact W.starProjection_eq_self_iff.mpr hx
-  have ht := hp.trace
-  rw [LinearMap.trace_eq_matrix_trace ℂ (EuclideanSpace.basisFun X ℂ).toBasis] at ht
-  simpa only [Matrix.trace,Matrix.diag_apply,LinearMap.toMatrix_apply,EuclideanSpace.basisFun_repr,
-    EuclideanSpace.basisFun_apply,OrthonormalBasis.coe_toBasis,
-    OrthonormalBasis.coe_toBasis_repr_apply,ContinuousLinearMap.coe_coe] using ht
-
-lemma projection_diagonal (W : Submodule ℂ (H X)) (x : X) :
-    (W.starProjection (EuclideanSpace.single x (1:ℂ))) x =
-      (‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 : ℝ) := by
-  classical
-  have hh := W.inner_starProjection_left_eq_right
-    (EuclideanSpace.single x (1:ℂ)) (W.starProjection (EuclideanSpace.single x (1:ℂ)))
-  have hf : W.starProjection (W.starProjection (EuclideanSpace.single x (1:ℂ))) =
-      W.starProjection (EuclideanSpace.single x (1:ℂ)) :=
-    W.starProjection_eq_self_iff.mpr (W.orthogonalProjectionOnto _).2
-  rw [hf] at hh
-  rw [inner_self_eq_norm_sq_to_K,EuclideanSpace.inner_single_left] at hh
-  simpa using hh.symm
-
-lemma projection_comm (W : Submodule ℂ (H X)) (p : Equiv.Perm X)
-    (hW : W.map (act p).toLinearEquiv.toLinearMap = W) (v : H X) :
-    act p (W.starProjection v) = W.starProjection (act p v) := by
-  have hh := (act p).toLinearIsometry.map_starProjection W v
-  change act p (W.starProjection v) =
-    (W.map (act p).toLinearEquiv.toLinearMap).starProjection (act p v) at hh
-  simpa only [hW] using hh
-
-/-- A transitive invariant subspace has constant diagonal orthogonal projection. -/
-lemma uniform_diagonal (W : Submodule ℂ (H X))
-    (htrans : ∀ x y : X, ∃ p : Equiv.Perm X, p x = y ∧
-      W.map (act p).toLinearEquiv.toLinearMap = W) (x : X) :
-    (Fintype.card X:ℝ) * ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 =
-      Module.finrank ℂ W := by
-  classical
-  have hh (y : X) : (W.starProjection (EuclideanSpace.single y (1:ℂ))) y =
-      (‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 : ℝ) := by
-    obtain ⟨p,hp,hW⟩ := htrans x y
-    subst y
-    rw [projection_diagonal,←act_single p x,←projection_comm W p hW,(act p).norm_map]
-  have ht := trace_projection W
-  simp_rw [hh] at ht
-  have hhreal : (∑ _y : X, ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2) =
-      (Module.finrank ℂ W:ℝ) := by exact_mod_cast ht
-  simpa using hhreal
-
-/-- The uncertainty bound, with no assumption of irreducibility. -/
-theorem uncertainty (W : Submodule ℂ (H X))
-    (htrans : ∀ x y : X, ∃ p : Equiv.Perm X, p x = y ∧
-      W.map (act p).toLinearEquiv.toLinearMap = W)
-    (v : H X) (hv : v ∈ W) (x : X) :
-    (Fintype.card X:ℝ) * ‖v x‖^2 ≤ (Module.finrank ℂ W:ℝ) * ‖v‖^2 := by
-  classical
-  have hh : inner ℂ (W.starProjection (EuclideanSpace.single x (1:ℂ))) v = v x := by
-    rw [W.inner_starProjection_left_eq_right,W.starProjection_eq_self_iff.mpr hv,
-      EuclideanSpace.inner_single_left]
-    simp
-  have hc := norm_inner_le_norm (𝕜 := ℂ) (W.starProjection (EuclideanSpace.single x (1:ℂ))) v
-  rw [hh] at hc
-  have hs : ‖v x‖^2 ≤ ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 * ‖v‖^2 := by
-    nlinarith [norm_nonneg (v x),norm_nonneg v,
-      norm_nonneg (W.starProjection (EuclideanSpace.single x (1:ℂ)))]
-  calc
-    (Fintype.card X:ℝ) * ‖v x‖^2 ≤
-      (Fintype.card X:ℝ) * (‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 * ‖v‖^2) :=
-      mul_le_mul_of_nonneg_left hs (Nat.cast_nonneg _)
-    _ = _ := by rw [←mul_assoc,uniform_diagonal W htrans x]
-
-end CubeShuffle.PermutationHilbert
-namespace CubeShuffle.PermutationHilbert
-open scoped BigOperators Classical
-
-variable {X : Type*} [Fintype X]
-
-lemma trace_projection_real (W : Submodule ℂ (H X)) :
-    ∑ x, ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 = (Module.finrank ℂ W:ℝ) := by
-  have hh := trace_projection W
-  simp_rw [projection_diagonal] at hh
-  exact_mod_cast hh
-
-/-- A coordinate footprint fixed by a subgroup supplies an unconditional
-lower bound on invariant multiplicity. This is the trace form of the branching
-estimate, valid for every transitive invariant submodule. -/
-theorem footprint_dimension (W K : Submodule ℂ (H X)) (hKW : K ≤ W)
-    (htrans : ∀ x y : X, ∃ p : Equiv.Perm X, p x = y ∧
-      W.map (act p).toLinearEquiv.toLinearMap = W)
-    (S : Finset X) (hS : ∀ x ∈ S, W.starProjection (EuclideanSpace.single x (1:ℂ)) ∈ K) :
-    (S.card:ℝ) * (Module.finrank ℂ W:ℝ) ≤
-      (Fintype.card X:ℝ) * (Module.finrank ℂ K:ℝ) := by
-  have he (x : X) (hx : x ∈ S) :
-      K.starProjection (EuclideanSpace.single x (1:ℂ)) =
-        W.starProjection (EuclideanSpace.single x (1:ℂ)) := by
-    apply K.eq_starProjection_of_mem_of_inner_eq_zero (hS x hx)
-    intro v hv
-    exact W.starProjection_inner_eq_zero _ v (hKW hv)
-  have hs : ∑ x ∈ S, ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 ≤
-      (Module.finrank ℂ K:ℝ) := by
-    rw [←trace_projection_real K]
-    calc
-      _ = ∑ x ∈ S, ‖K.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 := by
-        apply Finset.sum_congr rfl
-        intro x hx
-        rw [he x hx]
-      _ ≤ _ := Finset.sum_le_univ_sum_of_nonneg (fun x => sq_nonneg _)
-  have hmul := mul_le_mul_of_nonneg_left hs (Nat.cast_nonneg (Fintype.card X): (0:ℝ) ≤ _)
-  rw [Finset.mul_sum] at hmul
-  simp_rw [uniform_diagonal W htrans] at hmul
-  simpa using hmul
-
-end CubeShuffle.PermutationHilbert
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+namespace CubeShuffle.PermutationHilbert
+open scoped BigOperators ComplexConjugate Classical
+open Complex
+
+variable {X : Type*} [Fintype X]
+abbrev H (X : Type*) [Fintype X] := EuclideanSpace ℂ X
+
+noncomputable def act (p : Equiv.Perm X) : H X ≃ₗᵢ[ℂ] H X :=
+  LinearIsometryEquiv.piLpCongrLeft 2 ℂ ℂ p
+
+@[simp] lemma act_apply (p : Equiv.Perm X) (v : H X) (x : X) :
+    act p v x = v (p⁻¹ x) := rfl
+
+lemma act_single [DecidableEq X] (p : Equiv.Perm X) (x : X) :
+    act p (EuclideanSpace.single x (1:ℂ)) = EuclideanSpace.single (p x) 1 := by
+  exact EuclideanSpace.piLpCongrLeft_single p x 1
+
+lemma trace_projection (W : Submodule ℂ (H X)) :
+    ∑ x, (W.starProjection (EuclideanSpace.single x (1:ℂ))) x = Module.finrank ℂ W := by
+  classical
+  have hp : LinearMap.IsProj W W.starProjection.toLinearMap := by
+    constructor
+    · intro x; exact (W.orthogonalProjectionOnto x).2
+    · intro x hx; exact W.starProjection_eq_self_iff.mpr hx
+  have ht := hp.trace
+  rw [LinearMap.trace_eq_matrix_trace ℂ (EuclideanSpace.basisFun X ℂ).toBasis] at ht
+  simpa only [Matrix.trace,Matrix.diag_apply,LinearMap.toMatrix_apply,EuclideanSpace.basisFun_repr,
+    EuclideanSpace.basisFun_apply,OrthonormalBasis.coe_toBasis,
+    OrthonormalBasis.coe_toBasis_repr_apply,ContinuousLinearMap.coe_coe] using ht
+
+lemma projection_diagonal (W : Submodule ℂ (H X)) (x : X) :
+    (W.starProjection (EuclideanSpace.single x (1:ℂ))) x =
+      (‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 : ℝ) := by
+  classical
+  have hh := W.inner_starProjection_left_eq_right
+    (EuclideanSpace.single x (1:ℂ)) (W.starProjection (EuclideanSpace.single x (1:ℂ)))
+  have hf : W.starProjection (W.starProjection (EuclideanSpace.single x (1:ℂ))) =
+      W.starProjection (EuclideanSpace.single x (1:ℂ)) :=
+    W.starProjection_eq_self_iff.mpr (W.orthogonalProjectionOnto _).2
+  rw [hf] at hh
+  rw [inner_self_eq_norm_sq_to_K,EuclideanSpace.inner_single_left] at hh
+  simpa using hh.symm
+
+lemma projection_comm (W : Submodule ℂ (H X)) (p : Equiv.Perm X)
+    (hW : W.map (act p).toLinearEquiv.toLinearMap = W) (v : H X) :
+    act p (W.starProjection v) = W.starProjection (act p v) := by
+  have hh := (act p).toLinearIsometry.map_starProjection W v
+  change act p (W.starProjection v) =
+    (W.map (act p).toLinearEquiv.toLinearMap).starProjection (act p v) at hh
+  simpa only [hW] using hh
+
+/-- A transitive invariant subspace has constant diagonal orthogonal projection. -/
+lemma uniform_diagonal (W : Submodule ℂ (H X))
+    (htrans : ∀ x y : X, ∃ p : Equiv.Perm X, p x = y ∧
+      W.map (act p).toLinearEquiv.toLinearMap = W) (x : X) :
+    (Fintype.card X:ℝ) * ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 =
+      Module.finrank ℂ W := by
+  classical
+  have hh (y : X) : (W.starProjection (EuclideanSpace.single y (1:ℂ))) y =
+      (‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 : ℝ) := by
+    obtain ⟨p,hp,hW⟩ := htrans x y
+    subst y
+    rw [projection_diagonal,←act_single p x,←projection_comm W p hW,(act p).norm_map]
+  have ht := trace_projection W
+  simp_rw [hh] at ht
+  have hhreal : (∑ _y : X, ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2) =
+      (Module.finrank ℂ W:ℝ) := by exact_mod_cast ht
+  simpa using hhreal
+
+/-- The uncertainty bound, with no assumption of irreducibility. -/
+theorem uncertainty (W : Submodule ℂ (H X))
+    (htrans : ∀ x y : X, ∃ p : Equiv.Perm X, p x = y ∧
+      W.map (act p).toLinearEquiv.toLinearMap = W)
+    (v : H X) (hv : v ∈ W) (x : X) :
+    (Fintype.card X:ℝ) * ‖v x‖^2 ≤ (Module.finrank ℂ W:ℝ) * ‖v‖^2 := by
+  classical
+  have hh : inner ℂ (W.starProjection (EuclideanSpace.single x (1:ℂ))) v = v x := by
+    rw [W.inner_starProjection_left_eq_right,W.starProjection_eq_self_iff.mpr hv,
+      EuclideanSpace.inner_single_left]
+    simp
+  have hc := norm_inner_le_norm (𝕜 := ℂ) (W.starProjection (EuclideanSpace.single x (1:ℂ))) v
+  rw [hh] at hc
+  have hs : ‖v x‖^2 ≤ ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 * ‖v‖^2 := by
+    nlinarith [norm_nonneg (v x),norm_nonneg v,
+      norm_nonneg (W.starProjection (EuclideanSpace.single x (1:ℂ)))]
+  calc
+    (Fintype.card X:ℝ) * ‖v x‖^2 ≤
+      (Fintype.card X:ℝ) * (‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 * ‖v‖^2) :=
+      mul_le_mul_of_nonneg_left hs (Nat.cast_nonneg _)
+    _ = _ := by rw [←mul_assoc,uniform_diagonal W htrans x]
+
+end CubeShuffle.PermutationHilbert
+namespace CubeShuffle.PermutationHilbert
+open scoped BigOperators Classical
+
+variable {X : Type*} [Fintype X]
+
+lemma trace_projection_real (W : Submodule ℂ (H X)) :
+    ∑ x, ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 = (Module.finrank ℂ W:ℝ) := by
+  have hh := trace_projection W
+  simp_rw [projection_diagonal] at hh
+  exact_mod_cast hh
+
+/-- A coordinate footprint fixed by a subgroup supplies an unconditional
+lower bound on invariant multiplicity. This is the trace form of the branching
+estimate, valid for every transitive invariant submodule. -/
+theorem footprint_dimension (W K : Submodule ℂ (H X)) (hKW : K ≤ W)
+    (htrans : ∀ x y : X, ∃ p : Equiv.Perm X, p x = y ∧
+      W.map (act p).toLinearEquiv.toLinearMap = W)
+    (S : Finset X) (hS : ∀ x ∈ S, W.starProjection (EuclideanSpace.single x (1:ℂ)) ∈ K) :
+    (S.card:ℝ) * (Module.finrank ℂ W:ℝ) ≤
+      (Fintype.card X:ℝ) * (Module.finrank ℂ K:ℝ) := by
+  have he (x : X) (hx : x ∈ S) :
+      K.starProjection (EuclideanSpace.single x (1:ℂ)) =
+        W.starProjection (EuclideanSpace.single x (1:ℂ)) := by
+    apply K.eq_starProjection_of_mem_of_inner_eq_zero (hS x hx)
+    intro v hv
+    exact W.starProjection_inner_eq_zero _ v (hKW hv)
+  have hs : ∑ x ∈ S, ‖W.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 ≤
+      (Module.finrank ℂ K:ℝ) := by
+    rw [←trace_projection_real K]
+    calc
+      _ = ∑ x ∈ S, ‖K.starProjection (EuclideanSpace.single x (1:ℂ))‖^2 := by
+        apply Finset.sum_congr rfl
+        intro x hx
+        rw [he x hx]
+      _ ≤ _ := Finset.sum_le_univ_sum_of_nonneg (fun x => sq_nonneg _)
+  have hmul := mul_le_mul_of_nonneg_left hs (Nat.cast_nonneg (Fintype.card X): (0:ℝ) ≤ _)
+  rw [Finset.mul_sum] at hmul
+  simp_rw [uniform_diagonal W htrans] at hmul
+  simpa using hmul
+
+end CubeShuffle.PermutationHilbert
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe.lean b/lean/OAI/RepresentationTheory/FoulkesHowe.lean
index c70a81904384a674745e725e3590583dc7463327..37ced73b16d442f9f0dc0f7df21874454c26f13d 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe.lean
@@ -1,77 +1,77 @@
-import OAI.RepresentationTheory.FoulkesHowe.Model
-import OAI.RepresentationTheory.FoulkesHowe.LinearCase
-import OAI.RepresentationTheory.FoulkesHowe.Actions
-import OAI.RepresentationTheory.FoulkesHowe.Annihilator
-import OAI.RepresentationTheory.FoulkesHowe.BackwardCancellation
-import OAI.RepresentationTheory.FoulkesHowe.BinaryFactorization
-import OAI.RepresentationTheory.FoulkesHowe.BlockDerivation
-import OAI.RepresentationTheory.FoulkesHowe.BlockInjectivity
-import OAI.RepresentationTheory.FoulkesHowe.ShiftSequence
-import OAI.RepresentationTheory.FoulkesHowe.BlockWeight
-import OAI.RepresentationTheory.FoulkesHowe.CanonicalMap
-import OAI.RepresentationTheory.FoulkesHowe.Surjectivity
-import OAI.RepresentationTheory.FoulkesHowe.DualComparison
-import OAI.RepresentationTheory.FoulkesHowe.DegreeOne
-import OAI.RepresentationTheory.FoulkesHowe.DerivationInjectivity
-import OAI.RepresentationTheory.FoulkesHowe.DiagonalCoefficient
-import OAI.RepresentationTheory.FoulkesHowe.DiagonalPolarization
-import OAI.RepresentationTheory.FoulkesHowe.DualEmbedding
-import OAI.RepresentationTheory.FoulkesHowe.EncodedCancellation
-import OAI.RepresentationTheory.FoulkesHowe.EncodingHomogeneous
-import OAI.RepresentationTheory.FoulkesHowe.FiniteDimension
-import OAI.RepresentationTheory.FoulkesHowe.FirstCounts
-import OAI.RepresentationTheory.FoulkesHowe.FirstLabels
-import OAI.RepresentationTheory.FoulkesHowe.FirstPolynomial
-import OAI.RepresentationTheory.FoulkesHowe.FirstTransfer
-import OAI.RepresentationTheory.FoulkesHowe.TransferDerivative
-import OAI.RepresentationTheory.FoulkesHowe.TransferEndpoints
-import OAI.RepresentationTheory.FoulkesHowe.FixedSlot
-import OAI.RepresentationTheory.FoulkesHowe.FlattenedForm
-import OAI.RepresentationTheory.FoulkesHowe.FlattenedSwap
-import OAI.RepresentationTheory.FoulkesHowe.FoulkesDiagonal
-import OAI.RepresentationTheory.FoulkesHowe.FoulkesMultilinear
-import OAI.RepresentationTheory.FoulkesHowe.FoulkesSymmetries
-import OAI.RepresentationTheory.FoulkesHowe.Functoriality
-import OAI.RepresentationTheory.FoulkesHowe.GridCancellation
-import OAI.RepresentationTheory.FoulkesHowe.MonomialCut
-import OAI.RepresentationTheory.FoulkesHowe.MonomialMultilinear
-import OAI.RepresentationTheory.FoulkesHowe.MultilinearDescent
-import OAI.RepresentationTheory.FoulkesHowe.MultilinearPolynomial
-import OAI.RepresentationTheory.FoulkesHowe.Multiplication
-import OAI.RepresentationTheory.FoulkesHowe.Naturality
-import OAI.RepresentationTheory.FoulkesHowe.PairingConstruction
-import OAI.RepresentationTheory.FoulkesHowe.PairingPerfection
-import OAI.RepresentationTheory.FoulkesHowe.PermanentPairing
-import OAI.RepresentationTheory.FoulkesHowe.PlethysmDual
-import OAI.RepresentationTheory.FoulkesHowe.PolynomialShift
-import OAI.RepresentationTheory.FoulkesHowe.PowerSpan
-import OAI.RepresentationTheory.FoulkesHowe.Polarization
-import OAI.RepresentationTheory.FoulkesHowe.RowSwap
-import OAI.RepresentationTheory.FoulkesHowe.SecondTransfer
-import OAI.RepresentationTheory.FoulkesHowe.SecondCounts
-import OAI.RepresentationTheory.FoulkesHowe.SecondLabels
-import OAI.RepresentationTheory.FoulkesHowe.SecondEndpoints
-import OAI.RepresentationTheory.FoulkesHowe.SecondSwap
-import OAI.RepresentationTheory.FoulkesHowe.ShiftHomogeneity
-import OAI.RepresentationTheory.FoulkesHowe.ShiftInjectivity
-import OAI.RepresentationTheory.FoulkesHowe.ShiftNilpotence
-import OAI.RepresentationTheory.FoulkesHowe.SlotDerivative
-import OAI.RepresentationTheory.FoulkesHowe.Spanning
-import OAI.RepresentationTheory.FoulkesHowe.SpanningAgreement
-import OAI.RepresentationTheory.FoulkesHowe.StabilizationBounds
-import OAI.RepresentationTheory.FoulkesHowe.SumShift
-import OAI.RepresentationTheory.FoulkesHowe.AlgebraEvaluation
-import OAI.RepresentationTheory.FoulkesHowe.SymmetricDual
-import OAI.RepresentationTheory.FoulkesHowe.DualEquivalence
-import OAI.RepresentationTheory.FoulkesHowe.SymmetricPairing
-import OAI.RepresentationTheory.FoulkesHowe.PairingLift
-import OAI.RepresentationTheory.FoulkesHowe.GeneratorLift
-import OAI.RepresentationTheory.FoulkesHowe.SymmetricLift
-import OAI.RepresentationTheory.FoulkesHowe.TensorPresentation
-import OAI.RepresentationTheory.FoulkesHowe.Vanishing
-import OAI.RepresentationTheory.FoulkesHowe.BinaryVanishing
-import OAI.RepresentationTheory.FoulkesHowe.ProductVanishing
-import OAI.RepresentationTheory.FoulkesHowe.VanishingFirst
-import OAI.RepresentationTheory.FoulkesHowe.VanishingInduction
-import OAI.RepresentationTheory.FoulkesHowe.VanishingOne
-import OAI.RepresentationTheory.FoulkesHowe.Stabilization
+import OAI.RepresentationTheory.FoulkesHowe.Model
+import OAI.RepresentationTheory.FoulkesHowe.LinearCase
+import OAI.RepresentationTheory.FoulkesHowe.Actions
+import OAI.RepresentationTheory.FoulkesHowe.Annihilator
+import OAI.RepresentationTheory.FoulkesHowe.BackwardCancellation
+import OAI.RepresentationTheory.FoulkesHowe.BinaryFactorization
+import OAI.RepresentationTheory.FoulkesHowe.BlockDerivation
+import OAI.RepresentationTheory.FoulkesHowe.BlockInjectivity
+import OAI.RepresentationTheory.FoulkesHowe.ShiftSequence
+import OAI.RepresentationTheory.FoulkesHowe.BlockWeight
+import OAI.RepresentationTheory.FoulkesHowe.CanonicalMap
+import OAI.RepresentationTheory.FoulkesHowe.Surjectivity
+import OAI.RepresentationTheory.FoulkesHowe.DualComparison
+import OAI.RepresentationTheory.FoulkesHowe.DegreeOne
+import OAI.RepresentationTheory.FoulkesHowe.DerivationInjectivity
+import OAI.RepresentationTheory.FoulkesHowe.DiagonalCoefficient
+import OAI.RepresentationTheory.FoulkesHowe.DiagonalPolarization
+import OAI.RepresentationTheory.FoulkesHowe.DualEmbedding
+import OAI.RepresentationTheory.FoulkesHowe.EncodedCancellation
+import OAI.RepresentationTheory.FoulkesHowe.EncodingHomogeneous
+import OAI.RepresentationTheory.FoulkesHowe.FiniteDimension
+import OAI.RepresentationTheory.FoulkesHowe.FirstCounts
+import OAI.RepresentationTheory.FoulkesHowe.FirstLabels
+import OAI.RepresentationTheory.FoulkesHowe.FirstPolynomial
+import OAI.RepresentationTheory.FoulkesHowe.FirstTransfer
+import OAI.RepresentationTheory.FoulkesHowe.TransferDerivative
+import OAI.RepresentationTheory.FoulkesHowe.TransferEndpoints
+import OAI.RepresentationTheory.FoulkesHowe.FixedSlot
+import OAI.RepresentationTheory.FoulkesHowe.FlattenedForm
+import OAI.RepresentationTheory.FoulkesHowe.FlattenedSwap
+import OAI.RepresentationTheory.FoulkesHowe.FoulkesDiagonal
+import OAI.RepresentationTheory.FoulkesHowe.FoulkesMultilinear
+import OAI.RepresentationTheory.FoulkesHowe.FoulkesSymmetries
+import OAI.RepresentationTheory.FoulkesHowe.Functoriality
+import OAI.RepresentationTheory.FoulkesHowe.GridCancellation
+import OAI.RepresentationTheory.FoulkesHowe.MonomialCut
+import OAI.RepresentationTheory.FoulkesHowe.MonomialMultilinear
+import OAI.RepresentationTheory.FoulkesHowe.MultilinearDescent
+import OAI.RepresentationTheory.FoulkesHowe.MultilinearPolynomial
+import OAI.RepresentationTheory.FoulkesHowe.Multiplication
+import OAI.RepresentationTheory.FoulkesHowe.Naturality
+import OAI.RepresentationTheory.FoulkesHowe.PairingConstruction
+import OAI.RepresentationTheory.FoulkesHowe.PairingPerfection
+import OAI.RepresentationTheory.FoulkesHowe.PermanentPairing
+import OAI.RepresentationTheory.FoulkesHowe.PlethysmDual
+import OAI.RepresentationTheory.FoulkesHowe.PolynomialShift
+import OAI.RepresentationTheory.FoulkesHowe.PowerSpan
+import OAI.RepresentationTheory.FoulkesHowe.Polarization
+import OAI.RepresentationTheory.FoulkesHowe.RowSwap
+import OAI.RepresentationTheory.FoulkesHowe.SecondTransfer
+import OAI.RepresentationTheory.FoulkesHowe.SecondCounts
+import OAI.RepresentationTheory.FoulkesHowe.SecondLabels
+import OAI.RepresentationTheory.FoulkesHowe.SecondEndpoints
+import OAI.RepresentationTheory.FoulkesHowe.SecondSwap
+import OAI.RepresentationTheory.FoulkesHowe.ShiftHomogeneity
+import OAI.RepresentationTheory.FoulkesHowe.ShiftInjectivity
+import OAI.RepresentationTheory.FoulkesHowe.ShiftNilpotence
+import OAI.RepresentationTheory.FoulkesHowe.SlotDerivative
+import OAI.RepresentationTheory.FoulkesHowe.Spanning
+import OAI.RepresentationTheory.FoulkesHowe.SpanningAgreement
+import OAI.RepresentationTheory.FoulkesHowe.StabilizationBounds
+import OAI.RepresentationTheory.FoulkesHowe.SumShift
+import OAI.RepresentationTheory.FoulkesHowe.AlgebraEvaluation
+import OAI.RepresentationTheory.FoulkesHowe.SymmetricDual
+import OAI.RepresentationTheory.FoulkesHowe.DualEquivalence
+import OAI.RepresentationTheory.FoulkesHowe.SymmetricPairing
+import OAI.RepresentationTheory.FoulkesHowe.PairingLift
+import OAI.RepresentationTheory.FoulkesHowe.GeneratorLift
+import OAI.RepresentationTheory.FoulkesHowe.SymmetricLift
+import OAI.RepresentationTheory.FoulkesHowe.TensorPresentation
+import OAI.RepresentationTheory.FoulkesHowe.Vanishing
+import OAI.RepresentationTheory.FoulkesHowe.BinaryVanishing
+import OAI.RepresentationTheory.FoulkesHowe.ProductVanishing
+import OAI.RepresentationTheory.FoulkesHowe.VanishingFirst
+import OAI.RepresentationTheory.FoulkesHowe.VanishingInduction
+import OAI.RepresentationTheory.FoulkesHowe.VanishingOne
+import OAI.RepresentationTheory.FoulkesHowe.Stabilization
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/Actions.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/Actions.lean
index 455f67d02e967d0f123c62d67fef9fbc301d4e14..693fbbd750b5e57b36f38fc05117f9d03a32546a 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/Actions.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/Actions.lean
@@ -1,119 +1,119 @@
-import OAI.RepresentationTheory.FoulkesHowe.Functoriality
-
-namespace OAI
-
-noncomputable section
-open scoped BigOperators
-universe u
-namespace Problem346
-
-variable {V : Type u} [AddCommGroup V] [Module ℂ V]
-
-/-- Any map with the prescribed plethysm action carries the Foulkes formula
-    to the formula on the transformed vector array. -/
-theorem map_foulkesFormula_of_isPlethysmAction
-    {a b : ℕ} {g : V ≃ₗ[ℂ] V}
-    {L : SymPow a (SymPow b V) →ₗ[ℂ] SymPow a (SymPow b V)}
-    (hL : IsPlethysmAction a b V g L) (v : Fin b → Fin a → V) :
-    L (foulkesFormula a b V v) =
-      foulkesFormula a b V (fun j i => g (v j i)) := by
-  classical
-  unfold foulkesFormula
-  rw [map_smul, map_sum]
-  congr 1
-  apply Finset.sum_congr rfl
-  intro σ _
-  exact hL (fun i j => v j ((σ j) i))
-
-/-- The canonical action of a linear automorphism on a plethysm. -/
-def plethysmEquiv (outer inner : ℕ) (g : V ≃ₗ[ℂ] V) :
-    SymPow outer (SymPow inner V) ≃ₗ[ℂ] SymPow outer (SymPow inner V) :=
-  symPowEquiv outer (symPowEquiv inner g)
-
-/-- The underlying endomorphism of the canonical plethysm action. -/
-def plethysmAction (outer inner : ℕ) (g : V ≃ₗ[ℂ] V) :
-    SymPow outer (SymPow inner V) →ₗ[ℂ] SymPow outer (SymPow inner V) :=
-  (plethysmEquiv outer inner g).toLinearMap
-
-@[simp] theorem plethysmAction_symMonomial (outer inner : ℕ) (g : V ≃ₗ[ℂ] V)
-    (x : Fin outer → Fin inner → V) :
-    plethysmAction outer inner g
-      (symMonomial outer (SymPow inner V) (fun i => symMonomial inner V (x i))) =
-    symMonomial outer (SymPow inner V)
-      (fun i => symMonomial inner V (fun j => g (x i j))) := by
-  simp [plethysmAction, plethysmEquiv]
-
-theorem plethysmAction_isPlethysmAction (outer inner : ℕ) (g : V ≃ₗ[ℂ] V) :
-    IsPlethysmAction outer inner V g (plethysmAction outer inner g) :=
-  plethysmAction_symMonomial outer inner g
-
-theorem plethysmAction_bijective (outer inner : ℕ) (g : V ≃ₗ[ℂ] V) :
-    Function.Bijective (plethysmAction outer inner g) :=
-  (plethysmEquiv outer inner g).bijective
-
-theorem isGLEquivariantEmbedding_of_commutes {a b : ℕ}
-    (ι : SymPow a (SymPow b V) →ₗ[ℂ] SymPow b (SymPow a V))
-    (hinj : Function.Injective ι)
-    (hcomm : ∀ g : V ≃ₗ[ℂ] V,
-      (plethysmAction b a g).comp ι = ι.comp (plethysmAction a b g)) :
-    IsGLEquivariantEmbedding a b V ι := by
-  refine ⟨hinj, ?_⟩
-  intro g
-  exact ⟨plethysmAction b a g, plethysmAction a b g,
-    plethysmAction_isPlethysmAction b a g,
-    plethysmAction_isPlethysmAction a b g,
-    plethysmAction_bijective b a g, plethysmAction_bijective a b g, hcomm g⟩
-
-@[simp] theorem plethysmAction_refl (outer inner : ℕ) :
-    plethysmAction outer inner (LinearEquiv.refl ℂ V) = LinearMap.id := by
-  simp [plethysmAction, plethysmEquiv]
-
-@[simp] theorem plethysmAction_trans (outer inner : ℕ) (g h : V ≃ₗ[ℂ] V) :
-    plethysmAction outer inner (g.trans h) =
-      (plethysmAction outer inner h).comp (plethysmAction outer inner g) := by
-  change symPowMap outer (symPowMap inner (g.trans h).toLinearMap) = _
-  change symPowMap outer (symPowMap inner (h.toLinearMap.comp g.toLinearMap)) = _
-  rw [symPowMap_comp, symPowMap_comp]
-  rfl
-
-@[simp] theorem plethysmAction_symm_comp (outer inner : ℕ) (g : V ≃ₗ[ℂ] V) :
-    (plethysmAction outer inner g.symm).comp (plethysmAction outer inner g) =
-      LinearMap.id := by
-  rw [← plethysmAction_trans]
-  simp
-
-@[simp] theorem plethysmAction_comp_symm (outer inner : ℕ) (g : V ≃ₗ[ℂ] V) :
-    (plethysmAction outer inner g).comp (plethysmAction outer inner g.symm) =
-      LinearMap.id := by
-  rw [← plethysmAction_trans]
-  simp
-
-/-- Conjugating by the natural GL actions preserves the defining canonical formula. -/
-theorem isFoulkesMap_conjugate {a b : ℕ}
-    {μ : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V)}
-    (hμ : IsFoulkesMap a b V μ) (g : V ≃ₗ[ℂ] V) :
-    IsFoulkesMap a b V
-      ((plethysmAction a b g.symm).comp (μ.comp (plethysmAction b a g))) := by
-  intro v
-  simp only [LinearMap.comp_apply, plethysmAction_symMonomial]
-  rw [hμ]
-  rw [map_foulkesFormula_of_isPlethysmAction (plethysmAction_isPlethysmAction a b g.symm)]
-  simp
-
-/-- Uniqueness of the canonical map implies its GL equivariance. -/
-theorem foulkesMap_commutes_of_unique {a b : ℕ}
-    {μ : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V)}
-    (hμ : IsFoulkesMap a b V μ)
-    (huniq : ∀ ν : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V),
-      IsFoulkesMap a b V ν → ν = μ)
-    (g : V ≃ₗ[ℂ] V) :
-    (plethysmAction a b g).comp μ = μ.comp (plethysmAction b a g) := by
-  have hc := huniq _ (isFoulkesMap_conjugate hμ g)
-  have he := congrArg (fun L => (plethysmAction a b g).comp L) hc
-  simpa only [← LinearMap.comp_assoc, plethysmAction_comp_symm, LinearMap.id_comp] using he.symm
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.Functoriality
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators
+universe u
+namespace Problem346
+
+variable {V : Type u} [AddCommGroup V] [Module ℂ V]
+
+/-- Any map with the prescribed plethysm action carries the Foulkes formula
+    to the formula on the transformed vector array. -/
+theorem map_foulkesFormula_of_isPlethysmAction
+    {a b : ℕ} {g : V ≃ₗ[ℂ] V}
+    {L : SymPow a (SymPow b V) →ₗ[ℂ] SymPow a (SymPow b V)}
+    (hL : IsPlethysmAction a b V g L) (v : Fin b → Fin a → V) :
+    L (foulkesFormula a b V v) =
+      foulkesFormula a b V (fun j i => g (v j i)) := by
+  classical
+  unfold foulkesFormula
+  rw [map_smul, map_sum]
+  congr 1
+  apply Finset.sum_congr rfl
+  intro σ _
+  exact hL (fun i j => v j ((σ j) i))
+
+/-- The canonical action of a linear automorphism on a plethysm. -/
+def plethysmEquiv (outer inner : ℕ) (g : V ≃ₗ[ℂ] V) :
+    SymPow outer (SymPow inner V) ≃ₗ[ℂ] SymPow outer (SymPow inner V) :=
+  symPowEquiv outer (symPowEquiv inner g)
+
+/-- The underlying endomorphism of the canonical plethysm action. -/
+def plethysmAction (outer inner : ℕ) (g : V ≃ₗ[ℂ] V) :
+    SymPow outer (SymPow inner V) →ₗ[ℂ] SymPow outer (SymPow inner V) :=
+  (plethysmEquiv outer inner g).toLinearMap
+
+@[simp] theorem plethysmAction_symMonomial (outer inner : ℕ) (g : V ≃ₗ[ℂ] V)
+    (x : Fin outer → Fin inner → V) :
+    plethysmAction outer inner g
+      (symMonomial outer (SymPow inner V) (fun i => symMonomial inner V (x i))) =
+    symMonomial outer (SymPow inner V)
+      (fun i => symMonomial inner V (fun j => g (x i j))) := by
+  simp [plethysmAction, plethysmEquiv]
+
+theorem plethysmAction_isPlethysmAction (outer inner : ℕ) (g : V ≃ₗ[ℂ] V) :
+    IsPlethysmAction outer inner V g (plethysmAction outer inner g) :=
+  plethysmAction_symMonomial outer inner g
+
+theorem plethysmAction_bijective (outer inner : ℕ) (g : V ≃ₗ[ℂ] V) :
+    Function.Bijective (plethysmAction outer inner g) :=
+  (plethysmEquiv outer inner g).bijective
+
+theorem isGLEquivariantEmbedding_of_commutes {a b : ℕ}
+    (ι : SymPow a (SymPow b V) →ₗ[ℂ] SymPow b (SymPow a V))
+    (hinj : Function.Injective ι)
+    (hcomm : ∀ g : V ≃ₗ[ℂ] V,
+      (plethysmAction b a g).comp ι = ι.comp (plethysmAction a b g)) :
+    IsGLEquivariantEmbedding a b V ι := by
+  refine ⟨hinj, ?_⟩
+  intro g
+  exact ⟨plethysmAction b a g, plethysmAction a b g,
+    plethysmAction_isPlethysmAction b a g,
+    plethysmAction_isPlethysmAction a b g,
+    plethysmAction_bijective b a g, plethysmAction_bijective a b g, hcomm g⟩
+
+@[simp] theorem plethysmAction_refl (outer inner : ℕ) :
+    plethysmAction outer inner (LinearEquiv.refl ℂ V) = LinearMap.id := by
+  simp [plethysmAction, plethysmEquiv]
+
+@[simp] theorem plethysmAction_trans (outer inner : ℕ) (g h : V ≃ₗ[ℂ] V) :
+    plethysmAction outer inner (g.trans h) =
+      (plethysmAction outer inner h).comp (plethysmAction outer inner g) := by
+  change symPowMap outer (symPowMap inner (g.trans h).toLinearMap) = _
+  change symPowMap outer (symPowMap inner (h.toLinearMap.comp g.toLinearMap)) = _
+  rw [symPowMap_comp, symPowMap_comp]
+  rfl
+
+@[simp] theorem plethysmAction_symm_comp (outer inner : ℕ) (g : V ≃ₗ[ℂ] V) :
+    (plethysmAction outer inner g.symm).comp (plethysmAction outer inner g) =
+      LinearMap.id := by
+  rw [← plethysmAction_trans]
+  simp
+
+@[simp] theorem plethysmAction_comp_symm (outer inner : ℕ) (g : V ≃ₗ[ℂ] V) :
+    (plethysmAction outer inner g).comp (plethysmAction outer inner g.symm) =
+      LinearMap.id := by
+  rw [← plethysmAction_trans]
+  simp
+
+/-- Conjugating by the natural GL actions preserves the defining canonical formula. -/
+theorem isFoulkesMap_conjugate {a b : ℕ}
+    {μ : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V)}
+    (hμ : IsFoulkesMap a b V μ) (g : V ≃ₗ[ℂ] V) :
+    IsFoulkesMap a b V
+      ((plethysmAction a b g.symm).comp (μ.comp (plethysmAction b a g))) := by
+  intro v
+  simp only [LinearMap.comp_apply, plethysmAction_symMonomial]
+  rw [hμ]
+  rw [map_foulkesFormula_of_isPlethysmAction (plethysmAction_isPlethysmAction a b g.symm)]
+  simp
+
+/-- Uniqueness of the canonical map implies its GL equivariance. -/
+theorem foulkesMap_commutes_of_unique {a b : ℕ}
+    {μ : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V)}
+    (hμ : IsFoulkesMap a b V μ)
+    (huniq : ∀ ν : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V),
+      IsFoulkesMap a b V ν → ν = μ)
+    (g : V ≃ₗ[ℂ] V) :
+    (plethysmAction a b g).comp μ = μ.comp (plethysmAction b a g) := by
+  have hc := huniq _ (isFoulkesMap_conjugate hμ g)
+  have he := congrArg (fun L => (plethysmAction a b g).comp L) hc
+  simpa only [← LinearMap.comp_assoc, plethysmAction_comp_symm, LinearMap.id_comp] using he.symm
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/AlgebraEvaluation.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/AlgebraEvaluation.lean
index e8ab5cbe96811b57424ec7db6683176d8936198a..9e5fcab2cbd95bcc50694dcef82bd1df4ce19675 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/AlgebraEvaluation.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/AlgebraEvaluation.lean
@@ -1,45 +1,45 @@
-import Mathlib
-
-namespace OAI
-
-noncomputable section
-
-universe u
-namespace Problem346
-
-variable (W : Type u) [AddCommGroup W] [Module ℂ W]
-
-/-- Linear functionals separate points of the symmetric algebra over an infinite field. -/
-theorem symAlg_eq_zero_of_lift_eq_zero (p : SymmetricAlgebra ℂ W)
-    (h : ∀ φ : Module.Dual ℂ W, SymmetricAlgebra.lift φ p = 0) : p = 0 := by
-  classical
-  let b := Module.Free.chooseBasis ℂ W
-  let e := SymmetricAlgebra.equivMvPolynomial b
-  apply e.injective
-  simp only [map_zero]
-  apply MvPolynomial.funext
-  intro x
-  have he : (MvPolynomial.aeval x).comp e.toAlgHom =
-      SymmetricAlgebra.lift (b.constr ℂ x) := by
-    apply SymmetricAlgebra.algHom_ext
-    apply b.ext
-    intro i
-    simp [e]
-  have hx := congrArg (fun f : SymmetricAlgebra ℂ W →ₐ[ℂ] ℂ => f p) he
-  simpa using hx.trans (h (b.constr ℂ x))
-
-/-- Evaluation at vectors separates the symmetric algebra on the dual of a finite-dimensional
-space. -/
-theorem symAlg_dual_eq_zero_of_eval_eq_zero [FiniteDimensional ℂ W]
-    (p : SymmetricAlgebra ℂ (Module.Dual ℂ W))
-    (h : ∀ x : W, SymmetricAlgebra.lift (Module.Dual.eval ℂ W x) p = 0) : p = 0 := by
-  apply symAlg_eq_zero_of_lift_eq_zero
-  intro φ
-  obtain ⟨x, rfl⟩ := (Module.evalEquiv ℂ W).surjective φ
-  exact h x
-
-end Problem346
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+
+universe u
+namespace Problem346
+
+variable (W : Type u) [AddCommGroup W] [Module ℂ W]
+
+/-- Linear functionals separate points of the symmetric algebra over an infinite field. -/
+theorem symAlg_eq_zero_of_lift_eq_zero (p : SymmetricAlgebra ℂ W)
+    (h : ∀ φ : Module.Dual ℂ W, SymmetricAlgebra.lift φ p = 0) : p = 0 := by
+  classical
+  let b := Module.Free.chooseBasis ℂ W
+  let e := SymmetricAlgebra.equivMvPolynomial b
+  apply e.injective
+  simp only [map_zero]
+  apply MvPolynomial.funext
+  intro x
+  have he : (MvPolynomial.aeval x).comp e.toAlgHom =
+      SymmetricAlgebra.lift (b.constr ℂ x) := by
+    apply SymmetricAlgebra.algHom_ext
+    apply b.ext
+    intro i
+    simp [e]
+  have hx := congrArg (fun f : SymmetricAlgebra ℂ W →ₐ[ℂ] ℂ => f p) he
+  simpa using hx.trans (h (b.constr ℂ x))
+
+/-- Evaluation at vectors separates the symmetric algebra on the dual of a finite-dimensional
+space. -/
+theorem symAlg_dual_eq_zero_of_eval_eq_zero [FiniteDimensional ℂ W]
+    (p : SymmetricAlgebra ℂ (Module.Dual ℂ W))
+    (h : ∀ x : W, SymmetricAlgebra.lift (Module.Dual.eval ℂ W x) p = 0) : p = 0 := by
+  apply symAlg_eq_zero_of_lift_eq_zero
+  intro φ
+  obtain ⟨x, rfl⟩ := (Module.evalEquiv ℂ W).surjective φ
+  exact h x
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/Annihilator.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/Annihilator.lean
index 66672592da800dc139ffabff576ff01b24439530..d823dbbab5386e986c4cca5a2943f99a5d73aa47 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/Annihilator.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/Annihilator.lean
@@ -1,40 +1,40 @@
-import OAI.RepresentationTheory.FoulkesHowe.MonomialMultilinear
-import OAI.RepresentationTheory.FoulkesHowe.FoulkesDiagonal
-
-namespace OAI
-
-noncomputable section
-
-universe u
-
-namespace Problem346
-
-/-- The diagonal vanishing criterion forces every canonical Foulkes map to be
-surjective. This dual separation argument does not require finite dimension. -/
-theorem IsFoulkesMap.surjective_of_vanishing (a b : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V]
-    (μ : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V))
-    (hμ : IsFoulkesMap a b V μ)
-    (hvan : ∀ T : SymmetricMultilinearForm a b V,
-      IsSymmetricMultilinearForm a b V T →
-      (∀ u : Fin b → V, T (fun _ => symMonomial b V u) = 0) → T = 0) :
-    Function.Surjective μ := by
-  apply LinearMap.dualMap_injective_iff.mp
-  rw [← LinearMap.ker_eq_bot]
-  apply le_antisymm ?_ bot_le
-  intro φ hφ
-  change φ = 0
-  rw [LinearMap.mem_ker] at hφ
-  apply (monomialFunctional_eq_zero_iff a b V φ).mp
-  apply hvan _ (monomialFunctional_symmetric a b V φ)
-  intro u
-  rw [monomialFunctional_apply, ← hμ.map_diagonal a b V μ u]
-  have h := congrArg (fun L : SymPow b (SymPow a V) →ₗ[ℂ] ℂ =>
-    L (symMonomial b (SymPow a V) (fun j => symMonomial a V (fun _ => u j)))) hφ
-  exact h
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.MonomialMultilinear
+import OAI.RepresentationTheory.FoulkesHowe.FoulkesDiagonal
+
+namespace OAI
+
+noncomputable section
+
+universe u
+
+namespace Problem346
+
+/-- The diagonal vanishing criterion forces every canonical Foulkes map to be
+surjective. This dual separation argument does not require finite dimension. -/
+theorem IsFoulkesMap.surjective_of_vanishing (a b : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V]
+    (μ : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V))
+    (hμ : IsFoulkesMap a b V μ)
+    (hvan : ∀ T : SymmetricMultilinearForm a b V,
+      IsSymmetricMultilinearForm a b V T →
+      (∀ u : Fin b → V, T (fun _ => symMonomial b V u) = 0) → T = 0) :
+    Function.Surjective μ := by
+  apply LinearMap.dualMap_injective_iff.mp
+  rw [← LinearMap.ker_eq_bot]
+  apply le_antisymm ?_ bot_le
+  intro φ hφ
+  change φ = 0
+  rw [LinearMap.mem_ker] at hφ
+  apply (monomialFunctional_eq_zero_iff a b V φ).mp
+  apply hvan _ (monomialFunctional_symmetric a b V φ)
+  intro u
+  rw [monomialFunctional_apply, ← hμ.map_diagonal a b V μ u]
+  have h := congrArg (fun L : SymPow b (SymPow a V) →ₗ[ℂ] ℂ =>
+    L (symMonomial b (SymPow a V) (fun j => symMonomial a V (fun _ => u j)))) hφ
+  exact h
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/BackwardCancellation.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/BackwardCancellation.lean
index 3e8e21d048367f5ed0b01f3c01b6bd22c7490568..354073be8886727d27ae593c72eac8e500155f59 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/BackwardCancellation.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/BackwardCancellation.lean
@@ -1,31 +1,31 @@
-import Mathlib
-
-namespace OAI
-
-namespace Problem346
-
-/-- A finite sequence of zero-reflecting transitions can be cancelled backwards. -/
-theorem eq_zero_of_backward_chain {M : Type*} [Zero M]
-    (P : ℕ → M) (m : ℕ)
-    (hstep : ∀ k < m, P (k+1) = 0 → P k = 0)
-    (hm : P m = 0) : P 0 = 0 := by
-  exact Nat.decreasingInduction (motive := fun k _ => P k = 0)
-    (fun k hk => hstep k (by omega)) hm (Nat.zero_le m)
-
-/-- Scalar factors in derivative-chain identities need not be divided out when
-propagating a zero endpoint backwards. Only zero reflection of each derivative
-on its current polynomial is required. -/
-theorem eq_zero_of_linear_chain {R M : Type*} [Semiring R]
-    [AddCommMonoid M] [Module R M]
-    (P : ℕ → M) (D : ℕ → Module.End R M) (c : ℕ → R) (m : ℕ)
-    (hchain : ∀ k < m, D k (P k) = c k • P (k+1))
-    (hreflect : ∀ k < m, D k (P k) = 0 → P k = 0)
-    (hm : P m = 0) : P 0 = 0 := by
-  apply eq_zero_of_backward_chain P m _ hm
-  intro k hk hz
-  apply hreflect k hk
-  rw [hchain k hk, hz, smul_zero]
-
-end Problem346
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+namespace Problem346
+
+/-- A finite sequence of zero-reflecting transitions can be cancelled backwards. -/
+theorem eq_zero_of_backward_chain {M : Type*} [Zero M]
+    (P : ℕ → M) (m : ℕ)
+    (hstep : ∀ k < m, P (k+1) = 0 → P k = 0)
+    (hm : P m = 0) : P 0 = 0 := by
+  exact Nat.decreasingInduction (motive := fun k _ => P k = 0)
+    (fun k hk => hstep k (by omega)) hm (Nat.zero_le m)
+
+/-- Scalar factors in derivative-chain identities need not be divided out when
+propagating a zero endpoint backwards. Only zero reflection of each derivative
+on its current polynomial is required. -/
+theorem eq_zero_of_linear_chain {R M : Type*} [Semiring R]
+    [AddCommMonoid M] [Module R M]
+    (P : ℕ → M) (D : ℕ → Module.End R M) (c : ℕ → R) (m : ℕ)
+    (hchain : ∀ k < m, D k (P k) = c k • P (k+1))
+    (hreflect : ∀ k < m, D k (P k) = 0 → P k = 0)
+    (hm : P m = 0) : P 0 = 0 := by
+  apply eq_zero_of_backward_chain P m _ hm
+  intro k hk hz
+  apply hreflect k hk
+  rw [hchain k hk, hz, smul_zero]
+
+end Problem346
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/BinaryFactorization.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/BinaryFactorization.lean
index 3ac67a36281f5ea3d97ddbaa380be0fa0df78554..494ee8b3c4a2f1796632306369e1dcffa1f36f3f 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/BinaryFactorization.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/BinaryFactorization.lean
@@ -1,80 +1,80 @@
-import OAI.RepresentationTheory.FoulkesHowe.Model
-
-namespace OAI
-
-noncomputable section
-open scoped BigOperators
-
-namespace Problem346
-
-/-- Binary forms used in the first step of the vanishing argument split into linear factors. -/
-theorem binary_power_factorization {A : Type*} [CommRing A] [Algebra ℂ A]
-    (r : ℕ) (hr : 0 < r) (x y : A) (c : ℂ) :
-    ∃ z : Fin r → ℂ, (∏ i : Fin r, (x - z i • y)) = x ^ r + c • y ^ r := by
-  classical
-  obtain ⟨α, hα⟩ := IsAlgClosed.exists_pow_nat_eq (-c) hr
-  let ζ : ℂ := Complex.exp (2 * Real.pi * Complex.I / r)
-  have hζ : IsPrimitiveRoot ζ r := Complex.isPrimitiveRoot_exp r hr.ne'
-  have hp : (Polynomial.X ^ r - Polynomial.C (-c) : Polynomial ℂ) =
-      ∏ i ∈ Finset.range r, (Polynomial.X - Polynomial.C (ζ ^ i * α)) :=
-    X_pow_sub_C_eq_prod hζ hr hα
-  have hh := congrArg (fun p : Polynomial ℂ => p.homogenize r) hp
-  have hd : ∀ i ∈ Finset.range r,
-      (Polynomial.X - Polynomial.C (ζ ^ i * α)).natDegree ≤ 1 := by
-    intro i hi
-    rw [Polynomial.natDegree_X_sub_C]
-  have hprod := Polynomial.homogenize_finsetProd (n := fun _ : ℕ => 1) hd
-  simp only [Finset.sum_const, Finset.card_range, smul_eq_mul, mul_one] at hprod
-  rw [hprod] at hh
-  simp only [Polynomial.homogenize_sub, Polynomial.homogenize_X_pow le_rfl,
-    Nat.sub_self, pow_zero, mul_one, Polynomial.homogenize_C,
-    Polynomial.homogenize_X (by decide : (1 : ℕ) ≠ 0), Nat.sub_self, pow_one] at hh
-  have he := congrArg (MvPolynomial.aeval ![x, y]) hh
-  simp only [map_sub, map_pow, map_mul, map_prod, MvPolynomial.aeval_X,
-    Matrix.cons_val_zero, Matrix.cons_val_one,
-    MvPolynomial.aeval_C] at he
-  refine ⟨fun i => ζ ^ (i : ℕ) * α, ?_⟩
-  change (∏ i : Fin r, (x - (ζ ^ (i : ℕ) * α) • y)) = _
-  rw [Fin.prod_univ_eq_prod_range (fun i => x - (ζ ^ i * α) • y)]
-  simpa only [Algebra.smul_def, map_neg, neg_mul, sub_neg_eq_add, map_mul, map_pow] using he.symm
-
-/-- In the symmetric algebra, a binary sum of pure powers is a product of vectors. -/
-theorem symMonomialRaw_binary_power (V : Type*) [AddCommGroup V] [Module ℂ V]
-    (r : ℕ) (hr : 0 < r) (x y : V) (c : ℂ) :
-    ∃ v : Fin r → V, symMonomialRaw r V v =
-      (SymmetricAlgebra.ι ℂ V x) ^ r + c • (SymmetricAlgebra.ι ℂ V y) ^ r := by
-  obtain ⟨z, hz⟩ := binary_power_factorization r hr
-    (SymmetricAlgebra.ι ℂ V x) (SymmetricAlgebra.ι ℂ V y) c
-  refine ⟨fun i => x - z i • y, ?_⟩
-  simpa only [symMonomialRaw, map_sub, map_smul] using hz
-
-/-- Multiplying a split binary power by a product remains a product, with the exact degree. -/
-theorem symMonomialRaw_binary_power_mul (V : Type*) [AddCommGroup V] [Module ℂ V]
-    (r m : ℕ) (hr : 0 < r) (x y : V) (c : ℂ) (u : Fin m → V) :
-    ∃ v : Fin (r + m) → V, symMonomialRaw (r + m) V v =
-      ((SymmetricAlgebra.ι ℂ V x) ^ r + c • (SymmetricAlgebra.ι ℂ V y) ^ r) *
-        symMonomialRaw m V u := by
-  obtain ⟨w, hw⟩ := symMonomialRaw_binary_power V r hr x y c
-  refine ⟨Fin.append w u, ?_⟩
-  rw [← hw]
-  simp only [symMonomialRaw, Fin.prod_univ_add, Fin.append_left, Fin.append_right]
-
-/-- The split binary form is a genuine product in the prescribed symmetric-power subtype. -/
-theorem symMonomial_binary_power_mul (V : Type*) [AddCommGroup V] [Module ℂ V]
-    (r m : ℕ) (hr : 0 < r) (x y : V) (c : ℂ) (u : Fin m → V) :
-    ∃ v : Fin (r + m) → V, symMonomial (r + m) V v =
-      symMonomial (r + m) V (Fin.append (fun _ : Fin r => x) u) +
-        c • symMonomial (r + m) V (Fin.append (fun _ : Fin r => y) u) := by
-  obtain ⟨v, hv⟩ := symMonomialRaw_binary_power_mul V r m hr x y c u
-  refine ⟨v, Subtype.ext ?_⟩
-  change symMonomialRaw (r + m) V v = _
-  rw [hv]
-  simp only [symMonomial, Submodule.coe_add, Submodule.coe_smul, symMonomialRaw,
-    Fin.prod_univ_add, Fin.append_left, Fin.append_right,
-    Finset.prod_const, Finset.card_univ, Fintype.card_fin, add_mul, smul_mul_assoc]
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.Model
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators
+
+namespace Problem346
+
+/-- Binary forms used in the first step of the vanishing argument split into linear factors. -/
+theorem binary_power_factorization {A : Type*} [CommRing A] [Algebra ℂ A]
+    (r : ℕ) (hr : 0 < r) (x y : A) (c : ℂ) :
+    ∃ z : Fin r → ℂ, (∏ i : Fin r, (x - z i • y)) = x ^ r + c • y ^ r := by
+  classical
+  obtain ⟨α, hα⟩ := IsAlgClosed.exists_pow_nat_eq (-c) hr
+  let ζ : ℂ := Complex.exp (2 * Real.pi * Complex.I / r)
+  have hζ : IsPrimitiveRoot ζ r := Complex.isPrimitiveRoot_exp r hr.ne'
+  have hp : (Polynomial.X ^ r - Polynomial.C (-c) : Polynomial ℂ) =
+      ∏ i ∈ Finset.range r, (Polynomial.X - Polynomial.C (ζ ^ i * α)) :=
+    X_pow_sub_C_eq_prod hζ hr hα
+  have hh := congrArg (fun p : Polynomial ℂ => p.homogenize r) hp
+  have hd : ∀ i ∈ Finset.range r,
+      (Polynomial.X - Polynomial.C (ζ ^ i * α)).natDegree ≤ 1 := by
+    intro i hi
+    rw [Polynomial.natDegree_X_sub_C]
+  have hprod := Polynomial.homogenize_finsetProd (n := fun _ : ℕ => 1) hd
+  simp only [Finset.sum_const, Finset.card_range, smul_eq_mul, mul_one] at hprod
+  rw [hprod] at hh
+  simp only [Polynomial.homogenize_sub, Polynomial.homogenize_X_pow le_rfl,
+    Nat.sub_self, pow_zero, mul_one, Polynomial.homogenize_C,
+    Polynomial.homogenize_X (by decide : (1 : ℕ) ≠ 0), Nat.sub_self, pow_one] at hh
+  have he := congrArg (MvPolynomial.aeval ![x, y]) hh
+  simp only [map_sub, map_pow, map_mul, map_prod, MvPolynomial.aeval_X,
+    Matrix.cons_val_zero, Matrix.cons_val_one,
+    MvPolynomial.aeval_C] at he
+  refine ⟨fun i => ζ ^ (i : ℕ) * α, ?_⟩
+  change (∏ i : Fin r, (x - (ζ ^ (i : ℕ) * α) • y)) = _
+  rw [Fin.prod_univ_eq_prod_range (fun i => x - (ζ ^ i * α) • y)]
+  simpa only [Algebra.smul_def, map_neg, neg_mul, sub_neg_eq_add, map_mul, map_pow] using he.symm
+
+/-- In the symmetric algebra, a binary sum of pure powers is a product of vectors. -/
+theorem symMonomialRaw_binary_power (V : Type*) [AddCommGroup V] [Module ℂ V]
+    (r : ℕ) (hr : 0 < r) (x y : V) (c : ℂ) :
+    ∃ v : Fin r → V, symMonomialRaw r V v =
+      (SymmetricAlgebra.ι ℂ V x) ^ r + c • (SymmetricAlgebra.ι ℂ V y) ^ r := by
+  obtain ⟨z, hz⟩ := binary_power_factorization r hr
+    (SymmetricAlgebra.ι ℂ V x) (SymmetricAlgebra.ι ℂ V y) c
+  refine ⟨fun i => x - z i • y, ?_⟩
+  simpa only [symMonomialRaw, map_sub, map_smul] using hz
+
+/-- Multiplying a split binary power by a product remains a product, with the exact degree. -/
+theorem symMonomialRaw_binary_power_mul (V : Type*) [AddCommGroup V] [Module ℂ V]
+    (r m : ℕ) (hr : 0 < r) (x y : V) (c : ℂ) (u : Fin m → V) :
+    ∃ v : Fin (r + m) → V, symMonomialRaw (r + m) V v =
+      ((SymmetricAlgebra.ι ℂ V x) ^ r + c • (SymmetricAlgebra.ι ℂ V y) ^ r) *
+        symMonomialRaw m V u := by
+  obtain ⟨w, hw⟩ := symMonomialRaw_binary_power V r hr x y c
+  refine ⟨Fin.append w u, ?_⟩
+  rw [← hw]
+  simp only [symMonomialRaw, Fin.prod_univ_add, Fin.append_left, Fin.append_right]
+
+/-- The split binary form is a genuine product in the prescribed symmetric-power subtype. -/
+theorem symMonomial_binary_power_mul (V : Type*) [AddCommGroup V] [Module ℂ V]
+    (r m : ℕ) (hr : 0 < r) (x y : V) (c : ℂ) (u : Fin m → V) :
+    ∃ v : Fin (r + m) → V, symMonomial (r + m) V v =
+      symMonomial (r + m) V (Fin.append (fun _ : Fin r => x) u) +
+        c • symMonomial (r + m) V (Fin.append (fun _ : Fin r => y) u) := by
+  obtain ⟨v, hv⟩ := symMonomialRaw_binary_power_mul V r m hr x y c u
+  refine ⟨v, Subtype.ext ?_⟩
+  change symMonomialRaw (r + m) V v = _
+  rw [hv]
+  simp only [symMonomial, Submodule.coe_add, Submodule.coe_smul, symMonomialRaw,
+    Fin.prod_univ_add, Fin.append_left, Fin.append_right,
+    Finset.prod_const, Finset.card_univ, Fintype.card_fin, add_mul, smul_mul_assoc]
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/BinaryVanishing.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/BinaryVanishing.lean
index f86465ff38cbf48bfa5e79cb4b230a210086acec..b62a3583a16912299ba8bcc52ba6e5a0fbe96c57 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/BinaryVanishing.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/BinaryVanishing.lean
@@ -1,36 +1,36 @@
-import OAI.RepresentationTheory.FoulkesHowe.BinaryFactorization
-import OAI.RepresentationTheory.FoulkesHowe.DiagonalCoefficient
-import OAI.RepresentationTheory.FoulkesHowe.Multiplication
-
-namespace OAI
-
-noncomputable section
-
-namespace Problem346
-
-universe u
-variable {V : Type u} [AddCommGroup V] [Module ℂ V]
-
-/-- The split-pencil consequence of diagonal vanishing, expressed using the
-graded multiplication and first-slot convention of the induction. -/
-theorem binary_diagonal_to_mixed_products (r m : ℕ) (hr : 0 < r)
-    (T : SymmetricMultilinearForm (r+1) (r+m) V)
-    (hT : IsSymmetricMultilinearForm (r+1) (r+m) V T)
-    (hdiag : ∀ y : Fin (r+m) → V, T (fun _ => symMonomial (r+m) V y) = 0)
-    (x t : V) (y : Fin m → V) :
-    T (Fin.cons
-      (symPowMul r m (symMonomial r V (fun _ => t)) (symMonomial m V y))
-      (fun _ : Fin r =>
-        symPowMul r m (symMonomial r V (fun _ => x)) (symMonomial m V y))) = 0 := by
-  apply symmetric_diagonal_cons_zero T hT
-  intro c
-  obtain ⟨v, hv⟩ := symMonomial_binary_power_mul V r m hr x t c y
-  simp only [symPowMul_symMonomial]
-  rw [← hv]
-  exact hdiag v
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.BinaryFactorization
+import OAI.RepresentationTheory.FoulkesHowe.DiagonalCoefficient
+import OAI.RepresentationTheory.FoulkesHowe.Multiplication
+
+namespace OAI
+
+noncomputable section
+
+namespace Problem346
+
+universe u
+variable {V : Type u} [AddCommGroup V] [Module ℂ V]
+
+/-- The split-pencil consequence of diagonal vanishing, expressed using the
+graded multiplication and first-slot convention of the induction. -/
+theorem binary_diagonal_to_mixed_products (r m : ℕ) (hr : 0 < r)
+    (T : SymmetricMultilinearForm (r+1) (r+m) V)
+    (hT : IsSymmetricMultilinearForm (r+1) (r+m) V T)
+    (hdiag : ∀ y : Fin (r+m) → V, T (fun _ => symMonomial (r+m) V y) = 0)
+    (x t : V) (y : Fin m → V) :
+    T (Fin.cons
+      (symPowMul r m (symMonomial r V (fun _ => t)) (symMonomial m V y))
+      (fun _ : Fin r =>
+        symPowMul r m (symMonomial r V (fun _ => x)) (symMonomial m V y))) = 0 := by
+  apply symmetric_diagonal_cons_zero T hT
+  intro c
+  obtain ⟨v, hv⟩ := symMonomial_binary_power_mul V r m hr x t c y
+  simp only [symPowMul_symMonomial]
+  rw [← hv]
+  exact hdiag v
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/BlockDerivation.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/BlockDerivation.lean
index ef1960a7c35fd72f2b9acd187190a1861940963a..ffa208d81ae52636b7fb1e0e539649acf9a6bdc4 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/BlockDerivation.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/BlockDerivation.lean
@@ -1,102 +1,102 @@
-import Mathlib
-
-namespace OAI
-
-/-! Coordinate-block derivations used in the differential-shift argument. -/
-
-noncomputable section
-
-namespace Problem346
-
-open MvPolynomial
-open scoped BigOperators
-
-variable (R : Type*) [CommRing R]
-variable {ι κ : Type*} [DecidableEq ι]
-
-/-- Move one unit of polynomial degree from block `u` to block `v`.
-The other coordinate blocks are spectators. -/
-def blockShift (u v : ι) :
-    Derivation R (MvPolynomial (ι × κ) R) (MvPolynomial (ι × κ) R) :=
-  MvPolynomial.mkDerivation R (fun k => if k.1 = u then X (v, k.2) else 0)
-
-@[simp] theorem blockShift_X (u v k : ι) (l : κ) :
-    blockShift (κ := κ) R u v (X (k, l)) = if k = u then X (v, l) else 0 :=
-  MvPolynomial.mkDerivation_X R _ _
-
-/-- The commutator of opposite block shifts is the difference of the two
-block Euler operators. This identity needs no finiteness assumption. -/
-theorem blockShift_commutator (u v : ι) :
-    ⁅blockShift (κ := κ) R v u, blockShift (κ := κ) R u v⁆ =
-      blockShift (κ := κ) R u u - blockShift (κ := κ) R v v := by
-  apply MvPolynomial.derivation_ext
-  rintro ⟨k, l⟩
-  simp only [Derivation.commutator_apply, Derivation.sub_apply, blockShift_X]
-  by_cases huv : u = v
-  · subst v
-    simp
-  · by_cases hu : k = u <;> by_cases hv : k = v <;>
-      simp_all
-
-/-- The weight operator raises the weight of the reverse shift by two. -/
-theorem blockShift_weight_commutator (u v : ι) (huv : u ≠ v) :
-    ⁅blockShift (κ := κ) R u u - blockShift (κ := κ) R v v, blockShift (κ := κ) R v u⁆ =
-      (2 : R) • blockShift (κ := κ) R v u := by
-  apply MvPolynomial.derivation_ext
-  rintro ⟨k, l⟩
-  simp only [Derivation.commutator_apply, Derivation.sub_apply,
-    Derivation.smul_apply, blockShift_X]
-  by_cases hu : k = u <;> by_cases hv : k = v <;>
-    simp [hu, hv, huv, Ne.symm huv, two_smul]
-
-/-- Pointwise version of the opposite-shift commutator. -/
-theorem blockShift_commutator_apply (u v : ι) (f : MvPolynomial (ι × κ) R) :
-    blockShift (κ := κ) R v u (blockShift (κ := κ) R u v f) -
-        blockShift (κ := κ) R u v (blockShift (κ := κ) R v u f) =
-      blockShift (κ := κ) R u u f - blockShift (κ := κ) R v v f := by
-  have h := congrArg (fun D : Derivation R (MvPolynomial (ι × κ) R)
-      (MvPolynomial (ι × κ) R) => D f) (blockShift_commutator R u v)
-  exact h
-
-/-- Pointwise version of the weight/raising commutator. -/
-theorem blockShift_weight_commutator_apply (u v : ι) (huv : u ≠ v)
-    (f : MvPolynomial (ι × κ) R) :
-    (blockShift (κ := κ) R u u - blockShift (κ := κ) R v v) (blockShift (κ := κ) R v u f) -
-      blockShift (κ := κ) R v u ((blockShift (κ := κ) R u u - blockShift (κ := κ) R v v) f) =
-      (2 : R) • blockShift (κ := κ) R v u f := by
-  have h := congrArg (fun D : Derivation R (MvPolynomial (ι × κ) R)
-      (MvPolynomial (ι × κ) R) => D f) (blockShift_weight_commutator R u v huv)
-  exact h
-
-/-- An off-diagonal block shift squares to zero on each coordinate generator. -/
-theorem blockShift_square_X (u v k : ι) (huv : u ≠ v) (l : κ) :
-    blockShift (κ := κ) R u v (blockShift R u v (X (k, l))) = 0 := by
-  by_cases h : k = u <;> simp [h, Ne.symm huv]
-
-/-- With finitely many coordinates, `blockShift` is the usual directional
-partial derivative, written as a sum of coordinate derivations. -/
-theorem blockShift_eq_sum [Fintype κ] (u v : ι) :
-    blockShift (κ := κ) R u v =
-      ∑ l : κ, (X (v, l) : MvPolynomial (ι × κ) R) • pderiv (u, l) := by
-  classical
-  apply MvPolynomial.derivation_ext
-  rintro ⟨k, m⟩
-  rw [blockShift_X]
-  have heval (ds : κ → Derivation R (MvPolynomial (ι × κ) R)
-      (MvPolynomial (ι × κ) R)) (f : MvPolynomial (ι × κ) R) :
-      (∑ l, ds l) f = ∑ l, ds l f := by
-    simpa only [Derivation.coeFnAddMonoidHom_apply, Finset.sum_apply] using
-      congrFun (map_sum Derivation.coeFnAddMonoidHom ds Finset.univ) f
-  rw [heval]
-  simp only [Derivation.smul_apply, smul_eq_mul, pderiv_X, Pi.single_apply,
-    Prod.mk.injEq]
-  by_cases h : k = u
-  · subst k
-    simp
-  · simp [h]
-
-end Problem346
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-! Coordinate-block derivations used in the differential-shift argument. -/
+
+noncomputable section
+
+namespace Problem346
+
+open MvPolynomial
+open scoped BigOperators
+
+variable (R : Type*) [CommRing R]
+variable {ι κ : Type*} [DecidableEq ι]
+
+/-- Move one unit of polynomial degree from block `u` to block `v`.
+The other coordinate blocks are spectators. -/
+def blockShift (u v : ι) :
+    Derivation R (MvPolynomial (ι × κ) R) (MvPolynomial (ι × κ) R) :=
+  MvPolynomial.mkDerivation R (fun k => if k.1 = u then X (v, k.2) else 0)
+
+@[simp] theorem blockShift_X (u v k : ι) (l : κ) :
+    blockShift (κ := κ) R u v (X (k, l)) = if k = u then X (v, l) else 0 :=
+  MvPolynomial.mkDerivation_X R _ _
+
+/-- The commutator of opposite block shifts is the difference of the two
+block Euler operators. This identity needs no finiteness assumption. -/
+theorem blockShift_commutator (u v : ι) :
+    ⁅blockShift (κ := κ) R v u, blockShift (κ := κ) R u v⁆ =
+      blockShift (κ := κ) R u u - blockShift (κ := κ) R v v := by
+  apply MvPolynomial.derivation_ext
+  rintro ⟨k, l⟩
+  simp only [Derivation.commutator_apply, Derivation.sub_apply, blockShift_X]
+  by_cases huv : u = v
+  · subst v
+    simp
+  · by_cases hu : k = u <;> by_cases hv : k = v <;>
+      simp_all
+
+/-- The weight operator raises the weight of the reverse shift by two. -/
+theorem blockShift_weight_commutator (u v : ι) (huv : u ≠ v) :
+    ⁅blockShift (κ := κ) R u u - blockShift (κ := κ) R v v, blockShift (κ := κ) R v u⁆ =
+      (2 : R) • blockShift (κ := κ) R v u := by
+  apply MvPolynomial.derivation_ext
+  rintro ⟨k, l⟩
+  simp only [Derivation.commutator_apply, Derivation.sub_apply,
+    Derivation.smul_apply, blockShift_X]
+  by_cases hu : k = u <;> by_cases hv : k = v <;>
+    simp [hu, hv, huv, Ne.symm huv, two_smul]
+
+/-- Pointwise version of the opposite-shift commutator. -/
+theorem blockShift_commutator_apply (u v : ι) (f : MvPolynomial (ι × κ) R) :
+    blockShift (κ := κ) R v u (blockShift (κ := κ) R u v f) -
+        blockShift (κ := κ) R u v (blockShift (κ := κ) R v u f) =
+      blockShift (κ := κ) R u u f - blockShift (κ := κ) R v v f := by
+  have h := congrArg (fun D : Derivation R (MvPolynomial (ι × κ) R)
+      (MvPolynomial (ι × κ) R) => D f) (blockShift_commutator R u v)
+  exact h
+
+/-- Pointwise version of the weight/raising commutator. -/
+theorem blockShift_weight_commutator_apply (u v : ι) (huv : u ≠ v)
+    (f : MvPolynomial (ι × κ) R) :
+    (blockShift (κ := κ) R u u - blockShift (κ := κ) R v v) (blockShift (κ := κ) R v u f) -
+      blockShift (κ := κ) R v u ((blockShift (κ := κ) R u u - blockShift (κ := κ) R v v) f) =
+      (2 : R) • blockShift (κ := κ) R v u f := by
+  have h := congrArg (fun D : Derivation R (MvPolynomial (ι × κ) R)
+      (MvPolynomial (ι × κ) R) => D f) (blockShift_weight_commutator R u v huv)
+  exact h
+
+/-- An off-diagonal block shift squares to zero on each coordinate generator. -/
+theorem blockShift_square_X (u v k : ι) (huv : u ≠ v) (l : κ) :
+    blockShift (κ := κ) R u v (blockShift R u v (X (k, l))) = 0 := by
+  by_cases h : k = u <;> simp [h, Ne.symm huv]
+
+/-- With finitely many coordinates, `blockShift` is the usual directional
+partial derivative, written as a sum of coordinate derivations. -/
+theorem blockShift_eq_sum [Fintype κ] (u v : ι) :
+    blockShift (κ := κ) R u v =
+      ∑ l : κ, (X (v, l) : MvPolynomial (ι × κ) R) • pderiv (u, l) := by
+  classical
+  apply MvPolynomial.derivation_ext
+  rintro ⟨k, m⟩
+  rw [blockShift_X]
+  have heval (ds : κ → Derivation R (MvPolynomial (ι × κ) R)
+      (MvPolynomial (ι × κ) R)) (f : MvPolynomial (ι × κ) R) :
+      (∑ l, ds l) f = ∑ l, ds l f := by
+    simpa only [Derivation.coeFnAddMonoidHom_apply, Finset.sum_apply] using
+      congrFun (map_sum Derivation.coeFnAddMonoidHom ds Finset.univ) f
+  rw [heval]
+  simp only [Derivation.smul_apply, smul_eq_mul, pderiv_X, Pi.single_apply,
+    Prod.mk.injEq]
+  by_cases h : k = u
+  · subst k
+    simp
+  · simp [h]
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/BlockInjectivity.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/BlockInjectivity.lean
index a04017f990477ce7fb396d051a3c214cae626bed..f348516a1b9a3e08a355c6d6e78396dc08de0b0b 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/BlockInjectivity.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/BlockInjectivity.lean
@@ -1,58 +1,58 @@
-import OAI.RepresentationTheory.FoulkesHowe.DerivationInjectivity
-import OAI.RepresentationTheory.FoulkesHowe.PolynomialShift
-import OAI.RepresentationTheory.FoulkesHowe.BlockWeight
-
-namespace OAI
-
-noncomputable section
-
-namespace Problem346
-namespace PolynomialShift
-
-open MvPolynomial PolynomialWeights
-
-variable {B I : Type*} [Fintype B] [Fintype I] [DecidableEq B]
-
-/-- A polynomial block shift is injective in strictly positive source-minus-destination
-weight. All other variable blocks are arbitrary spectators. -/
-theorem eq_zero_of_shift_eq_zero (src dst : B) (hsd : src ≠ dst)
-    {f : MvPolynomial (B × I) ℂ} {p q : ℕ} (hpq : q < p)
-    (hp : IsBlockHomogeneous src f p) (hq : IsBlockHomogeneous dst f q)
-    (hDf : shift src dst f = 0) : f = 0 := by
-  refine derivation_eq_zero_of_positive_weight
-    (derivation src dst) (derivation dst src)
-    (derivation src src - derivation dst dst)
-    (commutator src dst hsd) (weight_commutator src dst hsd) ?_
-    f p q hpq ?_ hDf
-  · rintro ⟨b, i⟩
-    by_cases hb : b = dst
-    · subst b
-      simp [hsd]
-    · simp [hb]
-  · apply block_difference_eigenvalue src dst _ _ _ _ hp hq
-    · intro b i
-      by_cases hb : b = src
-      · subst b; simp
-      · simp [hb]
-    · intro b i
-      by_cases hb : b = dst
-      · subst b; simp
-      · simp [hb]
-
-/-- Injectivity on the intersection of two specified block-homogeneous pieces. -/
-theorem shift_injOn (src dst : B) (hsd : src ≠ dst) (p q : ℕ) (hpq : q < p) :
-    Set.InjOn (shift (K := ℂ) (I := I) src dst)
-      {f | IsBlockHomogeneous src f p ∧ IsBlockHomogeneous dst f q} := by
-  intro f hf g hg hfg
-  apply sub_eq_zero.mp
-  apply eq_zero_of_shift_eq_zero src dst hsd hpq
-  · exact (weightedHomogeneousSubmodule ℂ (blockWeight src) p).sub_mem hf.1 hg.1
-  · exact (weightedHomogeneousSubmodule ℂ (blockWeight dst) q).sub_mem hf.2 hg.2
-  · rw [map_sub, hfg, sub_self]
-
-end PolynomialShift
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.DerivationInjectivity
+import OAI.RepresentationTheory.FoulkesHowe.PolynomialShift
+import OAI.RepresentationTheory.FoulkesHowe.BlockWeight
+
+namespace OAI
+
+noncomputable section
+
+namespace Problem346
+namespace PolynomialShift
+
+open MvPolynomial PolynomialWeights
+
+variable {B I : Type*} [Fintype B] [Fintype I] [DecidableEq B]
+
+/-- A polynomial block shift is injective in strictly positive source-minus-destination
+weight. All other variable blocks are arbitrary spectators. -/
+theorem eq_zero_of_shift_eq_zero (src dst : B) (hsd : src ≠ dst)
+    {f : MvPolynomial (B × I) ℂ} {p q : ℕ} (hpq : q < p)
+    (hp : IsBlockHomogeneous src f p) (hq : IsBlockHomogeneous dst f q)
+    (hDf : shift src dst f = 0) : f = 0 := by
+  refine derivation_eq_zero_of_positive_weight
+    (derivation src dst) (derivation dst src)
+    (derivation src src - derivation dst dst)
+    (commutator src dst hsd) (weight_commutator src dst hsd) ?_
+    f p q hpq ?_ hDf
+  · rintro ⟨b, i⟩
+    by_cases hb : b = dst
+    · subst b
+      simp [hsd]
+    · simp [hb]
+  · apply block_difference_eigenvalue src dst _ _ _ _ hp hq
+    · intro b i
+      by_cases hb : b = src
+      · subst b; simp
+      · simp [hb]
+    · intro b i
+      by_cases hb : b = dst
+      · subst b; simp
+      · simp [hb]
+
+/-- Injectivity on the intersection of two specified block-homogeneous pieces. -/
+theorem shift_injOn (src dst : B) (hsd : src ≠ dst) (p q : ℕ) (hpq : q < p) :
+    Set.InjOn (shift (K := ℂ) (I := I) src dst)
+      {f | IsBlockHomogeneous src f p ∧ IsBlockHomogeneous dst f q} := by
+  intro f hf g hg hfg
+  apply sub_eq_zero.mp
+  apply eq_zero_of_shift_eq_zero src dst hsd hpq
+  · exact (weightedHomogeneousSubmodule ℂ (blockWeight src) p).sub_mem hf.1 hg.1
+  · exact (weightedHomogeneousSubmodule ℂ (blockWeight dst) q).sub_mem hf.2 hg.2
+  · rw [map_sub, hfg, sub_self]
+
+end PolynomialShift
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/BlockWeight.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/BlockWeight.lean
index 673495439c60607f91a7d1c5f6fb89690ba59e64..7c6138151aba05b012f6e017fe8573860def7cd4 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/BlockWeight.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/BlockWeight.lean
@@ -1,169 +1,169 @@
-import Mathlib
-
-namespace OAI
-
-noncomputable section
-open scoped BigOperators
-
-namespace Problem346
-namespace PolynomialWeights
-
-open MvPolynomial
-
-variable {R σ : Type*} [CommSemiring R] [Fintype σ]
-
-/-- The Euler derivation associated with natural-number weights. -/
-def weightedEuler (w : σ → ℕ) :
-    Derivation R (MvPolynomial σ R) (MvPolynomial σ R) :=
-  ∑ i : σ, (w i) • ((X i : MvPolynomial σ R) • pderiv i)
-
-@[simp] theorem weightedEuler_apply (w : σ → ℕ) (f : MvPolynomial σ R) :
-    weightedEuler w f = ∑ i : σ, w i • (X i * pderiv i f) := by
-  classical
-  unfold weightedEuler
-  induction (Finset.univ : Finset σ) using Finset.induction_on with
-  | empty => simp
-  | @insert a s ha ih => simp [Finset.sum_insert ha, Derivation.smul_apply, ih, smul_eq_mul]
-
-@[simp] theorem weightedEuler_X (w : σ → ℕ) (i : σ) :
-    weightedEuler (R := R) w (X i) = w i • X i := by
-  classical
-  simp [weightedEuler_apply, Pi.single_apply]
-
-/-- Euler's identity, expressed using a bundled derivation. -/
-theorem weightedEuler_of_homogeneous (w : σ → ℕ) {n : ℕ}
-    {f : MvPolynomial σ R} (hf : f.IsWeightedHomogeneous w n) :
-    weightedEuler w f = n • f := by
-  rw [weightedEuler_apply]
-  exact hf.sum_weight_X_mul_pderiv
-
-/-- Any derivation with the prescribed diagonal action on generators is Euler. -/
-theorem derivation_eq_weightedEuler (w : σ → ℕ)
-    (N : Derivation R (MvPolynomial σ R) (MvPolynomial σ R))
-    (hN : ∀ i, N (X i) = w i • X i) : N = weightedEuler w := by
-  apply MvPolynomial.derivation_ext
-  intro i
-  rw [hN, weightedEuler_X]
-
-/-- A homogeneous polynomial is an eigenvector for its weighted Euler derivation. -/
-theorem derivation_homogeneous_eigenvalue (w : σ → ℕ)
-    (N : Derivation R (MvPolynomial σ R) (MvPolynomial σ R))
-    (hN : ∀ i, N (X i) = w i • X i)
-    {n : ℕ} {f : MvPolynomial σ R} (hf : f.IsWeightedHomogeneous w n) :
-    N f = n • f := by
-  rw [derivation_eq_weightedEuler w N hN]
-  exact weightedEuler_of_homogeneous w hf
-
-section Blocks
-
-variable {β ι : Type*} [Fintype β] [Fintype ι] [DecidableEq β]
-
-/-- Weight one on one coordinate block, and zero on all spectator blocks. -/
-def blockWeight (u : β) (ki : β × ι) : ℕ := if ki.1 = u then 1 else 0
-
-/-- A polynomial is homogeneous of the specified degree in one block. -/
-def IsBlockHomogeneous (u : β) (f : MvPolynomial (β × ι) R) (n : ℕ) : Prop :=
-  f.IsWeightedHomogeneous (blockWeight u) n
-
-omit [Fintype β] [Fintype ι] in
-@[simp] theorem blockWeight_apply (u k : β) (i : ι) :
-    blockWeight u (k, i) = if k = u then 1 else 0 := rfl
-
-theorem block_euler_eigenvalue (u : β)
-    (N : Derivation R (MvPolynomial (β × ι) R) (MvPolynomial (β × ι) R))
-    (hN : ∀ k i, N (X (k, i)) = if k = u then X (k, i) else 0)
-    {n : ℕ} {f : MvPolynomial (β × ι) R} (hf : IsBlockHomogeneous u f n) :
-    N f = n • f := by
-  apply derivation_homogeneous_eigenvalue (blockWeight u) N _ hf
-  rintro ⟨k, i⟩
-  simp only [hN, blockWeight_apply]
-  split_ifs <;> simp
-
-/-- The difference of block counting operators has the difference of the block degrees
-as eigenvalue. This is the weight used by the positive-weight shift lemma. -/
-theorem block_difference_eigenvalue (u v : β)
-    (Nu Nv : Derivation ℂ (MvPolynomial (β × ι) ℂ) (MvPolynomial (β × ι) ℂ))
-    (hu : ∀ k i, Nu (X (k, i)) = if k = u then X (k, i) else 0)
-    (hv : ∀ k i, Nv (X (k, i)) = if k = v then X (k, i) else 0)
-    {p q : ℕ} {f : MvPolynomial (β × ι) ℂ}
-    (hp : IsBlockHomogeneous u f p) (hq : IsBlockHomogeneous v f q) :
-    (Nu - Nv) f = ((p : ℂ) - (q : ℂ)) • f := by
-  change Nu f - Nv f = _
-  rw [block_euler_eigenvalue u Nu hu hp, block_euler_eigenvalue v Nv hv hq]
-  rw [sub_smul]
-  simp only [Nat.cast_smul_eq_nsmul]
-
-theorem block_difference_nat_eigenvalue (u v : β)
-    (Nu Nv : Derivation ℂ (MvPolynomial (β × ι) ℂ) (MvPolynomial (β × ι) ℂ))
-    (hu : ∀ k i, Nu (X (k, i)) = if k = u then X (k, i) else 0)
-    (hv : ∀ k i, Nv (X (k, i)) = if k = v then X (k, i) else 0)
-    {p q : ℕ} (hpq : q ≤ p) {f : MvPolynomial (β × ι) ℂ}
-    (hp : IsBlockHomogeneous u f p) (hq : IsBlockHomogeneous v f q) :
-    (Nu - Nv) f = ((p - q : ℕ) : ℂ) • f := by
-  rw [Nat.cast_sub hpq]
-  exact block_difference_eigenvalue u v Nu Nv hu hv hp hq
-
-omit [Fintype β] [Fintype ι] in
-/-- Differentiating in a block lowers its degree by one, provided that degree is positive. -/
-theorem block_pderiv_source {u : β} {n : ℕ} {f : MvPolynomial (β × ι) R}
-    (hf : IsBlockHomogeneous u f n) (hn : 0 < n) (i : ι) :
-    IsBlockHomogeneous u (pderiv (u, i) f) (n - 1) := by
-  apply IsWeightedHomogeneous.pderiv hf
-  simp only [blockWeight_apply]
-  exact Nat.sub_add_cancel hn
-
-omit [Fintype β] [Fintype ι] in
-/-- Differentiating in another block leaves the indicated block degree unchanged. -/
-theorem block_pderiv_other {u k : β} (hku : k ≠ u) {n : ℕ}
-    {f : MvPolynomial (β × ι) R} (hf : IsBlockHomogeneous u f n) (i : ι) :
-    IsBlockHomogeneous u (pderiv (k, i) f) n := by
-  apply IsWeightedHomogeneous.pderiv hf
-  simp [blockWeight_apply, hku]
-
-omit [Fintype β] in
-/-- A directional shift lowers its source block degree by one. -/
-theorem block_shift_degree_source {u v : β} (huv : u ≠ v) {n : ℕ}
-    {f : MvPolynomial (β × ι) R} (hf : IsBlockHomogeneous u f n) (hn : 0 < n) :
-    IsBlockHomogeneous u (∑ i : ι, X (v, i) * pderiv (u, i) f) (n - 1) := by
-  apply (weightedHomogeneousSubmodule R (blockWeight u) (n - 1)).sum_mem
-  intro i hi
-  have hx : (X (v, i) : MvPolynomial (β × ι) R).IsWeightedHomogeneous
-      (blockWeight u) 0 := by
-    simpa [blockWeight_apply, Ne.symm huv] using
-      (isWeightedHomogeneous_X (R := R) (blockWeight u) (v, i))
-  simpa using hx.mul (block_pderiv_source hf hn i)
-
-omit [Fintype β] in
-/-- A directional shift raises its destination block degree by one. -/
-theorem block_shift_degree_destination {u v : β} (huv : u ≠ v) {n : ℕ}
-    {f : MvPolynomial (β × ι) R} (hf : IsBlockHomogeneous v f n) :
-    IsBlockHomogeneous v (∑ i : ι, X (v, i) * pderiv (u, i) f) (n + 1) := by
-  apply (weightedHomogeneousSubmodule R (blockWeight v) (n + 1)).sum_mem
-  intro i hi
-  have hx : (X (v, i) : MvPolynomial (β × ι) R).IsWeightedHomogeneous
-      (blockWeight v) 1 := by
-    simpa [blockWeight_apply] using
-      (isWeightedHomogeneous_X (R := R) (blockWeight v) (v, i))
-  simpa [Nat.add_comm] using hx.mul (block_pderiv_other huv hf i)
-
-omit [Fintype β] in
-/-- A directional shift leaves every spectator block degree unchanged. -/
-theorem block_shift_degree_spectator {u v k : β} (huk : u ≠ k) (hvk : v ≠ k)
-    {n : ℕ} {f : MvPolynomial (β × ι) R} (hf : IsBlockHomogeneous k f n) :
-    IsBlockHomogeneous k (∑ i : ι, X (v, i) * pderiv (u, i) f) n := by
-  apply (weightedHomogeneousSubmodule R (blockWeight k) n).sum_mem
-  intro i hi
-  have hx : (X (v, i) : MvPolynomial (β × ι) R).IsWeightedHomogeneous
-      (blockWeight k) 0 := by
-    simpa [blockWeight_apply, hvk] using
-      (isWeightedHomogeneous_X (R := R) (blockWeight k) (v, i))
-  simpa using hx.mul (block_pderiv_other huk hf i)
-
-end Blocks
-
-end PolynomialWeights
-end Problem346
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators
+
+namespace Problem346
+namespace PolynomialWeights
+
+open MvPolynomial
+
+variable {R σ : Type*} [CommSemiring R] [Fintype σ]
+
+/-- The Euler derivation associated with natural-number weights. -/
+def weightedEuler (w : σ → ℕ) :
+    Derivation R (MvPolynomial σ R) (MvPolynomial σ R) :=
+  ∑ i : σ, (w i) • ((X i : MvPolynomial σ R) • pderiv i)
+
+@[simp] theorem weightedEuler_apply (w : σ → ℕ) (f : MvPolynomial σ R) :
+    weightedEuler w f = ∑ i : σ, w i • (X i * pderiv i f) := by
+  classical
+  unfold weightedEuler
+  induction (Finset.univ : Finset σ) using Finset.induction_on with
+  | empty => simp
+  | @insert a s ha ih => simp [Finset.sum_insert ha, Derivation.smul_apply, ih, smul_eq_mul]
+
+@[simp] theorem weightedEuler_X (w : σ → ℕ) (i : σ) :
+    weightedEuler (R := R) w (X i) = w i • X i := by
+  classical
+  simp [weightedEuler_apply, Pi.single_apply]
+
+/-- Euler's identity, expressed using a bundled derivation. -/
+theorem weightedEuler_of_homogeneous (w : σ → ℕ) {n : ℕ}
+    {f : MvPolynomial σ R} (hf : f.IsWeightedHomogeneous w n) :
+    weightedEuler w f = n • f := by
+  rw [weightedEuler_apply]
+  exact hf.sum_weight_X_mul_pderiv
+
+/-- Any derivation with the prescribed diagonal action on generators is Euler. -/
+theorem derivation_eq_weightedEuler (w : σ → ℕ)
+    (N : Derivation R (MvPolynomial σ R) (MvPolynomial σ R))
+    (hN : ∀ i, N (X i) = w i • X i) : N = weightedEuler w := by
+  apply MvPolynomial.derivation_ext
+  intro i
+  rw [hN, weightedEuler_X]
+
+/-- A homogeneous polynomial is an eigenvector for its weighted Euler derivation. -/
+theorem derivation_homogeneous_eigenvalue (w : σ → ℕ)
+    (N : Derivation R (MvPolynomial σ R) (MvPolynomial σ R))
+    (hN : ∀ i, N (X i) = w i • X i)
+    {n : ℕ} {f : MvPolynomial σ R} (hf : f.IsWeightedHomogeneous w n) :
+    N f = n • f := by
+  rw [derivation_eq_weightedEuler w N hN]
+  exact weightedEuler_of_homogeneous w hf
+
+section Blocks
+
+variable {β ι : Type*} [Fintype β] [Fintype ι] [DecidableEq β]
+
+/-- Weight one on one coordinate block, and zero on all spectator blocks. -/
+def blockWeight (u : β) (ki : β × ι) : ℕ := if ki.1 = u then 1 else 0
+
+/-- A polynomial is homogeneous of the specified degree in one block. -/
+def IsBlockHomogeneous (u : β) (f : MvPolynomial (β × ι) R) (n : ℕ) : Prop :=
+  f.IsWeightedHomogeneous (blockWeight u) n
+
+omit [Fintype β] [Fintype ι] in
+@[simp] theorem blockWeight_apply (u k : β) (i : ι) :
+    blockWeight u (k, i) = if k = u then 1 else 0 := rfl
+
+theorem block_euler_eigenvalue (u : β)
+    (N : Derivation R (MvPolynomial (β × ι) R) (MvPolynomial (β × ι) R))
+    (hN : ∀ k i, N (X (k, i)) = if k = u then X (k, i) else 0)
+    {n : ℕ} {f : MvPolynomial (β × ι) R} (hf : IsBlockHomogeneous u f n) :
+    N f = n • f := by
+  apply derivation_homogeneous_eigenvalue (blockWeight u) N _ hf
+  rintro ⟨k, i⟩
+  simp only [hN, blockWeight_apply]
+  split_ifs <;> simp
+
+/-- The difference of block counting operators has the difference of the block degrees
+as eigenvalue. This is the weight used by the positive-weight shift lemma. -/
+theorem block_difference_eigenvalue (u v : β)
+    (Nu Nv : Derivation ℂ (MvPolynomial (β × ι) ℂ) (MvPolynomial (β × ι) ℂ))
+    (hu : ∀ k i, Nu (X (k, i)) = if k = u then X (k, i) else 0)
+    (hv : ∀ k i, Nv (X (k, i)) = if k = v then X (k, i) else 0)
+    {p q : ℕ} {f : MvPolynomial (β × ι) ℂ}
+    (hp : IsBlockHomogeneous u f p) (hq : IsBlockHomogeneous v f q) :
+    (Nu - Nv) f = ((p : ℂ) - (q : ℂ)) • f := by
+  change Nu f - Nv f = _
+  rw [block_euler_eigenvalue u Nu hu hp, block_euler_eigenvalue v Nv hv hq]
+  rw [sub_smul]
+  simp only [Nat.cast_smul_eq_nsmul]
+
+theorem block_difference_nat_eigenvalue (u v : β)
+    (Nu Nv : Derivation ℂ (MvPolynomial (β × ι) ℂ) (MvPolynomial (β × ι) ℂ))
+    (hu : ∀ k i, Nu (X (k, i)) = if k = u then X (k, i) else 0)
+    (hv : ∀ k i, Nv (X (k, i)) = if k = v then X (k, i) else 0)
+    {p q : ℕ} (hpq : q ≤ p) {f : MvPolynomial (β × ι) ℂ}
+    (hp : IsBlockHomogeneous u f p) (hq : IsBlockHomogeneous v f q) :
+    (Nu - Nv) f = ((p - q : ℕ) : ℂ) • f := by
+  rw [Nat.cast_sub hpq]
+  exact block_difference_eigenvalue u v Nu Nv hu hv hp hq
+
+omit [Fintype β] [Fintype ι] in
+/-- Differentiating in a block lowers its degree by one, provided that degree is positive. -/
+theorem block_pderiv_source {u : β} {n : ℕ} {f : MvPolynomial (β × ι) R}
+    (hf : IsBlockHomogeneous u f n) (hn : 0 < n) (i : ι) :
+    IsBlockHomogeneous u (pderiv (u, i) f) (n - 1) := by
+  apply IsWeightedHomogeneous.pderiv hf
+  simp only [blockWeight_apply]
+  exact Nat.sub_add_cancel hn
+
+omit [Fintype β] [Fintype ι] in
+/-- Differentiating in another block leaves the indicated block degree unchanged. -/
+theorem block_pderiv_other {u k : β} (hku : k ≠ u) {n : ℕ}
+    {f : MvPolynomial (β × ι) R} (hf : IsBlockHomogeneous u f n) (i : ι) :
+    IsBlockHomogeneous u (pderiv (k, i) f) n := by
+  apply IsWeightedHomogeneous.pderiv hf
+  simp [blockWeight_apply, hku]
+
+omit [Fintype β] in
+/-- A directional shift lowers its source block degree by one. -/
+theorem block_shift_degree_source {u v : β} (huv : u ≠ v) {n : ℕ}
+    {f : MvPolynomial (β × ι) R} (hf : IsBlockHomogeneous u f n) (hn : 0 < n) :
+    IsBlockHomogeneous u (∑ i : ι, X (v, i) * pderiv (u, i) f) (n - 1) := by
+  apply (weightedHomogeneousSubmodule R (blockWeight u) (n - 1)).sum_mem
+  intro i hi
+  have hx : (X (v, i) : MvPolynomial (β × ι) R).IsWeightedHomogeneous
+      (blockWeight u) 0 := by
+    simpa [blockWeight_apply, Ne.symm huv] using
+      (isWeightedHomogeneous_X (R := R) (blockWeight u) (v, i))
+  simpa using hx.mul (block_pderiv_source hf hn i)
+
+omit [Fintype β] in
+/-- A directional shift raises its destination block degree by one. -/
+theorem block_shift_degree_destination {u v : β} (huv : u ≠ v) {n : ℕ}
+    {f : MvPolynomial (β × ι) R} (hf : IsBlockHomogeneous v f n) :
+    IsBlockHomogeneous v (∑ i : ι, X (v, i) * pderiv (u, i) f) (n + 1) := by
+  apply (weightedHomogeneousSubmodule R (blockWeight v) (n + 1)).sum_mem
+  intro i hi
+  have hx : (X (v, i) : MvPolynomial (β × ι) R).IsWeightedHomogeneous
+      (blockWeight v) 1 := by
+    simpa [blockWeight_apply] using
+      (isWeightedHomogeneous_X (R := R) (blockWeight v) (v, i))
+  simpa [Nat.add_comm] using hx.mul (block_pderiv_other huv hf i)
+
+omit [Fintype β] in
+/-- A directional shift leaves every spectator block degree unchanged. -/
+theorem block_shift_degree_spectator {u v k : β} (huk : u ≠ k) (hvk : v ≠ k)
+    {n : ℕ} {f : MvPolynomial (β × ι) R} (hf : IsBlockHomogeneous k f n) :
+    IsBlockHomogeneous k (∑ i : ι, X (v, i) * pderiv (u, i) f) n := by
+  apply (weightedHomogeneousSubmodule R (blockWeight k) n).sum_mem
+  intro i hi
+  have hx : (X (v, i) : MvPolynomial (β × ι) R).IsWeightedHomogeneous
+      (blockWeight k) 0 := by
+    simpa [blockWeight_apply, hvk] using
+      (isWeightedHomogeneous_X (R := R) (blockWeight k) (v, i))
+  simpa using hx.mul (block_pderiv_other huk hf i)
+
+end Blocks
+
+end PolynomialWeights
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/CanonicalMap.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/CanonicalMap.lean
index 26deb0de1138b7c8e08c67f368b05d91714b36c8..b46bddd45d438c70e90ce02257f63d7b0200b9b9 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/CanonicalMap.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/CanonicalMap.lean
@@ -1,82 +1,82 @@
-import OAI.RepresentationTheory.FoulkesHowe.GeneratorLift
-import OAI.RepresentationTheory.FoulkesHowe.FoulkesMultilinear
-import OAI.RepresentationTheory.FoulkesHowe.FoulkesSymmetries
-
-namespace OAI
-
-noncomputable section
-namespace Problem346
-
-private def rowIndexEquiv {I J : Type*} (i : I) :
-    {p : I × J // p.1 = i} ≃ J where
-  toFun p := p.1.2
-  invFun j := ⟨(i, j), rfl⟩
-  left_inv := by
-    rintro ⟨⟨i', j⟩, h⟩
-    simp only at h
-    subst i'
-    rfl
-  right_inv _ := rfl
-
-/-- Hold all but a single input row of the Foulkes formula fixed. -/
-def foulkesRowFormSpanning (a b : ℕ) (V : Type*) [AddCommGroup V] [Module ℂ V]
-    (v : Fin b → Fin a → V) (j : Fin b) :
-    MultilinearMap ℂ (fun _ : Fin a => V) (SymPow a (SymPow b V)) := by
-  classical
-  exact ((foulkesMultilinear a b V).domDomRestrict
-    (fun p : Fin b × Fin a => p.1 = j) (fun p => v p.1.1 p.1.2)).domDomCongr
-      (rowIndexEquiv j)
-
-@[simp] theorem foulkesRowFormSpanning_apply (a b : ℕ) (V : Type*)
-    [AddCommGroup V] [Module ℂ V] (v : Fin b → Fin a → V) (j : Fin b)
-    (u : Fin a → V) :
-    foulkesRowFormSpanning a b V v j u =
-      foulkesFormula a b V (Function.update v j u) := by
-  classical
-  change foulkesMultilinear a b V
-    (fun p => if h : p.1 = j then u p.2 else v p.1 p.2) = _
-  rw [foulkesMultilinear_apply]
-  congr 1
-  funext k i
-  by_cases h : k = j
-  · subst k
-    simp
-  · simp [h]
-
-theorem foulkesRowFormSpanning_symmetric (a b : ℕ) (V : Type*)
-    [AddCommGroup V] [Module ℂ V] (v : Fin b → Fin a → V) (j : Fin b)
-    (σ : Equiv.Perm (Fin a)) (u : Fin a → V) :
-    foulkesRowFormSpanning a b V v j (fun i => u (σ i)) =
-      foulkesRowFormSpanning a b V v j u := by
-  classical
-  rw [foulkesRowFormSpanning_apply, foulkesRowFormSpanning_apply]
-  have h := foulkesFormula_inner_permute a b V (Function.update v j u)
-    (Function.update (fun _ => Equiv.refl (Fin a)) j σ)
-  convert h using 1
-  congr 1
-  funext k i
-  by_cases hkj : k = j
-  · subst k
-    simp
-  · simp [Function.update_of_ne hkj]
-
-/-- The specified canonical formula defines a linear map in every bidegree. -/
-theorem exists_foulkesMap_from_spanning (a b : ℕ) (V : Type*)
-    [AddCommGroup V] [Module ℂ V] :
-    ∃ μ : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V),
-      IsFoulkesMap a b V μ := by
-  classical
-  apply exists_symPow_lift_on_spanning (symMonomial a V) (span_symMonomial a V)
-    (foulkesFormula a b V)
-  · intro v j
-    obtain ⟨L, hL⟩ := exists_symPow_lift (foulkesRowFormSpanning a b V v j)
-      (foulkesRowFormSpanning_symmetric a b V v j)
-    exact ⟨L, fun u => (hL u).trans (foulkesRowFormSpanning_apply a b V v j u)⟩
-  · intro σ v
-    exact foulkesFormula_outer_permute a b V v σ
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.GeneratorLift
+import OAI.RepresentationTheory.FoulkesHowe.FoulkesMultilinear
+import OAI.RepresentationTheory.FoulkesHowe.FoulkesSymmetries
+
+namespace OAI
+
+noncomputable section
+namespace Problem346
+
+private def rowIndexEquiv {I J : Type*} (i : I) :
+    {p : I × J // p.1 = i} ≃ J where
+  toFun p := p.1.2
+  invFun j := ⟨(i, j), rfl⟩
+  left_inv := by
+    rintro ⟨⟨i', j⟩, h⟩
+    simp only at h
+    subst i'
+    rfl
+  right_inv _ := rfl
+
+/-- Hold all but a single input row of the Foulkes formula fixed. -/
+def foulkesRowFormSpanning (a b : ℕ) (V : Type*) [AddCommGroup V] [Module ℂ V]
+    (v : Fin b → Fin a → V) (j : Fin b) :
+    MultilinearMap ℂ (fun _ : Fin a => V) (SymPow a (SymPow b V)) := by
+  classical
+  exact ((foulkesMultilinear a b V).domDomRestrict
+    (fun p : Fin b × Fin a => p.1 = j) (fun p => v p.1.1 p.1.2)).domDomCongr
+      (rowIndexEquiv j)
+
+@[simp] theorem foulkesRowFormSpanning_apply (a b : ℕ) (V : Type*)
+    [AddCommGroup V] [Module ℂ V] (v : Fin b → Fin a → V) (j : Fin b)
+    (u : Fin a → V) :
+    foulkesRowFormSpanning a b V v j u =
+      foulkesFormula a b V (Function.update v j u) := by
+  classical
+  change foulkesMultilinear a b V
+    (fun p => if h : p.1 = j then u p.2 else v p.1 p.2) = _
+  rw [foulkesMultilinear_apply]
+  congr 1
+  funext k i
+  by_cases h : k = j
+  · subst k
+    simp
+  · simp [h]
+
+theorem foulkesRowFormSpanning_symmetric (a b : ℕ) (V : Type*)
+    [AddCommGroup V] [Module ℂ V] (v : Fin b → Fin a → V) (j : Fin b)
+    (σ : Equiv.Perm (Fin a)) (u : Fin a → V) :
+    foulkesRowFormSpanning a b V v j (fun i => u (σ i)) =
+      foulkesRowFormSpanning a b V v j u := by
+  classical
+  rw [foulkesRowFormSpanning_apply, foulkesRowFormSpanning_apply]
+  have h := foulkesFormula_inner_permute a b V (Function.update v j u)
+    (Function.update (fun _ => Equiv.refl (Fin a)) j σ)
+  convert h using 1
+  congr 1
+  funext k i
+  by_cases hkj : k = j
+  · subst k
+    simp
+  · simp [Function.update_of_ne hkj]
+
+/-- The specified canonical formula defines a linear map in every bidegree. -/
+theorem exists_foulkesMap_from_spanning (a b : ℕ) (V : Type*)
+    [AddCommGroup V] [Module ℂ V] :
+    ∃ μ : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V),
+      IsFoulkesMap a b V μ := by
+  classical
+  apply exists_symPow_lift_on_spanning (symMonomial a V) (span_symMonomial a V)
+    (foulkesFormula a b V)
+  · intro v j
+    obtain ⟨L, hL⟩ := exists_symPow_lift (foulkesRowFormSpanning a b V v j)
+      (foulkesRowFormSpanning_symmetric a b V v j)
+    exact ⟨L, fun u => (hL u).trans (foulkesRowFormSpanning_apply a b V v j u)⟩
+  · intro σ v
+    exact foulkesFormula_outer_permute a b V v σ
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/DegreeOne.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/DegreeOne.lean
index a8ae6889954f2b144b7e650818026cca8b632828..25ba9aa3885f5f6e9e72ae587bc05d25881342f8 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/DegreeOne.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/DegreeOne.lean
@@ -1,79 +1,79 @@
-import OAI.RepresentationTheory.FoulkesHowe.Model
-
-namespace OAI
-
-noncomputable section
-universe u
-namespace Problem346
-
-variable (V : Type u) [AddCommGroup V] [Module ℂ V]
-
-lemma symmetricAlgebra_ι_injective :
-    Function.Injective (SymmetricAlgebra.ι ℂ V) := by
-  let : Module ℂᵐᵒᵖ V := Module.compHom _ ((RingHom.id ℂ).fromOpposite mul_comm)
-  have : IsCentralScalar ℂ V := ⟨fun _ _ => rfl⟩
-  intro x y h
-  have h' := congrArg (SymmetricAlgebra.lift (TrivSqZeroExt.inrHom ℂ V)) h
-  apply TrivSqZeroExt.inr_injective (R := ℂ)
-  simpa using h'
-
-@[simp] lemma symMonomialRaw_one (v : Fin 1 → V) :
-    symMonomialRaw 1 V v = SymmetricAlgebra.ι ℂ V (v 0) := by
-  simp [symMonomialRaw]
-
-lemma symPowSubmodule_one :
-    symPowSubmodule 1 V = LinearMap.range (SymmetricAlgebra.ι ℂ V) := by
-  have h : Set.range (symMonomialRaw 1 V) =
-      Set.range (SymmetricAlgebra.ι ℂ V) := by
-    ext x
-    constructor
-    · rintro ⟨v, rfl⟩
-      exact ⟨v 0, (symMonomialRaw_one V v).symm⟩
-    · rintro ⟨v, rfl⟩
-      exact ⟨fun _ => v, symMonomialRaw_one V (fun _ => v)⟩
-  rw [symPowSubmodule, h]
-  exact Submodule.span_eq (LinearMap.range (SymmetricAlgebra.ι ℂ V))
-
-/-- The canonical inclusion of a vector into the degree-one symmetric power. -/
-def symPowOneIntro : V →ₗ[ℂ] SymPow 1 V :=
-  (SymmetricAlgebra.ι ℂ V).codRestrict (symPowSubmodule 1 V) (by
-    intro v
-    rw [symPowSubmodule_one]
-    exact ⟨v, rfl⟩)
-
-@[simp] lemma symPowOneIntro_coe (v : V) :
-    (symPowOneIntro V v : SymmetricAlgebra ℂ V) = SymmetricAlgebra.ι ℂ V v := rfl
-
-lemma symPowOneIntro_bijective : Function.Bijective (symPowOneIntro V) := by
-  constructor
-  · intro x y h
-    apply symmetricAlgebra_ι_injective V
-    exact congrArg Subtype.val h
-  · intro x
-    have hx := x.property
-    have hx' : (x : SymmetricAlgebra ℂ V) ∈ LinearMap.range (SymmetricAlgebra.ι ℂ V) := by
-      simpa only [symPowSubmodule_one] using hx
-    rcases hx' with ⟨v, hv⟩
-    exact ⟨v, Subtype.ext hv⟩
-
-/-- The degree-one symmetric power is canonically the original vector space. -/
-def symPowOneEquiv : SymPow 1 V ≃ₗ[ℂ] V :=
-  (LinearEquiv.ofBijective (symPowOneIntro V) (symPowOneIntro_bijective V)).symm
-
-@[simp] lemma symPowOneEquiv_symm_apply (v : V) :
-    (symPowOneEquiv V).symm v = symMonomial 1 V (fun _ => v) := by
-  apply Subtype.ext
-  exact (symMonomialRaw_one V (fun _ => v)).symm
-
-@[simp] lemma symPowOneEquiv_symMonomial (v : Fin 1 → V) :
-    symPowOneEquiv V (symMonomial 1 V v) = v 0 := by
-  apply (symPowOneEquiv V).symm.injective
-  simp only [LinearEquiv.symm_apply_apply, symPowOneEquiv_symm_apply]
-  apply Subtype.ext
-  simp only [symMonomial, symMonomialRaw_one]
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.Model
+
+namespace OAI
+
+noncomputable section
+universe u
+namespace Problem346
+
+variable (V : Type u) [AddCommGroup V] [Module ℂ V]
+
+lemma symmetricAlgebra_ι_injective :
+    Function.Injective (SymmetricAlgebra.ι ℂ V) := by
+  let : Module ℂᵐᵒᵖ V := Module.compHom _ ((RingHom.id ℂ).fromOpposite mul_comm)
+  have : IsCentralScalar ℂ V := ⟨fun _ _ => rfl⟩
+  intro x y h
+  have h' := congrArg (SymmetricAlgebra.lift (TrivSqZeroExt.inrHom ℂ V)) h
+  apply TrivSqZeroExt.inr_injective (R := ℂ)
+  simpa using h'
+
+@[simp] lemma symMonomialRaw_one (v : Fin 1 → V) :
+    symMonomialRaw 1 V v = SymmetricAlgebra.ι ℂ V (v 0) := by
+  simp [symMonomialRaw]
+
+lemma symPowSubmodule_one :
+    symPowSubmodule 1 V = LinearMap.range (SymmetricAlgebra.ι ℂ V) := by
+  have h : Set.range (symMonomialRaw 1 V) =
+      Set.range (SymmetricAlgebra.ι ℂ V) := by
+    ext x
+    constructor
+    · rintro ⟨v, rfl⟩
+      exact ⟨v 0, (symMonomialRaw_one V v).symm⟩
+    · rintro ⟨v, rfl⟩
+      exact ⟨fun _ => v, symMonomialRaw_one V (fun _ => v)⟩
+  rw [symPowSubmodule, h]
+  exact Submodule.span_eq (LinearMap.range (SymmetricAlgebra.ι ℂ V))
+
+/-- The canonical inclusion of a vector into the degree-one symmetric power. -/
+def symPowOneIntro : V →ₗ[ℂ] SymPow 1 V :=
+  (SymmetricAlgebra.ι ℂ V).codRestrict (symPowSubmodule 1 V) (by
+    intro v
+    rw [symPowSubmodule_one]
+    exact ⟨v, rfl⟩)
+
+@[simp] lemma symPowOneIntro_coe (v : V) :
+    (symPowOneIntro V v : SymmetricAlgebra ℂ V) = SymmetricAlgebra.ι ℂ V v := rfl
+
+lemma symPowOneIntro_bijective : Function.Bijective (symPowOneIntro V) := by
+  constructor
+  · intro x y h
+    apply symmetricAlgebra_ι_injective V
+    exact congrArg Subtype.val h
+  · intro x
+    have hx := x.property
+    have hx' : (x : SymmetricAlgebra ℂ V) ∈ LinearMap.range (SymmetricAlgebra.ι ℂ V) := by
+      simpa only [symPowSubmodule_one] using hx
+    rcases hx' with ⟨v, hv⟩
+    exact ⟨v, Subtype.ext hv⟩
+
+/-- The degree-one symmetric power is canonically the original vector space. -/
+def symPowOneEquiv : SymPow 1 V ≃ₗ[ℂ] V :=
+  (LinearEquiv.ofBijective (symPowOneIntro V) (symPowOneIntro_bijective V)).symm
+
+@[simp] lemma symPowOneEquiv_symm_apply (v : V) :
+    (symPowOneEquiv V).symm v = symMonomial 1 V (fun _ => v) := by
+  apply Subtype.ext
+  exact (symMonomialRaw_one V (fun _ => v)).symm
+
+@[simp] lemma symPowOneEquiv_symMonomial (v : Fin 1 → V) :
+    symPowOneEquiv V (symMonomial 1 V v) = v 0 := by
+  apply (symPowOneEquiv V).symm.injective
+  simp only [LinearEquiv.symm_apply_apply, symPowOneEquiv_symm_apply]
+  apply Subtype.ext
+  simp only [symMonomial, symMonomialRaw_one]
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/DerivationInjectivity.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/DerivationInjectivity.lean
index 6071eb01ef5a4901e95f4f87ab24a8d4745c714a..8203658e3c941f965d12c78d129af57a5e8879de 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/DerivationInjectivity.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/DerivationInjectivity.lean
@@ -1,44 +1,44 @@
-import OAI.RepresentationTheory.FoulkesHowe.ShiftInjectivity
-import OAI.RepresentationTheory.FoulkesHowe.ShiftNilpotence
-
-namespace OAI
-
-noncomputable section
-
-namespace Problem346
-
-open MvPolynomial
-
-variable {K σ : Type*} [Field K] [CharZero K]
-
-/-- An algebraic form of the positive-weight injectivity criterion for polynomial
-derivations. The generator condition admits arbitrary spectator variables. -/
-theorem derivation_eq_zero_of_positive_weight
-    (D E H : Derivation K (MvPolynomial σ K) (MvPolynomial σ K))
-    (hED : ⁅E, D⁆ = H) (hHE : ⁅H, E⁆ = (2 : K) • E)
-    (hEX : ∀ i : σ, E (E (X i)) = 0)
-    (f : MvPolynomial σ K) (p q : ℕ) (hpq : q < p)
-    (hf : H f = ((p : K) - q) • f) (hDf : D f = 0) : f = 0 := by
-  refine ShiftInjectivity.eq_zero_of_positive_weight D.toLinearMap E.toLinearMap
-    H.toLinearMap ?_ ?_ f (p - q) (Nat.sub_pos_of_lt hpq) ?_ hDf ?_
-  · intro z
-    have hz := congrArg (fun F : Derivation K (MvPolynomial σ K)
-      (MvPolynomial σ K) => F z) hED
-    simp only [Derivation.commutator_apply] at hz
-    change D (E z) = E (D z) - H z
-    linear_combination -hz
-  · intro z
-    have hz := congrArg (fun F : Derivation K (MvPolynomial σ K)
-      (MvPolynomial σ K) => F z) hHE
-    simp only [Derivation.commutator_apply, Derivation.smul_apply] at hz
-    change H (E z) = E (H z) + (2 : K) • E z
-    linear_combination hz
-  · change H f = ((p - q : ℕ) : K) • f
-    simpa only [Nat.cast_sub (Nat.le_of_lt hpq)] using hf
-  · exact mvPolynomial_derivation_locally_nilpotent_of_square_zero_X E hEX f
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.ShiftInjectivity
+import OAI.RepresentationTheory.FoulkesHowe.ShiftNilpotence
+
+namespace OAI
+
+noncomputable section
+
+namespace Problem346
+
+open MvPolynomial
+
+variable {K σ : Type*} [Field K] [CharZero K]
+
+/-- An algebraic form of the positive-weight injectivity criterion for polynomial
+derivations. The generator condition admits arbitrary spectator variables. -/
+theorem derivation_eq_zero_of_positive_weight
+    (D E H : Derivation K (MvPolynomial σ K) (MvPolynomial σ K))
+    (hED : ⁅E, D⁆ = H) (hHE : ⁅H, E⁆ = (2 : K) • E)
+    (hEX : ∀ i : σ, E (E (X i)) = 0)
+    (f : MvPolynomial σ K) (p q : ℕ) (hpq : q < p)
+    (hf : H f = ((p : K) - q) • f) (hDf : D f = 0) : f = 0 := by
+  refine ShiftInjectivity.eq_zero_of_positive_weight D.toLinearMap E.toLinearMap
+    H.toLinearMap ?_ ?_ f (p - q) (Nat.sub_pos_of_lt hpq) ?_ hDf ?_
+  · intro z
+    have hz := congrArg (fun F : Derivation K (MvPolynomial σ K)
+      (MvPolynomial σ K) => F z) hED
+    simp only [Derivation.commutator_apply] at hz
+    change D (E z) = E (D z) - H z
+    linear_combination -hz
+  · intro z
+    have hz := congrArg (fun F : Derivation K (MvPolynomial σ K)
+      (MvPolynomial σ K) => F z) hHE
+    simp only [Derivation.commutator_apply, Derivation.smul_apply] at hz
+    change H (E z) = E (H z) + (2 : K) • E z
+    linear_combination hz
+  · change H f = ((p - q : ℕ) : K) • f
+    simpa only [Nat.cast_sub (Nat.le_of_lt hpq)] using hf
+  · exact mvPolynomial_derivation_locally_nilpotent_of_square_zero_X E hEX f
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/DiagonalCoefficient.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/DiagonalCoefficient.lean
index 890b4e1bb314dcba9c52405254f208c01916eac2..69a7570eed6c9aa3265c0a1ddaa78f017f998240 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/DiagonalCoefficient.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/DiagonalCoefficient.lean
@@ -1,126 +1,126 @@
-import Mathlib
-
-namespace OAI
-
-noncomputable section
-
-open scoped BigOperators
-
-namespace Problem346
-
-universe u
-
-variable {E : Type u} [AddCommGroup E] [Module ℂ E]
-
-/-- The one-variable polynomial obtained by restricting a multilinear form to an affine diagonal. -/
-def diagonalPolynomial {a : ℕ} (T : MultilinearMap ℂ (fun _ : Fin a => E) ℂ)
-    (x y : E) : Polynomial ℂ :=
-  ∑ s : Finset (Fin a),
-    Polynomial.C (T (s.piecewise (fun _ => y) (fun _ => x))) * Polynomial.X ^ s.card
-
-lemma diagonalPolynomial_eval {a : ℕ}
-    (T : MultilinearMap ℂ (fun _ : Fin a => E) ℂ) (x y : E) (c : ℂ) :
-    (diagonalPolynomial T x y).eval c = T (fun _ => x + c • y) := by
-  classical
-  unfold diagonalPolynomial
-  simp only [Polynomial.eval_finsetSum, Polynomial.eval_mul, Polynomial.eval_C,
-    Polynomial.eval_pow, Polynomial.eval_X]
-  rw [show (fun _ : Fin a => x + c • y) =
-    (fun _ => c • y) + (fun _ => x) by ext; simp [add_comm]]
-  rw [T.map_add_univ]
-  apply Finset.sum_congr rfl
-  intro s hs
-  have h := T.map_piecewise_smul (fun _ => c)
-    (s.piecewise (fun _ => y) (fun _ => x)) s
-  have heq : s.piecewise (fun i => c • s.piecewise (fun _ => y) (fun _ => x) i)
-      (s.piecewise (fun _ => y) (fun _ => x)) =
-      s.piecewise (fun _ => c • y) (fun _ => x) := by
-    ext i
-    by_cases hi : i ∈ s <;> simp [Finset.piecewise, hi]
-  rw [heq] at h
-  simpa [smul_eq_mul, mul_comm] using h.symm
-
-lemma diagonalPolynomial_coeff_one {a : ℕ}
-    (T : MultilinearMap ℂ (fun _ : Fin a => E) ℂ) (x y : E) :
-    (diagonalPolynomial T x y).coeff 1 =
-      ∑ i : Fin a, T (Function.update (fun _ => x) i y) := by
-  classical
-  unfold diagonalPolynomial
-  simp only [Polynomial.finsetSum_coeff, Polynomial.coeff_C_mul_X_pow]
-  simp only [eq_comm (a := 1)]
-  rw [← Finset.sum_filter]
-  rw [← Finset.powerset_univ, ← Finset.powersetCard_eq_filter]
-  simp [Finset.powersetCard_one, Finset.piecewise_singleton]
-
-/-- If a multilinear form vanishes on an affine diagonal line, its linear diagonal
-coefficient vanishes. This statement does not require symmetry. -/
-theorem diagonal_linear_coefficient_zero {a : ℕ}
-    (T : MultilinearMap ℂ (fun _ : Fin a => E) ℂ) (x y : E)
-    (h : ∀ c : ℂ, T (fun _ => x + c • y) = 0) :
-    (∑ i : Fin a, T (Function.update (fun _ => x) i y)) = 0 := by
-  have hp : diagonalPolynomial T x y = 0 :=
-    Polynomial.zero_of_eval_zero _ (fun c => by rw [diagonalPolynomial_eval, h])
-  rw [← diagonalPolynomial_coeff_one, hp]
-  simp
-
-/-- Symmetry identifies the terms of the linear diagonal coefficient. -/
-lemma symmetric_update_eq {a : ℕ}
-    (T : MultilinearMap ℂ (fun _ : Fin a => E) ℂ)
-    (hsym : ∀ (σ : Equiv.Perm (Fin a)) (v : Fin a → E),
-      T (fun k => v (σ k)) = T v)
-    (x y : E) (i j : Fin a) :
-    T (Function.update (fun _ => x) j y) =
-      T (Function.update (fun _ => x) i y) := by
-  have h := hsym (Equiv.swap i j) (Function.update (fun _ => x) i y)
-  convert h using 1
-  congr 1
-  ext k
-  by_cases hi : k = i
-  · subst k
-    by_cases hij : i = j
-    · subst j
-      rw [Equiv.swap_apply_left, Function.update_self]
-    · simp [hij, Ne.symm hij]
-  · by_cases hj : k = j
-    · subst k; simp
-    · simp [Equiv.swap_apply_of_ne_of_ne hi hj, Function.update_of_ne, hi, hj]
-
-/-- Coefficient extraction for a symmetric multilinear form along an affine diagonal.
-This is the coefficient-of-c argument in the split binary form step. -/
-theorem symmetric_diagonal_mixed_zero {a : ℕ}
-    (T : MultilinearMap ℂ (fun _ : Fin a => E) ℂ)
-    (hsym : ∀ (σ : Equiv.Perm (Fin a)) (v : Fin a → E),
-      T (fun k => v (σ k)) = T v)
-    (x y : E) (i : Fin a)
-    (h : ∀ c : ℂ, T (fun _ => x + c • y) = 0) :
-    T (Function.update (fun _ => x) i y) = 0 := by
-  have hz := diagonal_linear_coefficient_zero T x y h
-  have heq : (∑ j : Fin a, T (Function.update (fun _ => x) j y)) =
-      (a : ℂ) * T (Function.update (fun _ => x) i y) := by
-    calc
-      _ = ∑ _j : Fin a, T (Function.update (fun _ => x) i y) := by
-        apply Finset.sum_congr rfl
-        intro j hj
-        exact symmetric_update_eq T hsym x y i j
-      _ = _ := by simp [nsmul_eq_mul]
-  rw [heq] at hz
-  exact (mul_eq_zero.mp hz).resolve_left (by exact_mod_cast Nat.ne_of_gt (Nat.zero_lt_of_lt i.isLt))
-
-/-- The same coefficient extraction with the distinguished slot placed first. -/
-theorem symmetric_diagonal_cons_zero {r : ℕ}
-    (T : MultilinearMap ℂ (fun _ : Fin (r + 1) => E) ℂ)
-    (hsym : ∀ (σ : Equiv.Perm (Fin (r + 1))) (v : Fin (r + 1) → E),
-      T (fun k => v (σ k)) = T v)
-    (x y : E) (h : ∀ c : ℂ, T (fun _ => x + c • y) = 0) :
-    T (Fin.cons y (fun _ : Fin r => x)) = 0 := by
-  have hz := symmetric_diagonal_mixed_zero T hsym x y 0 h
-  convert hz using 1
-  congr 1
-  ext i
-  refine Fin.cases ?_ (fun j => ?_) i <;> simp
-
-end Problem346
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+
+open scoped BigOperators
+
+namespace Problem346
+
+universe u
+
+variable {E : Type u} [AddCommGroup E] [Module ℂ E]
+
+/-- The one-variable polynomial obtained by restricting a multilinear form to an affine diagonal. -/
+def diagonalPolynomial {a : ℕ} (T : MultilinearMap ℂ (fun _ : Fin a => E) ℂ)
+    (x y : E) : Polynomial ℂ :=
+  ∑ s : Finset (Fin a),
+    Polynomial.C (T (s.piecewise (fun _ => y) (fun _ => x))) * Polynomial.X ^ s.card
+
+lemma diagonalPolynomial_eval {a : ℕ}
+    (T : MultilinearMap ℂ (fun _ : Fin a => E) ℂ) (x y : E) (c : ℂ) :
+    (diagonalPolynomial T x y).eval c = T (fun _ => x + c • y) := by
+  classical
+  unfold diagonalPolynomial
+  simp only [Polynomial.eval_finsetSum, Polynomial.eval_mul, Polynomial.eval_C,
+    Polynomial.eval_pow, Polynomial.eval_X]
+  rw [show (fun _ : Fin a => x + c • y) =
+    (fun _ => c • y) + (fun _ => x) by ext; simp [add_comm]]
+  rw [T.map_add_univ]
+  apply Finset.sum_congr rfl
+  intro s hs
+  have h := T.map_piecewise_smul (fun _ => c)
+    (s.piecewise (fun _ => y) (fun _ => x)) s
+  have heq : s.piecewise (fun i => c • s.piecewise (fun _ => y) (fun _ => x) i)
+      (s.piecewise (fun _ => y) (fun _ => x)) =
+      s.piecewise (fun _ => c • y) (fun _ => x) := by
+    ext i
+    by_cases hi : i ∈ s <;> simp [Finset.piecewise, hi]
+  rw [heq] at h
+  simpa [smul_eq_mul, mul_comm] using h.symm
+
+lemma diagonalPolynomial_coeff_one {a : ℕ}
+    (T : MultilinearMap ℂ (fun _ : Fin a => E) ℂ) (x y : E) :
+    (diagonalPolynomial T x y).coeff 1 =
+      ∑ i : Fin a, T (Function.update (fun _ => x) i y) := by
+  classical
+  unfold diagonalPolynomial
+  simp only [Polynomial.finsetSum_coeff, Polynomial.coeff_C_mul_X_pow]
+  simp only [eq_comm (a := 1)]
+  rw [← Finset.sum_filter]
+  rw [← Finset.powerset_univ, ← Finset.powersetCard_eq_filter]
+  simp [Finset.powersetCard_one, Finset.piecewise_singleton]
+
+/-- If a multilinear form vanishes on an affine diagonal line, its linear diagonal
+coefficient vanishes. This statement does not require symmetry. -/
+theorem diagonal_linear_coefficient_zero {a : ℕ}
+    (T : MultilinearMap ℂ (fun _ : Fin a => E) ℂ) (x y : E)
+    (h : ∀ c : ℂ, T (fun _ => x + c • y) = 0) :
+    (∑ i : Fin a, T (Function.update (fun _ => x) i y)) = 0 := by
+  have hp : diagonalPolynomial T x y = 0 :=
+    Polynomial.zero_of_eval_zero _ (fun c => by rw [diagonalPolynomial_eval, h])
+  rw [← diagonalPolynomial_coeff_one, hp]
+  simp
+
+/-- Symmetry identifies the terms of the linear diagonal coefficient. -/
+lemma symmetric_update_eq {a : ℕ}
+    (T : MultilinearMap ℂ (fun _ : Fin a => E) ℂ)
+    (hsym : ∀ (σ : Equiv.Perm (Fin a)) (v : Fin a → E),
+      T (fun k => v (σ k)) = T v)
+    (x y : E) (i j : Fin a) :
+    T (Function.update (fun _ => x) j y) =
+      T (Function.update (fun _ => x) i y) := by
+  have h := hsym (Equiv.swap i j) (Function.update (fun _ => x) i y)
+  convert h using 1
+  congr 1
+  ext k
+  by_cases hi : k = i
+  · subst k
+    by_cases hij : i = j
+    · subst j
+      rw [Equiv.swap_apply_left, Function.update_self]
+    · simp [hij, Ne.symm hij]
+  · by_cases hj : k = j
+    · subst k; simp
+    · simp [Equiv.swap_apply_of_ne_of_ne hi hj, Function.update_of_ne, hi, hj]
+
+/-- Coefficient extraction for a symmetric multilinear form along an affine diagonal.
+This is the coefficient-of-c argument in the split binary form step. -/
+theorem symmetric_diagonal_mixed_zero {a : ℕ}
+    (T : MultilinearMap ℂ (fun _ : Fin a => E) ℂ)
+    (hsym : ∀ (σ : Equiv.Perm (Fin a)) (v : Fin a → E),
+      T (fun k => v (σ k)) = T v)
+    (x y : E) (i : Fin a)
+    (h : ∀ c : ℂ, T (fun _ => x + c • y) = 0) :
+    T (Function.update (fun _ => x) i y) = 0 := by
+  have hz := diagonal_linear_coefficient_zero T x y h
+  have heq : (∑ j : Fin a, T (Function.update (fun _ => x) j y)) =
+      (a : ℂ) * T (Function.update (fun _ => x) i y) := by
+    calc
+      _ = ∑ _j : Fin a, T (Function.update (fun _ => x) i y) := by
+        apply Finset.sum_congr rfl
+        intro j hj
+        exact symmetric_update_eq T hsym x y i j
+      _ = _ := by simp [nsmul_eq_mul]
+  rw [heq] at hz
+  exact (mul_eq_zero.mp hz).resolve_left (by exact_mod_cast Nat.ne_of_gt (Nat.zero_lt_of_lt i.isLt))
+
+/-- The same coefficient extraction with the distinguished slot placed first. -/
+theorem symmetric_diagonal_cons_zero {r : ℕ}
+    (T : MultilinearMap ℂ (fun _ : Fin (r + 1) => E) ℂ)
+    (hsym : ∀ (σ : Equiv.Perm (Fin (r + 1))) (v : Fin (r + 1) → E),
+      T (fun k => v (σ k)) = T v)
+    (x y : E) (h : ∀ c : ℂ, T (fun _ => x + c • y) = 0) :
+    T (Fin.cons y (fun _ : Fin r => x)) = 0 := by
+  have hz := symmetric_diagonal_mixed_zero T hsym x y 0 h
+  convert hz using 1
+  congr 1
+  ext i
+  refine Fin.cases ?_ (fun j => ?_) i <;> simp
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/DiagonalPolarization.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/DiagonalPolarization.lean
index 29d8147faf2667d793401f75c0b749c8f91ce96d..0573d16d1ed903c0dbb742f459a77cb2d927b351 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/DiagonalPolarization.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/DiagonalPolarization.lean
@@ -1,62 +1,62 @@
-import Mathlib
-
-namespace OAI
-
-noncomputable section
-open scoped BigOperators
-
-namespace Problem346
-
-/-- A symmetric continuous multilinear form is determined by its diagonal. -/
-theorem continuous_symmetric_multilinear_eq_zero_of_diagonal
-    {W : Type*} [NormedAddCommGroup W] [NormedSpace ℂ W] {n : ℕ}
-    (T : ContinuousMultilinearMap ℂ (fun _ : Fin n => W) ℂ)
-    (hsym : ∀ (σ : Equiv.Perm (Fin n)) (x : Fin n → W),
-      T (fun i => x (σ i)) = T x)
-    (hdiag : ∀ x : W, T (fun _ => x) = 0) : T = 0 := by
-  ext x
-  have h := T.iteratedFDeriv_comp_diagonal (0 : W) x
-  have hz : (fun x : W => T (fun _ => x)) = fun _ => 0 := funext hdiag
-  rw [hz] at h
-  simp only [iteratedFDeriv_fun_zero, Pi.zero_apply, zero_apply] at h
-  simp only [hsym, Finset.sum_const, Finset.card_univ, Fintype.card_perm,
-    Fintype.card_fin, nsmul_eq_mul] at h
-  exact (mul_eq_zero.mp h.symm).resolve_left (by exact_mod_cast Nat.factorial_ne_zero n)
-
-/-- Algebraic polarization, without a topology on the source vector space. -/
-theorem symmetric_multilinear_eq_zero_of_diagonal
-    {W : Type*} [AddCommGroup W] [Module ℂ W] {n : ℕ}
-    (T : MultilinearMap ℂ (fun _ : Fin n => W) ℂ)
-    (hsym : ∀ (σ : Equiv.Perm (Fin n)) (x : Fin n → W),
-      T (fun i => x (σ i)) = T x)
-    (hdiag : ∀ x : W, T (fun _ => x) = 0) : T = 0 := by
-  classical
-  ext x
-  let L : (Fin n → ℂ) →ₗ[ℂ] W :=
-    ∑ i : Fin n, (LinearMap.proj i).smulRight (x i)
-  let S := T.compLinearMap (fun _ => L)
-  have hL (z : Fin n → ℂ) : L z = ∑ i, z i • x i := by simp [L]
-  have hS (z : Fin n → Fin n → ℂ) :
-      S z = ∑ r : Fin n → Fin n, (∏ i, z i (r i)) • T (fun i => x (r i)) := by
-    simp only [S, MultilinearMap.compLinearMap_apply, hL]
-    rw [T.map_sum]
-    simp only [T.map_smul_univ]
-  have hc : Continuous S := by
-    have heq : (fun z => S z) = fun z =>
-        ∑ r : Fin n → Fin n, (∏ i, z i (r i)) • T (fun i => x (r i)) := funext hS
-    rw [show (S : (Fin n → Fin n → ℂ) → ℂ) = _ from heq]
-    fun_prop
-  let C : ContinuousMultilinearMap ℂ (fun _ : Fin n => Fin n → ℂ) ℂ := ⟨S, hc⟩
-  have hC : C = 0 := continuous_symmetric_multilinear_eq_zero_of_diagonal C
-    (by intro σ z; exact hsym σ (fun i => L (z i)))
-    (by intro z; exact hdiag (L z))
-  have hh := congrArg (fun f : ContinuousMultilinearMap ℂ
-      (fun _ : Fin n => Fin n → ℂ) ℂ => f (fun i => Pi.single i 1)) hC
-  change T (fun i => L (Pi.single i 1)) = 0 at hh
-  simpa [hL] using hh
-
-end Problem346
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators
+
+namespace Problem346
+
+/-- A symmetric continuous multilinear form is determined by its diagonal. -/
+theorem continuous_symmetric_multilinear_eq_zero_of_diagonal
+    {W : Type*} [NormedAddCommGroup W] [NormedSpace ℂ W] {n : ℕ}
+    (T : ContinuousMultilinearMap ℂ (fun _ : Fin n => W) ℂ)
+    (hsym : ∀ (σ : Equiv.Perm (Fin n)) (x : Fin n → W),
+      T (fun i => x (σ i)) = T x)
+    (hdiag : ∀ x : W, T (fun _ => x) = 0) : T = 0 := by
+  ext x
+  have h := T.iteratedFDeriv_comp_diagonal (0 : W) x
+  have hz : (fun x : W => T (fun _ => x)) = fun _ => 0 := funext hdiag
+  rw [hz] at h
+  simp only [iteratedFDeriv_fun_zero, Pi.zero_apply, zero_apply] at h
+  simp only [hsym, Finset.sum_const, Finset.card_univ, Fintype.card_perm,
+    Fintype.card_fin, nsmul_eq_mul] at h
+  exact (mul_eq_zero.mp h.symm).resolve_left (by exact_mod_cast Nat.factorial_ne_zero n)
+
+/-- Algebraic polarization, without a topology on the source vector space. -/
+theorem symmetric_multilinear_eq_zero_of_diagonal
+    {W : Type*} [AddCommGroup W] [Module ℂ W] {n : ℕ}
+    (T : MultilinearMap ℂ (fun _ : Fin n => W) ℂ)
+    (hsym : ∀ (σ : Equiv.Perm (Fin n)) (x : Fin n → W),
+      T (fun i => x (σ i)) = T x)
+    (hdiag : ∀ x : W, T (fun _ => x) = 0) : T = 0 := by
+  classical
+  ext x
+  let L : (Fin n → ℂ) →ₗ[ℂ] W :=
+    ∑ i : Fin n, (LinearMap.proj i).smulRight (x i)
+  let S := T.compLinearMap (fun _ => L)
+  have hL (z : Fin n → ℂ) : L z = ∑ i, z i • x i := by simp [L]
+  have hS (z : Fin n → Fin n → ℂ) :
+      S z = ∑ r : Fin n → Fin n, (∏ i, z i (r i)) • T (fun i => x (r i)) := by
+    simp only [S, MultilinearMap.compLinearMap_apply, hL]
+    rw [T.map_sum]
+    simp only [T.map_smul_univ]
+  have hc : Continuous S := by
+    have heq : (fun z => S z) = fun z =>
+        ∑ r : Fin n → Fin n, (∏ i, z i (r i)) • T (fun i => x (r i)) := funext hS
+    rw [show (S : (Fin n → Fin n → ℂ) → ℂ) = _ from heq]
+    fun_prop
+  let C : ContinuousMultilinearMap ℂ (fun _ : Fin n => Fin n → ℂ) ℂ := ⟨S, hc⟩
+  have hC : C = 0 := continuous_symmetric_multilinear_eq_zero_of_diagonal C
+    (by intro σ z; exact hsym σ (fun i => L (z i)))
+    (by intro z; exact hdiag (L z))
+  have hh := congrArg (fun f : ContinuousMultilinearMap ℂ
+      (fun _ : Fin n => Fin n → ℂ) ℂ => f (fun i => Pi.single i 1)) hC
+  change T (fun i => L (Pi.single i 1)) = 0 at hh
+  simpa [hL] using hh
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/DualComparison.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/DualComparison.lean
index 0090e6c57644653c40c4484fc9ff6ed78cc54bac..f3ba63dfdb6a504453d44f01901fc01c9c3b246d 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/DualComparison.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/DualComparison.lean
@@ -1,76 +1,76 @@
-import OAI.RepresentationTheory.FoulkesHowe.DualEquivalence
-import OAI.RepresentationTheory.FoulkesHowe.PlethysmDual
-
-namespace OAI
-
-noncomputable section
-namespace Problem346
-
-/-- The canonical symmetric-power pairing is natural for arbitrary linear maps. -/
-theorem canonicalSymPowDualMap_adjoint (n : ℕ) {V W : Type*}
-    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
-    (A : V →ₗ[ℂ] W) (p : SymPow n (Module.Dual ℂ W)) (q : SymPow n V) :
-    symPowDualMap n V (symPowMap n A.dualMap p) q =
-      symPowDualMap n W p (symPowMap n A q) := by
-  have h : (symPowDualMap n V).comp (symPowMap n A.dualMap) =
-      (symPowMap n A).dualMap.comp (symPowDualMap n W) := by
-    apply symPow_linearMap_ext n (Module.Dual ℂ W)
-    intro φ
-    apply symPow_linearMap_ext n V
-    intro x
-    simp only [LinearMap.comp_apply, LinearMap.dualMap_apply,
-      symPowMap_symMonomial, symPowDualMap_monomial]
-    rfl
-  exact congrArg (fun f : Module.Dual ℂ (SymPow n V) => f q)
-    (LinearMap.congr_fun h p)
-
-/-- The perfect-pairing equivalence has the same adjointness identity. -/
-theorem canonicalSymPowDualEquiv_adjoint (n : ℕ) {V W : Type*}
-    [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
-    [AddCommGroup W] [Module ℂ W] [FiniteDimensional ℂ W]
-    (A : V →ₗ[ℂ] W) (p : SymPow n (Module.Dual ℂ W)) (q : SymPow n V) :
-    symPowDualEquiv n V (symPowMap n A.dualMap p) q =
-      symPowDualEquiv n W p (symPowMap n A q) :=
-  canonicalSymPowDualMap_adjoint n A p q
-
-/-- Canonical duality on a nested symmetric power. -/
-def concretePlethysmDualEquiv (outer inner : ℕ) (V : Type*)
-    [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] :
-    SymPow outer (SymPow inner (Module.Dual ℂ V)) ≃ₗ[ℂ]
-      Module.Dual ℂ (SymPow outer (SymPow inner V)) :=
-  PlethysmDual.plethysmDualEquivOf outer inner
-    (symPowDualEquiv inner V) (symPowDualEquiv outer (SymPow inner V))
-
-theorem concretePlethysmDualEquiv_adjoint (outer inner : ℕ) (V : Type*)
-    [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
-    (g : V ≃ₗ[ℂ] V)
-    (p : SymPow outer (SymPow inner (Module.Dual ℂ V)))
-    (q : SymPow outer (SymPow inner V)) :
-    concretePlethysmDualEquiv outer inner V
-      (plethysmAction outer inner g.dualMap p) q =
-      concretePlethysmDualEquiv outer inner V p (plethysmAction outer inner g q) := by
-  apply PlethysmDual.plethysmDualEquivOf_adjoint
-  · intro x y
-    exact canonicalSymPowDualEquiv_adjoint inner g.toLinearMap x y
-  · intro x y
-    exact canonicalSymPowDualEquiv_adjoint outer (symPowMap inner g.toLinearMap) x y
-
-/-- A canonical surjection at the dual vector space supplies the required
-GL-equivariant comparison embedding, using the concrete perfect pairings. -/
-theorem comparison_embedding_of_dual_foulkes_surjection (a b : ℕ) (V : Type*)
-    [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
-    (hsurj : ∃ μ : SymPow b (SymPow a (Module.Dual ℂ V)) →ₗ[ℂ]
-      SymPow a (SymPow b (Module.Dual ℂ V)),
-      IsFoulkesMap a b (Module.Dual ℂ V) μ ∧ Function.Surjective μ) :
-    ∃ ι : SymPow a (SymPow b V) →ₗ[ℂ] SymPow b (SymPow a V),
-      IsGLEquivariantEmbedding a b V ι :=
-  PlethysmDual.comparison_of_dual_charts a b
-    (concretePlethysmDualEquiv b a V) (concretePlethysmDualEquiv a b V)
-    (concretePlethysmDualEquiv_adjoint b a V)
-    (concretePlethysmDualEquiv_adjoint a b V) hsurj
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.DualEquivalence
+import OAI.RepresentationTheory.FoulkesHowe.PlethysmDual
+
+namespace OAI
+
+noncomputable section
+namespace Problem346
+
+/-- The canonical symmetric-power pairing is natural for arbitrary linear maps. -/
+theorem canonicalSymPowDualMap_adjoint (n : ℕ) {V W : Type*}
+    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
+    (A : V →ₗ[ℂ] W) (p : SymPow n (Module.Dual ℂ W)) (q : SymPow n V) :
+    symPowDualMap n V (symPowMap n A.dualMap p) q =
+      symPowDualMap n W p (symPowMap n A q) := by
+  have h : (symPowDualMap n V).comp (symPowMap n A.dualMap) =
+      (symPowMap n A).dualMap.comp (symPowDualMap n W) := by
+    apply symPow_linearMap_ext n (Module.Dual ℂ W)
+    intro φ
+    apply symPow_linearMap_ext n V
+    intro x
+    simp only [LinearMap.comp_apply, LinearMap.dualMap_apply,
+      symPowMap_symMonomial, symPowDualMap_monomial]
+    rfl
+  exact congrArg (fun f : Module.Dual ℂ (SymPow n V) => f q)
+    (LinearMap.congr_fun h p)
+
+/-- The perfect-pairing equivalence has the same adjointness identity. -/
+theorem canonicalSymPowDualEquiv_adjoint (n : ℕ) {V W : Type*}
+    [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
+    [AddCommGroup W] [Module ℂ W] [FiniteDimensional ℂ W]
+    (A : V →ₗ[ℂ] W) (p : SymPow n (Module.Dual ℂ W)) (q : SymPow n V) :
+    symPowDualEquiv n V (symPowMap n A.dualMap p) q =
+      symPowDualEquiv n W p (symPowMap n A q) :=
+  canonicalSymPowDualMap_adjoint n A p q
+
+/-- Canonical duality on a nested symmetric power. -/
+def concretePlethysmDualEquiv (outer inner : ℕ) (V : Type*)
+    [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] :
+    SymPow outer (SymPow inner (Module.Dual ℂ V)) ≃ₗ[ℂ]
+      Module.Dual ℂ (SymPow outer (SymPow inner V)) :=
+  PlethysmDual.plethysmDualEquivOf outer inner
+    (symPowDualEquiv inner V) (symPowDualEquiv outer (SymPow inner V))
+
+theorem concretePlethysmDualEquiv_adjoint (outer inner : ℕ) (V : Type*)
+    [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
+    (g : V ≃ₗ[ℂ] V)
+    (p : SymPow outer (SymPow inner (Module.Dual ℂ V)))
+    (q : SymPow outer (SymPow inner V)) :
+    concretePlethysmDualEquiv outer inner V
+      (plethysmAction outer inner g.dualMap p) q =
+      concretePlethysmDualEquiv outer inner V p (plethysmAction outer inner g q) := by
+  apply PlethysmDual.plethysmDualEquivOf_adjoint
+  · intro x y
+    exact canonicalSymPowDualEquiv_adjoint inner g.toLinearMap x y
+  · intro x y
+    exact canonicalSymPowDualEquiv_adjoint outer (symPowMap inner g.toLinearMap) x y
+
+/-- A canonical surjection at the dual vector space supplies the required
+GL-equivariant comparison embedding, using the concrete perfect pairings. -/
+theorem comparison_embedding_of_dual_foulkes_surjection (a b : ℕ) (V : Type*)
+    [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
+    (hsurj : ∃ μ : SymPow b (SymPow a (Module.Dual ℂ V)) →ₗ[ℂ]
+      SymPow a (SymPow b (Module.Dual ℂ V)),
+      IsFoulkesMap a b (Module.Dual ℂ V) μ ∧ Function.Surjective μ) :
+    ∃ ι : SymPow a (SymPow b V) →ₗ[ℂ] SymPow b (SymPow a V),
+      IsGLEquivariantEmbedding a b V ι :=
+  PlethysmDual.comparison_of_dual_charts a b
+    (concretePlethysmDualEquiv b a V) (concretePlethysmDualEquiv a b V)
+    (concretePlethysmDualEquiv_adjoint b a V)
+    (concretePlethysmDualEquiv_adjoint a b V) hsurj
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/DualEmbedding.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/DualEmbedding.lean
index 6d33ba034b8353a022426e6de37640fb91eb6add..556c789abf50d87f9fa921cb9525dbaed1453d26 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/DualEmbedding.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/DualEmbedding.lean
@@ -1,91 +1,91 @@
-import Mathlib
-
-namespace OAI
-
-/-!
-# Transposed embeddings
-
-A surjective intertwiner yields an injective intertwiner in the opposite
- direction when both modules carry compatible perfect pairings.  This is an
- algebraic alternative to splitting a representation by complete reducibility.
--/
-
-noncomputable section
-
-namespace Problem346
-namespace DualEmbedding
-
-variable {R E F E' F' : Type*} [CommSemiring R]
-  [AddCommMonoid E] [Module R E] [AddCommMonoid F] [Module R F]
-  [AddCommMonoid E'] [Module R E'] [AddCommMonoid F'] [Module R F']
-
-/-- Reverse a perfect pairing when the first module is reflexive.  This turns
-    the usual chart `E' ≃ Dual E` into the orientation needed by `transpose`. -/
-def flipPairing [Module.IsReflexive R E] (e : E' ≃ₗ[R] Module.Dual R E) :
-    E ≃ₗ[R] Module.Dual R E' :=
-  (Module.evalEquiv R E).trans e.dualMap
-
-@[simp]
-theorem flipPairing_apply [Module.IsReflexive R E]
-    (e : E' ≃ₗ[R] Module.Dual R E) (x : E) (y : E') :
-    flipPairing e x y = e y x := rfl
-
-/-- Transport the dual of a linear map along two perfect pairings. -/
-def transpose (e : E ≃ₗ[R] Module.Dual R E')
-    (f : F ≃ₗ[R] Module.Dual R F') (μ : E' →ₗ[R] F') : F →ₗ[R] E :=
-  e.symm.toLinearMap.comp (μ.dualMap.comp f.toLinearMap)
-
-@[simp]
-theorem pairing_transpose (e : E ≃ₗ[R] Module.Dual R E')
-    (f : F ≃ₗ[R] Module.Dual R F') (μ : E' →ₗ[R] F') (y : F) (x : E') :
-    e (transpose e f μ y) x = f y (μ x) := by
-  simp [transpose]
-
-/-- Surjectivity of a map implies injectivity of its transported transpose. -/
-theorem transpose_injective (e : E ≃ₗ[R] Module.Dual R E')
-    (f : F ≃ₗ[R] Module.Dual R F') (μ : E' →ₗ[R] F')
-    (hμ : Function.Surjective μ) : Function.Injective (transpose e f μ) := by
-  exact e.symm.injective.comp ((LinearMap.dualMap_injective_of_surjective hμ).comp
-    f.injective)
-
-/-- Transposition reverses an intertwining identity, with the two adjoint
-    identities expressed directly as evaluations of the perfect pairings. -/
-theorem transpose_intertwines
-    (e : E ≃ₗ[R] Module.Dual R E') (f : F ≃ₗ[R] Module.Dual R F')
-    (μ : E' →ₗ[R] F') (A : E →ₗ[R] E) (A' : E' →ₗ[R] E')
-    (B : F →ₗ[R] F) (B' : F' →ₗ[R] F')
-    (he : ∀ x z, e (A x) z = e x (A' z))
-    (hf : ∀ y w, f (B y) w = f y (B' w))
-    (hμ : μ.comp A' = B'.comp μ) :
-    A.comp (transpose e f μ) = (transpose e f μ).comp B := by
-  ext y
-  apply e.injective
-  ext x
-  change e (A (transpose e f μ y)) x = e (transpose e f μ (B y)) x
-  rw [he, pairing_transpose, pairing_transpose, hf]
-  exact congrArg (f y) (LinearMap.congr_fun hμ x)
-
-/-- A family of compatible adjoint actions transports a surjective
-    equivariant map into an injective equivariant map in the reverse direction.
-    No group structure or finite-dimensionality is needed by this argument. -/
-theorem exists_intertwining_injection_of_surjective {G : Type*}
-    (e : E ≃ₗ[R] Module.Dual R E') (f : F ≃ₗ[R] Module.Dual R F')
-    (A : G → E →ₗ[R] E) (B : G → F →ₗ[R] F)
-    (A' : G → E' →ₗ[R] E') (B' : G → F' →ₗ[R] F')
-    (he : ∀ g x z, e (A g x) z = e x (A' g z))
-    (hf : ∀ g y w, f (B g y) w = f y (B' g w))
-    (μ : E' →ₗ[R] F') (hμ : Function.Surjective μ)
-    (hcomm : ∀ g, μ.comp (A' g) = (B' g).comp μ) :
-    ∃ ι : F →ₗ[R] E, Function.Injective ι ∧
-      ∀ g, (A g).comp ι = ι.comp (B g) := by
-  refine ⟨transpose e f μ, transpose_injective e f μ hμ, ?_⟩
-  intro g
-  exact transpose_intertwines e f μ (A g) (A' g) (B g) (B' g)
-    (he g) (hf g) (hcomm g)
-
-end DualEmbedding
-end Problem346
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-!
+# Transposed embeddings
+
+A surjective intertwiner yields an injective intertwiner in the opposite
+ direction when both modules carry compatible perfect pairings.  This is an
+ algebraic alternative to splitting a representation by complete reducibility.
+-/
+
+noncomputable section
+
+namespace Problem346
+namespace DualEmbedding
+
+variable {R E F E' F' : Type*} [CommSemiring R]
+  [AddCommMonoid E] [Module R E] [AddCommMonoid F] [Module R F]
+  [AddCommMonoid E'] [Module R E'] [AddCommMonoid F'] [Module R F']
+
+/-- Reverse a perfect pairing when the first module is reflexive.  This turns
+    the usual chart `E' ≃ Dual E` into the orientation needed by `transpose`. -/
+def flipPairing [Module.IsReflexive R E] (e : E' ≃ₗ[R] Module.Dual R E) :
+    E ≃ₗ[R] Module.Dual R E' :=
+  (Module.evalEquiv R E).trans e.dualMap
+
+@[simp]
+theorem flipPairing_apply [Module.IsReflexive R E]
+    (e : E' ≃ₗ[R] Module.Dual R E) (x : E) (y : E') :
+    flipPairing e x y = e y x := rfl
+
+/-- Transport the dual of a linear map along two perfect pairings. -/
+def transpose (e : E ≃ₗ[R] Module.Dual R E')
+    (f : F ≃ₗ[R] Module.Dual R F') (μ : E' →ₗ[R] F') : F →ₗ[R] E :=
+  e.symm.toLinearMap.comp (μ.dualMap.comp f.toLinearMap)
+
+@[simp]
+theorem pairing_transpose (e : E ≃ₗ[R] Module.Dual R E')
+    (f : F ≃ₗ[R] Module.Dual R F') (μ : E' →ₗ[R] F') (y : F) (x : E') :
+    e (transpose e f μ y) x = f y (μ x) := by
+  simp [transpose]
+
+/-- Surjectivity of a map implies injectivity of its transported transpose. -/
+theorem transpose_injective (e : E ≃ₗ[R] Module.Dual R E')
+    (f : F ≃ₗ[R] Module.Dual R F') (μ : E' →ₗ[R] F')
+    (hμ : Function.Surjective μ) : Function.Injective (transpose e f μ) := by
+  exact e.symm.injective.comp ((LinearMap.dualMap_injective_of_surjective hμ).comp
+    f.injective)
+
+/-- Transposition reverses an intertwining identity, with the two adjoint
+    identities expressed directly as evaluations of the perfect pairings. -/
+theorem transpose_intertwines
+    (e : E ≃ₗ[R] Module.Dual R E') (f : F ≃ₗ[R] Module.Dual R F')
+    (μ : E' →ₗ[R] F') (A : E →ₗ[R] E) (A' : E' →ₗ[R] E')
+    (B : F →ₗ[R] F) (B' : F' →ₗ[R] F')
+    (he : ∀ x z, e (A x) z = e x (A' z))
+    (hf : ∀ y w, f (B y) w = f y (B' w))
+    (hμ : μ.comp A' = B'.comp μ) :
+    A.comp (transpose e f μ) = (transpose e f μ).comp B := by
+  ext y
+  apply e.injective
+  ext x
+  change e (A (transpose e f μ y)) x = e (transpose e f μ (B y)) x
+  rw [he, pairing_transpose, pairing_transpose, hf]
+  exact congrArg (f y) (LinearMap.congr_fun hμ x)
+
+/-- A family of compatible adjoint actions transports a surjective
+    equivariant map into an injective equivariant map in the reverse direction.
+    No group structure or finite-dimensionality is needed by this argument. -/
+theorem exists_intertwining_injection_of_surjective {G : Type*}
+    (e : E ≃ₗ[R] Module.Dual R E') (f : F ≃ₗ[R] Module.Dual R F')
+    (A : G → E →ₗ[R] E) (B : G → F →ₗ[R] F)
+    (A' : G → E' →ₗ[R] E') (B' : G → F' →ₗ[R] F')
+    (he : ∀ g x z, e (A g x) z = e x (A' g z))
+    (hf : ∀ g y w, f (B g y) w = f y (B' g w))
+    (μ : E' →ₗ[R] F') (hμ : Function.Surjective μ)
+    (hcomm : ∀ g, μ.comp (A' g) = (B' g).comp μ) :
+    ∃ ι : F →ₗ[R] E, Function.Injective ι ∧
+      ∀ g, (A g).comp ι = ι.comp (B g) := by
+  refine ⟨transpose e f μ, transpose_injective e f μ hμ, ?_⟩
+  intro g
+  exact transpose_intertwines e f μ (A g) (A' g) (B g) (B' g)
+    (he g) (hf g) (hcomm g)
+
+end DualEmbedding
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/DualEquivalence.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/DualEquivalence.lean
index d243012450225cee4cf411bb94eeeb3d57251a92..51a694842eda938b51e5d699c6fd24fbe817ebde 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/DualEquivalence.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/DualEquivalence.lean
@@ -1,39 +1,39 @@
-import OAI.RepresentationTheory.FoulkesHowe.SymmetricPairing
-import OAI.RepresentationTheory.FoulkesHowe.PairingPerfection
-
-namespace OAI
-
-noncomputable section
-universe u
-namespace Problem346
-
-variable (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]
-    [FiniteDimensional ℂ V]
-
-/-- The canonical symmetric-power pairing is perfect in finite dimension. -/
-instance symPowDualMap_isPerfPair : (symPowDualMap n V).IsPerfPair :=
-  symPowDual_isPerfPair_of_pure_evaluation V n (symPowDualMap n V)
-    (symPowDualMap_pure_right n V) (symPowDualMap_pure_left n V)
-
-/-- Symmetric powers commute with algebraic duality in finite dimension. -/
-def symPowDualEquiv :
-    SymPow n (Module.Dual ℂ V) ≃ₗ[ℂ] Module.Dual ℂ (SymPow n V) :=
-  (symPowDualMap n V).toPerfPair
-
-@[simp] theorem symPowDualEquiv_toLinearMap :
-    (symPowDualEquiv n V).toLinearMap = symPowDualMap n V := rfl
-
-@[simp] theorem symPowDualEquiv_apply (p : SymPow n (Module.Dual ℂ V))
-    (q : SymPow n V) : symPowDualEquiv n V p q = symPowDualMap n V p q := rfl
-
-@[simp] theorem symPowDualEquiv_monomial (φ : Fin n → Module.Dual ℂ V)
-    (x : Fin n → V) :
-    symPowDualEquiv n V (symMonomial n (Module.Dual ℂ V) φ) (symMonomial n V x) =
-      permanentPairing n (fun φ x => φ x) φ x :=
-  symPowDualMap_monomial n V φ x
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.SymmetricPairing
+import OAI.RepresentationTheory.FoulkesHowe.PairingPerfection
+
+namespace OAI
+
+noncomputable section
+universe u
+namespace Problem346
+
+variable (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]
+    [FiniteDimensional ℂ V]
+
+/-- The canonical symmetric-power pairing is perfect in finite dimension. -/
+instance symPowDualMap_isPerfPair : (symPowDualMap n V).IsPerfPair :=
+  symPowDual_isPerfPair_of_pure_evaluation V n (symPowDualMap n V)
+    (symPowDualMap_pure_right n V) (symPowDualMap_pure_left n V)
+
+/-- Symmetric powers commute with algebraic duality in finite dimension. -/
+def symPowDualEquiv :
+    SymPow n (Module.Dual ℂ V) ≃ₗ[ℂ] Module.Dual ℂ (SymPow n V) :=
+  (symPowDualMap n V).toPerfPair
+
+@[simp] theorem symPowDualEquiv_toLinearMap :
+    (symPowDualEquiv n V).toLinearMap = symPowDualMap n V := rfl
+
+@[simp] theorem symPowDualEquiv_apply (p : SymPow n (Module.Dual ℂ V))
+    (q : SymPow n V) : symPowDualEquiv n V p q = symPowDualMap n V p q := rfl
+
+@[simp] theorem symPowDualEquiv_monomial (φ : Fin n → Module.Dual ℂ V)
+    (x : Fin n → V) :
+    symPowDualEquiv n V (symMonomial n (Module.Dual ℂ V) φ) (symMonomial n V x) =
+      permanentPairing n (fun φ x => φ x) φ x :=
+  symPowDualMap_monomial n V φ x
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/EncodedCancellation.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/EncodedCancellation.lean
index a604b6dd0033f57fe0c1c728779784917057c7d6..014bff158227dd76d1d6c35b6a369354f47bbad3 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/EncodedCancellation.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/EncodedCancellation.lean
@@ -1,58 +1,58 @@
-import OAI.RepresentationTheory.FoulkesHowe.MultilinearPolynomial
-import OAI.RepresentationTheory.FoulkesHowe.SlotDerivative
-import OAI.RepresentationTheory.FoulkesHowe.EncodingHomogeneous
-import OAI.RepresentationTheory.FoulkesHowe.BlockInjectivity
-
-namespace OAI
-
-noncomputable section
-open scoped BigOperators
-
-namespace Problem346
-
-/-- Cancel one directional derivative of a repeated-block multilinear polynomial.
-It suffices that every single source-slot replacement vanishes; no normalization
-or nonvanishing factorial is needed. -/
-theorem multilinearPolynomial_eq_zero_of_replacements
-    {V S I B : Type*} [AddCommGroup V] [Module ℂ V]
-    [Fintype S] [Fintype I] [Fintype B] [hS : DecidableEq S] [hB : DecidableEq B]
-    (L : MultilinearMap ℂ (fun _ : S => V) ℂ) (e : I → V)
-    (blocks : S → B) (src dst : B) (hsd : src ≠ dst)
-    (hdegree : (Finset.univ.filter (fun s => blocks s = dst)).card <
-      (Finset.univ.filter (fun s => blocks s = src)).card)
-    (hzero : ∀ s, blocks s = src →
-      multilinearPolynomial L e (Function.update blocks s dst) = 0) :
-    multilinearPolynomial L e blocks = 0 := by
-  classical
-  have hS_eq : hS = Classical.decEq S := Subsingleton.elim _ _
-  cases hS_eq
-  have hB_eq : hB = Classical.decEq B := Subsingleton.elim _ _
-  cases hB_eq
-  apply PolynomialShift.eq_zero_of_shift_eq_zero src dst hsd hdegree
-  · unfold PolynomialWeights.IsBlockHomogeneous PolynomialWeights.blockWeight
-    simpa only [multilinearPolynomial] using
-      EncodingDegree.blockHomogeneous_expansion
-        (fun k => L (fun s => e (k s))) blocks src
-  · unfold PolynomialWeights.IsBlockHomogeneous PolynomialWeights.blockWeight
-    simpa only [multilinearPolynomial] using
-      EncodingDegree.blockHomogeneous_expansion
-        (fun k => L (fun s => e (k s))) blocks dst
-  · have hd := SlotPolynomial.derivation_polynomial
-      (fun k => L (fun s => e (k s))) blocks src dst
-      (PolynomialShift.derivation src dst) (by intro b i; simp)
-    have hz : (∑ s ∈ Finset.univ.filter (fun s => blocks s = src),
-        SlotPolynomial.polynomial (fun k => L (fun s => e (k s)))
-          (Function.update blocks s dst)) = 0 := by
-      apply Finset.sum_eq_zero
-      intro s hs
-      simpa only [SlotPolynomial.polynomial, multilinearPolynomial] using
-        hzero s (Finset.mem_filter.mp hs).2
-    rw [hz] at hd
-    simpa only [PolynomialShift.shift_apply, SlotPolynomial.polynomial,
-      multilinearPolynomial] using hd
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.MultilinearPolynomial
+import OAI.RepresentationTheory.FoulkesHowe.SlotDerivative
+import OAI.RepresentationTheory.FoulkesHowe.EncodingHomogeneous
+import OAI.RepresentationTheory.FoulkesHowe.BlockInjectivity
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators
+
+namespace Problem346
+
+/-- Cancel one directional derivative of a repeated-block multilinear polynomial.
+It suffices that every single source-slot replacement vanishes; no normalization
+or nonvanishing factorial is needed. -/
+theorem multilinearPolynomial_eq_zero_of_replacements
+    {V S I B : Type*} [AddCommGroup V] [Module ℂ V]
+    [Fintype S] [Fintype I] [Fintype B] [hS : DecidableEq S] [hB : DecidableEq B]
+    (L : MultilinearMap ℂ (fun _ : S => V) ℂ) (e : I → V)
+    (blocks : S → B) (src dst : B) (hsd : src ≠ dst)
+    (hdegree : (Finset.univ.filter (fun s => blocks s = dst)).card <
+      (Finset.univ.filter (fun s => blocks s = src)).card)
+    (hzero : ∀ s, blocks s = src →
+      multilinearPolynomial L e (Function.update blocks s dst) = 0) :
+    multilinearPolynomial L e blocks = 0 := by
+  classical
+  have hS_eq : hS = Classical.decEq S := Subsingleton.elim _ _
+  cases hS_eq
+  have hB_eq : hB = Classical.decEq B := Subsingleton.elim _ _
+  cases hB_eq
+  apply PolynomialShift.eq_zero_of_shift_eq_zero src dst hsd hdegree
+  · unfold PolynomialWeights.IsBlockHomogeneous PolynomialWeights.blockWeight
+    simpa only [multilinearPolynomial] using
+      EncodingDegree.blockHomogeneous_expansion
+        (fun k => L (fun s => e (k s))) blocks src
+  · unfold PolynomialWeights.IsBlockHomogeneous PolynomialWeights.blockWeight
+    simpa only [multilinearPolynomial] using
+      EncodingDegree.blockHomogeneous_expansion
+        (fun k => L (fun s => e (k s))) blocks dst
+  · have hd := SlotPolynomial.derivation_polynomial
+      (fun k => L (fun s => e (k s))) blocks src dst
+      (PolynomialShift.derivation src dst) (by intro b i; simp)
+    have hz : (∑ s ∈ Finset.univ.filter (fun s => blocks s = src),
+        SlotPolynomial.polynomial (fun k => L (fun s => e (k s)))
+          (Function.update blocks s dst)) = 0 := by
+      apply Finset.sum_eq_zero
+      intro s hs
+      simpa only [SlotPolynomial.polynomial, multilinearPolynomial] using
+        hzero s (Finset.mem_filter.mp hs).2
+    rw [hz] at hd
+    simpa only [PolynomialShift.shift_apply, SlotPolynomial.polynomial,
+      multilinearPolynomial] using hd
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/EncodingHomogeneous.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/EncodingHomogeneous.lean
index a21af5f9b1508f1066eabbe8ca844dff20470fab..31dee8a4471f1da9a10c895fe086e164e0a8207e 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/EncodingHomogeneous.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/EncodingHomogeneous.lean
@@ -1,61 +1,61 @@
-import Mathlib
-
-namespace OAI
-
-noncomputable section
-open scoped BigOperators
-
-namespace Problem346.EncodingDegree
-
-variable {R : Type*} [CommSemiring R]
-variable {S K B : Type*} [Fintype S] [DecidableEq S] [Fintype K]
-
-/-- A coefficient expansion has weighted degree given by the sum of the weights
-of its assigned vector blocks. The coefficients can be arbitrary. -/
-theorem weightedHomogeneous_expansion (c : (S → K) → R) (blocks : S → B)
-    (w : B → ℕ) :
-    MvPolynomial.IsWeightedHomogeneous (fun x : B × K => w x.1)
-      (∑ k : S → K, MvPolynomial.C (c k) *
-        ∏ i : S, MvPolynomial.X (blocks i, k i))
-      (∑ i : S, w (blocks i)) := by
-  classical
-  apply MvPolynomial.IsWeightedHomogeneous.sum
-  intro k _
-  apply MvPolynomial.IsWeightedHomogeneous.C_mul
-  apply MvPolynomial.IsWeightedHomogeneous.prod
-  intro i _
-  exact MvPolynomial.isWeightedHomogeneous_X _ _ _
-
-/-- The degree in one block is exactly the number of occurrences of that block. -/
-theorem blockHomogeneous_expansion [DecidableEq B]
-    (c : (S → K) → R) (blocks : S → B) (b : B) :
-    MvPolynomial.IsWeightedHomogeneous
-      (fun x : B × K => if x.1 = b then 1 else 0)
-      (∑ k : S → K, MvPolynomial.C (c k) *
-        ∏ i : S, MvPolynomial.X (blocks i, k i))
-      (Finset.univ.filter (fun i => blocks i = b)).card := by
-  simpa only [Finset.sum_boole, Nat.cast_id] using
-    (weightedHomogeneous_expansion c blocks (fun x => if x = b then 1 else 0))
-
-/-- Changing a single assigned block changes the weight by exactly the
-corresponding source/destination contribution. This form avoids truncated subtraction. -/
-theorem sum_weight_update (blocks : S → B) (w : B → ℕ)
-    (i : S) (b : B) :
-    (∑ j : S, w ((Function.update blocks i b) j)) + w (blocks i) =
-      (∑ j : S, w (blocks j)) + w b := by
-  classical
-  have h1 := Finset.sum_erase_add (s := Finset.univ) (f := fun j => w ((Function.update blocks i b) j)) (Finset.mem_univ i)
-  have h2 := Finset.sum_erase_add (s := Finset.univ) (f := fun j => w (blocks j)) (Finset.mem_univ i)
-  have h : (∑ j ∈ Finset.univ.erase i, w ((Function.update blocks i b) j)) =
-      ∑ j ∈ Finset.univ.erase i, w (blocks j) := by
-    apply Finset.sum_congr rfl
-    intro j hj
-    rw [Function.update_of_ne (Finset.mem_erase.mp hj).1]
-  simp only [Function.update_self] at h1
-  omega
-
-end Problem346.EncodingDegree
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators
+
+namespace Problem346.EncodingDegree
+
+variable {R : Type*} [CommSemiring R]
+variable {S K B : Type*} [Fintype S] [DecidableEq S] [Fintype K]
+
+/-- A coefficient expansion has weighted degree given by the sum of the weights
+of its assigned vector blocks. The coefficients can be arbitrary. -/
+theorem weightedHomogeneous_expansion (c : (S → K) → R) (blocks : S → B)
+    (w : B → ℕ) :
+    MvPolynomial.IsWeightedHomogeneous (fun x : B × K => w x.1)
+      (∑ k : S → K, MvPolynomial.C (c k) *
+        ∏ i : S, MvPolynomial.X (blocks i, k i))
+      (∑ i : S, w (blocks i)) := by
+  classical
+  apply MvPolynomial.IsWeightedHomogeneous.sum
+  intro k _
+  apply MvPolynomial.IsWeightedHomogeneous.C_mul
+  apply MvPolynomial.IsWeightedHomogeneous.prod
+  intro i _
+  exact MvPolynomial.isWeightedHomogeneous_X _ _ _
+
+/-- The degree in one block is exactly the number of occurrences of that block. -/
+theorem blockHomogeneous_expansion [DecidableEq B]
+    (c : (S → K) → R) (blocks : S → B) (b : B) :
+    MvPolynomial.IsWeightedHomogeneous
+      (fun x : B × K => if x.1 = b then 1 else 0)
+      (∑ k : S → K, MvPolynomial.C (c k) *
+        ∏ i : S, MvPolynomial.X (blocks i, k i))
+      (Finset.univ.filter (fun i => blocks i = b)).card := by
+  simpa only [Finset.sum_boole, Nat.cast_id] using
+    (weightedHomogeneous_expansion c blocks (fun x => if x = b then 1 else 0))
+
+/-- Changing a single assigned block changes the weight by exactly the
+corresponding source/destination contribution. This form avoids truncated subtraction. -/
+theorem sum_weight_update (blocks : S → B) (w : B → ℕ)
+    (i : S) (b : B) :
+    (∑ j : S, w ((Function.update blocks i b) j)) + w (blocks i) =
+      (∑ j : S, w (blocks j)) + w b := by
+  classical
+  have h1 := Finset.sum_erase_add (s := Finset.univ) (f := fun j => w ((Function.update blocks i b) j)) (Finset.mem_univ i)
+  have h2 := Finset.sum_erase_add (s := Finset.univ) (f := fun j => w (blocks j)) (Finset.mem_univ i)
+  have h : (∑ j ∈ Finset.univ.erase i, w ((Function.update blocks i b) j)) =
+      ∑ j ∈ Finset.univ.erase i, w (blocks j) := by
+    apply Finset.sum_congr rfl
+    intro j hj
+    rw [Function.update_of_ne (Finset.mem_erase.mp hj).1]
+  simp only [Function.update_self] at h1
+  omega
+
+end Problem346.EncodingDegree
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/FiniteDimension.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/FiniteDimension.lean
index 62a7a5a50fd46e1cf877b55e6a4495704031b918..a9e1cae105e19ad160240ba8f20858c2581f04ff 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/FiniteDimension.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/FiniteDimension.lean
@@ -1,20 +1,20 @@
-import OAI.RepresentationTheory.FoulkesHowe.TensorPresentation
-
-namespace OAI
-
-noncomputable section
-universe u
-
-namespace Problem346
-
-/-- Every homogeneous symmetric power of a finite-dimensional complex space is
-finite dimensional. The proof uses its tensor-power presentation. -/
-instance finiteDimensional_symPow (n : ℕ) (V : Type u) [AddCommGroup V]
-    [Module ℂ V] [FiniteDimensional ℂ V] : FiniteDimensional ℂ (SymPow n V) :=
-  Module.Finite.of_surjective (tensorToSymPow n V) (tensorToSymPow_surjective n V)
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.TensorPresentation
+
+namespace OAI
+
+noncomputable section
+universe u
+
+namespace Problem346
+
+/-- Every homogeneous symmetric power of a finite-dimensional complex space is
+finite dimensional. The proof uses its tensor-power presentation. -/
+instance finiteDimensional_symPow (n : ℕ) (V : Type u) [AddCommGroup V]
+    [Module ℂ V] [FiniteDimensional ℂ V] : FiniteDimensional ℂ (SymPow n V) :=
+  Module.Finite.of_surjective (tensorToSymPow n V) (tensorToSymPow_surjective n V)
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/FirstCounts.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/FirstCounts.lean
index 3a1af1626fd7b0ea1bd7839c938497c01876739c..ddd058c8797a6ccc2f863023281d42b06ae0f6d0 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/FirstCounts.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/FirstCounts.lean
@@ -1,160 +1,160 @@
-import OAI.RepresentationTheory.FoulkesHowe.FirstLabels
-
-namespace OAI
-
-open scoped BigOperators
-
-namespace Problem346
-
-@[simp] theorem firstLabel_zero (r m k : ℕ) (j : Fin (r + m)) :
-    firstLabel r m k (0, j) =
-      if h : j.val < k ∧ j.val < m then some (some ⟨j.val, h.2⟩) else some none := by
-  simp [firstLabel]
-
-@[simp] theorem firstLabel_succ (r m k : ℕ) (i : Fin r) (j : Fin (r + m)) :
-    firstLabel r m k (i.succ, j) =
-      Fin.addCases (fun _ : Fin r => none) (fun j : Fin m => some (some j)) j := by
-  simp [firstLabel]
-
-@[simp] theorem firstLabel_zero_ne_x (r m k : ℕ) (j : Fin (r + m)) :
-    firstLabel r m k (0, j) ≠ none := by
-  simp only [firstLabel_zero]
-  split <;> simp
-
-theorem firstLabel_zero_eq_t_iff (r m k : ℕ) (hk : k ≤ m) (j : Fin (r + m)) :
-    firstLabel r m k (0, j) = some none ↔ k ≤ j.val := by
-  simp only [firstLabel_zero]
-  split <;> simp_all
-  omega
-
-theorem firstLabel_zero_eq_y_iff (r m k : ℕ) (_hk : k ≤ m)
-    (j : Fin (r + m)) (y : Fin m) :
-    firstLabel r m k (0, j) = some (some y) ↔ j.val = y.val ∧ y.val < k := by
-  simp only [firstLabel_zero]
-  split
-  · simp only [Option.some.injEq, Fin.ext_iff]
-    rename_i h
-    constructor
-    · intro h'
-      exact ⟨h', h' ▸ h.1⟩
-    · exact fun h' => h'.1
-  · rename_i h
-    simp only [Option.some.injEq, reduceCtorEq, false_iff]
-    omega
-
-/-- Number of individual factors labelled by a given vector block. -/
-def firstLabelCount (r m k : ℕ) (c : FirstBlock m) : ℕ :=
-  ∑ p : Fin (r + 1) × Fin (r + m), if firstLabel r m k p = c then 1 else 0
-
-
-theorem firstLabelCount_x (r m k : ℕ) : firstLabelCount r m k none = r * r := by
-  classical
-  simp only [firstLabelCount, Fintype.sum_prod_type, Fin.sum_univ_succ,
-    firstLabel_zero_ne_x, ite_false, Finset.sum_const_zero, zero_add, firstLabel_succ]
-  have hrow : (∑ j : Fin (r + m),
-      if Fin.addCases (fun _ : Fin r => none) (fun j : Fin m => some (some j)) j =
-        (none : FirstBlock m) then 1 else 0) = r := by
-    rw [Fin.sum_univ_add]
-    simp
-  simp only [hrow, Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul, Nat.cast_id]
-
- theorem firstLabelCount_t (r m k : ℕ) (hk : k ≤ m) :
-    firstLabelCount r m k (some none) = r + m - k := by
-  classical
-  rw [firstLabelCount, Fintype.sum_prod_type, Fin.sum_univ_succ]
-  have hrow : (∑ j : Fin (r + m),
-      if Fin.addCases (fun _ : Fin r => none) (fun j : Fin m => some (some j)) j =
-        (some none : FirstBlock m) then 1 else 0) = 0 := by
-    rw [Fin.sum_univ_add]
-    simp
-  simp only [firstLabel_succ, hrow, Finset.sum_const_zero, add_zero,
-    firstLabel_zero_eq_t_iff r m k hk]
-  rw [← Finset.card_filter]
-  have hc := Finset.card_filter_add_card_filter_not
-    (s := (Finset.univ : Finset (Fin (r + m)))) (fun j => j.val < k)
-  rw [Fin.card_filter_val_lt, Nat.min_eq_right (by omega)] at hc
-  simp only [not_lt, Finset.card_univ, Fintype.card_fin] at hc
-  omega
-
-
-theorem firstLabelCount_y (r m k : ℕ) (hk : k ≤ m) (y : Fin m) :
-    firstLabelCount r m k (some (some y)) = r + if y.val < k then 1 else 0 := by
-  classical
-  rw [firstLabelCount, Fintype.sum_prod_type, Fin.sum_univ_succ]
-  have hrow : (∑ j : Fin (r + m),
-      if Fin.addCases (fun _ : Fin r => none) (fun j : Fin m => some (some j)) j =
-        (some (some y) : FirstBlock m) then 1 else 0) = 1 := by
-    rw [Fin.sum_univ_add]
-    simp
-  simp only [firstLabel_succ, hrow, Finset.sum_const, Finset.card_univ,
-    Fintype.card_fin, smul_eq_mul, mul_one]
-  let y' : Fin (r + m) := ⟨y.val, by omega⟩
-  have hj : ∀ j : Fin (r + m), j.val = y.val ↔ j = y' := by
-    intro j
-    change j.val = y'.val ↔ j = y'
-    exact Fin.ext_iff.symm
-  simp only [firstLabel_zero_eq_y_iff r m k hk, hj]
-  by_cases hy : y.val < k <;> simp [hy, add_comm]
-
-
-theorem firstLabel_step (r m k : ℕ) (hk : k < m) :
-    firstLabel r m (k + 1) = Function.update (firstLabel r m k)
-      (0, ⟨k, Nat.lt_of_lt_of_le hk (Nat.le_add_left m r)⟩) (some (some ⟨k, hk⟩)) := by
-  classical
-  funext p
-  rcases p with ⟨i, j⟩
-  by_cases hi : i = 0
-  · subst i
-    by_cases hj : j.val = k
-    · have hjeq : j = (⟨k, Nat.lt_of_lt_of_le hk (Nat.le_add_left m r)⟩ : Fin (r + m)) := Fin.ext hj
-      rw [hjeq]
-      simp [firstLabel, hk]
-    · have hp : ((0 : Fin (r + 1)), j) ≠ (0, (⟨k, Nat.lt_of_lt_of_le hk (Nat.le_add_left m r)⟩ : Fin (r + m))) := by
-        intro h
-        exact hj (congrArg (fun p : Fin (r + 1) × Fin (r + m) => p.2.val) h)
-      rw [Function.update_of_ne hp]
-      have hlt : j.val < k + 1 ↔ j.val < k := by omega
-      simp only [firstLabel_zero, hlt]
-  · have hp : (i, j) ≠ (0, (⟨k, Nat.lt_of_lt_of_le hk (Nat.le_add_left m r)⟩ : Fin (r + m))) := by
-      intro h
-      exact hi (congrArg Prod.fst h)
-    rw [Function.update_of_ne hp]
-    simp [firstLabel, hi]
-
-
-/-- The sum-of-indicators count agrees with the cardinality used by polynomial
-block-degree statements. -/
-theorem firstLabel_card (r m k : ℕ) (c : FirstBlock m) :
-    (Finset.univ.filter (fun p => firstLabel r m k p = c)).card =
-      firstLabelCount r m k c := by
-  classical
-  exact Finset.card_filter _ _
-
-theorem firstLabel_card_t (r m k : ℕ) (hk : k ≤ m) :
-    (Finset.univ.filter (fun p => firstLabel r m k p = some none)).card = r + m - k := by
-  rw [firstLabel_card, firstLabelCount_t r m k hk]
-
-theorem firstLabel_card_y (r m k : ℕ) (hk : k ≤ m) (y : Fin m) :
-    (Finset.univ.filter (fun p => firstLabel r m k p = some (some y))).card =
-      r + if y.val < k then 1 else 0 := by
-  rw [firstLabel_card, firstLabelCount_y r m k hk]
-
-theorem firstLabel_card_y_next (r m k : ℕ) (hk : k < m) :
-    (Finset.univ.filter (fun p => firstLabel r m k p = some (some ⟨k, hk⟩))).card = r := by
-  rw [firstLabel_card_y r m k (Nat.le_of_lt hk)]
-  simp
-
-/-- Only unreplaced positions of the distinguished row carry the source label. -/
-theorem firstLabel_eq_t_iff (r m k : ℕ) (hk : k ≤ m)
-    (p : Fin (r + 1) × Fin (r + m)) :
-    firstLabel r m k p = some none ↔ p.1 = 0 ∧ k ≤ p.2.val := by
-  rcases p with ⟨i, j⟩
-  by_cases hi : i = 0
-  · subst i
-    simpa using firstLabel_zero_eq_t_iff r m k hk j
-  · simp only [firstLabel, hi, ite_false, false_and, iff_false]
-    refine Fin.addCases (fun x => ?_) (fun y => ?_) j <;> simp
-
-end Problem346
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.FirstLabels
+
+namespace OAI
+
+open scoped BigOperators
+
+namespace Problem346
+
+@[simp] theorem firstLabel_zero (r m k : ℕ) (j : Fin (r + m)) :
+    firstLabel r m k (0, j) =
+      if h : j.val < k ∧ j.val < m then some (some ⟨j.val, h.2⟩) else some none := by
+  simp [firstLabel]
+
+@[simp] theorem firstLabel_succ (r m k : ℕ) (i : Fin r) (j : Fin (r + m)) :
+    firstLabel r m k (i.succ, j) =
+      Fin.addCases (fun _ : Fin r => none) (fun j : Fin m => some (some j)) j := by
+  simp [firstLabel]
+
+@[simp] theorem firstLabel_zero_ne_x (r m k : ℕ) (j : Fin (r + m)) :
+    firstLabel r m k (0, j) ≠ none := by
+  simp only [firstLabel_zero]
+  split <;> simp
+
+theorem firstLabel_zero_eq_t_iff (r m k : ℕ) (hk : k ≤ m) (j : Fin (r + m)) :
+    firstLabel r m k (0, j) = some none ↔ k ≤ j.val := by
+  simp only [firstLabel_zero]
+  split <;> simp_all
+  omega
+
+theorem firstLabel_zero_eq_y_iff (r m k : ℕ) (_hk : k ≤ m)
+    (j : Fin (r + m)) (y : Fin m) :
+    firstLabel r m k (0, j) = some (some y) ↔ j.val = y.val ∧ y.val < k := by
+  simp only [firstLabel_zero]
+  split
+  · simp only [Option.some.injEq, Fin.ext_iff]
+    rename_i h
+    constructor
+    · intro h'
+      exact ⟨h', h' ▸ h.1⟩
+    · exact fun h' => h'.1
+  · rename_i h
+    simp only [Option.some.injEq, reduceCtorEq, false_iff]
+    omega
+
+/-- Number of individual factors labelled by a given vector block. -/
+def firstLabelCount (r m k : ℕ) (c : FirstBlock m) : ℕ :=
+  ∑ p : Fin (r + 1) × Fin (r + m), if firstLabel r m k p = c then 1 else 0
+
+
+theorem firstLabelCount_x (r m k : ℕ) : firstLabelCount r m k none = r * r := by
+  classical
+  simp only [firstLabelCount, Fintype.sum_prod_type, Fin.sum_univ_succ,
+    firstLabel_zero_ne_x, ite_false, Finset.sum_const_zero, zero_add, firstLabel_succ]
+  have hrow : (∑ j : Fin (r + m),
+      if Fin.addCases (fun _ : Fin r => none) (fun j : Fin m => some (some j)) j =
+        (none : FirstBlock m) then 1 else 0) = r := by
+    rw [Fin.sum_univ_add]
+    simp
+  simp only [hrow, Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul, Nat.cast_id]
+
+ theorem firstLabelCount_t (r m k : ℕ) (hk : k ≤ m) :
+    firstLabelCount r m k (some none) = r + m - k := by
+  classical
+  rw [firstLabelCount, Fintype.sum_prod_type, Fin.sum_univ_succ]
+  have hrow : (∑ j : Fin (r + m),
+      if Fin.addCases (fun _ : Fin r => none) (fun j : Fin m => some (some j)) j =
+        (some none : FirstBlock m) then 1 else 0) = 0 := by
+    rw [Fin.sum_univ_add]
+    simp
+  simp only [firstLabel_succ, hrow, Finset.sum_const_zero, add_zero,
+    firstLabel_zero_eq_t_iff r m k hk]
+  rw [← Finset.card_filter]
+  have hc := Finset.card_filter_add_card_filter_not
+    (s := (Finset.univ : Finset (Fin (r + m)))) (fun j => j.val < k)
+  rw [Fin.card_filter_val_lt, Nat.min_eq_right (by omega)] at hc
+  simp only [not_lt, Finset.card_univ, Fintype.card_fin] at hc
+  omega
+
+
+theorem firstLabelCount_y (r m k : ℕ) (hk : k ≤ m) (y : Fin m) :
+    firstLabelCount r m k (some (some y)) = r + if y.val < k then 1 else 0 := by
+  classical
+  rw [firstLabelCount, Fintype.sum_prod_type, Fin.sum_univ_succ]
+  have hrow : (∑ j : Fin (r + m),
+      if Fin.addCases (fun _ : Fin r => none) (fun j : Fin m => some (some j)) j =
+        (some (some y) : FirstBlock m) then 1 else 0) = 1 := by
+    rw [Fin.sum_univ_add]
+    simp
+  simp only [firstLabel_succ, hrow, Finset.sum_const, Finset.card_univ,
+    Fintype.card_fin, smul_eq_mul, mul_one]
+  let y' : Fin (r + m) := ⟨y.val, by omega⟩
+  have hj : ∀ j : Fin (r + m), j.val = y.val ↔ j = y' := by
+    intro j
+    change j.val = y'.val ↔ j = y'
+    exact Fin.ext_iff.symm
+  simp only [firstLabel_zero_eq_y_iff r m k hk, hj]
+  by_cases hy : y.val < k <;> simp [hy, add_comm]
+
+
+theorem firstLabel_step (r m k : ℕ) (hk : k < m) :
+    firstLabel r m (k + 1) = Function.update (firstLabel r m k)
+      (0, ⟨k, Nat.lt_of_lt_of_le hk (Nat.le_add_left m r)⟩) (some (some ⟨k, hk⟩)) := by
+  classical
+  funext p
+  rcases p with ⟨i, j⟩
+  by_cases hi : i = 0
+  · subst i
+    by_cases hj : j.val = k
+    · have hjeq : j = (⟨k, Nat.lt_of_lt_of_le hk (Nat.le_add_left m r)⟩ : Fin (r + m)) := Fin.ext hj
+      rw [hjeq]
+      simp [firstLabel, hk]
+    · have hp : ((0 : Fin (r + 1)), j) ≠ (0, (⟨k, Nat.lt_of_lt_of_le hk (Nat.le_add_left m r)⟩ : Fin (r + m))) := by
+        intro h
+        exact hj (congrArg (fun p : Fin (r + 1) × Fin (r + m) => p.2.val) h)
+      rw [Function.update_of_ne hp]
+      have hlt : j.val < k + 1 ↔ j.val < k := by omega
+      simp only [firstLabel_zero, hlt]
+  · have hp : (i, j) ≠ (0, (⟨k, Nat.lt_of_lt_of_le hk (Nat.le_add_left m r)⟩ : Fin (r + m))) := by
+      intro h
+      exact hi (congrArg Prod.fst h)
+    rw [Function.update_of_ne hp]
+    simp [firstLabel, hi]
+
+
+/-- The sum-of-indicators count agrees with the cardinality used by polynomial
+block-degree statements. -/
+theorem firstLabel_card (r m k : ℕ) (c : FirstBlock m) :
+    (Finset.univ.filter (fun p => firstLabel r m k p = c)).card =
+      firstLabelCount r m k c := by
+  classical
+  exact Finset.card_filter _ _
+
+theorem firstLabel_card_t (r m k : ℕ) (hk : k ≤ m) :
+    (Finset.univ.filter (fun p => firstLabel r m k p = some none)).card = r + m - k := by
+  rw [firstLabel_card, firstLabelCount_t r m k hk]
+
+theorem firstLabel_card_y (r m k : ℕ) (hk : k ≤ m) (y : Fin m) :
+    (Finset.univ.filter (fun p => firstLabel r m k p = some (some y))).card =
+      r + if y.val < k then 1 else 0 := by
+  rw [firstLabel_card, firstLabelCount_y r m k hk]
+
+theorem firstLabel_card_y_next (r m k : ℕ) (hk : k < m) :
+    (Finset.univ.filter (fun p => firstLabel r m k p = some (some ⟨k, hk⟩))).card = r := by
+  rw [firstLabel_card_y r m k (Nat.le_of_lt hk)]
+  simp
+
+/-- Only unreplaced positions of the distinguished row carry the source label. -/
+theorem firstLabel_eq_t_iff (r m k : ℕ) (hk : k ≤ m)
+    (p : Fin (r + 1) × Fin (r + m)) :
+    firstLabel r m k p = some none ↔ p.1 = 0 ∧ k ≤ p.2.val := by
+  rcases p with ⟨i, j⟩
+  by_cases hi : i = 0
+  · subst i
+    simpa using firstLabel_zero_eq_t_iff r m k hk j
+  · simp only [firstLabel, hi, ite_false, false_and, iff_false]
+    refine Fin.addCases (fun x => ?_) (fun y => ?_) j <;> simp
+
+end Problem346
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/FirstLabels.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/FirstLabels.lean
index f35011ab3dd59ef3ee3909c7729ffd5f5da69682..8924c13ef524852c4102925fa446ae0859c00914 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/FirstLabels.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/FirstLabels.lean
@@ -1,34 +1,34 @@
-import Mathlib
-
-namespace OAI
-
-namespace Problem346
-
-/-- Blocks for the first transfer: `none` is x, `some none` is t,
-and `some (some j)` is the j-th common product factor. -/
-abbrev FirstBlock (m : ℕ) := Option (Option (Fin m))
-
-/-- The slot labels after transferring the first k factors out of the t block. -/
-def firstLabel (r m k : ℕ) (p : Fin (r+1) × Fin (r+m)) : FirstBlock m :=
-  if p.1 = 0 then
-    if h : p.2.val < k ∧ p.2.val < m then some (some ⟨p.2.val, h.2⟩)
-    else some none
-  else Fin.addCases (fun _ : Fin r => none) (fun j : Fin m => some (some j)) p.2
-
-@[simp] theorem firstLabel_zero_row (r m k : ℕ) (j : Fin (r+m)) :
-    firstLabel r m k (0,j) =
-      if h : j.val < k ∧ j.val < m then some (some ⟨j.val,h.2⟩) else some none := by
-  simp [firstLabel]
-
-@[simp] theorem firstLabel_succ_row (r m k : ℕ) (i : Fin r) (j : Fin (r+m)) :
-    firstLabel r m k (i.succ,j) =
-      Fin.addCases (fun _ : Fin r => none) (fun j : Fin m => some (some j)) j := by
-  simp [firstLabel]
-
-@[simp] theorem firstLabel_initial_zero_row (r m : ℕ) (j : Fin (r+m)) :
-    firstLabel r m 0 (0,j) = some none := by
-  simp
-
-end Problem346
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+namespace Problem346
+
+/-- Blocks for the first transfer: `none` is x, `some none` is t,
+and `some (some j)` is the j-th common product factor. -/
+abbrev FirstBlock (m : ℕ) := Option (Option (Fin m))
+
+/-- The slot labels after transferring the first k factors out of the t block. -/
+def firstLabel (r m k : ℕ) (p : Fin (r+1) × Fin (r+m)) : FirstBlock m :=
+  if p.1 = 0 then
+    if h : p.2.val < k ∧ p.2.val < m then some (some ⟨p.2.val, h.2⟩)
+    else some none
+  else Fin.addCases (fun _ : Fin r => none) (fun j : Fin m => some (some j)) p.2
+
+@[simp] theorem firstLabel_zero_row (r m k : ℕ) (j : Fin (r+m)) :
+    firstLabel r m k (0,j) =
+      if h : j.val < k ∧ j.val < m then some (some ⟨j.val,h.2⟩) else some none := by
+  simp [firstLabel]
+
+@[simp] theorem firstLabel_succ_row (r m k : ℕ) (i : Fin r) (j : Fin (r+m)) :
+    firstLabel r m k (i.succ,j) =
+      Fin.addCases (fun _ : Fin r => none) (fun j : Fin m => some (some j)) j := by
+  simp [firstLabel]
+
+@[simp] theorem firstLabel_initial_zero_row (r m : ℕ) (j : Fin (r+m)) :
+    firstLabel r m 0 (0,j) = some none := by
+  simp
+
+end Problem346
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/FirstPolynomial.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/FirstPolynomial.lean
index 9a2dd697ae41c48b5bbd48882a18b828ad898c1e..805e28ca1d568ff4b07149b9eebe9d3065b277da 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/FirstPolynomial.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/FirstPolynomial.lean
@@ -1,39 +1,39 @@
-import OAI.RepresentationTheory.FoulkesHowe.FirstLabels
-import OAI.RepresentationTheory.FoulkesHowe.FlattenedForm
-import OAI.RepresentationTheory.FoulkesHowe.MultilinearPolynomial
-import OAI.RepresentationTheory.FoulkesHowe.EncodingHomogeneous
-import OAI.RepresentationTheory.FoulkesHowe.BlockWeight
-
-namespace OAI
-
-noncomputable section
-open scoped BigOperators
-namespace Problem346
-universe u v
-variable {V : Type u} [AddCommGroup V] [Module ℂ V]
-variable {κ : Type v} [Fintype κ]
-
-/-- The scalar polynomial at the k-th stage of the first transfer sequence. -/
-def firstPolynomial {r m : ℕ}
-    (T : SymmetricMultilinearForm (r+1) (r+m) V) (e : κ → V) (k : ℕ) :
-    MvPolynomial (FirstBlock m × κ) ℂ :=
-  multilinearPolynomial (flattenedForm T) e (firstLabel r m k)
-
-/-- Direct block degrees of every intermediate polynomial, by counting its labels. -/
-theorem firstPolynomial_blockHomogeneous {r m : ℕ}
-    (T : SymmetricMultilinearForm (r+1) (r+m) V) (e : κ → V)
-    (k : ℕ) (u : FirstBlock m) :
-    PolynomialWeights.IsBlockHomogeneous u (firstPolynomial T e k)
-      (Finset.univ.filter (fun p => firstLabel r m k p = u)).card := by
-  classical
-  let : DecidableEq (Fin (r+1) × Fin (r+m)) := Classical.decEq _
-  unfold PolynomialWeights.IsBlockHomogeneous PolynomialWeights.blockWeight
-  simpa only [firstPolynomial, multilinearPolynomial] using
-    (EncodingDegree.blockHomogeneous_expansion
-      (fun z => flattenedForm T (fun p => e (z p))) (firstLabel r m k) u)
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.FirstLabels
+import OAI.RepresentationTheory.FoulkesHowe.FlattenedForm
+import OAI.RepresentationTheory.FoulkesHowe.MultilinearPolynomial
+import OAI.RepresentationTheory.FoulkesHowe.EncodingHomogeneous
+import OAI.RepresentationTheory.FoulkesHowe.BlockWeight
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators
+namespace Problem346
+universe u v
+variable {V : Type u} [AddCommGroup V] [Module ℂ V]
+variable {κ : Type v} [Fintype κ]
+
+/-- The scalar polynomial at the k-th stage of the first transfer sequence. -/
+def firstPolynomial {r m : ℕ}
+    (T : SymmetricMultilinearForm (r+1) (r+m) V) (e : κ → V) (k : ℕ) :
+    MvPolynomial (FirstBlock m × κ) ℂ :=
+  multilinearPolynomial (flattenedForm T) e (firstLabel r m k)
+
+/-- Direct block degrees of every intermediate polynomial, by counting its labels. -/
+theorem firstPolynomial_blockHomogeneous {r m : ℕ}
+    (T : SymmetricMultilinearForm (r+1) (r+m) V) (e : κ → V)
+    (k : ℕ) (u : FirstBlock m) :
+    PolynomialWeights.IsBlockHomogeneous u (firstPolynomial T e k)
+      (Finset.univ.filter (fun p => firstLabel r m k p = u)).card := by
+  classical
+  let : DecidableEq (Fin (r+1) × Fin (r+m)) := Classical.decEq _
+  unfold PolynomialWeights.IsBlockHomogeneous PolynomialWeights.blockWeight
+  simpa only [firstPolynomial, multilinearPolynomial] using
+    (EncodingDegree.blockHomogeneous_expansion
+      (fun z => flattenedForm T (fun p => e (z p))) (firstLabel r m k) u)
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/FirstTransfer.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/FirstTransfer.lean
index 80947af41fd137b6e218cfe2bf04457f4c3eaba6..03d091a3b866ea35c26fba438b1efa936684a40f 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/FirstTransfer.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/FirstTransfer.lean
@@ -1,76 +1,76 @@
-import OAI.RepresentationTheory.FoulkesHowe.FirstPolynomial
-import OAI.RepresentationTheory.FoulkesHowe.BlockInjectivity
-import OAI.RepresentationTheory.FoulkesHowe.BackwardCancellation
-import OAI.RepresentationTheory.FoulkesHowe.TransferDerivative
-import OAI.RepresentationTheory.FoulkesHowe.TransferEndpoints
-
-namespace OAI
-
-noncomputable section
-
-namespace Problem346
-
-universe u v
-variable {V : Type u} [AddCommGroup V] [Module ℂ V]
-variable {κ : Type v} [Fintype κ]
-
-/-- Cancel the first transfer sequence from its zero endpoint. The homogeneous
-and derivative identities are kept separate from this zero-reflection argument. -/
-theorem firstPolynomial_zero_of_chain {r m : ℕ}
-    (T : SymmetricMultilinearForm (r+1) (r+m) V) (e : κ → V)
-    (ht : ∀ k < m, PolynomialWeights.IsBlockHomogeneous (some none)
-      (firstPolynomial T e k) (r+m-k))
-    (hy : ∀ (k : ℕ) (hk : k < m), PolynomialWeights.IsBlockHomogeneous
-      (some (some (⟨k,hk⟩ : Fin m))) (firstPolynomial T e k) r)
-    (hD : ∀ (k : ℕ) (hk : k < m),
-      PolynomialShift.shift (some none) (some (some (⟨k,hk⟩ : Fin m)))
-        (firstPolynomial T e k) = ((r+m-k : ℕ) : ℂ) • firstPolynomial T e (k+1))
-    (hm : firstPolynomial T e m = 0) : firstPolynomial T e 0 = 0 := by
-  classical
-  apply eq_zero_of_backward_chain (firstPolynomial T e) m _ hm
-  intro k hk hz
-  apply PolynomialShift.eq_zero_of_shift_eq_zero
-    (some none) (some (some (⟨k,hk⟩ : Fin m))) (by simp)
-    (show r < r+m-k by omega) (ht k hk) (hy k hk)
-  rw [hD k hk, hz, smul_zero]
-
-/-- Every terminal-zero polynomial in the concrete first-transfer family is zero
-at the initial stage. All degree counts and derivative identities are discharged. -/
-theorem firstPolynomial_zero_of_final {r m : ℕ}
-    (T : SymmetricMultilinearForm (r+1) (r+m) V) (e : κ → V)
-    (hm : firstPolynomial T e m = 0) : firstPolynomial T e 0 = 0 := by
-  classical
-  apply firstPolynomial_zero_of_chain T e _ _ _ hm
-  · intro k hk
-    simpa only [firstLabel_card_t r m k (Nat.le_of_lt hk)] using
-      firstPolynomial_blockHomogeneous T e k (some none)
-  · intro k hk
-    simpa only [firstLabel_card_y_next r m k hk] using
-      firstPolynomial_blockHomogeneous T e k (some (some (⟨k,hk⟩ : Fin m)))
-  · intro k hk
-    simpa only [firstPolynomial, PolynomialShift.shift_apply] using
-      (firstLabel_derivative r m k hk V κ T e
-        (PolynomialShift.derivation (some none) (some (some (⟨k,hk⟩ : Fin m))))
-        (fun q j => by simp))
-
-/-- The first transfer in the stabilization argument: the common product can be
-removed from the distinguished argument, while the other arguments remain fixed.
-No outer symmetry or positive-degree assumptions are needed for this step. -/
-theorem first_transfer [FiniteDimensional ℂ V] {r m : ℕ}
-    (T : SymmetricMultilinearForm (r+1) (r+m) V)
-    (h : ∀ (x t : V) (y : Fin m → V),
-      T (Fin.cons (symPowMul r m (symMonomial r V (fun _ => t)) (symMonomial m V y))
-        (fun _ => symPowMul r m (symMonomial r V (fun _ => x)) (symMonomial m V y))) = 0) :
-    ∀ (x t : V) (y : Fin m → V),
-      T (Fin.cons (symMonomial (r+m) V (fun _ => t))
-        (fun _ => symPowMul r m (symMonomial r V (fun _ => x)) (symMonomial m V y))) = 0 := by
-  let e := Module.finBasis ℂ V
-  have hm := firstPolynomial_final_eq_zero T e h
-  have hzero := firstPolynomial_zero_of_final T e hm
-  exact firstPolynomial_initial_vanishing T e hzero
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.FirstPolynomial
+import OAI.RepresentationTheory.FoulkesHowe.BlockInjectivity
+import OAI.RepresentationTheory.FoulkesHowe.BackwardCancellation
+import OAI.RepresentationTheory.FoulkesHowe.TransferDerivative
+import OAI.RepresentationTheory.FoulkesHowe.TransferEndpoints
+
+namespace OAI
+
+noncomputable section
+
+namespace Problem346
+
+universe u v
+variable {V : Type u} [AddCommGroup V] [Module ℂ V]
+variable {κ : Type v} [Fintype κ]
+
+/-- Cancel the first transfer sequence from its zero endpoint. The homogeneous
+and derivative identities are kept separate from this zero-reflection argument. -/
+theorem firstPolynomial_zero_of_chain {r m : ℕ}
+    (T : SymmetricMultilinearForm (r+1) (r+m) V) (e : κ → V)
+    (ht : ∀ k < m, PolynomialWeights.IsBlockHomogeneous (some none)
+      (firstPolynomial T e k) (r+m-k))
+    (hy : ∀ (k : ℕ) (hk : k < m), PolynomialWeights.IsBlockHomogeneous
+      (some (some (⟨k,hk⟩ : Fin m))) (firstPolynomial T e k) r)
+    (hD : ∀ (k : ℕ) (hk : k < m),
+      PolynomialShift.shift (some none) (some (some (⟨k,hk⟩ : Fin m)))
+        (firstPolynomial T e k) = ((r+m-k : ℕ) : ℂ) • firstPolynomial T e (k+1))
+    (hm : firstPolynomial T e m = 0) : firstPolynomial T e 0 = 0 := by
+  classical
+  apply eq_zero_of_backward_chain (firstPolynomial T e) m _ hm
+  intro k hk hz
+  apply PolynomialShift.eq_zero_of_shift_eq_zero
+    (some none) (some (some (⟨k,hk⟩ : Fin m))) (by simp)
+    (show r < r+m-k by omega) (ht k hk) (hy k hk)
+  rw [hD k hk, hz, smul_zero]
+
+/-- Every terminal-zero polynomial in the concrete first-transfer family is zero
+at the initial stage. All degree counts and derivative identities are discharged. -/
+theorem firstPolynomial_zero_of_final {r m : ℕ}
+    (T : SymmetricMultilinearForm (r+1) (r+m) V) (e : κ → V)
+    (hm : firstPolynomial T e m = 0) : firstPolynomial T e 0 = 0 := by
+  classical
+  apply firstPolynomial_zero_of_chain T e _ _ _ hm
+  · intro k hk
+    simpa only [firstLabel_card_t r m k (Nat.le_of_lt hk)] using
+      firstPolynomial_blockHomogeneous T e k (some none)
+  · intro k hk
+    simpa only [firstLabel_card_y_next r m k hk] using
+      firstPolynomial_blockHomogeneous T e k (some (some (⟨k,hk⟩ : Fin m)))
+  · intro k hk
+    simpa only [firstPolynomial, PolynomialShift.shift_apply] using
+      (firstLabel_derivative r m k hk V κ T e
+        (PolynomialShift.derivation (some none) (some (some (⟨k,hk⟩ : Fin m))))
+        (fun q j => by simp))
+
+/-- The first transfer in the stabilization argument: the common product can be
+removed from the distinguished argument, while the other arguments remain fixed.
+No outer symmetry or positive-degree assumptions are needed for this step. -/
+theorem first_transfer [FiniteDimensional ℂ V] {r m : ℕ}
+    (T : SymmetricMultilinearForm (r+1) (r+m) V)
+    (h : ∀ (x t : V) (y : Fin m → V),
+      T (Fin.cons (symPowMul r m (symMonomial r V (fun _ => t)) (symMonomial m V y))
+        (fun _ => symPowMul r m (symMonomial r V (fun _ => x)) (symMonomial m V y))) = 0) :
+    ∀ (x t : V) (y : Fin m → V),
+      T (Fin.cons (symMonomial (r+m) V (fun _ => t))
+        (fun _ => symPowMul r m (symMonomial r V (fun _ => x)) (symMonomial m V y))) = 0 := by
+  let e := Module.finBasis ℂ V
+  have hm := firstPolynomial_final_eq_zero T e h
+  have hzero := firstPolynomial_zero_of_final T e hm
+  exact firstPolynomial_initial_vanishing T e hzero
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/FixedSlot.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/FixedSlot.lean
index d8809a17eded4f52a1f3a7d88f047e5593bc198c..e68953572f18c12addede3006634c3465a43c0f1 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/FixedSlot.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/FixedSlot.lean
@@ -1,74 +1,74 @@
-import OAI.RepresentationTheory.FoulkesHowe.FlattenedSwap
-import OAI.RepresentationTheory.FoulkesHowe.Multiplication
-
-namespace OAI
-
-noncomputable section
-namespace Problem346
-universe u
-variable {r b : ℕ} {V : Type u} [AddCommGroup V] [Module ℂ V]
-
-/-- Flatten the remaining rows after fixing the first slot to a pure power. -/
-def fixedSlotFlatten (T : SymmetricMultilinearForm (r+1) b V) (t : V) :
-    MultilinearMap ℂ (fun _ : Fin r × Fin b => V) ℂ :=
-  flattenedSymmetricForm (T.curryLeft (symMonomial b V (fun _ => t)))
-
-@[simp] theorem fixedSlotFlatten_apply
-    (T : SymmetricMultilinearForm (r+1) b V) (t : V) (v : Fin r × Fin b → V) :
-    fixedSlotFlatten T t v =
-      T (Fin.cons (symMonomial b V (fun _ => t))
-        (fun i => symMonomial b V (fun j => v (i,j)))) := rfl
-
-/-- Every row remains symmetric in its factors after fixing the first slot. -/
-theorem fixedSlotFlatten_inner_permute
-    (T : SymmetricMultilinearForm (r+1) b V) (t : V)
-    (σ : Fin r → Equiv.Perm (Fin b)) (v : Fin r × Fin b → V) :
-    fixedSlotFlatten T t (fun p => v (p.1, σ p.1 p.2)) = fixedSlotFlatten T t v :=
-  flattenedSymmetricForm_inner_permute _ σ v
-
-/-- Repeated vectors in the remaining rows are literal pure powers. -/
-@[simp] theorem fixedSlotFlatten_constant_rows
-    (T : SymmetricMultilinearForm (r+1) b V) (t : V) (v : Fin r → V) :
-    fixedSlotFlatten T t (fun p => v p.1) =
-      T (Fin.cons (symMonomial b V (fun _ => t))
-        (fun i => symMonomial b V (fun _ => v i))) := rfl
-
-/-- Concatenating factors in every unfixed row computes graded multiplication. -/
-theorem fixedSlotFlatten_append {m n : ℕ}
-    (T : SymmetricMultilinearForm (r+1) (m+n) V) (t : V)
-    (v : Fin r → Fin m → V) (w : Fin r → Fin n → V) :
-    fixedSlotFlatten T t (fun p => Fin.append (v p.1) (w p.1) p.2) =
-      T (Fin.cons (symMonomial (m+n) V (fun _ => t))
-        (fun i => symPowMul m n (symMonomial m V (v i)) (symMonomial n V (w i)))) := by
-  simp only [fixedSlotFlatten_apply, symPowMul_symMonomial]
-
-/-- The endpoint of the second shift chain has a common power factor in every row. -/
-theorem fixedSlotFlatten_common_power {m : ℕ}
-    (T : SymmetricMultilinearForm (r+1) (r+m) V) (x t : V) (v : Fin r → V) :
-    fixedSlotFlatten T t
-      (fun p => Fin.append (fun _ : Fin r => x) (fun _ : Fin m => v p.1) p.2) =
-      T (Fin.cons (symMonomial (r+m) V (fun _ => t))
-        (fun i => symPowMul r m (symMonomial r V (fun _ => x))
-          (symMonomial m V (fun _ => v i)))) := by
-  simp only [fixedSlotFlatten_apply, symPowMul_symMonomial]
-
-/-- Swapping two factor slots in the same unfixed row changes no value. -/
-theorem fixedSlotFlatten_swap_same_row
-    (T : SymmetricMultilinearForm (r+1) b V) (t : V)
-    (p q : Fin r × Fin b) (hpq : p.1=q.1) (v : Fin r × Fin b → V) :
-    fixedSlotFlatten T t (v ∘ Equiv.swap p q) = fixedSlotFlatten T t v :=
-  flattenedSymmetricForm_swap_same_row
-    (T.curryLeft (symMonomial b V (fun _ => t))) p q hpq v
-
-/-- The fixed-slot coefficient array meets the row-swap hypothesis of shift recurrence. -/
-theorem fixedSlotFlatten_coeff_swap_same_row {I : Type*}
-    (T : SymmetricMultilinearForm (r+1) b V) (t : V) (e : I → V)
-    (p q : Fin r × Fin b) (hpq : p.1=q.1) (k : Fin r × Fin b → I) :
-    fixedSlotFlatten T t (e ∘ (k ∘ Equiv.swap p q)) = fixedSlotFlatten T t (e ∘ k) :=
-  fixedSlotFlatten_swap_same_row T t p q hpq (e ∘ k)
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.FlattenedSwap
+import OAI.RepresentationTheory.FoulkesHowe.Multiplication
+
+namespace OAI
+
+noncomputable section
+namespace Problem346
+universe u
+variable {r b : ℕ} {V : Type u} [AddCommGroup V] [Module ℂ V]
+
+/-- Flatten the remaining rows after fixing the first slot to a pure power. -/
+def fixedSlotFlatten (T : SymmetricMultilinearForm (r+1) b V) (t : V) :
+    MultilinearMap ℂ (fun _ : Fin r × Fin b => V) ℂ :=
+  flattenedSymmetricForm (T.curryLeft (symMonomial b V (fun _ => t)))
+
+@[simp] theorem fixedSlotFlatten_apply
+    (T : SymmetricMultilinearForm (r+1) b V) (t : V) (v : Fin r × Fin b → V) :
+    fixedSlotFlatten T t v =
+      T (Fin.cons (symMonomial b V (fun _ => t))
+        (fun i => symMonomial b V (fun j => v (i,j)))) := rfl
+
+/-- Every row remains symmetric in its factors after fixing the first slot. -/
+theorem fixedSlotFlatten_inner_permute
+    (T : SymmetricMultilinearForm (r+1) b V) (t : V)
+    (σ : Fin r → Equiv.Perm (Fin b)) (v : Fin r × Fin b → V) :
+    fixedSlotFlatten T t (fun p => v (p.1, σ p.1 p.2)) = fixedSlotFlatten T t v :=
+  flattenedSymmetricForm_inner_permute _ σ v
+
+/-- Repeated vectors in the remaining rows are literal pure powers. -/
+@[simp] theorem fixedSlotFlatten_constant_rows
+    (T : SymmetricMultilinearForm (r+1) b V) (t : V) (v : Fin r → V) :
+    fixedSlotFlatten T t (fun p => v p.1) =
+      T (Fin.cons (symMonomial b V (fun _ => t))
+        (fun i => symMonomial b V (fun _ => v i))) := rfl
+
+/-- Concatenating factors in every unfixed row computes graded multiplication. -/
+theorem fixedSlotFlatten_append {m n : ℕ}
+    (T : SymmetricMultilinearForm (r+1) (m+n) V) (t : V)
+    (v : Fin r → Fin m → V) (w : Fin r → Fin n → V) :
+    fixedSlotFlatten T t (fun p => Fin.append (v p.1) (w p.1) p.2) =
+      T (Fin.cons (symMonomial (m+n) V (fun _ => t))
+        (fun i => symPowMul m n (symMonomial m V (v i)) (symMonomial n V (w i)))) := by
+  simp only [fixedSlotFlatten_apply, symPowMul_symMonomial]
+
+/-- The endpoint of the second shift chain has a common power factor in every row. -/
+theorem fixedSlotFlatten_common_power {m : ℕ}
+    (T : SymmetricMultilinearForm (r+1) (r+m) V) (x t : V) (v : Fin r → V) :
+    fixedSlotFlatten T t
+      (fun p => Fin.append (fun _ : Fin r => x) (fun _ : Fin m => v p.1) p.2) =
+      T (Fin.cons (symMonomial (r+m) V (fun _ => t))
+        (fun i => symPowMul r m (symMonomial r V (fun _ => x))
+          (symMonomial m V (fun _ => v i)))) := by
+  simp only [fixedSlotFlatten_apply, symPowMul_symMonomial]
+
+/-- Swapping two factor slots in the same unfixed row changes no value. -/
+theorem fixedSlotFlatten_swap_same_row
+    (T : SymmetricMultilinearForm (r+1) b V) (t : V)
+    (p q : Fin r × Fin b) (hpq : p.1=q.1) (v : Fin r × Fin b → V) :
+    fixedSlotFlatten T t (v ∘ Equiv.swap p q) = fixedSlotFlatten T t v :=
+  flattenedSymmetricForm_swap_same_row
+    (T.curryLeft (symMonomial b V (fun _ => t))) p q hpq v
+
+/-- The fixed-slot coefficient array meets the row-swap hypothesis of shift recurrence. -/
+theorem fixedSlotFlatten_coeff_swap_same_row {I : Type*}
+    (T : SymmetricMultilinearForm (r+1) b V) (t : V) (e : I → V)
+    (p q : Fin r × Fin b) (hpq : p.1=q.1) (k : Fin r × Fin b → I) :
+    fixedSlotFlatten T t (e ∘ (k ∘ Equiv.swap p q)) = fixedSlotFlatten T t (e ∘ k) :=
+  fixedSlotFlatten_swap_same_row T t p q hpq (e ∘ k)
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/FlattenedForm.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/FlattenedForm.lean
index 855d740171ecf08df5fbbb177896b562bd41b066..7b3b24a3755b347963591b0b367f4a177eaaf4f2 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/FlattenedForm.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/FlattenedForm.lean
@@ -1,109 +1,109 @@
-import OAI.RepresentationTheory.FoulkesHowe.MonomialMultilinear
-
-namespace OAI
-
-noncomputable section
-
-namespace Problem346
-
-universe u
-
-variable {a b : ℕ} {V : Type u} [AddCommGroup V] [Module ℂ V]
-
-/-- A form on symmetric powers, evaluated on products, is multilinear in all
-of the underlying vector factors. -/
-def flattenedSymmetricForm (T : SymmetricMultilinearForm a b V) :
-    MultilinearMap ℂ (fun _ : Fin a × Fin b => V) ℂ :=
-  (T.compMultilinearMap (fun _ => symMonomialMultilinear b V)).domDomCongr
-    (Equiv.sigmaEquivProd (Fin a) (Fin b))
-
-@[simp]
-theorem flattenedSymmetricForm_apply (T : SymmetricMultilinearForm a b V)
-    (v : Fin a × Fin b → V) :
-    flattenedSymmetricForm T v =
-      T (fun i => symMonomial b V (fun j => v (i, j))) := rfl
-
-/-- Independent permutations within the rows preserve the flattened form. -/
-theorem flattenedSymmetricForm_inner_permute
-    (T : SymmetricMultilinearForm a b V)
-    (σ : Fin a → Equiv.Perm (Fin b)) (v : Fin a × Fin b → V) :
-    flattenedSymmetricForm T (fun p => v (p.1, σ p.1 p.2)) =
-      flattenedSymmetricForm T v := by
-  apply congrArg T
-  funext i
-  exact symMonomial_permute b V (σ i) (fun j => v (i, j))
-
-/-- Symmetry of the original form permits permutation of the rows. -/
-theorem flattenedSymmetricForm_outer_permute
-    (T : SymmetricMultilinearForm a b V)
-    (hT : IsSymmetricMultilinearForm a b V T)
-    (σ : Equiv.Perm (Fin a)) (v : Fin a × Fin b → V) :
-    flattenedSymmetricForm T (fun p => v (σ p.1, p.2)) =
-      flattenedSymmetricForm T v := by
-  exact hT σ (fun i => symMonomial b V (fun j => v (i, j)))
-
-/-- Compatibility API for the same flattened form, with inferred degrees. -/
-abbrev flattenedForm {a b : ℕ} {V : Type u} [AddCommGroup V] [Module ℂ V]
-    (T : SymmetricMultilinearForm a b V) :
-    MultilinearMap ℂ (fun _ : Fin a × Fin b => V) ℂ :=
-  flattenedSymmetricForm T
-
-@[simp]
-theorem flattenedForm_apply {a b : ℕ} {V : Type u} [AddCommGroup V] [Module ℂ V]
-    (T : SymmetricMultilinearForm a b V) (x : Fin a × Fin b → V) :
-    flattenedForm T x = T (fun i => symMonomial b V (fun j => x (i, j))) := rfl
-
-theorem flattenedForm_permute_factors (a b : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V] (T : SymmetricMultilinearForm a b V)
-    (σ : Fin a → Equiv.Perm (Fin b)) (x : Fin a × Fin b → V) :
-    flattenedForm T (fun p => x (p.1, σ p.1 p.2)) =
-      flattenedForm T x :=
-  flattenedSymmetricForm_inner_permute T σ x
-
-theorem flattenedForm_permute_slots (a b : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V] (T : SymmetricMultilinearForm a b V)
-    (hT : IsSymmetricMultilinearForm a b V T)
-    (σ : Equiv.Perm (Fin a)) (x : Fin a × Fin b → V) :
-    flattenedForm T (fun p => x (σ p.1, p.2)) =
-      flattenedForm T x :=
-  flattenedSymmetricForm_outer_permute T hT σ x
-
-/-- Simultaneous row and independent within-row permutation invariance. -/
-theorem flattenedForm_permute (a b : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V] (T : SymmetricMultilinearForm a b V)
-    (hT : IsSymmetricMultilinearForm a b V T)
-    (σ : Equiv.Perm (Fin a)) (τ : Fin a → Equiv.Perm (Fin b))
-    (x : Fin a × Fin b → V) :
-    flattenedForm T (fun p => x (σ p.1, τ p.1 p.2)) =
-      flattenedForm T x :=
-  (flattenedForm_permute_factors a b V T τ (fun p => x (σ p.1, p.2))).trans
-    (flattenedForm_permute_slots a b V T hT σ x)
-
-/-- Implicit-parameter factor symmetry used by the transfer sequences. -/
-theorem flattenedForm_factor_perm (T : SymmetricMultilinearForm a b V)
-    (σ : Fin a → Equiv.Perm (Fin b)) (v : Fin a × Fin b → V) :
-    flattenedForm T (fun p => v (p.1, σ p.1 p.2)) = flattenedForm T v :=
-  flattenedSymmetricForm_inner_permute T σ v
-
-/-- Implicit-parameter row symmetry used by the transfer sequences. -/
-theorem flattenedForm_row_perm (T : SymmetricMultilinearForm a b V)
-    (hT : IsSymmetricMultilinearForm a b V T) (σ : Equiv.Perm (Fin a))
-    (v : Fin a × Fin b → V) :
-    flattenedForm T (fun p => v (σ p.1, p.2)) = flattenedForm T v :=
-  flattenedSymmetricForm_outer_permute T hT σ v
-
-@[simp] theorem flattenedForm_constant_rows (T : SymmetricMultilinearForm a b V)
-    (v : Fin a → V) :
-    flattenedForm T (fun p => v p.1) =
-      T (fun i => symMonomial b V (fun _ => v i)) := rfl
-
-@[simp] theorem flattenedForm_constant_columns (T : SymmetricMultilinearForm a b V)
-    (v : Fin b → V) :
-    flattenedForm T (fun p => v p.2) =
-      T (fun _ => symMonomial b V v) := rfl
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.MonomialMultilinear
+
+namespace OAI
+
+noncomputable section
+
+namespace Problem346
+
+universe u
+
+variable {a b : ℕ} {V : Type u} [AddCommGroup V] [Module ℂ V]
+
+/-- A form on symmetric powers, evaluated on products, is multilinear in all
+of the underlying vector factors. -/
+def flattenedSymmetricForm (T : SymmetricMultilinearForm a b V) :
+    MultilinearMap ℂ (fun _ : Fin a × Fin b => V) ℂ :=
+  (T.compMultilinearMap (fun _ => symMonomialMultilinear b V)).domDomCongr
+    (Equiv.sigmaEquivProd (Fin a) (Fin b))
+
+@[simp]
+theorem flattenedSymmetricForm_apply (T : SymmetricMultilinearForm a b V)
+    (v : Fin a × Fin b → V) :
+    flattenedSymmetricForm T v =
+      T (fun i => symMonomial b V (fun j => v (i, j))) := rfl
+
+/-- Independent permutations within the rows preserve the flattened form. -/
+theorem flattenedSymmetricForm_inner_permute
+    (T : SymmetricMultilinearForm a b V)
+    (σ : Fin a → Equiv.Perm (Fin b)) (v : Fin a × Fin b → V) :
+    flattenedSymmetricForm T (fun p => v (p.1, σ p.1 p.2)) =
+      flattenedSymmetricForm T v := by
+  apply congrArg T
+  funext i
+  exact symMonomial_permute b V (σ i) (fun j => v (i, j))
+
+/-- Symmetry of the original form permits permutation of the rows. -/
+theorem flattenedSymmetricForm_outer_permute
+    (T : SymmetricMultilinearForm a b V)
+    (hT : IsSymmetricMultilinearForm a b V T)
+    (σ : Equiv.Perm (Fin a)) (v : Fin a × Fin b → V) :
+    flattenedSymmetricForm T (fun p => v (σ p.1, p.2)) =
+      flattenedSymmetricForm T v := by
+  exact hT σ (fun i => symMonomial b V (fun j => v (i, j)))
+
+/-- Compatibility API for the same flattened form, with inferred degrees. -/
+abbrev flattenedForm {a b : ℕ} {V : Type u} [AddCommGroup V] [Module ℂ V]
+    (T : SymmetricMultilinearForm a b V) :
+    MultilinearMap ℂ (fun _ : Fin a × Fin b => V) ℂ :=
+  flattenedSymmetricForm T
+
+@[simp]
+theorem flattenedForm_apply {a b : ℕ} {V : Type u} [AddCommGroup V] [Module ℂ V]
+    (T : SymmetricMultilinearForm a b V) (x : Fin a × Fin b → V) :
+    flattenedForm T x = T (fun i => symMonomial b V (fun j => x (i, j))) := rfl
+
+theorem flattenedForm_permute_factors (a b : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V] (T : SymmetricMultilinearForm a b V)
+    (σ : Fin a → Equiv.Perm (Fin b)) (x : Fin a × Fin b → V) :
+    flattenedForm T (fun p => x (p.1, σ p.1 p.2)) =
+      flattenedForm T x :=
+  flattenedSymmetricForm_inner_permute T σ x
+
+theorem flattenedForm_permute_slots (a b : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V] (T : SymmetricMultilinearForm a b V)
+    (hT : IsSymmetricMultilinearForm a b V T)
+    (σ : Equiv.Perm (Fin a)) (x : Fin a × Fin b → V) :
+    flattenedForm T (fun p => x (σ p.1, p.2)) =
+      flattenedForm T x :=
+  flattenedSymmetricForm_outer_permute T hT σ x
+
+/-- Simultaneous row and independent within-row permutation invariance. -/
+theorem flattenedForm_permute (a b : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V] (T : SymmetricMultilinearForm a b V)
+    (hT : IsSymmetricMultilinearForm a b V T)
+    (σ : Equiv.Perm (Fin a)) (τ : Fin a → Equiv.Perm (Fin b))
+    (x : Fin a × Fin b → V) :
+    flattenedForm T (fun p => x (σ p.1, τ p.1 p.2)) =
+      flattenedForm T x :=
+  (flattenedForm_permute_factors a b V T τ (fun p => x (σ p.1, p.2))).trans
+    (flattenedForm_permute_slots a b V T hT σ x)
+
+/-- Implicit-parameter factor symmetry used by the transfer sequences. -/
+theorem flattenedForm_factor_perm (T : SymmetricMultilinearForm a b V)
+    (σ : Fin a → Equiv.Perm (Fin b)) (v : Fin a × Fin b → V) :
+    flattenedForm T (fun p => v (p.1, σ p.1 p.2)) = flattenedForm T v :=
+  flattenedSymmetricForm_inner_permute T σ v
+
+/-- Implicit-parameter row symmetry used by the transfer sequences. -/
+theorem flattenedForm_row_perm (T : SymmetricMultilinearForm a b V)
+    (hT : IsSymmetricMultilinearForm a b V T) (σ : Equiv.Perm (Fin a))
+    (v : Fin a × Fin b → V) :
+    flattenedForm T (fun p => v (σ p.1, p.2)) = flattenedForm T v :=
+  flattenedSymmetricForm_outer_permute T hT σ v
+
+@[simp] theorem flattenedForm_constant_rows (T : SymmetricMultilinearForm a b V)
+    (v : Fin a → V) :
+    flattenedForm T (fun p => v p.1) =
+      T (fun i => symMonomial b V (fun _ => v i)) := rfl
+
+@[simp] theorem flattenedForm_constant_columns (T : SymmetricMultilinearForm a b V)
+    (v : Fin b → V) :
+    flattenedForm T (fun p => v p.2) =
+      T (fun _ => symMonomial b V v) := rfl
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/FlattenedSwap.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/FlattenedSwap.lean
index e8c5bf731edef51df9eacfa107b0b389fc36e8a8..7348bc09b2e568d5d8443b19fd4f41cb7dca3e60 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/FlattenedSwap.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/FlattenedSwap.lean
@@ -1,36 +1,36 @@
-import OAI.RepresentationTheory.FoulkesHowe.FlattenedForm
-import OAI.RepresentationTheory.FoulkesHowe.RowSwap
-
-namespace OAI
-
-noncomputable section
-
-namespace Problem346
-
-/-- Flattened monomial forms are invariant under any transposition whose slots have the same row. -/
-theorem flattenedSymmetricForm_swap_same_row {a b : ℕ} {V : Type*}
-    [AddCommGroup V] [Module ℂ V] (T : SymmetricMultilinearForm a b V)
-    (p q : Fin a × Fin b) (hpq : p.1 = q.1) (v : Fin a × Fin b → V) :
-    flattenedSymmetricForm T (v ∘ Equiv.swap p q) = flattenedSymmetricForm T v := by
-  rcases p with ⟨i, j⟩
-  rcases q with ⟨i', k⟩
-  dsimp at hpq
-  subst i'
-  exact rowwise_invariant_swap (flattenedSymmetricForm T)
-    (flattenedSymmetricForm_inner_permute T) i j k v
-
-/-- Coefficient arrays of flattened forms satisfy exactly the row-swap condition
-required to collapse a directional derivative to a multiplicity times one term. -/
-theorem flattenedSymmetricForm_coeff_swap_same_row {a b : ℕ} {V I : Type*}
-    [AddCommGroup V] [Module ℂ V] (T : SymmetricMultilinearForm a b V)
-    (e : I → V) (p q : Fin a × Fin b) (hpq : p.1 = q.1)
-    (k : Fin a × Fin b → I) :
-    flattenedSymmetricForm T (e ∘ (k ∘ Equiv.swap p q)) =
-      flattenedSymmetricForm T (e ∘ k) := by
-  exact flattenedSymmetricForm_swap_same_row T p q hpq (e ∘ k)
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.FlattenedForm
+import OAI.RepresentationTheory.FoulkesHowe.RowSwap
+
+namespace OAI
+
+noncomputable section
+
+namespace Problem346
+
+/-- Flattened monomial forms are invariant under any transposition whose slots have the same row. -/
+theorem flattenedSymmetricForm_swap_same_row {a b : ℕ} {V : Type*}
+    [AddCommGroup V] [Module ℂ V] (T : SymmetricMultilinearForm a b V)
+    (p q : Fin a × Fin b) (hpq : p.1 = q.1) (v : Fin a × Fin b → V) :
+    flattenedSymmetricForm T (v ∘ Equiv.swap p q) = flattenedSymmetricForm T v := by
+  rcases p with ⟨i, j⟩
+  rcases q with ⟨i', k⟩
+  dsimp at hpq
+  subst i'
+  exact rowwise_invariant_swap (flattenedSymmetricForm T)
+    (flattenedSymmetricForm_inner_permute T) i j k v
+
+/-- Coefficient arrays of flattened forms satisfy exactly the row-swap condition
+required to collapse a directional derivative to a multiplicity times one term. -/
+theorem flattenedSymmetricForm_coeff_swap_same_row {a b : ℕ} {V I : Type*}
+    [AddCommGroup V] [Module ℂ V] (T : SymmetricMultilinearForm a b V)
+    (e : I → V) (p q : Fin a × Fin b) (hpq : p.1 = q.1)
+    (k : Fin a × Fin b → I) :
+    flattenedSymmetricForm T (e ∘ (k ∘ Equiv.swap p q)) =
+      flattenedSymmetricForm T (e ∘ k) := by
+  exact flattenedSymmetricForm_swap_same_row T p q hpq (e ∘ k)
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/FoulkesDiagonal.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/FoulkesDiagonal.lean
index 0c767c3b7fe4f04e4e8ed1423e9aef16d335444c..0de7f1e0664b54cf7aed752a222430540e7be3b9 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/FoulkesDiagonal.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/FoulkesDiagonal.lean
@@ -1,30 +1,30 @@
-import OAI.RepresentationTheory.FoulkesHowe.Model
-
-namespace OAI
-
-noncomputable section
-open scoped BigOperators
-universe u
-namespace Problem346
-
-variable (a b : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]
-
- theorem foulkesFormula_diagonal (u : Fin b → V) :
-    foulkesFormula a b V (fun j _ => u j) =
-      symMonomial a (SymPow b V) (fun _ => symMonomial b V u) := by
-  classical
-  simp [foulkesFormula, Fintype.card_perm, ← Nat.cast_smul_eq_nsmul ℂ, smul_smul,
-    Nat.factorial_ne_zero]
-
- theorem IsFoulkesMap.map_diagonal
-    (μ : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V))
-    (hμ : IsFoulkesMap a b V μ) (u : Fin b → V) :
-    μ (symMonomial b (SymPow a V) (fun j => symMonomial a V (fun _ => u j))) =
-      symMonomial a (SymPow b V) (fun _ => symMonomial b V u) := by
-  rw [hμ, foulkesFormula_diagonal]
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.Model
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators
+universe u
+namespace Problem346
+
+variable (a b : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]
+
+ theorem foulkesFormula_diagonal (u : Fin b → V) :
+    foulkesFormula a b V (fun j _ => u j) =
+      symMonomial a (SymPow b V) (fun _ => symMonomial b V u) := by
+  classical
+  simp [foulkesFormula, Fintype.card_perm, ← Nat.cast_smul_eq_nsmul ℂ, smul_smul,
+    Nat.factorial_ne_zero]
+
+ theorem IsFoulkesMap.map_diagonal
+    (μ : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V))
+    (hμ : IsFoulkesMap a b V μ) (u : Fin b → V) :
+    μ (symMonomial b (SymPow a V) (fun j => symMonomial a V (fun _ => u j))) =
+      symMonomial a (SymPow b V) (fun _ => symMonomial b V u) := by
+  rw [hμ, foulkesFormula_diagonal]
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/FoulkesMultilinear.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/FoulkesMultilinear.lean
index cc637cb15b4c6eac16eeadd64e2f7fb17d7929a7..b3460ae4b55da9dece179fb724ce8d5f4769d8b9 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/FoulkesMultilinear.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/FoulkesMultilinear.lean
@@ -1,66 +1,66 @@
-import OAI.RepresentationTheory.FoulkesHowe.Model
-
-namespace OAI
-
-noncomputable section
-
-open scoped BigOperators
-universe u
-
-namespace Problem346
-
-private def monomialMultilinearAux (n : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V] :
-    MultilinearMap ℂ (fun _ : Fin n => V) (SymPow n V) :=
-  ((MultilinearMap.mkPiAlgebra ℂ (Fin n) (SymmetricAlgebra ℂ V)).compLinearMap
-    (fun _ => SymmetricAlgebra.ι ℂ V)).codRestrict (symPowSubmodule n V)
-      (fun v => Submodule.subset_span (Set.mem_range_self v))
-
-private theorem monomialMultilinearAux_apply (n : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V] (v : Fin n → V) :
-    monomialMultilinearAux n V v = symMonomial n V v := rfl
-
-/-- Reindex the slots in a transposed grid, allowing an independent permutation
-in each original row. -/
-def foulkesIndexEquiv (a b : ℕ) (σ : Fin b → Equiv.Perm (Fin a)) :
-    (Σ _ : Fin a, Fin b) ≃ (Fin b × Fin a) where
-  toFun p := (p.2, σ p.2 p.1)
-  invFun p := ⟨(σ p.1).symm p.2, p.1⟩
-  left_inv p := by cases p; simp
-  right_inv p := by cases p; simp
-
-/-- A summand of the Foulkes--Howe formula, viewed as a multilinear map in
-all entries of the input grid. -/
-def foulkesSummandMultilinear (a b : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V] (σ : Fin b → Equiv.Perm (Fin a)) :
-    MultilinearMap ℂ (fun _ : Fin b × Fin a => V) (SymPow a (SymPow b V)) :=
-  ((monomialMultilinearAux a (SymPow b V)).compMultilinearMap
-      (fun _ => monomialMultilinearAux b V)).domDomCongr (foulkesIndexEquiv a b σ)
-
-@[simp] theorem foulkesSummandMultilinear_apply (a b : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V] (σ : Fin b → Equiv.Perm (Fin a))
-    (v : Fin b × Fin a → V) :
-    foulkesSummandMultilinear a b V σ v =
-      symMonomial a (SymPow b V)
-        (fun i => symMonomial b V (fun j => v (j, σ j i))) := rfl
-
-/-- The canonical Foulkes--Howe expression is multilinear in its `a * b`
-individual vector arguments. -/
-def foulkesMultilinear (a b : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V] :
-    MultilinearMap ℂ (fun _ : Fin b × Fin a => V) (SymPow a (SymPow b V)) := by
-  classical
-  exact ((a.factorial : ℂ) ^ b)⁻¹ •
-    ∑ σ : Fin b → Equiv.Perm (Fin a), foulkesSummandMultilinear a b V σ
-
-@[simp] theorem foulkesMultilinear_apply (a b : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V] (v : Fin b × Fin a → V) :
-    foulkesMultilinear a b V v = foulkesFormula a b V (fun j i => v (j, i)) := by
-  classical
-  simp [foulkesMultilinear, foulkesFormula]
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.Model
+
+namespace OAI
+
+noncomputable section
+
+open scoped BigOperators
+universe u
+
+namespace Problem346
+
+private def monomialMultilinearAux (n : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V] :
+    MultilinearMap ℂ (fun _ : Fin n => V) (SymPow n V) :=
+  ((MultilinearMap.mkPiAlgebra ℂ (Fin n) (SymmetricAlgebra ℂ V)).compLinearMap
+    (fun _ => SymmetricAlgebra.ι ℂ V)).codRestrict (symPowSubmodule n V)
+      (fun v => Submodule.subset_span (Set.mem_range_self v))
+
+private theorem monomialMultilinearAux_apply (n : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V] (v : Fin n → V) :
+    monomialMultilinearAux n V v = symMonomial n V v := rfl
+
+/-- Reindex the slots in a transposed grid, allowing an independent permutation
+in each original row. -/
+def foulkesIndexEquiv (a b : ℕ) (σ : Fin b → Equiv.Perm (Fin a)) :
+    (Σ _ : Fin a, Fin b) ≃ (Fin b × Fin a) where
+  toFun p := (p.2, σ p.2 p.1)
+  invFun p := ⟨(σ p.1).symm p.2, p.1⟩
+  left_inv p := by cases p; simp
+  right_inv p := by cases p; simp
+
+/-- A summand of the Foulkes--Howe formula, viewed as a multilinear map in
+all entries of the input grid. -/
+def foulkesSummandMultilinear (a b : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V] (σ : Fin b → Equiv.Perm (Fin a)) :
+    MultilinearMap ℂ (fun _ : Fin b × Fin a => V) (SymPow a (SymPow b V)) :=
+  ((monomialMultilinearAux a (SymPow b V)).compMultilinearMap
+      (fun _ => monomialMultilinearAux b V)).domDomCongr (foulkesIndexEquiv a b σ)
+
+@[simp] theorem foulkesSummandMultilinear_apply (a b : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V] (σ : Fin b → Equiv.Perm (Fin a))
+    (v : Fin b × Fin a → V) :
+    foulkesSummandMultilinear a b V σ v =
+      symMonomial a (SymPow b V)
+        (fun i => symMonomial b V (fun j => v (j, σ j i))) := rfl
+
+/-- The canonical Foulkes--Howe expression is multilinear in its `a * b`
+individual vector arguments. -/
+def foulkesMultilinear (a b : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V] :
+    MultilinearMap ℂ (fun _ : Fin b × Fin a => V) (SymPow a (SymPow b V)) := by
+  classical
+  exact ((a.factorial : ℂ) ^ b)⁻¹ •
+    ∑ σ : Fin b → Equiv.Perm (Fin a), foulkesSummandMultilinear a b V σ
+
+@[simp] theorem foulkesMultilinear_apply (a b : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V] (v : Fin b × Fin a → V) :
+    foulkesMultilinear a b V v = foulkesFormula a b V (fun j i => v (j, i)) := by
+  classical
+  simp [foulkesMultilinear, foulkesFormula]
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/FoulkesSymmetries.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/FoulkesSymmetries.lean
index 559cc3862d9b1a307b4cd376b4aca4f5dd26d0da..c9a23af70cc10c02e719f33e2b6d36b04873da66 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/FoulkesSymmetries.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/FoulkesSymmetries.lean
@@ -1,53 +1,53 @@
-import OAI.RepresentationTheory.FoulkesHowe.Model
-
-namespace OAI
-
-noncomputable section
-
-open scoped BigOperators
-universe u
-
-namespace Problem346
-
-/-- Reindexing the factors of a symmetric monomial does not change it. -/
-theorem symMonomial_perm_reindex (n : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V] (v : Fin n → V)
-    (τ : Equiv.Perm (Fin n)) :
-    symMonomial n V (fun i => v (τ i)) = symMonomial n V v := by
-  apply Subtype.ext
-  exact Equiv.prod_comp τ (fun i => SymmetricAlgebra.ι ℂ V (v i))
-
-/-- The canonical averaging formula is invariant under independent permutations
-of the entries in each row. -/
-theorem foulkesFormula_inner_permute (a b : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V] (v : Fin b → Fin a → V)
-    (τ : Fin b → Equiv.Perm (Fin a)) :
-    foulkesFormula a b V (fun j i => v j (τ j i)) =
-      foulkesFormula a b V v := by
-  classical
-  unfold foulkesFormula
-  congr 1
-  exact Fintype.sum_equiv
-    (Equiv.piCongrRight (fun j => Equiv.mulLeft (τ j))) _ _ (fun σ => rfl)
-
-/-- The canonical averaging formula is invariant under permutation of rows. -/
-theorem foulkesFormula_outer_permute (a b : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V] (v : Fin b → Fin a → V)
-    (τ : Equiv.Perm (Fin b)) :
-    foulkesFormula a b V (fun j i => v (τ j) i) =
-      foulkesFormula a b V v := by
-  classical
-  unfold foulkesFormula
-  congr 1
-  apply Fintype.sum_equiv (Equiv.arrowCongr τ (Equiv.refl (Equiv.Perm (Fin a))))
-  intro σ
-  congr 1
-  funext i
-  simpa using symMonomial_perm_reindex b V
-    (fun j => v j (σ (τ.symm j) i)) τ
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.Model
+
+namespace OAI
+
+noncomputable section
+
+open scoped BigOperators
+universe u
+
+namespace Problem346
+
+/-- Reindexing the factors of a symmetric monomial does not change it. -/
+theorem symMonomial_perm_reindex (n : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V] (v : Fin n → V)
+    (τ : Equiv.Perm (Fin n)) :
+    symMonomial n V (fun i => v (τ i)) = symMonomial n V v := by
+  apply Subtype.ext
+  exact Equiv.prod_comp τ (fun i => SymmetricAlgebra.ι ℂ V (v i))
+
+/-- The canonical averaging formula is invariant under independent permutations
+of the entries in each row. -/
+theorem foulkesFormula_inner_permute (a b : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V] (v : Fin b → Fin a → V)
+    (τ : Fin b → Equiv.Perm (Fin a)) :
+    foulkesFormula a b V (fun j i => v j (τ j i)) =
+      foulkesFormula a b V v := by
+  classical
+  unfold foulkesFormula
+  congr 1
+  exact Fintype.sum_equiv
+    (Equiv.piCongrRight (fun j => Equiv.mulLeft (τ j))) _ _ (fun σ => rfl)
+
+/-- The canonical averaging formula is invariant under permutation of rows. -/
+theorem foulkesFormula_outer_permute (a b : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V] (v : Fin b → Fin a → V)
+    (τ : Equiv.Perm (Fin b)) :
+    foulkesFormula a b V (fun j i => v (τ j) i) =
+      foulkesFormula a b V v := by
+  classical
+  unfold foulkesFormula
+  congr 1
+  apply Fintype.sum_equiv (Equiv.arrowCongr τ (Equiv.refl (Equiv.Perm (Fin a))))
+  intro σ
+  congr 1
+  funext i
+  simpa using symMonomial_perm_reindex b V
+    (fun j => v j (σ (τ.symm j) i)) τ
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/Functoriality.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/Functoriality.lean
index 59e3082f32ed35d3a8e71cb42712d3b8fec4fa2d..df3f57cf51fdf9ec8f6056c0bf0df39baf984f33 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/Functoriality.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/Functoriality.lean
@@ -1,128 +1,128 @@
-import OAI.RepresentationTheory.FoulkesHowe.Model
-
-namespace OAI
-
-noncomputable section
-open scoped BigOperators
-universe u v w
-namespace Problem346
-
-variable {V : Type u} {W : Type v} {U : Type w}
-  [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
-  [AddCommGroup U] [Module ℂ U]
-
-/-- The algebra map induced by a complex-linear map. -/
-def symAlgMap (f : V →ₗ[ℂ] W) : SymmetricAlgebra ℂ V →ₐ[ℂ] SymmetricAlgebra ℂ W :=
-  SymmetricAlgebra.lift ((SymmetricAlgebra.ι ℂ W).comp f)
-
-@[simp] theorem symAlgMap_ι (f : V →ₗ[ℂ] W) (x : V) :
-    symAlgMap f (SymmetricAlgebra.ι ℂ V x) = SymmetricAlgebra.ι ℂ W (f x) := by
-  simp [symAlgMap]
-
-@[simp] theorem symAlgMap_symMonomialRaw (n : ℕ) (f : V →ₗ[ℂ] W) (x : Fin n → V) :
-    symAlgMap f (symMonomialRaw n V x) = symMonomialRaw n W (fun i => f (x i)) := by
-  simp [symMonomialRaw]
-
-/-- Symmetric powers act covariantly on linear maps. -/
-def symPowMap (n : ℕ) (f : V →ₗ[ℂ] W) : SymPow n V →ₗ[ℂ] SymPow n W :=
-  ((symAlgMap f).toLinearMap.comp (symPowSubmodule n V).subtype).codRestrict
-    (symPowSubmodule n W) (by
-      have hle : symPowSubmodule n V ≤
-          (symPowSubmodule n W).comap (symAlgMap f).toLinearMap := by
-        apply Submodule.span_le.mpr
-        rintro x ⟨v, rfl⟩
-        change symAlgMap f (symMonomialRaw n V v) ∈ symPowSubmodule n W
-        rw [symAlgMap_symMonomialRaw]
-        exact Submodule.subset_span (Set.mem_range_self _)
-      intro x
-      exact hle x.property)
-
-@[simp] theorem symPowMap_symMonomial (n : ℕ) (f : V →ₗ[ℂ] W) (x : Fin n → V) :
-    symPowMap n f (symMonomial n V x) = symMonomial n W (fun i => f (x i)) := by
-  apply Subtype.ext
-  exact symAlgMap_symMonomialRaw n f x
-
-@[simp] theorem symAlgMap_id :
-    symAlgMap (LinearMap.id : V →ₗ[ℂ] V) = AlgHom.id ℂ (SymmetricAlgebra ℂ V) := by
-  ext x
-  simp
-
-@[simp] theorem symAlgMap_comp (g : W →ₗ[ℂ] U) (f : V →ₗ[ℂ] W) :
-    symAlgMap (g.comp f) = (symAlgMap g).comp (symAlgMap f) := by
-  ext x
-  simp
-
-@[simp] theorem symPowMap_coe (n : ℕ) (f : V →ₗ[ℂ] W) (x : SymPow n V) :
-    (symPowMap n f x : SymmetricAlgebra ℂ W) = symAlgMap f x := rfl
-
-@[simp] theorem symPowMap_id (n : ℕ) :
-    symPowMap n (LinearMap.id : V →ₗ[ℂ] V) = LinearMap.id := by
-  apply LinearMap.ext
-  intro x
-  apply Subtype.ext
-  simp
-
-@[simp] theorem symPowMap_comp (n : ℕ) (g : W →ₗ[ℂ] U) (f : V →ₗ[ℂ] W) :
-    symPowMap n (g.comp f) = (symPowMap n g).comp (symPowMap n f) := by
-  apply LinearMap.ext
-  intro x
-  apply Subtype.ext
-  simp
-
-/-- A linear equivalence induces an equivalence on each symmetric power. -/
-def symPowEquiv (n : ℕ) (e : V ≃ₗ[ℂ] W) : SymPow n V ≃ₗ[ℂ] SymPow n W where
-  toLinearMap := symPowMap n e.toLinearMap
-  invFun := symPowMap n e.symm.toLinearMap
-  left_inv x := by
-    change ((symPowMap n e.symm.toLinearMap).comp (symPowMap n e.toLinearMap)) x = x
-    rw [← symPowMap_comp]
-    simp
-  right_inv x := by
-    change ((symPowMap n e.toLinearMap).comp (symPowMap n e.symm.toLinearMap)) x = x
-    rw [← symPowMap_comp]
-    simp
-
-@[simp] theorem symPowEquiv_symMonomial (n : ℕ) (e : V ≃ₗ[ℂ] W) (x : Fin n → V) :
-    symPowEquiv n e (symMonomial n V x) = symMonomial n W (fun i => e (x i)) :=
-  symPowMap_symMonomial n e.toLinearMap x
-
-@[simp] theorem symPowEquiv_toLinearMap (n : ℕ) (e : V ≃ₗ[ℂ] W) :
-    (symPowEquiv n e).toLinearMap = symPowMap n e.toLinearMap := rfl
-
-@[simp] theorem symPowEquiv_refl (n : ℕ) :
-    symPowEquiv n (LinearEquiv.refl ℂ V) = LinearEquiv.refl ℂ (SymPow n V) := by
-  apply LinearEquiv.toLinearMap_injective
-  exact symPowMap_id n
-
-@[simp] theorem symPowEquiv_trans (n : ℕ) (e : V ≃ₗ[ℂ] W) (f : W ≃ₗ[ℂ] U) :
-    symPowEquiv n (e.trans f) = (symPowEquiv n e).trans (symPowEquiv n f) := by
-  apply LinearEquiv.toLinearMap_injective
-  exact symPowMap_comp n f.toLinearMap e.toLinearMap
-
-@[simp] theorem symPowEquiv_symm (n : ℕ) (e : V ≃ₗ[ℂ] W) :
-    symPowEquiv n e.symm = (symPowEquiv n e).symm := by
-  rfl
-
-theorem symPowMap_injective (n : ℕ) (f : V →ₗ[ℂ] W) (hf : Function.Injective f) :
-    Function.Injective (symPowMap n f) := by
-  obtain ⟨g, hg⟩ := f.exists_leftInverse_of_injective (LinearMap.ker_eq_bot.mpr hf)
-  have hgf : (symPowMap n g).comp (symPowMap n f) = LinearMap.id := by
-    rw [← symPowMap_comp, hg, symPowMap_id]
-  have hleft : Function.LeftInverse (symPowMap n g) (symPowMap n f) :=
-    fun x => LinearMap.congr_fun hgf x
-  exact hleft.injective
-
-theorem symPowMap_surjective (n : ℕ) (f : V →ₗ[ℂ] W) (hf : Function.Surjective f) :
-    Function.Surjective (symPowMap n f) := by
-  obtain ⟨g, hg⟩ := f.exists_rightInverse_of_surjective (LinearMap.range_eq_top.mpr hf)
-  have hfg : (symPowMap n f).comp (symPowMap n g) = LinearMap.id := by
-    rw [← symPowMap_comp, hg, symPowMap_id]
-  have hright : Function.RightInverse (symPowMap n g) (symPowMap n f) :=
-    fun x => LinearMap.congr_fun hfg x
-  exact hright.surjective
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.Model
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators
+universe u v w
+namespace Problem346
+
+variable {V : Type u} {W : Type v} {U : Type w}
+  [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
+  [AddCommGroup U] [Module ℂ U]
+
+/-- The algebra map induced by a complex-linear map. -/
+def symAlgMap (f : V →ₗ[ℂ] W) : SymmetricAlgebra ℂ V →ₐ[ℂ] SymmetricAlgebra ℂ W :=
+  SymmetricAlgebra.lift ((SymmetricAlgebra.ι ℂ W).comp f)
+
+@[simp] theorem symAlgMap_ι (f : V →ₗ[ℂ] W) (x : V) :
+    symAlgMap f (SymmetricAlgebra.ι ℂ V x) = SymmetricAlgebra.ι ℂ W (f x) := by
+  simp [symAlgMap]
+
+@[simp] theorem symAlgMap_symMonomialRaw (n : ℕ) (f : V →ₗ[ℂ] W) (x : Fin n → V) :
+    symAlgMap f (symMonomialRaw n V x) = symMonomialRaw n W (fun i => f (x i)) := by
+  simp [symMonomialRaw]
+
+/-- Symmetric powers act covariantly on linear maps. -/
+def symPowMap (n : ℕ) (f : V →ₗ[ℂ] W) : SymPow n V →ₗ[ℂ] SymPow n W :=
+  ((symAlgMap f).toLinearMap.comp (symPowSubmodule n V).subtype).codRestrict
+    (symPowSubmodule n W) (by
+      have hle : symPowSubmodule n V ≤
+          (symPowSubmodule n W).comap (symAlgMap f).toLinearMap := by
+        apply Submodule.span_le.mpr
+        rintro x ⟨v, rfl⟩
+        change symAlgMap f (symMonomialRaw n V v) ∈ symPowSubmodule n W
+        rw [symAlgMap_symMonomialRaw]
+        exact Submodule.subset_span (Set.mem_range_self _)
+      intro x
+      exact hle x.property)
+
+@[simp] theorem symPowMap_symMonomial (n : ℕ) (f : V →ₗ[ℂ] W) (x : Fin n → V) :
+    symPowMap n f (symMonomial n V x) = symMonomial n W (fun i => f (x i)) := by
+  apply Subtype.ext
+  exact symAlgMap_symMonomialRaw n f x
+
+@[simp] theorem symAlgMap_id :
+    symAlgMap (LinearMap.id : V →ₗ[ℂ] V) = AlgHom.id ℂ (SymmetricAlgebra ℂ V) := by
+  ext x
+  simp
+
+@[simp] theorem symAlgMap_comp (g : W →ₗ[ℂ] U) (f : V →ₗ[ℂ] W) :
+    symAlgMap (g.comp f) = (symAlgMap g).comp (symAlgMap f) := by
+  ext x
+  simp
+
+@[simp] theorem symPowMap_coe (n : ℕ) (f : V →ₗ[ℂ] W) (x : SymPow n V) :
+    (symPowMap n f x : SymmetricAlgebra ℂ W) = symAlgMap f x := rfl
+
+@[simp] theorem symPowMap_id (n : ℕ) :
+    symPowMap n (LinearMap.id : V →ₗ[ℂ] V) = LinearMap.id := by
+  apply LinearMap.ext
+  intro x
+  apply Subtype.ext
+  simp
+
+@[simp] theorem symPowMap_comp (n : ℕ) (g : W →ₗ[ℂ] U) (f : V →ₗ[ℂ] W) :
+    symPowMap n (g.comp f) = (symPowMap n g).comp (symPowMap n f) := by
+  apply LinearMap.ext
+  intro x
+  apply Subtype.ext
+  simp
+
+/-- A linear equivalence induces an equivalence on each symmetric power. -/
+def symPowEquiv (n : ℕ) (e : V ≃ₗ[ℂ] W) : SymPow n V ≃ₗ[ℂ] SymPow n W where
+  toLinearMap := symPowMap n e.toLinearMap
+  invFun := symPowMap n e.symm.toLinearMap
+  left_inv x := by
+    change ((symPowMap n e.symm.toLinearMap).comp (symPowMap n e.toLinearMap)) x = x
+    rw [← symPowMap_comp]
+    simp
+  right_inv x := by
+    change ((symPowMap n e.toLinearMap).comp (symPowMap n e.symm.toLinearMap)) x = x
+    rw [← symPowMap_comp]
+    simp
+
+@[simp] theorem symPowEquiv_symMonomial (n : ℕ) (e : V ≃ₗ[ℂ] W) (x : Fin n → V) :
+    symPowEquiv n e (symMonomial n V x) = symMonomial n W (fun i => e (x i)) :=
+  symPowMap_symMonomial n e.toLinearMap x
+
+@[simp] theorem symPowEquiv_toLinearMap (n : ℕ) (e : V ≃ₗ[ℂ] W) :
+    (symPowEquiv n e).toLinearMap = symPowMap n e.toLinearMap := rfl
+
+@[simp] theorem symPowEquiv_refl (n : ℕ) :
+    symPowEquiv n (LinearEquiv.refl ℂ V) = LinearEquiv.refl ℂ (SymPow n V) := by
+  apply LinearEquiv.toLinearMap_injective
+  exact symPowMap_id n
+
+@[simp] theorem symPowEquiv_trans (n : ℕ) (e : V ≃ₗ[ℂ] W) (f : W ≃ₗ[ℂ] U) :
+    symPowEquiv n (e.trans f) = (symPowEquiv n e).trans (symPowEquiv n f) := by
+  apply LinearEquiv.toLinearMap_injective
+  exact symPowMap_comp n f.toLinearMap e.toLinearMap
+
+@[simp] theorem symPowEquiv_symm (n : ℕ) (e : V ≃ₗ[ℂ] W) :
+    symPowEquiv n e.symm = (symPowEquiv n e).symm := by
+  rfl
+
+theorem symPowMap_injective (n : ℕ) (f : V →ₗ[ℂ] W) (hf : Function.Injective f) :
+    Function.Injective (symPowMap n f) := by
+  obtain ⟨g, hg⟩ := f.exists_leftInverse_of_injective (LinearMap.ker_eq_bot.mpr hf)
+  have hgf : (symPowMap n g).comp (symPowMap n f) = LinearMap.id := by
+    rw [← symPowMap_comp, hg, symPowMap_id]
+  have hleft : Function.LeftInverse (symPowMap n g) (symPowMap n f) :=
+    fun x => LinearMap.congr_fun hgf x
+  exact hleft.injective
+
+theorem symPowMap_surjective (n : ℕ) (f : V →ₗ[ℂ] W) (hf : Function.Surjective f) :
+    Function.Surjective (symPowMap n f) := by
+  obtain ⟨g, hg⟩ := f.exists_rightInverse_of_surjective (LinearMap.range_eq_top.mpr hf)
+  have hfg : (symPowMap n f).comp (symPowMap n g) = LinearMap.id := by
+    rw [← symPowMap_comp, hg, symPowMap_id]
+  have hright : Function.RightInverse (symPowMap n g) (symPowMap n f) :=
+    fun x => LinearMap.congr_fun hfg x
+  exact hright.surjective
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/GeneratorLift.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/GeneratorLift.lean
index 2d7a142c3a99b5d4c85ff7ac129ca17649fa3436..b0a7a47a70d4fc5a94f62600ee9396eb1d258eef 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/GeneratorLift.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/GeneratorLift.lean
@@ -1,46 +1,46 @@
-import OAI.RepresentationTheory.FoulkesHowe.SpanningAgreement
-import OAI.RepresentationTheory.FoulkesHowe.SymmetricLift
-import OAI.RepresentationTheory.FoulkesHowe.Spanning
-
-namespace OAI
-
-noncomputable section
-namespace Problem346
-
-/-- A separately linearly extendible symmetric formula on spanning generators
-induces a linear map out of the homogeneous symmetric power. -/
-theorem exists_symPow_lift_on_spanning
-    {n : ℕ} {X V W : Type*} [AddCommGroup V] [Module ℂ V]
-    [AddCommGroup W] [Module ℂ W]
-    (s : X → V) (hs : Submodule.span ℂ (Set.range s) = ⊤)
-    (F : (Fin n → X) → W)
-    (hlinear : ∀ x : Fin n → X, ∀ i : Fin n,
-      ∃ L : V →ₗ[ℂ] W, ∀ y, L (s y) = F (Function.update x i y))
-    (hsym : ∀ (σ : Equiv.Perm (Fin n)) (x : Fin n → X),
-      F (fun i => x (σ i)) = F x) :
-    ∃ L : SymPow n V →ₗ[ℂ] W, ∀ x : Fin n → X,
-      L (symMonomial n V (fun i => s (x i))) = F x := by
-  classical
-  obtain ⟨G, hG⟩ := exists_multilinear_extension_of_spanning
-    (fun _ : Fin n => s) (fun _ => hs) F hlinear
-  have hGs : ∀ (σ : Equiv.Perm (Fin n)) (v : Fin n → V),
-      G (fun i => v (σ i)) = G v := by
-    intro σ
-    have h : G.domDomCongr σ = G := by
-      apply multilinear_ext_of_span hs n
-      intro v hv
-      choose x hx using hv
-      have he : v = fun i => s (x i) := funext (fun i => (hx i).symm)
-      subst v
-      change G (fun i => s (x (σ i))) = G (fun i => s (x i))
-      rw [← hG, ← hG, hsym]
-    intro v
-    exact congrArg (fun G : MultilinearMap ℂ (fun _ : Fin n => V) W => G v) h
-  obtain ⟨L, hL⟩ := exists_symPow_lift G hGs
-  exact ⟨L, fun x => (hL _).trans (hG x).symm⟩
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.SpanningAgreement
+import OAI.RepresentationTheory.FoulkesHowe.SymmetricLift
+import OAI.RepresentationTheory.FoulkesHowe.Spanning
+
+namespace OAI
+
+noncomputable section
+namespace Problem346
+
+/-- A separately linearly extendible symmetric formula on spanning generators
+induces a linear map out of the homogeneous symmetric power. -/
+theorem exists_symPow_lift_on_spanning
+    {n : ℕ} {X V W : Type*} [AddCommGroup V] [Module ℂ V]
+    [AddCommGroup W] [Module ℂ W]
+    (s : X → V) (hs : Submodule.span ℂ (Set.range s) = ⊤)
+    (F : (Fin n → X) → W)
+    (hlinear : ∀ x : Fin n → X, ∀ i : Fin n,
+      ∃ L : V →ₗ[ℂ] W, ∀ y, L (s y) = F (Function.update x i y))
+    (hsym : ∀ (σ : Equiv.Perm (Fin n)) (x : Fin n → X),
+      F (fun i => x (σ i)) = F x) :
+    ∃ L : SymPow n V →ₗ[ℂ] W, ∀ x : Fin n → X,
+      L (symMonomial n V (fun i => s (x i))) = F x := by
+  classical
+  obtain ⟨G, hG⟩ := exists_multilinear_extension_of_spanning
+    (fun _ : Fin n => s) (fun _ => hs) F hlinear
+  have hGs : ∀ (σ : Equiv.Perm (Fin n)) (v : Fin n → V),
+      G (fun i => v (σ i)) = G v := by
+    intro σ
+    have h : G.domDomCongr σ = G := by
+      apply multilinear_ext_of_span hs n
+      intro v hv
+      choose x hx using hv
+      have he : v = fun i => s (x i) := funext (fun i => (hx i).symm)
+      subst v
+      change G (fun i => s (x (σ i))) = G (fun i => s (x i))
+      rw [← hG, ← hG, hsym]
+    intro v
+    exact congrArg (fun G : MultilinearMap ℂ (fun _ : Fin n => V) W => G v) h
+  obtain ⟨L, hL⟩ := exists_symPow_lift G hGs
+  exact ⟨L, fun x => (hL _).trans (hG x).symm⟩
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/GridCancellation.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/GridCancellation.lean
index f56d73df128d097764e171ecd46ca84609b061ed..a06b4ae5b0a5b108fd83a3710809982e79965eb4 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/GridCancellation.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/GridCancellation.lean
@@ -1,56 +1,56 @@
-import Mathlib
-
-namespace OAI
-
-open scoped BigOperators
-
-namespace Problem346
-
-/-- Reverse induction over a finite rectangular grid.  A zero at its upper corner
-propagates to every point when a one-coordinate upward step can be cancelled. -/
-theorem zero_of_upward_grid {ι M : Type*} [Fintype ι] [DecidableEq ι] [Zero M]
-    (r : ℕ) (F : (ι → ℕ) → M)
-    (hend : F (fun _ => r) = 0)
-    (hstep : ∀ (c : ι → ℕ), (∀ i, c i ≤ r) → ∀ i, c i < r →
-      F (Function.update c i (c i + 1)) = 0 → F c = 0)
-    (c : ι → ℕ) (hc : ∀ i, c i ≤ r) : F c = 0 := by
-  classical
-  have aux : ∀ n : ℕ, ∀ c : ι → ℕ, (∀ i, c i ≤ r) →
-      (∑ i, (r - c i)) = n → F c = 0 := by
-    intro n
-    induction n using Nat.strong_induction_on with
-    | h n ih =>
-      intro c hc hn
-      by_cases htop : ∀ i, c i = r
-      · have heq : c = fun _ => r := funext htop
-        simpa only [heq] using hend
-      · push Not at htop
-        obtain ⟨i, hi⟩ := htop
-        have hir : c i < r := lt_of_le_of_ne (hc i) hi
-        apply hstep c hc i hir
-        let c' := Function.update c i (c i + 1)
-        have hc' : ∀ j, c' j ≤ r := by
-          intro j
-          by_cases hji : j = i
-          · subst j
-            simp only [c', Function.update_self]
-            omega
-          · simpa only [c', Function.update_of_ne hji] using hc j
-        have hlt : (∑ j, (r - c' j)) < n := by
-          rw [← hn]
-          apply Finset.sum_lt_sum
-          · intro j hj
-            by_cases hji : j = i
-            · subst j
-              simp only [c', Function.update_self]
-              omega
-            · simp only [c', Function.update_of_ne hji, le_refl]
-          · refine ⟨i, Finset.mem_univ i, ?_⟩
-            simp only [c', Function.update_self]
-            omega
-        exact ih _ hlt c' hc' rfl
-  exact aux _ c hc rfl
-
-end Problem346
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+open scoped BigOperators
+
+namespace Problem346
+
+/-- Reverse induction over a finite rectangular grid.  A zero at its upper corner
+propagates to every point when a one-coordinate upward step can be cancelled. -/
+theorem zero_of_upward_grid {ι M : Type*} [Fintype ι] [DecidableEq ι] [Zero M]
+    (r : ℕ) (F : (ι → ℕ) → M)
+    (hend : F (fun _ => r) = 0)
+    (hstep : ∀ (c : ι → ℕ), (∀ i, c i ≤ r) → ∀ i, c i < r →
+      F (Function.update c i (c i + 1)) = 0 → F c = 0)
+    (c : ι → ℕ) (hc : ∀ i, c i ≤ r) : F c = 0 := by
+  classical
+  have aux : ∀ n : ℕ, ∀ c : ι → ℕ, (∀ i, c i ≤ r) →
+      (∑ i, (r - c i)) = n → F c = 0 := by
+    intro n
+    induction n using Nat.strong_induction_on with
+    | h n ih =>
+      intro c hc hn
+      by_cases htop : ∀ i, c i = r
+      · have heq : c = fun _ => r := funext htop
+        simpa only [heq] using hend
+      · push Not at htop
+        obtain ⟨i, hi⟩ := htop
+        have hir : c i < r := lt_of_le_of_ne (hc i) hi
+        apply hstep c hc i hir
+        let c' := Function.update c i (c i + 1)
+        have hc' : ∀ j, c' j ≤ r := by
+          intro j
+          by_cases hji : j = i
+          · subst j
+            simp only [c', Function.update_self]
+            omega
+          · simpa only [c', Function.update_of_ne hji] using hc j
+        have hlt : (∑ j, (r - c' j)) < n := by
+          rw [← hn]
+          apply Finset.sum_lt_sum
+          · intro j hj
+            by_cases hji : j = i
+            · subst j
+              simp only [c', Function.update_self]
+              omega
+            · simp only [c', Function.update_of_ne hji, le_refl]
+          · refine ⟨i, Finset.mem_univ i, ?_⟩
+            simp only [c', Function.update_self]
+            omega
+        exact ih _ hlt c' hc' rfl
+  exact aux _ c hc rfl
+
+end Problem346
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/LinearCase.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/LinearCase.lean
index 69ad043e4e78339c2b4ab62862a138e2e2eb7170..a3641ddffba2f9cac31fe1d2233dd010871ad91a 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/LinearCase.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/LinearCase.lean
@@ -1,52 +1,52 @@
-import OAI.RepresentationTheory.FoulkesHowe.DegreeOne
-import OAI.RepresentationTheory.FoulkesHowe.Functoriality
-
-namespace OAI
-
-noncomputable section
-open scoped BigOperators
-universe u
-namespace Problem346
-
-@[simp] theorem foulkesFormula_one (b : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V] (v : Fin b → Fin 1 → V) :
-    foulkesFormula 1 b V v =
-      symMonomial 1 (SymPow b V) (fun _ => symMonomial b V (fun j => v j 0)) := by
-  classical
-  simp [foulkesFormula, Fin.eq_zero]
-
-/-- The canonical Foulkes--Howe equivalence when the inner degree is one. -/
-def foulkesOneEquiv (b : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] :
-    SymPow b (SymPow 1 V) ≃ₗ[ℂ] SymPow 1 (SymPow b V) :=
-  (symPowEquiv b (symPowOneEquiv V)).trans (symPowOneEquiv (SymPow b V)).symm
-
-theorem foulkesOneEquiv_isFoulkesMap (b : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V] :
-    IsFoulkesMap 1 b V (foulkesOneEquiv b V).toLinearMap := by
-  intro v
-  simp [foulkesOneEquiv]
-
-/-- Degree-one stabilization, with no finite-dimensionality requirement. -/
-theorem canonical_foulkes_howe_a_one_aux (V : Type u)
-    [AddCommGroup V] [Module ℂ V] (b : ℕ) :
-    ∃ μ : SymPow b (SymPow 1 V) →ₗ[ℂ] SymPow 1 (SymPow b V),
-      IsFoulkesMap 1 b V μ ∧ Function.Bijective μ ∧
-      ∀ ν : SymPow b (SymPow 1 V) →ₗ[ℂ] SymPow 1 (SymPow b V),
-        IsFoulkesMap 1 b V ν → ν = μ := by
-  refine ⟨(foulkesOneEquiv b V).toLinearMap, foulkesOneEquiv_isFoulkesMap b V,
-    (foulkesOneEquiv b V).bijective, ?_⟩
-  intro ν hν
-  have hspan : Submodule.span ℂ (Set.range (symMonomial b (SymPow 1 V))) = ⊤ := by
-    exact (Submodule.span_range_subtype_eq_top_iff (symPowSubmodule b (SymPow 1 V))
-      (fun v => Submodule.subset_span (Set.mem_range_self v))).mpr rfl
-  apply LinearMap.ext_on_range hspan
-  intro v
-  have h := (hν (fun j _ => symPowOneEquiv V (v j))).trans
-    ((foulkesOneEquiv_isFoulkesMap b V (fun j _ => symPowOneEquiv V (v j))).symm)
-  simpa only [← symPowOneEquiv_symm_apply, LinearEquiv.symm_apply_apply] using h
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.DegreeOne
+import OAI.RepresentationTheory.FoulkesHowe.Functoriality
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators
+universe u
+namespace Problem346
+
+@[simp] theorem foulkesFormula_one (b : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V] (v : Fin b → Fin 1 → V) :
+    foulkesFormula 1 b V v =
+      symMonomial 1 (SymPow b V) (fun _ => symMonomial b V (fun j => v j 0)) := by
+  classical
+  simp [foulkesFormula, Fin.eq_zero]
+
+/-- The canonical Foulkes--Howe equivalence when the inner degree is one. -/
+def foulkesOneEquiv (b : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] :
+    SymPow b (SymPow 1 V) ≃ₗ[ℂ] SymPow 1 (SymPow b V) :=
+  (symPowEquiv b (symPowOneEquiv V)).trans (symPowOneEquiv (SymPow b V)).symm
+
+theorem foulkesOneEquiv_isFoulkesMap (b : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V] :
+    IsFoulkesMap 1 b V (foulkesOneEquiv b V).toLinearMap := by
+  intro v
+  simp [foulkesOneEquiv]
+
+/-- Degree-one stabilization, with no finite-dimensionality requirement. -/
+theorem canonical_foulkes_howe_a_one_aux (V : Type u)
+    [AddCommGroup V] [Module ℂ V] (b : ℕ) :
+    ∃ μ : SymPow b (SymPow 1 V) →ₗ[ℂ] SymPow 1 (SymPow b V),
+      IsFoulkesMap 1 b V μ ∧ Function.Bijective μ ∧
+      ∀ ν : SymPow b (SymPow 1 V) →ₗ[ℂ] SymPow 1 (SymPow b V),
+        IsFoulkesMap 1 b V ν → ν = μ := by
+  refine ⟨(foulkesOneEquiv b V).toLinearMap, foulkesOneEquiv_isFoulkesMap b V,
+    (foulkesOneEquiv b V).bijective, ?_⟩
+  intro ν hν
+  have hspan : Submodule.span ℂ (Set.range (symMonomial b (SymPow 1 V))) = ⊤ := by
+    exact (Submodule.span_range_subtype_eq_top_iff (symPowSubmodule b (SymPow 1 V))
+      (fun v => Submodule.subset_span (Set.mem_range_self v))).mpr rfl
+  apply LinearMap.ext_on_range hspan
+  intro v
+  have h := (hν (fun j _ => symPowOneEquiv V (v j))).trans
+    ((foulkesOneEquiv_isFoulkesMap b V (fun j _ => symPowOneEquiv V (v j))).symm)
+  simpa only [← symPowOneEquiv_symm_apply, LinearEquiv.symm_apply_apply] using h
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/Model.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/Model.lean
index b6c17cabc8f6df5a26cbe7257731a7403fdb37d3..b221a3b4d50c9ac4bea70c805700082495af0551 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/Model.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/Model.lean
@@ -1,79 +1,79 @@
-import Mathlib
-
-namespace OAI
-
-noncomputable section
-
-open scoped BigOperators
-universe u
-
-namespace Problem346
-
-def symMonomialRaw (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]
-    (v : Fin n → V) : SymmetricAlgebra ℂ V :=
-  ∏ i : Fin n, SymmetricAlgebra.ι ℂ V (v i)
-
-def symPowSubmodule (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] :
-    Submodule ℂ (SymmetricAlgebra ℂ V) :=
-  Submodule.span ℂ (Set.range (symMonomialRaw n V))
-
-abbrev SymPow (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] :=
-  ↥(symPowSubmodule n V)
-
-def symMonomial (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]
-    (v : Fin n → V) : SymPow n V :=
-  ⟨symMonomialRaw n V v, Submodule.subset_span (Set.mem_range_self v)⟩
-
-def foulkesFormula (a b : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]
-    (v : Fin b → Fin a → V) : SymPow a (SymPow b V) := by
-  classical
-  exact
-    ((a.factorial : ℂ) ^ b)⁻¹ •
-      ∑ σ : Fin b → Equiv.Perm (Fin a),
-        symMonomial a (SymPow b V)
-          (fun i => symMonomial b V (fun j => v j ((σ j) i)))
-
-def IsFoulkesMap (a b : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]
-    (μ : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V)) : Prop :=
-  ∀ v : Fin b → Fin a → V,
-    μ (symMonomial b (SymPow a V) (fun j => symMonomial a V (v j))) =
-      foulkesFormula a b V v
-
-def IsPlethysmAction (outer inner : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V] (g : V ≃ₗ[ℂ] V)
-    (L : SymPow outer (SymPow inner V) →ₗ[ℂ]
-      SymPow outer (SymPow inner V)) : Prop :=
-  ∀ v : Fin outer → Fin inner → V,
-    L (symMonomial outer (SymPow inner V)
-        (fun i => symMonomial inner V (v i))) =
-      symMonomial outer (SymPow inner V)
-        (fun i => symMonomial inner V (fun j => g (v i j)))
-
-def IsGLEquivariantEmbedding (a b : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V]
-    (ι : SymPow a (SymPow b V) →ₗ[ℂ] SymPow b (SymPow a V)) : Prop :=
-  Function.Injective ι ∧
-    ∀ g : V ≃ₗ[ℂ] V,
-      ∃ Lsource : SymPow b (SymPow a V) →ₗ[ℂ] SymPow b (SymPow a V),
-      ∃ Ltarget : SymPow a (SymPow b V) →ₗ[ℂ] SymPow a (SymPow b V),
-        IsPlethysmAction b a V g Lsource ∧
-        IsPlethysmAction a b V g Ltarget ∧
-        Function.Bijective Lsource ∧
-        Function.Bijective Ltarget ∧
-        Lsource.comp ι = ι.comp Ltarget
-
-abbrev SymmetricMultilinearForm (a b : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V] :=
-  MultilinearMap ℂ (fun _ : Fin a => SymPow b V) ℂ
-
-def IsSymmetricMultilinearForm (a b : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V]
-    (T : SymmetricMultilinearForm a b V) : Prop :=
-  ∀ (σ : Equiv.Perm (Fin a)) (x : Fin a → SymPow b V),
-    T (fun i => x (σ i)) = T x
-
-end Problem346
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+
+open scoped BigOperators
+universe u
+
+namespace Problem346
+
+def symMonomialRaw (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]
+    (v : Fin n → V) : SymmetricAlgebra ℂ V :=
+  ∏ i : Fin n, SymmetricAlgebra.ι ℂ V (v i)
+
+def symPowSubmodule (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] :
+    Submodule ℂ (SymmetricAlgebra ℂ V) :=
+  Submodule.span ℂ (Set.range (symMonomialRaw n V))
+
+abbrev SymPow (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] :=
+  ↥(symPowSubmodule n V)
+
+def symMonomial (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]
+    (v : Fin n → V) : SymPow n V :=
+  ⟨symMonomialRaw n V v, Submodule.subset_span (Set.mem_range_self v)⟩
+
+def foulkesFormula (a b : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]
+    (v : Fin b → Fin a → V) : SymPow a (SymPow b V) := by
+  classical
+  exact
+    ((a.factorial : ℂ) ^ b)⁻¹ •
+      ∑ σ : Fin b → Equiv.Perm (Fin a),
+        symMonomial a (SymPow b V)
+          (fun i => symMonomial b V (fun j => v j ((σ j) i)))
+
+def IsFoulkesMap (a b : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]
+    (μ : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V)) : Prop :=
+  ∀ v : Fin b → Fin a → V,
+    μ (symMonomial b (SymPow a V) (fun j => symMonomial a V (v j))) =
+      foulkesFormula a b V v
+
+def IsPlethysmAction (outer inner : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V] (g : V ≃ₗ[ℂ] V)
+    (L : SymPow outer (SymPow inner V) →ₗ[ℂ]
+      SymPow outer (SymPow inner V)) : Prop :=
+  ∀ v : Fin outer → Fin inner → V,
+    L (symMonomial outer (SymPow inner V)
+        (fun i => symMonomial inner V (v i))) =
+      symMonomial outer (SymPow inner V)
+        (fun i => symMonomial inner V (fun j => g (v i j)))
+
+def IsGLEquivariantEmbedding (a b : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V]
+    (ι : SymPow a (SymPow b V) →ₗ[ℂ] SymPow b (SymPow a V)) : Prop :=
+  Function.Injective ι ∧
+    ∀ g : V ≃ₗ[ℂ] V,
+      ∃ Lsource : SymPow b (SymPow a V) →ₗ[ℂ] SymPow b (SymPow a V),
+      ∃ Ltarget : SymPow a (SymPow b V) →ₗ[ℂ] SymPow a (SymPow b V),
+        IsPlethysmAction b a V g Lsource ∧
+        IsPlethysmAction a b V g Ltarget ∧
+        Function.Bijective Lsource ∧
+        Function.Bijective Ltarget ∧
+        Lsource.comp ι = ι.comp Ltarget
+
+abbrev SymmetricMultilinearForm (a b : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V] :=
+  MultilinearMap ℂ (fun _ : Fin a => SymPow b V) ℂ
+
+def IsSymmetricMultilinearForm (a b : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V]
+    (T : SymmetricMultilinearForm a b V) : Prop :=
+  ∀ (σ : Equiv.Perm (Fin a)) (x : Fin a → SymPow b V),
+    T (fun i => x (σ i)) = T x
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/MonomialCut.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/MonomialCut.lean
index fb5ddf2fc7e90dcb2d8856bc5eb1c6383bb74635..8b9f7460614d3a4118c74e6315f08c6fe2b19d3e 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/MonomialCut.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/MonomialCut.lean
@@ -1,59 +1,59 @@
-import OAI.RepresentationTheory.FoulkesHowe.Multiplication
-
-namespace OAI
-
-noncomputable section
-
-namespace Problem346
-
-universe u
-variable {V : Type u} [AddCommGroup V] [Module ℂ V]
-
-/-- Splitting a list of equal factors at a cutoff is graded multiplication of two pure powers. -/
-theorem symMonomial_cut (r m : ℕ) (x y : V) :
-    symMonomial (r+m) V (fun j => if j.val < r then x else y) =
-      symPowMul r m (symMonomial r V (fun _ => x))
-        (symMonomial m V (fun _ => y)) := by
-  rw [symPowMul_symMonomial]
-  congr 1
-  ext j
-  refine Fin.addCases (fun i => ?_) (fun i => ?_) j
-  · simp
-  · simp
-
-/-- Cast-free raw form of the cutoff identity. -/
-theorem symMonomialRaw_cut (r m : ℕ) (x y : V) :
-    symMonomialRaw (r+m) V (fun j => if j.val < r then x else y) =
-      SymmetricAlgebra.ι ℂ V x ^ r * SymmetricAlgebra.ι ℂ V y ^ m := by
-  have h := congrArg (fun z : SymPow (r+m) V => (z : SymmetricAlgebra ℂ V))
-    (symMonomial_cut r m x y)
-  simpa [symMonomial, symPowMul_coe, symMonomialRaw_const] using h
-
-/-- The all-row terminal cutoff in the second transfer matches its common-power product form. -/
-theorem multilinear_cut_endpoint (r m : ℕ)
-    (T : SymmetricMultilinearForm (r+1) (r+m) V)
-    (x t : V) (v : Fin r → V) :
-    T (Fin.cons (symMonomial (r+m) V (fun _ => t))
-      (fun i => symMonomial (r+m) V (fun j => if j.val < r then x else v i))) =
-    T (Fin.cons (symMonomial (r+m) V (fun _ => t))
-      (fun i => symPowMul r m (symMonomial r V (fun _ => x))
-        (symMonomial m V (fun _ => v i)))) := by
-  simp only [symMonomial_cut]
-
-/-- Vanishing on arbitrary residual products gives vanishing on the terminal cutoff. -/
-theorem multilinear_cut_endpoint_zero (r m : ℕ)
-    (T : SymmetricMultilinearForm (r+1) (r+m) V)
-    (h : ∀ x t : V, ∀ Q : Fin r → SymPow m V,
-      T (Fin.cons (symMonomial (r+m) V (fun _ => t))
-        (fun i => symPowMul r m (symMonomial r V (fun _ => x)) (Q i))) = 0)
-    (x t : V) (v : Fin r → V) :
-    T (Fin.cons (symMonomial (r+m) V (fun _ => t))
-      (fun i => symMonomial (r+m) V (fun j => if j.val < r then x else v i))) = 0 := by
-  rw [multilinear_cut_endpoint]
-  exact h x t (fun i => symMonomial m V (fun _ => v i))
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.Multiplication
+
+namespace OAI
+
+noncomputable section
+
+namespace Problem346
+
+universe u
+variable {V : Type u} [AddCommGroup V] [Module ℂ V]
+
+/-- Splitting a list of equal factors at a cutoff is graded multiplication of two pure powers. -/
+theorem symMonomial_cut (r m : ℕ) (x y : V) :
+    symMonomial (r+m) V (fun j => if j.val < r then x else y) =
+      symPowMul r m (symMonomial r V (fun _ => x))
+        (symMonomial m V (fun _ => y)) := by
+  rw [symPowMul_symMonomial]
+  congr 1
+  ext j
+  refine Fin.addCases (fun i => ?_) (fun i => ?_) j
+  · simp
+  · simp
+
+/-- Cast-free raw form of the cutoff identity. -/
+theorem symMonomialRaw_cut (r m : ℕ) (x y : V) :
+    symMonomialRaw (r+m) V (fun j => if j.val < r then x else y) =
+      SymmetricAlgebra.ι ℂ V x ^ r * SymmetricAlgebra.ι ℂ V y ^ m := by
+  have h := congrArg (fun z : SymPow (r+m) V => (z : SymmetricAlgebra ℂ V))
+    (symMonomial_cut r m x y)
+  simpa [symMonomial, symPowMul_coe, symMonomialRaw_const] using h
+
+/-- The all-row terminal cutoff in the second transfer matches its common-power product form. -/
+theorem multilinear_cut_endpoint (r m : ℕ)
+    (T : SymmetricMultilinearForm (r+1) (r+m) V)
+    (x t : V) (v : Fin r → V) :
+    T (Fin.cons (symMonomial (r+m) V (fun _ => t))
+      (fun i => symMonomial (r+m) V (fun j => if j.val < r then x else v i))) =
+    T (Fin.cons (symMonomial (r+m) V (fun _ => t))
+      (fun i => symPowMul r m (symMonomial r V (fun _ => x))
+        (symMonomial m V (fun _ => v i)))) := by
+  simp only [symMonomial_cut]
+
+/-- Vanishing on arbitrary residual products gives vanishing on the terminal cutoff. -/
+theorem multilinear_cut_endpoint_zero (r m : ℕ)
+    (T : SymmetricMultilinearForm (r+1) (r+m) V)
+    (h : ∀ x t : V, ∀ Q : Fin r → SymPow m V,
+      T (Fin.cons (symMonomial (r+m) V (fun _ => t))
+        (fun i => symPowMul r m (symMonomial r V (fun _ => x)) (Q i))) = 0)
+    (x t : V) (v : Fin r → V) :
+    T (Fin.cons (symMonomial (r+m) V (fun _ => t))
+      (fun i => symMonomial (r+m) V (fun j => if j.val < r then x else v i))) = 0 := by
+  rw [multilinear_cut_endpoint]
+  exact h x t (fun i => symMonomial m V (fun _ => v i))
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/MonomialMultilinear.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/MonomialMultilinear.lean
index fd15ab38d55b87a8092663d5e9d7ddf0f2f27a1c..ade61821a9a8a167d751e989ee3c4f599407c372 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/MonomialMultilinear.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/MonomialMultilinear.lean
@@ -1,117 +1,117 @@
-import OAI.RepresentationTheory.FoulkesHowe.Model
-
-namespace OAI
-
-noncomputable section
-
-open scoped BigOperators
-
-universe u v
-
-namespace Problem346
-
-/-- The product of degree-one generators, with its multilinearity bundled. -/
-def symMonomialMultilinear (n : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V] :
-    MultilinearMap ℂ (fun _ : Fin n => V) (SymPow n V) :=
-  ((MultilinearMap.mkPiAlgebra ℂ (Fin n) (SymmetricAlgebra ℂ V)).compLinearMap
-    (fun _ => SymmetricAlgebra.ι ℂ V)).codRestrict (symPowSubmodule n V)
-      (fun x => Submodule.subset_span (Set.mem_range_self x))
-
-@[simp]
-theorem symMonomialMultilinear_apply (n : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V] (x : Fin n → V) :
-    symMonomialMultilinear n V x = symMonomial n V x := rfl
-
-theorem symMonomial_permute (n : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V] (σ : Equiv.Perm (Fin n)) (x : Fin n → V) :
-    symMonomial n V (fun i => x (σ i)) = symMonomial n V x := by
-  apply Subtype.ext
-  exact Equiv.prod_comp σ (fun i => SymmetricAlgebra.ι ℂ V (x i))
-
-/-- A functional on a symmetric power induces a symmetric multilinear form. -/
-def monomialFunctional (a b : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V]
-    (φ : SymPow a (SymPow b V) →ₗ[ℂ] ℂ) : SymmetricMultilinearForm a b V :=
-  φ.compMultilinearMap (symMonomialMultilinear a (SymPow b V))
-
-@[simp]
-theorem monomialFunctional_apply (a b : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V]
-    (φ : SymPow a (SymPow b V) →ₗ[ℂ] ℂ) (x : Fin a → SymPow b V) :
-    monomialFunctional a b V φ x = φ (symMonomial a (SymPow b V) x) := rfl
-
-theorem monomialFunctional_symmetric (a b : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V]
-    (φ : SymPow a (SymPow b V) →ₗ[ℂ] ℂ) :
-    IsSymmetricMultilinearForm a b V (monomialFunctional a b V φ) := by
-  intro σ x
-  simp only [monomialFunctional_apply, symMonomial_permute]
-
-/-- Monomial values determine every linear map out of a symmetric power. -/
-theorem linearMap_eq_zero_of_symMonomial (n : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V] {W : Type v} [AddCommGroup W] [Module ℂ W]
-    (φ : SymPow n V →ₗ[ℂ] W)
-    (hφ : ∀ x : Fin n → V, φ (symMonomial n V x) = 0) : φ = 0 := by
-  ext x
-  change φ x = 0
-  obtain ⟨x, hx⟩ := x
-  induction hx using Submodule.span_induction with
-  | mem y hy =>
-      obtain ⟨v, rfl⟩ := hy
-      exact hφ v
-  | zero => exact map_zero φ
-  | add x y hx hy ihx ihy =>
-      change φ (⟨x, hx⟩ + ⟨y, hy⟩) = 0
-      rw [map_add, ihx, ihy, add_zero]
-  | smul c x hx ih =>
-      change φ (c • (⟨x, hx⟩ : SymPow n V)) = 0
-      rw [map_smul, ih, smul_zero]
-
-theorem monomialFunctional_eq_zero_iff (a b : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V]
-    (φ : SymPow a (SymPow b V) →ₗ[ℂ] ℂ) :
-    monomialFunctional a b V φ = 0 ↔ φ = 0 := by
-  constructor
-  · intro h
-    apply linearMap_eq_zero_of_symMonomial
-    intro x
-    exact congrArg (fun T : SymmetricMultilinearForm a b V => T x) h
-  · rintro rfl
-    rfl
-
-/-- Flatten the `a` rows of `b` vector arguments of a form on `SymPow b V`. -/
-def plethysmFormMultilinear (a b : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V] (T : SymmetricMultilinearForm a b V) :
-    MultilinearMap ℂ (fun _ : Σ _ : Fin a, Fin b => V) ℂ :=
-  T.compMultilinearMap (fun _ => symMonomialMultilinear b V)
-
-@[simp]
-theorem plethysmFormMultilinear_apply (a b : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V] (T : SymmetricMultilinearForm a b V)
-    (x : (Σ _ : Fin a, Fin b) → V) :
-    plethysmFormMultilinear a b V T x =
-      T (fun i => symMonomial b V (fun j => x ⟨i,j⟩)) := rfl
-
-theorem plethysmFormMultilinear_inner_permute (a b : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V] (T : SymmetricMultilinearForm a b V)
-    (σ : Fin a → Equiv.Perm (Fin b)) (x : (Σ _ : Fin a, Fin b) → V) :
-    plethysmFormMultilinear a b V T (fun ij => x ⟨ij.1, σ ij.1 ij.2⟩) =
-      plethysmFormMultilinear a b V T x := by
-  apply congrArg T
-  funext i
-  exact symMonomial_permute b V (σ i) (fun j => x ⟨i,j⟩)
-
-theorem plethysmFormMultilinear_outer_permute (a b : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V] (T : SymmetricMultilinearForm a b V)
-    (hT : IsSymmetricMultilinearForm a b V T)
-    (σ : Equiv.Perm (Fin a)) (x : (Σ _ : Fin a, Fin b) → V) :
-    plethysmFormMultilinear a b V T (fun ij => x ⟨σ ij.1, ij.2⟩) =
-      plethysmFormMultilinear a b V T x := by
-  exact hT σ (fun i => symMonomial b V (fun j => x ⟨i,j⟩))
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.Model
+
+namespace OAI
+
+noncomputable section
+
+open scoped BigOperators
+
+universe u v
+
+namespace Problem346
+
+/-- The product of degree-one generators, with its multilinearity bundled. -/
+def symMonomialMultilinear (n : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V] :
+    MultilinearMap ℂ (fun _ : Fin n => V) (SymPow n V) :=
+  ((MultilinearMap.mkPiAlgebra ℂ (Fin n) (SymmetricAlgebra ℂ V)).compLinearMap
+    (fun _ => SymmetricAlgebra.ι ℂ V)).codRestrict (symPowSubmodule n V)
+      (fun x => Submodule.subset_span (Set.mem_range_self x))
+
+@[simp]
+theorem symMonomialMultilinear_apply (n : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V] (x : Fin n → V) :
+    symMonomialMultilinear n V x = symMonomial n V x := rfl
+
+theorem symMonomial_permute (n : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V] (σ : Equiv.Perm (Fin n)) (x : Fin n → V) :
+    symMonomial n V (fun i => x (σ i)) = symMonomial n V x := by
+  apply Subtype.ext
+  exact Equiv.prod_comp σ (fun i => SymmetricAlgebra.ι ℂ V (x i))
+
+/-- A functional on a symmetric power induces a symmetric multilinear form. -/
+def monomialFunctional (a b : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V]
+    (φ : SymPow a (SymPow b V) →ₗ[ℂ] ℂ) : SymmetricMultilinearForm a b V :=
+  φ.compMultilinearMap (symMonomialMultilinear a (SymPow b V))
+
+@[simp]
+theorem monomialFunctional_apply (a b : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V]
+    (φ : SymPow a (SymPow b V) →ₗ[ℂ] ℂ) (x : Fin a → SymPow b V) :
+    monomialFunctional a b V φ x = φ (symMonomial a (SymPow b V) x) := rfl
+
+theorem monomialFunctional_symmetric (a b : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V]
+    (φ : SymPow a (SymPow b V) →ₗ[ℂ] ℂ) :
+    IsSymmetricMultilinearForm a b V (monomialFunctional a b V φ) := by
+  intro σ x
+  simp only [monomialFunctional_apply, symMonomial_permute]
+
+/-- Monomial values determine every linear map out of a symmetric power. -/
+theorem linearMap_eq_zero_of_symMonomial (n : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V] {W : Type v} [AddCommGroup W] [Module ℂ W]
+    (φ : SymPow n V →ₗ[ℂ] W)
+    (hφ : ∀ x : Fin n → V, φ (symMonomial n V x) = 0) : φ = 0 := by
+  ext x
+  change φ x = 0
+  obtain ⟨x, hx⟩ := x
+  induction hx using Submodule.span_induction with
+  | mem y hy =>
+      obtain ⟨v, rfl⟩ := hy
+      exact hφ v
+  | zero => exact map_zero φ
+  | add x y hx hy ihx ihy =>
+      change φ (⟨x, hx⟩ + ⟨y, hy⟩) = 0
+      rw [map_add, ihx, ihy, add_zero]
+  | smul c x hx ih =>
+      change φ (c • (⟨x, hx⟩ : SymPow n V)) = 0
+      rw [map_smul, ih, smul_zero]
+
+theorem monomialFunctional_eq_zero_iff (a b : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V]
+    (φ : SymPow a (SymPow b V) →ₗ[ℂ] ℂ) :
+    monomialFunctional a b V φ = 0 ↔ φ = 0 := by
+  constructor
+  · intro h
+    apply linearMap_eq_zero_of_symMonomial
+    intro x
+    exact congrArg (fun T : SymmetricMultilinearForm a b V => T x) h
+  · rintro rfl
+    rfl
+
+/-- Flatten the `a` rows of `b` vector arguments of a form on `SymPow b V`. -/
+def plethysmFormMultilinear (a b : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V] (T : SymmetricMultilinearForm a b V) :
+    MultilinearMap ℂ (fun _ : Σ _ : Fin a, Fin b => V) ℂ :=
+  T.compMultilinearMap (fun _ => symMonomialMultilinear b V)
+
+@[simp]
+theorem plethysmFormMultilinear_apply (a b : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V] (T : SymmetricMultilinearForm a b V)
+    (x : (Σ _ : Fin a, Fin b) → V) :
+    plethysmFormMultilinear a b V T x =
+      T (fun i => symMonomial b V (fun j => x ⟨i,j⟩)) := rfl
+
+theorem plethysmFormMultilinear_inner_permute (a b : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V] (T : SymmetricMultilinearForm a b V)
+    (σ : Fin a → Equiv.Perm (Fin b)) (x : (Σ _ : Fin a, Fin b) → V) :
+    plethysmFormMultilinear a b V T (fun ij => x ⟨ij.1, σ ij.1 ij.2⟩) =
+      plethysmFormMultilinear a b V T x := by
+  apply congrArg T
+  funext i
+  exact symMonomial_permute b V (σ i) (fun j => x ⟨i,j⟩)
+
+theorem plethysmFormMultilinear_outer_permute (a b : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V] (T : SymmetricMultilinearForm a b V)
+    (hT : IsSymmetricMultilinearForm a b V T)
+    (σ : Equiv.Perm (Fin a)) (x : (Σ _ : Fin a, Fin b) → V) :
+    plethysmFormMultilinear a b V T (fun ij => x ⟨σ ij.1, ij.2⟩) =
+      plethysmFormMultilinear a b V T x := by
+  exact hT σ (fun i => symMonomial b V (fun j => x ⟨i,j⟩))
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/MultilinearDescent.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/MultilinearDescent.lean
index fa55dc6e4bc160e2d11eb00639e5a8db3ba2d29d..601913643f31af27fa55855edc6aed1a48f85cb7 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/MultilinearDescent.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/MultilinearDescent.lean
@@ -1,95 +1,95 @@
-import Mathlib
-
-namespace OAI
-
-noncomputable section
-
-namespace Problem346
-
-open Function
-
-universe uR uI uM uN uW
-
-variable {R : Type uR} [Semiring R] {I : Type uI}
-    {M : I → Type uM} {N : I → Type uN} {W : Type uW}
-    [∀ i, AddCommMonoid (M i)] [∀ i, Module R (M i)]
-    [∀ i, AddCommMonoid (N i)] [∀ i, Module R (N i)]
-    [AddCommMonoid W] [Module R W]
-
-/-- A multilinear map constant on the fibers of coordinatewise surjections descends. -/
-def multilinearDescend
-    (q : ∀ i, M i →ₗ[R] N i) (hq : ∀ i, Surjective (q i))
-    (f : MultilinearMap R M W)
-    (hf : ∀ x y, (∀ i, q i (x i) = q i (y i)) → f x = f y) :
-    MultilinearMap R N W where
-  toFun y := f (fun i => surjInv (hq i) (y i))
-  map_update_add' y i x z := by
-    let s : ∀ i, N i → M i := fun i => surjInv (hq i)
-    have hs : ∀ i x, q i (s i x) = x := fun i x => surjInv_eq (hq i) x
-    have h (t : N i) :
-        f (fun j => s j (update y i t j)) = f (update (fun j => s j (y j)) i (s i t)) := by
-      apply hf
-      intro j
-      by_cases hji : j = i
-      · subst j; simp [hs]
-      · simp [hji, hs]
-    change f (fun j => s j (update y i (x + z) j)) =
-      f (fun j => s j (update y i x j)) + f (fun j => s j (update y i z j))
-    rw [h, h, h]
-    have hadd :
-        f (update (fun j => s j (y j)) i (s i (x + z))) =
-        f (update (fun j => s j (y j)) i (s i x + s i z)) := by
-      apply hf
-      intro j
-      by_cases hji : j = i
-      · subst j; simp [hs]
-      · simp [hji]
-    rw [hadd, f.map_update_add]
-  map_update_smul' y i c x := by
-    let s : ∀ i, N i → M i := fun i => surjInv (hq i)
-    have hs : ∀ i x, q i (s i x) = x := fun i x => surjInv_eq (hq i) x
-    have h (t : N i) :
-        f (fun j => s j (update y i t j)) = f (update (fun j => s j (y j)) i (s i t)) := by
-      apply hf
-      intro j
-      by_cases hji : j = i
-      · subst j; simp [hs]
-      · simp [hji, hs]
-    change f (fun j => s j (update y i (c • x) j)) =
-      c • f (fun j => s j (update y i x j))
-    rw [h, h]
-    have hsmul :
-        f (update (fun j => s j (y j)) i (s i (c • x))) =
-        f (update (fun j => s j (y j)) i (c • s i x)) := by
-      apply hf
-      intro j
-      by_cases hji : j = i
-      · subst j; simp [hs]
-      · simp [hji]
-    rw [hsmul, f.map_update_smul]
-
-@[simp]
-theorem multilinearDescend_apply_image
-    (q : ∀ i, M i →ₗ[R] N i) (hq : ∀ i, Surjective (q i))
-    (f : MultilinearMap R M W)
-    (hf : ∀ x y, (∀ i, q i (x i) = q i (y i)) → f x = f y)
-    (x : ∀ i, M i) :
-    multilinearDescend q hq f hf (fun i => q i (x i)) = f x := by
-  apply hf
-  intro i
-  exact surjInv_eq (hq i) _
-
-@[simp]
-theorem multilinearDescend_compLinearMap
-    (q : ∀ i, M i →ₗ[R] N i) (hq : ∀ i, Surjective (q i))
-    (f : MultilinearMap R M W)
-    (hf : ∀ x y, (∀ i, q i (x i) = q i (y i)) → f x = f y) :
-    (multilinearDescend q hq f hf).compLinearMap q = f := by
-  ext x
-  exact multilinearDescend_apply_image q hq f hf x
-
-end Problem346
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+
+namespace Problem346
+
+open Function
+
+universe uR uI uM uN uW
+
+variable {R : Type uR} [Semiring R] {I : Type uI}
+    {M : I → Type uM} {N : I → Type uN} {W : Type uW}
+    [∀ i, AddCommMonoid (M i)] [∀ i, Module R (M i)]
+    [∀ i, AddCommMonoid (N i)] [∀ i, Module R (N i)]
+    [AddCommMonoid W] [Module R W]
+
+/-- A multilinear map constant on the fibers of coordinatewise surjections descends. -/
+def multilinearDescend
+    (q : ∀ i, M i →ₗ[R] N i) (hq : ∀ i, Surjective (q i))
+    (f : MultilinearMap R M W)
+    (hf : ∀ x y, (∀ i, q i (x i) = q i (y i)) → f x = f y) :
+    MultilinearMap R N W where
+  toFun y := f (fun i => surjInv (hq i) (y i))
+  map_update_add' y i x z := by
+    let s : ∀ i, N i → M i := fun i => surjInv (hq i)
+    have hs : ∀ i x, q i (s i x) = x := fun i x => surjInv_eq (hq i) x
+    have h (t : N i) :
+        f (fun j => s j (update y i t j)) = f (update (fun j => s j (y j)) i (s i t)) := by
+      apply hf
+      intro j
+      by_cases hji : j = i
+      · subst j; simp [hs]
+      · simp [hji, hs]
+    change f (fun j => s j (update y i (x + z) j)) =
+      f (fun j => s j (update y i x j)) + f (fun j => s j (update y i z j))
+    rw [h, h, h]
+    have hadd :
+        f (update (fun j => s j (y j)) i (s i (x + z))) =
+        f (update (fun j => s j (y j)) i (s i x + s i z)) := by
+      apply hf
+      intro j
+      by_cases hji : j = i
+      · subst j; simp [hs]
+      · simp [hji]
+    rw [hadd, f.map_update_add]
+  map_update_smul' y i c x := by
+    let s : ∀ i, N i → M i := fun i => surjInv (hq i)
+    have hs : ∀ i x, q i (s i x) = x := fun i x => surjInv_eq (hq i) x
+    have h (t : N i) :
+        f (fun j => s j (update y i t j)) = f (update (fun j => s j (y j)) i (s i t)) := by
+      apply hf
+      intro j
+      by_cases hji : j = i
+      · subst j; simp [hs]
+      · simp [hji, hs]
+    change f (fun j => s j (update y i (c • x) j)) =
+      c • f (fun j => s j (update y i x j))
+    rw [h, h]
+    have hsmul :
+        f (update (fun j => s j (y j)) i (s i (c • x))) =
+        f (update (fun j => s j (y j)) i (c • s i x)) := by
+      apply hf
+      intro j
+      by_cases hji : j = i
+      · subst j; simp [hs]
+      · simp [hji]
+    rw [hsmul, f.map_update_smul]
+
+@[simp]
+theorem multilinearDescend_apply_image
+    (q : ∀ i, M i →ₗ[R] N i) (hq : ∀ i, Surjective (q i))
+    (f : MultilinearMap R M W)
+    (hf : ∀ x y, (∀ i, q i (x i) = q i (y i)) → f x = f y)
+    (x : ∀ i, M i) :
+    multilinearDescend q hq f hf (fun i => q i (x i)) = f x := by
+  apply hf
+  intro i
+  exact surjInv_eq (hq i) _
+
+@[simp]
+theorem multilinearDescend_compLinearMap
+    (q : ∀ i, M i →ₗ[R] N i) (hq : ∀ i, Surjective (q i))
+    (f : MultilinearMap R M W)
+    (hf : ∀ x y, (∀ i, q i (x i) = q i (y i)) → f x = f y) :
+    (multilinearDescend q hq f hf).compLinearMap q = f := by
+  ext x
+  exact multilinearDescend_apply_image q hq f hf x
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/MultilinearPolynomial.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/MultilinearPolynomial.lean
index 54aa7d5100ae801575af0401ceda7c8c5704b58c..ad97de6b5550866f3a3a48c01cdd0fca0f00785a 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/MultilinearPolynomial.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/MultilinearPolynomial.lean
@@ -1,132 +1,132 @@
-import Mathlib
-
-namespace OAI
-
-noncomputable section
-open scoped BigOperators
-
-namespace Problem346
-
-universe uι uκ uβ uV
-variable {ι : Type uι} {κ : Type uκ} {β : Type uβ} {V : Type uV}
-variable [Fintype ι] [Fintype κ] [AddCommGroup V] [Module ℂ V]
-
-/-- The scalar polynomial obtained by feeding a multilinear form vector variables,
-with `block i` specifying which vector variable occurs in its `i`th argument. -/
-def multilinearPolynomial (L : MultilinearMap ℂ (fun _ : ι => V) ℂ)
-    (e : κ → V) (block : ι → β) : MvPolynomial (β × κ) ℂ := by
-  classical
-  exact ∑ k : ι → κ,
-    MvPolynomial.C (L (fun i => e (k i))) *
-      ∏ i, MvPolynomial.X (block i, k i)
-
-/-- Evaluation of the multilinear polynomial on arbitrary coordinate assignments. -/
-theorem eval_multilinearPolynomial
-    (L : MultilinearMap ℂ (fun _ : ι => V) ℂ)
-    (e : κ → V) (block : ι → β) (z : β × κ → ℂ) :
-    MvPolynomial.eval z (multilinearPolynomial L e block) =
-      L (fun i => ∑ k, z (block i, k) • e k) := by
-  classical
-  simp only [multilinearPolynomial, map_sum, map_mul, MvPolynomial.eval_C,
-    map_prod, MvPolynomial.eval_X]
-  rw [L.map_sum]
-  apply Finset.sum_congr rfl
-  intro k hk
-  rw [L.map_smul_univ]
-  simp only [smul_eq_mul]
-  exact mul_comm _ _
-
-/-- With basis coordinates, evaluation is literal evaluation on the chosen vector blocks. -/
-theorem eval_multilinearPolynomial_basis
-    (L : MultilinearMap ℂ (fun _ : ι => V) ℂ)
-    (e : Module.Basis κ ℂ V) (block : ι → β) (v : β → V) :
-    MvPolynomial.eval (fun z : β × κ => e.repr (v z.1) z.2)
-      (multilinearPolynomial L e block) = L (fun i => v (block i)) := by
-  rw [eval_multilinearPolynomial]
-  simp only [e.sum_repr]
-
-/-- Pointwise vanishing on vector blocks implies the corresponding polynomial identity. -/
-theorem multilinearPolynomial_eq_zero
-    (L : MultilinearMap ℂ (fun _ : ι => V) ℂ)
-    (e : κ → V) (block : ι → β)
-    (h : ∀ v : β → V, L (fun i => v (block i)) = 0) :
-    multilinearPolynomial L e block = 0 := by
-  apply MvPolynomial.funext
-  intro z
-  rw [eval_multilinearPolynomial, map_zero]
-  exact h (fun j => ∑ k, z (j, k) • e k)
-
-/-- Conversely, a polynomial identity gives the vector-valued identity by basis evaluation. -/
-theorem multilinearPolynomial_eq_zero_iff
-    (L : MultilinearMap ℂ (fun _ : ι => V) ℂ)
-    (e : Module.Basis κ ℂ V) (block : ι → β) :
-    multilinearPolynomial L e block = 0 ↔
-      ∀ v : β → V, L (fun i => v (block i)) = 0 := by
-  constructor
-  · intro h v
-    rw [← eval_multilinearPolynomial_basis L e block v, h, map_zero]
-  · exact multilinearPolynomial_eq_zero L e block
-
-/-- Reindexing the labels by a symmetry of the multilinear form leaves its polynomial unchanged. -/
-theorem multilinearPolynomial_perm
-    (L : MultilinearMap ℂ (fun _ : ι => V) ℂ)
-    (e : κ → V) (block : ι → β) (σ : Equiv.Perm ι)
-    (hσ : ∀ x : ι → V, L (fun i => x (σ i)) = L x) :
-    multilinearPolynomial L e (fun i => block (σ i)) =
-      multilinearPolynomial L e block := by
-  apply MvPolynomial.funext
-  intro z
-  rw [eval_multilinearPolynomial, eval_multilinearPolynomial]
-  exact hσ (fun i => ∑ k, z (block i, k) • e k)
-
-/-- Equal labels make replacing either of two symmetric slots give the same polynomial. -/
-theorem multilinearPolynomial_update_eq [DecidableEq ι]
-    (L : MultilinearMap ℂ (fun _ : ι => V) ℂ)
-    (e : κ → V) (block : ι → β) (i j : ι) (dst : β)
-    (hij : block i = block j)
-    (hsym : ∀ x : ι → V, L (fun k => x (Equiv.swap i j k)) = L x) :
-    multilinearPolynomial L e (Function.update block i dst) =
-      multilinearPolynomial L e (Function.update block j dst) := by
-  classical
-  have hp := multilinearPolynomial_perm L e (Function.update block i dst)
-    (Equiv.swap i j) hsym
-  have hlabels :
-      (fun k => Function.update block i dst (Equiv.swap i j k)) =
-        Function.update block j dst := by
-    funext k
-    by_cases hi : k = i
-    · subst k
-      by_cases heq : i = j
-      · subst j
-        rw [Equiv.swap_apply_left, Function.update_self]
-      · simp [heq, Ne.symm heq, hij]
-    · by_cases hj : k = j
-      · subst k
-        simp
-      · simp [Equiv.swap_apply_of_ne_of_ne hi hj, hi, hj]
-  rw [hlabels] at hp
-  exact hp.symm
-
-/-- The derivative's sum over interchangeable source slots is a scalar multiple of one term. -/
-theorem multilinearPolynomial_sum_updates [DecidableEq ι]
-    (L : MultilinearMap ℂ (fun _ : ι => V) ℂ)
-    (e : κ → V) (block : ι → β) (s : Finset ι) (j : ι) (dst : β)
-    (hlabel : ∀ i ∈ s, block i = block j)
-    (hsym : ∀ i ∈ s, ∀ x : ι → V,
-      L (fun k => x (Equiv.swap i j k)) = L x) :
-    (∑ i ∈ s, multilinearPolynomial L e (Function.update block i dst)) =
-      (s.card : ℂ) • multilinearPolynomial L e (Function.update block j dst) := by
-  calc
-    (∑ i ∈ s, multilinearPolynomial L e (Function.update block i dst)) =
-        ∑ _i ∈ s, multilinearPolynomial L e (Function.update block j dst) := by
-      apply Finset.sum_congr rfl
-      intro i hi
-      exact multilinearPolynomial_update_eq L e block i j dst (hlabel i hi) (hsym i hi)
-    _ = _ := by
-      rw [Finset.sum_const, Nat.cast_smul_eq_nsmul]
-
-end Problem346
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators
+
+namespace Problem346
+
+universe uι uκ uβ uV
+variable {ι : Type uι} {κ : Type uκ} {β : Type uβ} {V : Type uV}
+variable [Fintype ι] [Fintype κ] [AddCommGroup V] [Module ℂ V]
+
+/-- The scalar polynomial obtained by feeding a multilinear form vector variables,
+with `block i` specifying which vector variable occurs in its `i`th argument. -/
+def multilinearPolynomial (L : MultilinearMap ℂ (fun _ : ι => V) ℂ)
+    (e : κ → V) (block : ι → β) : MvPolynomial (β × κ) ℂ := by
+  classical
+  exact ∑ k : ι → κ,
+    MvPolynomial.C (L (fun i => e (k i))) *
+      ∏ i, MvPolynomial.X (block i, k i)
+
+/-- Evaluation of the multilinear polynomial on arbitrary coordinate assignments. -/
+theorem eval_multilinearPolynomial
+    (L : MultilinearMap ℂ (fun _ : ι => V) ℂ)
+    (e : κ → V) (block : ι → β) (z : β × κ → ℂ) :
+    MvPolynomial.eval z (multilinearPolynomial L e block) =
+      L (fun i => ∑ k, z (block i, k) • e k) := by
+  classical
+  simp only [multilinearPolynomial, map_sum, map_mul, MvPolynomial.eval_C,
+    map_prod, MvPolynomial.eval_X]
+  rw [L.map_sum]
+  apply Finset.sum_congr rfl
+  intro k hk
+  rw [L.map_smul_univ]
+  simp only [smul_eq_mul]
+  exact mul_comm _ _
+
+/-- With basis coordinates, evaluation is literal evaluation on the chosen vector blocks. -/
+theorem eval_multilinearPolynomial_basis
+    (L : MultilinearMap ℂ (fun _ : ι => V) ℂ)
+    (e : Module.Basis κ ℂ V) (block : ι → β) (v : β → V) :
+    MvPolynomial.eval (fun z : β × κ => e.repr (v z.1) z.2)
+      (multilinearPolynomial L e block) = L (fun i => v (block i)) := by
+  rw [eval_multilinearPolynomial]
+  simp only [e.sum_repr]
+
+/-- Pointwise vanishing on vector blocks implies the corresponding polynomial identity. -/
+theorem multilinearPolynomial_eq_zero
+    (L : MultilinearMap ℂ (fun _ : ι => V) ℂ)
+    (e : κ → V) (block : ι → β)
+    (h : ∀ v : β → V, L (fun i => v (block i)) = 0) :
+    multilinearPolynomial L e block = 0 := by
+  apply MvPolynomial.funext
+  intro z
+  rw [eval_multilinearPolynomial, map_zero]
+  exact h (fun j => ∑ k, z (j, k) • e k)
+
+/-- Conversely, a polynomial identity gives the vector-valued identity by basis evaluation. -/
+theorem multilinearPolynomial_eq_zero_iff
+    (L : MultilinearMap ℂ (fun _ : ι => V) ℂ)
+    (e : Module.Basis κ ℂ V) (block : ι → β) :
+    multilinearPolynomial L e block = 0 ↔
+      ∀ v : β → V, L (fun i => v (block i)) = 0 := by
+  constructor
+  · intro h v
+    rw [← eval_multilinearPolynomial_basis L e block v, h, map_zero]
+  · exact multilinearPolynomial_eq_zero L e block
+
+/-- Reindexing the labels by a symmetry of the multilinear form leaves its polynomial unchanged. -/
+theorem multilinearPolynomial_perm
+    (L : MultilinearMap ℂ (fun _ : ι => V) ℂ)
+    (e : κ → V) (block : ι → β) (σ : Equiv.Perm ι)
+    (hσ : ∀ x : ι → V, L (fun i => x (σ i)) = L x) :
+    multilinearPolynomial L e (fun i => block (σ i)) =
+      multilinearPolynomial L e block := by
+  apply MvPolynomial.funext
+  intro z
+  rw [eval_multilinearPolynomial, eval_multilinearPolynomial]
+  exact hσ (fun i => ∑ k, z (block i, k) • e k)
+
+/-- Equal labels make replacing either of two symmetric slots give the same polynomial. -/
+theorem multilinearPolynomial_update_eq [DecidableEq ι]
+    (L : MultilinearMap ℂ (fun _ : ι => V) ℂ)
+    (e : κ → V) (block : ι → β) (i j : ι) (dst : β)
+    (hij : block i = block j)
+    (hsym : ∀ x : ι → V, L (fun k => x (Equiv.swap i j k)) = L x) :
+    multilinearPolynomial L e (Function.update block i dst) =
+      multilinearPolynomial L e (Function.update block j dst) := by
+  classical
+  have hp := multilinearPolynomial_perm L e (Function.update block i dst)
+    (Equiv.swap i j) hsym
+  have hlabels :
+      (fun k => Function.update block i dst (Equiv.swap i j k)) =
+        Function.update block j dst := by
+    funext k
+    by_cases hi : k = i
+    · subst k
+      by_cases heq : i = j
+      · subst j
+        rw [Equiv.swap_apply_left, Function.update_self]
+      · simp [heq, Ne.symm heq, hij]
+    · by_cases hj : k = j
+      · subst k
+        simp
+      · simp [Equiv.swap_apply_of_ne_of_ne hi hj, hi, hj]
+  rw [hlabels] at hp
+  exact hp.symm
+
+/-- The derivative's sum over interchangeable source slots is a scalar multiple of one term. -/
+theorem multilinearPolynomial_sum_updates [DecidableEq ι]
+    (L : MultilinearMap ℂ (fun _ : ι => V) ℂ)
+    (e : κ → V) (block : ι → β) (s : Finset ι) (j : ι) (dst : β)
+    (hlabel : ∀ i ∈ s, block i = block j)
+    (hsym : ∀ i ∈ s, ∀ x : ι → V,
+      L (fun k => x (Equiv.swap i j k)) = L x) :
+    (∑ i ∈ s, multilinearPolynomial L e (Function.update block i dst)) =
+      (s.card : ℂ) • multilinearPolynomial L e (Function.update block j dst) := by
+  calc
+    (∑ i ∈ s, multilinearPolynomial L e (Function.update block i dst)) =
+        ∑ _i ∈ s, multilinearPolynomial L e (Function.update block j dst) := by
+      apply Finset.sum_congr rfl
+      intro i hi
+      exact multilinearPolynomial_update_eq L e block i j dst (hlabel i hi) (hsym i hi)
+    _ = _ := by
+      rw [Finset.sum_const, Nat.cast_smul_eq_nsmul]
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/Multiplication.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/Multiplication.lean
index 5e8ccdb61389b4c80bd5f44a95cdf8b78a92f080..55cad99f85699b865fbadb990d786949d2d942c8 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/Multiplication.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/Multiplication.lean
@@ -1,95 +1,95 @@
-import OAI.RepresentationTheory.FoulkesHowe.Model
-
-namespace OAI
-
-noncomputable section
-open scoped BigOperators Pointwise
-universe u
-
-namespace Problem346
-
-variable {V : Type u} [AddCommGroup V] [Module ℂ V]
-
-/-- Concatenating two lists of factors multiplies their symmetric monomials. -/
-theorem symMonomialRaw_append (m n : ℕ) (v : Fin m → V) (w : Fin n → V) :
-    symMonomialRaw (m + n) V (Fin.append v w) =
-      symMonomialRaw m V v * symMonomialRaw n V w := by
-  simp [symMonomialRaw, Fin.prod_univ_add]
-
-/-- Products of homogeneous spans have the sum of their degrees. -/
-theorem symPowSubmodule_mul (m n : ℕ) :
-    symPowSubmodule m V * symPowSubmodule n V = symPowSubmodule (m + n) V := by
-  unfold symPowSubmodule
-  rw [Submodule.span_mul_span]
-  apply congrArg (Submodule.span ℂ)
-  ext z
-  constructor
-  · rintro ⟨x, ⟨v, rfl⟩, y, ⟨w, rfl⟩, rfl⟩
-    exact ⟨Fin.append v w, symMonomialRaw_append m n v w⟩
-  · rintro ⟨v, rfl⟩
-    refine ⟨symMonomialRaw m V (fun i => v (Fin.castAdd n i)),
-      ⟨_, rfl⟩, symMonomialRaw n V (fun i => v (Fin.natAdd m i)), ⟨_, rfl⟩, ?_⟩
-    simp [symMonomialRaw, Fin.prod_univ_add]
-
-/-- The product of two elements of the indicated homogeneous pieces. -/
-theorem symPow_mul_mem (m n : ℕ) (x : SymPow m V) (y : SymPow n V) :
-    (x : SymmetricAlgebra ℂ V) * (y : SymmetricAlgebra ℂ V) ∈
-      symPowSubmodule (m+n) V := by
-  rw [← symPowSubmodule_mul]
-  exact Submodule.mul_mem_mul x.property y.property
-
-/-- Graded multiplication in the symmetric algebra, as a bilinear map. -/
-def symPowMul (m n : ℕ) :
-    SymPow m V →ₗ[ℂ] SymPow n V →ₗ[ℂ] SymPow (m+n) V :=
-  LinearMap.mk₂ ℂ
-    (fun x y => ⟨(x : SymmetricAlgebra ℂ V) * (y : SymmetricAlgebra ℂ V),
-      symPow_mul_mem m n x y⟩)
-    (by intros; apply Subtype.ext; exact add_mul _ _ _)
-    (by intros; apply Subtype.ext; exact smul_mul_assoc _ _ _)
-    (by intros; apply Subtype.ext; exact mul_add _ _ _)
-    (by intros; apply Subtype.ext; exact mul_smul_comm _ _ _)
-
-@[simp] theorem symPowMul_coe (m n : ℕ) (x : SymPow m V) (y : SymPow n V) :
-    (symPowMul m n x y : SymmetricAlgebra ℂ V) =
-      (x : SymmetricAlgebra ℂ V) * (y : SymmetricAlgebra ℂ V) := rfl
-
-@[simp] theorem symPowMul_symMonomial (m n : ℕ) (v : Fin m → V) (w : Fin n → V) :
-    symPowMul m n (symMonomial m V v) (symMonomial n V w) =
-      symMonomial (m+n) V (Fin.append v w) := by
-  apply Subtype.ext
-  exact (symMonomialRaw_append m n v w).symm
-
-/-- Transport a homogeneous element across an equality of degrees. -/
-def symPowCast {m n : ℕ} (h : m = n) : SymPow m V ≃ₗ[ℂ] SymPow n V :=
-  LinearEquiv.ofEq _ _ (congrArg (fun k => symPowSubmodule k V) h)
-
-@[simp] theorem symPowCast_coe {m n : ℕ} (h : m = n) (x : SymPow m V) :
-    (symPowCast h x : SymmetricAlgebra ℂ V) = x := rfl
-
-@[simp] theorem symPowCast_symMonomial {m n : ℕ} (h : m = n) (v : Fin m → V) :
-    symPowCast h (symMonomial m V v) =
-      symMonomial n V (fun i => v (Fin.cast h.symm i)) := by
-  subst h
-  rfl
-
-/-- Multiplication followed by transport to a specified total degree. -/
-def symPowMulCast (m n b : ℕ) (h : m+n=b) :
-    SymPow m V →ₗ[ℂ] SymPow n V →ₗ[ℂ] SymPow b V :=
-  (LinearMap.llcomp ℂ (SymPow n V) (SymPow (m+n) V) (SymPow b V)
-    (symPowCast h).toLinearMap).comp (symPowMul m n)
-
-@[simp] theorem symPowMulCast_coe (m n b : ℕ) (h : m+n=b)
-    (x : SymPow m V) (y : SymPow n V) :
-    (symPowMulCast m n b h x y : SymmetricAlgebra ℂ V) =
-      (x : SymmetricAlgebra ℂ V) * (y : SymmetricAlgebra ℂ V) := rfl
-
-/-- The raw monomial of a constant tuple is a power. -/
-theorem symMonomialRaw_const (n : ℕ) (v : V) :
-    symMonomialRaw n V (fun _ => v) = SymmetricAlgebra.ι ℂ V v ^ n := by
-  simp [symMonomialRaw]
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.Model
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators Pointwise
+universe u
+
+namespace Problem346
+
+variable {V : Type u} [AddCommGroup V] [Module ℂ V]
+
+/-- Concatenating two lists of factors multiplies their symmetric monomials. -/
+theorem symMonomialRaw_append (m n : ℕ) (v : Fin m → V) (w : Fin n → V) :
+    symMonomialRaw (m + n) V (Fin.append v w) =
+      symMonomialRaw m V v * symMonomialRaw n V w := by
+  simp [symMonomialRaw, Fin.prod_univ_add]
+
+/-- Products of homogeneous spans have the sum of their degrees. -/
+theorem symPowSubmodule_mul (m n : ℕ) :
+    symPowSubmodule m V * symPowSubmodule n V = symPowSubmodule (m + n) V := by
+  unfold symPowSubmodule
+  rw [Submodule.span_mul_span]
+  apply congrArg (Submodule.span ℂ)
+  ext z
+  constructor
+  · rintro ⟨x, ⟨v, rfl⟩, y, ⟨w, rfl⟩, rfl⟩
+    exact ⟨Fin.append v w, symMonomialRaw_append m n v w⟩
+  · rintro ⟨v, rfl⟩
+    refine ⟨symMonomialRaw m V (fun i => v (Fin.castAdd n i)),
+      ⟨_, rfl⟩, symMonomialRaw n V (fun i => v (Fin.natAdd m i)), ⟨_, rfl⟩, ?_⟩
+    simp [symMonomialRaw, Fin.prod_univ_add]
+
+/-- The product of two elements of the indicated homogeneous pieces. -/
+theorem symPow_mul_mem (m n : ℕ) (x : SymPow m V) (y : SymPow n V) :
+    (x : SymmetricAlgebra ℂ V) * (y : SymmetricAlgebra ℂ V) ∈
+      symPowSubmodule (m+n) V := by
+  rw [← symPowSubmodule_mul]
+  exact Submodule.mul_mem_mul x.property y.property
+
+/-- Graded multiplication in the symmetric algebra, as a bilinear map. -/
+def symPowMul (m n : ℕ) :
+    SymPow m V →ₗ[ℂ] SymPow n V →ₗ[ℂ] SymPow (m+n) V :=
+  LinearMap.mk₂ ℂ
+    (fun x y => ⟨(x : SymmetricAlgebra ℂ V) * (y : SymmetricAlgebra ℂ V),
+      symPow_mul_mem m n x y⟩)
+    (by intros; apply Subtype.ext; exact add_mul _ _ _)
+    (by intros; apply Subtype.ext; exact smul_mul_assoc _ _ _)
+    (by intros; apply Subtype.ext; exact mul_add _ _ _)
+    (by intros; apply Subtype.ext; exact mul_smul_comm _ _ _)
+
+@[simp] theorem symPowMul_coe (m n : ℕ) (x : SymPow m V) (y : SymPow n V) :
+    (symPowMul m n x y : SymmetricAlgebra ℂ V) =
+      (x : SymmetricAlgebra ℂ V) * (y : SymmetricAlgebra ℂ V) := rfl
+
+@[simp] theorem symPowMul_symMonomial (m n : ℕ) (v : Fin m → V) (w : Fin n → V) :
+    symPowMul m n (symMonomial m V v) (symMonomial n V w) =
+      symMonomial (m+n) V (Fin.append v w) := by
+  apply Subtype.ext
+  exact (symMonomialRaw_append m n v w).symm
+
+/-- Transport a homogeneous element across an equality of degrees. -/
+def symPowCast {m n : ℕ} (h : m = n) : SymPow m V ≃ₗ[ℂ] SymPow n V :=
+  LinearEquiv.ofEq _ _ (congrArg (fun k => symPowSubmodule k V) h)
+
+@[simp] theorem symPowCast_coe {m n : ℕ} (h : m = n) (x : SymPow m V) :
+    (symPowCast h x : SymmetricAlgebra ℂ V) = x := rfl
+
+@[simp] theorem symPowCast_symMonomial {m n : ℕ} (h : m = n) (v : Fin m → V) :
+    symPowCast h (symMonomial m V v) =
+      symMonomial n V (fun i => v (Fin.cast h.symm i)) := by
+  subst h
+  rfl
+
+/-- Multiplication followed by transport to a specified total degree. -/
+def symPowMulCast (m n b : ℕ) (h : m+n=b) :
+    SymPow m V →ₗ[ℂ] SymPow n V →ₗ[ℂ] SymPow b V :=
+  (LinearMap.llcomp ℂ (SymPow n V) (SymPow (m+n) V) (SymPow b V)
+    (symPowCast h).toLinearMap).comp (symPowMul m n)
+
+@[simp] theorem symPowMulCast_coe (m n b : ℕ) (h : m+n=b)
+    (x : SymPow m V) (y : SymPow n V) :
+    (symPowMulCast m n b h x y : SymmetricAlgebra ℂ V) =
+      (x : SymmetricAlgebra ℂ V) * (y : SymmetricAlgebra ℂ V) := rfl
+
+/-- The raw monomial of a constant tuple is a power. -/
+theorem symMonomialRaw_const (n : ℕ) (v : V) :
+    symMonomialRaw n V (fun _ => v) = SymmetricAlgebra.ι ℂ V v ^ n := by
+  simp [symMonomialRaw]
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/Naturality.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/Naturality.lean
index bb04f2848d7f43eb7a25e2f3ee5001630a0def3e..d6e22e8f8c5456c03ddb163f70a79d065d42a6ab 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/Naturality.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/Naturality.lean
@@ -1,76 +1,76 @@
-import OAI.RepresentationTheory.FoulkesHowe.Actions
-import OAI.RepresentationTheory.FoulkesHowe.Spanning
-
-namespace OAI
-
-noncomputable section
-open scoped BigOperators
-universe u v w
-namespace Problem346
-
-variable {V : Type u} {W : Type v} {U : Type w}
-  [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
-  [AddCommGroup U] [Module ℂ U]
-
-/-- The map on a plethysm induced by a linear map on the original space. -/
-def plethysmMap (outer inner : ℕ) (f : V →ₗ[ℂ] W) :
-    SymPow outer (SymPow inner V) →ₗ[ℂ] SymPow outer (SymPow inner W) :=
-  symPowMap outer (symPowMap inner f)
-
-@[simp] theorem plethysmMap_symMonomial (outer inner : ℕ) (f : V →ₗ[ℂ] W)
-    (x : Fin outer → Fin inner → V) :
-    plethysmMap outer inner f
-        (symMonomial outer (SymPow inner V) (fun i => symMonomial inner V (x i))) =
-      symMonomial outer (SymPow inner W)
-        (fun i => symMonomial inner W (fun j => f (x i j))) := by
-  simp [plethysmMap]
-
-@[simp] theorem plethysmMap_id (outer inner : ℕ) :
-    plethysmMap outer inner (LinearMap.id : V →ₗ[ℂ] V) = LinearMap.id := by
-  simp [plethysmMap]
-
-@[simp] theorem plethysmMap_comp (outer inner : ℕ) (g : W →ₗ[ℂ] U) (f : V →ₗ[ℂ] W) :
-    plethysmMap outer inner (g.comp f) =
-      (plethysmMap outer inner g).comp (plethysmMap outer inner f) := by
-  simp [plethysmMap]
-
-@[simp] theorem plethysmMap_equiv (outer inner : ℕ) (g : V ≃ₗ[ℂ] V) :
-    plethysmMap outer inner g.toLinearMap = plethysmAction outer inner g := rfl
-
-/-- The explicitly normalized Foulkes formula is natural in the vector space. -/
-theorem plethysmMap_foulkesFormula (a b : ℕ) (f : V →ₗ[ℂ] W)
-    (x : Fin b → Fin a → V) :
-    plethysmMap a b f (foulkesFormula a b V x) =
-      foulkesFormula a b W (fun j i => f (x j i)) := by
-  classical
-  unfold foulkesFormula
-  rw [map_smul, map_sum]
-  congr 1
-  apply Finset.sum_congr rfl
-  intro σ _
-  exact plethysmMap_symMonomial a b f (fun i j => x j ((σ j) i))
-
-/-- Any two maps satisfying the canonical formula form a natural square. -/
-theorem IsFoulkesMap.natural {a b : ℕ}
-    {μV : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V)}
-    {μW : SymPow b (SymPow a W) →ₗ[ℂ] SymPow a (SymPow b W)}
-    (hV : IsFoulkesMap a b V μV) (hW : IsFoulkesMap a b W μW)
-    (f : V →ₗ[ℂ] W) :
-    (plethysmMap a b f).comp μV = μW.comp (plethysmMap b a f) := by
-  apply plethysm_linearMap_ext a b V
-  intro x
-  simp only [LinearMap.comp_apply, plethysmMap_symMonomial]
-  rw [hV, hW, plethysmMap_foulkesFormula]
-
-/-- Every canonical Foulkes map is GL-equivariant. -/
-theorem IsFoulkesMap.commutes {a b : ℕ}
-    {μ : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V)}
-    (hμ : IsFoulkesMap a b V μ) (g : V ≃ₗ[ℂ] V) :
-    (plethysmAction a b g).comp μ = μ.comp (plethysmAction b a g) :=
-  hμ.natural hμ g.toLinearMap
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.Actions
+import OAI.RepresentationTheory.FoulkesHowe.Spanning
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators
+universe u v w
+namespace Problem346
+
+variable {V : Type u} {W : Type v} {U : Type w}
+  [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
+  [AddCommGroup U] [Module ℂ U]
+
+/-- The map on a plethysm induced by a linear map on the original space. -/
+def plethysmMap (outer inner : ℕ) (f : V →ₗ[ℂ] W) :
+    SymPow outer (SymPow inner V) →ₗ[ℂ] SymPow outer (SymPow inner W) :=
+  symPowMap outer (symPowMap inner f)
+
+@[simp] theorem plethysmMap_symMonomial (outer inner : ℕ) (f : V →ₗ[ℂ] W)
+    (x : Fin outer → Fin inner → V) :
+    plethysmMap outer inner f
+        (symMonomial outer (SymPow inner V) (fun i => symMonomial inner V (x i))) =
+      symMonomial outer (SymPow inner W)
+        (fun i => symMonomial inner W (fun j => f (x i j))) := by
+  simp [plethysmMap]
+
+@[simp] theorem plethysmMap_id (outer inner : ℕ) :
+    plethysmMap outer inner (LinearMap.id : V →ₗ[ℂ] V) = LinearMap.id := by
+  simp [plethysmMap]
+
+@[simp] theorem plethysmMap_comp (outer inner : ℕ) (g : W →ₗ[ℂ] U) (f : V →ₗ[ℂ] W) :
+    plethysmMap outer inner (g.comp f) =
+      (plethysmMap outer inner g).comp (plethysmMap outer inner f) := by
+  simp [plethysmMap]
+
+@[simp] theorem plethysmMap_equiv (outer inner : ℕ) (g : V ≃ₗ[ℂ] V) :
+    plethysmMap outer inner g.toLinearMap = plethysmAction outer inner g := rfl
+
+/-- The explicitly normalized Foulkes formula is natural in the vector space. -/
+theorem plethysmMap_foulkesFormula (a b : ℕ) (f : V →ₗ[ℂ] W)
+    (x : Fin b → Fin a → V) :
+    plethysmMap a b f (foulkesFormula a b V x) =
+      foulkesFormula a b W (fun j i => f (x j i)) := by
+  classical
+  unfold foulkesFormula
+  rw [map_smul, map_sum]
+  congr 1
+  apply Finset.sum_congr rfl
+  intro σ _
+  exact plethysmMap_symMonomial a b f (fun i j => x j ((σ j) i))
+
+/-- Any two maps satisfying the canonical formula form a natural square. -/
+theorem IsFoulkesMap.natural {a b : ℕ}
+    {μV : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V)}
+    {μW : SymPow b (SymPow a W) →ₗ[ℂ] SymPow a (SymPow b W)}
+    (hV : IsFoulkesMap a b V μV) (hW : IsFoulkesMap a b W μW)
+    (f : V →ₗ[ℂ] W) :
+    (plethysmMap a b f).comp μV = μW.comp (plethysmMap b a f) := by
+  apply plethysm_linearMap_ext a b V
+  intro x
+  simp only [LinearMap.comp_apply, plethysmMap_symMonomial]
+  rw [hV, hW, plethysmMap_foulkesFormula]
+
+/-- Every canonical Foulkes map is GL-equivariant. -/
+theorem IsFoulkesMap.commutes {a b : ℕ}
+    {μ : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V)}
+    (hμ : IsFoulkesMap a b V μ) (g : V ≃ₗ[ℂ] V) :
+    (plethysmAction a b g).comp μ = μ.comp (plethysmAction b a g) :=
+  hμ.natural hμ g.toLinearMap
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/PairingConstruction.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/PairingConstruction.lean
index a06fe05b9f5ff97ce0ed05d6b71a63b12f319ddd..05696ba3b3d6dfc565952903a8bd6cb88b22f5ae 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/PairingConstruction.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/PairingConstruction.lean
@@ -1,149 +1,149 @@
-import OAI.RepresentationTheory.FoulkesHowe.PermanentPairing
-import OAI.RepresentationTheory.FoulkesHowe.SymmetricLift
-import OAI.RepresentationTheory.FoulkesHowe.SymmetricDual
-import OAI.RepresentationTheory.FoulkesHowe.FiniteDimension
-import OAI.RepresentationTheory.FoulkesHowe.Functoriality
-import OAI.RepresentationTheory.FoulkesHowe.Spanning
-import OAI.RepresentationTheory.FoulkesHowe.PlethysmDual
-
-namespace OAI
-
-noncomputable section
-open scoped BigOperators
-namespace Problem346.PairingConstruction
-
-universe u v
-variable {X : Type u} {Y : Type v}
-  [AddCommGroup X] [Module ℂ X] [AddCommGroup Y] [Module ℂ Y]
-
-/-- Descend the right-hand family of the normalized permanent. -/
-def leftMultilinear (n : ℕ) (B : X →ₗ[ℂ] Y →ₗ[ℂ] ℂ) :
-    MultilinearMap ℂ (fun _ : Fin n => X) (Module.Dual ℂ (SymPow n Y)) :=
-  (symPowLiftLinear n Y ℂ).compMultilinearMap
-    ((permanentPairingMultilinear n B).codRestrict
-      (symmetricMultilinearSubmodule n Y ℂ) (by
-        intro x σ y
-        exact permanentPairingMultilinear_perm_right n B x y σ))
-
-@[simp] theorem leftMultilinear_apply_monomial (n : ℕ)
-    (B : X →ₗ[ℂ] Y →ₗ[ℂ] ℂ) (x : Fin n → X) (y : Fin n → Y) :
-    leftMultilinear n B x (symMonomial n Y y) =
-      permanentPairing n (fun x y => B x y) x y := by
-  simp only [leftMultilinear, LinearMap.compMultilinearMap_apply,
-    symPowLiftLinear_symMonomial]
-  exact permanentPairingMultilinear_apply n B x y
-
-theorem leftMultilinear_symmetric (n : ℕ) (B : X →ₗ[ℂ] Y →ₗ[ℂ] ℂ)
-    (σ : Equiv.Perm (Fin n)) (x : Fin n → X) :
-    leftMultilinear n B (fun i => x (σ i)) = leftMultilinear n B x := by
-  apply symPow_linearMap_ext n Y
-  intro y
-  simp only [leftMultilinear_apply_monomial]
-  exact permanentPairing_perm_left n (fun x y => B x y) x y σ
-
-/-- The canonical bilinear pairing induced on homogeneous symmetric powers. -/
-def pairing (n : ℕ) (B : X →ₗ[ℂ] Y →ₗ[ℂ] ℂ) :
-    SymPow n X →ₗ[ℂ] Module.Dual ℂ (SymPow n Y) :=
-  symPowLift (leftMultilinear n B) (leftMultilinear_symmetric n B)
-
-@[simp] theorem pairing_monomials (n : ℕ) (B : X →ₗ[ℂ] Y →ₗ[ℂ] ℂ)
-    (x : Fin n → X) (y : Fin n → Y) :
-    pairing n B (symMonomial n X x) (symMonomial n Y y) =
-      permanentPairing n (fun x y => B x y) x y := by
-  simp [pairing]
-
-variable (V : Type u) [AddCommGroup V] [Module ℂ V]
-
-/-- The evaluation pairing of a symmetric power and the symmetric power of its dual. -/
-def dualMap (n : ℕ) :
-    SymPow n (Module.Dual ℂ V) →ₗ[ℂ] Module.Dual ℂ (SymPow n V) :=
-  pairing n (LinearMap.id : Module.Dual ℂ V →ₗ[ℂ] Module.Dual ℂ V)
-
-@[simp] theorem dualMap_monomials (n : ℕ)
-    (x : Fin n → Module.Dual ℂ V) (y : Fin n → V) :
-    dualMap V n (symMonomial n (Module.Dual ℂ V) x) (symMonomial n V y) =
-      permanentPairing n (fun f v => f v) x y := by
-  exact pairing_monomials n LinearMap.id x y
-
-theorem dualMap_pure (n : ℕ) (p : SymPow n (Module.Dual ℂ V)) (x : V) :
-    dualMap V n p (symMonomial n V (fun _ => x)) =
-      SymmetricAlgebra.lift (Module.Dual.eval ℂ V x) p.val := by
-  have h : (LinearMap.applyₗ (symMonomial n V (fun _ => x))).comp (dualMap V n) =
-      (SymmetricAlgebra.lift (Module.Dual.eval ℂ V x)).toLinearMap.domRestrict
-        (symPowSubmodule n (Module.Dual ℂ V)) := by
-    apply symPow_linearMap_ext n (Module.Dual ℂ V)
-    intro f
-    change dualMap V n (symMonomial n (Module.Dual ℂ V) f)
-      (symMonomial n V (fun _ => x)) = _
-    rw [dualMap_monomials, permanentPairing_const_right]
-    simp [symMonomial, symMonomialRaw]
-  exact LinearMap.congr_fun h p
-
-theorem dualMap_injective [FiniteDimensional ℂ V] (n : ℕ) :
-    Function.Injective (dualMap V n) :=
-  symPowDual_injective_of_pure_evaluation V n (dualMap V n) (dualMap_pure V n)
-
-theorem dual_finrank (n : ℕ) [FiniteDimensional ℂ V] :
-    Module.finrank ℂ (SymPow n (Module.Dual ℂ V)) =
-      Module.finrank ℂ (Module.Dual ℂ (SymPow n V)) := by
-  rw [Subspace.dual_finrank_eq]
-  let e : Module.Dual ℂ V ≃ₗ[ℂ] V :=
-    LinearEquiv.ofFinrankEq _ _ Subspace.dual_finrank_eq
-  exact (symPowEquiv n e).finrank_eq
-
-/-- Perfect duality for homogeneous symmetric powers in finite dimension. -/
-def dualEquiv [FiniteDimensional ℂ V] (n : ℕ) :
-    SymPow n (Module.Dual ℂ V) ≃ₗ[ℂ] Module.Dual ℂ (SymPow n V) :=
-  (dualMap V n).linearEquivOfInjective (dualMap_injective V n) (dual_finrank V n)
-
-@[simp] theorem dualEquiv_apply [FiniteDimensional ℂ V] (n : ℕ)
-    (p : SymPow n (Module.Dual ℂ V)) : dualEquiv V n p = dualMap V n p := rfl
-
-theorem dualMap_natural (n : ℕ) (f : V →ₗ[ℂ] V)
-    (p : SymPow n (Module.Dual ℂ V)) (q : SymPow n V) :
-    dualMap V n (symPowMap n f.dualMap p) q =
-      dualMap V n p (symPowMap n f q) := by
-  have h : (dualMap V n).comp (symPowMap n f.dualMap) =
-      (symPowMap n f).dualMap.comp (dualMap V n) := by
-    apply symPow_linearMap_ext n (Module.Dual ℂ V)
-    intro x
-    apply symPow_linearMap_ext n V
-    intro y
-    simp only [LinearMap.comp_apply, symPowMap_symMonomial,
-      LinearMap.dualMap_apply, dualMap_monomials]
-    rfl
-  exact LinearMap.congr_fun (LinearMap.congr_fun h p) q
-
-theorem dualEquiv_natural [FiniteDimensional ℂ V] (n : ℕ) (f : V →ₗ[ℂ] V)
-    (p : SymPow n (Module.Dual ℂ V)) (q : SymPow n V) :
-    dualEquiv V n (symPowMap n f.dualMap p) q =
-      dualEquiv V n p (symPowMap n f q) :=
-  dualMap_natural V n f p q
-
-/-- Canonical surjectivity on the dual space gives the equivariant comparison. -/
-theorem comparison_of_surjection [FiniteDimensional ℂ V] (a b : ℕ)
-    (hsurj : ∃ μ : SymPow b (SymPow a (Module.Dual ℂ V)) →ₗ[ℂ]
-      SymPow a (SymPow b (Module.Dual ℂ V)),
-      IsFoulkesMap a b (Module.Dual ℂ V) μ ∧ Function.Surjective μ) :
-    ∃ ι : SymPow a (SymPow b V) →ₗ[ℂ] SymPow b (SymPow a V),
-      IsGLEquivariantEmbedding a b V ι := by
-  apply PlethysmDual.comparison_of_dual_charts a b
-    (PlethysmDual.plethysmDualEquivOf b a (dualEquiv V a) (dualEquiv (SymPow a V) b))
-    (PlethysmDual.plethysmDualEquivOf a b (dualEquiv V b) (dualEquiv (SymPow b V) a))
-  · intro g
-    exact PlethysmDual.plethysmDualEquivOf_adjoint b a
-      (dualEquiv V a) (dualEquiv (SymPow a V) b) g
-      (dualEquiv_natural V a g.toLinearMap)
-      (dualEquiv_natural (SymPow a V) b (symPowMap a g.toLinearMap))
-  · intro g
-    exact PlethysmDual.plethysmDualEquivOf_adjoint a b
-      (dualEquiv V b) (dualEquiv (SymPow b V) a) g
-      (dualEquiv_natural V b g.toLinearMap)
-      (dualEquiv_natural (SymPow b V) a (symPowMap b g.toLinearMap))
-  · exact hsurj
-
-end Problem346.PairingConstruction
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.PermanentPairing
+import OAI.RepresentationTheory.FoulkesHowe.SymmetricLift
+import OAI.RepresentationTheory.FoulkesHowe.SymmetricDual
+import OAI.RepresentationTheory.FoulkesHowe.FiniteDimension
+import OAI.RepresentationTheory.FoulkesHowe.Functoriality
+import OAI.RepresentationTheory.FoulkesHowe.Spanning
+import OAI.RepresentationTheory.FoulkesHowe.PlethysmDual
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators
+namespace Problem346.PairingConstruction
+
+universe u v
+variable {X : Type u} {Y : Type v}
+  [AddCommGroup X] [Module ℂ X] [AddCommGroup Y] [Module ℂ Y]
+
+/-- Descend the right-hand family of the normalized permanent. -/
+def leftMultilinear (n : ℕ) (B : X →ₗ[ℂ] Y →ₗ[ℂ] ℂ) :
+    MultilinearMap ℂ (fun _ : Fin n => X) (Module.Dual ℂ (SymPow n Y)) :=
+  (symPowLiftLinear n Y ℂ).compMultilinearMap
+    ((permanentPairingMultilinear n B).codRestrict
+      (symmetricMultilinearSubmodule n Y ℂ) (by
+        intro x σ y
+        exact permanentPairingMultilinear_perm_right n B x y σ))
+
+@[simp] theorem leftMultilinear_apply_monomial (n : ℕ)
+    (B : X →ₗ[ℂ] Y →ₗ[ℂ] ℂ) (x : Fin n → X) (y : Fin n → Y) :
+    leftMultilinear n B x (symMonomial n Y y) =
+      permanentPairing n (fun x y => B x y) x y := by
+  simp only [leftMultilinear, LinearMap.compMultilinearMap_apply,
+    symPowLiftLinear_symMonomial]
+  exact permanentPairingMultilinear_apply n B x y
+
+theorem leftMultilinear_symmetric (n : ℕ) (B : X →ₗ[ℂ] Y →ₗ[ℂ] ℂ)
+    (σ : Equiv.Perm (Fin n)) (x : Fin n → X) :
+    leftMultilinear n B (fun i => x (σ i)) = leftMultilinear n B x := by
+  apply symPow_linearMap_ext n Y
+  intro y
+  simp only [leftMultilinear_apply_monomial]
+  exact permanentPairing_perm_left n (fun x y => B x y) x y σ
+
+/-- The canonical bilinear pairing induced on homogeneous symmetric powers. -/
+def pairing (n : ℕ) (B : X →ₗ[ℂ] Y →ₗ[ℂ] ℂ) :
+    SymPow n X →ₗ[ℂ] Module.Dual ℂ (SymPow n Y) :=
+  symPowLift (leftMultilinear n B) (leftMultilinear_symmetric n B)
+
+@[simp] theorem pairing_monomials (n : ℕ) (B : X →ₗ[ℂ] Y →ₗ[ℂ] ℂ)
+    (x : Fin n → X) (y : Fin n → Y) :
+    pairing n B (symMonomial n X x) (symMonomial n Y y) =
+      permanentPairing n (fun x y => B x y) x y := by
+  simp [pairing]
+
+variable (V : Type u) [AddCommGroup V] [Module ℂ V]
+
+/-- The evaluation pairing of a symmetric power and the symmetric power of its dual. -/
+def dualMap (n : ℕ) :
+    SymPow n (Module.Dual ℂ V) →ₗ[ℂ] Module.Dual ℂ (SymPow n V) :=
+  pairing n (LinearMap.id : Module.Dual ℂ V →ₗ[ℂ] Module.Dual ℂ V)
+
+@[simp] theorem dualMap_monomials (n : ℕ)
+    (x : Fin n → Module.Dual ℂ V) (y : Fin n → V) :
+    dualMap V n (symMonomial n (Module.Dual ℂ V) x) (symMonomial n V y) =
+      permanentPairing n (fun f v => f v) x y := by
+  exact pairing_monomials n LinearMap.id x y
+
+theorem dualMap_pure (n : ℕ) (p : SymPow n (Module.Dual ℂ V)) (x : V) :
+    dualMap V n p (symMonomial n V (fun _ => x)) =
+      SymmetricAlgebra.lift (Module.Dual.eval ℂ V x) p.val := by
+  have h : (LinearMap.applyₗ (symMonomial n V (fun _ => x))).comp (dualMap V n) =
+      (SymmetricAlgebra.lift (Module.Dual.eval ℂ V x)).toLinearMap.domRestrict
+        (symPowSubmodule n (Module.Dual ℂ V)) := by
+    apply symPow_linearMap_ext n (Module.Dual ℂ V)
+    intro f
+    change dualMap V n (symMonomial n (Module.Dual ℂ V) f)
+      (symMonomial n V (fun _ => x)) = _
+    rw [dualMap_monomials, permanentPairing_const_right]
+    simp [symMonomial, symMonomialRaw]
+  exact LinearMap.congr_fun h p
+
+theorem dualMap_injective [FiniteDimensional ℂ V] (n : ℕ) :
+    Function.Injective (dualMap V n) :=
+  symPowDual_injective_of_pure_evaluation V n (dualMap V n) (dualMap_pure V n)
+
+theorem dual_finrank (n : ℕ) [FiniteDimensional ℂ V] :
+    Module.finrank ℂ (SymPow n (Module.Dual ℂ V)) =
+      Module.finrank ℂ (Module.Dual ℂ (SymPow n V)) := by
+  rw [Subspace.dual_finrank_eq]
+  let e : Module.Dual ℂ V ≃ₗ[ℂ] V :=
+    LinearEquiv.ofFinrankEq _ _ Subspace.dual_finrank_eq
+  exact (symPowEquiv n e).finrank_eq
+
+/-- Perfect duality for homogeneous symmetric powers in finite dimension. -/
+def dualEquiv [FiniteDimensional ℂ V] (n : ℕ) :
+    SymPow n (Module.Dual ℂ V) ≃ₗ[ℂ] Module.Dual ℂ (SymPow n V) :=
+  (dualMap V n).linearEquivOfInjective (dualMap_injective V n) (dual_finrank V n)
+
+@[simp] theorem dualEquiv_apply [FiniteDimensional ℂ V] (n : ℕ)
+    (p : SymPow n (Module.Dual ℂ V)) : dualEquiv V n p = dualMap V n p := rfl
+
+theorem dualMap_natural (n : ℕ) (f : V →ₗ[ℂ] V)
+    (p : SymPow n (Module.Dual ℂ V)) (q : SymPow n V) :
+    dualMap V n (symPowMap n f.dualMap p) q =
+      dualMap V n p (symPowMap n f q) := by
+  have h : (dualMap V n).comp (symPowMap n f.dualMap) =
+      (symPowMap n f).dualMap.comp (dualMap V n) := by
+    apply symPow_linearMap_ext n (Module.Dual ℂ V)
+    intro x
+    apply symPow_linearMap_ext n V
+    intro y
+    simp only [LinearMap.comp_apply, symPowMap_symMonomial,
+      LinearMap.dualMap_apply, dualMap_monomials]
+    rfl
+  exact LinearMap.congr_fun (LinearMap.congr_fun h p) q
+
+theorem dualEquiv_natural [FiniteDimensional ℂ V] (n : ℕ) (f : V →ₗ[ℂ] V)
+    (p : SymPow n (Module.Dual ℂ V)) (q : SymPow n V) :
+    dualEquiv V n (symPowMap n f.dualMap p) q =
+      dualEquiv V n p (symPowMap n f q) :=
+  dualMap_natural V n f p q
+
+/-- Canonical surjectivity on the dual space gives the equivariant comparison. -/
+theorem comparison_of_surjection [FiniteDimensional ℂ V] (a b : ℕ)
+    (hsurj : ∃ μ : SymPow b (SymPow a (Module.Dual ℂ V)) →ₗ[ℂ]
+      SymPow a (SymPow b (Module.Dual ℂ V)),
+      IsFoulkesMap a b (Module.Dual ℂ V) μ ∧ Function.Surjective μ) :
+    ∃ ι : SymPow a (SymPow b V) →ₗ[ℂ] SymPow b (SymPow a V),
+      IsGLEquivariantEmbedding a b V ι := by
+  apply PlethysmDual.comparison_of_dual_charts a b
+    (PlethysmDual.plethysmDualEquivOf b a (dualEquiv V a) (dualEquiv (SymPow a V) b))
+    (PlethysmDual.plethysmDualEquivOf a b (dualEquiv V b) (dualEquiv (SymPow b V) a))
+  · intro g
+    exact PlethysmDual.plethysmDualEquivOf_adjoint b a
+      (dualEquiv V a) (dualEquiv (SymPow a V) b) g
+      (dualEquiv_natural V a g.toLinearMap)
+      (dualEquiv_natural (SymPow a V) b (symPowMap a g.toLinearMap))
+  · intro g
+    exact PlethysmDual.plethysmDualEquivOf_adjoint a b
+      (dualEquiv V b) (dualEquiv (SymPow b V) a) g
+      (dualEquiv_natural V b g.toLinearMap)
+      (dualEquiv_natural (SymPow b V) a (symPowMap b g.toLinearMap))
+  · exact hsurj
+
+end Problem346.PairingConstruction
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/PairingLift.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/PairingLift.lean
index e0f4c89b9fb2729905ba020e4d9a8b2c4e6cdb46..5879df7252210e2b539d2a8f785d9bc514676fee 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/PairingLift.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/PairingLift.lean
@@ -1,54 +1,54 @@
-import OAI.RepresentationTheory.FoulkesHowe.SymmetricLift
-
-namespace OAI
-
-noncomputable section
-universe u v w
-namespace Problem346
-
-/-- Descend a separately symmetric multilinear pairing in both blocks. -/
-theorem exists_symPow_bilinear_lift (n m : ℕ) (V : Type u) (W : Type v)
-    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
-    (Q : MultilinearMap ℂ (fun _ : Fin n => V)
-      (MultilinearMap ℂ (fun _ : Fin m => W) ℂ))
-    (hleft : ∀ (σ : Equiv.Perm (Fin n)) x,
-      Q (fun i => x (σ i)) = Q x)
-    (hright : ∀ x (σ : Equiv.Perm (Fin m)) y,
-      Q x (fun i => y (σ i)) = Q x y) :
-    ∃ B : SymPow n V →ₗ[ℂ] Module.Dual ℂ (SymPow m W),
-      ∀ x y, B (symMonomial n V x) (symMonomial m W y) = Q x y := by
-  classical
-  choose G hG using fun x => exists_symPow_lift (Q x) (hright x)
-  let F : MultilinearMap ℂ (fun _ : Fin n => V) (Module.Dual ℂ (SymPow m W)) :=
-    { toFun := G
-      map_update_add' := by
-        intro _ x i a b
-        apply SymmetricLift.linearMap_ext
-        intro y
-        simp only [LinearMap.add_apply, hG]
-        exact congrArg (fun T : MultilinearMap ℂ (fun _ : Fin m => W) ℂ => T y)
-          (Q.map_update_add x i a b)
-      map_update_smul' := by
-        intro _ x i c a
-        apply SymmetricLift.linearMap_ext
-        intro y
-        simp only [LinearMap.smul_apply, hG]
-        exact congrArg (fun T : MultilinearMap ℂ (fun _ : Fin m => W) ℂ => T y)
-          (Q.map_update_smul x i c a) }
-  have hF : ∀ (σ : Equiv.Perm (Fin n)) x, F (fun i => x (σ i)) = F x := by
-    intro σ x
-    apply SymmetricLift.linearMap_ext
-    intro y
-    change G (fun i => x (σ i)) (symMonomial m W y) = G x (symMonomial m W y)
-    rw [hG, hG, hleft]
-  obtain ⟨B, hB⟩ := exists_symPow_lift F hF
-  refine ⟨B, ?_⟩
-  intro x y
-  rw [hB]
-  exact hG x y
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.SymmetricLift
+
+namespace OAI
+
+noncomputable section
+universe u v w
+namespace Problem346
+
+/-- Descend a separately symmetric multilinear pairing in both blocks. -/
+theorem exists_symPow_bilinear_lift (n m : ℕ) (V : Type u) (W : Type v)
+    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
+    (Q : MultilinearMap ℂ (fun _ : Fin n => V)
+      (MultilinearMap ℂ (fun _ : Fin m => W) ℂ))
+    (hleft : ∀ (σ : Equiv.Perm (Fin n)) x,
+      Q (fun i => x (σ i)) = Q x)
+    (hright : ∀ x (σ : Equiv.Perm (Fin m)) y,
+      Q x (fun i => y (σ i)) = Q x y) :
+    ∃ B : SymPow n V →ₗ[ℂ] Module.Dual ℂ (SymPow m W),
+      ∀ x y, B (symMonomial n V x) (symMonomial m W y) = Q x y := by
+  classical
+  choose G hG using fun x => exists_symPow_lift (Q x) (hright x)
+  let F : MultilinearMap ℂ (fun _ : Fin n => V) (Module.Dual ℂ (SymPow m W)) :=
+    { toFun := G
+      map_update_add' := by
+        intro _ x i a b
+        apply SymmetricLift.linearMap_ext
+        intro y
+        simp only [LinearMap.add_apply, hG]
+        exact congrArg (fun T : MultilinearMap ℂ (fun _ : Fin m => W) ℂ => T y)
+          (Q.map_update_add x i a b)
+      map_update_smul' := by
+        intro _ x i c a
+        apply SymmetricLift.linearMap_ext
+        intro y
+        simp only [LinearMap.smul_apply, hG]
+        exact congrArg (fun T : MultilinearMap ℂ (fun _ : Fin m => W) ℂ => T y)
+          (Q.map_update_smul x i c a) }
+  have hF : ∀ (σ : Equiv.Perm (Fin n)) x, F (fun i => x (σ i)) = F x := by
+    intro σ x
+    apply SymmetricLift.linearMap_ext
+    intro y
+    change G (fun i => x (σ i)) (symMonomial m W y) = G x (symMonomial m W y)
+    rw [hG, hG, hleft]
+  obtain ⟨B, hB⟩ := exists_symPow_lift F hF
+  refine ⟨B, ?_⟩
+  intro x y
+  rw [hB]
+  exact hG x y
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/PairingPerfection.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/PairingPerfection.lean
index ff55cd523a953dc33ece8e1a35e59904a3643e9b..8ff2cc8e89b631f54fe3db0ef25e9a0601149f6f 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/PairingPerfection.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/PairingPerfection.lean
@@ -1,47 +1,47 @@
-import OAI.RepresentationTheory.FoulkesHowe.SymmetricDual
-import OAI.RepresentationTheory.FoulkesHowe.FiniteDimension
-
-namespace OAI
-
-noncomputable section
-universe u
-namespace Problem346
-
-variable (V : Type u) [AddCommGroup V] [Module ℂ V]
-
-/-- Evaluation on pure dual powers detects the second variable of a pairing. -/
-theorem symPowDual_flip_injective_of_pure_evaluation (n : ℕ)
-    (B : SymPow n (Module.Dual ℂ V) →ₗ[ℂ] Module.Dual ℂ (SymPow n V))
-    (hB : ∀ (φ : Module.Dual ℂ V) (p : SymPow n V),
-      B (symMonomial n (Module.Dual ℂ V) (fun _ => φ)) p =
-        SymmetricAlgebra.lift φ p.val) :
-    Function.Injective B.flip := by
-  intro p q hpq
-  apply Subtype.ext
-  apply sub_eq_zero.mp
-  apply symAlg_eq_zero_of_lift_eq_zero V
-  intro φ
-  rw [map_sub, ← hB φ p, ← hB φ q]
-  have h := congrArg
-    (fun f : Module.Dual ℂ (SymPow n (Module.Dual ℂ V)) =>
-      f (symMonomial n (Module.Dual ℂ V) (fun _ => φ))) hpq
-  exact sub_eq_zero.mpr h
-
-/-- Pure-power evaluation identities characterize a perfect symmetric-power pairing. -/
-theorem symPowDual_isPerfPair_of_pure_evaluation [FiniteDimensional ℂ V] (n : ℕ)
-    (B : SymPow n (Module.Dual ℂ V) →ₗ[ℂ] Module.Dual ℂ (SymPow n V))
-    (hRight : ∀ (p : SymPow n (Module.Dual ℂ V)) (x : V),
-      B p (symMonomial n V (fun _ => x)) =
-        SymmetricAlgebra.lift (Module.Dual.eval ℂ V x) p.val)
-    (hLeft : ∀ (φ : Module.Dual ℂ V) (p : SymPow n V),
-      B (symMonomial n (Module.Dual ℂ V) (fun _ => φ)) p =
-        SymmetricAlgebra.lift φ p.val) : B.IsPerfPair :=
-  LinearMap.IsPerfPair.of_injective'
-    (symPowDual_injective_of_pure_evaluation V n B hRight)
-    (symPowDual_flip_injective_of_pure_evaluation V n B hLeft)
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.SymmetricDual
+import OAI.RepresentationTheory.FoulkesHowe.FiniteDimension
+
+namespace OAI
+
+noncomputable section
+universe u
+namespace Problem346
+
+variable (V : Type u) [AddCommGroup V] [Module ℂ V]
+
+/-- Evaluation on pure dual powers detects the second variable of a pairing. -/
+theorem symPowDual_flip_injective_of_pure_evaluation (n : ℕ)
+    (B : SymPow n (Module.Dual ℂ V) →ₗ[ℂ] Module.Dual ℂ (SymPow n V))
+    (hB : ∀ (φ : Module.Dual ℂ V) (p : SymPow n V),
+      B (symMonomial n (Module.Dual ℂ V) (fun _ => φ)) p =
+        SymmetricAlgebra.lift φ p.val) :
+    Function.Injective B.flip := by
+  intro p q hpq
+  apply Subtype.ext
+  apply sub_eq_zero.mp
+  apply symAlg_eq_zero_of_lift_eq_zero V
+  intro φ
+  rw [map_sub, ← hB φ p, ← hB φ q]
+  have h := congrArg
+    (fun f : Module.Dual ℂ (SymPow n (Module.Dual ℂ V)) =>
+      f (symMonomial n (Module.Dual ℂ V) (fun _ => φ))) hpq
+  exact sub_eq_zero.mpr h
+
+/-- Pure-power evaluation identities characterize a perfect symmetric-power pairing. -/
+theorem symPowDual_isPerfPair_of_pure_evaluation [FiniteDimensional ℂ V] (n : ℕ)
+    (B : SymPow n (Module.Dual ℂ V) →ₗ[ℂ] Module.Dual ℂ (SymPow n V))
+    (hRight : ∀ (p : SymPow n (Module.Dual ℂ V)) (x : V),
+      B p (symMonomial n V (fun _ => x)) =
+        SymmetricAlgebra.lift (Module.Dual.eval ℂ V x) p.val)
+    (hLeft : ∀ (φ : Module.Dual ℂ V) (p : SymPow n V),
+      B (symMonomial n (Module.Dual ℂ V) (fun _ => φ)) p =
+        SymmetricAlgebra.lift φ p.val) : B.IsPerfPair :=
+  LinearMap.IsPerfPair.of_injective'
+    (symPowDual_injective_of_pure_evaluation V n B hRight)
+    (symPowDual_flip_injective_of_pure_evaluation V n B hLeft)
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/PermanentPairing.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/PermanentPairing.lean
index b82db84e5d3767398296108886999b3b3211d9b1..78f1b1f905bf9756dc18d4326b5ee59711176c9f 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/PermanentPairing.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/PermanentPairing.lean
@@ -1,167 +1,167 @@
-import Mathlib
-
-namespace OAI
-
-noncomputable section
-
-open scoped BigOperators
-
-namespace Problem346
-
-universe u v w z
-
-/-- The normalized permanent of a pairing on two indexed families. -/
-def permanentPairing {X : Type u} {Y : Type v} (n : ℕ)
-    (B : X → Y → ℂ) (x : Fin n → X) (y : Fin n → Y) : ℂ :=
-  (n.factorial : ℂ)⁻¹ * ∑ σ : Equiv.Perm (Fin n), ∏ i, B (x i) (y (σ i))
-
-theorem permanentPairing_perm_right {X : Type u} {Y : Type v} (n : ℕ)
-    (B : X → Y → ℂ) (x : Fin n → X) (y : Fin n → Y)
-    (τ : Equiv.Perm (Fin n)) :
-    permanentPairing n B x (fun i => y (τ i)) = permanentPairing n B x y := by
-  classical
-  unfold permanentPairing
-  congr 1
-  simpa only [Equiv.coe_mulLeft, Equiv.Perm.mul_apply] using
-    (Equiv.sum_comp (Equiv.mulLeft τ)
-      (fun σ : Equiv.Perm (Fin n) => ∏ i, B (x i) (y (σ i))))
-
-theorem permanentPairing_perm_left {X : Type u} {Y : Type v} (n : ℕ)
-    (B : X → Y → ℂ) (x : Fin n → X) (y : Fin n → Y)
-    (τ : Equiv.Perm (Fin n)) :
-    permanentPairing n B (fun i => x (τ i)) y = permanentPairing n B x y := by
-  classical
-  unfold permanentPairing
-  congr 1
-  calc
-    (∑ σ : Equiv.Perm (Fin n), ∏ i, B (x (τ i)) (y (σ i))) =
-        ∑ σ : Equiv.Perm (Fin n), ∏ i, B (x i) (y ((σ * τ⁻¹) i)) := by
-      apply Finset.sum_congr rfl
-      intro σ _
-      simpa only [Equiv.Perm.mul_apply, Equiv.Perm.inv_def, Equiv.symm_apply_apply] using
-        (Equiv.prod_comp τ (fun i => B (x i) (y ((σ * τ⁻¹) i))))
-    _ = ∑ σ : Equiv.Perm (Fin n), ∏ i, B (x i) (y (σ i)) := by
-      simpa only [Equiv.coe_mulRight] using
-        (Equiv.sum_comp (Equiv.mulRight τ⁻¹)
-          (fun σ : Equiv.Perm (Fin n) => ∏ i, B (x i) (y (σ i))))
-
-/-- Transposing the underlying pairing transposes the two families. -/
-theorem permanentPairing_flip {X : Type u} {Y : Type v} (n : ℕ)
-    (B : X → Y → ℂ) (x : Fin n → X) (y : Fin n → Y) :
-    permanentPairing n (fun y x => B x y) y x = permanentPairing n B x y := by
-  classical
-  unfold permanentPairing
-  congr 1
-  calc
-    (∑ σ : Equiv.Perm (Fin n), ∏ i, B (x (σ i)) (y i)) =
-        ∑ σ : Equiv.Perm (Fin n), ∏ i, B (x i) (y (σ⁻¹ i)) := by
-      apply Finset.sum_congr rfl
-      intro σ _
-      simpa only [Equiv.Perm.inv_def, Equiv.symm_apply_apply] using
-        (Equiv.prod_comp σ (fun i => B (x i) (y (σ⁻¹ i))))
-    _ = ∑ σ : Equiv.Perm (Fin n), ∏ i, B (x i) (y (σ i)) := by
-      exact Equiv.sum_comp (Equiv.inv (Equiv.Perm (Fin n)))
-        (fun σ : Equiv.Perm (Fin n) => ∏ i, B (x i) (y (σ i)))
-
-@[simp] theorem permanentPairing_const_right {X : Type u} {Y : Type v} (n : ℕ)
-    (B : X → Y → ℂ) (x : Fin n → X) (y : Y) :
-    permanentPairing n B x (fun _ => y) = ∏ i, B (x i) y := by
-  classical
-  have hn : (n.factorial : ℂ) ≠ 0 := by exact_mod_cast Nat.factorial_ne_zero n
-  simp [permanentPairing, Fintype.card_perm, hn]
-
-@[simp] theorem permanentPairing_const_left {X : Type u} {Y : Type v} (n : ℕ)
-    (B : X → Y → ℂ) (x : X) (y : Fin n → Y) :
-    permanentPairing n B (fun _ => x) y = ∏ i, B x (y i) := by
-  rw [← permanentPairing_flip n B]
-  exact permanentPairing_const_right n (fun y x => B x y) y x
-
-/-- Moving a map across each scalar pairing moves it across the normalized permanent. -/
-theorem permanentPairing_natural {X : Type u} {Y : Type v}
-    {X' : Type w} {Y' : Type z} (n : ℕ)
-    (B : X → Y → ℂ) (B' : X' → Y' → ℂ)
-    (f : X → X') (g : Y' → Y)
-    (h : ∀ x y, B' (f x) y = B x (g y))
-    (x : Fin n → X) (y : Fin n → Y') :
-    permanentPairing n B' (fun i => f (x i)) y =
-      permanentPairing n B x (fun i => g (y i)) := by
-  classical
-  simp only [permanentPairing, h]
-
-section Multilinear
-
-variable {X : Type u} {Y : Type v}
-  [AddCommGroup X] [Module ℂ X] [AddCommGroup Y] [Module ℂ Y]
-
-/-- The permanent pairing as a multilinear map in its right family. -/
-def permanentPairingRight (n : ℕ) (B : X →ₗ[ℂ] Y →ₗ[ℂ] ℂ)
-    (x : Fin n → X) : MultilinearMap ℂ (fun _ : Fin n => Y) ℂ :=
-  (n.factorial : ℂ)⁻¹ •
-    ∑ σ : Equiv.Perm (Fin n),
-      ((MultilinearMap.mkPiAlgebra ℂ (Fin n) ℂ).compLinearMap (fun i => B (x i))).domDomCongr σ
-
-@[simp] theorem permanentPairingRight_apply (n : ℕ)
-    (B : X →ₗ[ℂ] Y →ₗ[ℂ] ℂ) (x : Fin n → X) (y : Fin n → Y) :
-    permanentPairingRight n B x y = permanentPairing n (fun x y => B x y) x y := by
-  classical
-  simp [permanentPairingRight, permanentPairing]
-
-/-- The permanent pairing as a multilinear map in its left family. -/
-def permanentPairingLeft (n : ℕ) (B : X →ₗ[ℂ] Y →ₗ[ℂ] ℂ)
-    (y : Fin n → Y) : MultilinearMap ℂ (fun _ : Fin n => X) ℂ :=
-  (n.factorial : ℂ)⁻¹ •
-    ∑ σ : Equiv.Perm (Fin n),
-      (MultilinearMap.mkPiAlgebra ℂ (Fin n) ℂ).compLinearMap
-        (fun i => B.flip (y (σ i)))
-
-@[simp] theorem permanentPairingLeft_apply (n : ℕ)
-    (B : X →ₗ[ℂ] Y →ₗ[ℂ] ℂ) (x : Fin n → X) (y : Fin n → Y) :
-    permanentPairingLeft n B y x = permanentPairing n (fun x y => B x y) x y := by
-  classical
-  simp [permanentPairingLeft, permanentPairing]
-
-/-- Both families of the permanent pairing are multilinear. -/
-def permanentPairingMultilinear (n : ℕ) (B : X →ₗ[ℂ] Y →ₗ[ℂ] ℂ) :
-    MultilinearMap ℂ (fun _ : Fin n => X)
-      (MultilinearMap ℂ (fun _ : Fin n => Y) ℂ) where
-  toFun := permanentPairingRight n B
-  map_update_add' x i a b := by
-    ext y
-    simpa only [add_apply, permanentPairingRight_apply,
-      ← permanentPairingLeft_apply] using
-      (permanentPairingLeft n B y).map_update_add x i a b
-  map_update_smul' x i c a := by
-    ext y
-    simpa only [smul_apply, permanentPairingRight_apply,
-      ← permanentPairingLeft_apply] using
-      (permanentPairingLeft n B y).map_update_smul x i c a
-
-@[simp] theorem permanentPairingMultilinear_apply (n : ℕ)
-    (B : X →ₗ[ℂ] Y →ₗ[ℂ] ℂ) (x : Fin n → X) (y : Fin n → Y) :
-    permanentPairingMultilinear n B x y =
-      permanentPairing n (fun x y => B x y) x y :=
-  permanentPairingRight_apply n B x y
-
-theorem permanentPairingMultilinear_perm_left (n : ℕ)
-    (B : X →ₗ[ℂ] Y →ₗ[ℂ] ℂ) (x : Fin n → X) (τ : Equiv.Perm (Fin n)) :
-    permanentPairingMultilinear n B (fun i => x (τ i)) =
-      permanentPairingMultilinear n B x := by
-  ext y
-  simpa only [permanentPairingMultilinear_apply] using
-    permanentPairing_perm_left n (fun x y => B x y) x y τ
-
-theorem permanentPairingMultilinear_perm_right (n : ℕ)
-    (B : X →ₗ[ℂ] Y →ₗ[ℂ] ℂ) (x : Fin n → X) (y : Fin n → Y)
-    (τ : Equiv.Perm (Fin n)) :
-    permanentPairingMultilinear n B x (fun i => y (τ i)) =
-      permanentPairingMultilinear n B x y := by
-  simpa only [permanentPairingMultilinear_apply] using
-    permanentPairing_perm_right n (fun x y => B x y) x y τ
-
-end Multilinear
-
-end Problem346
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+
+open scoped BigOperators
+
+namespace Problem346
+
+universe u v w z
+
+/-- The normalized permanent of a pairing on two indexed families. -/
+def permanentPairing {X : Type u} {Y : Type v} (n : ℕ)
+    (B : X → Y → ℂ) (x : Fin n → X) (y : Fin n → Y) : ℂ :=
+  (n.factorial : ℂ)⁻¹ * ∑ σ : Equiv.Perm (Fin n), ∏ i, B (x i) (y (σ i))
+
+theorem permanentPairing_perm_right {X : Type u} {Y : Type v} (n : ℕ)
+    (B : X → Y → ℂ) (x : Fin n → X) (y : Fin n → Y)
+    (τ : Equiv.Perm (Fin n)) :
+    permanentPairing n B x (fun i => y (τ i)) = permanentPairing n B x y := by
+  classical
+  unfold permanentPairing
+  congr 1
+  simpa only [Equiv.coe_mulLeft, Equiv.Perm.mul_apply] using
+    (Equiv.sum_comp (Equiv.mulLeft τ)
+      (fun σ : Equiv.Perm (Fin n) => ∏ i, B (x i) (y (σ i))))
+
+theorem permanentPairing_perm_left {X : Type u} {Y : Type v} (n : ℕ)
+    (B : X → Y → ℂ) (x : Fin n → X) (y : Fin n → Y)
+    (τ : Equiv.Perm (Fin n)) :
+    permanentPairing n B (fun i => x (τ i)) y = permanentPairing n B x y := by
+  classical
+  unfold permanentPairing
+  congr 1
+  calc
+    (∑ σ : Equiv.Perm (Fin n), ∏ i, B (x (τ i)) (y (σ i))) =
+        ∑ σ : Equiv.Perm (Fin n), ∏ i, B (x i) (y ((σ * τ⁻¹) i)) := by
+      apply Finset.sum_congr rfl
+      intro σ _
+      simpa only [Equiv.Perm.mul_apply, Equiv.Perm.inv_def, Equiv.symm_apply_apply] using
+        (Equiv.prod_comp τ (fun i => B (x i) (y ((σ * τ⁻¹) i))))
+    _ = ∑ σ : Equiv.Perm (Fin n), ∏ i, B (x i) (y (σ i)) := by
+      simpa only [Equiv.coe_mulRight] using
+        (Equiv.sum_comp (Equiv.mulRight τ⁻¹)
+          (fun σ : Equiv.Perm (Fin n) => ∏ i, B (x i) (y (σ i))))
+
+/-- Transposing the underlying pairing transposes the two families. -/
+theorem permanentPairing_flip {X : Type u} {Y : Type v} (n : ℕ)
+    (B : X → Y → ℂ) (x : Fin n → X) (y : Fin n → Y) :
+    permanentPairing n (fun y x => B x y) y x = permanentPairing n B x y := by
+  classical
+  unfold permanentPairing
+  congr 1
+  calc
+    (∑ σ : Equiv.Perm (Fin n), ∏ i, B (x (σ i)) (y i)) =
+        ∑ σ : Equiv.Perm (Fin n), ∏ i, B (x i) (y (σ⁻¹ i)) := by
+      apply Finset.sum_congr rfl
+      intro σ _
+      simpa only [Equiv.Perm.inv_def, Equiv.symm_apply_apply] using
+        (Equiv.prod_comp σ (fun i => B (x i) (y (σ⁻¹ i))))
+    _ = ∑ σ : Equiv.Perm (Fin n), ∏ i, B (x i) (y (σ i)) := by
+      exact Equiv.sum_comp (Equiv.inv (Equiv.Perm (Fin n)))
+        (fun σ : Equiv.Perm (Fin n) => ∏ i, B (x i) (y (σ i)))
+
+@[simp] theorem permanentPairing_const_right {X : Type u} {Y : Type v} (n : ℕ)
+    (B : X → Y → ℂ) (x : Fin n → X) (y : Y) :
+    permanentPairing n B x (fun _ => y) = ∏ i, B (x i) y := by
+  classical
+  have hn : (n.factorial : ℂ) ≠ 0 := by exact_mod_cast Nat.factorial_ne_zero n
+  simp [permanentPairing, Fintype.card_perm, hn]
+
+@[simp] theorem permanentPairing_const_left {X : Type u} {Y : Type v} (n : ℕ)
+    (B : X → Y → ℂ) (x : X) (y : Fin n → Y) :
+    permanentPairing n B (fun _ => x) y = ∏ i, B x (y i) := by
+  rw [← permanentPairing_flip n B]
+  exact permanentPairing_const_right n (fun y x => B x y) y x
+
+/-- Moving a map across each scalar pairing moves it across the normalized permanent. -/
+theorem permanentPairing_natural {X : Type u} {Y : Type v}
+    {X' : Type w} {Y' : Type z} (n : ℕ)
+    (B : X → Y → ℂ) (B' : X' → Y' → ℂ)
+    (f : X → X') (g : Y' → Y)
+    (h : ∀ x y, B' (f x) y = B x (g y))
+    (x : Fin n → X) (y : Fin n → Y') :
+    permanentPairing n B' (fun i => f (x i)) y =
+      permanentPairing n B x (fun i => g (y i)) := by
+  classical
+  simp only [permanentPairing, h]
+
+section Multilinear
+
+variable {X : Type u} {Y : Type v}
+  [AddCommGroup X] [Module ℂ X] [AddCommGroup Y] [Module ℂ Y]
+
+/-- The permanent pairing as a multilinear map in its right family. -/
+def permanentPairingRight (n : ℕ) (B : X →ₗ[ℂ] Y →ₗ[ℂ] ℂ)
+    (x : Fin n → X) : MultilinearMap ℂ (fun _ : Fin n => Y) ℂ :=
+  (n.factorial : ℂ)⁻¹ •
+    ∑ σ : Equiv.Perm (Fin n),
+      ((MultilinearMap.mkPiAlgebra ℂ (Fin n) ℂ).compLinearMap (fun i => B (x i))).domDomCongr σ
+
+@[simp] theorem permanentPairingRight_apply (n : ℕ)
+    (B : X →ₗ[ℂ] Y →ₗ[ℂ] ℂ) (x : Fin n → X) (y : Fin n → Y) :
+    permanentPairingRight n B x y = permanentPairing n (fun x y => B x y) x y := by
+  classical
+  simp [permanentPairingRight, permanentPairing]
+
+/-- The permanent pairing as a multilinear map in its left family. -/
+def permanentPairingLeft (n : ℕ) (B : X →ₗ[ℂ] Y →ₗ[ℂ] ℂ)
+    (y : Fin n → Y) : MultilinearMap ℂ (fun _ : Fin n => X) ℂ :=
+  (n.factorial : ℂ)⁻¹ •
+    ∑ σ : Equiv.Perm (Fin n),
+      (MultilinearMap.mkPiAlgebra ℂ (Fin n) ℂ).compLinearMap
+        (fun i => B.flip (y (σ i)))
+
+@[simp] theorem permanentPairingLeft_apply (n : ℕ)
+    (B : X →ₗ[ℂ] Y →ₗ[ℂ] ℂ) (x : Fin n → X) (y : Fin n → Y) :
+    permanentPairingLeft n B y x = permanentPairing n (fun x y => B x y) x y := by
+  classical
+  simp [permanentPairingLeft, permanentPairing]
+
+/-- Both families of the permanent pairing are multilinear. -/
+def permanentPairingMultilinear (n : ℕ) (B : X →ₗ[ℂ] Y →ₗ[ℂ] ℂ) :
+    MultilinearMap ℂ (fun _ : Fin n => X)
+      (MultilinearMap ℂ (fun _ : Fin n => Y) ℂ) where
+  toFun := permanentPairingRight n B
+  map_update_add' x i a b := by
+    ext y
+    simpa only [add_apply, permanentPairingRight_apply,
+      ← permanentPairingLeft_apply] using
+      (permanentPairingLeft n B y).map_update_add x i a b
+  map_update_smul' x i c a := by
+    ext y
+    simpa only [smul_apply, permanentPairingRight_apply,
+      ← permanentPairingLeft_apply] using
+      (permanentPairingLeft n B y).map_update_smul x i c a
+
+@[simp] theorem permanentPairingMultilinear_apply (n : ℕ)
+    (B : X →ₗ[ℂ] Y →ₗ[ℂ] ℂ) (x : Fin n → X) (y : Fin n → Y) :
+    permanentPairingMultilinear n B x y =
+      permanentPairing n (fun x y => B x y) x y :=
+  permanentPairingRight_apply n B x y
+
+theorem permanentPairingMultilinear_perm_left (n : ℕ)
+    (B : X →ₗ[ℂ] Y →ₗ[ℂ] ℂ) (x : Fin n → X) (τ : Equiv.Perm (Fin n)) :
+    permanentPairingMultilinear n B (fun i => x (τ i)) =
+      permanentPairingMultilinear n B x := by
+  ext y
+  simpa only [permanentPairingMultilinear_apply] using
+    permanentPairing_perm_left n (fun x y => B x y) x y τ
+
+theorem permanentPairingMultilinear_perm_right (n : ℕ)
+    (B : X →ₗ[ℂ] Y →ₗ[ℂ] ℂ) (x : Fin n → X) (y : Fin n → Y)
+    (τ : Equiv.Perm (Fin n)) :
+    permanentPairingMultilinear n B x (fun i => y (τ i)) =
+      permanentPairingMultilinear n B x y := by
+  simpa only [permanentPairingMultilinear_apply] using
+    permanentPairing_perm_right n (fun x y => B x y) x y τ
+
+end Multilinear
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/PlethysmDual.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/PlethysmDual.lean
index c6d36a6219bfcd1e0f1f951d0f2ceeded8ebb6d5..f08d574c4904b0df0e30269d0e5b78eef34d32be 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/PlethysmDual.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/PlethysmDual.lean
@@ -1,148 +1,148 @@
-import OAI.RepresentationTheory.FoulkesHowe.Naturality
-import OAI.RepresentationTheory.FoulkesHowe.FiniteDimension
-import OAI.RepresentationTheory.FoulkesHowe.DualEmbedding
-
-namespace OAI
-
-/-!
-# Lifting perfect duality through a symmetric power
-
-Composition of perfect pairings is separate from construction of the one-level pairing.  Supplying the one-level duality
-charts and their naturality gives the duality and adjoint-action identities for
-nested symmetric powers.
--/
-
-noncomputable section
-
-namespace Problem346
-namespace PlethysmDual
-
-universe u v
-variable {V : Type u} {W : Type v}
-  [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
-
-/-- Lift a perfect-duality chart through a symmetric power using the canonical
-    duality chart for that symmetric power. -/
-def liftDualEquiv (n : ℕ) (e : W ≃ₗ[ℂ] Module.Dual ℂ V)
-    (d : SymPow n (Module.Dual ℂ V) ≃ₗ[ℂ] Module.Dual ℂ (SymPow n V)) :
-    SymPow n W ≃ₗ[ℂ] Module.Dual ℂ (SymPow n V) :=
-  (symPowEquiv n e).trans d
-
-/-- Adjointness is preserved when a perfect pairing is lifted through a
-    symmetric power whose duality chart is natural for the given map. -/
-theorem liftDualEquiv_adjoint (n : ℕ) (e : W ≃ₗ[ℂ] Module.Dual ℂ V)
-    (d : SymPow n (Module.Dual ℂ V) ≃ₗ[ℂ] Module.Dual ℂ (SymPow n V))
-    (A : V →ₗ[ℂ] V) (A' : W →ₗ[ℂ] W)
-    (he : ∀ x y, e (A' x) y = e x (A y))
-    (hd : ∀ x y, d (symPowMap n A.dualMap x) y = d x (symPowMap n A y))
-    (x : SymPow n W) (y : SymPow n V) :
-    liftDualEquiv n e d (symPowMap n A' x) y =
-      liftDualEquiv n e d x (symPowMap n A y) := by
-  have heq : e.toLinearMap.comp A' = A.dualMap.comp e.toLinearMap := by
-    ext x y
-    exact he x y
-  have hlift : (symPowMap n e.toLinearMap).comp (symPowMap n A') =
-      (symPowMap n A.dualMap).comp (symPowMap n e.toLinearMap) := by
-    rw [← symPowMap_comp, ← symPowMap_comp, heq]
-  change d (symPowMap n e.toLinearMap (symPowMap n A' x)) y =
-    d (symPowMap n e.toLinearMap x) (symPowMap n A y)
-  have hpoint := LinearMap.congr_fun hlift x
-  simp only [LinearMap.comp_apply] at hpoint
-  rw [hpoint]
-  exact hd _ _
-
-/-- The duality chart for a plethysm is the composite of its inner and outer
-    one-level duality charts. -/
-def plethysmDualEquivOf (outer inner : ℕ)
-    (e : SymPow inner (Module.Dual ℂ V) ≃ₗ[ℂ] Module.Dual ℂ (SymPow inner V))
-    (d : SymPow outer (Module.Dual ℂ (SymPow inner V)) ≃ₗ[ℂ]
-      Module.Dual ℂ (SymPow outer (SymPow inner V))) :
-    SymPow outer (SymPow inner (Module.Dual ℂ V)) ≃ₗ[ℂ]
-      Module.Dual ℂ (SymPow outer (SymPow inner V)) :=
-  liftDualEquiv outer e d
-
-/-- Naturality of each level gives the adjoint-action identity for the
-    plethysm pairing.  The dual automorphism here is the ordinary dual map,
-    not the inverse contragredient; this is precisely the adjoint identity. -/
-theorem plethysmDualEquivOf_adjoint (outer inner : ℕ)
-    (e : SymPow inner (Module.Dual ℂ V) ≃ₗ[ℂ] Module.Dual ℂ (SymPow inner V))
-    (d : SymPow outer (Module.Dual ℂ (SymPow inner V)) ≃ₗ[ℂ]
-      Module.Dual ℂ (SymPow outer (SymPow inner V)))
-    (g : V ≃ₗ[ℂ] V)
-    (he : ∀ x y, e (symPowMap inner g.dualMap.toLinearMap x) y =
-      e x (symPowMap inner g.toLinearMap y))
-    (hd : ∀ x y,
-      d (symPowMap outer (symPowMap inner g.toLinearMap).dualMap x) y =
-      d x (symPowMap outer (symPowMap inner g.toLinearMap) y))
-    (x : SymPow outer (SymPow inner (Module.Dual ℂ V)))
-    (y : SymPow outer (SymPow inner V)) :
-    plethysmDualEquivOf outer inner e d
-      (plethysmAction outer inner g.dualMap x) y =
-    plethysmDualEquivOf outer inner e d x (plethysmAction outer inner g y) := by
-  exact liftDualEquiv_adjoint outer e d
-    (symPowMap inner g.toLinearMap) (symPowMap inner g.dualMap.toLinearMap)
-    he hd x y
-
-/-- The comparison embedding follows formally from a surjection on the dual
-    vector space and compatible perfect pairings on the two plethysms. -/
-theorem exists_embedding_of_dual_surjection (a b : ℕ)
-    [Module.IsReflexive ℂ (SymPow b (SymPow a V))]
-    [Module.IsReflexive ℂ (SymPow a (SymPow b V))]
-    (e : SymPow b (SymPow a (Module.Dual ℂ V)) ≃ₗ[ℂ]
-      Module.Dual ℂ (SymPow b (SymPow a V)))
-    (f : SymPow a (SymPow b (Module.Dual ℂ V)) ≃ₗ[ℂ]
-      Module.Dual ℂ (SymPow a (SymPow b V)))
-    (he : ∀ (g : V ≃ₗ[ℂ] V) x y,
-      e (plethysmAction b a g.dualMap x) y = e x (plethysmAction b a g y))
-    (hf : ∀ (g : V ≃ₗ[ℂ] V) x y,
-      f (plethysmAction a b g.dualMap x) y = f x (plethysmAction a b g y))
-    (μ : SymPow b (SymPow a (Module.Dual ℂ V)) →ₗ[ℂ]
-      SymPow a (SymPow b (Module.Dual ℂ V)))
-    (hμ : Function.Surjective μ)
-    (hcomm : ∀ g : V ≃ₗ[ℂ] V,
-      μ.comp (plethysmAction b a g.dualMap) =
-        (plethysmAction a b g.dualMap).comp μ) :
-    ∃ ι : SymPow a (SymPow b V) →ₗ[ℂ] SymPow b (SymPow a V),
-      IsGLEquivariantEmbedding a b V ι := by
-  let e' := DualEmbedding.flipPairing e
-  let f' := DualEmbedding.flipPairing f
-  let ι := DualEmbedding.transpose e' f' μ
-  refine ⟨ι, isGLEquivariantEmbedding_of_commutes ι
-    (DualEmbedding.transpose_injective e' f' μ hμ) ?_⟩
-  intro g
-  apply DualEmbedding.transpose_intertwines e' f' μ
-    (plethysmAction b a g) (plethysmAction b a g.dualMap)
-    (plethysmAction a b g) (plethysmAction a b g.dualMap)
-  · intro x z
-    exact (he g z x).symm
-  · intro y w
-    exact (hf g w y).symm
-  · exact hcomm g
-
-/-- With finite-dimensionality and canonical-map naturality available, only the
-    two perfect-duality charts and their adjointness remain to construct the
-    comparison embedding from canonical surjectivity on the dual space. -/
-theorem comparison_of_dual_charts (a b : ℕ) [FiniteDimensional ℂ V]
-    (e : SymPow b (SymPow a (Module.Dual ℂ V)) ≃ₗ[ℂ]
-      Module.Dual ℂ (SymPow b (SymPow a V)))
-    (f : SymPow a (SymPow b (Module.Dual ℂ V)) ≃ₗ[ℂ]
-      Module.Dual ℂ (SymPow a (SymPow b V)))
-    (he : ∀ (g : V ≃ₗ[ℂ] V) x y,
-      e (plethysmAction b a g.dualMap x) y = e x (plethysmAction b a g y))
-    (hf : ∀ (g : V ≃ₗ[ℂ] V) x y,
-      f (plethysmAction a b g.dualMap x) y = f x (plethysmAction a b g y))
-    (hsurj : ∃ μ : SymPow b (SymPow a (Module.Dual ℂ V)) →ₗ[ℂ]
-      SymPow a (SymPow b (Module.Dual ℂ V)),
-      IsFoulkesMap a b (Module.Dual ℂ V) μ ∧ Function.Surjective μ) :
-    ∃ ι : SymPow a (SymPow b V) →ₗ[ℂ] SymPow b (SymPow a V),
-      IsGLEquivariantEmbedding a b V ι := by
-  obtain ⟨μ, hformula, hsurj⟩ := hsurj
-  exact exists_embedding_of_dual_surjection a b e f he hf μ hsurj
-    (fun g => (hformula.commutes g.dualMap).symm)
-
-end PlethysmDual
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.Naturality
+import OAI.RepresentationTheory.FoulkesHowe.FiniteDimension
+import OAI.RepresentationTheory.FoulkesHowe.DualEmbedding
+
+namespace OAI
+
+/-!
+# Lifting perfect duality through a symmetric power
+
+Composition of perfect pairings is separate from construction of the one-level pairing.  Supplying the one-level duality
+charts and their naturality gives the duality and adjoint-action identities for
+nested symmetric powers.
+-/
+
+noncomputable section
+
+namespace Problem346
+namespace PlethysmDual
+
+universe u v
+variable {V : Type u} {W : Type v}
+  [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
+
+/-- Lift a perfect-duality chart through a symmetric power using the canonical
+    duality chart for that symmetric power. -/
+def liftDualEquiv (n : ℕ) (e : W ≃ₗ[ℂ] Module.Dual ℂ V)
+    (d : SymPow n (Module.Dual ℂ V) ≃ₗ[ℂ] Module.Dual ℂ (SymPow n V)) :
+    SymPow n W ≃ₗ[ℂ] Module.Dual ℂ (SymPow n V) :=
+  (symPowEquiv n e).trans d
+
+/-- Adjointness is preserved when a perfect pairing is lifted through a
+    symmetric power whose duality chart is natural for the given map. -/
+theorem liftDualEquiv_adjoint (n : ℕ) (e : W ≃ₗ[ℂ] Module.Dual ℂ V)
+    (d : SymPow n (Module.Dual ℂ V) ≃ₗ[ℂ] Module.Dual ℂ (SymPow n V))
+    (A : V →ₗ[ℂ] V) (A' : W →ₗ[ℂ] W)
+    (he : ∀ x y, e (A' x) y = e x (A y))
+    (hd : ∀ x y, d (symPowMap n A.dualMap x) y = d x (symPowMap n A y))
+    (x : SymPow n W) (y : SymPow n V) :
+    liftDualEquiv n e d (symPowMap n A' x) y =
+      liftDualEquiv n e d x (symPowMap n A y) := by
+  have heq : e.toLinearMap.comp A' = A.dualMap.comp e.toLinearMap := by
+    ext x y
+    exact he x y
+  have hlift : (symPowMap n e.toLinearMap).comp (symPowMap n A') =
+      (symPowMap n A.dualMap).comp (symPowMap n e.toLinearMap) := by
+    rw [← symPowMap_comp, ← symPowMap_comp, heq]
+  change d (symPowMap n e.toLinearMap (symPowMap n A' x)) y =
+    d (symPowMap n e.toLinearMap x) (symPowMap n A y)
+  have hpoint := LinearMap.congr_fun hlift x
+  simp only [LinearMap.comp_apply] at hpoint
+  rw [hpoint]
+  exact hd _ _
+
+/-- The duality chart for a plethysm is the composite of its inner and outer
+    one-level duality charts. -/
+def plethysmDualEquivOf (outer inner : ℕ)
+    (e : SymPow inner (Module.Dual ℂ V) ≃ₗ[ℂ] Module.Dual ℂ (SymPow inner V))
+    (d : SymPow outer (Module.Dual ℂ (SymPow inner V)) ≃ₗ[ℂ]
+      Module.Dual ℂ (SymPow outer (SymPow inner V))) :
+    SymPow outer (SymPow inner (Module.Dual ℂ V)) ≃ₗ[ℂ]
+      Module.Dual ℂ (SymPow outer (SymPow inner V)) :=
+  liftDualEquiv outer e d
+
+/-- Naturality of each level gives the adjoint-action identity for the
+    plethysm pairing.  The dual automorphism here is the ordinary dual map,
+    not the inverse contragredient; this is precisely the adjoint identity. -/
+theorem plethysmDualEquivOf_adjoint (outer inner : ℕ)
+    (e : SymPow inner (Module.Dual ℂ V) ≃ₗ[ℂ] Module.Dual ℂ (SymPow inner V))
+    (d : SymPow outer (Module.Dual ℂ (SymPow inner V)) ≃ₗ[ℂ]
+      Module.Dual ℂ (SymPow outer (SymPow inner V)))
+    (g : V ≃ₗ[ℂ] V)
+    (he : ∀ x y, e (symPowMap inner g.dualMap.toLinearMap x) y =
+      e x (symPowMap inner g.toLinearMap y))
+    (hd : ∀ x y,
+      d (symPowMap outer (symPowMap inner g.toLinearMap).dualMap x) y =
+      d x (symPowMap outer (symPowMap inner g.toLinearMap) y))
+    (x : SymPow outer (SymPow inner (Module.Dual ℂ V)))
+    (y : SymPow outer (SymPow inner V)) :
+    plethysmDualEquivOf outer inner e d
+      (plethysmAction outer inner g.dualMap x) y =
+    plethysmDualEquivOf outer inner e d x (plethysmAction outer inner g y) := by
+  exact liftDualEquiv_adjoint outer e d
+    (symPowMap inner g.toLinearMap) (symPowMap inner g.dualMap.toLinearMap)
+    he hd x y
+
+/-- The comparison embedding follows formally from a surjection on the dual
+    vector space and compatible perfect pairings on the two plethysms. -/
+theorem exists_embedding_of_dual_surjection (a b : ℕ)
+    [Module.IsReflexive ℂ (SymPow b (SymPow a V))]
+    [Module.IsReflexive ℂ (SymPow a (SymPow b V))]
+    (e : SymPow b (SymPow a (Module.Dual ℂ V)) ≃ₗ[ℂ]
+      Module.Dual ℂ (SymPow b (SymPow a V)))
+    (f : SymPow a (SymPow b (Module.Dual ℂ V)) ≃ₗ[ℂ]
+      Module.Dual ℂ (SymPow a (SymPow b V)))
+    (he : ∀ (g : V ≃ₗ[ℂ] V) x y,
+      e (plethysmAction b a g.dualMap x) y = e x (plethysmAction b a g y))
+    (hf : ∀ (g : V ≃ₗ[ℂ] V) x y,
+      f (plethysmAction a b g.dualMap x) y = f x (plethysmAction a b g y))
+    (μ : SymPow b (SymPow a (Module.Dual ℂ V)) →ₗ[ℂ]
+      SymPow a (SymPow b (Module.Dual ℂ V)))
+    (hμ : Function.Surjective μ)
+    (hcomm : ∀ g : V ≃ₗ[ℂ] V,
+      μ.comp (plethysmAction b a g.dualMap) =
+        (plethysmAction a b g.dualMap).comp μ) :
+    ∃ ι : SymPow a (SymPow b V) →ₗ[ℂ] SymPow b (SymPow a V),
+      IsGLEquivariantEmbedding a b V ι := by
+  let e' := DualEmbedding.flipPairing e
+  let f' := DualEmbedding.flipPairing f
+  let ι := DualEmbedding.transpose e' f' μ
+  refine ⟨ι, isGLEquivariantEmbedding_of_commutes ι
+    (DualEmbedding.transpose_injective e' f' μ hμ) ?_⟩
+  intro g
+  apply DualEmbedding.transpose_intertwines e' f' μ
+    (plethysmAction b a g) (plethysmAction b a g.dualMap)
+    (plethysmAction a b g) (plethysmAction a b g.dualMap)
+  · intro x z
+    exact (he g z x).symm
+  · intro y w
+    exact (hf g w y).symm
+  · exact hcomm g
+
+/-- With finite-dimensionality and canonical-map naturality available, only the
+    two perfect-duality charts and their adjointness remain to construct the
+    comparison embedding from canonical surjectivity on the dual space. -/
+theorem comparison_of_dual_charts (a b : ℕ) [FiniteDimensional ℂ V]
+    (e : SymPow b (SymPow a (Module.Dual ℂ V)) ≃ₗ[ℂ]
+      Module.Dual ℂ (SymPow b (SymPow a V)))
+    (f : SymPow a (SymPow b (Module.Dual ℂ V)) ≃ₗ[ℂ]
+      Module.Dual ℂ (SymPow a (SymPow b V)))
+    (he : ∀ (g : V ≃ₗ[ℂ] V) x y,
+      e (plethysmAction b a g.dualMap x) y = e x (plethysmAction b a g y))
+    (hf : ∀ (g : V ≃ₗ[ℂ] V) x y,
+      f (plethysmAction a b g.dualMap x) y = f x (plethysmAction a b g y))
+    (hsurj : ∃ μ : SymPow b (SymPow a (Module.Dual ℂ V)) →ₗ[ℂ]
+      SymPow a (SymPow b (Module.Dual ℂ V)),
+      IsFoulkesMap a b (Module.Dual ℂ V) μ ∧ Function.Surjective μ) :
+    ∃ ι : SymPow a (SymPow b V) →ₗ[ℂ] SymPow b (SymPow a V),
+      IsGLEquivariantEmbedding a b V ι := by
+  obtain ⟨μ, hformula, hsurj⟩ := hsurj
+  exact exists_embedding_of_dual_surjection a b e f he hf μ hsurj
+    (fun g => (hformula.commutes g.dualMap).symm)
+
+end PlethysmDual
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/Polarization.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/Polarization.lean
index d39b65da9c33a986ad8a8e34be4168af28b1a139..aad4b7bceedd1d84e5ef2c3ffe1eb7b368a48635 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/Polarization.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/Polarization.lean
@@ -1,46 +1,46 @@
-import Mathlib
-
-namespace OAI
-
-noncomputable section
-
-namespace Problem346
-
-universe u v
-
-/-- A scalar-valued diagonal-detection principle upgrades to a spanning identity
-for the diagonal values of any symmetric multilinear map. -/
-theorem span_diagonal_eq_span_range_of_detection
-    {V : Type u} {W : Type v} [AddCommGroup V] [Module ℂ V]
-    [AddCommGroup W] [Module ℂ W] (n : ℕ)
-    (detect : ∀ T : MultilinearMap ℂ (fun _ : Fin n => V) ℂ,
-      (∀ (σ : Equiv.Perm (Fin n)) (x : Fin n → V), T (fun i => x (σ i)) = T x) →
-      (∀ x : V, T (fun _ => x) = 0) → T = 0)
-    (m : MultilinearMap ℂ (fun _ : Fin n => V) W)
-    (hsym : ∀ (σ : Equiv.Perm (Fin n)) (x : Fin n → V),
-      m (fun i => x (σ i)) = m x) :
-    Submodule.span ℂ (Set.range (fun x : V => m (fun _ => x))) =
-      Submodule.span ℂ (Set.range m) := by
-  apply le_antisymm
-  · apply Submodule.span_mono
-    rintro _ ⟨x, rfl⟩
-    exact ⟨fun _ => x, rfl⟩
-  · apply Submodule.span_le.mpr
-    rintro _ ⟨x, rfl⟩
-    apply (Subspace.forall_mem_dualAnnihilator_apply_eq_zero_iff _ _).mp
-    intro f hf
-    have hz : f.compMultilinearMap m = 0 := by
-      apply detect
-      · intro σ y
-        change f (m (fun i => y (σ i))) = f (m y)
-        rw [hsym]
-      · intro y
-        exact (Submodule.mem_dualAnnihilator f).mp hf _
-          (Submodule.subset_span ⟨y, rfl⟩)
-    exact DFunLike.congr_fun hz x
-
-end Problem346
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+
+namespace Problem346
+
+universe u v
+
+/-- A scalar-valued diagonal-detection principle upgrades to a spanning identity
+for the diagonal values of any symmetric multilinear map. -/
+theorem span_diagonal_eq_span_range_of_detection
+    {V : Type u} {W : Type v} [AddCommGroup V] [Module ℂ V]
+    [AddCommGroup W] [Module ℂ W] (n : ℕ)
+    (detect : ∀ T : MultilinearMap ℂ (fun _ : Fin n => V) ℂ,
+      (∀ (σ : Equiv.Perm (Fin n)) (x : Fin n → V), T (fun i => x (σ i)) = T x) →
+      (∀ x : V, T (fun _ => x) = 0) → T = 0)
+    (m : MultilinearMap ℂ (fun _ : Fin n => V) W)
+    (hsym : ∀ (σ : Equiv.Perm (Fin n)) (x : Fin n → V),
+      m (fun i => x (σ i)) = m x) :
+    Submodule.span ℂ (Set.range (fun x : V => m (fun _ => x))) =
+      Submodule.span ℂ (Set.range m) := by
+  apply le_antisymm
+  · apply Submodule.span_mono
+    rintro _ ⟨x, rfl⟩
+    exact ⟨fun _ => x, rfl⟩
+  · apply Submodule.span_le.mpr
+    rintro _ ⟨x, rfl⟩
+    apply (Subspace.forall_mem_dualAnnihilator_apply_eq_zero_iff _ _).mp
+    intro f hf
+    have hz : f.compMultilinearMap m = 0 := by
+      apply detect
+      · intro σ y
+        change f (m (fun i => y (σ i))) = f (m y)
+        rw [hsym]
+      · intro y
+        exact (Submodule.mem_dualAnnihilator f).mp hf _
+          (Submodule.subset_span ⟨y, rfl⟩)
+    exact DFunLike.congr_fun hz x
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/PolynomialShift.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/PolynomialShift.lean
index bb5e82b522185d47eea10b395961364b8dbea889..d0f7126649159c348b0ab3241c21b0fc3f3df684 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/PolynomialShift.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/PolynomialShift.lean
@@ -1,74 +1,74 @@
-import Mathlib
-
-namespace OAI
-
-noncomputable section
-
-open scoped BigOperators
-
-namespace Problem346
-namespace PolynomialShift
-
-variable {K B I : Type*} [CommRing K]
-
-attribute [local instance] Classical.propDecidable
-
-/-- Transfer one unit of polynomial degree from `src` to `dst`. Other blocks
-are spectator variables. -/
-def derivation (src dst : B) :
-    Derivation K (MvPolynomial (B × I) K) (MvPolynomial (B × I) K) := by
-  classical
-  exact MvPolynomial.mkDerivation K
-    (fun v => if v.1 = src then MvPolynomial.X (dst, v.2) else 0)
-
-@[simp]
-theorem derivation_X (src dst block : B) (i : I) :
-    derivation (K := K) src dst (MvPolynomial.X (block, i)) =
-      if block = src then MvPolynomial.X (dst, i) else 0 := by
-  classical
-  simp [derivation]
-
-/-- The directional shift as a linear endomorphism. -/
-def shift (src dst : B) : Module.End K (MvPolynomial (B × I) K) :=
-  (derivation src dst).toLinearMap
-
-@[simp]
-theorem shift_apply (src dst : B) (p : MvPolynomial (B × I) K) :
-    shift src dst p = derivation src dst p := rfl
-
-/-- Euler's operator for one block. -/
-def euler (block : B) : Module.End K (MvPolynomial (B × I) K) :=
-  shift block block
-
-theorem commutator (src dst : B) (h : src ≠ dst) :
-    ⁅derivation (K := K) (I := I) dst src, derivation (K := K) (I := I) src dst⁆ =
-      derivation (K := K) (I := I) src src - derivation (K := K) (I := I) dst dst := by
-  classical
-  apply MvPolynomial.derivation_ext
-  rintro ⟨block, i⟩
-  by_cases hs : block = src <;> by_cases hd : block = dst <;>
-    simp_all [Derivation.commutator_apply]
-
-theorem commutator_apply (src dst : B) (h : src ≠ dst)
-    (p : MvPolynomial (B × I) K) :
-    shift dst src (shift src dst p) - shift src dst (shift dst src p) =
-      euler src p - euler dst p := by
-  have h' := congrArg (fun d : Derivation K (MvPolynomial (B × I) K)
-    (MvPolynomial (B × I) K) => d p) (commutator (K := K) (I := I) src dst h)
-  simpa [Derivation.commutator_apply, euler] using h'
-
-theorem weight_commutator (src dst : B) (h : src ≠ dst) :
-    ⁅derivation (K := K) (I := I) src src - derivation (K := K) (I := I) dst dst,
-      derivation (K := K) (I := I) dst src⁆ = (2 : K) • derivation (K := K) (I := I) dst src := by
-  classical
-  apply MvPolynomial.derivation_ext
-  rintro ⟨block, i⟩
-  by_cases hs : block = src <;> by_cases hd : block = dst <;>
-    simp_all [Derivation.commutator_apply, two_smul]
-
-end PolynomialShift
-end Problem346
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+
+open scoped BigOperators
+
+namespace Problem346
+namespace PolynomialShift
+
+variable {K B I : Type*} [CommRing K]
+
+attribute [local instance] Classical.propDecidable
+
+/-- Transfer one unit of polynomial degree from `src` to `dst`. Other blocks
+are spectator variables. -/
+def derivation (src dst : B) :
+    Derivation K (MvPolynomial (B × I) K) (MvPolynomial (B × I) K) := by
+  classical
+  exact MvPolynomial.mkDerivation K
+    (fun v => if v.1 = src then MvPolynomial.X (dst, v.2) else 0)
+
+@[simp]
+theorem derivation_X (src dst block : B) (i : I) :
+    derivation (K := K) src dst (MvPolynomial.X (block, i)) =
+      if block = src then MvPolynomial.X (dst, i) else 0 := by
+  classical
+  simp [derivation]
+
+/-- The directional shift as a linear endomorphism. -/
+def shift (src dst : B) : Module.End K (MvPolynomial (B × I) K) :=
+  (derivation src dst).toLinearMap
+
+@[simp]
+theorem shift_apply (src dst : B) (p : MvPolynomial (B × I) K) :
+    shift src dst p = derivation src dst p := rfl
+
+/-- Euler's operator for one block. -/
+def euler (block : B) : Module.End K (MvPolynomial (B × I) K) :=
+  shift block block
+
+theorem commutator (src dst : B) (h : src ≠ dst) :
+    ⁅derivation (K := K) (I := I) dst src, derivation (K := K) (I := I) src dst⁆ =
+      derivation (K := K) (I := I) src src - derivation (K := K) (I := I) dst dst := by
+  classical
+  apply MvPolynomial.derivation_ext
+  rintro ⟨block, i⟩
+  by_cases hs : block = src <;> by_cases hd : block = dst <;>
+    simp_all [Derivation.commutator_apply]
+
+theorem commutator_apply (src dst : B) (h : src ≠ dst)
+    (p : MvPolynomial (B × I) K) :
+    shift dst src (shift src dst p) - shift src dst (shift dst src p) =
+      euler src p - euler dst p := by
+  have h' := congrArg (fun d : Derivation K (MvPolynomial (B × I) K)
+    (MvPolynomial (B × I) K) => d p) (commutator (K := K) (I := I) src dst h)
+  simpa [Derivation.commutator_apply, euler] using h'
+
+theorem weight_commutator (src dst : B) (h : src ≠ dst) :
+    ⁅derivation (K := K) (I := I) src src - derivation (K := K) (I := I) dst dst,
+      derivation (K := K) (I := I) dst src⁆ = (2 : K) • derivation (K := K) (I := I) dst src := by
+  classical
+  apply MvPolynomial.derivation_ext
+  rintro ⟨block, i⟩
+  by_cases hs : block = src <;> by_cases hd : block = dst <;>
+    simp_all [Derivation.commutator_apply, two_smul]
+
+end PolynomialShift
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/PowerSpan.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/PowerSpan.lean
index 846073b42f498a28ff3062dfc80ce9da99d0ed1a..7f61a7ae145a2b3bfd5e0885b4f9fde605d8b0b1 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/PowerSpan.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/PowerSpan.lean
@@ -1,57 +1,57 @@
-import OAI.RepresentationTheory.FoulkesHowe.DiagonalPolarization
-import OAI.RepresentationTheory.FoulkesHowe.MonomialMultilinear
-import OAI.RepresentationTheory.FoulkesHowe.Polarization
-
-namespace OAI
-
-noncomputable section
-
-namespace Problem346
-
-universe u v
-
-/-- Pure powers span each symmetric power, including degree zero. -/
-theorem span_purePowers_eq_top (b : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V] :
-    Submodule.span ℂ (Set.range (fun x : V => symMonomial b V (fun _ => x))) = ⊤ := by
-  apply Submodule.dualAnnihilator_eq_bot_iff.mp
-  apply le_antisymm
-  · intro f hf
-    change f = 0
-    apply linearMap_eq_zero_of_symMonomial b V f
-    have hz : f.compMultilinearMap (symMonomialMultilinear b V) = 0 := by
-      apply symmetric_multilinear_eq_zero_of_diagonal
-      · intro σ x
-        change f (symMonomial b V (fun i => x (σ i))) = f (symMonomial b V x)
-        rw [symMonomial_permute]
-      · intro x
-        exact (Submodule.mem_dualAnnihilator f).mp hf _
-          (Submodule.subset_span ⟨x, rfl⟩)
-    intro x
-    exact DFunLike.congr_fun hz x
-  · exact bot_le
-
-/-- Multilinearity extends vanishing on independent pure powers to vanishing everywhere. -/
-theorem multilinear_eq_zero_of_pure_powers (a b : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V] {W : Type v} [AddCommGroup W] [Module ℂ W]
-    (T : MultilinearMap ℂ (fun _ : Fin a => SymPow b V) W)
-    (hT : ∀ x : Fin a → V, T (fun i => symMonomial b V (fun _ => x i)) = 0) :
-    T = 0 := by
-  apply MultilinearMap.ext_of_span_eq_top
-    (g := fun {_ : Fin a} (x : V) => symMonomial b V (fun _ => x))
-    (fun _ => span_purePowers_eq_top b V)
-  exact hT
-
-/-- Independent pure powers detect a symmetric multilinear form. -/
-theorem symmetricMultilinearForm_eq_zero_of_pure_powers (a b : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V]
-    (T : SymmetricMultilinearForm a b V)
-    (hT : ∀ x : Fin a → V, T (fun i => symMonomial b V (fun _ => x i)) = 0) :
-    T = 0 :=
-  multilinear_eq_zero_of_pure_powers a b V T hT
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.DiagonalPolarization
+import OAI.RepresentationTheory.FoulkesHowe.MonomialMultilinear
+import OAI.RepresentationTheory.FoulkesHowe.Polarization
+
+namespace OAI
+
+noncomputable section
+
+namespace Problem346
+
+universe u v
+
+/-- Pure powers span each symmetric power, including degree zero. -/
+theorem span_purePowers_eq_top (b : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V] :
+    Submodule.span ℂ (Set.range (fun x : V => symMonomial b V (fun _ => x))) = ⊤ := by
+  apply Submodule.dualAnnihilator_eq_bot_iff.mp
+  apply le_antisymm
+  · intro f hf
+    change f = 0
+    apply linearMap_eq_zero_of_symMonomial b V f
+    have hz : f.compMultilinearMap (symMonomialMultilinear b V) = 0 := by
+      apply symmetric_multilinear_eq_zero_of_diagonal
+      · intro σ x
+        change f (symMonomial b V (fun i => x (σ i))) = f (symMonomial b V x)
+        rw [symMonomial_permute]
+      · intro x
+        exact (Submodule.mem_dualAnnihilator f).mp hf _
+          (Submodule.subset_span ⟨x, rfl⟩)
+    intro x
+    exact DFunLike.congr_fun hz x
+  · exact bot_le
+
+/-- Multilinearity extends vanishing on independent pure powers to vanishing everywhere. -/
+theorem multilinear_eq_zero_of_pure_powers (a b : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V] {W : Type v} [AddCommGroup W] [Module ℂ W]
+    (T : MultilinearMap ℂ (fun _ : Fin a => SymPow b V) W)
+    (hT : ∀ x : Fin a → V, T (fun i => symMonomial b V (fun _ => x i)) = 0) :
+    T = 0 := by
+  apply MultilinearMap.ext_of_span_eq_top
+    (g := fun {_ : Fin a} (x : V) => symMonomial b V (fun _ => x))
+    (fun _ => span_purePowers_eq_top b V)
+  exact hT
+
+/-- Independent pure powers detect a symmetric multilinear form. -/
+theorem symmetricMultilinearForm_eq_zero_of_pure_powers (a b : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V]
+    (T : SymmetricMultilinearForm a b V)
+    (hT : ∀ x : Fin a → V, T (fun i => symMonomial b V (fun _ => x i)) = 0) :
+    T = 0 :=
+  multilinear_eq_zero_of_pure_powers a b V T hT
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/ProductVanishing.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/ProductVanishing.lean
index 7d10505a990d5671982d1f5f322dc02d16941367..bb245da0531a4c7c949178dce6369cefab35dd6d 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/ProductVanishing.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/ProductVanishing.lean
@@ -1,28 +1,28 @@
-import OAI.RepresentationTheory.FoulkesHowe.VanishingFirst
-import OAI.RepresentationTheory.FoulkesHowe.SecondTransfer
-
-namespace OAI
-
-noncomputable section
-
-namespace Problem346
-
-universe u
-
-/-- Quadratic stabilization of diagonal vanishing on products. Both differential
-transfers and all of the induction hypotheses are discharged in this theorem. -/
-theorem vanishing_on_products_aux (V : Type u)
-    [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
-    (a b : ℕ) (ha : 1 ≤ a) (hb : 1 ≤ b) (hab : a*(a-1) ≤ b)
-    (T : SymmetricMultilinearForm a b V)
-    (hT : IsSymmetricMultilinearForm a b V T)
-    (hdiag : ∀ y : Fin b → V, T (fun _ => symMonomial b V y) = 0) : T = 0 := by
-  refine vanishing_on_products_of_second_transfer (V := V) ?_ a b ha hb hab T hT hdiag
-  intro r m _ _ hbound S _ hrest
-  exact SecondTransfer.second_transfer r m (by simpa only [Nat.mul_comm] using hbound) S hrest
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.VanishingFirst
+import OAI.RepresentationTheory.FoulkesHowe.SecondTransfer
+
+namespace OAI
+
+noncomputable section
+
+namespace Problem346
+
+universe u
+
+/-- Quadratic stabilization of diagonal vanishing on products. Both differential
+transfers and all of the induction hypotheses are discharged in this theorem. -/
+theorem vanishing_on_products_aux (V : Type u)
+    [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
+    (a b : ℕ) (ha : 1 ≤ a) (hb : 1 ≤ b) (hab : a*(a-1) ≤ b)
+    (T : SymmetricMultilinearForm a b V)
+    (hT : IsSymmetricMultilinearForm a b V T)
+    (hdiag : ∀ y : Fin b → V, T (fun _ => symMonomial b V y) = 0) : T = 0 := by
+  refine vanishing_on_products_of_second_transfer (V := V) ?_ a b ha hb hab T hT hdiag
+  intro r m _ _ hbound S _ hrest
+  exact SecondTransfer.second_transfer r m (by simpa only [Nat.mul_comm] using hbound) S hrest
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/RowSwap.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/RowSwap.lean
index 882754184a5b6197c2f768f436775124ea0ab7b3..f491d51c86d20186762dee417d771c84ea6fcaed 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/RowSwap.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/RowSwap.lean
@@ -1,67 +1,67 @@
-import Mathlib
-
-namespace OAI
-
-/-!
-# From rowwise permutation invariance to individual slot swaps
-
-The flattened plethysm forms are invariant under independent permutations
-inside each row.  The polynomial derivative formulas instead ask for
-invariance under a swap of two individual slots.  These lemmas provide that
-conversion for any realization of symmetric powers.
--/
-
-namespace Problem346
-
-variable {α β V W : Type*}
-
-/-- A transposition within one row acts by the transposition on that row
-and by the identity on every other row. -/
-theorem swap_same_row_apply [DecidableEq α] [DecidableEq β]
-    (r : α) (j k : β) (p : α × β) :
-    Equiv.swap (r, j) (r, k) p =
-      (p.1, if p.1 = r then Equiv.swap j k p.2 else p.2) := by
-  rcases p with ⟨i, l⟩
-  by_cases hi : i = r
-  · subst i
-    by_cases hj : l = j
-    · subst l
-      simp
-    by_cases hk : l = k
-    · subst l
-      simp
-    · simp [Equiv.swap_apply_of_ne_of_ne, hj, hk]
-  · have hj : (i, l) ≠ (r, j) := fun h => hi (congrArg Prod.fst h)
-    have hk : (i, l) ≠ (r, k) := fun h => hi (congrArg Prod.fst h)
-    simp [Equiv.swap_apply_of_ne_of_ne hj hk, hi]
-
-/-- Independent rowwise permutation invariance implies invariance under a
-swap of any two slots in a fixed row. -/
-theorem rowwise_invariant_swap [DecidableEq α] [DecidableEq β]
-    (F : (α × β → V) → W)
-    (hF : ∀ (σ : α → Equiv.Perm β) (x : α × β → V),
-      F (fun p => x (p.1, σ p.1 p.2)) = F x)
-    (r : α) (j k : β) (x : α × β → V) :
-    F (fun p => x (Equiv.swap (r, j) (r, k) p)) = F x := by
-  have h := hF (fun i => if i = r then Equiv.swap j k else Equiv.refl β) x
-  convert h using 1
-  congr 1
-  funext p
-  rw [swap_same_row_apply]
-  by_cases hp : p.1 = r <;> simp [hp]
-
-/-- Coefficient-array version of the preceding lemma.  This is the swap
-invariance hypothesis required by the grouped polynomial derivative rule. -/
-theorem rowwise_coefficients_swap {κ : Type*}
-    [DecidableEq α] [DecidableEq β]
-    (F : (α × β → V) → W)
-    (hF : ∀ (σ : α → Equiv.Perm β) (x : α × β → V),
-      F (fun p => x (p.1, σ p.1 p.2)) = F x)
-    (e : κ → V) (r : α) (j k : β) (c : α × β → κ) :
-    F (fun p => e ((c ∘ Equiv.swap (r, j) (r, k)) p)) =
-      F (fun p => e (c p)) := by
-  exact rowwise_invariant_swap F hF r j k (fun p => e (c p))
-
-end Problem346
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-!
+# From rowwise permutation invariance to individual slot swaps
+
+The flattened plethysm forms are invariant under independent permutations
+inside each row.  The polynomial derivative formulas instead ask for
+invariance under a swap of two individual slots.  These lemmas provide that
+conversion for any realization of symmetric powers.
+-/
+
+namespace Problem346
+
+variable {α β V W : Type*}
+
+/-- A transposition within one row acts by the transposition on that row
+and by the identity on every other row. -/
+theorem swap_same_row_apply [DecidableEq α] [DecidableEq β]
+    (r : α) (j k : β) (p : α × β) :
+    Equiv.swap (r, j) (r, k) p =
+      (p.1, if p.1 = r then Equiv.swap j k p.2 else p.2) := by
+  rcases p with ⟨i, l⟩
+  by_cases hi : i = r
+  · subst i
+    by_cases hj : l = j
+    · subst l
+      simp
+    by_cases hk : l = k
+    · subst l
+      simp
+    · simp [Equiv.swap_apply_of_ne_of_ne, hj, hk]
+  · have hj : (i, l) ≠ (r, j) := fun h => hi (congrArg Prod.fst h)
+    have hk : (i, l) ≠ (r, k) := fun h => hi (congrArg Prod.fst h)
+    simp [Equiv.swap_apply_of_ne_of_ne hj hk, hi]
+
+/-- Independent rowwise permutation invariance implies invariance under a
+swap of any two slots in a fixed row. -/
+theorem rowwise_invariant_swap [DecidableEq α] [DecidableEq β]
+    (F : (α × β → V) → W)
+    (hF : ∀ (σ : α → Equiv.Perm β) (x : α × β → V),
+      F (fun p => x (p.1, σ p.1 p.2)) = F x)
+    (r : α) (j k : β) (x : α × β → V) :
+    F (fun p => x (Equiv.swap (r, j) (r, k) p)) = F x := by
+  have h := hF (fun i => if i = r then Equiv.swap j k else Equiv.refl β) x
+  convert h using 1
+  congr 1
+  funext p
+  rw [swap_same_row_apply]
+  by_cases hp : p.1 = r <;> simp [hp]
+
+/-- Coefficient-array version of the preceding lemma.  This is the swap
+invariance hypothesis required by the grouped polynomial derivative rule. -/
+theorem rowwise_coefficients_swap {κ : Type*}
+    [DecidableEq α] [DecidableEq β]
+    (F : (α × β → V) → W)
+    (hF : ∀ (σ : α → Equiv.Perm β) (x : α × β → V),
+      F (fun p => x (p.1, σ p.1 p.2)) = F x)
+    (e : κ → V) (r : α) (j k : β) (c : α × β → κ) :
+    F (fun p => e ((c ∘ Equiv.swap (r, j) (r, k)) p)) =
+      F (fun p => e (c p)) := by
+  exact rowwise_invariant_swap F hF r j k (fun p => e (c p))
+
+end Problem346
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/SecondCounts.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/SecondCounts.lean
index 491879030eae3d8d153451495408620b91eacb93..47417d9e6e8ebe139ffb0442815abfb5fd13fc76 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/SecondCounts.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/SecondCounts.lean
@@ -1,134 +1,134 @@
-import OAI.RepresentationTheory.FoulkesHowe.SecondLabels
-
-namespace OAI
-
-noncomputable section
-open scoped BigOperators
-
-namespace Problem346.SecondTransfer
-
-/-- A source occurrence lies in exactly one nonzero row and beyond its
-already-transferred prefix. -/
-theorem label_eq_source_iff {r b : ℕ} (k : Fin r → ℕ)
-    (i : Fin r) (p : Fin (r+1) × Fin b) :
-    label k p = some i.succ ↔ p.1 = i.succ ∧ k i ≤ p.2.val := by
-  rcases p with ⟨l,j⟩
-  refine Fin.cases ?_ (fun l => ?_) l
-  · simp [Ne.symm (Fin.succ_ne_zero i)]
-  · by_cases h : l = i
-    · subst l
-      simp [not_lt]
-    · simp [label_succ_row, Fin.succ_inj, h]
-
-private theorem sum_fin_ite_lt (b k : ℕ) (hk : k ≤ b) :
-    (∑ j : Fin b, if j.val < k then 1 else 0) = k := by
-  classical
-  by_cases h : k < b
-  · let c : Fin b := ⟨k,h⟩
-    have hs : Finset.univ.filter (fun j : Fin b => j.val < k) = Finset.Iio c := by
-      ext j
-      simp [c, Fin.lt_def]
-    rw [← Finset.card_filter, hs, Fin.card_Iio]
-  · have hkb : k = b := by omega
-    subst k
-    simp
-
-private theorem sum_fin_ite_le (b k : ℕ) (hk : k ≤ b) :
-    (∑ j : Fin b, if k ≤ j.val then 1 else 0) = b-k := by
-  classical
-  have h := sum_fin_ite_lt b k hk
-  have hs : (∑ j : Fin b, if j.val < k then 1 else 0) +
-      (∑ j : Fin b, if k ≤ j.val then 1 else 0) = b := by
-    rw [← Finset.sum_add_distrib]
-    have : ∀ j : Fin b, (if j.val < k then 1 else 0) +
-        (if k ≤ j.val then 1 else 0) = (1 : ℕ) := by
-      intro j
-      split_ifs <;> omega
-    simp_rw [this]
-    simp
-  omega
-
-/-- Number of occurrences of the still-independent vector in row `i`. -/
-theorem source_occurrences {r b : ℕ} (k : Fin r → ℕ)
-    (hk : ∀ i, k i ≤ b) (i : Fin r) :
-    (∑ p : Fin (r+1) × Fin b, if label k p = some i.succ then 1 else 0) = b-k i := by
-  classical
-  simp_rw [label_eq_source_iff]
-  rw [Fintype.sum_prod_type]
-  dsimp only
-  simp only [ite_and]
-  rw [Finset.sum_comm]
-  calc
-    _ = ∑ j : Fin b, if k i ≤ j.val then 1 else 0 := by
-      apply Finset.sum_congr rfl
-      intro j hj
-      rw [Finset.sum_eq_single i.succ]
-      · simp
-      · intro l hl hli
-        simp [hli]
-      · simp
-    _ = b-k i := sum_fin_ite_le b (k i) (hk i)
-
-/-- Number of occurrences of the shared destination vector. -/
-theorem destination_occurrences {r b : ℕ} (k : Fin r → ℕ)
-    (hk : ∀ i, k i ≤ b) :
-    (∑ p : Fin (r+1) × Fin b, if label k p = none then 1 else 0) = ∑ i, k i := by
-  classical
-  rw [Fintype.sum_prod_type, Fin.sum_univ_succ]
-  simp only [label_zero_row, reduceCtorEq, ite_false, Finset.sum_const_zero, zero_add,
-    label_succ_row]
-  apply Finset.sum_congr rfl
-  intro i hi
-  convert sum_fin_ite_lt b (k i) (hk i) using 1
-  apply Finset.sum_congr rfl
-  intro j hj
-  split_ifs <;> simp_all
-
-/-- Cardinality version of the source occurrence formula. -/
-theorem source_card {r b : ℕ} (k : Fin r → ℕ)
-    (hk : ∀ i, k i ≤ b) (i : Fin r) :
-    (Finset.univ.filter (fun p : Fin (r+1) × Fin b => label k p = some i.succ)).card =
-      b-k i := by
-  classical
-  rw [Finset.card_filter]
-  exact source_occurrences k hk i
-
-/-- Cardinality version of the destination occurrence formula. -/
-theorem destination_card {r b : ℕ} (k : Fin r → ℕ)
-    (hk : ∀ i, k i ≤ b) :
-    (Finset.univ.filter (fun p : Fin (r+1) × Fin b => label k p = none)).card =
-      ∑ i, k i := by
-  classical
-  rw [Finset.card_filter]
-  exact destination_occurrences k hk
-
-/-- At every unfinished row, the quadratic threshold makes the source degree
-strictly larger than the accumulated shared-destination degree. -/
-theorem source_degree_gt_destination {r b : ℕ} (k : Fin r → ℕ)
-    (hb : r * (r+1) ≤ b) (hk : ∀ j, k j ≤ r)
-    (i : Fin r) (hi : k i < r) :
-    (∑ j, k j) < b - k i := by
-  have hsum : (∑ j, k j) ≤ r*r := by
-    calc
-      _ ≤ ∑ _ : Fin r, r := Finset.sum_le_sum (fun j _ => hk j)
-      _ = r*r := by simp
-  have hquad : r*r+r ≤ b := by nlinarith
-  omega
-
-/-- The actual occurrence counts satisfy the positive-weight hypothesis
-used to cancel a shift at any unfinished row. -/
-theorem destination_occurrences_lt_source {r b : ℕ} (k : Fin r → ℕ)
-    (hb : r * (r+1) ≤ b) (hk : ∀ j, k j ≤ r)
-    (i : Fin r) (hi : k i < r) :
-    (∑ p : Fin (r+1) × Fin b, if label k p = none then 1 else 0) <
-      ∑ p : Fin (r+1) × Fin b, if label k p = some i.succ then 1 else 0 := by
-  have hrb : r ≤ b := by nlinarith
-  have hkb : ∀ j, k j ≤ b := fun j => (hk j).trans hrb
-  rw [destination_occurrences k hkb, source_occurrences k hkb]
-  exact source_degree_gt_destination k hb hk i hi
-
-end Problem346.SecondTransfer
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.SecondLabels
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators
+
+namespace Problem346.SecondTransfer
+
+/-- A source occurrence lies in exactly one nonzero row and beyond its
+already-transferred prefix. -/
+theorem label_eq_source_iff {r b : ℕ} (k : Fin r → ℕ)
+    (i : Fin r) (p : Fin (r+1) × Fin b) :
+    label k p = some i.succ ↔ p.1 = i.succ ∧ k i ≤ p.2.val := by
+  rcases p with ⟨l,j⟩
+  refine Fin.cases ?_ (fun l => ?_) l
+  · simp [Ne.symm (Fin.succ_ne_zero i)]
+  · by_cases h : l = i
+    · subst l
+      simp [not_lt]
+    · simp [label_succ_row, Fin.succ_inj, h]
+
+private theorem sum_fin_ite_lt (b k : ℕ) (hk : k ≤ b) :
+    (∑ j : Fin b, if j.val < k then 1 else 0) = k := by
+  classical
+  by_cases h : k < b
+  · let c : Fin b := ⟨k,h⟩
+    have hs : Finset.univ.filter (fun j : Fin b => j.val < k) = Finset.Iio c := by
+      ext j
+      simp [c, Fin.lt_def]
+    rw [← Finset.card_filter, hs, Fin.card_Iio]
+  · have hkb : k = b := by omega
+    subst k
+    simp
+
+private theorem sum_fin_ite_le (b k : ℕ) (hk : k ≤ b) :
+    (∑ j : Fin b, if k ≤ j.val then 1 else 0) = b-k := by
+  classical
+  have h := sum_fin_ite_lt b k hk
+  have hs : (∑ j : Fin b, if j.val < k then 1 else 0) +
+      (∑ j : Fin b, if k ≤ j.val then 1 else 0) = b := by
+    rw [← Finset.sum_add_distrib]
+    have : ∀ j : Fin b, (if j.val < k then 1 else 0) +
+        (if k ≤ j.val then 1 else 0) = (1 : ℕ) := by
+      intro j
+      split_ifs <;> omega
+    simp_rw [this]
+    simp
+  omega
+
+/-- Number of occurrences of the still-independent vector in row `i`. -/
+theorem source_occurrences {r b : ℕ} (k : Fin r → ℕ)
+    (hk : ∀ i, k i ≤ b) (i : Fin r) :
+    (∑ p : Fin (r+1) × Fin b, if label k p = some i.succ then 1 else 0) = b-k i := by
+  classical
+  simp_rw [label_eq_source_iff]
+  rw [Fintype.sum_prod_type]
+  dsimp only
+  simp only [ite_and]
+  rw [Finset.sum_comm]
+  calc
+    _ = ∑ j : Fin b, if k i ≤ j.val then 1 else 0 := by
+      apply Finset.sum_congr rfl
+      intro j hj
+      rw [Finset.sum_eq_single i.succ]
+      · simp
+      · intro l hl hli
+        simp [hli]
+      · simp
+    _ = b-k i := sum_fin_ite_le b (k i) (hk i)
+
+/-- Number of occurrences of the shared destination vector. -/
+theorem destination_occurrences {r b : ℕ} (k : Fin r → ℕ)
+    (hk : ∀ i, k i ≤ b) :
+    (∑ p : Fin (r+1) × Fin b, if label k p = none then 1 else 0) = ∑ i, k i := by
+  classical
+  rw [Fintype.sum_prod_type, Fin.sum_univ_succ]
+  simp only [label_zero_row, reduceCtorEq, ite_false, Finset.sum_const_zero, zero_add,
+    label_succ_row]
+  apply Finset.sum_congr rfl
+  intro i hi
+  convert sum_fin_ite_lt b (k i) (hk i) using 1
+  apply Finset.sum_congr rfl
+  intro j hj
+  split_ifs <;> simp_all
+
+/-- Cardinality version of the source occurrence formula. -/
+theorem source_card {r b : ℕ} (k : Fin r → ℕ)
+    (hk : ∀ i, k i ≤ b) (i : Fin r) :
+    (Finset.univ.filter (fun p : Fin (r+1) × Fin b => label k p = some i.succ)).card =
+      b-k i := by
+  classical
+  rw [Finset.card_filter]
+  exact source_occurrences k hk i
+
+/-- Cardinality version of the destination occurrence formula. -/
+theorem destination_card {r b : ℕ} (k : Fin r → ℕ)
+    (hk : ∀ i, k i ≤ b) :
+    (Finset.univ.filter (fun p : Fin (r+1) × Fin b => label k p = none)).card =
+      ∑ i, k i := by
+  classical
+  rw [Finset.card_filter]
+  exact destination_occurrences k hk
+
+/-- At every unfinished row, the quadratic threshold makes the source degree
+strictly larger than the accumulated shared-destination degree. -/
+theorem source_degree_gt_destination {r b : ℕ} (k : Fin r → ℕ)
+    (hb : r * (r+1) ≤ b) (hk : ∀ j, k j ≤ r)
+    (i : Fin r) (hi : k i < r) :
+    (∑ j, k j) < b - k i := by
+  have hsum : (∑ j, k j) ≤ r*r := by
+    calc
+      _ ≤ ∑ _ : Fin r, r := Finset.sum_le_sum (fun j _ => hk j)
+      _ = r*r := by simp
+  have hquad : r*r+r ≤ b := by nlinarith
+  omega
+
+/-- The actual occurrence counts satisfy the positive-weight hypothesis
+used to cancel a shift at any unfinished row. -/
+theorem destination_occurrences_lt_source {r b : ℕ} (k : Fin r → ℕ)
+    (hb : r * (r+1) ≤ b) (hk : ∀ j, k j ≤ r)
+    (i : Fin r) (hi : k i < r) :
+    (∑ p : Fin (r+1) × Fin b, if label k p = none then 1 else 0) <
+      ∑ p : Fin (r+1) × Fin b, if label k p = some i.succ then 1 else 0 := by
+  have hrb : r ≤ b := by nlinarith
+  have hkb : ∀ j, k j ≤ b := fun j => (hk j).trans hrb
+  rw [destination_occurrences k hkb, source_occurrences k hkb]
+  exact source_degree_gt_destination k hb hk i hi
+
+end Problem346.SecondTransfer
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/SecondEndpoints.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/SecondEndpoints.lean
index 2de5ac1f9ac4d54aa7a224c6417264ab1727d9ff..c62f9a73c3247a24f510b9406e0c5b0a92db447d 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/SecondEndpoints.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/SecondEndpoints.lean
@@ -1,75 +1,75 @@
-import OAI.RepresentationTheory.FoulkesHowe.SecondLabels
-import OAI.RepresentationTheory.FoulkesHowe.FlattenedForm
-import OAI.RepresentationTheory.FoulkesHowe.MonomialCut
-import OAI.RepresentationTheory.FoulkesHowe.MultilinearPolynomial
-
-namespace OAI
-
-noncomputable section
-
-namespace Problem346.SecondTransfer
-
-universe u
-variable {V : Type u} [AddCommGroup V] [Module ℂ V]
-
-/-- At the zero count vector, every row consists entirely of its independent vector. -/
-theorem initial_eval {r b : ℕ} (T : SymmetricMultilinearForm (r+1) b V)
-    (v : Block r → V) :
-    flattenedSymmetricForm T (fun p => v (label (fun _ : Fin r => 0) p)) =
-      T (fun i => symMonomial b V (fun _ => v (some i))) := by
-  simp only [flattenedSymmetricForm_apply, label_zero]
-
-/-- At the terminal count vector, the first `r` factors in every moving row
-are the common vector. This is precisely the product form in the induction hypothesis. -/
-theorem terminal_eval (r m : ℕ)
-    (T : SymmetricMultilinearForm (r+1) (r+m) V) (v : Block r → V) :
-    flattenedSymmetricForm T (fun p => v (label (fun _ : Fin r => r) p)) =
-      T (Fin.cons (symMonomial (r+m) V (fun _ => v (some 0)))
-        (fun i => symPowMul r m (symMonomial r V (fun _ => v none))
-          (symMonomial m V (fun _ => v (some i.succ))))) := by
-  rw [flattenedSymmetricForm_apply]
-  apply congrArg T
-  funext i
-  refine Fin.cases ?_ (fun i => ?_) i
-  · simp only [label_zero_row, Fin.cons_zero]
-  · simp only [label_succ_row, Fin.cons_succ, apply_ite]
-    exact symMonomial_cut r m (v none) (v (some i.succ))
-
-/-- The known common-power vanishing implies the terminal labelled expression vanishes. -/
-theorem terminal_eval_zero (r m : ℕ)
-    (T : SymmetricMultilinearForm (r+1) (r+m) V)
-    (h : ∀ x t : V, ∀ Q : Fin r → SymPow m V,
-      T (Fin.cons (symMonomial (r+m) V (fun _ => t))
-        (fun i => symPowMul r m (symMonomial r V (fun _ => x)) (Q i))) = 0)
-    (v : Block r → V) :
-    flattenedSymmetricForm T (fun p => v (label (fun _ : Fin r => r) p)) = 0 := by
-  rw [terminal_eval]
-  exact h (v none) (v (some 0)) (fun i => symMonomial m V (fun _ => v (some i.succ)))
-
-/-- The terminal labelled polynomial is zero, before any degree-shift cancellation. -/
-theorem terminal_polynomial_zero {κ : Type*} [Fintype κ] (r m : ℕ)
-    (T : SymmetricMultilinearForm (r+1) (r+m) V) (e : κ → V)
-    (h : ∀ x t : V, ∀ Q : Fin r → SymPow m V,
-      T (Fin.cons (symMonomial (r+m) V (fun _ => t))
-        (fun i => symPowMul r m (symMonomial r V (fun _ => x)) (Q i))) = 0) :
-    multilinearPolynomial (flattenedSymmetricForm T) e (label (fun _ : Fin r => r)) = 0 := by
-  apply multilinearPolynomial_eq_zero
-  exact terminal_eval_zero r m T h
-
-/-- After cancelling all shifts, the initial polynomial identity evaluates to
-vanishing on independent pure powers in every slot. -/
-theorem independent_powers_zero_of_initial_polynomial_zero
-    {κ : Type*} [Fintype κ] {r b : ℕ}
-    (T : SymmetricMultilinearForm (r+1) b V) (e : Module.Basis κ ℂ V)
-    (h : multilinearPolynomial (flattenedSymmetricForm T) e
-      (label (fun _ : Fin r => 0)) = 0) (v : Fin (r+1) → V) :
-    T (fun i => symMonomial b V (fun _ => v i)) = 0 := by
-  have hv := (multilinearPolynomial_eq_zero_iff (flattenedSymmetricForm T) e
-    (label (fun _ : Fin r => 0))).mp h (fun o => o.elim 0 v)
-  simpa only [flattenedSymmetricForm_apply, label_zero, Option.elim_some] using hv
-
-end Problem346.SecondTransfer
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.SecondLabels
+import OAI.RepresentationTheory.FoulkesHowe.FlattenedForm
+import OAI.RepresentationTheory.FoulkesHowe.MonomialCut
+import OAI.RepresentationTheory.FoulkesHowe.MultilinearPolynomial
+
+namespace OAI
+
+noncomputable section
+
+namespace Problem346.SecondTransfer
+
+universe u
+variable {V : Type u} [AddCommGroup V] [Module ℂ V]
+
+/-- At the zero count vector, every row consists entirely of its independent vector. -/
+theorem initial_eval {r b : ℕ} (T : SymmetricMultilinearForm (r+1) b V)
+    (v : Block r → V) :
+    flattenedSymmetricForm T (fun p => v (label (fun _ : Fin r => 0) p)) =
+      T (fun i => symMonomial b V (fun _ => v (some i))) := by
+  simp only [flattenedSymmetricForm_apply, label_zero]
+
+/-- At the terminal count vector, the first `r` factors in every moving row
+are the common vector. This is precisely the product form in the induction hypothesis. -/
+theorem terminal_eval (r m : ℕ)
+    (T : SymmetricMultilinearForm (r+1) (r+m) V) (v : Block r → V) :
+    flattenedSymmetricForm T (fun p => v (label (fun _ : Fin r => r) p)) =
+      T (Fin.cons (symMonomial (r+m) V (fun _ => v (some 0)))
+        (fun i => symPowMul r m (symMonomial r V (fun _ => v none))
+          (symMonomial m V (fun _ => v (some i.succ))))) := by
+  rw [flattenedSymmetricForm_apply]
+  apply congrArg T
+  funext i
+  refine Fin.cases ?_ (fun i => ?_) i
+  · simp only [label_zero_row, Fin.cons_zero]
+  · simp only [label_succ_row, Fin.cons_succ, apply_ite]
+    exact symMonomial_cut r m (v none) (v (some i.succ))
+
+/-- The known common-power vanishing implies the terminal labelled expression vanishes. -/
+theorem terminal_eval_zero (r m : ℕ)
+    (T : SymmetricMultilinearForm (r+1) (r+m) V)
+    (h : ∀ x t : V, ∀ Q : Fin r → SymPow m V,
+      T (Fin.cons (symMonomial (r+m) V (fun _ => t))
+        (fun i => symPowMul r m (symMonomial r V (fun _ => x)) (Q i))) = 0)
+    (v : Block r → V) :
+    flattenedSymmetricForm T (fun p => v (label (fun _ : Fin r => r) p)) = 0 := by
+  rw [terminal_eval]
+  exact h (v none) (v (some 0)) (fun i => symMonomial m V (fun _ => v (some i.succ)))
+
+/-- The terminal labelled polynomial is zero, before any degree-shift cancellation. -/
+theorem terminal_polynomial_zero {κ : Type*} [Fintype κ] (r m : ℕ)
+    (T : SymmetricMultilinearForm (r+1) (r+m) V) (e : κ → V)
+    (h : ∀ x t : V, ∀ Q : Fin r → SymPow m V,
+      T (Fin.cons (symMonomial (r+m) V (fun _ => t))
+        (fun i => symPowMul r m (symMonomial r V (fun _ => x)) (Q i))) = 0) :
+    multilinearPolynomial (flattenedSymmetricForm T) e (label (fun _ : Fin r => r)) = 0 := by
+  apply multilinearPolynomial_eq_zero
+  exact terminal_eval_zero r m T h
+
+/-- After cancelling all shifts, the initial polynomial identity evaluates to
+vanishing on independent pure powers in every slot. -/
+theorem independent_powers_zero_of_initial_polynomial_zero
+    {κ : Type*} [Fintype κ] {r b : ℕ}
+    (T : SymmetricMultilinearForm (r+1) b V) (e : Module.Basis κ ℂ V)
+    (h : multilinearPolynomial (flattenedSymmetricForm T) e
+      (label (fun _ : Fin r => 0)) = 0) (v : Fin (r+1) → V) :
+    T (fun i => symMonomial b V (fun _ => v i)) = 0 := by
+  have hv := (multilinearPolynomial_eq_zero_iff (flattenedSymmetricForm T) e
+    (label (fun _ : Fin r => 0))).mp h (fun o => o.elim 0 v)
+  simpa only [flattenedSymmetricForm_apply, label_zero, Option.elim_some] using hv
+
+end Problem346.SecondTransfer
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/SecondLabels.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/SecondLabels.lean
index 5acfc25f706a3113bd236bc83592657004a25b90..e11fff64d071c6d1a6a5bfbc8d1690c456cd0d8a 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/SecondLabels.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/SecondLabels.lean
@@ -1,34 +1,34 @@
-import Mathlib
-
-namespace OAI
-
-noncomputable section
-open scoped BigOperators
-
-namespace Problem346.SecondTransfer
-
-/-- Blocks: `none` is the common vector, `some 0` is the fixed first row,
-and `some i.succ` is the independent vector in row `i`. -/
-abbrev Block (r : ℕ) := Option (Fin (r+1))
-
-/-- In each nonzero row, its first `k i` factors use the common block. -/
-def label {r b : ℕ} (k : Fin r → ℕ) (p : Fin (r+1) × Fin b) : Block r :=
-  Fin.cases (some 0) (fun i => if p.2.val < k i then none else some i.succ) p.1
-
-@[simp] theorem label_zero_row {r b : ℕ} (k : Fin r → ℕ) (j : Fin b) :
-    label k (0,j) = some 0 := rfl
-
-@[simp] theorem label_succ_row {r b : ℕ} (k : Fin r → ℕ)
-    (i : Fin r) (j : Fin b) :
-    label k (i.succ,j) = if j.val < k i then none else some i.succ := rfl
-
-@[simp] theorem label_zero {r b : ℕ} (p : Fin (r+1) × Fin b) :
-    label (fun _ : Fin r => 0) p = some p.1 := by
-  rcases p with ⟨i,j⟩
-  refine Fin.cases ?_ (fun i => ?_) i <;> simp
-
-end Problem346.SecondTransfer
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators
+
+namespace Problem346.SecondTransfer
+
+/-- Blocks: `none` is the common vector, `some 0` is the fixed first row,
+and `some i.succ` is the independent vector in row `i`. -/
+abbrev Block (r : ℕ) := Option (Fin (r+1))
+
+/-- In each nonzero row, its first `k i` factors use the common block. -/
+def label {r b : ℕ} (k : Fin r → ℕ) (p : Fin (r+1) × Fin b) : Block r :=
+  Fin.cases (some 0) (fun i => if p.2.val < k i then none else some i.succ) p.1
+
+@[simp] theorem label_zero_row {r b : ℕ} (k : Fin r → ℕ) (j : Fin b) :
+    label k (0,j) = some 0 := rfl
+
+@[simp] theorem label_succ_row {r b : ℕ} (k : Fin r → ℕ)
+    (i : Fin r) (j : Fin b) :
+    label k (i.succ,j) = if j.val < k i then none else some i.succ := rfl
+
+@[simp] theorem label_zero {r b : ℕ} (p : Fin (r+1) × Fin b) :
+    label (fun _ : Fin r => 0) p = some p.1 := by
+  rcases p with ⟨i,j⟩
+  refine Fin.cases ?_ (fun i => ?_) i <;> simp
+
+end Problem346.SecondTransfer
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/SecondSwap.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/SecondSwap.lean
index 206506bce4cfdf797f25dae6c293f282ba836fba..ad1f4dd29d4745d488a261726800a3dbfede4c75 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/SecondSwap.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/SecondSwap.lean
@@ -1,75 +1,75 @@
-import OAI.RepresentationTheory.FoulkesHowe.SecondLabels
-
-namespace OAI
-
-noncomputable section
-namespace Problem346.SecondTransfer
-
-/-- Swap two factor positions in one nonzero row, leaving all other rows fixed. -/
-def rowFactorSwap {r b : ℕ} (i : Fin r) (j l : Fin b) :
-    Fin (r+1) → Equiv.Perm (Fin b) :=
-  fun a => if a = i.succ then Equiv.swap j l else Equiv.refl _
-
-/-- Changing any remaining source factor to the common block agrees with progressing
-the prefix count, after a permutation internal to that row. -/
-theorem update_label_eq_progress_rowSwap
-    {r b : ℕ} (k : Fin r → ℕ) (i : Fin r) (hi : k i < b)
-    (j : Fin b) (hj : k i ≤ j.val) (p : Fin (r+1) × Fin b) :
-    Function.update (label k) (i.succ, j) none p =
-      label (Function.update k i (k i+1))
-        (p.1, rowFactorSwap i j ⟨k i, hi⟩ p.1 p.2) := by
-  classical
-  rcases p with ⟨a,l⟩
-  refine Fin.cases ?_ (fun a => ?_) a
-  · simp [Function.update, rowFactorSwap, label]
-    intro h
-    have hv := congrArg Fin.val h
-    simp at hv
-  · by_cases hai : a = i
-    · subst a
-      by_cases hlj : l = j
-      · subst l
-        simp [rowFactorSwap, label]
-      · by_cases hlk : l = (⟨k i, hi⟩ : Fin b)
-        · subst l
-          have hjval : k i < j.val := by
-            have hne : j.val ≠ k i := by
-              intro he
-              apply hlj
-              exact Fin.ext he.symm
-            omega
-          simp [rowFactorSwap, label, hlj, Nat.not_lt.mpr (by omega : k i+1 ≤ j.val)]
-        · have hval : l.val ≠ k i := by
-            intro he
-            apply hlk
-            exact Fin.ext he
-          have hlt : l.val < k i+1 ↔ l.val < k i := by omega
-          simp [rowFactorSwap, label, hlj, hlt,
-            Equiv.swap_apply_of_ne_of_ne hlj hlk]
-    · simp [rowFactorSwap, label, hai]
-
-/-- Any expression invariant under permutations of factors within each row takes the
-same value on a derivative label update and the progressed prefix labels. -/
-theorem invariant_update_label_eq_progress
-    {r b : ℕ} {A : Type*}
-    (F : ((Fin (r+1) × Fin b) → Block r) → A)
-    (hF : ∀ (l : (Fin (r+1) × Fin b) → Block r)
-      (σ : Fin (r+1) → Equiv.Perm (Fin b)),
-      F (fun p => l (p.1, σ p.1 p.2)) = F l)
-    (k : Fin r → ℕ) (i : Fin r) (hi : k i < b)
-    (j : Fin b) (hj : k i ≤ j.val) :
-    F (Function.update (label k) (i.succ,j) none) =
-      F (label (Function.update k i (k i+1))) := by
-  have heq : Function.update (label k) (i.succ,j) none =
-      fun p => label (Function.update k i (k i+1))
-        (p.1, rowFactorSwap i j ⟨k i, hi⟩ p.1 p.2) := by
-    funext p
-    exact update_label_eq_progress_rowSwap k i hi j hj p
-  rw [heq]
-  exact hF _ _
-
-end Problem346.SecondTransfer
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.SecondLabels
+
+namespace OAI
+
+noncomputable section
+namespace Problem346.SecondTransfer
+
+/-- Swap two factor positions in one nonzero row, leaving all other rows fixed. -/
+def rowFactorSwap {r b : ℕ} (i : Fin r) (j l : Fin b) :
+    Fin (r+1) → Equiv.Perm (Fin b) :=
+  fun a => if a = i.succ then Equiv.swap j l else Equiv.refl _
+
+/-- Changing any remaining source factor to the common block agrees with progressing
+the prefix count, after a permutation internal to that row. -/
+theorem update_label_eq_progress_rowSwap
+    {r b : ℕ} (k : Fin r → ℕ) (i : Fin r) (hi : k i < b)
+    (j : Fin b) (hj : k i ≤ j.val) (p : Fin (r+1) × Fin b) :
+    Function.update (label k) (i.succ, j) none p =
+      label (Function.update k i (k i+1))
+        (p.1, rowFactorSwap i j ⟨k i, hi⟩ p.1 p.2) := by
+  classical
+  rcases p with ⟨a,l⟩
+  refine Fin.cases ?_ (fun a => ?_) a
+  · simp [Function.update, rowFactorSwap, label]
+    intro h
+    have hv := congrArg Fin.val h
+    simp at hv
+  · by_cases hai : a = i
+    · subst a
+      by_cases hlj : l = j
+      · subst l
+        simp [rowFactorSwap, label]
+      · by_cases hlk : l = (⟨k i, hi⟩ : Fin b)
+        · subst l
+          have hjval : k i < j.val := by
+            have hne : j.val ≠ k i := by
+              intro he
+              apply hlj
+              exact Fin.ext he.symm
+            omega
+          simp [rowFactorSwap, label, hlj, Nat.not_lt.mpr (by omega : k i+1 ≤ j.val)]
+        · have hval : l.val ≠ k i := by
+            intro he
+            apply hlk
+            exact Fin.ext he
+          have hlt : l.val < k i+1 ↔ l.val < k i := by omega
+          simp [rowFactorSwap, label, hlj, hlt,
+            Equiv.swap_apply_of_ne_of_ne hlj hlk]
+    · simp [rowFactorSwap, label, hai]
+
+/-- Any expression invariant under permutations of factors within each row takes the
+same value on a derivative label update and the progressed prefix labels. -/
+theorem invariant_update_label_eq_progress
+    {r b : ℕ} {A : Type*}
+    (F : ((Fin (r+1) × Fin b) → Block r) → A)
+    (hF : ∀ (l : (Fin (r+1) × Fin b) → Block r)
+      (σ : Fin (r+1) → Equiv.Perm (Fin b)),
+      F (fun p => l (p.1, σ p.1 p.2)) = F l)
+    (k : Fin r → ℕ) (i : Fin r) (hi : k i < b)
+    (j : Fin b) (hj : k i ≤ j.val) :
+    F (Function.update (label k) (i.succ,j) none) =
+      F (label (Function.update k i (k i+1))) := by
+  have heq : Function.update (label k) (i.succ,j) none =
+      fun p => label (Function.update k i (k i+1))
+        (p.1, rowFactorSwap i j ⟨k i, hi⟩ p.1 p.2) := by
+    funext p
+    exact update_label_eq_progress_rowSwap k i hi j hj p
+  rw [heq]
+  exact hF _ _
+
+end Problem346.SecondTransfer
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/SecondTransfer.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/SecondTransfer.lean
index 321cbf1ebba28c67af8aa7f73af899d3735e7133..cc7bc93d8df0edf2b19447fceea6440ebb5fc7c5 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/SecondTransfer.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/SecondTransfer.lean
@@ -1,83 +1,83 @@
-import OAI.RepresentationTheory.FoulkesHowe.EncodedCancellation
-import OAI.RepresentationTheory.FoulkesHowe.SecondCounts
-import OAI.RepresentationTheory.FoulkesHowe.SecondSwap
-import OAI.RepresentationTheory.FoulkesHowe.SecondEndpoints
-import OAI.RepresentationTheory.FoulkesHowe.GridCancellation
-
-namespace OAI
-
-noncomputable section
-open scoped BigOperators
-
-namespace Problem346.SecondTransfer
-
-universe u
-variable {V : Type u} [AddCommGroup V] [Module ℂ V]
-
-/-- Polynomial encoding preserves the independent factor permutations of each row. -/
-theorem polynomial_row_permute {κ : Type*} [Fintype κ] {r b : ℕ}
-    (T : SymmetricMultilinearForm (r+1) b V) (e : κ → V)
-    (l : Fin (r+1) × Fin b → Block r)
-    (σ : Fin (r+1) → Equiv.Perm (Fin b)) :
-    multilinearPolynomial (flattenedSymmetricForm T) e
-        (fun p => l (p.1, σ p.1 p.2)) =
-      multilinearPolynomial (flattenedSymmetricForm T) e l := by
-  apply MvPolynomial.funext
-  intro z
-  simp only [eval_multilinearPolynomial]
-  exact flattenedSymmetricForm_inner_permute T σ
-    (fun p => ∑ j : κ, z (l p,j) • e j)
-
-/-- A single unfinished row can be recovered from its one-step progressed state. -/
-theorem polynomial_step {κ : Type*} [Fintype κ] {r b : ℕ}
-    (T : SymmetricMultilinearForm (r+1) b V) (e : κ → V)
-    (hb : r * (r+1) ≤ b) (k : Fin r → ℕ) (hk : ∀ j, k j ≤ r)
-    (i : Fin r) (hi : k i < r)
-    (hnext : multilinearPolynomial (flattenedSymmetricForm T) e
-      (label (Function.update k i (k i+1))) = 0) :
-    multilinearPolynomial (flattenedSymmetricForm T) e (label k) = 0 := by
-  classical
-  apply multilinearPolynomial_eq_zero_of_replacements
-    (flattenedSymmetricForm T) e (label k) (some i.succ) none (by simp)
-  · have hd := destination_occurrences_lt_source (b := b) k hb hk i hi
-    simpa only [Finset.card_filter] using hd
-  · rintro ⟨row,j⟩ hp
-    obtain ⟨hrow,hj⟩ := (label_eq_source_iff k i (row,j)).mp hp
-    change row = i.succ at hrow
-    change k i ≤ j.val at hj
-    subst row
-    have hrb : r ≤ b := by nlinarith
-    have hki : k i < b := lt_of_lt_of_le hi hrb
-    have heq := invariant_update_label_eq_progress
-      (fun l => multilinearPolynomial (flattenedSymmetricForm T) e l)
-      (polynomial_row_permute T e) k i hki j hj
-    exact heq.trans hnext
-
-/-- The second differential transfer: remove the common degree-`r` power from
-all `r` moving arguments. Count-vector induction allows the shifts in any order. -/
-theorem second_transfer [FiniteDimensional ℂ V] (r m : ℕ)
-    (hb : r * (r+1) ≤ r+m)
-    (T : SymmetricMultilinearForm (r+1) (r+m) V)
-    (h : ∀ x t : V, ∀ Q : Fin r → SymPow m V,
-      T (Fin.cons (symMonomial (r+m) V (fun _ => t))
-        (fun i => symPowMul r m (symMonomial r V (fun _ => x)) (Q i))) = 0) :
-    ∀ v : Fin (r+1) → V,
-      T (fun i => symMonomial (r+m) V (fun _ => v i)) = 0 := by
-  classical
-  let e := Module.finBasis ℂ V
-  let P (k : Fin r → ℕ) :=
-    multilinearPolynomial (flattenedSymmetricForm T) e (label k)
-  have hend : P (fun _ => r) = 0 := terminal_polynomial_zero r m T e h
-  have hzero : P (fun _ => 0) = 0 := by
-    apply zero_of_upward_grid r P hend
-    · intro k hk i hi hnext
-      exact polynomial_step T e hb k hk i hi hnext
-    · intro i
-      exact Nat.zero_le r
-  exact independent_powers_zero_of_initial_polynomial_zero T e hzero
-
-end Problem346.SecondTransfer
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.EncodedCancellation
+import OAI.RepresentationTheory.FoulkesHowe.SecondCounts
+import OAI.RepresentationTheory.FoulkesHowe.SecondSwap
+import OAI.RepresentationTheory.FoulkesHowe.SecondEndpoints
+import OAI.RepresentationTheory.FoulkesHowe.GridCancellation
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators
+
+namespace Problem346.SecondTransfer
+
+universe u
+variable {V : Type u} [AddCommGroup V] [Module ℂ V]
+
+/-- Polynomial encoding preserves the independent factor permutations of each row. -/
+theorem polynomial_row_permute {κ : Type*} [Fintype κ] {r b : ℕ}
+    (T : SymmetricMultilinearForm (r+1) b V) (e : κ → V)
+    (l : Fin (r+1) × Fin b → Block r)
+    (σ : Fin (r+1) → Equiv.Perm (Fin b)) :
+    multilinearPolynomial (flattenedSymmetricForm T) e
+        (fun p => l (p.1, σ p.1 p.2)) =
+      multilinearPolynomial (flattenedSymmetricForm T) e l := by
+  apply MvPolynomial.funext
+  intro z
+  simp only [eval_multilinearPolynomial]
+  exact flattenedSymmetricForm_inner_permute T σ
+    (fun p => ∑ j : κ, z (l p,j) • e j)
+
+/-- A single unfinished row can be recovered from its one-step progressed state. -/
+theorem polynomial_step {κ : Type*} [Fintype κ] {r b : ℕ}
+    (T : SymmetricMultilinearForm (r+1) b V) (e : κ → V)
+    (hb : r * (r+1) ≤ b) (k : Fin r → ℕ) (hk : ∀ j, k j ≤ r)
+    (i : Fin r) (hi : k i < r)
+    (hnext : multilinearPolynomial (flattenedSymmetricForm T) e
+      (label (Function.update k i (k i+1))) = 0) :
+    multilinearPolynomial (flattenedSymmetricForm T) e (label k) = 0 := by
+  classical
+  apply multilinearPolynomial_eq_zero_of_replacements
+    (flattenedSymmetricForm T) e (label k) (some i.succ) none (by simp)
+  · have hd := destination_occurrences_lt_source (b := b) k hb hk i hi
+    simpa only [Finset.card_filter] using hd
+  · rintro ⟨row,j⟩ hp
+    obtain ⟨hrow,hj⟩ := (label_eq_source_iff k i (row,j)).mp hp
+    change row = i.succ at hrow
+    change k i ≤ j.val at hj
+    subst row
+    have hrb : r ≤ b := by nlinarith
+    have hki : k i < b := lt_of_lt_of_le hi hrb
+    have heq := invariant_update_label_eq_progress
+      (fun l => multilinearPolynomial (flattenedSymmetricForm T) e l)
+      (polynomial_row_permute T e) k i hki j hj
+    exact heq.trans hnext
+
+/-- The second differential transfer: remove the common degree-`r` power from
+all `r` moving arguments. Count-vector induction allows the shifts in any order. -/
+theorem second_transfer [FiniteDimensional ℂ V] (r m : ℕ)
+    (hb : r * (r+1) ≤ r+m)
+    (T : SymmetricMultilinearForm (r+1) (r+m) V)
+    (h : ∀ x t : V, ∀ Q : Fin r → SymPow m V,
+      T (Fin.cons (symMonomial (r+m) V (fun _ => t))
+        (fun i => symPowMul r m (symMonomial r V (fun _ => x)) (Q i))) = 0) :
+    ∀ v : Fin (r+1) → V,
+      T (fun i => symMonomial (r+m) V (fun _ => v i)) = 0 := by
+  classical
+  let e := Module.finBasis ℂ V
+  let P (k : Fin r → ℕ) :=
+    multilinearPolynomial (flattenedSymmetricForm T) e (label k)
+  have hend : P (fun _ => r) = 0 := terminal_polynomial_zero r m T e h
+  have hzero : P (fun _ => 0) = 0 := by
+    apply zero_of_upward_grid r P hend
+    · intro k hk i hi hnext
+      exact polynomial_step T e hb k hk i hi hnext
+    · intro i
+      exact Nat.zero_le r
+  exact independent_powers_zero_of_initial_polynomial_zero T e hzero
+
+end Problem346.SecondTransfer
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/ShiftHomogeneity.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/ShiftHomogeneity.lean
index 52608db123a73d7bc68944e91438361a04670076..8bd48b0ac21cffeb381dc177bcb1aa0ecaefcc78 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/ShiftHomogeneity.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/ShiftHomogeneity.lean
@@ -1,158 +1,158 @@
-import OAI.RepresentationTheory.FoulkesHowe.BlockDerivation
-
-namespace OAI
-
-noncomputable section
-
-open scoped BigOperators
-open MvPolynomial
-
-namespace Problem346
-
-/-- Differentiating a weighted homogeneous polynomial lowers its weighted degree
-by the weight of the differentiated variable. The statement also covers degree
-zero, since a zero derivative is homogeneous in every degree. -/
-theorem weightedHomogeneous_pderiv
-    {σ R : Type*} [CommSemiring R] {w : σ → ℕ}
-    {f : MvPolynomial σ R} {m : ℕ}
-    (hf : IsWeightedHomogeneous w f m) (i : σ) :
-    IsWeightedHomogeneous w (pderiv i f) (m - w i) := by
-  classical
-  intro d hd
-  rw [coeff_pderiv] at hd
-  have h := hf (left_ne_zero_of_mul hd)
-  rw [map_add, Finsupp.weight_single, one_smul] at h
-  omega
-
-/-- A weighted directional derivative has the expected homogeneous degree
-when all source variables and all destination variables have common weights. -/
-theorem weightedHomogeneous_directional
-    {σ κ R : Type*} [Fintype κ] [CommSemiring R]
-    {w : σ → ℕ} {f : MvPolynomial σ R} {m a b : ℕ}
-    (hf : IsWeightedHomogeneous w f m) (s t : κ → σ)
-    (hs : ∀ i, w (s i) = a) (ht : ∀ i, w (t i) = b) :
-    IsWeightedHomogeneous w (∑ i, X (t i) * pderiv (s i) f) (b + (m - a)) := by
-  apply IsWeightedHomogeneous.sum
-  intro i hi
-  simpa [hs i, ht i] using
-    (isWeightedHomogeneous_X R w (t i)).mul (weightedHomogeneous_pderiv hf (s i))
-
-private theorem derivation_sum_apply
-    {σ κ R : Type*} [CommRing R] (s : Finset κ)
-    (D : κ → Derivation R (MvPolynomial σ R) (MvPolynomial σ R))
-    (f : MvPolynomial σ R) : (∑ i ∈ s, D i) f = ∑ i ∈ s, D i f := by
-  change Derivation.coeFnAddMonoidHom (∑ i ∈ s, D i) f = _
-  rw [map_sum]
-  simp
-
-/-- The generator-defined block shift agrees with the usual sum of partial
- derivatives over its coordinate block. -/
-theorem blockShift_eq_sum_pderiv
-    {ι κ R : Type*} [DecidableEq ι] [Fintype κ] [CommRing R] (u v : ι) :
-    blockShift (κ := κ) R u v =
-      ∑ i : κ, (X (v, i) : MvPolynomial (ι × κ) R) • pderiv (u, i) := by
-  classical
-  apply MvPolynomial.derivation_ext
-  rintro ⟨k, l⟩
-  simp only [blockShift_X, derivation_sum_apply, Derivation.smul_apply, pderiv_X]
-  by_cases h : k = u
-  · subst k
-    simp [Pi.single_apply, Prod.mk.injEq]
-  · simp [Prod.mk.injEq, h]
-
-@[simp] theorem blockShift_apply_eq_sum
-    {ι κ R : Type*} [DecidableEq ι] [Fintype κ] [CommRing R]
-    (u v : ι) (f : MvPolynomial (ι × κ) R) :
-    blockShift R u v f = ∑ i : κ, X (v, i) * pderiv (u, i) f := by
-  rw [blockShift_eq_sum_pderiv]
-  simp only [derivation_sum_apply, Derivation.smul_apply, smul_eq_mul]
-
-/-- A block shift changes the degree in an observed block by adding one for
-the destination and subtracting one for the source. -/
-theorem blockShift_isWeightedHomogeneous
-    {ι κ R : Type*} [DecidableEq ι] [Fintype κ] [CommRing R]
-    (u v k : ι) {f : MvPolynomial (ι × κ) R} {m : ℕ}
-    (hf : IsWeightedHomogeneous (fun z : ι × κ => if z.1 = k then 1 else 0) f m) :
-    IsWeightedHomogeneous (fun z : ι × κ => if z.1 = k then 1 else 0)
-      (blockShift R u v f)
-      ((if v = k then 1 else 0) + (m - (if u = k then 1 else 0))) := by
-  rw [blockShift_apply_eq_sum]
-  exact weightedHomogeneous_directional hf (fun i => (u, i)) (fun i => (v, i))
-    (fun _ => rfl) (fun _ => rfl)
-
-/-- The source block loses one unit of degree under a directional shift. -/
-theorem blockShift_source_homogeneous
-    {ι κ R : Type*} [DecidableEq ι] [Fintype κ] [CommRing R]
-    (u v : ι) (huv : u ≠ v) {f : MvPolynomial (ι × κ) R} {m : ℕ}
-    (hf : IsWeightedHomogeneous (fun z : ι × κ => if z.1 = u then 1 else 0) f m) :
-    IsWeightedHomogeneous (fun z : ι × κ => if z.1 = u then 1 else 0)
-      (blockShift R u v f) (m - 1) := by
-  simpa [huv, Ne.symm huv] using blockShift_isWeightedHomogeneous u v u hf
-
-/-- The destination block gains one unit of degree under a directional shift. -/
-theorem blockShift_destination_homogeneous
-    {ι κ R : Type*} [DecidableEq ι] [Fintype κ] [CommRing R]
-    (u v : ι) (huv : u ≠ v) {f : MvPolynomial (ι × κ) R} {m : ℕ}
-    (hf : IsWeightedHomogeneous (fun z : ι × κ => if z.1 = v then 1 else 0) f m) :
-    IsWeightedHomogeneous (fun z : ι × κ => if z.1 = v then 1 else 0)
-      (blockShift R u v f) (m + 1) := by
-  simpa [huv, Nat.add_comm] using blockShift_isWeightedHomogeneous u v v hf
-
-/-- Spectator block degrees are unchanged by a directional shift. -/
-theorem blockShift_spectator_homogeneous
-    {ι κ R : Type*} [DecidableEq ι] [Fintype κ] [CommRing R]
-    (u v k : ι) (hu : u ≠ k) (hv : v ≠ k)
-    {f : MvPolynomial (ι × κ) R} {m : ℕ}
-    (hf : IsWeightedHomogeneous (fun z : ι × κ => if z.1 = k then 1 else 0) f m) :
-    IsWeightedHomogeneous (fun z : ι × κ => if z.1 = k then 1 else 0)
-      (blockShift R u v f) m := by
-  simpa [hu, hv] using blockShift_isWeightedHomogeneous u v k hf
-
-/-- Repeated shifts lower the source degree by the number of operations. -/
-theorem blockShift_pow_source_homogeneous
-    {ι κ R : Type*} [DecidableEq ι] [Fintype κ] [CommRing R]
-    (u v : ι) (huv : u ≠ v) {f : MvPolynomial (ι × κ) R} {m : ℕ}
-    (hf : IsWeightedHomogeneous (fun z : ι × κ => if z.1 = u then 1 else 0) f m)
-    (n : ℕ) :
-    IsWeightedHomogeneous (fun z : ι × κ => if z.1 = u then 1 else 0)
-      (((blockShift R u v).toLinearMap ^ n) f) (m - n) := by
-  induction n with
-  | zero => simpa using hf
-  | succ n ih =>
-    rw [pow_succ', Module.End.mul_apply]
-    simpa [Nat.sub_sub] using blockShift_source_homogeneous u v huv ih
-
-/-- Repeated shifts increase the destination degree by the number of operations. -/
-theorem blockShift_pow_destination_homogeneous
-    {ι κ R : Type*} [DecidableEq ι] [Fintype κ] [CommRing R]
-    (u v : ι) (huv : u ≠ v) {f : MvPolynomial (ι × κ) R} {m : ℕ}
-    (hf : IsWeightedHomogeneous (fun z : ι × κ => if z.1 = v then 1 else 0) f m)
-    (n : ℕ) :
-    IsWeightedHomogeneous (fun z : ι × κ => if z.1 = v then 1 else 0)
-      (((blockShift R u v).toLinearMap ^ n) f) (m + n) := by
-  induction n with
-  | zero => simpa using hf
-  | succ n ih =>
-    rw [pow_succ', Module.End.mul_apply]
-    simpa [Nat.add_assoc] using blockShift_destination_homogeneous u v huv ih
-
-/-- Repeated shifts leave all spectator block degrees unchanged. -/
-theorem blockShift_pow_spectator_homogeneous
-    {ι κ R : Type*} [DecidableEq ι] [Fintype κ] [CommRing R]
-    (u v k : ι) (hu : u ≠ k) (hv : v ≠ k)
-    {f : MvPolynomial (ι × κ) R} {m : ℕ}
-    (hf : IsWeightedHomogeneous (fun z : ι × κ => if z.1 = k then 1 else 0) f m)
-    (n : ℕ) :
-    IsWeightedHomogeneous (fun z : ι × κ => if z.1 = k then 1 else 0)
-      (((blockShift R u v).toLinearMap ^ n) f) m := by
-  induction n with
-  | zero => simpa using hf
-  | succ n ih =>
-    rw [pow_succ', Module.End.mul_apply]
-    exact blockShift_spectator_homogeneous u v k hu hv ih
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.BlockDerivation
+
+namespace OAI
+
+noncomputable section
+
+open scoped BigOperators
+open MvPolynomial
+
+namespace Problem346
+
+/-- Differentiating a weighted homogeneous polynomial lowers its weighted degree
+by the weight of the differentiated variable. The statement also covers degree
+zero, since a zero derivative is homogeneous in every degree. -/
+theorem weightedHomogeneous_pderiv
+    {σ R : Type*} [CommSemiring R] {w : σ → ℕ}
+    {f : MvPolynomial σ R} {m : ℕ}
+    (hf : IsWeightedHomogeneous w f m) (i : σ) :
+    IsWeightedHomogeneous w (pderiv i f) (m - w i) := by
+  classical
+  intro d hd
+  rw [coeff_pderiv] at hd
+  have h := hf (left_ne_zero_of_mul hd)
+  rw [map_add, Finsupp.weight_single, one_smul] at h
+  omega
+
+/-- A weighted directional derivative has the expected homogeneous degree
+when all source variables and all destination variables have common weights. -/
+theorem weightedHomogeneous_directional
+    {σ κ R : Type*} [Fintype κ] [CommSemiring R]
+    {w : σ → ℕ} {f : MvPolynomial σ R} {m a b : ℕ}
+    (hf : IsWeightedHomogeneous w f m) (s t : κ → σ)
+    (hs : ∀ i, w (s i) = a) (ht : ∀ i, w (t i) = b) :
+    IsWeightedHomogeneous w (∑ i, X (t i) * pderiv (s i) f) (b + (m - a)) := by
+  apply IsWeightedHomogeneous.sum
+  intro i hi
+  simpa [hs i, ht i] using
+    (isWeightedHomogeneous_X R w (t i)).mul (weightedHomogeneous_pderiv hf (s i))
+
+private theorem derivation_sum_apply
+    {σ κ R : Type*} [CommRing R] (s : Finset κ)
+    (D : κ → Derivation R (MvPolynomial σ R) (MvPolynomial σ R))
+    (f : MvPolynomial σ R) : (∑ i ∈ s, D i) f = ∑ i ∈ s, D i f := by
+  change Derivation.coeFnAddMonoidHom (∑ i ∈ s, D i) f = _
+  rw [map_sum]
+  simp
+
+/-- The generator-defined block shift agrees with the usual sum of partial
+ derivatives over its coordinate block. -/
+theorem blockShift_eq_sum_pderiv
+    {ι κ R : Type*} [DecidableEq ι] [Fintype κ] [CommRing R] (u v : ι) :
+    blockShift (κ := κ) R u v =
+      ∑ i : κ, (X (v, i) : MvPolynomial (ι × κ) R) • pderiv (u, i) := by
+  classical
+  apply MvPolynomial.derivation_ext
+  rintro ⟨k, l⟩
+  simp only [blockShift_X, derivation_sum_apply, Derivation.smul_apply, pderiv_X]
+  by_cases h : k = u
+  · subst k
+    simp [Pi.single_apply, Prod.mk.injEq]
+  · simp [Prod.mk.injEq, h]
+
+@[simp] theorem blockShift_apply_eq_sum
+    {ι κ R : Type*} [DecidableEq ι] [Fintype κ] [CommRing R]
+    (u v : ι) (f : MvPolynomial (ι × κ) R) :
+    blockShift R u v f = ∑ i : κ, X (v, i) * pderiv (u, i) f := by
+  rw [blockShift_eq_sum_pderiv]
+  simp only [derivation_sum_apply, Derivation.smul_apply, smul_eq_mul]
+
+/-- A block shift changes the degree in an observed block by adding one for
+the destination and subtracting one for the source. -/
+theorem blockShift_isWeightedHomogeneous
+    {ι κ R : Type*} [DecidableEq ι] [Fintype κ] [CommRing R]
+    (u v k : ι) {f : MvPolynomial (ι × κ) R} {m : ℕ}
+    (hf : IsWeightedHomogeneous (fun z : ι × κ => if z.1 = k then 1 else 0) f m) :
+    IsWeightedHomogeneous (fun z : ι × κ => if z.1 = k then 1 else 0)
+      (blockShift R u v f)
+      ((if v = k then 1 else 0) + (m - (if u = k then 1 else 0))) := by
+  rw [blockShift_apply_eq_sum]
+  exact weightedHomogeneous_directional hf (fun i => (u, i)) (fun i => (v, i))
+    (fun _ => rfl) (fun _ => rfl)
+
+/-- The source block loses one unit of degree under a directional shift. -/
+theorem blockShift_source_homogeneous
+    {ι κ R : Type*} [DecidableEq ι] [Fintype κ] [CommRing R]
+    (u v : ι) (huv : u ≠ v) {f : MvPolynomial (ι × κ) R} {m : ℕ}
+    (hf : IsWeightedHomogeneous (fun z : ι × κ => if z.1 = u then 1 else 0) f m) :
+    IsWeightedHomogeneous (fun z : ι × κ => if z.1 = u then 1 else 0)
+      (blockShift R u v f) (m - 1) := by
+  simpa [huv, Ne.symm huv] using blockShift_isWeightedHomogeneous u v u hf
+
+/-- The destination block gains one unit of degree under a directional shift. -/
+theorem blockShift_destination_homogeneous
+    {ι κ R : Type*} [DecidableEq ι] [Fintype κ] [CommRing R]
+    (u v : ι) (huv : u ≠ v) {f : MvPolynomial (ι × κ) R} {m : ℕ}
+    (hf : IsWeightedHomogeneous (fun z : ι × κ => if z.1 = v then 1 else 0) f m) :
+    IsWeightedHomogeneous (fun z : ι × κ => if z.1 = v then 1 else 0)
+      (blockShift R u v f) (m + 1) := by
+  simpa [huv, Nat.add_comm] using blockShift_isWeightedHomogeneous u v v hf
+
+/-- Spectator block degrees are unchanged by a directional shift. -/
+theorem blockShift_spectator_homogeneous
+    {ι κ R : Type*} [DecidableEq ι] [Fintype κ] [CommRing R]
+    (u v k : ι) (hu : u ≠ k) (hv : v ≠ k)
+    {f : MvPolynomial (ι × κ) R} {m : ℕ}
+    (hf : IsWeightedHomogeneous (fun z : ι × κ => if z.1 = k then 1 else 0) f m) :
+    IsWeightedHomogeneous (fun z : ι × κ => if z.1 = k then 1 else 0)
+      (blockShift R u v f) m := by
+  simpa [hu, hv] using blockShift_isWeightedHomogeneous u v k hf
+
+/-- Repeated shifts lower the source degree by the number of operations. -/
+theorem blockShift_pow_source_homogeneous
+    {ι κ R : Type*} [DecidableEq ι] [Fintype κ] [CommRing R]
+    (u v : ι) (huv : u ≠ v) {f : MvPolynomial (ι × κ) R} {m : ℕ}
+    (hf : IsWeightedHomogeneous (fun z : ι × κ => if z.1 = u then 1 else 0) f m)
+    (n : ℕ) :
+    IsWeightedHomogeneous (fun z : ι × κ => if z.1 = u then 1 else 0)
+      (((blockShift R u v).toLinearMap ^ n) f) (m - n) := by
+  induction n with
+  | zero => simpa using hf
+  | succ n ih =>
+    rw [pow_succ', Module.End.mul_apply]
+    simpa [Nat.sub_sub] using blockShift_source_homogeneous u v huv ih
+
+/-- Repeated shifts increase the destination degree by the number of operations. -/
+theorem blockShift_pow_destination_homogeneous
+    {ι κ R : Type*} [DecidableEq ι] [Fintype κ] [CommRing R]
+    (u v : ι) (huv : u ≠ v) {f : MvPolynomial (ι × κ) R} {m : ℕ}
+    (hf : IsWeightedHomogeneous (fun z : ι × κ => if z.1 = v then 1 else 0) f m)
+    (n : ℕ) :
+    IsWeightedHomogeneous (fun z : ι × κ => if z.1 = v then 1 else 0)
+      (((blockShift R u v).toLinearMap ^ n) f) (m + n) := by
+  induction n with
+  | zero => simpa using hf
+  | succ n ih =>
+    rw [pow_succ', Module.End.mul_apply]
+    simpa [Nat.add_assoc] using blockShift_destination_homogeneous u v huv ih
+
+/-- Repeated shifts leave all spectator block degrees unchanged. -/
+theorem blockShift_pow_spectator_homogeneous
+    {ι κ R : Type*} [DecidableEq ι] [Fintype κ] [CommRing R]
+    (u v k : ι) (hu : u ≠ k) (hv : v ≠ k)
+    {f : MvPolynomial (ι × κ) R} {m : ℕ}
+    (hf : IsWeightedHomogeneous (fun z : ι × κ => if z.1 = k then 1 else 0) f m)
+    (n : ℕ) :
+    IsWeightedHomogeneous (fun z : ι × κ => if z.1 = k then 1 else 0)
+      (((blockShift R u v).toLinearMap ^ n) f) m := by
+  induction n with
+  | zero => simpa using hf
+  | succ n ih =>
+    rw [pow_succ', Module.End.mul_apply]
+    exact blockShift_spectator_homogeneous u v k hu hv ih
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/ShiftInjectivity.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/ShiftInjectivity.lean
index 792094fc925d001ff4017f863128eb7ceb7a738c..a5937b898b3c450f1ff4c9854fc5bac50d1abb08 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/ShiftInjectivity.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/ShiftInjectivity.lean
@@ -1,73 +1,73 @@
-import Mathlib
-
-namespace OAI
-
-noncomputable section
-
-namespace Problem346
-namespace ShiftInjectivity
-
-variable {K M : Type*} [Field K] [CharZero K]
-  [AddCommGroup M] [Module K M]
-
-/-- A weight-raising operator increases the weight of its iterates by two. -/
-theorem weight_pow (E H : Module.End K M)
-    (hHE : ∀ z, H (E z) = E (H z) + (2 : K) • E z)
-    (x : M) (w : K) (hx : H x = w • x) (n : ℕ) :
-    H ((E ^ n) x) = (w + 2 * (n : K)) • (E ^ n) x := by
-  induction n with
-  | zero => simpa using hx
-  | succ n ih =>
-    rw [pow_succ', Module.End.mul_apply, hHE, ih, map_smul,
-      ← add_smul]
-    congr 1
-    push_cast
-    ring
-
-/-- The lowering operator on a string generated from a vector in its kernel. -/
-theorem lowering_pow (D E H : Module.End K M)
-    (hDE : ∀ z, D (E z) = E (D z) - H z)
-    (hHE : ∀ z, H (E z) = E (H z) + (2 : K) • E z)
-    (x : M) (w : K) (hx : H x = w • x) (hDx : D x = 0) (n : ℕ) :
-    D ((E ^ (n + 1)) x) =
-      (-((n + 1 : K) * (w + n))) • (E ^ n) x := by
-  induction n with
-  | zero =>
-    simp [hDE, hDx, hx]
-  | succ n ih =>
-    rw [pow_succ', Module.End.mul_apply, hDE, ih,
-      weight_pow E H hHE x w hx (n + 1), map_smul]
-    rw [← Module.End.mul_apply, ← pow_succ', ← sub_smul]
-    congr 1
-    push_cast
-    ring
-
-/-- Positive-weight vectors killed by a lowering operator vanish provided that the
-opposite operator acts nilpotently on that vector. This is the algebraic form of
-positive-weight differential injectivity and requires no inner product. -/
-theorem eq_zero_of_positive_weight (D E H : Module.End K M)
-    (hDE : ∀ z, D (E z) = E (D z) - H z)
-    (hHE : ∀ z, H (E z) = E (H z) + (2 : K) • E z)
-    (x : M) (w : ℕ) (hw : 0 < w) (hx : H x = (w : K) • x)
-    (hDx : D x = 0) (hnil : ∃ N : ℕ, (E ^ N) x = 0) : x = 0 := by
-  obtain ⟨N, hN⟩ := hnil
-  induction N with
-  | zero => simpa using hN
-  | succ n ih =>
-    apply ih
-    have h := lowering_pow D E H hDE hHE x (w : K) hx hDx n
-    rw [hN, map_zero] at h
-    have hc : (-((n + 1 : K) * ((w : K) + n))) ≠ 0 := by
-      apply neg_ne_zero.mpr
-      apply mul_ne_zero
-      · exact_mod_cast (Nat.succ_ne_zero n)
-      · have : w + n ≠ 0 := by omega
-        exact_mod_cast this
-    exact (smul_eq_zero.mp h.symm).resolve_left hc
-
-end ShiftInjectivity
-end Problem346
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+
+namespace Problem346
+namespace ShiftInjectivity
+
+variable {K M : Type*} [Field K] [CharZero K]
+  [AddCommGroup M] [Module K M]
+
+/-- A weight-raising operator increases the weight of its iterates by two. -/
+theorem weight_pow (E H : Module.End K M)
+    (hHE : ∀ z, H (E z) = E (H z) + (2 : K) • E z)
+    (x : M) (w : K) (hx : H x = w • x) (n : ℕ) :
+    H ((E ^ n) x) = (w + 2 * (n : K)) • (E ^ n) x := by
+  induction n with
+  | zero => simpa using hx
+  | succ n ih =>
+    rw [pow_succ', Module.End.mul_apply, hHE, ih, map_smul,
+      ← add_smul]
+    congr 1
+    push_cast
+    ring
+
+/-- The lowering operator on a string generated from a vector in its kernel. -/
+theorem lowering_pow (D E H : Module.End K M)
+    (hDE : ∀ z, D (E z) = E (D z) - H z)
+    (hHE : ∀ z, H (E z) = E (H z) + (2 : K) • E z)
+    (x : M) (w : K) (hx : H x = w • x) (hDx : D x = 0) (n : ℕ) :
+    D ((E ^ (n + 1)) x) =
+      (-((n + 1 : K) * (w + n))) • (E ^ n) x := by
+  induction n with
+  | zero =>
+    simp [hDE, hDx, hx]
+  | succ n ih =>
+    rw [pow_succ', Module.End.mul_apply, hDE, ih,
+      weight_pow E H hHE x w hx (n + 1), map_smul]
+    rw [← Module.End.mul_apply, ← pow_succ', ← sub_smul]
+    congr 1
+    push_cast
+    ring
+
+/-- Positive-weight vectors killed by a lowering operator vanish provided that the
+opposite operator acts nilpotently on that vector. This is the algebraic form of
+positive-weight differential injectivity and requires no inner product. -/
+theorem eq_zero_of_positive_weight (D E H : Module.End K M)
+    (hDE : ∀ z, D (E z) = E (D z) - H z)
+    (hHE : ∀ z, H (E z) = E (H z) + (2 : K) • E z)
+    (x : M) (w : ℕ) (hw : 0 < w) (hx : H x = (w : K) • x)
+    (hDx : D x = 0) (hnil : ∃ N : ℕ, (E ^ N) x = 0) : x = 0 := by
+  obtain ⟨N, hN⟩ := hnil
+  induction N with
+  | zero => simpa using hN
+  | succ n ih =>
+    apply ih
+    have h := lowering_pow D E H hDE hHE x (w : K) hx hDx n
+    rw [hN, map_zero] at h
+    have hc : (-((n + 1 : K) * ((w : K) + n))) ≠ 0 := by
+      apply neg_ne_zero.mpr
+      apply mul_ne_zero
+      · exact_mod_cast (Nat.succ_ne_zero n)
+      · have : w + n ≠ 0 := by omega
+        exact_mod_cast this
+    exact (smul_eq_zero.mp h.symm).resolve_left hc
+
+end ShiftInjectivity
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/ShiftNilpotence.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/ShiftNilpotence.lean
index 2ac784ad743999c1c566b4280251caacff68385b..5eca41b66b46733799fc12e776c873a36226eca6 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/ShiftNilpotence.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/ShiftNilpotence.lean
@@ -1,109 +1,109 @@
-import Mathlib
-
-namespace OAI
-
-noncomputable section
-
-namespace Problem346
-
-section Derivation
-
-variable {R A : Type*} [CommRing R] [CommRing A] [Algebra R A]
-
-/-- A derivation is locally nilpotent at a product when it is locally nilpotent
-at each factor. The deliberately loose exponent avoids binomial coefficients. -/
-theorem derivation_pow_mul_eq_zero (D : Derivation R A A)
-    (m n : ℕ) (a b : A)
-    (ha : (D.toLinearMap ^ m) a = 0)
-    (hb : (D.toLinearMap ^ n) b = 0) :
-    (D.toLinearMap ^ (m + n)) (a * b) = 0 := by
-  induction m generalizing n a b with
-  | zero =>
-      simp only [pow_zero, Module.End.one_apply] at ha
-      simp [ha]
-  | succ m hm =>
-      induction n generalizing a b with
-      | zero =>
-          simp only [pow_zero, Module.End.one_apply] at hb
-          simp [hb]
-      | succ n hn =>
-          have hDa : (D.toLinearMap ^ m) (D a) = 0 := by
-            rw [pow_succ, Module.End.mul_apply] at ha
-            exact ha
-          have hDb : (D.toLinearMap ^ n) (D b) = 0 := by
-            rw [pow_succ, Module.End.mul_apply] at hb
-            exact hb
-          have hp := hm (n + 1) (D a) b hDa hb
-          have hq := hn a (D b) ha hDb
-          have he : m + 1 + (n + 1) = (m + (n + 1)) + 1 := by omega
-          rw [he, pow_succ, Module.End.mul_apply]
-          change (D.toLinearMap ^ (m + (n + 1))) (D (a * b)) = 0
-          rw [D.leibniz, map_add]
-          have hp' : (D.toLinearMap ^ (m + (n + 1))) (b * D a) = 0 := by
-            simpa only [mul_comm] using hp
-          have hq' : (D.toLinearMap ^ (m + (n + 1))) (a * D b) = 0 := by
-            simpa only [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm] using hq
-          simp only [smul_eq_mul, hp', hq', add_zero]
-
-
-/-- Pointwise local nilpotence of a derivation is closed under addition. -/
-theorem derivation_exists_pow_add_eq_zero (D : Derivation R A A) (a b : A)
-    (ha : ∃ m : ℕ, (D.toLinearMap ^ m) a = 0)
-    (hb : ∃ n : ℕ, (D.toLinearMap ^ n) b = 0) :
-    ∃ n : ℕ, (D.toLinearMap ^ n) (a + b) = 0 := by
-  obtain ⟨m, hm⟩ := ha
-  obtain ⟨n, hn⟩ := hb
-  refine ⟨m + n, ?_⟩
-  rw [map_add, Module.End.pow_map_zero_of_le (Nat.le_add_right m n) hm,
-    Module.End.pow_map_zero_of_le (Nat.le_add_left n m) hn, add_zero]
-
-/-- Pointwise local nilpotence of a derivation is closed under multiplication. -/
-theorem derivation_exists_pow_mul_eq_zero (D : Derivation R A A) (a b : A)
-    (ha : ∃ m : ℕ, (D.toLinearMap ^ m) a = 0)
-    (hb : ∃ n : ℕ, (D.toLinearMap ^ n) b = 0) :
-    ∃ n : ℕ, (D.toLinearMap ^ n) (a * b) = 0 := by
-  obtain ⟨m, hm⟩ := ha
-  obtain ⟨n, hn⟩ := hb
-  exact ⟨m + n, derivation_pow_mul_eq_zero D m n a b hm hn⟩
-
-end Derivation
-
-section MvPolynomial
-
-variable {R σ : Type*} [CommRing R]
-
-/-- Local nilpotence on polynomial generators implies local nilpotence everywhere.
-No finiteness assumption on the variable type is needed, so spectator variables
-can be retained without changing the argument. -/
-theorem mvPolynomial_derivation_locally_nilpotent
-    (D : Derivation R (MvPolynomial σ R) (MvPolynomial σ R))
-    (hX : ∀ i : σ, ∃ n : ℕ, (D.toLinearMap ^ n) (MvPolynomial.X i) = 0)
-    (f : MvPolynomial σ R) : ∃ n : ℕ, (D.toLinearMap ^ n) f = 0 := by
-  induction f using MvPolynomial.induction_on with
-  | C r =>
-      refine ⟨1, ?_⟩
-      rw [pow_one]
-      exact MvPolynomial.derivation_C D r
-  | add p q hp hq =>
-      exact derivation_exists_pow_add_eq_zero D p q hp hq
-  | mul_X p i hp =>
-      exact derivation_exists_pow_mul_eq_zero D p (MvPolynomial.X i) hp (hX i)
-
-/-- The square-zero-on-generators criterion used for directional shifts. -/
-theorem mvPolynomial_derivation_locally_nilpotent_of_square_zero_X
-    (D : Derivation R (MvPolynomial σ R) (MvPolynomial σ R))
-    (hX : ∀ i : σ, D (D (MvPolynomial.X i)) = 0)
-    (f : MvPolynomial σ R) : ∃ n : ℕ, (D.toLinearMap ^ n) f = 0 := by
-  apply mvPolynomial_derivation_locally_nilpotent D _ f
-  intro i
-  refine ⟨2, ?_⟩
-  rw [pow_two, Module.End.mul_apply]
-  exact hX i
-
-end MvPolynomial
-
-end Problem346
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+
+namespace Problem346
+
+section Derivation
+
+variable {R A : Type*} [CommRing R] [CommRing A] [Algebra R A]
+
+/-- A derivation is locally nilpotent at a product when it is locally nilpotent
+at each factor. The deliberately loose exponent avoids binomial coefficients. -/
+theorem derivation_pow_mul_eq_zero (D : Derivation R A A)
+    (m n : ℕ) (a b : A)
+    (ha : (D.toLinearMap ^ m) a = 0)
+    (hb : (D.toLinearMap ^ n) b = 0) :
+    (D.toLinearMap ^ (m + n)) (a * b) = 0 := by
+  induction m generalizing n a b with
+  | zero =>
+      simp only [pow_zero, Module.End.one_apply] at ha
+      simp [ha]
+  | succ m hm =>
+      induction n generalizing a b with
+      | zero =>
+          simp only [pow_zero, Module.End.one_apply] at hb
+          simp [hb]
+      | succ n hn =>
+          have hDa : (D.toLinearMap ^ m) (D a) = 0 := by
+            rw [pow_succ, Module.End.mul_apply] at ha
+            exact ha
+          have hDb : (D.toLinearMap ^ n) (D b) = 0 := by
+            rw [pow_succ, Module.End.mul_apply] at hb
+            exact hb
+          have hp := hm (n + 1) (D a) b hDa hb
+          have hq := hn a (D b) ha hDb
+          have he : m + 1 + (n + 1) = (m + (n + 1)) + 1 := by omega
+          rw [he, pow_succ, Module.End.mul_apply]
+          change (D.toLinearMap ^ (m + (n + 1))) (D (a * b)) = 0
+          rw [D.leibniz, map_add]
+          have hp' : (D.toLinearMap ^ (m + (n + 1))) (b * D a) = 0 := by
+            simpa only [mul_comm] using hp
+          have hq' : (D.toLinearMap ^ (m + (n + 1))) (a * D b) = 0 := by
+            simpa only [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm] using hq
+          simp only [smul_eq_mul, hp', hq', add_zero]
+
+
+/-- Pointwise local nilpotence of a derivation is closed under addition. -/
+theorem derivation_exists_pow_add_eq_zero (D : Derivation R A A) (a b : A)
+    (ha : ∃ m : ℕ, (D.toLinearMap ^ m) a = 0)
+    (hb : ∃ n : ℕ, (D.toLinearMap ^ n) b = 0) :
+    ∃ n : ℕ, (D.toLinearMap ^ n) (a + b) = 0 := by
+  obtain ⟨m, hm⟩ := ha
+  obtain ⟨n, hn⟩ := hb
+  refine ⟨m + n, ?_⟩
+  rw [map_add, Module.End.pow_map_zero_of_le (Nat.le_add_right m n) hm,
+    Module.End.pow_map_zero_of_le (Nat.le_add_left n m) hn, add_zero]
+
+/-- Pointwise local nilpotence of a derivation is closed under multiplication. -/
+theorem derivation_exists_pow_mul_eq_zero (D : Derivation R A A) (a b : A)
+    (ha : ∃ m : ℕ, (D.toLinearMap ^ m) a = 0)
+    (hb : ∃ n : ℕ, (D.toLinearMap ^ n) b = 0) :
+    ∃ n : ℕ, (D.toLinearMap ^ n) (a * b) = 0 := by
+  obtain ⟨m, hm⟩ := ha
+  obtain ⟨n, hn⟩ := hb
+  exact ⟨m + n, derivation_pow_mul_eq_zero D m n a b hm hn⟩
+
+end Derivation
+
+section MvPolynomial
+
+variable {R σ : Type*} [CommRing R]
+
+/-- Local nilpotence on polynomial generators implies local nilpotence everywhere.
+No finiteness assumption on the variable type is needed, so spectator variables
+can be retained without changing the argument. -/
+theorem mvPolynomial_derivation_locally_nilpotent
+    (D : Derivation R (MvPolynomial σ R) (MvPolynomial σ R))
+    (hX : ∀ i : σ, ∃ n : ℕ, (D.toLinearMap ^ n) (MvPolynomial.X i) = 0)
+    (f : MvPolynomial σ R) : ∃ n : ℕ, (D.toLinearMap ^ n) f = 0 := by
+  induction f using MvPolynomial.induction_on with
+  | C r =>
+      refine ⟨1, ?_⟩
+      rw [pow_one]
+      exact MvPolynomial.derivation_C D r
+  | add p q hp hq =>
+      exact derivation_exists_pow_add_eq_zero D p q hp hq
+  | mul_X p i hp =>
+      exact derivation_exists_pow_mul_eq_zero D p (MvPolynomial.X i) hp (hX i)
+
+/-- The square-zero-on-generators criterion used for directional shifts. -/
+theorem mvPolynomial_derivation_locally_nilpotent_of_square_zero_X
+    (D : Derivation R (MvPolynomial σ R) (MvPolynomial σ R))
+    (hX : ∀ i : σ, D (D (MvPolynomial.X i)) = 0)
+    (f : MvPolynomial σ R) : ∃ n : ℕ, (D.toLinearMap ^ n) f = 0 := by
+  apply mvPolynomial_derivation_locally_nilpotent D _ f
+  intro i
+  refine ⟨2, ?_⟩
+  rw [pow_two, Module.End.mul_apply]
+  exact hX i
+
+end MvPolynomial
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/ShiftSequence.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/ShiftSequence.lean
index a1ada5ce1e85bf463819b6893be22c351a94b93b..2364c04efa412cbc44d66a2a3e291d734c0912b0 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/ShiftSequence.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/ShiftSequence.lean
@@ -1,65 +1,65 @@
-import OAI.RepresentationTheory.FoulkesHowe.DerivationInjectivity
-import OAI.RepresentationTheory.FoulkesHowe.BlockDerivation
-import OAI.RepresentationTheory.FoulkesHowe.BlockWeight
-import OAI.RepresentationTheory.FoulkesHowe.ShiftHomogeneity
-
-namespace OAI
-
-noncomputable section
-
-namespace Problem346
-
-open MvPolynomial PolynomialWeights
-
-variable {B I : Type*} [Fintype B] [Fintype I] [DecidableEq B]
-
-/-- Positive-weight injectivity for the shared `blockShift` derivation API. -/
-theorem blockShift_eq_zero_of_pos_weight (src dst : B) (hsd : src ≠ dst)
-    {f : MvPolynomial (B × I) ℂ} {p q : ℕ} (hpq : q < p)
-    (hp : IsBlockHomogeneous src f p) (hq : IsBlockHomogeneous dst f q)
-    (hDf : blockShift ℂ src dst f = 0) : f = 0 := by
-  refine derivation_eq_zero_of_positive_weight
-    (blockShift ℂ src dst) (blockShift ℂ dst src)
-    (blockShift ℂ src src - blockShift ℂ dst dst)
-    (blockShift_commutator ℂ src dst) (blockShift_weight_commutator ℂ src dst hsd)
-    ?_ f p q hpq ?_ hDf
-  · rintro ⟨b, i⟩
-    exact blockShift_square_X ℂ dst src b hsd.symm i
-  · apply block_difference_eigenvalue src dst _ _ _ _ hp hq
-    · intro b i
-      by_cases hb : b = src
-      · subst b; simp only [blockShift_X]
-      · simp only [blockShift_X, ite_eq_right hb]
-    · intro b i
-      by_cases hb : b = dst
-      · subst b; simp only [blockShift_X]
-      · simp only [blockShift_X, ite_eq_right hb]
-
-/-- A chain of repeated shifts can be cancelled as long as its least intermediate
-source-minus-destination weight is positive. The convenient sufficient bound
-`q + 2*n ≤ p` is exactly what is used for each block in the quadratic argument. -/
-theorem blockShift_pow_eq_zero_of_degree_bound (src dst : B) (hsd : src ≠ dst)
-    {f : MvPolynomial (B × I) ℂ} {p q : ℕ}
-    (hp : IsBlockHomogeneous src f p) (hq : IsBlockHomogeneous dst f q)
-    (n : ℕ) : q + 2 * n ≤ p →
-    (((blockShift ℂ src dst).toLinearMap ^ n) f = 0) → f = 0 := by
-  induction n with
-  | zero =>
-    intro _ hz
-    simpa using hz
-  | succ n ih =>
-    intro hbound hz
-    apply ih (by omega)
-    apply blockShift_eq_zero_of_pos_weight src dst hsd (p := p - n) (q := q + n)
-      (by omega)
-    · exact blockShift_pow_source_homogeneous src dst hsd hp n
-    · exact blockShift_pow_destination_homogeneous src dst hsd hq n
-    · change (blockShift ℂ src dst).toLinearMap
-        (((blockShift ℂ src dst).toLinearMap ^ n) f) = 0
-      simpa only [pow_succ', Module.End.mul_apply] using hz
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.DerivationInjectivity
+import OAI.RepresentationTheory.FoulkesHowe.BlockDerivation
+import OAI.RepresentationTheory.FoulkesHowe.BlockWeight
+import OAI.RepresentationTheory.FoulkesHowe.ShiftHomogeneity
+
+namespace OAI
+
+noncomputable section
+
+namespace Problem346
+
+open MvPolynomial PolynomialWeights
+
+variable {B I : Type*} [Fintype B] [Fintype I] [DecidableEq B]
+
+/-- Positive-weight injectivity for the shared `blockShift` derivation API. -/
+theorem blockShift_eq_zero_of_pos_weight (src dst : B) (hsd : src ≠ dst)
+    {f : MvPolynomial (B × I) ℂ} {p q : ℕ} (hpq : q < p)
+    (hp : IsBlockHomogeneous src f p) (hq : IsBlockHomogeneous dst f q)
+    (hDf : blockShift ℂ src dst f = 0) : f = 0 := by
+  refine derivation_eq_zero_of_positive_weight
+    (blockShift ℂ src dst) (blockShift ℂ dst src)
+    (blockShift ℂ src src - blockShift ℂ dst dst)
+    (blockShift_commutator ℂ src dst) (blockShift_weight_commutator ℂ src dst hsd)
+    ?_ f p q hpq ?_ hDf
+  · rintro ⟨b, i⟩
+    exact blockShift_square_X ℂ dst src b hsd.symm i
+  · apply block_difference_eigenvalue src dst _ _ _ _ hp hq
+    · intro b i
+      by_cases hb : b = src
+      · subst b; simp only [blockShift_X]
+      · simp only [blockShift_X, ite_eq_right hb]
+    · intro b i
+      by_cases hb : b = dst
+      · subst b; simp only [blockShift_X]
+      · simp only [blockShift_X, ite_eq_right hb]
+
+/-- A chain of repeated shifts can be cancelled as long as its least intermediate
+source-minus-destination weight is positive. The convenient sufficient bound
+`q + 2*n ≤ p` is exactly what is used for each block in the quadratic argument. -/
+theorem blockShift_pow_eq_zero_of_degree_bound (src dst : B) (hsd : src ≠ dst)
+    {f : MvPolynomial (B × I) ℂ} {p q : ℕ}
+    (hp : IsBlockHomogeneous src f p) (hq : IsBlockHomogeneous dst f q)
+    (n : ℕ) : q + 2 * n ≤ p →
+    (((blockShift ℂ src dst).toLinearMap ^ n) f = 0) → f = 0 := by
+  induction n with
+  | zero =>
+    intro _ hz
+    simpa using hz
+  | succ n ih =>
+    intro hbound hz
+    apply ih (by omega)
+    apply blockShift_eq_zero_of_pos_weight src dst hsd (p := p - n) (q := q + n)
+      (by omega)
+    · exact blockShift_pow_source_homogeneous src dst hsd hp n
+    · exact blockShift_pow_destination_homogeneous src dst hsd hq n
+    · change (blockShift ℂ src dst).toLinearMap
+        (((blockShift ℂ src dst).toLinearMap ^ n) f) = 0
+      simpa only [pow_succ', Module.End.mul_apply] using hz
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/SlotDerivative.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/SlotDerivative.lean
index eeaa7a6195de24c0a67b2d32cf2211845c043439..31ded81881d254178c7ce369978e22eb6d3b45d8 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/SlotDerivative.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/SlotDerivative.lean
@@ -1,150 +1,150 @@
-import Mathlib
-
-namespace OAI
-
-noncomputable section
-open scoped BigOperators
-
-namespace Problem346.SlotPolynomial
-
-variable {R A S I B : Type*} [CommRing R] [CommRing A] [Algebra R A] [DecidableEq S]
-
-/-- Leibniz rule for a finite product, expressed by replacing one factor. -/
-theorem derivation_prod (D : Derivation R A A) (s : Finset S) (f : S → A) :
-    D (∏ i ∈ s, f i) =
-      ∑ i ∈ s, ∏ j ∈ s, Function.update f i (D (f i)) j := by
-  classical
-  induction s using Finset.induction_on with
-  | empty => simp
-  | @insert a s ha ih =>
-    rw [Finset.prod_insert ha, D.leibniz, ih, Finset.sum_insert ha]
-    simp only [Finset.prod_insert ha, Function.update_self, smul_eq_mul]
-    have h₁ : (∏ j ∈ s, Function.update f a (D (f a)) j) = ∏ j ∈ s, f j :=
-      Finset.prod_update_of_notMem ha _ _
-    rw [h₁, Finset.mul_sum]
-    have h₂ : (∑ i ∈ s, Function.update f i (D (f i)) a *
-          ∏ j ∈ s, Function.update f i (D (f i)) j) =
-        ∑ i ∈ s, f a * ∏ j ∈ s, Function.update f i (D (f i)) j := by
-      apply Finset.sum_congr rfl
-      intro i hi
-      have hai : a ≠ i := by
-        intro h
-        subst i
-        exact ha hi
-      rw [Function.update_of_ne hai]
-    rw [h₂]
-    ring
-
-variable [Fintype S] [Fintype I]
-
-/-- Coordinate polynomial of a coefficient array and an assignment of slots to blocks. -/
-def polynomial (c : (S → I) → R) (l : S → B) : MvPolynomial (B × I) R :=
-  ∑ k : S → I, MvPolynomial.C (c k) * ∏ s : S, MvPolynomial.X (l s, k s)
-
-/-- A directional block derivation replaces each occurrence of the source block in turn. -/
-theorem derivation_polynomial [DecidableEq B]
-    (c : (S → I) → R) (l : S → B) (src dst : B)
-    (D : Derivation R (MvPolynomial (B × I) R) (MvPolynomial (B × I) R))
-    (hD : ∀ b i, D (MvPolynomial.X (b, i)) =
-      if b = src then MvPolynomial.X (dst, i) else 0) :
-    D (polynomial c l) =
-      ∑ s ∈ Finset.univ.filter (fun s => l s = src),
-        polynomial c (Function.update l s dst) := by
-  classical
-  unfold polynomial
-  rw [map_sum]
-  simp_rw [D.leibniz, MvPolynomial.derivation_C, smul_zero, add_zero,
-    smul_eq_mul, derivation_prod]
-  simp_rw [Finset.mul_sum]
-  rw [Finset.sum_comm]
-  rw [Finset.sum_filter]
-  apply Finset.sum_congr rfl
-  intro s hs
-  by_cases h : l s = src
-  · simp only [h, ite_true]
-    apply Finset.sum_congr rfl
-    intro k hk
-    congr 1
-    apply Finset.prod_congr rfl
-    intro j hj
-    by_cases hjs : j = s
-    · subst j
-      simp [hD]
-    · simp [Function.update_of_ne hjs]
-  · simp only [h, ite_false]
-    apply Finset.sum_eq_zero
-    intro k hk
-    have hz : (∏ j : S, Function.update (fun s => MvPolynomial.X (l s, k s))
-        s (D (MvPolynomial.X (l s, k s))) j) = 0 := by
-      apply Finset.prod_eq_zero (Finset.mem_univ s)
-      simp [hD, h]
-    rw [hz, mul_zero]
-
-/-- Reindexing slots moves the same permutation onto the coefficient array. -/
-theorem polynomial_reindex (c : (S → I) → R) (l : S → B) (e : Equiv.Perm S) :
-    polynomial c (l ∘ e) = polynomial (fun k => c (k ∘ e)) l := by
-  classical
-  unfold polynomial
-  apply Fintype.sum_equiv (Equiv.arrowCongr e (Equiv.refl I))
-  intro k
-  congr 1
-  · congr 1
-    change c k = c _
-    congr 1
-    funext s
-    simp
-  · apply Fintype.prod_equiv e
-    intro s
-    simp
-
-/-- Symmetry of coefficients induces invariance under the same permutation of slot labels. -/
-theorem polynomial_reindex_of_invariant (c : (S → I) → R) (l : S → B)
-    (e : Equiv.Perm S) (hc : ∀ k, c (k ∘ e) = c k) :
-    polynomial c (l ∘ e) = polynomial c l := by
-  rw [polynomial_reindex]
-  congr 1
-  funext k
-  exact hc k
-
-/-- Swapping two equal-labelled slots shows that replacing either label gives the same polynomial. -/
-theorem polynomial_update_eq (c : (S → I) → R) (l : S → B) (s t : S) (dst : B)
-    (hl : l s = l t)
-    (hc : ∀ k, c (k ∘ Equiv.swap s t) = c k) :
-    polynomial c (Function.update l s dst) = polynomial c (Function.update l t dst) := by
-  classical
-  have hlabels : Function.update l s dst =
-      Function.update l t dst ∘ Equiv.swap s t := by
-    funext j
-    by_cases hjs : j = s
-    · subst j
-      simp
-    by_cases hjt : j = t
-    · subst j
-      simp [hjs, Ne.symm hjs, hl]
-    · simp [Equiv.swap_apply_of_ne_of_ne hjs hjt, hjs, hjt]
-  rw [hlabels]
-  exact polynomial_reindex_of_invariant c _ _ hc
-
-/-- If all source-labelled slots are interchangeable, the derivative is a single replacement
-multiplied by their number. -/
-theorem derivation_polynomial_eq_card_smul [DecidableEq B]
-    (c : (S → I) → R) (l : S → B) (src dst : B) (t : S) (ht : l t = src)
-    (D : Derivation R (MvPolynomial (B × I) R) (MvPolynomial (B × I) R))
-    (hD : ∀ b i, D (MvPolynomial.X (b, i)) =
-      if b = src then MvPolynomial.X (dst, i) else 0)
-    (hc : ∀ s, l s = src → ∀ k, c (k ∘ Equiv.swap s t) = c k) :
-    D (polynomial c l) =
-      (Finset.univ.filter (fun s => l s = src)).card •
-        polynomial c (Function.update l t dst) := by
-  classical
-  rw [derivation_polynomial c l src dst D hD, ← Finset.sum_const]
-  apply Finset.sum_congr rfl
-  intro s hs
-  have hs' := (Finset.mem_filter.mp hs).2
-  exact polynomial_update_eq c l s t dst (hs'.trans ht.symm) (hc s hs')
-
-end Problem346.SlotPolynomial
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators
+
+namespace Problem346.SlotPolynomial
+
+variable {R A S I B : Type*} [CommRing R] [CommRing A] [Algebra R A] [DecidableEq S]
+
+/-- Leibniz rule for a finite product, expressed by replacing one factor. -/
+theorem derivation_prod (D : Derivation R A A) (s : Finset S) (f : S → A) :
+    D (∏ i ∈ s, f i) =
+      ∑ i ∈ s, ∏ j ∈ s, Function.update f i (D (f i)) j := by
+  classical
+  induction s using Finset.induction_on with
+  | empty => simp
+  | @insert a s ha ih =>
+    rw [Finset.prod_insert ha, D.leibniz, ih, Finset.sum_insert ha]
+    simp only [Finset.prod_insert ha, Function.update_self, smul_eq_mul]
+    have h₁ : (∏ j ∈ s, Function.update f a (D (f a)) j) = ∏ j ∈ s, f j :=
+      Finset.prod_update_of_notMem ha _ _
+    rw [h₁, Finset.mul_sum]
+    have h₂ : (∑ i ∈ s, Function.update f i (D (f i)) a *
+          ∏ j ∈ s, Function.update f i (D (f i)) j) =
+        ∑ i ∈ s, f a * ∏ j ∈ s, Function.update f i (D (f i)) j := by
+      apply Finset.sum_congr rfl
+      intro i hi
+      have hai : a ≠ i := by
+        intro h
+        subst i
+        exact ha hi
+      rw [Function.update_of_ne hai]
+    rw [h₂]
+    ring
+
+variable [Fintype S] [Fintype I]
+
+/-- Coordinate polynomial of a coefficient array and an assignment of slots to blocks. -/
+def polynomial (c : (S → I) → R) (l : S → B) : MvPolynomial (B × I) R :=
+  ∑ k : S → I, MvPolynomial.C (c k) * ∏ s : S, MvPolynomial.X (l s, k s)
+
+/-- A directional block derivation replaces each occurrence of the source block in turn. -/
+theorem derivation_polynomial [DecidableEq B]
+    (c : (S → I) → R) (l : S → B) (src dst : B)
+    (D : Derivation R (MvPolynomial (B × I) R) (MvPolynomial (B × I) R))
+    (hD : ∀ b i, D (MvPolynomial.X (b, i)) =
+      if b = src then MvPolynomial.X (dst, i) else 0) :
+    D (polynomial c l) =
+      ∑ s ∈ Finset.univ.filter (fun s => l s = src),
+        polynomial c (Function.update l s dst) := by
+  classical
+  unfold polynomial
+  rw [map_sum]
+  simp_rw [D.leibniz, MvPolynomial.derivation_C, smul_zero, add_zero,
+    smul_eq_mul, derivation_prod]
+  simp_rw [Finset.mul_sum]
+  rw [Finset.sum_comm]
+  rw [Finset.sum_filter]
+  apply Finset.sum_congr rfl
+  intro s hs
+  by_cases h : l s = src
+  · simp only [h, ite_true]
+    apply Finset.sum_congr rfl
+    intro k hk
+    congr 1
+    apply Finset.prod_congr rfl
+    intro j hj
+    by_cases hjs : j = s
+    · subst j
+      simp [hD]
+    · simp [Function.update_of_ne hjs]
+  · simp only [h, ite_false]
+    apply Finset.sum_eq_zero
+    intro k hk
+    have hz : (∏ j : S, Function.update (fun s => MvPolynomial.X (l s, k s))
+        s (D (MvPolynomial.X (l s, k s))) j) = 0 := by
+      apply Finset.prod_eq_zero (Finset.mem_univ s)
+      simp [hD, h]
+    rw [hz, mul_zero]
+
+/-- Reindexing slots moves the same permutation onto the coefficient array. -/
+theorem polynomial_reindex (c : (S → I) → R) (l : S → B) (e : Equiv.Perm S) :
+    polynomial c (l ∘ e) = polynomial (fun k => c (k ∘ e)) l := by
+  classical
+  unfold polynomial
+  apply Fintype.sum_equiv (Equiv.arrowCongr e (Equiv.refl I))
+  intro k
+  congr 1
+  · congr 1
+    change c k = c _
+    congr 1
+    funext s
+    simp
+  · apply Fintype.prod_equiv e
+    intro s
+    simp
+
+/-- Symmetry of coefficients induces invariance under the same permutation of slot labels. -/
+theorem polynomial_reindex_of_invariant (c : (S → I) → R) (l : S → B)
+    (e : Equiv.Perm S) (hc : ∀ k, c (k ∘ e) = c k) :
+    polynomial c (l ∘ e) = polynomial c l := by
+  rw [polynomial_reindex]
+  congr 1
+  funext k
+  exact hc k
+
+/-- Swapping two equal-labelled slots shows that replacing either label gives the same polynomial. -/
+theorem polynomial_update_eq (c : (S → I) → R) (l : S → B) (s t : S) (dst : B)
+    (hl : l s = l t)
+    (hc : ∀ k, c (k ∘ Equiv.swap s t) = c k) :
+    polynomial c (Function.update l s dst) = polynomial c (Function.update l t dst) := by
+  classical
+  have hlabels : Function.update l s dst =
+      Function.update l t dst ∘ Equiv.swap s t := by
+    funext j
+    by_cases hjs : j = s
+    · subst j
+      simp
+    by_cases hjt : j = t
+    · subst j
+      simp [hjs, Ne.symm hjs, hl]
+    · simp [Equiv.swap_apply_of_ne_of_ne hjs hjt, hjs, hjt]
+  rw [hlabels]
+  exact polynomial_reindex_of_invariant c _ _ hc
+
+/-- If all source-labelled slots are interchangeable, the derivative is a single replacement
+multiplied by their number. -/
+theorem derivation_polynomial_eq_card_smul [DecidableEq B]
+    (c : (S → I) → R) (l : S → B) (src dst : B) (t : S) (ht : l t = src)
+    (D : Derivation R (MvPolynomial (B × I) R) (MvPolynomial (B × I) R))
+    (hD : ∀ b i, D (MvPolynomial.X (b, i)) =
+      if b = src then MvPolynomial.X (dst, i) else 0)
+    (hc : ∀ s, l s = src → ∀ k, c (k ∘ Equiv.swap s t) = c k) :
+    D (polynomial c l) =
+      (Finset.univ.filter (fun s => l s = src)).card •
+        polynomial c (Function.update l t dst) := by
+  classical
+  rw [derivation_polynomial c l src dst D hD, ← Finset.sum_const]
+  apply Finset.sum_congr rfl
+  intro s hs
+  have hs' := (Finset.mem_filter.mp hs).2
+  exact polynomial_update_eq c l s t dst (hs'.trans ht.symm) (hc s hs')
+
+end Problem346.SlotPolynomial
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/Spanning.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/Spanning.lean
index 6f042a72082ec8ddb081a29d8898fb597b223c31..5fe81e917178777099b97815b353cb60d8b2e141 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/Spanning.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/Spanning.lean
@@ -1,113 +1,113 @@
-import OAI.RepresentationTheory.FoulkesHowe.MonomialMultilinear
-
-namespace OAI
-
-noncomputable section
-open scoped BigOperators
-universe u v
-namespace Problem346
-
-/-- Monomials span the symmetric power, in its subtype realization. -/
-theorem span_symMonomial (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] :
-    Submodule.span ℂ (Set.range (symMonomial n V)) = ⊤ := by
-  apply (Submodule.span_range_subtype_eq_top_iff (symPowSubmodule n V)
-    (fun x => Submodule.subset_span (Set.mem_range_self x))).2
-  rfl
-
-/-- Maps out of a symmetric power are determined by monomials. -/
-theorem symPow_linearMap_ext (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]
-    {W : Type v} [AddCommGroup W] [Module ℂ W]
-    (f g : SymPow n V →ₗ[ℂ] W)
-    (h : ∀ x : Fin n → V, f (symMonomial n V x) = g (symMonomial n V x)) : f = g := by
-  apply (Submodule.linearMap_eq_iff_of_span_eq_top f g (span_symMonomial n V)).2
-  rintro ⟨_, x, rfl⟩
-  exact h x
-
-/-- A multilinear map vanishing on tuples from a spanning set is zero. -/
-theorem multilinear_eq_zero_of_span
-    {M : Type u} [AddCommGroup M] [Module ℂ M]
-    {N : Type v} [AddCommGroup N] [Module ℂ N]
-    {s : Set M} (hs : Submodule.span ℂ s = ⊤) (n : ℕ) :
-    ∀ F : MultilinearMap ℂ (fun _ : Fin n => M) N,
-      (∀ x, (∀ i, x i ∈ s) → F x = 0) → F = 0 := by
-  induction n with
-  | zero =>
-      intro F hF
-      ext x
-      exact hF x (fun i => Fin.elim0 i)
-  | succ n ih =>
-      intro F hF
-      have hc : F.curryLeft = 0 := by
-        apply (Submodule.linearMap_eq_zero_iff_of_span_eq_top F.curryLeft hs).2
-        intro x
-        apply ih
-        intro y hy
-        apply hF
-        exact Fin.cases x.property hy
-      ext x
-      have h := congrArg (fun f : M →ₗ[ℂ] MultilinearMap ℂ (fun _ : Fin n => M) N =>
-        f (x 0) (Fin.tail x)) hc
-      simpa using h
-
-/-- Multilinear maps agree if they agree on tuples from a spanning set. -/
-theorem multilinear_ext_of_span
-    {M : Type u} [AddCommGroup M] [Module ℂ M]
-    {N : Type v} [AddCommGroup N] [Module ℂ N]
-    {s : Set M} (hs : Submodule.span ℂ s = ⊤) (n : ℕ)
-    (F G : MultilinearMap ℂ (fun _ : Fin n => M) N)
-    (h : ∀ x, (∀ i, x i ∈ s) → F x = G x) : F = G := by
-  apply sub_eq_zero.mp
-  apply multilinear_eq_zero_of_span hs n
-  intro x hx
-  simp only [sub_apply, h x hx, sub_self]
-
-/-- Linear maps out of a plethysm are determined by double monomials. -/
-theorem plethysm_linearMap_ext (a b : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V]
-    {W : Type v} [AddCommGroup W] [Module ℂ W]
-    (f g : SymPow b (SymPow a V) →ₗ[ℂ] W)
-    (h : ∀ x : Fin b → Fin a → V,
-      f (symMonomial b (SymPow a V) (fun j => symMonomial a V (x j))) =
-      g (symMonomial b (SymPow a V) (fun j => symMonomial a V (x j)))) : f = g := by
-  classical
-  apply symPow_linearMap_ext b (SymPow a V)
-  have hm : f.compMultilinearMap (symMonomialMultilinear b (SymPow a V)) =
-      g.compMultilinearMap (symMonomialMultilinear b (SymPow a V)) := by
-    apply multilinear_ext_of_span (span_symMonomial a V) b
-    intro x hx
-    choose v hv using hx
-    have hx' : x = fun j => symMonomial a V (v j) := funext (fun j => (hv j).symm)
-    subst x
-    exact h v
-  intro x
-  exact congrArg (fun F => F x) hm
-
-/-- Double monomials span the plethysm. -/
-theorem span_plethysmMonomial (a b : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V] :
-    Submodule.span ℂ (Set.range (fun x : Fin b → Fin a → V =>
-      symMonomial b (SymPow a V) (fun j => symMonomial a V (x j)))) = ⊤ := by
-  let p := Submodule.span ℂ (Set.range (fun x : Fin b → Fin a → V =>
-    symMonomial b (SymPow a V) (fun j => symMonomial a V (x j))))
-  change p = ⊤
-  have hp : p.mkQ = 0 := by
-    apply plethysm_linearMap_ext a b V
-    intro x
-    apply (Submodule.Quotient.mk_eq_zero p).mpr
-    exact Submodule.subset_span (Set.mem_range_self x)
-  rw [← Submodule.ker_mkQ p, hp, LinearMap.ker_zero]
-
-/-- The defining formula determines a Foulkes map uniquely. -/
-theorem IsFoulkesMap.unique (a b : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V]
-    {f g : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V)}
-    (hf : IsFoulkesMap a b V f) (hg : IsFoulkesMap a b V g) : f = g := by
-  apply plethysm_linearMap_ext a b V
-  intro x
-  exact (hf x).trans (hg x).symm
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.MonomialMultilinear
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators
+universe u v
+namespace Problem346
+
+/-- Monomials span the symmetric power, in its subtype realization. -/
+theorem span_symMonomial (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] :
+    Submodule.span ℂ (Set.range (symMonomial n V)) = ⊤ := by
+  apply (Submodule.span_range_subtype_eq_top_iff (symPowSubmodule n V)
+    (fun x => Submodule.subset_span (Set.mem_range_self x))).2
+  rfl
+
+/-- Maps out of a symmetric power are determined by monomials. -/
+theorem symPow_linearMap_ext (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]
+    {W : Type v} [AddCommGroup W] [Module ℂ W]
+    (f g : SymPow n V →ₗ[ℂ] W)
+    (h : ∀ x : Fin n → V, f (symMonomial n V x) = g (symMonomial n V x)) : f = g := by
+  apply (Submodule.linearMap_eq_iff_of_span_eq_top f g (span_symMonomial n V)).2
+  rintro ⟨_, x, rfl⟩
+  exact h x
+
+/-- A multilinear map vanishing on tuples from a spanning set is zero. -/
+theorem multilinear_eq_zero_of_span
+    {M : Type u} [AddCommGroup M] [Module ℂ M]
+    {N : Type v} [AddCommGroup N] [Module ℂ N]
+    {s : Set M} (hs : Submodule.span ℂ s = ⊤) (n : ℕ) :
+    ∀ F : MultilinearMap ℂ (fun _ : Fin n => M) N,
+      (∀ x, (∀ i, x i ∈ s) → F x = 0) → F = 0 := by
+  induction n with
+  | zero =>
+      intro F hF
+      ext x
+      exact hF x (fun i => Fin.elim0 i)
+  | succ n ih =>
+      intro F hF
+      have hc : F.curryLeft = 0 := by
+        apply (Submodule.linearMap_eq_zero_iff_of_span_eq_top F.curryLeft hs).2
+        intro x
+        apply ih
+        intro y hy
+        apply hF
+        exact Fin.cases x.property hy
+      ext x
+      have h := congrArg (fun f : M →ₗ[ℂ] MultilinearMap ℂ (fun _ : Fin n => M) N =>
+        f (x 0) (Fin.tail x)) hc
+      simpa using h
+
+/-- Multilinear maps agree if they agree on tuples from a spanning set. -/
+theorem multilinear_ext_of_span
+    {M : Type u} [AddCommGroup M] [Module ℂ M]
+    {N : Type v} [AddCommGroup N] [Module ℂ N]
+    {s : Set M} (hs : Submodule.span ℂ s = ⊤) (n : ℕ)
+    (F G : MultilinearMap ℂ (fun _ : Fin n => M) N)
+    (h : ∀ x, (∀ i, x i ∈ s) → F x = G x) : F = G := by
+  apply sub_eq_zero.mp
+  apply multilinear_eq_zero_of_span hs n
+  intro x hx
+  simp only [sub_apply, h x hx, sub_self]
+
+/-- Linear maps out of a plethysm are determined by double monomials. -/
+theorem plethysm_linearMap_ext (a b : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V]
+    {W : Type v} [AddCommGroup W] [Module ℂ W]
+    (f g : SymPow b (SymPow a V) →ₗ[ℂ] W)
+    (h : ∀ x : Fin b → Fin a → V,
+      f (symMonomial b (SymPow a V) (fun j => symMonomial a V (x j))) =
+      g (symMonomial b (SymPow a V) (fun j => symMonomial a V (x j)))) : f = g := by
+  classical
+  apply symPow_linearMap_ext b (SymPow a V)
+  have hm : f.compMultilinearMap (symMonomialMultilinear b (SymPow a V)) =
+      g.compMultilinearMap (symMonomialMultilinear b (SymPow a V)) := by
+    apply multilinear_ext_of_span (span_symMonomial a V) b
+    intro x hx
+    choose v hv using hx
+    have hx' : x = fun j => symMonomial a V (v j) := funext (fun j => (hv j).symm)
+    subst x
+    exact h v
+  intro x
+  exact congrArg (fun F => F x) hm
+
+/-- Double monomials span the plethysm. -/
+theorem span_plethysmMonomial (a b : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V] :
+    Submodule.span ℂ (Set.range (fun x : Fin b → Fin a → V =>
+      symMonomial b (SymPow a V) (fun j => symMonomial a V (x j)))) = ⊤ := by
+  let p := Submodule.span ℂ (Set.range (fun x : Fin b → Fin a → V =>
+    symMonomial b (SymPow a V) (fun j => symMonomial a V (x j))))
+  change p = ⊤
+  have hp : p.mkQ = 0 := by
+    apply plethysm_linearMap_ext a b V
+    intro x
+    apply (Submodule.Quotient.mk_eq_zero p).mpr
+    exact Submodule.subset_span (Set.mem_range_self x)
+  rw [← Submodule.ker_mkQ p, hp, LinearMap.ker_zero]
+
+/-- The defining formula determines a Foulkes map uniquely. -/
+theorem IsFoulkesMap.unique (a b : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V]
+    {f g : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V)}
+    (hf : IsFoulkesMap a b V f) (hg : IsFoulkesMap a b V g) : f = g := by
+  apply plethysm_linearMap_ext a b V
+  intro x
+  exact (hf x).trans (hg x).symm
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/SpanningAgreement.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/SpanningAgreement.lean
index c5ae50a0f4dbc411ac34edff71972a5ab4643d4b..4422f0fa5a44005389f1ac8a39b5a19c9a4b545e 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/SpanningAgreement.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/SpanningAgreement.lean
@@ -1,106 +1,106 @@
-import Mathlib
-
-namespace OAI
-
-noncomputable section
-open scoped BigOperators
-open Module
-
-namespace Problem346
-
-/-- Equality on chosen basis generators extends to all generator tuples when each
-coordinate of the generator function has a linear extension. -/
-theorem multilinear_agreement_of_basis
-    {R I W : Type*} [Field R] [Fintype I] [DecidableEq I]
-    {M X K : I → Type*} [∀ i, AddCommGroup (M i)] [∀ i, Module R (M i)]
-    [AddCommGroup W] [Module R W]
-    (s : ∀ i, X i → M i) (B : ∀ i, Basis (K i) R (M i))
-    (t : ∀ i, K i → X i) (ht : ∀ i k, s i (t i k) = B i k)
-    (F : (∀ i, X i) → W) (G : MultilinearMap R M W)
-    (hlinear : ∀ x : ∀ i, X i, ∀ i,
-      ∃ L : M i →ₗ[R] W, ∀ y, L (s i y) = F (Function.update x i y))
-    (hbase : ∀ k : ∀ i, K i, F (fun i => t i (k i)) = G (fun i => B i (k i))) :
-    ∀ x : ∀ i, X i, F x = G (fun i => s i (x i)) := by
-  classical
-  have aux : ∀ (S : Finset I) (x : ∀ i, X i),
-      (∀ i, i ∉ S → ∃ k, x i = t i k) → F x = G (fun i => s i (x i)) := by
-    intro S
-    induction S using Finset.induction_on with
-    | empty =>
-      intro x hx
-      choose k hk using fun i => hx i (Finset.notMem_empty i)
-      have he : x = fun i => t i (k i) := funext hk
-      rw [he, hbase]
-      simp only [ht]
-    | @insert i S hi ih =>
-      intro x hx
-      obtain ⟨L, hL⟩ := hlinear x i
-      have heq : L = G.toLinearMap (fun j => s j (x j)) i := by
-        apply (B i).ext
-        intro k
-        rw [← ht i k, hL]
-        rw [ih (Function.update x i (t i k))]
-        · simp only [MultilinearMap.toLinearMap_apply]
-          congr 1
-          funext j
-          by_cases hji : j = i
-          · subst j
-            simp [ht]
-          · simp [Function.update_of_ne hji]
-        · intro j hj
-          by_cases hji : j = i
-          · subst j
-            exact ⟨k, by simp⟩
-          · obtain ⟨l, hl⟩ := hx j (by simp [hji, hj])
-            exact ⟨l, by simpa [Function.update_of_ne hji] using hl⟩
-      have h := congrArg (fun L : M i →ₗ[R] W => L (s i (x i))) heq
-      simpa only [hL, Function.update_eq_self, MultilinearMap.toLinearMap_apply] using h
-  intro x
-  exact aux Finset.univ x (by simp)
-
-/-- Construct the extension when a basis consists of specified generators. -/
-theorem exists_multilinear_extension_of_basis
-    {R I W : Type*} [Field R] [Fintype I] [DecidableEq I]
-    {M X K : I → Type*} [∀ i, AddCommGroup (M i)] [∀ i, Module R (M i)]
-    [AddCommGroup W] [Module R W]
-    (s : ∀ i, X i → M i) (B : ∀ i, Basis (K i) R (M i))
-    (t : ∀ i, K i → X i) (ht : ∀ i k, s i (t i k) = B i k)
-    (F : (∀ i, X i) → W)
-    (hlinear : ∀ x : ∀ i, X i, ∀ i,
-      ∃ L : M i →ₗ[R] W, ∀ y, L (s i y) = F (Function.update x i y)) :
-    ∃ G : MultilinearMap R M W, ∀ x : ∀ i, X i, F x = G (fun i => s i (x i)) := by
-  let L : (PiTensorProduct R M) →ₗ[R] W :=
-    (Basis.piTensorProduct B).constr R (fun k => F (fun i => t i (k i)))
-  let G : MultilinearMap R M W := L.compMultilinearMap (PiTensorProduct.tprod R)
-  refine ⟨G, multilinear_agreement_of_basis s B t ht F G hlinear ?_⟩
-  intro k
-  change F (fun i => t i (k i)) = L (PiTensorProduct.tprod R (fun i => B i (k i)))
-  rw [← Basis.piTensorProduct_apply]
-  dsimp only [L]
-  rw [Basis.constr_basis]
-
-/-- Separate linear extendibility on spanning generators gives a multilinear extension. -/
-theorem exists_multilinear_extension_of_spanning
-    {R I W : Type*} [Field R] [Fintype I] [DecidableEq I]
-    {M X : I → Type*} [∀ i, AddCommGroup (M i)] [∀ i, Module R (M i)]
-    [AddCommGroup W] [Module R W]
-    (s : ∀ i, X i → M i)
-    (hspan : ∀ i, Submodule.span R (Set.range (s i)) = ⊤)
-    (F : (∀ i, X i) → W)
-    (hlinear : ∀ x : ∀ i, X i, ∀ i,
-      ∃ L : M i →ₗ[R] W, ∀ y, L (s i y) = F (Function.update x i y)) :
-    ∃ G : MultilinearMap R M W, ∀ x : ∀ i, X i, F x = G (fun i => s i (x i)) := by
-  classical
-  have hs (i : I) : ⊤ ≤ Submodule.span R (Set.range (s i)) := by rw [hspan i]
-  let B := fun i => Basis.ofSpan (hs i)
-  have hpre : ∀ i k, ∃ x, s i x = B i k := by
-    intro i k
-    exact Basis.ofSpan_subset (hs i) (Set.mem_range_self k)
-  choose t ht using hpre
-  exact exists_multilinear_extension_of_basis s B t ht F hlinear
-
-end Problem346
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators
+open Module
+
+namespace Problem346
+
+/-- Equality on chosen basis generators extends to all generator tuples when each
+coordinate of the generator function has a linear extension. -/
+theorem multilinear_agreement_of_basis
+    {R I W : Type*} [Field R] [Fintype I] [DecidableEq I]
+    {M X K : I → Type*} [∀ i, AddCommGroup (M i)] [∀ i, Module R (M i)]
+    [AddCommGroup W] [Module R W]
+    (s : ∀ i, X i → M i) (B : ∀ i, Basis (K i) R (M i))
+    (t : ∀ i, K i → X i) (ht : ∀ i k, s i (t i k) = B i k)
+    (F : (∀ i, X i) → W) (G : MultilinearMap R M W)
+    (hlinear : ∀ x : ∀ i, X i, ∀ i,
+      ∃ L : M i →ₗ[R] W, ∀ y, L (s i y) = F (Function.update x i y))
+    (hbase : ∀ k : ∀ i, K i, F (fun i => t i (k i)) = G (fun i => B i (k i))) :
+    ∀ x : ∀ i, X i, F x = G (fun i => s i (x i)) := by
+  classical
+  have aux : ∀ (S : Finset I) (x : ∀ i, X i),
+      (∀ i, i ∉ S → ∃ k, x i = t i k) → F x = G (fun i => s i (x i)) := by
+    intro S
+    induction S using Finset.induction_on with
+    | empty =>
+      intro x hx
+      choose k hk using fun i => hx i (Finset.notMem_empty i)
+      have he : x = fun i => t i (k i) := funext hk
+      rw [he, hbase]
+      simp only [ht]
+    | @insert i S hi ih =>
+      intro x hx
+      obtain ⟨L, hL⟩ := hlinear x i
+      have heq : L = G.toLinearMap (fun j => s j (x j)) i := by
+        apply (B i).ext
+        intro k
+        rw [← ht i k, hL]
+        rw [ih (Function.update x i (t i k))]
+        · simp only [MultilinearMap.toLinearMap_apply]
+          congr 1
+          funext j
+          by_cases hji : j = i
+          · subst j
+            simp [ht]
+          · simp [Function.update_of_ne hji]
+        · intro j hj
+          by_cases hji : j = i
+          · subst j
+            exact ⟨k, by simp⟩
+          · obtain ⟨l, hl⟩ := hx j (by simp [hji, hj])
+            exact ⟨l, by simpa [Function.update_of_ne hji] using hl⟩
+      have h := congrArg (fun L : M i →ₗ[R] W => L (s i (x i))) heq
+      simpa only [hL, Function.update_eq_self, MultilinearMap.toLinearMap_apply] using h
+  intro x
+  exact aux Finset.univ x (by simp)
+
+/-- Construct the extension when a basis consists of specified generators. -/
+theorem exists_multilinear_extension_of_basis
+    {R I W : Type*} [Field R] [Fintype I] [DecidableEq I]
+    {M X K : I → Type*} [∀ i, AddCommGroup (M i)] [∀ i, Module R (M i)]
+    [AddCommGroup W] [Module R W]
+    (s : ∀ i, X i → M i) (B : ∀ i, Basis (K i) R (M i))
+    (t : ∀ i, K i → X i) (ht : ∀ i k, s i (t i k) = B i k)
+    (F : (∀ i, X i) → W)
+    (hlinear : ∀ x : ∀ i, X i, ∀ i,
+      ∃ L : M i →ₗ[R] W, ∀ y, L (s i y) = F (Function.update x i y)) :
+    ∃ G : MultilinearMap R M W, ∀ x : ∀ i, X i, F x = G (fun i => s i (x i)) := by
+  let L : (PiTensorProduct R M) →ₗ[R] W :=
+    (Basis.piTensorProduct B).constr R (fun k => F (fun i => t i (k i)))
+  let G : MultilinearMap R M W := L.compMultilinearMap (PiTensorProduct.tprod R)
+  refine ⟨G, multilinear_agreement_of_basis s B t ht F G hlinear ?_⟩
+  intro k
+  change F (fun i => t i (k i)) = L (PiTensorProduct.tprod R (fun i => B i (k i)))
+  rw [← Basis.piTensorProduct_apply]
+  dsimp only [L]
+  rw [Basis.constr_basis]
+
+/-- Separate linear extendibility on spanning generators gives a multilinear extension. -/
+theorem exists_multilinear_extension_of_spanning
+    {R I W : Type*} [Field R] [Fintype I] [DecidableEq I]
+    {M X : I → Type*} [∀ i, AddCommGroup (M i)] [∀ i, Module R (M i)]
+    [AddCommGroup W] [Module R W]
+    (s : ∀ i, X i → M i)
+    (hspan : ∀ i, Submodule.span R (Set.range (s i)) = ⊤)
+    (F : (∀ i, X i) → W)
+    (hlinear : ∀ x : ∀ i, X i, ∀ i,
+      ∃ L : M i →ₗ[R] W, ∀ y, L (s i y) = F (Function.update x i y)) :
+    ∃ G : MultilinearMap R M W, ∀ x : ∀ i, X i, F x = G (fun i => s i (x i)) := by
+  classical
+  have hs (i : I) : ⊤ ≤ Submodule.span R (Set.range (s i)) := by rw [hspan i]
+  let B := fun i => Basis.ofSpan (hs i)
+  have hpre : ∀ i k, ∃ x, s i x = B i k := by
+    intro i k
+    exact Basis.ofSpan_subset (hs i) (Set.mem_range_self k)
+  choose t ht using hpre
+  exact exists_multilinear_extension_of_basis s B t ht F hlinear
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/Stabilization.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/Stabilization.lean
index 108d8167d0698a771d64969578b8a5b81e14afce..6846928c65adf7c450d4b50d57d75625c1bb1d2d 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/Stabilization.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/Stabilization.lean
@@ -1,51 +1,51 @@
-import OAI.RepresentationTheory.FoulkesHowe.Model
-import OAI.RepresentationTheory.FoulkesHowe.LinearCase
-import OAI.RepresentationTheory.FoulkesHowe.PairingConstruction
-import OAI.RepresentationTheory.FoulkesHowe.Surjectivity
-import OAI.RepresentationTheory.FoulkesHowe.ProductVanishing
-
-namespace OAI
-
-noncomputable section
-
-universe u
-
-namespace Problem346
-
-theorem canonical_foulkes_howe_surjective :
-    ∀ (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] (a b : ℕ), 2 ≤ a → a * (a - 1) ≤ b → ∃ μ : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V), IsFoulkesMap a b V μ ∧ Function.Surjective μ ∧ ∀ ν : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V), IsFoulkesMap a b V ν → ν = μ := by
-  intro V _ _ _ a b ha hab
-  apply canonical_surjective_unique_of_vanishing a b V
-  have hb : 1 ≤ b := by
-    have hpred : 1 ≤ a - 1 := by omega
-    have hprod := Nat.mul_le_mul ha hpred
-    omega
-  exact vanishing_on_products_aux V a b (by omega) hb hab
-
-theorem canonical_foulkes_howe_a_one :
-    ∀ (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] (b : ℕ), ∃ μ : SymPow b (SymPow 1 V) →ₗ[ℂ] SymPow 1 (SymPow b V), IsFoulkesMap 1 b V μ ∧ Function.Bijective μ ∧ ∀ ν : SymPow b (SymPow 1 V) →ₗ[ℂ] SymPow 1 (SymPow b V), IsFoulkesMap 1 b V ν → ν = μ := by
-  intro V _ _ _ b
-  exact canonical_foulkes_howe_a_one_aux V b
-
-theorem vanishing_on_products :
-    ∀ (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] (a b : ℕ), 1 ≤ a → 1 ≤ b → a * (a - 1) ≤ b → ∀ T : SymmetricMultilinearForm a b V, IsSymmetricMultilinearForm a b V T → (∀ u : Fin b → V, T (fun _ => symMonomial b V u) = 0) → T = 0 := by
-  intro V _ _ _ a b ha hb hab T hT hdiag
-  exact vanishing_on_products_aux V a b ha hb hab T hT hdiag
-
-theorem foulkes_comparison_embedding :
-    ∀ (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] (a b : ℕ), 2 ≤ a → a * (a - 1) ≤ b → ∃ ι : SymPow a (SymPow b V) →ₗ[ℂ] SymPow b (SymPow a V), IsGLEquivariantEmbedding a b V ι := by
-  intro V _ _ _ a b ha hab
-  obtain ⟨μ, hμ, hsurj, _⟩ :=
-    canonical_foulkes_howe_surjective (Module.Dual ℂ V) a b ha hab
-  exact PairingConstruction.comparison_of_surjection V a b ⟨μ, hμ, hsurj⟩
-
-theorem sixth_symmetric_power_comparison :
-    ∀ (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] (b : ℕ), 30 ≤ b → ∃ ι : SymPow 6 (SymPow b V) →ₗ[ℂ] SymPow b (SymPow 6 V), IsGLEquivariantEmbedding 6 b V ι := by
-  intro V _ _ _ b hb
-  exact foulkes_comparison_embedding V 6 b (by norm_num) (by simpa using hb)
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.Model
+import OAI.RepresentationTheory.FoulkesHowe.LinearCase
+import OAI.RepresentationTheory.FoulkesHowe.PairingConstruction
+import OAI.RepresentationTheory.FoulkesHowe.Surjectivity
+import OAI.RepresentationTheory.FoulkesHowe.ProductVanishing
+
+namespace OAI
+
+noncomputable section
+
+universe u
+
+namespace Problem346
+
+theorem canonical_foulkes_howe_surjective :
+    ∀ (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] (a b : ℕ), 2 ≤ a → a * (a - 1) ≤ b → ∃ μ : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V), IsFoulkesMap a b V μ ∧ Function.Surjective μ ∧ ∀ ν : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V), IsFoulkesMap a b V ν → ν = μ := by
+  intro V _ _ _ a b ha hab
+  apply canonical_surjective_unique_of_vanishing a b V
+  have hb : 1 ≤ b := by
+    have hpred : 1 ≤ a - 1 := by omega
+    have hprod := Nat.mul_le_mul ha hpred
+    omega
+  exact vanishing_on_products_aux V a b (by omega) hb hab
+
+theorem canonical_foulkes_howe_a_one :
+    ∀ (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] (b : ℕ), ∃ μ : SymPow b (SymPow 1 V) →ₗ[ℂ] SymPow 1 (SymPow b V), IsFoulkesMap 1 b V μ ∧ Function.Bijective μ ∧ ∀ ν : SymPow b (SymPow 1 V) →ₗ[ℂ] SymPow 1 (SymPow b V), IsFoulkesMap 1 b V ν → ν = μ := by
+  intro V _ _ _ b
+  exact canonical_foulkes_howe_a_one_aux V b
+
+theorem vanishing_on_products :
+    ∀ (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] (a b : ℕ), 1 ≤ a → 1 ≤ b → a * (a - 1) ≤ b → ∀ T : SymmetricMultilinearForm a b V, IsSymmetricMultilinearForm a b V T → (∀ u : Fin b → V, T (fun _ => symMonomial b V u) = 0) → T = 0 := by
+  intro V _ _ _ a b ha hb hab T hT hdiag
+  exact vanishing_on_products_aux V a b ha hb hab T hT hdiag
+
+theorem foulkes_comparison_embedding :
+    ∀ (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] (a b : ℕ), 2 ≤ a → a * (a - 1) ≤ b → ∃ ι : SymPow a (SymPow b V) →ₗ[ℂ] SymPow b (SymPow a V), IsGLEquivariantEmbedding a b V ι := by
+  intro V _ _ _ a b ha hab
+  obtain ⟨μ, hμ, hsurj, _⟩ :=
+    canonical_foulkes_howe_surjective (Module.Dual ℂ V) a b ha hab
+  exact PairingConstruction.comparison_of_surjection V a b ⟨μ, hμ, hsurj⟩
+
+theorem sixth_symmetric_power_comparison :
+    ∀ (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] (b : ℕ), 30 ≤ b → ∃ ι : SymPow 6 (SymPow b V) →ₗ[ℂ] SymPow b (SymPow 6 V), IsGLEquivariantEmbedding 6 b V ι := by
+  intro V _ _ _ b hb
+  exact foulkes_comparison_embedding V 6 b (by norm_num) (by simpa using hb)
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/StabilizationBounds.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/StabilizationBounds.lean
index cbde88b8b963563fe94b62c41b241f89d1b61148..e349a0fcc5cbaf6fc3c2d2adcaade9254418add8 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/StabilizationBounds.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/StabilizationBounds.lean
@@ -1,70 +1,70 @@
-import Mathlib
-
-namespace OAI
-
-namespace Problem346
-
-open scoped BigOperators
-
-/-- The parameter reduction in the induction on the outer symmetric degree. -/
-theorem stabilization_reduction {a b : ℕ} (ha : 2 ≤ a)
-    (hab : a * (a - 1) ≤ b) :
-    let r := a - 1
-    let m := b - r
-    1 ≤ r ∧ r * (r + 1) ≤ b ∧ r * r ≤ m ∧
-      1 ≤ m ∧ r * (r - 1) ≤ m := by
-  dsimp
-  have ha' : a - 1 + 1 = a := Nat.sub_add_cancel (by omega)
-  have hr : 1 ≤ a - 1 := by omega
-  have hb : (a - 1) * ((a - 1) + 1) ≤ b := by
-    rw [ha', Nat.mul_comm]
-    exact hab
-  have hm : (a - 1) * (a - 1) ≤ b - (a - 1) := by
-    apply Nat.le_sub_of_add_le
-    nlinarith
-  have hmpos : 1 ≤ b - (a - 1) := by nlinarith
-  refine ⟨hr, hb, hm, hmpos, ?_⟩
-  exact (Nat.mul_le_mul_left (a - 1) (Nat.sub_le (a - 1) 1)).trans hm
-
-/-- The first sequence of shifts has positive source-minus-destination weight. -/
-theorem first_shift_weight {b r i : ℕ} (hir : i ≤ b - r)
-    (hi : 1 ≤ i) : r < b - i + 1 := by
-  omega
-
-/-- The shared destination variable remains at least two degrees below the source. -/
-theorem second_shift_weight_gap {b r i j : ℕ}
-    (hr : 1 ≤ r) (hb : r * (r + 1) ≤ b)
-    (hi : 1 ≤ i) (hir : i ≤ r) (hj : j < r) :
-    (i - 1) * r + j + 2 ≤ b - j := by
-  have hi' : i - 1 ≤ r - 1 := by omega
-  have hmul : (i - 1) * r ≤ (r - 1) * r := Nat.mul_le_mul_right r hi'
-  have hr' : r - 1 + 1 = r := Nat.sub_add_cancel hr
-  have hj' : j ≤ r - 1 := by omega
-  have hbound : (i - 1) * r + j + 2 + j ≤ b := by nlinarith
-  exact Nat.le_sub_of_add_le hbound
-
-/-- The second sequence of shifts satisfies the strict injectivity inequality. -/
-theorem second_shift_weight {b r i j : ℕ}
-    (hr : 1 ≤ r) (hb : r * (r + 1) ≤ b)
-    (hi : 1 ≤ i) (hir : i ≤ r) (hj : j < r) :
-    (i - 1) * r + j < b - j := by
-  have h := second_shift_weight_gap hr hb hi hir hj
-  omega
-
-/-- Arbitrary progress vectors, rather than only lexicographic schedules, satisfy the
-positive-weight inequality at every unfinished slot. -/
-theorem grid_shift_weight {b r : ℕ} (hb : r * (r + 1) ≤ b)
-    (counts : Fin r → ℕ) (hcounts : ∀ j, counts j ≤ r)
-    (i : Fin r) (hi : counts i < r) :
-    (∑ j, counts j) < b - counts i := by
-  have hsum : (∑ j, counts j) ≤ r * r := by
-    calc
-      (∑ j, counts j) ≤ ∑ _j : Fin r, r :=
-        Finset.sum_le_sum (fun j _ => hcounts j)
-      _ = r * r := by simp
-  have hbound : (∑ j, counts j) + counts i < b := by nlinarith
-  omega
-
-end Problem346
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+namespace Problem346
+
+open scoped BigOperators
+
+/-- The parameter reduction in the induction on the outer symmetric degree. -/
+theorem stabilization_reduction {a b : ℕ} (ha : 2 ≤ a)
+    (hab : a * (a - 1) ≤ b) :
+    let r := a - 1
+    let m := b - r
+    1 ≤ r ∧ r * (r + 1) ≤ b ∧ r * r ≤ m ∧
+      1 ≤ m ∧ r * (r - 1) ≤ m := by
+  dsimp
+  have ha' : a - 1 + 1 = a := Nat.sub_add_cancel (by omega)
+  have hr : 1 ≤ a - 1 := by omega
+  have hb : (a - 1) * ((a - 1) + 1) ≤ b := by
+    rw [ha', Nat.mul_comm]
+    exact hab
+  have hm : (a - 1) * (a - 1) ≤ b - (a - 1) := by
+    apply Nat.le_sub_of_add_le
+    nlinarith
+  have hmpos : 1 ≤ b - (a - 1) := by nlinarith
+  refine ⟨hr, hb, hm, hmpos, ?_⟩
+  exact (Nat.mul_le_mul_left (a - 1) (Nat.sub_le (a - 1) 1)).trans hm
+
+/-- The first sequence of shifts has positive source-minus-destination weight. -/
+theorem first_shift_weight {b r i : ℕ} (hir : i ≤ b - r)
+    (hi : 1 ≤ i) : r < b - i + 1 := by
+  omega
+
+/-- The shared destination variable remains at least two degrees below the source. -/
+theorem second_shift_weight_gap {b r i j : ℕ}
+    (hr : 1 ≤ r) (hb : r * (r + 1) ≤ b)
+    (hi : 1 ≤ i) (hir : i ≤ r) (hj : j < r) :
+    (i - 1) * r + j + 2 ≤ b - j := by
+  have hi' : i - 1 ≤ r - 1 := by omega
+  have hmul : (i - 1) * r ≤ (r - 1) * r := Nat.mul_le_mul_right r hi'
+  have hr' : r - 1 + 1 = r := Nat.sub_add_cancel hr
+  have hj' : j ≤ r - 1 := by omega
+  have hbound : (i - 1) * r + j + 2 + j ≤ b := by nlinarith
+  exact Nat.le_sub_of_add_le hbound
+
+/-- The second sequence of shifts satisfies the strict injectivity inequality. -/
+theorem second_shift_weight {b r i j : ℕ}
+    (hr : 1 ≤ r) (hb : r * (r + 1) ≤ b)
+    (hi : 1 ≤ i) (hir : i ≤ r) (hj : j < r) :
+    (i - 1) * r + j < b - j := by
+  have h := second_shift_weight_gap hr hb hi hir hj
+  omega
+
+/-- Arbitrary progress vectors, rather than only lexicographic schedules, satisfy the
+positive-weight inequality at every unfinished slot. -/
+theorem grid_shift_weight {b r : ℕ} (hb : r * (r + 1) ≤ b)
+    (counts : Fin r → ℕ) (hcounts : ∀ j, counts j ≤ r)
+    (i : Fin r) (hi : counts i < r) :
+    (∑ j, counts j) < b - counts i := by
+  have hsum : (∑ j, counts j) ≤ r * r := by
+    calc
+      (∑ j, counts j) ≤ ∑ _j : Fin r, r :=
+        Finset.sum_le_sum (fun j _ => hcounts j)
+      _ = r * r := by simp
+  have hbound : (∑ j, counts j) + counts i < b := by nlinarith
+  omega
+
+end Problem346
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/SumShift.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/SumShift.lean
index ac6bf3dd60dad9c77897710ddb5d66453864ad34..f4d19ab441be39ec1c40914e6e821d03aaeb96bb 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/SumShift.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/SumShift.lean
@@ -1,105 +1,105 @@
-import Mathlib
-
-namespace OAI
-
-noncomputable section
-
-namespace Problem346.PolynomialShift
-
-open MvPolynomial
-
-universe u v w
-
-abbrev Vars (ι : Type u) (σ : Type v) := ι ⊕ (ι ⊕ σ)
-abbrev Poly (R : Type w) (ι : Type u) (σ : Type v) [CommRing R] :=
-  MvPolynomial (Vars ι σ) R
-
-variable (R : Type w) (ι : Type u) (σ : Type v) [CommRing R]
-
-/-- Shift one unit of degree from the first coordinate block to the second. -/
-def down : Derivation R (Poly R ι σ) (Poly R ι σ) :=
-  mkDerivation R (Sum.elim (fun i => X (Sum.inr (Sum.inl i))) (fun _ => 0))
-
-/-- The opposite shift. -/
-def up : Derivation R (Poly R ι σ) (Poly R ι σ) :=
-  mkDerivation R (Sum.elim (fun _ => 0)
-    (Sum.elim (fun i => X (Sum.inl i)) (fun _ => 0)))
-
-/-- Euler operator of the first coordinate block. -/
-def sourceEuler : Derivation R (Poly R ι σ) (Poly R ι σ) :=
-  mkDerivation R (Sum.elim (fun i => X (Sum.inl i)) (fun _ => 0))
-
-/-- Euler operator of the second coordinate block. -/
-def targetEuler : Derivation R (Poly R ι σ) (Poly R ι σ) :=
-  mkDerivation R (Sum.elim (fun _ => 0)
-    (Sum.elim (fun i => X (Sum.inr (Sum.inl i))) (fun _ => 0)))
-
-/-- Source degree minus destination degree. -/
-def weight : Derivation R (Poly R ι σ) (Poly R ι σ) :=
-  sourceEuler R ι σ - targetEuler R ι σ
-
-@[simp] theorem down_source (i : ι) :
-    down R ι σ (X (Sum.inl i)) = X (Sum.inr (Sum.inl i)) := by
-  simp [down]
-
-@[simp] theorem down_target (i : ι) :
-    down R ι σ (X (Sum.inr (Sum.inl i))) = 0 := by
-  simp [down]
-
-@[simp] theorem down_spectator (i : σ) :
-    down R ι σ (X (Sum.inr (Sum.inr i))) = 0 := by
-  simp [down]
-
-@[simp] theorem up_source (i : ι) :
-    up R ι σ (X (Sum.inl i)) = 0 := by
-  simp [up]
-
-@[simp] theorem up_target (i : ι) :
-    up R ι σ (X (Sum.inr (Sum.inl i))) = X (Sum.inl i) := by
-  simp [up]
-
-@[simp] theorem up_spectator (i : σ) :
-    up R ι σ (X (Sum.inr (Sum.inr i))) = 0 := by
-  simp [up]
-
-@[simp] theorem weight_source (i : ι) :
-    weight R ι σ (X (Sum.inl i)) = X (Sum.inl i) := by
-  simp [weight, sourceEuler, targetEuler]
-
-@[simp] theorem weight_target (i : ι) :
-    weight R ι σ (X (Sum.inr (Sum.inl i))) = -X (Sum.inr (Sum.inl i)) := by
-  simp [weight, sourceEuler, targetEuler]
-
-@[simp] theorem weight_spectator (i : σ) :
-    weight R ι σ (X (Sum.inr (Sum.inr i))) = 0 := by
-  simp [weight, sourceEuler, targetEuler]
-
-/-- The fundamental commutator, with arbitrary spectator variables. -/
-theorem up_down_commutator :
-    ⁅up R ι σ, down R ι σ⁆ = weight R ι σ := by
-  apply MvPolynomial.derivation_ext
-  intro i
-  rcases i with i | (i | i) <;> simp [Derivation.commutator_apply]
-
-/-- The opposite shift raises the source-minus-destination weight by two. -/
-theorem weight_up_commutator :
-    ⁅weight R ι σ, up R ι σ⁆ = (2 : R) • up R ι σ := by
-  apply MvPolynomial.derivation_ext
-  intro i
-  rcases i with i | (i | i) <;>
-    simp [Derivation.commutator_apply, two_smul]
-
-/-- The forward shift lowers the source-minus-destination weight by two. -/
-theorem weight_down_commutator :
-    ⁅weight R ι σ, down R ι σ⁆ = -(2 : R) • down R ι σ := by
-  apply MvPolynomial.derivation_ext
-  intro i
-  rcases i with i | (i | i) <;>
-    simp [Derivation.commutator_apply, two_smul]
-  abel
-
-end Problem346.PolynomialShift
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+
+namespace Problem346.PolynomialShift
+
+open MvPolynomial
+
+universe u v w
+
+abbrev Vars (ι : Type u) (σ : Type v) := ι ⊕ (ι ⊕ σ)
+abbrev Poly (R : Type w) (ι : Type u) (σ : Type v) [CommRing R] :=
+  MvPolynomial (Vars ι σ) R
+
+variable (R : Type w) (ι : Type u) (σ : Type v) [CommRing R]
+
+/-- Shift one unit of degree from the first coordinate block to the second. -/
+def down : Derivation R (Poly R ι σ) (Poly R ι σ) :=
+  mkDerivation R (Sum.elim (fun i => X (Sum.inr (Sum.inl i))) (fun _ => 0))
+
+/-- The opposite shift. -/
+def up : Derivation R (Poly R ι σ) (Poly R ι σ) :=
+  mkDerivation R (Sum.elim (fun _ => 0)
+    (Sum.elim (fun i => X (Sum.inl i)) (fun _ => 0)))
+
+/-- Euler operator of the first coordinate block. -/
+def sourceEuler : Derivation R (Poly R ι σ) (Poly R ι σ) :=
+  mkDerivation R (Sum.elim (fun i => X (Sum.inl i)) (fun _ => 0))
+
+/-- Euler operator of the second coordinate block. -/
+def targetEuler : Derivation R (Poly R ι σ) (Poly R ι σ) :=
+  mkDerivation R (Sum.elim (fun _ => 0)
+    (Sum.elim (fun i => X (Sum.inr (Sum.inl i))) (fun _ => 0)))
+
+/-- Source degree minus destination degree. -/
+def weight : Derivation R (Poly R ι σ) (Poly R ι σ) :=
+  sourceEuler R ι σ - targetEuler R ι σ
+
+@[simp] theorem down_source (i : ι) :
+    down R ι σ (X (Sum.inl i)) = X (Sum.inr (Sum.inl i)) := by
+  simp [down]
+
+@[simp] theorem down_target (i : ι) :
+    down R ι σ (X (Sum.inr (Sum.inl i))) = 0 := by
+  simp [down]
+
+@[simp] theorem down_spectator (i : σ) :
+    down R ι σ (X (Sum.inr (Sum.inr i))) = 0 := by
+  simp [down]
+
+@[simp] theorem up_source (i : ι) :
+    up R ι σ (X (Sum.inl i)) = 0 := by
+  simp [up]
+
+@[simp] theorem up_target (i : ι) :
+    up R ι σ (X (Sum.inr (Sum.inl i))) = X (Sum.inl i) := by
+  simp [up]
+
+@[simp] theorem up_spectator (i : σ) :
+    up R ι σ (X (Sum.inr (Sum.inr i))) = 0 := by
+  simp [up]
+
+@[simp] theorem weight_source (i : ι) :
+    weight R ι σ (X (Sum.inl i)) = X (Sum.inl i) := by
+  simp [weight, sourceEuler, targetEuler]
+
+@[simp] theorem weight_target (i : ι) :
+    weight R ι σ (X (Sum.inr (Sum.inl i))) = -X (Sum.inr (Sum.inl i)) := by
+  simp [weight, sourceEuler, targetEuler]
+
+@[simp] theorem weight_spectator (i : σ) :
+    weight R ι σ (X (Sum.inr (Sum.inr i))) = 0 := by
+  simp [weight, sourceEuler, targetEuler]
+
+/-- The fundamental commutator, with arbitrary spectator variables. -/
+theorem up_down_commutator :
+    ⁅up R ι σ, down R ι σ⁆ = weight R ι σ := by
+  apply MvPolynomial.derivation_ext
+  intro i
+  rcases i with i | (i | i) <;> simp [Derivation.commutator_apply]
+
+/-- The opposite shift raises the source-minus-destination weight by two. -/
+theorem weight_up_commutator :
+    ⁅weight R ι σ, up R ι σ⁆ = (2 : R) • up R ι σ := by
+  apply MvPolynomial.derivation_ext
+  intro i
+  rcases i with i | (i | i) <;>
+    simp [Derivation.commutator_apply, two_smul]
+
+/-- The forward shift lowers the source-minus-destination weight by two. -/
+theorem weight_down_commutator :
+    ⁅weight R ι σ, down R ι σ⁆ = -(2 : R) • down R ι σ := by
+  apply MvPolynomial.derivation_ext
+  intro i
+  rcases i with i | (i | i) <;>
+    simp [Derivation.commutator_apply, two_smul]
+  abel
+
+end Problem346.PolynomialShift
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/Surjectivity.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/Surjectivity.lean
index 4e64c2e8765ff1abd323e279047ac3a36bb1684d..0ee873722e7078bd9a231282cd5fab30baeed959 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/Surjectivity.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/Surjectivity.lean
@@ -1,27 +1,27 @@
-import OAI.RepresentationTheory.FoulkesHowe.CanonicalMap
-import OAI.RepresentationTheory.FoulkesHowe.Annihilator
-
-namespace OAI
-
-noncomputable section
-namespace Problem346
-
-/-- Existence, surjectivity, and uniqueness reduce entirely to the vanishing theorem. -/
-theorem canonical_surjective_unique_of_vanishing (a b : ℕ) (V : Type*)
-    [AddCommGroup V] [Module ℂ V]
-    (hvan : ∀ T : SymmetricMultilinearForm a b V,
-      IsSymmetricMultilinearForm a b V T →
-      (∀ u : Fin b → V, T (fun _ => symMonomial b V u) = 0) → T = 0) :
-    ∃ μ : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V),
-      IsFoulkesMap a b V μ ∧ Function.Surjective μ ∧
-      ∀ ν : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V),
-        IsFoulkesMap a b V ν → ν = μ := by
-  obtain ⟨μ, hμ⟩ := exists_foulkesMap_from_spanning a b V
-  exact ⟨μ, hμ, IsFoulkesMap.surjective_of_vanishing a b V μ hμ hvan,
-    fun ν hν => IsFoulkesMap.unique a b V hν hμ⟩
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.CanonicalMap
+import OAI.RepresentationTheory.FoulkesHowe.Annihilator
+
+namespace OAI
+
+noncomputable section
+namespace Problem346
+
+/-- Existence, surjectivity, and uniqueness reduce entirely to the vanishing theorem. -/
+theorem canonical_surjective_unique_of_vanishing (a b : ℕ) (V : Type*)
+    [AddCommGroup V] [Module ℂ V]
+    (hvan : ∀ T : SymmetricMultilinearForm a b V,
+      IsSymmetricMultilinearForm a b V T →
+      (∀ u : Fin b → V, T (fun _ => symMonomial b V u) = 0) → T = 0) :
+    ∃ μ : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V),
+      IsFoulkesMap a b V μ ∧ Function.Surjective μ ∧
+      ∀ ν : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V),
+        IsFoulkesMap a b V ν → ν = μ := by
+  obtain ⟨μ, hμ⟩ := exists_foulkesMap_from_spanning a b V
+  exact ⟨μ, hμ, IsFoulkesMap.surjective_of_vanishing a b V μ hμ hvan,
+    fun ν hν => IsFoulkesMap.unique a b V hν hμ⟩
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/SymmetricDual.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/SymmetricDual.lean
index 23aba7ac4546cd6801831781e3b243369fa694eb..fb85ee43f05ee8ca914d02a475d5d4b1957c5b95 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/SymmetricDual.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/SymmetricDual.lean
@@ -1,31 +1,31 @@
-import OAI.RepresentationTheory.FoulkesHowe.Model
-import OAI.RepresentationTheory.FoulkesHowe.AlgebraEvaluation
-
-namespace OAI
-
-noncomputable section
-universe u
-namespace Problem346
-
-variable (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
-
-/-- A symmetric-power duality map is injective as soon as evaluation on pure powers agrees
-with evaluation in the symmetric algebra. -/
-theorem symPowDual_injective_of_pure_evaluation (n : ℕ)
-    (B : SymPow n (Module.Dual ℂ V) →ₗ[ℂ] Module.Dual ℂ (SymPow n V))
-    (hB : ∀ (p : SymPow n (Module.Dual ℂ V)) (x : V),
-      B p (symMonomial n V (fun _ => x)) =
-        SymmetricAlgebra.lift (Module.Dual.eval ℂ V x) p.val) :
-    Function.Injective B := by
-  intro p q hpq
-  apply Subtype.ext
-  apply sub_eq_zero.mp
-  apply symAlg_dual_eq_zero_of_eval_eq_zero V
-  intro x
-  rw [map_sub, ← hB p x, ← hB q x, hpq, sub_self]
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.Model
+import OAI.RepresentationTheory.FoulkesHowe.AlgebraEvaluation
+
+namespace OAI
+
+noncomputable section
+universe u
+namespace Problem346
+
+variable (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
+
+/-- A symmetric-power duality map is injective as soon as evaluation on pure powers agrees
+with evaluation in the symmetric algebra. -/
+theorem symPowDual_injective_of_pure_evaluation (n : ℕ)
+    (B : SymPow n (Module.Dual ℂ V) →ₗ[ℂ] Module.Dual ℂ (SymPow n V))
+    (hB : ∀ (p : SymPow n (Module.Dual ℂ V)) (x : V),
+      B p (symMonomial n V (fun _ => x)) =
+        SymmetricAlgebra.lift (Module.Dual.eval ℂ V x) p.val) :
+    Function.Injective B := by
+  intro p q hpq
+  apply Subtype.ext
+  apply sub_eq_zero.mp
+  apply symAlg_dual_eq_zero_of_eval_eq_zero V
+  intro x
+  rw [map_sub, ← hB p x, ← hB q x, hpq, sub_self]
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/SymmetricLift.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/SymmetricLift.lean
index cdf386a6bbe9463210b4ab02f3dc2c4714e468ad..55b58094f973bfb363548be780147812522645d3 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/SymmetricLift.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/SymmetricLift.lean
@@ -1,173 +1,173 @@
-import OAI.RepresentationTheory.FoulkesHowe.Model
-
-namespace OAI
-
-noncomputable section
-open scoped BigOperators Classical
-open Module
-
-namespace Problem346
-namespace SymmetricLift
-
-variable {κ : Type*} {n : ℕ}
-
-private def tupleExponent (f : Fin n → κ) : κ →₀ ℕ :=
-  ∑ i, Finsupp.single (f i) 1
-
-private lemma tupleExponent_apply (f : Fin n → κ) (k : κ) :
-    tupleExponent f k = Fintype.card {i // f i = k} := by
-  classical
-  rw [Fintype.card_subtype]
-  simp only [tupleExponent, Finsupp.coe_finsetSum, Finset.sum_apply, Finsupp.single_apply, Finset.sum_boole, Nat.cast_id]
-
-private lemma exists_perm_of_exponent_eq (f g : Fin n → κ)
-    (h : tupleExponent f = tupleExponent g) :
-    ∃ σ : Equiv.Perm (Fin n), ∀ i, g (σ i) = f i := by
-  classical
-  have hc (k : κ) : Fintype.card {i // f i = k} = Fintype.card {i // g i = k} := by
-    simpa only [tupleExponent_apply] using congrArg (fun d : κ →₀ ℕ => d k) h
-  let e (k : κ) : {i // f i = k} ≃ {i // g i = k} :=
-    Fintype.equivOfCardEq (hc k)
-  exact ⟨Equiv.ofFiberEquiv e, Equiv.ofFiberEquiv_map e⟩
-
-variable {V W : Type*} [AddCommGroup V] [Module ℂ V]
-    [AddCommGroup W] [Module ℂ W]
-
-private def liftValues (b : Basis κ ℂ V)
-    (T : MultilinearMap ℂ (fun _ : Fin n => V) W) (d : κ →₀ ℕ) : W :=
-  if h : ∃ f : Fin n → κ, tupleExponent f = d then
-    T (fun i => b (Classical.choose h i)) else 0
-
-private lemma liftValues_tuple (b : Basis κ ℂ V)
-    (T : MultilinearMap ℂ (fun _ : Fin n => V) W)
-    (hT : ∀ (σ : Equiv.Perm (Fin n)) (v : Fin n → V), T (fun i => v (σ i)) = T v)
-    (f : Fin n → κ) : liftValues b T (tupleExponent f) = T (fun i => b (f i)) := by
-  let h : ∃ g : Fin n → κ, tupleExponent g = tupleExponent f := ⟨f, rfl⟩
-  rw [liftValues, dite_eq_left h]
-  obtain ⟨σ, hσ⟩ := exists_perm_of_exponent_eq (Classical.choose h) f (Classical.choose_spec h)
-  convert hT σ (fun i => b (f i)) using 1
-  congr 1
-  funext i
-  rw [hσ]
-
-private def liftOnAlgebra (b : Basis κ ℂ V)
-    (T : MultilinearMap ℂ (fun _ : Fin n => V) W) :
-    SymmetricAlgebra ℂ V →ₗ[ℂ] W :=
-  ((MvPolynomial.basisMonomials κ ℂ).constr ℂ (liftValues b T)).comp
-    (SymmetricAlgebra.equivMvPolynomial b).toLinearMap
-
-private lemma liftOnAlgebra_basis (b : Basis κ ℂ V)
-    (T : MultilinearMap ℂ (fun _ : Fin n => V) W)
-    (hT : ∀ (σ : Equiv.Perm (Fin n)) (v : Fin n → V), T (fun i => v (σ i)) = T v)
-    (f : Fin n → κ) :
-    liftOnAlgebra b T (symMonomialRaw n V (fun i => b (f i))) = T (fun i => b (f i)) := by
-  have hp : (∏ i, MvPolynomial.X (f i) : MvPolynomial κ ℂ) =
-      MvPolynomial.monomial (tupleExponent f) 1 := by
-    simp only [tupleExponent, MvPolynomial.monomial_sum_one, MvPolynomial.X]
-  change ((MvPolynomial.basisMonomials κ ℂ).constr ℂ (liftValues b T))
-    ((SymmetricAlgebra.equivMvPolynomial b) (∏ i, SymmetricAlgebra.ι ℂ V (b (f i)))) = _
-  rw [map_prod]
-  simp only [SymmetricAlgebra.equivMvPolynomial_ι_apply]
-  rw [hp]
-  change ((MvPolynomial.basisMonomials κ ℂ).constr ℂ (liftValues b T))
-    ((MvPolynomial.basisMonomials κ ℂ) (tupleExponent f)) = _
-  rw [Basis.constr_basis, liftValues_tuple b T hT]
-
-private lemma liftOnAlgebra_monomial (b : Basis κ ℂ V)
-    (T : MultilinearMap ℂ (fun _ : Fin n => V) W)
-    (hT : ∀ (σ : Equiv.Perm (Fin n)) (v : Fin n → V), T (fun i => v (σ i)) = T v)
-    (v : Fin n → V) :
-    liftOnAlgebra b T (symMonomialRaw n V v) = T v := by
-  let M : MultilinearMap ℂ (fun _ : Fin n => V) (SymmetricAlgebra ℂ V) :=
-    (MultilinearMap.mkPiAlgebra ℂ (Fin n) (SymmetricAlgebra ℂ V)).compLinearMap
-      (fun _ => SymmetricAlgebra.ι ℂ V)
-  have h : (liftOnAlgebra b T).compMultilinearMap M = T := by
-    apply Basis.ext_multilinear (fun _ => b)
-    intro f
-    exact liftOnAlgebra_basis b T hT f
-  exact congrArg (fun S : MultilinearMap ℂ (fun _ : Fin n => V) W => S v) h
-
-end SymmetricLift
-
-/-- A symmetric multilinear map descends to the homogeneous symmetric-power submodule. -/
-theorem exists_symPow_lift {n : ℕ} {V W : Type*}
-    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
-    (T : MultilinearMap ℂ (fun _ : Fin n => V) W)
-    (hT : ∀ (σ : Equiv.Perm (Fin n)) (v : Fin n → V), T (fun i => v (σ i)) = T v) :
-    ∃ L : SymPow n V →ₗ[ℂ] W, ∀ v, L (symMonomial n V v) = T v := by
-  let b := Module.Free.chooseBasis ℂ V
-  refine ⟨(SymmetricLift.liftOnAlgebra b T).domRestrict (symPowSubmodule n V), ?_⟩
-  intro v
-  exact SymmetricLift.liftOnAlgebra_monomial b T hT v
-
-/-- A chosen linear descent of a symmetric multilinear map. -/
-def symPowLift {n : ℕ} {V W : Type*}
-    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
-    (T : MultilinearMap ℂ (fun _ : Fin n => V) W)
-    (hT : ∀ (σ : Equiv.Perm (Fin n)) (v : Fin n → V), T (fun i => v (σ i)) = T v) :
-    SymPow n V →ₗ[ℂ] W :=
-  Classical.choose (exists_symPow_lift T hT)
-
-@[simp] theorem symPowLift_symMonomial {n : ℕ} {V W : Type*}
-    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
-    (T : MultilinearMap ℂ (fun _ : Fin n => V) W)
-    (hT : ∀ (σ : Equiv.Perm (Fin n)) (v : Fin n → V), T (fun i => v (σ i)) = T v)
-    (v : Fin n → V) : symPowLift T hT (symMonomial n V v) = T v :=
-  Classical.choose_spec (exists_symPow_lift T hT) v
-
-namespace SymmetricLift
-
-/-- Linear maps out of a symmetric power are determined by products. -/
-theorem linearMap_ext {n : ℕ} {V W : Type*}
-    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
-    {f g : SymPow n V →ₗ[ℂ] W}
-    (h : ∀ v, f (symMonomial n V v) = g (symMonomial n V v)) : f = g := by
-  apply (Submodule.linearMap_eq_iff_of_eq_span f g rfl).2
-  rintro ⟨x, ⟨v, rfl⟩⟩
-  exact h v
-
-end SymmetricLift
-
-/-- The submodule of multilinear maps invariant under permutations of their arguments. -/
-def symmetricMultilinearSubmodule (n : ℕ) (V W : Type*)
-    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W] :
-    Submodule ℂ (MultilinearMap ℂ (fun _ : Fin n => V) W) where
-  carrier := {T | ∀ (σ : Equiv.Perm (Fin n)) (v : Fin n → V),
-    T (fun i => v (σ i)) = T v}
-  zero_mem' := by simp
-  add_mem' := by
-    intro T U hT hU σ v
-    simp only [add_apply, hT σ v, hU σ v]
-  smul_mem' := by
-    intro c T hT σ v
-    simp only [smul_apply, hT σ v]
-
-/-- Descent from symmetric multilinear maps is linear in the map being descended. -/
-def symPowLiftLinear (n : ℕ) (V W : Type*)
-    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W] :
-    symmetricMultilinearSubmodule n V W →ₗ[ℂ] (SymPow n V →ₗ[ℂ] W) where
-  toFun T := symPowLift T.val T.property
-  map_add' T U := by
-    apply SymmetricLift.linearMap_ext
-    intro v
-    exact (symPowLift_symMonomial (T + U).val (T + U).property v).trans
-      (congrArg₂ (fun left right : W => left + right)
-        (symPowLift_symMonomial T.val T.property v).symm
-        (symPowLift_symMonomial U.val U.property v).symm)
-  map_smul' c T := by
-    apply SymmetricLift.linearMap_ext
-    intro v
-    exact (symPowLift_symMonomial (c • T).val (c • T).property v).trans
-      (congrArg (c • ·) (symPowLift_symMonomial T.val T.property v).symm)
-
-@[simp] theorem symPowLiftLinear_symMonomial {n : ℕ} {V W : Type*}
-    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
-    (T : symmetricMultilinearSubmodule n V W) (v : Fin n → V) :
-    symPowLiftLinear n V W T (symMonomial n V v) = T.val v :=
-  symPowLift_symMonomial T.val T.property v
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.Model
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators Classical
+open Module
+
+namespace Problem346
+namespace SymmetricLift
+
+variable {κ : Type*} {n : ℕ}
+
+private def tupleExponent (f : Fin n → κ) : κ →₀ ℕ :=
+  ∑ i, Finsupp.single (f i) 1
+
+private lemma tupleExponent_apply (f : Fin n → κ) (k : κ) :
+    tupleExponent f k = Fintype.card {i // f i = k} := by
+  classical
+  rw [Fintype.card_subtype]
+  simp only [tupleExponent, Finsupp.coe_finsetSum, Finset.sum_apply, Finsupp.single_apply, Finset.sum_boole, Nat.cast_id]
+
+private lemma exists_perm_of_exponent_eq (f g : Fin n → κ)
+    (h : tupleExponent f = tupleExponent g) :
+    ∃ σ : Equiv.Perm (Fin n), ∀ i, g (σ i) = f i := by
+  classical
+  have hc (k : κ) : Fintype.card {i // f i = k} = Fintype.card {i // g i = k} := by
+    simpa only [tupleExponent_apply] using congrArg (fun d : κ →₀ ℕ => d k) h
+  let e (k : κ) : {i // f i = k} ≃ {i // g i = k} :=
+    Fintype.equivOfCardEq (hc k)
+  exact ⟨Equiv.ofFiberEquiv e, Equiv.ofFiberEquiv_map e⟩
+
+variable {V W : Type*} [AddCommGroup V] [Module ℂ V]
+    [AddCommGroup W] [Module ℂ W]
+
+private def liftValues (b : Basis κ ℂ V)
+    (T : MultilinearMap ℂ (fun _ : Fin n => V) W) (d : κ →₀ ℕ) : W :=
+  if h : ∃ f : Fin n → κ, tupleExponent f = d then
+    T (fun i => b (Classical.choose h i)) else 0
+
+private lemma liftValues_tuple (b : Basis κ ℂ V)
+    (T : MultilinearMap ℂ (fun _ : Fin n => V) W)
+    (hT : ∀ (σ : Equiv.Perm (Fin n)) (v : Fin n → V), T (fun i => v (σ i)) = T v)
+    (f : Fin n → κ) : liftValues b T (tupleExponent f) = T (fun i => b (f i)) := by
+  let h : ∃ g : Fin n → κ, tupleExponent g = tupleExponent f := ⟨f, rfl⟩
+  rw [liftValues, dite_eq_left h]
+  obtain ⟨σ, hσ⟩ := exists_perm_of_exponent_eq (Classical.choose h) f (Classical.choose_spec h)
+  convert hT σ (fun i => b (f i)) using 1
+  congr 1
+  funext i
+  rw [hσ]
+
+private def liftOnAlgebra (b : Basis κ ℂ V)
+    (T : MultilinearMap ℂ (fun _ : Fin n => V) W) :
+    SymmetricAlgebra ℂ V →ₗ[ℂ] W :=
+  ((MvPolynomial.basisMonomials κ ℂ).constr ℂ (liftValues b T)).comp
+    (SymmetricAlgebra.equivMvPolynomial b).toLinearMap
+
+private lemma liftOnAlgebra_basis (b : Basis κ ℂ V)
+    (T : MultilinearMap ℂ (fun _ : Fin n => V) W)
+    (hT : ∀ (σ : Equiv.Perm (Fin n)) (v : Fin n → V), T (fun i => v (σ i)) = T v)
+    (f : Fin n → κ) :
+    liftOnAlgebra b T (symMonomialRaw n V (fun i => b (f i))) = T (fun i => b (f i)) := by
+  have hp : (∏ i, MvPolynomial.X (f i) : MvPolynomial κ ℂ) =
+      MvPolynomial.monomial (tupleExponent f) 1 := by
+    simp only [tupleExponent, MvPolynomial.monomial_sum_one, MvPolynomial.X]
+  change ((MvPolynomial.basisMonomials κ ℂ).constr ℂ (liftValues b T))
+    ((SymmetricAlgebra.equivMvPolynomial b) (∏ i, SymmetricAlgebra.ι ℂ V (b (f i)))) = _
+  rw [map_prod]
+  simp only [SymmetricAlgebra.equivMvPolynomial_ι_apply]
+  rw [hp]
+  change ((MvPolynomial.basisMonomials κ ℂ).constr ℂ (liftValues b T))
+    ((MvPolynomial.basisMonomials κ ℂ) (tupleExponent f)) = _
+  rw [Basis.constr_basis, liftValues_tuple b T hT]
+
+private lemma liftOnAlgebra_monomial (b : Basis κ ℂ V)
+    (T : MultilinearMap ℂ (fun _ : Fin n => V) W)
+    (hT : ∀ (σ : Equiv.Perm (Fin n)) (v : Fin n → V), T (fun i => v (σ i)) = T v)
+    (v : Fin n → V) :
+    liftOnAlgebra b T (symMonomialRaw n V v) = T v := by
+  let M : MultilinearMap ℂ (fun _ : Fin n => V) (SymmetricAlgebra ℂ V) :=
+    (MultilinearMap.mkPiAlgebra ℂ (Fin n) (SymmetricAlgebra ℂ V)).compLinearMap
+      (fun _ => SymmetricAlgebra.ι ℂ V)
+  have h : (liftOnAlgebra b T).compMultilinearMap M = T := by
+    apply Basis.ext_multilinear (fun _ => b)
+    intro f
+    exact liftOnAlgebra_basis b T hT f
+  exact congrArg (fun S : MultilinearMap ℂ (fun _ : Fin n => V) W => S v) h
+
+end SymmetricLift
+
+/-- A symmetric multilinear map descends to the homogeneous symmetric-power submodule. -/
+theorem exists_symPow_lift {n : ℕ} {V W : Type*}
+    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
+    (T : MultilinearMap ℂ (fun _ : Fin n => V) W)
+    (hT : ∀ (σ : Equiv.Perm (Fin n)) (v : Fin n → V), T (fun i => v (σ i)) = T v) :
+    ∃ L : SymPow n V →ₗ[ℂ] W, ∀ v, L (symMonomial n V v) = T v := by
+  let b := Module.Free.chooseBasis ℂ V
+  refine ⟨(SymmetricLift.liftOnAlgebra b T).domRestrict (symPowSubmodule n V), ?_⟩
+  intro v
+  exact SymmetricLift.liftOnAlgebra_monomial b T hT v
+
+/-- A chosen linear descent of a symmetric multilinear map. -/
+def symPowLift {n : ℕ} {V W : Type*}
+    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
+    (T : MultilinearMap ℂ (fun _ : Fin n => V) W)
+    (hT : ∀ (σ : Equiv.Perm (Fin n)) (v : Fin n → V), T (fun i => v (σ i)) = T v) :
+    SymPow n V →ₗ[ℂ] W :=
+  Classical.choose (exists_symPow_lift T hT)
+
+@[simp] theorem symPowLift_symMonomial {n : ℕ} {V W : Type*}
+    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
+    (T : MultilinearMap ℂ (fun _ : Fin n => V) W)
+    (hT : ∀ (σ : Equiv.Perm (Fin n)) (v : Fin n → V), T (fun i => v (σ i)) = T v)
+    (v : Fin n → V) : symPowLift T hT (symMonomial n V v) = T v :=
+  Classical.choose_spec (exists_symPow_lift T hT) v
+
+namespace SymmetricLift
+
+/-- Linear maps out of a symmetric power are determined by products. -/
+theorem linearMap_ext {n : ℕ} {V W : Type*}
+    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
+    {f g : SymPow n V →ₗ[ℂ] W}
+    (h : ∀ v, f (symMonomial n V v) = g (symMonomial n V v)) : f = g := by
+  apply (Submodule.linearMap_eq_iff_of_eq_span f g rfl).2
+  rintro ⟨x, ⟨v, rfl⟩⟩
+  exact h v
+
+end SymmetricLift
+
+/-- The submodule of multilinear maps invariant under permutations of their arguments. -/
+def symmetricMultilinearSubmodule (n : ℕ) (V W : Type*)
+    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W] :
+    Submodule ℂ (MultilinearMap ℂ (fun _ : Fin n => V) W) where
+  carrier := {T | ∀ (σ : Equiv.Perm (Fin n)) (v : Fin n → V),
+    T (fun i => v (σ i)) = T v}
+  zero_mem' := by simp
+  add_mem' := by
+    intro T U hT hU σ v
+    simp only [add_apply, hT σ v, hU σ v]
+  smul_mem' := by
+    intro c T hT σ v
+    simp only [smul_apply, hT σ v]
+
+/-- Descent from symmetric multilinear maps is linear in the map being descended. -/
+def symPowLiftLinear (n : ℕ) (V W : Type*)
+    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W] :
+    symmetricMultilinearSubmodule n V W →ₗ[ℂ] (SymPow n V →ₗ[ℂ] W) where
+  toFun T := symPowLift T.val T.property
+  map_add' T U := by
+    apply SymmetricLift.linearMap_ext
+    intro v
+    exact (symPowLift_symMonomial (T + U).val (T + U).property v).trans
+      (congrArg₂ (fun left right : W => left + right)
+        (symPowLift_symMonomial T.val T.property v).symm
+        (symPowLift_symMonomial U.val U.property v).symm)
+  map_smul' c T := by
+    apply SymmetricLift.linearMap_ext
+    intro v
+    exact (symPowLift_symMonomial (c • T).val (c • T).property v).trans
+      (congrArg (c • ·) (symPowLift_symMonomial T.val T.property v).symm)
+
+@[simp] theorem symPowLiftLinear_symMonomial {n : ℕ} {V W : Type*}
+    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
+    (T : symmetricMultilinearSubmodule n V W) (v : Fin n → V) :
+    symPowLiftLinear n V W T (symMonomial n V v) = T.val v :=
+  symPowLift_symMonomial T.val T.property v
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/SymmetricPairing.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/SymmetricPairing.lean
index 5906c02c7363fe5f520b90f5974fc8029cb90e8a..67de70c4bdf4a55198a4b6c108304907c8909b28 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/SymmetricPairing.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/SymmetricPairing.lean
@@ -1,78 +1,78 @@
-import OAI.RepresentationTheory.FoulkesHowe.PermanentPairing
-import OAI.RepresentationTheory.FoulkesHowe.PairingLift
-
-namespace OAI
-
-noncomputable section
-universe u v
-namespace Problem346
-
-variable {X : Type u} {Y : Type v}
-    [AddCommGroup X] [Module ℂ X] [AddCommGroup Y] [Module ℂ Y]
-
-/-- The normalized symmetric-power pairing induced by a bilinear pairing. -/
-def symPowPairing (n : ℕ) (B : X →ₗ[ℂ] Y →ₗ[ℂ] ℂ) :
-    SymPow n X →ₗ[ℂ] Module.Dual ℂ (SymPow n Y) :=
-  Classical.choose (exists_symPow_bilinear_lift n n X Y
-    (permanentPairingMultilinear n B)
-    (fun σ x => permanentPairingMultilinear_perm_left n B x σ)
-    (fun x σ y => permanentPairingMultilinear_perm_right n B x y σ))
-
-@[simp] theorem symPowPairing_monomial (n : ℕ) (B : X →ₗ[ℂ] Y →ₗ[ℂ] ℂ)
-    (x : Fin n → X) (y : Fin n → Y) :
-    symPowPairing n B (symMonomial n X x) (symMonomial n Y y) =
-      permanentPairing n (fun x y => B x y) x y := by
-  simpa only [symPowPairing, permanentPairingMultilinear_apply] using
-    Classical.choose_spec (exists_symPow_bilinear_lift n n X Y
-    (permanentPairingMultilinear n B)
-    (fun σ x => permanentPairingMultilinear_perm_left n B x σ)
-    (fun x σ y => permanentPairingMultilinear_perm_right n B x y σ)) x y
-
-/-- The canonical pairing from the symmetric power of the dual space. -/
-def symPowDualMap (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] :
-    SymPow n (Module.Dual ℂ V) →ₗ[ℂ] Module.Dual ℂ (SymPow n V) :=
-  symPowPairing n (LinearMap.id : Module.Dual ℂ V →ₗ[ℂ] Module.Dual ℂ V)
-
-@[simp] theorem symPowDualMap_monomial (n : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V]
-    (φ : Fin n → Module.Dual ℂ V) (x : Fin n → V) :
-    symPowDualMap n V (symMonomial n (Module.Dual ℂ V) φ) (symMonomial n V x) =
-      permanentPairing n (fun φ x => φ x) φ x :=
-  symPowPairing_monomial n (LinearMap.id : Module.Dual ℂ V →ₗ[ℂ] Module.Dual ℂ V) φ x
-
-/-- On pure powers, the pairing is ordinary algebra evaluation. -/
-theorem symPowDualMap_pure_right (n : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V]
-    (p : SymPow n (Module.Dual ℂ V)) (x : V) :
-    symPowDualMap n V p (symMonomial n V (fun _ => x)) =
-      SymmetricAlgebra.lift (Module.Dual.eval ℂ V x) p.val := by
-  have h : (symPowDualMap n V).flip (symMonomial n V (fun _ => x)) =
-      (SymmetricAlgebra.lift (Module.Dual.eval ℂ V x)).toLinearMap.domRestrict
-        (symPowSubmodule n (Module.Dual ℂ V)) := by
-    apply SymmetricLift.linearMap_ext
-    intro φ
-    change symPowDualMap n V (symMonomial n (Module.Dual ℂ V) φ)
-      (symMonomial n V (fun _ => x)) = _
-    rw [symPowDualMap_monomial]
-    simp [symMonomial, symMonomialRaw]
-  exact congrArg (fun L : SymPow n (Module.Dual ℂ V) →ₗ[ℂ] ℂ => L p) h
-
-/-- The analogous pure-power evaluation in the dual variables. -/
-theorem symPowDualMap_pure_left (n : ℕ) (V : Type u)
-    [AddCommGroup V] [Module ℂ V]
-    (φ : Module.Dual ℂ V) (p : SymPow n V) :
-    symPowDualMap n V (symMonomial n (Module.Dual ℂ V) (fun _ => φ)) p =
-      SymmetricAlgebra.lift φ p.val := by
-  have h : symPowDualMap n V (symMonomial n (Module.Dual ℂ V) (fun _ => φ)) =
-      (SymmetricAlgebra.lift φ).toLinearMap.domRestrict (symPowSubmodule n V) := by
-    apply SymmetricLift.linearMap_ext
-    intro x
-    rw [symPowDualMap_monomial]
-    simp [symMonomial, symMonomialRaw]
-  exact congrArg (fun L : SymPow n V →ₗ[ℂ] ℂ => L p) h
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.PermanentPairing
+import OAI.RepresentationTheory.FoulkesHowe.PairingLift
+
+namespace OAI
+
+noncomputable section
+universe u v
+namespace Problem346
+
+variable {X : Type u} {Y : Type v}
+    [AddCommGroup X] [Module ℂ X] [AddCommGroup Y] [Module ℂ Y]
+
+/-- The normalized symmetric-power pairing induced by a bilinear pairing. -/
+def symPowPairing (n : ℕ) (B : X →ₗ[ℂ] Y →ₗ[ℂ] ℂ) :
+    SymPow n X →ₗ[ℂ] Module.Dual ℂ (SymPow n Y) :=
+  Classical.choose (exists_symPow_bilinear_lift n n X Y
+    (permanentPairingMultilinear n B)
+    (fun σ x => permanentPairingMultilinear_perm_left n B x σ)
+    (fun x σ y => permanentPairingMultilinear_perm_right n B x y σ))
+
+@[simp] theorem symPowPairing_monomial (n : ℕ) (B : X →ₗ[ℂ] Y →ₗ[ℂ] ℂ)
+    (x : Fin n → X) (y : Fin n → Y) :
+    symPowPairing n B (symMonomial n X x) (symMonomial n Y y) =
+      permanentPairing n (fun x y => B x y) x y := by
+  simpa only [symPowPairing, permanentPairingMultilinear_apply] using
+    Classical.choose_spec (exists_symPow_bilinear_lift n n X Y
+    (permanentPairingMultilinear n B)
+    (fun σ x => permanentPairingMultilinear_perm_left n B x σ)
+    (fun x σ y => permanentPairingMultilinear_perm_right n B x y σ)) x y
+
+/-- The canonical pairing from the symmetric power of the dual space. -/
+def symPowDualMap (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] :
+    SymPow n (Module.Dual ℂ V) →ₗ[ℂ] Module.Dual ℂ (SymPow n V) :=
+  symPowPairing n (LinearMap.id : Module.Dual ℂ V →ₗ[ℂ] Module.Dual ℂ V)
+
+@[simp] theorem symPowDualMap_monomial (n : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V]
+    (φ : Fin n → Module.Dual ℂ V) (x : Fin n → V) :
+    symPowDualMap n V (symMonomial n (Module.Dual ℂ V) φ) (symMonomial n V x) =
+      permanentPairing n (fun φ x => φ x) φ x :=
+  symPowPairing_monomial n (LinearMap.id : Module.Dual ℂ V →ₗ[ℂ] Module.Dual ℂ V) φ x
+
+/-- On pure powers, the pairing is ordinary algebra evaluation. -/
+theorem symPowDualMap_pure_right (n : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V]
+    (p : SymPow n (Module.Dual ℂ V)) (x : V) :
+    symPowDualMap n V p (symMonomial n V (fun _ => x)) =
+      SymmetricAlgebra.lift (Module.Dual.eval ℂ V x) p.val := by
+  have h : (symPowDualMap n V).flip (symMonomial n V (fun _ => x)) =
+      (SymmetricAlgebra.lift (Module.Dual.eval ℂ V x)).toLinearMap.domRestrict
+        (symPowSubmodule n (Module.Dual ℂ V)) := by
+    apply SymmetricLift.linearMap_ext
+    intro φ
+    change symPowDualMap n V (symMonomial n (Module.Dual ℂ V) φ)
+      (symMonomial n V (fun _ => x)) = _
+    rw [symPowDualMap_monomial]
+    simp [symMonomial, symMonomialRaw]
+  exact congrArg (fun L : SymPow n (Module.Dual ℂ V) →ₗ[ℂ] ℂ => L p) h
+
+/-- The analogous pure-power evaluation in the dual variables. -/
+theorem symPowDualMap_pure_left (n : ℕ) (V : Type u)
+    [AddCommGroup V] [Module ℂ V]
+    (φ : Module.Dual ℂ V) (p : SymPow n V) :
+    symPowDualMap n V (symMonomial n (Module.Dual ℂ V) (fun _ => φ)) p =
+      SymmetricAlgebra.lift φ p.val := by
+  have h : symPowDualMap n V (symMonomial n (Module.Dual ℂ V) (fun _ => φ)) =
+      (SymmetricAlgebra.lift φ).toLinearMap.domRestrict (symPowSubmodule n V) := by
+    apply SymmetricLift.linearMap_ext
+    intro x
+    rw [symPowDualMap_monomial]
+    simp [symMonomial, symMonomialRaw]
+  exact congrArg (fun L : SymPow n V →ₗ[ℂ] ℂ => L p) h
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/TensorPresentation.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/TensorPresentation.lean
index 886483f8e475216841472b0a427b1eee0ae98e5a..2fc796556b7634212311926c8ce2bda3122bc500 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/TensorPresentation.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/TensorPresentation.lean
@@ -1,61 +1,61 @@
-import OAI.RepresentationTheory.FoulkesHowe.Model
-
-namespace OAI
-
-noncomputable section
-open scoped BigOperators TensorProduct
-
-universe u
-
-namespace Problem346
-
-variable (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]
-
-/-- Multiplication of degree-one generators as a multilinear map. -/
-def symMonomialMultilinearRaw :
-    MultilinearMap ℂ (fun _ : Fin n => V) (SymmetricAlgebra ℂ V) :=
-  (MultilinearMap.mkPiAlgebra ℂ (Fin n) (SymmetricAlgebra ℂ V)).compLinearMap
-    (fun _ => SymmetricAlgebra.ι ℂ V)
-
-@[simp] theorem symMonomialMultilinearRaw_apply (v : Fin n → V) :
-    symMonomialMultilinearRaw n V v = symMonomialRaw n V v := rfl
-
-/-- The tensor-power presentation of the homogeneous symmetric power. -/
-def tensorToSymPowRaw :
-    (⨂[ℂ] _ : Fin n, V) →ₗ[ℂ] SymmetricAlgebra ℂ V :=
-  PiTensorProduct.lift (symMonomialMultilinearRaw n V)
-
-@[simp] theorem tensorToSymPowRaw_tprod (v : Fin n → V) :
-    tensorToSymPowRaw n V (PiTensorProduct.tprod ℂ v) = symMonomialRaw n V v :=
-  PiTensorProduct.lift.tprod _
-
-theorem tensorToSymPowRaw_range :
-    LinearMap.range (tensorToSymPowRaw n V) = symPowSubmodule n V := by
-  rw [← Submodule.map_top, ← PiTensorProduct.span_tprod_eq_top,
-    Submodule.map_span, ← Set.range_comp]
-  simp only [Function.comp_def, tensorToSymPowRaw_tprod, symPowSubmodule]
-
-/-- Multiplication of tensors, with codomain restricted to degree `n`. -/
-def tensorToSymPow : (⨂[ℂ] _ : Fin n, V) →ₗ[ℂ] SymPow n V :=
-  (tensorToSymPowRaw n V).codRestrict (symPowSubmodule n V) (fun x => by
-    rw [← tensorToSymPowRaw_range]
-    exact LinearMap.mem_range_self _ x)
-
-@[simp] theorem tensorToSymPow_tprod (v : Fin n → V) :
-    tensorToSymPow n V (PiTensorProduct.tprod ℂ v) = symMonomial n V v := by
-  apply Subtype.ext
-  exact tensorToSymPowRaw_tprod n V v
-
-theorem tensorToSymPow_surjective : Function.Surjective (tensorToSymPow n V) := by
-  intro x
-  have hx : x.val ∈ LinearMap.range (tensorToSymPowRaw n V) := by
-    rw [tensorToSymPowRaw_range]
-    exact x.property
-  obtain ⟨y, hy⟩ := hx
-  exact ⟨y, Subtype.ext hy⟩
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.Model
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators TensorProduct
+
+universe u
+
+namespace Problem346
+
+variable (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]
+
+/-- Multiplication of degree-one generators as a multilinear map. -/
+def symMonomialMultilinearRaw :
+    MultilinearMap ℂ (fun _ : Fin n => V) (SymmetricAlgebra ℂ V) :=
+  (MultilinearMap.mkPiAlgebra ℂ (Fin n) (SymmetricAlgebra ℂ V)).compLinearMap
+    (fun _ => SymmetricAlgebra.ι ℂ V)
+
+@[simp] theorem symMonomialMultilinearRaw_apply (v : Fin n → V) :
+    symMonomialMultilinearRaw n V v = symMonomialRaw n V v := rfl
+
+/-- The tensor-power presentation of the homogeneous symmetric power. -/
+def tensorToSymPowRaw :
+    (⨂[ℂ] _ : Fin n, V) →ₗ[ℂ] SymmetricAlgebra ℂ V :=
+  PiTensorProduct.lift (symMonomialMultilinearRaw n V)
+
+@[simp] theorem tensorToSymPowRaw_tprod (v : Fin n → V) :
+    tensorToSymPowRaw n V (PiTensorProduct.tprod ℂ v) = symMonomialRaw n V v :=
+  PiTensorProduct.lift.tprod _
+
+theorem tensorToSymPowRaw_range :
+    LinearMap.range (tensorToSymPowRaw n V) = symPowSubmodule n V := by
+  rw [← Submodule.map_top, ← PiTensorProduct.span_tprod_eq_top,
+    Submodule.map_span, ← Set.range_comp]
+  simp only [Function.comp_def, tensorToSymPowRaw_tprod, symPowSubmodule]
+
+/-- Multiplication of tensors, with codomain restricted to degree `n`. -/
+def tensorToSymPow : (⨂[ℂ] _ : Fin n, V) →ₗ[ℂ] SymPow n V :=
+  (tensorToSymPowRaw n V).codRestrict (symPowSubmodule n V) (fun x => by
+    rw [← tensorToSymPowRaw_range]
+    exact LinearMap.mem_range_self _ x)
+
+@[simp] theorem tensorToSymPow_tprod (v : Fin n → V) :
+    tensorToSymPow n V (PiTensorProduct.tprod ℂ v) = symMonomial n V v := by
+  apply Subtype.ext
+  exact tensorToSymPowRaw_tprod n V v
+
+theorem tensorToSymPow_surjective : Function.Surjective (tensorToSymPow n V) := by
+  intro x
+  have hx : x.val ∈ LinearMap.range (tensorToSymPowRaw n V) := by
+    rw [tensorToSymPowRaw_range]
+    exact x.property
+  obtain ⟨y, hy⟩ := hx
+  exact ⟨y, Subtype.ext hy⟩
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/TransferDerivative.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/TransferDerivative.lean
index a67a8506a8d68514bf20ea1c4f7533d016640dec..6ee6d471e26ff2942bce6cac9d15270a286c6c67 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/TransferDerivative.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/TransferDerivative.lean
@@ -1,121 +1,121 @@
-import OAI.RepresentationTheory.FoulkesHowe.FirstCounts
-import OAI.RepresentationTheory.FoulkesHowe.FlattenedForm
-import OAI.RepresentationTheory.FoulkesHowe.MultilinearPolynomial
-import OAI.RepresentationTheory.FoulkesHowe.SlotDerivative
-
-namespace OAI
-
-noncomputable section
-open scoped BigOperators
-
-namespace Problem346
-
-theorem multilinearPolynomial_eq_slotPolynomial
-    {S K B V : Type*} [Fintype S] [DecidableEq S] [Fintype K]
-    [AddCommGroup V] [Module ℂ V]
-    (T : MultilinearMap ℂ (fun _ : S => V) ℂ) (e : K → V) (l : S → B) :
-    multilinearPolynomial T e l =
-      SlotPolynomial.polynomial (fun k => T (fun s => e (k s))) l := by
-  classical
-  unfold multilinearPolynomial SlotPolynomial.polynomial
-  congr 1
-  ext k
-  simp
-
-private theorem swap_pair_same_row {A B : Type*} [DecidableEq A] [DecidableEq B]
-    (i : A) (j k : B) (p : A × B) :
-    Equiv.swap (i, j) (i, k) p =
-      (p.1, if p.1 = i then Equiv.swap j k p.2 else p.2) := by
-  rcases p with ⟨q, t⟩
-  by_cases hq : q = i
-  · subst q
-    by_cases htj : t = j
-    · subst t
-      simp
-    · by_cases htk : t = k
-      · subst t
-        simp
-      · simp [Equiv.swap_apply_of_ne_of_ne, htj, htk]
-  · have h1 : (q,t) ≠ (i,j) := by intro h; exact hq (congrArg Prod.fst h)
-    have h2 : (q,t) ≠ (i,k) := by intro h; exact hq (congrArg Prod.fst h)
-    simp [Equiv.swap_apply_of_ne_of_ne h1 h2, hq]
-
-/-- The flattened form is invariant under swapping two factors in a single row. -/
-theorem flattenedForm_swap_row (a b : ℕ) (V : Type*)
-    [AddCommGroup V] [Module ℂ V] (T : SymmetricMultilinearForm a b V)
-    (i : Fin a) (j k : Fin b) (x : Fin a × Fin b → V) :
-    flattenedForm T (x ∘ Equiv.swap (i,j) (i,k)) =
-      flattenedForm T x := by
-  classical
-  have h := flattenedForm_factor_perm T
-    (fun q => if q = i then Equiv.swap j k else Equiv.refl _) x
-  convert h using 1
-  congr 1
-  funext p
-  rw [Function.comp_apply, swap_pair_same_row]
-  by_cases hp : p.1 = i <;> simp [hp]
-
-/-- If the source label occurs only in one row, every occurrence gives the same
-coordinate-polynomial derivative term, by symmetry of factors in that row. -/
-theorem flattened_derivative_eq_card_smul
-    (a b : ℕ) (V K B : Type*) [AddCommGroup V] [Module ℂ V]
-    [Fintype K] [DecidableEq B]
-    (T : SymmetricMultilinearForm a b V) (e : K → V)
-    (l : Fin a × Fin b → B) (src dst : B) (i : Fin a) (j : Fin b)
-    (hj : l (i,j) = src)
-    (hrow : ∀ p, l p = src → p.1 = i)
-    (D : Derivation ℂ (MvPolynomial (B × K) ℂ) (MvPolynomial (B × K) ℂ))
-    (hD : ∀ q k, D (MvPolynomial.X (q,k)) =
-      if q = src then MvPolynomial.X (dst,k) else 0) :
-    D (multilinearPolynomial (flattenedForm T) e l) =
-      (Finset.univ.filter (fun p => l p = src)).card •
-        multilinearPolynomial (flattenedForm T) e
-          (Function.update l (i,j) dst) := by
-  classical
-  let c : ((Fin a × Fin b) → K) → ℂ :=
-    fun k => flattenedForm T (fun p => e (k p))
-  have hc : ∀ p, l p = src → ∀ k, c (k ∘ Equiv.swap p (i,j)) = c k := by
-    intro p hp k
-    rcases p with ⟨q,t⟩
-    have hq := hrow (q,t) hp
-    dsimp at hq
-    subst q
-    exact flattenedForm_swap_row a b V T i t j (fun p => e (k p))
-  have h := SlotPolynomial.derivation_polynomial_eq_card_smul
-    c l src dst (i,j) hj D hD hc
-  rw [multilinearPolynomial_eq_slotPolynomial, multilinearPolynomial_eq_slotPolynomial]
-  exact h
-
-/-- The concrete one-step identity in the first transfer. The source is the
-`t` block; its remaining occurrences all lie in the distinguished row. -/
-theorem firstLabel_derivative
-    (r m k : ℕ) (hk : k < m) (V K : Type*)
-    [AddCommGroup V] [Module ℂ V] [Fintype K]
-    (T : SymmetricMultilinearForm (r+1) (r+m) V) (e : K → V)
-    (D : Derivation ℂ (MvPolynomial (FirstBlock m × K) ℂ)
-      (MvPolynomial (FirstBlock m × K) ℂ))
-    (hD : ∀ q j, D (MvPolynomial.X (q,j)) =
-      if q = some none then MvPolynomial.X (some (some ⟨k,hk⟩),j) else 0) :
-    D (multilinearPolynomial (flattenedForm T) e
-        (firstLabel r m k)) =
-      ((r+m-k : ℕ) : ℂ) •
-        multilinearPolynomial (flattenedForm T) e
-          (firstLabel r m (k+1)) := by
-  classical
-  have hj : firstLabel r m k
-      (0, ⟨k, Nat.lt_of_lt_of_le hk (Nat.le_add_left m r)⟩) = some none := by
-    simp [firstLabel]
-  have hrow : ∀ p, firstLabel r m k p = some none → p.1 = 0 := by
-    intro p hp
-    exact ((firstLabel_eq_t_iff r m k (Nat.le_of_lt hk) p).mp hp).1
-  have h := flattened_derivative_eq_card_smul (r+1) (r+m) V K (FirstBlock m)
-    T e (firstLabel r m k) (some none) (some (some ⟨k,hk⟩))
-    0 ⟨k, Nat.lt_of_lt_of_le hk (Nat.le_add_left m r)⟩ hj hrow D hD
-  rw [firstLabel_card_t r m k (Nat.le_of_lt hk), ← firstLabel_step r m k hk] at h
-  simpa only [Nat.cast_smul_eq_nsmul] using h
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.FirstCounts
+import OAI.RepresentationTheory.FoulkesHowe.FlattenedForm
+import OAI.RepresentationTheory.FoulkesHowe.MultilinearPolynomial
+import OAI.RepresentationTheory.FoulkesHowe.SlotDerivative
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators
+
+namespace Problem346
+
+theorem multilinearPolynomial_eq_slotPolynomial
+    {S K B V : Type*} [Fintype S] [DecidableEq S] [Fintype K]
+    [AddCommGroup V] [Module ℂ V]
+    (T : MultilinearMap ℂ (fun _ : S => V) ℂ) (e : K → V) (l : S → B) :
+    multilinearPolynomial T e l =
+      SlotPolynomial.polynomial (fun k => T (fun s => e (k s))) l := by
+  classical
+  unfold multilinearPolynomial SlotPolynomial.polynomial
+  congr 1
+  ext k
+  simp
+
+private theorem swap_pair_same_row {A B : Type*} [DecidableEq A] [DecidableEq B]
+    (i : A) (j k : B) (p : A × B) :
+    Equiv.swap (i, j) (i, k) p =
+      (p.1, if p.1 = i then Equiv.swap j k p.2 else p.2) := by
+  rcases p with ⟨q, t⟩
+  by_cases hq : q = i
+  · subst q
+    by_cases htj : t = j
+    · subst t
+      simp
+    · by_cases htk : t = k
+      · subst t
+        simp
+      · simp [Equiv.swap_apply_of_ne_of_ne, htj, htk]
+  · have h1 : (q,t) ≠ (i,j) := by intro h; exact hq (congrArg Prod.fst h)
+    have h2 : (q,t) ≠ (i,k) := by intro h; exact hq (congrArg Prod.fst h)
+    simp [Equiv.swap_apply_of_ne_of_ne h1 h2, hq]
+
+/-- The flattened form is invariant under swapping two factors in a single row. -/
+theorem flattenedForm_swap_row (a b : ℕ) (V : Type*)
+    [AddCommGroup V] [Module ℂ V] (T : SymmetricMultilinearForm a b V)
+    (i : Fin a) (j k : Fin b) (x : Fin a × Fin b → V) :
+    flattenedForm T (x ∘ Equiv.swap (i,j) (i,k)) =
+      flattenedForm T x := by
+  classical
+  have h := flattenedForm_factor_perm T
+    (fun q => if q = i then Equiv.swap j k else Equiv.refl _) x
+  convert h using 1
+  congr 1
+  funext p
+  rw [Function.comp_apply, swap_pair_same_row]
+  by_cases hp : p.1 = i <;> simp [hp]
+
+/-- If the source label occurs only in one row, every occurrence gives the same
+coordinate-polynomial derivative term, by symmetry of factors in that row. -/
+theorem flattened_derivative_eq_card_smul
+    (a b : ℕ) (V K B : Type*) [AddCommGroup V] [Module ℂ V]
+    [Fintype K] [DecidableEq B]
+    (T : SymmetricMultilinearForm a b V) (e : K → V)
+    (l : Fin a × Fin b → B) (src dst : B) (i : Fin a) (j : Fin b)
+    (hj : l (i,j) = src)
+    (hrow : ∀ p, l p = src → p.1 = i)
+    (D : Derivation ℂ (MvPolynomial (B × K) ℂ) (MvPolynomial (B × K) ℂ))
+    (hD : ∀ q k, D (MvPolynomial.X (q,k)) =
+      if q = src then MvPolynomial.X (dst,k) else 0) :
+    D (multilinearPolynomial (flattenedForm T) e l) =
+      (Finset.univ.filter (fun p => l p = src)).card •
+        multilinearPolynomial (flattenedForm T) e
+          (Function.update l (i,j) dst) := by
+  classical
+  let c : ((Fin a × Fin b) → K) → ℂ :=
+    fun k => flattenedForm T (fun p => e (k p))
+  have hc : ∀ p, l p = src → ∀ k, c (k ∘ Equiv.swap p (i,j)) = c k := by
+    intro p hp k
+    rcases p with ⟨q,t⟩
+    have hq := hrow (q,t) hp
+    dsimp at hq
+    subst q
+    exact flattenedForm_swap_row a b V T i t j (fun p => e (k p))
+  have h := SlotPolynomial.derivation_polynomial_eq_card_smul
+    c l src dst (i,j) hj D hD hc
+  rw [multilinearPolynomial_eq_slotPolynomial, multilinearPolynomial_eq_slotPolynomial]
+  exact h
+
+/-- The concrete one-step identity in the first transfer. The source is the
+`t` block; its remaining occurrences all lie in the distinguished row. -/
+theorem firstLabel_derivative
+    (r m k : ℕ) (hk : k < m) (V K : Type*)
+    [AddCommGroup V] [Module ℂ V] [Fintype K]
+    (T : SymmetricMultilinearForm (r+1) (r+m) V) (e : K → V)
+    (D : Derivation ℂ (MvPolynomial (FirstBlock m × K) ℂ)
+      (MvPolynomial (FirstBlock m × K) ℂ))
+    (hD : ∀ q j, D (MvPolynomial.X (q,j)) =
+      if q = some none then MvPolynomial.X (some (some ⟨k,hk⟩),j) else 0) :
+    D (multilinearPolynomial (flattenedForm T) e
+        (firstLabel r m k)) =
+      ((r+m-k : ℕ) : ℂ) •
+        multilinearPolynomial (flattenedForm T) e
+          (firstLabel r m (k+1)) := by
+  classical
+  have hj : firstLabel r m k
+      (0, ⟨k, Nat.lt_of_lt_of_le hk (Nat.le_add_left m r)⟩) = some none := by
+    simp [firstLabel]
+  have hrow : ∀ p, firstLabel r m k p = some none → p.1 = 0 := by
+    intro p hp
+    exact ((firstLabel_eq_t_iff r m k (Nat.le_of_lt hk) p).mp hp).1
+  have h := flattened_derivative_eq_card_smul (r+1) (r+m) V K (FirstBlock m)
+    T e (firstLabel r m k) (some none) (some (some ⟨k,hk⟩))
+    0 ⟨k, Nat.lt_of_lt_of_le hk (Nat.le_add_left m r)⟩ hj hrow D hD
+  rw [firstLabel_card_t r m k (Nat.le_of_lt hk), ← firstLabel_step r m k hk] at h
+  simpa only [Nat.cast_smul_eq_nsmul] using h
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/TransferEndpoints.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/TransferEndpoints.lean
index 26eff67b7595405ef46a4436fa236efaf7309df6..4173211c5df01a9451018ddded9136807c833837 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/TransferEndpoints.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/TransferEndpoints.lean
@@ -1,133 +1,133 @@
-import OAI.RepresentationTheory.FoulkesHowe.FirstPolynomial
-import OAI.RepresentationTheory.FoulkesHowe.Multiplication
-
-namespace OAI
-
-noncomputable section
-open scoped BigOperators
-namespace Problem346
-universe u v
-variable {V : Type u} [AddCommGroup V] [Module ℂ V]
-variable {κ : Type v} [Fintype κ]
-
-/-- Assign vector values to the three kinds of blocks in the first transfer. -/
-def firstVectors {m : ℕ} (x t : V) (y : Fin m → V) : FirstBlock m → V
-  | none => x
-  | some none => t
-  | some (some j) => y j
-
-omit [AddCommGroup V] [Module ℂ V] in
-@[simp] theorem firstVectors_x {m : ℕ} (x t : V) (y : Fin m → V) :
-    firstVectors x t y none = x := rfl
-omit [AddCommGroup V] [Module ℂ V] in
-@[simp] theorem firstVectors_t {m : ℕ} (x t : V) (y : Fin m → V) :
-    firstVectors x t y (some none) = t := rfl
-omit [AddCommGroup V] [Module ℂ V] in
-@[simp] theorem firstVectors_y {m : ℕ} (x t : V) (y : Fin m → V) (j : Fin m) :
-    firstVectors x t y (some (some j)) = y j := rfl
-
-@[simp] theorem firstLabel_initial_monomial (r m : ℕ) (x t : V) (y : Fin m → V) :
-    symMonomial (r+m) V (fun j => firstVectors x t y (firstLabel r m 0 (0,j))) =
-      symMonomial (r+m) V (fun _ => t) := by
-  simp
-
-@[simp] theorem firstLabel_other_monomial (r m k : ℕ) (x t : V)
-    (y : Fin m → V) (i : Fin r) :
-    symMonomial (r+m) V (fun j => firstVectors x t y (firstLabel r m k (i.succ,j))) =
-      symPowMul r m (symMonomial r V (fun _ => x)) (symMonomial m V y) := by
-  rw [symPowMul_symMonomial]
-  congr 1
-  funext j
-  refine Fin.addCases ?_ ?_ j
-  · intro j
-    simp [Fin.append]
-  · intro j
-    simp [Fin.append]
-
-@[simp] theorem firstLabel_final_monomial (r m : ℕ) (x t : V) (y : Fin m → V) :
-    symMonomial (r+m) V (fun j => firstVectors x t y (firstLabel r m m (0,j))) =
-      symPowMul r m (symMonomial r V (fun _ => t)) (symMonomial m V y) := by
-  apply Subtype.ext
-  change (∏ j : Fin (r+m), SymmetricAlgebra.ι ℂ V
-      (firstVectors x t y (firstLabel r m m (0,j)))) =
-    (∏ _ : Fin r, SymmetricAlgebra.ι ℂ V t) * ∏ j : Fin m, SymmetricAlgebra.ι ℂ V (y j)
-  rw [← (finCongr (Nat.add_comm m r)).prod_comp
-    (fun j : Fin (r+m) => SymmetricAlgebra.ι ℂ V
-      (firstVectors x t y (firstLabel r m m (0,j))))]
-  rw [Fin.prod_univ_add]
-  have hleft : (∏ j : Fin m, SymmetricAlgebra.ι ℂ V
-      (firstVectors x t y (firstLabel r m m
-        (0, (finCongr (Nat.add_comm m r)) (Fin.castAdd r j))))) =
-      ∏ j : Fin m, SymmetricAlgebra.ι ℂ V (y j) := by
-    apply Finset.prod_congr rfl
-    intro j _
-    simp [firstLabel]
-  have hright : (∏ j : Fin r, SymmetricAlgebra.ι ℂ V
-      (firstVectors x t y (firstLabel r m m
-        (0, (finCongr (Nat.add_comm m r)) (Fin.natAdd m j))))) =
-      ∏ _ : Fin r, SymmetricAlgebra.ι ℂ V t := by
-    apply Finset.prod_congr rfl
-    intro j _
-    simp [firstLabel]
-  rw [hleft, hright, mul_comm]
-
-/-- Evaluation of the initial label configuration. -/
-theorem firstLabel_initial_form {r m : ℕ}
-    (T : SymmetricMultilinearForm (r+1) (r+m) V) (x t : V) (y : Fin m → V) :
-    flattenedForm T (fun p => firstVectors x t y (firstLabel r m 0 p)) =
-      T (Fin.cons (symMonomial (r+m) V (fun _ => t))
-        (fun _ => symPowMul r m (symMonomial r V (fun _ => x)) (symMonomial m V y))) := by
-  rw [flattenedForm_apply]
-  congr 1
-  funext i
-  cases i using Fin.cases with
-  | zero => exact firstLabel_initial_monomial r m x t y
-  | succ i => exact firstLabel_other_monomial r m 0 x t y i
-
-/-- Evaluation of the terminal label configuration, with t^r times the common product. -/
-theorem firstLabel_final_form {r m : ℕ}
-    (T : SymmetricMultilinearForm (r+1) (r+m) V) (x t : V) (y : Fin m → V) :
-    flattenedForm T (fun p => firstVectors x t y (firstLabel r m m p)) =
-      T (Fin.cons (symPowMul r m (symMonomial r V (fun _ => t)) (symMonomial m V y))
-        (fun _ => symPowMul r m (symMonomial r V (fun _ => x)) (symMonomial m V y))) := by
-  rw [flattenedForm_apply]
-  congr 1
-  funext i
-  cases i using Fin.cases with
-  | zero => exact firstLabel_final_monomial r m x t y
-  | succ i => exact firstLabel_other_monomial r m m x t y i
-
-/-- The binary identity makes the terminal polynomial zero. -/
-theorem firstPolynomial_final_eq_zero {r m : ℕ}
-    (T : SymmetricMultilinearForm (r+1) (r+m) V) (e : κ → V)
-    (h : ∀ (x t : V) (y : Fin m → V),
-      T (Fin.cons (symPowMul r m (symMonomial r V (fun _ => t)) (symMonomial m V y))
-        (fun _ => symPowMul r m (symMonomial r V (fun _ => x)) (symMonomial m V y))) = 0) :
-    firstPolynomial T e m = 0 := by
-  apply multilinearPolynomial_eq_zero
-  intro v
-  have hv : firstVectors (v none) (v (some none)) (fun j => v (some (some j))) = v := by
-    funext b
-    cases b with
-    | none => rfl
-    | some b => cases b <;> rfl
-  rw [← hv]
-  rw [firstLabel_final_form]
-  exact h _ _ _
-
-/-- A zero initial polynomial gives the freed-last-slot identity. -/
-theorem firstPolynomial_initial_vanishing {r m : ℕ}
-    (T : SymmetricMultilinearForm (r+1) (r+m) V) (e : Module.Basis κ ℂ V)
-    (h : firstPolynomial T e 0 = 0) (x t : V) (y : Fin m → V) :
-    T (Fin.cons (symMonomial (r+m) V (fun _ => t))
-      (fun _ => symPowMul r m (symMonomial r V (fun _ => x)) (symMonomial m V y))) = 0 := by
-  rw [← firstLabel_initial_form]
-  have hp := (multilinearPolynomial_eq_zero_iff (flattenedForm T) e
-    (firstLabel r m 0)).mp h
-  exact hp (firstVectors x t y)
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.FirstPolynomial
+import OAI.RepresentationTheory.FoulkesHowe.Multiplication
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators
+namespace Problem346
+universe u v
+variable {V : Type u} [AddCommGroup V] [Module ℂ V]
+variable {κ : Type v} [Fintype κ]
+
+/-- Assign vector values to the three kinds of blocks in the first transfer. -/
+def firstVectors {m : ℕ} (x t : V) (y : Fin m → V) : FirstBlock m → V
+  | none => x
+  | some none => t
+  | some (some j) => y j
+
+omit [AddCommGroup V] [Module ℂ V] in
+@[simp] theorem firstVectors_x {m : ℕ} (x t : V) (y : Fin m → V) :
+    firstVectors x t y none = x := rfl
+omit [AddCommGroup V] [Module ℂ V] in
+@[simp] theorem firstVectors_t {m : ℕ} (x t : V) (y : Fin m → V) :
+    firstVectors x t y (some none) = t := rfl
+omit [AddCommGroup V] [Module ℂ V] in
+@[simp] theorem firstVectors_y {m : ℕ} (x t : V) (y : Fin m → V) (j : Fin m) :
+    firstVectors x t y (some (some j)) = y j := rfl
+
+@[simp] theorem firstLabel_initial_monomial (r m : ℕ) (x t : V) (y : Fin m → V) :
+    symMonomial (r+m) V (fun j => firstVectors x t y (firstLabel r m 0 (0,j))) =
+      symMonomial (r+m) V (fun _ => t) := by
+  simp
+
+@[simp] theorem firstLabel_other_monomial (r m k : ℕ) (x t : V)
+    (y : Fin m → V) (i : Fin r) :
+    symMonomial (r+m) V (fun j => firstVectors x t y (firstLabel r m k (i.succ,j))) =
+      symPowMul r m (symMonomial r V (fun _ => x)) (symMonomial m V y) := by
+  rw [symPowMul_symMonomial]
+  congr 1
+  funext j
+  refine Fin.addCases ?_ ?_ j
+  · intro j
+    simp [Fin.append]
+  · intro j
+    simp [Fin.append]
+
+@[simp] theorem firstLabel_final_monomial (r m : ℕ) (x t : V) (y : Fin m → V) :
+    symMonomial (r+m) V (fun j => firstVectors x t y (firstLabel r m m (0,j))) =
+      symPowMul r m (symMonomial r V (fun _ => t)) (symMonomial m V y) := by
+  apply Subtype.ext
+  change (∏ j : Fin (r+m), SymmetricAlgebra.ι ℂ V
+      (firstVectors x t y (firstLabel r m m (0,j)))) =
+    (∏ _ : Fin r, SymmetricAlgebra.ι ℂ V t) * ∏ j : Fin m, SymmetricAlgebra.ι ℂ V (y j)
+  rw [← (finCongr (Nat.add_comm m r)).prod_comp
+    (fun j : Fin (r+m) => SymmetricAlgebra.ι ℂ V
+      (firstVectors x t y (firstLabel r m m (0,j))))]
+  rw [Fin.prod_univ_add]
+  have hleft : (∏ j : Fin m, SymmetricAlgebra.ι ℂ V
+      (firstVectors x t y (firstLabel r m m
+        (0, (finCongr (Nat.add_comm m r)) (Fin.castAdd r j))))) =
+      ∏ j : Fin m, SymmetricAlgebra.ι ℂ V (y j) := by
+    apply Finset.prod_congr rfl
+    intro j _
+    simp [firstLabel]
+  have hright : (∏ j : Fin r, SymmetricAlgebra.ι ℂ V
+      (firstVectors x t y (firstLabel r m m
+        (0, (finCongr (Nat.add_comm m r)) (Fin.natAdd m j))))) =
+      ∏ _ : Fin r, SymmetricAlgebra.ι ℂ V t := by
+    apply Finset.prod_congr rfl
+    intro j _
+    simp [firstLabel]
+  rw [hleft, hright, mul_comm]
+
+/-- Evaluation of the initial label configuration. -/
+theorem firstLabel_initial_form {r m : ℕ}
+    (T : SymmetricMultilinearForm (r+1) (r+m) V) (x t : V) (y : Fin m → V) :
+    flattenedForm T (fun p => firstVectors x t y (firstLabel r m 0 p)) =
+      T (Fin.cons (symMonomial (r+m) V (fun _ => t))
+        (fun _ => symPowMul r m (symMonomial r V (fun _ => x)) (symMonomial m V y))) := by
+  rw [flattenedForm_apply]
+  congr 1
+  funext i
+  cases i using Fin.cases with
+  | zero => exact firstLabel_initial_monomial r m x t y
+  | succ i => exact firstLabel_other_monomial r m 0 x t y i
+
+/-- Evaluation of the terminal label configuration, with t^r times the common product. -/
+theorem firstLabel_final_form {r m : ℕ}
+    (T : SymmetricMultilinearForm (r+1) (r+m) V) (x t : V) (y : Fin m → V) :
+    flattenedForm T (fun p => firstVectors x t y (firstLabel r m m p)) =
+      T (Fin.cons (symPowMul r m (symMonomial r V (fun _ => t)) (symMonomial m V y))
+        (fun _ => symPowMul r m (symMonomial r V (fun _ => x)) (symMonomial m V y))) := by
+  rw [flattenedForm_apply]
+  congr 1
+  funext i
+  cases i using Fin.cases with
+  | zero => exact firstLabel_final_monomial r m x t y
+  | succ i => exact firstLabel_other_monomial r m m x t y i
+
+/-- The binary identity makes the terminal polynomial zero. -/
+theorem firstPolynomial_final_eq_zero {r m : ℕ}
+    (T : SymmetricMultilinearForm (r+1) (r+m) V) (e : κ → V)
+    (h : ∀ (x t : V) (y : Fin m → V),
+      T (Fin.cons (symPowMul r m (symMonomial r V (fun _ => t)) (symMonomial m V y))
+        (fun _ => symPowMul r m (symMonomial r V (fun _ => x)) (symMonomial m V y))) = 0) :
+    firstPolynomial T e m = 0 := by
+  apply multilinearPolynomial_eq_zero
+  intro v
+  have hv : firstVectors (v none) (v (some none)) (fun j => v (some (some j))) = v := by
+    funext b
+    cases b with
+    | none => rfl
+    | some b => cases b <;> rfl
+  rw [← hv]
+  rw [firstLabel_final_form]
+  exact h _ _ _
+
+/-- A zero initial polynomial gives the freed-last-slot identity. -/
+theorem firstPolynomial_initial_vanishing {r m : ℕ}
+    (T : SymmetricMultilinearForm (r+1) (r+m) V) (e : Module.Basis κ ℂ V)
+    (h : firstPolynomial T e 0 = 0) (x t : V) (y : Fin m → V) :
+    T (Fin.cons (symMonomial (r+m) V (fun _ => t))
+      (fun _ => symPowMul r m (symMonomial r V (fun _ => x)) (symMonomial m V y))) = 0 := by
+  rw [← firstLabel_initial_form]
+  have hp := (multilinearPolynomial_eq_zero_iff (flattenedForm T) e
+    (firstLabel r m 0)).mp h
+  exact hp (firstVectors x t y)
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/Vanishing.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/Vanishing.lean
index d13b653542df17b5d9c3c1edde4a65ac8b58e787..a4182db3245e778bf573a73bf9472e5c9d81adbb 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/Vanishing.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/Vanishing.lean
@@ -1,72 +1,72 @@
-import OAI.RepresentationTheory.FoulkesHowe.VanishingOne
-import OAI.RepresentationTheory.FoulkesHowe.VanishingInduction
-
-namespace OAI
-
-noncomputable section
-
-namespace Problem346
-
-universe u
-variable {V : Type u} [AddCommGroup V] [Module ℂ V]
-
-/-- The induction on the number of slots, isolated from the two polynomial
-transfer arguments. The first argument packages the binary-pencil identity and
-the first transfer; the second is the common-power-removal transfer. -/
-theorem vanishing_induction_from_transfer_steps
-    (firstTransfer : ∀ r m : ℕ, 0 < r → 0 < m → (r+1)*r ≤ r+m →
-      ∀ T : SymmetricMultilinearForm (r+1) (r+m) V,
-      IsSymmetricMultilinearForm (r+1) (r+m) V T →
-      (∀ y : Fin (r+m) → V, T (fun _ => symMonomial (r+m) V y) = 0) →
-      ∀ x t : V, ∀ y : Fin m → V,
-        lowerProductForm r m T x t (fun _ => symMonomial m V y) = 0)
-    (secondTransfer : ∀ r m : ℕ, 0 < r → 0 < m → (r+1)*r ≤ r+m →
-      ∀ T : SymmetricMultilinearForm (r+1) (r+m) V,
-      IsSymmetricMultilinearForm (r+1) (r+m) V T →
-      (∀ x t : V, ∀ Q : Fin r → SymPow m V, lowerProductForm r m T x t Q = 0) →
-      ∀ v : Fin (r+1) → V,
-        T (fun i => symMonomial (r+m) V (fun _ => v i)) = 0)
-    (purePowerExt : ∀ a b : ℕ, ∀ T : SymmetricMultilinearForm a b V,
-      (∀ v : Fin a → V, T (fun i => symMonomial b V (fun _ => v i)) = 0) → T = 0) :
-    ∀ a b : ℕ, 1 ≤ a → 1 ≤ b → a*(a-1) ≤ b →
-      ∀ T : SymmetricMultilinearForm a b V,
-      IsSymmetricMultilinearForm a b V T →
-      (∀ y : Fin b → V, T (fun _ => symMonomial b V y) = 0) → T = 0 := by
-  intro a
-  induction a with
-  | zero =>
-      intro b ha
-      omega
-  | succ r ih =>
-      intro b ha hb hab T hT hdiag
-      by_cases hr : r = 0
-      · subst r
-        exact vanishing_on_products_one V b T hdiag
-      have hrpos : 0 < r := Nat.pos_of_ne_zero hr
-      have hrb : r ≤ b := by
-        have hmul : r ≤ (r+1)*r := by nlinarith
-        simpa only [Nat.succ_eq_add_one, Nat.add_sub_cancel] using hmul.trans hab
-      obtain ⟨m, rfl⟩ := Nat.exists_eq_add_of_le hrb
-      have hbound : (r+1)*r ≤ r+m := by simpa only [Nat.succ_eq_add_one,
-        Nat.add_sub_cancel] using hab
-      have hm : 0 < m := by nlinarith
-      have hmrange : r*(r-1) ≤ m := by
-        have hsub : r-1 ≤ r := Nat.sub_le ..
-        have hmul : r*(r-1) ≤ r*r := Nat.mul_le_mul_left r hsub
-        nlinarith
-      have hrestricted : ∀ x t : V, lowerProductForm r m T x t = 0 := by
-        intro x t
-        exact ih m hrpos hm hmrange (lowerProductForm r m T x t)
-          (lowerProductForm_symmetric r m T hT x t)
-          (firstTransfer r m hrpos hm hbound T hT hdiag x t)
-      apply purePowerExt (r+1) (r+m) T
-      apply secondTransfer r m hrpos hm hbound T hT
-      intro x t Q
-      rw [hrestricted x t]
-      rfl
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.VanishingOne
+import OAI.RepresentationTheory.FoulkesHowe.VanishingInduction
+
+namespace OAI
+
+noncomputable section
+
+namespace Problem346
+
+universe u
+variable {V : Type u} [AddCommGroup V] [Module ℂ V]
+
+/-- The induction on the number of slots, isolated from the two polynomial
+transfer arguments. The first argument packages the binary-pencil identity and
+the first transfer; the second is the common-power-removal transfer. -/
+theorem vanishing_induction_from_transfer_steps
+    (firstTransfer : ∀ r m : ℕ, 0 < r → 0 < m → (r+1)*r ≤ r+m →
+      ∀ T : SymmetricMultilinearForm (r+1) (r+m) V,
+      IsSymmetricMultilinearForm (r+1) (r+m) V T →
+      (∀ y : Fin (r+m) → V, T (fun _ => symMonomial (r+m) V y) = 0) →
+      ∀ x t : V, ∀ y : Fin m → V,
+        lowerProductForm r m T x t (fun _ => symMonomial m V y) = 0)
+    (secondTransfer : ∀ r m : ℕ, 0 < r → 0 < m → (r+1)*r ≤ r+m →
+      ∀ T : SymmetricMultilinearForm (r+1) (r+m) V,
+      IsSymmetricMultilinearForm (r+1) (r+m) V T →
+      (∀ x t : V, ∀ Q : Fin r → SymPow m V, lowerProductForm r m T x t Q = 0) →
+      ∀ v : Fin (r+1) → V,
+        T (fun i => symMonomial (r+m) V (fun _ => v i)) = 0)
+    (purePowerExt : ∀ a b : ℕ, ∀ T : SymmetricMultilinearForm a b V,
+      (∀ v : Fin a → V, T (fun i => symMonomial b V (fun _ => v i)) = 0) → T = 0) :
+    ∀ a b : ℕ, 1 ≤ a → 1 ≤ b → a*(a-1) ≤ b →
+      ∀ T : SymmetricMultilinearForm a b V,
+      IsSymmetricMultilinearForm a b V T →
+      (∀ y : Fin b → V, T (fun _ => symMonomial b V y) = 0) → T = 0 := by
+  intro a
+  induction a with
+  | zero =>
+      intro b ha
+      omega
+  | succ r ih =>
+      intro b ha hb hab T hT hdiag
+      by_cases hr : r = 0
+      · subst r
+        exact vanishing_on_products_one V b T hdiag
+      have hrpos : 0 < r := Nat.pos_of_ne_zero hr
+      have hrb : r ≤ b := by
+        have hmul : r ≤ (r+1)*r := by nlinarith
+        simpa only [Nat.succ_eq_add_one, Nat.add_sub_cancel] using hmul.trans hab
+      obtain ⟨m, rfl⟩ := Nat.exists_eq_add_of_le hrb
+      have hbound : (r+1)*r ≤ r+m := by simpa only [Nat.succ_eq_add_one,
+        Nat.add_sub_cancel] using hab
+      have hm : 0 < m := by nlinarith
+      have hmrange : r*(r-1) ≤ m := by
+        have hsub : r-1 ≤ r := Nat.sub_le ..
+        have hmul : r*(r-1) ≤ r*r := Nat.mul_le_mul_left r hsub
+        nlinarith
+      have hrestricted : ∀ x t : V, lowerProductForm r m T x t = 0 := by
+        intro x t
+        exact ih m hrpos hm hmrange (lowerProductForm r m T x t)
+          (lowerProductForm_symmetric r m T hT x t)
+          (firstTransfer r m hrpos hm hbound T hT hdiag x t)
+      apply purePowerExt (r+1) (r+m) T
+      apply secondTransfer r m hrpos hm hbound T hT
+      intro x t Q
+      rw [hrestricted x t]
+      rfl
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/VanishingFirst.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/VanishingFirst.lean
index 3b107a0b5a748e3c7e47e1391af9b1bbf4ce3cdb..cd8d1b0c0754f1b68913b2f713aaac63e0ef067d 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/VanishingFirst.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/VanishingFirst.lean
@@ -1,49 +1,49 @@
-import OAI.RepresentationTheory.FoulkesHowe.Vanishing
-import OAI.RepresentationTheory.FoulkesHowe.BinaryVanishing
-import OAI.RepresentationTheory.FoulkesHowe.FirstTransfer
-import OAI.RepresentationTheory.FoulkesHowe.PowerSpan
-
-namespace OAI
-
-noncomputable section
-
-namespace Problem346
-
-universe u
-variable {V : Type u} [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
-
-/-- Diagonal vanishing gives the diagonal vanishing of the lower-arity form.
-This combines binary splitting, coefficient extraction, and the first transfer. -/
-theorem lowerProductForm_diagonal_zero (r m : ℕ) (hr : 0 < r)
-    (T : SymmetricMultilinearForm (r+1) (r+m) V)
-    (hT : IsSymmetricMultilinearForm (r+1) (r+m) V T)
-    (hdiag : ∀ y : Fin (r+m) → V, T (fun _ => symMonomial (r+m) V y) = 0)
-    (x t : V) (y : Fin m → V) :
-    lowerProductForm r m T x t (fun _ => symMonomial m V y) = 0 := by
-  exact first_transfer T (binary_diagonal_to_mixed_products r m hr T hT hdiag) x t y
-
-/-- With the first transfer and pure-power detection established, the common-power
-removal transfer is the sole remaining input to the vanishing induction. -/
-theorem vanishing_on_products_of_second_transfer
-    (secondTransfer : ∀ r m : ℕ, 0 < r → 0 < m → (r+1)*r ≤ r+m →
-      ∀ T : SymmetricMultilinearForm (r+1) (r+m) V,
-      IsSymmetricMultilinearForm (r+1) (r+m) V T →
-      (∀ x t : V, ∀ Q : Fin r → SymPow m V, lowerProductForm r m T x t Q = 0) →
-      ∀ v : Fin (r+1) → V,
-        T (fun i => symMonomial (r+m) V (fun _ => v i)) = 0) :
-    ∀ a b : ℕ, 1 ≤ a → 1 ≤ b → a*(a-1) ≤ b →
-      ∀ T : SymmetricMultilinearForm a b V,
-      IsSymmetricMultilinearForm a b V T →
-      (∀ y : Fin b → V, T (fun _ => symMonomial b V y) = 0) → T = 0 := by
-  apply vanishing_induction_from_transfer_steps
-  · intro r m hr _ _ T hT hdiag
-    exact lowerProductForm_diagonal_zero r m hr T hT hdiag
-  · exact secondTransfer
-  · intro a b T hT
-    exact symmetricMultilinearForm_eq_zero_of_pure_powers a b V T hT
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.Vanishing
+import OAI.RepresentationTheory.FoulkesHowe.BinaryVanishing
+import OAI.RepresentationTheory.FoulkesHowe.FirstTransfer
+import OAI.RepresentationTheory.FoulkesHowe.PowerSpan
+
+namespace OAI
+
+noncomputable section
+
+namespace Problem346
+
+universe u
+variable {V : Type u} [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
+
+/-- Diagonal vanishing gives the diagonal vanishing of the lower-arity form.
+This combines binary splitting, coefficient extraction, and the first transfer. -/
+theorem lowerProductForm_diagonal_zero (r m : ℕ) (hr : 0 < r)
+    (T : SymmetricMultilinearForm (r+1) (r+m) V)
+    (hT : IsSymmetricMultilinearForm (r+1) (r+m) V T)
+    (hdiag : ∀ y : Fin (r+m) → V, T (fun _ => symMonomial (r+m) V y) = 0)
+    (x t : V) (y : Fin m → V) :
+    lowerProductForm r m T x t (fun _ => symMonomial m V y) = 0 := by
+  exact first_transfer T (binary_diagonal_to_mixed_products r m hr T hT hdiag) x t y
+
+/-- With the first transfer and pure-power detection established, the common-power
+removal transfer is the sole remaining input to the vanishing induction. -/
+theorem vanishing_on_products_of_second_transfer
+    (secondTransfer : ∀ r m : ℕ, 0 < r → 0 < m → (r+1)*r ≤ r+m →
+      ∀ T : SymmetricMultilinearForm (r+1) (r+m) V,
+      IsSymmetricMultilinearForm (r+1) (r+m) V T →
+      (∀ x t : V, ∀ Q : Fin r → SymPow m V, lowerProductForm r m T x t Q = 0) →
+      ∀ v : Fin (r+1) → V,
+        T (fun i => symMonomial (r+m) V (fun _ => v i)) = 0) :
+    ∀ a b : ℕ, 1 ≤ a → 1 ≤ b → a*(a-1) ≤ b →
+      ∀ T : SymmetricMultilinearForm a b V,
+      IsSymmetricMultilinearForm a b V T →
+      (∀ y : Fin b → V, T (fun _ => symMonomial b V y) = 0) → T = 0 := by
+  apply vanishing_induction_from_transfer_steps
+  · intro r m hr _ _ T hT hdiag
+    exact lowerProductForm_diagonal_zero r m hr T hT hdiag
+  · exact secondTransfer
+  · intro a b T hT
+    exact symmetricMultilinearForm_eq_zero_of_pure_powers a b V T hT
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/VanishingInduction.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/VanishingInduction.lean
index 809632ac43d5b20a9f906f898417aadea286b4f7..cb076328c3bc665bd93fda862357b684ccdb79ed 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/VanishingInduction.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/VanishingInduction.lean
@@ -1,53 +1,53 @@
-import OAI.RepresentationTheory.FoulkesHowe.Multiplication
-
-namespace OAI
-
-noncomputable section
-open scoped BigOperators
-universe u
-namespace Problem346
-
-variable {V : Type u} [AddCommGroup V] [Module ℂ V]
-
-/-- Fixing the first argument and pulling back all other arguments preserves symmetry. -/
-theorem symmetric_form_curry_comp {r b m : ℕ}
-    (T : SymmetricMultilinearForm (r+1) b V)
-    (hT : IsSymmetricMultilinearForm (r+1) b V T)
-    (z : SymPow b V) (L : SymPow m V →ₗ[ℂ] SymPow b V) :
-    IsSymmetricMultilinearForm r m V ((T.curryLeft z).compLinearMap (fun _ => L)) := by
-  intro σ x
-  let τ : Equiv.Perm (Fin (r+1)) :=
-    (finSuccEquiv r).trans ((Equiv.optionCongr σ).trans (finSuccEquiv r).symm)
-  have heq : (fun i : Fin (r+1) => (Fin.cons z (fun j => L (x j)) :
-      Fin (r+1) → SymPow b V) (τ i)) =
-      (Fin.cons z (fun i => L (x (σ i))) : Fin (r+1) → SymPow b V) := by
-    funext i
-    cases i using Fin.cases <;> simp [τ]
-  simpa only [MultilinearMap.compLinearMap_apply, MultilinearMap.curryLeft_apply, heq]
-    using hT τ (Fin.cons z (fun i => L (x i)))
-
-/-- The lower-arity form used in the stabilization induction. -/
-def lowerProductForm (r m : ℕ)
-    (T : SymmetricMultilinearForm (r+1) (r+m) V) (x t : V) :
-    SymmetricMultilinearForm r m V :=
-  (T.curryLeft (symMonomial (r+m) V (fun _ => t))).compLinearMap
-    (fun _ => symPowMul r m (symMonomial r V (fun _ => x)))
-
-@[simp] theorem lowerProductForm_apply (r m : ℕ)
-    (T : SymmetricMultilinearForm (r+1) (r+m) V) (x t : V)
-    (Q : Fin r → SymPow m V) :
-    lowerProductForm r m T x t Q =
-      T (Fin.cons (symMonomial (r+m) V (fun _ => t))
-        (fun i => symPowMul r m (symMonomial r V (fun _ => x)) (Q i))) := rfl
-
-theorem lowerProductForm_symmetric (r m : ℕ)
-    (T : SymmetricMultilinearForm (r+1) (r+m) V)
-    (hT : IsSymmetricMultilinearForm (r+1) (r+m) V T) (x t : V) :
-    IsSymmetricMultilinearForm r m V (lowerProductForm r m T x t) :=
-  symmetric_form_curry_comp T hT _ _
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.Multiplication
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators
+universe u
+namespace Problem346
+
+variable {V : Type u} [AddCommGroup V] [Module ℂ V]
+
+/-- Fixing the first argument and pulling back all other arguments preserves symmetry. -/
+theorem symmetric_form_curry_comp {r b m : ℕ}
+    (T : SymmetricMultilinearForm (r+1) b V)
+    (hT : IsSymmetricMultilinearForm (r+1) b V T)
+    (z : SymPow b V) (L : SymPow m V →ₗ[ℂ] SymPow b V) :
+    IsSymmetricMultilinearForm r m V ((T.curryLeft z).compLinearMap (fun _ => L)) := by
+  intro σ x
+  let τ : Equiv.Perm (Fin (r+1)) :=
+    (finSuccEquiv r).trans ((Equiv.optionCongr σ).trans (finSuccEquiv r).symm)
+  have heq : (fun i : Fin (r+1) => (Fin.cons z (fun j => L (x j)) :
+      Fin (r+1) → SymPow b V) (τ i)) =
+      (Fin.cons z (fun i => L (x (σ i))) : Fin (r+1) → SymPow b V) := by
+    funext i
+    cases i using Fin.cases <;> simp [τ]
+  simpa only [MultilinearMap.compLinearMap_apply, MultilinearMap.curryLeft_apply, heq]
+    using hT τ (Fin.cons z (fun i => L (x i)))
+
+/-- The lower-arity form used in the stabilization induction. -/
+def lowerProductForm (r m : ℕ)
+    (T : SymmetricMultilinearForm (r+1) (r+m) V) (x t : V) :
+    SymmetricMultilinearForm r m V :=
+  (T.curryLeft (symMonomial (r+m) V (fun _ => t))).compLinearMap
+    (fun _ => symPowMul r m (symMonomial r V (fun _ => x)))
+
+@[simp] theorem lowerProductForm_apply (r m : ℕ)
+    (T : SymmetricMultilinearForm (r+1) (r+m) V) (x t : V)
+    (Q : Fin r → SymPow m V) :
+    lowerProductForm r m T x t Q =
+      T (Fin.cons (symMonomial (r+m) V (fun _ => t))
+        (fun i => symPowMul r m (symMonomial r V (fun _ => x)) (Q i))) := rfl
+
+theorem lowerProductForm_symmetric (r m : ℕ)
+    (T : SymmetricMultilinearForm (r+1) (r+m) V)
+    (hT : IsSymmetricMultilinearForm (r+1) (r+m) V T) (x t : V) :
+    IsSymmetricMultilinearForm r m V (lowerProductForm r m T x t) :=
+  symmetric_form_curry_comp T hT _ _
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesHowe/VanishingOne.lean b/lean/OAI/RepresentationTheory/FoulkesHowe/VanishingOne.lean
index 4e7b8e9599d779f1de2c539d3de03ffcc4a2efa5..2d6ffd54f83fd8dbbb4a88d5475be6ce79378882 100644
--- a/lean/OAI/RepresentationTheory/FoulkesHowe/VanishingOne.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesHowe/VanishingOne.lean
@@ -1,29 +1,29 @@
-import OAI.RepresentationTheory.FoulkesHowe.Model
-
-namespace OAI
-
-noncomputable section
-universe u
-namespace Problem346
-
-theorem vanishing_on_products_one (V : Type u) [AddCommGroup V] [Module ℂ V]
-    (b : ℕ) (T : SymmetricMultilinearForm 1 b V)
-    (h : ∀ u : Fin b → V, T (fun _ => symMonomial b V u) = 0) : T = 0 := by
-  have hspan : Submodule.span ℂ (Set.range (symMonomial b V)) = ⊤ := by
-    exact (Submodule.span_range_subtype_eq_top_iff (symPowSubmodule b V)
-      (fun v => Submodule.subset_span (Set.mem_range_self v))).mpr rfl
-  apply MultilinearMap.ext_of_span_eq_top (g := fun (_ : Fin 1) => symMonomial b V)
-    (fun _ => hspan)
-  intro v
-  have hv : (fun i : Fin 1 => symMonomial b V (v i)) =
-      (fun _ : Fin 1 => symMonomial b V (v 0)) := by
-    funext i
-    rw [Fin.eq_zero i]
-  rw [hv]
-  exact h (v 0)
-
-end Problem346
-
-end
-
-end OAI
+import OAI.RepresentationTheory.FoulkesHowe.Model
+
+namespace OAI
+
+noncomputable section
+universe u
+namespace Problem346
+
+theorem vanishing_on_products_one (V : Type u) [AddCommGroup V] [Module ℂ V]
+    (b : ℕ) (T : SymmetricMultilinearForm 1 b V)
+    (h : ∀ u : Fin b → V, T (fun _ => symMonomial b V u) = 0) : T = 0 := by
+  have hspan : Submodule.span ℂ (Set.range (symMonomial b V)) = ⊤ := by
+    exact (Submodule.span_range_subtype_eq_top_iff (symPowSubmodule b V)
+      (fun v => Submodule.subset_span (Set.mem_range_self v))).mpr rfl
+  apply MultilinearMap.ext_of_span_eq_top (g := fun (_ : Fin 1) => symMonomial b V)
+    (fun _ => hspan)
+  intro v
+  have hv : (fun i : Fin 1 => symMonomial b V (v i)) =
+      (fun _ : Fin 1 => symMonomial b V (v 0)) := by
+    funext i
+    rw [Fin.eq_zero i]
+  rw [hv]
+  exact h (v 0)
+
+end Problem346
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/AlternantExpansion.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/AlternantExpansion.lean
index e75465a9b57a54ece9cd239eec48aed99f9b7021..237c95b43a72fb35ba8acd893a276ef328357484 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/AlternantExpansion.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/AlternantExpansion.lean
@@ -1,149 +1,149 @@
-import Mathlib
-import OAI.RepresentationTheory.FoulkesSixth.LookupExtraction
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.Lookup
-open MvPolynomial Finset Foulkes.Strips
-
-lemma sorted_dominance {n : ℕ} (l w : Fin n → ℕ) (hl : Antitone l)
-    (hw : ∀ j, w j ≤ l j) (j : Fin n) : w (sortDesc w j) ≤ l j := by
-  classical
-  by_contra h
-  let e : Fin (j.val+1) → Fin n := fun c => ⟨c.val, by omega⟩
-  have hrow (c : Fin (j.val+1)) : (sortDesc w (e c)).val < j.val := by
-    by_contra hh
-    have hw' : w (sortDesc w j) ≤ w (sortDesc w (e c)) :=
-      sortDesc_antitone w (by change c.val ≤ j.val; omega)
-    have hl' : l (sortDesc w (e c)) ≤ l j := hl (by change j.val ≤ _; omega)
-    have hc := hw (sortDesc w (e c))
-    omega
-  let f : Fin (j.val+1) → Fin j.val := fun c => ⟨(sortDesc w (e c)).val, hrow c⟩
-  have hf : Function.Injective f := by
-    intro c d hcd
-    have hv := congrArg (fun z : Fin j.val => z.val) hcd
-    have hh : sortDesc w (e c) = sortDesc w (e d) := Fin.ext hv
-    exact Fin.ext (congrArg (fun z : Fin n => z.val) ((sortDesc w).injective hh))
-  have hn := Fintype.card_le_of_injective f hf
-  simp only [Fintype.card_fin] at hn
-  omega
-
-theorem previous_partition_le {n : ℕ} (mu : Fin n → ℕ) (hm : Antitone mu)
-    (i : ℕ) (d : Fin n →₀ ℕ)
-    (_hn : ∀ j, 0 ≤ (shifted mu j : ℤ) - (i : ℤ)*d j) :
-    ∀ j, sortedPartition (fun j => ((shifted mu j : ℤ) - (i : ℤ)*d j).toNat) j ≤ mu j := by
-  intro j
-  let w := fun j => ((shifted mu j : ℤ) - (i : ℤ)*d j).toNat
-  have hw : ∀ j, w j ≤ shifted mu j := by
-    intro j
-    dsimp [w]
-    exact Int.toNat_le.mpr (sub_le_self _ (by positivity))
-  have hh := sorted_dominance (shifted mu) w (shifted_antitone hm) hw j
-  change w (sortDesc w j) - staircase n j ≤ mu j
-  change w (sortDesc w j) ≤ mu j + staircase n j at hh
-  omega
-
-lemma shifted_strictAnti {n : ℕ} {mu : Fin n → ℕ} (hm : Antitone mu) :
-    StrictAnti (shifted mu) := by
-  intro j k hjk
-  have hh := hm (le_of_lt hjk)
-  have hjn := j.isLt
-  have hkn := k.isLt
-  change mu k + (n-1-k.val) < mu j + (n-1-j.val)
-  change j.val < k.val at hjk
-  omega
-
-lemma eq_of_strict_coefficients {n : ℕ} {p q : MvPolynomial (Fin n) ℤ}
-    (hp : Alternating p) (hq : Alternating q)
-    (h : ∀ v : Fin n → ℕ, StrictAnti v → p.coeff (expVector v) = q.coeff (expVector v)) : p = q := by
-  classical
-  ext d
-  let v : Fin n → ℕ := fun j => d j
-  have hd : expVector v = d := by ext j; rfl
-  rw [← hd]
-  by_cases hv : Function.Injective v
-  · rw [coeff_eq_sign_mul_comp hp v (sortDesc v), coeff_eq_sign_mul_comp hq v (sortDesc v),
-      h (v ∘ sortDesc v) (sortDesc_strict v hv)]
-  · obtain ⟨j,k,hjk,he⟩ : ∃ j k, j ≠ k ∧ v j = v k := by
-      simpa only [Function.Injective, not_forall, exists_prop, Classical.not_imp, ne_eq,
-        and_comm] using hv
-    rw [coeff_repeated hp v hjk he, coeff_repeated hq v hjk he]
-
-lemma coeff_alternant_strict {n : ℕ} (m v : Fin n → ℕ)
-    (hm : StrictAnti m) (hv : StrictAnti v) :
-    (alternant m).coeff (expVector v) = if m = v then 1 else 0 := by
-  classical
-  have hper (σ : Equiv.Perm (Fin n)) :
-      expVector (m ∘ σ.symm) = expVector v ↔ m = v ∧ σ = 1 := by
-    constructor
-    · intro he
-      have he' : m ∘ σ.symm = v := Finsupp.equivFunOnFinite.symm.injective he
-      have hs : m ∘ σ.symm = m := Tuple.unique_antitone
-        (he'.symm ▸ hv.antitone) (show Antitone (m ∘ (1 : Equiv.Perm (Fin n))) from hm.antitone)
-      have hid : σ.symm = 1 := by
-        apply Equiv.ext
-        intro j
-        exact hm.injective (congrFun hs j)
-      have hσ : σ = 1 := by
-        change σ⁻¹ = 1 at hid
-        exact inv_eq_one.mp hid
-      exact ⟨hs.symm.trans he',hσ⟩
-    · rintro ⟨rfl,rfl⟩; rfl
-  have ha : alternant m = ∑ σ : Equiv.Perm (Fin n),
-      Equiv.Perm.sign σ • ∏ c, (X (σ c) : MvPolynomial (Fin n) ℤ) ^ m c :=
-    Matrix.det_apply _
-  rw [ha, coeff_sum]
-  simp_rw [coeff_smul, perm_prod_monomial]
-  change (∑ σ : Equiv.Perm (Fin n), (Equiv.Perm.sign σ) •
-    (monomial (expVector (m ∘ σ.symm)) (1 : ℤ)).coeff (expVector v)) = _
-  simp_rw [coeff_monomial, hper]
-  by_cases he : m = v
-  · simp [he]
-  · simp [he]
-
-lemma alternating_sum {α : Type*} {n : ℕ} (s : Finset α)
-    (p : α → MvPolynomial (Fin n) ℤ) (hp : ∀ a ∈ s, Alternating (p a)) :
-    Alternating (∑ a ∈ s, p a) := by
-  intro σ
-  simp only [map_sum, Finset.smul_sum]
-  exact Finset.sum_congr rfl (fun a ha => hp a ha σ)
-
-lemma alternating_smul {n : ℕ} (z : ℤ) {p : MvPolynomial (Fin n) ℤ}
-    (hp : Alternating p) : Alternating (z • p) := by
-  intro σ
-  rw [map_smul, hp σ, smul_comm]
-
-theorem alternant_expansion {n : ℕ} (p : MvPolynomial (Fin n) ℤ) (hp : Alternating p) :
-    p = ∑ d ∈ p.support.filter (fun d => StrictAnti (fun j => d j)),
-      p.coeff d • alternant (fun j => d j) := by
-  classical
-  let s := p.support.filter (fun d => StrictAnti (fun j => d j))
-  apply eq_of_strict_coefficients hp
-    (alternating_sum s _ (fun _ _ => alternating_smul _ (alternant_alternating _)))
-  intro v hv
-  rw [coeff_sum]
-  have hd (d : Fin n →₀ ℕ) (hd : d ∈ s) :
-      (p.coeff d • alternant (fun j => d j)).coeff (expVector v) =
-        if d = expVector v then p.coeff d else 0 := by
-    rw [coeff_smul, coeff_alternant_strict _ _ (mem_filter.mp hd).2 hv]
-    have he : (fun j => d j) = v ↔ d = expVector v := by
-      constructor
-      · intro h; ext j; exact congrFun h j
-      · intro h; subst d; rfl
-    simp [he]
-  change p.coeff (expVector v) = ∑ d ∈ s, _
-  rw [Finset.sum_congr rfl hd]
-  by_cases hmem : expVector v ∈ s
-  · simp [hmem]
-  · have hc : p.coeff (expVector v) = 0 := by
-      have hh : expVector v ∉ p.support := fun hh => hmem (by exact mem_filter.mpr ⟨hh,hv⟩)
-      exact notMem_support_iff.mp hh
-    simp [hmem,hc]
-
-end Foulkes.Lookup
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.FoulkesSixth.LookupExtraction
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.Lookup
+open MvPolynomial Finset Foulkes.Strips
+
+lemma sorted_dominance {n : ℕ} (l w : Fin n → ℕ) (hl : Antitone l)
+    (hw : ∀ j, w j ≤ l j) (j : Fin n) : w (sortDesc w j) ≤ l j := by
+  classical
+  by_contra h
+  let e : Fin (j.val+1) → Fin n := fun c => ⟨c.val, by omega⟩
+  have hrow (c : Fin (j.val+1)) : (sortDesc w (e c)).val < j.val := by
+    by_contra hh
+    have hw' : w (sortDesc w j) ≤ w (sortDesc w (e c)) :=
+      sortDesc_antitone w (by change c.val ≤ j.val; omega)
+    have hl' : l (sortDesc w (e c)) ≤ l j := hl (by change j.val ≤ _; omega)
+    have hc := hw (sortDesc w (e c))
+    omega
+  let f : Fin (j.val+1) → Fin j.val := fun c => ⟨(sortDesc w (e c)).val, hrow c⟩
+  have hf : Function.Injective f := by
+    intro c d hcd
+    have hv := congrArg (fun z : Fin j.val => z.val) hcd
+    have hh : sortDesc w (e c) = sortDesc w (e d) := Fin.ext hv
+    exact Fin.ext (congrArg (fun z : Fin n => z.val) ((sortDesc w).injective hh))
+  have hn := Fintype.card_le_of_injective f hf
+  simp only [Fintype.card_fin] at hn
+  omega
+
+theorem previous_partition_le {n : ℕ} (mu : Fin n → ℕ) (hm : Antitone mu)
+    (i : ℕ) (d : Fin n →₀ ℕ)
+    (_hn : ∀ j, 0 ≤ (shifted mu j : ℤ) - (i : ℤ)*d j) :
+    ∀ j, sortedPartition (fun j => ((shifted mu j : ℤ) - (i : ℤ)*d j).toNat) j ≤ mu j := by
+  intro j
+  let w := fun j => ((shifted mu j : ℤ) - (i : ℤ)*d j).toNat
+  have hw : ∀ j, w j ≤ shifted mu j := by
+    intro j
+    dsimp [w]
+    exact Int.toNat_le.mpr (sub_le_self _ (by positivity))
+  have hh := sorted_dominance (shifted mu) w (shifted_antitone hm) hw j
+  change w (sortDesc w j) - staircase n j ≤ mu j
+  change w (sortDesc w j) ≤ mu j + staircase n j at hh
+  omega
+
+lemma shifted_strictAnti {n : ℕ} {mu : Fin n → ℕ} (hm : Antitone mu) :
+    StrictAnti (shifted mu) := by
+  intro j k hjk
+  have hh := hm (le_of_lt hjk)
+  have hjn := j.isLt
+  have hkn := k.isLt
+  change mu k + (n-1-k.val) < mu j + (n-1-j.val)
+  change j.val < k.val at hjk
+  omega
+
+lemma eq_of_strict_coefficients {n : ℕ} {p q : MvPolynomial (Fin n) ℤ}
+    (hp : Alternating p) (hq : Alternating q)
+    (h : ∀ v : Fin n → ℕ, StrictAnti v → p.coeff (expVector v) = q.coeff (expVector v)) : p = q := by
+  classical
+  ext d
+  let v : Fin n → ℕ := fun j => d j
+  have hd : expVector v = d := by ext j; rfl
+  rw [← hd]
+  by_cases hv : Function.Injective v
+  · rw [coeff_eq_sign_mul_comp hp v (sortDesc v), coeff_eq_sign_mul_comp hq v (sortDesc v),
+      h (v ∘ sortDesc v) (sortDesc_strict v hv)]
+  · obtain ⟨j,k,hjk,he⟩ : ∃ j k, j ≠ k ∧ v j = v k := by
+      simpa only [Function.Injective, not_forall, exists_prop, Classical.not_imp, ne_eq,
+        and_comm] using hv
+    rw [coeff_repeated hp v hjk he, coeff_repeated hq v hjk he]
+
+lemma coeff_alternant_strict {n : ℕ} (m v : Fin n → ℕ)
+    (hm : StrictAnti m) (hv : StrictAnti v) :
+    (alternant m).coeff (expVector v) = if m = v then 1 else 0 := by
+  classical
+  have hper (σ : Equiv.Perm (Fin n)) :
+      expVector (m ∘ σ.symm) = expVector v ↔ m = v ∧ σ = 1 := by
+    constructor
+    · intro he
+      have he' : m ∘ σ.symm = v := Finsupp.equivFunOnFinite.symm.injective he
+      have hs : m ∘ σ.symm = m := Tuple.unique_antitone
+        (he'.symm ▸ hv.antitone) (show Antitone (m ∘ (1 : Equiv.Perm (Fin n))) from hm.antitone)
+      have hid : σ.symm = 1 := by
+        apply Equiv.ext
+        intro j
+        exact hm.injective (congrFun hs j)
+      have hσ : σ = 1 := by
+        change σ⁻¹ = 1 at hid
+        exact inv_eq_one.mp hid
+      exact ⟨hs.symm.trans he',hσ⟩
+    · rintro ⟨rfl,rfl⟩; rfl
+  have ha : alternant m = ∑ σ : Equiv.Perm (Fin n),
+      Equiv.Perm.sign σ • ∏ c, (X (σ c) : MvPolynomial (Fin n) ℤ) ^ m c :=
+    Matrix.det_apply _
+  rw [ha, coeff_sum]
+  simp_rw [coeff_smul, perm_prod_monomial]
+  change (∑ σ : Equiv.Perm (Fin n), (Equiv.Perm.sign σ) •
+    (monomial (expVector (m ∘ σ.symm)) (1 : ℤ)).coeff (expVector v)) = _
+  simp_rw [coeff_monomial, hper]
+  by_cases he : m = v
+  · simp [he]
+  · simp [he]
+
+lemma alternating_sum {α : Type*} {n : ℕ} (s : Finset α)
+    (p : α → MvPolynomial (Fin n) ℤ) (hp : ∀ a ∈ s, Alternating (p a)) :
+    Alternating (∑ a ∈ s, p a) := by
+  intro σ
+  simp only [map_sum, Finset.smul_sum]
+  exact Finset.sum_congr rfl (fun a ha => hp a ha σ)
+
+lemma alternating_smul {n : ℕ} (z : ℤ) {p : MvPolynomial (Fin n) ℤ}
+    (hp : Alternating p) : Alternating (z • p) := by
+  intro σ
+  rw [map_smul, hp σ, smul_comm]
+
+theorem alternant_expansion {n : ℕ} (p : MvPolynomial (Fin n) ℤ) (hp : Alternating p) :
+    p = ∑ d ∈ p.support.filter (fun d => StrictAnti (fun j => d j)),
+      p.coeff d • alternant (fun j => d j) := by
+  classical
+  let s := p.support.filter (fun d => StrictAnti (fun j => d j))
+  apply eq_of_strict_coefficients hp
+    (alternating_sum s _ (fun _ _ => alternating_smul _ (alternant_alternating _)))
+  intro v hv
+  rw [coeff_sum]
+  have hd (d : Fin n →₀ ℕ) (hd : d ∈ s) :
+      (p.coeff d • alternant (fun j => d j)).coeff (expVector v) =
+        if d = expVector v then p.coeff d else 0 := by
+    rw [coeff_smul, coeff_alternant_strict _ _ (mem_filter.mp hd).2 hv]
+    have he : (fun j => d j) = v ↔ d = expVector v := by
+      constructor
+      · intro h; ext j; exact congrFun h j
+      · intro h; subst d; rfl
+    simp [he]
+  change p.coeff (expVector v) = ∑ d ∈ s, _
+  rw [Finset.sum_congr rfl hd]
+  by_cases hmem : expVector v ∈ s
+  · simp [hmem]
+  · have hc : p.coeff (expVector v) = 0 := by
+      have hh : expVector v ∉ p.support := fun hh => hmem (by exact mem_filter.mpr ⟨hh,hv⟩)
+      exact notMem_support_iff.mp hh
+    simp [hmem,hc]
+
+end Foulkes.Lookup
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/BlockInjection.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/BlockInjection.lean
index 8eb1f4898843bc9b73b7650cfe7d4fefa5e39146..5b09770c876e0939bd7f38bb1508f2d64a1314a9 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/BlockInjection.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/BlockInjection.lean
@@ -1,78 +1,78 @@
-import Mathlib
-import OAI.RepresentationTheory.FoulkesSixth.QuadraticInjection
-import OAI.RepresentationTheory.FoulkesSixth.BlockVanishing
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.Canonical
-universe u
-open Module MvPolynomial SymmetricTensor Polarization
-variable {V : Type u} [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
-variable {n : ℕ} (B : Basis (Fin n) ℂ V) (a b : ℕ)
-theorem productPolynomial_range_block (ha : 1 ≤ a) (hb : (a-1)^2 ≤ b) :
-    (productPolynomial B a b).range = R a n b := by
-  classical
-  apply le_antisymm
-  · rintro _ ⟨x, rfl⟩
-    exact productPolynomial_mem B a b x
-  · intro p hp
-    by_contra hnot
-    obtain ⟨l, hlp, hlrange⟩ := Submodule.exists_dual_map_eq_bot_of_notMem hnot inferInstance
-    have hkill (f : P a n) (hf : f ∈ (productPolynomial B a b).range) : l f = 0 := by
-      have hh : l f ∈ ((productPolynomial B a b).range).map l := Submodule.mem_map.mpr ⟨f,hf,rfl⟩
-      rwa [hlrange, Submodule.mem_bot] at hh
-    let lav := l.comp (Polarization.average a n)
-    have hs : Vanishing.Symmetric lav := by
-      intro σ f
-      change l (Polarization.average a n (renameRow σ f)) = l (Polarization.average a n f)
-      rw [Vanishing.average_renameRow]
-    have hz : Vanishing.ProductsVanish lav b := by
-      intro y
-      change l (Polarization.average a n (∏ j, z (y j))) = 0
-      rw [Polarization.average_eq (show (∏ j, z (a := a) (y j)) ∈ rowInvariants a n from by
-        intro σ; simp)]
-      exact hkill _ (products_in_range B a b y)
-    have hh := Vanishing.block_vanishing a ha hb hs hz p hp.1
-    change l (Polarization.average a n p) = 0 at hh
-    rw [Polarization.average_eq hp.2] at hh
-    exact hlp hh
-
-theorem surjective_block (ha : 1 ≤ a) (hb : (a-1)^2 ≤ b) :
-    Function.Surjective (map B a b) := by
-  intro x
-  have hp : Rows.invariantMap B a b x ∈ (productPolynomial B a b).range := by
-    rw [productPolynomial_range_block B a b ha hb]
-    exact (Rows.equiv B a b x).property
-  obtain ⟨y, hy⟩ := hp
-  refine ⟨y, (Rows.invariantMap_injective B a b) ?_⟩
-  rw [map_polynomial, hy]
-
-end Foulkes.Canonical
-
-namespace Foulkes.Injection
-universe u
-open Module
-variable (a b : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
-
-lemma injective_block (ha : 1 ≤ a) (hb : (a-1)^2 ≤ b) :
-    Function.Injective (map a b V) :=
-  (PlethDual.flipEquiv b a V).symm.injective.comp
-    ((LinearMap.dualMap_injective_of_surjective
-      (Canonical.surjective_block (Module.finBasis ℂ (Dual ℂ V)) a b ha hb)).comp
-      (PlethDual.flipEquiv a b V).injective)
-
-theorem sixth_from_25 (b : ℕ) (hb : 25 ≤ b)
-    (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] :
-    ∃ i : Sym 6 (Sym b V) →ₗ[ℂ] Sym b (Sym 6 V), Function.Injective i ∧
-      ∀ (g : V ≃ₗ[ℂ] V) (x : Sym 6 (Sym b V)),
-        i (symMap 6 (symMap b g.toLinearMap) x) = symMap b (symMap 6 g.toLinearMap) (i x) :=
-  ⟨map 6 b V, injective_block 6 b V (by omega) (by simpa using hb),
-    fun g x => natural 6 b V g.toLinearMap x⟩
-
-end Foulkes.Injection
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.FoulkesSixth.QuadraticInjection
+import OAI.RepresentationTheory.FoulkesSixth.BlockVanishing
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.Canonical
+universe u
+open Module MvPolynomial SymmetricTensor Polarization
+variable {V : Type u} [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
+variable {n : ℕ} (B : Basis (Fin n) ℂ V) (a b : ℕ)
+theorem productPolynomial_range_block (ha : 1 ≤ a) (hb : (a-1)^2 ≤ b) :
+    (productPolynomial B a b).range = R a n b := by
+  classical
+  apply le_antisymm
+  · rintro _ ⟨x, rfl⟩
+    exact productPolynomial_mem B a b x
+  · intro p hp
+    by_contra hnot
+    obtain ⟨l, hlp, hlrange⟩ := Submodule.exists_dual_map_eq_bot_of_notMem hnot inferInstance
+    have hkill (f : P a n) (hf : f ∈ (productPolynomial B a b).range) : l f = 0 := by
+      have hh : l f ∈ ((productPolynomial B a b).range).map l := Submodule.mem_map.mpr ⟨f,hf,rfl⟩
+      rwa [hlrange, Submodule.mem_bot] at hh
+    let lav := l.comp (Polarization.average a n)
+    have hs : Vanishing.Symmetric lav := by
+      intro σ f
+      change l (Polarization.average a n (renameRow σ f)) = l (Polarization.average a n f)
+      rw [Vanishing.average_renameRow]
+    have hz : Vanishing.ProductsVanish lav b := by
+      intro y
+      change l (Polarization.average a n (∏ j, z (y j))) = 0
+      rw [Polarization.average_eq (show (∏ j, z (a := a) (y j)) ∈ rowInvariants a n from by
+        intro σ; simp)]
+      exact hkill _ (products_in_range B a b y)
+    have hh := Vanishing.block_vanishing a ha hb hs hz p hp.1
+    change l (Polarization.average a n p) = 0 at hh
+    rw [Polarization.average_eq hp.2] at hh
+    exact hlp hh
+
+theorem surjective_block (ha : 1 ≤ a) (hb : (a-1)^2 ≤ b) :
+    Function.Surjective (map B a b) := by
+  intro x
+  have hp : Rows.invariantMap B a b x ∈ (productPolynomial B a b).range := by
+    rw [productPolynomial_range_block B a b ha hb]
+    exact (Rows.equiv B a b x).property
+  obtain ⟨y, hy⟩ := hp
+  refine ⟨y, (Rows.invariantMap_injective B a b) ?_⟩
+  rw [map_polynomial, hy]
+
+end Foulkes.Canonical
+
+namespace Foulkes.Injection
+universe u
+open Module
+variable (a b : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
+
+lemma injective_block (ha : 1 ≤ a) (hb : (a-1)^2 ≤ b) :
+    Function.Injective (map a b V) :=
+  (PlethDual.flipEquiv b a V).symm.injective.comp
+    ((LinearMap.dualMap_injective_of_surjective
+      (Canonical.surjective_block (Module.finBasis ℂ (Dual ℂ V)) a b ha hb)).comp
+      (PlethDual.flipEquiv a b V).injective)
+
+theorem sixth_from_25 (b : ℕ) (hb : 25 ≤ b)
+    (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] :
+    ∃ i : Sym 6 (Sym b V) →ₗ[ℂ] Sym b (Sym 6 V), Function.Injective i ∧
+      ∀ (g : V ≃ₗ[ℂ] V) (x : Sym 6 (Sym b V)),
+        i (symMap 6 (symMap b g.toLinearMap) x) = symMap b (symMap 6 g.toLinearMap) (i x) :=
+  ⟨map 6 b V, injective_block 6 b V (by omega) (by simpa using hb),
+    fun g x => natural 6 b V g.toLinearMap x⟩
+
+end Foulkes.Injection
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/BlockVanishing.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/BlockVanishing.lean
index 73be636e329c9f5cf44bed5467bec898c6a415f6..0c72e1c70fc8050910227618989c2657d682db24 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/BlockVanishing.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/BlockVanishing.lean
@@ -1,64 +1,64 @@
-import Mathlib
-import OAI.RepresentationTheory.FoulkesSixth.Vanishing
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.Vanishing
-open MvPolynomial Foulkes.Polarization Foulkes.PolynomialSpan
-
-lemma vanishing_zero {a n : ℕ} {l : P a n →ₗ[ℂ] ℂ} (hz : ProductsVanish l 0)
-    {f : P a n} (hf : f ∈ W a n 0) : l f = 0 := by
-  have h1 : l 1 = 0 := by simpa using hz (fun i => Fin.elim0 i)
-  have hW : W a n 0 ≤ LinearMap.ker l := by
-    rw [W_eq_span_pureRows]
-    apply Submodule.span_le.mpr
-    rintro _ ⟨v, rfl⟩
-    simpa [pureRows] using h1
-  exact hW hf
-
-theorem block_vanishing (a : ℕ) : ∀ {n b : ℕ}, 1 ≤ a → (a-1)^2 ≤ b →
-    ∀ {l : P a n →ₗ[ℂ] ℂ}, Symmetric l → ProductsVanish l b →
-      ∀ f : P a n, f ∈ W a n b → l f = 0 := by
-  induction a with
-  | zero => intro n b ha; omega
-  | succ r ih =>
-    intro n b ha hb l hs hz f hf
-    have hpure : ∀ v : Fin (r+1) → Fin n → ℂ, l (pureRows b v) = 0 := by
-      intro v
-      by_cases hr : r = 0
-      · subst r
-        have hh := hz (fun _ => v 0)
-        simpa [pureRows, z, Fin.prod_univ_one] using hh
-      · have hrpos : 0 < r := Nat.pos_of_ne_zero hr
-        have hrb : r ≤ b := by
-          simp only [Nat.add_sub_cancel] at hb
-          nlinarith
-        obtain ⟨m, rfl⟩ := Nat.exists_eq_add_of_le hrb
-        have hb' : r*r ≤ r+m := by simpa [pow_two] using hb
-        have hrestrict (x t : Fin n → ℂ) (g : P r n) (hg : g ∈ W r n m) :
-            restrictionFunctional l x t (r+m) g = 0 := by
-          by_cases hm : m = 0
-          · subst m
-            exact vanishing_zero (restriction_products hrpos hs hz x t) hg
-          · apply ih (by omega) ?_ (restriction_symmetric hs x t)
-              (restriction_products hrpos hs hz x t) g hg
-            have he : (r-1)+1 = r := by omega
-            nlinarith
-        have hh := remove_common_power hb'
-          (v (Fin.last r)) (fun x g hg => hrestrict x (v (Fin.last r)) g hg)
-          (fun i => v i.castSucc)
-        simpa only [pureRows, Fin.prod_univ_castSucc] using hh
-    have hW : W (r+1) n b ≤ LinearMap.ker l := by
-      rw [W_eq_span_pureRows]
-      apply Submodule.span_le.mpr
-      rintro _ ⟨v,rfl⟩
-      exact hpure v
-    exact hW hf
-
-end Foulkes.Vanishing
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.FoulkesSixth.Vanishing
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.Vanishing
+open MvPolynomial Foulkes.Polarization Foulkes.PolynomialSpan
+
+lemma vanishing_zero {a n : ℕ} {l : P a n →ₗ[ℂ] ℂ} (hz : ProductsVanish l 0)
+    {f : P a n} (hf : f ∈ W a n 0) : l f = 0 := by
+  have h1 : l 1 = 0 := by simpa using hz (fun i => Fin.elim0 i)
+  have hW : W a n 0 ≤ LinearMap.ker l := by
+    rw [W_eq_span_pureRows]
+    apply Submodule.span_le.mpr
+    rintro _ ⟨v, rfl⟩
+    simpa [pureRows] using h1
+  exact hW hf
+
+theorem block_vanishing (a : ℕ) : ∀ {n b : ℕ}, 1 ≤ a → (a-1)^2 ≤ b →
+    ∀ {l : P a n →ₗ[ℂ] ℂ}, Symmetric l → ProductsVanish l b →
+      ∀ f : P a n, f ∈ W a n b → l f = 0 := by
+  induction a with
+  | zero => intro n b ha; omega
+  | succ r ih =>
+    intro n b ha hb l hs hz f hf
+    have hpure : ∀ v : Fin (r+1) → Fin n → ℂ, l (pureRows b v) = 0 := by
+      intro v
+      by_cases hr : r = 0
+      · subst r
+        have hh := hz (fun _ => v 0)
+        simpa [pureRows, z, Fin.prod_univ_one] using hh
+      · have hrpos : 0 < r := Nat.pos_of_ne_zero hr
+        have hrb : r ≤ b := by
+          simp only [Nat.add_sub_cancel] at hb
+          nlinarith
+        obtain ⟨m, rfl⟩ := Nat.exists_eq_add_of_le hrb
+        have hb' : r*r ≤ r+m := by simpa [pow_two] using hb
+        have hrestrict (x t : Fin n → ℂ) (g : P r n) (hg : g ∈ W r n m) :
+            restrictionFunctional l x t (r+m) g = 0 := by
+          by_cases hm : m = 0
+          · subst m
+            exact vanishing_zero (restriction_products hrpos hs hz x t) hg
+          · apply ih (by omega) ?_ (restriction_symmetric hs x t)
+              (restriction_products hrpos hs hz x t) g hg
+            have he : (r-1)+1 = r := by omega
+            nlinarith
+        have hh := remove_common_power hb'
+          (v (Fin.last r)) (fun x g hg => hrestrict x (v (Fin.last r)) g hg)
+          (fun i => v i.castSucc)
+        simpa only [pureRows, Fin.prod_univ_castSucc] using hh
+    have hW : W (r+1) n b ≤ LinearMap.ker l := by
+      rw [W_eq_span_pureRows]
+      apply Submodule.span_le.mpr
+      rintro _ ⟨v,rfl⟩
+      exact hpure v
+    exact hW hf
+
+end Foulkes.Vanishing
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/Collision.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/Collision.lean
index 50dd39d09f78570cb335816bd395f2f84a852605..ee6ea4ec767f4172419012a6afd0198e93fcaa62 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/Collision.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/Collision.lean
@@ -1,105 +1,105 @@
-import Mathlib
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.Collision
-open Finset
-open scoped Classical
-variable {A B α : Type*} [CommRing A] [CommRing B] [Algebra ℂ A] [Algebra ℂ B]
-  [Fintype α]
-
-def distinctSum (φ : α → A →ₐ[ℂ] B) {h : ℕ} (q : Fin h → A) : B :=
-  ∑ e : Fin h ↪ α, ∏ j : Fin h, φ (e j) (q j)
-
-omit [Fintype α] in
-lemma prod_update (φ : α → A →ₐ[ℂ] B) {h : ℕ} (q : Fin h → A)
-    (q₀ : A) (e : Fin h ↪ α) (r : Fin h) :
-    (∏ j : Fin h, φ (e j) (Function.update q r (q₀ * q r) j)) =
-      φ (e r) q₀ * ∏ j : Fin h, φ (e j) (q j) := by
-  classical
-  have heq (j : Fin h) : φ (e j) (Function.update q r (q₀ * q r) j) =
-      (if j = r then φ (e j) q₀ else 1) * φ (e j) (q j) := by
-    by_cases hj : j = r
-    · subst j; simp
-    · simp [hj]
-  simp_rw [heq]
-  rw [Finset.prod_mul_distrib]
-  simp
-
-lemma distinctSum_cons (φ : α → A →ₐ[ℂ] B) {h : ℕ} (q : Fin h → A) (q₀ : A) :
-    distinctSum φ (Fin.cons q₀ q) =
-      ∑ e : Fin h ↪ α, ∑ i : {i // i ∉ Set.range e},
-        φ i q₀ * ∏ j : Fin h, φ (e j) (q j) := by
-  classical
-  rw [distinctSum, ← Equiv.sum_comp (Equiv.embeddingFinSucc h α).symm,
-    Fintype.sum_sigma]
-  apply Finset.sum_congr rfl
-  intro e _
-  apply Finset.sum_congr rfl
-  intro i _
-  simp [Equiv.coe_embeddingFinSucc_symm, Fin.prod_univ_succ]
-
-lemma recurrence (φ : α → A →ₐ[ℂ] B) {h : ℕ} (q : Fin h → A) (q₀ : A) :
-    distinctSum φ (Fin.cons q₀ q) =
-      (∑ i : α, φ i q₀) * distinctSum φ q -
-        ∑ r : Fin h, distinctSum φ (Function.update q r (q₀ * q r)) := by
-  classical
-  have hsplit (e : Fin h ↪ α) :
-      (∑ i : α, φ i q₀) =
-        (∑ i : {i // i ∉ Set.range e}, φ i q₀) + ∑ r : Fin h, φ (e r) q₀ := by
-    have hh := Fintype.sum_subtype_add_sum_subtype (fun i : α => i ∉ Set.range e)
-      (fun i => φ i q₀)
-    rw [← hh]
-    congr 1
-    calc
-      _ = ∑ i : Set.range e, φ i q₀ :=
-        Equiv.sum_comp (Equiv.subtypeEquivRight (fun _ => not_not))
-          (fun i : Set.range e => φ i q₀)
-      _ = _ := (Equiv.sum_comp (Equiv.ofInjective e e.injective)
-        (fun i : Set.range e => φ i q₀)).symm
-  rw [eq_sub_iff_add_eq, distinctSum_cons]
-  simp only [distinctSum, prod_update]
-  rw [Finset.sum_comm (f := fun (r : Fin h) (e : Fin h ↪ α) =>
-    φ (e r) q₀ * ∏ j : Fin h, φ (e j) (q j)), ← Finset.sum_add_distrib,
-    Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro e _
-  rw [hsplit e, add_mul, Finset.sum_mul, Finset.sum_mul]
-
-end Foulkes.Collision
-
-
-noncomputable section
-
-open scoped Classical
-
-example : ((Tuple.sort (α := ℕᵒᵈ) ![1,7,4]) 0).val = 1 := by
-  rw [← show Equiv.swap (0 : Fin 3) 2 * Equiv.swap 0 1 =
-      Tuple.sort (α := ℕᵒᵈ) ![1,7,4] from Tuple.eq_sort_iff.mpr (by decide +kernel)]
-  rfl
-example : (Equiv.Perm.sign (Tuple.sort (α := ℕᵒᵈ) ![1,7,4]) : ℤ) = 1 := by
-  rw [← show Equiv.swap (0 : Fin 3) 2 * Equiv.swap 0 1 =
-      Tuple.sort (α := ℕᵒᵈ) ![1,7,4] from Tuple.eq_sort_iff.mpr (by decide +kernel)]
-  norm_num [map_mul, Equiv.Perm.sign_swap]
-
-
-noncomputable section
-
-example (i b : ℕ) : i ∈ List.range' 1 b ↔ 1 ≤ i ∧ i ≤ b := by
-  simp only [List.mem_range', Nat.one_mul]
-  constructor
-  · rintro ⟨index, hindex, rfl⟩
-    omega
-  · rintro ⟨hlower, hupper⟩
-    exact ⟨i - 1, by omega, by omega⟩
-example {B : Type*} [Fintype B] (nu : B → ℕ) (b : B) :
-    (Finsupp.equivFunOnFinite.symm nu) b = nu b := by
-  rfl
-
-end
-end
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.Collision
+open Finset
+open scoped Classical
+variable {A B α : Type*} [CommRing A] [CommRing B] [Algebra ℂ A] [Algebra ℂ B]
+  [Fintype α]
+
+def distinctSum (φ : α → A →ₐ[ℂ] B) {h : ℕ} (q : Fin h → A) : B :=
+  ∑ e : Fin h ↪ α, ∏ j : Fin h, φ (e j) (q j)
+
+omit [Fintype α] in
+lemma prod_update (φ : α → A →ₐ[ℂ] B) {h : ℕ} (q : Fin h → A)
+    (q₀ : A) (e : Fin h ↪ α) (r : Fin h) :
+    (∏ j : Fin h, φ (e j) (Function.update q r (q₀ * q r) j)) =
+      φ (e r) q₀ * ∏ j : Fin h, φ (e j) (q j) := by
+  classical
+  have heq (j : Fin h) : φ (e j) (Function.update q r (q₀ * q r) j) =
+      (if j = r then φ (e j) q₀ else 1) * φ (e j) (q j) := by
+    by_cases hj : j = r
+    · subst j; simp
+    · simp [hj]
+  simp_rw [heq]
+  rw [Finset.prod_mul_distrib]
+  simp
+
+lemma distinctSum_cons (φ : α → A →ₐ[ℂ] B) {h : ℕ} (q : Fin h → A) (q₀ : A) :
+    distinctSum φ (Fin.cons q₀ q) =
+      ∑ e : Fin h ↪ α, ∑ i : {i // i ∉ Set.range e},
+        φ i q₀ * ∏ j : Fin h, φ (e j) (q j) := by
+  classical
+  rw [distinctSum, ← Equiv.sum_comp (Equiv.embeddingFinSucc h α).symm,
+    Fintype.sum_sigma]
+  apply Finset.sum_congr rfl
+  intro e _
+  apply Finset.sum_congr rfl
+  intro i _
+  simp [Equiv.coe_embeddingFinSucc_symm, Fin.prod_univ_succ]
+
+lemma recurrence (φ : α → A →ₐ[ℂ] B) {h : ℕ} (q : Fin h → A) (q₀ : A) :
+    distinctSum φ (Fin.cons q₀ q) =
+      (∑ i : α, φ i q₀) * distinctSum φ q -
+        ∑ r : Fin h, distinctSum φ (Function.update q r (q₀ * q r)) := by
+  classical
+  have hsplit (e : Fin h ↪ α) :
+      (∑ i : α, φ i q₀) =
+        (∑ i : {i // i ∉ Set.range e}, φ i q₀) + ∑ r : Fin h, φ (e r) q₀ := by
+    have hh := Fintype.sum_subtype_add_sum_subtype (fun i : α => i ∉ Set.range e)
+      (fun i => φ i q₀)
+    rw [← hh]
+    congr 1
+    calc
+      _ = ∑ i : Set.range e, φ i q₀ :=
+        Equiv.sum_comp (Equiv.subtypeEquivRight (fun _ => not_not))
+          (fun i : Set.range e => φ i q₀)
+      _ = _ := (Equiv.sum_comp (Equiv.ofInjective e e.injective)
+        (fun i : Set.range e => φ i q₀)).symm
+  rw [eq_sub_iff_add_eq, distinctSum_cons]
+  simp only [distinctSum, prod_update]
+  rw [Finset.sum_comm (f := fun (r : Fin h) (e : Fin h ↪ α) =>
+    φ (e r) q₀ * ∏ j : Fin h, φ (e j) (q j)), ← Finset.sum_add_distrib,
+    Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro e _
+  rw [hsplit e, add_mul, Finset.sum_mul, Finset.sum_mul]
+
+end Foulkes.Collision
+
+
+noncomputable section
+
+open scoped Classical
+
+example : ((Tuple.sort (α := ℕᵒᵈ) ![1,7,4]) 0).val = 1 := by
+  rw [← show Equiv.swap (0 : Fin 3) 2 * Equiv.swap 0 1 =
+      Tuple.sort (α := ℕᵒᵈ) ![1,7,4] from Tuple.eq_sort_iff.mpr (by decide +kernel)]
+  rfl
+example : (Equiv.Perm.sign (Tuple.sort (α := ℕᵒᵈ) ![1,7,4]) : ℤ) = 1 := by
+  rw [← show Equiv.swap (0 : Fin 3) 2 * Equiv.swap 0 1 =
+      Tuple.sort (α := ℕᵒᵈ) ![1,7,4] from Tuple.eq_sort_iff.mpr (by decide +kernel)]
+  norm_num [map_mul, Equiv.Perm.sign_swap]
+
+
+noncomputable section
+
+example (i b : ℕ) : i ∈ List.range' 1 b ↔ 1 ≤ i ∧ i ≤ b := by
+  simp only [List.mem_range', Nat.one_mul]
+  constructor
+  · rintro ⟨index, hindex, rfl⟩
+    omega
+  · rintro ⟨hlower, hupper⟩
+    exact ⟨i - 1, by omega, by omega⟩
+example {B : Type*} [Fintype B] (nu : B → ℕ) (b : B) :
+    (Finsupp.equivFunOnFinite.symm nu) b = nu b := by
+  rfl
+
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/ColorOrbits.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/ColorOrbits.lean
index cb5845d6ee5afc586e7699cff557c64624264eaa..03de4afabb3bd9128adcdb51084e1b30932cf64a 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/ColorOrbits.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/ColorOrbits.lean
@@ -1,193 +1,193 @@
-import Mathlib
-import OAI.RepresentationTheory.FoulkesSixth.YoungModules
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.Young
-open Finset
-
-section Counts
-variable {D A B : Type*} [Fintype D] [Fintype A] [Fintype B]
-
-def counts (f : D → A) (a : A) : ℕ := Nat.card {i // f i = a}
-
-omit [Fintype D] [Fintype A] in
-lemma counts_reindex (f : D → A) (σ : Equiv.Perm D) : counts (f ∘ σ) = counts f := by
-  funext a
-  exact Nat.card_congr (Equiv.subtypeEquiv σ (fun i => Iff.rfl))
-
-omit [Fintype A] in
-lemma exists_perm_of_counts_eq (f g : D → A) (h : counts f = counts g) :
-    ∃ σ : Equiv.Perm D, f ∘ σ = g := by
-  classical
-  let e (a : A) : {i // g i = a} ≃ {i // f i = a} :=
-    Fintype.equivOfCardEq (by simpa [counts, Nat.card_eq_fintype_card] using (congrFun h a).symm)
-  let σ : Equiv.Perm D := (Equiv.sigmaFiberEquiv g).symm.trans
-    ((Equiv.sigmaCongrRight e).trans (Equiv.sigmaFiberEquiv f))
-  refine ⟨σ, ?_⟩
-  funext i
-  exact (e (g i) ⟨i,rfl⟩).property
-
-lemma counts_total (f : D → A) : ∑ a, counts f a = Fintype.card D := by
-  classical
-  simp only [counts, Nat.card_eq_fintype_card]
-  rw [← Fintype.card_sigma]
-  exact Fintype.card_congr (Equiv.sigmaFiberEquiv f)
-
-omit [Fintype A] in
-lemma counts_pair_fst (f : D → A) (g : D → B) (a : A) :
-    ∑ b, counts (fun i => (f i,g i)) (a,b) = counts f a := by
-  classical
-  let he (b : B) : {z : {i // f i = a} // g z.val = b} ≃
-      {i // (f i,g i) = (a,b)} := {
-    toFun z := ⟨z.val.val, Prod.ext z.val.property z.property⟩
-    invFun z := ⟨⟨z.val, congrArg Prod.fst z.property⟩, congrArg Prod.snd z.property⟩
-    left_inv z := rfl
-    right_inv z := rfl }
-  have hh := counts_total (fun z : {i // f i = a} => g z.val)
-  simp only [counts, Nat.card_eq_fintype_card] at hh ⊢
-  rw [← hh]
-  exact Finset.sum_congr rfl (fun b hb => (Fintype.card_congr (he b)).symm)
-
-omit [Fintype B] in
-lemma counts_pair_snd (f : D → A) (g : D → B) (b : B) :
-    ∑ a, counts (fun i => (f i,g i)) (a,b) = counts g b := by
-  classical
-  let he (a : A) : {z : {i // g i = b} // f z.val = a} ≃
-      {i // (f i,g i) = (a,b)} := {
-    toFun z := ⟨z.val.val, Prod.ext z.property z.val.property⟩
-    invFun z := ⟨⟨z.val, congrArg Prod.snd z.property⟩, congrArg Prod.fst z.property⟩
-    left_inv z := rfl
-    right_inv z := rfl }
-  have hh := counts_total (fun z : {i // g i = b} => f z.val)
-  simp only [counts, Nat.card_eq_fintype_card] at hh ⊢
-  rw [← hh]
-  exact Finset.sum_congr rfl (fun a ha => (Fintype.card_congr (he a)).symm)
-
-lemma exists_coloring (mu : A → ℕ) (hmu : ∑ a, mu a = Fintype.card D) :
-    ∃ f : D → A, counts f = mu := by
-  classical
-  let e : D ≃ (Σ a, Fin (mu a)) := Fintype.equivOfCardEq (by simp [hmu])
-  let f : D → A := fun i => (e i).1
-  let ea (a : A) : {t : (Σ a, Fin (mu a)) // t.1 = a} ≃ Fin (mu a) := {
-    toFun t := t.property ▸ t.val.2
-    invFun j := ⟨⟨a,j⟩,rfl⟩
-    left_inv t := by rcases t with ⟨⟨b,j⟩,h⟩; dsimp at h; subst b; rfl
-    right_inv j := rfl }
-  refine ⟨f, funext fun a => ?_⟩
-  have hcard := Nat.card_congr ((Equiv.subtypeEquiv e (fun i => Iff.rfl)).trans (ea a))
-  simpa [counts, f] using hcard
-
-end Counts
-
-def Coloring (D A : Type*) [Fintype D] (mu : A → ℕ) :=
-  {f : D → A // counts f = mu}
-
-instance {D A : Type*} [Fintype D] [Fintype A] (mu : A → ℕ) :
-    Fintype (Coloring D A mu) := by
-  classical
-  unfold Coloring
-  infer_instance
-
-instance coloringAction {D A : Type*} [Fintype D] [Fintype A] (mu : A → ℕ) :
-    MulAction (Equiv.Perm D) (Coloring D A mu) where
-  smul σ f := ⟨f.val ∘ σ.symm, (counts_reindex f.val σ.symm).trans f.property⟩
-  one_smul f := by apply Subtype.ext; rfl
-  mul_smul σ τ f := by apply Subtype.ext; rfl
-
-@[simp] lemma coloring_smul_apply {D A : Type*} [Fintype D] [Fintype A] (mu : A → ℕ)
-    (σ : Equiv.Perm D) (f : Coloring D A mu) (i : D) :
-    (σ • f).val i = f.val (σ.symm i) := rfl
-
-def Contingency {A B : Type*} [Fintype A] [Fintype B] (mu : A → ℕ) (nu : B → ℕ) :=
-  {M : Matrix A B ℕ // (∀ a, ∑ b, M a b = mu a) ∧ (∀ b, ∑ a, M a b = nu b)}
-
-section ColorPairs
-variable {D A B : Type*} [Fintype D] [Fintype A] [Fintype B]
-    (mu : A → ℕ) (nu : B → ℕ)
-
-def intersectionCounts (p : Coloring D A mu × Coloring D B nu) : Contingency mu nu :=
-  ⟨fun a b => counts (fun i => (p.1.val i,p.2.val i)) (a,b),
-    (fun a => (counts_pair_fst p.1.val p.2.val a).trans (congrFun p.1.property a)),
-    fun b => (counts_pair_snd p.1.val p.2.val b).trans (congrFun p.2.property b)⟩
-
-lemma intersectionCounts_smul (σ : Equiv.Perm D)
-    (p : Coloring D A mu × Coloring D B nu) :
-    intersectionCounts mu nu (σ • p) = intersectionCounts mu nu p := by
-  apply Subtype.ext
-  ext a b
-  exact congrFun (counts_reindex (fun i => (p.1.val i,p.2.val i)) σ.symm) (a,b)
-
-lemma orbit_iff_intersectionCounts (p q : Coloring D A mu × Coloring D B nu) :
-    MulAction.orbitRel (Equiv.Perm D) _ p q ↔
-      intersectionCounts mu nu p = intersectionCounts mu nu q := by
-  constructor
-  · rintro ⟨σ,rfl⟩
-    exact intersectionCounts_smul mu nu σ q
-  · intro h
-    have he : counts (fun i => (q.1.val i,q.2.val i)) =
-        counts (fun i => (p.1.val i,p.2.val i)) := by
-      funext ab
-      exact (congrFun (congrFun (congrArg Subtype.val h) ab.1) ab.2).symm
-    obtain ⟨σ,hσ⟩ := exists_perm_of_counts_eq _ _ he
-    refine ⟨σ.symm, ?_⟩
-    apply Prod.ext
-    · apply Subtype.ext; funext i; exact congrArg Prod.fst (congrFun hσ i)
-    · apply Subtype.ext; funext i; exact congrArg Prod.snd (congrFun hσ i)
-
-lemma intersectionCounts_surjective (hmu : ∑ a, mu a = Fintype.card D) :
-    Function.Surjective (intersectionCounts (D := D) mu nu) := by
-  intro M
-  have ht : ∑ ab : A × B, M.val ab.1 ab.2 = Fintype.card D := by
-    rw [Fintype.sum_prod_type]
-    simp only [M.property.1]
-    exact hmu
-  obtain ⟨f,hf⟩ := exists_coloring (D := D) (fun ab : A × B => M.val ab.1 ab.2) ht
-  have ha : counts (fun i => (f i).1) = mu := by
-    funext a
-    rw [← counts_pair_fst (fun i => (f i).1) (fun i => (f i).2) a]
-    simpa only [Prod.eta, hf] using M.property.1 a
-  have hb : counts (fun i => (f i).2) = nu := by
-    funext b
-    rw [← counts_pair_snd (fun i => (f i).1) (fun i => (f i).2) b]
-    simpa only [Prod.eta, hf] using M.property.2 b
-  refine ⟨(⟨fun i => (f i).1,ha⟩,⟨fun i => (f i).2,hb⟩), ?_⟩
-  apply Subtype.ext
-  ext a b
-  exact congrFun hf (a,b)
-
-def orbitContingencyEquiv (hmu : ∑ a, mu a = Fintype.card D) :
-    MulAction.orbitRel.Quotient (Equiv.Perm D) (Coloring D A mu × Coloring D B nu) ≃
-      Contingency mu nu := by
-  let f := Quotient.lift (intersectionCounts (D := D) mu nu)
-    (fun p q h => (orbit_iff_intersectionCounts mu nu p q).mp h)
-  apply Equiv.ofBijective f
-  constructor
-  · intro p q h
-    induction p using Quotient.inductionOn with | _ p =>
-      induction q using Quotient.inductionOn with | _ q =>
-        exact Quotient.sound ((orbit_iff_intersectionCounts mu nu p q).mpr h)
-  · intro M
-    obtain ⟨p,hp⟩ := intersectionCounts_surjective mu nu hmu M
-    exact ⟨Quotient.mk _ p,hp⟩
-
-end ColorPairs
-end Foulkes.Young
-
-namespace Foulkes.Young
-open Module Finset
-
-theorem young_hom_finrank {K D A B : Type*} [Field K]
-    [Fintype D] [Fintype A] [Fintype B] (mu : A → ℕ) (nu : B → ℕ)
-    (hnu : ∑ b, nu b = Fintype.card D) :
-    finrank K ((permRep K (Equiv.Perm D) (Coloring D A mu)).IntertwiningMap
-      (permRep K (Equiv.Perm D) (Coloring D B nu))) = Nat.card (Contingency nu mu) := by
-  rw [perm_hom_finrank]
-  exact Nat.card_congr (orbitContingencyEquiv nu mu hnu)
-end Foulkes.Young
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.FoulkesSixth.YoungModules
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.Young
+open Finset
+
+section Counts
+variable {D A B : Type*} [Fintype D] [Fintype A] [Fintype B]
+
+def counts (f : D → A) (a : A) : ℕ := Nat.card {i // f i = a}
+
+omit [Fintype D] [Fintype A] in
+lemma counts_reindex (f : D → A) (σ : Equiv.Perm D) : counts (f ∘ σ) = counts f := by
+  funext a
+  exact Nat.card_congr (Equiv.subtypeEquiv σ (fun i => Iff.rfl))
+
+omit [Fintype A] in
+lemma exists_perm_of_counts_eq (f g : D → A) (h : counts f = counts g) :
+    ∃ σ : Equiv.Perm D, f ∘ σ = g := by
+  classical
+  let e (a : A) : {i // g i = a} ≃ {i // f i = a} :=
+    Fintype.equivOfCardEq (by simpa [counts, Nat.card_eq_fintype_card] using (congrFun h a).symm)
+  let σ : Equiv.Perm D := (Equiv.sigmaFiberEquiv g).symm.trans
+    ((Equiv.sigmaCongrRight e).trans (Equiv.sigmaFiberEquiv f))
+  refine ⟨σ, ?_⟩
+  funext i
+  exact (e (g i) ⟨i,rfl⟩).property
+
+lemma counts_total (f : D → A) : ∑ a, counts f a = Fintype.card D := by
+  classical
+  simp only [counts, Nat.card_eq_fintype_card]
+  rw [← Fintype.card_sigma]
+  exact Fintype.card_congr (Equiv.sigmaFiberEquiv f)
+
+omit [Fintype A] in
+lemma counts_pair_fst (f : D → A) (g : D → B) (a : A) :
+    ∑ b, counts (fun i => (f i,g i)) (a,b) = counts f a := by
+  classical
+  let he (b : B) : {z : {i // f i = a} // g z.val = b} ≃
+      {i // (f i,g i) = (a,b)} := {
+    toFun z := ⟨z.val.val, Prod.ext z.val.property z.property⟩
+    invFun z := ⟨⟨z.val, congrArg Prod.fst z.property⟩, congrArg Prod.snd z.property⟩
+    left_inv z := rfl
+    right_inv z := rfl }
+  have hh := counts_total (fun z : {i // f i = a} => g z.val)
+  simp only [counts, Nat.card_eq_fintype_card] at hh ⊢
+  rw [← hh]
+  exact Finset.sum_congr rfl (fun b hb => (Fintype.card_congr (he b)).symm)
+
+omit [Fintype B] in
+lemma counts_pair_snd (f : D → A) (g : D → B) (b : B) :
+    ∑ a, counts (fun i => (f i,g i)) (a,b) = counts g b := by
+  classical
+  let he (a : A) : {z : {i // g i = b} // f z.val = a} ≃
+      {i // (f i,g i) = (a,b)} := {
+    toFun z := ⟨z.val.val, Prod.ext z.property z.val.property⟩
+    invFun z := ⟨⟨z.val, congrArg Prod.snd z.property⟩, congrArg Prod.fst z.property⟩
+    left_inv z := rfl
+    right_inv z := rfl }
+  have hh := counts_total (fun z : {i // g i = b} => f z.val)
+  simp only [counts, Nat.card_eq_fintype_card] at hh ⊢
+  rw [← hh]
+  exact Finset.sum_congr rfl (fun a ha => (Fintype.card_congr (he a)).symm)
+
+lemma exists_coloring (mu : A → ℕ) (hmu : ∑ a, mu a = Fintype.card D) :
+    ∃ f : D → A, counts f = mu := by
+  classical
+  let e : D ≃ (Σ a, Fin (mu a)) := Fintype.equivOfCardEq (by simp [hmu])
+  let f : D → A := fun i => (e i).1
+  let ea (a : A) : {t : (Σ a, Fin (mu a)) // t.1 = a} ≃ Fin (mu a) := {
+    toFun t := t.property ▸ t.val.2
+    invFun j := ⟨⟨a,j⟩,rfl⟩
+    left_inv t := by rcases t with ⟨⟨b,j⟩,h⟩; dsimp at h; subst b; rfl
+    right_inv j := rfl }
+  refine ⟨f, funext fun a => ?_⟩
+  have hcard := Nat.card_congr ((Equiv.subtypeEquiv e (fun i => Iff.rfl)).trans (ea a))
+  simpa [counts, f] using hcard
+
+end Counts
+
+def Coloring (D A : Type*) [Fintype D] (mu : A → ℕ) :=
+  {f : D → A // counts f = mu}
+
+instance {D A : Type*} [Fintype D] [Fintype A] (mu : A → ℕ) :
+    Fintype (Coloring D A mu) := by
+  classical
+  unfold Coloring
+  infer_instance
+
+instance coloringAction {D A : Type*} [Fintype D] [Fintype A] (mu : A → ℕ) :
+    MulAction (Equiv.Perm D) (Coloring D A mu) where
+  smul σ f := ⟨f.val ∘ σ.symm, (counts_reindex f.val σ.symm).trans f.property⟩
+  one_smul f := by apply Subtype.ext; rfl
+  mul_smul σ τ f := by apply Subtype.ext; rfl
+
+@[simp] lemma coloring_smul_apply {D A : Type*} [Fintype D] [Fintype A] (mu : A → ℕ)
+    (σ : Equiv.Perm D) (f : Coloring D A mu) (i : D) :
+    (σ • f).val i = f.val (σ.symm i) := rfl
+
+def Contingency {A B : Type*} [Fintype A] [Fintype B] (mu : A → ℕ) (nu : B → ℕ) :=
+  {M : Matrix A B ℕ // (∀ a, ∑ b, M a b = mu a) ∧ (∀ b, ∑ a, M a b = nu b)}
+
+section ColorPairs
+variable {D A B : Type*} [Fintype D] [Fintype A] [Fintype B]
+    (mu : A → ℕ) (nu : B → ℕ)
+
+def intersectionCounts (p : Coloring D A mu × Coloring D B nu) : Contingency mu nu :=
+  ⟨fun a b => counts (fun i => (p.1.val i,p.2.val i)) (a,b),
+    (fun a => (counts_pair_fst p.1.val p.2.val a).trans (congrFun p.1.property a)),
+    fun b => (counts_pair_snd p.1.val p.2.val b).trans (congrFun p.2.property b)⟩
+
+lemma intersectionCounts_smul (σ : Equiv.Perm D)
+    (p : Coloring D A mu × Coloring D B nu) :
+    intersectionCounts mu nu (σ • p) = intersectionCounts mu nu p := by
+  apply Subtype.ext
+  ext a b
+  exact congrFun (counts_reindex (fun i => (p.1.val i,p.2.val i)) σ.symm) (a,b)
+
+lemma orbit_iff_intersectionCounts (p q : Coloring D A mu × Coloring D B nu) :
+    MulAction.orbitRel (Equiv.Perm D) _ p q ↔
+      intersectionCounts mu nu p = intersectionCounts mu nu q := by
+  constructor
+  · rintro ⟨σ,rfl⟩
+    exact intersectionCounts_smul mu nu σ q
+  · intro h
+    have he : counts (fun i => (q.1.val i,q.2.val i)) =
+        counts (fun i => (p.1.val i,p.2.val i)) := by
+      funext ab
+      exact (congrFun (congrFun (congrArg Subtype.val h) ab.1) ab.2).symm
+    obtain ⟨σ,hσ⟩ := exists_perm_of_counts_eq _ _ he
+    refine ⟨σ.symm, ?_⟩
+    apply Prod.ext
+    · apply Subtype.ext; funext i; exact congrArg Prod.fst (congrFun hσ i)
+    · apply Subtype.ext; funext i; exact congrArg Prod.snd (congrFun hσ i)
+
+lemma intersectionCounts_surjective (hmu : ∑ a, mu a = Fintype.card D) :
+    Function.Surjective (intersectionCounts (D := D) mu nu) := by
+  intro M
+  have ht : ∑ ab : A × B, M.val ab.1 ab.2 = Fintype.card D := by
+    rw [Fintype.sum_prod_type]
+    simp only [M.property.1]
+    exact hmu
+  obtain ⟨f,hf⟩ := exists_coloring (D := D) (fun ab : A × B => M.val ab.1 ab.2) ht
+  have ha : counts (fun i => (f i).1) = mu := by
+    funext a
+    rw [← counts_pair_fst (fun i => (f i).1) (fun i => (f i).2) a]
+    simpa only [Prod.eta, hf] using M.property.1 a
+  have hb : counts (fun i => (f i).2) = nu := by
+    funext b
+    rw [← counts_pair_snd (fun i => (f i).1) (fun i => (f i).2) b]
+    simpa only [Prod.eta, hf] using M.property.2 b
+  refine ⟨(⟨fun i => (f i).1,ha⟩,⟨fun i => (f i).2,hb⟩), ?_⟩
+  apply Subtype.ext
+  ext a b
+  exact congrFun hf (a,b)
+
+def orbitContingencyEquiv (hmu : ∑ a, mu a = Fintype.card D) :
+    MulAction.orbitRel.Quotient (Equiv.Perm D) (Coloring D A mu × Coloring D B nu) ≃
+      Contingency mu nu := by
+  let f := Quotient.lift (intersectionCounts (D := D) mu nu)
+    (fun p q h => (orbit_iff_intersectionCounts mu nu p q).mp h)
+  apply Equiv.ofBijective f
+  constructor
+  · intro p q h
+    induction p using Quotient.inductionOn with | _ p =>
+      induction q using Quotient.inductionOn with | _ q =>
+        exact Quotient.sound ((orbit_iff_intersectionCounts mu nu p q).mpr h)
+  · intro M
+    obtain ⟨p,hp⟩ := intersectionCounts_surjective mu nu hmu M
+    exact ⟨Quotient.mk _ p,hp⟩
+
+end ColorPairs
+end Foulkes.Young
+
+namespace Foulkes.Young
+open Module Finset
+
+theorem young_hom_finrank {K D A B : Type*} [Field K]
+    [Fintype D] [Fintype A] [Fintype B] (mu : A → ℕ) (nu : B → ℕ)
+    (hnu : ∑ b, nu b = Fintype.card D) :
+    finrank K ((permRep K (Equiv.Perm D) (Coloring D A mu)).IntertwiningMap
+      (permRep K (Equiv.Perm D) (Coloring D B nu))) = Nat.card (Contingency nu mu) := by
+  rw [perm_hom_finrank]
+  exact Nat.card_congr (orbitContingencyEquiv nu mu hnu)
+end Foulkes.Young
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/CompleteNewton.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/CompleteNewton.lean
index ff937372e167adec1756a8515935cbf0eb914629..549c230f043a9a89fd1635ea9d9414a205b97c67 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/CompleteNewton.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/CompleteNewton.lean
@@ -1,130 +1,130 @@
-import Mathlib
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.Complete
-open MvPolynomial Finset
-
-abbrev Degree (α : Type*) (b : ℕ) := {d : α →₀ ℕ // d.degree = b}
-instance {α : Type*} [Finite α] (b : ℕ) : Fintype (Degree α b) :=
-  (Finsupp.finite_of_degree_eq (σ := α) b).fintype
-
-def h (α : Type*) [Finite α] (b : ℕ) : MvPolynomial α ℤ :=
-  ∑ d : Degree α b, monomial d.val 1
-
-@[simp] theorem coeff_h {α : Type*} [Finite α] (b : ℕ) (d : α →₀ ℕ) :
-    (h α b).coeff d = if d.degree = b then 1 else 0 := by
-  classical
-  simp only [h, coeff_sum, coeff_monomial]
-  by_cases hd : d.degree = b
-  · rw [ite_eq_left hd, Finset.sum_eq_single (⟨d,hd⟩ : Degree α b)]
-    · simp
-    · intro e he hne
-      have hneq : e.val ≠ d := fun hh => hne (Subtype.ext hh)
-      simp [hneq]
-    · simp
-  · rw [ite_eq_right hd]
-    apply Finset.sum_eq_zero
-    intro e he
-    have hneq : e.val ≠ d := fun hh => hd (hh ▸ e.property)
-    simp [hneq]
-
-@[simp] theorem h_zero (α : Type*) [Finite α] : h α 0 = 1 := by
-  classical
-  ext d
-  rw [coeff_h]
-  by_cases hd : d = 0
-  · simp [hd]
-  · have hh : d.degree ≠ 0 := by
-      intro hh
-      apply hd
-      ext j
-      have hj := Finsupp.le_degree j d
-      simp only [hh, Nat.le_zero] at hj
-      simpa using hj
-    simp [hh, coeff_one, Ne.symm hd]
-
-lemma coeff_X_pow_h {α : Type*} [Finite α] (b k : ℕ) (hkb : k ≤ b)
-    (j : α) (d : α →₀ ℕ) :
-    ((X j : MvPolynomial α ℤ)^k * h α (b-k)).coeff d =
-      if d.degree = b ∧ k ≤ d j then 1 else 0 := by
-  classical
-  rw [X_pow_eq_monomial, coeff_monomial_mul', coeff_h]
-  simp only [one_mul, Finsupp.single_le_iff]
-  by_cases hk : k ≤ d j
-  · rw [ite_eq_left hk]
-    have hs : Finsupp.single j k ≤ d := Finsupp.single_le_iff.mpr hk
-    have he : (d - Finsupp.single j k).degree + k = d.degree := by
-      have he := congrArg Finsupp.degree (tsub_add_cancel_of_le hs)
-      simpa using he
-    have hd : (d - Finsupp.single j k).degree = b-k ↔ d.degree = b := by omega
-    simp only [hd, hk, and_true]
-  · simp [hk]
-
-theorem newton {α : Type*} [Fintype α] (b : ℕ) :
-    (b : ℤ) • h α b =
-      ∑ k ∈ Finset.Icc 1 b, (∑ j : α, (X j : MvPolynomial α ℤ)^k) * h α (b-k) := by
-  classical
-  ext d
-  simp only [coeff_smul, coeff_h, coeff_sum, Finset.sum_mul]
-  rw [Finset.sum_comm]
-  by_cases hd : d.degree = b
-  · have hterm (j : α) :
-        ∑ k ∈ Finset.Icc 1 b, ((X j : MvPolynomial α ℤ)^k * h α (b-k)).coeff d = (d j : ℤ) := by
-      have hj : d j ≤ b := by simpa [hd] using Finsupp.le_degree j d
-      have he : (Finset.Icc 1 b).filter (fun k => k ≤ d j) = Finset.Icc 1 (d j) := by
-        ext k
-        simp only [mem_filter, mem_Icc]
-        omega
-      calc
-        _ = ∑ k ∈ Finset.Icc 1 b, if k ≤ d j then (1 : ℤ) else 0 := by
-          apply Finset.sum_congr rfl
-          intro k hk
-          rw [coeff_X_pow_h b k (mem_Icc.mp hk).2]
-          simp only [hd, true_and]
-        _ = ((Finset.Icc 1 (d j)).card : ℤ) := by
-          rw [← Finset.sum_filter, he]
-          simp
-        _ = (d j : ℤ) := by simp
-    simp only [hterm, ite_eq_left hd, zsmul_eq_mul, mul_one]
-    simpa using
-      congrArg (fun z : ℕ => (z : ℤ)) (hd.symm.trans (Finsupp.degree_eq_sum d))
-  · have hterm (j : α) :
-        ∑ k ∈ Finset.Icc 1 b, ((X j : MvPolynomial α ℤ)^k * h α (b-k)).coeff d = 0 := by
-      apply Finset.sum_eq_zero
-      intro k hk
-      rw [coeff_X_pow_h b k (mem_Icc.mp hk).2]
-      simp [hd]
-    simp [hterm, hd]
-
-def alphabetH {α σ : Type*} [Finite α] (x : α → MvPolynomial σ ℤ) (b : ℕ) : MvPolynomial σ ℤ :=
-  eval₂Hom (Int.castRingHom _) x (h α b)
-
-@[simp] theorem alphabetH_zero {α σ : Type*} [Finite α] (x : α → MvPolynomial σ ℤ) :
-    alphabetH x 0 = 1 := by simp [alphabetH]
-
-theorem alphabet_newton {α σ : Type*} [Fintype α] (x : α → MvPolynomial σ ℤ) (b : ℕ) :
-    (b : ℤ) • alphabetH x b =
-      ∑ k ∈ Finset.Icc 1 b, (∑ j : α, x j ^ k) * alphabetH x (b-k) := by
-  have hh := congrArg (eval₂Hom (Int.castRingHom (MvPolynomial σ ℤ)) x) (newton (α := α) b)
-  simpa [alphabetH] using hh
-
-def pleth (n b a : ℕ) : MvPolynomial (Fin n) ℤ :=
-  alphabetH (fun d : Degree (Fin n) a => monomial d.val 1) b
-
-@[simp] theorem pleth_zero (n a : ℕ) : pleth n 0 a = 1 := alphabetH_zero _
-
-theorem pleth_newton (n b a : ℕ) :
-    (b : ℤ) • pleth n b a =
-      ∑ k ∈ Finset.Icc 1 b,
-        (∑ d : Degree (Fin n) a, monomial (k • d.val) (1 : ℤ)) * pleth n (b-k) a := by
-  simpa [pleth, monomial_pow] using
-    (alphabet_newton (fun d : Degree (Fin n) a => monomial d.val (1 : ℤ)) b)
-
-end Foulkes.Complete
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.Complete
+open MvPolynomial Finset
+
+abbrev Degree (α : Type*) (b : ℕ) := {d : α →₀ ℕ // d.degree = b}
+instance {α : Type*} [Finite α] (b : ℕ) : Fintype (Degree α b) :=
+  (Finsupp.finite_of_degree_eq (σ := α) b).fintype
+
+def h (α : Type*) [Finite α] (b : ℕ) : MvPolynomial α ℤ :=
+  ∑ d : Degree α b, monomial d.val 1
+
+@[simp] theorem coeff_h {α : Type*} [Finite α] (b : ℕ) (d : α →₀ ℕ) :
+    (h α b).coeff d = if d.degree = b then 1 else 0 := by
+  classical
+  simp only [h, coeff_sum, coeff_monomial]
+  by_cases hd : d.degree = b
+  · rw [ite_eq_left hd, Finset.sum_eq_single (⟨d,hd⟩ : Degree α b)]
+    · simp
+    · intro e he hne
+      have hneq : e.val ≠ d := fun hh => hne (Subtype.ext hh)
+      simp [hneq]
+    · simp
+  · rw [ite_eq_right hd]
+    apply Finset.sum_eq_zero
+    intro e he
+    have hneq : e.val ≠ d := fun hh => hd (hh ▸ e.property)
+    simp [hneq]
+
+@[simp] theorem h_zero (α : Type*) [Finite α] : h α 0 = 1 := by
+  classical
+  ext d
+  rw [coeff_h]
+  by_cases hd : d = 0
+  · simp [hd]
+  · have hh : d.degree ≠ 0 := by
+      intro hh
+      apply hd
+      ext j
+      have hj := Finsupp.le_degree j d
+      simp only [hh, Nat.le_zero] at hj
+      simpa using hj
+    simp [hh, coeff_one, Ne.symm hd]
+
+lemma coeff_X_pow_h {α : Type*} [Finite α] (b k : ℕ) (hkb : k ≤ b)
+    (j : α) (d : α →₀ ℕ) :
+    ((X j : MvPolynomial α ℤ)^k * h α (b-k)).coeff d =
+      if d.degree = b ∧ k ≤ d j then 1 else 0 := by
+  classical
+  rw [X_pow_eq_monomial, coeff_monomial_mul', coeff_h]
+  simp only [one_mul, Finsupp.single_le_iff]
+  by_cases hk : k ≤ d j
+  · rw [ite_eq_left hk]
+    have hs : Finsupp.single j k ≤ d := Finsupp.single_le_iff.mpr hk
+    have he : (d - Finsupp.single j k).degree + k = d.degree := by
+      have he := congrArg Finsupp.degree (tsub_add_cancel_of_le hs)
+      simpa using he
+    have hd : (d - Finsupp.single j k).degree = b-k ↔ d.degree = b := by omega
+    simp only [hd, hk, and_true]
+  · simp [hk]
+
+theorem newton {α : Type*} [Fintype α] (b : ℕ) :
+    (b : ℤ) • h α b =
+      ∑ k ∈ Finset.Icc 1 b, (∑ j : α, (X j : MvPolynomial α ℤ)^k) * h α (b-k) := by
+  classical
+  ext d
+  simp only [coeff_smul, coeff_h, coeff_sum, Finset.sum_mul]
+  rw [Finset.sum_comm]
+  by_cases hd : d.degree = b
+  · have hterm (j : α) :
+        ∑ k ∈ Finset.Icc 1 b, ((X j : MvPolynomial α ℤ)^k * h α (b-k)).coeff d = (d j : ℤ) := by
+      have hj : d j ≤ b := by simpa [hd] using Finsupp.le_degree j d
+      have he : (Finset.Icc 1 b).filter (fun k => k ≤ d j) = Finset.Icc 1 (d j) := by
+        ext k
+        simp only [mem_filter, mem_Icc]
+        omega
+      calc
+        _ = ∑ k ∈ Finset.Icc 1 b, if k ≤ d j then (1 : ℤ) else 0 := by
+          apply Finset.sum_congr rfl
+          intro k hk
+          rw [coeff_X_pow_h b k (mem_Icc.mp hk).2]
+          simp only [hd, true_and]
+        _ = ((Finset.Icc 1 (d j)).card : ℤ) := by
+          rw [← Finset.sum_filter, he]
+          simp
+        _ = (d j : ℤ) := by simp
+    simp only [hterm, ite_eq_left hd, zsmul_eq_mul, mul_one]
+    simpa using
+      congrArg (fun z : ℕ => (z : ℤ)) (hd.symm.trans (Finsupp.degree_eq_sum d))
+  · have hterm (j : α) :
+        ∑ k ∈ Finset.Icc 1 b, ((X j : MvPolynomial α ℤ)^k * h α (b-k)).coeff d = 0 := by
+      apply Finset.sum_eq_zero
+      intro k hk
+      rw [coeff_X_pow_h b k (mem_Icc.mp hk).2]
+      simp [hd]
+    simp [hterm, hd]
+
+def alphabetH {α σ : Type*} [Finite α] (x : α → MvPolynomial σ ℤ) (b : ℕ) : MvPolynomial σ ℤ :=
+  eval₂Hom (Int.castRingHom _) x (h α b)
+
+@[simp] theorem alphabetH_zero {α σ : Type*} [Finite α] (x : α → MvPolynomial σ ℤ) :
+    alphabetH x 0 = 1 := by simp [alphabetH]
+
+theorem alphabet_newton {α σ : Type*} [Fintype α] (x : α → MvPolynomial σ ℤ) (b : ℕ) :
+    (b : ℤ) • alphabetH x b =
+      ∑ k ∈ Finset.Icc 1 b, (∑ j : α, x j ^ k) * alphabetH x (b-k) := by
+  have hh := congrArg (eval₂Hom (Int.castRingHom (MvPolynomial σ ℤ)) x) (newton (α := α) b)
+  simpa [alphabetH] using hh
+
+def pleth (n b a : ℕ) : MvPolynomial (Fin n) ℤ :=
+  alphabetH (fun d : Degree (Fin n) a => monomial d.val 1) b
+
+@[simp] theorem pleth_zero (n a : ℕ) : pleth n 0 a = 1 := alphabetH_zero _
+
+theorem pleth_newton (n b a : ℕ) :
+    (b : ℤ) • pleth n b a =
+      ∑ k ∈ Finset.Icc 1 b,
+        (∑ d : Degree (Fin n) a, monomial (k • d.val) (1 : ℤ)) * pleth n (b-k) a := by
+  simpa [pleth, monomial_pow] using
+    (alphabet_newton (fun d : Degree (Fin n) a => monomial d.val (1 : ℤ)) b)
+
+end Foulkes.Complete
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/CompositionEnumeration.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/CompositionEnumeration.lean
index ef732617b7df4f6b550e7418639137729d419c69..dd4bbe182d130686d9ead11d0307949a454ced16 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/CompositionEnumeration.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/CompositionEnumeration.lean
@@ -1,80 +1,80 @@
-import Mathlib
-import OAI.RepresentationTheory.FoulkesSixth.FastLookup
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.Enumeration
-open Finset
-
-def compositions : (n d : ℕ) → List (Fin n → ℕ)
-  | 0, d => if d = 0 then [Fin.elim0] else []
-  | n+1, d => (List.range (d+1)).flatMap fun k =>
-      (compositions n (d-k)).map (Fin.cons k)
-
-lemma mem_compositions (n d : ℕ) (v : Fin n → ℕ) :
-    v ∈ compositions n d ↔ ∑ j, v j = d := by
-  induction n generalizing d with
-  | zero =>
-    have hv : v = Fin.elim0 := by funext j; exact Fin.elim0 j
-    subst v
-    by_cases hd : d = 0
-    · simp [compositions, hd]
-    · simp [compositions, hd, Ne.symm hd]
-  | succ n ih =>
-    rw [compositions, List.mem_flatMap]
-    constructor
-    · rintro ⟨k,hk,hv⟩
-      obtain ⟨w,hw,rfl⟩ := List.mem_map.mp hv
-      have hw' := (ih (d-k) w).mp hw
-      have hk' : k ≤ d := by simpa using List.mem_range.mp hk
-      rw [Fin.sum_univ_succ]
-      simp only [Fin.cons_zero, Fin.cons_succ]
-      omega
-    · intro hv
-      refine ⟨v 0, List.mem_range.mpr ?_, List.mem_map.mpr ⟨Fin.tail v, ?_, ?_⟩⟩
-      · rw [Fin.sum_univ_succ] at hv
-        omega
-      · apply (ih (d-v 0) (Fin.tail v)).mpr
-        rw [Fin.sum_univ_succ] at hv
-        dsimp [Fin.tail]
-        omega
-      · exact Fin.cons_self_tail v
-
-lemma nodup_compositions (n d : ℕ) : (compositions n d).Nodup := by
-  induction n generalizing d with
-  | zero => unfold compositions; split_ifs <;> simp
-  | succ n ih =>
-    rw [compositions, List.nodup_flatMap]
-    refine ⟨fun k hk => (ih _).map ?_, (List.nodup_range (n := d+1)).imp ?_⟩
-    · intro v w h
-      funext j
-      exact congrFun h j.succ
-    · intro x y hxy v hx hy
-      obtain ⟨vx,hvx,hex⟩ := List.mem_map.mp hx
-      obtain ⟨vy,hvy,hey⟩ := List.mem_map.mp hy
-      have hh := congrFun (hex.trans hey.symm) 0
-      exact hxy hh
-
-def compositionSet (n d : ℕ) : Finset (Fin n → ℕ) := (compositions n d).toFinset
-
-@[simp] lemma mem_compositionSet (n d : ℕ) (v : Fin n → ℕ) :
-    v ∈ compositionSet n d ↔ ∑ j, v j = d := by
-  simp [compositionSet, mem_compositions]
-
-def degreeComposition (n d : ℕ) : Foulkes.Complete.Degree (Fin n) d ≃
-    {v : Fin n → ℕ // v ∈ compositionSet n d} where
-  toFun a := ⟨fun j => a.val j, (mem_compositionSet n d _).mpr (by
-    rw [← Finsupp.degree_eq_sum]; exact a.property)⟩
-  invFun v := ⟨Foulkes.Strips.expVector v.val, by
-    rw [Foulkes.Lookup.degree_expVector]
-    exact (mem_compositionSet n d _).mp v.property⟩
-  left_inv a := by apply Subtype.ext; ext j; rfl
-  right_inv v := rfl
-
-end Foulkes.Enumeration
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.FoulkesSixth.FastLookup
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.Enumeration
+open Finset
+
+def compositions : (n d : ℕ) → List (Fin n → ℕ)
+  | 0, d => if d = 0 then [Fin.elim0] else []
+  | n+1, d => (List.range (d+1)).flatMap fun k =>
+      (compositions n (d-k)).map (Fin.cons k)
+
+lemma mem_compositions (n d : ℕ) (v : Fin n → ℕ) :
+    v ∈ compositions n d ↔ ∑ j, v j = d := by
+  induction n generalizing d with
+  | zero =>
+    have hv : v = Fin.elim0 := by funext j; exact Fin.elim0 j
+    subst v
+    by_cases hd : d = 0
+    · simp [compositions, hd]
+    · simp [compositions, hd, Ne.symm hd]
+  | succ n ih =>
+    rw [compositions, List.mem_flatMap]
+    constructor
+    · rintro ⟨k,hk,hv⟩
+      obtain ⟨w,hw,rfl⟩ := List.mem_map.mp hv
+      have hw' := (ih (d-k) w).mp hw
+      have hk' : k ≤ d := by simpa using List.mem_range.mp hk
+      rw [Fin.sum_univ_succ]
+      simp only [Fin.cons_zero, Fin.cons_succ]
+      omega
+    · intro hv
+      refine ⟨v 0, List.mem_range.mpr ?_, List.mem_map.mpr ⟨Fin.tail v, ?_, ?_⟩⟩
+      · rw [Fin.sum_univ_succ] at hv
+        omega
+      · apply (ih (d-v 0) (Fin.tail v)).mpr
+        rw [Fin.sum_univ_succ] at hv
+        dsimp [Fin.tail]
+        omega
+      · exact Fin.cons_self_tail v
+
+lemma nodup_compositions (n d : ℕ) : (compositions n d).Nodup := by
+  induction n generalizing d with
+  | zero => unfold compositions; split_ifs <;> simp
+  | succ n ih =>
+    rw [compositions, List.nodup_flatMap]
+    refine ⟨fun k hk => (ih _).map ?_, (List.nodup_range (n := d+1)).imp ?_⟩
+    · intro v w h
+      funext j
+      exact congrFun h j.succ
+    · intro x y hxy v hx hy
+      obtain ⟨vx,hvx,hex⟩ := List.mem_map.mp hx
+      obtain ⟨vy,hvy,hey⟩ := List.mem_map.mp hy
+      have hh := congrFun (hex.trans hey.symm) 0
+      exact hxy hh
+
+def compositionSet (n d : ℕ) : Finset (Fin n → ℕ) := (compositions n d).toFinset
+
+@[simp] lemma mem_compositionSet (n d : ℕ) (v : Fin n → ℕ) :
+    v ∈ compositionSet n d ↔ ∑ j, v j = d := by
+  simp [compositionSet, mem_compositions]
+
+def degreeComposition (n d : ℕ) : Foulkes.Complete.Degree (Fin n) d ≃
+    {v : Fin n → ℕ // v ∈ compositionSet n d} where
+  toFun a := ⟨fun j => a.val j, (mem_compositionSet n d _).mpr (by
+    rw [← Finsupp.degree_eq_sum]; exact a.property)⟩
+  invFun v := ⟨Foulkes.Strips.expVector v.val, by
+    rw [Foulkes.Lookup.degree_expVector]
+    exact (mem_compositionSet n d _).mp v.property⟩
+  left_inv a := by apply Subtype.ext; ext j; rfl
+  right_inv v := rfl
+
+end Foulkes.Enumeration
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/Diagonal.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/Diagonal.lean
index 57155511b3fb6a133a703f734519764094aad476..2c9ad915f8171f2ed620f4db23045bcb2ac4e9a7 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/Diagonal.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/Diagonal.lean
@@ -1,89 +1,89 @@
-import Mathlib
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.Diagonal
-open MvPolynomial
-
-variable {ι V : Type*} [Fintype ι] [AddCommGroup V] [Module ℂ V]
-
-def exponent (f : ι → ι) : ι →₀ ℕ := ∑ i, Finsupp.single (f i) 1
-
-lemma exponent_id : exponent (id : ι → ι) = ∑ i, Finsupp.single i 1 := rfl
-
-lemma bijective_of_exponent_eq (f : ι → ι)
-    (h : exponent f = exponent (id : ι → ι)) : Function.Bijective f := by
-  classical
-  apply (Fintype.bijective_iff_surjective_and_card _).mpr
-  constructor
-  · intro k
-    by_contra hk
-    have hn : ∀ i, f i ≠ k := by simpa using hk
-    have hh := congrArg (fun d : ι →₀ ℕ => d k) h
-    have hz : exponent f k = 0 := by
-      simp [exponent, hn]
-    have ho : exponent (id : ι → ι) k = 1 := by
-      simp [exponent, Finsupp.single_apply]
-    rw [hz, ho] at hh
-    omega
-  · rfl
-
-def expansion (T : MultilinearMap ℂ (fun _ : ι => V) ℂ) (v : ι → V) :
-    MvPolynomial ι ℂ := by
-  classical
-  exact ∑ f : ι → ι, monomial (exponent f) (T (fun i => v (f i)))
-
-lemma eval_expansion (T : MultilinearMap ℂ (fun _ : ι => V) ℂ)
-    (v : ι → V) (x : ι → ℂ) :
-    eval x (expansion T v) = T (fun _ => ∑ j, x j • v j) := by
-  classical
-  rw [T.map_sum]
-  simp only [expansion, map_sum]
-  apply Finset.sum_congr rfl
-  intro f hf
-  rw [T.map_smul_univ]
-  have hm : (monomial (exponent f) 1 : MvPolynomial ι ℂ) = ∏ i, X (f i) := by
-    simpa only [exponent, X] using
-      (monomial_sum_one (R := ℂ) Finset.univ (fun i => Finsupp.single (f i) 1))
-  calc
-    eval x (monomial (exponent f) (T (fun i => v (f i)))) =
-        T (fun i => v (f i)) * eval x (monomial (exponent f) 1) := by
-      simp [eval_monomial]
-    _ = (∏ i, x (f i)) • T (fun i => v (f i)) := by
-      rw [hm]
-      simp [mul_comm]
-
-theorem eq_zero_of_symmetric_of_diagonal
-    (T : MultilinearMap ℂ (fun _ : ι => V) ℂ)
-    (hs : ∀ (σ : Equiv.Perm ι) (v : ι → V), T (fun i => v (σ i)) = T v)
-    (hd : ∀ v : V, T (fun _ => v) = 0) : T = 0 := by
-  classical
-  ext v
-  have hp : expansion T v = 0 := by
-    apply MvPolynomial.funext
-    intro x
-    rw [eval_expansion, hd, map_zero]
-  let G : Finset (ι → ι) := Finset.univ.filter (fun f => exponent f = exponent (id : ι → ι))
-  have hG : G.Nonempty := ⟨id, by simp [G]⟩
-  have hn : (G.card : ℂ) ≠ 0 := by exact_mod_cast hG.card_pos.ne'
-  have hc := congrArg (fun polynomial => polynomial.coeff (exponent (id : ι → ι))) hp
-  have he : (expansion T v).coeff (exponent (id : ι → ι)) = (G.card : ℂ) * T v := by
-    simp only [expansion, coeff_sum, coeff_monomial]
-    rw [← Finset.sum_filter]
-    change (∑ f ∈ G, T (fun i => v (f i))) = _
-    calc
-      _ = ∑ _f ∈ G, T v := by
-        apply Finset.sum_congr rfl
-        intro f hf
-        exact hs (Equiv.ofBijective f (bijective_of_exponent_eq f (Finset.mem_filter.mp hf).2)) v
-      _ = (G.card : ℂ) * T v := by simp
-  rw [he, AddMonoidAlgebra.coeff_zero, Finsupp.zero_apply] at hc
-  exact (mul_eq_zero.mp hc).resolve_left hn
-
-end Foulkes.Diagonal
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.Diagonal
+open MvPolynomial
+
+variable {ι V : Type*} [Fintype ι] [AddCommGroup V] [Module ℂ V]
+
+def exponent (f : ι → ι) : ι →₀ ℕ := ∑ i, Finsupp.single (f i) 1
+
+lemma exponent_id : exponent (id : ι → ι) = ∑ i, Finsupp.single i 1 := rfl
+
+lemma bijective_of_exponent_eq (f : ι → ι)
+    (h : exponent f = exponent (id : ι → ι)) : Function.Bijective f := by
+  classical
+  apply (Fintype.bijective_iff_surjective_and_card _).mpr
+  constructor
+  · intro k
+    by_contra hk
+    have hn : ∀ i, f i ≠ k := by simpa using hk
+    have hh := congrArg (fun d : ι →₀ ℕ => d k) h
+    have hz : exponent f k = 0 := by
+      simp [exponent, hn]
+    have ho : exponent (id : ι → ι) k = 1 := by
+      simp [exponent, Finsupp.single_apply]
+    rw [hz, ho] at hh
+    omega
+  · rfl
+
+def expansion (T : MultilinearMap ℂ (fun _ : ι => V) ℂ) (v : ι → V) :
+    MvPolynomial ι ℂ := by
+  classical
+  exact ∑ f : ι → ι, monomial (exponent f) (T (fun i => v (f i)))
+
+lemma eval_expansion (T : MultilinearMap ℂ (fun _ : ι => V) ℂ)
+    (v : ι → V) (x : ι → ℂ) :
+    eval x (expansion T v) = T (fun _ => ∑ j, x j • v j) := by
+  classical
+  rw [T.map_sum]
+  simp only [expansion, map_sum]
+  apply Finset.sum_congr rfl
+  intro f hf
+  rw [T.map_smul_univ]
+  have hm : (monomial (exponent f) 1 : MvPolynomial ι ℂ) = ∏ i, X (f i) := by
+    simpa only [exponent, X] using
+      (monomial_sum_one (R := ℂ) Finset.univ (fun i => Finsupp.single (f i) 1))
+  calc
+    eval x (monomial (exponent f) (T (fun i => v (f i)))) =
+        T (fun i => v (f i)) * eval x (monomial (exponent f) 1) := by
+      simp [eval_monomial]
+    _ = (∏ i, x (f i)) • T (fun i => v (f i)) := by
+      rw [hm]
+      simp [mul_comm]
+
+theorem eq_zero_of_symmetric_of_diagonal
+    (T : MultilinearMap ℂ (fun _ : ι => V) ℂ)
+    (hs : ∀ (σ : Equiv.Perm ι) (v : ι → V), T (fun i => v (σ i)) = T v)
+    (hd : ∀ v : V, T (fun _ => v) = 0) : T = 0 := by
+  classical
+  ext v
+  have hp : expansion T v = 0 := by
+    apply MvPolynomial.funext
+    intro x
+    rw [eval_expansion, hd, map_zero]
+  let G : Finset (ι → ι) := Finset.univ.filter (fun f => exponent f = exponent (id : ι → ι))
+  have hG : G.Nonempty := ⟨id, by simp [G]⟩
+  have hn : (G.card : ℂ) ≠ 0 := by exact_mod_cast hG.card_pos.ne'
+  have hc := congrArg (fun polynomial => polynomial.coeff (exponent (id : ι → ι))) hp
+  have he : (expansion T v).coeff (exponent (id : ι → ι)) = (G.card : ℂ) * T v := by
+    simp only [expansion, coeff_sum, coeff_monomial]
+    rw [← Finset.sum_filter]
+    change (∑ f ∈ G, T (fun i => v (f i))) = _
+    calc
+      _ = ∑ _f ∈ G, T v := by
+        apply Finset.sum_congr rfl
+        intro f hf
+        exact hs (Equiv.ofBijective f (bijective_of_exponent_eq f (Finset.mem_filter.mp hf).2)) v
+      _ = (G.card : ℂ) * T v := by simp
+  rw [he, AddMonoidAlgebra.coeff_zero, Finsupp.zero_apply] at hc
+  exact (mul_eq_zero.mp hc).resolve_left hn
+
+end Foulkes.Diagonal
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/Differential.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/Differential.lean
index af65c39e5f579c84f1aabb371246ebdde72db256..d53d84d271cc903586715bf71d979d381c67957f 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/Differential.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/Differential.lean
@@ -1,330 +1,330 @@
-import Mathlib
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.Differential
-open MvPolynomial
-
-abbrev Poly (m n : ℕ) := MvPolynomial (Fin m × Fin n) ℂ
-
-def shift {m n : ℕ} (u v : Fin m) : Derivation ℂ (Poly m n) (Poly m n) :=
-  MvPolynomial.mkDerivation ℂ fun ij => if ij.1 = u then X (v, ij.2) else 0
-
-@[simp] theorem shift_X {m n : ℕ} (u v : Fin m) (ij : Fin m × Fin n) :
-    shift (n := n) u v (X ij) = if ij.1 = u then X (v, ij.2) else 0 := by
-  simp [shift]
-
-@[simp] theorem derivation_sum_apply {m n : ℕ} {ι : Type*}
-    (s : Finset ι) (D : ι → Derivation ℂ (Poly m n) (Poly m n)) (f : Poly m n) :
-    (∑ k ∈ s, D k) f = ∑ k ∈ s, D k f := by
-  classical
-  induction s using Finset.induction_on with
-  | empty => simp
-  | insert x s hx ih => simp [Finset.sum_insert hx, ih]
-
-theorem shift_eq_sum {m n : ℕ} (u v : Fin m) :
-    shift (n := n) u v = ∑ k : Fin n, (X (v, k) : Poly m n) •
-      (pderiv (u, k) : Derivation ℂ (Poly m n) (Poly m n)) := by
-  apply MvPolynomial.derivation_ext
-  rintro ⟨r, k⟩
-  simp only [shift_X, derivation_sum_apply, Derivation.smul_apply, pderiv_X]
-  by_cases h : r = u
-  · subst r
-    simp [Pi.single_apply, Prod.mk.injEq]
-  · simp [Prod.mk.injEq, h]
-
-theorem shift_commutator {m n : ℕ} {u v : Fin m} (_hne : u ≠ v) :
-    ⁅shift (n := n) v u, shift (n := n) u v⁆ =
-      shift (n := n) u u - shift (n := n) v v := by
-  apply MvPolynomial.derivation_ext
-  rintro ⟨r, k⟩
-  simp only [Derivation.commutator_apply, Derivation.sub_apply, shift_X]
-  split_ifs <;> simp_all [sub_eq_add_neg]
-
-def rowWeight {m n : ℕ} (u : Fin m) (ij : Fin m × Fin n) : ℕ :=
-  if ij.1 = u then 1 else 0
-
-theorem euler {m n : ℕ} (u : Fin m) {f : Poly m n} {p : ℕ}
-    (hf : f.IsWeightedHomogeneous (rowWeight u) p) :
-    shift (n := n) u u f = (p : ℂ) • f := by
-  have h := hf.sum_weight_X_mul_pderiv
-  rw [shift_eq_sum]
-  simp only [derivation_sum_apply, Derivation.smul_apply, smul_eq_mul]
-  have hs : (∑ i : Fin m × Fin n, rowWeight u i • (X i * pderiv i f)) =
-      ∑ k : Fin n, X (u, k) * pderiv (u, k) f := by
-    rw [Fintype.sum_prod_type, Finset.sum_comm]
-    simp [rowWeight]
-  rw [hs] at h
-  simpa only [Nat.cast_smul_eq_nsmul] using h
-
-theorem shift_lowers {m n : ℕ} {u v : Fin m} (huv : u ≠ v)
-    {f : Poly m n} {p : ℕ}
-    (hf : f.IsWeightedHomogeneous (rowWeight u) (p + 1)) :
-    (shift u v f).IsWeightedHomogeneous (rowWeight u) p := by
-  rw [shift_eq_sum, derivation_sum_apply]
-  apply MvPolynomial.IsWeightedHomogeneous.sum
-  intro k _
-  simp only [Derivation.smul_apply, smul_eq_mul]
-  have hd : (pderiv (u, k) f).IsWeightedHomogeneous (rowWeight u) p :=
-    hf.pderiv (by simp [rowWeight])
-  have hx : (X (v, k) : Poly m n).IsWeightedHomogeneous (rowWeight u) 0 := by
-    simpa [rowWeight, Ne.symm huv] using
-      (MvPolynomial.isWeightedHomogeneous_X (R := ℂ) (rowWeight u) (v, k))
-  simpa using hx.mul hd
-
-theorem pderiv_zero_degree {m n : ℕ} {u : Fin m} {f : Poly m n}
-    (hf : f.IsWeightedHomogeneous (rowWeight u) 0) (k : Fin n) :
-    pderiv (u, k) f = 0 := by
-  ext d
-  rw [coeff_pderiv, AddMonoidAlgebra.coeff_zero]
-  have hc : f.coeff (d + Finsupp.single (u, k) 1) = 0 :=
-    hf.coeff_eq_zero _ (by simp [map_add, Finsupp.weight_single, rowWeight])
-  simp [hc]
-
-theorem shift_zero_degree {m n : ℕ} {u v : Fin m} {f : Poly m n}
-    (hf : f.IsWeightedHomogeneous (rowWeight u) 0) : shift u v f = 0 := by
-  rw [shift_eq_sum, derivation_sum_apply]
-  simp [pderiv_zero_degree hf]
-
-theorem shift_nilpotent {m n : ℕ} {u v : Fin m} (huv : u ≠ v)
-    {f : Poly m n} {p : ℕ} (hf : f.IsWeightedHomogeneous (rowWeight u) p) :
-    ((shift u v).toLinearMap ^ (p + 1)) f = 0 := by
-  induction p generalizing f with
-  | zero => simpa using shift_zero_degree (v := v) hf
-  | succ p ih =>
-    rw [pow_succ, Module.End.mul_apply]
-    exact ih (shift_lowers huv hf)
-
-theorem shift_triple {m n : ℕ} {u v : Fin m} (huv : u ≠ v)
-    (hH : shift (n := n) v v - shift (n := n) u u ≠ 0) :
-    IsSl2Triple (shift (n := n) v v - shift (n := n) u u)
-      (shift (n := n) u v) (shift (n := n) v u) where
-  h_ne_zero := hH
-  lie_e_f := shift_commutator (Ne.symm huv)
-  lie_h_e_nsmul := by
-    have hvu := Ne.symm huv
-    apply MvPolynomial.derivation_ext
-    rintro ⟨r, k⟩
-    simp only [Derivation.commutator_apply, Derivation.sub_apply, shift_X, map_sub,
-      two_nsmul, Derivation.add_apply]
-    split_ifs <;> simp_all [sub_eq_add_neg]
-  lie_h_f_nsmul := by
-    have hvu := Ne.symm huv
-    apply MvPolynomial.derivation_ext
-    rintro ⟨r, k⟩
-    simp only [Derivation.commutator_apply, Derivation.sub_apply, shift_X, map_sub,
-      two_nsmul, Derivation.add_apply, Derivation.neg_apply]
-    split_ifs <;> simp_all [sub_eq_add_neg]
-
-theorem shift_kernel_zero {m n : ℕ} {u v : Fin m} (huv : u ≠ v)
-    {f : Poly m n} {p q : ℕ}
-    (hp : f.IsWeightedHomogeneous (rowWeight u) p)
-    (hq : f.IsWeightedHomogeneous (rowWeight v) q)
-    (hpq : q < p) (hDf : shift u v f = 0) : f = 0 := by
-  by_contra hne
-  have hweight : (shift (n := n) v v - shift (n := n) u u) f =
-      ((q : ℂ) - p) • f := by
-    rw [Derivation.sub_apply, euler v hq, euler u hp, sub_smul]
-  have hH : shift (n := n) v v - shift (n := n) u u ≠ 0 := by
-    intro hzero
-    have hscalar : (q : ℂ) - p ≠ 0 := by
-      rw [sub_ne_zero]
-      exact_mod_cast (Nat.ne_of_lt hpq)
-    rw [hzero, Derivation.zero_apply] at hweight
-    exact (smul_ne_zero hscalar hne) hweight.symm
-  let t := shift_triple huv hH
-  have hprimitive : t.HasPrimitiveVectorWith f ((q : ℂ) - p) :=
-    { ne_zero := hne, lie_h := hweight, lie_e := hDf }
-  let E := LieModule.toEnd ℂ (Derivation ℂ (Poly m n) (Poly m n)) (Poly m n)
-    (shift (n := n) v u)
-  have hnil : ∃ k : ℕ, (E ^ k) f = 0 :=
-    ⟨q + 1, shift_nilpotent (Ne.symm huv) hq⟩
-  obtain ⟨k, hk₁, hk₂⟩ := Nat.exists_not_and_succ_of_not_zero_of_exists hne hnil
-  have hrec := hprimitive.lie_e_pow_succ_toEnd_f k
-  change shift u v ((E ^ (k + 1)) f) =
-    ((k + 1 : ℂ) * ((q : ℂ) - p - k)) • ((E ^ k) f) at hrec
-  rw [hk₂, map_zero, eq_comm, smul_eq_zero_iff_left hk₁, mul_eq_zero,
-    sub_eq_zero] at hrec
-  have heq : (q : ℂ) - p = k := hrec.resolve_left (Nat.cast_add_one_ne_zero k)
-  have heq' : q = k + p := by exact_mod_cast (sub_eq_iff_eq_add.mp heq)
-  omega
-
-theorem shift_injOn {m n : ℕ} {u v : Fin m} (huv : u ≠ v) {p q : ℕ}
-    (hpq : q < p) :
-    Set.InjOn (shift (n := n) u v)
-      {f | f.IsWeightedHomogeneous (rowWeight u) p ∧
-        f.IsWeightedHomogeneous (rowWeight v) q} := by
-  intro f hf g hg hfg
-  apply sub_eq_zero.mp
-  exact shift_kernel_zero huv (hf.1.sub hg.1) (hf.2.sub hg.2) hpq
-    (by rw [map_sub, hfg, sub_self])
-
-theorem nilpotence_raise {M : Type*} [AddCommGroup M] [Module ℂ M]
-    (D E H : Module.End ℂ M)
-    (hDE : ∀ x, D (E x) = E (D x) - H x)
-    (hHD : ∀ x, H (D x) = D (H x) - (2 : ℂ) • D x)
-    {x : M} {μ : ℂ} (hx : H x = μ • x) {L : ℕ}
-    (hL : (D ^ L) x = 0) : (D ^ (L + 1)) (E x) = 0 := by
-  induction L generalizing x μ with
-  | zero =>
-    simp only [pow_zero, Module.End.one_apply] at hL
-    simp [hL]
-  | succ L ih =>
-    have hxD : H (D x) = (μ - 2) • D x := by
-      rw [hHD, hx, map_smul, sub_smul]
-    have hLD : (D ^ L) (D x) = 0 := by
-      simpa only [pow_succ, Module.End.mul_apply] using hL
-    have hED := ih hxD hLD
-    rw [pow_succ, Module.End.mul_apply, hDE, map_sub, hx, map_smul]
-    change (D ^ (L + 1)) (E (D x)) - μ • (D ^ (L + 1)) x = 0
-    rw [hED, hL, smul_zero, sub_self]
-
-theorem shift_power_kernel_zero {m n : ℕ} {u v : Fin m} (huv : u ≠ v)
-    {f : Poly m n} {p q L : ℕ}
-    (hp : f.IsWeightedHomogeneous (rowWeight u) p)
-    (hq : f.IsWeightedHomogeneous (rowWeight v) q)
-    (hpq : q + L ≤ p) (hDf : ((shift u v).toLinearMap ^ L) f = 0) : f = 0 := by
-  by_cases hL : L = 0
-  · simpa [hL] using hDf
-  by_contra hne
-  have hpq' : q < p := by omega
-  have hweight : (shift (n := n) u u - shift (n := n) v v) f =
-      ((p - q : ℕ) : ℂ) • f := by
-    rw [Derivation.sub_apply, euler u hp, euler v hq,
-      Nat.cast_sub (Nat.le_of_lt hpq'), sub_smul]
-  have hH : shift (n := n) u u - shift (n := n) v v ≠ 0 := by
-    intro hzero
-    have hscalar : ((p - q : ℕ) : ℂ) ≠ 0 := by
-      exact_mod_cast (Nat.sub_ne_zero_of_lt hpq')
-    rw [hzero, Derivation.zero_apply] at hweight
-    exact (smul_ne_zero hscalar hne) hweight.symm
-  let t := shift_triple (Ne.symm huv) hH
-  let E := (shift (n := n) v u).toLinearMap
-  let D := (shift (n := n) u v).toLinearMap
-  let H := (shift (n := n) u u - shift (n := n) v v).toLinearMap
-  have hDE (g : Poly m n) : D (E g) = E (D g) - H g := by
-    have hh := DFunLike.congr_fun t.lie_e_f g
-    change E (D g) - D (E g) = H g at hh
-    apply eq_sub_iff_add_eq.mpr
-    rw [← hh]
-    abel
-  have hHD (g : Poly m n) : H (D g) = D (H g) - (2 : ℂ) • D g := by
-    have hh := DFunLike.congr_fun t.lie_h_f_nsmul g
-    simp only [Derivation.commutator_apply, Derivation.neg_apply, two_nsmul,
-      Derivation.add_apply] at hh
-    change H (D g) - D (H g) = -(D g + D g) at hh
-    rw [sub_eq_iff_eq_add] at hh
-    rw [hh]
-    module
-  have hnil : ∃ k : ℕ, (E ^ k) f = 0 :=
-    ⟨q + 1, shift_nilpotent (Ne.symm huv) hq⟩
-  obtain ⟨k, hk₁, hk₂⟩ := Nat.exists_not_and_succ_of_not_zero_of_exists hne hnil
-  have hweights (j : ℕ) : H ((E ^ j) f) =
-      (((p - q : ℕ) : ℂ) + 2 * j) • ((E ^ j) f) :=
-    t.lie_h_pow_toEnd_e hweight j
-  have hkernel (j : ℕ) : (D ^ (L + j)) ((E ^ j) f) = 0 := by
-    induction j with
-    | zero => simpa using hDf
-    | succ j ih =>
-      have h := nilpotence_raise D E H hDE hHD (hweights j) ih
-      simpa only [pow_succ', Module.End.mul_apply, ← Nat.add_assoc] using h
-  have hprimitive : t.HasPrimitiveVectorWith ((E ^ k) f)
-      (((p - q + 2 * k : ℕ)) : ℂ) := by
-    refine ⟨hk₁, ?_, ?_⟩
-    · change H ((E ^ k) f) = (((p - q + 2 * k : ℕ)) : ℂ) • ((E ^ k) f)
-      simpa only [Nat.cast_add, Nat.cast_mul, Nat.cast_ofNat] using hweights k
-    · change E ((E ^ k) f) = 0
-      simpa only [pow_succ', Module.End.mul_apply] using hk₂
-  have hbound : L + k ≤ p - q + 2 * k := by omega
-  exact (hprimitive.pow_toEnd_f_ne_zero_of_eq_nat rfl hbound) (hkernel k)
-
-theorem shift_raises {m n : ℕ} {u v : Fin m} (huv : u ≠ v)
-    {f : Poly m n} {q : ℕ}
-    (hf : f.IsWeightedHomogeneous (rowWeight v) q) :
-    (shift u v f).IsWeightedHomogeneous (rowWeight v) (q + 1) := by
-  rw [shift_eq_sum, derivation_sum_apply]
-  apply MvPolynomial.IsWeightedHomogeneous.sum
-  intro k _
-  simp only [Derivation.smul_apply, smul_eq_mul]
-  have hd : (pderiv (u, k) f).IsWeightedHomogeneous (rowWeight v) q :=
-    hf.pderiv (by simp [rowWeight, huv])
-  have hx : (X (v, k) : Poly m n).IsWeightedHomogeneous (rowWeight v) 1 := by
-    simpa [rowWeight] using
-      (MvPolynomial.isWeightedHomogeneous_X (R := ℂ) (rowWeight v) (v, k))
-  simpa [add_comm] using hx.mul hd
-
-theorem shift_spectator {m n : ℕ} {u v w : Fin m}
-    (huw : u ≠ w) (hvw : v ≠ w) {f : Poly m n} {q : ℕ}
-    (hf : f.IsWeightedHomogeneous (rowWeight w) q) :
-    (shift u v f).IsWeightedHomogeneous (rowWeight w) q := by
-  rw [shift_eq_sum, derivation_sum_apply]
-  apply MvPolynomial.IsWeightedHomogeneous.sum
-  intro k _
-  simp only [Derivation.smul_apply, smul_eq_mul]
-  have hd : (pderiv (u, k) f).IsWeightedHomogeneous (rowWeight w) q :=
-    hf.pderiv (by simp [rowWeight, huw])
-  have hx : (X (v, k) : Poly m n).IsWeightedHomogeneous (rowWeight w) 0 := by
-    simpa [rowWeight, hvw] using
-      (MvPolynomial.isWeightedHomogeneous_X (R := ℂ) (rowWeight w) (v, k))
-  simpa using hx.mul hd
-
-theorem shift_pow_raises {m n : ℕ} {u v : Fin m} (huv : u ≠ v)
-    {f : Poly m n} {q : ℕ}
-    (hf : f.IsWeightedHomogeneous (rowWeight v) q) (L : ℕ) :
-    (((shift u v).toLinearMap ^ L) f).IsWeightedHomogeneous (rowWeight v) (q + L) := by
-  induction L with
-  | zero => simpa using hf
-  | succ L ih =>
-    rw [pow_succ', Module.End.mul_apply]
-    simpa [Nat.add_assoc] using shift_raises huv ih
-
-theorem shift_pow_spectator {m n : ℕ} {u v w : Fin m}
-    (huw : u ≠ w) (hvw : v ≠ w) {f : Poly m n} {q : ℕ}
-    (hf : f.IsWeightedHomogeneous (rowWeight w) q) (L : ℕ) :
-    (((shift u v).toLinearMap ^ L) f).IsWeightedHomogeneous (rowWeight w) q := by
-  induction L with
-  | zero => simpa using hf
-  | succ L ih =>
-    rw [pow_succ', Module.End.mul_apply]
-    exact shift_spectator huw hvw ih
-
-def shiftSequence {m n : ℕ} (v : Fin m) (L : ℕ) (us : List (Fin m)) :
-    Poly m n →ₗ[ℂ] Poly m n :=
-  match us with
-  | [] => LinearMap.id
-  | u :: us => (shiftSequence v L us).comp ((shift u v).toLinearMap ^ L)
-
-theorem shiftSequence_kernel_zero {m n : ℕ} {us : List (Fin m)} {v : Fin m}
-    (hnd : us.Nodup) (hv : v ∉ us) {L j q : ℕ} {f : Poly m n}
-    (hp : ∀ u ∈ us, f.IsWeightedHomogeneous (rowWeight u) j)
-    (hq : f.IsWeightedHomogeneous (rowWeight v) q)
-    (hbound : q + L * us.length ≤ j)
-    (hz : shiftSequence v L us f = 0) : f = 0 := by
-  induction us generalizing f q with
-  | nil => simpa [shiftSequence] using hz
-  | cons u us ih =>
-    have huv : u ≠ v := by
-      intro h
-      exact hv (by simp [h])
-    have hnd' := List.nodup_cons.mp hnd
-    have hv' : v ∉ us := fun h => hv (List.mem_cons_of_mem u h)
-    have hs : ((shift u v).toLinearMap ^ L) f = 0 := by
-      apply ih hnd'.2 hv' (q := q + L)
-      · intro w hw
-        apply shift_pow_spectator (u := u) (v := v) (w := w) (fun h => hnd'.1 (h.symm ▸ hw))
-          (fun h => hv' (h.symm ▸ hw)) (hp w (List.mem_cons_of_mem u hw)) L
-      · exact shift_pow_raises huv hq L
-      · simp only [List.length_cons, Nat.mul_succ] at hbound
-        omega
-      · exact hz
-    apply shift_power_kernel_zero huv (hp u (by simp)) hq _ hs
-    simp only [List.length_cons, Nat.mul_succ] at hbound
-    omega
-
-end Foulkes.Differential
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.Differential
+open MvPolynomial
+
+abbrev Poly (m n : ℕ) := MvPolynomial (Fin m × Fin n) ℂ
+
+def shift {m n : ℕ} (u v : Fin m) : Derivation ℂ (Poly m n) (Poly m n) :=
+  MvPolynomial.mkDerivation ℂ fun ij => if ij.1 = u then X (v, ij.2) else 0
+
+@[simp] theorem shift_X {m n : ℕ} (u v : Fin m) (ij : Fin m × Fin n) :
+    shift (n := n) u v (X ij) = if ij.1 = u then X (v, ij.2) else 0 := by
+  simp [shift]
+
+@[simp] theorem derivation_sum_apply {m n : ℕ} {ι : Type*}
+    (s : Finset ι) (D : ι → Derivation ℂ (Poly m n) (Poly m n)) (f : Poly m n) :
+    (∑ k ∈ s, D k) f = ∑ k ∈ s, D k f := by
+  classical
+  induction s using Finset.induction_on with
+  | empty => simp
+  | insert x s hx ih => simp [Finset.sum_insert hx, ih]
+
+theorem shift_eq_sum {m n : ℕ} (u v : Fin m) :
+    shift (n := n) u v = ∑ k : Fin n, (X (v, k) : Poly m n) •
+      (pderiv (u, k) : Derivation ℂ (Poly m n) (Poly m n)) := by
+  apply MvPolynomial.derivation_ext
+  rintro ⟨r, k⟩
+  simp only [shift_X, derivation_sum_apply, Derivation.smul_apply, pderiv_X]
+  by_cases h : r = u
+  · subst r
+    simp [Pi.single_apply, Prod.mk.injEq]
+  · simp [Prod.mk.injEq, h]
+
+theorem shift_commutator {m n : ℕ} {u v : Fin m} (_hne : u ≠ v) :
+    ⁅shift (n := n) v u, shift (n := n) u v⁆ =
+      shift (n := n) u u - shift (n := n) v v := by
+  apply MvPolynomial.derivation_ext
+  rintro ⟨r, k⟩
+  simp only [Derivation.commutator_apply, Derivation.sub_apply, shift_X]
+  split_ifs <;> simp_all [sub_eq_add_neg]
+
+def rowWeight {m n : ℕ} (u : Fin m) (ij : Fin m × Fin n) : ℕ :=
+  if ij.1 = u then 1 else 0
+
+theorem euler {m n : ℕ} (u : Fin m) {f : Poly m n} {p : ℕ}
+    (hf : f.IsWeightedHomogeneous (rowWeight u) p) :
+    shift (n := n) u u f = (p : ℂ) • f := by
+  have h := hf.sum_weight_X_mul_pderiv
+  rw [shift_eq_sum]
+  simp only [derivation_sum_apply, Derivation.smul_apply, smul_eq_mul]
+  have hs : (∑ i : Fin m × Fin n, rowWeight u i • (X i * pderiv i f)) =
+      ∑ k : Fin n, X (u, k) * pderiv (u, k) f := by
+    rw [Fintype.sum_prod_type, Finset.sum_comm]
+    simp [rowWeight]
+  rw [hs] at h
+  simpa only [Nat.cast_smul_eq_nsmul] using h
+
+theorem shift_lowers {m n : ℕ} {u v : Fin m} (huv : u ≠ v)
+    {f : Poly m n} {p : ℕ}
+    (hf : f.IsWeightedHomogeneous (rowWeight u) (p + 1)) :
+    (shift u v f).IsWeightedHomogeneous (rowWeight u) p := by
+  rw [shift_eq_sum, derivation_sum_apply]
+  apply MvPolynomial.IsWeightedHomogeneous.sum
+  intro k _
+  simp only [Derivation.smul_apply, smul_eq_mul]
+  have hd : (pderiv (u, k) f).IsWeightedHomogeneous (rowWeight u) p :=
+    hf.pderiv (by simp [rowWeight])
+  have hx : (X (v, k) : Poly m n).IsWeightedHomogeneous (rowWeight u) 0 := by
+    simpa [rowWeight, Ne.symm huv] using
+      (MvPolynomial.isWeightedHomogeneous_X (R := ℂ) (rowWeight u) (v, k))
+  simpa using hx.mul hd
+
+theorem pderiv_zero_degree {m n : ℕ} {u : Fin m} {f : Poly m n}
+    (hf : f.IsWeightedHomogeneous (rowWeight u) 0) (k : Fin n) :
+    pderiv (u, k) f = 0 := by
+  ext d
+  rw [coeff_pderiv, AddMonoidAlgebra.coeff_zero]
+  have hc : f.coeff (d + Finsupp.single (u, k) 1) = 0 :=
+    hf.coeff_eq_zero _ (by simp [map_add, Finsupp.weight_single, rowWeight])
+  simp [hc]
+
+theorem shift_zero_degree {m n : ℕ} {u v : Fin m} {f : Poly m n}
+    (hf : f.IsWeightedHomogeneous (rowWeight u) 0) : shift u v f = 0 := by
+  rw [shift_eq_sum, derivation_sum_apply]
+  simp [pderiv_zero_degree hf]
+
+theorem shift_nilpotent {m n : ℕ} {u v : Fin m} (huv : u ≠ v)
+    {f : Poly m n} {p : ℕ} (hf : f.IsWeightedHomogeneous (rowWeight u) p) :
+    ((shift u v).toLinearMap ^ (p + 1)) f = 0 := by
+  induction p generalizing f with
+  | zero => simpa using shift_zero_degree (v := v) hf
+  | succ p ih =>
+    rw [pow_succ, Module.End.mul_apply]
+    exact ih (shift_lowers huv hf)
+
+theorem shift_triple {m n : ℕ} {u v : Fin m} (huv : u ≠ v)
+    (hH : shift (n := n) v v - shift (n := n) u u ≠ 0) :
+    IsSl2Triple (shift (n := n) v v - shift (n := n) u u)
+      (shift (n := n) u v) (shift (n := n) v u) where
+  h_ne_zero := hH
+  lie_e_f := shift_commutator (Ne.symm huv)
+  lie_h_e_nsmul := by
+    have hvu := Ne.symm huv
+    apply MvPolynomial.derivation_ext
+    rintro ⟨r, k⟩
+    simp only [Derivation.commutator_apply, Derivation.sub_apply, shift_X, map_sub,
+      two_nsmul, Derivation.add_apply]
+    split_ifs <;> simp_all [sub_eq_add_neg]
+  lie_h_f_nsmul := by
+    have hvu := Ne.symm huv
+    apply MvPolynomial.derivation_ext
+    rintro ⟨r, k⟩
+    simp only [Derivation.commutator_apply, Derivation.sub_apply, shift_X, map_sub,
+      two_nsmul, Derivation.add_apply, Derivation.neg_apply]
+    split_ifs <;> simp_all [sub_eq_add_neg]
+
+theorem shift_kernel_zero {m n : ℕ} {u v : Fin m} (huv : u ≠ v)
+    {f : Poly m n} {p q : ℕ}
+    (hp : f.IsWeightedHomogeneous (rowWeight u) p)
+    (hq : f.IsWeightedHomogeneous (rowWeight v) q)
+    (hpq : q < p) (hDf : shift u v f = 0) : f = 0 := by
+  by_contra hne
+  have hweight : (shift (n := n) v v - shift (n := n) u u) f =
+      ((q : ℂ) - p) • f := by
+    rw [Derivation.sub_apply, euler v hq, euler u hp, sub_smul]
+  have hH : shift (n := n) v v - shift (n := n) u u ≠ 0 := by
+    intro hzero
+    have hscalar : (q : ℂ) - p ≠ 0 := by
+      rw [sub_ne_zero]
+      exact_mod_cast (Nat.ne_of_lt hpq)
+    rw [hzero, Derivation.zero_apply] at hweight
+    exact (smul_ne_zero hscalar hne) hweight.symm
+  let t := shift_triple huv hH
+  have hprimitive : t.HasPrimitiveVectorWith f ((q : ℂ) - p) :=
+    { ne_zero := hne, lie_h := hweight, lie_e := hDf }
+  let E := LieModule.toEnd ℂ (Derivation ℂ (Poly m n) (Poly m n)) (Poly m n)
+    (shift (n := n) v u)
+  have hnil : ∃ k : ℕ, (E ^ k) f = 0 :=
+    ⟨q + 1, shift_nilpotent (Ne.symm huv) hq⟩
+  obtain ⟨k, hk₁, hk₂⟩ := Nat.exists_not_and_succ_of_not_zero_of_exists hne hnil
+  have hrec := hprimitive.lie_e_pow_succ_toEnd_f k
+  change shift u v ((E ^ (k + 1)) f) =
+    ((k + 1 : ℂ) * ((q : ℂ) - p - k)) • ((E ^ k) f) at hrec
+  rw [hk₂, map_zero, eq_comm, smul_eq_zero_iff_left hk₁, mul_eq_zero,
+    sub_eq_zero] at hrec
+  have heq : (q : ℂ) - p = k := hrec.resolve_left (Nat.cast_add_one_ne_zero k)
+  have heq' : q = k + p := by exact_mod_cast (sub_eq_iff_eq_add.mp heq)
+  omega
+
+theorem shift_injOn {m n : ℕ} {u v : Fin m} (huv : u ≠ v) {p q : ℕ}
+    (hpq : q < p) :
+    Set.InjOn (shift (n := n) u v)
+      {f | f.IsWeightedHomogeneous (rowWeight u) p ∧
+        f.IsWeightedHomogeneous (rowWeight v) q} := by
+  intro f hf g hg hfg
+  apply sub_eq_zero.mp
+  exact shift_kernel_zero huv (hf.1.sub hg.1) (hf.2.sub hg.2) hpq
+    (by rw [map_sub, hfg, sub_self])
+
+theorem nilpotence_raise {M : Type*} [AddCommGroup M] [Module ℂ M]
+    (D E H : Module.End ℂ M)
+    (hDE : ∀ x, D (E x) = E (D x) - H x)
+    (hHD : ∀ x, H (D x) = D (H x) - (2 : ℂ) • D x)
+    {x : M} {μ : ℂ} (hx : H x = μ • x) {L : ℕ}
+    (hL : (D ^ L) x = 0) : (D ^ (L + 1)) (E x) = 0 := by
+  induction L generalizing x μ with
+  | zero =>
+    simp only [pow_zero, Module.End.one_apply] at hL
+    simp [hL]
+  | succ L ih =>
+    have hxD : H (D x) = (μ - 2) • D x := by
+      rw [hHD, hx, map_smul, sub_smul]
+    have hLD : (D ^ L) (D x) = 0 := by
+      simpa only [pow_succ, Module.End.mul_apply] using hL
+    have hED := ih hxD hLD
+    rw [pow_succ, Module.End.mul_apply, hDE, map_sub, hx, map_smul]
+    change (D ^ (L + 1)) (E (D x)) - μ • (D ^ (L + 1)) x = 0
+    rw [hED, hL, smul_zero, sub_self]
+
+theorem shift_power_kernel_zero {m n : ℕ} {u v : Fin m} (huv : u ≠ v)
+    {f : Poly m n} {p q L : ℕ}
+    (hp : f.IsWeightedHomogeneous (rowWeight u) p)
+    (hq : f.IsWeightedHomogeneous (rowWeight v) q)
+    (hpq : q + L ≤ p) (hDf : ((shift u v).toLinearMap ^ L) f = 0) : f = 0 := by
+  by_cases hL : L = 0
+  · simpa [hL] using hDf
+  by_contra hne
+  have hpq' : q < p := by omega
+  have hweight : (shift (n := n) u u - shift (n := n) v v) f =
+      ((p - q : ℕ) : ℂ) • f := by
+    rw [Derivation.sub_apply, euler u hp, euler v hq,
+      Nat.cast_sub (Nat.le_of_lt hpq'), sub_smul]
+  have hH : shift (n := n) u u - shift (n := n) v v ≠ 0 := by
+    intro hzero
+    have hscalar : ((p - q : ℕ) : ℂ) ≠ 0 := by
+      exact_mod_cast (Nat.sub_ne_zero_of_lt hpq')
+    rw [hzero, Derivation.zero_apply] at hweight
+    exact (smul_ne_zero hscalar hne) hweight.symm
+  let t := shift_triple (Ne.symm huv) hH
+  let E := (shift (n := n) v u).toLinearMap
+  let D := (shift (n := n) u v).toLinearMap
+  let H := (shift (n := n) u u - shift (n := n) v v).toLinearMap
+  have hDE (g : Poly m n) : D (E g) = E (D g) - H g := by
+    have hh := DFunLike.congr_fun t.lie_e_f g
+    change E (D g) - D (E g) = H g at hh
+    apply eq_sub_iff_add_eq.mpr
+    rw [← hh]
+    abel
+  have hHD (g : Poly m n) : H (D g) = D (H g) - (2 : ℂ) • D g := by
+    have hh := DFunLike.congr_fun t.lie_h_f_nsmul g
+    simp only [Derivation.commutator_apply, Derivation.neg_apply, two_nsmul,
+      Derivation.add_apply] at hh
+    change H (D g) - D (H g) = -(D g + D g) at hh
+    rw [sub_eq_iff_eq_add] at hh
+    rw [hh]
+    module
+  have hnil : ∃ k : ℕ, (E ^ k) f = 0 :=
+    ⟨q + 1, shift_nilpotent (Ne.symm huv) hq⟩
+  obtain ⟨k, hk₁, hk₂⟩ := Nat.exists_not_and_succ_of_not_zero_of_exists hne hnil
+  have hweights (j : ℕ) : H ((E ^ j) f) =
+      (((p - q : ℕ) : ℂ) + 2 * j) • ((E ^ j) f) :=
+    t.lie_h_pow_toEnd_e hweight j
+  have hkernel (j : ℕ) : (D ^ (L + j)) ((E ^ j) f) = 0 := by
+    induction j with
+    | zero => simpa using hDf
+    | succ j ih =>
+      have h := nilpotence_raise D E H hDE hHD (hweights j) ih
+      simpa only [pow_succ', Module.End.mul_apply, ← Nat.add_assoc] using h
+  have hprimitive : t.HasPrimitiveVectorWith ((E ^ k) f)
+      (((p - q + 2 * k : ℕ)) : ℂ) := by
+    refine ⟨hk₁, ?_, ?_⟩
+    · change H ((E ^ k) f) = (((p - q + 2 * k : ℕ)) : ℂ) • ((E ^ k) f)
+      simpa only [Nat.cast_add, Nat.cast_mul, Nat.cast_ofNat] using hweights k
+    · change E ((E ^ k) f) = 0
+      simpa only [pow_succ', Module.End.mul_apply] using hk₂
+  have hbound : L + k ≤ p - q + 2 * k := by omega
+  exact (hprimitive.pow_toEnd_f_ne_zero_of_eq_nat rfl hbound) (hkernel k)
+
+theorem shift_raises {m n : ℕ} {u v : Fin m} (huv : u ≠ v)
+    {f : Poly m n} {q : ℕ}
+    (hf : f.IsWeightedHomogeneous (rowWeight v) q) :
+    (shift u v f).IsWeightedHomogeneous (rowWeight v) (q + 1) := by
+  rw [shift_eq_sum, derivation_sum_apply]
+  apply MvPolynomial.IsWeightedHomogeneous.sum
+  intro k _
+  simp only [Derivation.smul_apply, smul_eq_mul]
+  have hd : (pderiv (u, k) f).IsWeightedHomogeneous (rowWeight v) q :=
+    hf.pderiv (by simp [rowWeight, huv])
+  have hx : (X (v, k) : Poly m n).IsWeightedHomogeneous (rowWeight v) 1 := by
+    simpa [rowWeight] using
+      (MvPolynomial.isWeightedHomogeneous_X (R := ℂ) (rowWeight v) (v, k))
+  simpa [add_comm] using hx.mul hd
+
+theorem shift_spectator {m n : ℕ} {u v w : Fin m}
+    (huw : u ≠ w) (hvw : v ≠ w) {f : Poly m n} {q : ℕ}
+    (hf : f.IsWeightedHomogeneous (rowWeight w) q) :
+    (shift u v f).IsWeightedHomogeneous (rowWeight w) q := by
+  rw [shift_eq_sum, derivation_sum_apply]
+  apply MvPolynomial.IsWeightedHomogeneous.sum
+  intro k _
+  simp only [Derivation.smul_apply, smul_eq_mul]
+  have hd : (pderiv (u, k) f).IsWeightedHomogeneous (rowWeight w) q :=
+    hf.pderiv (by simp [rowWeight, huw])
+  have hx : (X (v, k) : Poly m n).IsWeightedHomogeneous (rowWeight w) 0 := by
+    simpa [rowWeight, hvw] using
+      (MvPolynomial.isWeightedHomogeneous_X (R := ℂ) (rowWeight w) (v, k))
+  simpa using hx.mul hd
+
+theorem shift_pow_raises {m n : ℕ} {u v : Fin m} (huv : u ≠ v)
+    {f : Poly m n} {q : ℕ}
+    (hf : f.IsWeightedHomogeneous (rowWeight v) q) (L : ℕ) :
+    (((shift u v).toLinearMap ^ L) f).IsWeightedHomogeneous (rowWeight v) (q + L) := by
+  induction L with
+  | zero => simpa using hf
+  | succ L ih =>
+    rw [pow_succ', Module.End.mul_apply]
+    simpa [Nat.add_assoc] using shift_raises huv ih
+
+theorem shift_pow_spectator {m n : ℕ} {u v w : Fin m}
+    (huw : u ≠ w) (hvw : v ≠ w) {f : Poly m n} {q : ℕ}
+    (hf : f.IsWeightedHomogeneous (rowWeight w) q) (L : ℕ) :
+    (((shift u v).toLinearMap ^ L) f).IsWeightedHomogeneous (rowWeight w) q := by
+  induction L with
+  | zero => simpa using hf
+  | succ L ih =>
+    rw [pow_succ', Module.End.mul_apply]
+    exact shift_spectator huw hvw ih
+
+def shiftSequence {m n : ℕ} (v : Fin m) (L : ℕ) (us : List (Fin m)) :
+    Poly m n →ₗ[ℂ] Poly m n :=
+  match us with
+  | [] => LinearMap.id
+  | u :: us => (shiftSequence v L us).comp ((shift u v).toLinearMap ^ L)
+
+theorem shiftSequence_kernel_zero {m n : ℕ} {us : List (Fin m)} {v : Fin m}
+    (hnd : us.Nodup) (hv : v ∉ us) {L j q : ℕ} {f : Poly m n}
+    (hp : ∀ u ∈ us, f.IsWeightedHomogeneous (rowWeight u) j)
+    (hq : f.IsWeightedHomogeneous (rowWeight v) q)
+    (hbound : q + L * us.length ≤ j)
+    (hz : shiftSequence v L us f = 0) : f = 0 := by
+  induction us generalizing f q with
+  | nil => simpa [shiftSequence] using hz
+  | cons u us ih =>
+    have huv : u ≠ v := by
+      intro h
+      exact hv (by simp [h])
+    have hnd' := List.nodup_cons.mp hnd
+    have hv' : v ∉ us := fun h => hv (List.mem_cons_of_mem u h)
+    have hs : ((shift u v).toLinearMap ^ L) f = 0 := by
+      apply ih hnd'.2 hv' (q := q + L)
+      · intro w hw
+        apply shift_pow_spectator (u := u) (v := v) (w := w) (fun h => hnd'.1 (h.symm ▸ hw))
+          (fun h => hv' (h.symm ▸ hw)) (hp w (List.mem_cons_of_mem u hw)) L
+      · exact shift_pow_raises huv hq L
+      · simp only [List.length_cons, Nat.mul_succ] at hbound
+        omega
+      · exact hz
+    apply shift_power_kernel_zero huv (hp u (by simp)) hq _ hs
+    simp only [List.length_cons, Nat.mul_succ] at hbound
+    omega
+
+end Foulkes.Differential
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/FastLookup.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/FastLookup.lean
index 37705eca435acc547db430a70eeea91980f0cf34..e076e43cb403e86093bf1f53515c3f7d9b2433e7 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/FastLookup.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/FastLookup.lean
@@ -1,124 +1,124 @@
-import Mathlib
-import OAI.RepresentationTheory.FoulkesSixth.HomogeneousPlethysm
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.FastLookup
-open MvPolynomial Foulkes.Strips Foulkes.Lookup
-
-lemma coeffZ_comp {n : ℕ} {p : MvPolynomial (Fin n) ℤ} (hp : Alternating p)
-    (v : Fin n → ℤ) (σ : Equiv.Perm (Fin n)) :
-    coeffZ (v ∘ σ) p = (Equiv.Perm.sign σ : ℤ) * coeffZ v p := by
-  classical
-  have hh : (∀ j, 0 ≤ v (σ j)) ↔ ∀ j, 0 ≤ v j :=
-    ⟨fun h j => by simpa using h (σ.symm j), fun h j => h (σ j)⟩
-  by_cases hn : ∀ j, 0 ≤ v j
-  · simp only [coeffZ, Function.comp_apply, ite_eq_left hn, ite_eq_left (hh.mpr hn)]
-    exact coeff_comp hp (fun j => (v j).toNat) σ
-  · simp only [coeffZ, Function.comp_apply, ite_eq_right hn, ite_eq_right (mt hh.mp hn), mul_zero]
-
-abbrev State (n : ℕ) := ℤ × (Fin n → ℤ)
-
-def compareStep {n : ℕ} (ij : Fin n × Fin n) (s : State n) : State n :=
-  if s.2 ij.1 < s.2 ij.2 then (-s.1, s.2 ∘ Equiv.swap ij.1 ij.2) else s
-
-lemma compareStep_sound {n : ℕ} {p : MvPolynomial (Fin n) ℤ} (hp : Alternating p)
-    (ij : Fin n × Fin n) (s : State n) :
-    (compareStep ij s).1 * coeffZ (compareStep ij s).2 p = s.1 * coeffZ s.2 p := by
-  unfold compareStep
-  split_ifs with h
-  · have hij : ij.1 ≠ ij.2 := fun heq => by simp [heq] at h
-    dsimp only
-    rw [coeffZ_comp hp, Equiv.Perm.sign_swap hij]
-    simp
-  · rfl
-
-def run {n : ℕ} (steps : List (Fin n × Fin n)) (s : State n) : State n :=
-  steps.foldl (fun s ij => compareStep ij s) s
-
-lemma run_sound {n : ℕ} {p : MvPolynomial (Fin n) ℤ} (hp : Alternating p)
-    (steps : List (Fin n × Fin n)) (s : State n) :
-    (run steps s).1 * coeffZ (run steps s).2 p = s.1 * coeffZ s.2 p := by
-  induction steps generalizing s with
-  | nil => rfl
-  | cons ij steps ih =>
-    exact (ih (compareStep ij s)).trans (compareStep_sound hp ij s)
-
-def network (n : ℕ) : List (Fin n × Fin n) :=
-  (List.finRange n).flatMap fun i =>
-    ((List.finRange n).filter (fun j => i < j)).map fun j => (i,j)
-
-def finish {n : ℕ} (f : (Fin n → ℕ) → ℤ) (s : State n) : Option ℤ :=
-  if ∀ j, 0 ≤ s.2 j then
-    if ∀ i j, i < j → (s.2 j).toNat < (s.2 i).toNat then
-      some (s.1 * f (fun j => (s.2 j).toNat - staircase n j))
-    else if ∃ i j, i ≠ j ∧ s.2 i = s.2 j then some 0 else none
-  else some 0
-
-lemma finish_sound {n : ℕ} (p : MvPolynomial (Fin n) ℤ)
-    (hp : ∀ σ : Equiv.Perm (Fin n), rename σ p = p)
-    (f : (Fin n → ℕ) → ℤ)
-    (hf : ∀ mu, Antitone mu → f mu = schurCoeff p mu)
-    (s : State n) (z : ℤ) (hz : finish f s = some z) :
-    z = s.1 * coeffZ s.2 (p * alternant (staircase n)) := by
-  classical
-  unfold finish at hz
-  split_ifs at hz with hn hs hr
-  · have hstrict : StrictAnti (fun j => (s.2 j).toNat) := hs
-    have ha := unshift_antitone hstrict
-    have he := shifted_unshift hstrict
-    have hval : f (fun j => (s.2 j).toNat - staircase n j) =
-        (p * alternant (staircase n)).coeff (expVector (fun j => (s.2 j).toNat)) := by
-      change f (unshift (fun j => (s.2 j).toNat)) = _
-      rw [hf _ ha]
-      unfold schurCoeff
-      rw [he]
-    have hz' := Option.some.inj hz
-    rw [← hz', hval]
-    simp only [coeffZ, ite_eq_left hn]
-  · have hz' := Option.some.inj hz
-    obtain ⟨i,j,hij,he⟩ := hr
-    have ha := alternating_mul_symmetric hp (alternant_alternating (staircase n))
-    have hc := coeff_repeated ha (fun j => (s.2 j).toNat) hij (congrArg Int.toNat he)
-    rw [← hz']
-    simp only [coeffZ, ite_eq_left hn, hc, mul_zero]
-  · have hz' := Option.some.inj hz
-    rw [← hz']
-    simp only [coeffZ, ite_eq_right hn, mul_zero]
-
-def checkedLookup {n : ℕ} (f : (Fin n → ℕ) → ℤ) (v : Fin n → ℤ) : Option ℤ :=
-  finish f (run (network n) (1,v))
-
-theorem checkedLookup_sound {n : ℕ} (p : MvPolynomial (Fin n) ℤ)
-    (hp : ∀ σ : Equiv.Perm (Fin n), rename σ p = p)
-    (f : (Fin n → ℕ) → ℤ)
-    (hf : ∀ mu, Antitone mu → f mu = schurCoeff p mu)
-    (v : Fin n → ℤ) (z : ℤ) (hz : checkedLookup f v = some z) :
-    z = signedLookup (schurCoeff p) v := by
-  rw [signedLookup_eq_coeffZ p hp]
-  have ha := alternating_mul_symmetric hp (alternant_alternating (staircase n))
-  calc
-    z = _ := finish_sound p hp f hf _ z hz
-    _ = _ := by simpa only [one_mul] using run_sound ha (network n) (1,v)
-
-end Foulkes.FastLookup
-
-
-noncomputable section
-
-namespace Foulkes.FastLookup
-
-example : checkedLookup (fun (_ : Fin 6 → ℕ) => (7 : ℤ)) ![1,8,5,3,2,10] = some (-7) := by
-  decide +kernel
-example : checkedLookup (fun (_ : Fin 6 → ℕ) => (7 : ℤ)) ![1,8,5,3,2,1] = some 0 := by
-  decide +kernel
-example : checkedLookup (fun (_ : Fin 6 → ℕ) => (7 : ℤ)) ![1,8,5,-3,2,10] = some 0 := by
-  decide +kernel
-end Foulkes.FastLookup
-
-end
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.FoulkesSixth.HomogeneousPlethysm
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.FastLookup
+open MvPolynomial Foulkes.Strips Foulkes.Lookup
+
+lemma coeffZ_comp {n : ℕ} {p : MvPolynomial (Fin n) ℤ} (hp : Alternating p)
+    (v : Fin n → ℤ) (σ : Equiv.Perm (Fin n)) :
+    coeffZ (v ∘ σ) p = (Equiv.Perm.sign σ : ℤ) * coeffZ v p := by
+  classical
+  have hh : (∀ j, 0 ≤ v (σ j)) ↔ ∀ j, 0 ≤ v j :=
+    ⟨fun h j => by simpa using h (σ.symm j), fun h j => h (σ j)⟩
+  by_cases hn : ∀ j, 0 ≤ v j
+  · simp only [coeffZ, Function.comp_apply, ite_eq_left hn, ite_eq_left (hh.mpr hn)]
+    exact coeff_comp hp (fun j => (v j).toNat) σ
+  · simp only [coeffZ, Function.comp_apply, ite_eq_right hn, ite_eq_right (mt hh.mp hn), mul_zero]
+
+abbrev State (n : ℕ) := ℤ × (Fin n → ℤ)
+
+def compareStep {n : ℕ} (ij : Fin n × Fin n) (s : State n) : State n :=
+  if s.2 ij.1 < s.2 ij.2 then (-s.1, s.2 ∘ Equiv.swap ij.1 ij.2) else s
+
+lemma compareStep_sound {n : ℕ} {p : MvPolynomial (Fin n) ℤ} (hp : Alternating p)
+    (ij : Fin n × Fin n) (s : State n) :
+    (compareStep ij s).1 * coeffZ (compareStep ij s).2 p = s.1 * coeffZ s.2 p := by
+  unfold compareStep
+  split_ifs with h
+  · have hij : ij.1 ≠ ij.2 := fun heq => by simp [heq] at h
+    dsimp only
+    rw [coeffZ_comp hp, Equiv.Perm.sign_swap hij]
+    simp
+  · rfl
+
+def run {n : ℕ} (steps : List (Fin n × Fin n)) (s : State n) : State n :=
+  steps.foldl (fun s ij => compareStep ij s) s
+
+lemma run_sound {n : ℕ} {p : MvPolynomial (Fin n) ℤ} (hp : Alternating p)
+    (steps : List (Fin n × Fin n)) (s : State n) :
+    (run steps s).1 * coeffZ (run steps s).2 p = s.1 * coeffZ s.2 p := by
+  induction steps generalizing s with
+  | nil => rfl
+  | cons ij steps ih =>
+    exact (ih (compareStep ij s)).trans (compareStep_sound hp ij s)
+
+def network (n : ℕ) : List (Fin n × Fin n) :=
+  (List.finRange n).flatMap fun i =>
+    ((List.finRange n).filter (fun j => i < j)).map fun j => (i,j)
+
+def finish {n : ℕ} (f : (Fin n → ℕ) → ℤ) (s : State n) : Option ℤ :=
+  if ∀ j, 0 ≤ s.2 j then
+    if ∀ i j, i < j → (s.2 j).toNat < (s.2 i).toNat then
+      some (s.1 * f (fun j => (s.2 j).toNat - staircase n j))
+    else if ∃ i j, i ≠ j ∧ s.2 i = s.2 j then some 0 else none
+  else some 0
+
+lemma finish_sound {n : ℕ} (p : MvPolynomial (Fin n) ℤ)
+    (hp : ∀ σ : Equiv.Perm (Fin n), rename σ p = p)
+    (f : (Fin n → ℕ) → ℤ)
+    (hf : ∀ mu, Antitone mu → f mu = schurCoeff p mu)
+    (s : State n) (z : ℤ) (hz : finish f s = some z) :
+    z = s.1 * coeffZ s.2 (p * alternant (staircase n)) := by
+  classical
+  unfold finish at hz
+  split_ifs at hz with hn hs hr
+  · have hstrict : StrictAnti (fun j => (s.2 j).toNat) := hs
+    have ha := unshift_antitone hstrict
+    have he := shifted_unshift hstrict
+    have hval : f (fun j => (s.2 j).toNat - staircase n j) =
+        (p * alternant (staircase n)).coeff (expVector (fun j => (s.2 j).toNat)) := by
+      change f (unshift (fun j => (s.2 j).toNat)) = _
+      rw [hf _ ha]
+      unfold schurCoeff
+      rw [he]
+    have hz' := Option.some.inj hz
+    rw [← hz', hval]
+    simp only [coeffZ, ite_eq_left hn]
+  · have hz' := Option.some.inj hz
+    obtain ⟨i,j,hij,he⟩ := hr
+    have ha := alternating_mul_symmetric hp (alternant_alternating (staircase n))
+    have hc := coeff_repeated ha (fun j => (s.2 j).toNat) hij (congrArg Int.toNat he)
+    rw [← hz']
+    simp only [coeffZ, ite_eq_left hn, hc, mul_zero]
+  · have hz' := Option.some.inj hz
+    rw [← hz']
+    simp only [coeffZ, ite_eq_right hn, mul_zero]
+
+def checkedLookup {n : ℕ} (f : (Fin n → ℕ) → ℤ) (v : Fin n → ℤ) : Option ℤ :=
+  finish f (run (network n) (1,v))
+
+theorem checkedLookup_sound {n : ℕ} (p : MvPolynomial (Fin n) ℤ)
+    (hp : ∀ σ : Equiv.Perm (Fin n), rename σ p = p)
+    (f : (Fin n → ℕ) → ℤ)
+    (hf : ∀ mu, Antitone mu → f mu = schurCoeff p mu)
+    (v : Fin n → ℤ) (z : ℤ) (hz : checkedLookup f v = some z) :
+    z = signedLookup (schurCoeff p) v := by
+  rw [signedLookup_eq_coeffZ p hp]
+  have ha := alternating_mul_symmetric hp (alternant_alternating (staircase n))
+  calc
+    z = _ := finish_sound p hp f hf _ z hz
+    _ = _ := by simpa only [one_mul] using run_sound ha (network n) (1,v)
+
+end Foulkes.FastLookup
+
+
+noncomputable section
+
+namespace Foulkes.FastLookup
+
+example : checkedLookup (fun (_ : Fin 6 → ℕ) => (7 : ℤ)) ![1,8,5,3,2,10] = some (-7) := by
+  decide +kernel
+example : checkedLookup (fun (_ : Fin 6 → ℕ) => (7 : ℤ)) ![1,8,5,3,2,1] = some 0 := by
+  decide +kernel
+example : checkedLookup (fun (_ : Fin 6 → ℕ) => (7 : ℤ)) ![1,8,5,-3,2,10] = some 0 := by
+  decide +kernel
+end Foulkes.FastLookup
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/FastLookupExact.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/FastLookupExact.lean
index ce4ef96fb75593d7b7bbca367579cebc25842163..92fa562aee729f30fd9cd2df4d806562a818f159 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/FastLookupExact.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/FastLookupExact.lean
@@ -1,87 +1,87 @@
-import Mathlib
-import OAI.RepresentationTheory.FoulkesSixth.FastLookup
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.FastLookup
-open MvPolynomial Foulkes.Strips Foulkes.Lookup
-
-lemma compareStep_perm {n : ℕ} (ij : Fin n × Fin n) (s : State n) :
-    ∃ σ : Equiv.Perm (Fin n), compareStep ij s =
-      (s.1 * (Equiv.Perm.sign σ : ℤ), s.2 ∘ σ) := by
-  unfold compareStep
-  split_ifs with h
-  · refine ⟨Equiv.swap ij.1 ij.2, ?_⟩
-    have hij : ij.1 ≠ ij.2 := fun heq => by simp [heq] at h
-    simp [Equiv.Perm.sign_swap hij]
-  · refine ⟨1, ?_⟩
-    simp
-
-lemma run_perm {n : ℕ} (steps : List (Fin n × Fin n)) (s : State n) :
-    ∃ σ : Equiv.Perm (Fin n), run steps s =
-      (s.1 * (Equiv.Perm.sign σ : ℤ), s.2 ∘ σ) := by
-  induction steps generalizing s with
-  | nil => exact ⟨1, by simp [run]⟩
-  | cons ij steps ih =>
-    obtain ⟨τ,hτ⟩ := compareStep_perm ij s
-    obtain ⟨σ,hσ⟩ := ih (compareStep ij s)
-    refine ⟨τ * σ, ?_⟩
-    change run steps (compareStep ij s) = _
-    rw [hσ, hτ]
-    simp only [map_mul, Units.val_mul, mul_assoc, Equiv.Perm.coe_mul]
-    rfl
-
-lemma finish_perm_exact {n : ℕ} (f : (Fin n → ℕ) → ℤ)
-    (v : Fin n → ℤ) (σ : Equiv.Perm (Fin n)) (z : ℤ)
-    (hz : finish f ((Equiv.Perm.sign σ : ℤ), v ∘ σ) = some z) :
-    z = signedLookup f v := by
-  classical
-  have hnon : (∀ j, 0 ≤ v (σ j)) ↔ ∀ j, 0 ≤ v j :=
-    ⟨fun h j => by simpa using h (σ.symm j), fun h j => h (σ j)⟩
-  unfold finish at hz
-  dsimp only [Function.comp_apply] at hz
-  split_ifs at hz with hn hs hr
-  · let w : Fin n → ℕ := fun j => (v j).toNat
-    have hstrict : StrictAnti (w ∘ σ) := hs
-    have hw : Function.Injective w := by
-      intro i j hij
-      have hh : σ.symm i = σ.symm j := hstrict.injective (by simpa using hij)
-      exact σ.symm.injective hh
-    have he : w ∘ σ = w ∘ sortDesc w :=
-      Tuple.unique_antitone hstrict.antitone (sortDesc_antitone w)
-    have hσ : σ = sortDesc w := Equiv.ext fun j => hw (congrFun he j)
-    have hv : (fun j => (v (σ j)).toNat - staircase n j) = sortedPartition w := by
-      rw [hσ]
-      rfl
-    have hz' := Option.some.inj hz
-    rw [← hz', hv, hσ]
-    simp only [signedLookup, dite_eq_left (hnon.mp hn)]
-    change _ = if Function.Injective w then (Equiv.Perm.sign (sortDesc w) : ℤ) * f (sortedPartition w) else 0
-    rw [ite_eq_left hw]
-  · have hz' := Option.some.inj hz
-    obtain ⟨i,j,hij,he⟩ := hr
-    have hw : ¬Function.Injective (fun j => (v j).toNat) := by
-      intro hw
-      exact hij (σ.injective (hw (congrArg Int.toNat he)))
-    rw [← hz']
-    simp only [signedLookup, dite_eq_left (hnon.mp hn), ite_eq_right hw]
-  · have hz' := Option.some.inj hz
-    rw [← hz']
-    simp only [signedLookup, dite_eq_right (mt hnon.mpr hn)]
-
-theorem checkedLookup_exact {n : ℕ} (f : (Fin n → ℕ) → ℤ)
-    (v : Fin n → ℤ) (z : ℤ) (hz : checkedLookup f v = some z) :
-    z = signedLookup f v := by
-  obtain ⟨σ,hσ⟩ := run_perm (network n) (1,v)
-  unfold checkedLookup at hz
-  rw [hσ] at hz
-  simp only [one_mul] at hz
-  exact finish_perm_exact f v σ z hz
-
-end Foulkes.FastLookup
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.FoulkesSixth.FastLookup
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.FastLookup
+open MvPolynomial Foulkes.Strips Foulkes.Lookup
+
+lemma compareStep_perm {n : ℕ} (ij : Fin n × Fin n) (s : State n) :
+    ∃ σ : Equiv.Perm (Fin n), compareStep ij s =
+      (s.1 * (Equiv.Perm.sign σ : ℤ), s.2 ∘ σ) := by
+  unfold compareStep
+  split_ifs with h
+  · refine ⟨Equiv.swap ij.1 ij.2, ?_⟩
+    have hij : ij.1 ≠ ij.2 := fun heq => by simp [heq] at h
+    simp [Equiv.Perm.sign_swap hij]
+  · refine ⟨1, ?_⟩
+    simp
+
+lemma run_perm {n : ℕ} (steps : List (Fin n × Fin n)) (s : State n) :
+    ∃ σ : Equiv.Perm (Fin n), run steps s =
+      (s.1 * (Equiv.Perm.sign σ : ℤ), s.2 ∘ σ) := by
+  induction steps generalizing s with
+  | nil => exact ⟨1, by simp [run]⟩
+  | cons ij steps ih =>
+    obtain ⟨τ,hτ⟩ := compareStep_perm ij s
+    obtain ⟨σ,hσ⟩ := ih (compareStep ij s)
+    refine ⟨τ * σ, ?_⟩
+    change run steps (compareStep ij s) = _
+    rw [hσ, hτ]
+    simp only [map_mul, Units.val_mul, mul_assoc, Equiv.Perm.coe_mul]
+    rfl
+
+lemma finish_perm_exact {n : ℕ} (f : (Fin n → ℕ) → ℤ)
+    (v : Fin n → ℤ) (σ : Equiv.Perm (Fin n)) (z : ℤ)
+    (hz : finish f ((Equiv.Perm.sign σ : ℤ), v ∘ σ) = some z) :
+    z = signedLookup f v := by
+  classical
+  have hnon : (∀ j, 0 ≤ v (σ j)) ↔ ∀ j, 0 ≤ v j :=
+    ⟨fun h j => by simpa using h (σ.symm j), fun h j => h (σ j)⟩
+  unfold finish at hz
+  dsimp only [Function.comp_apply] at hz
+  split_ifs at hz with hn hs hr
+  · let w : Fin n → ℕ := fun j => (v j).toNat
+    have hstrict : StrictAnti (w ∘ σ) := hs
+    have hw : Function.Injective w := by
+      intro i j hij
+      have hh : σ.symm i = σ.symm j := hstrict.injective (by simpa using hij)
+      exact σ.symm.injective hh
+    have he : w ∘ σ = w ∘ sortDesc w :=
+      Tuple.unique_antitone hstrict.antitone (sortDesc_antitone w)
+    have hσ : σ = sortDesc w := Equiv.ext fun j => hw (congrFun he j)
+    have hv : (fun j => (v (σ j)).toNat - staircase n j) = sortedPartition w := by
+      rw [hσ]
+      rfl
+    have hz' := Option.some.inj hz
+    rw [← hz', hv, hσ]
+    simp only [signedLookup, dite_eq_left (hnon.mp hn)]
+    change _ = if Function.Injective w then (Equiv.Perm.sign (sortDesc w) : ℤ) * f (sortedPartition w) else 0
+    rw [ite_eq_left hw]
+  · have hz' := Option.some.inj hz
+    obtain ⟨i,j,hij,he⟩ := hr
+    have hw : ¬Function.Injective (fun j => (v j).toNat) := by
+      intro hw
+      exact hij (σ.injective (hw (congrArg Int.toNat he)))
+    rw [← hz']
+    simp only [signedLookup, dite_eq_left (hnon.mp hn), ite_eq_right hw]
+  · have hz' := Option.some.inj hz
+    rw [← hz']
+    simp only [signedLookup, dite_eq_right (mt hnon.mpr hn)]
+
+theorem checkedLookup_exact {n : ℕ} (f : (Fin n → ℕ) → ℤ)
+    (v : Fin n → ℤ) (z : ℤ) (hz : checkedLookup f v = some z) :
+    z = signedLookup f v := by
+  obtain ⟨σ,hσ⟩ := run_perm (network n) (1,v)
+  unfold checkedLookup at hz
+  rw [hσ] at hz
+  simp only [one_mul] at hz
+  exact finish_perm_exact f v σ z hz
+
+end Foulkes.FastLookup
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/GenericShift.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/GenericShift.lean
index 255b6dc3380451acab6d52b9550c4dc5b66ec8a0..a9d40dd0d1237c5e23ec0480090001caa3b78c31 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/GenericShift.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/GenericShift.lean
@@ -1,214 +1,214 @@
-import Mathlib
-import OAI.RepresentationTheory.FoulkesSixth.PolynomialSpan
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.GenericShift
-open MvPolynomial
-open Foulkes.PolynomialSpan
-
-variable {A : Type*} [CommRing A]
-
-abbrev Poly (A : Type*) [CommRing A] (m n : ℕ) := MvPolynomial (Fin m × Fin n) A
-
-def shift {m n : ℕ} (u v : Fin m) : Derivation A (Poly A m n) (Poly A m n) :=
-  mkDerivation A fun ij => if ij.1 = u then X (v, ij.2) else 0
-
-@[simp] lemma shift_X {m n : ℕ} (u v : Fin m) (ij : Fin m × Fin n) :
-    shift (A := A) u v (X ij) = if ij.1 = u then X (v, ij.2) else 0 := by
-  simp [shift]
-
-lemma derivation_sum_apply {ι : Type*} {m n : ℕ}
-    (s : Finset ι) (D : ι → Derivation A (Poly A m n) (Poly A m n)) (f : Poly A m n) :
-    (∑ k ∈ s, D k) f = ∑ k ∈ s, D k f := by
-  classical
-  induction s using Finset.induction_on with
-  | empty => simp
-  | insert x s hx ih => simp [Finset.sum_insert hx, ih]
-
-lemma shift_eq_sum {m n : ℕ} (u v : Fin m) :
-    shift (A := A) (n := n) u v = ∑ k : Fin n, (X (v, k) : Poly A m n) •
-      (pderiv (u, k) : Derivation A (Poly A m n) (Poly A m n)) := by
-  apply MvPolynomial.derivation_ext
-  rintro ⟨r, k⟩
-  simp only [shift_X, derivation_sum_apply, Derivation.smul_apply, pderiv_X]
-  by_cases h : r = u
-  · subst r
-    simp [Pi.single_apply, Prod.mk.injEq]
-  · simp [Prod.mk.injEq, h]
-
-lemma mapCoeffs_shift [Algebra ℂ A] {m n : ℕ}
-    (l : A →ₗ[ℂ] ℂ) (u v : Fin m) (f : Poly A m n) :
-    mapCoeffs l (shift u v f) = shift u v (mapCoeffs l f) := by
-  simp only [shift_eq_sum, derivation_sum_apply, Derivation.smul_apply,
-    smul_eq_mul, mapCoeffs_sum, mapCoeffs_X_mul, mapCoeffs_pderiv]
-
-lemma mapCoeffs_shift_pow [Algebra ℂ A] {m n : ℕ}
-    (l : A →ₗ[ℂ] ℂ) (u v : Fin m) (f : Poly A m n) (L : ℕ) :
-    mapCoeffs l (((shift u v).toLinearMap ^ L) f) =
-      ((shift u v).toLinearMap ^ L) (mapCoeffs l f) := by
-  induction L with
-  | zero => rfl
-  | succ L ih =>
-    simpa only [pow_succ', Module.End.mul_apply, Derivation.coeFn_coe, mapCoeffs_shift] using
-      congrArg (shift u v) ih
-
-def rowLinear {m n : ℕ} (u : Fin m) (c : Fin n → A) : Poly A m n :=
-  ∑ k, c k • X (u, k)
-
-@[simp] lemma shift_rowLinear {m n : ℕ} (u v w : Fin m) (c : Fin n → A) :
-    shift u v (rowLinear w c) = if w = u then rowLinear v c else 0 := by
-  simp [rowLinear, map_sum, shift_X]
-
-lemma pow_apply_mul_of_left_zero {B : Type*} [CommRing B] [Algebra A B]
-    (D : Derivation A B B) {g : B} (hg : D g = 0) (f : B) (L : ℕ) :
-    (D.toLinearMap ^ L) (g * f) = g * (D.toLinearMap ^ L) f := by
-  induction L with
-  | zero => rfl
-  | succ L ih =>
-    simp only [pow_succ', Module.End.mul_apply, ih, Derivation.coeFn_coe, D.leibniz, hg,
-      smul_eq_mul, mul_zero, add_zero]
-
-lemma pow_apply_pow {B : Type*} [CommRing B] [Algebra A B]
-    (D : Derivation A B B) {f g : B} (hf : D f = g) (hg : D g = 0)
-    (j L : ℕ) (hL : L ≤ j) :
-    (D.toLinearMap ^ L) (f ^ j) = j.descFactorial L • (g ^ L * f ^ (j - L)) := by
-  induction L with
-  | zero => simp
-  | succ L ih =>
-    have hLj : L ≤ j := Nat.le_trans (Nat.le_succ L) hL
-    rw [pow_succ', Module.End.mul_apply, ih hLj, Derivation.coeFn_coe, map_nsmul]
-    have hkill : D (g ^ L) = 0 := by simp [D.leibniz_pow, hg]
-    rw [D.leibniz, hkill, smul_zero, add_zero, D.leibniz_pow, hf]
-    simp only [smul_eq_mul, nsmul_eq_mul]
-    rw [Nat.descFactorial_succ]
-    have hsub : j - L - 1 = j - (L + 1) := by omega
-    rw [hsub, pow_succ]
-    push_cast
-    ring
-
-lemma derivation_prod_zero {B ι : Type*} [CommRing B] [Algebra A B]
-    (D : Derivation A B B) (s : Finset ι) (f : ι → B)
-    (h : ∀ i ∈ s, D (f i) = 0) : D (∏ i ∈ s, f i) = 0 := by
-  classical
-  induction s using Finset.induction_on with
-  | empty => simp
-  | insert i s hi ih =>
-    rw [Finset.prod_insert hi, D.leibniz, h i (by simp),
-      ih (fun k hk => h k (Finset.mem_insert_of_mem hk))]
-    simp
-
-def mixedProduct {B ι : Type*} [CommRing B] [Fintype ι] [DecidableEq ι]
-    (x y : ι → B) (j L : ℕ) (s : Finset ι) : B :=
-  ∏ i, if i ∈ s then y i ^ L * x i ^ (j - L) else x i ^ j
-
-lemma pow_apply_mixedProduct {B ι : Type*} [CommRing B] [Algebra A B] [Fintype ι] [DecidableEq ι]
-    (D : Derivation A B B) (x y : ι → B) (j L : ℕ) (s : Finset ι) (u : ι)
-    (hu : u ∉ s) (hL : L ≤ j)
-    (hx : ∀ i, D (x i) = if i = u then y i else 0)
-    (hy : ∀ i, D (y i) = 0) :
-    (D.toLinearMap ^ L) (mixedProduct x y j L s) =
-      j.descFactorial L • mixedProduct x y j L (insert u s) := by
-  classical
-  let F (i : ι) := if i ∈ s then y i ^ L * x i ^ (j - L) else x i ^ j
-  let H : B := ∏ i ∈ Finset.univ.erase u, F i
-  have hH : D H = 0 := by
-    apply derivation_prod_zero
-    intro i hi
-    have hne : i ≠ u := (Finset.mem_erase.mp hi).1
-    dsimp [F]
-    split_ifs <;> simp [D.leibniz, D.leibniz_pow, hx, hy, hne]
-  have he : mixedProduct x y j L s = H * (x u) ^ j := by
-    rw [mixedProduct, ← Finset.mul_prod_erase Finset.univ F (a := u) (by simp)]
-    simp [F, hu, H, mul_comm]
-  have hf : mixedProduct x y j L (insert u s) = H * ((y u) ^ L * (x u) ^ (j - L)) := by
-    let G (i : ι) := if i ∈ insert u s then y i ^ L * x i ^ (j - L) else x i ^ j
-    rw [mixedProduct, ← Finset.mul_prod_erase Finset.univ G (a := u) (by simp)]
-    have hge : (∏ i ∈ Finset.univ.erase u, G i) = H := by
-      apply Finset.prod_congr rfl
-      intro i hi
-      have hne : i ≠ u := (Finset.mem_erase.mp hi).1
-      simp [F, G, hne]
-    rw [hge]
-    simp [G, mul_comm]
-  rw [he, pow_apply_mul_of_left_zero D hH,
-    pow_apply_pow D (f := x u) (g := y u) (by simp [hx]) (hy u) j L hL, hf]
-  simp only [nsmul_eq_mul]
-  ring
-
-def rowWeight {m n : ℕ} (u : Fin m) (ij : Fin m × Fin n) : ℕ :=
-  if ij.1 = u then 1 else 0
-
-lemma rowLinear_homogeneous {m n : ℕ} (u w : Fin m) (c : Fin n → A) :
-    (rowLinear u c).IsWeightedHomogeneous (rowWeight w) (if u = w then 1 else 0) := by
-  apply MvPolynomial.IsWeightedHomogeneous.sum
-  intro k _
-  exact (weightedHomogeneousSubmodule A (rowWeight w) _).smul_mem _
-    (isWeightedHomogeneous_X (R := A) (rowWeight w) (u, k))
-
-def family {a n : ℕ} (c : Fin a → Fin n → A) (j L : ℕ) (s : Finset (Fin a)) :
-    Poly A (a + 1) n :=
-  mixedProduct (fun i => rowLinear i.castSucc (c i))
-    (fun i => rowLinear (Fin.last a) (c i)) j L s
-
-lemma shift_family {a n : ℕ} (c : Fin a → Fin n → A) (j L : ℕ)
-    (s : Finset (Fin a)) (u : Fin a) (hu : u ∉ s) (hL : L ≤ j) :
-    ((shift (A := A) u.castSucc (Fin.last a)).toLinearMap ^ L) (family c j L s) =
-      j.descFactorial L • family c j L (insert u s) := by
-  apply pow_apply_mixedProduct _ _ _ _ _ _ _ hu hL
-  · intro i
-    simp
-  · intro i
-    simp [ne_comm]
-
-lemma family_degree_dest {a n : ℕ} (c : Fin a → Fin n → A) (j L : ℕ)
-    (s : Finset (Fin a)) :
-    (family c j L s).IsWeightedHomogeneous (rowWeight (Fin.last a)) (s.card * L) := by
-  have hX (i : Fin a) : (rowLinear i.castSucc (c i)).IsWeightedHomogeneous
-      (rowWeight (Fin.last a)) 0 := by simpa using rowLinear_homogeneous i.castSucc (Fin.last a) (c i)
-  have hY (i : Fin a) : (rowLinear (Fin.last a) (c i)).IsWeightedHomogeneous
-      (rowWeight (Fin.last a)) 1 := by simpa using rowLinear_homogeneous (Fin.last a) (Fin.last a) (c i)
-  have hF (i : Fin a) : (if i ∈ s then rowLinear (Fin.last a) (c i) ^ L *
-      rowLinear i.castSucc (c i) ^ (j - L) else rowLinear i.castSucc (c i) ^ j).IsWeightedHomogeneous (rowWeight (Fin.last a)) (if i ∈ s then L else 0) := by
-    split_ifs
-    · simpa using ((hY i).pow L).mul ((hX i).pow (j - L))
-    · simpa using (hX i).pow j
-  have hh := MvPolynomial.IsWeightedHomogeneous.prod Finset.univ _ _ (fun i _ => hF i)
-  simpa [family, mixedProduct] using hh
-
-lemma family_degree_source {a n : ℕ} (c : Fin a → Fin n → A) (j L : ℕ)
-    (s : Finset (Fin a)) (u : Fin a) (hu : u ∉ s) :
-    (family c j L s).IsWeightedHomogeneous (rowWeight u.castSucc) j := by
-  have hX (i : Fin a) : (rowLinear i.castSucc (c i)).IsWeightedHomogeneous
-      (rowWeight u.castSucc) (if i = u then 1 else 0) := by
-    simpa using rowLinear_homogeneous i.castSucc u.castSucc (c i)
-  have hY (i : Fin a) : (rowLinear (Fin.last a) (c i)).IsWeightedHomogeneous
-      (rowWeight u.castSucc) 0 := by
-    simpa [Ne.symm (Fin.castSucc_ne_last u)] using
-      rowLinear_homogeneous (Fin.last a) u.castSucc (c i)
-  have hF (i : Fin a) : (if i ∈ s then rowLinear (Fin.last a) (c i) ^ L *
-      rowLinear i.castSucc (c i) ^ (j - L) else rowLinear i.castSucc (c i) ^ j).IsWeightedHomogeneous (rowWeight u.castSucc) (if i = u then j else 0) := by
-    by_cases hi : i = u
-    · subst i
-      simp only [hu, ↓reduceIte]
-      simpa using (hX u).pow j
-    · have hX' : (rowLinear i.castSucc (c i)).IsWeightedHomogeneous
-          (rowWeight u.castSucc) 0 := by simpa [hi] using hX i
-      by_cases his : i ∈ s
-      · simpa [his, hi] using ((hY i).pow L).mul (hX'.pow (j - L))
-      · simpa [his, hi] using hX'.pow j
-  have hh := MvPolynomial.IsWeightedHomogeneous.prod Finset.univ _ _ (fun i _ => hF i)
-  simpa [family, mixedProduct] using hh
-
-lemma eval_rowLinear [Algebra ℂ A] {m n : ℕ} (u : Fin m) (c : Fin n → A)
-    (x : Fin m × Fin n → ℂ) :
-    eval (fun ij => algebraMap ℂ A (x ij)) (rowLinear u c) = ∑ k, x (u, k) • c k := by
-  simp [rowLinear, Algebra.smul_def, mul_comm]
-
-end Foulkes.GenericShift
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.FoulkesSixth.PolynomialSpan
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.GenericShift
+open MvPolynomial
+open Foulkes.PolynomialSpan
+
+variable {A : Type*} [CommRing A]
+
+abbrev Poly (A : Type*) [CommRing A] (m n : ℕ) := MvPolynomial (Fin m × Fin n) A
+
+def shift {m n : ℕ} (u v : Fin m) : Derivation A (Poly A m n) (Poly A m n) :=
+  mkDerivation A fun ij => if ij.1 = u then X (v, ij.2) else 0
+
+@[simp] lemma shift_X {m n : ℕ} (u v : Fin m) (ij : Fin m × Fin n) :
+    shift (A := A) u v (X ij) = if ij.1 = u then X (v, ij.2) else 0 := by
+  simp [shift]
+
+lemma derivation_sum_apply {ι : Type*} {m n : ℕ}
+    (s : Finset ι) (D : ι → Derivation A (Poly A m n) (Poly A m n)) (f : Poly A m n) :
+    (∑ k ∈ s, D k) f = ∑ k ∈ s, D k f := by
+  classical
+  induction s using Finset.induction_on with
+  | empty => simp
+  | insert x s hx ih => simp [Finset.sum_insert hx, ih]
+
+lemma shift_eq_sum {m n : ℕ} (u v : Fin m) :
+    shift (A := A) (n := n) u v = ∑ k : Fin n, (X (v, k) : Poly A m n) •
+      (pderiv (u, k) : Derivation A (Poly A m n) (Poly A m n)) := by
+  apply MvPolynomial.derivation_ext
+  rintro ⟨r, k⟩
+  simp only [shift_X, derivation_sum_apply, Derivation.smul_apply, pderiv_X]
+  by_cases h : r = u
+  · subst r
+    simp [Pi.single_apply, Prod.mk.injEq]
+  · simp [Prod.mk.injEq, h]
+
+lemma mapCoeffs_shift [Algebra ℂ A] {m n : ℕ}
+    (l : A →ₗ[ℂ] ℂ) (u v : Fin m) (f : Poly A m n) :
+    mapCoeffs l (shift u v f) = shift u v (mapCoeffs l f) := by
+  simp only [shift_eq_sum, derivation_sum_apply, Derivation.smul_apply,
+    smul_eq_mul, mapCoeffs_sum, mapCoeffs_X_mul, mapCoeffs_pderiv]
+
+lemma mapCoeffs_shift_pow [Algebra ℂ A] {m n : ℕ}
+    (l : A →ₗ[ℂ] ℂ) (u v : Fin m) (f : Poly A m n) (L : ℕ) :
+    mapCoeffs l (((shift u v).toLinearMap ^ L) f) =
+      ((shift u v).toLinearMap ^ L) (mapCoeffs l f) := by
+  induction L with
+  | zero => rfl
+  | succ L ih =>
+    simpa only [pow_succ', Module.End.mul_apply, Derivation.coeFn_coe, mapCoeffs_shift] using
+      congrArg (shift u v) ih
+
+def rowLinear {m n : ℕ} (u : Fin m) (c : Fin n → A) : Poly A m n :=
+  ∑ k, c k • X (u, k)
+
+@[simp] lemma shift_rowLinear {m n : ℕ} (u v w : Fin m) (c : Fin n → A) :
+    shift u v (rowLinear w c) = if w = u then rowLinear v c else 0 := by
+  simp [rowLinear, map_sum, shift_X]
+
+lemma pow_apply_mul_of_left_zero {B : Type*} [CommRing B] [Algebra A B]
+    (D : Derivation A B B) {g : B} (hg : D g = 0) (f : B) (L : ℕ) :
+    (D.toLinearMap ^ L) (g * f) = g * (D.toLinearMap ^ L) f := by
+  induction L with
+  | zero => rfl
+  | succ L ih =>
+    simp only [pow_succ', Module.End.mul_apply, ih, Derivation.coeFn_coe, D.leibniz, hg,
+      smul_eq_mul, mul_zero, add_zero]
+
+lemma pow_apply_pow {B : Type*} [CommRing B] [Algebra A B]
+    (D : Derivation A B B) {f g : B} (hf : D f = g) (hg : D g = 0)
+    (j L : ℕ) (hL : L ≤ j) :
+    (D.toLinearMap ^ L) (f ^ j) = j.descFactorial L • (g ^ L * f ^ (j - L)) := by
+  induction L with
+  | zero => simp
+  | succ L ih =>
+    have hLj : L ≤ j := Nat.le_trans (Nat.le_succ L) hL
+    rw [pow_succ', Module.End.mul_apply, ih hLj, Derivation.coeFn_coe, map_nsmul]
+    have hkill : D (g ^ L) = 0 := by simp [D.leibniz_pow, hg]
+    rw [D.leibniz, hkill, smul_zero, add_zero, D.leibniz_pow, hf]
+    simp only [smul_eq_mul, nsmul_eq_mul]
+    rw [Nat.descFactorial_succ]
+    have hsub : j - L - 1 = j - (L + 1) := by omega
+    rw [hsub, pow_succ]
+    push_cast
+    ring
+
+lemma derivation_prod_zero {B ι : Type*} [CommRing B] [Algebra A B]
+    (D : Derivation A B B) (s : Finset ι) (f : ι → B)
+    (h : ∀ i ∈ s, D (f i) = 0) : D (∏ i ∈ s, f i) = 0 := by
+  classical
+  induction s using Finset.induction_on with
+  | empty => simp
+  | insert i s hi ih =>
+    rw [Finset.prod_insert hi, D.leibniz, h i (by simp),
+      ih (fun k hk => h k (Finset.mem_insert_of_mem hk))]
+    simp
+
+def mixedProduct {B ι : Type*} [CommRing B] [Fintype ι] [DecidableEq ι]
+    (x y : ι → B) (j L : ℕ) (s : Finset ι) : B :=
+  ∏ i, if i ∈ s then y i ^ L * x i ^ (j - L) else x i ^ j
+
+lemma pow_apply_mixedProduct {B ι : Type*} [CommRing B] [Algebra A B] [Fintype ι] [DecidableEq ι]
+    (D : Derivation A B B) (x y : ι → B) (j L : ℕ) (s : Finset ι) (u : ι)
+    (hu : u ∉ s) (hL : L ≤ j)
+    (hx : ∀ i, D (x i) = if i = u then y i else 0)
+    (hy : ∀ i, D (y i) = 0) :
+    (D.toLinearMap ^ L) (mixedProduct x y j L s) =
+      j.descFactorial L • mixedProduct x y j L (insert u s) := by
+  classical
+  let F (i : ι) := if i ∈ s then y i ^ L * x i ^ (j - L) else x i ^ j
+  let H : B := ∏ i ∈ Finset.univ.erase u, F i
+  have hH : D H = 0 := by
+    apply derivation_prod_zero
+    intro i hi
+    have hne : i ≠ u := (Finset.mem_erase.mp hi).1
+    dsimp [F]
+    split_ifs <;> simp [D.leibniz, D.leibniz_pow, hx, hy, hne]
+  have he : mixedProduct x y j L s = H * (x u) ^ j := by
+    rw [mixedProduct, ← Finset.mul_prod_erase Finset.univ F (a := u) (by simp)]
+    simp [F, hu, H, mul_comm]
+  have hf : mixedProduct x y j L (insert u s) = H * ((y u) ^ L * (x u) ^ (j - L)) := by
+    let G (i : ι) := if i ∈ insert u s then y i ^ L * x i ^ (j - L) else x i ^ j
+    rw [mixedProduct, ← Finset.mul_prod_erase Finset.univ G (a := u) (by simp)]
+    have hge : (∏ i ∈ Finset.univ.erase u, G i) = H := by
+      apply Finset.prod_congr rfl
+      intro i hi
+      have hne : i ≠ u := (Finset.mem_erase.mp hi).1
+      simp [F, G, hne]
+    rw [hge]
+    simp [G, mul_comm]
+  rw [he, pow_apply_mul_of_left_zero D hH,
+    pow_apply_pow D (f := x u) (g := y u) (by simp [hx]) (hy u) j L hL, hf]
+  simp only [nsmul_eq_mul]
+  ring
+
+def rowWeight {m n : ℕ} (u : Fin m) (ij : Fin m × Fin n) : ℕ :=
+  if ij.1 = u then 1 else 0
+
+lemma rowLinear_homogeneous {m n : ℕ} (u w : Fin m) (c : Fin n → A) :
+    (rowLinear u c).IsWeightedHomogeneous (rowWeight w) (if u = w then 1 else 0) := by
+  apply MvPolynomial.IsWeightedHomogeneous.sum
+  intro k _
+  exact (weightedHomogeneousSubmodule A (rowWeight w) _).smul_mem _
+    (isWeightedHomogeneous_X (R := A) (rowWeight w) (u, k))
+
+def family {a n : ℕ} (c : Fin a → Fin n → A) (j L : ℕ) (s : Finset (Fin a)) :
+    Poly A (a + 1) n :=
+  mixedProduct (fun i => rowLinear i.castSucc (c i))
+    (fun i => rowLinear (Fin.last a) (c i)) j L s
+
+lemma shift_family {a n : ℕ} (c : Fin a → Fin n → A) (j L : ℕ)
+    (s : Finset (Fin a)) (u : Fin a) (hu : u ∉ s) (hL : L ≤ j) :
+    ((shift (A := A) u.castSucc (Fin.last a)).toLinearMap ^ L) (family c j L s) =
+      j.descFactorial L • family c j L (insert u s) := by
+  apply pow_apply_mixedProduct _ _ _ _ _ _ _ hu hL
+  · intro i
+    simp
+  · intro i
+    simp [ne_comm]
+
+lemma family_degree_dest {a n : ℕ} (c : Fin a → Fin n → A) (j L : ℕ)
+    (s : Finset (Fin a)) :
+    (family c j L s).IsWeightedHomogeneous (rowWeight (Fin.last a)) (s.card * L) := by
+  have hX (i : Fin a) : (rowLinear i.castSucc (c i)).IsWeightedHomogeneous
+      (rowWeight (Fin.last a)) 0 := by simpa using rowLinear_homogeneous i.castSucc (Fin.last a) (c i)
+  have hY (i : Fin a) : (rowLinear (Fin.last a) (c i)).IsWeightedHomogeneous
+      (rowWeight (Fin.last a)) 1 := by simpa using rowLinear_homogeneous (Fin.last a) (Fin.last a) (c i)
+  have hF (i : Fin a) : (if i ∈ s then rowLinear (Fin.last a) (c i) ^ L *
+      rowLinear i.castSucc (c i) ^ (j - L) else rowLinear i.castSucc (c i) ^ j).IsWeightedHomogeneous (rowWeight (Fin.last a)) (if i ∈ s then L else 0) := by
+    split_ifs
+    · simpa using ((hY i).pow L).mul ((hX i).pow (j - L))
+    · simpa using (hX i).pow j
+  have hh := MvPolynomial.IsWeightedHomogeneous.prod Finset.univ _ _ (fun i _ => hF i)
+  simpa [family, mixedProduct] using hh
+
+lemma family_degree_source {a n : ℕ} (c : Fin a → Fin n → A) (j L : ℕ)
+    (s : Finset (Fin a)) (u : Fin a) (hu : u ∉ s) :
+    (family c j L s).IsWeightedHomogeneous (rowWeight u.castSucc) j := by
+  have hX (i : Fin a) : (rowLinear i.castSucc (c i)).IsWeightedHomogeneous
+      (rowWeight u.castSucc) (if i = u then 1 else 0) := by
+    simpa using rowLinear_homogeneous i.castSucc u.castSucc (c i)
+  have hY (i : Fin a) : (rowLinear (Fin.last a) (c i)).IsWeightedHomogeneous
+      (rowWeight u.castSucc) 0 := by
+    simpa [Ne.symm (Fin.castSucc_ne_last u)] using
+      rowLinear_homogeneous (Fin.last a) u.castSucc (c i)
+  have hF (i : Fin a) : (if i ∈ s then rowLinear (Fin.last a) (c i) ^ L *
+      rowLinear i.castSucc (c i) ^ (j - L) else rowLinear i.castSucc (c i) ^ j).IsWeightedHomogeneous (rowWeight u.castSucc) (if i = u then j else 0) := by
+    by_cases hi : i = u
+    · subst i
+      simp only [hu, ↓reduceIte]
+      simpa using (hX u).pow j
+    · have hX' : (rowLinear i.castSucc (c i)).IsWeightedHomogeneous
+          (rowWeight u.castSucc) 0 := by simpa [hi] using hX i
+      by_cases his : i ∈ s
+      · simpa [his, hi] using ((hY i).pow L).mul (hX'.pow (j - L))
+      · simpa [his, hi] using hX'.pow j
+  have hh := MvPolynomial.IsWeightedHomogeneous.prod Finset.univ _ _ (fun i _ => hF i)
+  simpa [family, mixedProduct] using hh
+
+lemma eval_rowLinear [Algebra ℂ A] {m n : ℕ} (u : Fin m) (c : Fin n → A)
+    (x : Fin m × Fin n → ℂ) :
+    eval (fun ij => algebraMap ℂ A (x ij)) (rowLinear u c) = ∑ k, x (u, k) • c k := by
+  simp [rowLinear, Algebra.smul_def, mul_comm]
+
+end Foulkes.GenericShift
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/HomogeneousPlethysm.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/HomogeneousPlethysm.lean
index 0cb1eeca245bcc2206ababd5fa769e5a00b5590f..b8b98bd4a3d3ce0e1b883ef68b4604d7596fbd78 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/HomogeneousPlethysm.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/HomogeneousPlethysm.lean
@@ -1,148 +1,148 @@
-import Mathlib
-import OAI.RepresentationTheory.FoulkesSixth.AlternantExpansion
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.Complete
-open MvPolynomial Finset
-
-lemma homogeneous_h {α : Type*} [Finite α] (b : ℕ) : (h α b).IsHomogeneous b := by
-  classical
-  apply IsHomogeneous.sum
-  intro d hd
-  exact isHomogeneous_monomial 1 d.property
-
-lemma homogeneous_pleth (n b a : ℕ) : (pleth n b a).IsHomogeneous (a*b) := by
-  unfold pleth alphabetH
-  apply (homogeneous_h b).eval₂
-  · intro r
-    change (C r : MvPolynomial (Fin n) ℤ).IsHomogeneous 0
-    exact isHomogeneous_C _ _
-  · intro d
-    exact isHomogeneous_monomial 1 d.property
-
-end Foulkes.Complete
-
-namespace Foulkes.Lookup
-open MvPolynomial Finset Foulkes.Strips
-
-lemma degree_expVector {n : ℕ} (v : Fin n → ℕ) :
-    (expVector v).degree = ∑ j, v j := by
-  simpa using Finsupp.degree_eq_sum (expVector v)
-
-lemma homogeneous_alternant {n : ℕ} (m : Fin n → ℕ) :
-    (alternant m).IsHomogeneous (∑ j, m j) := by
-  classical
-  have ha : alternant m = ∑ σ : Equiv.Perm (Fin n),
-      Equiv.Perm.sign σ • ∏ c, (X (σ c) : MvPolynomial (Fin n) ℤ) ^ m c :=
-    Matrix.det_apply _
-  rw [ha]
-  apply IsHomogeneous.sum
-  intro σ hσ
-  rw [perm_prod_monomial]
-  change ((Equiv.Perm.sign σ : ℤ) • monomial (expVector (fun r => m (σ.symm r))) 1).IsHomogeneous _
-  rw [smul_monomial]
-  apply isHomogeneous_monomial
-  rw [degree_expVector, Equiv.sum_comp σ.symm m]
-
-abbrev Partition (n d : ℕ) :=
-  {e : Complete.Degree (Fin n) d // Antitone (fun j => e.val j)}
-
-instance (n d : ℕ) : Fintype (Partition n d) := by
-  classical
-  infer_instance
-
-def partVec {n d : ℕ} (mu : Partition n d) (j : Fin n) : ℕ := mu.val.val j
-
-lemma partVec_antitone {n d : ℕ} (mu : Partition n d) : Antitone (partVec mu) := mu.property
-
-lemma partVec_sum {n d : ℕ} (mu : Partition n d) : ∑ j, partVec mu j = d := by
-  change (∑ j, mu.val.val j) = d
-  rw [← Finsupp.degree_eq_sum]
-  exact mu.val.property
-
-def unshift {n : ℕ} (v : Fin n → ℕ) (j : Fin n) : ℕ := v j - staircase n j
-
-lemma unshift_antitone {n : ℕ} {v : Fin n → ℕ} (hv : StrictAnti v) :
-    Antitone (unshift v) := by
-  intro j k hjk
-  have hg := strictAnti_gap hv j k hjk
-  have hj := staircase_le_of_strictAnti hv j
-  have hk := staircase_le_of_strictAnti hv k
-  simp only [unshift, staircase] at *
-  have hjn := j.isLt
-  have hkn := k.isLt
-  omega
-
-lemma shifted_unshift {n : ℕ} {v : Fin n → ℕ} (hv : StrictAnti v) :
-    shifted (unshift v) = v := by
-  funext j
-  exact Nat.sub_add_cancel (staircase_le_of_strictAnti hv j)
-
-lemma unshift_sum {n d : ℕ} {v : Fin n → ℕ} (hv : StrictAnti v)
-    (hd : ∑ j, v j = d + ∑ j, staircase n j) : ∑ j, unshift v j = d := by
-  have hh : (∑ j, unshift v j) + ∑ j, staircase n j = ∑ j, v j := by
-    rw [← Finset.sum_add_distrib]
-    exact Finset.sum_congr rfl (fun j _ => congrFun (shifted_unshift hv) j)
-  omega
-
-def unshiftPartition {n d : ℕ} {v : Fin n → ℕ} (hv : StrictAnti v)
-    (hd : ∑ j, v j = d + ∑ j, staircase n j) : Partition n d :=
-  ⟨⟨expVector (unshift v), by rw [degree_expVector]; exact unshift_sum hv hd⟩,
-    unshift_antitone hv⟩
-
-lemma partVec_injective {n d : ℕ} : Function.Injective (@partVec n d) := by
-  intro v w h
-  apply Subtype.ext
-  apply Subtype.ext
-  ext j
-  exact congrFun h j
-
-lemma shifted_injective {n : ℕ} : Function.Injective (@shifted n) := by
-  intro v w h
-  funext j
-  exact Nat.add_right_cancel (congrFun h j)
-
-theorem homogeneous_schur_expansion {n d : ℕ} (p : MvPolynomial (Fin n) ℤ)
-    (hp : ∀ σ : Equiv.Perm (Fin n), rename σ p = p) (hd : p.IsHomogeneous d) :
-    p * alternant (staircase n) =
-      ∑ mu : Partition n d, schurCoeff p (partVec mu) • alternant (shifted (partVec mu)) := by
-  classical
-  apply eq_of_strict_coefficients
-    (alternating_mul_symmetric hp (alternant_alternating _))
-    (alternating_sum _ _ (fun _ _ => alternating_smul _ (alternant_alternating _)))
-  intro v hv
-  simp only [coeff_sum, coeff_smul]
-  by_cases hsize : ∑ j, v j = d + ∑ j, staircase n j
-  · let mu : Partition n d := unshiftPartition hv hsize
-    have he : shifted (partVec mu) = v := shifted_unshift hv
-    rw [Finset.sum_eq_single mu]
-    · rw [coeff_alternant_strict _ _ (shifted_strictAnti (partVec_antitone mu)) hv, ite_eq_left he]
-      simp only [smul_eq_mul, mul_one, schurCoeff, he]
-    · intro nu hnu hne
-      rw [coeff_alternant_strict _ _ (shifted_strictAnti (partVec_antitone nu)) hv]
-      have hn : shifted (partVec nu) ≠ v := by
-        intro hh
-        exact hne (partVec_injective (shifted_injective (hh.trans he.symm)))
-      simp [hn]
-    · simp
-  · have hgrad := hd.mul (homogeneous_alternant (staircase n))
-    rw [hgrad.coeff_eq_zero (by simpa [degree_expVector] using hsize)]
-    apply Eq.symm
-    apply Finset.sum_eq_zero
-    intro mu hmu
-    rw [coeff_alternant_strict _ _ (shifted_strictAnti (partVec_antitone mu)) hv]
-    have hn : shifted (partVec mu) ≠ v := by
-      intro hh
-      apply hsize
-      rw [← hh]
-      simp only [shifted, Finset.sum_add_distrib, partVec_sum]
-    simp [hn]
-
-end Foulkes.Lookup
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.FoulkesSixth.AlternantExpansion
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.Complete
+open MvPolynomial Finset
+
+lemma homogeneous_h {α : Type*} [Finite α] (b : ℕ) : (h α b).IsHomogeneous b := by
+  classical
+  apply IsHomogeneous.sum
+  intro d hd
+  exact isHomogeneous_monomial 1 d.property
+
+lemma homogeneous_pleth (n b a : ℕ) : (pleth n b a).IsHomogeneous (a*b) := by
+  unfold pleth alphabetH
+  apply (homogeneous_h b).eval₂
+  · intro r
+    change (C r : MvPolynomial (Fin n) ℤ).IsHomogeneous 0
+    exact isHomogeneous_C _ _
+  · intro d
+    exact isHomogeneous_monomial 1 d.property
+
+end Foulkes.Complete
+
+namespace Foulkes.Lookup
+open MvPolynomial Finset Foulkes.Strips
+
+lemma degree_expVector {n : ℕ} (v : Fin n → ℕ) :
+    (expVector v).degree = ∑ j, v j := by
+  simpa using Finsupp.degree_eq_sum (expVector v)
+
+lemma homogeneous_alternant {n : ℕ} (m : Fin n → ℕ) :
+    (alternant m).IsHomogeneous (∑ j, m j) := by
+  classical
+  have ha : alternant m = ∑ σ : Equiv.Perm (Fin n),
+      Equiv.Perm.sign σ • ∏ c, (X (σ c) : MvPolynomial (Fin n) ℤ) ^ m c :=
+    Matrix.det_apply _
+  rw [ha]
+  apply IsHomogeneous.sum
+  intro σ hσ
+  rw [perm_prod_monomial]
+  change ((Equiv.Perm.sign σ : ℤ) • monomial (expVector (fun r => m (σ.symm r))) 1).IsHomogeneous _
+  rw [smul_monomial]
+  apply isHomogeneous_monomial
+  rw [degree_expVector, Equiv.sum_comp σ.symm m]
+
+abbrev Partition (n d : ℕ) :=
+  {e : Complete.Degree (Fin n) d // Antitone (fun j => e.val j)}
+
+instance (n d : ℕ) : Fintype (Partition n d) := by
+  classical
+  infer_instance
+
+def partVec {n d : ℕ} (mu : Partition n d) (j : Fin n) : ℕ := mu.val.val j
+
+lemma partVec_antitone {n d : ℕ} (mu : Partition n d) : Antitone (partVec mu) := mu.property
+
+lemma partVec_sum {n d : ℕ} (mu : Partition n d) : ∑ j, partVec mu j = d := by
+  change (∑ j, mu.val.val j) = d
+  rw [← Finsupp.degree_eq_sum]
+  exact mu.val.property
+
+def unshift {n : ℕ} (v : Fin n → ℕ) (j : Fin n) : ℕ := v j - staircase n j
+
+lemma unshift_antitone {n : ℕ} {v : Fin n → ℕ} (hv : StrictAnti v) :
+    Antitone (unshift v) := by
+  intro j k hjk
+  have hg := strictAnti_gap hv j k hjk
+  have hj := staircase_le_of_strictAnti hv j
+  have hk := staircase_le_of_strictAnti hv k
+  simp only [unshift, staircase] at *
+  have hjn := j.isLt
+  have hkn := k.isLt
+  omega
+
+lemma shifted_unshift {n : ℕ} {v : Fin n → ℕ} (hv : StrictAnti v) :
+    shifted (unshift v) = v := by
+  funext j
+  exact Nat.sub_add_cancel (staircase_le_of_strictAnti hv j)
+
+lemma unshift_sum {n d : ℕ} {v : Fin n → ℕ} (hv : StrictAnti v)
+    (hd : ∑ j, v j = d + ∑ j, staircase n j) : ∑ j, unshift v j = d := by
+  have hh : (∑ j, unshift v j) + ∑ j, staircase n j = ∑ j, v j := by
+    rw [← Finset.sum_add_distrib]
+    exact Finset.sum_congr rfl (fun j _ => congrFun (shifted_unshift hv) j)
+  omega
+
+def unshiftPartition {n d : ℕ} {v : Fin n → ℕ} (hv : StrictAnti v)
+    (hd : ∑ j, v j = d + ∑ j, staircase n j) : Partition n d :=
+  ⟨⟨expVector (unshift v), by rw [degree_expVector]; exact unshift_sum hv hd⟩,
+    unshift_antitone hv⟩
+
+lemma partVec_injective {n d : ℕ} : Function.Injective (@partVec n d) := by
+  intro v w h
+  apply Subtype.ext
+  apply Subtype.ext
+  ext j
+  exact congrFun h j
+
+lemma shifted_injective {n : ℕ} : Function.Injective (@shifted n) := by
+  intro v w h
+  funext j
+  exact Nat.add_right_cancel (congrFun h j)
+
+theorem homogeneous_schur_expansion {n d : ℕ} (p : MvPolynomial (Fin n) ℤ)
+    (hp : ∀ σ : Equiv.Perm (Fin n), rename σ p = p) (hd : p.IsHomogeneous d) :
+    p * alternant (staircase n) =
+      ∑ mu : Partition n d, schurCoeff p (partVec mu) • alternant (shifted (partVec mu)) := by
+  classical
+  apply eq_of_strict_coefficients
+    (alternating_mul_symmetric hp (alternant_alternating _))
+    (alternating_sum _ _ (fun _ _ => alternating_smul _ (alternant_alternating _)))
+  intro v hv
+  simp only [coeff_sum, coeff_smul]
+  by_cases hsize : ∑ j, v j = d + ∑ j, staircase n j
+  · let mu : Partition n d := unshiftPartition hv hsize
+    have he : shifted (partVec mu) = v := shifted_unshift hv
+    rw [Finset.sum_eq_single mu]
+    · rw [coeff_alternant_strict _ _ (shifted_strictAnti (partVec_antitone mu)) hv, ite_eq_left he]
+      simp only [smul_eq_mul, mul_one, schurCoeff, he]
+    · intro nu hnu hne
+      rw [coeff_alternant_strict _ _ (shifted_strictAnti (partVec_antitone nu)) hv]
+      have hn : shifted (partVec nu) ≠ v := by
+        intro hh
+        exact hne (partVec_injective (shifted_injective (hh.trans he.symm)))
+      simp [hn]
+    · simp
+  · have hgrad := hd.mul (homogeneous_alternant (staircase n))
+    rw [hgrad.coeff_eq_zero (by simpa [degree_expVector] using hsize)]
+    apply Eq.symm
+    apply Finset.sum_eq_zero
+    intro mu hmu
+    rw [coeff_alternant_strict _ _ (shifted_strictAnti (partVec_antitone mu)) hv]
+    have hn : shifted (partVec mu) ≠ v := by
+      intro hh
+      apply hsize
+      rw [← hh]
+      simp only [shifted, Finset.sum_add_distrib, partVec_sum]
+    simp [hn]
+
+end Foulkes.Lookup
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/IndependentChart.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/IndependentChart.lean
index f9c83eb127d6aeb6c70b5e9899b7acfed5633667..8b85524e8c1b16376aaabe62470619ca9d61ca47 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/IndependentChart.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/IndependentChart.lean
@@ -1,521 +1,521 @@
-import Mathlib
-import OAI.RepresentationTheory.FoulkesSixth.InvariantPolarization
-import OAI.RepresentationTheory.FoulkesSixth.Collision
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.Chart
-open MvPolynomial Foulkes.Polarization Foulkes.PolynomialSpan
-
-def dehom (a n : ℕ) : P a (n + 1) →ₐ[ℂ] P a n :=
-  aeval (fun ij : Fin a × Fin (n + 1) => Fin.cases 1 (fun k => X (ij.1, k)) ij.2)
-
-@[simp] lemma dehom_X_zero {a n : ℕ} (i : Fin a) :
-    dehom a n (X (i, 0)) = 1 := by simp [dehom]
-@[simp] lemma dehom_X_succ {a n : ℕ} (i : Fin a) (k : Fin n) :
-    dehom a n (X (i, k.succ)) = X (i, k) := by simp [dehom]
-
-lemma dehom_rowForm {a n : ℕ} (i : Fin a) (v : Fin (n + 1) → ℂ) :
-    dehom a n (rowForm i v) = algebraMap ℂ (P a n) (v 0) + rowForm i (fun k => v k.succ) := by
-  simp [rowForm, Fin.sum_univ_succ, Algebra.smul_def]
-
-lemma dehom_z_cons {a n : ℕ} (s : ℂ) (u : Fin n → ℂ) :
-    dehom a n (z (Fin.cons s u)) = ∏ i : Fin a, (algebraMap ℂ (P a n) s + rowForm i u) := by
-  simp [z, dehom_rowForm]
-
-lemma dehom_renameRow {a n : ℕ} (σ : Equiv.Perm (Fin a)) (f : P a (n+1)) :
-    dehom a n (renameRow σ f) = renameRow σ (dehom a n f) := by
-  have hh : (dehom a n).comp (renameRow σ) = (renameRow σ).comp (dehom a n) := by
-    ext ij
-    rcases ij with ⟨i, k⟩
-    refine Fin.cases ?_ (fun k => ?_) k <;> simp [renameRow]
-  exact DFunLike.congr_fun hh f
-
-def E (a n : ℕ) : Submodule ℂ (P a n) := (R a (n+1) 1).map (dehom a n).toLinearMap
-
-lemma dehom_z_mem_E {a n : ℕ} (v : Fin (n+1) → ℂ) :
-    dehom a n (z v) ∈ E a n := Submodule.mem_map.mpr ⟨z v, z_mem_R v, rfl⟩
-
-lemma one_mem_E (a n : ℕ) : (1 : P a n) ∈ E a n := by
-  have hh := dehom_z_mem_E (a := a) (Fin.cons (1 : ℂ) (fun _ : Fin n => 0))
-  simpa [dehom_z_cons, rowForm] using hh
-
-lemma pow_mono {a n k l : ℕ} (h : k ≤ l) : E a n ^ k ≤ E a n ^ l := by
-  exact pow_le_pow_right₀ (Submodule.one_le.mpr (one_mem_E a n)) h
-
-lemma elementary_mem_E {a n : ℕ} (u : Fin n → ℂ) (h : ℕ) :
-    aeval (fun i : Fin a => rowForm i u) (esymm (Fin a) ℂ h) ∈ E a n := by
-  classical
-  let G : MvPolynomial Unit (P a n) := ∏ i : Fin a, (X () * C (rowForm i u) + 1)
-  have heval (s : Unit → ℂ) : eval (fun i => algebraMap ℂ (P a n) (s i)) G ∈ E a n := by
-    have hh := dehom_z_mem_E (a := a) (Fin.cons (1 : ℂ) (fun k => s () * u k))
-    convert hh using 1
-    simp only [G, eval_prod, eval_add, eval_mul, eval_X, eval_C, map_one,
-      dehom_z_cons, map_one, rowForm,
-      Algebra.smul_def]
-    apply Finset.prod_congr rfl
-    intro i _
-    simp [map_mul, Finset.mul_sum, mul_assoc, add_comm]
-  have hc := coeff_mem_of_evaluations G (E a n) heval (Finsupp.single () h)
-  have hg : G = ∑ s ∈ (Finset.univ : Finset (Fin a)).powerset,
-      (X () : MvPolynomial Unit (P a n)) ^ s.card * C (∏ i ∈ s, rowForm i u) := by
-    simp only [G, Finset.prod_add, Finset.prod_const_one, mul_one, Finset.prod_mul_distrib,
-      Finset.prod_const, ← map_prod]
-  rw [hg] at hc
-  have hcoeff (s : Finset (Fin a)) :
-      (X () ^ s.card * C (∏ i ∈ s, rowForm i u)).coeff (Finsupp.single () h) =
-        if s.card = h then ∏ i ∈ s, rowForm i u else 0 := by
-    rw [mul_comm, coeff_C_mul, X_pow_eq_monomial]
-    simp only [coeff_monomial]
-    simp [eq_comm]
-  simp only [coeff_sum, hcoeff] at hc
-  simpa only [esymm, map_sum, map_prod, aeval_X, Finset.powersetCard_eq_filter,
-    Finset.sum_filter, apply_ite, map_prod, aeval_X, map_zero] using hc
-
-lemma powersum_mem {a n : ℕ} (u : Fin n → ℂ) (d : ℕ) :
-    (∑ i : Fin a, rowForm i u ^ d) ∈ E a n ^ d := by
-  classical
-  induction d using Nat.strong_induction_on with
-  | h d ih =>
-    by_cases hd : d = 0
-    · subst d
-      simpa using (Submodule.sum_mem (1 : Submodule ℂ (P a n))
-        (fun (i : Fin a) (_ : i ∈ Finset.univ) => (Submodule.one_le.mp le_rfl)))
-    have hdpos : 0 < d := Nat.pos_of_ne_zero hd
-    have hn := congrArg (aeval (fun i : Fin a => rowForm i u))
-      (MvPolynomial.psum_eq_mul_esymm_sub_sum (Fin a) ℂ d hdpos)
-    simp only [psum, map_sum, map_pow, aeval_X, map_sub, map_mul, map_neg,
-      map_one, map_natCast] at hn
-    rw [hn]
-    apply Submodule.sub_mem
-    · have hh := (pow_mono (a := a) (n := n) hdpos)
-        (show aeval (fun i : Fin a => rowForm i u) (esymm (Fin a) ℂ d) ∈ E a n ^ 1 by
-          simpa using elementary_mem_E u d)
-      convert (E a n ^ d).smul_mem (((-1 : ℂ) ^ (d+1)) * (d : ℂ)) hh using 1
-      simp [Algebra.smul_def, mul_assoc]
-    · apply Submodule.sum_mem
-      intro ij hij
-      obtain ⟨hij, hi⟩ := Finset.mem_filter.mp hij
-      have hsum : ij.1 + ij.2 = d := Finset.HasAntidiagonal.mem_antidiagonal.mp hij
-      have hi' : 0 < ij.1 ∧ ij.1 < d := hi
-      have hj : ij.2 < d := by omega
-      have he := (pow_mono (a := a) (n := n) hi'.1)
-        (show aeval (fun i : Fin a => rowForm i u) (esymm (Fin a) ℂ ij.1) ∈ E a n ^ 1 by
-          simpa using elementary_mem_E u ij.1)
-      have hp := Submodule.mul_mem_mul he (ih ij.2 hj)
-      rw [← pow_add, hsum] at hp
-      convert (E a n ^ d).smul_mem ((-1 : ℂ) ^ ij.1) hp using 1
-      simp [Algebra.smul_def, mul_assoc]
-
-def inRow {a n : ℕ} (i : Fin a) : MvPolynomial (Fin n) ℂ →ₐ[ℂ] P a n :=
-  rename (fun k => (i, k))
-
-def pSum {a n : ℕ} : MvPolynomial (Fin n) ℂ →ₗ[ℂ] P a n :=
-  ∑ i : Fin a, (inRow i).toLinearMap
-
-lemma pSum_apply {a n : ℕ} (f : MvPolynomial (Fin n) ℂ) :
-    pSum (a := a) f = ∑ i : Fin a, inRow i f := by
-  simp [pSum, LinearMap.sum_apply]
-
-lemma pSum_homogeneous_mem {a n d : ℕ} {f : MvPolynomial (Fin n) ℂ}
-    (hf : f.IsHomogeneous d) : pSum (a := a) f ∈ E a n ^ d := by
-  have hle : homogeneousSubmodule (Fin n) ℂ d ≤ (E a n ^ d).comap pSum := by
-    rw [homogeneous_eq_span_powers]
-    apply Submodule.span_le.mpr
-    rintro _ ⟨u, rfl⟩
-    change pSum (a := a) ((∑ k : Fin n, u k • X k) ^ d) ∈ E a n ^ d
-    simpa [pSum_apply, inRow, rowForm] using powersum_mem (a := a) u d
-  exact hle hf
-
-end Foulkes.Chart
-
-namespace Foulkes.Chart
-open MvPolynomial Foulkes.Polarization Foulkes.Collision
-
-lemma distinctSum_homogeneous_mem {a n h : ℕ} (q : Fin h → MvPolynomial (Fin n) ℂ)
-    (d : Fin h → ℕ) (hq : ∀ r, (q r).IsHomogeneous (d r)) :
-    distinctSum (fun i : Fin a => inRow i) q ∈ E a n ^ (∑ r, d r) := by
-  classical
-  induction h with
-  | zero =>
-    simp only [distinctSum, Fin.prod_univ_zero, Fintype.sum_unique, Fin.sum_univ_zero, pow_zero]
-    exact Submodule.one_le.mp le_rfl
-  | succ h ih =>
-    rw [← Fin.cons_self_tail q, recurrence, Fin.sum_univ_succ]
-    apply Submodule.sub_mem
-    · have hh := Submodule.mul_mem_mul (pSum_homogeneous_mem (a := a) (hq 0))
-        (ih (Fin.tail q) (Fin.tail d) (fun r => hq r.succ))
-      rw [← pow_add] at hh
-      simpa only [pSum_apply, Fin.tail] using hh
-    · apply Submodule.sum_mem
-      intro r _
-      have hh := ih (Function.update (Fin.tail q) r (q 0 * q r.succ))
-        (Function.update (Fin.tail d) r (d 0 + d r.succ)) (by
-          intro j
-          by_cases hj : j = r
-          · subst j; simpa using (hq 0).mul (hq r.succ)
-          · simpa [hj, Fin.tail] using hq j.succ)
-      have hsum : (∑ j : Fin h, Function.update (Fin.tail d) r (d 0 + d r.succ) j) =
-          d 0 + ∑ j : Fin h, d j.succ := by
-        rw [Finset.sum_update_of_mem (Finset.mem_univ r)]
-        change d 0 + Fin.tail d r + ∑ j ∈ Finset.univ \ {r}, Fin.tail d j =
-          d 0 + ∑ j, Fin.tail d j
-        rw [Finset.sdiff_singleton_eq_erase, add_assoc,
-          Finset.add_sum_erase Finset.univ (Fin.tail d) (Finset.mem_univ r)]
-      rw [hsum] at hh
-      exact hh
-
-def rowExp {a n : ℕ} (d : (Fin a × Fin n) →₀ ℕ) (i : Fin a) : Fin n →₀ ℕ :=
-  Finsupp.equivFunOnFinite.symm (fun k => d (i,k))
-
-@[simp] lemma rowExp_apply {a n : ℕ} (d : (Fin a × Fin n) →₀ ℕ) (i : Fin a) (k : Fin n) :
-    rowExp d i k = d (i,k) := by simp [rowExp]
-
-lemma monomial_eq_prod_rows {a n : ℕ} (d : (Fin a × Fin n) →₀ ℕ) :
-    (monomial d 1 : P a n) = ∏ i : Fin a, inRow i (monomial (rowExp d i) 1) := by
-  classical
-  simp [inRow, monomial_eq, Finsupp.prod_fintype _ _ (fun _ => pow_zero _),
-    Fintype.prod_prod_type, rowExp]
-
-lemma renameRow_inRow {a n : ℕ} (σ : Equiv.Perm (Fin a)) (i : Fin a)
-    (f : MvPolynomial (Fin n) ℂ) : renameRow σ (inRow i f) = inRow (σ i) f := by
-  simp [renameRow, inRow, rename_rename, Function.comp_def]
-
-lemma permSum_monomial {a n : ℕ} (d : (Fin a × Fin n) →₀ ℕ) :
-    (∑ σ : Equiv.Perm (Fin a), renameRow σ (monomial d 1 : P a n)) =
-      distinctSum (fun i : Fin a => inRow i) (fun i => monomial (rowExp d i) 1) := by
-  classical
-  simp only [monomial_eq_prod_rows, map_prod, renameRow_inRow, distinctSum]
-  exact (Equiv.sum_comp (Equiv.embeddingEquivOfFinite (Fin a))
-    (fun σ => ∏ i : Fin a, inRow (σ i) (monomial (rowExp d i) 1))).symm
-
-lemma average_monomial_mem {a n : ℕ} (d : (Fin a × Fin n) →₀ ℕ) (c : ℂ) :
-    average a n (monomial d c) ∈ E a n ^ (∑ ij, d ij) := by
-  classical
-  have hhom (i : Fin a) : (monomial (rowExp d i) (1 : ℂ)).IsHomogeneous (∑ k, d (i,k)) := by
-    apply isWeightedHomogeneous_monomial
-    simp [Finsupp.weight_apply, Finsupp.sum_fintype, rowExp]
-  have hh := distinctSum_homogeneous_mem (a := a)
-    (fun i => (monomial (rowExp d i) (1 : ℂ))) (fun i => ∑ k, d (i,k)) hhom
-  rw [← Fintype.sum_prod_type, ← permSum_monomial] at hh
-  have hs : (monomial d c : P a n) = c • monomial d 1 := by simp [smul_monomial]
-  rw [hs, map_smul, average_apply]
-  apply Submodule.smul_mem
-  exact Submodule.smul_mem _ _ hh
-
-end Foulkes.Chart
-
-namespace Foulkes.Chart
-open MvPolynomial Foulkes.Polarization
-
-def tailExp {a n : ℕ} (d : (Fin a × Fin (n+1)) →₀ ℕ) : (Fin a × Fin n) →₀ ℕ :=
-  Finsupp.equivFunOnFinite.symm (fun ij => d (ij.1, ij.2.succ))
-
-@[simp] lemma tailExp_apply {a n : ℕ} (d : (Fin a × Fin (n+1)) →₀ ℕ) (ij : Fin a × Fin n) :
-    tailExp d ij = d (ij.1, ij.2.succ) := by simp [tailExp]
-
-def rehomExp {a n : ℕ} (k : ℕ) (d : (Fin a × Fin n) →₀ ℕ) :
-    (Fin a × Fin (n+1)) →₀ ℕ :=
-  Finsupp.equivFunOnFinite.symm (fun ij =>
-    Fin.cases (k - ∑ r : Fin n, d (ij.1, r)) (fun r => d (ij.1, r)) ij.2)
-
-lemma rehomExp_tailExp {a n k : ℕ} {d : (Fin a × Fin (n+1)) →₀ ℕ}
-    (hd : Finsupp.weight (weight a (n+1)) d = fun _ => k) : rehomExp k (tailExp d) = d := by
-  ext ij
-  rcases ij with ⟨i,r⟩
-  have hi := congrFun hd i
-  rw [weight_apply_row, Fin.sum_univ_succ] at hi
-  refine Fin.cases ?_ (fun r => ?_) r
-  · simp only [rehomExp, Finsupp.equivFunOnFinite_symm_apply_apply, Fin.cases_zero, tailExp_apply]
-    omega
-  · simp [rehomExp]
-
-lemma dehom_monomial {a n : ℕ} (d : (Fin a × Fin (n+1)) →₀ ℕ) (c : ℂ) :
-    dehom a n (monomial d c) = monomial (tailExp d) c := by
-  classical
-  simp only [monomial_eq, Finsupp.prod_fintype _ _ (fun _ => pow_zero _),
-    Fintype.prod_prod_type, map_mul, map_prod, map_pow]
-  rw [show (dehom a n) (C c) = C c from (dehom a n).commutes c]
-  congr 1
-  apply Finset.prod_congr rfl
-  intro i _
-  simp only [Fin.prod_univ_succ, dehom_X_zero, dehom_X_succ, one_pow, one_mul, tailExp_apply]
-
-def rehom (a n k : ℕ) : P a n →ₗ[ℂ] P a (n+1) :=
-  AddMonoidAlgebra.mapDomainLinearMap ℂ ℂ (rehomExp k)
-
-@[simp] lemma rehom_monomial {a n k : ℕ} (d : (Fin a × Fin n) →₀ ℕ) (c : ℂ) :
-    rehom a n k (monomial d c) = monomial (rehomExp k d) c := by
-  exact AddMonoidAlgebra.mapDomainLinearMap_single _ _ _
-
-lemma rehom_dehom {a n k : ℕ} {f : P a (n+1)} (hf : f ∈ W a (n+1) k) :
-    rehom a n k (dehom a n f) = f := by
-  induction hf using MvPolynomial.IsWeightedHomogeneous.induction_on with
-  | zero => simp
-  | add f g hf hg ihf ihg => simp [ihf, ihg]
-  | monomial d c hd => rw [dehom_monomial, rehom_monomial, rehomExp_tailExp hd]
-
-lemma dehom_injective_W {a n k : ℕ} {f g : P a (n+1)} (hf : f ∈ W a (n+1) k)
-    (hg : g ∈ W a (n+1) k) (hfg : dehom a n f = dehom a n g) : f = g := by
-  rw [← rehom_dehom hf, ← rehom_dehom hg, hfg]
-
-end Foulkes.Chart
-
-namespace Foulkes.Chart
-open MvPolynomial Foulkes.Polarization Foulkes.Collision
-
-lemma mul_mem_W {a n j k : ℕ} {f g : P a n} (hf : f ∈ W a n j) (hg : g ∈ W a n k) :
-    f * g ∈ W a n (j+k) := by
-  exact MvPolynomial.IsWeightedHomogeneous.mul hf hg
-
-lemma one_mem_W (a n : ℕ) : (1 : P a n) ∈ W a n 0 :=
-  MvPolynomial.isWeightedHomogeneous_one ℂ (weight a n)
-
-theorem invariant_mem_E_pow {a n t : ℕ} {f : P a n} (hf : f ∈ rowInvariants a n)
-    (ht : f.totalDegree ≤ t) : f ∈ E a n ^ t := by
-  classical
-  rw [← average_eq hf, f.as_sum]
-  simp only [map_sum]
-  apply Submodule.sum_mem
-  intro d hd
-  apply pow_mono _ (average_monomial_mem d _)
-  have hh := (MvPolynomial.le_totalDegree hd).trans ht
-  rw [Finsupp.sum_fintype _ _ (fun _ => rfl)] at hh
-  exact hh
-
-lemma dehom_average {a n : ℕ} (f : P a (n+1)) :
-    dehom a n (average a (n+1) f) = average a n (dehom a n f) := by
-  simp only [average_apply, map_smul, map_sum, dehom_renameRow]
-
-lemma dehom_R_mem_E_pow {a n k : ℕ} {f : P a (n+1)} (hf : f ∈ R a (n+1) k) :
-    dehom a n f ∈ E a n ^ (a*k) := by
-  classical
-  have haux {g : P a (n+1)} (hg : g ∈ W a (n+1) k) :
-      dehom a n (average a (n+1) g) ∈ E a n ^ (a*k) := by
-    induction hg using MvPolynomial.IsWeightedHomogeneous.induction_on with
-    | zero => simp
-    | add f g hf hg ihf ihg => simpa using (E a n ^ (a*k)).add_mem ihf ihg
-    | monomial d c hd =>
-      rw [dehom_average, dehom_monomial]
-      apply pow_mono _ (average_monomial_mem (tailExp d) c)
-      rw [Fintype.sum_prod_type]
-      calc
-        ∑ i : Fin a, ∑ r : Fin n, tailExp d (i,r) ≤ ∑ _i : Fin a, k := by
-          apply Finset.sum_le_sum
-          intro i _
-          have hi := congrFun hd i
-          rw [weight_apply_row, Fin.sum_univ_succ] at hi
-          simp only [tailExp_apply]
-          omega
-        _ = a*k := by simp
-  simpa only [average_eq hf.2] using haux hf.1
-
-def component (a n k : ℕ) : Submodule ℂ (P a n) := W a n k ⊓ (A a n).toSubmodule
-
-lemma E_pow_le_component_image (a n m : ℕ) :
-    E a n ^ m ≤ (component a (n+1) m).map (dehom a n).toLinearMap := by
-  induction m with
-  | zero =>
-    rw [pow_zero, Submodule.one_le]
-    exact Submodule.mem_map.mpr ⟨1, ⟨one_mem_W _ _, (A _ _).one_mem⟩, map_one (dehom a n)⟩
-  | succ m ih =>
-    intro f hf
-    rw [pow_succ] at hf
-    refine Submodule.mul_induction_on hf ?_ ?_
-    · intro x hx y hy
-      obtain ⟨g, hg, rfl⟩ := Submodule.mem_map.mp (ih hx)
-      obtain ⟨h, hh, rfl⟩ := Submodule.mem_map.mp hy
-      exact Submodule.mem_map.mpr ⟨g*h,
-        ⟨mul_mem_W hg.1 hh.1, (A _ _).mul_mem hg.2 (Algebra.subset_adjoin hh)⟩,
-        map_mul (dehom a n) g h⟩
-    · intro x y hx hy
-      exact Submodule.add_mem _ hx hy
-
-theorem chart_standard {a n k L : ℕ} {f : P a (n+1)} (hf : f ∈ R a (n+1) k)
-    (hL : a*k ≤ k+L) : z (Fin.cons (1 : ℂ) (fun _ : Fin n => 0)) ^ L * f ∈ A a (n+1) := by
-  have hd := pow_mono hL (dehom_R_mem_E_pow hf)
-  obtain ⟨g, hg, hgeq⟩ := Submodule.mem_map.mp (E_pow_le_component_image a n (k+L) hd)
-  have hzW : z (Fin.cons (1 : ℂ) (fun _ : Fin n => 0)) ^ L * f ∈ W a (n+1) (k+L) := by
-    simpa only [add_comm] using mul_mem_W (z_pow_mem _) hf.1
-  have heq : g = z (Fin.cons (1 : ℂ) (fun _ : Fin n => 0)) ^ L * f := by
-    apply dehom_injective_W hg.1 hzW
-    simpa [dehom_z_cons, rowForm] using hgeq
-  rw [← heq]
-  exact hg.2
-
-end Foulkes.Chart
-
-namespace Foulkes.Chart
-open MvPolynomial Foulkes.Polarization
-
-def columns {a n m : ℕ} (g : (Fin n → ℂ) →ₗ[ℂ] (Fin m → ℂ)) : P a n →ₐ[ℂ] P a m :=
-  aeval (fun ij : Fin a × Fin n => rowForm ij.1 (g (Pi.single ij.2 1)))
-
-def rowLinear {a n : ℕ} (i : Fin a) : (Fin n → ℂ) →ₗ[ℂ] P a n where
-  toFun := rowForm i
-  map_add' u v := by simp [rowForm, add_smul, Finset.sum_add_distrib]
-  map_smul' c u := by simp [rowForm, mul_smul, Finset.smul_sum]
-
-@[simp] lemma rowLinear_apply {a n : ℕ} (i : Fin a) (v : Fin n → ℂ) :
-    rowLinear i v = rowForm i v := rfl
-
-@[simp] lemma columns_X {a n m : ℕ} (g : (Fin n → ℂ) →ₗ[ℂ] (Fin m → ℂ))
-    (ij : Fin a × Fin n) : columns g (X ij) = rowForm ij.1 (g (Pi.single ij.2 1)) := by
-  simp [columns]
-
-lemma rowForm_single {a n : ℕ} (i : Fin a) (k : Fin n) :
-    rowForm i (Pi.single k 1) = (X (i,k) : P a n) := by
-  classical
-  simp [rowForm, Pi.single_apply]
-
-lemma columns_rowForm {a n m : ℕ} (g : (Fin n → ℂ) →ₗ[ℂ] (Fin m → ℂ))
-    (i : Fin a) (v : Fin n → ℂ) : columns g (rowForm i v) = rowForm i (g v) := by
-  classical
-  have hv : ∑ k : Fin n, v k • (Pi.single k (1 : ℂ) : Fin n → ℂ) = v := by
-    ext k
-    simp [Pi.single_apply]
-  calc
-    columns g (rowForm i v) = ∑ k : Fin n, v k • rowLinear i (g (Pi.single k 1)) := by
-      simp only [rowForm, map_sum, map_smul, columns_X, rowLinear_apply]
-    _ = rowLinear i (g (∑ k : Fin n, v k • Pi.single k 1)) := by simp
-    _ = rowForm i (g v) := by rw [hv]; rfl
-
-lemma columns_pureRows {a n m j : ℕ} (g : (Fin n → ℂ) →ₗ[ℂ] (Fin m → ℂ))
-    (v : Fin a → Fin n → ℂ) : columns g (pureRows j v) = pureRows j (fun i => g (v i)) := by
-  simp only [pureRows, map_prod, map_pow, columns_rowForm]
-
-lemma columns_mem_W {a n m j : ℕ} (g : (Fin n → ℂ) →ₗ[ℂ] (Fin m → ℂ))
-    {f : P a n} (hf : f ∈ W a n j) : columns g f ∈ W a m j := by
-  have hh : W a n j ≤ (W a m j).comap (columns g).toLinearMap := by
-    rw [W_eq_span_pureRows]
-    apply Submodule.span_le.mpr
-    rintro _ ⟨v, rfl⟩
-    change columns g (pureRows j v) ∈ W a m j
-    rw [columns_pureRows]
-    exact pureRows_mem _
-  exact hh hf
-
-lemma columns_renameRow {a n m : ℕ} (g : (Fin n → ℂ) →ₗ[ℂ] (Fin m → ℂ))
-    (σ : Equiv.Perm (Fin a)) (f : P a n) :
-    columns g (renameRow σ f) = renameRow σ (columns g f) := by
-  have hh : (columns g).comp (renameRow σ) = (renameRow σ).comp (columns g) := by
-    ext ij
-    simp [columns, renameRow, rowForm]
-  exact DFunLike.congr_fun hh f
-
-lemma columns_mem_R {a n m j : ℕ} (g : (Fin n → ℂ) →ₗ[ℂ] (Fin m → ℂ))
-    {f : P a n} (hf : f ∈ R a n j) : columns g f ∈ R a m j := by
-  refine ⟨columns_mem_W g hf.1, fun σ => ?_⟩
-  rw [← columns_renameRow, hf.2 σ]
-
-lemma columns_mem_A {a n m : ℕ} (g : (Fin n → ℂ) →ₗ[ℂ] (Fin m → ℂ))
-    {f : P a n} (hf : f ∈ A a n) : columns g f ∈ A a m := by
-  have hh : A a n ≤ (A a m).comap (columns g) := by
-    apply Algebra.adjoin_le
-    intro h hh
-    exact Algebra.subset_adjoin (columns_mem_R g hh)
-  exact hh hf
-
-lemma columns_z {a n m : ℕ} (g : (Fin n → ℂ) →ₗ[ℂ] (Fin m → ℂ)) (v : Fin n → ℂ) :
-    columns (a := a) g (z v) = z (g v) := by simp [z, columns_rowForm]
-
-lemma columns_inverse {a n : ℕ} (g : (Fin n → ℂ) ≃ₗ[ℂ] (Fin n → ℂ)) (f : P a n) :
-    columns g.toLinearMap (columns g.symm.toLinearMap f) = f := by
-  have hh : (columns (a := a) g.toLinearMap).comp (columns g.symm.toLinearMap) =
-      AlgHom.id ℂ (P a n) := by
-    apply MvPolynomial.algHom_ext
-    intro ij
-    simp only [AlgHom.comp_apply, columns_X, columns_rowForm, LinearEquiv.coe_coe,
-      LinearEquiv.apply_symm_apply, AlgHom.id_apply, rowForm_single]
-  exact DFunLike.congr_fun hh f
-
-lemma exists_linearEquiv_send {n : ℕ} {v w : Fin n → ℂ} (hv : v ≠ 0) (hw : w ≠ 0) :
-    ∃ g : (Fin n → ℂ) ≃ₗ[ℂ] (Fin n → ℂ), g v = w := by
-  let ev := LinearEquiv.toSpanNonzeroSingleton ℂ (Fin n → ℂ) v hv
-  let ew := LinearEquiv.toSpanNonzeroSingleton ℂ (Fin n → ℂ) w hw
-  obtain ⟨g, hg⟩ := Submodule.exists_linearEquiv_restrict_eq (ev.symm.trans ew)
-  refine ⟨g, ?_⟩
-  have hh := hg (ev 1)
-  change (ew (ev.symm (ev 1)) : Fin n → ℂ) = g (ev 1) at hh
-  rw [ev.symm_apply_apply] at hh
-  have hev : (ev 1 : Fin n → ℂ) = v := by simp [ev]
-  have hew : (ew 1 : Fin n → ℂ) = w := by simp [ew]
-  rw [hev, hew] at hh
-  exact hh.symm
-
-theorem chart {a n k L : ℕ} {f : P a n} (hf : f ∈ R a n k)
-    (v : Fin n → ℂ) (hv : v ≠ 0) (hL : a*k ≤ k+L) : z v ^ L * f ∈ A a n := by
-  cases n with
-  | zero =>
-    exfalso
-    apply hv
-    ext i
-    exact Fin.elim0 i
-  | succ n =>
-    let e : Fin (n+1) → ℂ := Fin.cons 1 (fun _ => 0)
-    have he : e ≠ 0 := by
-      intro hh
-      have := congrFun hh 0
-      simp [e] at this
-    obtain ⟨g, hg⟩ := exists_linearEquiv_send he hv
-    have hsource := chart_standard (columns_mem_R g.symm.toLinearMap hf) hL
-    have htarget := columns_mem_A g.toLinearMap hsource
-    change g (Fin.cons 1 (fun _ : Fin n => 0)) = v at hg
-    simpa only [map_mul, map_pow, columns_z, columns_inverse, LinearEquiv.coe_coe, hg] using htarget
-
-end Foulkes.Chart
-
-namespace Foulkes.Chart
-open MvPolynomial Foulkes.Polarization
-
-lemma z_zero {a n : ℕ} (ha : 0 < a) : z (a := a) (0 : Fin n → ℂ) = 0 := by
-  simp [z, rowForm, zero_pow ha.ne']
-
-theorem independent_chart_bound (a n b : ℕ) (ha : 2 ≤ a) (hb : a*(a-1)^2 ≤ b) :
-    R a n b = component a n b := by
-  classical
-  let L := (a-1)^2
-  have ha1 : 1 ≤ a := by omega
-  have hapos : 0 < a := by omega
-  have hLpos : 0 < L := by
-    have hs : 0 < a-1 := by omega
-    exact pow_pos hs 2
-  have hlow (v : Fin n → ℂ) (k : ℕ) (hk : k < a) {f : P a n} (hf : f ∈ R a n k) :
-      z v ^ L * f ∈ A a n := by
-    by_cases hv : v = 0
-    · simp only [hv, z_zero hapos, zero_pow hLpos.ne', zero_mul]
-      exact (A a n).zero_mem
-    · apply chart hf v hv
-      have hka : k ≤ a-1 := by omega
-      have hm := Nat.mul_le_mul_left (a-1) hka
-      have hs : a-1+1 = a := Nat.sub_add_cancel ha1
-      dsimp [L]
-      nlinarith
-  have hmodule (v : Fin n → ℂ) {f : P a n} (hf : f ∈ lowDegreeModule a n) :
-      z v ^ L * f ∈ A a n := by
-    induction hf using Submodule.span_induction with
-    | mem f hf =>
-      obtain ⟨k,hk,hf⟩ := hf
-      exact hlow v k hk hf
-    | zero => simp
-    | add f g hf hg ihf ihg => simpa [mul_add] using (A a n).add_mem ihf ihg
-    | smul c f hf ih =>
-      change z v ^ L * ((c : P a n) * f) ∈ A a n
-      simpa [mul_left_comm, mul_assoc] using (A a n).mul_mem c.property ih
-  have hspan : invariantPolarizedSpan a n b L ≤ (A a n).toSubmodule := by
-    apply Submodule.span_le.mpr
-    rintro _ ⟨v, f, hf, rfl⟩
-    exact hmodule v (module_generation ha1 _ hf)
-  apply le_antisymm
-  · intro f hf
-    refine ⟨hf.1, ?_⟩
-    exact hspan ((R_polarization ha1 hb).le hf)
-  · exact AComponent_le_R a n b
-
-end Foulkes.Chart
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.FoulkesSixth.InvariantPolarization
+import OAI.RepresentationTheory.FoulkesSixth.Collision
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.Chart
+open MvPolynomial Foulkes.Polarization Foulkes.PolynomialSpan
+
+def dehom (a n : ℕ) : P a (n + 1) →ₐ[ℂ] P a n :=
+  aeval (fun ij : Fin a × Fin (n + 1) => Fin.cases 1 (fun k => X (ij.1, k)) ij.2)
+
+@[simp] lemma dehom_X_zero {a n : ℕ} (i : Fin a) :
+    dehom a n (X (i, 0)) = 1 := by simp [dehom]
+@[simp] lemma dehom_X_succ {a n : ℕ} (i : Fin a) (k : Fin n) :
+    dehom a n (X (i, k.succ)) = X (i, k) := by simp [dehom]
+
+lemma dehom_rowForm {a n : ℕ} (i : Fin a) (v : Fin (n + 1) → ℂ) :
+    dehom a n (rowForm i v) = algebraMap ℂ (P a n) (v 0) + rowForm i (fun k => v k.succ) := by
+  simp [rowForm, Fin.sum_univ_succ, Algebra.smul_def]
+
+lemma dehom_z_cons {a n : ℕ} (s : ℂ) (u : Fin n → ℂ) :
+    dehom a n (z (Fin.cons s u)) = ∏ i : Fin a, (algebraMap ℂ (P a n) s + rowForm i u) := by
+  simp [z, dehom_rowForm]
+
+lemma dehom_renameRow {a n : ℕ} (σ : Equiv.Perm (Fin a)) (f : P a (n+1)) :
+    dehom a n (renameRow σ f) = renameRow σ (dehom a n f) := by
+  have hh : (dehom a n).comp (renameRow σ) = (renameRow σ).comp (dehom a n) := by
+    ext ij
+    rcases ij with ⟨i, k⟩
+    refine Fin.cases ?_ (fun k => ?_) k <;> simp [renameRow]
+  exact DFunLike.congr_fun hh f
+
+def E (a n : ℕ) : Submodule ℂ (P a n) := (R a (n+1) 1).map (dehom a n).toLinearMap
+
+lemma dehom_z_mem_E {a n : ℕ} (v : Fin (n+1) → ℂ) :
+    dehom a n (z v) ∈ E a n := Submodule.mem_map.mpr ⟨z v, z_mem_R v, rfl⟩
+
+lemma one_mem_E (a n : ℕ) : (1 : P a n) ∈ E a n := by
+  have hh := dehom_z_mem_E (a := a) (Fin.cons (1 : ℂ) (fun _ : Fin n => 0))
+  simpa [dehom_z_cons, rowForm] using hh
+
+lemma pow_mono {a n k l : ℕ} (h : k ≤ l) : E a n ^ k ≤ E a n ^ l := by
+  exact pow_le_pow_right₀ (Submodule.one_le.mpr (one_mem_E a n)) h
+
+lemma elementary_mem_E {a n : ℕ} (u : Fin n → ℂ) (h : ℕ) :
+    aeval (fun i : Fin a => rowForm i u) (esymm (Fin a) ℂ h) ∈ E a n := by
+  classical
+  let G : MvPolynomial Unit (P a n) := ∏ i : Fin a, (X () * C (rowForm i u) + 1)
+  have heval (s : Unit → ℂ) : eval (fun i => algebraMap ℂ (P a n) (s i)) G ∈ E a n := by
+    have hh := dehom_z_mem_E (a := a) (Fin.cons (1 : ℂ) (fun k => s () * u k))
+    convert hh using 1
+    simp only [G, eval_prod, eval_add, eval_mul, eval_X, eval_C, map_one,
+      dehom_z_cons, map_one, rowForm,
+      Algebra.smul_def]
+    apply Finset.prod_congr rfl
+    intro i _
+    simp [map_mul, Finset.mul_sum, mul_assoc, add_comm]
+  have hc := coeff_mem_of_evaluations G (E a n) heval (Finsupp.single () h)
+  have hg : G = ∑ s ∈ (Finset.univ : Finset (Fin a)).powerset,
+      (X () : MvPolynomial Unit (P a n)) ^ s.card * C (∏ i ∈ s, rowForm i u) := by
+    simp only [G, Finset.prod_add, Finset.prod_const_one, mul_one, Finset.prod_mul_distrib,
+      Finset.prod_const, ← map_prod]
+  rw [hg] at hc
+  have hcoeff (s : Finset (Fin a)) :
+      (X () ^ s.card * C (∏ i ∈ s, rowForm i u)).coeff (Finsupp.single () h) =
+        if s.card = h then ∏ i ∈ s, rowForm i u else 0 := by
+    rw [mul_comm, coeff_C_mul, X_pow_eq_monomial]
+    simp only [coeff_monomial]
+    simp [eq_comm]
+  simp only [coeff_sum, hcoeff] at hc
+  simpa only [esymm, map_sum, map_prod, aeval_X, Finset.powersetCard_eq_filter,
+    Finset.sum_filter, apply_ite, map_prod, aeval_X, map_zero] using hc
+
+lemma powersum_mem {a n : ℕ} (u : Fin n → ℂ) (d : ℕ) :
+    (∑ i : Fin a, rowForm i u ^ d) ∈ E a n ^ d := by
+  classical
+  induction d using Nat.strong_induction_on with
+  | h d ih =>
+    by_cases hd : d = 0
+    · subst d
+      simpa using (Submodule.sum_mem (1 : Submodule ℂ (P a n))
+        (fun (i : Fin a) (_ : i ∈ Finset.univ) => (Submodule.one_le.mp le_rfl)))
+    have hdpos : 0 < d := Nat.pos_of_ne_zero hd
+    have hn := congrArg (aeval (fun i : Fin a => rowForm i u))
+      (MvPolynomial.psum_eq_mul_esymm_sub_sum (Fin a) ℂ d hdpos)
+    simp only [psum, map_sum, map_pow, aeval_X, map_sub, map_mul, map_neg,
+      map_one, map_natCast] at hn
+    rw [hn]
+    apply Submodule.sub_mem
+    · have hh := (pow_mono (a := a) (n := n) hdpos)
+        (show aeval (fun i : Fin a => rowForm i u) (esymm (Fin a) ℂ d) ∈ E a n ^ 1 by
+          simpa using elementary_mem_E u d)
+      convert (E a n ^ d).smul_mem (((-1 : ℂ) ^ (d+1)) * (d : ℂ)) hh using 1
+      simp [Algebra.smul_def, mul_assoc]
+    · apply Submodule.sum_mem
+      intro ij hij
+      obtain ⟨hij, hi⟩ := Finset.mem_filter.mp hij
+      have hsum : ij.1 + ij.2 = d := Finset.HasAntidiagonal.mem_antidiagonal.mp hij
+      have hi' : 0 < ij.1 ∧ ij.1 < d := hi
+      have hj : ij.2 < d := by omega
+      have he := (pow_mono (a := a) (n := n) hi'.1)
+        (show aeval (fun i : Fin a => rowForm i u) (esymm (Fin a) ℂ ij.1) ∈ E a n ^ 1 by
+          simpa using elementary_mem_E u ij.1)
+      have hp := Submodule.mul_mem_mul he (ih ij.2 hj)
+      rw [← pow_add, hsum] at hp
+      convert (E a n ^ d).smul_mem ((-1 : ℂ) ^ ij.1) hp using 1
+      simp [Algebra.smul_def, mul_assoc]
+
+def inRow {a n : ℕ} (i : Fin a) : MvPolynomial (Fin n) ℂ →ₐ[ℂ] P a n :=
+  rename (fun k => (i, k))
+
+def pSum {a n : ℕ} : MvPolynomial (Fin n) ℂ →ₗ[ℂ] P a n :=
+  ∑ i : Fin a, (inRow i).toLinearMap
+
+lemma pSum_apply {a n : ℕ} (f : MvPolynomial (Fin n) ℂ) :
+    pSum (a := a) f = ∑ i : Fin a, inRow i f := by
+  simp [pSum, LinearMap.sum_apply]
+
+lemma pSum_homogeneous_mem {a n d : ℕ} {f : MvPolynomial (Fin n) ℂ}
+    (hf : f.IsHomogeneous d) : pSum (a := a) f ∈ E a n ^ d := by
+  have hle : homogeneousSubmodule (Fin n) ℂ d ≤ (E a n ^ d).comap pSum := by
+    rw [homogeneous_eq_span_powers]
+    apply Submodule.span_le.mpr
+    rintro _ ⟨u, rfl⟩
+    change pSum (a := a) ((∑ k : Fin n, u k • X k) ^ d) ∈ E a n ^ d
+    simpa [pSum_apply, inRow, rowForm] using powersum_mem (a := a) u d
+  exact hle hf
+
+end Foulkes.Chart
+
+namespace Foulkes.Chart
+open MvPolynomial Foulkes.Polarization Foulkes.Collision
+
+lemma distinctSum_homogeneous_mem {a n h : ℕ} (q : Fin h → MvPolynomial (Fin n) ℂ)
+    (d : Fin h → ℕ) (hq : ∀ r, (q r).IsHomogeneous (d r)) :
+    distinctSum (fun i : Fin a => inRow i) q ∈ E a n ^ (∑ r, d r) := by
+  classical
+  induction h with
+  | zero =>
+    simp only [distinctSum, Fin.prod_univ_zero, Fintype.sum_unique, Fin.sum_univ_zero, pow_zero]
+    exact Submodule.one_le.mp le_rfl
+  | succ h ih =>
+    rw [← Fin.cons_self_tail q, recurrence, Fin.sum_univ_succ]
+    apply Submodule.sub_mem
+    · have hh := Submodule.mul_mem_mul (pSum_homogeneous_mem (a := a) (hq 0))
+        (ih (Fin.tail q) (Fin.tail d) (fun r => hq r.succ))
+      rw [← pow_add] at hh
+      simpa only [pSum_apply, Fin.tail] using hh
+    · apply Submodule.sum_mem
+      intro r _
+      have hh := ih (Function.update (Fin.tail q) r (q 0 * q r.succ))
+        (Function.update (Fin.tail d) r (d 0 + d r.succ)) (by
+          intro j
+          by_cases hj : j = r
+          · subst j; simpa using (hq 0).mul (hq r.succ)
+          · simpa [hj, Fin.tail] using hq j.succ)
+      have hsum : (∑ j : Fin h, Function.update (Fin.tail d) r (d 0 + d r.succ) j) =
+          d 0 + ∑ j : Fin h, d j.succ := by
+        rw [Finset.sum_update_of_mem (Finset.mem_univ r)]
+        change d 0 + Fin.tail d r + ∑ j ∈ Finset.univ \ {r}, Fin.tail d j =
+          d 0 + ∑ j, Fin.tail d j
+        rw [Finset.sdiff_singleton_eq_erase, add_assoc,
+          Finset.add_sum_erase Finset.univ (Fin.tail d) (Finset.mem_univ r)]
+      rw [hsum] at hh
+      exact hh
+
+def rowExp {a n : ℕ} (d : (Fin a × Fin n) →₀ ℕ) (i : Fin a) : Fin n →₀ ℕ :=
+  Finsupp.equivFunOnFinite.symm (fun k => d (i,k))
+
+@[simp] lemma rowExp_apply {a n : ℕ} (d : (Fin a × Fin n) →₀ ℕ) (i : Fin a) (k : Fin n) :
+    rowExp d i k = d (i,k) := by simp [rowExp]
+
+lemma monomial_eq_prod_rows {a n : ℕ} (d : (Fin a × Fin n) →₀ ℕ) :
+    (monomial d 1 : P a n) = ∏ i : Fin a, inRow i (monomial (rowExp d i) 1) := by
+  classical
+  simp [inRow, monomial_eq, Finsupp.prod_fintype _ _ (fun _ => pow_zero _),
+    Fintype.prod_prod_type, rowExp]
+
+lemma renameRow_inRow {a n : ℕ} (σ : Equiv.Perm (Fin a)) (i : Fin a)
+    (f : MvPolynomial (Fin n) ℂ) : renameRow σ (inRow i f) = inRow (σ i) f := by
+  simp [renameRow, inRow, rename_rename, Function.comp_def]
+
+lemma permSum_monomial {a n : ℕ} (d : (Fin a × Fin n) →₀ ℕ) :
+    (∑ σ : Equiv.Perm (Fin a), renameRow σ (monomial d 1 : P a n)) =
+      distinctSum (fun i : Fin a => inRow i) (fun i => monomial (rowExp d i) 1) := by
+  classical
+  simp only [monomial_eq_prod_rows, map_prod, renameRow_inRow, distinctSum]
+  exact (Equiv.sum_comp (Equiv.embeddingEquivOfFinite (Fin a))
+    (fun σ => ∏ i : Fin a, inRow (σ i) (monomial (rowExp d i) 1))).symm
+
+lemma average_monomial_mem {a n : ℕ} (d : (Fin a × Fin n) →₀ ℕ) (c : ℂ) :
+    average a n (monomial d c) ∈ E a n ^ (∑ ij, d ij) := by
+  classical
+  have hhom (i : Fin a) : (monomial (rowExp d i) (1 : ℂ)).IsHomogeneous (∑ k, d (i,k)) := by
+    apply isWeightedHomogeneous_monomial
+    simp [Finsupp.weight_apply, Finsupp.sum_fintype, rowExp]
+  have hh := distinctSum_homogeneous_mem (a := a)
+    (fun i => (monomial (rowExp d i) (1 : ℂ))) (fun i => ∑ k, d (i,k)) hhom
+  rw [← Fintype.sum_prod_type, ← permSum_monomial] at hh
+  have hs : (monomial d c : P a n) = c • monomial d 1 := by simp [smul_monomial]
+  rw [hs, map_smul, average_apply]
+  apply Submodule.smul_mem
+  exact Submodule.smul_mem _ _ hh
+
+end Foulkes.Chart
+
+namespace Foulkes.Chart
+open MvPolynomial Foulkes.Polarization
+
+def tailExp {a n : ℕ} (d : (Fin a × Fin (n+1)) →₀ ℕ) : (Fin a × Fin n) →₀ ℕ :=
+  Finsupp.equivFunOnFinite.symm (fun ij => d (ij.1, ij.2.succ))
+
+@[simp] lemma tailExp_apply {a n : ℕ} (d : (Fin a × Fin (n+1)) →₀ ℕ) (ij : Fin a × Fin n) :
+    tailExp d ij = d (ij.1, ij.2.succ) := by simp [tailExp]
+
+def rehomExp {a n : ℕ} (k : ℕ) (d : (Fin a × Fin n) →₀ ℕ) :
+    (Fin a × Fin (n+1)) →₀ ℕ :=
+  Finsupp.equivFunOnFinite.symm (fun ij =>
+    Fin.cases (k - ∑ r : Fin n, d (ij.1, r)) (fun r => d (ij.1, r)) ij.2)
+
+lemma rehomExp_tailExp {a n k : ℕ} {d : (Fin a × Fin (n+1)) →₀ ℕ}
+    (hd : Finsupp.weight (weight a (n+1)) d = fun _ => k) : rehomExp k (tailExp d) = d := by
+  ext ij
+  rcases ij with ⟨i,r⟩
+  have hi := congrFun hd i
+  rw [weight_apply_row, Fin.sum_univ_succ] at hi
+  refine Fin.cases ?_ (fun r => ?_) r
+  · simp only [rehomExp, Finsupp.equivFunOnFinite_symm_apply_apply, Fin.cases_zero, tailExp_apply]
+    omega
+  · simp [rehomExp]
+
+lemma dehom_monomial {a n : ℕ} (d : (Fin a × Fin (n+1)) →₀ ℕ) (c : ℂ) :
+    dehom a n (monomial d c) = monomial (tailExp d) c := by
+  classical
+  simp only [monomial_eq, Finsupp.prod_fintype _ _ (fun _ => pow_zero _),
+    Fintype.prod_prod_type, map_mul, map_prod, map_pow]
+  rw [show (dehom a n) (C c) = C c from (dehom a n).commutes c]
+  congr 1
+  apply Finset.prod_congr rfl
+  intro i _
+  simp only [Fin.prod_univ_succ, dehom_X_zero, dehom_X_succ, one_pow, one_mul, tailExp_apply]
+
+def rehom (a n k : ℕ) : P a n →ₗ[ℂ] P a (n+1) :=
+  AddMonoidAlgebra.mapDomainLinearMap ℂ ℂ (rehomExp k)
+
+@[simp] lemma rehom_monomial {a n k : ℕ} (d : (Fin a × Fin n) →₀ ℕ) (c : ℂ) :
+    rehom a n k (monomial d c) = monomial (rehomExp k d) c := by
+  exact AddMonoidAlgebra.mapDomainLinearMap_single _ _ _
+
+lemma rehom_dehom {a n k : ℕ} {f : P a (n+1)} (hf : f ∈ W a (n+1) k) :
+    rehom a n k (dehom a n f) = f := by
+  induction hf using MvPolynomial.IsWeightedHomogeneous.induction_on with
+  | zero => simp
+  | add f g hf hg ihf ihg => simp [ihf, ihg]
+  | monomial d c hd => rw [dehom_monomial, rehom_monomial, rehomExp_tailExp hd]
+
+lemma dehom_injective_W {a n k : ℕ} {f g : P a (n+1)} (hf : f ∈ W a (n+1) k)
+    (hg : g ∈ W a (n+1) k) (hfg : dehom a n f = dehom a n g) : f = g := by
+  rw [← rehom_dehom hf, ← rehom_dehom hg, hfg]
+
+end Foulkes.Chart
+
+namespace Foulkes.Chart
+open MvPolynomial Foulkes.Polarization Foulkes.Collision
+
+lemma mul_mem_W {a n j k : ℕ} {f g : P a n} (hf : f ∈ W a n j) (hg : g ∈ W a n k) :
+    f * g ∈ W a n (j+k) := by
+  exact MvPolynomial.IsWeightedHomogeneous.mul hf hg
+
+lemma one_mem_W (a n : ℕ) : (1 : P a n) ∈ W a n 0 :=
+  MvPolynomial.isWeightedHomogeneous_one ℂ (weight a n)
+
+theorem invariant_mem_E_pow {a n t : ℕ} {f : P a n} (hf : f ∈ rowInvariants a n)
+    (ht : f.totalDegree ≤ t) : f ∈ E a n ^ t := by
+  classical
+  rw [← average_eq hf, f.as_sum]
+  simp only [map_sum]
+  apply Submodule.sum_mem
+  intro d hd
+  apply pow_mono _ (average_monomial_mem d _)
+  have hh := (MvPolynomial.le_totalDegree hd).trans ht
+  rw [Finsupp.sum_fintype _ _ (fun _ => rfl)] at hh
+  exact hh
+
+lemma dehom_average {a n : ℕ} (f : P a (n+1)) :
+    dehom a n (average a (n+1) f) = average a n (dehom a n f) := by
+  simp only [average_apply, map_smul, map_sum, dehom_renameRow]
+
+lemma dehom_R_mem_E_pow {a n k : ℕ} {f : P a (n+1)} (hf : f ∈ R a (n+1) k) :
+    dehom a n f ∈ E a n ^ (a*k) := by
+  classical
+  have haux {g : P a (n+1)} (hg : g ∈ W a (n+1) k) :
+      dehom a n (average a (n+1) g) ∈ E a n ^ (a*k) := by
+    induction hg using MvPolynomial.IsWeightedHomogeneous.induction_on with
+    | zero => simp
+    | add f g hf hg ihf ihg => simpa using (E a n ^ (a*k)).add_mem ihf ihg
+    | monomial d c hd =>
+      rw [dehom_average, dehom_monomial]
+      apply pow_mono _ (average_monomial_mem (tailExp d) c)
+      rw [Fintype.sum_prod_type]
+      calc
+        ∑ i : Fin a, ∑ r : Fin n, tailExp d (i,r) ≤ ∑ _i : Fin a, k := by
+          apply Finset.sum_le_sum
+          intro i _
+          have hi := congrFun hd i
+          rw [weight_apply_row, Fin.sum_univ_succ] at hi
+          simp only [tailExp_apply]
+          omega
+        _ = a*k := by simp
+  simpa only [average_eq hf.2] using haux hf.1
+
+def component (a n k : ℕ) : Submodule ℂ (P a n) := W a n k ⊓ (A a n).toSubmodule
+
+lemma E_pow_le_component_image (a n m : ℕ) :
+    E a n ^ m ≤ (component a (n+1) m).map (dehom a n).toLinearMap := by
+  induction m with
+  | zero =>
+    rw [pow_zero, Submodule.one_le]
+    exact Submodule.mem_map.mpr ⟨1, ⟨one_mem_W _ _, (A _ _).one_mem⟩, map_one (dehom a n)⟩
+  | succ m ih =>
+    intro f hf
+    rw [pow_succ] at hf
+    refine Submodule.mul_induction_on hf ?_ ?_
+    · intro x hx y hy
+      obtain ⟨g, hg, rfl⟩ := Submodule.mem_map.mp (ih hx)
+      obtain ⟨h, hh, rfl⟩ := Submodule.mem_map.mp hy
+      exact Submodule.mem_map.mpr ⟨g*h,
+        ⟨mul_mem_W hg.1 hh.1, (A _ _).mul_mem hg.2 (Algebra.subset_adjoin hh)⟩,
+        map_mul (dehom a n) g h⟩
+    · intro x y hx hy
+      exact Submodule.add_mem _ hx hy
+
+theorem chart_standard {a n k L : ℕ} {f : P a (n+1)} (hf : f ∈ R a (n+1) k)
+    (hL : a*k ≤ k+L) : z (Fin.cons (1 : ℂ) (fun _ : Fin n => 0)) ^ L * f ∈ A a (n+1) := by
+  have hd := pow_mono hL (dehom_R_mem_E_pow hf)
+  obtain ⟨g, hg, hgeq⟩ := Submodule.mem_map.mp (E_pow_le_component_image a n (k+L) hd)
+  have hzW : z (Fin.cons (1 : ℂ) (fun _ : Fin n => 0)) ^ L * f ∈ W a (n+1) (k+L) := by
+    simpa only [add_comm] using mul_mem_W (z_pow_mem _) hf.1
+  have heq : g = z (Fin.cons (1 : ℂ) (fun _ : Fin n => 0)) ^ L * f := by
+    apply dehom_injective_W hg.1 hzW
+    simpa [dehom_z_cons, rowForm] using hgeq
+  rw [← heq]
+  exact hg.2
+
+end Foulkes.Chart
+
+namespace Foulkes.Chart
+open MvPolynomial Foulkes.Polarization
+
+def columns {a n m : ℕ} (g : (Fin n → ℂ) →ₗ[ℂ] (Fin m → ℂ)) : P a n →ₐ[ℂ] P a m :=
+  aeval (fun ij : Fin a × Fin n => rowForm ij.1 (g (Pi.single ij.2 1)))
+
+def rowLinear {a n : ℕ} (i : Fin a) : (Fin n → ℂ) →ₗ[ℂ] P a n where
+  toFun := rowForm i
+  map_add' u v := by simp [rowForm, add_smul, Finset.sum_add_distrib]
+  map_smul' c u := by simp [rowForm, mul_smul, Finset.smul_sum]
+
+@[simp] lemma rowLinear_apply {a n : ℕ} (i : Fin a) (v : Fin n → ℂ) :
+    rowLinear i v = rowForm i v := rfl
+
+@[simp] lemma columns_X {a n m : ℕ} (g : (Fin n → ℂ) →ₗ[ℂ] (Fin m → ℂ))
+    (ij : Fin a × Fin n) : columns g (X ij) = rowForm ij.1 (g (Pi.single ij.2 1)) := by
+  simp [columns]
+
+lemma rowForm_single {a n : ℕ} (i : Fin a) (k : Fin n) :
+    rowForm i (Pi.single k 1) = (X (i,k) : P a n) := by
+  classical
+  simp [rowForm, Pi.single_apply]
+
+lemma columns_rowForm {a n m : ℕ} (g : (Fin n → ℂ) →ₗ[ℂ] (Fin m → ℂ))
+    (i : Fin a) (v : Fin n → ℂ) : columns g (rowForm i v) = rowForm i (g v) := by
+  classical
+  have hv : ∑ k : Fin n, v k • (Pi.single k (1 : ℂ) : Fin n → ℂ) = v := by
+    ext k
+    simp [Pi.single_apply]
+  calc
+    columns g (rowForm i v) = ∑ k : Fin n, v k • rowLinear i (g (Pi.single k 1)) := by
+      simp only [rowForm, map_sum, map_smul, columns_X, rowLinear_apply]
+    _ = rowLinear i (g (∑ k : Fin n, v k • Pi.single k 1)) := by simp
+    _ = rowForm i (g v) := by rw [hv]; rfl
+
+lemma columns_pureRows {a n m j : ℕ} (g : (Fin n → ℂ) →ₗ[ℂ] (Fin m → ℂ))
+    (v : Fin a → Fin n → ℂ) : columns g (pureRows j v) = pureRows j (fun i => g (v i)) := by
+  simp only [pureRows, map_prod, map_pow, columns_rowForm]
+
+lemma columns_mem_W {a n m j : ℕ} (g : (Fin n → ℂ) →ₗ[ℂ] (Fin m → ℂ))
+    {f : P a n} (hf : f ∈ W a n j) : columns g f ∈ W a m j := by
+  have hh : W a n j ≤ (W a m j).comap (columns g).toLinearMap := by
+    rw [W_eq_span_pureRows]
+    apply Submodule.span_le.mpr
+    rintro _ ⟨v, rfl⟩
+    change columns g (pureRows j v) ∈ W a m j
+    rw [columns_pureRows]
+    exact pureRows_mem _
+  exact hh hf
+
+lemma columns_renameRow {a n m : ℕ} (g : (Fin n → ℂ) →ₗ[ℂ] (Fin m → ℂ))
+    (σ : Equiv.Perm (Fin a)) (f : P a n) :
+    columns g (renameRow σ f) = renameRow σ (columns g f) := by
+  have hh : (columns g).comp (renameRow σ) = (renameRow σ).comp (columns g) := by
+    ext ij
+    simp [columns, renameRow, rowForm]
+  exact DFunLike.congr_fun hh f
+
+lemma columns_mem_R {a n m j : ℕ} (g : (Fin n → ℂ) →ₗ[ℂ] (Fin m → ℂ))
+    {f : P a n} (hf : f ∈ R a n j) : columns g f ∈ R a m j := by
+  refine ⟨columns_mem_W g hf.1, fun σ => ?_⟩
+  rw [← columns_renameRow, hf.2 σ]
+
+lemma columns_mem_A {a n m : ℕ} (g : (Fin n → ℂ) →ₗ[ℂ] (Fin m → ℂ))
+    {f : P a n} (hf : f ∈ A a n) : columns g f ∈ A a m := by
+  have hh : A a n ≤ (A a m).comap (columns g) := by
+    apply Algebra.adjoin_le
+    intro h hh
+    exact Algebra.subset_adjoin (columns_mem_R g hh)
+  exact hh hf
+
+lemma columns_z {a n m : ℕ} (g : (Fin n → ℂ) →ₗ[ℂ] (Fin m → ℂ)) (v : Fin n → ℂ) :
+    columns (a := a) g (z v) = z (g v) := by simp [z, columns_rowForm]
+
+lemma columns_inverse {a n : ℕ} (g : (Fin n → ℂ) ≃ₗ[ℂ] (Fin n → ℂ)) (f : P a n) :
+    columns g.toLinearMap (columns g.symm.toLinearMap f) = f := by
+  have hh : (columns (a := a) g.toLinearMap).comp (columns g.symm.toLinearMap) =
+      AlgHom.id ℂ (P a n) := by
+    apply MvPolynomial.algHom_ext
+    intro ij
+    simp only [AlgHom.comp_apply, columns_X, columns_rowForm, LinearEquiv.coe_coe,
+      LinearEquiv.apply_symm_apply, AlgHom.id_apply, rowForm_single]
+  exact DFunLike.congr_fun hh f
+
+lemma exists_linearEquiv_send {n : ℕ} {v w : Fin n → ℂ} (hv : v ≠ 0) (hw : w ≠ 0) :
+    ∃ g : (Fin n → ℂ) ≃ₗ[ℂ] (Fin n → ℂ), g v = w := by
+  let ev := LinearEquiv.toSpanNonzeroSingleton ℂ (Fin n → ℂ) v hv
+  let ew := LinearEquiv.toSpanNonzeroSingleton ℂ (Fin n → ℂ) w hw
+  obtain ⟨g, hg⟩ := Submodule.exists_linearEquiv_restrict_eq (ev.symm.trans ew)
+  refine ⟨g, ?_⟩
+  have hh := hg (ev 1)
+  change (ew (ev.symm (ev 1)) : Fin n → ℂ) = g (ev 1) at hh
+  rw [ev.symm_apply_apply] at hh
+  have hev : (ev 1 : Fin n → ℂ) = v := by simp [ev]
+  have hew : (ew 1 : Fin n → ℂ) = w := by simp [ew]
+  rw [hev, hew] at hh
+  exact hh.symm
+
+theorem chart {a n k L : ℕ} {f : P a n} (hf : f ∈ R a n k)
+    (v : Fin n → ℂ) (hv : v ≠ 0) (hL : a*k ≤ k+L) : z v ^ L * f ∈ A a n := by
+  cases n with
+  | zero =>
+    exfalso
+    apply hv
+    ext i
+    exact Fin.elim0 i
+  | succ n =>
+    let e : Fin (n+1) → ℂ := Fin.cons 1 (fun _ => 0)
+    have he : e ≠ 0 := by
+      intro hh
+      have := congrFun hh 0
+      simp [e] at this
+    obtain ⟨g, hg⟩ := exists_linearEquiv_send he hv
+    have hsource := chart_standard (columns_mem_R g.symm.toLinearMap hf) hL
+    have htarget := columns_mem_A g.toLinearMap hsource
+    change g (Fin.cons 1 (fun _ : Fin n => 0)) = v at hg
+    simpa only [map_mul, map_pow, columns_z, columns_inverse, LinearEquiv.coe_coe, hg] using htarget
+
+end Foulkes.Chart
+
+namespace Foulkes.Chart
+open MvPolynomial Foulkes.Polarization
+
+lemma z_zero {a n : ℕ} (ha : 0 < a) : z (a := a) (0 : Fin n → ℂ) = 0 := by
+  simp [z, rowForm, zero_pow ha.ne']
+
+theorem independent_chart_bound (a n b : ℕ) (ha : 2 ≤ a) (hb : a*(a-1)^2 ≤ b) :
+    R a n b = component a n b := by
+  classical
+  let L := (a-1)^2
+  have ha1 : 1 ≤ a := by omega
+  have hapos : 0 < a := by omega
+  have hLpos : 0 < L := by
+    have hs : 0 < a-1 := by omega
+    exact pow_pos hs 2
+  have hlow (v : Fin n → ℂ) (k : ℕ) (hk : k < a) {f : P a n} (hf : f ∈ R a n k) :
+      z v ^ L * f ∈ A a n := by
+    by_cases hv : v = 0
+    · simp only [hv, z_zero hapos, zero_pow hLpos.ne', zero_mul]
+      exact (A a n).zero_mem
+    · apply chart hf v hv
+      have hka : k ≤ a-1 := by omega
+      have hm := Nat.mul_le_mul_left (a-1) hka
+      have hs : a-1+1 = a := Nat.sub_add_cancel ha1
+      dsimp [L]
+      nlinarith
+  have hmodule (v : Fin n → ℂ) {f : P a n} (hf : f ∈ lowDegreeModule a n) :
+      z v ^ L * f ∈ A a n := by
+    induction hf using Submodule.span_induction with
+    | mem f hf =>
+      obtain ⟨k,hk,hf⟩ := hf
+      exact hlow v k hk hf
+    | zero => simp
+    | add f g hf hg ihf ihg => simpa [mul_add] using (A a n).add_mem ihf ihg
+    | smul c f hf ih =>
+      change z v ^ L * ((c : P a n) * f) ∈ A a n
+      simpa [mul_left_comm, mul_assoc] using (A a n).mul_mem c.property ih
+  have hspan : invariantPolarizedSpan a n b L ≤ (A a n).toSubmodule := by
+    apply Submodule.span_le.mpr
+    rintro _ ⟨v, f, hf, rfl⟩
+    exact hmodule v (module_generation ha1 _ hf)
+  apply le_antisymm
+  · intro f hf
+    refine ⟨hf.1, ?_⟩
+    exact hspan ((R_polarization ha1 hb).le hf)
+  · exact AComponent_le_R a n b
+
+end Foulkes.Chart
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/InvariantPolarization.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/InvariantPolarization.lean
index e1edb67f0b0469fc02a9e72406288e3220167e2a..939ddb321145157ba4e19fa1876ec638cf5c1f92 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/InvariantPolarization.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/InvariantPolarization.lean
@@ -1,164 +1,164 @@
-import Mathlib
-import OAI.RepresentationTheory.FoulkesSixth.TensorPolarization
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.Polarization
-open MvPolynomial
-
-def renameRow {a n : ℕ} (σ : Equiv.Perm (Fin a)) : P a n →ₐ[ℂ] P a n :=
-  MvPolynomial.rename (fun ij : Fin a × Fin n => (σ ij.1, ij.2))
-
-@[simp] lemma renameRow_comp {a n : ℕ} (σ τ : Equiv.Perm (Fin a)) (f : P a n) :
-    renameRow σ (renameRow τ f) = renameRow (σ * τ) f := by
-  simp [renameRow, rename_rename, Function.comp_def, Equiv.Perm.mul_apply]
-
-@[simp] lemma renameRow_rowForm {a n : ℕ} (σ : Equiv.Perm (Fin a)) (i : Fin a)
-    (v : Fin n → ℂ) : renameRow σ (rowForm i v) = rowForm (σ i) v := by
-  simp [renameRow, rowForm]
-
-lemma renameRow_pureRows {a n j : ℕ} (σ : Equiv.Perm (Fin a)) (v : Fin a → Fin n → ℂ) :
-    renameRow σ (pureRows j v) = pureRows j (fun i => v (σ.symm i)) := by
-  simp only [pureRows, map_prod, map_pow, renameRow_rowForm]
-  apply Fintype.prod_equiv σ
-  intro i
-  simp
-
-lemma renameRow_mem_W {a n j : ℕ} (σ : Equiv.Perm (Fin a)) {f : P a n}
-    (hf : f ∈ W a n j) : renameRow σ f ∈ W a n j := by
-  have hle : Submodule.span ℂ (Set.range (pureRows (a := a) (n := n) j)) ≤
-      (W a n j).comap (renameRow σ).toLinearMap := by
-    apply Submodule.span_le.mpr
-    rintro _ ⟨v, rfl⟩
-    change renameRow σ (pureRows j v) ∈ W a n j
-    rw [renameRow_pureRows]
-    exact pureRows_mem _
-  apply hle
-  rwa [← W_eq_span_pureRows]
-
-@[simp] lemma renameRow_z {a n : ℕ} (σ : Equiv.Perm (Fin a)) (v : Fin n → ℂ) :
-    renameRow σ (z (a := a) v) = z v := by
-  simp only [z, map_prod, renameRow_rowForm]
-  exact Equiv.prod_comp σ (fun i => rowForm i v)
-
-def rowInvariants (a n : ℕ) : Subalgebra ℂ (P a n) where
-  carrier := {f | ∀ σ : Equiv.Perm (Fin a), renameRow σ f = f}
-  zero_mem' := by intro σ; simp
-  add_mem' := by intro f g hf hg σ; simp only [map_add, hf σ, hg σ]
-  one_mem' := by intro σ; simp
-  mul_mem' := by intro f g hf hg σ; simp only [map_mul, hf σ, hg σ]
-  algebraMap_mem' := by intro c σ; simp
-
-def R (a n j : ℕ) : Submodule ℂ (P a n) := W a n j ⊓ (rowInvariants a n).toSubmodule
-
-def A (a n : ℕ) : Subalgebra ℂ (P a n) := Algebra.adjoin ℂ (R a n 1 : Set (P a n))
-
-def average (a n : ℕ) : P a n →ₗ[ℂ] P a n :=
-  (a.factorial : ℂ)⁻¹ • ∑ σ : Equiv.Perm (Fin a), (renameRow σ).toLinearMap
-
-lemma average_apply {a n : ℕ} (f : P a n) :
-    average a n f = (a.factorial : ℂ)⁻¹ • ∑ σ : Equiv.Perm (Fin a), renameRow σ f := by
-  simp [average, LinearMap.sum_apply]
-
-lemma average_invariant {a n : ℕ} (f : P a n) (τ : Equiv.Perm (Fin a)) :
-    renameRow τ (average a n f) = average a n f := by
-  simp only [average_apply, map_smul, map_sum, renameRow_comp]
-  congr 1
-  exact Equiv.sum_comp (Equiv.mulLeft τ) (fun σ => renameRow σ f)
-
-lemma average_mem_R {a n j : ℕ} {f : P a n} (hf : f ∈ W a n j) :
-    average a n f ∈ R a n j := by
-  constructor
-  · rw [average_apply]
-    apply Submodule.smul_mem
-    apply Submodule.sum_mem
-    intro σ _
-    exact renameRow_mem_W σ hf
-  · exact average_invariant f
-
-lemma average_eq {a n : ℕ} {f : P a n}
-    (hf : f ∈ rowInvariants a n) : average a n f = f := by
-  change ∀ σ : Equiv.Perm (Fin a), renameRow σ f = f at hf
-  have ha : (a.factorial : ℂ) ≠ 0 := by exact_mod_cast Nat.factorial_ne_zero a
-  simp only [average_apply, hf, Finset.sum_const, Finset.card_univ,
-    Fintype.card_perm, Fintype.card_fin]
-  rw [← Nat.cast_smul_eq_nsmul ℂ, smul_smul, inv_mul_cancel₀ ha, one_smul]
-
-lemma average_zpow_mul {a n L : ℕ} (v : Fin n → ℂ) (f : P a n) :
-    average a n (z v ^ L * f) = z v ^ L * average a n f := by
-  simp only [average_apply, map_mul, map_pow, renameRow_z, ← Finset.mul_sum]
-  rw [mul_smul_comm]
-
-lemma z_mem_R {a n : ℕ} (v : Fin n → ℂ) : z (a := a) v ∈ R a n 1 := by
-  constructor
-  · simpa using z_pow_mem (a := a) (L := 1) v
-  · exact fun σ => renameRow_z σ v
-
-lemma z_mem_A {a n : ℕ} (v : Fin n → ℂ) : z (a := a) v ∈ A a n :=
-  Algebra.subset_adjoin (z_mem_R v)
-
-def invariantPolarizedSpan (a n j L : ℕ) : Submodule ℂ (P a n) :=
-  Submodule.span ℂ {p | ∃ (v : Fin n → ℂ) (f : P a n),
-    f ∈ R a n (j - L) ∧ z v ^ L * f = p}
-
-theorem R_polarization {a n j L : ℕ} (ha : 1 ≤ a) (hj : a * L ≤ j) :
-    R a n j = invariantPolarizedSpan a n j L := by
-  classical
-  apply le_antisymm
-  · intro f hf
-    have hle : polarizedSpan a n j L ≤
-        (invariantPolarizedSpan a n j L).comap (average a n) := by
-      apply Submodule.span_le.mpr
-      rintro _ ⟨v, g, hg, rfl⟩
-      change average a n (z v ^ L * g) ∈ invariantPolarizedSpan a n j L
-      rw [average_zpow_mul]
-      exact Submodule.subset_span ⟨v, average a n g, average_mem_R hg, rfl⟩
-    have hmem := hle ((W_polarization ha hj).le hf.1)
-    change average a n f ∈ invariantPolarizedSpan a n j L at hmem
-    rwa [average_eq hf.2] at hmem
-  · apply Submodule.span_le.mpr
-    rintro _ ⟨v, f, hf, rfl⟩
-    constructor
-    · apply (W_polarization ha hj).ge
-      exact Submodule.subset_span ⟨v, f, hf.1, rfl⟩
-    · intro σ
-      simp only [map_mul, map_pow, renameRow_z, hf.2 σ]
-
-def lowDegreeModule (a n : ℕ) : Submodule (A a n) (P a n) :=
-  Submodule.span (A a n) {f | ∃ k : ℕ, k < a ∧ f ∈ R a n k}
-
-theorem module_generation {a n : ℕ} (ha : 1 ≤ a) (j : ℕ) {f : P a n}
-    (hf : f ∈ R a n j) : f ∈ lowDegreeModule a n := by
-  classical
-  induction j using Nat.strong_induction_on generalizing f with
-  | h j ih =>
-    by_cases hj : j < a
-    · exact Submodule.subset_span ⟨j, hj, hf⟩
-    have hja : a ≤ j := Nat.le_of_not_lt hj
-    have hsub : j - 1 < j := by omega
-    have hle : invariantPolarizedSpan a n j 1 ≤ (lowDegreeModule a n).restrictScalars ℂ := by
-      apply Submodule.span_le.mpr
-      rintro _ ⟨v, g, hg, rfl⟩
-      change z v ^ 1 * g ∈ lowDegreeModule a n
-      rw [pow_one]
-      exact (lowDegreeModule a n).smul_mem (⟨z v, z_mem_A v⟩ : A a n) (ih (j - 1) hsub hg)
-    apply hle
-    apply (R_polarization ha (by simpa using hja)).le hf
-
-lemma A_le_rowInvariants (a n : ℕ) : A a n ≤ rowInvariants a n := by
-  apply Algebra.adjoin_le
-  intro f hf
-  exact hf.2
-
-lemma AComponent_le_R (a n j : ℕ) :
-    W a n j ⊓ (A a n).toSubmodule ≤ R a n j := by
-  intro f hf
-  exact ⟨hf.1, A_le_rowInvariants a n hf.2⟩
-
-end Foulkes.Polarization
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.FoulkesSixth.TensorPolarization
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.Polarization
+open MvPolynomial
+
+def renameRow {a n : ℕ} (σ : Equiv.Perm (Fin a)) : P a n →ₐ[ℂ] P a n :=
+  MvPolynomial.rename (fun ij : Fin a × Fin n => (σ ij.1, ij.2))
+
+@[simp] lemma renameRow_comp {a n : ℕ} (σ τ : Equiv.Perm (Fin a)) (f : P a n) :
+    renameRow σ (renameRow τ f) = renameRow (σ * τ) f := by
+  simp [renameRow, rename_rename, Function.comp_def, Equiv.Perm.mul_apply]
+
+@[simp] lemma renameRow_rowForm {a n : ℕ} (σ : Equiv.Perm (Fin a)) (i : Fin a)
+    (v : Fin n → ℂ) : renameRow σ (rowForm i v) = rowForm (σ i) v := by
+  simp [renameRow, rowForm]
+
+lemma renameRow_pureRows {a n j : ℕ} (σ : Equiv.Perm (Fin a)) (v : Fin a → Fin n → ℂ) :
+    renameRow σ (pureRows j v) = pureRows j (fun i => v (σ.symm i)) := by
+  simp only [pureRows, map_prod, map_pow, renameRow_rowForm]
+  apply Fintype.prod_equiv σ
+  intro i
+  simp
+
+lemma renameRow_mem_W {a n j : ℕ} (σ : Equiv.Perm (Fin a)) {f : P a n}
+    (hf : f ∈ W a n j) : renameRow σ f ∈ W a n j := by
+  have hle : Submodule.span ℂ (Set.range (pureRows (a := a) (n := n) j)) ≤
+      (W a n j).comap (renameRow σ).toLinearMap := by
+    apply Submodule.span_le.mpr
+    rintro _ ⟨v, rfl⟩
+    change renameRow σ (pureRows j v) ∈ W a n j
+    rw [renameRow_pureRows]
+    exact pureRows_mem _
+  apply hle
+  rwa [← W_eq_span_pureRows]
+
+@[simp] lemma renameRow_z {a n : ℕ} (σ : Equiv.Perm (Fin a)) (v : Fin n → ℂ) :
+    renameRow σ (z (a := a) v) = z v := by
+  simp only [z, map_prod, renameRow_rowForm]
+  exact Equiv.prod_comp σ (fun i => rowForm i v)
+
+def rowInvariants (a n : ℕ) : Subalgebra ℂ (P a n) where
+  carrier := {f | ∀ σ : Equiv.Perm (Fin a), renameRow σ f = f}
+  zero_mem' := by intro σ; simp
+  add_mem' := by intro f g hf hg σ; simp only [map_add, hf σ, hg σ]
+  one_mem' := by intro σ; simp
+  mul_mem' := by intro f g hf hg σ; simp only [map_mul, hf σ, hg σ]
+  algebraMap_mem' := by intro c σ; simp
+
+def R (a n j : ℕ) : Submodule ℂ (P a n) := W a n j ⊓ (rowInvariants a n).toSubmodule
+
+def A (a n : ℕ) : Subalgebra ℂ (P a n) := Algebra.adjoin ℂ (R a n 1 : Set (P a n))
+
+def average (a n : ℕ) : P a n →ₗ[ℂ] P a n :=
+  (a.factorial : ℂ)⁻¹ • ∑ σ : Equiv.Perm (Fin a), (renameRow σ).toLinearMap
+
+lemma average_apply {a n : ℕ} (f : P a n) :
+    average a n f = (a.factorial : ℂ)⁻¹ • ∑ σ : Equiv.Perm (Fin a), renameRow σ f := by
+  simp [average, LinearMap.sum_apply]
+
+lemma average_invariant {a n : ℕ} (f : P a n) (τ : Equiv.Perm (Fin a)) :
+    renameRow τ (average a n f) = average a n f := by
+  simp only [average_apply, map_smul, map_sum, renameRow_comp]
+  congr 1
+  exact Equiv.sum_comp (Equiv.mulLeft τ) (fun σ => renameRow σ f)
+
+lemma average_mem_R {a n j : ℕ} {f : P a n} (hf : f ∈ W a n j) :
+    average a n f ∈ R a n j := by
+  constructor
+  · rw [average_apply]
+    apply Submodule.smul_mem
+    apply Submodule.sum_mem
+    intro σ _
+    exact renameRow_mem_W σ hf
+  · exact average_invariant f
+
+lemma average_eq {a n : ℕ} {f : P a n}
+    (hf : f ∈ rowInvariants a n) : average a n f = f := by
+  change ∀ σ : Equiv.Perm (Fin a), renameRow σ f = f at hf
+  have ha : (a.factorial : ℂ) ≠ 0 := by exact_mod_cast Nat.factorial_ne_zero a
+  simp only [average_apply, hf, Finset.sum_const, Finset.card_univ,
+    Fintype.card_perm, Fintype.card_fin]
+  rw [← Nat.cast_smul_eq_nsmul ℂ, smul_smul, inv_mul_cancel₀ ha, one_smul]
+
+lemma average_zpow_mul {a n L : ℕ} (v : Fin n → ℂ) (f : P a n) :
+    average a n (z v ^ L * f) = z v ^ L * average a n f := by
+  simp only [average_apply, map_mul, map_pow, renameRow_z, ← Finset.mul_sum]
+  rw [mul_smul_comm]
+
+lemma z_mem_R {a n : ℕ} (v : Fin n → ℂ) : z (a := a) v ∈ R a n 1 := by
+  constructor
+  · simpa using z_pow_mem (a := a) (L := 1) v
+  · exact fun σ => renameRow_z σ v
+
+lemma z_mem_A {a n : ℕ} (v : Fin n → ℂ) : z (a := a) v ∈ A a n :=
+  Algebra.subset_adjoin (z_mem_R v)
+
+def invariantPolarizedSpan (a n j L : ℕ) : Submodule ℂ (P a n) :=
+  Submodule.span ℂ {p | ∃ (v : Fin n → ℂ) (f : P a n),
+    f ∈ R a n (j - L) ∧ z v ^ L * f = p}
+
+theorem R_polarization {a n j L : ℕ} (ha : 1 ≤ a) (hj : a * L ≤ j) :
+    R a n j = invariantPolarizedSpan a n j L := by
+  classical
+  apply le_antisymm
+  · intro f hf
+    have hle : polarizedSpan a n j L ≤
+        (invariantPolarizedSpan a n j L).comap (average a n) := by
+      apply Submodule.span_le.mpr
+      rintro _ ⟨v, g, hg, rfl⟩
+      change average a n (z v ^ L * g) ∈ invariantPolarizedSpan a n j L
+      rw [average_zpow_mul]
+      exact Submodule.subset_span ⟨v, average a n g, average_mem_R hg, rfl⟩
+    have hmem := hle ((W_polarization ha hj).le hf.1)
+    change average a n f ∈ invariantPolarizedSpan a n j L at hmem
+    rwa [average_eq hf.2] at hmem
+  · apply Submodule.span_le.mpr
+    rintro _ ⟨v, f, hf, rfl⟩
+    constructor
+    · apply (W_polarization ha hj).ge
+      exact Submodule.subset_span ⟨v, f, hf.1, rfl⟩
+    · intro σ
+      simp only [map_mul, map_pow, renameRow_z, hf.2 σ]
+
+def lowDegreeModule (a n : ℕ) : Submodule (A a n) (P a n) :=
+  Submodule.span (A a n) {f | ∃ k : ℕ, k < a ∧ f ∈ R a n k}
+
+theorem module_generation {a n : ℕ} (ha : 1 ≤ a) (j : ℕ) {f : P a n}
+    (hf : f ∈ R a n j) : f ∈ lowDegreeModule a n := by
+  classical
+  induction j using Nat.strong_induction_on generalizing f with
+  | h j ih =>
+    by_cases hj : j < a
+    · exact Submodule.subset_span ⟨j, hj, hf⟩
+    have hja : a ≤ j := Nat.le_of_not_lt hj
+    have hsub : j - 1 < j := by omega
+    have hle : invariantPolarizedSpan a n j 1 ≤ (lowDegreeModule a n).restrictScalars ℂ := by
+      apply Submodule.span_le.mpr
+      rintro _ ⟨v, g, hg, rfl⟩
+      change z v ^ 1 * g ∈ lowDegreeModule a n
+      rw [pow_one]
+      exact (lowDegreeModule a n).smul_mem (⟨z v, z_mem_A v⟩ : A a n) (ih (j - 1) hsub hg)
+    apply hle
+    apply (R_polarization ha (by simpa using hja)).le hf
+
+lemma A_le_rowInvariants (a n : ℕ) : A a n ≤ rowInvariants a n := by
+  apply Algebra.adjoin_le
+  intro f hf
+  exact hf.2
+
+lemma AComponent_le_R (a n j : ℕ) :
+    W a n j ⊓ (A a n).toSubmodule ≤ R a n j := by
+  intro f hf
+  exact ⟨hf.1, A_le_rowInvariants a n hf.2⟩
+
+end Foulkes.Polarization
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/Lookup.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/Lookup.lean
index 1d96cb750613bb0032a1d3281286762509aff043..766d99e10841257d00b5215c471f515daa24234d 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/Lookup.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/Lookup.lean
@@ -1,138 +1,138 @@
-import Mathlib
-import OAI.RepresentationTheory.FoulkesSixth.CompleteNewton
-import OAI.RepresentationTheory.FoulkesSixth.SignedStrips
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.Complete
-open MvPolynomial Finset
-
-def degreeEquiv {α β : Type*} (e : α ≃ β) (b : ℕ) : Degree α b ≃ Degree β b where
-  toFun d := ⟨d.val.mapDomain e, by simpa using d.property⟩
-  invFun d := ⟨d.val.mapDomain e.symm, by simpa using d.property⟩
-  left_inv d := by
-    apply Subtype.ext
-    ext j
-    simp [Finsupp.mapDomain_equiv_apply]
-  right_inv d := by
-    apply Subtype.ext
-    ext j
-    simp [Finsupp.mapDomain_equiv_apply]
-
-theorem rename_h {α β : Type*} [Finite α] [Finite β] (e : α ≃ β) (b : ℕ) :
-    rename e (h α b) = h β b := by
-  simp only [h, map_sum, rename_monomial]
-  exact Equiv.sum_comp (degreeEquiv e b) (fun d => monomial d.val 1)
-
-theorem alphabetH_reindex {α β σ : Type*} [Finite α] [Finite β]
-    (e : α ≃ β) (x : β → MvPolynomial σ ℤ) (b : ℕ) :
-    alphabetH (x ∘ e) b = alphabetH x b := by
-  have hh := congrArg (eval₂Hom (Int.castRingHom (MvPolynomial σ ℤ)) x) (rename_h e b)
-  simpa only [eval₂Hom_rename, alphabetH] using hh
-
-theorem rename_pleth (n b a : ℕ) (σ : Equiv.Perm (Fin n)) :
-    rename σ (pleth n b a) = pleth n b a := by
-  unfold pleth alphabetH
-  change (rename σ).toRingHom _ = _
-  rw [MvPolynomial.map_eval₂Hom]
-  have hh : (rename σ).toRingHom.comp (Int.castRingHom (MvPolynomial (Fin n) ℤ)) =
-      Int.castRingHom (MvPolynomial (Fin n) ℤ) := by ext; simp
-  rw [hh]
-  simpa only [alphabetH, Function.comp_def, degreeEquiv, Equiv.coe_fn_mk, AlgHom.toRingHom_eq_coe, RingHom.coe_coe, rename_monomial] using
-    alphabetH_reindex (degreeEquiv σ a) (fun d => monomial d.val (1 : ℤ)) b
-
-end Foulkes.Complete
-
-namespace Foulkes.Lookup
-open MvPolynomial Finset Foulkes.Strips
-
-def Alternating {n : ℕ} (p : MvPolynomial (Fin n) ℤ) : Prop :=
-  ∀ σ : Equiv.Perm (Fin n), rename σ p = (Equiv.Perm.sign σ : ℤ) • p
-
-lemma alternant_alternating {n : ℕ} (m : Fin n → ℕ) : Alternating (alternant m) := by
-  intro σ
-  let M : Matrix (Fin n) (Fin n) (MvPolynomial (Fin n) ℤ) := Matrix.of (fun r c => X r ^ m c)
-  have hm : (rename σ).toRingHom.mapMatrix M = M.submatrix σ id := by
-    ext r c
-    simp [M, RingHom.mapMatrix_apply]
-  have hh := (rename σ).toRingHom.map_det M
-  rw [hm, Matrix.det_permute] at hh
-  change rename σ (alternant m) = _ at hh
-  have he : M.det = alternant m := rfl
-  rw [he] at hh
-  simpa only [zsmul_eq_mul] using hh
-
-lemma alternating_mul_symmetric {n : ℕ} {p q : MvPolynomial (Fin n) ℤ}
-    (hp : ∀ σ : Equiv.Perm (Fin n), rename σ p = p) (hq : Alternating q) :
-    Alternating (p*q) := by
-  intro σ
-  simp only [map_mul, hp σ, hq σ, mul_smul_comm]
-
-lemma mapDomain_expVector {n : ℕ} (σ : Equiv.Perm (Fin n)) (v : Fin n → ℕ) :
-    (expVector v).mapDomain σ = expVector (v ∘ σ.symm) := by
-  ext j
-  simp [Finsupp.mapDomain_equiv_apply, Function.comp_def]
-
-lemma coeff_comp {n : ℕ} {p : MvPolynomial (Fin n) ℤ} (hp : Alternating p)
-    (v : Fin n → ℕ) (σ : Equiv.Perm (Fin n)) :
-    p.coeff (expVector (v ∘ σ)) = (Equiv.Perm.sign σ : ℤ) * p.coeff (expVector v) := by
-  have hm : (expVector (v ∘ σ)).mapDomain σ = expVector v := by
-    rw [mapDomain_expVector]
-    congr 1
-    funext j
-    simp
-  have hh := coeff_rename_mapDomain σ σ.injective p (expVector (v ∘ σ))
-  rw [hm, hp σ, coeff_smul] at hh
-  simpa [zsmul_eq_mul] using hh.symm
-
-lemma coeff_eq_sign_mul_comp {n : ℕ} {p : MvPolynomial (Fin n) ℤ} (hp : Alternating p)
-    (v : Fin n → ℕ) (σ : Equiv.Perm (Fin n)) :
-    p.coeff (expVector v) = (Equiv.Perm.sign σ : ℤ) * p.coeff (expVector (v ∘ σ)) := by
-  have hh := coeff_comp hp (v ∘ σ) σ.symm
-  have he : (v ∘ σ) ∘ σ.symm = v := by funext j; simp
-  simpa only [he, Equiv.Perm.sign_symm] using hh
-
-lemma coeff_repeated {n : ℕ} {p : MvPolynomial (Fin n) ℤ} (hp : Alternating p)
-    (v : Fin n → ℕ) {j k : Fin n} (hjk : j ≠ k) (he : v j = v k) :
-    p.coeff (expVector v) = 0 := by
-  classical
-  have hs : v ∘ Equiv.swap j k = v := by
-    funext r
-    by_cases hrj : r = j
-    · subst r; simp [he]
-    by_cases hrk : r = k
-    · subst r; simp [he]
-    simp [Equiv.swap_apply_of_ne_of_ne hrj hrk]
-  have hh := coeff_comp hp v (Equiv.swap j k)
-  rw [hs, Equiv.Perm.sign_swap hjk] at hh
-  simp only [Units.val_neg, Units.val_one, neg_one_mul] at hh
-  omega
-
-def sortDesc {n : ℕ} (v : Fin n → ℕ) : Equiv.Perm (Fin n) :=
-  Tuple.sort (α := ℕᵒᵈ) (fun j => v j)
-
-lemma sortDesc_antitone {n : ℕ} (v : Fin n → ℕ) : Antitone (v ∘ sortDesc v) :=
-by
-  intro j k hjk
-  exact Tuple.monotone_sort (α := ℕᵒᵈ) (fun j => v j) hjk
-
-lemma sortDesc_strict {n : ℕ} (v : Fin n → ℕ) (hv : Function.Injective v) :
-    StrictAnti (v ∘ sortDesc v) :=
-  (sortDesc_antitone v).strictAnti_of_injective (hv.comp (sortDesc v).injective)
-
-def sortedPartition {n : ℕ} (v : Fin n → ℕ) (j : Fin n) : ℕ :=
-  v (sortDesc v j) - staircase n j
-
-def schurCoeff {n : ℕ} (p : MvPolynomial (Fin n) ℤ) (mu : Fin n → ℕ) : ℤ :=
-  (p * alternant (staircase n)).coeff (expVector (shifted mu))
-
-def Q (n b : ℕ) (mu : Fin n → ℕ) : ℤ := schurCoeff (Complete.pleth n b 6) mu
-def F (n b : ℕ) (mu : Fin n → ℕ) : ℤ := schurCoeff (Complete.pleth n 6 b) mu
-
-end Foulkes.Lookup
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.FoulkesSixth.CompleteNewton
+import OAI.RepresentationTheory.FoulkesSixth.SignedStrips
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.Complete
+open MvPolynomial Finset
+
+def degreeEquiv {α β : Type*} (e : α ≃ β) (b : ℕ) : Degree α b ≃ Degree β b where
+  toFun d := ⟨d.val.mapDomain e, by simpa using d.property⟩
+  invFun d := ⟨d.val.mapDomain e.symm, by simpa using d.property⟩
+  left_inv d := by
+    apply Subtype.ext
+    ext j
+    simp [Finsupp.mapDomain_equiv_apply]
+  right_inv d := by
+    apply Subtype.ext
+    ext j
+    simp [Finsupp.mapDomain_equiv_apply]
+
+theorem rename_h {α β : Type*} [Finite α] [Finite β] (e : α ≃ β) (b : ℕ) :
+    rename e (h α b) = h β b := by
+  simp only [h, map_sum, rename_monomial]
+  exact Equiv.sum_comp (degreeEquiv e b) (fun d => monomial d.val 1)
+
+theorem alphabetH_reindex {α β σ : Type*} [Finite α] [Finite β]
+    (e : α ≃ β) (x : β → MvPolynomial σ ℤ) (b : ℕ) :
+    alphabetH (x ∘ e) b = alphabetH x b := by
+  have hh := congrArg (eval₂Hom (Int.castRingHom (MvPolynomial σ ℤ)) x) (rename_h e b)
+  simpa only [eval₂Hom_rename, alphabetH] using hh
+
+theorem rename_pleth (n b a : ℕ) (σ : Equiv.Perm (Fin n)) :
+    rename σ (pleth n b a) = pleth n b a := by
+  unfold pleth alphabetH
+  change (rename σ).toRingHom _ = _
+  rw [MvPolynomial.map_eval₂Hom]
+  have hh : (rename σ).toRingHom.comp (Int.castRingHom (MvPolynomial (Fin n) ℤ)) =
+      Int.castRingHom (MvPolynomial (Fin n) ℤ) := by ext; simp
+  rw [hh]
+  simpa only [alphabetH, Function.comp_def, degreeEquiv, Equiv.coe_fn_mk, AlgHom.toRingHom_eq_coe, RingHom.coe_coe, rename_monomial] using
+    alphabetH_reindex (degreeEquiv σ a) (fun d => monomial d.val (1 : ℤ)) b
+
+end Foulkes.Complete
+
+namespace Foulkes.Lookup
+open MvPolynomial Finset Foulkes.Strips
+
+def Alternating {n : ℕ} (p : MvPolynomial (Fin n) ℤ) : Prop :=
+  ∀ σ : Equiv.Perm (Fin n), rename σ p = (Equiv.Perm.sign σ : ℤ) • p
+
+lemma alternant_alternating {n : ℕ} (m : Fin n → ℕ) : Alternating (alternant m) := by
+  intro σ
+  let M : Matrix (Fin n) (Fin n) (MvPolynomial (Fin n) ℤ) := Matrix.of (fun r c => X r ^ m c)
+  have hm : (rename σ).toRingHom.mapMatrix M = M.submatrix σ id := by
+    ext r c
+    simp [M, RingHom.mapMatrix_apply]
+  have hh := (rename σ).toRingHom.map_det M
+  rw [hm, Matrix.det_permute] at hh
+  change rename σ (alternant m) = _ at hh
+  have he : M.det = alternant m := rfl
+  rw [he] at hh
+  simpa only [zsmul_eq_mul] using hh
+
+lemma alternating_mul_symmetric {n : ℕ} {p q : MvPolynomial (Fin n) ℤ}
+    (hp : ∀ σ : Equiv.Perm (Fin n), rename σ p = p) (hq : Alternating q) :
+    Alternating (p*q) := by
+  intro σ
+  simp only [map_mul, hp σ, hq σ, mul_smul_comm]
+
+lemma mapDomain_expVector {n : ℕ} (σ : Equiv.Perm (Fin n)) (v : Fin n → ℕ) :
+    (expVector v).mapDomain σ = expVector (v ∘ σ.symm) := by
+  ext j
+  simp [Finsupp.mapDomain_equiv_apply, Function.comp_def]
+
+lemma coeff_comp {n : ℕ} {p : MvPolynomial (Fin n) ℤ} (hp : Alternating p)
+    (v : Fin n → ℕ) (σ : Equiv.Perm (Fin n)) :
+    p.coeff (expVector (v ∘ σ)) = (Equiv.Perm.sign σ : ℤ) * p.coeff (expVector v) := by
+  have hm : (expVector (v ∘ σ)).mapDomain σ = expVector v := by
+    rw [mapDomain_expVector]
+    congr 1
+    funext j
+    simp
+  have hh := coeff_rename_mapDomain σ σ.injective p (expVector (v ∘ σ))
+  rw [hm, hp σ, coeff_smul] at hh
+  simpa [zsmul_eq_mul] using hh.symm
+
+lemma coeff_eq_sign_mul_comp {n : ℕ} {p : MvPolynomial (Fin n) ℤ} (hp : Alternating p)
+    (v : Fin n → ℕ) (σ : Equiv.Perm (Fin n)) :
+    p.coeff (expVector v) = (Equiv.Perm.sign σ : ℤ) * p.coeff (expVector (v ∘ σ)) := by
+  have hh := coeff_comp hp (v ∘ σ) σ.symm
+  have he : (v ∘ σ) ∘ σ.symm = v := by funext j; simp
+  simpa only [he, Equiv.Perm.sign_symm] using hh
+
+lemma coeff_repeated {n : ℕ} {p : MvPolynomial (Fin n) ℤ} (hp : Alternating p)
+    (v : Fin n → ℕ) {j k : Fin n} (hjk : j ≠ k) (he : v j = v k) :
+    p.coeff (expVector v) = 0 := by
+  classical
+  have hs : v ∘ Equiv.swap j k = v := by
+    funext r
+    by_cases hrj : r = j
+    · subst r; simp [he]
+    by_cases hrk : r = k
+    · subst r; simp [he]
+    simp [Equiv.swap_apply_of_ne_of_ne hrj hrk]
+  have hh := coeff_comp hp v (Equiv.swap j k)
+  rw [hs, Equiv.Perm.sign_swap hjk] at hh
+  simp only [Units.val_neg, Units.val_one, neg_one_mul] at hh
+  omega
+
+def sortDesc {n : ℕ} (v : Fin n → ℕ) : Equiv.Perm (Fin n) :=
+  Tuple.sort (α := ℕᵒᵈ) (fun j => v j)
+
+lemma sortDesc_antitone {n : ℕ} (v : Fin n → ℕ) : Antitone (v ∘ sortDesc v) :=
+by
+  intro j k hjk
+  exact Tuple.monotone_sort (α := ℕᵒᵈ) (fun j => v j) hjk
+
+lemma sortDesc_strict {n : ℕ} (v : Fin n → ℕ) (hv : Function.Injective v) :
+    StrictAnti (v ∘ sortDesc v) :=
+  (sortDesc_antitone v).strictAnti_of_injective (hv.comp (sortDesc v).injective)
+
+def sortedPartition {n : ℕ} (v : Fin n → ℕ) (j : Fin n) : ℕ :=
+  v (sortDesc v j) - staircase n j
+
+def schurCoeff {n : ℕ} (p : MvPolynomial (Fin n) ℤ) (mu : Fin n → ℕ) : ℤ :=
+  (p * alternant (staircase n)).coeff (expVector (shifted mu))
+
+def Q (n b : ℕ) (mu : Fin n → ℕ) : ℤ := schurCoeff (Complete.pleth n b 6) mu
+def F (n b : ℕ) (mu : Fin n → ℕ) : ℤ := schurCoeff (Complete.pleth n 6 b) mu
+
+end Foulkes.Lookup
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/LookupExtraction.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/LookupExtraction.lean
index c743e4e4dadaa9fb2c7c0e802f60b0bcf45f7cd7..3d99719fb1e37553274a3b66414588da73f0e0ba 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/LookupExtraction.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/LookupExtraction.lean
@@ -1,132 +1,132 @@
-import Mathlib
-import OAI.RepresentationTheory.FoulkesSixth.Lookup
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.Lookup
-open MvPolynomial Finset Foulkes.Strips
-
-lemma strictAnti_gap {n : ℕ} {v : Fin n → ℕ} (hv : StrictAnti v)
-    (j k : Fin n) (hjk : j ≤ k) : v k + k.val ≤ v j + j.val := by
-  have hh : ∀ t, j.val ≤ t → ∀ ht : t < n, v ⟨t,ht⟩ + t ≤ v j + j.val := by
-    intro t hjt
-    induction t, hjt using Nat.le_induction with
-    | base => intro ht; simp
-    | succ t hjt ih =>
-      intro ht
-      have ht' : t < n := by omega
-      have hstep := hv (show (⟨t,ht'⟩ : Fin n) < ⟨t+1,ht⟩ from by simp)
-      have hprev := ih ht'
-      omega
-  exact hh k.val hjk k.isLt
-
-lemma staircase_le_of_strictAnti {n : ℕ} {v : Fin n → ℕ} (hv : StrictAnti v)
-    (j : Fin n) : staircase n j ≤ v j := by
-  have hn : 0 < n := Nat.zero_lt_of_lt j.isLt
-  let k : Fin n := ⟨n-1, by omega⟩
-  have hjk : j ≤ k := by change j.val ≤ n-1; omega
-  have hh := strictAnti_gap hv j k hjk
-  simp only [k, staircase] at *
-  omega
-
-lemma sortedPartition_antitone {n : ℕ} (v : Fin n → ℕ) (hv : Function.Injective v) :
-    Antitone (sortedPartition v) := by
-  intro j k hjk
-  have hs := sortDesc_strict v hv
-  have hg := strictAnti_gap hs j k hjk
-  have hj := staircase_le_of_strictAnti hs j
-  have hk := staircase_le_of_strictAnti hs k
-  simp only [sortedPartition, staircase, Function.comp_apply] at *
-  have hjn := j.isLt
-  have hkn := k.isLt
-  omega
-
-lemma shifted_sortedPartition {n : ℕ} (v : Fin n → ℕ) (hv : Function.Injective v) :
-    shifted (sortedPartition v) = v ∘ sortDesc v := by
-  funext j
-  exact Nat.sub_add_cancel (staircase_le_of_strictAnti (sortDesc_strict v hv) j)
-
-def signedLookup {n : ℕ} (f : (Fin n → ℕ) → ℤ) (v : Fin n → ℤ) : ℤ := by
-  classical
-  exact if hn : ∀ j, 0 ≤ v j then
-    let w := fun j => (v j).toNat
-    if Function.Injective w then
-      (Equiv.Perm.sign (sortDesc w) : ℤ) * f (sortedPartition w)
-    else 0
-  else 0
-
-def coeffZ {n : ℕ} (v : Fin n → ℤ) (p : MvPolynomial (Fin n) ℤ) : ℤ := by
-  classical
-  exact if ∀ j, 0 ≤ v j then p.coeff (expVector (fun j => (v j).toNat)) else 0
-
-theorem signedLookup_eq_coeffZ {n : ℕ} (p : MvPolynomial (Fin n) ℤ)
-    (hp : ∀ σ : Equiv.Perm (Fin n), rename σ p = p) (v : Fin n → ℤ) :
-    signedLookup (schurCoeff p) v = coeffZ v (p * alternant (staircase n)) := by
-  classical
-  unfold signedLookup coeffZ
-  by_cases hn : ∀ j, 0 ≤ v j
-  · simp only [dite_eq_left hn, ite_eq_left hn]
-    let w := fun j => (v j).toNat
-    change (if Function.Injective w then _ else _) = _
-    have ha := alternating_mul_symmetric hp (alternant_alternating (staircase n))
-    by_cases hw : Function.Injective w
-    · rw [ite_eq_left hw]
-      unfold schurCoeff
-      rw [shifted_sortedPartition w hw]
-      exact (coeff_eq_sign_mul_comp ha w (sortDesc w)).symm
-    · rw [ite_eq_right hw]
-      obtain ⟨j,k,he,hjk⟩ := Function.not_injective_iff.mp hw
-      exact (coeff_repeated ha w hjk he).symm
-  · simp only [dite_eq_right hn, ite_eq_right hn]
-
-lemma coeffZ_monomial_mul {n : ℕ} (l : Fin n → ℕ) (d : Fin n →₀ ℕ)
-    (p : MvPolynomial (Fin n) ℤ) :
-    (monomial d 1 * p).coeff (expVector l) =
-      coeffZ (fun j => (l j : ℤ) - d j) p := by
-  classical
-  rw [coeff_monomial_mul']
-  unfold coeffZ
-  have hiff : d ≤ expVector l ↔ ∀ j, 0 ≤ (l j : ℤ) - d j := by
-    simp [Finsupp.le_def, expVector]
-  by_cases hd : d ≤ expVector l
-  · rw [ite_eq_left hd, ite_eq_left (hiff.mp hd), one_mul]
-    congr 1
-    ext j
-    simp [expVector]
-  · rw [ite_eq_right hd, ite_eq_right (mt hiff.mpr hd)]
-
-theorem target_newton (n b : ℕ) (mu : Fin n → ℕ) :
-    (b : ℤ) * Q n b mu =
-      ∑ i ∈ Finset.Icc 1 b, ∑ d : Complete.Degree (Fin n) 6,
-        signedLookup (Q n (b-i))
-          (fun j => (shifted mu j : ℤ) - (i : ℤ) * d.val j) := by
-  have hh := congrArg
-    (fun p : MvPolynomial (Fin n) ℤ =>
-      (p * alternant (staircase n)).coeff (expVector (shifted mu)))
-    (Complete.pleth_newton n b 6)
-  change (((b : ℤ) • Complete.pleth n b 6) * alternant (staircase n)).coeff
-      (expVector (shifted mu)) = _ at hh
-  rw [smul_mul_assoc, coeff_smul] at hh
-  simp only [Finset.sum_mul, coeff_sum, zsmul_eq_mul] at hh
-  change (b : ℤ) * Q n b mu = _ at hh
-  rw [hh]
-  apply Finset.sum_congr rfl
-  intro i hi
-  apply Finset.sum_congr rfl
-  intro d hd
-  rw [mul_assoc, coeffZ_monomial_mul]
-  have he : (fun j : Fin n => (shifted mu j : ℤ) - ((i • d.val) j : ℕ)) =
-      (fun j => (shifted mu j : ℤ) - (i : ℤ) * d.val j) := by
-    funext j
-    simp
-  rw [he]
-  exact (signedLookup_eq_coeffZ (Complete.pleth n (b-i) 6)
-    (Complete.rename_pleth n (b-i) 6) _).symm
-
-end Foulkes.Lookup
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.FoulkesSixth.Lookup
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.Lookup
+open MvPolynomial Finset Foulkes.Strips
+
+lemma strictAnti_gap {n : ℕ} {v : Fin n → ℕ} (hv : StrictAnti v)
+    (j k : Fin n) (hjk : j ≤ k) : v k + k.val ≤ v j + j.val := by
+  have hh : ∀ t, j.val ≤ t → ∀ ht : t < n, v ⟨t,ht⟩ + t ≤ v j + j.val := by
+    intro t hjt
+    induction t, hjt using Nat.le_induction with
+    | base => intro ht; simp
+    | succ t hjt ih =>
+      intro ht
+      have ht' : t < n := by omega
+      have hstep := hv (show (⟨t,ht'⟩ : Fin n) < ⟨t+1,ht⟩ from by simp)
+      have hprev := ih ht'
+      omega
+  exact hh k.val hjk k.isLt
+
+lemma staircase_le_of_strictAnti {n : ℕ} {v : Fin n → ℕ} (hv : StrictAnti v)
+    (j : Fin n) : staircase n j ≤ v j := by
+  have hn : 0 < n := Nat.zero_lt_of_lt j.isLt
+  let k : Fin n := ⟨n-1, by omega⟩
+  have hjk : j ≤ k := by change j.val ≤ n-1; omega
+  have hh := strictAnti_gap hv j k hjk
+  simp only [k, staircase] at *
+  omega
+
+lemma sortedPartition_antitone {n : ℕ} (v : Fin n → ℕ) (hv : Function.Injective v) :
+    Antitone (sortedPartition v) := by
+  intro j k hjk
+  have hs := sortDesc_strict v hv
+  have hg := strictAnti_gap hs j k hjk
+  have hj := staircase_le_of_strictAnti hs j
+  have hk := staircase_le_of_strictAnti hs k
+  simp only [sortedPartition, staircase, Function.comp_apply] at *
+  have hjn := j.isLt
+  have hkn := k.isLt
+  omega
+
+lemma shifted_sortedPartition {n : ℕ} (v : Fin n → ℕ) (hv : Function.Injective v) :
+    shifted (sortedPartition v) = v ∘ sortDesc v := by
+  funext j
+  exact Nat.sub_add_cancel (staircase_le_of_strictAnti (sortDesc_strict v hv) j)
+
+def signedLookup {n : ℕ} (f : (Fin n → ℕ) → ℤ) (v : Fin n → ℤ) : ℤ := by
+  classical
+  exact if hn : ∀ j, 0 ≤ v j then
+    let w := fun j => (v j).toNat
+    if Function.Injective w then
+      (Equiv.Perm.sign (sortDesc w) : ℤ) * f (sortedPartition w)
+    else 0
+  else 0
+
+def coeffZ {n : ℕ} (v : Fin n → ℤ) (p : MvPolynomial (Fin n) ℤ) : ℤ := by
+  classical
+  exact if ∀ j, 0 ≤ v j then p.coeff (expVector (fun j => (v j).toNat)) else 0
+
+theorem signedLookup_eq_coeffZ {n : ℕ} (p : MvPolynomial (Fin n) ℤ)
+    (hp : ∀ σ : Equiv.Perm (Fin n), rename σ p = p) (v : Fin n → ℤ) :
+    signedLookup (schurCoeff p) v = coeffZ v (p * alternant (staircase n)) := by
+  classical
+  unfold signedLookup coeffZ
+  by_cases hn : ∀ j, 0 ≤ v j
+  · simp only [dite_eq_left hn, ite_eq_left hn]
+    let w := fun j => (v j).toNat
+    change (if Function.Injective w then _ else _) = _
+    have ha := alternating_mul_symmetric hp (alternant_alternating (staircase n))
+    by_cases hw : Function.Injective w
+    · rw [ite_eq_left hw]
+      unfold schurCoeff
+      rw [shifted_sortedPartition w hw]
+      exact (coeff_eq_sign_mul_comp ha w (sortDesc w)).symm
+    · rw [ite_eq_right hw]
+      obtain ⟨j,k,he,hjk⟩ := Function.not_injective_iff.mp hw
+      exact (coeff_repeated ha w hjk he).symm
+  · simp only [dite_eq_right hn, ite_eq_right hn]
+
+lemma coeffZ_monomial_mul {n : ℕ} (l : Fin n → ℕ) (d : Fin n →₀ ℕ)
+    (p : MvPolynomial (Fin n) ℤ) :
+    (monomial d 1 * p).coeff (expVector l) =
+      coeffZ (fun j => (l j : ℤ) - d j) p := by
+  classical
+  rw [coeff_monomial_mul']
+  unfold coeffZ
+  have hiff : d ≤ expVector l ↔ ∀ j, 0 ≤ (l j : ℤ) - d j := by
+    simp [Finsupp.le_def, expVector]
+  by_cases hd : d ≤ expVector l
+  · rw [ite_eq_left hd, ite_eq_left (hiff.mp hd), one_mul]
+    congr 1
+    ext j
+    simp [expVector]
+  · rw [ite_eq_right hd, ite_eq_right (mt hiff.mpr hd)]
+
+theorem target_newton (n b : ℕ) (mu : Fin n → ℕ) :
+    (b : ℤ) * Q n b mu =
+      ∑ i ∈ Finset.Icc 1 b, ∑ d : Complete.Degree (Fin n) 6,
+        signedLookup (Q n (b-i))
+          (fun j => (shifted mu j : ℤ) - (i : ℤ) * d.val j) := by
+  have hh := congrArg
+    (fun p : MvPolynomial (Fin n) ℤ =>
+      (p * alternant (staircase n)).coeff (expVector (shifted mu)))
+    (Complete.pleth_newton n b 6)
+  change (((b : ℤ) • Complete.pleth n b 6) * alternant (staircase n)).coeff
+      (expVector (shifted mu)) = _ at hh
+  rw [smul_mul_assoc, coeff_smul] at hh
+  simp only [Finset.sum_mul, coeff_sum, zsmul_eq_mul] at hh
+  change (b : ℤ) * Q n b mu = _ at hh
+  rw [hh]
+  apply Finset.sum_congr rfl
+  intro i hi
+  apply Finset.sum_congr rfl
+  intro d hd
+  rw [mul_assoc, coeffZ_monomial_mul]
+  have he : (fun j : Fin n => (shifted mu j : ℤ) - ((i • d.val) j : ℕ)) =
+      (fun j => (shifted mu j : ℤ) - (i : ℤ) * d.val j) := by
+    funext j
+    simp
+  rw [he]
+  exact (signedLookup_eq_coeffZ (Complete.pleth n (b-i) 6)
+    (Complete.rename_pleth n (b-i) 6) _).symm
+
+end Foulkes.Lookup
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/Main.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/Main.lean
index b1344cb77010993963d9f688060b3fab4127ae3a..c843368fa1c0d5f4fc9238b5bbf796fd7bb770a7 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/Main.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/Main.lean
@@ -1,40 +1,40 @@
-import OAI.RepresentationTheory.FoulkesSixth.Collision
-import OAI.RepresentationTheory.FoulkesSixth.Diagonal
-import OAI.RepresentationTheory.FoulkesSixth.StripSupport
-import OAI.RepresentationTheory.FoulkesSixth.CompleteNewton
-import OAI.RepresentationTheory.FoulkesSixth.YoungModules
-import OAI.RepresentationTheory.FoulkesSixth.PolynomialSpan
-import OAI.RepresentationTheory.FoulkesSixth.SymTensors
-import OAI.RepresentationTheory.FoulkesSixth.StripCoefficient
-import OAI.RepresentationTheory.FoulkesSixth.SemisimpleBridge
-import OAI.RepresentationTheory.FoulkesSixth.ColorOrbits
-import OAI.RepresentationTheory.FoulkesSixth.SymDual
-import OAI.RepresentationTheory.FoulkesSixth.Differential
-import OAI.RepresentationTheory.FoulkesSixth.StripInterlace
-import OAI.RepresentationTheory.FoulkesSixth.GenericShift
-import OAI.RepresentationTheory.FoulkesSixth.SymOperations
-import OAI.RepresentationTheory.FoulkesSixth.SignedStrips
-import OAI.RepresentationTheory.FoulkesSixth.PlethDual
-import OAI.RepresentationTheory.FoulkesSixth.SymPolynomial
-import OAI.RepresentationTheory.FoulkesSixth.Lookup
-import OAI.RepresentationTheory.FoulkesSixth.LookupExtraction
-import OAI.RepresentationTheory.FoulkesSixth.AlternantExpansion
-import OAI.RepresentationTheory.FoulkesSixth.TensorPolarization
-import OAI.RepresentationTheory.FoulkesSixth.HomogeneousPlethysm
-import OAI.RepresentationTheory.FoulkesSixth.InvariantPolarization
-import OAI.RepresentationTheory.FoulkesSixth.SourceNewton
-import OAI.RepresentationTheory.FoulkesSixth.FastLookup
-import OAI.RepresentationTheory.FoulkesSixth.SixCycles
-import OAI.RepresentationTheory.FoulkesSixth.CompositionEnumeration
-import OAI.RepresentationTheory.FoulkesSixth.FastLookupExact
-import OAI.RepresentationTheory.FoulkesSixth.NewtonUniqueness
-import OAI.RepresentationTheory.FoulkesSixth.YoungGram
-import OAI.RepresentationTheory.FoulkesSixth.Recurrence
-import OAI.RepresentationTheory.FoulkesSixth.VanishingSteps
-import OAI.RepresentationTheory.FoulkesSixth.Rows
-import OAI.RepresentationTheory.FoulkesSixth.IndependentChart
-import OAI.RepresentationTheory.FoulkesSixth.Vanishing
-import OAI.RepresentationTheory.FoulkesSixth.BlockVanishing
-import OAI.RepresentationTheory.FoulkesSixth.QuadraticInjection
-import OAI.RepresentationTheory.FoulkesSixth.BlockInjection
-import OAI.RepresentationTheory.FoulkesSixth.RangeReduction
+import OAI.RepresentationTheory.FoulkesSixth.Collision
+import OAI.RepresentationTheory.FoulkesSixth.Diagonal
+import OAI.RepresentationTheory.FoulkesSixth.StripSupport
+import OAI.RepresentationTheory.FoulkesSixth.CompleteNewton
+import OAI.RepresentationTheory.FoulkesSixth.YoungModules
+import OAI.RepresentationTheory.FoulkesSixth.PolynomialSpan
+import OAI.RepresentationTheory.FoulkesSixth.SymTensors
+import OAI.RepresentationTheory.FoulkesSixth.StripCoefficient
+import OAI.RepresentationTheory.FoulkesSixth.SemisimpleBridge
+import OAI.RepresentationTheory.FoulkesSixth.ColorOrbits
+import OAI.RepresentationTheory.FoulkesSixth.SymDual
+import OAI.RepresentationTheory.FoulkesSixth.Differential
+import OAI.RepresentationTheory.FoulkesSixth.StripInterlace
+import OAI.RepresentationTheory.FoulkesSixth.GenericShift
+import OAI.RepresentationTheory.FoulkesSixth.SymOperations
+import OAI.RepresentationTheory.FoulkesSixth.SignedStrips
+import OAI.RepresentationTheory.FoulkesSixth.PlethDual
+import OAI.RepresentationTheory.FoulkesSixth.SymPolynomial
+import OAI.RepresentationTheory.FoulkesSixth.Lookup
+import OAI.RepresentationTheory.FoulkesSixth.LookupExtraction
+import OAI.RepresentationTheory.FoulkesSixth.AlternantExpansion
+import OAI.RepresentationTheory.FoulkesSixth.TensorPolarization
+import OAI.RepresentationTheory.FoulkesSixth.HomogeneousPlethysm
+import OAI.RepresentationTheory.FoulkesSixth.InvariantPolarization
+import OAI.RepresentationTheory.FoulkesSixth.SourceNewton
+import OAI.RepresentationTheory.FoulkesSixth.FastLookup
+import OAI.RepresentationTheory.FoulkesSixth.SixCycles
+import OAI.RepresentationTheory.FoulkesSixth.CompositionEnumeration
+import OAI.RepresentationTheory.FoulkesSixth.FastLookupExact
+import OAI.RepresentationTheory.FoulkesSixth.NewtonUniqueness
+import OAI.RepresentationTheory.FoulkesSixth.YoungGram
+import OAI.RepresentationTheory.FoulkesSixth.Recurrence
+import OAI.RepresentationTheory.FoulkesSixth.VanishingSteps
+import OAI.RepresentationTheory.FoulkesSixth.Rows
+import OAI.RepresentationTheory.FoulkesSixth.IndependentChart
+import OAI.RepresentationTheory.FoulkesSixth.Vanishing
+import OAI.RepresentationTheory.FoulkesSixth.BlockVanishing
+import OAI.RepresentationTheory.FoulkesSixth.QuadraticInjection
+import OAI.RepresentationTheory.FoulkesSixth.BlockInjection
+import OAI.RepresentationTheory.FoulkesSixth.RangeReduction
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/NewtonUniqueness.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/NewtonUniqueness.lean
index 858a7c76e325c00b292311cc1ff0554cc8487dd6..a9717a6741aeca45d08cc422ada55a708c896b20 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/NewtonUniqueness.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/NewtonUniqueness.lean
@@ -1,96 +1,96 @@
-import Mathlib
-import OAI.RepresentationTheory.FoulkesSixth.SixCycles
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.Lookup
-open MvPolynomial Finset Foulkes.Strips
-
-lemma sortedPartition_sum {n : ℕ} (v : Fin n → ℕ) (hv : Function.Injective v) :
-    (∑ j, sortedPartition v j) + ∑ j, staircase n j = ∑ j, v j := by
-  rw [← Finset.sum_add_distrib]
-  calc
-    _ = ∑ j, v (sortDesc v j) := by
-      apply Finset.sum_congr rfl
-      intro j hj
-      exact congrFun (shifted_sortedPartition v hv) j
-    _ = _ := Equiv.sum_comp (sortDesc v) v
-
-lemma previous_partition_sum {n b : ℕ} (mu : Fin n → ℕ)
-    (hmu : ∑ j, mu j = 6*b) (i : ℕ) (hi : i ≤ b)
-    (d : Complete.Degree (Fin n) 6)
-    (hn : ∀ j, 0 ≤ (shifted mu j : ℤ) - (i : ℤ)*d.val j)
-    (hw : Function.Injective (fun j => ((shifted mu j : ℤ) - (i : ℤ)*d.val j).toNat)) :
-    ∑ j, sortedPartition (fun j => ((shifted mu j : ℤ) - (i : ℤ)*d.val j).toNat) j = 6*(b-i) := by
-  have hd : ∑ j, d.val j = 6 := (Finsupp.degree_eq_sum d.val).symm.trans d.property
-  have hs := sortedPartition_sum _ hw
-  have hh : (∑ j, (((shifted mu j : ℤ) - (i : ℤ)*d.val j).toNat : ℕ) : ℤ) =
-      (6*b : ℕ) + (∑ j, staircase n j : ℕ) - (i : ℤ)*6 := by
-    push_cast
-    simp only [Int.toNat_of_nonneg (hn _), Finset.sum_sub_distrib, ← Finset.mul_sum]
-    simp only [shifted, Nat.cast_add, Finset.sum_add_distrib]
-    simp only [← Nat.cast_sum, hmu, hd]
-    norm_num
-  have hs' := congrArg (fun z : ℕ => (z : ℤ)) hs
-  rw [← Nat.cast_sum] at hh
-  rw [hh] at hs'
-  simp only [Nat.cast_add, Nat.cast_mul, Nat.cast_ofNat] at hs'
-  omega
-
-lemma Q_zero_of_partition {n : ℕ} (mu : Fin n → ℕ) (hmu : ∑ j, mu j = 0) :
-    Q n 0 mu = 1 := by
-  have hm : mu = 0 := by
-    funext j
-    have hj := Finset.single_le_sum (f := mu) (fun k _ => Nat.zero_le _) (Finset.mem_univ j)
-    simpa [hmu] using hj
-  subst mu
-  unfold Q schurCoeff
-  rw [Complete.pleth_zero, one_mul]
-  have hstrict : StrictAnti (staircase n) := by
-    intro j k hjk
-    have hj := j.isLt
-    have hk := k.isLt
-    simp only [staircase]
-    change j.val < k.val at hjk
-    omega
-  have he : shifted (0 : Fin n → ℕ) = staircase n := by ext j; simp [shifted]
-  rw [he, coeff_alternant_strict _ _ hstrict hstrict, ite_eq_left rfl]
-
-theorem target_table_unique (n B : ℕ) (T : ℕ → (Fin n → ℕ) → ℤ)
-    (hzero : ∀ mu, Antitone mu → (∑ j, mu j = 0) → T 0 mu = 1)
-    (hstep : ∀ b, 1 ≤ b → b ≤ B → ∀ mu, Antitone mu → (∑ j, mu j = 6*b) →
-      (b : ℤ)*T b mu = ∑ i ∈ Finset.Icc 1 b, ∑ d : Complete.Degree (Fin n) 6,
-        signedLookup (T (b-i)) (fun j => (shifted mu j : ℤ) - (i : ℤ)*d.val j)) :
-    ∀ b, b ≤ B → ∀ mu, Antitone mu → (∑ j, mu j = 6*b) → T b mu = Q n b mu := by
-  classical
-  intro b
-  induction b using Nat.strong_induction_on with
-  | h b ih =>
-    intro hb mu hm hs
-    by_cases h0 : b = 0
-    · subst b
-      rw [hzero mu hm (by simpa using hs), Q_zero_of_partition mu (by simpa using hs)]
-    have hbpos : 1 ≤ b := by omega
-    apply (mul_left_cancel₀ (show (b : ℤ) ≠ 0 by exact_mod_cast h0))
-    rw [hstep b hbpos hb mu hm hs, target_newton]
-    apply Finset.sum_congr rfl
-    intro i hi
-    apply Finset.sum_congr rfl
-    intro d hd
-    unfold signedLookup
-    dsimp only
-    split_ifs with hn hw
-    · congr 1
-      exact ih (b-i) (by have := (Finset.mem_Icc.mp hi).1; omega)
-        (by omega) _ (sortedPartition_antitone _ hw)
-        (previous_partition_sum mu hs i (Finset.mem_Icc.mp hi).2 d hn hw)
-    · rfl
-    · rfl
-
-end Foulkes.Lookup
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.FoulkesSixth.SixCycles
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.Lookup
+open MvPolynomial Finset Foulkes.Strips
+
+lemma sortedPartition_sum {n : ℕ} (v : Fin n → ℕ) (hv : Function.Injective v) :
+    (∑ j, sortedPartition v j) + ∑ j, staircase n j = ∑ j, v j := by
+  rw [← Finset.sum_add_distrib]
+  calc
+    _ = ∑ j, v (sortDesc v j) := by
+      apply Finset.sum_congr rfl
+      intro j hj
+      exact congrFun (shifted_sortedPartition v hv) j
+    _ = _ := Equiv.sum_comp (sortDesc v) v
+
+lemma previous_partition_sum {n b : ℕ} (mu : Fin n → ℕ)
+    (hmu : ∑ j, mu j = 6*b) (i : ℕ) (hi : i ≤ b)
+    (d : Complete.Degree (Fin n) 6)
+    (hn : ∀ j, 0 ≤ (shifted mu j : ℤ) - (i : ℤ)*d.val j)
+    (hw : Function.Injective (fun j => ((shifted mu j : ℤ) - (i : ℤ)*d.val j).toNat)) :
+    ∑ j, sortedPartition (fun j => ((shifted mu j : ℤ) - (i : ℤ)*d.val j).toNat) j = 6*(b-i) := by
+  have hd : ∑ j, d.val j = 6 := (Finsupp.degree_eq_sum d.val).symm.trans d.property
+  have hs := sortedPartition_sum _ hw
+  have hh : (∑ j, (((shifted mu j : ℤ) - (i : ℤ)*d.val j).toNat : ℕ) : ℤ) =
+      (6*b : ℕ) + (∑ j, staircase n j : ℕ) - (i : ℤ)*6 := by
+    push_cast
+    simp only [Int.toNat_of_nonneg (hn _), Finset.sum_sub_distrib, ← Finset.mul_sum]
+    simp only [shifted, Nat.cast_add, Finset.sum_add_distrib]
+    simp only [← Nat.cast_sum, hmu, hd]
+    norm_num
+  have hs' := congrArg (fun z : ℕ => (z : ℤ)) hs
+  rw [← Nat.cast_sum] at hh
+  rw [hh] at hs'
+  simp only [Nat.cast_add, Nat.cast_mul, Nat.cast_ofNat] at hs'
+  omega
+
+lemma Q_zero_of_partition {n : ℕ} (mu : Fin n → ℕ) (hmu : ∑ j, mu j = 0) :
+    Q n 0 mu = 1 := by
+  have hm : mu = 0 := by
+    funext j
+    have hj := Finset.single_le_sum (f := mu) (fun k _ => Nat.zero_le _) (Finset.mem_univ j)
+    simpa [hmu] using hj
+  subst mu
+  unfold Q schurCoeff
+  rw [Complete.pleth_zero, one_mul]
+  have hstrict : StrictAnti (staircase n) := by
+    intro j k hjk
+    have hj := j.isLt
+    have hk := k.isLt
+    simp only [staircase]
+    change j.val < k.val at hjk
+    omega
+  have he : shifted (0 : Fin n → ℕ) = staircase n := by ext j; simp [shifted]
+  rw [he, coeff_alternant_strict _ _ hstrict hstrict, ite_eq_left rfl]
+
+theorem target_table_unique (n B : ℕ) (T : ℕ → (Fin n → ℕ) → ℤ)
+    (hzero : ∀ mu, Antitone mu → (∑ j, mu j = 0) → T 0 mu = 1)
+    (hstep : ∀ b, 1 ≤ b → b ≤ B → ∀ mu, Antitone mu → (∑ j, mu j = 6*b) →
+      (b : ℤ)*T b mu = ∑ i ∈ Finset.Icc 1 b, ∑ d : Complete.Degree (Fin n) 6,
+        signedLookup (T (b-i)) (fun j => (shifted mu j : ℤ) - (i : ℤ)*d.val j)) :
+    ∀ b, b ≤ B → ∀ mu, Antitone mu → (∑ j, mu j = 6*b) → T b mu = Q n b mu := by
+  classical
+  intro b
+  induction b using Nat.strong_induction_on with
+  | h b ih =>
+    intro hb mu hm hs
+    by_cases h0 : b = 0
+    · subst b
+      rw [hzero mu hm (by simpa using hs), Q_zero_of_partition mu (by simpa using hs)]
+    have hbpos : 1 ≤ b := by omega
+    apply (mul_left_cancel₀ (show (b : ℤ) ≠ 0 by exact_mod_cast h0))
+    rw [hstep b hbpos hb mu hm hs, target_newton]
+    apply Finset.sum_congr rfl
+    intro i hi
+    apply Finset.sum_congr rfl
+    intro d hd
+    unfold signedLookup
+    dsimp only
+    split_ifs with hn hw
+    · congr 1
+      exact ih (b-i) (by have := (Finset.mem_Icc.mp hi).1; omega)
+        (by omega) _ (sortedPartition_antitone _ hw)
+        (previous_partition_sum mu hs i (Finset.mem_Icc.mp hi).2 d hn hw)
+    · rfl
+    · rfl
+
+end Foulkes.Lookup
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/PlethDual.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/PlethDual.lean
index 9b1591a6530dcc0518274ca03859e2f7efee115f..4e85b9d8bde20aac9d99808ae32947c8df8f94d7 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/PlethDual.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/PlethDual.lean
@@ -1,54 +1,54 @@
-import Mathlib
-import OAI.RepresentationTheory.FoulkesSixth.SymOperations
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.PlethDual
-universe u
-open Module SymmetricTensor
-variable (a b : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
-
-local instance plethDualSymAddCommGroup (degree : ℕ) (space : Type u)
-    [AddCommGroup space] [Module ℂ space] : AddCommGroup (Sym degree space) :=
-  (Sym degree space).addCommGroup
-
-def equiv : Sym a (Sym b (Dual ℂ V)) ≃ₗ[ℂ] Dual ℂ (Sym a (Sym b V)) :=
-  (symEquiv a (dualEquiv b (V := V))).trans (dualEquiv a (V := Sym b V))
-
-def flipEquiv : Sym a (Sym b V) ≃ₗ[ℂ] Dual ℂ (Sym a (Sym b (Dual ℂ V))) :=
-  (Module.evalEquiv ℂ (Sym a (Sym b V))).trans (equiv a b V).dualMap
-
-@[simp] lemma flipEquiv_apply (x : Sym a (Sym b V)) (d : Sym a (Sym b (Dual ℂ V))) :
-    flipEquiv a b V x d = equiv a b V d x := rfl
-
-@[simp] lemma equiv_apply (d : Sym a (Sym b (Dual ℂ V))) (x : Sym a (Sym b V)) :
-    equiv a b V d x = pairing a (symMap a (dualEquiv b (V := V)).toLinearMap d) x := rfl
-
-lemma natural (g : V →ₗ[ℂ] V) (d : Sym a (Sym b (Dual ℂ V))) (x : Sym a (Sym b V)) :
-    equiv a b V (symMap a (symMap b g.dualMap) d) x =
-      equiv a b V d (symMap a (symMap b g) x) := by
-  have hi : (dualEquiv b (V := V)).toLinearMap.comp (symMap b g.dualMap) =
-      (symMap b g).dualMap.comp (dualEquiv b (V := V)).toLinearMap := by
-    apply LinearMap.ext
-    intro q
-    apply LinearMap.ext
-    intro v
-    exact pairing_map b g q v
-  have hs : symMap a (dualEquiv b (V := V)).toLinearMap
-      (symMap a (V := Sym b (Dual ℂ V)) (W := Sym b (Dual ℂ V)) (symMap b g.dualMap) d) =
-      symMap a (symMap b g).dualMap (symMap a (dualEquiv b (V := V)).toLinearMap d) := by
-    change ((symMap a (dualEquiv b (V := V)).toLinearMap).comp
-      (symMap a (symMap b g.dualMap))) d =
-      ((symMap a (symMap b g).dualMap).comp
-        (symMap a (dualEquiv b (V := V)).toLinearMap)) d
-    rw [← symMap_comp, hi, symMap_comp]
-  simp only [equiv_apply, hs]
-  exact pairing_map a (symMap b g) _ x
-
-end Foulkes.PlethDual
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.FoulkesSixth.SymOperations
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.PlethDual
+universe u
+open Module SymmetricTensor
+variable (a b : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
+
+local instance plethDualSymAddCommGroup (degree : ℕ) (space : Type u)
+    [AddCommGroup space] [Module ℂ space] : AddCommGroup (Sym degree space) :=
+  (Sym degree space).addCommGroup
+
+def equiv : Sym a (Sym b (Dual ℂ V)) ≃ₗ[ℂ] Dual ℂ (Sym a (Sym b V)) :=
+  (symEquiv a (dualEquiv b (V := V))).trans (dualEquiv a (V := Sym b V))
+
+def flipEquiv : Sym a (Sym b V) ≃ₗ[ℂ] Dual ℂ (Sym a (Sym b (Dual ℂ V))) :=
+  (Module.evalEquiv ℂ (Sym a (Sym b V))).trans (equiv a b V).dualMap
+
+@[simp] lemma flipEquiv_apply (x : Sym a (Sym b V)) (d : Sym a (Sym b (Dual ℂ V))) :
+    flipEquiv a b V x d = equiv a b V d x := rfl
+
+@[simp] lemma equiv_apply (d : Sym a (Sym b (Dual ℂ V))) (x : Sym a (Sym b V)) :
+    equiv a b V d x = pairing a (symMap a (dualEquiv b (V := V)).toLinearMap d) x := rfl
+
+lemma natural (g : V →ₗ[ℂ] V) (d : Sym a (Sym b (Dual ℂ V))) (x : Sym a (Sym b V)) :
+    equiv a b V (symMap a (symMap b g.dualMap) d) x =
+      equiv a b V d (symMap a (symMap b g) x) := by
+  have hi : (dualEquiv b (V := V)).toLinearMap.comp (symMap b g.dualMap) =
+      (symMap b g).dualMap.comp (dualEquiv b (V := V)).toLinearMap := by
+    apply LinearMap.ext
+    intro q
+    apply LinearMap.ext
+    intro v
+    exact pairing_map b g q v
+  have hs : symMap a (dualEquiv b (V := V)).toLinearMap
+      (symMap a (V := Sym b (Dual ℂ V)) (W := Sym b (Dual ℂ V)) (symMap b g.dualMap) d) =
+      symMap a (symMap b g).dualMap (symMap a (dualEquiv b (V := V)).toLinearMap d) := by
+    change ((symMap a (dualEquiv b (V := V)).toLinearMap).comp
+      (symMap a (symMap b g.dualMap))) d =
+      ((symMap a (symMap b g).dualMap).comp
+        (symMap a (dualEquiv b (V := V)).toLinearMap)) d
+    rw [← symMap_comp, hi, symMap_comp]
+  simp only [equiv_apply, hs]
+  exact pairing_map a (symMap b g) _ x
+
+end Foulkes.PlethDual
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/PolynomialSpan.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/PolynomialSpan.lean
index 0fcc7cced85c9f51e8e08d9a136d9b1dcf5ded70..c307590dbb64e6029e2ef9c71a8ab07d7ef9e391 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/PolynomialSpan.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/PolynomialSpan.lean
@@ -1,181 +1,181 @@
-import Mathlib
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.PolynomialSpan
-open MvPolynomial
-
-variable {σ A : Type*} [CommRing A] [Algebra ℂ A]
-
-def mapCoeffs (l : A →ₗ[ℂ] ℂ) (p : MvPolynomial σ A) : MvPolynomial σ ℂ :=
-  ∑ d ∈ p.support, monomial d (l (p.coeff d))
-
-@[simp] lemma coeff_mapCoeffs (l : A →ₗ[ℂ] ℂ) (p : MvPolynomial σ A) (d : σ →₀ ℕ) :
-    (mapCoeffs l p).coeff d = l (p.coeff d) := by
-  classical
-  by_cases hd : d ∈ p.support
-  · simp [mapCoeffs, coeff_monomial, hd]
-  · simp [mapCoeffs, coeff_monomial, hd, notMem_support_iff.mp hd]
-
-@[simp] lemma mapCoeffs_monomial (l : A →ₗ[ℂ] ℂ) (d : σ →₀ ℕ) (a : A) :
-    mapCoeffs l (monomial d a) = monomial d (l a) := by
-  classical
-  ext e
-  simp only [coeff_mapCoeffs, coeff_monomial]
-  split_ifs <;> simp
-
-@[simp] lemma mapCoeffs_add (l : A →ₗ[ℂ] ℂ) (p q : MvPolynomial σ A) :
-    mapCoeffs l (p + q) = mapCoeffs l p + mapCoeffs l q := by
-  ext d
-  simp
-
-lemma eval_mapCoeffs (l : A →ₗ[ℂ] ℂ) (p : MvPolynomial σ A) (x : σ → ℂ) :
-    eval x (mapCoeffs l p) = l (eval (fun i => algebraMap ℂ A (x i)) p) := by
-  classical
-  induction p using MvPolynomial.induction_on' with
-  | add p q hp hq => simp [hp, hq]
-  | monomial d a =>
-    simp only [mapCoeffs_monomial, eval_monomial]
-    have hh : d.prod (fun i k => (algebraMap ℂ A) (x i) ^ k) =
-        algebraMap ℂ A (d.prod (fun i k => x i ^ k)) := by
-      simp [Finsupp.prod, map_prod, map_pow]
-    rw [hh, mul_comm a, ← Algebra.smul_def, map_smul]
-    simp [mul_comm]
-
-@[simp] lemma mapCoeffs_zero (l : A →ₗ[ℂ] ℂ) :
-    mapCoeffs l (0 : MvPolynomial σ A) = 0 := by
-  ext d
-  simp
-
-@[simp] lemma mapCoeffs_smul (l : A →ₗ[ℂ] ℂ) (c : ℂ) (p : MvPolynomial σ A) :
-    mapCoeffs l (c • p) = c • mapCoeffs l p := by
-  ext d
-  simp
-
-def mapCoeffsLM (l : A →ₗ[ℂ] ℂ) : MvPolynomial σ A →ₗ[ℂ] MvPolynomial σ ℂ where
-  toFun := mapCoeffs l
-  map_add' := mapCoeffs_add l
-  map_smul' := mapCoeffs_smul l
-
-@[simp] lemma mapCoeffs_sum {ι : Type*} (l : A →ₗ[ℂ] ℂ)
-    (s : Finset ι) (p : ι → MvPolynomial σ A) :
-    mapCoeffs l (∑ i ∈ s, p i) = ∑ i ∈ s, mapCoeffs l (p i) :=
-  map_sum (mapCoeffsLM l) _ _
-
-@[simp] lemma mapCoeffs_pderiv (l : A →ₗ[ℂ] ℂ) (i : σ) (p : MvPolynomial σ A) :
-    mapCoeffs l (pderiv i p) = pderiv i (mapCoeffs l p) := by
-  ext d
-  simp only [coeff_mapCoeffs, coeff_pderiv]
-  simpa [Algebra.smul_def, Nat.cast_add, mul_comm] using
-    l.map_smul ((d i + 1 : ℕ) : ℂ) (p.coeff (d + Finsupp.single i 1))
-
-@[simp] lemma mapCoeffs_X_mul (l : A →ₗ[ℂ] ℂ) (i : σ) (p : MvPolynomial σ A) :
-    mapCoeffs l (X i * p) = X i * mapCoeffs l p := by
-  induction p using MvPolynomial.induction_on' with
-  | add p q hp hq => simp [mul_add, hp, hq]
-  | monomial d a =>
-    simp [X]
-
-lemma homogeneous_mapCoeffs {M : Type*} [AddCommMonoid M]
-    (l : A →ₗ[ℂ] ℂ) (w : σ → M) {p : MvPolynomial σ A} {d : M}
-    (hp : p.IsWeightedHomogeneous w d) : (mapCoeffs l p).IsWeightedHomogeneous w d := by
-  intro m hm
-  exact hp (fun h => hm (by simp [h]))
-
-theorem coeff_mem_of_evaluations (p : MvPolynomial σ A) (S : Submodule ℂ A)
-    (h : ∀ x : σ → ℂ, eval (fun i => algebraMap ℂ A (x i)) p ∈ S)
-    (d : σ →₀ ℕ) : p.coeff d ∈ S := by
-  by_contra hd
-  obtain ⟨l, hld, hlS⟩ := Submodule.exists_dual_map_eq_bot_of_notMem hd inferInstance
-  have hl (a : A) (ha : a ∈ S) : l a = 0 := by
-    have hh : l a ∈ S.map l := Submodule.mem_map.mpr ⟨a, ha, rfl⟩
-    rwa [hlS, Submodule.mem_bot] at hh
-  have hz : mapCoeffs l p = 0 := by
-    apply MvPolynomial.funext
-    intro x
-    rw [eval_mapCoeffs, hl _ (h x), map_zero]
-  have hc := congrArg (fun polynomial => polynomial.coeff d) hz
-  exact hld (by simpa using hc)
-
-theorem prod_pow_mem_span_powers [Fintype σ] (a : σ → A) (d : σ →₀ ℕ)
-    (n : ℕ) (hd : d.sum (fun _ k => k) = n) :
-    d.prod (fun i k => a i ^ k) ∈
-      Submodule.span ℂ (Set.range fun x : σ → ℂ => (∑ i, x i • a i) ^ n) := by
-  classical
-  let S := Submodule.span ℂ (Set.range fun x : σ → ℂ => (∑ i, x i • a i) ^ n)
-  let G : MvPolynomial σ A := (∑ i, a i • X i) ^ n
-  have heval (x : σ → ℂ) : eval (fun i => algebraMap ℂ A (x i)) G =
-      (∑ i, x i • a i) ^ n := by
-    simp [G, Algebra.smul_def, mul_comm]
-  have hcoeff : G.coeff d ∈ S := coeff_mem_of_evaluations G S (fun x => by
-    rw [heval]
-    exact Submodule.subset_span ⟨x, rfl⟩) d
-  change ((∑ i, a i • X i : MvPolynomial σ A) ^ n).coeff d ∈ S at hcoeff
-  rw [coeff_linearCombination_X_pow_of_fintype, ite_eq_left hd] at hcoeff
-  have hn : (d.multinomial : ℂ) ≠ 0 := by
-    exact_mod_cast (Nat.ne_of_gt (Nat.multinomial_pos (s := d.support) (f := d)))
-  apply (Submodule.smul_mem_iff S hn).mp
-  simpa only [Algebra.smul_def, map_natCast] using hcoeff
-
-theorem homogeneous_eq_span_powers [Fintype σ] (n : ℕ) :
-    homogeneousSubmodule σ ℂ n =
-      Submodule.span ℂ (Set.range fun x : σ → ℂ => (∑ i, x i • X i) ^ n) := by
-  classical
-  apply le_antisymm
-  · intro p hp
-    induction hp using MvPolynomial.IsWeightedHomogeneous.induction_on with
-    | zero => exact Submodule.zero_mem _
-    | add p q hp hq ihp ihq => exact Submodule.add_mem _ ihp ihq
-    | monomial d c hd =>
-      rw [monomial_eq, ← MvPolynomial.smul_eq_C_mul]
-      apply Submodule.smul_mem
-      apply prod_pow_mem_span_powers
-      simpa [Finsupp.weight_apply, Finsupp.sum, smul_eq_mul] using hd
-  · apply Submodule.span_le.mpr
-    rintro _ ⟨x, rfl⟩
-    have hh : (∑ i, x i • (X i : MvPolynomial σ ℂ)).IsHomogeneous 1 := by
-      apply MvPolynomial.IsWeightedHomogeneous.sum
-      intro i _
-      exact (homogeneousSubmodule σ ℂ 1).smul_mem _ (isHomogeneous_X ℂ i)
-    simpa using hh.pow n
-
-theorem prod_mem_span_products {ι : Type*} [Fintype ι]
-    {S : ι → Set A} {x : ι → A}
-    (hx : ∀ i, x i ∈ Submodule.span ℂ (S i)) :
-    (∏ i, x i) ∈ Submodule.span ℂ
-      {p | ∃ y : ι → A, (∀ i, y i ∈ S i) ∧ ∏ i, y i = p} := by
-  classical
-  let T := Submodule.span ℂ
-      {p | ∃ y : ι → A, (∀ i, y i ∈ S i) ∧ ∏ i, y i = p}
-  have hspan : (∏ i, Submodule.span ℂ (S i)) ≤ T := by
-    rw [Submodule.prod_span]
-    apply Submodule.span_le.mpr
-    intro p hp
-    obtain ⟨y, hy, he⟩ := (Set.mem_fintype_prod S p).mp hp
-    exact Submodule.subset_span ⟨y, hy, he⟩
-  apply hspan
-  have hprod (s : Finset ι) : (∏ i ∈ s, x i) ∈ ∏ i ∈ s, Submodule.span ℂ (S i) := by
-    induction s using Finset.induction_on with
-    | empty => simpa using (Submodule.algebraMap_mem (R := ℂ) (A := A) 1)
-    | insert i s hi ih =>
-      simp only [Finset.prod_insert hi]
-      exact Submodule.mul_mem_mul (hx i) ih
-  exact hprod Finset.univ
-
-theorem prod_mem_span_range {ι : Type*} [Fintype ι] {β : ι → Type*}
-    (f : ∀ i, β i → A) {x : ι → A}
-    (hx : ∀ i, x i ∈ Submodule.span ℂ (Set.range (f i))) :
-    (∏ i, x i) ∈ Submodule.span ℂ (Set.range fun y : ∀ i, β i => ∏ i, f i (y i)) := by
-  apply (Submodule.span_mono (t := Set.range fun y : ∀ i, β i => ∏ i, f i (y i))
-    (by
-      rintro p ⟨y, hy, rfl⟩
-      choose z hz using hy
-      exact ⟨z, by simp [hz]⟩)) (prod_mem_span_products hx)
-
-end Foulkes.PolynomialSpan
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.PolynomialSpan
+open MvPolynomial
+
+variable {σ A : Type*} [CommRing A] [Algebra ℂ A]
+
+def mapCoeffs (l : A →ₗ[ℂ] ℂ) (p : MvPolynomial σ A) : MvPolynomial σ ℂ :=
+  ∑ d ∈ p.support, monomial d (l (p.coeff d))
+
+@[simp] lemma coeff_mapCoeffs (l : A →ₗ[ℂ] ℂ) (p : MvPolynomial σ A) (d : σ →₀ ℕ) :
+    (mapCoeffs l p).coeff d = l (p.coeff d) := by
+  classical
+  by_cases hd : d ∈ p.support
+  · simp [mapCoeffs, coeff_monomial, hd]
+  · simp [mapCoeffs, coeff_monomial, hd, notMem_support_iff.mp hd]
+
+@[simp] lemma mapCoeffs_monomial (l : A →ₗ[ℂ] ℂ) (d : σ →₀ ℕ) (a : A) :
+    mapCoeffs l (monomial d a) = monomial d (l a) := by
+  classical
+  ext e
+  simp only [coeff_mapCoeffs, coeff_monomial]
+  split_ifs <;> simp
+
+@[simp] lemma mapCoeffs_add (l : A →ₗ[ℂ] ℂ) (p q : MvPolynomial σ A) :
+    mapCoeffs l (p + q) = mapCoeffs l p + mapCoeffs l q := by
+  ext d
+  simp
+
+lemma eval_mapCoeffs (l : A →ₗ[ℂ] ℂ) (p : MvPolynomial σ A) (x : σ → ℂ) :
+    eval x (mapCoeffs l p) = l (eval (fun i => algebraMap ℂ A (x i)) p) := by
+  classical
+  induction p using MvPolynomial.induction_on' with
+  | add p q hp hq => simp [hp, hq]
+  | monomial d a =>
+    simp only [mapCoeffs_monomial, eval_monomial]
+    have hh : d.prod (fun i k => (algebraMap ℂ A) (x i) ^ k) =
+        algebraMap ℂ A (d.prod (fun i k => x i ^ k)) := by
+      simp [Finsupp.prod, map_prod, map_pow]
+    rw [hh, mul_comm a, ← Algebra.smul_def, map_smul]
+    simp [mul_comm]
+
+@[simp] lemma mapCoeffs_zero (l : A →ₗ[ℂ] ℂ) :
+    mapCoeffs l (0 : MvPolynomial σ A) = 0 := by
+  ext d
+  simp
+
+@[simp] lemma mapCoeffs_smul (l : A →ₗ[ℂ] ℂ) (c : ℂ) (p : MvPolynomial σ A) :
+    mapCoeffs l (c • p) = c • mapCoeffs l p := by
+  ext d
+  simp
+
+def mapCoeffsLM (l : A →ₗ[ℂ] ℂ) : MvPolynomial σ A →ₗ[ℂ] MvPolynomial σ ℂ where
+  toFun := mapCoeffs l
+  map_add' := mapCoeffs_add l
+  map_smul' := mapCoeffs_smul l
+
+@[simp] lemma mapCoeffs_sum {ι : Type*} (l : A →ₗ[ℂ] ℂ)
+    (s : Finset ι) (p : ι → MvPolynomial σ A) :
+    mapCoeffs l (∑ i ∈ s, p i) = ∑ i ∈ s, mapCoeffs l (p i) :=
+  map_sum (mapCoeffsLM l) _ _
+
+@[simp] lemma mapCoeffs_pderiv (l : A →ₗ[ℂ] ℂ) (i : σ) (p : MvPolynomial σ A) :
+    mapCoeffs l (pderiv i p) = pderiv i (mapCoeffs l p) := by
+  ext d
+  simp only [coeff_mapCoeffs, coeff_pderiv]
+  simpa [Algebra.smul_def, Nat.cast_add, mul_comm] using
+    l.map_smul ((d i + 1 : ℕ) : ℂ) (p.coeff (d + Finsupp.single i 1))
+
+@[simp] lemma mapCoeffs_X_mul (l : A →ₗ[ℂ] ℂ) (i : σ) (p : MvPolynomial σ A) :
+    mapCoeffs l (X i * p) = X i * mapCoeffs l p := by
+  induction p using MvPolynomial.induction_on' with
+  | add p q hp hq => simp [mul_add, hp, hq]
+  | monomial d a =>
+    simp [X]
+
+lemma homogeneous_mapCoeffs {M : Type*} [AddCommMonoid M]
+    (l : A →ₗ[ℂ] ℂ) (w : σ → M) {p : MvPolynomial σ A} {d : M}
+    (hp : p.IsWeightedHomogeneous w d) : (mapCoeffs l p).IsWeightedHomogeneous w d := by
+  intro m hm
+  exact hp (fun h => hm (by simp [h]))
+
+theorem coeff_mem_of_evaluations (p : MvPolynomial σ A) (S : Submodule ℂ A)
+    (h : ∀ x : σ → ℂ, eval (fun i => algebraMap ℂ A (x i)) p ∈ S)
+    (d : σ →₀ ℕ) : p.coeff d ∈ S := by
+  by_contra hd
+  obtain ⟨l, hld, hlS⟩ := Submodule.exists_dual_map_eq_bot_of_notMem hd inferInstance
+  have hl (a : A) (ha : a ∈ S) : l a = 0 := by
+    have hh : l a ∈ S.map l := Submodule.mem_map.mpr ⟨a, ha, rfl⟩
+    rwa [hlS, Submodule.mem_bot] at hh
+  have hz : mapCoeffs l p = 0 := by
+    apply MvPolynomial.funext
+    intro x
+    rw [eval_mapCoeffs, hl _ (h x), map_zero]
+  have hc := congrArg (fun polynomial => polynomial.coeff d) hz
+  exact hld (by simpa using hc)
+
+theorem prod_pow_mem_span_powers [Fintype σ] (a : σ → A) (d : σ →₀ ℕ)
+    (n : ℕ) (hd : d.sum (fun _ k => k) = n) :
+    d.prod (fun i k => a i ^ k) ∈
+      Submodule.span ℂ (Set.range fun x : σ → ℂ => (∑ i, x i • a i) ^ n) := by
+  classical
+  let S := Submodule.span ℂ (Set.range fun x : σ → ℂ => (∑ i, x i • a i) ^ n)
+  let G : MvPolynomial σ A := (∑ i, a i • X i) ^ n
+  have heval (x : σ → ℂ) : eval (fun i => algebraMap ℂ A (x i)) G =
+      (∑ i, x i • a i) ^ n := by
+    simp [G, Algebra.smul_def, mul_comm]
+  have hcoeff : G.coeff d ∈ S := coeff_mem_of_evaluations G S (fun x => by
+    rw [heval]
+    exact Submodule.subset_span ⟨x, rfl⟩) d
+  change ((∑ i, a i • X i : MvPolynomial σ A) ^ n).coeff d ∈ S at hcoeff
+  rw [coeff_linearCombination_X_pow_of_fintype, ite_eq_left hd] at hcoeff
+  have hn : (d.multinomial : ℂ) ≠ 0 := by
+    exact_mod_cast (Nat.ne_of_gt (Nat.multinomial_pos (s := d.support) (f := d)))
+  apply (Submodule.smul_mem_iff S hn).mp
+  simpa only [Algebra.smul_def, map_natCast] using hcoeff
+
+theorem homogeneous_eq_span_powers [Fintype σ] (n : ℕ) :
+    homogeneousSubmodule σ ℂ n =
+      Submodule.span ℂ (Set.range fun x : σ → ℂ => (∑ i, x i • X i) ^ n) := by
+  classical
+  apply le_antisymm
+  · intro p hp
+    induction hp using MvPolynomial.IsWeightedHomogeneous.induction_on with
+    | zero => exact Submodule.zero_mem _
+    | add p q hp hq ihp ihq => exact Submodule.add_mem _ ihp ihq
+    | monomial d c hd =>
+      rw [monomial_eq, ← MvPolynomial.smul_eq_C_mul]
+      apply Submodule.smul_mem
+      apply prod_pow_mem_span_powers
+      simpa [Finsupp.weight_apply, Finsupp.sum, smul_eq_mul] using hd
+  · apply Submodule.span_le.mpr
+    rintro _ ⟨x, rfl⟩
+    have hh : (∑ i, x i • (X i : MvPolynomial σ ℂ)).IsHomogeneous 1 := by
+      apply MvPolynomial.IsWeightedHomogeneous.sum
+      intro i _
+      exact (homogeneousSubmodule σ ℂ 1).smul_mem _ (isHomogeneous_X ℂ i)
+    simpa using hh.pow n
+
+theorem prod_mem_span_products {ι : Type*} [Fintype ι]
+    {S : ι → Set A} {x : ι → A}
+    (hx : ∀ i, x i ∈ Submodule.span ℂ (S i)) :
+    (∏ i, x i) ∈ Submodule.span ℂ
+      {p | ∃ y : ι → A, (∀ i, y i ∈ S i) ∧ ∏ i, y i = p} := by
+  classical
+  let T := Submodule.span ℂ
+      {p | ∃ y : ι → A, (∀ i, y i ∈ S i) ∧ ∏ i, y i = p}
+  have hspan : (∏ i, Submodule.span ℂ (S i)) ≤ T := by
+    rw [Submodule.prod_span]
+    apply Submodule.span_le.mpr
+    intro p hp
+    obtain ⟨y, hy, he⟩ := (Set.mem_fintype_prod S p).mp hp
+    exact Submodule.subset_span ⟨y, hy, he⟩
+  apply hspan
+  have hprod (s : Finset ι) : (∏ i ∈ s, x i) ∈ ∏ i ∈ s, Submodule.span ℂ (S i) := by
+    induction s using Finset.induction_on with
+    | empty => simpa using (Submodule.algebraMap_mem (R := ℂ) (A := A) 1)
+    | insert i s hi ih =>
+      simp only [Finset.prod_insert hi]
+      exact Submodule.mul_mem_mul (hx i) ih
+  exact hprod Finset.univ
+
+theorem prod_mem_span_range {ι : Type*} [Fintype ι] {β : ι → Type*}
+    (f : ∀ i, β i → A) {x : ι → A}
+    (hx : ∀ i, x i ∈ Submodule.span ℂ (Set.range (f i))) :
+    (∏ i, x i) ∈ Submodule.span ℂ (Set.range fun y : ∀ i, β i => ∏ i, f i (y i)) := by
+  apply (Submodule.span_mono (t := Set.range fun y : ∀ i, β i => ∏ i, f i (y i))
+    (by
+      rintro p ⟨y, hy, rfl⟩
+      choose z hz using hy
+      exact ⟨z, by simp [hz]⟩)) (prod_mem_span_products hx)
+
+end Foulkes.PolynomialSpan
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/QuadraticInjection.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/QuadraticInjection.lean
index 8ffcb44e87f0f23bc39a4144f018d3813e266b72..22743788044931c310cee3bf56806835545f9530 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/QuadraticInjection.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/QuadraticInjection.lean
@@ -1,316 +1,316 @@
-import Mathlib
-import OAI.RepresentationTheory.FoulkesSixth.PlethDual
-import OAI.RepresentationTheory.FoulkesSixth.Vanishing
-import OAI.RepresentationTheory.FoulkesSixth.Rows
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes
-universe u
-variable (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]
-
-instance symAddCommGroup : AddCommGroup (Sym n V) := (Sym n V).addCommGroup
-instance symModule : Module ℂ (Sym n V) := (Sym n V).module
-instance symFinite [FiniteDimensional ℂ V] : FiniteDimensional ℂ (Sym n V) :=
-  inferInstance
-instance symFree : Module.Free ℂ (Sym n V) := inferInstance
-
-end Foulkes
-
-namespace Foulkes.Canonical
-universe u
-open Module MvPolynomial SymmetricTensor Polarization
-variable {V : Type u} [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
-variable {n : ℕ} (B : Basis (Fin n) ℂ V) (a b : ℕ)
-
-def firstEquiv : Sym a V ≃ₗ[ℂ] R a n 1 :=
-  (symEquiv a (oneEquiv V).symm).trans (Rows.equiv B a 1)
-
-def firstMap : Sym a V →ₗ[ℂ] P a n :=
-  (R a n 1).subtype.comp (firstEquiv B a).toLinearMap
-
-lemma firstMap_mem (x : Sym a V) : firstMap B a x ∈ R a n 1 :=
-  (firstEquiv B a x).property
-
-@[simp] lemma firstMap_power (v : V) : firstMap B a (power a V v) = z (a := a) (B.equivFun v) := by
-  change Rows.invariantMap B a 1
-    (symMap a (oneEquiv V).symm.toLinearMap (power a V v)) = _
-  rw [symMap_power, LinearEquiv.coe_coe, oneEquiv_symm]
-  change Rows.tensorMap B a 1
-    (PiTensorProduct.tprod ℂ (fun _ : Fin a => power 1 V v)) = _
-  rw [Rows.tensorMap_powers]
-  simp only [pureRows, pow_one, z]
-
-def productPolynomial : Sym b (Sym a V) →ₗ[ℂ] P a n :=
-  SymAlgebra.multiply b (firstMap B a)
-
-@[simp] lemma productPolynomial_power (q : Sym a V) :
-    productPolynomial B a b (power b (Sym a V) q) = firstMap B a q ^ b :=
-  SymAlgebra.multiply_power b (firstMap B a) q
-
-lemma productPolynomial_mem (x : Sym b (Sym a V)) : productPolynomial B a b x ∈ R a n b := by
-  have h : Submodule.span ℂ (Set.range (power b (Sym a V))) ≤
-      (R a n b).comap (productPolynomial B a b) := by
-    apply Submodule.span_le.mpr
-    rintro _ ⟨q, rfl⟩
-    change productPolynomial B a b (power b (Sym a V) q) ∈ R a n b
-    rw [productPolynomial_power]
-    constructor
-    · have hq := (firstMap_mem B a q).1.pow b
-      simpa [W, Pi.smul_def] using hq
-    · intro σ
-      rw [map_pow, (firstMap_mem B a q).2 σ]
-  rw [span_power] at h
-  exact h (Submodule.mem_top : x ∈ ⊤)
-
-lemma productPolynomial_project (q : Fin b → Sym a V) :
-    productPolynomial B a b (project b (Sym a V) (PiTensorProduct.tprod ℂ q)) =
-      ∏ i, firstMap B a (q i) := by
-  exact (SymAlgebra.multiply_project b (firstMap B a) _).trans
-    (SymAlgebra.tensorMul_tprod b (firstMap B a) q)
-
-lemma products_in_range (y : Fin b → Fin n → ℂ) :
-    (∏ j, z (a := a) (y j)) ∈ (productPolynomial B a b).range := by
-  refine ⟨project b (Sym a V) (PiTensorProduct.tprod ℂ
-    (fun j => power a V (B.equivFun.symm (y j)))), ?_⟩
-  rw [productPolynomial_project]
-  simp only [firstMap_power, LinearEquiv.apply_symm_apply]
-
-theorem productPolynomial_range (ha : 1 ≤ a) (hb0 : 1 ≤ b) (hb : a*(a-1) ≤ b) :
-    (productPolynomial B a b).range = R a n b := by
-  classical
-  apply le_antisymm
-  · rintro _ ⟨x, rfl⟩
-    exact productPolynomial_mem B a b x
-  · intro p hp
-    by_contra hnot
-    obtain ⟨l, hlp, hlrange⟩ := Submodule.exists_dual_map_eq_bot_of_notMem hnot inferInstance
-    have hkill (f : P a n) (hf : f ∈ (productPolynomial B a b).range) : l f = 0 := by
-      have hh : l f ∈ ((productPolynomial B a b).range).map l := Submodule.mem_map.mpr ⟨f,hf,rfl⟩
-      rwa [hlrange, Submodule.mem_bot] at hh
-    let lav := l.comp (Polarization.average a n)
-    have hs : Vanishing.Symmetric lav := by
-      intro σ f
-      change l (Polarization.average a n (renameRow σ f)) = l (Polarization.average a n f)
-      rw [Vanishing.average_renameRow]
-    have hz : Vanishing.ProductsVanish lav b := by
-      intro y
-      change l (Polarization.average a n (∏ j, z (y j))) = 0
-      rw [Polarization.average_eq (show (∏ j, z (a := a) (y j)) ∈ rowInvariants a n from by
-        intro σ; simp)]
-      exact hkill _ (products_in_range B a b y)
-    have hh := Vanishing.vanishing a ha hb0 hb hs hz p hp.1
-    change l (Polarization.average a n p) = 0 at hh
-    rw [Polarization.average_eq hp.2] at hh
-    exact hlp hh
-
-def map : Sym b (Sym a V) →ₗ[ℂ] Sym a (Sym b V) :=
-  (Rows.equiv B a b).symm.toLinearMap.comp
-    ((productPolynomial B a b).codRestrict (R a n b) (productPolynomial_mem B a b))
-
-@[simp] lemma map_polynomial (x : Sym b (Sym a V)) :
-    Rows.invariantMap B a b (map B a b x) = productPolynomial B a b x := by
-  change (Rows.equiv B a b
-    ((Rows.equiv B a b).symm ⟨productPolynomial B a b x, productPolynomial_mem B a b x⟩) : P a n) = _
-  rw [LinearEquiv.apply_symm_apply]
-
-theorem surjective (ha : 1 ≤ a) (hb0 : 1 ≤ b) (hb : a*(a-1) ≤ b) :
-    Function.Surjective (map B a b) := by
-  intro x
-  have hp : Rows.invariantMap B a b x ∈ (productPolynomial B a b).range := by
-    rw [productPolynomial_range B a b ha hb0 hb]
-    exact (Rows.equiv B a b x).property
-  obtain ⟨y, hy⟩ := hp
-  refine ⟨y, (Rows.invariantMap_injective B a b) ?_⟩
-  rw [map_polynomial, hy]
-
-end Foulkes.Canonical
-
-namespace Foulkes.Canonical
-universe u
-open Module MvPolynomial SymmetricTensor Polarization
-variable {V : Type u} [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
-variable {n : ℕ} (B : Basis (Fin n) ℂ V) (a b : ℕ)
-
-omit [FiniteDimensional ℂ V] in
-section
-
-def substitution (g : V →ₗ[ℂ] V) : P a n →ₐ[ℂ] P a n :=
-  aeval (fun ik : Fin a × Fin n =>
-    Rows.row ik.1 (SymPolynomial.linearPolynomial B (g (B ik.2))))
-
-lemma linearPolynomial_basis (k : Fin n) : SymPolynomial.linearPolynomial B (B k) = X k := by
-  classical
-  simp only [SymPolynomial.linearPolynomial_apply, B.equivFun_self]
-  simp
-
-lemma substitution_row_linear (g : V →ₗ[ℂ] V) (i : Fin a) (v : V) :
-    substitution B a g (Rows.row i (SymPolynomial.linearPolynomial B v)) =
-      Rows.row i (SymPolynomial.linearPolynomial B (g v)) := by
-  have hh : (substitution B a g).toLinearMap.comp
-      ((Rows.row i).toLinearMap.comp (SymPolynomial.linearPolynomial B)) =
-      (Rows.row i).toLinearMap.comp ((SymPolynomial.linearPolynomial B).comp g) := by
-    apply B.ext
-    intro k
-    simp only [LinearMap.comp_apply, AlgHom.toLinearMap_apply, linearPolynomial_basis]
-    simp [Rows.row, substitution]
-  exact DFunLike.congr_fun hh v
-
-lemma substitution_rowForm (g : V →ₗ[ℂ] V) (i : Fin a) (v : V) :
-    substitution B a g (rowForm i (B.equivFun v)) = rowForm i (B.equivFun (g v)) := by
-  simpa only [Rows.row_linear] using substitution_row_linear B a g i v
-
-lemma substitution_z (g : V →ₗ[ℂ] V) (v : V) :
-    substitution B a g (z (a := a) (B.equivFun v)) = z (a := a) (B.equivFun (g v)) := by
-  simp only [z, map_prod, substitution_rowForm]
-
-lemma row_polynomial_natural (g : V →ₗ[ℂ] V) (i : Fin a) (q : Sym b V) :
-    Rows.row i (SymPolynomial.polynomial B b (symMap b g q)) =
-      substitution B a g (Rows.row i (SymPolynomial.polynomial B b q)) := by
-  have hh : (Rows.row i).toLinearMap.comp ((SymPolynomial.polynomial B b).comp (symMap b g)) =
-      (substitution B a g).toLinearMap.comp
-        ((Rows.row i).toLinearMap.comp (SymPolynomial.polynomial B b)) := by
-    apply symMap_ext
-    intro v
-    simp only [LinearMap.comp_apply, AlgHom.toLinearMap_apply, symMap_power,
-      SymPolynomial.polynomial_power, map_pow, substitution_row_linear]
-  exact DFunLike.congr_fun hh q
-
-lemma tensorMap_natural (g : V →ₗ[ℂ] V) (x : T a (Sym b V)) :
-    Rows.tensorMap B a b (PiTensorProduct.map (fun _ : Fin a => symMap b g) x) =
-      substitution B a g (Rows.tensorMap B a b x) := by
-  have hh : (Rows.tensorMap B a b).comp (PiTensorProduct.map (fun _ : Fin a => symMap b g)) =
-      (substitution B a g).toLinearMap.comp (Rows.tensorMap B a b) := by
-    apply PiTensorProduct.ext
-    apply MultilinearMap.ext
-    intro v
-    simp only [LinearMap.compMultilinearMap_apply, LinearMap.comp_apply, PiTensorProduct.map_tprod,
-      Rows.tensorMap_tprod, AlgHom.toLinearMap_apply, map_prod, row_polynomial_natural]
-  exact DFunLike.congr_fun hh x
-
-lemma invariantMap_natural (g : V →ₗ[ℂ] V) (x : Sym a (Sym b V)) :
-    Rows.invariantMap B a b (symMap a (symMap b g) x) =
-      substitution B a g (Rows.invariantMap B a b x) :=
-  tensorMap_natural B a b g x
-
-end
-
-lemma firstMap_natural (g : V →ₗ[ℂ] V) (q : Sym a V) :
-    firstMap B a (symMap a g q) = substitution B a g (firstMap B a q) := by
-  have hh : (firstMap B a).comp (symMap a g) =
-      (substitution B a g).toLinearMap.comp (firstMap B a) := by
-    apply symMap_ext
-    intro v
-    simp only [LinearMap.comp_apply, AlgHom.toLinearMap_apply, symMap_power,
-      firstMap_power, substitution_z]
-  exact DFunLike.congr_fun hh q
-
-lemma productPolynomial_natural (g : V →ₗ[ℂ] V) (x : Sym b (Sym a V)) :
-    productPolynomial B a b (symMap b (symMap a g) x) =
-      substitution B a g (productPolynomial B a b x) := by
-  have hh : (productPolynomial B a b).comp (symMap b (symMap a g)) =
-      (substitution B a g).toLinearMap.comp (productPolynomial B a b) := by
-    apply symMap_ext
-    intro q
-    simp only [LinearMap.comp_apply, AlgHom.toLinearMap_apply, symMap_power,
-      productPolynomial_power, firstMap_natural, map_pow]
-  exact DFunLike.congr_fun hh x
-
-theorem natural (g : V →ₗ[ℂ] V) (x : Sym b (Sym a V)) :
-    map B a b (symMap b (symMap a g) x) = symMap a (symMap b g) (map B a b x) := by
-  apply Rows.invariantMap_injective B a b
-  rw [map_polynomial, invariantMap_natural, map_polynomial, productPolynomial_natural]
-
-end Foulkes.Canonical
-
-namespace Foulkes.PlethDual
-universe u
-open Module SymmetricTensor
-variable (a b : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
-
-lemma natural_congr (g : V →ₗ[ℂ] V) (d : Sym a (Sym b (Dual ℂ V))) (x : Sym a (Sym b V)) :
-    equiv a b V (symMap a (symMap b g.dualMap) d) x =
-      equiv a b V d (symMap a (symMap b g) x) := by
-  have hi : (dualEquiv b (V := V)).toLinearMap.comp (symMap b g.dualMap) =
-      (symMap b g).dualMap.comp (dualEquiv b (V := V)).toLinearMap := by
-    apply LinearMap.ext
-    intro q
-    apply LinearMap.ext
-    intro v
-    exact pairing_map b g q v
-  have hs := congrArg (fun (f : Sym b (Dual ℂ V) →ₗ[ℂ] Dual ℂ (Sym b V)) =>
-    symMap a f d) hi
-  simp only [symMap_comp, LinearMap.comp_apply] at hs
-  simp only [equiv_apply, hs]
-  exact pairing_map a (symMap b g) _ x
-
-end Foulkes.PlethDual
-
-namespace Foulkes.Injection
-universe u
-open Module
-variable (a b : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
-
-def map : Sym a (Sym b V) →ₗ[ℂ] Sym b (Sym a V) :=
-  (PlethDual.flipEquiv b a V).symm.toLinearMap.comp
-    ((Canonical.map (Module.finBasis ℂ (Dual ℂ V)) a b).dualMap.comp
-      (PlethDual.flipEquiv a b V).toLinearMap)
-
-lemma adjoint (x : Sym a (Sym b V)) (d : Sym b (Sym a (Dual ℂ V))) :
-    PlethDual.equiv b a V d (map a b V x) =
-      PlethDual.equiv a b V (Canonical.map (Module.finBasis ℂ (Dual ℂ V)) a b d) x := by
-  change PlethDual.flipEquiv b a V
-    ((PlethDual.flipEquiv b a V).symm
-      ((Canonical.map (Module.finBasis ℂ (Dual ℂ V)) a b).dualMap
-        (PlethDual.flipEquiv a b V x))) d = _
-  rw [LinearEquiv.apply_symm_apply]
-  rfl
-
-lemma injective (ha : 1 ≤ a) (hb0 : 1 ≤ b) (hb : a*(a-1) ≤ b) :
-    Function.Injective (map a b V) :=
-  (PlethDual.flipEquiv b a V).symm.injective.comp
-    ((LinearMap.dualMap_injective_of_surjective
-      (Canonical.surjective (Module.finBasis ℂ (Dual ℂ V)) a b ha hb0 hb)).comp
-      (PlethDual.flipEquiv a b V).injective)
-
-theorem natural (g : V →ₗ[ℂ] V) (x : Sym a (Sym b V)) :
-    map a b V (symMap a (symMap b g) x) = symMap b (symMap a g) (map a b V x) := by
-  apply (PlethDual.flipEquiv b a V).injective
-  apply LinearMap.ext
-  intro d
-  change PlethDual.equiv b a V d (map a b V (symMap a (symMap b g) x)) =
-    PlethDual.equiv b a V d (symMap b (symMap a g) (map a b V x))
-  calc
-    _ = PlethDual.equiv a b V (Canonical.map (Module.finBasis ℂ (Dual ℂ V)) a b d)
-        (symMap a (symMap b g) x) := adjoint a b V _ d
-    _ = PlethDual.equiv a b V
-        (symMap a (symMap b g.dualMap) (Canonical.map (Module.finBasis ℂ (Dual ℂ V)) a b d)) x :=
-      (PlethDual.natural a b V g _ x).symm
-    _ = PlethDual.equiv a b V
-        (Canonical.map (Module.finBasis ℂ (Dual ℂ V)) a b (symMap b (symMap a g.dualMap) d)) x := by
-      rw [Canonical.natural]
-    _ = PlethDual.equiv b a V (symMap b (symMap a g.dualMap) d) (map a b V x) :=
-      (adjoint a b V x _).symm
-    _ = _ := PlethDual.natural b a V g d (map a b V x)
-
-theorem quadratic_comparison (ha : 2 ≤ a) (hb : a*(a-1) ≤ b) :
-    ∃ i : Sym a (Sym b V) →ₗ[ℂ] Sym b (Sym a V), Function.Injective i ∧
-      ∀ (g : V ≃ₗ[ℂ] V) (x : Sym a (Sym b V)),
-        i (symMap a (symMap b g.toLinearMap) x) = symMap b (symMap a g.toLinearMap) (i x) := by
-  have hb0 : 1 ≤ b := le_trans (Nat.mul_pos (by omega) (by omega)) hb
-  exact ⟨map a b V, injective a b V (by omega) hb0 hb, fun g x => natural a b V g.toLinearMap x⟩
-
-theorem sixth_large (b : ℕ) (hb : 30 ≤ b)
-    (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] :
-    ∃ i : Sym 6 (Sym b V) →ₗ[ℂ] Sym b (Sym 6 V), Function.Injective i ∧
-      ∀ (g : V ≃ₗ[ℂ] V) (x : Sym 6 (Sym b V)),
-        i (symMap 6 (symMap b g.toLinearMap) x) = symMap b (symMap 6 g.toLinearMap) (i x) :=
-  quadratic_comparison 6 b V (by omega) (by omega)
-
-end Foulkes.Injection
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.FoulkesSixth.PlethDual
+import OAI.RepresentationTheory.FoulkesSixth.Vanishing
+import OAI.RepresentationTheory.FoulkesSixth.Rows
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes
+universe u
+variable (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]
+
+instance symAddCommGroup : AddCommGroup (Sym n V) := (Sym n V).addCommGroup
+instance symModule : Module ℂ (Sym n V) := (Sym n V).module
+instance symFinite [FiniteDimensional ℂ V] : FiniteDimensional ℂ (Sym n V) :=
+  inferInstance
+instance symFree : Module.Free ℂ (Sym n V) := inferInstance
+
+end Foulkes
+
+namespace Foulkes.Canonical
+universe u
+open Module MvPolynomial SymmetricTensor Polarization
+variable {V : Type u} [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
+variable {n : ℕ} (B : Basis (Fin n) ℂ V) (a b : ℕ)
+
+def firstEquiv : Sym a V ≃ₗ[ℂ] R a n 1 :=
+  (symEquiv a (oneEquiv V).symm).trans (Rows.equiv B a 1)
+
+def firstMap : Sym a V →ₗ[ℂ] P a n :=
+  (R a n 1).subtype.comp (firstEquiv B a).toLinearMap
+
+lemma firstMap_mem (x : Sym a V) : firstMap B a x ∈ R a n 1 :=
+  (firstEquiv B a x).property
+
+@[simp] lemma firstMap_power (v : V) : firstMap B a (power a V v) = z (a := a) (B.equivFun v) := by
+  change Rows.invariantMap B a 1
+    (symMap a (oneEquiv V).symm.toLinearMap (power a V v)) = _
+  rw [symMap_power, LinearEquiv.coe_coe, oneEquiv_symm]
+  change Rows.tensorMap B a 1
+    (PiTensorProduct.tprod ℂ (fun _ : Fin a => power 1 V v)) = _
+  rw [Rows.tensorMap_powers]
+  simp only [pureRows, pow_one, z]
+
+def productPolynomial : Sym b (Sym a V) →ₗ[ℂ] P a n :=
+  SymAlgebra.multiply b (firstMap B a)
+
+@[simp] lemma productPolynomial_power (q : Sym a V) :
+    productPolynomial B a b (power b (Sym a V) q) = firstMap B a q ^ b :=
+  SymAlgebra.multiply_power b (firstMap B a) q
+
+lemma productPolynomial_mem (x : Sym b (Sym a V)) : productPolynomial B a b x ∈ R a n b := by
+  have h : Submodule.span ℂ (Set.range (power b (Sym a V))) ≤
+      (R a n b).comap (productPolynomial B a b) := by
+    apply Submodule.span_le.mpr
+    rintro _ ⟨q, rfl⟩
+    change productPolynomial B a b (power b (Sym a V) q) ∈ R a n b
+    rw [productPolynomial_power]
+    constructor
+    · have hq := (firstMap_mem B a q).1.pow b
+      simpa [W, Pi.smul_def] using hq
+    · intro σ
+      rw [map_pow, (firstMap_mem B a q).2 σ]
+  rw [span_power] at h
+  exact h (Submodule.mem_top : x ∈ ⊤)
+
+lemma productPolynomial_project (q : Fin b → Sym a V) :
+    productPolynomial B a b (project b (Sym a V) (PiTensorProduct.tprod ℂ q)) =
+      ∏ i, firstMap B a (q i) := by
+  exact (SymAlgebra.multiply_project b (firstMap B a) _).trans
+    (SymAlgebra.tensorMul_tprod b (firstMap B a) q)
+
+lemma products_in_range (y : Fin b → Fin n → ℂ) :
+    (∏ j, z (a := a) (y j)) ∈ (productPolynomial B a b).range := by
+  refine ⟨project b (Sym a V) (PiTensorProduct.tprod ℂ
+    (fun j => power a V (B.equivFun.symm (y j)))), ?_⟩
+  rw [productPolynomial_project]
+  simp only [firstMap_power, LinearEquiv.apply_symm_apply]
+
+theorem productPolynomial_range (ha : 1 ≤ a) (hb0 : 1 ≤ b) (hb : a*(a-1) ≤ b) :
+    (productPolynomial B a b).range = R a n b := by
+  classical
+  apply le_antisymm
+  · rintro _ ⟨x, rfl⟩
+    exact productPolynomial_mem B a b x
+  · intro p hp
+    by_contra hnot
+    obtain ⟨l, hlp, hlrange⟩ := Submodule.exists_dual_map_eq_bot_of_notMem hnot inferInstance
+    have hkill (f : P a n) (hf : f ∈ (productPolynomial B a b).range) : l f = 0 := by
+      have hh : l f ∈ ((productPolynomial B a b).range).map l := Submodule.mem_map.mpr ⟨f,hf,rfl⟩
+      rwa [hlrange, Submodule.mem_bot] at hh
+    let lav := l.comp (Polarization.average a n)
+    have hs : Vanishing.Symmetric lav := by
+      intro σ f
+      change l (Polarization.average a n (renameRow σ f)) = l (Polarization.average a n f)
+      rw [Vanishing.average_renameRow]
+    have hz : Vanishing.ProductsVanish lav b := by
+      intro y
+      change l (Polarization.average a n (∏ j, z (y j))) = 0
+      rw [Polarization.average_eq (show (∏ j, z (a := a) (y j)) ∈ rowInvariants a n from by
+        intro σ; simp)]
+      exact hkill _ (products_in_range B a b y)
+    have hh := Vanishing.vanishing a ha hb0 hb hs hz p hp.1
+    change l (Polarization.average a n p) = 0 at hh
+    rw [Polarization.average_eq hp.2] at hh
+    exact hlp hh
+
+def map : Sym b (Sym a V) →ₗ[ℂ] Sym a (Sym b V) :=
+  (Rows.equiv B a b).symm.toLinearMap.comp
+    ((productPolynomial B a b).codRestrict (R a n b) (productPolynomial_mem B a b))
+
+@[simp] lemma map_polynomial (x : Sym b (Sym a V)) :
+    Rows.invariantMap B a b (map B a b x) = productPolynomial B a b x := by
+  change (Rows.equiv B a b
+    ((Rows.equiv B a b).symm ⟨productPolynomial B a b x, productPolynomial_mem B a b x⟩) : P a n) = _
+  rw [LinearEquiv.apply_symm_apply]
+
+theorem surjective (ha : 1 ≤ a) (hb0 : 1 ≤ b) (hb : a*(a-1) ≤ b) :
+    Function.Surjective (map B a b) := by
+  intro x
+  have hp : Rows.invariantMap B a b x ∈ (productPolynomial B a b).range := by
+    rw [productPolynomial_range B a b ha hb0 hb]
+    exact (Rows.equiv B a b x).property
+  obtain ⟨y, hy⟩ := hp
+  refine ⟨y, (Rows.invariantMap_injective B a b) ?_⟩
+  rw [map_polynomial, hy]
+
+end Foulkes.Canonical
+
+namespace Foulkes.Canonical
+universe u
+open Module MvPolynomial SymmetricTensor Polarization
+variable {V : Type u} [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
+variable {n : ℕ} (B : Basis (Fin n) ℂ V) (a b : ℕ)
+
+omit [FiniteDimensional ℂ V] in
+section
+
+def substitution (g : V →ₗ[ℂ] V) : P a n →ₐ[ℂ] P a n :=
+  aeval (fun ik : Fin a × Fin n =>
+    Rows.row ik.1 (SymPolynomial.linearPolynomial B (g (B ik.2))))
+
+lemma linearPolynomial_basis (k : Fin n) : SymPolynomial.linearPolynomial B (B k) = X k := by
+  classical
+  simp only [SymPolynomial.linearPolynomial_apply, B.equivFun_self]
+  simp
+
+lemma substitution_row_linear (g : V →ₗ[ℂ] V) (i : Fin a) (v : V) :
+    substitution B a g (Rows.row i (SymPolynomial.linearPolynomial B v)) =
+      Rows.row i (SymPolynomial.linearPolynomial B (g v)) := by
+  have hh : (substitution B a g).toLinearMap.comp
+      ((Rows.row i).toLinearMap.comp (SymPolynomial.linearPolynomial B)) =
+      (Rows.row i).toLinearMap.comp ((SymPolynomial.linearPolynomial B).comp g) := by
+    apply B.ext
+    intro k
+    simp only [LinearMap.comp_apply, AlgHom.toLinearMap_apply, linearPolynomial_basis]
+    simp [Rows.row, substitution]
+  exact DFunLike.congr_fun hh v
+
+lemma substitution_rowForm (g : V →ₗ[ℂ] V) (i : Fin a) (v : V) :
+    substitution B a g (rowForm i (B.equivFun v)) = rowForm i (B.equivFun (g v)) := by
+  simpa only [Rows.row_linear] using substitution_row_linear B a g i v
+
+lemma substitution_z (g : V →ₗ[ℂ] V) (v : V) :
+    substitution B a g (z (a := a) (B.equivFun v)) = z (a := a) (B.equivFun (g v)) := by
+  simp only [z, map_prod, substitution_rowForm]
+
+lemma row_polynomial_natural (g : V →ₗ[ℂ] V) (i : Fin a) (q : Sym b V) :
+    Rows.row i (SymPolynomial.polynomial B b (symMap b g q)) =
+      substitution B a g (Rows.row i (SymPolynomial.polynomial B b q)) := by
+  have hh : (Rows.row i).toLinearMap.comp ((SymPolynomial.polynomial B b).comp (symMap b g)) =
+      (substitution B a g).toLinearMap.comp
+        ((Rows.row i).toLinearMap.comp (SymPolynomial.polynomial B b)) := by
+    apply symMap_ext
+    intro v
+    simp only [LinearMap.comp_apply, AlgHom.toLinearMap_apply, symMap_power,
+      SymPolynomial.polynomial_power, map_pow, substitution_row_linear]
+  exact DFunLike.congr_fun hh q
+
+lemma tensorMap_natural (g : V →ₗ[ℂ] V) (x : T a (Sym b V)) :
+    Rows.tensorMap B a b (PiTensorProduct.map (fun _ : Fin a => symMap b g) x) =
+      substitution B a g (Rows.tensorMap B a b x) := by
+  have hh : (Rows.tensorMap B a b).comp (PiTensorProduct.map (fun _ : Fin a => symMap b g)) =
+      (substitution B a g).toLinearMap.comp (Rows.tensorMap B a b) := by
+    apply PiTensorProduct.ext
+    apply MultilinearMap.ext
+    intro v
+    simp only [LinearMap.compMultilinearMap_apply, LinearMap.comp_apply, PiTensorProduct.map_tprod,
+      Rows.tensorMap_tprod, AlgHom.toLinearMap_apply, map_prod, row_polynomial_natural]
+  exact DFunLike.congr_fun hh x
+
+lemma invariantMap_natural (g : V →ₗ[ℂ] V) (x : Sym a (Sym b V)) :
+    Rows.invariantMap B a b (symMap a (symMap b g) x) =
+      substitution B a g (Rows.invariantMap B a b x) :=
+  tensorMap_natural B a b g x
+
+end
+
+lemma firstMap_natural (g : V →ₗ[ℂ] V) (q : Sym a V) :
+    firstMap B a (symMap a g q) = substitution B a g (firstMap B a q) := by
+  have hh : (firstMap B a).comp (symMap a g) =
+      (substitution B a g).toLinearMap.comp (firstMap B a) := by
+    apply symMap_ext
+    intro v
+    simp only [LinearMap.comp_apply, AlgHom.toLinearMap_apply, symMap_power,
+      firstMap_power, substitution_z]
+  exact DFunLike.congr_fun hh q
+
+lemma productPolynomial_natural (g : V →ₗ[ℂ] V) (x : Sym b (Sym a V)) :
+    productPolynomial B a b (symMap b (symMap a g) x) =
+      substitution B a g (productPolynomial B a b x) := by
+  have hh : (productPolynomial B a b).comp (symMap b (symMap a g)) =
+      (substitution B a g).toLinearMap.comp (productPolynomial B a b) := by
+    apply symMap_ext
+    intro q
+    simp only [LinearMap.comp_apply, AlgHom.toLinearMap_apply, symMap_power,
+      productPolynomial_power, firstMap_natural, map_pow]
+  exact DFunLike.congr_fun hh x
+
+theorem natural (g : V →ₗ[ℂ] V) (x : Sym b (Sym a V)) :
+    map B a b (symMap b (symMap a g) x) = symMap a (symMap b g) (map B a b x) := by
+  apply Rows.invariantMap_injective B a b
+  rw [map_polynomial, invariantMap_natural, map_polynomial, productPolynomial_natural]
+
+end Foulkes.Canonical
+
+namespace Foulkes.PlethDual
+universe u
+open Module SymmetricTensor
+variable (a b : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
+
+lemma natural_congr (g : V →ₗ[ℂ] V) (d : Sym a (Sym b (Dual ℂ V))) (x : Sym a (Sym b V)) :
+    equiv a b V (symMap a (symMap b g.dualMap) d) x =
+      equiv a b V d (symMap a (symMap b g) x) := by
+  have hi : (dualEquiv b (V := V)).toLinearMap.comp (symMap b g.dualMap) =
+      (symMap b g).dualMap.comp (dualEquiv b (V := V)).toLinearMap := by
+    apply LinearMap.ext
+    intro q
+    apply LinearMap.ext
+    intro v
+    exact pairing_map b g q v
+  have hs := congrArg (fun (f : Sym b (Dual ℂ V) →ₗ[ℂ] Dual ℂ (Sym b V)) =>
+    symMap a f d) hi
+  simp only [symMap_comp, LinearMap.comp_apply] at hs
+  simp only [equiv_apply, hs]
+  exact pairing_map a (symMap b g) _ x
+
+end Foulkes.PlethDual
+
+namespace Foulkes.Injection
+universe u
+open Module
+variable (a b : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
+
+def map : Sym a (Sym b V) →ₗ[ℂ] Sym b (Sym a V) :=
+  (PlethDual.flipEquiv b a V).symm.toLinearMap.comp
+    ((Canonical.map (Module.finBasis ℂ (Dual ℂ V)) a b).dualMap.comp
+      (PlethDual.flipEquiv a b V).toLinearMap)
+
+lemma adjoint (x : Sym a (Sym b V)) (d : Sym b (Sym a (Dual ℂ V))) :
+    PlethDual.equiv b a V d (map a b V x) =
+      PlethDual.equiv a b V (Canonical.map (Module.finBasis ℂ (Dual ℂ V)) a b d) x := by
+  change PlethDual.flipEquiv b a V
+    ((PlethDual.flipEquiv b a V).symm
+      ((Canonical.map (Module.finBasis ℂ (Dual ℂ V)) a b).dualMap
+        (PlethDual.flipEquiv a b V x))) d = _
+  rw [LinearEquiv.apply_symm_apply]
+  rfl
+
+lemma injective (ha : 1 ≤ a) (hb0 : 1 ≤ b) (hb : a*(a-1) ≤ b) :
+    Function.Injective (map a b V) :=
+  (PlethDual.flipEquiv b a V).symm.injective.comp
+    ((LinearMap.dualMap_injective_of_surjective
+      (Canonical.surjective (Module.finBasis ℂ (Dual ℂ V)) a b ha hb0 hb)).comp
+      (PlethDual.flipEquiv a b V).injective)
+
+theorem natural (g : V →ₗ[ℂ] V) (x : Sym a (Sym b V)) :
+    map a b V (symMap a (symMap b g) x) = symMap b (symMap a g) (map a b V x) := by
+  apply (PlethDual.flipEquiv b a V).injective
+  apply LinearMap.ext
+  intro d
+  change PlethDual.equiv b a V d (map a b V (symMap a (symMap b g) x)) =
+    PlethDual.equiv b a V d (symMap b (symMap a g) (map a b V x))
+  calc
+    _ = PlethDual.equiv a b V (Canonical.map (Module.finBasis ℂ (Dual ℂ V)) a b d)
+        (symMap a (symMap b g) x) := adjoint a b V _ d
+    _ = PlethDual.equiv a b V
+        (symMap a (symMap b g.dualMap) (Canonical.map (Module.finBasis ℂ (Dual ℂ V)) a b d)) x :=
+      (PlethDual.natural a b V g _ x).symm
+    _ = PlethDual.equiv a b V
+        (Canonical.map (Module.finBasis ℂ (Dual ℂ V)) a b (symMap b (symMap a g.dualMap) d)) x := by
+      rw [Canonical.natural]
+    _ = PlethDual.equiv b a V (symMap b (symMap a g.dualMap) d) (map a b V x) :=
+      (adjoint a b V x _).symm
+    _ = _ := PlethDual.natural b a V g d (map a b V x)
+
+theorem quadratic_comparison (ha : 2 ≤ a) (hb : a*(a-1) ≤ b) :
+    ∃ i : Sym a (Sym b V) →ₗ[ℂ] Sym b (Sym a V), Function.Injective i ∧
+      ∀ (g : V ≃ₗ[ℂ] V) (x : Sym a (Sym b V)),
+        i (symMap a (symMap b g.toLinearMap) x) = symMap b (symMap a g.toLinearMap) (i x) := by
+  have hb0 : 1 ≤ b := le_trans (Nat.mul_pos (by omega) (by omega)) hb
+  exact ⟨map a b V, injective a b V (by omega) hb0 hb, fun g x => natural a b V g.toLinearMap x⟩
+
+theorem sixth_large (b : ℕ) (hb : 30 ≤ b)
+    (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] :
+    ∃ i : Sym 6 (Sym b V) →ₗ[ℂ] Sym b (Sym 6 V), Function.Injective i ∧
+      ∀ (g : V ≃ₗ[ℂ] V) (x : Sym 6 (Sym b V)),
+        i (symMap 6 (symMap b g.toLinearMap) x) = symMap b (symMap 6 g.toLinearMap) (i x) :=
+  quadratic_comparison 6 b V (by omega) (by omega)
+
+end Foulkes.Injection
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/RangeReduction.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/RangeReduction.lean
index 6814a543d01f73f6ff157f524f94ba07b2165661..08450db41dd5dddb12668fb2fc408ff853806ee1 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/RangeReduction.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/RangeReduction.lean
@@ -1,116 +1,116 @@
-import Mathlib
-import OAI.RepresentationTheory.FoulkesSixth.IndependentChart
-import OAI.RepresentationTheory.FoulkesSixth.BlockInjection
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes
-universe u
-
-open scoped TensorProduct
-
-def SixthPowerComparison : Prop :=
-  ∀ (b : ℕ), 6 ≤ b →
-  ∀ (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V],
-    ∃ (i : Sym 6 (Sym b V) →ₗ[ℂ] Sym b (Sym 6 V)),
-      Function.Injective i ∧
-        ∀ (g : V ≃ₗ[ℂ] V) (x : Sym 6 (Sym b V)),
-          i (symMap 6 (symMap b g.toLinearMap) x) =
-            symMap b (symMap 6 g.toLinearMap) (i x)
-
-def normalizedAverage (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] :
-    (⨂[ℂ] (_ : Fin n), V) →ₗ[ℂ] (⨂[ℂ] (_ : Fin n), V) :=
-  (n.factorial : ℂ)⁻¹ • ∑ σ : Equiv.Perm (Fin n),
-    (PiTensorProduct.reindex ℂ (fun _ : Fin n => V) σ).toLinearMap
-
-namespace PolynomialModel
-
-abbrev P (a n : ℕ) := MvPolynomial (Fin a × Fin n) ℂ
-
-def rowWeight (a n : ℕ) : Fin a × Fin n → (Fin a → ℕ) :=
-  fun ij => Pi.single ij.1 1
-
-def W (a n j : ℕ) : Submodule ℂ (P a n) :=
-  MvPolynomial.weightedHomogeneousSubmodule ℂ (rowWeight a n) (fun _ => j)
-
-def rowInvariants (a n : ℕ) : Subalgebra ℂ (P a n) where
-  carrier := {f | ∀ σ : Equiv.Perm (Fin a),
-    MvPolynomial.rename (fun ij : Fin a × Fin n => (σ ij.1, ij.2)) f = f}
-  zero_mem' := by intro σ; simp
-  add_mem' := by intro f g hf hg σ; simp only [map_add, hf σ, hg σ]
-  one_mem' := by intro σ; simp
-  mul_mem' := by intro f g hf hg σ; simp only [map_mul, hf σ, hg σ]
-  algebraMap_mem' := by intro c σ; simp
-
-def R (a n j : ℕ) : Submodule ℂ (P a n) :=
-  W a n j ⊓ (rowInvariants a n).toSubmodule
-
-def A (a n : ℕ) : Subalgebra ℂ (P a n) :=
-  Algebra.adjoin ℂ (R a n 1 : Set (P a n))
-
-def AComponent (a n b : ℕ) : Submodule ℂ (P a n) :=
-  W a n b ⊓ (A a n).toSubmodule
-
-def IndependentChartBound : Prop :=
-  ∀ (a n b : ℕ), 2 ≤ a → a * (a - 1)^2 ≤ b → R a n b = AComponent a n b
-
-end PolynomialModel
-theorem independent_chart : PolynomialModel.IndependentChartBound := by
-  intro a n b ha hb
-  exact Chart.independent_chart_bound a n b ha hb
-
-theorem sixth_equal (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] :
-    ∃ i : Sym 6 (Sym 6 V) →ₗ[ℂ] Sym 6 (Sym 6 V), Function.Injective i ∧
-      ∀ (g : V ≃ₗ[ℂ] V) (x : Sym 6 (Sym 6 V)),
-        i (symMap 6 (symMap 6 g.toLinearMap) x) = symMap 6 (symMap 6 g.toLinearMap) (i x) :=
-  ⟨LinearMap.id, fun _ _ h => h, fun _ _ => rfl⟩
-
-def FiniteSixthPowerComparison : Prop :=
-  ∀ (b : ℕ), 7 ≤ b → b ≤ 29 →
-  ∀ (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V],
-    ∃ (i : Sym 6 (Sym b V) →ₗ[ℂ] Sym b (Sym 6 V)),
-      Function.Injective i ∧
-        ∀ (g : V ≃ₗ[ℂ] V) (x : Sym 6 (Sym b V)),
-          i (symMap 6 (symMap b g.toLinearMap) x) =
-            symMap b (symMap 6 g.toLinearMap) (i x)
-
-theorem full_iff_finite : SixthPowerComparison.{u} ↔ FiniteSixthPowerComparison.{u} := by
-  constructor
-  · intro h b hb _ V _ _ _
-    exact h b (by omega) V
-  · intro h b hb V _ _ _
-    by_cases he : b = 6
-    · subst b
-      exact sixth_equal V
-    · by_cases hbig : 30 ≤ b
-      · exact Injection.sixth_large b hbig V
-      · exact h b (by omega) (by omega) V
-
-def BaseSixthPowerComparison : Prop :=
-  ∀ (b : ℕ), 7 ≤ b → b ≤ 24 →
-  ∀ (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V],
-    ∃ (i : Sym 6 (Sym b V) →ₗ[ℂ] Sym b (Sym 6 V)),
-      Function.Injective i ∧
-        ∀ (g : V ≃ₗ[ℂ] V) (x : Sym 6 (Sym b V)),
-          i (symMap 6 (symMap b g.toLinearMap) x) =
-            symMap b (symMap 6 g.toLinearMap) (i x)
-
-theorem full_iff_base : SixthPowerComparison.{u} ↔ BaseSixthPowerComparison.{u} := by
-  constructor
-  · intro h b hb _ V _ _ _
-    exact h b (by omega) V
-  · intro h b hb V _ _ _
-    by_cases he : b = 6
-    · subst b
-      exact sixth_equal V
-    · by_cases hbig : 25 ≤ b
-      · exact Injection.sixth_from_25 b hbig V
-      · exact h b (by omega) (by omega) V
-
-end Foulkes
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.FoulkesSixth.IndependentChart
+import OAI.RepresentationTheory.FoulkesSixth.BlockInjection
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes
+universe u
+
+open scoped TensorProduct
+
+def SixthPowerComparison : Prop :=
+  ∀ (b : ℕ), 6 ≤ b →
+  ∀ (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V],
+    ∃ (i : Sym 6 (Sym b V) →ₗ[ℂ] Sym b (Sym 6 V)),
+      Function.Injective i ∧
+        ∀ (g : V ≃ₗ[ℂ] V) (x : Sym 6 (Sym b V)),
+          i (symMap 6 (symMap b g.toLinearMap) x) =
+            symMap b (symMap 6 g.toLinearMap) (i x)
+
+def normalizedAverage (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] :
+    (⨂[ℂ] (_ : Fin n), V) →ₗ[ℂ] (⨂[ℂ] (_ : Fin n), V) :=
+  (n.factorial : ℂ)⁻¹ • ∑ σ : Equiv.Perm (Fin n),
+    (PiTensorProduct.reindex ℂ (fun _ : Fin n => V) σ).toLinearMap
+
+namespace PolynomialModel
+
+abbrev P (a n : ℕ) := MvPolynomial (Fin a × Fin n) ℂ
+
+def rowWeight (a n : ℕ) : Fin a × Fin n → (Fin a → ℕ) :=
+  fun ij => Pi.single ij.1 1
+
+def W (a n j : ℕ) : Submodule ℂ (P a n) :=
+  MvPolynomial.weightedHomogeneousSubmodule ℂ (rowWeight a n) (fun _ => j)
+
+def rowInvariants (a n : ℕ) : Subalgebra ℂ (P a n) where
+  carrier := {f | ∀ σ : Equiv.Perm (Fin a),
+    MvPolynomial.rename (fun ij : Fin a × Fin n => (σ ij.1, ij.2)) f = f}
+  zero_mem' := by intro σ; simp
+  add_mem' := by intro f g hf hg σ; simp only [map_add, hf σ, hg σ]
+  one_mem' := by intro σ; simp
+  mul_mem' := by intro f g hf hg σ; simp only [map_mul, hf σ, hg σ]
+  algebraMap_mem' := by intro c σ; simp
+
+def R (a n j : ℕ) : Submodule ℂ (P a n) :=
+  W a n j ⊓ (rowInvariants a n).toSubmodule
+
+def A (a n : ℕ) : Subalgebra ℂ (P a n) :=
+  Algebra.adjoin ℂ (R a n 1 : Set (P a n))
+
+def AComponent (a n b : ℕ) : Submodule ℂ (P a n) :=
+  W a n b ⊓ (A a n).toSubmodule
+
+def IndependentChartBound : Prop :=
+  ∀ (a n b : ℕ), 2 ≤ a → a * (a - 1)^2 ≤ b → R a n b = AComponent a n b
+
+end PolynomialModel
+theorem independent_chart : PolynomialModel.IndependentChartBound := by
+  intro a n b ha hb
+  exact Chart.independent_chart_bound a n b ha hb
+
+theorem sixth_equal (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] :
+    ∃ i : Sym 6 (Sym 6 V) →ₗ[ℂ] Sym 6 (Sym 6 V), Function.Injective i ∧
+      ∀ (g : V ≃ₗ[ℂ] V) (x : Sym 6 (Sym 6 V)),
+        i (symMap 6 (symMap 6 g.toLinearMap) x) = symMap 6 (symMap 6 g.toLinearMap) (i x) :=
+  ⟨LinearMap.id, fun _ _ h => h, fun _ _ => rfl⟩
+
+def FiniteSixthPowerComparison : Prop :=
+  ∀ (b : ℕ), 7 ≤ b → b ≤ 29 →
+  ∀ (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V],
+    ∃ (i : Sym 6 (Sym b V) →ₗ[ℂ] Sym b (Sym 6 V)),
+      Function.Injective i ∧
+        ∀ (g : V ≃ₗ[ℂ] V) (x : Sym 6 (Sym b V)),
+          i (symMap 6 (symMap b g.toLinearMap) x) =
+            symMap b (symMap 6 g.toLinearMap) (i x)
+
+theorem full_iff_finite : SixthPowerComparison.{u} ↔ FiniteSixthPowerComparison.{u} := by
+  constructor
+  · intro h b hb _ V _ _ _
+    exact h b (by omega) V
+  · intro h b hb V _ _ _
+    by_cases he : b = 6
+    · subst b
+      exact sixth_equal V
+    · by_cases hbig : 30 ≤ b
+      · exact Injection.sixth_large b hbig V
+      · exact h b (by omega) (by omega) V
+
+def BaseSixthPowerComparison : Prop :=
+  ∀ (b : ℕ), 7 ≤ b → b ≤ 24 →
+  ∀ (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V],
+    ∃ (i : Sym 6 (Sym b V) →ₗ[ℂ] Sym b (Sym 6 V)),
+      Function.Injective i ∧
+        ∀ (g : V ≃ₗ[ℂ] V) (x : Sym 6 (Sym b V)),
+          i (symMap 6 (symMap b g.toLinearMap) x) =
+            symMap b (symMap 6 g.toLinearMap) (i x)
+
+theorem full_iff_base : SixthPowerComparison.{u} ↔ BaseSixthPowerComparison.{u} := by
+  constructor
+  · intro h b hb _ V _ _ _
+    exact h b (by omega) V
+  · intro h b hb V _ _ _
+    by_cases he : b = 6
+    · subst b
+      exact sixth_equal V
+    · by_cases hbig : 25 ≤ b
+      · exact Injection.sixth_from_25 b hbig V
+      · exact h b (by omega) (by omega) V
+
+end Foulkes
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/Recurrence.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/Recurrence.lean
index 8e98dd5f6ca56ff6142d419e0d8437d42b14fce6..6b4e484b2b36b0ff004d3299dc8962483556d406 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/Recurrence.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/Recurrence.lean
@@ -1,101 +1,101 @@
-import Mathlib
-import OAI.RepresentationTheory.FoulkesSixth.NewtonUniqueness
-import OAI.RepresentationTheory.FoulkesSixth.FastLookupExact
-import OAI.RepresentationTheory.FoulkesSixth.CompositionEnumeration
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.Checker
-open Finset Foulkes.Enumeration Foulkes.FastLookup Foulkes.Lookup Foulkes.Strips
-
-def checkedSum {α : Type*} (f : α → Option ℤ) : List α → Option ℤ
-  | [] => some 0
-  | a::l => do
-      let x ← f a
-      let y ← checkedSum f l
-      some (x+y)
-
-lemma checkedSum_sound {α : Type*} (f : α → Option ℤ) (g : α → ℤ)
-    (l : List α) (h : ∀ a ∈ l, ∀ z, f a = some z → z = g a)
-    (z : ℤ) (hz : checkedSum f l = some z) : z = (l.map g).sum := by
-  induction l generalizing z with
-  | nil => simpa [checkedSum] using hz.symm
-  | cons a l ih =>
-    cases hx : f a with
-    | none => simp [checkedSum, hx] at hz
-    | some x =>
-      cases hy : checkedSum f l with
-      | none => simp [checkedSum, hx, hy] at hz
-      | some y =>
-        have hh := ih (fun b hb => h b (List.mem_cons_of_mem a hb)) y hy
-        have ha := h a (List.mem_cons_self ..) x hx
-        simpa [checkedSum, hx, hy, hh, ha] using hz.symm
-
-lemma sum_compositions (n d : ℕ) (f : (Fin n → ℕ) → ℤ) :
-    ((compositions n d).map f).sum = ∑ a : Foulkes.Complete.Degree (Fin n) d,
-      f (fun j => a.val j) := by
-  classical
-  rw [← List.sum_toFinset f (nodup_compositions n d)]
-  change (∑ v ∈ compositionSet n d, f v) = _
-  rw [← Finset.sum_attach (compositionSet n d) f]
-  exact (Equiv.sum_comp (degreeComposition n d) (fun v => f v.val)).symm
-
-def numerator {n : ℕ} (T : ℕ → (Fin n → ℕ) → ℤ) (b : ℕ) (mu : Fin n → ℕ) : Option ℤ :=
-  checkedSum (fun i => checkedSum (fun v => checkedLookup (T (b-i))
-    (fun j => (shifted mu j : ℤ) - (i : ℤ)*v j)) (compositions n 6)) (List.range' 1 b)
-
-lemma numerator_sound {n : ℕ} (T : ℕ → (Fin n → ℕ) → ℤ)
-    (b : ℕ) (mu : Fin n → ℕ) (z : ℤ) (hz : numerator T b mu = some z) :
-    z = ∑ i ∈ Finset.Icc 1 b, ∑ d : Foulkes.Complete.Degree (Fin n) 6,
-      signedLookup (T (b-i)) (fun j => (shifted mu j : ℤ) - (i : ℤ)*d.val j) := by
-  have hinner (i : ℕ) (w : ℤ)
-      (hw : checkedSum (fun v => checkedLookup (T (b-i))
-        (fun j => (shifted mu j : ℤ) - (i : ℤ)*v j)) (compositions n 6) = some w) :
-      w = ∑ d : Foulkes.Complete.Degree (Fin n) 6,
-        signedLookup (T (b-i)) (fun j => (shifted mu j : ℤ) - (i : ℤ)*d.val j) := by
-    calc
-      w = ((compositions n 6).map (fun v => signedLookup (T (b-i))
-        (fun j => (shifted mu j : ℤ) - (i : ℤ)*v j))).sum :=
-          checkedSum_sound _ _ _ (fun v hv w hw => checkedLookup_exact _ _ w hw) w hw
-      _ = _ := sum_compositions n 6 _
-  have hh := checkedSum_sound _ _ (List.range' 1 b) (fun i hi => hinner i) z hz
-  rw [← List.sum_toFinset _ (List.nodup_range' (s := 1) (n := b))] at hh
-  have hr : (List.range' 1 b).toFinset = Finset.Icc 1 b := by
-    ext i
-    simp only [List.mem_toFinset, List.mem_range'_1, Finset.mem_Icc]
-    omega
-  rw [hr] at hh
-  exact hh
-
-def checkEntry {n : ℕ} (T : ℕ → (Fin n → ℕ) → ℤ) (b : ℕ) (mu : Fin n → ℕ) : Bool :=
-  match numerator T b mu with
-  | none => false
-  | some z => decide ((b : ℤ)*T b mu = z)
-
-lemma checkEntry_sound {n : ℕ} (T : ℕ → (Fin n → ℕ) → ℤ)
-    (b : ℕ) (mu : Fin n → ℕ) (h : checkEntry T b mu = true) :
-    (b : ℤ)*T b mu = ∑ i ∈ Finset.Icc 1 b, ∑ d : Foulkes.Complete.Degree (Fin n) 6,
-      signedLookup (T (b-i)) (fun j => (shifted mu j : ℤ) - (i : ℤ)*d.val j) := by
-  unfold checkEntry at h
-  cases hn : numerator T b mu with
-  | none => simp [hn] at h
-  | some z =>
-    rw [hn, decide_eq_true_eq] at h
-    exact h.trans (numerator_sound T b mu z hn)
-
-theorem checked_table_unique (n B : ℕ) (T : ℕ → (Fin n → ℕ) → ℤ)
-    (hzero : ∀ mu, Antitone mu → (∑ j, mu j = 0) → T 0 mu = 1)
-    (hcheck : ∀ b, 1 ≤ b → b ≤ B → ∀ mu, Antitone mu → (∑ j, mu j = 6*b) →
-      checkEntry T b mu = true) :
-    ∀ b, b ≤ B → ∀ mu, Antitone mu → (∑ j, mu j = 6*b) → T b mu = Q n b mu := by
-  apply target_table_unique n B T hzero
-  intro b hb hB mu hm hs
-  exact checkEntry_sound T b mu (hcheck b hb hB mu hm hs)
-
-end Foulkes.Checker
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.FoulkesSixth.NewtonUniqueness
+import OAI.RepresentationTheory.FoulkesSixth.FastLookupExact
+import OAI.RepresentationTheory.FoulkesSixth.CompositionEnumeration
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.Checker
+open Finset Foulkes.Enumeration Foulkes.FastLookup Foulkes.Lookup Foulkes.Strips
+
+def checkedSum {α : Type*} (f : α → Option ℤ) : List α → Option ℤ
+  | [] => some 0
+  | a::l => do
+      let x ← f a
+      let y ← checkedSum f l
+      some (x+y)
+
+lemma checkedSum_sound {α : Type*} (f : α → Option ℤ) (g : α → ℤ)
+    (l : List α) (h : ∀ a ∈ l, ∀ z, f a = some z → z = g a)
+    (z : ℤ) (hz : checkedSum f l = some z) : z = (l.map g).sum := by
+  induction l generalizing z with
+  | nil => simpa [checkedSum] using hz.symm
+  | cons a l ih =>
+    cases hx : f a with
+    | none => simp [checkedSum, hx] at hz
+    | some x =>
+      cases hy : checkedSum f l with
+      | none => simp [checkedSum, hx, hy] at hz
+      | some y =>
+        have hh := ih (fun b hb => h b (List.mem_cons_of_mem a hb)) y hy
+        have ha := h a (List.mem_cons_self ..) x hx
+        simpa [checkedSum, hx, hy, hh, ha] using hz.symm
+
+lemma sum_compositions (n d : ℕ) (f : (Fin n → ℕ) → ℤ) :
+    ((compositions n d).map f).sum = ∑ a : Foulkes.Complete.Degree (Fin n) d,
+      f (fun j => a.val j) := by
+  classical
+  rw [← List.sum_toFinset f (nodup_compositions n d)]
+  change (∑ v ∈ compositionSet n d, f v) = _
+  rw [← Finset.sum_attach (compositionSet n d) f]
+  exact (Equiv.sum_comp (degreeComposition n d) (fun v => f v.val)).symm
+
+def numerator {n : ℕ} (T : ℕ → (Fin n → ℕ) → ℤ) (b : ℕ) (mu : Fin n → ℕ) : Option ℤ :=
+  checkedSum (fun i => checkedSum (fun v => checkedLookup (T (b-i))
+    (fun j => (shifted mu j : ℤ) - (i : ℤ)*v j)) (compositions n 6)) (List.range' 1 b)
+
+lemma numerator_sound {n : ℕ} (T : ℕ → (Fin n → ℕ) → ℤ)
+    (b : ℕ) (mu : Fin n → ℕ) (z : ℤ) (hz : numerator T b mu = some z) :
+    z = ∑ i ∈ Finset.Icc 1 b, ∑ d : Foulkes.Complete.Degree (Fin n) 6,
+      signedLookup (T (b-i)) (fun j => (shifted mu j : ℤ) - (i : ℤ)*d.val j) := by
+  have hinner (i : ℕ) (w : ℤ)
+      (hw : checkedSum (fun v => checkedLookup (T (b-i))
+        (fun j => (shifted mu j : ℤ) - (i : ℤ)*v j)) (compositions n 6) = some w) :
+      w = ∑ d : Foulkes.Complete.Degree (Fin n) 6,
+        signedLookup (T (b-i)) (fun j => (shifted mu j : ℤ) - (i : ℤ)*d.val j) := by
+    calc
+      w = ((compositions n 6).map (fun v => signedLookup (T (b-i))
+        (fun j => (shifted mu j : ℤ) - (i : ℤ)*v j))).sum :=
+          checkedSum_sound _ _ _ (fun v hv w hw => checkedLookup_exact _ _ w hw) w hw
+      _ = _ := sum_compositions n 6 _
+  have hh := checkedSum_sound _ _ (List.range' 1 b) (fun i hi => hinner i) z hz
+  rw [← List.sum_toFinset _ (List.nodup_range' (s := 1) (n := b))] at hh
+  have hr : (List.range' 1 b).toFinset = Finset.Icc 1 b := by
+    ext i
+    simp only [List.mem_toFinset, List.mem_range'_1, Finset.mem_Icc]
+    omega
+  rw [hr] at hh
+  exact hh
+
+def checkEntry {n : ℕ} (T : ℕ → (Fin n → ℕ) → ℤ) (b : ℕ) (mu : Fin n → ℕ) : Bool :=
+  match numerator T b mu with
+  | none => false
+  | some z => decide ((b : ℤ)*T b mu = z)
+
+lemma checkEntry_sound {n : ℕ} (T : ℕ → (Fin n → ℕ) → ℤ)
+    (b : ℕ) (mu : Fin n → ℕ) (h : checkEntry T b mu = true) :
+    (b : ℤ)*T b mu = ∑ i ∈ Finset.Icc 1 b, ∑ d : Foulkes.Complete.Degree (Fin n) 6,
+      signedLookup (T (b-i)) (fun j => (shifted mu j : ℤ) - (i : ℤ)*d.val j) := by
+  unfold checkEntry at h
+  cases hn : numerator T b mu with
+  | none => simp [hn] at h
+  | some z =>
+    rw [hn, decide_eq_true_eq] at h
+    exact h.trans (numerator_sound T b mu z hn)
+
+theorem checked_table_unique (n B : ℕ) (T : ℕ → (Fin n → ℕ) → ℤ)
+    (hzero : ∀ mu, Antitone mu → (∑ j, mu j = 0) → T 0 mu = 1)
+    (hcheck : ∀ b, 1 ≤ b → b ≤ B → ∀ mu, Antitone mu → (∑ j, mu j = 6*b) →
+      checkEntry T b mu = true) :
+    ∀ b, b ≤ B → ∀ mu, Antitone mu → (∑ j, mu j = 6*b) → T b mu = Q n b mu := by
+  apply target_table_unique n B T hzero
+  intro b hb hB mu hm hs
+  exact checkEntry_sound T b mu (hcheck b hb hB mu hm hs)
+
+end Foulkes.Checker
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/Rows.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/Rows.lean
index bfc79d2350b770cfc42d0df70f44de311cedbe15..1975bcc149038a3d56f4ce33e0e6b7c581bf6914 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/Rows.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/Rows.lean
@@ -1,165 +1,165 @@
-import Mathlib
-import OAI.RepresentationTheory.FoulkesSixth.InvariantPolarization
-import OAI.RepresentationTheory.FoulkesSixth.SymPolynomial
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.Rows
-universe u
-open Module MvPolynomial SymmetricTensor Polarization
-variable {V : Type u} [AddCommGroup V] [Module ℂ V]
-variable {n : ℕ} (B : Basis (Fin n) ℂ V) (a b : ℕ)
-
-def row {a n : ℕ} (i : Fin a) : SymPolynomial.Poly n →ₐ[ℂ] P a n :=
-  rename (fun k => (i,k))
-
-@[simp] lemma row_linear (i : Fin a) (v : V) :
-    row i (SymPolynomial.linearPolynomial B v) = rowForm i (B.equivFun v) := by
-  simp [row, SymPolynomial.linearPolynomial, rowForm]
-
-@[simp] lemma renameRow_row (σ : Equiv.Perm (Fin a)) (i : Fin a) (p : SymPolynomial.Poly n) :
-    renameRow σ (row i p) = row (σ i) p := by
-  simp [row, renameRow, rename_rename, Function.comp_def]
-
-def tensorMap : T a (Sym b V) →ₗ[ℂ] P a n :=
-  PiTensorProduct.lift ((MultilinearMap.mkPiAlgebra ℂ (Fin a) (P a n)).compLinearMap
-    (fun i => (row i).toLinearMap.comp (SymPolynomial.polynomial B b)))
-
-@[simp] lemma tensorMap_tprod (v : Fin a → Sym b V) :
-    tensorMap B a b (PiTensorProduct.tprod ℂ v) =
-      ∏ i, row i (SymPolynomial.polynomial B b (v i)) := by
-  simp [tensorMap]
-
-@[simp] lemma tensorMap_powers (v : Fin a → V) :
-    tensorMap B a b (PiTensorProduct.tprod ℂ (fun i => power b V (v i))) =
-      pureRows b (fun i => B.equivFun (v i)) := by
-  simp only [tensorMap_tprod, SymPolynomial.polynomial_power, map_pow, row_linear, pureRows]
-
-def evaluation (w : Fin n → ℂ) : Dual ℂ (Sym b V) :=
-  (aeval w).toLinearMap.comp (SymPolynomial.polynomial B b)
-
-@[simp] lemma evaluation_apply (w : Fin n → ℂ) (v : Sym b V) :
-    evaluation B b w v = eval w (SymPolynomial.polynomial B b v) := rfl
-
-variable [FiniteDimensional ℂ V]
-
-lemma evaluation_span : Submodule.span ℂ (Set.range (evaluation B b)) = ⊤ := by
-  apply Submodule.span_eq_top_of_ne_zero
-  intro v hv
-  by_contra! h
-  have hp : SymPolynomial.polynomial B b v = 0 := by
-    apply MvPolynomial.funext
-    intro w
-    simpa using h (evaluation B b w) ⟨w, rfl⟩
-  exact hv ((SymPolynomial.polynomial_injective B b) (hp.trans (map_zero _).symm))
-
-omit [FiniteDimensional ℂ V] in
-lemma eval_tensorMap (w : Fin a × Fin n → ℂ) (x : T a (Sym b V)) :
-    eval w (tensorMap B a b x) =
-      PiTensorProduct.dualDistrib
-        (PiTensorProduct.tprod ℂ (fun i : Fin a => evaluation B b (fun k => w (i,k)))) x := by
-  have h : (aeval w).toLinearMap.comp (tensorMap B a b) =
-      PiTensorProduct.dualDistrib
-        (PiTensorProduct.tprod ℂ (fun i : Fin a => evaluation B b (fun k => w (i,k)))) := by
-    ext v
-    simp [row, MvPolynomial.eval_rename, Function.comp_def]
-  exact DFunLike.congr_fun h x
-
-lemma tensorMap_injective : Function.Injective (tensorMap B a b) := by
-  refine (LinearMap.ker_eq_bot (R := ℂ) (R₂ := ℂ) (M := T a (Sym b V)) (M₂ := P a n)
-    (f := tensorMap B a b)).mp ?_
-  refine (LinearMap.ker_eq_bot' (R := ℂ) (R₂ := ℂ) (M := T a (Sym b V)) (M₂ := P a n)
-    (f := tensorMap B a b)).mpr ?_
-  intro x hx
-  have hzero : (LinearMap.applyₗ x).comp
-      (PiTensorProduct.dualDistrib (R := ℂ) (M := fun _ : Fin a => Sym b V)) = 0 := by
-    apply PiTensorProduct.ext_of_span_eq_top
-      (g := fun {_ : Fin a} w => evaluation B b w) (fun _ => evaluation_span B b)
-    intro w
-    change PiTensorProduct.dualDistrib (PiTensorProduct.tprod ℂ
-      (fun i => evaluation B b (w i))) x = 0
-    rw [← eval_tensorMap B a b (fun ik => w ik.1 ik.2), hx, map_zero]
-  let : Module.Free ℂ (T a (Sym b V)) := Module.Free.of_basis
-    (Basis.piTensorProduct (fun _ : Fin a => Module.Free.chooseBasis ℂ (Sym b V)))
-  refine (Module.forall_dual_apply_eq_zero_iff ℂ x).mp ?_
-  intro l
-  obtain ⟨d, hd⟩ := (PiTensorProduct.dualDistribEquiv (R := ℂ)
-      (M := fun _ : Fin a => Sym b V)).surjective l
-  have hh := DFunLike.congr_fun hzero d
-  change (PiTensorProduct.dualDistribEquiv d) x = 0 at hh
-  rwa [hd] at hh
-
-omit [FiniteDimensional ℂ V] in
-lemma tensorMap_range : (tensorMap B a b).range = W a n b := by
-  have hspan : Submodule.span ℂ (Set.range (fun v : Fin a → V =>
-      PiTensorProduct.tprod ℂ (fun i => power b V (v i)))) = ⊤ :=
-    PiTensorProduct.submodule_span_eq_top (g := fun {_ : Fin a} v => power b V v)
-      (fun _ => span_power b V)
-  rw [← Submodule.map_top, ← hspan, LinearMap.map_span, W_eq_span_pureRows]
-  congr 1
-  ext p
-  constructor
-  · rintro ⟨_, ⟨v, rfl⟩, rfl⟩
-    exact ⟨fun i => B.equivFun (v i), (tensorMap_powers B a b v).symm⟩
-  · rintro ⟨v, rfl⟩
-    refine ⟨PiTensorProduct.tprod ℂ (fun i => power b V (B.equivFun.symm (v i))),
-      ⟨fun i => B.equivFun.symm (v i), rfl⟩, ?_⟩
-    simp only [tensorMap_powers, LinearEquiv.apply_symm_apply]
-
-omit [FiniteDimensional ℂ V] in
-lemma tensorMap_perm (σ : Equiv.Perm (Fin a)) (x : T a (Sym b V)) :
-    tensorMap B a b (perm a (Sym b V) σ x) = renameRow σ (tensorMap B a b x) := by
-  have h : (tensorMap B a b).comp (perm a (Sym b V) σ).toLinearMap =
-      (renameRow σ).toLinearMap.comp (tensorMap B a b) := by
-    apply PiTensorProduct.ext
-    apply MultilinearMap.ext
-    intro v
-    simp only [LinearMap.compMultilinearMap_apply, LinearMap.comp_apply, LinearEquiv.coe_coe,
-      perm, PiTensorProduct.reindex_tprod, tensorMap_tprod, AlgHom.toLinearMap_apply,
-      map_prod, renameRow_row]
-    apply Fintype.prod_equiv σ.symm
-    intro i
-    simp
-  exact DFunLike.congr_fun h x
-
-def invariantMap : Sym a (Sym b V) →ₗ[ℂ] P a n :=
-  (tensorMap B a b).comp (symmetricTensors a (Sym b V)).subtype
-
-lemma invariantMap_injective : Function.Injective (invariantMap B a b) :=
-  (tensorMap_injective B a b).comp Subtype.val_injective
-
-lemma invariantMap_range : (invariantMap B a b).range = R a n b := by
-  apply le_antisymm
-  · rintro p ⟨x, rfl⟩
-    constructor
-    · rw [← tensorMap_range B a b]
-      exact ⟨x, rfl⟩
-    · intro σ
-      change renameRow σ (tensorMap B a b x) = tensorMap B a b x
-      rw [← tensorMap_perm]
-      exact congrArg (tensorMap B a b) (x.property σ)
-  · intro p hp
-    have hW : p ∈ (tensorMap B a b).range := by rw [tensorMap_range]; exact hp.1
-    obtain ⟨x, hx⟩ := hW
-    have hs : x ∈ Sym a (Sym b V) := by
-      intro σ
-      apply tensorMap_injective B a b
-      change tensorMap B a b (perm a (Sym b V) σ x) = tensorMap B a b x
-      rw [tensorMap_perm, hx]
-      exact hp.2 σ
-    exact ⟨⟨x, hs⟩, hx⟩
-
-def equiv : Sym a (Sym b V) ≃ₗ[ℂ] R a n b :=
-  (LinearEquiv.ofInjective (invariantMap B a b) (invariantMap_injective B a b)).trans
-    (LinearEquiv.ofEq _ _ (invariantMap_range B a b))
-
-@[simp] lemma equiv_coe (x : Sym a (Sym b V)) :
-    (equiv B a b x : P a n) = invariantMap B a b x := rfl
-
-end Foulkes.Rows
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.FoulkesSixth.InvariantPolarization
+import OAI.RepresentationTheory.FoulkesSixth.SymPolynomial
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.Rows
+universe u
+open Module MvPolynomial SymmetricTensor Polarization
+variable {V : Type u} [AddCommGroup V] [Module ℂ V]
+variable {n : ℕ} (B : Basis (Fin n) ℂ V) (a b : ℕ)
+
+def row {a n : ℕ} (i : Fin a) : SymPolynomial.Poly n →ₐ[ℂ] P a n :=
+  rename (fun k => (i,k))
+
+@[simp] lemma row_linear (i : Fin a) (v : V) :
+    row i (SymPolynomial.linearPolynomial B v) = rowForm i (B.equivFun v) := by
+  simp [row, SymPolynomial.linearPolynomial, rowForm]
+
+@[simp] lemma renameRow_row (σ : Equiv.Perm (Fin a)) (i : Fin a) (p : SymPolynomial.Poly n) :
+    renameRow σ (row i p) = row (σ i) p := by
+  simp [row, renameRow, rename_rename, Function.comp_def]
+
+def tensorMap : T a (Sym b V) →ₗ[ℂ] P a n :=
+  PiTensorProduct.lift ((MultilinearMap.mkPiAlgebra ℂ (Fin a) (P a n)).compLinearMap
+    (fun i => (row i).toLinearMap.comp (SymPolynomial.polynomial B b)))
+
+@[simp] lemma tensorMap_tprod (v : Fin a → Sym b V) :
+    tensorMap B a b (PiTensorProduct.tprod ℂ v) =
+      ∏ i, row i (SymPolynomial.polynomial B b (v i)) := by
+  simp [tensorMap]
+
+@[simp] lemma tensorMap_powers (v : Fin a → V) :
+    tensorMap B a b (PiTensorProduct.tprod ℂ (fun i => power b V (v i))) =
+      pureRows b (fun i => B.equivFun (v i)) := by
+  simp only [tensorMap_tprod, SymPolynomial.polynomial_power, map_pow, row_linear, pureRows]
+
+def evaluation (w : Fin n → ℂ) : Dual ℂ (Sym b V) :=
+  (aeval w).toLinearMap.comp (SymPolynomial.polynomial B b)
+
+@[simp] lemma evaluation_apply (w : Fin n → ℂ) (v : Sym b V) :
+    evaluation B b w v = eval w (SymPolynomial.polynomial B b v) := rfl
+
+variable [FiniteDimensional ℂ V]
+
+lemma evaluation_span : Submodule.span ℂ (Set.range (evaluation B b)) = ⊤ := by
+  apply Submodule.span_eq_top_of_ne_zero
+  intro v hv
+  by_contra! h
+  have hp : SymPolynomial.polynomial B b v = 0 := by
+    apply MvPolynomial.funext
+    intro w
+    simpa using h (evaluation B b w) ⟨w, rfl⟩
+  exact hv ((SymPolynomial.polynomial_injective B b) (hp.trans (map_zero _).symm))
+
+omit [FiniteDimensional ℂ V] in
+lemma eval_tensorMap (w : Fin a × Fin n → ℂ) (x : T a (Sym b V)) :
+    eval w (tensorMap B a b x) =
+      PiTensorProduct.dualDistrib
+        (PiTensorProduct.tprod ℂ (fun i : Fin a => evaluation B b (fun k => w (i,k)))) x := by
+  have h : (aeval w).toLinearMap.comp (tensorMap B a b) =
+      PiTensorProduct.dualDistrib
+        (PiTensorProduct.tprod ℂ (fun i : Fin a => evaluation B b (fun k => w (i,k)))) := by
+    ext v
+    simp [row, MvPolynomial.eval_rename, Function.comp_def]
+  exact DFunLike.congr_fun h x
+
+lemma tensorMap_injective : Function.Injective (tensorMap B a b) := by
+  refine (LinearMap.ker_eq_bot (R := ℂ) (R₂ := ℂ) (M := T a (Sym b V)) (M₂ := P a n)
+    (f := tensorMap B a b)).mp ?_
+  refine (LinearMap.ker_eq_bot' (R := ℂ) (R₂ := ℂ) (M := T a (Sym b V)) (M₂ := P a n)
+    (f := tensorMap B a b)).mpr ?_
+  intro x hx
+  have hzero : (LinearMap.applyₗ x).comp
+      (PiTensorProduct.dualDistrib (R := ℂ) (M := fun _ : Fin a => Sym b V)) = 0 := by
+    apply PiTensorProduct.ext_of_span_eq_top
+      (g := fun {_ : Fin a} w => evaluation B b w) (fun _ => evaluation_span B b)
+    intro w
+    change PiTensorProduct.dualDistrib (PiTensorProduct.tprod ℂ
+      (fun i => evaluation B b (w i))) x = 0
+    rw [← eval_tensorMap B a b (fun ik => w ik.1 ik.2), hx, map_zero]
+  let : Module.Free ℂ (T a (Sym b V)) := Module.Free.of_basis
+    (Basis.piTensorProduct (fun _ : Fin a => Module.Free.chooseBasis ℂ (Sym b V)))
+  refine (Module.forall_dual_apply_eq_zero_iff ℂ x).mp ?_
+  intro l
+  obtain ⟨d, hd⟩ := (PiTensorProduct.dualDistribEquiv (R := ℂ)
+      (M := fun _ : Fin a => Sym b V)).surjective l
+  have hh := DFunLike.congr_fun hzero d
+  change (PiTensorProduct.dualDistribEquiv d) x = 0 at hh
+  rwa [hd] at hh
+
+omit [FiniteDimensional ℂ V] in
+lemma tensorMap_range : (tensorMap B a b).range = W a n b := by
+  have hspan : Submodule.span ℂ (Set.range (fun v : Fin a → V =>
+      PiTensorProduct.tprod ℂ (fun i => power b V (v i)))) = ⊤ :=
+    PiTensorProduct.submodule_span_eq_top (g := fun {_ : Fin a} v => power b V v)
+      (fun _ => span_power b V)
+  rw [← Submodule.map_top, ← hspan, LinearMap.map_span, W_eq_span_pureRows]
+  congr 1
+  ext p
+  constructor
+  · rintro ⟨_, ⟨v, rfl⟩, rfl⟩
+    exact ⟨fun i => B.equivFun (v i), (tensorMap_powers B a b v).symm⟩
+  · rintro ⟨v, rfl⟩
+    refine ⟨PiTensorProduct.tprod ℂ (fun i => power b V (B.equivFun.symm (v i))),
+      ⟨fun i => B.equivFun.symm (v i), rfl⟩, ?_⟩
+    simp only [tensorMap_powers, LinearEquiv.apply_symm_apply]
+
+omit [FiniteDimensional ℂ V] in
+lemma tensorMap_perm (σ : Equiv.Perm (Fin a)) (x : T a (Sym b V)) :
+    tensorMap B a b (perm a (Sym b V) σ x) = renameRow σ (tensorMap B a b x) := by
+  have h : (tensorMap B a b).comp (perm a (Sym b V) σ).toLinearMap =
+      (renameRow σ).toLinearMap.comp (tensorMap B a b) := by
+    apply PiTensorProduct.ext
+    apply MultilinearMap.ext
+    intro v
+    simp only [LinearMap.compMultilinearMap_apply, LinearMap.comp_apply, LinearEquiv.coe_coe,
+      perm, PiTensorProduct.reindex_tprod, tensorMap_tprod, AlgHom.toLinearMap_apply,
+      map_prod, renameRow_row]
+    apply Fintype.prod_equiv σ.symm
+    intro i
+    simp
+  exact DFunLike.congr_fun h x
+
+def invariantMap : Sym a (Sym b V) →ₗ[ℂ] P a n :=
+  (tensorMap B a b).comp (symmetricTensors a (Sym b V)).subtype
+
+lemma invariantMap_injective : Function.Injective (invariantMap B a b) :=
+  (tensorMap_injective B a b).comp Subtype.val_injective
+
+lemma invariantMap_range : (invariantMap B a b).range = R a n b := by
+  apply le_antisymm
+  · rintro p ⟨x, rfl⟩
+    constructor
+    · rw [← tensorMap_range B a b]
+      exact ⟨x, rfl⟩
+    · intro σ
+      change renameRow σ (tensorMap B a b x) = tensorMap B a b x
+      rw [← tensorMap_perm]
+      exact congrArg (tensorMap B a b) (x.property σ)
+  · intro p hp
+    have hW : p ∈ (tensorMap B a b).range := by rw [tensorMap_range]; exact hp.1
+    obtain ⟨x, hx⟩ := hW
+    have hs : x ∈ Sym a (Sym b V) := by
+      intro σ
+      apply tensorMap_injective B a b
+      change tensorMap B a b (perm a (Sym b V) σ x) = tensorMap B a b x
+      rw [tensorMap_perm, hx]
+      exact hp.2 σ
+    exact ⟨⟨x, hs⟩, hx⟩
+
+def equiv : Sym a (Sym b V) ≃ₗ[ℂ] R a n b :=
+  (LinearEquiv.ofInjective (invariantMap B a b) (invariantMap_injective B a b)).trans
+    (LinearEquiv.ofEq _ _ (invariantMap_range B a b))
+
+@[simp] lemma equiv_coe (x : Sym a (Sym b V)) :
+    (equiv B a b x : P a n) = invariantMap B a b x := rfl
+
+end Foulkes.Rows
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/SemisimpleBridge.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/SemisimpleBridge.lean
index 777ebe156fcb3b64f7b179c995c395c76b0df029..1fced9968f494669e43f1168f9389d0fe4836243 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/SemisimpleBridge.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/SemisimpleBridge.lean
@@ -1,232 +1,232 @@
-import Mathlib
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.SemisimpleBridge
-open Module
-open scoped Classical
-
-section Algebra
-variable {K R A B C : Type*} [Field K] [Ring R] [Algebra K R]
-  [AddCommGroup A] [Module K A] [Module R A] [IsScalarTower K R A]
-  [AddCommGroup B] [Module K B] [Module R B] [IsScalarTower K R B]
-  [AddCommGroup C] [Module K C] [Module R C] [IsScalarTower K R C]
-
-def homCongrRight (e : B ≃ₗ[R] C) : (A →ₗ[R] B) ≃ₗ[K] (A →ₗ[R] C) where
-  toFun f := e.toLinearMap.comp f
-  invFun f := e.symm.toLinearMap.comp f
-  left_inv f := by ext x; simp
-  right_inv f := by ext x; simp
-  map_add' f g := by ext x; simp
-  map_smul' k f := by
-    ext x
-    exact e.toLinearMap.map_smul_of_tower k (f x)
-
-lemma finite_hom [FiniteDimensional K A] [FiniteDimensional K B] :
-    FiniteDimensional K (A →ₗ[R] B) :=
-  Module.Finite.of_injective (LinearMap.restrictScalarsₗ K R A B K)
-    (LinearMap.restrictScalars_injective K)
-
-lemma simple_hom_finrank [IsAlgClosed K] [FiniteDimensional K A]
-    [FiniteDimensional K B] [IsSimpleModule R A] [IsSimpleModule R B] :
-    finrank K (A →ₗ[R] B) = if Nonempty (A ≃ₗ[R] B) then 1 else 0 := by
-  classical
-  by_cases h : Nonempty (A ≃ₗ[R] B)
-  · rw [ite_eq_left h]
-    have he := homCongrRight (K := K) (A := A) h.some.symm
-    rw [he.finrank_eq]
-    have hb := IsSimpleModule.algebraMap_end_bijective_of_isAlgClosed (A := R) (V := A) K
-    have hh := (LinearEquiv.ofBijective (Algebra.linearMap K (Module.End R A)) hb).finrank_eq
-    simpa using hh.symm
-  · rw [ite_eq_right h]
-    have hz (f : A →ₗ[R] B) : f = 0 := by
-      rcases LinearMap.bijective_or_eq_zero f with hbij | hzero
-      · exact (h ⟨LinearEquiv.ofBijective f hbij⟩).elim
-      · exact hzero
-    let : Subsingleton (A →ₗ[R] B) := ⟨fun f g => (hz f).trans (hz g).symm⟩
-    exact finrank_zero_of_subsingleton
-
-end Algebra
-end Foulkes.SemisimpleBridge
-
-namespace Foulkes.SemisimpleBridge
-open scoped Classical
-open Module Finset
-
-def fibreEmbedding {ι κ C : Type*} [Fintype ι] [Fintype κ]
-    (f : ι → C) (g : κ → C)
-    (hc : ∀ c, Fintype.card {i // f i = c} ≤ Fintype.card {j // g j = c}) : ι ↪ κ := by
-  let e (c : C) : {i // f i = c} ↪ {j // g j = c} :=
-    (Fintype.equivFin _).toEmbedding.trans
-      ((Fin.castLEEmb (hc c)).trans (Fintype.equivFin _).symm.toEmbedding)
-  exact (Equiv.sigmaFiberEquiv f).symm.toEmbedding.trans
-    ((Function.Embedding.sigmaMap (Function.Embedding.refl C) e).trans
-      (Equiv.sigmaFiberEquiv g).toEmbedding)
-
-lemma fibreEmbedding_label {ι κ C : Type*} [Fintype ι] [Fintype κ]
-    (f : ι → C) (g : κ → C)
-    (hc : ∀ c, Fintype.card {i // f i = c} ≤ Fintype.card {j // g j = c}) (i : ι) :
-    g (fibreEmbedding f g hc i) = f i := by
-  exact (show {j // g j = f i} from
-    ((Fintype.equivFin {i' // f i' = f i}).toEmbedding.trans
-      ((Fin.castLEEmb (hc (f i))).trans
-        (Fintype.equivFin {j // g j = f i}).symm.toEmbedding)) ⟨i,rfl⟩).property
-
-section PiHom
-variable {K R A : Type*} [Field K] [Ring R] [Algebra K R]
-  [IsAlgClosed K] [AddCommGroup A] [Module K A] [Module R A]
-  [IsScalarTower K R A] [FiniteDimensional K A] [IsSimpleModule R A]
-  {ι : Type*} [Fintype ι] (P : ι → Type*) [∀ i, AddCommGroup (P i)]
-  [∀ i, Module K (P i)] [∀ i, Module R (P i)] [∀ i, IsScalarTower K R (P i)]
-  [∀ i, FiniteDimensional K (P i)] [∀ i, IsSimpleModule R (P i)]
-
-lemma hom_pi_finrank :
-    finrank K (A →ₗ[R] Π i, P i) = Fintype.card {i // Nonempty (A ≃ₗ[R] P i)} := by
-  let (i : ι) : FiniteDimensional K (A →ₗ[R] P i) := finite_hom
-  rw [← (LinearEquiv.linearMapPi (R := R) (M₂ := A) (φ := P) K).finrank_eq]
-  rw [Module.finrank_pi_fintype]
-  simp only [simple_hom_finrank]
-  simp [Fintype.card_subtype]
-
-end PiHom
-
-section PiEmbedding
-variable {R : Type*} [Ring R] {ι κ : Type*} [Fintype ι] [Fintype κ]
-  (A : ι → Type*) (B : κ → Type*) [∀ i, AddCommGroup (A i)]
-  [∀ j, AddCommGroup (B j)] [∀ i, Module R (A i)] [∀ j, Module R (B j)]
-
-def piMapOfMatching (f : ι ↪ κ) (e : ∀ i, A i ≃ₗ[R] B (f i)) :
-    (Π i, A i) →ₗ[R] (Π j, B j) :=
-  ∑ i, (LinearMap.single _ _ (f i)).comp
-    ((e i).toLinearMap.comp (LinearMap.proj i))
-
-omit [Fintype κ] in
-lemma piMapOfMatching_apply (f : ι ↪ κ) (e : ∀ i, A i ≃ₗ[R] B (f i))
-    (x : Π i, A i) (i : ι) : piMapOfMatching A B f e x (f i) = e i (x i) := by
-  classical
-  simp only [piMapOfMatching, LinearMap.sum_apply, Finset.sum_apply, LinearMap.comp_apply,
-    LinearEquiv.coe_coe, LinearMap.proj_apply]
-  rw [Finset.sum_eq_single i]
-  · simp
-  · intro j hj hji
-    have hf : f j ≠ f i := fun hh => hji (f.injective hh)
-    simp [LinearMap.single_apply, Pi.single_eq_of_ne (Ne.symm hf)]
-  · simp
-
-omit [Fintype κ] in
-lemma piMapOfMatching_injective (f : ι ↪ κ) (e : ∀ i, A i ≃ₗ[R] B (f i)) :
-    Function.Injective (piMapOfMatching A B f e) := by
-  intro x y h
-  funext i
-  apply (e i).injective
-  simpa only [piMapOfMatching_apply] using congrFun h (f i)
-
-end PiEmbedding
-end Foulkes.SemisimpleBridge
-
-namespace Foulkes.SemisimpleBridge
-universe v
-open scoped Classical
-
-private def classSet {X : Type*} (s : Setoid X) (x : X) : Set X := {y | s.r x y}
-
-private lemma classSet_eq_iff {X : Type*} (s : Setoid X) (x y : X) :
-    classSet s x = classSet s y ↔ s.r x y := by
-  constructor
-  · intro h
-    have hy : y ∈ classSet s y := s.refl y
-    rw [← h] at hy
-    exact hy
-  · intro h
-    ext z
-    exact ⟨fun hz => s.trans (s.symm h) hz, fun hz => s.trans h hz⟩
-
-theorem matching_of_equivalence_counts {ι κ X : Type*} [Fintype ι] [Fintype κ]
-    (s : Setoid X) (a : ι → X) (b : κ → X)
-    (hc : ∀ i, Fintype.card {i' // s.r (a i) (a i')} ≤
-      Fintype.card {j // s.r (a i) (b j)}) :
-    ∃ f : ι ↪ κ, ∀ i, s.r (a i) (b (f i)) := by
-  classical
-  let fa := classSet s ∘ a
-  let fb := classSet s ∘ b
-  have hcard : ∀ c, Fintype.card {i // fa i = c} ≤ Fintype.card {j // fb j = c} := by
-    intro c
-    by_cases hc' : ∃ i, fa i = c
-    · obtain ⟨i,rfl⟩ := hc'
-      have equivA : {i' // fa i' = fa i} ≃ {i' // s.r (a i) (a i')} :=
-        Equiv.subtypeEquivRight fun i' =>
-          (classSet_eq_iff s (a i') (a i)).trans ⟨fun h => s.symm h, fun h => s.symm h⟩
-      have equivB : {j // fb j = fa i} ≃ {j // s.r (a i) (b j)} :=
-        Equiv.subtypeEquivRight fun j =>
-          (classSet_eq_iff s (b j) (a i)).trans ⟨fun h => s.symm h, fun h => s.symm h⟩
-      rw [Fintype.card_congr equivA, Fintype.card_congr equivB]
-      exact hc i
-    · have : IsEmpty {i // fa i = c} := ⟨fun i => hc' ⟨i.val,i.property⟩⟩
-      simp
-  refine ⟨fibreEmbedding fa fb hcard, fun i => ?_⟩
-  exact s.symm ((classSet_eq_iff s _ _).mp (fibreEmbedding_label fa fb hcard i))
-
-private def moduleIsoSetoid (R : Type*) [Ring R] : Setoid (ModuleCat R) where
-  r M N := Nonempty (M ≃ₗ[R] N)
-  iseqv := ⟨fun M => ⟨LinearEquiv.refl R M⟩,
-    fun h => ⟨h.some.symm⟩, fun h h' => ⟨h.some.trans h'.some⟩⟩
-
-theorem pi_embedding_of_iso_counts {R : Type*} [Ring R]
-    {ι κ : Type*} [Fintype ι] [Fintype κ]
-    (A : ι → Type v) (B : κ → Type v) [∀ i, AddCommGroup (A i)]
-    [∀ j, AddCommGroup (B j)] [∀ i, Module R (A i)] [∀ j, Module R (B j)]
-    (hc : ∀ i, Fintype.card {i' // Nonempty (A i ≃ₗ[R] A i')} ≤
-      Fintype.card {j // Nonempty (A i ≃ₗ[R] B j)}) :
-    ∃ f : (Π i, A i) →ₗ[R] (Π j, B j), Function.Injective f := by
-  obtain ⟨f,hf⟩ := matching_of_equivalence_counts (moduleIsoSetoid R)
-    (fun i => ModuleCat.of R (A i)) (fun j => ModuleCat.of R (B j)) hc
-  let e (i : ι) : A i ≃ₗ[R] B (f i) := (hf i).some
-  exact ⟨piMapOfMatching A B f e, piMapOfMatching_injective A B f e⟩
-
-end Foulkes.SemisimpleBridge
-
-namespace Foulkes.SemisimpleBridge
-universe u
-open Module
-
-theorem embedding_of_simple_hom_le {K R : Type*} [Field K] [IsAlgClosed K]
-    [Ring R] [Algebra K R] (A B : Type u)
-    [AddCommGroup A] [Module K A] [Module R A] [IsScalarTower K R A]
-    [AddCommGroup B] [Module K B] [Module R B] [IsScalarTower K R B]
-    [FiniteDimensional K A] [FiniteDimensional K B]
-    [IsSemisimpleModule R A] [IsSemisimpleModule R B]
-    (h : ∀ (T : Type u) [AddCommGroup T] [Module K T] [Module R T]
-      [IsScalarTower K R T] [FiniteDimensional K T] [IsSimpleModule R T],
-      finrank K (T →ₗ[R] A) ≤ finrank K (T →ₗ[R] B)) :
-    ∃ f : A →ₗ[R] B, Function.Injective f := by
-  classical
-  let : Module.Finite R A := Module.Finite.of_restrictScalars_finite K R A
-  let : Module.Finite R B := Module.Finite.of_restrictScalars_finite K R B
-  obtain ⟨n,S,eA,hS⟩ := IsSemisimpleModule.exists_linearEquiv_fin_dfinsupp R A
-  obtain ⟨m,T,eB,hT⟩ := IsSemisimpleModule.exists_linearEquiv_fin_dfinsupp R B
-  let (i : Fin n) : IsSimpleModule R (S i) := hS i
-  let (j : Fin m) : IsSimpleModule R (T j) := hT j
-  let (i : Fin n) : FiniteDimensional K (S i) :=
-    Module.Finite.of_injective ((S i).subtype.restrictScalars K) (S i).subtype_injective
-  let (j : Fin m) : FiniteDimensional K (T j) :=
-    Module.Finite.of_injective ((T j).subtype.restrictScalars K) (T j).subtype_injective
-  let eA' : A ≃ₗ[R] Π i, S i := eA.trans DFinsupp.linearEquivFunOnFintype
-  let eB' : B ≃ₗ[R] Π j, T j := eB.trans DFinsupp.linearEquivFunOnFintype
-  have hc (i : Fin n) : Fintype.card {i' // Nonempty (S i ≃ₗ[R] S i')} ≤
-      Fintype.card {j // Nonempty (S i ≃ₗ[R] T j)} := by
-    rw [← hom_pi_finrank (K := K) (R := R) (A := S i) (fun i' => S i'),
-      ← hom_pi_finrank (K := K) (R := R) (A := S i) (fun j => T j)]
-    rw [← (homCongrRight (K := K) (A := S i) eA').finrank_eq,
-      ← (homCongrRight (K := K) (A := S i) eB').finrank_eq]
-    exact h (S i)
-  obtain ⟨f,hf⟩ := pi_embedding_of_iso_counts (R := R) (fun i => S i) (fun j => T j) hc
-  refine ⟨eB'.symm.toLinearMap.comp (f.comp eA'.toLinearMap), ?_⟩
-  exact eB'.symm.injective.comp (hf.comp eA'.injective)
-
-end Foulkes.SemisimpleBridge
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.SemisimpleBridge
+open Module
+open scoped Classical
+
+section Algebra
+variable {K R A B C : Type*} [Field K] [Ring R] [Algebra K R]
+  [AddCommGroup A] [Module K A] [Module R A] [IsScalarTower K R A]
+  [AddCommGroup B] [Module K B] [Module R B] [IsScalarTower K R B]
+  [AddCommGroup C] [Module K C] [Module R C] [IsScalarTower K R C]
+
+def homCongrRight (e : B ≃ₗ[R] C) : (A →ₗ[R] B) ≃ₗ[K] (A →ₗ[R] C) where
+  toFun f := e.toLinearMap.comp f
+  invFun f := e.symm.toLinearMap.comp f
+  left_inv f := by ext x; simp
+  right_inv f := by ext x; simp
+  map_add' f g := by ext x; simp
+  map_smul' k f := by
+    ext x
+    exact e.toLinearMap.map_smul_of_tower k (f x)
+
+lemma finite_hom [FiniteDimensional K A] [FiniteDimensional K B] :
+    FiniteDimensional K (A →ₗ[R] B) :=
+  Module.Finite.of_injective (LinearMap.restrictScalarsₗ K R A B K)
+    (LinearMap.restrictScalars_injective K)
+
+lemma simple_hom_finrank [IsAlgClosed K] [FiniteDimensional K A]
+    [FiniteDimensional K B] [IsSimpleModule R A] [IsSimpleModule R B] :
+    finrank K (A →ₗ[R] B) = if Nonempty (A ≃ₗ[R] B) then 1 else 0 := by
+  classical
+  by_cases h : Nonempty (A ≃ₗ[R] B)
+  · rw [ite_eq_left h]
+    have he := homCongrRight (K := K) (A := A) h.some.symm
+    rw [he.finrank_eq]
+    have hb := IsSimpleModule.algebraMap_end_bijective_of_isAlgClosed (A := R) (V := A) K
+    have hh := (LinearEquiv.ofBijective (Algebra.linearMap K (Module.End R A)) hb).finrank_eq
+    simpa using hh.symm
+  · rw [ite_eq_right h]
+    have hz (f : A →ₗ[R] B) : f = 0 := by
+      rcases LinearMap.bijective_or_eq_zero f with hbij | hzero
+      · exact (h ⟨LinearEquiv.ofBijective f hbij⟩).elim
+      · exact hzero
+    let : Subsingleton (A →ₗ[R] B) := ⟨fun f g => (hz f).trans (hz g).symm⟩
+    exact finrank_zero_of_subsingleton
+
+end Algebra
+end Foulkes.SemisimpleBridge
+
+namespace Foulkes.SemisimpleBridge
+open scoped Classical
+open Module Finset
+
+def fibreEmbedding {ι κ C : Type*} [Fintype ι] [Fintype κ]
+    (f : ι → C) (g : κ → C)
+    (hc : ∀ c, Fintype.card {i // f i = c} ≤ Fintype.card {j // g j = c}) : ι ↪ κ := by
+  let e (c : C) : {i // f i = c} ↪ {j // g j = c} :=
+    (Fintype.equivFin _).toEmbedding.trans
+      ((Fin.castLEEmb (hc c)).trans (Fintype.equivFin _).symm.toEmbedding)
+  exact (Equiv.sigmaFiberEquiv f).symm.toEmbedding.trans
+    ((Function.Embedding.sigmaMap (Function.Embedding.refl C) e).trans
+      (Equiv.sigmaFiberEquiv g).toEmbedding)
+
+lemma fibreEmbedding_label {ι κ C : Type*} [Fintype ι] [Fintype κ]
+    (f : ι → C) (g : κ → C)
+    (hc : ∀ c, Fintype.card {i // f i = c} ≤ Fintype.card {j // g j = c}) (i : ι) :
+    g (fibreEmbedding f g hc i) = f i := by
+  exact (show {j // g j = f i} from
+    ((Fintype.equivFin {i' // f i' = f i}).toEmbedding.trans
+      ((Fin.castLEEmb (hc (f i))).trans
+        (Fintype.equivFin {j // g j = f i}).symm.toEmbedding)) ⟨i,rfl⟩).property
+
+section PiHom
+variable {K R A : Type*} [Field K] [Ring R] [Algebra K R]
+  [IsAlgClosed K] [AddCommGroup A] [Module K A] [Module R A]
+  [IsScalarTower K R A] [FiniteDimensional K A] [IsSimpleModule R A]
+  {ι : Type*} [Fintype ι] (P : ι → Type*) [∀ i, AddCommGroup (P i)]
+  [∀ i, Module K (P i)] [∀ i, Module R (P i)] [∀ i, IsScalarTower K R (P i)]
+  [∀ i, FiniteDimensional K (P i)] [∀ i, IsSimpleModule R (P i)]
+
+lemma hom_pi_finrank :
+    finrank K (A →ₗ[R] Π i, P i) = Fintype.card {i // Nonempty (A ≃ₗ[R] P i)} := by
+  let (i : ι) : FiniteDimensional K (A →ₗ[R] P i) := finite_hom
+  rw [← (LinearEquiv.linearMapPi (R := R) (M₂ := A) (φ := P) K).finrank_eq]
+  rw [Module.finrank_pi_fintype]
+  simp only [simple_hom_finrank]
+  simp [Fintype.card_subtype]
+
+end PiHom
+
+section PiEmbedding
+variable {R : Type*} [Ring R] {ι κ : Type*} [Fintype ι] [Fintype κ]
+  (A : ι → Type*) (B : κ → Type*) [∀ i, AddCommGroup (A i)]
+  [∀ j, AddCommGroup (B j)] [∀ i, Module R (A i)] [∀ j, Module R (B j)]
+
+def piMapOfMatching (f : ι ↪ κ) (e : ∀ i, A i ≃ₗ[R] B (f i)) :
+    (Π i, A i) →ₗ[R] (Π j, B j) :=
+  ∑ i, (LinearMap.single _ _ (f i)).comp
+    ((e i).toLinearMap.comp (LinearMap.proj i))
+
+omit [Fintype κ] in
+lemma piMapOfMatching_apply (f : ι ↪ κ) (e : ∀ i, A i ≃ₗ[R] B (f i))
+    (x : Π i, A i) (i : ι) : piMapOfMatching A B f e x (f i) = e i (x i) := by
+  classical
+  simp only [piMapOfMatching, LinearMap.sum_apply, Finset.sum_apply, LinearMap.comp_apply,
+    LinearEquiv.coe_coe, LinearMap.proj_apply]
+  rw [Finset.sum_eq_single i]
+  · simp
+  · intro j hj hji
+    have hf : f j ≠ f i := fun hh => hji (f.injective hh)
+    simp [LinearMap.single_apply, Pi.single_eq_of_ne (Ne.symm hf)]
+  · simp
+
+omit [Fintype κ] in
+lemma piMapOfMatching_injective (f : ι ↪ κ) (e : ∀ i, A i ≃ₗ[R] B (f i)) :
+    Function.Injective (piMapOfMatching A B f e) := by
+  intro x y h
+  funext i
+  apply (e i).injective
+  simpa only [piMapOfMatching_apply] using congrFun h (f i)
+
+end PiEmbedding
+end Foulkes.SemisimpleBridge
+
+namespace Foulkes.SemisimpleBridge
+universe v
+open scoped Classical
+
+private def classSet {X : Type*} (s : Setoid X) (x : X) : Set X := {y | s.r x y}
+
+private lemma classSet_eq_iff {X : Type*} (s : Setoid X) (x y : X) :
+    classSet s x = classSet s y ↔ s.r x y := by
+  constructor
+  · intro h
+    have hy : y ∈ classSet s y := s.refl y
+    rw [← h] at hy
+    exact hy
+  · intro h
+    ext z
+    exact ⟨fun hz => s.trans (s.symm h) hz, fun hz => s.trans h hz⟩
+
+theorem matching_of_equivalence_counts {ι κ X : Type*} [Fintype ι] [Fintype κ]
+    (s : Setoid X) (a : ι → X) (b : κ → X)
+    (hc : ∀ i, Fintype.card {i' // s.r (a i) (a i')} ≤
+      Fintype.card {j // s.r (a i) (b j)}) :
+    ∃ f : ι ↪ κ, ∀ i, s.r (a i) (b (f i)) := by
+  classical
+  let fa := classSet s ∘ a
+  let fb := classSet s ∘ b
+  have hcard : ∀ c, Fintype.card {i // fa i = c} ≤ Fintype.card {j // fb j = c} := by
+    intro c
+    by_cases hc' : ∃ i, fa i = c
+    · obtain ⟨i,rfl⟩ := hc'
+      have equivA : {i' // fa i' = fa i} ≃ {i' // s.r (a i) (a i')} :=
+        Equiv.subtypeEquivRight fun i' =>
+          (classSet_eq_iff s (a i') (a i)).trans ⟨fun h => s.symm h, fun h => s.symm h⟩
+      have equivB : {j // fb j = fa i} ≃ {j // s.r (a i) (b j)} :=
+        Equiv.subtypeEquivRight fun j =>
+          (classSet_eq_iff s (b j) (a i)).trans ⟨fun h => s.symm h, fun h => s.symm h⟩
+      rw [Fintype.card_congr equivA, Fintype.card_congr equivB]
+      exact hc i
+    · have : IsEmpty {i // fa i = c} := ⟨fun i => hc' ⟨i.val,i.property⟩⟩
+      simp
+  refine ⟨fibreEmbedding fa fb hcard, fun i => ?_⟩
+  exact s.symm ((classSet_eq_iff s _ _).mp (fibreEmbedding_label fa fb hcard i))
+
+private def moduleIsoSetoid (R : Type*) [Ring R] : Setoid (ModuleCat R) where
+  r M N := Nonempty (M ≃ₗ[R] N)
+  iseqv := ⟨fun M => ⟨LinearEquiv.refl R M⟩,
+    fun h => ⟨h.some.symm⟩, fun h h' => ⟨h.some.trans h'.some⟩⟩
+
+theorem pi_embedding_of_iso_counts {R : Type*} [Ring R]
+    {ι κ : Type*} [Fintype ι] [Fintype κ]
+    (A : ι → Type v) (B : κ → Type v) [∀ i, AddCommGroup (A i)]
+    [∀ j, AddCommGroup (B j)] [∀ i, Module R (A i)] [∀ j, Module R (B j)]
+    (hc : ∀ i, Fintype.card {i' // Nonempty (A i ≃ₗ[R] A i')} ≤
+      Fintype.card {j // Nonempty (A i ≃ₗ[R] B j)}) :
+    ∃ f : (Π i, A i) →ₗ[R] (Π j, B j), Function.Injective f := by
+  obtain ⟨f,hf⟩ := matching_of_equivalence_counts (moduleIsoSetoid R)
+    (fun i => ModuleCat.of R (A i)) (fun j => ModuleCat.of R (B j)) hc
+  let e (i : ι) : A i ≃ₗ[R] B (f i) := (hf i).some
+  exact ⟨piMapOfMatching A B f e, piMapOfMatching_injective A B f e⟩
+
+end Foulkes.SemisimpleBridge
+
+namespace Foulkes.SemisimpleBridge
+universe u
+open Module
+
+theorem embedding_of_simple_hom_le {K R : Type*} [Field K] [IsAlgClosed K]
+    [Ring R] [Algebra K R] (A B : Type u)
+    [AddCommGroup A] [Module K A] [Module R A] [IsScalarTower K R A]
+    [AddCommGroup B] [Module K B] [Module R B] [IsScalarTower K R B]
+    [FiniteDimensional K A] [FiniteDimensional K B]
+    [IsSemisimpleModule R A] [IsSemisimpleModule R B]
+    (h : ∀ (T : Type u) [AddCommGroup T] [Module K T] [Module R T]
+      [IsScalarTower K R T] [FiniteDimensional K T] [IsSimpleModule R T],
+      finrank K (T →ₗ[R] A) ≤ finrank K (T →ₗ[R] B)) :
+    ∃ f : A →ₗ[R] B, Function.Injective f := by
+  classical
+  let : Module.Finite R A := Module.Finite.of_restrictScalars_finite K R A
+  let : Module.Finite R B := Module.Finite.of_restrictScalars_finite K R B
+  obtain ⟨n,S,eA,hS⟩ := IsSemisimpleModule.exists_linearEquiv_fin_dfinsupp R A
+  obtain ⟨m,T,eB,hT⟩ := IsSemisimpleModule.exists_linearEquiv_fin_dfinsupp R B
+  let (i : Fin n) : IsSimpleModule R (S i) := hS i
+  let (j : Fin m) : IsSimpleModule R (T j) := hT j
+  let (i : Fin n) : FiniteDimensional K (S i) :=
+    Module.Finite.of_injective ((S i).subtype.restrictScalars K) (S i).subtype_injective
+  let (j : Fin m) : FiniteDimensional K (T j) :=
+    Module.Finite.of_injective ((T j).subtype.restrictScalars K) (T j).subtype_injective
+  let eA' : A ≃ₗ[R] Π i, S i := eA.trans DFinsupp.linearEquivFunOnFintype
+  let eB' : B ≃ₗ[R] Π j, T j := eB.trans DFinsupp.linearEquivFunOnFintype
+  have hc (i : Fin n) : Fintype.card {i' // Nonempty (S i ≃ₗ[R] S i')} ≤
+      Fintype.card {j // Nonempty (S i ≃ₗ[R] T j)} := by
+    rw [← hom_pi_finrank (K := K) (R := R) (A := S i) (fun i' => S i'),
+      ← hom_pi_finrank (K := K) (R := R) (A := S i) (fun j => T j)]
+    rw [← (homCongrRight (K := K) (A := S i) eA').finrank_eq,
+      ← (homCongrRight (K := K) (A := S i) eB').finrank_eq]
+    exact h (S i)
+  obtain ⟨f,hf⟩ := pi_embedding_of_iso_counts (R := R) (fun i => S i) (fun j => T j) hc
+  refine ⟨eB'.symm.toLinearMap.comp (f.comp eA'.toLinearMap), ?_⟩
+  exact eB'.symm.injective.comp (hf.comp eA'.injective)
+
+end Foulkes.SemisimpleBridge
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/SignedStrips.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/SignedStrips.lean
index e8e0a696465ad769c84a6d00ae8310e49651a2d8..098bab7ac92c45a4db90a07f250f10d0e10a30a7 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/SignedStrips.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/SignedStrips.lean
@@ -1,67 +1,67 @@
-import Mathlib
-import OAI.RepresentationTheory.FoulkesSixth.StripInterlace
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.Strips
-open MvPolynomial Finset
-
-def staircase (n : ℕ) (j : Fin n) : ℕ := n-1-j.val
-
-def shifted {n : ℕ} (γ : Fin n → ℕ) (j : Fin n) : ℕ := γ j + staircase n j
-
-lemma staircase_antitone (n : ℕ) : Antitone (staircase n) := by
-  intro j k hjk
-  simp only [staircase]
-  exact Nat.sub_le_sub_left hjk (n-1)
-
-lemma shifted_antitone {n : ℕ} {γ : Fin n → ℕ} (hγ : Antitone γ) : Antitone (shifted γ) := by
-  intro j k hjk
-  exact add_le_add (hγ hjk) (staircase_antitone n hjk)
-
-lemma shifted_sum {n i b : ℕ} (γ μ : Fin n → ℕ)
-    (hs : ∑ j, γ j = (∑ j, μ j) + i*b) :
-    ∑ j, shifted γ j = (∑ j, shifted μ j) + i*b := by
-  simp only [shifted, Finset.sum_add_distrib]
-  omega
-
-def stripCoeff {n : ℕ} (i b : ℕ) (γ μ : Fin n → ℕ) : ℤ :=
-  (completePowers n i b * alternant (shifted μ)).coeff (expVector (shifted γ))
-
-theorem signed_strips {n i b : ℕ} (hi : 1 ≤ i) (hb : 1 ≤ b)
-    (γ μ : Fin n → ℕ) (hγ : Antitone γ) (hμ : Antitone μ)
-    (hsize : (∑ j, γ j) - (∑ j, μ j) = i*b) :
-    (stripCoeff i b γ μ = 0 ∨ stripCoeff i b γ μ = 1 ∨ stripCoeff i b γ μ = -1) ∧
-    (stripCoeff i b γ μ ≠ 0 → ∀ j : Fin n, μ j ≤ γ j ∧
-      ∀ hji : j.val+i < n, γ ⟨j.val+i,hji⟩ ≤ μ j) := by
-  have hip : 0 < i := by omega
-  have hbp : 0 < b := by omega
-  have hprod : 0 < i*b := Nat.mul_pos hip hbp
-  have hs : ∑ j, γ j = (∑ j, μ j) + i*b := by omega
-  have hbound := signed_strip_abs_le_one hip (shifted γ) (shifted μ) (shifted_sum γ μ hs)
-  change Int.natAbs (stripCoeff i b γ μ) ≤ 1 at hbound
-  constructor
-  · omega
-  · intro hnon j
-    have heq := coefficient_eq_assignment_det hip (shifted γ) (shifted μ) (shifted_sum γ μ hs)
-    change stripCoeff i b γ μ = _ at heq
-    have hd : (incidence (assignments i (shifted γ) (shifted μ))).det ≠ 0 := by
-      rwa [← heq]
-    constructor
-    · have hc := containment_of_det_ne_zero (shifted γ) (shifted μ)
-        (shifted_antitone hγ) (shifted_antitone hμ) hd j
-      simpa only [shifted, Nat.add_le_add_iff_right] using hc
-    · intro hji
-      have hc := shifted_containment_of_det_ne_zero hip (shifted γ) (shifted μ)
-        (shifted_antitone hγ) (shifted_antitone hμ) hd j hji
-      simp only [shifted, staircase] at hc
-      have hjn := j.isLt
-      omega
-
-end Foulkes.Strips
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.FoulkesSixth.StripInterlace
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.Strips
+open MvPolynomial Finset
+
+def staircase (n : ℕ) (j : Fin n) : ℕ := n-1-j.val
+
+def shifted {n : ℕ} (γ : Fin n → ℕ) (j : Fin n) : ℕ := γ j + staircase n j
+
+lemma staircase_antitone (n : ℕ) : Antitone (staircase n) := by
+  intro j k hjk
+  simp only [staircase]
+  exact Nat.sub_le_sub_left hjk (n-1)
+
+lemma shifted_antitone {n : ℕ} {γ : Fin n → ℕ} (hγ : Antitone γ) : Antitone (shifted γ) := by
+  intro j k hjk
+  exact add_le_add (hγ hjk) (staircase_antitone n hjk)
+
+lemma shifted_sum {n i b : ℕ} (γ μ : Fin n → ℕ)
+    (hs : ∑ j, γ j = (∑ j, μ j) + i*b) :
+    ∑ j, shifted γ j = (∑ j, shifted μ j) + i*b := by
+  simp only [shifted, Finset.sum_add_distrib]
+  omega
+
+def stripCoeff {n : ℕ} (i b : ℕ) (γ μ : Fin n → ℕ) : ℤ :=
+  (completePowers n i b * alternant (shifted μ)).coeff (expVector (shifted γ))
+
+theorem signed_strips {n i b : ℕ} (hi : 1 ≤ i) (hb : 1 ≤ b)
+    (γ μ : Fin n → ℕ) (hγ : Antitone γ) (hμ : Antitone μ)
+    (hsize : (∑ j, γ j) - (∑ j, μ j) = i*b) :
+    (stripCoeff i b γ μ = 0 ∨ stripCoeff i b γ μ = 1 ∨ stripCoeff i b γ μ = -1) ∧
+    (stripCoeff i b γ μ ≠ 0 → ∀ j : Fin n, μ j ≤ γ j ∧
+      ∀ hji : j.val+i < n, γ ⟨j.val+i,hji⟩ ≤ μ j) := by
+  have hip : 0 < i := by omega
+  have hbp : 0 < b := by omega
+  have hprod : 0 < i*b := Nat.mul_pos hip hbp
+  have hs : ∑ j, γ j = (∑ j, μ j) + i*b := by omega
+  have hbound := signed_strip_abs_le_one hip (shifted γ) (shifted μ) (shifted_sum γ μ hs)
+  change Int.natAbs (stripCoeff i b γ μ) ≤ 1 at hbound
+  constructor
+  · omega
+  · intro hnon j
+    have heq := coefficient_eq_assignment_det hip (shifted γ) (shifted μ) (shifted_sum γ μ hs)
+    change stripCoeff i b γ μ = _ at heq
+    have hd : (incidence (assignments i (shifted γ) (shifted μ))).det ≠ 0 := by
+      rwa [← heq]
+    constructor
+    · have hc := containment_of_det_ne_zero (shifted γ) (shifted μ)
+        (shifted_antitone hγ) (shifted_antitone hμ) hd j
+      simpa only [shifted, Nat.add_le_add_iff_right] using hc
+    · intro hji
+      have hc := shifted_containment_of_det_ne_zero hip (shifted γ) (shifted μ)
+        (shifted_antitone hγ) (shifted_antitone hμ) hd j hji
+      simp only [shifted, staircase] at hc
+      have hjn := j.isLt
+      omega
+
+end Foulkes.Strips
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/SixCycles.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/SixCycles.lean
index aeee4a63317b44509b2fc73c872c069ab3c0fa2d..b314e0b44caa7864ead42bc9d2e8b6fe8fa9cf78 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/SixCycles.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/SixCycles.lean
@@ -1,52 +1,52 @@
-import Mathlib
-import OAI.RepresentationTheory.FoulkesSixth.SourceNewton
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.Complete
-open MvPolynomial Finset
-
-def cycleSix {R : Type*} [CommRing R] (p : ℕ → R) : R :=
-  p 1 ^ 6 + 15*p 1^4*p 2 + 45*p 1^2*p 2^2 + 15*p 2^3 +
-  40*p 1^3*p 3 + 120*p 1*p 2*p 3 + 40*p 3^2 +
-  90*p 1^2*p 4 + 90*p 2*p 4 + 144*p 1*p 5 + 120*p 6
-
-theorem cycleSix_of_newton {R : Type*} [CommRing R] (h p : ℕ → R)
-    (h0 : h 0 = 1)
-    (hn : ∀ j : ℕ, (j : R)*h j = ∑ i ∈ Finset.Icc 1 j, p i*h (j-i)) :
-    720*h 6 = cycleSix p := by
-  have n1 := hn 1
-  have n2 := hn 2
-  have n3 := hn 3
-  have n4 := hn 4
-  have n5 := hn 5
-  have n6 := hn 6
-  norm_num [Finset.sum_Icc_succ_top, h0] at n1 n2 n3 n4 n5 n6
-  have e2 : 2*h 2 = p 1^2+p 2 := by
-    linear_combination n2 + p 1*n1
-  have e3 : 6*h 3 = p 1^3+3*p 1*p 2+2*p 3 := by
-    linear_combination 2*n3 + p 1*e2 + 2*p 2*n1
-  have e4 : 24*h 4 = p 1^4+6*p 1^2*p 2+3*p 2^2+8*p 1*p 3+6*p 4 := by
-    linear_combination 6*n4 + p 1*e3 + 3*p 2*e2 + 6*p 3*n1
-  have e5 : 120*h 5 = p 1^5+10*p 1^3*p 2+15*p 1*p 2^2+
-      20*p 1^2*p 3+20*p 2*p 3+30*p 1*p 4+24*p 5 := by
-    linear_combination 24*n5 + p 1*e4 + 4*p 2*e3 + 12*p 3*e2 + 24*p 4*n1
-  unfold cycleSix
-  linear_combination 120*n6 + p 1*e5 + 5*p 2*e4 + 20*p 3*e3 + 60*p 4*e2 + 120*p 5*n1
-
-theorem pleth_cycleSix (n b : ℕ) :
-    (720 : ℤ) • pleth n 6 b =
-      cycleSix (fun i => Foulkes.Strips.completePowers n i b) := by
-  norm_num only [zsmul_eq_mul, Int.cast_ofNat]
-  apply cycleSix_of_newton (h := fun j => pleth n j b)
-  · exact pleth_zero n b
-  · intro j
-    simpa only [zsmul_eq_mul, Int.cast_natCast] using pleth_newton_powers n j b
-
-end Foulkes.Complete
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.FoulkesSixth.SourceNewton
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.Complete
+open MvPolynomial Finset
+
+def cycleSix {R : Type*} [CommRing R] (p : ℕ → R) : R :=
+  p 1 ^ 6 + 15*p 1^4*p 2 + 45*p 1^2*p 2^2 + 15*p 2^3 +
+  40*p 1^3*p 3 + 120*p 1*p 2*p 3 + 40*p 3^2 +
+  90*p 1^2*p 4 + 90*p 2*p 4 + 144*p 1*p 5 + 120*p 6
+
+theorem cycleSix_of_newton {R : Type*} [CommRing R] (h p : ℕ → R)
+    (h0 : h 0 = 1)
+    (hn : ∀ j : ℕ, (j : R)*h j = ∑ i ∈ Finset.Icc 1 j, p i*h (j-i)) :
+    720*h 6 = cycleSix p := by
+  have n1 := hn 1
+  have n2 := hn 2
+  have n3 := hn 3
+  have n4 := hn 4
+  have n5 := hn 5
+  have n6 := hn 6
+  norm_num [Finset.sum_Icc_succ_top, h0] at n1 n2 n3 n4 n5 n6
+  have e2 : 2*h 2 = p 1^2+p 2 := by
+    linear_combination n2 + p 1*n1
+  have e3 : 6*h 3 = p 1^3+3*p 1*p 2+2*p 3 := by
+    linear_combination 2*n3 + p 1*e2 + 2*p 2*n1
+  have e4 : 24*h 4 = p 1^4+6*p 1^2*p 2+3*p 2^2+8*p 1*p 3+6*p 4 := by
+    linear_combination 6*n4 + p 1*e3 + 3*p 2*e2 + 6*p 3*n1
+  have e5 : 120*h 5 = p 1^5+10*p 1^3*p 2+15*p 1*p 2^2+
+      20*p 1^2*p 3+20*p 2*p 3+30*p 1*p 4+24*p 5 := by
+    linear_combination 24*n5 + p 1*e4 + 4*p 2*e3 + 12*p 3*e2 + 24*p 4*n1
+  unfold cycleSix
+  linear_combination 120*n6 + p 1*e5 + 5*p 2*e4 + 20*p 3*e3 + 60*p 4*e2 + 120*p 5*n1
+
+theorem pleth_cycleSix (n b : ℕ) :
+    (720 : ℤ) • pleth n 6 b =
+      cycleSix (fun i => Foulkes.Strips.completePowers n i b) := by
+  norm_num only [zsmul_eq_mul, Int.cast_ofNat]
+  apply cycleSix_of_newton (h := fun j => pleth n j b)
+  · exact pleth_zero n b
+  · intro j
+    simpa only [zsmul_eq_mul, Int.cast_natCast] using pleth_newton_powers n j b
+
+end Foulkes.Complete
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/SourceNewton.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/SourceNewton.lean
index 2c70d24717646b094a3ab11081ff1ae76fc7616c..93aa65b5fa2bcb36baf4928aff7f052c9bad7c98 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/SourceNewton.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/SourceNewton.lean
@@ -1,82 +1,82 @@
-import Mathlib
-import OAI.RepresentationTheory.FoulkesSixth.HomogeneousPlethysm
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.Complete
-open MvPolynomial Finset Foulkes.Strips
-
-def degreeCompositionEquiv (n b : ℕ) : Degree (Fin n) b ≃ Composition n b where
-  toFun d := ⟨fun j => ⟨d.val j, by
-      have hj := Finsupp.le_degree j d.val
-      omega⟩, by simpa using (Finsupp.degree_eq_sum d.val).symm.trans d.property⟩
-  invFun v := ⟨expVector (fun j => (v.val j).val), by
-    rw [Lookup.degree_expVector]
-    exact v.property⟩
-  left_inv d := by apply Subtype.ext; ext j; rfl
-  right_inv v := by apply Subtype.ext; funext j; apply Fin.ext; rfl
-
-lemma powerAlphabet_eq_completePowers (n i b : ℕ) :
-    (∑ d : Degree (Fin n) b,
-      monomial (i • d.val) (1 : ℤ)) = completePowers n i b := by
-  classical
-  unfold completePowers
-  apply Fintype.sum_equiv (degreeCompositionEquiv n b)
-  intro d
-  have he : i • d.val = Finsupp.equivFunOnFinite.symm
-      (fun j => i*((degreeCompositionEquiv n b d).val j).val) := by
-    ext j
-    change i * d.val j = i * d.val j
-    rfl
-  rw [he]
-
-lemma pleth_newton_powers (n j b : ℕ) :
-    (j : ℤ) • pleth n j b =
-      ∑ i ∈ Finset.Icc 1 j, completePowers n i b * pleth n (j-i) b := by
-  simpa only [powerAlphabet_eq_completePowers] using pleth_newton n j b
-
-end Foulkes.Complete
-
-namespace Foulkes.Lookup
-open MvPolynomial Finset Foulkes.Strips
-
-def stripCoeff {n : ℕ} (b i : ℕ) (gamma mu : Fin n → ℕ) : ℤ :=
-  (completePowers n i b * alternant (shifted mu)).coeff (expVector (shifted gamma))
-
-theorem source_newton (n j b : ℕ) (gamma : Fin n → ℕ) :
-    (j : ℤ) * schurCoeff (Complete.pleth n j b) gamma =
-      ∑ i ∈ Finset.Icc 1 j,
-        ∑ mu : Partition n ((j-i)*b),
-          stripCoeff b i gamma (partVec mu) *
-            schurCoeff (Complete.pleth n (j-i) b) (partVec mu) := by
-  classical
-  have hh := congrArg (fun p : MvPolynomial (Fin n) ℤ =>
-    (p * alternant (staircase n)).coeff (expVector (shifted gamma)))
-      (Complete.pleth_newton_powers n j b)
-  change (((j : ℤ) • Complete.pleth n j b) * alternant (staircase n)).coeff
-      (expVector (shifted gamma)) = _ at hh
-  rw [smul_mul_assoc, coeff_smul] at hh
-  simp only [zsmul_eq_mul, Finset.sum_mul, coeff_sum] at hh
-  change (j : ℤ) * schurCoeff (Complete.pleth n j b) gamma = _ at hh
-  rw [hh]
-  apply Finset.sum_congr rfl
-  intro i hi
-  rw [mul_assoc]
-  rw [homogeneous_schur_expansion (d := (j-i)*b) _ (Complete.rename_pleth n (j-i) b)
-    (by simpa [Nat.mul_comm] using Complete.homogeneous_pleth n (j-i) b)]
-  simp only [Finset.mul_sum, coeff_sum]
-  apply Finset.sum_congr rfl
-  intro mu hmu
-  change (completePowers n i b *
-    (schurCoeff (Complete.pleth n (j-i) b) (partVec mu) •
-      alternant (shifted (partVec mu)))).coeff _ = _
-  rw [mul_smul_comm, coeff_smul]
-  exact mul_comm _ _
-
-end Foulkes.Lookup
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.FoulkesSixth.HomogeneousPlethysm
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.Complete
+open MvPolynomial Finset Foulkes.Strips
+
+def degreeCompositionEquiv (n b : ℕ) : Degree (Fin n) b ≃ Composition n b where
+  toFun d := ⟨fun j => ⟨d.val j, by
+      have hj := Finsupp.le_degree j d.val
+      omega⟩, by simpa using (Finsupp.degree_eq_sum d.val).symm.trans d.property⟩
+  invFun v := ⟨expVector (fun j => (v.val j).val), by
+    rw [Lookup.degree_expVector]
+    exact v.property⟩
+  left_inv d := by apply Subtype.ext; ext j; rfl
+  right_inv v := by apply Subtype.ext; funext j; apply Fin.ext; rfl
+
+lemma powerAlphabet_eq_completePowers (n i b : ℕ) :
+    (∑ d : Degree (Fin n) b,
+      monomial (i • d.val) (1 : ℤ)) = completePowers n i b := by
+  classical
+  unfold completePowers
+  apply Fintype.sum_equiv (degreeCompositionEquiv n b)
+  intro d
+  have he : i • d.val = Finsupp.equivFunOnFinite.symm
+      (fun j => i*((degreeCompositionEquiv n b d).val j).val) := by
+    ext j
+    change i * d.val j = i * d.val j
+    rfl
+  rw [he]
+
+lemma pleth_newton_powers (n j b : ℕ) :
+    (j : ℤ) • pleth n j b =
+      ∑ i ∈ Finset.Icc 1 j, completePowers n i b * pleth n (j-i) b := by
+  simpa only [powerAlphabet_eq_completePowers] using pleth_newton n j b
+
+end Foulkes.Complete
+
+namespace Foulkes.Lookup
+open MvPolynomial Finset Foulkes.Strips
+
+def stripCoeff {n : ℕ} (b i : ℕ) (gamma mu : Fin n → ℕ) : ℤ :=
+  (completePowers n i b * alternant (shifted mu)).coeff (expVector (shifted gamma))
+
+theorem source_newton (n j b : ℕ) (gamma : Fin n → ℕ) :
+    (j : ℤ) * schurCoeff (Complete.pleth n j b) gamma =
+      ∑ i ∈ Finset.Icc 1 j,
+        ∑ mu : Partition n ((j-i)*b),
+          stripCoeff b i gamma (partVec mu) *
+            schurCoeff (Complete.pleth n (j-i) b) (partVec mu) := by
+  classical
+  have hh := congrArg (fun p : MvPolynomial (Fin n) ℤ =>
+    (p * alternant (staircase n)).coeff (expVector (shifted gamma)))
+      (Complete.pleth_newton_powers n j b)
+  change (((j : ℤ) • Complete.pleth n j b) * alternant (staircase n)).coeff
+      (expVector (shifted gamma)) = _ at hh
+  rw [smul_mul_assoc, coeff_smul] at hh
+  simp only [zsmul_eq_mul, Finset.sum_mul, coeff_sum] at hh
+  change (j : ℤ) * schurCoeff (Complete.pleth n j b) gamma = _ at hh
+  rw [hh]
+  apply Finset.sum_congr rfl
+  intro i hi
+  rw [mul_assoc]
+  rw [homogeneous_schur_expansion (d := (j-i)*b) _ (Complete.rename_pleth n (j-i) b)
+    (by simpa [Nat.mul_comm] using Complete.homogeneous_pleth n (j-i) b)]
+  simp only [Finset.mul_sum, coeff_sum]
+  apply Finset.sum_congr rfl
+  intro mu hmu
+  change (completePowers n i b *
+    (schurCoeff (Complete.pleth n (j-i) b) (partVec mu) •
+      alternant (shifted (partVec mu)))).coeff _ = _
+  rw [mul_smul_comm, coeff_smul]
+  exact mul_comm _ _
+
+end Foulkes.Lookup
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/StripCoefficient.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/StripCoefficient.lean
index 4b6cc136e8b2932876fd169797a9aa6fdfef81f9..85db034b4b6dbcd8510bb61214b0e7fb57315d66 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/StripCoefficient.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/StripCoefficient.lean
@@ -1,141 +1,141 @@
-import Mathlib
-import OAI.RepresentationTheory.FoulkesSixth.StripSupport
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.Strips
-open MvPolynomial Finset Matrix
-
-def Composition (n b : ℕ) := {v : Fin n → Fin (b+1) // ∑ j, (v j).val = b}
-instance (n b : ℕ) : Fintype (Composition n b) :=
-  inferInstanceAs (Fintype {_composition : Fin n → Fin (b+1) // _})
-
-def completePowers (n i b : ℕ) : MvPolynomial (Fin n) ℤ :=
-  ∑ v : Composition n b,
-    monomial (Finsupp.equivFunOnFinite.symm (fun j => i*(v.val j).val)) 1
-
-def alternant {n : ℕ} (m : Fin n → ℕ) : MvPolynomial (Fin n) ℤ :=
-  Matrix.det fun r c => (X r : MvPolynomial (Fin n) ℤ) ^ m c
-
-abbrev expVector {n : ℕ} (v : Fin n → ℕ) : Fin n →₀ ℕ := Finsupp.equivFunOnFinite.symm v
-
-@[simp] lemma expVector_apply {n : ℕ} (v : Fin n → ℕ) (j : Fin n) : expVector v j = v j := by
-  simp [expVector]
-
-lemma exponent_injective {n b i : ℕ} (hi : 0 < i) (d : Fin n → ℕ) :
-    Function.Injective (fun v : Composition n b => expVector (fun j => i*(v.val j).val) + expVector d) := by
-  intro v w h
-  apply Subtype.ext
-  funext j
-  apply Fin.ext
-  have hj := DFunLike.congr_fun h j
-  simp only [Finsupp.add_apply, expVector_apply] at hj
-  exact Nat.eq_of_mul_eq_mul_left hi (Nat.add_right_cancel hj)
-
-lemma coeff_complete_monomial {n b i : ℕ} (hi : 0 < i) (l d : Fin n → ℕ) :
-    (completePowers n i b * monomial (expVector d) 1).coeff (expVector l) =
-      if ∃ v : Composition n b, expVector (fun j => i*(v.val j).val) + expVector d = expVector l
-      then 1 else 0 := by
-  classical
-  simp only [completePowers, Finset.sum_mul, coeff_sum, monomial_mul_monomial, one_mul, coeff_monomial]
-  split_ifs with h
-  · obtain ⟨v,hv⟩ := h
-    rw [Finset.sum_eq_single v]
-    · simp [hv]
-    · intro w hw hne
-      have hh : expVector (fun j => i*(w.val j).val) + expVector d ≠ expVector l := by
-        intro hw
-        exact hne (exponent_injective hi d (hw.trans hv.symm))
-      simp [hh]
-    · simp
-  · apply Finset.sum_eq_zero
-    intro v hv
-    simp only [ite_eq_right (fun hh => h ⟨v,hh⟩)]
-
-lemma perm_prod_monomial {n : ℕ} (m : Fin n → ℕ) (σ : Equiv.Perm (Fin n)) :
-    (∏ c, (X (σ c) : MvPolynomial (Fin n) ℤ) ^ m c) =
-      monomial (expVector (fun r => m (σ.symm r))) 1 := by
-  rw [monic_monomial_eq, Finsupp.prod_fintype _ _ (fun _ => pow_zero _)]
-  simpa using Equiv.prod_comp σ (fun r => (X r : MvPolynomial (Fin n) ℤ) ^ m (σ.symm r))
-
-lemma exists_composition_iff {n b i : ℕ} (hi : 0 < i) (l d : Fin n → ℕ)
-    (htotal : ∑ r, l r = (∑ r, d r) + i*b) :
-    (∃ v : Composition n b, expVector (fun j => i*(v.val j).val) + expVector d = expVector l) ↔
-      ∀ r, d r ≤ l r ∧ d r % i = l r % i := by
-  classical
-  constructor
-  · rintro ⟨v,hv⟩ r
-    have hr := DFunLike.congr_fun hv r
-    simp only [Finsupp.add_apply, expVector_apply] at hr
-    refine ⟨by omega, ?_⟩
-    rw [← hr]
-    simp
-  · intro h
-    let q : Fin n → ℕ := fun r => (l r - d r) / i
-    have hq (r : Fin n) : i*q r + d r = l r := by
-      have hdvd : i ∣ l r-d r := (Nat.modEq_iff_dvd' (h r).1).mp (h r).2
-      dsimp [q]
-      rw [Nat.mul_div_cancel' hdvd, Nat.sub_add_cancel (h r).1]
-    have hsum : ∑ r, q r = b := by
-      have hs : i*(∑ r, q r) + ∑ r, d r = ∑ r, l r := by
-        rw [Finset.mul_sum, ← Finset.sum_add_distrib]
-        exact Finset.sum_congr rfl (fun r hr => hq r)
-      have he : i*(∑ r, q r) = i*b := by omega
-      exact Nat.eq_of_mul_eq_mul_left hi he
-    have hb (r : Fin n) : q r < b+1 := by
-      have hh : q r ≤ ∑ j, q j := Finset.single_le_sum (fun j hj => Nat.zero_le _) (mem_univ r)
-      omega
-    let v : Composition n b := ⟨fun r => ⟨q r, hb r⟩, hsum⟩
-    refine ⟨v, ?_⟩
-    ext r
-    exact hq r
-
-lemma assignment_product {n : ℕ} (i : ℕ) (l m : Fin n → ℕ) (σ : Equiv.Perm (Fin n)) :
-    (∏ c, incidence (assignments i l m) (σ c) c) =
-      if ∀ r, m (σ.symm r) ≤ l r ∧ m (σ.symm r) % i = l r % i then 1 else 0 := by
-  classical
-  by_cases h : ∀ r, m (σ.symm r) ≤ l r ∧ m (σ.symm r) % i = l r % i
-  · rw [ite_eq_left h]
-    apply Finset.prod_eq_one
-    intro c hc
-    have hh := h (σ c)
-    simp only [incidence, assignments, mem_filter, mem_univ, true_and]
-    rw [ite_eq_left (by simpa using hh)]
-  · rw [ite_eq_right h]
-    push Not at h
-    obtain ⟨r,hr⟩ := h
-    apply Finset.prod_eq_zero (mem_univ (σ.symm r))
-    simp only [incidence, assignments, mem_filter, mem_univ, true_and, Equiv.apply_symm_apply]
-    exact ite_eq_right (fun hh => hr hh.1 hh.2)
-
-theorem coefficient_eq_assignment_det {n b i : ℕ} (hi : 0 < i) (l m : Fin n → ℕ)
-    (htotal : ∑ r, l r = (∑ r, m r) + i*b) :
-    (completePowers n i b * alternant m).coeff (expVector l) =
-      (incidence (assignments i l m)).det := by
-  classical
-  have ha : alternant m = ∑ σ : Equiv.Perm (Fin n),
-      Equiv.Perm.sign σ • ∏ c, (X (σ c) : MvPolynomial (Fin n) ℤ) ^ m c :=
-    Matrix.det_apply _
-  rw [ha, Matrix.det_apply, Finset.mul_sum, coeff_sum]
-  apply Finset.sum_congr rfl
-  intro σ hσ
-  rw [mul_smul_comm, coeff_smul, perm_prod_monomial, coeff_complete_monomial hi,
-    assignment_product]
-  have ht : ∑ r, l r = (∑ r, m (σ.symm r)) + i*b := by
-    rw [Equiv.sum_comp σ.symm m]
-    exact htotal
-  simp only [exists_composition_iff hi l _ ht]
-
-theorem signed_strip_abs_le_one {n b i : ℕ} (hi : 0 < i) (l m : Fin n → ℕ)
-    (htotal : ∑ r, l r = (∑ r, m r) + i*b) :
-    Int.natAbs ((completePowers n i b * alternant m).coeff (expVector l)) ≤ 1 := by
-  rw [coefficient_eq_assignment_det hi l m htotal]
-  exact assignment_det_bound i l m
-
-end Foulkes.Strips
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.FoulkesSixth.StripSupport
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.Strips
+open MvPolynomial Finset Matrix
+
+def Composition (n b : ℕ) := {v : Fin n → Fin (b+1) // ∑ j, (v j).val = b}
+instance (n b : ℕ) : Fintype (Composition n b) :=
+  inferInstanceAs (Fintype {_composition : Fin n → Fin (b+1) // _})
+
+def completePowers (n i b : ℕ) : MvPolynomial (Fin n) ℤ :=
+  ∑ v : Composition n b,
+    monomial (Finsupp.equivFunOnFinite.symm (fun j => i*(v.val j).val)) 1
+
+def alternant {n : ℕ} (m : Fin n → ℕ) : MvPolynomial (Fin n) ℤ :=
+  Matrix.det fun r c => (X r : MvPolynomial (Fin n) ℤ) ^ m c
+
+abbrev expVector {n : ℕ} (v : Fin n → ℕ) : Fin n →₀ ℕ := Finsupp.equivFunOnFinite.symm v
+
+@[simp] lemma expVector_apply {n : ℕ} (v : Fin n → ℕ) (j : Fin n) : expVector v j = v j := by
+  simp [expVector]
+
+lemma exponent_injective {n b i : ℕ} (hi : 0 < i) (d : Fin n → ℕ) :
+    Function.Injective (fun v : Composition n b => expVector (fun j => i*(v.val j).val) + expVector d) := by
+  intro v w h
+  apply Subtype.ext
+  funext j
+  apply Fin.ext
+  have hj := DFunLike.congr_fun h j
+  simp only [Finsupp.add_apply, expVector_apply] at hj
+  exact Nat.eq_of_mul_eq_mul_left hi (Nat.add_right_cancel hj)
+
+lemma coeff_complete_monomial {n b i : ℕ} (hi : 0 < i) (l d : Fin n → ℕ) :
+    (completePowers n i b * monomial (expVector d) 1).coeff (expVector l) =
+      if ∃ v : Composition n b, expVector (fun j => i*(v.val j).val) + expVector d = expVector l
+      then 1 else 0 := by
+  classical
+  simp only [completePowers, Finset.sum_mul, coeff_sum, monomial_mul_monomial, one_mul, coeff_monomial]
+  split_ifs with h
+  · obtain ⟨v,hv⟩ := h
+    rw [Finset.sum_eq_single v]
+    · simp [hv]
+    · intro w hw hne
+      have hh : expVector (fun j => i*(w.val j).val) + expVector d ≠ expVector l := by
+        intro hw
+        exact hne (exponent_injective hi d (hw.trans hv.symm))
+      simp [hh]
+    · simp
+  · apply Finset.sum_eq_zero
+    intro v hv
+    simp only [ite_eq_right (fun hh => h ⟨v,hh⟩)]
+
+lemma perm_prod_monomial {n : ℕ} (m : Fin n → ℕ) (σ : Equiv.Perm (Fin n)) :
+    (∏ c, (X (σ c) : MvPolynomial (Fin n) ℤ) ^ m c) =
+      monomial (expVector (fun r => m (σ.symm r))) 1 := by
+  rw [monic_monomial_eq, Finsupp.prod_fintype _ _ (fun _ => pow_zero _)]
+  simpa using Equiv.prod_comp σ (fun r => (X r : MvPolynomial (Fin n) ℤ) ^ m (σ.symm r))
+
+lemma exists_composition_iff {n b i : ℕ} (hi : 0 < i) (l d : Fin n → ℕ)
+    (htotal : ∑ r, l r = (∑ r, d r) + i*b) :
+    (∃ v : Composition n b, expVector (fun j => i*(v.val j).val) + expVector d = expVector l) ↔
+      ∀ r, d r ≤ l r ∧ d r % i = l r % i := by
+  classical
+  constructor
+  · rintro ⟨v,hv⟩ r
+    have hr := DFunLike.congr_fun hv r
+    simp only [Finsupp.add_apply, expVector_apply] at hr
+    refine ⟨by omega, ?_⟩
+    rw [← hr]
+    simp
+  · intro h
+    let q : Fin n → ℕ := fun r => (l r - d r) / i
+    have hq (r : Fin n) : i*q r + d r = l r := by
+      have hdvd : i ∣ l r-d r := (Nat.modEq_iff_dvd' (h r).1).mp (h r).2
+      dsimp [q]
+      rw [Nat.mul_div_cancel' hdvd, Nat.sub_add_cancel (h r).1]
+    have hsum : ∑ r, q r = b := by
+      have hs : i*(∑ r, q r) + ∑ r, d r = ∑ r, l r := by
+        rw [Finset.mul_sum, ← Finset.sum_add_distrib]
+        exact Finset.sum_congr rfl (fun r hr => hq r)
+      have he : i*(∑ r, q r) = i*b := by omega
+      exact Nat.eq_of_mul_eq_mul_left hi he
+    have hb (r : Fin n) : q r < b+1 := by
+      have hh : q r ≤ ∑ j, q j := Finset.single_le_sum (fun j hj => Nat.zero_le _) (mem_univ r)
+      omega
+    let v : Composition n b := ⟨fun r => ⟨q r, hb r⟩, hsum⟩
+    refine ⟨v, ?_⟩
+    ext r
+    exact hq r
+
+lemma assignment_product {n : ℕ} (i : ℕ) (l m : Fin n → ℕ) (σ : Equiv.Perm (Fin n)) :
+    (∏ c, incidence (assignments i l m) (σ c) c) =
+      if ∀ r, m (σ.symm r) ≤ l r ∧ m (σ.symm r) % i = l r % i then 1 else 0 := by
+  classical
+  by_cases h : ∀ r, m (σ.symm r) ≤ l r ∧ m (σ.symm r) % i = l r % i
+  · rw [ite_eq_left h]
+    apply Finset.prod_eq_one
+    intro c hc
+    have hh := h (σ c)
+    simp only [incidence, assignments, mem_filter, mem_univ, true_and]
+    rw [ite_eq_left (by simpa using hh)]
+  · rw [ite_eq_right h]
+    push Not at h
+    obtain ⟨r,hr⟩ := h
+    apply Finset.prod_eq_zero (mem_univ (σ.symm r))
+    simp only [incidence, assignments, mem_filter, mem_univ, true_and, Equiv.apply_symm_apply]
+    exact ite_eq_right (fun hh => hr hh.1 hh.2)
+
+theorem coefficient_eq_assignment_det {n b i : ℕ} (hi : 0 < i) (l m : Fin n → ℕ)
+    (htotal : ∑ r, l r = (∑ r, m r) + i*b) :
+    (completePowers n i b * alternant m).coeff (expVector l) =
+      (incidence (assignments i l m)).det := by
+  classical
+  have ha : alternant m = ∑ σ : Equiv.Perm (Fin n),
+      Equiv.Perm.sign σ • ∏ c, (X (σ c) : MvPolynomial (Fin n) ℤ) ^ m c :=
+    Matrix.det_apply _
+  rw [ha, Matrix.det_apply, Finset.mul_sum, coeff_sum]
+  apply Finset.sum_congr rfl
+  intro σ hσ
+  rw [mul_smul_comm, coeff_smul, perm_prod_monomial, coeff_complete_monomial hi,
+    assignment_product]
+  have ht : ∑ r, l r = (∑ r, m (σ.symm r)) + i*b := by
+    rw [Equiv.sum_comp σ.symm m]
+    exact htotal
+  simp only [exists_composition_iff hi l _ ht]
+
+theorem signed_strip_abs_le_one {n b i : ℕ} (hi : 0 < i) (l m : Fin n → ℕ)
+    (htotal : ∑ r, l r = (∑ r, m r) + i*b) :
+    Int.natAbs ((completePowers n i b * alternant m).coeff (expVector l)) ≤ 1 := by
+  rw [coefficient_eq_assignment_det hi l m htotal]
+  exact assignment_det_bound i l m
+
+end Foulkes.Strips
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/StripInterlace.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/StripInterlace.lean
index cf00b266cf9e4b1fe13838ae4fb09a44ece676ed..738329a27ec23696305f0a3f47f4a0a2917dd03b 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/StripInterlace.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/StripInterlace.lean
@@ -1,141 +1,141 @@
-import Mathlib
-import OAI.RepresentationTheory.FoulkesSixth.StripCoefficient
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.Strips
-open Matrix Finset Submodule
-
-lemma matching_of_det_ne_zero {n i : ℕ} (l m : Fin n → ℕ)
-    (hd : (incidence (assignments i l m)).det ≠ 0) :
-    ∃ σ : Equiv.Perm (Fin n), ∀ c, m c ≤ l (σ c) ∧ m c % i = l (σ c) % i := by
-  classical
-  rw [Matrix.det_apply] at hd
-  obtain ⟨σ, hσ, ht⟩ := Finset.exists_ne_zero_of_sum_ne_zero hd
-  have hp : (∏ c, incidence (assignments i l m) (σ c) c) ≠ 0 := by
-    intro hh
-    exact ht (by rw [hh, smul_zero])
-  refine ⟨σ, fun c => ?_⟩
-  have hc := (Finset.prod_ne_zero_iff.mp hp) c (mem_univ c)
-  simpa only [incidence, assignments, mem_filter, mem_univ, true_and,
-    ne_eq, ite_eq_right_iff, one_ne_zero, imp_false, not_not] using hc
-
-lemma containment_of_det_ne_zero {n i : ℕ} (l m : Fin n → ℕ)
-    (hl : Antitone l) (hm : Antitone m)
-    (hd : (incidence (assignments i l m)).det ≠ 0) (j : Fin n) : m j ≤ l j := by
-  classical
-  obtain ⟨σ,hσ⟩ := matching_of_det_ne_zero l m hd
-  by_contra h
-  let e : Fin (j.val+1) → Fin n := fun c => ⟨c.val, by omega⟩
-  have hrow (c : Fin (j.val+1)) : (σ (e c)).val < j.val := by
-    by_contra hh
-    have hm' : m j ≤ m (e c) := hm (by change c.val ≤ j.val; omega)
-    have hl' : l (σ (e c)) ≤ l j := hl (by change j.val ≤ (σ (e c)).val; omega)
-    have hc := (hσ (e c)).1
-    omega
-  let f : Fin (j.val+1) → Fin j.val := fun c => ⟨(σ (e c)).val, hrow c⟩
-  have hf : Function.Injective f := by
-    intro c d hcd
-    have hv : (σ (e c)).val = (σ (e d)).val := congrArg (fun z : Fin j.val => z.val) hcd
-    have hh : σ (e c) = σ (e d) := Fin.ext hv
-    have hc : c.val = d.val := congrArg (fun z : Fin n => z.val) (σ.injective hh)
-    exact Fin.ext hc
-  have hn := Fintype.card_le_of_injective f hf
-  simp only [Fintype.card_fin] at hn
-  omega
-
-lemma le_of_residue_eq_of_lt_add {i x y : ℕ} (_hi : 0 < i)
-    (hres : x % i = y % i) (hxy : x < y+i) : x ≤ y := by
-  by_contra h
-  have hyx : y < x := Nat.lt_of_not_ge h
-  have hd : i ∣ x-y := (Nat.modEq_iff_dvd' (Nat.le_of_lt hyx)).mp hres.symm
-  have hh := Nat.le_of_dvd (Nat.sub_pos_of_lt hyx) hd
-  omega
-
-lemma shifted_containment_of_det_ne_zero {n i : ℕ} (hi : 0 < i) (l m : Fin n → ℕ)
-    (hl : Antitone l) (hm : Antitone m)
-    (hd : (incidence (assignments i l m)).det ≠ 0)
-    (j : Fin n) (hji : j.val+i < n) : l ⟨j.val+i,hji⟩ + i ≤ m j := by
-  classical
-  by_contra h
-  have hjm : m j < l ⟨j.val+i,hji⟩ + i := Nat.lt_of_not_ge h
-  let A : Matrix (Fin n) (Fin n) ℚ := (incidence (assignments i l m)).map Int.cast
-  have hA : A.det ≠ 0 := by
-    rw [show A = (incidence (assignments i l m)).map Int.cast from rfl, ← Int.cast_det]
-    exact_mod_cast hd
-  have hind := Matrix.linearIndependent_rows_of_det_ne_zero hA
-  let embed : Fin (j.val+i+1) → Fin n := fun r => ⟨r.val, by omega⟩
-  have hembed : Function.Injective embed := by
-    intro r s hrs
-    exact Fin.ext (congrArg (fun z : Fin n => z.val) hrs)
-  have hlin : LinearIndependent ℚ (fun r : Fin (j.val+i+1) => A (embed r)) :=
-    hind.comp embed hembed
-  let gen : Fin j.val ⊕ Fin i → Fin n → ℚ := Sum.elim
-    (fun c => Pi.single (⟨c.val, by omega⟩ : Fin n) 1)
-    (fun u c => if j.val ≤ c.val ∧ m c % i = u.val then 1 else 0)
-  let U : Submodule ℚ (Fin n → ℚ) := Submodule.span ℚ (Set.range gen)
-  have hgen (q : Fin j.val ⊕ Fin i) : gen q ∈ U :=
-    Submodule.subset_span ⟨q,rfl⟩
-  have hrow (r : Fin (j.val+i+1)) : A (embed r) ∈ U := by
-    let u : Fin i := ⟨l (embed r) % i, Nat.mod_lt _ hi⟩
-    have he : A (embed r) =
-        (∑ c : Fin j.val, A (embed r) ⟨c.val, by omega⟩ • gen (Sum.inl c)) + gen (Sum.inr u) := by
-      funext c
-      have hlr : l ⟨j.val+i,hji⟩ ≤ l (embed r) := hl (show embed r ≤ ⟨j.val+i,hji⟩ by
-        exact Nat.le_of_lt_succ r.isLt)
-      by_cases hc : c.val < j.val
-      · have hcc : (⟨c.val,hc⟩ : Fin j.val).val = c.val := rfl
-        simp only [Pi.add_apply, Finset.sum_apply, Pi.smul_apply, smul_eq_mul]
-        have hsum : (∑ d : Fin j.val, A (embed r) ⟨d.val, by omega⟩ * gen (Sum.inl d) c) =
-            A (embed r) c := by
-          rw [Finset.sum_eq_single ⟨c.val,hc⟩]
-          · simp [gen]
-          · intro d hd hne
-            have hdc : (⟨d.val, by omega⟩ : Fin n) ≠ c := by
-              intro he
-              apply hne
-              exact Fin.ext (congrArg (fun z : Fin n => z.val) he)
-            simp [gen, Ne.symm hdc]
-          · simp
-        rw [hsum]
-        have hnc : ¬j ≤ c := by change ¬j.val ≤ c.val; omega
-        simp [gen, u, hnc]
-      · have hsum : (∑ d : Fin j.val, A (embed r) ⟨d.val, by omega⟩ • gen (Sum.inl d)) c = 0 := by
-          simp only [Finset.sum_apply, Pi.smul_apply, smul_eq_mul]
-          apply Finset.sum_eq_zero
-          intro d hd
-          have hdc : (⟨d.val, by omega⟩ : Fin n) ≠ c := by
-            intro he
-            have hv : d.val = c.val := congrArg (fun z : Fin n => z.val) he
-            have hdlt := d.isLt
-            omega
-          simp [gen, Ne.symm hdc]
-        rw [Pi.add_apply, hsum, zero_add]
-        have hmj : m c ≤ m j := hm (show j ≤ c by exact Nat.le_of_not_gt hc)
-        have hlt : m c < l (embed r)+i := by omega
-        have iffcond : (m c ≤ l (embed r) ∧ m c % i = l (embed r) % i) ↔
-            m c % i = l (embed r) % i :=
-          ⟨And.right, fun hr => ⟨le_of_residue_eq_of_lt_add hi hr hlt, hr⟩⟩
-        simp only [A, Matrix.map_apply, incidence, assignments, mem_filter, mem_univ, true_and,
-          gen, Sum.elim_inr, u, Nat.le_of_not_gt hc, true_and]
-        simp only [iffcond, Int.cast_ite, Int.cast_one, Int.cast_zero]
-    rw [he]
-    exact U.add_mem (U.sum_mem (fun c hc => U.smul_mem _ (hgen (Sum.inl c)))) (hgen (Sum.inr u))
-  have hsub : Submodule.span ℚ (Set.range (fun r : Fin (j.val+i+1) => A (embed r))) ≤ U := by
-    apply Submodule.span_le.mpr
-    rintro _ ⟨r,rfl⟩
-    exact hrow r
-  have hdim := Submodule.finrank_mono hsub
-  have heq := finrank_span_eq_card hlin
-  have hbound : Module.finrank ℚ U ≤ j.val+i := by
-    simpa [U, Set.finrank] using (finrank_range_le_card (R := ℚ) gen)
-  rw [heq, Fintype.card_fin] at hdim
-  omega
-
-end Foulkes.Strips
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.FoulkesSixth.StripCoefficient
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.Strips
+open Matrix Finset Submodule
+
+lemma matching_of_det_ne_zero {n i : ℕ} (l m : Fin n → ℕ)
+    (hd : (incidence (assignments i l m)).det ≠ 0) :
+    ∃ σ : Equiv.Perm (Fin n), ∀ c, m c ≤ l (σ c) ∧ m c % i = l (σ c) % i := by
+  classical
+  rw [Matrix.det_apply] at hd
+  obtain ⟨σ, hσ, ht⟩ := Finset.exists_ne_zero_of_sum_ne_zero hd
+  have hp : (∏ c, incidence (assignments i l m) (σ c) c) ≠ 0 := by
+    intro hh
+    exact ht (by rw [hh, smul_zero])
+  refine ⟨σ, fun c => ?_⟩
+  have hc := (Finset.prod_ne_zero_iff.mp hp) c (mem_univ c)
+  simpa only [incidence, assignments, mem_filter, mem_univ, true_and,
+    ne_eq, ite_eq_right_iff, one_ne_zero, imp_false, not_not] using hc
+
+lemma containment_of_det_ne_zero {n i : ℕ} (l m : Fin n → ℕ)
+    (hl : Antitone l) (hm : Antitone m)
+    (hd : (incidence (assignments i l m)).det ≠ 0) (j : Fin n) : m j ≤ l j := by
+  classical
+  obtain ⟨σ,hσ⟩ := matching_of_det_ne_zero l m hd
+  by_contra h
+  let e : Fin (j.val+1) → Fin n := fun c => ⟨c.val, by omega⟩
+  have hrow (c : Fin (j.val+1)) : (σ (e c)).val < j.val := by
+    by_contra hh
+    have hm' : m j ≤ m (e c) := hm (by change c.val ≤ j.val; omega)
+    have hl' : l (σ (e c)) ≤ l j := hl (by change j.val ≤ (σ (e c)).val; omega)
+    have hc := (hσ (e c)).1
+    omega
+  let f : Fin (j.val+1) → Fin j.val := fun c => ⟨(σ (e c)).val, hrow c⟩
+  have hf : Function.Injective f := by
+    intro c d hcd
+    have hv : (σ (e c)).val = (σ (e d)).val := congrArg (fun z : Fin j.val => z.val) hcd
+    have hh : σ (e c) = σ (e d) := Fin.ext hv
+    have hc : c.val = d.val := congrArg (fun z : Fin n => z.val) (σ.injective hh)
+    exact Fin.ext hc
+  have hn := Fintype.card_le_of_injective f hf
+  simp only [Fintype.card_fin] at hn
+  omega
+
+lemma le_of_residue_eq_of_lt_add {i x y : ℕ} (_hi : 0 < i)
+    (hres : x % i = y % i) (hxy : x < y+i) : x ≤ y := by
+  by_contra h
+  have hyx : y < x := Nat.lt_of_not_ge h
+  have hd : i ∣ x-y := (Nat.modEq_iff_dvd' (Nat.le_of_lt hyx)).mp hres.symm
+  have hh := Nat.le_of_dvd (Nat.sub_pos_of_lt hyx) hd
+  omega
+
+lemma shifted_containment_of_det_ne_zero {n i : ℕ} (hi : 0 < i) (l m : Fin n → ℕ)
+    (hl : Antitone l) (hm : Antitone m)
+    (hd : (incidence (assignments i l m)).det ≠ 0)
+    (j : Fin n) (hji : j.val+i < n) : l ⟨j.val+i,hji⟩ + i ≤ m j := by
+  classical
+  by_contra h
+  have hjm : m j < l ⟨j.val+i,hji⟩ + i := Nat.lt_of_not_ge h
+  let A : Matrix (Fin n) (Fin n) ℚ := (incidence (assignments i l m)).map Int.cast
+  have hA : A.det ≠ 0 := by
+    rw [show A = (incidence (assignments i l m)).map Int.cast from rfl, ← Int.cast_det]
+    exact_mod_cast hd
+  have hind := Matrix.linearIndependent_rows_of_det_ne_zero hA
+  let embed : Fin (j.val+i+1) → Fin n := fun r => ⟨r.val, by omega⟩
+  have hembed : Function.Injective embed := by
+    intro r s hrs
+    exact Fin.ext (congrArg (fun z : Fin n => z.val) hrs)
+  have hlin : LinearIndependent ℚ (fun r : Fin (j.val+i+1) => A (embed r)) :=
+    hind.comp embed hembed
+  let gen : Fin j.val ⊕ Fin i → Fin n → ℚ := Sum.elim
+    (fun c => Pi.single (⟨c.val, by omega⟩ : Fin n) 1)
+    (fun u c => if j.val ≤ c.val ∧ m c % i = u.val then 1 else 0)
+  let U : Submodule ℚ (Fin n → ℚ) := Submodule.span ℚ (Set.range gen)
+  have hgen (q : Fin j.val ⊕ Fin i) : gen q ∈ U :=
+    Submodule.subset_span ⟨q,rfl⟩
+  have hrow (r : Fin (j.val+i+1)) : A (embed r) ∈ U := by
+    let u : Fin i := ⟨l (embed r) % i, Nat.mod_lt _ hi⟩
+    have he : A (embed r) =
+        (∑ c : Fin j.val, A (embed r) ⟨c.val, by omega⟩ • gen (Sum.inl c)) + gen (Sum.inr u) := by
+      funext c
+      have hlr : l ⟨j.val+i,hji⟩ ≤ l (embed r) := hl (show embed r ≤ ⟨j.val+i,hji⟩ by
+        exact Nat.le_of_lt_succ r.isLt)
+      by_cases hc : c.val < j.val
+      · have hcc : (⟨c.val,hc⟩ : Fin j.val).val = c.val := rfl
+        simp only [Pi.add_apply, Finset.sum_apply, Pi.smul_apply, smul_eq_mul]
+        have hsum : (∑ d : Fin j.val, A (embed r) ⟨d.val, by omega⟩ * gen (Sum.inl d) c) =
+            A (embed r) c := by
+          rw [Finset.sum_eq_single ⟨c.val,hc⟩]
+          · simp [gen]
+          · intro d hd hne
+            have hdc : (⟨d.val, by omega⟩ : Fin n) ≠ c := by
+              intro he
+              apply hne
+              exact Fin.ext (congrArg (fun z : Fin n => z.val) he)
+            simp [gen, Ne.symm hdc]
+          · simp
+        rw [hsum]
+        have hnc : ¬j ≤ c := by change ¬j.val ≤ c.val; omega
+        simp [gen, u, hnc]
+      · have hsum : (∑ d : Fin j.val, A (embed r) ⟨d.val, by omega⟩ • gen (Sum.inl d)) c = 0 := by
+          simp only [Finset.sum_apply, Pi.smul_apply, smul_eq_mul]
+          apply Finset.sum_eq_zero
+          intro d hd
+          have hdc : (⟨d.val, by omega⟩ : Fin n) ≠ c := by
+            intro he
+            have hv : d.val = c.val := congrArg (fun z : Fin n => z.val) he
+            have hdlt := d.isLt
+            omega
+          simp [gen, Ne.symm hdc]
+        rw [Pi.add_apply, hsum, zero_add]
+        have hmj : m c ≤ m j := hm (show j ≤ c by exact Nat.le_of_not_gt hc)
+        have hlt : m c < l (embed r)+i := by omega
+        have iffcond : (m c ≤ l (embed r) ∧ m c % i = l (embed r) % i) ↔
+            m c % i = l (embed r) % i :=
+          ⟨And.right, fun hr => ⟨le_of_residue_eq_of_lt_add hi hr hlt, hr⟩⟩
+        simp only [A, Matrix.map_apply, incidence, assignments, mem_filter, mem_univ, true_and,
+          gen, Sum.elim_inr, u, Nat.le_of_not_gt hc, true_and]
+        simp only [iffcond, Int.cast_ite, Int.cast_one, Int.cast_zero]
+    rw [he]
+    exact U.add_mem (U.sum_mem (fun c hc => U.smul_mem _ (hgen (Sum.inl c)))) (hgen (Sum.inr u))
+  have hsub : Submodule.span ℚ (Set.range (fun r : Fin (j.val+i+1) => A (embed r))) ≤ U := by
+    apply Submodule.span_le.mpr
+    rintro _ ⟨r,rfl⟩
+    exact hrow r
+  have hdim := Submodule.finrank_mono hsub
+  have heq := finrank_span_eq_card hlin
+  have hbound : Module.finrank ℚ U ≤ j.val+i := by
+    simpa [U, Set.finrank] using (finrank_range_le_card (R := ℚ) gen)
+  rw [heq, Fintype.card_fin] at hdim
+  omega
+
+end Foulkes.Strips
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/StripSupport.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/StripSupport.lean
index 78667767daa8da45f9a0eae40ea7e3f0d388125c..6e6ef61c8769d3a3c02c8a9b3f3cd8a93dbaa56b 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/StripSupport.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/StripSupport.lean
@@ -1,106 +1,106 @@
-import Mathlib
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.Strips
-open Finset Matrix
-
-def incidence {n : ℕ} (S : Fin n → Finset (Fin n)) : Matrix (Fin n) (Fin n) ℤ :=
-  fun r c => if c ∈ S r then 1 else 0
-
-def Laminar {n : ℕ} (S : Fin n → Finset (Fin n)) : Prop :=
-  ∀ r s, S r ⊆ S s ∨ S s ⊆ S r ∨ Disjoint (S r) (S s)
-
-theorem incidence_det_bound (n : ℕ) (S : Fin n → Finset (Fin n)) (hS : Laminar S) :
-    Int.natAbs (incidence S).det ≤ 1 := by
-  classical
-  induction n with
-  | zero => simp
-  | succ n ih =>
-    by_cases hd : (incidence S).det = 0
-    · simp [hd]
-    have hnon (r : Fin (n+1)) : (S r).Nonempty := by
-      by_contra h
-      have he : S r = ∅ := Finset.not_nonempty_iff_eq_empty.mp h
-      apply hd
-      apply Matrix.det_eq_zero_of_row_eq_zero r
-      intro c
-      simp [incidence, he]
-    obtain ⟨r, hr, hmin⟩ := Finset.exists_min_image univ (fun r : Fin (n+1) => (S r).card)
-      (Finset.univ_nonempty)
-    have hcols {j k : Fin (n+1)} (hj : j ∈ S r) (hk : k ∈ S r) : j = k := by
-      by_contra hjk
-      apply hd
-      apply Matrix.det_zero_of_column_eq hjk
-      intro s
-      rcases hS r s with hrs | hsr | hdj
-      · simp [incidence, hrs hj, hrs hk]
-      · have he : S s = S r := Finset.eq_of_subset_of_card_le hsr (hmin s (mem_univ s))
-        simp [incidence, he, hj, hk]
-      · have hj' : j ∉ S s := fun hs => Finset.disjoint_left.mp hdj hj hs
-        have hk' : k ∉ S s := fun hs => Finset.disjoint_left.mp hdj hk hs
-        simp [incidence, hj', hk']
-    obtain ⟨c, hc⟩ := hnon r
-    have hrow : S r = {c} := by
-      ext j
-      simp only [mem_singleton]
-      exact ⟨fun hj => hcols hj hc, fun h => h ▸ hc⟩
-    let S' : Fin n → Finset (Fin n) := fun i => univ.filter fun j => c.succAbove j ∈ S (r.succAbove i)
-    have hlam : Laminar S' := by
-      intro i j
-      rcases hS (r.succAbove i) (r.succAbove j) with hij | hji | hdj
-      · left
-        intro k hk
-        simpa [S'] using hij (by simpa [S'] using hk)
-      · right; left
-        intro k hk
-        simpa [S'] using hji (by simpa [S'] using hk)
-      · right; right
-        apply Finset.disjoint_left.mpr
-        intro k hi hj
-        exact Finset.disjoint_left.mp hdj (by simpa [S'] using hi) (by simpa [S'] using hj)
-    have hminor : (incidence S).submatrix r.succAbove c.succAbove = incidence S' := by
-      ext i j
-      simp [incidence, S']
-    have he : (incidence S).det = (-1 : ℤ) ^ (r.val+c.val) * (incidence S').det := by
-      rw [Matrix.det_succ_row _ r]
-      simp only [incidence, hrow, mem_singleton]
-      simp only [mul_ite, mul_one, mul_zero, ite_mul, zero_mul]
-      rw [Finset.sum_ite_eq' univ c]
-      simp only [mem_univ, ite_true, hminor]
-    rw [he, Int.natAbs_mul, Int.natAbs_pow]
-    simpa using ih S' hlam
-
-def assignments {n : ℕ} (i : ℕ) (l m : Fin n → ℕ) : Fin n → Finset (Fin n) :=
-  fun r => univ.filter fun c => m c ≤ l r ∧ m c % i = l r % i
-
-lemma assignments_laminar {n : ℕ} (i : ℕ) (l m : Fin n → ℕ) :
-    Laminar (assignments i l m) := by
-  intro r s
-  by_cases he : l r % i = l s % i
-  · rcases le_total (l r) (l s) with hrs | hsr
-    · left
-      intro c hc
-      simp only [assignments, mem_filter, mem_univ, true_and] at hc ⊢
-      exact ⟨hc.1.trans hrs, hc.2.trans he⟩
-    · right; left
-      intro c hc
-      simp only [assignments, mem_filter, mem_univ, true_and] at hc ⊢
-      exact ⟨hc.1.trans hsr, hc.2.trans he.symm⟩
-  · right; right
-    apply Finset.disjoint_left.mpr
-    intro c hr hs
-    simp only [assignments, mem_filter, mem_univ, true_and] at hr hs
-    exact he (hr.2.symm.trans hs.2)
-
-theorem assignment_det_bound {n : ℕ} (i : ℕ) (l m : Fin n → ℕ) :
-    Int.natAbs (incidence (assignments i l m)).det ≤ 1 :=
-  incidence_det_bound n _ (assignments_laminar i l m)
-
-end Foulkes.Strips
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.Strips
+open Finset Matrix
+
+def incidence {n : ℕ} (S : Fin n → Finset (Fin n)) : Matrix (Fin n) (Fin n) ℤ :=
+  fun r c => if c ∈ S r then 1 else 0
+
+def Laminar {n : ℕ} (S : Fin n → Finset (Fin n)) : Prop :=
+  ∀ r s, S r ⊆ S s ∨ S s ⊆ S r ∨ Disjoint (S r) (S s)
+
+theorem incidence_det_bound (n : ℕ) (S : Fin n → Finset (Fin n)) (hS : Laminar S) :
+    Int.natAbs (incidence S).det ≤ 1 := by
+  classical
+  induction n with
+  | zero => simp
+  | succ n ih =>
+    by_cases hd : (incidence S).det = 0
+    · simp [hd]
+    have hnon (r : Fin (n+1)) : (S r).Nonempty := by
+      by_contra h
+      have he : S r = ∅ := Finset.not_nonempty_iff_eq_empty.mp h
+      apply hd
+      apply Matrix.det_eq_zero_of_row_eq_zero r
+      intro c
+      simp [incidence, he]
+    obtain ⟨r, hr, hmin⟩ := Finset.exists_min_image univ (fun r : Fin (n+1) => (S r).card)
+      (Finset.univ_nonempty)
+    have hcols {j k : Fin (n+1)} (hj : j ∈ S r) (hk : k ∈ S r) : j = k := by
+      by_contra hjk
+      apply hd
+      apply Matrix.det_zero_of_column_eq hjk
+      intro s
+      rcases hS r s with hrs | hsr | hdj
+      · simp [incidence, hrs hj, hrs hk]
+      · have he : S s = S r := Finset.eq_of_subset_of_card_le hsr (hmin s (mem_univ s))
+        simp [incidence, he, hj, hk]
+      · have hj' : j ∉ S s := fun hs => Finset.disjoint_left.mp hdj hj hs
+        have hk' : k ∉ S s := fun hs => Finset.disjoint_left.mp hdj hk hs
+        simp [incidence, hj', hk']
+    obtain ⟨c, hc⟩ := hnon r
+    have hrow : S r = {c} := by
+      ext j
+      simp only [mem_singleton]
+      exact ⟨fun hj => hcols hj hc, fun h => h ▸ hc⟩
+    let S' : Fin n → Finset (Fin n) := fun i => univ.filter fun j => c.succAbove j ∈ S (r.succAbove i)
+    have hlam : Laminar S' := by
+      intro i j
+      rcases hS (r.succAbove i) (r.succAbove j) with hij | hji | hdj
+      · left
+        intro k hk
+        simpa [S'] using hij (by simpa [S'] using hk)
+      · right; left
+        intro k hk
+        simpa [S'] using hji (by simpa [S'] using hk)
+      · right; right
+        apply Finset.disjoint_left.mpr
+        intro k hi hj
+        exact Finset.disjoint_left.mp hdj (by simpa [S'] using hi) (by simpa [S'] using hj)
+    have hminor : (incidence S).submatrix r.succAbove c.succAbove = incidence S' := by
+      ext i j
+      simp [incidence, S']
+    have he : (incidence S).det = (-1 : ℤ) ^ (r.val+c.val) * (incidence S').det := by
+      rw [Matrix.det_succ_row _ r]
+      simp only [incidence, hrow, mem_singleton]
+      simp only [mul_ite, mul_one, mul_zero, ite_mul, zero_mul]
+      rw [Finset.sum_ite_eq' univ c]
+      simp only [mem_univ, ite_true, hminor]
+    rw [he, Int.natAbs_mul, Int.natAbs_pow]
+    simpa using ih S' hlam
+
+def assignments {n : ℕ} (i : ℕ) (l m : Fin n → ℕ) : Fin n → Finset (Fin n) :=
+  fun r => univ.filter fun c => m c ≤ l r ∧ m c % i = l r % i
+
+lemma assignments_laminar {n : ℕ} (i : ℕ) (l m : Fin n → ℕ) :
+    Laminar (assignments i l m) := by
+  intro r s
+  by_cases he : l r % i = l s % i
+  · rcases le_total (l r) (l s) with hrs | hsr
+    · left
+      intro c hc
+      simp only [assignments, mem_filter, mem_univ, true_and] at hc ⊢
+      exact ⟨hc.1.trans hrs, hc.2.trans he⟩
+    · right; left
+      intro c hc
+      simp only [assignments, mem_filter, mem_univ, true_and] at hc ⊢
+      exact ⟨hc.1.trans hsr, hc.2.trans he.symm⟩
+  · right; right
+    apply Finset.disjoint_left.mpr
+    intro c hr hs
+    simp only [assignments, mem_filter, mem_univ, true_and] at hr hs
+    exact he (hr.2.symm.trans hs.2)
+
+theorem assignment_det_bound {n : ℕ} (i : ℕ) (l m : Fin n → ℕ) :
+    Int.natAbs (incidence (assignments i l m)).det ≤ 1 :=
+  incidence_det_bound n _ (assignments_laminar i l m)
+
+end Foulkes.Strips
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/SymDual.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/SymDual.lean
index bb480a22ac7375ab6f4470d28f14e5e7ab8dcec3..84e8642e5b5ec37514d52994f6ca3dac98aa16fc 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/SymDual.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/SymDual.lean
@@ -1,120 +1,120 @@
-import Mathlib
-import OAI.RepresentationTheory.FoulkesSixth.SymTensors
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.SymmetricTensor
-universe u v
-open scoped TensorProduct
-open Module
-variable (n : ℕ) {V : Type u} {W : Type v}
-  [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
-
-lemma pair_perm (σ : Equiv.Perm (Fin n)) (d : T n (Dual ℂ V)) (x : T n V) :
-    PiTensorProduct.dualDistrib (perm n (Dual ℂ V) σ d) x =
-    PiTensorProduct.dualDistrib d (perm n V σ.symm x) := by
-  have h : (PiTensorProduct.dualDistrib (R := ℂ) (M := fun _ : Fin n => V)).comp
-      (perm n (Dual ℂ V) σ).toLinearMap =
-      (perm n V σ.symm).toLinearMap.dualMap.comp PiTensorProduct.dualDistrib := by
-    ext f x
-    simp only [LinearMap.compMultilinearMap_apply, LinearMap.comp_apply, LinearEquiv.coe_coe, perm,
-      PiTensorProduct.reindex_tprod, PiTensorProduct.dualDistrib_apply,
-      LinearMap.dualMap_apply, Equiv.symm_symm]
-    have hp := Equiv.prod_comp σ.symm (fun i => f i (x (σ i)))
-    convert hp using 1
-    apply Finset.prod_congr rfl
-    intro i hi
-    rw [σ.apply_symm_apply]
-  exact DFunLike.congr_fun (DFunLike.congr_fun h d) x
-
-lemma pair_map (f : V →ₗ[ℂ] W) (d : T n (Dual ℂ W)) (x : T n V) :
-    PiTensorProduct.dualDistrib (PiTensorProduct.map (fun _ : Fin n => f.dualMap) d) x =
-    PiTensorProduct.dualDistrib d (PiTensorProduct.map (fun _ : Fin n => f) x) := by
-  have h : (PiTensorProduct.dualDistrib (R := ℂ) (M := fun _ : Fin n => V)).comp
-      (PiTensorProduct.map (fun _ : Fin n => f.dualMap)) =
-      (PiTensorProduct.map (fun _ : Fin n => f)).dualMap.comp PiTensorProduct.dualDistrib := by
-    ext d x
-    simp
-  exact DFunLike.congr_fun (DFunLike.congr_fun h d) x
-
-def pairing : Sym n (Dual ℂ V) →ₗ[ℂ] Dual ℂ (Sym n V) :=
-  (symmetricTensors n V).subtype.dualMap.comp
-    (PiTensorProduct.dualDistrib.comp (symmetricTensors n (Dual ℂ V)).subtype)
-
-@[simp] lemma pairing_apply (d : Sym n (Dual ℂ V)) (x : Sym n V) :
-    pairing n d x = PiTensorProduct.dualDistrib (d : T n (Dual ℂ V)) (x : T n V) := rfl
-
-@[simp] lemma pairing_power (d : Dual ℂ V) (v : V) :
-    pairing n (power n (Dual ℂ V) d) (power n V v) = d v ^ n := by
-  simp [pairing_apply, PiTensorProduct.dualDistrib_apply]
-
-lemma pairing_map (f : V →ₗ[ℂ] W) (d : Sym n (Dual ℂ W)) (x : Sym n V) :
-    pairing n (symMap n f.dualMap d) x = pairing n d (symMap n f x) :=
-  pair_map n f d x
-
-variable [FiniteDimensional ℂ V]
-
-def pairingInv (l : Dual ℂ (Sym n V)) : Sym n (Dual ℂ V) :=
-  ⟨PiTensorProduct.dualDistribEquiv.symm (l.comp (project n V)), by
-    intro σ
-    apply PiTensorProduct.dualDistribEquiv.injective
-    apply LinearMap.ext
-    intro x
-    change PiTensorProduct.dualDistrib
-      (perm n (Dual ℂ V) σ (PiTensorProduct.dualDistribEquiv.symm (l.comp (project n V)))) x =
-      PiTensorProduct.dualDistrib
-        (PiTensorProduct.dualDistribEquiv.symm (l.comp (project n V))) x
-    rw [pair_perm]
-    change PiTensorProduct.dualDistribEquiv
-      (PiTensorProduct.dualDistribEquiv.symm (l.comp (project n V))) (perm n V σ.symm x) =
-      PiTensorProduct.dualDistribEquiv
-        (PiTensorProduct.dualDistribEquiv.symm (l.comp (project n V))) x
-    simp only [LinearEquiv.apply_symm_apply, LinearMap.comp_apply, project_perm]⟩
-
-lemma pairing_pairingInv (l : Dual ℂ (Sym n V)) : pairing n (pairingInv n l) = l := by
-  ext x
-  change PiTensorProduct.dualDistribEquiv
-    (PiTensorProduct.dualDistribEquiv.symm (l.comp (project n V))) (x : T n V) = l x
-  rw [LinearEquiv.apply_symm_apply]
-  exact congrArg l (project_coe n V x)
-
-omit [FiniteDimensional ℂ V] in
-lemma pair_project (d : Sym n (Dual ℂ V)) (x : T n V) :
-    pairing n d (project n V x) = PiTensorProduct.dualDistrib (d : T n (Dual ℂ V)) x := by
-  have hinv (σ : Equiv.Perm (Fin n)) :
-      PiTensorProduct.dualDistrib (d : T n (Dual ℂ V)) (perm n V σ x) =
-      PiTensorProduct.dualDistrib (d : T n (Dual ℂ V)) x := by
-    have hh := pair_perm n σ.symm (d : T n (Dual ℂ V)) x
-    rw [show perm n (Dual ℂ V) σ.symm d = d from d.property σ.symm] at hh
-    simpa using hh.symm
-  change PiTensorProduct.dualDistrib (d : T n (Dual ℂ V)) (average n V x) = _
-  rw [average_apply, map_smul, map_sum]
-  simp only [hinv, Finset.sum_const, Finset.card_univ, Fintype.card_perm, Fintype.card_fin]
-  have hn : (n.factorial : ℂ) ≠ 0 := by exact_mod_cast Nat.factorial_ne_zero n
-  rw [← Nat.cast_smul_eq_nsmul ℂ, smul_smul, inv_mul_cancel₀ hn, one_smul]
-
-lemma pairing_injective : Function.Injective (pairing n (V := V)) := by
-  intro d e h
-  apply Subtype.ext
-  apply PiTensorProduct.dualDistribEquiv.injective
-  apply LinearMap.ext
-  intro x
-  have hh := DFunLike.congr_fun h (project n V x)
-  change PiTensorProduct.dualDistrib (d : T n (Dual ℂ V)) x =
-    PiTensorProduct.dualDistrib (e : T n (Dual ℂ V)) x
-  simpa only [pair_project] using hh
-
-def dualEquiv : Sym n (Dual ℂ V) ≃ₗ[ℂ] Dual ℂ (Sym n V) :=
-  LinearEquiv.ofBijective (pairing n) ⟨pairing_injective n,
-    fun l => ⟨pairingInv n l, pairing_pairingInv n l⟩⟩
-
-@[simp] lemma dualEquiv_apply (d : Sym n (Dual ℂ V)) (x : Sym n V) :
-    dualEquiv n d x = pairing n d x := rfl
-
-end Foulkes.SymmetricTensor
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.FoulkesSixth.SymTensors
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.SymmetricTensor
+universe u v
+open scoped TensorProduct
+open Module
+variable (n : ℕ) {V : Type u} {W : Type v}
+  [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
+
+lemma pair_perm (σ : Equiv.Perm (Fin n)) (d : T n (Dual ℂ V)) (x : T n V) :
+    PiTensorProduct.dualDistrib (perm n (Dual ℂ V) σ d) x =
+    PiTensorProduct.dualDistrib d (perm n V σ.symm x) := by
+  have h : (PiTensorProduct.dualDistrib (R := ℂ) (M := fun _ : Fin n => V)).comp
+      (perm n (Dual ℂ V) σ).toLinearMap =
+      (perm n V σ.symm).toLinearMap.dualMap.comp PiTensorProduct.dualDistrib := by
+    ext f x
+    simp only [LinearMap.compMultilinearMap_apply, LinearMap.comp_apply, LinearEquiv.coe_coe, perm,
+      PiTensorProduct.reindex_tprod, PiTensorProduct.dualDistrib_apply,
+      LinearMap.dualMap_apply, Equiv.symm_symm]
+    have hp := Equiv.prod_comp σ.symm (fun i => f i (x (σ i)))
+    convert hp using 1
+    apply Finset.prod_congr rfl
+    intro i hi
+    rw [σ.apply_symm_apply]
+  exact DFunLike.congr_fun (DFunLike.congr_fun h d) x
+
+lemma pair_map (f : V →ₗ[ℂ] W) (d : T n (Dual ℂ W)) (x : T n V) :
+    PiTensorProduct.dualDistrib (PiTensorProduct.map (fun _ : Fin n => f.dualMap) d) x =
+    PiTensorProduct.dualDistrib d (PiTensorProduct.map (fun _ : Fin n => f) x) := by
+  have h : (PiTensorProduct.dualDistrib (R := ℂ) (M := fun _ : Fin n => V)).comp
+      (PiTensorProduct.map (fun _ : Fin n => f.dualMap)) =
+      (PiTensorProduct.map (fun _ : Fin n => f)).dualMap.comp PiTensorProduct.dualDistrib := by
+    ext d x
+    simp
+  exact DFunLike.congr_fun (DFunLike.congr_fun h d) x
+
+def pairing : Sym n (Dual ℂ V) →ₗ[ℂ] Dual ℂ (Sym n V) :=
+  (symmetricTensors n V).subtype.dualMap.comp
+    (PiTensorProduct.dualDistrib.comp (symmetricTensors n (Dual ℂ V)).subtype)
+
+@[simp] lemma pairing_apply (d : Sym n (Dual ℂ V)) (x : Sym n V) :
+    pairing n d x = PiTensorProduct.dualDistrib (d : T n (Dual ℂ V)) (x : T n V) := rfl
+
+@[simp] lemma pairing_power (d : Dual ℂ V) (v : V) :
+    pairing n (power n (Dual ℂ V) d) (power n V v) = d v ^ n := by
+  simp [pairing_apply, PiTensorProduct.dualDistrib_apply]
+
+lemma pairing_map (f : V →ₗ[ℂ] W) (d : Sym n (Dual ℂ W)) (x : Sym n V) :
+    pairing n (symMap n f.dualMap d) x = pairing n d (symMap n f x) :=
+  pair_map n f d x
+
+variable [FiniteDimensional ℂ V]
+
+def pairingInv (l : Dual ℂ (Sym n V)) : Sym n (Dual ℂ V) :=
+  ⟨PiTensorProduct.dualDistribEquiv.symm (l.comp (project n V)), by
+    intro σ
+    apply PiTensorProduct.dualDistribEquiv.injective
+    apply LinearMap.ext
+    intro x
+    change PiTensorProduct.dualDistrib
+      (perm n (Dual ℂ V) σ (PiTensorProduct.dualDistribEquiv.symm (l.comp (project n V)))) x =
+      PiTensorProduct.dualDistrib
+        (PiTensorProduct.dualDistribEquiv.symm (l.comp (project n V))) x
+    rw [pair_perm]
+    change PiTensorProduct.dualDistribEquiv
+      (PiTensorProduct.dualDistribEquiv.symm (l.comp (project n V))) (perm n V σ.symm x) =
+      PiTensorProduct.dualDistribEquiv
+        (PiTensorProduct.dualDistribEquiv.symm (l.comp (project n V))) x
+    simp only [LinearEquiv.apply_symm_apply, LinearMap.comp_apply, project_perm]⟩
+
+lemma pairing_pairingInv (l : Dual ℂ (Sym n V)) : pairing n (pairingInv n l) = l := by
+  ext x
+  change PiTensorProduct.dualDistribEquiv
+    (PiTensorProduct.dualDistribEquiv.symm (l.comp (project n V))) (x : T n V) = l x
+  rw [LinearEquiv.apply_symm_apply]
+  exact congrArg l (project_coe n V x)
+
+omit [FiniteDimensional ℂ V] in
+lemma pair_project (d : Sym n (Dual ℂ V)) (x : T n V) :
+    pairing n d (project n V x) = PiTensorProduct.dualDistrib (d : T n (Dual ℂ V)) x := by
+  have hinv (σ : Equiv.Perm (Fin n)) :
+      PiTensorProduct.dualDistrib (d : T n (Dual ℂ V)) (perm n V σ x) =
+      PiTensorProduct.dualDistrib (d : T n (Dual ℂ V)) x := by
+    have hh := pair_perm n σ.symm (d : T n (Dual ℂ V)) x
+    rw [show perm n (Dual ℂ V) σ.symm d = d from d.property σ.symm] at hh
+    simpa using hh.symm
+  change PiTensorProduct.dualDistrib (d : T n (Dual ℂ V)) (average n V x) = _
+  rw [average_apply, map_smul, map_sum]
+  simp only [hinv, Finset.sum_const, Finset.card_univ, Fintype.card_perm, Fintype.card_fin]
+  have hn : (n.factorial : ℂ) ≠ 0 := by exact_mod_cast Nat.factorial_ne_zero n
+  rw [← Nat.cast_smul_eq_nsmul ℂ, smul_smul, inv_mul_cancel₀ hn, one_smul]
+
+lemma pairing_injective : Function.Injective (pairing n (V := V)) := by
+  intro d e h
+  apply Subtype.ext
+  apply PiTensorProduct.dualDistribEquiv.injective
+  apply LinearMap.ext
+  intro x
+  have hh := DFunLike.congr_fun h (project n V x)
+  change PiTensorProduct.dualDistrib (d : T n (Dual ℂ V)) x =
+    PiTensorProduct.dualDistrib (e : T n (Dual ℂ V)) x
+  simpa only [pair_project] using hh
+
+def dualEquiv : Sym n (Dual ℂ V) ≃ₗ[ℂ] Dual ℂ (Sym n V) :=
+  LinearEquiv.ofBijective (pairing n) ⟨pairing_injective n,
+    fun l => ⟨pairingInv n l, pairing_pairingInv n l⟩⟩
+
+@[simp] lemma dualEquiv_apply (d : Sym n (Dual ℂ V)) (x : Sym n V) :
+    dualEquiv n d x = pairing n d x := rfl
+
+end Foulkes.SymmetricTensor
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/SymOperations.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/SymOperations.lean
index 04566fbdd66060810ee0a8c50b28dac0bc11645e..db222369f20021b0b1ef3d6abd14cd6984583200 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/SymOperations.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/SymOperations.lean
@@ -1,86 +1,86 @@
-import Mathlib
-import OAI.RepresentationTheory.FoulkesSixth.SymDual
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes
-universe u v
-open scoped TensorProduct
-open SymmetricTensor
-
-variable {V : Type u} {W : Type v} [AddCommGroup V] [Module ℂ V]
-  [AddCommGroup W] [Module ℂ W]
-
-def symEquiv (n : ℕ) (e : V ≃ₗ[ℂ] W) : Sym n V ≃ₗ[ℂ] Sym n W :=
-  LinearEquiv.ofLinearMap (symMap n e.toLinearMap) (symMap n e.symm.toLinearMap)
-    (by apply symMap_ext; intro v; simp)
-    (by apply symMap_ext; intro v; simp)
-
-@[simp] lemma symEquiv_apply (n : ℕ) (e : V ≃ₗ[ℂ] W) (x : Sym n V) :
-    symEquiv n e x = symMap n e.toLinearMap x := rfl
-
-namespace SymmetricTensor
-
-def oneEquiv (V : Type u) [AddCommGroup V] [Module ℂ V] : Sym 1 V ≃ₗ[ℂ] V :=
-  LinearEquiv.ofBijective
-    ((PiTensorProduct.subsingletonEquiv (R := ℂ) (s := fun _ : Fin 1 => V) 0).toLinearMap.comp
-      (symmetricTensors 1 V).subtype)
-    ⟨(PiTensorProduct.subsingletonEquiv (R := ℂ) (s := fun _ : Fin 1 => V) 0).injective.comp
-        Subtype.val_injective,
-      fun v => ⟨power 1 V v, by simp [power]⟩⟩
-
-@[simp] lemma oneEquiv_power (v : V) : oneEquiv V (power 1 V v) = v := by
-  change PiTensorProduct.subsingletonEquiv (R := ℂ) (s := fun _ : Fin 1 => V) 0
-    (PiTensorProduct.tprod ℂ (fun _ : Fin 1 => v)) = v
-  simp
-
-@[simp] lemma oneEquiv_symm (v : V) : (oneEquiv V).symm v = power 1 V v := by
-  apply (oneEquiv V).injective
-  simp
-
-end SymmetricTensor
-
-namespace SymAlgebra
-variable {A : Type*} [CommRing A] [Algebra ℂ A]
-
-def tensorMul (b : ℕ) (q : V →ₗ[ℂ] A) : T b V →ₗ[ℂ] A :=
-  PiTensorProduct.lift ((MultilinearMap.mkPiAlgebra ℂ (Fin b) A).compLinearMap (fun _ => q))
-
-@[simp] lemma tensorMul_tprod (b : ℕ) (q : V →ₗ[ℂ] A) (v : Fin b → V) :
-    tensorMul b q (PiTensorProduct.tprod ℂ v) = ∏ i, q (v i) := by
-  simp [tensorMul]
-
-lemma tensorMul_perm (b : ℕ) (q : V →ₗ[ℂ] A) (σ : Equiv.Perm (Fin b)) (x : T b V) :
-    tensorMul b q (perm b V σ x) = tensorMul b q x := by
-  have h : (tensorMul b q).comp (perm b V σ).toLinearMap = tensorMul b q := by
-    apply PiTensorProduct.ext
-    apply MultilinearMap.ext
-    intro v
-    simp only [LinearMap.compMultilinearMap_apply, LinearMap.comp_apply, LinearEquiv.coe_coe,
-      perm, PiTensorProduct.reindex_tprod, tensorMul_tprod]
-    exact Equiv.prod_comp σ.symm (fun i => q (v i))
-  exact DFunLike.congr_fun h x
-
-def multiply (b : ℕ) (q : V →ₗ[ℂ] A) : Sym b V →ₗ[ℂ] A :=
-  (tensorMul b q).comp (symmetricTensors b V).subtype
-
-@[simp] lemma multiply_power (b : ℕ) (q : V →ₗ[ℂ] A) (v : V) :
-    multiply b q (power b V v) = q v ^ b := by
-  simp [multiply, power]
-
-lemma multiply_project (b : ℕ) (q : V →ₗ[ℂ] A) (x : T b V) :
-    multiply b q (project b V x) = tensorMul b q x := by
-  change tensorMul b q (average b V x) = _
-  simp only [average_apply, map_smul, map_sum, tensorMul_perm, Finset.sum_const,
-    Finset.card_univ, Fintype.card_perm, Fintype.card_fin]
-  have hb : (b.factorial : ℂ) ≠ 0 := by exact_mod_cast Nat.factorial_ne_zero b
-  rw [← Nat.cast_smul_eq_nsmul ℂ, smul_smul, inv_mul_cancel₀ hb, one_smul]
-
-end SymAlgebra
-end Foulkes
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.FoulkesSixth.SymDual
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes
+universe u v
+open scoped TensorProduct
+open SymmetricTensor
+
+variable {V : Type u} {W : Type v} [AddCommGroup V] [Module ℂ V]
+  [AddCommGroup W] [Module ℂ W]
+
+def symEquiv (n : ℕ) (e : V ≃ₗ[ℂ] W) : Sym n V ≃ₗ[ℂ] Sym n W :=
+  LinearEquiv.ofLinearMap (symMap n e.toLinearMap) (symMap n e.symm.toLinearMap)
+    (by apply symMap_ext; intro v; simp)
+    (by apply symMap_ext; intro v; simp)
+
+@[simp] lemma symEquiv_apply (n : ℕ) (e : V ≃ₗ[ℂ] W) (x : Sym n V) :
+    symEquiv n e x = symMap n e.toLinearMap x := rfl
+
+namespace SymmetricTensor
+
+def oneEquiv (V : Type u) [AddCommGroup V] [Module ℂ V] : Sym 1 V ≃ₗ[ℂ] V :=
+  LinearEquiv.ofBijective
+    ((PiTensorProduct.subsingletonEquiv (R := ℂ) (s := fun _ : Fin 1 => V) 0).toLinearMap.comp
+      (symmetricTensors 1 V).subtype)
+    ⟨(PiTensorProduct.subsingletonEquiv (R := ℂ) (s := fun _ : Fin 1 => V) 0).injective.comp
+        Subtype.val_injective,
+      fun v => ⟨power 1 V v, by simp [power]⟩⟩
+
+@[simp] lemma oneEquiv_power (v : V) : oneEquiv V (power 1 V v) = v := by
+  change PiTensorProduct.subsingletonEquiv (R := ℂ) (s := fun _ : Fin 1 => V) 0
+    (PiTensorProduct.tprod ℂ (fun _ : Fin 1 => v)) = v
+  simp
+
+@[simp] lemma oneEquiv_symm (v : V) : (oneEquiv V).symm v = power 1 V v := by
+  apply (oneEquiv V).injective
+  simp
+
+end SymmetricTensor
+
+namespace SymAlgebra
+variable {A : Type*} [CommRing A] [Algebra ℂ A]
+
+def tensorMul (b : ℕ) (q : V →ₗ[ℂ] A) : T b V →ₗ[ℂ] A :=
+  PiTensorProduct.lift ((MultilinearMap.mkPiAlgebra ℂ (Fin b) A).compLinearMap (fun _ => q))
+
+@[simp] lemma tensorMul_tprod (b : ℕ) (q : V →ₗ[ℂ] A) (v : Fin b → V) :
+    tensorMul b q (PiTensorProduct.tprod ℂ v) = ∏ i, q (v i) := by
+  simp [tensorMul]
+
+lemma tensorMul_perm (b : ℕ) (q : V →ₗ[ℂ] A) (σ : Equiv.Perm (Fin b)) (x : T b V) :
+    tensorMul b q (perm b V σ x) = tensorMul b q x := by
+  have h : (tensorMul b q).comp (perm b V σ).toLinearMap = tensorMul b q := by
+    apply PiTensorProduct.ext
+    apply MultilinearMap.ext
+    intro v
+    simp only [LinearMap.compMultilinearMap_apply, LinearMap.comp_apply, LinearEquiv.coe_coe,
+      perm, PiTensorProduct.reindex_tprod, tensorMul_tprod]
+    exact Equiv.prod_comp σ.symm (fun i => q (v i))
+  exact DFunLike.congr_fun h x
+
+def multiply (b : ℕ) (q : V →ₗ[ℂ] A) : Sym b V →ₗ[ℂ] A :=
+  (tensorMul b q).comp (symmetricTensors b V).subtype
+
+@[simp] lemma multiply_power (b : ℕ) (q : V →ₗ[ℂ] A) (v : V) :
+    multiply b q (power b V v) = q v ^ b := by
+  simp [multiply, power]
+
+lemma multiply_project (b : ℕ) (q : V →ₗ[ℂ] A) (x : T b V) :
+    multiply b q (project b V x) = tensorMul b q x := by
+  change tensorMul b q (average b V x) = _
+  simp only [average_apply, map_smul, map_sum, tensorMul_perm, Finset.sum_const,
+    Finset.card_univ, Fintype.card_perm, Fintype.card_fin]
+  have hb : (b.factorial : ℂ) ≠ 0 := by exact_mod_cast Nat.factorial_ne_zero b
+  rw [← Nat.cast_smul_eq_nsmul ℂ, smul_smul, inv_mul_cancel₀ hb, one_smul]
+
+end SymAlgebra
+end Foulkes
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/SymPolynomial.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/SymPolynomial.lean
index 93369db7622ba32769790e3f3e2d744f472bbaa4..77a053dff71eb07bde0394a0b9819ddda6c57760 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/SymPolynomial.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/SymPolynomial.lean
@@ -1,101 +1,101 @@
-import Mathlib
-import OAI.RepresentationTheory.FoulkesSixth.PolynomialSpan
-import OAI.RepresentationTheory.FoulkesSixth.SymDual
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.SymPolynomial
-universe u
-open Module MvPolynomial SymmetricTensor
-variable {V : Type u} [AddCommGroup V] [Module ℂ V]
-variable {n : ℕ} (B : Basis (Fin n) ℂ V) (b : ℕ)
-
-abbrev Poly (n : ℕ) := MvPolynomial (Fin n) ℂ
-abbrev H (n b : ℕ) := homogeneousSubmodule (Fin n) ℂ b
-
-def linearPolynomial : V →ₗ[ℂ] Poly n where
-  toFun v := ∑ i, B.equivFun v i • X i
-  map_add' v w := by simp [add_smul, Finset.sum_add_distrib]
-  map_smul' c v := by simp [smul_smul, Finset.smul_sum]
-
-@[simp] lemma linearPolynomial_apply (v : V) :
-    linearPolynomial B v = ∑ i, B.equivFun v i • X i := rfl
-
-lemma eval_linearPolynomial (x : Fin n → ℂ) (v : V) :
-    eval x (linearPolynomial B v) = B.constr ℂ x v := by
-  have h : B.constr ℂ x v = ∑ i, B.equivFun v i * x i := by
-    conv_lhs => rw [← B.sum_equivFun v]
-    simp
-  rw [h]
-  simp [linearPolynomial, MvPolynomial.smul_eq_C_mul]
-
-def tensorPolynomial : T b V →ₗ[ℂ] Poly n :=
-  PiTensorProduct.lift ((MultilinearMap.mkPiAlgebra ℂ (Fin b) (Poly n)).compLinearMap
-    (fun _ => linearPolynomial B))
-
-@[simp] lemma tensorPolynomial_tprod (v : Fin b → V) :
-    tensorPolynomial B b (PiTensorProduct.tprod ℂ v) = ∏ i, linearPolynomial B (v i) := by
-  simp [tensorPolynomial]
-
-def polynomial : Sym b V →ₗ[ℂ] Poly n :=
-  (tensorPolynomial B b).comp (symmetricTensors b V).subtype
-
-@[simp] lemma polynomial_power (v : V) :
-    polynomial B b (power b V v) = linearPolynomial B v ^ b := by
-  simp [polynomial, power]
-
-lemma polynomial_eval (x : Fin n → ℂ) (v : Sym b V) :
-    eval x (polynomial B b v) = pairing b (power b (Dual ℂ V) (B.constr ℂ x)) v := by
-  have hh : (aeval x).toLinearMap.comp (polynomial B b) =
-      pairing b (power b (Dual ℂ V) (B.constr ℂ x)) := by
-    apply symMap_ext
-    intro v
-    change eval x (polynomial B b (power b V v)) = _
-    rw [polynomial_power, eval_pow, eval_linearPolynomial, pairing_power]
-  exact DFunLike.congr_fun hh v
-
-variable [FiniteDimensional ℂ V]
-
-lemma polynomial_injective : Function.Injective (polynomial B b) := by
-  apply (LinearMap.ker_eq_bot).mp
-  apply (LinearMap.ker_eq_bot').mpr
-  intro v hv
-  apply (Module.forall_dual_apply_eq_zero_iff ℂ v).mp
-  intro l
-  have hp : (pairing b (V := V)).flip v = 0 := by
-    apply symMap_ext
-    intro d
-    obtain ⟨x, rfl⟩ := (B.constr ℂ).surjective d
-    change pairing b (power b (Dual ℂ V) (B.constr ℂ x)) v = 0
-    rw [← polynomial_eval, hv, map_zero]
-  have hh := DFunLike.congr_fun hp ((dualEquiv b (V := V)).symm l)
-  change dualEquiv b ((dualEquiv b (V := V)).symm l) v = 0 at hh
-  rwa [LinearEquiv.apply_symm_apply] at hh
-
-omit [FiniteDimensional ℂ V] in
-lemma polynomial_range : (polynomial B b).range = H n b := by
-  change (polynomial B b).range = homogeneousSubmodule (Fin n) ℂ b
-  rw [← Submodule.map_top, ← span_power b V, LinearMap.map_span,
-    PolynomialSpan.homogeneous_eq_span_powers]
-  congr 1
-  ext p
-  constructor
-  · rintro ⟨_, ⟨v, rfl⟩, rfl⟩
-    exact ⟨B.equivFun v, (polynomial_power B b v).symm⟩
-  · rintro ⟨x, rfl⟩
-    refine ⟨power b V (B.equivFun.symm x), ⟨_, rfl⟩, ?_⟩
-    simp
-
-def equiv : Sym b V ≃ₗ[ℂ] H n b :=
-  (LinearEquiv.ofInjective (polynomial B b) (polynomial_injective B b)).trans
-    (LinearEquiv.ofEq _ _ (polynomial_range B b))
-
-@[simp] lemma equiv_coe (v : Sym b V) : (equiv B b v : Poly n) = polynomial B b v := rfl
-
-end Foulkes.SymPolynomial
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.FoulkesSixth.PolynomialSpan
+import OAI.RepresentationTheory.FoulkesSixth.SymDual
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.SymPolynomial
+universe u
+open Module MvPolynomial SymmetricTensor
+variable {V : Type u} [AddCommGroup V] [Module ℂ V]
+variable {n : ℕ} (B : Basis (Fin n) ℂ V) (b : ℕ)
+
+abbrev Poly (n : ℕ) := MvPolynomial (Fin n) ℂ
+abbrev H (n b : ℕ) := homogeneousSubmodule (Fin n) ℂ b
+
+def linearPolynomial : V →ₗ[ℂ] Poly n where
+  toFun v := ∑ i, B.equivFun v i • X i
+  map_add' v w := by simp [add_smul, Finset.sum_add_distrib]
+  map_smul' c v := by simp [smul_smul, Finset.smul_sum]
+
+@[simp] lemma linearPolynomial_apply (v : V) :
+    linearPolynomial B v = ∑ i, B.equivFun v i • X i := rfl
+
+lemma eval_linearPolynomial (x : Fin n → ℂ) (v : V) :
+    eval x (linearPolynomial B v) = B.constr ℂ x v := by
+  have h : B.constr ℂ x v = ∑ i, B.equivFun v i * x i := by
+    conv_lhs => rw [← B.sum_equivFun v]
+    simp
+  rw [h]
+  simp [linearPolynomial, MvPolynomial.smul_eq_C_mul]
+
+def tensorPolynomial : T b V →ₗ[ℂ] Poly n :=
+  PiTensorProduct.lift ((MultilinearMap.mkPiAlgebra ℂ (Fin b) (Poly n)).compLinearMap
+    (fun _ => linearPolynomial B))
+
+@[simp] lemma tensorPolynomial_tprod (v : Fin b → V) :
+    tensorPolynomial B b (PiTensorProduct.tprod ℂ v) = ∏ i, linearPolynomial B (v i) := by
+  simp [tensorPolynomial]
+
+def polynomial : Sym b V →ₗ[ℂ] Poly n :=
+  (tensorPolynomial B b).comp (symmetricTensors b V).subtype
+
+@[simp] lemma polynomial_power (v : V) :
+    polynomial B b (power b V v) = linearPolynomial B v ^ b := by
+  simp [polynomial, power]
+
+lemma polynomial_eval (x : Fin n → ℂ) (v : Sym b V) :
+    eval x (polynomial B b v) = pairing b (power b (Dual ℂ V) (B.constr ℂ x)) v := by
+  have hh : (aeval x).toLinearMap.comp (polynomial B b) =
+      pairing b (power b (Dual ℂ V) (B.constr ℂ x)) := by
+    apply symMap_ext
+    intro v
+    change eval x (polynomial B b (power b V v)) = _
+    rw [polynomial_power, eval_pow, eval_linearPolynomial, pairing_power]
+  exact DFunLike.congr_fun hh v
+
+variable [FiniteDimensional ℂ V]
+
+lemma polynomial_injective : Function.Injective (polynomial B b) := by
+  apply (LinearMap.ker_eq_bot).mp
+  apply (LinearMap.ker_eq_bot').mpr
+  intro v hv
+  apply (Module.forall_dual_apply_eq_zero_iff ℂ v).mp
+  intro l
+  have hp : (pairing b (V := V)).flip v = 0 := by
+    apply symMap_ext
+    intro d
+    obtain ⟨x, rfl⟩ := (B.constr ℂ).surjective d
+    change pairing b (power b (Dual ℂ V) (B.constr ℂ x)) v = 0
+    rw [← polynomial_eval, hv, map_zero]
+  have hh := DFunLike.congr_fun hp ((dualEquiv b (V := V)).symm l)
+  change dualEquiv b ((dualEquiv b (V := V)).symm l) v = 0 at hh
+  rwa [LinearEquiv.apply_symm_apply] at hh
+
+omit [FiniteDimensional ℂ V] in
+lemma polynomial_range : (polynomial B b).range = H n b := by
+  change (polynomial B b).range = homogeneousSubmodule (Fin n) ℂ b
+  rw [← Submodule.map_top, ← span_power b V, LinearMap.map_span,
+    PolynomialSpan.homogeneous_eq_span_powers]
+  congr 1
+  ext p
+  constructor
+  · rintro ⟨_, ⟨v, rfl⟩, rfl⟩
+    exact ⟨B.equivFun v, (polynomial_power B b v).symm⟩
+  · rintro ⟨x, rfl⟩
+    refine ⟨power b V (B.equivFun.symm x), ⟨_, rfl⟩, ?_⟩
+    simp
+
+def equiv : Sym b V ≃ₗ[ℂ] H n b :=
+  (LinearEquiv.ofInjective (polynomial B b) (polynomial_injective B b)).trans
+    (LinearEquiv.ofEq _ _ (polynomial_range B b))
+
+@[simp] lemma equiv_coe (v : Sym b V) : (equiv B b v : Poly n) = polynomial B b v := rfl
+
+end Foulkes.SymPolynomial
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/SymTensors.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/SymTensors.lean
index c0ffb69f50c1d45322f57a5e48d6c2a95b009a70..4102c7a1dc48ca9e8d4f33182a7943f123a0b5a8 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/SymTensors.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/SymTensors.lean
@@ -1,153 +1,153 @@
-import Mathlib
-import OAI.RepresentationTheory.FoulkesSixth.Diagonal
-
-namespace OAI
-
-noncomputable section
-
-open scoped TensorProduct
-namespace Foulkes
-universe u v w
-
-def symmetricTensors (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] :
-    Submodule ℂ (⨂[ℂ] (_ : Fin n), V) where
-  carrier := {x | ∀ σ : Equiv.Perm (Fin n),
-    PiTensorProduct.reindex ℂ (fun _ : Fin n => V) σ x = x}
-  zero_mem' := by intro σ; simp
-  add_mem' := by intro x y hx hy σ; simp only [map_add, hx σ, hy σ]
-  smul_mem' := by intro c x hx σ; simp only [map_smul, hx σ]
-
-abbrev Sym (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] :=
-  symmetricTensors n V
-
-def symMap (n : ℕ) {V : Type u} {W : Type v}
-    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
-    (f : V →ₗ[ℂ] W) : Sym n V →ₗ[ℂ] Sym n W :=
-  (PiTensorProduct.map (fun _ : Fin n => f)).restrict fun x hx σ => by
-    rw [← PiTensorProduct.map_reindex]
-    rw [hx σ]
-
-namespace SymmetricTensor
-
-variable (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]
-
-abbrev T := ⨂[ℂ] (_ : Fin n), V
-
-def perm (σ : Equiv.Perm (Fin n)) : T n V ≃ₗ[ℂ] T n V :=
-  PiTensorProduct.reindex ℂ (fun _ : Fin n => V) σ
-
-lemma perm_mul (σ τ : Equiv.Perm (Fin n)) (x : T n V) :
-    perm n V σ (perm n V τ x) = perm n V (σ * τ) x := by
-  exact PiTensorProduct.reindex_reindex _ _ _
-
-def average : T n V →ₗ[ℂ] T n V :=
-  (n.factorial : ℂ)⁻¹ • ∑ σ : Equiv.Perm (Fin n), (perm n V σ).toLinearMap
-
-lemma average_apply (x : T n V) :
-    average n V x = (n.factorial : ℂ)⁻¹ • ∑ σ : Equiv.Perm (Fin n), perm n V σ x := by
-  simp [average]
-
-lemma average_invariant (x : T n V) : average n V x ∈ Sym n V := by
-  intro τ
-  change perm n V τ (average n V x) = average n V x
-  simp only [average_apply, map_smul, map_sum, perm_mul]
-  congr 1
-  exact Equiv.sum_comp (Equiv.mulLeft τ) (fun σ => perm n V σ x)
-
-lemma average_perm (σ : Equiv.Perm (Fin n)) (x : T n V) :
-    average n V (perm n V σ x) = average n V x := by
-  simp only [average_apply, perm_mul]
-  congr 1
-  exact Equiv.sum_comp (Equiv.mulRight σ) (fun τ => perm n V τ x)
-
-lemma average_eq {x : T n V} (hx : x ∈ Sym n V) : average n V x = x := by
-  change ∀ σ : Equiv.Perm (Fin n), perm n V σ x = x at hx
-  have hn : (n.factorial : ℂ) ≠ 0 := by exact_mod_cast Nat.factorial_ne_zero n
-  simp only [average_apply, hx, Finset.sum_const, Finset.card_univ,
-    Fintype.card_perm, Fintype.card_fin]
-  rw [← Nat.cast_smul_eq_nsmul ℂ, smul_smul, inv_mul_cancel₀ hn, one_smul]
-
-def project : T n V →ₗ[ℂ] Sym n V :=
-  (average n V).codRestrict _ (average_invariant n V)
-
-@[simp] lemma project_coe (x : Sym n V) : project n V x = x := by
-  apply Subtype.ext
-  exact average_eq n V x.property
-
-@[simp] lemma project_perm (σ : Equiv.Perm (Fin n)) (x : T n V) :
-    project n V (perm n V σ x) = project n V x := by
-  apply Subtype.ext
-  exact average_perm n V σ x
-
-def power (v : V) : Sym n V :=
-  ⟨PiTensorProduct.tprod ℂ (fun _ : Fin n => v), by intro σ; simp⟩
-
-@[simp] lemma coe_power (v : V) :
-    (power n V v : T n V) = PiTensorProduct.tprod ℂ (fun _ : Fin n => v) := rfl
-
-@[simp] lemma project_power (v : V) :
-    project n V (PiTensorProduct.tprod ℂ (fun _ : Fin n => v)) = power n V v :=
-  project_coe n V (power n V v)
-
-theorem span_power : Submodule.span ℂ (Set.range (power n V)) = ⊤ := by
-  apply top_unique
-  intro x hx
-  by_contra hnot
-  obtain ⟨l, hne, hl⟩ := Submodule.exists_dual_map_eq_bot_of_notMem hnot inferInstance
-  have hlpower (v : V) : l (power n V v) = 0 := by
-    have hh : l (power n V v) ∈ (Submodule.span ℂ (Set.range (power n V))).map l :=
-      Submodule.mem_map.mpr ⟨_, Submodule.subset_span ⟨v, rfl⟩, rfl⟩
-    rwa [hl, Submodule.mem_bot] at hh
-  let F : MultilinearMap ℂ (fun _ : Fin n => V) ℂ :=
-    (l.comp (project n V)).compMultilinearMap (PiTensorProduct.tprod ℂ)
-  have hF : F = 0 := by
-    apply Diagonal.eq_zero_of_symmetric_of_diagonal
-    · intro σ v
-      change l (project n V (PiTensorProduct.tprod ℂ (fun i => v (σ i)))) =
-        l (project n V (PiTensorProduct.tprod ℂ v))
-      simpa [perm] using congrArg l
-        (project_perm n V σ.symm (PiTensorProduct.tprod ℂ v))
-    · intro v
-      change l (project n V (PiTensorProduct.tprod ℂ (fun _ : Fin n => v))) = 0
-      rw [project_power, hlpower]
-  have hlin : l.comp (project n V) = 0 := by
-    apply PiTensorProduct.ext
-    exact hF
-  exact hne (by have hh := DFunLike.congr_fun hlin (x : T n V); simpa using hh)
-
-end SymmetricTensor
-
-@[simp] lemma symMap_power (n : ℕ) {V : Type u} {W : Type v}
-    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
-    (f : V →ₗ[ℂ] W) (v : V) :
-    symMap n f (SymmetricTensor.power n V v) = SymmetricTensor.power n W (f v) := by
-  apply Subtype.ext
-  exact PiTensorProduct.map_tprod _ _
-
-lemma symMap_ext (n : ℕ) {V : Type u} {W : Type v}
-    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
-    {f g : Sym n V →ₗ[ℂ] W}
-    (h : ∀ v : V, f (SymmetricTensor.power n V v) = g (SymmetricTensor.power n V v)) : f = g := by
-  apply (Submodule.linearMap_eq_iff_of_span_eq_top _ _ (SymmetricTensor.span_power n V)).mpr
-  rintro ⟨_, ⟨v, rfl⟩⟩
-  exact h v
-
-@[simp] lemma symMap_comp (n : ℕ) {V : Type u} {W : Type v} {U : Type w}
-    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
-    [AddCommGroup U] [Module ℂ U] (g : W →ₗ[ℂ] U) (f : V →ₗ[ℂ] W) :
-    symMap n (g.comp f) = (symMap n g).comp (symMap n f) := by
-  apply symMap_ext
-  intro v
-  simp
-
-@[simp] lemma symMap_id (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] :
-    symMap n (LinearMap.id : V →ₗ[ℂ] V) = LinearMap.id := by
-  apply symMap_ext
-  intro v
-  simp
-
-end Foulkes
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.FoulkesSixth.Diagonal
+
+namespace OAI
+
+noncomputable section
+
+open scoped TensorProduct
+namespace Foulkes
+universe u v w
+
+def symmetricTensors (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] :
+    Submodule ℂ (⨂[ℂ] (_ : Fin n), V) where
+  carrier := {x | ∀ σ : Equiv.Perm (Fin n),
+    PiTensorProduct.reindex ℂ (fun _ : Fin n => V) σ x = x}
+  zero_mem' := by intro σ; simp
+  add_mem' := by intro x y hx hy σ; simp only [map_add, hx σ, hy σ]
+  smul_mem' := by intro c x hx σ; simp only [map_smul, hx σ]
+
+abbrev Sym (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] :=
+  symmetricTensors n V
+
+def symMap (n : ℕ) {V : Type u} {W : Type v}
+    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
+    (f : V →ₗ[ℂ] W) : Sym n V →ₗ[ℂ] Sym n W :=
+  (PiTensorProduct.map (fun _ : Fin n => f)).restrict fun x hx σ => by
+    rw [← PiTensorProduct.map_reindex]
+    rw [hx σ]
+
+namespace SymmetricTensor
+
+variable (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]
+
+abbrev T := ⨂[ℂ] (_ : Fin n), V
+
+def perm (σ : Equiv.Perm (Fin n)) : T n V ≃ₗ[ℂ] T n V :=
+  PiTensorProduct.reindex ℂ (fun _ : Fin n => V) σ
+
+lemma perm_mul (σ τ : Equiv.Perm (Fin n)) (x : T n V) :
+    perm n V σ (perm n V τ x) = perm n V (σ * τ) x := by
+  exact PiTensorProduct.reindex_reindex _ _ _
+
+def average : T n V →ₗ[ℂ] T n V :=
+  (n.factorial : ℂ)⁻¹ • ∑ σ : Equiv.Perm (Fin n), (perm n V σ).toLinearMap
+
+lemma average_apply (x : T n V) :
+    average n V x = (n.factorial : ℂ)⁻¹ • ∑ σ : Equiv.Perm (Fin n), perm n V σ x := by
+  simp [average]
+
+lemma average_invariant (x : T n V) : average n V x ∈ Sym n V := by
+  intro τ
+  change perm n V τ (average n V x) = average n V x
+  simp only [average_apply, map_smul, map_sum, perm_mul]
+  congr 1
+  exact Equiv.sum_comp (Equiv.mulLeft τ) (fun σ => perm n V σ x)
+
+lemma average_perm (σ : Equiv.Perm (Fin n)) (x : T n V) :
+    average n V (perm n V σ x) = average n V x := by
+  simp only [average_apply, perm_mul]
+  congr 1
+  exact Equiv.sum_comp (Equiv.mulRight σ) (fun τ => perm n V τ x)
+
+lemma average_eq {x : T n V} (hx : x ∈ Sym n V) : average n V x = x := by
+  change ∀ σ : Equiv.Perm (Fin n), perm n V σ x = x at hx
+  have hn : (n.factorial : ℂ) ≠ 0 := by exact_mod_cast Nat.factorial_ne_zero n
+  simp only [average_apply, hx, Finset.sum_const, Finset.card_univ,
+    Fintype.card_perm, Fintype.card_fin]
+  rw [← Nat.cast_smul_eq_nsmul ℂ, smul_smul, inv_mul_cancel₀ hn, one_smul]
+
+def project : T n V →ₗ[ℂ] Sym n V :=
+  (average n V).codRestrict _ (average_invariant n V)
+
+@[simp] lemma project_coe (x : Sym n V) : project n V x = x := by
+  apply Subtype.ext
+  exact average_eq n V x.property
+
+@[simp] lemma project_perm (σ : Equiv.Perm (Fin n)) (x : T n V) :
+    project n V (perm n V σ x) = project n V x := by
+  apply Subtype.ext
+  exact average_perm n V σ x
+
+def power (v : V) : Sym n V :=
+  ⟨PiTensorProduct.tprod ℂ (fun _ : Fin n => v), by intro σ; simp⟩
+
+@[simp] lemma coe_power (v : V) :
+    (power n V v : T n V) = PiTensorProduct.tprod ℂ (fun _ : Fin n => v) := rfl
+
+@[simp] lemma project_power (v : V) :
+    project n V (PiTensorProduct.tprod ℂ (fun _ : Fin n => v)) = power n V v :=
+  project_coe n V (power n V v)
+
+theorem span_power : Submodule.span ℂ (Set.range (power n V)) = ⊤ := by
+  apply top_unique
+  intro x hx
+  by_contra hnot
+  obtain ⟨l, hne, hl⟩ := Submodule.exists_dual_map_eq_bot_of_notMem hnot inferInstance
+  have hlpower (v : V) : l (power n V v) = 0 := by
+    have hh : l (power n V v) ∈ (Submodule.span ℂ (Set.range (power n V))).map l :=
+      Submodule.mem_map.mpr ⟨_, Submodule.subset_span ⟨v, rfl⟩, rfl⟩
+    rwa [hl, Submodule.mem_bot] at hh
+  let F : MultilinearMap ℂ (fun _ : Fin n => V) ℂ :=
+    (l.comp (project n V)).compMultilinearMap (PiTensorProduct.tprod ℂ)
+  have hF : F = 0 := by
+    apply Diagonal.eq_zero_of_symmetric_of_diagonal
+    · intro σ v
+      change l (project n V (PiTensorProduct.tprod ℂ (fun i => v (σ i)))) =
+        l (project n V (PiTensorProduct.tprod ℂ v))
+      simpa [perm] using congrArg l
+        (project_perm n V σ.symm (PiTensorProduct.tprod ℂ v))
+    · intro v
+      change l (project n V (PiTensorProduct.tprod ℂ (fun _ : Fin n => v))) = 0
+      rw [project_power, hlpower]
+  have hlin : l.comp (project n V) = 0 := by
+    apply PiTensorProduct.ext
+    exact hF
+  exact hne (by have hh := DFunLike.congr_fun hlin (x : T n V); simpa using hh)
+
+end SymmetricTensor
+
+@[simp] lemma symMap_power (n : ℕ) {V : Type u} {W : Type v}
+    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
+    (f : V →ₗ[ℂ] W) (v : V) :
+    symMap n f (SymmetricTensor.power n V v) = SymmetricTensor.power n W (f v) := by
+  apply Subtype.ext
+  exact PiTensorProduct.map_tprod _ _
+
+lemma symMap_ext (n : ℕ) {V : Type u} {W : Type v}
+    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
+    {f g : Sym n V →ₗ[ℂ] W}
+    (h : ∀ v : V, f (SymmetricTensor.power n V v) = g (SymmetricTensor.power n V v)) : f = g := by
+  apply (Submodule.linearMap_eq_iff_of_span_eq_top _ _ (SymmetricTensor.span_power n V)).mpr
+  rintro ⟨_, ⟨v, rfl⟩⟩
+  exact h v
+
+@[simp] lemma symMap_comp (n : ℕ) {V : Type u} {W : Type v} {U : Type w}
+    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
+    [AddCommGroup U] [Module ℂ U] (g : W →ₗ[ℂ] U) (f : V →ₗ[ℂ] W) :
+    symMap n (g.comp f) = (symMap n g).comp (symMap n f) := by
+  apply symMap_ext
+  intro v
+  simp
+
+@[simp] lemma symMap_id (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] :
+    symMap n (LinearMap.id : V →ₗ[ℂ] V) = LinearMap.id := by
+  apply symMap_ext
+  intro v
+  simp
+
+end Foulkes
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/TensorPolarization.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/TensorPolarization.lean
index b5b5ebd062dffdc5011effd2a65a8020645e52e2..68bd0ad75eb8bec768c97bdca91f4fba59928e35 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/TensorPolarization.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/TensorPolarization.lean
@@ -1,201 +1,201 @@
-import Mathlib
-import OAI.RepresentationTheory.FoulkesSixth.Differential
-import OAI.RepresentationTheory.FoulkesSixth.GenericShift
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.Polarization
-open MvPolynomial
-open Foulkes.PolynomialSpan
-
-abbrev P (a n : ℕ) := MvPolynomial (Fin a × Fin n) ℂ
-
-def weight (a n : ℕ) (ij : Fin a × Fin n) : Fin a → ℕ := Pi.single ij.1 1
-
-def W (a n j : ℕ) : Submodule ℂ (P a n) :=
-  weightedHomogeneousSubmodule ℂ (weight a n) (fun _ => j)
-
-def rowForm {a n : ℕ} (i : Fin a) (x : Fin n → ℂ) : P a n :=
-  ∑ k, x k • X (i, k)
-
-def pureRows {a n : ℕ} (j : ℕ) (x : Fin a → Fin n → ℂ) : P a n :=
-  ∏ i, (rowForm i (x i)) ^ j
-
-lemma weight_apply_row {a n : ℕ} (d : (Fin a × Fin n) →₀ ℕ) (i : Fin a) :
-    Finsupp.weight (weight a n) d i = ∑ k, d (i, k) := by
-  classical
-  rw [Finsupp.weight_eq_sum]
-  simp only [weight, Finset.sum_apply, Fintype.sum_prod_type, Pi.smul_apply,
-    Pi.single_apply, smul_eq_mul, mul_ite, mul_one, mul_zero]
-  rw [Finset.sum_comm]
-  simp
-
-lemma pureRows_mem {a n j : ℕ} (x : Fin a → Fin n → ℂ) :
-    pureRows j x ∈ W a n j := by
-  classical
-  have hrow (i : Fin a) : (rowForm i (x i)).IsWeightedHomogeneous
-      (weight a n) (Pi.single i 1) := by
-    apply MvPolynomial.IsWeightedHomogeneous.sum
-    intro k _
-    exact (weightedHomogeneousSubmodule ℂ (weight a n) (Pi.single i 1)).smul_mem _
-      (isWeightedHomogeneous_X (R := ℂ) (weight a n) (i, k))
-  have hprod := MvPolynomial.IsWeightedHomogeneous.prod Finset.univ
-    (fun i => (rowForm i (x i)) ^ j) (fun i => j • Pi.single i 1)
-    (fun i _ => (hrow i).pow j)
-  have hdeg : (∑ i : Fin a, j • (Pi.single i 1 : Fin a → ℕ)) = fun _ => j := by
-    ext r
-    simp [Pi.single_apply]
-  rw [hdeg] at hprod
-  exact hprod
-
-theorem W_eq_span_pureRows (a n j : ℕ) :
-    W a n j = Submodule.span ℂ (Set.range (pureRows (a := a) (n := n) j)) := by
-  classical
-  apply le_antisymm
-  · intro p hp
-    induction hp using MvPolynomial.IsWeightedHomogeneous.induction_on with
-    | zero => exact Submodule.zero_mem _
-    | add p q hp hq ihp ihq => exact Submodule.add_mem _ ihp ihq
-    | monomial d c hd =>
-      have hdrow (i : Fin a) : ∑ k, d (i, k) = j := by
-        simpa [weight_apply_row] using congrFun hd i
-      have hx (i : Fin a) : (∏ k : Fin n, (X (i, k) : P a n) ^ d (i, k)) ∈
-          Submodule.span ℂ (Set.range fun x : Fin n → ℂ => (rowForm i x) ^ j) := by
-        let e : Fin n →₀ ℕ := Finsupp.equivFunOnFinite.symm (fun k => d (i, k))
-        have he : e.sum (fun _ v => v) = j := by
-          rw [Finsupp.sum_fintype _ _ (by simp)]
-          exact hdrow i
-        have hh := prod_pow_mem_span_powers (fun k => (X (i, k) : P a n)) e j he
-        rw [Finsupp.prod_fintype _ _ (fun _ => pow_zero _)] at hh
-        simpa [e, rowForm] using hh
-      have hprod := prod_mem_span_range (fun i (x : Fin n → ℂ) => (rowForm i x) ^ j) hx
-      have hm : (monomial d c : P a n) = c • ∏ i : Fin a, ∏ k : Fin n,
-          (X (i, k) : P a n) ^ d (i, k) := by
-        rw [monomial_eq, ← MvPolynomial.smul_eq_C_mul,
-          Finsupp.prod_fintype _ _ (fun _ => pow_zero _), Fintype.prod_prod_type]
-      rw [hm]
-      exact Submodule.smul_mem _ _ hprod
-  · apply Submodule.span_le.mpr
-    rintro _ ⟨x, rfl⟩
-    exact pureRows_mem x
-
-lemma family_zero_of_all_shifted {A : Type*} [CommRing A] [Algebra ℂ A]
-    {a n j L : ℕ} (c : Fin a → Fin n → A) (l : A →ₗ[ℂ] ℂ)
-    (hj : a * L ≤ j)
-    (hz : mapCoeffs l (GenericShift.family c j L Finset.univ) = 0) :
-    mapCoeffs l (GenericShift.family c j L ∅) = 0 := by
-  classical
-  have hc (t : Finset (Fin a)) :
-      mapCoeffs l (GenericShift.family c j L (Finset.univ \ t)) = 0 := by
-    induction t using Finset.induction_on with
-    | empty => simpa using hz
-    | insert u t hu ih =>
-      let s : Finset (Fin a) := Finset.univ \ insert u t
-      have hus : u ∉ s := by simp [s]
-      have hse : insert u s = Finset.univ \ t := by
-        ext i
-        by_cases hi : i = u <;> simp [s, hi, hu]
-      have hcard : s.card + 1 ≤ a := by
-        calc
-          s.card + 1 = (insert u s).card := (Finset.card_insert_of_notMem hus).symm
-          _ ≤ (Finset.univ : Finset (Fin a)).card := Finset.card_le_card (Finset.subset_univ _)
-          _ = a := by simp
-      have hbound : s.card * L + L ≤ j := by
-        have := Nat.mul_le_mul_right L hcard
-        nlinarith
-      have hL : L ≤ j := by omega
-      change mapCoeffs l (GenericShift.family c j L s) = 0
-      apply Differential.shift_power_kernel_zero (u := u.castSucc) (v := Fin.last a)
-        (by simp)
-        (homogeneous_mapCoeffs l _ (GenericShift.family_degree_source c j L s u hus))
-        (homogeneous_mapCoeffs l _ (GenericShift.family_degree_dest c j L s)) hbound
-      change ((GenericShift.shift u.castSucc (Fin.last a)).toLinearMap ^ L)
-        (mapCoeffs l (GenericShift.family c j L s)) = 0
-      rw [← GenericShift.mapCoeffs_shift_pow, GenericShift.shift_family c j L s u hus hL, hse]
-      change (mapCoeffsLM l) (j.descFactorial L • GenericShift.family c j L (Finset.univ \ t)) = 0
-      rw [map_nsmul]
-      change j.descFactorial L • mapCoeffs l (GenericShift.family c j L (Finset.univ \ t)) = 0
-      rw [ih, smul_zero]
-  simpa using hc Finset.univ
-
-def z {a n : ℕ} (v : Fin n → ℂ) : P a n := ∏ i, rowForm i v
-
-def polarizedSpan (a n j L : ℕ) : Submodule ℂ (P a n) :=
-  Submodule.span ℂ {p | ∃ (v : Fin n → ℂ) (f : P a n),
-    f ∈ W a n (j - L) ∧ z v ^ L * f = p}
-
-lemma eval_original_rowLinear {a n : ℕ} (u : Fin (a + 1)) (i : Fin a)
-    (x : Fin (a + 1) × Fin n → ℂ) :
-    eval (fun ij => algebraMap ℂ (P a n) (x ij))
-      (GenericShift.rowLinear u (fun k => (X (i, k) : P a n))) =
-        rowForm i (fun k => x (u, k)) :=
-  GenericShift.eval_rowLinear u (fun k => (X (i, k) : P a n)) x
-
-lemma eval_family_empty {a n j L : ℕ} (x : Fin (a + 1) × Fin n → ℂ) :
-    eval (fun ij => algebraMap ℂ (P a n) (x ij))
-      (GenericShift.family (fun i k => (X (i, k) : P a n)) j L ∅) =
-        pureRows j (fun i k => x (i.castSucc, k)) := by
-  simp only [GenericShift.family, GenericShift.mixedProduct, Finset.notMem_empty,
-    ite_false, map_prod, map_pow, eval_original_rowLinear, pureRows]
-
-lemma eval_family_univ {a n j L : ℕ} (x : Fin (a + 1) × Fin n → ℂ) :
-    eval (fun ij => algebraMap ℂ (P a n) (x ij))
-      (GenericShift.family (fun i k => (X (i, k) : P a n)) j L Finset.univ) =
-        z (fun k => x (Fin.last a, k)) ^ L *
-          pureRows (j - L) (fun i k => x (i.castSucc, k)) := by
-  simp only [GenericShift.family, GenericShift.mixedProduct, Finset.mem_univ,
-    ite_true, map_prod, map_mul, map_pow, eval_original_rowLinear, pureRows, z]
-  rw [Finset.prod_mul_distrib, Finset.prod_pow]
-
-lemma z_pow_mem {a n L : ℕ} (v : Fin n → ℂ) : z (a := a) v ^ L ∈ W a n L := by
-  have h : (z (a := a) v).IsWeightedHomogeneous (weight a n) (fun _ => 1) := by
-    simpa [W, z, pureRows] using pureRows_mem (j := 1) (fun _ => v)
-  have hp := h.pow L
-  simpa [W, Pi.smul_def] using hp
-
-theorem W_polarization {a n j L : ℕ} (ha : 1 ≤ a) (hj : a * L ≤ j) :
-    W a n j = polarizedSpan a n j L := by
-  classical
-  apply le_antisymm
-  · intro f hf
-    by_contra hnot
-    obtain ⟨l, hlf, hlS⟩ := Submodule.exists_dual_map_eq_bot_of_notMem hnot inferInstance
-    have hl (p : P a n) (hp : p ∈ polarizedSpan a n j L) : l p = 0 := by
-      have hmem : l p ∈ (polarizedSpan a n j L).map l := Submodule.mem_map.mpr ⟨p, hp, rfl⟩
-      rwa [hlS, Submodule.mem_bot] at hmem
-    have hfull : mapCoeffs l (GenericShift.family
-        (fun i k => (X (i, k) : P a n)) j L Finset.univ) = 0 := by
-      apply MvPolynomial.funext
-      intro x
-      rw [eval_mapCoeffs, eval_family_univ, map_zero]
-      apply hl
-      exact Submodule.subset_span ⟨_, _, pureRows_mem _, rfl⟩
-    have hzero := family_zero_of_all_shifted (fun i k => (X (i, k) : P a n)) l hj hfull
-    have hpure (v : Fin a → Fin n → ℂ) : l (pureRows j v) = 0 := by
-      let x : Fin (a + 1) × Fin n → ℂ :=
-        fun ij => Fin.lastCases 0 (fun i => v i ij.2) ij.1
-      have he := congrArg (eval x) hzero
-      rw [eval_mapCoeffs, eval_family_empty, map_zero] at he
-      simpa [x] using he
-    have hW : W a n j ≤ LinearMap.ker l := by
-      rw [W_eq_span_pureRows]
-      apply Submodule.span_le.mpr
-      rintro _ ⟨v, rfl⟩
-      exact hpure v
-    exact hlf (hW hf)
-  · apply Submodule.span_le.mpr
-    rintro _ ⟨v, f, hf, rfl⟩
-    have hL : L ≤ j := le_trans (Nat.le_mul_of_pos_left L ha) hj
-    have hzm := z_pow_mem (a := a) (L := L) v
-    change (z v ^ L).IsWeightedHomogeneous (weight a n) (fun _ => L) at hzm
-    change f.IsWeightedHomogeneous (weight a n) (fun _ => j - L) at hf
-    have hprod := hzm.mul hf
-    simpa [W, Pi.add_def, Nat.add_sub_of_le hL] using hprod
-
-end Foulkes.Polarization
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.FoulkesSixth.Differential
+import OAI.RepresentationTheory.FoulkesSixth.GenericShift
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.Polarization
+open MvPolynomial
+open Foulkes.PolynomialSpan
+
+abbrev P (a n : ℕ) := MvPolynomial (Fin a × Fin n) ℂ
+
+def weight (a n : ℕ) (ij : Fin a × Fin n) : Fin a → ℕ := Pi.single ij.1 1
+
+def W (a n j : ℕ) : Submodule ℂ (P a n) :=
+  weightedHomogeneousSubmodule ℂ (weight a n) (fun _ => j)
+
+def rowForm {a n : ℕ} (i : Fin a) (x : Fin n → ℂ) : P a n :=
+  ∑ k, x k • X (i, k)
+
+def pureRows {a n : ℕ} (j : ℕ) (x : Fin a → Fin n → ℂ) : P a n :=
+  ∏ i, (rowForm i (x i)) ^ j
+
+lemma weight_apply_row {a n : ℕ} (d : (Fin a × Fin n) →₀ ℕ) (i : Fin a) :
+    Finsupp.weight (weight a n) d i = ∑ k, d (i, k) := by
+  classical
+  rw [Finsupp.weight_eq_sum]
+  simp only [weight, Finset.sum_apply, Fintype.sum_prod_type, Pi.smul_apply,
+    Pi.single_apply, smul_eq_mul, mul_ite, mul_one, mul_zero]
+  rw [Finset.sum_comm]
+  simp
+
+lemma pureRows_mem {a n j : ℕ} (x : Fin a → Fin n → ℂ) :
+    pureRows j x ∈ W a n j := by
+  classical
+  have hrow (i : Fin a) : (rowForm i (x i)).IsWeightedHomogeneous
+      (weight a n) (Pi.single i 1) := by
+    apply MvPolynomial.IsWeightedHomogeneous.sum
+    intro k _
+    exact (weightedHomogeneousSubmodule ℂ (weight a n) (Pi.single i 1)).smul_mem _
+      (isWeightedHomogeneous_X (R := ℂ) (weight a n) (i, k))
+  have hprod := MvPolynomial.IsWeightedHomogeneous.prod Finset.univ
+    (fun i => (rowForm i (x i)) ^ j) (fun i => j • Pi.single i 1)
+    (fun i _ => (hrow i).pow j)
+  have hdeg : (∑ i : Fin a, j • (Pi.single i 1 : Fin a → ℕ)) = fun _ => j := by
+    ext r
+    simp [Pi.single_apply]
+  rw [hdeg] at hprod
+  exact hprod
+
+theorem W_eq_span_pureRows (a n j : ℕ) :
+    W a n j = Submodule.span ℂ (Set.range (pureRows (a := a) (n := n) j)) := by
+  classical
+  apply le_antisymm
+  · intro p hp
+    induction hp using MvPolynomial.IsWeightedHomogeneous.induction_on with
+    | zero => exact Submodule.zero_mem _
+    | add p q hp hq ihp ihq => exact Submodule.add_mem _ ihp ihq
+    | monomial d c hd =>
+      have hdrow (i : Fin a) : ∑ k, d (i, k) = j := by
+        simpa [weight_apply_row] using congrFun hd i
+      have hx (i : Fin a) : (∏ k : Fin n, (X (i, k) : P a n) ^ d (i, k)) ∈
+          Submodule.span ℂ (Set.range fun x : Fin n → ℂ => (rowForm i x) ^ j) := by
+        let e : Fin n →₀ ℕ := Finsupp.equivFunOnFinite.symm (fun k => d (i, k))
+        have he : e.sum (fun _ v => v) = j := by
+          rw [Finsupp.sum_fintype _ _ (by simp)]
+          exact hdrow i
+        have hh := prod_pow_mem_span_powers (fun k => (X (i, k) : P a n)) e j he
+        rw [Finsupp.prod_fintype _ _ (fun _ => pow_zero _)] at hh
+        simpa [e, rowForm] using hh
+      have hprod := prod_mem_span_range (fun i (x : Fin n → ℂ) => (rowForm i x) ^ j) hx
+      have hm : (monomial d c : P a n) = c • ∏ i : Fin a, ∏ k : Fin n,
+          (X (i, k) : P a n) ^ d (i, k) := by
+        rw [monomial_eq, ← MvPolynomial.smul_eq_C_mul,
+          Finsupp.prod_fintype _ _ (fun _ => pow_zero _), Fintype.prod_prod_type]
+      rw [hm]
+      exact Submodule.smul_mem _ _ hprod
+  · apply Submodule.span_le.mpr
+    rintro _ ⟨x, rfl⟩
+    exact pureRows_mem x
+
+lemma family_zero_of_all_shifted {A : Type*} [CommRing A] [Algebra ℂ A]
+    {a n j L : ℕ} (c : Fin a → Fin n → A) (l : A →ₗ[ℂ] ℂ)
+    (hj : a * L ≤ j)
+    (hz : mapCoeffs l (GenericShift.family c j L Finset.univ) = 0) :
+    mapCoeffs l (GenericShift.family c j L ∅) = 0 := by
+  classical
+  have hc (t : Finset (Fin a)) :
+      mapCoeffs l (GenericShift.family c j L (Finset.univ \ t)) = 0 := by
+    induction t using Finset.induction_on with
+    | empty => simpa using hz
+    | insert u t hu ih =>
+      let s : Finset (Fin a) := Finset.univ \ insert u t
+      have hus : u ∉ s := by simp [s]
+      have hse : insert u s = Finset.univ \ t := by
+        ext i
+        by_cases hi : i = u <;> simp [s, hi, hu]
+      have hcard : s.card + 1 ≤ a := by
+        calc
+          s.card + 1 = (insert u s).card := (Finset.card_insert_of_notMem hus).symm
+          _ ≤ (Finset.univ : Finset (Fin a)).card := Finset.card_le_card (Finset.subset_univ _)
+          _ = a := by simp
+      have hbound : s.card * L + L ≤ j := by
+        have := Nat.mul_le_mul_right L hcard
+        nlinarith
+      have hL : L ≤ j := by omega
+      change mapCoeffs l (GenericShift.family c j L s) = 0
+      apply Differential.shift_power_kernel_zero (u := u.castSucc) (v := Fin.last a)
+        (by simp)
+        (homogeneous_mapCoeffs l _ (GenericShift.family_degree_source c j L s u hus))
+        (homogeneous_mapCoeffs l _ (GenericShift.family_degree_dest c j L s)) hbound
+      change ((GenericShift.shift u.castSucc (Fin.last a)).toLinearMap ^ L)
+        (mapCoeffs l (GenericShift.family c j L s)) = 0
+      rw [← GenericShift.mapCoeffs_shift_pow, GenericShift.shift_family c j L s u hus hL, hse]
+      change (mapCoeffsLM l) (j.descFactorial L • GenericShift.family c j L (Finset.univ \ t)) = 0
+      rw [map_nsmul]
+      change j.descFactorial L • mapCoeffs l (GenericShift.family c j L (Finset.univ \ t)) = 0
+      rw [ih, smul_zero]
+  simpa using hc Finset.univ
+
+def z {a n : ℕ} (v : Fin n → ℂ) : P a n := ∏ i, rowForm i v
+
+def polarizedSpan (a n j L : ℕ) : Submodule ℂ (P a n) :=
+  Submodule.span ℂ {p | ∃ (v : Fin n → ℂ) (f : P a n),
+    f ∈ W a n (j - L) ∧ z v ^ L * f = p}
+
+lemma eval_original_rowLinear {a n : ℕ} (u : Fin (a + 1)) (i : Fin a)
+    (x : Fin (a + 1) × Fin n → ℂ) :
+    eval (fun ij => algebraMap ℂ (P a n) (x ij))
+      (GenericShift.rowLinear u (fun k => (X (i, k) : P a n))) =
+        rowForm i (fun k => x (u, k)) :=
+  GenericShift.eval_rowLinear u (fun k => (X (i, k) : P a n)) x
+
+lemma eval_family_empty {a n j L : ℕ} (x : Fin (a + 1) × Fin n → ℂ) :
+    eval (fun ij => algebraMap ℂ (P a n) (x ij))
+      (GenericShift.family (fun i k => (X (i, k) : P a n)) j L ∅) =
+        pureRows j (fun i k => x (i.castSucc, k)) := by
+  simp only [GenericShift.family, GenericShift.mixedProduct, Finset.notMem_empty,
+    ite_false, map_prod, map_pow, eval_original_rowLinear, pureRows]
+
+lemma eval_family_univ {a n j L : ℕ} (x : Fin (a + 1) × Fin n → ℂ) :
+    eval (fun ij => algebraMap ℂ (P a n) (x ij))
+      (GenericShift.family (fun i k => (X (i, k) : P a n)) j L Finset.univ) =
+        z (fun k => x (Fin.last a, k)) ^ L *
+          pureRows (j - L) (fun i k => x (i.castSucc, k)) := by
+  simp only [GenericShift.family, GenericShift.mixedProduct, Finset.mem_univ,
+    ite_true, map_prod, map_mul, map_pow, eval_original_rowLinear, pureRows, z]
+  rw [Finset.prod_mul_distrib, Finset.prod_pow]
+
+lemma z_pow_mem {a n L : ℕ} (v : Fin n → ℂ) : z (a := a) v ^ L ∈ W a n L := by
+  have h : (z (a := a) v).IsWeightedHomogeneous (weight a n) (fun _ => 1) := by
+    simpa [W, z, pureRows] using pureRows_mem (j := 1) (fun _ => v)
+  have hp := h.pow L
+  simpa [W, Pi.smul_def] using hp
+
+theorem W_polarization {a n j L : ℕ} (ha : 1 ≤ a) (hj : a * L ≤ j) :
+    W a n j = polarizedSpan a n j L := by
+  classical
+  apply le_antisymm
+  · intro f hf
+    by_contra hnot
+    obtain ⟨l, hlf, hlS⟩ := Submodule.exists_dual_map_eq_bot_of_notMem hnot inferInstance
+    have hl (p : P a n) (hp : p ∈ polarizedSpan a n j L) : l p = 0 := by
+      have hmem : l p ∈ (polarizedSpan a n j L).map l := Submodule.mem_map.mpr ⟨p, hp, rfl⟩
+      rwa [hlS, Submodule.mem_bot] at hmem
+    have hfull : mapCoeffs l (GenericShift.family
+        (fun i k => (X (i, k) : P a n)) j L Finset.univ) = 0 := by
+      apply MvPolynomial.funext
+      intro x
+      rw [eval_mapCoeffs, eval_family_univ, map_zero]
+      apply hl
+      exact Submodule.subset_span ⟨_, _, pureRows_mem _, rfl⟩
+    have hzero := family_zero_of_all_shifted (fun i k => (X (i, k) : P a n)) l hj hfull
+    have hpure (v : Fin a → Fin n → ℂ) : l (pureRows j v) = 0 := by
+      let x : Fin (a + 1) × Fin n → ℂ :=
+        fun ij => Fin.lastCases 0 (fun i => v i ij.2) ij.1
+      have he := congrArg (eval x) hzero
+      rw [eval_mapCoeffs, eval_family_empty, map_zero] at he
+      simpa [x] using he
+    have hW : W a n j ≤ LinearMap.ker l := by
+      rw [W_eq_span_pureRows]
+      apply Submodule.span_le.mpr
+      rintro _ ⟨v, rfl⟩
+      exact hpure v
+    exact hlf (hW hf)
+  · apply Submodule.span_le.mpr
+    rintro _ ⟨v, f, hf, rfl⟩
+    have hL : L ≤ j := le_trans (Nat.le_mul_of_pos_left L ha) hj
+    have hzm := z_pow_mem (a := a) (L := L) v
+    change (z v ^ L).IsWeightedHomogeneous (weight a n) (fun _ => L) at hzm
+    change f.IsWeightedHomogeneous (weight a n) (fun _ => j - L) at hf
+    have hprod := hzm.mul hf
+    simpa [W, Pi.add_def, Nat.add_sub_of_le hL] using hprod
+
+end Foulkes.Polarization
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/Vanishing.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/Vanishing.lean
index 2711ce3f4808c594f0cfc1f378826e9be819d858..c5b44c72ee304797d496129d8cf209d82938287f 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/Vanishing.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/Vanishing.lean
@@ -1,215 +1,215 @@
-import Mathlib
-import OAI.RepresentationTheory.FoulkesSixth.VanishingSteps
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.Vanishing
-open MvPolynomial Foulkes.Polarization Foulkes.PolynomialSpan
-
-def extendPerm {r : ℕ} (σ : Equiv.Perm (Fin r)) : Equiv.Perm (Fin (r+1)) where
-  toFun := Fin.lastCases (Fin.last r) (fun i => (σ i).castSucc)
-  invFun := Fin.lastCases (Fin.last r) (fun i => (σ.symm i).castSucc)
-  left_inv := by intro i; refine Fin.lastCases ?_ (fun j => ?_) i <;> simp
-  right_inv := by intro i; refine Fin.lastCases ?_ (fun j => ?_) i <;> simp
-
-@[simp] lemma extendPerm_castSucc {r : ℕ} (σ : Equiv.Perm (Fin r)) (i : Fin r) :
-    extendPerm σ i.castSucc = (σ i).castSucc := by simp [extendPerm]
-
-@[simp] lemma extendPerm_last {r : ℕ} (σ : Equiv.Perm (Fin r)) :
-    extendPerm σ (Fin.last r) = Fin.last r := by simp [extendPerm]
-
-def embedRows {r n : ℕ} : P r n →ₐ[ℂ] P (r+1) n :=
-  rename (fun ij : Fin r × Fin n => (ij.1.castSucc,ij.2))
-
-lemma embedRows_rowLift {r n : ℕ} (i : Fin r) (p : S n) :
-    embedRows (rowLift i p) = rowLift i.castSucc p := by
-  simp [embedRows, rowLift, rename_rename, Function.comp_def]
-
-@[simp] lemma embedRows_rowForm {r n : ℕ} (i : Fin r) (v : Fin n → ℂ) :
-    embedRows (rowForm i v) = rowForm i.castSucc v := by simp [embedRows, rowForm]
-
-lemma embedRows_pureRows {r n b : ℕ} (v : Fin r → Fin n → ℂ) :
-    embedRows (pureRows b v) = ∏ i, rowForm i.castSucc (v i) ^ b := by
-  simp only [pureRows, map_prod, map_pow, embedRows_rowForm]
-
-lemma embedRows_tensorRows {r n : ℕ} (p : Fin r → S n) :
-    embedRows (tensorRows p) = ∏ i, rowLift i.castSucc (p i) := by
-  simp only [tensorRows, map_prod, embedRows_rowLift]
-
-lemma renameRow_embedRows {r n : ℕ} (σ : Equiv.Perm (Fin r)) (f : P r n) :
-    renameRow (extendPerm σ) (embedRows f) = embedRows (renameRow σ f) := by
-  simp [renameRow, embedRows, rename_rename, Function.comp_def]
-
-lemma renameRow_commonPower {r n : ℕ} (σ : Equiv.Perm (Fin r)) (x : Fin n → ℂ) :
-    renameRow (extendPerm σ) (commonPower x) = commonPower x := by
-  simp only [commonPower, map_prod, map_pow, renameRow_rowForm, extendPerm_castSucc]
-  exact Equiv.prod_comp σ (fun i => rowForm i.castSucc x ^ r)
-
-def restrictionFunctional {r n : ℕ} (l : P (r+1) n →ₗ[ℂ] ℂ)
-    (x t : Fin n → ℂ) (b : ℕ) : P r n →ₗ[ℂ] ℂ :=
-  (l.comp (LinearMap.mulLeft ℂ (commonPower x * rowForm (Fin.last r) t ^ b))).comp embedRows.toLinearMap
-
-@[simp] lemma restrictionFunctional_apply {r n : ℕ} (l : P (r+1) n →ₗ[ℂ] ℂ)
-    (x t : Fin n → ℂ) (b : ℕ) (f : P r n) :
-    restrictionFunctional l x t b f =
-      l ((commonPower x * rowForm (Fin.last r) t ^ b) * embedRows f) := rfl
-
-lemma restriction_symmetric {r n b : ℕ} {l : P (r+1) n →ₗ[ℂ] ℂ}
-    (hs : Symmetric l) (x t : Fin n → ℂ) : Symmetric (restrictionFunctional l x t b) := by
-  intro σ f
-  simp only [restrictionFunctional_apply, ← renameRow_embedRows]
-  have hh := hs (extendPerm σ) ((commonPower x * rowForm (Fin.last r) t ^ b) * embedRows f)
-  simpa only [map_mul, map_pow, renameRow_commonPower, renameRow_rowForm, extendPerm_last] using hh
-
-lemma restriction_products {r m n : ℕ} (hr : 0 < r) {l : P (r+1) n →ₗ[ℂ] ℂ}
-    (hs : Symmetric l) (hz : ProductsVanish l (r+m)) (x t : Fin n → ℂ) :
-    ProductsVanish (restrictionFunctional l x t (r+m)) m := by
-  intro y
-  rw [← tensorRows_diag_product, restrictionFunctional_apply, embedRows_tensorRows]
-  have hh := free_last_slot hr hs hz x t y
-  simp only [map_mul, map_pow, rowLift_lin, Finset.prod_mul_distrib] at hh
-  simpa only [commonPower, mul_assoc, mul_left_comm, mul_comm] using hh
-
-lemma eval_cast_rowLinear {r n : ℕ} (u : Fin (r+1)) (i : Fin r)
-    (w : Fin (r+1) × Fin n → ℂ) :
-    eval (fun ij => algebraMap ℂ (P (r+1) n) (w ij))
-      (GenericShift.rowLinear u (fun k => (X (i.castSucc,k) : P (r+1) n))) =
-      rowForm i.castSucc (fun k => w (u,k)) :=
-  GenericShift.eval_rowLinear u _ w
-
-lemma eval_induction_family_univ {r n j : ℕ} (w : Fin (r+1) × Fin n → ℂ) :
-    eval (fun ij => algebraMap ℂ (P (r+1) n) (w ij))
-      (GenericShift.family (fun i : Fin r => fun k => (X (i.castSucc,k) : P (r+1) n))
-        j r Finset.univ) =
-      commonPower (fun k => w (Fin.last r,k)) *
-        embedRows (pureRows (j-r) (fun i k => w (i.castSucc,k))) := by
-  simp only [GenericShift.family, GenericShift.mixedProduct, Finset.mem_univ, ite_true,
-    map_prod, map_mul, map_pow, eval_cast_rowLinear, embedRows_pureRows, commonPower]
-  exact Finset.prod_mul_distrib
-
-lemma eval_induction_family_empty {r n j : ℕ} (w : Fin (r+1) × Fin n → ℂ) :
-    eval (fun ij => algebraMap ℂ (P (r+1) n) (w ij))
-      (GenericShift.family (fun i : Fin r => fun k => (X (i.castSucc,k) : P (r+1) n))
-        j r ∅) = embedRows (pureRows j (fun i k => w (i.castSucc,k))) := by
-  simp only [GenericShift.family, GenericShift.mixedProduct, Finset.notMem_empty, ite_false,
-    map_prod, map_pow, eval_cast_rowLinear, embedRows_pureRows]
-
-lemma remove_common_power {r m n : ℕ} (hb : r*r ≤ r+m)
-    {l : P (r+1) n →ₗ[ℂ] ℂ} (t : Fin n → ℂ)
-    (hz : ∀ (x : Fin n → ℂ) (f : P r n), f ∈ W r n m →
-      restrictionFunctional l x t (r+m) f = 0)
-    (v : Fin r → Fin n → ℂ) :
-    l ((∏ i : Fin r, rowForm i.castSucc (v i) ^ (r+m)) *
-      rowForm (Fin.last r) t ^ (r+m)) = 0 := by
-  let c := fun i : Fin r => fun k : Fin n => (X (i.castSucc,k) : P (r+1) n)
-  let lt : P (r+1) n →ₗ[ℂ] ℂ :=
-    l.comp (LinearMap.mulLeft ℂ (rowForm (Fin.last r) t ^ (r+m)))
-  have hfull : mapCoeffs lt (GenericShift.family c (r+m) r Finset.univ) = 0 := by
-    apply MvPolynomial.funext
-    intro w
-    rw [eval_mapCoeffs, eval_induction_family_univ, Nat.add_sub_cancel_left, map_zero]
-    have hh := hz (fun k => w (Fin.last r,k))
-      (pureRows m (fun i k => w (i.castSucc,k))) (pureRows_mem _)
-    change l ((commonPower (fun k => w (Fin.last r,k)) * rowForm (Fin.last r) t ^ (r+m)) *
-      embedRows (pureRows m (fun i k => w (i.castSucc,k)))) = 0 at hh
-    change l (rowForm (Fin.last r) t ^ (r+m) * (commonPower (fun k => w (Fin.last r,k)) *
-      embedRows (pureRows m (fun i k => w (i.castSucc,k))))) = 0
-    rw [← mul_assoc, mul_comm (rowForm (Fin.last r) t ^ (r+m))]
-    exact hh
-  have hzero := family_zero_of_all_shifted c lt hb hfull
-  let w : Fin (r+1) × Fin n → ℂ := fun ij => Fin.lastCases 0 (fun i => v i ij.2) ij.1
-  have hh := congrArg (eval w) hzero
-  rw [eval_mapCoeffs, eval_induction_family_empty, map_zero] at hh
-  change l (rowForm (Fin.last r) t ^ (r+m) * embedRows (pureRows (r+m) _)) = 0 at hh
-  rw [embedRows_pureRows, mul_comm] at hh
-  simpa only [w, Fin.lastCases_castSucc] using hh
-
-theorem vanishing (a : ℕ) : ∀ {n b : ℕ}, 1 ≤ a → 1 ≤ b → a*(a-1) ≤ b →
-    ∀ {l : P a n →ₗ[ℂ] ℂ}, Symmetric l → ProductsVanish l b →
-      ∀ f : P a n, f ∈ W a n b → l f = 0 := by
-  induction a with
-  | zero => intro n b ha; omega
-  | succ r ih =>
-    intro n b ha hb0 hb l hs hz f hf
-    have hpure : ∀ v : Fin (r+1) → Fin n → ℂ, l (pureRows b v) = 0 := by
-      intro v
-      by_cases hr : r = 0
-      · subst r
-        have hh := hz (fun _ => v 0)
-        simpa [pureRows, z, Fin.prod_univ_one] using hh
-      · have hrpos : 0 < r := Nat.pos_of_ne_zero hr
-        have hrb : r ≤ b := by
-          simp only [Nat.add_sub_cancel] at hb
-          nlinarith
-        obtain ⟨m, rfl⟩ := Nat.exists_eq_add_of_le hrb
-        have hm : r*r ≤ m := by
-          simp only [Nat.add_sub_cancel] at hb
-          nlinarith
-        have hrestrict (x t : Fin n → ℂ) (g : P r n) (hg : g ∈ W r n m) :
-            restrictionFunctional l x t (r+m) g = 0 := by
-          apply ih (by omega) (by nlinarith) ?_ (restriction_symmetric hs x t)
-            (restriction_products hrpos hs hz x t) g hg
-          have hpred : r-1 ≤ r := Nat.sub_le _ _
-          exact (Nat.mul_le_mul_left r hpred).trans hm
-        have hh := remove_common_power (by nlinarith : r*r ≤ r+m)
-          (v (Fin.last r)) (fun x g hg => hrestrict x (v (Fin.last r)) g hg)
-          (fun i => v i.castSucc)
-        simpa only [pureRows, Fin.prod_univ_castSucc] using hh
-    have hW : W (r+1) n b ≤ LinearMap.ker l := by
-      rw [W_eq_span_pureRows]
-      apply Submodule.span_le.mpr
-      rintro _ ⟨v,rfl⟩
-      exact hpure v
-    exact hW hf
-
-end Foulkes.Vanishing
-
-namespace Foulkes.Vanishing
-open MvPolynomial Foulkes.Polarization
-
-lemma average_renameRow {a n : ℕ} (τ : Equiv.Perm (Fin a)) (f : P a n) :
-    average a n (renameRow τ f) = average a n f := by
-  simp only [average_apply, renameRow_comp]
-  congr 1
-  exact Equiv.sum_comp (Equiv.mulRight τ) (fun σ => renameRow σ f)
-
-theorem quadratic_bound (a n b : ℕ) (ha : 1 ≤ a) (hb0 : 1 ≤ b) (hb : a*(a-1) ≤ b) :
-    R a n b = W a n b ⊓ (A a n).toSubmodule := by
-  classical
-  apply le_antisymm
-  · intro f hf
-    refine ⟨hf.1, ?_⟩
-    by_contra hnot
-    obtain ⟨l, hlf, hlA⟩ := Submodule.exists_dual_map_eq_bot_of_notMem hnot inferInstance
-    have hkill (p : P a n) (hp : p ∈ A a n) : l p = 0 := by
-      have hh : l p ∈ (A a n).toSubmodule.map l := Submodule.mem_map.mpr ⟨p,hp,rfl⟩
-      rwa [hlA, Submodule.mem_bot] at hh
-    let lav := l.comp (average a n)
-    have hs : Symmetric lav := by
-      intro σ p
-      change l (average a n (renameRow σ p)) = l (average a n p)
-      rw [average_renameRow]
-    have hz : ProductsVanish lav b := by
-      intro y
-      have hp : (∏ j : Fin b, z (y j)) ∈ A a n :=
-        (A a n).prod_mem (fun j _ => z_mem_A (y j))
-      change l (average a n (∏ j, z (y j))) = 0
-      rw [average_eq (A_le_rowInvariants a n hp)]
-      exact hkill _ hp
-    have hh := vanishing a ha hb0 hb hs hz f hf.1
-    change l (average a n f) = 0 at hh
-    rw [average_eq hf.2] at hh
-    exact hlf hh
-  · exact AComponent_le_R a n b
-
-theorem sixth_quadratic_bound (n b : ℕ) (hb : 30 ≤ b) :
-    R 6 n b = W 6 n b ⊓ (A 6 n).toSubmodule :=
-  quadratic_bound 6 n b (by omega) (by omega) (by omega)
-
-end Foulkes.Vanishing
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.FoulkesSixth.VanishingSteps
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.Vanishing
+open MvPolynomial Foulkes.Polarization Foulkes.PolynomialSpan
+
+def extendPerm {r : ℕ} (σ : Equiv.Perm (Fin r)) : Equiv.Perm (Fin (r+1)) where
+  toFun := Fin.lastCases (Fin.last r) (fun i => (σ i).castSucc)
+  invFun := Fin.lastCases (Fin.last r) (fun i => (σ.symm i).castSucc)
+  left_inv := by intro i; refine Fin.lastCases ?_ (fun j => ?_) i <;> simp
+  right_inv := by intro i; refine Fin.lastCases ?_ (fun j => ?_) i <;> simp
+
+@[simp] lemma extendPerm_castSucc {r : ℕ} (σ : Equiv.Perm (Fin r)) (i : Fin r) :
+    extendPerm σ i.castSucc = (σ i).castSucc := by simp [extendPerm]
+
+@[simp] lemma extendPerm_last {r : ℕ} (σ : Equiv.Perm (Fin r)) :
+    extendPerm σ (Fin.last r) = Fin.last r := by simp [extendPerm]
+
+def embedRows {r n : ℕ} : P r n →ₐ[ℂ] P (r+1) n :=
+  rename (fun ij : Fin r × Fin n => (ij.1.castSucc,ij.2))
+
+lemma embedRows_rowLift {r n : ℕ} (i : Fin r) (p : S n) :
+    embedRows (rowLift i p) = rowLift i.castSucc p := by
+  simp [embedRows, rowLift, rename_rename, Function.comp_def]
+
+@[simp] lemma embedRows_rowForm {r n : ℕ} (i : Fin r) (v : Fin n → ℂ) :
+    embedRows (rowForm i v) = rowForm i.castSucc v := by simp [embedRows, rowForm]
+
+lemma embedRows_pureRows {r n b : ℕ} (v : Fin r → Fin n → ℂ) :
+    embedRows (pureRows b v) = ∏ i, rowForm i.castSucc (v i) ^ b := by
+  simp only [pureRows, map_prod, map_pow, embedRows_rowForm]
+
+lemma embedRows_tensorRows {r n : ℕ} (p : Fin r → S n) :
+    embedRows (tensorRows p) = ∏ i, rowLift i.castSucc (p i) := by
+  simp only [tensorRows, map_prod, embedRows_rowLift]
+
+lemma renameRow_embedRows {r n : ℕ} (σ : Equiv.Perm (Fin r)) (f : P r n) :
+    renameRow (extendPerm σ) (embedRows f) = embedRows (renameRow σ f) := by
+  simp [renameRow, embedRows, rename_rename, Function.comp_def]
+
+lemma renameRow_commonPower {r n : ℕ} (σ : Equiv.Perm (Fin r)) (x : Fin n → ℂ) :
+    renameRow (extendPerm σ) (commonPower x) = commonPower x := by
+  simp only [commonPower, map_prod, map_pow, renameRow_rowForm, extendPerm_castSucc]
+  exact Equiv.prod_comp σ (fun i => rowForm i.castSucc x ^ r)
+
+def restrictionFunctional {r n : ℕ} (l : P (r+1) n →ₗ[ℂ] ℂ)
+    (x t : Fin n → ℂ) (b : ℕ) : P r n →ₗ[ℂ] ℂ :=
+  (l.comp (LinearMap.mulLeft ℂ (commonPower x * rowForm (Fin.last r) t ^ b))).comp embedRows.toLinearMap
+
+@[simp] lemma restrictionFunctional_apply {r n : ℕ} (l : P (r+1) n →ₗ[ℂ] ℂ)
+    (x t : Fin n → ℂ) (b : ℕ) (f : P r n) :
+    restrictionFunctional l x t b f =
+      l ((commonPower x * rowForm (Fin.last r) t ^ b) * embedRows f) := rfl
+
+lemma restriction_symmetric {r n b : ℕ} {l : P (r+1) n →ₗ[ℂ] ℂ}
+    (hs : Symmetric l) (x t : Fin n → ℂ) : Symmetric (restrictionFunctional l x t b) := by
+  intro σ f
+  simp only [restrictionFunctional_apply, ← renameRow_embedRows]
+  have hh := hs (extendPerm σ) ((commonPower x * rowForm (Fin.last r) t ^ b) * embedRows f)
+  simpa only [map_mul, map_pow, renameRow_commonPower, renameRow_rowForm, extendPerm_last] using hh
+
+lemma restriction_products {r m n : ℕ} (hr : 0 < r) {l : P (r+1) n →ₗ[ℂ] ℂ}
+    (hs : Symmetric l) (hz : ProductsVanish l (r+m)) (x t : Fin n → ℂ) :
+    ProductsVanish (restrictionFunctional l x t (r+m)) m := by
+  intro y
+  rw [← tensorRows_diag_product, restrictionFunctional_apply, embedRows_tensorRows]
+  have hh := free_last_slot hr hs hz x t y
+  simp only [map_mul, map_pow, rowLift_lin, Finset.prod_mul_distrib] at hh
+  simpa only [commonPower, mul_assoc, mul_left_comm, mul_comm] using hh
+
+lemma eval_cast_rowLinear {r n : ℕ} (u : Fin (r+1)) (i : Fin r)
+    (w : Fin (r+1) × Fin n → ℂ) :
+    eval (fun ij => algebraMap ℂ (P (r+1) n) (w ij))
+      (GenericShift.rowLinear u (fun k => (X (i.castSucc,k) : P (r+1) n))) =
+      rowForm i.castSucc (fun k => w (u,k)) :=
+  GenericShift.eval_rowLinear u _ w
+
+lemma eval_induction_family_univ {r n j : ℕ} (w : Fin (r+1) × Fin n → ℂ) :
+    eval (fun ij => algebraMap ℂ (P (r+1) n) (w ij))
+      (GenericShift.family (fun i : Fin r => fun k => (X (i.castSucc,k) : P (r+1) n))
+        j r Finset.univ) =
+      commonPower (fun k => w (Fin.last r,k)) *
+        embedRows (pureRows (j-r) (fun i k => w (i.castSucc,k))) := by
+  simp only [GenericShift.family, GenericShift.mixedProduct, Finset.mem_univ, ite_true,
+    map_prod, map_mul, map_pow, eval_cast_rowLinear, embedRows_pureRows, commonPower]
+  exact Finset.prod_mul_distrib
+
+lemma eval_induction_family_empty {r n j : ℕ} (w : Fin (r+1) × Fin n → ℂ) :
+    eval (fun ij => algebraMap ℂ (P (r+1) n) (w ij))
+      (GenericShift.family (fun i : Fin r => fun k => (X (i.castSucc,k) : P (r+1) n))
+        j r ∅) = embedRows (pureRows j (fun i k => w (i.castSucc,k))) := by
+  simp only [GenericShift.family, GenericShift.mixedProduct, Finset.notMem_empty, ite_false,
+    map_prod, map_pow, eval_cast_rowLinear, embedRows_pureRows]
+
+lemma remove_common_power {r m n : ℕ} (hb : r*r ≤ r+m)
+    {l : P (r+1) n →ₗ[ℂ] ℂ} (t : Fin n → ℂ)
+    (hz : ∀ (x : Fin n → ℂ) (f : P r n), f ∈ W r n m →
+      restrictionFunctional l x t (r+m) f = 0)
+    (v : Fin r → Fin n → ℂ) :
+    l ((∏ i : Fin r, rowForm i.castSucc (v i) ^ (r+m)) *
+      rowForm (Fin.last r) t ^ (r+m)) = 0 := by
+  let c := fun i : Fin r => fun k : Fin n => (X (i.castSucc,k) : P (r+1) n)
+  let lt : P (r+1) n →ₗ[ℂ] ℂ :=
+    l.comp (LinearMap.mulLeft ℂ (rowForm (Fin.last r) t ^ (r+m)))
+  have hfull : mapCoeffs lt (GenericShift.family c (r+m) r Finset.univ) = 0 := by
+    apply MvPolynomial.funext
+    intro w
+    rw [eval_mapCoeffs, eval_induction_family_univ, Nat.add_sub_cancel_left, map_zero]
+    have hh := hz (fun k => w (Fin.last r,k))
+      (pureRows m (fun i k => w (i.castSucc,k))) (pureRows_mem _)
+    change l ((commonPower (fun k => w (Fin.last r,k)) * rowForm (Fin.last r) t ^ (r+m)) *
+      embedRows (pureRows m (fun i k => w (i.castSucc,k)))) = 0 at hh
+    change l (rowForm (Fin.last r) t ^ (r+m) * (commonPower (fun k => w (Fin.last r,k)) *
+      embedRows (pureRows m (fun i k => w (i.castSucc,k))))) = 0
+    rw [← mul_assoc, mul_comm (rowForm (Fin.last r) t ^ (r+m))]
+    exact hh
+  have hzero := family_zero_of_all_shifted c lt hb hfull
+  let w : Fin (r+1) × Fin n → ℂ := fun ij => Fin.lastCases 0 (fun i => v i ij.2) ij.1
+  have hh := congrArg (eval w) hzero
+  rw [eval_mapCoeffs, eval_induction_family_empty, map_zero] at hh
+  change l (rowForm (Fin.last r) t ^ (r+m) * embedRows (pureRows (r+m) _)) = 0 at hh
+  rw [embedRows_pureRows, mul_comm] at hh
+  simpa only [w, Fin.lastCases_castSucc] using hh
+
+theorem vanishing (a : ℕ) : ∀ {n b : ℕ}, 1 ≤ a → 1 ≤ b → a*(a-1) ≤ b →
+    ∀ {l : P a n →ₗ[ℂ] ℂ}, Symmetric l → ProductsVanish l b →
+      ∀ f : P a n, f ∈ W a n b → l f = 0 := by
+  induction a with
+  | zero => intro n b ha; omega
+  | succ r ih =>
+    intro n b ha hb0 hb l hs hz f hf
+    have hpure : ∀ v : Fin (r+1) → Fin n → ℂ, l (pureRows b v) = 0 := by
+      intro v
+      by_cases hr : r = 0
+      · subst r
+        have hh := hz (fun _ => v 0)
+        simpa [pureRows, z, Fin.prod_univ_one] using hh
+      · have hrpos : 0 < r := Nat.pos_of_ne_zero hr
+        have hrb : r ≤ b := by
+          simp only [Nat.add_sub_cancel] at hb
+          nlinarith
+        obtain ⟨m, rfl⟩ := Nat.exists_eq_add_of_le hrb
+        have hm : r*r ≤ m := by
+          simp only [Nat.add_sub_cancel] at hb
+          nlinarith
+        have hrestrict (x t : Fin n → ℂ) (g : P r n) (hg : g ∈ W r n m) :
+            restrictionFunctional l x t (r+m) g = 0 := by
+          apply ih (by omega) (by nlinarith) ?_ (restriction_symmetric hs x t)
+            (restriction_products hrpos hs hz x t) g hg
+          have hpred : r-1 ≤ r := Nat.sub_le _ _
+          exact (Nat.mul_le_mul_left r hpred).trans hm
+        have hh := remove_common_power (by nlinarith : r*r ≤ r+m)
+          (v (Fin.last r)) (fun x g hg => hrestrict x (v (Fin.last r)) g hg)
+          (fun i => v i.castSucc)
+        simpa only [pureRows, Fin.prod_univ_castSucc] using hh
+    have hW : W (r+1) n b ≤ LinearMap.ker l := by
+      rw [W_eq_span_pureRows]
+      apply Submodule.span_le.mpr
+      rintro _ ⟨v,rfl⟩
+      exact hpure v
+    exact hW hf
+
+end Foulkes.Vanishing
+
+namespace Foulkes.Vanishing
+open MvPolynomial Foulkes.Polarization
+
+lemma average_renameRow {a n : ℕ} (τ : Equiv.Perm (Fin a)) (f : P a n) :
+    average a n (renameRow τ f) = average a n f := by
+  simp only [average_apply, renameRow_comp]
+  congr 1
+  exact Equiv.sum_comp (Equiv.mulRight τ) (fun σ => renameRow σ f)
+
+theorem quadratic_bound (a n b : ℕ) (ha : 1 ≤ a) (hb0 : 1 ≤ b) (hb : a*(a-1) ≤ b) :
+    R a n b = W a n b ⊓ (A a n).toSubmodule := by
+  classical
+  apply le_antisymm
+  · intro f hf
+    refine ⟨hf.1, ?_⟩
+    by_contra hnot
+    obtain ⟨l, hlf, hlA⟩ := Submodule.exists_dual_map_eq_bot_of_notMem hnot inferInstance
+    have hkill (p : P a n) (hp : p ∈ A a n) : l p = 0 := by
+      have hh : l p ∈ (A a n).toSubmodule.map l := Submodule.mem_map.mpr ⟨p,hp,rfl⟩
+      rwa [hlA, Submodule.mem_bot] at hh
+    let lav := l.comp (average a n)
+    have hs : Symmetric lav := by
+      intro σ p
+      change l (average a n (renameRow σ p)) = l (average a n p)
+      rw [average_renameRow]
+    have hz : ProductsVanish lav b := by
+      intro y
+      have hp : (∏ j : Fin b, z (y j)) ∈ A a n :=
+        (A a n).prod_mem (fun j _ => z_mem_A (y j))
+      change l (average a n (∏ j, z (y j))) = 0
+      rw [average_eq (A_le_rowInvariants a n hp)]
+      exact hkill _ hp
+    have hh := vanishing a ha hb0 hb hs hz f hf.1
+    change l (average a n f) = 0 at hh
+    rw [average_eq hf.2] at hh
+    exact hlf hh
+  · exact AComponent_le_R a n b
+
+theorem sixth_quadratic_bound (n b : ℕ) (hb : 30 ≤ b) :
+    R 6 n b = W 6 n b ⊓ (A 6 n).toSubmodule :=
+  quadratic_bound 6 n b (by omega) (by omega) (by omega)
+
+end Foulkes.Vanishing
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/VanishingSteps.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/VanishingSteps.lean
index 09b9f87a6f9c2a203ab5ea97ad000b65d6a13b31..67cc68f8ca4200b0b36de497f54d530d6ecbe01a 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/VanishingSteps.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/VanishingSteps.lean
@@ -1,401 +1,401 @@
-import Mathlib
-import OAI.RepresentationTheory.FoulkesSixth.InvariantPolarization
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.Vanishing
-open MvPolynomial Foulkes.Polarization Foulkes.PolynomialSpan
-
-lemma binary_split {a n r : ℕ} (hr : 0 < r) (c : ℂ) (x t : Fin n → ℂ) :
-    ∃ v : Fin r → Fin n → ℂ, ∀ i : Fin a,
-      rowForm i x ^ r + c • rowForm i t ^ r = ∏ j, rowForm i (v j) := by
-  classical
-  obtain ⟨α, hα⟩ := IsAlgClosed.exists_pow_nat_eq (-c) hr
-  let ζ := Complex.exp (2 * Real.pi * Complex.I / r)
-  have hζ : IsPrimitiveRoot ζ r := Complex.isPrimitiveRoot_exp r hr.ne'
-  let S := Polynomial.nthRootsFinset r (1 : ℂ)
-  have hS : Fintype.card S = r := by
-    rw [Fintype.card_coe]
-    exact hζ.card_nthRootsFinset
-  let e : Fin r ≃ S := (Fintype.equivFinOfCardEq hS).symm
-  refine ⟨fun j k => x k - (α * (e j : ℂ)) * t k, ?_⟩
-  intro i
-  apply MvPolynomial.funext
-  intro w
-  let X₀ := ∑ k, x k * w (i,k)
-  let Y₀ := ∑ k, t k * w (i,k)
-  have heval (j : Fin r) :
-      eval w (rowForm i (fun k => x k - α * (e j : ℂ) * t k)) =
-        X₀ - (e j : ℂ) * (α * Y₀) := by
-    simp only [rowForm, eval_sum, MvPolynomial.smul_eq_C_mul, eval_mul, eval_C, eval_X]
-    simp only [sub_mul, Finset.sum_sub_distrib]
-    dsimp [X₀, Y₀]
-    simp only [Finset.mul_sum]
-    congr 1
-    apply Finset.sum_congr rfl
-    intro k _
-    ring
-  have hprod : (∏ j : Fin r, (X₀ - (e j : ℂ) * (α * Y₀))) =
-      ∏ z ∈ S, (X₀ - z * (α * Y₀)) := by
-    calc
-      _ = ∏ z : S, (X₀ - (z : ℂ) * (α * Y₀)) :=
-        Fintype.prod_equiv e _ _ (fun _ => rfl)
-      _ = _ := Finset.prod_coe_sort S (fun z : ℂ => X₀ - z * (α * Y₀))
-  simp only [eval_add, eval_pow, MvPolynomial.smul_eq_C_mul, eval_mul, eval_C, eval_prod, heval]
-  have hx : eval w (rowForm i x) = X₀ := by simp [rowForm, X₀, MvPolynomial.smul_eq_C_mul]
-  have hy : eval w (rowForm i t) = Y₀ := by simp [rowForm, Y₀, MvPolynomial.smul_eq_C_mul]
-  rw [hx, hy]
-  rw [hprod, ← hζ.pow_sub_pow_eq_prod_sub_mul X₀ (α * Y₀) hr, mul_pow, hα]
-  ring
-
-abbrev S (n : ℕ) := MvPolynomial (Fin n) ℂ
-
-def rowLift {a n : ℕ} (i : Fin a) : S n →ₐ[ℂ] P a n :=
-  rename (fun k => (i,k))
-
-def tensorRows {a n : ℕ} (p : Fin a → S n) : P a n := ∏ i, rowLift i (p i)
-
-def lin {n : ℕ} (x : Fin n → ℂ) : S n := ∑ k, x k • X k
-
-@[simp] lemma rowLift_lin {a n : ℕ} (i : Fin a) (x : Fin n → ℂ) :
-    rowLift i (lin x) = rowForm i x := by simp [rowLift, lin, rowForm]
-
-@[simp] lemma renameRow_rowLift {a n : ℕ} (σ : Equiv.Perm (Fin a))
-    (i : Fin a) (p : S n) : renameRow σ (rowLift i p) = rowLift (σ i) p := by
-  simp [rowLift, renameRow, rename_rename, Function.comp_def]
-
-lemma renameRow_tensorRows {a n : ℕ} (σ : Equiv.Perm (Fin a)) (p : Fin a → S n) :
-    renameRow σ (tensorRows p) = tensorRows (fun i => p (σ.symm i)) := by
-  simp only [tensorRows, map_prod, renameRow_rowLift]
-  apply Fintype.prod_equiv σ
-  intro i
-  simp
-
-def Symmetric {a n : ℕ} (l : P a n →ₗ[ℂ] ℂ) : Prop :=
-  ∀ (σ : Equiv.Perm (Fin a)) (f : P a n), l (renameRow σ f) = l f
-
-lemma symmetric_tensorRows {a n : ℕ} {l : P a n →ₗ[ℂ] ℂ} (hl : Symmetric l)
-    (σ : Equiv.Perm (Fin a)) (p : Fin a → S n) :
-    l (tensorRows (fun i => p (σ.symm i))) = l (tensorRows p) := by
-  rw [← renameRow_tensorRows, hl]
-
-def oneSlot {a n : ℕ} (p q : S n) (i : Fin a) : P a n :=
-  tensorRows (fun j => if j = i then q else p)
-
-lemma oneSlot_eq {a n : ℕ} {l : P a n →ₗ[ℂ] ℂ} (hl : Symmetric l)
-    (p q : S n) (i j : Fin a) : l (oneSlot p q i) = l (oneSlot p q j) := by
-  classical
-  have hh := symmetric_tensorRows hl (Equiv.swap i j) (fun k => if k = j then q else p)
-  have he : (fun k => if (Equiv.swap i j).symm k = j then q else p) =
-      (fun k => if k = i then q else p) := by
-    funext k
-    simp [Equiv.swap_apply_eq_iff]
-  rw [he] at hh
-  exact hh
-
-lemma eval_pderiv_prod {A ι σ : Type*} [CommRing A] [Fintype ι] [DecidableEq ι]
-    (v : σ) (w : σ → A) (f : ι → MvPolynomial σ A) :
-    eval w (pderiv v (∏ i, f i)) =
-      ∑ i, eval w (pderiv v (f i)) * ∏ j ∈ Finset.univ.erase i, eval w (f j) := by
-  classical
-  have hs (s : Finset ι) : eval w (pderiv v (∏ i ∈ s, f i)) =
-      ∑ i ∈ s, eval w (pderiv v (f i)) * ∏ j ∈ s.erase i, eval w (f j) := by
-    induction s using Finset.induction_on with
-    | empty => simp
-    | insert u s hu ih =>
-      rw [Finset.prod_insert hu, pderiv_mul, eval_add, eval_mul, eval_mul, ih,
-        Finset.sum_insert hu]
-      rw [Finset.erase_insert hu]
-      simp only [eval_prod, Finset.mul_sum]
-      congr 1
-      apply Finset.sum_congr rfl
-      intro i hi
-      have hne : i ≠ u := ne_of_mem_of_not_mem hi hu
-      rw [Finset.erase_insert_of_ne hne.symm, Finset.prod_insert (by simp [hu])]
-      ring
-  exact hs Finset.univ
-
-lemma binary_coefficient {a n : ℕ} (ha : 0 < a) {l : P a n →ₗ[ℂ] ℂ}
-    (hl : Symmetric l) (p q : S n)
-    (hz : ∀ c : ℂ, l (tensorRows (fun _ => p + c • q)) = 0) (i : Fin a) :
-    l (oneSlot p q i) = 0 := by
-  classical
-  let F : MvPolynomial Unit (P a n) :=
-    ∏ j, (C (rowLift j p) + X () * C (rowLift j q))
-  have heval (c : ℂ) : eval (fun _ : Unit => algebraMap ℂ (P a n) c) F =
-      tensorRows (fun _ => p + c • q) := by
-    simp [F, tensorRows, rowLift, Algebra.smul_def]
-  have hF : mapCoeffs l F = 0 := by
-    apply MvPolynomial.funext
-    intro c
-    rw [eval_mapCoeffs]
-    have hc : c = fun _ : Unit => c () := funext (fun u => by cases u; rfl)
-    rw [hc, heval, hz, map_zero]
-  have hd := congrArg (eval (fun _ : Unit => (0 : ℂ)))
-    (congrArg (pderiv ()) hF)
-  rw [← mapCoeffs_pderiv, eval_mapCoeffs] at hd
-  have hD : eval (fun _ : Unit => algebraMap ℂ (P a n) 0) (pderiv () F) =
-      ∑ j, oneSlot p q j := by
-    dsimp only [F]
-    rw [eval_pderiv_prod]
-    apply Finset.sum_congr rfl
-    intro j _
-    simp only [map_add, pderiv_C, pderiv_mul, pderiv_X_self, mul_zero,
-      add_zero, zero_add, eval_C, eval_mul, eval_X, map_zero, zero_mul]
-    rw [oneSlot, tensorRows, ← Finset.mul_prod_erase Finset.univ _ (a := j) (by simp)]
-    simp only [ite_true, map_one, one_mul]
-    congr 1
-    apply Finset.prod_congr rfl
-    intro k hk
-    rw [ite_eq_right (Finset.mem_erase.mp hk).1]
-  rw [hD, map_sum] at hd
-  have hsum : (∑ j : Fin a, l (oneSlot p q j)) = (a : ℂ) * l (oneSlot p q i) := by
-    simp_rw [fun j => oneSlot_eq hl p q j i]
-    simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul]
-  rw [hsum] at hd
-  have haC : (a : ℂ) ≠ 0 := by exact_mod_cast ha.ne'
-  exact (mul_eq_zero.mp (by simpa using hd)).resolve_left haC
-
-end Foulkes.Vanishing
-
-namespace Foulkes.Vanishing
-open MvPolynomial Foulkes.PolynomialSpan Foulkes.GenericShift
-
-variable {A : Type*} [CommRing A]
-
-def freeBase {r m n : ℕ} (c : Fin r → Fin n → A) : Poly A (m+1) n :=
-  ∏ j : Fin m, ∏ i : Fin r, rowLinear j.castSucc (c i)
-
-def freeFamily {r m n : ℕ} (c : Fin r → Fin n → A) (d : Fin n → A)
-    (b : ℕ) (s : Finset (Fin m)) : Poly A (m+1) n :=
-  (freeBase c * ∏ j ∈ s, rowLinear j.castSucc d) * rowLinear (Fin.last m) d ^ (b-s.card)
-
-lemma shift_freeBase {r m n : ℕ} (c : Fin r → Fin n → A) (u : Fin m) :
-    shift (Fin.last m) u.castSucc (freeBase c) = 0 := by
-  apply derivation_prod_zero
-  intro j _
-  apply derivation_prod_zero
-  intro i _
-  simp
-
-lemma freeBase_degree_last {r m n : ℕ} (c : Fin r → Fin n → A) :
-    (freeBase (m := m) c).IsWeightedHomogeneous (rowWeight (Fin.last m)) 0 := by
-  have hi (j : Fin m) (i : Fin r) : (rowLinear j.castSucc (c i)).IsWeightedHomogeneous
-      (rowWeight (Fin.last m)) 0 := by simpa using rowLinear_homogeneous j.castSucc (Fin.last m) (c i)
-  have hj (j : Fin m) : (∏ i : Fin r, rowLinear j.castSucc (c i)).IsWeightedHomogeneous
-      (rowWeight (Fin.last m)) 0 := by
-    simpa using MvPolynomial.IsWeightedHomogeneous.prod Finset.univ _ _ (fun i _ => hi j i)
-  simpa [freeBase] using MvPolynomial.IsWeightedHomogeneous.prod Finset.univ _ _ (fun j _ => hj j)
-
-lemma freeBase_degree_cast {r m n : ℕ} (c : Fin r → Fin n → A) (u : Fin m) :
-    (freeBase c).IsWeightedHomogeneous (rowWeight u.castSucc) r := by
-  classical
-  have hi (j : Fin m) (i : Fin r) : (rowLinear j.castSucc (c i)).IsWeightedHomogeneous
-      (rowWeight u.castSucc) (if j = u then 1 else 0) := by
-    simpa using rowLinear_homogeneous j.castSucc u.castSucc (c i)
-  have hj (j : Fin m) : (∏ i : Fin r, rowLinear j.castSucc (c i)).IsWeightedHomogeneous
-      (rowWeight u.castSucc) (if j = u then r else 0) := by
-    have hh := MvPolynomial.IsWeightedHomogeneous.prod Finset.univ _ _ (fun i _ => hi j i)
-    split_ifs at hh ⊢ <;> simpa using hh
-  simpa [freeBase] using MvPolynomial.IsWeightedHomogeneous.prod Finset.univ _ _ (fun j _ => hj j)
-
-lemma freeFamily_degree_last {r m n : ℕ} (c : Fin r → Fin n → A) (d : Fin n → A)
-    (b : ℕ) (s : Finset (Fin m)) :
-    (freeFamily c d b s).IsWeightedHomogeneous (rowWeight (Fin.last m)) (b-s.card) := by
-  have hi (j : Fin m) : (rowLinear j.castSucc d).IsWeightedHomogeneous
-      (rowWeight (Fin.last m)) 0 := by simpa using rowLinear_homogeneous j.castSucc (Fin.last m) d
-  have hs : (∏ j ∈ s, rowLinear j.castSucc d).IsWeightedHomogeneous (rowWeight (Fin.last m)) 0 := by
-    simpa using MvPolynomial.IsWeightedHomogeneous.prod s _ _ (fun j _ => hi j)
-  have ht : (rowLinear (Fin.last m) d).IsWeightedHomogeneous (rowWeight (Fin.last m)) 1 := by
-    simpa using rowLinear_homogeneous (Fin.last m) (Fin.last m) d
-  simpa [freeFamily] using ((freeBase_degree_last c).mul hs).mul (ht.pow (b-s.card))
-
-lemma freeFamily_degree_cast {r m n : ℕ} (c : Fin r → Fin n → A) (d : Fin n → A)
-    (b : ℕ) (s : Finset (Fin m)) (u : Fin m) (hu : u ∉ s) :
-    (freeFamily c d b s).IsWeightedHomogeneous (rowWeight u.castSucc) r := by
-  have hi (j : Fin m) (hj : j ∈ s) : (rowLinear j.castSucc d).IsWeightedHomogeneous
-      (rowWeight u.castSucc) 0 := by
-    have hne : j ≠ u := ne_of_mem_of_not_mem hj hu
-    simpa [hne] using rowLinear_homogeneous j.castSucc u.castSucc d
-  have hs : (∏ j ∈ s, rowLinear j.castSucc d).IsWeightedHomogeneous (rowWeight u.castSucc) 0 := by
-    simpa using MvPolynomial.IsWeightedHomogeneous.prod s _ _ hi
-  have ht : (rowLinear (Fin.last m) d).IsWeightedHomogeneous (rowWeight u.castSucc) 0 := by
-    simpa [Ne.symm (Fin.castSucc_ne_last u)] using rowLinear_homogeneous (Fin.last m) u.castSucc d
-  simpa [freeFamily] using ((freeBase_degree_cast c u).mul hs).mul (ht.pow (b-s.card))
-
-lemma shift_freeFamily {r m n : ℕ} (c : Fin r → Fin n → A) (d : Fin n → A)
-    (b : ℕ) (s : Finset (Fin m)) (u : Fin m) (hu : u ∉ s) (hs : s.card < b) :
-    shift (Fin.last m) u.castSucc (freeFamily c d b s) =
-      (b-s.card) • freeFamily c d b (insert u s) := by
-  classical
-  let D := shift (A := A) (n := n) (Fin.last m) u.castSucc
-  have hG : D (freeBase c * ∏ j ∈ s, rowLinear j.castSucc d) = 0 := by
-    rw [D.leibniz, shift_freeBase]
-    have hz : D (∏ j ∈ s, rowLinear j.castSucc d) = 0 := by
-      apply derivation_prod_zero
-      intro j _
-      simp [D]
-    rw [hz]
-    simp
-  have hder := pow_apply_pow D (f := rowLinear (Fin.last m) d)
-    (g := rowLinear u.castSucc d) (by simp [D]) (by simp [D]) (b-s.card) 1 (by omega)
-  simp only [pow_one, Derivation.coeFn_coe, Nat.descFactorial_one] at hder
-  change D (_ * _) = _
-  rw [D.leibniz, hG, smul_zero, add_zero, hder]
-  simp only [freeFamily, Finset.prod_insert hu, Finset.card_insert_of_notMem hu,
-    smul_eq_mul, nsmul_eq_mul]
-  have he : b-s.card-1 = b-(s.card+1) := by omega
-  rw [he]
-  ring
-
-theorem freeFamily_zero [Algebra ℂ A] {r m n b : ℕ}
-    (c : Fin r → Fin n → A) (d : Fin n → A) (l : A →ₗ[ℂ] ℂ)
-    (hb : m+r ≤ b) (hz : mapCoeffs l (freeFamily (m := m) c d b Finset.univ) = 0) :
-    mapCoeffs l (freeFamily (m := m) c d b ∅) = 0 := by
-  classical
-  have ht (t : Finset (Fin m)) :
-      mapCoeffs l (freeFamily c d b (Finset.univ \ t)) = 0 := by
-    induction t using Finset.induction_on with
-    | empty => simpa using hz
-    | insert u t hu ih =>
-      let s : Finset (Fin m) := Finset.univ \ insert u t
-      have hus : u ∉ s := by simp [s]
-      have hse : insert u s = Finset.univ \ t := by
-        ext i
-        by_cases hi : i = u <;> simp [s, hi, hu]
-      have hcard : s.card + 1 ≤ m := by
-        calc
-          s.card + 1 = (insert u s).card := (Finset.card_insert_of_notMem hus).symm
-          _ ≤ (Finset.univ : Finset (Fin m)).card := Finset.card_le_card (Finset.subset_univ _)
-          _ = m := by simp
-      change mapCoeffs l (freeFamily c d b s) = 0
-      apply Differential.shift_kernel_zero (u := Fin.last m) (v := u.castSucc) (Ne.symm (Fin.castSucc_ne_last u))
-        (homogeneous_mapCoeffs l _ (freeFamily_degree_last c d b s))
-        (homogeneous_mapCoeffs l _ (freeFamily_degree_cast c d b s u hus)) (by omega)
-      change shift (Fin.last m) u.castSucc (mapCoeffs l (freeFamily c d b s)) = 0
-      rw [← mapCoeffs_shift, shift_freeFamily c d b s u hus (by omega), hse]
-      change (mapCoeffsLM l) ((b-s.card) • freeFamily c d b (Finset.univ \ t)) = 0
-      rw [map_nsmul]
-      change (b-s.card) • mapCoeffs l (freeFamily c d b (Finset.univ \ t)) = 0
-      rw [ih, smul_zero]
-  simpa using ht Finset.univ
-
-end Foulkes.Vanishing
-
-namespace Foulkes.Vanishing
-open MvPolynomial Foulkes.Polarization Foulkes.PolynomialSpan
-
-lemma lin_binary_split {n r : ℕ} (hr : 0 < r) (c : ℂ) (x t : Fin n → ℂ) :
-    ∃ v : Fin r → Fin n → ℂ, lin x ^ r + c • lin t ^ r = ∏ j, lin (v j) := by
-  obtain ⟨v,hv⟩ := binary_split (a := 1) hr c x t
-  refine ⟨v, ?_⟩
-  have hh := congrArg (rename (Prod.snd : Fin 1 × Fin n → Fin n)) (hv 0)
-  simpa [rowForm, lin] using hh
-
-def vecProduct {m n : ℕ} (y : Fin m → Fin n → ℂ) : S n := ∏ j, lin (y j)
-
-def ProductsVanish {a n : ℕ} (l : P a n →ₗ[ℂ] ℂ) (b : ℕ) : Prop :=
-  ∀ y : Fin b → Fin n → ℂ, l (∏ j, z (y j)) = 0
-
-lemma tensorRows_diag_product {a m n : ℕ} (y : Fin m → Fin n → ℂ) :
-    tensorRows (a := a) (fun _ => vecProduct y) = ∏ j, z (y j) := by
-  simp only [tensorRows, vecProduct, map_prod, rowLift_lin, z]
-  exact Finset.prod_comm
-
-lemma productsVanish_mul {a n r m : ℕ} {l : P a n →ₗ[ℂ] ℂ}
-    (hl : ProductsVanish l (r+m)) (u : Fin r → Fin n → ℂ) (v : Fin m → Fin n → ℂ) :
-    l (tensorRows (fun _ => vecProduct u * vecProduct v)) = 0 := by
-  have hh := hl (Fin.addCases u v)
-  rw [Fin.prod_univ_add] at hh
-  simp only [Fin.addCases_left, Fin.addCases_right] at hh
-  have he : tensorRows (a := a) (fun _ => vecProduct u * vecProduct v) =
-      (∏ i, z (u i)) * ∏ j, z (v j) := by
-    simp only [tensorRows, map_mul, Finset.prod_mul_distrib]
-    rw [← tensorRows, ← tensorRows, tensorRows_diag_product, tensorRows_diag_product]
-  rwa [he]
-
-lemma binary_step {r m n : ℕ} (hr : 0 < r) {l : P (r+1) n →ₗ[ℂ] ℂ}
-    (hs : Symmetric l) (hz : ProductsVanish l (r+m))
-    (x t : Fin n → ℂ) (y : Fin m → Fin n → ℂ) :
-    l (oneSlot (lin x ^ r * vecProduct y) (lin t ^ r * vecProduct y) (Fin.last r)) = 0 := by
-  apply binary_coefficient (by omega) hs
-  intro c
-  obtain ⟨u, hu⟩ := lin_binary_split hr c x t
-  have he : lin x ^ r * vecProduct y + c • (lin t ^ r * vecProduct y) =
-      vecProduct u * vecProduct y := by
-    rw [← smul_mul_assoc, ← add_mul, hu]
-    rfl
-  rw [he]
-  exact productsVanish_mul hz u y
-
-lemma oneSlot_last {r n : ℕ} (p q : S n) :
-    oneSlot p q (Fin.last r) = (∏ i : Fin r, rowLift i.castSucc p) * rowLift (Fin.last r) q := by
-  rw [oneSlot, tensorRows, Fin.prod_univ_castSucc]
-  simp only [Fin.castSucc_ne_last, ite_false, ite_true]
-
-def commonPower {r n : ℕ} (x : Fin n → ℂ) : P (r+1) n :=
-  ∏ i : Fin r, rowForm i.castSucc x ^ r
-
-def fixedFunctional {r n : ℕ} (l : P (r+1) n →ₗ[ℂ] ℂ) (x : Fin n → ℂ) :
-    P (r+1) n →ₗ[ℂ] ℂ := l.comp (LinearMap.mulLeft ℂ (commonPower x))
-
-@[simp] lemma fixedFunctional_apply {r n : ℕ} (l : P (r+1) n →ₗ[ℂ] ℂ)
-    (x : Fin n → ℂ) (f : P (r+1) n) :
-    fixedFunctional l x f = l (commonPower x * f) := rfl
-
-lemma eval_freeFamily {r m n b : ℕ} (s : Finset (Fin m))
-    (w : Fin (m+1) × Fin n → ℂ) :
-    eval (fun ij => algebraMap ℂ (P (r+1) n) (w ij))
-      (freeFamily (fun i : Fin r => fun k => (X (i.castSucc,k) : P (r+1) n))
-        (fun k => X (Fin.last r,k)) b s) =
-      ((∏ j : Fin m, ∏ i : Fin r, rowForm i.castSucc (fun k => w (j.castSucc,k))) *
-        ∏ j ∈ s, rowForm (Fin.last r) (fun k => w (j.castSucc,k))) *
-          rowForm (Fin.last r) (fun k => w (Fin.last m,k)) ^ (b-s.card) := by
-  simp only [freeFamily, freeBase, map_mul, map_prod, map_pow, GenericShift.eval_rowLinear, rowForm]
-
-lemma free_last_slot {r m n : ℕ} (hr : 0 < r) {l : P (r+1) n →ₗ[ℂ] ℂ}
-    (hs : Symmetric l) (hz : ProductsVanish l (r+m))
-    (x t : Fin n → ℂ) (y : Fin m → Fin n → ℂ) :
-    l ((∏ i : Fin r, rowLift i.castSucc (lin x ^ r * vecProduct y)) *
-      rowForm (Fin.last r) t ^ (r+m)) = 0 := by
-  let c := fun i : Fin r => fun k : Fin n => (X (i.castSucc,k) : P (r+1) n)
-  let d := fun k : Fin n => (X (Fin.last r,k) : P (r+1) n)
-  have hfull : mapCoeffs (fixedFunctional l x)
-      (freeFamily (m := m) c d (r+m) Finset.univ) = 0 := by
-    apply MvPolynomial.funext
-    intro w
-    rw [eval_mapCoeffs]
-    simp only [c, d, eval_freeFamily, Finset.card_univ, Fintype.card_fin,
-      Nat.add_sub_cancel_right, fixedFunctional_apply, map_zero]
-    have hh := binary_step hr hs hz x (fun k => w (Fin.last m,k))
-      (fun j k => w (j.castSucc,k))
-    rw [oneSlot_last] at hh
-    simp only [map_mul, map_pow, rowLift_lin, vecProduct, map_prod,
-      Finset.prod_mul_distrib] at hh
-    have hcomm : (∏ i : Fin r, ∏ j : Fin m, rowForm i.castSucc (fun k => w (j.castSucc,k))) =
-        ∏ j : Fin m, ∏ i : Fin r, rowForm i.castSucc (fun k => w (j.castSucc,k)) := Finset.prod_comm
-    rw [hcomm] at hh
-    convert hh using 1
-    dsimp only [commonPower]
-    congr 1
-    ring
-  have hzero := freeFamily_zero c d (fixedFunctional l x) (by omega : m+r ≤ r+m) hfull
-  let w : Fin (m+1) × Fin n → ℂ := fun ij => Fin.lastCases (t ij.2) (fun j => y j ij.2) ij.1
-  have hh := congrArg (eval w) hzero
-  rw [eval_mapCoeffs] at hh
-  simp only [c, d, eval_freeFamily, Finset.prod_empty, Finset.card_empty, Nat.sub_zero,
-    mul_one, fixedFunctional_apply, map_zero, w, Fin.lastCases_castSucc, Fin.lastCases_last] at hh
-  simp only [commonPower] at hh
-  simp only [map_mul, map_pow, rowLift_lin, vecProduct, map_prod, Finset.prod_mul_distrib]
-  have hcomm : (∏ i : Fin r, ∏ j : Fin m, rowForm i.castSucc (y j)) =
-      ∏ j : Fin m, ∏ i : Fin r, rowForm i.castSucc (y j) := Finset.prod_comm
-  rw [hcomm]
-  simpa only [mul_assoc] using hh
-
-end Foulkes.Vanishing
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.FoulkesSixth.InvariantPolarization
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.Vanishing
+open MvPolynomial Foulkes.Polarization Foulkes.PolynomialSpan
+
+lemma binary_split {a n r : ℕ} (hr : 0 < r) (c : ℂ) (x t : Fin n → ℂ) :
+    ∃ v : Fin r → Fin n → ℂ, ∀ i : Fin a,
+      rowForm i x ^ r + c • rowForm i t ^ r = ∏ j, rowForm i (v j) := by
+  classical
+  obtain ⟨α, hα⟩ := IsAlgClosed.exists_pow_nat_eq (-c) hr
+  let ζ := Complex.exp (2 * Real.pi * Complex.I / r)
+  have hζ : IsPrimitiveRoot ζ r := Complex.isPrimitiveRoot_exp r hr.ne'
+  let S := Polynomial.nthRootsFinset r (1 : ℂ)
+  have hS : Fintype.card S = r := by
+    rw [Fintype.card_coe]
+    exact hζ.card_nthRootsFinset
+  let e : Fin r ≃ S := (Fintype.equivFinOfCardEq hS).symm
+  refine ⟨fun j k => x k - (α * (e j : ℂ)) * t k, ?_⟩
+  intro i
+  apply MvPolynomial.funext
+  intro w
+  let X₀ := ∑ k, x k * w (i,k)
+  let Y₀ := ∑ k, t k * w (i,k)
+  have heval (j : Fin r) :
+      eval w (rowForm i (fun k => x k - α * (e j : ℂ) * t k)) =
+        X₀ - (e j : ℂ) * (α * Y₀) := by
+    simp only [rowForm, eval_sum, MvPolynomial.smul_eq_C_mul, eval_mul, eval_C, eval_X]
+    simp only [sub_mul, Finset.sum_sub_distrib]
+    dsimp [X₀, Y₀]
+    simp only [Finset.mul_sum]
+    congr 1
+    apply Finset.sum_congr rfl
+    intro k _
+    ring
+  have hprod : (∏ j : Fin r, (X₀ - (e j : ℂ) * (α * Y₀))) =
+      ∏ z ∈ S, (X₀ - z * (α * Y₀)) := by
+    calc
+      _ = ∏ z : S, (X₀ - (z : ℂ) * (α * Y₀)) :=
+        Fintype.prod_equiv e _ _ (fun _ => rfl)
+      _ = _ := Finset.prod_coe_sort S (fun z : ℂ => X₀ - z * (α * Y₀))
+  simp only [eval_add, eval_pow, MvPolynomial.smul_eq_C_mul, eval_mul, eval_C, eval_prod, heval]
+  have hx : eval w (rowForm i x) = X₀ := by simp [rowForm, X₀, MvPolynomial.smul_eq_C_mul]
+  have hy : eval w (rowForm i t) = Y₀ := by simp [rowForm, Y₀, MvPolynomial.smul_eq_C_mul]
+  rw [hx, hy]
+  rw [hprod, ← hζ.pow_sub_pow_eq_prod_sub_mul X₀ (α * Y₀) hr, mul_pow, hα]
+  ring
+
+abbrev S (n : ℕ) := MvPolynomial (Fin n) ℂ
+
+def rowLift {a n : ℕ} (i : Fin a) : S n →ₐ[ℂ] P a n :=
+  rename (fun k => (i,k))
+
+def tensorRows {a n : ℕ} (p : Fin a → S n) : P a n := ∏ i, rowLift i (p i)
+
+def lin {n : ℕ} (x : Fin n → ℂ) : S n := ∑ k, x k • X k
+
+@[simp] lemma rowLift_lin {a n : ℕ} (i : Fin a) (x : Fin n → ℂ) :
+    rowLift i (lin x) = rowForm i x := by simp [rowLift, lin, rowForm]
+
+@[simp] lemma renameRow_rowLift {a n : ℕ} (σ : Equiv.Perm (Fin a))
+    (i : Fin a) (p : S n) : renameRow σ (rowLift i p) = rowLift (σ i) p := by
+  simp [rowLift, renameRow, rename_rename, Function.comp_def]
+
+lemma renameRow_tensorRows {a n : ℕ} (σ : Equiv.Perm (Fin a)) (p : Fin a → S n) :
+    renameRow σ (tensorRows p) = tensorRows (fun i => p (σ.symm i)) := by
+  simp only [tensorRows, map_prod, renameRow_rowLift]
+  apply Fintype.prod_equiv σ
+  intro i
+  simp
+
+def Symmetric {a n : ℕ} (l : P a n →ₗ[ℂ] ℂ) : Prop :=
+  ∀ (σ : Equiv.Perm (Fin a)) (f : P a n), l (renameRow σ f) = l f
+
+lemma symmetric_tensorRows {a n : ℕ} {l : P a n →ₗ[ℂ] ℂ} (hl : Symmetric l)
+    (σ : Equiv.Perm (Fin a)) (p : Fin a → S n) :
+    l (tensorRows (fun i => p (σ.symm i))) = l (tensorRows p) := by
+  rw [← renameRow_tensorRows, hl]
+
+def oneSlot {a n : ℕ} (p q : S n) (i : Fin a) : P a n :=
+  tensorRows (fun j => if j = i then q else p)
+
+lemma oneSlot_eq {a n : ℕ} {l : P a n →ₗ[ℂ] ℂ} (hl : Symmetric l)
+    (p q : S n) (i j : Fin a) : l (oneSlot p q i) = l (oneSlot p q j) := by
+  classical
+  have hh := symmetric_tensorRows hl (Equiv.swap i j) (fun k => if k = j then q else p)
+  have he : (fun k => if (Equiv.swap i j).symm k = j then q else p) =
+      (fun k => if k = i then q else p) := by
+    funext k
+    simp [Equiv.swap_apply_eq_iff]
+  rw [he] at hh
+  exact hh
+
+lemma eval_pderiv_prod {A ι σ : Type*} [CommRing A] [Fintype ι] [DecidableEq ι]
+    (v : σ) (w : σ → A) (f : ι → MvPolynomial σ A) :
+    eval w (pderiv v (∏ i, f i)) =
+      ∑ i, eval w (pderiv v (f i)) * ∏ j ∈ Finset.univ.erase i, eval w (f j) := by
+  classical
+  have hs (s : Finset ι) : eval w (pderiv v (∏ i ∈ s, f i)) =
+      ∑ i ∈ s, eval w (pderiv v (f i)) * ∏ j ∈ s.erase i, eval w (f j) := by
+    induction s using Finset.induction_on with
+    | empty => simp
+    | insert u s hu ih =>
+      rw [Finset.prod_insert hu, pderiv_mul, eval_add, eval_mul, eval_mul, ih,
+        Finset.sum_insert hu]
+      rw [Finset.erase_insert hu]
+      simp only [eval_prod, Finset.mul_sum]
+      congr 1
+      apply Finset.sum_congr rfl
+      intro i hi
+      have hne : i ≠ u := ne_of_mem_of_not_mem hi hu
+      rw [Finset.erase_insert_of_ne hne.symm, Finset.prod_insert (by simp [hu])]
+      ring
+  exact hs Finset.univ
+
+lemma binary_coefficient {a n : ℕ} (ha : 0 < a) {l : P a n →ₗ[ℂ] ℂ}
+    (hl : Symmetric l) (p q : S n)
+    (hz : ∀ c : ℂ, l (tensorRows (fun _ => p + c • q)) = 0) (i : Fin a) :
+    l (oneSlot p q i) = 0 := by
+  classical
+  let F : MvPolynomial Unit (P a n) :=
+    ∏ j, (C (rowLift j p) + X () * C (rowLift j q))
+  have heval (c : ℂ) : eval (fun _ : Unit => algebraMap ℂ (P a n) c) F =
+      tensorRows (fun _ => p + c • q) := by
+    simp [F, tensorRows, rowLift, Algebra.smul_def]
+  have hF : mapCoeffs l F = 0 := by
+    apply MvPolynomial.funext
+    intro c
+    rw [eval_mapCoeffs]
+    have hc : c = fun _ : Unit => c () := funext (fun u => by cases u; rfl)
+    rw [hc, heval, hz, map_zero]
+  have hd := congrArg (eval (fun _ : Unit => (0 : ℂ)))
+    (congrArg (pderiv ()) hF)
+  rw [← mapCoeffs_pderiv, eval_mapCoeffs] at hd
+  have hD : eval (fun _ : Unit => algebraMap ℂ (P a n) 0) (pderiv () F) =
+      ∑ j, oneSlot p q j := by
+    dsimp only [F]
+    rw [eval_pderiv_prod]
+    apply Finset.sum_congr rfl
+    intro j _
+    simp only [map_add, pderiv_C, pderiv_mul, pderiv_X_self, mul_zero,
+      add_zero, zero_add, eval_C, eval_mul, eval_X, map_zero, zero_mul]
+    rw [oneSlot, tensorRows, ← Finset.mul_prod_erase Finset.univ _ (a := j) (by simp)]
+    simp only [ite_true, map_one, one_mul]
+    congr 1
+    apply Finset.prod_congr rfl
+    intro k hk
+    rw [ite_eq_right (Finset.mem_erase.mp hk).1]
+  rw [hD, map_sum] at hd
+  have hsum : (∑ j : Fin a, l (oneSlot p q j)) = (a : ℂ) * l (oneSlot p q i) := by
+    simp_rw [fun j => oneSlot_eq hl p q j i]
+    simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul]
+  rw [hsum] at hd
+  have haC : (a : ℂ) ≠ 0 := by exact_mod_cast ha.ne'
+  exact (mul_eq_zero.mp (by simpa using hd)).resolve_left haC
+
+end Foulkes.Vanishing
+
+namespace Foulkes.Vanishing
+open MvPolynomial Foulkes.PolynomialSpan Foulkes.GenericShift
+
+variable {A : Type*} [CommRing A]
+
+def freeBase {r m n : ℕ} (c : Fin r → Fin n → A) : Poly A (m+1) n :=
+  ∏ j : Fin m, ∏ i : Fin r, rowLinear j.castSucc (c i)
+
+def freeFamily {r m n : ℕ} (c : Fin r → Fin n → A) (d : Fin n → A)
+    (b : ℕ) (s : Finset (Fin m)) : Poly A (m+1) n :=
+  (freeBase c * ∏ j ∈ s, rowLinear j.castSucc d) * rowLinear (Fin.last m) d ^ (b-s.card)
+
+lemma shift_freeBase {r m n : ℕ} (c : Fin r → Fin n → A) (u : Fin m) :
+    shift (Fin.last m) u.castSucc (freeBase c) = 0 := by
+  apply derivation_prod_zero
+  intro j _
+  apply derivation_prod_zero
+  intro i _
+  simp
+
+lemma freeBase_degree_last {r m n : ℕ} (c : Fin r → Fin n → A) :
+    (freeBase (m := m) c).IsWeightedHomogeneous (rowWeight (Fin.last m)) 0 := by
+  have hi (j : Fin m) (i : Fin r) : (rowLinear j.castSucc (c i)).IsWeightedHomogeneous
+      (rowWeight (Fin.last m)) 0 := by simpa using rowLinear_homogeneous j.castSucc (Fin.last m) (c i)
+  have hj (j : Fin m) : (∏ i : Fin r, rowLinear j.castSucc (c i)).IsWeightedHomogeneous
+      (rowWeight (Fin.last m)) 0 := by
+    simpa using MvPolynomial.IsWeightedHomogeneous.prod Finset.univ _ _ (fun i _ => hi j i)
+  simpa [freeBase] using MvPolynomial.IsWeightedHomogeneous.prod Finset.univ _ _ (fun j _ => hj j)
+
+lemma freeBase_degree_cast {r m n : ℕ} (c : Fin r → Fin n → A) (u : Fin m) :
+    (freeBase c).IsWeightedHomogeneous (rowWeight u.castSucc) r := by
+  classical
+  have hi (j : Fin m) (i : Fin r) : (rowLinear j.castSucc (c i)).IsWeightedHomogeneous
+      (rowWeight u.castSucc) (if j = u then 1 else 0) := by
+    simpa using rowLinear_homogeneous j.castSucc u.castSucc (c i)
+  have hj (j : Fin m) : (∏ i : Fin r, rowLinear j.castSucc (c i)).IsWeightedHomogeneous
+      (rowWeight u.castSucc) (if j = u then r else 0) := by
+    have hh := MvPolynomial.IsWeightedHomogeneous.prod Finset.univ _ _ (fun i _ => hi j i)
+    split_ifs at hh ⊢ <;> simpa using hh
+  simpa [freeBase] using MvPolynomial.IsWeightedHomogeneous.prod Finset.univ _ _ (fun j _ => hj j)
+
+lemma freeFamily_degree_last {r m n : ℕ} (c : Fin r → Fin n → A) (d : Fin n → A)
+    (b : ℕ) (s : Finset (Fin m)) :
+    (freeFamily c d b s).IsWeightedHomogeneous (rowWeight (Fin.last m)) (b-s.card) := by
+  have hi (j : Fin m) : (rowLinear j.castSucc d).IsWeightedHomogeneous
+      (rowWeight (Fin.last m)) 0 := by simpa using rowLinear_homogeneous j.castSucc (Fin.last m) d
+  have hs : (∏ j ∈ s, rowLinear j.castSucc d).IsWeightedHomogeneous (rowWeight (Fin.last m)) 0 := by
+    simpa using MvPolynomial.IsWeightedHomogeneous.prod s _ _ (fun j _ => hi j)
+  have ht : (rowLinear (Fin.last m) d).IsWeightedHomogeneous (rowWeight (Fin.last m)) 1 := by
+    simpa using rowLinear_homogeneous (Fin.last m) (Fin.last m) d
+  simpa [freeFamily] using ((freeBase_degree_last c).mul hs).mul (ht.pow (b-s.card))
+
+lemma freeFamily_degree_cast {r m n : ℕ} (c : Fin r → Fin n → A) (d : Fin n → A)
+    (b : ℕ) (s : Finset (Fin m)) (u : Fin m) (hu : u ∉ s) :
+    (freeFamily c d b s).IsWeightedHomogeneous (rowWeight u.castSucc) r := by
+  have hi (j : Fin m) (hj : j ∈ s) : (rowLinear j.castSucc d).IsWeightedHomogeneous
+      (rowWeight u.castSucc) 0 := by
+    have hne : j ≠ u := ne_of_mem_of_not_mem hj hu
+    simpa [hne] using rowLinear_homogeneous j.castSucc u.castSucc d
+  have hs : (∏ j ∈ s, rowLinear j.castSucc d).IsWeightedHomogeneous (rowWeight u.castSucc) 0 := by
+    simpa using MvPolynomial.IsWeightedHomogeneous.prod s _ _ hi
+  have ht : (rowLinear (Fin.last m) d).IsWeightedHomogeneous (rowWeight u.castSucc) 0 := by
+    simpa [Ne.symm (Fin.castSucc_ne_last u)] using rowLinear_homogeneous (Fin.last m) u.castSucc d
+  simpa [freeFamily] using ((freeBase_degree_cast c u).mul hs).mul (ht.pow (b-s.card))
+
+lemma shift_freeFamily {r m n : ℕ} (c : Fin r → Fin n → A) (d : Fin n → A)
+    (b : ℕ) (s : Finset (Fin m)) (u : Fin m) (hu : u ∉ s) (hs : s.card < b) :
+    shift (Fin.last m) u.castSucc (freeFamily c d b s) =
+      (b-s.card) • freeFamily c d b (insert u s) := by
+  classical
+  let D := shift (A := A) (n := n) (Fin.last m) u.castSucc
+  have hG : D (freeBase c * ∏ j ∈ s, rowLinear j.castSucc d) = 0 := by
+    rw [D.leibniz, shift_freeBase]
+    have hz : D (∏ j ∈ s, rowLinear j.castSucc d) = 0 := by
+      apply derivation_prod_zero
+      intro j _
+      simp [D]
+    rw [hz]
+    simp
+  have hder := pow_apply_pow D (f := rowLinear (Fin.last m) d)
+    (g := rowLinear u.castSucc d) (by simp [D]) (by simp [D]) (b-s.card) 1 (by omega)
+  simp only [pow_one, Derivation.coeFn_coe, Nat.descFactorial_one] at hder
+  change D (_ * _) = _
+  rw [D.leibniz, hG, smul_zero, add_zero, hder]
+  simp only [freeFamily, Finset.prod_insert hu, Finset.card_insert_of_notMem hu,
+    smul_eq_mul, nsmul_eq_mul]
+  have he : b-s.card-1 = b-(s.card+1) := by omega
+  rw [he]
+  ring
+
+theorem freeFamily_zero [Algebra ℂ A] {r m n b : ℕ}
+    (c : Fin r → Fin n → A) (d : Fin n → A) (l : A →ₗ[ℂ] ℂ)
+    (hb : m+r ≤ b) (hz : mapCoeffs l (freeFamily (m := m) c d b Finset.univ) = 0) :
+    mapCoeffs l (freeFamily (m := m) c d b ∅) = 0 := by
+  classical
+  have ht (t : Finset (Fin m)) :
+      mapCoeffs l (freeFamily c d b (Finset.univ \ t)) = 0 := by
+    induction t using Finset.induction_on with
+    | empty => simpa using hz
+    | insert u t hu ih =>
+      let s : Finset (Fin m) := Finset.univ \ insert u t
+      have hus : u ∉ s := by simp [s]
+      have hse : insert u s = Finset.univ \ t := by
+        ext i
+        by_cases hi : i = u <;> simp [s, hi, hu]
+      have hcard : s.card + 1 ≤ m := by
+        calc
+          s.card + 1 = (insert u s).card := (Finset.card_insert_of_notMem hus).symm
+          _ ≤ (Finset.univ : Finset (Fin m)).card := Finset.card_le_card (Finset.subset_univ _)
+          _ = m := by simp
+      change mapCoeffs l (freeFamily c d b s) = 0
+      apply Differential.shift_kernel_zero (u := Fin.last m) (v := u.castSucc) (Ne.symm (Fin.castSucc_ne_last u))
+        (homogeneous_mapCoeffs l _ (freeFamily_degree_last c d b s))
+        (homogeneous_mapCoeffs l _ (freeFamily_degree_cast c d b s u hus)) (by omega)
+      change shift (Fin.last m) u.castSucc (mapCoeffs l (freeFamily c d b s)) = 0
+      rw [← mapCoeffs_shift, shift_freeFamily c d b s u hus (by omega), hse]
+      change (mapCoeffsLM l) ((b-s.card) • freeFamily c d b (Finset.univ \ t)) = 0
+      rw [map_nsmul]
+      change (b-s.card) • mapCoeffs l (freeFamily c d b (Finset.univ \ t)) = 0
+      rw [ih, smul_zero]
+  simpa using ht Finset.univ
+
+end Foulkes.Vanishing
+
+namespace Foulkes.Vanishing
+open MvPolynomial Foulkes.Polarization Foulkes.PolynomialSpan
+
+lemma lin_binary_split {n r : ℕ} (hr : 0 < r) (c : ℂ) (x t : Fin n → ℂ) :
+    ∃ v : Fin r → Fin n → ℂ, lin x ^ r + c • lin t ^ r = ∏ j, lin (v j) := by
+  obtain ⟨v,hv⟩ := binary_split (a := 1) hr c x t
+  refine ⟨v, ?_⟩
+  have hh := congrArg (rename (Prod.snd : Fin 1 × Fin n → Fin n)) (hv 0)
+  simpa [rowForm, lin] using hh
+
+def vecProduct {m n : ℕ} (y : Fin m → Fin n → ℂ) : S n := ∏ j, lin (y j)
+
+def ProductsVanish {a n : ℕ} (l : P a n →ₗ[ℂ] ℂ) (b : ℕ) : Prop :=
+  ∀ y : Fin b → Fin n → ℂ, l (∏ j, z (y j)) = 0
+
+lemma tensorRows_diag_product {a m n : ℕ} (y : Fin m → Fin n → ℂ) :
+    tensorRows (a := a) (fun _ => vecProduct y) = ∏ j, z (y j) := by
+  simp only [tensorRows, vecProduct, map_prod, rowLift_lin, z]
+  exact Finset.prod_comm
+
+lemma productsVanish_mul {a n r m : ℕ} {l : P a n →ₗ[ℂ] ℂ}
+    (hl : ProductsVanish l (r+m)) (u : Fin r → Fin n → ℂ) (v : Fin m → Fin n → ℂ) :
+    l (tensorRows (fun _ => vecProduct u * vecProduct v)) = 0 := by
+  have hh := hl (Fin.addCases u v)
+  rw [Fin.prod_univ_add] at hh
+  simp only [Fin.addCases_left, Fin.addCases_right] at hh
+  have he : tensorRows (a := a) (fun _ => vecProduct u * vecProduct v) =
+      (∏ i, z (u i)) * ∏ j, z (v j) := by
+    simp only [tensorRows, map_mul, Finset.prod_mul_distrib]
+    rw [← tensorRows, ← tensorRows, tensorRows_diag_product, tensorRows_diag_product]
+  rwa [he]
+
+lemma binary_step {r m n : ℕ} (hr : 0 < r) {l : P (r+1) n →ₗ[ℂ] ℂ}
+    (hs : Symmetric l) (hz : ProductsVanish l (r+m))
+    (x t : Fin n → ℂ) (y : Fin m → Fin n → ℂ) :
+    l (oneSlot (lin x ^ r * vecProduct y) (lin t ^ r * vecProduct y) (Fin.last r)) = 0 := by
+  apply binary_coefficient (by omega) hs
+  intro c
+  obtain ⟨u, hu⟩ := lin_binary_split hr c x t
+  have he : lin x ^ r * vecProduct y + c • (lin t ^ r * vecProduct y) =
+      vecProduct u * vecProduct y := by
+    rw [← smul_mul_assoc, ← add_mul, hu]
+    rfl
+  rw [he]
+  exact productsVanish_mul hz u y
+
+lemma oneSlot_last {r n : ℕ} (p q : S n) :
+    oneSlot p q (Fin.last r) = (∏ i : Fin r, rowLift i.castSucc p) * rowLift (Fin.last r) q := by
+  rw [oneSlot, tensorRows, Fin.prod_univ_castSucc]
+  simp only [Fin.castSucc_ne_last, ite_false, ite_true]
+
+def commonPower {r n : ℕ} (x : Fin n → ℂ) : P (r+1) n :=
+  ∏ i : Fin r, rowForm i.castSucc x ^ r
+
+def fixedFunctional {r n : ℕ} (l : P (r+1) n →ₗ[ℂ] ℂ) (x : Fin n → ℂ) :
+    P (r+1) n →ₗ[ℂ] ℂ := l.comp (LinearMap.mulLeft ℂ (commonPower x))
+
+@[simp] lemma fixedFunctional_apply {r n : ℕ} (l : P (r+1) n →ₗ[ℂ] ℂ)
+    (x : Fin n → ℂ) (f : P (r+1) n) :
+    fixedFunctional l x f = l (commonPower x * f) := rfl
+
+lemma eval_freeFamily {r m n b : ℕ} (s : Finset (Fin m))
+    (w : Fin (m+1) × Fin n → ℂ) :
+    eval (fun ij => algebraMap ℂ (P (r+1) n) (w ij))
+      (freeFamily (fun i : Fin r => fun k => (X (i.castSucc,k) : P (r+1) n))
+        (fun k => X (Fin.last r,k)) b s) =
+      ((∏ j : Fin m, ∏ i : Fin r, rowForm i.castSucc (fun k => w (j.castSucc,k))) *
+        ∏ j ∈ s, rowForm (Fin.last r) (fun k => w (j.castSucc,k))) *
+          rowForm (Fin.last r) (fun k => w (Fin.last m,k)) ^ (b-s.card) := by
+  simp only [freeFamily, freeBase, map_mul, map_prod, map_pow, GenericShift.eval_rowLinear, rowForm]
+
+lemma free_last_slot {r m n : ℕ} (hr : 0 < r) {l : P (r+1) n →ₗ[ℂ] ℂ}
+    (hs : Symmetric l) (hz : ProductsVanish l (r+m))
+    (x t : Fin n → ℂ) (y : Fin m → Fin n → ℂ) :
+    l ((∏ i : Fin r, rowLift i.castSucc (lin x ^ r * vecProduct y)) *
+      rowForm (Fin.last r) t ^ (r+m)) = 0 := by
+  let c := fun i : Fin r => fun k : Fin n => (X (i.castSucc,k) : P (r+1) n)
+  let d := fun k : Fin n => (X (Fin.last r,k) : P (r+1) n)
+  have hfull : mapCoeffs (fixedFunctional l x)
+      (freeFamily (m := m) c d (r+m) Finset.univ) = 0 := by
+    apply MvPolynomial.funext
+    intro w
+    rw [eval_mapCoeffs]
+    simp only [c, d, eval_freeFamily, Finset.card_univ, Fintype.card_fin,
+      Nat.add_sub_cancel_right, fixedFunctional_apply, map_zero]
+    have hh := binary_step hr hs hz x (fun k => w (Fin.last m,k))
+      (fun j k => w (j.castSucc,k))
+    rw [oneSlot_last] at hh
+    simp only [map_mul, map_pow, rowLift_lin, vecProduct, map_prod,
+      Finset.prod_mul_distrib] at hh
+    have hcomm : (∏ i : Fin r, ∏ j : Fin m, rowForm i.castSucc (fun k => w (j.castSucc,k))) =
+        ∏ j : Fin m, ∏ i : Fin r, rowForm i.castSucc (fun k => w (j.castSucc,k)) := Finset.prod_comm
+    rw [hcomm] at hh
+    convert hh using 1
+    dsimp only [commonPower]
+    congr 1
+    ring
+  have hzero := freeFamily_zero c d (fixedFunctional l x) (by omega : m+r ≤ r+m) hfull
+  let w : Fin (m+1) × Fin n → ℂ := fun ij => Fin.lastCases (t ij.2) (fun j => y j ij.2) ij.1
+  have hh := congrArg (eval w) hzero
+  rw [eval_mapCoeffs] at hh
+  simp only [c, d, eval_freeFamily, Finset.prod_empty, Finset.card_empty, Nat.sub_zero,
+    mul_one, fixedFunctional_apply, map_zero, w, Fin.lastCases_castSucc, Fin.lastCases_last] at hh
+  simp only [commonPower] at hh
+  simp only [map_mul, map_pow, rowLift_lin, vecProduct, map_prod, Finset.prod_mul_distrib]
+  have hcomm : (∏ i : Fin r, ∏ j : Fin m, rowForm i.castSucc (y j)) =
+      ∏ j : Fin m, ∏ i : Fin r, rowForm i.castSucc (y j) := Finset.prod_comm
+  rw [hcomm]
+  simpa only [mul_assoc] using hh
+
+end Foulkes.Vanishing
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/YoungGram.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/YoungGram.lean
index 4c6ce0249a92cbbf18fe7180dfc0b57d57ba54e2..ed7d689d197c46ec3affb8292e497236eb1826c0 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/YoungGram.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/YoungGram.lean
@@ -1,72 +1,72 @@
-import Mathlib
-import OAI.RepresentationTheory.FoulkesSixth.ColorOrbits
-import OAI.RepresentationTheory.FoulkesSixth.HomogeneousPlethysm
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.Young
-open MvPolynomial Foulkes.Complete Finset Module
-
-instance finiteContingency {A B : Type*} [Fintype A] [Fintype B]
-    (mu : A → ℕ) (nu : B → ℕ) : Finite (Contingency mu nu) := by
-  classical
-  let f : Contingency mu nu → (∀ a : A, B → Fin (mu a + 1)) := fun M a b =>
-    ⟨M.val a b, by
-      have hh := Finset.single_le_sum (f := M.val a) (fun _ _ => Nat.zero_le _) (Finset.mem_univ b)
-      rw [M.property.1 a] at hh
-      omega⟩
-  apply Finite.of_injective f
-  intro M N h
-  apply Subtype.ext
-  ext a b
-  exact congrArg Fin.val (congrFun (congrFun h a) b)
-
-def rowChoicesEquiv {A B : Type*} [Fintype A] [Fintype B]
-    (mu : A → ℕ) (nu : B → ℕ) :
-    {r : ∀ a, Degree B (mu a) // (∑ a, (r a).val) = Finsupp.equivFunOnFinite.symm nu} ≃
-      Contingency mu nu where
-  toFun r := ⟨fun a b => (r.val a).val b,
-    (fun a => (Finsupp.degree_eq_sum _).symm.trans (r.val a).property),
-    fun b => by simpa only [Finsupp.coe_finsetSum, Finset.sum_apply, Finsupp.coe_equivFunOnFinite_symm]
-      using congrArg (fun f : B →₀ ℕ => f b) r.property⟩
-  invFun M := ⟨fun a => ⟨Finsupp.equivFunOnFinite.symm (M.val a), by
-      rw [Finsupp.degree_eq_sum]
-      exact M.property.1 a⟩, by
-    ext b
-    rw [Finsupp.finsetSum_apply]
-    exact M.property.2 b⟩
-  left_inv r := by apply Subtype.ext; funext a; apply Subtype.ext; ext b; rfl
-  right_inv M := rfl
-
-theorem coeff_product_h {A B : Type*} [Fintype A] [Fintype B]
-    (mu : A → ℕ) (nu : B → ℕ) :
-    (∏ a, h B (mu a)).coeff (Finsupp.equivFunOnFinite.symm nu) =
-      (Nat.card (Contingency mu nu) : ℤ) := by
-  classical
-  have he : (∏ a, h B (mu a)) = ∑ r : ∀ a, Degree B (mu a),
-      monomial (∑ a, (r a).val) (1 : ℤ) := by
-    simp only [h, Fintype.prod_sum, ← monomial_sum_one]
-  rw [he, coeff_sum]
-  simp only [coeff_monomial]
-  have hh : (∑ r : ∀ a, Degree B (mu a),
-      if (∑ a, (r a).val) = Finsupp.equivFunOnFinite.symm nu then (1 : ℤ) else 0) =
-      (Nat.card {r : ∀ a, Degree B (mu a) //
-        (∑ a, (r a).val) = Finsupp.equivFunOnFinite.symm nu} : ℤ) := by
-    simp [Nat.card_eq_fintype_card, Fintype.card_subtype]
-  rw [hh, Nat.card_congr (rowChoicesEquiv mu nu)]
-
-theorem young_hom_as_coefficient {K D A B : Type*} [Field K]
-    [Fintype D] [Fintype A] [Fintype B] (mu : A → ℕ) (nu : B → ℕ)
-    (hnu : ∑ b, nu b = Fintype.card D) :
-    (finrank K ((permRep K (Equiv.Perm D) (Coloring D A mu)).IntertwiningMap
-      (permRep K (Equiv.Perm D) (Coloring D B nu))) : ℤ) =
-      (∏ b, h A (nu b)).coeff (Finsupp.equivFunOnFinite.symm mu) := by
-  rw [young_hom_finrank mu nu hnu, coeff_product_h]
-
-end Foulkes.Young
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.FoulkesSixth.ColorOrbits
+import OAI.RepresentationTheory.FoulkesSixth.HomogeneousPlethysm
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.Young
+open MvPolynomial Foulkes.Complete Finset Module
+
+instance finiteContingency {A B : Type*} [Fintype A] [Fintype B]
+    (mu : A → ℕ) (nu : B → ℕ) : Finite (Contingency mu nu) := by
+  classical
+  let f : Contingency mu nu → (∀ a : A, B → Fin (mu a + 1)) := fun M a b =>
+    ⟨M.val a b, by
+      have hh := Finset.single_le_sum (f := M.val a) (fun _ _ => Nat.zero_le _) (Finset.mem_univ b)
+      rw [M.property.1 a] at hh
+      omega⟩
+  apply Finite.of_injective f
+  intro M N h
+  apply Subtype.ext
+  ext a b
+  exact congrArg Fin.val (congrFun (congrFun h a) b)
+
+def rowChoicesEquiv {A B : Type*} [Fintype A] [Fintype B]
+    (mu : A → ℕ) (nu : B → ℕ) :
+    {r : ∀ a, Degree B (mu a) // (∑ a, (r a).val) = Finsupp.equivFunOnFinite.symm nu} ≃
+      Contingency mu nu where
+  toFun r := ⟨fun a b => (r.val a).val b,
+    (fun a => (Finsupp.degree_eq_sum _).symm.trans (r.val a).property),
+    fun b => by simpa only [Finsupp.coe_finsetSum, Finset.sum_apply, Finsupp.coe_equivFunOnFinite_symm]
+      using congrArg (fun f : B →₀ ℕ => f b) r.property⟩
+  invFun M := ⟨fun a => ⟨Finsupp.equivFunOnFinite.symm (M.val a), by
+      rw [Finsupp.degree_eq_sum]
+      exact M.property.1 a⟩, by
+    ext b
+    rw [Finsupp.finsetSum_apply]
+    exact M.property.2 b⟩
+  left_inv r := by apply Subtype.ext; funext a; apply Subtype.ext; ext b; rfl
+  right_inv M := rfl
+
+theorem coeff_product_h {A B : Type*} [Fintype A] [Fintype B]
+    (mu : A → ℕ) (nu : B → ℕ) :
+    (∏ a, h B (mu a)).coeff (Finsupp.equivFunOnFinite.symm nu) =
+      (Nat.card (Contingency mu nu) : ℤ) := by
+  classical
+  have he : (∏ a, h B (mu a)) = ∑ r : ∀ a, Degree B (mu a),
+      monomial (∑ a, (r a).val) (1 : ℤ) := by
+    simp only [h, Fintype.prod_sum, ← monomial_sum_one]
+  rw [he, coeff_sum]
+  simp only [coeff_monomial]
+  have hh : (∑ r : ∀ a, Degree B (mu a),
+      if (∑ a, (r a).val) = Finsupp.equivFunOnFinite.symm nu then (1 : ℤ) else 0) =
+      (Nat.card {r : ∀ a, Degree B (mu a) //
+        (∑ a, (r a).val) = Finsupp.equivFunOnFinite.symm nu} : ℤ) := by
+    simp [Nat.card_eq_fintype_card, Fintype.card_subtype]
+  rw [hh, Nat.card_congr (rowChoicesEquiv mu nu)]
+
+theorem young_hom_as_coefficient {K D A B : Type*} [Field K]
+    [Fintype D] [Fintype A] [Fintype B] (mu : A → ℕ) (nu : B → ℕ)
+    (hnu : ∑ b, nu b = Fintype.card D) :
+    (finrank K ((permRep K (Equiv.Perm D) (Coloring D A mu)).IntertwiningMap
+      (permRep K (Equiv.Perm D) (Coloring D B nu))) : ℤ) =
+      (∏ b, h A (nu b)).coeff (Finsupp.equivFunOnFinite.symm mu) := by
+  rw [young_hom_finrank mu nu hnu, coeff_product_h]
+
+end Foulkes.Young
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/FoulkesSixth/YoungModules.lean b/lean/OAI/RepresentationTheory/FoulkesSixth/YoungModules.lean
index 3861e577b3d2a8de2bbe4256354ff2b5794f9a2e..e10f5cde7537d6822df78c62f76b353fc1b99613 100644
--- a/lean/OAI/RepresentationTheory/FoulkesSixth/YoungModules.lean
+++ b/lean/OAI/RepresentationTheory/FoulkesSixth/YoungModules.lean
@@ -1,129 +1,129 @@
-import Mathlib
-
-namespace OAI
-
-noncomputable section
-
-namespace Foulkes.Young
-open Module Representation Finset
-
-section Permutation
-variable {K G X Y : Type*} [Field K] [Group G]
-  [Fintype X] [Fintype Y] [MulAction G X] [MulAction G Y]
-
-def permRep (K G X : Type*) [Field K] [Group G] [MulAction G X] :
-    Representation K G (X → K) where
-  toFun g := LinearMap.pi fun x => LinearMap.proj (g⁻¹ • x)
-  map_one' := by ext f x; simp
-  map_mul' g h := by ext f x; simp [mul_smul]
-
-omit [Fintype X] in
-@[simp] lemma permRep_apply (g : G) (f : X → K) (x : X) :
-    permRep K G X g f x = f (g⁻¹ • x) := rfl
-
-omit [Fintype X] in
-@[simp] lemma permRep_single [DecidableEq X] (g : G) (x : X) (r : K) :
-    permRep K G X g (Pi.single x r) = Pi.single (g • x) r := by
-  ext y
-  by_cases h : y = g • x
-  · subst y
-    simp
-  · have h' : g⁻¹ • y ≠ x := by
-      intro he
-      apply h
-      have hh := congrArg (fun z => g • z) he
-      simpa using hh
-    simp [Pi.single_eq_of_ne h, Pi.single_eq_of_ne h']
-
-def matrixInvariants (K G X Y : Type*) [Field K] [Group G]
-    [MulAction G X] [MulAction G Y] : Submodule K (Matrix Y X K) where
-  carrier := {M | ∀ g : G, ∀ y x, M (g • y) (g • x) = M y x}
-  zero_mem' := by intro g y x; rfl
-  add_mem' := by intro M N hM hN g y x; exact congrArg₂ (· + ·) (hM g y x) (hN g y x)
-  smul_mem' := by intro r M hM g y x; exact congrArg (r * ·) (hM g y x)
-
-omit [Fintype Y] in
-lemma matrix_intertwining_iff [DecidableEq X] [DecidableEq Y] (M : Matrix Y X K) :
-    (∀ g : G, (Matrix.toLin' M).comp (permRep K G X g) =
-      (permRep K G Y g).comp (Matrix.toLin' M)) ↔ M ∈ matrixInvariants K G X Y := by
-  constructor
-  · intro h g y x
-    have hh := congrFun (LinearMap.congr_fun (h g) (Pi.single x 1)) (g • y)
-    simpa [Matrix.toLin'_apply, Matrix.mulVec, dotProduct, Pi.single_apply, mul_ite] using hh
-  · intro h g
-    apply LinearMap.toMatrix'.injective
-    ext y x
-    simp only [LinearMap.toMatrix'_apply, LinearMap.comp_apply, permRep_single,
-      permRep_apply, Matrix.toLin'_apply, Matrix.mulVec, dotProduct]
-    simp only [Pi.single_apply, mul_ite, mul_one, mul_zero, Finset.sum_ite_eq',
-      Finset.mem_univ, ite_true]
-    simpa using h g (g⁻¹ • y) x
-
-def intertwinerMatrix [DecidableEq X] [DecidableEq Y] :
-    (permRep K G X).IntertwiningMap (permRep K G Y) ≃ₗ[K] matrixInvariants K G X Y where
-  toFun f := ⟨LinearMap.toMatrix' f.toLinearMap,
-    (matrix_intertwining_iff (LinearMap.toMatrix' f.toLinearMap)).mp (by
-      simpa using f.isIntertwining')⟩
-  invFun M := ⟨Matrix.toLin' M.val, (matrix_intertwining_iff M.val).mpr M.property⟩
-  left_inv f := by ext x; simp
-  right_inv M := by apply Subtype.ext; simp
-  map_add' f g := by apply Subtype.ext; simp
-  map_smul' r f := by apply Subtype.ext; simp
-
-end Permutation
-
-section Quotient
-variable {K G Z : Type*} [Field K] [Group G] [MulAction G Z]
-
-def orbitFunctions (K G Z : Type*) [Field K] [Group G] [MulAction G Z] :
-    Submodule K (Z → K) where
-  carrier := {f | ∀ g : G, ∀ z, f (g • z) = f z}
-  zero_mem' := by intro g z; rfl
-  add_mem' := by intro f h hf hh g z; exact congrArg₂ (· + ·) (hf g z) (hh g z)
-  smul_mem' := by intro r f hf g z; exact congrArg (r * ·) (hf g z)
-
-def orbitFunctionsEquiv : orbitFunctions K G Z ≃ₗ[K] (MulAction.orbitRel.Quotient G Z → K) where
-  toFun f := Quotient.lift f.val (by
-    intro x y h
-    obtain ⟨g,hg⟩ := h
-    rw [← hg]
-    exact f.property g y)
-  invFun f := ⟨fun z => f (Quotient.mk (MulAction.orbitRel G Z) z), by
-    intro g z
-    apply congrArg f
-    exact Quotient.sound (show MulAction.orbitRel G Z (g • z) z from ⟨g,rfl⟩)⟩
-  left_inv f := rfl
-  right_inv f := by funext q; induction q using Quotient.inductionOn; rfl
-  map_add' f h := by funext q; induction q using Quotient.inductionOn; rfl
-  map_smul' r f := by funext q; induction q using Quotient.inductionOn; rfl
-
-end Quotient
-
-section OrbitGram
-variable {K G X Y : Type*} [Field K] [Group G]
-  [Fintype X] [Fintype Y] [MulAction G X] [MulAction G Y]
-
-def matrixOrbitEquiv : matrixInvariants K G X Y ≃ₗ[K] orbitFunctions K G (Y × X) where
-  toFun M := ⟨fun z => M.val z.1 z.2, fun g z => M.property g z.1 z.2⟩
-  invFun f := ⟨fun y x => f.val (y,x), fun g y x => f.property g (y,x)⟩
-  left_inv _ := rfl
-  right_inv _ := rfl
-  map_add' _ _ := rfl
-  map_smul' _ _ := rfl
-
-theorem perm_hom_finrank :
-    finrank K ((permRep K G X).IntertwiningMap (permRep K G Y)) =
-      Nat.card (MulAction.orbitRel.Quotient G (Y × X)) := by
-  classical
-  let e := (intertwinerMatrix (K := K) (G := G) (X := X) (Y := Y)).trans
-    (matrixOrbitEquiv.trans orbitFunctionsEquiv)
-  rw [e.finrank_eq]
-  let := Fintype.ofFinite (MulAction.orbitRel.Quotient G (Y × X))
-  simp [Nat.card_eq_fintype_card]
-
-end OrbitGram
-end Foulkes.Young
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+
+namespace Foulkes.Young
+open Module Representation Finset
+
+section Permutation
+variable {K G X Y : Type*} [Field K] [Group G]
+  [Fintype X] [Fintype Y] [MulAction G X] [MulAction G Y]
+
+def permRep (K G X : Type*) [Field K] [Group G] [MulAction G X] :
+    Representation K G (X → K) where
+  toFun g := LinearMap.pi fun x => LinearMap.proj (g⁻¹ • x)
+  map_one' := by ext f x; simp
+  map_mul' g h := by ext f x; simp [mul_smul]
+
+omit [Fintype X] in
+@[simp] lemma permRep_apply (g : G) (f : X → K) (x : X) :
+    permRep K G X g f x = f (g⁻¹ • x) := rfl
+
+omit [Fintype X] in
+@[simp] lemma permRep_single [DecidableEq X] (g : G) (x : X) (r : K) :
+    permRep K G X g (Pi.single x r) = Pi.single (g • x) r := by
+  ext y
+  by_cases h : y = g • x
+  · subst y
+    simp
+  · have h' : g⁻¹ • y ≠ x := by
+      intro he
+      apply h
+      have hh := congrArg (fun z => g • z) he
+      simpa using hh
+    simp [Pi.single_eq_of_ne h, Pi.single_eq_of_ne h']
+
+def matrixInvariants (K G X Y : Type*) [Field K] [Group G]
+    [MulAction G X] [MulAction G Y] : Submodule K (Matrix Y X K) where
+  carrier := {M | ∀ g : G, ∀ y x, M (g • y) (g • x) = M y x}
+  zero_mem' := by intro g y x; rfl
+  add_mem' := by intro M N hM hN g y x; exact congrArg₂ (· + ·) (hM g y x) (hN g y x)
+  smul_mem' := by intro r M hM g y x; exact congrArg (r * ·) (hM g y x)
+
+omit [Fintype Y] in
+lemma matrix_intertwining_iff [DecidableEq X] [DecidableEq Y] (M : Matrix Y X K) :
+    (∀ g : G, (Matrix.toLin' M).comp (permRep K G X g) =
+      (permRep K G Y g).comp (Matrix.toLin' M)) ↔ M ∈ matrixInvariants K G X Y := by
+  constructor
+  · intro h g y x
+    have hh := congrFun (LinearMap.congr_fun (h g) (Pi.single x 1)) (g • y)
+    simpa [Matrix.toLin'_apply, Matrix.mulVec, dotProduct, Pi.single_apply, mul_ite] using hh
+  · intro h g
+    apply LinearMap.toMatrix'.injective
+    ext y x
+    simp only [LinearMap.toMatrix'_apply, LinearMap.comp_apply, permRep_single,
+      permRep_apply, Matrix.toLin'_apply, Matrix.mulVec, dotProduct]
+    simp only [Pi.single_apply, mul_ite, mul_one, mul_zero, Finset.sum_ite_eq',
+      Finset.mem_univ, ite_true]
+    simpa using h g (g⁻¹ • y) x
+
+def intertwinerMatrix [DecidableEq X] [DecidableEq Y] :
+    (permRep K G X).IntertwiningMap (permRep K G Y) ≃ₗ[K] matrixInvariants K G X Y where
+  toFun f := ⟨LinearMap.toMatrix' f.toLinearMap,
+    (matrix_intertwining_iff (LinearMap.toMatrix' f.toLinearMap)).mp (by
+      simpa using f.isIntertwining')⟩
+  invFun M := ⟨Matrix.toLin' M.val, (matrix_intertwining_iff M.val).mpr M.property⟩
+  left_inv f := by ext x; simp
+  right_inv M := by apply Subtype.ext; simp
+  map_add' f g := by apply Subtype.ext; simp
+  map_smul' r f := by apply Subtype.ext; simp
+
+end Permutation
+
+section Quotient
+variable {K G Z : Type*} [Field K] [Group G] [MulAction G Z]
+
+def orbitFunctions (K G Z : Type*) [Field K] [Group G] [MulAction G Z] :
+    Submodule K (Z → K) where
+  carrier := {f | ∀ g : G, ∀ z, f (g • z) = f z}
+  zero_mem' := by intro g z; rfl
+  add_mem' := by intro f h hf hh g z; exact congrArg₂ (· + ·) (hf g z) (hh g z)
+  smul_mem' := by intro r f hf g z; exact congrArg (r * ·) (hf g z)
+
+def orbitFunctionsEquiv : orbitFunctions K G Z ≃ₗ[K] (MulAction.orbitRel.Quotient G Z → K) where
+  toFun f := Quotient.lift f.val (by
+    intro x y h
+    obtain ⟨g,hg⟩ := h
+    rw [← hg]
+    exact f.property g y)
+  invFun f := ⟨fun z => f (Quotient.mk (MulAction.orbitRel G Z) z), by
+    intro g z
+    apply congrArg f
+    exact Quotient.sound (show MulAction.orbitRel G Z (g • z) z from ⟨g,rfl⟩)⟩
+  left_inv f := rfl
+  right_inv f := by funext q; induction q using Quotient.inductionOn; rfl
+  map_add' f h := by funext q; induction q using Quotient.inductionOn; rfl
+  map_smul' r f := by funext q; induction q using Quotient.inductionOn; rfl
+
+end Quotient
+
+section OrbitGram
+variable {K G X Y : Type*} [Field K] [Group G]
+  [Fintype X] [Fintype Y] [MulAction G X] [MulAction G Y]
+
+def matrixOrbitEquiv : matrixInvariants K G X Y ≃ₗ[K] orbitFunctions K G (Y × X) where
+  toFun M := ⟨fun z => M.val z.1 z.2, fun g z => M.property g z.1 z.2⟩
+  invFun f := ⟨fun y x => f.val (y,x), fun g y x => f.property g (y,x)⟩
+  left_inv _ := rfl
+  right_inv _ := rfl
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+
+theorem perm_hom_finrank :
+    finrank K ((permRep K G X).IntertwiningMap (permRep K G Y)) =
+      Nat.card (MulAction.orbitRel.Quotient G (Y × X)) := by
+  classical
+  let e := (intertwinerMatrix (K := K) (G := G) (X := X) (Y := Y)).trans
+    (matrixOrbitEquiv.trans orbitFunctionsEquiv)
+  rw [e.finrank_eq]
+  let := Fintype.ofFinite (MulAction.orbitRel.Quotient G (Y × X))
+  simp [Nat.card_eq_fintype_card]
+
+end OrbitGram
+end Foulkes.Young
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/GraphModels/Bundles.lean b/lean/OAI/RepresentationTheory/GraphModels/Bundles.lean
index 0f9520c8dc947b3ce8c80b74cdd0c5c6167cf290..e25182c4438d43cb0d8bd2961cb7160dcead69f0 100644
--- a/lean/OAI/RepresentationTheory/GraphModels/Bundles.lean
+++ b/lean/OAI/RepresentationTheory/GraphModels/Bundles.lean
@@ -1,477 +1,477 @@
-import Mathlib
-import OAI.GroupTheory.ArithmeticGerms.Basic
-
-namespace OAI
-
-/-! Projective Hilbert bundles and exact induction. -/
-
-namespace GermBundles
-attribute [local instance] Classical.propDecidable
-open scoped BigOperators InnerProductSpace
-
-structure ScalarCocycle (K : Type*) [Group K] where
-  val : K → K → Circle
-  mul_assoc : ∀ g h j, val g h * val (g * h) j = val h j * val g (h * j)
-  one_left : ∀ g, val 1 g = 1
-  one_right : ∀ g, val g 1 = 1
-
-section Bundles
-
-variable {K X : Type*} [Group K] [MulAction K X]
-variable (α : ScalarCocycle K) (E : X → Type*)
-variable [∀ x, NormedAddCommGroup (E x)] [∀ x, InnerProductSpace ℂ (E x)]
-
-attribute [local instance] Classical.propDecidable
-
-structure ProjectiveBundle where
-  action : K → lp E 2 ≃ₗᵢ[ℂ] lp E 2
-  fibre : ∀ (k : K) (x y : X), k • x = y → E x ≃ₗᵢ[ℂ] E y
-  action_single : ∀ (k : K) (x y : X) (h : k • x = y) (v : E x),
-    action k (lp.single 2 x v) = lp.single 2 y (fibre k x y h v)
-  action_one : ∀ v, action 1 v = v
-  action_mul : ∀ g h v, action g (action h v) =
-    (α.val g h : ℂ) • action (g * h) v
-
-variable {α E}
-
-variable {F : X → Type*}
-variable [∀ x, NormedAddCommGroup (F x)] [∀ x, InnerProductSpace ℂ (F x)]
-
-def Intertwines (B : ProjectiveBundle α E) (C : ProjectiveBundle α F)
-    (T : lp E 2 →L[ℂ] lp F 2) : Prop :=
-  ∀ k v, T (B.action k v) = C.action k (T v)
-
-end Bundles
-
-section BuildBundle
-
-variable {X : Type*} {E F : X → Type*}
-variable [∀ x, NormedAddCommGroup (E x)] [∀ x, InnerProductSpace ℂ (E x)]
-variable [∀ x, NormedAddCommGroup (F x)] [∀ x, InnerProductSpace ℂ (F x)]
-
-private theorem relabel_pi_apply (e : X ≃ X)
-    (u : ∀ x y, e x = y → E x ≃ₗᵢ[ℂ] F y) (f : ∀ x, E x) (x : X) :
-    u (e.symm (e x)) (e x) (e.apply_symm_apply (e x)) (f (e.symm (e x))) =
-      u x (e x) rfl (f x) := by
-  have haux (z : X) (hz : z = x) (he : e z = e x) :
-      u z (e x) he (f z) = u x (e x) rfl (f x) := by
-    subst z
-    rfl
-  exact haux (e.symm (e x)) (e.symm_apply_apply x) (e.apply_symm_apply (e x))
-
-private theorem relabel_pi_right (e : X ≃ X)
-    (u : ∀ x y, e x = y → E x ≃ₗᵢ[ℂ] F y) (f : ∀ x, F x) (y : X) :
-    u (e.symm y) y (e.apply_symm_apply y)
-      ((u (e.symm y) (e (e.symm y)) rfl).symm (f (e (e.symm y)))) = f y := by
-  have haux (x y : X) (he : e x = y) :
-      u x y he ((u x (e x) rfl).symm (f (e x))) = f y := by
-    subst y
-    exact (u x (e x) rfl).apply_symm_apply _
-  exact haux (e.symm y) y (e.apply_symm_apply y)
-
-noncomputable def relabel (e : X ≃ X)
-    (u : ∀ x y, e x = y → E x ≃ₗᵢ[ℂ] F y) : lp E 2 ≃ₗᵢ[ℂ] lp F 2 where
-  toLinearEquiv :=
-    { toFun := fun v => ⟨fun y => u (e.symm y) y (e.apply_symm_apply y) (v (e.symm y)), by
-        apply memℓp_gen
-        convert! ((lp.memℓp v).summable (by norm_num)).comp_injective e.symm.injective using 1
-        all_goals first | rfl | (simp only [LinearIsometryEquiv.norm_map]; rfl)⟩
-      invFun := fun v => ⟨fun x => (u x (e x) rfl).symm (v (e x)), by
-        apply memℓp_gen
-        convert! ((lp.memℓp v).summable (by norm_num)).comp_injective e.injective using 1
-        all_goals first | rfl | (simp only [LinearIsometryEquiv.norm_map]; rfl)⟩
-      left_inv := by
-        intro v
-        apply lp.ext
-        funext x
-        change (u x (e x) rfl).symm
-          (u (e.symm (e x)) (e x) (e.apply_symm_apply (e x)) (v.val (e.symm (e x)))) = v.val x
-        exact (congrArg (u x (e x) rfl).symm (relabel_pi_apply e u (fun i => v i) x)).trans
-          ((u x (e x) rfl).symm_apply_apply (v x))
-      right_inv := by
-        intro v
-        apply lp.ext
-        funext y
-        exact relabel_pi_right e u (fun i => v i) y
-      map_add' := by
-        intro v w
-        apply lp.ext
-        funext y
-        change u _ _ _ (v _ + w _) = u _ _ _ (v _) + u _ _ _ (w _)
-        exact map_add _ _ _
-      map_smul' := by
-        intro a v
-        apply lp.ext
-        funext y
-        change u _ _ _ (a • v _) = a • u _ _ _ (v _)
-        exact map_smul _ _ _ }
-  norm_map' := by
-    intro v
-    rw [lp.norm_eq_tsum_rpow (by norm_num), lp.norm_eq_tsum_rpow (by norm_num)]
-    congr 1
-    change (∑' y, ‖u (e.symm y) y (e.apply_symm_apply y) (v (e.symm y))‖ ^ (2 : ENNReal).toReal) = _
-    simp only [LinearIsometryEquiv.norm_map]
-    convert! e.symm.tsum_eq (fun x => ‖v x‖ ^ (2 : ENNReal).toReal) using 1
-
-theorem relabel_apply (e : X ≃ X) (u : ∀ x y, e x = y → E x ≃ₗᵢ[ℂ] F y)
-    (v : lp E 2) (x y : X) (h : e x = y) : relabel e u v y = u x y h (v x) := by
-  subst y
-  change u (e.symm (e x)) (e x) (e.apply_symm_apply (e x)) (v.val (e.symm (e x))) = u x (e x) rfl (v.val x)
-  exact relabel_pi_apply e u (fun i => v i) x
-
-theorem relabel_single (e : X ≃ X) (u : ∀ x y, e x = y → E x ≃ₗᵢ[ℂ] F y)
-    (x y : X) (h : e x = y) (v : E x) :
-    relabel e u (lp.single 2 x v) = lp.single 2 y (u x y h v) := by
-  apply lp.ext
-  funext z
-  by_cases hz : z = y
-  · subst z
-    rw [relabel_apply e u _ x y h, lp.single_apply_self, lp.single_apply_self]
-  · have hn : e.symm z ≠ x := by
-      intro he
-      apply hz
-      rw [← h, ← he, e.apply_symm_apply]
-    change u (e.symm z) z (e.apply_symm_apply z) ((lp.single 2 x v) (e.symm z)) = _
-    rw [lp.single_apply_ne 2 x v hn, map_zero, lp.single_apply_ne 2 y _ hz]
-
-end BuildBundle
-
-section FibreSystem
-
-variable {K X : Type*} [Group K] [MulAction K X]
-variable (α : ScalarCocycle K) (E : X → Type*)
-variable [∀ x, NormedAddCommGroup (E x)] [∀ x, InnerProductSpace ℂ (E x)]
-
-structure FibreSystem where
-  unitary : ∀ (k : K) (x y : X), k • x = y → E x ≃ₗᵢ[ℂ] E y
-  one : ∀ x v, unitary 1 x x (one_smul K x) v = v
-  mul : ∀ g h x y z (hy : h • x = y) (hz : g • y = z) (hgh : (g * h) • x = z) v,
-    unitary g y z hz (unitary h x y hy v) = (α.val g h : ℂ) • unitary (g * h) x z hgh v
-
-variable {α E}
-
-noncomputable def FibreSystem.toBundle (U : FibreSystem α E) : ProjectiveBundle α E where
-  action k := relabel (MulAction.toPerm k) (U.unitary k)
-  fibre := U.unitary
-  action_single k x y h v := relabel_single (MulAction.toPerm k) (U.unitary k) x y h v
-  action_one := by
-    intro v
-    have he : (relabel (MulAction.toPerm (1 : K)) (U.unitary 1)).toContinuousLinearEquiv.toContinuousLinearMap =
-        ContinuousLinearMap.id ℂ (lp E 2) := by
-      apply lp.ext_continuousLinearMap (by norm_num)
-      intro x
-      apply ContinuousLinearMap.ext
-      intro w
-      change relabel (MulAction.toPerm (1 : K)) (U.unitary 1) (lp.single 2 x w) = lp.single 2 x w
-      rw [relabel_single (MulAction.toPerm (1 : K)) (U.unitary 1) x x (one_smul K x), U.one]
-    exact DFunLike.congr_fun he v
-  action_mul := by
-    intro g h v
-    have he :
-        (relabel (MulAction.toPerm g) (U.unitary g)).toContinuousLinearEquiv.toContinuousLinearMap.comp
-          (relabel (MulAction.toPerm h) (U.unitary h)).toContinuousLinearEquiv.toContinuousLinearMap =
-        (α.val g h : ℂ) • (relabel (MulAction.toPerm (g * h)) (U.unitary (g * h))).toContinuousLinearEquiv.toContinuousLinearMap := by
-      apply lp.ext_continuousLinearMap (by norm_num)
-      intro x
-      apply ContinuousLinearMap.ext
-      intro w
-      change relabel (MulAction.toPerm g) (U.unitary g)
-        (relabel (MulAction.toPerm h) (U.unitary h) (lp.single 2 x w)) =
-        (α.val g h : ℂ) • relabel (MulAction.toPerm (g * h)) (U.unitary (g * h)) (lp.single 2 x w)
-      rw [relabel_single (MulAction.toPerm h) (U.unitary h) x (h • x) rfl,
-        relabel_single (MulAction.toPerm g) (U.unitary g) (h • x) ((g * h) • x) (mul_smul g h x).symm,
-        relabel_single (MulAction.toPerm (g * h)) (U.unitary (g * h)) x ((g * h) • x) rfl]
-      erw [U.mul g h x (h • x) ((g * h) • x) rfl (mul_smul g h x).symm rfl w,
-        lp.single_smul]
-    exact DFunLike.congr_fun he v
-
-end FibreSystem
-
-noncomputable section ExactInduction
-
-attribute [local instance] Classical.propDecidable
-
-variable {K : Type*} [Group K]
-
-def ScalarCocycle.restrict (α : ScalarCocycle K) (C : Subgroup K) : ScalarCocycle C where
-  val g h := α.val g h
-  mul_assoc g h j := α.mul_assoc g h j
-  one_left g := α.one_left g
-  one_right g := α.one_right g
-
-structure ProjectiveRepresentation (α : ScalarCocycle K) (V : Type*)
-    [NormedAddCommGroup V] [InnerProductSpace ℂ V] where
-  unitary : K → V ≃ₗᵢ[ℂ] V
-  one : ∀ v, unitary 1 v = v
-  mul : ∀ g h v, unitary g (unitary h v) = (α.val g h : ℂ) • unitary (g * h) v
-
-variable (C : Subgroup K)
-
-structure CosetSection where
-  repr : K ⧸ C → K
-  repr_spec : ∀ x, QuotientGroup.mk (repr x) = x
-  repr_one : repr (QuotientGroup.mk 1) = 1
-
-noncomputable def chooseCosetSection : CosetSection C where
-  repr x := if x = QuotientGroup.mk 1 then 1 else x.out
-  repr_spec x := by
-    split_ifs with h
-    · exact h.symm
-    · exact QuotientGroup.out_eq' x
-  repr_one := by simp
-
-variable {C}
-
-def CosetSection.correction (r : CosetSection C) (s : K) (x : K ⧸ C) : C :=
-  ⟨(r.repr (s • x))⁻¹ * s * r.repr x, by
-    rw [mul_assoc]
-    apply QuotientGroup.eq.mp
-    rw [r.repr_spec, ← smul_eq_mul, ← MulAction.Quotient.smul_mk, r.repr_spec]⟩
-
-@[simp] theorem CosetSection.correction_val (r : CosetSection C) (s : K) (x : K ⧸ C) :
-    (r.correction s x : K) = (r.repr (s • x))⁻¹ * s * r.repr x := rfl
-
-@[simp] theorem CosetSection.repr_mul_correction (r : CosetSection C) (s : K) (x : K ⧸ C) :
-    r.repr (s • x) * (r.correction s x : K) = s * r.repr x := by
-  rw [correction_val]
-  group
-
-@[simp] theorem CosetSection.correction_one (r : CosetSection C) (x : K ⧸ C) :
-    r.correction 1 x = 1 := by
-  apply Subtype.ext
-  simp [correction]
-
-@[simp] theorem CosetSection.correction_mul (r : CosetSection C) (s t : K) (x : K ⧸ C) :
-    r.correction s (t • x) * r.correction t x = r.correction (s * t) x := by
-  apply Subtype.ext
-  simp only [Subgroup.coe_mul, correction_val, mul_smul]
-  group
-
-def CosetSection.phase (r : CosetSection C) (α : ScalarCocycle K) (s : K) (x : K ⧸ C) : Circle :=
-  α.val s (r.repr x) / α.val (r.repr (s • x)) (r.correction s x)
-
-@[simp] theorem CosetSection.phase_one (r : CosetSection C) (α : ScalarCocycle K) (x : K ⧸ C) :
-    r.phase α 1 x = 1 := by
-  simp [phase, α.one_left, α.one_right]
-
-theorem cocycle_transport_phase (α : ScalarCocycle K) (s t x y z a b : K)
-    (ha : z * a = s * y) (hb : y * b = t * x) :
-    (α.val s y / α.val z a) * (α.val t x / α.val y b) * α.val a b =
-      α.val s t * (α.val (s * t) x / α.val z (a * b)) := by
-  have hid : α.val s y * α.val t x * α.val a b * α.val z (a * b) =
-      α.val s t * α.val (s * t) x * (α.val z a * α.val y b) := by
-    calc
-      _ = α.val t x * α.val s y * (α.val a b * α.val z (a * b)) := by ac_rfl
-      _ = α.val t x * α.val s y * (α.val z a * α.val (z * a) b) := by
-        rw [α.mul_assoc z a b]
-      _ = α.val z a * α.val t x * (α.val s y * α.val (s * y) b) := by
-        rw [ha]
-        ac_rfl
-      _ = α.val z a * α.val t x * (α.val y b * α.val s (y * b)) := by
-        rw [α.mul_assoc s y b]
-      _ = (α.val t x * α.val s (t * x)) * (α.val z a * α.val y b) := by
-        rw [hb]
-        ac_rfl
-      _ = _ := by rw [← α.mul_assoc s t x]
-  rw [div_mul_div_comm, div_mul_eq_mul_div, mul_div_assoc]
-  simpa only [mul_div_assoc] using div_eq_div_iff_mul_eq_mul.mpr hid
-
-theorem CosetSection.phase_mul (r : CosetSection C) (α : ScalarCocycle K) (s t : K)
-    (x : K ⧸ C) :
-    r.phase α s (t • x) * r.phase α t x * α.val (r.correction s (t • x)) (r.correction t x) =
-      α.val s t * r.phase α (s * t) x := by
-  unfold phase
-  have h := cocycle_transport_phase α s t (r.repr x) (r.repr (t • x))
-    (r.repr (s • t • x)) (r.correction s (t • x)) (r.correction t x)
-    (r.repr_mul_correction s (t • x)) (r.repr_mul_correction t x)
-  simpa only [← Subgroup.coe_mul, correction_mul, mul_smul] using h
-
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-
-def circleUnitary (z : Circle) : V ≃ₗᵢ[ℂ] V where
-  toLinearEquiv :=
-    { toFun := fun v => (z : ℂ) • v
-      invFun := fun v => ((z⁻¹ : Circle) : ℂ) • v
-      left_inv := by intro v; simp [smul_smul]
-      right_inv := by intro v; simp [smul_smul]
-      map_add' := by intro v w; exact smul_add _ _ _
-      map_smul' := by intro a v; exact smul_comm _ _ _ }
-  norm_map' := by intro v; simp [norm_smul, Circle.norm_coe]
-
-@[simp] theorem circleUnitary_apply (z : Circle) (v : V) : circleUnitary z v = (z : ℂ) • v := rfl
-
-def CosetSection.inducedUnitary (r : CosetSection C) (α : ScalarCocycle K)
-    (γ : ProjectiveRepresentation (α.restrict C) V) (s : K) (x : K ⧸ C) : V ≃ₗᵢ[ℂ] V :=
-  (γ.unitary (r.correction s x)).trans (circleUnitary (r.phase α s x))
-
-@[simp] theorem CosetSection.inducedUnitary_apply (r : CosetSection C) (α : ScalarCocycle K)
-    (γ : ProjectiveRepresentation (α.restrict C) V) (s : K) (x : K ⧸ C) (v : V) :
-    r.inducedUnitary α γ s x v = (r.phase α s x : ℂ) • γ.unitary (r.correction s x) v := rfl
-
-@[simp] theorem CosetSection.inducedUnitary_one (r : CosetSection C) (α : ScalarCocycle K)
-    (γ : ProjectiveRepresentation (α.restrict C) V) (x : K ⧸ C) (v : V) :
-    r.inducedUnitary α γ 1 x v = v := by
-  simp [γ.one]
-
-theorem CosetSection.inducedUnitary_mul (r : CosetSection C) (α : ScalarCocycle K)
-    (γ : ProjectiveRepresentation (α.restrict C) V) (s t : K) (x : K ⧸ C) (v : V) :
-    r.inducedUnitary α γ s (t • x) (r.inducedUnitary α γ t x v) =
-      (α.val s t : ℂ) • r.inducedUnitary α γ (s * t) x v := by
-  simp only [inducedUnitary_apply, map_smul, γ.mul, smul_smul]
-  rw [correction_mul]
-  congr 1
-  have hp := congrArg (fun z : Circle => (z : ℂ)) (r.phase_mul α s t x)
-  simpa only [Circle.coe_mul, ScalarCocycle.restrict, mul_assoc, mul_left_comm] using hp
-
-def CosetSection.change (r r' : CosetSection C) (x : K ⧸ C) : C :=
-  ⟨(r.repr x)⁻¹ * r'.repr x, QuotientGroup.eq.mp ((r.repr_spec x).trans (r'.repr_spec x).symm)⟩
-
-@[simp] theorem CosetSection.repr_mul_change (r r' : CosetSection C) (x : K ⧸ C) :
-    r.repr x * (r.change r' x : K) = r'.repr x := by
-  simp [change]
-
-theorem CosetSection.change_correction (r r' : CosetSection C) (s : K) (x : K ⧸ C) :
-    r.change r' (s • x) * r'.correction s x = r.correction s x * r.change r' x := by
-  apply Subtype.ext
-  simp only [Subgroup.coe_mul, change, correction]
-  group
-
-theorem cocycle_change_phase (α : ScalarCocycle K) (s x y c k l c' : K)
-    (hc : y * c = s * x) (hk : l * c' = c * k) :
-    (α.val y l)⁻¹ * (α.val s (x * k) / α.val (y * l) c') * α.val l c' =
-      (α.val s x / α.val y c) * (α.val x k)⁻¹ * α.val c k := by
-  have hid : α.val s (x * k) * α.val l c' * (α.val y c * α.val x k) =
-      α.val s x * α.val c k * (α.val (y * l) c' * α.val y l) := by
-    calc
-      _ = α.val l c' * α.val y c * (α.val x k * α.val s (x * k)) := by ac_rfl
-      _ = α.val l c' * α.val y c * (α.val s x * α.val (s * x) k) := by
-        rw [α.mul_assoc s x k]
-      _ = α.val s x * α.val l c' * (α.val y c * α.val (y * c) k) := by
-        rw [← hc]
-        ac_rfl
-      _ = α.val s x * α.val l c' * (α.val c k * α.val y (c * k)) := by
-        rw [α.mul_assoc y c k]
-      _ = α.val s x * α.val c k * (α.val l c' * α.val y (l * c')) := by
-        rw [← hk]
-        ac_rfl
-      _ = _ := by rw [← α.mul_assoc y l c']; ac_rfl
-  have hr := div_eq_div_iff_mul_eq_mul.mpr hid
-  convert hr using 1 <;> simp only [div_eq_mul_inv, mul_inv_rev, mul_comm, mul_left_comm, mul_assoc]
-
-def CosetSection.changeUnitary (r r' : CosetSection C) (α : ScalarCocycle K)
-    (γ : ProjectiveRepresentation (α.restrict C) V) (x : K ⧸ C) : V ≃ₗᵢ[ℂ] V :=
-  (γ.unitary (r.change r' x)).trans
-    (circleUnitary (α.val (r.repr x) (r.change r' x))⁻¹)
-
-@[simp] theorem CosetSection.changeUnitary_apply (r r' : CosetSection C) (α : ScalarCocycle K)
-    (γ : ProjectiveRepresentation (α.restrict C) V) (x : K ⧸ C) (v : V) :
-    r.changeUnitary r' α γ x v = ((α.val (r.repr x) (r.change r' x))⁻¹ : Circle) •
-      γ.unitary (r.change r' x) v := rfl
-
-theorem CosetSection.change_phase (r r' : CosetSection C) (α : ScalarCocycle K)
-    (s : K) (x : K ⧸ C) :
-    (α.val (r.repr (s • x)) (r.change r' (s • x)))⁻¹ * r'.phase α s x *
-        α.val (r.change r' (s • x)) (r'.correction s x) =
-      r.phase α s x * (α.val (r.repr x) (r.change r' x))⁻¹ *
-        α.val (r.correction s x) (r.change r' x) := by
-  have h := cocycle_change_phase α s (r.repr x) (r.repr (s • x))
-    (r.correction s x) (r.change r' x) (r.change r' (s • x)) (r'.correction s x)
-    (r.repr_mul_correction s x) (congrArg Subtype.val (r.change_correction r' s x))
-  simpa only [phase, repr_mul_change] using h
-
-theorem CosetSection.changeUnitary_intertwines (r r' : CosetSection C) (α : ScalarCocycle K)
-    (γ : ProjectiveRepresentation (α.restrict C) V) (s : K) (x : K ⧸ C) (v : V) :
-    r.changeUnitary r' α γ (s • x) (r'.inducedUnitary α γ s x v) =
-      r.inducedUnitary α γ s x (r.changeUnitary r' α γ x v) := by
-  change ((α.val (r.repr (s • x)) (r.change r' (s • x)))⁻¹ : Circle) •
-    γ.unitary (r.change r' (s • x)) ((r'.phase α s x : ℂ) • γ.unitary (r'.correction s x) v) =
-    (r.phase α s x : ℂ) • γ.unitary (r.correction s x)
-      (((α.val (r.repr x) (r.change r' x))⁻¹ : Circle) • γ.unitary (r.change r' x) v)
-  simp only [Circle.smul_def, map_smul, γ.mul, smul_smul]
-  rw [change_correction]
-  congr 1
-  have hp := congrArg (fun z : Circle => (z : ℂ)) (r.change_phase r' α s x)
-  simpa only [Circle.coe_mul, ScalarCocycle.restrict, mul_assoc] using hp
-
-end ExactInduction
-
-noncomputable section InducedBundle
-
-attribute [local instance] Classical.propDecidable
-
-variable {K : Type*} [Group K] {C : Subgroup K}
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-
-local instance : DecidableEq (K ⧸ C) := Classical.decEq _
-
-def CosetSection.inducedFibreSystem (r : CosetSection C) (α : ScalarCocycle K)
-    (γ : ProjectiveRepresentation (α.restrict C) V) :
-    FibreSystem α (fun _ : K ⧸ C => V) where
-  unitary s x _ _ := r.inducedUnitary α γ s x
-  one x v := r.inducedUnitary_one α γ x v
-  mul s t x y z hy hz hst v := by
-    subst y
-    exact r.inducedUnitary_mul α γ s t x v
-
-def CosetSection.inducedBundle (r : CosetSection C) (α : ScalarCocycle K)
-    (γ : ProjectiveRepresentation (α.restrict C) V) :
-    ProjectiveBundle α (fun _ : K ⧸ C => V) :=
-  (r.inducedFibreSystem α γ).toBundle
-
-theorem CosetSection.inducedBundle_single (r : CosetSection C) (α : ScalarCocycle K)
-    (γ : ProjectiveRepresentation (α.restrict C) V) (s : K) (x : K ⧸ C) (v : V) :
-    (r.inducedBundle α γ).action s (lp.single 2 x v) =
-      lp.single 2 (s • x) ((r.phase α s x : ℂ) • γ.unitary (r.correction s x) v) := by
-  exact (r.inducedBundle α γ).action_single s x (s • x) rfl v
-
-def CosetSection.changeBundleUnitary (r r' : CosetSection C) (α : ScalarCocycle K)
-    (γ : ProjectiveRepresentation (α.restrict C) V) :
-    lp (fun _ : K ⧸ C => V) 2 ≃ₗᵢ[ℂ] lp (fun _ : K ⧸ C => V) 2 :=
-  relabel (Equiv.refl _) (fun x _ _ => r.changeUnitary r' α γ x)
-
-@[simp] theorem CosetSection.changeBundleUnitary_single (r r' : CosetSection C) (α : ScalarCocycle K)
-    (γ : ProjectiveRepresentation (α.restrict C) V) (x : K ⧸ C) (v : V) :
-    r.changeBundleUnitary r' α γ (lp.single 2 x v) =
-      lp.single 2 x (r.changeUnitary r' α γ x v) := by
-  exact relabel_single (Equiv.refl _) (fun x _ _ => r.changeUnitary r' α γ x) x x rfl v
-
-theorem CosetSection.changeBundleUnitary_intertwines (r r' : CosetSection C) (α : ScalarCocycle K)
-    (γ : ProjectiveRepresentation (α.restrict C) V) :
-    Intertwines (r'.inducedBundle α γ) (r.inducedBundle α γ)
-      (r.changeBundleUnitary r' α γ).toContinuousLinearEquiv.toContinuousLinearMap := by
-  intro s v
-  have he :
-      (r.changeBundleUnitary r' α γ).toContinuousLinearEquiv.toContinuousLinearMap.comp
-        ((r'.inducedBundle α γ).action s).toContinuousLinearEquiv.toContinuousLinearMap =
-      ((r.inducedBundle α γ).action s).toContinuousLinearEquiv.toContinuousLinearMap.comp
-        (r.changeBundleUnitary r' α γ).toContinuousLinearEquiv.toContinuousLinearMap := by
-    apply lp.ext_continuousLinearMap (by norm_num)
-    intro x
-    apply ContinuousLinearMap.ext
-    intro w
-    change r.changeBundleUnitary r' α γ ((r'.inducedBundle α γ).action s (lp.single 2 x w)) =
-      (r.inducedBundle α γ).action s (r.changeBundleUnitary r' α γ (lp.single 2 x w))
-    rw [(r'.inducedBundle α γ).action_single s x (s • x) rfl,
-      changeBundleUnitary_single, changeBundleUnitary_single,
-      (r.inducedBundle α γ).action_single s x (s • x) rfl]
-    exact congrArg (lp.single 2 (s • x)) (r.changeUnitary_intertwines r' α γ s x w)
-  exact DFunLike.congr_fun he v
-
-theorem exact_projective_induction (α : ScalarCocycle K)
-    (γ : ProjectiveRepresentation (α.restrict C) V) :
-    (∃ B : ProjectiveBundle α (fun _ : K ⧸ C => V),
-      ∀ s x v, B.action s (lp.single 2 x v) =
-        lp.single 2 (s • x) (((chooseCosetSection C).phase α s x : ℂ) •
-          γ.unitary ((chooseCosetSection C).correction s x) v)) ∧
-    ∀ r r' : CosetSection C,
-      Intertwines (r'.inducedBundle α γ) (r.inducedBundle α γ)
-        (r.changeBundleUnitary r' α γ).toContinuousLinearEquiv.toContinuousLinearMap := by
-  refine ⟨⟨(chooseCosetSection C).inducedBundle α γ, ?_⟩, ?_⟩
-  · exact (chooseCosetSection C).inducedBundle_single α γ
-  · exact fun r r' => r.changeBundleUnitary_intertwines r' α γ
-
-end InducedBundle
-
-end GermBundles
-
-end OAI
+import Mathlib
+import OAI.GroupTheory.ArithmeticGerms.Basic
+
+namespace OAI
+
+/-! Projective Hilbert bundles and exact induction. -/
+
+namespace GermBundles
+attribute [local instance] Classical.propDecidable
+open scoped BigOperators InnerProductSpace
+
+structure ScalarCocycle (K : Type*) [Group K] where
+  val : K → K → Circle
+  mul_assoc : ∀ g h j, val g h * val (g * h) j = val h j * val g (h * j)
+  one_left : ∀ g, val 1 g = 1
+  one_right : ∀ g, val g 1 = 1
+
+section Bundles
+
+variable {K X : Type*} [Group K] [MulAction K X]
+variable (α : ScalarCocycle K) (E : X → Type*)
+variable [∀ x, NormedAddCommGroup (E x)] [∀ x, InnerProductSpace ℂ (E x)]
+
+attribute [local instance] Classical.propDecidable
+
+structure ProjectiveBundle where
+  action : K → lp E 2 ≃ₗᵢ[ℂ] lp E 2
+  fibre : ∀ (k : K) (x y : X), k • x = y → E x ≃ₗᵢ[ℂ] E y
+  action_single : ∀ (k : K) (x y : X) (h : k • x = y) (v : E x),
+    action k (lp.single 2 x v) = lp.single 2 y (fibre k x y h v)
+  action_one : ∀ v, action 1 v = v
+  action_mul : ∀ g h v, action g (action h v) =
+    (α.val g h : ℂ) • action (g * h) v
+
+variable {α E}
+
+variable {F : X → Type*}
+variable [∀ x, NormedAddCommGroup (F x)] [∀ x, InnerProductSpace ℂ (F x)]
+
+def Intertwines (B : ProjectiveBundle α E) (C : ProjectiveBundle α F)
+    (T : lp E 2 →L[ℂ] lp F 2) : Prop :=
+  ∀ k v, T (B.action k v) = C.action k (T v)
+
+end Bundles
+
+section BuildBundle
+
+variable {X : Type*} {E F : X → Type*}
+variable [∀ x, NormedAddCommGroup (E x)] [∀ x, InnerProductSpace ℂ (E x)]
+variable [∀ x, NormedAddCommGroup (F x)] [∀ x, InnerProductSpace ℂ (F x)]
+
+private theorem relabel_pi_apply (e : X ≃ X)
+    (u : ∀ x y, e x = y → E x ≃ₗᵢ[ℂ] F y) (f : ∀ x, E x) (x : X) :
+    u (e.symm (e x)) (e x) (e.apply_symm_apply (e x)) (f (e.symm (e x))) =
+      u x (e x) rfl (f x) := by
+  have haux (z : X) (hz : z = x) (he : e z = e x) :
+      u z (e x) he (f z) = u x (e x) rfl (f x) := by
+    subst z
+    rfl
+  exact haux (e.symm (e x)) (e.symm_apply_apply x) (e.apply_symm_apply (e x))
+
+private theorem relabel_pi_right (e : X ≃ X)
+    (u : ∀ x y, e x = y → E x ≃ₗᵢ[ℂ] F y) (f : ∀ x, F x) (y : X) :
+    u (e.symm y) y (e.apply_symm_apply y)
+      ((u (e.symm y) (e (e.symm y)) rfl).symm (f (e (e.symm y)))) = f y := by
+  have haux (x y : X) (he : e x = y) :
+      u x y he ((u x (e x) rfl).symm (f (e x))) = f y := by
+    subst y
+    exact (u x (e x) rfl).apply_symm_apply _
+  exact haux (e.symm y) y (e.apply_symm_apply y)
+
+noncomputable def relabel (e : X ≃ X)
+    (u : ∀ x y, e x = y → E x ≃ₗᵢ[ℂ] F y) : lp E 2 ≃ₗᵢ[ℂ] lp F 2 where
+  toLinearEquiv :=
+    { toFun := fun v => ⟨fun y => u (e.symm y) y (e.apply_symm_apply y) (v (e.symm y)), by
+        apply memℓp_gen
+        convert! ((lp.memℓp v).summable (by norm_num)).comp_injective e.symm.injective using 1
+        all_goals first | rfl | (simp only [LinearIsometryEquiv.norm_map]; rfl)⟩
+      invFun := fun v => ⟨fun x => (u x (e x) rfl).symm (v (e x)), by
+        apply memℓp_gen
+        convert! ((lp.memℓp v).summable (by norm_num)).comp_injective e.injective using 1
+        all_goals first | rfl | (simp only [LinearIsometryEquiv.norm_map]; rfl)⟩
+      left_inv := by
+        intro v
+        apply lp.ext
+        funext x
+        change (u x (e x) rfl).symm
+          (u (e.symm (e x)) (e x) (e.apply_symm_apply (e x)) (v.val (e.symm (e x)))) = v.val x
+        exact (congrArg (u x (e x) rfl).symm (relabel_pi_apply e u (fun i => v i) x)).trans
+          ((u x (e x) rfl).symm_apply_apply (v x))
+      right_inv := by
+        intro v
+        apply lp.ext
+        funext y
+        exact relabel_pi_right e u (fun i => v i) y
+      map_add' := by
+        intro v w
+        apply lp.ext
+        funext y
+        change u _ _ _ (v _ + w _) = u _ _ _ (v _) + u _ _ _ (w _)
+        exact map_add _ _ _
+      map_smul' := by
+        intro a v
+        apply lp.ext
+        funext y
+        change u _ _ _ (a • v _) = a • u _ _ _ (v _)
+        exact map_smul _ _ _ }
+  norm_map' := by
+    intro v
+    rw [lp.norm_eq_tsum_rpow (by norm_num), lp.norm_eq_tsum_rpow (by norm_num)]
+    congr 1
+    change (∑' y, ‖u (e.symm y) y (e.apply_symm_apply y) (v (e.symm y))‖ ^ (2 : ENNReal).toReal) = _
+    simp only [LinearIsometryEquiv.norm_map]
+    convert! e.symm.tsum_eq (fun x => ‖v x‖ ^ (2 : ENNReal).toReal) using 1
+
+theorem relabel_apply (e : X ≃ X) (u : ∀ x y, e x = y → E x ≃ₗᵢ[ℂ] F y)
+    (v : lp E 2) (x y : X) (h : e x = y) : relabel e u v y = u x y h (v x) := by
+  subst y
+  change u (e.symm (e x)) (e x) (e.apply_symm_apply (e x)) (v.val (e.symm (e x))) = u x (e x) rfl (v.val x)
+  exact relabel_pi_apply e u (fun i => v i) x
+
+theorem relabel_single (e : X ≃ X) (u : ∀ x y, e x = y → E x ≃ₗᵢ[ℂ] F y)
+    (x y : X) (h : e x = y) (v : E x) :
+    relabel e u (lp.single 2 x v) = lp.single 2 y (u x y h v) := by
+  apply lp.ext
+  funext z
+  by_cases hz : z = y
+  · subst z
+    rw [relabel_apply e u _ x y h, lp.single_apply_self, lp.single_apply_self]
+  · have hn : e.symm z ≠ x := by
+      intro he
+      apply hz
+      rw [← h, ← he, e.apply_symm_apply]
+    change u (e.symm z) z (e.apply_symm_apply z) ((lp.single 2 x v) (e.symm z)) = _
+    rw [lp.single_apply_ne 2 x v hn, map_zero, lp.single_apply_ne 2 y _ hz]
+
+end BuildBundle
+
+section FibreSystem
+
+variable {K X : Type*} [Group K] [MulAction K X]
+variable (α : ScalarCocycle K) (E : X → Type*)
+variable [∀ x, NormedAddCommGroup (E x)] [∀ x, InnerProductSpace ℂ (E x)]
+
+structure FibreSystem where
+  unitary : ∀ (k : K) (x y : X), k • x = y → E x ≃ₗᵢ[ℂ] E y
+  one : ∀ x v, unitary 1 x x (one_smul K x) v = v
+  mul : ∀ g h x y z (hy : h • x = y) (hz : g • y = z) (hgh : (g * h) • x = z) v,
+    unitary g y z hz (unitary h x y hy v) = (α.val g h : ℂ) • unitary (g * h) x z hgh v
+
+variable {α E}
+
+noncomputable def FibreSystem.toBundle (U : FibreSystem α E) : ProjectiveBundle α E where
+  action k := relabel (MulAction.toPerm k) (U.unitary k)
+  fibre := U.unitary
+  action_single k x y h v := relabel_single (MulAction.toPerm k) (U.unitary k) x y h v
+  action_one := by
+    intro v
+    have he : (relabel (MulAction.toPerm (1 : K)) (U.unitary 1)).toContinuousLinearEquiv.toContinuousLinearMap =
+        ContinuousLinearMap.id ℂ (lp E 2) := by
+      apply lp.ext_continuousLinearMap (by norm_num)
+      intro x
+      apply ContinuousLinearMap.ext
+      intro w
+      change relabel (MulAction.toPerm (1 : K)) (U.unitary 1) (lp.single 2 x w) = lp.single 2 x w
+      rw [relabel_single (MulAction.toPerm (1 : K)) (U.unitary 1) x x (one_smul K x), U.one]
+    exact DFunLike.congr_fun he v
+  action_mul := by
+    intro g h v
+    have he :
+        (relabel (MulAction.toPerm g) (U.unitary g)).toContinuousLinearEquiv.toContinuousLinearMap.comp
+          (relabel (MulAction.toPerm h) (U.unitary h)).toContinuousLinearEquiv.toContinuousLinearMap =
+        (α.val g h : ℂ) • (relabel (MulAction.toPerm (g * h)) (U.unitary (g * h))).toContinuousLinearEquiv.toContinuousLinearMap := by
+      apply lp.ext_continuousLinearMap (by norm_num)
+      intro x
+      apply ContinuousLinearMap.ext
+      intro w
+      change relabel (MulAction.toPerm g) (U.unitary g)
+        (relabel (MulAction.toPerm h) (U.unitary h) (lp.single 2 x w)) =
+        (α.val g h : ℂ) • relabel (MulAction.toPerm (g * h)) (U.unitary (g * h)) (lp.single 2 x w)
+      rw [relabel_single (MulAction.toPerm h) (U.unitary h) x (h • x) rfl,
+        relabel_single (MulAction.toPerm g) (U.unitary g) (h • x) ((g * h) • x) (mul_smul g h x).symm,
+        relabel_single (MulAction.toPerm (g * h)) (U.unitary (g * h)) x ((g * h) • x) rfl]
+      erw [U.mul g h x (h • x) ((g * h) • x) rfl (mul_smul g h x).symm rfl w,
+        lp.single_smul]
+    exact DFunLike.congr_fun he v
+
+end FibreSystem
+
+noncomputable section ExactInduction
+
+attribute [local instance] Classical.propDecidable
+
+variable {K : Type*} [Group K]
+
+def ScalarCocycle.restrict (α : ScalarCocycle K) (C : Subgroup K) : ScalarCocycle C where
+  val g h := α.val g h
+  mul_assoc g h j := α.mul_assoc g h j
+  one_left g := α.one_left g
+  one_right g := α.one_right g
+
+structure ProjectiveRepresentation (α : ScalarCocycle K) (V : Type*)
+    [NormedAddCommGroup V] [InnerProductSpace ℂ V] where
+  unitary : K → V ≃ₗᵢ[ℂ] V
+  one : ∀ v, unitary 1 v = v
+  mul : ∀ g h v, unitary g (unitary h v) = (α.val g h : ℂ) • unitary (g * h) v
+
+variable (C : Subgroup K)
+
+structure CosetSection where
+  repr : K ⧸ C → K
+  repr_spec : ∀ x, QuotientGroup.mk (repr x) = x
+  repr_one : repr (QuotientGroup.mk 1) = 1
+
+noncomputable def chooseCosetSection : CosetSection C where
+  repr x := if x = QuotientGroup.mk 1 then 1 else x.out
+  repr_spec x := by
+    split_ifs with h
+    · exact h.symm
+    · exact QuotientGroup.out_eq' x
+  repr_one := by simp
+
+variable {C}
+
+def CosetSection.correction (r : CosetSection C) (s : K) (x : K ⧸ C) : C :=
+  ⟨(r.repr (s • x))⁻¹ * s * r.repr x, by
+    rw [mul_assoc]
+    apply QuotientGroup.eq.mp
+    rw [r.repr_spec, ← smul_eq_mul, ← MulAction.Quotient.smul_mk, r.repr_spec]⟩
+
+@[simp] theorem CosetSection.correction_val (r : CosetSection C) (s : K) (x : K ⧸ C) :
+    (r.correction s x : K) = (r.repr (s • x))⁻¹ * s * r.repr x := rfl
+
+@[simp] theorem CosetSection.repr_mul_correction (r : CosetSection C) (s : K) (x : K ⧸ C) :
+    r.repr (s • x) * (r.correction s x : K) = s * r.repr x := by
+  rw [correction_val]
+  group
+
+@[simp] theorem CosetSection.correction_one (r : CosetSection C) (x : K ⧸ C) :
+    r.correction 1 x = 1 := by
+  apply Subtype.ext
+  simp [correction]
+
+@[simp] theorem CosetSection.correction_mul (r : CosetSection C) (s t : K) (x : K ⧸ C) :
+    r.correction s (t • x) * r.correction t x = r.correction (s * t) x := by
+  apply Subtype.ext
+  simp only [Subgroup.coe_mul, correction_val, mul_smul]
+  group
+
+def CosetSection.phase (r : CosetSection C) (α : ScalarCocycle K) (s : K) (x : K ⧸ C) : Circle :=
+  α.val s (r.repr x) / α.val (r.repr (s • x)) (r.correction s x)
+
+@[simp] theorem CosetSection.phase_one (r : CosetSection C) (α : ScalarCocycle K) (x : K ⧸ C) :
+    r.phase α 1 x = 1 := by
+  simp [phase, α.one_left, α.one_right]
+
+theorem cocycle_transport_phase (α : ScalarCocycle K) (s t x y z a b : K)
+    (ha : z * a = s * y) (hb : y * b = t * x) :
+    (α.val s y / α.val z a) * (α.val t x / α.val y b) * α.val a b =
+      α.val s t * (α.val (s * t) x / α.val z (a * b)) := by
+  have hid : α.val s y * α.val t x * α.val a b * α.val z (a * b) =
+      α.val s t * α.val (s * t) x * (α.val z a * α.val y b) := by
+    calc
+      _ = α.val t x * α.val s y * (α.val a b * α.val z (a * b)) := by ac_rfl
+      _ = α.val t x * α.val s y * (α.val z a * α.val (z * a) b) := by
+        rw [α.mul_assoc z a b]
+      _ = α.val z a * α.val t x * (α.val s y * α.val (s * y) b) := by
+        rw [ha]
+        ac_rfl
+      _ = α.val z a * α.val t x * (α.val y b * α.val s (y * b)) := by
+        rw [α.mul_assoc s y b]
+      _ = (α.val t x * α.val s (t * x)) * (α.val z a * α.val y b) := by
+        rw [hb]
+        ac_rfl
+      _ = _ := by rw [← α.mul_assoc s t x]
+  rw [div_mul_div_comm, div_mul_eq_mul_div, mul_div_assoc]
+  simpa only [mul_div_assoc] using div_eq_div_iff_mul_eq_mul.mpr hid
+
+theorem CosetSection.phase_mul (r : CosetSection C) (α : ScalarCocycle K) (s t : K)
+    (x : K ⧸ C) :
+    r.phase α s (t • x) * r.phase α t x * α.val (r.correction s (t • x)) (r.correction t x) =
+      α.val s t * r.phase α (s * t) x := by
+  unfold phase
+  have h := cocycle_transport_phase α s t (r.repr x) (r.repr (t • x))
+    (r.repr (s • t • x)) (r.correction s (t • x)) (r.correction t x)
+    (r.repr_mul_correction s (t • x)) (r.repr_mul_correction t x)
+  simpa only [← Subgroup.coe_mul, correction_mul, mul_smul] using h
+
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+
+def circleUnitary (z : Circle) : V ≃ₗᵢ[ℂ] V where
+  toLinearEquiv :=
+    { toFun := fun v => (z : ℂ) • v
+      invFun := fun v => ((z⁻¹ : Circle) : ℂ) • v
+      left_inv := by intro v; simp [smul_smul]
+      right_inv := by intro v; simp [smul_smul]
+      map_add' := by intro v w; exact smul_add _ _ _
+      map_smul' := by intro a v; exact smul_comm _ _ _ }
+  norm_map' := by intro v; simp [norm_smul, Circle.norm_coe]
+
+@[simp] theorem circleUnitary_apply (z : Circle) (v : V) : circleUnitary z v = (z : ℂ) • v := rfl
+
+def CosetSection.inducedUnitary (r : CosetSection C) (α : ScalarCocycle K)
+    (γ : ProjectiveRepresentation (α.restrict C) V) (s : K) (x : K ⧸ C) : V ≃ₗᵢ[ℂ] V :=
+  (γ.unitary (r.correction s x)).trans (circleUnitary (r.phase α s x))
+
+@[simp] theorem CosetSection.inducedUnitary_apply (r : CosetSection C) (α : ScalarCocycle K)
+    (γ : ProjectiveRepresentation (α.restrict C) V) (s : K) (x : K ⧸ C) (v : V) :
+    r.inducedUnitary α γ s x v = (r.phase α s x : ℂ) • γ.unitary (r.correction s x) v := rfl
+
+@[simp] theorem CosetSection.inducedUnitary_one (r : CosetSection C) (α : ScalarCocycle K)
+    (γ : ProjectiveRepresentation (α.restrict C) V) (x : K ⧸ C) (v : V) :
+    r.inducedUnitary α γ 1 x v = v := by
+  simp [γ.one]
+
+theorem CosetSection.inducedUnitary_mul (r : CosetSection C) (α : ScalarCocycle K)
+    (γ : ProjectiveRepresentation (α.restrict C) V) (s t : K) (x : K ⧸ C) (v : V) :
+    r.inducedUnitary α γ s (t • x) (r.inducedUnitary α γ t x v) =
+      (α.val s t : ℂ) • r.inducedUnitary α γ (s * t) x v := by
+  simp only [inducedUnitary_apply, map_smul, γ.mul, smul_smul]
+  rw [correction_mul]
+  congr 1
+  have hp := congrArg (fun z : Circle => (z : ℂ)) (r.phase_mul α s t x)
+  simpa only [Circle.coe_mul, ScalarCocycle.restrict, mul_assoc, mul_left_comm] using hp
+
+def CosetSection.change (r r' : CosetSection C) (x : K ⧸ C) : C :=
+  ⟨(r.repr x)⁻¹ * r'.repr x, QuotientGroup.eq.mp ((r.repr_spec x).trans (r'.repr_spec x).symm)⟩
+
+@[simp] theorem CosetSection.repr_mul_change (r r' : CosetSection C) (x : K ⧸ C) :
+    r.repr x * (r.change r' x : K) = r'.repr x := by
+  simp [change]
+
+theorem CosetSection.change_correction (r r' : CosetSection C) (s : K) (x : K ⧸ C) :
+    r.change r' (s • x) * r'.correction s x = r.correction s x * r.change r' x := by
+  apply Subtype.ext
+  simp only [Subgroup.coe_mul, change, correction]
+  group
+
+theorem cocycle_change_phase (α : ScalarCocycle K) (s x y c k l c' : K)
+    (hc : y * c = s * x) (hk : l * c' = c * k) :
+    (α.val y l)⁻¹ * (α.val s (x * k) / α.val (y * l) c') * α.val l c' =
+      (α.val s x / α.val y c) * (α.val x k)⁻¹ * α.val c k := by
+  have hid : α.val s (x * k) * α.val l c' * (α.val y c * α.val x k) =
+      α.val s x * α.val c k * (α.val (y * l) c' * α.val y l) := by
+    calc
+      _ = α.val l c' * α.val y c * (α.val x k * α.val s (x * k)) := by ac_rfl
+      _ = α.val l c' * α.val y c * (α.val s x * α.val (s * x) k) := by
+        rw [α.mul_assoc s x k]
+      _ = α.val s x * α.val l c' * (α.val y c * α.val (y * c) k) := by
+        rw [← hc]
+        ac_rfl
+      _ = α.val s x * α.val l c' * (α.val c k * α.val y (c * k)) := by
+        rw [α.mul_assoc y c k]
+      _ = α.val s x * α.val c k * (α.val l c' * α.val y (l * c')) := by
+        rw [← hk]
+        ac_rfl
+      _ = _ := by rw [← α.mul_assoc y l c']; ac_rfl
+  have hr := div_eq_div_iff_mul_eq_mul.mpr hid
+  convert hr using 1 <;> simp only [div_eq_mul_inv, mul_inv_rev, mul_comm, mul_left_comm, mul_assoc]
+
+def CosetSection.changeUnitary (r r' : CosetSection C) (α : ScalarCocycle K)
+    (γ : ProjectiveRepresentation (α.restrict C) V) (x : K ⧸ C) : V ≃ₗᵢ[ℂ] V :=
+  (γ.unitary (r.change r' x)).trans
+    (circleUnitary (α.val (r.repr x) (r.change r' x))⁻¹)
+
+@[simp] theorem CosetSection.changeUnitary_apply (r r' : CosetSection C) (α : ScalarCocycle K)
+    (γ : ProjectiveRepresentation (α.restrict C) V) (x : K ⧸ C) (v : V) :
+    r.changeUnitary r' α γ x v = ((α.val (r.repr x) (r.change r' x))⁻¹ : Circle) •
+      γ.unitary (r.change r' x) v := rfl
+
+theorem CosetSection.change_phase (r r' : CosetSection C) (α : ScalarCocycle K)
+    (s : K) (x : K ⧸ C) :
+    (α.val (r.repr (s • x)) (r.change r' (s • x)))⁻¹ * r'.phase α s x *
+        α.val (r.change r' (s • x)) (r'.correction s x) =
+      r.phase α s x * (α.val (r.repr x) (r.change r' x))⁻¹ *
+        α.val (r.correction s x) (r.change r' x) := by
+  have h := cocycle_change_phase α s (r.repr x) (r.repr (s • x))
+    (r.correction s x) (r.change r' x) (r.change r' (s • x)) (r'.correction s x)
+    (r.repr_mul_correction s x) (congrArg Subtype.val (r.change_correction r' s x))
+  simpa only [phase, repr_mul_change] using h
+
+theorem CosetSection.changeUnitary_intertwines (r r' : CosetSection C) (α : ScalarCocycle K)
+    (γ : ProjectiveRepresentation (α.restrict C) V) (s : K) (x : K ⧸ C) (v : V) :
+    r.changeUnitary r' α γ (s • x) (r'.inducedUnitary α γ s x v) =
+      r.inducedUnitary α γ s x (r.changeUnitary r' α γ x v) := by
+  change ((α.val (r.repr (s • x)) (r.change r' (s • x)))⁻¹ : Circle) •
+    γ.unitary (r.change r' (s • x)) ((r'.phase α s x : ℂ) • γ.unitary (r'.correction s x) v) =
+    (r.phase α s x : ℂ) • γ.unitary (r.correction s x)
+      (((α.val (r.repr x) (r.change r' x))⁻¹ : Circle) • γ.unitary (r.change r' x) v)
+  simp only [Circle.smul_def, map_smul, γ.mul, smul_smul]
+  rw [change_correction]
+  congr 1
+  have hp := congrArg (fun z : Circle => (z : ℂ)) (r.change_phase r' α s x)
+  simpa only [Circle.coe_mul, ScalarCocycle.restrict, mul_assoc] using hp
+
+end ExactInduction
+
+noncomputable section InducedBundle
+
+attribute [local instance] Classical.propDecidable
+
+variable {K : Type*} [Group K] {C : Subgroup K}
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+
+local instance : DecidableEq (K ⧸ C) := Classical.decEq _
+
+def CosetSection.inducedFibreSystem (r : CosetSection C) (α : ScalarCocycle K)
+    (γ : ProjectiveRepresentation (α.restrict C) V) :
+    FibreSystem α (fun _ : K ⧸ C => V) where
+  unitary s x _ _ := r.inducedUnitary α γ s x
+  one x v := r.inducedUnitary_one α γ x v
+  mul s t x y z hy hz hst v := by
+    subst y
+    exact r.inducedUnitary_mul α γ s t x v
+
+def CosetSection.inducedBundle (r : CosetSection C) (α : ScalarCocycle K)
+    (γ : ProjectiveRepresentation (α.restrict C) V) :
+    ProjectiveBundle α (fun _ : K ⧸ C => V) :=
+  (r.inducedFibreSystem α γ).toBundle
+
+theorem CosetSection.inducedBundle_single (r : CosetSection C) (α : ScalarCocycle K)
+    (γ : ProjectiveRepresentation (α.restrict C) V) (s : K) (x : K ⧸ C) (v : V) :
+    (r.inducedBundle α γ).action s (lp.single 2 x v) =
+      lp.single 2 (s • x) ((r.phase α s x : ℂ) • γ.unitary (r.correction s x) v) := by
+  exact (r.inducedBundle α γ).action_single s x (s • x) rfl v
+
+def CosetSection.changeBundleUnitary (r r' : CosetSection C) (α : ScalarCocycle K)
+    (γ : ProjectiveRepresentation (α.restrict C) V) :
+    lp (fun _ : K ⧸ C => V) 2 ≃ₗᵢ[ℂ] lp (fun _ : K ⧸ C => V) 2 :=
+  relabel (Equiv.refl _) (fun x _ _ => r.changeUnitary r' α γ x)
+
+@[simp] theorem CosetSection.changeBundleUnitary_single (r r' : CosetSection C) (α : ScalarCocycle K)
+    (γ : ProjectiveRepresentation (α.restrict C) V) (x : K ⧸ C) (v : V) :
+    r.changeBundleUnitary r' α γ (lp.single 2 x v) =
+      lp.single 2 x (r.changeUnitary r' α γ x v) := by
+  exact relabel_single (Equiv.refl _) (fun x _ _ => r.changeUnitary r' α γ x) x x rfl v
+
+theorem CosetSection.changeBundleUnitary_intertwines (r r' : CosetSection C) (α : ScalarCocycle K)
+    (γ : ProjectiveRepresentation (α.restrict C) V) :
+    Intertwines (r'.inducedBundle α γ) (r.inducedBundle α γ)
+      (r.changeBundleUnitary r' α γ).toContinuousLinearEquiv.toContinuousLinearMap := by
+  intro s v
+  have he :
+      (r.changeBundleUnitary r' α γ).toContinuousLinearEquiv.toContinuousLinearMap.comp
+        ((r'.inducedBundle α γ).action s).toContinuousLinearEquiv.toContinuousLinearMap =
+      ((r.inducedBundle α γ).action s).toContinuousLinearEquiv.toContinuousLinearMap.comp
+        (r.changeBundleUnitary r' α γ).toContinuousLinearEquiv.toContinuousLinearMap := by
+    apply lp.ext_continuousLinearMap (by norm_num)
+    intro x
+    apply ContinuousLinearMap.ext
+    intro w
+    change r.changeBundleUnitary r' α γ ((r'.inducedBundle α γ).action s (lp.single 2 x w)) =
+      (r.inducedBundle α γ).action s (r.changeBundleUnitary r' α γ (lp.single 2 x w))
+    rw [(r'.inducedBundle α γ).action_single s x (s • x) rfl,
+      changeBundleUnitary_single, changeBundleUnitary_single,
+      (r.inducedBundle α γ).action_single s x (s • x) rfl]
+    exact congrArg (lp.single 2 (s • x)) (r.changeUnitary_intertwines r' α γ s x w)
+  exact DFunLike.congr_fun he v
+
+theorem exact_projective_induction (α : ScalarCocycle K)
+    (γ : ProjectiveRepresentation (α.restrict C) V) :
+    (∃ B : ProjectiveBundle α (fun _ : K ⧸ C => V),
+      ∀ s x v, B.action s (lp.single 2 x v) =
+        lp.single 2 (s • x) (((chooseCosetSection C).phase α s x : ℂ) •
+          γ.unitary ((chooseCosetSection C).correction s x) v)) ∧
+    ∀ r r' : CosetSection C,
+      Intertwines (r'.inducedBundle α γ) (r.inducedBundle α γ)
+        (r.changeBundleUnitary r' α γ).toContinuousLinearEquiv.toContinuousLinearMap := by
+  refine ⟨⟨(chooseCosetSection C).inducedBundle α γ, ?_⟩, ?_⟩
+  · exact (chooseCosetSection C).inducedBundle_single α γ
+  · exact fun r r' => r.changeBundleUnitary_intertwines r' α γ
+
+end InducedBundle
+
+end GermBundles
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/GraphModels/NormalActions.lean b/lean/OAI/RepresentationTheory/GraphModels/NormalActions.lean
index 85aeb26eead74b3ce91fbbcd23fe508df6eb5bb4..9afd4c91f68d8fa2808d25862931ca8b58cbac7f 100644
--- a/lean/OAI/RepresentationTheory/GraphModels/NormalActions.lean
+++ b/lean/OAI/RepresentationTheory/GraphModels/NormalActions.lean
@@ -1,598 +1,598 @@
-import Mathlib
-import OAI.RepresentationTheory.GraphModels.Regular
-
-namespace OAI
-
-noncomputable section
-
-/-! Commuting normal actions and regular multiplicities. -/
-
-namespace GraphNormality
-section
-open ArithmeticGerms GermBundles
-open scoped InnerProductSpace
-section WeakOperatorAlgebras
-variable {L W : Type*}
-variable [NormedAddCommGroup L] [InnerProductSpace ℂ L] [CompleteSpace L]
-variable [NormedAddCommGroup W] [InnerProductSpace ℂ W] [CompleteSpace W]
-
-def wotLift (F : (L →L[ℂ] L) →⋆ₐ[ℂ] (W →L[ℂ] W)) :
-    (L →WOT[ℂ] L) →⋆ₐ[ℂ] (W →WOT[ℂ] W) where
-  toFun T := ContinuousLinearMapWOT.ofCLM (F T.toCLM)
-  map_one' := congrArg ContinuousLinearMapWOT.ofCLM (map_one F)
-  map_mul' T S := congrArg ContinuousLinearMapWOT.ofCLM (map_mul F T.toCLM S.toCLM)
-  map_add' T S := congrArg ContinuousLinearMapWOT.ofCLM (map_add F T.toCLM S.toCLM)
-  map_zero' := congrArg ContinuousLinearMapWOT.ofCLM (map_zero F)
-  commutes' z := congrArg ContinuousLinearMapWOT.ofCLM (F.commutes z)
-  map_star' T := congrArg ContinuousLinearMapWOT.ofCLM (map_star F T.toCLM)
-
-theorem closed_generated_commute (s t : Set (W →WOT[ℂ] W))
-    (hst : s ⊆ StarSubalgebra.centralizer ℂ t)
-    {x y : W →WOT[ℂ] W}
-    (hx : x ∈ (StarAlgebra.adjoin ℂ s).topologicalClosure)
-    (hy : y ∈ (StarAlgebra.adjoin ℂ t).topologicalClosure) : x * y = y * x := by
-  have hs : (StarAlgebra.adjoin ℂ s).topologicalClosure ≤
-      StarSubalgebra.centralizer ℂ t :=
-    StarSubalgebra.topologicalClosure_minimal (StarAlgebra.adjoin_le hst)
-      (Set.isClosed_centralizer _)
-  have ht := StarSubalgebra.topologicalClosure_adjoin_le_centralizer_centralizer ℂ t hy
-  exact ((StarSubalgebra.mem_centralizer_iff ℂ).mp ht x (hs hx)).1
-
-theorem map_mem_closed_generated (F : (L →WOT[ℂ] L) →⋆ₐ[ℂ] (W →WOT[ℂ] W))
-    (hF : Continuous F) (s : Set (L →WOT[ℂ] L)) {x : L →WOT[ℂ] L}
-    (hx : x ∈ (StarAlgebra.adjoin ℂ s).topologicalClosure) :
-    F x ∈ (StarAlgebra.adjoin ℂ (F '' s)).topologicalClosure := by
-  have h := StarSubalgebra.map_topologicalClosure_le (StarAlgebra.adjoin ℂ s) F hF
-  rw [StarAlgHom.map_adjoin] at h
-  exact h ⟨x, hx, rfl⟩
-
-theorem isometries_commute_star_right (U V : W ≃ₗᵢ[ℂ] W)
-    (h : ∀ x, U (V x) = V (U x)) :
-    U.toContinuousLinearEquiv.toContinuousLinearMap *
-        star V.toContinuousLinearEquiv.toContinuousLinearMap =
-      star V.toContinuousLinearEquiv.toContinuousLinearMap *
-        U.toContinuousLinearEquiv.toContinuousLinearMap := by
-  rw [LinearIsometryEquiv.star_eq_symm]
-  apply ContinuousLinearMap.ext
-  intro x
-  change U (V.symm x) = V.symm (U x)
-  apply V.injective
-  rw [← h, V.apply_symm_apply, V.apply_symm_apply]
-
-end WeakOperatorAlgebras
-
-variable {G : Type*} [Group G]
-
-def regularGenerator (μ : ScalarCocycle G) (g : G) :
-    lp (fun _ : G => ℂ) 2 →WOT[ℂ] lp (fun _ : G => ℂ) 2 :=
-  ContinuousLinearMapWOT.ofCLM (scalarRegular μ g).toContinuousLinearEquiv.toContinuousLinearMap
-
-def twistedGroupAlgebra (μ : ScalarCocycle G) :
-    StarSubalgebra ℂ (lp (fun _ : G => ℂ) 2 →WOT[ℂ] lp (fun _ : G => ℂ) 2) :=
-  (StarAlgebra.adjoin ℂ (Set.range (regularGenerator μ))).topologicalClosure
-
-theorem regularGenerator_mem (μ : ScalarCocycle G) (g : G) :
-    regularGenerator μ g ∈ twistedGroupAlgebra μ :=
-  StarSubalgebra.le_topologicalClosure _ (StarAlgebra.subset_adjoin ℂ _ (Set.mem_range_self g))
-
-end
-
-section
-open ArithmeticGerms GermBundles
-variable {G H V J : Type*} [Group G] [Group H] [Fintype J]
-variable [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
-variable (C : VirtualGraph G H)
-local instance normalActionDomainFintype : Fintype (G ⧸ C.domain) :=
-  Subgroup.fintypeQuotientOfFiniteIndex
-local instance normalActionCodomainFintype : Fintype (H ⧸ C.codomain) :=
-  Subgroup.fintypeQuotientOfFiniteIndex
-variable (μ : ScalarCocycle G) (ω : ScalarCocycle H)
-variable (B : ProjectiveBundle (prodRatio μ ω) (fun _ : (G × H) ⧸ C.graph => V))
-variable (b : OrthonormalBasis J ℂ V)
-
-def coordinateOperator (s : G × H) : lp (fun _ : (G × H) ⧸ C.graph => V) 2 →WOT[ℂ]
-    lp (fun _ : (G × H) ⧸ C.graph => V) 2 :=
-  ContinuousLinearMapWOT.ofCLM (B.action s).toContinuousLinearEquiv.toContinuousLinearMap
-
-theorem coordinateOperator_centralizer :
-    Set.range (fun g : G => coordinateOperator C μ ω B (g,1)) ⊆
-      StarSubalgebra.centralizer ℂ
-        (Set.range (fun h : H => coordinateOperator C μ ω B (1,h))) := by
-  rintro _ ⟨g,rfl⟩
-  apply (StarSubalgebra.mem_centralizer_iff ℂ).mpr
-  rintro _ ⟨h,rfl⟩
-  constructor
-  · apply ContinuousLinearMapWOT.ext
-    intro v
-    exact (coordinate_actions_commute C μ ω B g h v).symm
-  · exact congrArg ContinuousLinearMapWOT.ofCLM
-      (isometries_commute_star_right (B.action (g,1)) (B.action (1,h))
-        (coordinate_actions_commute C μ ω B g h)).symm
-
-theorem firstExtension_generating_image :
-    (wotLift (firstExtension C μ ω B b)) '' Set.range (regularGenerator μ) =
-      Set.range (fun g : G => coordinateOperator C μ ω B (g,1)) := by
-  rw [← Set.range_comp']
-  congr 1
-  funext g
-  exact congrArg ContinuousLinearMapWOT.ofCLM (firstExtension_generator C μ ω B b g)
-
-theorem secondExtension_generating_image :
-    (wotLift (secondExtension C μ ω B b)) '' Set.range (regularGenerator (inverseCocycle ω)) =
-      Set.range (fun h : H => coordinateOperator C μ ω B (1,h)) := by
-  rw [← Set.range_comp']
-  congr 1
-  funext h
-  exact congrArg ContinuousLinearMapWOT.ofCLM (secondExtension_generator C μ ω B b h)
-
-def leftNormalAction : twistedGroupAlgebra μ →⋆ₐ[ℂ]
-    (lp (fun _ : (G × H) ⧸ C.graph => V) 2 →WOT[ℂ]
-      lp (fun _ : (G × H) ⧸ C.graph => V) 2) :=
-  (wotLift (firstExtension C μ ω B b)).comp (twistedGroupAlgebra μ).subtype
-
-def rightAdjointNormalAction : twistedGroupAlgebra (inverseCocycle ω) →⋆ₐ[ℂ]
-    (lp (fun _ : (G × H) ⧸ C.graph => V) 2 →WOT[ℂ]
-      lp (fun _ : (G × H) ⧸ C.graph => V) 2) :=
-  (wotLift (secondExtension C μ ω B b)).comp (twistedGroupAlgebra (inverseCocycle ω)).subtype
-
-theorem leftNormalAction_continuous : Continuous (leftNormalAction C μ ω B b) :=
-  (firstExtension_wot_continuous C μ ω B b).comp continuous_subtype_val
-
-theorem rightAdjointNormalAction_continuous : Continuous (rightAdjointNormalAction C μ ω B b) :=
-  (secondExtension_wot_continuous C μ ω B b).comp continuous_subtype_val
-
-theorem leftNormalAction_generator (g : G) :
-    leftNormalAction C μ ω B b ⟨regularGenerator μ g, regularGenerator_mem μ g⟩ =
-      coordinateOperator C μ ω B (g,1) :=
-  congrArg ContinuousLinearMapWOT.ofCLM (firstExtension_generator C μ ω B b g)
-
-theorem rightAdjointNormalAction_generator (h : H) :
-    rightAdjointNormalAction C μ ω B b
-      ⟨regularGenerator (inverseCocycle ω) h, regularGenerator_mem (inverseCocycle ω) h⟩ =
-      coordinateOperator C μ ω B (1,h) :=
-  congrArg ContinuousLinearMapWOT.ofCLM (secondExtension_generator C μ ω B b h)
-
-theorem normalActions_commute (T : twistedGroupAlgebra μ)
-    (S : twistedGroupAlgebra (inverseCocycle ω)) :
-    leftNormalAction C μ ω B b T * rightAdjointNormalAction C μ ω B b S =
-      rightAdjointNormalAction C μ ω B b S * leftNormalAction C μ ω B b T := by
-  have hT := map_mem_closed_generated (wotLift (firstExtension C μ ω B b))
-    (firstExtension_wot_continuous C μ ω B b) (Set.range (regularGenerator μ)) T.property
-  have hS := map_mem_closed_generated (wotLift (secondExtension C μ ω B b))
-    (secondExtension_wot_continuous C μ ω B b)
-    (Set.range (regularGenerator (inverseCocycle ω))) S.property
-  rw [firstExtension_generating_image] at hT
-  rw [secondExtension_generating_image] at hS
-  exact closed_generated_commute _ _ (coordinateOperator_centralizer C μ ω B) hT hS
-
-end
-
-section
-open scoped InnerProductSpace
-variable {I : Type*}
-
-lemma scalarLp_inner_star_star (x y : lp (fun _ : I => ℂ) 2) :
-    ⟪star x, star y⟫_ℂ = ⟪y, x⟫_ℂ := by
-  rw [lp.inner_eq_tsum, lp.inner_eq_tsum]
-  apply tsum_congr
-  intro i
-  simp [RCLike.inner_apply, mul_comm]
-
-lemma scalarLp_inner_star_comm (x y : lp (fun _ : I => ℂ) 2) :
-    ⟪x, star y⟫_ℂ = ⟪y, star x⟫_ℂ := by
-  simpa only [star_star] using scalarLp_inner_star_star (star x) y
-
-def operatorTranspose (T : lp (fun _ : I => ℂ) 2 →L[ℂ] lp (fun _ : I => ℂ) 2) :
-    lp (fun _ : I => ℂ) 2 →L[ℂ] lp (fun _ : I => ℂ) 2 where
-  toFun x := star ((star T) (star x))
-  map_add' x y := by simp
-  map_smul' z x := by simp
-  cont := continuous_star.comp ((star T).continuous.comp continuous_star)
-
-lemma operatorTranspose_inner (T : lp (fun _ : I => ℂ) 2 →L[ℂ] lp (fun _ : I => ℂ) 2)
-    (x y : lp (fun _ : I => ℂ) 2) :
-    ⟪y, operatorTranspose T x⟫_ℂ = ⟪star x, T (star y)⟫_ℂ := by
-  change ⟪y, star ((star T) (star x))⟫_ℂ = _
-  rw [scalarLp_inner_star_comm, ContinuousLinearMap.star_eq_adjoint,
-    ContinuousLinearMap.adjoint_inner_left]
-
-lemma operatorTranspose_add (T S : lp (fun _ : I => ℂ) 2 →L[ℂ] lp (fun _ : I => ℂ) 2) :
-    operatorTranspose (T + S) = operatorTranspose T + operatorTranspose S := by
-  apply ContinuousLinearMap.ext
-  intro x
-  simp [operatorTranspose, ContinuousLinearMap.star_eq_adjoint]
-
-lemma operatorTranspose_smul (z : ℂ)
-    (T : lp (fun _ : I => ℂ) 2 →L[ℂ] lp (fun _ : I => ℂ) 2) :
-    operatorTranspose (z • T) = z • operatorTranspose T := by
-  apply ContinuousLinearMap.ext
-  intro x
-  simp [operatorTranspose]
-
-lemma operatorTranspose_mul (T S : lp (fun _ : I => ℂ) 2 →L[ℂ] lp (fun _ : I => ℂ) 2) :
-    operatorTranspose (T * S) = operatorTranspose S * operatorTranspose T := by
-  apply ContinuousLinearMap.ext
-  intro x
-  simp [operatorTranspose]
-
-lemma operatorTranspose_star (T : lp (fun _ : I => ℂ) 2 →L[ℂ] lp (fun _ : I => ℂ) 2) :
-    operatorTranspose (star T) = star (operatorTranspose T) := by
-  rw [ContinuousLinearMap.star_eq_adjoint (operatorTranspose T)]
-  apply (ContinuousLinearMap.eq_adjoint_iff _ _).mpr
-  intro x y
-  rw [← inner_conj_symm, operatorTranspose_inner, ContinuousLinearMap.star_eq_adjoint,
-    ContinuousLinearMap.adjoint_inner_right, inner_conj_symm, operatorTranspose_inner]
-
-lemma operatorTranspose_involutive
-    (T : lp (fun _ : I => ℂ) 2 →L[ℂ] lp (fun _ : I => ℂ) 2) :
-    operatorTranspose (operatorTranspose T) = T := by
-  apply ContinuousLinearMap.ext
-  intro x
-  apply ext_inner_left ℂ
-  intro y
-  rw [operatorTranspose_inner, operatorTranspose_inner, star_star, star_star]
-
-lemma operatorTranspose_one :
-    operatorTranspose (1 : lp (fun _ : I => ℂ) 2 →L[ℂ] lp (fun _ : I => ℂ) 2) = 1 := by
-  apply ContinuousLinearMap.ext
-  intro x
-  simp [operatorTranspose]
-
-def operatorOppositeEquiv :
-    (lp (fun _ : I => ℂ) 2 →L[ℂ] lp (fun _ : I => ℂ) 2)ᵐᵒᵖ ≃⋆ₐ[ℂ]
-      (lp (fun _ : I => ℂ) 2 →L[ℂ] lp (fun _ : I => ℂ) 2) where
-  toFun T := operatorTranspose T.unop
-  invFun T := MulOpposite.op (operatorTranspose T)
-  left_inv T := by simp [operatorTranspose_involutive]
-  right_inv T := operatorTranspose_involutive T
-  map_mul' T S := operatorTranspose_mul S.unop T.unop
-  map_add' T S := operatorTranspose_add T.unop S.unop
-  map_smul' z T := operatorTranspose_smul z T.unop
-  map_star' T := operatorTranspose_star T.unop
-
-lemma operatorTranspose_wot_continuous :
-    Continuous (fun T : lp (fun _ : I => ℂ) 2 →WOT[ℂ] lp (fun _ : I => ℂ) 2 =>
-      ContinuousLinearMapWOT.ofCLM (operatorTranspose T.toCLM)) := by
-  apply ContinuousLinearMapWOT.continuous_iff.mpr
-  intro x y
-  change Continuous (fun T : lp (fun _ : I => ℂ) 2 →WOT[ℂ] lp (fun _ : I => ℂ) 2 =>
-    ⟪y, operatorTranspose T.toCLM x⟫_ℂ)
-  simp_rw [operatorTranspose_inner]
-  exact ContinuousLinearMapWOT.continuous_iff.mp continuous_id (star y) (star x)
-
-end
-
-section
-open GermBundles
-variable {G : Type*}
-
-def wotTranspose (T : lp (fun _ : G => ℂ) 2 →WOT[ℂ] lp (fun _ : G => ℂ) 2) :
-    lp (fun _ : G => ℂ) 2 →WOT[ℂ] lp (fun _ : G => ℂ) 2 :=
-  ContinuousLinearMapWOT.ofCLM (operatorTranspose T.toCLM)
-
-lemma wotTranspose_continuous : Continuous (wotTranspose (G := G)) :=
-  operatorTranspose_wot_continuous
-
-lemma wotTranspose_add (T S : lp (fun _ : G => ℂ) 2 →WOT[ℂ] lp (fun _ : G => ℂ) 2) :
-    wotTranspose (T + S) = wotTranspose T + wotTranspose S :=
-  congrArg ContinuousLinearMapWOT.ofCLM (operatorTranspose_add T.toCLM S.toCLM)
-
-lemma wotTranspose_mul (T S : lp (fun _ : G => ℂ) 2 →WOT[ℂ] lp (fun _ : G => ℂ) 2) :
-    wotTranspose (T * S) = wotTranspose S * wotTranspose T :=
-  congrArg ContinuousLinearMapWOT.ofCLM (operatorTranspose_mul T.toCLM S.toCLM)
-
-lemma wotTranspose_smul (z : ℂ)
-    (T : lp (fun _ : G => ℂ) 2 →WOT[ℂ] lp (fun _ : G => ℂ) 2) :
-    wotTranspose (z • T) = z • wotTranspose T :=
-  congrArg ContinuousLinearMapWOT.ofCLM (operatorTranspose_smul z T.toCLM)
-
-lemma wotTranspose_star (T : lp (fun _ : G => ℂ) 2 →WOT[ℂ] lp (fun _ : G => ℂ) 2) :
-    wotTranspose (star T) = star (wotTranspose T) :=
-  congrArg ContinuousLinearMapWOT.ofCLM (operatorTranspose_star T.toCLM)
-
-lemma wotTranspose_one :
-    wotTranspose (1 : lp (fun _ : G => ℂ) 2 →WOT[ℂ] lp (fun _ : G => ℂ) 2) = 1 :=
-  congrArg ContinuousLinearMapWOT.ofCLM operatorTranspose_one
-
-lemma wotTranspose_involutive
-    (T : lp (fun _ : G => ℂ) 2 →WOT[ℂ] lp (fun _ : G => ℂ) 2) :
-    wotTranspose (wotTranspose T) = T :=
-  congrArg ContinuousLinearMapWOT.ofCLM (operatorTranspose_involutive T.toCLM)
-
-lemma wotTranspose_zero :
-    wotTranspose (0 : lp (fun _ : G => ℂ) 2 →WOT[ℂ] lp (fun _ : G => ℂ) 2) = 0 := by
-  apply ContinuousLinearMapWOT.ext
-  intro x
-  change star ((star (0 : lp (fun _ : G => ℂ) 2 →L[ℂ] lp (fun _ : G => ℂ) 2)) (star x)) = 0
-  simp [ContinuousLinearMap.star_eq_adjoint]
-
-variable [Group G]
-
-lemma wotTranspose_star_regularGenerator (ω : ScalarCocycle G) (g : G) :
-    wotTranspose (star (regularGenerator ω g)) = regularGenerator (inverseCocycle ω) g := by
-  apply ContinuousLinearMapWOT.ext
-  intro x
-  apply lp.ext
-  funext k
-  obtain ⟨h,rfl⟩ := (Equiv.mulLeft g).surjective k
-  change (star ((star (star (scalarRegular ω g).toContinuousLinearEquiv.toContinuousLinearMap))
-    (star x))) (g*h) = scalarRegular (inverseCocycle ω) g x (g*h)
-  rw [star_star]
-  change star (scalarRegular ω g (star x) (g*h)) = _
-  rw [scalarRegular_apply, scalarRegular_apply]
-  simp only [inverseCocycle, star_mul, lp.star_apply, star_star]
-  rw [Circle.coe_inv_eq_conj]
-  exact mul_comm _ _
-
-lemma wotTranspose_regularGenerator (ω : ScalarCocycle G) (g : G) :
-    wotTranspose (regularGenerator ω g) = star (regularGenerator (inverseCocycle ω) g) := by
-  rw [← wotTranspose_star_regularGenerator, wotTranspose_star, star_star]
-
-def transposePreimage
-    (S : StarSubalgebra ℂ (lp (fun _ : G => ℂ) 2 →WOT[ℂ] lp (fun _ : G => ℂ) 2)) :
-    StarSubalgebra ℂ (lp (fun _ : G => ℂ) 2 →WOT[ℂ] lp (fun _ : G => ℂ) 2) where
-  carrier := {T | wotTranspose T ∈ S}
-  zero_mem' := by
-    change wotTranspose 0 ∈ S
-    rw [wotTranspose_zero]
-    exact zero_mem S
-  one_mem' := by
-    change wotTranspose 1 ∈ S
-    rw [wotTranspose_one]
-    exact one_mem S
-  add_mem' hT hU := by
-    change wotTranspose _ ∈ S
-    rw [wotTranspose_add]
-    exact add_mem hT hU
-  mul_mem' hT hU := by
-    change wotTranspose _ ∈ S
-    rw [wotTranspose_mul]
-    exact mul_mem hU hT
-  algebraMap_mem' z := by
-    change wotTranspose (algebraMap ℂ _ z) ∈ S
-    rw [Algebra.algebraMap_eq_smul_one, wotTranspose_smul, wotTranspose_one]
-    exact S.smul_mem S.one_mem z
-  star_mem' hT := by
-    change wotTranspose _ ∈ S
-    rw [wotTranspose_star]
-    exact star_mem hT
-
-lemma wotTranspose_mem_twistedGroupAlgebra (ω : ScalarCocycle G)
-    {T : lp (fun _ : G => ℂ) 2 →WOT[ℂ] lp (fun _ : G => ℂ) 2}
-    (hT : T ∈ twistedGroupAlgebra ω) :
-    wotTranspose T ∈ twistedGroupAlgebra (inverseCocycle ω) := by
-  have hgen : Set.range (regularGenerator ω) ⊆
-      transposePreimage (twistedGroupAlgebra (inverseCocycle ω)) := by
-    rintro _ ⟨g,rfl⟩
-    change wotTranspose (regularGenerator ω g) ∈ twistedGroupAlgebra (inverseCocycle ω)
-    rw [wotTranspose_regularGenerator]
-    exact star_mem (regularGenerator_mem _ g)
-  have hclosed : IsClosed (transposePreimage (twistedGroupAlgebra (inverseCocycle ω)) :
-      Set (lp (fun _ : G => ℂ) 2 →WOT[ℂ] lp (fun _ : G => ℂ) 2)) := by
-    change IsClosed (wotTranspose ⁻¹' (twistedGroupAlgebra (inverseCocycle ω)).carrier)
-    exact (StarSubalgebra.isClosed_topologicalClosure _).preimage wotTranspose_continuous
-  exact StarSubalgebra.topologicalClosure_minimal (StarAlgebra.adjoin_le hgen) hclosed hT
-
-def opTransposeOnTwisted (ω : ScalarCocycle G) :
-    (twistedGroupAlgebra ω)ᵐᵒᵖ →⋆ₐ[ℂ] twistedGroupAlgebra (inverseCocycle ω) where
-  toFun T := ⟨wotTranspose T.unop.val, wotTranspose_mem_twistedGroupAlgebra ω T.unop.property⟩
-  map_one' := Subtype.ext wotTranspose_one
-  map_mul' T S := Subtype.ext (wotTranspose_mul S.unop.val T.unop.val)
-  map_add' T S := Subtype.ext (wotTranspose_add T.unop.val S.unop.val)
-  map_zero' := Subtype.ext wotTranspose_zero
-  commutes' z := by
-    apply Subtype.ext
-    change wotTranspose (algebraMap ℂ _ z) = algebraMap ℂ _ z
-    rw [Algebra.algebraMap_eq_smul_one, wotTranspose_smul, wotTranspose_one]
-  map_star' T := Subtype.ext (wotTranspose_star T.unop.val)
-
-lemma opTransposeOnTwisted_continuous (ω : ScalarCocycle G) :
-    Continuous (opTransposeOnTwisted ω) :=
-  (wotTranspose_continuous.comp (continuous_subtype_val.comp
-    MulOpposite.continuous_unop)).subtype_mk _
-
-end
-
-section
-open ArithmeticGerms GermBundles
-section RightAction
-variable {G H V J : Type*} [Group G] [Group H] [Fintype J]
-variable [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
-variable (C : VirtualGraph G H)
-local instance rightNormalActionDomainFintype : Fintype (G ⧸ C.domain) :=
-  Subgroup.fintypeQuotientOfFiniteIndex
-local instance rightNormalActionCodomainFintype : Fintype (H ⧸ C.codomain) :=
-  Subgroup.fintypeQuotientOfFiniteIndex
-variable (μ : ScalarCocycle G) (ω : ScalarCocycle H)
-variable (B : ProjectiveBundle (prodRatio μ ω) (fun _ : (G × H) ⧸ C.graph => V))
-variable (b : OrthonormalBasis J ℂ V)
-
-def rightNormalAction : (twistedGroupAlgebra ω)ᵐᵒᵖ →⋆ₐ[ℂ]
-    (lp (fun _ : (G × H) ⧸ C.graph => V) 2 →WOT[ℂ]
-      lp (fun _ : (G × H) ⧸ C.graph => V) 2) :=
-  (rightAdjointNormalAction C μ ω B b).comp (opTransposeOnTwisted ω)
-
-theorem rightNormalAction_continuous : Continuous (rightNormalAction C μ ω B b) :=
-  (rightAdjointNormalAction_continuous C μ ω B b).comp
-    (opTransposeOnTwisted_continuous ω)
-
-theorem rightNormalAction_adjoint_generator (h : H) :
-    rightNormalAction C μ ω B b
-      (MulOpposite.op (star (⟨regularGenerator ω h, regularGenerator_mem ω h⟩ :
-        twistedGroupAlgebra ω))) = coordinateOperator C μ ω B (1,h) := by
-  have ht : opTransposeOnTwisted ω
-      (MulOpposite.op (star (⟨regularGenerator ω h, regularGenerator_mem ω h⟩ :
-        twistedGroupAlgebra ω))) =
-      ⟨regularGenerator (inverseCocycle ω) h, regularGenerator_mem (inverseCocycle ω) h⟩ :=
-    Subtype.ext (wotTranspose_star_regularGenerator ω h)
-  exact (congrArg (rightAdjointNormalAction C μ ω B b) ht).trans
-    (rightAdjointNormalAction_generator C μ ω B b h)
-
-theorem left_rightNormalActions_commute (T : twistedGroupAlgebra μ)
-    (S : (twistedGroupAlgebra ω)ᵐᵒᵖ) :
-    leftNormalAction C μ ω B b T * rightNormalAction C μ ω B b S =
-      rightNormalAction C μ ω B b S * leftNormalAction C μ ω B b T :=
-  normalActions_commute C μ ω B b T (opTransposeOnTwisted ω S)
-
-end RightAction
-
-variable {G H V : Type*} [Group G] [Group H]
-variable [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-variable (C : VirtualGraph G H)
-local instance inducedGraphDomainFintype : Fintype (G ⧸ C.domain) :=
-  Subgroup.fintypeQuotientOfFiniteIndex
-local instance inducedGraphCodomainFintype : Fintype (H ⧸ C.codomain) :=
-  Subgroup.fintypeQuotientOfFiniteIndex
-variable (μ : ScalarCocycle G) (ω : ScalarCocycle H)
-variable (r : CosetSection C.graph)
-variable (γ : ProjectiveRepresentation ((prodRatio μ ω).restrict C.graph) V)
-
-theorem induced_graph_normal_actions :
-    let B := r.inducedBundle (prodRatio μ ω) γ
-    ∃ (L : twistedGroupAlgebra μ →⋆ₐ[ℂ]
-          (lp (fun _ : (G × H) ⧸ C.graph => V) 2 →WOT[ℂ]
-            lp (fun _ : (G × H) ⧸ C.graph => V) 2))
-      (R : (twistedGroupAlgebra ω)ᵐᵒᵖ →⋆ₐ[ℂ]
-          (lp (fun _ : (G × H) ⧸ C.graph => V) 2 →WOT[ℂ]
-            lp (fun _ : (G × H) ⧸ C.graph => V) 2))
-      (UL : PiLp 2 (fun _ : (H ⧸ C.codomain) × Fin (Module.finrank ℂ V) =>
-          lp (fun _ : G => ℂ) 2) ≃ₗᵢ[ℂ] lp (fun _ : (G × H) ⧸ C.graph => V) 2)
-      (UR : PiLp 2 (fun _ : (G ⧸ C.domain) × Fin (Module.finrank ℂ V) =>
-          lp (fun _ : H => ℂ) 2) ≃ₗᵢ[ℂ] lp (fun _ : (G × H) ⧸ C.graph => V) 2),
-      Continuous L ∧ Continuous R ∧
-      (∀ T S, L T * R S = R S * L T) ∧
-      (∀ g, L ⟨regularGenerator μ g, regularGenerator_mem μ g⟩ =
-        coordinateOperator C μ ω B (g,1)) ∧
-      (∀ h, R (MulOpposite.op (star (⟨regularGenerator ω h, regularGenerator_mem ω h⟩ :
-        twistedGroupAlgebra ω))) = coordinateOperator C μ ω B (1,h)) ∧
-      (∀ g v, B.action (g,1) (UL v) = UL
-        (amplify (scalarRegular μ g).toContinuousLinearEquiv.toContinuousLinearMap v)) ∧
-      (∀ h v, B.action (1,h) (UR v) = UR
-        (amplify (scalarRegular (inverseCocycle ω) h).toContinuousLinearEquiv.toContinuousLinearMap v)) := by
-  let B := r.inducedBundle (prodRatio μ ω) γ
-  let b := stdOrthonormalBasis ℂ V
-  refine ⟨leftNormalAction C μ ω B b, rightNormalAction C μ ω B b,
-    firstFiniteRegularUnitary C μ ω B b, secondFiniteRegularUnitary C μ ω B b,
-    leftNormalAction_continuous C μ ω B b, rightNormalAction_continuous C μ ω B b, ?_⟩
-  exact ⟨left_rightNormalActions_commute C μ ω B b,
-    leftNormalAction_generator C μ ω B b, rightNormalAction_adjoint_generator C μ ω B b,
-    firstFiniteRegularUnitary_intertwines C μ ω B b,
-    secondFiniteRegularUnitary_intertwines C μ ω B b⟩
-
-end
-
-section
-open GermBundles
-variable {G : Type*} [Group G]
-
-def standardRightRegular (ω : ScalarCocycle G) (h : G) :
-    lp (fun _ : G => ℂ) 2 ≃ₗᵢ[ℂ] lp (fun _ : G => ℂ) 2 :=
-  transportLp (Equiv.mulRight h) (fun g => circleUnitary (ω.val g h))
-
-lemma standardRightRegular_apply (ω : ScalarCocycle G) (h : G)
-    (x : lp (fun _ : G => ℂ) 2) (g : G) :
-    standardRightRegular ω h x (g*h) = (ω.val g h : ℂ) * x g :=
-  transportLp_apply (Equiv.mulRight h) (fun g => circleUnitary (ω.val g h)) x g
-
-def oppositeToStandard (ω : ScalarCocycle G) :
-    lp (fun _ : G => ℂ) 2 ≃ₗᵢ[ℂ] lp (fun _ : G => ℂ) 2 :=
-  transportLp (Equiv.inv G) (fun g => circleUnitary ((ω.val g⁻¹ g)⁻¹))
-
-lemma oppositeToStandard_apply (ω : ScalarCocycle G)
-    (x : lp (fun _ : G => ℂ) 2) (g : G) :
-    oppositeToStandard ω x g⁻¹ = ((ω.val g⁻¹ g)⁻¹ : Circle) * x g := by
-  exact transportLp_apply (Equiv.inv G) (fun g => circleUnitary ((ω.val g⁻¹ g)⁻¹)) x g
-
-lemma rightStandard_phase (ω : ScalarCocycle G) (h g : G) :
-    ω.val (h*g)⁻¹ h * (ω.val h g)⁻¹ * (ω.val (h*g)⁻¹ (h*g))⁻¹ =
-      (ω.val g⁻¹ g)⁻¹ := by
-  have hh := ω.mul_assoc (h*g)⁻¹ h g
-  have hg : (h*g)⁻¹ * h = g⁻¹ := by group
-  rw [hg] at hh
-  have he : ω.val (h*g)⁻¹ h =
-      (ω.val h g * ω.val (h*g)⁻¹ (h*g)) * (ω.val g⁻¹ g)⁻¹ :=
-    (eq_mul_inv_iff_mul_eq).mpr hh
-  rw [he]
-  simp [mul_comm, mul_left_comm, mul_assoc]
-
-lemma oppositeToStandard_intertwines (ω : ScalarCocycle G) (h : G)
-    (x : lp (fun _ : G => ℂ) 2) :
-    oppositeToStandard ω (scalarRegular (inverseCocycle ω) h x) =
-      (standardRightRegular ω h).symm (oppositeToStandard ω x) := by
-  apply (standardRightRegular ω h).injective
-  rw [LinearIsometryEquiv.apply_symm_apply]
-  apply lp.ext
-  funext k
-  obtain ⟨g,rfl⟩ := (Equiv.inv G).surjective k
-  change standardRightRegular ω h (oppositeToStandard ω (scalarRegular (inverseCocycle ω) h x)) g⁻¹ =
-    oppositeToStandard ω x g⁻¹
-  have hi : (h*g)⁻¹ * h = g⁻¹ := by group
-  rw [← hi, standardRightRegular_apply, oppositeToStandard_apply, scalarRegular_apply,
-    hi, oppositeToStandard_apply]
-  change (ω.val (h*g)⁻¹ h : ℂ) *
-    (((ω.val (h*g)⁻¹ (h*g))⁻¹ : Circle) * (((ω.val h g)⁻¹ : Circle) * x g)) =
-      ((ω.val g⁻¹ g)⁻¹ : Circle) * x g
-  have hp := congrArg (fun z : Circle => (z : ℂ)) (rightStandard_phase ω h g)
-  simp only [Circle.coe_mul] at hp
-  calc
-    _ = ((ω.val (h*g)⁻¹ h : ℂ) * ((ω.val h g)⁻¹ : Circle) *
-        ((ω.val (h*g)⁻¹ (h*g))⁻¹ : Circle)) * x g := by ring
-    _ = _ := congrArg (fun z : ℂ => z * x g) hp
-
-end
-
-open ArithmeticGerms GermBundles
-section EntireRegularActions
-variable {G H V J : Type*} [Group G] [Group H] [Fintype J]
-variable [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
-variable (C : VirtualGraph G H)
-local instance regularActionDomainFintype : Fintype (G ⧸ C.domain) :=
-  Subgroup.fintypeQuotientOfFiniteIndex
-local instance regularActionCodomainFintype : Fintype (H ⧸ C.codomain) :=
-  Subgroup.fintypeQuotientOfFiniteIndex
-variable (μ : ScalarCocycle G) (ω : ScalarCocycle H)
-variable (B : ProjectiveBundle (prodRatio μ ω) (fun _ : (G × H) ⧸ C.graph => V))
-variable (b : OrthonormalBasis J ℂ V)
-
-lemma leftNormalAction_finite_regular (T : twistedGroupAlgebra μ)
-    (v : PiLp 2 (fun _ : (H ⧸ C.codomain) × J => lp (fun _ : G => ℂ) 2)) :
-    leftNormalAction C μ ω B b T (firstFiniteRegularUnitary C μ ω B b v) =
-      firstFiniteRegularUnitary C μ ω B b (amplify T.val.toCLM v) := by
-  change firstFiniteRegularUnitary C μ ω B b
-    (amplify T.val.toCLM ((firstFiniteRegularUnitary C μ ω B b).symm
-      (firstFiniteRegularUnitary C μ ω B b v))) = _
-  rw [LinearIsometryEquiv.symm_apply_apply]
-
-lemma rightNormalAction_finite_regular (S : (twistedGroupAlgebra ω)ᵐᵒᵖ)
-    (v : PiLp 2 (fun _ : (G ⧸ C.domain) × J => lp (fun _ : H => ℂ) 2)) :
-    rightNormalAction C μ ω B b S (secondFiniteRegularUnitary C μ ω B b v) =
-      secondFiniteRegularUnitary C μ ω B b (amplify
-        (wotTranspose S.unop.val).toCLM v) := by
-  change secondFiniteRegularUnitary C μ ω B b
-    (amplify (wotTranspose S.unop.val).toCLM ((secondFiniteRegularUnitary C μ ω B b).symm
-      (secondFiniteRegularUnitary C μ ω B b v))) = _
-  rw [LinearIsometryEquiv.symm_apply_apply]
-
-end EntireRegularActions
-
-variable {G H V : Type*} [Group G] [Group H]
-variable [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-variable (C : VirtualGraph G H)
-local instance multiplicityDomainFintype : Fintype (G ⧸ C.domain) :=
-  Subgroup.fintypeQuotientOfFiniteIndex
-local instance multiplicityCodomainFintype : Fintype (H ⧸ C.codomain) :=
-  Subgroup.fintypeQuotientOfFiniteIndex
-
-omit [FiniteDimensional ℂ V] in
-lemma graph_regular_multiplicities :
-    Fintype.card ((H ⧸ C.codomain) × Fin (Module.finrank ℂ V)) =
-      Module.finrank ℂ V * C.codomain.index ∧
-    Fintype.card ((G ⧸ C.domain) × Fin (Module.finrank ℂ V)) =
-      Module.finrank ℂ V * C.domain.index := by
-  simp [Fintype.card_prod, Subgroup.index_eq_card, Nat.card_eq_fintype_card, mul_comm]
-
-end GraphNormality
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.GraphModels.Regular
+
+namespace OAI
+
+noncomputable section
+
+/-! Commuting normal actions and regular multiplicities. -/
+
+namespace GraphNormality
+section
+open ArithmeticGerms GermBundles
+open scoped InnerProductSpace
+section WeakOperatorAlgebras
+variable {L W : Type*}
+variable [NormedAddCommGroup L] [InnerProductSpace ℂ L] [CompleteSpace L]
+variable [NormedAddCommGroup W] [InnerProductSpace ℂ W] [CompleteSpace W]
+
+def wotLift (F : (L →L[ℂ] L) →⋆ₐ[ℂ] (W →L[ℂ] W)) :
+    (L →WOT[ℂ] L) →⋆ₐ[ℂ] (W →WOT[ℂ] W) where
+  toFun T := ContinuousLinearMapWOT.ofCLM (F T.toCLM)
+  map_one' := congrArg ContinuousLinearMapWOT.ofCLM (map_one F)
+  map_mul' T S := congrArg ContinuousLinearMapWOT.ofCLM (map_mul F T.toCLM S.toCLM)
+  map_add' T S := congrArg ContinuousLinearMapWOT.ofCLM (map_add F T.toCLM S.toCLM)
+  map_zero' := congrArg ContinuousLinearMapWOT.ofCLM (map_zero F)
+  commutes' z := congrArg ContinuousLinearMapWOT.ofCLM (F.commutes z)
+  map_star' T := congrArg ContinuousLinearMapWOT.ofCLM (map_star F T.toCLM)
+
+theorem closed_generated_commute (s t : Set (W →WOT[ℂ] W))
+    (hst : s ⊆ StarSubalgebra.centralizer ℂ t)
+    {x y : W →WOT[ℂ] W}
+    (hx : x ∈ (StarAlgebra.adjoin ℂ s).topologicalClosure)
+    (hy : y ∈ (StarAlgebra.adjoin ℂ t).topologicalClosure) : x * y = y * x := by
+  have hs : (StarAlgebra.adjoin ℂ s).topologicalClosure ≤
+      StarSubalgebra.centralizer ℂ t :=
+    StarSubalgebra.topologicalClosure_minimal (StarAlgebra.adjoin_le hst)
+      (Set.isClosed_centralizer _)
+  have ht := StarSubalgebra.topologicalClosure_adjoin_le_centralizer_centralizer ℂ t hy
+  exact ((StarSubalgebra.mem_centralizer_iff ℂ).mp ht x (hs hx)).1
+
+theorem map_mem_closed_generated (F : (L →WOT[ℂ] L) →⋆ₐ[ℂ] (W →WOT[ℂ] W))
+    (hF : Continuous F) (s : Set (L →WOT[ℂ] L)) {x : L →WOT[ℂ] L}
+    (hx : x ∈ (StarAlgebra.adjoin ℂ s).topologicalClosure) :
+    F x ∈ (StarAlgebra.adjoin ℂ (F '' s)).topologicalClosure := by
+  have h := StarSubalgebra.map_topologicalClosure_le (StarAlgebra.adjoin ℂ s) F hF
+  rw [StarAlgHom.map_adjoin] at h
+  exact h ⟨x, hx, rfl⟩
+
+theorem isometries_commute_star_right (U V : W ≃ₗᵢ[ℂ] W)
+    (h : ∀ x, U (V x) = V (U x)) :
+    U.toContinuousLinearEquiv.toContinuousLinearMap *
+        star V.toContinuousLinearEquiv.toContinuousLinearMap =
+      star V.toContinuousLinearEquiv.toContinuousLinearMap *
+        U.toContinuousLinearEquiv.toContinuousLinearMap := by
+  rw [LinearIsometryEquiv.star_eq_symm]
+  apply ContinuousLinearMap.ext
+  intro x
+  change U (V.symm x) = V.symm (U x)
+  apply V.injective
+  rw [← h, V.apply_symm_apply, V.apply_symm_apply]
+
+end WeakOperatorAlgebras
+
+variable {G : Type*} [Group G]
+
+def regularGenerator (μ : ScalarCocycle G) (g : G) :
+    lp (fun _ : G => ℂ) 2 →WOT[ℂ] lp (fun _ : G => ℂ) 2 :=
+  ContinuousLinearMapWOT.ofCLM (scalarRegular μ g).toContinuousLinearEquiv.toContinuousLinearMap
+
+def twistedGroupAlgebra (μ : ScalarCocycle G) :
+    StarSubalgebra ℂ (lp (fun _ : G => ℂ) 2 →WOT[ℂ] lp (fun _ : G => ℂ) 2) :=
+  (StarAlgebra.adjoin ℂ (Set.range (regularGenerator μ))).topologicalClosure
+
+theorem regularGenerator_mem (μ : ScalarCocycle G) (g : G) :
+    regularGenerator μ g ∈ twistedGroupAlgebra μ :=
+  StarSubalgebra.le_topologicalClosure _ (StarAlgebra.subset_adjoin ℂ _ (Set.mem_range_self g))
+
+end
+
+section
+open ArithmeticGerms GermBundles
+variable {G H V J : Type*} [Group G] [Group H] [Fintype J]
+variable [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
+variable (C : VirtualGraph G H)
+local instance normalActionDomainFintype : Fintype (G ⧸ C.domain) :=
+  Subgroup.fintypeQuotientOfFiniteIndex
+local instance normalActionCodomainFintype : Fintype (H ⧸ C.codomain) :=
+  Subgroup.fintypeQuotientOfFiniteIndex
+variable (μ : ScalarCocycle G) (ω : ScalarCocycle H)
+variable (B : ProjectiveBundle (prodRatio μ ω) (fun _ : (G × H) ⧸ C.graph => V))
+variable (b : OrthonormalBasis J ℂ V)
+
+def coordinateOperator (s : G × H) : lp (fun _ : (G × H) ⧸ C.graph => V) 2 →WOT[ℂ]
+    lp (fun _ : (G × H) ⧸ C.graph => V) 2 :=
+  ContinuousLinearMapWOT.ofCLM (B.action s).toContinuousLinearEquiv.toContinuousLinearMap
+
+theorem coordinateOperator_centralizer :
+    Set.range (fun g : G => coordinateOperator C μ ω B (g,1)) ⊆
+      StarSubalgebra.centralizer ℂ
+        (Set.range (fun h : H => coordinateOperator C μ ω B (1,h))) := by
+  rintro _ ⟨g,rfl⟩
+  apply (StarSubalgebra.mem_centralizer_iff ℂ).mpr
+  rintro _ ⟨h,rfl⟩
+  constructor
+  · apply ContinuousLinearMapWOT.ext
+    intro v
+    exact (coordinate_actions_commute C μ ω B g h v).symm
+  · exact congrArg ContinuousLinearMapWOT.ofCLM
+      (isometries_commute_star_right (B.action (g,1)) (B.action (1,h))
+        (coordinate_actions_commute C μ ω B g h)).symm
+
+theorem firstExtension_generating_image :
+    (wotLift (firstExtension C μ ω B b)) '' Set.range (regularGenerator μ) =
+      Set.range (fun g : G => coordinateOperator C μ ω B (g,1)) := by
+  rw [← Set.range_comp']
+  congr 1
+  funext g
+  exact congrArg ContinuousLinearMapWOT.ofCLM (firstExtension_generator C μ ω B b g)
+
+theorem secondExtension_generating_image :
+    (wotLift (secondExtension C μ ω B b)) '' Set.range (regularGenerator (inverseCocycle ω)) =
+      Set.range (fun h : H => coordinateOperator C μ ω B (1,h)) := by
+  rw [← Set.range_comp']
+  congr 1
+  funext h
+  exact congrArg ContinuousLinearMapWOT.ofCLM (secondExtension_generator C μ ω B b h)
+
+def leftNormalAction : twistedGroupAlgebra μ →⋆ₐ[ℂ]
+    (lp (fun _ : (G × H) ⧸ C.graph => V) 2 →WOT[ℂ]
+      lp (fun _ : (G × H) ⧸ C.graph => V) 2) :=
+  (wotLift (firstExtension C μ ω B b)).comp (twistedGroupAlgebra μ).subtype
+
+def rightAdjointNormalAction : twistedGroupAlgebra (inverseCocycle ω) →⋆ₐ[ℂ]
+    (lp (fun _ : (G × H) ⧸ C.graph => V) 2 →WOT[ℂ]
+      lp (fun _ : (G × H) ⧸ C.graph => V) 2) :=
+  (wotLift (secondExtension C μ ω B b)).comp (twistedGroupAlgebra (inverseCocycle ω)).subtype
+
+theorem leftNormalAction_continuous : Continuous (leftNormalAction C μ ω B b) :=
+  (firstExtension_wot_continuous C μ ω B b).comp continuous_subtype_val
+
+theorem rightAdjointNormalAction_continuous : Continuous (rightAdjointNormalAction C μ ω B b) :=
+  (secondExtension_wot_continuous C μ ω B b).comp continuous_subtype_val
+
+theorem leftNormalAction_generator (g : G) :
+    leftNormalAction C μ ω B b ⟨regularGenerator μ g, regularGenerator_mem μ g⟩ =
+      coordinateOperator C μ ω B (g,1) :=
+  congrArg ContinuousLinearMapWOT.ofCLM (firstExtension_generator C μ ω B b g)
+
+theorem rightAdjointNormalAction_generator (h : H) :
+    rightAdjointNormalAction C μ ω B b
+      ⟨regularGenerator (inverseCocycle ω) h, regularGenerator_mem (inverseCocycle ω) h⟩ =
+      coordinateOperator C μ ω B (1,h) :=
+  congrArg ContinuousLinearMapWOT.ofCLM (secondExtension_generator C μ ω B b h)
+
+theorem normalActions_commute (T : twistedGroupAlgebra μ)
+    (S : twistedGroupAlgebra (inverseCocycle ω)) :
+    leftNormalAction C μ ω B b T * rightAdjointNormalAction C μ ω B b S =
+      rightAdjointNormalAction C μ ω B b S * leftNormalAction C μ ω B b T := by
+  have hT := map_mem_closed_generated (wotLift (firstExtension C μ ω B b))
+    (firstExtension_wot_continuous C μ ω B b) (Set.range (regularGenerator μ)) T.property
+  have hS := map_mem_closed_generated (wotLift (secondExtension C μ ω B b))
+    (secondExtension_wot_continuous C μ ω B b)
+    (Set.range (regularGenerator (inverseCocycle ω))) S.property
+  rw [firstExtension_generating_image] at hT
+  rw [secondExtension_generating_image] at hS
+  exact closed_generated_commute _ _ (coordinateOperator_centralizer C μ ω B) hT hS
+
+end
+
+section
+open scoped InnerProductSpace
+variable {I : Type*}
+
+lemma scalarLp_inner_star_star (x y : lp (fun _ : I => ℂ) 2) :
+    ⟪star x, star y⟫_ℂ = ⟪y, x⟫_ℂ := by
+  rw [lp.inner_eq_tsum, lp.inner_eq_tsum]
+  apply tsum_congr
+  intro i
+  simp [RCLike.inner_apply, mul_comm]
+
+lemma scalarLp_inner_star_comm (x y : lp (fun _ : I => ℂ) 2) :
+    ⟪x, star y⟫_ℂ = ⟪y, star x⟫_ℂ := by
+  simpa only [star_star] using scalarLp_inner_star_star (star x) y
+
+def operatorTranspose (T : lp (fun _ : I => ℂ) 2 →L[ℂ] lp (fun _ : I => ℂ) 2) :
+    lp (fun _ : I => ℂ) 2 →L[ℂ] lp (fun _ : I => ℂ) 2 where
+  toFun x := star ((star T) (star x))
+  map_add' x y := by simp
+  map_smul' z x := by simp
+  cont := continuous_star.comp ((star T).continuous.comp continuous_star)
+
+lemma operatorTranspose_inner (T : lp (fun _ : I => ℂ) 2 →L[ℂ] lp (fun _ : I => ℂ) 2)
+    (x y : lp (fun _ : I => ℂ) 2) :
+    ⟪y, operatorTranspose T x⟫_ℂ = ⟪star x, T (star y)⟫_ℂ := by
+  change ⟪y, star ((star T) (star x))⟫_ℂ = _
+  rw [scalarLp_inner_star_comm, ContinuousLinearMap.star_eq_adjoint,
+    ContinuousLinearMap.adjoint_inner_left]
+
+lemma operatorTranspose_add (T S : lp (fun _ : I => ℂ) 2 →L[ℂ] lp (fun _ : I => ℂ) 2) :
+    operatorTranspose (T + S) = operatorTranspose T + operatorTranspose S := by
+  apply ContinuousLinearMap.ext
+  intro x
+  simp [operatorTranspose, ContinuousLinearMap.star_eq_adjoint]
+
+lemma operatorTranspose_smul (z : ℂ)
+    (T : lp (fun _ : I => ℂ) 2 →L[ℂ] lp (fun _ : I => ℂ) 2) :
+    operatorTranspose (z • T) = z • operatorTranspose T := by
+  apply ContinuousLinearMap.ext
+  intro x
+  simp [operatorTranspose]
+
+lemma operatorTranspose_mul (T S : lp (fun _ : I => ℂ) 2 →L[ℂ] lp (fun _ : I => ℂ) 2) :
+    operatorTranspose (T * S) = operatorTranspose S * operatorTranspose T := by
+  apply ContinuousLinearMap.ext
+  intro x
+  simp [operatorTranspose]
+
+lemma operatorTranspose_star (T : lp (fun _ : I => ℂ) 2 →L[ℂ] lp (fun _ : I => ℂ) 2) :
+    operatorTranspose (star T) = star (operatorTranspose T) := by
+  rw [ContinuousLinearMap.star_eq_adjoint (operatorTranspose T)]
+  apply (ContinuousLinearMap.eq_adjoint_iff _ _).mpr
+  intro x y
+  rw [← inner_conj_symm, operatorTranspose_inner, ContinuousLinearMap.star_eq_adjoint,
+    ContinuousLinearMap.adjoint_inner_right, inner_conj_symm, operatorTranspose_inner]
+
+lemma operatorTranspose_involutive
+    (T : lp (fun _ : I => ℂ) 2 →L[ℂ] lp (fun _ : I => ℂ) 2) :
+    operatorTranspose (operatorTranspose T) = T := by
+  apply ContinuousLinearMap.ext
+  intro x
+  apply ext_inner_left ℂ
+  intro y
+  rw [operatorTranspose_inner, operatorTranspose_inner, star_star, star_star]
+
+lemma operatorTranspose_one :
+    operatorTranspose (1 : lp (fun _ : I => ℂ) 2 →L[ℂ] lp (fun _ : I => ℂ) 2) = 1 := by
+  apply ContinuousLinearMap.ext
+  intro x
+  simp [operatorTranspose]
+
+def operatorOppositeEquiv :
+    (lp (fun _ : I => ℂ) 2 →L[ℂ] lp (fun _ : I => ℂ) 2)ᵐᵒᵖ ≃⋆ₐ[ℂ]
+      (lp (fun _ : I => ℂ) 2 →L[ℂ] lp (fun _ : I => ℂ) 2) where
+  toFun T := operatorTranspose T.unop
+  invFun T := MulOpposite.op (operatorTranspose T)
+  left_inv T := by simp [operatorTranspose_involutive]
+  right_inv T := operatorTranspose_involutive T
+  map_mul' T S := operatorTranspose_mul S.unop T.unop
+  map_add' T S := operatorTranspose_add T.unop S.unop
+  map_smul' z T := operatorTranspose_smul z T.unop
+  map_star' T := operatorTranspose_star T.unop
+
+lemma operatorTranspose_wot_continuous :
+    Continuous (fun T : lp (fun _ : I => ℂ) 2 →WOT[ℂ] lp (fun _ : I => ℂ) 2 =>
+      ContinuousLinearMapWOT.ofCLM (operatorTranspose T.toCLM)) := by
+  apply ContinuousLinearMapWOT.continuous_iff.mpr
+  intro x y
+  change Continuous (fun T : lp (fun _ : I => ℂ) 2 →WOT[ℂ] lp (fun _ : I => ℂ) 2 =>
+    ⟪y, operatorTranspose T.toCLM x⟫_ℂ)
+  simp_rw [operatorTranspose_inner]
+  exact ContinuousLinearMapWOT.continuous_iff.mp continuous_id (star y) (star x)
+
+end
+
+section
+open GermBundles
+variable {G : Type*}
+
+def wotTranspose (T : lp (fun _ : G => ℂ) 2 →WOT[ℂ] lp (fun _ : G => ℂ) 2) :
+    lp (fun _ : G => ℂ) 2 →WOT[ℂ] lp (fun _ : G => ℂ) 2 :=
+  ContinuousLinearMapWOT.ofCLM (operatorTranspose T.toCLM)
+
+lemma wotTranspose_continuous : Continuous (wotTranspose (G := G)) :=
+  operatorTranspose_wot_continuous
+
+lemma wotTranspose_add (T S : lp (fun _ : G => ℂ) 2 →WOT[ℂ] lp (fun _ : G => ℂ) 2) :
+    wotTranspose (T + S) = wotTranspose T + wotTranspose S :=
+  congrArg ContinuousLinearMapWOT.ofCLM (operatorTranspose_add T.toCLM S.toCLM)
+
+lemma wotTranspose_mul (T S : lp (fun _ : G => ℂ) 2 →WOT[ℂ] lp (fun _ : G => ℂ) 2) :
+    wotTranspose (T * S) = wotTranspose S * wotTranspose T :=
+  congrArg ContinuousLinearMapWOT.ofCLM (operatorTranspose_mul T.toCLM S.toCLM)
+
+lemma wotTranspose_smul (z : ℂ)
+    (T : lp (fun _ : G => ℂ) 2 →WOT[ℂ] lp (fun _ : G => ℂ) 2) :
+    wotTranspose (z • T) = z • wotTranspose T :=
+  congrArg ContinuousLinearMapWOT.ofCLM (operatorTranspose_smul z T.toCLM)
+
+lemma wotTranspose_star (T : lp (fun _ : G => ℂ) 2 →WOT[ℂ] lp (fun _ : G => ℂ) 2) :
+    wotTranspose (star T) = star (wotTranspose T) :=
+  congrArg ContinuousLinearMapWOT.ofCLM (operatorTranspose_star T.toCLM)
+
+lemma wotTranspose_one :
+    wotTranspose (1 : lp (fun _ : G => ℂ) 2 →WOT[ℂ] lp (fun _ : G => ℂ) 2) = 1 :=
+  congrArg ContinuousLinearMapWOT.ofCLM operatorTranspose_one
+
+lemma wotTranspose_involutive
+    (T : lp (fun _ : G => ℂ) 2 →WOT[ℂ] lp (fun _ : G => ℂ) 2) :
+    wotTranspose (wotTranspose T) = T :=
+  congrArg ContinuousLinearMapWOT.ofCLM (operatorTranspose_involutive T.toCLM)
+
+lemma wotTranspose_zero :
+    wotTranspose (0 : lp (fun _ : G => ℂ) 2 →WOT[ℂ] lp (fun _ : G => ℂ) 2) = 0 := by
+  apply ContinuousLinearMapWOT.ext
+  intro x
+  change star ((star (0 : lp (fun _ : G => ℂ) 2 →L[ℂ] lp (fun _ : G => ℂ) 2)) (star x)) = 0
+  simp [ContinuousLinearMap.star_eq_adjoint]
+
+variable [Group G]
+
+lemma wotTranspose_star_regularGenerator (ω : ScalarCocycle G) (g : G) :
+    wotTranspose (star (regularGenerator ω g)) = regularGenerator (inverseCocycle ω) g := by
+  apply ContinuousLinearMapWOT.ext
+  intro x
+  apply lp.ext
+  funext k
+  obtain ⟨h,rfl⟩ := (Equiv.mulLeft g).surjective k
+  change (star ((star (star (scalarRegular ω g).toContinuousLinearEquiv.toContinuousLinearMap))
+    (star x))) (g*h) = scalarRegular (inverseCocycle ω) g x (g*h)
+  rw [star_star]
+  change star (scalarRegular ω g (star x) (g*h)) = _
+  rw [scalarRegular_apply, scalarRegular_apply]
+  simp only [inverseCocycle, star_mul, lp.star_apply, star_star]
+  rw [Circle.coe_inv_eq_conj]
+  exact mul_comm _ _
+
+lemma wotTranspose_regularGenerator (ω : ScalarCocycle G) (g : G) :
+    wotTranspose (regularGenerator ω g) = star (regularGenerator (inverseCocycle ω) g) := by
+  rw [← wotTranspose_star_regularGenerator, wotTranspose_star, star_star]
+
+def transposePreimage
+    (S : StarSubalgebra ℂ (lp (fun _ : G => ℂ) 2 →WOT[ℂ] lp (fun _ : G => ℂ) 2)) :
+    StarSubalgebra ℂ (lp (fun _ : G => ℂ) 2 →WOT[ℂ] lp (fun _ : G => ℂ) 2) where
+  carrier := {T | wotTranspose T ∈ S}
+  zero_mem' := by
+    change wotTranspose 0 ∈ S
+    rw [wotTranspose_zero]
+    exact zero_mem S
+  one_mem' := by
+    change wotTranspose 1 ∈ S
+    rw [wotTranspose_one]
+    exact one_mem S
+  add_mem' hT hU := by
+    change wotTranspose _ ∈ S
+    rw [wotTranspose_add]
+    exact add_mem hT hU
+  mul_mem' hT hU := by
+    change wotTranspose _ ∈ S
+    rw [wotTranspose_mul]
+    exact mul_mem hU hT
+  algebraMap_mem' z := by
+    change wotTranspose (algebraMap ℂ _ z) ∈ S
+    rw [Algebra.algebraMap_eq_smul_one, wotTranspose_smul, wotTranspose_one]
+    exact S.smul_mem S.one_mem z
+  star_mem' hT := by
+    change wotTranspose _ ∈ S
+    rw [wotTranspose_star]
+    exact star_mem hT
+
+lemma wotTranspose_mem_twistedGroupAlgebra (ω : ScalarCocycle G)
+    {T : lp (fun _ : G => ℂ) 2 →WOT[ℂ] lp (fun _ : G => ℂ) 2}
+    (hT : T ∈ twistedGroupAlgebra ω) :
+    wotTranspose T ∈ twistedGroupAlgebra (inverseCocycle ω) := by
+  have hgen : Set.range (regularGenerator ω) ⊆
+      transposePreimage (twistedGroupAlgebra (inverseCocycle ω)) := by
+    rintro _ ⟨g,rfl⟩
+    change wotTranspose (regularGenerator ω g) ∈ twistedGroupAlgebra (inverseCocycle ω)
+    rw [wotTranspose_regularGenerator]
+    exact star_mem (regularGenerator_mem _ g)
+  have hclosed : IsClosed (transposePreimage (twistedGroupAlgebra (inverseCocycle ω)) :
+      Set (lp (fun _ : G => ℂ) 2 →WOT[ℂ] lp (fun _ : G => ℂ) 2)) := by
+    change IsClosed (wotTranspose ⁻¹' (twistedGroupAlgebra (inverseCocycle ω)).carrier)
+    exact (StarSubalgebra.isClosed_topologicalClosure _).preimage wotTranspose_continuous
+  exact StarSubalgebra.topologicalClosure_minimal (StarAlgebra.adjoin_le hgen) hclosed hT
+
+def opTransposeOnTwisted (ω : ScalarCocycle G) :
+    (twistedGroupAlgebra ω)ᵐᵒᵖ →⋆ₐ[ℂ] twistedGroupAlgebra (inverseCocycle ω) where
+  toFun T := ⟨wotTranspose T.unop.val, wotTranspose_mem_twistedGroupAlgebra ω T.unop.property⟩
+  map_one' := Subtype.ext wotTranspose_one
+  map_mul' T S := Subtype.ext (wotTranspose_mul S.unop.val T.unop.val)
+  map_add' T S := Subtype.ext (wotTranspose_add T.unop.val S.unop.val)
+  map_zero' := Subtype.ext wotTranspose_zero
+  commutes' z := by
+    apply Subtype.ext
+    change wotTranspose (algebraMap ℂ _ z) = algebraMap ℂ _ z
+    rw [Algebra.algebraMap_eq_smul_one, wotTranspose_smul, wotTranspose_one]
+  map_star' T := Subtype.ext (wotTranspose_star T.unop.val)
+
+lemma opTransposeOnTwisted_continuous (ω : ScalarCocycle G) :
+    Continuous (opTransposeOnTwisted ω) :=
+  (wotTranspose_continuous.comp (continuous_subtype_val.comp
+    MulOpposite.continuous_unop)).subtype_mk _
+
+end
+
+section
+open ArithmeticGerms GermBundles
+section RightAction
+variable {G H V J : Type*} [Group G] [Group H] [Fintype J]
+variable [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
+variable (C : VirtualGraph G H)
+local instance rightNormalActionDomainFintype : Fintype (G ⧸ C.domain) :=
+  Subgroup.fintypeQuotientOfFiniteIndex
+local instance rightNormalActionCodomainFintype : Fintype (H ⧸ C.codomain) :=
+  Subgroup.fintypeQuotientOfFiniteIndex
+variable (μ : ScalarCocycle G) (ω : ScalarCocycle H)
+variable (B : ProjectiveBundle (prodRatio μ ω) (fun _ : (G × H) ⧸ C.graph => V))
+variable (b : OrthonormalBasis J ℂ V)
+
+def rightNormalAction : (twistedGroupAlgebra ω)ᵐᵒᵖ →⋆ₐ[ℂ]
+    (lp (fun _ : (G × H) ⧸ C.graph => V) 2 →WOT[ℂ]
+      lp (fun _ : (G × H) ⧸ C.graph => V) 2) :=
+  (rightAdjointNormalAction C μ ω B b).comp (opTransposeOnTwisted ω)
+
+theorem rightNormalAction_continuous : Continuous (rightNormalAction C μ ω B b) :=
+  (rightAdjointNormalAction_continuous C μ ω B b).comp
+    (opTransposeOnTwisted_continuous ω)
+
+theorem rightNormalAction_adjoint_generator (h : H) :
+    rightNormalAction C μ ω B b
+      (MulOpposite.op (star (⟨regularGenerator ω h, regularGenerator_mem ω h⟩ :
+        twistedGroupAlgebra ω))) = coordinateOperator C μ ω B (1,h) := by
+  have ht : opTransposeOnTwisted ω
+      (MulOpposite.op (star (⟨regularGenerator ω h, regularGenerator_mem ω h⟩ :
+        twistedGroupAlgebra ω))) =
+      ⟨regularGenerator (inverseCocycle ω) h, regularGenerator_mem (inverseCocycle ω) h⟩ :=
+    Subtype.ext (wotTranspose_star_regularGenerator ω h)
+  exact (congrArg (rightAdjointNormalAction C μ ω B b) ht).trans
+    (rightAdjointNormalAction_generator C μ ω B b h)
+
+theorem left_rightNormalActions_commute (T : twistedGroupAlgebra μ)
+    (S : (twistedGroupAlgebra ω)ᵐᵒᵖ) :
+    leftNormalAction C μ ω B b T * rightNormalAction C μ ω B b S =
+      rightNormalAction C μ ω B b S * leftNormalAction C μ ω B b T :=
+  normalActions_commute C μ ω B b T (opTransposeOnTwisted ω S)
+
+end RightAction
+
+variable {G H V : Type*} [Group G] [Group H]
+variable [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+variable (C : VirtualGraph G H)
+local instance inducedGraphDomainFintype : Fintype (G ⧸ C.domain) :=
+  Subgroup.fintypeQuotientOfFiniteIndex
+local instance inducedGraphCodomainFintype : Fintype (H ⧸ C.codomain) :=
+  Subgroup.fintypeQuotientOfFiniteIndex
+variable (μ : ScalarCocycle G) (ω : ScalarCocycle H)
+variable (r : CosetSection C.graph)
+variable (γ : ProjectiveRepresentation ((prodRatio μ ω).restrict C.graph) V)
+
+theorem induced_graph_normal_actions :
+    let B := r.inducedBundle (prodRatio μ ω) γ
+    ∃ (L : twistedGroupAlgebra μ →⋆ₐ[ℂ]
+          (lp (fun _ : (G × H) ⧸ C.graph => V) 2 →WOT[ℂ]
+            lp (fun _ : (G × H) ⧸ C.graph => V) 2))
+      (R : (twistedGroupAlgebra ω)ᵐᵒᵖ →⋆ₐ[ℂ]
+          (lp (fun _ : (G × H) ⧸ C.graph => V) 2 →WOT[ℂ]
+            lp (fun _ : (G × H) ⧸ C.graph => V) 2))
+      (UL : PiLp 2 (fun _ : (H ⧸ C.codomain) × Fin (Module.finrank ℂ V) =>
+          lp (fun _ : G => ℂ) 2) ≃ₗᵢ[ℂ] lp (fun _ : (G × H) ⧸ C.graph => V) 2)
+      (UR : PiLp 2 (fun _ : (G ⧸ C.domain) × Fin (Module.finrank ℂ V) =>
+          lp (fun _ : H => ℂ) 2) ≃ₗᵢ[ℂ] lp (fun _ : (G × H) ⧸ C.graph => V) 2),
+      Continuous L ∧ Continuous R ∧
+      (∀ T S, L T * R S = R S * L T) ∧
+      (∀ g, L ⟨regularGenerator μ g, regularGenerator_mem μ g⟩ =
+        coordinateOperator C μ ω B (g,1)) ∧
+      (∀ h, R (MulOpposite.op (star (⟨regularGenerator ω h, regularGenerator_mem ω h⟩ :
+        twistedGroupAlgebra ω))) = coordinateOperator C μ ω B (1,h)) ∧
+      (∀ g v, B.action (g,1) (UL v) = UL
+        (amplify (scalarRegular μ g).toContinuousLinearEquiv.toContinuousLinearMap v)) ∧
+      (∀ h v, B.action (1,h) (UR v) = UR
+        (amplify (scalarRegular (inverseCocycle ω) h).toContinuousLinearEquiv.toContinuousLinearMap v)) := by
+  let B := r.inducedBundle (prodRatio μ ω) γ
+  let b := stdOrthonormalBasis ℂ V
+  refine ⟨leftNormalAction C μ ω B b, rightNormalAction C μ ω B b,
+    firstFiniteRegularUnitary C μ ω B b, secondFiniteRegularUnitary C μ ω B b,
+    leftNormalAction_continuous C μ ω B b, rightNormalAction_continuous C μ ω B b, ?_⟩
+  exact ⟨left_rightNormalActions_commute C μ ω B b,
+    leftNormalAction_generator C μ ω B b, rightNormalAction_adjoint_generator C μ ω B b,
+    firstFiniteRegularUnitary_intertwines C μ ω B b,
+    secondFiniteRegularUnitary_intertwines C μ ω B b⟩
+
+end
+
+section
+open GermBundles
+variable {G : Type*} [Group G]
+
+def standardRightRegular (ω : ScalarCocycle G) (h : G) :
+    lp (fun _ : G => ℂ) 2 ≃ₗᵢ[ℂ] lp (fun _ : G => ℂ) 2 :=
+  transportLp (Equiv.mulRight h) (fun g => circleUnitary (ω.val g h))
+
+lemma standardRightRegular_apply (ω : ScalarCocycle G) (h : G)
+    (x : lp (fun _ : G => ℂ) 2) (g : G) :
+    standardRightRegular ω h x (g*h) = (ω.val g h : ℂ) * x g :=
+  transportLp_apply (Equiv.mulRight h) (fun g => circleUnitary (ω.val g h)) x g
+
+def oppositeToStandard (ω : ScalarCocycle G) :
+    lp (fun _ : G => ℂ) 2 ≃ₗᵢ[ℂ] lp (fun _ : G => ℂ) 2 :=
+  transportLp (Equiv.inv G) (fun g => circleUnitary ((ω.val g⁻¹ g)⁻¹))
+
+lemma oppositeToStandard_apply (ω : ScalarCocycle G)
+    (x : lp (fun _ : G => ℂ) 2) (g : G) :
+    oppositeToStandard ω x g⁻¹ = ((ω.val g⁻¹ g)⁻¹ : Circle) * x g := by
+  exact transportLp_apply (Equiv.inv G) (fun g => circleUnitary ((ω.val g⁻¹ g)⁻¹)) x g
+
+lemma rightStandard_phase (ω : ScalarCocycle G) (h g : G) :
+    ω.val (h*g)⁻¹ h * (ω.val h g)⁻¹ * (ω.val (h*g)⁻¹ (h*g))⁻¹ =
+      (ω.val g⁻¹ g)⁻¹ := by
+  have hh := ω.mul_assoc (h*g)⁻¹ h g
+  have hg : (h*g)⁻¹ * h = g⁻¹ := by group
+  rw [hg] at hh
+  have he : ω.val (h*g)⁻¹ h =
+      (ω.val h g * ω.val (h*g)⁻¹ (h*g)) * (ω.val g⁻¹ g)⁻¹ :=
+    (eq_mul_inv_iff_mul_eq).mpr hh
+  rw [he]
+  simp [mul_comm, mul_left_comm, mul_assoc]
+
+lemma oppositeToStandard_intertwines (ω : ScalarCocycle G) (h : G)
+    (x : lp (fun _ : G => ℂ) 2) :
+    oppositeToStandard ω (scalarRegular (inverseCocycle ω) h x) =
+      (standardRightRegular ω h).symm (oppositeToStandard ω x) := by
+  apply (standardRightRegular ω h).injective
+  rw [LinearIsometryEquiv.apply_symm_apply]
+  apply lp.ext
+  funext k
+  obtain ⟨g,rfl⟩ := (Equiv.inv G).surjective k
+  change standardRightRegular ω h (oppositeToStandard ω (scalarRegular (inverseCocycle ω) h x)) g⁻¹ =
+    oppositeToStandard ω x g⁻¹
+  have hi : (h*g)⁻¹ * h = g⁻¹ := by group
+  rw [← hi, standardRightRegular_apply, oppositeToStandard_apply, scalarRegular_apply,
+    hi, oppositeToStandard_apply]
+  change (ω.val (h*g)⁻¹ h : ℂ) *
+    (((ω.val (h*g)⁻¹ (h*g))⁻¹ : Circle) * (((ω.val h g)⁻¹ : Circle) * x g)) =
+      ((ω.val g⁻¹ g)⁻¹ : Circle) * x g
+  have hp := congrArg (fun z : Circle => (z : ℂ)) (rightStandard_phase ω h g)
+  simp only [Circle.coe_mul] at hp
+  calc
+    _ = ((ω.val (h*g)⁻¹ h : ℂ) * ((ω.val h g)⁻¹ : Circle) *
+        ((ω.val (h*g)⁻¹ (h*g))⁻¹ : Circle)) * x g := by ring
+    _ = _ := congrArg (fun z : ℂ => z * x g) hp
+
+end
+
+open ArithmeticGerms GermBundles
+section EntireRegularActions
+variable {G H V J : Type*} [Group G] [Group H] [Fintype J]
+variable [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
+variable (C : VirtualGraph G H)
+local instance regularActionDomainFintype : Fintype (G ⧸ C.domain) :=
+  Subgroup.fintypeQuotientOfFiniteIndex
+local instance regularActionCodomainFintype : Fintype (H ⧸ C.codomain) :=
+  Subgroup.fintypeQuotientOfFiniteIndex
+variable (μ : ScalarCocycle G) (ω : ScalarCocycle H)
+variable (B : ProjectiveBundle (prodRatio μ ω) (fun _ : (G × H) ⧸ C.graph => V))
+variable (b : OrthonormalBasis J ℂ V)
+
+lemma leftNormalAction_finite_regular (T : twistedGroupAlgebra μ)
+    (v : PiLp 2 (fun _ : (H ⧸ C.codomain) × J => lp (fun _ : G => ℂ) 2)) :
+    leftNormalAction C μ ω B b T (firstFiniteRegularUnitary C μ ω B b v) =
+      firstFiniteRegularUnitary C μ ω B b (amplify T.val.toCLM v) := by
+  change firstFiniteRegularUnitary C μ ω B b
+    (amplify T.val.toCLM ((firstFiniteRegularUnitary C μ ω B b).symm
+      (firstFiniteRegularUnitary C μ ω B b v))) = _
+  rw [LinearIsometryEquiv.symm_apply_apply]
+
+lemma rightNormalAction_finite_regular (S : (twistedGroupAlgebra ω)ᵐᵒᵖ)
+    (v : PiLp 2 (fun _ : (G ⧸ C.domain) × J => lp (fun _ : H => ℂ) 2)) :
+    rightNormalAction C μ ω B b S (secondFiniteRegularUnitary C μ ω B b v) =
+      secondFiniteRegularUnitary C μ ω B b (amplify
+        (wotTranspose S.unop.val).toCLM v) := by
+  change secondFiniteRegularUnitary C μ ω B b
+    (amplify (wotTranspose S.unop.val).toCLM ((secondFiniteRegularUnitary C μ ω B b).symm
+      (secondFiniteRegularUnitary C μ ω B b v))) = _
+  rw [LinearIsometryEquiv.symm_apply_apply]
+
+end EntireRegularActions
+
+variable {G H V : Type*} [Group G] [Group H]
+variable [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+variable (C : VirtualGraph G H)
+local instance multiplicityDomainFintype : Fintype (G ⧸ C.domain) :=
+  Subgroup.fintypeQuotientOfFiniteIndex
+local instance multiplicityCodomainFintype : Fintype (H ⧸ C.codomain) :=
+  Subgroup.fintypeQuotientOfFiniteIndex
+
+omit [FiniteDimensional ℂ V] in
+lemma graph_regular_multiplicities :
+    Fintype.card ((H ⧸ C.codomain) × Fin (Module.finrank ℂ V)) =
+      Module.finrank ℂ V * C.codomain.index ∧
+    Fintype.card ((G ⧸ C.domain) × Fin (Module.finrank ℂ V)) =
+      Module.finrank ℂ V * C.domain.index := by
+  simp [Fintype.card_prod, Subgroup.index_eq_card, Nat.card_eq_fintype_card, mul_comm]
+
+end GraphNormality
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/GraphModels/Regular.lean b/lean/OAI/RepresentationTheory/GraphModels/Regular.lean
index 8557b869ed7783be4d0d31f31e912b54fff1c3eb..87aa8bc38581282ba1936a0b8eb54dab16717fa5 100644
--- a/lean/OAI/RepresentationTheory/GraphModels/Regular.lean
+++ b/lean/OAI/RepresentationTheory/GraphModels/Regular.lean
@@ -1,679 +1,679 @@
-import Mathlib
-import OAI.RepresentationTheory.GraphModels.Bundles
-
-namespace OAI
-
-noncomputable section
-
-/-! Regular coordinates and finite amplification of graph models. -/
-
-namespace GraphNormality
-section
-open ArithmeticGerms GermBundles
-variable {G H : Type*} [Group G] [Group H]
-variable (C : VirtualGraph G H) (r : CosetSection C.codomain)
-
-def firstCoordinateMap (x : (H ⧸ C.codomain) × G) : (G × H) ⧸ C.graph :=
-  QuotientGroup.mk (x.2, r.repr x.1)
-
-theorem firstCoordinateMap_injective : Function.Injective (firstCoordinateMap C r) := by
-  rintro ⟨i, g⟩ ⟨j, g'⟩ he
-  have hm : (g⁻¹ * g', (r.repr i)⁻¹ * r.repr j) ∈ C.graph :=
-    QuotientGroup.eq.mp he
-  have hb : (r.repr i)⁻¹ * r.repr j ∈ C.codomain :=
-    ⟨_, hm, rfl⟩
-  have hij : i = j := by
-    rw [← r.repr_spec i, ← r.repr_spec j]
-    exact QuotientGroup.eq.mpr hb
-  subst j
-  have hg : g⁻¹ * g' = 1 := by
-    have h := C.snd_injective (a₁ := ⟨_, hm⟩) (a₂ := 1) (by simp)
-    exact congrArg (fun x : C.graph => x.val.1) h
-  exact Prod.ext rfl (inv_mul_eq_one.mp hg)
-
-theorem firstCoordinateMap_surjective : Function.Surjective (firstCoordinateMap C r) := by
-  intro x
-  induction x using QuotientGroup.induction_on with
-  | H x =>
-    let q : H ⧸ C.codomain := QuotientGroup.mk x.2
-    have hb : (r.repr q)⁻¹ * x.2 ∈ C.codomain :=
-      QuotientGroup.eq.mp (r.repr_spec q)
-    obtain ⟨c, hc, he⟩ := hb
-    refine ⟨(q, x.1 * c.1⁻¹), ?_⟩
-    apply QuotientGroup.eq.mpr
-    change (((x.1 * c.1⁻¹)⁻¹ * x.1), (r.repr q)⁻¹ * x.2) ∈ C.graph
-    have hf : (x.1 * c.1⁻¹)⁻¹ * x.1 = c.1 := by group
-    rw [hf, ← he]
-    exact hc
-
-def firstCoordinates : ((H ⧸ C.codomain) × G) ≃ ((G × H) ⧸ C.graph) :=
-  Equiv.ofBijective (firstCoordinateMap C r)
-    ⟨firstCoordinateMap_injective C r, firstCoordinateMap_surjective C r⟩
-
-theorem firstCoordinates_smul (a : G) (x : (H ⧸ C.codomain) × G) :
-    (a, (1 : H)) • firstCoordinates C r x = firstCoordinates C r (x.1, a * x.2) := by
-  change QuotientGroup.mk ((a, (1 : H)) * (x.2, r.repr x.1)) =
-    QuotientGroup.mk (a * x.2, r.repr x.1)
-  simp
-
-end
-
-section
-section
-attribute [local instance] Classical.propDecidable
-
-local instance (priority := 2000) (X : Type*) : DecidableEq X := Classical.decEq X
-
-section Reindex
-variable {X Y V W : Type*}
-variable [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-variable [NormedAddCommGroup W] [InnerProductSpace ℂ W]
-
-def transportLp (e : X ≃ Y) (u : X → V ≃ₗᵢ[ℂ] W) :
-    lp (fun _ : X => V) 2 ≃ₗᵢ[ℂ] lp (fun _ : Y => W) 2 where
-  toLinearEquiv :=
-    { toFun := fun v => ⟨fun y => u (e.symm y) (v (e.symm y)), by
-        apply memℓp_gen
-        convert! ((lp.memℓp v).summable (by norm_num)).comp_injective e.symm.injective using 1
-        simp only [LinearIsometryEquiv.norm_map]; rfl⟩
-      invFun := fun v => ⟨fun x => (u x).symm (v (e x)), by
-        apply memℓp_gen
-        convert! ((lp.memℓp v).summable (by norm_num)).comp_injective e.injective using 1
-        simp only [LinearIsometryEquiv.norm_map]; rfl⟩
-      left_inv := by
-        intro v
-        apply lp.ext
-        funext x
-        change (u x).symm (u (e.symm (e x)) (v (e.symm (e x)))) = _
-        simp
-      right_inv := by
-        intro v
-        apply lp.ext
-        funext y
-        change u (e.symm y) ((u (e.symm y)).symm (v (e (e.symm y)))) = _
-        simp
-      map_add' := by
-        intro v w
-        apply lp.ext
-        funext y
-        exact map_add _ _ _
-      map_smul' := by
-        intro a v
-        apply lp.ext
-        funext y
-        exact map_smul _ _ _ }
-  norm_map' := by
-    intro v
-    rw [lp.norm_eq_tsum_rpow (by norm_num), lp.norm_eq_tsum_rpow (by norm_num)]
-    congr 1
-    change (∑' y, ‖u (e.symm y) (v (e.symm y))‖ ^ (2 : ENNReal).toReal) = _
-    simp only [LinearIsometryEquiv.norm_map]
-    exact e.symm.tsum_eq (fun x => ‖v x‖ ^ (2 : ENNReal).toReal)
-
-theorem transportLp_apply (e : X ≃ Y) (u : X → V ≃ₗᵢ[ℂ] W)
-    (v : lp (fun _ : X => V) 2) (x : X) :
-    transportLp e u v (e x) = u x (v x) := by
-  change u (e.symm (e x)) (v (e.symm (e x))) = _
-  simp
-
-theorem transportLp_single (e : X ≃ Y) (u : X → V ≃ₗᵢ[ℂ] W)
-    (x : X) (v : V) :
-    transportLp e u (lp.single 2 x v) = lp.single 2 (e x) (u x v) := by
-  apply lp.ext
-  funext y
-  by_cases hy : y = e x
-  · subst y
-    rw [transportLp_apply, lp.single_apply_self, lp.single_apply_self]
-  · have hxy : e.symm y ≠ x := by
-      intro h
-      exact hy ((e.apply_symm_apply y).symm.trans (congrArg e h))
-    change u (e.symm y) ((lp.single 2 x v : lp (fun _ : X => V) 2) (e.symm y)) = _
-    rw [lp.single_apply_ne (E := fun _ : X => V) 2 x v hxy, map_zero,
-      lp.single_apply_ne (E := fun _ : Y => W) 2 (e x) _ hy]
-end Reindex
-
-open ArithmeticGerms GermBundles
-variable {G H I V : Type*} [Group G] [Group H]
-variable [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-
-def prodRatio (μ : ScalarCocycle G) (ω : ScalarCocycle H) : ScalarCocycle (G × H) where
-  val g h := μ.val g.1 h.1 / ω.val g.2 h.2
-  mul_assoc g h j := by
-    dsimp
-    rw [div_mul_div_comm, div_mul_div_comm, μ.mul_assoc, ω.mul_assoc]
-  one_left g := by simp [μ.one_left, ω.one_left]
-  one_right g := by simp [μ.one_right, ω.one_right]
-
-def regularPerm (a : G) : I × G ≃ I × G where
-  toFun x := (x.1, a * x.2)
-  invFun x := (x.1, a⁻¹ * x.2)
-  left_inv x := by simp
-  right_inv x := by simp
-
-def regularAction (μ : ScalarCocycle G) (a : G) :
-    lp (fun _ : I × G => V) 2 ≃ₗᵢ[ℂ] lp (fun _ : I × G => V) 2 :=
-  transportLp (regularPerm a) (fun x => circleUnitary (μ.val a x.2))
-
-theorem regularAction_single (μ : ScalarCocycle G) (a : G) (i : I) (g : G) (v : V) :
-    regularAction μ a (lp.single 2 (i,g) v) =
-      lp.single 2 (i,a*g) ((μ.val a g : ℂ) • v) := by
-  exact transportLp_single (regularPerm (I := I) a)
-    (fun x => circleUnitary (V := V) (μ.val a x.2)) (i, g) v
-
-variable (C : VirtualGraph G H) (r : CosetSection C.codomain)
-variable (μ : ScalarCocycle G) (ω : ScalarCocycle H)
-variable (B : ProjectiveBundle (prodRatio μ ω) (fun _ : (G × H) ⧸ C.graph => V))
-
-def firstRegularUnitary :
-    lp (fun _ : (H ⧸ C.codomain) × G => V) 2 ≃ₗᵢ[ℂ]
-      lp (fun _ : (G × H) ⧸ C.graph => V) 2 :=
-  transportLp (firstCoordinates C r) (fun x =>
-    B.fibre (x.2, 1) (firstCoordinates C r (x.1, 1)) (firstCoordinates C r x) (by
-      simpa using firstCoordinates_smul C r x.2 (x.1, 1)))
-
-theorem firstRegularUnitary_single (i : H ⧸ C.codomain) (g : G) (v : V) :
-    firstRegularUnitary C r μ ω B (lp.single 2 (i,g) v) =
-      B.action (g,1) (lp.single 2 (firstCoordinates C r (i,1)) v) := by
-  rw [firstRegularUnitary, transportLp_single]
-  exact (B.action_single (g,1) _ _ _ v).symm
-
-theorem firstRegularUnitary_intertwines (a : G)
-    (v : lp (fun _ : (H ⧸ C.codomain) × G => V) 2) :
-    B.action (a,1) (firstRegularUnitary C r μ ω B v) =
-      firstRegularUnitary C r μ ω B (regularAction μ a v) := by
-  have he :
-      (B.action (a,1)).toContinuousLinearEquiv.toContinuousLinearMap.comp
-        (firstRegularUnitary C r μ ω B).toContinuousLinearEquiv.toContinuousLinearMap =
-      (firstRegularUnitary C r μ ω B).toContinuousLinearEquiv.toContinuousLinearMap.comp
-        (regularAction μ a).toContinuousLinearEquiv.toContinuousLinearMap := by
-    apply lp.ext_continuousLinearMap (by norm_num)
-    rintro ⟨i,g⟩
-    apply ContinuousLinearMap.ext
-    intro w
-    change B.action (a,1) (firstRegularUnitary C r μ ω B (lp.single 2 (i,g) w)) =
-      firstRegularUnitary C r μ ω B (regularAction μ a (lp.single 2 (i,g) w))
-    rw [firstRegularUnitary_single, B.action_mul, regularAction_single,
-      firstRegularUnitary_single, lp.single_smul, map_smul]
-    simp [prodRatio, ω.one_left]
-  exact DFunLike.congr_fun he v
-
-end
-
-open ArithmeticGerms GermBundles
-local instance (priority := 2000) (X : Type*) : DecidableEq X := Classical.decEq X
-
-variable {G H V : Type*} [Group G] [Group H]
-variable [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-variable (C : VirtualGraph G H) (r : CosetSection C.domain)
-
-def secondCoordinateMap (x : (G ⧸ C.domain) × H) : (G × H) ⧸ C.graph :=
-  QuotientGroup.mk (r.repr x.1, x.2)
-
-theorem secondCoordinateMap_injective : Function.Injective (secondCoordinateMap C r) := by
-  rintro ⟨i, h⟩ ⟨j, h'⟩ he
-  have hm : ((r.repr i)⁻¹ * r.repr j, h⁻¹ * h') ∈ C.graph :=
-    QuotientGroup.eq.mp he
-  have hb : (r.repr i)⁻¹ * r.repr j ∈ C.domain := ⟨_, hm, rfl⟩
-  have hij : i = j := by
-    rw [← r.repr_spec i, ← r.repr_spec j]
-    exact QuotientGroup.eq.mpr hb
-  subst j
-  have hh : h⁻¹ * h' = 1 := by
-    have hh := C.fst_injective (a₁ := ⟨_, hm⟩) (a₂ := 1) (by simp)
-    exact congrArg (fun x : C.graph => x.val.2) hh
-  exact Prod.ext rfl (inv_mul_eq_one.mp hh)
-
-theorem secondCoordinateMap_surjective : Function.Surjective (secondCoordinateMap C r) := by
-  intro x
-  induction x using QuotientGroup.induction_on with
-  | H x =>
-    let q : G ⧸ C.domain := QuotientGroup.mk x.1
-    have hb : (r.repr q)⁻¹ * x.1 ∈ C.domain :=
-      QuotientGroup.eq.mp (r.repr_spec q)
-    obtain ⟨c, hc, he⟩ := hb
-    refine ⟨(q, x.2 * c.2⁻¹), ?_⟩
-    apply QuotientGroup.eq.mpr
-    change ((r.repr q)⁻¹ * x.1, ((x.2 * c.2⁻¹)⁻¹ * x.2)) ∈ C.graph
-    have hf : (x.2 * c.2⁻¹)⁻¹ * x.2 = c.2 := by group
-    rw [hf, ← he]
-    exact hc
-
-def secondCoordinates : ((G ⧸ C.domain) × H) ≃ ((G × H) ⧸ C.graph) :=
-  Equiv.ofBijective (secondCoordinateMap C r)
-    ⟨secondCoordinateMap_injective C r, secondCoordinateMap_surjective C r⟩
-
-theorem secondCoordinates_smul (a : H) (x : (G ⧸ C.domain) × H) :
-    ((1 : G), a) • secondCoordinates C r x = secondCoordinates C r (x.1, a * x.2) := by
-  change QuotientGroup.mk (((1 : G), a) * (r.repr x.1, x.2)) =
-    QuotientGroup.mk (r.repr x.1, a * x.2)
-  simp
-
-def inverseCocycle (ω : ScalarCocycle H) : ScalarCocycle H where
-  val h k := (ω.val h k)⁻¹
-  mul_assoc h k l := by
-    simpa only [mul_inv] using congrArg Inv.inv (ω.mul_assoc h k l)
-  one_left h := by simp [ω.one_left]
-  one_right h := by simp [ω.one_right]
-
-variable (μ : ScalarCocycle G) (ω : ScalarCocycle H)
-variable (B : ProjectiveBundle (prodRatio μ ω) (fun _ : (G × H) ⧸ C.graph => V))
-
-def secondRegularUnitary :
-    lp (fun _ : (G ⧸ C.domain) × H => V) 2 ≃ₗᵢ[ℂ]
-      lp (fun _ : (G × H) ⧸ C.graph => V) 2 :=
-  transportLp (secondCoordinates C r) (fun x =>
-    B.fibre (1, x.2) (secondCoordinates C r (x.1, 1)) (secondCoordinates C r x) (by
-      simpa using secondCoordinates_smul C r x.2 (x.1, 1)))
-
-theorem secondRegularUnitary_single (i : G ⧸ C.domain) (h : H) (v : V) :
-    secondRegularUnitary C r μ ω B (lp.single 2 (i,h) v) =
-      B.action (1,h) (lp.single 2 (secondCoordinates C r (i,1)) v) := by
-  rw [secondRegularUnitary, transportLp_single]
-  exact (B.action_single (1,h) _ _ _ v).symm
-
-theorem secondRegularUnitary_intertwines (a : H)
-    (v : lp (fun _ : (G ⧸ C.domain) × H => V) 2) :
-    B.action (1,a) (secondRegularUnitary C r μ ω B v) =
-      secondRegularUnitary C r μ ω B (regularAction (inverseCocycle ω) a v) := by
-  have he :
-      (B.action (1,a)).toContinuousLinearEquiv.toContinuousLinearMap.comp
-        (secondRegularUnitary C r μ ω B).toContinuousLinearEquiv.toContinuousLinearMap =
-      (secondRegularUnitary C r μ ω B).toContinuousLinearEquiv.toContinuousLinearMap.comp
-        (regularAction (inverseCocycle ω) a).toContinuousLinearEquiv.toContinuousLinearMap := by
-    apply lp.ext_continuousLinearMap (by norm_num)
-    rintro ⟨i,h⟩
-    apply ContinuousLinearMap.ext
-    intro w
-    change B.action (1,a) (secondRegularUnitary C r μ ω B (lp.single 2 (i,h) w)) =
-      secondRegularUnitary C r μ ω B (regularAction (inverseCocycle ω) a (lp.single 2 (i,h) w))
-    rw [secondRegularUnitary_single, B.action_mul, regularAction_single,
-      secondRegularUnitary_single, lp.single_smul, map_smul]
-    simp [prodRatio, inverseCocycle, μ.one_left]
-  exact DFunLike.congr_fun he v
-
-theorem coordinate_actions_commute (g : G) (h : H)
-    (v : lp (fun _ : (G × H) ⧸ C.graph => V) 2) :
-    B.action (g,1) (B.action (1,h) v) = B.action (1,h) (B.action (g,1) v) := by
-  rw [B.action_mul, B.action_mul]
-  simp [prodRatio, μ.one_left, μ.one_right, ω.one_left, ω.one_right]
-
-end
-
-section
-open GermBundles
-open scoped InnerProductSpace
-local instance (priority := 2000) (X : Type*) : DecidableEq X := Classical.decEq X
-
-section ScalarMultiplicity
-variable {I J G V : Type*} [Fintype J]
-variable [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
-variable (b : OrthonormalBasis J ℂ V)
-
-def bundleBasisVector (x : (I × J) × G) : lp (fun _ : I × G => V) 2 :=
-  lp.single 2 (x.1.1, x.2) (b x.1.2)
-
-omit [CompleteSpace V] in
-theorem bundleBasisVector_orthonormal : Orthonormal ℂ (bundleBasisVector (I := I) (G := G) b) := by
-  rw [orthonormal_iff_ite]
-  rintro ⟨⟨i,j⟩,g⟩ ⟨⟨k,l⟩,h⟩
-  change ⟪lp.single (E := fun _ : I × G => V) 2 (i,g) (b j),
-    lp.single (E := fun _ : I × G => V) 2 (k,h) (b l)⟫_ℂ = _
-  rw [lp.inner_single_left]
-  by_cases hi : i = k
-  · subst k
-    by_cases hg : g = h
-    · subst h
-      rw [lp.single_apply_self]
-      simpa only [Prod.mk.injEq, true_and, and_true] using
-        (orthonormal_iff_ite.mp b.orthonormal j l)
-    · rw [lp.single_apply_ne (E := fun _ : I × G => V) 2 (i,h) (b l)
-          (fun hh => hg (congrArg Prod.snd hh))]
-      simp [hg]
-  · rw [lp.single_apply_ne (E := fun _ : I × G => V) 2 (k,h) (b l)
-        (fun hh => hi (congrArg Prod.fst hh))]
-    simp [hi]
-
-omit [CompleteSpace V] in
-theorem bundleBasisVector_total :
-    (Submodule.span ℂ (Set.range (bundleBasisVector (I := I) (G := G) b)))ᗮ = ⊥ := by
-  apply le_antisymm _ bot_le
-  intro v hv
-  change v = 0
-  apply lp.ext
-  funext x
-  change v x = 0
-  apply b.repr.injective
-  ext j
-  rw [b.repr_apply_apply, map_zero]
-  have h := Submodule.inner_right_of_mem_orthogonal
-    (Submodule.subset_span (Set.mem_range_self ((x.1,j),x.2))) hv
-  simpa only [bundleBasisVector, Prod.eta, lp.inner_single_left, PiLp.zero_apply] using h
-
-def regularHilbertBasis : HilbertBasis ((I × J) × G) ℂ (lp (fun _ : I × G => V) 2) :=
-  HilbertBasis.mkOfOrthogonalEqBot (bundleBasisVector_orthonormal b) (bundleBasisVector_total b)
-
-@[simp] theorem regularHilbertBasis_apply (x : (I × J) × G) :
-    regularHilbertBasis b x = bundleBasisVector b x := by
-  exact congrFun (HilbertBasis.coe_mkOfOrthogonalEqBot _ _) x
-
-def scalarization : lp (fun _ : I × G => V) 2 ≃ₗᵢ[ℂ] lp (fun _ : (I × J) × G => ℂ) 2 :=
-  (regularHilbertBasis b).repr
-
-theorem scalarization_apply (v : lp (fun _ : I × G => V) 2) (i : I) (j : J) (g : G) :
-    scalarization b v ((i,j),g) = ⟪b j, v (i,g)⟫_ℂ := by
-  change (regularHilbertBasis b).repr v ((i,j),g) = _
-  rw [HilbertBasis.repr_apply_apply, regularHilbertBasis_apply]
-  exact lp.inner_single_left _ _ _
-
-end ScalarMultiplicity
-
-section ScalarRegular
-variable {I G V : Type*} [Group G]
-variable [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-
-theorem regularAction_apply (μ : ScalarCocycle G) (a : G)
-    (v : lp (fun _ : I × G => V) 2) (i : I) (g : G) :
-    regularAction μ a v (i,a*g) = (μ.val a g : ℂ) • v (i,g) := by
-  exact transportLp_apply (regularPerm a) (fun x => circleUnitary (μ.val a x.2)) v (i,g)
-
-end ScalarRegular
-
-variable {I J G V : Type*} [Fintype J] [Group G]
-variable [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
-variable (b : OrthonormalBasis J ℂ V)
-
-theorem scalarization_intertwines (μ : ScalarCocycle G) (a : G)
-    (v : lp (fun _ : I × G => V) 2) :
-    scalarization b (regularAction μ a v) = regularAction μ a (scalarization b v) := by
-  apply lp.ext
-  funext x
-  obtain ⟨y,rfl⟩ := (regularPerm (I := I × J) a).surjective x
-  rcases y with ⟨⟨i,j⟩,g⟩
-  change scalarization b (regularAction μ a v) ((i,j),a*g) =
-    regularAction μ a (scalarization b v) ((i,j),a*g)
-  rw [scalarization_apply, regularAction_apply, inner_smul_right,
-    regularAction_apply, scalarization_apply]
-  rfl
-
-end
-
-section
-open scoped InnerProductSpace
-local instance (priority := 2000) (X : Type*) : DecidableEq X := Classical.decEq X
-
-variable {K G : Type*} [Fintype K]
-
-def finiteSumBasisVector (x : K × G) : PiLp 2 (fun _ : K => lp (fun _ : G => ℂ) 2) :=
-  PiLp.single 2 x.1 (lp.single 2 x.2 1)
-
-theorem finiteSumBasisVector_inner (x : K × G)
-    (v : PiLp 2 (fun _ : K => lp (fun _ : G => ℂ) 2)) :
-    ⟪finiteSumBasisVector x, v⟫_ℂ = v x.1 x.2 := by
-  rw [PiLp.inner_apply, Finset.sum_eq_single x.1]
-  · simp [finiteSumBasisVector, lp.inner_single_left]
-  · intro i hi hix
-    simp [finiteSumBasisVector, hix]
-  · simp
-
-theorem finiteSumBasisVector_orthonormal :
-    Orthonormal ℂ (finiteSumBasisVector (K := K) (G := G)) := by
-  rw [orthonormal_iff_ite]
-  rintro ⟨i,g⟩ ⟨j,h⟩
-  rw [finiteSumBasisVector_inner]
-  by_cases hi : i = j
-  · subst j
-    simp [finiteSumBasisVector, lp.single_apply, Pi.single_apply]
-  · simp [finiteSumBasisVector, hi]
-
-theorem finiteSumBasisVector_total :
-    (Submodule.span ℂ (Set.range (finiteSumBasisVector (K := K) (G := G))))ᗮ = ⊥ := by
-  apply le_antisymm _ bot_le
-  intro v hv
-  change v = 0
-  ext i g
-  change v i g = 0
-  have h := Submodule.inner_right_of_mem_orthogonal
-    (Submodule.subset_span (Set.mem_range_self (i,g))) hv
-  simpa only [finiteSumBasisVector_inner] using h
-
-def finiteSumHilbertBasis :
-    HilbertBasis (K × G) ℂ (PiLp 2 (fun _ : K => lp (fun _ : G => ℂ) 2)) :=
-  HilbertBasis.mkOfOrthogonalEqBot finiteSumBasisVector_orthonormal finiteSumBasisVector_total
-
-@[simp] theorem finiteSumHilbertBasis_apply (x : K × G) :
-    finiteSumHilbertBasis x = finiteSumBasisVector x := by
-  exact congrFun (HilbertBasis.coe_mkOfOrthogonalEqBot _ _) x
-
-def finiteRegrouping : lp (fun _ : K × G => ℂ) 2 ≃ₗᵢ[ℂ]
-    PiLp 2 (fun _ : K => lp (fun _ : G => ℂ) 2) :=
-  finiteSumHilbertBasis.repr.symm
-
-theorem finiteRegrouping_apply (v : lp (fun _ : K × G => ℂ) 2) (i : K) (g : G) :
-    finiteRegrouping v i g = v (i,g) := by
-  have h := congrArg (fun f : lp (fun _ : K × G => ℂ) 2 => f (i,g))
-    (finiteSumHilbertBasis.repr.apply_symm_apply v)
-  rw [HilbertBasis.repr_apply_apply, finiteSumHilbertBasis_apply,
-    finiteSumBasisVector_inner] at h
-  exact h
-
-end
-
-section
-open scoped InnerProductSpace
-variable {K L : Type*} [Fintype K]
-variable [NormedAddCommGroup L] [InnerProductSpace ℂ L] [CompleteSpace L]
-
-def amplify (T : L →L[ℂ] L) :
-    PiLp 2 (fun _ : K => L) →L[ℂ] PiLp 2 (fun _ : K => L) :=
-  (PiLp.continuousLinearEquiv 2 ℂ (fun _ : K => L)).symm.toContinuousLinearMap.comp
-    ((ContinuousLinearMap.pi (fun i : K => T.comp (ContinuousLinearMap.proj i))).comp
-      (PiLp.continuousLinearEquiv 2 ℂ (fun _ : K => L)).toContinuousLinearMap)
-
-omit [Fintype K] [CompleteSpace L] in
-@[simp] theorem amplify_apply (T : L →L[ℂ] L) (v : PiLp 2 (fun _ : K => L)) (i : K) :
-    amplify T v i = T (v i) := rfl
-
-def amplification : (L →L[ℂ] L) →⋆ₐ[ℂ]
-    (PiLp 2 (fun _ : K => L) →L[ℂ] PiLp 2 (fun _ : K => L)) where
-  toFun := amplify
-  map_one' := by ext v i; rfl
-  map_mul' T S := by ext v i; rfl
-  map_add' T S := by ext v i; rfl
-  map_zero' := by ext v i; rfl
-  commutes' z := by ext v i; rfl
-  map_star' T := by
-    change amplify (K := K) (ContinuousLinearMap.adjoint T) =
-      ContinuousLinearMap.adjoint (amplify (K := K) T)
-    apply (ContinuousLinearMap.eq_adjoint_iff _ _).mpr
-    intro v w
-    simp only [PiLp.inner_apply, amplify_apply, ContinuousLinearMap.adjoint_inner_left]
-
-theorem amplification_wot_continuous :
-    Continuous (fun T : L →WOT[ℂ] L =>
-      ContinuousLinearMapWOT.ofCLM (amplification (K := K) (L := L) (ContinuousLinearMapWOT.toCLM T))) := by
-  apply ContinuousLinearMapWOT.continuous_iff.mpr
-  intro v w
-  change Continuous (fun T : L →WOT[ℂ] L => ∑ i : K, ⟪w i, T (v i)⟫_ℂ)
-  apply continuous_finsetSum
-  intro i hi
-  exact ContinuousLinearMapWOT.continuous_iff.mp continuous_id (v i) (w i)
-
-end
-
-section
-open scoped InnerProductSpace
-local instance (priority := 2000) (X : Type*) : DecidableEq X := Classical.decEq X
-
-open GermBundles
-
-section ScalarTwistedRegular
-variable {G K : Type*} [Group G] [Fintype K]
-
-def scalarRegular (μ : ScalarCocycle G) (a : G) :
-    lp (fun _ : G => ℂ) 2 ≃ₗᵢ[ℂ] lp (fun _ : G => ℂ) 2 :=
-  transportLp (Equiv.mulLeft a) (fun g => circleUnitary (μ.val a g))
-
-theorem scalarRegular_apply (μ : ScalarCocycle G) (a : G)
-    (v : lp (fun _ : G => ℂ) 2) (g : G) :
-    scalarRegular μ a v (a*g) = (μ.val a g : ℂ) * v g := by
-  exact transportLp_apply (Equiv.mulLeft a) (fun g => circleUnitary (μ.val a g)) v g
-
-theorem finiteRegrouping_intertwines (μ : ScalarCocycle G) (a : G)
-    (v : lp (fun _ : K × G => ℂ) 2) :
-    finiteRegrouping (regularAction μ a v) =
-      amplify (scalarRegular μ a).toContinuousLinearEquiv.toContinuousLinearMap
-        (finiteRegrouping v) := by
-  ext i g
-  obtain ⟨h,rfl⟩ := (Equiv.mulLeft a).surjective g
-  change finiteRegrouping (regularAction μ a v) i (a*h) =
-    scalarRegular μ a (finiteRegrouping v i) (a*h)
-  rw [finiteRegrouping_apply, regularAction_apply, scalarRegular_apply,
-    finiteRegrouping_apply]
-  rfl
-
-end ScalarTwistedRegular
-
-variable {L W : Type*}
-variable [NormedAddCommGroup L] [InnerProductSpace ℂ L] [CompleteSpace L]
-variable [NormedAddCommGroup W] [InnerProductSpace ℂ W] [CompleteSpace W]
-
-theorem unitaryConjugation_wot_continuous (U : L ≃ₗᵢ[ℂ] W) :
-    Continuous (fun T : L →WOT[ℂ] L =>
-      ContinuousLinearMapWOT.ofCLM (U.conjStarAlgEquiv (ContinuousLinearMapWOT.toCLM T))) := by
-  apply ContinuousLinearMapWOT.continuous_iff.mpr
-  intro v w
-  have h : (fun T : L →WOT[ℂ] L =>
-      ⟪w, U.conjStarAlgEquiv (ContinuousLinearMapWOT.toCLM T) v⟫_ℂ) =
-      (fun T : L →WOT[ℂ] L => ⟪U.symm w, T (U.symm v)⟫_ℂ) := by
-    funext T
-    rw [LinearIsometryEquiv.conjStarAlgEquiv_apply_apply]
-    conv_lhs => lhs; rw [← U.apply_symm_apply w]
-    exact U.inner_map_map _ _
-  change Continuous (fun T : L →WOT[ℂ] L =>
-    ⟪w, U.conjStarAlgEquiv (ContinuousLinearMapWOT.toCLM T) v⟫_ℂ)
-  rw [h]
-  exact ContinuousLinearMapWOT.continuous_iff.mp continuous_id (U.symm v) (U.symm w)
-
-end
-
-open ArithmeticGerms GermBundles
-local instance (priority := 2000) (X : Type*) : DecidableEq X := Classical.decEq X
-
-section FiniteRegularModules
-variable {I J G V : Type*} [Fintype I] [Fintype J] [Group G]
-variable [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
-variable (b : OrthonormalBasis J ℂ V)
-
-def regularToFinite : lp (fun _ : I × G => V) 2 ≃ₗᵢ[ℂ]
-    PiLp 2 (fun _ : I × J => lp (fun _ : G => ℂ) 2) :=
-  (scalarization b).trans finiteRegrouping
-
-theorem regularToFinite_intertwines (μ : ScalarCocycle G) (a : G)
-    (v : lp (fun _ : I × G => V) 2) :
-    regularToFinite b (regularAction μ a v) =
-      amplify (scalarRegular μ a).toContinuousLinearEquiv.toContinuousLinearMap
-        (regularToFinite b v) := by
-  change finiteRegrouping (scalarization b (regularAction μ a v)) = _
-  rw [scalarization_intertwines, finiteRegrouping_intertwines]
-  rfl
-
-end FiniteRegularModules
-
-variable {G H V J : Type*} [Group G] [Group H] [Fintype J]
-variable [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
-variable (C : VirtualGraph G H)
-local instance : Fintype (G ⧸ C.domain) := Subgroup.fintypeQuotientOfFiniteIndex
-local instance : Fintype (H ⧸ C.codomain) := Subgroup.fintypeQuotientOfFiniteIndex
-
-variable (μ : ScalarCocycle G) (ω : ScalarCocycle H)
-variable (B : ProjectiveBundle (prodRatio μ ω) (fun _ : (G × H) ⧸ C.graph => V))
-variable (b : OrthonormalBasis J ℂ V)
-
-def firstFiniteRegularUnitary :
-    PiLp 2 (fun _ : (H ⧸ C.codomain) × J => lp (fun _ : G => ℂ) 2) ≃ₗᵢ[ℂ]
-      lp (fun _ : (G × H) ⧸ C.graph => V) 2 :=
-  (regularToFinite b).symm.trans
-    (firstRegularUnitary C (chooseCosetSection C.codomain) μ ω B)
-
-theorem firstFiniteRegularUnitary_intertwines (a : G)
-    (v : PiLp 2 (fun _ : (H ⧸ C.codomain) × J => lp (fun _ : G => ℂ) 2)) :
-    B.action (a,1) (firstFiniteRegularUnitary C μ ω B b v) =
-      firstFiniteRegularUnitary C μ ω B b
-        (amplify (scalarRegular μ a).toContinuousLinearEquiv.toContinuousLinearMap v) := by
-  change B.action (a,1) (firstRegularUnitary C _ μ ω B ((regularToFinite b).symm v)) =
-    firstRegularUnitary C _ μ ω B ((regularToFinite b).symm _)
-  rw [firstRegularUnitary_intertwines]
-  congr 1
-  apply (regularToFinite b).injective
-  rw [LinearIsometryEquiv.apply_symm_apply]
-  simpa only [LinearIsometryEquiv.apply_symm_apply] using
-    regularToFinite_intertwines b μ a ((regularToFinite b).symm v)
-
-def secondFiniteRegularUnitary :
-    PiLp 2 (fun _ : (G ⧸ C.domain) × J => lp (fun _ : H => ℂ) 2) ≃ₗᵢ[ℂ]
-      lp (fun _ : (G × H) ⧸ C.graph => V) 2 :=
-  (regularToFinite b).symm.trans
-    (secondRegularUnitary C (chooseCosetSection C.domain) μ ω B)
-
-theorem secondFiniteRegularUnitary_intertwines (a : H)
-    (v : PiLp 2 (fun _ : (G ⧸ C.domain) × J => lp (fun _ : H => ℂ) 2)) :
-    B.action (1,a) (secondFiniteRegularUnitary C μ ω B b v) =
-      secondFiniteRegularUnitary C μ ω B b
-        (amplify (scalarRegular (inverseCocycle ω) a).toContinuousLinearEquiv.toContinuousLinearMap v) := by
-  change B.action (1,a) (secondRegularUnitary C _ μ ω B ((regularToFinite b).symm v)) =
-    secondRegularUnitary C _ μ ω B ((regularToFinite b).symm _)
-  rw [secondRegularUnitary_intertwines]
-  congr 1
-  apply (regularToFinite b).injective
-  rw [LinearIsometryEquiv.apply_symm_apply]
-  simpa only [LinearIsometryEquiv.apply_symm_apply] using
-    regularToFinite_intertwines b (inverseCocycle ω) a ((regularToFinite b).symm v)
-
-def firstExtension : (lp (fun _ : G => ℂ) 2 →L[ℂ] lp (fun _ : G => ℂ) 2) →⋆ₐ[ℂ]
-    (lp (fun _ : (G × H) ⧸ C.graph => V) 2 →L[ℂ]
-      lp (fun _ : (G × H) ⧸ C.graph => V) 2) :=
-  (firstFiniteRegularUnitary C μ ω B b).conjStarAlgEquiv.toStarAlgHom.comp
-    (amplification (K := (H ⧸ C.codomain) × J))
-
-def secondExtension : (lp (fun _ : H => ℂ) 2 →L[ℂ] lp (fun _ : H => ℂ) 2) →⋆ₐ[ℂ]
-    (lp (fun _ : (G × H) ⧸ C.graph => V) 2 →L[ℂ]
-      lp (fun _ : (G × H) ⧸ C.graph => V) 2) :=
-  (secondFiniteRegularUnitary C μ ω B b).conjStarAlgEquiv.toStarAlgHom.comp
-    (amplification (K := (G ⧸ C.domain) × J))
-
-theorem firstExtension_generator (a : G) :
-    firstExtension C μ ω B b (scalarRegular μ a).toContinuousLinearEquiv.toContinuousLinearMap =
-      (B.action (a,1)).toContinuousLinearEquiv.toContinuousLinearMap := by
-  apply ContinuousLinearMap.ext
-  intro v
-  change firstFiniteRegularUnitary C μ ω B b
-    (amplify _ ((firstFiniteRegularUnitary C μ ω B b).symm v)) = B.action (a,1) v
-  rw [← firstFiniteRegularUnitary_intertwines, LinearIsometryEquiv.apply_symm_apply]
-
-theorem secondExtension_generator (a : H) :
-    secondExtension C μ ω B b
-      (scalarRegular (inverseCocycle ω) a).toContinuousLinearEquiv.toContinuousLinearMap =
-      (B.action (1,a)).toContinuousLinearEquiv.toContinuousLinearMap := by
-  apply ContinuousLinearMap.ext
-  intro v
-  change secondFiniteRegularUnitary C μ ω B b
-    (amplify _ ((secondFiniteRegularUnitary C μ ω B b).symm v)) = B.action (1,a) v
-  rw [← secondFiniteRegularUnitary_intertwines, LinearIsometryEquiv.apply_symm_apply]
-
-theorem firstExtension_wot_continuous :
-    Continuous (fun T : lp (fun _ : G => ℂ) 2 →WOT[ℂ] lp (fun _ : G => ℂ) 2 =>
-      ContinuousLinearMapWOT.ofCLM (firstExtension C μ ω B b T.toCLM)) :=
-  (unitaryConjugation_wot_continuous (firstFiniteRegularUnitary C μ ω B b)).comp
-    (amplification_wot_continuous (K := (H ⧸ C.codomain) × J))
-
-theorem secondExtension_wot_continuous :
-    Continuous (fun T : lp (fun _ : H => ℂ) 2 →WOT[ℂ] lp (fun _ : H => ℂ) 2 =>
-      ContinuousLinearMapWOT.ofCLM (secondExtension C μ ω B b T.toCLM)) :=
-  (unitaryConjugation_wot_continuous (secondFiniteRegularUnitary C μ ω B b)).comp
-    (amplification_wot_continuous (K := (G ⧸ C.domain) × J))
-
-end GraphNormality
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.GraphModels.Bundles
+
+namespace OAI
+
+noncomputable section
+
+/-! Regular coordinates and finite amplification of graph models. -/
+
+namespace GraphNormality
+section
+open ArithmeticGerms GermBundles
+variable {G H : Type*} [Group G] [Group H]
+variable (C : VirtualGraph G H) (r : CosetSection C.codomain)
+
+def firstCoordinateMap (x : (H ⧸ C.codomain) × G) : (G × H) ⧸ C.graph :=
+  QuotientGroup.mk (x.2, r.repr x.1)
+
+theorem firstCoordinateMap_injective : Function.Injective (firstCoordinateMap C r) := by
+  rintro ⟨i, g⟩ ⟨j, g'⟩ he
+  have hm : (g⁻¹ * g', (r.repr i)⁻¹ * r.repr j) ∈ C.graph :=
+    QuotientGroup.eq.mp he
+  have hb : (r.repr i)⁻¹ * r.repr j ∈ C.codomain :=
+    ⟨_, hm, rfl⟩
+  have hij : i = j := by
+    rw [← r.repr_spec i, ← r.repr_spec j]
+    exact QuotientGroup.eq.mpr hb
+  subst j
+  have hg : g⁻¹ * g' = 1 := by
+    have h := C.snd_injective (a₁ := ⟨_, hm⟩) (a₂ := 1) (by simp)
+    exact congrArg (fun x : C.graph => x.val.1) h
+  exact Prod.ext rfl (inv_mul_eq_one.mp hg)
+
+theorem firstCoordinateMap_surjective : Function.Surjective (firstCoordinateMap C r) := by
+  intro x
+  induction x using QuotientGroup.induction_on with
+  | H x =>
+    let q : H ⧸ C.codomain := QuotientGroup.mk x.2
+    have hb : (r.repr q)⁻¹ * x.2 ∈ C.codomain :=
+      QuotientGroup.eq.mp (r.repr_spec q)
+    obtain ⟨c, hc, he⟩ := hb
+    refine ⟨(q, x.1 * c.1⁻¹), ?_⟩
+    apply QuotientGroup.eq.mpr
+    change (((x.1 * c.1⁻¹)⁻¹ * x.1), (r.repr q)⁻¹ * x.2) ∈ C.graph
+    have hf : (x.1 * c.1⁻¹)⁻¹ * x.1 = c.1 := by group
+    rw [hf, ← he]
+    exact hc
+
+def firstCoordinates : ((H ⧸ C.codomain) × G) ≃ ((G × H) ⧸ C.graph) :=
+  Equiv.ofBijective (firstCoordinateMap C r)
+    ⟨firstCoordinateMap_injective C r, firstCoordinateMap_surjective C r⟩
+
+theorem firstCoordinates_smul (a : G) (x : (H ⧸ C.codomain) × G) :
+    (a, (1 : H)) • firstCoordinates C r x = firstCoordinates C r (x.1, a * x.2) := by
+  change QuotientGroup.mk ((a, (1 : H)) * (x.2, r.repr x.1)) =
+    QuotientGroup.mk (a * x.2, r.repr x.1)
+  simp
+
+end
+
+section
+section
+attribute [local instance] Classical.propDecidable
+
+local instance (priority := 2000) (X : Type*) : DecidableEq X := Classical.decEq X
+
+section Reindex
+variable {X Y V W : Type*}
+variable [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+variable [NormedAddCommGroup W] [InnerProductSpace ℂ W]
+
+def transportLp (e : X ≃ Y) (u : X → V ≃ₗᵢ[ℂ] W) :
+    lp (fun _ : X => V) 2 ≃ₗᵢ[ℂ] lp (fun _ : Y => W) 2 where
+  toLinearEquiv :=
+    { toFun := fun v => ⟨fun y => u (e.symm y) (v (e.symm y)), by
+        apply memℓp_gen
+        convert! ((lp.memℓp v).summable (by norm_num)).comp_injective e.symm.injective using 1
+        simp only [LinearIsometryEquiv.norm_map]; rfl⟩
+      invFun := fun v => ⟨fun x => (u x).symm (v (e x)), by
+        apply memℓp_gen
+        convert! ((lp.memℓp v).summable (by norm_num)).comp_injective e.injective using 1
+        simp only [LinearIsometryEquiv.norm_map]; rfl⟩
+      left_inv := by
+        intro v
+        apply lp.ext
+        funext x
+        change (u x).symm (u (e.symm (e x)) (v (e.symm (e x)))) = _
+        simp
+      right_inv := by
+        intro v
+        apply lp.ext
+        funext y
+        change u (e.symm y) ((u (e.symm y)).symm (v (e (e.symm y)))) = _
+        simp
+      map_add' := by
+        intro v w
+        apply lp.ext
+        funext y
+        exact map_add _ _ _
+      map_smul' := by
+        intro a v
+        apply lp.ext
+        funext y
+        exact map_smul _ _ _ }
+  norm_map' := by
+    intro v
+    rw [lp.norm_eq_tsum_rpow (by norm_num), lp.norm_eq_tsum_rpow (by norm_num)]
+    congr 1
+    change (∑' y, ‖u (e.symm y) (v (e.symm y))‖ ^ (2 : ENNReal).toReal) = _
+    simp only [LinearIsometryEquiv.norm_map]
+    exact e.symm.tsum_eq (fun x => ‖v x‖ ^ (2 : ENNReal).toReal)
+
+theorem transportLp_apply (e : X ≃ Y) (u : X → V ≃ₗᵢ[ℂ] W)
+    (v : lp (fun _ : X => V) 2) (x : X) :
+    transportLp e u v (e x) = u x (v x) := by
+  change u (e.symm (e x)) (v (e.symm (e x))) = _
+  simp
+
+theorem transportLp_single (e : X ≃ Y) (u : X → V ≃ₗᵢ[ℂ] W)
+    (x : X) (v : V) :
+    transportLp e u (lp.single 2 x v) = lp.single 2 (e x) (u x v) := by
+  apply lp.ext
+  funext y
+  by_cases hy : y = e x
+  · subst y
+    rw [transportLp_apply, lp.single_apply_self, lp.single_apply_self]
+  · have hxy : e.symm y ≠ x := by
+      intro h
+      exact hy ((e.apply_symm_apply y).symm.trans (congrArg e h))
+    change u (e.symm y) ((lp.single 2 x v : lp (fun _ : X => V) 2) (e.symm y)) = _
+    rw [lp.single_apply_ne (E := fun _ : X => V) 2 x v hxy, map_zero,
+      lp.single_apply_ne (E := fun _ : Y => W) 2 (e x) _ hy]
+end Reindex
+
+open ArithmeticGerms GermBundles
+variable {G H I V : Type*} [Group G] [Group H]
+variable [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+
+def prodRatio (μ : ScalarCocycle G) (ω : ScalarCocycle H) : ScalarCocycle (G × H) where
+  val g h := μ.val g.1 h.1 / ω.val g.2 h.2
+  mul_assoc g h j := by
+    dsimp
+    rw [div_mul_div_comm, div_mul_div_comm, μ.mul_assoc, ω.mul_assoc]
+  one_left g := by simp [μ.one_left, ω.one_left]
+  one_right g := by simp [μ.one_right, ω.one_right]
+
+def regularPerm (a : G) : I × G ≃ I × G where
+  toFun x := (x.1, a * x.2)
+  invFun x := (x.1, a⁻¹ * x.2)
+  left_inv x := by simp
+  right_inv x := by simp
+
+def regularAction (μ : ScalarCocycle G) (a : G) :
+    lp (fun _ : I × G => V) 2 ≃ₗᵢ[ℂ] lp (fun _ : I × G => V) 2 :=
+  transportLp (regularPerm a) (fun x => circleUnitary (μ.val a x.2))
+
+theorem regularAction_single (μ : ScalarCocycle G) (a : G) (i : I) (g : G) (v : V) :
+    regularAction μ a (lp.single 2 (i,g) v) =
+      lp.single 2 (i,a*g) ((μ.val a g : ℂ) • v) := by
+  exact transportLp_single (regularPerm (I := I) a)
+    (fun x => circleUnitary (V := V) (μ.val a x.2)) (i, g) v
+
+variable (C : VirtualGraph G H) (r : CosetSection C.codomain)
+variable (μ : ScalarCocycle G) (ω : ScalarCocycle H)
+variable (B : ProjectiveBundle (prodRatio μ ω) (fun _ : (G × H) ⧸ C.graph => V))
+
+def firstRegularUnitary :
+    lp (fun _ : (H ⧸ C.codomain) × G => V) 2 ≃ₗᵢ[ℂ]
+      lp (fun _ : (G × H) ⧸ C.graph => V) 2 :=
+  transportLp (firstCoordinates C r) (fun x =>
+    B.fibre (x.2, 1) (firstCoordinates C r (x.1, 1)) (firstCoordinates C r x) (by
+      simpa using firstCoordinates_smul C r x.2 (x.1, 1)))
+
+theorem firstRegularUnitary_single (i : H ⧸ C.codomain) (g : G) (v : V) :
+    firstRegularUnitary C r μ ω B (lp.single 2 (i,g) v) =
+      B.action (g,1) (lp.single 2 (firstCoordinates C r (i,1)) v) := by
+  rw [firstRegularUnitary, transportLp_single]
+  exact (B.action_single (g,1) _ _ _ v).symm
+
+theorem firstRegularUnitary_intertwines (a : G)
+    (v : lp (fun _ : (H ⧸ C.codomain) × G => V) 2) :
+    B.action (a,1) (firstRegularUnitary C r μ ω B v) =
+      firstRegularUnitary C r μ ω B (regularAction μ a v) := by
+  have he :
+      (B.action (a,1)).toContinuousLinearEquiv.toContinuousLinearMap.comp
+        (firstRegularUnitary C r μ ω B).toContinuousLinearEquiv.toContinuousLinearMap =
+      (firstRegularUnitary C r μ ω B).toContinuousLinearEquiv.toContinuousLinearMap.comp
+        (regularAction μ a).toContinuousLinearEquiv.toContinuousLinearMap := by
+    apply lp.ext_continuousLinearMap (by norm_num)
+    rintro ⟨i,g⟩
+    apply ContinuousLinearMap.ext
+    intro w
+    change B.action (a,1) (firstRegularUnitary C r μ ω B (lp.single 2 (i,g) w)) =
+      firstRegularUnitary C r μ ω B (regularAction μ a (lp.single 2 (i,g) w))
+    rw [firstRegularUnitary_single, B.action_mul, regularAction_single,
+      firstRegularUnitary_single, lp.single_smul, map_smul]
+    simp [prodRatio, ω.one_left]
+  exact DFunLike.congr_fun he v
+
+end
+
+open ArithmeticGerms GermBundles
+local instance (priority := 2000) (X : Type*) : DecidableEq X := Classical.decEq X
+
+variable {G H V : Type*} [Group G] [Group H]
+variable [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+variable (C : VirtualGraph G H) (r : CosetSection C.domain)
+
+def secondCoordinateMap (x : (G ⧸ C.domain) × H) : (G × H) ⧸ C.graph :=
+  QuotientGroup.mk (r.repr x.1, x.2)
+
+theorem secondCoordinateMap_injective : Function.Injective (secondCoordinateMap C r) := by
+  rintro ⟨i, h⟩ ⟨j, h'⟩ he
+  have hm : ((r.repr i)⁻¹ * r.repr j, h⁻¹ * h') ∈ C.graph :=
+    QuotientGroup.eq.mp he
+  have hb : (r.repr i)⁻¹ * r.repr j ∈ C.domain := ⟨_, hm, rfl⟩
+  have hij : i = j := by
+    rw [← r.repr_spec i, ← r.repr_spec j]
+    exact QuotientGroup.eq.mpr hb
+  subst j
+  have hh : h⁻¹ * h' = 1 := by
+    have hh := C.fst_injective (a₁ := ⟨_, hm⟩) (a₂ := 1) (by simp)
+    exact congrArg (fun x : C.graph => x.val.2) hh
+  exact Prod.ext rfl (inv_mul_eq_one.mp hh)
+
+theorem secondCoordinateMap_surjective : Function.Surjective (secondCoordinateMap C r) := by
+  intro x
+  induction x using QuotientGroup.induction_on with
+  | H x =>
+    let q : G ⧸ C.domain := QuotientGroup.mk x.1
+    have hb : (r.repr q)⁻¹ * x.1 ∈ C.domain :=
+      QuotientGroup.eq.mp (r.repr_spec q)
+    obtain ⟨c, hc, he⟩ := hb
+    refine ⟨(q, x.2 * c.2⁻¹), ?_⟩
+    apply QuotientGroup.eq.mpr
+    change ((r.repr q)⁻¹ * x.1, ((x.2 * c.2⁻¹)⁻¹ * x.2)) ∈ C.graph
+    have hf : (x.2 * c.2⁻¹)⁻¹ * x.2 = c.2 := by group
+    rw [hf, ← he]
+    exact hc
+
+def secondCoordinates : ((G ⧸ C.domain) × H) ≃ ((G × H) ⧸ C.graph) :=
+  Equiv.ofBijective (secondCoordinateMap C r)
+    ⟨secondCoordinateMap_injective C r, secondCoordinateMap_surjective C r⟩
+
+theorem secondCoordinates_smul (a : H) (x : (G ⧸ C.domain) × H) :
+    ((1 : G), a) • secondCoordinates C r x = secondCoordinates C r (x.1, a * x.2) := by
+  change QuotientGroup.mk (((1 : G), a) * (r.repr x.1, x.2)) =
+    QuotientGroup.mk (r.repr x.1, a * x.2)
+  simp
+
+def inverseCocycle (ω : ScalarCocycle H) : ScalarCocycle H where
+  val h k := (ω.val h k)⁻¹
+  mul_assoc h k l := by
+    simpa only [mul_inv] using congrArg Inv.inv (ω.mul_assoc h k l)
+  one_left h := by simp [ω.one_left]
+  one_right h := by simp [ω.one_right]
+
+variable (μ : ScalarCocycle G) (ω : ScalarCocycle H)
+variable (B : ProjectiveBundle (prodRatio μ ω) (fun _ : (G × H) ⧸ C.graph => V))
+
+def secondRegularUnitary :
+    lp (fun _ : (G ⧸ C.domain) × H => V) 2 ≃ₗᵢ[ℂ]
+      lp (fun _ : (G × H) ⧸ C.graph => V) 2 :=
+  transportLp (secondCoordinates C r) (fun x =>
+    B.fibre (1, x.2) (secondCoordinates C r (x.1, 1)) (secondCoordinates C r x) (by
+      simpa using secondCoordinates_smul C r x.2 (x.1, 1)))
+
+theorem secondRegularUnitary_single (i : G ⧸ C.domain) (h : H) (v : V) :
+    secondRegularUnitary C r μ ω B (lp.single 2 (i,h) v) =
+      B.action (1,h) (lp.single 2 (secondCoordinates C r (i,1)) v) := by
+  rw [secondRegularUnitary, transportLp_single]
+  exact (B.action_single (1,h) _ _ _ v).symm
+
+theorem secondRegularUnitary_intertwines (a : H)
+    (v : lp (fun _ : (G ⧸ C.domain) × H => V) 2) :
+    B.action (1,a) (secondRegularUnitary C r μ ω B v) =
+      secondRegularUnitary C r μ ω B (regularAction (inverseCocycle ω) a v) := by
+  have he :
+      (B.action (1,a)).toContinuousLinearEquiv.toContinuousLinearMap.comp
+        (secondRegularUnitary C r μ ω B).toContinuousLinearEquiv.toContinuousLinearMap =
+      (secondRegularUnitary C r μ ω B).toContinuousLinearEquiv.toContinuousLinearMap.comp
+        (regularAction (inverseCocycle ω) a).toContinuousLinearEquiv.toContinuousLinearMap := by
+    apply lp.ext_continuousLinearMap (by norm_num)
+    rintro ⟨i,h⟩
+    apply ContinuousLinearMap.ext
+    intro w
+    change B.action (1,a) (secondRegularUnitary C r μ ω B (lp.single 2 (i,h) w)) =
+      secondRegularUnitary C r μ ω B (regularAction (inverseCocycle ω) a (lp.single 2 (i,h) w))
+    rw [secondRegularUnitary_single, B.action_mul, regularAction_single,
+      secondRegularUnitary_single, lp.single_smul, map_smul]
+    simp [prodRatio, inverseCocycle, μ.one_left]
+  exact DFunLike.congr_fun he v
+
+theorem coordinate_actions_commute (g : G) (h : H)
+    (v : lp (fun _ : (G × H) ⧸ C.graph => V) 2) :
+    B.action (g,1) (B.action (1,h) v) = B.action (1,h) (B.action (g,1) v) := by
+  rw [B.action_mul, B.action_mul]
+  simp [prodRatio, μ.one_left, μ.one_right, ω.one_left, ω.one_right]
+
+end
+
+section
+open GermBundles
+open scoped InnerProductSpace
+local instance (priority := 2000) (X : Type*) : DecidableEq X := Classical.decEq X
+
+section ScalarMultiplicity
+variable {I J G V : Type*} [Fintype J]
+variable [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
+variable (b : OrthonormalBasis J ℂ V)
+
+def bundleBasisVector (x : (I × J) × G) : lp (fun _ : I × G => V) 2 :=
+  lp.single 2 (x.1.1, x.2) (b x.1.2)
+
+omit [CompleteSpace V] in
+theorem bundleBasisVector_orthonormal : Orthonormal ℂ (bundleBasisVector (I := I) (G := G) b) := by
+  rw [orthonormal_iff_ite]
+  rintro ⟨⟨i,j⟩,g⟩ ⟨⟨k,l⟩,h⟩
+  change ⟪lp.single (E := fun _ : I × G => V) 2 (i,g) (b j),
+    lp.single (E := fun _ : I × G => V) 2 (k,h) (b l)⟫_ℂ = _
+  rw [lp.inner_single_left]
+  by_cases hi : i = k
+  · subst k
+    by_cases hg : g = h
+    · subst h
+      rw [lp.single_apply_self]
+      simpa only [Prod.mk.injEq, true_and, and_true] using
+        (orthonormal_iff_ite.mp b.orthonormal j l)
+    · rw [lp.single_apply_ne (E := fun _ : I × G => V) 2 (i,h) (b l)
+          (fun hh => hg (congrArg Prod.snd hh))]
+      simp [hg]
+  · rw [lp.single_apply_ne (E := fun _ : I × G => V) 2 (k,h) (b l)
+        (fun hh => hi (congrArg Prod.fst hh))]
+    simp [hi]
+
+omit [CompleteSpace V] in
+theorem bundleBasisVector_total :
+    (Submodule.span ℂ (Set.range (bundleBasisVector (I := I) (G := G) b)))ᗮ = ⊥ := by
+  apply le_antisymm _ bot_le
+  intro v hv
+  change v = 0
+  apply lp.ext
+  funext x
+  change v x = 0
+  apply b.repr.injective
+  ext j
+  rw [b.repr_apply_apply, map_zero]
+  have h := Submodule.inner_right_of_mem_orthogonal
+    (Submodule.subset_span (Set.mem_range_self ((x.1,j),x.2))) hv
+  simpa only [bundleBasisVector, Prod.eta, lp.inner_single_left, PiLp.zero_apply] using h
+
+def regularHilbertBasis : HilbertBasis ((I × J) × G) ℂ (lp (fun _ : I × G => V) 2) :=
+  HilbertBasis.mkOfOrthogonalEqBot (bundleBasisVector_orthonormal b) (bundleBasisVector_total b)
+
+@[simp] theorem regularHilbertBasis_apply (x : (I × J) × G) :
+    regularHilbertBasis b x = bundleBasisVector b x := by
+  exact congrFun (HilbertBasis.coe_mkOfOrthogonalEqBot _ _) x
+
+def scalarization : lp (fun _ : I × G => V) 2 ≃ₗᵢ[ℂ] lp (fun _ : (I × J) × G => ℂ) 2 :=
+  (regularHilbertBasis b).repr
+
+theorem scalarization_apply (v : lp (fun _ : I × G => V) 2) (i : I) (j : J) (g : G) :
+    scalarization b v ((i,j),g) = ⟪b j, v (i,g)⟫_ℂ := by
+  change (regularHilbertBasis b).repr v ((i,j),g) = _
+  rw [HilbertBasis.repr_apply_apply, regularHilbertBasis_apply]
+  exact lp.inner_single_left _ _ _
+
+end ScalarMultiplicity
+
+section ScalarRegular
+variable {I G V : Type*} [Group G]
+variable [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+
+theorem regularAction_apply (μ : ScalarCocycle G) (a : G)
+    (v : lp (fun _ : I × G => V) 2) (i : I) (g : G) :
+    regularAction μ a v (i,a*g) = (μ.val a g : ℂ) • v (i,g) := by
+  exact transportLp_apply (regularPerm a) (fun x => circleUnitary (μ.val a x.2)) v (i,g)
+
+end ScalarRegular
+
+variable {I J G V : Type*} [Fintype J] [Group G]
+variable [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
+variable (b : OrthonormalBasis J ℂ V)
+
+theorem scalarization_intertwines (μ : ScalarCocycle G) (a : G)
+    (v : lp (fun _ : I × G => V) 2) :
+    scalarization b (regularAction μ a v) = regularAction μ a (scalarization b v) := by
+  apply lp.ext
+  funext x
+  obtain ⟨y,rfl⟩ := (regularPerm (I := I × J) a).surjective x
+  rcases y with ⟨⟨i,j⟩,g⟩
+  change scalarization b (regularAction μ a v) ((i,j),a*g) =
+    regularAction μ a (scalarization b v) ((i,j),a*g)
+  rw [scalarization_apply, regularAction_apply, inner_smul_right,
+    regularAction_apply, scalarization_apply]
+  rfl
+
+end
+
+section
+open scoped InnerProductSpace
+local instance (priority := 2000) (X : Type*) : DecidableEq X := Classical.decEq X
+
+variable {K G : Type*} [Fintype K]
+
+def finiteSumBasisVector (x : K × G) : PiLp 2 (fun _ : K => lp (fun _ : G => ℂ) 2) :=
+  PiLp.single 2 x.1 (lp.single 2 x.2 1)
+
+theorem finiteSumBasisVector_inner (x : K × G)
+    (v : PiLp 2 (fun _ : K => lp (fun _ : G => ℂ) 2)) :
+    ⟪finiteSumBasisVector x, v⟫_ℂ = v x.1 x.2 := by
+  rw [PiLp.inner_apply, Finset.sum_eq_single x.1]
+  · simp [finiteSumBasisVector, lp.inner_single_left]
+  · intro i hi hix
+    simp [finiteSumBasisVector, hix]
+  · simp
+
+theorem finiteSumBasisVector_orthonormal :
+    Orthonormal ℂ (finiteSumBasisVector (K := K) (G := G)) := by
+  rw [orthonormal_iff_ite]
+  rintro ⟨i,g⟩ ⟨j,h⟩
+  rw [finiteSumBasisVector_inner]
+  by_cases hi : i = j
+  · subst j
+    simp [finiteSumBasisVector, lp.single_apply, Pi.single_apply]
+  · simp [finiteSumBasisVector, hi]
+
+theorem finiteSumBasisVector_total :
+    (Submodule.span ℂ (Set.range (finiteSumBasisVector (K := K) (G := G))))ᗮ = ⊥ := by
+  apply le_antisymm _ bot_le
+  intro v hv
+  change v = 0
+  ext i g
+  change v i g = 0
+  have h := Submodule.inner_right_of_mem_orthogonal
+    (Submodule.subset_span (Set.mem_range_self (i,g))) hv
+  simpa only [finiteSumBasisVector_inner] using h
+
+def finiteSumHilbertBasis :
+    HilbertBasis (K × G) ℂ (PiLp 2 (fun _ : K => lp (fun _ : G => ℂ) 2)) :=
+  HilbertBasis.mkOfOrthogonalEqBot finiteSumBasisVector_orthonormal finiteSumBasisVector_total
+
+@[simp] theorem finiteSumHilbertBasis_apply (x : K × G) :
+    finiteSumHilbertBasis x = finiteSumBasisVector x := by
+  exact congrFun (HilbertBasis.coe_mkOfOrthogonalEqBot _ _) x
+
+def finiteRegrouping : lp (fun _ : K × G => ℂ) 2 ≃ₗᵢ[ℂ]
+    PiLp 2 (fun _ : K => lp (fun _ : G => ℂ) 2) :=
+  finiteSumHilbertBasis.repr.symm
+
+theorem finiteRegrouping_apply (v : lp (fun _ : K × G => ℂ) 2) (i : K) (g : G) :
+    finiteRegrouping v i g = v (i,g) := by
+  have h := congrArg (fun f : lp (fun _ : K × G => ℂ) 2 => f (i,g))
+    (finiteSumHilbertBasis.repr.apply_symm_apply v)
+  rw [HilbertBasis.repr_apply_apply, finiteSumHilbertBasis_apply,
+    finiteSumBasisVector_inner] at h
+  exact h
+
+end
+
+section
+open scoped InnerProductSpace
+variable {K L : Type*} [Fintype K]
+variable [NormedAddCommGroup L] [InnerProductSpace ℂ L] [CompleteSpace L]
+
+def amplify (T : L →L[ℂ] L) :
+    PiLp 2 (fun _ : K => L) →L[ℂ] PiLp 2 (fun _ : K => L) :=
+  (PiLp.continuousLinearEquiv 2 ℂ (fun _ : K => L)).symm.toContinuousLinearMap.comp
+    ((ContinuousLinearMap.pi (fun i : K => T.comp (ContinuousLinearMap.proj i))).comp
+      (PiLp.continuousLinearEquiv 2 ℂ (fun _ : K => L)).toContinuousLinearMap)
+
+omit [Fintype K] [CompleteSpace L] in
+@[simp] theorem amplify_apply (T : L →L[ℂ] L) (v : PiLp 2 (fun _ : K => L)) (i : K) :
+    amplify T v i = T (v i) := rfl
+
+def amplification : (L →L[ℂ] L) →⋆ₐ[ℂ]
+    (PiLp 2 (fun _ : K => L) →L[ℂ] PiLp 2 (fun _ : K => L)) where
+  toFun := amplify
+  map_one' := by ext v i; rfl
+  map_mul' T S := by ext v i; rfl
+  map_add' T S := by ext v i; rfl
+  map_zero' := by ext v i; rfl
+  commutes' z := by ext v i; rfl
+  map_star' T := by
+    change amplify (K := K) (ContinuousLinearMap.adjoint T) =
+      ContinuousLinearMap.adjoint (amplify (K := K) T)
+    apply (ContinuousLinearMap.eq_adjoint_iff _ _).mpr
+    intro v w
+    simp only [PiLp.inner_apply, amplify_apply, ContinuousLinearMap.adjoint_inner_left]
+
+theorem amplification_wot_continuous :
+    Continuous (fun T : L →WOT[ℂ] L =>
+      ContinuousLinearMapWOT.ofCLM (amplification (K := K) (L := L) (ContinuousLinearMapWOT.toCLM T))) := by
+  apply ContinuousLinearMapWOT.continuous_iff.mpr
+  intro v w
+  change Continuous (fun T : L →WOT[ℂ] L => ∑ i : K, ⟪w i, T (v i)⟫_ℂ)
+  apply continuous_finsetSum
+  intro i hi
+  exact ContinuousLinearMapWOT.continuous_iff.mp continuous_id (v i) (w i)
+
+end
+
+section
+open scoped InnerProductSpace
+local instance (priority := 2000) (X : Type*) : DecidableEq X := Classical.decEq X
+
+open GermBundles
+
+section ScalarTwistedRegular
+variable {G K : Type*} [Group G] [Fintype K]
+
+def scalarRegular (μ : ScalarCocycle G) (a : G) :
+    lp (fun _ : G => ℂ) 2 ≃ₗᵢ[ℂ] lp (fun _ : G => ℂ) 2 :=
+  transportLp (Equiv.mulLeft a) (fun g => circleUnitary (μ.val a g))
+
+theorem scalarRegular_apply (μ : ScalarCocycle G) (a : G)
+    (v : lp (fun _ : G => ℂ) 2) (g : G) :
+    scalarRegular μ a v (a*g) = (μ.val a g : ℂ) * v g := by
+  exact transportLp_apply (Equiv.mulLeft a) (fun g => circleUnitary (μ.val a g)) v g
+
+theorem finiteRegrouping_intertwines (μ : ScalarCocycle G) (a : G)
+    (v : lp (fun _ : K × G => ℂ) 2) :
+    finiteRegrouping (regularAction μ a v) =
+      amplify (scalarRegular μ a).toContinuousLinearEquiv.toContinuousLinearMap
+        (finiteRegrouping v) := by
+  ext i g
+  obtain ⟨h,rfl⟩ := (Equiv.mulLeft a).surjective g
+  change finiteRegrouping (regularAction μ a v) i (a*h) =
+    scalarRegular μ a (finiteRegrouping v i) (a*h)
+  rw [finiteRegrouping_apply, regularAction_apply, scalarRegular_apply,
+    finiteRegrouping_apply]
+  rfl
+
+end ScalarTwistedRegular
+
+variable {L W : Type*}
+variable [NormedAddCommGroup L] [InnerProductSpace ℂ L] [CompleteSpace L]
+variable [NormedAddCommGroup W] [InnerProductSpace ℂ W] [CompleteSpace W]
+
+theorem unitaryConjugation_wot_continuous (U : L ≃ₗᵢ[ℂ] W) :
+    Continuous (fun T : L →WOT[ℂ] L =>
+      ContinuousLinearMapWOT.ofCLM (U.conjStarAlgEquiv (ContinuousLinearMapWOT.toCLM T))) := by
+  apply ContinuousLinearMapWOT.continuous_iff.mpr
+  intro v w
+  have h : (fun T : L →WOT[ℂ] L =>
+      ⟪w, U.conjStarAlgEquiv (ContinuousLinearMapWOT.toCLM T) v⟫_ℂ) =
+      (fun T : L →WOT[ℂ] L => ⟪U.symm w, T (U.symm v)⟫_ℂ) := by
+    funext T
+    rw [LinearIsometryEquiv.conjStarAlgEquiv_apply_apply]
+    conv_lhs => lhs; rw [← U.apply_symm_apply w]
+    exact U.inner_map_map _ _
+  change Continuous (fun T : L →WOT[ℂ] L =>
+    ⟪w, U.conjStarAlgEquiv (ContinuousLinearMapWOT.toCLM T) v⟫_ℂ)
+  rw [h]
+  exact ContinuousLinearMapWOT.continuous_iff.mp continuous_id (U.symm v) (U.symm w)
+
+end
+
+open ArithmeticGerms GermBundles
+local instance (priority := 2000) (X : Type*) : DecidableEq X := Classical.decEq X
+
+section FiniteRegularModules
+variable {I J G V : Type*} [Fintype I] [Fintype J] [Group G]
+variable [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
+variable (b : OrthonormalBasis J ℂ V)
+
+def regularToFinite : lp (fun _ : I × G => V) 2 ≃ₗᵢ[ℂ]
+    PiLp 2 (fun _ : I × J => lp (fun _ : G => ℂ) 2) :=
+  (scalarization b).trans finiteRegrouping
+
+theorem regularToFinite_intertwines (μ : ScalarCocycle G) (a : G)
+    (v : lp (fun _ : I × G => V) 2) :
+    regularToFinite b (regularAction μ a v) =
+      amplify (scalarRegular μ a).toContinuousLinearEquiv.toContinuousLinearMap
+        (regularToFinite b v) := by
+  change finiteRegrouping (scalarization b (regularAction μ a v)) = _
+  rw [scalarization_intertwines, finiteRegrouping_intertwines]
+  rfl
+
+end FiniteRegularModules
+
+variable {G H V J : Type*} [Group G] [Group H] [Fintype J]
+variable [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
+variable (C : VirtualGraph G H)
+local instance : Fintype (G ⧸ C.domain) := Subgroup.fintypeQuotientOfFiniteIndex
+local instance : Fintype (H ⧸ C.codomain) := Subgroup.fintypeQuotientOfFiniteIndex
+
+variable (μ : ScalarCocycle G) (ω : ScalarCocycle H)
+variable (B : ProjectiveBundle (prodRatio μ ω) (fun _ : (G × H) ⧸ C.graph => V))
+variable (b : OrthonormalBasis J ℂ V)
+
+def firstFiniteRegularUnitary :
+    PiLp 2 (fun _ : (H ⧸ C.codomain) × J => lp (fun _ : G => ℂ) 2) ≃ₗᵢ[ℂ]
+      lp (fun _ : (G × H) ⧸ C.graph => V) 2 :=
+  (regularToFinite b).symm.trans
+    (firstRegularUnitary C (chooseCosetSection C.codomain) μ ω B)
+
+theorem firstFiniteRegularUnitary_intertwines (a : G)
+    (v : PiLp 2 (fun _ : (H ⧸ C.codomain) × J => lp (fun _ : G => ℂ) 2)) :
+    B.action (a,1) (firstFiniteRegularUnitary C μ ω B b v) =
+      firstFiniteRegularUnitary C μ ω B b
+        (amplify (scalarRegular μ a).toContinuousLinearEquiv.toContinuousLinearMap v) := by
+  change B.action (a,1) (firstRegularUnitary C _ μ ω B ((regularToFinite b).symm v)) =
+    firstRegularUnitary C _ μ ω B ((regularToFinite b).symm _)
+  rw [firstRegularUnitary_intertwines]
+  congr 1
+  apply (regularToFinite b).injective
+  rw [LinearIsometryEquiv.apply_symm_apply]
+  simpa only [LinearIsometryEquiv.apply_symm_apply] using
+    regularToFinite_intertwines b μ a ((regularToFinite b).symm v)
+
+def secondFiniteRegularUnitary :
+    PiLp 2 (fun _ : (G ⧸ C.domain) × J => lp (fun _ : H => ℂ) 2) ≃ₗᵢ[ℂ]
+      lp (fun _ : (G × H) ⧸ C.graph => V) 2 :=
+  (regularToFinite b).symm.trans
+    (secondRegularUnitary C (chooseCosetSection C.domain) μ ω B)
+
+theorem secondFiniteRegularUnitary_intertwines (a : H)
+    (v : PiLp 2 (fun _ : (G ⧸ C.domain) × J => lp (fun _ : H => ℂ) 2)) :
+    B.action (1,a) (secondFiniteRegularUnitary C μ ω B b v) =
+      secondFiniteRegularUnitary C μ ω B b
+        (amplify (scalarRegular (inverseCocycle ω) a).toContinuousLinearEquiv.toContinuousLinearMap v) := by
+  change B.action (1,a) (secondRegularUnitary C _ μ ω B ((regularToFinite b).symm v)) =
+    secondRegularUnitary C _ μ ω B ((regularToFinite b).symm _)
+  rw [secondRegularUnitary_intertwines]
+  congr 1
+  apply (regularToFinite b).injective
+  rw [LinearIsometryEquiv.apply_symm_apply]
+  simpa only [LinearIsometryEquiv.apply_symm_apply] using
+    regularToFinite_intertwines b (inverseCocycle ω) a ((regularToFinite b).symm v)
+
+def firstExtension : (lp (fun _ : G => ℂ) 2 →L[ℂ] lp (fun _ : G => ℂ) 2) →⋆ₐ[ℂ]
+    (lp (fun _ : (G × H) ⧸ C.graph => V) 2 →L[ℂ]
+      lp (fun _ : (G × H) ⧸ C.graph => V) 2) :=
+  (firstFiniteRegularUnitary C μ ω B b).conjStarAlgEquiv.toStarAlgHom.comp
+    (amplification (K := (H ⧸ C.codomain) × J))
+
+def secondExtension : (lp (fun _ : H => ℂ) 2 →L[ℂ] lp (fun _ : H => ℂ) 2) →⋆ₐ[ℂ]
+    (lp (fun _ : (G × H) ⧸ C.graph => V) 2 →L[ℂ]
+      lp (fun _ : (G × H) ⧸ C.graph => V) 2) :=
+  (secondFiniteRegularUnitary C μ ω B b).conjStarAlgEquiv.toStarAlgHom.comp
+    (amplification (K := (G ⧸ C.domain) × J))
+
+theorem firstExtension_generator (a : G) :
+    firstExtension C μ ω B b (scalarRegular μ a).toContinuousLinearEquiv.toContinuousLinearMap =
+      (B.action (a,1)).toContinuousLinearEquiv.toContinuousLinearMap := by
+  apply ContinuousLinearMap.ext
+  intro v
+  change firstFiniteRegularUnitary C μ ω B b
+    (amplify _ ((firstFiniteRegularUnitary C μ ω B b).symm v)) = B.action (a,1) v
+  rw [← firstFiniteRegularUnitary_intertwines, LinearIsometryEquiv.apply_symm_apply]
+
+theorem secondExtension_generator (a : H) :
+    secondExtension C μ ω B b
+      (scalarRegular (inverseCocycle ω) a).toContinuousLinearEquiv.toContinuousLinearMap =
+      (B.action (1,a)).toContinuousLinearEquiv.toContinuousLinearMap := by
+  apply ContinuousLinearMap.ext
+  intro v
+  change secondFiniteRegularUnitary C μ ω B b
+    (amplify _ ((secondFiniteRegularUnitary C μ ω B b).symm v)) = B.action (1,a) v
+  rw [← secondFiniteRegularUnitary_intertwines, LinearIsometryEquiv.apply_symm_apply]
+
+theorem firstExtension_wot_continuous :
+    Continuous (fun T : lp (fun _ : G => ℂ) 2 →WOT[ℂ] lp (fun _ : G => ℂ) 2 =>
+      ContinuousLinearMapWOT.ofCLM (firstExtension C μ ω B b T.toCLM)) :=
+  (unitaryConjugation_wot_continuous (firstFiniteRegularUnitary C μ ω B b)).comp
+    (amplification_wot_continuous (K := (H ⧸ C.codomain) × J))
+
+theorem secondExtension_wot_continuous :
+    Continuous (fun T : lp (fun _ : H => ℂ) 2 →WOT[ℂ] lp (fun _ : H => ℂ) 2 =>
+      ContinuousLinearMapWOT.ofCLM (secondExtension C μ ω B b T.toCLM)) :=
+  (unitaryConjugation_wot_continuous (secondFiniteRegularUnitary C μ ω B b)).comp
+    (amplification_wot_continuous (K := (G ⧸ C.domain) × J))
+
+end GraphNormality
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/KazhdanLusztig/BalancedPairings.lean b/lean/OAI/RepresentationTheory/KazhdanLusztig/BalancedPairings.lean
index 5c4d1d17ead09a398407e08a45911b1eae863489..ca248c00dc3ac5cf84b6311cfa4bd36a46da04d4 100644
--- a/lean/OAI/RepresentationTheory/KazhdanLusztig/BalancedPairings.lean
+++ b/lean/OAI/RepresentationTheory/KazhdanLusztig/BalancedPairings.lean
@@ -1,2111 +1,2111 @@
-import OAI.RepresentationTheory.KazhdanLusztig.IntegralPairings
-
-namespace OAI
-
-/-!
-Graded stalk duality, integral perfect balanced all-sections pairings, recognition and top grading.
--/
-
-section
-
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- theorem bottPairing_dualToSupport (l : List I) (x : W)
-    (lam : Dual (SymmetricCoefficient (M := M)) (bottStalk cs l x))
-    (m : BottSamelsonModule cs l) :
-    bottSamelsonDuality cs l (bottStalkDualToSupport cs l x lam).val m =
-      lam (bottStalkProjection cs l x m) := by
-  change bottSamelsonDuality cs l
-    ((bottSamelsonDuality cs l).symm ((bottStalkProjection cs l x).dualMap lam)) m = _
-  rw [LinearEquiv.apply_symm_apply]
-  rfl
-
- theorem bottStalkDuality_apply_projection (l : List I) (x : W)
-    (s : bottWeightSupported cs l x) (m : BottSamelsonModule cs l) :
-    bottStalkDuality cs l x s (bottStalkProjection cs l x m) =
-      bottSamelsonDuality cs l s.val m := by
-  have he : bottStalkDualToSupport cs l x (bottStalkDuality cs l x s)=s :=
-    (LinearEquiv.ofBijective (bottStalkDualToSupport cs l x)
-      (bottStalkDualToSupport_bijective cs l x)).apply_symm_apply s
-  calc
-    _ = bottSamelsonDuality cs l
-        (bottStalkDualToSupport cs l x (bottStalkDuality cs l x s)).val m :=
-      (bottPairing_dualToSupport cs l x _ m).symm
-    _ = _ := by rw [he]
-
- theorem bottStalkDuality_homogeneous (l : List I) (x : W) (n : ℤ)
-    (s : bottWeightSupported cs l x) (hs : s ∈ bottSupportPiece cs basis l x n)
-    (j : ℤ) (v : bottStalk cs l x) (hv : v ∈ bottStalkPiece cs basis l x j) :
-    bottStalkDuality cs l x s v ∈ symmetricGrading basis (n+j-l.length) := by
-  obtain ⟨m,hm,rfl⟩ := bottStalk_homogeneous_lift cs basis l x j v hv
-  rw [bottStalkDuality_apply_projection]
-  exact bottDuality_homogeneous cs basis l hs hm
-
- theorem bottDualToSupport_homogeneous (l : List I) (x : W) (r : ℤ)
-    (lam : Dual (SymmetricCoefficient (M := M)) (bottStalk cs l x))
-    (hlam : ∀ j v, v ∈ bottStalkPiece cs basis l x j →
-      lam v ∈ symmetricGrading basis (j+r)) :
-    bottStalkDualToSupport cs l x lam ∈ bottSupportPiece cs basis l x (r+l.length) := by
-  change (bottSamelsonDuality cs l).symm ((bottStalkProjection cs l x).dualMap lam) ∈
-    bottPiece cs basis l (r+l.length)
-  apply perfect_inverse_of_homogeneous_functional (symmetricGrading basis) (bottPiece cs basis l)
-    (bottSamelsonDuality cs l) l.length
-    (fun _ _ _ _ hm hn => bottDuality_homogeneous cs basis l hm hn) r
-  intro j m hm
-  exact hlam j _ (bottStalkProjection_homogeneous cs basis l x j m hm)
-
- theorem bottStalkDuality_homogeneous_iff (l : List I) (x : W) (n : ℤ)
-    (s : bottWeightSupported cs l x) :
-    (∀ j v, v ∈ bottStalkPiece cs basis l x j →
-      bottStalkDuality cs l x s v ∈ symmetricGrading basis (n+j-l.length)) ↔
-        s ∈ bottSupportPiece cs basis l x n := by
-  constructor
-  · intro h
-    have hh := bottDualToSupport_homogeneous cs basis l x (n-l.length)
-      (bottStalkDuality cs l x s) (by
-        intro j v hv
-        have he : n+j-(l.length:ℤ) = j+(n-l.length) := by omega
-        rw [← he]
-        exact h j v hv)
-    have he : bottStalkDualToSupport cs l x (bottStalkDuality cs l x s)=s :=
-      (LinearEquiv.ofBijective (bottStalkDualToSupport cs l x)
-        (bottStalkDualToSupport_bijective cs l x)).apply_symm_apply s
-    simpa only [he,sub_add_cancel] using hh
-  · intro h
-    exact bottStalkDuality_homogeneous cs basis l x n s h
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! The actual word adjoint preserves homogeneity. Projection kernels are
-annihilators of the actual supported lattices, integrally, with no localization
-or freeness assumption on the projection lattice. -/
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
- theorem bott_projection_zero_iff_annihilates_support (l : List I) (x : W)
-    (m : BottSamelsonModule cs l) :
-    bottStalkProjection cs l x m=0 ↔ ∀ s : bottWeightSupported cs l x,
-      bottSamelsonDuality cs l s.val m=0 := by
-  constructor
-  · intro hm s
-    exact bott_supported_annihilates cs l x s.property m
-      ((bottStalkProjection_eq_zero cs l x m).mp hm)
-  · intro h
-    apply Subtype.ext
-    funext e
-    let lam : Dual (SymmetricCoefficient (M := M)) (bottStalk cs l x) :=
-      (LinearMap.proj e).comp (bottStalk cs l x).subtype
-    have he := h (bottStalkDualToSupport cs l x lam)
-    rw [bottPairing_dualToSupport] at he
-    exact he
-
- def bottAdjoint (l : List I) (f : End (SymmetricCoefficient (M := M))
-    (BottSamelsonModule cs l)) := PerfectAdjoint.adjoint (bottSamelsonDuality cs l) f
-
- theorem bottAdjoint_preserves_kernel (l : List I) (x : W)
-    (f : End (SymmetricCoefficient (M := M)) (BottSamelsonModule cs l))
-    (hf : ∀ s, s ∈ bottWeightSupported cs l x → f s ∈ bottWeightSupported cs l x)
-    (m : BottSamelsonModule cs l) (hm : bottStalkProjection cs l x m=0) :
-    bottStalkProjection cs l x (bottAdjoint cs l f m)=0 := by
-  apply (bott_projection_zero_iff_annihilates_support cs l x _).mpr
-  intro s
-  change bottSamelsonDuality cs l s.val (PerfectAdjoint.adjoint (bottSamelsonDuality cs l) f m)=0
-  rw [PerfectAdjoint.symmetric_pairing (bottSamelsonDuality cs l)
-    (bottSamelson_symmetric cs l)]
-  exact (bott_projection_zero_iff_annihilates_support cs l x m).mp hm ⟨f s.val,hf s.val s.property⟩
-
- theorem bottAdjoint_preserves_support (l : List I) (x : W)
-    (f : End (SymmetricCoefficient (M := M)) (BottSamelsonModule cs l))
-    (hf : ∀ m, bottStalkProjection cs l x m=0 → bottStalkProjection cs l x (f m)=0)
-    (m : BottSamelsonModule cs l) (hm : m ∈ bottWeightSupported cs l x) :
-    bottAdjoint cs l f m ∈ bottWeightSupported cs l x := by
-  apply bott_annihilates_supported cs l x
-  intro n hn
-  change bottSamelsonDuality cs l (PerfectAdjoint.adjoint (bottSamelsonDuality cs l) f m) n=0
-  rw [PerfectAdjoint.pairing]
-  apply bott_supported_annihilates cs l x hm
-  exact (bottStalkProjection_eq_zero cs l x (f n)).mp
-    (hf n ((bottStalkProjection_eq_zero cs l x n).mpr hn))
-
- theorem bottAdjoint_homogeneous
-    {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-    (l : List I) (f : End (SymmetricCoefficient (M := M)) (BottSamelsonModule cs l))
-    (hf : ∀ n m, m ∈ bottPiece cs basis l n → f m ∈ bottPiece cs basis l n)
-    (n : ℤ) (m : BottSamelsonModule cs l) (hm : m ∈ bottPiece cs basis l n) :
-    bottAdjoint cs l f m ∈ bottPiece cs basis l n :=
-  adjoint_homogeneous (symmetricGrading basis) (bottPiece cs basis l)
-    (bottSamelsonDuality cs l) l.length
-    (fun _ _ _ _ h1 h2 => bottDuality_homogeneous cs basis l h1 h2) f hf n m hm
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Integral realization of tensor words as ALL sections of their actual
-pair-projection sheaf. This assertion alone is not a BMP summand theorem,
-flabbiness/standard-filtration theorem, or EW character theorem. -/
-namespace KLInvariance.TitsSpace
-open Module MomentGraph BruhatGraph FiniteCharacterLocalization
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
- def bottVertexProjection (l : List I) (u b : W) : BottSamelsonModule cs l →ₗ[
-    SymmetricCoefficient (M := M)]
-    (∀ x : Interval cs u b, bottStalk cs l x.val) :=
-  LinearMap.pi (fun x => bottStalkProjection cs l x.val)
-
- theorem bottVertexProjection_injective (l : List I) (hl : cs.IsReduced l) :
-    Function.Injective (bottVertexProjection cs l 1 (cs.wordProd l)) := by
-  intro m n h
-  apply bottSection_injective cs l hl
-  exact Subtype.ext h
-
- theorem bottVertexProjection_stable (l : List I) (u b : W)
-    (a : SymmetricCoefficient (M := M)) :
-    (fun x : Interval cs u b => coefficientAction cs x.val a) ∈
-      stabilizer (LinearMap.range (bottVertexProjection cs l u b)) := by
-  rintro z ⟨m,rfl⟩
-  refine ⟨bottSamelsonRight cs l a m,?_⟩
-  funext x
-  exact bottStalkProjection_right cs l x.val a m
-
- theorem bottVertexProjection_scaled (l : List I) (hl : cs.IsReduced l)
-    (z : ∀ x : Interval cs 1 (cs.wordProd l), bottStalk cs l x.val) :
-    bottCoordinateDenominator cs l • z ∈
-      LinearMap.range (bottVertexProjection cs l 1 (cs.wordProd l)) := by
-  let c (e : BottIndex l) : SymmetricCoefficient (M := M) :=
-    (z ⟨bottWeight cs l e,one_bruhat cs _,bottWeight_bruhat cs l hl e⟩).val ⟨e,rfl⟩
-  obtain ⟨m,hm⟩ := bottEvaluation_contains_scaled_coordinates cs l c
-  refine ⟨m,?_⟩
-  funext x
-  apply Subtype.ext
-  funext e
-  rcases x with ⟨x,hx⟩
-  rcases e with ⟨e,he⟩
-  change bottWeight cs l e=x at he
-  subst x
-  exact hm e
-
-
- theorem bottSection_pair_lift (l : List I) (u b : W)
-    (s : (bottProjectionSheaf cs l u b).sections Set.univ)
-    (x y : Interval cs u b) (t : W) (ht : cs.IsReflection t) (he : t*x.val=y.val) :
-    ∃ m : BottSamelsonModule cs l,
-      bottStalkProjection cs l x.val m=s.val x ∧ bottStalkProjection cs l y.val m=s.val y := by
-  have he' : t*y.val=x.val := by
-    rw [← he,← mul_assoc,ht.mul_self,one_mul]
-  have hne : cs.length y.val ≠ cs.length x.val := by
-    rw [← he]
-    exact ht.length_mul_right_ne x.val
-  rcases lt_or_gt_of_ne hne with hlt | hlt
-  · let e : Edge cs u b := ⟨(y,x),hlt,⟨t,ht,he'.symm⟩⟩
-    have hs := s.property e (Set.mem_univ _) (Set.mem_univ _)
-    obtain ⟨m,hm,hn⟩ := (PairProjection.compatible_iff _ _ _ _).mp hs
-    exact ⟨m,hn,hm⟩
-  · let e : Edge cs u b := ⟨(x,y),hlt,⟨t,ht,he.symm⟩⟩
-    have hs := s.property e (Set.mem_univ _) (Set.mem_univ _)
-    exact (PairProjection.compatible_iff _ _ _ _).mp hs
-
- theorem bottSection_prime_lift (l : List I) (u b : W)
-    (s : (bottProjectionSheaf cs l u b).sections Set.univ)
-    (p : SymmetricCoefficient (M := M)) (hp : Prime p) (x : Interval cs u b) :
-    ∃ m : BottSamelsonModule cs l, ∀ y : Interval cs u b,
-      (∀ a, p ∣ coefficientAction cs x.val a-coefficientAction cs y.val a) →
-      bottStalkProjection cs l y.val m=s.val y := by
-  classical
-  by_cases hx : ∃ y : Interval cs u b, x ≠ y ∧
-      ∀ a, p ∣ coefficientAction cs x.val a-coefficientAction cs y.val a
-  · obtain ⟨y,hxy,hcon⟩ := hx
-    obtain ⟨t,ht,he,hclass⟩ := prime_character_class cs p hp
-      (fun h => hxy (Subtype.ext h)) hcon
-    obtain ⟨m,hm,hn⟩ := bottSection_pair_lift cs l u b s x y t ht he.symm
-    refine ⟨m,?_⟩
-    intro z hz
-    rcases (hclass z.val).mp hz with hxz | hzy
-    · have h : z=x := Subtype.ext hxz.symm
-      subst z
-      exact hm
-    · have h : z=y := Subtype.ext (hzy.trans he.symm)
-      subst z
-      exact hn
-  · obtain ⟨m,hm⟩ := bottStalkProjection_surjective cs l x.val (s.val x)
-    refine ⟨m,?_⟩
-    intro y hy
-    have h : y=x := by
-      by_contra h
-      exact hx ⟨y,Ne.symm h,hy⟩
-    subst y
-    exact hm
-
- theorem bottSection_local_denominator (l : List I) (u b : W)
-    (s : (bottProjectionSheaf cs l u b).sections Set.univ)
-    (p : SymmetricCoefficient (M := M)) (hp : Prime p) :
-    ∃ a : SymmetricCoefficient (M := M), ¬ p ∣ a ∧ a • s.val ∈
-      LinearMap.range (bottVertexProjection cs l u b) := by
-  classical
-  let : Finite (Interval cs u b) := interval_finite cs u b
-  let : Fintype (Interval cs u b) := Fintype.ofFinite _
-  apply character_local_denominator (E := fun x : Interval cs u b => bottStalk cs l x.val)
-    (LinearMap.range (bottVertexProjection cs l u b)) p hp
-    (fun a x => coefficientAction cs x.val a) (fun a => bottVertexProjection_stable cs l u b a) s.val
-  intro x
-  obtain ⟨m,hm⟩ := bottSection_prime_lift cs l u b s p hp x
-  exact ⟨bottVertexProjection cs l u b m,⟨m,rfl⟩,hm⟩
-
- theorem bottSection_surjective (l : List I) (hl : cs.IsReduced l) :
-    Function.Surjective (bottSection cs l 1 (cs.wordProd l)) := by
-  intro s
-  let L := LinearMap.range (bottVertexProjection cs l 1 (cs.wordProd l))
-  let : Module.Free (SymmetricCoefficient (M := M)) L :=
-    Module.Free.of_equiv (LinearEquiv.ofInjective _ (bottVertexProjection_injective cs l hl))
-  let : UniqueFactorizationMonoid (SymmetricCoefficient (M := M)) :=
-    (SymmetricAlgebra.equivMvPolynomial (Module.Free.chooseBasis ℝ (Extended M))).symm.toRingEquiv.toMulEquiv.uniqueFactorizationMonoid inferInstance
-  have h : s.val ∈ L := EdgeLattice.mem_of_prime_local_membership L s.val
-    ⟨bottCoordinateDenominator cs l,bottCoordinateDenominator_ne_zero cs l,
-      bottVertexProjection_scaled cs l hl s.val⟩
-    (fun p hp => bottSection_local_denominator cs l 1 (cs.wordProd l) s p hp)
-  obtain ⟨m,hm⟩ := h
-  exact ⟨m,Subtype.ext hm⟩
-
- def bottSectionEquiv (l : List I) (hl : cs.IsReduced l) :
-    BottSamelsonModule cs l ≃ₗ[SymmetricCoefficient (M := M)]
-      (bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ :=
-  LinearEquiv.ofBijective (bottSection cs l 1 (cs.wordProd l))
-    ⟨bottSection_injective cs l hl,bottSection_surjective cs l hl⟩
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-namespace TransportPerfectPairing
-open Module
-variable {A M N : Type*} [CommRing A] [AddCommGroup M] [Module A M]
-  [AddCommGroup N] [Module A N]
-
-theorem balanced (e : M ≃ₗ[A] N) (D : M ≃ₗ[A] Dual A M) (a : End A M)
-    (h : ∀ x y, D (a x) y = D x (a y)) (s z : N) :
-    (e.symm.trans (D.trans e.symm.dualMap))
-      ((e.toLinearMap.comp (a.comp e.symm.toLinearMap)) s) z =
-    (e.symm.trans (D.trans e.symm.dualMap)) s
-      ((e.toLinearMap.comp (a.comp e.symm.toLinearMap)) z) := by
-  change D (e.symm (e (a (e.symm s)))) (e.symm z) =
-    D (e.symm s) (e.symm (e (a (e.symm z))))
-  simp only [LinearEquiv.symm_apply_apply]
-  exact h _ _
-
-theorem symmetric (e : M ≃ₗ[A] N) (D : M ≃ₗ[A] Dual A M)
-    (h : ∀ x y, D x y = D y x) (s z : N) :
-    (e.symm.trans (D.trans e.symm.dualMap)) s z =
-    (e.symm.trans (D.trans e.symm.dualMap)) z s :=
-  h (e.symm s) (e.symm z)
-
-end TransportPerfectPairing
-
-end
-
-
-section
-
-/-! Integral perfect balanced self-duality on all sections of the genuine
-word projection sheaf. This is the word object, NOT the indecomposable BMP
-object. Selection of a BMP summand with the required normalization is not
-made or assumed here. -/
-namespace KLInvariance.TitsSpace
-open Module MomentGraph BruhatGraph
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
- def bottSectionDuality (l : List I) (hl : cs.IsReduced l) :
-    (bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ ≃ₗ[
-      SymmetricCoefficient (M := M)]
-      Dual (SymmetricCoefficient (M := M))
-        ((bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ) :=
-  (bottSectionEquiv cs l hl).symm.trans
-    ((bottSamelsonDuality cs l).trans (bottSectionEquiv cs l hl).symm.dualMap)
-
- theorem bottSectionDuality_apply (l : List I) (hl : cs.IsReduced l)
-    (m n : BottSamelsonModule cs l) :
-    bottSectionDuality cs l hl (bottSection cs l 1 (cs.wordProd l) m)
-      (bottSection cs l 1 (cs.wordProd l) n) = bottSamelsonDuality cs l m n := by
-  change bottSamelsonDuality cs l
-    ((bottSectionEquiv cs l hl).symm ((bottSectionEquiv cs l hl) m))
-    ((bottSectionEquiv cs l hl).symm ((bottSectionEquiv cs l hl) n))= _
-  simp only [LinearEquiv.symm_apply_apply]
-
- def bottSectionRight (l : List I) (hl : cs.IsReduced l)
-    (a : SymmetricCoefficient (M := M)) :
-    End (SymmetricCoefficient (M := M))
-      ((bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ) :=
-  (bottSectionEquiv cs l hl).toLinearMap.comp
-    ((bottSamelsonRight cs l a).comp (bottSectionEquiv cs l hl).symm.toLinearMap)
-
- theorem bottSectionRight_apply (l : List I) (hl : cs.IsReduced l)
-    (a : SymmetricCoefficient (M := M)) (m : BottSamelsonModule cs l) :
-    bottSectionRight cs l hl a (bottSection cs l 1 (cs.wordProd l) m) =
-      bottSection cs l 1 (cs.wordProd l) (bottSamelsonRight cs l a m) := by
-  change (bottSectionEquiv cs l hl) (bottSamelsonRight cs l a
-    ((bottSectionEquiv cs l hl).symm ((bottSectionEquiv cs l hl) m))) = _
-  rw [LinearEquiv.symm_apply_apply]
-  rfl
-
- theorem bottSectionRight_coordinates (l : List I) (hl : cs.IsReduced l)
-    (a : SymmetricCoefficient (M := M))
-    (s : (bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ)
-    (x : Interval cs 1 (cs.wordProd l)) :
-    (bottSectionRight cs l hl a s).val x=coefficientAction cs x.val a • s.val x := by
-  obtain ⟨m,rfl⟩ := bottSection_surjective cs l hl s
-  rw [bottSectionRight_apply]
-  exact bottStalkProjection_right cs l x.val a m
-
- theorem bottSection_balanced (l : List I) (hl : cs.IsReduced l)
-    (a : SymmetricCoefficient (M := M))
-    (s z : (bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ) :
-    bottSectionDuality cs l hl (bottSectionRight cs l hl a s) z =
-      bottSectionDuality cs l hl s (bottSectionRight cs l hl a z) := by
-  exact TransportPerfectPairing.balanced (bottSectionEquiv cs l hl)
-    (bottSamelsonDuality cs l) (bottSamelsonRight cs l a)
-    (bottSamelson_balanced cs l a) s z
-
- theorem bottSection_symmetric (l : List I) (hl : cs.IsReduced l)
-    (s z : (bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ) :
-    bottSectionDuality cs l hl s z=bottSectionDuality cs l hl z s := by
-  exact TransportPerfectPairing.symmetric (bottSectionEquiv cs l hl)
-    (bottSamelsonDuality cs l) (bottSamelson_symmetric cs l) s z
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! The word SECTION pairing has precisely the word-length shift in the
-actual coordinatewise sheaf grading. It is not an asserted pairing on the
-indecomposable BMP boundary sheaf. -/
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- theorem bottPiece_iff_stalks (l : List I) (hl : cs.IsReduced l) (n : ℤ)
-    (m : BottSamelsonModule cs l) : m ∈ bottPiece cs basis l n ↔
-      ∀ x : Interval cs 1 (cs.wordProd l),
-        bottStalkProjection cs l x.val m ∈ bottStalkPiece cs basis l x.val n := by
-  constructor
-  · intro hm x
-    exact bottStalkProjection_homogeneous cs basis l x.val n m hm
-  · intro hm
-    have he : component (bottPiece cs basis l) n m=m := by
-      apply bottVertexProjection_injective cs l hl
-      funext x
-      have hh := map_component (bottPiece cs basis l) (bottStalkPiece cs basis l x.val)
-        (bottStalkProjection cs l x.val) 0 (by
-          simpa only [add_zero] using bottStalkProjection_homogeneous cs basis l x.val) n m
-      rw [sub_zero,show component (bottStalkPiece cs basis l x.val) n
-        (bottStalkProjection cs l x.val m)=bottStalkProjection cs l x.val m from
-        DirectSum.decompose_of_mem_same _ (hm x)] at hh
-      exact hh.symm
-    rw [← he]
-    exact (DirectSum.decompose (bottPiece cs basis l) m n).property
-
- theorem bottSectionDuality_homogeneous (l : List I) (hl : cs.IsReduced l)
-    (i j : ℤ)
-    (s z : (bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ)
-    (hs : ∀ x, s.val x ∈ bottStalkPiece cs basis l x.val i)
-    (hz : ∀ x, z.val x ∈ bottStalkPiece cs basis l x.val j) :
-    bottSectionDuality cs l hl s z ∈ symmetricGrading basis (i+j-l.length) := by
-  obtain ⟨m,rfl⟩ := bottSection_surjective cs l hl s
-  obtain ⟨n,rfl⟩ := bottSection_surjective cs l hl z
-  rw [bottSectionDuality_apply]
-  exact bottDuality_homogeneous cs basis l ((bottPiece_iff_stalks cs basis l hl i m).mpr hs)
-    ((bottPiece_iff_stalks cs basis l hl j n).mpr hz)
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Section maps for the actual graded sheaf morphisms. A sheaf splitting
-really gives an integral graded section splitting; it is not replaced by a
-generic-fibre splitting. -/
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
-  {G : OrderedGraph V E} {B C : GradedSheaf (𝓐 := 𝓐) G}
-noncomputable section
-
- def Hom.assignmentMap (h : Hom B C) : B.forget.Assignment →ₗ[A] C.forget.Assignment where
-  toFun f x := (h.vertex x).map (f x)
-  map_add' f g := by funext x; exact map_add (h.vertex x).map (f x) (g x)
-  map_smul' r f := by funext x; exact map_smul (h.vertex x).map r (f x)
-
- theorem Hom.assignmentMap_compatible (h : Hom B C) (Ω : Set V)
-    (f : B.forget.Assignment) (hf : B.forget.CompatibleOn Ω f) :
-    C.forget.CompatibleOn Ω (h.assignmentMap f) := by
-  intro e hs ht
-  change (C.lower e).map ((h.vertex _).map (f _)) =
-    (C.upper e).map ((h.vertex _).map (f _))
-  exact (h.lower e (f (G.source e))).symm.trans
-    ((congrArg (h.edge e).map (hf e hs ht)).trans (h.upper e (f (G.target e))))
-
- def Hom.sectionMap (h : Hom B C) (Ω : Set V) :
-    B.forget.sections Ω →ₗ[A] C.forget.sections Ω :=
-  (h.assignmentMap.domRestrict (B.forget.sections Ω)).codRestrict (C.forget.sections Ω)
-    (fun f => h.assignmentMap_compatible Ω f.val f.property)
-
- theorem Hom.sectionMap_apply (h : Hom B C) (Ω : Set V)
-    (f : B.forget.sections Ω) (x : V) :
-    (h.sectionMap Ω f).val x=(h.vertex x).map (f.val x) := rfl
-
- variable [Fintype V] [Fintype E] [IsNoetherianRing A]
-
- def Hom.gradedSectionMap (h : Hom B C) (Ω : Set V) :
-    ModuleData.Hom (B.sections Ω) (C.sections Ω) where
-  map := h.sectionMap Ω
-  graded := by
-    intro n f hf x hx
-    exact (h.vertex x).graded n (f.val x) (hf x hx)
-
-omit [Fintype V] [Fintype E] [IsNoetherianRing A] in
- theorem SplitIn.sections_split (s : SplitIn B C) (Ω : Set V)
-    (f : B.forget.sections Ω) :
-    s.projection.sectionMap Ω (s.inclusion.sectionMap Ω f)=f := by
-  apply Subtype.ext
-  funext x
-  exact s.vertex_split x (f.val x)
-
-end
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-/-! Endomorphisms of the ACTUAL graded word sheaf act on its integral tensor
-module and preserve every stalk kernel and single-weight support lattice. -/
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
-universe u us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  (l : List I) (hl : cs.IsReduced l)
-noncomputable section
-
- def bottEndOfSheaf (h : GradedSheaf.Hom
-    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
-    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))) :
-    End (SymmetricCoefficient (M := M)) (BottSamelsonModule cs l) :=
-  (bottSectionEquiv cs l hl).symm.toLinearMap.comp
-    ((h.sectionMap Set.univ).comp (bottSectionEquiv cs l hl).toLinearMap)
-
- theorem bottEndOfSheaf_coordinates (h : GradedSheaf.Hom
-    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
-    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
-    (m : BottSamelsonModule cs l) (x : Interval cs 1 (cs.wordProd l)) :
-    bottStalkProjection cs l x.val (bottEndOfSheaf cs basis l hl h m)=
-      (h.vertex x).map (bottStalkProjection cs l x.val m) := by
-  have he := congrArg (fun s : (bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ =>
-    s.val x) ((bottSectionEquiv cs l hl).apply_symm_apply
-      (h.sectionMap Set.univ ((bottSectionEquiv cs l hl) m)))
-  exact he
-
- theorem bottEndOfSheaf_homogeneous (h : GradedSheaf.Hom
-    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
-    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
-    (n : ℤ) (m : BottSamelsonModule cs l) (hm : m ∈ bottPiece cs basis l n) :
-    bottEndOfSheaf cs basis l hl h m ∈ bottPiece cs basis l n := by
-  apply (bottPiece_iff_stalks cs basis l hl n _).mpr
-  intro x
-  rw [bottEndOfSheaf_coordinates]
-  exact (h.vertex x).graded n _ (bottStalkProjection_homogeneous cs basis l x.val n m hm)
-
- theorem bottEndOfSheaf_preserves_kernel (h : GradedSheaf.Hom
-    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
-    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
-    (x : Interval cs 1 (cs.wordProd l)) (m : BottSamelsonModule cs l)
-    (hm : bottStalkProjection cs l x.val m=0) :
-    bottStalkProjection cs l x.val (bottEndOfSheaf cs basis l hl h m)=0 := by
-  rw [bottEndOfSheaf_coordinates,hm]
-  exact (h.vertex x).map.map_zero
-
- theorem bottEndOfSheaf_preserves_support (h : GradedSheaf.Hom
-    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
-    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
-    (x : Interval cs 1 (cs.wordProd l)) (m : BottSamelsonModule cs l)
-    (hm : m ∈ bottWeightSupported cs l x.val) :
-    bottEndOfSheaf cs basis l hl h m ∈ bottWeightSupported cs l x.val := by
-  intro e he
-  let y : Interval cs 1 (cs.wordProd l) :=
-    ⟨bottWeight cs l e,one_bruhat cs _,bottWeight_bruhat cs l hl e⟩
-  have hmy : bottStalkProjection cs l y.val m=0 := by
-    apply Subtype.ext
-    funext f
-    exact hm f.val (fun hx => he (f.property.symm.trans hx))
-  have hfy := bottEndOfSheaf_preserves_kernel cs basis l hl h y m hmy
-  exact congrFun (congrArg Subtype.val hfy) ⟨e,rfl⟩
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-namespace LinearDescent
-universe ur um ux
-variable {R : Type ur} [CommRing R]
-  {M : Type um} [AddCommGroup M] [Module R M]
-  {X : Type ux} [AddCommGroup X] [Module R X]
-  (f : M →ₗ[R] X) (hf : Function.Surjective f) (e : Module.End R M)
-  (he : ∀ m, f m=0 → f (e m)=0)
-noncomputable section
-
- def descend : Module.End R X :=
-  ((LinearMap.ker f).liftQ (f.comp e) he).comp
-    (f.quotKerEquivOfSurjective hf).symm.toLinearMap
-
- theorem descend_apply (m : M) : descend f hf e he (f m)=f (e m) := by
-  change (LinearMap.ker f).liftQ (f.comp e) he
-    ((f.quotKerEquivOfSurjective hf).symm (f m))=f (e m)
-  rw [LinearMap.quotKerEquivOfSurjective_symm_apply]
-  rfl
-
- theorem descend_idempotent (hee : ∀ m, e (e m)=e m) (x : X) :
-    descend f hf e he (descend f hf e he x)=descend f hf e he x := by
-  obtain ⟨m,rfl⟩ := hf x
-  rw [descend_apply,descend_apply,hee]
-
-end
-end LinearDescent
-
-end
-
-
-section
-
-namespace KLInvariance.Graded.ModuleData
-open PairProjection LinearDescent
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} {M X Y : ModuleData.{uk,ua,um} 𝓐}
-  (e : Hom M M) (f : Hom M X) (g : Hom M Y)
-  (hf : Function.Surjective f.map) (hg : Function.Surjective g.map)
-  (hef : ∀ m, f.map m=0 → f.map (e.map m)=0)
-  (heg : ∀ m, g.map m=0 → g.map (e.map m)=0)
-noncomputable section
-
- def descended : Hom X X where
-  map := descend f.map hf e.map hef
-  graded := by
-    intro n x hx
-    obtain ⟨m,hm,rfl⟩ := f.homogeneous_lift hf n x hx
-    rw [descend_apply]
-    exact f.graded n _ (e.graded n m hm)
-
-include heg in
- theorem lower_kernel_preserved (x : X) (hx : lower f.map g.map x=0) :
-    lower f.map g.map ((descended e f hf hef).map x)=0 := by
-  obtain ⟨m,hm,hn⟩ := (kernel_lower f.map g.map x).mp hx
-  apply (kernel_lower f.map g.map _).mpr
-  refine ⟨e.map m,?_,heg m hn⟩
-  rw [← hm]
-  exact (descend_apply f.map hf e.map hef m).symm
-
- def descendedEdge : Hom (pairEdge f g hg) (pairEdge f g hg) where
-  map := descend (lower f.map g.map) (lower_surjective f.map g.map hg)
-    (descended e f hf hef).map (lower_kernel_preserved e f g hf hef heg)
-  graded := by
-    intro n z hz
-    obtain ⟨x,hx,rfl⟩ := hz
-    refine ⟨(descended e f hf hef).map x,(descended e f hf hef).graded n x hx,?_⟩
-    exact (descend_apply (lower f.map g.map) (lower_surjective f.map g.map hg)
-      (descended e f hf hef).map (lower_kernel_preserved e f g hf hef heg) x).symm
-
- theorem descendedEdge_lower (x : X) :
-    (descendedEdge e f g hf hg hef heg).map (lower f.map g.map x)=
-      lower f.map g.map ((descended e f hf hef).map x) :=
-  descend_apply _ _ _ _ x
-
- theorem descendedEdge_upper (y : Y) :
-    (descendedEdge e f g hf hg hef heg).map (upper f.map g.map y)=
-      upper f.map g.map ((descended e g hg heg).map y) := by
-  obtain ⟨m,rfl⟩ := hg y
-  rw [← compatible f.map g.map m,descendedEdge_lower]
-  change lower f.map g.map (descend f.map hf e.map hef (f.map m)) =
-    upper f.map g.map (descend g.map hg e.map heg (g.map m))
-  rw [descend_apply,descend_apply]
-  exact compatible f.map g.map (e.map m)
-
-end
-end KLInvariance.Graded.ModuleData
-
-end
-
-
-section
-
-/-! The adjoint of an actual graded word-sheaf endomorphism is again an
-integral graded word-sheaf endomorphism. No stalk freeness or BMP character
-is used or asserted. This supplies the correct adjoint route for a future
-BMP summand, instead of restricting a form to an arbitrary summand. -/
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
-universe u us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  (l : List I) (hl : cs.IsReduced l)
-noncomputable section
-
- def bottAdjointWordHom (h : GradedSheaf.Hom
-    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
-    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))) :
-    ModuleData.Hom (bottWordData cs basis l) (bottWordData cs basis l) where
-  map := bottAdjoint cs l (bottEndOfSheaf cs basis l hl h)
-  graded := bottAdjoint_homogeneous cs basis l _
-    (bottEndOfSheaf_homogeneous cs basis l hl h)
-
- theorem bottAdjointWordHom_kernel (h : GradedSheaf.Hom
-    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
-    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
-    (x : Interval cs 1 (cs.wordProd l)) (m : BottSamelsonModule cs l)
-    (hm : bottStalkProjection cs l x.val m=0) :
-    bottStalkProjection cs l x.val ((bottAdjointWordHom cs basis l hl h).map m)=0 :=
-  bottAdjoint_preserves_kernel cs l x.val _
-    (bottEndOfSheaf_preserves_support cs basis l hl h x) m hm
-
- def bottAdjointSheaf (h : GradedSheaf.Hom
-    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
-    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))) : GradedSheaf.Hom
-    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
-    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)) where
-  vertex x := ModuleData.descended (bottAdjointWordHom cs basis l hl h)
-    (bottStalkHom cs basis l x.val) (bottStalkHom_surjective cs basis l x.val)
-    (bottAdjointWordHom_kernel cs basis l hl h x)
-  edge e := ModuleData.descendedEdge (bottAdjointWordHom cs basis l hl h)
-    (bottStalkHom cs basis l (source cs 1 (cs.wordProd l) e).val)
-    (bottStalkHom cs basis l (target cs 1 (cs.wordProd l) e).val)
-    (bottStalkHom_surjective cs basis l _)
-    (bottStalkHom_surjective cs basis l _)
-    (bottAdjointWordHom_kernel cs basis l hl h _)
-    (bottAdjointWordHom_kernel cs basis l hl h _)
-  lower _ x := ModuleData.descendedEdge_lower _ _ _ _ _ _ _ x
-  upper _ y := ModuleData.descendedEdge_upper _ _ _ _ _ _ _ y
-
- theorem bottAdjointSheaf_vertex (h : GradedSheaf.Hom
-    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
-    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
-    (x : Interval cs 1 (cs.wordProd l)) (m : BottSamelsonModule cs l) :
-    ((bottAdjointSheaf cs basis l hl h).vertex x).map (bottStalkProjection cs l x.val m)=
-      bottStalkProjection cs l x.val (bottAdjoint cs l (bottEndOfSheaf cs basis l hl h) m) :=
-  LinearDescent.descend_apply _ _ _ _ m
-
- theorem bottEndOfAdjointSheaf (h : GradedSheaf.Hom
-    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
-    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))) :
-    bottEndOfSheaf cs basis l hl (bottAdjointSheaf cs basis l hl h)=
-      bottAdjoint cs l (bottEndOfSheaf cs basis l hl h) := by
-  apply LinearMap.ext
-  intro m
-  apply bottVertexProjection_injective cs l hl
-  funext x
-  exact (bottEndOfSheaf_coordinates cs basis l hl _ m x).trans
-    (bottAdjointSheaf_vertex cs basis l hl h x m)
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Integral graded adjoint WORD projectors. The assumption below is only
-that the GIVEN endomorphism is a projector; no existence of a BMP projector
-or identification of its image with its adjoint is asserted. -/
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
-universe u us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  (l : List I) (hl : cs.IsReduced l)
-  (h : GradedSheaf.Hom
-    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
-    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
-noncomputable section
-
- theorem bottEndOfSheaf_idempotent
-    (hh : ∀ x v, (h.vertex x).map ((h.vertex x).map v)=(h.vertex x).map v)
-    (m : BottSamelsonModule cs l) :
-    bottEndOfSheaf cs basis l hl h (bottEndOfSheaf cs basis l hl h m)=
-      bottEndOfSheaf cs basis l hl h m := by
-  apply bottVertexProjection_injective cs l hl
-  funext x
-  change bottStalkProjection cs l x.val _=bottStalkProjection cs l x.val _
-  simp only [bottEndOfSheaf_coordinates]
-  exact hh x _
-
- theorem bottAdjointSheaf_vertex_idempotent
-    (hh : ∀ x v, (h.vertex x).map ((h.vertex x).map v)=(h.vertex x).map v)
-    (x : Interval cs 1 (cs.wordProd l))
-    (v : bottStalk cs l x.val) :
-    ((bottAdjointSheaf cs basis l hl h).vertex x).map
-      (((bottAdjointSheaf cs basis l hl h).vertex x).map v)=
-      ((bottAdjointSheaf cs basis l hl h).vertex x).map v := by
-  obtain ⟨m,rfl⟩ := bottStalkProjection_surjective cs l x.val v
-  simp only [bottAdjointSheaf_vertex]
-  exact congrArg (bottStalkProjection cs l x.val)
-    (PerfectAdjoint.adjoint_idempotent (bottSamelsonDuality cs l) _
-      (bottEndOfSheaf_idempotent cs basis l hl h hh) m)
-
- theorem bottAdjointSheaf_edge_idempotent
-    (hh : ∀ x v, (h.vertex x).map ((h.vertex x).map v)=(h.vertex x).map v)
-    (e : Edge cs 1 (cs.wordProd l))
-    (v : (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)).edge e) :
-    ((bottAdjointSheaf cs basis l hl h).edge e).map
-      (((bottAdjointSheaf cs basis l hl h).edge e).map v)=
-      ((bottAdjointSheaf cs basis l hl h).edge e).map v := by
-  have hs : Function.Surjective
-      ((bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)).lower e).map :=
-    bottProjectionSheaf_lower_surjective cs l 1 (cs.wordProd l) e
-  obtain ⟨m,rfl⟩ := hs v
-  simp only [(bottAdjointSheaf cs basis l hl h).lower]
-  exact congrArg ((bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)).lower e).map
-    (bottAdjointSheaf_vertex_idempotent cs basis l hl h hh _ m)
-
- theorem bottAdjointSheaf_twice_vertex (x : Interval cs 1 (cs.wordProd l))
-    (v : bottStalk cs l x.val) :
-    ((bottAdjointSheaf cs basis l hl (bottAdjointSheaf cs basis l hl h)).vertex x).map v=
-      (h.vertex x).map v := by
-  obtain ⟨m,rfl⟩ := bottStalkProjection_surjective cs l x.val v
-  rw [bottAdjointSheaf_vertex,bottEndOfAdjointSheaf]
-  change bottStalkProjection cs l x.val
-    (PerfectAdjoint.adjoint (bottSamelsonDuality cs l)
-      (PerfectAdjoint.adjoint (bottSamelsonDuality cs l) (bottEndOfSheaf cs basis l hl h)) m)=_
-  rw [PerfectAdjoint.adjoint_adjoint _ (bottSamelson_symmetric cs l)]
-  exact bottEndOfSheaf_coordinates cs basis l hl h m x
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! The literal integral image sheaf of a graded projector, with its actual
-splitting. Used for word-adjoint projectors; no BMP summand is assumed. -/
-namespace KLInvariance.Graded.ModuleData
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} [IsNoetherianRing A]
-  {M N : ModuleData.{uk,ua,um} 𝓐}
-noncomputable section
-
- def rangeMap (e : Hom M M) (f : Hom N N) (g : Hom M N)
-    (h : ∀ m, f.map (g.map m)=g.map (e.map m)) : Hom (range e) (range f) where
-  map := (g.map.domRestrict e.map.range).codRestrict f.map.range (by
-    rintro ⟨_,m,rfl⟩
-    exact ⟨g.map m,h m⟩)
-  graded n m hm := g.graded n m.val hm
-
- def rangeInclusion (e : Hom M M) : Hom (range e) M where
-  map := e.map.range.subtype
-  graded _ _ hm := hm
-
- def rangeProjection (e : Hom M M) : Hom M (range e) where
-  map := e.map.rangeRestrict
-  graded n m hm := e.graded n m hm
-
-end
-end KLInvariance.Graded.ModuleData
-
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} [IsNoetherianRing A]
-  {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
-  {B : GradedSheaf (𝓐 := 𝓐) G}
-noncomputable section
-
- def Hom.image (h : Hom B B) : GradedSheaf (𝓐 := 𝓐) G where
-  vertex x := ModuleData.range (h.vertex x)
-  edge e := ModuleData.range (h.edge e)
-  lower e := ModuleData.rangeMap (h.vertex (G.source e)) (h.edge e) (B.lower e) (h.lower e)
-  upper e := ModuleData.rangeMap (h.vertex (G.target e)) (h.edge e) (B.upper e) (h.upper e)
-
- def Hom.imageInclusion (h : Hom B B) : Hom h.image B where
-  vertex x := ModuleData.rangeInclusion (h.vertex x)
-  edge e := ModuleData.rangeInclusion (h.edge e)
-  lower _ _ := rfl
-  upper _ _ := rfl
-
- def Hom.imageProjection (h : Hom B B) : Hom B h.image where
-  vertex x := ModuleData.rangeProjection (h.vertex x)
-  edge e := ModuleData.rangeProjection (h.edge e)
-  lower e v := Subtype.ext (h.lower e v)
-  upper e v := Subtype.ext (h.upper e v)
-
- def Hom.imageSplit (h : Hom B B)
-    (hv : ∀ x v, (h.vertex x).map ((h.vertex x).map v)=(h.vertex x).map v)
-    (he : ∀ e v, (h.edge e).map ((h.edge e).map v)=(h.edge e).map v) : SplitIn h.image B where
-  inclusion := h.imageInclusion
-  projection := h.imageProjection
-  vertex_split x v := by
-    obtain ⟨m,hm⟩ := v.property
-    apply Subtype.ext
-    change (h.vertex x).map v.val=v.val
-    rw [← hm,hv]
-  edge_split e v := by
-    obtain ⟨m,hm⟩ := v.property
-    apply Subtype.ext
-    change (h.edge e).map v.val=v.val
-    rw [← hm,he]
-
-end
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-/-! All integral sections of an idempotent image sheaf are the image of the
-actual section projector; this is stronger than equality over the fraction
-field and requires the actual idempotence, not merely pointwise surjectivity. -/
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} [IsNoetherianRing A]
-  {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
-  {B : GradedSheaf (𝓐 := 𝓐) G}
-noncomputable section
-
- theorem Hom.imageSection_fixed (h : Hom B B)
-    (hv : ∀ x v, (h.vertex x).map ((h.vertex x).map v)=(h.vertex x).map v)
-    (Ω : Set V) (f : h.image.forget.sections Ω) :
-    h.sectionMap Ω (h.imageInclusion.sectionMap Ω f)=h.imageInclusion.sectionMap Ω f := by
-  apply Subtype.ext
-  funext x
-  obtain ⟨v,hv'⟩ := (f.val x).property
-  change (h.vertex x).map (f.val x).val=(f.val x).val
-  rw [← hv',hv]
-
- def Hom.imageSectionMap (h : Hom B B)
-    (hv : ∀ x v, (h.vertex x).map ((h.vertex x).map v)=(h.vertex x).map v)
-    (Ω : Set V) : h.image.forget.sections Ω →ₗ[A] LinearMap.range (h.sectionMap Ω) :=
-  (h.imageInclusion.sectionMap Ω).codRestrict _ (fun f =>
-    ⟨h.imageInclusion.sectionMap Ω f,h.imageSection_fixed hv Ω f⟩)
-
- theorem Hom.imageSectionMap_bijective (h : Hom B B)
-    (hv : ∀ x v, (h.vertex x).map ((h.vertex x).map v)=(h.vertex x).map v)
-    (Ω : Set V) : Function.Bijective (h.imageSectionMap hv Ω) := by
-  constructor
-  · intro f g hfg
-    apply Subtype.ext
-    funext x
-    apply Subtype.ext
-    exact congrFun (congrArg Subtype.val (congrArg Subtype.val hfg)) x
-  · rintro ⟨s,m,rfl⟩
-    refine ⟨h.imageProjection.sectionMap Ω m,?_⟩
-    apply Subtype.ext
-    apply Subtype.ext
-    rfl
-
- def Hom.imageSectionEquiv (h : Hom B B)
-    (hv : ∀ x v, (h.vertex x).map ((h.vertex x).map v)=(h.vertex x).map v)
-    (Ω : Set V) : h.image.forget.sections Ω ≃ₗ[A] LinearMap.range (h.sectionMap Ω) :=
-  LinearEquiv.ofBijective (h.imageSectionMap hv Ω) (h.imageSectionMap_bijective hv Ω)
-
-end
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-namespace ConjugateRange
-open Module
-universe ur um un
-variable {R : Type ur} [CommRing R]
-  {M : Type um} [AddCommGroup M] [Module R M]
-  {N : Type un} [AddCommGroup N] [Module R N]
-  (e : M ≃ₗ[R] N) (f : End R N)
-noncomputable section
-
- def conjugate : End R M := e.symm.toLinearMap.comp (f.comp e.toLinearMap)
-
- def map : LinearMap.range f →ₗ[R] LinearMap.range (conjugate e f) :=
-  (e.symm.toLinearMap.domRestrict f.range).codRestrict _ (by
-    rintro ⟨_,n,rfl⟩
-    refine ⟨e.symm n,?_⟩
-    change e.symm (f (e (e.symm n)))=e.symm (f n)
-    rw [LinearEquiv.apply_symm_apply])
-
- theorem map_bijective : Function.Bijective (map e f) := by
-  constructor
-  · intro m n h
-    apply Subtype.ext
-    apply e.symm.injective
-    exact congrArg Subtype.val h
-  · rintro ⟨_,m,rfl⟩
-    refine ⟨⟨f (e m),e m,rfl⟩,?_⟩
-    rfl
-
- def equiv : LinearMap.range f ≃ₗ[R] LinearMap.range (conjugate e f) :=
-  LinearEquiv.ofBijective (map e f) (map_bijective e f)
-
- theorem equiv_apply (n : LinearMap.range f) : (equiv e f n).val=e.symm n.val := rfl
-
-end
-end ConjugateRange
-
-end
-
-
-section
-
-/-! The correct integral summand duality. A chosen summand of a perfectly
-paired module is paired with its ADJOINT summand, not with itself by fiat.
-This is the algebraic connector for the unresolved BMP summand recognition.
-No existence of a BMP summand, or isomorphism with its adjoint, is asserted. -/
-namespace AdjointSummandDuality
-universe ur um un
-variable {R : Type ur} [CommRing R]
-  {M : Type um} [AddCommGroup M] [Module R M]
-  {N : Type un} [AddCommGroup N] [Module R N]
-  (d : M ≃ₗ[R] Module.Dual R M) (i : N →ₗ[R] M) (p : M →ₗ[R] N)
-  (hpi : ∀ n, p (i n)=n)
-noncomputable section
-
- def dualSection : Module.Dual R N →ₗ[R] M :=
-  d.symm.toLinearMap.comp p.dualMap
-
- def dualProjection : M →ₗ[R] Module.Dual R N := i.dualMap.comp d.toLinearMap
-
-include hpi in
- theorem projection_section (lam : Module.Dual R N) :
-    dualProjection d i (dualSection d p lam)=lam := by
-  apply LinearMap.ext
-  intro n
-  change d (d.symm (p.dualMap lam)) (i n)=lam n
-  rw [LinearEquiv.apply_symm_apply]
-  change lam (p (i n))=lam n
-  rw [hpi]
-
- def adjointProjector : Module.End R M := (dualSection d p).comp (dualProjection d i)
-
-include hpi in
- theorem adjointProjector_idempotent (m : M) :
-    adjointProjector d i p (adjointProjector d i p m)=adjointProjector d i p m := by
-  change dualSection d p (dualProjection d i (dualSection d p (dualProjection d i m))) = _
-  rw [projection_section d i p hpi]
-  rfl
-
- def adjointSummand : Submodule R M := LinearMap.range (dualSection d p)
-
-include hpi in
- theorem dualProjection_injective_on_adjoint :
-    Function.Injective ((dualProjection d i).domRestrict (adjointSummand d p)) := by
-  rintro ⟨m,lam,rfl⟩ ⟨n,mu,rfl⟩ h
-  apply Subtype.ext
-  change dualSection d p lam=dualSection d p mu
-  change dualProjection d i (dualSection d p lam)=dualProjection d i (dualSection d p mu) at h
-  rw [projection_section d i p hpi,projection_section d i p hpi] at h
-  rw [h]
-
-include hpi in
- theorem dualProjection_surjective_on_adjoint :
-    Function.Surjective ((dualProjection d i).domRestrict (adjointSummand d p)) := by
-  intro lam
-  exact ⟨⟨dualSection d p lam,lam,rfl⟩,projection_section d i p hpi lam⟩
-
- def adjointEquivDual : adjointSummand d p ≃ₗ[R] Module.Dual R N :=
-  LinearEquiv.ofBijective ((dualProjection d i).domRestrict (adjointSummand d p))
-    ⟨dualProjection_injective_on_adjoint d i p hpi,dualProjection_surjective_on_adjoint d i p hpi⟩
-
- theorem adjointEquivDual_apply (m : adjointSummand d p) (n : N) :
-    adjointEquivDual d i p hpi m n=d m.val (i n) := rfl
-
-include hpi in
- theorem adjointProjector_range :
-    LinearMap.range (adjointProjector d i p)=adjointSummand d p := by
-  apply le_antisymm
-  · rintro m ⟨n,rfl⟩
-    exact ⟨dualProjection d i n,rfl⟩
-  · rintro m ⟨lam,rfl⟩
-    refine ⟨dualSection d p lam,?_⟩
-    change dualSection d p (dualProjection d i (dualSection d p lam))=dualSection d p lam
-    rw [projection_section d i p hpi]
-
-variable (f : Module.End R M) (g : Module.End R N)
-  (hbal : ∀ m n, d (f m) n=d m (f n))
-  (hi : ∀ n, f (i n)=i (g n)) (hp : ∀ m, p (f m)=g (p m))
-
-include hbal hi in
- theorem projection_intertwines (m : M) :
-    dualProjection d i (f m)=g.dualMap (dualProjection d i m) := by
-  apply LinearMap.ext
-  intro n
-  change d (f m) (i n)=d m (i (g n))
-  rw [hbal,hi]
-
-include hbal hp in
- theorem section_intertwines (lam : Module.Dual R N) :
-    f (dualSection d p lam)=dualSection d p (g.dualMap lam) := by
-  apply d.injective
-  apply LinearMap.ext
-  intro n
-  change d (f (d.symm (p.dualMap lam))) n=
-    d (d.symm (p.dualMap (g.dualMap lam))) n
-  rw [hbal,LinearEquiv.apply_symm_apply,LinearEquiv.apply_symm_apply]
-  change lam (p (f n))=lam (g (p n))
-  rw [hp]
-
-include hbal hi hp in
- theorem adjointProjector_commutes (m : M) :
-    f (adjointProjector d i p m)=adjointProjector d i p (f m) := by
-  change f (dualSection d p (dualProjection d i m)) =
-    dualSection d p (dualProjection d i (f m))
-  rw [section_intertwines d p f g hbal hp,projection_intertwines d i f g hbal hi]
-
-end
-end AdjointSummandDuality
-
-end
-
-
-section
-
-/-! The range of the genuine adjoint projector is integrally the dual of
-the original summand, with no assumed self-duality of that summand. -/
-namespace PerfectAdjoint
-open Module
-universe ur um un
-variable {R : Type ur} [CommRing R]
-  {M : Type um} [AddCommGroup M] [Module R M]
-  {N : Type un} [AddCommGroup N] [Module R N]
-  (d : M ≃ₗ[R] Dual R M) (i : N →ₗ[R] M) (p : M →ₗ[R] N)
-  (hpi : ∀ n, p (i n)=n)
-noncomputable section
-
- theorem adjoint_projector_eq : adjoint d (i.comp p)=
-    AdjointSummandDuality.adjointProjector d i p := by
-  apply LinearMap.ext
-  intro m
-  rfl
-
- def rangeAdjointEquivDual : LinearMap.range (adjoint d (i.comp p)) ≃ₗ[R] Dual R N :=
-  (LinearEquiv.ofEq _ _ (by
-    rw [adjoint_projector_eq,AdjointSummandDuality.adjointProjector_range d i p hpi])).trans
-    (AdjointSummandDuality.adjointEquivDual d i p hpi)
-
- theorem rangeAdjointEquivDual_apply (m : LinearMap.range (adjoint d (i.comp p))) (n : N) :
-    rangeAdjointEquivDual d i p hpi m n=d m.val (i n) := rfl
-
-end
-end PerfectAdjoint
-
-end
-
-
-section
-
-namespace PerfectAdjoint
-open Module
-universe ur um
-variable {R : Type ur} [CommRing R]
-  {M : Type um} [AddCommGroup M] [Module R M]
-  (d : M ≃ₗ[R] Dual R M) (f : End R M) (hf : ∀ m, f (f m)=f m)
-noncomputable section
-
-include hf in
- theorem range_projector_split (m : f.range) : f.rangeRestrict (f.range.subtype m)=m := by
-  apply Subtype.ext
-  obtain ⟨n,hn⟩ := m.property
-  change f m.val=m.val
-  rw [← hn,hf]
-
- def rangeAdjointEquivDualRange : LinearMap.range (adjoint d f) ≃ₗ[R] Dual R f.range :=
-  rangeAdjointEquivDual d f.range.subtype f.rangeRestrict (range_projector_split f hf)
-
- theorem rangeAdjointEquivDualRange_apply (m : LinearMap.range (adjoint d f)) (n : f.range) :
-    rangeAdjointEquivDualRange d f hf m n=d m.val n.val := rfl
-
-end
-end PerfectAdjoint
-
-end
-
-
-section
-
-namespace KLInvariance.TitsSpace
-open Module
-universe u us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-
-include basis in
- theorem symmetricCoefficientNoetherian : IsNoetherianRing (SymmetricCoefficient (M := M)) :=
-  isNoetherianRing_of_surjective (MvPolynomial σ ℝ) (SymmetricCoefficient (M := M))
-    (SymmetricAlgebra.equivMvPolynomial basis).symm.toRingHom
-    (SymmetricAlgebra.equivMvPolynomial basis).symm.surjective
-
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! The actual adjoint IMAGE sheaf has all integral sections dual to those
-of the original projector image. This does not identify those two image
-sheaves or assert that the word sheaf satisfies the BMP axioms. -/
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
-universe u us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  (l : List I) (hl : cs.IsReduced l)
-  (h : GradedSheaf.Hom
-    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
-    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
-noncomputable section
-
- def bottImageSheaf : GradedSheaf (𝓐 := symmetricGrading basis)
-    (graph cs 1 (cs.wordProd l)) := by
-  letI := symmetricCoefficientNoetherian basis
-  exact h.image
-
- def bottImageSectionWordEquiv
-    (hh : ∀ x v, (h.vertex x).map ((h.vertex x).map v)=(h.vertex x).map v) :
-    (bottImageSheaf cs basis l h).forget.sections Set.univ ≃ₗ[
-      SymmetricCoefficient (M := M)] LinearMap.range (bottEndOfSheaf cs basis l hl h) := by
-  letI := symmetricCoefficientNoetherian basis
-  exact (h.imageSectionEquiv hh Set.univ).trans
-    (ConjugateRange.equiv (bottSectionEquiv cs l hl) (h.sectionMap Set.univ))
-
- def bottAdjointImageEquivDual
-    (hh : ∀ x v, (h.vertex x).map ((h.vertex x).map v)=(h.vertex x).map v) :
-    (bottImageSheaf cs basis l (bottAdjointSheaf cs basis l hl h)).forget.sections Set.univ ≃ₗ[
-      SymmetricCoefficient (M := M)]
-      Dual (SymmetricCoefficient (M := M))
-        ((bottImageSheaf cs basis l h).forget.sections Set.univ) :=
-  (bottImageSectionWordEquiv cs basis l hl (bottAdjointSheaf cs basis l hl h)
-      (bottAdjointSheaf_vertex_idempotent cs basis l hl h hh)).trans
-    ((LinearEquiv.ofEq _ _ (congrArg LinearMap.range
-        (bottEndOfAdjointSheaf cs basis l hl h))).trans
-      ((PerfectAdjoint.rangeAdjointEquivDualRange (bottSamelsonDuality cs l)
-          (bottEndOfSheaf cs basis l hl h)
-          (bottEndOfSheaf_idempotent cs basis l hl h hh)).trans
-        (bottImageSectionWordEquiv cs basis l hl h hh).dualMap))
-
- theorem bottImageSectionWordEquiv_coordinates
-    (hh : ∀ x v, (h.vertex x).map ((h.vertex x).map v)=(h.vertex x).map v)
-    (m : (bottImageSheaf cs basis l h).forget.sections Set.univ)
-    (x : Interval cs 1 (cs.wordProd l)) :
-    bottStalkProjection cs l x.val ((bottImageSectionWordEquiv cs basis l hl h hh m).val)=
-      (m.val x).val := by
-  let := symmetricCoefficientNoetherian basis
-  exact congrArg (fun s : (bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ =>
-    s.val x) ((bottSectionEquiv cs l hl).apply_symm_apply
-      (h.imageInclusion.sectionMap Set.univ m))
-
- theorem bottImageSectionWordEquiv_homogeneous
-    (hh : ∀ x v, (h.vertex x).map ((h.vertex x).map v)=(h.vertex x).map v)
-    (n : ℤ) (m : (bottImageSheaf cs basis l h).forget.sections Set.univ)
-    (hm : ∀ x, m.val x ∈ ((bottImageSheaf cs basis l h).vertex x).piece n) :
-    (bottImageSectionWordEquiv cs basis l hl h hh m).val ∈ bottPiece cs basis l n := by
-  apply (bottPiece_iff_stalks cs basis l hl n _).mpr
-  intro x
-  rw [bottImageSectionWordEquiv_coordinates]
-  exact hm x
-
- theorem bottAdjointImageEquivDual_apply
-    (hh : ∀ x v, (h.vertex x).map ((h.vertex x).map v)=(h.vertex x).map v)
-    (m : (bottImageSheaf cs basis l (bottAdjointSheaf cs basis l hl h)).forget.sections Set.univ)
-    (n : (bottImageSheaf cs basis l h).forget.sections Set.univ) :
-    bottAdjointImageEquivDual cs basis l hl h hh m n=bottSamelsonDuality cs l
-      (bottImageSectionWordEquiv cs basis l hl (bottAdjointSheaf cs basis l hl h)
-        (bottAdjointSheaf_vertex_idempotent cs basis l hl h hh) m).val
-      (bottImageSectionWordEquiv cs basis l hl h hh n).val := rfl
-
- theorem bottAdjointImagePairing_homogeneous
-    (hh : ∀ x v, (h.vertex x).map ((h.vertex x).map v)=(h.vertex x).map v)
-    (i j : ℤ)
-    (m : (bottImageSheaf cs basis l (bottAdjointSheaf cs basis l hl h)).forget.sections Set.univ)
-    (n : (bottImageSheaf cs basis l h).forget.sections Set.univ)
-    (hm : ∀ x, m.val x ∈
-      ((bottImageSheaf cs basis l (bottAdjointSheaf cs basis l hl h)).vertex x).piece i)
-    (hn : ∀ x, n.val x ∈ ((bottImageSheaf cs basis l h).vertex x).piece j) :
-    bottAdjointImageEquivDual cs basis l hl h hh m n ∈ symmetricGrading basis (i+j-(l.length : ℤ)) := by
-  rw [bottAdjointImageEquivDual_apply]
-  exact bottDuality_homogeneous cs basis l
-    (bottImageSectionWordEquiv_homogeneous cs basis l hl _
-      (bottAdjointSheaf_vertex_idempotent cs basis l hl h hh) i m hm)
-    (bottImageSectionWordEquiv_homogeneous cs basis l hl h hh j n hn)
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! A specified integral graded sheaf splitting identifies its source with
-its literal projector image. No existence of a word/BMP splitting is assumed
-outside each explicitly parametric helper. -/
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} [IsNoetherianRing A]
-  {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
-  {B C : GradedSheaf (𝓐 := 𝓐) G} (s : SplitIn B C)
-noncomputable section
-
- def SplitIn.projector : Hom C C := s.inclusion.comp s.projection
-
-omit [IsNoetherianRing A] in
- theorem SplitIn.projector_vertex (x : V) (v : C.vertex x) :
-    (s.projector.vertex x).map ((s.projector.vertex x).map v)=(s.projector.vertex x).map v := by
-  change (s.inclusion.vertex x).map ((s.projection.vertex x).map
-    ((s.inclusion.vertex x).map ((s.projection.vertex x).map v)))=_
-  rw [s.vertex_split]
-  rfl
-
-omit [IsNoetherianRing A] in
- theorem SplitIn.projector_edge (e : E) (v : C.edge e) :
-    (s.projector.edge e).map ((s.projector.edge e).map v)=(s.projector.edge e).map v := by
-  change (s.inclusion.edge e).map ((s.projection.edge e).map
-    ((s.inclusion.edge e).map ((s.projection.edge e).map v)))=_
-  rw [s.edge_split]
-  rfl
-
- def SplitIn.imageTo : Hom s.projector.image B := s.projection.comp s.projector.imageInclusion
- def SplitIn.toImage : Hom B s.projector.image := s.projector.imageProjection.comp s.inclusion
-
- theorem SplitIn.imageTo_toImage_vertex (x : V) (v : B.vertex x) :
-    (s.imageTo.vertex x).map ((s.toImage.vertex x).map v)=v := by
-  change (s.projection.vertex x).map ((s.inclusion.vertex x).map
-    ((s.projection.vertex x).map ((s.inclusion.vertex x).map v)))=v
-  rw [s.vertex_split,s.vertex_split]
-
- theorem SplitIn.toImage_imageTo_vertex (x : V) (v : s.projector.image.vertex x) :
-    (s.toImage.vertex x).map ((s.imageTo.vertex x).map v)=v := by
-  apply Subtype.ext
-  obtain ⟨m,hm⟩ := v.property
-  change (s.projector.vertex x).map ((s.projector.vertex x).map v.val)=v.val
-  rw [s.projector_vertex,← hm,s.projector_vertex]
-
- theorem SplitIn.imageTo_toImage_edge (e : E) (v : B.edge e) :
-    (s.imageTo.edge e).map ((s.toImage.edge e).map v)=v := by
-  change (s.projection.edge e).map ((s.inclusion.edge e).map
-    ((s.projection.edge e).map ((s.inclusion.edge e).map v)))=v
-  rw [s.edge_split,s.edge_split]
-
- theorem SplitIn.toImage_imageTo_edge (e : E) (v : s.projector.image.edge e) :
-    (s.toImage.edge e).map ((s.imageTo.edge e).map v)=v := by
-  apply Subtype.ext
-  obtain ⟨m,hm⟩ := v.property
-  change (s.projector.edge e).map ((s.projector.edge e).map v.val)=v.val
-  rw [s.projector_edge,← hm,s.projector_edge]
-
- theorem SplitIn.imageSection_bijective (Ω : Set V) :
-    Function.Bijective (s.imageTo.sectionMap Ω) := by
-  constructor
-  · intro m n he
-    apply Subtype.ext
-    funext x
-    have hx := congrFun (congrArg Subtype.val he) x
-    have hy := congrArg (s.toImage.vertex x).map hx
-    exact (s.toImage_imageTo_vertex x (m.val x)).symm.trans
-      (hy.trans (s.toImage_imageTo_vertex x (n.val x)))
-  · intro m
-    refine ⟨s.toImage.sectionMap Ω m,?_⟩
-    apply Subtype.ext
-    funext x
-    exact s.imageTo_toImage_vertex x (m.val x)
-
- def SplitIn.imageSectionEquiv (Ω : Set V) :
-    s.projector.image.forget.sections Ω ≃ₗ[A] B.forget.sections Ω :=
-  LinearEquiv.ofBijective (s.imageTo.sectionMap Ω) (s.imageSection_bijective Ω)
-
- theorem SplitIn.imageSectionEquiv_symm_apply (Ω : Set V) (m : B.forget.sections Ω) :
-    (s.imageSectionEquiv Ω).symm m=s.toImage.sectionMap Ω m := by
-  apply (s.imageSectionEquiv Ω).injective
-  rw [LinearEquiv.apply_symm_apply]
-  apply Subtype.ext
-  funext x
-  exact (s.imageTo_toImage_vertex x (m.val x)).symm
-
- theorem SplitIn.imageSectionEquiv_symm_homogeneous (Ω : Set V) (n : ℤ)
-    (m : B.forget.sections Ω) (hm : ∀ x, m.val x ∈ (B.vertex x).piece n) (x : V) :
-    ((s.imageSectionEquiv Ω).symm m).val x ∈ (s.projector.image.vertex x).piece n := by
-  rw [s.imageSectionEquiv_symm_apply]
-  exact (s.toImage.vertex x).graded n (m.val x) (hm x)
-
-end
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-/-! Correct integral graded duality for a GIVEN word-sheaf summand: the dual
-is realized by its ADJOINT image, not identified with the original summand.
-The existence and BMP recognition of that summand remain separate. -/
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
-universe u us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  (l : List I) (hl : cs.IsReduced l)
-  (B : GradedSheaf (𝓐 := symmetricGrading basis) (graph cs 1 (cs.wordProd l)))
-  (s : GradedSheaf.SplitIn B (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
-noncomputable section
-
- def bottSplitAdjointSheaf : GradedSheaf (𝓐 := symmetricGrading basis)
-    (graph cs 1 (cs.wordProd l)) :=
-  bottImageSheaf cs basis l (bottAdjointSheaf cs basis l hl s.projector)
-
- def bottSplitAdjointEquivDual :
-    (bottSplitAdjointSheaf cs basis l hl B s).forget.sections Set.univ ≃ₗ[
-      SymmetricCoefficient (M := M)]
-      Dual (SymmetricCoefficient (M := M)) (B.forget.sections Set.univ) := by
-  letI := symmetricCoefficientNoetherian basis
-  exact (bottAdjointImageEquivDual cs basis l hl s.projector s.projector_vertex).trans
-    (s.imageSectionEquiv Set.univ).symm.dualMap
-
- theorem bottSplitAdjointPairing_homogeneous
-    (i j : ℤ)
-    (m : (bottSplitAdjointSheaf cs basis l hl B s).forget.sections Set.univ)
-    (n : B.forget.sections Set.univ)
-    (hm : ∀ x, m.val x ∈ ((bottSplitAdjointSheaf cs basis l hl B s).vertex x).piece i)
-    (hn : ∀ x, n.val x ∈ (B.vertex x).piece j) :
-    bottSplitAdjointEquivDual cs basis l hl B s m n ∈
-      symmetricGrading basis (i+j-(l.length : ℤ)) := by
-  let := symmetricCoefficientNoetherian basis
-  exact bottAdjointImagePairing_homogeneous cs basis l hl s.projector s.projector_vertex i j m
-    ((s.imageSectionEquiv Set.univ).symm n) hm
-    (s.imageSectionEquiv_symm_homogeneous Set.univ j n hn)
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! An actual word-sheaf endomorphism that is the identity at one stalk has
-adjoint equal to the identity at that same stalk. In particular, an identity
-TOP projector stays normalized on taking the adjoint. This alone does not
-supply free stalks, flabbiness, or the BMP character. -/
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
-universe u us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  (l : List I) (hl : cs.IsReduced l)
-  (h : GradedSheaf.Hom
-    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
-    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
-noncomputable section
-
- theorem bottEndOfSheaf_fixed_support (x : Interval cs 1 (cs.wordProd l))
-    (hx : ∀ v, (h.vertex x).map v=v)
-    (m : BottSamelsonModule cs l) (hm : m ∈ bottWeightSupported cs l x.val) :
-    bottEndOfSheaf cs basis l hl h m=m := by
-  apply bottEvaluation_injective cs l
-  funext e
-  by_cases he : bottWeight cs l e=x.val
-  · have ho := bottEndOfSheaf_coordinates cs basis l hl h m x
-    exact congrFun (congrArg Subtype.val (ho.trans (hx _))) ⟨e,he⟩
-  · exact (bottEndOfSheaf_preserves_support cs basis l hl h x m hm e he).trans
-      (hm e he).symm
-
- theorem bottAdjointSheaf_fixed_stalk (x : Interval cs 1 (cs.wordProd l))
-    (hx : ∀ v, (h.vertex x).map v=v) (v : bottStalk cs l x.val) :
-    ((bottAdjointSheaf cs basis l hl h).vertex x).map v=v := by
-  obtain ⟨m,rfl⟩ := bottStalkProjection_surjective cs l x.val v
-  rw [bottAdjointSheaf_vertex]
-  have hz : bottStalkProjection cs l x.val
-      (bottAdjoint cs l (bottEndOfSheaf cs basis l hl h) m-m)=0 := by
-    apply (bott_projection_zero_iff_annihilates_support cs l x.val _).mpr
-    intro z
-    change bottSamelsonDuality cs l z.val
-      (PerfectAdjoint.adjoint (bottSamelsonDuality cs l)
-        (bottEndOfSheaf cs basis l hl h) m-m)=0
-    rw [map_sub,PerfectAdjoint.symmetric_pairing _ (bottSamelson_symmetric cs l)]
-    rw [bottEndOfSheaf_fixed_support cs basis l hl h x hx z.val z.property,sub_self]
-  rw [map_sub] at hz
-  exact sub_eq_zero.mp hz
-
- theorem bottAdjointSheaf_fixed_stalk_iff (x : Interval cs 1 (cs.wordProd l)) :
-    (∀ v, ((bottAdjointSheaf cs basis l hl h).vertex x).map v=v) ↔
-      (∀ v, (h.vertex x).map v=v) := by
-  constructor
-  · intro hx v
-    have h' := bottAdjointSheaf_fixed_stalk cs basis l hl
-      (bottAdjointSheaf cs basis l hl h) x hx v
-    exact (bottAdjointSheaf_twice_vertex cs basis l hl h x v).symm.trans h'
-  · exact fun hx v => bottAdjointSheaf_fixed_stalk cs basis l hl h x hx v
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-namespace KLInvariance.Graded.ModuleData
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} {M N : ModuleData.{uk,ua,um} 𝓐}
- theorem Hom.eq_of_map_eq (f g : Hom M N) (h : f.map=g.map) : f=g := by
-  cases f
-  cases g
-  cases h
-  rfl
-end KLInvariance.Graded.ModuleData
-
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
-  {B C : GradedSheaf (𝓐 := 𝓐) G}
-
- theorem Hom.eq_of_maps (f g : Hom B C)
-    (hv : ∀ x v, (f.vertex x).map v=(g.vertex x).map v)
-    (he : ∀ e v, (f.edge e).map v=(g.edge e).map v) : f=g := by
-  have hv' : f.vertex=g.vertex := funext (fun x =>
-    ModuleData.Hom.eq_of_map_eq _ _ (LinearMap.ext (hv x)))
-  have he' : f.edge=g.edge := funext (fun e =>
-    ModuleData.Hom.eq_of_map_eq _ _ (LinearMap.ext (he e)))
-  cases f
-  cases g
-  cases hv'
-  cases he'
-  rfl
-
- theorem Hom.eq_of_vertex (f g : Hom B C)
-    (hs : ∀ e, Function.Surjective (B.lower e).map)
-    (hv : ∀ x v, (f.vertex x).map v=(g.vertex x).map v) : f=g := by
-  apply f.eq_of_maps g hv
-  intro e v
-  obtain ⟨w,rfl⟩ := hs e v
-  rw [f.lower,g.lower,hv]
-
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-/-! Integral anti-involution on the actual graded word-sheaf endomorphisms.
-This is the algebra required to compare an indecomposable BMP summand to
-its adjoint, not an assertion that such a summand is already recognized. -/
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
-universe u us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  (l : List I) (hl : cs.IsReduced l)
-noncomputable section
-
- theorem bottEndOfSheaf_comp
-    (h g : GradedSheaf.Hom
-      (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
-      (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))) :
-    bottEndOfSheaf cs basis l hl (h.comp g)=
-      (bottEndOfSheaf cs basis l hl h).comp (bottEndOfSheaf cs basis l hl g) := by
-  apply LinearMap.ext
-  intro m
-  apply bottVertexProjection_injective cs l hl
-  funext x
-  exact (bottEndOfSheaf_coordinates cs basis l hl (h.comp g) m x).trans
-    ((congrArg (h.vertex x).map (bottEndOfSheaf_coordinates cs basis l hl g m x)).symm.trans
-      (bottEndOfSheaf_coordinates cs basis l hl h (bottEndOfSheaf cs basis l hl g m) x).symm)
-
- theorem bottAdjointSheaf_comp
-    (h g : GradedSheaf.Hom
-      (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
-      (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))) :
-    bottAdjointSheaf cs basis l hl (h.comp g)=
-      (bottAdjointSheaf cs basis l hl g).comp (bottAdjointSheaf cs basis l hl h) := by
-  apply GradedSheaf.Hom.eq_of_vertex
-  · exact fun e => bottProjectionSheaf_lower_surjective cs l 1 (cs.wordProd l) e
-  · intro x v
-    have hs : Function.Surjective (bottStalkProjection cs l x.val) :=
-      bottStalkProjection_surjective cs l x.val
-    obtain ⟨m,rfl⟩ := hs v
-    change ((bottAdjointSheaf cs basis l hl (h.comp g)).vertex x).map
-      (bottStalkProjection cs l x.val m)=
-        ((bottAdjointSheaf cs basis l hl g).vertex x).map
-          (((bottAdjointSheaf cs basis l hl h).vertex x).map (bottStalkProjection cs l x.val m))
-    simp only [bottAdjointSheaf_vertex]
-    change bottStalkProjection cs l x.val
-      (PerfectAdjoint.adjoint (bottSamelsonDuality cs l) (bottEndOfSheaf cs basis l hl (h.comp g)) m)=
-      bottStalkProjection cs l x.val
-        (PerfectAdjoint.adjoint (bottSamelsonDuality cs l) (bottEndOfSheaf cs basis l hl g)
-          (PerfectAdjoint.adjoint (bottSamelsonDuality cs l) (bottEndOfSheaf cs basis l hl h) m))
-    rw [bottEndOfSheaf_comp,PerfectAdjoint.adjoint_comp]
-    rfl
-
- theorem bottAdjointSheaf_involutive : Function.Involutive (bottAdjointSheaf cs basis l hl) := by
-  intro h
-  apply GradedSheaf.Hom.eq_of_vertex
-  · exact fun e => bottProjectionSheaf_lower_surjective cs l 1 (cs.wordProd l) e
-  · exact bottAdjointSheaf_twice_vertex cs basis l hl h
-
- theorem bottAdjointSheaf_injective : Function.Injective (bottAdjointSheaf cs basis l hl) :=
-  (bottAdjointSheaf_involutive cs basis l hl).injective
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! The endomorphism-corner formulation of a primitive graded sheaf
-projector. This is a criterion, not an assertion of a BMP splitting. -/
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
-  {B C : GradedSheaf (𝓐 := 𝓐) G}
-
- def Hom.zero : Hom B C where
-  vertex x := ⟨0,fun n _ _ => (C.vertex x).piece n |>.zero_mem⟩
-  edge e := ⟨0,fun n _ _ => (C.edge e).piece n |>.zero_mem⟩
-  lower e v := by simp
-  upper e v := by simp
-
- def Hom.id (B : GradedSheaf (𝓐 := 𝓐) G) : Hom B B where
-  vertex x := ModuleData.Hom.id (B.vertex x)
-  edge e := ModuleData.Hom.id (B.edge e)
-  lower _ _ := rfl
-  upper _ _ := rfl
-
- def Hom.PrimitiveProjector (p : Hom B B) : Prop :=
-  p.comp p=p ∧ ∀ f : Hom B B, f.comp f=f → f.comp p=f → p.comp f=f →
-    f=Hom.zero ∨ f=p
-
- theorem Hom.primitive_of_antiInvolution
-    (a : Hom B B → Hom B B)
-    (ha : Function.Involutive a)
-    (hac : ∀ f g, a (f.comp g)=(a g).comp (a f))
-    (haz : a Hom.zero=Hom.zero)
-    (p : Hom B B) (hp : p.PrimitiveProjector) :
-    (a p).PrimitiveProjector := by
-  constructor
-  · exact (hac p p).symm.trans (congrArg a hp.1)
-  · intro f hf hfp hpf
-    have hi : (a f).comp (a f)=a f := (hac f f).symm.trans (congrArg a hf)
-    have hleft : (a f).comp p=a f := by
-      have hh := congrArg a hpf
-      rw [hac,ha] at hh
-      exact hh
-    have hright : p.comp (a f)=a f := by
-      have hh := congrArg a hfp
-      rw [hac,ha] at hh
-      exact hh
-    rcases hp.2 (a f) hi hleft hright with hz | he
-    · left
-      have hh := congrArg a hz
-      rw [ha,haz] at hh
-      exact hh
-    · right
-      exact (ha f).symm.trans (congrArg a he)
-
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-
-namespace PerfectAdjoint
-open Module
-universe ur um
-variable {R : Type ur} [CommRing R] {N : Type um} [AddCommGroup N] [Module R N]
- theorem adjoint_zero (d : N ≃ₗ[R] Dual R N) : adjoint d (0 : End R N)=0 := by
-  apply LinearMap.ext
-  intro m
-  apply d.injective
-  apply LinearMap.ext
-  intro n
-  rw [pairing]
-  simp
-end PerfectAdjoint
-
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
-universe u us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  (l : List I) (hl : cs.IsReduced l)
-noncomputable section
-
- theorem bottEndOfSheaf_zero :
-    bottEndOfSheaf cs basis l hl GradedSheaf.Hom.zero=0 := by
-  apply LinearMap.ext
-  intro m
-  apply bottVertexProjection_injective cs l hl
-  funext x
-  change bottStalkProjection cs l x.val _=bottStalkProjection cs l x.val 0
-  rw [map_zero]
-  exact bottEndOfSheaf_coordinates cs basis l hl GradedSheaf.Hom.zero m x
-
- theorem bottAdjointSheaf_zero :
-    bottAdjointSheaf cs basis l hl GradedSheaf.Hom.zero=GradedSheaf.Hom.zero := by
-  apply GradedSheaf.Hom.eq_of_vertex
-  · exact fun e => bottProjectionSheaf_lower_surjective cs l 1 (cs.wordProd l) e
-  · intro x v
-    have hs : Function.Surjective (bottStalkProjection cs l x.val) :=
-      bottStalkProjection_surjective cs l x.val
-    obtain ⟨m,rfl⟩ := hs v
-    change ((bottAdjointSheaf cs basis l hl GradedSheaf.Hom.zero).vertex x).map
-      (bottStalkProjection cs l x.val m)=0
-    rw [bottAdjointSheaf_vertex,bottEndOfSheaf_zero]
-    change bottStalkProjection cs l x.val
-      (PerfectAdjoint.adjoint (bottSamelsonDuality cs l) 0 m)=0
-    rw [PerfectAdjoint.adjoint_zero]
-    exact map_zero _
-
- theorem bottAdjointSheaf_primitive
-    (p : GradedSheaf.Hom
-      (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
-      (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
-    (hp : p.PrimitiveProjector) :
-    (bottAdjointSheaf cs basis l hl p).PrimitiveProjector :=
-  GradedSheaf.Hom.primitive_of_antiInvolution _
-    (bottAdjointSheaf_involutive cs basis l hl)
-    (bottAdjointSheaf_comp cs basis l hl)
-    (bottAdjointSheaf_zero cs basis l hl) p hp
-
- theorem bottAdjointSheaf_primitive_iff
-    (p : GradedSheaf.Hom
-      (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
-      (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))) :
-    (bottAdjointSheaf cs basis l hl p).PrimitiveProjector ↔ p.PrimitiveProjector := by
-  constructor
-  · intro hp
-    have hh := bottAdjointSheaf_primitive cs basis l hl _ hp
-    rw [bottAdjointSheaf_involutive cs basis l hl p] at hh
-    exact hh
-  · exact bottAdjointSheaf_primitive cs basis l hl p
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Primitive-corner recognition for genuine graded sheaf splittings.
-This supplies the algebraic recognition step for a prospective BMP summand;
-it neither postulates the splitting nor the word/BMP character. -/
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
-  {B C : GradedSheaf (𝓐 := 𝓐) G}
-noncomputable section
-
- def SplitIn.liftEnd (s : SplitIn B C) (f : Hom B B) : Hom C C :=
-  s.inclusion.comp (f.comp s.projection)
-
- theorem SplitIn.liftEnd_injective (s : SplitIn B C) : Function.Injective s.liftEnd := by
-  intro f g h
-  apply Hom.eq_of_maps
-  · intro x v
-    have hh := congrArg (fun q : Hom C C => (s.projection.vertex x).map
-      ((q.vertex x).map ((s.inclusion.vertex x).map v))) h
-    change (s.projection.vertex x).map ((s.inclusion.vertex x).map
-      ((f.vertex x).map ((s.projection.vertex x).map ((s.inclusion.vertex x).map v))))=
-      (s.projection.vertex x).map ((s.inclusion.vertex x).map
-      ((g.vertex x).map ((s.projection.vertex x).map ((s.inclusion.vertex x).map v)))) at hh
-    simpa only [s.vertex_split] using hh
-  · intro e v
-    have hh := congrArg (fun q : Hom C C => (s.projection.edge e).map
-      ((q.edge e).map ((s.inclusion.edge e).map v))) h
-    change (s.projection.edge e).map ((s.inclusion.edge e).map
-      ((f.edge e).map ((s.projection.edge e).map ((s.inclusion.edge e).map v))))=
-      (s.projection.edge e).map ((s.inclusion.edge e).map
-      ((g.edge e).map ((s.projection.edge e).map ((s.inclusion.edge e).map v)))) at hh
-    simpa only [s.edge_split] using hh
-
- theorem SplitIn.liftEnd_comp (s : SplitIn B C) (f g : Hom B B) :
-    s.liftEnd (f.comp g)=(s.liftEnd f).comp (s.liftEnd g) := by
-  apply Hom.eq_of_maps
-  · intro x v
-    change (s.inclusion.vertex x).map ((f.vertex x).map ((g.vertex x).map
-      ((s.projection.vertex x).map v)))=
-      (s.inclusion.vertex x).map ((f.vertex x).map ((s.projection.vertex x).map
-      ((s.inclusion.vertex x).map ((g.vertex x).map ((s.projection.vertex x).map v)))))
-    rw [s.vertex_split]
-  · intro e v
-    change (s.inclusion.edge e).map ((f.edge e).map ((g.edge e).map
-      ((s.projection.edge e).map v)))=
-      (s.inclusion.edge e).map ((f.edge e).map ((s.projection.edge e).map
-      ((s.inclusion.edge e).map ((g.edge e).map ((s.projection.edge e).map v)))))
-    rw [s.edge_split]
-
- theorem SplitIn.liftEnd_zero (s : SplitIn B C) : s.liftEnd Hom.zero=Hom.zero := by
-  apply Hom.eq_of_maps
-  · intro x v
-    change (s.inclusion.vertex x).map 0=0
-    exact map_zero _
-  · intro e v
-    change (s.inclusion.edge e).map 0=0
-    exact map_zero _
-
- theorem SplitIn.primitive_id_of_projector (s : SplitIn B C)
-    (h : s.projector.PrimitiveProjector) : (Hom.id B).PrimitiveProjector := by
-  constructor
-  · apply Hom.eq_of_maps <;> intros <;> rfl
-  · intro f hf _ _
-    have hc : (s.liftEnd f).comp (s.liftEnd f)=s.liftEnd f :=
-      (s.liftEnd_comp f f).symm.trans (congrArg s.liftEnd hf)
-    have hi : s.liftEnd (Hom.id B)=s.projector := by
-      apply Hom.eq_of_maps <;> intros <;> rfl
-    have hleft : (s.liftEnd f).comp s.projector=s.liftEnd f := by
-      rw [← hi,← s.liftEnd_comp]
-      congr 1
-    have hright : s.projector.comp (s.liftEnd f)=s.liftEnd f := by
-      rw [← hi,← s.liftEnd_comp]
-      congr 1
-    rcases h.2 (s.liftEnd f) hc hleft hright with hz | he
-    · exact Or.inl (s.liftEnd_injective (hz.trans s.liftEnd_zero.symm))
-    · exact Or.inr (s.liftEnd_injective (he.trans hi.symm))
-
- theorem SplitIn.projector_eq_id_of_primitive (s : SplitIn B C)
-    (h : (Hom.id C).PrimitiveProjector)
-    (x : V) [Nontrivial (B.vertex x)] : s.projector=Hom.id C := by
-  have hi : s.projector.comp s.projector=s.projector := by
-    apply Hom.eq_of_maps
-    · exact s.projector_vertex
-    · exact s.projector_edge
-  have hl : s.projector.comp (Hom.id C)=s.projector := by
-    apply Hom.eq_of_maps <;> intros <;> rfl
-  have hr : (Hom.id C).comp s.projector=s.projector := by
-    apply Hom.eq_of_maps <;> intros <;> rfl
-  rcases h.2 s.projector hi hl hr with hz | he
-  · obtain ⟨v,hv⟩ := exists_ne (0 : B.vertex x)
-    exfalso
-    apply hv
-    have hh := congrArg (fun q : Hom C C => (s.projection.vertex x).map
-      ((q.vertex x).map ((s.inclusion.vertex x).map v))) hz
-    change (s.projection.vertex x).map ((s.inclusion.vertex x).map
-      ((s.projection.vertex x).map ((s.inclusion.vertex x).map v)))=
-      (s.projection.vertex x).map 0 at hh
-    simpa only [s.vertex_split,map_zero] using hh
-  · exact he
-
- def SplitIn.isoOfProjectorEqId (s : SplitIn B C) (h : s.projector=Hom.id C) : Iso B C where
-  vertex x :=
-    { toLinearMap := (s.inclusion.vertex x).map
-      invFun := (s.projection.vertex x).map
-      left_inv := s.vertex_split x
-      right_inv := fun v => congrArg (fun q : Hom C C => (q.vertex x).map v) h }
-  edge e :=
-    { toLinearMap := (s.inclusion.edge e).map
-      invFun := (s.projection.edge e).map
-      left_inv := s.edge_split e
-      right_inv := fun v => congrArg (fun q : Hom C C => (q.edge e).map v) h }
-  lower e v := s.inclusion.lower e v
-  upper e v := s.inclusion.upper e v
-  vertex_graded x := (s.inclusion.vertex x).graded
-  edge_graded e := (s.inclusion.edge e).graded
-
- def SplitIn.isoOfPrimitive (s : SplitIn B C)
-    (h : (Hom.id C).PrimitiveProjector)
-    (x : V) [Nontrivial (B.vertex x)] : Iso B C :=
-  s.isoOfProjectorEqId (s.projector_eq_id_of_primitive h x)
-
- variable [IsNoetherianRing A]
-
- theorem Hom.primitive_image (p : Hom B B) (hp : p.PrimitiveProjector) :
-    (Hom.id p.image).PrimitiveProjector := by
-  have hv : ∀ x v, (p.vertex x).map ((p.vertex x).map v)=(p.vertex x).map v := by
-    intro x v
-    exact congrArg (fun q : Hom B B => (q.vertex x).map v) hp.1
-  have he : ∀ e v, (p.edge e).map ((p.edge e).map v)=(p.edge e).map v := by
-    intro e v
-    exact congrArg (fun q : Hom B B => (q.edge e).map v) hp.1
-  let s := p.imageSplit hv he
-  have hs : s.projector=p := by
-    apply Hom.eq_of_maps <;> intros <;> rfl
-  apply s.primitive_id_of_projector
-  rw [hs]
-  exact hp
-
-end
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-/-! A normalized minimal boundary sheaf is primitive, and that property
-passes to its projector in every ACTUAL graded splitting. No word splitting
-is asserted here. -/
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
-  {B C : GradedSheaf (𝓐 := 𝓐) G}
-noncomputable section
-
- def SplitIn.downEnd (s : SplitIn B C) (f : Hom C C) : Hom B B :=
-  s.projection.comp (f.comp s.inclusion)
-
- theorem SplitIn.lift_down (s : SplitIn B C) (f : Hom C C)
-    (hf : f.comp s.projector=f) (hf' : s.projector.comp f=f) :
-    s.liftEnd (s.downEnd f)=f := by
-  have heq : s.liftEnd (s.downEnd f)=s.projector.comp (f.comp s.projector) := by
-    apply Hom.eq_of_maps <;> intros <;> rfl
-  rw [heq,hf,hf']
-
- theorem SplitIn.downEnd_idempotent (s : SplitIn B C) (f : Hom C C)
-    (hf : f.comp f=f) (hf' : s.projector.comp f=f) :
-    (s.downEnd f).comp (s.downEnd f)=s.downEnd f := by
-  have heq : (s.downEnd f).comp (s.downEnd f)=s.downEnd (f.comp (s.projector.comp f)) := by
-    apply Hom.eq_of_maps <;> intros <;> rfl
-  rw [heq,hf',hf]
-
- theorem SplitIn.primitive_projector_of_id (s : SplitIn B C)
-    (h : (Hom.id B).PrimitiveProjector) : s.projector.PrimitiveProjector := by
-  constructor
-  · apply Hom.eq_of_maps
-    · exact s.projector_vertex
-    · exact s.projector_edge
-  · intro f hf hfp hpf
-    have hg := s.downEnd_idempotent f hf hpf
-    have hl : (s.downEnd f).comp (Hom.id B)=s.downEnd f := by
-      apply Hom.eq_of_maps <;> intros <;> rfl
-    have hr : (Hom.id B).comp (s.downEnd f)=s.downEnd f := by
-      apply Hom.eq_of_maps <;> intros <;> rfl
-    rcases h.2 (s.downEnd f) hg hl hr with hz | he
-    · left
-      exact (s.lift_down f hfp hpf).symm.trans ((congrArg s.liftEnd hz).trans s.liftEnd_zero)
-    · right
-      have hi : s.liftEnd (Hom.id B)=s.projector := by
-        apply Hom.eq_of_maps <;> intros <;> rfl
-      exact (s.lift_down f hfp hpf).symm.trans ((congrArg s.liftEnd he).trans hi)
-
-end
-end KLInvariance.MomentGraph.GradedSheaf
-
-namespace KLInvariance.MomentGraph.BoundarySheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-  {hneg : ∀ n < 0, 𝓐 n=⊥} {V E : Type um} [PartialOrder V] [Finite V]
-  {G : OrderedGraph V E} {α : E → A} {b : V} [IsDomain A]
-  (B : BoundarySheaf hneg G α b)
-
- theorem primitive_identity : (GradedSheaf.Hom.id B.sheaf).PrimitiveProjector := by
-  constructor
-  · apply GradedSheaf.Hom.eq_of_maps <;> intros <;> rfl
-  · intro f hf _ _
-    have hi : ∀ x v, (f.toEndomorphism.vertex x).map ((f.toEndomorphism.vertex x).map v)=
-        (f.toEndomorphism.vertex x).map v := by
-      intro x v
-      exact congrArg (fun q : GradedSheaf.Hom B.sheaf B.sheaf => (q.vertex x).map v) hf
-    rcases B.idempotent_trivial f.toEndomorphism hi with hz | hi
-    · left
-      apply GradedSheaf.Hom.eq_of_maps
-      · intro x v
-        exact LinearMap.congr_fun (hz.1 x) v
-      · intro e v
-        exact LinearMap.congr_fun (hz.2 e) v
-    · right
-      apply GradedSheaf.Hom.eq_of_maps
-      · intro x v
-        exact LinearMap.congr_fun (hi.1 x) v
-      · intro e v
-        exact LinearMap.congr_fun (hi.2 e) v
-
- theorem split_projector_primitive
-    {C : GradedSheaf (𝓐 := 𝓐) G} (s : GradedSheaf.SplitIn B.sheaf C) :
-    s.projector.PrimitiveProjector :=
-  s.primitive_projector_of_id B.primitive_identity
-
-end KLInvariance.MomentGraph.BoundarySheaf
-
-end
-
-
-section
-
-/-! For a supplied normalized boundary sheaf and an ACTUAL word splitting,
-the adjoint image is primitively indecomposable. The splitting's existence
-and the BMP axioms on that image still require word recognition. -/
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
-universe u us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  (l : List I) (hl : cs.IsReduced l)
-  (α : Edge cs 1 (cs.wordProd l) → SymmetricCoefficient (M := M))
-  (B : BoundarySheaf (symmetricGrading_negative basis)
-    (graph cs 1 (cs.wordProd l)) α
-    ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩)
-  (s : GradedSheaf.SplitIn B.sheaf (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
-noncomputable section
-
- theorem bottBoundarySplitAdjoint_primitive :
-    (GradedSheaf.Hom.id (bottSplitAdjointSheaf cs basis l hl B.sheaf s)).PrimitiveProjector := by
-  let := symmetricCoefficientNoetherian basis
-  let := interval_finite cs 1 (cs.wordProd l)
-  exact GradedSheaf.Hom.primitive_image _ (bottAdjointSheaf_primitive cs basis l hl
-    s.projector (B.split_projector_primitive s))
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! The actual reduced word's top stalk is A in degree zero, integrally and
-in BOTH grading directions. This is the normalization needed to start a BMP
-summand construction, not a claim about the other stalks. -/
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded
-universe u us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  (l : List I) (hl : cs.IsReduced l)
-noncomputable section
-
-omit [Fintype σ] in
- theorem bottTopStalkEquiv_homogeneous_iff (n : ℤ)
-    (s : bottStalk cs l (cs.wordProd l)) :
-    bottTopStalkEquiv cs l hl s ∈ symmetricGrading basis n ↔
-      s ∈ bottStalkPiece cs basis l (cs.wordProd l) n := by
-  constructor
-  · intro hs e _
-    have he : e=⟨bottFullIndex l,bottWeight_full cs l⟩ :=
-      Subtype.ext (bottWeight_eq_top cs l hl e.val e.property)
-    subst e
-    exact hs
-  · intro hs
-    exact hs ⟨bottFullIndex l,bottWeight_full cs l⟩ (Set.mem_univ _)
-
- def bottTopHom : ModuleData.Hom (bottStalkData cs basis l (cs.wordProd l))
-    (ModuleData.unit (symmetricGrading_negative basis)) where
-  map := LinearMap.pi (fun _ => (bottTopStalkEquiv cs l hl).toLinearMap)
-  graded := by
-    intro n s hs j
-    change bottTopStalkEquiv cs l hl s ∈ symmetricGrading basis (n-0)
-    simpa only [sub_zero] using
-      (bottTopStalkEquiv_homogeneous_iff cs basis l hl n s).mpr hs
-
- def bottTopInvHom : ModuleData.Hom (ModuleData.unit (symmetricGrading_negative basis))
-    (bottStalkData cs basis l (cs.wordProd l)) where
-  map := (bottTopStalkEquiv cs l hl).symm.toLinearMap.comp (LinearMap.proj PUnit.unit)
-  graded := by
-    intro n a ha
-    apply (bottTopStalkEquiv_homogeneous_iff cs basis l hl n _).mp
-    change bottTopStalkEquiv cs l hl ((bottTopStalkEquiv cs l hl).symm (a PUnit.unit)) ∈ _
-    rw [LinearEquiv.apply_symm_apply]
-    have h := ha PUnit.unit
-    change a PUnit.unit ∈ symmetricGrading basis (n-0) at h
-    simpa only [sub_zero] using h
-
- theorem bottTopHom_inv (a : ModuleData.unit (symmetricGrading_negative basis)) :
-    (bottTopHom cs basis l hl).map ((bottTopInvHom cs basis l hl).map a)=a := by
-  funext j
-  change bottTopStalkEquiv cs l hl ((bottTopStalkEquiv cs l hl).symm (a PUnit.unit))=a j
-  rw [LinearEquiv.apply_symm_apply]
-
- theorem bottTopInvHom_top (s : bottStalk cs l (cs.wordProd l)) :
-    (bottTopInvHom cs basis l hl).map ((bottTopHom cs basis l hl).map s)=s :=
-  (bottTopStalkEquiv cs l hl).symm_apply_apply s
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! A split pair between rank-one normalized degree-zero modules reverses.
-This uses commutativity of the genuine coefficient ring, not finite-dimensional
-vector-space reasoning or a restricted perfect form. -/
-
-end
-
-end OAI
+import OAI.RepresentationTheory.KazhdanLusztig.IntegralPairings
+
+namespace OAI
+
+/-!
+Graded stalk duality, integral perfect balanced all-sections pairings, recognition and top grading.
+-/
+
+section
+
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ theorem bottPairing_dualToSupport (l : List I) (x : W)
+    (lam : Dual (SymmetricCoefficient (M := M)) (bottStalk cs l x))
+    (m : BottSamelsonModule cs l) :
+    bottSamelsonDuality cs l (bottStalkDualToSupport cs l x lam).val m =
+      lam (bottStalkProjection cs l x m) := by
+  change bottSamelsonDuality cs l
+    ((bottSamelsonDuality cs l).symm ((bottStalkProjection cs l x).dualMap lam)) m = _
+  rw [LinearEquiv.apply_symm_apply]
+  rfl
+
+ theorem bottStalkDuality_apply_projection (l : List I) (x : W)
+    (s : bottWeightSupported cs l x) (m : BottSamelsonModule cs l) :
+    bottStalkDuality cs l x s (bottStalkProjection cs l x m) =
+      bottSamelsonDuality cs l s.val m := by
+  have he : bottStalkDualToSupport cs l x (bottStalkDuality cs l x s)=s :=
+    (LinearEquiv.ofBijective (bottStalkDualToSupport cs l x)
+      (bottStalkDualToSupport_bijective cs l x)).apply_symm_apply s
+  calc
+    _ = bottSamelsonDuality cs l
+        (bottStalkDualToSupport cs l x (bottStalkDuality cs l x s)).val m :=
+      (bottPairing_dualToSupport cs l x _ m).symm
+    _ = _ := by rw [he]
+
+ theorem bottStalkDuality_homogeneous (l : List I) (x : W) (n : ℤ)
+    (s : bottWeightSupported cs l x) (hs : s ∈ bottSupportPiece cs basis l x n)
+    (j : ℤ) (v : bottStalk cs l x) (hv : v ∈ bottStalkPiece cs basis l x j) :
+    bottStalkDuality cs l x s v ∈ symmetricGrading basis (n+j-l.length) := by
+  obtain ⟨m,hm,rfl⟩ := bottStalk_homogeneous_lift cs basis l x j v hv
+  rw [bottStalkDuality_apply_projection]
+  exact bottDuality_homogeneous cs basis l hs hm
+
+ theorem bottDualToSupport_homogeneous (l : List I) (x : W) (r : ℤ)
+    (lam : Dual (SymmetricCoefficient (M := M)) (bottStalk cs l x))
+    (hlam : ∀ j v, v ∈ bottStalkPiece cs basis l x j →
+      lam v ∈ symmetricGrading basis (j+r)) :
+    bottStalkDualToSupport cs l x lam ∈ bottSupportPiece cs basis l x (r+l.length) := by
+  change (bottSamelsonDuality cs l).symm ((bottStalkProjection cs l x).dualMap lam) ∈
+    bottPiece cs basis l (r+l.length)
+  apply perfect_inverse_of_homogeneous_functional (symmetricGrading basis) (bottPiece cs basis l)
+    (bottSamelsonDuality cs l) l.length
+    (fun _ _ _ _ hm hn => bottDuality_homogeneous cs basis l hm hn) r
+  intro j m hm
+  exact hlam j _ (bottStalkProjection_homogeneous cs basis l x j m hm)
+
+ theorem bottStalkDuality_homogeneous_iff (l : List I) (x : W) (n : ℤ)
+    (s : bottWeightSupported cs l x) :
+    (∀ j v, v ∈ bottStalkPiece cs basis l x j →
+      bottStalkDuality cs l x s v ∈ symmetricGrading basis (n+j-l.length)) ↔
+        s ∈ bottSupportPiece cs basis l x n := by
+  constructor
+  · intro h
+    have hh := bottDualToSupport_homogeneous cs basis l x (n-l.length)
+      (bottStalkDuality cs l x s) (by
+        intro j v hv
+        have he : n+j-(l.length:ℤ) = j+(n-l.length) := by omega
+        rw [← he]
+        exact h j v hv)
+    have he : bottStalkDualToSupport cs l x (bottStalkDuality cs l x s)=s :=
+      (LinearEquiv.ofBijective (bottStalkDualToSupport cs l x)
+        (bottStalkDualToSupport_bijective cs l x)).apply_symm_apply s
+    simpa only [he,sub_add_cancel] using hh
+  · intro h
+    exact bottStalkDuality_homogeneous cs basis l x n s h
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! The actual word adjoint preserves homogeneity. Projection kernels are
+annihilators of the actual supported lattices, integrally, with no localization
+or freeness assumption on the projection lattice. -/
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+ theorem bott_projection_zero_iff_annihilates_support (l : List I) (x : W)
+    (m : BottSamelsonModule cs l) :
+    bottStalkProjection cs l x m=0 ↔ ∀ s : bottWeightSupported cs l x,
+      bottSamelsonDuality cs l s.val m=0 := by
+  constructor
+  · intro hm s
+    exact bott_supported_annihilates cs l x s.property m
+      ((bottStalkProjection_eq_zero cs l x m).mp hm)
+  · intro h
+    apply Subtype.ext
+    funext e
+    let lam : Dual (SymmetricCoefficient (M := M)) (bottStalk cs l x) :=
+      (LinearMap.proj e).comp (bottStalk cs l x).subtype
+    have he := h (bottStalkDualToSupport cs l x lam)
+    rw [bottPairing_dualToSupport] at he
+    exact he
+
+ def bottAdjoint (l : List I) (f : End (SymmetricCoefficient (M := M))
+    (BottSamelsonModule cs l)) := PerfectAdjoint.adjoint (bottSamelsonDuality cs l) f
+
+ theorem bottAdjoint_preserves_kernel (l : List I) (x : W)
+    (f : End (SymmetricCoefficient (M := M)) (BottSamelsonModule cs l))
+    (hf : ∀ s, s ∈ bottWeightSupported cs l x → f s ∈ bottWeightSupported cs l x)
+    (m : BottSamelsonModule cs l) (hm : bottStalkProjection cs l x m=0) :
+    bottStalkProjection cs l x (bottAdjoint cs l f m)=0 := by
+  apply (bott_projection_zero_iff_annihilates_support cs l x _).mpr
+  intro s
+  change bottSamelsonDuality cs l s.val (PerfectAdjoint.adjoint (bottSamelsonDuality cs l) f m)=0
+  rw [PerfectAdjoint.symmetric_pairing (bottSamelsonDuality cs l)
+    (bottSamelson_symmetric cs l)]
+  exact (bott_projection_zero_iff_annihilates_support cs l x m).mp hm ⟨f s.val,hf s.val s.property⟩
+
+ theorem bottAdjoint_preserves_support (l : List I) (x : W)
+    (f : End (SymmetricCoefficient (M := M)) (BottSamelsonModule cs l))
+    (hf : ∀ m, bottStalkProjection cs l x m=0 → bottStalkProjection cs l x (f m)=0)
+    (m : BottSamelsonModule cs l) (hm : m ∈ bottWeightSupported cs l x) :
+    bottAdjoint cs l f m ∈ bottWeightSupported cs l x := by
+  apply bott_annihilates_supported cs l x
+  intro n hn
+  change bottSamelsonDuality cs l (PerfectAdjoint.adjoint (bottSamelsonDuality cs l) f m) n=0
+  rw [PerfectAdjoint.pairing]
+  apply bott_supported_annihilates cs l x hm
+  exact (bottStalkProjection_eq_zero cs l x (f n)).mp
+    (hf n ((bottStalkProjection_eq_zero cs l x n).mpr hn))
+
+ theorem bottAdjoint_homogeneous
+    {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+    (l : List I) (f : End (SymmetricCoefficient (M := M)) (BottSamelsonModule cs l))
+    (hf : ∀ n m, m ∈ bottPiece cs basis l n → f m ∈ bottPiece cs basis l n)
+    (n : ℤ) (m : BottSamelsonModule cs l) (hm : m ∈ bottPiece cs basis l n) :
+    bottAdjoint cs l f m ∈ bottPiece cs basis l n :=
+  adjoint_homogeneous (symmetricGrading basis) (bottPiece cs basis l)
+    (bottSamelsonDuality cs l) l.length
+    (fun _ _ _ _ h1 h2 => bottDuality_homogeneous cs basis l h1 h2) f hf n m hm
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Integral realization of tensor words as ALL sections of their actual
+pair-projection sheaf. This assertion alone is not a BMP summand theorem,
+flabbiness/standard-filtration theorem, or EW character theorem. -/
+namespace KLInvariance.TitsSpace
+open Module MomentGraph BruhatGraph FiniteCharacterLocalization
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+ def bottVertexProjection (l : List I) (u b : W) : BottSamelsonModule cs l →ₗ[
+    SymmetricCoefficient (M := M)]
+    (∀ x : Interval cs u b, bottStalk cs l x.val) :=
+  LinearMap.pi (fun x => bottStalkProjection cs l x.val)
+
+ theorem bottVertexProjection_injective (l : List I) (hl : cs.IsReduced l) :
+    Function.Injective (bottVertexProjection cs l 1 (cs.wordProd l)) := by
+  intro m n h
+  apply bottSection_injective cs l hl
+  exact Subtype.ext h
+
+ theorem bottVertexProjection_stable (l : List I) (u b : W)
+    (a : SymmetricCoefficient (M := M)) :
+    (fun x : Interval cs u b => coefficientAction cs x.val a) ∈
+      stabilizer (LinearMap.range (bottVertexProjection cs l u b)) := by
+  rintro z ⟨m,rfl⟩
+  refine ⟨bottSamelsonRight cs l a m,?_⟩
+  funext x
+  exact bottStalkProjection_right cs l x.val a m
+
+ theorem bottVertexProjection_scaled (l : List I) (hl : cs.IsReduced l)
+    (z : ∀ x : Interval cs 1 (cs.wordProd l), bottStalk cs l x.val) :
+    bottCoordinateDenominator cs l • z ∈
+      LinearMap.range (bottVertexProjection cs l 1 (cs.wordProd l)) := by
+  let c (e : BottIndex l) : SymmetricCoefficient (M := M) :=
+    (z ⟨bottWeight cs l e,one_bruhat cs _,bottWeight_bruhat cs l hl e⟩).val ⟨e,rfl⟩
+  obtain ⟨m,hm⟩ := bottEvaluation_contains_scaled_coordinates cs l c
+  refine ⟨m,?_⟩
+  funext x
+  apply Subtype.ext
+  funext e
+  rcases x with ⟨x,hx⟩
+  rcases e with ⟨e,he⟩
+  change bottWeight cs l e=x at he
+  subst x
+  exact hm e
+
+
+ theorem bottSection_pair_lift (l : List I) (u b : W)
+    (s : (bottProjectionSheaf cs l u b).sections Set.univ)
+    (x y : Interval cs u b) (t : W) (ht : cs.IsReflection t) (he : t*x.val=y.val) :
+    ∃ m : BottSamelsonModule cs l,
+      bottStalkProjection cs l x.val m=s.val x ∧ bottStalkProjection cs l y.val m=s.val y := by
+  have he' : t*y.val=x.val := by
+    rw [← he,← mul_assoc,ht.mul_self,one_mul]
+  have hne : cs.length y.val ≠ cs.length x.val := by
+    rw [← he]
+    exact ht.length_mul_right_ne x.val
+  rcases lt_or_gt_of_ne hne with hlt | hlt
+  · let e : Edge cs u b := ⟨(y,x),hlt,⟨t,ht,he'.symm⟩⟩
+    have hs := s.property e (Set.mem_univ _) (Set.mem_univ _)
+    obtain ⟨m,hm,hn⟩ := (PairProjection.compatible_iff _ _ _ _).mp hs
+    exact ⟨m,hn,hm⟩
+  · let e : Edge cs u b := ⟨(x,y),hlt,⟨t,ht,he.symm⟩⟩
+    have hs := s.property e (Set.mem_univ _) (Set.mem_univ _)
+    exact (PairProjection.compatible_iff _ _ _ _).mp hs
+
+ theorem bottSection_prime_lift (l : List I) (u b : W)
+    (s : (bottProjectionSheaf cs l u b).sections Set.univ)
+    (p : SymmetricCoefficient (M := M)) (hp : Prime p) (x : Interval cs u b) :
+    ∃ m : BottSamelsonModule cs l, ∀ y : Interval cs u b,
+      (∀ a, p ∣ coefficientAction cs x.val a-coefficientAction cs y.val a) →
+      bottStalkProjection cs l y.val m=s.val y := by
+  classical
+  by_cases hx : ∃ y : Interval cs u b, x ≠ y ∧
+      ∀ a, p ∣ coefficientAction cs x.val a-coefficientAction cs y.val a
+  · obtain ⟨y,hxy,hcon⟩ := hx
+    obtain ⟨t,ht,he,hclass⟩ := prime_character_class cs p hp
+      (fun h => hxy (Subtype.ext h)) hcon
+    obtain ⟨m,hm,hn⟩ := bottSection_pair_lift cs l u b s x y t ht he.symm
+    refine ⟨m,?_⟩
+    intro z hz
+    rcases (hclass z.val).mp hz with hxz | hzy
+    · have h : z=x := Subtype.ext hxz.symm
+      subst z
+      exact hm
+    · have h : z=y := Subtype.ext (hzy.trans he.symm)
+      subst z
+      exact hn
+  · obtain ⟨m,hm⟩ := bottStalkProjection_surjective cs l x.val (s.val x)
+    refine ⟨m,?_⟩
+    intro y hy
+    have h : y=x := by
+      by_contra h
+      exact hx ⟨y,Ne.symm h,hy⟩
+    subst y
+    exact hm
+
+ theorem bottSection_local_denominator (l : List I) (u b : W)
+    (s : (bottProjectionSheaf cs l u b).sections Set.univ)
+    (p : SymmetricCoefficient (M := M)) (hp : Prime p) :
+    ∃ a : SymmetricCoefficient (M := M), ¬ p ∣ a ∧ a • s.val ∈
+      LinearMap.range (bottVertexProjection cs l u b) := by
+  classical
+  let : Finite (Interval cs u b) := interval_finite cs u b
+  let : Fintype (Interval cs u b) := Fintype.ofFinite _
+  apply character_local_denominator (E := fun x : Interval cs u b => bottStalk cs l x.val)
+    (LinearMap.range (bottVertexProjection cs l u b)) p hp
+    (fun a x => coefficientAction cs x.val a) (fun a => bottVertexProjection_stable cs l u b a) s.val
+  intro x
+  obtain ⟨m,hm⟩ := bottSection_prime_lift cs l u b s p hp x
+  exact ⟨bottVertexProjection cs l u b m,⟨m,rfl⟩,hm⟩
+
+ theorem bottSection_surjective (l : List I) (hl : cs.IsReduced l) :
+    Function.Surjective (bottSection cs l 1 (cs.wordProd l)) := by
+  intro s
+  let L := LinearMap.range (bottVertexProjection cs l 1 (cs.wordProd l))
+  let : Module.Free (SymmetricCoefficient (M := M)) L :=
+    Module.Free.of_equiv (LinearEquiv.ofInjective _ (bottVertexProjection_injective cs l hl))
+  let : UniqueFactorizationMonoid (SymmetricCoefficient (M := M)) :=
+    (SymmetricAlgebra.equivMvPolynomial (Module.Free.chooseBasis ℝ (Extended M))).symm.toRingEquiv.toMulEquiv.uniqueFactorizationMonoid inferInstance
+  have h : s.val ∈ L := EdgeLattice.mem_of_prime_local_membership L s.val
+    ⟨bottCoordinateDenominator cs l,bottCoordinateDenominator_ne_zero cs l,
+      bottVertexProjection_scaled cs l hl s.val⟩
+    (fun p hp => bottSection_local_denominator cs l 1 (cs.wordProd l) s p hp)
+  obtain ⟨m,hm⟩ := h
+  exact ⟨m,Subtype.ext hm⟩
+
+ def bottSectionEquiv (l : List I) (hl : cs.IsReduced l) :
+    BottSamelsonModule cs l ≃ₗ[SymmetricCoefficient (M := M)]
+      (bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ :=
+  LinearEquiv.ofBijective (bottSection cs l 1 (cs.wordProd l))
+    ⟨bottSection_injective cs l hl,bottSection_surjective cs l hl⟩
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+namespace TransportPerfectPairing
+open Module
+variable {A M N : Type*} [CommRing A] [AddCommGroup M] [Module A M]
+  [AddCommGroup N] [Module A N]
+
+theorem balanced (e : M ≃ₗ[A] N) (D : M ≃ₗ[A] Dual A M) (a : End A M)
+    (h : ∀ x y, D (a x) y = D x (a y)) (s z : N) :
+    (e.symm.trans (D.trans e.symm.dualMap))
+      ((e.toLinearMap.comp (a.comp e.symm.toLinearMap)) s) z =
+    (e.symm.trans (D.trans e.symm.dualMap)) s
+      ((e.toLinearMap.comp (a.comp e.symm.toLinearMap)) z) := by
+  change D (e.symm (e (a (e.symm s)))) (e.symm z) =
+    D (e.symm s) (e.symm (e (a (e.symm z))))
+  simp only [LinearEquiv.symm_apply_apply]
+  exact h _ _
+
+theorem symmetric (e : M ≃ₗ[A] N) (D : M ≃ₗ[A] Dual A M)
+    (h : ∀ x y, D x y = D y x) (s z : N) :
+    (e.symm.trans (D.trans e.symm.dualMap)) s z =
+    (e.symm.trans (D.trans e.symm.dualMap)) z s :=
+  h (e.symm s) (e.symm z)
+
+end TransportPerfectPairing
+
+end
+
+
+section
+
+/-! Integral perfect balanced self-duality on all sections of the genuine
+word projection sheaf. This is the word object, NOT the indecomposable BMP
+object. Selection of a BMP summand with the required normalization is not
+made or assumed here. -/
+namespace KLInvariance.TitsSpace
+open Module MomentGraph BruhatGraph
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+ def bottSectionDuality (l : List I) (hl : cs.IsReduced l) :
+    (bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ ≃ₗ[
+      SymmetricCoefficient (M := M)]
+      Dual (SymmetricCoefficient (M := M))
+        ((bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ) :=
+  (bottSectionEquiv cs l hl).symm.trans
+    ((bottSamelsonDuality cs l).trans (bottSectionEquiv cs l hl).symm.dualMap)
+
+ theorem bottSectionDuality_apply (l : List I) (hl : cs.IsReduced l)
+    (m n : BottSamelsonModule cs l) :
+    bottSectionDuality cs l hl (bottSection cs l 1 (cs.wordProd l) m)
+      (bottSection cs l 1 (cs.wordProd l) n) = bottSamelsonDuality cs l m n := by
+  change bottSamelsonDuality cs l
+    ((bottSectionEquiv cs l hl).symm ((bottSectionEquiv cs l hl) m))
+    ((bottSectionEquiv cs l hl).symm ((bottSectionEquiv cs l hl) n))= _
+  simp only [LinearEquiv.symm_apply_apply]
+
+ def bottSectionRight (l : List I) (hl : cs.IsReduced l)
+    (a : SymmetricCoefficient (M := M)) :
+    End (SymmetricCoefficient (M := M))
+      ((bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ) :=
+  (bottSectionEquiv cs l hl).toLinearMap.comp
+    ((bottSamelsonRight cs l a).comp (bottSectionEquiv cs l hl).symm.toLinearMap)
+
+ theorem bottSectionRight_apply (l : List I) (hl : cs.IsReduced l)
+    (a : SymmetricCoefficient (M := M)) (m : BottSamelsonModule cs l) :
+    bottSectionRight cs l hl a (bottSection cs l 1 (cs.wordProd l) m) =
+      bottSection cs l 1 (cs.wordProd l) (bottSamelsonRight cs l a m) := by
+  change (bottSectionEquiv cs l hl) (bottSamelsonRight cs l a
+    ((bottSectionEquiv cs l hl).symm ((bottSectionEquiv cs l hl) m))) = _
+  rw [LinearEquiv.symm_apply_apply]
+  rfl
+
+ theorem bottSectionRight_coordinates (l : List I) (hl : cs.IsReduced l)
+    (a : SymmetricCoefficient (M := M))
+    (s : (bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ)
+    (x : Interval cs 1 (cs.wordProd l)) :
+    (bottSectionRight cs l hl a s).val x=coefficientAction cs x.val a • s.val x := by
+  obtain ⟨m,rfl⟩ := bottSection_surjective cs l hl s
+  rw [bottSectionRight_apply]
+  exact bottStalkProjection_right cs l x.val a m
+
+ theorem bottSection_balanced (l : List I) (hl : cs.IsReduced l)
+    (a : SymmetricCoefficient (M := M))
+    (s z : (bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ) :
+    bottSectionDuality cs l hl (bottSectionRight cs l hl a s) z =
+      bottSectionDuality cs l hl s (bottSectionRight cs l hl a z) := by
+  exact TransportPerfectPairing.balanced (bottSectionEquiv cs l hl)
+    (bottSamelsonDuality cs l) (bottSamelsonRight cs l a)
+    (bottSamelson_balanced cs l a) s z
+
+ theorem bottSection_symmetric (l : List I) (hl : cs.IsReduced l)
+    (s z : (bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ) :
+    bottSectionDuality cs l hl s z=bottSectionDuality cs l hl z s := by
+  exact TransportPerfectPairing.symmetric (bottSectionEquiv cs l hl)
+    (bottSamelsonDuality cs l) (bottSamelson_symmetric cs l) s z
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! The word SECTION pairing has precisely the word-length shift in the
+actual coordinatewise sheaf grading. It is not an asserted pairing on the
+indecomposable BMP boundary sheaf. -/
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ theorem bottPiece_iff_stalks (l : List I) (hl : cs.IsReduced l) (n : ℤ)
+    (m : BottSamelsonModule cs l) : m ∈ bottPiece cs basis l n ↔
+      ∀ x : Interval cs 1 (cs.wordProd l),
+        bottStalkProjection cs l x.val m ∈ bottStalkPiece cs basis l x.val n := by
+  constructor
+  · intro hm x
+    exact bottStalkProjection_homogeneous cs basis l x.val n m hm
+  · intro hm
+    have he : component (bottPiece cs basis l) n m=m := by
+      apply bottVertexProjection_injective cs l hl
+      funext x
+      have hh := map_component (bottPiece cs basis l) (bottStalkPiece cs basis l x.val)
+        (bottStalkProjection cs l x.val) 0 (by
+          simpa only [add_zero] using bottStalkProjection_homogeneous cs basis l x.val) n m
+      rw [sub_zero,show component (bottStalkPiece cs basis l x.val) n
+        (bottStalkProjection cs l x.val m)=bottStalkProjection cs l x.val m from
+        DirectSum.decompose_of_mem_same _ (hm x)] at hh
+      exact hh.symm
+    rw [← he]
+    exact (DirectSum.decompose (bottPiece cs basis l) m n).property
+
+ theorem bottSectionDuality_homogeneous (l : List I) (hl : cs.IsReduced l)
+    (i j : ℤ)
+    (s z : (bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ)
+    (hs : ∀ x, s.val x ∈ bottStalkPiece cs basis l x.val i)
+    (hz : ∀ x, z.val x ∈ bottStalkPiece cs basis l x.val j) :
+    bottSectionDuality cs l hl s z ∈ symmetricGrading basis (i+j-l.length) := by
+  obtain ⟨m,rfl⟩ := bottSection_surjective cs l hl s
+  obtain ⟨n,rfl⟩ := bottSection_surjective cs l hl z
+  rw [bottSectionDuality_apply]
+  exact bottDuality_homogeneous cs basis l ((bottPiece_iff_stalks cs basis l hl i m).mpr hs)
+    ((bottPiece_iff_stalks cs basis l hl j n).mpr hz)
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Section maps for the actual graded sheaf morphisms. A sheaf splitting
+really gives an integral graded section splitting; it is not replaced by a
+generic-fibre splitting. -/
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
+  {G : OrderedGraph V E} {B C : GradedSheaf (𝓐 := 𝓐) G}
+noncomputable section
+
+ def Hom.assignmentMap (h : Hom B C) : B.forget.Assignment →ₗ[A] C.forget.Assignment where
+  toFun f x := (h.vertex x).map (f x)
+  map_add' f g := by funext x; exact map_add (h.vertex x).map (f x) (g x)
+  map_smul' r f := by funext x; exact map_smul (h.vertex x).map r (f x)
+
+ theorem Hom.assignmentMap_compatible (h : Hom B C) (Ω : Set V)
+    (f : B.forget.Assignment) (hf : B.forget.CompatibleOn Ω f) :
+    C.forget.CompatibleOn Ω (h.assignmentMap f) := by
+  intro e hs ht
+  change (C.lower e).map ((h.vertex _).map (f _)) =
+    (C.upper e).map ((h.vertex _).map (f _))
+  exact (h.lower e (f (G.source e))).symm.trans
+    ((congrArg (h.edge e).map (hf e hs ht)).trans (h.upper e (f (G.target e))))
+
+ def Hom.sectionMap (h : Hom B C) (Ω : Set V) :
+    B.forget.sections Ω →ₗ[A] C.forget.sections Ω :=
+  (h.assignmentMap.domRestrict (B.forget.sections Ω)).codRestrict (C.forget.sections Ω)
+    (fun f => h.assignmentMap_compatible Ω f.val f.property)
+
+ theorem Hom.sectionMap_apply (h : Hom B C) (Ω : Set V)
+    (f : B.forget.sections Ω) (x : V) :
+    (h.sectionMap Ω f).val x=(h.vertex x).map (f.val x) := rfl
+
+ variable [Fintype V] [Fintype E] [IsNoetherianRing A]
+
+ def Hom.gradedSectionMap (h : Hom B C) (Ω : Set V) :
+    ModuleData.Hom (B.sections Ω) (C.sections Ω) where
+  map := h.sectionMap Ω
+  graded := by
+    intro n f hf x hx
+    exact (h.vertex x).graded n (f.val x) (hf x hx)
+
+omit [Fintype V] [Fintype E] [IsNoetherianRing A] in
+ theorem SplitIn.sections_split (s : SplitIn B C) (Ω : Set V)
+    (f : B.forget.sections Ω) :
+    s.projection.sectionMap Ω (s.inclusion.sectionMap Ω f)=f := by
+  apply Subtype.ext
+  funext x
+  exact s.vertex_split x (f.val x)
+
+end
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+/-! Endomorphisms of the ACTUAL graded word sheaf act on its integral tensor
+module and preserve every stalk kernel and single-weight support lattice. -/
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
+universe u us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  (l : List I) (hl : cs.IsReduced l)
+noncomputable section
+
+ def bottEndOfSheaf (h : GradedSheaf.Hom
+    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
+    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))) :
+    End (SymmetricCoefficient (M := M)) (BottSamelsonModule cs l) :=
+  (bottSectionEquiv cs l hl).symm.toLinearMap.comp
+    ((h.sectionMap Set.univ).comp (bottSectionEquiv cs l hl).toLinearMap)
+
+ theorem bottEndOfSheaf_coordinates (h : GradedSheaf.Hom
+    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
+    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
+    (m : BottSamelsonModule cs l) (x : Interval cs 1 (cs.wordProd l)) :
+    bottStalkProjection cs l x.val (bottEndOfSheaf cs basis l hl h m)=
+      (h.vertex x).map (bottStalkProjection cs l x.val m) := by
+  have he := congrArg (fun s : (bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ =>
+    s.val x) ((bottSectionEquiv cs l hl).apply_symm_apply
+      (h.sectionMap Set.univ ((bottSectionEquiv cs l hl) m)))
+  exact he
+
+ theorem bottEndOfSheaf_homogeneous (h : GradedSheaf.Hom
+    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
+    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
+    (n : ℤ) (m : BottSamelsonModule cs l) (hm : m ∈ bottPiece cs basis l n) :
+    bottEndOfSheaf cs basis l hl h m ∈ bottPiece cs basis l n := by
+  apply (bottPiece_iff_stalks cs basis l hl n _).mpr
+  intro x
+  rw [bottEndOfSheaf_coordinates]
+  exact (h.vertex x).graded n _ (bottStalkProjection_homogeneous cs basis l x.val n m hm)
+
+ theorem bottEndOfSheaf_preserves_kernel (h : GradedSheaf.Hom
+    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
+    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
+    (x : Interval cs 1 (cs.wordProd l)) (m : BottSamelsonModule cs l)
+    (hm : bottStalkProjection cs l x.val m=0) :
+    bottStalkProjection cs l x.val (bottEndOfSheaf cs basis l hl h m)=0 := by
+  rw [bottEndOfSheaf_coordinates,hm]
+  exact (h.vertex x).map.map_zero
+
+ theorem bottEndOfSheaf_preserves_support (h : GradedSheaf.Hom
+    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
+    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
+    (x : Interval cs 1 (cs.wordProd l)) (m : BottSamelsonModule cs l)
+    (hm : m ∈ bottWeightSupported cs l x.val) :
+    bottEndOfSheaf cs basis l hl h m ∈ bottWeightSupported cs l x.val := by
+  intro e he
+  let y : Interval cs 1 (cs.wordProd l) :=
+    ⟨bottWeight cs l e,one_bruhat cs _,bottWeight_bruhat cs l hl e⟩
+  have hmy : bottStalkProjection cs l y.val m=0 := by
+    apply Subtype.ext
+    funext f
+    exact hm f.val (fun hx => he (f.property.symm.trans hx))
+  have hfy := bottEndOfSheaf_preserves_kernel cs basis l hl h y m hmy
+  exact congrFun (congrArg Subtype.val hfy) ⟨e,rfl⟩
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+namespace LinearDescent
+universe ur um ux
+variable {R : Type ur} [CommRing R]
+  {M : Type um} [AddCommGroup M] [Module R M]
+  {X : Type ux} [AddCommGroup X] [Module R X]
+  (f : M →ₗ[R] X) (hf : Function.Surjective f) (e : Module.End R M)
+  (he : ∀ m, f m=0 → f (e m)=0)
+noncomputable section
+
+ def descend : Module.End R X :=
+  ((LinearMap.ker f).liftQ (f.comp e) he).comp
+    (f.quotKerEquivOfSurjective hf).symm.toLinearMap
+
+ theorem descend_apply (m : M) : descend f hf e he (f m)=f (e m) := by
+  change (LinearMap.ker f).liftQ (f.comp e) he
+    ((f.quotKerEquivOfSurjective hf).symm (f m))=f (e m)
+  rw [LinearMap.quotKerEquivOfSurjective_symm_apply]
+  rfl
+
+ theorem descend_idempotent (hee : ∀ m, e (e m)=e m) (x : X) :
+    descend f hf e he (descend f hf e he x)=descend f hf e he x := by
+  obtain ⟨m,rfl⟩ := hf x
+  rw [descend_apply,descend_apply,hee]
+
+end
+end LinearDescent
+
+end
+
+
+section
+
+namespace KLInvariance.Graded.ModuleData
+open PairProjection LinearDescent
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} {M X Y : ModuleData.{uk,ua,um} 𝓐}
+  (e : Hom M M) (f : Hom M X) (g : Hom M Y)
+  (hf : Function.Surjective f.map) (hg : Function.Surjective g.map)
+  (hef : ∀ m, f.map m=0 → f.map (e.map m)=0)
+  (heg : ∀ m, g.map m=0 → g.map (e.map m)=0)
+noncomputable section
+
+ def descended : Hom X X where
+  map := descend f.map hf e.map hef
+  graded := by
+    intro n x hx
+    obtain ⟨m,hm,rfl⟩ := f.homogeneous_lift hf n x hx
+    rw [descend_apply]
+    exact f.graded n _ (e.graded n m hm)
+
+include heg in
+ theorem lower_kernel_preserved (x : X) (hx : lower f.map g.map x=0) :
+    lower f.map g.map ((descended e f hf hef).map x)=0 := by
+  obtain ⟨m,hm,hn⟩ := (kernel_lower f.map g.map x).mp hx
+  apply (kernel_lower f.map g.map _).mpr
+  refine ⟨e.map m,?_,heg m hn⟩
+  rw [← hm]
+  exact (descend_apply f.map hf e.map hef m).symm
+
+ def descendedEdge : Hom (pairEdge f g hg) (pairEdge f g hg) where
+  map := descend (lower f.map g.map) (lower_surjective f.map g.map hg)
+    (descended e f hf hef).map (lower_kernel_preserved e f g hf hef heg)
+  graded := by
+    intro n z hz
+    obtain ⟨x,hx,rfl⟩ := hz
+    refine ⟨(descended e f hf hef).map x,(descended e f hf hef).graded n x hx,?_⟩
+    exact (descend_apply (lower f.map g.map) (lower_surjective f.map g.map hg)
+      (descended e f hf hef).map (lower_kernel_preserved e f g hf hef heg) x).symm
+
+ theorem descendedEdge_lower (x : X) :
+    (descendedEdge e f g hf hg hef heg).map (lower f.map g.map x)=
+      lower f.map g.map ((descended e f hf hef).map x) :=
+  descend_apply _ _ _ _ x
+
+ theorem descendedEdge_upper (y : Y) :
+    (descendedEdge e f g hf hg hef heg).map (upper f.map g.map y)=
+      upper f.map g.map ((descended e g hg heg).map y) := by
+  obtain ⟨m,rfl⟩ := hg y
+  rw [← compatible f.map g.map m,descendedEdge_lower]
+  change lower f.map g.map (descend f.map hf e.map hef (f.map m)) =
+    upper f.map g.map (descend g.map hg e.map heg (g.map m))
+  rw [descend_apply,descend_apply]
+  exact compatible f.map g.map (e.map m)
+
+end
+end KLInvariance.Graded.ModuleData
+
+end
+
+
+section
+
+/-! The adjoint of an actual graded word-sheaf endomorphism is again an
+integral graded word-sheaf endomorphism. No stalk freeness or BMP character
+is used or asserted. This supplies the correct adjoint route for a future
+BMP summand, instead of restricting a form to an arbitrary summand. -/
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
+universe u us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  (l : List I) (hl : cs.IsReduced l)
+noncomputable section
+
+ def bottAdjointWordHom (h : GradedSheaf.Hom
+    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
+    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))) :
+    ModuleData.Hom (bottWordData cs basis l) (bottWordData cs basis l) where
+  map := bottAdjoint cs l (bottEndOfSheaf cs basis l hl h)
+  graded := bottAdjoint_homogeneous cs basis l _
+    (bottEndOfSheaf_homogeneous cs basis l hl h)
+
+ theorem bottAdjointWordHom_kernel (h : GradedSheaf.Hom
+    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
+    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
+    (x : Interval cs 1 (cs.wordProd l)) (m : BottSamelsonModule cs l)
+    (hm : bottStalkProjection cs l x.val m=0) :
+    bottStalkProjection cs l x.val ((bottAdjointWordHom cs basis l hl h).map m)=0 :=
+  bottAdjoint_preserves_kernel cs l x.val _
+    (bottEndOfSheaf_preserves_support cs basis l hl h x) m hm
+
+ def bottAdjointSheaf (h : GradedSheaf.Hom
+    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
+    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))) : GradedSheaf.Hom
+    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
+    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)) where
+  vertex x := ModuleData.descended (bottAdjointWordHom cs basis l hl h)
+    (bottStalkHom cs basis l x.val) (bottStalkHom_surjective cs basis l x.val)
+    (bottAdjointWordHom_kernel cs basis l hl h x)
+  edge e := ModuleData.descendedEdge (bottAdjointWordHom cs basis l hl h)
+    (bottStalkHom cs basis l (source cs 1 (cs.wordProd l) e).val)
+    (bottStalkHom cs basis l (target cs 1 (cs.wordProd l) e).val)
+    (bottStalkHom_surjective cs basis l _)
+    (bottStalkHom_surjective cs basis l _)
+    (bottAdjointWordHom_kernel cs basis l hl h _)
+    (bottAdjointWordHom_kernel cs basis l hl h _)
+  lower _ x := ModuleData.descendedEdge_lower _ _ _ _ _ _ _ x
+  upper _ y := ModuleData.descendedEdge_upper _ _ _ _ _ _ _ y
+
+ theorem bottAdjointSheaf_vertex (h : GradedSheaf.Hom
+    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
+    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
+    (x : Interval cs 1 (cs.wordProd l)) (m : BottSamelsonModule cs l) :
+    ((bottAdjointSheaf cs basis l hl h).vertex x).map (bottStalkProjection cs l x.val m)=
+      bottStalkProjection cs l x.val (bottAdjoint cs l (bottEndOfSheaf cs basis l hl h) m) :=
+  LinearDescent.descend_apply _ _ _ _ m
+
+ theorem bottEndOfAdjointSheaf (h : GradedSheaf.Hom
+    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
+    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))) :
+    bottEndOfSheaf cs basis l hl (bottAdjointSheaf cs basis l hl h)=
+      bottAdjoint cs l (bottEndOfSheaf cs basis l hl h) := by
+  apply LinearMap.ext
+  intro m
+  apply bottVertexProjection_injective cs l hl
+  funext x
+  exact (bottEndOfSheaf_coordinates cs basis l hl _ m x).trans
+    (bottAdjointSheaf_vertex cs basis l hl h x m)
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Integral graded adjoint WORD projectors. The assumption below is only
+that the GIVEN endomorphism is a projector; no existence of a BMP projector
+or identification of its image with its adjoint is asserted. -/
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
+universe u us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  (l : List I) (hl : cs.IsReduced l)
+  (h : GradedSheaf.Hom
+    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
+    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
+noncomputable section
+
+ theorem bottEndOfSheaf_idempotent
+    (hh : ∀ x v, (h.vertex x).map ((h.vertex x).map v)=(h.vertex x).map v)
+    (m : BottSamelsonModule cs l) :
+    bottEndOfSheaf cs basis l hl h (bottEndOfSheaf cs basis l hl h m)=
+      bottEndOfSheaf cs basis l hl h m := by
+  apply bottVertexProjection_injective cs l hl
+  funext x
+  change bottStalkProjection cs l x.val _=bottStalkProjection cs l x.val _
+  simp only [bottEndOfSheaf_coordinates]
+  exact hh x _
+
+ theorem bottAdjointSheaf_vertex_idempotent
+    (hh : ∀ x v, (h.vertex x).map ((h.vertex x).map v)=(h.vertex x).map v)
+    (x : Interval cs 1 (cs.wordProd l))
+    (v : bottStalk cs l x.val) :
+    ((bottAdjointSheaf cs basis l hl h).vertex x).map
+      (((bottAdjointSheaf cs basis l hl h).vertex x).map v)=
+      ((bottAdjointSheaf cs basis l hl h).vertex x).map v := by
+  obtain ⟨m,rfl⟩ := bottStalkProjection_surjective cs l x.val v
+  simp only [bottAdjointSheaf_vertex]
+  exact congrArg (bottStalkProjection cs l x.val)
+    (PerfectAdjoint.adjoint_idempotent (bottSamelsonDuality cs l) _
+      (bottEndOfSheaf_idempotent cs basis l hl h hh) m)
+
+ theorem bottAdjointSheaf_edge_idempotent
+    (hh : ∀ x v, (h.vertex x).map ((h.vertex x).map v)=(h.vertex x).map v)
+    (e : Edge cs 1 (cs.wordProd l))
+    (v : (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)).edge e) :
+    ((bottAdjointSheaf cs basis l hl h).edge e).map
+      (((bottAdjointSheaf cs basis l hl h).edge e).map v)=
+      ((bottAdjointSheaf cs basis l hl h).edge e).map v := by
+  have hs : Function.Surjective
+      ((bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)).lower e).map :=
+    bottProjectionSheaf_lower_surjective cs l 1 (cs.wordProd l) e
+  obtain ⟨m,rfl⟩ := hs v
+  simp only [(bottAdjointSheaf cs basis l hl h).lower]
+  exact congrArg ((bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)).lower e).map
+    (bottAdjointSheaf_vertex_idempotent cs basis l hl h hh _ m)
+
+ theorem bottAdjointSheaf_twice_vertex (x : Interval cs 1 (cs.wordProd l))
+    (v : bottStalk cs l x.val) :
+    ((bottAdjointSheaf cs basis l hl (bottAdjointSheaf cs basis l hl h)).vertex x).map v=
+      (h.vertex x).map v := by
+  obtain ⟨m,rfl⟩ := bottStalkProjection_surjective cs l x.val v
+  rw [bottAdjointSheaf_vertex,bottEndOfAdjointSheaf]
+  change bottStalkProjection cs l x.val
+    (PerfectAdjoint.adjoint (bottSamelsonDuality cs l)
+      (PerfectAdjoint.adjoint (bottSamelsonDuality cs l) (bottEndOfSheaf cs basis l hl h)) m)=_
+  rw [PerfectAdjoint.adjoint_adjoint _ (bottSamelson_symmetric cs l)]
+  exact bottEndOfSheaf_coordinates cs basis l hl h m x
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! The literal integral image sheaf of a graded projector, with its actual
+splitting. Used for word-adjoint projectors; no BMP summand is assumed. -/
+namespace KLInvariance.Graded.ModuleData
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} [IsNoetherianRing A]
+  {M N : ModuleData.{uk,ua,um} 𝓐}
+noncomputable section
+
+ def rangeMap (e : Hom M M) (f : Hom N N) (g : Hom M N)
+    (h : ∀ m, f.map (g.map m)=g.map (e.map m)) : Hom (range e) (range f) where
+  map := (g.map.domRestrict e.map.range).codRestrict f.map.range (by
+    rintro ⟨_,m,rfl⟩
+    exact ⟨g.map m,h m⟩)
+  graded n m hm := g.graded n m.val hm
+
+ def rangeInclusion (e : Hom M M) : Hom (range e) M where
+  map := e.map.range.subtype
+  graded _ _ hm := hm
+
+ def rangeProjection (e : Hom M M) : Hom M (range e) where
+  map := e.map.rangeRestrict
+  graded n m hm := e.graded n m hm
+
+end
+end KLInvariance.Graded.ModuleData
+
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} [IsNoetherianRing A]
+  {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
+  {B : GradedSheaf (𝓐 := 𝓐) G}
+noncomputable section
+
+ def Hom.image (h : Hom B B) : GradedSheaf (𝓐 := 𝓐) G where
+  vertex x := ModuleData.range (h.vertex x)
+  edge e := ModuleData.range (h.edge e)
+  lower e := ModuleData.rangeMap (h.vertex (G.source e)) (h.edge e) (B.lower e) (h.lower e)
+  upper e := ModuleData.rangeMap (h.vertex (G.target e)) (h.edge e) (B.upper e) (h.upper e)
+
+ def Hom.imageInclusion (h : Hom B B) : Hom h.image B where
+  vertex x := ModuleData.rangeInclusion (h.vertex x)
+  edge e := ModuleData.rangeInclusion (h.edge e)
+  lower _ _ := rfl
+  upper _ _ := rfl
+
+ def Hom.imageProjection (h : Hom B B) : Hom B h.image where
+  vertex x := ModuleData.rangeProjection (h.vertex x)
+  edge e := ModuleData.rangeProjection (h.edge e)
+  lower e v := Subtype.ext (h.lower e v)
+  upper e v := Subtype.ext (h.upper e v)
+
+ def Hom.imageSplit (h : Hom B B)
+    (hv : ∀ x v, (h.vertex x).map ((h.vertex x).map v)=(h.vertex x).map v)
+    (he : ∀ e v, (h.edge e).map ((h.edge e).map v)=(h.edge e).map v) : SplitIn h.image B where
+  inclusion := h.imageInclusion
+  projection := h.imageProjection
+  vertex_split x v := by
+    obtain ⟨m,hm⟩ := v.property
+    apply Subtype.ext
+    change (h.vertex x).map v.val=v.val
+    rw [← hm,hv]
+  edge_split e v := by
+    obtain ⟨m,hm⟩ := v.property
+    apply Subtype.ext
+    change (h.edge e).map v.val=v.val
+    rw [← hm,he]
+
+end
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+/-! All integral sections of an idempotent image sheaf are the image of the
+actual section projector; this is stronger than equality over the fraction
+field and requires the actual idempotence, not merely pointwise surjectivity. -/
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} [IsNoetherianRing A]
+  {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
+  {B : GradedSheaf (𝓐 := 𝓐) G}
+noncomputable section
+
+ theorem Hom.imageSection_fixed (h : Hom B B)
+    (hv : ∀ x v, (h.vertex x).map ((h.vertex x).map v)=(h.vertex x).map v)
+    (Ω : Set V) (f : h.image.forget.sections Ω) :
+    h.sectionMap Ω (h.imageInclusion.sectionMap Ω f)=h.imageInclusion.sectionMap Ω f := by
+  apply Subtype.ext
+  funext x
+  obtain ⟨v,hv'⟩ := (f.val x).property
+  change (h.vertex x).map (f.val x).val=(f.val x).val
+  rw [← hv',hv]
+
+ def Hom.imageSectionMap (h : Hom B B)
+    (hv : ∀ x v, (h.vertex x).map ((h.vertex x).map v)=(h.vertex x).map v)
+    (Ω : Set V) : h.image.forget.sections Ω →ₗ[A] LinearMap.range (h.sectionMap Ω) :=
+  (h.imageInclusion.sectionMap Ω).codRestrict _ (fun f =>
+    ⟨h.imageInclusion.sectionMap Ω f,h.imageSection_fixed hv Ω f⟩)
+
+ theorem Hom.imageSectionMap_bijective (h : Hom B B)
+    (hv : ∀ x v, (h.vertex x).map ((h.vertex x).map v)=(h.vertex x).map v)
+    (Ω : Set V) : Function.Bijective (h.imageSectionMap hv Ω) := by
+  constructor
+  · intro f g hfg
+    apply Subtype.ext
+    funext x
+    apply Subtype.ext
+    exact congrFun (congrArg Subtype.val (congrArg Subtype.val hfg)) x
+  · rintro ⟨s,m,rfl⟩
+    refine ⟨h.imageProjection.sectionMap Ω m,?_⟩
+    apply Subtype.ext
+    apply Subtype.ext
+    rfl
+
+ def Hom.imageSectionEquiv (h : Hom B B)
+    (hv : ∀ x v, (h.vertex x).map ((h.vertex x).map v)=(h.vertex x).map v)
+    (Ω : Set V) : h.image.forget.sections Ω ≃ₗ[A] LinearMap.range (h.sectionMap Ω) :=
+  LinearEquiv.ofBijective (h.imageSectionMap hv Ω) (h.imageSectionMap_bijective hv Ω)
+
+end
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+namespace ConjugateRange
+open Module
+universe ur um un
+variable {R : Type ur} [CommRing R]
+  {M : Type um} [AddCommGroup M] [Module R M]
+  {N : Type un} [AddCommGroup N] [Module R N]
+  (e : M ≃ₗ[R] N) (f : End R N)
+noncomputable section
+
+ def conjugate : End R M := e.symm.toLinearMap.comp (f.comp e.toLinearMap)
+
+ def map : LinearMap.range f →ₗ[R] LinearMap.range (conjugate e f) :=
+  (e.symm.toLinearMap.domRestrict f.range).codRestrict _ (by
+    rintro ⟨_,n,rfl⟩
+    refine ⟨e.symm n,?_⟩
+    change e.symm (f (e (e.symm n)))=e.symm (f n)
+    rw [LinearEquiv.apply_symm_apply])
+
+ theorem map_bijective : Function.Bijective (map e f) := by
+  constructor
+  · intro m n h
+    apply Subtype.ext
+    apply e.symm.injective
+    exact congrArg Subtype.val h
+  · rintro ⟨_,m,rfl⟩
+    refine ⟨⟨f (e m),e m,rfl⟩,?_⟩
+    rfl
+
+ def equiv : LinearMap.range f ≃ₗ[R] LinearMap.range (conjugate e f) :=
+  LinearEquiv.ofBijective (map e f) (map_bijective e f)
+
+ theorem equiv_apply (n : LinearMap.range f) : (equiv e f n).val=e.symm n.val := rfl
+
+end
+end ConjugateRange
+
+end
+
+
+section
+
+/-! The correct integral summand duality. A chosen summand of a perfectly
+paired module is paired with its ADJOINT summand, not with itself by fiat.
+This is the algebraic connector for the unresolved BMP summand recognition.
+No existence of a BMP summand, or isomorphism with its adjoint, is asserted. -/
+namespace AdjointSummandDuality
+universe ur um un
+variable {R : Type ur} [CommRing R]
+  {M : Type um} [AddCommGroup M] [Module R M]
+  {N : Type un} [AddCommGroup N] [Module R N]
+  (d : M ≃ₗ[R] Module.Dual R M) (i : N →ₗ[R] M) (p : M →ₗ[R] N)
+  (hpi : ∀ n, p (i n)=n)
+noncomputable section
+
+ def dualSection : Module.Dual R N →ₗ[R] M :=
+  d.symm.toLinearMap.comp p.dualMap
+
+ def dualProjection : M →ₗ[R] Module.Dual R N := i.dualMap.comp d.toLinearMap
+
+include hpi in
+ theorem projection_section (lam : Module.Dual R N) :
+    dualProjection d i (dualSection d p lam)=lam := by
+  apply LinearMap.ext
+  intro n
+  change d (d.symm (p.dualMap lam)) (i n)=lam n
+  rw [LinearEquiv.apply_symm_apply]
+  change lam (p (i n))=lam n
+  rw [hpi]
+
+ def adjointProjector : Module.End R M := (dualSection d p).comp (dualProjection d i)
+
+include hpi in
+ theorem adjointProjector_idempotent (m : M) :
+    adjointProjector d i p (adjointProjector d i p m)=adjointProjector d i p m := by
+  change dualSection d p (dualProjection d i (dualSection d p (dualProjection d i m))) = _
+  rw [projection_section d i p hpi]
+  rfl
+
+ def adjointSummand : Submodule R M := LinearMap.range (dualSection d p)
+
+include hpi in
+ theorem dualProjection_injective_on_adjoint :
+    Function.Injective ((dualProjection d i).domRestrict (adjointSummand d p)) := by
+  rintro ⟨m,lam,rfl⟩ ⟨n,mu,rfl⟩ h
+  apply Subtype.ext
+  change dualSection d p lam=dualSection d p mu
+  change dualProjection d i (dualSection d p lam)=dualProjection d i (dualSection d p mu) at h
+  rw [projection_section d i p hpi,projection_section d i p hpi] at h
+  rw [h]
+
+include hpi in
+ theorem dualProjection_surjective_on_adjoint :
+    Function.Surjective ((dualProjection d i).domRestrict (adjointSummand d p)) := by
+  intro lam
+  exact ⟨⟨dualSection d p lam,lam,rfl⟩,projection_section d i p hpi lam⟩
+
+ def adjointEquivDual : adjointSummand d p ≃ₗ[R] Module.Dual R N :=
+  LinearEquiv.ofBijective ((dualProjection d i).domRestrict (adjointSummand d p))
+    ⟨dualProjection_injective_on_adjoint d i p hpi,dualProjection_surjective_on_adjoint d i p hpi⟩
+
+ theorem adjointEquivDual_apply (m : adjointSummand d p) (n : N) :
+    adjointEquivDual d i p hpi m n=d m.val (i n) := rfl
+
+include hpi in
+ theorem adjointProjector_range :
+    LinearMap.range (adjointProjector d i p)=adjointSummand d p := by
+  apply le_antisymm
+  · rintro m ⟨n,rfl⟩
+    exact ⟨dualProjection d i n,rfl⟩
+  · rintro m ⟨lam,rfl⟩
+    refine ⟨dualSection d p lam,?_⟩
+    change dualSection d p (dualProjection d i (dualSection d p lam))=dualSection d p lam
+    rw [projection_section d i p hpi]
+
+variable (f : Module.End R M) (g : Module.End R N)
+  (hbal : ∀ m n, d (f m) n=d m (f n))
+  (hi : ∀ n, f (i n)=i (g n)) (hp : ∀ m, p (f m)=g (p m))
+
+include hbal hi in
+ theorem projection_intertwines (m : M) :
+    dualProjection d i (f m)=g.dualMap (dualProjection d i m) := by
+  apply LinearMap.ext
+  intro n
+  change d (f m) (i n)=d m (i (g n))
+  rw [hbal,hi]
+
+include hbal hp in
+ theorem section_intertwines (lam : Module.Dual R N) :
+    f (dualSection d p lam)=dualSection d p (g.dualMap lam) := by
+  apply d.injective
+  apply LinearMap.ext
+  intro n
+  change d (f (d.symm (p.dualMap lam))) n=
+    d (d.symm (p.dualMap (g.dualMap lam))) n
+  rw [hbal,LinearEquiv.apply_symm_apply,LinearEquiv.apply_symm_apply]
+  change lam (p (f n))=lam (g (p n))
+  rw [hp]
+
+include hbal hi hp in
+ theorem adjointProjector_commutes (m : M) :
+    f (adjointProjector d i p m)=adjointProjector d i p (f m) := by
+  change f (dualSection d p (dualProjection d i m)) =
+    dualSection d p (dualProjection d i (f m))
+  rw [section_intertwines d p f g hbal hp,projection_intertwines d i f g hbal hi]
+
+end
+end AdjointSummandDuality
+
+end
+
+
+section
+
+/-! The range of the genuine adjoint projector is integrally the dual of
+the original summand, with no assumed self-duality of that summand. -/
+namespace PerfectAdjoint
+open Module
+universe ur um un
+variable {R : Type ur} [CommRing R]
+  {M : Type um} [AddCommGroup M] [Module R M]
+  {N : Type un} [AddCommGroup N] [Module R N]
+  (d : M ≃ₗ[R] Dual R M) (i : N →ₗ[R] M) (p : M →ₗ[R] N)
+  (hpi : ∀ n, p (i n)=n)
+noncomputable section
+
+ theorem adjoint_projector_eq : adjoint d (i.comp p)=
+    AdjointSummandDuality.adjointProjector d i p := by
+  apply LinearMap.ext
+  intro m
+  rfl
+
+ def rangeAdjointEquivDual : LinearMap.range (adjoint d (i.comp p)) ≃ₗ[R] Dual R N :=
+  (LinearEquiv.ofEq _ _ (by
+    rw [adjoint_projector_eq,AdjointSummandDuality.adjointProjector_range d i p hpi])).trans
+    (AdjointSummandDuality.adjointEquivDual d i p hpi)
+
+ theorem rangeAdjointEquivDual_apply (m : LinearMap.range (adjoint d (i.comp p))) (n : N) :
+    rangeAdjointEquivDual d i p hpi m n=d m.val (i n) := rfl
+
+end
+end PerfectAdjoint
+
+end
+
+
+section
+
+namespace PerfectAdjoint
+open Module
+universe ur um
+variable {R : Type ur} [CommRing R]
+  {M : Type um} [AddCommGroup M] [Module R M]
+  (d : M ≃ₗ[R] Dual R M) (f : End R M) (hf : ∀ m, f (f m)=f m)
+noncomputable section
+
+include hf in
+ theorem range_projector_split (m : f.range) : f.rangeRestrict (f.range.subtype m)=m := by
+  apply Subtype.ext
+  obtain ⟨n,hn⟩ := m.property
+  change f m.val=m.val
+  rw [← hn,hf]
+
+ def rangeAdjointEquivDualRange : LinearMap.range (adjoint d f) ≃ₗ[R] Dual R f.range :=
+  rangeAdjointEquivDual d f.range.subtype f.rangeRestrict (range_projector_split f hf)
+
+ theorem rangeAdjointEquivDualRange_apply (m : LinearMap.range (adjoint d f)) (n : f.range) :
+    rangeAdjointEquivDualRange d f hf m n=d m.val n.val := rfl
+
+end
+end PerfectAdjoint
+
+end
+
+
+section
+
+namespace KLInvariance.TitsSpace
+open Module
+universe u us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+
+include basis in
+ theorem symmetricCoefficientNoetherian : IsNoetherianRing (SymmetricCoefficient (M := M)) :=
+  isNoetherianRing_of_surjective (MvPolynomial σ ℝ) (SymmetricCoefficient (M := M))
+    (SymmetricAlgebra.equivMvPolynomial basis).symm.toRingHom
+    (SymmetricAlgebra.equivMvPolynomial basis).symm.surjective
+
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! The actual adjoint IMAGE sheaf has all integral sections dual to those
+of the original projector image. This does not identify those two image
+sheaves or assert that the word sheaf satisfies the BMP axioms. -/
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
+universe u us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  (l : List I) (hl : cs.IsReduced l)
+  (h : GradedSheaf.Hom
+    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
+    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
+noncomputable section
+
+ def bottImageSheaf : GradedSheaf (𝓐 := symmetricGrading basis)
+    (graph cs 1 (cs.wordProd l)) := by
+  letI := symmetricCoefficientNoetherian basis
+  exact h.image
+
+ def bottImageSectionWordEquiv
+    (hh : ∀ x v, (h.vertex x).map ((h.vertex x).map v)=(h.vertex x).map v) :
+    (bottImageSheaf cs basis l h).forget.sections Set.univ ≃ₗ[
+      SymmetricCoefficient (M := M)] LinearMap.range (bottEndOfSheaf cs basis l hl h) := by
+  letI := symmetricCoefficientNoetherian basis
+  exact (h.imageSectionEquiv hh Set.univ).trans
+    (ConjugateRange.equiv (bottSectionEquiv cs l hl) (h.sectionMap Set.univ))
+
+ def bottAdjointImageEquivDual
+    (hh : ∀ x v, (h.vertex x).map ((h.vertex x).map v)=(h.vertex x).map v) :
+    (bottImageSheaf cs basis l (bottAdjointSheaf cs basis l hl h)).forget.sections Set.univ ≃ₗ[
+      SymmetricCoefficient (M := M)]
+      Dual (SymmetricCoefficient (M := M))
+        ((bottImageSheaf cs basis l h).forget.sections Set.univ) :=
+  (bottImageSectionWordEquiv cs basis l hl (bottAdjointSheaf cs basis l hl h)
+      (bottAdjointSheaf_vertex_idempotent cs basis l hl h hh)).trans
+    ((LinearEquiv.ofEq _ _ (congrArg LinearMap.range
+        (bottEndOfAdjointSheaf cs basis l hl h))).trans
+      ((PerfectAdjoint.rangeAdjointEquivDualRange (bottSamelsonDuality cs l)
+          (bottEndOfSheaf cs basis l hl h)
+          (bottEndOfSheaf_idempotent cs basis l hl h hh)).trans
+        (bottImageSectionWordEquiv cs basis l hl h hh).dualMap))
+
+ theorem bottImageSectionWordEquiv_coordinates
+    (hh : ∀ x v, (h.vertex x).map ((h.vertex x).map v)=(h.vertex x).map v)
+    (m : (bottImageSheaf cs basis l h).forget.sections Set.univ)
+    (x : Interval cs 1 (cs.wordProd l)) :
+    bottStalkProjection cs l x.val ((bottImageSectionWordEquiv cs basis l hl h hh m).val)=
+      (m.val x).val := by
+  let := symmetricCoefficientNoetherian basis
+  exact congrArg (fun s : (bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ =>
+    s.val x) ((bottSectionEquiv cs l hl).apply_symm_apply
+      (h.imageInclusion.sectionMap Set.univ m))
+
+ theorem bottImageSectionWordEquiv_homogeneous
+    (hh : ∀ x v, (h.vertex x).map ((h.vertex x).map v)=(h.vertex x).map v)
+    (n : ℤ) (m : (bottImageSheaf cs basis l h).forget.sections Set.univ)
+    (hm : ∀ x, m.val x ∈ ((bottImageSheaf cs basis l h).vertex x).piece n) :
+    (bottImageSectionWordEquiv cs basis l hl h hh m).val ∈ bottPiece cs basis l n := by
+  apply (bottPiece_iff_stalks cs basis l hl n _).mpr
+  intro x
+  rw [bottImageSectionWordEquiv_coordinates]
+  exact hm x
+
+ theorem bottAdjointImageEquivDual_apply
+    (hh : ∀ x v, (h.vertex x).map ((h.vertex x).map v)=(h.vertex x).map v)
+    (m : (bottImageSheaf cs basis l (bottAdjointSheaf cs basis l hl h)).forget.sections Set.univ)
+    (n : (bottImageSheaf cs basis l h).forget.sections Set.univ) :
+    bottAdjointImageEquivDual cs basis l hl h hh m n=bottSamelsonDuality cs l
+      (bottImageSectionWordEquiv cs basis l hl (bottAdjointSheaf cs basis l hl h)
+        (bottAdjointSheaf_vertex_idempotent cs basis l hl h hh) m).val
+      (bottImageSectionWordEquiv cs basis l hl h hh n).val := rfl
+
+ theorem bottAdjointImagePairing_homogeneous
+    (hh : ∀ x v, (h.vertex x).map ((h.vertex x).map v)=(h.vertex x).map v)
+    (i j : ℤ)
+    (m : (bottImageSheaf cs basis l (bottAdjointSheaf cs basis l hl h)).forget.sections Set.univ)
+    (n : (bottImageSheaf cs basis l h).forget.sections Set.univ)
+    (hm : ∀ x, m.val x ∈
+      ((bottImageSheaf cs basis l (bottAdjointSheaf cs basis l hl h)).vertex x).piece i)
+    (hn : ∀ x, n.val x ∈ ((bottImageSheaf cs basis l h).vertex x).piece j) :
+    bottAdjointImageEquivDual cs basis l hl h hh m n ∈ symmetricGrading basis (i+j-(l.length : ℤ)) := by
+  rw [bottAdjointImageEquivDual_apply]
+  exact bottDuality_homogeneous cs basis l
+    (bottImageSectionWordEquiv_homogeneous cs basis l hl _
+      (bottAdjointSheaf_vertex_idempotent cs basis l hl h hh) i m hm)
+    (bottImageSectionWordEquiv_homogeneous cs basis l hl h hh j n hn)
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! A specified integral graded sheaf splitting identifies its source with
+its literal projector image. No existence of a word/BMP splitting is assumed
+outside each explicitly parametric helper. -/
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} [IsNoetherianRing A]
+  {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
+  {B C : GradedSheaf (𝓐 := 𝓐) G} (s : SplitIn B C)
+noncomputable section
+
+ def SplitIn.projector : Hom C C := s.inclusion.comp s.projection
+
+omit [IsNoetherianRing A] in
+ theorem SplitIn.projector_vertex (x : V) (v : C.vertex x) :
+    (s.projector.vertex x).map ((s.projector.vertex x).map v)=(s.projector.vertex x).map v := by
+  change (s.inclusion.vertex x).map ((s.projection.vertex x).map
+    ((s.inclusion.vertex x).map ((s.projection.vertex x).map v)))=_
+  rw [s.vertex_split]
+  rfl
+
+omit [IsNoetherianRing A] in
+ theorem SplitIn.projector_edge (e : E) (v : C.edge e) :
+    (s.projector.edge e).map ((s.projector.edge e).map v)=(s.projector.edge e).map v := by
+  change (s.inclusion.edge e).map ((s.projection.edge e).map
+    ((s.inclusion.edge e).map ((s.projection.edge e).map v)))=_
+  rw [s.edge_split]
+  rfl
+
+ def SplitIn.imageTo : Hom s.projector.image B := s.projection.comp s.projector.imageInclusion
+ def SplitIn.toImage : Hom B s.projector.image := s.projector.imageProjection.comp s.inclusion
+
+ theorem SplitIn.imageTo_toImage_vertex (x : V) (v : B.vertex x) :
+    (s.imageTo.vertex x).map ((s.toImage.vertex x).map v)=v := by
+  change (s.projection.vertex x).map ((s.inclusion.vertex x).map
+    ((s.projection.vertex x).map ((s.inclusion.vertex x).map v)))=v
+  rw [s.vertex_split,s.vertex_split]
+
+ theorem SplitIn.toImage_imageTo_vertex (x : V) (v : s.projector.image.vertex x) :
+    (s.toImage.vertex x).map ((s.imageTo.vertex x).map v)=v := by
+  apply Subtype.ext
+  obtain ⟨m,hm⟩ := v.property
+  change (s.projector.vertex x).map ((s.projector.vertex x).map v.val)=v.val
+  rw [s.projector_vertex,← hm,s.projector_vertex]
+
+ theorem SplitIn.imageTo_toImage_edge (e : E) (v : B.edge e) :
+    (s.imageTo.edge e).map ((s.toImage.edge e).map v)=v := by
+  change (s.projection.edge e).map ((s.inclusion.edge e).map
+    ((s.projection.edge e).map ((s.inclusion.edge e).map v)))=v
+  rw [s.edge_split,s.edge_split]
+
+ theorem SplitIn.toImage_imageTo_edge (e : E) (v : s.projector.image.edge e) :
+    (s.toImage.edge e).map ((s.imageTo.edge e).map v)=v := by
+  apply Subtype.ext
+  obtain ⟨m,hm⟩ := v.property
+  change (s.projector.edge e).map ((s.projector.edge e).map v.val)=v.val
+  rw [s.projector_edge,← hm,s.projector_edge]
+
+ theorem SplitIn.imageSection_bijective (Ω : Set V) :
+    Function.Bijective (s.imageTo.sectionMap Ω) := by
+  constructor
+  · intro m n he
+    apply Subtype.ext
+    funext x
+    have hx := congrFun (congrArg Subtype.val he) x
+    have hy := congrArg (s.toImage.vertex x).map hx
+    exact (s.toImage_imageTo_vertex x (m.val x)).symm.trans
+      (hy.trans (s.toImage_imageTo_vertex x (n.val x)))
+  · intro m
+    refine ⟨s.toImage.sectionMap Ω m,?_⟩
+    apply Subtype.ext
+    funext x
+    exact s.imageTo_toImage_vertex x (m.val x)
+
+ def SplitIn.imageSectionEquiv (Ω : Set V) :
+    s.projector.image.forget.sections Ω ≃ₗ[A] B.forget.sections Ω :=
+  LinearEquiv.ofBijective (s.imageTo.sectionMap Ω) (s.imageSection_bijective Ω)
+
+ theorem SplitIn.imageSectionEquiv_symm_apply (Ω : Set V) (m : B.forget.sections Ω) :
+    (s.imageSectionEquiv Ω).symm m=s.toImage.sectionMap Ω m := by
+  apply (s.imageSectionEquiv Ω).injective
+  rw [LinearEquiv.apply_symm_apply]
+  apply Subtype.ext
+  funext x
+  exact (s.imageTo_toImage_vertex x (m.val x)).symm
+
+ theorem SplitIn.imageSectionEquiv_symm_homogeneous (Ω : Set V) (n : ℤ)
+    (m : B.forget.sections Ω) (hm : ∀ x, m.val x ∈ (B.vertex x).piece n) (x : V) :
+    ((s.imageSectionEquiv Ω).symm m).val x ∈ (s.projector.image.vertex x).piece n := by
+  rw [s.imageSectionEquiv_symm_apply]
+  exact (s.toImage.vertex x).graded n (m.val x) (hm x)
+
+end
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+/-! Correct integral graded duality for a GIVEN word-sheaf summand: the dual
+is realized by its ADJOINT image, not identified with the original summand.
+The existence and BMP recognition of that summand remain separate. -/
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
+universe u us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  (l : List I) (hl : cs.IsReduced l)
+  (B : GradedSheaf (𝓐 := symmetricGrading basis) (graph cs 1 (cs.wordProd l)))
+  (s : GradedSheaf.SplitIn B (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
+noncomputable section
+
+ def bottSplitAdjointSheaf : GradedSheaf (𝓐 := symmetricGrading basis)
+    (graph cs 1 (cs.wordProd l)) :=
+  bottImageSheaf cs basis l (bottAdjointSheaf cs basis l hl s.projector)
+
+ def bottSplitAdjointEquivDual :
+    (bottSplitAdjointSheaf cs basis l hl B s).forget.sections Set.univ ≃ₗ[
+      SymmetricCoefficient (M := M)]
+      Dual (SymmetricCoefficient (M := M)) (B.forget.sections Set.univ) := by
+  letI := symmetricCoefficientNoetherian basis
+  exact (bottAdjointImageEquivDual cs basis l hl s.projector s.projector_vertex).trans
+    (s.imageSectionEquiv Set.univ).symm.dualMap
+
+ theorem bottSplitAdjointPairing_homogeneous
+    (i j : ℤ)
+    (m : (bottSplitAdjointSheaf cs basis l hl B s).forget.sections Set.univ)
+    (n : B.forget.sections Set.univ)
+    (hm : ∀ x, m.val x ∈ ((bottSplitAdjointSheaf cs basis l hl B s).vertex x).piece i)
+    (hn : ∀ x, n.val x ∈ (B.vertex x).piece j) :
+    bottSplitAdjointEquivDual cs basis l hl B s m n ∈
+      symmetricGrading basis (i+j-(l.length : ℤ)) := by
+  let := symmetricCoefficientNoetherian basis
+  exact bottAdjointImagePairing_homogeneous cs basis l hl s.projector s.projector_vertex i j m
+    ((s.imageSectionEquiv Set.univ).symm n) hm
+    (s.imageSectionEquiv_symm_homogeneous Set.univ j n hn)
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! An actual word-sheaf endomorphism that is the identity at one stalk has
+adjoint equal to the identity at that same stalk. In particular, an identity
+TOP projector stays normalized on taking the adjoint. This alone does not
+supply free stalks, flabbiness, or the BMP character. -/
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
+universe u us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  (l : List I) (hl : cs.IsReduced l)
+  (h : GradedSheaf.Hom
+    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
+    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
+noncomputable section
+
+ theorem bottEndOfSheaf_fixed_support (x : Interval cs 1 (cs.wordProd l))
+    (hx : ∀ v, (h.vertex x).map v=v)
+    (m : BottSamelsonModule cs l) (hm : m ∈ bottWeightSupported cs l x.val) :
+    bottEndOfSheaf cs basis l hl h m=m := by
+  apply bottEvaluation_injective cs l
+  funext e
+  by_cases he : bottWeight cs l e=x.val
+  · have ho := bottEndOfSheaf_coordinates cs basis l hl h m x
+    exact congrFun (congrArg Subtype.val (ho.trans (hx _))) ⟨e,he⟩
+  · exact (bottEndOfSheaf_preserves_support cs basis l hl h x m hm e he).trans
+      (hm e he).symm
+
+ theorem bottAdjointSheaf_fixed_stalk (x : Interval cs 1 (cs.wordProd l))
+    (hx : ∀ v, (h.vertex x).map v=v) (v : bottStalk cs l x.val) :
+    ((bottAdjointSheaf cs basis l hl h).vertex x).map v=v := by
+  obtain ⟨m,rfl⟩ := bottStalkProjection_surjective cs l x.val v
+  rw [bottAdjointSheaf_vertex]
+  have hz : bottStalkProjection cs l x.val
+      (bottAdjoint cs l (bottEndOfSheaf cs basis l hl h) m-m)=0 := by
+    apply (bott_projection_zero_iff_annihilates_support cs l x.val _).mpr
+    intro z
+    change bottSamelsonDuality cs l z.val
+      (PerfectAdjoint.adjoint (bottSamelsonDuality cs l)
+        (bottEndOfSheaf cs basis l hl h) m-m)=0
+    rw [map_sub,PerfectAdjoint.symmetric_pairing _ (bottSamelson_symmetric cs l)]
+    rw [bottEndOfSheaf_fixed_support cs basis l hl h x hx z.val z.property,sub_self]
+  rw [map_sub] at hz
+  exact sub_eq_zero.mp hz
+
+ theorem bottAdjointSheaf_fixed_stalk_iff (x : Interval cs 1 (cs.wordProd l)) :
+    (∀ v, ((bottAdjointSheaf cs basis l hl h).vertex x).map v=v) ↔
+      (∀ v, (h.vertex x).map v=v) := by
+  constructor
+  · intro hx v
+    have h' := bottAdjointSheaf_fixed_stalk cs basis l hl
+      (bottAdjointSheaf cs basis l hl h) x hx v
+    exact (bottAdjointSheaf_twice_vertex cs basis l hl h x v).symm.trans h'
+  · exact fun hx v => bottAdjointSheaf_fixed_stalk cs basis l hl h x hx v
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+namespace KLInvariance.Graded.ModuleData
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} {M N : ModuleData.{uk,ua,um} 𝓐}
+ theorem Hom.eq_of_map_eq (f g : Hom M N) (h : f.map=g.map) : f=g := by
+  cases f
+  cases g
+  cases h
+  rfl
+end KLInvariance.Graded.ModuleData
+
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
+  {B C : GradedSheaf (𝓐 := 𝓐) G}
+
+ theorem Hom.eq_of_maps (f g : Hom B C)
+    (hv : ∀ x v, (f.vertex x).map v=(g.vertex x).map v)
+    (he : ∀ e v, (f.edge e).map v=(g.edge e).map v) : f=g := by
+  have hv' : f.vertex=g.vertex := funext (fun x =>
+    ModuleData.Hom.eq_of_map_eq _ _ (LinearMap.ext (hv x)))
+  have he' : f.edge=g.edge := funext (fun e =>
+    ModuleData.Hom.eq_of_map_eq _ _ (LinearMap.ext (he e)))
+  cases f
+  cases g
+  cases hv'
+  cases he'
+  rfl
+
+ theorem Hom.eq_of_vertex (f g : Hom B C)
+    (hs : ∀ e, Function.Surjective (B.lower e).map)
+    (hv : ∀ x v, (f.vertex x).map v=(g.vertex x).map v) : f=g := by
+  apply f.eq_of_maps g hv
+  intro e v
+  obtain ⟨w,rfl⟩ := hs e v
+  rw [f.lower,g.lower,hv]
+
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+/-! Integral anti-involution on the actual graded word-sheaf endomorphisms.
+This is the algebra required to compare an indecomposable BMP summand to
+its adjoint, not an assertion that such a summand is already recognized. -/
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
+universe u us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  (l : List I) (hl : cs.IsReduced l)
+noncomputable section
+
+ theorem bottEndOfSheaf_comp
+    (h g : GradedSheaf.Hom
+      (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
+      (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))) :
+    bottEndOfSheaf cs basis l hl (h.comp g)=
+      (bottEndOfSheaf cs basis l hl h).comp (bottEndOfSheaf cs basis l hl g) := by
+  apply LinearMap.ext
+  intro m
+  apply bottVertexProjection_injective cs l hl
+  funext x
+  exact (bottEndOfSheaf_coordinates cs basis l hl (h.comp g) m x).trans
+    ((congrArg (h.vertex x).map (bottEndOfSheaf_coordinates cs basis l hl g m x)).symm.trans
+      (bottEndOfSheaf_coordinates cs basis l hl h (bottEndOfSheaf cs basis l hl g m) x).symm)
+
+ theorem bottAdjointSheaf_comp
+    (h g : GradedSheaf.Hom
+      (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
+      (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))) :
+    bottAdjointSheaf cs basis l hl (h.comp g)=
+      (bottAdjointSheaf cs basis l hl g).comp (bottAdjointSheaf cs basis l hl h) := by
+  apply GradedSheaf.Hom.eq_of_vertex
+  · exact fun e => bottProjectionSheaf_lower_surjective cs l 1 (cs.wordProd l) e
+  · intro x v
+    have hs : Function.Surjective (bottStalkProjection cs l x.val) :=
+      bottStalkProjection_surjective cs l x.val
+    obtain ⟨m,rfl⟩ := hs v
+    change ((bottAdjointSheaf cs basis l hl (h.comp g)).vertex x).map
+      (bottStalkProjection cs l x.val m)=
+        ((bottAdjointSheaf cs basis l hl g).vertex x).map
+          (((bottAdjointSheaf cs basis l hl h).vertex x).map (bottStalkProjection cs l x.val m))
+    simp only [bottAdjointSheaf_vertex]
+    change bottStalkProjection cs l x.val
+      (PerfectAdjoint.adjoint (bottSamelsonDuality cs l) (bottEndOfSheaf cs basis l hl (h.comp g)) m)=
+      bottStalkProjection cs l x.val
+        (PerfectAdjoint.adjoint (bottSamelsonDuality cs l) (bottEndOfSheaf cs basis l hl g)
+          (PerfectAdjoint.adjoint (bottSamelsonDuality cs l) (bottEndOfSheaf cs basis l hl h) m))
+    rw [bottEndOfSheaf_comp,PerfectAdjoint.adjoint_comp]
+    rfl
+
+ theorem bottAdjointSheaf_involutive : Function.Involutive (bottAdjointSheaf cs basis l hl) := by
+  intro h
+  apply GradedSheaf.Hom.eq_of_vertex
+  · exact fun e => bottProjectionSheaf_lower_surjective cs l 1 (cs.wordProd l) e
+  · exact bottAdjointSheaf_twice_vertex cs basis l hl h
+
+ theorem bottAdjointSheaf_injective : Function.Injective (bottAdjointSheaf cs basis l hl) :=
+  (bottAdjointSheaf_involutive cs basis l hl).injective
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! The endomorphism-corner formulation of a primitive graded sheaf
+projector. This is a criterion, not an assertion of a BMP splitting. -/
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
+  {B C : GradedSheaf (𝓐 := 𝓐) G}
+
+ def Hom.zero : Hom B C where
+  vertex x := ⟨0,fun n _ _ => (C.vertex x).piece n |>.zero_mem⟩
+  edge e := ⟨0,fun n _ _ => (C.edge e).piece n |>.zero_mem⟩
+  lower e v := by simp
+  upper e v := by simp
+
+ def Hom.id (B : GradedSheaf (𝓐 := 𝓐) G) : Hom B B where
+  vertex x := ModuleData.Hom.id (B.vertex x)
+  edge e := ModuleData.Hom.id (B.edge e)
+  lower _ _ := rfl
+  upper _ _ := rfl
+
+ def Hom.PrimitiveProjector (p : Hom B B) : Prop :=
+  p.comp p=p ∧ ∀ f : Hom B B, f.comp f=f → f.comp p=f → p.comp f=f →
+    f=Hom.zero ∨ f=p
+
+ theorem Hom.primitive_of_antiInvolution
+    (a : Hom B B → Hom B B)
+    (ha : Function.Involutive a)
+    (hac : ∀ f g, a (f.comp g)=(a g).comp (a f))
+    (haz : a Hom.zero=Hom.zero)
+    (p : Hom B B) (hp : p.PrimitiveProjector) :
+    (a p).PrimitiveProjector := by
+  constructor
+  · exact (hac p p).symm.trans (congrArg a hp.1)
+  · intro f hf hfp hpf
+    have hi : (a f).comp (a f)=a f := (hac f f).symm.trans (congrArg a hf)
+    have hleft : (a f).comp p=a f := by
+      have hh := congrArg a hpf
+      rw [hac,ha] at hh
+      exact hh
+    have hright : p.comp (a f)=a f := by
+      have hh := congrArg a hfp
+      rw [hac,ha] at hh
+      exact hh
+    rcases hp.2 (a f) hi hleft hright with hz | he
+    · left
+      have hh := congrArg a hz
+      rw [ha,haz] at hh
+      exact hh
+    · right
+      exact (ha f).symm.trans (congrArg a he)
+
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+
+namespace PerfectAdjoint
+open Module
+universe ur um
+variable {R : Type ur} [CommRing R] {N : Type um} [AddCommGroup N] [Module R N]
+ theorem adjoint_zero (d : N ≃ₗ[R] Dual R N) : adjoint d (0 : End R N)=0 := by
+  apply LinearMap.ext
+  intro m
+  apply d.injective
+  apply LinearMap.ext
+  intro n
+  rw [pairing]
+  simp
+end PerfectAdjoint
+
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
+universe u us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  (l : List I) (hl : cs.IsReduced l)
+noncomputable section
+
+ theorem bottEndOfSheaf_zero :
+    bottEndOfSheaf cs basis l hl GradedSheaf.Hom.zero=0 := by
+  apply LinearMap.ext
+  intro m
+  apply bottVertexProjection_injective cs l hl
+  funext x
+  change bottStalkProjection cs l x.val _=bottStalkProjection cs l x.val 0
+  rw [map_zero]
+  exact bottEndOfSheaf_coordinates cs basis l hl GradedSheaf.Hom.zero m x
+
+ theorem bottAdjointSheaf_zero :
+    bottAdjointSheaf cs basis l hl GradedSheaf.Hom.zero=GradedSheaf.Hom.zero := by
+  apply GradedSheaf.Hom.eq_of_vertex
+  · exact fun e => bottProjectionSheaf_lower_surjective cs l 1 (cs.wordProd l) e
+  · intro x v
+    have hs : Function.Surjective (bottStalkProjection cs l x.val) :=
+      bottStalkProjection_surjective cs l x.val
+    obtain ⟨m,rfl⟩ := hs v
+    change ((bottAdjointSheaf cs basis l hl GradedSheaf.Hom.zero).vertex x).map
+      (bottStalkProjection cs l x.val m)=0
+    rw [bottAdjointSheaf_vertex,bottEndOfSheaf_zero]
+    change bottStalkProjection cs l x.val
+      (PerfectAdjoint.adjoint (bottSamelsonDuality cs l) 0 m)=0
+    rw [PerfectAdjoint.adjoint_zero]
+    exact map_zero _
+
+ theorem bottAdjointSheaf_primitive
+    (p : GradedSheaf.Hom
+      (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
+      (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
+    (hp : p.PrimitiveProjector) :
+    (bottAdjointSheaf cs basis l hl p).PrimitiveProjector :=
+  GradedSheaf.Hom.primitive_of_antiInvolution _
+    (bottAdjointSheaf_involutive cs basis l hl)
+    (bottAdjointSheaf_comp cs basis l hl)
+    (bottAdjointSheaf_zero cs basis l hl) p hp
+
+ theorem bottAdjointSheaf_primitive_iff
+    (p : GradedSheaf.Hom
+      (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
+      (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))) :
+    (bottAdjointSheaf cs basis l hl p).PrimitiveProjector ↔ p.PrimitiveProjector := by
+  constructor
+  · intro hp
+    have hh := bottAdjointSheaf_primitive cs basis l hl _ hp
+    rw [bottAdjointSheaf_involutive cs basis l hl p] at hh
+    exact hh
+  · exact bottAdjointSheaf_primitive cs basis l hl p
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Primitive-corner recognition for genuine graded sheaf splittings.
+This supplies the algebraic recognition step for a prospective BMP summand;
+it neither postulates the splitting nor the word/BMP character. -/
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
+  {B C : GradedSheaf (𝓐 := 𝓐) G}
+noncomputable section
+
+ def SplitIn.liftEnd (s : SplitIn B C) (f : Hom B B) : Hom C C :=
+  s.inclusion.comp (f.comp s.projection)
+
+ theorem SplitIn.liftEnd_injective (s : SplitIn B C) : Function.Injective s.liftEnd := by
+  intro f g h
+  apply Hom.eq_of_maps
+  · intro x v
+    have hh := congrArg (fun q : Hom C C => (s.projection.vertex x).map
+      ((q.vertex x).map ((s.inclusion.vertex x).map v))) h
+    change (s.projection.vertex x).map ((s.inclusion.vertex x).map
+      ((f.vertex x).map ((s.projection.vertex x).map ((s.inclusion.vertex x).map v))))=
+      (s.projection.vertex x).map ((s.inclusion.vertex x).map
+      ((g.vertex x).map ((s.projection.vertex x).map ((s.inclusion.vertex x).map v)))) at hh
+    simpa only [s.vertex_split] using hh
+  · intro e v
+    have hh := congrArg (fun q : Hom C C => (s.projection.edge e).map
+      ((q.edge e).map ((s.inclusion.edge e).map v))) h
+    change (s.projection.edge e).map ((s.inclusion.edge e).map
+      ((f.edge e).map ((s.projection.edge e).map ((s.inclusion.edge e).map v))))=
+      (s.projection.edge e).map ((s.inclusion.edge e).map
+      ((g.edge e).map ((s.projection.edge e).map ((s.inclusion.edge e).map v)))) at hh
+    simpa only [s.edge_split] using hh
+
+ theorem SplitIn.liftEnd_comp (s : SplitIn B C) (f g : Hom B B) :
+    s.liftEnd (f.comp g)=(s.liftEnd f).comp (s.liftEnd g) := by
+  apply Hom.eq_of_maps
+  · intro x v
+    change (s.inclusion.vertex x).map ((f.vertex x).map ((g.vertex x).map
+      ((s.projection.vertex x).map v)))=
+      (s.inclusion.vertex x).map ((f.vertex x).map ((s.projection.vertex x).map
+      ((s.inclusion.vertex x).map ((g.vertex x).map ((s.projection.vertex x).map v)))))
+    rw [s.vertex_split]
+  · intro e v
+    change (s.inclusion.edge e).map ((f.edge e).map ((g.edge e).map
+      ((s.projection.edge e).map v)))=
+      (s.inclusion.edge e).map ((f.edge e).map ((s.projection.edge e).map
+      ((s.inclusion.edge e).map ((g.edge e).map ((s.projection.edge e).map v)))))
+    rw [s.edge_split]
+
+ theorem SplitIn.liftEnd_zero (s : SplitIn B C) : s.liftEnd Hom.zero=Hom.zero := by
+  apply Hom.eq_of_maps
+  · intro x v
+    change (s.inclusion.vertex x).map 0=0
+    exact map_zero _
+  · intro e v
+    change (s.inclusion.edge e).map 0=0
+    exact map_zero _
+
+ theorem SplitIn.primitive_id_of_projector (s : SplitIn B C)
+    (h : s.projector.PrimitiveProjector) : (Hom.id B).PrimitiveProjector := by
+  constructor
+  · apply Hom.eq_of_maps <;> intros <;> rfl
+  · intro f hf _ _
+    have hc : (s.liftEnd f).comp (s.liftEnd f)=s.liftEnd f :=
+      (s.liftEnd_comp f f).symm.trans (congrArg s.liftEnd hf)
+    have hi : s.liftEnd (Hom.id B)=s.projector := by
+      apply Hom.eq_of_maps <;> intros <;> rfl
+    have hleft : (s.liftEnd f).comp s.projector=s.liftEnd f := by
+      rw [← hi,← s.liftEnd_comp]
+      congr 1
+    have hright : s.projector.comp (s.liftEnd f)=s.liftEnd f := by
+      rw [← hi,← s.liftEnd_comp]
+      congr 1
+    rcases h.2 (s.liftEnd f) hc hleft hright with hz | he
+    · exact Or.inl (s.liftEnd_injective (hz.trans s.liftEnd_zero.symm))
+    · exact Or.inr (s.liftEnd_injective (he.trans hi.symm))
+
+ theorem SplitIn.projector_eq_id_of_primitive (s : SplitIn B C)
+    (h : (Hom.id C).PrimitiveProjector)
+    (x : V) [Nontrivial (B.vertex x)] : s.projector=Hom.id C := by
+  have hi : s.projector.comp s.projector=s.projector := by
+    apply Hom.eq_of_maps
+    · exact s.projector_vertex
+    · exact s.projector_edge
+  have hl : s.projector.comp (Hom.id C)=s.projector := by
+    apply Hom.eq_of_maps <;> intros <;> rfl
+  have hr : (Hom.id C).comp s.projector=s.projector := by
+    apply Hom.eq_of_maps <;> intros <;> rfl
+  rcases h.2 s.projector hi hl hr with hz | he
+  · obtain ⟨v,hv⟩ := exists_ne (0 : B.vertex x)
+    exfalso
+    apply hv
+    have hh := congrArg (fun q : Hom C C => (s.projection.vertex x).map
+      ((q.vertex x).map ((s.inclusion.vertex x).map v))) hz
+    change (s.projection.vertex x).map ((s.inclusion.vertex x).map
+      ((s.projection.vertex x).map ((s.inclusion.vertex x).map v)))=
+      (s.projection.vertex x).map 0 at hh
+    simpa only [s.vertex_split,map_zero] using hh
+  · exact he
+
+ def SplitIn.isoOfProjectorEqId (s : SplitIn B C) (h : s.projector=Hom.id C) : Iso B C where
+  vertex x :=
+    { toLinearMap := (s.inclusion.vertex x).map
+      invFun := (s.projection.vertex x).map
+      left_inv := s.vertex_split x
+      right_inv := fun v => congrArg (fun q : Hom C C => (q.vertex x).map v) h }
+  edge e :=
+    { toLinearMap := (s.inclusion.edge e).map
+      invFun := (s.projection.edge e).map
+      left_inv := s.edge_split e
+      right_inv := fun v => congrArg (fun q : Hom C C => (q.edge e).map v) h }
+  lower e v := s.inclusion.lower e v
+  upper e v := s.inclusion.upper e v
+  vertex_graded x := (s.inclusion.vertex x).graded
+  edge_graded e := (s.inclusion.edge e).graded
+
+ def SplitIn.isoOfPrimitive (s : SplitIn B C)
+    (h : (Hom.id C).PrimitiveProjector)
+    (x : V) [Nontrivial (B.vertex x)] : Iso B C :=
+  s.isoOfProjectorEqId (s.projector_eq_id_of_primitive h x)
+
+ variable [IsNoetherianRing A]
+
+ theorem Hom.primitive_image (p : Hom B B) (hp : p.PrimitiveProjector) :
+    (Hom.id p.image).PrimitiveProjector := by
+  have hv : ∀ x v, (p.vertex x).map ((p.vertex x).map v)=(p.vertex x).map v := by
+    intro x v
+    exact congrArg (fun q : Hom B B => (q.vertex x).map v) hp.1
+  have he : ∀ e v, (p.edge e).map ((p.edge e).map v)=(p.edge e).map v := by
+    intro e v
+    exact congrArg (fun q : Hom B B => (q.edge e).map v) hp.1
+  let s := p.imageSplit hv he
+  have hs : s.projector=p := by
+    apply Hom.eq_of_maps <;> intros <;> rfl
+  apply s.primitive_id_of_projector
+  rw [hs]
+  exact hp
+
+end
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+/-! A normalized minimal boundary sheaf is primitive, and that property
+passes to its projector in every ACTUAL graded splitting. No word splitting
+is asserted here. -/
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
+  {B C : GradedSheaf (𝓐 := 𝓐) G}
+noncomputable section
+
+ def SplitIn.downEnd (s : SplitIn B C) (f : Hom C C) : Hom B B :=
+  s.projection.comp (f.comp s.inclusion)
+
+ theorem SplitIn.lift_down (s : SplitIn B C) (f : Hom C C)
+    (hf : f.comp s.projector=f) (hf' : s.projector.comp f=f) :
+    s.liftEnd (s.downEnd f)=f := by
+  have heq : s.liftEnd (s.downEnd f)=s.projector.comp (f.comp s.projector) := by
+    apply Hom.eq_of_maps <;> intros <;> rfl
+  rw [heq,hf,hf']
+
+ theorem SplitIn.downEnd_idempotent (s : SplitIn B C) (f : Hom C C)
+    (hf : f.comp f=f) (hf' : s.projector.comp f=f) :
+    (s.downEnd f).comp (s.downEnd f)=s.downEnd f := by
+  have heq : (s.downEnd f).comp (s.downEnd f)=s.downEnd (f.comp (s.projector.comp f)) := by
+    apply Hom.eq_of_maps <;> intros <;> rfl
+  rw [heq,hf',hf]
+
+ theorem SplitIn.primitive_projector_of_id (s : SplitIn B C)
+    (h : (Hom.id B).PrimitiveProjector) : s.projector.PrimitiveProjector := by
+  constructor
+  · apply Hom.eq_of_maps
+    · exact s.projector_vertex
+    · exact s.projector_edge
+  · intro f hf hfp hpf
+    have hg := s.downEnd_idempotent f hf hpf
+    have hl : (s.downEnd f).comp (Hom.id B)=s.downEnd f := by
+      apply Hom.eq_of_maps <;> intros <;> rfl
+    have hr : (Hom.id B).comp (s.downEnd f)=s.downEnd f := by
+      apply Hom.eq_of_maps <;> intros <;> rfl
+    rcases h.2 (s.downEnd f) hg hl hr with hz | he
+    · left
+      exact (s.lift_down f hfp hpf).symm.trans ((congrArg s.liftEnd hz).trans s.liftEnd_zero)
+    · right
+      have hi : s.liftEnd (Hom.id B)=s.projector := by
+        apply Hom.eq_of_maps <;> intros <;> rfl
+      exact (s.lift_down f hfp hpf).symm.trans ((congrArg s.liftEnd he).trans hi)
+
+end
+end KLInvariance.MomentGraph.GradedSheaf
+
+namespace KLInvariance.MomentGraph.BoundarySheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+  {hneg : ∀ n < 0, 𝓐 n=⊥} {V E : Type um} [PartialOrder V] [Finite V]
+  {G : OrderedGraph V E} {α : E → A} {b : V} [IsDomain A]
+  (B : BoundarySheaf hneg G α b)
+
+ theorem primitive_identity : (GradedSheaf.Hom.id B.sheaf).PrimitiveProjector := by
+  constructor
+  · apply GradedSheaf.Hom.eq_of_maps <;> intros <;> rfl
+  · intro f hf _ _
+    have hi : ∀ x v, (f.toEndomorphism.vertex x).map ((f.toEndomorphism.vertex x).map v)=
+        (f.toEndomorphism.vertex x).map v := by
+      intro x v
+      exact congrArg (fun q : GradedSheaf.Hom B.sheaf B.sheaf => (q.vertex x).map v) hf
+    rcases B.idempotent_trivial f.toEndomorphism hi with hz | hi
+    · left
+      apply GradedSheaf.Hom.eq_of_maps
+      · intro x v
+        exact LinearMap.congr_fun (hz.1 x) v
+      · intro e v
+        exact LinearMap.congr_fun (hz.2 e) v
+    · right
+      apply GradedSheaf.Hom.eq_of_maps
+      · intro x v
+        exact LinearMap.congr_fun (hi.1 x) v
+      · intro e v
+        exact LinearMap.congr_fun (hi.2 e) v
+
+ theorem split_projector_primitive
+    {C : GradedSheaf (𝓐 := 𝓐) G} (s : GradedSheaf.SplitIn B.sheaf C) :
+    s.projector.PrimitiveProjector :=
+  s.primitive_projector_of_id B.primitive_identity
+
+end KLInvariance.MomentGraph.BoundarySheaf
+
+end
+
+
+section
+
+/-! For a supplied normalized boundary sheaf and an ACTUAL word splitting,
+the adjoint image is primitively indecomposable. The splitting's existence
+and the BMP axioms on that image still require word recognition. -/
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
+universe u us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  (l : List I) (hl : cs.IsReduced l)
+  (α : Edge cs 1 (cs.wordProd l) → SymmetricCoefficient (M := M))
+  (B : BoundarySheaf (symmetricGrading_negative basis)
+    (graph cs 1 (cs.wordProd l)) α
+    ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩)
+  (s : GradedSheaf.SplitIn B.sheaf (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
+noncomputable section
+
+ theorem bottBoundarySplitAdjoint_primitive :
+    (GradedSheaf.Hom.id (bottSplitAdjointSheaf cs basis l hl B.sheaf s)).PrimitiveProjector := by
+  let := symmetricCoefficientNoetherian basis
+  let := interval_finite cs 1 (cs.wordProd l)
+  exact GradedSheaf.Hom.primitive_image _ (bottAdjointSheaf_primitive cs basis l hl
+    s.projector (B.split_projector_primitive s))
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! The actual reduced word's top stalk is A in degree zero, integrally and
+in BOTH grading directions. This is the normalization needed to start a BMP
+summand construction, not a claim about the other stalks. -/
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded
+universe u us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  (l : List I) (hl : cs.IsReduced l)
+noncomputable section
+
+omit [Fintype σ] in
+ theorem bottTopStalkEquiv_homogeneous_iff (n : ℤ)
+    (s : bottStalk cs l (cs.wordProd l)) :
+    bottTopStalkEquiv cs l hl s ∈ symmetricGrading basis n ↔
+      s ∈ bottStalkPiece cs basis l (cs.wordProd l) n := by
+  constructor
+  · intro hs e _
+    have he : e=⟨bottFullIndex l,bottWeight_full cs l⟩ :=
+      Subtype.ext (bottWeight_eq_top cs l hl e.val e.property)
+    subst e
+    exact hs
+  · intro hs
+    exact hs ⟨bottFullIndex l,bottWeight_full cs l⟩ (Set.mem_univ _)
+
+ def bottTopHom : ModuleData.Hom (bottStalkData cs basis l (cs.wordProd l))
+    (ModuleData.unit (symmetricGrading_negative basis)) where
+  map := LinearMap.pi (fun _ => (bottTopStalkEquiv cs l hl).toLinearMap)
+  graded := by
+    intro n s hs j
+    change bottTopStalkEquiv cs l hl s ∈ symmetricGrading basis (n-0)
+    simpa only [sub_zero] using
+      (bottTopStalkEquiv_homogeneous_iff cs basis l hl n s).mpr hs
+
+ def bottTopInvHom : ModuleData.Hom (ModuleData.unit (symmetricGrading_negative basis))
+    (bottStalkData cs basis l (cs.wordProd l)) where
+  map := (bottTopStalkEquiv cs l hl).symm.toLinearMap.comp (LinearMap.proj PUnit.unit)
+  graded := by
+    intro n a ha
+    apply (bottTopStalkEquiv_homogeneous_iff cs basis l hl n _).mp
+    change bottTopStalkEquiv cs l hl ((bottTopStalkEquiv cs l hl).symm (a PUnit.unit)) ∈ _
+    rw [LinearEquiv.apply_symm_apply]
+    have h := ha PUnit.unit
+    change a PUnit.unit ∈ symmetricGrading basis (n-0) at h
+    simpa only [sub_zero] using h
+
+ theorem bottTopHom_inv (a : ModuleData.unit (symmetricGrading_negative basis)) :
+    (bottTopHom cs basis l hl).map ((bottTopInvHom cs basis l hl).map a)=a := by
+  funext j
+  change bottTopStalkEquiv cs l hl ((bottTopStalkEquiv cs l hl).symm (a PUnit.unit))=a j
+  rw [LinearEquiv.apply_symm_apply]
+
+ theorem bottTopInvHom_top (s : bottStalk cs l (cs.wordProd l)) :
+    (bottTopInvHom cs basis l hl).map ((bottTopHom cs basis l hl).map s)=s :=
+  (bottTopStalkEquiv cs l hl).symm_apply_apply s
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! A split pair between rank-one normalized degree-zero modules reverses.
+This uses commutativity of the genuine coefficient ring, not finite-dimensional
+vector-space reasoning or a restricted perfect form. -/
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/KazhdanLusztig/BottSamelson.lean b/lean/OAI/RepresentationTheory/KazhdanLusztig/BottSamelson.lean
index cfd6f6480758e4d8c10e5cdc48954415f133335a..7eedc87f3ca659a09e7ad60ce3df1a497d04aae2 100644
--- a/lean/OAI/RepresentationTheory/KazhdanLusztig/BottSamelson.lean
+++ b/lean/OAI/RepresentationTheory/KazhdanLusztig/BottSamelson.lean
@@ -1,2111 +1,2111 @@
-import OAI.RepresentationTheory.KazhdanLusztig.Frobenius
-
-namespace OAI
-
-/-!
-Bott-Samelson tensor ranges, actual sections, grading and right kernels.
--/
-
-section
-
-namespace KLInvariance
-open TensorProduct
-universe uc ua un uv up
-variable {C : Type uc} [CommRing C] {A : Type ua} [CommRing A] [Algebra C A]
-  {N : Type un} [AddCommGroup N] [Module C N]
-  {V : Type uv} [AddCommGroup V] [Module A V]
-  {P : Type up} [AddCommGroup P] [Module A P]
- theorem tensor_range_le_of_one_tmul (F : (A ⊗[C] N) →ₗ[A] V) (G : P →ₗ[A] V)
-    (h : ∀ n : N, F ((1 : A) ⊗ₜ[C] n) ∈ LinearMap.range G) :
-    LinearMap.range F ≤ LinearMap.range G := by
-  rintro _ ⟨m,rfl⟩
-  induction m using TensorProduct.inductionOn with
-  | add m n hm hn => rw [map_add]; exact (LinearMap.range G).add_mem hm hn
-  | tmul a n =>
-    have he : a ⊗ₜ[C] n = a • ((1 : A) ⊗ₜ[C] n) := by
-      rw [TensorProduct.smul_tmul', smul_eq_mul,mul_one]
-    rw [he,map_smul]
-    exact (LinearMap.range G).smul_mem a (h n)
- theorem tensor_maps_eq_of_one_tmul (F G : (A ⊗[C] N) →ₗ[A] V)
-    (h : ∀ n : N, F ((1 : A) ⊗ₜ[C] n)=G ((1 : A) ⊗ₜ[C] n)) : F=G := by
-  apply LinearMap.ext
-  intro m
-  induction m using TensorProduct.inductionOn with
-  | add m n hm hn => rw [map_add,map_add,hm,hn]
-  | tmul a n =>
-    have he : a ⊗ₜ[C] n = a • ((1 : A) ⊗ₜ[C] n) := by
-      rw [TensorProduct.smul_tmul',smul_eq_mul,mul_one]
-    rw [he,map_smul,map_smul,h n]
-
- theorem codRestrict_surjective_of_range_le (G : P →ₗ[A] V) (S : Submodule A V)
-    (hG : ∀ p, G p ∈ S) (hS : S ≤ LinearMap.range G) :
-    Function.Surjective (LinearMap.codRestrict S G hG) := by
-  intro z
-  obtain ⟨p,hp⟩ := hS z.property
-  exact ⟨p,Subtype.ext hp⟩
-end KLInvariance
-
-end
-
-
-section
-
-namespace KLInvariance.TitsSpace
-open Module TensorProduct FrobeniusDuality ReflectionFrobenius
-universe u v up
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-attribute [local instance] restrictedModule restrictedTower restrictedFree restrictedFinite
- theorem bottWeightProjection_range_le (i : I) (l : List I) (x : W)
-    {P : Type up} [AddCommGroup P] [Module (SymmetricCoefficient (M := M)) P]
-    (G : P →ₗ[SymmetricCoefficient (M := M)]
-      ({e : BottIndex (i::l) // bottWeight cs (i::l) e=x} → SymmetricCoefficient (M := M)))
-    (h : ∀ n : BottSamelsonModule cs l,
-      bottWeightProjection cs (i::l) x ((1 : SymmetricCoefficient (M := M)) ⊗ₜ[simpleFixed cs i] n) ∈ LinearMap.range G) :
-    LinearMap.range (bottWeightProjection cs (i::l) x) ≤ LinearMap.range G := by
-  exact tensor_range_le_of_one_tmul (C := simpleFixed cs i)
-    (A := SymmetricCoefficient (M := M)) (N := BottSamelsonModule cs l)
-    (bottWeightProjection cs (i::l) x) G h
- theorem bottTensor_maps_eq (i : I) (l : List I)
-    {P : Type up} [AddCommGroup P] [Module (SymmetricCoefficient (M := M)) P]
-    (F G : BottSamelsonModule cs (i::l) →ₗ[SymmetricCoefficient (M := M)] P)
-    (h : ∀ n : BottSamelsonModule cs l,
-      F ((1 : SymmetricCoefficient (M := M)) ⊗ₜ[simpleFixed cs i] n)=
-      G ((1 : SymmetricCoefficient (M := M)) ⊗ₜ[simpleFixed cs i] n)) : F=G :=
-  tensor_maps_eq_of_one_tmul (C := simpleFixed cs i)
-    (A := SymmetricCoefficient (M := M)) (N := BottSamelsonModule cs l) F G h
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! The genuine head-word stalk is an integral reflected pair lattice of the
-tail word. This supplies the stalk-freeness step once tail edge exactness is
-proved; it does not assume the desired main conclusion. -/
-namespace KLInvariance.TitsSpace
-open Module TensorProduct FrobeniusDuality ReflectionFrobenius ReflectionPairTranslation
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-attribute [local instance] restrictedModule restrictedTower restrictedFree restrictedFinite
-
- omit [Fintype I] in
- theorem bottWeight_tail_zero (i : I) (l : List I) (x : W)
-    (e : {e : BottIndex (i::l) // bottWeight cs (i::l) e=x}) (h : e.val.1=0) :
-    bottWeight cs l e.val.2=x := by
-  simpa only [bottWeight,ite_eq_left h] using e.property
-
- omit [Fintype I] in
- theorem bottWeight_tail_one (i : I) (l : List I) (x : W)
-    (e : {e : BottIndex (i::l) // bottWeight cs (i::l) e=x}) (h : e.val.1≠0) :
-    bottWeight cs l e.val.2=cs.simple i*x := by
-  have he : cs.simple i*bottWeight cs l e.val.2=x := by
-    simpa only [bottWeight,ite_eq_right h] using e.property
-  simpa only [cs.simple_mul_simple_cancel_left] using (congrArg (cs.simple i * ·) he)
-
- def bottHeadPairCoordinates (i : I) (l : List I) (x : W) :
-    (bottStalk cs l x × Twist (coefficientAction cs (cs.simple i)).toRingEquiv
-      (bottStalk cs l (cs.simple i*x))) →ₗ[SymmetricCoefficient (M := M)]
-      ({e : BottIndex (i::l) // bottWeight cs (i::l) e=x} → SymmetricCoefficient (M := M)) where
-  toFun p e := if h : e.val.1=0 then
-      p.1.val ⟨e.val.2,bottWeight_tail_zero cs i l x e h⟩
-    else coefficientAction cs (cs.simple i)
-      (p.2.val.val ⟨e.val.2,bottWeight_tail_one cs i l x e h⟩)
-  map_add' p q := by
-    funext e
-    by_cases h : e.val.1=0
-    · simp only [Pi.add_apply,dite_eq_left h]; rfl
-    · simp only [Pi.add_apply,dite_eq_right h]
-      exact map_add (coefficientAction cs (cs.simple i)) _ _
-  map_smul' r p := by
-    funext e
-    dsimp
-    split_ifs with h
-    · rfl
-    · change coefficientAction cs (cs.simple i)
-        (coefficientAction cs (cs.simple i) r * _) = r * coefficientAction cs (cs.simple i) _
-      rw [map_mul,simple_involutive]; rfl
-
- theorem bottHeadPairCoordinates_injective (i : I) (l : List I) (x : W) :
-    Function.Injective (bottHeadPairCoordinates cs i l x) := by
-  rintro ⟨p,q⟩ ⟨p',q'⟩ h
-  apply Prod.ext
-  · apply Subtype.ext
-    funext e
-    have hh := congrFun h ⟨(0,e.val),by simpa only [bottWeight,ite_true] using e.property⟩
-    simpa only [bottHeadPairCoordinates,LinearMap.coe_mk,AddHom.coe_mk,dite_eq_left rfl,dite_true] using hh
-  · apply Twist.ext
-    apply Subtype.ext
-    funext e
-    have he : bottWeight cs (i::l) (1,e.val)=x := by
-      change (if (1 : Fin 2)=0 then _ else cs.simple i*bottWeight cs l e.val)=x
-      rw [ite_eq_right (by decide),e.property,cs.simple_mul_simple_cancel_left]
-    have hh := congrFun h ⟨(1,e.val),he⟩
-    have hh' : coefficientAction cs (cs.simple i) (q.val.val e) =
-        coefficientAction cs (cs.simple i) (q'.val.val e) := by
-      simpa only [bottHeadPairCoordinates,LinearMap.coe_mk,AddHom.coe_mk,
-        dite_eq_right (show (1 : Fin 2)≠0 by decide)] using hh
-    exact (coefficientAction cs (cs.simple i)).injective hh'
-
- theorem simpleReverseDivisibility (i : I) (a : SymmetricCoefficient (M := M)) :
-    simpleRootPolynomial (M := M) i ∣ coefficientAction cs (cs.simple i) a-a :=
-  (dvd_sub_comm).mpr (simple_divisible cs i a)
-
- variable (i : I) (l : List I) (x : W)
-
- def bottHeadPairLattice := lattice (bottStalkProjection cs l x)
-    (bottStalkProjection cs l (cs.simple i*x)) (coefficientAction cs (cs.simple i)).toRingEquiv
-    (simpleRootPolynomial (M := M) i) (simpleReverseDivisibility cs i) (bottSimpleEdge_kernel_inclusion cs i l x)
-
- instance bottHeadPairAddCommGroup : AddCommGroup (bottHeadPairLattice cs i l x) :=
-  Submodule.addCommGroup (bottHeadPairLattice cs i l x)
-
- instance bottHeadPairModule : Module (SymmetricCoefficient (M := M)) (bottHeadPairLattice cs i l x) :=
-  Submodule.module (bottHeadPairLattice cs i l x)
-
- theorem bottHeadPairCoordinates_seed (n : BottSamelsonModule cs l) :
-    bottHeadPairCoordinates cs i l x
-      (bottStalkProjection cs l x n,⟨bottStalkProjection cs l (cs.simple i*x) n⟩) =
-    bottWeightProjection cs (i::l) x ((1 : SymmetricCoefficient (M := M)) ⊗ₜ[simpleFixed cs i] n) := by
-  funext e
-  change (if h : e.val.1=0 then _ else _) = (1 : SymmetricCoefficient (M := M))*(if e.val.1=0 then _ else _)
-  rw [one_mul]
-  split_ifs <;> rfl
-
- def bottHeadPairToStalk : bottHeadPairLattice cs i l x →ₗ[
-     SymmetricCoefficient (M := M)] bottStalk cs (i::l) x :=
-  LinearMap.codRestrict (bottStalk cs (i::l) x)
-    ((bottHeadPairCoordinates cs i l x).comp (bottHeadPairLattice cs i l x).subtype) (by
-      rintro ⟨⟨p,q⟩,n,hn,hn'⟩
-      refine ⟨(1 : SymmetricCoefficient (M := M)) ⊗ₜ[simpleFixed cs i] n,?_⟩
-      change bottWeightProjection cs (i::l) x _=bottHeadPairCoordinates cs i l x (p,q)
-      rw [← bottHeadPairCoordinates_seed]
-      congr 1
-      exact Prod.ext hn (Twist.ext hn'))
-
- theorem bottHeadPairToStalk_injective : Function.Injective (bottHeadPairToStalk cs i l x) := by
-  intro p q h
-  apply Subtype.ext
-  exact bottHeadPairCoordinates_injective cs i l x (congrArg Subtype.val h)
-
- theorem bottHeadPair_range :
-    LinearMap.range (bottWeightProjection cs (i::l) x) ≤
-      LinearMap.range ((bottHeadPairCoordinates cs i l x).comp
-        (bottHeadPairLattice cs i l x).subtype) := by
-  refine bottWeightProjection_range_le cs i l x (P := bottHeadPairLattice cs i l x)
-    ((bottHeadPairCoordinates cs i l x).comp (bottHeadPairLattice cs i l x).subtype) ?_
-  intro n
-  refine ⟨⟨(bottStalkProjection cs l x n,⟨bottStalkProjection cs l (cs.simple i*x) n⟩),n,rfl,rfl⟩,?_⟩
-  exact bottHeadPairCoordinates_seed cs i l x n
-
- theorem bottHeadPairToStalk_surjective : Function.Surjective (bottHeadPairToStalk cs i l x) := by
-  exact codRestrict_surjective_of_range_le (A := SymmetricCoefficient (M := M))
-    (P := bottHeadPairLattice cs i l x)
-    (V := {e : BottIndex (i::l) // bottWeight cs (i::l) e=x} → SymmetricCoefficient (M := M))
-    ((bottHeadPairCoordinates cs i l x).comp (bottHeadPairLattice cs i l x).subtype)
-    (bottStalk cs (i::l) x) _ (bottHeadPair_range cs i l x)
-
- def bottHeadPairEquiv : bottHeadPairLattice cs i l x ≃ₗ[
-     SymmetricCoefficient (M := M)] bottStalk cs (i::l) x :=
-  LinearEquiv.ofBijective (bottHeadPairToStalk cs i l x)
-    ⟨bottHeadPairToStalk_injective cs i l x,bottHeadPairToStalk_surjective cs i l x⟩
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! The top projection lattice of a reduced actual tensor word is exactly A.
-This verifies the genuine rank-one top used in the BMP summand construction;
-it does not assert that the other projection lattices form a flabby sheaf. -/
-namespace KLInvariance.TitsSpace
-open Module TensorProduct
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
- def bottFullIndex : (l : List I) → BottIndex l
-  | [] => PUnit.unit
-  | _::l => (1,bottFullIndex l)
-
-omit [Fintype I] in
- theorem bottWeight_full (l : List I) :
-    bottWeight cs l (bottFullIndex l)=cs.wordProd l := by
-  induction l with
-  | nil => exact cs.wordProd_nil.symm
-  | cons i l ih =>
-    simp only [bottFullIndex,bottWeight,show (1:Fin 2) ≠ 0 by decide,ite_false,
-      cs.wordProd_cons,ih]
-
-omit [Fintype I] in
- theorem bottSubword_length_eq_full (l : List I) (e : BottIndex l)
-    (h : (bottSubword l e).length=l.length) : e=bottFullIndex l := by
-  induction l with
-  | nil =>
-    change PUnit at e
-    change e=(PUnit.unit : PUnit)
-    exact Subsingleton.elim _ _
-  | cons i l ih =>
-    change Fin 2 × BottIndex l at e
-    by_cases he : e.1=0
-    · simp only [bottSubword,ite_eq_left he,List.length_cons] at h
-      have hn := (bottSubword_sublist l e.2).length_le
-      omega
-    · have he1 : e.1=1 := by omega
-      simp only [bottSubword,ite_eq_right he,List.length_cons] at h
-      exact Prod.ext he1 (ih e.2 (Nat.succ.inj h))
-
-omit [Fintype I] in
- theorem bottWeight_eq_top (l : List I) (hl : cs.IsReduced l) (e : BottIndex l)
-    (he : bottWeight cs l e=cs.wordProd l) : e=bottFullIndex l := by
-  apply bottSubword_length_eq_full
-  apply Nat.le_antisymm (bottSubword_sublist l e).length_le
-  have hh := cs.length_wordProd_le (bottSubword l e)
-  rwa [bottSubword_wordProd,he,hl.eq] at hh
-
- def bottUnit : (l : List I) → BottSamelsonModule cs l
-  | [] => (1 : SymmetricCoefficient (M := M))
-  | i::l => 1 ⊗ₜ[simpleFixed cs i] bottUnit l
-
- theorem bottEvaluation_unit (l : List I) (e : BottIndex l) :
-    bottEvaluation cs l e (bottUnit cs l)=1 := by
-  induction l with
-  | nil => rfl
-  | cons i l ih =>
-    change bottEvaluation cs (i::l) e (1 ⊗ₜ[simpleFixed cs i] bottUnit cs l)=1
-    rw [bottEvaluation_tmul,ih]
-    split_ifs <;> simp only [map_one,mul_one]
-
- theorem bottEvaluation_surjective (l : List I) (e : BottIndex l) :
-    Function.Surjective (bottEvaluation cs l e) := by
-  intro a
-  refine ⟨a • bottUnit cs l,?_⟩
-  rw [map_smul,bottEvaluation_unit,smul_eq_mul,mul_one]
-
- def bottTopCoordinate (l : List I) : bottStalk cs l (cs.wordProd l) →ₗ[
-    SymmetricCoefficient (M := M)] SymmetricCoefficient (M := M) :=
-  (LinearMap.proj ⟨bottFullIndex l,bottWeight_full cs l⟩).comp
-    (bottStalk cs l (cs.wordProd l)).subtype
-
- theorem bottTopCoordinate_bijective (l : List I) (hl : cs.IsReduced l) :
-    Function.Bijective (bottTopCoordinate cs l) := by
-  constructor
-  · intro a b hab
-    apply Subtype.ext
-    funext e
-    have he : e=⟨bottFullIndex l,bottWeight_full cs l⟩ :=
-      Subtype.ext (bottWeight_eq_top cs l hl e.val e.property)
-    subst e
-    exact hab
-  · intro a
-    obtain ⟨m,hm⟩ := bottEvaluation_surjective cs l (bottFullIndex l) a
-    exact ⟨bottStalkProjection cs l (cs.wordProd l) m,hm⟩
-
- def bottTopStalkEquiv (l : List I) (hl : cs.IsReduced l) :
-    bottStalk cs l (cs.wordProd l) ≃ₗ[SymmetricCoefficient (M := M)]
-      SymmetricCoefficient (M := M) :=
-  LinearEquiv.ofBijective (bottTopCoordinate cs l) (bottTopCoordinate_bijective cs l hl)
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! The genuine subexpression projections preserve the actual word grading.
-In particular the top projection of a reduced tensor word is a graded copy of
-A, rather than just an ungraded rank-one lattice. -/
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- theorem bottEvaluation_basis_homogeneous (l : List I) (e f : BottIndex l) :
-    bottEvaluationMatrix cs l e f ∈ symmetricGrading basis (bottDegree l f) := by
-  induction l with
-  | nil =>
-    change PUnit at e f
-    change (Basis.singleton PUnit (SymmetricCoefficient (M := M)) f) ∈
-      symmetricGrading basis 0
-    simp only [Basis.singleton_apply]
-    exact SetLike.one_mem_graded _
-  | cons i l ih =>
-    change Fin 2 × BottIndex l at e f
-    rcases e with ⟨r,j⟩
-    rcases f with ⟨c,k⟩
-    rw [bottEvaluationMatrix_cons]
-    by_cases hc : c=0
-    · subst c
-      simp only [ite_true,bottDegree,Fin.val_zero,Nat.cast_zero,zero_add]
-      split_ifs
-      · exact ih j k
-      · exact coefficientAction_homogeneous cs basis _ _ _ (ih j k)
-    · have hc1 : c=1 := by omega
-      subst c
-      simp only [show (1:Fin 2) ≠ 0 by decide,ite_false,bottDegree,Fin.val_one,Nat.cast_one]
-      split_ifs
-      · exact SetLike.GradedMul.mul_mem (simpleRoot_homogeneous basis i) (ih j k)
-      · exact SetLike.GradedMul.mul_mem ((symmetricGrading basis 1).neg_mem
-          (simpleRoot_homogeneous basis i))
-          (coefficientAction_homogeneous cs basis _ _ _ (ih j k))
-
- theorem bottEvaluation_homogeneous (l : List I) (e : BottIndex l)
-    (n : ℤ) (m : BottSamelsonModule cs l) (hm : m ∈ bottPiece cs basis l n) :
-    bottEvaluation cs l e m ∈ symmetricGrading basis n := by
-  classical
-  have hexp : bottEvaluation cs l e m = ∑ f : BottIndex l,
-      (bottBasis cs l).repr m f * bottEvaluationMatrix cs l e f := by
-    conv_lhs => rw [← (bottBasis cs l).sum_repr m]
-    simp only [map_sum,map_smul,smul_eq_mul,bottEvaluationMatrix]
-  rw [hexp]
-  apply Submodule.sum_mem
-  intro f _
-  change ∀ f, (bottBasis cs l).repr m f ∈
-    symmetricGrading basis (n-bottDegree l f) at hm
-  have h := SetLike.GradedMul.mul_mem (hm f)
-    (bottEvaluation_basis_homogeneous cs basis l e f)
-  simpa only [sub_add_cancel] using h
-
- theorem bottEvaluation_component (l : List I) (e : BottIndex l)
-    (n : ℤ) (m : BottSamelsonModule cs l) :
-    component (symmetricGrading basis) n (bottEvaluation cs l e m) =
-      bottEvaluation cs l e (component (bottPiece cs basis l) n m) := by
-  simpa only [sub_zero] using map_component (bottPiece cs basis l) (symmetricGrading basis)
-    (bottEvaluation cs l e) 0 (by simpa only [add_zero] using bottEvaluation_homogeneous cs basis l e) n m
-
- theorem bottUnit_homogeneous (l : List I) : bottUnit cs l ∈ bottPiece cs basis l 0 := by
-  have he : component (bottPiece cs basis l) 0 (bottUnit cs l)=bottUnit cs l := by
-    apply bottEvaluation_injective cs l
-    funext e
-    change bottEvaluation cs l e (component (bottPiece cs basis l) 0 (bottUnit cs l)) =
-      bottEvaluation cs l e (bottUnit cs l)
-    rw [← bottEvaluation_component,bottEvaluation_unit]
-    exact DirectSum.decompose_of_mem_same (symmetricGrading basis) (SetLike.one_mem_graded _)
-  rw [← he]
-  exact (DirectSum.decompose (bottPiece cs basis l) (bottUnit cs l) 0).property
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Actual integral projection and support lattices inherit their grading from
-the genuine tensor word. In particular a homogeneous stalk vector has a
-homogeneous tensor lift. No freeness/character claim is assumed. -/
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- def bottCoordinatePiece (l : List I) (x : W) : ℤ → Submodule ℝ
-    ({e : BottIndex l // bottWeight cs l e=x} → SymmetricCoefficient (M := M)) :=
-  piPiece (fun _ => symmetricGrading basis)
-
- instance bottCoordinatePiece_decomposition (l : List I) (x : W) :
-    DirectSum.Decomposition (bottCoordinatePiece cs basis l x) := by
-  classical
-  exact (pi_isInternal (fun _ : {e : BottIndex l // bottWeight cs l e=x} =>
-    symmetricGrading basis)).chooseDecomposition
-
- instance bottCoordinatePiece_graded (l : List I) (x : W) :
-    SetLike.GradedSMul (symmetricGrading basis) (bottCoordinatePiece cs basis l x) :=
-  inferInstanceAs (SetLike.GradedSMul (symmetricGrading basis)
-    (piPiece (fun _ : {e : BottIndex l // bottWeight cs l e=x} => symmetricGrading basis)))
-
- theorem bottWeightProjection_homogeneous (l : List I) (x : W) (n : ℤ)
-    (m : BottSamelsonModule cs l) (hm : m ∈ bottPiece cs basis l n) :
-    bottWeightProjection cs l x m ∈ bottCoordinatePiece cs basis l x n := by
-  intro e _
-  exact bottEvaluation_homogeneous cs basis l e.val n m hm
-
- theorem bottStalk_homogeneous (l : List I) (x : W) :
-    ∀ n f, f ∈ bottStalk cs l x →
-      component (bottCoordinatePiece cs basis l x) n f ∈ bottStalk cs l x := by
-  exact range_homogeneous (bottPiece cs basis l) (bottCoordinatePiece cs basis l x)
-    (bottWeightProjection cs l x) 0 (by
-      simpa only [add_zero] using bottWeightProjection_homogeneous cs basis l x)
-
- def bottStalkPiece (l : List I) (x : W) : ℤ → Submodule ℝ (bottStalk cs l x) :=
-  subPiece (bottCoordinatePiece cs basis l x) (bottStalk cs l x)
-
- instance bottStalkPiece_decomposition (l : List I) (x : W) :
-    DirectSum.Decomposition (bottStalkPiece cs basis l x) :=
-  (sub_isInternal (bottCoordinatePiece cs basis l x) (bottStalk cs l x)
-    (bottStalk_homogeneous cs basis l x)).chooseDecomposition
-
- instance bottStalkPiece_graded (l : List I) (x : W) :
-    SetLike.GradedSMul (symmetricGrading basis) (bottStalkPiece cs basis l x) :=
-  inferInstanceAs (SetLike.GradedSMul (symmetricGrading basis)
-    (subPiece (bottCoordinatePiece cs basis l x) (bottStalk cs l x)))
-
- theorem bottStalkProjection_homogeneous (l : List I) (x : W) (n : ℤ)
-    (m : BottSamelsonModule cs l) (hm : m ∈ bottPiece cs basis l n) :
-    bottStalkProjection cs l x m ∈ bottStalkPiece cs basis l x n :=
-  bottWeightProjection_homogeneous cs basis l x n m hm
-
- theorem bottStalk_homogeneous_lift (l : List I) (x : W) (n : ℤ)
-    (s : bottStalk cs l x) (hs : s ∈ bottStalkPiece cs basis l x n) :
-    ∃ m : BottSamelsonModule cs l, m ∈ bottPiece cs basis l n ∧
-      bottStalkProjection cs l x m=s := by
-  obtain ⟨m,hm⟩ := bottStalkProjection_surjective cs l x s
-  refine ⟨component (bottPiece cs basis l) n m,
-    (DirectSum.decompose (bottPiece cs basis l) m n).property,?_⟩
-  have he := map_component (bottPiece cs basis l) (bottStalkPiece cs basis l x)
-    (bottStalkProjection cs l x) 0 (by
-      simpa only [add_zero] using bottStalkProjection_homogeneous cs basis l x) n m
-  rw [sub_zero,hm,show component (bottStalkPiece cs basis l x) n s=s from
-    DirectSum.decompose_of_mem_same _ hs] at he
-  exact he.symm
-
- theorem bottWeightSupported_homogeneous (l : List I) (x : W) :
-    ∀ n m, m ∈ bottWeightSupported cs l x →
-      component (bottPiece cs basis l) n m ∈ bottWeightSupported cs l x := by
-  intro n m hm e he
-  rw [← bottEvaluation_component,hm e he,map_zero]
-
- def bottSupportPiece (l : List I) (x : W) :
-    ℤ → Submodule ℝ (bottWeightSupported cs l x) :=
-  subPiece (bottPiece cs basis l) (bottWeightSupported cs l x)
-
- instance bottSupportPiece_decomposition (l : List I) (x : W) :
-    DirectSum.Decomposition (bottSupportPiece cs basis l x) :=
-  (sub_isInternal (bottPiece cs basis l) (bottWeightSupported cs l x)
-    (bottWeightSupported_homogeneous cs basis l x)).chooseDecomposition
-
- instance bottSupportPiece_graded (l : List I) (x : W) :
-    SetLike.GradedSMul (symmetricGrading basis) (bottSupportPiece cs basis l x) :=
-  inferInstanceAs (SetLike.GradedSMul (symmetricGrading basis)
-    (subPiece (bottPiece cs basis l) (bottWeightSupported cs l x)))
-
-omit [Fintype σ] in
- theorem bottTopStalkEquiv_homogeneous (l : List I) (hl : cs.IsReduced l)
-    (n : ℤ) (s : bottStalk cs l (cs.wordProd l)) :
-    bottTopStalkEquiv cs l hl s ∈ symmetricGrading basis n ↔
-      s ∈ bottStalkPiece cs basis l (cs.wordProd l) n := by
-  constructor
-  · intro h e _
-    have he : e=⟨bottFullIndex l,bottWeight_full cs l⟩ :=
-      Subtype.ext (bottWeight_eq_top cs l hl e.val e.property)
-    subst e
-    exact h
-  · intro h
-    exact h ⟨bottFullIndex l,bottWeight_full cs l⟩ (Set.mem_univ _)
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Restriction of the honest reflection-twisted module to the fixed base field.
-The automorphism fixes the base field, so no scalar information is discarded. -/
-namespace KLInvariance.ReflectionPairTranslation.Twist
-universe uk ur uy
-variable {k : Type uk} [Field k] {R : Type ur} [CommRing R] [Algebra k R]
-  (τ : R ≃ₐ[k] R) (Y : Type uy) [AddCommGroup Y] [Module R Y] [Module k Y]
-  [IsScalarTower k R Y]
-
-instance scalarModule : Module k (Twist τ.toRingEquiv Y) :=
-  ({ equivVal τ.toRingEquiv Y with map_add' := fun _ _ => rfl } :
-    Twist τ.toRingEquiv Y ≃+ Y).module k
-
-instance scalarTower : IsScalarTower k R (Twist τ.toRingEquiv Y) where
-  smul_assoc c r y := by
-    apply Twist.ext
-    change τ (c • r) • y.val=c • (τ r • y.val)
-    rw [map_smul,smul_assoc]
-
-def scalarEquiv : Twist τ.toRingEquiv Y ≃ₗ[k] Y where
-  toFun := val
-  invFun := mk
-  left_inv _ := rfl
-  right_inv _ := rfl
-  map_add' _ _ := rfl
-  map_smul' _ _ := rfl
-
-end KLInvariance.ReflectionPairTranslation.Twist
-
-end
-
-
-section
-
-/-! Homogeneous perfect pairings on the actual integral free modules.  The
-inverse grading is proved, not inferred merely after fraction-field extension. -/
-namespace KLInvariance.Graded
-open Module
-universe uk ua um un ui
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {N : Type um} [AddCommGroup N] [Module A N] [Module k N] [IsScalarTower k A N]
-  {P : Type un} [AddCommGroup P] [Module A P] [Module k P] [IsScalarTower k A P]
-noncomputable section
-
-omit [Algebra k A] [IsScalarTower k A N] [IsScalarTower k A P] in
- theorem equiv_inverse_homogeneous
-    (𝓝 : ℤ → Submodule k N) (𝓟 : ℤ → Submodule k P)
-    [DirectSum.Decomposition 𝓝] [DirectSum.Decomposition 𝓟]
-    (e : N ≃ₗ[A] P) (d : ℤ)
-    (he : ∀ n x, x ∈ 𝓝 n → e x ∈ 𝓟 (n+d))
-    (n : ℤ) (y : P) (hy : y ∈ 𝓟 n) : e.symm y ∈ 𝓝 (n-d) := by
-  have hc : component 𝓝 (n-d) (e.symm y) = e.symm y := by
-    apply e.injective
-    change e.toLinearMap (component 𝓝 (n-d) (e.symm y)) = e.toLinearMap (e.symm y)
-    rw [← map_component 𝓝 𝓟 e.toLinearMap d he n (e.symm y)]
-    simp only [LinearEquiv.coe_coe,LinearEquiv.apply_symm_apply]
-    exact DirectSum.decompose_of_mem_same 𝓟 hy
-  rw [← hc]
-  exact (DirectSum.decompose 𝓝 (e.symm y) (n-d)).property
-
-variable (𝓐 : ℤ → Submodule k A) [DirectSum.Decomposition 𝓐]
-  [SetLike.GradedMonoid 𝓐] {ι : Type ui} [Fintype ι] [DecidableEq ι]
-  (b : Basis ι A N) (d : ι → ℤ)
-
-omit [DirectSum.Decomposition 𝓐] in
- theorem pairing_to_dual_homogeneous (p : N →ₗ[A] Dual A N) (l : ℤ)
-    (hp : ∀ i j, p (b i) (b j) ∈ 𝓐 (d i+d j-l))
-    {n : ℤ} {x : N} (hx : x ∈ basisPiece 𝓐 b d n) :
-    p x ∈ basisPiece 𝓐 b.dualBasis (fun i => -d i) (n-l) := by
-  rw [mem_basisPiece]
-  intro i
-  rw [Basis.dualBasis_repr]
-  have h := pairing_homogeneous 𝓐 b d p l hp hx (basis_mem_piece 𝓐 b d i)
-  convert h using 1
-  congr 1
-  omega
-
- theorem perfectPairing_inverse_homogeneous (p : N ≃ₗ[A] Dual A N) (l : ℤ)
-    (hp : ∀ i j, p (b i) (b j) ∈ 𝓐 (d i+d j-l))
-    {n : ℤ} {y : Dual A N}
-    (hy : y ∈ basisPiece 𝓐 b.dualBasis (fun i => -d i) n) :
-    p.symm y ∈ basisPiece 𝓐 b d (n+l) := by
-  have he : ∀ m x, x ∈ basisPiece 𝓐 b d m →
-      p x ∈ basisPiece 𝓐 b.dualBasis (fun i => -d i) (m+(-l)) := by
-    intro m x hx
-    exact pairing_to_dual_homogeneous 𝓐 b d p.toLinearMap l hp hx
-  simpa only [sub_neg_eq_add] using equiv_inverse_homogeneous
-    (basisPiece 𝓐 b d) (basisPiece 𝓐 b.dualBasis (fun i => -d i)) p (-l) he n y hy
-
-end
-end KLInvariance.Graded
-
-end
-
-
-section
-
-/-! The genuine head-word projection maps, on the fixed real field, preserve
-homogeneous degrees. These maps are not abstract character operators. -/
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded ReflectionPairTranslation
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- def bottHeadLeft (i : I) (l : List I) (x : W) :
-    bottStalk cs (i::l) x →ₗ[SymmetricCoefficient (M := M)] bottStalk cs l x :=
-  (ReflectionPairTranslation.left (bottStalkProjection cs l x)
-    (bottStalkProjection cs l (cs.simple i*x)) (coefficientAction cs (cs.simple i)).toRingEquiv
-    (simpleRootPolynomial (M := M) i) (simpleReverseDivisibility cs i)
-    (bottSimpleEdge_kernel_inclusion cs i l x)).comp (bottHeadPairEquiv cs i l x).symm.toLinearMap
-
- def bottHeadRight (i : I) (l : List I) (x : W) :
-    bottStalk cs (i::l) x →ₗ[ℝ] bottStalk cs l (cs.simple i*x) :=
-  (Twist.scalarEquiv (coefficientAction cs (cs.simple i)) _).toLinearMap.comp
-    (((LinearMap.snd (SymmetricCoefficient (M := M)) (bottStalk cs l x)
-      (Twist (coefficientAction cs (cs.simple i)).toRingEquiv (bottStalk cs l (cs.simple i*x)))).comp
-        ((bottHeadPairLattice cs i l x).subtype.comp
-          (bottHeadPairEquiv cs i l x).symm.toLinearMap)).restrictScalars ℝ)
-
- theorem bottHeadLeft_surjective (i : I) (l : List I) (x : W) :
-    Function.Surjective (bottHeadLeft cs i l x) :=
-  (left_surjective _ _ _ _ _ _ (bottStalkProjection_surjective cs l x)).comp
-    (bottHeadPairEquiv cs i l x).symm.surjective
-
- theorem bottHeadPairEquiv_coordinates (i : I) (l : List I) (x : W)
-    (m : bottStalk cs (i::l) x) :
-    bottHeadPairCoordinates cs i l x ((bottHeadPairEquiv cs i l x).symm m).val=m.val :=
-  congrArg Subtype.val ((bottHeadPairEquiv cs i l x).apply_symm_apply m)
-
-omit [Fintype σ] in
- theorem bottHeadLeft_homogeneous (i : I) (l : List I) (x : W) (n : ℤ)
-    (m : bottStalk cs (i::l) x) (hm : m ∈ bottStalkPiece cs basis (i::l) x n) :
-    bottHeadLeft cs i l x m ∈ bottStalkPiece cs basis l x n := by
-  intro e _
-  change (bottHeadLeft cs i l x m).val e ∈ symmetricGrading basis n
-  let E : {e : BottIndex (i::l) // bottWeight cs (i::l) e=x} :=
-    ⟨(0,e.val),by simpa only [bottWeight,ite_true] using e.property⟩
-  have he := congrFun (bottHeadPairEquiv_coordinates cs i l x m) E
-  have hh : (bottHeadLeft cs i l x m).val e=m.val E := by
-    change ((bottHeadPairEquiv cs i l x).symm m).val.1.val e=m.val E at he
-    exact he
-  rw [hh]
-  exact hm E (Set.mem_univ E)
-
- theorem bottHeadRight_homogeneous (i : I) (l : List I) (x : W) (n : ℤ)
-    (m : bottStalk cs (i::l) x) (hm : m ∈ bottStalkPiece cs basis (i::l) x n) :
-    bottHeadRight cs i l x m ∈ bottStalkPiece cs basis l (cs.simple i*x) n := by
-  intro e _
-  change (bottHeadRight cs i l x m).val e ∈ symmetricGrading basis n
-  let E : {e : BottIndex (i::l) // bottWeight cs (i::l) e=x} :=
-    ⟨(1,e.val),by simp only [bottWeight,ite_eq_right (show (1:Fin 2)≠0 by decide),
-      e.property,cs.simple_mul_simple_cancel_left]⟩
-  have he := congrFun (bottHeadPairEquiv_coordinates cs i l x m) E
-  have hh : coefficientAction cs (cs.simple i) ((bottHeadRight cs i l x m).val e)=m.val E := by
-    change coefficientAction cs (cs.simple i)
-      (((bottHeadPairEquiv cs i l x).symm m).val.2.val.val e)=m.val E at he
-    exact he
-  have h := coefficientAction_homogeneous cs basis (cs.simple i) n (m.val E)
-    (hm E (Set.mem_univ E))
-  rw [← hh,simple_involutive] at h
-  exact h
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! The two common tail components of a reflected tensor. They are defined
-on the actual tensor, so divisibility assertions at different weights use the
-same tail vectors rather than unrelated existential lifts. -/
-namespace KLInvariance.ReflectionPairTranslation
-open TensorProduct
-universe uc ua un
-variable {C : Type uc} [CommRing C] {A : Type ua} [CommRing A] [Algebra C A]
-  {N : Type un} [AddCommGroup N] [Module A N] [Module C N] [IsScalarTower C A N]
-  (τ : A ≃+* A) (hfix : ∀ c : C, τ (algebraMap C A c)=algebraMap C A c)
-noncomputable section
-
- def tensorFirst : (A ⊗[C] N) →ₗ[A] N :=
-  AlgebraTensorModule.lift ((LinearMap.id : A →ₗ[A] A).smulRight
-    (LinearMap.id : N →ₗ[C] N))
-
- @[simp] theorem tensorFirst_tmul (a : A) (n : N) :
-    tensorFirst (C := C) (a ⊗ₜ[C] n)=a • n := rfl
-
- def tensorSecond : (A ⊗[C] N) →ₗ[A] Twist τ N := by
-  letI : Module C (Twist τ N) := Module.compHom _ (algebraMap C A)
-  letI : IsScalarTower C A (Twist τ N) :=
-    IsScalarTower.of_algebraMap_smul fun _ _ => rfl
-  let F : N →ₗ[C] Twist τ N := {
-    toFun := Twist.mk
-    map_add' := fun _ _ => rfl
-    map_smul' := fun c n => by
-      apply Twist.ext
-      change c • n=τ (algebraMap C A c) • n
-      rw [hfix,algebraMap_smul] }
-  exact AlgebraTensorModule.lift ((LinearMap.id : A →ₗ[A] A).smulRight F)
-
- @[simp] theorem tensorSecond_tmul (a : A) (n : N) :
-    tensorSecond τ hfix (a ⊗ₜ[C] n)=Twist.mk (τ a • n) := rfl
-
- def twistMap {P : Type*} [AddCommGroup P] [Module A P] (f : N →ₗ[A] P) :
-    Twist τ N →ₗ[A] Twist τ P where
-  toFun m := ⟨f m.val⟩
-  map_add' m n := Twist.ext (map_add f m.val n.val)
-  map_smul' a m := Twist.ext (map_smul f (τ a) m.val)
-end
-end KLInvariance.ReflectionPairTranslation
-
-end
-
-
-section
-
-namespace KLInvariance.TitsSpace
-open Module TensorProduct FrobeniusDuality ReflectionFrobenius ReflectionPairTranslation
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-attribute [local instance] restrictedModule restrictedTower restrictedFree restrictedFinite
-
- def bottHeadTail (i : I) (l : List I) : BottSamelsonModule cs (i::l) →ₗ[
-    SymmetricCoefficient (M := M)] BottSamelsonModule cs l :=
-  tensorFirst (C := simpleFixed cs i)
-
- def bottHeadReflectedTail (i : I) (l : List I) : BottSamelsonModule cs (i::l) →ₗ[
-    SymmetricCoefficient (M := M)] Twist (coefficientAction cs (cs.simple i)).toRingEquiv
-      (BottSamelsonModule cs l) :=
-  tensorSecond (C := simpleFixed cs i) (coefficientAction cs (cs.simple i)).toRingEquiv
-    (fun c => c.property)
-
- theorem bottHeadTail_seed (i : I) (l : List I) (n : BottSamelsonModule cs l) :
-    bottHeadTail cs i l ((1 : SymmetricCoefficient (M := M)) ⊗ₜ[simpleFixed cs i] n)=n := by
-  exact one_smul _ n
-
- theorem bottHeadReflectedTail_seed (i : I) (l : List I) (n : BottSamelsonModule cs l) :
-    bottHeadReflectedTail cs i l ((1 : SymmetricCoefficient (M := M)) ⊗ₜ[simpleFixed cs i] n)=Twist.mk n := by
-  change Twist.mk ((coefficientAction cs (cs.simple i)) 1 • n)=Twist.mk n
-  rw [map_one,one_smul]
-
- def bottHeadLocalPair (i : I) (l : List I) (x : W) : BottSamelsonModule cs (i::l) →ₗ[
-    SymmetricCoefficient (M := M)]
-    (bottStalk cs l x × Twist (coefficientAction cs (cs.simple i)).toRingEquiv
-      (bottStalk cs l (cs.simple i*x))) :=
-  ((bottStalkProjection cs l x).comp (bottHeadTail cs i l)).prod
-    ((twistMap (coefficientAction cs (cs.simple i)).toRingEquiv
-      (bottStalkProjection cs l (cs.simple i*x))).comp (bottHeadReflectedTail cs i l))
-
- theorem bottHeadLocalPair_seed (i : I) (l : List I) (x : W) (n : BottSamelsonModule cs l) :
-    bottHeadLocalPair cs i l x ((1 : SymmetricCoefficient (M := M)) ⊗ₜ[simpleFixed cs i] n)=
-      (bottStalkProjection cs l x n,Twist.mk (bottStalkProjection cs l (cs.simple i*x) n)) := by
-  change ((bottStalkProjection cs l x) (bottHeadTail cs i l _),
-    (twistMap (coefficientAction cs (cs.simple i)).toRingEquiv
-      (bottStalkProjection cs l (cs.simple i*x))) (bottHeadReflectedTail cs i l _)) = _
-  rw [bottHeadTail_seed,bottHeadReflectedTail_seed]
-  rfl
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-namespace KLInvariance
-universe ua up uv un
-variable {A : Type ua} [CommRing A]
-  {P : Type up} [AddCommGroup P] [Module A P]
-  {V : Type uv} [AddCommGroup V] [Module A V]
-  {N : Type un} [AddCommGroup N] [Module A N]
- theorem linearMap_mem_of_coordinates (S : Submodule A P)
-    (F : P →ₗ[A] V) (G : N →ₗ[A] P) (E : N →ₗ[A] V)
-    (hF : Function.Injective F) (heq : F.comp G = E)
-    (hrange : LinearMap.range E ≤ LinearMap.range (F.comp S.subtype)) (n : N) : G n ∈ S := by
-  obtain ⟨p,hp⟩ := hrange (show E n ∈ LinearMap.range E from ⟨n,rfl⟩)
-  have he : (p : P) = G n := hF (hp.trans (LinearMap.congr_fun heq n).symm)
-  exact he ▸ p.property
-end KLInvariance
-
-end
-
-
-section
-
-namespace KLInvariance.TitsSpace
-open Module TensorProduct FrobeniusDuality ReflectionFrobenius ReflectionPairTranslation
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-attribute [local instance] restrictedModule restrictedTower restrictedFree restrictedFinite
-
- theorem bottHeadLocalPair_coordinates (i : I) (l : List I) (x : W) :
-    (bottHeadPairCoordinates cs i l x).comp (bottHeadLocalPair cs i l x) =
-      bottWeightProjection cs (i::l) x := by
-  apply bottTensor_maps_eq cs i l
-  intro n
-  change bottHeadPairCoordinates cs i l x (bottHeadLocalPair cs i l x _) = _
-  rw [bottHeadLocalPair_seed,bottHeadPairCoordinates_seed]
-
- theorem bottHeadLocalPair_mem (i : I) (l : List I) (x : W)
-    (m : BottSamelsonModule cs (i::l)) :
-    bottHeadLocalPair cs i l x m ∈ bottHeadPairLattice cs i l x := by
-  exact linearMap_mem_of_coordinates (bottHeadPairLattice cs i l x)
-    (bottHeadPairCoordinates cs i l x) (bottHeadLocalPair cs i l x)
-    (bottWeightProjection cs (i::l) x) (bottHeadPairCoordinates_injective cs i l x)
-    (bottHeadLocalPair_coordinates cs i l x) (bottHeadPair_range cs i l x) m
-
- theorem bottHeadLocalPair_zero_iff (i : I) (l : List I) (x : W)
-    (m : BottSamelsonModule cs (i::l)) :
-    bottStalkProjection cs (i::l) x m=0 ↔ bottHeadLocalPair cs i l x m=0 := by
-  constructor
-  · intro h
-    apply bottHeadPairCoordinates_injective cs i l x
-    rw [map_zero]
-    have hh := congrArg Subtype.val h
-    change bottWeightProjection cs (i::l) x m=0 at hh
-    exact (LinearMap.congr_fun (bottHeadLocalPair_coordinates cs i l x) m).trans hh
-  · intro h
-    apply Subtype.ext
-    change bottWeightProjection cs (i::l) x m=0
-    rw [← LinearMap.congr_fun (bottHeadLocalPair_coordinates cs i l x) m]
-    change bottHeadPairCoordinates cs i l x (bottHeadLocalPair cs i l x m)=0
-    rw [h,map_zero]
-
- theorem bottHeadLocalPair_value (i : I) (l : List I) (x : W)
-    (m : BottSamelsonModule cs (i::l)) :
-    bottHeadLocalPair cs i l x m =
-      (bottStalkProjection cs l x (bottHeadTail cs i l m),
-       Twist.mk (bottStalkProjection cs l (cs.simple i*x) (bottHeadReflectedTail cs i l m).val)) := rfl
-
- theorem bottHeadLocalPair_toStalk (i : I) (l : List I) (x : W)
-    (m : BottSamelsonModule cs (i::l)) :
-    bottHeadPairToStalk cs i l x ⟨bottHeadLocalPair cs i l x m,bottHeadLocalPair_mem cs i l x m⟩ =
-      bottStalkProjection cs (i::l) x m := by
-  apply Subtype.ext
-  exact LinearMap.congr_fun (bottHeadLocalPair_coordinates cs i l x) m
-
- theorem bottHeadStalk_divisible_of_pair (i : I) (l : List I) (x : W)
-    (m : BottSamelsonModule cs (i::l)) (β : SymmetricCoefficient (M := M))
-    (p : bottStalk cs l x × Twist (coefficientAction cs (cs.simple i)).toRingEquiv
-      (bottStalk cs l (cs.simple i*x)))
-    (hm : bottHeadLocalPair cs i l x m=β • p) (hp : p ∈ bottHeadPairLattice cs i l x) :
-    ∃ z : bottStalk cs (i::l) x, bottStalkProjection cs (i::l) x m=β • z := by
-  let z : bottHeadPairLattice cs i l x := ⟨p,hp⟩
-  refine ⟨bottHeadPairToStalk cs i l x z,?_⟩
-  have he : (⟨bottHeadLocalPair cs i l x m,bottHeadLocalPair_mem cs i l x m⟩ :
-      bottHeadPairLattice cs i l x) = β • z := Subtype.ext hm
-  calc
-    bottStalkProjection cs (i::l) x m =
-        bottHeadPairToStalk cs i l x ⟨bottHeadLocalPair cs i l x m,bottHeadLocalPair_mem cs i l x m⟩ :=
-      (bottHeadLocalPair_toStalk cs i l x m).symm
-    _ = bottHeadPairToStalk cs i l x (β • z) := congrArg (bottHeadPairToStalk cs i l x) he
-    _ = β • bottHeadPairToStalk cs i l x z := map_smul _ _ _
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-namespace KLInvariance.FiniteFreeScalar
-universe ur um un ui
-variable {R : Type ur} [CommRing R]
-  {M : Type um} [AddCommGroup M] [Module R M]
-  {ι : Type ui} [Fintype ι] (b : Module.Basis ι R M)
-
-
- theorem exists_smul_iff (r : R) (m : M) :
-    (∃ n, r • n = m) ↔ ∀ i, r ∣ b.equivFun m i := by
-  classical
-  constructor
-  · rintro ⟨n,rfl⟩ i
-    exact ⟨b.equivFun n i, by simp⟩
-  · intro h
-    choose n hn using h
-    refine ⟨b.equivFun.symm n,b.equivFun.injective ?_⟩
-    ext i
-    simp only [map_smul,LinearEquiv.apply_symm_apply,Pi.smul_apply,smul_eq_mul]
-    exact (hn i).symm
-
-include b in
- theorem exists_prod_smul [DecompositionMonoid R] {J : Type*} (s : Finset J) (r : J → R)
-    (hrel : (s : Set J).Pairwise (Function.onFun IsRelPrime r)) (m : M)
-    (h : ∀ j ∈ s, ∃ n, r j • n = m) : ∃ n, (∏ j ∈ s, r j) • n = m := by
-  apply (exists_smul_iff b _ m).mpr
-  intro i
-  apply Finset.prod_dvd_of_isRelPrime hrel
-  intro j hj
-  exact (exists_smul_iff b _ _).mp (h j hj) i
-
-include b in
-
-
- theorem scalar_quotient_regular
-    {N : Type un} [AddCommGroup N] [Module R N]
-    (q : M →ₗ[R] N) (hq : Function.Surjective q)
-    (a r : R) (ha : Prime a) (hr : ¬ a ∣ r)
-    (hker : ∀ m, q m = 0 ↔ ∃ n, m=a • n) :
-    Function.Injective (fun v : N => r • v) := by
-  suffices hh : ∀ v : N, r • v = 0 → v = 0 by
-    intro v w hvw
-    change r • v = r • w at hvw
-    have hv : r • (v-w) = 0 := by rw [smul_sub,hvw,sub_self]
-    exact sub_eq_zero.mp (hh _ hv)
-  intro v hv
-  obtain ⟨m,rfl⟩ := hq v
-  have h0 : q (r • m) = 0 := by simpa only [map_smul] using hv
-  obtain ⟨n,hn⟩ := (hker _).mp h0
-  have hdiv : ∀ i, a ∣ b.equivFun m i := by
-    intro i
-    have hc := congrArg (fun z => b.equivFun z i) hn
-    simp only [map_smul,Pi.smul_apply,smul_eq_mul] at hc
-    apply (ha.dvd_mul.mp ?_).resolve_left hr
-    exact ⟨b.equivFun n i,hc⟩
-  obtain ⟨z,hz⟩ := (exists_smul_iff b a m).mpr hdiv
-  exact (hker m).mpr ⟨z,hz.symm⟩
-
-end KLInvariance.FiniteFreeScalar
-
-end
-
-
-section
-
-/-! Intermediate standard-filtration predicates on the genuine tensor word.
-These are proved by word induction; they are not main-theorem assumptions. -/
-namespace KLInvariance.TitsSpace
-open Module ReflectionFrobenius
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
- def BottKernelExact (l : List I) : Prop :=
-  ∀ (x t : W) (a : I → ℝ), Represents M cs t a →
-    cs.length x < cs.length (t*x) →
-    ∀ m : BottSamelsonModule cs l, bottStalkProjection cs l x m=0 →
-      ∃ z : bottStalk cs l (t*x), bottStalkProjection cs l (t*x) m=
-        SymmetricAlgebra.ι ℝ (Extended M) (embed M a) • z
-
- theorem simple_represents (i : I) : Represents M cs (cs.simple i) (root i) := by
-  refine ⟨1,i,?_,?_⟩
-  · simp
-  · simp
-
- theorem represents_polynomial_prime {t : W} {a : I → ℝ} (ha : Represents M cs t a) :
-    Prime (SymmetricAlgebra.ι ℝ (Extended M) (embed M a)) := by
-  apply EdgeAlgebra.symmetric_ι_prime
-  intro h
-  exact ha.isRoot.ne_zero (congrArg Prod.fst h)
-
- theorem represents_polynomial_not_dvd {t s : W} {a b : I → ℝ}
-    (ha : Represents M cs t a) (hb : Represents M cs s b) (hne : t≠s) :
-    ¬ SymmetricAlgebra.ι ℝ (Extended M) (embed M a) ∣
-      SymmetricAlgebra.ι ℝ (Extended M) (embed M b) := by
-  rw [EdgeAlgebra.symmetric_ι_dvd_iff]
-  intro h
-  have hi := hb.extended_independent_of_ne ha (Ne.symm hne)
-  obtain ⟨r,hr⟩ := Submodule.mem_span_singleton.mp h
-  exact (linearIndependent_fin2.mp hi).2 r hr
-
- theorem bottStalk_smul_injective (l : List I) (x : W)
-    {r : SymmetricCoefficient (M := M)} (hr : r≠0) :
-    Function.Injective (fun z : bottStalk cs l x => r • z) :=
-  smul_right_injective (bottStalk cs l x) hr
-
- instance bottStalkFinite (l : List I) (x : W) :
-    Module.Finite (SymmetricCoefficient (M := M)) (bottStalk cs l x) :=
-  Module.Finite.of_surjective (bottStalkProjection cs l x) (bottStalkProjection_surjective cs l x)
-
- theorem bottReflectionEdge_annihilated (l : List I) (x t : W) {a : I → ℝ}
-    (ha : Represents M cs t a)
-    (z : PairProjection.Edge (bottStalkProjection cs l x) (bottStalkProjection cs l (t*x))) :
-    SymmetricAlgebra.ι ℝ (Extended M) (embed M a) • z=0 := by
-  obtain ⟨p,hp⟩ := reflectionCoefficient_difference cs ha x
-  rw [← hp]
-  exact PairProjection.difference_annihilates _ _
-    (bottStalkProjection_surjective cs _ _) (bottStalkProjection_surjective cs _ _)
-    (bottSamelsonRight cs l p) _ _
-    (bottStalkProjection_right cs _ _ p) (bottStalkProjection_right cs _ _ p) z
-
- theorem BottKernelExact.upper_kernel {l : List I} (H : BottKernelExact cs l)
-    (x t : W) {a : I → ℝ} (ha : Represents M cs t a)
-    (hlt : cs.length x < cs.length (t*x)) (z : bottStalk cs l (t*x)) :
-    PairProjection.upper (bottStalkProjection cs l x) (bottStalkProjection cs l (t*x)) z=0 ↔
-      ∃ w, z=SymmetricAlgebra.ι ℝ (Extended M) (embed M a) • w :=
-  PairProjection.upper_kernel_of_divisibility _ _ _ (H x t a ha hlt)
-    (bottReflectionEdge_annihilated cs l x t ha) z
-
- theorem BottKernelExact.kernel_iff {l : List I} (H : BottKernelExact cs l)
-    (x t : W) {a : I → ℝ} (ha : Represents M cs t a)
-    (hlt : cs.length x < cs.length (t*x)) (z : bottStalk cs l (t*x)) :
-    (∃ m, bottStalkProjection cs l x m=0 ∧ bottStalkProjection cs l (t*x) m=z) ↔
-      ∃ w, z=SymmetricAlgebra.ι ℝ (Extended M) (embed M a) • w := by
-  rw [← PairProjection.kernel_upper]
-  exact H.upper_kernel cs x t ha hlt z
-
- theorem BottKernelExact.kernel_iff_of_mul {l : List I} (H : BottKernelExact cs l)
-    (x y t : W) {a : I → ℝ} (ha : Represents M cs t a) (hxy : t*x=y)
-    (hlt : cs.length x < cs.length y) (z : bottStalk cs l y) :
-    (∃ m, bottStalkProjection cs l x m=0 ∧ bottStalkProjection cs l y m=z) ↔
-      ∃ w, z=SymmetricAlgebra.ι ℝ (Extended M) (embed M a) • w := by
-  subst y
-  exact H.kernel_iff cs x t ha hlt z
-
- theorem BottKernelExact.projection_of_mul {l : List I} (H : BottKernelExact cs l)
-    (x y t : W) {a : I → ℝ} (ha : Represents M cs t a) (hxy : t*x=y)
-    (hlt : cs.length x < cs.length y) (m : BottSamelsonModule cs l)
-    (hm : bottStalkProjection cs l x m=0) :
-    ∃ z : bottStalk cs l y, bottStalkProjection cs l y m=
-      SymmetricAlgebra.ι ℝ (Extended M) (embed M a) • z := by
-  subst y
-  exact H x t a ha hlt m hm
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-namespace KLInvariance.TitsSpace
-open Module ReflectionFrobenius ReflectionPairTranslation
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
- theorem bottSimpleEdge_regular (i : I) (l : List I)
-    (H : BottKernelExact cs l)
-    (F : ∀ x : W, Module.Free (SymmetricCoefficient (M := M)) (bottStalk cs l x))
-    (x : W) (β : SymmetricCoefficient (M := M)) (hβ : ¬simpleRootPolynomial (M := M) i ∣ β) :
-    Function.Injective (fun e : PairProjection.Edge (bottStalkProjection cs l x)
-      (bottStalkProjection cs l (cs.simple i*x)) => β • e) := by
-  have hα : Prime (simpleRootPolynomial (M := M) i) :=
-    represents_polynomial_prime cs (simple_represents cs i)
-  rcases lt_or_gt_of_ne (cs.length_simple_mul_ne x i) with hdesc | hasc
-  · let := F x
-    apply FiniteFreeScalar.scalar_quotient_regular
-      (Module.Free.chooseBasis (SymmetricCoefficient (M := M)) (bottStalk cs l x))
-      (PairProjection.lower (bottStalkProjection cs l x) (bottStalkProjection cs l (cs.simple i*x)))
-      (PairProjection.lower_surjective _ _ (bottStalkProjection_surjective cs l (cs.simple i*x)))
-      (simpleRootPolynomial (M := M) i) β hα hβ
-    intro z
-    rw [PairProjection.kernel_lower]
-    have hh := H.kernel_iff_of_mul cs (cs.simple i*x) x (cs.simple i)
-      (simple_represents cs i) (cs.simple_mul_simple_cancel_left i) hdesc z
-    simpa only [and_comm,simpleRootPolynomial,extendedRoot] using hh
-  · let := F (cs.simple i*x)
-    exact FiniteFreeScalar.scalar_quotient_regular
-      (Module.Free.chooseBasis (SymmetricCoefficient (M := M)) (bottStalk cs l (cs.simple i*x)))
-      (PairProjection.upper (bottStalkProjection cs l x) (bottStalkProjection cs l (cs.simple i*x)))
-      (PairProjection.upper_surjective _ _ (bottStalkProjection_surjective cs l x))
-      (simpleRootPolynomial (M := M) i) β hα hβ
-      (H.upper_kernel cs x (cs.simple i) (simple_represents cs i) hasc)
-
- theorem bottHeadPairLattice_saturated (i : I) (l : List I)
-    (H : BottKernelExact cs l)
-    (F : ∀ x : W, Module.Free (SymmetricCoefficient (M := M)) (bottStalk cs l x))
-    (x : W) (β : SymmetricCoefficient (M := M)) (hβ : ¬simpleRootPolynomial (M := M) i ∣ β)
-    (p : bottStalk cs l x × Twist (coefficientAction cs (cs.simple i)).toRingEquiv
-      (bottStalk cs l (cs.simple i*x))) :
-    β • p ∈ bottHeadPairLattice cs i l x ↔ p ∈ bottHeadPairLattice cs i l x :=
-  lattice_saturated (bottStalkProjection cs l x) (bottStalkProjection cs l (cs.simple i*x))
-    (coefficientAction cs (cs.simple i)).toRingEquiv (simpleRootPolynomial (M := M) i)
-    (simpleReverseDivisibility cs i) (bottSimpleEdge_kernel_inclusion cs i l x)
-    β (bottSimpleEdge_annihilated cs i l x) (bottSimpleEdge_regular cs i l H F x β hβ) p
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-
-namespace KLInvariance.TitsSpace
-open Module ReflectionFrobenius ReflectionPairTranslation
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
- theorem bottHeadStalk_divisible (i : I) (l : List I)
-    (H : BottKernelExact cs l)
-    (F : ∀ x : W, Module.Free (SymmetricCoefficient (M := M)) (bottStalk cs l x))
-    (x : W) (m : BottSamelsonModule cs (i::l)) (β : SymmetricCoefficient (M := M))
-    (hβ : ¬simpleRootPolynomial (M := M) i ∣ β)
-    (p : bottStalk cs l x × Twist (coefficientAction cs (cs.simple i)).toRingEquiv
-      (bottStalk cs l (cs.simple i*x)))
-    (hm : bottHeadLocalPair cs i l x m=β • p) :
-    ∃ z : bottStalk cs (i::l) x, bottStalkProjection cs (i::l) x m=β • z := by
-  have hp := bottHeadLocalPair_mem cs i l x m
-  have hp' : β • p ∈ bottHeadPairLattice cs i l x := hm ▸ hp
-  exact bottHeadStalk_divisible_of_pair cs i l x m β p hm
-    ((bottHeadPairLattice_saturated cs i l H F x β hβ p).mp hp')
-
- theorem bottKernelExact_cons_nonhead (i : I) (l : List I)
-    (H : BottKernelExact cs l)
-    (F : ∀ x : W, Module.Free (SymmetricCoefficient (M := M)) (bottStalk cs l x))
-    (x t : W) (a : I → ℝ) (ha : Represents M cs t a)
-    (hlt : cs.length x < cs.length (t*x)) (hts : t≠cs.simple i)
-    (m : BottSamelsonModule cs (i::l)) (hm : bottStalkProjection cs (i::l) x m=0) :
-    ∃ z : bottStalk cs (i::l) (t*x), bottStalkProjection cs (i::l) (t*x) m=
-      SymmetricAlgebra.ι ℝ (Extended M) (embed M a) • z := by
-  have he : t*x≠cs.simple i*x := fun h => hts (mul_right_cancel h)
-  have hstep : BruhatStep cs x (t*x) := ⟨hlt,t,ha.isReflection,rfl⟩
-  have hstep' := bruhatStep_simple_mul cs hstep i he
-  have hconj : (cs.simple i*t*(cs.simple i)⁻¹)*(cs.simple i*x)=cs.simple i*(t*x) := by group
-  have hzero := (bottHeadLocalPair_zero_iff cs i l x m).mp hm
-  have hμ : bottStalkProjection cs l x (bottHeadTail cs i l m)=0 := congrArg Prod.fst hzero
-  have hν : bottStalkProjection cs l (cs.simple i*x) (bottHeadReflectedTail cs i l m).val=0 :=
-    congrArg (fun p => p.2.val) hzero
-  obtain ⟨p,hp⟩ := H x t a ha hlt (bottHeadTail cs i l m) hμ
-  obtain ⟨q,hq⟩ := H.projection_of_mul cs (cs.simple i*x) (cs.simple i*(t*x))
-    (cs.simple i*t*(cs.simple i)⁻¹) (ha.act (cs.simple i)) hconj hstep'.1
-    (bottHeadReflectedTail cs i l m).val hν
-  let β : SymmetricCoefficient (M := M) := SymmetricAlgebra.ι ℝ (Extended M) (embed M a)
-  let p' : bottStalk cs l (t*x) × Twist (coefficientAction cs (cs.simple i)).toRingEquiv
-      (bottStalk cs l (cs.simple i*(t*x))) := (p,⟨q⟩)
-  have hβ' : coefficientAction cs (cs.simple i) β =
-      SymmetricAlgebra.ι ℝ (Extended M) (embed M (action M cs (cs.simple i) a)) := by
-    rw [coefficientAction_ι,extendedAction_restrict]
-  have heq : bottHeadLocalPair cs i l (t*x) m=β • p' := by
-    apply Prod.ext
-    · exact hp
-    · apply Twist.ext
-      change bottStalkProjection cs l (cs.simple i*(t*x))
-        (bottHeadReflectedTail cs i l m).val = coefficientAction cs (cs.simple i) β • q
-      rw [hβ']
-      exact hq
-  have hn : ¬simpleRootPolynomial (M := M) i ∣ β :=
-    represents_polynomial_not_dvd cs (simple_represents cs i) ha (Ne.symm hts)
-  exact bottHeadStalk_divisible cs i l H F (t*x) m β hn p' heq
-
- theorem bottKernelExact_cons (i : I) (l : List I)
-    (H : BottKernelExact cs l)
-    (F : ∀ x : W, Module.Free (SymmetricCoefficient (M := M)) (bottStalk cs l x)) :
-    BottKernelExact cs (i::l) := by
-  intro x t a ha hlt m hm
-  by_cases hts : t=cs.simple i
-  · subst t
-    obtain ⟨z,hz⟩ := bottSimpleWall_kernel_projection cs i l x hm
-    rcases ha.unique_up_to_sign (simple_represents cs i) with h|h
-    · subst a
-      exact ⟨z,hz⟩
-    · subst a
-      refine ⟨-z,?_⟩
-      rw [hz]
-      change simpleRootPolynomial (M := M) i • z =
-        SymmetricAlgebra.ι ℝ (Extended M) (embed M (-root i)) • (-z)
-      have he : embed M (-root i) = -embed M (root i) := by simp [embed]
-      have hn : SymmetricAlgebra.ι ℝ (Extended M) (embed M (-root i)) =
-          -simpleRootPolynomial (M := M) i := by
-        rw [he]
-        exact (SymmetricAlgebra.ι ℝ (Extended M)).map_neg _
-      rw [hn]
-      exact (neg_smul_neg _ z).symm
-  · exact bottKernelExact_cons_nonhead cs i l H F x t a ha hlt hts m hm
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-namespace KLInvariance.ReflectionPairTranslation
-universe u v w z
-variable {R : Type u} [CommRing R]
-  {N : Type v} [AddCommGroup N] [Module R N]
-  {X : Type w} [AddCommGroup X] [Module R X]
-  {Y : Type z} [AddCommGroup Y] [Module R Y]
-  (f : N →ₗ[R] X) (g : N →ₗ[R] Y) (τ : R ≃+* R) (α : R)
-  (hdiv : ∀ r : R, α ∣ τ r-r)
-  (hinc : ∀ y : Y, ∃ n : N, f n=0 ∧ g n=α • y)
-  (hker : ∀ x : X, (∃ n : N, f n=x ∧ g n=0) ↔ ∃ z : X, x=α • z)
-
- def right : lattice f g τ α hdiv hinc →ₗ[R] Twist τ Y :=
-  (LinearMap.snd R X (Twist τ Y)).comp (lattice f g τ α hdiv hinc).subtype
-
- theorem right_surjective (hg : Function.Surjective g) :
-    Function.Surjective (right f g τ α hdiv hinc) := by
-  intro y
-  obtain ⟨n,hn⟩ := hg y.val
-  exact ⟨⟨(f n,y),n,rfl,hn⟩,rfl⟩
-
- def rightKernelMap : X →ₗ[R] LinearMap.ker (right f g τ α hdiv hinc) where
-  toFun x := ⟨⟨(α • x,⟨0⟩),(hker (α • x)).mpr ⟨x,rfl⟩⟩,rfl⟩
-  map_add' x y := by
-    apply Subtype.ext
-    apply Subtype.ext
-    apply Prod.ext
-    · exact smul_add α x y
-    · apply Twist.ext
-      exact (zero_add (0 : Y)).symm
-  map_smul' r x := by
-    apply Subtype.ext
-    apply Subtype.ext
-    apply Prod.ext
-    · exact smul_comm α r x
-    · apply Twist.ext
-      exact (smul_zero _).symm
-
- theorem rightKernelMap_surjective :
-    Function.Surjective (rightKernelMap f g τ α hdiv hinc hker) := by
-  rintro ⟨⟨⟨x,y⟩,n,hn,hn'⟩,hy⟩
-  have hy' : y.val=0 := congrArg Twist.val hy
-  obtain ⟨z,hz⟩ := (hker x).mp ⟨n,hn,hn'.trans hy'⟩
-  refine ⟨z,?_⟩
-  apply Subtype.ext
-  apply Subtype.ext
-  exact Prod.ext hz.symm (Twist.ext hy'.symm)
-
- theorem rightKernelMap_injective (hinj : Function.Injective (fun x : X => α • x)) :
-    Function.Injective (rightKernelMap f g τ α hdiv hinc hker) := by
-  intro x y h
-  exact hinj (congrArg (fun p => p.val.val.1) h)
-
- noncomputable def rightKernelEquiv (hinj : Function.Injective (fun x : X => α • x)) :
-    X ≃ₗ[R] LinearMap.ker (right f g τ α hdiv hinc) :=
-  LinearEquiv.ofBijective (rightKernelMap f g τ α hdiv hinc hker)
-    ⟨rightKernelMap_injective f g τ α hdiv hinc hker hinj,
-      rightKernelMap_surjective f g τ α hdiv hinc hker⟩
-
- include hker in
- theorem lattice_free_right (hg : Function.Surjective g)
-    (hinj : Function.Injective (fun x : X => α • x))
-    [Module.Free R X] [Module.Free R Y] :
-    Module.Free R (lattice f g τ α hdiv hinc) := by
-  let : Module.Free R (LinearMap.ker (right f g τ α hdiv hinc)) :=
-    Module.Free.of_equiv (rightKernelEquiv f g τ α hdiv hinc hker hinj)
-  exact free_of_surjective_of_ker_free (right f g τ α hdiv hinc)
-    (right_surjective f g τ α hdiv hinc hg)
-end KLInvariance.ReflectionPairTranslation
-
-end
-
-
-section
-
-namespace KLInvariance.TitsSpace
-open Module ReflectionFrobenius ReflectionPairTranslation
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
- theorem bottStalk_free_cons (i : I) (l : List I)
-    (H : BottKernelExact cs l)
-    (F : ∀ x : W, Module.Free (SymmetricCoefficient (M := M)) (bottStalk cs l x))
-    (x : W) : Module.Free (SymmetricCoefficient (M := M)) (bottStalk cs (i::l) x) := by
-  let := F x
-  let := F (cs.simple i*x)
-  have hα : simpleRootPolynomial (M := M) i ≠ 0 :=
-    (represents_polynomial_prime cs (simple_represents cs i)).ne_zero
-  have hlattice : Module.Free (SymmetricCoefficient (M := M)) (bottHeadPairLattice cs i l x) := by
-    rcases lt_or_gt_of_ne (cs.length_simple_mul_ne x i) with hdesc | hasc
-    · apply lattice_free_right
-        (bottStalkProjection cs l x) (bottStalkProjection cs l (cs.simple i*x))
-        (coefficientAction cs (cs.simple i)).toRingEquiv (simpleRootPolynomial (M := M) i)
-        (simpleReverseDivisibility cs i) (bottSimpleEdge_kernel_inclusion cs i l x)
-      · intro z
-        have hh := H.kernel_iff_of_mul cs (cs.simple i*x) x (cs.simple i)
-          (simple_represents cs i) (cs.simple_mul_simple_cancel_left i) hdesc z
-        simpa only [and_comm,simpleRootPolynomial,extendedRoot] using hh
-      · exact bottStalkProjection_surjective cs l (cs.simple i*x)
-      · exact bottStalk_smul_injective cs l x hα
-    · exact lattice_free (bottStalkProjection cs l x) (bottStalkProjection cs l (cs.simple i*x))
-        (coefficientAction cs (cs.simple i)).toRingEquiv (simpleRootPolynomial (M := M) i)
-        (simpleReverseDivisibility cs i) (bottSimpleEdge_kernel_inclusion cs i l x)
-        (H.kernel_iff cs x (cs.simple i) (simple_represents cs i) hasc)
-        (bottStalkProjection_surjective cs l x)
-        (bottStalk_smul_injective cs l (cs.simple i*x) hα)
-  let := hlattice
-  exact Module.Free.of_equiv (bottHeadPairEquiv cs i l x)
-
- theorem bottStalk_nil_subsingleton (x : W) (hx : x≠1) : Subsingleton (bottStalk cs [] x) := by
-  constructor
-  intro p q
-  apply Subtype.ext
-  funext e
-  exact False.elim (hx e.property.symm)
-
- theorem bottStalk_nil_free (x : W) :
-    Module.Free (SymmetricCoefficient (M := M)) (bottStalk cs [] x) := by
-  by_cases hx : x=1
-  · subst x
-    exact Module.Free.of_equiv (bottTopStalkEquiv cs [] (by simp [CoxeterSystem.IsReduced])).symm
-  · let := bottStalk_nil_subsingleton cs x hx
-    infer_instance
-
- theorem bottKernelExact_nil : BottKernelExact cs [] := by
-  intro x t a ha hlt m hm
-  have hx : t*x≠1 := by
-    intro h
-    rw [h] at hlt
-    rw [cs.length_one] at hlt
-    exact Nat.not_lt_zero _ hlt
-  let := bottStalk_nil_subsingleton cs (t*x) hx
-  exact ⟨0,Subsingleton.elim _ _⟩
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Pair-projection edges of actual finite graded modules. The hypotheses say
-only that the two actual weight projections are homogeneous and surjective. -/
-namespace KLInvariance.Graded.ModuleData
-open PairProjection
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} {M X Y : ModuleData.{uk,ua,um} 𝓐}
-  (f : Hom M X) (g : Hom M Y)
-noncomputable section
-
- theorem pairLower_kernel_homogeneous (n : ℤ) (z : X)
-    (hz : lower f.map g.map z=0) :
-    lower f.map g.map (component X.piece n z)=0 := by
-  obtain ⟨m,hm,hy⟩ := (kernel_lower f.map g.map z).mp hz
-  apply (kernel_lower f.map g.map _).mpr
-  refine ⟨component M.piece n m,?_,?_⟩
-  · rw [← f.component,hm]
-  · rw [← g.component,hy,map_zero]
-
- def pairEdgePiece (n : ℤ) : Submodule k (Edge f.map g.map) :=
-  (X.piece n).map ((lower f.map g.map).restrictScalars k)
-
- def pairEdge (hg : Function.Surjective g.map) : ModuleData.{uk,ua,um} 𝓐 where
-  obj := ModuleCat.of A (Edge f.map g.map)
-  kModule := inferInstance
-  tower := inferInstance
-  piece := pairEdgePiece f g
-  decomposition := (image_isInternal X.piece ((lower f.map g.map).restrictScalars k)
-    (lower_surjective f.map g.map hg) (pairLower_kernel_homogeneous f g)).chooseDecomposition
-  graded := ⟨by
-    rintro i j a z ha ⟨w,hw,rfl⟩
-    exact ⟨a • w,SetLike.GradedSMul.smul_mem ha hw,map_smul (lower f.map g.map) a w⟩⟩
-  nonneg := by
-    intro n hn
-    unfold pairEdgePiece
-    rw [X.nonneg n hn,Submodule.map_bot]
-  finite := Module.Finite.of_surjective (lower f.map g.map) (lower_surjective f.map g.map hg)
-
- def pairLower (hg : Function.Surjective g.map) : Hom X (pairEdge f g hg) where
-  map := lower f.map g.map
-  graded := fun _ z hz => ⟨z,hz,rfl⟩
-
- theorem Hom.homogeneous_lift (hg : Function.Surjective g.map) (n : ℤ)
-    (y : Y) (hy : y ∈ Y.piece n) : ∃ m : M, m ∈ M.piece n ∧ g.map m=y := by
-  obtain ⟨m,hm⟩ := hg y
-  refine ⟨Graded.component M.piece n m,(DirectSum.decompose M.piece m n).property,?_⟩
-  rw [← g.component,hm]
-  exact DirectSum.decompose_of_mem_same Y.piece hy
-
- def pairUpper (hg : Function.Surjective g.map) : Hom Y (pairEdge f g hg) where
-  map := upper f.map g.map
-  graded := by
-    intro n y hy
-    obtain ⟨m,hm,rfl⟩ := g.homogeneous_lift hg n y hy
-    exact ⟨f.map m,f.graded n m hm,compatible f.map g.map m⟩
-
-end
-end KLInvariance.Graded.ModuleData
-
-end
-
-
-section
-
-/-! The genuine projection WORD sheaf with its nonnegative grading and finite
-integral stalk/edge modules. No free-stalk, flabbiness, or KL character claim
-is smuggled into the grading. The universe here contains both the finite
-Coxeter matrix and its standard Coxeter group, as needed by the main reduction. -/
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
-universe u us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
-omit [Fintype σ] in
- theorem bottStalkPiece_negative (l : List I) (x : W) (n : ℤ) (hn : n<0) :
-    bottStalkPiece cs basis l x n=⊥ := by
-  apply subPiece_eq_bot
-  exact piPiece_eq_bot (fun _ => symmetricGrading basis) n
-    (fun _ => symmetricGrading_negative basis n hn)
-
- def bottStalkData (l : List I) (x : W) : ModuleData.{0,u,u} (symmetricGrading basis) where
-  obj := ModuleCat.of _ (bottStalk cs l x)
-  kModule := inferInstance
-  tower := inferInstance
-  piece := bottStalkPiece cs basis l x
-  decomposition := inferInstance
-  graded := inferInstance
-  nonneg := bottStalkPiece_negative cs basis l x
-  finite := Module.Finite.of_surjective (bottStalkProjection cs l x)
-    (bottStalkProjection_surjective cs l x)
-
- theorem bottPiece_negative (l : List I) (n : ℤ) (hn : n<0) :
-    bottPiece cs basis l n=⊥ := by
-  apply bot_unique
-  intro m hm
-  change m=0
-  apply bottEvaluation_injective cs l
-  funext e
-  have h := bottEvaluation_homogeneous cs basis l e n m hm
-  rw [symmetricGrading_negative basis n hn,Submodule.mem_bot] at h
-  exact h.trans (map_zero (bottEvaluation cs l e)).symm
-
- def bottWordData (l : List I) : ModuleData.{0,u,u} (symmetricGrading basis) where
-  obj := ModuleCat.of _ (BottSamelsonModule cs l)
-  kModule := inferInstance
-  tower := inferInstance
-  piece := bottPiece cs basis l
-  decomposition := inferInstance
-  graded := inferInstance
-  nonneg := bottPiece_negative cs basis l
-  finite := inferInstance
-
- def bottStalkHom (l : List I) (x : W) :
-    ModuleData.Hom (bottWordData cs basis l) (bottStalkData cs basis l x) where
-  map := bottStalkProjection cs l x
-  graded := bottStalkProjection_homogeneous cs basis l x
-
- theorem bottStalkHom_surjective (l : List I) (x : W) :
-    Function.Surjective (bottStalkHom cs basis l x).map :=
-  bottStalkProjection_surjective cs l x
-
- def bottEdgeData (l : List I) (x y : W) : ModuleData.{0,u,u} (symmetricGrading basis) :=
-  ModuleData.pairEdge (bottStalkHom cs basis l x) (bottStalkHom cs basis l y)
-    (bottStalkHom_surjective cs basis l y)
-
- def bottLowerHom (l : List I) (x y : W) :
-    ModuleData.Hom (bottStalkData cs basis l x) (bottEdgeData cs basis l x y) :=
-  ModuleData.pairLower (bottStalkHom cs basis l x) (bottStalkHom cs basis l y)
-    (bottStalkHom_surjective cs basis l y)
-
- def bottUpperHom (l : List I) (x y : W) :
-    ModuleData.Hom (bottStalkData cs basis l y) (bottEdgeData cs basis l x y) :=
-  ModuleData.pairUpper (bottStalkHom cs basis l x) (bottStalkHom cs basis l y)
-    (bottStalkHom_surjective cs basis l y)
-
- def bottGradedProjectionSheaf (l : List I) (u b : W) :
-    GradedSheaf (𝓐 := symmetricGrading basis) (graph cs u b) where
-  vertex x := bottStalkData cs basis l x.val
-  edge e := bottEdgeData cs basis l (source cs u b e).val (target cs u b e).val
-  lower e := bottLowerHom cs basis l (source cs u b e).val (target cs u b e).val
-  upper e := bottUpperHom cs basis l (source cs u b e).val (target cs u b e).val
-
- theorem bottGradedProjectionSheaf_forget (l : List I) (u b : W) :
-    (bottGradedProjectionSheaf cs basis l u b).forget=bottProjectionSheaf cs l u b := rfl
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Unconditional integral free stalks and exact oriented edge kernels for
-all tensor words. These are the local parts of BMP recognition; flabbiness
-and the KL character are not asserted here. -/
-namespace KLInvariance.TitsSpace
-open Module MomentGraph BruhatGraph
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
- theorem bottStandardStalks (l : List I) :
-    BottKernelExact cs l ∧ ∀ x : W,
-      Module.Free (SymmetricCoefficient (M := M)) (bottStalk cs l x) := by
-  induction l with
-  | nil => exact ⟨bottKernelExact_nil cs,bottStalk_nil_free cs⟩
-  | cons i l h => exact ⟨bottKernelExact_cons cs i l h.1 h.2,
-      bottStalk_free_cons cs i l h.1 h.2⟩
-
- theorem bottKernelExact (l : List I) : BottKernelExact cs l := (bottStandardStalks cs l).1
-
- instance bottStalkFree (l : List I) (x : W) :
-    Module.Free (SymmetricCoefficient (M := M)) (bottStalk cs l x) :=
-  (bottStandardStalks cs l).2 x
-
- theorem bottProjectionSheaf_upper_kernel (l : List I) (u b : W) (e : Edge cs u b)
-    {a : I → ℝ} (ha : Represents M cs (label cs u b e) a)
-    (z : (bottProjectionSheaf cs l u b).vertex (target cs u b e)) :
-    (bottProjectionSheaf cs l u b).upper e z=0 ↔
-      ∃ w, z=SymmetricAlgebra.ι ℝ (Extended M) (embed M a) • w := by
-  change PairProjection.upper (bottStalkProjection cs l (source cs u b e).val)
-      (bottStalkProjection cs l (target cs u b e).val) z=0 ↔ _
-  exact (PairProjection.kernel_upper _ _ z).trans
-    ((bottKernelExact cs l).kernel_iff_of_mul cs _ _ _ ha
-      (label_mul_source cs u b e) e.property.1 z)
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Transport of actual linear-map kernels along a source isomorphism. -/
-namespace KLInvariance
-universe ur um un up
-variable {R : Type ur} [Ring R]
-  {M : Type um} [AddCommGroup M] [Module R M]
-  {N : Type un} [AddCommGroup N] [Module R N]
-  {P : Type up} [AddCommGroup P] [Module R P]
-
-def kernelSourceEquiv (f : M →ₗ[R] P) (e : M ≃ₗ[R] N) :
-    LinearMap.ker f ≃ₗ[R] LinearMap.ker (f.comp e.symm.toLinearMap) where
-  toFun x := ⟨e x.val,by simpa only [LinearMap.mem_ker,LinearMap.comp_apply,
-    LinearEquiv.coe_coe,LinearEquiv.symm_apply_apply] using x.property⟩
-  invFun y := ⟨e.symm y.val,y.property⟩
-  left_inv x := Subtype.ext (e.symm_apply_apply x.val)
-  right_inv y := Subtype.ext (e.apply_symm_apply y.val)
-  map_add' x y := Subtype.ext (map_add e x.val y.val)
-  map_smul' r x := Subtype.ext (map_smul e r x.val)
-
-end KLInvariance
-
-end
-
-
-section
-
-/-! The actual ascent short exact sequence of word stalks has kernel the
-upper tail stalk shifted by one. This is the degree-one Hecke normalization. -/
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded ReflectionPairTranslation
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- def bottHeadKernelEquiv (i : I) (l : List I) (x : W)
-    (h : cs.length x<cs.length (cs.simple i*x)) :
-    bottStalk cs l (cs.simple i*x) ≃ₗ[ℝ] LinearMap.ker (bottHeadLeft cs i l x) :=
-  (Twist.scalarEquiv (coefficientAction cs (cs.simple i)) _).symm.trans
-    (((kernelEquiv (bottStalkProjection cs l x) (bottStalkProjection cs l (cs.simple i*x))
-      (coefficientAction cs (cs.simple i)).toRingEquiv (simpleRootPolynomial (M := M) i)
-      (simpleReverseDivisibility cs i) (bottSimpleEdge_kernel_inclusion cs i l x)
-      ((bottKernelExact cs l).kernel_iff cs x (cs.simple i) (simple_represents cs i) h)
-      (bottStalk_smul_injective cs l (cs.simple i*x) (simpleRootPolynomial_ne_zero i))).trans
-      (kernelSourceEquiv _ (bottHeadPairEquiv cs i l x))).restrictScalars ℝ)
-
- theorem bottHeadKernelEquiv_eval (i : I) (l : List I) (x : W)
-    (h : cs.length x<cs.length (cs.simple i*x))
-    (y : bottStalk cs l (cs.simple i*x))
-    (e : {e : BottIndex (i::l) // bottWeight cs (i::l) e=x}) :
-    (bottHeadKernelEquiv cs i l x h y).val.val e=
-      if he : e.val.1=0 then 0 else coefficientAction cs (cs.simple i)
-        (simpleRootPolynomial (M := M) i * y.val ⟨e.val.2,bottWeight_tail_one cs i l x e he⟩) := rfl
-
- def bottHeadKernelPiece (i : I) (l : List I) (x : W) :
-    ℤ → Submodule ℝ (LinearMap.ker (bottHeadLeft cs i l x)) :=
-  subPiece (bottStalkPiece cs basis (i::l) x) (LinearMap.ker (bottHeadLeft cs i l x))
-
- theorem bottHeadLeft_kernel_homogeneous (i : I) (l : List I) (x : W) :
-    ∀ n m, m ∈ LinearMap.ker (bottHeadLeft cs i l x) →
-      component (bottStalkPiece cs basis (i::l) x) n m ∈ LinearMap.ker (bottHeadLeft cs i l x) := by
-  intro n m hm
-  have h := map_component (bottStalkPiece cs basis (i::l) x) (bottStalkPiece cs basis l x)
-    (bottHeadLeft cs i l x) 0 (by
-      intro j z hz
-      simpa only [add_zero] using (bottHeadLeft_homogeneous cs basis i l x j z hz)) n m
-  change bottHeadLeft cs i l x _=0
-  rw [sub_zero] at h
-  rw [← h,LinearMap.mem_ker.mp hm,map_zero]
-
- instance bottHeadKernelPiece_decomposition (i : I) (l : List I) (x : W) :
-    DirectSum.Decomposition (bottHeadKernelPiece cs basis i l x) :=
-  (sub_isInternal (bottStalkPiece cs basis (i::l) x) (LinearMap.ker (bottHeadLeft cs i l x))
-    (bottHeadLeft_kernel_homogeneous cs basis i l x)).chooseDecomposition
-
- theorem bottHeadKernelEquiv_homogeneous (i : I) (l : List I) (x : W)
-    (h : cs.length x<cs.length (cs.simple i*x)) (n : ℤ)
-    (y : bottStalk cs l (cs.simple i*x)) (hy : y ∈ bottStalkPiece cs basis l (cs.simple i*x) n) :
-    bottHeadKernelEquiv cs i l x h y ∈ bottHeadKernelPiece cs basis i l x (n+1) := by
-  intro e _
-  change (bottHeadKernelEquiv cs i l x h y).val.val e ∈ symmetricGrading basis (n+1)
-  rw [bottHeadKernelEquiv_eval]
-  split_ifs with he
-  · exact Submodule.zero_mem _
-  · apply coefficientAction_homogeneous cs basis
-    have hy' : y.val ⟨e.val.2,bottWeight_tail_one cs i l x e he⟩ ∈ symmetricGrading basis n :=
-      hy ⟨e.val.2,bottWeight_tail_one cs i l x e he⟩ (Set.mem_univ _)
-    have hp := SetLike.GradedMul.mul_mem (simpleRoot_homogeneous basis i) hy'
-    simpa only [add_comm] using hp
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! A homogeneous integral equivalence induces genuine shifted degreewise
-vector-space equivalences, including its inverse grading. -/
-namespace KLInvariance.Graded
-universe uk um un
-variable {k : Type uk} [Field k]
-  {M : Type um} [AddCommGroup M] [Module k M]
-  {N : Type un} [AddCommGroup N] [Module k N]
-  (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
-  [DirectSum.Decomposition 𝓜] [DirectSum.Decomposition 𝓝]
-  (e : M ≃ₗ[k] N) (d : ℤ)
-  (he : ∀ n m, m ∈ 𝓜 n → e m ∈ 𝓝 (n+d))
-noncomputable section
-
- def homogeneousPieceEquiv (n : ℤ) : 𝓜 n ≃ₗ[k] 𝓝 (n+d) where
-  toFun m := ⟨e m.val,he n m.val m.property⟩
-  invFun v := ⟨e.symm v.val,by
-    simpa only [add_sub_cancel_right] using
-      equiv_inverse_homogeneous 𝓜 𝓝 e d he (n+d) v.val v.property⟩
-  left_inv v := Subtype.ext (e.symm_apply_apply v.val)
-  right_inv v := Subtype.ext (e.apply_symm_apply v.val)
-  map_add' v w := Subtype.ext (map_add e v.val w.val)
-  map_smul' r v := Subtype.ext (map_smul e r v.val)
-
-include e he in
- theorem finrank_homogeneousEquiv (n : ℤ) :
-    Module.finrank k (𝓝 n)=Module.finrank k (𝓜 (n-d)) := by
-  have h := (homogeneousPieceEquiv 𝓜 𝓝 e d he (n-d)).finrank_eq
-  have hn : n-d+d=n := by omega
-  rw [hn] at h
-  exact h.symm
-
-end
-end KLInvariance.Graded
-
-end
-
-
-section
-
-/-! Finiteness of true submodule and finite product homogeneous components. -/
-namespace KLInvariance.Graded
-universe uk ua um ui
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {M : Type um} [AddCommGroup M] [Module k M] [Module A M] [IsScalarTower k A M]
-
- instance subPiece_finite (𝓜 : ℤ → Submodule k M) (S : Submodule A M) (n : ℤ)
-    [Module.Finite k (𝓜 n)] : Module.Finite k (subPiece 𝓜 S n) := by
-  let f : subPiece 𝓜 S n →ₗ[k] 𝓜 n :=
-    { toFun := fun v => ⟨v.val.val,v.property⟩
-      map_add' := fun _ _ => rfl
-      map_smul' := fun _ _ => rfl }
-  apply Module.Finite.of_injective f
-  intro v w h
-  apply Subtype.ext
-  apply Subtype.ext
-  exact congrArg (fun z : 𝓜 n => z.val) h
-
- variable {ι : Type ui} [Finite ι] {P : ι → Type um}
-   [∀ i, AddCommGroup (P i)] [∀ i, Module k (P i)]
-
- instance piPiece_finite (𝓟 : ∀ i, ℤ → Submodule k (P i)) (n : ℤ)
-    [∀ i, Module.Finite k (𝓟 i n)] : Module.Finite k (piPiece 𝓟 n) := by
-  let f : piPiece 𝓟 n →ₗ[k] (∀ i, 𝓟 i n) :=
-    { toFun := fun v i => ⟨v.val i,v.property i (Set.mem_univ i)⟩
-      map_add' := fun _ _ => rfl
-      map_smul' := fun _ _ => rfl }
-  apply Module.Finite.of_injective f
-  intro v w h
-  apply Subtype.ext
-  funext i
-  exact congrArg Subtype.val (congrFun h i)
-
-end KLInvariance.Graded
-
-end
-
-
-section
-
-/-! The genuine Bott--Samelson stalk Hilbert coefficients satisfy the simple
-reflection ascent recurrence. Both exactness and the degree-one kernel shift
-were constructed integrally above; no KL character theorem is assumed. -/
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- instance symmetricGrading_finite (n : ℤ) : Module.Finite ℝ (symmetricGrading basis n) :=
-  inferInstanceAs (Module.Finite ℝ ((polynomialGrading ℝ σ n).map
-    (SymmetricAlgebra.equivMvPolynomial basis).symm.toLinearMap))
-
- instance bottCoordinatePiece_finite (l : List I) (x : W) (n : ℤ) :
-    Module.Finite ℝ (bottCoordinatePiece cs basis l x n) := by
-  classical
-  exact piPiece_finite _ n
-
- instance bottStalkPiece_finite (l : List I) (x : W) (n : ℤ) :
-    Module.Finite ℝ (bottStalkPiece cs basis l x n) :=
-  inferInstanceAs (Module.Finite ℝ (subPiece (bottCoordinatePiece cs basis l x) (bottStalk cs l x) n))
-
- theorem bottStalk_finrank_ascent (i : I) (l : List I) (x : W)
-    (h : cs.length x<cs.length (cs.simple i*x)) (n : ℤ) :
-    Module.finrank ℝ (bottStalkPiece cs basis (i::l) x n)=
-      Module.finrank ℝ (bottStalkPiece cs basis l x n)+
-      Module.finrank ℝ (bottStalkPiece cs basis l (cs.simple i*x) (n-1)) := by
-  have hdim := Hilbert.finrank_exact (bottStalkPiece cs basis (i::l) x)
-    (bottStalkPiece cs basis l x) (bottHeadLeft cs i l x)
-    (bottHeadLeft_homogeneous cs basis i l x) (bottHeadLeft_surjective cs i l x) n
-  have hk := finrank_homogeneousEquiv (k := ℝ)
-    (N := LinearMap.ker (bottHeadLeft cs i l x)) (bottStalkPiece cs basis l (cs.simple i*x))
-    (bottHeadKernelPiece cs basis i l x) (bottHeadKernelEquiv cs i l x h) 1
-    (bottHeadKernelEquiv_homogeneous cs basis i l x h) n
-  change Module.finrank ℝ (bottStalkPiece cs basis (i::l) x n)=
-    Module.finrank ℝ (bottHeadKernelPiece cs basis i l x n)+
-      Module.finrank ℝ (bottStalkPiece cs basis l x n) at hdim
-  rw [hk] at hdim
-  omega
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Honest change of the coefficient algebra along an algebra equivalence.
-Both scalar actions and semilinear identifications are explicit. -/
-namespace KLInvariance
-universe uk ua ub um un
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {B : Type ub} [CommRing B] [Algebra k B]
-
-@[ext] structure CoefficientChange (_e : A ≃ₐ[k] B) (M : Type um) where
-  val : M
-
-namespace CoefficientChange
-variable (e : A ≃ₐ[k] B) (M : Type um) [AddCommGroup M] [Module A M] [Module k M]
-  [IsScalarTower k A M]
-
- def equivVal : CoefficientChange e M ≃ M where
-  toFun := val
-  invFun := mk
-  left_inv _ := rfl
-  right_inv _ := rfl
-
- instance : AddCommGroup (CoefficientChange e M) := (equivVal e M).addCommGroup
- instance : Module B (CoefficientChange e M) := by
-   letI : Module B M := Module.compHom M e.symm.toRingHom
-   exact ({ equivVal e M with map_add' := fun _ _ => rfl } : CoefficientChange e M ≃+ M).module B
- instance : Module k (CoefficientChange e M) :=
-   ({ equivVal e M with map_add' := fun _ _ => rfl } : CoefficientChange e M ≃+ M).module k
- instance : IsScalarTower k B (CoefficientChange e M) where
-   smul_assoc c b m := by
-     apply CoefficientChange.ext
-     change e.symm (c • b) • m.val=c • (e.symm b • m.val)
-     rw [map_smul,smul_assoc]
-
- local instance : RingHomInvPair e.symm.toRingHom e.toRingHom := RingHomInvPair.of_ringEquiv_symm e.toRingEquiv
- local instance : RingHomInvPair e.toRingHom e.symm.toRingHom := RingHomInvPair.of_ringEquiv e.toRingEquiv
-
- def equiv : M ≃ₛₗ[e.toRingHom] CoefficientChange e M where
-   toFun := mk
-   invFun := val
-   left_inv _ := rfl
-   right_inv _ := rfl
-   map_add' _ _ := rfl
-   map_smul' a m := by
-     change mk (a • m)=mk (e.symm (e a) • m)
-     rw [e.symm_apply_apply]
-
- def kEquiv : M ≃ₗ[k] CoefficientChange e M where
-   toFun := mk
-   invFun := val
-   left_inv _ := rfl
-   right_inv _ := rfl
-   map_add' _ _ := rfl
-   map_smul' _ _ := rfl
-
- instance [Module.Free A M] : Module.Free B (CoefficientChange e M) :=
-   Module.Free.of_equiv (equiv e M)
- instance [Module.Finite A M] : Module.Finite B (CoefficientChange e M) :=
-   Module.Finite.of_surjective (equiv e M).toLinearMap (equiv e M).surjective
-
- variable {M} {N : Type un} [AddCommGroup N] [Module A N] [Module k N] [IsScalarTower k A N]
- def map (f : M →ₗ[A] N) : CoefficientChange e M →ₗ[B] CoefficientChange e N where
-   toFun m := mk (f m.val)
-   map_add' m n := CoefficientChange.ext (map_add f m.val n.val)
-   map_smul' b m := CoefficientChange.ext (map_smul f (e.symm b) m.val)
-
- omit [Module k M] [IsScalarTower k A M] [Module k N] [IsScalarTower k A N] in
- theorem map_injective (f : M →ₗ[A] N) (hf : Function.Injective f) :
-     Function.Injective (map e f) := fun _m _n h => CoefficientChange.ext (hf (congrArg val h))
- omit [Module k M] [IsScalarTower k A M] [Module k N] [IsScalarTower k A N] in
- theorem map_surjective (f : M →ₗ[A] N) (hf : Function.Surjective f) :
-     Function.Surjective (map e f) := by
-   intro n
-   obtain ⟨m,hm⟩ := hf n.val
-   exact ⟨mk m,CoefficientChange.ext hm⟩
-
-end CoefficientChange
-end KLInvariance
-
-end
-
-
-section
-
-/-! Grading-preserving coefficient change on the actual finite modules. -/
-namespace KLInvariance.Graded.ModuleData
-universe uk ua um
-variable {k : Type uk} [Field k] {A B : Type ua} [CommRing A] [CommRing B]
-  [Algebra k A] [Algebra k B]
-  {𝓐 : ℤ → Submodule k A} {𝓑 : ℤ → Submodule k B}
-  (e : A ≃ₐ[k] B) (he : ∀ n a, a∈𝓐 n ↔ e a∈𝓑 n)
-noncomputable section
-
- def changePiece (M : ModuleData.{uk,ua,um} 𝓐) : ℤ → Submodule k (CoefficientChange e M) :=
-  fun n => (M.piece n).map (CoefficientChange.kEquiv e M).toLinearMap
-
- theorem mem_changePiece (M : ModuleData.{uk,ua,um} 𝓐) (n : ℤ) (m : CoefficientChange e M) :
-    m ∈ changePiece e M n ↔ m.val ∈ M.piece n := by
-  constructor
-  · rintro ⟨x,hx,hxm⟩
-    change CoefficientChange.mk x=m at hxm
-    exact (congrArg CoefficientChange.val hxm) ▸ hx
-  · intro hm
-    exact ⟨m.val,hm,rfl⟩
-
- def change (M : ModuleData.{uk,ua,um} 𝓐) : ModuleData.{uk,ua,um} 𝓑 where
-  obj := ModuleCat.of B (CoefficientChange e M)
-  kModule := inferInstance
-  tower := inferInstance
-  piece := changePiece e M
-  decomposition := by
-    apply DirectSum.IsInternal.chooseDecomposition
-    apply image_isInternal M.piece (CoefficientChange.kEquiv e M).toLinearMap
-      (CoefficientChange.kEquiv e M).surjective
-    intro n m hm
-    have hz : m=0 := congrArg CoefficientChange.val hm
-    subst m
-    simp only [map_zero]
-  graded := by
-    constructor
-    intro i j b m hb hm
-    apply (mem_changePiece e M (i+j) _).mpr
-    change e.symm b • m.val ∈ M.piece (i+j)
-    have ha : e.symm b ∈ 𝓐 i := (he i _).mpr (by simpa using hb)
-    exact SetLike.GradedSMul.smul_mem ha ((mem_changePiece e M j m).mp hm)
-  nonneg n hn := by simp only [changePiece,M.nonneg n hn,Submodule.map_bot]
-  finite := inferInstance
-
- def changeHom {M N : ModuleData.{uk,ua,um} 𝓐} (f : Hom M N) :
-    Hom (change e he M) (change e he N) where
-  map := CoefficientChange.map e f.map
-  graded n m hm := (mem_changePiece e N n _).mpr
-    (f.graded n m.val ((mem_changePiece e M n m).mp hm))
-
- theorem change_free (M : ModuleData.{uk,ua,um} 𝓐) [Module.Free A M] :
-    Module.Free B (change e he M) := inferInstanceAs (Module.Free B (CoefficientChange e M))
-
-end
-end KLInvariance.Graded.ModuleData
-
-end
-
-
-section
-
-/-! The genuine nonnegative polynomial generator character of word stalks,
-with its convergent Hilbert-series interpretation. No KL identification is
-asserted or included in the definition. -/
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded
-universe u
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- def bottPolynomialStalkData (l : List I) (x : W) :
-    ModuleData.{0,u,u} (polynomialGrading ℝ σ) :=
-  ModuleData.change (SymmetricAlgebra.equivMvPolynomial basis) (mem_symmetricGrading basis)
-    (bottStalkData cs basis l x)
-
- instance bottPolynomialStalkFree (l : List I) (x : W) :
-    Module.Free (MvPolynomial σ ℝ) (bottPolynomialStalkData cs basis l x) :=
-  inferInstanceAs (Module.Free (MvPolynomial σ ℝ)
-    (CoefficientChange (SymmetricAlgebra.equivMvPolynomial basis) (bottStalk cs l x)))
-
- def bottStalkCharacter (l : List I) (x : W) : Polynomial ℤ :=
-  Graded.character (σ := σ) (bottPolynomialStalkData cs basis l x).piece
-    (bottPolynomialStalkData cs basis l x).nonneg
-
- theorem bottPolynomialStalk_finrank (l : List I) (x : W) (n : ℤ) :
-    Module.finrank ℝ ((bottPolynomialStalkData cs basis l x).piece n)=
-      Module.finrank ℝ (bottStalkPiece cs basis l x n) :=
-  LinearEquiv.finrank_map_eq
-    (CoefficientChange.kEquiv (SymmetricAlgebra.equivMvPolynomial basis)
-      (bottStalkData cs basis l x)) _
-
- theorem bottStalkCharacter_hasSum (l : List I) (x : W) {q : ℝ}
-    (hq : 0<q) (hq1 : q<1) :
-    HasSum (fun n : ℤ => (Module.finrank ℝ (bottStalkPiece cs basis l x n) : ℝ)*q^n)
-      (Polynomial.eval₂ (Int.castRingHom ℝ) q (bottStalkCharacter cs basis l x) *
-        (1-q)⁻¹ ^ Fintype.card σ) := by
-  have hh := (exists_nonnegativeBasis (k := ℝ) (σ := σ) (𝓜 := (bottPolynomialStalkData cs basis l x).piece)
-    (bottPolynomialStalkData cs basis l x).nonneg).some.hasSum hq hq1
-  apply hh.congr_fun
-  intro n
-  rw [bottPolynomialStalk_finrank]
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-namespace KLInvariance
-universe ur um un up
-variable {R : Type ur} [Ring R]
-  {M : Type um} [AddCommGroup M] [Module R M]
-  {N : Type un} [AddCommGroup N] [Module R N]
-  {P : Type up} [AddCommGroup P] [Module R P]
-
-def kernelTargetEquiv (f : M →ₗ[R] N) (e : N ≃ₗ[R] P) :
-    LinearMap.ker f ≃ₗ[R] LinearMap.ker (e.toLinearMap.comp f) where
-  toFun x := ⟨x.val, by
-    change e (f x.val)=0
-    rw [LinearMap.mem_ker.mp x.property,map_zero]⟩
-  invFun y := ⟨y.val, by
-    apply e.injective
-    simpa only [LinearMap.comp_apply,LinearEquiv.coe_coe,map_zero] using (LinearMap.mem_ker.mp y.property)⟩
-  left_inv _ := rfl
-  right_inv _ := rfl
-  map_add' _ _ := rfl
-  map_smul' _ _ := rfl
-
-end KLInvariance
-
-end
-
-
-section
-
-namespace KLInvariance
-universe uk ua um un up uq
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {M : Type um} [AddCommGroup M] [Module k M] [Module A M] [IsScalarTower k A M]
-  {N : Type un} [AddCommGroup N] [Module k N] [Module A N] [IsScalarTower k A N]
-  {P : Type up} [AddCommGroup P] [Module k P] [Module A P] [IsScalarTower k A P]
-  {Q : Type uq} [AddCommGroup Q] [Module k Q]
-
-def kernelMixedEquiv (f : M →ₗ[A] P) (e : M ≃ₗ[A] N) (g : P ≃ₗ[k] Q) :
-    LinearMap.ker f ≃ₗ[k] LinearMap.ker
-      (g.toLinearMap.comp ((f.comp e.symm.toLinearMap).restrictScalars k)) where
-  toFun x := ⟨e x.val, by
-    change g (f (e.symm (e x.val)))=0
-    rw [e.symm_apply_apply,LinearMap.mem_ker.mp x.property,map_zero]⟩
-  invFun y := ⟨e.symm y.val, by
-    apply g.injective
-    have hy : g (f (e.symm y.val))=0 := y.property
-    simpa only [map_zero] using hy⟩
-  left_inv x := Subtype.ext (e.symm_apply_apply x.val)
-  right_inv y := Subtype.ext (e.apply_symm_apply y.val)
-  map_add' x y := Subtype.ext (map_add e x.val y.val)
-  map_smul' c x := Subtype.ext (map_smul (e.restrictScalars k) c x.val)
-
-end KLInvariance
-
-end
-
-
-section
-
-/-! The descent short exact sequence for the actual word projection lattice.
-Its kernel has the genuine degree-one shift, over the unchanged realization. -/
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded ReflectionPairTranslation
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- theorem bottHeadRight_surjective (i : I) (l : List I) (x : W) :
-    Function.Surjective (bottHeadRight cs i l x) :=
-  (Twist.scalarEquiv (coefficientAction cs (cs.simple i)) _).surjective.comp
-    ((right_surjective _ _ _ _ _ _ (bottStalkProjection_surjective cs l (cs.simple i*x))).comp
-      (bottHeadPairEquiv cs i l x).symm.surjective)
-
- theorem bottDescentKernel (i : I) (l : List I) (x : W)
-    (h : cs.length (cs.simple i*x)<cs.length x) (z : bottStalk cs l x) :
-    (∃ n, bottStalkProjection cs l x n=z ∧ bottStalkProjection cs l (cs.simple i*x) n=0) ↔
-      ∃ w, z=simpleRootPolynomial (M := M) i • w := by
-  have hh := (bottKernelExact cs l).kernel_iff_of_mul cs (cs.simple i*x) x (cs.simple i)
-    (simple_represents cs i) (cs.simple_mul_simple_cancel_left i) h z
-  simpa only [and_comm,simpleRootPolynomial,embed,extendedRoot] using hh
-
- def bottHeadRightKernelEquiv (i : I) (l : List I) (x : W)
-    (h : cs.length (cs.simple i*x)<cs.length x) :
-    bottStalk cs l x ≃ₗ[ℝ] LinearMap.ker (bottHeadRight cs i l x) :=
-  ((rightKernelEquiv (bottStalkProjection cs l x) (bottStalkProjection cs l (cs.simple i*x))
-      (coefficientAction cs (cs.simple i)).toRingEquiv (simpleRootPolynomial (M := M) i)
-      (simpleReverseDivisibility cs i) (bottSimpleEdge_kernel_inclusion cs i l x)
-      (bottDescentKernel cs i l x h)
-      (bottStalk_smul_injective cs l x (simpleRootPolynomial_ne_zero i))).restrictScalars ℝ).trans
-    (kernelMixedEquiv _ (bottHeadPairEquiv cs i l x)
-      (Twist.scalarEquiv (coefficientAction cs (cs.simple i)) _))
-
- theorem bottHeadRightKernelEquiv_eval (i : I) (l : List I) (x : W)
-    (h : cs.length (cs.simple i*x)<cs.length x) (y : bottStalk cs l x)
-    (e : {e : BottIndex (i::l) // bottWeight cs (i::l) e=x}) :
-    (bottHeadRightKernelEquiv cs i l x h y).val.val e=
-      if he : e.val.1=0 then simpleRootPolynomial (M := M) i *
-        y.val ⟨e.val.2,bottWeight_tail_zero cs i l x e he⟩ else 0 := by
-  change (if he : e.val.1=0 then _ else coefficientAction cs (cs.simple i) 0)=_
-  split_ifs
-  · rfl
-  · exact map_zero _
-
- def bottHeadRightKernelPiece (i : I) (l : List I) (x : W) :
-    ℤ → Submodule ℝ (LinearMap.ker (bottHeadRight cs i l x)) :=
-  subPiece (bottStalkPiece cs basis (i::l) x) (LinearMap.ker (bottHeadRight cs i l x))
-
- theorem bottHeadRight_kernel_homogeneous (i : I) (l : List I) (x : W) :
-    ∀ n m, m ∈ LinearMap.ker (bottHeadRight cs i l x) →
-      component (bottStalkPiece cs basis (i::l) x) n m ∈ LinearMap.ker (bottHeadRight cs i l x) := by
-  exact kernel_homogeneous (A := ℝ) (bottStalkPiece cs basis (i::l) x)
-    (bottStalkPiece cs basis l (cs.simple i*x)) (bottHeadRight cs i l x) 0 (by
-      intro j z hz
-      simpa only [add_zero] using (bottHeadRight_homogeneous cs basis i l x j z hz))
-
- instance bottHeadRightKernelPiece_decomposition (i : I) (l : List I) (x : W) :
-    DirectSum.Decomposition (bottHeadRightKernelPiece cs basis i l x) :=
-  (sub_isInternal (bottStalkPiece cs basis (i::l) x) (LinearMap.ker (bottHeadRight cs i l x))
-    (bottHeadRight_kernel_homogeneous cs basis i l x)).chooseDecomposition
-
- theorem bottHeadRightKernelEquiv_homogeneous (i : I) (l : List I) (x : W)
-    (h : cs.length (cs.simple i*x)<cs.length x) (n : ℤ)
-    (y : bottStalk cs l x) (hy : y ∈ bottStalkPiece cs basis l x n) :
-    bottHeadRightKernelEquiv cs i l x h y ∈ bottHeadRightKernelPiece cs basis i l x (n+1) := by
-  intro e _
-  change (bottHeadRightKernelEquiv cs i l x h y).val.val e ∈ symmetricGrading basis (n+1)
-  rw [bottHeadRightKernelEquiv_eval]
-  split_ifs with he
-  · have hy' : y.val ⟨e.val.2,bottWeight_tail_zero cs i l x e he⟩ ∈ symmetricGrading basis n :=
-      hy ⟨e.val.2,bottWeight_tail_zero cs i l x e he⟩ (Set.mem_univ _)
-    have hp := SetLike.GradedMul.mul_mem (simpleRoot_homogeneous basis i) hy'
-    simpa only [add_comm] using hp
-  · exact Submodule.zero_mem _
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Uniqueness and degree-one exact recurrences of actual polynomial Hilbert
-numerators, with no character-theorem assumptions. -/
-
-end
-
-end OAI
+import OAI.RepresentationTheory.KazhdanLusztig.Frobenius
+
+namespace OAI
+
+/-!
+Bott-Samelson tensor ranges, actual sections, grading and right kernels.
+-/
+
+section
+
+namespace KLInvariance
+open TensorProduct
+universe uc ua un uv up
+variable {C : Type uc} [CommRing C] {A : Type ua} [CommRing A] [Algebra C A]
+  {N : Type un} [AddCommGroup N] [Module C N]
+  {V : Type uv} [AddCommGroup V] [Module A V]
+  {P : Type up} [AddCommGroup P] [Module A P]
+ theorem tensor_range_le_of_one_tmul (F : (A ⊗[C] N) →ₗ[A] V) (G : P →ₗ[A] V)
+    (h : ∀ n : N, F ((1 : A) ⊗ₜ[C] n) ∈ LinearMap.range G) :
+    LinearMap.range F ≤ LinearMap.range G := by
+  rintro _ ⟨m,rfl⟩
+  induction m using TensorProduct.inductionOn with
+  | add m n hm hn => rw [map_add]; exact (LinearMap.range G).add_mem hm hn
+  | tmul a n =>
+    have he : a ⊗ₜ[C] n = a • ((1 : A) ⊗ₜ[C] n) := by
+      rw [TensorProduct.smul_tmul', smul_eq_mul,mul_one]
+    rw [he,map_smul]
+    exact (LinearMap.range G).smul_mem a (h n)
+ theorem tensor_maps_eq_of_one_tmul (F G : (A ⊗[C] N) →ₗ[A] V)
+    (h : ∀ n : N, F ((1 : A) ⊗ₜ[C] n)=G ((1 : A) ⊗ₜ[C] n)) : F=G := by
+  apply LinearMap.ext
+  intro m
+  induction m using TensorProduct.inductionOn with
+  | add m n hm hn => rw [map_add,map_add,hm,hn]
+  | tmul a n =>
+    have he : a ⊗ₜ[C] n = a • ((1 : A) ⊗ₜ[C] n) := by
+      rw [TensorProduct.smul_tmul',smul_eq_mul,mul_one]
+    rw [he,map_smul,map_smul,h n]
+
+ theorem codRestrict_surjective_of_range_le (G : P →ₗ[A] V) (S : Submodule A V)
+    (hG : ∀ p, G p ∈ S) (hS : S ≤ LinearMap.range G) :
+    Function.Surjective (LinearMap.codRestrict S G hG) := by
+  intro z
+  obtain ⟨p,hp⟩ := hS z.property
+  exact ⟨p,Subtype.ext hp⟩
+end KLInvariance
+
+end
+
+
+section
+
+namespace KLInvariance.TitsSpace
+open Module TensorProduct FrobeniusDuality ReflectionFrobenius
+universe u v up
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+attribute [local instance] restrictedModule restrictedTower restrictedFree restrictedFinite
+ theorem bottWeightProjection_range_le (i : I) (l : List I) (x : W)
+    {P : Type up} [AddCommGroup P] [Module (SymmetricCoefficient (M := M)) P]
+    (G : P →ₗ[SymmetricCoefficient (M := M)]
+      ({e : BottIndex (i::l) // bottWeight cs (i::l) e=x} → SymmetricCoefficient (M := M)))
+    (h : ∀ n : BottSamelsonModule cs l,
+      bottWeightProjection cs (i::l) x ((1 : SymmetricCoefficient (M := M)) ⊗ₜ[simpleFixed cs i] n) ∈ LinearMap.range G) :
+    LinearMap.range (bottWeightProjection cs (i::l) x) ≤ LinearMap.range G := by
+  exact tensor_range_le_of_one_tmul (C := simpleFixed cs i)
+    (A := SymmetricCoefficient (M := M)) (N := BottSamelsonModule cs l)
+    (bottWeightProjection cs (i::l) x) G h
+ theorem bottTensor_maps_eq (i : I) (l : List I)
+    {P : Type up} [AddCommGroup P] [Module (SymmetricCoefficient (M := M)) P]
+    (F G : BottSamelsonModule cs (i::l) →ₗ[SymmetricCoefficient (M := M)] P)
+    (h : ∀ n : BottSamelsonModule cs l,
+      F ((1 : SymmetricCoefficient (M := M)) ⊗ₜ[simpleFixed cs i] n)=
+      G ((1 : SymmetricCoefficient (M := M)) ⊗ₜ[simpleFixed cs i] n)) : F=G :=
+  tensor_maps_eq_of_one_tmul (C := simpleFixed cs i)
+    (A := SymmetricCoefficient (M := M)) (N := BottSamelsonModule cs l) F G h
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! The genuine head-word stalk is an integral reflected pair lattice of the
+tail word. This supplies the stalk-freeness step once tail edge exactness is
+proved; it does not assume the desired main conclusion. -/
+namespace KLInvariance.TitsSpace
+open Module TensorProduct FrobeniusDuality ReflectionFrobenius ReflectionPairTranslation
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+attribute [local instance] restrictedModule restrictedTower restrictedFree restrictedFinite
+
+ omit [Fintype I] in
+ theorem bottWeight_tail_zero (i : I) (l : List I) (x : W)
+    (e : {e : BottIndex (i::l) // bottWeight cs (i::l) e=x}) (h : e.val.1=0) :
+    bottWeight cs l e.val.2=x := by
+  simpa only [bottWeight,ite_eq_left h] using e.property
+
+ omit [Fintype I] in
+ theorem bottWeight_tail_one (i : I) (l : List I) (x : W)
+    (e : {e : BottIndex (i::l) // bottWeight cs (i::l) e=x}) (h : e.val.1≠0) :
+    bottWeight cs l e.val.2=cs.simple i*x := by
+  have he : cs.simple i*bottWeight cs l e.val.2=x := by
+    simpa only [bottWeight,ite_eq_right h] using e.property
+  simpa only [cs.simple_mul_simple_cancel_left] using (congrArg (cs.simple i * ·) he)
+
+ def bottHeadPairCoordinates (i : I) (l : List I) (x : W) :
+    (bottStalk cs l x × Twist (coefficientAction cs (cs.simple i)).toRingEquiv
+      (bottStalk cs l (cs.simple i*x))) →ₗ[SymmetricCoefficient (M := M)]
+      ({e : BottIndex (i::l) // bottWeight cs (i::l) e=x} → SymmetricCoefficient (M := M)) where
+  toFun p e := if h : e.val.1=0 then
+      p.1.val ⟨e.val.2,bottWeight_tail_zero cs i l x e h⟩
+    else coefficientAction cs (cs.simple i)
+      (p.2.val.val ⟨e.val.2,bottWeight_tail_one cs i l x e h⟩)
+  map_add' p q := by
+    funext e
+    by_cases h : e.val.1=0
+    · simp only [Pi.add_apply,dite_eq_left h]; rfl
+    · simp only [Pi.add_apply,dite_eq_right h]
+      exact map_add (coefficientAction cs (cs.simple i)) _ _
+  map_smul' r p := by
+    funext e
+    dsimp
+    split_ifs with h
+    · rfl
+    · change coefficientAction cs (cs.simple i)
+        (coefficientAction cs (cs.simple i) r * _) = r * coefficientAction cs (cs.simple i) _
+      rw [map_mul,simple_involutive]; rfl
+
+ theorem bottHeadPairCoordinates_injective (i : I) (l : List I) (x : W) :
+    Function.Injective (bottHeadPairCoordinates cs i l x) := by
+  rintro ⟨p,q⟩ ⟨p',q'⟩ h
+  apply Prod.ext
+  · apply Subtype.ext
+    funext e
+    have hh := congrFun h ⟨(0,e.val),by simpa only [bottWeight,ite_true] using e.property⟩
+    simpa only [bottHeadPairCoordinates,LinearMap.coe_mk,AddHom.coe_mk,dite_eq_left rfl,dite_true] using hh
+  · apply Twist.ext
+    apply Subtype.ext
+    funext e
+    have he : bottWeight cs (i::l) (1,e.val)=x := by
+      change (if (1 : Fin 2)=0 then _ else cs.simple i*bottWeight cs l e.val)=x
+      rw [ite_eq_right (by decide),e.property,cs.simple_mul_simple_cancel_left]
+    have hh := congrFun h ⟨(1,e.val),he⟩
+    have hh' : coefficientAction cs (cs.simple i) (q.val.val e) =
+        coefficientAction cs (cs.simple i) (q'.val.val e) := by
+      simpa only [bottHeadPairCoordinates,LinearMap.coe_mk,AddHom.coe_mk,
+        dite_eq_right (show (1 : Fin 2)≠0 by decide)] using hh
+    exact (coefficientAction cs (cs.simple i)).injective hh'
+
+ theorem simpleReverseDivisibility (i : I) (a : SymmetricCoefficient (M := M)) :
+    simpleRootPolynomial (M := M) i ∣ coefficientAction cs (cs.simple i) a-a :=
+  (dvd_sub_comm).mpr (simple_divisible cs i a)
+
+ variable (i : I) (l : List I) (x : W)
+
+ def bottHeadPairLattice := lattice (bottStalkProjection cs l x)
+    (bottStalkProjection cs l (cs.simple i*x)) (coefficientAction cs (cs.simple i)).toRingEquiv
+    (simpleRootPolynomial (M := M) i) (simpleReverseDivisibility cs i) (bottSimpleEdge_kernel_inclusion cs i l x)
+
+ instance bottHeadPairAddCommGroup : AddCommGroup (bottHeadPairLattice cs i l x) :=
+  Submodule.addCommGroup (bottHeadPairLattice cs i l x)
+
+ instance bottHeadPairModule : Module (SymmetricCoefficient (M := M)) (bottHeadPairLattice cs i l x) :=
+  Submodule.module (bottHeadPairLattice cs i l x)
+
+ theorem bottHeadPairCoordinates_seed (n : BottSamelsonModule cs l) :
+    bottHeadPairCoordinates cs i l x
+      (bottStalkProjection cs l x n,⟨bottStalkProjection cs l (cs.simple i*x) n⟩) =
+    bottWeightProjection cs (i::l) x ((1 : SymmetricCoefficient (M := M)) ⊗ₜ[simpleFixed cs i] n) := by
+  funext e
+  change (if h : e.val.1=0 then _ else _) = (1 : SymmetricCoefficient (M := M))*(if e.val.1=0 then _ else _)
+  rw [one_mul]
+  split_ifs <;> rfl
+
+ def bottHeadPairToStalk : bottHeadPairLattice cs i l x →ₗ[
+     SymmetricCoefficient (M := M)] bottStalk cs (i::l) x :=
+  LinearMap.codRestrict (bottStalk cs (i::l) x)
+    ((bottHeadPairCoordinates cs i l x).comp (bottHeadPairLattice cs i l x).subtype) (by
+      rintro ⟨⟨p,q⟩,n,hn,hn'⟩
+      refine ⟨(1 : SymmetricCoefficient (M := M)) ⊗ₜ[simpleFixed cs i] n,?_⟩
+      change bottWeightProjection cs (i::l) x _=bottHeadPairCoordinates cs i l x (p,q)
+      rw [← bottHeadPairCoordinates_seed]
+      congr 1
+      exact Prod.ext hn (Twist.ext hn'))
+
+ theorem bottHeadPairToStalk_injective : Function.Injective (bottHeadPairToStalk cs i l x) := by
+  intro p q h
+  apply Subtype.ext
+  exact bottHeadPairCoordinates_injective cs i l x (congrArg Subtype.val h)
+
+ theorem bottHeadPair_range :
+    LinearMap.range (bottWeightProjection cs (i::l) x) ≤
+      LinearMap.range ((bottHeadPairCoordinates cs i l x).comp
+        (bottHeadPairLattice cs i l x).subtype) := by
+  refine bottWeightProjection_range_le cs i l x (P := bottHeadPairLattice cs i l x)
+    ((bottHeadPairCoordinates cs i l x).comp (bottHeadPairLattice cs i l x).subtype) ?_
+  intro n
+  refine ⟨⟨(bottStalkProjection cs l x n,⟨bottStalkProjection cs l (cs.simple i*x) n⟩),n,rfl,rfl⟩,?_⟩
+  exact bottHeadPairCoordinates_seed cs i l x n
+
+ theorem bottHeadPairToStalk_surjective : Function.Surjective (bottHeadPairToStalk cs i l x) := by
+  exact codRestrict_surjective_of_range_le (A := SymmetricCoefficient (M := M))
+    (P := bottHeadPairLattice cs i l x)
+    (V := {e : BottIndex (i::l) // bottWeight cs (i::l) e=x} → SymmetricCoefficient (M := M))
+    ((bottHeadPairCoordinates cs i l x).comp (bottHeadPairLattice cs i l x).subtype)
+    (bottStalk cs (i::l) x) _ (bottHeadPair_range cs i l x)
+
+ def bottHeadPairEquiv : bottHeadPairLattice cs i l x ≃ₗ[
+     SymmetricCoefficient (M := M)] bottStalk cs (i::l) x :=
+  LinearEquiv.ofBijective (bottHeadPairToStalk cs i l x)
+    ⟨bottHeadPairToStalk_injective cs i l x,bottHeadPairToStalk_surjective cs i l x⟩
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! The top projection lattice of a reduced actual tensor word is exactly A.
+This verifies the genuine rank-one top used in the BMP summand construction;
+it does not assert that the other projection lattices form a flabby sheaf. -/
+namespace KLInvariance.TitsSpace
+open Module TensorProduct
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+ def bottFullIndex : (l : List I) → BottIndex l
+  | [] => PUnit.unit
+  | _::l => (1,bottFullIndex l)
+
+omit [Fintype I] in
+ theorem bottWeight_full (l : List I) :
+    bottWeight cs l (bottFullIndex l)=cs.wordProd l := by
+  induction l with
+  | nil => exact cs.wordProd_nil.symm
+  | cons i l ih =>
+    simp only [bottFullIndex,bottWeight,show (1:Fin 2) ≠ 0 by decide,ite_false,
+      cs.wordProd_cons,ih]
+
+omit [Fintype I] in
+ theorem bottSubword_length_eq_full (l : List I) (e : BottIndex l)
+    (h : (bottSubword l e).length=l.length) : e=bottFullIndex l := by
+  induction l with
+  | nil =>
+    change PUnit at e
+    change e=(PUnit.unit : PUnit)
+    exact Subsingleton.elim _ _
+  | cons i l ih =>
+    change Fin 2 × BottIndex l at e
+    by_cases he : e.1=0
+    · simp only [bottSubword,ite_eq_left he,List.length_cons] at h
+      have hn := (bottSubword_sublist l e.2).length_le
+      omega
+    · have he1 : e.1=1 := by omega
+      simp only [bottSubword,ite_eq_right he,List.length_cons] at h
+      exact Prod.ext he1 (ih e.2 (Nat.succ.inj h))
+
+omit [Fintype I] in
+ theorem bottWeight_eq_top (l : List I) (hl : cs.IsReduced l) (e : BottIndex l)
+    (he : bottWeight cs l e=cs.wordProd l) : e=bottFullIndex l := by
+  apply bottSubword_length_eq_full
+  apply Nat.le_antisymm (bottSubword_sublist l e).length_le
+  have hh := cs.length_wordProd_le (bottSubword l e)
+  rwa [bottSubword_wordProd,he,hl.eq] at hh
+
+ def bottUnit : (l : List I) → BottSamelsonModule cs l
+  | [] => (1 : SymmetricCoefficient (M := M))
+  | i::l => 1 ⊗ₜ[simpleFixed cs i] bottUnit l
+
+ theorem bottEvaluation_unit (l : List I) (e : BottIndex l) :
+    bottEvaluation cs l e (bottUnit cs l)=1 := by
+  induction l with
+  | nil => rfl
+  | cons i l ih =>
+    change bottEvaluation cs (i::l) e (1 ⊗ₜ[simpleFixed cs i] bottUnit cs l)=1
+    rw [bottEvaluation_tmul,ih]
+    split_ifs <;> simp only [map_one,mul_one]
+
+ theorem bottEvaluation_surjective (l : List I) (e : BottIndex l) :
+    Function.Surjective (bottEvaluation cs l e) := by
+  intro a
+  refine ⟨a • bottUnit cs l,?_⟩
+  rw [map_smul,bottEvaluation_unit,smul_eq_mul,mul_one]
+
+ def bottTopCoordinate (l : List I) : bottStalk cs l (cs.wordProd l) →ₗ[
+    SymmetricCoefficient (M := M)] SymmetricCoefficient (M := M) :=
+  (LinearMap.proj ⟨bottFullIndex l,bottWeight_full cs l⟩).comp
+    (bottStalk cs l (cs.wordProd l)).subtype
+
+ theorem bottTopCoordinate_bijective (l : List I) (hl : cs.IsReduced l) :
+    Function.Bijective (bottTopCoordinate cs l) := by
+  constructor
+  · intro a b hab
+    apply Subtype.ext
+    funext e
+    have he : e=⟨bottFullIndex l,bottWeight_full cs l⟩ :=
+      Subtype.ext (bottWeight_eq_top cs l hl e.val e.property)
+    subst e
+    exact hab
+  · intro a
+    obtain ⟨m,hm⟩ := bottEvaluation_surjective cs l (bottFullIndex l) a
+    exact ⟨bottStalkProjection cs l (cs.wordProd l) m,hm⟩
+
+ def bottTopStalkEquiv (l : List I) (hl : cs.IsReduced l) :
+    bottStalk cs l (cs.wordProd l) ≃ₗ[SymmetricCoefficient (M := M)]
+      SymmetricCoefficient (M := M) :=
+  LinearEquiv.ofBijective (bottTopCoordinate cs l) (bottTopCoordinate_bijective cs l hl)
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! The genuine subexpression projections preserve the actual word grading.
+In particular the top projection of a reduced tensor word is a graded copy of
+A, rather than just an ungraded rank-one lattice. -/
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ theorem bottEvaluation_basis_homogeneous (l : List I) (e f : BottIndex l) :
+    bottEvaluationMatrix cs l e f ∈ symmetricGrading basis (bottDegree l f) := by
+  induction l with
+  | nil =>
+    change PUnit at e f
+    change (Basis.singleton PUnit (SymmetricCoefficient (M := M)) f) ∈
+      symmetricGrading basis 0
+    simp only [Basis.singleton_apply]
+    exact SetLike.one_mem_graded _
+  | cons i l ih =>
+    change Fin 2 × BottIndex l at e f
+    rcases e with ⟨r,j⟩
+    rcases f with ⟨c,k⟩
+    rw [bottEvaluationMatrix_cons]
+    by_cases hc : c=0
+    · subst c
+      simp only [ite_true,bottDegree,Fin.val_zero,Nat.cast_zero,zero_add]
+      split_ifs
+      · exact ih j k
+      · exact coefficientAction_homogeneous cs basis _ _ _ (ih j k)
+    · have hc1 : c=1 := by omega
+      subst c
+      simp only [show (1:Fin 2) ≠ 0 by decide,ite_false,bottDegree,Fin.val_one,Nat.cast_one]
+      split_ifs
+      · exact SetLike.GradedMul.mul_mem (simpleRoot_homogeneous basis i) (ih j k)
+      · exact SetLike.GradedMul.mul_mem ((symmetricGrading basis 1).neg_mem
+          (simpleRoot_homogeneous basis i))
+          (coefficientAction_homogeneous cs basis _ _ _ (ih j k))
+
+ theorem bottEvaluation_homogeneous (l : List I) (e : BottIndex l)
+    (n : ℤ) (m : BottSamelsonModule cs l) (hm : m ∈ bottPiece cs basis l n) :
+    bottEvaluation cs l e m ∈ symmetricGrading basis n := by
+  classical
+  have hexp : bottEvaluation cs l e m = ∑ f : BottIndex l,
+      (bottBasis cs l).repr m f * bottEvaluationMatrix cs l e f := by
+    conv_lhs => rw [← (bottBasis cs l).sum_repr m]
+    simp only [map_sum,map_smul,smul_eq_mul,bottEvaluationMatrix]
+  rw [hexp]
+  apply Submodule.sum_mem
+  intro f _
+  change ∀ f, (bottBasis cs l).repr m f ∈
+    symmetricGrading basis (n-bottDegree l f) at hm
+  have h := SetLike.GradedMul.mul_mem (hm f)
+    (bottEvaluation_basis_homogeneous cs basis l e f)
+  simpa only [sub_add_cancel] using h
+
+ theorem bottEvaluation_component (l : List I) (e : BottIndex l)
+    (n : ℤ) (m : BottSamelsonModule cs l) :
+    component (symmetricGrading basis) n (bottEvaluation cs l e m) =
+      bottEvaluation cs l e (component (bottPiece cs basis l) n m) := by
+  simpa only [sub_zero] using map_component (bottPiece cs basis l) (symmetricGrading basis)
+    (bottEvaluation cs l e) 0 (by simpa only [add_zero] using bottEvaluation_homogeneous cs basis l e) n m
+
+ theorem bottUnit_homogeneous (l : List I) : bottUnit cs l ∈ bottPiece cs basis l 0 := by
+  have he : component (bottPiece cs basis l) 0 (bottUnit cs l)=bottUnit cs l := by
+    apply bottEvaluation_injective cs l
+    funext e
+    change bottEvaluation cs l e (component (bottPiece cs basis l) 0 (bottUnit cs l)) =
+      bottEvaluation cs l e (bottUnit cs l)
+    rw [← bottEvaluation_component,bottEvaluation_unit]
+    exact DirectSum.decompose_of_mem_same (symmetricGrading basis) (SetLike.one_mem_graded _)
+  rw [← he]
+  exact (DirectSum.decompose (bottPiece cs basis l) (bottUnit cs l) 0).property
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Actual integral projection and support lattices inherit their grading from
+the genuine tensor word. In particular a homogeneous stalk vector has a
+homogeneous tensor lift. No freeness/character claim is assumed. -/
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ def bottCoordinatePiece (l : List I) (x : W) : ℤ → Submodule ℝ
+    ({e : BottIndex l // bottWeight cs l e=x} → SymmetricCoefficient (M := M)) :=
+  piPiece (fun _ => symmetricGrading basis)
+
+ instance bottCoordinatePiece_decomposition (l : List I) (x : W) :
+    DirectSum.Decomposition (bottCoordinatePiece cs basis l x) := by
+  classical
+  exact (pi_isInternal (fun _ : {e : BottIndex l // bottWeight cs l e=x} =>
+    symmetricGrading basis)).chooseDecomposition
+
+ instance bottCoordinatePiece_graded (l : List I) (x : W) :
+    SetLike.GradedSMul (symmetricGrading basis) (bottCoordinatePiece cs basis l x) :=
+  inferInstanceAs (SetLike.GradedSMul (symmetricGrading basis)
+    (piPiece (fun _ : {e : BottIndex l // bottWeight cs l e=x} => symmetricGrading basis)))
+
+ theorem bottWeightProjection_homogeneous (l : List I) (x : W) (n : ℤ)
+    (m : BottSamelsonModule cs l) (hm : m ∈ bottPiece cs basis l n) :
+    bottWeightProjection cs l x m ∈ bottCoordinatePiece cs basis l x n := by
+  intro e _
+  exact bottEvaluation_homogeneous cs basis l e.val n m hm
+
+ theorem bottStalk_homogeneous (l : List I) (x : W) :
+    ∀ n f, f ∈ bottStalk cs l x →
+      component (bottCoordinatePiece cs basis l x) n f ∈ bottStalk cs l x := by
+  exact range_homogeneous (bottPiece cs basis l) (bottCoordinatePiece cs basis l x)
+    (bottWeightProjection cs l x) 0 (by
+      simpa only [add_zero] using bottWeightProjection_homogeneous cs basis l x)
+
+ def bottStalkPiece (l : List I) (x : W) : ℤ → Submodule ℝ (bottStalk cs l x) :=
+  subPiece (bottCoordinatePiece cs basis l x) (bottStalk cs l x)
+
+ instance bottStalkPiece_decomposition (l : List I) (x : W) :
+    DirectSum.Decomposition (bottStalkPiece cs basis l x) :=
+  (sub_isInternal (bottCoordinatePiece cs basis l x) (bottStalk cs l x)
+    (bottStalk_homogeneous cs basis l x)).chooseDecomposition
+
+ instance bottStalkPiece_graded (l : List I) (x : W) :
+    SetLike.GradedSMul (symmetricGrading basis) (bottStalkPiece cs basis l x) :=
+  inferInstanceAs (SetLike.GradedSMul (symmetricGrading basis)
+    (subPiece (bottCoordinatePiece cs basis l x) (bottStalk cs l x)))
+
+ theorem bottStalkProjection_homogeneous (l : List I) (x : W) (n : ℤ)
+    (m : BottSamelsonModule cs l) (hm : m ∈ bottPiece cs basis l n) :
+    bottStalkProjection cs l x m ∈ bottStalkPiece cs basis l x n :=
+  bottWeightProjection_homogeneous cs basis l x n m hm
+
+ theorem bottStalk_homogeneous_lift (l : List I) (x : W) (n : ℤ)
+    (s : bottStalk cs l x) (hs : s ∈ bottStalkPiece cs basis l x n) :
+    ∃ m : BottSamelsonModule cs l, m ∈ bottPiece cs basis l n ∧
+      bottStalkProjection cs l x m=s := by
+  obtain ⟨m,hm⟩ := bottStalkProjection_surjective cs l x s
+  refine ⟨component (bottPiece cs basis l) n m,
+    (DirectSum.decompose (bottPiece cs basis l) m n).property,?_⟩
+  have he := map_component (bottPiece cs basis l) (bottStalkPiece cs basis l x)
+    (bottStalkProjection cs l x) 0 (by
+      simpa only [add_zero] using bottStalkProjection_homogeneous cs basis l x) n m
+  rw [sub_zero,hm,show component (bottStalkPiece cs basis l x) n s=s from
+    DirectSum.decompose_of_mem_same _ hs] at he
+  exact he.symm
+
+ theorem bottWeightSupported_homogeneous (l : List I) (x : W) :
+    ∀ n m, m ∈ bottWeightSupported cs l x →
+      component (bottPiece cs basis l) n m ∈ bottWeightSupported cs l x := by
+  intro n m hm e he
+  rw [← bottEvaluation_component,hm e he,map_zero]
+
+ def bottSupportPiece (l : List I) (x : W) :
+    ℤ → Submodule ℝ (bottWeightSupported cs l x) :=
+  subPiece (bottPiece cs basis l) (bottWeightSupported cs l x)
+
+ instance bottSupportPiece_decomposition (l : List I) (x : W) :
+    DirectSum.Decomposition (bottSupportPiece cs basis l x) :=
+  (sub_isInternal (bottPiece cs basis l) (bottWeightSupported cs l x)
+    (bottWeightSupported_homogeneous cs basis l x)).chooseDecomposition
+
+ instance bottSupportPiece_graded (l : List I) (x : W) :
+    SetLike.GradedSMul (symmetricGrading basis) (bottSupportPiece cs basis l x) :=
+  inferInstanceAs (SetLike.GradedSMul (symmetricGrading basis)
+    (subPiece (bottPiece cs basis l) (bottWeightSupported cs l x)))
+
+omit [Fintype σ] in
+ theorem bottTopStalkEquiv_homogeneous (l : List I) (hl : cs.IsReduced l)
+    (n : ℤ) (s : bottStalk cs l (cs.wordProd l)) :
+    bottTopStalkEquiv cs l hl s ∈ symmetricGrading basis n ↔
+      s ∈ bottStalkPiece cs basis l (cs.wordProd l) n := by
+  constructor
+  · intro h e _
+    have he : e=⟨bottFullIndex l,bottWeight_full cs l⟩ :=
+      Subtype.ext (bottWeight_eq_top cs l hl e.val e.property)
+    subst e
+    exact h
+  · intro h
+    exact h ⟨bottFullIndex l,bottWeight_full cs l⟩ (Set.mem_univ _)
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Restriction of the honest reflection-twisted module to the fixed base field.
+The automorphism fixes the base field, so no scalar information is discarded. -/
+namespace KLInvariance.ReflectionPairTranslation.Twist
+universe uk ur uy
+variable {k : Type uk} [Field k] {R : Type ur} [CommRing R] [Algebra k R]
+  (τ : R ≃ₐ[k] R) (Y : Type uy) [AddCommGroup Y] [Module R Y] [Module k Y]
+  [IsScalarTower k R Y]
+
+instance scalarModule : Module k (Twist τ.toRingEquiv Y) :=
+  ({ equivVal τ.toRingEquiv Y with map_add' := fun _ _ => rfl } :
+    Twist τ.toRingEquiv Y ≃+ Y).module k
+
+instance scalarTower : IsScalarTower k R (Twist τ.toRingEquiv Y) where
+  smul_assoc c r y := by
+    apply Twist.ext
+    change τ (c • r) • y.val=c • (τ r • y.val)
+    rw [map_smul,smul_assoc]
+
+def scalarEquiv : Twist τ.toRingEquiv Y ≃ₗ[k] Y where
+  toFun := val
+  invFun := mk
+  left_inv _ := rfl
+  right_inv _ := rfl
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+
+end KLInvariance.ReflectionPairTranslation.Twist
+
+end
+
+
+section
+
+/-! Homogeneous perfect pairings on the actual integral free modules.  The
+inverse grading is proved, not inferred merely after fraction-field extension. -/
+namespace KLInvariance.Graded
+open Module
+universe uk ua um un ui
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {N : Type um} [AddCommGroup N] [Module A N] [Module k N] [IsScalarTower k A N]
+  {P : Type un} [AddCommGroup P] [Module A P] [Module k P] [IsScalarTower k A P]
+noncomputable section
+
+omit [Algebra k A] [IsScalarTower k A N] [IsScalarTower k A P] in
+ theorem equiv_inverse_homogeneous
+    (𝓝 : ℤ → Submodule k N) (𝓟 : ℤ → Submodule k P)
+    [DirectSum.Decomposition 𝓝] [DirectSum.Decomposition 𝓟]
+    (e : N ≃ₗ[A] P) (d : ℤ)
+    (he : ∀ n x, x ∈ 𝓝 n → e x ∈ 𝓟 (n+d))
+    (n : ℤ) (y : P) (hy : y ∈ 𝓟 n) : e.symm y ∈ 𝓝 (n-d) := by
+  have hc : component 𝓝 (n-d) (e.symm y) = e.symm y := by
+    apply e.injective
+    change e.toLinearMap (component 𝓝 (n-d) (e.symm y)) = e.toLinearMap (e.symm y)
+    rw [← map_component 𝓝 𝓟 e.toLinearMap d he n (e.symm y)]
+    simp only [LinearEquiv.coe_coe,LinearEquiv.apply_symm_apply]
+    exact DirectSum.decompose_of_mem_same 𝓟 hy
+  rw [← hc]
+  exact (DirectSum.decompose 𝓝 (e.symm y) (n-d)).property
+
+variable (𝓐 : ℤ → Submodule k A) [DirectSum.Decomposition 𝓐]
+  [SetLike.GradedMonoid 𝓐] {ι : Type ui} [Fintype ι] [DecidableEq ι]
+  (b : Basis ι A N) (d : ι → ℤ)
+
+omit [DirectSum.Decomposition 𝓐] in
+ theorem pairing_to_dual_homogeneous (p : N →ₗ[A] Dual A N) (l : ℤ)
+    (hp : ∀ i j, p (b i) (b j) ∈ 𝓐 (d i+d j-l))
+    {n : ℤ} {x : N} (hx : x ∈ basisPiece 𝓐 b d n) :
+    p x ∈ basisPiece 𝓐 b.dualBasis (fun i => -d i) (n-l) := by
+  rw [mem_basisPiece]
+  intro i
+  rw [Basis.dualBasis_repr]
+  have h := pairing_homogeneous 𝓐 b d p l hp hx (basis_mem_piece 𝓐 b d i)
+  convert h using 1
+  congr 1
+  omega
+
+ theorem perfectPairing_inverse_homogeneous (p : N ≃ₗ[A] Dual A N) (l : ℤ)
+    (hp : ∀ i j, p (b i) (b j) ∈ 𝓐 (d i+d j-l))
+    {n : ℤ} {y : Dual A N}
+    (hy : y ∈ basisPiece 𝓐 b.dualBasis (fun i => -d i) n) :
+    p.symm y ∈ basisPiece 𝓐 b d (n+l) := by
+  have he : ∀ m x, x ∈ basisPiece 𝓐 b d m →
+      p x ∈ basisPiece 𝓐 b.dualBasis (fun i => -d i) (m+(-l)) := by
+    intro m x hx
+    exact pairing_to_dual_homogeneous 𝓐 b d p.toLinearMap l hp hx
+  simpa only [sub_neg_eq_add] using equiv_inverse_homogeneous
+    (basisPiece 𝓐 b d) (basisPiece 𝓐 b.dualBasis (fun i => -d i)) p (-l) he n y hy
+
+end
+end KLInvariance.Graded
+
+end
+
+
+section
+
+/-! The genuine head-word projection maps, on the fixed real field, preserve
+homogeneous degrees. These maps are not abstract character operators. -/
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded ReflectionPairTranslation
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ def bottHeadLeft (i : I) (l : List I) (x : W) :
+    bottStalk cs (i::l) x →ₗ[SymmetricCoefficient (M := M)] bottStalk cs l x :=
+  (ReflectionPairTranslation.left (bottStalkProjection cs l x)
+    (bottStalkProjection cs l (cs.simple i*x)) (coefficientAction cs (cs.simple i)).toRingEquiv
+    (simpleRootPolynomial (M := M) i) (simpleReverseDivisibility cs i)
+    (bottSimpleEdge_kernel_inclusion cs i l x)).comp (bottHeadPairEquiv cs i l x).symm.toLinearMap
+
+ def bottHeadRight (i : I) (l : List I) (x : W) :
+    bottStalk cs (i::l) x →ₗ[ℝ] bottStalk cs l (cs.simple i*x) :=
+  (Twist.scalarEquiv (coefficientAction cs (cs.simple i)) _).toLinearMap.comp
+    (((LinearMap.snd (SymmetricCoefficient (M := M)) (bottStalk cs l x)
+      (Twist (coefficientAction cs (cs.simple i)).toRingEquiv (bottStalk cs l (cs.simple i*x)))).comp
+        ((bottHeadPairLattice cs i l x).subtype.comp
+          (bottHeadPairEquiv cs i l x).symm.toLinearMap)).restrictScalars ℝ)
+
+ theorem bottHeadLeft_surjective (i : I) (l : List I) (x : W) :
+    Function.Surjective (bottHeadLeft cs i l x) :=
+  (left_surjective _ _ _ _ _ _ (bottStalkProjection_surjective cs l x)).comp
+    (bottHeadPairEquiv cs i l x).symm.surjective
+
+ theorem bottHeadPairEquiv_coordinates (i : I) (l : List I) (x : W)
+    (m : bottStalk cs (i::l) x) :
+    bottHeadPairCoordinates cs i l x ((bottHeadPairEquiv cs i l x).symm m).val=m.val :=
+  congrArg Subtype.val ((bottHeadPairEquiv cs i l x).apply_symm_apply m)
+
+omit [Fintype σ] in
+ theorem bottHeadLeft_homogeneous (i : I) (l : List I) (x : W) (n : ℤ)
+    (m : bottStalk cs (i::l) x) (hm : m ∈ bottStalkPiece cs basis (i::l) x n) :
+    bottHeadLeft cs i l x m ∈ bottStalkPiece cs basis l x n := by
+  intro e _
+  change (bottHeadLeft cs i l x m).val e ∈ symmetricGrading basis n
+  let E : {e : BottIndex (i::l) // bottWeight cs (i::l) e=x} :=
+    ⟨(0,e.val),by simpa only [bottWeight,ite_true] using e.property⟩
+  have he := congrFun (bottHeadPairEquiv_coordinates cs i l x m) E
+  have hh : (bottHeadLeft cs i l x m).val e=m.val E := by
+    change ((bottHeadPairEquiv cs i l x).symm m).val.1.val e=m.val E at he
+    exact he
+  rw [hh]
+  exact hm E (Set.mem_univ E)
+
+ theorem bottHeadRight_homogeneous (i : I) (l : List I) (x : W) (n : ℤ)
+    (m : bottStalk cs (i::l) x) (hm : m ∈ bottStalkPiece cs basis (i::l) x n) :
+    bottHeadRight cs i l x m ∈ bottStalkPiece cs basis l (cs.simple i*x) n := by
+  intro e _
+  change (bottHeadRight cs i l x m).val e ∈ symmetricGrading basis n
+  let E : {e : BottIndex (i::l) // bottWeight cs (i::l) e=x} :=
+    ⟨(1,e.val),by simp only [bottWeight,ite_eq_right (show (1:Fin 2)≠0 by decide),
+      e.property,cs.simple_mul_simple_cancel_left]⟩
+  have he := congrFun (bottHeadPairEquiv_coordinates cs i l x m) E
+  have hh : coefficientAction cs (cs.simple i) ((bottHeadRight cs i l x m).val e)=m.val E := by
+    change coefficientAction cs (cs.simple i)
+      (((bottHeadPairEquiv cs i l x).symm m).val.2.val.val e)=m.val E at he
+    exact he
+  have h := coefficientAction_homogeneous cs basis (cs.simple i) n (m.val E)
+    (hm E (Set.mem_univ E))
+  rw [← hh,simple_involutive] at h
+  exact h
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! The two common tail components of a reflected tensor. They are defined
+on the actual tensor, so divisibility assertions at different weights use the
+same tail vectors rather than unrelated existential lifts. -/
+namespace KLInvariance.ReflectionPairTranslation
+open TensorProduct
+universe uc ua un
+variable {C : Type uc} [CommRing C] {A : Type ua} [CommRing A] [Algebra C A]
+  {N : Type un} [AddCommGroup N] [Module A N] [Module C N] [IsScalarTower C A N]
+  (τ : A ≃+* A) (hfix : ∀ c : C, τ (algebraMap C A c)=algebraMap C A c)
+noncomputable section
+
+ def tensorFirst : (A ⊗[C] N) →ₗ[A] N :=
+  AlgebraTensorModule.lift ((LinearMap.id : A →ₗ[A] A).smulRight
+    (LinearMap.id : N →ₗ[C] N))
+
+ @[simp] theorem tensorFirst_tmul (a : A) (n : N) :
+    tensorFirst (C := C) (a ⊗ₜ[C] n)=a • n := rfl
+
+ def tensorSecond : (A ⊗[C] N) →ₗ[A] Twist τ N := by
+  letI : Module C (Twist τ N) := Module.compHom _ (algebraMap C A)
+  letI : IsScalarTower C A (Twist τ N) :=
+    IsScalarTower.of_algebraMap_smul fun _ _ => rfl
+  let F : N →ₗ[C] Twist τ N := {
+    toFun := Twist.mk
+    map_add' := fun _ _ => rfl
+    map_smul' := fun c n => by
+      apply Twist.ext
+      change c • n=τ (algebraMap C A c) • n
+      rw [hfix,algebraMap_smul] }
+  exact AlgebraTensorModule.lift ((LinearMap.id : A →ₗ[A] A).smulRight F)
+
+ @[simp] theorem tensorSecond_tmul (a : A) (n : N) :
+    tensorSecond τ hfix (a ⊗ₜ[C] n)=Twist.mk (τ a • n) := rfl
+
+ def twistMap {P : Type*} [AddCommGroup P] [Module A P] (f : N →ₗ[A] P) :
+    Twist τ N →ₗ[A] Twist τ P where
+  toFun m := ⟨f m.val⟩
+  map_add' m n := Twist.ext (map_add f m.val n.val)
+  map_smul' a m := Twist.ext (map_smul f (τ a) m.val)
+end
+end KLInvariance.ReflectionPairTranslation
+
+end
+
+
+section
+
+namespace KLInvariance.TitsSpace
+open Module TensorProduct FrobeniusDuality ReflectionFrobenius ReflectionPairTranslation
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+attribute [local instance] restrictedModule restrictedTower restrictedFree restrictedFinite
+
+ def bottHeadTail (i : I) (l : List I) : BottSamelsonModule cs (i::l) →ₗ[
+    SymmetricCoefficient (M := M)] BottSamelsonModule cs l :=
+  tensorFirst (C := simpleFixed cs i)
+
+ def bottHeadReflectedTail (i : I) (l : List I) : BottSamelsonModule cs (i::l) →ₗ[
+    SymmetricCoefficient (M := M)] Twist (coefficientAction cs (cs.simple i)).toRingEquiv
+      (BottSamelsonModule cs l) :=
+  tensorSecond (C := simpleFixed cs i) (coefficientAction cs (cs.simple i)).toRingEquiv
+    (fun c => c.property)
+
+ theorem bottHeadTail_seed (i : I) (l : List I) (n : BottSamelsonModule cs l) :
+    bottHeadTail cs i l ((1 : SymmetricCoefficient (M := M)) ⊗ₜ[simpleFixed cs i] n)=n := by
+  exact one_smul _ n
+
+ theorem bottHeadReflectedTail_seed (i : I) (l : List I) (n : BottSamelsonModule cs l) :
+    bottHeadReflectedTail cs i l ((1 : SymmetricCoefficient (M := M)) ⊗ₜ[simpleFixed cs i] n)=Twist.mk n := by
+  change Twist.mk ((coefficientAction cs (cs.simple i)) 1 • n)=Twist.mk n
+  rw [map_one,one_smul]
+
+ def bottHeadLocalPair (i : I) (l : List I) (x : W) : BottSamelsonModule cs (i::l) →ₗ[
+    SymmetricCoefficient (M := M)]
+    (bottStalk cs l x × Twist (coefficientAction cs (cs.simple i)).toRingEquiv
+      (bottStalk cs l (cs.simple i*x))) :=
+  ((bottStalkProjection cs l x).comp (bottHeadTail cs i l)).prod
+    ((twistMap (coefficientAction cs (cs.simple i)).toRingEquiv
+      (bottStalkProjection cs l (cs.simple i*x))).comp (bottHeadReflectedTail cs i l))
+
+ theorem bottHeadLocalPair_seed (i : I) (l : List I) (x : W) (n : BottSamelsonModule cs l) :
+    bottHeadLocalPair cs i l x ((1 : SymmetricCoefficient (M := M)) ⊗ₜ[simpleFixed cs i] n)=
+      (bottStalkProjection cs l x n,Twist.mk (bottStalkProjection cs l (cs.simple i*x) n)) := by
+  change ((bottStalkProjection cs l x) (bottHeadTail cs i l _),
+    (twistMap (coefficientAction cs (cs.simple i)).toRingEquiv
+      (bottStalkProjection cs l (cs.simple i*x))) (bottHeadReflectedTail cs i l _)) = _
+  rw [bottHeadTail_seed,bottHeadReflectedTail_seed]
+  rfl
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+namespace KLInvariance
+universe ua up uv un
+variable {A : Type ua} [CommRing A]
+  {P : Type up} [AddCommGroup P] [Module A P]
+  {V : Type uv} [AddCommGroup V] [Module A V]
+  {N : Type un} [AddCommGroup N] [Module A N]
+ theorem linearMap_mem_of_coordinates (S : Submodule A P)
+    (F : P →ₗ[A] V) (G : N →ₗ[A] P) (E : N →ₗ[A] V)
+    (hF : Function.Injective F) (heq : F.comp G = E)
+    (hrange : LinearMap.range E ≤ LinearMap.range (F.comp S.subtype)) (n : N) : G n ∈ S := by
+  obtain ⟨p,hp⟩ := hrange (show E n ∈ LinearMap.range E from ⟨n,rfl⟩)
+  have he : (p : P) = G n := hF (hp.trans (LinearMap.congr_fun heq n).symm)
+  exact he ▸ p.property
+end KLInvariance
+
+end
+
+
+section
+
+namespace KLInvariance.TitsSpace
+open Module TensorProduct FrobeniusDuality ReflectionFrobenius ReflectionPairTranslation
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+attribute [local instance] restrictedModule restrictedTower restrictedFree restrictedFinite
+
+ theorem bottHeadLocalPair_coordinates (i : I) (l : List I) (x : W) :
+    (bottHeadPairCoordinates cs i l x).comp (bottHeadLocalPair cs i l x) =
+      bottWeightProjection cs (i::l) x := by
+  apply bottTensor_maps_eq cs i l
+  intro n
+  change bottHeadPairCoordinates cs i l x (bottHeadLocalPair cs i l x _) = _
+  rw [bottHeadLocalPair_seed,bottHeadPairCoordinates_seed]
+
+ theorem bottHeadLocalPair_mem (i : I) (l : List I) (x : W)
+    (m : BottSamelsonModule cs (i::l)) :
+    bottHeadLocalPair cs i l x m ∈ bottHeadPairLattice cs i l x := by
+  exact linearMap_mem_of_coordinates (bottHeadPairLattice cs i l x)
+    (bottHeadPairCoordinates cs i l x) (bottHeadLocalPair cs i l x)
+    (bottWeightProjection cs (i::l) x) (bottHeadPairCoordinates_injective cs i l x)
+    (bottHeadLocalPair_coordinates cs i l x) (bottHeadPair_range cs i l x) m
+
+ theorem bottHeadLocalPair_zero_iff (i : I) (l : List I) (x : W)
+    (m : BottSamelsonModule cs (i::l)) :
+    bottStalkProjection cs (i::l) x m=0 ↔ bottHeadLocalPair cs i l x m=0 := by
+  constructor
+  · intro h
+    apply bottHeadPairCoordinates_injective cs i l x
+    rw [map_zero]
+    have hh := congrArg Subtype.val h
+    change bottWeightProjection cs (i::l) x m=0 at hh
+    exact (LinearMap.congr_fun (bottHeadLocalPair_coordinates cs i l x) m).trans hh
+  · intro h
+    apply Subtype.ext
+    change bottWeightProjection cs (i::l) x m=0
+    rw [← LinearMap.congr_fun (bottHeadLocalPair_coordinates cs i l x) m]
+    change bottHeadPairCoordinates cs i l x (bottHeadLocalPair cs i l x m)=0
+    rw [h,map_zero]
+
+ theorem bottHeadLocalPair_value (i : I) (l : List I) (x : W)
+    (m : BottSamelsonModule cs (i::l)) :
+    bottHeadLocalPair cs i l x m =
+      (bottStalkProjection cs l x (bottHeadTail cs i l m),
+       Twist.mk (bottStalkProjection cs l (cs.simple i*x) (bottHeadReflectedTail cs i l m).val)) := rfl
+
+ theorem bottHeadLocalPair_toStalk (i : I) (l : List I) (x : W)
+    (m : BottSamelsonModule cs (i::l)) :
+    bottHeadPairToStalk cs i l x ⟨bottHeadLocalPair cs i l x m,bottHeadLocalPair_mem cs i l x m⟩ =
+      bottStalkProjection cs (i::l) x m := by
+  apply Subtype.ext
+  exact LinearMap.congr_fun (bottHeadLocalPair_coordinates cs i l x) m
+
+ theorem bottHeadStalk_divisible_of_pair (i : I) (l : List I) (x : W)
+    (m : BottSamelsonModule cs (i::l)) (β : SymmetricCoefficient (M := M))
+    (p : bottStalk cs l x × Twist (coefficientAction cs (cs.simple i)).toRingEquiv
+      (bottStalk cs l (cs.simple i*x)))
+    (hm : bottHeadLocalPair cs i l x m=β • p) (hp : p ∈ bottHeadPairLattice cs i l x) :
+    ∃ z : bottStalk cs (i::l) x, bottStalkProjection cs (i::l) x m=β • z := by
+  let z : bottHeadPairLattice cs i l x := ⟨p,hp⟩
+  refine ⟨bottHeadPairToStalk cs i l x z,?_⟩
+  have he : (⟨bottHeadLocalPair cs i l x m,bottHeadLocalPair_mem cs i l x m⟩ :
+      bottHeadPairLattice cs i l x) = β • z := Subtype.ext hm
+  calc
+    bottStalkProjection cs (i::l) x m =
+        bottHeadPairToStalk cs i l x ⟨bottHeadLocalPair cs i l x m,bottHeadLocalPair_mem cs i l x m⟩ :=
+      (bottHeadLocalPair_toStalk cs i l x m).symm
+    _ = bottHeadPairToStalk cs i l x (β • z) := congrArg (bottHeadPairToStalk cs i l x) he
+    _ = β • bottHeadPairToStalk cs i l x z := map_smul _ _ _
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+namespace KLInvariance.FiniteFreeScalar
+universe ur um un ui
+variable {R : Type ur} [CommRing R]
+  {M : Type um} [AddCommGroup M] [Module R M]
+  {ι : Type ui} [Fintype ι] (b : Module.Basis ι R M)
+
+
+ theorem exists_smul_iff (r : R) (m : M) :
+    (∃ n, r • n = m) ↔ ∀ i, r ∣ b.equivFun m i := by
+  classical
+  constructor
+  · rintro ⟨n,rfl⟩ i
+    exact ⟨b.equivFun n i, by simp⟩
+  · intro h
+    choose n hn using h
+    refine ⟨b.equivFun.symm n,b.equivFun.injective ?_⟩
+    ext i
+    simp only [map_smul,LinearEquiv.apply_symm_apply,Pi.smul_apply,smul_eq_mul]
+    exact (hn i).symm
+
+include b in
+ theorem exists_prod_smul [DecompositionMonoid R] {J : Type*} (s : Finset J) (r : J → R)
+    (hrel : (s : Set J).Pairwise (Function.onFun IsRelPrime r)) (m : M)
+    (h : ∀ j ∈ s, ∃ n, r j • n = m) : ∃ n, (∏ j ∈ s, r j) • n = m := by
+  apply (exists_smul_iff b _ m).mpr
+  intro i
+  apply Finset.prod_dvd_of_isRelPrime hrel
+  intro j hj
+  exact (exists_smul_iff b _ _).mp (h j hj) i
+
+include b in
+
+
+ theorem scalar_quotient_regular
+    {N : Type un} [AddCommGroup N] [Module R N]
+    (q : M →ₗ[R] N) (hq : Function.Surjective q)
+    (a r : R) (ha : Prime a) (hr : ¬ a ∣ r)
+    (hker : ∀ m, q m = 0 ↔ ∃ n, m=a • n) :
+    Function.Injective (fun v : N => r • v) := by
+  suffices hh : ∀ v : N, r • v = 0 → v = 0 by
+    intro v w hvw
+    change r • v = r • w at hvw
+    have hv : r • (v-w) = 0 := by rw [smul_sub,hvw,sub_self]
+    exact sub_eq_zero.mp (hh _ hv)
+  intro v hv
+  obtain ⟨m,rfl⟩ := hq v
+  have h0 : q (r • m) = 0 := by simpa only [map_smul] using hv
+  obtain ⟨n,hn⟩ := (hker _).mp h0
+  have hdiv : ∀ i, a ∣ b.equivFun m i := by
+    intro i
+    have hc := congrArg (fun z => b.equivFun z i) hn
+    simp only [map_smul,Pi.smul_apply,smul_eq_mul] at hc
+    apply (ha.dvd_mul.mp ?_).resolve_left hr
+    exact ⟨b.equivFun n i,hc⟩
+  obtain ⟨z,hz⟩ := (exists_smul_iff b a m).mpr hdiv
+  exact (hker m).mpr ⟨z,hz.symm⟩
+
+end KLInvariance.FiniteFreeScalar
+
+end
+
+
+section
+
+/-! Intermediate standard-filtration predicates on the genuine tensor word.
+These are proved by word induction; they are not main-theorem assumptions. -/
+namespace KLInvariance.TitsSpace
+open Module ReflectionFrobenius
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+ def BottKernelExact (l : List I) : Prop :=
+  ∀ (x t : W) (a : I → ℝ), Represents M cs t a →
+    cs.length x < cs.length (t*x) →
+    ∀ m : BottSamelsonModule cs l, bottStalkProjection cs l x m=0 →
+      ∃ z : bottStalk cs l (t*x), bottStalkProjection cs l (t*x) m=
+        SymmetricAlgebra.ι ℝ (Extended M) (embed M a) • z
+
+ theorem simple_represents (i : I) : Represents M cs (cs.simple i) (root i) := by
+  refine ⟨1,i,?_,?_⟩
+  · simp
+  · simp
+
+ theorem represents_polynomial_prime {t : W} {a : I → ℝ} (ha : Represents M cs t a) :
+    Prime (SymmetricAlgebra.ι ℝ (Extended M) (embed M a)) := by
+  apply EdgeAlgebra.symmetric_ι_prime
+  intro h
+  exact ha.isRoot.ne_zero (congrArg Prod.fst h)
+
+ theorem represents_polynomial_not_dvd {t s : W} {a b : I → ℝ}
+    (ha : Represents M cs t a) (hb : Represents M cs s b) (hne : t≠s) :
+    ¬ SymmetricAlgebra.ι ℝ (Extended M) (embed M a) ∣
+      SymmetricAlgebra.ι ℝ (Extended M) (embed M b) := by
+  rw [EdgeAlgebra.symmetric_ι_dvd_iff]
+  intro h
+  have hi := hb.extended_independent_of_ne ha (Ne.symm hne)
+  obtain ⟨r,hr⟩ := Submodule.mem_span_singleton.mp h
+  exact (linearIndependent_fin2.mp hi).2 r hr
+
+ theorem bottStalk_smul_injective (l : List I) (x : W)
+    {r : SymmetricCoefficient (M := M)} (hr : r≠0) :
+    Function.Injective (fun z : bottStalk cs l x => r • z) :=
+  smul_right_injective (bottStalk cs l x) hr
+
+ instance bottStalkFinite (l : List I) (x : W) :
+    Module.Finite (SymmetricCoefficient (M := M)) (bottStalk cs l x) :=
+  Module.Finite.of_surjective (bottStalkProjection cs l x) (bottStalkProjection_surjective cs l x)
+
+ theorem bottReflectionEdge_annihilated (l : List I) (x t : W) {a : I → ℝ}
+    (ha : Represents M cs t a)
+    (z : PairProjection.Edge (bottStalkProjection cs l x) (bottStalkProjection cs l (t*x))) :
+    SymmetricAlgebra.ι ℝ (Extended M) (embed M a) • z=0 := by
+  obtain ⟨p,hp⟩ := reflectionCoefficient_difference cs ha x
+  rw [← hp]
+  exact PairProjection.difference_annihilates _ _
+    (bottStalkProjection_surjective cs _ _) (bottStalkProjection_surjective cs _ _)
+    (bottSamelsonRight cs l p) _ _
+    (bottStalkProjection_right cs _ _ p) (bottStalkProjection_right cs _ _ p) z
+
+ theorem BottKernelExact.upper_kernel {l : List I} (H : BottKernelExact cs l)
+    (x t : W) {a : I → ℝ} (ha : Represents M cs t a)
+    (hlt : cs.length x < cs.length (t*x)) (z : bottStalk cs l (t*x)) :
+    PairProjection.upper (bottStalkProjection cs l x) (bottStalkProjection cs l (t*x)) z=0 ↔
+      ∃ w, z=SymmetricAlgebra.ι ℝ (Extended M) (embed M a) • w :=
+  PairProjection.upper_kernel_of_divisibility _ _ _ (H x t a ha hlt)
+    (bottReflectionEdge_annihilated cs l x t ha) z
+
+ theorem BottKernelExact.kernel_iff {l : List I} (H : BottKernelExact cs l)
+    (x t : W) {a : I → ℝ} (ha : Represents M cs t a)
+    (hlt : cs.length x < cs.length (t*x)) (z : bottStalk cs l (t*x)) :
+    (∃ m, bottStalkProjection cs l x m=0 ∧ bottStalkProjection cs l (t*x) m=z) ↔
+      ∃ w, z=SymmetricAlgebra.ι ℝ (Extended M) (embed M a) • w := by
+  rw [← PairProjection.kernel_upper]
+  exact H.upper_kernel cs x t ha hlt z
+
+ theorem BottKernelExact.kernel_iff_of_mul {l : List I} (H : BottKernelExact cs l)
+    (x y t : W) {a : I → ℝ} (ha : Represents M cs t a) (hxy : t*x=y)
+    (hlt : cs.length x < cs.length y) (z : bottStalk cs l y) :
+    (∃ m, bottStalkProjection cs l x m=0 ∧ bottStalkProjection cs l y m=z) ↔
+      ∃ w, z=SymmetricAlgebra.ι ℝ (Extended M) (embed M a) • w := by
+  subst y
+  exact H.kernel_iff cs x t ha hlt z
+
+ theorem BottKernelExact.projection_of_mul {l : List I} (H : BottKernelExact cs l)
+    (x y t : W) {a : I → ℝ} (ha : Represents M cs t a) (hxy : t*x=y)
+    (hlt : cs.length x < cs.length y) (m : BottSamelsonModule cs l)
+    (hm : bottStalkProjection cs l x m=0) :
+    ∃ z : bottStalk cs l y, bottStalkProjection cs l y m=
+      SymmetricAlgebra.ι ℝ (Extended M) (embed M a) • z := by
+  subst y
+  exact H x t a ha hlt m hm
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+namespace KLInvariance.TitsSpace
+open Module ReflectionFrobenius ReflectionPairTranslation
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+ theorem bottSimpleEdge_regular (i : I) (l : List I)
+    (H : BottKernelExact cs l)
+    (F : ∀ x : W, Module.Free (SymmetricCoefficient (M := M)) (bottStalk cs l x))
+    (x : W) (β : SymmetricCoefficient (M := M)) (hβ : ¬simpleRootPolynomial (M := M) i ∣ β) :
+    Function.Injective (fun e : PairProjection.Edge (bottStalkProjection cs l x)
+      (bottStalkProjection cs l (cs.simple i*x)) => β • e) := by
+  have hα : Prime (simpleRootPolynomial (M := M) i) :=
+    represents_polynomial_prime cs (simple_represents cs i)
+  rcases lt_or_gt_of_ne (cs.length_simple_mul_ne x i) with hdesc | hasc
+  · let := F x
+    apply FiniteFreeScalar.scalar_quotient_regular
+      (Module.Free.chooseBasis (SymmetricCoefficient (M := M)) (bottStalk cs l x))
+      (PairProjection.lower (bottStalkProjection cs l x) (bottStalkProjection cs l (cs.simple i*x)))
+      (PairProjection.lower_surjective _ _ (bottStalkProjection_surjective cs l (cs.simple i*x)))
+      (simpleRootPolynomial (M := M) i) β hα hβ
+    intro z
+    rw [PairProjection.kernel_lower]
+    have hh := H.kernel_iff_of_mul cs (cs.simple i*x) x (cs.simple i)
+      (simple_represents cs i) (cs.simple_mul_simple_cancel_left i) hdesc z
+    simpa only [and_comm,simpleRootPolynomial,extendedRoot] using hh
+  · let := F (cs.simple i*x)
+    exact FiniteFreeScalar.scalar_quotient_regular
+      (Module.Free.chooseBasis (SymmetricCoefficient (M := M)) (bottStalk cs l (cs.simple i*x)))
+      (PairProjection.upper (bottStalkProjection cs l x) (bottStalkProjection cs l (cs.simple i*x)))
+      (PairProjection.upper_surjective _ _ (bottStalkProjection_surjective cs l x))
+      (simpleRootPolynomial (M := M) i) β hα hβ
+      (H.upper_kernel cs x (cs.simple i) (simple_represents cs i) hasc)
+
+ theorem bottHeadPairLattice_saturated (i : I) (l : List I)
+    (H : BottKernelExact cs l)
+    (F : ∀ x : W, Module.Free (SymmetricCoefficient (M := M)) (bottStalk cs l x))
+    (x : W) (β : SymmetricCoefficient (M := M)) (hβ : ¬simpleRootPolynomial (M := M) i ∣ β)
+    (p : bottStalk cs l x × Twist (coefficientAction cs (cs.simple i)).toRingEquiv
+      (bottStalk cs l (cs.simple i*x))) :
+    β • p ∈ bottHeadPairLattice cs i l x ↔ p ∈ bottHeadPairLattice cs i l x :=
+  lattice_saturated (bottStalkProjection cs l x) (bottStalkProjection cs l (cs.simple i*x))
+    (coefficientAction cs (cs.simple i)).toRingEquiv (simpleRootPolynomial (M := M) i)
+    (simpleReverseDivisibility cs i) (bottSimpleEdge_kernel_inclusion cs i l x)
+    β (bottSimpleEdge_annihilated cs i l x) (bottSimpleEdge_regular cs i l H F x β hβ) p
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+
+namespace KLInvariance.TitsSpace
+open Module ReflectionFrobenius ReflectionPairTranslation
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+ theorem bottHeadStalk_divisible (i : I) (l : List I)
+    (H : BottKernelExact cs l)
+    (F : ∀ x : W, Module.Free (SymmetricCoefficient (M := M)) (bottStalk cs l x))
+    (x : W) (m : BottSamelsonModule cs (i::l)) (β : SymmetricCoefficient (M := M))
+    (hβ : ¬simpleRootPolynomial (M := M) i ∣ β)
+    (p : bottStalk cs l x × Twist (coefficientAction cs (cs.simple i)).toRingEquiv
+      (bottStalk cs l (cs.simple i*x)))
+    (hm : bottHeadLocalPair cs i l x m=β • p) :
+    ∃ z : bottStalk cs (i::l) x, bottStalkProjection cs (i::l) x m=β • z := by
+  have hp := bottHeadLocalPair_mem cs i l x m
+  have hp' : β • p ∈ bottHeadPairLattice cs i l x := hm ▸ hp
+  exact bottHeadStalk_divisible_of_pair cs i l x m β p hm
+    ((bottHeadPairLattice_saturated cs i l H F x β hβ p).mp hp')
+
+ theorem bottKernelExact_cons_nonhead (i : I) (l : List I)
+    (H : BottKernelExact cs l)
+    (F : ∀ x : W, Module.Free (SymmetricCoefficient (M := M)) (bottStalk cs l x))
+    (x t : W) (a : I → ℝ) (ha : Represents M cs t a)
+    (hlt : cs.length x < cs.length (t*x)) (hts : t≠cs.simple i)
+    (m : BottSamelsonModule cs (i::l)) (hm : bottStalkProjection cs (i::l) x m=0) :
+    ∃ z : bottStalk cs (i::l) (t*x), bottStalkProjection cs (i::l) (t*x) m=
+      SymmetricAlgebra.ι ℝ (Extended M) (embed M a) • z := by
+  have he : t*x≠cs.simple i*x := fun h => hts (mul_right_cancel h)
+  have hstep : BruhatStep cs x (t*x) := ⟨hlt,t,ha.isReflection,rfl⟩
+  have hstep' := bruhatStep_simple_mul cs hstep i he
+  have hconj : (cs.simple i*t*(cs.simple i)⁻¹)*(cs.simple i*x)=cs.simple i*(t*x) := by group
+  have hzero := (bottHeadLocalPair_zero_iff cs i l x m).mp hm
+  have hμ : bottStalkProjection cs l x (bottHeadTail cs i l m)=0 := congrArg Prod.fst hzero
+  have hν : bottStalkProjection cs l (cs.simple i*x) (bottHeadReflectedTail cs i l m).val=0 :=
+    congrArg (fun p => p.2.val) hzero
+  obtain ⟨p,hp⟩ := H x t a ha hlt (bottHeadTail cs i l m) hμ
+  obtain ⟨q,hq⟩ := H.projection_of_mul cs (cs.simple i*x) (cs.simple i*(t*x))
+    (cs.simple i*t*(cs.simple i)⁻¹) (ha.act (cs.simple i)) hconj hstep'.1
+    (bottHeadReflectedTail cs i l m).val hν
+  let β : SymmetricCoefficient (M := M) := SymmetricAlgebra.ι ℝ (Extended M) (embed M a)
+  let p' : bottStalk cs l (t*x) × Twist (coefficientAction cs (cs.simple i)).toRingEquiv
+      (bottStalk cs l (cs.simple i*(t*x))) := (p,⟨q⟩)
+  have hβ' : coefficientAction cs (cs.simple i) β =
+      SymmetricAlgebra.ι ℝ (Extended M) (embed M (action M cs (cs.simple i) a)) := by
+    rw [coefficientAction_ι,extendedAction_restrict]
+  have heq : bottHeadLocalPair cs i l (t*x) m=β • p' := by
+    apply Prod.ext
+    · exact hp
+    · apply Twist.ext
+      change bottStalkProjection cs l (cs.simple i*(t*x))
+        (bottHeadReflectedTail cs i l m).val = coefficientAction cs (cs.simple i) β • q
+      rw [hβ']
+      exact hq
+  have hn : ¬simpleRootPolynomial (M := M) i ∣ β :=
+    represents_polynomial_not_dvd cs (simple_represents cs i) ha (Ne.symm hts)
+  exact bottHeadStalk_divisible cs i l H F (t*x) m β hn p' heq
+
+ theorem bottKernelExact_cons (i : I) (l : List I)
+    (H : BottKernelExact cs l)
+    (F : ∀ x : W, Module.Free (SymmetricCoefficient (M := M)) (bottStalk cs l x)) :
+    BottKernelExact cs (i::l) := by
+  intro x t a ha hlt m hm
+  by_cases hts : t=cs.simple i
+  · subst t
+    obtain ⟨z,hz⟩ := bottSimpleWall_kernel_projection cs i l x hm
+    rcases ha.unique_up_to_sign (simple_represents cs i) with h|h
+    · subst a
+      exact ⟨z,hz⟩
+    · subst a
+      refine ⟨-z,?_⟩
+      rw [hz]
+      change simpleRootPolynomial (M := M) i • z =
+        SymmetricAlgebra.ι ℝ (Extended M) (embed M (-root i)) • (-z)
+      have he : embed M (-root i) = -embed M (root i) := by simp [embed]
+      have hn : SymmetricAlgebra.ι ℝ (Extended M) (embed M (-root i)) =
+          -simpleRootPolynomial (M := M) i := by
+        rw [he]
+        exact (SymmetricAlgebra.ι ℝ (Extended M)).map_neg _
+      rw [hn]
+      exact (neg_smul_neg _ z).symm
+  · exact bottKernelExact_cons_nonhead cs i l H F x t a ha hlt hts m hm
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+namespace KLInvariance.ReflectionPairTranslation
+universe u v w z
+variable {R : Type u} [CommRing R]
+  {N : Type v} [AddCommGroup N] [Module R N]
+  {X : Type w} [AddCommGroup X] [Module R X]
+  {Y : Type z} [AddCommGroup Y] [Module R Y]
+  (f : N →ₗ[R] X) (g : N →ₗ[R] Y) (τ : R ≃+* R) (α : R)
+  (hdiv : ∀ r : R, α ∣ τ r-r)
+  (hinc : ∀ y : Y, ∃ n : N, f n=0 ∧ g n=α • y)
+  (hker : ∀ x : X, (∃ n : N, f n=x ∧ g n=0) ↔ ∃ z : X, x=α • z)
+
+ def right : lattice f g τ α hdiv hinc →ₗ[R] Twist τ Y :=
+  (LinearMap.snd R X (Twist τ Y)).comp (lattice f g τ α hdiv hinc).subtype
+
+ theorem right_surjective (hg : Function.Surjective g) :
+    Function.Surjective (right f g τ α hdiv hinc) := by
+  intro y
+  obtain ⟨n,hn⟩ := hg y.val
+  exact ⟨⟨(f n,y),n,rfl,hn⟩,rfl⟩
+
+ def rightKernelMap : X →ₗ[R] LinearMap.ker (right f g τ α hdiv hinc) where
+  toFun x := ⟨⟨(α • x,⟨0⟩),(hker (α • x)).mpr ⟨x,rfl⟩⟩,rfl⟩
+  map_add' x y := by
+    apply Subtype.ext
+    apply Subtype.ext
+    apply Prod.ext
+    · exact smul_add α x y
+    · apply Twist.ext
+      exact (zero_add (0 : Y)).symm
+  map_smul' r x := by
+    apply Subtype.ext
+    apply Subtype.ext
+    apply Prod.ext
+    · exact smul_comm α r x
+    · apply Twist.ext
+      exact (smul_zero _).symm
+
+ theorem rightKernelMap_surjective :
+    Function.Surjective (rightKernelMap f g τ α hdiv hinc hker) := by
+  rintro ⟨⟨⟨x,y⟩,n,hn,hn'⟩,hy⟩
+  have hy' : y.val=0 := congrArg Twist.val hy
+  obtain ⟨z,hz⟩ := (hker x).mp ⟨n,hn,hn'.trans hy'⟩
+  refine ⟨z,?_⟩
+  apply Subtype.ext
+  apply Subtype.ext
+  exact Prod.ext hz.symm (Twist.ext hy'.symm)
+
+ theorem rightKernelMap_injective (hinj : Function.Injective (fun x : X => α • x)) :
+    Function.Injective (rightKernelMap f g τ α hdiv hinc hker) := by
+  intro x y h
+  exact hinj (congrArg (fun p => p.val.val.1) h)
+
+ noncomputable def rightKernelEquiv (hinj : Function.Injective (fun x : X => α • x)) :
+    X ≃ₗ[R] LinearMap.ker (right f g τ α hdiv hinc) :=
+  LinearEquiv.ofBijective (rightKernelMap f g τ α hdiv hinc hker)
+    ⟨rightKernelMap_injective f g τ α hdiv hinc hker hinj,
+      rightKernelMap_surjective f g τ α hdiv hinc hker⟩
+
+ include hker in
+ theorem lattice_free_right (hg : Function.Surjective g)
+    (hinj : Function.Injective (fun x : X => α • x))
+    [Module.Free R X] [Module.Free R Y] :
+    Module.Free R (lattice f g τ α hdiv hinc) := by
+  let : Module.Free R (LinearMap.ker (right f g τ α hdiv hinc)) :=
+    Module.Free.of_equiv (rightKernelEquiv f g τ α hdiv hinc hker hinj)
+  exact free_of_surjective_of_ker_free (right f g τ α hdiv hinc)
+    (right_surjective f g τ α hdiv hinc hg)
+end KLInvariance.ReflectionPairTranslation
+
+end
+
+
+section
+
+namespace KLInvariance.TitsSpace
+open Module ReflectionFrobenius ReflectionPairTranslation
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+ theorem bottStalk_free_cons (i : I) (l : List I)
+    (H : BottKernelExact cs l)
+    (F : ∀ x : W, Module.Free (SymmetricCoefficient (M := M)) (bottStalk cs l x))
+    (x : W) : Module.Free (SymmetricCoefficient (M := M)) (bottStalk cs (i::l) x) := by
+  let := F x
+  let := F (cs.simple i*x)
+  have hα : simpleRootPolynomial (M := M) i ≠ 0 :=
+    (represents_polynomial_prime cs (simple_represents cs i)).ne_zero
+  have hlattice : Module.Free (SymmetricCoefficient (M := M)) (bottHeadPairLattice cs i l x) := by
+    rcases lt_or_gt_of_ne (cs.length_simple_mul_ne x i) with hdesc | hasc
+    · apply lattice_free_right
+        (bottStalkProjection cs l x) (bottStalkProjection cs l (cs.simple i*x))
+        (coefficientAction cs (cs.simple i)).toRingEquiv (simpleRootPolynomial (M := M) i)
+        (simpleReverseDivisibility cs i) (bottSimpleEdge_kernel_inclusion cs i l x)
+      · intro z
+        have hh := H.kernel_iff_of_mul cs (cs.simple i*x) x (cs.simple i)
+          (simple_represents cs i) (cs.simple_mul_simple_cancel_left i) hdesc z
+        simpa only [and_comm,simpleRootPolynomial,extendedRoot] using hh
+      · exact bottStalkProjection_surjective cs l (cs.simple i*x)
+      · exact bottStalk_smul_injective cs l x hα
+    · exact lattice_free (bottStalkProjection cs l x) (bottStalkProjection cs l (cs.simple i*x))
+        (coefficientAction cs (cs.simple i)).toRingEquiv (simpleRootPolynomial (M := M) i)
+        (simpleReverseDivisibility cs i) (bottSimpleEdge_kernel_inclusion cs i l x)
+        (H.kernel_iff cs x (cs.simple i) (simple_represents cs i) hasc)
+        (bottStalkProjection_surjective cs l x)
+        (bottStalk_smul_injective cs l (cs.simple i*x) hα)
+  let := hlattice
+  exact Module.Free.of_equiv (bottHeadPairEquiv cs i l x)
+
+ theorem bottStalk_nil_subsingleton (x : W) (hx : x≠1) : Subsingleton (bottStalk cs [] x) := by
+  constructor
+  intro p q
+  apply Subtype.ext
+  funext e
+  exact False.elim (hx e.property.symm)
+
+ theorem bottStalk_nil_free (x : W) :
+    Module.Free (SymmetricCoefficient (M := M)) (bottStalk cs [] x) := by
+  by_cases hx : x=1
+  · subst x
+    exact Module.Free.of_equiv (bottTopStalkEquiv cs [] (by simp [CoxeterSystem.IsReduced])).symm
+  · let := bottStalk_nil_subsingleton cs x hx
+    infer_instance
+
+ theorem bottKernelExact_nil : BottKernelExact cs [] := by
+  intro x t a ha hlt m hm
+  have hx : t*x≠1 := by
+    intro h
+    rw [h] at hlt
+    rw [cs.length_one] at hlt
+    exact Nat.not_lt_zero _ hlt
+  let := bottStalk_nil_subsingleton cs (t*x) hx
+  exact ⟨0,Subsingleton.elim _ _⟩
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Pair-projection edges of actual finite graded modules. The hypotheses say
+only that the two actual weight projections are homogeneous and surjective. -/
+namespace KLInvariance.Graded.ModuleData
+open PairProjection
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} {M X Y : ModuleData.{uk,ua,um} 𝓐}
+  (f : Hom M X) (g : Hom M Y)
+noncomputable section
+
+ theorem pairLower_kernel_homogeneous (n : ℤ) (z : X)
+    (hz : lower f.map g.map z=0) :
+    lower f.map g.map (component X.piece n z)=0 := by
+  obtain ⟨m,hm,hy⟩ := (kernel_lower f.map g.map z).mp hz
+  apply (kernel_lower f.map g.map _).mpr
+  refine ⟨component M.piece n m,?_,?_⟩
+  · rw [← f.component,hm]
+  · rw [← g.component,hy,map_zero]
+
+ def pairEdgePiece (n : ℤ) : Submodule k (Edge f.map g.map) :=
+  (X.piece n).map ((lower f.map g.map).restrictScalars k)
+
+ def pairEdge (hg : Function.Surjective g.map) : ModuleData.{uk,ua,um} 𝓐 where
+  obj := ModuleCat.of A (Edge f.map g.map)
+  kModule := inferInstance
+  tower := inferInstance
+  piece := pairEdgePiece f g
+  decomposition := (image_isInternal X.piece ((lower f.map g.map).restrictScalars k)
+    (lower_surjective f.map g.map hg) (pairLower_kernel_homogeneous f g)).chooseDecomposition
+  graded := ⟨by
+    rintro i j a z ha ⟨w,hw,rfl⟩
+    exact ⟨a • w,SetLike.GradedSMul.smul_mem ha hw,map_smul (lower f.map g.map) a w⟩⟩
+  nonneg := by
+    intro n hn
+    unfold pairEdgePiece
+    rw [X.nonneg n hn,Submodule.map_bot]
+  finite := Module.Finite.of_surjective (lower f.map g.map) (lower_surjective f.map g.map hg)
+
+ def pairLower (hg : Function.Surjective g.map) : Hom X (pairEdge f g hg) where
+  map := lower f.map g.map
+  graded := fun _ z hz => ⟨z,hz,rfl⟩
+
+ theorem Hom.homogeneous_lift (hg : Function.Surjective g.map) (n : ℤ)
+    (y : Y) (hy : y ∈ Y.piece n) : ∃ m : M, m ∈ M.piece n ∧ g.map m=y := by
+  obtain ⟨m,hm⟩ := hg y
+  refine ⟨Graded.component M.piece n m,(DirectSum.decompose M.piece m n).property,?_⟩
+  rw [← g.component,hm]
+  exact DirectSum.decompose_of_mem_same Y.piece hy
+
+ def pairUpper (hg : Function.Surjective g.map) : Hom Y (pairEdge f g hg) where
+  map := upper f.map g.map
+  graded := by
+    intro n y hy
+    obtain ⟨m,hm,rfl⟩ := g.homogeneous_lift hg n y hy
+    exact ⟨f.map m,f.graded n m hm,compatible f.map g.map m⟩
+
+end
+end KLInvariance.Graded.ModuleData
+
+end
+
+
+section
+
+/-! The genuine projection WORD sheaf with its nonnegative grading and finite
+integral stalk/edge modules. No free-stalk, flabbiness, or KL character claim
+is smuggled into the grading. The universe here contains both the finite
+Coxeter matrix and its standard Coxeter group, as needed by the main reduction. -/
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
+universe u us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+omit [Fintype σ] in
+ theorem bottStalkPiece_negative (l : List I) (x : W) (n : ℤ) (hn : n<0) :
+    bottStalkPiece cs basis l x n=⊥ := by
+  apply subPiece_eq_bot
+  exact piPiece_eq_bot (fun _ => symmetricGrading basis) n
+    (fun _ => symmetricGrading_negative basis n hn)
+
+ def bottStalkData (l : List I) (x : W) : ModuleData.{0,u,u} (symmetricGrading basis) where
+  obj := ModuleCat.of _ (bottStalk cs l x)
+  kModule := inferInstance
+  tower := inferInstance
+  piece := bottStalkPiece cs basis l x
+  decomposition := inferInstance
+  graded := inferInstance
+  nonneg := bottStalkPiece_negative cs basis l x
+  finite := Module.Finite.of_surjective (bottStalkProjection cs l x)
+    (bottStalkProjection_surjective cs l x)
+
+ theorem bottPiece_negative (l : List I) (n : ℤ) (hn : n<0) :
+    bottPiece cs basis l n=⊥ := by
+  apply bot_unique
+  intro m hm
+  change m=0
+  apply bottEvaluation_injective cs l
+  funext e
+  have h := bottEvaluation_homogeneous cs basis l e n m hm
+  rw [symmetricGrading_negative basis n hn,Submodule.mem_bot] at h
+  exact h.trans (map_zero (bottEvaluation cs l e)).symm
+
+ def bottWordData (l : List I) : ModuleData.{0,u,u} (symmetricGrading basis) where
+  obj := ModuleCat.of _ (BottSamelsonModule cs l)
+  kModule := inferInstance
+  tower := inferInstance
+  piece := bottPiece cs basis l
+  decomposition := inferInstance
+  graded := inferInstance
+  nonneg := bottPiece_negative cs basis l
+  finite := inferInstance
+
+ def bottStalkHom (l : List I) (x : W) :
+    ModuleData.Hom (bottWordData cs basis l) (bottStalkData cs basis l x) where
+  map := bottStalkProjection cs l x
+  graded := bottStalkProjection_homogeneous cs basis l x
+
+ theorem bottStalkHom_surjective (l : List I) (x : W) :
+    Function.Surjective (bottStalkHom cs basis l x).map :=
+  bottStalkProjection_surjective cs l x
+
+ def bottEdgeData (l : List I) (x y : W) : ModuleData.{0,u,u} (symmetricGrading basis) :=
+  ModuleData.pairEdge (bottStalkHom cs basis l x) (bottStalkHom cs basis l y)
+    (bottStalkHom_surjective cs basis l y)
+
+ def bottLowerHom (l : List I) (x y : W) :
+    ModuleData.Hom (bottStalkData cs basis l x) (bottEdgeData cs basis l x y) :=
+  ModuleData.pairLower (bottStalkHom cs basis l x) (bottStalkHom cs basis l y)
+    (bottStalkHom_surjective cs basis l y)
+
+ def bottUpperHom (l : List I) (x y : W) :
+    ModuleData.Hom (bottStalkData cs basis l y) (bottEdgeData cs basis l x y) :=
+  ModuleData.pairUpper (bottStalkHom cs basis l x) (bottStalkHom cs basis l y)
+    (bottStalkHom_surjective cs basis l y)
+
+ def bottGradedProjectionSheaf (l : List I) (u b : W) :
+    GradedSheaf (𝓐 := symmetricGrading basis) (graph cs u b) where
+  vertex x := bottStalkData cs basis l x.val
+  edge e := bottEdgeData cs basis l (source cs u b e).val (target cs u b e).val
+  lower e := bottLowerHom cs basis l (source cs u b e).val (target cs u b e).val
+  upper e := bottUpperHom cs basis l (source cs u b e).val (target cs u b e).val
+
+ theorem bottGradedProjectionSheaf_forget (l : List I) (u b : W) :
+    (bottGradedProjectionSheaf cs basis l u b).forget=bottProjectionSheaf cs l u b := rfl
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Unconditional integral free stalks and exact oriented edge kernels for
+all tensor words. These are the local parts of BMP recognition; flabbiness
+and the KL character are not asserted here. -/
+namespace KLInvariance.TitsSpace
+open Module MomentGraph BruhatGraph
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+ theorem bottStandardStalks (l : List I) :
+    BottKernelExact cs l ∧ ∀ x : W,
+      Module.Free (SymmetricCoefficient (M := M)) (bottStalk cs l x) := by
+  induction l with
+  | nil => exact ⟨bottKernelExact_nil cs,bottStalk_nil_free cs⟩
+  | cons i l h => exact ⟨bottKernelExact_cons cs i l h.1 h.2,
+      bottStalk_free_cons cs i l h.1 h.2⟩
+
+ theorem bottKernelExact (l : List I) : BottKernelExact cs l := (bottStandardStalks cs l).1
+
+ instance bottStalkFree (l : List I) (x : W) :
+    Module.Free (SymmetricCoefficient (M := M)) (bottStalk cs l x) :=
+  (bottStandardStalks cs l).2 x
+
+ theorem bottProjectionSheaf_upper_kernel (l : List I) (u b : W) (e : Edge cs u b)
+    {a : I → ℝ} (ha : Represents M cs (label cs u b e) a)
+    (z : (bottProjectionSheaf cs l u b).vertex (target cs u b e)) :
+    (bottProjectionSheaf cs l u b).upper e z=0 ↔
+      ∃ w, z=SymmetricAlgebra.ι ℝ (Extended M) (embed M a) • w := by
+  change PairProjection.upper (bottStalkProjection cs l (source cs u b e).val)
+      (bottStalkProjection cs l (target cs u b e).val) z=0 ↔ _
+  exact (PairProjection.kernel_upper _ _ z).trans
+    ((bottKernelExact cs l).kernel_iff_of_mul cs _ _ _ ha
+      (label_mul_source cs u b e) e.property.1 z)
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Transport of actual linear-map kernels along a source isomorphism. -/
+namespace KLInvariance
+universe ur um un up
+variable {R : Type ur} [Ring R]
+  {M : Type um} [AddCommGroup M] [Module R M]
+  {N : Type un} [AddCommGroup N] [Module R N]
+  {P : Type up} [AddCommGroup P] [Module R P]
+
+def kernelSourceEquiv (f : M →ₗ[R] P) (e : M ≃ₗ[R] N) :
+    LinearMap.ker f ≃ₗ[R] LinearMap.ker (f.comp e.symm.toLinearMap) where
+  toFun x := ⟨e x.val,by simpa only [LinearMap.mem_ker,LinearMap.comp_apply,
+    LinearEquiv.coe_coe,LinearEquiv.symm_apply_apply] using x.property⟩
+  invFun y := ⟨e.symm y.val,y.property⟩
+  left_inv x := Subtype.ext (e.symm_apply_apply x.val)
+  right_inv y := Subtype.ext (e.apply_symm_apply y.val)
+  map_add' x y := Subtype.ext (map_add e x.val y.val)
+  map_smul' r x := Subtype.ext (map_smul e r x.val)
+
+end KLInvariance
+
+end
+
+
+section
+
+/-! The actual ascent short exact sequence of word stalks has kernel the
+upper tail stalk shifted by one. This is the degree-one Hecke normalization. -/
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded ReflectionPairTranslation
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ def bottHeadKernelEquiv (i : I) (l : List I) (x : W)
+    (h : cs.length x<cs.length (cs.simple i*x)) :
+    bottStalk cs l (cs.simple i*x) ≃ₗ[ℝ] LinearMap.ker (bottHeadLeft cs i l x) :=
+  (Twist.scalarEquiv (coefficientAction cs (cs.simple i)) _).symm.trans
+    (((kernelEquiv (bottStalkProjection cs l x) (bottStalkProjection cs l (cs.simple i*x))
+      (coefficientAction cs (cs.simple i)).toRingEquiv (simpleRootPolynomial (M := M) i)
+      (simpleReverseDivisibility cs i) (bottSimpleEdge_kernel_inclusion cs i l x)
+      ((bottKernelExact cs l).kernel_iff cs x (cs.simple i) (simple_represents cs i) h)
+      (bottStalk_smul_injective cs l (cs.simple i*x) (simpleRootPolynomial_ne_zero i))).trans
+      (kernelSourceEquiv _ (bottHeadPairEquiv cs i l x))).restrictScalars ℝ)
+
+ theorem bottHeadKernelEquiv_eval (i : I) (l : List I) (x : W)
+    (h : cs.length x<cs.length (cs.simple i*x))
+    (y : bottStalk cs l (cs.simple i*x))
+    (e : {e : BottIndex (i::l) // bottWeight cs (i::l) e=x}) :
+    (bottHeadKernelEquiv cs i l x h y).val.val e=
+      if he : e.val.1=0 then 0 else coefficientAction cs (cs.simple i)
+        (simpleRootPolynomial (M := M) i * y.val ⟨e.val.2,bottWeight_tail_one cs i l x e he⟩) := rfl
+
+ def bottHeadKernelPiece (i : I) (l : List I) (x : W) :
+    ℤ → Submodule ℝ (LinearMap.ker (bottHeadLeft cs i l x)) :=
+  subPiece (bottStalkPiece cs basis (i::l) x) (LinearMap.ker (bottHeadLeft cs i l x))
+
+ theorem bottHeadLeft_kernel_homogeneous (i : I) (l : List I) (x : W) :
+    ∀ n m, m ∈ LinearMap.ker (bottHeadLeft cs i l x) →
+      component (bottStalkPiece cs basis (i::l) x) n m ∈ LinearMap.ker (bottHeadLeft cs i l x) := by
+  intro n m hm
+  have h := map_component (bottStalkPiece cs basis (i::l) x) (bottStalkPiece cs basis l x)
+    (bottHeadLeft cs i l x) 0 (by
+      intro j z hz
+      simpa only [add_zero] using (bottHeadLeft_homogeneous cs basis i l x j z hz)) n m
+  change bottHeadLeft cs i l x _=0
+  rw [sub_zero] at h
+  rw [← h,LinearMap.mem_ker.mp hm,map_zero]
+
+ instance bottHeadKernelPiece_decomposition (i : I) (l : List I) (x : W) :
+    DirectSum.Decomposition (bottHeadKernelPiece cs basis i l x) :=
+  (sub_isInternal (bottStalkPiece cs basis (i::l) x) (LinearMap.ker (bottHeadLeft cs i l x))
+    (bottHeadLeft_kernel_homogeneous cs basis i l x)).chooseDecomposition
+
+ theorem bottHeadKernelEquiv_homogeneous (i : I) (l : List I) (x : W)
+    (h : cs.length x<cs.length (cs.simple i*x)) (n : ℤ)
+    (y : bottStalk cs l (cs.simple i*x)) (hy : y ∈ bottStalkPiece cs basis l (cs.simple i*x) n) :
+    bottHeadKernelEquiv cs i l x h y ∈ bottHeadKernelPiece cs basis i l x (n+1) := by
+  intro e _
+  change (bottHeadKernelEquiv cs i l x h y).val.val e ∈ symmetricGrading basis (n+1)
+  rw [bottHeadKernelEquiv_eval]
+  split_ifs with he
+  · exact Submodule.zero_mem _
+  · apply coefficientAction_homogeneous cs basis
+    have hy' : y.val ⟨e.val.2,bottWeight_tail_one cs i l x e he⟩ ∈ symmetricGrading basis n :=
+      hy ⟨e.val.2,bottWeight_tail_one cs i l x e he⟩ (Set.mem_univ _)
+    have hp := SetLike.GradedMul.mul_mem (simpleRoot_homogeneous basis i) hy'
+    simpa only [add_comm] using hp
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! A homogeneous integral equivalence induces genuine shifted degreewise
+vector-space equivalences, including its inverse grading. -/
+namespace KLInvariance.Graded
+universe uk um un
+variable {k : Type uk} [Field k]
+  {M : Type um} [AddCommGroup M] [Module k M]
+  {N : Type un} [AddCommGroup N] [Module k N]
+  (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
+  [DirectSum.Decomposition 𝓜] [DirectSum.Decomposition 𝓝]
+  (e : M ≃ₗ[k] N) (d : ℤ)
+  (he : ∀ n m, m ∈ 𝓜 n → e m ∈ 𝓝 (n+d))
+noncomputable section
+
+ def homogeneousPieceEquiv (n : ℤ) : 𝓜 n ≃ₗ[k] 𝓝 (n+d) where
+  toFun m := ⟨e m.val,he n m.val m.property⟩
+  invFun v := ⟨e.symm v.val,by
+    simpa only [add_sub_cancel_right] using
+      equiv_inverse_homogeneous 𝓜 𝓝 e d he (n+d) v.val v.property⟩
+  left_inv v := Subtype.ext (e.symm_apply_apply v.val)
+  right_inv v := Subtype.ext (e.apply_symm_apply v.val)
+  map_add' v w := Subtype.ext (map_add e v.val w.val)
+  map_smul' r v := Subtype.ext (map_smul e r v.val)
+
+include e he in
+ theorem finrank_homogeneousEquiv (n : ℤ) :
+    Module.finrank k (𝓝 n)=Module.finrank k (𝓜 (n-d)) := by
+  have h := (homogeneousPieceEquiv 𝓜 𝓝 e d he (n-d)).finrank_eq
+  have hn : n-d+d=n := by omega
+  rw [hn] at h
+  exact h.symm
+
+end
+end KLInvariance.Graded
+
+end
+
+
+section
+
+/-! Finiteness of true submodule and finite product homogeneous components. -/
+namespace KLInvariance.Graded
+universe uk ua um ui
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {M : Type um} [AddCommGroup M] [Module k M] [Module A M] [IsScalarTower k A M]
+
+ instance subPiece_finite (𝓜 : ℤ → Submodule k M) (S : Submodule A M) (n : ℤ)
+    [Module.Finite k (𝓜 n)] : Module.Finite k (subPiece 𝓜 S n) := by
+  let f : subPiece 𝓜 S n →ₗ[k] 𝓜 n :=
+    { toFun := fun v => ⟨v.val.val,v.property⟩
+      map_add' := fun _ _ => rfl
+      map_smul' := fun _ _ => rfl }
+  apply Module.Finite.of_injective f
+  intro v w h
+  apply Subtype.ext
+  apply Subtype.ext
+  exact congrArg (fun z : 𝓜 n => z.val) h
+
+ variable {ι : Type ui} [Finite ι] {P : ι → Type um}
+   [∀ i, AddCommGroup (P i)] [∀ i, Module k (P i)]
+
+ instance piPiece_finite (𝓟 : ∀ i, ℤ → Submodule k (P i)) (n : ℤ)
+    [∀ i, Module.Finite k (𝓟 i n)] : Module.Finite k (piPiece 𝓟 n) := by
+  let f : piPiece 𝓟 n →ₗ[k] (∀ i, 𝓟 i n) :=
+    { toFun := fun v i => ⟨v.val i,v.property i (Set.mem_univ i)⟩
+      map_add' := fun _ _ => rfl
+      map_smul' := fun _ _ => rfl }
+  apply Module.Finite.of_injective f
+  intro v w h
+  apply Subtype.ext
+  funext i
+  exact congrArg Subtype.val (congrFun h i)
+
+end KLInvariance.Graded
+
+end
+
+
+section
+
+/-! The genuine Bott--Samelson stalk Hilbert coefficients satisfy the simple
+reflection ascent recurrence. Both exactness and the degree-one kernel shift
+were constructed integrally above; no KL character theorem is assumed. -/
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ instance symmetricGrading_finite (n : ℤ) : Module.Finite ℝ (symmetricGrading basis n) :=
+  inferInstanceAs (Module.Finite ℝ ((polynomialGrading ℝ σ n).map
+    (SymmetricAlgebra.equivMvPolynomial basis).symm.toLinearMap))
+
+ instance bottCoordinatePiece_finite (l : List I) (x : W) (n : ℤ) :
+    Module.Finite ℝ (bottCoordinatePiece cs basis l x n) := by
+  classical
+  exact piPiece_finite _ n
+
+ instance bottStalkPiece_finite (l : List I) (x : W) (n : ℤ) :
+    Module.Finite ℝ (bottStalkPiece cs basis l x n) :=
+  inferInstanceAs (Module.Finite ℝ (subPiece (bottCoordinatePiece cs basis l x) (bottStalk cs l x) n))
+
+ theorem bottStalk_finrank_ascent (i : I) (l : List I) (x : W)
+    (h : cs.length x<cs.length (cs.simple i*x)) (n : ℤ) :
+    Module.finrank ℝ (bottStalkPiece cs basis (i::l) x n)=
+      Module.finrank ℝ (bottStalkPiece cs basis l x n)+
+      Module.finrank ℝ (bottStalkPiece cs basis l (cs.simple i*x) (n-1)) := by
+  have hdim := Hilbert.finrank_exact (bottStalkPiece cs basis (i::l) x)
+    (bottStalkPiece cs basis l x) (bottHeadLeft cs i l x)
+    (bottHeadLeft_homogeneous cs basis i l x) (bottHeadLeft_surjective cs i l x) n
+  have hk := finrank_homogeneousEquiv (k := ℝ)
+    (N := LinearMap.ker (bottHeadLeft cs i l x)) (bottStalkPiece cs basis l (cs.simple i*x))
+    (bottHeadKernelPiece cs basis i l x) (bottHeadKernelEquiv cs i l x h) 1
+    (bottHeadKernelEquiv_homogeneous cs basis i l x h) n
+  change Module.finrank ℝ (bottStalkPiece cs basis (i::l) x n)=
+    Module.finrank ℝ (bottHeadKernelPiece cs basis i l x n)+
+      Module.finrank ℝ (bottStalkPiece cs basis l x n) at hdim
+  rw [hk] at hdim
+  omega
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Honest change of the coefficient algebra along an algebra equivalence.
+Both scalar actions and semilinear identifications are explicit. -/
+namespace KLInvariance
+universe uk ua ub um un
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {B : Type ub} [CommRing B] [Algebra k B]
+
+@[ext] structure CoefficientChange (_e : A ≃ₐ[k] B) (M : Type um) where
+  val : M
+
+namespace CoefficientChange
+variable (e : A ≃ₐ[k] B) (M : Type um) [AddCommGroup M] [Module A M] [Module k M]
+  [IsScalarTower k A M]
+
+ def equivVal : CoefficientChange e M ≃ M where
+  toFun := val
+  invFun := mk
+  left_inv _ := rfl
+  right_inv _ := rfl
+
+ instance : AddCommGroup (CoefficientChange e M) := (equivVal e M).addCommGroup
+ instance : Module B (CoefficientChange e M) := by
+   letI : Module B M := Module.compHom M e.symm.toRingHom
+   exact ({ equivVal e M with map_add' := fun _ _ => rfl } : CoefficientChange e M ≃+ M).module B
+ instance : Module k (CoefficientChange e M) :=
+   ({ equivVal e M with map_add' := fun _ _ => rfl } : CoefficientChange e M ≃+ M).module k
+ instance : IsScalarTower k B (CoefficientChange e M) where
+   smul_assoc c b m := by
+     apply CoefficientChange.ext
+     change e.symm (c • b) • m.val=c • (e.symm b • m.val)
+     rw [map_smul,smul_assoc]
+
+ local instance : RingHomInvPair e.symm.toRingHom e.toRingHom := RingHomInvPair.of_ringEquiv_symm e.toRingEquiv
+ local instance : RingHomInvPair e.toRingHom e.symm.toRingHom := RingHomInvPair.of_ringEquiv e.toRingEquiv
+
+ def equiv : M ≃ₛₗ[e.toRingHom] CoefficientChange e M where
+   toFun := mk
+   invFun := val
+   left_inv _ := rfl
+   right_inv _ := rfl
+   map_add' _ _ := rfl
+   map_smul' a m := by
+     change mk (a • m)=mk (e.symm (e a) • m)
+     rw [e.symm_apply_apply]
+
+ def kEquiv : M ≃ₗ[k] CoefficientChange e M where
+   toFun := mk
+   invFun := val
+   left_inv _ := rfl
+   right_inv _ := rfl
+   map_add' _ _ := rfl
+   map_smul' _ _ := rfl
+
+ instance [Module.Free A M] : Module.Free B (CoefficientChange e M) :=
+   Module.Free.of_equiv (equiv e M)
+ instance [Module.Finite A M] : Module.Finite B (CoefficientChange e M) :=
+   Module.Finite.of_surjective (equiv e M).toLinearMap (equiv e M).surjective
+
+ variable {M} {N : Type un} [AddCommGroup N] [Module A N] [Module k N] [IsScalarTower k A N]
+ def map (f : M →ₗ[A] N) : CoefficientChange e M →ₗ[B] CoefficientChange e N where
+   toFun m := mk (f m.val)
+   map_add' m n := CoefficientChange.ext (map_add f m.val n.val)
+   map_smul' b m := CoefficientChange.ext (map_smul f (e.symm b) m.val)
+
+ omit [Module k M] [IsScalarTower k A M] [Module k N] [IsScalarTower k A N] in
+ theorem map_injective (f : M →ₗ[A] N) (hf : Function.Injective f) :
+     Function.Injective (map e f) := fun _m _n h => CoefficientChange.ext (hf (congrArg val h))
+ omit [Module k M] [IsScalarTower k A M] [Module k N] [IsScalarTower k A N] in
+ theorem map_surjective (f : M →ₗ[A] N) (hf : Function.Surjective f) :
+     Function.Surjective (map e f) := by
+   intro n
+   obtain ⟨m,hm⟩ := hf n.val
+   exact ⟨mk m,CoefficientChange.ext hm⟩
+
+end CoefficientChange
+end KLInvariance
+
+end
+
+
+section
+
+/-! Grading-preserving coefficient change on the actual finite modules. -/
+namespace KLInvariance.Graded.ModuleData
+universe uk ua um
+variable {k : Type uk} [Field k] {A B : Type ua} [CommRing A] [CommRing B]
+  [Algebra k A] [Algebra k B]
+  {𝓐 : ℤ → Submodule k A} {𝓑 : ℤ → Submodule k B}
+  (e : A ≃ₐ[k] B) (he : ∀ n a, a∈𝓐 n ↔ e a∈𝓑 n)
+noncomputable section
+
+ def changePiece (M : ModuleData.{uk,ua,um} 𝓐) : ℤ → Submodule k (CoefficientChange e M) :=
+  fun n => (M.piece n).map (CoefficientChange.kEquiv e M).toLinearMap
+
+ theorem mem_changePiece (M : ModuleData.{uk,ua,um} 𝓐) (n : ℤ) (m : CoefficientChange e M) :
+    m ∈ changePiece e M n ↔ m.val ∈ M.piece n := by
+  constructor
+  · rintro ⟨x,hx,hxm⟩
+    change CoefficientChange.mk x=m at hxm
+    exact (congrArg CoefficientChange.val hxm) ▸ hx
+  · intro hm
+    exact ⟨m.val,hm,rfl⟩
+
+ def change (M : ModuleData.{uk,ua,um} 𝓐) : ModuleData.{uk,ua,um} 𝓑 where
+  obj := ModuleCat.of B (CoefficientChange e M)
+  kModule := inferInstance
+  tower := inferInstance
+  piece := changePiece e M
+  decomposition := by
+    apply DirectSum.IsInternal.chooseDecomposition
+    apply image_isInternal M.piece (CoefficientChange.kEquiv e M).toLinearMap
+      (CoefficientChange.kEquiv e M).surjective
+    intro n m hm
+    have hz : m=0 := congrArg CoefficientChange.val hm
+    subst m
+    simp only [map_zero]
+  graded := by
+    constructor
+    intro i j b m hb hm
+    apply (mem_changePiece e M (i+j) _).mpr
+    change e.symm b • m.val ∈ M.piece (i+j)
+    have ha : e.symm b ∈ 𝓐 i := (he i _).mpr (by simpa using hb)
+    exact SetLike.GradedSMul.smul_mem ha ((mem_changePiece e M j m).mp hm)
+  nonneg n hn := by simp only [changePiece,M.nonneg n hn,Submodule.map_bot]
+  finite := inferInstance
+
+ def changeHom {M N : ModuleData.{uk,ua,um} 𝓐} (f : Hom M N) :
+    Hom (change e he M) (change e he N) where
+  map := CoefficientChange.map e f.map
+  graded n m hm := (mem_changePiece e N n _).mpr
+    (f.graded n m.val ((mem_changePiece e M n m).mp hm))
+
+ theorem change_free (M : ModuleData.{uk,ua,um} 𝓐) [Module.Free A M] :
+    Module.Free B (change e he M) := inferInstanceAs (Module.Free B (CoefficientChange e M))
+
+end
+end KLInvariance.Graded.ModuleData
+
+end
+
+
+section
+
+/-! The genuine nonnegative polynomial generator character of word stalks,
+with its convergent Hilbert-series interpretation. No KL identification is
+asserted or included in the definition. -/
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded
+universe u
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ def bottPolynomialStalkData (l : List I) (x : W) :
+    ModuleData.{0,u,u} (polynomialGrading ℝ σ) :=
+  ModuleData.change (SymmetricAlgebra.equivMvPolynomial basis) (mem_symmetricGrading basis)
+    (bottStalkData cs basis l x)
+
+ instance bottPolynomialStalkFree (l : List I) (x : W) :
+    Module.Free (MvPolynomial σ ℝ) (bottPolynomialStalkData cs basis l x) :=
+  inferInstanceAs (Module.Free (MvPolynomial σ ℝ)
+    (CoefficientChange (SymmetricAlgebra.equivMvPolynomial basis) (bottStalk cs l x)))
+
+ def bottStalkCharacter (l : List I) (x : W) : Polynomial ℤ :=
+  Graded.character (σ := σ) (bottPolynomialStalkData cs basis l x).piece
+    (bottPolynomialStalkData cs basis l x).nonneg
+
+ theorem bottPolynomialStalk_finrank (l : List I) (x : W) (n : ℤ) :
+    Module.finrank ℝ ((bottPolynomialStalkData cs basis l x).piece n)=
+      Module.finrank ℝ (bottStalkPiece cs basis l x n) :=
+  LinearEquiv.finrank_map_eq
+    (CoefficientChange.kEquiv (SymmetricAlgebra.equivMvPolynomial basis)
+      (bottStalkData cs basis l x)) _
+
+ theorem bottStalkCharacter_hasSum (l : List I) (x : W) {q : ℝ}
+    (hq : 0<q) (hq1 : q<1) :
+    HasSum (fun n : ℤ => (Module.finrank ℝ (bottStalkPiece cs basis l x n) : ℝ)*q^n)
+      (Polynomial.eval₂ (Int.castRingHom ℝ) q (bottStalkCharacter cs basis l x) *
+        (1-q)⁻¹ ^ Fintype.card σ) := by
+  have hh := (exists_nonnegativeBasis (k := ℝ) (σ := σ) (𝓜 := (bottPolynomialStalkData cs basis l x).piece)
+    (bottPolynomialStalkData cs basis l x).nonneg).some.hasSum hq hq1
+  apply hh.congr_fun
+  intro n
+  rw [bottPolynomialStalk_finrank]
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+namespace KLInvariance
+universe ur um un up
+variable {R : Type ur} [Ring R]
+  {M : Type um} [AddCommGroup M] [Module R M]
+  {N : Type un} [AddCommGroup N] [Module R N]
+  {P : Type up} [AddCommGroup P] [Module R P]
+
+def kernelTargetEquiv (f : M →ₗ[R] N) (e : N ≃ₗ[R] P) :
+    LinearMap.ker f ≃ₗ[R] LinearMap.ker (e.toLinearMap.comp f) where
+  toFun x := ⟨x.val, by
+    change e (f x.val)=0
+    rw [LinearMap.mem_ker.mp x.property,map_zero]⟩
+  invFun y := ⟨y.val, by
+    apply e.injective
+    simpa only [LinearMap.comp_apply,LinearEquiv.coe_coe,map_zero] using (LinearMap.mem_ker.mp y.property)⟩
+  left_inv _ := rfl
+  right_inv _ := rfl
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+
+end KLInvariance
+
+end
+
+
+section
+
+namespace KLInvariance
+universe uk ua um un up uq
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {M : Type um} [AddCommGroup M] [Module k M] [Module A M] [IsScalarTower k A M]
+  {N : Type un} [AddCommGroup N] [Module k N] [Module A N] [IsScalarTower k A N]
+  {P : Type up} [AddCommGroup P] [Module k P] [Module A P] [IsScalarTower k A P]
+  {Q : Type uq} [AddCommGroup Q] [Module k Q]
+
+def kernelMixedEquiv (f : M →ₗ[A] P) (e : M ≃ₗ[A] N) (g : P ≃ₗ[k] Q) :
+    LinearMap.ker f ≃ₗ[k] LinearMap.ker
+      (g.toLinearMap.comp ((f.comp e.symm.toLinearMap).restrictScalars k)) where
+  toFun x := ⟨e x.val, by
+    change g (f (e.symm (e x.val)))=0
+    rw [e.symm_apply_apply,LinearMap.mem_ker.mp x.property,map_zero]⟩
+  invFun y := ⟨e.symm y.val, by
+    apply g.injective
+    have hy : g (f (e.symm y.val))=0 := y.property
+    simpa only [map_zero] using hy⟩
+  left_inv x := Subtype.ext (e.symm_apply_apply x.val)
+  right_inv y := Subtype.ext (e.apply_symm_apply y.val)
+  map_add' x y := Subtype.ext (map_add e x.val y.val)
+  map_smul' c x := Subtype.ext (map_smul (e.restrictScalars k) c x.val)
+
+end KLInvariance
+
+end
+
+
+section
+
+/-! The descent short exact sequence for the actual word projection lattice.
+Its kernel has the genuine degree-one shift, over the unchanged realization. -/
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded ReflectionPairTranslation
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ theorem bottHeadRight_surjective (i : I) (l : List I) (x : W) :
+    Function.Surjective (bottHeadRight cs i l x) :=
+  (Twist.scalarEquiv (coefficientAction cs (cs.simple i)) _).surjective.comp
+    ((right_surjective _ _ _ _ _ _ (bottStalkProjection_surjective cs l (cs.simple i*x))).comp
+      (bottHeadPairEquiv cs i l x).symm.surjective)
+
+ theorem bottDescentKernel (i : I) (l : List I) (x : W)
+    (h : cs.length (cs.simple i*x)<cs.length x) (z : bottStalk cs l x) :
+    (∃ n, bottStalkProjection cs l x n=z ∧ bottStalkProjection cs l (cs.simple i*x) n=0) ↔
+      ∃ w, z=simpleRootPolynomial (M := M) i • w := by
+  have hh := (bottKernelExact cs l).kernel_iff_of_mul cs (cs.simple i*x) x (cs.simple i)
+    (simple_represents cs i) (cs.simple_mul_simple_cancel_left i) h z
+  simpa only [and_comm,simpleRootPolynomial,embed,extendedRoot] using hh
+
+ def bottHeadRightKernelEquiv (i : I) (l : List I) (x : W)
+    (h : cs.length (cs.simple i*x)<cs.length x) :
+    bottStalk cs l x ≃ₗ[ℝ] LinearMap.ker (bottHeadRight cs i l x) :=
+  ((rightKernelEquiv (bottStalkProjection cs l x) (bottStalkProjection cs l (cs.simple i*x))
+      (coefficientAction cs (cs.simple i)).toRingEquiv (simpleRootPolynomial (M := M) i)
+      (simpleReverseDivisibility cs i) (bottSimpleEdge_kernel_inclusion cs i l x)
+      (bottDescentKernel cs i l x h)
+      (bottStalk_smul_injective cs l x (simpleRootPolynomial_ne_zero i))).restrictScalars ℝ).trans
+    (kernelMixedEquiv _ (bottHeadPairEquiv cs i l x)
+      (Twist.scalarEquiv (coefficientAction cs (cs.simple i)) _))
+
+ theorem bottHeadRightKernelEquiv_eval (i : I) (l : List I) (x : W)
+    (h : cs.length (cs.simple i*x)<cs.length x) (y : bottStalk cs l x)
+    (e : {e : BottIndex (i::l) // bottWeight cs (i::l) e=x}) :
+    (bottHeadRightKernelEquiv cs i l x h y).val.val e=
+      if he : e.val.1=0 then simpleRootPolynomial (M := M) i *
+        y.val ⟨e.val.2,bottWeight_tail_zero cs i l x e he⟩ else 0 := by
+  change (if he : e.val.1=0 then _ else coefficientAction cs (cs.simple i) 0)=_
+  split_ifs
+  · rfl
+  · exact map_zero _
+
+ def bottHeadRightKernelPiece (i : I) (l : List I) (x : W) :
+    ℤ → Submodule ℝ (LinearMap.ker (bottHeadRight cs i l x)) :=
+  subPiece (bottStalkPiece cs basis (i::l) x) (LinearMap.ker (bottHeadRight cs i l x))
+
+ theorem bottHeadRight_kernel_homogeneous (i : I) (l : List I) (x : W) :
+    ∀ n m, m ∈ LinearMap.ker (bottHeadRight cs i l x) →
+      component (bottStalkPiece cs basis (i::l) x) n m ∈ LinearMap.ker (bottHeadRight cs i l x) := by
+  exact kernel_homogeneous (A := ℝ) (bottStalkPiece cs basis (i::l) x)
+    (bottStalkPiece cs basis l (cs.simple i*x)) (bottHeadRight cs i l x) 0 (by
+      intro j z hz
+      simpa only [add_zero] using (bottHeadRight_homogeneous cs basis i l x j z hz))
+
+ instance bottHeadRightKernelPiece_decomposition (i : I) (l : List I) (x : W) :
+    DirectSum.Decomposition (bottHeadRightKernelPiece cs basis i l x) :=
+  (sub_isInternal (bottStalkPiece cs basis (i::l) x) (LinearMap.ker (bottHeadRight cs i l x))
+    (bottHeadRight_kernel_homogeneous cs basis i l x)).chooseDecomposition
+
+ theorem bottHeadRightKernelEquiv_homogeneous (i : I) (l : List I) (x : W)
+    (h : cs.length (cs.simple i*x)<cs.length x) (n : ℤ)
+    (y : bottStalk cs l x) (hy : y ∈ bottStalkPiece cs basis l x n) :
+    bottHeadRightKernelEquiv cs i l x h y ∈ bottHeadRightKernelPiece cs basis i l x (n+1) := by
+  intro e _
+  change (bottHeadRightKernelEquiv cs i l x h y).val.val e ∈ symmetricGrading basis (n+1)
+  rw [bottHeadRightKernelEquiv_eval]
+  split_ifs with he
+  · have hy' : y.val ⟨e.val.2,bottWeight_tail_zero cs i l x e he⟩ ∈ symmetricGrading basis n :=
+      hy ⟨e.val.2,bottWeight_tail_zero cs i l x e he⟩ (Set.mem_univ _)
+    have hp := SetLike.GradedMul.mul_mem (simpleRoot_homogeneous basis i) hy'
+    simpa only [add_comm] using hp
+  · exact Submodule.zero_mem _
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Uniqueness and degree-one exact recurrences of actual polynomial Hilbert
+numerators, with no character-theorem assumptions. -/
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/KazhdanLusztig/Bruhat.lean b/lean/OAI/RepresentationTheory/KazhdanLusztig/Bruhat.lean
index 65dfba574760e427d799c068356c999e777dfeac..1e867b66126a2ebc14c822a98e955682bc484f7f 100644
--- a/lean/OAI/RepresentationTheory/KazhdanLusztig/Bruhat.lean
+++ b/lean/OAI/RepresentationTheory/KazhdanLusztig/Bruhat.lean
@@ -1,531 +1,531 @@
-import OAI.RepresentationTheory.KazhdanLusztig.Model
-import OAI.RepresentationTheory.KazhdanLusztig.Coxeter
-
-namespace OAI
-
-/-!
-Strong Bruhat order, subwords, length, finite lower ideals and arbitrary finite intervals; the selected R family.
--/
-
-namespace KLInvariance
-
-open Polynomial
-
-universe u v u' v'
-
-variable {B : Type u} {W : Type v} [Group W] {M : CoxeterMatrix B}
-
-
-theorem bruhat_refl (cs : CoxeterSystem M W) (x : W) : BruhatLE cs x x :=
-  Relation.ReflTransGen.refl
-
-theorem bruhat_trans (cs : CoxeterSystem M W) {x y z : W}
-    (hxy : BruhatLE cs x y) (hyz : BruhatLE cs y z) : BruhatLE cs x z :=
-  hxy.trans hyz
-
-theorem length_le_of_bruhat (cs : CoxeterSystem M W) {x y : W}
-    (h : BruhatLE cs x y) : cs.length x ≤ cs.length y := by
-  induction h using Relation.ReflTransGen.head_induction_on with
-  | refl => exact le_rfl
-  | head hstep _ ih => exact le_trans hstep.1.le ih
-
-theorem length_lt_of_bruhat_ne (cs : CoxeterSystem M W) {x y : W}
-    (h : BruhatLE cs x y) (hne : x ≠ y) : cs.length x < cs.length y := by
-  rcases h.cases_head with heq | ⟨z, hxz, hzy⟩
-  · exact (hne heq).elim
-  · exact lt_of_lt_of_le hxz.1 (length_le_of_bruhat cs hzy)
-
-theorem bruhat_antisymm (cs : CoxeterSystem M W) {x y : W}
-    (hxy : BruhatLE cs x y) (hyx : BruhatLE cs y x) : x = y := by
-  by_contra hne
-  exact (not_lt_of_ge (length_le_of_bruhat cs hyx)) (length_lt_of_bruhat_ne cs hxy hne)
-
-end KLInvariance
-
-namespace KLInvariance
-
-open Polynomial
-
-universe u v u' v'
-
-variable {B : Type u} {W : Type v} [Group W] {M : CoxeterMatrix B}
-
-
-noncomputable def rPolynomial (cs : CoxeterSystem M W) (x y : W) : ℤ[X] :=
-  (klFamilies cs).1 x y
-
-end KLInvariance
-
-section
-
-
-namespace KLInvariance
-
-open Polynomial
-
-universe u v u' v'
-
-variable {B : Type u} {W : Type v} [klPreservedInstance1 : Group W] {M : CoxeterMatrix B}
-
-
-/-- A vertex has only finitely many incoming strong Bruhat edges, using
-strong exchange proved from the Coxeter presentation above. -/
-theorem finite_bruhat_predecessors (cs : CoxeterSystem M W) (b : W) :
-    {x : W | BruhatStep cs x b}.Finite := by
-  classical
-  obtain ⟨ω, hω, heq⟩ := cs.exists_isReduced b
-  refine ((cs.leftInvSeq ω).finite_toSet.image (fun t => t * b)).subset ?_
-  rintro x ⟨hlen, t, href, hb⟩
-  have hx : t * b = x := by
-    rw [hb, ← mul_assoc, href.mul_self, one_mul]
-  refine ⟨t, ?_, hx⟩
-  apply CoxeterSupport.mem_leftInvSeq_of_inversion cs hω
-  exact ⟨href, by rw [← heq, hx]; exact hlen⟩
-
-/-- Finiteness of all lower Bruhat ideals, with no finite-rank assumption. -/
-theorem finite_bruhat_lower (cs : CoxeterSystem M W) (b : W) :
-    {x : W | BruhatLE cs x b}.Finite := by
-  classical
-  suffices ∀ n : ℕ, ∀ b : W, cs.length b = n → {x : W | BruhatLE cs x b}.Finite by
-    exact this (cs.length b) b rfl
-  intro n
-  induction n using Nat.strong_induction_on with
-  | h n ih =>
-    intro b hb
-    have hfinite : ({b} ∪ ⋃ z ∈ {z : W | BruhatStep cs z b},
-        {x : W | BruhatLE cs x z}).Finite :=
-      (Set.finite_singleton b).union ((finite_bruhat_predecessors cs b).biUnion
-        (fun z hz => ih (cs.length z) (hb ▸ hz.1) z rfl))
-    apply hfinite.subset
-    intro x hx
-    rcases hx.cases_tail with h | ⟨z, hxz, hzb⟩
-    · exact Or.inl (h.symm)
-    · exact Or.inr (Set.mem_iUnion.mpr ⟨z, Set.mem_iUnion.mpr ⟨hzb, hxz⟩⟩)
-
-/-- The finite-interval obligation in the faithful trusted main target. -/
-theorem interval_finite (cs : CoxeterSystem M W) (u b : W) :
-    Finite (Interval cs u b) :=
-  ((finite_bruhat_lower cs b).subset (fun _ hx => hx.2)).to_subtype
-
-/-- The exact R recursion determines at most one R family. -/
-theorem normalized_R_unique (cs : CoxeterSystem M W)
-    {RP SQ : PolynomialFamilies W} (hRP : NormalizedKL cs RP) (hSQ : NormalizedKL cs SQ) :
-    RP.1 = SQ.1 := by
-  classical
-  suffices ∀ n : ℕ, ∀ y : W, cs.length y = n → ∀ x, RP.1 x y = SQ.1 x y by
-    funext x y
-    exact this (cs.length y) y rfl x
-  intro n
-  induction n using Nat.strong_induction_on with
-  | h n ih =>
-    intro y hy x
-    by_cases h1 : y = 1
-    · subst y
-      by_cases hx : x = 1
-      · subst x
-        rw [hRP.R_diagonal, hSQ.R_diagonal]
-      · have hn : ¬ BruhatLE cs x 1 := by
-          intro h
-          have hl := length_le_of_bruhat cs h
-          have hx0 : cs.length x = 0 := by simpa using hl
-          exact hx (cs.length_eq_zero_iff.mp hx0)
-        rw [hRP.R_zero _ _ hn, hSQ.R_zero _ _ hn]
-    · obtain ⟨i, hi⟩ := cs.exists_leftDescent_of_ne_one h1
-      have hlt : cs.length (cs.simple i * y) < n := hy ▸ hi
-      rw [hRP.R_recursion x y i hi, hSQ.R_recursion x y i hi]
-      rw [ih _ hlt _ rfl x, ih _ hlt _ rfl (cs.simple i * x)]
-
-/-- The degree/reciprocity rigidity used both by normalization uniqueness and
-by the final main-theorem induction: no polynomial supported strictly below
-the midpoint can be self-reciprocal about that midpoint. -/
-theorem low_degree_reflect_eq_self (f : ℤ[X]) (d : ℕ)
-    (hd : 2 * f.natDegree < d) (hself : reflect d f = f) : f = 0 := by
-  by_contra hn
-  have hc := congrArg (fun p : ℤ[X] => p.coeff f.natDegree) hself
-  rw [coeff_reflect, revAt_le (by omega)] at hc
-  rw [coeff_eq_zero_of_natDegree_lt (by omega), coeff_natDegree] at hc
-  exact (leadingCoeff_ne_zero.mpr hn) hc.symm
-
-/-- The polynomial part of the manuscript's normalization is unique; the
-proof uses the actual finite strong-Bruhat intervals, not extra assumptions. -/
-theorem normalized_P_unique (cs : CoxeterSystem M W)
-    {RP SQ : PolynomialFamilies W} (hRP : NormalizedKL cs RP) (hSQ : NormalizedKL cs SQ) :
-    RP.2 = SQ.2 := by
-  classical
-  have hR := normalized_R_unique cs hRP hSQ
-  suffices ∀ n : ℕ, ∀ x y : W, rankDifference cs x y = n → RP.2 x y = SQ.2 x y by
-    funext x y
-    exact this (rankDifference cs x y) x y rfl
-  intro n
-  induction n using Nat.strong_induction_on with
-  | h n ih =>
-    intro x y hxy
-    by_cases hcomp : BruhatLE cs x y
-    · by_cases heq : x = y
-      · subst y
-        rw [hRP.P_diagonal, hSQ.P_diagonal]
-      · let : Finite (Interval cs x y) := interval_finite cs x y
-        let : Fintype (Interval cs x y) := Fintype.ofFinite _
-        let bottom : Interval cs x y := ⟨x, bruhat_refl cs x, hcomp⟩
-        have hsum : (∑ z : Interval cs x y,
-            RP.1 x z.val * (RP.2 z.val y - SQ.2 z.val y)) = RP.2 x y - SQ.2 x y := by
-          rw [Finset.sum_eq_single bottom]
-          · simp [bottom, hRP.R_diagonal]
-          · intro z _ hz
-            have hxz : x ≠ z.val := by
-              intro hc
-              apply hz
-              exact Subtype.ext hc.symm
-            have hlt := length_lt_of_bruhat_ne cs z.property.1 hxz
-            have hle := length_le_of_bruhat cs z.property.2
-            have hdiff : rankDifference cs z.val y < n := by
-              dsimp [rankDifference] at hxy ⊢
-              omega
-            rw [ih _ hdiff z.val y rfl, sub_self, mul_zero]
-          · simp
-        have href : reflect (rankDifference cs x y) (RP.2 x y - SQ.2 x y) =
-            RP.2 x y - SQ.2 x y := by
-          rw [reflect_sub, hRP.reciprocity x y hcomp, hSQ.reciprocity x y hcomp,
-            finsum_eq_sum_of_fintype, finsum_eq_sum_of_fintype, ← Finset.sum_sub_distrib]
-          rw [← hsum]
-          apply Finset.sum_congr rfl
-          intro z _
-          rw [← hR, mul_sub]
-        have hd1 := hRP.P_degree x y hcomp heq
-        have hd2 := hSQ.P_degree x y hcomp heq
-        have hdeg := natDegree_sub_le (RP.2 x y) (SQ.2 x y)
-        have hzero := low_degree_reflect_eq_self (RP.2 x y - SQ.2 x y)
-          (rankDifference cs x y) (by omega) href
-        exact sub_eq_zero.mp hzero
-    · rw [hRP.P_zero _ _ hcomp, hSQ.P_zero _ _ hcomp]
-
-/-- Unique existence is still a separate obligation: this proves its
-uniqueness half only, without postulating a normalized family. -/
-theorem normalizedKL_unique (cs : CoxeterSystem M W)
-    {RP SQ : PolynomialFamilies W} (hRP : NormalizedKL cs RP) (hSQ : NormalizedKL cs SQ) :
-    RP = SQ :=
-  Prod.ext (normalized_R_unique cs hRP hSQ) (normalized_P_unique cs hRP hSQ)
-
-
-/-! Exchange consequences required for the source's finite-rank reduction
-and for consistency of the equal-parameter Hecke recursion. -/
-
-/-- Every word contains a reduced subword with the same product. -/
-theorem exists_reduced_subword (cs : CoxeterSystem M W) (ω : List B) :
-    ∃ ν : List B, ν.Sublist ω ∧ cs.IsReduced ν ∧ cs.wordProd ν = cs.wordProd ω := by
-  induction ω with
-  | nil => exact ⟨[], .refl _, by simp [CoxeterSystem.IsReduced], rfl⟩
-  | cons i ω ih =>
-    obtain ⟨ν, hsub, hν, heq⟩ := ih
-    rcases cs.length_simple_mul (cs.wordProd ν) i with hasc | hdesc
-    · refine ⟨i :: ν, hsub.cons_cons i, ?_, ?_⟩
-      · simpa [CoxeterSystem.IsReduced, cs.wordProd_cons, hν.eq] using hasc
-      · simp [cs.wordProd_cons, heq]
-    · obtain ⟨j, hj, hdel⟩ := CoxeterSupport.strong_exchange cs hν
-        ⟨cs.isReflection_simple i, by omega⟩
-      refine ⟨ν.eraseIdx j, ((ν.eraseIdx_sublist j).trans hsub).cons i, ?_, ?_⟩
-      · dsimp [CoxeterSystem.IsReduced]
-        rw [← hdel, List.length_eraseIdx, ite_eq_left hj]
-        have := hν.eq
-        omega
-      · rw [← hdel, heq, cs.wordProd_cons]
-
-/-- A single downward reflection step can be realized inside any reduced word. -/
-theorem subword_of_bruhat_step (cs : CoxeterSystem M W) {x y : W}
-    (hxy : BruhatStep cs x y) {ω : List B} (hω : cs.IsReduced ω)
-    (hy : cs.wordProd ω = y) :
-    ∃ ν : List B, ν.Sublist ω ∧ cs.IsReduced ν ∧ cs.wordProd ν = x := by
-  obtain ⟨hlen, t, ht, rfl⟩ := hxy
-  have hx : t * cs.wordProd ω = x := by simp [hy, ← mul_assoc, ht.mul_self]
-  obtain ⟨j, hj, hdel⟩ := CoxeterSupport.strong_exchange cs hω
-    ⟨ht, by
-      rw [hx, hy]
-      exact hlen⟩
-  obtain ⟨ν, hsub, hν, heq⟩ := exists_reduced_subword cs (ω.eraseIdx j)
-  exact ⟨ν, hsub.trans (ω.eraseIdx_sublist j), hν, heq.trans (hdel.symm.trans hx)⟩
-
-/-- Forward subword property, for every chosen reduced expression of the top. -/
-theorem reduced_subword_of_bruhat (cs : CoxeterSystem M W) {x y : W}
-    (hxy : BruhatLE cs x y) {ω : List B} (hω : cs.IsReduced ω)
-    (hy : cs.wordProd ω = y) :
-    ∃ ν : List B, ν.Sublist ω ∧ cs.IsReduced ν ∧ cs.wordProd ν = x := by
-  induction hxy using Relation.ReflTransGen.head_induction_on generalizing ω with
-  | refl => exact ⟨ω, .refl _, hω, hy⟩
-  | head hstep _ ih =>
-    obtain ⟨ν, hsub, hν, heq⟩ := ih hω hy
-    obtain ⟨μ, hμsub, hμ, hμeq⟩ := subword_of_bruhat_step cs hstep hν heq
-    exact ⟨μ, hμsub.trans hsub, hμ, hμeq⟩
-
-
-/-- Simple left multiplication preserves the orientation of every reflection
-edge except the edge carrying that simple reflection itself. -/
-theorem bruhatStep_simple_mul (cs : CoxeterSystem M W) {x y : W}
-    (hxy : BruhatStep cs x y) (i : B) (hne : y ≠ cs.simple i * x) :
-    BruhatStep cs (cs.simple i * x) (cs.simple i * y) := by
-  obtain ⟨hlen, t, ht, heq⟩ := hxy
-  refine ⟨?_, cs.simple i * t * cs.simple i, by simpa using ht.conj (cs.simple i), ?_⟩
-  · rcases cs.length_simple_mul y i with hasc | hdesc
-    · have hx := cs.length_simple_mul x i
-      omega
-    · obtain ⟨ω, hω, hωeq⟩ := cs.exists_isReduced (cs.simple i * y)
-      have hprod : cs.wordProd (i :: ω) = y := by
-        rw [cs.wordProd_cons, ← hωeq, cs.simple_mul_simple_cancel_left]
-      have hr : cs.IsReduced (i :: ω) := by
-        dsimp [CoxeterSystem.IsReduced]
-        rw [hprod, ← hω.eq, ← hωeq]
-        exact hdesc.symm
-      have htx : t * cs.wordProd (i :: ω) = x := by
-        rw [hprod, heq, ← mul_assoc, ht.mul_self, one_mul]
-      obtain ⟨j, hj, hdel⟩ := CoxeterSupport.strong_exchange cs hr
-        ⟨ht, by rw [htx, hprod]; exact hlen⟩
-      cases j with
-      | zero =>
-        have hx : x = cs.wordProd ω := by simpa [htx] using hdel
-        exact (hne (by rw [hx, ← cs.wordProd_cons, hprod])).elim
-      | succ j =>
-        have hx : x = cs.simple i * cs.wordProd (ω.eraseIdx j) := by
-          rw [htx, List.eraseIdx_cons_succ, cs.wordProd_cons] at hdel
-          exact hdel
-        have hsx : cs.simple i * x = cs.wordProd (ω.eraseIdx j) := by simp [hx]
-        have hj' : j < ω.length := by simpa using hj
-        have hlen' := cs.length_wordProd_le (ω.eraseIdx j)
-        rw [List.length_eraseIdx, ite_eq_left hj'] at hlen'
-        rw [hsx, hωeq, hω.eq]
-        omega
-  · rw [heq]
-    simp [mul_assoc]
-
-/-- The smaller of `x` and `s*x` in Bruhat order. -/
-noncomputable def lowerSimple (cs : CoxeterSystem M W) (i : B) (x : W) : W :=
-  if cs.length (cs.simple i * x) < cs.length x then cs.simple i * x else x
-
-/-- The larger of `x` and `s*x` in Bruhat order. -/
-noncomputable def upperSimple (cs : CoxeterSystem M W) (i : B) (x : W) : W :=
-  if cs.length (cs.simple i * x) < cs.length x then x else cs.simple i * x
-
-theorem lowerSimple_le (cs : CoxeterSystem M W) (i : B) (x : W) :
-    BruhatLE cs (lowerSimple cs i x) x := by
-  classical
-  unfold lowerSimple
-  split_ifs with h
-  · exact .single ⟨h, cs.simple i, cs.isReflection_simple i, by simp⟩
-  · exact .refl
-
-theorem le_upperSimple (cs : CoxeterSystem M W) (i : B) (x : W) :
-    BruhatLE cs x (upperSimple cs i x) := by
-  classical
-  unfold upperSimple
-  split_ifs with h
-  · exact .refl
-  · exact .single ⟨by have := cs.length_simple_mul x i; omega,
-      cs.simple i, cs.isReflection_simple i, rfl⟩
-
-@[simp] theorem lowerSimple_mul (cs : CoxeterSystem M W) (i : B) (x : W) :
-    lowerSimple cs i (cs.simple i * x) = lowerSimple cs i x := by
-  classical
-  unfold lowerSimple
-  rw [cs.simple_mul_simple_cancel_left]
-  have := cs.length_simple_mul x i
-  split_ifs <;> first | rfl | omega
-
-@[simp] theorem upperSimple_mul (cs : CoxeterSystem M W) (i : B) (x : W) :
-    upperSimple cs i (cs.simple i * x) = upperSimple cs i x := by
-  classical
-  unfold upperSimple
-  rw [cs.simple_mul_simple_cancel_left]
-  have := cs.length_simple_mul x i
-  split_ifs <;> first | rfl | omega
-
-theorem lowerSimple_step_mono (cs : CoxeterSystem M W) (i : B) {x y : W}
-    (hxy : BruhatStep cs x y) :
-    BruhatLE cs (lowerSimple cs i x) (lowerSimple cs i y) := by
-  classical
-  by_cases heq : y = cs.simple i * x
-  · rw [heq, lowerSimple_mul]
-    exact .refl
-  · have hs := bruhatStep_simple_mul cs hxy i heq
-    by_cases hx : cs.length (cs.simple i * x) < cs.length x
-    · by_cases hy : cs.length (cs.simple i * y) < cs.length y
-      · simp only [lowerSimple, ite_eq_left hx, ite_eq_left hy]
-        exact .single hs
-      · simp only [lowerSimple, ite_eq_left hx, ite_eq_right hy]
-        have hlow := lowerSimple_le cs i x
-        simp only [lowerSimple, ite_eq_left hx] at hlow
-        exact hlow.trans (.single hxy)
-    · by_cases hy : cs.length (cs.simple i * y) < cs.length y
-      · simp only [lowerSimple, ite_eq_right hx, ite_eq_left hy]
-        have hupp := le_upperSimple cs i x
-        simp only [upperSimple, ite_eq_right hx] at hupp
-        exact hupp.trans (.single hs)
-      · simp only [lowerSimple, ite_eq_right hx, ite_eq_right hy]
-        exact .single hxy
-
-theorem upperSimple_step_mono (cs : CoxeterSystem M W) (i : B) {x y : W}
-    (hxy : BruhatStep cs x y) :
-    BruhatLE cs (upperSimple cs i x) (upperSimple cs i y) := by
-  classical
-  by_cases heq : y = cs.simple i * x
-  · rw [heq, upperSimple_mul]
-    exact .refl
-  · have hs := bruhatStep_simple_mul cs hxy i heq
-    by_cases hx : cs.length (cs.simple i * x) < cs.length x
-    · by_cases hy : cs.length (cs.simple i * y) < cs.length y
-      · simp only [upperSimple, ite_eq_left hx, ite_eq_left hy]
-        exact .single hxy
-      · simp only [upperSimple, ite_eq_left hx, ite_eq_right hy]
-        have hupp := le_upperSimple cs i y
-        simp only [upperSimple, ite_eq_right hy] at hupp
-        exact (Relation.ReflTransGen.single hxy).trans hupp
-    · by_cases hy : cs.length (cs.simple i * y) < cs.length y
-      · simp only [upperSimple, ite_eq_right hx, ite_eq_left hy]
-        have hlow := lowerSimple_le cs i y
-        simp only [lowerSimple, ite_eq_left hy] at hlow
-        exact (Relation.ReflTransGen.single hs).trans hlow
-      · simp only [upperSimple, ite_eq_right hx, ite_eq_right hy]
-        exact .single hs
-
-theorem lowerSimple_mono (cs : CoxeterSystem M W) (i : B) {x y : W}
-    (hxy : BruhatLE cs x y) :
-    BruhatLE cs (lowerSimple cs i x) (lowerSimple cs i y) := by
-  induction hxy using Relation.ReflTransGen.head_induction_on with
-  | refl => exact .refl
-  | head hstep _ ih => exact (lowerSimple_step_mono cs i hstep).trans ih
-
-theorem upperSimple_mono (cs : CoxeterSystem M W) (i : B) {x y : W}
-    (hxy : BruhatLE cs x y) :
-    BruhatLE cs (upperSimple cs i x) (upperSimple cs i y) := by
-  induction hxy using Relation.ReflTransGen.head_induction_on with
-  | refl => exact .refl
-  | head hstep _ ih => exact (upperSimple_step_mono cs i hstep).trans ih
-
-
-/-- The standard lifting property, now for the actual strong order. -/
-theorem bruhat_lifting (cs : CoxeterSystem M W) {x y : W}
-    (hxy : BruhatLE cs x y) (i : B)
-    (hx : cs.length x < cs.length (cs.simple i * x))
-    (hy : cs.length (cs.simple i * y) < cs.length y) :
-    BruhatLE cs x (cs.simple i * y) ∧ BruhatLE cs (cs.simple i * x) y := by
-  have hx' : ¬cs.length (cs.simple i * x) < cs.length x := by omega
-  constructor
-  · simpa only [lowerSimple, ite_eq_right hx', ite_eq_left hy] using lowerSimple_mono cs i hxy
-  · simpa only [upperSimple, ite_eq_right hx', ite_eq_left hy] using upperSimple_mono cs i hxy
-
-/-- Converse subword property; no labels are added to Bruhat intervals. -/
-theorem bruhat_of_subword (cs : CoxeterSystem M W) {ν ω : List B}
-    (hsub : ν.Sublist ω) (hω : cs.IsReduced ω) :
-    BruhatLE cs (cs.wordProd ν) (cs.wordProd ω) := by
-  induction ω generalizing ν with
-  | nil =>
-    have : ν = [] := List.sublist_nil.mp hsub
-    subst ν
-    exact .refl
-  | cons i ω ih =>
-    have hω' : cs.IsReduced ω := by simpa using hω.drop 1
-    have hasc : cs.length (cs.simple i * cs.wordProd ω) = cs.length (cs.wordProd ω) + 1 := by
-      simpa [CoxeterSystem.IsReduced, cs.wordProd_cons, hω'.eq] using hω.eq
-    have htop : upperSimple cs i (cs.wordProd ω) = cs.wordProd (i :: ω) := by
-      simp [upperSimple, show ¬cs.length (cs.simple i * cs.wordProd ω) <
-        cs.length (cs.wordProd ω) by omega, cs.wordProd_cons]
-    rcases List.sublist_cons_iff.mp hsub with hsub' | ⟨μ, rfl, hμsub⟩
-    · exact (ih hsub' hω').trans (htop ▸ le_upperSimple cs i (cs.wordProd ω))
-    · have hle := upperSimple_mono cs i (ih hμsub hω')
-      rw [htop] at hle
-      have hx := le_upperSimple cs i (cs.simple i * cs.wordProd μ)
-      rw [upperSimple_mul] at hx
-      rw [cs.wordProd_cons]
-      exact hx.trans hle
-
-/-- The subword criterion with an arbitrary fixed reduced word for the top. -/
-theorem bruhat_iff_reduced_subword (cs : CoxeterSystem M W) {x : W} {ω : List B}
-    (hω : cs.IsReduced ω) :
-    BruhatLE cs x (cs.wordProd ω) ↔
-      ∃ ν : List B, ν.Sublist ω ∧ cs.IsReduced ν ∧ cs.wordProd ν = x := by
-  constructor
-  · intro hx
-    exact reduced_subword_of_bruhat cs hx hω rfl
-  · rintro ⟨ν, hsub, _, rfl⟩
-    exact bruhat_of_subword cs hsub hω
-
-/-- Precisely the finite-support part used by the manuscript's finite-rank
-reduction: all lower elements are words in the generators of any reduced top
-word. Equality of parabolic polynomial normalizations remains a later step. -/
-theorem lower_ideal_finite_generator_support (cs : CoxeterSystem M W) (b : W) :
-    ∃ J : Set B, J.Finite ∧ ∀ x, BruhatLE cs x b →
-      ∃ ν : List B, (∀ i ∈ ν, i ∈ J) ∧ cs.wordProd ν = x := by
-  obtain ⟨ω, hω, heq⟩ := cs.exists_isReduced b
-  refine ⟨{i | i ∈ ω}, ω.finite_toSet, ?_⟩
-  intro x hx
-  obtain ⟨ν, hsub, _, hprod⟩ := reduced_subword_of_bruhat cs hx hω heq.symm
-  exact ⟨ν, fun i hi => hsub.subset hi, hprod⟩
-
-
-/-- Inversion preserves the actual strong Bruhat graph. -/
-theorem bruhatStep_inv (cs : CoxeterSystem M W) {x y : W}
-    (hxy : BruhatStep cs x y) : BruhatStep cs x⁻¹ y⁻¹ := by
-  obtain ⟨hlen, t, ht, heq⟩ := hxy
-  refine ⟨by simpa using hlen, x⁻¹ * t * x, by simpa using ht.conj x⁻¹, ?_⟩
-  rw [heq, mul_inv_rev, ht.inv]
-  group
-
-theorem bruhatStep_mul_simple (cs : CoxeterSystem M W) {x y : W}
-    (hxy : BruhatStep cs x y) (i : B) (hne : y ≠ x * cs.simple i) :
-    BruhatStep cs (x * cs.simple i) (y * cs.simple i) := by
-  have hinv := bruhatStep_inv cs hxy
-  have hne' : y⁻¹ ≠ cs.simple i * x⁻¹ := by
-    intro heq
-    apply hne
-    have := congrArg Inv.inv heq
-    simpa [mul_inv_rev] using this
-  have h := bruhatStep_inv cs (bruhatStep_simple_mul cs hinv i hne')
-  simpa [mul_inv_rev] using h
-
-/-- The noncollapsed left/right simple square has consistent orientations. -/
-theorem left_descent_mul_simple_iff (cs : CoxeterSystem M W) (i j : B) (x : W)
-    (hne : cs.simple i * x ≠ x * cs.simple j) :
-    cs.length (cs.simple i * (x * cs.simple j)) < cs.length (x * cs.simple j) ↔
-      cs.length (cs.simple i * x) < cs.length x := by
-  by_cases hx : cs.length (cs.simple i * x) < cs.length x
-  · have hedge : BruhatStep cs (cs.simple i * x) x :=
-      ⟨hx, cs.simple i, cs.isReflection_simple i, by simp⟩
-    have hne' : x ≠ cs.simple i * x * cs.simple j := by
-      intro heq
-      apply hne
-      have := congrArg (fun z => z * cs.simple j) heq
-      simpa using this.symm
-    have h := (bruhatStep_mul_simple cs hedge j hne').1
-    simpa only [mul_assoc, hx, iff_true] using h
-  · have hedge : BruhatStep cs x (cs.simple i * x) :=
-      ⟨by have := cs.length_simple_mul x i; omega, cs.simple i,
-        cs.isReflection_simple i, rfl⟩
-    have h := (bruhatStep_mul_simple cs hedge j hne).1
-    rw [mul_assoc] at h
-    exact iff_of_false (by omega) hx
-
-theorem right_descent_simple_mul_iff (cs : CoxeterSystem M W) (i j : B) (x : W)
-    (hne : cs.simple i * x ≠ x * cs.simple j) :
-    cs.length (cs.simple i * x * cs.simple j) < cs.length (cs.simple i * x) ↔
-      cs.length (x * cs.simple j) < cs.length x := by
-  have hne' : cs.simple j * x⁻¹ ≠ x⁻¹ * cs.simple i := by
-    intro heq
-    apply hne
-    have h := congrArg Inv.inv heq
-    simpa [mul_inv_rev] using h.symm
-  have h := left_descent_mul_simple_iff cs j i x⁻¹ hne'
-  have h1 : cs.length (cs.simple i * x * cs.simple j) =
-      cs.length (cs.simple j * (x⁻¹ * cs.simple i)) := by
-    rw [← cs.length_inv (cs.simple i * x * cs.simple j)]
-    simp [mul_inv_rev]
-  have h2 : cs.length (cs.simple i * x) = cs.length (x⁻¹ * cs.simple i) := by
-    rw [← cs.length_inv (cs.simple i * x)]
-    simp [mul_inv_rev]
-  have h3 : cs.length (x * cs.simple j) = cs.length (cs.simple j * x⁻¹) := by
-    rw [← cs.length_inv (x * cs.simple j)]
-    simp [mul_inv_rev]
-  rw [h1, h2, h3]
-  simpa using h
-
-
-end KLInvariance
-end
-
-end OAI
+import OAI.RepresentationTheory.KazhdanLusztig.Model
+import OAI.RepresentationTheory.KazhdanLusztig.Coxeter
+
+namespace OAI
+
+/-!
+Strong Bruhat order, subwords, length, finite lower ideals and arbitrary finite intervals; the selected R family.
+-/
+
+namespace KLInvariance
+
+open Polynomial
+
+universe u v u' v'
+
+variable {B : Type u} {W : Type v} [Group W] {M : CoxeterMatrix B}
+
+
+theorem bruhat_refl (cs : CoxeterSystem M W) (x : W) : BruhatLE cs x x :=
+  Relation.ReflTransGen.refl
+
+theorem bruhat_trans (cs : CoxeterSystem M W) {x y z : W}
+    (hxy : BruhatLE cs x y) (hyz : BruhatLE cs y z) : BruhatLE cs x z :=
+  hxy.trans hyz
+
+theorem length_le_of_bruhat (cs : CoxeterSystem M W) {x y : W}
+    (h : BruhatLE cs x y) : cs.length x ≤ cs.length y := by
+  induction h using Relation.ReflTransGen.head_induction_on with
+  | refl => exact le_rfl
+  | head hstep _ ih => exact le_trans hstep.1.le ih
+
+theorem length_lt_of_bruhat_ne (cs : CoxeterSystem M W) {x y : W}
+    (h : BruhatLE cs x y) (hne : x ≠ y) : cs.length x < cs.length y := by
+  rcases h.cases_head with heq | ⟨z, hxz, hzy⟩
+  · exact (hne heq).elim
+  · exact lt_of_lt_of_le hxz.1 (length_le_of_bruhat cs hzy)
+
+theorem bruhat_antisymm (cs : CoxeterSystem M W) {x y : W}
+    (hxy : BruhatLE cs x y) (hyx : BruhatLE cs y x) : x = y := by
+  by_contra hne
+  exact (not_lt_of_ge (length_le_of_bruhat cs hyx)) (length_lt_of_bruhat_ne cs hxy hne)
+
+end KLInvariance
+
+namespace KLInvariance
+
+open Polynomial
+
+universe u v u' v'
+
+variable {B : Type u} {W : Type v} [Group W] {M : CoxeterMatrix B}
+
+
+noncomputable def rPolynomial (cs : CoxeterSystem M W) (x y : W) : ℤ[X] :=
+  (klFamilies cs).1 x y
+
+end KLInvariance
+
+section
+
+
+namespace KLInvariance
+
+open Polynomial
+
+universe u v u' v'
+
+variable {B : Type u} {W : Type v} [klPreservedInstance1 : Group W] {M : CoxeterMatrix B}
+
+
+/-- A vertex has only finitely many incoming strong Bruhat edges, using
+strong exchange proved from the Coxeter presentation above. -/
+theorem finite_bruhat_predecessors (cs : CoxeterSystem M W) (b : W) :
+    {x : W | BruhatStep cs x b}.Finite := by
+  classical
+  obtain ⟨ω, hω, heq⟩ := cs.exists_isReduced b
+  refine ((cs.leftInvSeq ω).finite_toSet.image (fun t => t * b)).subset ?_
+  rintro x ⟨hlen, t, href, hb⟩
+  have hx : t * b = x := by
+    rw [hb, ← mul_assoc, href.mul_self, one_mul]
+  refine ⟨t, ?_, hx⟩
+  apply CoxeterSupport.mem_leftInvSeq_of_inversion cs hω
+  exact ⟨href, by rw [← heq, hx]; exact hlen⟩
+
+/-- Finiteness of all lower Bruhat ideals, with no finite-rank assumption. -/
+theorem finite_bruhat_lower (cs : CoxeterSystem M W) (b : W) :
+    {x : W | BruhatLE cs x b}.Finite := by
+  classical
+  suffices ∀ n : ℕ, ∀ b : W, cs.length b = n → {x : W | BruhatLE cs x b}.Finite by
+    exact this (cs.length b) b rfl
+  intro n
+  induction n using Nat.strong_induction_on with
+  | h n ih =>
+    intro b hb
+    have hfinite : ({b} ∪ ⋃ z ∈ {z : W | BruhatStep cs z b},
+        {x : W | BruhatLE cs x z}).Finite :=
+      (Set.finite_singleton b).union ((finite_bruhat_predecessors cs b).biUnion
+        (fun z hz => ih (cs.length z) (hb ▸ hz.1) z rfl))
+    apply hfinite.subset
+    intro x hx
+    rcases hx.cases_tail with h | ⟨z, hxz, hzb⟩
+    · exact Or.inl (h.symm)
+    · exact Or.inr (Set.mem_iUnion.mpr ⟨z, Set.mem_iUnion.mpr ⟨hzb, hxz⟩⟩)
+
+/-- The finite-interval obligation in the faithful trusted main target. -/
+theorem interval_finite (cs : CoxeterSystem M W) (u b : W) :
+    Finite (Interval cs u b) :=
+  ((finite_bruhat_lower cs b).subset (fun _ hx => hx.2)).to_subtype
+
+/-- The exact R recursion determines at most one R family. -/
+theorem normalized_R_unique (cs : CoxeterSystem M W)
+    {RP SQ : PolynomialFamilies W} (hRP : NormalizedKL cs RP) (hSQ : NormalizedKL cs SQ) :
+    RP.1 = SQ.1 := by
+  classical
+  suffices ∀ n : ℕ, ∀ y : W, cs.length y = n → ∀ x, RP.1 x y = SQ.1 x y by
+    funext x y
+    exact this (cs.length y) y rfl x
+  intro n
+  induction n using Nat.strong_induction_on with
+  | h n ih =>
+    intro y hy x
+    by_cases h1 : y = 1
+    · subst y
+      by_cases hx : x = 1
+      · subst x
+        rw [hRP.R_diagonal, hSQ.R_diagonal]
+      · have hn : ¬ BruhatLE cs x 1 := by
+          intro h
+          have hl := length_le_of_bruhat cs h
+          have hx0 : cs.length x = 0 := by simpa using hl
+          exact hx (cs.length_eq_zero_iff.mp hx0)
+        rw [hRP.R_zero _ _ hn, hSQ.R_zero _ _ hn]
+    · obtain ⟨i, hi⟩ := cs.exists_leftDescent_of_ne_one h1
+      have hlt : cs.length (cs.simple i * y) < n := hy ▸ hi
+      rw [hRP.R_recursion x y i hi, hSQ.R_recursion x y i hi]
+      rw [ih _ hlt _ rfl x, ih _ hlt _ rfl (cs.simple i * x)]
+
+/-- The degree/reciprocity rigidity used both by normalization uniqueness and
+by the final main-theorem induction: no polynomial supported strictly below
+the midpoint can be self-reciprocal about that midpoint. -/
+theorem low_degree_reflect_eq_self (f : ℤ[X]) (d : ℕ)
+    (hd : 2 * f.natDegree < d) (hself : reflect d f = f) : f = 0 := by
+  by_contra hn
+  have hc := congrArg (fun p : ℤ[X] => p.coeff f.natDegree) hself
+  rw [coeff_reflect, revAt_le (by omega)] at hc
+  rw [coeff_eq_zero_of_natDegree_lt (by omega), coeff_natDegree] at hc
+  exact (leadingCoeff_ne_zero.mpr hn) hc.symm
+
+/-- The polynomial part of the manuscript's normalization is unique; the
+proof uses the actual finite strong-Bruhat intervals, not extra assumptions. -/
+theorem normalized_P_unique (cs : CoxeterSystem M W)
+    {RP SQ : PolynomialFamilies W} (hRP : NormalizedKL cs RP) (hSQ : NormalizedKL cs SQ) :
+    RP.2 = SQ.2 := by
+  classical
+  have hR := normalized_R_unique cs hRP hSQ
+  suffices ∀ n : ℕ, ∀ x y : W, rankDifference cs x y = n → RP.2 x y = SQ.2 x y by
+    funext x y
+    exact this (rankDifference cs x y) x y rfl
+  intro n
+  induction n using Nat.strong_induction_on with
+  | h n ih =>
+    intro x y hxy
+    by_cases hcomp : BruhatLE cs x y
+    · by_cases heq : x = y
+      · subst y
+        rw [hRP.P_diagonal, hSQ.P_diagonal]
+      · let : Finite (Interval cs x y) := interval_finite cs x y
+        let : Fintype (Interval cs x y) := Fintype.ofFinite _
+        let bottom : Interval cs x y := ⟨x, bruhat_refl cs x, hcomp⟩
+        have hsum : (∑ z : Interval cs x y,
+            RP.1 x z.val * (RP.2 z.val y - SQ.2 z.val y)) = RP.2 x y - SQ.2 x y := by
+          rw [Finset.sum_eq_single bottom]
+          · simp [bottom, hRP.R_diagonal]
+          · intro z _ hz
+            have hxz : x ≠ z.val := by
+              intro hc
+              apply hz
+              exact Subtype.ext hc.symm
+            have hlt := length_lt_of_bruhat_ne cs z.property.1 hxz
+            have hle := length_le_of_bruhat cs z.property.2
+            have hdiff : rankDifference cs z.val y < n := by
+              dsimp [rankDifference] at hxy ⊢
+              omega
+            rw [ih _ hdiff z.val y rfl, sub_self, mul_zero]
+          · simp
+        have href : reflect (rankDifference cs x y) (RP.2 x y - SQ.2 x y) =
+            RP.2 x y - SQ.2 x y := by
+          rw [reflect_sub, hRP.reciprocity x y hcomp, hSQ.reciprocity x y hcomp,
+            finsum_eq_sum_of_fintype, finsum_eq_sum_of_fintype, ← Finset.sum_sub_distrib]
+          rw [← hsum]
+          apply Finset.sum_congr rfl
+          intro z _
+          rw [← hR, mul_sub]
+        have hd1 := hRP.P_degree x y hcomp heq
+        have hd2 := hSQ.P_degree x y hcomp heq
+        have hdeg := natDegree_sub_le (RP.2 x y) (SQ.2 x y)
+        have hzero := low_degree_reflect_eq_self (RP.2 x y - SQ.2 x y)
+          (rankDifference cs x y) (by omega) href
+        exact sub_eq_zero.mp hzero
+    · rw [hRP.P_zero _ _ hcomp, hSQ.P_zero _ _ hcomp]
+
+/-- Unique existence is still a separate obligation: this proves its
+uniqueness half only, without postulating a normalized family. -/
+theorem normalizedKL_unique (cs : CoxeterSystem M W)
+    {RP SQ : PolynomialFamilies W} (hRP : NormalizedKL cs RP) (hSQ : NormalizedKL cs SQ) :
+    RP = SQ :=
+  Prod.ext (normalized_R_unique cs hRP hSQ) (normalized_P_unique cs hRP hSQ)
+
+
+/-! Exchange consequences required for the source's finite-rank reduction
+and for consistency of the equal-parameter Hecke recursion. -/
+
+/-- Every word contains a reduced subword with the same product. -/
+theorem exists_reduced_subword (cs : CoxeterSystem M W) (ω : List B) :
+    ∃ ν : List B, ν.Sublist ω ∧ cs.IsReduced ν ∧ cs.wordProd ν = cs.wordProd ω := by
+  induction ω with
+  | nil => exact ⟨[], .refl _, by simp [CoxeterSystem.IsReduced], rfl⟩
+  | cons i ω ih =>
+    obtain ⟨ν, hsub, hν, heq⟩ := ih
+    rcases cs.length_simple_mul (cs.wordProd ν) i with hasc | hdesc
+    · refine ⟨i :: ν, hsub.cons_cons i, ?_, ?_⟩
+      · simpa [CoxeterSystem.IsReduced, cs.wordProd_cons, hν.eq] using hasc
+      · simp [cs.wordProd_cons, heq]
+    · obtain ⟨j, hj, hdel⟩ := CoxeterSupport.strong_exchange cs hν
+        ⟨cs.isReflection_simple i, by omega⟩
+      refine ⟨ν.eraseIdx j, ((ν.eraseIdx_sublist j).trans hsub).cons i, ?_, ?_⟩
+      · dsimp [CoxeterSystem.IsReduced]
+        rw [← hdel, List.length_eraseIdx, ite_eq_left hj]
+        have := hν.eq
+        omega
+      · rw [← hdel, heq, cs.wordProd_cons]
+
+/-- A single downward reflection step can be realized inside any reduced word. -/
+theorem subword_of_bruhat_step (cs : CoxeterSystem M W) {x y : W}
+    (hxy : BruhatStep cs x y) {ω : List B} (hω : cs.IsReduced ω)
+    (hy : cs.wordProd ω = y) :
+    ∃ ν : List B, ν.Sublist ω ∧ cs.IsReduced ν ∧ cs.wordProd ν = x := by
+  obtain ⟨hlen, t, ht, rfl⟩ := hxy
+  have hx : t * cs.wordProd ω = x := by simp [hy, ← mul_assoc, ht.mul_self]
+  obtain ⟨j, hj, hdel⟩ := CoxeterSupport.strong_exchange cs hω
+    ⟨ht, by
+      rw [hx, hy]
+      exact hlen⟩
+  obtain ⟨ν, hsub, hν, heq⟩ := exists_reduced_subword cs (ω.eraseIdx j)
+  exact ⟨ν, hsub.trans (ω.eraseIdx_sublist j), hν, heq.trans (hdel.symm.trans hx)⟩
+
+/-- Forward subword property, for every chosen reduced expression of the top. -/
+theorem reduced_subword_of_bruhat (cs : CoxeterSystem M W) {x y : W}
+    (hxy : BruhatLE cs x y) {ω : List B} (hω : cs.IsReduced ω)
+    (hy : cs.wordProd ω = y) :
+    ∃ ν : List B, ν.Sublist ω ∧ cs.IsReduced ν ∧ cs.wordProd ν = x := by
+  induction hxy using Relation.ReflTransGen.head_induction_on generalizing ω with
+  | refl => exact ⟨ω, .refl _, hω, hy⟩
+  | head hstep _ ih =>
+    obtain ⟨ν, hsub, hν, heq⟩ := ih hω hy
+    obtain ⟨μ, hμsub, hμ, hμeq⟩ := subword_of_bruhat_step cs hstep hν heq
+    exact ⟨μ, hμsub.trans hsub, hμ, hμeq⟩
+
+
+/-- Simple left multiplication preserves the orientation of every reflection
+edge except the edge carrying that simple reflection itself. -/
+theorem bruhatStep_simple_mul (cs : CoxeterSystem M W) {x y : W}
+    (hxy : BruhatStep cs x y) (i : B) (hne : y ≠ cs.simple i * x) :
+    BruhatStep cs (cs.simple i * x) (cs.simple i * y) := by
+  obtain ⟨hlen, t, ht, heq⟩ := hxy
+  refine ⟨?_, cs.simple i * t * cs.simple i, by simpa using ht.conj (cs.simple i), ?_⟩
+  · rcases cs.length_simple_mul y i with hasc | hdesc
+    · have hx := cs.length_simple_mul x i
+      omega
+    · obtain ⟨ω, hω, hωeq⟩ := cs.exists_isReduced (cs.simple i * y)
+      have hprod : cs.wordProd (i :: ω) = y := by
+        rw [cs.wordProd_cons, ← hωeq, cs.simple_mul_simple_cancel_left]
+      have hr : cs.IsReduced (i :: ω) := by
+        dsimp [CoxeterSystem.IsReduced]
+        rw [hprod, ← hω.eq, ← hωeq]
+        exact hdesc.symm
+      have htx : t * cs.wordProd (i :: ω) = x := by
+        rw [hprod, heq, ← mul_assoc, ht.mul_self, one_mul]
+      obtain ⟨j, hj, hdel⟩ := CoxeterSupport.strong_exchange cs hr
+        ⟨ht, by rw [htx, hprod]; exact hlen⟩
+      cases j with
+      | zero =>
+        have hx : x = cs.wordProd ω := by simpa [htx] using hdel
+        exact (hne (by rw [hx, ← cs.wordProd_cons, hprod])).elim
+      | succ j =>
+        have hx : x = cs.simple i * cs.wordProd (ω.eraseIdx j) := by
+          rw [htx, List.eraseIdx_cons_succ, cs.wordProd_cons] at hdel
+          exact hdel
+        have hsx : cs.simple i * x = cs.wordProd (ω.eraseIdx j) := by simp [hx]
+        have hj' : j < ω.length := by simpa using hj
+        have hlen' := cs.length_wordProd_le (ω.eraseIdx j)
+        rw [List.length_eraseIdx, ite_eq_left hj'] at hlen'
+        rw [hsx, hωeq, hω.eq]
+        omega
+  · rw [heq]
+    simp [mul_assoc]
+
+/-- The smaller of `x` and `s*x` in Bruhat order. -/
+noncomputable def lowerSimple (cs : CoxeterSystem M W) (i : B) (x : W) : W :=
+  if cs.length (cs.simple i * x) < cs.length x then cs.simple i * x else x
+
+/-- The larger of `x` and `s*x` in Bruhat order. -/
+noncomputable def upperSimple (cs : CoxeterSystem M W) (i : B) (x : W) : W :=
+  if cs.length (cs.simple i * x) < cs.length x then x else cs.simple i * x
+
+theorem lowerSimple_le (cs : CoxeterSystem M W) (i : B) (x : W) :
+    BruhatLE cs (lowerSimple cs i x) x := by
+  classical
+  unfold lowerSimple
+  split_ifs with h
+  · exact .single ⟨h, cs.simple i, cs.isReflection_simple i, by simp⟩
+  · exact .refl
+
+theorem le_upperSimple (cs : CoxeterSystem M W) (i : B) (x : W) :
+    BruhatLE cs x (upperSimple cs i x) := by
+  classical
+  unfold upperSimple
+  split_ifs with h
+  · exact .refl
+  · exact .single ⟨by have := cs.length_simple_mul x i; omega,
+      cs.simple i, cs.isReflection_simple i, rfl⟩
+
+@[simp] theorem lowerSimple_mul (cs : CoxeterSystem M W) (i : B) (x : W) :
+    lowerSimple cs i (cs.simple i * x) = lowerSimple cs i x := by
+  classical
+  unfold lowerSimple
+  rw [cs.simple_mul_simple_cancel_left]
+  have := cs.length_simple_mul x i
+  split_ifs <;> first | rfl | omega
+
+@[simp] theorem upperSimple_mul (cs : CoxeterSystem M W) (i : B) (x : W) :
+    upperSimple cs i (cs.simple i * x) = upperSimple cs i x := by
+  classical
+  unfold upperSimple
+  rw [cs.simple_mul_simple_cancel_left]
+  have := cs.length_simple_mul x i
+  split_ifs <;> first | rfl | omega
+
+theorem lowerSimple_step_mono (cs : CoxeterSystem M W) (i : B) {x y : W}
+    (hxy : BruhatStep cs x y) :
+    BruhatLE cs (lowerSimple cs i x) (lowerSimple cs i y) := by
+  classical
+  by_cases heq : y = cs.simple i * x
+  · rw [heq, lowerSimple_mul]
+    exact .refl
+  · have hs := bruhatStep_simple_mul cs hxy i heq
+    by_cases hx : cs.length (cs.simple i * x) < cs.length x
+    · by_cases hy : cs.length (cs.simple i * y) < cs.length y
+      · simp only [lowerSimple, ite_eq_left hx, ite_eq_left hy]
+        exact .single hs
+      · simp only [lowerSimple, ite_eq_left hx, ite_eq_right hy]
+        have hlow := lowerSimple_le cs i x
+        simp only [lowerSimple, ite_eq_left hx] at hlow
+        exact hlow.trans (.single hxy)
+    · by_cases hy : cs.length (cs.simple i * y) < cs.length y
+      · simp only [lowerSimple, ite_eq_right hx, ite_eq_left hy]
+        have hupp := le_upperSimple cs i x
+        simp only [upperSimple, ite_eq_right hx] at hupp
+        exact hupp.trans (.single hs)
+      · simp only [lowerSimple, ite_eq_right hx, ite_eq_right hy]
+        exact .single hxy
+
+theorem upperSimple_step_mono (cs : CoxeterSystem M W) (i : B) {x y : W}
+    (hxy : BruhatStep cs x y) :
+    BruhatLE cs (upperSimple cs i x) (upperSimple cs i y) := by
+  classical
+  by_cases heq : y = cs.simple i * x
+  · rw [heq, upperSimple_mul]
+    exact .refl
+  · have hs := bruhatStep_simple_mul cs hxy i heq
+    by_cases hx : cs.length (cs.simple i * x) < cs.length x
+    · by_cases hy : cs.length (cs.simple i * y) < cs.length y
+      · simp only [upperSimple, ite_eq_left hx, ite_eq_left hy]
+        exact .single hxy
+      · simp only [upperSimple, ite_eq_left hx, ite_eq_right hy]
+        have hupp := le_upperSimple cs i y
+        simp only [upperSimple, ite_eq_right hy] at hupp
+        exact (Relation.ReflTransGen.single hxy).trans hupp
+    · by_cases hy : cs.length (cs.simple i * y) < cs.length y
+      · simp only [upperSimple, ite_eq_right hx, ite_eq_left hy]
+        have hlow := lowerSimple_le cs i y
+        simp only [lowerSimple, ite_eq_left hy] at hlow
+        exact (Relation.ReflTransGen.single hs).trans hlow
+      · simp only [upperSimple, ite_eq_right hx, ite_eq_right hy]
+        exact .single hs
+
+theorem lowerSimple_mono (cs : CoxeterSystem M W) (i : B) {x y : W}
+    (hxy : BruhatLE cs x y) :
+    BruhatLE cs (lowerSimple cs i x) (lowerSimple cs i y) := by
+  induction hxy using Relation.ReflTransGen.head_induction_on with
+  | refl => exact .refl
+  | head hstep _ ih => exact (lowerSimple_step_mono cs i hstep).trans ih
+
+theorem upperSimple_mono (cs : CoxeterSystem M W) (i : B) {x y : W}
+    (hxy : BruhatLE cs x y) :
+    BruhatLE cs (upperSimple cs i x) (upperSimple cs i y) := by
+  induction hxy using Relation.ReflTransGen.head_induction_on with
+  | refl => exact .refl
+  | head hstep _ ih => exact (upperSimple_step_mono cs i hstep).trans ih
+
+
+/-- The standard lifting property, now for the actual strong order. -/
+theorem bruhat_lifting (cs : CoxeterSystem M W) {x y : W}
+    (hxy : BruhatLE cs x y) (i : B)
+    (hx : cs.length x < cs.length (cs.simple i * x))
+    (hy : cs.length (cs.simple i * y) < cs.length y) :
+    BruhatLE cs x (cs.simple i * y) ∧ BruhatLE cs (cs.simple i * x) y := by
+  have hx' : ¬cs.length (cs.simple i * x) < cs.length x := by omega
+  constructor
+  · simpa only [lowerSimple, ite_eq_right hx', ite_eq_left hy] using lowerSimple_mono cs i hxy
+  · simpa only [upperSimple, ite_eq_right hx', ite_eq_left hy] using upperSimple_mono cs i hxy
+
+/-- Converse subword property; no labels are added to Bruhat intervals. -/
+theorem bruhat_of_subword (cs : CoxeterSystem M W) {ν ω : List B}
+    (hsub : ν.Sublist ω) (hω : cs.IsReduced ω) :
+    BruhatLE cs (cs.wordProd ν) (cs.wordProd ω) := by
+  induction ω generalizing ν with
+  | nil =>
+    have : ν = [] := List.sublist_nil.mp hsub
+    subst ν
+    exact .refl
+  | cons i ω ih =>
+    have hω' : cs.IsReduced ω := by simpa using hω.drop 1
+    have hasc : cs.length (cs.simple i * cs.wordProd ω) = cs.length (cs.wordProd ω) + 1 := by
+      simpa [CoxeterSystem.IsReduced, cs.wordProd_cons, hω'.eq] using hω.eq
+    have htop : upperSimple cs i (cs.wordProd ω) = cs.wordProd (i :: ω) := by
+      simp [upperSimple, show ¬cs.length (cs.simple i * cs.wordProd ω) <
+        cs.length (cs.wordProd ω) by omega, cs.wordProd_cons]
+    rcases List.sublist_cons_iff.mp hsub with hsub' | ⟨μ, rfl, hμsub⟩
+    · exact (ih hsub' hω').trans (htop ▸ le_upperSimple cs i (cs.wordProd ω))
+    · have hle := upperSimple_mono cs i (ih hμsub hω')
+      rw [htop] at hle
+      have hx := le_upperSimple cs i (cs.simple i * cs.wordProd μ)
+      rw [upperSimple_mul] at hx
+      rw [cs.wordProd_cons]
+      exact hx.trans hle
+
+/-- The subword criterion with an arbitrary fixed reduced word for the top. -/
+theorem bruhat_iff_reduced_subword (cs : CoxeterSystem M W) {x : W} {ω : List B}
+    (hω : cs.IsReduced ω) :
+    BruhatLE cs x (cs.wordProd ω) ↔
+      ∃ ν : List B, ν.Sublist ω ∧ cs.IsReduced ν ∧ cs.wordProd ν = x := by
+  constructor
+  · intro hx
+    exact reduced_subword_of_bruhat cs hx hω rfl
+  · rintro ⟨ν, hsub, _, rfl⟩
+    exact bruhat_of_subword cs hsub hω
+
+/-- Precisely the finite-support part used by the manuscript's finite-rank
+reduction: all lower elements are words in the generators of any reduced top
+word. Equality of parabolic polynomial normalizations remains a later step. -/
+theorem lower_ideal_finite_generator_support (cs : CoxeterSystem M W) (b : W) :
+    ∃ J : Set B, J.Finite ∧ ∀ x, BruhatLE cs x b →
+      ∃ ν : List B, (∀ i ∈ ν, i ∈ J) ∧ cs.wordProd ν = x := by
+  obtain ⟨ω, hω, heq⟩ := cs.exists_isReduced b
+  refine ⟨{i | i ∈ ω}, ω.finite_toSet, ?_⟩
+  intro x hx
+  obtain ⟨ν, hsub, _, hprod⟩ := reduced_subword_of_bruhat cs hx hω heq.symm
+  exact ⟨ν, fun i hi => hsub.subset hi, hprod⟩
+
+
+/-- Inversion preserves the actual strong Bruhat graph. -/
+theorem bruhatStep_inv (cs : CoxeterSystem M W) {x y : W}
+    (hxy : BruhatStep cs x y) : BruhatStep cs x⁻¹ y⁻¹ := by
+  obtain ⟨hlen, t, ht, heq⟩ := hxy
+  refine ⟨by simpa using hlen, x⁻¹ * t * x, by simpa using ht.conj x⁻¹, ?_⟩
+  rw [heq, mul_inv_rev, ht.inv]
+  group
+
+theorem bruhatStep_mul_simple (cs : CoxeterSystem M W) {x y : W}
+    (hxy : BruhatStep cs x y) (i : B) (hne : y ≠ x * cs.simple i) :
+    BruhatStep cs (x * cs.simple i) (y * cs.simple i) := by
+  have hinv := bruhatStep_inv cs hxy
+  have hne' : y⁻¹ ≠ cs.simple i * x⁻¹ := by
+    intro heq
+    apply hne
+    have := congrArg Inv.inv heq
+    simpa [mul_inv_rev] using this
+  have h := bruhatStep_inv cs (bruhatStep_simple_mul cs hinv i hne')
+  simpa [mul_inv_rev] using h
+
+/-- The noncollapsed left/right simple square has consistent orientations. -/
+theorem left_descent_mul_simple_iff (cs : CoxeterSystem M W) (i j : B) (x : W)
+    (hne : cs.simple i * x ≠ x * cs.simple j) :
+    cs.length (cs.simple i * (x * cs.simple j)) < cs.length (x * cs.simple j) ↔
+      cs.length (cs.simple i * x) < cs.length x := by
+  by_cases hx : cs.length (cs.simple i * x) < cs.length x
+  · have hedge : BruhatStep cs (cs.simple i * x) x :=
+      ⟨hx, cs.simple i, cs.isReflection_simple i, by simp⟩
+    have hne' : x ≠ cs.simple i * x * cs.simple j := by
+      intro heq
+      apply hne
+      have := congrArg (fun z => z * cs.simple j) heq
+      simpa using this.symm
+    have h := (bruhatStep_mul_simple cs hedge j hne').1
+    simpa only [mul_assoc, hx, iff_true] using h
+  · have hedge : BruhatStep cs x (cs.simple i * x) :=
+      ⟨by have := cs.length_simple_mul x i; omega, cs.simple i,
+        cs.isReflection_simple i, rfl⟩
+    have h := (bruhatStep_mul_simple cs hedge j hne).1
+    rw [mul_assoc] at h
+    exact iff_of_false (by omega) hx
+
+theorem right_descent_simple_mul_iff (cs : CoxeterSystem M W) (i j : B) (x : W)
+    (hne : cs.simple i * x ≠ x * cs.simple j) :
+    cs.length (cs.simple i * x * cs.simple j) < cs.length (cs.simple i * x) ↔
+      cs.length (x * cs.simple j) < cs.length x := by
+  have hne' : cs.simple j * x⁻¹ ≠ x⁻¹ * cs.simple i := by
+    intro heq
+    apply hne
+    have h := congrArg Inv.inv heq
+    simpa [mul_inv_rev] using h.symm
+  have h := left_descent_mul_simple_iff cs j i x⁻¹ hne'
+  have h1 : cs.length (cs.simple i * x * cs.simple j) =
+      cs.length (cs.simple j * (x⁻¹ * cs.simple i)) := by
+    rw [← cs.length_inv (cs.simple i * x * cs.simple j)]
+    simp [mul_inv_rev]
+  have h2 : cs.length (cs.simple i * x) = cs.length (x⁻¹ * cs.simple i) := by
+    rw [← cs.length_inv (cs.simple i * x)]
+    simp [mul_inv_rev]
+  have h3 : cs.length (x * cs.simple j) = cs.length (cs.simple j * x⁻¹) := by
+    rw [← cs.length_inv (x * cs.simple j)]
+    simp [mul_inv_rev]
+  rw [h1, h2, h3]
+  simpa using h
+
+
+end KLInvariance
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/KazhdanLusztig/CanonicalKL.lean b/lean/OAI/RepresentationTheory/KazhdanLusztig/CanonicalKL.lean
index 1366e659787a636a42e5db7a24f2aa1eacb4f5cf..fa4df51c96f6884a9b1edb02a4b9246713536478 100644
--- a/lean/OAI/RepresentationTheory/KazhdanLusztig/CanonicalKL.lean
+++ b/lean/OAI/RepresentationTheory/KazhdanLusztig/CanonicalKL.lean
@@ -1,473 +1,473 @@
-import OAI.RepresentationTheory.KazhdanLusztig.OrdinaryR
-
-namespace OAI
-
-/-!
-Canonical low-part construction, strict halfdegree, reciprocity and uniqueness; certification of the selected polynomials and interval ranks.
--/
-
-section
-namespace KLInvariance
-
-open Polynomial
-
-universe u v u' v'
-
-variable {B : Type u} {W : Type v} [klPreservedInstance1 : Group W] {M : CoxeterMatrix B}
-
-noncomputable local instance : DecidableEq W := Classical.decEq W
-
-namespace CanonicalP
-
-/-- The strict lower half of a polynomial, used in the KL recursion. -/
-noncomputable def lowPart (d : ℕ) (p : ℤ[X]) : ℤ[X] :=
-  ∑ n ∈ Finset.range ((d + 1) / 2), Polynomial.monomial n (p.coeff n)
-
-@[simp] theorem coeff_lowPart (d : ℕ) (p : ℤ[X]) (n : ℕ) :
-    (lowPart d p).coeff n = if 2 * n < d then p.coeff n else 0 := by
-  classical
-  simp only [lowPart, Polynomial.finsetSum_coeff, Polynomial.coeff_monomial]
-  rw [Finset.sum_ite_eq']
-  have he : n < (d + 1) / 2 ↔ 2 * n < d := by omega
-  simp only [Finset.mem_range, he]
-
-theorem lowPart_degree (d : ℕ) (p : ℤ[X]) (hd : 0 < d) :
-    2 * (lowPart d p).natDegree < d := by
-  have h : (lowPart d p).natDegree ≤ (d - 1) / 2 := by
-    apply Polynomial.natDegree_le_iff_coeff_eq_zero.mpr
-    intro n hn
-    rw [coeff_lowPart, ite_eq_right (by omega)]
-  omega
-
-/-- Solve the anti-reciprocal defect with the required strict degree bound. -/
-theorem lowPart_solution (d : ℕ) (p : ℤ[X]) (hanti : reflect d p = -p) :
-    reflect d (-lowPart d p) = -lowPart d p + p := by
-  apply Polynomial.ext
-  intro n
-  have hc := congrArg (fun f : ℤ[X] => f.coeff n) hanti
-  simp only [Polynomial.coeff_reflect, Polynomial.coeff_neg] at hc
-  simp only [Polynomial.coeff_reflect, Polynomial.coeff_neg, Polynomial.coeff_add,
-    coeff_lowPart]
-  by_cases hn : n ≤ d
-  · rw [Polynomial.revAt_le hn] at hc ⊢
-    by_cases hlow : 2 * n < d
-    · rw [ite_eq_left hlow, ite_eq_right (by omega)]
-      ring
-    · rw [ite_eq_right hlow]
-      by_cases hmid : 2 * n = d
-      · have hm : d - n = n := by omega
-        rw [hm, ite_eq_right hlow]
-        rw [hm] at hc
-        linarith
-      · rw [ite_eq_left (by omega)]
-        linarith
-  · rw [Polynomial.revAt_eq_self_of_lt (by omega)] at hc ⊢
-    rw [ite_eq_right (by omega)]
-    linarith
-
-/-- The standard lower-half KL recursion, on the actual Bruhat interval.
-Its reciprocity will be proved from the genuine R-kernel, not stipulated. -/
-noncomputable def polynomial (cs : CoxeterSystem M W) (x y : W) : ℤ[X] := by
-  classical
-  exact if BruhatLE cs x y then
-    if x = y then 1 else
-      -lowPart (rankDifference cs x y)
-        (∑ z ∈ (properIntervalFinset cs x y).attach,
-          OrdinaryR.polynomial cs x z.val * polynomial cs z.val y)
-    else 0
-termination_by rankDifference cs x y
-decreasing_by
-  have hz := (mem_properIntervalFinset cs x y z.val).mp z.property
-  have hlt := length_lt_of_bruhat_ne cs hz.2.1 hz.2.2
-  have hle := length_le_of_bruhat cs hz.1
-  dsimp [rankDifference]
-  omega
-
-@[simp] theorem polynomial_diagonal (cs : CoxeterSystem M W) (x : W) :
-    polynomial cs x x = 1 := by
-  rw [polynomial]
-  simp [bruhat_refl]
-
-theorem polynomial_zero (cs : CoxeterSystem M W) (x y : W)
-    (hxy : ¬BruhatLE cs x y) : polynomial cs x y = 0 := by
-  rw [polynomial, ite_eq_right hxy]
-
-noncomputable def defect (cs : CoxeterSystem M W) (x y : W) : ℤ[X] :=
-  ∑ z ∈ properIntervalFinset cs x y, OrdinaryR.polynomial cs x z * polynomial cs z y
-
-theorem polynomial_recursion (cs : CoxeterSystem M W) (x y : W)
-    (hxy : BruhatLE cs x y) (hne : x ≠ y) :
-    polynomial cs x y = -lowPart (rankDifference cs x y) (defect cs x y) := by
-  rw [polynomial, ite_eq_left hxy, ite_eq_right hne]
-  rw [Finset.sum_attach _ (fun z => OrdinaryR.polynomial cs x z * polynomial cs z y), defect]
-
-theorem polynomial_strict_degree (cs : CoxeterSystem M W) (x y : W)
-    (hxy : BruhatLE cs x y) (hne : x ≠ y) :
-    2 * (polynomial cs x y).natDegree < rankDifference cs x y := by
-  rw [polynomial_recursion cs x y hxy hne, Polynomial.natDegree_neg]
-  apply lowPart_degree
-  have := length_lt_of_bruhat_ne cs hxy hne
-  dsimp [rankDifference]
-  omega
-
-theorem polynomial_degree (cs : CoxeterSystem M W) (x y : W) :
-    (polynomial cs x y).natDegree ≤ rankDifference cs x y := by
-  by_cases hxy : BruhatLE cs x y
-  · by_cases hne : x = y
-    · subst y
-      simp
-    · have := polynomial_strict_degree cs x y hxy hne
-      omega
-  · rw [polynomial_zero cs x y hxy]
-    simp
-
-theorem defect_full_sum (cs : CoxeterSystem M W) (x y : W)
-    (hxy : BruhatLE cs x y) :
-    defect cs x y = (∑ z ∈ lowerFinset cs y,
-      OrdinaryR.polynomial cs x z * polynomial cs z y) - polynomial cs x y := by
-  rw [defect, sum_proper_eq_sub cs x y hxy]
-  · simp
-  · intro z hz
-    rw [OrdinaryR.polynomial_zero cs x z hz, zero_mul]
-
-theorem reversed_kernel_on_lower (cs : CoxeterSystem M W) (x y t : W)
-    (hty : BruhatLE cs t y) :
-    ∑ z ∈ lowerFinset cs y,
-      reflect (rankDifference cs x z) (OrdinaryR.polynomial cs x z) *
-        OrdinaryR.polynomial cs z t = if x = t then 1 else 0 := by
-  classical
-  rw [← OrdinaryR.polynomial_kernel_reverse cs x t]
-  symm
-  apply Finset.sum_subset
-  · intro z hz
-    exact (mem_lowerFinset cs y z).mpr
-      (bruhat_trans cs ((mem_lowerFinset cs t z).mp hz) hty)
-  · intro z hz hnt
-    have hzt : ¬BruhatLE cs z t := fun h => hnt ((mem_lowerFinset cs t z).mpr h)
-    rw [OrdinaryR.polynomial_zero cs z t hzt, mul_zero]
-
-theorem reversed_kernel_on_proper (cs : CoxeterSystem M W) (x y t : W)
-    (hxy : BruhatLE cs x y) (hty : BruhatLE cs t y) :
-    ∑ z ∈ properIntervalFinset cs x y,
-      reflect (rankDifference cs x z) (OrdinaryR.polynomial cs x z) *
-        OrdinaryR.polynomial cs z t =
-      (if x = t then 1 else 0) - OrdinaryR.polynomial cs x t := by
-  rw [sum_proper_eq_sub cs x y hxy]
-  · rw [reversed_kernel_on_lower cs x y t hty]
-    simp [rankDifference]
-  · intro z hz
-    rw [OrdinaryR.polynomial_zero cs x z hz, reflect_zero, zero_mul]
-
-/-- The reciprocity equation is a theorem of the lower-half construction. -/
-theorem polynomial_reciprocity (cs : CoxeterSystem M W) (x y : W)
-    (hxy : BruhatLE cs x y) :
-    reflect (rankDifference cs x y) (polynomial cs x y) =
-      ∑ z ∈ lowerFinset cs y, OrdinaryR.polynomial cs x z * polynomial cs z y := by
-  classical
-  suffices ∀ n : ℕ, ∀ x y : W, rankDifference cs x y = n → BruhatLE cs x y →
-      reflect (rankDifference cs x y) (polynomial cs x y) =
-        ∑ z ∈ lowerFinset cs y, OrdinaryR.polynomial cs x z * polynomial cs z y by
-    exact this _ x y rfl hxy
-  intro n
-  induction n using Nat.strong_induction_on with
-  | h n ih =>
-    intro x y hn hxy
-    by_cases hne : x = y
-    · subst y
-      rw [polynomial_diagonal]
-      simp only [rankDifference, Nat.sub_self, Polynomial.reflect_one, pow_zero]
-      symm
-      rw [Finset.sum_eq_single x]
-      · simp
-      · intro z hz hzx
-        have hnxz : ¬BruhatLE cs x z := fun hxz =>
-          hzx (bruhat_antisymm cs ((mem_lowerFinset cs x z).mp hz) hxz)
-        rw [OrdinaryR.polynomial_zero cs x z hnxz, zero_mul]
-      · intro hmiss
-        exact (hmiss ((mem_lowerFinset cs x x).mpr (bruhat_refl cs x))).elim
-    · have hanti : reflect (rankDifference cs x y) (defect cs x y) = -defect cs x y := by
-        rw [defect, reflect_finset_sum]
-        have heq : ∀ z ∈ properIntervalFinset cs x y,
-            reflect (rankDifference cs x y)
-              (OrdinaryR.polynomial cs x z * polynomial cs z y) =
-            reflect (rankDifference cs x z) (OrdinaryR.polynomial cs x z) *
-              (∑ t ∈ lowerFinset cs y, OrdinaryR.polynomial cs z t * polynomial cs t y) := by
-          intro z hz
-          have hz := (mem_properIntervalFinset cs x y z).mp hz
-          have hlt : rankDifference cs z y < n := by
-            have := length_lt_of_bruhat_ne cs hz.2.1 hz.2.2
-            have := length_le_of_bruhat cs hz.1
-            dsimp [rankDifference] at hn ⊢
-            omega
-          rw [rankDifference_add cs hz.2.1 hz.1,
-            reflect_mul _ _ (OrdinaryR.polynomial_degree cs x z) (polynomial_degree cs z y),
-            ih _ hlt z y rfl hz.1]
-        rw [Finset.sum_congr rfl heq]
-        simp_rw [Finset.mul_sum, ← mul_assoc]
-        rw [Finset.sum_comm]
-        simp_rw [← Finset.sum_mul]
-        have hkernel : ∀ t ∈ lowerFinset cs y,
-            (∑ z ∈ properIntervalFinset cs x y,
-              reflect (rankDifference cs x z) (OrdinaryR.polynomial cs x z) *
-                OrdinaryR.polynomial cs z t) * polynomial cs t y =
-            ((if x = t then 1 else 0) - OrdinaryR.polynomial cs x t) * polynomial cs t y := by
-          intro t ht
-          rw [reversed_kernel_on_proper cs x y t hxy ((mem_lowerFinset cs y t).mp ht)]
-        rw [Finset.sum_congr rfl hkernel]
-        simp_rw [sub_mul, Finset.sum_sub_distrib]
-        have hdelta : (∑ t ∈ lowerFinset cs y,
-            (if x = t then (1 : ℤ[X]) else 0) * polynomial cs t y) = polynomial cs x y := by
-          rw [Finset.sum_eq_single x]
-          · simp
-          · intro t ht htx
-            simp [Ne.symm htx]
-          · intro hmiss
-            exact (hmiss ((mem_lowerFinset cs y x).mpr hxy)).elim
-        rw [hdelta]
-        change polynomial cs x y - (∑ z ∈ lowerFinset cs y,
-          OrdinaryR.polynomial cs x z * polynomial cs z y) = -defect cs x y
-        rw [defect_full_sum cs x y hxy]
-        ring
-      rw [polynomial_recursion cs x y hxy hne, lowPart_solution _ _ hanti]
-      rw [← polynomial_recursion cs x y hxy hne, defect_full_sum cs x y hxy]
-      ring
-
-end CanonicalP
-
-/-- Finitely supported interval sums agree with the lower-ideal sums used in
-our construction; all incomparable summands are literally zero. -/
-theorem normalized_interval_sum (cs : CoxeterSystem M W) (x y : W) :
-    (∑ᶠ z : Interval cs x y,
-      OrdinaryR.polynomial cs x z.val * CanonicalP.polynomial cs z.val y) =
-    ∑ z ∈ lowerFinset cs y,
-      OrdinaryR.polynomial cs x z * CanonicalP.polynomial cs z y := by
-  classical
-  let := interval_finite cs x y
-  let : Fintype (Interval cs x y) := Fintype.ofFinite _
-  rw [finsum_eq_sum_of_fintype]
-  have hsum := Finset.sum_subtype (F := (inferInstance : Fintype (Interval cs x y)))
-    ((lowerFinset cs y).filter (fun z => BruhatLE cs x z))
-    (fun z => by simp only [Finset.mem_filter, mem_lowerFinset]; exact and_comm)
-    (fun z => OrdinaryR.polynomial cs x z * CanonicalP.polynomial cs z y)
-  refine hsum.symm.trans ?_
-  apply Finset.sum_subset (Finset.filter_subset _ _)
-  intro z hz hnot
-  have hnxz : ¬BruhatLE cs x z := fun hxz => hnot (Finset.mem_filter.mpr ⟨hz, hxz⟩)
-  rw [OrdinaryR.polynomial_zero cs x z hnxz, zero_mul]
-
-/-- The selected polynomials really exist with the classical equal-parameter
-normalization. This is a construction, not an extra hypothesis. -/
-theorem constructed_normalizedKL (cs : CoxeterSystem M W) :
-    NormalizedKL cs (OrdinaryR.polynomial cs, CanonicalP.polynomial cs) where
-  R_diagonal := OrdinaryR.polynomial_diagonal cs
-  R_zero := OrdinaryR.polynomial_zero cs
-  R_recursion := OrdinaryR.polynomial_recursion cs
-  P_diagonal := CanonicalP.polynomial_diagonal cs
-  P_zero := CanonicalP.polynomial_zero cs
-  P_degree := CanonicalP.polynomial_strict_degree cs
-  reciprocity := by
-    intro x y hxy
-    rw [normalized_interval_sum]
-    exact CanonicalP.polynomial_reciprocity cs x y hxy
-
-theorem normalizedKL_existsUnique (cs : CoxeterSystem M W) :
-    ∃! RP : PolynomialFamilies W, NormalizedKL cs RP := by
-  refine ⟨(OrdinaryR.polynomial cs, CanonicalP.polynomial cs), constructed_normalizedKL cs, ?_⟩
-  intro RP hRP
-  exact normalizedKL_unique cs hRP (constructed_normalizedKL cs)
-
-theorem klFamilies_spec (cs : CoxeterSystem M W) : NormalizedKL cs (klFamilies cs) := by
-  exact Classical.epsilon_spec (normalizedKL_existsUnique cs).exists
-
-theorem klFamilies_eq_constructed (cs : CoxeterSystem M W) :
-    klFamilies cs = (OrdinaryR.polynomial cs, CanonicalP.polynomial cs) :=
-  normalizedKL_unique cs (klFamilies_spec cs) (constructed_normalizedKL cs)
-
-theorem klPolynomial_eq_constructed (cs : CoxeterSystem M W) (x y : W) :
-    klPolynomial cs x y = CanonicalP.polynomial cs x y := by
-  simp only [klPolynomial, klFamilies_eq_constructed]
-
-theorem rPolynomial_eq_constructed (cs : CoxeterSystem M W) (x y : W) :
-    rPolynomial cs x y = OrdinaryR.polynomial cs x y := by
-  simp only [rPolynomial, klFamilies_eq_constructed]
-
-
-/-- Every nontrivial Bruhat interval contains a first step of length one.
-This is the grading fact needed to transport rank differences without labels. -/
-theorem exists_bruhat_successor (cs : CoxeterSystem M W) {x y : W}
-    (hxy : BruhatLE cs x y) (hne : x ≠ y) :
-    ∃ z, BruhatLE cs x z ∧ BruhatLE cs z y ∧ cs.length z = cs.length x + 1 := by
-  classical
-  suffices ∀ n : ℕ, ∀ y : W, cs.length y = n → ∀ x : W,
-      BruhatLE cs x y → x ≠ y →
-      ∃ z, BruhatLE cs x z ∧ BruhatLE cs z y ∧ cs.length z = cs.length x + 1 by
-    exact this _ y rfl x hxy hne
-  intro n
-  induction n using Nat.strong_induction_on with
-  | h n ih =>
-    intro y hyn x hxy hne
-    have hy1 : y ≠ 1 := by
-      intro hy
-      subst y
-      have hlen := length_le_of_bruhat cs hxy
-      have hx : cs.length x = 0 := by simpa using hlen
-      exact hne (cs.length_eq_zero_iff.mp hx)
-    obtain ⟨i, hi⟩ := cs.exists_leftDescent_of_ne_one hy1
-    change cs.length (cs.simple i * y) < cs.length y at hi
-    have hylen : cs.length y = cs.length (cs.simple i * y) + 1 := by
-      have := cs.length_simple_mul y i
-      omega
-    have hsyn : cs.length (cs.simple i * y) < n := by omega
-    have hsyy : BruhatLE cs (cs.simple i * y) y := by
-      simpa only [lowerSimple, ite_eq_left hi] using lowerSimple_le cs i y
-    by_cases hxsy : BruhatLE cs x (cs.simple i * y)
-    · by_cases hxeq : x = cs.simple i * y
-      · exact ⟨y, hxy, bruhat_refl cs y, by simpa [hxeq] using hylen⟩
-      · obtain ⟨z, hxz, hzsy, hzlen⟩ := ih _ hsyn _ rfl x hxsy hxeq
-        exact ⟨z, hxz, bruhat_trans cs hzsy hsyy, hzlen⟩
-    · have hxi : cs.length (cs.simple i * x) < cs.length x := by
-        by_contra hnxi
-        have hxim : cs.length x < cs.length (cs.simple i * x) := by
-          have := cs.length_simple_mul x i
-          omega
-        exact hxsy (bruhat_lifting cs hxy i hxim hi).1
-      have hxlen : cs.length x = cs.length (cs.simple i * x) + 1 := by
-        have := cs.length_simple_mul x i
-        omega
-      have hsxsy : BruhatLE cs (cs.simple i * x) (cs.simple i * y) := by
-        simpa only [lowerSimple, ite_eq_left hxi, ite_eq_left hi] using lowerSimple_mono cs i hxy
-      have hne' : cs.simple i * x ≠ cs.simple i * y := fun he => hne (mul_left_cancel he)
-      obtain ⟨w, hsxw, hwsy, hwlen⟩ := ih _ hsyn _ rfl _ hsxsy hne'
-      have hux : upperSimple cs i (cs.simple i * x) = x := by
-        rw [upperSimple_mul]
-        simp only [upperSimple, ite_eq_left hxi]
-      have huy : upperSimple cs i (cs.simple i * y) = y := by
-        rw [upperSimple_mul]
-        simp only [upperSimple, ite_eq_left hi]
-      have hxz := upperSimple_mono cs i hsxw
-      have hzy := upperSimple_mono cs i hwsy
-      rw [hux] at hxz
-      rw [huy] at hzy
-      have hwlen' : cs.length w = cs.length x := by omega
-      have hwi : ¬cs.length (cs.simple i * w) < cs.length w := by
-        intro hwi
-        simp only [upperSimple, ite_eq_left hwi] at hxz
-        have hxw : x = w := by
-          by_contra hnxw
-          have := length_lt_of_bruhat_ne cs hxz hnxw
-          omega
-        exact hxsy (hxw ▸ hwsy)
-      refine ⟨upperSimple cs i w, hxz, hzy, ?_⟩
-      simp only [upperSimple, ite_eq_right hwi]
-      have := cs.length_simple_mul w i
-      omega
-
-
-/-- An order embedding between Bruhat intervals cannot shorten rank. -/
-theorem rankDifference_le_of_strictMono
-    {B' : Type u'} {W' : Type v'} [Group W'] {M' : CoxeterMatrix B'}
-    (cs : CoxeterSystem M W) (cs' : CoxeterSystem M' W')
-    {u b : W} {u' b' : W'}
-    (f : Interval cs u b → Interval cs' u' b') (hf : StrictMono f)
-    {x y : Interval cs u b} (hxy : x ≤ y) :
-    rankDifference cs x.val y.val ≤ rankDifference cs' (f x).val (f y).val := by
-  suffices ∀ n : ℕ, ∀ x y : Interval cs u b,
-      rankDifference cs x.val y.val = n → x ≤ y →
-      rankDifference cs x.val y.val ≤ rankDifference cs' (f x).val (f y).val by
-    exact this _ x y rfl hxy
-  intro n
-  induction n using Nat.strong_induction_on with
-  | h n ih =>
-    intro x y hn hxy
-    by_cases heq : x = y
-    · subst y
-      simp [rankDifference]
-    · have hne : x.val ≠ y.val := fun hv => heq (Subtype.ext hv)
-      obtain ⟨z, hxz, hzy, hzlen⟩ := exists_bruhat_successor cs hxy hne
-      let zI : Interval cs u b := ⟨z, bruhat_trans cs x.property.1 hxz,
-        bruhat_trans cs hzy y.property.2⟩
-      have hxylen := length_le_of_bruhat cs hxy
-      have hzylen := length_le_of_bruhat cs hzy
-      have hsmall : rankDifference cs zI.val y.val < n := by
-        dsimp [zI, rankDifference] at hn ⊢
-        omega
-      have hsub := ih _ hsmall zI y rfl hzy
-      have hxz' : x < zI := by
-        refine (lt_iff_le_and_ne).mpr ⟨hxz, ?_⟩
-        intro hx
-        have hv : x.val = z := congrArg Subtype.val hx
-        have := congrArg cs.length hv
-        omega
-      have hfxz := hf hxz'
-      have hfxzlen : cs'.length (f x).val < cs'.length (f zI).val := by
-        apply length_lt_of_bruhat_ne cs' hfxz.le
-        intro hv
-        exact hfxz.ne (Subtype.ext hv)
-      have hfzyle := length_le_of_bruhat cs' (hf.monotone (show zI ≤ y from hzy))
-      change cs.length y.val - cs.length z ≤
-        cs'.length (f y).val - cs'.length (f zI).val at hsub
-      change cs.length y.val - cs.length x.val ≤
-        cs'.length (f y).val - cs'.length (f x).val
-      omega
-
-/-- Rank is intrinsic to the abstract interval poset, with neither root
-labels nor simple generators preserved by the isomorphism. -/
-theorem intervalIso_rankDifference
-    {B' : Type u'} {W' : Type v'} [Group W'] {M' : CoxeterMatrix B'}
-    (cs : CoxeterSystem M W) (cs' : CoxeterSystem M' W')
-    {u b : W} {u' b' : W'}
-    (ι : Interval cs u b ≃o Interval cs' u' b')
-    {x y : Interval cs u b} (hxy : x ≤ y) :
-    rankDifference cs x.val y.val = rankDifference cs' (ι x).val (ι y).val := by
-  apply Nat.le_antisymm
-  · exact rankDifference_le_of_strictMono cs cs' ι ι.strictMono hxy
-  · have h := rankDifference_le_of_strictMono cs' cs ι.symm ι.symm.strictMono (ι.monotone hxy)
-    simpa only [OrderIso.symm_apply_apply] using h
-
-/-- Named endpoints for an actual nonempty interval. -/
-def intervalBottom (cs : CoxeterSystem M W) {u b : W} (hub : BruhatLE cs u b) :
-    Interval cs u b := ⟨u, bruhat_refl cs u, hub⟩
-
-def intervalTop (cs : CoxeterSystem M W) {u b : W} (hub : BruhatLE cs u b) :
-    Interval cs u b := ⟨b, hub, bruhat_refl cs b⟩
-
-theorem intervalIso_bottom
-    {B' : Type u'} {W' : Type v'} [Group W'] {M' : CoxeterMatrix B'}
-    (cs : CoxeterSystem M W) (cs' : CoxeterSystem M' W')
-    {u b : W} {u' b' : W'} (hub : BruhatLE cs u b) (hub' : BruhatLE cs' u' b')
-    (ι : Interval cs u b ≃o Interval cs' u' b') :
-    ι (intervalBottom cs hub) = intervalBottom cs' hub' := by
-  apply le_antisymm
-  · have h : intervalBottom cs hub ≤ ι.symm (intervalBottom cs' hub') :=
-      (ι.symm (intervalBottom cs' hub')).property.1
-    simpa only [OrderIso.apply_symm_apply] using ι.monotone h
-  · exact (ι (intervalBottom cs hub)).property.1
-
-theorem intervalIso_top
-    {B' : Type u'} {W' : Type v'} [Group W'] {M' : CoxeterMatrix B'}
-    (cs : CoxeterSystem M W) (cs' : CoxeterSystem M' W')
-    {u b : W} {u' b' : W'} (hub : BruhatLE cs u b) (hub' : BruhatLE cs' u' b')
-    (ι : Interval cs u b ≃o Interval cs' u' b') :
-    ι (intervalTop cs hub) = intervalTop cs' hub' := by
-  apply le_antisymm
-  · exact (ι (intervalTop cs hub)).property.2
-  · have h : ι.symm (intervalTop cs' hub') ≤ intervalTop cs hub :=
-      (ι.symm (intervalTop cs' hub')).property.2
-    simpa only [OrderIso.apply_symm_apply] using ι.monotone h
-
-theorem intervalIso_full_rankDifference
-    {B' : Type u'} {W' : Type v'} [Group W'] {M' : CoxeterMatrix B'}
-    (cs : CoxeterSystem M W) (cs' : CoxeterSystem M' W')
-    {u b : W} {u' b' : W'} (hub : BruhatLE cs u b) (hub' : BruhatLE cs' u' b')
-    (ι : Interval cs u b ≃o Interval cs' u' b') :
-    rankDifference cs u b = rankDifference cs' u' b' := by
-  have h := intervalIso_rankDifference cs cs' ι
-    (show intervalBottom cs hub ≤ intervalTop cs hub from hub)
-  rw [intervalIso_bottom cs cs' hub hub' ι, intervalIso_top cs cs' hub hub' ι] at h
-  exact h
-
-end KLInvariance
-
-end
-
-end OAI
+import OAI.RepresentationTheory.KazhdanLusztig.OrdinaryR
+
+namespace OAI
+
+/-!
+Canonical low-part construction, strict halfdegree, reciprocity and uniqueness; certification of the selected polynomials and interval ranks.
+-/
+
+section
+namespace KLInvariance
+
+open Polynomial
+
+universe u v u' v'
+
+variable {B : Type u} {W : Type v} [klPreservedInstance1 : Group W] {M : CoxeterMatrix B}
+
+noncomputable local instance : DecidableEq W := Classical.decEq W
+
+namespace CanonicalP
+
+/-- The strict lower half of a polynomial, used in the KL recursion. -/
+noncomputable def lowPart (d : ℕ) (p : ℤ[X]) : ℤ[X] :=
+  ∑ n ∈ Finset.range ((d + 1) / 2), Polynomial.monomial n (p.coeff n)
+
+@[simp] theorem coeff_lowPart (d : ℕ) (p : ℤ[X]) (n : ℕ) :
+    (lowPart d p).coeff n = if 2 * n < d then p.coeff n else 0 := by
+  classical
+  simp only [lowPart, Polynomial.finsetSum_coeff, Polynomial.coeff_monomial]
+  rw [Finset.sum_ite_eq']
+  have he : n < (d + 1) / 2 ↔ 2 * n < d := by omega
+  simp only [Finset.mem_range, he]
+
+theorem lowPart_degree (d : ℕ) (p : ℤ[X]) (hd : 0 < d) :
+    2 * (lowPart d p).natDegree < d := by
+  have h : (lowPart d p).natDegree ≤ (d - 1) / 2 := by
+    apply Polynomial.natDegree_le_iff_coeff_eq_zero.mpr
+    intro n hn
+    rw [coeff_lowPart, ite_eq_right (by omega)]
+  omega
+
+/-- Solve the anti-reciprocal defect with the required strict degree bound. -/
+theorem lowPart_solution (d : ℕ) (p : ℤ[X]) (hanti : reflect d p = -p) :
+    reflect d (-lowPart d p) = -lowPart d p + p := by
+  apply Polynomial.ext
+  intro n
+  have hc := congrArg (fun f : ℤ[X] => f.coeff n) hanti
+  simp only [Polynomial.coeff_reflect, Polynomial.coeff_neg] at hc
+  simp only [Polynomial.coeff_reflect, Polynomial.coeff_neg, Polynomial.coeff_add,
+    coeff_lowPart]
+  by_cases hn : n ≤ d
+  · rw [Polynomial.revAt_le hn] at hc ⊢
+    by_cases hlow : 2 * n < d
+    · rw [ite_eq_left hlow, ite_eq_right (by omega)]
+      ring
+    · rw [ite_eq_right hlow]
+      by_cases hmid : 2 * n = d
+      · have hm : d - n = n := by omega
+        rw [hm, ite_eq_right hlow]
+        rw [hm] at hc
+        linarith
+      · rw [ite_eq_left (by omega)]
+        linarith
+  · rw [Polynomial.revAt_eq_self_of_lt (by omega)] at hc ⊢
+    rw [ite_eq_right (by omega)]
+    linarith
+
+/-- The standard lower-half KL recursion, on the actual Bruhat interval.
+Its reciprocity will be proved from the genuine R-kernel, not stipulated. -/
+noncomputable def polynomial (cs : CoxeterSystem M W) (x y : W) : ℤ[X] := by
+  classical
+  exact if BruhatLE cs x y then
+    if x = y then 1 else
+      -lowPart (rankDifference cs x y)
+        (∑ z ∈ (properIntervalFinset cs x y).attach,
+          OrdinaryR.polynomial cs x z.val * polynomial cs z.val y)
+    else 0
+termination_by rankDifference cs x y
+decreasing_by
+  have hz := (mem_properIntervalFinset cs x y z.val).mp z.property
+  have hlt := length_lt_of_bruhat_ne cs hz.2.1 hz.2.2
+  have hle := length_le_of_bruhat cs hz.1
+  dsimp [rankDifference]
+  omega
+
+@[simp] theorem polynomial_diagonal (cs : CoxeterSystem M W) (x : W) :
+    polynomial cs x x = 1 := by
+  rw [polynomial]
+  simp [bruhat_refl]
+
+theorem polynomial_zero (cs : CoxeterSystem M W) (x y : W)
+    (hxy : ¬BruhatLE cs x y) : polynomial cs x y = 0 := by
+  rw [polynomial, ite_eq_right hxy]
+
+noncomputable def defect (cs : CoxeterSystem M W) (x y : W) : ℤ[X] :=
+  ∑ z ∈ properIntervalFinset cs x y, OrdinaryR.polynomial cs x z * polynomial cs z y
+
+theorem polynomial_recursion (cs : CoxeterSystem M W) (x y : W)
+    (hxy : BruhatLE cs x y) (hne : x ≠ y) :
+    polynomial cs x y = -lowPart (rankDifference cs x y) (defect cs x y) := by
+  rw [polynomial, ite_eq_left hxy, ite_eq_right hne]
+  rw [Finset.sum_attach _ (fun z => OrdinaryR.polynomial cs x z * polynomial cs z y), defect]
+
+theorem polynomial_strict_degree (cs : CoxeterSystem M W) (x y : W)
+    (hxy : BruhatLE cs x y) (hne : x ≠ y) :
+    2 * (polynomial cs x y).natDegree < rankDifference cs x y := by
+  rw [polynomial_recursion cs x y hxy hne, Polynomial.natDegree_neg]
+  apply lowPart_degree
+  have := length_lt_of_bruhat_ne cs hxy hne
+  dsimp [rankDifference]
+  omega
+
+theorem polynomial_degree (cs : CoxeterSystem M W) (x y : W) :
+    (polynomial cs x y).natDegree ≤ rankDifference cs x y := by
+  by_cases hxy : BruhatLE cs x y
+  · by_cases hne : x = y
+    · subst y
+      simp
+    · have := polynomial_strict_degree cs x y hxy hne
+      omega
+  · rw [polynomial_zero cs x y hxy]
+    simp
+
+theorem defect_full_sum (cs : CoxeterSystem M W) (x y : W)
+    (hxy : BruhatLE cs x y) :
+    defect cs x y = (∑ z ∈ lowerFinset cs y,
+      OrdinaryR.polynomial cs x z * polynomial cs z y) - polynomial cs x y := by
+  rw [defect, sum_proper_eq_sub cs x y hxy]
+  · simp
+  · intro z hz
+    rw [OrdinaryR.polynomial_zero cs x z hz, zero_mul]
+
+theorem reversed_kernel_on_lower (cs : CoxeterSystem M W) (x y t : W)
+    (hty : BruhatLE cs t y) :
+    ∑ z ∈ lowerFinset cs y,
+      reflect (rankDifference cs x z) (OrdinaryR.polynomial cs x z) *
+        OrdinaryR.polynomial cs z t = if x = t then 1 else 0 := by
+  classical
+  rw [← OrdinaryR.polynomial_kernel_reverse cs x t]
+  symm
+  apply Finset.sum_subset
+  · intro z hz
+    exact (mem_lowerFinset cs y z).mpr
+      (bruhat_trans cs ((mem_lowerFinset cs t z).mp hz) hty)
+  · intro z hz hnt
+    have hzt : ¬BruhatLE cs z t := fun h => hnt ((mem_lowerFinset cs t z).mpr h)
+    rw [OrdinaryR.polynomial_zero cs z t hzt, mul_zero]
+
+theorem reversed_kernel_on_proper (cs : CoxeterSystem M W) (x y t : W)
+    (hxy : BruhatLE cs x y) (hty : BruhatLE cs t y) :
+    ∑ z ∈ properIntervalFinset cs x y,
+      reflect (rankDifference cs x z) (OrdinaryR.polynomial cs x z) *
+        OrdinaryR.polynomial cs z t =
+      (if x = t then 1 else 0) - OrdinaryR.polynomial cs x t := by
+  rw [sum_proper_eq_sub cs x y hxy]
+  · rw [reversed_kernel_on_lower cs x y t hty]
+    simp [rankDifference]
+  · intro z hz
+    rw [OrdinaryR.polynomial_zero cs x z hz, reflect_zero, zero_mul]
+
+/-- The reciprocity equation is a theorem of the lower-half construction. -/
+theorem polynomial_reciprocity (cs : CoxeterSystem M W) (x y : W)
+    (hxy : BruhatLE cs x y) :
+    reflect (rankDifference cs x y) (polynomial cs x y) =
+      ∑ z ∈ lowerFinset cs y, OrdinaryR.polynomial cs x z * polynomial cs z y := by
+  classical
+  suffices ∀ n : ℕ, ∀ x y : W, rankDifference cs x y = n → BruhatLE cs x y →
+      reflect (rankDifference cs x y) (polynomial cs x y) =
+        ∑ z ∈ lowerFinset cs y, OrdinaryR.polynomial cs x z * polynomial cs z y by
+    exact this _ x y rfl hxy
+  intro n
+  induction n using Nat.strong_induction_on with
+  | h n ih =>
+    intro x y hn hxy
+    by_cases hne : x = y
+    · subst y
+      rw [polynomial_diagonal]
+      simp only [rankDifference, Nat.sub_self, Polynomial.reflect_one, pow_zero]
+      symm
+      rw [Finset.sum_eq_single x]
+      · simp
+      · intro z hz hzx
+        have hnxz : ¬BruhatLE cs x z := fun hxz =>
+          hzx (bruhat_antisymm cs ((mem_lowerFinset cs x z).mp hz) hxz)
+        rw [OrdinaryR.polynomial_zero cs x z hnxz, zero_mul]
+      · intro hmiss
+        exact (hmiss ((mem_lowerFinset cs x x).mpr (bruhat_refl cs x))).elim
+    · have hanti : reflect (rankDifference cs x y) (defect cs x y) = -defect cs x y := by
+        rw [defect, reflect_finset_sum]
+        have heq : ∀ z ∈ properIntervalFinset cs x y,
+            reflect (rankDifference cs x y)
+              (OrdinaryR.polynomial cs x z * polynomial cs z y) =
+            reflect (rankDifference cs x z) (OrdinaryR.polynomial cs x z) *
+              (∑ t ∈ lowerFinset cs y, OrdinaryR.polynomial cs z t * polynomial cs t y) := by
+          intro z hz
+          have hz := (mem_properIntervalFinset cs x y z).mp hz
+          have hlt : rankDifference cs z y < n := by
+            have := length_lt_of_bruhat_ne cs hz.2.1 hz.2.2
+            have := length_le_of_bruhat cs hz.1
+            dsimp [rankDifference] at hn ⊢
+            omega
+          rw [rankDifference_add cs hz.2.1 hz.1,
+            reflect_mul _ _ (OrdinaryR.polynomial_degree cs x z) (polynomial_degree cs z y),
+            ih _ hlt z y rfl hz.1]
+        rw [Finset.sum_congr rfl heq]
+        simp_rw [Finset.mul_sum, ← mul_assoc]
+        rw [Finset.sum_comm]
+        simp_rw [← Finset.sum_mul]
+        have hkernel : ∀ t ∈ lowerFinset cs y,
+            (∑ z ∈ properIntervalFinset cs x y,
+              reflect (rankDifference cs x z) (OrdinaryR.polynomial cs x z) *
+                OrdinaryR.polynomial cs z t) * polynomial cs t y =
+            ((if x = t then 1 else 0) - OrdinaryR.polynomial cs x t) * polynomial cs t y := by
+          intro t ht
+          rw [reversed_kernel_on_proper cs x y t hxy ((mem_lowerFinset cs y t).mp ht)]
+        rw [Finset.sum_congr rfl hkernel]
+        simp_rw [sub_mul, Finset.sum_sub_distrib]
+        have hdelta : (∑ t ∈ lowerFinset cs y,
+            (if x = t then (1 : ℤ[X]) else 0) * polynomial cs t y) = polynomial cs x y := by
+          rw [Finset.sum_eq_single x]
+          · simp
+          · intro t ht htx
+            simp [Ne.symm htx]
+          · intro hmiss
+            exact (hmiss ((mem_lowerFinset cs y x).mpr hxy)).elim
+        rw [hdelta]
+        change polynomial cs x y - (∑ z ∈ lowerFinset cs y,
+          OrdinaryR.polynomial cs x z * polynomial cs z y) = -defect cs x y
+        rw [defect_full_sum cs x y hxy]
+        ring
+      rw [polynomial_recursion cs x y hxy hne, lowPart_solution _ _ hanti]
+      rw [← polynomial_recursion cs x y hxy hne, defect_full_sum cs x y hxy]
+      ring
+
+end CanonicalP
+
+/-- Finitely supported interval sums agree with the lower-ideal sums used in
+our construction; all incomparable summands are literally zero. -/
+theorem normalized_interval_sum (cs : CoxeterSystem M W) (x y : W) :
+    (∑ᶠ z : Interval cs x y,
+      OrdinaryR.polynomial cs x z.val * CanonicalP.polynomial cs z.val y) =
+    ∑ z ∈ lowerFinset cs y,
+      OrdinaryR.polynomial cs x z * CanonicalP.polynomial cs z y := by
+  classical
+  let := interval_finite cs x y
+  let : Fintype (Interval cs x y) := Fintype.ofFinite _
+  rw [finsum_eq_sum_of_fintype]
+  have hsum := Finset.sum_subtype (F := (inferInstance : Fintype (Interval cs x y)))
+    ((lowerFinset cs y).filter (fun z => BruhatLE cs x z))
+    (fun z => by simp only [Finset.mem_filter, mem_lowerFinset]; exact and_comm)
+    (fun z => OrdinaryR.polynomial cs x z * CanonicalP.polynomial cs z y)
+  refine hsum.symm.trans ?_
+  apply Finset.sum_subset (Finset.filter_subset _ _)
+  intro z hz hnot
+  have hnxz : ¬BruhatLE cs x z := fun hxz => hnot (Finset.mem_filter.mpr ⟨hz, hxz⟩)
+  rw [OrdinaryR.polynomial_zero cs x z hnxz, zero_mul]
+
+/-- The selected polynomials really exist with the classical equal-parameter
+normalization. This is a construction, not an extra hypothesis. -/
+theorem constructed_normalizedKL (cs : CoxeterSystem M W) :
+    NormalizedKL cs (OrdinaryR.polynomial cs, CanonicalP.polynomial cs) where
+  R_diagonal := OrdinaryR.polynomial_diagonal cs
+  R_zero := OrdinaryR.polynomial_zero cs
+  R_recursion := OrdinaryR.polynomial_recursion cs
+  P_diagonal := CanonicalP.polynomial_diagonal cs
+  P_zero := CanonicalP.polynomial_zero cs
+  P_degree := CanonicalP.polynomial_strict_degree cs
+  reciprocity := by
+    intro x y hxy
+    rw [normalized_interval_sum]
+    exact CanonicalP.polynomial_reciprocity cs x y hxy
+
+theorem normalizedKL_existsUnique (cs : CoxeterSystem M W) :
+    ∃! RP : PolynomialFamilies W, NormalizedKL cs RP := by
+  refine ⟨(OrdinaryR.polynomial cs, CanonicalP.polynomial cs), constructed_normalizedKL cs, ?_⟩
+  intro RP hRP
+  exact normalizedKL_unique cs hRP (constructed_normalizedKL cs)
+
+theorem klFamilies_spec (cs : CoxeterSystem M W) : NormalizedKL cs (klFamilies cs) := by
+  exact Classical.epsilon_spec (normalizedKL_existsUnique cs).exists
+
+theorem klFamilies_eq_constructed (cs : CoxeterSystem M W) :
+    klFamilies cs = (OrdinaryR.polynomial cs, CanonicalP.polynomial cs) :=
+  normalizedKL_unique cs (klFamilies_spec cs) (constructed_normalizedKL cs)
+
+theorem klPolynomial_eq_constructed (cs : CoxeterSystem M W) (x y : W) :
+    klPolynomial cs x y = CanonicalP.polynomial cs x y := by
+  simp only [klPolynomial, klFamilies_eq_constructed]
+
+theorem rPolynomial_eq_constructed (cs : CoxeterSystem M W) (x y : W) :
+    rPolynomial cs x y = OrdinaryR.polynomial cs x y := by
+  simp only [rPolynomial, klFamilies_eq_constructed]
+
+
+/-- Every nontrivial Bruhat interval contains a first step of length one.
+This is the grading fact needed to transport rank differences without labels. -/
+theorem exists_bruhat_successor (cs : CoxeterSystem M W) {x y : W}
+    (hxy : BruhatLE cs x y) (hne : x ≠ y) :
+    ∃ z, BruhatLE cs x z ∧ BruhatLE cs z y ∧ cs.length z = cs.length x + 1 := by
+  classical
+  suffices ∀ n : ℕ, ∀ y : W, cs.length y = n → ∀ x : W,
+      BruhatLE cs x y → x ≠ y →
+      ∃ z, BruhatLE cs x z ∧ BruhatLE cs z y ∧ cs.length z = cs.length x + 1 by
+    exact this _ y rfl x hxy hne
+  intro n
+  induction n using Nat.strong_induction_on with
+  | h n ih =>
+    intro y hyn x hxy hne
+    have hy1 : y ≠ 1 := by
+      intro hy
+      subst y
+      have hlen := length_le_of_bruhat cs hxy
+      have hx : cs.length x = 0 := by simpa using hlen
+      exact hne (cs.length_eq_zero_iff.mp hx)
+    obtain ⟨i, hi⟩ := cs.exists_leftDescent_of_ne_one hy1
+    change cs.length (cs.simple i * y) < cs.length y at hi
+    have hylen : cs.length y = cs.length (cs.simple i * y) + 1 := by
+      have := cs.length_simple_mul y i
+      omega
+    have hsyn : cs.length (cs.simple i * y) < n := by omega
+    have hsyy : BruhatLE cs (cs.simple i * y) y := by
+      simpa only [lowerSimple, ite_eq_left hi] using lowerSimple_le cs i y
+    by_cases hxsy : BruhatLE cs x (cs.simple i * y)
+    · by_cases hxeq : x = cs.simple i * y
+      · exact ⟨y, hxy, bruhat_refl cs y, by simpa [hxeq] using hylen⟩
+      · obtain ⟨z, hxz, hzsy, hzlen⟩ := ih _ hsyn _ rfl x hxsy hxeq
+        exact ⟨z, hxz, bruhat_trans cs hzsy hsyy, hzlen⟩
+    · have hxi : cs.length (cs.simple i * x) < cs.length x := by
+        by_contra hnxi
+        have hxim : cs.length x < cs.length (cs.simple i * x) := by
+          have := cs.length_simple_mul x i
+          omega
+        exact hxsy (bruhat_lifting cs hxy i hxim hi).1
+      have hxlen : cs.length x = cs.length (cs.simple i * x) + 1 := by
+        have := cs.length_simple_mul x i
+        omega
+      have hsxsy : BruhatLE cs (cs.simple i * x) (cs.simple i * y) := by
+        simpa only [lowerSimple, ite_eq_left hxi, ite_eq_left hi] using lowerSimple_mono cs i hxy
+      have hne' : cs.simple i * x ≠ cs.simple i * y := fun he => hne (mul_left_cancel he)
+      obtain ⟨w, hsxw, hwsy, hwlen⟩ := ih _ hsyn _ rfl _ hsxsy hne'
+      have hux : upperSimple cs i (cs.simple i * x) = x := by
+        rw [upperSimple_mul]
+        simp only [upperSimple, ite_eq_left hxi]
+      have huy : upperSimple cs i (cs.simple i * y) = y := by
+        rw [upperSimple_mul]
+        simp only [upperSimple, ite_eq_left hi]
+      have hxz := upperSimple_mono cs i hsxw
+      have hzy := upperSimple_mono cs i hwsy
+      rw [hux] at hxz
+      rw [huy] at hzy
+      have hwlen' : cs.length w = cs.length x := by omega
+      have hwi : ¬cs.length (cs.simple i * w) < cs.length w := by
+        intro hwi
+        simp only [upperSimple, ite_eq_left hwi] at hxz
+        have hxw : x = w := by
+          by_contra hnxw
+          have := length_lt_of_bruhat_ne cs hxz hnxw
+          omega
+        exact hxsy (hxw ▸ hwsy)
+      refine ⟨upperSimple cs i w, hxz, hzy, ?_⟩
+      simp only [upperSimple, ite_eq_right hwi]
+      have := cs.length_simple_mul w i
+      omega
+
+
+/-- An order embedding between Bruhat intervals cannot shorten rank. -/
+theorem rankDifference_le_of_strictMono
+    {B' : Type u'} {W' : Type v'} [Group W'] {M' : CoxeterMatrix B'}
+    (cs : CoxeterSystem M W) (cs' : CoxeterSystem M' W')
+    {u b : W} {u' b' : W'}
+    (f : Interval cs u b → Interval cs' u' b') (hf : StrictMono f)
+    {x y : Interval cs u b} (hxy : x ≤ y) :
+    rankDifference cs x.val y.val ≤ rankDifference cs' (f x).val (f y).val := by
+  suffices ∀ n : ℕ, ∀ x y : Interval cs u b,
+      rankDifference cs x.val y.val = n → x ≤ y →
+      rankDifference cs x.val y.val ≤ rankDifference cs' (f x).val (f y).val by
+    exact this _ x y rfl hxy
+  intro n
+  induction n using Nat.strong_induction_on with
+  | h n ih =>
+    intro x y hn hxy
+    by_cases heq : x = y
+    · subst y
+      simp [rankDifference]
+    · have hne : x.val ≠ y.val := fun hv => heq (Subtype.ext hv)
+      obtain ⟨z, hxz, hzy, hzlen⟩ := exists_bruhat_successor cs hxy hne
+      let zI : Interval cs u b := ⟨z, bruhat_trans cs x.property.1 hxz,
+        bruhat_trans cs hzy y.property.2⟩
+      have hxylen := length_le_of_bruhat cs hxy
+      have hzylen := length_le_of_bruhat cs hzy
+      have hsmall : rankDifference cs zI.val y.val < n := by
+        dsimp [zI, rankDifference] at hn ⊢
+        omega
+      have hsub := ih _ hsmall zI y rfl hzy
+      have hxz' : x < zI := by
+        refine (lt_iff_le_and_ne).mpr ⟨hxz, ?_⟩
+        intro hx
+        have hv : x.val = z := congrArg Subtype.val hx
+        have := congrArg cs.length hv
+        omega
+      have hfxz := hf hxz'
+      have hfxzlen : cs'.length (f x).val < cs'.length (f zI).val := by
+        apply length_lt_of_bruhat_ne cs' hfxz.le
+        intro hv
+        exact hfxz.ne (Subtype.ext hv)
+      have hfzyle := length_le_of_bruhat cs' (hf.monotone (show zI ≤ y from hzy))
+      change cs.length y.val - cs.length z ≤
+        cs'.length (f y).val - cs'.length (f zI).val at hsub
+      change cs.length y.val - cs.length x.val ≤
+        cs'.length (f y).val - cs'.length (f x).val
+      omega
+
+/-- Rank is intrinsic to the abstract interval poset, with neither root
+labels nor simple generators preserved by the isomorphism. -/
+theorem intervalIso_rankDifference
+    {B' : Type u'} {W' : Type v'} [Group W'] {M' : CoxeterMatrix B'}
+    (cs : CoxeterSystem M W) (cs' : CoxeterSystem M' W')
+    {u b : W} {u' b' : W'}
+    (ι : Interval cs u b ≃o Interval cs' u' b')
+    {x y : Interval cs u b} (hxy : x ≤ y) :
+    rankDifference cs x.val y.val = rankDifference cs' (ι x).val (ι y).val := by
+  apply Nat.le_antisymm
+  · exact rankDifference_le_of_strictMono cs cs' ι ι.strictMono hxy
+  · have h := rankDifference_le_of_strictMono cs' cs ι.symm ι.symm.strictMono (ι.monotone hxy)
+    simpa only [OrderIso.symm_apply_apply] using h
+
+/-- Named endpoints for an actual nonempty interval. -/
+def intervalBottom (cs : CoxeterSystem M W) {u b : W} (hub : BruhatLE cs u b) :
+    Interval cs u b := ⟨u, bruhat_refl cs u, hub⟩
+
+def intervalTop (cs : CoxeterSystem M W) {u b : W} (hub : BruhatLE cs u b) :
+    Interval cs u b := ⟨b, hub, bruhat_refl cs b⟩
+
+theorem intervalIso_bottom
+    {B' : Type u'} {W' : Type v'} [Group W'] {M' : CoxeterMatrix B'}
+    (cs : CoxeterSystem M W) (cs' : CoxeterSystem M' W')
+    {u b : W} {u' b' : W'} (hub : BruhatLE cs u b) (hub' : BruhatLE cs' u' b')
+    (ι : Interval cs u b ≃o Interval cs' u' b') :
+    ι (intervalBottom cs hub) = intervalBottom cs' hub' := by
+  apply le_antisymm
+  · have h : intervalBottom cs hub ≤ ι.symm (intervalBottom cs' hub') :=
+      (ι.symm (intervalBottom cs' hub')).property.1
+    simpa only [OrderIso.apply_symm_apply] using ι.monotone h
+  · exact (ι (intervalBottom cs hub)).property.1
+
+theorem intervalIso_top
+    {B' : Type u'} {W' : Type v'} [Group W'] {M' : CoxeterMatrix B'}
+    (cs : CoxeterSystem M W) (cs' : CoxeterSystem M' W')
+    {u b : W} {u' b' : W'} (hub : BruhatLE cs u b) (hub' : BruhatLE cs' u' b')
+    (ι : Interval cs u b ≃o Interval cs' u' b') :
+    ι (intervalTop cs hub) = intervalTop cs' hub' := by
+  apply le_antisymm
+  · exact (ι (intervalTop cs hub)).property.2
+  · have h : ι.symm (intervalTop cs' hub') ≤ intervalTop cs hub :=
+      (ι.symm (intervalTop cs' hub')).property.2
+    simpa only [OrderIso.apply_symm_apply] using ι.monotone h
+
+theorem intervalIso_full_rankDifference
+    {B' : Type u'} {W' : Type v'} [Group W'] {M' : CoxeterMatrix B'}
+    (cs : CoxeterSystem M W) (cs' : CoxeterSystem M' W')
+    {u b : W} {u' b' : W'} (hub : BruhatLE cs u b) (hub' : BruhatLE cs' u' b')
+    (ι : Interval cs u b ≃o Interval cs' u' b') :
+    rankDifference cs u b = rankDifference cs' u' b' := by
+  have h := intervalIso_rankDifference cs cs' ι
+    (show intervalBottom cs hub ≤ intervalTop cs hub from hub)
+  rw [intervalIso_bottom cs cs' hub hub' ι, intervalIso_top cs cs' hub hub' ι] at h
+  exact h
+
+end KLInvariance
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/KazhdanLusztig/CharacterReciprocity.lean b/lean/OAI/RepresentationTheory/KazhdanLusztig/CharacterReciprocity.lean
index 112306998161484603bdfcdf051194305e1d4f2c..df5761f4de7f4a728c53da39d21853242748c33e 100644
--- a/lean/OAI/RepresentationTheory/KazhdanLusztig/CharacterReciprocity.lean
+++ b/lean/OAI/RepresentationTheory/KazhdanLusztig/CharacterReciprocity.lean
@@ -1,2201 +1,2201 @@
-import OAI.RepresentationTheory.KazhdanLusztig.Support
-
-namespace OAI
-
-/-!
-Conditional boundary-sheaf duality and stalk-character propositions, free sheaves, actual R-reciprocity and lower convolution. StalkCharacterObligation remains a Prop, not an unconditional character theorem or a premise of the R-first route.
--/
-
-section
-
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W) (b : W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-variable [Fintype (Edge cs 1 b)] [Fintype (Interval cs 1 b)] [DecidableEq (Interval cs 1 b)]
-noncomputable section
-
-def SectionDualityObligation : Prop :=
-  ∃ d : SectionDuality cs 1 b basis (one_bruhat cs b)
-    (boundarySheaf cs 1 b basis (one_bruhat cs b)),
-    ∀ i j (m n : (boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf.sections Set.univ),
-      m ∈ ((boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf.sections Set.univ).piece i →
-      n ∈ ((boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf.sections Set.univ).piece j →
-      d.pairing m n ∈ polynomialGrading ℝ σ (i+j-(cs.length b : ℤ))
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-/-! Balance of a two-sided perfect pairing needs orthogonality on both
-sides, not equality of its two integral embeddings into an ambient form. -/
-namespace BalancedCuts
-universe ur um ua
-variable {R : Type ur} [CommRing R] [IsDomain R]
-  {M : Type um} [AddCommGroup M] [Module R M]
-  {A : Type ua} [AddCommGroup A] [Module R A]
-  (d : M →ₗ[R] Module.Dual R M) (π : M →ₗ[R] A) (c : M →ₗ[R] M)
-  (p : R) (hp : p≠0) (hπ : ∀ m, π (c m)=p • π m)
-  (hleft : ∀ m n, π n=0 → d (c m) n=0)
-  (hright : ∀ m n, π m=0 → d m (c n)=0)
-
-include hp hπ hleft hright in
- theorem balanced_two (m n : M) : d (c m) n=d m (c n) := by
-  have hn : π (p • n-c n)=0 := by rw [map_sub,map_smul,hπ,sub_self]
-  have hm : π (p • m-c m)=0 := by rw [map_sub,map_smul,hπ,sub_self]
-  have en := hleft m _ hn
-  have em := hright _ n hm
-  simp only [map_sub,map_smul,LinearMap.sub_apply,LinearMap.smul_apply,smul_eq_mul,sub_eq_zero] at en em
-  exact mul_left_cancel₀ hp (en.trans em.symm)
-
-end BalancedCuts
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded Module
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
-  {G : OrderedGraph V E} [IsDomain A] [DecidableEq V]
-  {S : GradedSheaf (𝓐 := 𝓐) G} {d : ℤ} (p : PerfectPairing S d)
-  (a : A) (ha : a≠0)
-  (hcut : ∀ x (m : S.forget.sections Set.univ), a • Pi.single x (m.val x)∈S.forget.sections Set.univ)
-noncomputable section
-
- include ha in
- theorem PerfectPairing.cut_balanced (x : V) (m n : S.forget.sections Set.univ) :
-    p.pairing (IntegralSupport.cut _ a hcut x m) n =
-      p.pairing m (IntegralSupport.cut _ a hcut x n) := by
-  apply BalancedCuts.balanced_two p.pairing.toLinearMap
-    (IntegralSupport.projection (S.forget.sections Set.univ) x)
-    (IntegralSupport.cut _ a hcut x) a ha
-    (IntegralSupport.cut_self _ a hcut x)
-  · intro v w hw
-    exact p.left_orthogonal x _ w (fun y hy => IntegralSupport.cut_other _ a hcut x y hy v) hw
-  · intro v w hv
-    exact p.right_orthogonal x v _ hv (fun y hy => IntegralSupport.cut_other _ a hcut x y hy w)
-
-end
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W) (b : W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  [Fintype (Edge cs 1 b)] [Fintype (Interval cs 1 b)] [DecidableEq (Interval cs 1 b)]
-noncomputable section
-
- theorem sectionDualityObligation_of_perfectPairing
-    (p : GradedSheaf.PerfectPairing
-      (boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf (cs.length b : ℤ)) :
-    SectionDualityObligation cs b basis := by
-  let B := boundarySheaf cs 1 b basis (one_bruhat cs b)
-  refine ⟨⟨p.pairing,?_⟩,?_⟩
-  · intro x m n
-    exact p.cut_balanced (Sheaf.labelProduct (polynomialLabel cs 1 b basis))
-      (Sheaf.labelProduct_ne_zero _ (polynomialLabel_ne_zero cs 1 b basis))
-      (B.sheaf.forget.scaled_single_section
-        (B.sheaf.forget.labelProduct_annihilates (polynomialLabel cs 1 b basis)
-          (GradedSheaf.UpperQuotient.annihilates B.sheaf B.quotient))) x m n
-  · intro i j m n hm hn
-    exact p.homogeneous i j m n (fun x => hm x (Set.mem_univ x))
-      (fun x => hn x (Set.mem_univ x))
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-/-! Integral costalk duality only uses free stalks, the genuine edge quotients,
-generation and a perfect support-orthogonal section form. No top/minimality or
-KL character assumption is needed. This extension applies to genuine words. -/
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
-  {σ : Type us} (basis : Basis σ ℝ (Extended M))
-
-structure FreeSheaf where
-  sheaf : GradedSheaf (𝓐 := polynomialGrading ℝ σ) (graph cs u b)
-  free : sheaf.FreeStalks
-  quotient : sheaf.UpperQuotient (polynomialLabel cs u b basis)
-  generated : sheaf.forget.Generated
-
-namespace FreeSheaf
-variable (B : FreeSheaf cs u b basis)
-  [Fintype (Edge cs u b)] [DecidableEq (Interval cs u b)]
-noncomputable section
- def kernelEquivSingleSupport (x : Interval cs u b) :
-    LinearMap.ker (B.sheaf.forget.stalkMap x) ≃ₗ[Coefficient (σ := σ)]
-      IntegralSupport.supported (B.sheaf.forget.sections Set.univ) x := by
-  let _ (x : Interval cs u b) : Module.Free (Coefficient (σ := σ)) (B.sheaf.forget.vertex x) := B.free x
-  let _ (x : Interval cs u b) : Module.Finite (Coefficient (σ := σ)) (B.sheaf.forget.vertex x) :=
-    (B.sheaf.vertex x).finite
-  let _ (x : Interval cs u b) : Fintype (Module.Free.ChooseBasisIndex
-      (Coefficient (σ := σ)) (B.sheaf.forget.vertex x)) := Fintype.ofFinite _
-  let bs := fun x : Interval cs u b =>
-    Module.Free.chooseBasis (Coefficient (σ := σ)) (B.sheaf.forget.vertex x)
-  exact (B.sheaf.forget.kernelEquivSingleStalk (polynomialLabel cs u b basis)
-    (polynomialLabel_prime cs u b basis) (fun e => (B.quotient e).2) bs x
-    (incoming_pairwise_relPrime cs u b basis x)
-    (B.sheaf.forget.outgoing_product_regular (polynomialLabel cs u b basis)
-      (polynomialLabel_prime cs u b basis) (fun e => (B.quotient e).1)
-      (fun e => (B.quotient e).2) bs x
-      (outgoing_incoming_not_dvd cs u b basis x))).trans
-        (B.sheaf.forget.singleStalkEquivSupport x)
-
-
-structure SectionDuality where
-  pairing : B.sheaf.forget.sections Set.univ ≃ₗ[Coefficient (σ := σ)]
-    Dual (Coefficient (σ := σ)) (B.sheaf.forget.sections Set.univ)
-  balanced : ∀ x m n,
-    pairing (IntegralSupport.cut (B.sheaf.forget.sections Set.univ)
-      (Sheaf.labelProduct (polynomialLabel cs u b basis))
-      (B.sheaf.forget.scaled_single_section
-        (B.sheaf.forget.labelProduct_annihilates (polynomialLabel cs u b basis)
-          (GradedSheaf.UpperQuotient.annihilates B.sheaf B.quotient))) x m) n =
-    pairing m (IntegralSupport.cut (B.sheaf.forget.sections Set.univ)
-      (Sheaf.labelProduct (polynomialLabel cs u b basis))
-      (B.sheaf.forget.scaled_single_section
-        (B.sheaf.forget.labelProduct_annihilates (polynomialLabel cs u b basis)
-          (GradedSheaf.UpperQuotient.annihilates B.sheaf B.quotient))) x n)
-
- def costalkEquivDual (d : SectionDuality cs u b basis B) (x : Interval cs u b) :
-    LinearMap.ker (B.sheaf.forget.stalkMap x) ≃ₗ[Coefficient (σ := σ)]
-      Dual (Coefficient (σ := σ)) (B.sheaf.vertex x) := by
-  let _ (x : Interval cs u b) : Module.Free (Coefficient (σ := σ)) (B.sheaf.forget.vertex x) := B.free x
-  let _ (y : Interval cs u b) : Module.IsTorsionFree (Coefficient (σ := σ))
-      (B.sheaf.forget.vertex y) := inferInstance
-  exact (kernelEquivSingleSupport cs u b basis B x).trans
-    (B.sheaf.forget.singleSupportEquivDual (polynomialLabel cs u b basis)
-      (polynomialLabel_ne_zero cs u b basis)
-      (GradedSheaf.UpperQuotient.annihilates B.sheaf B.quotient)
-      d.pairing d.balanced B.generated x)
-
- theorem costalk_free_of_duality (d : SectionDuality cs u b basis B)
-    (x : Interval cs u b) :
-    Module.Free (Coefficient (σ := σ)) (LinearMap.ker (B.sheaf.forget.stalkMap x)) := by
-  let _ := B.free x
-  exact Module.Free.of_equiv (costalkEquivDual cs u b basis B d x).symm
-
- variable [Fintype σ] [Fintype (Interval cs u b)]
-
-omit [Fintype σ] [Fintype (Interval cs u b)] in
- theorem kernelEquivSingleSupport_apply (x : Interval cs u b)
-    (v : LinearMap.ker (B.sheaf.forget.stalkMap x)) :
-    ((kernelEquivSingleSupport cs u b basis B x) v).val.val x =
-      Sheaf.incomingProduct (G := graph cs u b) (polynomialLabel cs u b basis) x • v.val := by
-  change (Pi.single x
-    (Sheaf.incomingProduct (G := graph cs u b) (polynomialLabel cs u b basis) x • v.val) :
-      B.sheaf.forget.Assignment) x = _
-  exact Pi.single_eq_same _ _
-
- theorem costalkEquivDual_inverse_homogeneous
-    (d : SectionDuality cs u b basis B) (l : ℤ)
-    (hd : ∀ i j (m n : B.sheaf.sections Set.univ),
-      m ∈ (B.sheaf.sections Set.univ).piece i → n ∈ (B.sheaf.sections Set.univ).piece j →
-      d.pairing m n ∈ polynomialGrading ℝ σ (i+j-l))
-    (x : Interval cs u b) (r : ℤ) (lam : Dual (Coefficient (σ := σ)) (B.sheaf.vertex x))
-    (hlam : ∀ j (v : B.sheaf.vertex x), v ∈ (B.sheaf.vertex x).piece j →
-      lam v ∈ polynomialGrading ℝ σ (j+r)) :
-    ((costalkEquivDual cs u b basis B d x).symm lam).val ∈
-      (B.sheaf.vertex x).piece
-        (r+l-(Sheaf.incoming (G := graph cs u b) x).card) := by
-  let _ (y : Interval cs u b) : Module.Free (Coefficient (σ := σ))
-      (B.sheaf.forget.vertex y) := B.free y
-  let _ (y : Interval cs u b) : Module.IsTorsionFree (Coefficient (σ := σ))
-      (B.sheaf.forget.vertex y) := inferInstance
-  let S := B.sheaf.forget.singleSupportEquivDual (polynomialLabel cs u b basis)
-    (polynomialLabel_ne_zero cs u b basis)
-    (GradedSheaf.UpperQuotient.annihilates B.sheaf B.quotient)
-    d.pairing d.balanced B.generated x
-  have hs : (S.symm lam).val ∈ (B.sheaf.sections Set.univ).piece (r+l) := by
-    exact B.sheaf.dualToSupport_homogeneous (polynomialLabel cs u b basis)
-      (polynomialLabel_ne_zero cs u b basis)
-      (GradedSheaf.UpperQuotient.annihilates B.sheaf B.quotient)
-      d.pairing d.balanced l hd x r lam hlam
-  have hv := hs x (Set.mem_univ x)
-  have he : (S.symm lam).val.val x =
-      Sheaf.incomingProduct (G := graph cs u b) (polynomialLabel cs u b basis) x •
-      ((costalkEquivDual cs u b basis B d x).symm lam).val := by
-    rw [← kernelEquivSingleSupport_apply cs u b basis B x]
-    change (S.symm lam).val.val x =
-      ((kernelEquivSingleSupport cs u b basis B x)
-        ((kernelEquivSingleSupport cs u b basis B x).symm (S.symm lam))).val.val x
-    rw [LinearEquiv.apply_symm_apply]
-  change (S.symm lam).val.val x ∈ (B.sheaf.vertex x).piece (r+l) at hv
-  rw [he] at hv
-  apply mem_of_homogeneous_smul (polynomialGrading ℝ σ) (B.sheaf.vertex x).piece
-    ((Sheaf.incoming (G := graph cs u b) x).card : ℤ)
-    (Sheaf.incomingProduct (G := graph cs u b) (polynomialLabel cs u b basis) x)
-    (incomingProduct_homogeneous cs u b basis x) _ (r+l) _ hv
-  let _ := B.free x
-  exact smul_right_injective _ (Sheaf.incomingProduct_ne_zero
-    (polynomialLabel cs u b basis) (polynomialLabel_prime cs u b basis) x)
-
- def fullSubmodule (x : Interval cs u b) : Submodule (Coefficient (σ := σ)) (B.sheaf.vertex x) :=
-    LinearMap.ker (B.sheaf.forget.stalkMap x)
-
-omit [Fintype σ] [Fintype (Edge cs u b)] [Fintype (Interval cs u b)]
-  [DecidableEq (Interval cs u b)] in
- theorem fullKernel_eq (x : Interval cs u b) :
-    B.sheaf.upwardSubmodule x Set.univ = fullSubmodule cs u b basis B x := by
-  ext m
-  change (∀ e ∈ Set.univ, B.sheaf.forget.stalkMap x m e = 0) ↔ B.sheaf.forget.stalkMap x m = 0
-  simp only [Set.mem_univ,forall_const]
-  exact funext_iff.symm
-
- def fullKernel (x : Interval cs u b) : ModuleData (polynomialGrading ℝ σ) :=
-  ModuleData.sub (B.sheaf.vertex x) (fullSubmodule cs u b basis B x) (by
-    simpa only [fullKernel_eq cs u b basis B x] using
-      B.sheaf.upwardKernel_homogeneous x Set.univ)
-
- theorem costalk_hilbert_of_duality
-    (d : SectionDuality cs u b basis B) (l : ℤ)
-    (hd : ∀ i j (m n : B.sheaf.sections Set.univ),
-      m ∈ (B.sheaf.sections Set.univ).piece i → n ∈ (B.sheaf.sections Set.univ).piece j →
-      d.pairing m n ∈ polynomialGrading ℝ σ (i+j-l))
-    (x : Interval cs u b) (c : NonnegativeBasis (σ := σ) (B.sheaf.vertex x).piece)
-    {q : ℝ} (hq : 0 < q) (hq1 : q < 1) :
-    Hilbert.value (B.sheaf.vertex x).piece
-      ((fullSubmodule cs u b basis B x).restrictScalars ℝ) q =
-      q^(l-(Sheaf.incoming (G := graph cs u b) x).card) *
-        Polynomial.eval₂ (Int.castRingHom ℝ) q⁻¹ c.polynomial *
-        (1-q)⁻¹ ^ Fintype.card σ := by
-  classical
-  let K := fullKernel cs u b basis B x
-  let cb : Basis c.index (Coefficient (σ := σ)) K :=
-    c.basis.dualBasis.map (costalkEquivDual cs u b basis B d x).symm
-  have hcb (i : c.index) : cb i ∈ K.piece
-      (l-(Sheaf.incoming (G := graph cs u b) x).card-(c.degree i : ℤ)) := by
-    change ((costalkEquivDual cs u b basis B d x).symm (c.basis.dualBasis i)).val ∈
-      (B.sheaf.vertex x).piece _
-    have h := costalkEquivDual_inverse_homogeneous cs u b basis B d l hd x
-      (-(c.degree i : ℤ)) (c.basis.dualBasis i)
-      (dualBasis_homogeneous (polynomialGrading ℝ σ) (B.sheaf.vertex x).piece
-        c.basis (fun i => (c.degree i : ℤ)) c.homogeneous i)
-    simpa only [show -(c.degree i : ℤ)+l-(Sheaf.incoming (G := graph cs u b) x).card =
-      l-(Sheaf.incoming (G := graph cs u b) x).card-(c.degree i : ℤ) by omega] using h
-  have hs := Hilbert.hasSum_homogeneous_basis K.piece cb
-    (fun i => l-(Sheaf.incoming (G := graph cs u b) x).card-(c.degree i : ℤ)) hcb hq hq1
-  have hv : Hilbert.value K.piece ⊤ q =
-      (∑ i, q^(l-(Sheaf.incoming (G := graph cs u b) x).card-(c.degree i : ℤ))) *
-        (1-q)⁻¹ ^ Fintype.card σ := by
-    simpa only [Hilbert.value,Hilbert.term_top] using hs.tsum_eq
-  change Hilbert.value (B.sheaf.vertex x).piece
-    ((fullSubmodule cs u b basis B x).restrictScalars ℝ) q = _
-  rw [← Hilbert.value_subPiece]
-  change Hilbert.value K.piece ⊤ q = _
-  rw [hv,c.eval₂_eq,Finset.mul_sum]
-  congr 1
-  apply Finset.sum_congr rfl
-  intro i _
-  rw [zpow_sub₀ hq.ne',zpow_natCast,div_eq_mul_inv,inv_pow]
-
-
- def dualityOfPairing {l : ℤ} (p : GradedSheaf.PerfectPairing B.sheaf l) :
-    SectionDuality cs u b basis B where
-  pairing := p.pairing
-  balanced := p.cut_balanced
-    (Sheaf.labelProduct (polynomialLabel cs u b basis))
-    (Sheaf.labelProduct_ne_zero _ (polynomialLabel_ne_zero cs u b basis))
-    (B.sheaf.forget.scaled_single_section
-      (B.sheaf.forget.labelProduct_annihilates (polynomialLabel cs u b basis)
-        (GradedSheaf.UpperQuotient.annihilates B.sheaf B.quotient)))
-
-end
-end FreeSheaf
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-/-! The finite sums used by the actual local decomposition. The model criteria
-are kept explicit here; they are not asserted for the constructed BMP object
-or inserted into the main KL theorem. -/
-namespace KLInvariance.MomentGraph.Sheaf
-universe u v w z
-variable {R : Type u} [CommRing R] [IsLocalRing R]
-  {V : Type v} [PartialOrder V] {E : Type w} {G : OrderedGraph V E}
-
-
-instance zeroVertex_subsingleton (x : V) : Subsingleton ((zero (R := R) (G := G)).vertex x) :=
-  inferInstanceAs (Subsingleton PUnit)
-
-instance zeroEdge_subsingleton (e : E) : Subsingleton ((zero (R := R) (G := G)).edge e) :=
-  inferInstanceAs (Subsingleton PUnit)
-
-structure LocalModels (α : E → R) where
-  sheaf : V → Sheaf.{u,v,w,z} (R := R) G
-  free : ∀ c x, Module.Free R ((sheaf c).vertex x)
-  finite : ∀ c x, Module.Finite R ((sheaf c).vertex x)
-  quotient : ∀ c, (sheaf c).UpperQuotient α
-  generated : ∀ c, (sheaf c).Generated
-  flabby : ∀ c, (sheaf c).Flabby
-  support : ∀ c x, ¬x ≤ c → Subsingleton ((sheaf c).vertex x)
-  top : ∀ c, (sheaf c).vertex c ≃ₗ[R] R
-  minimal : ∀ c x, x ≠ c → LinearMap.ker ((sheaf c).stalkMap x) ≤
-    IsLocalRing.maximalIdeal R • (⊤ : Submodule R ((sheaf c).vertex x))
-
-inductive ModelSum (O : V → Sheaf.{u,v,w,z} (R := R) G) (U : Set V) :
-    Sheaf.{u,v,w,z} (R := R) G → Prop where
-  | zero : ModelSum O U zero
-  | add {B} : ModelSum O U B → ∀ c ∈ U, ∀ n : ℕ,
-      ModelSum O U (prod B (pi (fun _ : Fin n => O c)))
-
-namespace ModelSum
-variable {α : E → R} (O : LocalModels.{u,v,w,z} (G := G) α)
-  {U : Set V} {B : Sheaf.{u,v,w,z} (R := R) G}
-
-omit [IsLocalRing R] in
-lemma mono {O : V → Sheaf.{u,v,w,z} (R := R) G} (h : ModelSum O U B)
-    {T : Set V} (hUT : U ⊆ T) : ModelSum O T B := by
-  induction h with
-  | zero => exact .zero
-  | add _ c hc n ih => exact .add ih c (hUT hc) n
-
-lemma free (h : ModelSum O.sheaf U B) (x : V) : Module.Free R (B.vertex x) := by
-  induction h with
-  | zero => exact inferInstanceAs (Module.Free R PUnit.{z+1})
-  | @add C _ c _ n ih =>
-    let _ : Module.Free R (C.vertex x) := ih
-    let _ := O.free c x
-    change Module.Free R (C.vertex x × (Fin n → (O.sheaf c).vertex x))
-    infer_instance
-
-lemma finite (h : ModelSum O.sheaf U B) (x : V) : Module.Finite R (B.vertex x) := by
-  induction h with
-  | zero => exact inferInstanceAs (Module.Finite R PUnit.{z+1})
-  | @add C _ c _ n ih =>
-    let _ : Module.Finite R (C.vertex x) := ih
-    let _ := O.finite c x
-    change Module.Finite R (C.vertex x × (Fin n → (O.sheaf c).vertex x))
-    infer_instance
-
-lemma quotient (h : ModelSum O.sheaf U B) : B.UpperQuotient α := by
-  induction h with
-  | zero =>
-    intro e
-    exact ⟨fun v => ⟨0,Subsingleton.elim _ _⟩,fun v => ⟨fun _ => ⟨0,Subsingleton.elim _ _⟩,fun _ => Subsingleton.elim _ _⟩⟩
-  | add _ c _ n ih => exact prod_upperQuotient _ _ α ih (pi_upperQuotient _ α (fun _ => O.quotient c))
-
-lemma generated (h : ModelSum O.sheaf U B) : B.Generated := by
-  induction h with
-  | zero => exact fun x v => ⟨0,fun _ _ _ => rfl,Subsingleton.elim _ _⟩
-  | add _ c _ n ih => exact prod_generated _ _ ih (pi_generated _ (fun _ => O.generated c))
-
-lemma flabby (h : ModelSum O.sheaf U B) : B.Flabby := by
-  induction h with
-  | zero => exact fun U _ f _ => ⟨f,fun _ _ _ => rfl,fun _ _ => rfl⟩
-  | add _ c _ n ih => exact prod_flabby _ _ ih (pi_flabby _ (fun _ => O.flabby c))
-
-lemma minimal (h : ModelSum O.sheaf U B) (x : V) (hx : x ∉ U) :
-    LinearMap.ker (B.stalkMap x) ≤
-      IsLocalRing.maximalIdeal R • (⊤ : Submodule R (B.vertex x)) := by
-  induction h with
-  | zero =>
-    intro v _
-    rw [Subsingleton.elim v 0]
-    exact Submodule.zero_mem _
-  | add _ c hc n ih =>
-    exact prod_minimal _ _ x _ ih (pi_minimal _ x _
-      (fun _ => O.minimal c x (fun he => hx (he ▸ hc))))
-
-end ModelSum
-end KLInvariance.MomentGraph.Sheaf
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.Sheaf
-open scoped Pointwise
-universe ur uv ue uz
-variable {R : Type ur} [CommRing R] {V : Type uv} [PartialOrder V]
-  {E : Type ue} {G : OrderedGraph V E}
-
-/-- Equality of embedded images with a scalar on the upper image. -/
-def ScaledImages (B : Sheaf.{ur,uv,ue,uz} (R := R) G) (e : E)
-    (F : Set (Outgoing (G := G) (G.source e)))
-    (H : Set (Outgoing (G := G) (G.target e))) (r : R) : Prop :=
-  ∀ z : B.edge e,
-    (∃ v ∈ B.upwardKernel (G.source e) F, B.lower e v=z) ↔
-    ∃ w ∈ B.upwardKernel (G.target e) H, r • B.upper e w=z
-
-theorem scaledImages_iff (B : Sheaf.{ur,uv,ue,uz} (R := R) G) (e : E)
-    (F : Set (Outgoing (G := G) (G.source e)))
-    (H : Set (Outgoing (G := G) (G.target e))) (r : R) :
-    B.ScaledImages e F H r ↔
-      (B.upwardKernel (G.source e) F).map (B.lower e) =
-        r • (B.upwardKernel (G.target e) H).map (B.upper e) := by
-  constructor
-  · intro h
-    ext z
-    rw [Submodule.mem_smul_pointwise_iff_exists]
-    change (∃ v, v ∈ B.upwardKernel (G.source e) F ∧ B.lower e v=z) ↔ _
-    rw [h z]
-    constructor
-    · rintro ⟨w,hw,he⟩
-      exact ⟨B.upper e w,⟨w,hw,rfl⟩,he⟩
-    · rintro ⟨t,⟨w,hw,rfl⟩,he⟩
-      exact ⟨w,hw,he⟩
-  · intro h z
-    constructor
-    · rintro ⟨v,hv,rfl⟩
-      have hv' : B.lower e v ∈ r • (B.upwardKernel (G.target e) H).map (B.upper e) :=
-        h ▸ (show B.lower e v ∈ (B.upwardKernel (G.source e) F).map (B.lower e) from ⟨v,hv,rfl⟩)
-      obtain ⟨t,⟨w,hw,rfl⟩,he⟩ := (Submodule.mem_smul_pointwise_iff_exists _ _ _).mp hv'
-      exact ⟨w,hw,he⟩
-    · rintro ⟨w,hw,rfl⟩
-      have hw' : r • B.upper e w ∈ r • (B.upwardKernel (G.target e) H).map (B.upper e) :=
-        (Submodule.mem_smul_pointwise_iff_exists _ _ _).mpr ⟨B.upper e w,⟨w,hw,rfl⟩,rfl⟩
-      rw [← h] at hw'
-      exact hw'
-
-theorem prod_upwardKernel_iff (B C : Sheaf.{ur,uv,ue,uz} (R := R) G)
-    (x : V) (F : Set (Outgoing (G := G) x)) (v : (prod B C).vertex x) :
-    v ∈ (prod B C).upwardKernel x F ↔
-      v.1 ∈ B.upwardKernel x F ∧ v.2 ∈ C.upwardKernel x F := by
-  constructor
-  · intro h
-    constructor
-    · rintro ⟨e,heq⟩ he
-      subst x
-      exact congrArg Prod.fst (h ⟨e,rfl⟩ he)
-    · rintro ⟨e,heq⟩ he
-      subst x
-      exact congrArg Prod.snd (h ⟨e,rfl⟩ he)
-  · rintro ⟨hB,hC⟩ ⟨e,heq⟩ he
-    subst x
-    exact Prod.ext (hB ⟨e,rfl⟩ he) (hC ⟨e,rfl⟩ he)
-
-theorem pi_upwardKernel_iff {ι : Type*} (B : ι → Sheaf.{ur,uv,ue,uz} (R := R) G)
-    (x : V) (F : Set (Outgoing (G := G) x)) (v : (pi B).vertex x) :
-    v ∈ (pi B).upwardKernel x F ↔ ∀ i, v i ∈ (B i).upwardKernel x F := by
-  constructor
-  · rintro h i ⟨e,heq⟩ he
-    subst x
-    exact congrFun (h ⟨e,rfl⟩ he) i
-  · rintro h ⟨e,heq⟩ he
-    subst x
-    exact funext (fun i => h i ⟨e,rfl⟩ he)
-
-theorem prod_scaledImages (B C : Sheaf.{ur,uv,ue,uz} (R := R) G) (e : E)
-    (F : Set (Outgoing (G := G) (G.source e)))
-    (H : Set (Outgoing (G := G) (G.target e))) (r : R)
-    (hB : B.ScaledImages e F H r) (hC : C.ScaledImages e F H r) :
-    (prod B C).ScaledImages e F H r := by
-  intro z
-  constructor
-  · rintro ⟨v,hv,he⟩
-    have hv' := (prod_upwardKernel_iff B C _ F v).mp hv
-    obtain ⟨b,hb,hbe⟩ := (hB z.1).mp ⟨v.1,hv'.1,congrArg Prod.fst he⟩
-    obtain ⟨c,hc,hce⟩ := (hC z.2).mp ⟨v.2,hv'.2,congrArg Prod.snd he⟩
-    exact ⟨(b,c),(prod_upwardKernel_iff B C _ H _).mpr ⟨hb,hc⟩,Prod.ext hbe hce⟩
-  · rintro ⟨v,hv,he⟩
-    have hv' := (prod_upwardKernel_iff B C _ H v).mp hv
-    obtain ⟨b,hb,hbe⟩ := (hB z.1).mpr ⟨v.1,hv'.1,congrArg Prod.fst he⟩
-    obtain ⟨c,hc,hce⟩ := (hC z.2).mpr ⟨v.2,hv'.2,congrArg Prod.snd he⟩
-    exact ⟨(b,c),(prod_upwardKernel_iff B C _ F _).mpr ⟨hb,hc⟩,Prod.ext hbe hce⟩
-
-theorem pi_scaledImages {ι : Type*} (B : ι → Sheaf.{ur,uv,ue,uz} (R := R) G) (e : E)
-    (F : Set (Outgoing (G := G) (G.source e)))
-    (H : Set (Outgoing (G := G) (G.target e))) (r : R)
-    (hB : ∀ i, (B i).ScaledImages e F H r) : (pi B).ScaledImages e F H r := by
-  classical
-  intro z
-  constructor
-  · rintro ⟨v,hv,he⟩
-    have hv' := (pi_upwardKernel_iff B _ F v).mp hv
-    choose w hw hwe using fun i => (hB i (z i)).mp ⟨v i,hv' i,congrFun he i⟩
-    exact ⟨w,(pi_upwardKernel_iff B _ H _).mpr hw,funext hwe⟩
-  · rintro ⟨v,hv,he⟩
-    have hv' := (pi_upwardKernel_iff B _ H v).mp hv
-    choose w hw hwe using fun i => (hB i (z i)).mpr ⟨v i,hv' i,congrFun he i⟩
-    exact ⟨w,(pi_upwardKernel_iff B _ F _).mpr hw,funext hwe⟩
-
-theorem zero_scaledImages (e : E) (F : Set (Outgoing (G := G) (G.source e)))
-    (H : Set (Outgoing (G := G) (G.target e))) (r : R) :
-    (zero : Sheaf.{ur,uv,ue,uz} (R := R) G).ScaledImages e F H r := by
-  intro z
-  constructor <;> intro _ <;> exact ⟨0,Submodule.zero_mem _,Subsingleton.elim _ _⟩
-
-theorem ModelSum.scaledImages {O : V → Sheaf.{ur,uv,ue,uz} (R := R) G}
-    {U : Set V} {B : Sheaf.{ur,uv,ue,uz} (R := R) G} (h : ModelSum O U B)
-    (e : E) (F : Set (Outgoing (G := G) (G.source e)))
-    (H : Set (Outgoing (G := G) (G.target e))) (r : R)
-    (hO : ∀ c ∈ U, (O c).ScaledImages e F H r) : B.ScaledImages e F H r := by
-  induction h with
-  | zero => exact zero_scaledImages e F H r
-  | add _ c hc n ih =>
-    exact prod_scaledImages _ _ e F H r ih
-      (pi_scaledImages (fun _ : Fin n => O c) e F H r (fun _ => hO c hc))
-
-end KLInvariance.MomentGraph.Sheaf
-
-end
-
-
-section
-
-/-! Actual full upward kernels commute with finite graded sums, shifts and
-sheaf isomorphisms. No freeness is inferred without a free model. -/
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  [IsNoetherianRing A] {𝓐 : ℤ → Submodule k A}
-  {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
-noncomputable section
-
- abbrev kernelData (B : GradedSheaf (𝓐 := 𝓐) G) (x : V) :=
-  B.upwardKernel x Set.univ
-
- def kernelProdEquiv (B C : GradedSheaf (𝓐 := 𝓐) G) (x : V) :
-    kernelData (prod B C) x ≃ₗ[A] ((kernelData B x).prod (kernelData C x)) where
-  toFun m := ⟨⟨m.val.1,((Sheaf.prod_upwardKernel_iff B.forget C.forget x Set.univ m.val).mp m.property).1⟩,
-    ⟨m.val.2,((Sheaf.prod_upwardKernel_iff B.forget C.forget x Set.univ m.val).mp m.property).2⟩⟩
-  invFun m := ⟨⟨m.1.val,m.2.val⟩,(Sheaf.prod_upwardKernel_iff B.forget C.forget x Set.univ _).mpr
-    ⟨m.1.property,m.2.property⟩⟩
-  left_inv _ := rfl
-  right_inv _ := rfl
-  map_add' _ _ := rfl
-  map_smul' _ _ := rfl
-
- theorem kernelProdEquiv_graded (B C : GradedSheaf (𝓐 := 𝓐) G) (x : V)
-    (n : ℤ) (v : kernelData (prod B C) x) (hv : v∈(kernelData (prod B C) x).piece n) :
-    kernelProdEquiv B C x v∈((kernelData B x).prod (kernelData C x)).piece n := hv
-
- def kernelPiEquiv {ι : Type um} [Fintype ι] (B : ι → GradedSheaf (𝓐 := 𝓐) G) (x : V) :
-    kernelData (pi B) x ≃ₗ[A] ModuleData.pi (fun i => kernelData (B i) x) where
-  toFun m i := ⟨m.val i,(Sheaf.pi_upwardKernel_iff (fun i => (B i).forget) x Set.univ m.val).mp m.property i⟩
-  invFun m := ⟨(fun i => (m i).val),(Sheaf.pi_upwardKernel_iff (fun i => (B i).forget) x Set.univ _).mpr
-    (fun i => (m i).property)⟩
-  left_inv _ := rfl
-  right_inv _ := rfl
-  map_add' _ _ := rfl
-  map_smul' _ _ := rfl
-
- theorem kernelPiEquiv_graded {ι : Type um} [Fintype ι] (B : ι → GradedSheaf (𝓐 := 𝓐) G) (x : V)
-    (n : ℤ) (v : kernelData (pi B) x) (hv : v∈(kernelData (pi B) x).piece n) :
-    kernelPiEquiv B x v∈(ModuleData.pi (fun i => kernelData (B i) x)).piece n := hv
-
- def kernelShiftEquiv (B : GradedSheaf (𝓐 := 𝓐) G) (x : V) (d : ℕ) :
-    kernelData (B.shift d) x ≃ₗ[A] (kernelData B x).shift d := LinearEquiv.refl A _
-
- theorem kernelShiftEquiv_graded (B : GradedSheaf (𝓐 := 𝓐) G) (x : V) (d : ℕ)
-    (n : ℤ) (v : kernelData (B.shift d) x) (hv : v∈(kernelData (B.shift d) x).piece n) :
-    kernelShiftEquiv B x d v∈((kernelData B x).shift d).piece n := hv
-
- def Iso.kernelEquiv {B C : GradedSheaf (𝓐 := 𝓐) G} (j : Iso B C) (x : V) :
-    kernelData B x ≃ₗ[A] kernelData C x where
-  toFun m := ⟨j.vertex x m.val,(j.toIso.upwardKernel_iff x Set.univ m.val).mpr m.property⟩
-  invFun m := ⟨(j.vertex x).symm m.val,(j.toIso.symm.upwardKernel_iff x Set.univ m.val).mpr m.property⟩
-  left_inv m := Subtype.ext ((j.vertex x).symm_apply_apply m.val)
-  right_inv m := Subtype.ext ((j.vertex x).apply_symm_apply m.val)
-  map_add' m n := Subtype.ext (map_add (j.vertex x) m.val n.val)
-  map_smul' a m := Subtype.ext (map_smul (j.vertex x) a m.val)
-
- theorem Iso.kernelEquiv_graded {B C : GradedSheaf (𝓐 := 𝓐) G} (j : Iso B C) (x : V)
-    (n : ℤ) (v : kernelData B x) (hv : v∈(kernelData B x).piece n) :
-    j.kernelEquiv x v∈(kernelData C x).piece n := j.vertex_graded x n v.val hv
-
-end
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded Polynomial
-universe uk us um
-variable {k : Type uk} [Field k] {σ : Type us} [Finite σ]
-  {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
-noncomputable section
-
- instance kernelShiftFree (B : GradedSheaf (𝓐 := polynomialGrading k σ) G) (x : V) (d : ℕ)
-    [Module.Free (MvPolynomial σ k) (kernelData B x)] :
-    Module.Free (MvPolynomial σ k) (kernelData (B.shift d) x) :=
-  Module.Free.of_equiv (kernelShiftEquiv B x d).symm
-
- instance kernelProdFree (B C : GradedSheaf (𝓐 := polynomialGrading k σ) G) (x : V)
-    [Module.Free (MvPolynomial σ k) (kernelData B x)]
-    [Module.Free (MvPolynomial σ k) (kernelData C x)] :
-    Module.Free (MvPolynomial σ k) (kernelData (prod B C) x) :=
-  Module.Free.of_equiv (kernelProdEquiv B C x).symm
-
- instance kernelPiFree {ι : Type um} [Fintype ι]
-    (B : ι → GradedSheaf (𝓐 := polynomialGrading k σ) G) (x : V)
-    [∀ i, Module.Free (MvPolynomial σ k) (kernelData (B i) x)] :
-    Module.Free (MvPolynomial σ k) (kernelData (pi B) x) :=
-  Module.Free.of_equiv (kernelPiEquiv B x).symm
-
- instance kernelZeroFree (x : V) :
-    Module.Free (MvPolynomial σ k) (kernelData (zero (𝓐 := polynomialGrading k σ) (G := G)) x) := by
-  let : Subsingleton (kernelData (zero (𝓐 := polynomialGrading k σ) (G := G)) x) :=
-    ⟨fun x y => Subtype.ext (Subsingleton.elim _ _)⟩
-  infer_instance
-
- theorem kernelChar_shift (B : GradedSheaf (𝓐 := polynomialGrading k σ) G) (x : V) (d : ℕ)
-    [Module.Free (MvPolynomial σ k) (kernelData B x)] :
-    (kernelData (B.shift d) x).char=X^d*(kernelData B x).char := by
-  rw [← ModuleData.char_shift _ d]
-  exact ModuleData.character_eq_of_gradedEquiv (kernelShiftEquiv B x d)
-    (kernelShiftEquiv_graded B x d)
-
- theorem kernelChar_prod (B C : GradedSheaf (𝓐 := polynomialGrading k σ) G) (x : V)
-    [Module.Free (MvPolynomial σ k) (kernelData B x)]
-    [Module.Free (MvPolynomial σ k) (kernelData C x)] :
-    (kernelData (prod B C) x).char=(kernelData B x).char+(kernelData C x).char := by
-  rw [← ModuleData.char_prod]
-  exact ModuleData.character_eq_of_gradedEquiv (kernelProdEquiv B C x)
-    (kernelProdEquiv_graded B C x)
-
- theorem kernelChar_pi {ι : Type um} [Fintype ι]
-    (B : ι → GradedSheaf (𝓐 := polynomialGrading k σ) G) (x : V)
-    [∀ i, Module.Free (MvPolynomial σ k) (kernelData (B i) x)] :
-    (kernelData (pi B) x).char=∑ i, (kernelData (B i) x).char := by
-  rw [← ModuleData.char_pi]
-  exact ModuleData.character_eq_of_gradedEquiv (kernelPiEquiv B x) (kernelPiEquiv_graded B x)
-
- theorem kernelChar_zero (x : V) :
-    (kernelData (zero (𝓐 := polynomialGrading k σ) (G := G)) x).char=0 := by
-  let : Subsingleton (kernelData (zero (𝓐 := polynomialGrading k σ) (G := G)) x) :=
-    ⟨fun x y => Subtype.ext (Subsingleton.elim _ _)⟩
-  exact ModuleData.char_zero _
-
- theorem ModelSum.kernelFree {O : V → GradedSheaf (𝓐 := polynomialGrading k σ) G}
-    (hf : ∀ y x, Module.Free (MvPolynomial σ k) (kernelData (O y) x))
-    {U : Set V} {B : GradedSheaf (𝓐 := polynomialGrading k σ) G}
-    (h : ModelSum O U B) (x : V) : Module.Free (MvPolynomial σ k) (kernelData B x) := by
-  induction h with
-  | zero => infer_instance
-  | @add C _ c _ n d ih =>
-    let _ := ih
-    let _ := hf c x
-    infer_instance
-
- def freeKernelChar (B : GradedSheaf (𝓐 := polynomialGrading k σ) G)
-    (hf : ∀ x, Module.Free (MvPolynomial σ k) (kernelData B x)) (x : V) : Polynomial ℤ := by
-  let := hf x
-  exact (kernelData B x).char
-
- theorem ModelSum.character_both [Fintype V] {α : E → MvPolynomial σ k}
-    (O : BoundaryModels (polynomialGrading_negative k σ) (G := G) α)
-    (hk : ∀ y x, Module.Free (MvPolynomial σ k) (kernelData (O.sheaf y) x))
-    {U : Set V} {B : GradedSheaf (𝓐 := polynomialGrading k σ) G}
-    (h : ModelSum O.sheaf U B) :
-    ∃ m : V → Polynomial ℤ, (∀ y n, 0≤(m y).coeff n) ∧
-      (∀ x, freeStalkChar B (h.free O) x =
-        ∑ y, m y*freeStalkChar (O.sheaf y) (O.free y) x) ∧
-      (∀ x, freeKernelChar B (h.kernelFree hk) x =
-        ∑ y, m y*freeKernelChar (O.sheaf y) (hk y) x) := by
-  classical
-  induction h with
-  | zero =>
-    refine ⟨0,by simp,?_,?_⟩
-    · intro x
-      change ((GradedSheaf.zero (𝓐 := polynomialGrading k σ) (G := G)).vertex x).char = _
-      simp only [Pi.zero_apply,zero_mul,Finset.sum_const_zero]
-      exact ModuleData.char_zero _
-    · intro x
-      change (kernelData (GradedSheaf.zero (𝓐 := polynomialGrading k σ) (G := G)) x).char = _
-      simp only [Pi.zero_apply,zero_mul,Finset.sum_const_zero]
-      exact kernelChar_zero (σ := σ) (G := G) x
-  | @add B h c hc n d ih =>
-    obtain ⟨m,hm,heq,hkeq⟩ := ih
-    let a : Polynomial ℤ := ∑ i, X^(d i)
-    have ha (r : ℕ) : 0 ≤ a.coeff r := by
-      simp only [a,Polynomial.finsetSum_coeff,Polynomial.coeff_X_pow]
-      exact Finset.sum_nonneg (fun i _ => by split <;> norm_num)
-    refine ⟨fun y => m y+if y=c then a else 0,?_,?_,?_⟩
-    · intro y r
-      rw [Polynomial.coeff_add]
-      apply add_nonneg (hm y r)
-      split
-      · exact ha r
-      · simp
-    · intro x
-      let _ := h.free O x
-      let _ := O.free c x
-      change ((B.vertex x).prod (ModuleData.pi (fun i => ((O.sheaf c).vertex x).shift (d i)))).char = _
-      rw [ModuleData.char_prod,ModuleData.char_pi]
-      simp only [ModuleData.char_shift]
-      change freeStalkChar B (h.free O) x + _ = _
-      rw [heq x]
-      simp only [add_mul,Finset.sum_add_distrib]
-      congr 1
-      rw [← Finset.sum_mul]
-      simp only [ite_mul,zero_mul,Finset.sum_ite_eq',Finset.mem_univ,ite_true]
-      rfl
-    · intro x
-      let _ := h.kernelFree hk x
-      let _ := hk c x
-      change (kernelData (prod B (pi (fun i => (O.sheaf c).shift (d i)))) x).char = _
-      rw [kernelChar_prod,kernelChar_pi]
-      simp only [kernelChar_shift]
-      change freeKernelChar B (h.kernelFree hk) x + _ = _
-      rw [hkeq x]
-      simp only [add_mul,Finset.sum_add_distrib]
-      congr 1
-      rw [← Finset.sum_mul]
-      simp only [ite_mul,zero_mul,Finset.sum_ite_eq',Finset.mem_univ,ite_true]
-      rfl
-
-end
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  [IsNoetherianRing A] {𝓐 : ℤ → Submodule k A}
-  {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
-noncomputable section
- def kernelEquivFull (B : GradedSheaf (𝓐 := 𝓐) G) (x : V) :
-    kernelData B x ≃ₗ[A] LinearMap.ker (B.forget.stalkMap x) where
-  toFun m := ⟨m.val,funext (fun e => m.property e (Set.mem_univ e))⟩
-  invFun m := ⟨m.val,fun e _ => congrFun m.property e⟩
-  left_inv _ := rfl
-  right_inv _ := rfl
-  map_add' _ _ := rfl
-  map_smul' _ _ := rfl
-end
-end KLInvariance.MomentGraph.GradedSheaf
-
-namespace KLInvariance.BruhatGraph.FreeSheaf
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
-universe u
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W) (b : W)
-  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  (B : FreeSheaf cs 1 b basis)
-  [Fintype (Edge cs 1 b)] [Fintype (Interval cs 1 b)] [DecidableEq (Interval cs 1 b)]
-noncomputable section
-
- def kernelDual (d : SectionDuality cs 1 b basis B) (x : Interval cs 1 b) :
-    Dual (Coefficient (σ := σ)) (B.sheaf.vertex x) ≃ₗ[Coefficient (σ := σ)]
-      GradedSheaf.kernelData B.sheaf x :=
-  (costalkEquivDual cs 1 b basis B d x).symm.trans (B.sheaf.kernelEquivFull x).symm
-
-omit [Fintype (Interval cs 1 b)] in
- theorem kernel_free (d : SectionDuality cs 1 b basis B) (x : Interval cs 1 b) :
-    Module.Free (Coefficient (σ := σ)) (GradedSheaf.kernelData B.sheaf x) := by
-  let := B.free x
-  exact Module.Free.of_equiv (kernelDual cs b basis B d x)
-
- theorem kernelDual_homogeneous (d : SectionDuality cs 1 b basis B) (n : ℕ)
-    (hd : ∀ i j (m a : B.sheaf.sections Set.univ),
-      m∈(B.sheaf.sections Set.univ).piece i → a∈(B.sheaf.sections Set.univ).piece j →
-      d.pairing m a∈polynomialGrading ℝ σ (i+j-(n:ℤ)))
-    (x : Interval cs 1 b) (hn : cs.length x.val≤n) (r : ℤ)
-    (lam : Dual (Coefficient (σ := σ)) (B.sheaf.vertex x))
-    (hlam : ∀ j (v : B.sheaf.vertex x), v∈(B.sheaf.vertex x).piece j →
-      lam v∈polynomialGrading ℝ σ (j+r)) :
-    kernelDual cs b basis B d x lam∈(GradedSheaf.kernelData B.sheaf x).piece
-      (r+(n-cs.length x.val : ℕ)) := by
-  have h := costalkEquivDual_inverse_homogeneous cs 1 b basis B d n hd x r lam hlam
-  rw [incoming_card] at h
-  change ((costalkEquivDual cs 1 b basis B d x).symm lam).val∈
-    (B.sheaf.vertex x).piece (r+↑(n-cs.length x.val))
-  have he : r+(n:ℤ)-cs.length x.val=r+↑(n-cs.length x.val) := by omega
-  rwa [he] at h
-
- theorem kernel_character (d : SectionDuality cs 1 b basis B) (n : ℕ)
-    (hd : ∀ i j (m a : B.sheaf.sections Set.univ),
-      m∈(B.sheaf.sections Set.univ).piece i → a∈(B.sheaf.sections Set.univ).piece j →
-      d.pairing m a∈polynomialGrading ℝ σ (i+j-(n:ℤ)))
-    (hn : cs.length b≤n) (x : Interval cs 1 b) :
-    GradedSheaf.freeKernelChar B.sheaf (kernel_free cs b basis B d) x=
-      Polynomial.reflect (n-cs.length x.val) (GradedSheaf.freeStalkChar B.sheaf B.free x) := by
-  let := B.free x
-  let := kernel_free cs b basis B d x
-  exact ModuleData.char_of_dual_equiv _ _ (kernelDual cs b basis B d x) _
-    (kernelDual_homogeneous cs b basis B d n hd x ((length_le_of_bruhat cs x.property.2).trans hn))
-
-end
-end KLInvariance.BruhatGraph.FreeSheaf
-
-end
-
-
-section
-
-/-! Genuine word costalk character from the integral word pairing, transported
-along the fixed symmetric-algebra coordinate isomorphism. -/
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
-universe u
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- theorem bottSection_left_orthogonal (l : List I) (hl : cs.IsReduced l)
-    (x : Interval cs 1 (cs.wordProd l))
-    (m n : (bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ)
-    (hm : ∀ y, y≠x → m.val y=0) (hn : n.val x=0) :
-    bottSectionDuality cs l hl m n=0 := by
-  obtain ⟨a,rfl⟩ := bottSection_surjective cs l hl m
-  obtain ⟨b,rfl⟩ := bottSection_surjective cs l hl n
-  rw [bottSectionDuality_apply]
-  exact bottStalk_supported_annihilates cs l hl x a b hm hn
-
- theorem bottSection_right_orthogonal (l : List I) (hl : cs.IsReduced l)
-    (x : Interval cs 1 (cs.wordProd l))
-    (m n : (bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ)
-    (hm : m.val x=0) (hn : ∀ y, y≠x → n.val y=0) :
-    bottSectionDuality cs l hl m n=0 := by
-  rw [bottSection_symmetric]
-  exact bottSection_left_orthogonal cs l hl x n m hn hm
-
- def bottWordPerfectPairing (l : List I) (hl : cs.IsReduced l) :
-    GradedSheaf.PerfectPairing (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
-      (l.length : ℤ) where
-  pairing := bottSectionDuality cs l hl
-  homogeneous := bottSectionDuality_homogeneous cs basis l hl
-  left_orthogonal := bottSection_left_orthogonal cs l hl
-  right_orthogonal := bottSection_right_orthogonal cs l hl
-
- def bottPolynomialPerfectPairing (l : List I) (hl : cs.IsReduced l) :
-    GradedSheaf.PerfectPairing (bottPolynomialSheaf cs basis l 1 (cs.wordProd l)) (l.length : ℤ) :=
-  (bottWordPerfectPairing cs basis l hl).coefficientChange (SymmetricAlgebra.equivMvPolynomial basis)
-    (mem_symmetricGrading basis)
-
- def bottFreeSheaf (l : List I) : FreeSheaf cs 1 (cs.wordProd l) basis where
-  sheaf := bottPolynomialSheaf cs basis l 1 (cs.wordProd l)
-  free := bottPolynomialSheaf_free cs basis l 1 (cs.wordProd l)
-  quotient := bottPolynomialSheaf_quotient cs basis l 1 (cs.wordProd l)
-  generated := bottPolynomialSheaf_generated cs basis l 1 (cs.wordProd l)
-
-local instance finiteEdges (l : List I) : Fintype (Edge cs 1 (cs.wordProd l)) := Fintype.ofFinite _
-local instance (a : Type*) : DecidableEq a := Classical.decEq a
-
- def bottFreeDuality (l : List I) (hl : cs.IsReduced l) :
-    FreeSheaf.SectionDuality cs 1 (cs.wordProd l) basis (bottFreeSheaf cs basis l) :=
-  FreeSheaf.dualityOfPairing cs 1 (cs.wordProd l) basis _ (bottPolynomialPerfectPairing cs basis l hl)
-
- theorem bottKernelFree (l : List I) (hl : cs.IsReduced l) (x : Interval cs 1 (cs.wordProd l)) :
-    Module.Free (Coefficient (σ := σ))
-      (GradedSheaf.kernelData (bottPolynomialSheaf cs basis l 1 (cs.wordProd l)) x) :=
-  FreeSheaf.kernel_free cs (cs.wordProd l) basis _ (bottFreeDuality cs basis l hl) x
-
- theorem bottKernelCharacter (l : List I) (hl : cs.IsReduced l)
-    [Fintype (Interval cs 1 (cs.wordProd l))] (x : Interval cs 1 (cs.wordProd l)) :
-    GradedSheaf.freeKernelChar (bottPolynomialSheaf cs basis l 1 (cs.wordProd l))
-      (bottKernelFree cs basis l hl) x=Polynomial.reflect (rankDifference cs x.val (cs.wordProd l))
-        (bottStalkCharacter cs basis l x.val) := by
-  have he : cs.length (cs.wordProd l)=l.length := hl
-  change _=Polynomial.reflect (cs.length (cs.wordProd l)-cs.length x.val) _
-  rw [he]
-  apply FreeSheaf.kernel_character cs (cs.wordProd l) basis _ (bottFreeDuality cs basis l hl)
-    l.length _ he.le x
-  intro i j m a hm ha
-  exact (bottPolynomialPerfectPairing cs basis l hl).homogeneous i j m a
-    (fun x => hm x (Set.mem_univ x)) (fun x => ha x (Set.mem_univ x))
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Bar and the exact R-convolution for genuine positive Hecke words.
-No KL character or degree estimate is assumed. -/
-namespace KLInvariance.HeckeQ
-open Hecke (Space basis)
-variable {I W C : Type*} [Group W] {M : CoxeterMatrix I} [CommRing C]
-noncomputable section
-
- theorem one_add_leftInv (cs : CoxeterSystem M W) (q : Cˣ) (i : I) (v : Space W C) :
-    v+leftInv cs q i v=(↑q⁻¹:C) • (v+left cs (q:C) i v) := by
-  rw [leftInv_apply,smul_sub,smul_smul,smul_add]
-  have h : (↑q⁻¹:C)*((q:C)-1)=1-(↑q⁻¹:C) := by rw [mul_sub,q.inv_mul,mul_one]
-  rw [h,sub_smul,one_smul]
-  module
-
- theorem barLinear_positiveWord (cs : CoxeterSystem M W) (q : Cˣ) (l : List I) :
-    barLinear cs q (positiveWord cs (↑q⁻¹:C) l)=
-      (↑q⁻¹:C)^l.length • positiveWord cs (q:C) l := by
-  induction l with
-  | nil => simp [barBasis_one]
-  | cons i l ih =>
-    simp only [positiveWord,map_add]
-    have h := congrArg (fun f : Module.End C (Space W C) => f (positiveWord cs (↑q⁻¹:C) l))
-      (barLinear_left cs q i)
-    simp only [Module.End.mul_apply] at h
-    rw [h,ih,map_smul,← smul_add,one_add_leftInv,smul_smul,List.length_cons,pow_succ]
-
- theorem positiveWord_support (cs : CoxeterSystem M W) (q : C) (l : List I)
-    (hl : cs.IsReduced l) : (positiveWord cs q l).support ⊆ lowerFinset cs (cs.wordProd l) := by
-  classical
-  induction l with
-  | nil => simpa [positiveWord,Hecke.basis,lowerFinset] using
-      (show (Finsupp.single 1 (1:C)).support ⊆ lowerFinset cs 1 by
-        intro x hx
-        by_cases h : x=1
-        · subst x; exact (mem_lowerFinset cs 1 1).mpr .refl
-        · simp [Finsupp.mem_support_iff,Ne.symm h] at hx)
-  | cons i l ih =>
-    have ht : cs.IsReduced l := by simpa using hl.drop 1
-    have ha : ¬cs.length (cs.simple i*cs.wordProd l)<cs.length (cs.wordProd l) := by
-      have h := hl.eq
-      simp only [cs.wordProd_cons,List.length_cons,←ht.eq] at h
-      omega
-    intro x hx
-    apply (mem_lowerFinset cs (cs.wordProd (i::l)) x).mpr
-    have step (y : W) (hy : BruhatLE cs y (cs.wordProd l)) :
-        BruhatLE cs (upperSimple cs i y) (cs.wordProd (i::l)) := by
-      simpa only [upperSimple,ite_eq_right ha,cs.wordProd_cons] using upperSimple_mono cs i hy
-    have h : positiveWord cs q l x≠0 ∨ positiveWord cs q l (cs.simple i*x)≠0 := by
-      by_contra hn
-      push Not at hn
-      rw [Finsupp.mem_support_iff,positiveWord_cons_coeff,hn.1,hn.2] at hx
-      split_ifs at hx <;> simp_all
-    rcases h with h|h
-    · exact (le_upperSimple cs i x).trans
-        (step x ((mem_lowerFinset cs _ _).mp (ih ht (Finsupp.mem_support_iff.mpr h))))
-    · have hh := step (cs.simple i*x)
-        ((mem_lowerFinset cs _ _).mp (ih ht (Finsupp.mem_support_iff.mpr h)))
-      rw [upperSimple_mul] at hh
-      exact (le_upperSimple cs i x).trans hh
-
- theorem positiveWord_rValue (cs : CoxeterSystem M W) (q : Cˣ) (l : List I)
-    (hl : cs.IsReduced l) (x : W) :
-    ∑ z ∈ lowerFinset cs (cs.wordProd l), rValue cs q x z * positiveWord cs (q:C) l z=
-      (↑q⁻¹:C)^cs.length x * (q:C)^l.length * positiveWord cs (↑q⁻¹:C) l x := by
-  classical
-  have h := congrArg (fun v : Space W C => v x) (barLinear_positiveWord cs q⁻¹ l)
-  simp only [inv_inv] at h
-  rw [barLinear,Finsupp.linearCombination_apply,Finsupp.sum_apply,
-    Finsupp.sum_of_support_subset _ (positiveWord_support cs (q:C) l hl) _ (by simp)] at h
-  simp only [Finsupp.smul_apply,smul_eq_mul] at h
-  have hr (z : W) : rValue cs q x z=(↑q⁻¹:C)^cs.length x * barBasis cs q⁻¹ z x := by
-    simpa using rValue_inverse_eq_barBasis cs q⁻¹ x z
-  simp only [hr,mul_assoc,←Finset.mul_sum]
-  rw [show (∑ z ∈ lowerFinset cs (cs.wordProd l), barBasis cs q⁻¹ z x * positiveWord cs (q:C) l z)=
-      (q:C)^l.length * positiveWord cs (↑q⁻¹:C) l x by
-    simpa only [mul_comm] using h]
-
-end
-end KLInvariance.HeckeQ
-
-end
-
-
-section
-
-namespace KLInvariance.HeckeQ
-open Hecke (Space basis)
-variable {I W C : Type*} [Group W] {M : CoxeterMatrix I} [CommRing C]
-noncomputable section
-
- theorem eval_positiveWord (cs : CoxeterSystem M W) (q : C) (l : List I) (x : W) :
-    Polynomial.eval₂ (Int.castRingHom C) q
-      (positiveWord cs (Polynomial.X : Polynomial ℤ) l x)=positiveWord cs q l x := by
-  classical
-  induction l generalizing x with
-  | nil => simp only [positiveWord_nil,Hecke.basis,Finsupp.single_apply]; split_ifs <;> simp
-  | cons i l ih =>
-    rw [positiveWord_cons_coeff,positiveWord_cons_coeff]
-    split_ifs <;> simp only [Polynomial.eval₂_add,Polynomial.eval₂_mul,
-      Polynomial.eval₂_X,ih]
-
-end
-end KLInvariance.HeckeQ
-
-end
-
-
-section
-
-namespace KLInvariance.Graded.ModuleData
-open Polynomial Module
-universe uk us um
-variable {k : Type uk} [Field k] {σ : Type us} [Finite σ]
-  (M N : ModuleData.{uk,max uk us,um} (polynomialGrading k σ))
-noncomputable section
-
- theorem char_natDegree_le_of_dual_equiv [Module.Free (MvPolynomial σ k) M]
-    (e : Dual (MvPolynomial σ k) M ≃ₗ[MvPolynomial σ k] N) (l : ℕ)
-    (he : ∀ r lam, (∀ j (v : M), v∈M.piece j →
-      lam v∈polynomialGrading k σ (j+r)) → e lam∈N.piece (r+l)) :
-    M.char.natDegree≤l := by
-  classical
-  let c := (exists_nonnegativeBasis (σ := σ) M.piece M.nonneg).some
-  rw [char,character_eq _ _ c]
-  apply Polynomial.natDegree_sum_le_of_forall_le
-  intro i _
-  simpa only [Polynomial.natDegree_X_pow] using M.degree_le_of_dual_equiv N e l he c i
-
-end
-end KLInvariance.Graded.ModuleData
-
-end
-
-
-section
-
-/-! Genuine word stalk-to-costalk R-convolution, derived from the Hecke bar
-and the constructed integral perfect word pairing. -/
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph Polynomial
-universe u
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-noncomputable section
-local instance (a : Type*) : DecidableEq a := Classical.decEq a
-local instance rWordEdges (l : List I) : Fintype (Edge cs 1 (cs.wordProd l)) := Fintype.ofFinite _
-local instance rWordInterval (l : List I) : Fintype (Interval cs 1 (cs.wordProd l)) := by
-  let := interval_finite cs 1 (cs.wordProd l)
-  exact Fintype.ofFinite _
-
- theorem bottStalkCharacter_natDegree_le (l : List I) (hl : cs.IsReduced l)
-    (x : Interval cs 1 (cs.wordProd l)) :
-    (bottStalkCharacter cs basis l x.val).natDegree≤rankDifference cs x.val (cs.wordProd l) := by
-  let B := bottFreeSheaf cs basis l
-  let := B.free x
-  change (B.sheaf.vertex x).char.natDegree≤rankDifference cs x.val (cs.wordProd l)
-  apply ModuleData.char_natDegree_le_of_dual_equiv _ _
-    (FreeSheaf.kernelDual cs (cs.wordProd l) basis B (bottFreeDuality cs basis l hl) x)
-    (rankDifference cs x.val (cs.wordProd l))
-  have hd := (bottPolynomialPerfectPairing cs basis l hl).homogeneous
-  have he : cs.length (cs.wordProd l)=l.length := hl
-  apply FreeSheaf.kernelDual_homogeneous cs (cs.wordProd l) basis B (bottFreeDuality cs basis l hl)
-    (cs.length (cs.wordProd l)) _ x (length_le_of_bruhat cs x.property.2)
-  intro i j m a hm ha
-  rw [he]
-  exact hd i j m a (fun x => hm x (Set.mem_univ x)) (fun x => ha x (Set.mem_univ x))
-
- theorem bottStalkCharacter_reciprocity (l : List I) (hl : cs.IsReduced l)
-    (x : Interval cs 1 (cs.wordProd l)) :
-    Polynomial.reflect (rankDifference cs x.val (cs.wordProd l)) (bottStalkCharacter cs basis l x.val)=
-      ∑ z ∈ lowerFinset cs (cs.wordProd l),
-        OrdinaryR.polynomial cs x.val z*bottStalkCharacter cs basis l z := by
-  apply Polynomial.toLaurent_injective
-  let q := OrdinaryR.variableUnit
-  have h := HeckeQ.positiveWord_rValue cs q l hl x.val
-  have hlen : cs.length x.val≤l.length := hl ▸ length_le_of_bruhat cs x.property.2
-  have hp : (↑q⁻¹ : LaurentPolynomial ℤ)^cs.length x.val * (q:LaurentPolynomial ℤ)^l.length=
-      (q:LaurentPolynomial ℤ)^(rankDifference cs x.val (cs.wordProd l)) := by
-    have he : cs.length (cs.wordProd l)=l.length := hl
-    rw [rankDifference,he,← Nat.sub_add_cancel hlen,pow_add]
-    calc
-      _ = ((↑q⁻¹:LaurentPolynomial ℤ)*(q:LaurentPolynomial ℤ))^cs.length x.val *
-          (q:LaurentPolynomial ℤ)^(l.length-cs.length x.val) := by rw [mul_pow]; ring
-      _ = _ := by rw [q.inv_mul]; simp
-  rw [hp] at h
-  have hv (p : Polynomial ℤ) : Polynomial.toLaurent p=
-      Polynomial.eval₂ (Int.castRingHom (LaurentPolynomial ℤ)) (q:LaurentPolynomial ℤ) p := by
-    induction p using Polynomial.induction_on' with
-    | add p r hp hr => simp only [map_add,Polynomial.eval₂_add,hp,hr]
-    | monomial n a => simp [q,Polynomial.toLaurent_C_mul_T]
-  rw [hv,eval₂_reflect_unit q _ _ (bottStalkCharacter_natDegree_le cs basis l hl x)]
-  rw [hv]
-  simp only [Polynomial.eval₂_finsetSum,Polynomial.eval₂_mul,OrdinaryR.polynomial_eval,
-    bottStalkCharacter_eq_heckeWord,HeckeQ.eval_positiveWord]
-  exact h.symm
-
- theorem bottKernelCharacter_rConvolution (l : List I) (hl : cs.IsReduced l)
-    (x : Interval cs 1 (cs.wordProd l)) :
-    GradedSheaf.freeKernelChar (bottPolynomialSheaf cs basis l 1 (cs.wordProd l))
-      (bottKernelFree cs basis l hl) x=
-      ∑ z : Interval cs 1 (cs.wordProd l),
-        OrdinaryR.polynomial cs x.val z.val*bottStalkCharacter cs basis l z.val := by
-  rw [bottKernelCharacter cs basis l hl x,bottStalkCharacter_reciprocity cs basis l hl x]
-  exact Finset.sum_subtype (F := (inferInstance : Fintype (Interval cs 1 (cs.wordProd l))))
-    (lowerFinset cs (cs.wordProd l))
-    (fun z => by simp only [mem_lowerFinset]; exact ⟨fun h => ⟨one_bruhat cs z,h⟩,And.right⟩)
-    (fun z => OrdinaryR.polynomial cs x.val z*bottStalkCharacter cs basis l z)
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Recognition by the actual minimal-boundary object, not by an asserted
-character or by restricting a perfect form to an arbitrary summand. -/
-namespace KLInvariance.MomentGraph.BoundarySheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-  [IsDomain A] {hneg : ∀ n < 0, 𝓐 n=⊥}
-  {V E : Type um} [PartialOrder V] {G : OrderedGraph V E} {α : E → A} {b : V}
-  (B : BoundarySheaf hneg G α b)
-noncomputable section
-
- theorem top_nontrivial : Nontrivial (B.sheaf.vertex b) := by
-  let e := ModuleData.equivOfEq B.top
-  let z : ModuleData.unit.{uk,ua,um} hneg := fun _ => (0 : A)
-  let o : ModuleData.unit.{uk,ua,um} hneg := fun _ => (1 : A)
-  refine ⟨⟨e.symm z,e.symm o,?_⟩⟩
-  intro h
-  exact zero_ne_one (congrFun (e.symm.injective h) PUnit.unit)
-
- variable [IsNoetherianRing A] [Fintype V] [Fintype E]
-
- theorem iso_of_primitive_top_split
-    (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c=a)
-    (hb : ∀ x, x≤b) (C : GradedSheaf (𝓐 := 𝓐) G)
-    (hCq : C.UpperQuotient α) (hCfree : ∀ x, Module.Free A (C.vertex x))
-    (hCf : C.forget.Flabby) (hCg : C.forget.Generated)
-    (hCp : (GradedSheaf.Hom.id C).PrimitiveProjector)
-    (i : ModuleData.Hom (B.sheaf.vertex b) (C.vertex b))
-    (p : ModuleData.Hom (C.vertex b) (B.sheaf.vertex b))
-    (hpi : ∀ v, p.map (i.map v)=v) : Nonempty (GradedSheaf.Iso B.sheaf C) := by
-  let := B.top_nontrivial
-  obtain ⟨s⟩ := B.splitIn_of_top_split hzero hb C hCq hCfree hCf hCg i p hpi
-  exact ⟨s.isoOfPrimitive hCp b⟩
-
-end
-end KLInvariance.MomentGraph.BoundarySheaf
-
-end
-
-
-section
-
-/-! The actual symmetric-coefficient BMP boundary sheaf is isomorphic to
-its integral adjoint image. The splitting is constructed from the word's
-proved free-stalk, exact-edge and flabbiness properties. -/
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
-universe u us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  (l : List I) (hl : cs.IsReduced l)
-  (B : BoundarySheaf (symmetricGrading_negative basis)
-    (graph cs 1 (cs.wordProd l)) (symmetricLabel cs 1 (cs.wordProd l))
-    ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩)
-  (s : GradedSheaf.SplitIn B.sheaf (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
-noncomputable section
-
- def bottSplitAdjointSplit : GradedSheaf.SplitIn
-    (bottSplitAdjointSheaf cs basis l hl B.sheaf s)
-    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)) := by
-  let := symmetricCoefficientNoetherian basis
-  exact (bottAdjointSheaf cs basis l hl s.projector).imageSplit
-    (bottAdjointSheaf_vertex_idempotent cs basis l hl s.projector s.projector_vertex)
-    (bottAdjointSheaf_edge_idempotent cs basis l hl s.projector s.projector_vertex)
-
- theorem bottSplitAdjoint_free (x : Interval cs 1 (cs.wordProd l)) :
-    Module.Free (SymmetricCoefficient (M := M))
-      ((bottSplitAdjointSheaf cs basis l hl B.sheaf s).vertex x) := by
-  let := symmetricCoefficientNoetherian basis
-  let : Module.Free (SymmetricCoefficient (M := M))
-      ((bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)).vertex x) :=
-    bottStalkFree cs l x.val
-  exact (bottSplitAdjointSplit cs basis l hl B s).vertex_free
-    (symmetricGrading_negative basis) (symmetricGrading_zero basis) x
-
- def bottSplitAdjointTopInclusion : ModuleData.Hom
-    (B.sheaf.vertex ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩)
-    ((bottSplitAdjointSheaf cs basis l hl B.sheaf s).vertex
-      ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩) :=
-  ((bottSplitAdjointSplit cs basis l hl B s).projection.vertex _).comp
-    (bottBoundarySheafTopInclusion cs basis l hl B)
-
- def bottSplitAdjointTopProjection : ModuleData.Hom
-    ((bottSplitAdjointSheaf cs basis l hl B.sheaf s).vertex
-      ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩)
-    (B.sheaf.vertex ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩) :=
-  (bottBoundarySheafTopProjection cs basis l hl B).comp
-    ((bottSplitAdjointSplit cs basis l hl B s).inclusion.vertex _)
-
- theorem bottSplitAdjointTop_split
-    (v : B.sheaf.vertex ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩) :
-    (bottSplitAdjointTopProjection cs basis l hl B s).map
-      ((bottSplitAdjointTopInclusion cs basis l hl B s).map v)=v := by
-  change (bottBoundarySheafTopProjection cs basis l hl B).map
-    (((bottAdjointSheaf cs basis l hl s.projector).vertex
-      ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩).map
-      ((bottBoundarySheafTopInclusion cs basis l hl B).map v))=v
-  exact (congrArg (bottBoundarySheafTopProjection cs basis l hl B).map
-    (bottBoundarySplitAdjoint_top_fixed cs basis l hl _ B s
-      ((bottBoundarySheafTopInclusion cs basis l hl B).map v))).trans
-    (bottBoundarySheafTop_split cs basis l hl B v)
-
- theorem bottBoundarySheaf_iso_adjoint : Nonempty
-    (GradedSheaf.Iso B.sheaf (bottSplitAdjointSheaf cs basis l hl B.sheaf s)) := by
-  classical
-  let := symmetricCoefficientNoetherian basis
-  let := interval_finite cs 1 (cs.wordProd l)
-  let := Fintype.ofFinite (Interval cs 1 (cs.wordProd l))
-  let := Fintype.ofFinite (Edge cs 1 (cs.wordProd l))
-  let t := bottSplitAdjointSplit cs basis l hl B s
-  exact B.iso_of_primitive_top_split (symmetricGrading_zero basis) (fun x => x.property.2)
-    (bottSplitAdjointSheaf cs basis l hl B.sheaf s)
-    (t.upperQuotient (bottGradedProjectionSheaf_upperQuotient cs basis l 1 (cs.wordProd l)))
-    (bottSplitAdjoint_free cs basis l hl B s)
-    (t.flabby (bottProjectionSheaf_flabby cs l hl))
-    (t.generated (bottProjectionSheaf_generated cs l 1 (cs.wordProd l)))
-    (bottBoundarySplitAdjoint_primitive cs basis l hl _ B s)
-    (bottSplitAdjointTopInclusion cs basis l hl B s)
-    (bottSplitAdjointTopProjection cs basis l hl B s)
-    (bottSplitAdjointTop_split cs basis l hl B s)
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} [IsNoetherianRing A]
-  {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
-  {B C : GradedSheaf (𝓐 := 𝓐) G} (s : SplitIn B C)
-noncomputable section
-
- theorem SplitIn.imageSectionEquiv_symm_coordinates (U : Set V)
-    (n : B.forget.sections U) (x : V) :
-    (((s.imageSectionEquiv U).symm n).val x).val=(s.inclusion.vertex x).map (n.val x) := by
-  rw [s.imageSectionEquiv_symm_apply]
-  change (s.inclusion.vertex x).map ((s.projection.vertex x).map
-    ((s.inclusion.vertex x).map (n.val x)))=(s.inclusion.vertex x).map (n.val x)
-  exact congrArg (s.inclusion.vertex x).map (s.vertex_split x (n.val x))
-
- theorem SplitIn.imageSectionEquiv_symm_inclusion (U : Set V)
-    (n : B.forget.sections U) :
-    s.projector.imageInclusion.sectionMap U ((s.imageSectionEquiv U).symm n)=
-      s.inclusion.sectionMap U n := by
-  apply Subtype.ext
-  funext x
-  exact s.imageSectionEquiv_symm_coordinates U n x
-
-end
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-/-! The adjoint-summand pairing is the ambient word pairing on its two
-honest (generally different) integral sheaf embeddings. -/
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
-universe u us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  (l : List I) (hl : cs.IsReduced l)
-  (B : GradedSheaf (𝓐 := symmetricGrading basis) (graph cs 1 (cs.wordProd l)))
-  (s : GradedSheaf.SplitIn B (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
-noncomputable section
-
- def bottSplitRightWord (n : B.forget.sections Set.univ) : BottSamelsonModule cs l := by
-  let := symmetricCoefficientNoetherian basis
-  exact (bottImageSectionWordEquiv cs basis l hl s.projector s.projector_vertex
-    ((s.imageSectionEquiv Set.univ).symm n)).val
-
- def bottSplitLeftWord
-    (m : (bottSplitAdjointSheaf cs basis l hl B s).forget.sections Set.univ) :
-    BottSamelsonModule cs l := by
-  let := symmetricCoefficientNoetherian basis
-  exact (bottImageSectionWordEquiv cs basis l hl (bottAdjointSheaf cs basis l hl s.projector)
-    (bottAdjointSheaf_vertex_idempotent cs basis l hl s.projector s.projector_vertex) m).val
-
- theorem bottSplitRightWord_coordinates (n : B.forget.sections Set.univ)
-    (x : Interval cs 1 (cs.wordProd l)) :
-    bottStalkProjection cs l x.val (bottSplitRightWord cs basis l hl B s n)=
-      (s.inclusion.vertex x).map (n.val x) := by
-  let := symmetricCoefficientNoetherian basis
-  exact (bottImageSectionWordEquiv_coordinates cs basis l hl s.projector s.projector_vertex
-    ((s.imageSectionEquiv Set.univ).symm n) x).trans
-      (s.imageSectionEquiv_symm_coordinates Set.univ n x)
-
- theorem bottSplitLeftWord_coordinates
-    (m : (bottSplitAdjointSheaf cs basis l hl B s).forget.sections Set.univ)
-    (x : Interval cs 1 (cs.wordProd l)) :
-    bottStalkProjection cs l x.val (bottSplitLeftWord cs basis l hl B s m)=(m.val x).val := by
-  let := symmetricCoefficientNoetherian basis
-  exact bottImageSectionWordEquiv_coordinates cs basis l hl
-    (bottAdjointSheaf cs basis l hl s.projector)
-    (bottAdjointSheaf_vertex_idempotent cs basis l hl s.projector s.projector_vertex) m x
-
- theorem bottSplitAdjointWord_apply
-    (m : (bottSplitAdjointSheaf cs basis l hl B s).forget.sections Set.univ)
-    (n : B.forget.sections Set.univ) :
-    bottSplitAdjointEquivDual cs basis l hl B s m n=
-      bottSamelsonDuality cs l (bottSplitLeftWord cs basis l hl B s m)
-        (bottSplitRightWord cs basis l hl B s n) := by
-  let := symmetricCoefficientNoetherian basis
-  exact bottAdjointImageEquivDual_apply cs basis l hl s.projector s.projector_vertex m
-    ((s.imageSectionEquiv Set.univ).symm n)
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! An actual recognition isomorphism to the adjoint image gives the required
-perfect pairing. The two ambient embeddings remain distinct. -/
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
-universe u us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  (l : List I) (hl : cs.IsReduced l)
-  (B : GradedSheaf (𝓐 := symmetricGrading basis) (graph cs 1 (cs.wordProd l)))
-  (s : GradedSheaf.SplitIn B (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
-  (j : GradedSheaf.Iso B (bottSplitAdjointSheaf cs basis l hl B s))
-noncomputable section
-
- def bottRecognizedPerfectPairing : GradedSheaf.PerfectPairing B (l.length : ℤ) where
-  pairing := (j.toIso.sections Set.univ).trans (bottSplitAdjointEquivDual cs basis l hl B s)
-  homogeneous i q m n hm hn := bottSplitAdjointPairing_homogeneous cs basis l hl B s i q
-    (j.toIso.sections Set.univ m) n (fun x => j.vertex_graded x i _ (hm x)) hn
-  left_orthogonal x m n hm hn := by
-    change bottSplitAdjointEquivDual cs basis l hl B s (j.toIso.sections Set.univ m) n=0
-    rw [bottSplitAdjointWord_apply]
-    apply bottStalk_supported_annihilates cs l hl x
-    · intro y hy
-      rw [bottSplitLeftWord_coordinates]
-      change ((j.vertex y (m.val y)).val)=0
-      rw [hm y hy,map_zero]
-      rfl
-    · rw [bottSplitRightWord_coordinates]
-      change (s.inclusion.vertex x).map (n.val x)=0
-      rw [hn]
-      exact (s.inclusion.vertex x).map.map_zero
-  right_orthogonal x m n hm hn := by
-    change bottSplitAdjointEquivDual cs basis l hl B s (j.toIso.sections Set.univ m) n=0
-    rw [bottSplitAdjointWord_apply,bottSamelson_symmetric]
-    apply bottStalk_supported_annihilates cs l hl x
-    · intro y hy
-      rw [bottSplitRightWord_coordinates]
-      change (s.inclusion.vertex y).map (n.val y)=0
-      rw [hn y hy]
-      exact (s.inclusion.vertex y).map.map_zero
-    · rw [bottSplitLeftWord_coordinates]
-      change ((j.vertex x (m.val x)).val)=0
-      rw [hm,map_zero]
-      rfl
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! The same genuine minimal boundary construction over the symmetric
-coefficient algebra. Its polynomial-coordinate comparison is proved separately. -/
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
-universe u us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- omit [Fintype σ] in
- theorem symmetric_scalar_mod_positive (a : SymmetricCoefficient (M := M)) :
-    ∃ c : ℝ, a-algebraMap ℝ (SymmetricCoefficient (M := M)) c ∈ positiveIdeal (symmetricGrading basis) := by
-  obtain ⟨c,hc⟩ := symmetricGrading_zero basis _
-    (DirectSum.decompose (symmetricGrading basis) a 0).property
-  refine ⟨c,?_⟩
-  rw [positiveIdeal_eq_augmentation_ker _ (symmetricGrading_negative basis)]
-  change component (symmetricGrading basis) 0 (a-algebraMap ℝ (SymmetricCoefficient (M := M)) c)=0
-  rw [map_sub,hc]
-  rw [show component (symmetricGrading basis) 0
-    (DirectSum.decompose (symmetricGrading basis) a 0 : SymmetricCoefficient (M := M)) =
-    (DirectSum.decompose (symmetricGrading basis) a 0 : SymmetricCoefficient (M := M)) from
-    DirectSum.decompose_of_mem_same _ (DirectSum.decompose (symmetricGrading basis) a 0).property]
-  exact sub_self _
-
- theorem symmetricLabel_homogeneous (u b : W) (e : Edge cs u b) :
-    symmetricLabel cs u b e ∈ symmetricGrading basis 1 :=
-  (mem_symmetricGrading basis 1 _).mpr (polynomialLabel_homogeneous cs u b basis e)
-
- theorem exists_symmetricBoundarySheaf (b : W) :
-    Nonempty (BoundarySheaf (symmetricGrading_negative basis)
-      (graph cs 1 b) (symmetricLabel cs 1 b) ⟨b,one_bruhat cs b,bruhat_refl cs b⟩) := by
-  classical
-  let := symmetricCoefficientNoetherian basis
-  let := interval_finite cs 1 b
-  let := Fintype.ofFinite (Interval cs 1 b)
-  let := Fintype.ofFinite (Edge cs 1 b)
-  exact MomentGraph.exists_boundarySheaf (symmetricGrading_negative basis)
-    (graph cs 1 b) (symmetric_scalar_mod_positive basis)
-    ⟨b,one_bruhat cs b,bruhat_refl cs b⟩ (fun x => x.property.2)
-    (symmetricLabel cs 1 b) (symmetricLabel_homogeneous cs basis 1 b)
-
- def symmetricBoundarySheaf (b : W) :
-    BoundarySheaf (symmetricGrading_negative basis)
-      (graph cs 1 b) (symmetricLabel cs 1 b) ⟨b,one_bruhat cs b,bruhat_refl cs b⟩ :=
-  (exists_symmetricBoundarySheaf cs basis b).some
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Unconditional integral graded support-orthogonal duality for the actual
-normalized minimal-boundary sheaf over the symmetric coefficient algebra. -/
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
-universe u us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- theorem symmetricBoundarySheaf_perfectPairing_word (l : List I) (hl : cs.IsReduced l)
-    (B : BoundarySheaf (symmetricGrading_negative basis)
-      (graph cs 1 (cs.wordProd l)) (symmetricLabel cs 1 (cs.wordProd l))
-      ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩) :
-    Nonempty (GradedSheaf.PerfectPairing B.sheaf (cs.length (cs.wordProd l) : ℤ)) := by
-  obtain ⟨s⟩ := bottBoundarySheaf_splitIn cs basis l hl B
-  obtain ⟨j⟩ := bottBoundarySheaf_iso_adjoint cs basis l hl B s
-  have hd : cs.length (cs.wordProd l)=l.length := hl
-  rw [hd]
-  exact ⟨bottRecognizedPerfectPairing cs basis l hl B.sheaf s j⟩
-
- theorem symmetricBoundarySheaf_perfectPairing (b : W) :
-    Nonempty (GradedSheaf.PerfectPairing (symmetricBoundarySheaf cs basis b).sheaf (cs.length b : ℤ)) := by
-  obtain ⟨l,hl,he⟩ := cs.exists_isReduced b
-  subst b
-  exact symmetricBoundarySheaf_perfectPairing_word cs basis l hl _
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-namespace KLInvariance.Graded.ModuleData
-universe uk ua um
-variable {k : Type uk} [Field k] {A B : Type ua} [CommRing A] [CommRing B]
-  [Algebra k A] [Algebra k B]
-  {𝓐 : ℤ → Submodule k A} {𝓑 : ℤ → Submodule k B}
-  [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-  [DirectSum.Decomposition 𝓑] [SetLike.GradedMonoid 𝓑]
-  (e : A ≃ₐ[k] B) (he : ∀ n a, a∈𝓐 n ↔ e a∈𝓑 n)
-  (hA : ∀ n<0,𝓐 n=⊥) (hB : ∀ n<0,𝓑 n=⊥)
-noncomputable section
-
- def changeUnitHom : Hom (change e he (unit.{uk,ua,um} hA)) (unit hB) where
-  map :=
-    { toFun m j := e (m.val j)
-      map_add' m n := funext fun j => map_add e (m.val j) (n.val j)
-      map_smul' b m := by
-        funext j
-        change e (e.symm b * m.val j)=b * e (m.val j)
-        rw [map_mul,e.apply_symm_apply] }
-  graded := by
-    intro n m hm j
-    exact (he (n-0) _).mp ((mem_changePiece e (unit hA) n m).mp hm j)
-
- def changeUnitInvHom : Hom (unit.{uk,ua,um} hB) (change e he (unit hA)) where
-  map :=
-    { toFun m := CoefficientChange.mk (fun j => e.symm (m j))
-      map_add' m n := CoefficientChange.ext (funext fun j => map_add e.symm (m j) (n j))
-      map_smul' b m := by
-        apply CoefficientChange.ext
-        funext j
-        exact map_mul e.symm b (m j) }
-  graded := by
-    intro n m hm
-    apply (mem_changePiece e (unit hA) n _).mpr
-    intro j
-    apply (he (n-0) _).mpr
-    change e (e.symm (m j)) ∈ 𝓑 (n-0)
-    rw [e.apply_symm_apply]
-    exact hm j
-
- theorem changeUnitHom_inv (m : unit.{uk,ua,um} hB) :
-    (changeUnitHom e he hA hB).map ((changeUnitInvHom e he hA hB).map m)=m :=
-  funext fun j => e.apply_symm_apply (m j)
-
- theorem changeUnitInvHom_hom (m : change e he (unit.{uk,ua,um} hA)) :
-    (changeUnitInvHom e he hA hB).map ((changeUnitHom e he hA hB).map m)=m :=
-  CoefficientChange.ext (funext fun j => e.symm_apply_apply (m.val j))
-
-end
-end KLInvariance.Graded.ModuleData
-
-end
-
-
-section
-
-namespace KLInvariance.Graded.ModuleData
-universe uk ua um
-variable {k : Type uk} [Field k] {A B : Type ua} [CommRing A] [CommRing B]
-  [Algebra k A] [Algebra k B]
-  {𝓐 : ℤ → Submodule k A} {𝓑 : ℤ → Submodule k B}
-  (e : A ≃ₐ[k] B) (he : ∀ n a, a∈𝓐 n ↔ e a∈𝓑 n)
-noncomputable section
-
- def unchangeHom {M N : ModuleData.{uk,ua,um} 𝓐} (f : Hom (change e he M) (change e he N)) :
-    Hom M N where
-  map :=
-    { toFun m := (f.map (CoefficientChange.mk m)).val
-      map_add' m n := congrArg CoefficientChange.val (map_add f.map (CoefficientChange.mk m) (CoefficientChange.mk n))
-      map_smul' a m := by
-        have hh := congrArg CoefficientChange.val (map_smul f.map (e a) (CoefficientChange.mk m))
-        change (f.map (CoefficientChange.mk (e.symm (e a) • m))).val=
-          e.symm (e a) • (f.map (CoefficientChange.mk m)).val at hh
-        simpa only [e.symm_apply_apply, RingHom.id_apply] using hh }
-  graded n m hm := (mem_changePiece e N n _).mp
-    (f.graded n (CoefficientChange.mk m) ((mem_changePiece e M n _).mpr hm))
-
-end
-end KLInvariance.Graded.ModuleData
-
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A B : Type ua} [CommRing A] [CommRing B]
-  [Algebra k A] [Algebra k B]
-  {𝓐 : ℤ → Submodule k A} {𝓑 : ℤ → Submodule k B}
-  (e : A ≃ₐ[k] B) (he : ∀ n a, a∈𝓐 n ↔ e a∈𝓑 n)
-  {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
-noncomputable section
-
- def Hom.coefficientUnchange {S T : GradedSheaf (𝓐 := 𝓐) G}
-    (g : Hom (GradedSheaf.coefficientChange e he S) (GradedSheaf.coefficientChange e he T)) : Hom S T where
-  vertex x := ModuleData.unchangeHom e he (g.vertex x)
-  edge f := ModuleData.unchangeHom e he (g.edge f)
-  lower f m := congrArg CoefficientChange.val (g.lower f (CoefficientChange.mk m))
-  upper f m := congrArg CoefficientChange.val (g.upper f (CoefficientChange.mk m))
-
- theorem coefficientChange_primitive (S : GradedSheaf (𝓐 := 𝓐) G)
-    (hS : (Hom.id S).PrimitiveProjector) :
-    (Hom.id (GradedSheaf.coefficientChange e he S)).PrimitiveProjector := by
-  constructor
-  · apply Hom.eq_of_maps <;> intros <;> rfl
-  · intro f hf _ _
-    let g : Hom S S := f.coefficientUnchange e he
-    have hg : g.comp g=g := by
-      apply Hom.eq_of_maps
-      · intro x m
-        exact congrArg (fun q : Hom (GradedSheaf.coefficientChange e he S) (GradedSheaf.coefficientChange e he S) =>
-          ((q.vertex x).map (CoefficientChange.mk m)).val) hf
-      · intro x m
-        exact congrArg (fun q : Hom (GradedSheaf.coefficientChange e he S) (GradedSheaf.coefficientChange e he S) =>
-          ((q.edge x).map (CoefficientChange.mk m)).val) hf
-    have hgL : g.comp (Hom.id S)=g := by apply Hom.eq_of_maps <;> intros <;> rfl
-    have hgR : (Hom.id S).comp g=g := by apply Hom.eq_of_maps <;> intros <;> rfl
-    rcases hS.2 g hg hgL hgR with hz | hi
-    · left
-      apply Hom.eq_of_maps
-      · intro x m
-        apply CoefficientChange.ext
-        exact congrArg (fun q : Hom S S => (q.vertex x).map m.val) hz
-      · intro x m
-        apply CoefficientChange.ext
-        exact congrArg (fun q : Hom S S => (q.edge x).map m.val) hz
-    · right
-      apply Hom.eq_of_maps
-      · intro x m
-        apply CoefficientChange.ext
-        exact congrArg (fun q : Hom S S => (q.vertex x).map m.val) hi
-      · intro x m
-        apply CoefficientChange.ext
-        exact congrArg (fun q : Hom S S => (q.edge x).map m.val) hi
-
-end
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-namespace KLInvariance.Graded.ModuleData
-universe uk ua um
-variable {k : Type uk} [Field k] {A B : Type ua} [CommRing A] [CommRing B]
-  [Algebra k A] [Algebra k B]
-  {𝓐 : ℤ → Submodule k A} {𝓑 : ℤ → Submodule k B}
-  [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-  [DirectSum.Decomposition 𝓑] [SetLike.GradedMonoid 𝓑]
-  (e : A ≃ₐ[k] B) (he : ∀ n a, a∈𝓐 n ↔ e a∈𝓑 n)
-  {hA : ∀ n<0,𝓐 n=⊥} {hB : ∀ n<0,𝓑 n=⊥}
-  {M : ModuleData.{uk,ua,um} 𝓐} (hM : M=unit hA)
-  {N : ModuleData.{uk,ua,um} 𝓑} (hN : N=unit hB)
-noncomputable section
-
- def changeTopInclusion : Hom N (change e he M) :=
-  (changeHom e he (homOfEq hM.symm)).comp
-    ((changeUnitInvHom e he hA hB).comp (homOfEq hN))
-
- def changeTopProjection : Hom (change e he M) N :=
-  (homOfEq hN.symm).comp
-    ((changeUnitHom e he hA hB).comp (changeHom e he (homOfEq hM)))
-
- theorem changeTop_split (n : N) :
-    (changeTopProjection e he hM hN).map ((changeTopInclusion e he hM hN).map n)=n := by
-  subst M
-  subst N
-  change (changeUnitHom e he hA hB).map ((changeUnitInvHom e he hA hB).map n)=n
-  exact changeUnitHom_inv e he hA hB n
-
-end
-end KLInvariance.Graded.ModuleData
-
-namespace KLInvariance.MomentGraph.BoundarySheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A B : Type ua} [CommRing A] [CommRing B]
-  [Algebra k A] [Algebra k B]
-  {𝓐 : ℤ → Submodule k A} {𝓑 : ℤ → Submodule k B}
-  [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-  [DirectSum.Decomposition 𝓑] [SetLike.GradedMonoid 𝓑]
-  [IsDomain A] [IsDomain B] [IsNoetherianRing B]
-  (e : A ≃ₐ[k] B) (he : ∀ n a, a∈𝓐 n ↔ e a∈𝓑 n)
-  {hA : ∀ n<0,𝓐 n=⊥} {hB : ∀ n<0,𝓑 n=⊥}
-  {V E : Type um} [PartialOrder V] [Fintype V] [Fintype E]
-  {G : OrderedGraph V E} {α : E → A} {b : V}
-  (S : BoundarySheaf hA G α b) (T : BoundarySheaf hB G (fun q => e (α q)) b)
-noncomputable section
-
- theorem iso_coefficientChange
-    (hzero : ∀ a ∈ 𝓑 0, ∃ c : k, algebraMap k B c=a)
-    (hb : ∀ x, x≤b) : Nonempty
-      (GradedSheaf.Iso T.sheaf (GradedSheaf.coefficientChange e he S.sheaf)) := by
-  exact T.iso_of_primitive_top_split hzero hb (GradedSheaf.coefficientChange e he S.sheaf)
-    (GradedSheaf.coefficientChange_upperQuotient e he S.sheaf α S.quotient)
-    (GradedSheaf.coefficientChange_free e he S.sheaf S.free)
-    (GradedSheaf.coefficientChange_flabby e he S.sheaf S.flabby.flabby)
-    (GradedSheaf.coefficientChange_generated e he S.sheaf S.generated)
-    (GradedSheaf.coefficientChange_primitive e he S.sheaf S.primitive_identity)
-    (ModuleData.changeTopInclusion e he S.top T.top)
-    (ModuleData.changeTopProjection e he S.top T.top)
-    (ModuleData.changeTop_split e he S.top T.top)
-
-end
-end KLInvariance.MomentGraph.BoundarySheaf
-
-end
-
-
-section
-
-
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
-universe u
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W) (b : W)
-  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- theorem boundarySheaf_perfectPairing :
-    Nonempty (GradedSheaf.PerfectPairing
-      (boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf (cs.length b : ℤ)) := by
-  classical
-  let := interval_finite cs 1 b
-  let := Fintype.ofFinite (Interval cs 1 b)
-  let := Fintype.ofFinite (Edge cs 1 b)
-  let S := symmetricBoundarySheaf cs basis b
-  let T := boundarySheaf cs 1 b basis (one_bruhat cs b)
-  let e := SymmetricAlgebra.equivMvPolynomial basis
-  have he : ∀ n a, a∈symmetricGrading basis n ↔ e a∈polynomialGrading ℝ σ n :=
-    mem_symmetricGrading basis
-  obtain ⟨p⟩ := symmetricBoundarySheaf_perfectPairing cs basis b
-  obtain ⟨j⟩ := BoundarySheaf.iso_coefficientChange e he S T
-    (polynomialGrading_zero ℝ σ) (fun x => x.property.2)
-  exact ⟨(p.coefficientChange e he).pullback j⟩
-
- variable [Fintype (Edge cs 1 b)] [Fintype (Interval cs 1 b)] [DecidableEq (Interval cs 1 b)]
-
- theorem sectionDualityObligation : SectionDualityObligation cs b basis := by
-  obtain ⟨p⟩ := boundarySheaf_perfectPairing cs b basis
-  exact sectionDualityObligation_of_perfectPairing cs b basis p
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
-universe u
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W) (b : W)
-  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
-local instance finiteInterval : Fintype (Interval cs 1 b) := by
-  let := interval_finite cs 1 b
-  exact Fintype.ofFinite _
-local instance finiteEdges : Fintype (Edge cs 1 b) := Fintype.ofFinite _
-local instance (a : Type*) : DecidableEq a := Classical.decEq a
-
- def boundaryFreeSheaf : FreeSheaf cs 1 b basis where
-  sheaf := (boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf
-  free := (boundarySheaf cs 1 b basis (one_bruhat cs b)).free
-  quotient := (boundarySheaf cs 1 b basis (one_bruhat cs b)).quotient
-  generated := (boundarySheaf cs 1 b basis (one_bruhat cs b)).generated
-
- def boundaryPerfectPairing : GradedSheaf.PerfectPairing
-    (boundaryFreeSheaf cs b basis).sheaf (cs.length b : ℤ) :=
-  (boundarySheaf_perfectPairing cs b basis).some
-
- def boundaryFreeDuality : FreeSheaf.SectionDuality cs 1 b basis (boundaryFreeSheaf cs b basis) :=
-  FreeSheaf.dualityOfPairing cs 1 b basis _ (boundaryPerfectPairing cs b basis)
-
- theorem boundaryKernelFree (x : Interval cs 1 b) :
-    Module.Free (Coefficient (σ := σ))
-      (GradedSheaf.kernelData (boundaryFreeSheaf cs b basis).sheaf x) :=
-  FreeSheaf.kernel_free cs b basis _ (boundaryFreeDuality cs b basis) x
-
- def costalkCharacter (x : Interval cs 1 b) : Polynomial ℤ :=
-  GradedSheaf.freeKernelChar (boundaryFreeSheaf cs b basis).sheaf (boundaryKernelFree cs b basis) x
-
- theorem costalkCharacter_reflect (x : Interval cs 1 b) :
-    costalkCharacter cs b basis x=Polynomial.reflect (rankDifference cs x.val b)
-      (stalkCharacter cs 1 b basis (one_bruhat cs b) x) := by
-  apply FreeSheaf.kernel_character cs b basis _ (boundaryFreeDuality cs b basis) (cs.length b) _ le_rfl x
-  intro i j m a hm ha
-  exact (boundaryPerfectPairing cs b basis).homogeneous i j m a
-    (fun x => hm x (Set.mem_univ x)) (fun x => ha x (Set.mem_univ x))
-
- theorem stalkBasis_degree_le (x : Interval cs 1 b)
-    (c : NonnegativeBasis (σ := σ)
-      ((boundaryFreeSheaf cs b basis).sheaf.vertex x).piece) (i : c.index) :
-    c.degree i≤rankDifference cs x.val b := by
-  apply ModuleData.degree_le_of_dual_equiv _ _
-    (FreeSheaf.kernelDual cs b basis _ (boundaryFreeDuality cs b basis) x)
-    (rankDifference cs x.val b) _ c i
-  exact FreeSheaf.kernelDual_homogeneous cs b basis _ (boundaryFreeDuality cs b basis)
-    (cs.length b) (fun i j m a hm ha => (boundaryPerfectPairing cs b basis).homogeneous i j m a
-      (fun x => hm x (Set.mem_univ x)) (fun x => ha x (Set.mem_univ x)))
-    x (length_le_of_bruhat cs x.property.2)
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  [IsNoetherianRing A] {𝓐 : ℤ → Submodule k A}
-  {V E W F : Type um} [PartialOrder V] [PartialOrder W]
-  {G : OrderedGraph V E} {H : OrderedGraph W F}
-noncomputable section
-attribute [local instance] Classical.propDecidable
-variable {K : Set V} {hK : IsLowerSet K}
-
-omit [IsNoetherianRing A] in
- theorem lowerExtend_kernel_iff (B : GradedSheaf (𝓐 := 𝓐) (lowerGraph G K hK))
-    (x : K) (m : B.lowerExtend.vertex x.val) :
-    m∈B.lowerExtend.upwardSubmodule x.val Set.univ ↔
-      B.lowerExtendVertex x m∈B.upwardSubmodule x Set.univ := by
-  constructor
-  · intro hm e _
-    rcases e with ⟨⟨e,he⟩,hx⟩
-    cases hx
-    have hh := hm ⟨e,rfl⟩ (Set.mem_univ _)
-    exact congrFun hh ⟨⟨he⟩⟩
-  · intro hm e _
-    rcases x with ⟨x,hx⟩
-    rcases e with ⟨e,he⟩
-    change G.source e=x at he
-    subst x
-    apply funext
-    intro h
-    have hh := hm ⟨⟨e,h.down.down⟩,rfl⟩ (Set.mem_univ _)
-    exact hh
-
- def lowerExtendKernelEquiv (B : GradedSheaf (𝓐 := 𝓐) (lowerGraph G K hK))
-    (x : K) : kernelData B.lowerExtend x.val ≃ₗ[A] kernelData B x where
-  toFun m := ⟨B.lowerExtendVertex x m.val,(lowerExtend_kernel_iff B x m.val).mp m.property⟩
-  invFun m := ⟨(B.lowerExtendVertex x).symm m.val,(lowerExtend_kernel_iff B x _).mpr (by
-    simpa only [LinearEquiv.apply_symm_apply] using m.property)⟩
-  left_inv m := Subtype.ext ((B.lowerExtendVertex x).symm_apply_apply m.val)
-  right_inv m := Subtype.ext ((B.lowerExtendVertex x).apply_symm_apply m.val)
-  map_add' m n := Subtype.ext (map_add (B.lowerExtendVertex x) m.val n.val)
-  map_smul' a m := Subtype.ext (map_smul (B.lowerExtendVertex x) a m.val)
-
- theorem lowerExtendKernelEquiv_graded (B : GradedSheaf (𝓐 := 𝓐) (lowerGraph G K hK))
-    (x : K) (n : ℤ) (m : kernelData B.lowerExtend x.val)
-    (hm : m∈(kernelData B.lowerExtend x.val).piece n) :
-    B.lowerExtendKernelEquiv x m∈(kernelData B x).piece n :=
-  B.lowerExtendVertex_graded x n m.val hm
-
-omit [IsNoetherianRing A] in
- theorem fullKernel_pullback (B : GradedSheaf (𝓐 := 𝓐) H) (f : GraphMap G H)
-    (hlift : ∀ e x, f.vertex x=H.source e → ∃ a, f.edge a=e ∧ G.source a=x) (x : V) :
-    (B.pullback f).upwardSubmodule x Set.univ=B.upwardSubmodule (f.vertex x) Set.univ := by
-  exact B.forget.pullback_kernel f x Set.univ Set.univ
-    (fun e he => (he (Set.mem_univ e)).elim) (fun e _ he => hlift e x he)
-
- def pullbackKernelEquiv (B : GradedSheaf (𝓐 := 𝓐) H) (f : GraphMap G H)
-    (hlift : ∀ e x, f.vertex x=H.source e → ∃ a, f.edge a=e ∧ G.source a=x) (x : V) :
-    kernelData (B.pullback f) x ≃ₗ[A] kernelData B (f.vertex x) where
-  toFun m := ⟨m.val,fullKernel_pullback B f hlift x ▸ m.property⟩
-  invFun m := ⟨m.val,(fullKernel_pullback B f hlift x).symm ▸ m.property⟩
-  left_inv _ := rfl
-  right_inv _ := rfl
-  map_add' _ _ := rfl
-  map_smul' _ _ := rfl
-
- theorem pullbackKernelEquiv_graded (B : GradedSheaf (𝓐 := 𝓐) H) (f : GraphMap G H)
-    (hlift : ∀ e x, f.vertex x=H.source e → ∃ a, f.edge a=e ∧ G.source a=x) (x : V)
-    (n : ℤ) (m : kernelData (B.pullback f) x) (hm : m∈(kernelData (B.pullback f) x).piece n) :
-    B.pullbackKernelEquiv f hlift x m∈(kernelData B (f.vertex x)).piece n := hm
-
-end
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph Polynomial
-universe u
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W) (b : W)
-  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
- def lowerKernelEquiv (c x : Interval cs 1 b) (hx : x≤c) :
-    GradedSheaf.kernelData ((boundaryModels cs b basis).sheaf c) x ≃ₗ[Coefficient (σ := σ)]
-      GradedSheaf.kernelData (boundaryFreeSheaf cs c.val basis).sheaf ⟨x.val,x.property.1,hx⟩ :=
-  ((lowerIntervalBoundary cs b basis c).sheaf.lowerExtendKernelEquiv ⟨x,hx⟩).trans
-    ((boundaryFreeSheaf cs c.val basis).sheaf.pullbackKernelEquiv
-      (lowerIntervalMap cs b c) (lowerIntervalMap_outgoing cs b c) ⟨x,hx⟩)
-
- theorem lowerKernelEquiv_graded (c x : Interval cs 1 b) (hx : x≤c)
-    (n : ℤ) (v : GradedSheaf.kernelData ((boundaryModels cs b basis).sheaf c) x)
-    (hv : v∈(GradedSheaf.kernelData ((boundaryModels cs b basis).sheaf c) x).piece n) :
-    lowerKernelEquiv cs b basis c x hx v∈(GradedSheaf.kernelData
-      (boundaryFreeSheaf cs c.val basis).sheaf ⟨x.val,x.property.1,hx⟩).piece n :=
-  (lowerIntervalBoundary cs b basis c).sheaf.lowerExtendKernelEquiv_graded ⟨x,hx⟩ n v hv
-
- theorem boundaryModels_kernelFree (c x : Interval cs 1 b) :
-    Module.Free (Coefficient (σ := σ))
-      (GradedSheaf.kernelData ((boundaryModels cs b basis).sheaf c) x) := by
-  by_cases hx : x≤c
-  · let := boundaryKernelFree cs c.val basis ⟨x.val,x.property.1,hx⟩
-    exact Module.Free.of_equiv (lowerKernelEquiv cs b basis c x hx).symm
-  · let := (boundaryModels cs b basis).support c x hx
-    let : Subsingleton (GradedSheaf.kernelData ((boundaryModels cs b basis).sheaf c) x) :=
-      ⟨fun v w => Subtype.ext (Subsingleton.elim _ _)⟩
-    infer_instance
-
- theorem boundaryModels_kernelCharacter (c x : Interval cs 1 b) :
-    GradedSheaf.freeKernelChar ((boundaryModels cs b basis).sheaf c)
-      (boundaryModels_kernelFree cs b basis c) x=
-      Polynomial.reflect (rankDifference cs x.val c.val) (lowerCharacter cs b basis c x) := by
-  let := boundaryModels_kernelFree cs b basis c x
-  by_cases hx : x≤c
-  · rw [lowerCharacter,dite_eq_left hx,← costalkCharacter_reflect cs c.val basis ⟨x.val,x.property.1,hx⟩]
-    let := boundaryKernelFree cs c.val basis ⟨x.val,x.property.1,hx⟩
-    exact ModuleData.character_eq_of_gradedEquiv (lowerKernelEquiv cs b basis c x hx)
-      (lowerKernelEquiv_graded cs b basis c x hx)
-  · rw [lowerCharacter,dite_eq_right hx,Polynomial.reflect_zero]
-    let := (boundaryModels cs b basis).support c x hx
-    let : Subsingleton (GradedSheaf.kernelData ((boundaryModels cs b basis).sheaf c) x) :=
-      ⟨fun v w => Subtype.ext (Subsingleton.elim _ _)⟩
-    exact ModuleData.char_zero _
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded Polynomial
-universe uk us um
-variable {k : Type uk} [Field k] {σ : Type us} [Finite σ]
-  {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
-noncomputable section
-
- theorem exists_character_decomposition [Fintype V] [Fintype E]
-    {α : E → MvPolynomial σ k}
-    (O : BoundaryModels (polynomialGrading_negative k σ) (G := G) α)
-    (hk : ∀ y x, Module.Free (MvPolynomial σ k) (kernelData (O.sheaf y) x))
-    (B : GradedSheaf (𝓐 := polynomialGrading k σ) G)
-    (hf : ∀ x, Module.Free (MvPolynomial σ k) (B.vertex x))
-    (hkb : ∀ x, Module.Free (MvPolynomial σ k) (kernelData B x))
-    (hq : B.UpperQuotient α) (hfl : B.forget.Flabby) (hg : B.forget.Generated) :
-    ∃ m : V → Polynomial ℤ, (∀ y n, 0≤(m y).coeff n) ∧
-      (∀ x, freeStalkChar B hf x =
-        ∑ y, m y*freeStalkChar (O.sheaf y) (O.free y) x) ∧
-      (∀ x, freeKernelChar B hkb x =
-        ∑ y, m y*freeKernelChar (O.sheaf y) (hk y) x) := by
-  obtain ⟨C,hC,⟨j⟩⟩ := exists_modelSumIso (polynomialGrading_zero k σ) O B hf hq hfl hg
-  obtain ⟨m,hm,heq,hkeq⟩ := hC.character_both O hk
-  refine ⟨m,hm,fun x => ?_,fun x => ?_⟩
-  · rw [← heq x]
-    let _ := hf x
-    let _ := hC.free O x
-    exact ModuleData.character_eq_of_gradedEquiv (j.vertex x) (j.vertex_graded x)
-  · rw [← hkeq x]
-    let _ := hkb x
-    let _ := hC.kernelFree hk x
-    exact ModuleData.character_eq_of_gradedEquiv (j.kernelEquiv x) (j.kernelEquiv_graded x)
-
-end
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph Polynomial
-universe u
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- theorem bottCharacter_decomposition_both (l : List I) (hl : cs.IsReduced l)
-    [Fintype (Interval cs 1 (cs.wordProd l))] :
-    ∃ m : Interval cs 1 (cs.wordProd l) → Polynomial ℤ,
-      (∀ y n, 0≤(m y).coeff n) ∧
-      (∀ x, bottStalkCharacter cs basis l x.val=
-        ∑ y, m y*lowerCharacter cs (cs.wordProd l) basis y x) ∧
-      (∀ x, GradedSheaf.freeKernelChar (bottPolynomialSheaf cs basis l 1 (cs.wordProd l))
-        (bottKernelFree cs basis l hl) x=
-        ∑ y, m y*Polynomial.reflect (rankDifference cs x.val y.val)
-          (lowerCharacter cs (cs.wordProd l) basis y x)) := by
-  classical
-  let := Fintype.ofFinite (Edge cs 1 (cs.wordProd l))
-  obtain ⟨m,hm,he,hk⟩ := GradedSheaf.exists_character_decomposition
-    (boundaryModels cs (cs.wordProd l) basis) (boundaryModels_kernelFree cs (cs.wordProd l) basis)
-    (bottPolynomialSheaf cs basis l 1 (cs.wordProd l))
-    (bottPolynomialSheaf_free cs basis l 1 (cs.wordProd l)) (bottKernelFree cs basis l hl)
-    (bottPolynomialSheaf_quotient cs basis l 1 (cs.wordProd l))
-    (bottPolynomialSheaf_flabby cs basis l hl) (bottPolynomialSheaf_generated cs basis l 1 (cs.wordProd l))
-  refine ⟨m,hm,fun x => ?_,fun x => ?_⟩
-  · simpa only [bottPolynomialSheaf_character,boundaryModels_character] using he x
-  · simpa only [boundaryModels_kernelCharacter] using hk x
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-namespace KLInvariance.Graded.ModuleData
-open Polynomial Module
-universe uk us um
-variable {k : Type uk} [Field k] {σ : Type us} [Finite σ]
-noncomputable section
-
- def unitPolynomialBasis : NonnegativeBasis (σ := σ)
-    (unit.{uk,max uk us,um} (polynomialGrading_negative k σ)).piece where
-  index := PUnit.{max uk us um + 1}
-  finite := inferInstance
-  basis := Pi.basisFun (MvPolynomial σ k) _
-  degree _ := 0
-  homogeneous i := by
-    intro j
-    change (Pi.basisFun (MvPolynomial σ k) _ i) j ∈ polynomialGrading k σ (0-0)
-    cases i
-    cases j
-    simpa using SetLike.one_mem_graded (polynomialGrading k σ)
-
- theorem char_unit :
-    (unit.{uk,max uk us,um} (polynomialGrading_negative k σ)).char=1 := by
-  rw [char,character_eq _ _ (unitPolynomialBasis (k := k) (σ := σ))]
-  change (∑ i : PUnit.{max uk us um + 1}, (X : Polynomial ℤ)^0)=1
-  simp
-
-end
-end KLInvariance.Graded.ModuleData
-
-end
-
-
-section
-
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph Polynomial
-universe u
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W) (b : W)
-  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
- theorem stalkCharacter_top :
-    stalkCharacter cs 1 b basis (one_bruhat cs b) ⟨b,one_bruhat cs b,bruhat_refl cs b⟩=1 := by
-  let B := boundarySheaf cs 1 b basis (one_bruhat cs b)
-  let := B.free ⟨b,one_bruhat cs b,bruhat_refl cs b⟩
-  change (B.sheaf.vertex ⟨b,one_bruhat cs b,bruhat_refl cs b⟩).char=1
-  exact (ModuleData.character_eq_of_gradedEquiv (ModuleData.equivOfEq B.top)
-    (ModuleData.equivOfEq_graded B.top)).trans ModuleData.char_unit
-
- theorem lowerCharacter_diagonal (c : Interval cs 1 b) : lowerCharacter cs b basis c c=1 := by
-  rw [lowerCharacter,dite_eq_left le_rfl]
-  exact stalkCharacter_top cs c.val basis
-
- theorem lowerCharacter_at_top (c : Interval cs 1 b) :
-    lowerCharacter cs b basis c ⟨b,one_bruhat cs b,bruhat_refl cs b⟩=
-      if c.val=b then 1 else 0 := by
-  by_cases hc : c.val=b
-  · have he : c=⟨b,one_bruhat cs b,bruhat_refl cs b⟩ := Subtype.ext hc
-    rw [ite_eq_left hc,←he,lowerCharacter_diagonal]
-  · rw [ite_eq_right hc]
-    apply lowerCharacter_zero
-    intro h
-    exact hc (bruhat_antisymm cs c.property.2 h)
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph Polynomial
-universe u
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W) (b : W)
-  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-local instance lowerConvInterval (c : W) : Fintype (Interval cs 1 c) := by
-  let := interval_finite cs 1 c
-  exact Fintype.ofFinite _
-
- def lowerIntervalEquiv (c : Interval cs 1 b) : Set.Iic c ≃ Interval cs 1 c.val where
-  toFun := (lowerIntervalMap cs b c).vertex
-  invFun z := ⟨⟨z.val,z.property.1,bruhat_trans cs z.property.2 c.property.2⟩,z.property.2⟩
-  left_inv _ := rfl
-  right_inv _ := rfl
-
- theorem lowerCharacter_convolution (c x : Interval cs 1 b) :
-    (∑ z : Interval cs 1 b, OrdinaryR.polynomial cs x.val z.val*lowerCharacter cs b basis c z)=
-      ∑ z : Interval cs 1 c.val, OrdinaryR.polynomial cs x.val z.val*
-        stalkCharacter cs 1 c.val basis (one_bruhat cs c.val) z := by
-  classical
-  let f (z : Interval cs 1 b) := OrdinaryR.polynomial cs x.val z.val*lowerCharacter cs b basis c z
-  have hf : ∀ z : Interval cs 1 b, ¬z≤c → f z=0 := by
-    intro z hz
-    simp only [f,lowerCharacter_zero cs b basis c z hz,mul_zero]
-  have he : (∑ z : Interval cs 1 b, f z) = ∑ z ∈ Finset.univ.filter (fun z => z≤c), f z := by
-    symm
-    apply Finset.sum_subset (Finset.filter_subset _ _)
-    intro z _ hz
-    exact hf z (by simpa using hz)
-  change (∑ z, f z)=_
-  rw [he,Finset.sum_subtype (p := fun z : Interval cs 1 b => z≤c) _ (by simp)]
-  calc
-    _ = ∑ z : Set.Iic c, OrdinaryR.polynomial cs x.val z.val.val*
-        stalkCharacter cs 1 c.val basis (one_bruhat cs c.val) ((lowerIntervalEquiv cs b c) z) := by
-      apply Finset.sum_congr rfl
-      intro z _
-      simp only [f,lowerCharacter,dite_eq_left z.property]
-      rfl
-    _ = _ := Equiv.sum_comp (lowerIntervalEquiv cs b c) (fun z => OrdinaryR.polynomial cs x.val z.val*
-      stalkCharacter cs 1 c.val basis (one_bruhat cs c.val) z)
-
- theorem lowerCharacter_convolution_zero (c x : Interval cs 1 b) (hx : ¬x≤c) :
-    (∑ z : Interval cs 1 b, OrdinaryR.polynomial cs x.val z.val*lowerCharacter cs b basis c z)=0 := by
-  apply Finset.sum_eq_zero
-  intro z _
-  by_cases hz : z≤c
-  · rw [OrdinaryR.polynomial_zero cs x.val z.val (fun h => hx (bruhat_trans cs h hz)),zero_mul]
-  · rw [lowerCharacter_zero cs b basis c z hz,mul_zero]
-
- theorem lowerCharacter_reciprocity_lift (c : Interval cs 1 b)
-    (h : ∀ x : Interval cs 1 c.val,
-      reflect (rankDifference cs x.val c.val) (stalkCharacter cs 1 c.val basis (one_bruhat cs c.val) x)=
-        ∑ z : Interval cs 1 c.val, OrdinaryR.polynomial cs x.val z.val*
-          stalkCharacter cs 1 c.val basis (one_bruhat cs c.val) z)
-    (x : Interval cs 1 b) :
-      reflect (rankDifference cs x.val c.val) (lowerCharacter cs b basis c x)=
-        ∑ z : Interval cs 1 b, OrdinaryR.polynomial cs x.val z.val*lowerCharacter cs b basis c z := by
-  by_cases hx : x≤c
-  · rw [lowerCharacter_convolution,lowerCharacter,dite_eq_left hx]
-    exact h ⟨x.val,x.property.1,hx⟩
-  · rw [lowerCharacter_zero cs b basis c x hx,reflect_zero,lowerCharacter_convolution_zero cs b basis c x hx]
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-/-! Finite unipotent character cancellation. The transition data is actually
-constructed for word/BMP sheaves; this lemma assumes only the displayed
-linear identities and cannot supply the KL degree bound. -/
-
-end
-
-end OAI
+import OAI.RepresentationTheory.KazhdanLusztig.Support
+
+namespace OAI
+
+/-!
+Conditional boundary-sheaf duality and stalk-character propositions, free sheaves, actual R-reciprocity and lower convolution. StalkCharacterObligation remains a Prop, not an unconditional character theorem or a premise of the R-first route.
+-/
+
+section
+
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W) (b : W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+variable [Fintype (Edge cs 1 b)] [Fintype (Interval cs 1 b)] [DecidableEq (Interval cs 1 b)]
+noncomputable section
+
+def SectionDualityObligation : Prop :=
+  ∃ d : SectionDuality cs 1 b basis (one_bruhat cs b)
+    (boundarySheaf cs 1 b basis (one_bruhat cs b)),
+    ∀ i j (m n : (boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf.sections Set.univ),
+      m ∈ ((boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf.sections Set.univ).piece i →
+      n ∈ ((boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf.sections Set.univ).piece j →
+      d.pairing m n ∈ polynomialGrading ℝ σ (i+j-(cs.length b : ℤ))
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+/-! Balance of a two-sided perfect pairing needs orthogonality on both
+sides, not equality of its two integral embeddings into an ambient form. -/
+namespace BalancedCuts
+universe ur um ua
+variable {R : Type ur} [CommRing R] [IsDomain R]
+  {M : Type um} [AddCommGroup M] [Module R M]
+  {A : Type ua} [AddCommGroup A] [Module R A]
+  (d : M →ₗ[R] Module.Dual R M) (π : M →ₗ[R] A) (c : M →ₗ[R] M)
+  (p : R) (hp : p≠0) (hπ : ∀ m, π (c m)=p • π m)
+  (hleft : ∀ m n, π n=0 → d (c m) n=0)
+  (hright : ∀ m n, π m=0 → d m (c n)=0)
+
+include hp hπ hleft hright in
+ theorem balanced_two (m n : M) : d (c m) n=d m (c n) := by
+  have hn : π (p • n-c n)=0 := by rw [map_sub,map_smul,hπ,sub_self]
+  have hm : π (p • m-c m)=0 := by rw [map_sub,map_smul,hπ,sub_self]
+  have en := hleft m _ hn
+  have em := hright _ n hm
+  simp only [map_sub,map_smul,LinearMap.sub_apply,LinearMap.smul_apply,smul_eq_mul,sub_eq_zero] at en em
+  exact mul_left_cancel₀ hp (en.trans em.symm)
+
+end BalancedCuts
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded Module
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
+  {G : OrderedGraph V E} [IsDomain A] [DecidableEq V]
+  {S : GradedSheaf (𝓐 := 𝓐) G} {d : ℤ} (p : PerfectPairing S d)
+  (a : A) (ha : a≠0)
+  (hcut : ∀ x (m : S.forget.sections Set.univ), a • Pi.single x (m.val x)∈S.forget.sections Set.univ)
+noncomputable section
+
+ include ha in
+ theorem PerfectPairing.cut_balanced (x : V) (m n : S.forget.sections Set.univ) :
+    p.pairing (IntegralSupport.cut _ a hcut x m) n =
+      p.pairing m (IntegralSupport.cut _ a hcut x n) := by
+  apply BalancedCuts.balanced_two p.pairing.toLinearMap
+    (IntegralSupport.projection (S.forget.sections Set.univ) x)
+    (IntegralSupport.cut _ a hcut x) a ha
+    (IntegralSupport.cut_self _ a hcut x)
+  · intro v w hw
+    exact p.left_orthogonal x _ w (fun y hy => IntegralSupport.cut_other _ a hcut x y hy v) hw
+  · intro v w hv
+    exact p.right_orthogonal x v _ hv (fun y hy => IntegralSupport.cut_other _ a hcut x y hy w)
+
+end
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W) (b : W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  [Fintype (Edge cs 1 b)] [Fintype (Interval cs 1 b)] [DecidableEq (Interval cs 1 b)]
+noncomputable section
+
+ theorem sectionDualityObligation_of_perfectPairing
+    (p : GradedSheaf.PerfectPairing
+      (boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf (cs.length b : ℤ)) :
+    SectionDualityObligation cs b basis := by
+  let B := boundarySheaf cs 1 b basis (one_bruhat cs b)
+  refine ⟨⟨p.pairing,?_⟩,?_⟩
+  · intro x m n
+    exact p.cut_balanced (Sheaf.labelProduct (polynomialLabel cs 1 b basis))
+      (Sheaf.labelProduct_ne_zero _ (polynomialLabel_ne_zero cs 1 b basis))
+      (B.sheaf.forget.scaled_single_section
+        (B.sheaf.forget.labelProduct_annihilates (polynomialLabel cs 1 b basis)
+          (GradedSheaf.UpperQuotient.annihilates B.sheaf B.quotient))) x m n
+  · intro i j m n hm hn
+    exact p.homogeneous i j m n (fun x => hm x (Set.mem_univ x))
+      (fun x => hn x (Set.mem_univ x))
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+/-! Integral costalk duality only uses free stalks, the genuine edge quotients,
+generation and a perfect support-orthogonal section form. No top/minimality or
+KL character assumption is needed. This extension applies to genuine words. -/
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
+  {σ : Type us} (basis : Basis σ ℝ (Extended M))
+
+structure FreeSheaf where
+  sheaf : GradedSheaf (𝓐 := polynomialGrading ℝ σ) (graph cs u b)
+  free : sheaf.FreeStalks
+  quotient : sheaf.UpperQuotient (polynomialLabel cs u b basis)
+  generated : sheaf.forget.Generated
+
+namespace FreeSheaf
+variable (B : FreeSheaf cs u b basis)
+  [Fintype (Edge cs u b)] [DecidableEq (Interval cs u b)]
+noncomputable section
+ def kernelEquivSingleSupport (x : Interval cs u b) :
+    LinearMap.ker (B.sheaf.forget.stalkMap x) ≃ₗ[Coefficient (σ := σ)]
+      IntegralSupport.supported (B.sheaf.forget.sections Set.univ) x := by
+  let _ (x : Interval cs u b) : Module.Free (Coefficient (σ := σ)) (B.sheaf.forget.vertex x) := B.free x
+  let _ (x : Interval cs u b) : Module.Finite (Coefficient (σ := σ)) (B.sheaf.forget.vertex x) :=
+    (B.sheaf.vertex x).finite
+  let _ (x : Interval cs u b) : Fintype (Module.Free.ChooseBasisIndex
+      (Coefficient (σ := σ)) (B.sheaf.forget.vertex x)) := Fintype.ofFinite _
+  let bs := fun x : Interval cs u b =>
+    Module.Free.chooseBasis (Coefficient (σ := σ)) (B.sheaf.forget.vertex x)
+  exact (B.sheaf.forget.kernelEquivSingleStalk (polynomialLabel cs u b basis)
+    (polynomialLabel_prime cs u b basis) (fun e => (B.quotient e).2) bs x
+    (incoming_pairwise_relPrime cs u b basis x)
+    (B.sheaf.forget.outgoing_product_regular (polynomialLabel cs u b basis)
+      (polynomialLabel_prime cs u b basis) (fun e => (B.quotient e).1)
+      (fun e => (B.quotient e).2) bs x
+      (outgoing_incoming_not_dvd cs u b basis x))).trans
+        (B.sheaf.forget.singleStalkEquivSupport x)
+
+
+structure SectionDuality where
+  pairing : B.sheaf.forget.sections Set.univ ≃ₗ[Coefficient (σ := σ)]
+    Dual (Coefficient (σ := σ)) (B.sheaf.forget.sections Set.univ)
+  balanced : ∀ x m n,
+    pairing (IntegralSupport.cut (B.sheaf.forget.sections Set.univ)
+      (Sheaf.labelProduct (polynomialLabel cs u b basis))
+      (B.sheaf.forget.scaled_single_section
+        (B.sheaf.forget.labelProduct_annihilates (polynomialLabel cs u b basis)
+          (GradedSheaf.UpperQuotient.annihilates B.sheaf B.quotient))) x m) n =
+    pairing m (IntegralSupport.cut (B.sheaf.forget.sections Set.univ)
+      (Sheaf.labelProduct (polynomialLabel cs u b basis))
+      (B.sheaf.forget.scaled_single_section
+        (B.sheaf.forget.labelProduct_annihilates (polynomialLabel cs u b basis)
+          (GradedSheaf.UpperQuotient.annihilates B.sheaf B.quotient))) x n)
+
+ def costalkEquivDual (d : SectionDuality cs u b basis B) (x : Interval cs u b) :
+    LinearMap.ker (B.sheaf.forget.stalkMap x) ≃ₗ[Coefficient (σ := σ)]
+      Dual (Coefficient (σ := σ)) (B.sheaf.vertex x) := by
+  let _ (x : Interval cs u b) : Module.Free (Coefficient (σ := σ)) (B.sheaf.forget.vertex x) := B.free x
+  let _ (y : Interval cs u b) : Module.IsTorsionFree (Coefficient (σ := σ))
+      (B.sheaf.forget.vertex y) := inferInstance
+  exact (kernelEquivSingleSupport cs u b basis B x).trans
+    (B.sheaf.forget.singleSupportEquivDual (polynomialLabel cs u b basis)
+      (polynomialLabel_ne_zero cs u b basis)
+      (GradedSheaf.UpperQuotient.annihilates B.sheaf B.quotient)
+      d.pairing d.balanced B.generated x)
+
+ theorem costalk_free_of_duality (d : SectionDuality cs u b basis B)
+    (x : Interval cs u b) :
+    Module.Free (Coefficient (σ := σ)) (LinearMap.ker (B.sheaf.forget.stalkMap x)) := by
+  let _ := B.free x
+  exact Module.Free.of_equiv (costalkEquivDual cs u b basis B d x).symm
+
+ variable [Fintype σ] [Fintype (Interval cs u b)]
+
+omit [Fintype σ] [Fintype (Interval cs u b)] in
+ theorem kernelEquivSingleSupport_apply (x : Interval cs u b)
+    (v : LinearMap.ker (B.sheaf.forget.stalkMap x)) :
+    ((kernelEquivSingleSupport cs u b basis B x) v).val.val x =
+      Sheaf.incomingProduct (G := graph cs u b) (polynomialLabel cs u b basis) x • v.val := by
+  change (Pi.single x
+    (Sheaf.incomingProduct (G := graph cs u b) (polynomialLabel cs u b basis) x • v.val) :
+      B.sheaf.forget.Assignment) x = _
+  exact Pi.single_eq_same _ _
+
+ theorem costalkEquivDual_inverse_homogeneous
+    (d : SectionDuality cs u b basis B) (l : ℤ)
+    (hd : ∀ i j (m n : B.sheaf.sections Set.univ),
+      m ∈ (B.sheaf.sections Set.univ).piece i → n ∈ (B.sheaf.sections Set.univ).piece j →
+      d.pairing m n ∈ polynomialGrading ℝ σ (i+j-l))
+    (x : Interval cs u b) (r : ℤ) (lam : Dual (Coefficient (σ := σ)) (B.sheaf.vertex x))
+    (hlam : ∀ j (v : B.sheaf.vertex x), v ∈ (B.sheaf.vertex x).piece j →
+      lam v ∈ polynomialGrading ℝ σ (j+r)) :
+    ((costalkEquivDual cs u b basis B d x).symm lam).val ∈
+      (B.sheaf.vertex x).piece
+        (r+l-(Sheaf.incoming (G := graph cs u b) x).card) := by
+  let _ (y : Interval cs u b) : Module.Free (Coefficient (σ := σ))
+      (B.sheaf.forget.vertex y) := B.free y
+  let _ (y : Interval cs u b) : Module.IsTorsionFree (Coefficient (σ := σ))
+      (B.sheaf.forget.vertex y) := inferInstance
+  let S := B.sheaf.forget.singleSupportEquivDual (polynomialLabel cs u b basis)
+    (polynomialLabel_ne_zero cs u b basis)
+    (GradedSheaf.UpperQuotient.annihilates B.sheaf B.quotient)
+    d.pairing d.balanced B.generated x
+  have hs : (S.symm lam).val ∈ (B.sheaf.sections Set.univ).piece (r+l) := by
+    exact B.sheaf.dualToSupport_homogeneous (polynomialLabel cs u b basis)
+      (polynomialLabel_ne_zero cs u b basis)
+      (GradedSheaf.UpperQuotient.annihilates B.sheaf B.quotient)
+      d.pairing d.balanced l hd x r lam hlam
+  have hv := hs x (Set.mem_univ x)
+  have he : (S.symm lam).val.val x =
+      Sheaf.incomingProduct (G := graph cs u b) (polynomialLabel cs u b basis) x •
+      ((costalkEquivDual cs u b basis B d x).symm lam).val := by
+    rw [← kernelEquivSingleSupport_apply cs u b basis B x]
+    change (S.symm lam).val.val x =
+      ((kernelEquivSingleSupport cs u b basis B x)
+        ((kernelEquivSingleSupport cs u b basis B x).symm (S.symm lam))).val.val x
+    rw [LinearEquiv.apply_symm_apply]
+  change (S.symm lam).val.val x ∈ (B.sheaf.vertex x).piece (r+l) at hv
+  rw [he] at hv
+  apply mem_of_homogeneous_smul (polynomialGrading ℝ σ) (B.sheaf.vertex x).piece
+    ((Sheaf.incoming (G := graph cs u b) x).card : ℤ)
+    (Sheaf.incomingProduct (G := graph cs u b) (polynomialLabel cs u b basis) x)
+    (incomingProduct_homogeneous cs u b basis x) _ (r+l) _ hv
+  let _ := B.free x
+  exact smul_right_injective _ (Sheaf.incomingProduct_ne_zero
+    (polynomialLabel cs u b basis) (polynomialLabel_prime cs u b basis) x)
+
+ def fullSubmodule (x : Interval cs u b) : Submodule (Coefficient (σ := σ)) (B.sheaf.vertex x) :=
+    LinearMap.ker (B.sheaf.forget.stalkMap x)
+
+omit [Fintype σ] [Fintype (Edge cs u b)] [Fintype (Interval cs u b)]
+  [DecidableEq (Interval cs u b)] in
+ theorem fullKernel_eq (x : Interval cs u b) :
+    B.sheaf.upwardSubmodule x Set.univ = fullSubmodule cs u b basis B x := by
+  ext m
+  change (∀ e ∈ Set.univ, B.sheaf.forget.stalkMap x m e = 0) ↔ B.sheaf.forget.stalkMap x m = 0
+  simp only [Set.mem_univ,forall_const]
+  exact funext_iff.symm
+
+ def fullKernel (x : Interval cs u b) : ModuleData (polynomialGrading ℝ σ) :=
+  ModuleData.sub (B.sheaf.vertex x) (fullSubmodule cs u b basis B x) (by
+    simpa only [fullKernel_eq cs u b basis B x] using
+      B.sheaf.upwardKernel_homogeneous x Set.univ)
+
+ theorem costalk_hilbert_of_duality
+    (d : SectionDuality cs u b basis B) (l : ℤ)
+    (hd : ∀ i j (m n : B.sheaf.sections Set.univ),
+      m ∈ (B.sheaf.sections Set.univ).piece i → n ∈ (B.sheaf.sections Set.univ).piece j →
+      d.pairing m n ∈ polynomialGrading ℝ σ (i+j-l))
+    (x : Interval cs u b) (c : NonnegativeBasis (σ := σ) (B.sheaf.vertex x).piece)
+    {q : ℝ} (hq : 0 < q) (hq1 : q < 1) :
+    Hilbert.value (B.sheaf.vertex x).piece
+      ((fullSubmodule cs u b basis B x).restrictScalars ℝ) q =
+      q^(l-(Sheaf.incoming (G := graph cs u b) x).card) *
+        Polynomial.eval₂ (Int.castRingHom ℝ) q⁻¹ c.polynomial *
+        (1-q)⁻¹ ^ Fintype.card σ := by
+  classical
+  let K := fullKernel cs u b basis B x
+  let cb : Basis c.index (Coefficient (σ := σ)) K :=
+    c.basis.dualBasis.map (costalkEquivDual cs u b basis B d x).symm
+  have hcb (i : c.index) : cb i ∈ K.piece
+      (l-(Sheaf.incoming (G := graph cs u b) x).card-(c.degree i : ℤ)) := by
+    change ((costalkEquivDual cs u b basis B d x).symm (c.basis.dualBasis i)).val ∈
+      (B.sheaf.vertex x).piece _
+    have h := costalkEquivDual_inverse_homogeneous cs u b basis B d l hd x
+      (-(c.degree i : ℤ)) (c.basis.dualBasis i)
+      (dualBasis_homogeneous (polynomialGrading ℝ σ) (B.sheaf.vertex x).piece
+        c.basis (fun i => (c.degree i : ℤ)) c.homogeneous i)
+    simpa only [show -(c.degree i : ℤ)+l-(Sheaf.incoming (G := graph cs u b) x).card =
+      l-(Sheaf.incoming (G := graph cs u b) x).card-(c.degree i : ℤ) by omega] using h
+  have hs := Hilbert.hasSum_homogeneous_basis K.piece cb
+    (fun i => l-(Sheaf.incoming (G := graph cs u b) x).card-(c.degree i : ℤ)) hcb hq hq1
+  have hv : Hilbert.value K.piece ⊤ q =
+      (∑ i, q^(l-(Sheaf.incoming (G := graph cs u b) x).card-(c.degree i : ℤ))) *
+        (1-q)⁻¹ ^ Fintype.card σ := by
+    simpa only [Hilbert.value,Hilbert.term_top] using hs.tsum_eq
+  change Hilbert.value (B.sheaf.vertex x).piece
+    ((fullSubmodule cs u b basis B x).restrictScalars ℝ) q = _
+  rw [← Hilbert.value_subPiece]
+  change Hilbert.value K.piece ⊤ q = _
+  rw [hv,c.eval₂_eq,Finset.mul_sum]
+  congr 1
+  apply Finset.sum_congr rfl
+  intro i _
+  rw [zpow_sub₀ hq.ne',zpow_natCast,div_eq_mul_inv,inv_pow]
+
+
+ def dualityOfPairing {l : ℤ} (p : GradedSheaf.PerfectPairing B.sheaf l) :
+    SectionDuality cs u b basis B where
+  pairing := p.pairing
+  balanced := p.cut_balanced
+    (Sheaf.labelProduct (polynomialLabel cs u b basis))
+    (Sheaf.labelProduct_ne_zero _ (polynomialLabel_ne_zero cs u b basis))
+    (B.sheaf.forget.scaled_single_section
+      (B.sheaf.forget.labelProduct_annihilates (polynomialLabel cs u b basis)
+        (GradedSheaf.UpperQuotient.annihilates B.sheaf B.quotient)))
+
+end
+end FreeSheaf
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+/-! The finite sums used by the actual local decomposition. The model criteria
+are kept explicit here; they are not asserted for the constructed BMP object
+or inserted into the main KL theorem. -/
+namespace KLInvariance.MomentGraph.Sheaf
+universe u v w z
+variable {R : Type u} [CommRing R] [IsLocalRing R]
+  {V : Type v} [PartialOrder V] {E : Type w} {G : OrderedGraph V E}
+
+
+instance zeroVertex_subsingleton (x : V) : Subsingleton ((zero (R := R) (G := G)).vertex x) :=
+  inferInstanceAs (Subsingleton PUnit)
+
+instance zeroEdge_subsingleton (e : E) : Subsingleton ((zero (R := R) (G := G)).edge e) :=
+  inferInstanceAs (Subsingleton PUnit)
+
+structure LocalModels (α : E → R) where
+  sheaf : V → Sheaf.{u,v,w,z} (R := R) G
+  free : ∀ c x, Module.Free R ((sheaf c).vertex x)
+  finite : ∀ c x, Module.Finite R ((sheaf c).vertex x)
+  quotient : ∀ c, (sheaf c).UpperQuotient α
+  generated : ∀ c, (sheaf c).Generated
+  flabby : ∀ c, (sheaf c).Flabby
+  support : ∀ c x, ¬x ≤ c → Subsingleton ((sheaf c).vertex x)
+  top : ∀ c, (sheaf c).vertex c ≃ₗ[R] R
+  minimal : ∀ c x, x ≠ c → LinearMap.ker ((sheaf c).stalkMap x) ≤
+    IsLocalRing.maximalIdeal R • (⊤ : Submodule R ((sheaf c).vertex x))
+
+inductive ModelSum (O : V → Sheaf.{u,v,w,z} (R := R) G) (U : Set V) :
+    Sheaf.{u,v,w,z} (R := R) G → Prop where
+  | zero : ModelSum O U zero
+  | add {B} : ModelSum O U B → ∀ c ∈ U, ∀ n : ℕ,
+      ModelSum O U (prod B (pi (fun _ : Fin n => O c)))
+
+namespace ModelSum
+variable {α : E → R} (O : LocalModels.{u,v,w,z} (G := G) α)
+  {U : Set V} {B : Sheaf.{u,v,w,z} (R := R) G}
+
+omit [IsLocalRing R] in
+lemma mono {O : V → Sheaf.{u,v,w,z} (R := R) G} (h : ModelSum O U B)
+    {T : Set V} (hUT : U ⊆ T) : ModelSum O T B := by
+  induction h with
+  | zero => exact .zero
+  | add _ c hc n ih => exact .add ih c (hUT hc) n
+
+lemma free (h : ModelSum O.sheaf U B) (x : V) : Module.Free R (B.vertex x) := by
+  induction h with
+  | zero => exact inferInstanceAs (Module.Free R PUnit.{z+1})
+  | @add C _ c _ n ih =>
+    let _ : Module.Free R (C.vertex x) := ih
+    let _ := O.free c x
+    change Module.Free R (C.vertex x × (Fin n → (O.sheaf c).vertex x))
+    infer_instance
+
+lemma finite (h : ModelSum O.sheaf U B) (x : V) : Module.Finite R (B.vertex x) := by
+  induction h with
+  | zero => exact inferInstanceAs (Module.Finite R PUnit.{z+1})
+  | @add C _ c _ n ih =>
+    let _ : Module.Finite R (C.vertex x) := ih
+    let _ := O.finite c x
+    change Module.Finite R (C.vertex x × (Fin n → (O.sheaf c).vertex x))
+    infer_instance
+
+lemma quotient (h : ModelSum O.sheaf U B) : B.UpperQuotient α := by
+  induction h with
+  | zero =>
+    intro e
+    exact ⟨fun v => ⟨0,Subsingleton.elim _ _⟩,fun v => ⟨fun _ => ⟨0,Subsingleton.elim _ _⟩,fun _ => Subsingleton.elim _ _⟩⟩
+  | add _ c _ n ih => exact prod_upperQuotient _ _ α ih (pi_upperQuotient _ α (fun _ => O.quotient c))
+
+lemma generated (h : ModelSum O.sheaf U B) : B.Generated := by
+  induction h with
+  | zero => exact fun x v => ⟨0,fun _ _ _ => rfl,Subsingleton.elim _ _⟩
+  | add _ c _ n ih => exact prod_generated _ _ ih (pi_generated _ (fun _ => O.generated c))
+
+lemma flabby (h : ModelSum O.sheaf U B) : B.Flabby := by
+  induction h with
+  | zero => exact fun U _ f _ => ⟨f,fun _ _ _ => rfl,fun _ _ => rfl⟩
+  | add _ c _ n ih => exact prod_flabby _ _ ih (pi_flabby _ (fun _ => O.flabby c))
+
+lemma minimal (h : ModelSum O.sheaf U B) (x : V) (hx : x ∉ U) :
+    LinearMap.ker (B.stalkMap x) ≤
+      IsLocalRing.maximalIdeal R • (⊤ : Submodule R (B.vertex x)) := by
+  induction h with
+  | zero =>
+    intro v _
+    rw [Subsingleton.elim v 0]
+    exact Submodule.zero_mem _
+  | add _ c hc n ih =>
+    exact prod_minimal _ _ x _ ih (pi_minimal _ x _
+      (fun _ => O.minimal c x (fun he => hx (he ▸ hc))))
+
+end ModelSum
+end KLInvariance.MomentGraph.Sheaf
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.Sheaf
+open scoped Pointwise
+universe ur uv ue uz
+variable {R : Type ur} [CommRing R] {V : Type uv} [PartialOrder V]
+  {E : Type ue} {G : OrderedGraph V E}
+
+/-- Equality of embedded images with a scalar on the upper image. -/
+def ScaledImages (B : Sheaf.{ur,uv,ue,uz} (R := R) G) (e : E)
+    (F : Set (Outgoing (G := G) (G.source e)))
+    (H : Set (Outgoing (G := G) (G.target e))) (r : R) : Prop :=
+  ∀ z : B.edge e,
+    (∃ v ∈ B.upwardKernel (G.source e) F, B.lower e v=z) ↔
+    ∃ w ∈ B.upwardKernel (G.target e) H, r • B.upper e w=z
+
+theorem scaledImages_iff (B : Sheaf.{ur,uv,ue,uz} (R := R) G) (e : E)
+    (F : Set (Outgoing (G := G) (G.source e)))
+    (H : Set (Outgoing (G := G) (G.target e))) (r : R) :
+    B.ScaledImages e F H r ↔
+      (B.upwardKernel (G.source e) F).map (B.lower e) =
+        r • (B.upwardKernel (G.target e) H).map (B.upper e) := by
+  constructor
+  · intro h
+    ext z
+    rw [Submodule.mem_smul_pointwise_iff_exists]
+    change (∃ v, v ∈ B.upwardKernel (G.source e) F ∧ B.lower e v=z) ↔ _
+    rw [h z]
+    constructor
+    · rintro ⟨w,hw,he⟩
+      exact ⟨B.upper e w,⟨w,hw,rfl⟩,he⟩
+    · rintro ⟨t,⟨w,hw,rfl⟩,he⟩
+      exact ⟨w,hw,he⟩
+  · intro h z
+    constructor
+    · rintro ⟨v,hv,rfl⟩
+      have hv' : B.lower e v ∈ r • (B.upwardKernel (G.target e) H).map (B.upper e) :=
+        h ▸ (show B.lower e v ∈ (B.upwardKernel (G.source e) F).map (B.lower e) from ⟨v,hv,rfl⟩)
+      obtain ⟨t,⟨w,hw,rfl⟩,he⟩ := (Submodule.mem_smul_pointwise_iff_exists _ _ _).mp hv'
+      exact ⟨w,hw,he⟩
+    · rintro ⟨w,hw,rfl⟩
+      have hw' : r • B.upper e w ∈ r • (B.upwardKernel (G.target e) H).map (B.upper e) :=
+        (Submodule.mem_smul_pointwise_iff_exists _ _ _).mpr ⟨B.upper e w,⟨w,hw,rfl⟩,rfl⟩
+      rw [← h] at hw'
+      exact hw'
+
+theorem prod_upwardKernel_iff (B C : Sheaf.{ur,uv,ue,uz} (R := R) G)
+    (x : V) (F : Set (Outgoing (G := G) x)) (v : (prod B C).vertex x) :
+    v ∈ (prod B C).upwardKernel x F ↔
+      v.1 ∈ B.upwardKernel x F ∧ v.2 ∈ C.upwardKernel x F := by
+  constructor
+  · intro h
+    constructor
+    · rintro ⟨e,heq⟩ he
+      subst x
+      exact congrArg Prod.fst (h ⟨e,rfl⟩ he)
+    · rintro ⟨e,heq⟩ he
+      subst x
+      exact congrArg Prod.snd (h ⟨e,rfl⟩ he)
+  · rintro ⟨hB,hC⟩ ⟨e,heq⟩ he
+    subst x
+    exact Prod.ext (hB ⟨e,rfl⟩ he) (hC ⟨e,rfl⟩ he)
+
+theorem pi_upwardKernel_iff {ι : Type*} (B : ι → Sheaf.{ur,uv,ue,uz} (R := R) G)
+    (x : V) (F : Set (Outgoing (G := G) x)) (v : (pi B).vertex x) :
+    v ∈ (pi B).upwardKernel x F ↔ ∀ i, v i ∈ (B i).upwardKernel x F := by
+  constructor
+  · rintro h i ⟨e,heq⟩ he
+    subst x
+    exact congrFun (h ⟨e,rfl⟩ he) i
+  · rintro h ⟨e,heq⟩ he
+    subst x
+    exact funext (fun i => h i ⟨e,rfl⟩ he)
+
+theorem prod_scaledImages (B C : Sheaf.{ur,uv,ue,uz} (R := R) G) (e : E)
+    (F : Set (Outgoing (G := G) (G.source e)))
+    (H : Set (Outgoing (G := G) (G.target e))) (r : R)
+    (hB : B.ScaledImages e F H r) (hC : C.ScaledImages e F H r) :
+    (prod B C).ScaledImages e F H r := by
+  intro z
+  constructor
+  · rintro ⟨v,hv,he⟩
+    have hv' := (prod_upwardKernel_iff B C _ F v).mp hv
+    obtain ⟨b,hb,hbe⟩ := (hB z.1).mp ⟨v.1,hv'.1,congrArg Prod.fst he⟩
+    obtain ⟨c,hc,hce⟩ := (hC z.2).mp ⟨v.2,hv'.2,congrArg Prod.snd he⟩
+    exact ⟨(b,c),(prod_upwardKernel_iff B C _ H _).mpr ⟨hb,hc⟩,Prod.ext hbe hce⟩
+  · rintro ⟨v,hv,he⟩
+    have hv' := (prod_upwardKernel_iff B C _ H v).mp hv
+    obtain ⟨b,hb,hbe⟩ := (hB z.1).mpr ⟨v.1,hv'.1,congrArg Prod.fst he⟩
+    obtain ⟨c,hc,hce⟩ := (hC z.2).mpr ⟨v.2,hv'.2,congrArg Prod.snd he⟩
+    exact ⟨(b,c),(prod_upwardKernel_iff B C _ F _).mpr ⟨hb,hc⟩,Prod.ext hbe hce⟩
+
+theorem pi_scaledImages {ι : Type*} (B : ι → Sheaf.{ur,uv,ue,uz} (R := R) G) (e : E)
+    (F : Set (Outgoing (G := G) (G.source e)))
+    (H : Set (Outgoing (G := G) (G.target e))) (r : R)
+    (hB : ∀ i, (B i).ScaledImages e F H r) : (pi B).ScaledImages e F H r := by
+  classical
+  intro z
+  constructor
+  · rintro ⟨v,hv,he⟩
+    have hv' := (pi_upwardKernel_iff B _ F v).mp hv
+    choose w hw hwe using fun i => (hB i (z i)).mp ⟨v i,hv' i,congrFun he i⟩
+    exact ⟨w,(pi_upwardKernel_iff B _ H _).mpr hw,funext hwe⟩
+  · rintro ⟨v,hv,he⟩
+    have hv' := (pi_upwardKernel_iff B _ H v).mp hv
+    choose w hw hwe using fun i => (hB i (z i)).mpr ⟨v i,hv' i,congrFun he i⟩
+    exact ⟨w,(pi_upwardKernel_iff B _ F _).mpr hw,funext hwe⟩
+
+theorem zero_scaledImages (e : E) (F : Set (Outgoing (G := G) (G.source e)))
+    (H : Set (Outgoing (G := G) (G.target e))) (r : R) :
+    (zero : Sheaf.{ur,uv,ue,uz} (R := R) G).ScaledImages e F H r := by
+  intro z
+  constructor <;> intro _ <;> exact ⟨0,Submodule.zero_mem _,Subsingleton.elim _ _⟩
+
+theorem ModelSum.scaledImages {O : V → Sheaf.{ur,uv,ue,uz} (R := R) G}
+    {U : Set V} {B : Sheaf.{ur,uv,ue,uz} (R := R) G} (h : ModelSum O U B)
+    (e : E) (F : Set (Outgoing (G := G) (G.source e)))
+    (H : Set (Outgoing (G := G) (G.target e))) (r : R)
+    (hO : ∀ c ∈ U, (O c).ScaledImages e F H r) : B.ScaledImages e F H r := by
+  induction h with
+  | zero => exact zero_scaledImages e F H r
+  | add _ c hc n ih =>
+    exact prod_scaledImages _ _ e F H r ih
+      (pi_scaledImages (fun _ : Fin n => O c) e F H r (fun _ => hO c hc))
+
+end KLInvariance.MomentGraph.Sheaf
+
+end
+
+
+section
+
+/-! Actual full upward kernels commute with finite graded sums, shifts and
+sheaf isomorphisms. No freeness is inferred without a free model. -/
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  [IsNoetherianRing A] {𝓐 : ℤ → Submodule k A}
+  {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
+noncomputable section
+
+ abbrev kernelData (B : GradedSheaf (𝓐 := 𝓐) G) (x : V) :=
+  B.upwardKernel x Set.univ
+
+ def kernelProdEquiv (B C : GradedSheaf (𝓐 := 𝓐) G) (x : V) :
+    kernelData (prod B C) x ≃ₗ[A] ((kernelData B x).prod (kernelData C x)) where
+  toFun m := ⟨⟨m.val.1,((Sheaf.prod_upwardKernel_iff B.forget C.forget x Set.univ m.val).mp m.property).1⟩,
+    ⟨m.val.2,((Sheaf.prod_upwardKernel_iff B.forget C.forget x Set.univ m.val).mp m.property).2⟩⟩
+  invFun m := ⟨⟨m.1.val,m.2.val⟩,(Sheaf.prod_upwardKernel_iff B.forget C.forget x Set.univ _).mpr
+    ⟨m.1.property,m.2.property⟩⟩
+  left_inv _ := rfl
+  right_inv _ := rfl
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+
+ theorem kernelProdEquiv_graded (B C : GradedSheaf (𝓐 := 𝓐) G) (x : V)
+    (n : ℤ) (v : kernelData (prod B C) x) (hv : v∈(kernelData (prod B C) x).piece n) :
+    kernelProdEquiv B C x v∈((kernelData B x).prod (kernelData C x)).piece n := hv
+
+ def kernelPiEquiv {ι : Type um} [Fintype ι] (B : ι → GradedSheaf (𝓐 := 𝓐) G) (x : V) :
+    kernelData (pi B) x ≃ₗ[A] ModuleData.pi (fun i => kernelData (B i) x) where
+  toFun m i := ⟨m.val i,(Sheaf.pi_upwardKernel_iff (fun i => (B i).forget) x Set.univ m.val).mp m.property i⟩
+  invFun m := ⟨(fun i => (m i).val),(Sheaf.pi_upwardKernel_iff (fun i => (B i).forget) x Set.univ _).mpr
+    (fun i => (m i).property)⟩
+  left_inv _ := rfl
+  right_inv _ := rfl
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+
+ theorem kernelPiEquiv_graded {ι : Type um} [Fintype ι] (B : ι → GradedSheaf (𝓐 := 𝓐) G) (x : V)
+    (n : ℤ) (v : kernelData (pi B) x) (hv : v∈(kernelData (pi B) x).piece n) :
+    kernelPiEquiv B x v∈(ModuleData.pi (fun i => kernelData (B i) x)).piece n := hv
+
+ def kernelShiftEquiv (B : GradedSheaf (𝓐 := 𝓐) G) (x : V) (d : ℕ) :
+    kernelData (B.shift d) x ≃ₗ[A] (kernelData B x).shift d := LinearEquiv.refl A _
+
+ theorem kernelShiftEquiv_graded (B : GradedSheaf (𝓐 := 𝓐) G) (x : V) (d : ℕ)
+    (n : ℤ) (v : kernelData (B.shift d) x) (hv : v∈(kernelData (B.shift d) x).piece n) :
+    kernelShiftEquiv B x d v∈((kernelData B x).shift d).piece n := hv
+
+ def Iso.kernelEquiv {B C : GradedSheaf (𝓐 := 𝓐) G} (j : Iso B C) (x : V) :
+    kernelData B x ≃ₗ[A] kernelData C x where
+  toFun m := ⟨j.vertex x m.val,(j.toIso.upwardKernel_iff x Set.univ m.val).mpr m.property⟩
+  invFun m := ⟨(j.vertex x).symm m.val,(j.toIso.symm.upwardKernel_iff x Set.univ m.val).mpr m.property⟩
+  left_inv m := Subtype.ext ((j.vertex x).symm_apply_apply m.val)
+  right_inv m := Subtype.ext ((j.vertex x).apply_symm_apply m.val)
+  map_add' m n := Subtype.ext (map_add (j.vertex x) m.val n.val)
+  map_smul' a m := Subtype.ext (map_smul (j.vertex x) a m.val)
+
+ theorem Iso.kernelEquiv_graded {B C : GradedSheaf (𝓐 := 𝓐) G} (j : Iso B C) (x : V)
+    (n : ℤ) (v : kernelData B x) (hv : v∈(kernelData B x).piece n) :
+    j.kernelEquiv x v∈(kernelData C x).piece n := j.vertex_graded x n v.val hv
+
+end
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded Polynomial
+universe uk us um
+variable {k : Type uk} [Field k] {σ : Type us} [Finite σ]
+  {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
+noncomputable section
+
+ instance kernelShiftFree (B : GradedSheaf (𝓐 := polynomialGrading k σ) G) (x : V) (d : ℕ)
+    [Module.Free (MvPolynomial σ k) (kernelData B x)] :
+    Module.Free (MvPolynomial σ k) (kernelData (B.shift d) x) :=
+  Module.Free.of_equiv (kernelShiftEquiv B x d).symm
+
+ instance kernelProdFree (B C : GradedSheaf (𝓐 := polynomialGrading k σ) G) (x : V)
+    [Module.Free (MvPolynomial σ k) (kernelData B x)]
+    [Module.Free (MvPolynomial σ k) (kernelData C x)] :
+    Module.Free (MvPolynomial σ k) (kernelData (prod B C) x) :=
+  Module.Free.of_equiv (kernelProdEquiv B C x).symm
+
+ instance kernelPiFree {ι : Type um} [Fintype ι]
+    (B : ι → GradedSheaf (𝓐 := polynomialGrading k σ) G) (x : V)
+    [∀ i, Module.Free (MvPolynomial σ k) (kernelData (B i) x)] :
+    Module.Free (MvPolynomial σ k) (kernelData (pi B) x) :=
+  Module.Free.of_equiv (kernelPiEquiv B x).symm
+
+ instance kernelZeroFree (x : V) :
+    Module.Free (MvPolynomial σ k) (kernelData (zero (𝓐 := polynomialGrading k σ) (G := G)) x) := by
+  let : Subsingleton (kernelData (zero (𝓐 := polynomialGrading k σ) (G := G)) x) :=
+    ⟨fun x y => Subtype.ext (Subsingleton.elim _ _)⟩
+  infer_instance
+
+ theorem kernelChar_shift (B : GradedSheaf (𝓐 := polynomialGrading k σ) G) (x : V) (d : ℕ)
+    [Module.Free (MvPolynomial σ k) (kernelData B x)] :
+    (kernelData (B.shift d) x).char=X^d*(kernelData B x).char := by
+  rw [← ModuleData.char_shift _ d]
+  exact ModuleData.character_eq_of_gradedEquiv (kernelShiftEquiv B x d)
+    (kernelShiftEquiv_graded B x d)
+
+ theorem kernelChar_prod (B C : GradedSheaf (𝓐 := polynomialGrading k σ) G) (x : V)
+    [Module.Free (MvPolynomial σ k) (kernelData B x)]
+    [Module.Free (MvPolynomial σ k) (kernelData C x)] :
+    (kernelData (prod B C) x).char=(kernelData B x).char+(kernelData C x).char := by
+  rw [← ModuleData.char_prod]
+  exact ModuleData.character_eq_of_gradedEquiv (kernelProdEquiv B C x)
+    (kernelProdEquiv_graded B C x)
+
+ theorem kernelChar_pi {ι : Type um} [Fintype ι]
+    (B : ι → GradedSheaf (𝓐 := polynomialGrading k σ) G) (x : V)
+    [∀ i, Module.Free (MvPolynomial σ k) (kernelData (B i) x)] :
+    (kernelData (pi B) x).char=∑ i, (kernelData (B i) x).char := by
+  rw [← ModuleData.char_pi]
+  exact ModuleData.character_eq_of_gradedEquiv (kernelPiEquiv B x) (kernelPiEquiv_graded B x)
+
+ theorem kernelChar_zero (x : V) :
+    (kernelData (zero (𝓐 := polynomialGrading k σ) (G := G)) x).char=0 := by
+  let : Subsingleton (kernelData (zero (𝓐 := polynomialGrading k σ) (G := G)) x) :=
+    ⟨fun x y => Subtype.ext (Subsingleton.elim _ _)⟩
+  exact ModuleData.char_zero _
+
+ theorem ModelSum.kernelFree {O : V → GradedSheaf (𝓐 := polynomialGrading k σ) G}
+    (hf : ∀ y x, Module.Free (MvPolynomial σ k) (kernelData (O y) x))
+    {U : Set V} {B : GradedSheaf (𝓐 := polynomialGrading k σ) G}
+    (h : ModelSum O U B) (x : V) : Module.Free (MvPolynomial σ k) (kernelData B x) := by
+  induction h with
+  | zero => infer_instance
+  | @add C _ c _ n d ih =>
+    let _ := ih
+    let _ := hf c x
+    infer_instance
+
+ def freeKernelChar (B : GradedSheaf (𝓐 := polynomialGrading k σ) G)
+    (hf : ∀ x, Module.Free (MvPolynomial σ k) (kernelData B x)) (x : V) : Polynomial ℤ := by
+  let := hf x
+  exact (kernelData B x).char
+
+ theorem ModelSum.character_both [Fintype V] {α : E → MvPolynomial σ k}
+    (O : BoundaryModels (polynomialGrading_negative k σ) (G := G) α)
+    (hk : ∀ y x, Module.Free (MvPolynomial σ k) (kernelData (O.sheaf y) x))
+    {U : Set V} {B : GradedSheaf (𝓐 := polynomialGrading k σ) G}
+    (h : ModelSum O.sheaf U B) :
+    ∃ m : V → Polynomial ℤ, (∀ y n, 0≤(m y).coeff n) ∧
+      (∀ x, freeStalkChar B (h.free O) x =
+        ∑ y, m y*freeStalkChar (O.sheaf y) (O.free y) x) ∧
+      (∀ x, freeKernelChar B (h.kernelFree hk) x =
+        ∑ y, m y*freeKernelChar (O.sheaf y) (hk y) x) := by
+  classical
+  induction h with
+  | zero =>
+    refine ⟨0,by simp,?_,?_⟩
+    · intro x
+      change ((GradedSheaf.zero (𝓐 := polynomialGrading k σ) (G := G)).vertex x).char = _
+      simp only [Pi.zero_apply,zero_mul,Finset.sum_const_zero]
+      exact ModuleData.char_zero _
+    · intro x
+      change (kernelData (GradedSheaf.zero (𝓐 := polynomialGrading k σ) (G := G)) x).char = _
+      simp only [Pi.zero_apply,zero_mul,Finset.sum_const_zero]
+      exact kernelChar_zero (σ := σ) (G := G) x
+  | @add B h c hc n d ih =>
+    obtain ⟨m,hm,heq,hkeq⟩ := ih
+    let a : Polynomial ℤ := ∑ i, X^(d i)
+    have ha (r : ℕ) : 0 ≤ a.coeff r := by
+      simp only [a,Polynomial.finsetSum_coeff,Polynomial.coeff_X_pow]
+      exact Finset.sum_nonneg (fun i _ => by split <;> norm_num)
+    refine ⟨fun y => m y+if y=c then a else 0,?_,?_,?_⟩
+    · intro y r
+      rw [Polynomial.coeff_add]
+      apply add_nonneg (hm y r)
+      split
+      · exact ha r
+      · simp
+    · intro x
+      let _ := h.free O x
+      let _ := O.free c x
+      change ((B.vertex x).prod (ModuleData.pi (fun i => ((O.sheaf c).vertex x).shift (d i)))).char = _
+      rw [ModuleData.char_prod,ModuleData.char_pi]
+      simp only [ModuleData.char_shift]
+      change freeStalkChar B (h.free O) x + _ = _
+      rw [heq x]
+      simp only [add_mul,Finset.sum_add_distrib]
+      congr 1
+      rw [← Finset.sum_mul]
+      simp only [ite_mul,zero_mul,Finset.sum_ite_eq',Finset.mem_univ,ite_true]
+      rfl
+    · intro x
+      let _ := h.kernelFree hk x
+      let _ := hk c x
+      change (kernelData (prod B (pi (fun i => (O.sheaf c).shift (d i)))) x).char = _
+      rw [kernelChar_prod,kernelChar_pi]
+      simp only [kernelChar_shift]
+      change freeKernelChar B (h.kernelFree hk) x + _ = _
+      rw [hkeq x]
+      simp only [add_mul,Finset.sum_add_distrib]
+      congr 1
+      rw [← Finset.sum_mul]
+      simp only [ite_mul,zero_mul,Finset.sum_ite_eq',Finset.mem_univ,ite_true]
+      rfl
+
+end
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  [IsNoetherianRing A] {𝓐 : ℤ → Submodule k A}
+  {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
+noncomputable section
+ def kernelEquivFull (B : GradedSheaf (𝓐 := 𝓐) G) (x : V) :
+    kernelData B x ≃ₗ[A] LinearMap.ker (B.forget.stalkMap x) where
+  toFun m := ⟨m.val,funext (fun e => m.property e (Set.mem_univ e))⟩
+  invFun m := ⟨m.val,fun e _ => congrFun m.property e⟩
+  left_inv _ := rfl
+  right_inv _ := rfl
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+end
+end KLInvariance.MomentGraph.GradedSheaf
+
+namespace KLInvariance.BruhatGraph.FreeSheaf
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
+universe u
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W) (b : W)
+  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  (B : FreeSheaf cs 1 b basis)
+  [Fintype (Edge cs 1 b)] [Fintype (Interval cs 1 b)] [DecidableEq (Interval cs 1 b)]
+noncomputable section
+
+ def kernelDual (d : SectionDuality cs 1 b basis B) (x : Interval cs 1 b) :
+    Dual (Coefficient (σ := σ)) (B.sheaf.vertex x) ≃ₗ[Coefficient (σ := σ)]
+      GradedSheaf.kernelData B.sheaf x :=
+  (costalkEquivDual cs 1 b basis B d x).symm.trans (B.sheaf.kernelEquivFull x).symm
+
+omit [Fintype (Interval cs 1 b)] in
+ theorem kernel_free (d : SectionDuality cs 1 b basis B) (x : Interval cs 1 b) :
+    Module.Free (Coefficient (σ := σ)) (GradedSheaf.kernelData B.sheaf x) := by
+  let := B.free x
+  exact Module.Free.of_equiv (kernelDual cs b basis B d x)
+
+ theorem kernelDual_homogeneous (d : SectionDuality cs 1 b basis B) (n : ℕ)
+    (hd : ∀ i j (m a : B.sheaf.sections Set.univ),
+      m∈(B.sheaf.sections Set.univ).piece i → a∈(B.sheaf.sections Set.univ).piece j →
+      d.pairing m a∈polynomialGrading ℝ σ (i+j-(n:ℤ)))
+    (x : Interval cs 1 b) (hn : cs.length x.val≤n) (r : ℤ)
+    (lam : Dual (Coefficient (σ := σ)) (B.sheaf.vertex x))
+    (hlam : ∀ j (v : B.sheaf.vertex x), v∈(B.sheaf.vertex x).piece j →
+      lam v∈polynomialGrading ℝ σ (j+r)) :
+    kernelDual cs b basis B d x lam∈(GradedSheaf.kernelData B.sheaf x).piece
+      (r+(n-cs.length x.val : ℕ)) := by
+  have h := costalkEquivDual_inverse_homogeneous cs 1 b basis B d n hd x r lam hlam
+  rw [incoming_card] at h
+  change ((costalkEquivDual cs 1 b basis B d x).symm lam).val∈
+    (B.sheaf.vertex x).piece (r+↑(n-cs.length x.val))
+  have he : r+(n:ℤ)-cs.length x.val=r+↑(n-cs.length x.val) := by omega
+  rwa [he] at h
+
+ theorem kernel_character (d : SectionDuality cs 1 b basis B) (n : ℕ)
+    (hd : ∀ i j (m a : B.sheaf.sections Set.univ),
+      m∈(B.sheaf.sections Set.univ).piece i → a∈(B.sheaf.sections Set.univ).piece j →
+      d.pairing m a∈polynomialGrading ℝ σ (i+j-(n:ℤ)))
+    (hn : cs.length b≤n) (x : Interval cs 1 b) :
+    GradedSheaf.freeKernelChar B.sheaf (kernel_free cs b basis B d) x=
+      Polynomial.reflect (n-cs.length x.val) (GradedSheaf.freeStalkChar B.sheaf B.free x) := by
+  let := B.free x
+  let := kernel_free cs b basis B d x
+  exact ModuleData.char_of_dual_equiv _ _ (kernelDual cs b basis B d x) _
+    (kernelDual_homogeneous cs b basis B d n hd x ((length_le_of_bruhat cs x.property.2).trans hn))
+
+end
+end KLInvariance.BruhatGraph.FreeSheaf
+
+end
+
+
+section
+
+/-! Genuine word costalk character from the integral word pairing, transported
+along the fixed symmetric-algebra coordinate isomorphism. -/
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
+universe u
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ theorem bottSection_left_orthogonal (l : List I) (hl : cs.IsReduced l)
+    (x : Interval cs 1 (cs.wordProd l))
+    (m n : (bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ)
+    (hm : ∀ y, y≠x → m.val y=0) (hn : n.val x=0) :
+    bottSectionDuality cs l hl m n=0 := by
+  obtain ⟨a,rfl⟩ := bottSection_surjective cs l hl m
+  obtain ⟨b,rfl⟩ := bottSection_surjective cs l hl n
+  rw [bottSectionDuality_apply]
+  exact bottStalk_supported_annihilates cs l hl x a b hm hn
+
+ theorem bottSection_right_orthogonal (l : List I) (hl : cs.IsReduced l)
+    (x : Interval cs 1 (cs.wordProd l))
+    (m n : (bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ)
+    (hm : m.val x=0) (hn : ∀ y, y≠x → n.val y=0) :
+    bottSectionDuality cs l hl m n=0 := by
+  rw [bottSection_symmetric]
+  exact bottSection_left_orthogonal cs l hl x n m hn hm
+
+ def bottWordPerfectPairing (l : List I) (hl : cs.IsReduced l) :
+    GradedSheaf.PerfectPairing (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
+      (l.length : ℤ) where
+  pairing := bottSectionDuality cs l hl
+  homogeneous := bottSectionDuality_homogeneous cs basis l hl
+  left_orthogonal := bottSection_left_orthogonal cs l hl
+  right_orthogonal := bottSection_right_orthogonal cs l hl
+
+ def bottPolynomialPerfectPairing (l : List I) (hl : cs.IsReduced l) :
+    GradedSheaf.PerfectPairing (bottPolynomialSheaf cs basis l 1 (cs.wordProd l)) (l.length : ℤ) :=
+  (bottWordPerfectPairing cs basis l hl).coefficientChange (SymmetricAlgebra.equivMvPolynomial basis)
+    (mem_symmetricGrading basis)
+
+ def bottFreeSheaf (l : List I) : FreeSheaf cs 1 (cs.wordProd l) basis where
+  sheaf := bottPolynomialSheaf cs basis l 1 (cs.wordProd l)
+  free := bottPolynomialSheaf_free cs basis l 1 (cs.wordProd l)
+  quotient := bottPolynomialSheaf_quotient cs basis l 1 (cs.wordProd l)
+  generated := bottPolynomialSheaf_generated cs basis l 1 (cs.wordProd l)
+
+local instance finiteEdges (l : List I) : Fintype (Edge cs 1 (cs.wordProd l)) := Fintype.ofFinite _
+local instance (a : Type*) : DecidableEq a := Classical.decEq a
+
+ def bottFreeDuality (l : List I) (hl : cs.IsReduced l) :
+    FreeSheaf.SectionDuality cs 1 (cs.wordProd l) basis (bottFreeSheaf cs basis l) :=
+  FreeSheaf.dualityOfPairing cs 1 (cs.wordProd l) basis _ (bottPolynomialPerfectPairing cs basis l hl)
+
+ theorem bottKernelFree (l : List I) (hl : cs.IsReduced l) (x : Interval cs 1 (cs.wordProd l)) :
+    Module.Free (Coefficient (σ := σ))
+      (GradedSheaf.kernelData (bottPolynomialSheaf cs basis l 1 (cs.wordProd l)) x) :=
+  FreeSheaf.kernel_free cs (cs.wordProd l) basis _ (bottFreeDuality cs basis l hl) x
+
+ theorem bottKernelCharacter (l : List I) (hl : cs.IsReduced l)
+    [Fintype (Interval cs 1 (cs.wordProd l))] (x : Interval cs 1 (cs.wordProd l)) :
+    GradedSheaf.freeKernelChar (bottPolynomialSheaf cs basis l 1 (cs.wordProd l))
+      (bottKernelFree cs basis l hl) x=Polynomial.reflect (rankDifference cs x.val (cs.wordProd l))
+        (bottStalkCharacter cs basis l x.val) := by
+  have he : cs.length (cs.wordProd l)=l.length := hl
+  change _=Polynomial.reflect (cs.length (cs.wordProd l)-cs.length x.val) _
+  rw [he]
+  apply FreeSheaf.kernel_character cs (cs.wordProd l) basis _ (bottFreeDuality cs basis l hl)
+    l.length _ he.le x
+  intro i j m a hm ha
+  exact (bottPolynomialPerfectPairing cs basis l hl).homogeneous i j m a
+    (fun x => hm x (Set.mem_univ x)) (fun x => ha x (Set.mem_univ x))
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Bar and the exact R-convolution for genuine positive Hecke words.
+No KL character or degree estimate is assumed. -/
+namespace KLInvariance.HeckeQ
+open Hecke (Space basis)
+variable {I W C : Type*} [Group W] {M : CoxeterMatrix I} [CommRing C]
+noncomputable section
+
+ theorem one_add_leftInv (cs : CoxeterSystem M W) (q : Cˣ) (i : I) (v : Space W C) :
+    v+leftInv cs q i v=(↑q⁻¹:C) • (v+left cs (q:C) i v) := by
+  rw [leftInv_apply,smul_sub,smul_smul,smul_add]
+  have h : (↑q⁻¹:C)*((q:C)-1)=1-(↑q⁻¹:C) := by rw [mul_sub,q.inv_mul,mul_one]
+  rw [h,sub_smul,one_smul]
+  module
+
+ theorem barLinear_positiveWord (cs : CoxeterSystem M W) (q : Cˣ) (l : List I) :
+    barLinear cs q (positiveWord cs (↑q⁻¹:C) l)=
+      (↑q⁻¹:C)^l.length • positiveWord cs (q:C) l := by
+  induction l with
+  | nil => simp [barBasis_one]
+  | cons i l ih =>
+    simp only [positiveWord,map_add]
+    have h := congrArg (fun f : Module.End C (Space W C) => f (positiveWord cs (↑q⁻¹:C) l))
+      (barLinear_left cs q i)
+    simp only [Module.End.mul_apply] at h
+    rw [h,ih,map_smul,← smul_add,one_add_leftInv,smul_smul,List.length_cons,pow_succ]
+
+ theorem positiveWord_support (cs : CoxeterSystem M W) (q : C) (l : List I)
+    (hl : cs.IsReduced l) : (positiveWord cs q l).support ⊆ lowerFinset cs (cs.wordProd l) := by
+  classical
+  induction l with
+  | nil => simpa [positiveWord,Hecke.basis,lowerFinset] using
+      (show (Finsupp.single 1 (1:C)).support ⊆ lowerFinset cs 1 by
+        intro x hx
+        by_cases h : x=1
+        · subst x; exact (mem_lowerFinset cs 1 1).mpr .refl
+        · simp [Finsupp.mem_support_iff,Ne.symm h] at hx)
+  | cons i l ih =>
+    have ht : cs.IsReduced l := by simpa using hl.drop 1
+    have ha : ¬cs.length (cs.simple i*cs.wordProd l)<cs.length (cs.wordProd l) := by
+      have h := hl.eq
+      simp only [cs.wordProd_cons,List.length_cons,←ht.eq] at h
+      omega
+    intro x hx
+    apply (mem_lowerFinset cs (cs.wordProd (i::l)) x).mpr
+    have step (y : W) (hy : BruhatLE cs y (cs.wordProd l)) :
+        BruhatLE cs (upperSimple cs i y) (cs.wordProd (i::l)) := by
+      simpa only [upperSimple,ite_eq_right ha,cs.wordProd_cons] using upperSimple_mono cs i hy
+    have h : positiveWord cs q l x≠0 ∨ positiveWord cs q l (cs.simple i*x)≠0 := by
+      by_contra hn
+      push Not at hn
+      rw [Finsupp.mem_support_iff,positiveWord_cons_coeff,hn.1,hn.2] at hx
+      split_ifs at hx <;> simp_all
+    rcases h with h|h
+    · exact (le_upperSimple cs i x).trans
+        (step x ((mem_lowerFinset cs _ _).mp (ih ht (Finsupp.mem_support_iff.mpr h))))
+    · have hh := step (cs.simple i*x)
+        ((mem_lowerFinset cs _ _).mp (ih ht (Finsupp.mem_support_iff.mpr h)))
+      rw [upperSimple_mul] at hh
+      exact (le_upperSimple cs i x).trans hh
+
+ theorem positiveWord_rValue (cs : CoxeterSystem M W) (q : Cˣ) (l : List I)
+    (hl : cs.IsReduced l) (x : W) :
+    ∑ z ∈ lowerFinset cs (cs.wordProd l), rValue cs q x z * positiveWord cs (q:C) l z=
+      (↑q⁻¹:C)^cs.length x * (q:C)^l.length * positiveWord cs (↑q⁻¹:C) l x := by
+  classical
+  have h := congrArg (fun v : Space W C => v x) (barLinear_positiveWord cs q⁻¹ l)
+  simp only [inv_inv] at h
+  rw [barLinear,Finsupp.linearCombination_apply,Finsupp.sum_apply,
+    Finsupp.sum_of_support_subset _ (positiveWord_support cs (q:C) l hl) _ (by simp)] at h
+  simp only [Finsupp.smul_apply,smul_eq_mul] at h
+  have hr (z : W) : rValue cs q x z=(↑q⁻¹:C)^cs.length x * barBasis cs q⁻¹ z x := by
+    simpa using rValue_inverse_eq_barBasis cs q⁻¹ x z
+  simp only [hr,mul_assoc,←Finset.mul_sum]
+  rw [show (∑ z ∈ lowerFinset cs (cs.wordProd l), barBasis cs q⁻¹ z x * positiveWord cs (q:C) l z)=
+      (q:C)^l.length * positiveWord cs (↑q⁻¹:C) l x by
+    simpa only [mul_comm] using h]
+
+end
+end KLInvariance.HeckeQ
+
+end
+
+
+section
+
+namespace KLInvariance.HeckeQ
+open Hecke (Space basis)
+variable {I W C : Type*} [Group W] {M : CoxeterMatrix I} [CommRing C]
+noncomputable section
+
+ theorem eval_positiveWord (cs : CoxeterSystem M W) (q : C) (l : List I) (x : W) :
+    Polynomial.eval₂ (Int.castRingHom C) q
+      (positiveWord cs (Polynomial.X : Polynomial ℤ) l x)=positiveWord cs q l x := by
+  classical
+  induction l generalizing x with
+  | nil => simp only [positiveWord_nil,Hecke.basis,Finsupp.single_apply]; split_ifs <;> simp
+  | cons i l ih =>
+    rw [positiveWord_cons_coeff,positiveWord_cons_coeff]
+    split_ifs <;> simp only [Polynomial.eval₂_add,Polynomial.eval₂_mul,
+      Polynomial.eval₂_X,ih]
+
+end
+end KLInvariance.HeckeQ
+
+end
+
+
+section
+
+namespace KLInvariance.Graded.ModuleData
+open Polynomial Module
+universe uk us um
+variable {k : Type uk} [Field k] {σ : Type us} [Finite σ]
+  (M N : ModuleData.{uk,max uk us,um} (polynomialGrading k σ))
+noncomputable section
+
+ theorem char_natDegree_le_of_dual_equiv [Module.Free (MvPolynomial σ k) M]
+    (e : Dual (MvPolynomial σ k) M ≃ₗ[MvPolynomial σ k] N) (l : ℕ)
+    (he : ∀ r lam, (∀ j (v : M), v∈M.piece j →
+      lam v∈polynomialGrading k σ (j+r)) → e lam∈N.piece (r+l)) :
+    M.char.natDegree≤l := by
+  classical
+  let c := (exists_nonnegativeBasis (σ := σ) M.piece M.nonneg).some
+  rw [char,character_eq _ _ c]
+  apply Polynomial.natDegree_sum_le_of_forall_le
+  intro i _
+  simpa only [Polynomial.natDegree_X_pow] using M.degree_le_of_dual_equiv N e l he c i
+
+end
+end KLInvariance.Graded.ModuleData
+
+end
+
+
+section
+
+/-! Genuine word stalk-to-costalk R-convolution, derived from the Hecke bar
+and the constructed integral perfect word pairing. -/
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph Polynomial
+universe u
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+noncomputable section
+local instance (a : Type*) : DecidableEq a := Classical.decEq a
+local instance rWordEdges (l : List I) : Fintype (Edge cs 1 (cs.wordProd l)) := Fintype.ofFinite _
+local instance rWordInterval (l : List I) : Fintype (Interval cs 1 (cs.wordProd l)) := by
+  let := interval_finite cs 1 (cs.wordProd l)
+  exact Fintype.ofFinite _
+
+ theorem bottStalkCharacter_natDegree_le (l : List I) (hl : cs.IsReduced l)
+    (x : Interval cs 1 (cs.wordProd l)) :
+    (bottStalkCharacter cs basis l x.val).natDegree≤rankDifference cs x.val (cs.wordProd l) := by
+  let B := bottFreeSheaf cs basis l
+  let := B.free x
+  change (B.sheaf.vertex x).char.natDegree≤rankDifference cs x.val (cs.wordProd l)
+  apply ModuleData.char_natDegree_le_of_dual_equiv _ _
+    (FreeSheaf.kernelDual cs (cs.wordProd l) basis B (bottFreeDuality cs basis l hl) x)
+    (rankDifference cs x.val (cs.wordProd l))
+  have hd := (bottPolynomialPerfectPairing cs basis l hl).homogeneous
+  have he : cs.length (cs.wordProd l)=l.length := hl
+  apply FreeSheaf.kernelDual_homogeneous cs (cs.wordProd l) basis B (bottFreeDuality cs basis l hl)
+    (cs.length (cs.wordProd l)) _ x (length_le_of_bruhat cs x.property.2)
+  intro i j m a hm ha
+  rw [he]
+  exact hd i j m a (fun x => hm x (Set.mem_univ x)) (fun x => ha x (Set.mem_univ x))
+
+ theorem bottStalkCharacter_reciprocity (l : List I) (hl : cs.IsReduced l)
+    (x : Interval cs 1 (cs.wordProd l)) :
+    Polynomial.reflect (rankDifference cs x.val (cs.wordProd l)) (bottStalkCharacter cs basis l x.val)=
+      ∑ z ∈ lowerFinset cs (cs.wordProd l),
+        OrdinaryR.polynomial cs x.val z*bottStalkCharacter cs basis l z := by
+  apply Polynomial.toLaurent_injective
+  let q := OrdinaryR.variableUnit
+  have h := HeckeQ.positiveWord_rValue cs q l hl x.val
+  have hlen : cs.length x.val≤l.length := hl ▸ length_le_of_bruhat cs x.property.2
+  have hp : (↑q⁻¹ : LaurentPolynomial ℤ)^cs.length x.val * (q:LaurentPolynomial ℤ)^l.length=
+      (q:LaurentPolynomial ℤ)^(rankDifference cs x.val (cs.wordProd l)) := by
+    have he : cs.length (cs.wordProd l)=l.length := hl
+    rw [rankDifference,he,← Nat.sub_add_cancel hlen,pow_add]
+    calc
+      _ = ((↑q⁻¹:LaurentPolynomial ℤ)*(q:LaurentPolynomial ℤ))^cs.length x.val *
+          (q:LaurentPolynomial ℤ)^(l.length-cs.length x.val) := by rw [mul_pow]; ring
+      _ = _ := by rw [q.inv_mul]; simp
+  rw [hp] at h
+  have hv (p : Polynomial ℤ) : Polynomial.toLaurent p=
+      Polynomial.eval₂ (Int.castRingHom (LaurentPolynomial ℤ)) (q:LaurentPolynomial ℤ) p := by
+    induction p using Polynomial.induction_on' with
+    | add p r hp hr => simp only [map_add,Polynomial.eval₂_add,hp,hr]
+    | monomial n a => simp [q,Polynomial.toLaurent_C_mul_T]
+  rw [hv,eval₂_reflect_unit q _ _ (bottStalkCharacter_natDegree_le cs basis l hl x)]
+  rw [hv]
+  simp only [Polynomial.eval₂_finsetSum,Polynomial.eval₂_mul,OrdinaryR.polynomial_eval,
+    bottStalkCharacter_eq_heckeWord,HeckeQ.eval_positiveWord]
+  exact h.symm
+
+ theorem bottKernelCharacter_rConvolution (l : List I) (hl : cs.IsReduced l)
+    (x : Interval cs 1 (cs.wordProd l)) :
+    GradedSheaf.freeKernelChar (bottPolynomialSheaf cs basis l 1 (cs.wordProd l))
+      (bottKernelFree cs basis l hl) x=
+      ∑ z : Interval cs 1 (cs.wordProd l),
+        OrdinaryR.polynomial cs x.val z.val*bottStalkCharacter cs basis l z.val := by
+  rw [bottKernelCharacter cs basis l hl x,bottStalkCharacter_reciprocity cs basis l hl x]
+  exact Finset.sum_subtype (F := (inferInstance : Fintype (Interval cs 1 (cs.wordProd l))))
+    (lowerFinset cs (cs.wordProd l))
+    (fun z => by simp only [mem_lowerFinset]; exact ⟨fun h => ⟨one_bruhat cs z,h⟩,And.right⟩)
+    (fun z => OrdinaryR.polynomial cs x.val z*bottStalkCharacter cs basis l z)
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Recognition by the actual minimal-boundary object, not by an asserted
+character or by restricting a perfect form to an arbitrary summand. -/
+namespace KLInvariance.MomentGraph.BoundarySheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+  [IsDomain A] {hneg : ∀ n < 0, 𝓐 n=⊥}
+  {V E : Type um} [PartialOrder V] {G : OrderedGraph V E} {α : E → A} {b : V}
+  (B : BoundarySheaf hneg G α b)
+noncomputable section
+
+ theorem top_nontrivial : Nontrivial (B.sheaf.vertex b) := by
+  let e := ModuleData.equivOfEq B.top
+  let z : ModuleData.unit.{uk,ua,um} hneg := fun _ => (0 : A)
+  let o : ModuleData.unit.{uk,ua,um} hneg := fun _ => (1 : A)
+  refine ⟨⟨e.symm z,e.symm o,?_⟩⟩
+  intro h
+  exact zero_ne_one (congrFun (e.symm.injective h) PUnit.unit)
+
+ variable [IsNoetherianRing A] [Fintype V] [Fintype E]
+
+ theorem iso_of_primitive_top_split
+    (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c=a)
+    (hb : ∀ x, x≤b) (C : GradedSheaf (𝓐 := 𝓐) G)
+    (hCq : C.UpperQuotient α) (hCfree : ∀ x, Module.Free A (C.vertex x))
+    (hCf : C.forget.Flabby) (hCg : C.forget.Generated)
+    (hCp : (GradedSheaf.Hom.id C).PrimitiveProjector)
+    (i : ModuleData.Hom (B.sheaf.vertex b) (C.vertex b))
+    (p : ModuleData.Hom (C.vertex b) (B.sheaf.vertex b))
+    (hpi : ∀ v, p.map (i.map v)=v) : Nonempty (GradedSheaf.Iso B.sheaf C) := by
+  let := B.top_nontrivial
+  obtain ⟨s⟩ := B.splitIn_of_top_split hzero hb C hCq hCfree hCf hCg i p hpi
+  exact ⟨s.isoOfPrimitive hCp b⟩
+
+end
+end KLInvariance.MomentGraph.BoundarySheaf
+
+end
+
+
+section
+
+/-! The actual symmetric-coefficient BMP boundary sheaf is isomorphic to
+its integral adjoint image. The splitting is constructed from the word's
+proved free-stalk, exact-edge and flabbiness properties. -/
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
+universe u us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  (l : List I) (hl : cs.IsReduced l)
+  (B : BoundarySheaf (symmetricGrading_negative basis)
+    (graph cs 1 (cs.wordProd l)) (symmetricLabel cs 1 (cs.wordProd l))
+    ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩)
+  (s : GradedSheaf.SplitIn B.sheaf (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
+noncomputable section
+
+ def bottSplitAdjointSplit : GradedSheaf.SplitIn
+    (bottSplitAdjointSheaf cs basis l hl B.sheaf s)
+    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)) := by
+  let := symmetricCoefficientNoetherian basis
+  exact (bottAdjointSheaf cs basis l hl s.projector).imageSplit
+    (bottAdjointSheaf_vertex_idempotent cs basis l hl s.projector s.projector_vertex)
+    (bottAdjointSheaf_edge_idempotent cs basis l hl s.projector s.projector_vertex)
+
+ theorem bottSplitAdjoint_free (x : Interval cs 1 (cs.wordProd l)) :
+    Module.Free (SymmetricCoefficient (M := M))
+      ((bottSplitAdjointSheaf cs basis l hl B.sheaf s).vertex x) := by
+  let := symmetricCoefficientNoetherian basis
+  let : Module.Free (SymmetricCoefficient (M := M))
+      ((bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)).vertex x) :=
+    bottStalkFree cs l x.val
+  exact (bottSplitAdjointSplit cs basis l hl B s).vertex_free
+    (symmetricGrading_negative basis) (symmetricGrading_zero basis) x
+
+ def bottSplitAdjointTopInclusion : ModuleData.Hom
+    (B.sheaf.vertex ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩)
+    ((bottSplitAdjointSheaf cs basis l hl B.sheaf s).vertex
+      ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩) :=
+  ((bottSplitAdjointSplit cs basis l hl B s).projection.vertex _).comp
+    (bottBoundarySheafTopInclusion cs basis l hl B)
+
+ def bottSplitAdjointTopProjection : ModuleData.Hom
+    ((bottSplitAdjointSheaf cs basis l hl B.sheaf s).vertex
+      ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩)
+    (B.sheaf.vertex ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩) :=
+  (bottBoundarySheafTopProjection cs basis l hl B).comp
+    ((bottSplitAdjointSplit cs basis l hl B s).inclusion.vertex _)
+
+ theorem bottSplitAdjointTop_split
+    (v : B.sheaf.vertex ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩) :
+    (bottSplitAdjointTopProjection cs basis l hl B s).map
+      ((bottSplitAdjointTopInclusion cs basis l hl B s).map v)=v := by
+  change (bottBoundarySheafTopProjection cs basis l hl B).map
+    (((bottAdjointSheaf cs basis l hl s.projector).vertex
+      ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩).map
+      ((bottBoundarySheafTopInclusion cs basis l hl B).map v))=v
+  exact (congrArg (bottBoundarySheafTopProjection cs basis l hl B).map
+    (bottBoundarySplitAdjoint_top_fixed cs basis l hl _ B s
+      ((bottBoundarySheafTopInclusion cs basis l hl B).map v))).trans
+    (bottBoundarySheafTop_split cs basis l hl B v)
+
+ theorem bottBoundarySheaf_iso_adjoint : Nonempty
+    (GradedSheaf.Iso B.sheaf (bottSplitAdjointSheaf cs basis l hl B.sheaf s)) := by
+  classical
+  let := symmetricCoefficientNoetherian basis
+  let := interval_finite cs 1 (cs.wordProd l)
+  let := Fintype.ofFinite (Interval cs 1 (cs.wordProd l))
+  let := Fintype.ofFinite (Edge cs 1 (cs.wordProd l))
+  let t := bottSplitAdjointSplit cs basis l hl B s
+  exact B.iso_of_primitive_top_split (symmetricGrading_zero basis) (fun x => x.property.2)
+    (bottSplitAdjointSheaf cs basis l hl B.sheaf s)
+    (t.upperQuotient (bottGradedProjectionSheaf_upperQuotient cs basis l 1 (cs.wordProd l)))
+    (bottSplitAdjoint_free cs basis l hl B s)
+    (t.flabby (bottProjectionSheaf_flabby cs l hl))
+    (t.generated (bottProjectionSheaf_generated cs l 1 (cs.wordProd l)))
+    (bottBoundarySplitAdjoint_primitive cs basis l hl _ B s)
+    (bottSplitAdjointTopInclusion cs basis l hl B s)
+    (bottSplitAdjointTopProjection cs basis l hl B s)
+    (bottSplitAdjointTop_split cs basis l hl B s)
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} [IsNoetherianRing A]
+  {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
+  {B C : GradedSheaf (𝓐 := 𝓐) G} (s : SplitIn B C)
+noncomputable section
+
+ theorem SplitIn.imageSectionEquiv_symm_coordinates (U : Set V)
+    (n : B.forget.sections U) (x : V) :
+    (((s.imageSectionEquiv U).symm n).val x).val=(s.inclusion.vertex x).map (n.val x) := by
+  rw [s.imageSectionEquiv_symm_apply]
+  change (s.inclusion.vertex x).map ((s.projection.vertex x).map
+    ((s.inclusion.vertex x).map (n.val x)))=(s.inclusion.vertex x).map (n.val x)
+  exact congrArg (s.inclusion.vertex x).map (s.vertex_split x (n.val x))
+
+ theorem SplitIn.imageSectionEquiv_symm_inclusion (U : Set V)
+    (n : B.forget.sections U) :
+    s.projector.imageInclusion.sectionMap U ((s.imageSectionEquiv U).symm n)=
+      s.inclusion.sectionMap U n := by
+  apply Subtype.ext
+  funext x
+  exact s.imageSectionEquiv_symm_coordinates U n x
+
+end
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+/-! The adjoint-summand pairing is the ambient word pairing on its two
+honest (generally different) integral sheaf embeddings. -/
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
+universe u us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  (l : List I) (hl : cs.IsReduced l)
+  (B : GradedSheaf (𝓐 := symmetricGrading basis) (graph cs 1 (cs.wordProd l)))
+  (s : GradedSheaf.SplitIn B (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
+noncomputable section
+
+ def bottSplitRightWord (n : B.forget.sections Set.univ) : BottSamelsonModule cs l := by
+  let := symmetricCoefficientNoetherian basis
+  exact (bottImageSectionWordEquiv cs basis l hl s.projector s.projector_vertex
+    ((s.imageSectionEquiv Set.univ).symm n)).val
+
+ def bottSplitLeftWord
+    (m : (bottSplitAdjointSheaf cs basis l hl B s).forget.sections Set.univ) :
+    BottSamelsonModule cs l := by
+  let := symmetricCoefficientNoetherian basis
+  exact (bottImageSectionWordEquiv cs basis l hl (bottAdjointSheaf cs basis l hl s.projector)
+    (bottAdjointSheaf_vertex_idempotent cs basis l hl s.projector s.projector_vertex) m).val
+
+ theorem bottSplitRightWord_coordinates (n : B.forget.sections Set.univ)
+    (x : Interval cs 1 (cs.wordProd l)) :
+    bottStalkProjection cs l x.val (bottSplitRightWord cs basis l hl B s n)=
+      (s.inclusion.vertex x).map (n.val x) := by
+  let := symmetricCoefficientNoetherian basis
+  exact (bottImageSectionWordEquiv_coordinates cs basis l hl s.projector s.projector_vertex
+    ((s.imageSectionEquiv Set.univ).symm n) x).trans
+      (s.imageSectionEquiv_symm_coordinates Set.univ n x)
+
+ theorem bottSplitLeftWord_coordinates
+    (m : (bottSplitAdjointSheaf cs basis l hl B s).forget.sections Set.univ)
+    (x : Interval cs 1 (cs.wordProd l)) :
+    bottStalkProjection cs l x.val (bottSplitLeftWord cs basis l hl B s m)=(m.val x).val := by
+  let := symmetricCoefficientNoetherian basis
+  exact bottImageSectionWordEquiv_coordinates cs basis l hl
+    (bottAdjointSheaf cs basis l hl s.projector)
+    (bottAdjointSheaf_vertex_idempotent cs basis l hl s.projector s.projector_vertex) m x
+
+ theorem bottSplitAdjointWord_apply
+    (m : (bottSplitAdjointSheaf cs basis l hl B s).forget.sections Set.univ)
+    (n : B.forget.sections Set.univ) :
+    bottSplitAdjointEquivDual cs basis l hl B s m n=
+      bottSamelsonDuality cs l (bottSplitLeftWord cs basis l hl B s m)
+        (bottSplitRightWord cs basis l hl B s n) := by
+  let := symmetricCoefficientNoetherian basis
+  exact bottAdjointImageEquivDual_apply cs basis l hl s.projector s.projector_vertex m
+    ((s.imageSectionEquiv Set.univ).symm n)
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! An actual recognition isomorphism to the adjoint image gives the required
+perfect pairing. The two ambient embeddings remain distinct. -/
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
+universe u us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  (l : List I) (hl : cs.IsReduced l)
+  (B : GradedSheaf (𝓐 := symmetricGrading basis) (graph cs 1 (cs.wordProd l)))
+  (s : GradedSheaf.SplitIn B (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
+  (j : GradedSheaf.Iso B (bottSplitAdjointSheaf cs basis l hl B s))
+noncomputable section
+
+ def bottRecognizedPerfectPairing : GradedSheaf.PerfectPairing B (l.length : ℤ) where
+  pairing := (j.toIso.sections Set.univ).trans (bottSplitAdjointEquivDual cs basis l hl B s)
+  homogeneous i q m n hm hn := bottSplitAdjointPairing_homogeneous cs basis l hl B s i q
+    (j.toIso.sections Set.univ m) n (fun x => j.vertex_graded x i _ (hm x)) hn
+  left_orthogonal x m n hm hn := by
+    change bottSplitAdjointEquivDual cs basis l hl B s (j.toIso.sections Set.univ m) n=0
+    rw [bottSplitAdjointWord_apply]
+    apply bottStalk_supported_annihilates cs l hl x
+    · intro y hy
+      rw [bottSplitLeftWord_coordinates]
+      change ((j.vertex y (m.val y)).val)=0
+      rw [hm y hy,map_zero]
+      rfl
+    · rw [bottSplitRightWord_coordinates]
+      change (s.inclusion.vertex x).map (n.val x)=0
+      rw [hn]
+      exact (s.inclusion.vertex x).map.map_zero
+  right_orthogonal x m n hm hn := by
+    change bottSplitAdjointEquivDual cs basis l hl B s (j.toIso.sections Set.univ m) n=0
+    rw [bottSplitAdjointWord_apply,bottSamelson_symmetric]
+    apply bottStalk_supported_annihilates cs l hl x
+    · intro y hy
+      rw [bottSplitRightWord_coordinates]
+      change (s.inclusion.vertex y).map (n.val y)=0
+      rw [hn y hy]
+      exact (s.inclusion.vertex y).map.map_zero
+    · rw [bottSplitLeftWord_coordinates]
+      change ((j.vertex x (m.val x)).val)=0
+      rw [hm,map_zero]
+      rfl
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! The same genuine minimal boundary construction over the symmetric
+coefficient algebra. Its polynomial-coordinate comparison is proved separately. -/
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
+universe u us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ omit [Fintype σ] in
+ theorem symmetric_scalar_mod_positive (a : SymmetricCoefficient (M := M)) :
+    ∃ c : ℝ, a-algebraMap ℝ (SymmetricCoefficient (M := M)) c ∈ positiveIdeal (symmetricGrading basis) := by
+  obtain ⟨c,hc⟩ := symmetricGrading_zero basis _
+    (DirectSum.decompose (symmetricGrading basis) a 0).property
+  refine ⟨c,?_⟩
+  rw [positiveIdeal_eq_augmentation_ker _ (symmetricGrading_negative basis)]
+  change component (symmetricGrading basis) 0 (a-algebraMap ℝ (SymmetricCoefficient (M := M)) c)=0
+  rw [map_sub,hc]
+  rw [show component (symmetricGrading basis) 0
+    (DirectSum.decompose (symmetricGrading basis) a 0 : SymmetricCoefficient (M := M)) =
+    (DirectSum.decompose (symmetricGrading basis) a 0 : SymmetricCoefficient (M := M)) from
+    DirectSum.decompose_of_mem_same _ (DirectSum.decompose (symmetricGrading basis) a 0).property]
+  exact sub_self _
+
+ theorem symmetricLabel_homogeneous (u b : W) (e : Edge cs u b) :
+    symmetricLabel cs u b e ∈ symmetricGrading basis 1 :=
+  (mem_symmetricGrading basis 1 _).mpr (polynomialLabel_homogeneous cs u b basis e)
+
+ theorem exists_symmetricBoundarySheaf (b : W) :
+    Nonempty (BoundarySheaf (symmetricGrading_negative basis)
+      (graph cs 1 b) (symmetricLabel cs 1 b) ⟨b,one_bruhat cs b,bruhat_refl cs b⟩) := by
+  classical
+  let := symmetricCoefficientNoetherian basis
+  let := interval_finite cs 1 b
+  let := Fintype.ofFinite (Interval cs 1 b)
+  let := Fintype.ofFinite (Edge cs 1 b)
+  exact MomentGraph.exists_boundarySheaf (symmetricGrading_negative basis)
+    (graph cs 1 b) (symmetric_scalar_mod_positive basis)
+    ⟨b,one_bruhat cs b,bruhat_refl cs b⟩ (fun x => x.property.2)
+    (symmetricLabel cs 1 b) (symmetricLabel_homogeneous cs basis 1 b)
+
+ def symmetricBoundarySheaf (b : W) :
+    BoundarySheaf (symmetricGrading_negative basis)
+      (graph cs 1 b) (symmetricLabel cs 1 b) ⟨b,one_bruhat cs b,bruhat_refl cs b⟩ :=
+  (exists_symmetricBoundarySheaf cs basis b).some
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Unconditional integral graded support-orthogonal duality for the actual
+normalized minimal-boundary sheaf over the symmetric coefficient algebra. -/
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
+universe u us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ theorem symmetricBoundarySheaf_perfectPairing_word (l : List I) (hl : cs.IsReduced l)
+    (B : BoundarySheaf (symmetricGrading_negative basis)
+      (graph cs 1 (cs.wordProd l)) (symmetricLabel cs 1 (cs.wordProd l))
+      ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩) :
+    Nonempty (GradedSheaf.PerfectPairing B.sheaf (cs.length (cs.wordProd l) : ℤ)) := by
+  obtain ⟨s⟩ := bottBoundarySheaf_splitIn cs basis l hl B
+  obtain ⟨j⟩ := bottBoundarySheaf_iso_adjoint cs basis l hl B s
+  have hd : cs.length (cs.wordProd l)=l.length := hl
+  rw [hd]
+  exact ⟨bottRecognizedPerfectPairing cs basis l hl B.sheaf s j⟩
+
+ theorem symmetricBoundarySheaf_perfectPairing (b : W) :
+    Nonempty (GradedSheaf.PerfectPairing (symmetricBoundarySheaf cs basis b).sheaf (cs.length b : ℤ)) := by
+  obtain ⟨l,hl,he⟩ := cs.exists_isReduced b
+  subst b
+  exact symmetricBoundarySheaf_perfectPairing_word cs basis l hl _
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+namespace KLInvariance.Graded.ModuleData
+universe uk ua um
+variable {k : Type uk} [Field k] {A B : Type ua} [CommRing A] [CommRing B]
+  [Algebra k A] [Algebra k B]
+  {𝓐 : ℤ → Submodule k A} {𝓑 : ℤ → Submodule k B}
+  [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+  [DirectSum.Decomposition 𝓑] [SetLike.GradedMonoid 𝓑]
+  (e : A ≃ₐ[k] B) (he : ∀ n a, a∈𝓐 n ↔ e a∈𝓑 n)
+  (hA : ∀ n<0,𝓐 n=⊥) (hB : ∀ n<0,𝓑 n=⊥)
+noncomputable section
+
+ def changeUnitHom : Hom (change e he (unit.{uk,ua,um} hA)) (unit hB) where
+  map :=
+    { toFun m j := e (m.val j)
+      map_add' m n := funext fun j => map_add e (m.val j) (n.val j)
+      map_smul' b m := by
+        funext j
+        change e (e.symm b * m.val j)=b * e (m.val j)
+        rw [map_mul,e.apply_symm_apply] }
+  graded := by
+    intro n m hm j
+    exact (he (n-0) _).mp ((mem_changePiece e (unit hA) n m).mp hm j)
+
+ def changeUnitInvHom : Hom (unit.{uk,ua,um} hB) (change e he (unit hA)) where
+  map :=
+    { toFun m := CoefficientChange.mk (fun j => e.symm (m j))
+      map_add' m n := CoefficientChange.ext (funext fun j => map_add e.symm (m j) (n j))
+      map_smul' b m := by
+        apply CoefficientChange.ext
+        funext j
+        exact map_mul e.symm b (m j) }
+  graded := by
+    intro n m hm
+    apply (mem_changePiece e (unit hA) n _).mpr
+    intro j
+    apply (he (n-0) _).mpr
+    change e (e.symm (m j)) ∈ 𝓑 (n-0)
+    rw [e.apply_symm_apply]
+    exact hm j
+
+ theorem changeUnitHom_inv (m : unit.{uk,ua,um} hB) :
+    (changeUnitHom e he hA hB).map ((changeUnitInvHom e he hA hB).map m)=m :=
+  funext fun j => e.apply_symm_apply (m j)
+
+ theorem changeUnitInvHom_hom (m : change e he (unit.{uk,ua,um} hA)) :
+    (changeUnitInvHom e he hA hB).map ((changeUnitHom e he hA hB).map m)=m :=
+  CoefficientChange.ext (funext fun j => e.symm_apply_apply (m.val j))
+
+end
+end KLInvariance.Graded.ModuleData
+
+end
+
+
+section
+
+namespace KLInvariance.Graded.ModuleData
+universe uk ua um
+variable {k : Type uk} [Field k] {A B : Type ua} [CommRing A] [CommRing B]
+  [Algebra k A] [Algebra k B]
+  {𝓐 : ℤ → Submodule k A} {𝓑 : ℤ → Submodule k B}
+  (e : A ≃ₐ[k] B) (he : ∀ n a, a∈𝓐 n ↔ e a∈𝓑 n)
+noncomputable section
+
+ def unchangeHom {M N : ModuleData.{uk,ua,um} 𝓐} (f : Hom (change e he M) (change e he N)) :
+    Hom M N where
+  map :=
+    { toFun m := (f.map (CoefficientChange.mk m)).val
+      map_add' m n := congrArg CoefficientChange.val (map_add f.map (CoefficientChange.mk m) (CoefficientChange.mk n))
+      map_smul' a m := by
+        have hh := congrArg CoefficientChange.val (map_smul f.map (e a) (CoefficientChange.mk m))
+        change (f.map (CoefficientChange.mk (e.symm (e a) • m))).val=
+          e.symm (e a) • (f.map (CoefficientChange.mk m)).val at hh
+        simpa only [e.symm_apply_apply, RingHom.id_apply] using hh }
+  graded n m hm := (mem_changePiece e N n _).mp
+    (f.graded n (CoefficientChange.mk m) ((mem_changePiece e M n _).mpr hm))
+
+end
+end KLInvariance.Graded.ModuleData
+
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A B : Type ua} [CommRing A] [CommRing B]
+  [Algebra k A] [Algebra k B]
+  {𝓐 : ℤ → Submodule k A} {𝓑 : ℤ → Submodule k B}
+  (e : A ≃ₐ[k] B) (he : ∀ n a, a∈𝓐 n ↔ e a∈𝓑 n)
+  {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
+noncomputable section
+
+ def Hom.coefficientUnchange {S T : GradedSheaf (𝓐 := 𝓐) G}
+    (g : Hom (GradedSheaf.coefficientChange e he S) (GradedSheaf.coefficientChange e he T)) : Hom S T where
+  vertex x := ModuleData.unchangeHom e he (g.vertex x)
+  edge f := ModuleData.unchangeHom e he (g.edge f)
+  lower f m := congrArg CoefficientChange.val (g.lower f (CoefficientChange.mk m))
+  upper f m := congrArg CoefficientChange.val (g.upper f (CoefficientChange.mk m))
+
+ theorem coefficientChange_primitive (S : GradedSheaf (𝓐 := 𝓐) G)
+    (hS : (Hom.id S).PrimitiveProjector) :
+    (Hom.id (GradedSheaf.coefficientChange e he S)).PrimitiveProjector := by
+  constructor
+  · apply Hom.eq_of_maps <;> intros <;> rfl
+  · intro f hf _ _
+    let g : Hom S S := f.coefficientUnchange e he
+    have hg : g.comp g=g := by
+      apply Hom.eq_of_maps
+      · intro x m
+        exact congrArg (fun q : Hom (GradedSheaf.coefficientChange e he S) (GradedSheaf.coefficientChange e he S) =>
+          ((q.vertex x).map (CoefficientChange.mk m)).val) hf
+      · intro x m
+        exact congrArg (fun q : Hom (GradedSheaf.coefficientChange e he S) (GradedSheaf.coefficientChange e he S) =>
+          ((q.edge x).map (CoefficientChange.mk m)).val) hf
+    have hgL : g.comp (Hom.id S)=g := by apply Hom.eq_of_maps <;> intros <;> rfl
+    have hgR : (Hom.id S).comp g=g := by apply Hom.eq_of_maps <;> intros <;> rfl
+    rcases hS.2 g hg hgL hgR with hz | hi
+    · left
+      apply Hom.eq_of_maps
+      · intro x m
+        apply CoefficientChange.ext
+        exact congrArg (fun q : Hom S S => (q.vertex x).map m.val) hz
+      · intro x m
+        apply CoefficientChange.ext
+        exact congrArg (fun q : Hom S S => (q.edge x).map m.val) hz
+    · right
+      apply Hom.eq_of_maps
+      · intro x m
+        apply CoefficientChange.ext
+        exact congrArg (fun q : Hom S S => (q.vertex x).map m.val) hi
+      · intro x m
+        apply CoefficientChange.ext
+        exact congrArg (fun q : Hom S S => (q.edge x).map m.val) hi
+
+end
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+namespace KLInvariance.Graded.ModuleData
+universe uk ua um
+variable {k : Type uk} [Field k] {A B : Type ua} [CommRing A] [CommRing B]
+  [Algebra k A] [Algebra k B]
+  {𝓐 : ℤ → Submodule k A} {𝓑 : ℤ → Submodule k B}
+  [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+  [DirectSum.Decomposition 𝓑] [SetLike.GradedMonoid 𝓑]
+  (e : A ≃ₐ[k] B) (he : ∀ n a, a∈𝓐 n ↔ e a∈𝓑 n)
+  {hA : ∀ n<0,𝓐 n=⊥} {hB : ∀ n<0,𝓑 n=⊥}
+  {M : ModuleData.{uk,ua,um} 𝓐} (hM : M=unit hA)
+  {N : ModuleData.{uk,ua,um} 𝓑} (hN : N=unit hB)
+noncomputable section
+
+ def changeTopInclusion : Hom N (change e he M) :=
+  (changeHom e he (homOfEq hM.symm)).comp
+    ((changeUnitInvHom e he hA hB).comp (homOfEq hN))
+
+ def changeTopProjection : Hom (change e he M) N :=
+  (homOfEq hN.symm).comp
+    ((changeUnitHom e he hA hB).comp (changeHom e he (homOfEq hM)))
+
+ theorem changeTop_split (n : N) :
+    (changeTopProjection e he hM hN).map ((changeTopInclusion e he hM hN).map n)=n := by
+  subst M
+  subst N
+  change (changeUnitHom e he hA hB).map ((changeUnitInvHom e he hA hB).map n)=n
+  exact changeUnitHom_inv e he hA hB n
+
+end
+end KLInvariance.Graded.ModuleData
+
+namespace KLInvariance.MomentGraph.BoundarySheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A B : Type ua} [CommRing A] [CommRing B]
+  [Algebra k A] [Algebra k B]
+  {𝓐 : ℤ → Submodule k A} {𝓑 : ℤ → Submodule k B}
+  [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+  [DirectSum.Decomposition 𝓑] [SetLike.GradedMonoid 𝓑]
+  [IsDomain A] [IsDomain B] [IsNoetherianRing B]
+  (e : A ≃ₐ[k] B) (he : ∀ n a, a∈𝓐 n ↔ e a∈𝓑 n)
+  {hA : ∀ n<0,𝓐 n=⊥} {hB : ∀ n<0,𝓑 n=⊥}
+  {V E : Type um} [PartialOrder V] [Fintype V] [Fintype E]
+  {G : OrderedGraph V E} {α : E → A} {b : V}
+  (S : BoundarySheaf hA G α b) (T : BoundarySheaf hB G (fun q => e (α q)) b)
+noncomputable section
+
+ theorem iso_coefficientChange
+    (hzero : ∀ a ∈ 𝓑 0, ∃ c : k, algebraMap k B c=a)
+    (hb : ∀ x, x≤b) : Nonempty
+      (GradedSheaf.Iso T.sheaf (GradedSheaf.coefficientChange e he S.sheaf)) := by
+  exact T.iso_of_primitive_top_split hzero hb (GradedSheaf.coefficientChange e he S.sheaf)
+    (GradedSheaf.coefficientChange_upperQuotient e he S.sheaf α S.quotient)
+    (GradedSheaf.coefficientChange_free e he S.sheaf S.free)
+    (GradedSheaf.coefficientChange_flabby e he S.sheaf S.flabby.flabby)
+    (GradedSheaf.coefficientChange_generated e he S.sheaf S.generated)
+    (GradedSheaf.coefficientChange_primitive e he S.sheaf S.primitive_identity)
+    (ModuleData.changeTopInclusion e he S.top T.top)
+    (ModuleData.changeTopProjection e he S.top T.top)
+    (ModuleData.changeTop_split e he S.top T.top)
+
+end
+end KLInvariance.MomentGraph.BoundarySheaf
+
+end
+
+
+section
+
+
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
+universe u
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W) (b : W)
+  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ theorem boundarySheaf_perfectPairing :
+    Nonempty (GradedSheaf.PerfectPairing
+      (boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf (cs.length b : ℤ)) := by
+  classical
+  let := interval_finite cs 1 b
+  let := Fintype.ofFinite (Interval cs 1 b)
+  let := Fintype.ofFinite (Edge cs 1 b)
+  let S := symmetricBoundarySheaf cs basis b
+  let T := boundarySheaf cs 1 b basis (one_bruhat cs b)
+  let e := SymmetricAlgebra.equivMvPolynomial basis
+  have he : ∀ n a, a∈symmetricGrading basis n ↔ e a∈polynomialGrading ℝ σ n :=
+    mem_symmetricGrading basis
+  obtain ⟨p⟩ := symmetricBoundarySheaf_perfectPairing cs basis b
+  obtain ⟨j⟩ := BoundarySheaf.iso_coefficientChange e he S T
+    (polynomialGrading_zero ℝ σ) (fun x => x.property.2)
+  exact ⟨(p.coefficientChange e he).pullback j⟩
+
+ variable [Fintype (Edge cs 1 b)] [Fintype (Interval cs 1 b)] [DecidableEq (Interval cs 1 b)]
+
+ theorem sectionDualityObligation : SectionDualityObligation cs b basis := by
+  obtain ⟨p⟩ := boundarySheaf_perfectPairing cs b basis
+  exact sectionDualityObligation_of_perfectPairing cs b basis p
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
+universe u
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W) (b : W)
+  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+local instance finiteInterval : Fintype (Interval cs 1 b) := by
+  let := interval_finite cs 1 b
+  exact Fintype.ofFinite _
+local instance finiteEdges : Fintype (Edge cs 1 b) := Fintype.ofFinite _
+local instance (a : Type*) : DecidableEq a := Classical.decEq a
+
+ def boundaryFreeSheaf : FreeSheaf cs 1 b basis where
+  sheaf := (boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf
+  free := (boundarySheaf cs 1 b basis (one_bruhat cs b)).free
+  quotient := (boundarySheaf cs 1 b basis (one_bruhat cs b)).quotient
+  generated := (boundarySheaf cs 1 b basis (one_bruhat cs b)).generated
+
+ def boundaryPerfectPairing : GradedSheaf.PerfectPairing
+    (boundaryFreeSheaf cs b basis).sheaf (cs.length b : ℤ) :=
+  (boundarySheaf_perfectPairing cs b basis).some
+
+ def boundaryFreeDuality : FreeSheaf.SectionDuality cs 1 b basis (boundaryFreeSheaf cs b basis) :=
+  FreeSheaf.dualityOfPairing cs 1 b basis _ (boundaryPerfectPairing cs b basis)
+
+ theorem boundaryKernelFree (x : Interval cs 1 b) :
+    Module.Free (Coefficient (σ := σ))
+      (GradedSheaf.kernelData (boundaryFreeSheaf cs b basis).sheaf x) :=
+  FreeSheaf.kernel_free cs b basis _ (boundaryFreeDuality cs b basis) x
+
+ def costalkCharacter (x : Interval cs 1 b) : Polynomial ℤ :=
+  GradedSheaf.freeKernelChar (boundaryFreeSheaf cs b basis).sheaf (boundaryKernelFree cs b basis) x
+
+ theorem costalkCharacter_reflect (x : Interval cs 1 b) :
+    costalkCharacter cs b basis x=Polynomial.reflect (rankDifference cs x.val b)
+      (stalkCharacter cs 1 b basis (one_bruhat cs b) x) := by
+  apply FreeSheaf.kernel_character cs b basis _ (boundaryFreeDuality cs b basis) (cs.length b) _ le_rfl x
+  intro i j m a hm ha
+  exact (boundaryPerfectPairing cs b basis).homogeneous i j m a
+    (fun x => hm x (Set.mem_univ x)) (fun x => ha x (Set.mem_univ x))
+
+ theorem stalkBasis_degree_le (x : Interval cs 1 b)
+    (c : NonnegativeBasis (σ := σ)
+      ((boundaryFreeSheaf cs b basis).sheaf.vertex x).piece) (i : c.index) :
+    c.degree i≤rankDifference cs x.val b := by
+  apply ModuleData.degree_le_of_dual_equiv _ _
+    (FreeSheaf.kernelDual cs b basis _ (boundaryFreeDuality cs b basis) x)
+    (rankDifference cs x.val b) _ c i
+  exact FreeSheaf.kernelDual_homogeneous cs b basis _ (boundaryFreeDuality cs b basis)
+    (cs.length b) (fun i j m a hm ha => (boundaryPerfectPairing cs b basis).homogeneous i j m a
+      (fun x => hm x (Set.mem_univ x)) (fun x => ha x (Set.mem_univ x)))
+    x (length_le_of_bruhat cs x.property.2)
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  [IsNoetherianRing A] {𝓐 : ℤ → Submodule k A}
+  {V E W F : Type um} [PartialOrder V] [PartialOrder W]
+  {G : OrderedGraph V E} {H : OrderedGraph W F}
+noncomputable section
+attribute [local instance] Classical.propDecidable
+variable {K : Set V} {hK : IsLowerSet K}
+
+omit [IsNoetherianRing A] in
+ theorem lowerExtend_kernel_iff (B : GradedSheaf (𝓐 := 𝓐) (lowerGraph G K hK))
+    (x : K) (m : B.lowerExtend.vertex x.val) :
+    m∈B.lowerExtend.upwardSubmodule x.val Set.univ ↔
+      B.lowerExtendVertex x m∈B.upwardSubmodule x Set.univ := by
+  constructor
+  · intro hm e _
+    rcases e with ⟨⟨e,he⟩,hx⟩
+    cases hx
+    have hh := hm ⟨e,rfl⟩ (Set.mem_univ _)
+    exact congrFun hh ⟨⟨he⟩⟩
+  · intro hm e _
+    rcases x with ⟨x,hx⟩
+    rcases e with ⟨e,he⟩
+    change G.source e=x at he
+    subst x
+    apply funext
+    intro h
+    have hh := hm ⟨⟨e,h.down.down⟩,rfl⟩ (Set.mem_univ _)
+    exact hh
+
+ def lowerExtendKernelEquiv (B : GradedSheaf (𝓐 := 𝓐) (lowerGraph G K hK))
+    (x : K) : kernelData B.lowerExtend x.val ≃ₗ[A] kernelData B x where
+  toFun m := ⟨B.lowerExtendVertex x m.val,(lowerExtend_kernel_iff B x m.val).mp m.property⟩
+  invFun m := ⟨(B.lowerExtendVertex x).symm m.val,(lowerExtend_kernel_iff B x _).mpr (by
+    simpa only [LinearEquiv.apply_symm_apply] using m.property)⟩
+  left_inv m := Subtype.ext ((B.lowerExtendVertex x).symm_apply_apply m.val)
+  right_inv m := Subtype.ext ((B.lowerExtendVertex x).apply_symm_apply m.val)
+  map_add' m n := Subtype.ext (map_add (B.lowerExtendVertex x) m.val n.val)
+  map_smul' a m := Subtype.ext (map_smul (B.lowerExtendVertex x) a m.val)
+
+ theorem lowerExtendKernelEquiv_graded (B : GradedSheaf (𝓐 := 𝓐) (lowerGraph G K hK))
+    (x : K) (n : ℤ) (m : kernelData B.lowerExtend x.val)
+    (hm : m∈(kernelData B.lowerExtend x.val).piece n) :
+    B.lowerExtendKernelEquiv x m∈(kernelData B x).piece n :=
+  B.lowerExtendVertex_graded x n m.val hm
+
+omit [IsNoetherianRing A] in
+ theorem fullKernel_pullback (B : GradedSheaf (𝓐 := 𝓐) H) (f : GraphMap G H)
+    (hlift : ∀ e x, f.vertex x=H.source e → ∃ a, f.edge a=e ∧ G.source a=x) (x : V) :
+    (B.pullback f).upwardSubmodule x Set.univ=B.upwardSubmodule (f.vertex x) Set.univ := by
+  exact B.forget.pullback_kernel f x Set.univ Set.univ
+    (fun e he => (he (Set.mem_univ e)).elim) (fun e _ he => hlift e x he)
+
+ def pullbackKernelEquiv (B : GradedSheaf (𝓐 := 𝓐) H) (f : GraphMap G H)
+    (hlift : ∀ e x, f.vertex x=H.source e → ∃ a, f.edge a=e ∧ G.source a=x) (x : V) :
+    kernelData (B.pullback f) x ≃ₗ[A] kernelData B (f.vertex x) where
+  toFun m := ⟨m.val,fullKernel_pullback B f hlift x ▸ m.property⟩
+  invFun m := ⟨m.val,(fullKernel_pullback B f hlift x).symm ▸ m.property⟩
+  left_inv _ := rfl
+  right_inv _ := rfl
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+
+ theorem pullbackKernelEquiv_graded (B : GradedSheaf (𝓐 := 𝓐) H) (f : GraphMap G H)
+    (hlift : ∀ e x, f.vertex x=H.source e → ∃ a, f.edge a=e ∧ G.source a=x) (x : V)
+    (n : ℤ) (m : kernelData (B.pullback f) x) (hm : m∈(kernelData (B.pullback f) x).piece n) :
+    B.pullbackKernelEquiv f hlift x m∈(kernelData B (f.vertex x)).piece n := hm
+
+end
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph Polynomial
+universe u
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W) (b : W)
+  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+ def lowerKernelEquiv (c x : Interval cs 1 b) (hx : x≤c) :
+    GradedSheaf.kernelData ((boundaryModels cs b basis).sheaf c) x ≃ₗ[Coefficient (σ := σ)]
+      GradedSheaf.kernelData (boundaryFreeSheaf cs c.val basis).sheaf ⟨x.val,x.property.1,hx⟩ :=
+  ((lowerIntervalBoundary cs b basis c).sheaf.lowerExtendKernelEquiv ⟨x,hx⟩).trans
+    ((boundaryFreeSheaf cs c.val basis).sheaf.pullbackKernelEquiv
+      (lowerIntervalMap cs b c) (lowerIntervalMap_outgoing cs b c) ⟨x,hx⟩)
+
+ theorem lowerKernelEquiv_graded (c x : Interval cs 1 b) (hx : x≤c)
+    (n : ℤ) (v : GradedSheaf.kernelData ((boundaryModels cs b basis).sheaf c) x)
+    (hv : v∈(GradedSheaf.kernelData ((boundaryModels cs b basis).sheaf c) x).piece n) :
+    lowerKernelEquiv cs b basis c x hx v∈(GradedSheaf.kernelData
+      (boundaryFreeSheaf cs c.val basis).sheaf ⟨x.val,x.property.1,hx⟩).piece n :=
+  (lowerIntervalBoundary cs b basis c).sheaf.lowerExtendKernelEquiv_graded ⟨x,hx⟩ n v hv
+
+ theorem boundaryModels_kernelFree (c x : Interval cs 1 b) :
+    Module.Free (Coefficient (σ := σ))
+      (GradedSheaf.kernelData ((boundaryModels cs b basis).sheaf c) x) := by
+  by_cases hx : x≤c
+  · let := boundaryKernelFree cs c.val basis ⟨x.val,x.property.1,hx⟩
+    exact Module.Free.of_equiv (lowerKernelEquiv cs b basis c x hx).symm
+  · let := (boundaryModels cs b basis).support c x hx
+    let : Subsingleton (GradedSheaf.kernelData ((boundaryModels cs b basis).sheaf c) x) :=
+      ⟨fun v w => Subtype.ext (Subsingleton.elim _ _)⟩
+    infer_instance
+
+ theorem boundaryModels_kernelCharacter (c x : Interval cs 1 b) :
+    GradedSheaf.freeKernelChar ((boundaryModels cs b basis).sheaf c)
+      (boundaryModels_kernelFree cs b basis c) x=
+      Polynomial.reflect (rankDifference cs x.val c.val) (lowerCharacter cs b basis c x) := by
+  let := boundaryModels_kernelFree cs b basis c x
+  by_cases hx : x≤c
+  · rw [lowerCharacter,dite_eq_left hx,← costalkCharacter_reflect cs c.val basis ⟨x.val,x.property.1,hx⟩]
+    let := boundaryKernelFree cs c.val basis ⟨x.val,x.property.1,hx⟩
+    exact ModuleData.character_eq_of_gradedEquiv (lowerKernelEquiv cs b basis c x hx)
+      (lowerKernelEquiv_graded cs b basis c x hx)
+  · rw [lowerCharacter,dite_eq_right hx,Polynomial.reflect_zero]
+    let := (boundaryModels cs b basis).support c x hx
+    let : Subsingleton (GradedSheaf.kernelData ((boundaryModels cs b basis).sheaf c) x) :=
+      ⟨fun v w => Subtype.ext (Subsingleton.elim _ _)⟩
+    exact ModuleData.char_zero _
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded Polynomial
+universe uk us um
+variable {k : Type uk} [Field k] {σ : Type us} [Finite σ]
+  {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
+noncomputable section
+
+ theorem exists_character_decomposition [Fintype V] [Fintype E]
+    {α : E → MvPolynomial σ k}
+    (O : BoundaryModels (polynomialGrading_negative k σ) (G := G) α)
+    (hk : ∀ y x, Module.Free (MvPolynomial σ k) (kernelData (O.sheaf y) x))
+    (B : GradedSheaf (𝓐 := polynomialGrading k σ) G)
+    (hf : ∀ x, Module.Free (MvPolynomial σ k) (B.vertex x))
+    (hkb : ∀ x, Module.Free (MvPolynomial σ k) (kernelData B x))
+    (hq : B.UpperQuotient α) (hfl : B.forget.Flabby) (hg : B.forget.Generated) :
+    ∃ m : V → Polynomial ℤ, (∀ y n, 0≤(m y).coeff n) ∧
+      (∀ x, freeStalkChar B hf x =
+        ∑ y, m y*freeStalkChar (O.sheaf y) (O.free y) x) ∧
+      (∀ x, freeKernelChar B hkb x =
+        ∑ y, m y*freeKernelChar (O.sheaf y) (hk y) x) := by
+  obtain ⟨C,hC,⟨j⟩⟩ := exists_modelSumIso (polynomialGrading_zero k σ) O B hf hq hfl hg
+  obtain ⟨m,hm,heq,hkeq⟩ := hC.character_both O hk
+  refine ⟨m,hm,fun x => ?_,fun x => ?_⟩
+  · rw [← heq x]
+    let _ := hf x
+    let _ := hC.free O x
+    exact ModuleData.character_eq_of_gradedEquiv (j.vertex x) (j.vertex_graded x)
+  · rw [← hkeq x]
+    let _ := hkb x
+    let _ := hC.kernelFree hk x
+    exact ModuleData.character_eq_of_gradedEquiv (j.kernelEquiv x) (j.kernelEquiv_graded x)
+
+end
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph Polynomial
+universe u
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ theorem bottCharacter_decomposition_both (l : List I) (hl : cs.IsReduced l)
+    [Fintype (Interval cs 1 (cs.wordProd l))] :
+    ∃ m : Interval cs 1 (cs.wordProd l) → Polynomial ℤ,
+      (∀ y n, 0≤(m y).coeff n) ∧
+      (∀ x, bottStalkCharacter cs basis l x.val=
+        ∑ y, m y*lowerCharacter cs (cs.wordProd l) basis y x) ∧
+      (∀ x, GradedSheaf.freeKernelChar (bottPolynomialSheaf cs basis l 1 (cs.wordProd l))
+        (bottKernelFree cs basis l hl) x=
+        ∑ y, m y*Polynomial.reflect (rankDifference cs x.val y.val)
+          (lowerCharacter cs (cs.wordProd l) basis y x)) := by
+  classical
+  let := Fintype.ofFinite (Edge cs 1 (cs.wordProd l))
+  obtain ⟨m,hm,he,hk⟩ := GradedSheaf.exists_character_decomposition
+    (boundaryModels cs (cs.wordProd l) basis) (boundaryModels_kernelFree cs (cs.wordProd l) basis)
+    (bottPolynomialSheaf cs basis l 1 (cs.wordProd l))
+    (bottPolynomialSheaf_free cs basis l 1 (cs.wordProd l)) (bottKernelFree cs basis l hl)
+    (bottPolynomialSheaf_quotient cs basis l 1 (cs.wordProd l))
+    (bottPolynomialSheaf_flabby cs basis l hl) (bottPolynomialSheaf_generated cs basis l 1 (cs.wordProd l))
+  refine ⟨m,hm,fun x => ?_,fun x => ?_⟩
+  · simpa only [bottPolynomialSheaf_character,boundaryModels_character] using he x
+  · simpa only [boundaryModels_kernelCharacter] using hk x
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+namespace KLInvariance.Graded.ModuleData
+open Polynomial Module
+universe uk us um
+variable {k : Type uk} [Field k] {σ : Type us} [Finite σ]
+noncomputable section
+
+ def unitPolynomialBasis : NonnegativeBasis (σ := σ)
+    (unit.{uk,max uk us,um} (polynomialGrading_negative k σ)).piece where
+  index := PUnit.{max uk us um + 1}
+  finite := inferInstance
+  basis := Pi.basisFun (MvPolynomial σ k) _
+  degree _ := 0
+  homogeneous i := by
+    intro j
+    change (Pi.basisFun (MvPolynomial σ k) _ i) j ∈ polynomialGrading k σ (0-0)
+    cases i
+    cases j
+    simpa using SetLike.one_mem_graded (polynomialGrading k σ)
+
+ theorem char_unit :
+    (unit.{uk,max uk us,um} (polynomialGrading_negative k σ)).char=1 := by
+  rw [char,character_eq _ _ (unitPolynomialBasis (k := k) (σ := σ))]
+  change (∑ i : PUnit.{max uk us um + 1}, (X : Polynomial ℤ)^0)=1
+  simp
+
+end
+end KLInvariance.Graded.ModuleData
+
+end
+
+
+section
+
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph Polynomial
+universe u
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W) (b : W)
+  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+ theorem stalkCharacter_top :
+    stalkCharacter cs 1 b basis (one_bruhat cs b) ⟨b,one_bruhat cs b,bruhat_refl cs b⟩=1 := by
+  let B := boundarySheaf cs 1 b basis (one_bruhat cs b)
+  let := B.free ⟨b,one_bruhat cs b,bruhat_refl cs b⟩
+  change (B.sheaf.vertex ⟨b,one_bruhat cs b,bruhat_refl cs b⟩).char=1
+  exact (ModuleData.character_eq_of_gradedEquiv (ModuleData.equivOfEq B.top)
+    (ModuleData.equivOfEq_graded B.top)).trans ModuleData.char_unit
+
+ theorem lowerCharacter_diagonal (c : Interval cs 1 b) : lowerCharacter cs b basis c c=1 := by
+  rw [lowerCharacter,dite_eq_left le_rfl]
+  exact stalkCharacter_top cs c.val basis
+
+ theorem lowerCharacter_at_top (c : Interval cs 1 b) :
+    lowerCharacter cs b basis c ⟨b,one_bruhat cs b,bruhat_refl cs b⟩=
+      if c.val=b then 1 else 0 := by
+  by_cases hc : c.val=b
+  · have he : c=⟨b,one_bruhat cs b,bruhat_refl cs b⟩ := Subtype.ext hc
+    rw [ite_eq_left hc,←he,lowerCharacter_diagonal]
+  · rw [ite_eq_right hc]
+    apply lowerCharacter_zero
+    intro h
+    exact hc (bruhat_antisymm cs c.property.2 h)
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph Polynomial
+universe u
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W) (b : W)
+  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+local instance lowerConvInterval (c : W) : Fintype (Interval cs 1 c) := by
+  let := interval_finite cs 1 c
+  exact Fintype.ofFinite _
+
+ def lowerIntervalEquiv (c : Interval cs 1 b) : Set.Iic c ≃ Interval cs 1 c.val where
+  toFun := (lowerIntervalMap cs b c).vertex
+  invFun z := ⟨⟨z.val,z.property.1,bruhat_trans cs z.property.2 c.property.2⟩,z.property.2⟩
+  left_inv _ := rfl
+  right_inv _ := rfl
+
+ theorem lowerCharacter_convolution (c x : Interval cs 1 b) :
+    (∑ z : Interval cs 1 b, OrdinaryR.polynomial cs x.val z.val*lowerCharacter cs b basis c z)=
+      ∑ z : Interval cs 1 c.val, OrdinaryR.polynomial cs x.val z.val*
+        stalkCharacter cs 1 c.val basis (one_bruhat cs c.val) z := by
+  classical
+  let f (z : Interval cs 1 b) := OrdinaryR.polynomial cs x.val z.val*lowerCharacter cs b basis c z
+  have hf : ∀ z : Interval cs 1 b, ¬z≤c → f z=0 := by
+    intro z hz
+    simp only [f,lowerCharacter_zero cs b basis c z hz,mul_zero]
+  have he : (∑ z : Interval cs 1 b, f z) = ∑ z ∈ Finset.univ.filter (fun z => z≤c), f z := by
+    symm
+    apply Finset.sum_subset (Finset.filter_subset _ _)
+    intro z _ hz
+    exact hf z (by simpa using hz)
+  change (∑ z, f z)=_
+  rw [he,Finset.sum_subtype (p := fun z : Interval cs 1 b => z≤c) _ (by simp)]
+  calc
+    _ = ∑ z : Set.Iic c, OrdinaryR.polynomial cs x.val z.val.val*
+        stalkCharacter cs 1 c.val basis (one_bruhat cs c.val) ((lowerIntervalEquiv cs b c) z) := by
+      apply Finset.sum_congr rfl
+      intro z _
+      simp only [f,lowerCharacter,dite_eq_left z.property]
+      rfl
+    _ = _ := Equiv.sum_comp (lowerIntervalEquiv cs b c) (fun z => OrdinaryR.polynomial cs x.val z.val*
+      stalkCharacter cs 1 c.val basis (one_bruhat cs c.val) z)
+
+ theorem lowerCharacter_convolution_zero (c x : Interval cs 1 b) (hx : ¬x≤c) :
+    (∑ z : Interval cs 1 b, OrdinaryR.polynomial cs x.val z.val*lowerCharacter cs b basis c z)=0 := by
+  apply Finset.sum_eq_zero
+  intro z _
+  by_cases hz : z≤c
+  · rw [OrdinaryR.polynomial_zero cs x.val z.val (fun h => hx (bruhat_trans cs h hz)),zero_mul]
+  · rw [lowerCharacter_zero cs b basis c z hz,mul_zero]
+
+ theorem lowerCharacter_reciprocity_lift (c : Interval cs 1 b)
+    (h : ∀ x : Interval cs 1 c.val,
+      reflect (rankDifference cs x.val c.val) (stalkCharacter cs 1 c.val basis (one_bruhat cs c.val) x)=
+        ∑ z : Interval cs 1 c.val, OrdinaryR.polynomial cs x.val z.val*
+          stalkCharacter cs 1 c.val basis (one_bruhat cs c.val) z)
+    (x : Interval cs 1 b) :
+      reflect (rankDifference cs x.val c.val) (lowerCharacter cs b basis c x)=
+        ∑ z : Interval cs 1 b, OrdinaryR.polynomial cs x.val z.val*lowerCharacter cs b basis c z := by
+  by_cases hx : x≤c
+  · rw [lowerCharacter_convolution,lowerCharacter,dite_eq_left hx]
+    exact h ⟨x.val,x.property.1,hx⟩
+  · rw [lowerCharacter_zero cs b basis c x hx,reflect_zero,lowerCharacter_convolution_zero cs b basis c x hx]
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+/-! Finite unipotent character cancellation. The transition data is actually
+constructed for word/BMP sheaves; this lemma assumes only the displayed
+linear identities and cannot supply the KL degree bound. -/
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/KazhdanLusztig/Comparison.lean b/lean/OAI/RepresentationTheory/KazhdanLusztig/Comparison.lean
index 850ffe6d19b5bee99dec3cdee4122485ee14a6f0..37eb39d0f0e9033a1e7d978138a1edbc77208c20 100644
--- a/lean/OAI/RepresentationTheory/KazhdanLusztig/Comparison.lean
+++ b/lean/OAI/RepresentationTheory/KazhdanLusztig/Comparison.lean
@@ -1,149 +1,149 @@
-import Mathlib
-
-namespace OAI
-
-/-!
-Reciprocal low-degree rigidity and positive elementary sheaf-transfer inequalities.
--/
-
-section
-
-
-/-! The closing algebra of the admitted main proof, source §6. These are
-necessary dependencies, not a substitute (conditional or special-case) main
-invariance theorem. The sheaf edge-image comparison is NOT assumed as an
-instance or asserted here. -/
-
-namespace KLInvariance.Comparison
-
-open Polynomial
-
-/-- A polynomial supported strictly below the middle cannot equal its
-reciprocal on an open real interval. This discharges the final rigidity
-step after the source's two comparison inequalities have been combined. -/
-theorem low_degree_zero_of_eval_reciprocal (p : ℝ[X]) (d : ℕ)
-    (hp : 2 * p.natDegree < d)
-    (h : ∀ q : ℝ, 0 < q → q < 1 → p.eval q = q ^ d * p.eval q⁻¹) : p = 0 := by
-  have heq : p = reflect d p := by
-    apply p.eq_of_infinite_eval_eq
-    apply (Set.Ioo_infinite (show (0 : ℝ) < 1 by norm_num)).mono
-    intro q hq
-    change p.eval q = (reflect d p).eval q
-    let : Invertible q⁻¹ := invertibleOfNonzero (inv_ne_zero (ne_of_gt hq.1))
-    have hr := eval₂_reflect_mul_pow (RingHom.id ℝ) q⁻¹ d p (by omega)
-    simp only [eval₂_id, invOf_eq_inv, inv_inv] at hr
-    have hr' := congrArg (fun z : ℝ => z * q ^ d) hr
-    have hq0 : q ≠ 0 := ne_of_gt hq.1
-    simp only [mul_assoc, ← mul_pow, inv_mul_cancel₀ hq0, one_pow, mul_one] at hr'
-    calc
-      p.eval q = q ^ d * p.eval q⁻¹ := h q hq.1 hq.2
-      _ = (reflect d p).eval q := by
-        simpa only [mul_comm] using hr'.symm
-  by_contra hp0
-  have hc := congrArg (fun f : ℝ[X] => f.coeff p.natDegree) heq.symm
-  rw [coeff_reflect, revAt_le (by omega), coeff_eq_zero_of_natDegree_lt (by omega),
-    coeff_natDegree] at hc
-  exact (leadingCoeff_ne_zero.mpr hp0) hc.symm
-
-/-- The exact polynomial conclusion from reciprocal difference equality,
-without assuming positivity of KL coefficients. -/
-theorem eq_of_eval_difference_reciprocal (p p' : ℤ[X]) (d : ℕ)
-    (hp : 2 * p.natDegree < d) (hp' : 2 * p'.natDegree < d)
-    (h : ∀ q : ℝ, 0 < q → q < 1 →
-      p.eval₂ (Int.castRingHom ℝ) q - p'.eval₂ (Int.castRingHom ℝ) q =
-      q ^ d * (p.eval₂ (Int.castRingHom ℝ) q⁻¹ -
-        p'.eval₂ (Int.castRingHom ℝ) q⁻¹)) : p = p' := by
-  apply sub_eq_zero.mp
-  apply Polynomial.map_injective (f := Int.castRingHom ℝ) Int.cast_injective
-  rw [Polynomial.map_zero]
-  apply low_degree_zero_of_eval_reciprocal ((p - p').map (Int.castRingHom ℝ)) d
-  · have hdeg := Polynomial.natDegree_map_le (f := Int.castRingHom ℝ) (p := p - p')
-    have hsub := Polynomial.natDegree_sub_le p p'
-    omega
-  · intro q hq hq'
-    simpa only [eval_map, eval₂_sub] using h q hq hq'
-
-section Schedule
-
-variable {V : Type*}
-
-noncomputable local instance : DecidableEq V := Classical.decEq V
-
-/-- The genuine elementary transfer `id + T E_xy` from the source's edge
-schedule, expressed on vectors rather than with a choice of enumeration. -/
-noncomputable def transfer (T : ℝ) (x y : V) (v : V → ℝ) : V → ℝ :=
-  fun z => v z + if z = x then T * v y else 0
-
-theorem transfer_monotone (T : ℝ) (hT : 0 ≤ T) (x y : V) :
-    Monotone (transfer T x y) := by
-  intro v w hv z
-  dsimp [transfer]
-  split_ifs
-  · exact add_le_add (hv z) (mul_le_mul_of_nonneg_left (hv y) hT)
-  · simpa using hv z
-
-theorem transfer_injective (T : ℝ) (x y : V) (hxy : x ≠ y) :
-    Function.Injective (transfer T x y) := by
-  intro v w h
-  have hy := congrFun h y
-  simp only [transfer, Ne.symm hxy, ite_false, add_zero] at hy
-  funext z
-  have hz := congrFun h z
-  dsimp [transfer] at hz
-  split_ifs at hz with hz'
-  · rw [hy] at hz
-    exact add_right_cancel hz
-  · exact add_right_cancel hz
-
-noncomputable def idealState (T : ℝ) (edges : ℕ → V × V) (initial : V → ℝ) :
-    ℕ → V → ℝ
-  | 0 => initial
-  | n + 1 => transfer T (edges n).1 (edges n).2 (idealState T edges initial n)
-
-/-- Propagation of the actual Hilbert-series bound through all earlier edges.
-No freeness or positivity of the *vectors* is needed for this part. -/
-theorem state_le_ideal (T : ℝ) (hT : 0 ≤ T) (edges : ℕ → V × V)
-    (D : ℕ → V → ℝ) (n : ℕ)
-    (hstep : ∀ k < n, D (k + 1) ≤ transfer T (edges k).1 (edges k).2 (D k)) :
-    D n ≤ idealState T edges (D 0) n := by
-  induction n with
-  | zero => exact le_rfl
-  | succ n ih =>
-    exact (hstep n (Nat.lt_succ_self n)).trans
-      (transfer_monotone T hT _ _ (ih (fun k hk => hstep k (Nat.lt_succ_of_lt hk))))
-
-/-- Endpoint equality forces every edge-image deficit to vanish, including
-those invisible in codimension one. This uses actual inequalities and the
-injectivity of elementary edge transfers; it does not assume image equality.
-It is the no-deficit conclusion of source (6.8). -/
-theorem every_step_eq_of_endpoint_eq (T : ℝ) (hT : 0 ≤ T)
-    (edges : ℕ → V × V) (D : ℕ → V → ℝ) (n : ℕ)
-    (hedges : ∀ k < n, (edges k).1 ≠ (edges k).2)
-    (hstep : ∀ k < n, D (k + 1) ≤ transfer T (edges k).1 (edges k).2 (D k))
-    (hend : D n = idealState T edges (D 0) n) :
-    ∀ k < n, D (k + 1) = transfer T (edges k).1 (edges k).2 (D k) := by
-  induction n with
-  | zero => omega
-  | succ n ih =>
-    have hD : D n ≤ idealState T edges (D 0) n :=
-      state_le_ideal T hT edges D n (fun k hk => hstep k (Nat.lt_succ_of_lt hk))
-    have ht := transfer_monotone T hT (edges n).1 (edges n).2 hD
-    have hlast : D (n + 1) = transfer T (edges n).1 (edges n).2 (D n) := by
-      apply le_antisymm (hstep n (Nat.lt_succ_self n))
-      exact ht.trans_eq hend.symm
-    have heqn : D n = idealState T edges (D 0) n := by
-      apply transfer_injective T (edges n).1 (edges n).2
-        (hedges n (Nat.lt_succ_self n))
-      exact hlast.symm.trans hend
-    intro k hk
-    rcases Nat.lt_succ_iff_lt_or_eq.mp hk with hk | rfl
-    · exact ih (fun j hj => hedges j (Nat.lt_succ_of_lt hj))
-        (fun j hj => hstep j (Nat.lt_succ_of_lt hj)) heqn k hk
-    · exact hlast
-
-end Schedule
-end KLInvariance.Comparison
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-!
+Reciprocal low-degree rigidity and positive elementary sheaf-transfer inequalities.
+-/
+
+section
+
+
+/-! The closing algebra of the admitted main proof, source §6. These are
+necessary dependencies, not a substitute (conditional or special-case) main
+invariance theorem. The sheaf edge-image comparison is NOT assumed as an
+instance or asserted here. -/
+
+namespace KLInvariance.Comparison
+
+open Polynomial
+
+/-- A polynomial supported strictly below the middle cannot equal its
+reciprocal on an open real interval. This discharges the final rigidity
+step after the source's two comparison inequalities have been combined. -/
+theorem low_degree_zero_of_eval_reciprocal (p : ℝ[X]) (d : ℕ)
+    (hp : 2 * p.natDegree < d)
+    (h : ∀ q : ℝ, 0 < q → q < 1 → p.eval q = q ^ d * p.eval q⁻¹) : p = 0 := by
+  have heq : p = reflect d p := by
+    apply p.eq_of_infinite_eval_eq
+    apply (Set.Ioo_infinite (show (0 : ℝ) < 1 by norm_num)).mono
+    intro q hq
+    change p.eval q = (reflect d p).eval q
+    let : Invertible q⁻¹ := invertibleOfNonzero (inv_ne_zero (ne_of_gt hq.1))
+    have hr := eval₂_reflect_mul_pow (RingHom.id ℝ) q⁻¹ d p (by omega)
+    simp only [eval₂_id, invOf_eq_inv, inv_inv] at hr
+    have hr' := congrArg (fun z : ℝ => z * q ^ d) hr
+    have hq0 : q ≠ 0 := ne_of_gt hq.1
+    simp only [mul_assoc, ← mul_pow, inv_mul_cancel₀ hq0, one_pow, mul_one] at hr'
+    calc
+      p.eval q = q ^ d * p.eval q⁻¹ := h q hq.1 hq.2
+      _ = (reflect d p).eval q := by
+        simpa only [mul_comm] using hr'.symm
+  by_contra hp0
+  have hc := congrArg (fun f : ℝ[X] => f.coeff p.natDegree) heq.symm
+  rw [coeff_reflect, revAt_le (by omega), coeff_eq_zero_of_natDegree_lt (by omega),
+    coeff_natDegree] at hc
+  exact (leadingCoeff_ne_zero.mpr hp0) hc.symm
+
+/-- The exact polynomial conclusion from reciprocal difference equality,
+without assuming positivity of KL coefficients. -/
+theorem eq_of_eval_difference_reciprocal (p p' : ℤ[X]) (d : ℕ)
+    (hp : 2 * p.natDegree < d) (hp' : 2 * p'.natDegree < d)
+    (h : ∀ q : ℝ, 0 < q → q < 1 →
+      p.eval₂ (Int.castRingHom ℝ) q - p'.eval₂ (Int.castRingHom ℝ) q =
+      q ^ d * (p.eval₂ (Int.castRingHom ℝ) q⁻¹ -
+        p'.eval₂ (Int.castRingHom ℝ) q⁻¹)) : p = p' := by
+  apply sub_eq_zero.mp
+  apply Polynomial.map_injective (f := Int.castRingHom ℝ) Int.cast_injective
+  rw [Polynomial.map_zero]
+  apply low_degree_zero_of_eval_reciprocal ((p - p').map (Int.castRingHom ℝ)) d
+  · have hdeg := Polynomial.natDegree_map_le (f := Int.castRingHom ℝ) (p := p - p')
+    have hsub := Polynomial.natDegree_sub_le p p'
+    omega
+  · intro q hq hq'
+    simpa only [eval_map, eval₂_sub] using h q hq hq'
+
+section Schedule
+
+variable {V : Type*}
+
+noncomputable local instance : DecidableEq V := Classical.decEq V
+
+/-- The genuine elementary transfer `id + T E_xy` from the source's edge
+schedule, expressed on vectors rather than with a choice of enumeration. -/
+noncomputable def transfer (T : ℝ) (x y : V) (v : V → ℝ) : V → ℝ :=
+  fun z => v z + if z = x then T * v y else 0
+
+theorem transfer_monotone (T : ℝ) (hT : 0 ≤ T) (x y : V) :
+    Monotone (transfer T x y) := by
+  intro v w hv z
+  dsimp [transfer]
+  split_ifs
+  · exact add_le_add (hv z) (mul_le_mul_of_nonneg_left (hv y) hT)
+  · simpa using hv z
+
+theorem transfer_injective (T : ℝ) (x y : V) (hxy : x ≠ y) :
+    Function.Injective (transfer T x y) := by
+  intro v w h
+  have hy := congrFun h y
+  simp only [transfer, Ne.symm hxy, ite_false, add_zero] at hy
+  funext z
+  have hz := congrFun h z
+  dsimp [transfer] at hz
+  split_ifs at hz with hz'
+  · rw [hy] at hz
+    exact add_right_cancel hz
+  · exact add_right_cancel hz
+
+noncomputable def idealState (T : ℝ) (edges : ℕ → V × V) (initial : V → ℝ) :
+    ℕ → V → ℝ
+  | 0 => initial
+  | n + 1 => transfer T (edges n).1 (edges n).2 (idealState T edges initial n)
+
+/-- Propagation of the actual Hilbert-series bound through all earlier edges.
+No freeness or positivity of the *vectors* is needed for this part. -/
+theorem state_le_ideal (T : ℝ) (hT : 0 ≤ T) (edges : ℕ → V × V)
+    (D : ℕ → V → ℝ) (n : ℕ)
+    (hstep : ∀ k < n, D (k + 1) ≤ transfer T (edges k).1 (edges k).2 (D k)) :
+    D n ≤ idealState T edges (D 0) n := by
+  induction n with
+  | zero => exact le_rfl
+  | succ n ih =>
+    exact (hstep n (Nat.lt_succ_self n)).trans
+      (transfer_monotone T hT _ _ (ih (fun k hk => hstep k (Nat.lt_succ_of_lt hk))))
+
+/-- Endpoint equality forces every edge-image deficit to vanish, including
+those invisible in codimension one. This uses actual inequalities and the
+injectivity of elementary edge transfers; it does not assume image equality.
+It is the no-deficit conclusion of source (6.8). -/
+theorem every_step_eq_of_endpoint_eq (T : ℝ) (hT : 0 ≤ T)
+    (edges : ℕ → V × V) (D : ℕ → V → ℝ) (n : ℕ)
+    (hedges : ∀ k < n, (edges k).1 ≠ (edges k).2)
+    (hstep : ∀ k < n, D (k + 1) ≤ transfer T (edges k).1 (edges k).2 (D k))
+    (hend : D n = idealState T edges (D 0) n) :
+    ∀ k < n, D (k + 1) = transfer T (edges k).1 (edges k).2 (D k) := by
+  induction n with
+  | zero => omega
+  | succ n ih =>
+    have hD : D n ≤ idealState T edges (D 0) n :=
+      state_le_ideal T hT edges D n (fun k hk => hstep k (Nat.lt_succ_of_lt hk))
+    have ht := transfer_monotone T hT (edges n).1 (edges n).2 hD
+    have hlast : D (n + 1) = transfer T (edges n).1 (edges n).2 (D n) := by
+      apply le_antisymm (hstep n (Nat.lt_succ_self n))
+      exact ht.trans_eq hend.symm
+    have heqn : D n = idealState T edges (D 0) n := by
+      apply transfer_injective T (edges n).1 (edges n).2
+        (hedges n (Nat.lt_succ_self n))
+      exact hlast.symm.trans hend
+    intro k hk
+    rcases Nat.lt_succ_iff_lt_or_eq.mp hk with hk | rfl
+    · exact ih (fun j hj => hedges j (Nat.lt_succ_of_lt hj))
+        (fun j hj => hstep j (Nat.lt_succ_of_lt hj)) heqn k hk
+    · exact hlast
+
+end Schedule
+end KLInvariance.Comparison
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/KazhdanLusztig/Coxeter.lean b/lean/OAI/RepresentationTheory/KazhdanLusztig/Coxeter.lean
index 329fcd62caccfb9830cb1ca8e7d368fb7d8f4a30..4c72d9fe50e00af5e3eec787e01d1798613d334c 100644
--- a/lean/OAI/RepresentationTheory/KazhdanLusztig/Coxeter.lean
+++ b/lean/OAI/RepresentationTheory/KazhdanLusztig/Coxeter.lean
@@ -1,242 +1,242 @@
-import Mathlib
-
-namespace OAI
-
-/-!
-Signed reflection permutations and strong exchange for arbitrary Coxeter systems.
--/
-
-section
-
-
-namespace KLInvariance.CoxeterSupport
-
-open CoxeterSystem
-
-universe u v
-variable {B : Type u} {W : Type v} [Group W] {M : CoxeterMatrix B}
-variable (cs : CoxeterSystem M W)
-
-noncomputable local instance : DecidableEq W := Classical.decEq W
-
-noncomputable def simpleAction (i : B) : Equiv.Perm (W × ZMod 2) := by
-  classical
-  let f : W × ZMod 2 → W × ZMod 2 := fun p =>
-    (cs.simple i * p.1 * cs.simple i, p.2 + if p.1 = cs.simple i then 1 else 0)
-  have hf : Function.Involutive f := by
-    rintro ⟨t, a⟩
-    dsimp [f]
-    have heq : cs.simple i * t * cs.simple i = cs.simple i ↔ t = cs.simple i := by
-      constructor
-      · intro h
-        have := congrArg (fun z => cs.simple i * z * cs.simple i) h
-        simpa [mul_assoc] using this
-      · rintro rfl
-        simp
-    simp only [heq]
-    ext
-    · simp [mul_assoc]
-    · split_ifs
-      · simp only [add_assoc, ZModModule.add_self, add_zero]
-      · simp
-  exact ⟨f, f, hf, hf⟩
-
-@[simp] theorem simpleAction_apply (i : B) (t : W) (a : ZMod 2) :
-    simpleAction cs i (t, a) =
-      (cs.simple i * t * cs.simple i, a + if t = cs.simple i then 1 else 0) := by
-  classical
-  rfl
-
-noncomputable def wordAction (ω : List B) : Equiv.Perm (W × ZMod 2) :=
-  (ω.map (simpleAction cs)).prod
-
-@[simp] theorem wordAction_nil : wordAction cs [] = 1 := rfl
-
-theorem wordAction_concat (ω : List B) (i : B) :
-    wordAction cs (ω.concat i) = wordAction cs ω * simpleAction cs i := by
-  simp [wordAction]
-
-/-- Signed action records the parity of the genuine left inversion sequence. -/
-theorem wordAction_record (ω : List B) (t : W) (a : ZMod 2) :
-    wordAction cs ω ((cs.wordProd ω)⁻¹ * t * cs.wordProd ω, a) =
-      (t, a + ((cs.leftInvSeq ω).count t : ZMod 2)) := by
-  classical
-  induction ω using List.reverseRecOn generalizing a with
-  | nil => simp [wordAction]
-  | append_singleton ω i ih =>
-    rw [← List.concat_eq_append, wordAction_concat, Equiv.Perm.mul_apply,
-      cs.wordProd_concat, mul_inv_rev, cs.inv_simple, simpleAction_apply]
-    have hc : cs.simple i * ((cs.simple i * (cs.wordProd ω)⁻¹) * t *
-          (cs.wordProd ω * cs.simple i)) * cs.simple i =
-          (cs.wordProd ω)⁻¹ * t * cs.wordProd ω := by
-      simp [mul_assoc]
-    rw [hc, ih]
-    rw [cs.leftInvSeq_concat]
-    have heq : (cs.simple i * (cs.wordProd ω)⁻¹) * t *
-          (cs.wordProd ω * cs.simple i) = cs.simple i ↔
-          t = cs.wordProd ω * cs.simple i * (cs.wordProd ω)⁻¹ := by
-      constructor
-      · intro h
-        have := congrArg (fun z => cs.wordProd ω * cs.simple i * z *
-          cs.simple i * (cs.wordProd ω)⁻¹) h
-        simpa [mul_assoc] using this
-      · rintro rfl
-        simp [mul_assoc]
-    simp only [List.concat_eq_append, List.count_append, List.count_singleton,
-      Nat.cast_add, heq]
-    congr 1
-    split_ifs <;> simp_all [add_comm, add_left_comm]
-
-/-- The presentation relation has a doubled inversion sequence, so its
-signed action is trivial, including for no-relation entries `M i j = 0`. -/
-theorem leftInvSeq_alternating_formula (i j : B) (p : ℕ) :
-    cs.leftInvSeq (alternatingWord i j (2 * p)) =
-      (List.range (2 * p)).map (fun k => cs.simple i * (cs.simple j * cs.simple i) ^ k) := by
-  apply List.ext_getElem (by simp)
-  intro k hk hk'
-  have hk0 : k < 2 * p := by simpa using hk
-  rw [cs.getElem_leftInvSeq_alternatingWord i j p k hk0,
-    List.getElem_map, List.getElem_range]
-  rw [cs.prod_alternatingWord_eq_mul_pow]
-  simp only [Nat.not_even_iff_odd.mpr (show Odd (2 * k + 1) from ⟨k, rfl⟩),
-    ↓reduceIte]
-  congr 2
-  omega
-
-theorem leftInvSeq_relation_doubled (i j : B) :
-    ∃ l : List W, cs.leftInvSeq (alternatingWord i j (2 * M i j)) = l ++ l := by
-  let f : ℕ → W := fun k => cs.simple i * (cs.simple j * cs.simple i) ^ k
-  refine ⟨(List.range (M i j)).map f, ?_⟩
-  rw [leftInvSeq_alternating_formula, two_mul, List.range_add, List.map_append, List.map_map]
-  congr 1
-  apply List.map_congr_left
-  intro k _
-  dsimp [f]
-  rw [pow_add, cs.simple_mul_simple_pow']
-  simp
-
-theorem wordAction_alternating (i j : B) (p : ℕ) :
-    wordAction cs (alternatingWord i j (2 * p)) =
-      (simpleAction cs i * simpleAction cs j) ^ p := by
-  induction p with
-  | zero => simp [alternatingWord]
-  | succ p ih =>
-    rw [show 2 * (p + 1) = (2 * p + 1) + 1 by omega,
-      alternatingWord_succ', alternatingWord_succ']
-    simp only [Nat.not_even_iff_odd.mpr (show Odd (2 * p + 1) from ⟨p, rfl⟩),
-      ↓reduceIte, even_two, Even.mul_right]
-    change simpleAction cs i * (simpleAction cs j *
-      wordAction cs (alternatingWord i j (2 * p))) = _
-    rw [ih, ← mul_assoc, pow_succ']
-
-theorem simpleAction_liftable : M.IsLiftable (simpleAction cs) := by
-  intro i j
-  rw [← wordAction_alternating]
-  apply Equiv.ext
-  rintro ⟨t, a⟩
-  have h := wordAction_record cs (alternatingWord i j (2 * M i j)) t a
-  have hw : cs.wordProd (alternatingWord i j (2 * M i j)) = 1 := by
-    rw [cs.prod_alternatingWord_eq_mul_pow]
-    simp
-  obtain ⟨l, hl⟩ := leftInvSeq_relation_doubled cs i j
-  simpa [hw, hl, List.count_append, Nat.cast_add, ZModModule.add_self] using h
-
-noncomputable def reflectionAction : W →* Equiv.Perm (W × ZMod 2) :=
-  cs.lift ⟨simpleAction cs, simpleAction_liftable cs⟩
-
-@[simp] theorem reflectionAction_simple (i : B) :
-    reflectionAction cs (cs.simple i) = simpleAction cs i :=
-  cs.lift_apply_simple (simpleAction_liftable cs) i
-
-theorem reflectionAction_wordProd (ω : List B) :
-    reflectionAction cs (cs.wordProd ω) = wordAction cs ω := by
-  induction ω with
-  | nil => simp
-  | cons i ω ih =>
-    rw [cs.wordProd_cons, map_mul, reflectionAction_simple, ih]
-    rfl
-
-/-- The signed action is conjugation on labels and a translation on signs. -/
-theorem reflectionAction_translation (w t : W) :
-    ∃ δ : ZMod 2, ∀ a : ZMod 2,
-      reflectionAction cs w (t, a) = (w * t * w⁻¹, a + δ) := by
-  classical
-  obtain ⟨ω, rfl⟩ := cs.wordProd_surjective w
-  refine ⟨((cs.leftInvSeq ω).count (cs.wordProd ω * t * (cs.wordProd ω)⁻¹) : ZMod 2), ?_⟩
-  intro a
-  rw [reflectionAction_wordProd]
-  have h := wordAction_record cs ω (cs.wordProd ω * t * (cs.wordProd ω)⁻¹) a
-  simpa [mul_assoc] using h
-
-/-- A reflection changes the sign of its own label. This is the critical
-conjugation step making the representation yield *strong* exchange. -/
-theorem reflectionAction_reflection {t : W} (ht : cs.IsReflection t) (a : ZMod 2) :
-    reflectionAction cs t (t, a) = (t, a + 1) := by
-  obtain ⟨w, i, rfl⟩ := ht
-  obtain ⟨δ, hδ⟩ := reflectionAction_translation cs w⁻¹
-    (w * cs.simple i * w⁻¹)
-  obtain ⟨ε, hε⟩ := reflectionAction_translation cs w (cs.simple i)
-  have hδ' (a : ZMod 2) : reflectionAction cs w⁻¹
-      (w * cs.simple i * w⁻¹, a) = (cs.simple i, a + δ) := by
-    simpa [mul_assoc] using hδ a
-  have hz : δ + ε = 0 := by
-    have h := congrArg (fun p : W × ZMod 2 => p.2)
-      (Equiv.apply_symm_apply (reflectionAction cs w) (w * cs.simple i * w⁻¹, 0))
-    change ((reflectionAction cs w) ((reflectionAction cs w)⁻¹
-      (w * cs.simple i * w⁻¹, 0))).2 = 0 at h
-    rw [← map_inv, hδ', hε] at h
-    simpa using h
-  rw [map_mul, map_mul, reflectionAction_simple, Equiv.Perm.mul_apply,
-    Equiv.Perm.mul_apply, hδ', simpleAction_apply]
-  simp only [cs.simple_mul_simple_self, one_mul, ↓reduceIte]
-  rw [hε]
-  congr 1
-  calc
-    (a + δ + 1) + ε = (a + 1) + (δ + ε) := by ring
-    _ = a + 1 := by rw [hz, add_zero]
-
-
-theorem mem_leftInvSeq_of_inversion {ω : List B} (_hω : cs.IsReduced ω)
-    {t : W} (ht : cs.IsLeftInversion (cs.wordProd ω) t) :
-    t ∈ cs.leftInvSeq ω := by
-  classical
-  by_contra hnot
-  obtain ⟨ν, hν, heq⟩ := cs.exists_isReduced (t * cs.wordProd ω)
-  have hw := wordAction_record cs ω t (0 : ZMod 2)
-  rw [← reflectionAction_wordProd] at hw
-  simp only [List.count_eq_zero.mpr hnot, Nat.cast_zero, add_zero] at hw
-  have hv := wordAction_record cs ν t (0 : ZMod 2)
-  rw [← reflectionAction_wordProd, ← heq] at hv
-  have hin : (t * cs.wordProd ω)⁻¹ * t * (t * cs.wordProd ω) =
-      (cs.wordProd ω)⁻¹ * t * cs.wordProd ω := by
-    rw [mul_inv_rev, ht.1.inv]
-    simp [mul_assoc, ht.1.mul_self]
-  rw [hin, map_mul, Equiv.Perm.mul_apply, hw,
-    reflectionAction_reflection cs ht.1] at hv
-  have hmem : t ∈ cs.leftInvSeq ν := by
-    by_contra hnone
-    have h := congrArg Prod.snd hv
-    simp [List.count_eq_zero.mpr hnone] at h
-  have hlt := (cs.isLeftInversion_of_mem_leftInvSeq hν hmem).2
-  rw [← heq] at hlt
-  have hcancel : t * (t * cs.wordProd ω) = cs.wordProd ω := by
-    rw [← mul_assoc, ht.1.mul_self, one_mul]
-  rw [hcancel] at hlt
-  exact (not_lt_of_ge hlt.le) ht.2
-
-/-- Full strong exchange, with the deletion index in the supplied reduced word. -/
-theorem strong_exchange {ω : List B} (hω : cs.IsReduced ω) {t : W}
-    (ht : cs.IsLeftInversion (cs.wordProd ω) t) :
-    ∃ j : ℕ, j < ω.length ∧ t * cs.wordProd ω = cs.wordProd (ω.eraseIdx j) := by
-  obtain ⟨j, hj, heq⟩ := List.mem_iff_getElem.mp
-    (mem_leftInvSeq_of_inversion cs hω ht)
-  refine ⟨j, by simpa using hj, ?_⟩
-  rw [← heq, ← List.getD_eq_getElem _ 1 hj]
-  exact cs.getD_leftInvSeq_mul_wordProd ω j
-
-end KLInvariance.CoxeterSupport
-
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-!
+Signed reflection permutations and strong exchange for arbitrary Coxeter systems.
+-/
+
+section
+
+
+namespace KLInvariance.CoxeterSupport
+
+open CoxeterSystem
+
+universe u v
+variable {B : Type u} {W : Type v} [Group W] {M : CoxeterMatrix B}
+variable (cs : CoxeterSystem M W)
+
+noncomputable local instance : DecidableEq W := Classical.decEq W
+
+noncomputable def simpleAction (i : B) : Equiv.Perm (W × ZMod 2) := by
+  classical
+  let f : W × ZMod 2 → W × ZMod 2 := fun p =>
+    (cs.simple i * p.1 * cs.simple i, p.2 + if p.1 = cs.simple i then 1 else 0)
+  have hf : Function.Involutive f := by
+    rintro ⟨t, a⟩
+    dsimp [f]
+    have heq : cs.simple i * t * cs.simple i = cs.simple i ↔ t = cs.simple i := by
+      constructor
+      · intro h
+        have := congrArg (fun z => cs.simple i * z * cs.simple i) h
+        simpa [mul_assoc] using this
+      · rintro rfl
+        simp
+    simp only [heq]
+    ext
+    · simp [mul_assoc]
+    · split_ifs
+      · simp only [add_assoc, ZModModule.add_self, add_zero]
+      · simp
+  exact ⟨f, f, hf, hf⟩
+
+@[simp] theorem simpleAction_apply (i : B) (t : W) (a : ZMod 2) :
+    simpleAction cs i (t, a) =
+      (cs.simple i * t * cs.simple i, a + if t = cs.simple i then 1 else 0) := by
+  classical
+  rfl
+
+noncomputable def wordAction (ω : List B) : Equiv.Perm (W × ZMod 2) :=
+  (ω.map (simpleAction cs)).prod
+
+@[simp] theorem wordAction_nil : wordAction cs [] = 1 := rfl
+
+theorem wordAction_concat (ω : List B) (i : B) :
+    wordAction cs (ω.concat i) = wordAction cs ω * simpleAction cs i := by
+  simp [wordAction]
+
+/-- Signed action records the parity of the genuine left inversion sequence. -/
+theorem wordAction_record (ω : List B) (t : W) (a : ZMod 2) :
+    wordAction cs ω ((cs.wordProd ω)⁻¹ * t * cs.wordProd ω, a) =
+      (t, a + ((cs.leftInvSeq ω).count t : ZMod 2)) := by
+  classical
+  induction ω using List.reverseRecOn generalizing a with
+  | nil => simp [wordAction]
+  | append_singleton ω i ih =>
+    rw [← List.concat_eq_append, wordAction_concat, Equiv.Perm.mul_apply,
+      cs.wordProd_concat, mul_inv_rev, cs.inv_simple, simpleAction_apply]
+    have hc : cs.simple i * ((cs.simple i * (cs.wordProd ω)⁻¹) * t *
+          (cs.wordProd ω * cs.simple i)) * cs.simple i =
+          (cs.wordProd ω)⁻¹ * t * cs.wordProd ω := by
+      simp [mul_assoc]
+    rw [hc, ih]
+    rw [cs.leftInvSeq_concat]
+    have heq : (cs.simple i * (cs.wordProd ω)⁻¹) * t *
+          (cs.wordProd ω * cs.simple i) = cs.simple i ↔
+          t = cs.wordProd ω * cs.simple i * (cs.wordProd ω)⁻¹ := by
+      constructor
+      · intro h
+        have := congrArg (fun z => cs.wordProd ω * cs.simple i * z *
+          cs.simple i * (cs.wordProd ω)⁻¹) h
+        simpa [mul_assoc] using this
+      · rintro rfl
+        simp [mul_assoc]
+    simp only [List.concat_eq_append, List.count_append, List.count_singleton,
+      Nat.cast_add, heq]
+    congr 1
+    split_ifs <;> simp_all [add_comm, add_left_comm]
+
+/-- The presentation relation has a doubled inversion sequence, so its
+signed action is trivial, including for no-relation entries `M i j = 0`. -/
+theorem leftInvSeq_alternating_formula (i j : B) (p : ℕ) :
+    cs.leftInvSeq (alternatingWord i j (2 * p)) =
+      (List.range (2 * p)).map (fun k => cs.simple i * (cs.simple j * cs.simple i) ^ k) := by
+  apply List.ext_getElem (by simp)
+  intro k hk hk'
+  have hk0 : k < 2 * p := by simpa using hk
+  rw [cs.getElem_leftInvSeq_alternatingWord i j p k hk0,
+    List.getElem_map, List.getElem_range]
+  rw [cs.prod_alternatingWord_eq_mul_pow]
+  simp only [Nat.not_even_iff_odd.mpr (show Odd (2 * k + 1) from ⟨k, rfl⟩),
+    ↓reduceIte]
+  congr 2
+  omega
+
+theorem leftInvSeq_relation_doubled (i j : B) :
+    ∃ l : List W, cs.leftInvSeq (alternatingWord i j (2 * M i j)) = l ++ l := by
+  let f : ℕ → W := fun k => cs.simple i * (cs.simple j * cs.simple i) ^ k
+  refine ⟨(List.range (M i j)).map f, ?_⟩
+  rw [leftInvSeq_alternating_formula, two_mul, List.range_add, List.map_append, List.map_map]
+  congr 1
+  apply List.map_congr_left
+  intro k _
+  dsimp [f]
+  rw [pow_add, cs.simple_mul_simple_pow']
+  simp
+
+theorem wordAction_alternating (i j : B) (p : ℕ) :
+    wordAction cs (alternatingWord i j (2 * p)) =
+      (simpleAction cs i * simpleAction cs j) ^ p := by
+  induction p with
+  | zero => simp [alternatingWord]
+  | succ p ih =>
+    rw [show 2 * (p + 1) = (2 * p + 1) + 1 by omega,
+      alternatingWord_succ', alternatingWord_succ']
+    simp only [Nat.not_even_iff_odd.mpr (show Odd (2 * p + 1) from ⟨p, rfl⟩),
+      ↓reduceIte, even_two, Even.mul_right]
+    change simpleAction cs i * (simpleAction cs j *
+      wordAction cs (alternatingWord i j (2 * p))) = _
+    rw [ih, ← mul_assoc, pow_succ']
+
+theorem simpleAction_liftable : M.IsLiftable (simpleAction cs) := by
+  intro i j
+  rw [← wordAction_alternating]
+  apply Equiv.ext
+  rintro ⟨t, a⟩
+  have h := wordAction_record cs (alternatingWord i j (2 * M i j)) t a
+  have hw : cs.wordProd (alternatingWord i j (2 * M i j)) = 1 := by
+    rw [cs.prod_alternatingWord_eq_mul_pow]
+    simp
+  obtain ⟨l, hl⟩ := leftInvSeq_relation_doubled cs i j
+  simpa [hw, hl, List.count_append, Nat.cast_add, ZModModule.add_self] using h
+
+noncomputable def reflectionAction : W →* Equiv.Perm (W × ZMod 2) :=
+  cs.lift ⟨simpleAction cs, simpleAction_liftable cs⟩
+
+@[simp] theorem reflectionAction_simple (i : B) :
+    reflectionAction cs (cs.simple i) = simpleAction cs i :=
+  cs.lift_apply_simple (simpleAction_liftable cs) i
+
+theorem reflectionAction_wordProd (ω : List B) :
+    reflectionAction cs (cs.wordProd ω) = wordAction cs ω := by
+  induction ω with
+  | nil => simp
+  | cons i ω ih =>
+    rw [cs.wordProd_cons, map_mul, reflectionAction_simple, ih]
+    rfl
+
+/-- The signed action is conjugation on labels and a translation on signs. -/
+theorem reflectionAction_translation (w t : W) :
+    ∃ δ : ZMod 2, ∀ a : ZMod 2,
+      reflectionAction cs w (t, a) = (w * t * w⁻¹, a + δ) := by
+  classical
+  obtain ⟨ω, rfl⟩ := cs.wordProd_surjective w
+  refine ⟨((cs.leftInvSeq ω).count (cs.wordProd ω * t * (cs.wordProd ω)⁻¹) : ZMod 2), ?_⟩
+  intro a
+  rw [reflectionAction_wordProd]
+  have h := wordAction_record cs ω (cs.wordProd ω * t * (cs.wordProd ω)⁻¹) a
+  simpa [mul_assoc] using h
+
+/-- A reflection changes the sign of its own label. This is the critical
+conjugation step making the representation yield *strong* exchange. -/
+theorem reflectionAction_reflection {t : W} (ht : cs.IsReflection t) (a : ZMod 2) :
+    reflectionAction cs t (t, a) = (t, a + 1) := by
+  obtain ⟨w, i, rfl⟩ := ht
+  obtain ⟨δ, hδ⟩ := reflectionAction_translation cs w⁻¹
+    (w * cs.simple i * w⁻¹)
+  obtain ⟨ε, hε⟩ := reflectionAction_translation cs w (cs.simple i)
+  have hδ' (a : ZMod 2) : reflectionAction cs w⁻¹
+      (w * cs.simple i * w⁻¹, a) = (cs.simple i, a + δ) := by
+    simpa [mul_assoc] using hδ a
+  have hz : δ + ε = 0 := by
+    have h := congrArg (fun p : W × ZMod 2 => p.2)
+      (Equiv.apply_symm_apply (reflectionAction cs w) (w * cs.simple i * w⁻¹, 0))
+    change ((reflectionAction cs w) ((reflectionAction cs w)⁻¹
+      (w * cs.simple i * w⁻¹, 0))).2 = 0 at h
+    rw [← map_inv, hδ', hε] at h
+    simpa using h
+  rw [map_mul, map_mul, reflectionAction_simple, Equiv.Perm.mul_apply,
+    Equiv.Perm.mul_apply, hδ', simpleAction_apply]
+  simp only [cs.simple_mul_simple_self, one_mul, ↓reduceIte]
+  rw [hε]
+  congr 1
+  calc
+    (a + δ + 1) + ε = (a + 1) + (δ + ε) := by ring
+    _ = a + 1 := by rw [hz, add_zero]
+
+
+theorem mem_leftInvSeq_of_inversion {ω : List B} (_hω : cs.IsReduced ω)
+    {t : W} (ht : cs.IsLeftInversion (cs.wordProd ω) t) :
+    t ∈ cs.leftInvSeq ω := by
+  classical
+  by_contra hnot
+  obtain ⟨ν, hν, heq⟩ := cs.exists_isReduced (t * cs.wordProd ω)
+  have hw := wordAction_record cs ω t (0 : ZMod 2)
+  rw [← reflectionAction_wordProd] at hw
+  simp only [List.count_eq_zero.mpr hnot, Nat.cast_zero, add_zero] at hw
+  have hv := wordAction_record cs ν t (0 : ZMod 2)
+  rw [← reflectionAction_wordProd, ← heq] at hv
+  have hin : (t * cs.wordProd ω)⁻¹ * t * (t * cs.wordProd ω) =
+      (cs.wordProd ω)⁻¹ * t * cs.wordProd ω := by
+    rw [mul_inv_rev, ht.1.inv]
+    simp [mul_assoc, ht.1.mul_self]
+  rw [hin, map_mul, Equiv.Perm.mul_apply, hw,
+    reflectionAction_reflection cs ht.1] at hv
+  have hmem : t ∈ cs.leftInvSeq ν := by
+    by_contra hnone
+    have h := congrArg Prod.snd hv
+    simp [List.count_eq_zero.mpr hnone] at h
+  have hlt := (cs.isLeftInversion_of_mem_leftInvSeq hν hmem).2
+  rw [← heq] at hlt
+  have hcancel : t * (t * cs.wordProd ω) = cs.wordProd ω := by
+    rw [← mul_assoc, ht.1.mul_self, one_mul]
+  rw [hcancel] at hlt
+  exact (not_lt_of_ge hlt.le) ht.2
+
+/-- Full strong exchange, with the deletion index in the supplied reduced word. -/
+theorem strong_exchange {ω : List B} (hω : cs.IsReduced ω) {t : W}
+    (ht : cs.IsLeftInversion (cs.wordProd ω) t) :
+    ∃ j : ℕ, j < ω.length ∧ t * cs.wordProd ω = cs.wordProd (ω.eraseIdx j) := by
+  obtain ⟨j, hj, heq⟩ := List.mem_iff_getElem.mp
+    (mem_leftInvSeq_of_inversion cs hω ht)
+  refine ⟨j, by simpa using hj, ?_⟩
+  rw [← heq, ← List.getD_eq_getElem _ 1 hj]
+  exact cs.getD_leftInvSeq_mul_wordProd ω j
+
+end KLInvariance.CoxeterSupport
+
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/KazhdanLusztig/Decomposition.lean b/lean/OAI/RepresentationTheory/KazhdanLusztig/Decomposition.lean
index a69ad79d112dd911ec81590ae5f5b35470617216..c2dda4b37cefe2177a31347e0d4e80c91946cf91 100644
--- a/lean/OAI/RepresentationTheory/KazhdanLusztig/Decomposition.lean
+++ b/lean/OAI/RepresentationTheory/KazhdanLusztig/Decomposition.lean
@@ -1,1709 +1,1709 @@
-import OAI.RepresentationTheory.KazhdanLusztig.SheafMorphisms
-
-namespace OAI
-
-/-!
-Graded decomposition characters, reflection-plane algebras and reflection Frobenius structures.
--/
-
-section
-
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded Polynomial
-universe uk us um
-variable {k : Type uk} [Field k] {σ : Type us} [Finite σ]
-  {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
-noncomputable section
-
- def freeStalkChar (B : GradedSheaf (𝓐 := polynomialGrading k σ) G)
-    (hf : ∀ x, Module.Free (MvPolynomial σ k) (B.vertex x)) (x : V) : Polynomial ℤ := by
-  let := hf x
-  exact (B.vertex x).char
-
- theorem ModelSum.character {α : E → MvPolynomial σ k}
-    (O : BoundaryModels (polynomialGrading_negative k σ) (G := G) α)
-    [Fintype V] {U : Set V} {B : GradedSheaf (𝓐 := polynomialGrading k σ) G}
-    (h : ModelSum O.sheaf U B) :
-    ∃ hmul : V → Polynomial ℤ,
-      (∀ y n, 0 ≤ (hmul y).coeff n) ∧
-      (∀ y, y ∉ U → hmul y=0) ∧
-      ∀ x, freeStalkChar B (h.free O) x =
-        ∑ y, hmul y*freeStalkChar (O.sheaf y) (O.free y) x := by
-  classical
-  induction h with
-  | zero =>
-    refine ⟨0,by simp,by simp,?_⟩
-    intro x
-    simp only [Pi.zero_apply,zero_mul,Finset.sum_const_zero]
-    exact ModuleData.char_zero _
-  | @add B h c hc n d ih =>
-    obtain ⟨m,hm,hsupp,heq⟩ := ih
-    let a : Polynomial ℤ := ∑ i, X^(d i)
-    have ha (r : ℕ) : 0 ≤ a.coeff r := by
-      simp only [a,Polynomial.finsetSum_coeff,Polynomial.coeff_X_pow]
-      exact Finset.sum_nonneg (fun i _ => by split <;> norm_num)
-    refine ⟨fun y => m y+if y=c then a else 0,?_,?_,?_⟩
-    · intro y r
-      rw [Polynomial.coeff_add]
-      apply add_nonneg (hm y r)
-      split
-      · exact ha r
-      · simp
-    · intro y hy
-      have hne : y≠c := fun hh => hy (hh ▸ hc)
-      simp [hsupp y hy,hne]
-    · intro x
-      let _ := h.free O x
-      let _ := O.free c x
-      change ((B.vertex x).prod (ModuleData.pi (fun i => ((O.sheaf c).vertex x).shift (d i)))).char = _
-      rw [ModuleData.char_prod,ModuleData.char_pi]
-      simp only [ModuleData.char_shift]
-      change freeStalkChar B (h.free O) x + _ = _
-      rw [heq x]
-      simp only [add_mul,Finset.sum_add_distrib]
-      congr 1
-      rw [← Finset.sum_mul]
-      simp only [ite_mul,zero_mul,Finset.sum_ite_eq',Finset.mem_univ,ite_true]
-      rfl
-
- theorem exists_stalkCharacter_decomposition [Fintype V] [Fintype E]
-    {α : E → MvPolynomial σ k}
-    (O : BoundaryModels (polynomialGrading_negative k σ) (G := G) α)
-    (B : GradedSheaf (𝓐 := polynomialGrading k σ) G)
-    (hf : ∀ x, Module.Free (MvPolynomial σ k) (B.vertex x))
-    (hq : B.UpperQuotient α) (hfl : B.forget.Flabby) (hg : B.forget.Generated) :
-    ∃ hmul : V → Polynomial ℤ, (∀ y n, 0 ≤ (hmul y).coeff n) ∧
-      ∀ x, freeStalkChar B hf x =
-        ∑ y, hmul y*freeStalkChar (O.sheaf y) (O.free y) x := by
-  obtain ⟨C,hC,⟨j⟩⟩ := exists_modelSumIso (polynomialGrading_zero k σ) O B hf hq hfl hg
-  obtain ⟨m,hm,_,heq⟩ := hC.character O
-  refine ⟨m,hm,fun x => ?_⟩
-  rw [← heq x]
-  let _ := hf x
-  let _ := hC.free O x
-  exact ModuleData.character_eq_of_gradedEquiv (j.vertex x)
-    (j.vertex_graded x)
-
-end
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph Polynomial
-universe u
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W) (b : W)
-  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
- def lowerCharacter (c x : Interval cs 1 b) : Polynomial ℤ :=
-  if h : x≤c then stalkCharacter cs 1 c.val basis (one_bruhat cs c.val)
-    ⟨x.val,x.property.1,h⟩ else 0
-
- theorem boundaryModels_character (c x : Interval cs 1 b) :
-    GradedSheaf.freeStalkChar ((boundaryModels cs b basis).sheaf c)
-      ((boundaryModels cs b basis).free c) x=lowerCharacter cs b basis c x := by
-  let _ := (boundaryModels cs b basis).free c x
-  by_cases hx : x≤c
-  · rw [lowerCharacter,dite_eq_left hx]
-    let B := lowerIntervalBoundary cs b basis c
-    let _ := B.sheaf.lowerExtend_free B.free x
-    let _ := (boundarySheaf cs 1 c.val basis (one_bruhat cs c.val)).free ⟨x.val,x.property.1,hx⟩
-    exact ModuleData.character_eq_of_gradedEquiv (B.sheaf.lowerExtendVertex ⟨x,hx⟩)
-      (B.sheaf.lowerExtendVertex_graded ⟨x,hx⟩)
-  · rw [lowerCharacter,dite_eq_right hx]
-    let _ := (boundaryModels cs b basis).support c x hx
-    exact ModuleData.char_zero _
-
- theorem lowerCharacter_zero (c x : Interval cs 1 b) (hx : ¬x≤c) :
-    lowerCharacter cs b basis c x=0 := dite_eq_right hx
-
- theorem lowerCharacter_nonneg (c x : Interval cs 1 b) (n : ℕ) :
-    0≤(lowerCharacter cs b basis c x).coeff n := by
-  unfold lowerCharacter
-  split
-  · exact stalkCharacter_coeff_nonneg cs 1 c.val basis (one_bruhat cs c.val) _ n
-  · simp
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-/-! Integral pair-projection quotients. The purpose is the actual localization
-functor from a tensor bimodule to a moment-graph sheaf; no flabbiness, stalk
-freeness, BMP identification or character theorem is assumed or asserted. -/
-namespace KLInvariance.PairProjection
-universe u v w z
-variable {R : Type u} [CommRing R]
-  {M : Type v} [AddCommGroup M] [Module R M]
-  {X : Type w} [AddCommGroup X] [Module R X]
-  {Y : Type z} [AddCommGroup Y] [Module R Y]
-  (f : M →ₗ[R] X) (g : M →ₗ[R] Y)
-noncomputable section
-
-def relation : Submodule R (X × Y) := LinearMap.range (f.prod g)
-abbrev Edge := (X × Y) ⧸ relation f g
-
-def lower : X →ₗ[R] Edge f g := (relation f g).mkQ.comp (LinearMap.inl R X Y)
-def upper : Y →ₗ[R] Edge f g := -((relation f g).mkQ.comp (LinearMap.inr R X Y))
-
-theorem diagonal_zero (m : M) : (relation f g).mkQ (f m,g m)=0 := by
-  exact (Submodule.Quotient.mk_eq_zero _).mpr ⟨m,rfl⟩
-
-theorem compatible (m : M) : lower f g (f m)=upper f g (g m) := by
-  have h := diagonal_zero f g m
-  have hp : (f m,g m)=(f m,0)+(0,g m) := by simp
-  rw [hp,map_add] at h
-  exact eq_neg_of_add_eq_zero_left h
-
-theorem compatible_iff (x : X) (y : Y) :
-    lower f g x=upper f g y ↔ ∃ m : M, f m=x ∧ g m=y := by
-  change (relation f g).mkQ (x,0) = -((relation f g).mkQ (0,y)) ↔ _
-  rw [eq_neg_iff_add_eq_zero,← map_add]
-  rw [show ((x,0)+(0,y) : X × Y)=(x,y) by simp]
-  change (Submodule.Quotient.mk (x,y) : Edge f g)=0 ↔ _
-  rw [Submodule.Quotient.mk_eq_zero]
-  constructor
-  · rintro ⟨m,hm⟩
-    exact ⟨m,congrArg Prod.fst hm,congrArg Prod.snd hm⟩
-  · rintro ⟨m,hm,hn⟩
-    exact ⟨m,Prod.ext hm hn⟩
-
-theorem kernel_lower (x : X) : lower f g x=0 ↔ ∃ m : M, f m=x ∧ g m=0 := by
-  change (relation f g).mkQ (x,0)=0 ↔ _
-  change Submodule.Quotient.mk (x,0) = 0 ↔ _
-  rw [Submodule.Quotient.mk_eq_zero]
-  constructor
-  · rintro ⟨m,hm⟩
-    exact ⟨m,congrArg Prod.fst hm,congrArg Prod.snd hm⟩
-  · rintro ⟨m,hm,hn⟩
-    exact ⟨m,Prod.ext hm hn⟩
-
-theorem lower_surjective (hg : Function.Surjective g) :
-    Function.Surjective (lower f g) := by
-  intro v
-  obtain ⟨⟨x,y⟩,rfl⟩ := (relation f g).mkQ_surjective v
-  obtain ⟨m,rfl⟩ := hg y
-  refine ⟨x-f m,?_⟩
-  change (relation f g).mkQ (x-f m,0) = (relation f g).mkQ (x,g m)
-  have hp : (x-f m,0)=(x,g m)-(f m,g m) := by simp
-  rw [hp,map_sub,diagonal_zero,sub_zero]
-
-theorem upper_surjective (hf : Function.Surjective f) :
-    Function.Surjective (upper f g) := by
-  intro v
-  obtain ⟨⟨x,y⟩,rfl⟩ := (relation f g).mkQ_surjective v
-  obtain ⟨m,rfl⟩ := hf x
-  refine ⟨g m-y,?_⟩
-  change -((relation f g).mkQ (0,g m-y)) = (relation f g).mkQ (f m,y)
-  have hp : (0,g m-y)=(f m,g m)-(f m,y) := by simp
-  rw [hp,map_sub,diagonal_zero,zero_sub,neg_neg]
-
-/-- A genuine right-action eigenvalue difference annihilates the edge
-quotient. This uses only the integral module, not a fraction-field equality. -/
-theorem difference_annihilates (hf : Function.Surjective f)
-    (hg : Function.Surjective g) (T : M →ₗ[R] M) (r s : R)
-    (hr : ∀ m, f (T m)=r • f m) (hs : ∀ m, g (T m)=s • g m)
-    (v : Edge f g) : (r-s) • v=0 := by
-  obtain ⟨x,rfl⟩ := lower_surjective f g hg v
-  rw [← map_smul]
-  apply (kernel_lower f g _).mpr
-  obtain ⟨m,rfl⟩ := hf x
-  refine ⟨T m-s • m,?_,?_⟩
-  · rw [map_sub,map_smul,hr,sub_smul]
-  · rw [map_sub,map_smul,hs,sub_self]
-
-end
-end KLInvariance.PairProjection
-
-end
-
-
-section
-
-
-namespace KLInvariance.Dihedral
-noncomputable section
-
-variable {G : Type*} [Group G]
-
-/-- The matrix of a pair of distinct involutions. -/
-def pairMatrix (s t : G) (hs : s * s = 1) (hst : s ≠ t) : CoxeterMatrix Bool where
-  M i j := if i = j then 1 else orderOf (s * t)
-  isSymm := by
-    ext i j
-    change (if j = i then 1 else orderOf (s*t)) = if i = j then 1 else orderOf (s*t)
-    cases i <;> cases j <;> rfl
-  diagonal i := ite_eq_left rfl
-  off_diagonal i j hij := by
-    rw [ite_eq_right hij]
-    intro h
-    have he := orderOf_eq_one_iff.mp h
-    apply hst
-    calc
-      s = s * (s * t) := by rw [he, mul_one]
-      _ = t := by rw [← mul_assoc, hs, one_mul]
-
-/-- Alternating even words are powers of the product of the two generators. -/
-theorem wordProd_alt_even {N : CoxeterMatrix Bool} {H : Type*} [Group H]
-    (cs : CoxeterSystem N H) (i : Bool) (n : ℕ) :
-    cs.wordProd (alt i (2 * n)) = (cs.simple i * cs.simple (!i)) ^ n := by
-  induction n with
-  | zero => simp
-  | succ n ih =>
-    rw [show 2 * (n + 1) = 2 * n + 2 by omega, alt_add_two]
-    rw [cs.wordProd_cons, cs.wordProd_cons, ih, pow_succ']
-    exact (mul_assoc _ _ _).symm
-
-/-- Homomorphisms out of a rank-two Coxeter group that do not lower
-its product order and do not kill either generator are injective. -/
-theorem injective_of_simple_order {N : CoxeterMatrix Bool} {H : Type*} [Group H]
-    (cs : CoxeterSystem N H) (f : H →* G)
-    (hne : ∀ i, f (cs.simple i) ≠ 1)
-    (horder : ∀ i, orderOf (f (cs.simple i) * f (cs.simple (!i))) = N i (!i)) :
-    Function.Injective f := by
-  apply (injective_iff_map_eq_one f).mpr
-  intro w hw
-  rcases Nat.even_or_odd (cs.length w) with he | ho
-  · obtain ⟨n, hn⟩ := he
-    have hn' : cs.length w = 2 * n := by omega
-    obtain ⟨i, hi⟩ := exists_alt cs w
-    rw [hn', wordProd_alt_even] at hi
-    have hpow : (f (cs.simple i) * f (cs.simple (!i))) ^ n = 1 := by
-      simpa only [hi, map_pow, map_mul] using hw
-    have hdiv := orderOf_dvd_of_pow_eq_one hpow
-    rw [horder i] at hdiv
-    obtain ⟨k, hk⟩ := hdiv
-    rw [hi, hk, pow_mul, cs.simple_mul_simple_pow, one_pow]
-  · obtain ⟨a, i, ha⟩ := (reflection_iff_odd cs w).mpr ho
-    have h := hw
-    rw [ha, map_mul, map_mul, map_inv] at h
-    apply False.elim
-    apply hne i
-    have h' := congrArg (fun z => (f a)⁻¹ * z * f a) h
-    simpa only [mul_assoc, inv_mul_cancel_left, inv_mul_cancel, mul_one,
-      one_mul, mul_inv_cancel_left] using h'
-
-variable (s t : G) (hs : s * s = 1) (ht : t * t = 1) (hst : s ≠ t)
-
-/-- The map from the genuine presented rank-two Coxeter group. -/
-def pairLift : (pairMatrix s t hs hst).Group →* G :=
-  (pairMatrix s t hs hst).toCoxeterSystem.lift
-    ⟨(fun i => if i then t else s), by
-      intro i j
-      cases i <;> cases j
-      · simpa [pairMatrix] using hs
-      · simp only [pairMatrix, Bool.false_eq_true, ↓reduceIte]
-        exact pow_orderOf_eq_one (s*t)
-      · simp only [pairMatrix, Bool.true_eq_false, Bool.false_eq_true, ↓reduceIte]
-        have hinv : (s*t)⁻¹ = t*s := by
-          rw [mul_inv_rev, (inv_eq_of_mul_eq_one_right ht), (inv_eq_of_mul_eq_one_right hs)]
-        rw [← hinv, inv_pow, pow_orderOf_eq_one, inv_one]
-      · simpa [pairMatrix] using ht⟩
-
-@[simp] theorem pairLift_simple (i : Bool) :
-    pairLift s t hs ht hst ((pairMatrix s t hs hst).toCoxeterSystem.simple i) =
-      if i then t else s := by
-  apply CoxeterSystem.lift_apply_simple
-
-/-- No relation other than the Coxeter relations is introduced by the pair. -/
-theorem pairLift_injective (hs1 : s ≠ 1) (ht1 : t ≠ 1) :
-    Function.Injective (pairLift s t hs ht hst) := by
-  apply injective_of_simple_order (pairMatrix s t hs hst).toCoxeterSystem
-  · intro i
-    rw [pairLift_simple]
-    cases i <;> simp_all
-  · intro i
-    simp only [pairLift_simple]
-    cases i
-    · simp [pairMatrix]
-    · simp only [Bool.not_true, ↓reduceIte, pairMatrix, Bool.true_eq_false, Bool.false_eq_true]
-      have hinv : (s*t)⁻¹ = t*s := by
-        rw [mul_inv_rev, (inv_eq_of_mul_eq_one_right ht), (inv_eq_of_mul_eq_one_right hs)]
-      rw [← hinv, orderOf_inv]
-
-/-- The presented pair generates exactly the subgroup generated by its images. -/
-theorem pairLift_range :
-    (pairLift s t hs ht hst).range = Subgroup.closure {s, t} := by
-  apply le_antisymm
-  · rintro x ⟨w, rfl⟩
-    apply (pairMatrix s t hs hst).toCoxeterSystem.simple_induction w
-    · intro i
-      rw [pairLift_simple]
-      cases i <;> apply Subgroup.subset_closure <;> simp
-    · simp only [map_one, Subgroup.one_mem]
-    · intro a b ha hb
-      rw [map_mul]
-      exact Subgroup.mul_mem _ ha hb
-  · apply (Subgroup.closure_le _).mpr
-    intro x hx
-    rcases Set.mem_insert_iff.mp hx with hx | hx
-    · rw [hx]
-      exact ⟨(pairMatrix s t hs hst).toCoxeterSystem.simple false, pairLift_simple _ _ _ _ _ _⟩
-    · have : x = t := Set.mem_singleton_iff.mp hx
-      subst x
-      exact ⟨(pairMatrix s t hs hst).toCoxeterSystem.simple true, pairLift_simple _ _ _ _ _ _⟩
-
-/-- An actual Coxeter system on the generated subgroup, not an extra assumption. -/
-def pairSystem (hs1 : s ≠ 1) (ht1 : t ≠ 1) :
-    CoxeterSystem (pairMatrix s t hs hst) (Subgroup.closure {s, t}) :=
-  (pairMatrix s t hs hst).toCoxeterSystem.map
-    ((MulEquiv.ofBijective (pairLift s t hs ht hst).rangeRestrict
-      ⟨MonoidHom.rangeRestrict_injective_iff.mpr
-        (pairLift_injective s t hs ht hst hs1 ht1),
-       (pairLift s t hs ht hst).rangeRestrict_surjective⟩).trans
-      (MulEquiv.subgroupCongr (pairLift_range s t hs ht hst)))
-
-@[simp] theorem pairSystem_simple (hs1 : s ≠ 1) (ht1 : t ≠ 1) (i : Bool) :
-    ((pairSystem s t hs ht hst hs1 ht1).simple i : G) = if i then t else s := by
-  exact pairLift_simple s t hs ht hst i
-
-end
-end KLInvariance.Dihedral
-
-
-namespace KLInvariance.TitsSpace
-open Module
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-noncomputable section
-
-/-- Any two distinct positive roots in a two-dimensional root plane span it. -/
-theorem rootPlane_eq_of_mem (a b l r : PositiveRoot M cs) (hab : a ≠ b)
-    (hl : l.val ∈ rootPlane a b) (hr : r.val ∈ rootPlane a b) (hlr : l ≠ r) :
-    rootPlane l r = rootPlane a b := by
-  apply Submodule.eq_of_le_of_finrank_eq
-  · apply Submodule.span_le.mpr
-    intro z hz
-    change z ∈ rootPlane a b
-    rcases Set.mem_insert_iff.mp hz with hz | hz
-    · rw [hz]; exact hl
-    · rw [Set.mem_singleton_iff.mp hz]; exact hr
-  · rw [rootPlane_finrank _ _ hlr, rootPlane_finrank _ _ hab]
-
-/-- Slope betweenness detects the actual positive cone, not only one implication. -/
-theorem positive_combination_of_slope_between (a b c : PositiveRoot M cs)
-    (hc : c.val ∈ rootPlane a b)
-    (hac : geometricSlope a < geometricSlope c)
-    (hcb : geometricSlope c < geometricSlope b) :
-    ∃ r s : ℝ, 0 < r ∧ 0 < s ∧ c.val = r • a.val + s • b.val := by
-  let f := geometricFunctional (M := M) (cs := cs)
-  obtain ⟨r, s, he⟩ := Submodule.mem_span_pair.mp hc
-  refine ⟨r, s, ?_, ?_, he.symm⟩
-  all_goals
-    have ha := a.height_pos
-    have hb := b.height_pos
-    have hh := c.height_pos
-    have hfc := congrArg f he
-    have hhc := congrArg (height (I := I)) he
-    simp only [map_add, map_smul, smul_eq_mul] at hfc hhc
-    change f a.val / height a.val <
-      f c.val / height c.val at hac
-    change f c.val / height c.val <
-      f b.val / height b.val at hcb
-    have hab := lt_trans hac hcb
-    rw [div_lt_div_iff₀ ha hb] at hab
-    rw [div_lt_div_iff₀ ha hh] at hac
-    rw [div_lt_div_iff₀ hh hb] at hcb
-  · have heq : r * (f b.val * height a.val -
-        f a.val * height b.val) =
-        f b.val * height c.val -
-          f c.val * height b.val := by rw [← hfc, ← hhc]; ring
-    have hp : 0 < r * (f b.val * height a.val -
-        f a.val * height b.val) := by rw [heq]; linarith
-    exact (mul_pos_iff.mp hp).resolve_right (by intro h; linarith [h.2]) |>.1
-  · have heq : s * (f b.val * height a.val -
-        f a.val * height b.val) =
-        f c.val * height a.val -
-          f a.val * height c.val := by rw [← hfc, ← hhc]; ring
-    have hp : 0 < s * (f b.val * height a.val -
-        f a.val * height b.val) := by rw [heq]; linarith
-    exact (mul_pos_iff.mp hp).resolve_right (by intro h; linarith [h.2]) |>.1
-
-/-- Positivity is convex in the root cone. -/
-theorem noninverted_of_positive_combination (w : W) (a b c : PositiveRoot M cs)
-    {r s : ℝ} (hr : 0 ≤ r) (hs : 0 ≤ s)
-    (hc : c.val = r • a.val + s • b.val)
-    (ha : ¬ rootInverted w a) (hb : ¬ rootInverted w b) :
-    ¬ rootInverted w c := by
-  rw [not_rootInverted_iff] at ha hb ⊢
-  intro i
-  rw [hc, map_add, map_smul, map_smul]
-  exact add_nonneg (mul_nonneg hr (ha i)) (mul_nonneg hs (hb i))
-
-@[simp] theorem not_rootInverted_one (a : PositiveRoot M cs) :
-    ¬ rootInverted (1 : W) a := by
-  rw [not_rootInverted_iff]
-  simp only [inv_one, map_one]
-  exact a.property.2
-
-@[simp] theorem rootInverted_ref_self (a : PositiveRoot M cs) :
-    rootInverted (rootRef a) a := by
-  simpa only [mul_one] using (rootInverted_mul_self 1 a).mpr (not_rootInverted_one a)
-
-/-- A root plane has an extreme positive root. Finiteness of inversion sets
-replaces the false assertion that the (real-valued) root heights are discrete. -/
-theorem exists_plane_extreme_inverted (a b : PositiveRoot M cs) (hab : a ≠ b) :
-    ∃ p : PositiveRoot M cs, p.val ∈ rootPlane a b ∧ rootInverted (rootRef a) p ∧
-      ((∀ r : PositiveRoot M cs, r.val ∈ rootPlane a b →
-          geometricSlope p ≤ geometricSlope r) ∨
-       (∀ r : PositiveRoot M cs, r.val ∈ rootPlane a b →
-          geometricSlope r ≤ geometricSlope p)) := by
-  classical
-  let S := (finite_rootInverted (M := M) (cs := cs) (rootRef a)).toFinset.filter
-    (fun r => r.val ∈ rootPlane a b)
-  have hS (r : PositiveRoot M cs) : r ∈ S ↔
-      rootInverted (rootRef a) r ∧ r.val ∈ rootPlane a b := by
-    simp only [S, Finset.mem_filter, Set.Finite.mem_toFinset, Set.mem_ofPred_eq]
-  have haP : a.val ∈ rootPlane a b := Submodule.subset_span (Set.mem_insert _ _)
-  have haS : a ∈ S := (hS a).mpr ⟨rootInverted_ref_self a, haP⟩
-  obtain ⟨p, hp, hmin⟩ := S.exists_min_image geometricSlope ⟨a, haS⟩
-  obtain ⟨q, hq, hmax⟩ := S.exists_max_image geometricSlope ⟨a, haS⟩
-  by_cases hleft : ∀ r : PositiveRoot M cs, r.val ∈ rootPlane a b →
-      geometricSlope p ≤ geometricSlope r
-  · exact ⟨p, (hS p).mp hp |>.2, (hS p).mp hp |>.1, Or.inl hleft⟩
-  by_cases hright : ∀ r : PositiveRoot M cs, r.val ∈ rootPlane a b →
-      geometricSlope r ≤ geometricSlope q
-  · exact ⟨q, (hS q).mp hq |>.2, (hS q).mp hq |>.1, Or.inr hright⟩
-  push Not at hleft hright
-  obtain ⟨l, hlP, hlp⟩ := hleft
-  obtain ⟨r, hrP, hqr⟩ := hright
-  have hla := lt_of_lt_of_le hlp (hmin a haS)
-  have har := lt_of_le_of_lt (hmax a haS) hqr
-  have hlr : l ≠ r := fun h => (lt_trans hla har).ne (congrArg geometricSlope h)
-  have hplane := rootPlane_eq_of_mem a b l r hab hlP hrP hlr
-  have hlam : ¬ rootInverted (rootRef a) l := by
-    intro hi
-    exact (not_le_of_gt hlp) (hmin l ((hS l).mpr ⟨hi, hlP⟩))
-  have hram : ¬ rootInverted (rootRef a) r := by
-    intro hi
-    exact (not_le_of_gt hqr) (hmax r ((hS r).mpr ⟨hi, hrP⟩))
-  obtain ⟨c, d, hc, hd, he⟩ := positive_combination_of_slope_between l r a
-    (hplane.symm ▸ haP) hla har
-  exact False.elim ((noninverted_of_positive_combination (rootRef a) l r a
-    hc.le hd.le he hlam hram) (rootInverted_ref_self a))
-
-/-- The canonical simple-root criterion stated on the actual ambient roots. -/
-def PlaneSimple (P : Submodule ℝ (I → ℝ)) (a : PositiveRoot M cs) : Prop :=
-  a.val ∈ P ∧ ∀ b : PositiveRoot M cs, b.val ∈ P →
-    (rootInverted (rootRef a) b ↔ b = a)
-
-theorem rootInverted_ref_iff_flipped (a b : PositiveRoot M cs) :
-    rootInverted (rootRef a) b ↔ flipped a b := by
-  unfold rootInverted flipped
-  rw [(rootReflection M cs a).property.inv]
-
-/-- An angular extreme inverts exactly its own positive root. -/
-theorem planeSimple_of_extreme {P : Submodule ℝ (I → ℝ)}
-    (a : PositiveRoot M cs) (ha : a.val ∈ P)
-    (hext : (∀ r : PositiveRoot M cs, r.val ∈ P → geometricSlope a ≤ geometricSlope r) ∨
-      (∀ r : PositiveRoot M cs, r.val ∈ P → geometricSlope r ≤ geometricSlope a)) :
-    PlaneSimple P a := by
-  refine ⟨ha, fun b hb => ⟨?_, ?_⟩⟩
-  · intro hinv
-    by_contra hba
-    have hn := (rootInverted_ref_iff_flipped a b).mp hinv
-    have hp := partner_mem a b ha hb
-    have hs := negative_pair_sum a b hn
-    have hc := negative_pair_coefficient_pos a b hn
-    have he : a.val = (form M a.val b.val)⁻¹ • b.val +
-        (form M a.val b.val)⁻¹ • (partner a b).val := by
-      calc
-        a.val = (form M a.val b.val)⁻¹ • (form M a.val b.val • a.val) := by
-          rw [smul_smul, inv_mul_cancel₀ hc.ne', one_smul]
-        _ = _ := by rw [← hs, smul_add]
-    have hne : geometricSlope b ≠ geometricSlope (partner a b) := by
-      intro h
-      exact hba (negative_partner_fixed a b hn (geometricSlope_injective h).symm)
-    rcases lt_or_gt_of_ne hne with hlt | hlt
-    · have hh := geometricSlope_between b (partner a b) a
-        (inv_pos.mpr hc) (inv_pos.mpr hc) he hlt
-      rcases hext with hmin | hmax
-      · exact (not_lt_of_ge (hmin b hb)) hh.1
-      · exact (not_lt_of_ge (hmax (partner a b) hp)) hh.2
-    · have he' : a.val = (form M a.val b.val)⁻¹ • (partner a b).val +
-          (form M a.val b.val)⁻¹ • b.val := by rw [add_comm]; exact he
-      have hh := geometricSlope_between (partner a b) b a
-        (inv_pos.mpr hc) (inv_pos.mpr hc) he' hlt
-      rcases hext with hmin | hmax
-      · exact (not_lt_of_ge (hmin (partner a b) hp)) hh.1
-      · exact (not_lt_of_ge (hmax b hb)) hh.2
-  · intro h
-    rw [h]
-    exact rootInverted_ref_self a
-
-theorem PlaneSimple.partner_val {P : Submodule ℝ (I → ℝ)}
-    {a : PositiveRoot M cs} (ha : PlaneSimple P a)
-    {b : PositiveRoot M cs} (hb : b.val ∈ P) (hba : b ≠ a) :
-    (partner a b).val = action M cs (rootRef a) b.val := by
-  have hn : ¬ rootInverted (rootRef a) b := fun h => hba ((ha.2 b hb).mp h)
-  have hp := (not_rootInverted_iff (rootRef a) b).mp hn
-  rw [(rootReflection M cs a).property.inv] at hp
-  exact positiveAction_eq_of_nonneg _ _ hp
-
-theorem PlaneSimple.exists_nonpositive_pair {P : Submodule ℝ (I → ℝ)}
-    {a : PositiveRoot M cs} (ha : PlaneSimple P a)
-    {b : PositiveRoot M cs} (hb : b.val ∈ P) (hba : b ≠ a) :
-    ∃ c : PositiveRoot M cs, c.val ∈ P ∧ c ≠ a ∧ form M a.val c.val ≤ 0 := by
-  by_cases hn : form M a.val b.val ≤ 0
-  · exact ⟨b, hb, hba, hn⟩
-  · refine ⟨partner a b, partner_mem a b ha.1 hb,
-      fun h => hba ((partner_eq_self_of_eq a b).mp h), ?_⟩
-    rw [ha.partner_val hb hba, (rootReflection_represents M cs a).action_formula]
-    simp only [map_sub, map_smul, smul_eq_mul, a.property.1.norm]
-    linarith
-
-theorem noninverted_ref_of_pairing_nonpos (a b : PositiveRoot M cs)
-    (hab : form M a.val b.val ≤ 0) : ¬ rootInverted (rootRef b) a := by
-  rw [not_rootInverted_iff, (rootReflection M cs b).property.inv,
-    (rootReflection_represents M cs b).action_formula]
-  have hba : form M b.val a.val ≤ 0 := by rwa [(form_symm M).eq]
-  intro i
-  exact sub_nonneg.mpr ((mul_nonpos_of_nonpos_of_nonneg hba (b.property.2 i)).trans
-    (a.property.2 i))
-
-/-- The other endpoint exists even in an infinite or degenerate plane. -/
-theorem exists_opposite_plane_extreme (a b p : PositiveRoot M cs) (hab : a ≠ b)
-    (hp : p.val ∈ rootPlane a b)
-    (hext : (∀ r : PositiveRoot M cs, r.val ∈ rootPlane a b →
-        geometricSlope p ≤ geometricSlope r) ∨
-      (∀ r : PositiveRoot M cs, r.val ∈ rootPlane a b →
-        geometricSlope r ≤ geometricSlope p)) :
-    ∃ q : PositiveRoot M cs, q.val ∈ rootPlane a b ∧ q ≠ p ∧
-      (((∀ r : PositiveRoot M cs, r.val ∈ rootPlane a b →
-          geometricSlope p ≤ geometricSlope r) ∧
-        (∀ r : PositiveRoot M cs, r.val ∈ rootPlane a b →
-          geometricSlope r ≤ geometricSlope q)) ∨
-       ((∀ r : PositiveRoot M cs, r.val ∈ rootPlane a b →
-          geometricSlope q ≤ geometricSlope r) ∧
-        (∀ r : PositiveRoot M cs, r.val ∈ rootPlane a b →
-          geometricSlope r ≤ geometricSlope p))) := by
-  have hs := planeSimple_of_extreme p hp hext
-  have hother : ∃ r : PositiveRoot M cs, r.val ∈ rootPlane a b ∧ r ≠ p := by
-    by_cases hap : a = p
-    · refine ⟨b, Submodule.subset_span (Set.mem_insert_of_mem _ (Set.mem_singleton _)), ?_⟩
-      exact fun h => hab (hap.trans h.symm)
-    · exact ⟨a, Submodule.subset_span (Set.mem_insert _ _), hap⟩
-  obtain ⟨r, hrP, hrp⟩ := hother
-  obtain ⟨c, hcP, hcp, hcpair⟩ := hs.exists_nonpositive_pair hrP hrp
-  have hn := noninverted_ref_of_pairing_nonpos p c hcpair
-  have hplane := rootPlane_eq_of_mem a b c p hab hcP hp hcp
-  obtain ⟨q, hqP, hqinv, hqext⟩ := exists_plane_extreme_inverted c p hcp
-  rw [hplane] at hqP hqext
-  have hqp : q ≠ p := fun h => hn (h ▸ hqinv)
-  refine ⟨q, hqP, hqp, ?_⟩
-  rcases hext with hmin | hmax
-  · refine Or.inl ⟨hmin, ?_⟩
-    rcases hqext with hqmin | hqmax
-    · exact False.elim (hqp (geometricSlope_injective (le_antisymm
-        (hqmin p hp) (hmin q hqP))))
-    · exact hqmax
-  · refine Or.inr ⟨?_, hmax⟩
-    rcases hqext with hqmin | hqmax
-    · exact hqmin
-    · exact False.elim (hqp (geometricSlope_injective (le_antisymm
-        (hmax q hqP) (hqmax p hp))))
-
-/-- Canonical angular endpoints of a plane; no Coxeter-system assertion is
-built into this data. The latter will be proved from the inversion criterion. -/
-structure PlaneBasis (P : Submodule ℝ (I → ℝ)) where
-  left : PositiveRoot M cs
-  right : PositiveRoot M cs
-  distinct : left ≠ right
-  plane_eq : rootPlane left right = P
-  left_min : ∀ r : PositiveRoot M cs, r.val ∈ P → geometricSlope left ≤ geometricSlope r
-  right_max : ∀ r : PositiveRoot M cs, r.val ∈ P → geometricSlope r ≤ geometricSlope right
-
-theorem nonempty_planeBasis (a b : PositiveRoot M cs) (hab : a ≠ b) :
-    Nonempty (PlaneBasis (M := M) (cs := cs) (rootPlane a b)) := by
-  obtain ⟨p, hp, _, hext⟩ := exists_plane_extreme_inverted a b hab
-  obtain ⟨q, hq, hqp, hpair⟩ := exists_opposite_plane_extreme a b p hab hp hext
-  rcases hpair with ⟨hl, hr⟩ | ⟨hl, hr⟩
-  · exact ⟨⟨p, q, hqp.symm, rootPlane_eq_of_mem a b p q hab hp hq hqp.symm, hl, hr⟩⟩
-  · exact ⟨⟨q, p, hqp, rootPlane_eq_of_mem a b q p hab hq hp hqp, hl, hr⟩⟩
-
-namespace PlaneBasis
-variable {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-
-theorem left_mem : B.left.val ∈ P := by
-  apply B.plane_eq.le
-  exact Submodule.subset_span (Set.mem_insert _ _)
-
-theorem right_mem : B.right.val ∈ P := by
-  apply B.plane_eq.le
-  exact Submodule.subset_span (Set.mem_insert_of_mem _ (Set.mem_singleton _))
-
-theorem left_simple : PlaneSimple P B.left :=
-  planeSimple_of_extreme B.left B.left_mem (.inl B.left_min)
-
-theorem right_simple : PlaneSimple P B.right :=
-  planeSimple_of_extreme B.right B.right_mem (.inr B.right_max)
-
-/-- Every positive root in the plane is in its canonical nonnegative cone. -/
-theorem positive_cone (r : PositiveRoot M cs) (hr : r.val ∈ P) :
-    ∃ c d : ℝ, 0 ≤ c ∧ 0 ≤ d ∧ r.val = c • B.left.val + d • B.right.val := by
-  by_cases hl : r = B.left
-  · exact ⟨1, 0, zero_le_one, le_refl 0, by rw [hl]; simp⟩
-  by_cases hh : r = B.right
-  · exact ⟨0, 1, le_refl 0, zero_le_one, by rw [hh]; simp⟩
-  have hlr : geometricSlope B.left < geometricSlope r :=
-    lt_of_le_of_ne (B.left_min r hr) (fun h => hl (geometricSlope_injective h).symm)
-  have hrr : geometricSlope r < geometricSlope B.right :=
-    lt_of_le_of_ne (B.right_max r hr) (fun h => hh (geometricSlope_injective h))
-  obtain ⟨c, d, hc, hd, he⟩ := positive_combination_of_slope_between B.left B.right r
-    (B.plane_eq.ge hr) hlr hrr
-  exact ⟨c, d, hc.le, hd.le, he⟩
-
-end PlaneBasis
-
-
-/-- Canonical-simple reflections have no extra root pair in their own plane. -/
-theorem PlaneSimple.planeWeight_eq_zero {P : Submodule ℝ (I → ℝ)}
-    {a : PositiveRoot M cs} (ha : PlaneSimple P a) (x : W) :
-    planeWeight x a P = 0 := by
-  classical
-  rw [planeWeight, Finsupp.mapDomain, Finsupp.sum_apply]
-  apply Finset.sum_eq_zero
-  intro b _
-  change (Finsupp.single (rootPlane a b) (halfVector x a b)) P = 0
-  rw [Finsupp.single_apply]
-  by_cases hP : rootPlane a b = P
-  · rw [ite_eq_left hP]
-    simp only [halfVector, rootIndicator_apply]
-    apply ite_eq_right
-    intro hh
-    have hbP : b.val ∈ P := hP.le
-      (Submodule.subset_span (Set.mem_insert_of_mem _ (Set.mem_singleton _)))
-    exact hh.1.1 ((ha.2 b hbP).mp ((rootInverted_ref_iff_flipped a b).mpr hh.1.2.1))
-  · exact ite_eq_right hP
-
-/-- The relative inversion count has the genuine simple-reflection step. -/
-theorem PlaneSimple.planeLength_step {P : Submodule ℝ (I → ℝ)}
-    {a b : PositiveRoot M cs} (ha : PlaneSimple P a)
-    (hab : a ≠ b) (hP : rootPlane a b = P) (x : W)
-    (hx : ¬ rootInverted x a) :
-    planeLength (M := M) (cs := cs) P (rootRef a * x) =
-      planeLength (M := M) (cs := cs) P x + 1 := by
-  have hh := planeWeight_relative_degree x a b hab hx
-  rw [hP, ha.planeWeight_eq_zero] at hh
-  linarith
-
-theorem PlaneSimple.planeLength_step_down {P : Submodule ℝ (I → ℝ)}
-    {a b : PositiveRoot M cs} (ha : PlaneSimple P a)
-    (hab : a ≠ b) (hP : rootPlane a b = P) (x : W)
-    (hx : rootInverted x a) :
-    planeLength (M := M) (cs := cs) P (rootRef a * x) =
-      planeLength (M := M) (cs := cs) P x - 1 := by
-  have hy : ¬ rootInverted (rootRef a * x) a :=
-    fun h => (rootInverted_mul_self x a).mp h hx
-  have hh := ha.planeLength_step hab hP (rootRef a * x) hy
-  rw [← mul_assoc, (rootReflection M cs a).property.mul_self, one_mul] at hh
-  linarith
-
-/-- The usual inversion bijection, proved on the actual positive roots. -/
-theorem rootInverted_inverse_action (w : W) (a : PositiveRoot M cs) :
-    rootInverted w⁻¹ (positiveAction w⁻¹ a) ↔ rootInverted w a := by
-  have hw : action M cs w (action M cs w⁻¹ a.val) = a.val := by
-    rw [← LinearEquiv.mul_apply, ← map_mul, mul_inv_cancel, map_one]
-    rfl
-  by_cases hi : rootInverted w a
-  · apply iff_of_true ?_ hi
-    unfold rootInverted
-    rw [inv_inv, positiveAction_eq_of_nonpos _ _ hi, map_neg, hw]
-    exact fun i => neg_nonpos.mpr (a.property.2 i)
-  · apply iff_of_false ?_ hi
-    rw [not_rootInverted_iff, inv_inv,
-      positiveAction_eq_of_nonneg _ _ ((not_rootInverted_iff w a).mp hi), hw]
-    exact a.property.2
-
-/-- Reindexing the inversion indicator by the inverse root action. -/
-theorem inversionVector_inverse (w : W) :
-    inversionVector (M := M) (cs := cs) w⁻¹ =
-      Finsupp.mapDomain (positiveActionEquiv w⁻¹) (inversionVector w) := by
-  classical
-  apply (Finsupp.ext_iff).mpr
-  intro b
-  obtain ⟨a, rfl⟩ := (positiveActionEquiv (M := M) (cs := cs) w⁻¹).surjective b
-  rw [Finsupp.mapDomain_equiv_apply, Equiv.symm_apply_apply]
-  simp only [inversionVector_apply]
-  change (if rootInverted w⁻¹ (positiveAction w⁻¹ a) then 1 else 0) = _
-  rw [rootInverted_inverse_action]
-
-theorem positiveAction_mem_plane {P : Submodule ℝ (I → ℝ)}
-    {w : W} (hw : w ∈ planeSubgroup M cs P) (a : PositiveRoot M cs)
-    (ha : a.val ∈ P) : (positiveAction w a).val ∈ P := by
-  have hp := planeSubgroup_preserves hw ha
-  rcases (a.property.1.act w).sign with hs | hs
-  · exact (positiveAction_eq_of_nonneg w a hs).symm ▸ hp
-  · exact (positiveAction_eq_of_nonpos w a hs).symm ▸ P.neg_mem hp
-
-theorem positiveAction_mem_plane_iff {P : Submodule ℝ (I → ℝ)}
-    {w : W} (hw : w ∈ planeSubgroup M cs P) (a : PositiveRoot M cs) :
-    (positiveAction w a).val ∈ P ↔ a.val ∈ P := by
-  refine ⟨fun h => ?_, positiveAction_mem_plane hw a⟩
-  have hh := positiveAction_mem_plane ((planeSubgroup M cs P).inv_mem hw)
-    (positiveAction w a) h
-  rwa [← positiveAction_mul, inv_mul_cancel, positiveAction_one] at hh
-
-/-- Relative inversion count is invariant under inverse on the actual plane subgroup. -/
-theorem planeLength_inv {P : Submodule ℝ (I → ℝ)} {w : W}
-    (hw : w ∈ planeSubgroup M cs P) :
-    planeLength (M := M) (cs := cs) P w⁻¹ = planeLength (M := M) (cs := cs) P w := by
-  classical
-  rw [planeLength, inversionVector_inverse,
-    Finsupp.sum_mapDomain_index (by intros; split_ifs <;> rfl)
-      (by intros; split_ifs <;> simp)]
-  apply Finsupp.sum_congr
-  intro a _
-  change (if (positiveAction w⁻¹ a).val ∈ P then _ else _) = _
-  rw [positiveAction_mem_plane_iff ((planeSubgroup M cs P).inv_mem hw)]
-
-theorem planeLength_nonneg (P : Submodule ℝ (I → ℝ)) (w : W) :
-    0 ≤ planeLength (M := M) (cs := cs) P w := by
-  classical
-  apply Finset.sum_nonneg
-  intro a _
-  simp only [inversionVector_apply]
-  split_ifs <;> norm_num
-
-namespace PlaneBasis
-variable {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-
-/-- A plane reflection must invert at least one of the canonical endpoints.
-This uses convex positivity, not any assumed Coxeter structure on the subgroup. -/
-theorem reflection_inverts_endpoint (r : PositiveRoot M cs) (hr : r.val ∈ P) :
-    rootInverted (rootRef r) B.left ∨ rootInverted (rootRef r) B.right := by
-  by_contra h
-  push Not at h
-  obtain ⟨c, d, hc, hd, he⟩ := B.positive_cone r hr
-  exact (noninverted_of_positive_combination (rootRef r) B.left B.right r
-    hc hd he h.1 h.2) (rootInverted_ref_self r)
-
-end PlaneBasis
-
-
-/-- The second descent in a simple conjugation of a nonsimple reflection. -/
-theorem PlaneSimple.inverted_after_mul {P : Submodule ℝ (I → ℝ)}
-    {a r : PositiveRoot M cs} (ha : PlaneSimple P a) (hr : r.val ∈ P)
-    (hra : r ≠ a) (hi : rootInverted (rootRef r) a) :
-    rootInverted (rootRef r * rootRef a) a := by
-  have hn := (rootInverted_ref_iff_flipped r a).mp hi
-  apply (rootInverted_mul_of_nonpos (rootRef a) r a hn).mpr
-  intro hinv
-  have he : partner r a = a := (ha.2 (partner r a) (partner_mem r a hr ha.1)).mp hinv
-  exact hra (negative_partner_fixed r a hn he).symm
-
-/-- The honest relative length drops by two under the chosen simple conjugation. -/
-theorem PlaneSimple.planeLength_conj_drop {P : Submodule ℝ (I → ℝ)}
-    {a b r : PositiveRoot M cs} (ha : PlaneSimple P a)
-    (hab : a ≠ b) (hP : rootPlane a b = P) (hr : r.val ∈ P)
-    (hra : r ≠ a) (hi : rootInverted (rootRef r) a) :
-    planeLength (M := M) (cs := cs) P (rootRef (partner a r)) =
-      planeLength (M := M) (cs := cs) P (rootRef r) - 2 := by
-  have har : rootRef a ∈ planeSubgroup M cs P :=
-    ((rootReflection_represents M cs a).planeSubgroup_mem_iff M cs).mpr ha.1
-  have hrr : rootRef r ∈ planeSubgroup M cs P :=
-    ((rootReflection_represents M cs r).planeSubgroup_mem_iff M cs).mpr hr
-  have hmul := (planeSubgroup M cs P).mul_mem har hrr
-  have hd := ha.planeLength_step_down hab hP (rootRef r) hi
-  have hd' := ha.planeLength_step_down hab hP (rootRef r * rootRef a)
-    (ha.inverted_after_mul hr hra hi)
-  have hli := planeLength_inv hmul
-  rw [mul_inv_rev, (rootReflection M cs r).property.inv,
-    (rootReflection M cs a).property.inv] at hli
-  change planeLength (M := M) (cs := cs) P
-    (rootReflection M cs (positiveAction (rootRef a) r)).val = _
-  rw [rootReflection_positiveAction, (rootReflection M cs a).property.inv, mul_assoc]
-  linarith
-
-namespace PlaneBasis
-variable {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-
-/-- The canonical endpoint reflections generate every ambient reflection in the plane. -/
-theorem rootRef_mem_pairClosure (r : PositiveRoot M cs) (hr : r.val ∈ P) :
-    rootRef r ∈ Subgroup.closure {rootRef B.left, rootRef B.right} := by
-  classical
-  generalize hn : (planeLength (M := M) (cs := cs) P (rootRef r)).toNat = n
-  induction n using Nat.strong_induction_on generalizing r with
-  | h n ih =>
-    by_cases hl : r = B.left
-    · rw [hl]; exact Subgroup.subset_closure (Set.mem_insert _ _)
-    by_cases hh : r = B.right
-    · rw [hh]; exact Subgroup.subset_closure (Set.mem_insert_of_mem _ (Set.mem_singleton _))
-    have step (a b : PositiveRoot M cs) (ha : PlaneSimple P a)
-        (hab : a ≠ b) (hP : rootPlane a b = P) (hra : r ≠ a)
-        (hgen : rootRef a ∈ Subgroup.closure {rootRef B.left, rootRef B.right})
-        (hi : rootInverted (rootRef r) a) :
-        rootRef r ∈ Subgroup.closure {rootRef B.left, rootRef B.right} := by
-      have hp : (partner a r).val ∈ P := partner_mem a r ha.1 hr
-      have hdrop := ha.planeLength_conj_drop hab hP hr hra hi
-      have hless : (planeLength (M := M) (cs := cs) P (rootRef (partner a r))).toNat < n := by
-        have hnonneg := planeLength_nonneg (M := M) (cs := cs) P (rootRef (partner a r))
-        omega
-      have hind := ih _ hless (partner a r) hp rfl
-      have hlen := (Subgroup.closure {rootRef B.left, rootRef B.right}).mul_mem
-        ((Subgroup.closure {rootRef B.left, rootRef B.right}).mul_mem hgen hind) hgen
-      change rootRef a * (rootReflection M cs (positiveAction (rootRef a) r)).val * rootRef a ∈ _ at hlen
-      rw [rootReflection_positiveAction, (rootReflection M cs a).property.inv] at hlen
-      rw [← mul_assoc, ← mul_assoc, (rootReflection M cs a).property.mul_self,
-        one_mul, mul_assoc, (rootReflection M cs a).property.mul_self, mul_one] at hlen
-      exact hlen
-    rcases B.reflection_inverts_endpoint r hr with hi | hi
-    · exact step B.left B.right B.left_simple B.distinct B.plane_eq hl
-        (Subgroup.subset_closure (Set.mem_insert _ _)) hi
-    · exact step B.right B.left B.right_simple B.distinct.symm
-        (by rw [rootPlane, Set.pair_comm]; exact B.plane_eq) hh
-        (Subgroup.subset_closure (Set.mem_insert_of_mem _ (Set.mem_singleton _))) hi
-
-/-- Exact identification with the manuscript's maximal root-plane subgroup. -/
-theorem pairClosure_eq_planeSubgroup :
-    Subgroup.closure {rootRef B.left, rootRef B.right} = planeSubgroup M cs P := by
-  apply le_antisymm
-  · apply (Subgroup.closure_le _).mpr
-    intro t ht
-    rcases Set.mem_insert_iff.mp ht with he | he
-    · rw [he]
-      exact ((rootReflection_represents M cs B.left).planeSubgroup_mem_iff M cs).mpr B.left_mem
-    · rw [Set.mem_singleton_iff.mp he]
-      exact ((rootReflection_represents M cs B.right).planeSubgroup_mem_iff M cs).mpr B.right_mem
-  · apply (Subgroup.closure_le _).mpr
-    rintro t ⟨a, ha, haP⟩
-    let r := positiveRoot M cs ⟨t, ha.isReflection⟩
-    have hre : rootRef r = t := (rootReflection_represents M cs r).reflection_unique
-      (positiveRoot_represents M cs ⟨t, ha.isReflection⟩)
-    have hrP : r.val ∈ P := ((positiveRoot_represents M cs
-      ⟨t, ha.isReflection⟩).planeSubgroup_mem_iff M cs).mp (Subgroup.subset_closure ⟨a, ha, haP⟩)
-    rw [← hre]
-    exact B.rootRef_mem_pairClosure r hrP
-
-/-- Distinct positive roots give distinct actual reflections. -/
-theorem rootRef_distinct : rootRef B.left ≠ rootRef B.right := by
-  intro he
-  apply B.distinct
-  apply (positiveRootEquiv M cs).symm.injective
-  exact Subtype.ext he
-
-theorem rootRef_ne_one (a : PositiveRoot M cs) : rootRef a ≠ 1 := by
-  intro h
-  exact (not_rootInverted_one a) (h ▸ rootInverted_ref_self a)
-
-/-- The genuine dihedral matrix attached to the canonical plane endpoints. -/
-def matrix : CoxeterMatrix Bool :=
-  Dihedral.pairMatrix (rootRef B.left) (rootRef B.right)
-    (rootReflection M cs B.left).property.mul_self B.rootRef_distinct
-
-/-- A proved Coxeter system on the *actual maximal plane subgroup*. -/
-def system : CoxeterSystem B.matrix (planeSubgroup M cs P) :=
-  (Dihedral.pairSystem (rootRef B.left) (rootRef B.right)
-    (rootReflection M cs B.left).property.mul_self
-    (rootReflection M cs B.right).property.mul_self B.rootRef_distinct
-    (rootRef_ne_one B.left)
-    (rootRef_ne_one B.right)).map
-      (MulEquiv.subgroupCongr B.pairClosure_eq_planeSubgroup)
-
-/-- Canonical simple positive roots, indexed by Bool just like the system. -/
-def simpleRoot (i : Bool) : PositiveRoot M cs := if i then B.right else B.left
-
-theorem simpleRoot_mem (i : Bool) : (B.simpleRoot i).val ∈ P := by
-  cases i
-  · exact B.left_mem
-  · exact B.right_mem
-
-theorem simpleRoot_simple (i : Bool) : PlaneSimple P (B.simpleRoot i) := by
-  cases i
-  · exact B.left_simple
-  · exact B.right_simple
-
-@[simp] theorem system_simple (i : Bool) :
-    (B.system.simple i : W) = rootRef (B.simpleRoot i) := by
-  change ((Dihedral.pairSystem (rootRef B.left) (rootRef B.right)
-    (rootReflection M cs B.left).property.mul_self
-    (rootReflection M cs B.right).property.mul_self B.rootRef_distinct
-    (rootRef_ne_one B.left) (rootRef_ne_one B.right)).simple i : W) = _
-  rw [Dihedral.pairSystem_simple]
-  cases i <;> rfl
-
-/-- The positive simple-sign criterion for the induced plane root system. -/
-theorem simple_sign {t : planeSubgroup M cs P} {a : I → ℝ}
-    (ha : Represents M cs t.val a) (haP : a ∈ P) (hp : ∀ k, 0 ≤ a k) (i : Bool) :
-    ((∀ k, action M cs (B.system.simple i : W) a k ≤ 0) ↔ t = B.system.simple i) ∧
-    ((∀ k, 0 ≤ action M cs (B.system.simple i : W) a k) ↔ t ≠ B.system.simple i) := by
-  let r : PositiveRoot M cs := ⟨a, ha.isRoot, hp⟩
-  have hneg : (∀ k, action M cs (B.system.simple i : W) a k ≤ 0) ↔ t = B.system.simple i := by
-    rw [B.system_simple]
-    change flipped (B.simpleRoot i) r ↔ _
-    rw [← rootInverted_ref_iff_flipped, (B.simpleRoot_simple i).2 r haP]
-    constructor
-    · intro he
-      apply Subtype.ext
-      rw [B.system_simple]
-      exact ha.reflection_unique (he ▸ rootReflection_represents M cs r)
-    · intro he
-      apply Subtype.ext
-      have hr : Represents M cs (rootRef (B.simpleRoot i)) a := by
-        simpa only [he, B.system_simple] using ha
-      exact hr.unique_positive (rootReflection_represents M cs _) hp (B.simpleRoot i).property.2
-  refine ⟨hneg, ?_⟩
-  constructor
-  · intro h he
-    exact (ha.isRoot.act (B.system.simple i : W)).not_both h (hneg.mpr he)
-  · intro h
-    exact (ha.isRoot.act (B.system.simple i : W)).sign.resolve_right (fun hn => h (hneg.mp hn))
-
-/-- Signed roots for the induced system. The vectors are the original ambient
-roots, and membership in the actual plane is part of the relation. -/
-structure InducedSigned (p : planeSubgroup M cs P × ZMod 2) (a : I → ℝ) : Prop where
-  represents : Represents M cs p.1.val a
-  plane_mem : a ∈ P
-  sign : (p.2 = 0 ∧ ∀ i, 0 ≤ a i) ∨ (p.2 = 1 ∧ ∀ i, a i ≤ 0)
-
-theorem InducedSigned.simple {p : planeSubgroup M cs P × ZMod 2} {a : I → ℝ}
-    (ha : InducedSigned p a) (i : Bool) :
-    InducedSigned (CoxeterSupport.simpleAction B.system i p)
-      (action M cs (B.system.simple i : W) a) := by
-  rcases p with ⟨t,z⟩
-  rw [CoxeterSupport.simpleAction_apply]
-  refine ⟨?_, planeSubgroup_preserves (B.system.simple i).property ha.plane_mem, ?_⟩
-  · have h := ha.represents.act (B.system.simple i : W)
-    have he : (B.system.simple i : W)⁻¹ = (B.system.simple i : W) :=
-      congrArg Subtype.val (B.system.inv_simple i)
-    simpa only [he, Subgroup.coe_mul] using h
-  · rcases ha.sign with ⟨rfl,hp⟩ | ⟨rfl,hn⟩
-    · by_cases heq : t = B.system.simple i
-      · exact Or.inr ⟨by simp [heq],
-          (B.simple_sign ha.represents ha.plane_mem hp i).1.mpr heq⟩
-      · exact Or.inl ⟨by simp [heq, ZModModule.add_self],
-          (B.simple_sign ha.represents ha.plane_mem hp i).2.mpr heq⟩
-    · have hp : ∀ k, 0 ≤ (-a) k := fun k => neg_nonneg.mpr (hn k)
-      by_cases heq : t = B.system.simple i
-      · refine Or.inl ⟨by simp [heq, ZModModule.add_self], ?_⟩
-        have h := (B.simple_sign ha.represents.neg (P.neg_mem ha.plane_mem) hp i).1.mpr heq
-        simpa only [map_neg, Pi.neg_apply, neg_nonpos] using h
-      · refine Or.inr ⟨by simp [heq], ?_⟩
-        have h := (B.simple_sign ha.represents.neg (P.neg_mem ha.plane_mem) hp i).2.mpr heq
-        simpa only [map_neg, Pi.neg_apply, neg_nonneg] using h
-
-theorem InducedSigned.act {p : planeSubgroup M cs P × ZMod 2} {a : I → ℝ}
-    (ha : InducedSigned p a) (w : planeSubgroup M cs P) :
-    InducedSigned (CoxeterSupport.reflectionAction B.system w p)
-      (action M cs (w : W) a) := by
-  obtain ⟨ω,rfl⟩ := B.system.wordProd_surjective w
-  induction ω with
-  | nil => simpa using ha
-  | cons i ω ih =>
-    simpa only [B.system.wordProd_cons, map_mul, Equiv.Perm.mul_apply,
-      CoxeterSupport.reflectionAction_simple, Subgroup.coe_mul,
-      LinearEquiv.mul_apply] using ih.simple B i
-
-/-- The induced positive roots have precisely the intrinsic Coxeter inversion
-signs. No change of realization or extra canonical-system assumption is used. -/
-theorem inversion_iff {t : planeSubgroup M cs P} {a : I → ℝ}
-    (ha : Represents M cs t.val a) (haP : a ∈ P) (hp : ∀ i, 0 ≤ a i)
-    (w : planeSubgroup M cs P) :
-    (∀ i, action M cs (w : W)⁻¹ a i ≤ 0) ↔ B.system.IsLeftInversion w t := by
-  classical
-  let : BEq (planeSubgroup M cs P) := @instBEqOfDecidableEq _ (Classical.decEq _)
-  obtain ⟨ω,hω,heq⟩ := B.system.exists_isReduced w
-  rw [heq]
-  let c : ZMod 2 := if t ∈ B.system.leftInvSeq ω then 1 else 0
-  have hrec := CoxeterSupport.wordAction_record B.system ω t c
-  rw [← CoxeterSupport.reflectionAction_wordProd] at hrec
-  have hrec' : CoxeterSupport.reflectionAction B.system (B.system.wordProd ω)
-      ((B.system.wordProd ω)⁻¹ * t * B.system.wordProd ω, c) = (t,0) := by
-    by_cases hm : t ∈ B.system.leftInvSeq ω
-    · have hc : c = 1 := ite_eq_left hm
-      simpa only [hc, List.count_eq_one_of_mem hω.nodup_leftInvSeq hm,
-        Nat.cast_one, ZModModule.add_self] using hrec
-    · have hc : c = 0 := ite_eq_right hm
-      simpa only [hc, List.count_eq_zero.mpr hm, Nat.cast_zero, add_zero] using hrec
-  have hinv : CoxeterSupport.reflectionAction B.system (B.system.wordProd ω)⁻¹ (t,0) =
-      ((B.system.wordProd ω)⁻¹ * t * B.system.wordProd ω,c) := by
-    rw [map_inv, ← hrec']
-    exact Equiv.symm_apply_apply _ _
-  have hs : InducedSigned (t,0) a := ⟨ha,haP,Or.inl ⟨rfl,hp⟩⟩
-  have hsi := hs.act B (B.system.wordProd ω)⁻¹
-  rw [hinv] at hsi
-  have hcount : c = if t ∈ B.system.leftInvSeq ω then 1 else 0 := rfl
-  have hmem : (∀ i, action M cs (B.system.wordProd ω : W)⁻¹ a i ≤ 0) ↔
-      t ∈ B.system.leftInvSeq ω := by
-    by_cases hm : t ∈ B.system.leftInvSeq ω
-    · have hc1 : c=1 := by simpa [hm] using hcount
-      have hh := hsi.sign
-      simp only [hc1] at hh
-      rcases hh with ⟨hbad,_⟩ | ⟨_,hn⟩
-      · norm_num at hbad
-      · exact iff_of_true hn hm
-    · have hc0 : c=0 := by simpa [hm] using hcount
-      have hh := hsi.sign
-      simp only [hc0] at hh
-      rcases hh with ⟨_,hpos⟩ | ⟨hbad,_⟩
-      · exact iff_of_false (fun hn => (ha.isRoot.act _).not_both hpos hn) hm
-      · norm_num at hbad
-  rw [hmem]
-  exact ⟨B.system.isLeftInversion_of_mem_leftInvSeq hω,
-    CoxeterSupport.mem_leftInvSeq_of_inversion B.system hω⟩
-
-/-- The intrinsic reflections are exactly the ambient reflections in the
-actual maximal plane subgroup. -/
-theorem reflection_iff (t : planeSubgroup M cs P) :
-    B.system.IsReflection t ↔ cs.IsReflection (t : W) := by
-  constructor
-  · rintro ⟨w,i,he⟩
-    obtain ⟨z,j,hz⟩ := (rootReflection M cs (B.simpleRoot i)).property
-    refine ⟨(w : W) * z,j,?_⟩
-    have he' := congrArg Subtype.val he
-    simp only [Subgroup.coe_mul, Subgroup.coe_inv, B.system_simple] at he'
-    change rootRef (B.simpleRoot i) = _ at hz
-    rw [he', hz]
-    group
-  · intro ht
-    let a := positiveRoot M cs ⟨(t:W),ht⟩
-    have ha := positiveRoot_represents M cs ⟨(t:W),ht⟩
-    have haP : a.val ∈ P := (ha.planeSubgroup_mem_iff M cs).mp t.property
-    have hinv := (B.inversion_iff ha haP a.property.2 t).mp
-      ((ha.inversion_iff a.property.2 (t:W)).mpr
-        ⟨ht, by
-          rw [ht.mul_self, cs.length_one]
-          obtain ⟨k,hk⟩ := ht.odd_length
-          omega⟩)
-    exact hinv.1
-
-/-- The true ambient plane inversions are in bijection with the inversions
-of the canonical induced system. -/
-def inversionEquiv (w : planeSubgroup M cs P) :
-    {a : PositiveRoot M cs // a.val ∈ P ∧ rootInverted (w:W) a} ≃
-      {t : planeSubgroup M cs P // B.system.IsLeftInversion w t} where
-  toFun a := ⟨⟨rootRef a.val,
-    ((rootReflection_represents M cs a.val).planeSubgroup_mem_iff M cs).mpr a.property.1⟩,
-    (B.inversion_iff (rootReflection_represents M cs a.val) a.property.1
-      a.val.property.2 w).mp a.property.2⟩
-  invFun t :=
-    let a := positiveRoot M cs ⟨(t.val:W),(B.reflection_iff t.val).mp t.property.1⟩
-    let ha := positiveRoot_represents M cs
-      ⟨(t.val:W),(B.reflection_iff t.val).mp t.property.1⟩
-    let haP := (ha.planeSubgroup_mem_iff M cs).mp t.val.property
-    ⟨a,haP,(B.inversion_iff ha haP a.property.2 w).mpr t.property⟩
-  left_inv a := by
-    apply Subtype.ext
-    exact (positiveRootEquiv M cs).apply_symm_apply a.val
-  right_inv t := by
-    apply Subtype.ext
-    apply Subtype.ext
-    exact congrArg (fun r : {r : W // cs.IsReflection r} => r.val)
-      ((positiveRootEquiv M cs).symm_apply_apply
-        ⟨(t.val:W),(B.reflection_iff t.val).mp t.property.1⟩)
-
-/-- Relative inversion count is the actual Coxeter length, completing the
-length identification, not merely a rank-two presentation. -/
-theorem length_eq (w : planeSubgroup M cs P) :
-    planeLength (M := M) (cs := cs) P (w:W) = (B.system.length w : ℤ) := by
-  classical
-  let f := inversionVector (M := M) (cs := cs) (w:W)
-  let F := f.support.filter (fun a => a.val ∈ P)
-  have hF : (F : Set (PositiveRoot M cs)) =
-      {a | a.val ∈ P ∧ rootInverted (w:W) a} := by
-    ext a
-    simp only [F, Finset.mem_coe, Finset.mem_filter, Finsupp.mem_support_iff,
-      f, inversionVector_apply, ne_eq, ite_eq_right_iff, one_ne_zero, imp_false, not_not,
-      Set.mem_ofPred_eq, and_comm]
-  have hcard : F.card = B.system.length w := by
-    rw [← Set.ncard_coe_finset, hF, ← Nat.card_coe_set_eq]
-    change Nat.card {a : PositiveRoot M cs // a.val ∈ P ∧ rootInverted (w:W) a} = _
-    rw [Nat.card_congr (B.inversionEquiv w)]
-    obtain ⟨ω,hω,he⟩ := B.system.exists_isReduced w
-    rw [he, Nat.card_congr (leftInversionListEquiv B.matrix B.system hω),
-      ← Nat.card_congr (List.Nodup.getEquiv (B.system.leftInvSeq ω) hω.nodup_leftInvSeq),
-      Nat.card_fin, B.system.length_leftInvSeq, hω.eq]
-  change f.sum (fun a n => if a.val ∈ P then n else 0) = _
-  rw [Finsupp.sum, ← Finset.sum_filter]
-  calc
-    ∑ a ∈ F, f a = ∑ _a ∈ F, (1:ℤ) := by
-      apply Finset.sum_congr rfl
-      intro a ha
-      have hm : rootInverted (w:W) a := by
-        have hm := (Finset.mem_filter.mp ha).1
-        simpa only [Finsupp.mem_support_iff, f, inversionVector_apply,
-          ne_eq, ite_eq_right_iff, one_ne_zero, imp_false, not_not] using hm
-      simp [f,hm]
-    _ = _ := by simp [hcard]
-
-/-- Intrinsic and ambient left inversions coincide on the plane subgroup. -/
-theorem leftInversion_iff (w t : planeSubgroup M cs P) :
-    B.system.IsLeftInversion w t ↔ cs.IsLeftInversion (w:W) (t:W) := by
-  by_cases ht : cs.IsReflection (t:W)
-  · let a := positiveRoot M cs ⟨(t:W),ht⟩
-    have ha := positiveRoot_represents M cs ⟨(t:W),ht⟩
-    have haP := (ha.planeSubgroup_mem_iff M cs).mp t.property
-    exact (B.inversion_iff ha haP a.property.2 w).symm.trans
-      (ha.inversion_iff a.property.2 (w:W))
-  · exact iff_of_false (fun h => ht ((B.reflection_iff t).mp h.1)) (fun h => ht h.1)
-
-/-- The intrinsic directed dihedral graph is the actual ambient graph
-restricted to the maximal plane subgroup. -/
-theorem bruhatStep_iff (g h : planeSubgroup M cs P) :
-    BruhatStep cs (g:W) (h:W) ↔ BruhatStep B.system g h := by
-  constructor
-  · rintro ⟨hlen,t,ht,he⟩
-    have htP : t ∈ planeSubgroup M cs P := by
-      have he' : t = (h:W) * (g:W)⁻¹ := by rw [he]; group
-      rw [he']
-      exact (planeSubgroup M cs P).mul_mem h.property
-        ((planeSubgroup M cs P).inv_mem g.property)
-    let tr : planeSubgroup M cs P := ⟨t,htP⟩
-    have htr : B.system.IsReflection tr := (B.reflection_iff tr).mpr ht
-    have het : h=tr*g := Subtype.ext he
-    have htg : t*(h:W)=(g:W) := by rw [he, ← mul_assoc, ht.mul_self, one_mul]
-    have hi := (B.leftInversion_iff h tr).mpr ⟨ht,by
-      change cs.length (t*(h:W)) < cs.length (h:W)
-      simpa only [htg] using hlen⟩
-    have htg' : tr*h=g := by rw [het, ← mul_assoc, htr.mul_self, one_mul]
-    exact ⟨by simpa only [htg'] using hi.2,tr,htr,het⟩
-  · rintro ⟨hlen,t,ht,he⟩
-    have het : t*h=g := by rw [he, ← mul_assoc, ht.mul_self, one_mul]
-    have hi := (B.leftInversion_iff h t).mp ⟨ht,by simpa only [het] using hlen⟩
-    refine ⟨?_,(t:W),(B.reflection_iff t).mp ht,congrArg Subtype.val he⟩
-    simpa only [← Subgroup.coe_mul, het] using hi.2
-
-/-- The exact canonical coset graph theorem needed by the reconstruction. -/
-theorem coset_step_iff {a₀ : W}
-    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-    (g h : planeSubgroup M cs P) :
-    BruhatStep cs ((g:W)*a₀) ((h:W)*a₀) ↔ BruhatStep B.system g h :=
-  (bruhatStep_coset_iff hmin g.property h.property).trans (B.bruhatStep_iff g h)
-
-/-- Relative dihedral paths give actual ambient Bruhat paths in the coset. -/
-theorem coset_bruhat {a₀ : W}
-    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-    {g h : planeSubgroup M cs P} (hgh : BruhatLE B.system g h) :
-    BruhatLE cs ((g:W)*a₀) ((h:W)*a₀) := by
-  induction hgh with
-  | refl => exact Relation.ReflTransGen.refl
-  | @tail y z _ hyz ih =>
-    exact ih.tail ((B.coset_step_iff hmin y z).mpr hyz)
-
-/-- The source inversion-count characterization holds on every minimal
-representative coset, with its actual induced length. -/
-theorem coset_length_eq {a₀ : W}
-    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-    (w : planeSubgroup M cs P) :
-    planeLength (M := M) (cs := cs) P ((w:W)*a₀) = (B.system.length w : ℤ) := by
-  classical
-  rw [← B.length_eq w]
-  let f := inversionVector (M := M) (cs := cs) ((w:W)*a₀)
-  let g := inversionVector (M := M) (cs := cs) (w:W)
-  change f.sum (fun a n => if a.val ∈ P then n else 0) =
-    g.sum (fun a n => if a.val ∈ P then n else 0)
-  rw [Finsupp.sum_of_support_subset f Finset.subset_union_left _ (by intro a; simp),
-    Finsupp.sum_of_support_subset g Finset.subset_union_right _ (by intro a; simp)]
-  apply Finset.sum_congr rfl
-  intro a _
-  by_cases haP : a.val ∈ P
-  · simp only [ite_eq_left haP, f, g, inversionVector_apply]
-    congr 1
-    exact propext (root_inversion_coset_iff hmin w.property a.property.1 haP)
-  · simp [haP]
-
-end PlaneBasis
-
-/-- Every nonsaturated reflection edge has a genuine rank-two contribution
-whose induced relative edge gap is at least three. -/
-theorem exists_positive_plane (x : W) (a : PositiveRoot M cs)
-    (hx : ¬ rootInverted x a)
-    (hlen : cs.length x + 1 < cs.length (rootRef a*x)) :
-    ∃ b : PositiveRoot M cs, a ≠ b ∧ 0 < planeWeight x a (rootPlane a b) := by
-  classical
-  have hsum := sum_planeWeight x a hx
-  have hp : 0 < (planeWeight x a).sum (fun _ n => n) := by omega
-  obtain ⟨P,hP,hpos⟩ := (Finset.sum_pos_iff_of_nonneg
-    (fun P _ => planeWeight_nonneg x a P)).mp hp
-  obtain ⟨b,hb,hbP⟩ := Finset.mem_image.mp (Finsupp.mapDomain_support hP)
-  have hb' : halfRoot x a b := by
-    simpa only [Finsupp.mem_support_iff, halfVector, rootIndicator_apply,
-      ne_eq, ite_eq_right_iff, one_ne_zero, imp_false, not_not] using hb
-  exact ⟨b,hb'.1.1.symm,hbP ▸ hpos⟩
-
-
-end
-end KLInvariance.TitsSpace
-
-namespace KLInvariance.Dihedral
-variable {G : Type*} [Group G] {N : CoxeterMatrix Bool}
-
-/-- Every nonsaturated dihedral edge contains a full eight-edge rectangle. -/
-theorem edge_diamond (ds : CoxeterSystem N G) {x y : G}
-    (hxy : BruhatStep ds x y) (hlen : ds.length x + 1 < ds.length y) :
-    ∃ a b c d : G, a ≠ b ∧ c ≠ d ∧
-      BruhatStep ds x a ∧ BruhatStep ds x b ∧
-      BruhatStep ds a c ∧ BruhatStep ds a d ∧
-      BruhatStep ds b c ∧ BruhatStep ds b d ∧
-      BruhatStep ds c y ∧ BruhatStep ds d y := by
-  have hodd := ((step_iff_odd ds x y).mp hxy).2
-  have hthree : ds.length x+2 < ds.length y := by
-    rw [Nat.odd_iff] at hodd
-    omega
-  obtain ⟨a,b,hab,hlev₁⟩ := two_elements_length ds
-    (by omega : 0 < ds.length x+1) hlen
-  obtain ⟨c,d,hcd,hlev₂⟩ := two_elements_length ds
-    (by omega : 0 < ds.length x+2) hthree
-  have ha := (hlev₁ a).mpr (Or.inl rfl)
-  have hb := (hlev₁ b).mpr (Or.inr rfl)
-  have hc := (hlev₂ c).mpr (Or.inl rfl)
-  have hd := (hlev₂ d).mpr (Or.inr rfl)
-  have hsucc (p q : G) (h : ds.length q=ds.length p+1) : BruhatStep ds p q := by
-    rw [step_iff_odd, Nat.odd_iff]
-    constructor <;> omega
-  have htop (p : G) (hp : ds.length p=ds.length x+2) : BruhatStep ds p y := by
-    rw [step_iff_odd, Nat.odd_iff]
-    rw [Nat.odd_iff] at hodd
-    constructor <;> omega
-  exact ⟨a,b,c,d,hab,hcd,hsucc x a ha,hsucc x b hb,
-    hsucc a c (by omega),hsucc a d (by omega),
-    hsucc b c (by omega),hsucc b d (by omega),htop c hc,htop d hd⟩
-
-end KLInvariance.Dihedral
-
-namespace KLInvariance.TitsSpace
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-
-/-- Every genuine ambient reflection edge of length gap greater than one
-contains a full eight-edge diamond. The chosen plane comes from an actual
-positive inversion contribution, avoiding the false arbitrary-plane claim. -/
-theorem ambient_edge_diamond {x y : W} (hxy : BruhatStep cs x y)
-    (hlen : cs.length x+1 < cs.length y) :
-    ∃ a b c d : W, a ≠ b ∧ c ≠ d ∧
-      BruhatStep cs x a ∧ BruhatStep cs x b ∧
-      BruhatStep cs a c ∧ BruhatStep cs a d ∧
-      BruhatStep cs b c ∧ BruhatStep cs b d ∧
-      BruhatStep cs c y ∧ BruhatStep cs d y := by
-  rcases hxy.2 with ⟨t,ht,het⟩
-  let a := positiveRoot M cs ⟨t,ht⟩
-  have ha := positiveRoot_represents M cs ⟨t,ht⟩
-  have hta : rootRef a=t := congrArg (fun r : {r : W // cs.IsReflection r} => r.val)
-    ((positiveRootEquiv M cs).symm_apply_apply ⟨t,ht⟩)
-  have hx : ¬ rootInverted x a := by
-    intro hi
-    have h := (ha.inversion_iff a.property.2 x).mp hi
-    have hh : cs.length (t*x) < cs.length x := h.2
-    rw [← het] at hh
-    exact (not_lt_of_gt hxy.1) hh
-  have hae : rootRef a*x=y := by rw [hta]; exact het.symm
-  obtain ⟨b,hab,hpos⟩ := exists_positive_plane x a hx (by rwa [hae])
-  obtain ⟨B⟩ := nonempty_planeBasis a b hab
-  let P := rootPlane a b
-  have haP : a.val ∈ P := Submodule.subset_span (Set.mem_insert _ _)
-  obtain ⟨a₀,g,hg,hgx,hmin⟩ := planeCoset_exists_minimal M cs P x
-  let X : planeSubgroup M cs P := ⟨g,hg⟩
-  let T : planeSubgroup M cs P := ⟨rootRef a,
-    ((rootReflection_represents M cs a).planeSubgroup_mem_iff M cs).mpr haP⟩
-  let Y := T*X
-  have hxEq : (X:W)*a₀=x := hgx.symm
-  have hyEq : (Y:W)*a₀=y := by
-    change (rootRef a*g)*a₀=y
-    rw [mul_assoc, ← hgx, hae]
-  have hrel : BruhatStep B.system X Y := (B.coset_step_iff hmin X Y).mp
-    (by simpa only [hxEq,hyEq] using hxy)
-  have hLx := B.coset_length_eq hmin X
-  have hLy := B.coset_length_eq hmin Y
-  rw [hxEq] at hLx
-  rw [hyEq] at hLy
-  have hdegree := planeWeight_relative_degree x a b hab hx
-  rw [hae,hLx,hLy] at hdegree
-  have hlen' : B.system.length X+1 < B.system.length Y := by omega
-  obtain ⟨p,q,r,s,hpq,hrs,hXp,hXq,hpr,hps,hqr,hqs,hrY,hsY⟩ :=
-    Dihedral.edge_diamond B.system hrel hlen'
-  have hinj : Function.Injective (fun z : planeSubgroup M cs P => (z:W)*a₀) := by
-    intro p q hpq
-    exact Subtype.ext (mul_right_cancel hpq)
-  refine ⟨(p:W)*a₀,(q:W)*a₀,(r:W)*a₀,(s:W)*a₀,
-    fun h => hpq (hinj h),fun h => hrs (hinj h),?_,?_,?_,?_,?_,?_,?_,?_⟩
-  · simpa only [hxEq] using (B.coset_step_iff hmin X p).mpr hXp
-  · simpa only [hxEq] using (B.coset_step_iff hmin X q).mpr hXq
-  · exact (B.coset_step_iff hmin p r).mpr hpr
-  · exact (B.coset_step_iff hmin p s).mpr hps
-  · exact (B.coset_step_iff hmin q r).mpr hqr
-  · exact (B.coset_step_iff hmin q s).mpr hqs
-  · simpa only [hyEq] using (B.coset_step_iff hmin r Y).mpr hrY
-  · simpa only [hyEq] using (B.coset_step_iff hmin s Y).mpr hsY
-
-end KLInvariance.TitsSpace
-
-namespace KLInvariance
-variable {I W : Type*} [Fintype I] [Group W] {M : CoxeterMatrix I}
-
-/-- General unlabelled Bruhat graph reconstruction completeness, for genuine
-finite-rank Coxeter systems and arbitrary bottoms and interval ranks. -/
-theorem diamondEdges_of_bruhatStep (cs : CoxeterSystem M W)
-    {u b : W} {x y : Interval cs u b} (hxy : BruhatStep cs x.val y.val) :
-    DiamondEdges x y := by
-  suffices ∀ n : ℕ, ∀ x y : Interval cs u b,
-      cs.length y.val-cs.length x.val=n → BruhatStep cs x.val y.val →
-      DiamondEdges x y by
-    exact this _ x y rfl hxy
-  intro n
-  induction n using Nat.strong_induction_on with
-  | h n ih =>
-    intro x y hn hxy
-    by_cases he : cs.length y.val=cs.length x.val+1
-    · exact DiamondEdges.cover (interval_covBy_of_length_succ cs
-        (Relation.ReflTransGen.single hxy) he)
-    · have hn' : cs.length x.val+1 < cs.length y.val := by have := hxy.1; omega
-      obtain ⟨a,bm,c,d,hab,hcd,hxa,hxb,hac,had,hbc,hbd,hcy,hdy⟩ :=
-        TitsSpace.ambient_edge_diamond hxy hn'
-      have leOf {p q : W} (h : BruhatStep cs p q) : BruhatLE cs p q :=
-        Relation.ReflTransGen.single h
-      let A : Interval cs u b := ⟨a,bruhat_trans cs x.property.1 (leOf hxa),
-        bruhat_trans cs (leOf hac) (bruhat_trans cs (leOf hcy) y.property.2)⟩
-      let B : Interval cs u b := ⟨bm,bruhat_trans cs x.property.1 (leOf hxb),
-        bruhat_trans cs (leOf hbc) (bruhat_trans cs (leOf hcy) y.property.2)⟩
-      let C : Interval cs u b := ⟨c,
-        bruhat_trans cs x.property.1 (bruhat_trans cs (leOf hxa) (leOf hac)),
-        bruhat_trans cs (leOf hcy) y.property.2⟩
-      let D : Interval cs u b := ⟨d,
-        bruhat_trans cs x.property.1 (bruhat_trans cs (leOf hxa) (leOf had)),
-        bruhat_trans cs (leOf hdy) y.property.2⟩
-      have hsmall {p q : Interval cs u b} (hpq : BruhatStep cs p.val q.val)
-          (hp : cs.length x.val ≤ cs.length p.val)
-          (hq : cs.length q.val ≤ cs.length y.val)
-          (hs : cs.length x.val < cs.length p.val ∨ cs.length q.val < cs.length y.val) :
-          DiamondEdges p q := by
-        exact ih _ (by omega) p q rfl hpq
-      have h₁ := hxa.1
-      have h₂ := hxb.1
-      have h₃ := hac.1
-      have h₄ := had.1
-      have h₅ := hbc.1
-      have h₆ := hbd.1
-      have h₇ := hcy.1
-      have h₈ := hdy.1
-      refine DiamondEdges.diamond
-        (hsmall (p := x) (q := A) hxa ?_ ?_ ?_)
-        (hsmall (p := x) (q := B) hxb ?_ ?_ ?_)
-        (hsmall (p := A) (q := C) hac ?_ ?_ ?_)
-        (hsmall (p := A) (q := D) had ?_ ?_ ?_)
-        (hsmall (p := B) (q := C) hbc ?_ ?_ ?_)
-        (hsmall (p := B) (q := D) hbd ?_ ?_ ?_)
-        (hsmall (p := C) (q := y) hcy ?_ ?_ ?_)
-        (hsmall (p := D) (q := y) hdy ?_ ?_ ?_)
-        (fun h => hab (congrArg Subtype.val h))
-        (fun h => hcd (congrArg Subtype.val h))
-      all_goals first | omega | (dsimp only [A,B,C,D]; omega)
-
-/-- An abstract interval order isomorphism determines its actual directed
-reflection edges. No root labels, generators or geometric data are supplied to
-that isomorphism. This proves the finite-rank graph input of the source main. -/
-theorem intervalIso_bruhatStep_iff {I' W' : Type*} [Fintype I'] [Group W']
-    {M' : CoxeterMatrix I'} (cs : CoxeterSystem M W) (cs' : CoxeterSystem M' W')
-    {u b : W} {u' b' : W'} (e : Interval cs u b ≃o Interval cs' u' b')
-    (x y : Interval cs u b) :
-    BruhatStep cs' (e x).val (e y).val ↔ BruhatStep cs x.val y.val := by
-  constructor
-  · intro h
-    exact diamondEdges_bruhatStep cs ((DiamondEdges.map_iff e).mp
-      (diamondEdges_of_bruhatStep cs' h))
-  · intro h
-    exact diamondEdges_bruhatStep cs' ((diamondEdges_of_bruhatStep cs h).map e)
-
-end KLInvariance
-
-end
-
-
-section
-
-/-! Rank-two Frobenius extension for an actual reflection on a symmetric
-algebra. This supplies the elementary simple-reflection duality input; it
-is not a proof of BMP or Elias--Williamson self-duality/characters. -/
-namespace KLInvariance.ReflectionFrobenius
-universe uk ua uv
-variable {k : Type uk} [Field k] [CharZero k]
-  {A : Type ua} [CommRing A] [IsDomain A] [Algebra k A]
-
-noncomputable section
-
-def fixed (τ : A ≃ₐ[k] A) : Subalgebra k A :=
-  AlgHom.equalizer τ.toAlgHom (AlgHom.id k A)
-
-omit [CharZero k] [IsDomain A] in
-@[simp] theorem mem_fixed (τ : A ≃ₐ[k] A) (a : A) :
-    a ∈ fixed τ ↔ τ a = a := Iff.rfl
-
-omit [CharZero k] [IsDomain A] in
- theorem fixed_val (τ : A ≃ₐ[k] A) (a : fixed τ) : τ (a : A) = a := a.property
-
-variable (τ : A ≃ₐ[k] A) (α : A)
-
-def expand : (fixed τ × fixed τ) →ₗ[fixed τ] A where
-  toFun a := (a.1 : A) + α * (a.2 : A)
-  map_add' a b := by simp; ring
-  map_smul' c a := by simp only [Prod.smul_fst, Prod.smul_snd, smul_eq_mul,
-    Subalgebra.coe_mul, Subalgebra.smul_def, RingHom.id_apply]; ring
-
-omit [CharZero k] [IsDomain A] in
-@[simp] theorem expand_apply (a : fixed τ × fixed τ) :
-    expand τ α a = (a.1 : A) + α * (a.2 : A) := rfl
-
-variable (hinv : Function.Involutive τ) (hα : α ≠ 0) (hanti : τ α = -α)
-  (hdiv : ∀ a : A, α ∣ a - τ a)
-
-omit [CharZero k] in
-include hinv hα hanti in
- theorem quotient_fixed {a d : A} (hd : a - τ a = α * d) : τ d = d := by
-  have hh := congrArg τ hd
-  simp only [map_sub, map_mul, hinv a, hanti] at hh
-  apply mul_left_cancel₀ hα
-  linear_combination hh + hd
-
-include hα hanti in
- theorem expand_injective : Function.Injective (expand τ α) := by
-  intro a b hab
-  have ht := congrArg τ hab
-  simp only [expand_apply, map_add, map_mul, fixed_val, hanti] at ht
-  have he : (2 : A) * α * ((a.2 : A) - (b.2 : A)) = 0 := by
-    simp only [expand_apply] at hab
-    linear_combination hab - ht
-  have htwo : (2 : A) ≠ 0 := by
-    have hi : Function.Injective (algebraMap k A) := (algebraMap k A).injective
-    exact fun h => two_ne_zero (hi (by simpa only [map_ofNat, map_zero] using h : algebraMap k A (2:k) = algebraMap k A 0))
-  have h2 : (a.2 : A) = (b.2 : A) := sub_eq_zero.mp
-    ((mul_eq_zero.mp he).resolve_left (mul_ne_zero htwo hα))
-  have h1 : (a.1 : A) = (b.1 : A) := by
-    simpa only [expand_apply, h2, add_left_inj] using hab
-  exact Prod.ext (Subtype.ext h1) (Subtype.ext h2)
-
-include hinv hα hanti hdiv in
- theorem expand_surjective : Function.Surjective (expand τ α) := by
-  intro a
-  obtain ⟨d, hd⟩ := hdiv a
-  have hdf : τ d = d := quotient_fixed τ α hinv hα hanti hd
-  let h : k := (2:k)⁻¹
-  have htwo : (2:A) * algebraMap k A h = 1 := by
-    rw [show (2:A) = algebraMap k A (2:k) by exact (map_ofNat (algebraMap k A) 2).symm, ← map_mul]
-    simp [h]
-  let c₀ : fixed τ := ⟨algebraMap k A h * (a + τ a), by
-    simp only [mem_fixed, map_mul, AlgEquiv.commutes, map_add, hinv a]
-    ring⟩
-  let c₁ : fixed τ := ⟨algebraMap k A h * d, by
-    simp only [mem_fixed, map_mul, AlgEquiv.commutes, hdf]⟩
-  refine ⟨(c₀,c₁), ?_⟩
-  change algebraMap k A h * (a + τ a) + α * (algebraMap k A h * d) = a
-  linear_combination a * htwo - (algebraMap k A h) * hd
-
-/-- The actual invariant-ring basis `1, α`, with no finiteness assumption. -/
-def coordinates : A ≃ₗ[fixed τ] (fixed τ × fixed τ) :=
-  (LinearEquiv.ofBijective (expand τ α)
-    ⟨expand_injective τ α hα hanti, expand_surjective τ α hinv hα hanti hdiv⟩).symm
-
-@[simp] theorem expand_coordinates (a : A) :
-    expand τ α (coordinates τ α hinv hα hanti hdiv a) = a :=
-  (coordinates τ α hinv hα hanti hdiv).symm_apply_apply a
-
-@[simp] theorem coordinates_expand (a : fixed τ × fixed τ) :
-    coordinates τ α hinv hα hanti hdiv (expand τ α a) = a :=
-  (coordinates τ α hinv hα hanti hdiv).apply_symm_apply a
-
-
-def trace : A →ₗ[fixed τ] fixed τ :=
-  (LinearMap.snd _ _ _).comp (coordinates τ α hinv hα hanti hdiv).toLinearMap
-
- theorem trace_expand (a : fixed τ × fixed τ) :
-    trace τ α hinv hα hanti hdiv (expand τ α a) = a.2 := by
-  change (coordinates τ α hinv hα hanti hdiv (expand τ α a)).2 = a.2
-  rw [coordinates_expand]
-
-omit [CharZero k] [IsDomain A] in
-include hanti in
- theorem square_fixed : α * α ∈ fixed τ := by
-  simp only [mem_fixed, map_mul, hanti, neg_mul_neg]
-
- theorem trace_mul (a b : A) :
-    trace τ α hinv hα hanti hdiv (a*b) =
-      (coordinates τ α hinv hα hanti hdiv a).1 *
-        (coordinates τ α hinv hα hanti hdiv b).2 +
-      (coordinates τ α hinv hα hanti hdiv a).2 *
-        (coordinates τ α hinv hα hanti hdiv b).1 := by
-  let c := coordinates τ α hinv hα hanti hdiv a
-  let d := coordinates τ α hinv hα hanti hdiv b
-  let α₂ : fixed τ := ⟨α*α, square_fixed τ α hanti⟩
-  have he : a*b = expand τ α
-      (c.1*d.1+α₂*c.2*d.2, c.1*d.2+c.2*d.1) := by
-    rw [← expand_coordinates τ α hinv hα hanti hdiv a,
-      ← expand_coordinates τ α hinv hα hanti hdiv b]
-    change ((c.1:A)+α*(c.2:A))*((d.1:A)+α*(d.2:A)) =
-      (c.1:A)*(d.1:A)+(α*α)*(c.2:A)*(d.2:A)+
-        α*((c.1:A)*(d.2:A)+(c.2:A)*(d.1:A))
-    ring
-  rw [he,trace_expand]
-
-section Pair
-variable (R : Type*) [CommRing R]
-
-def crossPair : (R × R) →ₗ[R] Module.Dual R (R × R) where
-  toFun a :=
-    { toFun := fun b => a.1*b.2+a.2*b.1
-      map_add' := by intro b c; simp; ring
-      map_smul' := by intro r b; simp; ring }
-  map_add' := by intro a b; apply LinearMap.ext; intro c; simp; ring
-  map_smul' := by intro r a; apply LinearMap.ext; intro b; simp; ring
-
- theorem crossPair_bijective : Function.Bijective (crossPair R) := by
-  constructor
-  · intro a b hab
-    have h₁ := congrArg (fun f : Module.Dual R (R×R) => f (0,1)) hab
-    have h₂ := congrArg (fun f : Module.Dual R (R×R) => f (1,0)) hab
-    exact Prod.ext (by simpa [crossPair] using h₁) (by simpa [crossPair] using h₂)
-  · intro f
-    refine ⟨(f (0,1), f (1,0)), ?_⟩
-    apply LinearMap.ext
-    intro a
-    have ha : a = a.1 • (1,0) + a.2 • (0,1) := by simp
-    change f (0,1)*a.2 + f (1,0)*a.1 = f a
-    calc
-      _ = a.1 * f (1,0) + a.2 * f (0,1) := by ring
-      _ = f (a.1 • (1,0) + a.2 • (0,1)) := by rw [map_add,map_smul,map_smul]; rfl
-      _ = f a := congrArg f ha.symm
-
-def crossPairEquiv : (R × R) ≃ₗ[R] Module.Dual R (R × R) :=
-  LinearEquiv.ofBijective (crossPair R) (crossPair_bijective R)
-
-end Pair
-
-/-- The Frobenius duality over the invariant ring is integral and perfect. -/
-def duality : A ≃ₗ[fixed τ] Module.Dual (fixed τ) A :=
-  (coordinates τ α hinv hα hanti hdiv).trans
-    ((crossPairEquiv (fixed τ)).trans
-      (coordinates τ α hinv hα hanti hdiv).dualMap)
-
- theorem duality_apply (a b : A) :
-    duality τ α hinv hα hanti hdiv a b = trace τ α hinv hα hanti hdiv (a*b) := by
-  rw [trace_mul]
-  rfl
-
- theorem duality_balanced (a b c : A) :
-    duality τ α hinv hα hanti hdiv (c*a) b =
-      duality τ α hinv hα hanti hdiv a (c*b) := by
-  rw [duality_apply,duality_apply]
-  congr 1
-  ring
-
-
-section Symmetric
-variable {V : Type uv} [AddCommGroup V] [Module k V]
-
-def symmetricMap (e : V ≃ₗ[k] V) : SymmetricAlgebra k V →ₐ[k] SymmetricAlgebra k V :=
-  SymmetricAlgebra.lift ((SymmetricAlgebra.ι k V).comp e.toLinearMap)
-
-omit [CharZero k] in
-@[simp] theorem symmetricMap_ι (e : V ≃ₗ[k] V) (v : V) :
-    symmetricMap e (SymmetricAlgebra.ι k V v) = SymmetricAlgebra.ι k V (e v) := by
-  simp [symmetricMap]
-
-omit [CharZero k] in
- theorem symmetricMap_comp (e d : V ≃ₗ[k] V) :
-    (symmetricMap d).comp (symmetricMap e) = symmetricMap (e.trans d) := by
-  apply SymmetricAlgebra.algHom_ext
-  ext v
-  simp
-
-omit [CharZero k] in
- theorem symmetricMap_refl : symmetricMap (LinearEquiv.refl k V) =
-    AlgHom.id k (SymmetricAlgebra k V) := by
-  apply SymmetricAlgebra.algHom_ext
-  ext v
-  simp
-
-/-- The algebra action is induced by the linear reflection, not postulated. -/
-def symmetricAction (e : V ≃ₗ[k] V) : SymmetricAlgebra k V ≃ₐ[k] SymmetricAlgebra k V :=
-  AlgEquiv.ofAlgHom (symmetricMap e) (symmetricMap e.symm)
-    (by rw [symmetricMap_comp,LinearEquiv.symm_trans_self,symmetricMap_refl])
-    (by rw [symmetricMap_comp,LinearEquiv.self_trans_symm,symmetricMap_refl])
-
-omit [CharZero k] in
-@[simp] theorem symmetricAction_ι (e : V ≃ₗ[k] V) (v : V) :
-    symmetricAction e (SymmetricAlgebra.ι k V v) = SymmetricAlgebra.ι k V (e v) :=
-  symmetricMap_ι e v
-
-variable (root : V) (coroot : Module.Dual k V) (hroot : coroot root = 2)
-
-abbrev reflectionAction : SymmetricAlgebra k V ≃ₐ[k] SymmetricAlgebra k V :=
-  symmetricAction (Module.reflection hroot)
-
-omit [CharZero k] in
- theorem reflectionAction_involutive : Function.Involutive (reflectionAction root coroot hroot) := by
-  intro a
-  have he : (reflectionAction root coroot hroot).toAlgHom.comp
-      (reflectionAction root coroot hroot).toAlgHom =
-        AlgHom.id k (SymmetricAlgebra k V) := by
-    apply SymmetricAlgebra.algHom_ext
-    ext v
-    simp [Module.involutive_reflection hroot v]
-  exact congrArg (fun f : SymmetricAlgebra k V →ₐ[k] SymmetricAlgebra k V => f a) he
-
-omit [CharZero k] in
- theorem reflectionAction_anti :
-    reflectionAction root coroot hroot (SymmetricAlgebra.ι k V root) =
-      -SymmetricAlgebra.ι k V root := by
-  simp
-
-include hroot in
- theorem reflectionRoot_ne_zero : SymmetricAlgebra.ι k V root ≠ 0 := by
-  intro he
-  have hh := congrArg (SymmetricAlgebra.lift coroot) he
-  rw [SymmetricAlgebra.lift_ι_apply,map_zero,hroot] at hh
-  exact two_ne_zero hh
-
-omit [CharZero k] in
- theorem reflectionAction_divisible (a : SymmetricAlgebra k V) :
-    SymmetricAlgebra.ι k V root ∣ a-reflectionAction root coroot hroot a := by
-  induction a using SymmetricAlgebra.induction with
-  | algebraMap c => simp
-  | ι v =>
-    refine ⟨algebraMap k (SymmetricAlgebra k V) (coroot v), ?_⟩
-    rw [reflectionAction,symmetricAction_ι,Module.reflection_apply,map_sub,map_smul]
-    simp only [sub_sub_cancel,Algebra.smul_def]
-    ring
-  | mul a b ha hb =>
-    obtain ⟨c,hc⟩ := ha
-    obtain ⟨d,hd⟩ := hb
-    refine ⟨c*b+(reflectionAction root coroot hroot a)*d, ?_⟩
-    rw [map_mul]
-    linear_combination b*hc+(reflectionAction root coroot hroot a)*hd
-  | add a b ha hb =>
-    simpa only [map_add, add_sub_add_comm] using dvd_add ha hb
-
-/-- A genuine simple-reflection integral Frobenius extension in characteristic
-zero, for the symmetric algebra of any representation carrying the reflection. -/
-def reflectionDuality :
-    SymmetricAlgebra k V ≃ₗ[fixed (reflectionAction root coroot hroot)]
-      Module.Dual (fixed (reflectionAction root coroot hroot)) (SymmetricAlgebra k V) :=
-  duality (reflectionAction root coroot hroot) (SymmetricAlgebra.ι k V root)
-    (reflectionAction_involutive root coroot hroot) (reflectionRoot_ne_zero root coroot hroot)
-    (reflectionAction_anti root coroot hroot) (reflectionAction_divisible root coroot hroot)
-
-end Symmetric
-
-end
-end KLInvariance.ReflectionFrobenius
-
-end
-
-
-section
-
-/-! Frobenius extensions for the actual simple reflections of the extended
-theorem realization, not abstract reflection data carried by the target. -/
-
-end
-
-end OAI
+import OAI.RepresentationTheory.KazhdanLusztig.SheafMorphisms
+
+namespace OAI
+
+/-!
+Graded decomposition characters, reflection-plane algebras and reflection Frobenius structures.
+-/
+
+section
+
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded Polynomial
+universe uk us um
+variable {k : Type uk} [Field k] {σ : Type us} [Finite σ]
+  {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
+noncomputable section
+
+ def freeStalkChar (B : GradedSheaf (𝓐 := polynomialGrading k σ) G)
+    (hf : ∀ x, Module.Free (MvPolynomial σ k) (B.vertex x)) (x : V) : Polynomial ℤ := by
+  let := hf x
+  exact (B.vertex x).char
+
+ theorem ModelSum.character {α : E → MvPolynomial σ k}
+    (O : BoundaryModels (polynomialGrading_negative k σ) (G := G) α)
+    [Fintype V] {U : Set V} {B : GradedSheaf (𝓐 := polynomialGrading k σ) G}
+    (h : ModelSum O.sheaf U B) :
+    ∃ hmul : V → Polynomial ℤ,
+      (∀ y n, 0 ≤ (hmul y).coeff n) ∧
+      (∀ y, y ∉ U → hmul y=0) ∧
+      ∀ x, freeStalkChar B (h.free O) x =
+        ∑ y, hmul y*freeStalkChar (O.sheaf y) (O.free y) x := by
+  classical
+  induction h with
+  | zero =>
+    refine ⟨0,by simp,by simp,?_⟩
+    intro x
+    simp only [Pi.zero_apply,zero_mul,Finset.sum_const_zero]
+    exact ModuleData.char_zero _
+  | @add B h c hc n d ih =>
+    obtain ⟨m,hm,hsupp,heq⟩ := ih
+    let a : Polynomial ℤ := ∑ i, X^(d i)
+    have ha (r : ℕ) : 0 ≤ a.coeff r := by
+      simp only [a,Polynomial.finsetSum_coeff,Polynomial.coeff_X_pow]
+      exact Finset.sum_nonneg (fun i _ => by split <;> norm_num)
+    refine ⟨fun y => m y+if y=c then a else 0,?_,?_,?_⟩
+    · intro y r
+      rw [Polynomial.coeff_add]
+      apply add_nonneg (hm y r)
+      split
+      · exact ha r
+      · simp
+    · intro y hy
+      have hne : y≠c := fun hh => hy (hh ▸ hc)
+      simp [hsupp y hy,hne]
+    · intro x
+      let _ := h.free O x
+      let _ := O.free c x
+      change ((B.vertex x).prod (ModuleData.pi (fun i => ((O.sheaf c).vertex x).shift (d i)))).char = _
+      rw [ModuleData.char_prod,ModuleData.char_pi]
+      simp only [ModuleData.char_shift]
+      change freeStalkChar B (h.free O) x + _ = _
+      rw [heq x]
+      simp only [add_mul,Finset.sum_add_distrib]
+      congr 1
+      rw [← Finset.sum_mul]
+      simp only [ite_mul,zero_mul,Finset.sum_ite_eq',Finset.mem_univ,ite_true]
+      rfl
+
+ theorem exists_stalkCharacter_decomposition [Fintype V] [Fintype E]
+    {α : E → MvPolynomial σ k}
+    (O : BoundaryModels (polynomialGrading_negative k σ) (G := G) α)
+    (B : GradedSheaf (𝓐 := polynomialGrading k σ) G)
+    (hf : ∀ x, Module.Free (MvPolynomial σ k) (B.vertex x))
+    (hq : B.UpperQuotient α) (hfl : B.forget.Flabby) (hg : B.forget.Generated) :
+    ∃ hmul : V → Polynomial ℤ, (∀ y n, 0 ≤ (hmul y).coeff n) ∧
+      ∀ x, freeStalkChar B hf x =
+        ∑ y, hmul y*freeStalkChar (O.sheaf y) (O.free y) x := by
+  obtain ⟨C,hC,⟨j⟩⟩ := exists_modelSumIso (polynomialGrading_zero k σ) O B hf hq hfl hg
+  obtain ⟨m,hm,_,heq⟩ := hC.character O
+  refine ⟨m,hm,fun x => ?_⟩
+  rw [← heq x]
+  let _ := hf x
+  let _ := hC.free O x
+  exact ModuleData.character_eq_of_gradedEquiv (j.vertex x)
+    (j.vertex_graded x)
+
+end
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph Polynomial
+universe u
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W) (b : W)
+  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+ def lowerCharacter (c x : Interval cs 1 b) : Polynomial ℤ :=
+  if h : x≤c then stalkCharacter cs 1 c.val basis (one_bruhat cs c.val)
+    ⟨x.val,x.property.1,h⟩ else 0
+
+ theorem boundaryModels_character (c x : Interval cs 1 b) :
+    GradedSheaf.freeStalkChar ((boundaryModels cs b basis).sheaf c)
+      ((boundaryModels cs b basis).free c) x=lowerCharacter cs b basis c x := by
+  let _ := (boundaryModels cs b basis).free c x
+  by_cases hx : x≤c
+  · rw [lowerCharacter,dite_eq_left hx]
+    let B := lowerIntervalBoundary cs b basis c
+    let _ := B.sheaf.lowerExtend_free B.free x
+    let _ := (boundarySheaf cs 1 c.val basis (one_bruhat cs c.val)).free ⟨x.val,x.property.1,hx⟩
+    exact ModuleData.character_eq_of_gradedEquiv (B.sheaf.lowerExtendVertex ⟨x,hx⟩)
+      (B.sheaf.lowerExtendVertex_graded ⟨x,hx⟩)
+  · rw [lowerCharacter,dite_eq_right hx]
+    let _ := (boundaryModels cs b basis).support c x hx
+    exact ModuleData.char_zero _
+
+ theorem lowerCharacter_zero (c x : Interval cs 1 b) (hx : ¬x≤c) :
+    lowerCharacter cs b basis c x=0 := dite_eq_right hx
+
+ theorem lowerCharacter_nonneg (c x : Interval cs 1 b) (n : ℕ) :
+    0≤(lowerCharacter cs b basis c x).coeff n := by
+  unfold lowerCharacter
+  split
+  · exact stalkCharacter_coeff_nonneg cs 1 c.val basis (one_bruhat cs c.val) _ n
+  · simp
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+/-! Integral pair-projection quotients. The purpose is the actual localization
+functor from a tensor bimodule to a moment-graph sheaf; no flabbiness, stalk
+freeness, BMP identification or character theorem is assumed or asserted. -/
+namespace KLInvariance.PairProjection
+universe u v w z
+variable {R : Type u} [CommRing R]
+  {M : Type v} [AddCommGroup M] [Module R M]
+  {X : Type w} [AddCommGroup X] [Module R X]
+  {Y : Type z} [AddCommGroup Y] [Module R Y]
+  (f : M →ₗ[R] X) (g : M →ₗ[R] Y)
+noncomputable section
+
+def relation : Submodule R (X × Y) := LinearMap.range (f.prod g)
+abbrev Edge := (X × Y) ⧸ relation f g
+
+def lower : X →ₗ[R] Edge f g := (relation f g).mkQ.comp (LinearMap.inl R X Y)
+def upper : Y →ₗ[R] Edge f g := -((relation f g).mkQ.comp (LinearMap.inr R X Y))
+
+theorem diagonal_zero (m : M) : (relation f g).mkQ (f m,g m)=0 := by
+  exact (Submodule.Quotient.mk_eq_zero _).mpr ⟨m,rfl⟩
+
+theorem compatible (m : M) : lower f g (f m)=upper f g (g m) := by
+  have h := diagonal_zero f g m
+  have hp : (f m,g m)=(f m,0)+(0,g m) := by simp
+  rw [hp,map_add] at h
+  exact eq_neg_of_add_eq_zero_left h
+
+theorem compatible_iff (x : X) (y : Y) :
+    lower f g x=upper f g y ↔ ∃ m : M, f m=x ∧ g m=y := by
+  change (relation f g).mkQ (x,0) = -((relation f g).mkQ (0,y)) ↔ _
+  rw [eq_neg_iff_add_eq_zero,← map_add]
+  rw [show ((x,0)+(0,y) : X × Y)=(x,y) by simp]
+  change (Submodule.Quotient.mk (x,y) : Edge f g)=0 ↔ _
+  rw [Submodule.Quotient.mk_eq_zero]
+  constructor
+  · rintro ⟨m,hm⟩
+    exact ⟨m,congrArg Prod.fst hm,congrArg Prod.snd hm⟩
+  · rintro ⟨m,hm,hn⟩
+    exact ⟨m,Prod.ext hm hn⟩
+
+theorem kernel_lower (x : X) : lower f g x=0 ↔ ∃ m : M, f m=x ∧ g m=0 := by
+  change (relation f g).mkQ (x,0)=0 ↔ _
+  change Submodule.Quotient.mk (x,0) = 0 ↔ _
+  rw [Submodule.Quotient.mk_eq_zero]
+  constructor
+  · rintro ⟨m,hm⟩
+    exact ⟨m,congrArg Prod.fst hm,congrArg Prod.snd hm⟩
+  · rintro ⟨m,hm,hn⟩
+    exact ⟨m,Prod.ext hm hn⟩
+
+theorem lower_surjective (hg : Function.Surjective g) :
+    Function.Surjective (lower f g) := by
+  intro v
+  obtain ⟨⟨x,y⟩,rfl⟩ := (relation f g).mkQ_surjective v
+  obtain ⟨m,rfl⟩ := hg y
+  refine ⟨x-f m,?_⟩
+  change (relation f g).mkQ (x-f m,0) = (relation f g).mkQ (x,g m)
+  have hp : (x-f m,0)=(x,g m)-(f m,g m) := by simp
+  rw [hp,map_sub,diagonal_zero,sub_zero]
+
+theorem upper_surjective (hf : Function.Surjective f) :
+    Function.Surjective (upper f g) := by
+  intro v
+  obtain ⟨⟨x,y⟩,rfl⟩ := (relation f g).mkQ_surjective v
+  obtain ⟨m,rfl⟩ := hf x
+  refine ⟨g m-y,?_⟩
+  change -((relation f g).mkQ (0,g m-y)) = (relation f g).mkQ (f m,y)
+  have hp : (0,g m-y)=(f m,g m)-(f m,y) := by simp
+  rw [hp,map_sub,diagonal_zero,zero_sub,neg_neg]
+
+/-- A genuine right-action eigenvalue difference annihilates the edge
+quotient. This uses only the integral module, not a fraction-field equality. -/
+theorem difference_annihilates (hf : Function.Surjective f)
+    (hg : Function.Surjective g) (T : M →ₗ[R] M) (r s : R)
+    (hr : ∀ m, f (T m)=r • f m) (hs : ∀ m, g (T m)=s • g m)
+    (v : Edge f g) : (r-s) • v=0 := by
+  obtain ⟨x,rfl⟩ := lower_surjective f g hg v
+  rw [← map_smul]
+  apply (kernel_lower f g _).mpr
+  obtain ⟨m,rfl⟩ := hf x
+  refine ⟨T m-s • m,?_,?_⟩
+  · rw [map_sub,map_smul,hr,sub_smul]
+  · rw [map_sub,map_smul,hs,sub_self]
+
+end
+end KLInvariance.PairProjection
+
+end
+
+
+section
+
+
+namespace KLInvariance.Dihedral
+noncomputable section
+
+variable {G : Type*} [Group G]
+
+/-- The matrix of a pair of distinct involutions. -/
+def pairMatrix (s t : G) (hs : s * s = 1) (hst : s ≠ t) : CoxeterMatrix Bool where
+  M i j := if i = j then 1 else orderOf (s * t)
+  isSymm := by
+    ext i j
+    change (if j = i then 1 else orderOf (s*t)) = if i = j then 1 else orderOf (s*t)
+    cases i <;> cases j <;> rfl
+  diagonal i := ite_eq_left rfl
+  off_diagonal i j hij := by
+    rw [ite_eq_right hij]
+    intro h
+    have he := orderOf_eq_one_iff.mp h
+    apply hst
+    calc
+      s = s * (s * t) := by rw [he, mul_one]
+      _ = t := by rw [← mul_assoc, hs, one_mul]
+
+/-- Alternating even words are powers of the product of the two generators. -/
+theorem wordProd_alt_even {N : CoxeterMatrix Bool} {H : Type*} [Group H]
+    (cs : CoxeterSystem N H) (i : Bool) (n : ℕ) :
+    cs.wordProd (alt i (2 * n)) = (cs.simple i * cs.simple (!i)) ^ n := by
+  induction n with
+  | zero => simp
+  | succ n ih =>
+    rw [show 2 * (n + 1) = 2 * n + 2 by omega, alt_add_two]
+    rw [cs.wordProd_cons, cs.wordProd_cons, ih, pow_succ']
+    exact (mul_assoc _ _ _).symm
+
+/-- Homomorphisms out of a rank-two Coxeter group that do not lower
+its product order and do not kill either generator are injective. -/
+theorem injective_of_simple_order {N : CoxeterMatrix Bool} {H : Type*} [Group H]
+    (cs : CoxeterSystem N H) (f : H →* G)
+    (hne : ∀ i, f (cs.simple i) ≠ 1)
+    (horder : ∀ i, orderOf (f (cs.simple i) * f (cs.simple (!i))) = N i (!i)) :
+    Function.Injective f := by
+  apply (injective_iff_map_eq_one f).mpr
+  intro w hw
+  rcases Nat.even_or_odd (cs.length w) with he | ho
+  · obtain ⟨n, hn⟩ := he
+    have hn' : cs.length w = 2 * n := by omega
+    obtain ⟨i, hi⟩ := exists_alt cs w
+    rw [hn', wordProd_alt_even] at hi
+    have hpow : (f (cs.simple i) * f (cs.simple (!i))) ^ n = 1 := by
+      simpa only [hi, map_pow, map_mul] using hw
+    have hdiv := orderOf_dvd_of_pow_eq_one hpow
+    rw [horder i] at hdiv
+    obtain ⟨k, hk⟩ := hdiv
+    rw [hi, hk, pow_mul, cs.simple_mul_simple_pow, one_pow]
+  · obtain ⟨a, i, ha⟩ := (reflection_iff_odd cs w).mpr ho
+    have h := hw
+    rw [ha, map_mul, map_mul, map_inv] at h
+    apply False.elim
+    apply hne i
+    have h' := congrArg (fun z => (f a)⁻¹ * z * f a) h
+    simpa only [mul_assoc, inv_mul_cancel_left, inv_mul_cancel, mul_one,
+      one_mul, mul_inv_cancel_left] using h'
+
+variable (s t : G) (hs : s * s = 1) (ht : t * t = 1) (hst : s ≠ t)
+
+/-- The map from the genuine presented rank-two Coxeter group. -/
+def pairLift : (pairMatrix s t hs hst).Group →* G :=
+  (pairMatrix s t hs hst).toCoxeterSystem.lift
+    ⟨(fun i => if i then t else s), by
+      intro i j
+      cases i <;> cases j
+      · simpa [pairMatrix] using hs
+      · simp only [pairMatrix, Bool.false_eq_true, ↓reduceIte]
+        exact pow_orderOf_eq_one (s*t)
+      · simp only [pairMatrix, Bool.true_eq_false, Bool.false_eq_true, ↓reduceIte]
+        have hinv : (s*t)⁻¹ = t*s := by
+          rw [mul_inv_rev, (inv_eq_of_mul_eq_one_right ht), (inv_eq_of_mul_eq_one_right hs)]
+        rw [← hinv, inv_pow, pow_orderOf_eq_one, inv_one]
+      · simpa [pairMatrix] using ht⟩
+
+@[simp] theorem pairLift_simple (i : Bool) :
+    pairLift s t hs ht hst ((pairMatrix s t hs hst).toCoxeterSystem.simple i) =
+      if i then t else s := by
+  apply CoxeterSystem.lift_apply_simple
+
+/-- No relation other than the Coxeter relations is introduced by the pair. -/
+theorem pairLift_injective (hs1 : s ≠ 1) (ht1 : t ≠ 1) :
+    Function.Injective (pairLift s t hs ht hst) := by
+  apply injective_of_simple_order (pairMatrix s t hs hst).toCoxeterSystem
+  · intro i
+    rw [pairLift_simple]
+    cases i <;> simp_all
+  · intro i
+    simp only [pairLift_simple]
+    cases i
+    · simp [pairMatrix]
+    · simp only [Bool.not_true, ↓reduceIte, pairMatrix, Bool.true_eq_false, Bool.false_eq_true]
+      have hinv : (s*t)⁻¹ = t*s := by
+        rw [mul_inv_rev, (inv_eq_of_mul_eq_one_right ht), (inv_eq_of_mul_eq_one_right hs)]
+      rw [← hinv, orderOf_inv]
+
+/-- The presented pair generates exactly the subgroup generated by its images. -/
+theorem pairLift_range :
+    (pairLift s t hs ht hst).range = Subgroup.closure {s, t} := by
+  apply le_antisymm
+  · rintro x ⟨w, rfl⟩
+    apply (pairMatrix s t hs hst).toCoxeterSystem.simple_induction w
+    · intro i
+      rw [pairLift_simple]
+      cases i <;> apply Subgroup.subset_closure <;> simp
+    · simp only [map_one, Subgroup.one_mem]
+    · intro a b ha hb
+      rw [map_mul]
+      exact Subgroup.mul_mem _ ha hb
+  · apply (Subgroup.closure_le _).mpr
+    intro x hx
+    rcases Set.mem_insert_iff.mp hx with hx | hx
+    · rw [hx]
+      exact ⟨(pairMatrix s t hs hst).toCoxeterSystem.simple false, pairLift_simple _ _ _ _ _ _⟩
+    · have : x = t := Set.mem_singleton_iff.mp hx
+      subst x
+      exact ⟨(pairMatrix s t hs hst).toCoxeterSystem.simple true, pairLift_simple _ _ _ _ _ _⟩
+
+/-- An actual Coxeter system on the generated subgroup, not an extra assumption. -/
+def pairSystem (hs1 : s ≠ 1) (ht1 : t ≠ 1) :
+    CoxeterSystem (pairMatrix s t hs hst) (Subgroup.closure {s, t}) :=
+  (pairMatrix s t hs hst).toCoxeterSystem.map
+    ((MulEquiv.ofBijective (pairLift s t hs ht hst).rangeRestrict
+      ⟨MonoidHom.rangeRestrict_injective_iff.mpr
+        (pairLift_injective s t hs ht hst hs1 ht1),
+       (pairLift s t hs ht hst).rangeRestrict_surjective⟩).trans
+      (MulEquiv.subgroupCongr (pairLift_range s t hs ht hst)))
+
+@[simp] theorem pairSystem_simple (hs1 : s ≠ 1) (ht1 : t ≠ 1) (i : Bool) :
+    ((pairSystem s t hs ht hst hs1 ht1).simple i : G) = if i then t else s := by
+  exact pairLift_simple s t hs ht hst i
+
+end
+end KLInvariance.Dihedral
+
+
+namespace KLInvariance.TitsSpace
+open Module
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+noncomputable section
+
+/-- Any two distinct positive roots in a two-dimensional root plane span it. -/
+theorem rootPlane_eq_of_mem (a b l r : PositiveRoot M cs) (hab : a ≠ b)
+    (hl : l.val ∈ rootPlane a b) (hr : r.val ∈ rootPlane a b) (hlr : l ≠ r) :
+    rootPlane l r = rootPlane a b := by
+  apply Submodule.eq_of_le_of_finrank_eq
+  · apply Submodule.span_le.mpr
+    intro z hz
+    change z ∈ rootPlane a b
+    rcases Set.mem_insert_iff.mp hz with hz | hz
+    · rw [hz]; exact hl
+    · rw [Set.mem_singleton_iff.mp hz]; exact hr
+  · rw [rootPlane_finrank _ _ hlr, rootPlane_finrank _ _ hab]
+
+/-- Slope betweenness detects the actual positive cone, not only one implication. -/
+theorem positive_combination_of_slope_between (a b c : PositiveRoot M cs)
+    (hc : c.val ∈ rootPlane a b)
+    (hac : geometricSlope a < geometricSlope c)
+    (hcb : geometricSlope c < geometricSlope b) :
+    ∃ r s : ℝ, 0 < r ∧ 0 < s ∧ c.val = r • a.val + s • b.val := by
+  let f := geometricFunctional (M := M) (cs := cs)
+  obtain ⟨r, s, he⟩ := Submodule.mem_span_pair.mp hc
+  refine ⟨r, s, ?_, ?_, he.symm⟩
+  all_goals
+    have ha := a.height_pos
+    have hb := b.height_pos
+    have hh := c.height_pos
+    have hfc := congrArg f he
+    have hhc := congrArg (height (I := I)) he
+    simp only [map_add, map_smul, smul_eq_mul] at hfc hhc
+    change f a.val / height a.val <
+      f c.val / height c.val at hac
+    change f c.val / height c.val <
+      f b.val / height b.val at hcb
+    have hab := lt_trans hac hcb
+    rw [div_lt_div_iff₀ ha hb] at hab
+    rw [div_lt_div_iff₀ ha hh] at hac
+    rw [div_lt_div_iff₀ hh hb] at hcb
+  · have heq : r * (f b.val * height a.val -
+        f a.val * height b.val) =
+        f b.val * height c.val -
+          f c.val * height b.val := by rw [← hfc, ← hhc]; ring
+    have hp : 0 < r * (f b.val * height a.val -
+        f a.val * height b.val) := by rw [heq]; linarith
+    exact (mul_pos_iff.mp hp).resolve_right (by intro h; linarith [h.2]) |>.1
+  · have heq : s * (f b.val * height a.val -
+        f a.val * height b.val) =
+        f c.val * height a.val -
+          f a.val * height c.val := by rw [← hfc, ← hhc]; ring
+    have hp : 0 < s * (f b.val * height a.val -
+        f a.val * height b.val) := by rw [heq]; linarith
+    exact (mul_pos_iff.mp hp).resolve_right (by intro h; linarith [h.2]) |>.1
+
+/-- Positivity is convex in the root cone. -/
+theorem noninverted_of_positive_combination (w : W) (a b c : PositiveRoot M cs)
+    {r s : ℝ} (hr : 0 ≤ r) (hs : 0 ≤ s)
+    (hc : c.val = r • a.val + s • b.val)
+    (ha : ¬ rootInverted w a) (hb : ¬ rootInverted w b) :
+    ¬ rootInverted w c := by
+  rw [not_rootInverted_iff] at ha hb ⊢
+  intro i
+  rw [hc, map_add, map_smul, map_smul]
+  exact add_nonneg (mul_nonneg hr (ha i)) (mul_nonneg hs (hb i))
+
+@[simp] theorem not_rootInverted_one (a : PositiveRoot M cs) :
+    ¬ rootInverted (1 : W) a := by
+  rw [not_rootInverted_iff]
+  simp only [inv_one, map_one]
+  exact a.property.2
+
+@[simp] theorem rootInverted_ref_self (a : PositiveRoot M cs) :
+    rootInverted (rootRef a) a := by
+  simpa only [mul_one] using (rootInverted_mul_self 1 a).mpr (not_rootInverted_one a)
+
+/-- A root plane has an extreme positive root. Finiteness of inversion sets
+replaces the false assertion that the (real-valued) root heights are discrete. -/
+theorem exists_plane_extreme_inverted (a b : PositiveRoot M cs) (hab : a ≠ b) :
+    ∃ p : PositiveRoot M cs, p.val ∈ rootPlane a b ∧ rootInverted (rootRef a) p ∧
+      ((∀ r : PositiveRoot M cs, r.val ∈ rootPlane a b →
+          geometricSlope p ≤ geometricSlope r) ∨
+       (∀ r : PositiveRoot M cs, r.val ∈ rootPlane a b →
+          geometricSlope r ≤ geometricSlope p)) := by
+  classical
+  let S := (finite_rootInverted (M := M) (cs := cs) (rootRef a)).toFinset.filter
+    (fun r => r.val ∈ rootPlane a b)
+  have hS (r : PositiveRoot M cs) : r ∈ S ↔
+      rootInverted (rootRef a) r ∧ r.val ∈ rootPlane a b := by
+    simp only [S, Finset.mem_filter, Set.Finite.mem_toFinset, Set.mem_ofPred_eq]
+  have haP : a.val ∈ rootPlane a b := Submodule.subset_span (Set.mem_insert _ _)
+  have haS : a ∈ S := (hS a).mpr ⟨rootInverted_ref_self a, haP⟩
+  obtain ⟨p, hp, hmin⟩ := S.exists_min_image geometricSlope ⟨a, haS⟩
+  obtain ⟨q, hq, hmax⟩ := S.exists_max_image geometricSlope ⟨a, haS⟩
+  by_cases hleft : ∀ r : PositiveRoot M cs, r.val ∈ rootPlane a b →
+      geometricSlope p ≤ geometricSlope r
+  · exact ⟨p, (hS p).mp hp |>.2, (hS p).mp hp |>.1, Or.inl hleft⟩
+  by_cases hright : ∀ r : PositiveRoot M cs, r.val ∈ rootPlane a b →
+      geometricSlope r ≤ geometricSlope q
+  · exact ⟨q, (hS q).mp hq |>.2, (hS q).mp hq |>.1, Or.inr hright⟩
+  push Not at hleft hright
+  obtain ⟨l, hlP, hlp⟩ := hleft
+  obtain ⟨r, hrP, hqr⟩ := hright
+  have hla := lt_of_lt_of_le hlp (hmin a haS)
+  have har := lt_of_le_of_lt (hmax a haS) hqr
+  have hlr : l ≠ r := fun h => (lt_trans hla har).ne (congrArg geometricSlope h)
+  have hplane := rootPlane_eq_of_mem a b l r hab hlP hrP hlr
+  have hlam : ¬ rootInverted (rootRef a) l := by
+    intro hi
+    exact (not_le_of_gt hlp) (hmin l ((hS l).mpr ⟨hi, hlP⟩))
+  have hram : ¬ rootInverted (rootRef a) r := by
+    intro hi
+    exact (not_le_of_gt hqr) (hmax r ((hS r).mpr ⟨hi, hrP⟩))
+  obtain ⟨c, d, hc, hd, he⟩ := positive_combination_of_slope_between l r a
+    (hplane.symm ▸ haP) hla har
+  exact False.elim ((noninverted_of_positive_combination (rootRef a) l r a
+    hc.le hd.le he hlam hram) (rootInverted_ref_self a))
+
+/-- The canonical simple-root criterion stated on the actual ambient roots. -/
+def PlaneSimple (P : Submodule ℝ (I → ℝ)) (a : PositiveRoot M cs) : Prop :=
+  a.val ∈ P ∧ ∀ b : PositiveRoot M cs, b.val ∈ P →
+    (rootInverted (rootRef a) b ↔ b = a)
+
+theorem rootInverted_ref_iff_flipped (a b : PositiveRoot M cs) :
+    rootInverted (rootRef a) b ↔ flipped a b := by
+  unfold rootInverted flipped
+  rw [(rootReflection M cs a).property.inv]
+
+/-- An angular extreme inverts exactly its own positive root. -/
+theorem planeSimple_of_extreme {P : Submodule ℝ (I → ℝ)}
+    (a : PositiveRoot M cs) (ha : a.val ∈ P)
+    (hext : (∀ r : PositiveRoot M cs, r.val ∈ P → geometricSlope a ≤ geometricSlope r) ∨
+      (∀ r : PositiveRoot M cs, r.val ∈ P → geometricSlope r ≤ geometricSlope a)) :
+    PlaneSimple P a := by
+  refine ⟨ha, fun b hb => ⟨?_, ?_⟩⟩
+  · intro hinv
+    by_contra hba
+    have hn := (rootInverted_ref_iff_flipped a b).mp hinv
+    have hp := partner_mem a b ha hb
+    have hs := negative_pair_sum a b hn
+    have hc := negative_pair_coefficient_pos a b hn
+    have he : a.val = (form M a.val b.val)⁻¹ • b.val +
+        (form M a.val b.val)⁻¹ • (partner a b).val := by
+      calc
+        a.val = (form M a.val b.val)⁻¹ • (form M a.val b.val • a.val) := by
+          rw [smul_smul, inv_mul_cancel₀ hc.ne', one_smul]
+        _ = _ := by rw [← hs, smul_add]
+    have hne : geometricSlope b ≠ geometricSlope (partner a b) := by
+      intro h
+      exact hba (negative_partner_fixed a b hn (geometricSlope_injective h).symm)
+    rcases lt_or_gt_of_ne hne with hlt | hlt
+    · have hh := geometricSlope_between b (partner a b) a
+        (inv_pos.mpr hc) (inv_pos.mpr hc) he hlt
+      rcases hext with hmin | hmax
+      · exact (not_lt_of_ge (hmin b hb)) hh.1
+      · exact (not_lt_of_ge (hmax (partner a b) hp)) hh.2
+    · have he' : a.val = (form M a.val b.val)⁻¹ • (partner a b).val +
+          (form M a.val b.val)⁻¹ • b.val := by rw [add_comm]; exact he
+      have hh := geometricSlope_between (partner a b) b a
+        (inv_pos.mpr hc) (inv_pos.mpr hc) he' hlt
+      rcases hext with hmin | hmax
+      · exact (not_lt_of_ge (hmin (partner a b) hp)) hh.1
+      · exact (not_lt_of_ge (hmax b hb)) hh.2
+  · intro h
+    rw [h]
+    exact rootInverted_ref_self a
+
+theorem PlaneSimple.partner_val {P : Submodule ℝ (I → ℝ)}
+    {a : PositiveRoot M cs} (ha : PlaneSimple P a)
+    {b : PositiveRoot M cs} (hb : b.val ∈ P) (hba : b ≠ a) :
+    (partner a b).val = action M cs (rootRef a) b.val := by
+  have hn : ¬ rootInverted (rootRef a) b := fun h => hba ((ha.2 b hb).mp h)
+  have hp := (not_rootInverted_iff (rootRef a) b).mp hn
+  rw [(rootReflection M cs a).property.inv] at hp
+  exact positiveAction_eq_of_nonneg _ _ hp
+
+theorem PlaneSimple.exists_nonpositive_pair {P : Submodule ℝ (I → ℝ)}
+    {a : PositiveRoot M cs} (ha : PlaneSimple P a)
+    {b : PositiveRoot M cs} (hb : b.val ∈ P) (hba : b ≠ a) :
+    ∃ c : PositiveRoot M cs, c.val ∈ P ∧ c ≠ a ∧ form M a.val c.val ≤ 0 := by
+  by_cases hn : form M a.val b.val ≤ 0
+  · exact ⟨b, hb, hba, hn⟩
+  · refine ⟨partner a b, partner_mem a b ha.1 hb,
+      fun h => hba ((partner_eq_self_of_eq a b).mp h), ?_⟩
+    rw [ha.partner_val hb hba, (rootReflection_represents M cs a).action_formula]
+    simp only [map_sub, map_smul, smul_eq_mul, a.property.1.norm]
+    linarith
+
+theorem noninverted_ref_of_pairing_nonpos (a b : PositiveRoot M cs)
+    (hab : form M a.val b.val ≤ 0) : ¬ rootInverted (rootRef b) a := by
+  rw [not_rootInverted_iff, (rootReflection M cs b).property.inv,
+    (rootReflection_represents M cs b).action_formula]
+  have hba : form M b.val a.val ≤ 0 := by rwa [(form_symm M).eq]
+  intro i
+  exact sub_nonneg.mpr ((mul_nonpos_of_nonpos_of_nonneg hba (b.property.2 i)).trans
+    (a.property.2 i))
+
+/-- The other endpoint exists even in an infinite or degenerate plane. -/
+theorem exists_opposite_plane_extreme (a b p : PositiveRoot M cs) (hab : a ≠ b)
+    (hp : p.val ∈ rootPlane a b)
+    (hext : (∀ r : PositiveRoot M cs, r.val ∈ rootPlane a b →
+        geometricSlope p ≤ geometricSlope r) ∨
+      (∀ r : PositiveRoot M cs, r.val ∈ rootPlane a b →
+        geometricSlope r ≤ geometricSlope p)) :
+    ∃ q : PositiveRoot M cs, q.val ∈ rootPlane a b ∧ q ≠ p ∧
+      (((∀ r : PositiveRoot M cs, r.val ∈ rootPlane a b →
+          geometricSlope p ≤ geometricSlope r) ∧
+        (∀ r : PositiveRoot M cs, r.val ∈ rootPlane a b →
+          geometricSlope r ≤ geometricSlope q)) ∨
+       ((∀ r : PositiveRoot M cs, r.val ∈ rootPlane a b →
+          geometricSlope q ≤ geometricSlope r) ∧
+        (∀ r : PositiveRoot M cs, r.val ∈ rootPlane a b →
+          geometricSlope r ≤ geometricSlope p))) := by
+  have hs := planeSimple_of_extreme p hp hext
+  have hother : ∃ r : PositiveRoot M cs, r.val ∈ rootPlane a b ∧ r ≠ p := by
+    by_cases hap : a = p
+    · refine ⟨b, Submodule.subset_span (Set.mem_insert_of_mem _ (Set.mem_singleton _)), ?_⟩
+      exact fun h => hab (hap.trans h.symm)
+    · exact ⟨a, Submodule.subset_span (Set.mem_insert _ _), hap⟩
+  obtain ⟨r, hrP, hrp⟩ := hother
+  obtain ⟨c, hcP, hcp, hcpair⟩ := hs.exists_nonpositive_pair hrP hrp
+  have hn := noninverted_ref_of_pairing_nonpos p c hcpair
+  have hplane := rootPlane_eq_of_mem a b c p hab hcP hp hcp
+  obtain ⟨q, hqP, hqinv, hqext⟩ := exists_plane_extreme_inverted c p hcp
+  rw [hplane] at hqP hqext
+  have hqp : q ≠ p := fun h => hn (h ▸ hqinv)
+  refine ⟨q, hqP, hqp, ?_⟩
+  rcases hext with hmin | hmax
+  · refine Or.inl ⟨hmin, ?_⟩
+    rcases hqext with hqmin | hqmax
+    · exact False.elim (hqp (geometricSlope_injective (le_antisymm
+        (hqmin p hp) (hmin q hqP))))
+    · exact hqmax
+  · refine Or.inr ⟨?_, hmax⟩
+    rcases hqext with hqmin | hqmax
+    · exact hqmin
+    · exact False.elim (hqp (geometricSlope_injective (le_antisymm
+        (hmax q hqP) (hqmax p hp))))
+
+/-- Canonical angular endpoints of a plane; no Coxeter-system assertion is
+built into this data. The latter will be proved from the inversion criterion. -/
+structure PlaneBasis (P : Submodule ℝ (I → ℝ)) where
+  left : PositiveRoot M cs
+  right : PositiveRoot M cs
+  distinct : left ≠ right
+  plane_eq : rootPlane left right = P
+  left_min : ∀ r : PositiveRoot M cs, r.val ∈ P → geometricSlope left ≤ geometricSlope r
+  right_max : ∀ r : PositiveRoot M cs, r.val ∈ P → geometricSlope r ≤ geometricSlope right
+
+theorem nonempty_planeBasis (a b : PositiveRoot M cs) (hab : a ≠ b) :
+    Nonempty (PlaneBasis (M := M) (cs := cs) (rootPlane a b)) := by
+  obtain ⟨p, hp, _, hext⟩ := exists_plane_extreme_inverted a b hab
+  obtain ⟨q, hq, hqp, hpair⟩ := exists_opposite_plane_extreme a b p hab hp hext
+  rcases hpair with ⟨hl, hr⟩ | ⟨hl, hr⟩
+  · exact ⟨⟨p, q, hqp.symm, rootPlane_eq_of_mem a b p q hab hp hq hqp.symm, hl, hr⟩⟩
+  · exact ⟨⟨q, p, hqp, rootPlane_eq_of_mem a b q p hab hq hp hqp, hl, hr⟩⟩
+
+namespace PlaneBasis
+variable {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+
+theorem left_mem : B.left.val ∈ P := by
+  apply B.plane_eq.le
+  exact Submodule.subset_span (Set.mem_insert _ _)
+
+theorem right_mem : B.right.val ∈ P := by
+  apply B.plane_eq.le
+  exact Submodule.subset_span (Set.mem_insert_of_mem _ (Set.mem_singleton _))
+
+theorem left_simple : PlaneSimple P B.left :=
+  planeSimple_of_extreme B.left B.left_mem (.inl B.left_min)
+
+theorem right_simple : PlaneSimple P B.right :=
+  planeSimple_of_extreme B.right B.right_mem (.inr B.right_max)
+
+/-- Every positive root in the plane is in its canonical nonnegative cone. -/
+theorem positive_cone (r : PositiveRoot M cs) (hr : r.val ∈ P) :
+    ∃ c d : ℝ, 0 ≤ c ∧ 0 ≤ d ∧ r.val = c • B.left.val + d • B.right.val := by
+  by_cases hl : r = B.left
+  · exact ⟨1, 0, zero_le_one, le_refl 0, by rw [hl]; simp⟩
+  by_cases hh : r = B.right
+  · exact ⟨0, 1, le_refl 0, zero_le_one, by rw [hh]; simp⟩
+  have hlr : geometricSlope B.left < geometricSlope r :=
+    lt_of_le_of_ne (B.left_min r hr) (fun h => hl (geometricSlope_injective h).symm)
+  have hrr : geometricSlope r < geometricSlope B.right :=
+    lt_of_le_of_ne (B.right_max r hr) (fun h => hh (geometricSlope_injective h))
+  obtain ⟨c, d, hc, hd, he⟩ := positive_combination_of_slope_between B.left B.right r
+    (B.plane_eq.ge hr) hlr hrr
+  exact ⟨c, d, hc.le, hd.le, he⟩
+
+end PlaneBasis
+
+
+/-- Canonical-simple reflections have no extra root pair in their own plane. -/
+theorem PlaneSimple.planeWeight_eq_zero {P : Submodule ℝ (I → ℝ)}
+    {a : PositiveRoot M cs} (ha : PlaneSimple P a) (x : W) :
+    planeWeight x a P = 0 := by
+  classical
+  rw [planeWeight, Finsupp.mapDomain, Finsupp.sum_apply]
+  apply Finset.sum_eq_zero
+  intro b _
+  change (Finsupp.single (rootPlane a b) (halfVector x a b)) P = 0
+  rw [Finsupp.single_apply]
+  by_cases hP : rootPlane a b = P
+  · rw [ite_eq_left hP]
+    simp only [halfVector, rootIndicator_apply]
+    apply ite_eq_right
+    intro hh
+    have hbP : b.val ∈ P := hP.le
+      (Submodule.subset_span (Set.mem_insert_of_mem _ (Set.mem_singleton _)))
+    exact hh.1.1 ((ha.2 b hbP).mp ((rootInverted_ref_iff_flipped a b).mpr hh.1.2.1))
+  · exact ite_eq_right hP
+
+/-- The relative inversion count has the genuine simple-reflection step. -/
+theorem PlaneSimple.planeLength_step {P : Submodule ℝ (I → ℝ)}
+    {a b : PositiveRoot M cs} (ha : PlaneSimple P a)
+    (hab : a ≠ b) (hP : rootPlane a b = P) (x : W)
+    (hx : ¬ rootInverted x a) :
+    planeLength (M := M) (cs := cs) P (rootRef a * x) =
+      planeLength (M := M) (cs := cs) P x + 1 := by
+  have hh := planeWeight_relative_degree x a b hab hx
+  rw [hP, ha.planeWeight_eq_zero] at hh
+  linarith
+
+theorem PlaneSimple.planeLength_step_down {P : Submodule ℝ (I → ℝ)}
+    {a b : PositiveRoot M cs} (ha : PlaneSimple P a)
+    (hab : a ≠ b) (hP : rootPlane a b = P) (x : W)
+    (hx : rootInverted x a) :
+    planeLength (M := M) (cs := cs) P (rootRef a * x) =
+      planeLength (M := M) (cs := cs) P x - 1 := by
+  have hy : ¬ rootInverted (rootRef a * x) a :=
+    fun h => (rootInverted_mul_self x a).mp h hx
+  have hh := ha.planeLength_step hab hP (rootRef a * x) hy
+  rw [← mul_assoc, (rootReflection M cs a).property.mul_self, one_mul] at hh
+  linarith
+
+/-- The usual inversion bijection, proved on the actual positive roots. -/
+theorem rootInverted_inverse_action (w : W) (a : PositiveRoot M cs) :
+    rootInverted w⁻¹ (positiveAction w⁻¹ a) ↔ rootInverted w a := by
+  have hw : action M cs w (action M cs w⁻¹ a.val) = a.val := by
+    rw [← LinearEquiv.mul_apply, ← map_mul, mul_inv_cancel, map_one]
+    rfl
+  by_cases hi : rootInverted w a
+  · apply iff_of_true ?_ hi
+    unfold rootInverted
+    rw [inv_inv, positiveAction_eq_of_nonpos _ _ hi, map_neg, hw]
+    exact fun i => neg_nonpos.mpr (a.property.2 i)
+  · apply iff_of_false ?_ hi
+    rw [not_rootInverted_iff, inv_inv,
+      positiveAction_eq_of_nonneg _ _ ((not_rootInverted_iff w a).mp hi), hw]
+    exact a.property.2
+
+/-- Reindexing the inversion indicator by the inverse root action. -/
+theorem inversionVector_inverse (w : W) :
+    inversionVector (M := M) (cs := cs) w⁻¹ =
+      Finsupp.mapDomain (positiveActionEquiv w⁻¹) (inversionVector w) := by
+  classical
+  apply (Finsupp.ext_iff).mpr
+  intro b
+  obtain ⟨a, rfl⟩ := (positiveActionEquiv (M := M) (cs := cs) w⁻¹).surjective b
+  rw [Finsupp.mapDomain_equiv_apply, Equiv.symm_apply_apply]
+  simp only [inversionVector_apply]
+  change (if rootInverted w⁻¹ (positiveAction w⁻¹ a) then 1 else 0) = _
+  rw [rootInverted_inverse_action]
+
+theorem positiveAction_mem_plane {P : Submodule ℝ (I → ℝ)}
+    {w : W} (hw : w ∈ planeSubgroup M cs P) (a : PositiveRoot M cs)
+    (ha : a.val ∈ P) : (positiveAction w a).val ∈ P := by
+  have hp := planeSubgroup_preserves hw ha
+  rcases (a.property.1.act w).sign with hs | hs
+  · exact (positiveAction_eq_of_nonneg w a hs).symm ▸ hp
+  · exact (positiveAction_eq_of_nonpos w a hs).symm ▸ P.neg_mem hp
+
+theorem positiveAction_mem_plane_iff {P : Submodule ℝ (I → ℝ)}
+    {w : W} (hw : w ∈ planeSubgroup M cs P) (a : PositiveRoot M cs) :
+    (positiveAction w a).val ∈ P ↔ a.val ∈ P := by
+  refine ⟨fun h => ?_, positiveAction_mem_plane hw a⟩
+  have hh := positiveAction_mem_plane ((planeSubgroup M cs P).inv_mem hw)
+    (positiveAction w a) h
+  rwa [← positiveAction_mul, inv_mul_cancel, positiveAction_one] at hh
+
+/-- Relative inversion count is invariant under inverse on the actual plane subgroup. -/
+theorem planeLength_inv {P : Submodule ℝ (I → ℝ)} {w : W}
+    (hw : w ∈ planeSubgroup M cs P) :
+    planeLength (M := M) (cs := cs) P w⁻¹ = planeLength (M := M) (cs := cs) P w := by
+  classical
+  rw [planeLength, inversionVector_inverse,
+    Finsupp.sum_mapDomain_index (by intros; split_ifs <;> rfl)
+      (by intros; split_ifs <;> simp)]
+  apply Finsupp.sum_congr
+  intro a _
+  change (if (positiveAction w⁻¹ a).val ∈ P then _ else _) = _
+  rw [positiveAction_mem_plane_iff ((planeSubgroup M cs P).inv_mem hw)]
+
+theorem planeLength_nonneg (P : Submodule ℝ (I → ℝ)) (w : W) :
+    0 ≤ planeLength (M := M) (cs := cs) P w := by
+  classical
+  apply Finset.sum_nonneg
+  intro a _
+  simp only [inversionVector_apply]
+  split_ifs <;> norm_num
+
+namespace PlaneBasis
+variable {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+
+/-- A plane reflection must invert at least one of the canonical endpoints.
+This uses convex positivity, not any assumed Coxeter structure on the subgroup. -/
+theorem reflection_inverts_endpoint (r : PositiveRoot M cs) (hr : r.val ∈ P) :
+    rootInverted (rootRef r) B.left ∨ rootInverted (rootRef r) B.right := by
+  by_contra h
+  push Not at h
+  obtain ⟨c, d, hc, hd, he⟩ := B.positive_cone r hr
+  exact (noninverted_of_positive_combination (rootRef r) B.left B.right r
+    hc hd he h.1 h.2) (rootInverted_ref_self r)
+
+end PlaneBasis
+
+
+/-- The second descent in a simple conjugation of a nonsimple reflection. -/
+theorem PlaneSimple.inverted_after_mul {P : Submodule ℝ (I → ℝ)}
+    {a r : PositiveRoot M cs} (ha : PlaneSimple P a) (hr : r.val ∈ P)
+    (hra : r ≠ a) (hi : rootInverted (rootRef r) a) :
+    rootInverted (rootRef r * rootRef a) a := by
+  have hn := (rootInverted_ref_iff_flipped r a).mp hi
+  apply (rootInverted_mul_of_nonpos (rootRef a) r a hn).mpr
+  intro hinv
+  have he : partner r a = a := (ha.2 (partner r a) (partner_mem r a hr ha.1)).mp hinv
+  exact hra (negative_partner_fixed r a hn he).symm
+
+/-- The honest relative length drops by two under the chosen simple conjugation. -/
+theorem PlaneSimple.planeLength_conj_drop {P : Submodule ℝ (I → ℝ)}
+    {a b r : PositiveRoot M cs} (ha : PlaneSimple P a)
+    (hab : a ≠ b) (hP : rootPlane a b = P) (hr : r.val ∈ P)
+    (hra : r ≠ a) (hi : rootInverted (rootRef r) a) :
+    planeLength (M := M) (cs := cs) P (rootRef (partner a r)) =
+      planeLength (M := M) (cs := cs) P (rootRef r) - 2 := by
+  have har : rootRef a ∈ planeSubgroup M cs P :=
+    ((rootReflection_represents M cs a).planeSubgroup_mem_iff M cs).mpr ha.1
+  have hrr : rootRef r ∈ planeSubgroup M cs P :=
+    ((rootReflection_represents M cs r).planeSubgroup_mem_iff M cs).mpr hr
+  have hmul := (planeSubgroup M cs P).mul_mem har hrr
+  have hd := ha.planeLength_step_down hab hP (rootRef r) hi
+  have hd' := ha.planeLength_step_down hab hP (rootRef r * rootRef a)
+    (ha.inverted_after_mul hr hra hi)
+  have hli := planeLength_inv hmul
+  rw [mul_inv_rev, (rootReflection M cs r).property.inv,
+    (rootReflection M cs a).property.inv] at hli
+  change planeLength (M := M) (cs := cs) P
+    (rootReflection M cs (positiveAction (rootRef a) r)).val = _
+  rw [rootReflection_positiveAction, (rootReflection M cs a).property.inv, mul_assoc]
+  linarith
+
+namespace PlaneBasis
+variable {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+
+/-- The canonical endpoint reflections generate every ambient reflection in the plane. -/
+theorem rootRef_mem_pairClosure (r : PositiveRoot M cs) (hr : r.val ∈ P) :
+    rootRef r ∈ Subgroup.closure {rootRef B.left, rootRef B.right} := by
+  classical
+  generalize hn : (planeLength (M := M) (cs := cs) P (rootRef r)).toNat = n
+  induction n using Nat.strong_induction_on generalizing r with
+  | h n ih =>
+    by_cases hl : r = B.left
+    · rw [hl]; exact Subgroup.subset_closure (Set.mem_insert _ _)
+    by_cases hh : r = B.right
+    · rw [hh]; exact Subgroup.subset_closure (Set.mem_insert_of_mem _ (Set.mem_singleton _))
+    have step (a b : PositiveRoot M cs) (ha : PlaneSimple P a)
+        (hab : a ≠ b) (hP : rootPlane a b = P) (hra : r ≠ a)
+        (hgen : rootRef a ∈ Subgroup.closure {rootRef B.left, rootRef B.right})
+        (hi : rootInverted (rootRef r) a) :
+        rootRef r ∈ Subgroup.closure {rootRef B.left, rootRef B.right} := by
+      have hp : (partner a r).val ∈ P := partner_mem a r ha.1 hr
+      have hdrop := ha.planeLength_conj_drop hab hP hr hra hi
+      have hless : (planeLength (M := M) (cs := cs) P (rootRef (partner a r))).toNat < n := by
+        have hnonneg := planeLength_nonneg (M := M) (cs := cs) P (rootRef (partner a r))
+        omega
+      have hind := ih _ hless (partner a r) hp rfl
+      have hlen := (Subgroup.closure {rootRef B.left, rootRef B.right}).mul_mem
+        ((Subgroup.closure {rootRef B.left, rootRef B.right}).mul_mem hgen hind) hgen
+      change rootRef a * (rootReflection M cs (positiveAction (rootRef a) r)).val * rootRef a ∈ _ at hlen
+      rw [rootReflection_positiveAction, (rootReflection M cs a).property.inv] at hlen
+      rw [← mul_assoc, ← mul_assoc, (rootReflection M cs a).property.mul_self,
+        one_mul, mul_assoc, (rootReflection M cs a).property.mul_self, mul_one] at hlen
+      exact hlen
+    rcases B.reflection_inverts_endpoint r hr with hi | hi
+    · exact step B.left B.right B.left_simple B.distinct B.plane_eq hl
+        (Subgroup.subset_closure (Set.mem_insert _ _)) hi
+    · exact step B.right B.left B.right_simple B.distinct.symm
+        (by rw [rootPlane, Set.pair_comm]; exact B.plane_eq) hh
+        (Subgroup.subset_closure (Set.mem_insert_of_mem _ (Set.mem_singleton _))) hi
+
+/-- Exact identification with the manuscript's maximal root-plane subgroup. -/
+theorem pairClosure_eq_planeSubgroup :
+    Subgroup.closure {rootRef B.left, rootRef B.right} = planeSubgroup M cs P := by
+  apply le_antisymm
+  · apply (Subgroup.closure_le _).mpr
+    intro t ht
+    rcases Set.mem_insert_iff.mp ht with he | he
+    · rw [he]
+      exact ((rootReflection_represents M cs B.left).planeSubgroup_mem_iff M cs).mpr B.left_mem
+    · rw [Set.mem_singleton_iff.mp he]
+      exact ((rootReflection_represents M cs B.right).planeSubgroup_mem_iff M cs).mpr B.right_mem
+  · apply (Subgroup.closure_le _).mpr
+    rintro t ⟨a, ha, haP⟩
+    let r := positiveRoot M cs ⟨t, ha.isReflection⟩
+    have hre : rootRef r = t := (rootReflection_represents M cs r).reflection_unique
+      (positiveRoot_represents M cs ⟨t, ha.isReflection⟩)
+    have hrP : r.val ∈ P := ((positiveRoot_represents M cs
+      ⟨t, ha.isReflection⟩).planeSubgroup_mem_iff M cs).mp (Subgroup.subset_closure ⟨a, ha, haP⟩)
+    rw [← hre]
+    exact B.rootRef_mem_pairClosure r hrP
+
+/-- Distinct positive roots give distinct actual reflections. -/
+theorem rootRef_distinct : rootRef B.left ≠ rootRef B.right := by
+  intro he
+  apply B.distinct
+  apply (positiveRootEquiv M cs).symm.injective
+  exact Subtype.ext he
+
+theorem rootRef_ne_one (a : PositiveRoot M cs) : rootRef a ≠ 1 := by
+  intro h
+  exact (not_rootInverted_one a) (h ▸ rootInverted_ref_self a)
+
+/-- The genuine dihedral matrix attached to the canonical plane endpoints. -/
+def matrix : CoxeterMatrix Bool :=
+  Dihedral.pairMatrix (rootRef B.left) (rootRef B.right)
+    (rootReflection M cs B.left).property.mul_self B.rootRef_distinct
+
+/-- A proved Coxeter system on the *actual maximal plane subgroup*. -/
+def system : CoxeterSystem B.matrix (planeSubgroup M cs P) :=
+  (Dihedral.pairSystem (rootRef B.left) (rootRef B.right)
+    (rootReflection M cs B.left).property.mul_self
+    (rootReflection M cs B.right).property.mul_self B.rootRef_distinct
+    (rootRef_ne_one B.left)
+    (rootRef_ne_one B.right)).map
+      (MulEquiv.subgroupCongr B.pairClosure_eq_planeSubgroup)
+
+/-- Canonical simple positive roots, indexed by Bool just like the system. -/
+def simpleRoot (i : Bool) : PositiveRoot M cs := if i then B.right else B.left
+
+theorem simpleRoot_mem (i : Bool) : (B.simpleRoot i).val ∈ P := by
+  cases i
+  · exact B.left_mem
+  · exact B.right_mem
+
+theorem simpleRoot_simple (i : Bool) : PlaneSimple P (B.simpleRoot i) := by
+  cases i
+  · exact B.left_simple
+  · exact B.right_simple
+
+@[simp] theorem system_simple (i : Bool) :
+    (B.system.simple i : W) = rootRef (B.simpleRoot i) := by
+  change ((Dihedral.pairSystem (rootRef B.left) (rootRef B.right)
+    (rootReflection M cs B.left).property.mul_self
+    (rootReflection M cs B.right).property.mul_self B.rootRef_distinct
+    (rootRef_ne_one B.left) (rootRef_ne_one B.right)).simple i : W) = _
+  rw [Dihedral.pairSystem_simple]
+  cases i <;> rfl
+
+/-- The positive simple-sign criterion for the induced plane root system. -/
+theorem simple_sign {t : planeSubgroup M cs P} {a : I → ℝ}
+    (ha : Represents M cs t.val a) (haP : a ∈ P) (hp : ∀ k, 0 ≤ a k) (i : Bool) :
+    ((∀ k, action M cs (B.system.simple i : W) a k ≤ 0) ↔ t = B.system.simple i) ∧
+    ((∀ k, 0 ≤ action M cs (B.system.simple i : W) a k) ↔ t ≠ B.system.simple i) := by
+  let r : PositiveRoot M cs := ⟨a, ha.isRoot, hp⟩
+  have hneg : (∀ k, action M cs (B.system.simple i : W) a k ≤ 0) ↔ t = B.system.simple i := by
+    rw [B.system_simple]
+    change flipped (B.simpleRoot i) r ↔ _
+    rw [← rootInverted_ref_iff_flipped, (B.simpleRoot_simple i).2 r haP]
+    constructor
+    · intro he
+      apply Subtype.ext
+      rw [B.system_simple]
+      exact ha.reflection_unique (he ▸ rootReflection_represents M cs r)
+    · intro he
+      apply Subtype.ext
+      have hr : Represents M cs (rootRef (B.simpleRoot i)) a := by
+        simpa only [he, B.system_simple] using ha
+      exact hr.unique_positive (rootReflection_represents M cs _) hp (B.simpleRoot i).property.2
+  refine ⟨hneg, ?_⟩
+  constructor
+  · intro h he
+    exact (ha.isRoot.act (B.system.simple i : W)).not_both h (hneg.mpr he)
+  · intro h
+    exact (ha.isRoot.act (B.system.simple i : W)).sign.resolve_right (fun hn => h (hneg.mp hn))
+
+/-- Signed roots for the induced system. The vectors are the original ambient
+roots, and membership in the actual plane is part of the relation. -/
+structure InducedSigned (p : planeSubgroup M cs P × ZMod 2) (a : I → ℝ) : Prop where
+  represents : Represents M cs p.1.val a
+  plane_mem : a ∈ P
+  sign : (p.2 = 0 ∧ ∀ i, 0 ≤ a i) ∨ (p.2 = 1 ∧ ∀ i, a i ≤ 0)
+
+theorem InducedSigned.simple {p : planeSubgroup M cs P × ZMod 2} {a : I → ℝ}
+    (ha : InducedSigned p a) (i : Bool) :
+    InducedSigned (CoxeterSupport.simpleAction B.system i p)
+      (action M cs (B.system.simple i : W) a) := by
+  rcases p with ⟨t,z⟩
+  rw [CoxeterSupport.simpleAction_apply]
+  refine ⟨?_, planeSubgroup_preserves (B.system.simple i).property ha.plane_mem, ?_⟩
+  · have h := ha.represents.act (B.system.simple i : W)
+    have he : (B.system.simple i : W)⁻¹ = (B.system.simple i : W) :=
+      congrArg Subtype.val (B.system.inv_simple i)
+    simpa only [he, Subgroup.coe_mul] using h
+  · rcases ha.sign with ⟨rfl,hp⟩ | ⟨rfl,hn⟩
+    · by_cases heq : t = B.system.simple i
+      · exact Or.inr ⟨by simp [heq],
+          (B.simple_sign ha.represents ha.plane_mem hp i).1.mpr heq⟩
+      · exact Or.inl ⟨by simp [heq, ZModModule.add_self],
+          (B.simple_sign ha.represents ha.plane_mem hp i).2.mpr heq⟩
+    · have hp : ∀ k, 0 ≤ (-a) k := fun k => neg_nonneg.mpr (hn k)
+      by_cases heq : t = B.system.simple i
+      · refine Or.inl ⟨by simp [heq, ZModModule.add_self], ?_⟩
+        have h := (B.simple_sign ha.represents.neg (P.neg_mem ha.plane_mem) hp i).1.mpr heq
+        simpa only [map_neg, Pi.neg_apply, neg_nonpos] using h
+      · refine Or.inr ⟨by simp [heq], ?_⟩
+        have h := (B.simple_sign ha.represents.neg (P.neg_mem ha.plane_mem) hp i).2.mpr heq
+        simpa only [map_neg, Pi.neg_apply, neg_nonneg] using h
+
+theorem InducedSigned.act {p : planeSubgroup M cs P × ZMod 2} {a : I → ℝ}
+    (ha : InducedSigned p a) (w : planeSubgroup M cs P) :
+    InducedSigned (CoxeterSupport.reflectionAction B.system w p)
+      (action M cs (w : W) a) := by
+  obtain ⟨ω,rfl⟩ := B.system.wordProd_surjective w
+  induction ω with
+  | nil => simpa using ha
+  | cons i ω ih =>
+    simpa only [B.system.wordProd_cons, map_mul, Equiv.Perm.mul_apply,
+      CoxeterSupport.reflectionAction_simple, Subgroup.coe_mul,
+      LinearEquiv.mul_apply] using ih.simple B i
+
+/-- The induced positive roots have precisely the intrinsic Coxeter inversion
+signs. No change of realization or extra canonical-system assumption is used. -/
+theorem inversion_iff {t : planeSubgroup M cs P} {a : I → ℝ}
+    (ha : Represents M cs t.val a) (haP : a ∈ P) (hp : ∀ i, 0 ≤ a i)
+    (w : planeSubgroup M cs P) :
+    (∀ i, action M cs (w : W)⁻¹ a i ≤ 0) ↔ B.system.IsLeftInversion w t := by
+  classical
+  let : BEq (planeSubgroup M cs P) := @instBEqOfDecidableEq _ (Classical.decEq _)
+  obtain ⟨ω,hω,heq⟩ := B.system.exists_isReduced w
+  rw [heq]
+  let c : ZMod 2 := if t ∈ B.system.leftInvSeq ω then 1 else 0
+  have hrec := CoxeterSupport.wordAction_record B.system ω t c
+  rw [← CoxeterSupport.reflectionAction_wordProd] at hrec
+  have hrec' : CoxeterSupport.reflectionAction B.system (B.system.wordProd ω)
+      ((B.system.wordProd ω)⁻¹ * t * B.system.wordProd ω, c) = (t,0) := by
+    by_cases hm : t ∈ B.system.leftInvSeq ω
+    · have hc : c = 1 := ite_eq_left hm
+      simpa only [hc, List.count_eq_one_of_mem hω.nodup_leftInvSeq hm,
+        Nat.cast_one, ZModModule.add_self] using hrec
+    · have hc : c = 0 := ite_eq_right hm
+      simpa only [hc, List.count_eq_zero.mpr hm, Nat.cast_zero, add_zero] using hrec
+  have hinv : CoxeterSupport.reflectionAction B.system (B.system.wordProd ω)⁻¹ (t,0) =
+      ((B.system.wordProd ω)⁻¹ * t * B.system.wordProd ω,c) := by
+    rw [map_inv, ← hrec']
+    exact Equiv.symm_apply_apply _ _
+  have hs : InducedSigned (t,0) a := ⟨ha,haP,Or.inl ⟨rfl,hp⟩⟩
+  have hsi := hs.act B (B.system.wordProd ω)⁻¹
+  rw [hinv] at hsi
+  have hcount : c = if t ∈ B.system.leftInvSeq ω then 1 else 0 := rfl
+  have hmem : (∀ i, action M cs (B.system.wordProd ω : W)⁻¹ a i ≤ 0) ↔
+      t ∈ B.system.leftInvSeq ω := by
+    by_cases hm : t ∈ B.system.leftInvSeq ω
+    · have hc1 : c=1 := by simpa [hm] using hcount
+      have hh := hsi.sign
+      simp only [hc1] at hh
+      rcases hh with ⟨hbad,_⟩ | ⟨_,hn⟩
+      · norm_num at hbad
+      · exact iff_of_true hn hm
+    · have hc0 : c=0 := by simpa [hm] using hcount
+      have hh := hsi.sign
+      simp only [hc0] at hh
+      rcases hh with ⟨_,hpos⟩ | ⟨hbad,_⟩
+      · exact iff_of_false (fun hn => (ha.isRoot.act _).not_both hpos hn) hm
+      · norm_num at hbad
+  rw [hmem]
+  exact ⟨B.system.isLeftInversion_of_mem_leftInvSeq hω,
+    CoxeterSupport.mem_leftInvSeq_of_inversion B.system hω⟩
+
+/-- The intrinsic reflections are exactly the ambient reflections in the
+actual maximal plane subgroup. -/
+theorem reflection_iff (t : planeSubgroup M cs P) :
+    B.system.IsReflection t ↔ cs.IsReflection (t : W) := by
+  constructor
+  · rintro ⟨w,i,he⟩
+    obtain ⟨z,j,hz⟩ := (rootReflection M cs (B.simpleRoot i)).property
+    refine ⟨(w : W) * z,j,?_⟩
+    have he' := congrArg Subtype.val he
+    simp only [Subgroup.coe_mul, Subgroup.coe_inv, B.system_simple] at he'
+    change rootRef (B.simpleRoot i) = _ at hz
+    rw [he', hz]
+    group
+  · intro ht
+    let a := positiveRoot M cs ⟨(t:W),ht⟩
+    have ha := positiveRoot_represents M cs ⟨(t:W),ht⟩
+    have haP : a.val ∈ P := (ha.planeSubgroup_mem_iff M cs).mp t.property
+    have hinv := (B.inversion_iff ha haP a.property.2 t).mp
+      ((ha.inversion_iff a.property.2 (t:W)).mpr
+        ⟨ht, by
+          rw [ht.mul_self, cs.length_one]
+          obtain ⟨k,hk⟩ := ht.odd_length
+          omega⟩)
+    exact hinv.1
+
+/-- The true ambient plane inversions are in bijection with the inversions
+of the canonical induced system. -/
+def inversionEquiv (w : planeSubgroup M cs P) :
+    {a : PositiveRoot M cs // a.val ∈ P ∧ rootInverted (w:W) a} ≃
+      {t : planeSubgroup M cs P // B.system.IsLeftInversion w t} where
+  toFun a := ⟨⟨rootRef a.val,
+    ((rootReflection_represents M cs a.val).planeSubgroup_mem_iff M cs).mpr a.property.1⟩,
+    (B.inversion_iff (rootReflection_represents M cs a.val) a.property.1
+      a.val.property.2 w).mp a.property.2⟩
+  invFun t :=
+    let a := positiveRoot M cs ⟨(t.val:W),(B.reflection_iff t.val).mp t.property.1⟩
+    let ha := positiveRoot_represents M cs
+      ⟨(t.val:W),(B.reflection_iff t.val).mp t.property.1⟩
+    let haP := (ha.planeSubgroup_mem_iff M cs).mp t.val.property
+    ⟨a,haP,(B.inversion_iff ha haP a.property.2 w).mpr t.property⟩
+  left_inv a := by
+    apply Subtype.ext
+    exact (positiveRootEquiv M cs).apply_symm_apply a.val
+  right_inv t := by
+    apply Subtype.ext
+    apply Subtype.ext
+    exact congrArg (fun r : {r : W // cs.IsReflection r} => r.val)
+      ((positiveRootEquiv M cs).symm_apply_apply
+        ⟨(t.val:W),(B.reflection_iff t.val).mp t.property.1⟩)
+
+/-- Relative inversion count is the actual Coxeter length, completing the
+length identification, not merely a rank-two presentation. -/
+theorem length_eq (w : planeSubgroup M cs P) :
+    planeLength (M := M) (cs := cs) P (w:W) = (B.system.length w : ℤ) := by
+  classical
+  let f := inversionVector (M := M) (cs := cs) (w:W)
+  let F := f.support.filter (fun a => a.val ∈ P)
+  have hF : (F : Set (PositiveRoot M cs)) =
+      {a | a.val ∈ P ∧ rootInverted (w:W) a} := by
+    ext a
+    simp only [F, Finset.mem_coe, Finset.mem_filter, Finsupp.mem_support_iff,
+      f, inversionVector_apply, ne_eq, ite_eq_right_iff, one_ne_zero, imp_false, not_not,
+      Set.mem_ofPred_eq, and_comm]
+  have hcard : F.card = B.system.length w := by
+    rw [← Set.ncard_coe_finset, hF, ← Nat.card_coe_set_eq]
+    change Nat.card {a : PositiveRoot M cs // a.val ∈ P ∧ rootInverted (w:W) a} = _
+    rw [Nat.card_congr (B.inversionEquiv w)]
+    obtain ⟨ω,hω,he⟩ := B.system.exists_isReduced w
+    rw [he, Nat.card_congr (leftInversionListEquiv B.matrix B.system hω),
+      ← Nat.card_congr (List.Nodup.getEquiv (B.system.leftInvSeq ω) hω.nodup_leftInvSeq),
+      Nat.card_fin, B.system.length_leftInvSeq, hω.eq]
+  change f.sum (fun a n => if a.val ∈ P then n else 0) = _
+  rw [Finsupp.sum, ← Finset.sum_filter]
+  calc
+    ∑ a ∈ F, f a = ∑ _a ∈ F, (1:ℤ) := by
+      apply Finset.sum_congr rfl
+      intro a ha
+      have hm : rootInverted (w:W) a := by
+        have hm := (Finset.mem_filter.mp ha).1
+        simpa only [Finsupp.mem_support_iff, f, inversionVector_apply,
+          ne_eq, ite_eq_right_iff, one_ne_zero, imp_false, not_not] using hm
+      simp [f,hm]
+    _ = _ := by simp [hcard]
+
+/-- Intrinsic and ambient left inversions coincide on the plane subgroup. -/
+theorem leftInversion_iff (w t : planeSubgroup M cs P) :
+    B.system.IsLeftInversion w t ↔ cs.IsLeftInversion (w:W) (t:W) := by
+  by_cases ht : cs.IsReflection (t:W)
+  · let a := positiveRoot M cs ⟨(t:W),ht⟩
+    have ha := positiveRoot_represents M cs ⟨(t:W),ht⟩
+    have haP := (ha.planeSubgroup_mem_iff M cs).mp t.property
+    exact (B.inversion_iff ha haP a.property.2 w).symm.trans
+      (ha.inversion_iff a.property.2 (w:W))
+  · exact iff_of_false (fun h => ht ((B.reflection_iff t).mp h.1)) (fun h => ht h.1)
+
+/-- The intrinsic directed dihedral graph is the actual ambient graph
+restricted to the maximal plane subgroup. -/
+theorem bruhatStep_iff (g h : planeSubgroup M cs P) :
+    BruhatStep cs (g:W) (h:W) ↔ BruhatStep B.system g h := by
+  constructor
+  · rintro ⟨hlen,t,ht,he⟩
+    have htP : t ∈ planeSubgroup M cs P := by
+      have he' : t = (h:W) * (g:W)⁻¹ := by rw [he]; group
+      rw [he']
+      exact (planeSubgroup M cs P).mul_mem h.property
+        ((planeSubgroup M cs P).inv_mem g.property)
+    let tr : planeSubgroup M cs P := ⟨t,htP⟩
+    have htr : B.system.IsReflection tr := (B.reflection_iff tr).mpr ht
+    have het : h=tr*g := Subtype.ext he
+    have htg : t*(h:W)=(g:W) := by rw [he, ← mul_assoc, ht.mul_self, one_mul]
+    have hi := (B.leftInversion_iff h tr).mpr ⟨ht,by
+      change cs.length (t*(h:W)) < cs.length (h:W)
+      simpa only [htg] using hlen⟩
+    have htg' : tr*h=g := by rw [het, ← mul_assoc, htr.mul_self, one_mul]
+    exact ⟨by simpa only [htg'] using hi.2,tr,htr,het⟩
+  · rintro ⟨hlen,t,ht,he⟩
+    have het : t*h=g := by rw [he, ← mul_assoc, ht.mul_self, one_mul]
+    have hi := (B.leftInversion_iff h t).mp ⟨ht,by simpa only [het] using hlen⟩
+    refine ⟨?_,(t:W),(B.reflection_iff t).mp ht,congrArg Subtype.val he⟩
+    simpa only [← Subgroup.coe_mul, het] using hi.2
+
+/-- The exact canonical coset graph theorem needed by the reconstruction. -/
+theorem coset_step_iff {a₀ : W}
+    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+    (g h : planeSubgroup M cs P) :
+    BruhatStep cs ((g:W)*a₀) ((h:W)*a₀) ↔ BruhatStep B.system g h :=
+  (bruhatStep_coset_iff hmin g.property h.property).trans (B.bruhatStep_iff g h)
+
+/-- Relative dihedral paths give actual ambient Bruhat paths in the coset. -/
+theorem coset_bruhat {a₀ : W}
+    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+    {g h : planeSubgroup M cs P} (hgh : BruhatLE B.system g h) :
+    BruhatLE cs ((g:W)*a₀) ((h:W)*a₀) := by
+  induction hgh with
+  | refl => exact Relation.ReflTransGen.refl
+  | @tail y z _ hyz ih =>
+    exact ih.tail ((B.coset_step_iff hmin y z).mpr hyz)
+
+/-- The source inversion-count characterization holds on every minimal
+representative coset, with its actual induced length. -/
+theorem coset_length_eq {a₀ : W}
+    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+    (w : planeSubgroup M cs P) :
+    planeLength (M := M) (cs := cs) P ((w:W)*a₀) = (B.system.length w : ℤ) := by
+  classical
+  rw [← B.length_eq w]
+  let f := inversionVector (M := M) (cs := cs) ((w:W)*a₀)
+  let g := inversionVector (M := M) (cs := cs) (w:W)
+  change f.sum (fun a n => if a.val ∈ P then n else 0) =
+    g.sum (fun a n => if a.val ∈ P then n else 0)
+  rw [Finsupp.sum_of_support_subset f Finset.subset_union_left _ (by intro a; simp),
+    Finsupp.sum_of_support_subset g Finset.subset_union_right _ (by intro a; simp)]
+  apply Finset.sum_congr rfl
+  intro a _
+  by_cases haP : a.val ∈ P
+  · simp only [ite_eq_left haP, f, g, inversionVector_apply]
+    congr 1
+    exact propext (root_inversion_coset_iff hmin w.property a.property.1 haP)
+  · simp [haP]
+
+end PlaneBasis
+
+/-- Every nonsaturated reflection edge has a genuine rank-two contribution
+whose induced relative edge gap is at least three. -/
+theorem exists_positive_plane (x : W) (a : PositiveRoot M cs)
+    (hx : ¬ rootInverted x a)
+    (hlen : cs.length x + 1 < cs.length (rootRef a*x)) :
+    ∃ b : PositiveRoot M cs, a ≠ b ∧ 0 < planeWeight x a (rootPlane a b) := by
+  classical
+  have hsum := sum_planeWeight x a hx
+  have hp : 0 < (planeWeight x a).sum (fun _ n => n) := by omega
+  obtain ⟨P,hP,hpos⟩ := (Finset.sum_pos_iff_of_nonneg
+    (fun P _ => planeWeight_nonneg x a P)).mp hp
+  obtain ⟨b,hb,hbP⟩ := Finset.mem_image.mp (Finsupp.mapDomain_support hP)
+  have hb' : halfRoot x a b := by
+    simpa only [Finsupp.mem_support_iff, halfVector, rootIndicator_apply,
+      ne_eq, ite_eq_right_iff, one_ne_zero, imp_false, not_not] using hb
+  exact ⟨b,hb'.1.1.symm,hbP ▸ hpos⟩
+
+
+end
+end KLInvariance.TitsSpace
+
+namespace KLInvariance.Dihedral
+variable {G : Type*} [Group G] {N : CoxeterMatrix Bool}
+
+/-- Every nonsaturated dihedral edge contains a full eight-edge rectangle. -/
+theorem edge_diamond (ds : CoxeterSystem N G) {x y : G}
+    (hxy : BruhatStep ds x y) (hlen : ds.length x + 1 < ds.length y) :
+    ∃ a b c d : G, a ≠ b ∧ c ≠ d ∧
+      BruhatStep ds x a ∧ BruhatStep ds x b ∧
+      BruhatStep ds a c ∧ BruhatStep ds a d ∧
+      BruhatStep ds b c ∧ BruhatStep ds b d ∧
+      BruhatStep ds c y ∧ BruhatStep ds d y := by
+  have hodd := ((step_iff_odd ds x y).mp hxy).2
+  have hthree : ds.length x+2 < ds.length y := by
+    rw [Nat.odd_iff] at hodd
+    omega
+  obtain ⟨a,b,hab,hlev₁⟩ := two_elements_length ds
+    (by omega : 0 < ds.length x+1) hlen
+  obtain ⟨c,d,hcd,hlev₂⟩ := two_elements_length ds
+    (by omega : 0 < ds.length x+2) hthree
+  have ha := (hlev₁ a).mpr (Or.inl rfl)
+  have hb := (hlev₁ b).mpr (Or.inr rfl)
+  have hc := (hlev₂ c).mpr (Or.inl rfl)
+  have hd := (hlev₂ d).mpr (Or.inr rfl)
+  have hsucc (p q : G) (h : ds.length q=ds.length p+1) : BruhatStep ds p q := by
+    rw [step_iff_odd, Nat.odd_iff]
+    constructor <;> omega
+  have htop (p : G) (hp : ds.length p=ds.length x+2) : BruhatStep ds p y := by
+    rw [step_iff_odd, Nat.odd_iff]
+    rw [Nat.odd_iff] at hodd
+    constructor <;> omega
+  exact ⟨a,b,c,d,hab,hcd,hsucc x a ha,hsucc x b hb,
+    hsucc a c (by omega),hsucc a d (by omega),
+    hsucc b c (by omega),hsucc b d (by omega),htop c hc,htop d hd⟩
+
+end KLInvariance.Dihedral
+
+namespace KLInvariance.TitsSpace
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+
+/-- Every genuine ambient reflection edge of length gap greater than one
+contains a full eight-edge diamond. The chosen plane comes from an actual
+positive inversion contribution, avoiding the false arbitrary-plane claim. -/
+theorem ambient_edge_diamond {x y : W} (hxy : BruhatStep cs x y)
+    (hlen : cs.length x+1 < cs.length y) :
+    ∃ a b c d : W, a ≠ b ∧ c ≠ d ∧
+      BruhatStep cs x a ∧ BruhatStep cs x b ∧
+      BruhatStep cs a c ∧ BruhatStep cs a d ∧
+      BruhatStep cs b c ∧ BruhatStep cs b d ∧
+      BruhatStep cs c y ∧ BruhatStep cs d y := by
+  rcases hxy.2 with ⟨t,ht,het⟩
+  let a := positiveRoot M cs ⟨t,ht⟩
+  have ha := positiveRoot_represents M cs ⟨t,ht⟩
+  have hta : rootRef a=t := congrArg (fun r : {r : W // cs.IsReflection r} => r.val)
+    ((positiveRootEquiv M cs).symm_apply_apply ⟨t,ht⟩)
+  have hx : ¬ rootInverted x a := by
+    intro hi
+    have h := (ha.inversion_iff a.property.2 x).mp hi
+    have hh : cs.length (t*x) < cs.length x := h.2
+    rw [← het] at hh
+    exact (not_lt_of_gt hxy.1) hh
+  have hae : rootRef a*x=y := by rw [hta]; exact het.symm
+  obtain ⟨b,hab,hpos⟩ := exists_positive_plane x a hx (by rwa [hae])
+  obtain ⟨B⟩ := nonempty_planeBasis a b hab
+  let P := rootPlane a b
+  have haP : a.val ∈ P := Submodule.subset_span (Set.mem_insert _ _)
+  obtain ⟨a₀,g,hg,hgx,hmin⟩ := planeCoset_exists_minimal M cs P x
+  let X : planeSubgroup M cs P := ⟨g,hg⟩
+  let T : planeSubgroup M cs P := ⟨rootRef a,
+    ((rootReflection_represents M cs a).planeSubgroup_mem_iff M cs).mpr haP⟩
+  let Y := T*X
+  have hxEq : (X:W)*a₀=x := hgx.symm
+  have hyEq : (Y:W)*a₀=y := by
+    change (rootRef a*g)*a₀=y
+    rw [mul_assoc, ← hgx, hae]
+  have hrel : BruhatStep B.system X Y := (B.coset_step_iff hmin X Y).mp
+    (by simpa only [hxEq,hyEq] using hxy)
+  have hLx := B.coset_length_eq hmin X
+  have hLy := B.coset_length_eq hmin Y
+  rw [hxEq] at hLx
+  rw [hyEq] at hLy
+  have hdegree := planeWeight_relative_degree x a b hab hx
+  rw [hae,hLx,hLy] at hdegree
+  have hlen' : B.system.length X+1 < B.system.length Y := by omega
+  obtain ⟨p,q,r,s,hpq,hrs,hXp,hXq,hpr,hps,hqr,hqs,hrY,hsY⟩ :=
+    Dihedral.edge_diamond B.system hrel hlen'
+  have hinj : Function.Injective (fun z : planeSubgroup M cs P => (z:W)*a₀) := by
+    intro p q hpq
+    exact Subtype.ext (mul_right_cancel hpq)
+  refine ⟨(p:W)*a₀,(q:W)*a₀,(r:W)*a₀,(s:W)*a₀,
+    fun h => hpq (hinj h),fun h => hrs (hinj h),?_,?_,?_,?_,?_,?_,?_,?_⟩
+  · simpa only [hxEq] using (B.coset_step_iff hmin X p).mpr hXp
+  · simpa only [hxEq] using (B.coset_step_iff hmin X q).mpr hXq
+  · exact (B.coset_step_iff hmin p r).mpr hpr
+  · exact (B.coset_step_iff hmin p s).mpr hps
+  · exact (B.coset_step_iff hmin q r).mpr hqr
+  · exact (B.coset_step_iff hmin q s).mpr hqs
+  · simpa only [hyEq] using (B.coset_step_iff hmin r Y).mpr hrY
+  · simpa only [hyEq] using (B.coset_step_iff hmin s Y).mpr hsY
+
+end KLInvariance.TitsSpace
+
+namespace KLInvariance
+variable {I W : Type*} [Fintype I] [Group W] {M : CoxeterMatrix I}
+
+/-- General unlabelled Bruhat graph reconstruction completeness, for genuine
+finite-rank Coxeter systems and arbitrary bottoms and interval ranks. -/
+theorem diamondEdges_of_bruhatStep (cs : CoxeterSystem M W)
+    {u b : W} {x y : Interval cs u b} (hxy : BruhatStep cs x.val y.val) :
+    DiamondEdges x y := by
+  suffices ∀ n : ℕ, ∀ x y : Interval cs u b,
+      cs.length y.val-cs.length x.val=n → BruhatStep cs x.val y.val →
+      DiamondEdges x y by
+    exact this _ x y rfl hxy
+  intro n
+  induction n using Nat.strong_induction_on with
+  | h n ih =>
+    intro x y hn hxy
+    by_cases he : cs.length y.val=cs.length x.val+1
+    · exact DiamondEdges.cover (interval_covBy_of_length_succ cs
+        (Relation.ReflTransGen.single hxy) he)
+    · have hn' : cs.length x.val+1 < cs.length y.val := by have := hxy.1; omega
+      obtain ⟨a,bm,c,d,hab,hcd,hxa,hxb,hac,had,hbc,hbd,hcy,hdy⟩ :=
+        TitsSpace.ambient_edge_diamond hxy hn'
+      have leOf {p q : W} (h : BruhatStep cs p q) : BruhatLE cs p q :=
+        Relation.ReflTransGen.single h
+      let A : Interval cs u b := ⟨a,bruhat_trans cs x.property.1 (leOf hxa),
+        bruhat_trans cs (leOf hac) (bruhat_trans cs (leOf hcy) y.property.2)⟩
+      let B : Interval cs u b := ⟨bm,bruhat_trans cs x.property.1 (leOf hxb),
+        bruhat_trans cs (leOf hbc) (bruhat_trans cs (leOf hcy) y.property.2)⟩
+      let C : Interval cs u b := ⟨c,
+        bruhat_trans cs x.property.1 (bruhat_trans cs (leOf hxa) (leOf hac)),
+        bruhat_trans cs (leOf hcy) y.property.2⟩
+      let D : Interval cs u b := ⟨d,
+        bruhat_trans cs x.property.1 (bruhat_trans cs (leOf hxa) (leOf had)),
+        bruhat_trans cs (leOf hdy) y.property.2⟩
+      have hsmall {p q : Interval cs u b} (hpq : BruhatStep cs p.val q.val)
+          (hp : cs.length x.val ≤ cs.length p.val)
+          (hq : cs.length q.val ≤ cs.length y.val)
+          (hs : cs.length x.val < cs.length p.val ∨ cs.length q.val < cs.length y.val) :
+          DiamondEdges p q := by
+        exact ih _ (by omega) p q rfl hpq
+      have h₁ := hxa.1
+      have h₂ := hxb.1
+      have h₃ := hac.1
+      have h₄ := had.1
+      have h₅ := hbc.1
+      have h₆ := hbd.1
+      have h₇ := hcy.1
+      have h₈ := hdy.1
+      refine DiamondEdges.diamond
+        (hsmall (p := x) (q := A) hxa ?_ ?_ ?_)
+        (hsmall (p := x) (q := B) hxb ?_ ?_ ?_)
+        (hsmall (p := A) (q := C) hac ?_ ?_ ?_)
+        (hsmall (p := A) (q := D) had ?_ ?_ ?_)
+        (hsmall (p := B) (q := C) hbc ?_ ?_ ?_)
+        (hsmall (p := B) (q := D) hbd ?_ ?_ ?_)
+        (hsmall (p := C) (q := y) hcy ?_ ?_ ?_)
+        (hsmall (p := D) (q := y) hdy ?_ ?_ ?_)
+        (fun h => hab (congrArg Subtype.val h))
+        (fun h => hcd (congrArg Subtype.val h))
+      all_goals first | omega | (dsimp only [A,B,C,D]; omega)
+
+/-- An abstract interval order isomorphism determines its actual directed
+reflection edges. No root labels, generators or geometric data are supplied to
+that isomorphism. This proves the finite-rank graph input of the source main. -/
+theorem intervalIso_bruhatStep_iff {I' W' : Type*} [Fintype I'] [Group W']
+    {M' : CoxeterMatrix I'} (cs : CoxeterSystem M W) (cs' : CoxeterSystem M' W')
+    {u b : W} {u' b' : W'} (e : Interval cs u b ≃o Interval cs' u' b')
+    (x y : Interval cs u b) :
+    BruhatStep cs' (e x).val (e y).val ↔ BruhatStep cs x.val y.val := by
+  constructor
+  · intro h
+    exact diamondEdges_bruhatStep cs ((DiamondEdges.map_iff e).mp
+      (diamondEdges_of_bruhatStep cs' h))
+  · intro h
+    exact diamondEdges_bruhatStep cs' ((diamondEdges_of_bruhatStep cs h).map e)
+
+end KLInvariance
+
+end
+
+
+section
+
+/-! Rank-two Frobenius extension for an actual reflection on a symmetric
+algebra. This supplies the elementary simple-reflection duality input; it
+is not a proof of BMP or Elias--Williamson self-duality/characters. -/
+namespace KLInvariance.ReflectionFrobenius
+universe uk ua uv
+variable {k : Type uk} [Field k] [CharZero k]
+  {A : Type ua} [CommRing A] [IsDomain A] [Algebra k A]
+
+noncomputable section
+
+def fixed (τ : A ≃ₐ[k] A) : Subalgebra k A :=
+  AlgHom.equalizer τ.toAlgHom (AlgHom.id k A)
+
+omit [CharZero k] [IsDomain A] in
+@[simp] theorem mem_fixed (τ : A ≃ₐ[k] A) (a : A) :
+    a ∈ fixed τ ↔ τ a = a := Iff.rfl
+
+omit [CharZero k] [IsDomain A] in
+ theorem fixed_val (τ : A ≃ₐ[k] A) (a : fixed τ) : τ (a : A) = a := a.property
+
+variable (τ : A ≃ₐ[k] A) (α : A)
+
+def expand : (fixed τ × fixed τ) →ₗ[fixed τ] A where
+  toFun a := (a.1 : A) + α * (a.2 : A)
+  map_add' a b := by simp; ring
+  map_smul' c a := by simp only [Prod.smul_fst, Prod.smul_snd, smul_eq_mul,
+    Subalgebra.coe_mul, Subalgebra.smul_def, RingHom.id_apply]; ring
+
+omit [CharZero k] [IsDomain A] in
+@[simp] theorem expand_apply (a : fixed τ × fixed τ) :
+    expand τ α a = (a.1 : A) + α * (a.2 : A) := rfl
+
+variable (hinv : Function.Involutive τ) (hα : α ≠ 0) (hanti : τ α = -α)
+  (hdiv : ∀ a : A, α ∣ a - τ a)
+
+omit [CharZero k] in
+include hinv hα hanti in
+ theorem quotient_fixed {a d : A} (hd : a - τ a = α * d) : τ d = d := by
+  have hh := congrArg τ hd
+  simp only [map_sub, map_mul, hinv a, hanti] at hh
+  apply mul_left_cancel₀ hα
+  linear_combination hh + hd
+
+include hα hanti in
+ theorem expand_injective : Function.Injective (expand τ α) := by
+  intro a b hab
+  have ht := congrArg τ hab
+  simp only [expand_apply, map_add, map_mul, fixed_val, hanti] at ht
+  have he : (2 : A) * α * ((a.2 : A) - (b.2 : A)) = 0 := by
+    simp only [expand_apply] at hab
+    linear_combination hab - ht
+  have htwo : (2 : A) ≠ 0 := by
+    have hi : Function.Injective (algebraMap k A) := (algebraMap k A).injective
+    exact fun h => two_ne_zero (hi (by simpa only [map_ofNat, map_zero] using h : algebraMap k A (2:k) = algebraMap k A 0))
+  have h2 : (a.2 : A) = (b.2 : A) := sub_eq_zero.mp
+    ((mul_eq_zero.mp he).resolve_left (mul_ne_zero htwo hα))
+  have h1 : (a.1 : A) = (b.1 : A) := by
+    simpa only [expand_apply, h2, add_left_inj] using hab
+  exact Prod.ext (Subtype.ext h1) (Subtype.ext h2)
+
+include hinv hα hanti hdiv in
+ theorem expand_surjective : Function.Surjective (expand τ α) := by
+  intro a
+  obtain ⟨d, hd⟩ := hdiv a
+  have hdf : τ d = d := quotient_fixed τ α hinv hα hanti hd
+  let h : k := (2:k)⁻¹
+  have htwo : (2:A) * algebraMap k A h = 1 := by
+    rw [show (2:A) = algebraMap k A (2:k) by exact (map_ofNat (algebraMap k A) 2).symm, ← map_mul]
+    simp [h]
+  let c₀ : fixed τ := ⟨algebraMap k A h * (a + τ a), by
+    simp only [mem_fixed, map_mul, AlgEquiv.commutes, map_add, hinv a]
+    ring⟩
+  let c₁ : fixed τ := ⟨algebraMap k A h * d, by
+    simp only [mem_fixed, map_mul, AlgEquiv.commutes, hdf]⟩
+  refine ⟨(c₀,c₁), ?_⟩
+  change algebraMap k A h * (a + τ a) + α * (algebraMap k A h * d) = a
+  linear_combination a * htwo - (algebraMap k A h) * hd
+
+/-- The actual invariant-ring basis `1, α`, with no finiteness assumption. -/
+def coordinates : A ≃ₗ[fixed τ] (fixed τ × fixed τ) :=
+  (LinearEquiv.ofBijective (expand τ α)
+    ⟨expand_injective τ α hα hanti, expand_surjective τ α hinv hα hanti hdiv⟩).symm
+
+@[simp] theorem expand_coordinates (a : A) :
+    expand τ α (coordinates τ α hinv hα hanti hdiv a) = a :=
+  (coordinates τ α hinv hα hanti hdiv).symm_apply_apply a
+
+@[simp] theorem coordinates_expand (a : fixed τ × fixed τ) :
+    coordinates τ α hinv hα hanti hdiv (expand τ α a) = a :=
+  (coordinates τ α hinv hα hanti hdiv).apply_symm_apply a
+
+
+def trace : A →ₗ[fixed τ] fixed τ :=
+  (LinearMap.snd _ _ _).comp (coordinates τ α hinv hα hanti hdiv).toLinearMap
+
+ theorem trace_expand (a : fixed τ × fixed τ) :
+    trace τ α hinv hα hanti hdiv (expand τ α a) = a.2 := by
+  change (coordinates τ α hinv hα hanti hdiv (expand τ α a)).2 = a.2
+  rw [coordinates_expand]
+
+omit [CharZero k] [IsDomain A] in
+include hanti in
+ theorem square_fixed : α * α ∈ fixed τ := by
+  simp only [mem_fixed, map_mul, hanti, neg_mul_neg]
+
+ theorem trace_mul (a b : A) :
+    trace τ α hinv hα hanti hdiv (a*b) =
+      (coordinates τ α hinv hα hanti hdiv a).1 *
+        (coordinates τ α hinv hα hanti hdiv b).2 +
+      (coordinates τ α hinv hα hanti hdiv a).2 *
+        (coordinates τ α hinv hα hanti hdiv b).1 := by
+  let c := coordinates τ α hinv hα hanti hdiv a
+  let d := coordinates τ α hinv hα hanti hdiv b
+  let α₂ : fixed τ := ⟨α*α, square_fixed τ α hanti⟩
+  have he : a*b = expand τ α
+      (c.1*d.1+α₂*c.2*d.2, c.1*d.2+c.2*d.1) := by
+    rw [← expand_coordinates τ α hinv hα hanti hdiv a,
+      ← expand_coordinates τ α hinv hα hanti hdiv b]
+    change ((c.1:A)+α*(c.2:A))*((d.1:A)+α*(d.2:A)) =
+      (c.1:A)*(d.1:A)+(α*α)*(c.2:A)*(d.2:A)+
+        α*((c.1:A)*(d.2:A)+(c.2:A)*(d.1:A))
+    ring
+  rw [he,trace_expand]
+
+section Pair
+variable (R : Type*) [CommRing R]
+
+def crossPair : (R × R) →ₗ[R] Module.Dual R (R × R) where
+  toFun a :=
+    { toFun := fun b => a.1*b.2+a.2*b.1
+      map_add' := by intro b c; simp; ring
+      map_smul' := by intro r b; simp; ring }
+  map_add' := by intro a b; apply LinearMap.ext; intro c; simp; ring
+  map_smul' := by intro r a; apply LinearMap.ext; intro b; simp; ring
+
+ theorem crossPair_bijective : Function.Bijective (crossPair R) := by
+  constructor
+  · intro a b hab
+    have h₁ := congrArg (fun f : Module.Dual R (R×R) => f (0,1)) hab
+    have h₂ := congrArg (fun f : Module.Dual R (R×R) => f (1,0)) hab
+    exact Prod.ext (by simpa [crossPair] using h₁) (by simpa [crossPair] using h₂)
+  · intro f
+    refine ⟨(f (0,1), f (1,0)), ?_⟩
+    apply LinearMap.ext
+    intro a
+    have ha : a = a.1 • (1,0) + a.2 • (0,1) := by simp
+    change f (0,1)*a.2 + f (1,0)*a.1 = f a
+    calc
+      _ = a.1 * f (1,0) + a.2 * f (0,1) := by ring
+      _ = f (a.1 • (1,0) + a.2 • (0,1)) := by rw [map_add,map_smul,map_smul]; rfl
+      _ = f a := congrArg f ha.symm
+
+def crossPairEquiv : (R × R) ≃ₗ[R] Module.Dual R (R × R) :=
+  LinearEquiv.ofBijective (crossPair R) (crossPair_bijective R)
+
+end Pair
+
+/-- The Frobenius duality over the invariant ring is integral and perfect. -/
+def duality : A ≃ₗ[fixed τ] Module.Dual (fixed τ) A :=
+  (coordinates τ α hinv hα hanti hdiv).trans
+    ((crossPairEquiv (fixed τ)).trans
+      (coordinates τ α hinv hα hanti hdiv).dualMap)
+
+ theorem duality_apply (a b : A) :
+    duality τ α hinv hα hanti hdiv a b = trace τ α hinv hα hanti hdiv (a*b) := by
+  rw [trace_mul]
+  rfl
+
+ theorem duality_balanced (a b c : A) :
+    duality τ α hinv hα hanti hdiv (c*a) b =
+      duality τ α hinv hα hanti hdiv a (c*b) := by
+  rw [duality_apply,duality_apply]
+  congr 1
+  ring
+
+
+section Symmetric
+variable {V : Type uv} [AddCommGroup V] [Module k V]
+
+def symmetricMap (e : V ≃ₗ[k] V) : SymmetricAlgebra k V →ₐ[k] SymmetricAlgebra k V :=
+  SymmetricAlgebra.lift ((SymmetricAlgebra.ι k V).comp e.toLinearMap)
+
+omit [CharZero k] in
+@[simp] theorem symmetricMap_ι (e : V ≃ₗ[k] V) (v : V) :
+    symmetricMap e (SymmetricAlgebra.ι k V v) = SymmetricAlgebra.ι k V (e v) := by
+  simp [symmetricMap]
+
+omit [CharZero k] in
+ theorem symmetricMap_comp (e d : V ≃ₗ[k] V) :
+    (symmetricMap d).comp (symmetricMap e) = symmetricMap (e.trans d) := by
+  apply SymmetricAlgebra.algHom_ext
+  ext v
+  simp
+
+omit [CharZero k] in
+ theorem symmetricMap_refl : symmetricMap (LinearEquiv.refl k V) =
+    AlgHom.id k (SymmetricAlgebra k V) := by
+  apply SymmetricAlgebra.algHom_ext
+  ext v
+  simp
+
+/-- The algebra action is induced by the linear reflection, not postulated. -/
+def symmetricAction (e : V ≃ₗ[k] V) : SymmetricAlgebra k V ≃ₐ[k] SymmetricAlgebra k V :=
+  AlgEquiv.ofAlgHom (symmetricMap e) (symmetricMap e.symm)
+    (by rw [symmetricMap_comp,LinearEquiv.symm_trans_self,symmetricMap_refl])
+    (by rw [symmetricMap_comp,LinearEquiv.self_trans_symm,symmetricMap_refl])
+
+omit [CharZero k] in
+@[simp] theorem symmetricAction_ι (e : V ≃ₗ[k] V) (v : V) :
+    symmetricAction e (SymmetricAlgebra.ι k V v) = SymmetricAlgebra.ι k V (e v) :=
+  symmetricMap_ι e v
+
+variable (root : V) (coroot : Module.Dual k V) (hroot : coroot root = 2)
+
+abbrev reflectionAction : SymmetricAlgebra k V ≃ₐ[k] SymmetricAlgebra k V :=
+  symmetricAction (Module.reflection hroot)
+
+omit [CharZero k] in
+ theorem reflectionAction_involutive : Function.Involutive (reflectionAction root coroot hroot) := by
+  intro a
+  have he : (reflectionAction root coroot hroot).toAlgHom.comp
+      (reflectionAction root coroot hroot).toAlgHom =
+        AlgHom.id k (SymmetricAlgebra k V) := by
+    apply SymmetricAlgebra.algHom_ext
+    ext v
+    simp [Module.involutive_reflection hroot v]
+  exact congrArg (fun f : SymmetricAlgebra k V →ₐ[k] SymmetricAlgebra k V => f a) he
+
+omit [CharZero k] in
+ theorem reflectionAction_anti :
+    reflectionAction root coroot hroot (SymmetricAlgebra.ι k V root) =
+      -SymmetricAlgebra.ι k V root := by
+  simp
+
+include hroot in
+ theorem reflectionRoot_ne_zero : SymmetricAlgebra.ι k V root ≠ 0 := by
+  intro he
+  have hh := congrArg (SymmetricAlgebra.lift coroot) he
+  rw [SymmetricAlgebra.lift_ι_apply,map_zero,hroot] at hh
+  exact two_ne_zero hh
+
+omit [CharZero k] in
+ theorem reflectionAction_divisible (a : SymmetricAlgebra k V) :
+    SymmetricAlgebra.ι k V root ∣ a-reflectionAction root coroot hroot a := by
+  induction a using SymmetricAlgebra.induction with
+  | algebraMap c => simp
+  | ι v =>
+    refine ⟨algebraMap k (SymmetricAlgebra k V) (coroot v), ?_⟩
+    rw [reflectionAction,symmetricAction_ι,Module.reflection_apply,map_sub,map_smul]
+    simp only [sub_sub_cancel,Algebra.smul_def]
+    ring
+  | mul a b ha hb =>
+    obtain ⟨c,hc⟩ := ha
+    obtain ⟨d,hd⟩ := hb
+    refine ⟨c*b+(reflectionAction root coroot hroot a)*d, ?_⟩
+    rw [map_mul]
+    linear_combination b*hc+(reflectionAction root coroot hroot a)*hd
+  | add a b ha hb =>
+    simpa only [map_add, add_sub_add_comm] using dvd_add ha hb
+
+/-- A genuine simple-reflection integral Frobenius extension in characteristic
+zero, for the symmetric algebra of any representation carrying the reflection. -/
+def reflectionDuality :
+    SymmetricAlgebra k V ≃ₗ[fixed (reflectionAction root coroot hroot)]
+      Module.Dual (fixed (reflectionAction root coroot hroot)) (SymmetricAlgebra k V) :=
+  duality (reflectionAction root coroot hroot) (SymmetricAlgebra.ι k V root)
+    (reflectionAction_involutive root coroot hroot) (reflectionRoot_ne_zero root coroot hroot)
+    (reflectionAction_anti root coroot hroot) (reflectionAction_divisible root coroot hroot)
+
+end Symmetric
+
+end
+end KLInvariance.ReflectionFrobenius
+
+end
+
+
+section
+
+/-! Frobenius extensions for the actual simple reflections of the extended
+theorem realization, not abstract reflection data carried by the target. -/
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/KazhdanLusztig/Dihedral.lean b/lean/OAI/RepresentationTheory/KazhdanLusztig/Dihedral.lean
index 58955f15e443056ba3e02efc8726a7d12b56d85b..50f6d6250ec94d19017ad7e4ed3da29b70566ef9 100644
--- a/lean/OAI/RepresentationTheory/KazhdanLusztig/Dihedral.lean
+++ b/lean/OAI/RepresentationTheory/KazhdanLusztig/Dihedral.lean
@@ -1,803 +1,803 @@
-import OAI.RepresentationTheory.KazhdanLusztig.CanonicalKL
-
-namespace OAI
-
-/-!
-Interval restrictions, dihedral Coxeter and Hecke calculations, and the genuine square-parameter R normalization.
--/
-
-section
-
-namespace KLInvariance
-
-universe u₁ v₁ u₂ v₂
-variable {B : Type u₁} {W : Type v₁} [Group W] {M : CoxeterMatrix B}
-variable {B' : Type u₂} {W' : Type v₂} [Group W'] {M' : CoxeterMatrix B'}
-
-/-- Restriction of the actual interval isomorphism to any subinterval. This
-is the same unlabeled map, not a new isomorphism with extra structure. -/
-def intervalIso_restrict (cs : CoxeterSystem M W) (cs' : CoxeterSystem M' W')
-    {u b : W} {u' b' : W'} (ι : Interval cs u b ≃o Interval cs' u' b')
-    (x y : Interval cs u b) :
-    Interval cs x.val y.val ≃o Interval cs' (ι x).val (ι y).val where
-  toFun z :=
-    let zI : Interval cs u b := ⟨z.val, bruhat_trans cs x.property.1 z.property.1,
-      bruhat_trans cs z.property.2 y.property.2⟩
-    ⟨(ι zI).val, ι.monotone (show x ≤ zI from z.property.1),
-      ι.monotone (show zI ≤ y from z.property.2)⟩
-  invFun z :=
-    let zI : Interval cs' u' b' := ⟨z.val,
-      bruhat_trans cs' (ι x).property.1 z.property.1,
-      bruhat_trans cs' z.property.2 (ι y).property.2⟩
-    ⟨(ι.symm zI).val,
-      by
-        change x ≤ ι.symm zI
-        simpa only [OrderIso.symm_apply_apply] using
-          ι.symm.monotone (show ι x ≤ zI from z.property.1),
-      by
-        change ι.symm zI ≤ y
-        simpa only [OrderIso.symm_apply_apply] using
-          ι.symm.monotone (show zI ≤ ι y from z.property.2)⟩
-  left_inv z := by
-    apply Subtype.ext
-    let zI : Interval cs u b := ⟨z.val, bruhat_trans cs x.property.1 z.property.1,
-      bruhat_trans cs z.property.2 y.property.2⟩
-    change (ι.symm (ι zI)).val = zI.val
-    rw [ι.symm_apply_apply]
-  right_inv z := by
-    apply Subtype.ext
-    let zI : Interval cs' u' b' := ⟨z.val,
-      bruhat_trans cs' (ι x).property.1 z.property.1,
-      bruhat_trans cs' z.property.2 (ι y).property.2⟩
-    change (ι (ι.symm zI)).val = zI.val
-    rw [ι.apply_symm_apply]
-  map_rel_iff' := by
-    intro z w
-    exact ι.map_rel_iff
-
-@[simp] theorem intervalIso_restrict_val (cs : CoxeterSystem M W)
-    (cs' : CoxeterSystem M' W') {u b : W} {u' b' : W'}
-    (ι : Interval cs u b ≃o Interval cs' u' b') (x y : Interval cs u b)
-    (z : Interval cs x.val y.val) :
-    (intervalIso_restrict cs cs' ι x y z).val =
-      (ι ⟨z.val, bruhat_trans cs x.property.1 z.property.1,
-        bruhat_trans cs z.property.2 y.property.2⟩).val := rfl
-
-end KLInvariance
-
-namespace KLInvariance.Dihedral
-
-variable {W : Type*} [Group W] {M : CoxeterMatrix Bool}
-
-/-- The alternating expression starting with `i`, unlike Mathlib's
-`alternatingWord`, which specifies the final letter. -/
-def alt (i : Bool) : ℕ → List Bool
-  | 0 => []
-  | n + 1 => i :: alt (!i) n
-
-@[simp] theorem alt_zero (i : Bool) : alt i 0 = [] := rfl
-@[simp] theorem alt_succ (i : Bool) (n : ℕ) : alt i (n+1) = i :: alt (!i) n := rfl
-@[simp] theorem length_alt (i : Bool) (n : ℕ) : (alt i n).length = n := by
-  induction n generalizing i with
-  | zero => rfl
-  | succ n ih => simp [ih]
-
-@[simp] theorem take_alt (i : Bool) (k n : ℕ) : (alt i n).take k = alt i (min k n) := by
-  induction n generalizing i k with
-  | zero => simp
-  | succ n ih =>
-    cases k with
-    | zero => simp
-    | succ k => simp [ih]
-
-theorem reduced_cons_cons_ne (cs : CoxeterSystem M W) {i j : Bool} {ω : List Bool}
-    (h : cs.IsReduced (i :: j :: ω)) : i ≠ j := by
-  rintro rfl
-  have hlen := cs.length_wordProd_le ω
-  have heq := h.eq
-  simp only [cs.wordProd_cons, cs.simple_mul_simple_cancel_left, List.length_cons] at heq
-  omega
-
-theorem reduced_cons_eq_alt (cs : CoxeterSystem M W) (i : Bool) (ω : List Bool)
-    (h : cs.IsReduced (i :: ω)) : i :: ω = alt i (ω.length + 1) := by
-  induction ω generalizing i with
-  | nil => simp
-  | cons j ω ih =>
-    have hj : j = !i := by
-      have := reduced_cons_cons_ne cs h
-      cases i <;> cases j <;> simp_all
-    have hω : cs.IsReduced (j :: ω) := by simpa using h.drop 1
-    rw [show (j :: ω).length + 1 = (ω.length + 1) + 1 by rfl, alt_succ]
-    congr 1
-    rw [ih j hω, hj]
-
-theorem reduced_eq_alt (cs : CoxeterSystem M W) (ω : List Bool)
-    (h : cs.IsReduced ω) : ∃ i, ω = alt i ω.length := by
-  cases ω with
-  | nil => exact ⟨false, rfl⟩
-  | cons i ω => exact ⟨i, reduced_cons_eq_alt cs i ω h⟩
-
-theorem exists_alt (cs : CoxeterSystem M W) (w : W) :
-    ∃ i, w = cs.wordProd (alt i (cs.length w)) := by
-  obtain ⟨ω, hω, hw⟩ := cs.exists_isReduced w
-  obtain ⟨i, hi⟩ := reduced_eq_alt cs ω hω
-  refine ⟨i, ?_⟩
-  rw [hw, hω.eq, ← hi]
-
-theorem reduced_alt_take (cs : CoxeterSystem M W) {i : Bool} {k n : ℕ}
-    (h : cs.IsReduced (alt i n)) (hk : k ≤ n) : cs.IsReduced (alt i k) := by
-  simpa only [take_alt, Nat.min_eq_left hk] using h.take k
-
-theorem alt_sublist (i : Bool) {k n : ℕ} (h : k ≤ n) : (alt i k).Sublist (alt i n) := by
-  simpa only [take_alt, Nat.min_eq_left h] using List.take_sublist k (alt i n)
-
-end KLInvariance.Dihedral
-
-namespace KLInvariance.Dihedral
-
-noncomputable local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
-variable {W : Type*} [Group W] {M : CoxeterMatrix Bool}
-
-/-- In a rank-two Coxeter system, every strictly shorter element lies below
-every longer element. This uses the actual Bruhat subword theorem. -/
-theorem shorter_bruhat (cs : CoxeterSystem M W) {x w : W}
-    (h : cs.length x < cs.length w) : BruhatLE cs x w := by
-  obtain ⟨i, hi⟩ := exists_alt cs w
-  obtain ⟨j, hj⟩ := exists_alt cs x
-  have hr : cs.IsReduced (alt i (cs.length w)) := by
-    change cs.length (cs.wordProd (alt i (cs.length w))) = (alt i (cs.length w)).length
-    rw [← hi, length_alt]
-  obtain ⟨n, hn⟩ := Nat.exists_eq_succ_of_ne_zero (show cs.length w ≠ 0 by omega)
-  rw [hn] at hi hr h
-  rw [hi, hj]
-  apply bruhat_of_subword cs _ hr
-  by_cases hji : j = i
-  · subst j
-    exact alt_sublist i (by omega)
-  · have hj' : j = !i := by cases i <;> cases j <;> first | rfl | exact (hji rfl).elim
-    rw [hj', alt_succ]
-    exact (alt_sublist (!i) (by omega)).cons i
-
-theorem bruhat_iff_length_lt_or_eq (cs : CoxeterSystem M W) (x w : W) :
-    BruhatLE cs x w ↔ cs.length x < cs.length w ∨ x = w := by
-  constructor
-  · intro h
-    by_cases he : x = w
-    · exact Or.inr he
-    · exact Or.inl (length_lt_of_bruhat_ne cs h he)
-  · rintro (h | rfl)
-    · exact shorter_bruhat cs h
-    · exact bruhat_refl cs _
-
-theorem eq_alt_of_descent (cs : CoxeterSystem M W) (i : Bool) {w : W}
-    (h : cs.length (cs.simple i * w) < cs.length w) :
-    w = cs.wordProd (alt i (cs.length w)) := by
-  obtain ⟨ω, hω, hw⟩ := cs.exists_isReduced (cs.simple i * w)
-  have hlen : cs.length w = ω.length + 1 := by
-    have := cs.length_simple_mul w i
-    rw [hw, hω.eq] at this h
-    omega
-  have hp : cs.wordProd (i :: ω) = w := by
-    simp only [cs.wordProd_cons, ← hw, cs.simple_mul_simple_cancel_left]
-  have hr : cs.IsReduced (i :: ω) := by
-    simpa only [CoxeterSystem.IsReduced, hp, List.length_cons] using hlen
-  rw [hlen, ← reduced_cons_eq_alt cs i ω hr, hp]
-
-end KLInvariance.Dihedral
-
-namespace KLInvariance.Dihedral
-
-noncomputable local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
-open Hecke (Space basis)
-variable {W : Type*} [Group W] {M : CoxeterMatrix Bool}
-variable {C : Type*} [CommRing C]
-
-noncomputable def lowerSum (cs : CoxeterSystem M W) (w : W) : Space W C :=
-  ∑ z ∈ lowerFinset cs w, basis z
-
-@[simp] theorem lowerSum_coeff (cs : CoxeterSystem M W) (w x : W) :
-    (lowerSum cs w : Space W C) x = if BruhatLE cs x w then 1 else 0 := by
-  classical
-  simp [lowerSum, basis, Finsupp.single_apply, eq_comm]
-
-@[simp] theorem lowerSum_one (cs : CoxeterSystem M W) :
-    (lowerSum cs 1 : Space W C) = basis 1 := by
-  classical
-  ext x
-  simp only [lowerSum_coeff, basis, Finsupp.single_apply]
-  have h : BruhatLE cs x 1 ↔ 1 = x := by
-    rw [bruhat_iff_length_lt_or_eq]
-    simp [eq_comm]
-  rw [h]
-
-/-- The two members of an s-string enter a lower ideal together precisely
-on the common lower interval appearing in the source's rank-two product. -/
-theorem double_string (cs : CoxeterSystem M W) (i : Bool) (n : ℕ)
-    (hr : cs.IsReduced (alt i (n+2))) (x : W) :
-    (BruhatLE cs x (cs.wordProd (alt (!i) (n+1))) ∧
-      BruhatLE cs (cs.simple i * x) (cs.wordProd (alt (!i) (n+1)))) ↔
-    (0 < n ∧ BruhatLE cs x (cs.wordProd (alt i n))) := by
-  have hr' : cs.IsReduced (alt (!i) (n+1)) := by simpa using hr.drop 1
-  have hrp := reduced_alt_take cs hr (show n ≤ n+2 by omega)
-  have hlenw : cs.length (cs.wordProd (alt i (n+2))) = n+2 := by
-    simpa only [length_alt] using hr.eq
-  have hlenz : cs.length (cs.wordProd (alt (!i) (n+1))) = n+1 := by
-    simpa only [length_alt] using hr'.eq
-  have hlenp : cs.length (cs.wordProd (alt i n)) = n := by
-    simpa only [length_alt] using hrp.eq
-  have hsz : cs.simple i * cs.wordProd (alt (!i) (n+1)) =
-      cs.wordProd (alt i (n+2)) := by rw [alt_succ i (n+1), cs.wordProd_cons]
-  constructor
-  · rintro ⟨hx, hsx⟩
-    have hxlen := length_le_of_bruhat cs hx
-    have hsxlen := length_le_of_bruhat cs hsx
-    have hxl : cs.length x < n+1 := by
-      rcases (bruhat_iff_length_lt_or_eq cs _ _).mp hx with h | rfl
-      · omega
-      · rw [hsz, hlenw] at hsxlen
-        omega
-    have hsxl : cs.length (cs.simple i * x) < n+1 := by
-      rcases (bruhat_iff_length_lt_or_eq cs _ _).mp hsx with h | h
-      · omega
-      · have heq : x = cs.wordProd (alt i (n+2)) := by
-          rw [← hsz, ← h, cs.simple_mul_simple_cancel_left]
-        rw [heq, hlenw] at hxl
-        omega
-    have hchange := cs.length_simple_mul x i
-    have hn : 0 < n := by omega
-    refine ⟨hn, ?_⟩
-    by_cases hxlt : cs.length x < n
-    · exact shorter_bruhat cs (by omega)
-    · have hxi : cs.length x = n := by omega
-      have hdesc : cs.length (cs.simple i * x) < cs.length x := by omega
-      have heq := eq_alt_of_descent cs i hdesc
-      rw [hxi] at heq
-      rw [heq]
-      exact bruhat_refl cs _
-  · rintro ⟨hn, hx⟩
-    have hxlen := length_le_of_bruhat cs hx
-    constructor
-    · exact shorter_bruhat cs (by omega)
-    · by_cases hxlt : cs.length x < n
-      · have := cs.length_simple_mul x i
-        exact shorter_bruhat cs (by omega)
-      · have heq : x = cs.wordProd (alt i n) := by
-          rcases (bruhat_iff_length_lt_or_eq cs _ _).mp hx with h | h
-          · omega
-          · exact h
-        obtain ⟨k, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (show n ≠ 0 by omega)
-        have hrpt : cs.IsReduced (alt (!i) k) := by simpa using hrp.drop 1
-        have hpk : cs.length (cs.simple i * x) = k := by
-          rw [heq, alt_succ, cs.wordProd_cons, cs.simple_mul_simple_cancel_left,
-            hrpt.eq, length_alt]
-        exact shorter_bruhat cs (by omega)
-
-end KLInvariance.Dihedral
-
-namespace KLInvariance.Dihedral
-
-noncomputable local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
-open Hecke (Space basis)
-variable {W : Type*} [Group W] {M : CoxeterMatrix Bool}
-variable {C : Type*} [CommRing C]
-
-theorem left_plus_coeff (cs : CoxeterSystem M W) (q : Cˣ) (i : Bool)
-    (v : Space W C) (x : W) :
-    (v + HeckeQ.left cs (q : C) i v) x =
-      if cs.length (cs.simple i * x) < cs.length x then
-        v (cs.simple i * x) + (q : C) * v x
-      else v x + (q : C) * v (cs.simple i * x) := by
-  have h := congrArg (fun t : Space W C => (q : C) * t x)
-    (HeckeQ.leftInv_apply cs q i v)
-  simp only [Finsupp.smul_apply, smul_eq_mul, Finsupp.sub_apply, ← mul_assoc,
-    q.mul_inv, one_mul] at h
-  rw [HeckeQ.leftInv_coeff] at h
-  simp only [Finsupp.add_apply]
-  split_ifs at h ⊢
-  · linear_combination -h + v (cs.simple i * x) * q.mul_inv
-  · linear_combination -h + v x * q.mul_inv
-
-theorem union_string (cs : CoxeterSystem M W) (i : Bool) {z : W}
-    (hz : cs.length z < cs.length (cs.simple i * z)) (x : W) :
-    BruhatLE cs x (cs.simple i * z) ↔
-      BruhatLE cs x z ∨ BruhatLE cs (cs.simple i * x) z := by
-  have hsz : cs.length (cs.simple i * (cs.simple i * z)) <
-      cs.length (cs.simple i * z) := by simpa using hz
-  constructor
-  · intro h
-    have hh := lowerSimple_mono cs i h
-    simp only [lowerSimple, cs.simple_mul_simple_cancel_left] at hh
-    split_ifs at hh with hx
-    · exact Or.inr hh
-    · exact Or.inl hh
-  · rintro (h | h)
-    · apply bruhat_trans cs h
-      exact Relation.ReflTransGen.single ⟨hz, cs.simple i, cs.isReflection_simple i, rfl⟩
-    · simpa only [cs.simple_mul_simple_cancel_left] using
-        Hecke.simple_mul_le_ascent cs i h hz
-
-noncomputable def correction (cs : CoxeterSystem M W) (i : Bool) (n : ℕ) : Space W C :=
-  if n = 0 then 0 else lowerSum cs (cs.wordProd (alt i n))
-
-@[simp] theorem correction_coeff (cs : CoxeterSystem M W) (i : Bool) (n : ℕ) (x : W) :
-    (correction cs i n : Space W C) x =
-      if 0 < n ∧ BruhatLE cs x (cs.wordProd (alt i n)) then 1 else 0 := by
-  classical
-  by_cases h : n = 0
-  · simp [correction, h]
-  · have hn : 0 < n := Nat.pos_of_ne_zero h
-    simp [correction, h, hn]
-
-/-- The Hecke sum identity `loc:dihedral-product`, before normalization by
-q to a half-integer power. Every interval here is genuine strong Bruhat. -/
-theorem lowerSum_product (cs : CoxeterSystem M W) (q : Cˣ) (i : Bool) (n : ℕ)
-    (hr : cs.IsReduced (alt i (n+2))) :
-    lowerSum cs (cs.wordProd (alt (!i) (n+1))) +
-      HeckeQ.left cs (q : C) i (lowerSum cs (cs.wordProd (alt (!i) (n+1)))) =
-    lowerSum cs (cs.wordProd (alt i (n+2))) + (q : C) • correction cs i n := by
-  classical
-  have hr' : cs.IsReduced (alt (!i) (n+1)) := by simpa using hr.drop 1
-  have hz : cs.length (cs.wordProd (alt (!i) (n+1))) <
-      cs.length (cs.simple i * cs.wordProd (alt (!i) (n+1))) := by
-    rw [← cs.wordProd_cons, ← alt_succ, hr.eq, hr'.eq, length_alt, length_alt]
-    omega
-  ext x
-  rw [left_plus_coeff, Finsupp.add_apply, Finsupp.smul_apply, smul_eq_mul,
-    correction_coeff, lowerSum_coeff, lowerSum_coeff, lowerSum_coeff]
-  have hu := union_string cs i hz x
-  rw [← cs.wordProd_cons, ← alt_succ] at hu
-  have hd := double_string cs i n hr x
-  simp only [hu, ← hd]
-  by_cases hdesc : cs.length (cs.simple i * x) < cs.length x
-  · have hclosure : BruhatLE cs x (cs.wordProd (alt (!i) (n+1))) →
-        BruhatLE cs (cs.simple i * x) (cs.wordProd (alt (!i) (n+1))) := by
-      intro h
-      apply bruhat_trans cs _ h
-      exact Relation.ReflTransGen.single
-        ⟨hdesc, cs.simple i, cs.isReflection_simple i, by simp⟩
-    by_cases hx : BruhatLE cs x (cs.wordProd (alt (!i) (n+1)))
-    · simp [-alt_succ, hdesc, hx, hclosure hx]
-    · by_cases hs : BruhatLE cs (cs.simple i * x) (cs.wordProd (alt (!i) (n+1))) <;>
-        simp [-alt_succ, hdesc, hx, hs]
-  · have hasc : cs.length x < cs.length (cs.simple i * x) := by
-      have := cs.length_simple_mul x i
-      omega
-    have hclosure : BruhatLE cs (cs.simple i * x) (cs.wordProd (alt (!i) (n+1))) →
-        BruhatLE cs x (cs.wordProd (alt (!i) (n+1))) := by
-      intro h
-      apply bruhat_trans cs _ h
-      exact Relation.ReflTransGen.single
-        ⟨hasc, cs.simple i, cs.isReflection_simple i, rfl⟩
-    by_cases hs : BruhatLE cs (cs.simple i * x) (cs.wordProd (alt (!i) (n+1)))
-    · simp [-alt_succ, hdesc, hs, hclosure hs]
-    · by_cases hx : BruhatLE cs x (cs.wordProd (alt (!i) (n+1))) <;>
-        simp [-alt_succ, hdesc, hx, hs]
-
-end KLInvariance.Dihedral
-
-namespace KLInvariance.HeckeQ
-
-universe uR vR
-variable {B : Type uR} {W : Type vR} [Group W] {M : CoxeterMatrix B}
-variable {C : Type*} [CommRing C]
-
-noncomputable def dualR (cs : CoxeterSystem M W) (q : Cˣ) (x y : W) : C :=
-  (-(q : C)) ^ cs.length y * (-(↑q⁻¹ : C)) ^ cs.length x * rValue cs q⁻¹ x y
-
-@[simp] theorem dualR_diagonal (cs : CoxeterSystem M W) (q : Cˣ) (x : W) :
-    dualR cs q x x = 1 := by
-  rw [dualR, rValue_diagonal, mul_one, ← mul_pow]
-  simp
-
-theorem dualR_zero (cs : CoxeterSystem M W) (q : Cˣ) (x y : W)
-    (h : ¬BruhatLE cs x y) : dualR cs q x y = 0 := by
-  simp [dualR, rValue_zero cs q⁻¹ x y h]
-
-theorem dualR_recursion (cs : CoxeterSystem M W) (q : Cˣ) (x y : W) (i : B)
-    (hy : cs.length (cs.simple i * y) < cs.length y) :
-    dualR cs q x y =
-      if cs.length (cs.simple i * x) < cs.length x then
-        dualR cs q (cs.simple i * x) (cs.simple i * y)
-      else ((q : C) - 1) * dualR cs q x (cs.simple i * y) +
-        (q : C) * dualR cs q (cs.simple i * x) (cs.simple i * y) := by
-  have hyl : cs.length y = cs.length (cs.simple i * y) + 1 := by
-    have := cs.length_simple_mul y i
-    omega
-  unfold dualR
-  rw [rValue_recursion cs q⁻¹ x y i hy]
-  split_ifs with hx
-  · have hxl : cs.length x = cs.length (cs.simple i * x) + 1 := by
-      have := cs.length_simple_mul x i
-      omega
-    rw [hyl, hxl, pow_succ, pow_succ]
-    calc
-      _ = (-(q:C)) ^ cs.length (cs.simple i * y) *
-          (-(↑q⁻¹:C)) ^ cs.length (cs.simple i * x) *
-          rValue cs q⁻¹ (cs.simple i * x) (cs.simple i * y) *
-          ((q:C) * (↑q⁻¹:C)) := by ring
-      _ = _ := by rw [q.mul_inv, mul_one]
-  · have hxl : cs.length (cs.simple i * x) = cs.length x + 1 := by
-      have := cs.length_simple_mul x i
-      omega
-    rw [hyl, hxl, pow_succ, pow_succ]
-    have hq : (-(q:C)) * ((↑q⁻¹:C) - 1) = (q:C) - 1 := by
-      rw [mul_sub, neg_mul, q.mul_inv, mul_one]
-      ring
-    calc
-      _ = (-(q:C)) ^ cs.length (cs.simple i * y) *
-          (-(↑q⁻¹:C)) ^ cs.length x *
-          ((-(q:C)) * ((↑q⁻¹:C) - 1)) * rValue cs q⁻¹ x (cs.simple i * y) +
-          (q:C) * ((-(q:C)) ^ cs.length (cs.simple i * y) *
-          ((-(↑q⁻¹:C)) ^ cs.length x * (-(↑q⁻¹:C))) *
-          rValue cs q⁻¹ (cs.simple i * x) (cs.simple i * y)) := by ring
-      _ = _ := by rw [hq]; ring
-
-/-- Signed inversion symmetry of actual equal-parameter R-values, proved
-from all-descent recurrence, not imposed as an additional normalization. -/
-theorem rValue_eq_dualR (cs : CoxeterSystem M W) (q : Cˣ) (x y : W) :
-    rValue cs q x y = dualR cs q x y := by
-  suffices ∀ n : ℕ, ∀ y : W, cs.length y = n → ∀ x,
-      rValue cs q x y = dualR cs q x y by
-    exact this _ y rfl x
-  intro n
-  induction n using Nat.strong_induction_on with
-  | h n ih =>
-    intro y hy x
-    by_cases hy1 : y = 1
-    · subst y
-      by_cases hx1 : x = 1
-      · subst x
-        simp
-      · have hnot : ¬BruhatLE cs x 1 := by
-          intro h
-          have hl := length_le_of_bruhat cs h
-          have hl0 : cs.length x = 0 := by simpa using hl
-          exact hx1 (cs.length_eq_zero_iff.mp hl0)
-        rw [rValue_zero cs q x 1 hnot, dualR_zero cs q x 1 hnot]
-    · obtain ⟨i, hi⟩ := cs.exists_leftDescent_of_ne_one hy1
-      have hsmall : cs.length (cs.simple i * y) < n := hy ▸ hi
-      rw [rValue_recursion cs q x y i hi, dualR_recursion cs q x y i hi,
-        ih _ hsmall _ rfl x, ih _ hsmall _ rfl (cs.simple i * x)]
-
-/-- R at the inverse parameter is exactly the corresponding bar coefficient
-with the bottom q-power. This is the coefficient form used below. -/
-theorem rValue_inverse_eq_barBasis (cs : CoxeterSystem M W) (q : Cˣ) (x y : W) :
-    rValue cs q⁻¹ x y = (q:C) ^ cs.length x * barBasis cs q y x := by
-  rw [rValue_eq_dualR cs q⁻¹, dualR, inv_inv, rValue]
-  simp only [neg_pow (↑q⁻¹:C), neg_pow (q:C)]
-  have hsign (n : ℕ) : (-1:C)^n * (-1:C)^n = 1 := by
-    rw [← mul_pow]; simp
-  calc
-    _ = ((-1:C)^cs.length y * (-1:C)^cs.length y) *
-        ((-1:C)^cs.length x * (-1:C)^cs.length x) *
-        (((↑q⁻¹:C) * (q:C)) ^ cs.length y) *
-        (q:C)^cs.length x * barBasis cs q y x := by ring
-    _ = _ := by rw [hsign, hsign, q.inv_mul]; simp
-
-end KLInvariance.HeckeQ
-
-namespace KLInvariance.Dihedral
-
-noncomputable local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
-open Hecke (Space basis)
-variable {W : Type*} [Group W] {M : CoxeterMatrix Bool}
-variable {C : Type*} [CommRing C]
-
-@[simp] theorem lowerSum_simple (cs : CoxeterSystem M W) (i : Bool) :
-    (lowerSum cs (cs.simple i) : Space W C) = basis 1 + basis (cs.simple i) := by
-  classical
-  ext x
-  rw [lowerSum_coeff, bruhat_iff_length_lt_or_eq, cs.length_simple]
-  simp only [Nat.lt_one_iff, cs.length_eq_zero_iff, Finsupp.add_apply, basis,
-    Finsupp.single_apply]
-  have hi : cs.simple i ≠ 1 := by
-    intro h
-    have := cs.length_simple i
-    simp [h] at this
-  by_cases hx : x = 1
-  · subst x
-    simp [hi]
-  · by_cases hs : x = cs.simple i
-    · subst x
-      simp [hi, Ne.symm hi]
-    · simp [hx, hs, Ne.symm hx, Ne.symm hs]
-
-theorem plus_leftInv (cs : CoxeterSystem M W) (q : Cˣ) (i : Bool)
-    (v : Space W C) :
-    v + HeckeQ.leftInv cs q i v =
-      (↑q⁻¹:C) • (v + HeckeQ.left cs (q:C) i v) := by
-  rw [HeckeQ.leftInv_apply, smul_sub, smul_smul, mul_sub, q.inv_mul, mul_one]
-  module
-
-/-- Unnormalized form of bar invariance of the dihedral constant sums. -/
-theorem bar_lowerSum_alt (cs : CoxeterSystem M W) (q : Cˣ) (i : Bool) (n : ℕ)
-    (hr : cs.IsReduced (alt i n)) :
-    HeckeQ.barLinear cs q (lowerSum cs (cs.wordProd (alt i n))) =
-      (↑q⁻¹:C)^n • lowerSum cs (cs.wordProd (alt i n)) := by
-  induction n using Nat.strong_induction_on generalizing i with
-  | h n ih =>
-    rcases n with _ | (_ | n)
-    · simp
-    · have hasc : cs.length (1:W) < cs.length (cs.simple i * 1) := by simp
-      have hb := HeckeQ.barBasis_simple_mul_ascent cs q i 1 hasc
-      simp only [mul_one, HeckeQ.barBasis_one] at hb
-      simp only [alt_succ, alt_zero, cs.wordProd_singleton, lowerSum_simple,
-        map_add, HeckeQ.barLinear_basis, HeckeQ.barBasis_one, hb]
-      rw [plus_leftInv, HeckeQ.left_basis]
-      simp
-    · have hr' : cs.IsReduced (alt (!i) (n+1)) := by simpa using hr.drop 1
-      have hrp := reduced_alt_take cs hr (show n ≤ n+2 by omega)
-      have hz := ih (n+1) (by omega) (!i) hr'
-      have hc : HeckeQ.barLinear cs q (correction cs i n) =
-          (↑q⁻¹:C)^n • correction cs i n := by
-        by_cases hn : n = 0
-        · simp [correction, hn]
-        · simp only [correction, ite_eq_right hn]
-          exact ih n (by omega) i hrp
-      have hprod := lowerSum_product cs q i n hr
-      have hbar := congrArg (HeckeQ.barLinear cs q)
-        (lowerSum_product cs q⁻¹ i n hr)
-      have hinter := congrArg (fun f : Module.End C (Space W C) =>
-          f (lowerSum cs (cs.wordProd (alt (!i) (n+1)))))
-        (HeckeQ.barLinear_left cs q i)
-      simp only [map_add, map_smul, Module.End.mul_apply] at hbar hinter
-      rw [hinter, hz, map_smul, ← smul_add, plus_leftInv, hprod] at hbar
-      simp only [smul_add, smul_smul, hc] at hbar
-      have hq1 : (↑q⁻¹:C)^(n+1) * (↑q⁻¹:C) = (↑q⁻¹:C)^(n+2) := by
-        exact (pow_succ (↑q⁻¹:C) (n+1)).symm
-      have hq2 : (↑q⁻¹:C)^(n+1) * ((↑q⁻¹:C) * (q:C)) =
-          (↑q⁻¹:C) * (↑q⁻¹:C)^n := by rw [q.inv_mul, mul_one, pow_succ'];
-      rw [hq1, hq2] at hbar
-      exact add_right_cancel hbar.symm
-
-theorem bar_lowerSum (cs : CoxeterSystem M W) (q : Cˣ) (w : W) :
-    HeckeQ.barLinear cs q (lowerSum cs w) =
-      (↑q⁻¹:C)^cs.length w • lowerSum cs w := by
-  obtain ⟨i, hi⟩ := exists_alt cs w
-  have hr : cs.IsReduced (alt i (cs.length w)) := by
-    simp only [CoxeterSystem.IsReduced, ← hi, length_alt]
-  simpa only [← hi] using bar_lowerSum_alt cs q i (cs.length w) hr
-
-end KLInvariance.Dihedral
-
-namespace KLInvariance.Dihedral
-
-noncomputable local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
-open Hecke (Space basis)
-open Polynomial
-variable {W : Type*} [Group W] {M : CoxeterMatrix Bool}
-variable {C : Type*} [CommRing C]
-
-theorem rValue_sum (cs : CoxeterSystem M W) (q : Cˣ) (x y : W)
-    (hxy : BruhatLE cs x y) :
-    ∑ z ∈ lowerFinset cs y, HeckeQ.rValue cs q x z =
-      (q:C) ^ rankDifference cs x y := by
-  classical
-  have hb := congrArg (fun f : Space W C => f x) (bar_lowerSum cs q⁻¹ y)
-  simp only [Finsupp.smul_apply, smul_eq_mul, inv_inv, lowerSum_coeff,
-    ite_eq_left hxy, mul_one] at hb
-  rw [lowerSum, map_sum] at hb
-  simp only [Finsupp.coe_finsetSum, Finset.sum_apply, HeckeQ.barLinear_basis] at hb
-  have hlen : cs.length y = cs.length x + rankDifference cs x y := by
-    have := length_le_of_bruhat cs hxy
-    unfold rankDifference
-    omega
-  calc
-    _ = (↑q⁻¹:C)^cs.length x *
-        ∑ z ∈ lowerFinset cs y, HeckeQ.barBasis cs q⁻¹ z x := by
-      rw [Finset.mul_sum]
-      apply Finset.sum_congr rfl
-      intro z hz
-      simpa using HeckeQ.rValue_inverse_eq_barBasis cs q⁻¹ x z
-    _ = (↑q⁻¹:C)^cs.length x * (q:C)^cs.length y := by rw [hb]
-    _ = _ := by
-      rw [hlen, pow_add]
-      calc
-        _ = ((↑q⁻¹:C) * (q:C))^cs.length x * (q:C)^rankDifference cs x y := by
-          rw [mul_pow]; ring
-        _ = _ := by rw [q.inv_mul]; simp
-
-theorem rPolynomial_sum (cs : CoxeterSystem M W) (x y : W)
-    (hxy : BruhatLE cs x y) :
-    ∑ z ∈ lowerFinset cs y, OrdinaryR.polynomial cs x z =
-      (X:ℤ[X]) ^ rankDifference cs x y := by
-  apply Polynomial.toLaurent_injective
-  simpa using rValue_sum cs OrdinaryR.variableUnit x y hxy
-
-noncomputable def constantP (cs : CoxeterSystem M W) (x y : W) : ℤ[X] :=
-  if BruhatLE cs x y then 1 else 0
-
-theorem constant_normalizedKL (cs : CoxeterSystem M W) :
-    NormalizedKL cs (OrdinaryR.polynomial cs, constantP cs) where
-  R_diagonal := OrdinaryR.polynomial_diagonal cs
-  R_zero := OrdinaryR.polynomial_zero cs
-  R_recursion := OrdinaryR.polynomial_recursion cs
-  P_diagonal := by intro x; simp [constantP, bruhat_refl cs x]
-  P_zero := by intro x y h; simp [constantP, h]
-  P_degree := by
-    intro x y hxy hne
-    have := length_lt_of_bruhat_ne cs hxy hne
-    simp only [constantP, ite_eq_left hxy, natDegree_one, mul_zero]
-    unfold rankDifference
-    omega
-  reciprocity := by
-    intro x y hxy
-    simp only [constantP, ite_eq_left hxy, reflect_one]
-    classical
-    let := interval_finite cs x y
-    let : Fintype (Interval cs x y) := Fintype.ofFinite _
-    rw [finsum_eq_sum_of_fintype]
-    have heq : (∑ z : Interval cs x y,
-        OrdinaryR.polynomial cs x z.val * (if BruhatLE cs z.val y then 1 else 0)) =
-        ∑ z : Interval cs x y, OrdinaryR.polynomial cs x z.val := by
-      apply Finset.sum_congr rfl
-      intro z hz
-      rw [ite_eq_left z.property.2, mul_one]
-    rw [heq]
-    have hsum := Finset.sum_subtype (F := (inferInstance : Fintype (Interval cs x y)))
-      ((lowerFinset cs y).filter (fun z => BruhatLE cs x z))
-      (fun z => by simp only [Finset.mem_filter, mem_lowerFinset]; exact and_comm)
-      (fun z => OrdinaryR.polynomial cs x z)
-    symm
-    refine hsum.symm.trans (Eq.trans ?_ (rPolynomial_sum cs x y hxy))
-    apply Finset.sum_subset (Finset.filter_subset _ _)
-    intro z hz hnot
-    have hnxz : ¬BruhatLE cs x z := fun hxz => hnot (Finset.mem_filter.mpr ⟨hz, hxz⟩)
-    exact OrdinaryR.polynomial_zero cs x z hnxz
-
-/-- The manuscript's necessary dihedral calculation (`loc:dihedral-kl`),
-for the actual KL family, with the exact standard normalization. This is
-support for localization, not a substitute for general invariance. -/
-theorem klPolynomial_eq_one (cs : CoxeterSystem M W) (x y : W)
-    (hxy : BruhatLE cs x y) : klPolynomial cs x y = 1 := by
-  have heq := normalizedKL_unique cs (klFamilies_spec cs) (constant_normalizedKL cs)
-  have hp := congrArg (fun RP : PolynomialFamilies W => RP.2 x y) heq
-  simpa only [klPolynomial, constantP, ite_eq_left hxy] using hp
-
-end KLInvariance.Dihedral
-
-end
-
-section
-
-namespace KLInvariance
-open Polynomial
-universe u v
-variable {B : Type u} {W : Type v} [Group W] {M : CoxeterMatrix B}
-
-/-- The exact square-parameter normalization, proved from the two genuine
-Hecke recurrences. Integer exponents let the formula include incomparable
-pairs without silently changing the R-polynomial definition. -/
-theorem rPolynomial_normalization {C : Type*} [Field C]
-    (cs : CoxeterSystem M W) (v : C) (hv : v ≠ 0) (x y : W) :
-    eval₂ (Int.castRingHom C) (v^2) (rPolynomial cs x y) =
-      v ^ ((cs.length y : ℤ) - cs.length x) *
-        eval₂ (Int.castRingHom C) (v-v⁻¹) (Hecke.rTilde cs x y) := by
-  rw [rPolynomial_eq_constructed]
-  suffices ∀ n : ℕ, ∀ y : W, cs.length y = n → ∀ x,
-    eval₂ (Int.castRingHom C) (v^2) (OrdinaryR.polynomial cs x y) =
-      v ^ ((cs.length y : ℤ) - cs.length x) *
-        eval₂ (Int.castRingHom C) (v-v⁻¹) (Hecke.rTilde cs x y) from
-    this _ y rfl x
-  intro n
-  induction n using Nat.strong_induction_on with
-  | h n ih =>
-    intro y hy x
-    by_cases h1 : y=1
-    · subst y
-      by_cases hx : x=1
-      · subst x; simp
-      · have hn : ¬BruhatLE cs x 1 := by
-          intro h
-          have hl := length_le_of_bruhat cs h
-          have hx0 : cs.length x=0 := by simpa using hl
-          exact hx (cs.length_eq_zero_iff.mp hx0)
-        rw [OrdinaryR.polynomial_zero cs _ _ hn,Hecke.rTilde_zero cs _ _ hn]
-        simp
-    · obtain ⟨i,hi⟩ := cs.exists_leftDescent_of_ne_one h1
-      have hlt : cs.length (cs.simple i*y)<n := hy ▸ hi
-      have hylen : cs.length (cs.simple i*y)+1=cs.length y := by
-        have := cs.length_simple_mul y i; omega
-      rw [OrdinaryR.polynomial_recursion cs x y i hi,Hecke.rTilde_recursion cs x y i hi]
-      by_cases hx : cs.length (cs.simple i*x)<cs.length x
-      · rw [ite_eq_left hx,ite_eq_left hx,ih _ hlt _ rfl (cs.simple i*x)]
-        congr 2
-        have := cs.length_simple_mul x i
-        omega
-      · rw [ite_eq_right hx,ite_eq_right hx]
-        simp only [eval₂_add,eval₂_mul,eval₂_sub,eval₂_X,eval₂_one]
-        rw [ih _ hlt _ rfl x,ih _ hlt _ rfl (cs.simple i*x)]
-        have hxlen : cs.length (cs.simple i*x)=cs.length x+1 := by
-          have := cs.length_simple_mul x i; omega
-        have h1 : (cs.length (cs.simple i*y) : ℤ)-cs.length x =
-            ((cs.length y : ℤ)-cs.length x)-1 := by omega
-        have h2 : (cs.length (cs.simple i*y) : ℤ)-cs.length (cs.simple i*x) =
-            ((cs.length y : ℤ)-cs.length x)-2 := by omega
-        rw [h1,h2,zpow_sub₀ hv ((cs.length y : ℤ)-cs.length x) 1,
-          zpow_sub₀ hv ((cs.length y : ℤ)-cs.length x) 2]
-        simp only [zpow_ofNat]
-        field_simp [hv]
-        ; ring
-
-/-- The manuscript's q^{-d/2} P(q), written with t^2=q so no square-root
-branch is hidden. This is an evaluation of the already constructed genuine
-KL polynomial, not a new combinatorial invariant. -/
-noncomputable def scaledKL {C : Type*} [Field C]
-    (cs : CoxeterSystem M W) (t : C) (x b : W) : C :=
-  t ^ ((cs.length x : ℤ)-cs.length b) *
-    eval₂ (Int.castRingHom C) (t^2) (klPolynomial cs x b)
-
-/-- Exact normalized reciprocity in the direction useful for reversing the
-parameter. No path formula and no moment-graph assumption occurs here. -/
-theorem scaledKL_reciprocity_inverse {C : Type*} [Field C]
-    (cs : CoxeterSystem M W) (v : C) (hv : v ≠ 0)
-    (x b : W) (hxb : BruhatLE cs x b) :
-    scaledKL cs v⁻¹ x b = ∑ᶠ z : Interval cs x b,
-      eval₂ (Int.castRingHom C) (v-v⁻¹) (Hecke.rTilde cs x z.val) *
-        scaledKL cs v z.val b := by
-  classical
-  let : Finite (Interval cs x b) := interval_finite cs x b
-  let : Fintype (Interval cs x b) := Fintype.ofFinite _
-  have hrec := (klFamilies_spec cs).reciprocity x b hxb
-  change reflect (rankDifference cs x b) (klPolynomial cs x b) =
-      ∑ᶠ z : Interval cs x b, rPolynomial cs x z.val * klPolynomial cs z.val b at hrec
-  have hdeg : (klPolynomial cs x b).natDegree ≤ rankDifference cs x b := by
-    rw [klPolynomial_eq_constructed]
-    exact CanonicalP.polynomial_degree cs x b
-  let q : Cˣ := Units.mk0 (v^2) (pow_ne_zero 2 hv)
-  have heval := congrArg (eval₂ (Int.castRingHom C) (q : C)) hrec
-  rw [eval₂_reflect_unit q _ _ hdeg] at heval
-  simp only [finsum_eq_sum_of_fintype,eval₂_finsetSum,eval₂_mul] at heval ⊢
-  have hq : (q : C) = v^2 := rfl
-  have hqi : (↑q⁻¹ : C) = (v⁻¹)^2 := by simp [q,inv_pow]
-  rw [hq,hqi] at heval
-  have heq := congrArg (fun a : C => v ^ ((cs.length x : ℤ)-cs.length b)*a) heval
-  have hd : (rankDifference cs x b : ℤ) = (cs.length b : ℤ)-cs.length x := by
-    have hlen := length_le_of_bruhat cs hxb
-    simp [rankDifference,Int.ofNat_sub hlen]
-  have hscale : v ^ ((cs.length x : ℤ)-cs.length b) *
-      (v^2)^(rankDifference cs x b) =
-        (v⁻¹) ^ ((cs.length x : ℤ)-cs.length b) := by
-    rw [←pow_mul,←zpow_natCast,←zpow_add₀ hv,inv_zpow,←zpow_neg]
-    congr 1
-    push_cast
-    omega
-  rw [←mul_assoc,hscale,Finset.mul_sum] at heq
-  change (v⁻¹)^((cs.length x : ℤ)-cs.length b)*
-      eval₂ (Int.castRingHom C) ((v⁻¹)^2) (klPolynomial cs x b) = _
-  rw [heq]
-  apply Finset.sum_congr rfl
-  intro z _
-  rw [rPolynomial_normalization cs v hv]
-  unfold scaledKL
-  calc
-    _ = (v ^ ((cs.length x : ℤ)-cs.length b) *
-        v ^ ((cs.length z.val : ℤ)-cs.length x)) *
-        eval₂ (Int.castRingHom C) (v-v⁻¹) (Hecke.rTilde cs x z.val) *
-        eval₂ (Int.castRingHom C) (v^2) (klPolynomial cs z.val b) := by ring
-    _ = _ := by
-      rw [←zpow_add₀ hv,show (cs.length x : ℤ)-cs.length b+
-        ((cs.length z.val : ℤ)-cs.length x) =
-        (cs.length z.val : ℤ)-cs.length b by ring]
-      ring
-
-
-theorem scaledKL_reciprocity {C : Type*} [Field C]
-    (cs : CoxeterSystem M W) (v : C) (hv : v ≠ 0)
-    (x b : W) (hxb : BruhatLE cs x b) :
-    scaledKL cs v x b = ∑ᶠ z : Interval cs x b,
-      eval₂ (Int.castRingHom C) (v⁻¹-v) (Hecke.rTilde cs x z.val) *
-        scaledKL cs v⁻¹ z.val b := by
-  simpa only [inv_inv] using scaledKL_reciprocity_inverse cs v⁻¹ (inv_ne_zero hv) x b hxb
-
-end KLInvariance
-
-
-end
-
-end OAI
+import OAI.RepresentationTheory.KazhdanLusztig.CanonicalKL
+
+namespace OAI
+
+/-!
+Interval restrictions, dihedral Coxeter and Hecke calculations, and the genuine square-parameter R normalization.
+-/
+
+section
+
+namespace KLInvariance
+
+universe u₁ v₁ u₂ v₂
+variable {B : Type u₁} {W : Type v₁} [Group W] {M : CoxeterMatrix B}
+variable {B' : Type u₂} {W' : Type v₂} [Group W'] {M' : CoxeterMatrix B'}
+
+/-- Restriction of the actual interval isomorphism to any subinterval. This
+is the same unlabeled map, not a new isomorphism with extra structure. -/
+def intervalIso_restrict (cs : CoxeterSystem M W) (cs' : CoxeterSystem M' W')
+    {u b : W} {u' b' : W'} (ι : Interval cs u b ≃o Interval cs' u' b')
+    (x y : Interval cs u b) :
+    Interval cs x.val y.val ≃o Interval cs' (ι x).val (ι y).val where
+  toFun z :=
+    let zI : Interval cs u b := ⟨z.val, bruhat_trans cs x.property.1 z.property.1,
+      bruhat_trans cs z.property.2 y.property.2⟩
+    ⟨(ι zI).val, ι.monotone (show x ≤ zI from z.property.1),
+      ι.monotone (show zI ≤ y from z.property.2)⟩
+  invFun z :=
+    let zI : Interval cs' u' b' := ⟨z.val,
+      bruhat_trans cs' (ι x).property.1 z.property.1,
+      bruhat_trans cs' z.property.2 (ι y).property.2⟩
+    ⟨(ι.symm zI).val,
+      by
+        change x ≤ ι.symm zI
+        simpa only [OrderIso.symm_apply_apply] using
+          ι.symm.monotone (show ι x ≤ zI from z.property.1),
+      by
+        change ι.symm zI ≤ y
+        simpa only [OrderIso.symm_apply_apply] using
+          ι.symm.monotone (show zI ≤ ι y from z.property.2)⟩
+  left_inv z := by
+    apply Subtype.ext
+    let zI : Interval cs u b := ⟨z.val, bruhat_trans cs x.property.1 z.property.1,
+      bruhat_trans cs z.property.2 y.property.2⟩
+    change (ι.symm (ι zI)).val = zI.val
+    rw [ι.symm_apply_apply]
+  right_inv z := by
+    apply Subtype.ext
+    let zI : Interval cs' u' b' := ⟨z.val,
+      bruhat_trans cs' (ι x).property.1 z.property.1,
+      bruhat_trans cs' z.property.2 (ι y).property.2⟩
+    change (ι (ι.symm zI)).val = zI.val
+    rw [ι.apply_symm_apply]
+  map_rel_iff' := by
+    intro z w
+    exact ι.map_rel_iff
+
+@[simp] theorem intervalIso_restrict_val (cs : CoxeterSystem M W)
+    (cs' : CoxeterSystem M' W') {u b : W} {u' b' : W'}
+    (ι : Interval cs u b ≃o Interval cs' u' b') (x y : Interval cs u b)
+    (z : Interval cs x.val y.val) :
+    (intervalIso_restrict cs cs' ι x y z).val =
+      (ι ⟨z.val, bruhat_trans cs x.property.1 z.property.1,
+        bruhat_trans cs z.property.2 y.property.2⟩).val := rfl
+
+end KLInvariance
+
+namespace KLInvariance.Dihedral
+
+variable {W : Type*} [Group W] {M : CoxeterMatrix Bool}
+
+/-- The alternating expression starting with `i`, unlike Mathlib's
+`alternatingWord`, which specifies the final letter. -/
+def alt (i : Bool) : ℕ → List Bool
+  | 0 => []
+  | n + 1 => i :: alt (!i) n
+
+@[simp] theorem alt_zero (i : Bool) : alt i 0 = [] := rfl
+@[simp] theorem alt_succ (i : Bool) (n : ℕ) : alt i (n+1) = i :: alt (!i) n := rfl
+@[simp] theorem length_alt (i : Bool) (n : ℕ) : (alt i n).length = n := by
+  induction n generalizing i with
+  | zero => rfl
+  | succ n ih => simp [ih]
+
+@[simp] theorem take_alt (i : Bool) (k n : ℕ) : (alt i n).take k = alt i (min k n) := by
+  induction n generalizing i k with
+  | zero => simp
+  | succ n ih =>
+    cases k with
+    | zero => simp
+    | succ k => simp [ih]
+
+theorem reduced_cons_cons_ne (cs : CoxeterSystem M W) {i j : Bool} {ω : List Bool}
+    (h : cs.IsReduced (i :: j :: ω)) : i ≠ j := by
+  rintro rfl
+  have hlen := cs.length_wordProd_le ω
+  have heq := h.eq
+  simp only [cs.wordProd_cons, cs.simple_mul_simple_cancel_left, List.length_cons] at heq
+  omega
+
+theorem reduced_cons_eq_alt (cs : CoxeterSystem M W) (i : Bool) (ω : List Bool)
+    (h : cs.IsReduced (i :: ω)) : i :: ω = alt i (ω.length + 1) := by
+  induction ω generalizing i with
+  | nil => simp
+  | cons j ω ih =>
+    have hj : j = !i := by
+      have := reduced_cons_cons_ne cs h
+      cases i <;> cases j <;> simp_all
+    have hω : cs.IsReduced (j :: ω) := by simpa using h.drop 1
+    rw [show (j :: ω).length + 1 = (ω.length + 1) + 1 by rfl, alt_succ]
+    congr 1
+    rw [ih j hω, hj]
+
+theorem reduced_eq_alt (cs : CoxeterSystem M W) (ω : List Bool)
+    (h : cs.IsReduced ω) : ∃ i, ω = alt i ω.length := by
+  cases ω with
+  | nil => exact ⟨false, rfl⟩
+  | cons i ω => exact ⟨i, reduced_cons_eq_alt cs i ω h⟩
+
+theorem exists_alt (cs : CoxeterSystem M W) (w : W) :
+    ∃ i, w = cs.wordProd (alt i (cs.length w)) := by
+  obtain ⟨ω, hω, hw⟩ := cs.exists_isReduced w
+  obtain ⟨i, hi⟩ := reduced_eq_alt cs ω hω
+  refine ⟨i, ?_⟩
+  rw [hw, hω.eq, ← hi]
+
+theorem reduced_alt_take (cs : CoxeterSystem M W) {i : Bool} {k n : ℕ}
+    (h : cs.IsReduced (alt i n)) (hk : k ≤ n) : cs.IsReduced (alt i k) := by
+  simpa only [take_alt, Nat.min_eq_left hk] using h.take k
+
+theorem alt_sublist (i : Bool) {k n : ℕ} (h : k ≤ n) : (alt i k).Sublist (alt i n) := by
+  simpa only [take_alt, Nat.min_eq_left h] using List.take_sublist k (alt i n)
+
+end KLInvariance.Dihedral
+
+namespace KLInvariance.Dihedral
+
+noncomputable local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+variable {W : Type*} [Group W] {M : CoxeterMatrix Bool}
+
+/-- In a rank-two Coxeter system, every strictly shorter element lies below
+every longer element. This uses the actual Bruhat subword theorem. -/
+theorem shorter_bruhat (cs : CoxeterSystem M W) {x w : W}
+    (h : cs.length x < cs.length w) : BruhatLE cs x w := by
+  obtain ⟨i, hi⟩ := exists_alt cs w
+  obtain ⟨j, hj⟩ := exists_alt cs x
+  have hr : cs.IsReduced (alt i (cs.length w)) := by
+    change cs.length (cs.wordProd (alt i (cs.length w))) = (alt i (cs.length w)).length
+    rw [← hi, length_alt]
+  obtain ⟨n, hn⟩ := Nat.exists_eq_succ_of_ne_zero (show cs.length w ≠ 0 by omega)
+  rw [hn] at hi hr h
+  rw [hi, hj]
+  apply bruhat_of_subword cs _ hr
+  by_cases hji : j = i
+  · subst j
+    exact alt_sublist i (by omega)
+  · have hj' : j = !i := by cases i <;> cases j <;> first | rfl | exact (hji rfl).elim
+    rw [hj', alt_succ]
+    exact (alt_sublist (!i) (by omega)).cons i
+
+theorem bruhat_iff_length_lt_or_eq (cs : CoxeterSystem M W) (x w : W) :
+    BruhatLE cs x w ↔ cs.length x < cs.length w ∨ x = w := by
+  constructor
+  · intro h
+    by_cases he : x = w
+    · exact Or.inr he
+    · exact Or.inl (length_lt_of_bruhat_ne cs h he)
+  · rintro (h | rfl)
+    · exact shorter_bruhat cs h
+    · exact bruhat_refl cs _
+
+theorem eq_alt_of_descent (cs : CoxeterSystem M W) (i : Bool) {w : W}
+    (h : cs.length (cs.simple i * w) < cs.length w) :
+    w = cs.wordProd (alt i (cs.length w)) := by
+  obtain ⟨ω, hω, hw⟩ := cs.exists_isReduced (cs.simple i * w)
+  have hlen : cs.length w = ω.length + 1 := by
+    have := cs.length_simple_mul w i
+    rw [hw, hω.eq] at this h
+    omega
+  have hp : cs.wordProd (i :: ω) = w := by
+    simp only [cs.wordProd_cons, ← hw, cs.simple_mul_simple_cancel_left]
+  have hr : cs.IsReduced (i :: ω) := by
+    simpa only [CoxeterSystem.IsReduced, hp, List.length_cons] using hlen
+  rw [hlen, ← reduced_cons_eq_alt cs i ω hr, hp]
+
+end KLInvariance.Dihedral
+
+namespace KLInvariance.Dihedral
+
+noncomputable local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+open Hecke (Space basis)
+variable {W : Type*} [Group W] {M : CoxeterMatrix Bool}
+variable {C : Type*} [CommRing C]
+
+noncomputable def lowerSum (cs : CoxeterSystem M W) (w : W) : Space W C :=
+  ∑ z ∈ lowerFinset cs w, basis z
+
+@[simp] theorem lowerSum_coeff (cs : CoxeterSystem M W) (w x : W) :
+    (lowerSum cs w : Space W C) x = if BruhatLE cs x w then 1 else 0 := by
+  classical
+  simp [lowerSum, basis, Finsupp.single_apply, eq_comm]
+
+@[simp] theorem lowerSum_one (cs : CoxeterSystem M W) :
+    (lowerSum cs 1 : Space W C) = basis 1 := by
+  classical
+  ext x
+  simp only [lowerSum_coeff, basis, Finsupp.single_apply]
+  have h : BruhatLE cs x 1 ↔ 1 = x := by
+    rw [bruhat_iff_length_lt_or_eq]
+    simp [eq_comm]
+  rw [h]
+
+/-- The two members of an s-string enter a lower ideal together precisely
+on the common lower interval appearing in the source's rank-two product. -/
+theorem double_string (cs : CoxeterSystem M W) (i : Bool) (n : ℕ)
+    (hr : cs.IsReduced (alt i (n+2))) (x : W) :
+    (BruhatLE cs x (cs.wordProd (alt (!i) (n+1))) ∧
+      BruhatLE cs (cs.simple i * x) (cs.wordProd (alt (!i) (n+1)))) ↔
+    (0 < n ∧ BruhatLE cs x (cs.wordProd (alt i n))) := by
+  have hr' : cs.IsReduced (alt (!i) (n+1)) := by simpa using hr.drop 1
+  have hrp := reduced_alt_take cs hr (show n ≤ n+2 by omega)
+  have hlenw : cs.length (cs.wordProd (alt i (n+2))) = n+2 := by
+    simpa only [length_alt] using hr.eq
+  have hlenz : cs.length (cs.wordProd (alt (!i) (n+1))) = n+1 := by
+    simpa only [length_alt] using hr'.eq
+  have hlenp : cs.length (cs.wordProd (alt i n)) = n := by
+    simpa only [length_alt] using hrp.eq
+  have hsz : cs.simple i * cs.wordProd (alt (!i) (n+1)) =
+      cs.wordProd (alt i (n+2)) := by rw [alt_succ i (n+1), cs.wordProd_cons]
+  constructor
+  · rintro ⟨hx, hsx⟩
+    have hxlen := length_le_of_bruhat cs hx
+    have hsxlen := length_le_of_bruhat cs hsx
+    have hxl : cs.length x < n+1 := by
+      rcases (bruhat_iff_length_lt_or_eq cs _ _).mp hx with h | rfl
+      · omega
+      · rw [hsz, hlenw] at hsxlen
+        omega
+    have hsxl : cs.length (cs.simple i * x) < n+1 := by
+      rcases (bruhat_iff_length_lt_or_eq cs _ _).mp hsx with h | h
+      · omega
+      · have heq : x = cs.wordProd (alt i (n+2)) := by
+          rw [← hsz, ← h, cs.simple_mul_simple_cancel_left]
+        rw [heq, hlenw] at hxl
+        omega
+    have hchange := cs.length_simple_mul x i
+    have hn : 0 < n := by omega
+    refine ⟨hn, ?_⟩
+    by_cases hxlt : cs.length x < n
+    · exact shorter_bruhat cs (by omega)
+    · have hxi : cs.length x = n := by omega
+      have hdesc : cs.length (cs.simple i * x) < cs.length x := by omega
+      have heq := eq_alt_of_descent cs i hdesc
+      rw [hxi] at heq
+      rw [heq]
+      exact bruhat_refl cs _
+  · rintro ⟨hn, hx⟩
+    have hxlen := length_le_of_bruhat cs hx
+    constructor
+    · exact shorter_bruhat cs (by omega)
+    · by_cases hxlt : cs.length x < n
+      · have := cs.length_simple_mul x i
+        exact shorter_bruhat cs (by omega)
+      · have heq : x = cs.wordProd (alt i n) := by
+          rcases (bruhat_iff_length_lt_or_eq cs _ _).mp hx with h | h
+          · omega
+          · exact h
+        obtain ⟨k, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (show n ≠ 0 by omega)
+        have hrpt : cs.IsReduced (alt (!i) k) := by simpa using hrp.drop 1
+        have hpk : cs.length (cs.simple i * x) = k := by
+          rw [heq, alt_succ, cs.wordProd_cons, cs.simple_mul_simple_cancel_left,
+            hrpt.eq, length_alt]
+        exact shorter_bruhat cs (by omega)
+
+end KLInvariance.Dihedral
+
+namespace KLInvariance.Dihedral
+
+noncomputable local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+open Hecke (Space basis)
+variable {W : Type*} [Group W] {M : CoxeterMatrix Bool}
+variable {C : Type*} [CommRing C]
+
+theorem left_plus_coeff (cs : CoxeterSystem M W) (q : Cˣ) (i : Bool)
+    (v : Space W C) (x : W) :
+    (v + HeckeQ.left cs (q : C) i v) x =
+      if cs.length (cs.simple i * x) < cs.length x then
+        v (cs.simple i * x) + (q : C) * v x
+      else v x + (q : C) * v (cs.simple i * x) := by
+  have h := congrArg (fun t : Space W C => (q : C) * t x)
+    (HeckeQ.leftInv_apply cs q i v)
+  simp only [Finsupp.smul_apply, smul_eq_mul, Finsupp.sub_apply, ← mul_assoc,
+    q.mul_inv, one_mul] at h
+  rw [HeckeQ.leftInv_coeff] at h
+  simp only [Finsupp.add_apply]
+  split_ifs at h ⊢
+  · linear_combination -h + v (cs.simple i * x) * q.mul_inv
+  · linear_combination -h + v x * q.mul_inv
+
+theorem union_string (cs : CoxeterSystem M W) (i : Bool) {z : W}
+    (hz : cs.length z < cs.length (cs.simple i * z)) (x : W) :
+    BruhatLE cs x (cs.simple i * z) ↔
+      BruhatLE cs x z ∨ BruhatLE cs (cs.simple i * x) z := by
+  have hsz : cs.length (cs.simple i * (cs.simple i * z)) <
+      cs.length (cs.simple i * z) := by simpa using hz
+  constructor
+  · intro h
+    have hh := lowerSimple_mono cs i h
+    simp only [lowerSimple, cs.simple_mul_simple_cancel_left] at hh
+    split_ifs at hh with hx
+    · exact Or.inr hh
+    · exact Or.inl hh
+  · rintro (h | h)
+    · apply bruhat_trans cs h
+      exact Relation.ReflTransGen.single ⟨hz, cs.simple i, cs.isReflection_simple i, rfl⟩
+    · simpa only [cs.simple_mul_simple_cancel_left] using
+        Hecke.simple_mul_le_ascent cs i h hz
+
+noncomputable def correction (cs : CoxeterSystem M W) (i : Bool) (n : ℕ) : Space W C :=
+  if n = 0 then 0 else lowerSum cs (cs.wordProd (alt i n))
+
+@[simp] theorem correction_coeff (cs : CoxeterSystem M W) (i : Bool) (n : ℕ) (x : W) :
+    (correction cs i n : Space W C) x =
+      if 0 < n ∧ BruhatLE cs x (cs.wordProd (alt i n)) then 1 else 0 := by
+  classical
+  by_cases h : n = 0
+  · simp [correction, h]
+  · have hn : 0 < n := Nat.pos_of_ne_zero h
+    simp [correction, h, hn]
+
+/-- The Hecke sum identity `loc:dihedral-product`, before normalization by
+q to a half-integer power. Every interval here is genuine strong Bruhat. -/
+theorem lowerSum_product (cs : CoxeterSystem M W) (q : Cˣ) (i : Bool) (n : ℕ)
+    (hr : cs.IsReduced (alt i (n+2))) :
+    lowerSum cs (cs.wordProd (alt (!i) (n+1))) +
+      HeckeQ.left cs (q : C) i (lowerSum cs (cs.wordProd (alt (!i) (n+1)))) =
+    lowerSum cs (cs.wordProd (alt i (n+2))) + (q : C) • correction cs i n := by
+  classical
+  have hr' : cs.IsReduced (alt (!i) (n+1)) := by simpa using hr.drop 1
+  have hz : cs.length (cs.wordProd (alt (!i) (n+1))) <
+      cs.length (cs.simple i * cs.wordProd (alt (!i) (n+1))) := by
+    rw [← cs.wordProd_cons, ← alt_succ, hr.eq, hr'.eq, length_alt, length_alt]
+    omega
+  ext x
+  rw [left_plus_coeff, Finsupp.add_apply, Finsupp.smul_apply, smul_eq_mul,
+    correction_coeff, lowerSum_coeff, lowerSum_coeff, lowerSum_coeff]
+  have hu := union_string cs i hz x
+  rw [← cs.wordProd_cons, ← alt_succ] at hu
+  have hd := double_string cs i n hr x
+  simp only [hu, ← hd]
+  by_cases hdesc : cs.length (cs.simple i * x) < cs.length x
+  · have hclosure : BruhatLE cs x (cs.wordProd (alt (!i) (n+1))) →
+        BruhatLE cs (cs.simple i * x) (cs.wordProd (alt (!i) (n+1))) := by
+      intro h
+      apply bruhat_trans cs _ h
+      exact Relation.ReflTransGen.single
+        ⟨hdesc, cs.simple i, cs.isReflection_simple i, by simp⟩
+    by_cases hx : BruhatLE cs x (cs.wordProd (alt (!i) (n+1)))
+    · simp [-alt_succ, hdesc, hx, hclosure hx]
+    · by_cases hs : BruhatLE cs (cs.simple i * x) (cs.wordProd (alt (!i) (n+1))) <;>
+        simp [-alt_succ, hdesc, hx, hs]
+  · have hasc : cs.length x < cs.length (cs.simple i * x) := by
+      have := cs.length_simple_mul x i
+      omega
+    have hclosure : BruhatLE cs (cs.simple i * x) (cs.wordProd (alt (!i) (n+1))) →
+        BruhatLE cs x (cs.wordProd (alt (!i) (n+1))) := by
+      intro h
+      apply bruhat_trans cs _ h
+      exact Relation.ReflTransGen.single
+        ⟨hasc, cs.simple i, cs.isReflection_simple i, rfl⟩
+    by_cases hs : BruhatLE cs (cs.simple i * x) (cs.wordProd (alt (!i) (n+1)))
+    · simp [-alt_succ, hdesc, hs, hclosure hs]
+    · by_cases hx : BruhatLE cs x (cs.wordProd (alt (!i) (n+1))) <;>
+        simp [-alt_succ, hdesc, hx, hs]
+
+end KLInvariance.Dihedral
+
+namespace KLInvariance.HeckeQ
+
+universe uR vR
+variable {B : Type uR} {W : Type vR} [Group W] {M : CoxeterMatrix B}
+variable {C : Type*} [CommRing C]
+
+noncomputable def dualR (cs : CoxeterSystem M W) (q : Cˣ) (x y : W) : C :=
+  (-(q : C)) ^ cs.length y * (-(↑q⁻¹ : C)) ^ cs.length x * rValue cs q⁻¹ x y
+
+@[simp] theorem dualR_diagonal (cs : CoxeterSystem M W) (q : Cˣ) (x : W) :
+    dualR cs q x x = 1 := by
+  rw [dualR, rValue_diagonal, mul_one, ← mul_pow]
+  simp
+
+theorem dualR_zero (cs : CoxeterSystem M W) (q : Cˣ) (x y : W)
+    (h : ¬BruhatLE cs x y) : dualR cs q x y = 0 := by
+  simp [dualR, rValue_zero cs q⁻¹ x y h]
+
+theorem dualR_recursion (cs : CoxeterSystem M W) (q : Cˣ) (x y : W) (i : B)
+    (hy : cs.length (cs.simple i * y) < cs.length y) :
+    dualR cs q x y =
+      if cs.length (cs.simple i * x) < cs.length x then
+        dualR cs q (cs.simple i * x) (cs.simple i * y)
+      else ((q : C) - 1) * dualR cs q x (cs.simple i * y) +
+        (q : C) * dualR cs q (cs.simple i * x) (cs.simple i * y) := by
+  have hyl : cs.length y = cs.length (cs.simple i * y) + 1 := by
+    have := cs.length_simple_mul y i
+    omega
+  unfold dualR
+  rw [rValue_recursion cs q⁻¹ x y i hy]
+  split_ifs with hx
+  · have hxl : cs.length x = cs.length (cs.simple i * x) + 1 := by
+      have := cs.length_simple_mul x i
+      omega
+    rw [hyl, hxl, pow_succ, pow_succ]
+    calc
+      _ = (-(q:C)) ^ cs.length (cs.simple i * y) *
+          (-(↑q⁻¹:C)) ^ cs.length (cs.simple i * x) *
+          rValue cs q⁻¹ (cs.simple i * x) (cs.simple i * y) *
+          ((q:C) * (↑q⁻¹:C)) := by ring
+      _ = _ := by rw [q.mul_inv, mul_one]
+  · have hxl : cs.length (cs.simple i * x) = cs.length x + 1 := by
+      have := cs.length_simple_mul x i
+      omega
+    rw [hyl, hxl, pow_succ, pow_succ]
+    have hq : (-(q:C)) * ((↑q⁻¹:C) - 1) = (q:C) - 1 := by
+      rw [mul_sub, neg_mul, q.mul_inv, mul_one]
+      ring
+    calc
+      _ = (-(q:C)) ^ cs.length (cs.simple i * y) *
+          (-(↑q⁻¹:C)) ^ cs.length x *
+          ((-(q:C)) * ((↑q⁻¹:C) - 1)) * rValue cs q⁻¹ x (cs.simple i * y) +
+          (q:C) * ((-(q:C)) ^ cs.length (cs.simple i * y) *
+          ((-(↑q⁻¹:C)) ^ cs.length x * (-(↑q⁻¹:C))) *
+          rValue cs q⁻¹ (cs.simple i * x) (cs.simple i * y)) := by ring
+      _ = _ := by rw [hq]; ring
+
+/-- Signed inversion symmetry of actual equal-parameter R-values, proved
+from all-descent recurrence, not imposed as an additional normalization. -/
+theorem rValue_eq_dualR (cs : CoxeterSystem M W) (q : Cˣ) (x y : W) :
+    rValue cs q x y = dualR cs q x y := by
+  suffices ∀ n : ℕ, ∀ y : W, cs.length y = n → ∀ x,
+      rValue cs q x y = dualR cs q x y by
+    exact this _ y rfl x
+  intro n
+  induction n using Nat.strong_induction_on with
+  | h n ih =>
+    intro y hy x
+    by_cases hy1 : y = 1
+    · subst y
+      by_cases hx1 : x = 1
+      · subst x
+        simp
+      · have hnot : ¬BruhatLE cs x 1 := by
+          intro h
+          have hl := length_le_of_bruhat cs h
+          have hl0 : cs.length x = 0 := by simpa using hl
+          exact hx1 (cs.length_eq_zero_iff.mp hl0)
+        rw [rValue_zero cs q x 1 hnot, dualR_zero cs q x 1 hnot]
+    · obtain ⟨i, hi⟩ := cs.exists_leftDescent_of_ne_one hy1
+      have hsmall : cs.length (cs.simple i * y) < n := hy ▸ hi
+      rw [rValue_recursion cs q x y i hi, dualR_recursion cs q x y i hi,
+        ih _ hsmall _ rfl x, ih _ hsmall _ rfl (cs.simple i * x)]
+
+/-- R at the inverse parameter is exactly the corresponding bar coefficient
+with the bottom q-power. This is the coefficient form used below. -/
+theorem rValue_inverse_eq_barBasis (cs : CoxeterSystem M W) (q : Cˣ) (x y : W) :
+    rValue cs q⁻¹ x y = (q:C) ^ cs.length x * barBasis cs q y x := by
+  rw [rValue_eq_dualR cs q⁻¹, dualR, inv_inv, rValue]
+  simp only [neg_pow (↑q⁻¹:C), neg_pow (q:C)]
+  have hsign (n : ℕ) : (-1:C)^n * (-1:C)^n = 1 := by
+    rw [← mul_pow]; simp
+  calc
+    _ = ((-1:C)^cs.length y * (-1:C)^cs.length y) *
+        ((-1:C)^cs.length x * (-1:C)^cs.length x) *
+        (((↑q⁻¹:C) * (q:C)) ^ cs.length y) *
+        (q:C)^cs.length x * barBasis cs q y x := by ring
+    _ = _ := by rw [hsign, hsign, q.inv_mul]; simp
+
+end KLInvariance.HeckeQ
+
+namespace KLInvariance.Dihedral
+
+noncomputable local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+open Hecke (Space basis)
+variable {W : Type*} [Group W] {M : CoxeterMatrix Bool}
+variable {C : Type*} [CommRing C]
+
+@[simp] theorem lowerSum_simple (cs : CoxeterSystem M W) (i : Bool) :
+    (lowerSum cs (cs.simple i) : Space W C) = basis 1 + basis (cs.simple i) := by
+  classical
+  ext x
+  rw [lowerSum_coeff, bruhat_iff_length_lt_or_eq, cs.length_simple]
+  simp only [Nat.lt_one_iff, cs.length_eq_zero_iff, Finsupp.add_apply, basis,
+    Finsupp.single_apply]
+  have hi : cs.simple i ≠ 1 := by
+    intro h
+    have := cs.length_simple i
+    simp [h] at this
+  by_cases hx : x = 1
+  · subst x
+    simp [hi]
+  · by_cases hs : x = cs.simple i
+    · subst x
+      simp [hi, Ne.symm hi]
+    · simp [hx, hs, Ne.symm hx, Ne.symm hs]
+
+theorem plus_leftInv (cs : CoxeterSystem M W) (q : Cˣ) (i : Bool)
+    (v : Space W C) :
+    v + HeckeQ.leftInv cs q i v =
+      (↑q⁻¹:C) • (v + HeckeQ.left cs (q:C) i v) := by
+  rw [HeckeQ.leftInv_apply, smul_sub, smul_smul, mul_sub, q.inv_mul, mul_one]
+  module
+
+/-- Unnormalized form of bar invariance of the dihedral constant sums. -/
+theorem bar_lowerSum_alt (cs : CoxeterSystem M W) (q : Cˣ) (i : Bool) (n : ℕ)
+    (hr : cs.IsReduced (alt i n)) :
+    HeckeQ.barLinear cs q (lowerSum cs (cs.wordProd (alt i n))) =
+      (↑q⁻¹:C)^n • lowerSum cs (cs.wordProd (alt i n)) := by
+  induction n using Nat.strong_induction_on generalizing i with
+  | h n ih =>
+    rcases n with _ | (_ | n)
+    · simp
+    · have hasc : cs.length (1:W) < cs.length (cs.simple i * 1) := by simp
+      have hb := HeckeQ.barBasis_simple_mul_ascent cs q i 1 hasc
+      simp only [mul_one, HeckeQ.barBasis_one] at hb
+      simp only [alt_succ, alt_zero, cs.wordProd_singleton, lowerSum_simple,
+        map_add, HeckeQ.barLinear_basis, HeckeQ.barBasis_one, hb]
+      rw [plus_leftInv, HeckeQ.left_basis]
+      simp
+    · have hr' : cs.IsReduced (alt (!i) (n+1)) := by simpa using hr.drop 1
+      have hrp := reduced_alt_take cs hr (show n ≤ n+2 by omega)
+      have hz := ih (n+1) (by omega) (!i) hr'
+      have hc : HeckeQ.barLinear cs q (correction cs i n) =
+          (↑q⁻¹:C)^n • correction cs i n := by
+        by_cases hn : n = 0
+        · simp [correction, hn]
+        · simp only [correction, ite_eq_right hn]
+          exact ih n (by omega) i hrp
+      have hprod := lowerSum_product cs q i n hr
+      have hbar := congrArg (HeckeQ.barLinear cs q)
+        (lowerSum_product cs q⁻¹ i n hr)
+      have hinter := congrArg (fun f : Module.End C (Space W C) =>
+          f (lowerSum cs (cs.wordProd (alt (!i) (n+1)))))
+        (HeckeQ.barLinear_left cs q i)
+      simp only [map_add, map_smul, Module.End.mul_apply] at hbar hinter
+      rw [hinter, hz, map_smul, ← smul_add, plus_leftInv, hprod] at hbar
+      simp only [smul_add, smul_smul, hc] at hbar
+      have hq1 : (↑q⁻¹:C)^(n+1) * (↑q⁻¹:C) = (↑q⁻¹:C)^(n+2) := by
+        exact (pow_succ (↑q⁻¹:C) (n+1)).symm
+      have hq2 : (↑q⁻¹:C)^(n+1) * ((↑q⁻¹:C) * (q:C)) =
+          (↑q⁻¹:C) * (↑q⁻¹:C)^n := by rw [q.inv_mul, mul_one, pow_succ'];
+      rw [hq1, hq2] at hbar
+      exact add_right_cancel hbar.symm
+
+theorem bar_lowerSum (cs : CoxeterSystem M W) (q : Cˣ) (w : W) :
+    HeckeQ.barLinear cs q (lowerSum cs w) =
+      (↑q⁻¹:C)^cs.length w • lowerSum cs w := by
+  obtain ⟨i, hi⟩ := exists_alt cs w
+  have hr : cs.IsReduced (alt i (cs.length w)) := by
+    simp only [CoxeterSystem.IsReduced, ← hi, length_alt]
+  simpa only [← hi] using bar_lowerSum_alt cs q i (cs.length w) hr
+
+end KLInvariance.Dihedral
+
+namespace KLInvariance.Dihedral
+
+noncomputable local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+open Hecke (Space basis)
+open Polynomial
+variable {W : Type*} [Group W] {M : CoxeterMatrix Bool}
+variable {C : Type*} [CommRing C]
+
+theorem rValue_sum (cs : CoxeterSystem M W) (q : Cˣ) (x y : W)
+    (hxy : BruhatLE cs x y) :
+    ∑ z ∈ lowerFinset cs y, HeckeQ.rValue cs q x z =
+      (q:C) ^ rankDifference cs x y := by
+  classical
+  have hb := congrArg (fun f : Space W C => f x) (bar_lowerSum cs q⁻¹ y)
+  simp only [Finsupp.smul_apply, smul_eq_mul, inv_inv, lowerSum_coeff,
+    ite_eq_left hxy, mul_one] at hb
+  rw [lowerSum, map_sum] at hb
+  simp only [Finsupp.coe_finsetSum, Finset.sum_apply, HeckeQ.barLinear_basis] at hb
+  have hlen : cs.length y = cs.length x + rankDifference cs x y := by
+    have := length_le_of_bruhat cs hxy
+    unfold rankDifference
+    omega
+  calc
+    _ = (↑q⁻¹:C)^cs.length x *
+        ∑ z ∈ lowerFinset cs y, HeckeQ.barBasis cs q⁻¹ z x := by
+      rw [Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro z hz
+      simpa using HeckeQ.rValue_inverse_eq_barBasis cs q⁻¹ x z
+    _ = (↑q⁻¹:C)^cs.length x * (q:C)^cs.length y := by rw [hb]
+    _ = _ := by
+      rw [hlen, pow_add]
+      calc
+        _ = ((↑q⁻¹:C) * (q:C))^cs.length x * (q:C)^rankDifference cs x y := by
+          rw [mul_pow]; ring
+        _ = _ := by rw [q.inv_mul]; simp
+
+theorem rPolynomial_sum (cs : CoxeterSystem M W) (x y : W)
+    (hxy : BruhatLE cs x y) :
+    ∑ z ∈ lowerFinset cs y, OrdinaryR.polynomial cs x z =
+      (X:ℤ[X]) ^ rankDifference cs x y := by
+  apply Polynomial.toLaurent_injective
+  simpa using rValue_sum cs OrdinaryR.variableUnit x y hxy
+
+noncomputable def constantP (cs : CoxeterSystem M W) (x y : W) : ℤ[X] :=
+  if BruhatLE cs x y then 1 else 0
+
+theorem constant_normalizedKL (cs : CoxeterSystem M W) :
+    NormalizedKL cs (OrdinaryR.polynomial cs, constantP cs) where
+  R_diagonal := OrdinaryR.polynomial_diagonal cs
+  R_zero := OrdinaryR.polynomial_zero cs
+  R_recursion := OrdinaryR.polynomial_recursion cs
+  P_diagonal := by intro x; simp [constantP, bruhat_refl cs x]
+  P_zero := by intro x y h; simp [constantP, h]
+  P_degree := by
+    intro x y hxy hne
+    have := length_lt_of_bruhat_ne cs hxy hne
+    simp only [constantP, ite_eq_left hxy, natDegree_one, mul_zero]
+    unfold rankDifference
+    omega
+  reciprocity := by
+    intro x y hxy
+    simp only [constantP, ite_eq_left hxy, reflect_one]
+    classical
+    let := interval_finite cs x y
+    let : Fintype (Interval cs x y) := Fintype.ofFinite _
+    rw [finsum_eq_sum_of_fintype]
+    have heq : (∑ z : Interval cs x y,
+        OrdinaryR.polynomial cs x z.val * (if BruhatLE cs z.val y then 1 else 0)) =
+        ∑ z : Interval cs x y, OrdinaryR.polynomial cs x z.val := by
+      apply Finset.sum_congr rfl
+      intro z hz
+      rw [ite_eq_left z.property.2, mul_one]
+    rw [heq]
+    have hsum := Finset.sum_subtype (F := (inferInstance : Fintype (Interval cs x y)))
+      ((lowerFinset cs y).filter (fun z => BruhatLE cs x z))
+      (fun z => by simp only [Finset.mem_filter, mem_lowerFinset]; exact and_comm)
+      (fun z => OrdinaryR.polynomial cs x z)
+    symm
+    refine hsum.symm.trans (Eq.trans ?_ (rPolynomial_sum cs x y hxy))
+    apply Finset.sum_subset (Finset.filter_subset _ _)
+    intro z hz hnot
+    have hnxz : ¬BruhatLE cs x z := fun hxz => hnot (Finset.mem_filter.mpr ⟨hz, hxz⟩)
+    exact OrdinaryR.polynomial_zero cs x z hnxz
+
+/-- The manuscript's necessary dihedral calculation (`loc:dihedral-kl`),
+for the actual KL family, with the exact standard normalization. This is
+support for localization, not a substitute for general invariance. -/
+theorem klPolynomial_eq_one (cs : CoxeterSystem M W) (x y : W)
+    (hxy : BruhatLE cs x y) : klPolynomial cs x y = 1 := by
+  have heq := normalizedKL_unique cs (klFamilies_spec cs) (constant_normalizedKL cs)
+  have hp := congrArg (fun RP : PolynomialFamilies W => RP.2 x y) heq
+  simpa only [klPolynomial, constantP, ite_eq_left hxy] using hp
+
+end KLInvariance.Dihedral
+
+end
+
+section
+
+namespace KLInvariance
+open Polynomial
+universe u v
+variable {B : Type u} {W : Type v} [Group W] {M : CoxeterMatrix B}
+
+/-- The exact square-parameter normalization, proved from the two genuine
+Hecke recurrences. Integer exponents let the formula include incomparable
+pairs without silently changing the R-polynomial definition. -/
+theorem rPolynomial_normalization {C : Type*} [Field C]
+    (cs : CoxeterSystem M W) (v : C) (hv : v ≠ 0) (x y : W) :
+    eval₂ (Int.castRingHom C) (v^2) (rPolynomial cs x y) =
+      v ^ ((cs.length y : ℤ) - cs.length x) *
+        eval₂ (Int.castRingHom C) (v-v⁻¹) (Hecke.rTilde cs x y) := by
+  rw [rPolynomial_eq_constructed]
+  suffices ∀ n : ℕ, ∀ y : W, cs.length y = n → ∀ x,
+    eval₂ (Int.castRingHom C) (v^2) (OrdinaryR.polynomial cs x y) =
+      v ^ ((cs.length y : ℤ) - cs.length x) *
+        eval₂ (Int.castRingHom C) (v-v⁻¹) (Hecke.rTilde cs x y) from
+    this _ y rfl x
+  intro n
+  induction n using Nat.strong_induction_on with
+  | h n ih =>
+    intro y hy x
+    by_cases h1 : y=1
+    · subst y
+      by_cases hx : x=1
+      · subst x; simp
+      · have hn : ¬BruhatLE cs x 1 := by
+          intro h
+          have hl := length_le_of_bruhat cs h
+          have hx0 : cs.length x=0 := by simpa using hl
+          exact hx (cs.length_eq_zero_iff.mp hx0)
+        rw [OrdinaryR.polynomial_zero cs _ _ hn,Hecke.rTilde_zero cs _ _ hn]
+        simp
+    · obtain ⟨i,hi⟩ := cs.exists_leftDescent_of_ne_one h1
+      have hlt : cs.length (cs.simple i*y)<n := hy ▸ hi
+      have hylen : cs.length (cs.simple i*y)+1=cs.length y := by
+        have := cs.length_simple_mul y i; omega
+      rw [OrdinaryR.polynomial_recursion cs x y i hi,Hecke.rTilde_recursion cs x y i hi]
+      by_cases hx : cs.length (cs.simple i*x)<cs.length x
+      · rw [ite_eq_left hx,ite_eq_left hx,ih _ hlt _ rfl (cs.simple i*x)]
+        congr 2
+        have := cs.length_simple_mul x i
+        omega
+      · rw [ite_eq_right hx,ite_eq_right hx]
+        simp only [eval₂_add,eval₂_mul,eval₂_sub,eval₂_X,eval₂_one]
+        rw [ih _ hlt _ rfl x,ih _ hlt _ rfl (cs.simple i*x)]
+        have hxlen : cs.length (cs.simple i*x)=cs.length x+1 := by
+          have := cs.length_simple_mul x i; omega
+        have h1 : (cs.length (cs.simple i*y) : ℤ)-cs.length x =
+            ((cs.length y : ℤ)-cs.length x)-1 := by omega
+        have h2 : (cs.length (cs.simple i*y) : ℤ)-cs.length (cs.simple i*x) =
+            ((cs.length y : ℤ)-cs.length x)-2 := by omega
+        rw [h1,h2,zpow_sub₀ hv ((cs.length y : ℤ)-cs.length x) 1,
+          zpow_sub₀ hv ((cs.length y : ℤ)-cs.length x) 2]
+        simp only [zpow_ofNat]
+        field_simp [hv]
+        ; ring
+
+/-- The manuscript's q^{-d/2} P(q), written with t^2=q so no square-root
+branch is hidden. This is an evaluation of the already constructed genuine
+KL polynomial, not a new combinatorial invariant. -/
+noncomputable def scaledKL {C : Type*} [Field C]
+    (cs : CoxeterSystem M W) (t : C) (x b : W) : C :=
+  t ^ ((cs.length x : ℤ)-cs.length b) *
+    eval₂ (Int.castRingHom C) (t^2) (klPolynomial cs x b)
+
+/-- Exact normalized reciprocity in the direction useful for reversing the
+parameter. No path formula and no moment-graph assumption occurs here. -/
+theorem scaledKL_reciprocity_inverse {C : Type*} [Field C]
+    (cs : CoxeterSystem M W) (v : C) (hv : v ≠ 0)
+    (x b : W) (hxb : BruhatLE cs x b) :
+    scaledKL cs v⁻¹ x b = ∑ᶠ z : Interval cs x b,
+      eval₂ (Int.castRingHom C) (v-v⁻¹) (Hecke.rTilde cs x z.val) *
+        scaledKL cs v z.val b := by
+  classical
+  let : Finite (Interval cs x b) := interval_finite cs x b
+  let : Fintype (Interval cs x b) := Fintype.ofFinite _
+  have hrec := (klFamilies_spec cs).reciprocity x b hxb
+  change reflect (rankDifference cs x b) (klPolynomial cs x b) =
+      ∑ᶠ z : Interval cs x b, rPolynomial cs x z.val * klPolynomial cs z.val b at hrec
+  have hdeg : (klPolynomial cs x b).natDegree ≤ rankDifference cs x b := by
+    rw [klPolynomial_eq_constructed]
+    exact CanonicalP.polynomial_degree cs x b
+  let q : Cˣ := Units.mk0 (v^2) (pow_ne_zero 2 hv)
+  have heval := congrArg (eval₂ (Int.castRingHom C) (q : C)) hrec
+  rw [eval₂_reflect_unit q _ _ hdeg] at heval
+  simp only [finsum_eq_sum_of_fintype,eval₂_finsetSum,eval₂_mul] at heval ⊢
+  have hq : (q : C) = v^2 := rfl
+  have hqi : (↑q⁻¹ : C) = (v⁻¹)^2 := by simp [q,inv_pow]
+  rw [hq,hqi] at heval
+  have heq := congrArg (fun a : C => v ^ ((cs.length x : ℤ)-cs.length b)*a) heval
+  have hd : (rankDifference cs x b : ℤ) = (cs.length b : ℤ)-cs.length x := by
+    have hlen := length_le_of_bruhat cs hxb
+    simp [rankDifference,Int.ofNat_sub hlen]
+  have hscale : v ^ ((cs.length x : ℤ)-cs.length b) *
+      (v^2)^(rankDifference cs x b) =
+        (v⁻¹) ^ ((cs.length x : ℤ)-cs.length b) := by
+    rw [←pow_mul,←zpow_natCast,←zpow_add₀ hv,inv_zpow,←zpow_neg]
+    congr 1
+    push_cast
+    omega
+  rw [←mul_assoc,hscale,Finset.mul_sum] at heq
+  change (v⁻¹)^((cs.length x : ℤ)-cs.length b)*
+      eval₂ (Int.castRingHom C) ((v⁻¹)^2) (klPolynomial cs x b) = _
+  rw [heq]
+  apply Finset.sum_congr rfl
+  intro z _
+  rw [rPolynomial_normalization cs v hv]
+  unfold scaledKL
+  calc
+    _ = (v ^ ((cs.length x : ℤ)-cs.length b) *
+        v ^ ((cs.length z.val : ℤ)-cs.length x)) *
+        eval₂ (Int.castRingHom C) (v-v⁻¹) (Hecke.rTilde cs x z.val) *
+        eval₂ (Int.castRingHom C) (v^2) (klPolynomial cs z.val b) := by ring
+    _ = _ := by
+      rw [←zpow_add₀ hv,show (cs.length x : ℤ)-cs.length b+
+        ((cs.length z.val : ℤ)-cs.length x) =
+        (cs.length z.val : ℤ)-cs.length b by ring]
+      ring
+
+
+theorem scaledKL_reciprocity {C : Type*} [Field C]
+    (cs : CoxeterSystem M W) (v : C) (hv : v ≠ 0)
+    (x b : W) (hxb : BruhatLE cs x b) :
+    scaledKL cs v x b = ∑ᶠ z : Interval cs x b,
+      eval₂ (Int.castRingHom C) (v⁻¹-v) (Hecke.rTilde cs x z.val) *
+        scaledKL cs v⁻¹ z.val b := by
+  simpa only [inv_inv] using scaledKL_reciprocity_inverse cs v⁻¹ (inv_ne_zero hv) x b hxb
+
+end KLInvariance
+
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/KazhdanLusztig/DihedralIntervals.lean b/lean/OAI/RepresentationTheory/KazhdanLusztig/DihedralIntervals.lean
index 984958965b1c839031895ea0ab61a20073538658..baf37f5c606cc7deca0c51dc63d70aff81d31748 100644
--- a/lean/OAI/RepresentationTheory/KazhdanLusztig/DihedralIntervals.lean
+++ b/lean/OAI/RepresentationTheory/KazhdanLusztig/DihedralIntervals.lean
@@ -1,2274 +1,2274 @@
-import OAI.RepresentationTheory.KazhdanLusztig.ReflectionOrders
-
-namespace OAI
-
-/-!
-Signed dihedral coordinates, actual dihedral intervals, exact later-edge counts and root-plane calculations.
--/
-
-section
-
-namespace KLInvariance.Dihedral
-variable {W : Type*} [Group W] {M : CoxeterMatrix Bool}
-noncomputable section
-
- theorem wordProd_alt_odd (cs : CoxeterSystem M W) (i : Bool) (n : ℕ) :
-    cs.wordProd (alt i (2*n+1)) = (cs.simple i * cs.simple (!i)) ^ n * cs.simple i := by
-  induction n with
-  | zero => simp
-  | succ n ih =>
-    rw [show 2*(n+1)+1 = (2*n+1)+2 by omega, alt_add_two]
-    simp only [cs.wordProd_cons, ih, pow_succ']
-    group
-
- def rotation (cs : CoxeterSystem M W) : W := cs.simple false * cs.simple true
-
- theorem simple_rotation (cs : CoxeterSystem M W) :
-    cs.simple false * rotation cs = (rotation cs)⁻¹ * cs.simple false := by
-  simp only [rotation, mul_inv_rev, cs.inv_simple]
-  rw [cs.simple_mul_simple_cancel_left, mul_assoc, cs.simple_mul_simple_self, mul_one]
-
- theorem simple_rotation_zpow (cs : CoxeterSystem M W) (n : ℤ) :
-    cs.simple false * rotation cs ^ n = rotation cs ^ (-n) * cs.simple false := by
-  have h : SemiconjBy (cs.simple false) (rotation cs) (rotation cs)⁻¹ :=
-    simple_rotation cs
-  have hh := h.zpow_right n
-  change cs.simple false * rotation cs ^ n = ((rotation cs)⁻¹) ^ n * cs.simple false at hh
-  simpa only [inv_zpow, zpow_neg] using hh
-
-/-- The signed coordinate evaluates both alternating rays in the same group.
-The Euclidean quotient also gives the correct odd negative coordinates. -/
- def indexWord (cs : CoxeterSystem M W) (i : ℤ) : W :=
-  if i % 2 = 0 then rotation cs ^ (i/2)
-  else rotation cs ^ (i/2) * cs.simple false
-
- theorem indexWord_even (cs : CoxeterSystem M W) (n : ℤ) :
-    indexWord cs (2*n) = rotation cs ^ n := by simp [indexWord]
-
- theorem indexWord_odd (cs : CoxeterSystem M W) (n : ℤ) :
-    indexWord cs (2*n+1) = rotation cs ^ n * cs.simple false := by
-  simp only [indexWord, show (2*n+1)%2 = 1 by omega, show (1:ℤ) ≠ 0 by omega, ite_false,
-    show (2*n+1)/2 = n by omega]
-
- theorem indexWord_nat (cs : CoxeterSystem M W) (n : ℕ) :
-    indexWord cs n = cs.wordProd (alt false n) := by
-  rcases Nat.even_or_odd n with ⟨k,rfl⟩ | ⟨k,rfl⟩
-  · rw [show k+k = 2*k by omega, Nat.cast_mul, Nat.cast_ofNat, indexWord_even,
-      wordProd_alt_even]
-    simp only [rotation, Bool.not_false, zpow_natCast]
-  · rw [Nat.cast_add, Nat.cast_mul,
-      Nat.cast_ofNat, Nat.cast_one, indexWord_odd, wordProd_alt_odd]
-    simp only [rotation, Bool.not_false, zpow_natCast]
-
- theorem indexWord_neg_nat (cs : CoxeterSystem M W) (n : ℕ) :
-    indexWord cs (-(n:ℤ)) = cs.wordProd (alt true n) := by
-  have hr : cs.simple true * cs.simple false = (rotation cs)⁻¹ := by
-    simp only [rotation, mul_inv_rev, cs.inv_simple]
-  rcases Nat.even_or_odd n with ⟨k,rfl⟩ | ⟨k,rfl⟩
-  · rw [show k+k = 2*k by omega, Nat.cast_mul, Nat.cast_ofNat,
-      show -(2*(k:ℤ)) = 2*(-k) by ring, indexWord_even, wordProd_alt_even]
-    simp only [Bool.not_true, hr, zpow_neg, zpow_natCast, inv_pow]
-  · rw [Nat.cast_add, Nat.cast_mul,
-      Nat.cast_ofNat, Nat.cast_one,
-      show -(2*(k:ℤ)+1) = 2*(-(k:ℤ)-1)+1 by ring, indexWord_odd,
-      wordProd_alt_odd]
-    rw [show -(k:ℤ)-1 = -(k:ℤ)+(-1) by ring, zpow_add]
-    simp only [zpow_neg, zpow_natCast, zpow_one, Bool.not_true, hr, inv_pow]
-    simp only [rotation, mul_inv_rev, cs.inv_simple]
-    simp only [mul_assoc, cs.simple_mul_simple_self, mul_one]
-
- theorem exists_indexWord (cs : CoxeterSystem M W) (w : W) :
-    ∃ i : ℤ, |i| = (cs.length w : ℤ) ∧ indexWord cs i = w := by
-  obtain ⟨i,hi⟩ := exists_alt cs w
-  cases i
-  · exact ⟨cs.length w, by simp, (indexWord_nat cs _).trans hi.symm⟩
-  · exact ⟨-(cs.length w : ℤ), by simp, (indexWord_neg_nat cs _).trans hi.symm⟩
-
-/-- A reflection acts on the signed index line by reflection in its midpoint.
-Finite dihedral groups are handled by equality in the group, without choosing
-residue representatives here. -/
- theorem indexWord_reflection_mul (cs : CoxeterSystem M W) {a : ℤ} (ha : Odd a)
-    (i : ℤ) : indexWord cs a * indexWord cs i = indexWord cs (a-i) := by
-  obtain ⟨k,hk⟩ := ha
-  have hk' : a = 2*k+1 := by omega
-  rw [hk',indexWord_odd]
-  rcases Int.even_or_odd i with ⟨l,hl⟩ | ⟨l,hl⟩
-  · have hl' : i=2*l := by omega
-    rw [hl',indexWord_even,show 2*k+1-2*l=2*(k-l)+1 by ring,indexWord_odd]
-    rw [mul_assoc, simple_rotation_zpow, ← mul_assoc, ← zpow_add]
-    congr 2
-  · have hl' : i=2*l+1 := by omega
-    rw [hl',indexWord_odd,show 2*k+1-(2*l+1)=2*(k-l) by ring,indexWord_even]
-    calc
-      (rotation cs ^ k * cs.simple false) * (rotation cs ^ l * cs.simple false) =
-          rotation cs ^ k * (cs.simple false * rotation cs ^ l) * cs.simple false := by group
-      _ = rotation cs ^ k * (rotation cs ^ (-l) * cs.simple false) * cs.simple false := by
-        rw [simple_rotation_zpow]
-      _ = rotation cs ^ (k-l) := by
-        rw [← mul_assoc, mul_assoc (rotation cs ^ k * rotation cs ^ (-l)),
-          cs.simple_mul_simple_self, mul_one, ← zpow_add]
-        congr 1
-
- theorem indexWord_label (cs : CoxeterSystem M W) {i j : ℤ} (h : Odd (i+j)) :
-    indexWord cs j * (indexWord cs i)⁻¹ = indexWord cs (i+j) := by
-  have he := indexWord_reflection_mul cs h i
-  rw [show i+j-i=j by omega] at he
-  rw [← he]
-  group
-
-end
-end KLInvariance.Dihedral
-
-end
-
-
-section
-
-/-! Actual degree-one splines in the ambient extended Tits realization.
-They supply the interpolation factors needed for the relative rank-two model. -/
-namespace KLInvariance.TitsSpace
-open Module
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- def linearPolynomial : Extended M →ₗ[ℝ] MvPolynomial σ ℝ :=
-  (SymmetricAlgebra.equivMvPolynomial basis).toLinearMap.comp
-    (SymmetricAlgebra.ι ℝ (Extended M))
-
- def orbitSpline (y : W) (v₀ : Extended M) (v : W) : MvPolynomial σ ℝ :=
-  linearPolynomial basis (extendedAction M cs (v*y⁻¹) v₀ - v₀)
-
- theorem orbitSpline_self (y : W) (v₀ : Extended M) : orbitSpline cs basis y v₀ y = 0 := by
-  simp [orbitSpline]
-
- theorem orbitSpline_reflection {t : W} {a : I → ℝ}
-    (ha : Represents M cs t a) (y x : W) (v₀ : Extended M) :
-    linearPolynomial basis (embed M a) ∣
-      orbitSpline cs basis y v₀ x - orbitSpline cs basis y v₀ (t*x) := by
-  let c := extendedForm M (embed M a) (extendedAction M cs (x*y⁻¹) v₀)
-  have heq : extendedAction M cs (t*x*y⁻¹) v₀ =
-      extendedAction M cs (x*y⁻¹) v₀ - c • embed M a := by
-    rw [mul_assoc,map_mul,LinearEquiv.mul_apply,ha.formula]
-  unfold orbitSpline
-  rw [← map_sub,heq]
-  have hv : (extendedAction M cs (x*y⁻¹) v₀ - v₀) -
-      (extendedAction M cs (x*y⁻¹) v₀ - c • embed M a - v₀) = c • embed M a := by abel
-  rw [hv,map_smul]
-  exact ⟨algebraMap ℝ (MvPolynomial σ ℝ) c, by simp [Algebra.smul_def,mul_comm]⟩
-
- theorem exists_pair_vector {t r : W} {a d : I → ℝ}
-    (ha : Represents M cs t a) (hd : Represents M cs r d) (htr : t ≠ r) :
-    ∃ v₀ : Extended M, extendedForm M (embed M a) v₀ = -1 ∧
-      extendedForm M (embed M d) v₀ = 0 := by
-  have hli := (ha.extended_independent_of_ne hd htr).map'
-    (extendedForm M) (extendedForm_spec M).2.1.ker_eq_bot
-  have hli' : LinearIndependent ℝ ![extendedForm M (embed M a),
-      extendedForm M (embed M d)] := by
-    convert hli using 1
-    funext i
-    fin_cases i <;> rfl
-  obtain ⟨v₀,hv₀⟩ := ReflectionPlane.formPair_surjective _ _ hli' (-1,0)
-  exact ⟨v₀,congrArg Prod.fst hv₀,congrArg Prod.snd hv₀⟩
-
- theorem orbitSpline_pair_factor {x y z t r : W} {a d : I → ℝ}
-    (ha : Represents M cs t a) (hd : Represents M cs r d)
-    (hxy : x*y⁻¹=t) (hzy : z*y⁻¹=r) (hxz : x ≠ z) :
-    ∃ v₀ : Extended M,
-      orbitSpline cs basis y v₀ x = linearPolynomial basis (embed M a) ∧
-      orbitSpline cs basis y v₀ y = 0 ∧ orbitSpline cs basis y v₀ z = 0 := by
-  have htr : t ≠ r := by
-    intro he
-    apply hxz
-    exact mul_right_cancel (hxy.trans (he.trans hzy.symm))
-  obtain ⟨v₀,hv₀a,hv₀d⟩ := exists_pair_vector cs ha hd htr
-  refine ⟨v₀,?_,orbitSpline_self cs basis y v₀,?_⟩
-  · simp only [orbitSpline,hxy,ha.formula,hv₀a]
-    congr 1
-    module
-  · simp only [orbitSpline,hzy,hd.formula,hv₀d,zero_smul,sub_zero,sub_self,map_zero]
-
- theorem orbitSpline_single_factor {x y t : W} {a : I → ℝ}
-    (ha : Represents M cs t a) (hxy : x*y⁻¹=t) :
-    ∃ v₀ : Extended M,
-      orbitSpline cs basis y v₀ x = linearPolynomial basis (embed M a) ∧
-      orbitSpline cs basis y v₀ y = 0 := by
-  let v₀ : Extended M := (-1/2 : ℝ) • embed M a
-  have hv₀ : extendedForm M (embed M a) v₀ = -1 := by
-    simp only [v₀,map_smul,smul_eq_mul,embed,(extendedForm_spec M).2.2,ha.isRoot.norm]
-    norm_num
-  refine ⟨v₀,?_,orbitSpline_self cs basis y v₀⟩
-  simp only [orbitSpline,hxy,ha.formula,hv₀]
-  congr 1
-  module
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Polynomial interpolation on an actual dihedral reflection subgroup.
-The ambient realization, including all fixed directions and degenerate root
-planes, is retained; no character theorem is assumed. -/
-namespace KLInvariance.Dihedral
-open Module TitsSpace
-universe u v w us
-variable {H : Type w} [Group H] {N : CoxeterMatrix Bool} (ds : CoxeterSystem N H)
-  {I : Type u} [Fintype I] {M : CoxeterMatrix I} {W : Type v} [Group W]
-  (cs : CoxeterSystem M W) (φ : H →* W)
-  (hφ : ∀ t, ds.IsReflection t → cs.IsReflection (φ t))
-  {σ : Type us} (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- theorem reflection_of_parity {x y : H} (hp : ds.length x % 2 ≠ ds.length y % 2) :
-    ds.IsReflection (x*y⁻¹) := by
-  rw [reflection_iff_odd,Nat.odd_iff]
-  have h := ds.length_mul_mod_two x y⁻¹
-  rw [ds.length_inv] at h
-  omega
-
- def oppositeExternal (x c : H) : Finset H := by
-  classical
-  exact (externalLower ds x c).filter (fun z => ds.length z % 2 ≠ ds.length x % 2)
-
- def sameExternal (x c : H) : Finset H := by
-  classical
-  exact (externalLower ds x c).filter (fun z => ds.length z % 2 = ds.length x % 2)
-
- theorem oppositeExternal_reflection {x c y : H} (hy : y ∈ oppositeExternal ds x c) :
-    ds.IsReflection (y*x⁻¹) := by
-  classical
-  exact reflection_of_parity ds (Finset.mem_filter.mp hy).2
-
- def interpolationRoot (x c : H) (y : oppositeExternal ds x c) : PositiveRoot M cs :=
-  positiveRoot M cs ⟨φ (y.val*x⁻¹), hφ _ (oppositeExternal_reflection ds y.property)⟩
-
- theorem interpolation_exists (hφinj : Function.Injective φ) {x c : H}
-    (hxc : BruhatLE ds x c) : ∃ F : H → MvPolynomial σ ℝ,
-      (∀ t, ds.IsReflection t → ∀ (a : I → ℝ), Represents M cs (φ t) a → ∀ v,
-        linearPolynomial basis (embed M a) ∣ F v-F (t*v)) ∧
-      F x = ∏ y : oppositeExternal ds x c,
-        linearPolynomial basis (embed M (interpolationRoot ds cs φ hφ x c y).val) ∧
-      ∀ z, BruhatLE ds z c → ¬ BruhatLE ds z x → F z = 0 := by
-  classical
-  let Y := {y // y ∈ oppositeExternal ds x c}
-  let Z := {z // z ∈ sameExternal ds x c}
-  have hc : Fintype.card Z ≤ Fintype.card Y := by
-    change Fintype.card ↥(sameExternal ds x c) ≤ Fintype.card ↥(oppositeExternal ds x c)
-    rw [Fintype.card_coe,Fintype.card_coe]
-    exact external_parity_card_le ds hxc
-  obtain ⟨j⟩ := Function.Embedding.nonempty_of_card_le hc
-  have hfac (y : Y) : ∃ v₀ : Extended M,
-      orbitSpline cs basis (φ y.val) v₀ (φ x) =
-        linearPolynomial basis (embed M (interpolationRoot ds cs φ hφ x c y).val) ∧
-      orbitSpline cs basis (φ y.val) v₀ (φ y.val) = 0 ∧
-      ∀ z : Z, j z = y → orbitSpline cs basis (φ y.val) v₀ (φ z.val) = 0 := by
-    have ht := oppositeExternal_reflection ds y.property
-    have ha := positiveRoot_represents M cs
-      ⟨φ (y.val*x⁻¹),hφ _ ht⟩
-    have hxy₀ : x*y.val⁻¹=y.val*x⁻¹ := by simpa only [mul_inv_rev,inv_inv] using ht.inv
-    have hxy : φ x*(φ y.val)⁻¹=φ (y.val*x⁻¹) := by
-      rw [← map_inv,← map_mul,hxy₀]
-    by_cases hex : ∃ z : Z, j z = y
-    · obtain ⟨z,hz⟩ := hex
-      have hzprop := Finset.mem_filter.mp z.property
-      have hyprop := Finset.mem_filter.mp y.property
-      have hr : ds.IsReflection (z.val*y.val⁻¹) := reflection_of_parity ds (by omega)
-      let d := positiveRoot M cs ⟨φ (z.val*y.val⁻¹),hφ _ hr⟩
-      have hd : Represents M cs (φ (z.val*y.val⁻¹)) d.val := positiveRoot_represents M cs ⟨φ (z.val*y.val⁻¹),hφ _ hr⟩
-      have hxz : φ x ≠ φ z.val := by
-        intro he
-        have he' := hφinj he
-        have hh := (mem_externalLower ds x c z.val).mp hzprop.1
-        exact hh.2 (he' ▸ bruhat_refl ds x)
-      obtain ⟨v₀,ha',hy',hz0'⟩ := orbitSpline_pair_factor cs basis ha hd hxy
-        (by rw [map_mul,map_inv]) hxz
-      refine ⟨v₀,ha',hy',?_⟩
-      intro z' hz'
-      have he := j.injective (hz'.trans hz.symm)
-      simpa only [he] using hz0'
-    · obtain ⟨v₀,ha',hy'⟩ := orbitSpline_single_factor cs basis ha hxy
-      exact ⟨v₀,ha',hy',fun z hz => (hex ⟨z,hz⟩).elim⟩
-  choose v₀ hv₀ hy₀ hz₀ using hfac
-  let F : H → MvPolynomial σ ℝ := fun v => ∏ y : Y,
-    orbitSpline cs basis (φ y.val) (v₀ y) (φ v)
-  refine ⟨F,?_,?_,?_⟩
-  · intro t _ht a ha v
-    apply Ideal.mem_span_singleton.mp
-    apply Ideal.Quotient.eq.mp
-    change Ideal.Quotient.mk _ (∏ y : Y, orbitSpline cs basis (φ y.val) (v₀ y) (φ v)) =
-      Ideal.Quotient.mk _ (∏ y : Y, orbitSpline cs basis (φ y.val) (v₀ y) (φ (t*v)))
-    simp only [map_prod]
-    apply Finset.prod_congr rfl
-    intro y _
-    apply Ideal.Quotient.eq.mpr
-    apply Ideal.mem_span_singleton.mpr
-    rw [map_mul]
-    exact orbitSpline_reflection cs basis ha _ _ _
-  · exact Finset.prod_congr rfl (fun y _ => hv₀ y)
-  · intro z hzc hzx
-    have hz : z ∈ externalLower ds x c := (mem_externalLower ds x c z).mpr ⟨hzc,hzx⟩
-    by_cases hp : ds.length z % 2 = ds.length x % 2
-    · let z' : Z := ⟨z,Finset.mem_filter.mpr ⟨hz,hp⟩⟩
-      exact Finset.prod_eq_zero (Finset.mem_univ (j z')) (hz₀ (j z') z' rfl)
-    · let y : Y := ⟨z,Finset.mem_filter.mpr ⟨hz,hp⟩⟩
-      exact Finset.prod_eq_zero (Finset.mem_univ y) (hy₀ y)
-
-end
-end KLInvariance.Dihedral
-
-end
-
-
-section
-
-/-! An elementary flabbiness criterion for the exact rank-one model. Flow-down
-sections with the genuine product of outgoing labels suffice. The pairwise
-relative-primality condition will be instantiated by actual root labels. -/
-namespace KLInvariance.MomentGraph.Sheaf
-universe ur uv ue
-variable {R : Type ur} [CommRing R] [DecompositionMonoid R]
-  {V : Type uv} [PartialOrder V] {E : Type ue}
-  (G : OrderedGraph V E) (α : E → R) (c : V)
-noncomputable section
-variable [Fintype E] [DecidableEq V]
-
-def modelOutgoing (x : V) : Finset E := by
-  classical
-  exact Finset.univ.filter (fun e => G.source e = x ∧ G.target e ≤ c)
-
-@[simp] theorem mem_modelOutgoing (x : V) (e : E) :
-    e ∈ modelOutgoing G c x ↔ G.source e = x ∧ G.target e ≤ c := by
-  classical
-  simp [modelOutgoing]
-
-def modelProduct (x : V) : R := ∏ e ∈ modelOutgoing G c x, α e
-
-def HasFlowDown : Prop :=
-  ∀ x, x ≤ c → ∃ f : (structureSheaf G α c).Assignment,
-    (structureSheaf G α c).IsGlobal f ∧
-    (∀ h : PLift (x ≤ c), f x h = modelProduct G α c x) ∧
-    (∀ y, ¬ y ≤ x → f y = 0)
-
-omit [DecompositionMonoid R] [Fintype E] [DecidableEq V] in
- theorem structureCompatible_iff (Ω : Set V) (f : (structureSheaf G α c).Assignment) :
-    (structureSheaf G α c).CompatibleOn Ω f ↔
-      ∀ e, G.source e ∈ Ω → G.target e ∈ Ω → ∀ h : PLift (G.target e ≤ c),
-        α e ∣ f (G.source e) ⟨(G.increasing e).le.trans h.down⟩ - f (G.target e) h := by
-  constructor
-  · intro hf e hs ht h
-    exact Ideal.mem_span_singleton.mp
-      (Ideal.Quotient.eq.mp (congrFun (hf e hs ht) h))
-  · intro hf e hs ht
-    funext h
-    exact Ideal.Quotient.eq.mpr (Ideal.mem_span_singleton.mpr (hf e hs ht h))
-
-include G α c in
- theorem flowDown_flabby [Finite V]
-    (hrel : ∀ x, (modelOutgoing G c x : Set E).Pairwise (Function.onFun IsRelPrime α))
-    (hflow : HasFlowDown G α c) : (structureSheaf G α c).Flabby := by
-  classical
-  let B := structureSheaf G α c
-  let : Fintype V := Fintype.ofFinite V
-  suffices hh : ∀ s : Finset V, IsUpperSet (s : Set V) →
-      ∀ f : B.Assignment, B.CompatibleOn (s : Set V) f →
-        ∃ g : B.Assignment, B.IsGlobal g ∧ ∀ x ∈ s, g x = f x by
-    intro Ω hΩ f hf
-    have hset : (↑(Finset.univ.filter (· ∈ Ω)) : Set V) = Ω := by ext x; simp
-    obtain ⟨g,hg,hgf⟩ := hh (Finset.univ.filter (· ∈ Ω)) (by rwa [hset]) f (by rwa [hset])
-    exact ⟨g,hg,fun x hx => hgf x (by simpa using hx)⟩
-  intro s
-  refine Finset.strongInductionOn s ?_
-  intro s ih hs f hf
-  by_cases hempty : s = ∅
-  · subst s
-    refine ⟨0,?_,by simp⟩
-    exact (B.sections Set.univ).zero_mem
-  obtain ⟨a,ha,hmin⟩ := s.exists_minimal (Finset.nonempty_iff_ne_empty.mpr hempty)
-  have hu : IsUpperSet (↑(s.erase a) : Set V) := by
-    intro x y hxy hx
-    have hx' := Finset.mem_erase.mp hx
-    refine Finset.mem_erase.mpr ⟨?_,hs hxy hx'.2⟩
-    intro hya
-    subst y
-    exact hx'.1 (le_antisymm hxy (hmin hx'.2 hxy))
-  obtain ⟨g,hg,hgf⟩ := ih (s.erase a) (Finset.erase_ssubset ha) hu f
-    (fun e he ht => hf e (Finset.mem_of_mem_erase he) (Finset.mem_of_mem_erase ht))
-  by_cases hac : a ≤ c
-  · obtain ⟨t,ht,hta,htzero⟩ := hflow a hac
-    have hdiv : modelProduct G α c a ∣ f a ⟨hac⟩ - g a ⟨hac⟩ := by
-      apply Finset.prod_dvd_of_isRelPrime (hrel a)
-      intro e he
-      obtain ⟨hea,hec⟩ := (mem_modelOutgoing G c a e).mp he
-      have hsource : G.source e ∈ s := hea ▸ ha
-      have htarget : G.target e ∈ s.erase a := Finset.mem_erase.mpr
-        ⟨by simpa [← hea] using (G.increasing e).ne',hs (G.increasing e).le hsource⟩
-      have hf' := (structureCompatible_iff G α c (s : Set V) f).mp hf e hsource
-        (Finset.mem_of_mem_erase htarget) ⟨hec⟩
-      have hg' := (structureCompatible_iff G α c Set.univ g).mp hg e
-        (Set.mem_univ _) (Set.mem_univ _) ⟨hec⟩
-      rw [hgf _ htarget] at hg'
-      have hd := dvd_sub hf' hg'
-      rw [sub_sub_sub_cancel_right] at hd
-      subst a
-      exact hd
-    obtain ⟨r,hr⟩ := hdiv
-    refine ⟨g+r • t,(B.sections Set.univ).add_mem hg
-      ((B.sections Set.univ).smul_mem r ht),?_⟩
-    intro x hx
-    by_cases hxa : x = a
-    · subst x
-      funext h
-      change g a h + r * t a h = f a h
-      rw [hta h]
-      change f a h - g a h = modelProduct G α c a * r at hr
-      linear_combination -hr
-    · have hnot : ¬ x ≤ a := fun h => hxa (le_antisymm h (hmin hx h))
-      have ht0 := htzero x hnot
-      change g x + r • t x = f x
-      rw [ht0,smul_zero,add_zero]
-      exact hgf x (Finset.mem_erase.mpr ⟨hxa,hx⟩)
-  · refine ⟨g,hg,?_⟩
-    intro x hx
-    by_cases hxa : x = a
-    · subst x
-      let := structureSheaf_support G α c a hac
-      exact Subsingleton.elim _ _
-    · exact hgf x (Finset.mem_erase.mpr ⟨hxa,hx⟩)
-
-end
-end KLInvariance.MomentGraph.Sheaf
-
-end
-
-
-section
-
-/-! Flabbiness of the exact rank-one model with ambient labels, proved by
-interpolation rather than an assumed character theorem. -/
-namespace KLInvariance.Dihedral
-open Module TitsSpace BruhatGraph MomentGraph MomentGraph.Sheaf
-universe u v w us
-variable {H : Type w} [Group H] {N : CoxeterMatrix Bool} (ds : CoxeterSystem N H)
-  {I : Type u} [Fintype I] {M : CoxeterMatrix I} {W : Type v} [Group W]
-  (cs : CoxeterSystem M W) (φ : H →* W)
-  (hφ : ∀ t, ds.IsReflection t → cs.IsReflection (φ t))
-  {σ : Type us} (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- def inducedRoot (b : H) (e : Edge ds 1 b) : PositiveRoot M cs :=
-  positiveRoot M cs ⟨φ (label ds 1 b e),hφ _ (label_isReflection ds 1 b e)⟩
-
- def inducedLabel (b : H) (e : Edge ds 1 b) : MvPolynomial σ ℝ :=
-  linearPolynomial basis (embed M (inducedRoot ds cs φ hφ b e).val)
-
- theorem external_step {x c y : H} (hy : y ∈ oppositeExternal ds x c) :
-    BruhatStep ds x y := by
-  classical
-  obtain ⟨hm,hp⟩ := Finset.mem_filter.mp hy
-  obtain ⟨_,hn⟩ := (mem_externalLower ds x c y).mp hm
-  have hl : ds.length x < ds.length y := by
-    have hh : ¬ ds.length y < ds.length x := fun h => hn (shorter_bruhat ds h)
-    omega
-  exact ⟨hl,y*x⁻¹,oppositeExternal_reflection ds hy,by group⟩
-
- theorem step_external {x c y : H} (hy : BruhatStep ds x y) (hyc : BruhatLE ds y c) :
-    y ∈ oppositeExternal ds x c := by
-  classical
-  refine Finset.mem_filter.mpr ⟨(mem_externalLower ds x c y).mpr ⟨hyc,?_⟩,?_⟩
-  · exact fun h => (length_le_of_bruhat ds h).not_gt hy.1
-  · have hh := (step_iff_odd ds x y).mp hy
-    rw [Nat.odd_iff] at hh
-    omega
-
- variable (b : H) [Fintype (Edge ds 1 b)] [DecidableEq (Interval ds 1 b)]
-
- def externalEdge (x c : Interval ds 1 b) (y : oppositeExternal ds x.val c.val) : Edge ds 1 b :=
-  ⟨(x,⟨y.val,one_bruhat ds y.val,
-    bruhat_trans ds ((mem_externalLower ds x.val c.val y.val).mp
-      (Finset.mem_filter.mp y.property).1).1 c.property.2⟩),external_step ds y.property⟩
-
- theorem externalEdge_mem (x c : Interval ds 1 b) (y : oppositeExternal ds x.val c.val) :
-    externalEdge ds b x c y ∈ modelOutgoing (graph ds 1 b) c x := by
-  classical
-  apply (mem_modelOutgoing _ _ _ _).mpr
-  exact ⟨rfl,((mem_externalLower ds x.val c.val y.val).mp
-    (Finset.mem_filter.mp y.property).1).1⟩
-
- def outgoingEquiv (x c : Interval ds 1 b) :
-    ↥(oppositeExternal ds x.val c.val) ≃ ↥(modelOutgoing (graph ds 1 b) c x) where
-  toFun y := ⟨externalEdge ds b x c y,externalEdge_mem ds b x c y⟩
-  invFun e := ⟨(target ds 1 b e.val).val,by
-    obtain ⟨hs,ht⟩ := (mem_modelOutgoing _ _ _ _).mp e.property
-    have hh := e.val.property
-    change BruhatStep ds (source ds 1 b e.val).val (target ds 1 b e.val).val at hh
-    change source ds 1 b e.val = x at hs
-    rw [hs] at hh
-    exact step_external ds hh ht⟩
-  left_inv _ := rfl
-  right_inv e := by
-    apply Subtype.ext
-    apply Subtype.ext
-    apply Prod.ext
-    · exact ((mem_modelOutgoing _ _ _ _).mp e.property).1.symm
-    · rfl
-
- theorem interpolation_product (x c : Interval ds 1 b) :
-    (∏ y : oppositeExternal ds x.val c.val,
-      linearPolynomial basis (embed M (interpolationRoot ds cs φ hφ x.val c.val y).val)) =
-      modelProduct (graph ds 1 b) (inducedLabel ds cs φ hφ basis b) c x := by
-  classical
-  calc
-    _ = ∏ e : ↥(modelOutgoing (graph ds 1 b) c x), inducedLabel ds cs φ hφ basis b e.val :=
-      (outgoingEquiv ds b x c).prod_comp _
-    _ = _ := Finset.prod_coe_sort _ _
-
- theorem hasFlowDown (hφinj : Function.Injective φ) (c : Interval ds 1 b) :
-    HasFlowDown (graph ds 1 b) (inducedLabel ds cs φ hφ basis b) c := by
-  classical
-  intro x hxc
-  obtain ⟨F,hF,hFx,hFzero⟩ := interpolation_exists ds cs φ hφ basis hφinj hxc
-  let f : (structureSheaf (graph ds 1 b) (inducedLabel ds cs φ hφ basis b) c).Assignment :=
-    fun v _ => F v.val
-  refine ⟨f,?_,?_,?_⟩
-  · apply (structureCompatible_iff _ _ _ _ _).mpr
-    intro e _ _ h
-    have hdiv := hF (label ds 1 b e) (label_isReflection ds 1 b e)
-      (inducedRoot ds cs φ hφ b e).val (positiveRoot_represents M cs
-        ⟨φ (label ds 1 b e),hφ _ (label_isReflection ds 1 b e)⟩)
-      (source ds 1 b e).val
-    rw [label_mul_source] at hdiv
-    exact hdiv
-  · intro _
-    exact hFx.trans (interpolation_product ds cs φ hφ basis b x c)
-  · intro y hy
-    funext h
-    exact hFzero y.val h.down hy
-
-omit [Fintype (Edge ds 1 b)] [DecidableEq (Interval ds 1 b)] in
- theorem inducedLabel_prime (e : Edge ds 1 b) : Prime (inducedLabel ds cs φ hφ basis b e) := by
-  apply (MulEquiv.prime_iff (SymmetricAlgebra.equivMvPolynomial basis)).mpr
-  apply EdgeAlgebra.symmetric_ι_prime
-  intro h
-  exact (inducedRoot ds cs φ hφ b e).property.1.ne_zero (congrArg Prod.fst h)
-
-omit [Fintype (Edge ds 1 b)] [DecidableEq (Interval ds 1 b)] in
- theorem label_eq_of_induced_dvd (hφinj : Function.Injective φ) {e f : Edge ds 1 b}
-    (h : inducedLabel ds cs φ hφ basis b e ∣ inducedLabel ds cs φ hφ basis b f) :
-    label ds 1 b e = label ds 1 b f := by
-  apply hφinj
-  suffices he : inducedRoot ds cs φ hφ b e = inducedRoot ds cs φ hφ b f from
-    congrArg Subtype.val ((positiveRootEquiv M cs).injective he)
-  by_contra hn
-  have hd : SymmetricAlgebra.ι ℝ (Extended M) (embed M (inducedRoot ds cs φ hφ b e).val) ∣
-      SymmetricAlgebra.ι ℝ (Extended M) (embed M (inducedRoot ds cs φ hφ b f).val) := by
-    obtain ⟨z,hz⟩ := h
-    exact ⟨(SymmetricAlgebra.equivMvPolynomial basis).symm z,by
-      apply (SymmetricAlgebra.equivMvPolynomial basis).injective
-      simpa only [map_mul,AlgEquiv.apply_symm_apply,inducedLabel,linearPolynomial,LinearMap.comp_apply,AlgEquiv.toLinearMap_apply] using hz⟩
-  obtain ⟨r,hr⟩ := Submodule.mem_span_singleton.mp ((EdgeAlgebra.symmetric_ι_dvd_iff _ _).mp hd)
-  have he : r • (inducedRoot ds cs φ hφ b e).val = (1:ℝ) • (inducedRoot ds cs φ hφ b f).val := by
-    have hh := congrArg Prod.fst hr
-    exact hh.trans (one_smul ℝ _).symm
-  exact one_ne_zero ((positiveRoots_independent _ _ hn).eq_zero_of_pair' he).2
-
- theorem model_flabby (hφinj : Function.Injective φ) (c : Interval ds 1 b) :
-    (structureSheaf (graph ds 1 b) (inducedLabel ds cs φ hφ basis b) c).Flabby := by
-  classical
-  let : Finite (Interval ds 1 b) := interval_finite ds 1 b
-  apply flowDown_flabby
-  · intro x e he f hf hne
-    apply ((inducedLabel_prime ds cs φ hφ basis b e).irreducible.isRelPrime_iff_not_dvd).mpr
-    intro hdiv
-    apply hne
-    apply eq_of_label_source_eq ds 1 b (label_eq_of_induced_dvd ds cs φ hφ basis b hφinj hdiv)
-    exact ((mem_modelOutgoing _ _ _ _).mp he).1.trans ((mem_modelOutgoing _ _ _ _).mp hf).1.symm
-  · exact hasFlowDown ds cs φ hφ basis b hφinj c
-
-end
-end KLInvariance.Dihedral
-
-end
-
-
-section
-
-/-! Instantiation on the actual maximal subgroup of the ambient root plane.
-The exact canonical roots, including degenerate/infinite planes, are used. -/
-namespace KLInvariance.TitsSpace.PlaneBasis
-open Module BruhatGraph MomentGraph MomentGraph.Sheaf
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-  {σ : Type us} (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- def modelLabel (b : planeSubgroup M cs P) (e : Edge B.system 1 b) : MvPolynomial σ ℝ :=
-  Dihedral.inducedLabel B.system cs (planeSubgroup M cs P).subtype
-    (fun t => (B.reflection_iff t).mp) basis b e
-
- theorem model_flabby (b : planeSubgroup M cs P) [Fintype (Edge B.system 1 b)]
-    [DecidableEq (Interval B.system 1 b)] (c : Interval B.system 1 b) :
-    (structureSheaf (graph B.system 1 b) (B.modelLabel basis b) c).Flabby :=
-  Dihedral.model_flabby B.system cs (planeSubgroup M cs P).subtype
-    (fun t => (B.reflection_iff t).mp) basis b Subtype.val_injective c
-
- theorem model_generated (b : planeSubgroup M cs P) (c : Interval B.system 1 b) :
-    (structureSheaf (graph B.system 1 b) (B.modelLabel basis b) c).Generated :=
-  structureSheaf_generated _ _ _
-
- theorem model_quotient (b : planeSubgroup M cs P) (c : Interval B.system 1 b) :
-    (structureSheaf (graph B.system 1 b) (B.modelLabel basis b) c).UpperQuotient
-      (B.modelLabel basis b) := structureSheaf_upperQuotient _ _ _
-
-end
-end KLInvariance.TitsSpace.PlaneBasis
-
-end
-
-
-section
-
-/-! Each finite rank-two set fits in a principal Bruhat interval, even in
-an infinite dihedral group. No finite Coxeter-system hypothesis is used. -/
-namespace KLInvariance
-universe u v
-variable {I : Type u} {M : CoxeterMatrix I} {W : Type v} [Group W]
-
- theorem bruhat_top_of_all_descents (cs : CoxeterSystem M W) (w : W)
-    (hw : ∀ i, cs.length (cs.simple i * w) < cs.length w) (x : W) :
-    BruhatLE cs x w := by
-  classical
-  have hword : ∀ ω, BruhatLE cs (cs.wordProd ω) w := by
-    intro ω
-    induction ω with
-    | nil => simpa only [cs.wordProd_nil] using one_bruhat cs w
-    | cons i ω ih =>
-      have hu := upperSimple_mono cs i ih
-      have hz := le_upperSimple cs i (cs.simple i * cs.wordProd ω)
-      rw [upperSimple_mul] at hz
-      have he : upperSimple cs i w = w := by simp only [upperSimple,ite_eq_left (hw i)]
-      rw [he] at hu
-      change BruhatLE cs (cs.simple i * cs.wordProd ω) w
-      exact hz.trans hu
-  obtain ⟨ω,_,hx⟩ := cs.exists_isReduced x
-  rw [hx]
-  exact hword ω
-
-namespace Dihedral
-variable {N : CoxeterMatrix Bool} (ds : CoxeterSystem N W)
-
- theorem exists_common_upper (s : Finset W) : ∃ b, ∀ x ∈ s, BruhatLE ds x b := by
-  classical
-  by_cases hb : BddAbove (Set.range ds.length)
-  · have hf : (Set.range ds.length).Finite := hb.finite
-    obtain ⟨b,hb⟩ := (Set.range_nonempty ds.length).csSup_mem hf
-    have hmax (x : W) : ds.length x ≤ ds.length b := by
-      rw [hb]
-      exact le_csSup (hf.bddAbove) (Set.mem_range_self x)
-    refine ⟨b,fun x _ => bruhat_top_of_all_descents ds b ?_ x⟩
-    intro i
-    have hle := hmax (ds.simple i*b)
-    have hne := ds.length_simple_mul b i
-    omega
-  · have hn : ∀ n : ℕ, ∃ w, n < ds.length w := by
-      intro n
-      by_contra h
-      push Not at h
-      exact hb ⟨n,by rintro m ⟨x,rfl⟩; exact h x⟩
-    obtain ⟨b,hb⟩ := hn (s.sup ds.length)
-    exact ⟨b,fun x hx => shorter_bruhat ds ((Finset.le_sup hx).trans_lt hb)⟩
-
-end Dihedral
-end KLInvariance
-
-end
-
-
-section
-
-namespace KLInvariance.Dihedral
-variable {W : Type*} [Group W] {M : CoxeterMatrix Bool}
-noncomputable section
-
- theorem map_not_alt (i : Bool) (n : ℕ) : (alt i n).map Bool.not = alt (!i) n := by
-  induction n generalizing i with
-  | zero => rfl
-  | succ n ih => rw [alt_succ,List.map_cons,ih,alt_succ]
-
- theorem swap_alt (cs : CoxeterSystem M W) (i : Bool) (n : ℕ) :
-    swap cs (cs.wordProd (alt i n)) = cs.wordProd (alt (!i) n) := by
-  rw [swap_wordProd,map_not_alt]
-
- theorem length_alt_le_length (cs : CoxeterSystem M W) (i : Bool) {n : ℕ} {w : W}
-    (h : n ≤ cs.length w) : cs.length (cs.wordProd (alt i n)) = n := by
-  obtain ⟨j,hj⟩ := exists_alt cs w
-  have hr : cs.IsReduced (alt j (cs.length w)) := by
-    change cs.length (cs.wordProd _) = _
-    rw [← hj,length_alt]
-  have hn := (reduced_alt_take cs hr h).eq
-  rw [length_alt] at hn
-  have hi : i=j ∨ i=(!j) := by cases i <;> cases j <;> simp
-  rcases hi with rfl|rfl
-  · exact hn
-  · rw [← swap_alt,length_swap,hn]
-
- theorem length_indexWord_nat (cs : CoxeterSystem M W) {n : ℕ} {w : W}
-    (h : n ≤ cs.length w) : cs.length (indexWord cs n) = n := by
-  rw [indexWord_nat]
-  exact length_alt_le_length cs false h
-
- theorem length_indexWord_neg_nat (cs : CoxeterSystem M W) {n : ℕ} {w : W}
-    (h : n ≤ cs.length w) : cs.length (indexWord cs (-(n:ℤ))) = n := by
-  rw [indexWord_neg_nat]
-  exact length_alt_le_length cs true h
-
- theorem length_indexWord (cs : CoxeterSystem M W) {i : ℤ} {w : W}
-    (h : i.natAbs ≤ cs.length w) : cs.length (indexWord cs i) = i.natAbs := by
-  obtain hi|hi := le_total 0 i
-  · have he : i=(i.natAbs:ℤ) := by rw [Int.natCast_natAbs,abs_of_nonneg hi]
-    nth_rw 1 [he]
-    exact length_indexWord_nat cs h
-  · have he : i= -(i.natAbs:ℤ) := by rw [Int.natCast_natAbs,abs_of_nonpos hi,neg_neg]
-    nth_rw 1 [he]
-    exact length_indexWord_neg_nat cs h
-
- theorem leftDescent_alt (cs : CoxeterSystem M W) (i : Bool) {n : ℕ} {w : W}
-    (hn : 0<n) (h : n ≤ cs.length w) :
-    cs.IsLeftDescent (cs.wordProd (alt i n)) i := by
-  obtain ⟨k,rfl⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : n≠0)
-  unfold CoxeterSystem.IsLeftDescent
-  rw [length_alt_le_length cs i h,alt_succ,cs.wordProd_cons,cs.simple_mul_simple_cancel_left]
-  have hh := cs.length_wordProd_le (alt (!i) k)
-  rw [length_alt] at hh
-  omega
-
- theorem indexWord_add_even (cs : CoxeterSystem M W) (i n : ℤ) :
-    indexWord cs (i+2*n) = rotation cs ^ n * indexWord cs i := by
-  have hm : (i+2*n)%2=i%2 := by omega
-  have hd : (i+2*n)/2=i/2+n := by omega
-  simp only [indexWord,hm,hd,zpow_add]
-  split_ifs
-  · exact (Commute.zpow_zpow_self (rotation cs) (i/2) n).eq
-  · rw [← mul_assoc,(Commute.zpow_zpow_self (rotation cs) (i/2) n).eq,mul_assoc]
-
-/-- The maximal-length case supplies the precise signed period, without an
-extra finiteness hypothesis or a numerical cosine restriction. -/
- theorem rotation_pow_max (cs : CoxeterSystem M W) (w : W)
-    (hmax : ∀ z, cs.length z ≤ cs.length w) : rotation cs ^ cs.length w = 1 := by
-  have hd (i : Bool) : cs.IsLeftDescent w i := by
-    have hle := hmax (cs.simple i*w)
-    have hne := cs.length_simple_mul w i
-    unfold CoxeterSystem.IsLeftDescent
-    omega
-  have he : indexWord cs (cs.length w) = indexWord cs (-(cs.length w:ℤ)) := by
-    rw [indexWord_nat,indexWord_neg_nat,← eq_alt_of_descent cs false (hd false),
-      ← eq_alt_of_descent cs true (hd true)]
-  have hp := indexWord_add_even cs (-(cs.length w:ℤ)) (cs.length w)
-  rw [show -(cs.length w:ℤ)+2*(cs.length w:ℤ)=(cs.length w:ℤ) by ring,
-    zpow_natCast,he] at hp
-  exact mul_right_cancel (show rotation cs ^ cs.length w * indexWord cs (-(cs.length w:ℤ)) =
-      1 * indexWord cs (-(cs.length w:ℤ)) by simpa only [one_mul] using hp.symm)
-
- theorem indexWord_period (cs : CoxeterSystem M W) {N : ℕ}
-    (hN : rotation cs ^ N = 1) (i k : ℤ) :
-    indexWord cs (i+2*(N:ℤ)*k) = indexWord cs i := by
-  rw [show i+2*(N:ℤ)*k=i+2*((N:ℤ)*k) by ring,indexWord_add_even,zpow_mul,zpow_natCast,hN,
-    one_zpow,one_mul]
-
- theorem indexWord_mod_period (cs : CoxeterSystem M W) {N : ℕ}
-    (hN : rotation cs ^ N = 1) (i : ℤ) :
-    indexWord cs (i%(2*(N:ℤ))) = indexWord cs i := by
-  have hp := indexWord_period cs hN (i%(2*(N:ℤ))) (i/(2*(N:ℤ)))
-  rw [Int.emod_add_mul_ediv] at hp
-  exact hp.symm
-
-end
-end KLInvariance.Dihedral
-
-end
-
-
-section
-
-namespace KLInvariance.Dihedral
-variable {W : Type*} [Group W] {M : CoxeterMatrix Bool}
-
- theorem indexWord_step (cs : CoxeterSystem M W) {n : ℕ} {w : W}
-    (h : n+1 ≤ cs.length w) : BruhatStep cs (indexWord cs n) (indexWord cs (n+1:ℕ)) := by
-  rw [step_iff_odd,length_indexWord_nat cs (show n≤cs.length w by omega),
-    length_indexWord_nat cs h]
-  exact ⟨by omega,by simp⟩
-
- theorem neg_indexWord_step (cs : CoxeterSystem M W) {n : ℕ} {w : W}
-    (h : n+1 ≤ cs.length w) :
-    BruhatStep cs (indexWord cs (-(n:ℤ))) (indexWord cs (-((n+1:ℕ):ℤ))) := by
-  rw [step_iff_odd,length_indexWord_neg_nat cs (show n≤cs.length w by omega),
-    length_indexWord_neg_nat cs h]
-  exact ⟨by omega,by simp⟩
-
-end KLInvariance.Dihedral
-
-namespace KLInvariance.TitsSpace.PlaneBasis
-open KLInvariance.Dihedral
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-noncomputable section
-
- def prefixRoot (w : planeSubgroup M cs P) (n : ℕ) : PositiveRoot M cs :=
-  if h : n < B.system.length w then B.edgeRoot (indexWord_step B.system h) else B.left
-
- def negPrefixRoot (w : planeSubgroup M cs P) (n : ℕ) : PositiveRoot M cs :=
-  if h : n < B.system.length w then B.edgeRoot (neg_indexWord_step B.system h) else B.right
-
- theorem prefixRoot_strictMono (w : planeSubgroup M cs P) :
-    StrictMonoOn (fun n => geometricSlope (B.prefixRoot w n)) (Set.Iio (B.system.length w)) := by
-  apply strictMonoOn_of_lt_succ Set.ordConnected_Iio
-  intro n _ hn hn'
-  change n < B.system.length w at hn
-  change n+1 < B.system.length w at hn'
-  change geometricSlope (B.prefixRoot w n) < geometricSlope (B.prefixRoot w (n+1))
-  rw [prefixRoot,prefixRoot,dite_eq_left hn,dite_eq_left hn']
-  apply (B.edgeRoot_slopes_iff _ _).mpr
-  rw [indexWord_nat]
-  exact leftDescent_alt B.system false (by omega) hn'.le
-
- theorem negPrefixRoot_strictAnti (w : planeSubgroup M cs P) :
-    StrictAntiOn (fun n => geometricSlope (B.negPrefixRoot w n))
-      (Set.Iio (B.system.length w)) := by
-  apply strictAntiOn_of_succ_lt Set.ordConnected_Iio
-  intro n _ hn hn'
-  change n < B.system.length w at hn
-  change n+1 < B.system.length w at hn'
-  change geometricSlope (B.negPrefixRoot w (n+1)) < geometricSlope (B.negPrefixRoot w n)
-  rw [negPrefixRoot,negPrefixRoot,dite_eq_left hn,dite_eq_left hn']
-  have ht : B.system.IsLeftDescent (indexWord B.system (-((n+1:ℕ):ℤ))) true := by
-    rw [indexWord_neg_nat]
-    exact leftDescent_alt B.system true (by omega) hn'.le
-  have hlen := length_indexWord_neg_nat B.system hn'.le
-  have hu := unique_leftDescent B.system (w := indexWord B.system (-((n+1:ℕ):ℤ)))
-    (z := w) (by rw [hlen]; omega) (by rw [hlen]; omega)
-  have hf : ¬B.system.IsLeftDescent (indexWord B.system (-((n+1:ℕ):ℤ))) false := by
-    intro hf
-    obtain ⟨i,_,hi⟩ := hu
-    have h1 := hi true ht
-    have h2 := hi false hf
-    simp_all
-  have hnlt := (B.edgeRoot_slopes_iff (neg_indexWord_step B.system hn)
-    (neg_indexWord_step B.system hn')).not.mpr hf
-  apply lt_of_le_of_ne (le_of_not_gt hnlt)
-  intro he
-  have he' := geometricSlope_injective he
-  exact graphRoot_ne_of_path
-    ((B.bruhatStep_iff _ _).mpr (neg_indexWord_step B.system hn))
-    ((B.bruhatStep_iff _ _).mpr (neg_indexWord_step B.system hn')) he'.symm
-
- theorem edgeRoot_eq_of_label {x y z w : planeSubgroup M cs P}
-    (hxy : BruhatStep B.system x y) (hzw : BruhatStep B.system z w)
-    (he : y*x⁻¹=w*z⁻¹) : B.edgeRoot hxy = B.edgeRoot hzw := by
-  unfold edgeRoot graphRoot
-  apply congrArg (positiveRoot M cs)
-  apply Subtype.ext
-  exact congrArg (fun z : planeSubgroup M cs P => (z : W)) he
-
- theorem prefixRoot_eq_edgeRoot {x y w : planeSubgroup M cs P}
-    (hxy : BruhatStep B.system x y) {n : ℕ} (hn : n<B.system.length w)
-    (he : y*x⁻¹=indexWord B.system (2*(n:ℤ)+1)) :
-    B.prefixRoot w n = B.edgeRoot hxy := by
-  rw [prefixRoot,dite_eq_left hn]
-  apply B.edgeRoot_eq_of_label
-  rw [he,indexWord_label B.system (by rw [Int.odd_iff]; omega)]
-  congr 1
-  push_cast
-  ring
-
- theorem negPrefixRoot_eq_edgeRoot {x y w : planeSubgroup M cs P}
-    (hxy : BruhatStep B.system x y) {n : ℕ} (hn : n<B.system.length w)
-    (he : y*x⁻¹=indexWord B.system (-(2*(n:ℤ)+1))) :
-    B.negPrefixRoot w n = B.edgeRoot hxy := by
-  rw [negPrefixRoot,dite_eq_left hn]
-  apply B.edgeRoot_eq_of_label
-  rw [he,indexWord_label B.system (by rw [Int.odd_iff]; omega)]
-  congr 1
-  push_cast
-  ring
-
-end
-end KLInvariance.TitsSpace.PlaneBasis
-
-end
-
-
-section
-
-
-namespace KLInvariance.SignedDihedral
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
- def Before (a b : ℤ) : Prop :=
-  (0 < a ∧ 0 < b ∧ a < b) ∨ (a < 0 ∧ b < 0 ∧ a < b) ∨ (0 < a ∧ b < 0)
-
- theorem before_same_source_pos {i j z : ℤ} (hij : |i| < |j|)
-    (hiz : |i| < |z|) (hj : 0 < j) :
-    Before (i+j) (i+z) ↔ z < 0 ∨ j < z := by
-  rw [abs_of_pos hj, abs_lt] at hij
-  obtain hz | hz := le_total 0 z
-  · rw [abs_of_nonneg hz, abs_lt] at hiz
-    dsimp [Before]
-    omega
-  · rw [abs_of_nonpos hz, abs_lt] at hiz
-    dsimp [Before]
-    omega
-
- theorem before_same_source_neg {i j z : ℤ} (hij : |i| < |j|)
-    (hiz : |i| < |z|) (hj : j < 0) :
-    Before (i+j) (i+z) ↔ z < 0 ∧ j < z := by
-  rw [abs_of_neg hj, abs_lt] at hij
-  obtain hz | hz := le_total 0 z
-  · rw [abs_of_nonneg hz, abs_lt] at hiz
-    dsimp [Before]
-    omega
-  · rw [abs_of_nonpos hz, abs_lt] at hiz
-    dsimp [Before]
-    omega
-
- theorem before_next_source {i j z : ℤ} (hij : |i| < |j|)
-    (hjz : |j| < |z|) :
-    Before (i+j) (j+z) ↔ 0 < j := by
-  obtain hj | hj := le_total 0 j
-  · rw [abs_of_nonneg hj, abs_lt] at hij
-    obtain hz | hz := le_total 0 z
-    · rw [abs_of_nonneg hj, abs_of_nonneg hz] at hjz
-      dsimp [Before]
-      omega
-    · rw [abs_of_nonneg hj, abs_of_nonpos hz] at hjz
-      dsimp [Before]
-      omega
-  · rw [abs_of_nonpos hj, abs_lt] at hij
-    obtain hz | hz := le_total 0 z
-    · rw [abs_of_nonpos hj, abs_of_nonneg hz] at hjz
-      dsimp [Before]
-      omega
-    · rw [abs_of_nonpos hj, abs_of_nonpos hz] at hjz
-      dsimp [Before]
-      omega
-
- theorem midpoint_mod {m a : ℤ} (_hm : 0 < m) (ha : -2*m < a) (ha' : a < 2*m) :
-    a % (2*m) = if 0 ≤ a then a else 2*m+a := by
-  split_ifs with h
-  · exact Int.emod_eq_of_lt h ha'
-  · have he : a % (2*m) = (a+2*m) % (2*m) := by simp
-    rw [he, Int.emod_eq_of_lt (by omega) (by omega)]
-    omega
-
-/-- For two outgoing edges at the same vertex, reduction modulo the period
-has no wrap-around exception in the midpoint comparison table. -/
- theorem finite_before_same {m i j z : ℤ} (hm : 0 < m)
-    (hi : |i| < |j|) (hz : |i| < |z|)
-    (hjM : -m < j ∧ j ≤ m) (hzM : -m < z ∧ z ≤ m) :
-    (i+j) % (2*m) < (i+z) % (2*m) ↔ Before (i+j) (i+z) := by
-  have hjabs : |j| ≤ m := abs_le.mpr (by omega)
-  have hzabs : |z| ≤ m := abs_le.mpr (by omega)
-  have hiabs := abs_lt.mp (hi.trans_le hjabs)
-  rw [midpoint_mod hm (by omega) (by omega), midpoint_mod hm (by omega) (by omega)]
-  obtain hjs | hjs := le_total 0 j
-  · rw [abs_of_nonneg hjs, abs_lt] at hi
-    obtain hzs | hzs := le_total 0 z
-    · rw [abs_of_nonneg hzs, abs_lt] at hz
-      dsimp [Before]
-      split_ifs <;> omega
-    · rw [abs_of_nonpos hzs, abs_lt] at hz
-      dsimp [Before]
-      split_ifs <;> omega
-  · rw [abs_of_nonpos hjs, abs_lt] at hi
-    obtain hzs | hzs := le_total 0 z
-    · rw [abs_of_nonneg hzs, abs_lt] at hz
-      dsimp [Before]
-      split_ifs <;> omega
-    · rw [abs_of_nonpos hzs, abs_lt] at hz
-      dsimp [Before]
-      split_ifs <;> omega
-
- theorem finite_before_next {m i j z : ℤ} (hm : 0 < m)
-    (hi : |i| < |j|) (hz : |j| < |z|)
-    (hzM : -m < z ∧ z ≤ m) :
-    (i+j) % (2*m) < (j+z) % (2*m) ↔ Before (i+j) (j+z) := by
-  have hzabs : |z| ≤ m := abs_le.mpr (by omega)
-  have hjabs := abs_lt.mp (hz.trans_le hzabs)
-  have hiabs := abs_lt.mp ((hi.trans hz).trans_le hzabs)
-  rw [midpoint_mod hm (by omega) (by omega), midpoint_mod hm (by omega) (by omega)]
-  obtain hjs | hjs := le_total 0 j
-  · rw [abs_of_nonneg hjs, abs_lt] at hi
-    obtain hzs | hzs := le_total 0 z
-    · rw [abs_of_nonneg hjs, abs_of_nonneg hzs] at hz
-      dsimp [Before]
-      split_ifs <;> omega
-    · rw [abs_of_nonneg hjs, abs_of_nonpos hzs] at hz
-      dsimp [Before]
-      split_ifs <;> omega
-  · rw [abs_of_nonpos hjs, abs_lt] at hi
-    obtain hzs | hzs := le_total 0 z
-    · rw [abs_of_nonpos hjs, abs_of_nonneg hzs] at hz
-      dsimp [Before]
-      split_ifs <;> omega
-    · rw [abs_of_nonpos hjs, abs_of_nonpos hzs] at hz
-      dsimp [Before]
-      split_ifs <;> omega
-
- theorem eligible_ranks_card (L k : ℕ) :
-    ((Finset.range (L+2*k+1)).filter (fun r => L < r ∧ Odd (L+r))).card = k := by
-  classical
-  have hc : (Finset.range k).card =
-      ((Finset.range (L+2*k+1)).filter (fun r => L < r ∧ Odd (L+r))).card := by
-    apply Finset.card_bij (fun a _ => L+1+2*a)
-    · intro a ha
-      simp only [Finset.mem_range, Finset.mem_filter, Nat.odd_iff] at ha ⊢
-      omega
-    · intro a ha b hb he
-      omega
-    · intro b hb
-      simp only [Finset.mem_range, Finset.mem_filter, Nat.odd_iff] at hb
-      refine ⟨(b-L-1)/2, ?_, ?_⟩
-      · simp only [Finset.mem_range]
-        omega
-      · omega
-  simpa only [Finset.card_range] using hc.symm
-
- noncomputable def arrowAfter (a label z : ℤ) : ℕ :=
-  if |a| < |z| ∧ Odd (a+z) ∧ Before label (a+z) then 1 else 0
-
-/-- Double internal ranks and the singleton top, with no singleton bottom
-term because upward arrows never end at the bottom. -/
- noncomputable def countAfter (a label c : ℤ) : ℕ :=
-  (∑ r ∈ Finset.range c.natAbs, (arrowAfter a label r + arrowAfter a label (-r))) +
-    arrowAfter a label c
-
- theorem arrowAfter_of_not_lt {a label z : ℤ} (h : ¬ |a| < |z|) :
-    arrowAfter a label z = 0 := by simp [arrowAfter,h]
-
- theorem arrowAfter_self (i j : ℤ) : arrowAfter i (i+j) j = 0 := by
-  have hb : ¬ Before (i+j) (i+j) := by
-    dsimp [Before]
-    omega
-  simp only [arrowAfter, hb, and_false, ite_false]
-
-end
-end KLInvariance.SignedDihedral
-
-end
-
-
-section
-
-namespace KLInvariance.Dihedral
-variable {W : Type*} [Group W] {M : CoxeterMatrix Bool}
-
- theorem length_indexWord_unbounded (cs : CoxeterSystem M W)
-    (hu : ∀ n : ℕ, ∃ w, n ≤ cs.length w) (i : ℤ) :
-    cs.length (indexWord cs i) = i.natAbs := by
-  obtain ⟨w,hw⟩ := hu i.natAbs
-  exact length_indexWord cs hw
-
-end KLInvariance.Dihedral
-
-namespace KLInvariance.TitsSpace.PlaneBasis
-open KLInvariance.Dihedral
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-noncomputable section
-
- theorem edgeRoot_inverted_iff {x y : planeSubgroup M cs P}
-    (hxy : BruhatStep B.system x y) (w : planeSubgroup M cs P) :
-    rootInverted (w:W) (B.edgeRoot hxy) ↔ B.system.length ((y*x⁻¹)*w)<B.system.length w := by
-  have hh := B.inversion_iff (t := y*x⁻¹)
-    (graphRoot_represents ((B.bruhatStep_iff x y).mpr hxy))
-    (B.edgeRoot_plane hxy) (B.edgeRoot hxy).property.2 w
-  change rootInverted (w:W) (B.edgeRoot hxy) ↔ _ at hh
-  rw [hh]
-  exact and_iff_right hxy.label_reflection
-
- theorem edgeRoot_positive_lt_negative
-    (hu : ∀ n : ℕ, ∃ w, n ≤ B.system.length w)
-    {x y z t : planeSubgroup M cs P}
-    (hxy : BruhatStep B.system x y) (hzt : BruhatStep B.system z t)
-    (k l : ℕ)
-    (hx : y*x⁻¹=indexWord B.system (2*(k:ℤ)+1))
-    (hz : t*z⁻¹=indexWord B.system (-(2*(l:ℤ)+1))) :
-    geometricSlope (B.edgeRoot hxy) < geometricSlope (B.edgeRoot hzt) := by
-  let w := indexWord B.system ((k+1:ℕ):ℤ)
-  have hi : rootInverted (w:W) (B.edgeRoot hxy) := by
-    rw [B.edgeRoot_inverted_iff,hx]
-    unfold w
-    rw [indexWord_reflection_mul B.system (by rw [Int.odd_iff]; omega),
-      length_indexWord_unbounded B.system hu,length_indexWord_unbounded B.system hu]
-    have he : 2*(k:ℤ)+1-((k+1:ℕ):ℤ)=(k:ℤ) := by omega
-    rw [he]
-    simp only [Int.natAbs_natCast]
-    omega
-  have hn : ¬rootInverted (w:W) (B.edgeRoot hzt) := by
-    rw [B.edgeRoot_inverted_iff,hz]
-    unfold w
-    rw [indexWord_reflection_mul B.system (by rw [Int.odd_iff]; omega),
-      length_indexWord_unbounded B.system hu,length_indexWord_unbounded B.system hu]
-    have he : -(2*(l:ℤ)+1)-((k+1:ℕ):ℤ) = -((2*l+k+2:ℕ):ℤ) := by omega
-    rw [he]
-    simp only [Int.natAbs_neg,Int.natAbs_natCast]
-    omega
-  have hd : B.system.IsLeftDescent w false := by
-    obtain ⟨w',hw'⟩ := hu (k+1)
-    change B.system.IsLeftDescent (indexWord B.system ((k+1:ℕ):ℤ)) false
-    rw [indexWord_nat]
-    exact leftDescent_alt B.system false (by omega) hw'
-  have hm : ∀ z ∈ planeSubgroup M cs P, cs.length (1:W) ≤ cs.length (z*1) :=
-    fun z _ => by simp
-  have hl : rootInverted (w:W) B.left := by
-    have hh := (B.simpleRoot_inverted_coset_iff hm w false).mpr hd
-    simpa only [mul_one,simpleRoot,Bool.false_eq_true,↓reduceIte] using hh
-  exact B.slope_lt_of_inverted_left (w:W) hl _ _ (B.edgeRoot_plane hxy)
-    (B.edgeRoot_plane hzt) hi hn
-
-/-- The two angular rays of an infinite canonical dihedral subsystem, with
-actual ambient labels, have precisely the source's signed-midpoint order. -/
- theorem edgeRoot_infinite_order
-    (hu : ∀ n : ℕ, ∃ w, n ≤ B.system.length w)
-    {x y z t : planeSubgroup M cs P}
-    (hxy : BruhatStep B.system x y) (hzt : BruhatStep B.system z t)
-    {a b : ℤ} (ha : Odd a) (hb : Odd b)
-    (hx : y*x⁻¹=indexWord B.system a) (hz : t*z⁻¹=indexWord B.system b) :
-    geometricSlope (B.edgeRoot hxy) < geometricSlope (B.edgeRoot hzt) ↔
-      SignedDihedral.Before a b := by
-  have repr (q : ℤ) (hq : Odd q) :
-      ∃ k : ℕ, q=2*(k:ℤ)+1 ∨ q= -(2*(k:ℤ)+1) := by
-    have habs := Int.natCast_natAbs q
-    have ho : q.natAbs % 2=1 := by
-      rw [Int.odd_iff] at hq
-      rcases le_total 0 q with h|h
-      · rw [abs_of_nonneg h] at habs
-        omega
-      · rw [abs_of_nonpos h] at habs
-        omega
-    refine ⟨q.natAbs/2,?_⟩
-    rcases le_total 0 q with h|h
-    · left
-      rw [abs_of_nonneg h] at habs
-      omega
-    · right
-      rw [abs_of_nonpos h] at habs
-      omega
-  obtain ⟨k,hk|hk⟩ := repr a ha <;> obtain ⟨l,hl|hl⟩ := repr b hb
-  · obtain ⟨w,hw⟩ := hu (max k l+1)
-    have hk' : k<B.system.length w := by omega
-    have hl' : l<B.system.length w := by omega
-    rw [hk] at hx
-    rw [hl] at hz
-    rw [← B.prefixRoot_eq_edgeRoot hxy hk' hx,← B.prefixRoot_eq_edgeRoot hzt hl' hz]
-    have hs := B.prefixRoot_strictMono w
-    have he : geometricSlope (B.prefixRoot w k) < geometricSlope (B.prefixRoot w l) ↔ k<l :=
-      hs.lt_iff_lt hk' hl'
-    rw [he,hk,hl]
-    unfold SignedDihedral.Before
-    omega
-  · rw [hk] at hx
-    rw [hl] at hz
-    have hh := B.edgeRoot_positive_lt_negative hu hxy hzt k l hx hz
-    have hbefore : SignedDihedral.Before a b := by
-      rw [hk,hl]
-      unfold SignedDihedral.Before
-      omega
-    exact iff_of_true hh hbefore
-  · rw [hk] at hx
-    rw [hl] at hz
-    have hh := B.edgeRoot_positive_lt_negative hu hzt hxy l k hz hx
-    have hn := not_lt_of_gt hh
-    have hbefore : ¬SignedDihedral.Before a b := by
-      rw [hk,hl]
-      unfold SignedDihedral.Before
-      omega
-    exact iff_of_false hn hbefore
-  · obtain ⟨w,hw⟩ := hu (max k l+1)
-    have hk' : k<B.system.length w := by omega
-    have hl' : l<B.system.length w := by omega
-    rw [hk] at hx
-    rw [hl] at hz
-    rw [← B.negPrefixRoot_eq_edgeRoot hxy hk' hx,← B.negPrefixRoot_eq_edgeRoot hzt hl' hz]
-    have hs := B.negPrefixRoot_strictAnti w
-    have he : geometricSlope (B.negPrefixRoot w k) < geometricSlope (B.negPrefixRoot w l) ↔ l<k := by
-      constructor
-      · intro h
-        by_contra hn
-        rcases lt_or_eq_of_le (le_of_not_gt hn) with hkl | rfl
-        · exact (not_lt_of_gt (hs hk' hl' hkl)) h
-        · exact (lt_irrefl _) h
-      · exact hs hl' hk'
-    rw [he,hk,hl]
-    unfold SignedDihedral.Before
-    omega
-
-end
-end KLInvariance.TitsSpace.PlaneBasis
-
-end
-
-
-section
-
-namespace KLInvariance.TitsSpace.PlaneBasis
-open KLInvariance.Dihedral
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-noncomputable section
-
- theorem max_length_pos (w : planeSubgroup M cs P)
-    (hmax : ∀ z, B.system.length z ≤ B.system.length w) : 0 < B.system.length w := by
-  have h := hmax (B.system.simple false)
-  rw [B.system.length_simple] at h
-  omega
-
- theorem odd_residue_repr {N : ℕ} (hN : 0<N) {a : ℤ} (ha : Odd a) :
-    ∃ k : ℕ, k<N ∧ a % (2*(N:ℤ)) = 2*(k:ℤ)+1 := by
-  have hl := Int.emod_nonneg a (by omega : (2*(N:ℤ))≠0)
-  have hu := Int.emod_lt_of_pos a (by omega : 0<2*(N:ℤ))
-  have hp : (a%(2*(N:ℤ)))%2=1 := by
-    rw [Int.emod_emod_of_dvd a (by exact dvd_mul_right 2 (N:ℤ))]
-    exact Int.odd_iff.mp ha
-  refine ⟨(a%(2*(N:ℤ))).toNat/2, ?_, ?_⟩ <;> omega
-
-/-- The actual ambient angular order in a finite canonical plane subsystem
-is the increasing residue order on odd doubled midpoints. -/
- theorem edgeRoot_finite_order
-    (w : planeSubgroup M cs P) (hmax : ∀ z, B.system.length z ≤ B.system.length w)
-    {x y z t : planeSubgroup M cs P}
-    (hxy : BruhatStep B.system x y) (hzt : BruhatStep B.system z t)
-    {a b : ℤ} (ha : Odd a) (hb : Odd b)
-    (hx : y*x⁻¹=indexWord B.system a) (hz : t*z⁻¹=indexWord B.system b) :
-    geometricSlope (B.edgeRoot hxy) < geometricSlope (B.edgeRoot hzt) ↔
-      a % (2*(B.system.length w:ℤ)) < b % (2*(B.system.length w:ℤ)) := by
-  obtain ⟨k,hk,he⟩ := odd_residue_repr (B.max_length_pos w hmax) ha
-  obtain ⟨l,hl,hf⟩ := odd_residue_repr (B.max_length_pos w hmax) hb
-  have hp := rotation_pow_max B.system w hmax
-  rw [← indexWord_mod_period B.system hp a, he] at hx
-  rw [← indexWord_mod_period B.system hp b, hf] at hz
-  rw [← B.prefixRoot_eq_edgeRoot hxy hk hx,← B.prefixRoot_eq_edgeRoot hzt hl hz]
-  rw [(B.prefixRoot_strictMono w).lt_iff_lt hk hl,he,hf]
-  omega
-
-end
-end KLInvariance.TitsSpace.PlaneBasis
-
-end
-
-
-section
-
-namespace KLInvariance.Dihedral
-variable {W : Type*} [Group W] {M : CoxeterMatrix Bool}
-noncomputable section
-local instance : DecidableEq W := Classical.decEq W
-
- theorem indexWord_pos_ne_neg (cs : CoxeterSystem M W) {n : ℕ} {w : W}
-    (hn : 0<n) (hw : n<cs.length w) : indexWord cs n ≠ indexWord cs (-(n:ℤ)) := by
-  have hp := leftDescent_alt cs false hn hw.le
-  have hm := leftDescent_alt cs true hn hw.le
-  rw [← indexWord_nat] at hp
-  rw [← indexWord_neg_nat] at hm
-  intro he
-  rw [← he] at hm
-  have hl := length_indexWord_nat cs hw.le
-  obtain ⟨i,_,hi⟩ := unique_leftDescent cs (w := indexWord cs n) (z := w)
-    (by rw [hl]; exact hn) (by rw [hl]; exact hw)
-  have hf := hi false hp
-  have ht := hi true hm
-  simp_all
-
- theorem lower_length_fiber (cs : CoxeterSystem M W) (c : W) {n : ℕ}
-    (hc : n<cs.length c) :
-    (lowerFinset cs c).filter (fun z => cs.length z=n) =
-      {indexWord cs n,indexWord cs (-(n:ℤ))} := by
-  classical
-  ext z
-  simp only [Finset.mem_filter,mem_lowerFinset,Finset.mem_insert,Finset.mem_singleton]
-  constructor
-  · intro hz
-    obtain ⟨i,hi⟩ := exists_alt cs z
-    rw [hz.2] at hi
-    cases i
-    · exact Or.inl (hi.trans (indexWord_nat cs n).symm)
-    · exact Or.inr (hi.trans (indexWord_neg_nat cs n).symm)
-  · rintro (rfl|rfl)
-    · have hl := length_indexWord_nat cs hc.le
-      exact ⟨shorter_bruhat cs (by rw [hl]; exact hc),hl⟩
-    · have hl := length_indexWord_neg_nat cs hc.le
-      exact ⟨shorter_bruhat cs (by rw [hl]; exact hc),hl⟩
-
- theorem lower_zero_fiber (cs : CoxeterSystem M W) (c : W) :
-    (lowerFinset cs c).filter (fun z => cs.length z=0) = {1} := by
-  classical
-  ext z
-  simp only [Finset.mem_filter,mem_lowerFinset,Finset.mem_singleton]
-  constructor
-  · exact fun hz => cs.length_eq_zero_iff.mp hz.2
-  · rintro rfl
-    exact ⟨one_bruhat cs c,cs.length_one⟩
-
- theorem lower_top_fiber (cs : CoxeterSystem M W) (c : W) :
-    (lowerFinset cs c).filter (fun z => cs.length z=cs.length c) = {c} := by
-  classical
-  ext z
-  simp only [Finset.mem_filter,mem_lowerFinset,Finset.mem_singleton]
-  constructor
-  · intro hz
-    by_contra hne
-    have hl := length_lt_of_bruhat_ne cs hz.1 hne
-    omega
-  · rintro rfl
-    exact ⟨bruhat_refl cs _,rfl⟩
-
-/-- Actual finite dihedral lower ideals have two internal alternating rays
-and a singleton top. The doubled identity contributes zero. -/
- theorem sum_lower_signed (cs : CoxeterSystem M W) (c : W)
-    {A : Type*} [AddCommMonoid A] (f : W → A) (h1 : f 1=0) :
-    ∑ z ∈ lowerFinset cs c, f z =
-      (∑ n ∈ Finset.range (cs.length c),
-        (f (indexWord cs n)+f (indexWord cs (-(n:ℤ))))) + f c := by
-  classical
-  have hmap : ∀ z ∈ lowerFinset cs c, cs.length z ∈ Finset.range (cs.length c+1) := by
-    intro z hz
-    have hl := length_le_of_bruhat cs ((mem_lowerFinset cs c z).mp hz)
-    simp only [Finset.mem_range]
-    omega
-  rw [← Finset.sum_fiberwise_of_maps_to hmap f,Finset.sum_range_succ,lower_top_fiber,
-    Finset.sum_singleton]
-  congr 1
-  apply Finset.sum_congr rfl
-  intro n hn
-  have hc := Finset.mem_range.mp hn
-  by_cases hz : n=0
-  · subst n
-    rw [lower_zero_fiber]
-    simp only [Finset.sum_singleton,indexWord_nat,indexWord_neg_nat,alt_zero,
-      cs.wordProd_nil,h1,add_zero]
-  · rw [lower_length_fiber cs c hc,
-      Finset.sum_pair (indexWord_pos_ne_neg cs (by omega) hc)]
-
-end
-end KLInvariance.Dihedral
-
-end
-
-
-section
-
-namespace KLInvariance.Dihedral
-variable {W : Type*} [Group W] {M : CoxeterMatrix Bool}
-noncomputable section
-
- def GoodIndex (cs : CoxeterSystem M W) (i : ℤ) : Prop :=
-  ∀ w, (∀ z, cs.length z ≤ cs.length w) → -(cs.length w:ℤ) < i ∧ i ≤ (cs.length w:ℤ)
-
- theorem maximal_or_unbounded (cs : CoxeterSystem M W) :
-    (∃ w, ∀ z, cs.length z ≤ cs.length w) ∨ (∀ n : ℕ, ∃ w, n≤cs.length w) := by
-  by_cases hb : BddAbove (Set.range cs.length)
-  · have hf : (Set.range cs.length).Finite := hb.finite
-    obtain ⟨b,hb⟩ := (Set.range_nonempty cs.length).csSup_mem hf
-    left
-    refine ⟨b,fun z => ?_⟩
-    rw [hb]
-    exact le_csSup hf.bddAbove (Set.mem_range_self z)
-  · right
-    intro n
-    by_contra h
-    push Not at h
-    exact hb ⟨n,by rintro m ⟨z,rfl⟩; exact (h z).le⟩
-
- theorem max_length_positive (cs : CoxeterSystem M W) {w : W}
-    (hmax : ∀ z, cs.length z ≤ cs.length w) : 0<cs.length w := by
-  have h := hmax (cs.simple false)
-  rw [cs.length_simple] at h
-  omega
-
- theorem maximal_eq (cs : CoxeterSystem M W) {w z : W}
-    (hmax : ∀ x, cs.length x≤cs.length w) (hz : cs.length z=cs.length w) : z=w := by
-  have htop := bruhat_top_of_all_descents cs w (fun i => by
-    have h := hmax (cs.simple i*w)
-    have h' := cs.length_simple_mul w i
-    omega) z
-  by_contra hne
-  have hl := length_lt_of_bruhat_ne cs htop hne
-  omega
-
- theorem goodIndex_nat (cs : CoxeterSystem M W) {n : ℕ} {c : W}
-    (hc : n≤cs.length c) : GoodIndex cs (n:ℤ) := by
-  intro w hw
-  have h := hw c
-  have hp := max_length_positive cs hw
-  omega
-
- theorem goodIndex_neg_nat (cs : CoxeterSystem M W) {n : ℕ} {c : W}
-    (hc : n<cs.length c) : GoodIndex cs (-(n:ℤ)) := by
-  intro w hw
-  have h := hw c
-  omega
-
- theorem exists_good_index (cs : CoxeterSystem M W) (z : W) :
-    ∃ i : ℤ, GoodIndex cs i ∧ |i|=(cs.length z:ℤ) ∧ indexWord cs i=z := by
-  obtain ⟨i,hi,he⟩ := exists_indexWord cs z
-  by_cases hgood : GoodIndex cs i
-  · exact ⟨i,hgood,hi,he⟩
-  · have hbad : ∃ w, (∀ x, cs.length x≤cs.length w) ∧
-        ¬(-(cs.length w:ℤ) < i ∧ i ≤ (cs.length w:ℤ)) := by
-      simpa only [GoodIndex,not_forall,exists_prop] using hgood
-    obtain ⟨w,hw,hbad⟩ := hbad
-    have hl := hw z
-    have habs := abs_le.mp (show |i|≤(cs.length w:ℤ) by rw [hi]; exact_mod_cast hl)
-    have hi' : i= -(cs.length w:ℤ) := by omega
-    have hwz : cs.length w=cs.length z := by rw [hi'] at hi; simpa using hi
-    refine ⟨cs.length z,goodIndex_nat cs (c := z) le_rfl,by simp,?_⟩
-    have hp := indexWord_period cs (rotation_pow_max cs w hw) (-(cs.length w:ℤ)) 1
-    rw [show -(cs.length w:ℤ)+2*(cs.length w:ℤ)*1=(cs.length w:ℤ) by ring] at hp
-    rw [← hwz,hp,← hi',he]
-
-end
-end KLInvariance.Dihedral
-
-end
-
-
-section
-
-namespace KLInvariance.SignedDihedral
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
- theorem odd_add_natAbs (i : ℤ) (r : ℕ) : Odd (i+r) ↔ Odd (i.natAbs+r) := by
-  rw [Int.odd_iff, Nat.odd_iff]
-  have hi := Int.natCast_natAbs i
-  obtain h | h := le_total 0 i
-  · rw [abs_of_nonneg h] at hi
-    omega
-  · rw [abs_of_nonpos h] at hi
-    omega
-
- theorem odd_sub_natAbs (i : ℤ) (r : ℕ) : Odd (i-r) ↔ Odd (i.natAbs+r) := by
-  rw [← odd_add_natAbs, Int.odd_iff, Int.odd_iff]
-  omega
-
- theorem arrowAfter_pos {i j : ℤ} (hij : |i| < |j|) (r : ℕ) :
-    arrowAfter i (i+j) r =
-      if i.natAbs < r ∧ Odd (i.natAbs+r) ∧ 0 < j ∧ j.natAbs < r then 1 else 0 := by
-  have hir : |i| < |(r:ℤ)| ↔ i.natAbs < r := by
-    rw [Int.abs_natCast, ← Int.natCast_natAbs i, Nat.cast_lt]
-  by_cases hl : i.natAbs < r
-  · have hb : Before (i+j) (i+r) ↔ 0 < j ∧ j.natAbs < r := by
-      by_cases hj : 0 < j
-      · rw [before_same_source_pos hij (hir.mpr hl) hj]
-        have he : (j.natAbs : ℤ) = j := (Int.natCast_natAbs j).trans (abs_of_pos hj)
-        omega
-      · have hj' : j < 0 := by
-          have := abs_nonneg i
-          rw [abs_of_nonpos (by omega : j ≤ 0)] at hij
-          omega
-        rw [before_same_source_neg hij (hir.mpr hl) hj']
-        omega
-    simp only [arrowAfter, hir, odd_add_natAbs, hb]
-  · simp only [arrowAfter, hir, hl, false_and, ite_false]
-
- theorem arrowAfter_neg {i j : ℤ} (hij : |i| < |j|) (r : ℕ) :
-    arrowAfter i (i+j) (-r) =
-      if i.natAbs < r ∧ Odd (i.natAbs+r) ∧ (0 < j ∨ r < j.natAbs) then 1 else 0 := by
-  have hir : |i| < |-(r:ℤ)| ↔ i.natAbs < r := by
-    rw [abs_neg, Int.abs_natCast, ← Int.natCast_natAbs i, Nat.cast_lt]
-  by_cases hl : i.natAbs < r
-  · have hb : Before (i+j) (i+ -r) ↔ 0 < j ∨ r < j.natAbs := by
-      by_cases hj : 0 < j
-      · rw [before_same_source_pos hij (hir.mpr hl) hj]
-        omega
-      · have hj' : j < 0 := by
-          have := abs_nonneg i
-          rw [abs_of_nonpos (by omega : j ≤ 0)] at hij
-          omega
-        rw [before_same_source_neg hij (hir.mpr hl) hj']
-        have he : (j.natAbs : ℤ) = -j := (Int.natCast_natAbs j).trans (abs_of_neg hj')
-        omega
-    simp only [arrowAfter, hir, show i+ -(r:ℤ)=i-r by ring, odd_sub_natAbs]
-    rw [show i-(r:ℤ) = i+ -r by ring, hb]
-    split_ifs <;> rfl
-  · simp only [arrowAfter, hir, hl, false_and, ite_false]
-
- theorem arrowAfter_next {i j : ℤ} (hij : |i| < |j|) (z : ℤ) :
-    arrowAfter j (i+j) z =
-      if j.natAbs < z.natAbs ∧ Odd (j.natAbs+z.natAbs) ∧ 0 < j then 1 else 0 := by
-  have hlt : |j| < |z| ↔ j.natAbs < z.natAbs := by
-    rw [← Int.natCast_natAbs j, ← Int.natCast_natAbs z, Nat.cast_lt]
-  have ho : Odd (j+z) ↔ Odd (j.natAbs+z.natAbs) := by
-    have hj := Int.natCast_natAbs j
-    have hz := Int.natCast_natAbs z
-    rw [Int.odd_iff,Nat.odd_iff]
-    rcases le_total 0 j with hj'|hj' <;> rcases le_total 0 z with hz'|hz'
-    · rw [abs_of_nonneg hj'] at hj
-      rw [abs_of_nonneg hz'] at hz
-      omega
-    · rw [abs_of_nonneg hj'] at hj
-      rw [abs_of_nonpos hz'] at hz
-      omega
-    · rw [abs_of_nonpos hj'] at hj
-      rw [abs_of_nonneg hz'] at hz
-      omega
-    · rw [abs_of_nonpos hj'] at hj
-      rw [abs_of_nonpos hz'] at hz
-      omega
-  by_cases h : |j| < |z|
-  · simp only [arrowAfter, before_next_source hij h, ho, hlt]
-  · have h' := hlt.not.mp h
-    simp only [arrowAfter,h,h',false_and,ite_false]
-
- theorem countAfter_base {i j : ℤ} (hij : |i| < |j|)
-    (ho : Odd (i.natAbs+j.natAbs)) :
-    countAfter i (i+j) j = (j.natAbs-i.natAbs-1)/2 := by
-  unfold countAfter
-  rw [arrowAfter_self,add_zero]
-  have hs : ∀ r ∈ Finset.range j.natAbs,
-      arrowAfter i (i+j) r + arrowAfter i (i+j) (-r) =
-        if i.natAbs < r ∧ Odd (i.natAbs+r) then 1 else 0 := by
-    intro r hr
-    have hr' := Finset.mem_range.mp hr
-    rw [arrowAfter_pos hij,arrowAfter_neg hij]
-    have hn : ¬ j.natAbs < r := by omega
-    simp only [hn,and_false,ite_false,zero_add,hr',or_true,and_true]
-  rw [Finset.sum_congr rfl hs, ← Finset.sum_filter, Finset.sum_const,
-    nsmul_eq_mul, mul_one]
-  have hh : i.natAbs < j.natAbs := by
-    rw [← Nat.cast_lt (α := ℤ),Int.natCast_natAbs,Int.natCast_natAbs]
-    exact hij
-  have he : j.natAbs = i.natAbs+2*((j.natAbs-i.natAbs-1)/2)+1 := by
-    rw [Nat.odd_iff] at ho
-    omega
-  conv_lhs => rw [he]
-  exact eligible_ranks_card _ _
-
-/-- A double internal level together with the singleton top contributes one
-outgoing arrow per rank, independently of top parity. -/
- theorem total_outgoing_count (L C : ℕ) (h : L ≤ C) :
-    (∑ r ∈ Finset.range C, if L < r ∧ Odd (L+r) then 2 else 0) +
-      (if L < C ∧ Odd (L+C) then 1 else 0) = C-L := by
-  induction C, h using Nat.le_induction with
-  | base =>
-    have hz : ∀ r ∈ Finset.range L, ¬ L < r := by
-      intro r hr
-      have := Finset.mem_range.mp hr
-      omega
-    have hs : (∑ r ∈ Finset.range L, if L < r ∧ Odd (L+r) then 2 else 0) = 0 := by
-      apply Finset.sum_eq_zero
-      intro r hr
-      simp only [hz r hr, false_and, ite_false]
-    simp only [hs, lt_self_iff_false, false_and, ite_false, add_zero, Nat.sub_self]
-  | succ C h ih =>
-    rw [Finset.sum_range_succ]
-    by_cases hc : L<C
-    · have hc' : L<C+1 := by omega
-      by_cases he : Odd (L+C)
-      · have he' : ¬Odd (L+(C+1)) := by rw [Nat.odd_iff] at he ⊢; omega
-        simp only [hc,he,and_self,ite_true] at ih
-        simp only [hc,he,hc',he',and_true,and_false,ite_true,ite_false,add_zero]
-        omega
-      · have he' : Odd (L+(C+1)) := by rw [Nat.odd_iff] at he ⊢; omega
-        simp only [he,and_false,ite_false,add_zero] at ih
-        simp only [hc,he,hc',he',and_true,and_false,ite_true,ite_false,add_zero]
-        omega
-    · have heq : C=L := by omega
-      subst C
-      have he : ¬Odd (L+L) := by rw [Nat.odd_iff]; omega
-      have he' : Odd (L+(L+1)) := by rw [Nat.odd_iff]; omega
-      simp only [lt_self_iff_false,false_and,ite_false,add_zero] at ih
-      simp only [he,he',and_false,and_true,ite_false,Nat.lt_succ_self,ite_true,add_zero]
-      omega
-
-end
-end KLInvariance.SignedDihedral
-
-end
-
-
-section
-
-namespace KLInvariance.SignedDihedral
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
- theorem odd_add_abs (i z : ℤ) : Odd (i+z) ↔ Odd (i.natAbs+z.natAbs) := by
-  have hi := Int.natCast_natAbs i
-  have hz := Int.natCast_natAbs z
-  rw [Int.odd_iff,Nat.odd_iff]
-  rcases le_total 0 i with hi'|hi' <;> rcases le_total 0 z with hz'|hz'
-  · rw [abs_of_nonneg hi'] at hi
-    rw [abs_of_nonneg hz'] at hz
-    omega
-  · rw [abs_of_nonneg hi'] at hi
-    rw [abs_of_nonpos hz'] at hz
-    omega
-  · rw [abs_of_nonpos hi'] at hi
-    rw [abs_of_nonneg hz'] at hz
-    omega
-  · rw [abs_of_nonpos hi'] at hi
-    rw [abs_of_nonpos hz'] at hz
-    omega
-
- theorem arrowAfter_far {i j z : ℤ} (hij : |i| < |j|) (hjz : |j| < |z|) :
-    arrowAfter i (i+j) z = if Odd (i.natAbs+z.natAbs) ∧ 0 < j then 1 else 0 := by
-  have hiz := hij.trans hjz
-  have hb : Before (i+j) (i+z) ↔ 0 < j := by
-    by_cases hj : 0 < j
-    · rw [before_same_source_pos hij hiz hj]
-      rw [abs_of_pos hj] at hjz
-      rw [lt_abs] at hjz
-      omega
-    · have hj' : j < 0 := by
-        have := abs_nonneg i
-        rw [abs_of_nonpos (by omega : j ≤ 0)] at hij
-        omega
-      rw [before_same_source_neg hij hiz hj']
-      rw [abs_of_neg hj',lt_abs] at hjz
-      omega
-  simp only [arrowAfter,hiz,true_and,odd_add_abs,hb]
-
- def positiveCount (L M C : ℕ) : ℕ :=
-  (∑ r ∈ Finset.range C, if L < r ∧ Odd (L+r) then (if M < r then 2 else 1) else 0) +
-    (if M < C ∧ Odd (L+C) then 1 else 0)
-
- theorem positiveCount_formula (L M C : ℕ) (hLM : L < M) (ho : Odd (L+M))
-    (hMC : M ≤ C) : positiveCount L M C = (M-L-1)/2 + (C-M) := by
-  induction C, hMC using Nat.le_induction with
-  | base =>
-    have he : M = L+2*((M-L-1)/2)+1 := by
-      rw [Nat.odd_iff] at ho
-      omega
-    have hs : (∑ r ∈ Finset.range M,
-        if L < r ∧ Odd (L+r) then (if M < r then 2 else 1) else 0) =
-        (∑ r ∈ Finset.range M, if L < r ∧ Odd (L+r) then 1 else 0) := by
-      apply Finset.sum_congr rfl
-      intro r hr
-      have hn : ¬ M < r := by have := Finset.mem_range.mp hr; omega
-      simp only [hn,ite_false]
-    rw [positiveCount,hs,← Finset.sum_filter,Finset.sum_const,nsmul_eq_mul,mul_one]
-    simp only [lt_self_iff_false,false_and,ite_false,add_zero,Nat.sub_self]
-    conv_lhs => rw [he]
-    exact eligible_ranks_card _ _
-  | succ C hMC ih =>
-    rw [positiveCount,Finset.sum_range_succ]
-    unfold positiveCount at ih
-    by_cases h : M<C
-    · have h' : M<C+1 := by omega
-      have hl : L<C := by omega
-      by_cases hc : Odd (L+C)
-      · have hc' : ¬Odd (L+(C+1)) := by rw [Nat.odd_iff] at hc ⊢; omega
-        simp only [h,h',hl,hc,hc',and_true,and_false,ite_true,ite_false,add_zero] at ih ⊢
-        omega
-      · have hc' : Odd (L+(C+1)) := by rw [Nat.odd_iff] at hc ⊢; omega
-        simp only [h,h',hl,hc,hc',and_true,and_false,ite_true,ite_false,add_zero] at ih ⊢
-        omega
-    · have heq : C=M := by omega
-      subst C
-      have hc' : ¬Odd (L+(M+1)) := by rw [Nat.odd_iff] at ho ⊢; omega
-      simp only [lt_self_iff_false,false_and,ite_false,add_zero] at ih
-      simp only [hLM,ho,hc',and_self,and_false,ite_true,ite_false,lt_self_iff_false,add_zero]
-      omega
-
- theorem countAfter_pos_formula {i j c : ℤ} (hij : |i| < |j|)
-    (ho : Odd (i.natAbs+j.natAbs)) (hj : 0 < j)
-    (h : c=j ∨ |j| < |c|) :
-    countAfter i (i+j) c = (j.natAbs-i.natAbs-1)/2 + (c.natAbs-j.natAbs) := by
-  rcases h with rfl|h
-  · rw [countAfter_base hij ho,Nat.sub_self,add_zero]
-  have hMC : j.natAbs < c.natAbs := by
-    exact_mod_cast (show (j.natAbs:ℤ) < c.natAbs by simpa only [Int.natCast_natAbs] using h)
-  have hLM : i.natAbs < j.natAbs := by
-    exact_mod_cast (show (i.natAbs:ℤ) < j.natAbs by simpa only [Int.natCast_natAbs] using hij)
-  have hs : ∀ r ∈ Finset.range c.natAbs,
-      arrowAfter i (i+j) r + arrowAfter i (i+j) (-r) =
-      if i.natAbs < r ∧ Odd (i.natAbs+r) then (if j.natAbs < r then 2 else 1) else 0 := by
-    intro r _
-    rw [arrowAfter_pos hij,arrowAfter_neg hij]
-    simp only [hj,true_or,and_true,true_and]
-    by_cases h1 : i.natAbs < r ∧ Odd (i.natAbs+r)
-    · obtain ⟨h1,h2⟩ := h1
-      simp only [h1,h2,true_and,ite_true]
-      split_ifs <;> rfl
-    · have hh : ¬(i.natAbs < r ∧ Odd (i.natAbs+r) ∧ j.natAbs < r) := by tauto
-      simp only [h1,hh,ite_false,add_zero]
-  rw [countAfter,Finset.sum_congr rfl hs,arrowAfter_far hij h]
-  simp only [hj,and_true]
-  have heq : (if Odd (i.natAbs+c.natAbs) then 1 else 0) =
-      (if j.natAbs < c.natAbs ∧ Odd (i.natAbs+c.natAbs) then 1 else 0) := by
-    simp only [hMC,true_and]
-  rw [heq]
-  exact positiveCount_formula _ _ _ hLM ho hMC.le
-
- theorem countAfter_neg_formula {i j c : ℤ} (hij : |i| < |j|)
-    (ho : Odd (i.natAbs+j.natAbs)) (hj : j < 0)
-    (h : c=j ∨ |j| < |c|) :
-    countAfter i (i+j) c = (j.natAbs-i.natAbs-1)/2 := by
-  rcases h with rfl|h
-  · exact countAfter_base hij ho
-  have hMC : j.natAbs < c.natAbs := by
-    exact_mod_cast (show (j.natAbs:ℤ) < c.natAbs by simpa only [Int.natCast_natAbs] using h)
-  have hn : ¬0<j := by omega
-  have hs : (∑ r ∈ Finset.range c.natAbs, (arrowAfter i (i+j) r + arrowAfter i (i+j) (-r))) =
-      (∑ r ∈ Finset.range j.natAbs, (arrowAfter i (i+j) r + arrowAfter i (i+j) (-r))) := by
-    symm
-    apply Finset.sum_subset (Finset.range_mono hMC.le)
-    intro r _ hr
-    have hr' : ¬ r < j.natAbs := by simpa only [Finset.mem_range] using hr
-    rw [arrowAfter_pos hij,arrowAfter_neg hij]
-    simp only [hn,hr',false_or,and_false,false_and,ite_false,add_zero]
-  rw [countAfter,hs,arrowAfter_far hij h]
-  simp only [hn,and_false,ite_false,add_zero]
-  have hh := countAfter_base hij ho
-  rw [countAfter,arrowAfter_self,add_zero] at hh
-  exact hh
-
- theorem countAfter_next_formula {i j c : ℤ} (hij : |i| < |j|)
-    (h : c=j ∨ |j| < |c|) :
-    countAfter j (i+j) c = if 0<j then c.natAbs-j.natAbs else 0 := by
-  have hMC : j.natAbs ≤ c.natAbs := by
-    rcases h with rfl|h
-    · exact le_rfl
-    · exact_mod_cast (show (j.natAbs:ℤ) ≤ c.natAbs by simpa only [Int.natCast_natAbs] using h.le)
-  unfold countAfter
-  simp_rw [arrowAfter_next hij]
-  simp only [Int.natAbs_natCast,Int.natAbs_neg]
-  by_cases hj : 0<j
-  · simp only [hj,and_true,ite_true]
-    have hs : ∀ r ∈ Finset.range c.natAbs,
-        (if j.natAbs < r ∧ Odd (j.natAbs+r) then 1 else 0) +
-        (if j.natAbs < r ∧ Odd (j.natAbs+r) then 1 else 0) =
-        (if j.natAbs < r ∧ Odd (j.natAbs+r) then 2 else 0) := by
-      intro r _
-      split_ifs <;> rfl
-    rw [Finset.sum_congr rfl hs]
-    exact total_outgoing_count _ _ hMC
-  · simp only [hj,and_false,ite_false,add_zero,Finset.sum_const_zero]
-
-/-- The signed-index cardinality identity for the source's forward angular
-order. Its identification with actual Coxeter angular roots is separate. -/
- theorem countAfter_difference {i j c : ℤ} (hij : |i| < |j|)
-    (ho : Odd (i.natAbs+j.natAbs)) (h : c=j ∨ |j| < |c|) :
-    countAfter i (i+j) c = countAfter j (i+j) c + (j.natAbs-i.natAbs-1)/2 := by
-  rw [countAfter_next_formula hij h]
-  by_cases hj : 0<j
-  · rw [ite_eq_left hj,countAfter_pos_formula hij ho hj h,Nat.add_comm]
-  · have hj' : j<0 := by
-      have := abs_nonneg i
-      rw [abs_of_nonpos (by omega : j≤0)] at hij
-      omega
-    rw [ite_eq_right hj,zero_add,countAfter_neg_formula hij ho hj' h]
-
-end
-end KLInvariance.SignedDihedral
-
-end
-
-
-section
-
-namespace KLInvariance.Dihedral
-variable {W : Type*} [Group W] {M : CoxeterMatrix Bool}
-
- theorem natAbs_lt_iff (i j : ℤ) : i.natAbs<j.natAbs ↔ |i| < |j| := by
-  rw [← Int.natCast_natAbs,← Int.natCast_natAbs]
-  exact (Nat.cast_lt (α := ℤ)).symm
-
- theorem index_step_iff (cs : CoxeterSystem M W) {i j : ℤ}
-    (hi : cs.length (indexWord cs i)=i.natAbs)
-    (hj : cs.length (indexWord cs j)=j.natAbs) :
-    BruhatStep cs (indexWord cs i) (indexWord cs j) ↔ |i| < |j| ∧ Odd (i+j) := by
-  rw [step_iff_odd,hi,hj,natAbs_lt_iff]
-  apply and_congr_right
-  intro h
-  have hnat := (natAbs_lt_iff i j).mpr h
-  rw [SignedDihedral.odd_add_abs,Nat.odd_iff,Nat.odd_iff]
-  omega
-
-end KLInvariance.Dihedral
-
-namespace KLInvariance.TitsSpace.PlaneBasis
-open KLInvariance.Dihedral
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
- theorem edgeRoot_same_table {i j z : ℤ}
-    (hxy : BruhatStep B.system (indexWord B.system i) (indexWord B.system j))
-    (hxz : BruhatStep B.system (indexWord B.system i) (indexWord B.system z))
-    (hi : B.system.length (indexWord B.system i)=i.natAbs)
-    (hj : B.system.length (indexWord B.system j)=j.natAbs)
-    (hz : B.system.length (indexWord B.system z)=z.natAbs)
-    (gj : GoodIndex B.system j) (gz : GoodIndex B.system z) :
-    geometricSlope (B.edgeRoot hxy) < geometricSlope (B.edgeRoot hxz) ↔
-      SignedDihedral.Before (i+j) (i+z) := by
-  have hij := (index_step_iff B.system hi hj).mp hxy
-  have hiz := (index_step_iff B.system hi hz).mp hxz
-  have hlabel := indexWord_label B.system hij.2
-  have hlabel' := indexWord_label B.system hiz.2
-  rcases maximal_or_unbounded B.system with ⟨w,hw⟩ | hu
-  · rw [B.edgeRoot_finite_order w hw hxy hxz hij.2 hiz.2 hlabel hlabel']
-    exact SignedDihedral.finite_before_same (by exact_mod_cast max_length_positive B.system hw)
-      hij.1 hiz.1 (gj w hw) (gz w hw)
-  · exact B.edgeRoot_infinite_order hu hxy hxz hij.2 hiz.2 hlabel hlabel'
-
- theorem edgeRoot_next_table {i j z : ℤ}
-    (hxy : BruhatStep B.system (indexWord B.system i) (indexWord B.system j))
-    (hyz : BruhatStep B.system (indexWord B.system j) (indexWord B.system z))
-    (hi : B.system.length (indexWord B.system i)=i.natAbs)
-    (hj : B.system.length (indexWord B.system j)=j.natAbs)
-    (hz : B.system.length (indexWord B.system z)=z.natAbs)
-    (gz : GoodIndex B.system z) :
-    geometricSlope (B.edgeRoot hxy) < geometricSlope (B.edgeRoot hyz) ↔
-      SignedDihedral.Before (i+j) (j+z) := by
-  have hij := (index_step_iff B.system hi hj).mp hxy
-  have hjz := (index_step_iff B.system hj hz).mp hyz
-  have hlabel := indexWord_label B.system hij.2
-  have hlabel' := indexWord_label B.system hjz.2
-  rcases maximal_or_unbounded B.system with ⟨w,hw⟩ | hu
-  · rw [B.edgeRoot_finite_order w hw hxy hyz hij.2 hjz.2 hlabel hlabel']
-    exact SignedDihedral.finite_before_next (by exact_mod_cast max_length_positive B.system hw)
-      hij.1 hjz.1 (gz w hw)
-  · exact B.edgeRoot_infinite_order hu hxy hyz hij.2 hjz.2 hlabel hlabel'
-
- def afterIndicator (x : planeSubgroup M cs P) (α : PositiveRoot M cs)
-    (z : planeSubgroup M cs P) : ℕ :=
-  if h : BruhatStep B.system x z then
-    if geometricSlope α < geometricSlope (B.edgeRoot h) then 1 else 0
-  else 0
-
- theorem afterIndicator_one (x : planeSubgroup M cs P) (α : PositiveRoot M cs) :
-    B.afterIndicator x α 1=0 := by
-  have hn : ¬BruhatStep B.system x 1 := by
-    intro h
-    have := h.1
-    rw [B.system.length_one] at this
-    omega
-  simp only [afterIndicator,dite_eq_right hn]
-
- theorem afterIndicator_same {i j z : ℤ}
-    (hxy : BruhatStep B.system (indexWord B.system i) (indexWord B.system j))
-    (hi : B.system.length (indexWord B.system i)=i.natAbs)
-    (hj : B.system.length (indexWord B.system j)=j.natAbs)
-    (hz : B.system.length (indexWord B.system z)=z.natAbs)
-    (gj : GoodIndex B.system j) (gz : GoodIndex B.system z) :
-    B.afterIndicator (indexWord B.system i) (B.edgeRoot hxy) (indexWord B.system z) =
-      SignedDihedral.arrowAfter i (i+j) z := by
-  by_cases hstep : BruhatStep B.system (indexWord B.system i) (indexWord B.system z)
-  · have hs := (index_step_iff B.system hi hz).mp hstep
-    simp only [afterIndicator,dite_eq_left hstep,B.edgeRoot_same_table hxy hstep hi hj hz gj gz,
-      SignedDihedral.arrowAfter,hs.1,hs.2,true_and]
-  · have hs := (index_step_iff B.system hi hz).not.mp hstep
-    rw [afterIndicator,dite_eq_right hstep,SignedDihedral.arrowAfter]
-    split_ifs with h
-    · exact (hs ⟨h.1,h.2.1⟩).elim
-    · rfl
-
- theorem afterIndicator_next {i j z : ℤ}
-    (hxy : BruhatStep B.system (indexWord B.system i) (indexWord B.system j))
-    (hi : B.system.length (indexWord B.system i)=i.natAbs)
-    (hj : B.system.length (indexWord B.system j)=j.natAbs)
-    (hz : B.system.length (indexWord B.system z)=z.natAbs)
-    (gz : GoodIndex B.system z) :
-    B.afterIndicator (indexWord B.system j) (B.edgeRoot hxy) (indexWord B.system z) =
-      SignedDihedral.arrowAfter j (i+j) z := by
-  by_cases hstep : BruhatStep B.system (indexWord B.system j) (indexWord B.system z)
-  · have hs := (index_step_iff B.system hj hz).mp hstep
-    simp only [afterIndicator,dite_eq_left hstep,B.edgeRoot_next_table hxy hstep hi hj hz gz,
-      SignedDihedral.arrowAfter,hs.1,hs.2,true_and]
-  · have hs := (index_step_iff B.system hj hz).not.mp hstep
-    rw [afterIndicator,dite_eq_right hstep,SignedDihedral.arrowAfter]
-    split_ifs with h
-    · exact (hs ⟨h.1,h.2.1⟩).elim
-    · rfl
-
-end
-end KLInvariance.TitsSpace.PlaneBasis
-
-end
-
-
-section
-
-namespace KLInvariance.TitsSpace.PlaneBasis
-open KLInvariance.Dihedral
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-noncomputable section
-
- def laterCount (x : planeSubgroup M cs P) (α : PositiveRoot M cs)
-    (c : planeSubgroup M cs P) : ℕ :=
-  ∑ z ∈ lowerFinset B.system c, B.afterIndicator x α z
-
- theorem laterCount_same {i j k : ℤ}
-    (hxy : BruhatStep B.system (indexWord B.system i) (indexWord B.system j))
-    (hi : B.system.length (indexWord B.system i)=i.natAbs)
-    (hj : B.system.length (indexWord B.system j)=j.natAbs)
-    (hk : B.system.length (indexWord B.system k)=k.natAbs)
-    (gj : GoodIndex B.system j) (gk : GoodIndex B.system k) :
-    B.laterCount (indexWord B.system i) (B.edgeRoot hxy) (indexWord B.system k) =
-      SignedDihedral.countAfter i (i+j) k := by
-  rw [laterCount,sum_lower_signed B.system _ _ (B.afterIndicator_one _ _),hk,
-    B.afterIndicator_same hxy hi hj hk gj gk,SignedDihedral.countAfter]
-  congr 1
-  apply Finset.sum_congr rfl
-  intro n hn
-  have hnc : n<B.system.length (indexWord B.system k) := by
-    rw [hk]
-    exact Finset.mem_range.mp hn
-  have hp := length_indexWord_nat B.system hnc.le
-  have hm := length_indexWord_neg_nat B.system hnc.le
-  have hp' : B.system.length (indexWord B.system (n:ℤ))=(n:ℤ).natAbs := by
-    simpa only [Int.natAbs_natCast] using hp
-  have hm' : B.system.length (indexWord B.system (-(n:ℤ)))=(-(n:ℤ)).natAbs := by
-    simpa only [Int.natAbs_neg,Int.natAbs_natCast] using hm
-  rw [B.afterIndicator_same hxy hi hj hp' gj (goodIndex_nat B.system hnc.le),
-    B.afterIndicator_same hxy hi hj hm' gj (goodIndex_neg_nat B.system hnc)]
-
- theorem laterCount_next {i j k : ℤ}
-    (hxy : BruhatStep B.system (indexWord B.system i) (indexWord B.system j))
-    (hi : B.system.length (indexWord B.system i)=i.natAbs)
-    (hj : B.system.length (indexWord B.system j)=j.natAbs)
-    (hk : B.system.length (indexWord B.system k)=k.natAbs)
-    (gk : GoodIndex B.system k) :
-    B.laterCount (indexWord B.system j) (B.edgeRoot hxy) (indexWord B.system k) =
-      SignedDihedral.countAfter j (i+j) k := by
-  rw [laterCount,sum_lower_signed B.system _ _ (B.afterIndicator_one _ _),hk,
-    B.afterIndicator_next hxy hi hj hk gk,SignedDihedral.countAfter]
-  congr 1
-  apply Finset.sum_congr rfl
-  intro n hn
-  have hnc : n<B.system.length (indexWord B.system k) := by
-    rw [hk]
-    exact Finset.mem_range.mp hn
-  have hp := length_indexWord_nat B.system hnc.le
-  have hm := length_indexWord_neg_nat B.system hnc.le
-  have hp' : B.system.length (indexWord B.system (n:ℤ))=(n:ℤ).natAbs := by
-    simpa only [Int.natAbs_natCast] using hp
-  have hm' : B.system.length (indexWord B.system (-(n:ℤ)))=(-(n:ℤ)).natAbs := by
-    simpa only [Int.natAbs_neg,Int.natAbs_natCast] using hm
-  rw [B.afterIndicator_next hxy hi hj hp' (goodIndex_nat B.system hnc.le),
-    B.afterIndicator_next hxy hi hj hm' (goodIndex_neg_nat B.system hnc)]
-
-
- theorem laterCount_difference {x y c : planeSubgroup M cs P}
-    (hxy : BruhatStep B.system x y) (hyc : BruhatLE B.system y c) :
-    B.laterCount x (B.edgeRoot hxy) c = B.laterCount y (B.edgeRoot hxy) c +
-      (B.system.length y-B.system.length x-1)/2 := by
-  obtain ⟨i,_,hi,he⟩ := exists_good_index B.system x
-  obtain ⟨j,gj,hj,hf⟩ := exists_good_index B.system y
-  have hi' : i.natAbs=B.system.length x := by
-    exact_mod_cast (show (i.natAbs:ℤ)=(B.system.length x:ℤ) by rw [Int.natCast_natAbs,hi])
-  have hj' : j.natAbs=B.system.length y := by
-    exact_mod_cast (show (j.natAbs:ℤ)=(B.system.length y:ℤ) by rw [Int.natCast_natAbs,hj])
-  obtain ⟨k,gk,hk,hg,hcomp⟩ : ∃ k : ℤ, GoodIndex B.system k ∧
-      k.natAbs=B.system.length c ∧ indexWord B.system k=c ∧
-      (k=j ∨ |j| < |k|) := by
-    by_cases h : c=y
-    · subst c
-      exact ⟨j,gj,hj',hf,Or.inl rfl⟩
-    · obtain ⟨k,gk,hk,hg⟩ := exists_good_index B.system c
-      have hlen := length_lt_of_bruhat_ne B.system hyc (Ne.symm h)
-      refine ⟨k,gk,?_,hg,Or.inr ?_⟩
-      · exact_mod_cast (show (k.natAbs:ℤ)=(B.system.length c:ℤ) by rw [Int.natCast_natAbs,hk])
-      · rw [hj,hk]
-        exact_mod_cast hlen
-  have ilen : B.system.length (indexWord B.system i)=i.natAbs := by rw [he,hi']
-  have jlen : B.system.length (indexWord B.system j)=j.natAbs := by rw [hf,hj']
-  have klen : B.system.length (indexWord B.system k)=k.natAbs := by rw [hg,hk]
-  subst x y c
-  rw [B.laterCount_same hxy ilen jlen klen gj gk,
-    B.laterCount_next hxy ilen jlen klen gk,ilen,jlen]
-  have hs := (index_step_iff B.system ilen jlen).mp hxy
-  exact SignedDihedral.countAfter_difference hs.1
-    ((SignedDihedral.odd_add_abs i j).mp hs.2) hcomp
-
-end
-end KLInvariance.TitsSpace.PlaneBasis
-
-end
-
-
-section
-
-namespace KLInvariance.TitsSpace.PlaneBasis
-open KLInvariance.Dihedral
-universe u v u' v'
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-noncomputable section
-
- theorem afterIndicator_eq_zero {x z : planeSubgroup M cs P}
-    (α : PositiveRoot M cs) (h : ¬BruhatLE B.system x z) :
-    B.afterIndicator x α z = 0 := by
-  have hn : ¬BruhatStep B.system x z := fun hz => h (Relation.ReflTransGen.single hz)
-  simp only [afterIndicator,dite_eq_right hn]
-
-/-- Counting only the relative interval above a lower bound does not alter
-an outgoing count based at a vertex above that bound. -/
- theorem interval_laterCount {a x c : planeSubgroup M cs P}
-    (hax : BruhatLE B.system a x) (α : PositiveRoot M cs) :
-    (∑ᶠ z : Interval B.system a c, B.afterIndicator x α z.val) =
-      B.laterCount x α c := by
-  classical
-  let := interval_finite B.system a c
-  let : Fintype (Interval B.system a c) := Fintype.ofFinite _
-  rw [finsum_eq_sum_of_fintype,laterCount]
-  have hs := Finset.sum_subtype (F := (inferInstance : Fintype (Interval B.system a c)))
-    ((lowerFinset B.system c).filter (fun z => BruhatLE B.system a z))
-    (fun z => by simp only [Finset.mem_filter,mem_lowerFinset]; exact and_comm)
-    (fun z => B.afterIndicator x α z)
-  refine hs.symm.trans (Finset.sum_subset (Finset.filter_subset _ _) ?_)
-  intro z hz hn
-  apply B.afterIndicator_eq_zero
-  intro hxz
-  exact hn (Finset.mem_filter.mpr ⟨hz,bruhat_trans B.system hax hxz⟩)
-
- theorem edgeRoot_coset {a₀ : W}
-    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-    {x y : planeSubgroup M cs P} (hxy : BruhatStep B.system x y) :
-    B.edgeRoot hxy = graphRoot ((B.coset_step_iff hmin x y).mpr hxy) := by
-  unfold edgeRoot graphRoot
-  apply congrArg (positiveRoot M cs)
-  apply Subtype.ext
-  change (y:W)*(x:W)⁻¹=((y:W)*a₀)*((x:W)*a₀)⁻¹
-  group
-
-variable {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
-  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
-  {P' : Submodule ℝ (I' → ℝ)} (B' : PlaneBasis (M := M') (cs := cs') P')
-
-/-- A full relative interval transport carries the actual angular later-edge
-count. An arbitrary graph embedding would not suffice here. -/
- theorem interval_map_count {a c : planeSubgroup M cs P}
-    (hac : BruhatLE B.system a c)
-    (F : Interval B.system a c → planeSubgroup M' cs' P')
-    (hinj : Function.Injective F)
-    (hrange : Set.range F = {z | BruhatLE B'.system (F ⟨a,bruhat_refl B.system a,hac⟩) z ∧
-      BruhatLE B'.system z (F ⟨c,hac,bruhat_refl B.system c⟩)})
-    (x : Interval B.system a c) (α : PositiveRoot M' cs') :
-    (∑ᶠ z : Interval B.system a c, B'.afterIndicator (F x) α (F z)) =
-      B'.laterCount (F x) α (F ⟨c,hac,bruhat_refl B.system c⟩) := by
-  classical
-  let lo : Interval B.system a c := ⟨a,bruhat_refl B.system a,hac⟩
-  let hi : Interval B.system a c := ⟨c,hac,bruhat_refl B.system c⟩
-  have hmem (z : Interval B.system a c) : BruhatLE B'.system (F lo) (F z) ∧
-      BruhatLE B'.system (F z) (F hi) := by
-    exact (Set.ext_iff.mp hrange (F z)).mp ⟨z,rfl⟩
-  let f : Interval B.system a c → Interval B'.system (F lo) (F hi) :=
-    fun z => ⟨F z,hmem z⟩
-  have hf : Function.Bijective f := by
-    constructor
-    · intro z w h
-      exact hinj (congrArg Subtype.val h)
-    · intro z
-      obtain ⟨w,hw⟩ := (Set.ext_iff.mp hrange z.val).mpr z.property
-      exact ⟨w,Subtype.ext hw⟩
-  let E := Equiv.ofBijective f hf
-  have hs := finsum_comp_equiv E (f := fun z : Interval B'.system (F lo) (F hi) =>
-    B'.afterIndicator (F x) α z.val)
-  exact hs.trans (B'.interval_laterCount (hmem x).1 α)
-
-end
-end KLInvariance.TitsSpace.PlaneBasis
-
-end
-
-
-section
-
-namespace KLInvariance.TitsSpace
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-noncomputable section
-
- def ambientAfterIndicator (x : W) (α : PositiveRoot M cs) (z : W) : ℕ := by
-  classical
-  exact if h : BruhatStep cs x z then
-    if geometricSlope α < geometricSlope (graphRoot h) then 1 else 0
-  else 0
-
- theorem ambientAfterIndicator_of_le {x z : W} (α : PositiveRoot M cs)
-    (hz : BruhatLE cs z x) : ambientAfterIndicator x α z=0 := by
-  classical
-  have hn : ¬BruhatStep cs x z := fun h =>
-    (not_lt_of_ge (length_le_of_bruhat cs hz)) h.1
-  simp only [ambientAfterIndicator,dite_eq_right hn]
-
- theorem ambientAfterIndicator_edge_self {x y : W} (hxy : BruhatStep cs x y) :
-    ambientAfterIndicator x (graphRoot hxy) y=0 := by
-  classical
-  simp only [ambientAfterIndicator,dite_eq_left hxy,lt_self_iff_false,ite_false]
-
-namespace PlaneBasis
-variable {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-
- theorem ambientAfterIndicator_coset {a₀ : W}
-    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-    (x z : planeSubgroup M cs P) (α : PositiveRoot M cs) :
-    ambientAfterIndicator ((x:W)*a₀) α ((z:W)*a₀) = B.afterIndicator x α z := by
-  classical
-  by_cases h : BruhatStep B.system x z
-  · have h' := (B.coset_step_iff hmin x z).mpr h
-    simp only [ambientAfterIndicator,afterIndicator,dite_eq_left h,dite_eq_left h',B.edgeRoot_coset hmin h]
-  · have h' : ¬BruhatStep cs ((x:W)*a₀) ((z:W)*a₀) :=
-      fun hz => h ((B.coset_step_iff hmin x z).mp hz)
-    simp only [ambientAfterIndicator,afterIndicator,dite_eq_right h,dite_eq_right h']
-
-universe u' v'
-variable {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
-  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
-
-
- theorem transported_count_difference {a₀ u b : W} {u' b' : W'}
-    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-    {a c : planeSubgroup M cs P} (hac : BruhatLE B.system a c)
-    (hrank : B.system.length a+2 ≤ B.system.length c)
-    (hlo : BruhatLE cs u ((a:W)*a₀)) (hhi : BruhatLE cs ((c:W)*a₀) b)
-    (e : Interval cs u b ≃o Interval cs' u' b')
-    (y : Interval B.system a c) (hxy : BruhatStep B.system a y.val) :
-    let inc := B.cosetIntervalMap hmin hlo hhi
-    let lo : Interval B.system a c := ⟨a,bruhat_refl B.system a,hac⟩
-    let α := graphRoot ((intervalIso_bruhatStep_iff_general cs cs' e (inc lo) (inc y)).mpr
-      ((B.coset_step_iff hmin a y.val).mpr hxy))
-    (∑ᶠ z : Interval B.system a c,
-      ambientAfterIndicator (e (inc lo)).val α (e (inc z)).val) =
-    (∑ᶠ z : Interval B.system a c,
-      ambientAfterIndicator (e (inc y)).val α (e (inc z)).val) +
-      (B.system.length y.val-B.system.length a-1)/2 := by
-  classical
-  dsimp only
-  let inc := B.cosetIntervalMap hmin hlo hhi
-  let lo : Interval B.system a c := ⟨a,bruhat_refl B.system a,hac⟩
-  let hi : Interval B.system a c := ⟨c,hac,bruhat_refl B.system c⟩
-  have hedge := (intervalIso_bruhatStep_iff_general cs cs' e (inc lo) (inc y)).mpr
-    ((B.coset_step_iff hmin a y.val).mpr hxy)
-  change (∑ᶠ z : Interval B.system a c,
-      ambientAfterIndicator (e (inc lo)).val (graphRoot hedge) (e (inc z)).val) =
-    (∑ᶠ z : Interval B.system a c,
-      ambientAfterIndicator (e (inc y)).val (graphRoot hedge) (e (inc z)).val) + _
-  obtain ⟨P',B',a',F,hmin',hinj,hcoord,horder,hstep,hlen,hrange⟩ :=
-    B.transport_interval hmin hac hrank hlo hhi e
-  have hFxy : BruhatStep B'.system (F lo) (F y) := (hstep lo y).mpr hxy
-  have hFyc : BruhatLE B'.system (F y) (F hi) := (horder y hi).mpr y.property.2
-  have hroot : B'.edgeRoot hFxy=graphRoot hedge := by
-    rw [B'.edgeRoot_coset hmin' hFxy]
-    unfold graphRoot
-    apply congrArg (positiveRoot M' cs')
-    apply Subtype.ext
-    change ((F y:W')*a')*((F lo:W')*a')⁻¹=(e (inc y)).val*(e (inc lo)).val⁻¹
-    rw [hcoord y,hcoord lo]
-  have hsum (x : Interval B.system a c) :
-      (∑ᶠ z : Interval B.system a c,
-        ambientAfterIndicator (e (inc x)).val (graphRoot hedge) (e (inc z)).val) =
-      B'.laterCount (F x) (B'.edgeRoot hFxy) (F hi) := by
-    rw [← B.interval_map_count B' hac F hinj hrange x (B'.edgeRoot hFxy)]
-    apply finsum_congr
-    intro z
-    rw [← hcoord x,← hcoord z,← hroot]
-    exact B'.ambientAfterIndicator_coset hmin' (F x) (F z) _
-  rw [hsum lo,hsum y,B'.laterCount_difference hFxy hFyc]
-  congr 1
-  have hy := hlen y
-  change B'.system.length (F y)+B.system.length a =
-    B'.system.length (F lo)+B.system.length y.val at hy
-  omega
-
-/-- The same scheduled count with no restriction on the relative rank. -/
- theorem transported_count_difference_all {a₀ u b : W} {u' b' : W'}
-    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-    {a c : planeSubgroup M cs P} (hac : BruhatLE B.system a c)
-    (hlo : BruhatLE cs u ((a:W)*a₀)) (hhi : BruhatLE cs ((c:W)*a₀) b)
-    (e : Interval cs u b ≃o Interval cs' u' b')
-    (y : Interval B.system a c) (hxy : BruhatStep B.system a y.val) :
-    let inc := B.cosetIntervalMap hmin hlo hhi
-    let lo : Interval B.system a c := ⟨a,bruhat_refl B.system a,hac⟩
-    let α := graphRoot ((intervalIso_bruhatStep_iff_general cs cs' e (inc lo) (inc y)).mpr
-      ((B.coset_step_iff hmin a y.val).mpr hxy))
-    (∑ᶠ z : Interval B.system a c,
-      ambientAfterIndicator (e (inc lo)).val α (e (inc z)).val) =
-    (∑ᶠ z : Interval B.system a c,
-      ambientAfterIndicator (e (inc y)).val α (e (inc z)).val) +
-      (B.system.length y.val-B.system.length a-1)/2 := by
-  classical
-  by_cases hrank : B.system.length a+2 ≤ B.system.length c
-  · exact B.transported_count_difference hmin hac hrank hlo hhi e y hxy
-  let inc := B.cosetIntervalMap hmin hlo hhi
-  let lo : Interval B.system a c := ⟨a,bruhat_refl B.system a,hac⟩
-  have hedge := (intervalIso_bruhatStep_iff_general cs cs' e (inc lo) (inc y)).mpr
-    ((B.coset_step_iff hmin a y.val).mpr hxy)
-  change (∑ᶠ z : Interval B.system a c,
-      ambientAfterIndicator (e (inc lo)).val (graphRoot hedge) (e (inc z)).val) =
-    (∑ᶠ z : Interval B.system a c,
-      ambientAfterIndicator (e (inc y)).val (graphRoot hedge) (e (inc z)).val) + _
-  have hylen := hxy.1
-  have hyc := length_le_of_bruhat B.system y.property.2
-  have hlen : B.system.length y.val=B.system.length a+1 := by omega
-  have heyc : y.val=c := by
-    by_contra hn
-    have hlt := length_lt_of_bruhat_ne B.system y.property.2 hn
-    omega
-  have hverts (z : Interval B.system a c) : z=lo ∨ z=y := by
-    by_cases hz : z.val=a
-    · exact Or.inl (Subtype.ext hz)
-    · right
-      apply Subtype.ext
-      rw [heyc]
-      by_contra hzc
-      have hzlo := length_lt_of_bruhat_ne B.system z.property.1 (Ne.symm hz)
-      have hzhi := length_lt_of_bruhat_ne B.system z.property.2 hzc
-      omega
-  have hzero₁ : (∑ᶠ z : Interval B.system a c,
-      ambientAfterIndicator (e (inc lo)).val (graphRoot hedge) (e (inc z)).val)=0 := by
-    apply finsum_eq_zero_of_forall_eq_zero
-    intro z
-    rcases hverts z with rfl | rfl
-    · exact ambientAfterIndicator_of_le _ (bruhat_refl cs' _)
-    · exact ambientAfterIndicator_edge_self hedge
-  have hzero₂ : (∑ᶠ z : Interval B.system a c,
-      ambientAfterIndicator (e (inc y)).val (graphRoot hedge) (e (inc z)).val)=0 := by
-    apply finsum_eq_zero_of_forall_eq_zero
-    intro z
-    rcases hverts z with rfl | rfl
-    · exact ambientAfterIndicator_of_le _ (Relation.ReflTransGen.single hedge)
-    · exact ambientAfterIndicator_of_le _ (bruhat_refl cs' _)
-  rw [hzero₁,hzero₂,hlen]
-  omega
-
-end PlaneBasis
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Linear prime ideals in the exact symmetric algebra, including the actual
-root-plane coefficient primes. No local-ring or survival property is assumed. -/
-
-end
-
-end OAI
+import OAI.RepresentationTheory.KazhdanLusztig.ReflectionOrders
+
+namespace OAI
+
+/-!
+Signed dihedral coordinates, actual dihedral intervals, exact later-edge counts and root-plane calculations.
+-/
+
+section
+
+namespace KLInvariance.Dihedral
+variable {W : Type*} [Group W] {M : CoxeterMatrix Bool}
+noncomputable section
+
+ theorem wordProd_alt_odd (cs : CoxeterSystem M W) (i : Bool) (n : ℕ) :
+    cs.wordProd (alt i (2*n+1)) = (cs.simple i * cs.simple (!i)) ^ n * cs.simple i := by
+  induction n with
+  | zero => simp
+  | succ n ih =>
+    rw [show 2*(n+1)+1 = (2*n+1)+2 by omega, alt_add_two]
+    simp only [cs.wordProd_cons, ih, pow_succ']
+    group
+
+ def rotation (cs : CoxeterSystem M W) : W := cs.simple false * cs.simple true
+
+ theorem simple_rotation (cs : CoxeterSystem M W) :
+    cs.simple false * rotation cs = (rotation cs)⁻¹ * cs.simple false := by
+  simp only [rotation, mul_inv_rev, cs.inv_simple]
+  rw [cs.simple_mul_simple_cancel_left, mul_assoc, cs.simple_mul_simple_self, mul_one]
+
+ theorem simple_rotation_zpow (cs : CoxeterSystem M W) (n : ℤ) :
+    cs.simple false * rotation cs ^ n = rotation cs ^ (-n) * cs.simple false := by
+  have h : SemiconjBy (cs.simple false) (rotation cs) (rotation cs)⁻¹ :=
+    simple_rotation cs
+  have hh := h.zpow_right n
+  change cs.simple false * rotation cs ^ n = ((rotation cs)⁻¹) ^ n * cs.simple false at hh
+  simpa only [inv_zpow, zpow_neg] using hh
+
+/-- The signed coordinate evaluates both alternating rays in the same group.
+The Euclidean quotient also gives the correct odd negative coordinates. -/
+ def indexWord (cs : CoxeterSystem M W) (i : ℤ) : W :=
+  if i % 2 = 0 then rotation cs ^ (i/2)
+  else rotation cs ^ (i/2) * cs.simple false
+
+ theorem indexWord_even (cs : CoxeterSystem M W) (n : ℤ) :
+    indexWord cs (2*n) = rotation cs ^ n := by simp [indexWord]
+
+ theorem indexWord_odd (cs : CoxeterSystem M W) (n : ℤ) :
+    indexWord cs (2*n+1) = rotation cs ^ n * cs.simple false := by
+  simp only [indexWord, show (2*n+1)%2 = 1 by omega, show (1:ℤ) ≠ 0 by omega, ite_false,
+    show (2*n+1)/2 = n by omega]
+
+ theorem indexWord_nat (cs : CoxeterSystem M W) (n : ℕ) :
+    indexWord cs n = cs.wordProd (alt false n) := by
+  rcases Nat.even_or_odd n with ⟨k,rfl⟩ | ⟨k,rfl⟩
+  · rw [show k+k = 2*k by omega, Nat.cast_mul, Nat.cast_ofNat, indexWord_even,
+      wordProd_alt_even]
+    simp only [rotation, Bool.not_false, zpow_natCast]
+  · rw [Nat.cast_add, Nat.cast_mul,
+      Nat.cast_ofNat, Nat.cast_one, indexWord_odd, wordProd_alt_odd]
+    simp only [rotation, Bool.not_false, zpow_natCast]
+
+ theorem indexWord_neg_nat (cs : CoxeterSystem M W) (n : ℕ) :
+    indexWord cs (-(n:ℤ)) = cs.wordProd (alt true n) := by
+  have hr : cs.simple true * cs.simple false = (rotation cs)⁻¹ := by
+    simp only [rotation, mul_inv_rev, cs.inv_simple]
+  rcases Nat.even_or_odd n with ⟨k,rfl⟩ | ⟨k,rfl⟩
+  · rw [show k+k = 2*k by omega, Nat.cast_mul, Nat.cast_ofNat,
+      show -(2*(k:ℤ)) = 2*(-k) by ring, indexWord_even, wordProd_alt_even]
+    simp only [Bool.not_true, hr, zpow_neg, zpow_natCast, inv_pow]
+  · rw [Nat.cast_add, Nat.cast_mul,
+      Nat.cast_ofNat, Nat.cast_one,
+      show -(2*(k:ℤ)+1) = 2*(-(k:ℤ)-1)+1 by ring, indexWord_odd,
+      wordProd_alt_odd]
+    rw [show -(k:ℤ)-1 = -(k:ℤ)+(-1) by ring, zpow_add]
+    simp only [zpow_neg, zpow_natCast, zpow_one, Bool.not_true, hr, inv_pow]
+    simp only [rotation, mul_inv_rev, cs.inv_simple]
+    simp only [mul_assoc, cs.simple_mul_simple_self, mul_one]
+
+ theorem exists_indexWord (cs : CoxeterSystem M W) (w : W) :
+    ∃ i : ℤ, |i| = (cs.length w : ℤ) ∧ indexWord cs i = w := by
+  obtain ⟨i,hi⟩ := exists_alt cs w
+  cases i
+  · exact ⟨cs.length w, by simp, (indexWord_nat cs _).trans hi.symm⟩
+  · exact ⟨-(cs.length w : ℤ), by simp, (indexWord_neg_nat cs _).trans hi.symm⟩
+
+/-- A reflection acts on the signed index line by reflection in its midpoint.
+Finite dihedral groups are handled by equality in the group, without choosing
+residue representatives here. -/
+ theorem indexWord_reflection_mul (cs : CoxeterSystem M W) {a : ℤ} (ha : Odd a)
+    (i : ℤ) : indexWord cs a * indexWord cs i = indexWord cs (a-i) := by
+  obtain ⟨k,hk⟩ := ha
+  have hk' : a = 2*k+1 := by omega
+  rw [hk',indexWord_odd]
+  rcases Int.even_or_odd i with ⟨l,hl⟩ | ⟨l,hl⟩
+  · have hl' : i=2*l := by omega
+    rw [hl',indexWord_even,show 2*k+1-2*l=2*(k-l)+1 by ring,indexWord_odd]
+    rw [mul_assoc, simple_rotation_zpow, ← mul_assoc, ← zpow_add]
+    congr 2
+  · have hl' : i=2*l+1 := by omega
+    rw [hl',indexWord_odd,show 2*k+1-(2*l+1)=2*(k-l) by ring,indexWord_even]
+    calc
+      (rotation cs ^ k * cs.simple false) * (rotation cs ^ l * cs.simple false) =
+          rotation cs ^ k * (cs.simple false * rotation cs ^ l) * cs.simple false := by group
+      _ = rotation cs ^ k * (rotation cs ^ (-l) * cs.simple false) * cs.simple false := by
+        rw [simple_rotation_zpow]
+      _ = rotation cs ^ (k-l) := by
+        rw [← mul_assoc, mul_assoc (rotation cs ^ k * rotation cs ^ (-l)),
+          cs.simple_mul_simple_self, mul_one, ← zpow_add]
+        congr 1
+
+ theorem indexWord_label (cs : CoxeterSystem M W) {i j : ℤ} (h : Odd (i+j)) :
+    indexWord cs j * (indexWord cs i)⁻¹ = indexWord cs (i+j) := by
+  have he := indexWord_reflection_mul cs h i
+  rw [show i+j-i=j by omega] at he
+  rw [← he]
+  group
+
+end
+end KLInvariance.Dihedral
+
+end
+
+
+section
+
+/-! Actual degree-one splines in the ambient extended Tits realization.
+They supply the interpolation factors needed for the relative rank-two model. -/
+namespace KLInvariance.TitsSpace
+open Module
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ def linearPolynomial : Extended M →ₗ[ℝ] MvPolynomial σ ℝ :=
+  (SymmetricAlgebra.equivMvPolynomial basis).toLinearMap.comp
+    (SymmetricAlgebra.ι ℝ (Extended M))
+
+ def orbitSpline (y : W) (v₀ : Extended M) (v : W) : MvPolynomial σ ℝ :=
+  linearPolynomial basis (extendedAction M cs (v*y⁻¹) v₀ - v₀)
+
+ theorem orbitSpline_self (y : W) (v₀ : Extended M) : orbitSpline cs basis y v₀ y = 0 := by
+  simp [orbitSpline]
+
+ theorem orbitSpline_reflection {t : W} {a : I → ℝ}
+    (ha : Represents M cs t a) (y x : W) (v₀ : Extended M) :
+    linearPolynomial basis (embed M a) ∣
+      orbitSpline cs basis y v₀ x - orbitSpline cs basis y v₀ (t*x) := by
+  let c := extendedForm M (embed M a) (extendedAction M cs (x*y⁻¹) v₀)
+  have heq : extendedAction M cs (t*x*y⁻¹) v₀ =
+      extendedAction M cs (x*y⁻¹) v₀ - c • embed M a := by
+    rw [mul_assoc,map_mul,LinearEquiv.mul_apply,ha.formula]
+  unfold orbitSpline
+  rw [← map_sub,heq]
+  have hv : (extendedAction M cs (x*y⁻¹) v₀ - v₀) -
+      (extendedAction M cs (x*y⁻¹) v₀ - c • embed M a - v₀) = c • embed M a := by abel
+  rw [hv,map_smul]
+  exact ⟨algebraMap ℝ (MvPolynomial σ ℝ) c, by simp [Algebra.smul_def,mul_comm]⟩
+
+ theorem exists_pair_vector {t r : W} {a d : I → ℝ}
+    (ha : Represents M cs t a) (hd : Represents M cs r d) (htr : t ≠ r) :
+    ∃ v₀ : Extended M, extendedForm M (embed M a) v₀ = -1 ∧
+      extendedForm M (embed M d) v₀ = 0 := by
+  have hli := (ha.extended_independent_of_ne hd htr).map'
+    (extendedForm M) (extendedForm_spec M).2.1.ker_eq_bot
+  have hli' : LinearIndependent ℝ ![extendedForm M (embed M a),
+      extendedForm M (embed M d)] := by
+    convert hli using 1
+    funext i
+    fin_cases i <;> rfl
+  obtain ⟨v₀,hv₀⟩ := ReflectionPlane.formPair_surjective _ _ hli' (-1,0)
+  exact ⟨v₀,congrArg Prod.fst hv₀,congrArg Prod.snd hv₀⟩
+
+ theorem orbitSpline_pair_factor {x y z t r : W} {a d : I → ℝ}
+    (ha : Represents M cs t a) (hd : Represents M cs r d)
+    (hxy : x*y⁻¹=t) (hzy : z*y⁻¹=r) (hxz : x ≠ z) :
+    ∃ v₀ : Extended M,
+      orbitSpline cs basis y v₀ x = linearPolynomial basis (embed M a) ∧
+      orbitSpline cs basis y v₀ y = 0 ∧ orbitSpline cs basis y v₀ z = 0 := by
+  have htr : t ≠ r := by
+    intro he
+    apply hxz
+    exact mul_right_cancel (hxy.trans (he.trans hzy.symm))
+  obtain ⟨v₀,hv₀a,hv₀d⟩ := exists_pair_vector cs ha hd htr
+  refine ⟨v₀,?_,orbitSpline_self cs basis y v₀,?_⟩
+  · simp only [orbitSpline,hxy,ha.formula,hv₀a]
+    congr 1
+    module
+  · simp only [orbitSpline,hzy,hd.formula,hv₀d,zero_smul,sub_zero,sub_self,map_zero]
+
+ theorem orbitSpline_single_factor {x y t : W} {a : I → ℝ}
+    (ha : Represents M cs t a) (hxy : x*y⁻¹=t) :
+    ∃ v₀ : Extended M,
+      orbitSpline cs basis y v₀ x = linearPolynomial basis (embed M a) ∧
+      orbitSpline cs basis y v₀ y = 0 := by
+  let v₀ : Extended M := (-1/2 : ℝ) • embed M a
+  have hv₀ : extendedForm M (embed M a) v₀ = -1 := by
+    simp only [v₀,map_smul,smul_eq_mul,embed,(extendedForm_spec M).2.2,ha.isRoot.norm]
+    norm_num
+  refine ⟨v₀,?_,orbitSpline_self cs basis y v₀⟩
+  simp only [orbitSpline,hxy,ha.formula,hv₀]
+  congr 1
+  module
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Polynomial interpolation on an actual dihedral reflection subgroup.
+The ambient realization, including all fixed directions and degenerate root
+planes, is retained; no character theorem is assumed. -/
+namespace KLInvariance.Dihedral
+open Module TitsSpace
+universe u v w us
+variable {H : Type w} [Group H] {N : CoxeterMatrix Bool} (ds : CoxeterSystem N H)
+  {I : Type u} [Fintype I] {M : CoxeterMatrix I} {W : Type v} [Group W]
+  (cs : CoxeterSystem M W) (φ : H →* W)
+  (hφ : ∀ t, ds.IsReflection t → cs.IsReflection (φ t))
+  {σ : Type us} (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ theorem reflection_of_parity {x y : H} (hp : ds.length x % 2 ≠ ds.length y % 2) :
+    ds.IsReflection (x*y⁻¹) := by
+  rw [reflection_iff_odd,Nat.odd_iff]
+  have h := ds.length_mul_mod_two x y⁻¹
+  rw [ds.length_inv] at h
+  omega
+
+ def oppositeExternal (x c : H) : Finset H := by
+  classical
+  exact (externalLower ds x c).filter (fun z => ds.length z % 2 ≠ ds.length x % 2)
+
+ def sameExternal (x c : H) : Finset H := by
+  classical
+  exact (externalLower ds x c).filter (fun z => ds.length z % 2 = ds.length x % 2)
+
+ theorem oppositeExternal_reflection {x c y : H} (hy : y ∈ oppositeExternal ds x c) :
+    ds.IsReflection (y*x⁻¹) := by
+  classical
+  exact reflection_of_parity ds (Finset.mem_filter.mp hy).2
+
+ def interpolationRoot (x c : H) (y : oppositeExternal ds x c) : PositiveRoot M cs :=
+  positiveRoot M cs ⟨φ (y.val*x⁻¹), hφ _ (oppositeExternal_reflection ds y.property)⟩
+
+ theorem interpolation_exists (hφinj : Function.Injective φ) {x c : H}
+    (hxc : BruhatLE ds x c) : ∃ F : H → MvPolynomial σ ℝ,
+      (∀ t, ds.IsReflection t → ∀ (a : I → ℝ), Represents M cs (φ t) a → ∀ v,
+        linearPolynomial basis (embed M a) ∣ F v-F (t*v)) ∧
+      F x = ∏ y : oppositeExternal ds x c,
+        linearPolynomial basis (embed M (interpolationRoot ds cs φ hφ x c y).val) ∧
+      ∀ z, BruhatLE ds z c → ¬ BruhatLE ds z x → F z = 0 := by
+  classical
+  let Y := {y // y ∈ oppositeExternal ds x c}
+  let Z := {z // z ∈ sameExternal ds x c}
+  have hc : Fintype.card Z ≤ Fintype.card Y := by
+    change Fintype.card ↥(sameExternal ds x c) ≤ Fintype.card ↥(oppositeExternal ds x c)
+    rw [Fintype.card_coe,Fintype.card_coe]
+    exact external_parity_card_le ds hxc
+  obtain ⟨j⟩ := Function.Embedding.nonempty_of_card_le hc
+  have hfac (y : Y) : ∃ v₀ : Extended M,
+      orbitSpline cs basis (φ y.val) v₀ (φ x) =
+        linearPolynomial basis (embed M (interpolationRoot ds cs φ hφ x c y).val) ∧
+      orbitSpline cs basis (φ y.val) v₀ (φ y.val) = 0 ∧
+      ∀ z : Z, j z = y → orbitSpline cs basis (φ y.val) v₀ (φ z.val) = 0 := by
+    have ht := oppositeExternal_reflection ds y.property
+    have ha := positiveRoot_represents M cs
+      ⟨φ (y.val*x⁻¹),hφ _ ht⟩
+    have hxy₀ : x*y.val⁻¹=y.val*x⁻¹ := by simpa only [mul_inv_rev,inv_inv] using ht.inv
+    have hxy : φ x*(φ y.val)⁻¹=φ (y.val*x⁻¹) := by
+      rw [← map_inv,← map_mul,hxy₀]
+    by_cases hex : ∃ z : Z, j z = y
+    · obtain ⟨z,hz⟩ := hex
+      have hzprop := Finset.mem_filter.mp z.property
+      have hyprop := Finset.mem_filter.mp y.property
+      have hr : ds.IsReflection (z.val*y.val⁻¹) := reflection_of_parity ds (by omega)
+      let d := positiveRoot M cs ⟨φ (z.val*y.val⁻¹),hφ _ hr⟩
+      have hd : Represents M cs (φ (z.val*y.val⁻¹)) d.val := positiveRoot_represents M cs ⟨φ (z.val*y.val⁻¹),hφ _ hr⟩
+      have hxz : φ x ≠ φ z.val := by
+        intro he
+        have he' := hφinj he
+        have hh := (mem_externalLower ds x c z.val).mp hzprop.1
+        exact hh.2 (he' ▸ bruhat_refl ds x)
+      obtain ⟨v₀,ha',hy',hz0'⟩ := orbitSpline_pair_factor cs basis ha hd hxy
+        (by rw [map_mul,map_inv]) hxz
+      refine ⟨v₀,ha',hy',?_⟩
+      intro z' hz'
+      have he := j.injective (hz'.trans hz.symm)
+      simpa only [he] using hz0'
+    · obtain ⟨v₀,ha',hy'⟩ := orbitSpline_single_factor cs basis ha hxy
+      exact ⟨v₀,ha',hy',fun z hz => (hex ⟨z,hz⟩).elim⟩
+  choose v₀ hv₀ hy₀ hz₀ using hfac
+  let F : H → MvPolynomial σ ℝ := fun v => ∏ y : Y,
+    orbitSpline cs basis (φ y.val) (v₀ y) (φ v)
+  refine ⟨F,?_,?_,?_⟩
+  · intro t _ht a ha v
+    apply Ideal.mem_span_singleton.mp
+    apply Ideal.Quotient.eq.mp
+    change Ideal.Quotient.mk _ (∏ y : Y, orbitSpline cs basis (φ y.val) (v₀ y) (φ v)) =
+      Ideal.Quotient.mk _ (∏ y : Y, orbitSpline cs basis (φ y.val) (v₀ y) (φ (t*v)))
+    simp only [map_prod]
+    apply Finset.prod_congr rfl
+    intro y _
+    apply Ideal.Quotient.eq.mpr
+    apply Ideal.mem_span_singleton.mpr
+    rw [map_mul]
+    exact orbitSpline_reflection cs basis ha _ _ _
+  · exact Finset.prod_congr rfl (fun y _ => hv₀ y)
+  · intro z hzc hzx
+    have hz : z ∈ externalLower ds x c := (mem_externalLower ds x c z).mpr ⟨hzc,hzx⟩
+    by_cases hp : ds.length z % 2 = ds.length x % 2
+    · let z' : Z := ⟨z,Finset.mem_filter.mpr ⟨hz,hp⟩⟩
+      exact Finset.prod_eq_zero (Finset.mem_univ (j z')) (hz₀ (j z') z' rfl)
+    · let y : Y := ⟨z,Finset.mem_filter.mpr ⟨hz,hp⟩⟩
+      exact Finset.prod_eq_zero (Finset.mem_univ y) (hy₀ y)
+
+end
+end KLInvariance.Dihedral
+
+end
+
+
+section
+
+/-! An elementary flabbiness criterion for the exact rank-one model. Flow-down
+sections with the genuine product of outgoing labels suffice. The pairwise
+relative-primality condition will be instantiated by actual root labels. -/
+namespace KLInvariance.MomentGraph.Sheaf
+universe ur uv ue
+variable {R : Type ur} [CommRing R] [DecompositionMonoid R]
+  {V : Type uv} [PartialOrder V] {E : Type ue}
+  (G : OrderedGraph V E) (α : E → R) (c : V)
+noncomputable section
+variable [Fintype E] [DecidableEq V]
+
+def modelOutgoing (x : V) : Finset E := by
+  classical
+  exact Finset.univ.filter (fun e => G.source e = x ∧ G.target e ≤ c)
+
+@[simp] theorem mem_modelOutgoing (x : V) (e : E) :
+    e ∈ modelOutgoing G c x ↔ G.source e = x ∧ G.target e ≤ c := by
+  classical
+  simp [modelOutgoing]
+
+def modelProduct (x : V) : R := ∏ e ∈ modelOutgoing G c x, α e
+
+def HasFlowDown : Prop :=
+  ∀ x, x ≤ c → ∃ f : (structureSheaf G α c).Assignment,
+    (structureSheaf G α c).IsGlobal f ∧
+    (∀ h : PLift (x ≤ c), f x h = modelProduct G α c x) ∧
+    (∀ y, ¬ y ≤ x → f y = 0)
+
+omit [DecompositionMonoid R] [Fintype E] [DecidableEq V] in
+ theorem structureCompatible_iff (Ω : Set V) (f : (structureSheaf G α c).Assignment) :
+    (structureSheaf G α c).CompatibleOn Ω f ↔
+      ∀ e, G.source e ∈ Ω → G.target e ∈ Ω → ∀ h : PLift (G.target e ≤ c),
+        α e ∣ f (G.source e) ⟨(G.increasing e).le.trans h.down⟩ - f (G.target e) h := by
+  constructor
+  · intro hf e hs ht h
+    exact Ideal.mem_span_singleton.mp
+      (Ideal.Quotient.eq.mp (congrFun (hf e hs ht) h))
+  · intro hf e hs ht
+    funext h
+    exact Ideal.Quotient.eq.mpr (Ideal.mem_span_singleton.mpr (hf e hs ht h))
+
+include G α c in
+ theorem flowDown_flabby [Finite V]
+    (hrel : ∀ x, (modelOutgoing G c x : Set E).Pairwise (Function.onFun IsRelPrime α))
+    (hflow : HasFlowDown G α c) : (structureSheaf G α c).Flabby := by
+  classical
+  let B := structureSheaf G α c
+  let : Fintype V := Fintype.ofFinite V
+  suffices hh : ∀ s : Finset V, IsUpperSet (s : Set V) →
+      ∀ f : B.Assignment, B.CompatibleOn (s : Set V) f →
+        ∃ g : B.Assignment, B.IsGlobal g ∧ ∀ x ∈ s, g x = f x by
+    intro Ω hΩ f hf
+    have hset : (↑(Finset.univ.filter (· ∈ Ω)) : Set V) = Ω := by ext x; simp
+    obtain ⟨g,hg,hgf⟩ := hh (Finset.univ.filter (· ∈ Ω)) (by rwa [hset]) f (by rwa [hset])
+    exact ⟨g,hg,fun x hx => hgf x (by simpa using hx)⟩
+  intro s
+  refine Finset.strongInductionOn s ?_
+  intro s ih hs f hf
+  by_cases hempty : s = ∅
+  · subst s
+    refine ⟨0,?_,by simp⟩
+    exact (B.sections Set.univ).zero_mem
+  obtain ⟨a,ha,hmin⟩ := s.exists_minimal (Finset.nonempty_iff_ne_empty.mpr hempty)
+  have hu : IsUpperSet (↑(s.erase a) : Set V) := by
+    intro x y hxy hx
+    have hx' := Finset.mem_erase.mp hx
+    refine Finset.mem_erase.mpr ⟨?_,hs hxy hx'.2⟩
+    intro hya
+    subst y
+    exact hx'.1 (le_antisymm hxy (hmin hx'.2 hxy))
+  obtain ⟨g,hg,hgf⟩ := ih (s.erase a) (Finset.erase_ssubset ha) hu f
+    (fun e he ht => hf e (Finset.mem_of_mem_erase he) (Finset.mem_of_mem_erase ht))
+  by_cases hac : a ≤ c
+  · obtain ⟨t,ht,hta,htzero⟩ := hflow a hac
+    have hdiv : modelProduct G α c a ∣ f a ⟨hac⟩ - g a ⟨hac⟩ := by
+      apply Finset.prod_dvd_of_isRelPrime (hrel a)
+      intro e he
+      obtain ⟨hea,hec⟩ := (mem_modelOutgoing G c a e).mp he
+      have hsource : G.source e ∈ s := hea ▸ ha
+      have htarget : G.target e ∈ s.erase a := Finset.mem_erase.mpr
+        ⟨by simpa [← hea] using (G.increasing e).ne',hs (G.increasing e).le hsource⟩
+      have hf' := (structureCompatible_iff G α c (s : Set V) f).mp hf e hsource
+        (Finset.mem_of_mem_erase htarget) ⟨hec⟩
+      have hg' := (structureCompatible_iff G α c Set.univ g).mp hg e
+        (Set.mem_univ _) (Set.mem_univ _) ⟨hec⟩
+      rw [hgf _ htarget] at hg'
+      have hd := dvd_sub hf' hg'
+      rw [sub_sub_sub_cancel_right] at hd
+      subst a
+      exact hd
+    obtain ⟨r,hr⟩ := hdiv
+    refine ⟨g+r • t,(B.sections Set.univ).add_mem hg
+      ((B.sections Set.univ).smul_mem r ht),?_⟩
+    intro x hx
+    by_cases hxa : x = a
+    · subst x
+      funext h
+      change g a h + r * t a h = f a h
+      rw [hta h]
+      change f a h - g a h = modelProduct G α c a * r at hr
+      linear_combination -hr
+    · have hnot : ¬ x ≤ a := fun h => hxa (le_antisymm h (hmin hx h))
+      have ht0 := htzero x hnot
+      change g x + r • t x = f x
+      rw [ht0,smul_zero,add_zero]
+      exact hgf x (Finset.mem_erase.mpr ⟨hxa,hx⟩)
+  · refine ⟨g,hg,?_⟩
+    intro x hx
+    by_cases hxa : x = a
+    · subst x
+      let := structureSheaf_support G α c a hac
+      exact Subsingleton.elim _ _
+    · exact hgf x (Finset.mem_erase.mpr ⟨hxa,hx⟩)
+
+end
+end KLInvariance.MomentGraph.Sheaf
+
+end
+
+
+section
+
+/-! Flabbiness of the exact rank-one model with ambient labels, proved by
+interpolation rather than an assumed character theorem. -/
+namespace KLInvariance.Dihedral
+open Module TitsSpace BruhatGraph MomentGraph MomentGraph.Sheaf
+universe u v w us
+variable {H : Type w} [Group H] {N : CoxeterMatrix Bool} (ds : CoxeterSystem N H)
+  {I : Type u} [Fintype I] {M : CoxeterMatrix I} {W : Type v} [Group W]
+  (cs : CoxeterSystem M W) (φ : H →* W)
+  (hφ : ∀ t, ds.IsReflection t → cs.IsReflection (φ t))
+  {σ : Type us} (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ def inducedRoot (b : H) (e : Edge ds 1 b) : PositiveRoot M cs :=
+  positiveRoot M cs ⟨φ (label ds 1 b e),hφ _ (label_isReflection ds 1 b e)⟩
+
+ def inducedLabel (b : H) (e : Edge ds 1 b) : MvPolynomial σ ℝ :=
+  linearPolynomial basis (embed M (inducedRoot ds cs φ hφ b e).val)
+
+ theorem external_step {x c y : H} (hy : y ∈ oppositeExternal ds x c) :
+    BruhatStep ds x y := by
+  classical
+  obtain ⟨hm,hp⟩ := Finset.mem_filter.mp hy
+  obtain ⟨_,hn⟩ := (mem_externalLower ds x c y).mp hm
+  have hl : ds.length x < ds.length y := by
+    have hh : ¬ ds.length y < ds.length x := fun h => hn (shorter_bruhat ds h)
+    omega
+  exact ⟨hl,y*x⁻¹,oppositeExternal_reflection ds hy,by group⟩
+
+ theorem step_external {x c y : H} (hy : BruhatStep ds x y) (hyc : BruhatLE ds y c) :
+    y ∈ oppositeExternal ds x c := by
+  classical
+  refine Finset.mem_filter.mpr ⟨(mem_externalLower ds x c y).mpr ⟨hyc,?_⟩,?_⟩
+  · exact fun h => (length_le_of_bruhat ds h).not_gt hy.1
+  · have hh := (step_iff_odd ds x y).mp hy
+    rw [Nat.odd_iff] at hh
+    omega
+
+ variable (b : H) [Fintype (Edge ds 1 b)] [DecidableEq (Interval ds 1 b)]
+
+ def externalEdge (x c : Interval ds 1 b) (y : oppositeExternal ds x.val c.val) : Edge ds 1 b :=
+  ⟨(x,⟨y.val,one_bruhat ds y.val,
+    bruhat_trans ds ((mem_externalLower ds x.val c.val y.val).mp
+      (Finset.mem_filter.mp y.property).1).1 c.property.2⟩),external_step ds y.property⟩
+
+ theorem externalEdge_mem (x c : Interval ds 1 b) (y : oppositeExternal ds x.val c.val) :
+    externalEdge ds b x c y ∈ modelOutgoing (graph ds 1 b) c x := by
+  classical
+  apply (mem_modelOutgoing _ _ _ _).mpr
+  exact ⟨rfl,((mem_externalLower ds x.val c.val y.val).mp
+    (Finset.mem_filter.mp y.property).1).1⟩
+
+ def outgoingEquiv (x c : Interval ds 1 b) :
+    ↥(oppositeExternal ds x.val c.val) ≃ ↥(modelOutgoing (graph ds 1 b) c x) where
+  toFun y := ⟨externalEdge ds b x c y,externalEdge_mem ds b x c y⟩
+  invFun e := ⟨(target ds 1 b e.val).val,by
+    obtain ⟨hs,ht⟩ := (mem_modelOutgoing _ _ _ _).mp e.property
+    have hh := e.val.property
+    change BruhatStep ds (source ds 1 b e.val).val (target ds 1 b e.val).val at hh
+    change source ds 1 b e.val = x at hs
+    rw [hs] at hh
+    exact step_external ds hh ht⟩
+  left_inv _ := rfl
+  right_inv e := by
+    apply Subtype.ext
+    apply Subtype.ext
+    apply Prod.ext
+    · exact ((mem_modelOutgoing _ _ _ _).mp e.property).1.symm
+    · rfl
+
+ theorem interpolation_product (x c : Interval ds 1 b) :
+    (∏ y : oppositeExternal ds x.val c.val,
+      linearPolynomial basis (embed M (interpolationRoot ds cs φ hφ x.val c.val y).val)) =
+      modelProduct (graph ds 1 b) (inducedLabel ds cs φ hφ basis b) c x := by
+  classical
+  calc
+    _ = ∏ e : ↥(modelOutgoing (graph ds 1 b) c x), inducedLabel ds cs φ hφ basis b e.val :=
+      (outgoingEquiv ds b x c).prod_comp _
+    _ = _ := Finset.prod_coe_sort _ _
+
+ theorem hasFlowDown (hφinj : Function.Injective φ) (c : Interval ds 1 b) :
+    HasFlowDown (graph ds 1 b) (inducedLabel ds cs φ hφ basis b) c := by
+  classical
+  intro x hxc
+  obtain ⟨F,hF,hFx,hFzero⟩ := interpolation_exists ds cs φ hφ basis hφinj hxc
+  let f : (structureSheaf (graph ds 1 b) (inducedLabel ds cs φ hφ basis b) c).Assignment :=
+    fun v _ => F v.val
+  refine ⟨f,?_,?_,?_⟩
+  · apply (structureCompatible_iff _ _ _ _ _).mpr
+    intro e _ _ h
+    have hdiv := hF (label ds 1 b e) (label_isReflection ds 1 b e)
+      (inducedRoot ds cs φ hφ b e).val (positiveRoot_represents M cs
+        ⟨φ (label ds 1 b e),hφ _ (label_isReflection ds 1 b e)⟩)
+      (source ds 1 b e).val
+    rw [label_mul_source] at hdiv
+    exact hdiv
+  · intro _
+    exact hFx.trans (interpolation_product ds cs φ hφ basis b x c)
+  · intro y hy
+    funext h
+    exact hFzero y.val h.down hy
+
+omit [Fintype (Edge ds 1 b)] [DecidableEq (Interval ds 1 b)] in
+ theorem inducedLabel_prime (e : Edge ds 1 b) : Prime (inducedLabel ds cs φ hφ basis b e) := by
+  apply (MulEquiv.prime_iff (SymmetricAlgebra.equivMvPolynomial basis)).mpr
+  apply EdgeAlgebra.symmetric_ι_prime
+  intro h
+  exact (inducedRoot ds cs φ hφ b e).property.1.ne_zero (congrArg Prod.fst h)
+
+omit [Fintype (Edge ds 1 b)] [DecidableEq (Interval ds 1 b)] in
+ theorem label_eq_of_induced_dvd (hφinj : Function.Injective φ) {e f : Edge ds 1 b}
+    (h : inducedLabel ds cs φ hφ basis b e ∣ inducedLabel ds cs φ hφ basis b f) :
+    label ds 1 b e = label ds 1 b f := by
+  apply hφinj
+  suffices he : inducedRoot ds cs φ hφ b e = inducedRoot ds cs φ hφ b f from
+    congrArg Subtype.val ((positiveRootEquiv M cs).injective he)
+  by_contra hn
+  have hd : SymmetricAlgebra.ι ℝ (Extended M) (embed M (inducedRoot ds cs φ hφ b e).val) ∣
+      SymmetricAlgebra.ι ℝ (Extended M) (embed M (inducedRoot ds cs φ hφ b f).val) := by
+    obtain ⟨z,hz⟩ := h
+    exact ⟨(SymmetricAlgebra.equivMvPolynomial basis).symm z,by
+      apply (SymmetricAlgebra.equivMvPolynomial basis).injective
+      simpa only [map_mul,AlgEquiv.apply_symm_apply,inducedLabel,linearPolynomial,LinearMap.comp_apply,AlgEquiv.toLinearMap_apply] using hz⟩
+  obtain ⟨r,hr⟩ := Submodule.mem_span_singleton.mp ((EdgeAlgebra.symmetric_ι_dvd_iff _ _).mp hd)
+  have he : r • (inducedRoot ds cs φ hφ b e).val = (1:ℝ) • (inducedRoot ds cs φ hφ b f).val := by
+    have hh := congrArg Prod.fst hr
+    exact hh.trans (one_smul ℝ _).symm
+  exact one_ne_zero ((positiveRoots_independent _ _ hn).eq_zero_of_pair' he).2
+
+ theorem model_flabby (hφinj : Function.Injective φ) (c : Interval ds 1 b) :
+    (structureSheaf (graph ds 1 b) (inducedLabel ds cs φ hφ basis b) c).Flabby := by
+  classical
+  let : Finite (Interval ds 1 b) := interval_finite ds 1 b
+  apply flowDown_flabby
+  · intro x e he f hf hne
+    apply ((inducedLabel_prime ds cs φ hφ basis b e).irreducible.isRelPrime_iff_not_dvd).mpr
+    intro hdiv
+    apply hne
+    apply eq_of_label_source_eq ds 1 b (label_eq_of_induced_dvd ds cs φ hφ basis b hφinj hdiv)
+    exact ((mem_modelOutgoing _ _ _ _).mp he).1.trans ((mem_modelOutgoing _ _ _ _).mp hf).1.symm
+  · exact hasFlowDown ds cs φ hφ basis b hφinj c
+
+end
+end KLInvariance.Dihedral
+
+end
+
+
+section
+
+/-! Instantiation on the actual maximal subgroup of the ambient root plane.
+The exact canonical roots, including degenerate/infinite planes, are used. -/
+namespace KLInvariance.TitsSpace.PlaneBasis
+open Module BruhatGraph MomentGraph MomentGraph.Sheaf
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+  {σ : Type us} (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ def modelLabel (b : planeSubgroup M cs P) (e : Edge B.system 1 b) : MvPolynomial σ ℝ :=
+  Dihedral.inducedLabel B.system cs (planeSubgroup M cs P).subtype
+    (fun t => (B.reflection_iff t).mp) basis b e
+
+ theorem model_flabby (b : planeSubgroup M cs P) [Fintype (Edge B.system 1 b)]
+    [DecidableEq (Interval B.system 1 b)] (c : Interval B.system 1 b) :
+    (structureSheaf (graph B.system 1 b) (B.modelLabel basis b) c).Flabby :=
+  Dihedral.model_flabby B.system cs (planeSubgroup M cs P).subtype
+    (fun t => (B.reflection_iff t).mp) basis b Subtype.val_injective c
+
+ theorem model_generated (b : planeSubgroup M cs P) (c : Interval B.system 1 b) :
+    (structureSheaf (graph B.system 1 b) (B.modelLabel basis b) c).Generated :=
+  structureSheaf_generated _ _ _
+
+ theorem model_quotient (b : planeSubgroup M cs P) (c : Interval B.system 1 b) :
+    (structureSheaf (graph B.system 1 b) (B.modelLabel basis b) c).UpperQuotient
+      (B.modelLabel basis b) := structureSheaf_upperQuotient _ _ _
+
+end
+end KLInvariance.TitsSpace.PlaneBasis
+
+end
+
+
+section
+
+/-! Each finite rank-two set fits in a principal Bruhat interval, even in
+an infinite dihedral group. No finite Coxeter-system hypothesis is used. -/
+namespace KLInvariance
+universe u v
+variable {I : Type u} {M : CoxeterMatrix I} {W : Type v} [Group W]
+
+ theorem bruhat_top_of_all_descents (cs : CoxeterSystem M W) (w : W)
+    (hw : ∀ i, cs.length (cs.simple i * w) < cs.length w) (x : W) :
+    BruhatLE cs x w := by
+  classical
+  have hword : ∀ ω, BruhatLE cs (cs.wordProd ω) w := by
+    intro ω
+    induction ω with
+    | nil => simpa only [cs.wordProd_nil] using one_bruhat cs w
+    | cons i ω ih =>
+      have hu := upperSimple_mono cs i ih
+      have hz := le_upperSimple cs i (cs.simple i * cs.wordProd ω)
+      rw [upperSimple_mul] at hz
+      have he : upperSimple cs i w = w := by simp only [upperSimple,ite_eq_left (hw i)]
+      rw [he] at hu
+      change BruhatLE cs (cs.simple i * cs.wordProd ω) w
+      exact hz.trans hu
+  obtain ⟨ω,_,hx⟩ := cs.exists_isReduced x
+  rw [hx]
+  exact hword ω
+
+namespace Dihedral
+variable {N : CoxeterMatrix Bool} (ds : CoxeterSystem N W)
+
+ theorem exists_common_upper (s : Finset W) : ∃ b, ∀ x ∈ s, BruhatLE ds x b := by
+  classical
+  by_cases hb : BddAbove (Set.range ds.length)
+  · have hf : (Set.range ds.length).Finite := hb.finite
+    obtain ⟨b,hb⟩ := (Set.range_nonempty ds.length).csSup_mem hf
+    have hmax (x : W) : ds.length x ≤ ds.length b := by
+      rw [hb]
+      exact le_csSup (hf.bddAbove) (Set.mem_range_self x)
+    refine ⟨b,fun x _ => bruhat_top_of_all_descents ds b ?_ x⟩
+    intro i
+    have hle := hmax (ds.simple i*b)
+    have hne := ds.length_simple_mul b i
+    omega
+  · have hn : ∀ n : ℕ, ∃ w, n < ds.length w := by
+      intro n
+      by_contra h
+      push Not at h
+      exact hb ⟨n,by rintro m ⟨x,rfl⟩; exact h x⟩
+    obtain ⟨b,hb⟩ := hn (s.sup ds.length)
+    exact ⟨b,fun x hx => shorter_bruhat ds ((Finset.le_sup hx).trans_lt hb)⟩
+
+end Dihedral
+end KLInvariance
+
+end
+
+
+section
+
+namespace KLInvariance.Dihedral
+variable {W : Type*} [Group W] {M : CoxeterMatrix Bool}
+noncomputable section
+
+ theorem map_not_alt (i : Bool) (n : ℕ) : (alt i n).map Bool.not = alt (!i) n := by
+  induction n generalizing i with
+  | zero => rfl
+  | succ n ih => rw [alt_succ,List.map_cons,ih,alt_succ]
+
+ theorem swap_alt (cs : CoxeterSystem M W) (i : Bool) (n : ℕ) :
+    swap cs (cs.wordProd (alt i n)) = cs.wordProd (alt (!i) n) := by
+  rw [swap_wordProd,map_not_alt]
+
+ theorem length_alt_le_length (cs : CoxeterSystem M W) (i : Bool) {n : ℕ} {w : W}
+    (h : n ≤ cs.length w) : cs.length (cs.wordProd (alt i n)) = n := by
+  obtain ⟨j,hj⟩ := exists_alt cs w
+  have hr : cs.IsReduced (alt j (cs.length w)) := by
+    change cs.length (cs.wordProd _) = _
+    rw [← hj,length_alt]
+  have hn := (reduced_alt_take cs hr h).eq
+  rw [length_alt] at hn
+  have hi : i=j ∨ i=(!j) := by cases i <;> cases j <;> simp
+  rcases hi with rfl|rfl
+  · exact hn
+  · rw [← swap_alt,length_swap,hn]
+
+ theorem length_indexWord_nat (cs : CoxeterSystem M W) {n : ℕ} {w : W}
+    (h : n ≤ cs.length w) : cs.length (indexWord cs n) = n := by
+  rw [indexWord_nat]
+  exact length_alt_le_length cs false h
+
+ theorem length_indexWord_neg_nat (cs : CoxeterSystem M W) {n : ℕ} {w : W}
+    (h : n ≤ cs.length w) : cs.length (indexWord cs (-(n:ℤ))) = n := by
+  rw [indexWord_neg_nat]
+  exact length_alt_le_length cs true h
+
+ theorem length_indexWord (cs : CoxeterSystem M W) {i : ℤ} {w : W}
+    (h : i.natAbs ≤ cs.length w) : cs.length (indexWord cs i) = i.natAbs := by
+  obtain hi|hi := le_total 0 i
+  · have he : i=(i.natAbs:ℤ) := by rw [Int.natCast_natAbs,abs_of_nonneg hi]
+    nth_rw 1 [he]
+    exact length_indexWord_nat cs h
+  · have he : i= -(i.natAbs:ℤ) := by rw [Int.natCast_natAbs,abs_of_nonpos hi,neg_neg]
+    nth_rw 1 [he]
+    exact length_indexWord_neg_nat cs h
+
+ theorem leftDescent_alt (cs : CoxeterSystem M W) (i : Bool) {n : ℕ} {w : W}
+    (hn : 0<n) (h : n ≤ cs.length w) :
+    cs.IsLeftDescent (cs.wordProd (alt i n)) i := by
+  obtain ⟨k,rfl⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : n≠0)
+  unfold CoxeterSystem.IsLeftDescent
+  rw [length_alt_le_length cs i h,alt_succ,cs.wordProd_cons,cs.simple_mul_simple_cancel_left]
+  have hh := cs.length_wordProd_le (alt (!i) k)
+  rw [length_alt] at hh
+  omega
+
+ theorem indexWord_add_even (cs : CoxeterSystem M W) (i n : ℤ) :
+    indexWord cs (i+2*n) = rotation cs ^ n * indexWord cs i := by
+  have hm : (i+2*n)%2=i%2 := by omega
+  have hd : (i+2*n)/2=i/2+n := by omega
+  simp only [indexWord,hm,hd,zpow_add]
+  split_ifs
+  · exact (Commute.zpow_zpow_self (rotation cs) (i/2) n).eq
+  · rw [← mul_assoc,(Commute.zpow_zpow_self (rotation cs) (i/2) n).eq,mul_assoc]
+
+/-- The maximal-length case supplies the precise signed period, without an
+extra finiteness hypothesis or a numerical cosine restriction. -/
+ theorem rotation_pow_max (cs : CoxeterSystem M W) (w : W)
+    (hmax : ∀ z, cs.length z ≤ cs.length w) : rotation cs ^ cs.length w = 1 := by
+  have hd (i : Bool) : cs.IsLeftDescent w i := by
+    have hle := hmax (cs.simple i*w)
+    have hne := cs.length_simple_mul w i
+    unfold CoxeterSystem.IsLeftDescent
+    omega
+  have he : indexWord cs (cs.length w) = indexWord cs (-(cs.length w:ℤ)) := by
+    rw [indexWord_nat,indexWord_neg_nat,← eq_alt_of_descent cs false (hd false),
+      ← eq_alt_of_descent cs true (hd true)]
+  have hp := indexWord_add_even cs (-(cs.length w:ℤ)) (cs.length w)
+  rw [show -(cs.length w:ℤ)+2*(cs.length w:ℤ)=(cs.length w:ℤ) by ring,
+    zpow_natCast,he] at hp
+  exact mul_right_cancel (show rotation cs ^ cs.length w * indexWord cs (-(cs.length w:ℤ)) =
+      1 * indexWord cs (-(cs.length w:ℤ)) by simpa only [one_mul] using hp.symm)
+
+ theorem indexWord_period (cs : CoxeterSystem M W) {N : ℕ}
+    (hN : rotation cs ^ N = 1) (i k : ℤ) :
+    indexWord cs (i+2*(N:ℤ)*k) = indexWord cs i := by
+  rw [show i+2*(N:ℤ)*k=i+2*((N:ℤ)*k) by ring,indexWord_add_even,zpow_mul,zpow_natCast,hN,
+    one_zpow,one_mul]
+
+ theorem indexWord_mod_period (cs : CoxeterSystem M W) {N : ℕ}
+    (hN : rotation cs ^ N = 1) (i : ℤ) :
+    indexWord cs (i%(2*(N:ℤ))) = indexWord cs i := by
+  have hp := indexWord_period cs hN (i%(2*(N:ℤ))) (i/(2*(N:ℤ)))
+  rw [Int.emod_add_mul_ediv] at hp
+  exact hp.symm
+
+end
+end KLInvariance.Dihedral
+
+end
+
+
+section
+
+namespace KLInvariance.Dihedral
+variable {W : Type*} [Group W] {M : CoxeterMatrix Bool}
+
+ theorem indexWord_step (cs : CoxeterSystem M W) {n : ℕ} {w : W}
+    (h : n+1 ≤ cs.length w) : BruhatStep cs (indexWord cs n) (indexWord cs (n+1:ℕ)) := by
+  rw [step_iff_odd,length_indexWord_nat cs (show n≤cs.length w by omega),
+    length_indexWord_nat cs h]
+  exact ⟨by omega,by simp⟩
+
+ theorem neg_indexWord_step (cs : CoxeterSystem M W) {n : ℕ} {w : W}
+    (h : n+1 ≤ cs.length w) :
+    BruhatStep cs (indexWord cs (-(n:ℤ))) (indexWord cs (-((n+1:ℕ):ℤ))) := by
+  rw [step_iff_odd,length_indexWord_neg_nat cs (show n≤cs.length w by omega),
+    length_indexWord_neg_nat cs h]
+  exact ⟨by omega,by simp⟩
+
+end KLInvariance.Dihedral
+
+namespace KLInvariance.TitsSpace.PlaneBasis
+open KLInvariance.Dihedral
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+noncomputable section
+
+ def prefixRoot (w : planeSubgroup M cs P) (n : ℕ) : PositiveRoot M cs :=
+  if h : n < B.system.length w then B.edgeRoot (indexWord_step B.system h) else B.left
+
+ def negPrefixRoot (w : planeSubgroup M cs P) (n : ℕ) : PositiveRoot M cs :=
+  if h : n < B.system.length w then B.edgeRoot (neg_indexWord_step B.system h) else B.right
+
+ theorem prefixRoot_strictMono (w : planeSubgroup M cs P) :
+    StrictMonoOn (fun n => geometricSlope (B.prefixRoot w n)) (Set.Iio (B.system.length w)) := by
+  apply strictMonoOn_of_lt_succ Set.ordConnected_Iio
+  intro n _ hn hn'
+  change n < B.system.length w at hn
+  change n+1 < B.system.length w at hn'
+  change geometricSlope (B.prefixRoot w n) < geometricSlope (B.prefixRoot w (n+1))
+  rw [prefixRoot,prefixRoot,dite_eq_left hn,dite_eq_left hn']
+  apply (B.edgeRoot_slopes_iff _ _).mpr
+  rw [indexWord_nat]
+  exact leftDescent_alt B.system false (by omega) hn'.le
+
+ theorem negPrefixRoot_strictAnti (w : planeSubgroup M cs P) :
+    StrictAntiOn (fun n => geometricSlope (B.negPrefixRoot w n))
+      (Set.Iio (B.system.length w)) := by
+  apply strictAntiOn_of_succ_lt Set.ordConnected_Iio
+  intro n _ hn hn'
+  change n < B.system.length w at hn
+  change n+1 < B.system.length w at hn'
+  change geometricSlope (B.negPrefixRoot w (n+1)) < geometricSlope (B.negPrefixRoot w n)
+  rw [negPrefixRoot,negPrefixRoot,dite_eq_left hn,dite_eq_left hn']
+  have ht : B.system.IsLeftDescent (indexWord B.system (-((n+1:ℕ):ℤ))) true := by
+    rw [indexWord_neg_nat]
+    exact leftDescent_alt B.system true (by omega) hn'.le
+  have hlen := length_indexWord_neg_nat B.system hn'.le
+  have hu := unique_leftDescent B.system (w := indexWord B.system (-((n+1:ℕ):ℤ)))
+    (z := w) (by rw [hlen]; omega) (by rw [hlen]; omega)
+  have hf : ¬B.system.IsLeftDescent (indexWord B.system (-((n+1:ℕ):ℤ))) false := by
+    intro hf
+    obtain ⟨i,_,hi⟩ := hu
+    have h1 := hi true ht
+    have h2 := hi false hf
+    simp_all
+  have hnlt := (B.edgeRoot_slopes_iff (neg_indexWord_step B.system hn)
+    (neg_indexWord_step B.system hn')).not.mpr hf
+  apply lt_of_le_of_ne (le_of_not_gt hnlt)
+  intro he
+  have he' := geometricSlope_injective he
+  exact graphRoot_ne_of_path
+    ((B.bruhatStep_iff _ _).mpr (neg_indexWord_step B.system hn))
+    ((B.bruhatStep_iff _ _).mpr (neg_indexWord_step B.system hn')) he'.symm
+
+ theorem edgeRoot_eq_of_label {x y z w : planeSubgroup M cs P}
+    (hxy : BruhatStep B.system x y) (hzw : BruhatStep B.system z w)
+    (he : y*x⁻¹=w*z⁻¹) : B.edgeRoot hxy = B.edgeRoot hzw := by
+  unfold edgeRoot graphRoot
+  apply congrArg (positiveRoot M cs)
+  apply Subtype.ext
+  exact congrArg (fun z : planeSubgroup M cs P => (z : W)) he
+
+ theorem prefixRoot_eq_edgeRoot {x y w : planeSubgroup M cs P}
+    (hxy : BruhatStep B.system x y) {n : ℕ} (hn : n<B.system.length w)
+    (he : y*x⁻¹=indexWord B.system (2*(n:ℤ)+1)) :
+    B.prefixRoot w n = B.edgeRoot hxy := by
+  rw [prefixRoot,dite_eq_left hn]
+  apply B.edgeRoot_eq_of_label
+  rw [he,indexWord_label B.system (by rw [Int.odd_iff]; omega)]
+  congr 1
+  push_cast
+  ring
+
+ theorem negPrefixRoot_eq_edgeRoot {x y w : planeSubgroup M cs P}
+    (hxy : BruhatStep B.system x y) {n : ℕ} (hn : n<B.system.length w)
+    (he : y*x⁻¹=indexWord B.system (-(2*(n:ℤ)+1))) :
+    B.negPrefixRoot w n = B.edgeRoot hxy := by
+  rw [negPrefixRoot,dite_eq_left hn]
+  apply B.edgeRoot_eq_of_label
+  rw [he,indexWord_label B.system (by rw [Int.odd_iff]; omega)]
+  congr 1
+  push_cast
+  ring
+
+end
+end KLInvariance.TitsSpace.PlaneBasis
+
+end
+
+
+section
+
+
+namespace KLInvariance.SignedDihedral
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+ def Before (a b : ℤ) : Prop :=
+  (0 < a ∧ 0 < b ∧ a < b) ∨ (a < 0 ∧ b < 0 ∧ a < b) ∨ (0 < a ∧ b < 0)
+
+ theorem before_same_source_pos {i j z : ℤ} (hij : |i| < |j|)
+    (hiz : |i| < |z|) (hj : 0 < j) :
+    Before (i+j) (i+z) ↔ z < 0 ∨ j < z := by
+  rw [abs_of_pos hj, abs_lt] at hij
+  obtain hz | hz := le_total 0 z
+  · rw [abs_of_nonneg hz, abs_lt] at hiz
+    dsimp [Before]
+    omega
+  · rw [abs_of_nonpos hz, abs_lt] at hiz
+    dsimp [Before]
+    omega
+
+ theorem before_same_source_neg {i j z : ℤ} (hij : |i| < |j|)
+    (hiz : |i| < |z|) (hj : j < 0) :
+    Before (i+j) (i+z) ↔ z < 0 ∧ j < z := by
+  rw [abs_of_neg hj, abs_lt] at hij
+  obtain hz | hz := le_total 0 z
+  · rw [abs_of_nonneg hz, abs_lt] at hiz
+    dsimp [Before]
+    omega
+  · rw [abs_of_nonpos hz, abs_lt] at hiz
+    dsimp [Before]
+    omega
+
+ theorem before_next_source {i j z : ℤ} (hij : |i| < |j|)
+    (hjz : |j| < |z|) :
+    Before (i+j) (j+z) ↔ 0 < j := by
+  obtain hj | hj := le_total 0 j
+  · rw [abs_of_nonneg hj, abs_lt] at hij
+    obtain hz | hz := le_total 0 z
+    · rw [abs_of_nonneg hj, abs_of_nonneg hz] at hjz
+      dsimp [Before]
+      omega
+    · rw [abs_of_nonneg hj, abs_of_nonpos hz] at hjz
+      dsimp [Before]
+      omega
+  · rw [abs_of_nonpos hj, abs_lt] at hij
+    obtain hz | hz := le_total 0 z
+    · rw [abs_of_nonpos hj, abs_of_nonneg hz] at hjz
+      dsimp [Before]
+      omega
+    · rw [abs_of_nonpos hj, abs_of_nonpos hz] at hjz
+      dsimp [Before]
+      omega
+
+ theorem midpoint_mod {m a : ℤ} (_hm : 0 < m) (ha : -2*m < a) (ha' : a < 2*m) :
+    a % (2*m) = if 0 ≤ a then a else 2*m+a := by
+  split_ifs with h
+  · exact Int.emod_eq_of_lt h ha'
+  · have he : a % (2*m) = (a+2*m) % (2*m) := by simp
+    rw [he, Int.emod_eq_of_lt (by omega) (by omega)]
+    omega
+
+/-- For two outgoing edges at the same vertex, reduction modulo the period
+has no wrap-around exception in the midpoint comparison table. -/
+ theorem finite_before_same {m i j z : ℤ} (hm : 0 < m)
+    (hi : |i| < |j|) (hz : |i| < |z|)
+    (hjM : -m < j ∧ j ≤ m) (hzM : -m < z ∧ z ≤ m) :
+    (i+j) % (2*m) < (i+z) % (2*m) ↔ Before (i+j) (i+z) := by
+  have hjabs : |j| ≤ m := abs_le.mpr (by omega)
+  have hzabs : |z| ≤ m := abs_le.mpr (by omega)
+  have hiabs := abs_lt.mp (hi.trans_le hjabs)
+  rw [midpoint_mod hm (by omega) (by omega), midpoint_mod hm (by omega) (by omega)]
+  obtain hjs | hjs := le_total 0 j
+  · rw [abs_of_nonneg hjs, abs_lt] at hi
+    obtain hzs | hzs := le_total 0 z
+    · rw [abs_of_nonneg hzs, abs_lt] at hz
+      dsimp [Before]
+      split_ifs <;> omega
+    · rw [abs_of_nonpos hzs, abs_lt] at hz
+      dsimp [Before]
+      split_ifs <;> omega
+  · rw [abs_of_nonpos hjs, abs_lt] at hi
+    obtain hzs | hzs := le_total 0 z
+    · rw [abs_of_nonneg hzs, abs_lt] at hz
+      dsimp [Before]
+      split_ifs <;> omega
+    · rw [abs_of_nonpos hzs, abs_lt] at hz
+      dsimp [Before]
+      split_ifs <;> omega
+
+ theorem finite_before_next {m i j z : ℤ} (hm : 0 < m)
+    (hi : |i| < |j|) (hz : |j| < |z|)
+    (hzM : -m < z ∧ z ≤ m) :
+    (i+j) % (2*m) < (j+z) % (2*m) ↔ Before (i+j) (j+z) := by
+  have hzabs : |z| ≤ m := abs_le.mpr (by omega)
+  have hjabs := abs_lt.mp (hz.trans_le hzabs)
+  have hiabs := abs_lt.mp ((hi.trans hz).trans_le hzabs)
+  rw [midpoint_mod hm (by omega) (by omega), midpoint_mod hm (by omega) (by omega)]
+  obtain hjs | hjs := le_total 0 j
+  · rw [abs_of_nonneg hjs, abs_lt] at hi
+    obtain hzs | hzs := le_total 0 z
+    · rw [abs_of_nonneg hjs, abs_of_nonneg hzs] at hz
+      dsimp [Before]
+      split_ifs <;> omega
+    · rw [abs_of_nonneg hjs, abs_of_nonpos hzs] at hz
+      dsimp [Before]
+      split_ifs <;> omega
+  · rw [abs_of_nonpos hjs, abs_lt] at hi
+    obtain hzs | hzs := le_total 0 z
+    · rw [abs_of_nonpos hjs, abs_of_nonneg hzs] at hz
+      dsimp [Before]
+      split_ifs <;> omega
+    · rw [abs_of_nonpos hjs, abs_of_nonpos hzs] at hz
+      dsimp [Before]
+      split_ifs <;> omega
+
+ theorem eligible_ranks_card (L k : ℕ) :
+    ((Finset.range (L+2*k+1)).filter (fun r => L < r ∧ Odd (L+r))).card = k := by
+  classical
+  have hc : (Finset.range k).card =
+      ((Finset.range (L+2*k+1)).filter (fun r => L < r ∧ Odd (L+r))).card := by
+    apply Finset.card_bij (fun a _ => L+1+2*a)
+    · intro a ha
+      simp only [Finset.mem_range, Finset.mem_filter, Nat.odd_iff] at ha ⊢
+      omega
+    · intro a ha b hb he
+      omega
+    · intro b hb
+      simp only [Finset.mem_range, Finset.mem_filter, Nat.odd_iff] at hb
+      refine ⟨(b-L-1)/2, ?_, ?_⟩
+      · simp only [Finset.mem_range]
+        omega
+      · omega
+  simpa only [Finset.card_range] using hc.symm
+
+ noncomputable def arrowAfter (a label z : ℤ) : ℕ :=
+  if |a| < |z| ∧ Odd (a+z) ∧ Before label (a+z) then 1 else 0
+
+/-- Double internal ranks and the singleton top, with no singleton bottom
+term because upward arrows never end at the bottom. -/
+ noncomputable def countAfter (a label c : ℤ) : ℕ :=
+  (∑ r ∈ Finset.range c.natAbs, (arrowAfter a label r + arrowAfter a label (-r))) +
+    arrowAfter a label c
+
+ theorem arrowAfter_of_not_lt {a label z : ℤ} (h : ¬ |a| < |z|) :
+    arrowAfter a label z = 0 := by simp [arrowAfter,h]
+
+ theorem arrowAfter_self (i j : ℤ) : arrowAfter i (i+j) j = 0 := by
+  have hb : ¬ Before (i+j) (i+j) := by
+    dsimp [Before]
+    omega
+  simp only [arrowAfter, hb, and_false, ite_false]
+
+end
+end KLInvariance.SignedDihedral
+
+end
+
+
+section
+
+namespace KLInvariance.Dihedral
+variable {W : Type*} [Group W] {M : CoxeterMatrix Bool}
+
+ theorem length_indexWord_unbounded (cs : CoxeterSystem M W)
+    (hu : ∀ n : ℕ, ∃ w, n ≤ cs.length w) (i : ℤ) :
+    cs.length (indexWord cs i) = i.natAbs := by
+  obtain ⟨w,hw⟩ := hu i.natAbs
+  exact length_indexWord cs hw
+
+end KLInvariance.Dihedral
+
+namespace KLInvariance.TitsSpace.PlaneBasis
+open KLInvariance.Dihedral
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+noncomputable section
+
+ theorem edgeRoot_inverted_iff {x y : planeSubgroup M cs P}
+    (hxy : BruhatStep B.system x y) (w : planeSubgroup M cs P) :
+    rootInverted (w:W) (B.edgeRoot hxy) ↔ B.system.length ((y*x⁻¹)*w)<B.system.length w := by
+  have hh := B.inversion_iff (t := y*x⁻¹)
+    (graphRoot_represents ((B.bruhatStep_iff x y).mpr hxy))
+    (B.edgeRoot_plane hxy) (B.edgeRoot hxy).property.2 w
+  change rootInverted (w:W) (B.edgeRoot hxy) ↔ _ at hh
+  rw [hh]
+  exact and_iff_right hxy.label_reflection
+
+ theorem edgeRoot_positive_lt_negative
+    (hu : ∀ n : ℕ, ∃ w, n ≤ B.system.length w)
+    {x y z t : planeSubgroup M cs P}
+    (hxy : BruhatStep B.system x y) (hzt : BruhatStep B.system z t)
+    (k l : ℕ)
+    (hx : y*x⁻¹=indexWord B.system (2*(k:ℤ)+1))
+    (hz : t*z⁻¹=indexWord B.system (-(2*(l:ℤ)+1))) :
+    geometricSlope (B.edgeRoot hxy) < geometricSlope (B.edgeRoot hzt) := by
+  let w := indexWord B.system ((k+1:ℕ):ℤ)
+  have hi : rootInverted (w:W) (B.edgeRoot hxy) := by
+    rw [B.edgeRoot_inverted_iff,hx]
+    unfold w
+    rw [indexWord_reflection_mul B.system (by rw [Int.odd_iff]; omega),
+      length_indexWord_unbounded B.system hu,length_indexWord_unbounded B.system hu]
+    have he : 2*(k:ℤ)+1-((k+1:ℕ):ℤ)=(k:ℤ) := by omega
+    rw [he]
+    simp only [Int.natAbs_natCast]
+    omega
+  have hn : ¬rootInverted (w:W) (B.edgeRoot hzt) := by
+    rw [B.edgeRoot_inverted_iff,hz]
+    unfold w
+    rw [indexWord_reflection_mul B.system (by rw [Int.odd_iff]; omega),
+      length_indexWord_unbounded B.system hu,length_indexWord_unbounded B.system hu]
+    have he : -(2*(l:ℤ)+1)-((k+1:ℕ):ℤ) = -((2*l+k+2:ℕ):ℤ) := by omega
+    rw [he]
+    simp only [Int.natAbs_neg,Int.natAbs_natCast]
+    omega
+  have hd : B.system.IsLeftDescent w false := by
+    obtain ⟨w',hw'⟩ := hu (k+1)
+    change B.system.IsLeftDescent (indexWord B.system ((k+1:ℕ):ℤ)) false
+    rw [indexWord_nat]
+    exact leftDescent_alt B.system false (by omega) hw'
+  have hm : ∀ z ∈ planeSubgroup M cs P, cs.length (1:W) ≤ cs.length (z*1) :=
+    fun z _ => by simp
+  have hl : rootInverted (w:W) B.left := by
+    have hh := (B.simpleRoot_inverted_coset_iff hm w false).mpr hd
+    simpa only [mul_one,simpleRoot,Bool.false_eq_true,↓reduceIte] using hh
+  exact B.slope_lt_of_inverted_left (w:W) hl _ _ (B.edgeRoot_plane hxy)
+    (B.edgeRoot_plane hzt) hi hn
+
+/-- The two angular rays of an infinite canonical dihedral subsystem, with
+actual ambient labels, have precisely the source's signed-midpoint order. -/
+ theorem edgeRoot_infinite_order
+    (hu : ∀ n : ℕ, ∃ w, n ≤ B.system.length w)
+    {x y z t : planeSubgroup M cs P}
+    (hxy : BruhatStep B.system x y) (hzt : BruhatStep B.system z t)
+    {a b : ℤ} (ha : Odd a) (hb : Odd b)
+    (hx : y*x⁻¹=indexWord B.system a) (hz : t*z⁻¹=indexWord B.system b) :
+    geometricSlope (B.edgeRoot hxy) < geometricSlope (B.edgeRoot hzt) ↔
+      SignedDihedral.Before a b := by
+  have repr (q : ℤ) (hq : Odd q) :
+      ∃ k : ℕ, q=2*(k:ℤ)+1 ∨ q= -(2*(k:ℤ)+1) := by
+    have habs := Int.natCast_natAbs q
+    have ho : q.natAbs % 2=1 := by
+      rw [Int.odd_iff] at hq
+      rcases le_total 0 q with h|h
+      · rw [abs_of_nonneg h] at habs
+        omega
+      · rw [abs_of_nonpos h] at habs
+        omega
+    refine ⟨q.natAbs/2,?_⟩
+    rcases le_total 0 q with h|h
+    · left
+      rw [abs_of_nonneg h] at habs
+      omega
+    · right
+      rw [abs_of_nonpos h] at habs
+      omega
+  obtain ⟨k,hk|hk⟩ := repr a ha <;> obtain ⟨l,hl|hl⟩ := repr b hb
+  · obtain ⟨w,hw⟩ := hu (max k l+1)
+    have hk' : k<B.system.length w := by omega
+    have hl' : l<B.system.length w := by omega
+    rw [hk] at hx
+    rw [hl] at hz
+    rw [← B.prefixRoot_eq_edgeRoot hxy hk' hx,← B.prefixRoot_eq_edgeRoot hzt hl' hz]
+    have hs := B.prefixRoot_strictMono w
+    have he : geometricSlope (B.prefixRoot w k) < geometricSlope (B.prefixRoot w l) ↔ k<l :=
+      hs.lt_iff_lt hk' hl'
+    rw [he,hk,hl]
+    unfold SignedDihedral.Before
+    omega
+  · rw [hk] at hx
+    rw [hl] at hz
+    have hh := B.edgeRoot_positive_lt_negative hu hxy hzt k l hx hz
+    have hbefore : SignedDihedral.Before a b := by
+      rw [hk,hl]
+      unfold SignedDihedral.Before
+      omega
+    exact iff_of_true hh hbefore
+  · rw [hk] at hx
+    rw [hl] at hz
+    have hh := B.edgeRoot_positive_lt_negative hu hzt hxy l k hz hx
+    have hn := not_lt_of_gt hh
+    have hbefore : ¬SignedDihedral.Before a b := by
+      rw [hk,hl]
+      unfold SignedDihedral.Before
+      omega
+    exact iff_of_false hn hbefore
+  · obtain ⟨w,hw⟩ := hu (max k l+1)
+    have hk' : k<B.system.length w := by omega
+    have hl' : l<B.system.length w := by omega
+    rw [hk] at hx
+    rw [hl] at hz
+    rw [← B.negPrefixRoot_eq_edgeRoot hxy hk' hx,← B.negPrefixRoot_eq_edgeRoot hzt hl' hz]
+    have hs := B.negPrefixRoot_strictAnti w
+    have he : geometricSlope (B.negPrefixRoot w k) < geometricSlope (B.negPrefixRoot w l) ↔ l<k := by
+      constructor
+      · intro h
+        by_contra hn
+        rcases lt_or_eq_of_le (le_of_not_gt hn) with hkl | rfl
+        · exact (not_lt_of_gt (hs hk' hl' hkl)) h
+        · exact (lt_irrefl _) h
+      · exact hs hl' hk'
+    rw [he,hk,hl]
+    unfold SignedDihedral.Before
+    omega
+
+end
+end KLInvariance.TitsSpace.PlaneBasis
+
+end
+
+
+section
+
+namespace KLInvariance.TitsSpace.PlaneBasis
+open KLInvariance.Dihedral
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+noncomputable section
+
+ theorem max_length_pos (w : planeSubgroup M cs P)
+    (hmax : ∀ z, B.system.length z ≤ B.system.length w) : 0 < B.system.length w := by
+  have h := hmax (B.system.simple false)
+  rw [B.system.length_simple] at h
+  omega
+
+ theorem odd_residue_repr {N : ℕ} (hN : 0<N) {a : ℤ} (ha : Odd a) :
+    ∃ k : ℕ, k<N ∧ a % (2*(N:ℤ)) = 2*(k:ℤ)+1 := by
+  have hl := Int.emod_nonneg a (by omega : (2*(N:ℤ))≠0)
+  have hu := Int.emod_lt_of_pos a (by omega : 0<2*(N:ℤ))
+  have hp : (a%(2*(N:ℤ)))%2=1 := by
+    rw [Int.emod_emod_of_dvd a (by exact dvd_mul_right 2 (N:ℤ))]
+    exact Int.odd_iff.mp ha
+  refine ⟨(a%(2*(N:ℤ))).toNat/2, ?_, ?_⟩ <;> omega
+
+/-- The actual ambient angular order in a finite canonical plane subsystem
+is the increasing residue order on odd doubled midpoints. -/
+ theorem edgeRoot_finite_order
+    (w : planeSubgroup M cs P) (hmax : ∀ z, B.system.length z ≤ B.system.length w)
+    {x y z t : planeSubgroup M cs P}
+    (hxy : BruhatStep B.system x y) (hzt : BruhatStep B.system z t)
+    {a b : ℤ} (ha : Odd a) (hb : Odd b)
+    (hx : y*x⁻¹=indexWord B.system a) (hz : t*z⁻¹=indexWord B.system b) :
+    geometricSlope (B.edgeRoot hxy) < geometricSlope (B.edgeRoot hzt) ↔
+      a % (2*(B.system.length w:ℤ)) < b % (2*(B.system.length w:ℤ)) := by
+  obtain ⟨k,hk,he⟩ := odd_residue_repr (B.max_length_pos w hmax) ha
+  obtain ⟨l,hl,hf⟩ := odd_residue_repr (B.max_length_pos w hmax) hb
+  have hp := rotation_pow_max B.system w hmax
+  rw [← indexWord_mod_period B.system hp a, he] at hx
+  rw [← indexWord_mod_period B.system hp b, hf] at hz
+  rw [← B.prefixRoot_eq_edgeRoot hxy hk hx,← B.prefixRoot_eq_edgeRoot hzt hl hz]
+  rw [(B.prefixRoot_strictMono w).lt_iff_lt hk hl,he,hf]
+  omega
+
+end
+end KLInvariance.TitsSpace.PlaneBasis
+
+end
+
+
+section
+
+namespace KLInvariance.Dihedral
+variable {W : Type*} [Group W] {M : CoxeterMatrix Bool}
+noncomputable section
+local instance : DecidableEq W := Classical.decEq W
+
+ theorem indexWord_pos_ne_neg (cs : CoxeterSystem M W) {n : ℕ} {w : W}
+    (hn : 0<n) (hw : n<cs.length w) : indexWord cs n ≠ indexWord cs (-(n:ℤ)) := by
+  have hp := leftDescent_alt cs false hn hw.le
+  have hm := leftDescent_alt cs true hn hw.le
+  rw [← indexWord_nat] at hp
+  rw [← indexWord_neg_nat] at hm
+  intro he
+  rw [← he] at hm
+  have hl := length_indexWord_nat cs hw.le
+  obtain ⟨i,_,hi⟩ := unique_leftDescent cs (w := indexWord cs n) (z := w)
+    (by rw [hl]; exact hn) (by rw [hl]; exact hw)
+  have hf := hi false hp
+  have ht := hi true hm
+  simp_all
+
+ theorem lower_length_fiber (cs : CoxeterSystem M W) (c : W) {n : ℕ}
+    (hc : n<cs.length c) :
+    (lowerFinset cs c).filter (fun z => cs.length z=n) =
+      {indexWord cs n,indexWord cs (-(n:ℤ))} := by
+  classical
+  ext z
+  simp only [Finset.mem_filter,mem_lowerFinset,Finset.mem_insert,Finset.mem_singleton]
+  constructor
+  · intro hz
+    obtain ⟨i,hi⟩ := exists_alt cs z
+    rw [hz.2] at hi
+    cases i
+    · exact Or.inl (hi.trans (indexWord_nat cs n).symm)
+    · exact Or.inr (hi.trans (indexWord_neg_nat cs n).symm)
+  · rintro (rfl|rfl)
+    · have hl := length_indexWord_nat cs hc.le
+      exact ⟨shorter_bruhat cs (by rw [hl]; exact hc),hl⟩
+    · have hl := length_indexWord_neg_nat cs hc.le
+      exact ⟨shorter_bruhat cs (by rw [hl]; exact hc),hl⟩
+
+ theorem lower_zero_fiber (cs : CoxeterSystem M W) (c : W) :
+    (lowerFinset cs c).filter (fun z => cs.length z=0) = {1} := by
+  classical
+  ext z
+  simp only [Finset.mem_filter,mem_lowerFinset,Finset.mem_singleton]
+  constructor
+  · exact fun hz => cs.length_eq_zero_iff.mp hz.2
+  · rintro rfl
+    exact ⟨one_bruhat cs c,cs.length_one⟩
+
+ theorem lower_top_fiber (cs : CoxeterSystem M W) (c : W) :
+    (lowerFinset cs c).filter (fun z => cs.length z=cs.length c) = {c} := by
+  classical
+  ext z
+  simp only [Finset.mem_filter,mem_lowerFinset,Finset.mem_singleton]
+  constructor
+  · intro hz
+    by_contra hne
+    have hl := length_lt_of_bruhat_ne cs hz.1 hne
+    omega
+  · rintro rfl
+    exact ⟨bruhat_refl cs _,rfl⟩
+
+/-- Actual finite dihedral lower ideals have two internal alternating rays
+and a singleton top. The doubled identity contributes zero. -/
+ theorem sum_lower_signed (cs : CoxeterSystem M W) (c : W)
+    {A : Type*} [AddCommMonoid A] (f : W → A) (h1 : f 1=0) :
+    ∑ z ∈ lowerFinset cs c, f z =
+      (∑ n ∈ Finset.range (cs.length c),
+        (f (indexWord cs n)+f (indexWord cs (-(n:ℤ))))) + f c := by
+  classical
+  have hmap : ∀ z ∈ lowerFinset cs c, cs.length z ∈ Finset.range (cs.length c+1) := by
+    intro z hz
+    have hl := length_le_of_bruhat cs ((mem_lowerFinset cs c z).mp hz)
+    simp only [Finset.mem_range]
+    omega
+  rw [← Finset.sum_fiberwise_of_maps_to hmap f,Finset.sum_range_succ,lower_top_fiber,
+    Finset.sum_singleton]
+  congr 1
+  apply Finset.sum_congr rfl
+  intro n hn
+  have hc := Finset.mem_range.mp hn
+  by_cases hz : n=0
+  · subst n
+    rw [lower_zero_fiber]
+    simp only [Finset.sum_singleton,indexWord_nat,indexWord_neg_nat,alt_zero,
+      cs.wordProd_nil,h1,add_zero]
+  · rw [lower_length_fiber cs c hc,
+      Finset.sum_pair (indexWord_pos_ne_neg cs (by omega) hc)]
+
+end
+end KLInvariance.Dihedral
+
+end
+
+
+section
+
+namespace KLInvariance.Dihedral
+variable {W : Type*} [Group W] {M : CoxeterMatrix Bool}
+noncomputable section
+
+ def GoodIndex (cs : CoxeterSystem M W) (i : ℤ) : Prop :=
+  ∀ w, (∀ z, cs.length z ≤ cs.length w) → -(cs.length w:ℤ) < i ∧ i ≤ (cs.length w:ℤ)
+
+ theorem maximal_or_unbounded (cs : CoxeterSystem M W) :
+    (∃ w, ∀ z, cs.length z ≤ cs.length w) ∨ (∀ n : ℕ, ∃ w, n≤cs.length w) := by
+  by_cases hb : BddAbove (Set.range cs.length)
+  · have hf : (Set.range cs.length).Finite := hb.finite
+    obtain ⟨b,hb⟩ := (Set.range_nonempty cs.length).csSup_mem hf
+    left
+    refine ⟨b,fun z => ?_⟩
+    rw [hb]
+    exact le_csSup hf.bddAbove (Set.mem_range_self z)
+  · right
+    intro n
+    by_contra h
+    push Not at h
+    exact hb ⟨n,by rintro m ⟨z,rfl⟩; exact (h z).le⟩
+
+ theorem max_length_positive (cs : CoxeterSystem M W) {w : W}
+    (hmax : ∀ z, cs.length z ≤ cs.length w) : 0<cs.length w := by
+  have h := hmax (cs.simple false)
+  rw [cs.length_simple] at h
+  omega
+
+ theorem maximal_eq (cs : CoxeterSystem M W) {w z : W}
+    (hmax : ∀ x, cs.length x≤cs.length w) (hz : cs.length z=cs.length w) : z=w := by
+  have htop := bruhat_top_of_all_descents cs w (fun i => by
+    have h := hmax (cs.simple i*w)
+    have h' := cs.length_simple_mul w i
+    omega) z
+  by_contra hne
+  have hl := length_lt_of_bruhat_ne cs htop hne
+  omega
+
+ theorem goodIndex_nat (cs : CoxeterSystem M W) {n : ℕ} {c : W}
+    (hc : n≤cs.length c) : GoodIndex cs (n:ℤ) := by
+  intro w hw
+  have h := hw c
+  have hp := max_length_positive cs hw
+  omega
+
+ theorem goodIndex_neg_nat (cs : CoxeterSystem M W) {n : ℕ} {c : W}
+    (hc : n<cs.length c) : GoodIndex cs (-(n:ℤ)) := by
+  intro w hw
+  have h := hw c
+  omega
+
+ theorem exists_good_index (cs : CoxeterSystem M W) (z : W) :
+    ∃ i : ℤ, GoodIndex cs i ∧ |i|=(cs.length z:ℤ) ∧ indexWord cs i=z := by
+  obtain ⟨i,hi,he⟩ := exists_indexWord cs z
+  by_cases hgood : GoodIndex cs i
+  · exact ⟨i,hgood,hi,he⟩
+  · have hbad : ∃ w, (∀ x, cs.length x≤cs.length w) ∧
+        ¬(-(cs.length w:ℤ) < i ∧ i ≤ (cs.length w:ℤ)) := by
+      simpa only [GoodIndex,not_forall,exists_prop] using hgood
+    obtain ⟨w,hw,hbad⟩ := hbad
+    have hl := hw z
+    have habs := abs_le.mp (show |i|≤(cs.length w:ℤ) by rw [hi]; exact_mod_cast hl)
+    have hi' : i= -(cs.length w:ℤ) := by omega
+    have hwz : cs.length w=cs.length z := by rw [hi'] at hi; simpa using hi
+    refine ⟨cs.length z,goodIndex_nat cs (c := z) le_rfl,by simp,?_⟩
+    have hp := indexWord_period cs (rotation_pow_max cs w hw) (-(cs.length w:ℤ)) 1
+    rw [show -(cs.length w:ℤ)+2*(cs.length w:ℤ)*1=(cs.length w:ℤ) by ring] at hp
+    rw [← hwz,hp,← hi',he]
+
+end
+end KLInvariance.Dihedral
+
+end
+
+
+section
+
+namespace KLInvariance.SignedDihedral
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+ theorem odd_add_natAbs (i : ℤ) (r : ℕ) : Odd (i+r) ↔ Odd (i.natAbs+r) := by
+  rw [Int.odd_iff, Nat.odd_iff]
+  have hi := Int.natCast_natAbs i
+  obtain h | h := le_total 0 i
+  · rw [abs_of_nonneg h] at hi
+    omega
+  · rw [abs_of_nonpos h] at hi
+    omega
+
+ theorem odd_sub_natAbs (i : ℤ) (r : ℕ) : Odd (i-r) ↔ Odd (i.natAbs+r) := by
+  rw [← odd_add_natAbs, Int.odd_iff, Int.odd_iff]
+  omega
+
+ theorem arrowAfter_pos {i j : ℤ} (hij : |i| < |j|) (r : ℕ) :
+    arrowAfter i (i+j) r =
+      if i.natAbs < r ∧ Odd (i.natAbs+r) ∧ 0 < j ∧ j.natAbs < r then 1 else 0 := by
+  have hir : |i| < |(r:ℤ)| ↔ i.natAbs < r := by
+    rw [Int.abs_natCast, ← Int.natCast_natAbs i, Nat.cast_lt]
+  by_cases hl : i.natAbs < r
+  · have hb : Before (i+j) (i+r) ↔ 0 < j ∧ j.natAbs < r := by
+      by_cases hj : 0 < j
+      · rw [before_same_source_pos hij (hir.mpr hl) hj]
+        have he : (j.natAbs : ℤ) = j := (Int.natCast_natAbs j).trans (abs_of_pos hj)
+        omega
+      · have hj' : j < 0 := by
+          have := abs_nonneg i
+          rw [abs_of_nonpos (by omega : j ≤ 0)] at hij
+          omega
+        rw [before_same_source_neg hij (hir.mpr hl) hj']
+        omega
+    simp only [arrowAfter, hir, odd_add_natAbs, hb]
+  · simp only [arrowAfter, hir, hl, false_and, ite_false]
+
+ theorem arrowAfter_neg {i j : ℤ} (hij : |i| < |j|) (r : ℕ) :
+    arrowAfter i (i+j) (-r) =
+      if i.natAbs < r ∧ Odd (i.natAbs+r) ∧ (0 < j ∨ r < j.natAbs) then 1 else 0 := by
+  have hir : |i| < |-(r:ℤ)| ↔ i.natAbs < r := by
+    rw [abs_neg, Int.abs_natCast, ← Int.natCast_natAbs i, Nat.cast_lt]
+  by_cases hl : i.natAbs < r
+  · have hb : Before (i+j) (i+ -r) ↔ 0 < j ∨ r < j.natAbs := by
+      by_cases hj : 0 < j
+      · rw [before_same_source_pos hij (hir.mpr hl) hj]
+        omega
+      · have hj' : j < 0 := by
+          have := abs_nonneg i
+          rw [abs_of_nonpos (by omega : j ≤ 0)] at hij
+          omega
+        rw [before_same_source_neg hij (hir.mpr hl) hj']
+        have he : (j.natAbs : ℤ) = -j := (Int.natCast_natAbs j).trans (abs_of_neg hj')
+        omega
+    simp only [arrowAfter, hir, show i+ -(r:ℤ)=i-r by ring, odd_sub_natAbs]
+    rw [show i-(r:ℤ) = i+ -r by ring, hb]
+    split_ifs <;> rfl
+  · simp only [arrowAfter, hir, hl, false_and, ite_false]
+
+ theorem arrowAfter_next {i j : ℤ} (hij : |i| < |j|) (z : ℤ) :
+    arrowAfter j (i+j) z =
+      if j.natAbs < z.natAbs ∧ Odd (j.natAbs+z.natAbs) ∧ 0 < j then 1 else 0 := by
+  have hlt : |j| < |z| ↔ j.natAbs < z.natAbs := by
+    rw [← Int.natCast_natAbs j, ← Int.natCast_natAbs z, Nat.cast_lt]
+  have ho : Odd (j+z) ↔ Odd (j.natAbs+z.natAbs) := by
+    have hj := Int.natCast_natAbs j
+    have hz := Int.natCast_natAbs z
+    rw [Int.odd_iff,Nat.odd_iff]
+    rcases le_total 0 j with hj'|hj' <;> rcases le_total 0 z with hz'|hz'
+    · rw [abs_of_nonneg hj'] at hj
+      rw [abs_of_nonneg hz'] at hz
+      omega
+    · rw [abs_of_nonneg hj'] at hj
+      rw [abs_of_nonpos hz'] at hz
+      omega
+    · rw [abs_of_nonpos hj'] at hj
+      rw [abs_of_nonneg hz'] at hz
+      omega
+    · rw [abs_of_nonpos hj'] at hj
+      rw [abs_of_nonpos hz'] at hz
+      omega
+  by_cases h : |j| < |z|
+  · simp only [arrowAfter, before_next_source hij h, ho, hlt]
+  · have h' := hlt.not.mp h
+    simp only [arrowAfter,h,h',false_and,ite_false]
+
+ theorem countAfter_base {i j : ℤ} (hij : |i| < |j|)
+    (ho : Odd (i.natAbs+j.natAbs)) :
+    countAfter i (i+j) j = (j.natAbs-i.natAbs-1)/2 := by
+  unfold countAfter
+  rw [arrowAfter_self,add_zero]
+  have hs : ∀ r ∈ Finset.range j.natAbs,
+      arrowAfter i (i+j) r + arrowAfter i (i+j) (-r) =
+        if i.natAbs < r ∧ Odd (i.natAbs+r) then 1 else 0 := by
+    intro r hr
+    have hr' := Finset.mem_range.mp hr
+    rw [arrowAfter_pos hij,arrowAfter_neg hij]
+    have hn : ¬ j.natAbs < r := by omega
+    simp only [hn,and_false,ite_false,zero_add,hr',or_true,and_true]
+  rw [Finset.sum_congr rfl hs, ← Finset.sum_filter, Finset.sum_const,
+    nsmul_eq_mul, mul_one]
+  have hh : i.natAbs < j.natAbs := by
+    rw [← Nat.cast_lt (α := ℤ),Int.natCast_natAbs,Int.natCast_natAbs]
+    exact hij
+  have he : j.natAbs = i.natAbs+2*((j.natAbs-i.natAbs-1)/2)+1 := by
+    rw [Nat.odd_iff] at ho
+    omega
+  conv_lhs => rw [he]
+  exact eligible_ranks_card _ _
+
+/-- A double internal level together with the singleton top contributes one
+outgoing arrow per rank, independently of top parity. -/
+ theorem total_outgoing_count (L C : ℕ) (h : L ≤ C) :
+    (∑ r ∈ Finset.range C, if L < r ∧ Odd (L+r) then 2 else 0) +
+      (if L < C ∧ Odd (L+C) then 1 else 0) = C-L := by
+  induction C, h using Nat.le_induction with
+  | base =>
+    have hz : ∀ r ∈ Finset.range L, ¬ L < r := by
+      intro r hr
+      have := Finset.mem_range.mp hr
+      omega
+    have hs : (∑ r ∈ Finset.range L, if L < r ∧ Odd (L+r) then 2 else 0) = 0 := by
+      apply Finset.sum_eq_zero
+      intro r hr
+      simp only [hz r hr, false_and, ite_false]
+    simp only [hs, lt_self_iff_false, false_and, ite_false, add_zero, Nat.sub_self]
+  | succ C h ih =>
+    rw [Finset.sum_range_succ]
+    by_cases hc : L<C
+    · have hc' : L<C+1 := by omega
+      by_cases he : Odd (L+C)
+      · have he' : ¬Odd (L+(C+1)) := by rw [Nat.odd_iff] at he ⊢; omega
+        simp only [hc,he,and_self,ite_true] at ih
+        simp only [hc,he,hc',he',and_true,and_false,ite_true,ite_false,add_zero]
+        omega
+      · have he' : Odd (L+(C+1)) := by rw [Nat.odd_iff] at he ⊢; omega
+        simp only [he,and_false,ite_false,add_zero] at ih
+        simp only [hc,he,hc',he',and_true,and_false,ite_true,ite_false,add_zero]
+        omega
+    · have heq : C=L := by omega
+      subst C
+      have he : ¬Odd (L+L) := by rw [Nat.odd_iff]; omega
+      have he' : Odd (L+(L+1)) := by rw [Nat.odd_iff]; omega
+      simp only [lt_self_iff_false,false_and,ite_false,add_zero] at ih
+      simp only [he,he',and_false,and_true,ite_false,Nat.lt_succ_self,ite_true,add_zero]
+      omega
+
+end
+end KLInvariance.SignedDihedral
+
+end
+
+
+section
+
+namespace KLInvariance.SignedDihedral
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+ theorem odd_add_abs (i z : ℤ) : Odd (i+z) ↔ Odd (i.natAbs+z.natAbs) := by
+  have hi := Int.natCast_natAbs i
+  have hz := Int.natCast_natAbs z
+  rw [Int.odd_iff,Nat.odd_iff]
+  rcases le_total 0 i with hi'|hi' <;> rcases le_total 0 z with hz'|hz'
+  · rw [abs_of_nonneg hi'] at hi
+    rw [abs_of_nonneg hz'] at hz
+    omega
+  · rw [abs_of_nonneg hi'] at hi
+    rw [abs_of_nonpos hz'] at hz
+    omega
+  · rw [abs_of_nonpos hi'] at hi
+    rw [abs_of_nonneg hz'] at hz
+    omega
+  · rw [abs_of_nonpos hi'] at hi
+    rw [abs_of_nonpos hz'] at hz
+    omega
+
+ theorem arrowAfter_far {i j z : ℤ} (hij : |i| < |j|) (hjz : |j| < |z|) :
+    arrowAfter i (i+j) z = if Odd (i.natAbs+z.natAbs) ∧ 0 < j then 1 else 0 := by
+  have hiz := hij.trans hjz
+  have hb : Before (i+j) (i+z) ↔ 0 < j := by
+    by_cases hj : 0 < j
+    · rw [before_same_source_pos hij hiz hj]
+      rw [abs_of_pos hj] at hjz
+      rw [lt_abs] at hjz
+      omega
+    · have hj' : j < 0 := by
+        have := abs_nonneg i
+        rw [abs_of_nonpos (by omega : j ≤ 0)] at hij
+        omega
+      rw [before_same_source_neg hij hiz hj']
+      rw [abs_of_neg hj',lt_abs] at hjz
+      omega
+  simp only [arrowAfter,hiz,true_and,odd_add_abs,hb]
+
+ def positiveCount (L M C : ℕ) : ℕ :=
+  (∑ r ∈ Finset.range C, if L < r ∧ Odd (L+r) then (if M < r then 2 else 1) else 0) +
+    (if M < C ∧ Odd (L+C) then 1 else 0)
+
+ theorem positiveCount_formula (L M C : ℕ) (hLM : L < M) (ho : Odd (L+M))
+    (hMC : M ≤ C) : positiveCount L M C = (M-L-1)/2 + (C-M) := by
+  induction C, hMC using Nat.le_induction with
+  | base =>
+    have he : M = L+2*((M-L-1)/2)+1 := by
+      rw [Nat.odd_iff] at ho
+      omega
+    have hs : (∑ r ∈ Finset.range M,
+        if L < r ∧ Odd (L+r) then (if M < r then 2 else 1) else 0) =
+        (∑ r ∈ Finset.range M, if L < r ∧ Odd (L+r) then 1 else 0) := by
+      apply Finset.sum_congr rfl
+      intro r hr
+      have hn : ¬ M < r := by have := Finset.mem_range.mp hr; omega
+      simp only [hn,ite_false]
+    rw [positiveCount,hs,← Finset.sum_filter,Finset.sum_const,nsmul_eq_mul,mul_one]
+    simp only [lt_self_iff_false,false_and,ite_false,add_zero,Nat.sub_self]
+    conv_lhs => rw [he]
+    exact eligible_ranks_card _ _
+  | succ C hMC ih =>
+    rw [positiveCount,Finset.sum_range_succ]
+    unfold positiveCount at ih
+    by_cases h : M<C
+    · have h' : M<C+1 := by omega
+      have hl : L<C := by omega
+      by_cases hc : Odd (L+C)
+      · have hc' : ¬Odd (L+(C+1)) := by rw [Nat.odd_iff] at hc ⊢; omega
+        simp only [h,h',hl,hc,hc',and_true,and_false,ite_true,ite_false,add_zero] at ih ⊢
+        omega
+      · have hc' : Odd (L+(C+1)) := by rw [Nat.odd_iff] at hc ⊢; omega
+        simp only [h,h',hl,hc,hc',and_true,and_false,ite_true,ite_false,add_zero] at ih ⊢
+        omega
+    · have heq : C=M := by omega
+      subst C
+      have hc' : ¬Odd (L+(M+1)) := by rw [Nat.odd_iff] at ho ⊢; omega
+      simp only [lt_self_iff_false,false_and,ite_false,add_zero] at ih
+      simp only [hLM,ho,hc',and_self,and_false,ite_true,ite_false,lt_self_iff_false,add_zero]
+      omega
+
+ theorem countAfter_pos_formula {i j c : ℤ} (hij : |i| < |j|)
+    (ho : Odd (i.natAbs+j.natAbs)) (hj : 0 < j)
+    (h : c=j ∨ |j| < |c|) :
+    countAfter i (i+j) c = (j.natAbs-i.natAbs-1)/2 + (c.natAbs-j.natAbs) := by
+  rcases h with rfl|h
+  · rw [countAfter_base hij ho,Nat.sub_self,add_zero]
+  have hMC : j.natAbs < c.natAbs := by
+    exact_mod_cast (show (j.natAbs:ℤ) < c.natAbs by simpa only [Int.natCast_natAbs] using h)
+  have hLM : i.natAbs < j.natAbs := by
+    exact_mod_cast (show (i.natAbs:ℤ) < j.natAbs by simpa only [Int.natCast_natAbs] using hij)
+  have hs : ∀ r ∈ Finset.range c.natAbs,
+      arrowAfter i (i+j) r + arrowAfter i (i+j) (-r) =
+      if i.natAbs < r ∧ Odd (i.natAbs+r) then (if j.natAbs < r then 2 else 1) else 0 := by
+    intro r _
+    rw [arrowAfter_pos hij,arrowAfter_neg hij]
+    simp only [hj,true_or,and_true,true_and]
+    by_cases h1 : i.natAbs < r ∧ Odd (i.natAbs+r)
+    · obtain ⟨h1,h2⟩ := h1
+      simp only [h1,h2,true_and,ite_true]
+      split_ifs <;> rfl
+    · have hh : ¬(i.natAbs < r ∧ Odd (i.natAbs+r) ∧ j.natAbs < r) := by tauto
+      simp only [h1,hh,ite_false,add_zero]
+  rw [countAfter,Finset.sum_congr rfl hs,arrowAfter_far hij h]
+  simp only [hj,and_true]
+  have heq : (if Odd (i.natAbs+c.natAbs) then 1 else 0) =
+      (if j.natAbs < c.natAbs ∧ Odd (i.natAbs+c.natAbs) then 1 else 0) := by
+    simp only [hMC,true_and]
+  rw [heq]
+  exact positiveCount_formula _ _ _ hLM ho hMC.le
+
+ theorem countAfter_neg_formula {i j c : ℤ} (hij : |i| < |j|)
+    (ho : Odd (i.natAbs+j.natAbs)) (hj : j < 0)
+    (h : c=j ∨ |j| < |c|) :
+    countAfter i (i+j) c = (j.natAbs-i.natAbs-1)/2 := by
+  rcases h with rfl|h
+  · exact countAfter_base hij ho
+  have hMC : j.natAbs < c.natAbs := by
+    exact_mod_cast (show (j.natAbs:ℤ) < c.natAbs by simpa only [Int.natCast_natAbs] using h)
+  have hn : ¬0<j := by omega
+  have hs : (∑ r ∈ Finset.range c.natAbs, (arrowAfter i (i+j) r + arrowAfter i (i+j) (-r))) =
+      (∑ r ∈ Finset.range j.natAbs, (arrowAfter i (i+j) r + arrowAfter i (i+j) (-r))) := by
+    symm
+    apply Finset.sum_subset (Finset.range_mono hMC.le)
+    intro r _ hr
+    have hr' : ¬ r < j.natAbs := by simpa only [Finset.mem_range] using hr
+    rw [arrowAfter_pos hij,arrowAfter_neg hij]
+    simp only [hn,hr',false_or,and_false,false_and,ite_false,add_zero]
+  rw [countAfter,hs,arrowAfter_far hij h]
+  simp only [hn,and_false,ite_false,add_zero]
+  have hh := countAfter_base hij ho
+  rw [countAfter,arrowAfter_self,add_zero] at hh
+  exact hh
+
+ theorem countAfter_next_formula {i j c : ℤ} (hij : |i| < |j|)
+    (h : c=j ∨ |j| < |c|) :
+    countAfter j (i+j) c = if 0<j then c.natAbs-j.natAbs else 0 := by
+  have hMC : j.natAbs ≤ c.natAbs := by
+    rcases h with rfl|h
+    · exact le_rfl
+    · exact_mod_cast (show (j.natAbs:ℤ) ≤ c.natAbs by simpa only [Int.natCast_natAbs] using h.le)
+  unfold countAfter
+  simp_rw [arrowAfter_next hij]
+  simp only [Int.natAbs_natCast,Int.natAbs_neg]
+  by_cases hj : 0<j
+  · simp only [hj,and_true,ite_true]
+    have hs : ∀ r ∈ Finset.range c.natAbs,
+        (if j.natAbs < r ∧ Odd (j.natAbs+r) then 1 else 0) +
+        (if j.natAbs < r ∧ Odd (j.natAbs+r) then 1 else 0) =
+        (if j.natAbs < r ∧ Odd (j.natAbs+r) then 2 else 0) := by
+      intro r _
+      split_ifs <;> rfl
+    rw [Finset.sum_congr rfl hs]
+    exact total_outgoing_count _ _ hMC
+  · simp only [hj,and_false,ite_false,add_zero,Finset.sum_const_zero]
+
+/-- The signed-index cardinality identity for the source's forward angular
+order. Its identification with actual Coxeter angular roots is separate. -/
+ theorem countAfter_difference {i j c : ℤ} (hij : |i| < |j|)
+    (ho : Odd (i.natAbs+j.natAbs)) (h : c=j ∨ |j| < |c|) :
+    countAfter i (i+j) c = countAfter j (i+j) c + (j.natAbs-i.natAbs-1)/2 := by
+  rw [countAfter_next_formula hij h]
+  by_cases hj : 0<j
+  · rw [ite_eq_left hj,countAfter_pos_formula hij ho hj h,Nat.add_comm]
+  · have hj' : j<0 := by
+      have := abs_nonneg i
+      rw [abs_of_nonpos (by omega : j≤0)] at hij
+      omega
+    rw [ite_eq_right hj,zero_add,countAfter_neg_formula hij ho hj' h]
+
+end
+end KLInvariance.SignedDihedral
+
+end
+
+
+section
+
+namespace KLInvariance.Dihedral
+variable {W : Type*} [Group W] {M : CoxeterMatrix Bool}
+
+ theorem natAbs_lt_iff (i j : ℤ) : i.natAbs<j.natAbs ↔ |i| < |j| := by
+  rw [← Int.natCast_natAbs,← Int.natCast_natAbs]
+  exact (Nat.cast_lt (α := ℤ)).symm
+
+ theorem index_step_iff (cs : CoxeterSystem M W) {i j : ℤ}
+    (hi : cs.length (indexWord cs i)=i.natAbs)
+    (hj : cs.length (indexWord cs j)=j.natAbs) :
+    BruhatStep cs (indexWord cs i) (indexWord cs j) ↔ |i| < |j| ∧ Odd (i+j) := by
+  rw [step_iff_odd,hi,hj,natAbs_lt_iff]
+  apply and_congr_right
+  intro h
+  have hnat := (natAbs_lt_iff i j).mpr h
+  rw [SignedDihedral.odd_add_abs,Nat.odd_iff,Nat.odd_iff]
+  omega
+
+end KLInvariance.Dihedral
+
+namespace KLInvariance.TitsSpace.PlaneBasis
+open KLInvariance.Dihedral
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+ theorem edgeRoot_same_table {i j z : ℤ}
+    (hxy : BruhatStep B.system (indexWord B.system i) (indexWord B.system j))
+    (hxz : BruhatStep B.system (indexWord B.system i) (indexWord B.system z))
+    (hi : B.system.length (indexWord B.system i)=i.natAbs)
+    (hj : B.system.length (indexWord B.system j)=j.natAbs)
+    (hz : B.system.length (indexWord B.system z)=z.natAbs)
+    (gj : GoodIndex B.system j) (gz : GoodIndex B.system z) :
+    geometricSlope (B.edgeRoot hxy) < geometricSlope (B.edgeRoot hxz) ↔
+      SignedDihedral.Before (i+j) (i+z) := by
+  have hij := (index_step_iff B.system hi hj).mp hxy
+  have hiz := (index_step_iff B.system hi hz).mp hxz
+  have hlabel := indexWord_label B.system hij.2
+  have hlabel' := indexWord_label B.system hiz.2
+  rcases maximal_or_unbounded B.system with ⟨w,hw⟩ | hu
+  · rw [B.edgeRoot_finite_order w hw hxy hxz hij.2 hiz.2 hlabel hlabel']
+    exact SignedDihedral.finite_before_same (by exact_mod_cast max_length_positive B.system hw)
+      hij.1 hiz.1 (gj w hw) (gz w hw)
+  · exact B.edgeRoot_infinite_order hu hxy hxz hij.2 hiz.2 hlabel hlabel'
+
+ theorem edgeRoot_next_table {i j z : ℤ}
+    (hxy : BruhatStep B.system (indexWord B.system i) (indexWord B.system j))
+    (hyz : BruhatStep B.system (indexWord B.system j) (indexWord B.system z))
+    (hi : B.system.length (indexWord B.system i)=i.natAbs)
+    (hj : B.system.length (indexWord B.system j)=j.natAbs)
+    (hz : B.system.length (indexWord B.system z)=z.natAbs)
+    (gz : GoodIndex B.system z) :
+    geometricSlope (B.edgeRoot hxy) < geometricSlope (B.edgeRoot hyz) ↔
+      SignedDihedral.Before (i+j) (j+z) := by
+  have hij := (index_step_iff B.system hi hj).mp hxy
+  have hjz := (index_step_iff B.system hj hz).mp hyz
+  have hlabel := indexWord_label B.system hij.2
+  have hlabel' := indexWord_label B.system hjz.2
+  rcases maximal_or_unbounded B.system with ⟨w,hw⟩ | hu
+  · rw [B.edgeRoot_finite_order w hw hxy hyz hij.2 hjz.2 hlabel hlabel']
+    exact SignedDihedral.finite_before_next (by exact_mod_cast max_length_positive B.system hw)
+      hij.1 hjz.1 (gz w hw)
+  · exact B.edgeRoot_infinite_order hu hxy hyz hij.2 hjz.2 hlabel hlabel'
+
+ def afterIndicator (x : planeSubgroup M cs P) (α : PositiveRoot M cs)
+    (z : planeSubgroup M cs P) : ℕ :=
+  if h : BruhatStep B.system x z then
+    if geometricSlope α < geometricSlope (B.edgeRoot h) then 1 else 0
+  else 0
+
+ theorem afterIndicator_one (x : planeSubgroup M cs P) (α : PositiveRoot M cs) :
+    B.afterIndicator x α 1=0 := by
+  have hn : ¬BruhatStep B.system x 1 := by
+    intro h
+    have := h.1
+    rw [B.system.length_one] at this
+    omega
+  simp only [afterIndicator,dite_eq_right hn]
+
+ theorem afterIndicator_same {i j z : ℤ}
+    (hxy : BruhatStep B.system (indexWord B.system i) (indexWord B.system j))
+    (hi : B.system.length (indexWord B.system i)=i.natAbs)
+    (hj : B.system.length (indexWord B.system j)=j.natAbs)
+    (hz : B.system.length (indexWord B.system z)=z.natAbs)
+    (gj : GoodIndex B.system j) (gz : GoodIndex B.system z) :
+    B.afterIndicator (indexWord B.system i) (B.edgeRoot hxy) (indexWord B.system z) =
+      SignedDihedral.arrowAfter i (i+j) z := by
+  by_cases hstep : BruhatStep B.system (indexWord B.system i) (indexWord B.system z)
+  · have hs := (index_step_iff B.system hi hz).mp hstep
+    simp only [afterIndicator,dite_eq_left hstep,B.edgeRoot_same_table hxy hstep hi hj hz gj gz,
+      SignedDihedral.arrowAfter,hs.1,hs.2,true_and]
+  · have hs := (index_step_iff B.system hi hz).not.mp hstep
+    rw [afterIndicator,dite_eq_right hstep,SignedDihedral.arrowAfter]
+    split_ifs with h
+    · exact (hs ⟨h.1,h.2.1⟩).elim
+    · rfl
+
+ theorem afterIndicator_next {i j z : ℤ}
+    (hxy : BruhatStep B.system (indexWord B.system i) (indexWord B.system j))
+    (hi : B.system.length (indexWord B.system i)=i.natAbs)
+    (hj : B.system.length (indexWord B.system j)=j.natAbs)
+    (hz : B.system.length (indexWord B.system z)=z.natAbs)
+    (gz : GoodIndex B.system z) :
+    B.afterIndicator (indexWord B.system j) (B.edgeRoot hxy) (indexWord B.system z) =
+      SignedDihedral.arrowAfter j (i+j) z := by
+  by_cases hstep : BruhatStep B.system (indexWord B.system j) (indexWord B.system z)
+  · have hs := (index_step_iff B.system hj hz).mp hstep
+    simp only [afterIndicator,dite_eq_left hstep,B.edgeRoot_next_table hxy hstep hi hj hz gz,
+      SignedDihedral.arrowAfter,hs.1,hs.2,true_and]
+  · have hs := (index_step_iff B.system hj hz).not.mp hstep
+    rw [afterIndicator,dite_eq_right hstep,SignedDihedral.arrowAfter]
+    split_ifs with h
+    · exact (hs ⟨h.1,h.2.1⟩).elim
+    · rfl
+
+end
+end KLInvariance.TitsSpace.PlaneBasis
+
+end
+
+
+section
+
+namespace KLInvariance.TitsSpace.PlaneBasis
+open KLInvariance.Dihedral
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+noncomputable section
+
+ def laterCount (x : planeSubgroup M cs P) (α : PositiveRoot M cs)
+    (c : planeSubgroup M cs P) : ℕ :=
+  ∑ z ∈ lowerFinset B.system c, B.afterIndicator x α z
+
+ theorem laterCount_same {i j k : ℤ}
+    (hxy : BruhatStep B.system (indexWord B.system i) (indexWord B.system j))
+    (hi : B.system.length (indexWord B.system i)=i.natAbs)
+    (hj : B.system.length (indexWord B.system j)=j.natAbs)
+    (hk : B.system.length (indexWord B.system k)=k.natAbs)
+    (gj : GoodIndex B.system j) (gk : GoodIndex B.system k) :
+    B.laterCount (indexWord B.system i) (B.edgeRoot hxy) (indexWord B.system k) =
+      SignedDihedral.countAfter i (i+j) k := by
+  rw [laterCount,sum_lower_signed B.system _ _ (B.afterIndicator_one _ _),hk,
+    B.afterIndicator_same hxy hi hj hk gj gk,SignedDihedral.countAfter]
+  congr 1
+  apply Finset.sum_congr rfl
+  intro n hn
+  have hnc : n<B.system.length (indexWord B.system k) := by
+    rw [hk]
+    exact Finset.mem_range.mp hn
+  have hp := length_indexWord_nat B.system hnc.le
+  have hm := length_indexWord_neg_nat B.system hnc.le
+  have hp' : B.system.length (indexWord B.system (n:ℤ))=(n:ℤ).natAbs := by
+    simpa only [Int.natAbs_natCast] using hp
+  have hm' : B.system.length (indexWord B.system (-(n:ℤ)))=(-(n:ℤ)).natAbs := by
+    simpa only [Int.natAbs_neg,Int.natAbs_natCast] using hm
+  rw [B.afterIndicator_same hxy hi hj hp' gj (goodIndex_nat B.system hnc.le),
+    B.afterIndicator_same hxy hi hj hm' gj (goodIndex_neg_nat B.system hnc)]
+
+ theorem laterCount_next {i j k : ℤ}
+    (hxy : BruhatStep B.system (indexWord B.system i) (indexWord B.system j))
+    (hi : B.system.length (indexWord B.system i)=i.natAbs)
+    (hj : B.system.length (indexWord B.system j)=j.natAbs)
+    (hk : B.system.length (indexWord B.system k)=k.natAbs)
+    (gk : GoodIndex B.system k) :
+    B.laterCount (indexWord B.system j) (B.edgeRoot hxy) (indexWord B.system k) =
+      SignedDihedral.countAfter j (i+j) k := by
+  rw [laterCount,sum_lower_signed B.system _ _ (B.afterIndicator_one _ _),hk,
+    B.afterIndicator_next hxy hi hj hk gk,SignedDihedral.countAfter]
+  congr 1
+  apply Finset.sum_congr rfl
+  intro n hn
+  have hnc : n<B.system.length (indexWord B.system k) := by
+    rw [hk]
+    exact Finset.mem_range.mp hn
+  have hp := length_indexWord_nat B.system hnc.le
+  have hm := length_indexWord_neg_nat B.system hnc.le
+  have hp' : B.system.length (indexWord B.system (n:ℤ))=(n:ℤ).natAbs := by
+    simpa only [Int.natAbs_natCast] using hp
+  have hm' : B.system.length (indexWord B.system (-(n:ℤ)))=(-(n:ℤ)).natAbs := by
+    simpa only [Int.natAbs_neg,Int.natAbs_natCast] using hm
+  rw [B.afterIndicator_next hxy hi hj hp' (goodIndex_nat B.system hnc.le),
+    B.afterIndicator_next hxy hi hj hm' (goodIndex_neg_nat B.system hnc)]
+
+
+ theorem laterCount_difference {x y c : planeSubgroup M cs P}
+    (hxy : BruhatStep B.system x y) (hyc : BruhatLE B.system y c) :
+    B.laterCount x (B.edgeRoot hxy) c = B.laterCount y (B.edgeRoot hxy) c +
+      (B.system.length y-B.system.length x-1)/2 := by
+  obtain ⟨i,_,hi,he⟩ := exists_good_index B.system x
+  obtain ⟨j,gj,hj,hf⟩ := exists_good_index B.system y
+  have hi' : i.natAbs=B.system.length x := by
+    exact_mod_cast (show (i.natAbs:ℤ)=(B.system.length x:ℤ) by rw [Int.natCast_natAbs,hi])
+  have hj' : j.natAbs=B.system.length y := by
+    exact_mod_cast (show (j.natAbs:ℤ)=(B.system.length y:ℤ) by rw [Int.natCast_natAbs,hj])
+  obtain ⟨k,gk,hk,hg,hcomp⟩ : ∃ k : ℤ, GoodIndex B.system k ∧
+      k.natAbs=B.system.length c ∧ indexWord B.system k=c ∧
+      (k=j ∨ |j| < |k|) := by
+    by_cases h : c=y
+    · subst c
+      exact ⟨j,gj,hj',hf,Or.inl rfl⟩
+    · obtain ⟨k,gk,hk,hg⟩ := exists_good_index B.system c
+      have hlen := length_lt_of_bruhat_ne B.system hyc (Ne.symm h)
+      refine ⟨k,gk,?_,hg,Or.inr ?_⟩
+      · exact_mod_cast (show (k.natAbs:ℤ)=(B.system.length c:ℤ) by rw [Int.natCast_natAbs,hk])
+      · rw [hj,hk]
+        exact_mod_cast hlen
+  have ilen : B.system.length (indexWord B.system i)=i.natAbs := by rw [he,hi']
+  have jlen : B.system.length (indexWord B.system j)=j.natAbs := by rw [hf,hj']
+  have klen : B.system.length (indexWord B.system k)=k.natAbs := by rw [hg,hk]
+  subst x y c
+  rw [B.laterCount_same hxy ilen jlen klen gj gk,
+    B.laterCount_next hxy ilen jlen klen gk,ilen,jlen]
+  have hs := (index_step_iff B.system ilen jlen).mp hxy
+  exact SignedDihedral.countAfter_difference hs.1
+    ((SignedDihedral.odd_add_abs i j).mp hs.2) hcomp
+
+end
+end KLInvariance.TitsSpace.PlaneBasis
+
+end
+
+
+section
+
+namespace KLInvariance.TitsSpace.PlaneBasis
+open KLInvariance.Dihedral
+universe u v u' v'
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+noncomputable section
+
+ theorem afterIndicator_eq_zero {x z : planeSubgroup M cs P}
+    (α : PositiveRoot M cs) (h : ¬BruhatLE B.system x z) :
+    B.afterIndicator x α z = 0 := by
+  have hn : ¬BruhatStep B.system x z := fun hz => h (Relation.ReflTransGen.single hz)
+  simp only [afterIndicator,dite_eq_right hn]
+
+/-- Counting only the relative interval above a lower bound does not alter
+an outgoing count based at a vertex above that bound. -/
+ theorem interval_laterCount {a x c : planeSubgroup M cs P}
+    (hax : BruhatLE B.system a x) (α : PositiveRoot M cs) :
+    (∑ᶠ z : Interval B.system a c, B.afterIndicator x α z.val) =
+      B.laterCount x α c := by
+  classical
+  let := interval_finite B.system a c
+  let : Fintype (Interval B.system a c) := Fintype.ofFinite _
+  rw [finsum_eq_sum_of_fintype,laterCount]
+  have hs := Finset.sum_subtype (F := (inferInstance : Fintype (Interval B.system a c)))
+    ((lowerFinset B.system c).filter (fun z => BruhatLE B.system a z))
+    (fun z => by simp only [Finset.mem_filter,mem_lowerFinset]; exact and_comm)
+    (fun z => B.afterIndicator x α z)
+  refine hs.symm.trans (Finset.sum_subset (Finset.filter_subset _ _) ?_)
+  intro z hz hn
+  apply B.afterIndicator_eq_zero
+  intro hxz
+  exact hn (Finset.mem_filter.mpr ⟨hz,bruhat_trans B.system hax hxz⟩)
+
+ theorem edgeRoot_coset {a₀ : W}
+    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+    {x y : planeSubgroup M cs P} (hxy : BruhatStep B.system x y) :
+    B.edgeRoot hxy = graphRoot ((B.coset_step_iff hmin x y).mpr hxy) := by
+  unfold edgeRoot graphRoot
+  apply congrArg (positiveRoot M cs)
+  apply Subtype.ext
+  change (y:W)*(x:W)⁻¹=((y:W)*a₀)*((x:W)*a₀)⁻¹
+  group
+
+variable {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
+  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
+  {P' : Submodule ℝ (I' → ℝ)} (B' : PlaneBasis (M := M') (cs := cs') P')
+
+/-- A full relative interval transport carries the actual angular later-edge
+count. An arbitrary graph embedding would not suffice here. -/
+ theorem interval_map_count {a c : planeSubgroup M cs P}
+    (hac : BruhatLE B.system a c)
+    (F : Interval B.system a c → planeSubgroup M' cs' P')
+    (hinj : Function.Injective F)
+    (hrange : Set.range F = {z | BruhatLE B'.system (F ⟨a,bruhat_refl B.system a,hac⟩) z ∧
+      BruhatLE B'.system z (F ⟨c,hac,bruhat_refl B.system c⟩)})
+    (x : Interval B.system a c) (α : PositiveRoot M' cs') :
+    (∑ᶠ z : Interval B.system a c, B'.afterIndicator (F x) α (F z)) =
+      B'.laterCount (F x) α (F ⟨c,hac,bruhat_refl B.system c⟩) := by
+  classical
+  let lo : Interval B.system a c := ⟨a,bruhat_refl B.system a,hac⟩
+  let hi : Interval B.system a c := ⟨c,hac,bruhat_refl B.system c⟩
+  have hmem (z : Interval B.system a c) : BruhatLE B'.system (F lo) (F z) ∧
+      BruhatLE B'.system (F z) (F hi) := by
+    exact (Set.ext_iff.mp hrange (F z)).mp ⟨z,rfl⟩
+  let f : Interval B.system a c → Interval B'.system (F lo) (F hi) :=
+    fun z => ⟨F z,hmem z⟩
+  have hf : Function.Bijective f := by
+    constructor
+    · intro z w h
+      exact hinj (congrArg Subtype.val h)
+    · intro z
+      obtain ⟨w,hw⟩ := (Set.ext_iff.mp hrange z.val).mpr z.property
+      exact ⟨w,Subtype.ext hw⟩
+  let E := Equiv.ofBijective f hf
+  have hs := finsum_comp_equiv E (f := fun z : Interval B'.system (F lo) (F hi) =>
+    B'.afterIndicator (F x) α z.val)
+  exact hs.trans (B'.interval_laterCount (hmem x).1 α)
+
+end
+end KLInvariance.TitsSpace.PlaneBasis
+
+end
+
+
+section
+
+namespace KLInvariance.TitsSpace
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+noncomputable section
+
+ def ambientAfterIndicator (x : W) (α : PositiveRoot M cs) (z : W) : ℕ := by
+  classical
+  exact if h : BruhatStep cs x z then
+    if geometricSlope α < geometricSlope (graphRoot h) then 1 else 0
+  else 0
+
+ theorem ambientAfterIndicator_of_le {x z : W} (α : PositiveRoot M cs)
+    (hz : BruhatLE cs z x) : ambientAfterIndicator x α z=0 := by
+  classical
+  have hn : ¬BruhatStep cs x z := fun h =>
+    (not_lt_of_ge (length_le_of_bruhat cs hz)) h.1
+  simp only [ambientAfterIndicator,dite_eq_right hn]
+
+ theorem ambientAfterIndicator_edge_self {x y : W} (hxy : BruhatStep cs x y) :
+    ambientAfterIndicator x (graphRoot hxy) y=0 := by
+  classical
+  simp only [ambientAfterIndicator,dite_eq_left hxy,lt_self_iff_false,ite_false]
+
+namespace PlaneBasis
+variable {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+
+ theorem ambientAfterIndicator_coset {a₀ : W}
+    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+    (x z : planeSubgroup M cs P) (α : PositiveRoot M cs) :
+    ambientAfterIndicator ((x:W)*a₀) α ((z:W)*a₀) = B.afterIndicator x α z := by
+  classical
+  by_cases h : BruhatStep B.system x z
+  · have h' := (B.coset_step_iff hmin x z).mpr h
+    simp only [ambientAfterIndicator,afterIndicator,dite_eq_left h,dite_eq_left h',B.edgeRoot_coset hmin h]
+  · have h' : ¬BruhatStep cs ((x:W)*a₀) ((z:W)*a₀) :=
+      fun hz => h ((B.coset_step_iff hmin x z).mp hz)
+    simp only [ambientAfterIndicator,afterIndicator,dite_eq_right h,dite_eq_right h']
+
+universe u' v'
+variable {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
+  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
+
+
+ theorem transported_count_difference {a₀ u b : W} {u' b' : W'}
+    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+    {a c : planeSubgroup M cs P} (hac : BruhatLE B.system a c)
+    (hrank : B.system.length a+2 ≤ B.system.length c)
+    (hlo : BruhatLE cs u ((a:W)*a₀)) (hhi : BruhatLE cs ((c:W)*a₀) b)
+    (e : Interval cs u b ≃o Interval cs' u' b')
+    (y : Interval B.system a c) (hxy : BruhatStep B.system a y.val) :
+    let inc := B.cosetIntervalMap hmin hlo hhi
+    let lo : Interval B.system a c := ⟨a,bruhat_refl B.system a,hac⟩
+    let α := graphRoot ((intervalIso_bruhatStep_iff_general cs cs' e (inc lo) (inc y)).mpr
+      ((B.coset_step_iff hmin a y.val).mpr hxy))
+    (∑ᶠ z : Interval B.system a c,
+      ambientAfterIndicator (e (inc lo)).val α (e (inc z)).val) =
+    (∑ᶠ z : Interval B.system a c,
+      ambientAfterIndicator (e (inc y)).val α (e (inc z)).val) +
+      (B.system.length y.val-B.system.length a-1)/2 := by
+  classical
+  dsimp only
+  let inc := B.cosetIntervalMap hmin hlo hhi
+  let lo : Interval B.system a c := ⟨a,bruhat_refl B.system a,hac⟩
+  let hi : Interval B.system a c := ⟨c,hac,bruhat_refl B.system c⟩
+  have hedge := (intervalIso_bruhatStep_iff_general cs cs' e (inc lo) (inc y)).mpr
+    ((B.coset_step_iff hmin a y.val).mpr hxy)
+  change (∑ᶠ z : Interval B.system a c,
+      ambientAfterIndicator (e (inc lo)).val (graphRoot hedge) (e (inc z)).val) =
+    (∑ᶠ z : Interval B.system a c,
+      ambientAfterIndicator (e (inc y)).val (graphRoot hedge) (e (inc z)).val) + _
+  obtain ⟨P',B',a',F,hmin',hinj,hcoord,horder,hstep,hlen,hrange⟩ :=
+    B.transport_interval hmin hac hrank hlo hhi e
+  have hFxy : BruhatStep B'.system (F lo) (F y) := (hstep lo y).mpr hxy
+  have hFyc : BruhatLE B'.system (F y) (F hi) := (horder y hi).mpr y.property.2
+  have hroot : B'.edgeRoot hFxy=graphRoot hedge := by
+    rw [B'.edgeRoot_coset hmin' hFxy]
+    unfold graphRoot
+    apply congrArg (positiveRoot M' cs')
+    apply Subtype.ext
+    change ((F y:W')*a')*((F lo:W')*a')⁻¹=(e (inc y)).val*(e (inc lo)).val⁻¹
+    rw [hcoord y,hcoord lo]
+  have hsum (x : Interval B.system a c) :
+      (∑ᶠ z : Interval B.system a c,
+        ambientAfterIndicator (e (inc x)).val (graphRoot hedge) (e (inc z)).val) =
+      B'.laterCount (F x) (B'.edgeRoot hFxy) (F hi) := by
+    rw [← B.interval_map_count B' hac F hinj hrange x (B'.edgeRoot hFxy)]
+    apply finsum_congr
+    intro z
+    rw [← hcoord x,← hcoord z,← hroot]
+    exact B'.ambientAfterIndicator_coset hmin' (F x) (F z) _
+  rw [hsum lo,hsum y,B'.laterCount_difference hFxy hFyc]
+  congr 1
+  have hy := hlen y
+  change B'.system.length (F y)+B.system.length a =
+    B'.system.length (F lo)+B.system.length y.val at hy
+  omega
+
+/-- The same scheduled count with no restriction on the relative rank. -/
+ theorem transported_count_difference_all {a₀ u b : W} {u' b' : W'}
+    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+    {a c : planeSubgroup M cs P} (hac : BruhatLE B.system a c)
+    (hlo : BruhatLE cs u ((a:W)*a₀)) (hhi : BruhatLE cs ((c:W)*a₀) b)
+    (e : Interval cs u b ≃o Interval cs' u' b')
+    (y : Interval B.system a c) (hxy : BruhatStep B.system a y.val) :
+    let inc := B.cosetIntervalMap hmin hlo hhi
+    let lo : Interval B.system a c := ⟨a,bruhat_refl B.system a,hac⟩
+    let α := graphRoot ((intervalIso_bruhatStep_iff_general cs cs' e (inc lo) (inc y)).mpr
+      ((B.coset_step_iff hmin a y.val).mpr hxy))
+    (∑ᶠ z : Interval B.system a c,
+      ambientAfterIndicator (e (inc lo)).val α (e (inc z)).val) =
+    (∑ᶠ z : Interval B.system a c,
+      ambientAfterIndicator (e (inc y)).val α (e (inc z)).val) +
+      (B.system.length y.val-B.system.length a-1)/2 := by
+  classical
+  by_cases hrank : B.system.length a+2 ≤ B.system.length c
+  · exact B.transported_count_difference hmin hac hrank hlo hhi e y hxy
+  let inc := B.cosetIntervalMap hmin hlo hhi
+  let lo : Interval B.system a c := ⟨a,bruhat_refl B.system a,hac⟩
+  have hedge := (intervalIso_bruhatStep_iff_general cs cs' e (inc lo) (inc y)).mpr
+    ((B.coset_step_iff hmin a y.val).mpr hxy)
+  change (∑ᶠ z : Interval B.system a c,
+      ambientAfterIndicator (e (inc lo)).val (graphRoot hedge) (e (inc z)).val) =
+    (∑ᶠ z : Interval B.system a c,
+      ambientAfterIndicator (e (inc y)).val (graphRoot hedge) (e (inc z)).val) + _
+  have hylen := hxy.1
+  have hyc := length_le_of_bruhat B.system y.property.2
+  have hlen : B.system.length y.val=B.system.length a+1 := by omega
+  have heyc : y.val=c := by
+    by_contra hn
+    have hlt := length_lt_of_bruhat_ne B.system y.property.2 hn
+    omega
+  have hverts (z : Interval B.system a c) : z=lo ∨ z=y := by
+    by_cases hz : z.val=a
+    · exact Or.inl (Subtype.ext hz)
+    · right
+      apply Subtype.ext
+      rw [heyc]
+      by_contra hzc
+      have hzlo := length_lt_of_bruhat_ne B.system z.property.1 (Ne.symm hz)
+      have hzhi := length_lt_of_bruhat_ne B.system z.property.2 hzc
+      omega
+  have hzero₁ : (∑ᶠ z : Interval B.system a c,
+      ambientAfterIndicator (e (inc lo)).val (graphRoot hedge) (e (inc z)).val)=0 := by
+    apply finsum_eq_zero_of_forall_eq_zero
+    intro z
+    rcases hverts z with rfl | rfl
+    · exact ambientAfterIndicator_of_le _ (bruhat_refl cs' _)
+    · exact ambientAfterIndicator_edge_self hedge
+  have hzero₂ : (∑ᶠ z : Interval B.system a c,
+      ambientAfterIndicator (e (inc y)).val (graphRoot hedge) (e (inc z)).val)=0 := by
+    apply finsum_eq_zero_of_forall_eq_zero
+    intro z
+    rcases hverts z with rfl | rfl
+    · exact ambientAfterIndicator_of_le _ (Relation.ReflTransGen.single hedge)
+    · exact ambientAfterIndicator_of_le _ (bruhat_refl cs' _)
+  rw [hzero₁,hzero₂,hlen]
+  omega
+
+end PlaneBasis
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Linear prime ideals in the exact symmetric algebra, including the actual
+root-plane coefficient primes. No local-ring or survival property is assumed. -/
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/KazhdanLusztig/DyerPaths.lean b/lean/OAI/RepresentationTheory/KazhdanLusztig/DyerPaths.lean
index 8aa65f6f13e5ae89bf13a1eba853d3c198a8eed9..80af4ce3f182aba0624693113288fa11e5e149c7 100644
--- a/lean/OAI/RepresentationTheory/KazhdanLusztig/DyerPaths.lean
+++ b/lean/OAI/RepresentationTheory/KazhdanLusztig/DyerPaths.lean
@@ -1,2507 +1,2507 @@
-import OAI.RepresentationTheory.KazhdanLusztig.CharacterReciprocity
-
-namespace OAI
-
-/-!
-Finite character cancellation, reflection-ratio orders and adapted Dyer decreasing paths.
--/
-
-section
-
-namespace KLInvariance
-universe u v
-variable {V : Type u} [Fintype V] {C : Type v} [CommRing C]
-
- theorem finite_character_cancellation (t : V) (m : V → C) (r : V → C)
-    (p : V → V → C) (q : V → C) (hm : m t=1)
-    (hword : (∑ y, m y*q y)=∑ z, r z*(∑ y, m y*p y z))
-    (hlower : ∀ y, y≠t → q y=∑ z, r z*p y z) : q t=∑ z, r z*p t z := by
-  classical
-  have hexpand : (∑ z, r z*(∑ y, m y*p y z))=∑ y, m y*(∑ z, r z*p y z) := by
-    simp only [Finset.mul_sum]
-    rw [Finset.sum_comm]
-    apply Finset.sum_congr rfl
-    intro y _
-    apply Finset.sum_congr rfl
-    intro z _
-    ring
-  rw [hexpand] at hword
-  have hzero : (∑ y, m y*(q y-∑ z, r z*p y z))=0 := by
-    simp only [mul_sub,Finset.sum_sub_distrib,hword,sub_self]
-  have hsingle : (∑ y, m y*(q y-∑ z, r z*p y z))=q t-∑ z, r z*p t z := by
-    rw [Finset.sum_eq_single t]
-    · rw [hm,one_mul]
-    · intro y _ hy
-      rw [hlower y hy,sub_self,mul_zero]
-    · simp
-  rw [hsingle] at hzero
-  exact sub_eq_zero.mp hzero
-
-end KLInvariance
-
-end
-
-
-section
-
-/-! R-reciprocity for the ACTUAL minimal-boundary BMP character, obtained
-by simultaneous word stalk/costalk decomposition and unipotent cancellation.
-This is not a character identification: strict halfdegree remains separate. -/
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph Polynomial
-universe u
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-local instance bmpRecInterval (b : W) : Fintype (Interval cs 1 b) := by
-  let := interval_finite cs 1 b
-  exact Fintype.ofFinite _
-
- theorem stalkCharacter_reciprocity (b : W) (x : Interval cs 1 b) :
-    reflect (rankDifference cs x.val b) (stalkCharacter cs 1 b basis (one_bruhat cs b) x)=
-      ∑ z : Interval cs 1 b, OrdinaryR.polynomial cs x.val z.val*
-        stalkCharacter cs 1 b basis (one_bruhat cs b) z := by
-  generalize hn : cs.length b=n
-  induction n using Nat.strong_induction_on generalizing b with
-  | h n ih =>
-    obtain ⟨l,hl,rfl⟩ := cs.exists_isReduced b
-    let b := cs.wordProd l
-    let t : Interval cs 1 b := ⟨b,one_bruhat cs b,bruhat_refl cs b⟩
-    obtain ⟨m,_hm,hs,hk⟩ := bottCharacter_decomposition_both cs basis l hl
-    have hmt : m t=1 := by
-      have hh := hs t
-      rw [bottStalkCharacter_top cs basis l hl] at hh
-      rw [Finset.sum_eq_single t] at hh
-      · simpa only [lowerCharacter_diagonal,mul_one] using hh.symm
-      · intro y _ hy
-        have hv : y.val≠b := fun h => hy (Subtype.ext h)
-        rw [lowerCharacter_at_top,ite_eq_right hv,mul_zero]
-      · simp
-    have ht (z : Interval cs 1 b) : lowerCharacter cs b basis t z=
-        stalkCharacter cs 1 b basis (one_bruhat cs b) z := dite_eq_left z.property.2
-    have hw : (∑ y, m y*reflect (rankDifference cs x.val y.val) (lowerCharacter cs b basis y x))=
-        ∑ z : Interval cs 1 b, OrdinaryR.polynomial cs x.val z.val*
-          (∑ y, m y*lowerCharacter cs b basis y z) := by
-      rw [←hk x,bottKernelCharacter_rConvolution cs basis l hl x]
-      simp only [hs]
-      rfl
-    have hh := finite_character_cancellation t m
-      (fun z : Interval cs 1 b => OrdinaryR.polynomial cs x.val z.val)
-      (fun y z => lowerCharacter cs b basis y z)
-      (fun y => reflect (rankDifference cs x.val y.val) (lowerCharacter cs b basis y x))
-      hmt hw (fun y hy => ?_)
-    · simpa only [ht] using hh
-    · apply lowerCharacter_reciprocity_lift
-      intro z
-      apply ih (cs.length y.val) _ y.val z rfl
-      rw [←hn]
-      exact length_lt_of_bruhat_ne cs y.property.2 (fun he => hy (Subtype.ext he))
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-/-! Removing the finite-rank restriction from the actual graph bridge,
-using the already established standard-parabolic embedding. -/
-namespace KLInvariance.StandardParabolic
-open CoxeterSystem
-universe u v
-variable {I : Type u} {W : Type v} [Group W] {M : CoxeterMatrix I}
-variable (cs : CoxeterSystem M W) (J : Set I)
-noncomputable section
-
-theorem leftInvSeq_inclusion (ω : List J) :
-    ((system (M := M) J).leftInvSeq ω).map (inclusion cs J) =
-      cs.leftInvSeq (ω.map Subtype.val) := by
-  induction ω with
-  | nil => rfl
-  | cons i ω ih =>
-    simp only [CoxeterSystem.leftInvSeq, List.map_cons, List.map_map,
-      inclusion_simple]
-    rw [← ih, List.map_map]
-    congr 1
-    apply List.map_congr_left
-    intro t _
-    simp only [Function.comp_apply, MulAut.conj_apply, map_mul, map_inv,
-      inclusion_simple]
-
-theorem reflection_inclusion {t : (matrix (M := M) J).Group}
-    (ht : (system (M := M) J).IsReflection t) :
-    cs.IsReflection (inclusion cs J t) := by
-  obtain ⟨w, i, rfl⟩ := ht
-  exact ⟨inclusion cs J w, i.val, by simp only [map_mul, map_inv, inclusion_simple]⟩
-
-theorem bruhatStep_inclusion_iff [Fintype J] (x y : (matrix (M := M) J).Group) :
-    BruhatStep cs (inclusion cs J x) (inclusion cs J y) ↔
-      BruhatStep (system (M := M) J) x y := by
-  constructor
-  · rintro ⟨hlen, t, ht, heq⟩
-    obtain ⟨ω, hω, hprod⟩ := (system (M := M) J).exists_isReduced y
-    have hi : cs.IsLeftInversion (inclusion cs J y) t := by
-      refine ⟨ht, ?_⟩
-      rw [heq, ← mul_assoc, ht.mul_self, one_mul]
-      simpa only [heq] using hlen
-    have hr := (reduced_map_iff cs J ω).2 hω
-    have hw : cs.wordProd (ω.map Subtype.val) = inclusion cs J y := by
-      rw [← inclusion_word, hprod]
-    have hm := CoxeterSupport.mem_leftInvSeq_of_inversion cs hr (hw ▸ hi)
-    rw [← leftInvSeq_inclusion] at hm
-    obtain ⟨r, hr, hrt⟩ := List.mem_map.mp hm
-    refine ⟨by simpa only [length_inclusion] using hlen,
-      r, (system (M := M) J).isReflection_of_mem_leftInvSeq ω hr, ?_⟩
-    apply inclusion_injective cs J
-    simpa only [map_mul, hrt] using heq
-  · rintro ⟨hlen, t, ht, heq⟩
-    refine ⟨by simpa only [length_inclusion] using hlen,
-      inclusion cs J t, reflection_inclusion cs J ht, ?_⟩
-    rw [heq, map_mul]
-
-end
-end KLInvariance.StandardParabolic
-
-namespace KLInvariance
-universe u v u' v'
-variable {I : Type u} {W : Type v} [Group W] {M : CoxeterMatrix I}
-
-/-- The graph reconstruction input in its source scope: the Coxeter system
-can have infinitely many generators. The interval determines every reflection
-edge without being supplied the reflection labels. -/
-theorem diamondEdges_of_bruhatStep_general (cs : CoxeterSystem M W)
-    {u b : W} {x y : Interval cs u b} (hxy : BruhatStep cs x.val y.val) :
-    DiamondEdges x y := by
-  classical
-  obtain ⟨J, hJ, hwords⟩ := lower_ideal_finite_generator_support cs b
-  let : Fintype J := hJ.fintype
-  obtain ⟨ω, hω, hprod⟩ := hwords b (bruhat_refl cs b)
-  obtain ⟨ν, rfl⟩ := StandardParabolic.lift_word J hω
-  let bb := (StandardParabolic.system (M := M) J).wordProd ν
-  have hb : StandardParabolic.inclusion cs J bb = b :=
-    (StandardParabolic.inclusion_word cs J ν).trans hprod
-  have hub : BruhatLE cs u b := (bruhat_trans cs x.property.1 x.property.2)
-  obtain ⟨ub, hu⟩ := StandardParabolic.lower_range cs J bb (hb ▸ hub)
-  subst u
-  clear hprod
-  subst b
-  let e := StandardParabolic.intervalIso cs J ub bb
-  let xx := e.symm x
-  let yy := e.symm y
-  have hx : StandardParabolic.inclusion cs J xx.val = x.val :=
-    congrArg Subtype.val (e.apply_symm_apply x)
-  have hy : StandardParabolic.inclusion cs J yy.val = y.val :=
-    congrArg Subtype.val (e.apply_symm_apply y)
-  have hh : BruhatStep (StandardParabolic.system (M := M) J) xx.val yy.val := by
-    apply (StandardParabolic.bruhatStep_inclusion_iff cs J _ _).mp
-    rwa [hx, hy]
-  have hg := (diamondEdges_of_bruhatStep (StandardParabolic.system (M := M) J) hh).map e
-  simpa only [xx, yy, e.apply_symm_apply] using hg
-
-theorem diamondEdges_bruhatStep_general (cs : CoxeterSystem M W)
-    {u b : W} {x y : Interval cs u b} (hxy : DiamondEdges x y) :
-    BruhatStep cs x.val y.val := by
-  classical
-  obtain ⟨J, hJ, hwords⟩ := lower_ideal_finite_generator_support cs b
-  let : Fintype J := hJ.fintype
-  obtain ⟨ω, hω, hprod⟩ := hwords b (bruhat_refl cs b)
-  obtain ⟨ν, rfl⟩ := StandardParabolic.lift_word J hω
-  let bb := (StandardParabolic.system (M := M) J).wordProd ν
-  have hb : StandardParabolic.inclusion cs J bb = b :=
-    (StandardParabolic.inclusion_word cs J ν).trans hprod
-  have hub : BruhatLE cs u b := (bruhat_trans cs x.property.1 x.property.2)
-  obtain ⟨ub, hu⟩ := StandardParabolic.lower_range cs J bb (hb ▸ hub)
-  subst u
-  clear hprod
-  subst b
-  let e := StandardParabolic.intervalIso cs J ub bb
-  let xx := e.symm x
-  let yy := e.symm y
-  have hx : StandardParabolic.inclusion cs J xx.val = x.val :=
-    congrArg Subtype.val (e.apply_symm_apply x)
-  have hy : StandardParabolic.inclusion cs J yy.val = y.val :=
-    congrArg Subtype.val (e.apply_symm_apply y)
-  have hh : DiamondEdges xx yy := hxy.map e.symm
-  have hg := diamondEdges_bruhatStep (StandardParabolic.system (M := M) J) hh
-  have hs := (StandardParabolic.bruhatStep_inclusion_iff cs J xx.val yy.val).mpr hg
-  rwa [hx, hy] at hs
-
-/-- Exact unlabelled directed graph invariance, also for arbitrary infinite
-Coxeter systems. This is not yet KL polynomial invariance. -/
-theorem intervalIso_bruhatStep_iff_general
-    {I' : Type u'} {W' : Type v'} [Group W'] {M' : CoxeterMatrix I'}
-    (cs : CoxeterSystem M W) (cs' : CoxeterSystem M' W')
-    {u b : W} {u' b' : W'} (e : Interval cs u b ≃o Interval cs' u' b')
-    (x y : Interval cs u b) :
-    BruhatStep cs' (e x).val (e y).val ↔ BruhatStep cs x.val y.val := by
-  constructor
-  · intro h
-    exact diamondEdges_bruhatStep_general cs ((DiamondEdges.map_iff e).mp
-      (diamondEdges_of_bruhatStep_general cs' h))
-  · intro h
-    exact diamondEdges_bruhatStep_general cs' ((diamondEdges_of_bruhatStep_general cs h).map e)
-
-end KLInvariance
-
-end
-
-
-section
-
-
-namespace KLInvariance.TitsSpace
-open Module
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-noncomputable section
-
-namespace PlaneBasis
-variable {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-
-theorem graphRoot_mem {a₀ : W} {g h : planeSubgroup M cs P}
-    (he : BruhatStep cs ((g : W) * a₀) ((h : W) * a₀)) :
-    (graphRoot he).val ∈ P := by
-  exact reflection_between_coset_mem M cs (a := a₀) g.property h.property (by group)
-    (graphRoot_represents he)
-
-include B
-
-/-- The two labels on a composable coset path span the whole actual plane. -/
-theorem graphRoot_path_plane {a₀ : W} {g h k : planeSubgroup M cs P}
-    (he : BruhatStep cs ((g : W) * a₀) ((h : W) * a₀))
-    (hf : BruhatStep cs ((h : W) * a₀) ((k : W) * a₀)) :
-    rootPlane (graphRoot he) (graphRoot hf) = P := by
-  exact (rootPlane_eq_of_mem B.left B.right (graphRoot he) (graphRoot hf)
-    B.distinct (B.plane_eq.ge (graphRoot_mem he)) (B.plane_eq.ge (graphRoot_mem hf))
-    (graphRoot_ne_of_path he hf)).trans B.plane_eq
-
-
-theorem twoStep_mem_coset {a₀ : W} {g h k : planeSubgroup M cs P}
-    (he : BruhatStep cs ((g : W) * a₀) ((h : W) * a₀))
-    (hf : BruhatStep cs ((h : W) * a₀) ((k : W) * a₀))
-    {z : W} (hez : BruhatStep cs ((g : W) * a₀) z)
-    (hzf : BruhatStep cs z ((k : W) * a₀)) :
-    z * a₀⁻¹ ∈ planeSubgroup M cs P := by
-  by_cases hz : (h : W) * a₀ = z
-  · rw [← hz, mul_inv_cancel_right]
-    exact h.property
-  have hplane := graphRoot_square_plane he hez hf hzf hz
-  have hroot : (graphRoot hez).val ∈ P := by
-    apply (hplane.symm.trans (B.graphRoot_path_plane he hf)).le
-    exact Submodule.subset_span (Set.mem_insert_of_mem _ (Set.mem_singleton _))
-  have href : z * ((g : W) * a₀)⁻¹ ∈ planeSubgroup M cs P :=
-    (Represents.planeSubgroup_mem_iff M cs (graphRoot_represents hez)).mpr hroot
-  convert (planeSubgroup M cs P).mul_mem href g.property using 1; group
-
-/-- A relative rank-two interval has exactly its two ambient two-edge paths,
-with no assumption that the two ambient and relative orders coincide. -/
-theorem ambient_two_step_rank_two {a₀ : W}
-    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z * a₀))
-    {g k : planeSubgroup M cs P} (hlen : B.system.length k = B.system.length g + 2) :
-    ∃ a b : W, a ≠ b ∧ ∀ z : W,
-      (BruhatStep cs ((g : W) * a₀) z ∧ BruhatStep cs z ((k : W) * a₀)) ↔
-        z = a ∨ z = b := by
-  obtain ⟨a,b,hab,hpaths⟩ := Dihedral.two_step_rank_two B.system hlen
-  have ha := (hpaths a).mpr (Or.inl rfl)
-  have he := (B.coset_step_iff hmin g a).mpr ha.1
-  have hf := (B.coset_step_iff hmin a k).mpr ha.2
-  refine ⟨(a : W)*a₀,(b : W)*a₀,?_,fun z => ?_⟩
-  · intro heq
-    exact hab (Subtype.ext (mul_right_cancel heq))
-  constructor
-  · intro hz
-    let z' : planeSubgroup M cs P := ⟨z*a₀⁻¹,B.twoStep_mem_coset he hf hz.1 hz.2⟩
-    have hz' : (z' : W)*a₀=z := by dsimp [z']; group
-    have hp : BruhatStep B.system g z' ∧ BruhatStep B.system z' k := by
-      constructor
-      · exact (B.coset_step_iff hmin g z').mp (hz' ▸ hz.1)
-      · exact (B.coset_step_iff hmin z' k).mp (hz' ▸ hz.2)
-    rcases (hpaths z').mp hp with hza | hzb
-    · exact Or.inl (hz'.symm.trans (congrArg (fun q : planeSubgroup M cs P => (q:W)*a₀) hza))
-    · exact Or.inr (hz'.symm.trans (congrArg (fun q : planeSubgroup M cs P => (q:W)*a₀) hzb))
-  · rintro (rfl | rfl)
-    · exact ⟨he,hf⟩
-    · have hb := (hpaths b).mpr (Or.inr rfl)
-      exact ⟨(B.coset_step_iff hmin g b).mpr hb.1,
-        (B.coset_step_iff hmin b k).mpr hb.2⟩
-
-end PlaneBasis
-end
-end KLInvariance.TitsSpace
-
-namespace KLInvariance.Dihedral
-variable {G : Type*} [Group G] {N : CoxeterMatrix Bool}
-
-/-- Exactly two intermediate vertices characterize relative rank two as soon
-as one two-edge path exists. Used to rule out skipped ranks after transport. -/
-theorem rank_two_of_two_step_bound (ds : CoxeterSystem N G) {x y a b : G}
-    (ha : BruhatStep ds x a) (hay : BruhatStep ds a y)
-    (hbound : ∀ z, BruhatStep ds x z → BruhatStep ds z y → z = a ∨ z = b) :
-    ds.length y = ds.length x + 2 := by
-  have hxa := (step_iff_odd ds x a).mp ha
-  have hay' := (step_iff_odd ds a y).mp hay
-  rw [Nat.odd_iff] at hxa hay'
-  have heven : (ds.length y - ds.length x) % 2 = 0 := by omega
-  by_contra hlen
-  have hlarge : ds.length x + 3 < ds.length y := by omega
-  obtain ⟨p,q,hpq,hlevel⟩ := two_elements_length ds
-    (by omega : 0 < ds.length x+1) (w := y) (by omega)
-  have hp := (hlevel p).mpr (Or.inl rfl)
-  have hq := (hlevel q).mpr (Or.inr rfl)
-  obtain ⟨r,s,hrs,hlevel'⟩ := two_elements_length ds
-    (by omega : 0 < ds.length x+3) hlarge
-  have hr := (hlevel' r).mpr (Or.inl rfl)
-  have paths : ∀ z, ds.length z = ds.length x+1 ∨ ds.length z = ds.length x+3 →
-      BruhatStep ds x z ∧ BruhatStep ds z y := by
-    intro z hz
-    rw [step_iff_odd,step_iff_odd,Nat.odd_iff,Nat.odd_iff]
-    rcases hz with hz | hz <;> omega
-  have pp := paths p (Or.inl hp)
-  have pq := paths q (Or.inl hq)
-  have pr := paths r (Or.inr hr)
-  have hpr : p ≠ r := by intro he; rw [he] at hp; omega
-  have hqr : q ≠ r := by intro he; rw [he] at hq; omega
-  rcases hbound p pp.1 pp.2 with hp' | hp' <;>
-    rcases hbound q pq.1 pq.2 with hq' | hq' <;>
-    rcases hbound r pr.1 pr.2 with hr' | hr' <;> grind
-
-end KLInvariance.Dihedral
-
-namespace KLInvariance.TitsSpace
-open Module
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-noncomputable section
-
-theorem graphRoot_mem_of_coset {P : Submodule ℝ (I → ℝ)} {a₀ x y : W}
-    (h : BruhatStep cs x y) (hx : x*a₀⁻¹ ∈ planeSubgroup M cs P)
-    (hy : y*a₀⁻¹ ∈ planeSubgroup M cs P) : (graphRoot h).val ∈ P := by
-  apply (Represents.planeSubgroup_mem_iff M cs (graphRoot_represents h)).mp
-  convert (planeSubgroup M cs P).mul_mem hy ((planeSubgroup M cs P).inv_mem hx) using 1
-  group
-
-theorem mem_coset_of_graphRoot {P : Submodule ℝ (I → ℝ)} {a₀ x y : W}
-    (h : BruhatStep cs x y) (hx : x*a₀⁻¹ ∈ planeSubgroup M cs P)
-    (hy : (graphRoot h).val ∈ P) : y*a₀⁻¹ ∈ planeSubgroup M cs P := by
-  have hm := (Represents.planeSubgroup_mem_iff M cs (graphRoot_represents h)).mpr hy
-  convert (planeSubgroup M cs P).mul_mem hm hx using 1
-  group
-
-variable {D : Type*} [Group D] {N : CoxeterMatrix Bool} (ds : CoxeterSystem N D)
-  {l r : D}
-
-/-- The image of a full relative dihedral interval lies in one maximal plane
-coset. This is derived just from the directed graph and distinct vertices. -/
-theorem dihedral_image_in_plane
-    (f : Interval ds l r → W) (hinj : Function.Injective f)
-    (hgraph : ∀ a b : Interval ds l r,
-      BruhatStep ds a.val b.val → BruhatStep cs (f a) (f b))
-    (hlr : BruhatLE ds l r) (hrank : ds.length l + 2 ≤ ds.length r) :
-    ∃ a b : PositiveRoot M cs, a ≠ b ∧
-      ∀ z : Interval ds l r,
-        f z * (f ⟨l,bruhat_refl ds l,hlr⟩)⁻¹ ∈ planeSubgroup M cs (rootPlane a b) := by
-  let lo : Interval ds l r := ⟨l,bruhat_refl ds l,hlr⟩
-  let hi : Interval ds l r := ⟨r,hlr,bruhat_refl ds r⟩
-  obtain ⟨a,b,hab,ha,hb⟩ := Dihedral.two_interval_elements_length ds lo hi
-    (n := ds.length l+1) (by dsimp [lo]; omega) (by dsimp [hi]; omega)
-  have cover : ∀ p q : Interval ds l r, ds.length q.val = ds.length p.val+1 →
-      BruhatStep cs (f p) (f q) := by
-    intro p q hlen
-    apply hgraph
-    rw [Dihedral.step_iff_odd,Nat.odd_iff]
-    omega
-  have hla := cover lo a ha
-  have hlb := cover lo b hb
-  have hrootne := graphRoot_ne_of_same_source hla hlb (fun h => hab (hinj h))
-  let P := rootPlane (graphRoot hla) (graphRoot hlb)
-  refine ⟨graphRoot hla,graphRoot hlb,hrootne,?_⟩
-  change ∀ z : Interval ds l r, f z * (f lo)⁻¹ ∈ planeSubgroup M cs P
-  have hlo : f lo * (f lo)⁻¹ ∈ planeSubgroup M cs P := by simp
-  have hat : f a * (f lo)⁻¹ ∈ planeSubgroup M cs P := by
-    exact mem_coset_of_graphRoot hla hlo (Submodule.subset_span (Set.mem_insert _ _))
-  have hbt : f b * (f lo)⁻¹ ∈ planeSubgroup M cs P := by
-    exact mem_coset_of_graphRoot hlb hlo
-      (Submodule.subset_span (Set.mem_insert_of_mem _ (Set.mem_singleton _)))
-  intro z
-  suffices ∀ n, ∀ z : Interval ds l r, ds.length z.val = n →
-      f z * (f lo)⁻¹ ∈ planeSubgroup M cs P from this _ z rfl
-  intro n
-  induction n using Nat.strong_induction_on with
-  | h n ih =>
-    intro z hz
-    have hzmin := length_le_of_bruhat ds z.property.1
-    by_cases hzlo : z = lo
-    · simpa only [hzlo] using hlo
-    have hzgt : ds.length l < ds.length z.val :=
-      length_lt_of_bruhat_ne ds z.property.1 (fun h => hzlo (Subtype.ext h.symm))
-    by_cases hzone : ds.length z.val = ds.length l+1
-    · obtain ⟨p,q,hpq,hlev⟩ := Dihedral.two_elements_length ds
-        (by omega : 0 < ds.length l+1) (w := r) (by omega)
-      have ap := (hlev a.val).mp ha
-      have bp := (hlev b.val).mp hb
-      have zp := (hlev z.val).mp hzone
-      have habval : a.val ≠ b.val := fun h => hab (Subtype.ext h)
-      have hza : z = a ∨ z = b := by
-        apply (show z.val = a.val ∨ z.val = b.val → z = a ∨ z = b from
-          fun h => h.elim (fun h => Or.inl (Subtype.ext h)) (fun h => Or.inr (Subtype.ext h)))
-        grind
-      rcases hza with rfl | rfl <;> assumption
-    have htwo : ds.length l+2 ≤ ds.length z.val := by omega
-    have hexp : ∃ p : Interval ds l r, ds.length p.val+2=ds.length z.val := by
-      by_cases he : ds.length z.val = ds.length l+2
-      · exact ⟨lo,he.symm⟩
-      · obtain ⟨p,q,_,hp,_⟩ := Dihedral.two_interval_elements_length ds lo z
-          (n := ds.length z.val-2) (by dsimp [lo]; omega) (by omega)
-        exact ⟨p,by omega⟩
-    obtain ⟨p,hp⟩ := hexp
-    obtain ⟨q,s,hqs,hq,hs⟩ := Dihedral.two_interval_elements_length ds lo z
-      (n := ds.length z.val-1) (by dsimp [lo]; omega) (by omega)
-    have hp' := ih (ds.length p.val) (by omega) p rfl
-    have hq' := ih (ds.length q.val) (by omega) q rfl
-    have hs' := ih (ds.length s.val) (by omega) s rfl
-    have hpq := cover p q (by omega)
-    have hps := cover p s (by omega)
-    have hqz := cover q z (by omega)
-    have hsz := cover s z (by omega)
-    have hplane : rootPlane (graphRoot hpq) (graphRoot hps) = P :=
-      rootPlane_eq_of_mem _ _ _ _ hrootne (graphRoot_mem_of_coset hpq hp' hq')
-        (graphRoot_mem_of_coset hps hp' hs')
-        (graphRoot_ne_of_same_source hpq hps (fun h => hqs (hinj h)))
-    have hsquare := graphRoot_square_plane hpq hps hqz hsz (fun h => hqs (hinj h))
-    apply mem_coset_of_graphRoot hqz hq'
-    apply (hsquare.trans hplane).le
-    exact Submodule.subset_span (Set.mem_insert_of_mem _ (Set.mem_singleton _))
-
-end
-end KLInvariance.TitsSpace
-
-namespace KLInvariance.Dihedral
-universe u₁ u₂
-variable {D : Type u₁} [Group D] {N : CoxeterMatrix Bool} (ds : CoxeterSystem N D)
-  {l r : D}
-
-theorem exists_interval_length (hlr : BruhatLE ds l r) {n : ℕ}
-    (hnl : ds.length l ≤ n) (hnr : n ≤ ds.length r) :
-    ∃ z : Interval ds l r, ds.length z.val = n := by
-  by_cases hl : n = ds.length l
-  · exact ⟨⟨l,bruhat_refl ds l,hlr⟩,hl.symm⟩
-  by_cases hr : n = ds.length r
-  · exact ⟨⟨r,hlr,bruhat_refl ds r⟩,hr.symm⟩
-  obtain ⟨a,b,_,ha,_⟩ := two_interval_elements_length ds
-    (⟨l,bruhat_refl ds l,hlr⟩ : Interval ds l r)
-    (⟨r,hlr,bruhat_refl ds r⟩ : Interval ds l r)
-    (n := n) (by dsimp; omega) (by dsimp; omega)
-  exact ⟨a,ha⟩
-
-variable {D' : Type u₂} [Group D'] {N' : CoxeterMatrix Bool}
-  (ds' : CoxeterSystem N' D')
-
-theorem map_cover_length
-    (f : Interval ds l r → D')
-    (hgraph : ∀ x y : Interval ds l r, BruhatStep ds x.val y.val →
-      BruhatStep ds' (f x) (f y))
-    (htwo : ∀ x y : Interval ds l r, ds.length y.val = ds.length x.val+2 →
-      ds'.length (f y) = ds'.length (f x)+2)
-    (hlr : BruhatLE ds l r) (hrank : ds.length l+2 ≤ ds.length r)
-    {x y : Interval ds l r} (hxy : ds.length y.val = ds.length x.val+1) :
-    ds'.length (f y) = ds'.length (f x)+1 := by
-  have hstep : ∀ p q : Interval ds l r, ds.length q.val = ds.length p.val+1 →
-      ds'.length (f p) < ds'.length (f q) := by
-    intro p q hpq
-    apply (hgraph p q ?_).1
-    rw [step_iff_odd,Nat.odd_iff]
-    omega
-  have hxy' := hstep x y hxy
-  have hyr := length_le_of_bruhat ds y.property.2
-  by_cases hy : ds.length y.val < ds.length r
-  · obtain ⟨z,hz⟩ := exists_interval_length ds hlr
-      (n := ds.length y.val+1)
-      (by have := length_le_of_bruhat ds y.property.1; omega) (by omega)
-    have hyz := hstep y z hz
-    have hxz := htwo x z (by omega)
-    omega
-  · obtain ⟨z,hz⟩ := exists_interval_length ds hlr
-      (n := ds.length x.val-1) (by omega) (by omega)
-    have hzx := hstep z x (by omega)
-    have hzy := htwo z y (by omega)
-    omega
-
-/-- A directed dihedral-interval graph injection which does not stretch any
-rank-two interval preserves all relative ranks. -/
-theorem map_length_affine
-    (f : Interval ds l r → D')
-    (hgraph : ∀ x y : Interval ds l r, BruhatStep ds x.val y.val →
-      BruhatStep ds' (f x) (f y))
-    (htwo : ∀ x y : Interval ds l r, ds.length y.val = ds.length x.val+2 →
-      ds'.length (f y) = ds'.length (f x)+2)
-    (hlr : BruhatLE ds l r) (hrank : ds.length l+2 ≤ ds.length r)
-    (z : Interval ds l r) :
-    ds'.length (f z) + ds.length l =
-      ds'.length (f ⟨l,bruhat_refl ds l,hlr⟩) + ds.length z.val := by
-  let lo : Interval ds l r := ⟨l,bruhat_refl ds l,hlr⟩
-  change ds'.length (f z)+ds.length l=ds'.length (f lo)+ds.length z.val
-  suffices ∀ n, ∀ z : Interval ds l r, ds.length z.val=n →
-      ds'.length (f z)+ds.length l=ds'.length (f lo)+ds.length z.val from this _ z rfl
-  intro n
-  induction n using Nat.strong_induction_on with
-  | h n ih =>
-    intro z hz
-    by_cases he : z=lo
-    · simp [he,lo]
-    have hlt : ds.length l < ds.length z.val :=
-      length_lt_of_bruhat_ne ds z.property.1 (fun h => he (Subtype.ext h.symm))
-    obtain ⟨p,hp⟩ := exists_interval_length ds hlr (n := ds.length z.val-1)
-      (by omega) (by have := length_le_of_bruhat ds z.property.2; omega)
-    have ihp := ih (ds.length p.val) (by omega) p rfl
-    have hc := map_cover_length ds ds' f hgraph htwo hlr hrank (x := p) (y := z)
-      (by omega)
-    omega
-
-/-- Once relative covers are recovered, the image exhausts rather than merely
-embeds in the target dihedral interval. -/
-theorem range_eq_interval_of_rank_two
-    (f : Interval ds l r → D') (hinj : Function.Injective f)
-    (hgraph : ∀ x y : Interval ds l r, BruhatStep ds x.val y.val →
-      BruhatStep ds' (f x) (f y))
-    (htwo : ∀ x y : Interval ds l r, ds.length y.val = ds.length x.val+2 →
-      ds'.length (f y) = ds'.length (f x)+2)
-    (hlr : BruhatLE ds l r) (hrank : ds.length l+2 ≤ ds.length r) :
-    Set.range f = {z | BruhatLE ds' (f ⟨l,bruhat_refl ds l,hlr⟩) z ∧
-      BruhatLE ds' z (f ⟨r,hlr,bruhat_refl ds r⟩)} := by
-  let lo : Interval ds l r := ⟨l,bruhat_refl ds l,hlr⟩
-  let hi : Interval ds l r := ⟨r,hlr,bruhat_refl ds r⟩
-  have hlen (z : Interval ds l r) : ds'.length (f z)+ds.length l =
-      ds'.length (f lo)+ds.length z.val :=
-    map_length_affine ds ds' f hgraph htwo hlr hrank z
-  have hhi : ds'.length (f hi)+ds.length l=ds'.length (f lo)+ds.length r := hlen hi
-  have horder (x y : Interval ds l r) :
-      BruhatLE ds' (f x) (f y) ↔ x ≤ y := by
-    change BruhatLE ds' (f x) (f y) ↔ BruhatLE ds x.val y.val
-    rw [bruhat_iff_length_lt_or_eq,bruhat_iff_length_lt_or_eq]
-    have hx := hlen x
-    have hy := hlen y
-    constructor
-    · rintro (h | h)
-      · left; omega
-      · exact Or.inr (congrArg Subtype.val (hinj h))
-    · rintro (h | h)
-      · left; omega
-      · exact Or.inr (congrArg f (Subtype.ext h))
-  apply Set.ext
-  intro z
-  change (∃ x, f x=z) ↔ BruhatLE ds' (f lo) z ∧ BruhatLE ds' z (f hi)
-  constructor
-  · rintro ⟨x,rfl⟩
-    exact ⟨(horder lo x).mpr x.property.1,(horder x hi).mpr x.property.2⟩
-  · intro hz
-    by_cases hlo : z=f lo
-    · exact ⟨lo,hlo.symm⟩
-    by_cases htop : z=f hi
-    · exact ⟨hi,htop.symm⟩
-    have hzlo := length_lt_of_bruhat_ne ds' hz.1 (Ne.symm hlo)
-    have hzhi := length_lt_of_bruhat_ne ds' hz.2 htop
-    have hnlo : ds.length l < ds.length l+(ds'.length z-ds'.length (f lo)) := by omega
-    have hnhi : ds.length l+(ds'.length z-ds'.length (f lo)) < ds.length r := by omega
-    obtain ⟨a,b,hab,ha,hb⟩ := two_interval_elements_length ds lo hi hnlo hnhi
-    have hfa : ds'.length (f a)=ds'.length z := by have := hlen a; omega
-    have hfb : ds'.length (f b)=ds'.length z := by have := hlen b; omega
-    obtain ⟨p,q,hpq,helements⟩ := two_elements_length ds' (by omega : 0<ds'.length z) hzhi
-    have hp := (helements (f a)).mp hfa
-    have hq := (helements (f b)).mp hfb
-    have hr := (helements z).mp rfl
-    have hab' : f a ≠ f b := fun h => hab (hinj h)
-    have he : z=f a ∨ z=f b := by grind
-    rcases he with he | he
-    · exact ⟨a,he.symm⟩
-    · exact ⟨b,he.symm⟩
-
-end KLInvariance.Dihedral
-
-namespace KLInvariance
-universe u v u' v'
-variable {I : Type u} {W : Type v} [Group W] {M : CoxeterMatrix I}
-  {I' : Type u'} {W' : Type v'} [Group W'] {M' : CoxeterMatrix I'}
-  {cs : CoxeterSystem M W} {cs' : CoxeterSystem M' W'}
-  {u b : W} {u' b' : W'}
-
-/-- An interval graph isomorphism preserves the exact two-edge intermediate
-set, including ambient intermediates not specified as interval data. -/
-theorem intervalIso_twoStep_iff
-    (e : Interval cs u b ≃o Interval cs' u' b') (x y a c : Interval cs u b)
-    (hpaths : ∀ z : W, (BruhatStep cs x.val z ∧ BruhatStep cs z y.val) ↔
-      z=a.val ∨ z=c.val) (z : W') :
-    (BruhatStep cs' (e x).val z ∧ BruhatStep cs' z (e y).val) ↔
-      z=(e a).val ∨ z=(e c).val := by
-  constructor
-  · intro hz
-    let z' : Interval cs' u' b' := ⟨z,
-      bruhat_trans cs' (e x).property.1 (Relation.ReflTransGen.single hz.1),
-      bruhat_trans cs' (Relation.ReflTransGen.single hz.2) (e y).property.2⟩
-    have hz₁ : BruhatStep cs x.val (e.symm z').val := by
-      apply (intervalIso_bruhatStep_iff_general cs cs' e x (e.symm z')).mp
-      simpa only [e.apply_symm_apply] using hz.1
-    have hz₂ : BruhatStep cs (e.symm z').val y.val := by
-      apply (intervalIso_bruhatStep_iff_general cs cs' e (e.symm z') y).mp
-      simpa only [e.apply_symm_apply] using hz.2
-    rcases (hpaths (e.symm z').val).mp ⟨hz₁,hz₂⟩ with ha | hc
-    · have he : e.symm z'=a := Subtype.ext ha
-      have he' : z'=e a := by rw [← he,e.apply_symm_apply]
-      exact Or.inl (congrArg Subtype.val he')
-    · have he : e.symm z'=c := Subtype.ext hc
-      have he' : z'=e c := by rw [← he,e.apply_symm_apply]
-      exact Or.inr (congrArg Subtype.val he')
-  · rintro (rfl | rfl)
-    · have h := (hpaths a.val).mpr (Or.inl rfl)
-      exact ⟨(intervalIso_bruhatStep_iff_general cs cs' e x a).mpr h.1,
-        (intervalIso_bruhatStep_iff_general cs cs' e a y).mpr h.2⟩
-    · have h := (hpaths c.val).mpr (Or.inr rfl)
-      exact ⟨(intervalIso_bruhatStep_iff_general cs cs' e x c).mpr h.1,
-        (intervalIso_bruhatStep_iff_general cs cs' e c y).mpr h.2⟩
-
-end KLInvariance
-
-namespace KLInvariance.TitsSpace.PlaneBasis
-open Module
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-noncomputable section
-
-def cosetIntervalMap {a₀ u b : W}
-    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-    {g k : planeSubgroup M cs P}
-    (hlo : BruhatLE cs u ((g : W)*a₀)) (hhi : BruhatLE cs ((k : W)*a₀) b) :
-    Interval B.system g k ↪ Interval cs u b where
-  toFun z := ⟨(z.val : W)*a₀,bruhat_trans cs hlo (B.coset_bruhat hmin z.property.1),
-    bruhat_trans cs (B.coset_bruhat hmin z.property.2) hhi⟩
-  inj' := by
-    intro x y he
-    exact Subtype.ext (Subtype.ext (mul_right_cancel (congrArg Subtype.val he)))
-
-theorem twoStep_interval_iff {a₀ : W}
-    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-    {g k : planeSubgroup M cs P} (x y : Interval B.system g k)
-    (hlen : B.system.length y.val=B.system.length x.val+2) :
-    ∃ a b : Interval B.system g k, a ≠ b ∧ ∀ z : W,
-      (BruhatStep cs ((x.val : W)*a₀) z ∧ BruhatStep cs z ((y.val : W)*a₀)) ↔
-        z=(a.val : W)*a₀ ∨ z=(b.val : W)*a₀ := by
-  obtain ⟨a,b,hab,hpaths⟩ := Dihedral.two_step_rank_two B.system hlen
-  have ha := (hpaths a).mpr (Or.inl rfl)
-  have hb := (hpaths b).mpr (Or.inr rfl)
-  let ai : Interval B.system g k := ⟨a,
-    bruhat_trans B.system x.property.1 (Relation.ReflTransGen.single ha.1),
-    bruhat_trans B.system (Relation.ReflTransGen.single ha.2) y.property.2⟩
-  let bi : Interval B.system g k := ⟨b,
-    bruhat_trans B.system x.property.1 (Relation.ReflTransGen.single hb.1),
-    bruhat_trans B.system (Relation.ReflTransGen.single hb.2) y.property.2⟩
-  have he := (B.coset_step_iff hmin x.val a).mpr ha.1
-  have hf := (B.coset_step_iff hmin a y.val).mpr ha.2
-  refine ⟨ai,bi,fun h => hab (congrArg Subtype.val h),fun z => ?_⟩
-  constructor
-  · intro hz
-    let z' : planeSubgroup M cs P := ⟨z*a₀⁻¹,B.twoStep_mem_coset he hf hz.1 hz.2⟩
-    have hz' : (z' : W)*a₀=z := by dsimp [z']; group
-    have hp : BruhatStep B.system x.val z' ∧ BruhatStep B.system z' y.val := by
-      constructor
-      · exact (B.coset_step_iff hmin x.val z').mp (hz' ▸ hz.1)
-      · exact (B.coset_step_iff hmin z' y.val).mp (hz' ▸ hz.2)
-    rcases (hpaths z').mp hp with hza | hzb
-    · exact Or.inl (hz'.symm.trans (congrArg (fun q : planeSubgroup M cs P => (q:W)*a₀) hza))
-    · exact Or.inr (hz'.symm.trans (congrArg (fun q : planeSubgroup M cs P => (q:W)*a₀) hzb))
-  · rintro (rfl | rfl)
-    · exact ⟨he,hf⟩
-    · exact ⟨(B.coset_step_iff hmin x.val b).mpr hb.1,
-        (B.coset_step_iff hmin b y.val).mpr hb.2⟩
-
-end
-end KLInvariance.TitsSpace.PlaneBasis
-
-namespace KLInvariance.TitsSpace.PlaneBasis
-open Module
-universe u v u' v'
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
-  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
-  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-noncomputable section
-
-
-theorem transport_interval {a₀ u b : W} {u' b' : W'}
-    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-    {g k : planeSubgroup M cs P} (hgk : BruhatLE B.system g k)
-    (hrank : B.system.length g+2 ≤ B.system.length k)
-    (hlo : BruhatLE cs u ((g : W)*a₀)) (hhi : BruhatLE cs ((k : W)*a₀) b)
-    (e : Interval cs u b ≃o Interval cs' u' b') :
-    ∃ (P' : Submodule ℝ (I' → ℝ)) (B' : PlaneBasis (M := M') (cs := cs') P')
-      (a' : W') (F : Interval B.system g k → planeSubgroup M' cs' P'),
-      (∀ z ∈ planeSubgroup M' cs' P', cs'.length a' ≤ cs'.length (z*a')) ∧
-      Function.Injective F ∧
-      (∀ z, (F z : W')*a' = (e (B.cosetIntervalMap hmin hlo hhi z)).val) ∧
-      (∀ x y, BruhatLE B'.system (F x) (F y) ↔ x ≤ y) ∧
-      (∀ x y, BruhatStep B'.system (F x) (F y) ↔ BruhatStep B.system x.val y.val) ∧
-      (∀ z, B'.system.length (F z)+B.system.length g =
-        B'.system.length (F ⟨g,bruhat_refl B.system g,hgk⟩)+B.system.length z.val) ∧
-      Set.range F = {z | BruhatLE B'.system (F ⟨g,bruhat_refl B.system g,hgk⟩) z ∧
-        BruhatLE B'.system z (F ⟨k,hgk,bruhat_refl B.system k⟩)} := by
-  let inc := B.cosetIntervalMap hmin hlo hhi
-  let f : Interval B.system g k → W' := fun z => (e (inc z)).val
-  let lo : Interval B.system g k := ⟨g,bruhat_refl B.system g,hgk⟩
-  have hinj : Function.Injective f := by
-    intro x y h
-    exact inc.injective (e.injective (Subtype.ext h))
-  have hgraph (x y : Interval B.system g k) (h : BruhatStep B.system x.val y.val) :
-      BruhatStep cs' (f x) (f y) := by
-    exact (intervalIso_bruhatStep_iff_general cs cs' e (inc x) (inc y)).mpr
-      ((B.coset_step_iff hmin x.val y.val).mpr h)
-  obtain ⟨α,β,hαβ,hplane⟩ := dihedral_image_in_plane B.system f hinj hgraph hgk hrank
-  let P' := rootPlane α β
-  obtain ⟨B'⟩ := nonempty_planeBasis α β hαβ
-  obtain ⟨a',g₀,hg₀,hg₀a,hmin'⟩ := planeCoset_exists_minimal M' cs' P' (f lo)
-  have hmem (z : Interval B.system g k) : f z*a'⁻¹ ∈ planeSubgroup M' cs' P' := by
-    convert (planeSubgroup M' cs' P').mul_mem (hplane z) hg₀ using 1
-    change f z*a'⁻¹=f z*(f lo)⁻¹*g₀
-    rw [hg₀a]
-    group
-  let F : Interval B.system g k → planeSubgroup M' cs' P' := fun z => ⟨f z*a'⁻¹,hmem z⟩
-  have hF (z : Interval B.system g k) : (F z : W')*a'=f z := by
-    dsimp [F]; group
-  have hFinj : Function.Injective F := by
-    intro x y h
-    apply hinj
-    rw [← hF x,← hF y,h]
-  have hFgraph (x y : Interval B.system g k) (h : BruhatStep B.system x.val y.val) :
-      BruhatStep B'.system (F x) (F y) := by
-    apply (B'.coset_step_iff hmin' (F x) (F y)).mp
-    rw [hF x,hF y]
-    exact hgraph x y h
-  have htwo (x y : Interval B.system g k)
-      (hlen : B.system.length y.val=B.system.length x.val+2) :
-      B'.system.length (F y)=B'.system.length (F x)+2 := by
-    obtain ⟨a,c,hac,hpaths⟩ := B.twoStep_interval_iff hmin x y hlen
-    have hpaths' (z : W') : (BruhatStep cs' (f x) z ∧ BruhatStep cs' z (f y)) ↔
-        z=f a ∨ z=f c := intervalIso_twoStep_iff e (inc x) (inc y) (inc a) (inc c) hpaths z
-    have hpa := (hpaths' (f a)).mpr (Or.inl rfl)
-    have hxa : BruhatStep B'.system (F x) (F a) := by
-      apply (B'.coset_step_iff hmin' (F x) (F a)).mp
-      simpa only [hF] using hpa.1
-    have hay : BruhatStep B'.system (F a) (F y) := by
-      apply (B'.coset_step_iff hmin' (F a) (F y)).mp
-      simpa only [hF] using hpa.2
-    apply Dihedral.rank_two_of_two_step_bound B'.system hxa hay (b := F c)
-    intro z hxz hzy
-    have hp₁ := (B'.coset_step_iff hmin' (F x) z).mpr hxz
-    have hp₂ := (B'.coset_step_iff hmin' z (F y)).mpr hzy
-    rw [hF x] at hp₁
-    rw [hF y] at hp₂
-    rcases (hpaths' ((z : W')*a')).mp ⟨hp₁,hp₂⟩ with he | he
-    · exact Or.inl (Subtype.ext (mul_right_cancel (he.trans (hF a).symm)))
-    · exact Or.inr (Subtype.ext (mul_right_cancel (he.trans (hF c).symm)))
-  have hlen := Dihedral.map_length_affine B.system B'.system F hFgraph htwo hgk hrank
-  have horder (x y : Interval B.system g k) : BruhatLE B'.system (F x) (F y) ↔ x ≤ y := by
-    change BruhatLE B'.system (F x) (F y) ↔ BruhatLE B.system x.val y.val
-    rw [Dihedral.bruhat_iff_length_lt_or_eq,Dihedral.bruhat_iff_length_lt_or_eq]
-    have hx := hlen x
-    have hy := hlen y
-    constructor
-    · rintro (h | h)
-      · left; omega
-      · exact Or.inr (congrArg Subtype.val (hFinj h))
-    · rintro (h | h)
-      · left; omega
-      · exact Or.inr (congrArg F (Subtype.ext h))
-  refine ⟨P',B',a',F,hmin',hFinj,hF,horder,?_,hlen,?_⟩
-  · intro x y
-    rw [Dihedral.step_iff_odd,Dihedral.step_iff_odd,Nat.odd_iff,Nat.odd_iff]
-    have hx := hlen x
-    have hy := hlen y
-    omega
-  · exact Dihedral.range_eq_interval_of_rank_two B.system B'.system F hFinj
-      hFgraph htwo hgk hrank
-
-end
-end KLInvariance.TitsSpace.PlaneBasis
-
-end
-
-
-section
-
-
-namespace KLInvariance.TitsSpace
-open Module
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-noncomputable section
-
-theorem inverted_of_positive_combination (w : W) (a b c : PositiveRoot M cs)
-    {r s : ℝ} (hr : 0 ≤ r) (hs : 0 ≤ s)
-    (hc : c.val = r • a.val + s • b.val)
-    (ha : rootInverted w a) (hb : rootInverted w b) :
-    rootInverted w c := by
-  intro i
-  rw [hc, map_add, map_smul, map_smul]
-  exact add_nonpos (mul_nonpos_of_nonneg_of_nonpos hr (ha i))
-    (mul_nonpos_of_nonneg_of_nonpos hs (hb i))
-
-/-- Incoming graph labels are actual inversions at their upper vertex. -/
-theorem graphRoot_inverted_target {x y : W} (h : BruhatStep cs x y) :
-    rootInverted y (graphRoot h) := by
-  apply ((graphRoot_represents h).inversion_iff (graphRoot h).property.2 y).mpr
-  refine ⟨h.label_reflection, ?_⟩
-  have he : (y*x⁻¹)*y=x := by
-    calc
-      (y*x⁻¹)*y = (y*x⁻¹)*((y*x⁻¹)*x) := by group
-      _ = x := by rw [← mul_assoc,h.label_reflection.mul_self,one_mul]
-  simpa only [he] using h.1
-
-/-- Outgoing graph labels are actual noninversions at their lower vertex. -/
-theorem graphRoot_not_inverted_source {x y : W} (h : BruhatStep cs x y) :
-    ¬ rootInverted x (graphRoot h) := by
-  intro hi
-  have hh := ((graphRoot_represents h).inversion_iff (graphRoot h).property.2 x).mp hi
-  have he : (y*x⁻¹)*x=y := by group
-  have hh' := hh.2
-  rw [he] at hh'
-  exact lt_asymm h.1 hh'
-
-namespace PlaneBasis
-variable {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-
-/-- In a plane with its first simple root inverted, every noninverted root
-is later than every inverted root. This uses convexity, not a signed-index
-formula for a particular realization. -/
-theorem slope_lt_of_inverted_left (w : W) (hl : rootInverted w B.left)
-    (a b : PositiveRoot M cs) (haP : a.val ∈ P) (hbP : b.val ∈ P)
-    (ha : rootInverted w a) (hb : ¬ rootInverted w b) :
-    geometricSlope a < geometricSlope b := by
-  by_contra h
-  have hba : geometricSlope b < geometricSlope a :=
-    lt_of_le_of_ne (le_of_not_gt h) (fun he => hb (geometricSlope_injective he ▸ ha))
-  have hlb : geometricSlope B.left < geometricSlope b :=
-    lt_of_le_of_ne (B.left_min b hbP) (fun he => hb (geometricSlope_injective he ▸ hl))
-  have heq : rootPlane B.left a = P :=
-    rootPlane_eq_of_mem B.left B.right B.left a B.distinct
-      (Submodule.subset_span (Set.mem_insert _ _))
-      (B.plane_eq.ge haP) (fun he => by rw [he] at hlb; exact (lt_asymm hlb hba)) |>.trans B.plane_eq
-  obtain ⟨r,s,hr,hs,hc⟩ := positive_combination_of_slope_between B.left a b
-    (heq.symm ▸ hbP) hlb hba
-  exact hb (inverted_of_positive_combination w B.left a b hr.le hs.le hc hl ha)
-
-/-- The other angular cut, when the last simple root is inverted. -/
-theorem slope_lt_of_inverted_right (w : W) (hr : rootInverted w B.right)
-    (a b : PositiveRoot M cs) (haP : a.val ∈ P) (hbP : b.val ∈ P)
-    (ha : rootInverted w a) (hb : ¬ rootInverted w b) :
-    geometricSlope b < geometricSlope a := by
-  by_contra h
-  have hab : geometricSlope a < geometricSlope b :=
-    lt_of_le_of_ne (le_of_not_gt h) (fun he => hb (geometricSlope_injective he ▸ ha))
-  have hbr : geometricSlope b < geometricSlope B.right :=
-    lt_of_le_of_ne (B.right_max b hbP) (fun he => hb (geometricSlope_injective he ▸ hr))
-  have har : a ≠ B.right := by
-    intro he
-    rw [he] at hab
-    exact lt_asymm hbr hab
-  have heq : rootPlane a B.right = P :=
-    (rootPlane_eq_of_mem B.left B.right a B.right B.distinct
-      (B.plane_eq.ge haP) (Submodule.subset_span (Set.mem_insert_of_mem _ (Set.mem_singleton _)))
-      har).trans B.plane_eq
-  obtain ⟨r,s,hrr,hss,hc⟩ := positive_combination_of_slope_between a B.right b
-    (heq.symm ▸ hbP) hab hbr
-  exact hb (inverted_of_positive_combination w a B.right b hrr.le hss.le hc ha hr)
-
-/-- In the mixed-sign case the angular cut is determined by the first
-canonical simple root. In particular the comparison of consecutive path
-labels depends only on their middle vertex. -/
-theorem slope_lt_iff_inverted_left (w : W)
-    (a b : PositiveRoot M cs) (haP : a.val ∈ P) (hbP : b.val ∈ P)
-    (ha : rootInverted w a) (hb : ¬ rootInverted w b) :
-    geometricSlope a < geometricSlope b ↔ rootInverted w B.left := by
-  constructor
-  · intro hab
-    by_contra hl
-    have hr : rootInverted w B.right := by
-      by_contra hr
-      obtain ⟨r,s,hrr,hss,he⟩ := B.positive_cone a haP
-      exact noninverted_of_positive_combination w B.left B.right a hrr hss he hl hr ha
-    exact lt_asymm hab (B.slope_lt_of_inverted_right w hr a b haP hbP ha hb)
-  · intro hl
-    exact B.slope_lt_of_inverted_left w hl a b haP hbP ha hb
-
-/-- The exact middle-vertex test for two ascending graph arrows with both
-labels in this root plane. -/
-theorem consecutive_slopes_iff {x y z : W}
-    (hxy : BruhatStep cs x y) (hyz : BruhatStep cs y z)
-    (ha : (graphRoot hxy).val ∈ P) (hb : (graphRoot hyz).val ∈ P) :
-    geometricSlope (graphRoot hxy) < geometricSlope (graphRoot hyz) ↔
-      rootInverted y B.left :=
-  B.slope_lt_iff_inverted_left y _ _ ha hb
-    (graphRoot_inverted_target hxy) (graphRoot_not_inverted_source hyz)
-
-end PlaneBasis
-end
-end KLInvariance.TitsSpace
-
-namespace KLInvariance.Dihedral
-variable {W : Type*} [Group W] {M : CoxeterMatrix Bool}
-noncomputable section
-
-/-- The genuine diagram automorphism of an arbitrary rank-two Coxeter system,
-including the infinite one. -/
-def swapHom (cs : CoxeterSystem M W) : W →* W :=
-  cs.lift ⟨fun i => cs.simple (!i), by
-    intro i j
-    have he : M i j = M (!i) (!j) := by
-      cases i <;> cases j
-      · simp
-      · exact M.symmetric _ _
-      · exact M.symmetric _ _
-      · simp
-    rw [he]
-    exact cs.simple_mul_simple_pow _ _⟩
-
-@[simp] theorem swapHom_simple (cs : CoxeterSystem M W) (i : Bool) :
-    swapHom cs (cs.simple i) = cs.simple (!i) := cs.lift_apply_simple _ i
-
-@[simp] theorem swapHom_involutive (cs : CoxeterSystem M W) :
-    Function.Involutive (swapHom cs) := by
-  intro w
-  apply cs.simple_induction w
-  · intro i
-    simp
-  · simp
-  · intro a b ha hb
-    simp only [map_mul,ha,hb]
-
-def swap (cs : CoxeterSystem M W) : W ≃* W where
-  toFun := swapHom cs
-  invFun := swapHom cs
-  left_inv := swapHom_involutive cs
-  right_inv := swapHom_involutive cs
-  map_mul' := map_mul (swapHom cs)
-
-@[simp] theorem swap_simple (cs : CoxeterSystem M W) (i : Bool) :
-    swap cs (cs.simple i) = cs.simple (!i) := swapHom_simple cs i
-
-@[simp] theorem swap_swap (cs : CoxeterSystem M W) (w : W) :
-    swap cs (swap cs w) = w := swapHom_involutive cs w
-
-theorem swap_wordProd (cs : CoxeterSystem M W) (ω : List Bool) :
-    swap cs (cs.wordProd ω) = cs.wordProd (ω.map Bool.not) := by
-  induction ω with
-  | nil => simp
-  | cons i ω ih =>
-    simp only [cs.wordProd_cons, map_mul, swap_simple, List.map_cons, ih]
-
-@[simp] theorem length_swap (cs : CoxeterSystem M W) (w : W) :
-    cs.length (swap cs w) = cs.length w := by
-  have hle (z : W) : cs.length (swap cs z) ≤ cs.length z := by
-    obtain ⟨ω,hω,rfl⟩ := cs.exists_isReduced z
-    rw [swap_wordProd, hω.eq]
-    simpa using cs.length_wordProd_le (ω.map Bool.not)
-  exact le_antisymm (hle w) (by simpa using hle (swap cs w))
-
-@[simp] theorem descent_swap (cs : CoxeterSystem M W) (w : W) (i : Bool) :
-    cs.IsLeftDescent (swap cs w) i ↔ cs.IsLeftDescent w (!i) := by
-  unfold CoxeterSystem.IsLeftDescent
-  rw [length_swap, ← Bool.not_not i, ← swap_simple, ← map_mul, length_swap]
-  simp
-
-/-- A nonzero length below an existing greater length has exactly one left
-simple descent. This is not asserted for a finite dihedral longest element. -/
-theorem unique_leftDescent (cs : CoxeterSystem M W) {w z : W}
-    (hw : 0 < cs.length w) (hz : cs.length w < cs.length z) :
-    ∃! i : Bool, cs.IsLeftDescent w i := by
-  obtain ⟨i,hi⟩ := cs.exists_leftDescent_of_ne_one (w := w) (by
-    intro he
-    rw [he,cs.length_one] at hw
-    omega)
-  refine ⟨i,hi,?_⟩
-  intro j hj
-  by_contra hji
-  have hj' : j = !i := by cases i <;> cases j <;> simp_all
-  have he : cs.wordProd (alt i (cs.length w)) =
-      cs.wordProd (alt (!i) (cs.length w)) := by
-    rw [← eq_alt_of_descent cs i hi, ← hj', ← eq_alt_of_descent cs j hj]
-  obtain ⟨a,b,hab,hab'⟩ := two_elements_length cs hw hz
-  have all_eq (v : W) (hv : cs.length v = cs.length w) : v=w := by
-    obtain ⟨k,hk⟩ := exists_alt cs v
-    rw [hv] at hk
-    have hki : k=i ∨ k=(!i) := by cases k <;> cases i <;> simp
-    rcases hki with hki | hki
-    · rw [hki] at hk
-      exact hk.trans (eq_alt_of_descent cs i hi).symm
-    · rw [hki] at hk
-      exact hk.trans (he.symm.trans (eq_alt_of_descent cs i hi).symm)
-  exact hab ((all_eq a ((hab' a).mpr (.inl rfl))).trans
-    (all_eq b ((hab' b).mpr (.inr rfl))).symm)
-
-/-- Internally, the diagram involution interchanges the two unique-descent
-vertices of each rank. -/
-theorem descent_swap_false_iff_not (cs : CoxeterSystem M W) {w z : W}
-    (hw : 0 < cs.length w) (hz : cs.length w < cs.length z) :
-    cs.IsLeftDescent (swap cs w) false ↔ ¬ cs.IsLeftDescent w false := by
-  rw [descent_swap]
-  obtain ⟨i,hi,hu⟩ := unique_leftDescent cs hw hz
-  cases i
-  · constructor
-    · intro h
-      have hh := hu true h
-      contradiction
-    · exact fun h => (h hi).elim
-  · constructor
-    · intro _ h
-      have hh := hu false h
-      contradiction
-    · exact fun _ => hi
-
-end
-end KLInvariance.Dihedral
-
-namespace KLInvariance.TitsSpace
-open Module
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-noncomputable section
-namespace PlaneBasis
-variable {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-
-/-- The ambient sign of a canonical plane root in a minimal coset is the
-intrinsic simple-descent sign, with the original left labels. -/
-theorem simpleRoot_inverted_coset_iff {a₀ : W}
-    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-    (w : planeSubgroup M cs P) (i : Bool) :
-    rootInverted ((w : W)*a₀) (B.simpleRoot i) ↔ B.system.IsLeftDescent w i := by
-  change (∀ k, action M cs ((w : W)*a₀)⁻¹ (B.simpleRoot i).val k ≤ 0) ↔ _
-  rw [root_inversion_coset_iff hmin w.property (B.simpleRoot i).property.1
-    (B.simpleRoot_mem i)]
-  have hr : Represents M cs (B.system.simple i : W) (B.simpleRoot i).val := by
-    rw [B.system_simple]
-    exact rootReflection_represents M cs _
-  rw [B.inversion_iff hr (B.simpleRoot_mem i) (B.simpleRoot i).property.2 w]
-  exact B.system.isLeftInversion_simple_iff_isLeftDescent w i
-
-/-- Geometric label of an intrinsic graph edge, taken in the actual ambient
-root plane rather than an abstract alternate representation. -/
-def edgeRoot {x y : planeSubgroup M cs P} (h : BruhatStep B.system x y) :
-    PositiveRoot M cs := graphRoot ((B.bruhatStep_iff x y).mpr h)
-
-theorem edgeRoot_plane {x y : planeSubgroup M cs P} (h : BruhatStep B.system x y) :
-    (B.edgeRoot h).val ∈ P := by
-  apply (Represents.planeSubgroup_mem_iff M cs (graphRoot_represents
-    ((B.bruhatStep_iff x y).mpr h))).mp
-  exact (planeSubgroup M cs P).mul_mem y.property
-    ((planeSubgroup M cs P).inv_mem x.property)
-
-/-- In the intrinsic system, the middle-vertex test uses its actual left
-simple generator; it therefore applies to any induced dihedral realization. -/
-theorem edgeRoot_slopes_iff {x y z : planeSubgroup M cs P}
-    (hxy : BruhatStep B.system x y) (hyz : BruhatStep B.system y z) :
-    geometricSlope (B.edgeRoot hxy) < geometricSlope (B.edgeRoot hyz) ↔
-      B.system.IsLeftDescent y false := by
-  unfold edgeRoot
-  rw [B.consecutive_slopes_iff ((B.bruhatStep_iff x y).mpr hxy)
-    ((B.bruhatStep_iff y z).mpr hyz) (B.edgeRoot_plane hxy) (B.edgeRoot_plane hyz)]
-  have hmin : ∀ z ∈ planeSubgroup M cs P, cs.length (1:W) ≤ cs.length (z*1) :=
-    fun z _ => by simp
-  simpa only [mul_one, simpleRoot, Bool.false_eq_true, ↓reduceIte] using
-    B.simpleRoot_inverted_coset_iff hmin y false
-
-end PlaneBasis
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Geometric orders adapted to a fixed upper vertex. These are honest
-ratios of linear functionals, with a strictly positive denominator. They
-will be used only as a permissible choice of the source's schedule. -/
-namespace KLInvariance.TitsSpace
-open Module
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
-def simplePositive (i : I) : PositiveRoot M cs :=
-  ⟨root i, ⟨1,i,by simp⟩, root_nonneg i⟩
-
-@[simp] theorem simplePositive_val (i : I) :
-    (simplePositive (M := M) (cs := cs) i).val = root i := rfl
-
-@[simp] theorem rootRef_simplePositive (i : I) :
-    rootRef (simplePositive (M := M) (cs := cs) i) = cs.simple i := by
-  exact (rootReflection_represents M cs _).reflection_unique ⟨1,i,by simp,by simp⟩
-
-theorem simplePositive_injective : Function.Injective (simplePositive (M := M) (cs := cs)) := by
-  intro i j hij
-  have he := congrArg (fun a : PositiveRoot M cs => a.val i) hij
-  by_contra hne
-  simp [simplePositive, root, hne] at he
-
-theorem coordinate_sum (a : I → ℝ) : ∑ i, a i • root i = a := by
-  ext j
-  simp [root, Pi.single_apply, Finset.sum_apply]
-
-theorem dual_coordinate_sum (h : Module.Dual ℝ (I → ℝ)) (a : I → ℝ) :
-    h a = ∑ i, a i * h (root i) := by
-  conv_lhs => rw [← coordinate_sum a]
-  simp only [map_sum,map_smul,smul_eq_mul]
-
-def PositiveFunctional (h : Module.Dual ℝ (I → ℝ)) : Prop := ∀ i, 0 < h (root i)
-
-theorem PositiveFunctional.root_pos {h : Module.Dual ℝ (I → ℝ)}
-    (hh : PositiveFunctional h) (a : PositiveRoot M cs) : 0 < h a.val := by
-  rw [dual_coordinate_sum]
-  obtain ⟨i,hi⟩ : ∃ i, a.val i ≠ 0 := by
-    by_contra hn
-    push Not at hn
-    exact a.property.1.ne_zero (funext hn)
-  exact Finset.sum_pos' (fun j _ => mul_nonneg (a.property.2 j) (hh j).le)
-    ⟨i,Finset.mem_univ _,mul_pos (lt_of_le_of_ne' (a.property.2 i) hi) (hh i)⟩
-
-theorem height_positive : PositiveFunctional (height (I := I)) := by
-  intro i
-  simp [height,root]
-
-/-- Both functionals are actual dual vectors; the root order is not a
-label-carrying addition to the interval isomorphism. -/
-structure RatioOrder where
-  denominator : Module.Dual ℝ (I → ℝ)
-  numerator : Module.Dual ℝ (I → ℝ)
-  positive : PositiveFunctional denominator
-  generic : Function.Injective (fun a : PositiveRoot M cs => numerator a.val / denominator a.val)
-
-namespace RatioOrder
-variable (O : RatioOrder (M := M) (cs := cs))
-
-def slope (a : PositiveRoot M cs) : ℝ := O.numerator a.val / O.denominator a.val
-
-theorem denominator_pos (a : PositiveRoot M cs) : 0 < O.denominator a.val :=
-  O.positive.root_pos a
-
-theorem slope_injective : Function.Injective O.slope := O.generic
-
-/-- The cut at zero is the genuine inversion set of the upper vertex. -/
-def Adapted (y : W) : Prop := ∀ a : PositiveRoot M cs,
-  (O.slope a < 0 ↔ rootInverted y a) ∧ O.slope a ≠ 0
-
-theorem simple_lower_bound (i : I)
-    (hi : ∀ j, O.slope (simplePositive i) ≤ O.slope (simplePositive j))
-    (a : PositiveRoot M cs) : O.slope (simplePositive i) ≤ O.slope a := by
-  change O.slope (simplePositive i) ≤ O.numerator a.val / O.denominator a.val
-  rw [le_div_iff₀ (O.denominator_pos a)]
-  rw [dual_coordinate_sum O.numerator a.val, dual_coordinate_sum O.denominator a.val,
-    Finset.mul_sum]
-  apply Finset.sum_le_sum
-  intro j _
-  have hj := (le_div_iff₀ (O.positive j)).mp (hi j)
-  change O.slope (simplePositive i) * O.denominator (root j) ≤ O.numerator (root j) at hj
-  nlinarith [mul_nonneg (a.property.2 j) (sub_nonneg.mpr hj)]
-
-theorem exists_least_simple [Nonempty I] : ∃ i : I,
-    ∀ a : PositiveRoot M cs, O.slope (simplePositive i) ≤ O.slope a := by
-  obtain ⟨i,_,hi⟩ := Finset.exists_min_image (Finset.univ : Finset I)
-    (fun i => O.slope (simplePositive i)) Finset.univ_nonempty
-  exact ⟨i, O.simple_lower_bound i (fun j => hi j (Finset.mem_univ j))⟩
-
-theorem least_simple_descent {y : W} (hy : y ≠ 1) (ha : O.Adapted y) :
-    ∃ i : I, cs.IsLeftDescent y i ∧
-      O.slope (simplePositive i) < 0 ∧
-      ∀ a : PositiveRoot M cs, O.slope (simplePositive i) ≤ O.slope a := by
-  obtain ⟨j,hj⟩ := cs.exists_leftDescent_of_ne_one hy
-  let : Nonempty I := ⟨j⟩
-  obtain ⟨i,hi⟩ := O.exists_least_simple
-  have hjr : rootInverted y (simplePositive (M := M) (cs := cs) j) := by
-    apply ((rootReflection_represents M cs _).inversion_iff (simplePositive j).property.2 y).mpr
-    simpa only [rootRef_simplePositive] using (show cs.IsLeftInversion y (cs.simple j) from
-      ⟨cs.isReflection_simple j,hj⟩)
-  have hi0 := lt_of_le_of_lt (hi (simplePositive j)) ((ha _).1.mpr hjr)
-  refine ⟨i,?_,hi0,hi⟩
-  have hir := ((rootReflection_represents M cs _).inversion_iff
-    (simplePositive i).property.2 y).mp ((ha _).1.mp hi0)
-  simpa only [rootRef_simplePositive, CoxeterSystem.IsLeftDescent] using hir.2
-
-
-/-- A functional in the chamber indexed by y has exactly Inv(y) as its
-negative roots. This is stronger data used only to construct the order. -/
-def Chamber (y : W) : Prop := ∀ i, 0 < O.numerator (action M cs y (root i))
-
-theorem adapted_of_chamber {y : W} (hc : O.Chamber y) : O.Adapted y := by
-  let f : Module.Dual ℝ (I → ℝ) := O.numerator.comp (action M cs y).toLinearMap
-  have hf : PositiveFunctional f := hc
-  intro a
-  have he (v : I → ℝ) : f (action M cs y⁻¹ v) = O.numerator v := by
-    change O.numerator (action M cs y (action M cs y⁻¹ v)) = _
-    rw [← LinearEquiv.mul_apply, ← map_mul, mul_inv_cancel, map_one]
-    rfl
-  by_cases ha : rootInverted y a
-  · have hv := hf.root_pos (positiveAction y⁻¹ a)
-    rw [positiveAction_eq_of_nonpos _ _ ha, map_neg, he] at hv
-    have hn : O.slope a < 0 := div_neg_of_neg_of_pos (by linarith) (O.denominator_pos a)
-    exact ⟨iff_of_true hn ha, hn.ne⟩
-  · have hp := (not_rootInverted_iff y a).mp ha
-    have hv := hf.root_pos (positiveAction y⁻¹ a)
-    rw [positiveAction_eq_of_nonneg _ _ hp, he] at hv
-    have hn : 0 < O.slope a := div_pos hv (O.denominator_pos a)
-    exact ⟨iff_of_false (not_lt.mpr hn.le) ha, hn.ne'⟩
-
-end RatioOrder
-
-/-- The generic functional can be chosen in any prescribed Coxeter chamber,
-not just outside the countably many equality hyperplanes. The explicit
-polynomial perturbation makes both requirements simultaneous. -/
-theorem exists_ratioOrder_chamber_height (y : W) :
-    ∃ O : RatioOrder (M := M) (cs := cs), O.Chamber y ∧ O.denominator=height := by
-  classical
-  let := countable_Positiveroot M cs
-  let f₀ : Module.Dual ℝ (I → ℝ) := height.comp (action M cs y⁻¹).toLinearMap
-  let f : ℝ → Module.Dual ℝ (I → ℝ) := fun t =>
-    f₀ + t • ReflectionSchedules.momentFunctional t
-  have hf (t : ℝ) (v : I → ℝ) : f t v = f₀ v +
-      t * (ReflectionSchedules.coordinatePolynomial v).eval t := by
-    simp [f,ReflectionSchedules.coordinatePolynomial_eval]
-  let U : Set ℝ := {t | ∀ i, 0 < f t (action M cs y (root i))}
-  have hu : IsOpen U := by
-    change IsOpen {t | ∀ i, 0 < f t (action M cs y (root i))}
-    rw [Set.ofPred_forall]
-    apply isOpen_iInter_of_finite
-    intro i
-    apply isOpen_lt continuous_const
-    simp only [hf]
-    exact continuous_const.add (continuous_id.mul (Polynomial.continuous _))
-  have h0 : 0 ∈ U := by
-    intro i
-    simp only [f,zero_smul,add_zero]
-    change 0 < height (action M cs y⁻¹ (action M cs y (root i)))
-    rw [← LinearEquiv.mul_apply,← map_mul,inv_mul_cancel,map_one]
-    exact height_positive i
-  let P := {p : PositiveRoot M cs × PositiveRoot M cs // p.1 ≠ p.2}
-  let v : P → (I → ℝ) := fun p => normalizedRoot p.val.1 - normalizedRoot p.val.2
-  let d : P → Polynomial ℝ := fun p => Polynomial.C (f₀ (v p)) +
-    Polynomial.X * ReflectionSchedules.coordinatePolynomial (v p)
-  have hd : ∀ p, d p ≠ 0 := by
-    intro p hp
-    have hz := congrArg (fun q : Polynomial ℝ => q.eval 0) hp
-    have hz' : f₀ (v p) = 0 := by simpa [d] using hz
-    have hq : ReflectionSchedules.coordinatePolynomial (v p) = 0 := by
-      simpa only [d,hz',Polynomial.C_0,zero_add,mul_eq_zero,Polynomial.X_ne_zero,false_or] using hp
-    have hv : v p = 0 := ReflectionSchedules.coordinatePolynomial_injective (by simpa [ReflectionSchedules.coordinatePolynomial] using hq)
-    exact p.property (normalizedRoot_injective (sub_eq_zero.mp hv))
-  let bad : Set ℝ := ⋃ p : P, {t | (d p).IsRoot t}
-  have hbad : bad.Countable := Set.countable_iUnion
-    (fun p => (Polynomial.finite_setOfPred_isRoot (hd p)).countable)
-  obtain ⟨t,htU,ht⟩ := (hbad.dense_compl ℝ).inter_open_nonempty U hu ⟨0,h0⟩
-  have hinj : Function.Injective (fun a : PositiveRoot M cs => f t a.val / height a.val) := by
-    intro a b heq
-    by_contra hab
-    apply ht
-    apply Set.mem_iUnion.mpr
-    refine ⟨⟨(a,b),hab⟩,?_⟩
-    change (d ⟨(a,b),hab⟩).eval t = 0
-    dsimp only [d]
-    rw [Polynomial.eval_add,Polynomial.eval_C,Polynomial.eval_mul,Polynomial.eval_X,← hf]
-    change f t (normalizedRoot a - normalizedRoot b) = 0
-    rw [map_sub]
-    apply sub_eq_zero.mpr
-    simpa only [normalizedRoot,map_smul,smul_eq_mul,div_eq_mul_inv,mul_comm] using heq
-  exact ⟨⟨height,f t,height_positive,hinj⟩,htU,rfl⟩
-
-theorem exists_ratioOrder_chamber (y : W) :
-    ∃ O : RatioOrder (M := M) (cs := cs), O.Chamber y := by
-  obtain ⟨O,hO,_⟩ := exists_ratioOrder_chamber_height (M := M) (cs := cs) y
-  exact ⟨O,hO⟩
-
-@[simp] theorem positiveAction_simplePositive (i : I) :
-    positiveAction (cs.simple i) (simplePositive (M := M) (cs := cs) i) = simplePositive i := by
-  apply Subtype.ext
-  rw [positiveAction_eq_of_nonpos]
-  · simp only [simplePositive_val,action_simple_root,neg_neg]
-  · simpa only [simplePositive_val,action_simple_root,Pi.neg_apply,neg_nonpos] using root_nonneg i
-
-theorem simple_action_nonneg (i : I) (a : PositiveRoot M cs)
-    (ha : a ≠ simplePositive i) : ∀ k, 0 ≤ action M cs (cs.simple i) a.val k := by
-  apply a.property.1.act (cs.simple i) |>.sign.resolve_right
-  intro hn
-  exact ha (Subtype.ext (a.property.1.eq_simple_of_nonpos a.property.2 i hn))
-
-theorem simple_positiveAction_val (i : I) (a : PositiveRoot M cs)
-    (ha : a ≠ simplePositive i) :
-    (positiveAction (cs.simple i) a).val = action M cs (cs.simple i) a.val :=
-  positiveAction_eq_of_nonneg _ _ (simple_action_nonneg i a ha)
-
-theorem simple_positiveAction_ne (i : I) (a : PositiveRoot M cs)
-    (ha : a ≠ simplePositive i) : positiveAction (cs.simple i) a ≠ simplePositive i := by
-  intro he
-  apply ha
-  exact (positiveActionEquiv (cs.simple i)).injective (he.trans (positiveAction_simplePositive i).symm)
-
-theorem exists_uniform_pos_lt (v : I → ℝ) (hv : ∀ i, 0 < v i) :
-    ∃ ε : ℝ, 0 < ε ∧ ∀ i, ε < v i := by
-  have aux (s : Finset I) : ∃ ε : ℝ, 0 < ε ∧ ∀ i ∈ s, ε < v i := by
-    induction s using Finset.induction_on with
-    | empty => exact ⟨1,by norm_num,by simp⟩
-    | @insert a s ha ih =>
-      obtain ⟨ε,hε,hlt⟩ := ih
-      refine ⟨min ε (v a / 2),lt_min hε (half_pos (hv a)),?_⟩
-      intro i hi
-      rcases Finset.mem_insert.mp hi with rfl|hi
-      · exact lt_of_le_of_lt (min_le_right _ _) (half_lt_self (hv i))
-      · exact lt_of_le_of_lt (min_le_left _ _) (hlt i hi)
-  obtain ⟨ε,hε,hlt⟩ := aux Finset.univ
-  exact ⟨ε,hε,fun i => hlt i (Finset.mem_univ i)⟩
-
-namespace RatioOrder
-variable (O : RatioOrder (M := M) (cs := cs))
-
-def turnNumerator (i : I) : Module.Dual ℝ (I → ℝ) :=
-  O.numerator.comp (action M cs (cs.simple i)).toLinearMap
-
-def turnDenominator (i : I) (k : ℝ) : Module.Dual ℝ (I → ℝ) :=
-  (O.numerator - k • O.denominator).comp (action M cs (cs.simple i)).toLinearMap
-
-theorem exists_turn_parameter (i : I) (hi0 : O.slope (simplePositive i) < 0)
-    (hi : ∀ a : PositiveRoot M cs, O.slope (simplePositive i) ≤ O.slope a) :
-    ∃ k : ℝ, k < 0 ∧ O.slope (simplePositive i) < k ∧
-      PositiveFunctional (O.turnDenominator i k) := by
-  let c := O.slope (simplePositive i)
-  let v : I → ℝ := fun j => if j=i then -c else
-    O.slope (positiveAction (cs.simple i) (simplePositive j)) - c
-  have hv : ∀ j, 0 < v j := by
-    intro j
-    by_cases hj : j=i
-    · simp only [v,ite_eq_left hj]
-      exact neg_pos.mpr hi0
-    · simp only [v,ite_eq_right hj]
-      apply sub_pos.mpr
-      apply lt_of_le_of_ne (hi _)
-      intro he
-      exact simple_positiveAction_ne i (simplePositive j)
-        (fun hh => hj (simplePositive_injective hh)) (O.slope_injective he.symm)
-  obtain ⟨ε,hε,hεv⟩ := exists_uniform_pos_lt v hv
-  have hεc : ε < -c := by simpa only [v,ite_eq_left rfl] using hεv i
-  refine ⟨c+ε,by linarith,by dsimp [c]; linarith,?_⟩
-  intro j
-  change 0 < O.numerator (action M cs (cs.simple i) (root j)) -
-    (c+ε)*O.denominator (action M cs (cs.simple i) (root j))
-  by_cases hj : j=i
-  · subst j
-    rw [action_simple_root,map_neg,map_neg]
-    have hc : c * O.denominator (root i) = O.numerator (root i) := by
-      exact div_mul_cancel₀ _ (O.positive i).ne'
-    nlinarith [mul_pos hε (O.positive i)]
-  · let a := positiveAction (cs.simple i) (simplePositive (M := M) (cs := cs) j)
-    have ha : a.val = action M cs (cs.simple i) (root j) :=
-      simple_positiveAction_val i (simplePositive j) (fun hh => hj (simplePositive_injective hh))
-    rw [← ha]
-    have hvj : ε < O.slope a - c := by simpa only [v,ite_eq_right hj] using hεv j
-    have hlt : c+ε < O.slope a := by linarith
-    exact sub_pos.mpr ((lt_div_iff₀ (O.denominator_pos a)).mp hlt)
-
-theorem turn_ratio_injective (i : I) (k : ℝ) (hk : k ≠ 0)
-    (hp : PositiveFunctional (O.turnDenominator i k)) :
-    Function.Injective (fun a : PositiveRoot M cs =>
-      O.turnNumerator i a.val / O.turnDenominator i k a.val) := by
-  intro a b hab
-  have hh (a : PositiveRoot M cs) : O.denominator (action M cs (cs.simple i) a.val) ≠ 0 := by
-    rcases (a.property.1.act (cs.simple i)).sign with hpos|hneg
-    · have h := O.denominator_pos (positiveAction (cs.simple i) a)
-      rw [positiveAction_eq_of_nonneg _ _ hpos] at h
-      exact h.ne'
-    · have h := O.denominator_pos (positiveAction (cs.simple i) a)
-      rw [positiveAction_eq_of_nonpos _ _ hneg,map_neg] at h
-      exact (neg_pos.mp h).ne
-  have hc := (div_eq_div_iff (hp.root_pos a).ne' (hp.root_pos b).ne').mp hab
-  change O.numerator (action M cs (cs.simple i) a.val) *
-      (O.numerator (action M cs (cs.simple i) b.val) -
-        k*O.denominator (action M cs (cs.simple i) b.val)) =
-    O.numerator (action M cs (cs.simple i) b.val) *
-      (O.numerator (action M cs (cs.simple i) a.val) -
-        k*O.denominator (action M cs (cs.simple i) a.val)) at hc
-  have hcross : O.numerator (action M cs (cs.simple i) a.val) *
-      O.denominator (action M cs (cs.simple i) b.val) =
-    O.numerator (action M cs (cs.simple i) b.val) *
-      O.denominator (action M cs (cs.simple i) a.val) := by
-    apply sub_eq_zero.mp
-    apply (mul_eq_zero.mp (show k * (O.numerator (action M cs (cs.simple i) a.val) *
-      O.denominator (action M cs (cs.simple i) b.val) -
-        O.numerator (action M cs (cs.simple i) b.val) *
-      O.denominator (action M cs (cs.simple i) a.val)) = 0 by nlinarith [hc])).resolve_left hk
-  have he := (div_eq_div_iff (hh a) (hh b)).mpr hcross
-  have hval (a : PositiveRoot M cs) :
-      O.numerator (action M cs (cs.simple i) a.val) /
-          O.denominator (action M cs (cs.simple i) a.val) =
-        O.slope (positiveAction (cs.simple i) a) := by
-    unfold slope
-    rcases (a.property.1.act (cs.simple i)).sign with hpos|hneg
-    · rw [positiveAction_eq_of_nonneg _ _ hpos]
-    · rw [positiveAction_eq_of_nonpos _ _ hneg,map_neg,map_neg,neg_div_neg_eq]
-  rw [hval a,hval b] at he
-  exact (positiveActionEquiv (cs.simple i)).injective (O.slope_injective he)
-
-def turn (i : I) (k : ℝ) (hk : k ≠ 0)
-    (hp : PositiveFunctional (O.turnDenominator i k)) : RatioOrder (M := M) (cs := cs) :=
-  ⟨O.turnDenominator i k,O.turnNumerator i,hp,O.turn_ratio_injective i k hk hp⟩
-
-
-theorem turn_chamber (i : I) (k : ℝ) (hk : k ≠ 0)
-    (hp : PositiveFunctional (O.turnDenominator i k)) {y : W} (hy : O.Chamber y) :
-    (O.turn i k hk hp).Chamber (cs.simple i * y) := by
-  intro j
-  change 0 < O.numerator (action M cs (cs.simple i)
-    (action M cs (cs.simple i * y) (root j)))
-  rw [← LinearEquiv.mul_apply,← map_mul,cs.simple_mul_simple_cancel_left]
-  exact hy j
-
-theorem turn_simple_greatest (i : I) (k : ℝ) (hk : k < 0)
-    (hp : PositiveFunctional (O.turnDenominator i k)) (a : PositiveRoot M cs)
-    (ha : a ≠ simplePositive i) :
-    (O.turn i k hk.ne hp).slope a < (O.turn i k hk.ne hp).slope (simplePositive i) := by
-  have hlt : (O.turn i k hk.ne hp).slope a < 1 := by
-    apply (div_lt_iff₀ (hp.root_pos a)).mpr
-    change O.numerator (action M cs (cs.simple i) a.val) <
-      1 * (O.numerator (action M cs (cs.simple i) a.val) -
-        k * O.denominator (action M cs (cs.simple i) a.val))
-    have hap := O.denominator_pos (positiveAction (cs.simple i) a)
-    rw [simple_positiveAction_val i a ha] at hap
-    nlinarith [mul_neg_of_neg_of_pos hk hap]
-  have hgt : 1 < (O.turn i k hk.ne hp).slope (simplePositive i) := by
-    apply (lt_div_iff₀ (hp.root_pos (simplePositive i))).mpr
-    change 1 * (O.numerator (action M cs (cs.simple i) (root i)) -
-       k * O.denominator (action M cs (cs.simple i) (root i))) <
-      O.numerator (action M cs (cs.simple i) (root i))
-    rw [action_simple_root,map_neg,map_neg]
-    nlinarith [mul_neg_of_neg_of_pos hk (O.positive i)]
-  exact hlt.trans hgt
-
-theorem turn_slope_lt_iff (i : I) (k : ℝ) (hk : k < 0)
-    (hp : PositiveFunctional (O.turnDenominator i k))
-    (a b : PositiveRoot M cs) (ha : a ≠ simplePositive i) (hb : b ≠ simplePositive i) :
-    (O.turn i k hk.ne hp).slope (positiveAction (cs.simple i) a) <
-      (O.turn i k hk.ne hp).slope (positiveAction (cs.simple i) b) ↔ O.slope a < O.slope b := by
-  have hact (a : PositiveRoot M cs) (ha : a ≠ simplePositive i) :
-      action M cs (cs.simple i) (positiveAction (cs.simple i) a).val = a.val := by
-    rw [simple_positiveAction_val i a ha,← LinearEquiv.mul_apply,← map_mul,
-      (cs.isReflection_simple i).mul_self,map_one]
-    rfl
-  have haD := hp.root_pos (positiveAction (cs.simple i) a)
-  have hbD := hp.root_pos (positiveAction (cs.simple i) b)
-  change 0 < O.numerator (action M cs (cs.simple i) (positiveAction (cs.simple i) a).val) -
-    k * O.denominator (action M cs (cs.simple i) (positiveAction (cs.simple i) a).val) at haD
-  change 0 < O.numerator (action M cs (cs.simple i) (positiveAction (cs.simple i) b).val) -
-    k * O.denominator (action M cs (cs.simple i) (positiveAction (cs.simple i) b).val) at hbD
-  rw [hact a ha] at haD
-  rw [hact b hb] at hbD
-  change O.numerator (action M cs (cs.simple i) (positiveAction (cs.simple i) a).val) /
-    (O.numerator (action M cs (cs.simple i) (positiveAction (cs.simple i) a).val) -
-      k * O.denominator (action M cs (cs.simple i) (positiveAction (cs.simple i) a).val)) <
-    O.numerator (action M cs (cs.simple i) (positiveAction (cs.simple i) b).val) /
-    (O.numerator (action M cs (cs.simple i) (positiveAction (cs.simple i) b).val) -
-      k * O.denominator (action M cs (cs.simple i) (positiveAction (cs.simple i) b).val)) ↔
-    O.numerator a.val / O.denominator a.val < O.numerator b.val / O.denominator b.val
-  rw [hact a ha,hact b hb,div_lt_div_iff₀ haD hbD,
-    div_lt_div_iff₀ (O.denominator_pos a) (O.denominator_pos b)]
-  constructor
-  · intro h
-    by_contra hn
-    have := mul_nonpos_of_nonpos_of_nonneg hk.le (sub_nonneg.mpr (le_of_not_gt hn))
-    nlinarith
-  · intro h
-    have := mul_neg_of_neg_of_pos hk (sub_pos.mpr h)
-    nlinarith
-
-/-- The actual geometric rotation needed for the Hecke recursion. It moves
-the least simple root to last and preserves all other root comparisons. -/
-theorem exists_rotation {y : W} (hy : y ≠ 1) (hc : O.Chamber y) :
-    ∃ (i : I) (O' : RatioOrder (M := M) (cs := cs)),
-      cs.IsLeftDescent y i ∧ O'.Chamber (cs.simple i * y) ∧
-      (∀ a : PositiveRoot M cs, a ≠ simplePositive i →
-        O.slope (simplePositive i) < O.slope a ∧ O'.slope a < O'.slope (simplePositive i)) ∧
-      (∀ a b : PositiveRoot M cs, a ≠ simplePositive i → b ≠ simplePositive i →
-        (O'.slope (positiveAction (cs.simple i) a) < O'.slope (positiveAction (cs.simple i) b) ↔
-          O.slope a < O.slope b)) := by
-  obtain ⟨i,hi,hi0,hmin⟩ := O.least_simple_descent hy (O.adapted_of_chamber hc)
-  obtain ⟨k,hk,_,hpos⟩ := O.exists_turn_parameter i hi0 hmin
-  refine ⟨i,O.turn i k hk.ne hpos,hi,O.turn_chamber i k hk.ne hpos hc,?_,?_⟩
-  · intro a ha
-    exact ⟨lt_of_le_of_ne (hmin a) (fun he => ha (O.slope_injective he).symm),
-      O.turn_simple_greatest i k hk hpos a ha⟩
-  · exact O.turn_slope_lt_iff i k hk hpos
-
-end RatioOrder
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Rank-two path reversal in the actual induced root plane, for every finite
-full interval. This is a required Dyer wall-crossing relation, not the Dyer
-identification itself. -/
-namespace KLInvariance.Dihedral
-variable {W : Type*} [Group W] {M : CoxeterMatrix Bool}
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
-/-- Switch the two vertices at every strictly internal rank, leaving all
-other ranks fixed. The diagram involution supplies the intrinsic switch. -/
-def rankFlip (cs : CoxeterSystem M W) (L H : ℕ) (w : W) : W :=
-  if L < cs.length w ∧ cs.length w < H then swap cs w else w
-
-@[simp] theorem length_rankFlip (cs : CoxeterSystem M W) (L H : ℕ) (w : W) :
-    cs.length (rankFlip cs L H w) = cs.length w := by
-  unfold rankFlip
-  split <;> simp
-
-@[simp] theorem rankFlip_involutive (cs : CoxeterSystem M W) (L H : ℕ) :
-    Function.Involutive (rankFlip cs L H) := by
-  intro w
-  unfold rankFlip
-  split <;> simp_all
-
-@[simp] theorem rankFlip_lower (cs : CoxeterSystem M W) (x y : W) :
-    rankFlip cs (cs.length x) (cs.length y) x = x := by simp [rankFlip]
-
-@[simp] theorem rankFlip_upper (cs : CoxeterSystem M W) (x y : W) :
-    rankFlip cs (cs.length x) (cs.length y) y = y := by simp [rankFlip]
-
-@[simp] theorem rankFlip_bruhat_iff (cs : CoxeterSystem M W) (L H : ℕ) (x y : W) :
-    BruhatLE cs (rankFlip cs L H x) (rankFlip cs L H y) ↔ BruhatLE cs x y := by
-  simp only [bruhat_iff_length_lt_or_eq, length_rankFlip,
-    (rankFlip_involutive cs L H).injective.eq_iff]
-
-@[simp] theorem rankFlip_step_iff (cs : CoxeterSystem M W) (L H : ℕ) (x y : W) :
-    BruhatStep cs (rankFlip cs L H x) (rankFlip cs L H y) ↔ BruhatStep cs x y := by
-  simp only [step_iff_odd, length_rankFlip]
-
-/-- An automorphism of the unlabelled full dihedral interval, fixing its ends.
-It need not arise from a Coxeter-system automorphism at those ends. -/
-def intervalFlip (cs : CoxeterSystem M W) (lo hi : W) :
-    Interval cs lo hi ≃o Interval cs lo hi where
-  toFun x := ⟨rankFlip cs (cs.length lo) (cs.length hi) x.val, by
-    constructor
-    · simpa only [rankFlip_lower] using
-        (rankFlip_bruhat_iff cs (cs.length lo) (cs.length hi) lo x.val).mpr x.property.1
-    · simpa only [rankFlip_upper] using
-        (rankFlip_bruhat_iff cs (cs.length lo) (cs.length hi) x.val hi).mpr x.property.2⟩
-  invFun x := ⟨rankFlip cs (cs.length lo) (cs.length hi) x.val, by
-    constructor
-    · simpa only [rankFlip_lower] using
-        (rankFlip_bruhat_iff cs (cs.length lo) (cs.length hi) lo x.val).mpr x.property.1
-    · simpa only [rankFlip_upper] using
-        (rankFlip_bruhat_iff cs (cs.length lo) (cs.length hi) x.val hi).mpr x.property.2⟩
-  left_inv x := Subtype.ext (rankFlip_involutive cs _ _ x.val)
-  right_inv x := Subtype.ext (rankFlip_involutive cs _ _ x.val)
-  map_rel_iff' := rankFlip_bruhat_iff cs _ _ _ _
-
-@[simp] theorem intervalFlip_val (cs : CoxeterSystem M W) (lo hi : W)
-    (x : Interval cs lo hi) :
-    (intervalFlip cs lo hi x).val = rankFlip cs (cs.length lo) (cs.length hi) x.val := rfl
-
-@[simp] theorem intervalFlip_involutive (cs : CoxeterSystem M W) (lo hi : W) :
-    Function.Involutive (intervalFlip cs lo hi) :=
-  fun x => Subtype.ext (rankFlip_involutive cs _ _ x.val)
-
-def edgeFlip (cs : CoxeterSystem M W) (lo hi : W) :
-    BruhatGraph.Edge cs lo hi ≃ BruhatGraph.Edge cs lo hi where
-  toFun e := ⟨(intervalFlip cs lo hi e.val.1, intervalFlip cs lo hi e.val.2),
-    (rankFlip_step_iff cs _ _ _ _).mpr e.property⟩
-  invFun e := ⟨(intervalFlip cs lo hi e.val.1, intervalFlip cs lo hi e.val.2),
-    (rankFlip_step_iff cs _ _ _ _).mpr e.property⟩
-  left_inv e := Subtype.ext (Prod.ext
-    (intervalFlip_involutive cs lo hi e.val.1)
-    (intervalFlip_involutive cs lo hi e.val.2))
-  right_inv e := Subtype.ext (Prod.ext
-    (intervalFlip_involutive cs lo hi e.val.1)
-    (intervalFlip_involutive cs lo hi e.val.2))
-
-@[simp] theorem edgeFlip_source (cs : CoxeterSystem M W) (lo hi : W)
-    (e : BruhatGraph.Edge cs lo hi) :
-    BruhatGraph.source cs lo hi (edgeFlip cs lo hi e) =
-      intervalFlip cs lo hi (BruhatGraph.source cs lo hi e) := rfl
-
-@[simp] theorem edgeFlip_target (cs : CoxeterSystem M W) (lo hi : W)
-    (e : BruhatGraph.Edge cs lo hi) :
-    BruhatGraph.target cs lo hi (edgeFlip cs lo hi e) =
-      intervalFlip cs lo hi (BruhatGraph.target cs lo hi e) := rfl
-
-@[simp] theorem edgeFlip_involutive (cs : CoxeterSystem M W) (lo hi : W) :
-    Function.Involutive (edgeFlip cs lo hi) := by
-  intro e
-  apply Subtype.ext
-  change (intervalFlip cs lo hi (intervalFlip cs lo hi e.val.1),
-    intervalFlip cs lo hi (intervalFlip cs lo hi e.val.2)) = e.val
-  exact Prod.ext (intervalFlip_involutive cs lo hi e.val.1)
-    (intervalFlip_involutive cs lo hi e.val.2)
-
-end
-end KLInvariance.Dihedral
-
-namespace KLInvariance.SchedulePaths
-variable {V E V' E' : Type*} (source target : E → V) (source' target' : E' → V')
-
-/-- Relabelling graph vertices and edges preserves the actual path predicate. -/
-theorem IsPath.map {x y : V} {p : List E}
-    (hp : IsPath source target x y p) (f : V → V') (g : E → E')
-    (hs : ∀ e, source' (g e) = f (source e))
-    (ht : ∀ e, target' (g e) = f (target e)) :
-    IsPath source' target' (f x) (f y) (p.map g) := by
-  induction p generalizing x with
-  | nil => exact congrArg f hp
-  | cons e p ih =>
-    exact ⟨(hs e).trans (congrArg f hp.1), ht e ▸ ih hp.2⟩
-
-/-- Every adjacent pair in a genuine path is composable. -/
-theorem IsPath.isChain {x y : V} {p : List E} (hp : IsPath source target x y p) :
-    p.IsChain (fun e f => target e = source f) := by
-  induction p generalizing x with
-  | nil => exact .nil
-  | cons e p ih =>
-    cases p with
-    | nil => exact .singleton e
-    | cons f p =>
-      apply List.isChain_cons_cons.mpr
-      exact ⟨hp.2.1.symm, ih hp.2⟩
-
-/-- A path's adjacent comparisons suffice for a transitive label order. The
-statement allows the comparison to change under an edge map, provided the
-local relation is checked only on composable edges. -/
-theorem IsPath.pairwise_map_of_local {Λ Λ' : Type*} [LinearOrder Λ] [LinearOrder Λ']
-    {x y : V} {p : List E} (hp : IsPath source target x y p)
-    (label : E → Λ) (label' : E' → Λ') (g : E → E')
-    (hlocal : ∀ e f, target e = source f →
-      label e < label f → label' (g e) < label' (g f))
-    (horder : p.Pairwise (fun e f => label e < label f)) :
-    (p.map g).Pairwise (fun e f => label' e < label' f) := by
-  rw [← List.isChain_iff_pairwise] at horder ⊢
-  rw [List.isChain_map]
-  induction p generalizing x with
-  | nil => exact .nil
-  | cons e p ih =>
-    cases p with
-    | nil => exact .singleton e
-    | cons f p =>
-      rw [List.isChain_cons_cons] at horder ⊢
-      exact ⟨hlocal e f hp.2.1.symm horder.1, ih hp.2 horder.2⟩
-
-end KLInvariance.SchedulePaths
-
-namespace KLInvariance.TitsSpace
-open Module
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-noncomputable section
-namespace PlaneBasis
-variable {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-open Dihedral BruhatGraph
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
-theorem edgeRoot_ne_of_path {x y z : planeSubgroup M cs P}
-    (hxy : BruhatStep B.system x y) (hyz : BruhatStep B.system y z) :
-    B.edgeRoot hxy ≠ B.edgeRoot hyz :=
-  graphRoot_ne_of_path ((B.bruhatStep_iff x y).mpr hxy) ((B.bruhatStep_iff y z).mpr hyz)
-
-/-- At every internal vertex, the cut switches side under the path involution.
-This is the local relation behind rank-two path-product reversal. -/
-theorem edgeRoot_slopes_rankFlip {x y z top : planeSubgroup M cs P} {L : ℕ}
-    (hxy : BruhatStep B.system x y) (hyz : BruhatStep B.system y z)
-    (hL : L < B.system.length y) (hH : B.system.length y < B.system.length top) :
-    let hx := (rankFlip_step_iff B.system L (B.system.length top) x y).mpr hxy
-    let hz := (rankFlip_step_iff B.system L (B.system.length top) y z).mpr hyz
-    geometricSlope (B.edgeRoot hxy) < geometricSlope (B.edgeRoot hyz) ↔
-      geometricSlope (B.edgeRoot hz) < geometricSlope (B.edgeRoot hx) := by
-  dsimp only
-  let hx := (rankFlip_step_iff B.system L (B.system.length top) x y).mpr hxy
-  let hz := (rankFlip_step_iff B.system L (B.system.length top) y z).mpr hyz
-  have hsign : B.system.IsLeftDescent
-      (rankFlip B.system L (B.system.length top) y) false ↔
-      ¬ B.system.IsLeftDescent y false := by
-    rw [rankFlip,ite_eq_left ⟨hL,hH⟩]
-    exact descent_swap_false_iff_not B.system (by omega) hH
-  have hne : geometricSlope (B.edgeRoot hx) ≠ geometricSlope (B.edgeRoot hz) :=
-    fun he => B.edgeRoot_ne_of_path hx hz (geometricSlope_injective he)
-  rw [B.edgeRoot_slopes_iff]
-  constructor
-  · intro hi
-    have hn : ¬ geometricSlope (B.edgeRoot hx) < geometricSlope (B.edgeRoot hz) := by
-      rw [B.edgeRoot_slopes_iff,hsign]
-      exact not_not.mpr hi
-    exact lt_of_le_of_ne (le_of_not_gt hn) hne.symm
-  · intro hi
-    by_contra hn
-    have hlt := (B.edgeRoot_slopes_iff hx hz).mpr (hsign.mpr hn)
-    exact lt_asymm hlt hi
-
-/-- Restriction of the actual geometric root slope to a full plane interval. -/
-def intervalEdgeSlope (lo hi : planeSubgroup M cs P) (e : Edge B.system lo hi) : ℝ :=
-  geometricSlope (B.edgeRoot e.property)
-
-/-- The global edge map reverses the comparison for every composable pair.
-No statement is asserted for disjoint edge labels, which can share ties. -/
-theorem intervalEdgeSlope_flip {lo hi : planeSubgroup M cs P}
-    (e f : Edge B.system lo hi)
-    (hjoin : target B.system lo hi e = source B.system lo hi f) :
-    B.intervalEdgeSlope lo hi e < B.intervalEdgeSlope lo hi f ↔
-      B.intervalEdgeSlope lo hi (edgeFlip B.system lo hi f) <
-        B.intervalEdgeSlope lo hi (edgeFlip B.system lo hi e) := by
-  rcases e with ⟨⟨ex,ey⟩,he⟩
-  rcases f with ⟨⟨fx,fy⟩,hf⟩
-  change ey = fx at hjoin
-  subst fx
-  have hl : B.system.length lo < B.system.length ey.val :=
-    lt_of_le_of_lt (length_le_of_bruhat B.system ex.property.1) he.1
-  have hh : B.system.length ey.val < B.system.length hi :=
-    lt_of_lt_of_le hf.1 (length_le_of_bruhat B.system fy.property.2)
-  exact B.edgeRoot_slopes_rankFlip he hf hl hh
-
-end PlaneBasis
-end
-end KLInvariance.TitsSpace
-
-namespace KLInvariance.SchedulePaths
-open scoped BigOperators
-variable {V E R : Type*} (source target : E → V)
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
-/-- The finite set of genuine directed paths selected by an edge schedule. -/
-def subpaths (schedule : List E) (x y : V) : Finset (List E) :=
-  schedule.sublists'.toFinset.filter (IsPath source target x y)
-
-@[simp] theorem mem_subpaths (schedule : List E) (x y : V) (p : List E) :
-    p ∈ subpaths source target schedule x y ↔
-      p.Sublist schedule ∧ IsPath source target x y p := by
-  simp [subpaths, List.mem_sublists']
-
-variable [Semiring R]
-
-theorem dispatch_eq_subpaths_sum (T : R) (schedule : List E) (hnd : schedule.Nodup)
-    (x y : V) :
-    dispatch source target T (fun z => if z=y then 1 else 0) schedule x =
-      ∑ p ∈ subpaths source target schedule x y, T ^ p.length := by
-  rw [entry_eq_weighted_paths, subpaths, Finset.sum_filter,
-    List.sum_toFinset _ (List.nodup_sublists'.mpr hnd)]
-
-/-- A length-preserving involution on the actual schedule-path sets proves
-an operator-entry relation, with no Hecke polynomial used in the assumptions. -/
-theorem dispatch_eq_of_path_involution (T : R) (schedule schedule' : List E)
-    (hnd : schedule.Nodup) (hnd' : schedule'.Nodup) (x y : V) (g : E → E)
-    (hg : Function.Involutive g)
-    (hforward : ∀ p ∈ subpaths source target schedule x y,
-      p.map g ∈ subpaths source target schedule' x y)
-    (hbackward : ∀ p ∈ subpaths source target schedule' x y,
-      p.map g ∈ subpaths source target schedule x y) :
-    dispatch source target T (fun z => if z=y then 1 else 0) schedule x =
-      dispatch source target T (fun z => if z=y then 1 else 0) schedule' x := by
-  rw [dispatch_eq_subpaths_sum source target T schedule hnd,
-    dispatch_eq_subpaths_sum source target T schedule' hnd']
-  exact Finset.sum_nbij' (List.map g) (List.map g) hforward hbackward
-    (fun p _ => List.map_involutive_iff.mpr hg p)
-    (fun p _ => List.map_involutive_iff.mpr hg p) (fun p _ => by simp)
-
-end
-end KLInvariance.SchedulePaths
-
-namespace KLInvariance.SchedulePaths
-variable {V E : Type*} (source target : E → V)
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
-/-- Sorted full schedules select precisely their strictly monotone paths;
-an actual graph map satisfying the adjacent comparison test transports them. -/
-theorem subpaths_map_of_sorted (label label' : E → ℝ)
-    (schedule schedule' : List E) (x y : V)
-    (hs : schedule.Pairwise (fun e f => label f ≤ label e))
-    (hs' : schedule'.Pairwise (fun e f => label' f ≤ label' e))
-    (hfull' : ∀ e, e ∈ schedule')
-    (hties : ∀ e f, label e = label f → target e ≠ source f)
-    (f : V → V) (g : E → E)
-    (hsource : ∀ e, source (g e) = f (source e))
-    (htarget : ∀ e, target (g e) = f (target e))
-    (hx : f x=x) (hy : f y=y)
-    (hlocal : ∀ e f, target e=source f → label f<label e → label' (g f)<label' (g e))
-    (p : List E) (hp : p ∈ subpaths source target schedule x y) :
-    p.map g ∈ subpaths source target schedule' x y := by
-  obtain ⟨hsub,hp⟩ := (mem_subpaths source target schedule x y p).mp hp
-  have hstrict := decreasing_of_schedule_path source target label hties hp (hs.sublist hsub)
-  have horder := hp.pairwise_map_of_local source target
-    (Λ := OrderDual ℝ) (Λ' := OrderDual ℝ)
-    (fun e => (label e : OrderDual ℝ)) (fun e => (label' e : OrderDual ℝ)) g hlocal hstrict
-  have hp' := hp.map source target source target f g hsource htarget
-  rw [hx,hy] at hp'
-  exact (mem_subpaths source target schedule' x y (p.map g)).mpr
-    ⟨sublist_of_decreasing_subset label' horder hs' (fun _ _ => hfull' _), hp'⟩
-
-end
-end KLInvariance.SchedulePaths
-
-namespace KLInvariance.TitsSpace
-open Module
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-noncomputable section
-namespace PlaneBasis
-variable {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-open Dihedral BruhatGraph SchedulePaths
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
-theorem intervalEdgeSlope_ties {lo hi : planeSubgroup M cs P}
-    (e f : Edge B.system lo hi) (heq : B.intervalEdgeSlope lo hi e = B.intervalEdgeSlope lo hi f) :
-    target B.system lo hi e ≠ source B.system lo hi f := by
-  intro hjoin
-  rcases e with ⟨⟨ex,ey⟩,he⟩
-  rcases f with ⟨⟨fx,fy⟩,hf⟩
-  change ey = fx at hjoin
-  subst fx
-  exact B.edgeRoot_ne_of_path he hf (geometricSlope_injective heq)
-
-theorem intervalEdgeSlope_flip_reverse {lo hi : planeSubgroup M cs P}
-    (e f : Edge B.system lo hi)
-    (hjoin : target B.system lo hi e = source B.system lo hi f) :
-    B.intervalEdgeSlope lo hi f < B.intervalEdgeSlope lo hi e ↔
-      B.intervalEdgeSlope lo hi (edgeFlip B.system lo hi e) <
-        B.intervalEdgeSlope lo hi (edgeFlip B.system lo hi f) := by
-  have hj : target B.system lo hi (edgeFlip B.system lo hi e) =
-      source B.system lo hi (edgeFlip B.system lo hi f) := by
-    simpa only [edgeFlip_source,edgeFlip_target] using congrArg (intervalFlip B.system lo hi) hjoin
-  simpa only [edgeFlip_involutive B.system lo hi e, edgeFlip_involutive B.system lo hi f] using
-    (B.intervalEdgeSlope_flip (edgeFlip B.system lo hi e) (edgeFlip B.system lo hi f) hj).symm
-
-/-- Rank-two reversal of the genuine path-product entry between the two
-ends of any full interval. The order is the actual induced angular order. -/
-theorem dispatch_reverse_endpoints {R : Type*} [Semiring R]
-    {lo hi : planeSubgroup M cs P} (hlohi : BruhatLE B.system lo hi)
-    (schedule : List (Edge B.system lo hi)) (hnd : schedule.Nodup)
-    (hfull : ∀ e, e ∈ schedule)
-    (hs : schedule.Pairwise (fun e f => B.intervalEdgeSlope lo hi f ≤ B.intervalEdgeSlope lo hi e))
-    (T : R) :
-    let x : Interval B.system lo hi := ⟨lo,bruhat_refl B.system lo,hlohi⟩
-    let y : Interval B.system lo hi := ⟨hi,hlohi,bruhat_refl B.system hi⟩
-    dispatch (source B.system lo hi) (target B.system lo hi) T
-      (fun z => if z=y then 1 else 0) schedule x =
-    dispatch (source B.system lo hi) (target B.system lo hi) T
-      (fun z => if z=y then 1 else 0) schedule.reverse x := by
-  dsimp only
-  let x : Interval B.system lo hi := ⟨lo,bruhat_refl B.system lo,hlohi⟩
-  let y : Interval B.system lo hi := ⟨hi,hlohi,bruhat_refl B.system hi⟩
-  have hx : intervalFlip B.system lo hi x=x := Subtype.ext (rankFlip_lower B.system lo hi)
-  have hy : intervalFlip B.system lo hi y=y := Subtype.ext (rankFlip_upper B.system lo hi)
-  have hrs : schedule.reverse.Pairwise
-      (fun e f => -B.intervalEdgeSlope lo hi f ≤ -B.intervalEdgeSlope lo hi e) := by
-    rw [List.pairwise_reverse]
-    exact hs.imp (fun h => neg_le_neg h)
-  have hrfull : ∀ e, e ∈ schedule.reverse := fun e => List.mem_reverse.mpr (hfull e)
-  apply dispatch_eq_of_path_involution _ _ T schedule schedule.reverse hnd (List.nodup_reverse.mpr hnd) x y
-    (edgeFlip B.system lo hi) (edgeFlip_involutive B.system lo hi)
-  · apply subpaths_map_of_sorted _ _ (B.intervalEdgeSlope lo hi)
-      (fun e => -B.intervalEdgeSlope lo hi e) schedule schedule.reverse x y hs hrs hrfull
-      (B.intervalEdgeSlope_ties) (intervalFlip B.system lo hi) (edgeFlip B.system lo hi)
-      (edgeFlip_source B.system lo hi) (edgeFlip_target B.system lo hi) hx hy
-    intro e f hj hlt
-    exact neg_lt_neg ((B.intervalEdgeSlope_flip_reverse e f hj).mp hlt)
-  · apply subpaths_map_of_sorted _ _ (fun e => -B.intervalEdgeSlope lo hi e)
-      (B.intervalEdgeSlope lo hi) schedule.reverse schedule x y hrs hs hfull
-      (fun e f he => B.intervalEdgeSlope_ties e f (neg_injective he))
-      (intervalFlip B.system lo hi) (edgeFlip B.system lo hi)
-      (edgeFlip_source B.system lo hi) (edgeFlip_target B.system lo hi) hx hy
-    intro e f hj hlt
-    exact (B.intervalEdgeSlope_flip e f hj).mp (neg_lt_neg_iff.mp hlt)
-
-end PlaneBasis
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Genuine Dyer identification for geometric orders adapted to a fixed top.
-The path values below are defined from actual reflection edges; equality with
-`Hecke.rTilde` will be proved by its genuine simple-descent recurrence. -/
-namespace KLInvariance.AdaptedDyer
-open TitsSpace SchedulePaths
-open scoped BigOperators
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
-abbrev Edge := {e : W × W // BruhatStep cs e.1 e.2}
-def source (e : Edge cs) : W := e.val.1
-def target (e : Edge cs) : W := e.val.2
-def rootLabel (e : Edge cs) : PositiveRoot M cs := graphRoot e.property
-
-@[simp] theorem rootRef_rootLabel (e : Edge cs) :
-    rootRef (rootLabel cs e) = target cs e * (source cs e)⁻¹ := by
-  exact congrArg Subtype.val ((positiveRootEquiv M cs).left_inv
-    ⟨e.val.2 * e.val.1⁻¹,e.property.label_reflection⟩)
-
-theorem target_eq_root (e : Edge cs) :
-    target cs e = rootRef (rootLabel cs e) * source cs e := by
-  rw [rootRef_rootLabel]
-  group
-
-theorem rootLabel_ne_of_join (e f : Edge cs) (h : target cs e = source cs f) :
-    rootLabel cs e ≠ rootLabel cs f := by
-  rcases e with ⟨⟨x,y⟩,he⟩
-  rcases f with ⟨⟨z,w⟩,hf⟩
-  change y=z at h
-  subst z
-  exact graphRoot_ne_of_path he hf
-
-variable {cs}
-def score (O : RatioOrder (M := M) (cs := cs)) (e : Edge cs) : ℝ :=
-  O.slope (rootLabel cs e)
-
-theorem score_ties (O : RatioOrder (M := M) (cs := cs)) (e f : Edge cs)
-    (h : score O e = score O f) : target cs e ≠ source cs f :=
-  fun hj => rootLabel_ne_of_join cs e f hj (O.slope_injective h)
-
-variable (cs)
-def edgeFinset (b : W) : Finset (Edge cs) :=
-  let P := ((lowerFinset cs b).product (lowerFinset cs b)).filter
-    (fun e => BruhatStep cs e.1 e.2)
-  P.attach.image (fun e => (⟨e.val,by
-    exact (Finset.mem_filter.mp e.property).2⟩ : Edge cs))
-
-omit [Fintype I] in
-@[simp] theorem mem_edgeFinset (b : W) (e : Edge cs) :
-    e ∈ edgeFinset cs b ↔ BruhatLE cs (target cs e) b := by
-  unfold edgeFinset
-  constructor
-  · intro h
-    obtain ⟨p,_,he⟩ := Finset.mem_image.mp h
-    subst e
-    exact (mem_lowerFinset cs b p.val.2).mp
-      (Finset.mem_product.mp (Finset.mem_filter.mp p.property).1).2
-  · intro h
-    apply Finset.mem_image.mpr
-    have hx : BruhatLE cs (source cs e) b := (show BruhatLE cs (source cs e) (target cs e) from .single e.property).trans h
-    refine ⟨⟨e.val,Finset.mem_filter.mpr ⟨Finset.mem_product.mpr
-      ⟨(mem_lowerFinset cs b _).mpr hx,(mem_lowerFinset cs b _).mpr h⟩,e.property⟩⟩,
-      Finset.mem_attach _ _,?_⟩
-    rfl
-
-variable {cs}
-def schedule (O : RatioOrder (M := M) (cs := cs)) (b : W) : List (Edge cs) :=
-  (edgeFinset cs b).toList.mergeSort (fun e f => score O e ≤ score O f)
-
-@[simp] theorem mem_schedule (O : RatioOrder (M := M) (cs := cs)) (b : W) (e : Edge cs) :
-    e ∈ schedule O b ↔ BruhatLE cs (target cs e) b := by
-  simp [schedule]
-
-theorem schedule_nodup (O : RatioOrder (M := M) (cs := cs)) (b : W) :
-    (schedule O b).Nodup :=
-  (List.mergeSort_perm _ _).nodup_iff.mpr (Finset.nodup_toList _)
-
-theorem schedule_increasing (O : RatioOrder (M := M) (cs := cs)) (b : W) :
-    (schedule O b).Pairwise (fun e f => score O e ≤ score O f) := by
-  unfold schedule
-  apply List.pairwise_mergeSort'
-
-variable (cs)
-omit [Fintype I] in
-theorem path_le {x y : W} {p : List (Edge cs)}
-    (hp : IsPath (source cs) (target cs) x y p) : BruhatLE cs x y := by
-  induction p generalizing x with
-  | nil => exact hp ▸ .refl
-  | cons e p ih =>
-    exact hp.1 ▸ (show BruhatLE cs (source cs e) (target cs e) from .single e.property).trans (ih hp.2)
-
-omit [Fintype I] in
-theorem path_targets_le {x y : W} {p : List (Edge cs)}
-    (hp : IsPath (source cs) (target cs) x y p) :
-    ∀ e ∈ p, BruhatLE cs (target cs e) y := by
-  induction p generalizing x with
-  | nil => simp
-  | cons e p ih =>
-    intro f hf
-    rcases List.mem_cons.mp hf with rfl | hf
-    · exact path_le cs hp.2
-    · exact ih hp.2 f hf
-
-variable {cs}
-def paths (O : RatioOrder (M := M) (cs := cs)) (x y : W) : Finset (List (Edge cs)) :=
-  subpaths (source cs) (target cs) (schedule O y) x y
-
-@[simp] theorem mem_paths (O : RatioOrder (M := M) (cs := cs)) (x y : W) (p : List (Edge cs)) :
-    p ∈ paths O x y ↔ IsPath (source cs) (target cs) x y p ∧
-      p.Pairwise (fun e f => score O e < score O f) := by
-  rw [paths,mem_subpaths]
-  constructor
-  · rintro ⟨hsub,hp⟩
-    refine ⟨hp,?_⟩
-    exact decreasing_of_schedule_path (source cs) (target cs)
-      (Λ := OrderDual ℝ) (fun e => (score O e : OrderDual ℝ))
-      (fun e f h => score_ties O e f h) hp ((schedule_increasing O y).sublist hsub)
-  · rintro ⟨hp,hord⟩
-    refine ⟨?_,hp⟩
-    apply sublist_of_decreasing_subset (Λ := OrderDual ℝ)
-      (fun e => (score O e : OrderDual ℝ)) hord (schedule_increasing O y)
-    intro e he
-    exact (mem_schedule O y e).mpr (path_targets_le cs hp e he)
-
-def polynomial (O : RatioOrder (M := M) (cs := cs)) (x y : W) : Polynomial ℤ :=
-  ∑ p ∈ paths O x y, Polynomial.X ^ p.length
-
-theorem polynomial_dispatch (O : RatioOrder (M := M) (cs := cs)) (x y : W) :
-    polynomial O x y = dispatch (source cs) (target cs) Polynomial.X
-      (fun z => if z=y then 1 else 0) (schedule O y) x :=
-  (dispatch_eq_subpaths_sum _ _ _ _ (schedule_nodup O y) x y).symm
-
-
-variable (cs)
-theorem rootRef_injective : Function.Injective (rootRef (M := M) (cs := cs)) := by
-  intro a b h
-  apply (positiveRootEquiv M cs).symm.injective
-  exact Subtype.ext h
-
-theorem rootLabel_simple_iff (i : I) (e : Edge cs) :
-    rootLabel cs e = simplePositive i ↔ target cs e = cs.simple i * source cs e := by
-  constructor
-  · intro h
-    rw [target_eq_root cs e,h,rootRef_simplePositive]
-  · intro h
-    apply rootRef_injective cs
-    rw [rootRef_rootLabel,rootRef_simplePositive,h]
-    group
-
-/-- On a non-simple-labelled edge this is actual left multiplication by s.
-Simple-labelled edges are fixed merely to make the extension a total involution;
-all path uses below exclude them before applying this map. -/
-def rotateEdge (i : I) (e : Edge cs) : Edge cs :=
-  if h : target cs e = cs.simple i * source cs e then e else
-    ⟨(cs.simple i * source cs e,cs.simple i * target cs e),
-      bruhatStep_simple_mul cs e.property i h⟩
-
-theorem rotateEdge_val (i : I) (e : Edge cs) (h : rootLabel cs e ≠ simplePositive i) :
-    (rotateEdge cs i e).val = (cs.simple i * source cs e,cs.simple i * target cs e) := by
-  rw [rotateEdge,dite_eq_right (fun hh => h ((rootLabel_simple_iff cs i e).mpr hh))]
-
-theorem rotateEdge_root (i : I) (e : Edge cs) (h : rootLabel cs e ≠ simplePositive i) :
-    rootLabel cs (rotateEdge cs i e) = positiveAction (cs.simple i) (rootLabel cs e) := by
-  apply rootRef_injective cs
-  rw [rootRef_rootLabel]
-  change (rotateEdge cs i e).val.2 * (rotateEdge cs i e).val.1⁻¹ =
-    rootRef (positiveAction (cs.simple i) (rootLabel cs e))
-  have hc : rootRef (positiveAction (cs.simple i) (rootLabel cs e)) =
-      cs.simple i * rootRef (rootLabel cs e) * (cs.simple i)⁻¹ :=
-    rootReflection_positiveAction _ _
-  rw [rotateEdge_val cs i e h,hc,rootRef_rootLabel]
-  group
-
-theorem rotateEdge_nonsimple (i : I) (e : Edge cs) (h : rootLabel cs e ≠ simplePositive i) :
-    rootLabel cs (rotateEdge cs i e) ≠ simplePositive i := by
-  rw [rotateEdge_root cs i e h]
-  exact simple_positiveAction_ne i _ h
-
-@[simp] theorem rotateEdge_involutive (i : I) : Function.Involutive (rotateEdge cs i) := by
-  intro e
-  by_cases h : rootLabel cs e = simplePositive i
-  · have he : rotateEdge cs i e = e := by
-      rw [rotateEdge,dite_eq_left ((rootLabel_simple_iff cs i e).mp h)]
-    rw [he,he]
-  · apply Subtype.ext
-    rw [rotateEdge_val cs i _ (rotateEdge_nonsimple cs i e h)]
-    change (cs.simple i * (rotateEdge cs i e).val.1,
-      cs.simple i * (rotateEdge cs i e).val.2) = e.val
-    rw [rotateEdge_val cs i e h]
-    simp only [cs.simple_mul_simple_cancel_left]
-    rfl
-
-omit [Fintype I] in
-theorem path_map_of_mem {E E' V V' : Type*} (src tgt : E → V) (src' tgt' : E' → V')
-    {p : List E} {x y : V} (hp : IsPath src tgt x y p) (f : V → V') (g : E → E')
-    (hs : ∀ e ∈ p, src' (g e) = f (src e)) (ht : ∀ e ∈ p, tgt' (g e) = f (tgt e)) :
-    IsPath src' tgt' (f x) (f y) (p.map g) := by
-  induction p generalizing x with
-  | nil => exact congrArg f hp
-  | cons e p ih =>
-    refine ⟨(hs e List.mem_cons_self).trans (congrArg f hp.1),?_⟩
-    rw [ht e List.mem_cons_self]
-    exact ih hp.2 (fun a ha => hs a (List.mem_cons_of_mem e ha))
-      (fun a ha => ht a (List.mem_cons_of_mem e ha))
-
-theorem path_rotate {x y : W} {p : List (Edge cs)} (i : I)
-    (hp : IsPath (source cs) (target cs) x y p)
-    (hn : ∀ e ∈ p, rootLabel cs e ≠ simplePositive i) :
-    IsPath (source cs) (target cs) (cs.simple i*x) (cs.simple i*y) (p.map (rotateEdge cs i)) := by
-  apply path_map_of_mem (source cs) (target cs) (source cs) (target cs)
-    hp (fun z => cs.simple i*z) (rotateEdge cs i)
-  · intro e he
-    exact congrArg Prod.fst (rotateEdge_val cs i e (hn e he))
-  · intro e he
-    exact congrArg Prod.snd (rotateEdge_val cs i e (hn e he))
-
-
-@[simp] theorem rotateEdge_twice (i : I) (e : Edge cs) :
-    rotateEdge cs i (rotateEdge cs i e) = e := rotateEdge_involutive cs i e
-
-variable {cs}
-def Nonsimple (i : I) (p : List (Edge cs)) : Prop :=
-  ∀ e ∈ p, rootLabel cs e ≠ simplePositive i
-
-@[simp] theorem nonsimple_map (i : I) {p : List (Edge cs)} (hn : Nonsimple i p) :
-    Nonsimple i (p.map (rotateEdge cs i)) := by
-  intro e he
-  obtain ⟨f,hf,rfl⟩ := List.mem_map.mp he
-  exact rotateEdge_nonsimple cs i f (hn f hf)
-
-@[simp] theorem path_map_twice (i : I) (p : List (Edge cs)) :
-    (p.map (rotateEdge cs i)).map (rotateEdge cs i) = p := by
-  simp only [List.map_map,Function.comp_def,rotateEdge_twice]
-  exact List.map_id p
-
-theorem path_rotate_iff (i : I) {x y : W} {p : List (Edge cs)} (hn : Nonsimple i p) :
-    IsPath (source cs) (target cs) (cs.simple i*x) (cs.simple i*y)
-      (p.map (rotateEdge cs i)) ↔ IsPath (source cs) (target cs) x y p := by
-  constructor
-  · intro hp
-    simpa only [cs.simple_mul_simple_cancel_left,path_map_twice] using
-      path_rotate cs i hp (nonsimple_map i hn)
-  · intro hp
-    exact path_rotate cs i hp hn
-
-theorem paths_rotate_iff (O O' : RatioOrder (M := M) (cs := cs)) (i : I)
-    (hcmp : ∀ a b : PositiveRoot M cs, a ≠ simplePositive i → b ≠ simplePositive i →
-      (O'.slope (positiveAction (cs.simple i) a) < O'.slope (positiveAction (cs.simple i) b) ↔
-        O.slope a < O.slope b))
-    {x y : W} {p : List (Edge cs)} (hn : Nonsimple i p) :
-    p.map (rotateEdge cs i) ∈ paths O' (cs.simple i*x) (cs.simple i*y) ↔ p ∈ paths O x y := by
-  rw [mem_paths,mem_paths,path_rotate_iff i hn,List.pairwise_map]
-  apply and_congr_right
-  intro _
-  apply List.Pairwise.iff_of_mem
-  intro e f he hf
-  simpa only [score,rotateEdge_root cs i e (hn e he),rotateEdge_root cs i f (hn f hf)] using
-    hcmp (rootLabel cs e) (rootLabel cs f) (hn e he) (hn f hf)
-
-theorem nonsimple_of_greatest (O : RatioOrder (M := M) (cs := cs)) (i : I)
-    (hmax : ∀ a : PositiveRoot M cs, a ≠ simplePositive i →
-      O.slope a < O.slope (simplePositive i))
-    {x y : W} (hy : ¬cs.length (cs.simple i*y) < cs.length y)
-    {p : List (Edge cs)} (hp : p ∈ paths O x y) : Nonsimple i p := by
-  rw [mem_paths] at hp
-  rcases hp with ⟨hp,ho⟩
-  induction p generalizing x with
-  | nil => simp [Nonsimple]
-  | cons e p ih =>
-    have he : rootLabel cs e ≠ simplePositive i := by
-      intro he
-      cases p with
-      | nil =>
-        have hy' : target cs e = y := hp.2
-        have hr := (rootLabel_simple_iff cs i e).mp he
-        have hx' : source cs e = cs.simple i*y := by
-          rw [← hy',hr,cs.simple_mul_simple_cancel_left]
-        have hl := e.property.1
-        change cs.length (source cs e) < cs.length (target cs e) at hl
-        rw [hy',hx'] at hl
-        exact hy hl
-      | cons f p =>
-        have hlt := (List.pairwise_cons.mp ho).1 f List.mem_cons_self
-        have hle : score O f ≤ O.slope (simplePositive i) := by
-          by_cases hf : rootLabel cs f = simplePositive i
-          · exact le_of_eq (congrArg O.slope hf)
-          · exact (hmax _ hf).le
-        have hes : score O e = O.slope (simplePositive i) := congrArg O.slope he
-        exact (not_lt_of_ge hle) (hes ▸ hlt)
-    intro f hf
-    rcases List.mem_cons.mp hf with rfl|hf
-    · exact he
-    · exact ih hp.2 ho.tail f hf
-
-theorem nonsimple_tail_of_least (O : RatioOrder (M := M) (cs := cs)) (i : I)
-    (hmin : ∀ a : PositiveRoot M cs, a ≠ simplePositive i →
-      O.slope (simplePositive i) < O.slope a)
-    {e : Edge cs} {p : List (Edge cs)}
-    (ho : (e::p).Pairwise (fun e f => score O e < score O f)) : Nonsimple i p := by
-  intro f hf heq
-  have hef := (List.pairwise_cons.mp ho).1 f hf
-  have he : O.slope (simplePositive i) ≤ score O e := by
-    by_cases he : rootLabel cs e = simplePositive i
-    · exact le_of_eq (congrArg O.slope he).symm
-    · exact (hmin _ he).le
-  have hf' : score O f = O.slope (simplePositive i) := congrArg O.slope heq
-  exact (not_lt_of_ge he) (hf' ▸ hef)
-
-theorem nonsimple_of_least_descent (O : RatioOrder (M := M) (cs := cs)) (i : I)
-    (hmin : ∀ a : PositiveRoot M cs, a ≠ simplePositive i →
-      O.slope (simplePositive i) < O.slope a)
-    {x y : W} (hx : cs.length (cs.simple i*x) < cs.length x)
-    {p : List (Edge cs)} (hp : p ∈ paths O x y) : Nonsimple i p := by
-  obtain ⟨hp,ho⟩ := (mem_paths O x y p).mp hp
-  cases p with
-  | nil => simp [Nonsimple]
-  | cons e p =>
-    have he : rootLabel cs e ≠ simplePositive i := by
-      intro he
-      have hr := (rootLabel_simple_iff cs i e).mp he
-      have hl := e.property.1
-      change cs.length (source cs e) < cs.length (target cs e) at hl
-      rw [hr,hp.1] at hl
-      exact (lt_asymm hx hl)
-    intro f hf
-    rcases List.mem_cons.mp hf with rfl|hf
-    · exact he
-    · exact nonsimple_tail_of_least O i hmin ho f hf
-
-def nonsimplePaths (O : RatioOrder (M := M) (cs := cs)) (i : I) (x y : W) :
-    Finset (List (Edge cs)) := (paths O x y).filter (Nonsimple i)
-
-@[simp] theorem mem_nonsimplePaths (O : RatioOrder (M := M) (cs := cs)) (i : I)
-    (x y : W) (p : List (Edge cs)) :
-    p ∈ nonsimplePaths O i x y ↔ p ∈ paths O x y ∧ Nonsimple i p := by
-  simp [nonsimplePaths]
-
-theorem sum_nonsimple_rotate (O O' : RatioOrder (M := M) (cs := cs)) (i : I)
-    (hmax : ∀ a : PositiveRoot M cs, a ≠ simplePositive i →
-      O'.slope a < O'.slope (simplePositive i))
-    (hcmp : ∀ a b : PositiveRoot M cs, a ≠ simplePositive i → b ≠ simplePositive i →
-      (O'.slope (positiveAction (cs.simple i) a) < O'.slope (positiveAction (cs.simple i) b) ↔
-        O.slope a < O.slope b))
-    (x y : W) (hy : cs.length (cs.simple i*y) < cs.length y) :
-    ∑ p ∈ nonsimplePaths O i x y, (Polynomial.X : Polynomial ℤ)^p.length =
-      polynomial O' (cs.simple i*x) (cs.simple i*y) := by
-  unfold polynomial
-  have hasc : ¬cs.length (cs.simple i*(cs.simple i*y)) < cs.length (cs.simple i*y) := by
-    rw [cs.simple_mul_simple_cancel_left]
-    omega
-  apply Finset.sum_bij (fun p _ => p.map (rotateEdge cs i))
-  · intro p hp
-    obtain ⟨hp,hn⟩ := (mem_nonsimplePaths O i x y p).mp hp
-    exact (paths_rotate_iff O O' i hcmp hn).mpr hp
-  · intro p hp q hq he
-    have := congrArg (List.map (rotateEdge cs i)) he
-    simpa only [path_map_twice] using this
-  · intro q hq
-    have hn := nonsimple_of_greatest O' i hmax hasc hq
-    refine ⟨q.map (rotateEdge cs i),?_,path_map_twice i q⟩
-    have hn' := nonsimple_map i hn
-    apply (mem_nonsimplePaths O i x y _).mpr
-    refine ⟨?_,hn'⟩
-    apply (paths_rotate_iff O O' i hcmp hn').mp
-    simpa only [path_map_twice] using hq
-  · intro p hp
-    simp only [List.length_map]
-
-
-variable (cs)
-def simpleEdge (i : I) (x : W) (hx : cs.length x < cs.length (cs.simple i*x)) : Edge cs :=
-  ⟨(x,cs.simple i*x),hx,cs.simple i,cs.isReflection_simple i,rfl⟩
-
-omit [Fintype I] in
-@[simp] theorem simpleEdge_source (i : I) (x : W) (hx : cs.length x < cs.length (cs.simple i*x)) :
-    source cs (simpleEdge cs i x hx) = x := rfl
-
-omit [Fintype I] in
-@[simp] theorem simpleEdge_target (i : I) (x : W) (hx : cs.length x < cs.length (cs.simple i*x)) :
-    target cs (simpleEdge cs i x hx) = cs.simple i*x := rfl
-
-@[simp] theorem simpleEdge_root (i : I) (x : W) (hx : cs.length x < cs.length (cs.simple i*x)) :
-    rootLabel cs (simpleEdge cs i x hx) = simplePositive i :=
-  (rootLabel_simple_iff cs i _).mpr rfl
-
-variable {cs}
-theorem cons_simple_mem (O : RatioOrder (M := M) (cs := cs)) (i : I)
-    (hmin : ∀ a : PositiveRoot M cs, a ≠ simplePositive i →
-      O.slope (simplePositive i) < O.slope a)
-    {x y : W} (hx : cs.length x < cs.length (cs.simple i*x))
-    {p : List (Edge cs)} (hp : p ∈ nonsimplePaths O i (cs.simple i*x) y) :
-    simpleEdge cs i x hx :: p ∈ paths O x y := by
-  obtain ⟨hp,hn⟩ := (mem_nonsimplePaths O i _ _ _).mp hp
-  obtain ⟨hp,ho⟩ := (mem_paths O _ _ _).mp hp
-  apply (mem_paths O _ _ _).mpr
-  refine ⟨⟨rfl,hp⟩,List.pairwise_cons.mpr ⟨?_,ho⟩⟩
-  intro e he
-  change O.slope (rootLabel cs (simpleEdge cs i x hx)) < O.slope (rootLabel cs e)
-  rw [simpleEdge_root]
-  exact hmin _ (hn e he)
-
-theorem not_nonsimple_iff_cons (O : RatioOrder (M := M) (cs := cs)) (i : I)
-    (hmin : ∀ a : PositiveRoot M cs, a ≠ simplePositive i →
-      O.slope (simplePositive i) < O.slope a)
-    {x y : W} (hx : cs.length x < cs.length (cs.simple i*x)) (p : List (Edge cs)) :
-    p ∈ (paths O x y).filter (fun p => ¬Nonsimple i p) ↔
-      ∃ q ∈ nonsimplePaths O i (cs.simple i*x) y, p = simpleEdge cs i x hx :: q := by
-  constructor
-  · intro hp
-    obtain ⟨hp,hn⟩ := Finset.mem_filter.mp hp
-    obtain ⟨hp,ho⟩ := (mem_paths O _ _ _).mp hp
-    cases p with
-    | nil => exact (hn (by simp [Nonsimple])).elim
-    | cons e p =>
-      have hnp : Nonsimple i p := nonsimple_tail_of_least O i hmin ho
-      have he : rootLabel cs e = simplePositive i := by
-        by_contra he
-        apply hn
-        intro f hf
-        rcases List.mem_cons.mp hf with rfl|hf
-        · exact he
-        · exact hnp f hf
-      have heq : e = simpleEdge cs i x hx := by
-        apply Subtype.ext
-        apply Prod.ext
-        · exact hp.1
-        · change target cs e = cs.simple i*x
-          rw [(rootLabel_simple_iff cs i e).mp he,hp.1]
-      subst e
-      refine ⟨p,(mem_nonsimplePaths O i _ _ _).mpr ⟨?_,hnp⟩,rfl⟩
-      exact (mem_paths O _ _ _).mpr ⟨hp.2,ho.tail⟩
-  · rintro ⟨q,hq,rfl⟩
-    apply Finset.mem_filter.mpr
-    refine ⟨cons_simple_mem O i hmin hx hq,?_⟩
-    intro hn
-    exact hn _ List.mem_cons_self (simpleEdge_root cs i x hx)
-
-theorem polynomial_split (O : RatioOrder (M := M) (cs := cs)) (i : I)
-    (hmin : ∀ a : PositiveRoot M cs, a ≠ simplePositive i →
-      O.slope (simplePositive i) < O.slope a)
-    {x y : W} (hx : cs.length x < cs.length (cs.simple i*x)) :
-    polynomial O x y =
-      (∑ p ∈ nonsimplePaths O i x y, (Polynomial.X : Polynomial ℤ)^p.length) +
-      Polynomial.X * (∑ p ∈ nonsimplePaths O i (cs.simple i*x) y,
-        (Polynomial.X : Polynomial ℤ)^p.length) := by
-  unfold polynomial
-  rw [← Finset.sum_filter_add_sum_filter_not (paths O x y) (Nonsimple i)]
-  congr 1
-  rw [Finset.mul_sum]
-  symm
-  apply Finset.sum_bij (fun p _ => simpleEdge cs i x hx :: p)
-  · intro p hp
-    exact (not_nonsimple_iff_cons O i hmin hx _).mpr ⟨p,hp,rfl⟩
-  · intro p hp q hq he
-    exact List.cons.inj he |>.2
-  · intro q hq
-    obtain ⟨p,hp,rfl⟩ := (not_nonsimple_iff_cons O i hmin hx q).mp hq
-    exact ⟨p,hp,rfl⟩
-  · intro p hp
-    simp only [List.length_cons,pow_succ']
-
-/-- The path expansion satisfies exactly the genuine normalized Hecke
-recursion, because the chosen geometric order can be rotated at a descent. -/
-theorem polynomial_recursion (O O' : RatioOrder (M := M) (cs := cs)) (i : I)
-    (hmin : ∀ a : PositiveRoot M cs, a ≠ simplePositive i →
-      O.slope (simplePositive i) < O.slope a)
-    (hmax : ∀ a : PositiveRoot M cs, a ≠ simplePositive i →
-      O'.slope a < O'.slope (simplePositive i))
-    (hcmp : ∀ a b : PositiveRoot M cs, a ≠ simplePositive i → b ≠ simplePositive i →
-      (O'.slope (positiveAction (cs.simple i) a) < O'.slope (positiveAction (cs.simple i) b) ↔
-        O.slope a < O.slope b))
-    (x y : W) (hy : cs.length (cs.simple i*y) < cs.length y) :
-    polynomial O x y = if cs.length (cs.simple i*x) < cs.length x then
-      polynomial O' (cs.simple i*x) (cs.simple i*y) else
-      polynomial O' (cs.simple i*x) (cs.simple i*y) +
-        Polynomial.X * polynomial O' x (cs.simple i*y) := by
-  split_ifs with hx
-  · have heq : nonsimplePaths O i x y = paths O x y := by
-      ext p
-      rw [mem_nonsimplePaths]
-      exact and_iff_left_of_imp (nonsimple_of_least_descent O i hmin hx)
-    rw [← sum_nonsimple_rotate O O' i hmax hcmp x y hy,heq]
-    rfl
-  · have ha : cs.length x < cs.length (cs.simple i*x) := by
-      have := cs.length_simple_mul x i
-      omega
-    rw [polynomial_split O i hmin ha,
-      sum_nonsimple_rotate O O' i hmax hcmp x y hy,
-      sum_nonsimple_rotate O O' i hmax hcmp (cs.simple i*x) y hy,
-      cs.simple_mul_simple_cancel_left]
-
-
-@[simp] theorem paths_diagonal (O : RatioOrder (M := M) (cs := cs)) (x : W) :
-    paths O x x = {[]} := by
-  ext p
-  rw [mem_paths,Finset.mem_singleton]
-  cases p with
-  | nil => simp [IsPath]
-  | cons e p =>
-    constructor
-    · rintro ⟨hp,_⟩
-      have hl := length_le_of_bruhat cs (path_le cs hp.2)
-      have he := e.property.1
-      change cs.length (source cs e) < cs.length (target cs e) at he
-      rw [hp.1] at he
-      omega
-    · simp
-
-@[simp] theorem polynomial_diagonal (O : RatioOrder (M := M) (cs := cs)) (x : W) :
-    polynomial O x x = 1 := by
-  simp [polynomial]
-
-theorem polynomial_zero (O : RatioOrder (M := M) (cs := cs)) (x y : W)
-    (h : ¬BruhatLE cs x y) : polynomial O x y = 0 := by
-  have hp : paths O x y = ∅ := by
-    apply Finset.eq_empty_iff_forall_notMem.mpr
-    intro p hp
-    exact h (path_le cs ((mem_paths O x y p).mp hp).1)
-  simp [polynomial,hp]
-
-/-- Dyer's increasing-path formula for a genuine generic geometric order
-whose numerator is in the top chamber. The target is the Hecke-defined
-normalized R-polynomial, not a renamed path polynomial. -/
-theorem polynomial_eq_rTilde (O : RatioOrder (M := M) (cs := cs)) (x y : W)
-    (hc : O.Chamber y) : polynomial O x y = Hecke.rTilde cs x y := by
-  generalize hn : cs.length y = n
-  induction n using Nat.strong_induction_on generalizing O x y with
-  | h n ih =>
-    by_cases hy : y=1
-    · subst y
-      by_cases hx : x=1
-      · subst x
-        rw [polynomial_diagonal,Hecke.rTilde_diagonal]
-      · have hxy : ¬ BruhatLE cs x 1 := by
-          intro h
-          have hl := length_le_of_bruhat cs h
-          rw [cs.length_one] at hl
-          exact hx (cs.length_eq_zero_iff.mp (Nat.eq_zero_of_le_zero hl))
-        rw [polynomial_zero O x 1 hxy,Hecke.rTilde_zero cs x 1 hxy]
-    · obtain ⟨i,O',hi,hc',hends,hcmp⟩ := O.exists_rotation hy hc
-      have hi' : cs.length (cs.simple i*y) < n := hn ▸ hi
-      have hsx := ih _ hi' O' (cs.simple i*x) (cs.simple i*y) hc' rfl
-      have hx := ih _ hi' O' x (cs.simple i*y) hc' rfl
-      rw [polynomial_recursion O O' i (fun a ha => (hends a ha).1)
-        (fun a ha => (hends a ha).2) hcmp x y hi,
-        Hecke.rTilde_recursion cs x y i hi,hsx,hx]
-
-/-- An adapted geometric order exists for every top, without adding labels
-or geometric data to the given interval isomorphism. -/
-theorem exists_dyer_schedule (y : W) :
-    ∃ O : RatioOrder (M := M) (cs := cs), O.Chamber y ∧
-      ∀ x : W, polynomial O x y = Hecke.rTilde cs x y := by
-  obtain ⟨O,hO⟩ := exists_ratioOrder_chamber (M := M) (cs := cs) y
-  exact ⟨O,hO,fun x => polynomial_eq_rTilde O x y hO⟩
-
-end
-end KLInvariance.AdaptedDyer
-
-end
-
-
-section
-
-/-! Finite edge-product algebra needed for geometric wall crossing. Every
-operator is the actual `1 + T E_xy`; no R/KL identity is assumed here. -/
-
-end
-
-end OAI
+import OAI.RepresentationTheory.KazhdanLusztig.CharacterReciprocity
+
+namespace OAI
+
+/-!
+Finite character cancellation, reflection-ratio orders and adapted Dyer decreasing paths.
+-/
+
+section
+
+namespace KLInvariance
+universe u v
+variable {V : Type u} [Fintype V] {C : Type v} [CommRing C]
+
+ theorem finite_character_cancellation (t : V) (m : V → C) (r : V → C)
+    (p : V → V → C) (q : V → C) (hm : m t=1)
+    (hword : (∑ y, m y*q y)=∑ z, r z*(∑ y, m y*p y z))
+    (hlower : ∀ y, y≠t → q y=∑ z, r z*p y z) : q t=∑ z, r z*p t z := by
+  classical
+  have hexpand : (∑ z, r z*(∑ y, m y*p y z))=∑ y, m y*(∑ z, r z*p y z) := by
+    simp only [Finset.mul_sum]
+    rw [Finset.sum_comm]
+    apply Finset.sum_congr rfl
+    intro y _
+    apply Finset.sum_congr rfl
+    intro z _
+    ring
+  rw [hexpand] at hword
+  have hzero : (∑ y, m y*(q y-∑ z, r z*p y z))=0 := by
+    simp only [mul_sub,Finset.sum_sub_distrib,hword,sub_self]
+  have hsingle : (∑ y, m y*(q y-∑ z, r z*p y z))=q t-∑ z, r z*p t z := by
+    rw [Finset.sum_eq_single t]
+    · rw [hm,one_mul]
+    · intro y _ hy
+      rw [hlower y hy,sub_self,mul_zero]
+    · simp
+  rw [hsingle] at hzero
+  exact sub_eq_zero.mp hzero
+
+end KLInvariance
+
+end
+
+
+section
+
+/-! R-reciprocity for the ACTUAL minimal-boundary BMP character, obtained
+by simultaneous word stalk/costalk decomposition and unipotent cancellation.
+This is not a character identification: strict halfdegree remains separate. -/
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph Polynomial
+universe u
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+local instance bmpRecInterval (b : W) : Fintype (Interval cs 1 b) := by
+  let := interval_finite cs 1 b
+  exact Fintype.ofFinite _
+
+ theorem stalkCharacter_reciprocity (b : W) (x : Interval cs 1 b) :
+    reflect (rankDifference cs x.val b) (stalkCharacter cs 1 b basis (one_bruhat cs b) x)=
+      ∑ z : Interval cs 1 b, OrdinaryR.polynomial cs x.val z.val*
+        stalkCharacter cs 1 b basis (one_bruhat cs b) z := by
+  generalize hn : cs.length b=n
+  induction n using Nat.strong_induction_on generalizing b with
+  | h n ih =>
+    obtain ⟨l,hl,rfl⟩ := cs.exists_isReduced b
+    let b := cs.wordProd l
+    let t : Interval cs 1 b := ⟨b,one_bruhat cs b,bruhat_refl cs b⟩
+    obtain ⟨m,_hm,hs,hk⟩ := bottCharacter_decomposition_both cs basis l hl
+    have hmt : m t=1 := by
+      have hh := hs t
+      rw [bottStalkCharacter_top cs basis l hl] at hh
+      rw [Finset.sum_eq_single t] at hh
+      · simpa only [lowerCharacter_diagonal,mul_one] using hh.symm
+      · intro y _ hy
+        have hv : y.val≠b := fun h => hy (Subtype.ext h)
+        rw [lowerCharacter_at_top,ite_eq_right hv,mul_zero]
+      · simp
+    have ht (z : Interval cs 1 b) : lowerCharacter cs b basis t z=
+        stalkCharacter cs 1 b basis (one_bruhat cs b) z := dite_eq_left z.property.2
+    have hw : (∑ y, m y*reflect (rankDifference cs x.val y.val) (lowerCharacter cs b basis y x))=
+        ∑ z : Interval cs 1 b, OrdinaryR.polynomial cs x.val z.val*
+          (∑ y, m y*lowerCharacter cs b basis y z) := by
+      rw [←hk x,bottKernelCharacter_rConvolution cs basis l hl x]
+      simp only [hs]
+      rfl
+    have hh := finite_character_cancellation t m
+      (fun z : Interval cs 1 b => OrdinaryR.polynomial cs x.val z.val)
+      (fun y z => lowerCharacter cs b basis y z)
+      (fun y => reflect (rankDifference cs x.val y.val) (lowerCharacter cs b basis y x))
+      hmt hw (fun y hy => ?_)
+    · simpa only [ht] using hh
+    · apply lowerCharacter_reciprocity_lift
+      intro z
+      apply ih (cs.length y.val) _ y.val z rfl
+      rw [←hn]
+      exact length_lt_of_bruhat_ne cs y.property.2 (fun he => hy (Subtype.ext he))
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+/-! Removing the finite-rank restriction from the actual graph bridge,
+using the already established standard-parabolic embedding. -/
+namespace KLInvariance.StandardParabolic
+open CoxeterSystem
+universe u v
+variable {I : Type u} {W : Type v} [Group W] {M : CoxeterMatrix I}
+variable (cs : CoxeterSystem M W) (J : Set I)
+noncomputable section
+
+theorem leftInvSeq_inclusion (ω : List J) :
+    ((system (M := M) J).leftInvSeq ω).map (inclusion cs J) =
+      cs.leftInvSeq (ω.map Subtype.val) := by
+  induction ω with
+  | nil => rfl
+  | cons i ω ih =>
+    simp only [CoxeterSystem.leftInvSeq, List.map_cons, List.map_map,
+      inclusion_simple]
+    rw [← ih, List.map_map]
+    congr 1
+    apply List.map_congr_left
+    intro t _
+    simp only [Function.comp_apply, MulAut.conj_apply, map_mul, map_inv,
+      inclusion_simple]
+
+theorem reflection_inclusion {t : (matrix (M := M) J).Group}
+    (ht : (system (M := M) J).IsReflection t) :
+    cs.IsReflection (inclusion cs J t) := by
+  obtain ⟨w, i, rfl⟩ := ht
+  exact ⟨inclusion cs J w, i.val, by simp only [map_mul, map_inv, inclusion_simple]⟩
+
+theorem bruhatStep_inclusion_iff [Fintype J] (x y : (matrix (M := M) J).Group) :
+    BruhatStep cs (inclusion cs J x) (inclusion cs J y) ↔
+      BruhatStep (system (M := M) J) x y := by
+  constructor
+  · rintro ⟨hlen, t, ht, heq⟩
+    obtain ⟨ω, hω, hprod⟩ := (system (M := M) J).exists_isReduced y
+    have hi : cs.IsLeftInversion (inclusion cs J y) t := by
+      refine ⟨ht, ?_⟩
+      rw [heq, ← mul_assoc, ht.mul_self, one_mul]
+      simpa only [heq] using hlen
+    have hr := (reduced_map_iff cs J ω).2 hω
+    have hw : cs.wordProd (ω.map Subtype.val) = inclusion cs J y := by
+      rw [← inclusion_word, hprod]
+    have hm := CoxeterSupport.mem_leftInvSeq_of_inversion cs hr (hw ▸ hi)
+    rw [← leftInvSeq_inclusion] at hm
+    obtain ⟨r, hr, hrt⟩ := List.mem_map.mp hm
+    refine ⟨by simpa only [length_inclusion] using hlen,
+      r, (system (M := M) J).isReflection_of_mem_leftInvSeq ω hr, ?_⟩
+    apply inclusion_injective cs J
+    simpa only [map_mul, hrt] using heq
+  · rintro ⟨hlen, t, ht, heq⟩
+    refine ⟨by simpa only [length_inclusion] using hlen,
+      inclusion cs J t, reflection_inclusion cs J ht, ?_⟩
+    rw [heq, map_mul]
+
+end
+end KLInvariance.StandardParabolic
+
+namespace KLInvariance
+universe u v u' v'
+variable {I : Type u} {W : Type v} [Group W] {M : CoxeterMatrix I}
+
+/-- The graph reconstruction input in its source scope: the Coxeter system
+can have infinitely many generators. The interval determines every reflection
+edge without being supplied the reflection labels. -/
+theorem diamondEdges_of_bruhatStep_general (cs : CoxeterSystem M W)
+    {u b : W} {x y : Interval cs u b} (hxy : BruhatStep cs x.val y.val) :
+    DiamondEdges x y := by
+  classical
+  obtain ⟨J, hJ, hwords⟩ := lower_ideal_finite_generator_support cs b
+  let : Fintype J := hJ.fintype
+  obtain ⟨ω, hω, hprod⟩ := hwords b (bruhat_refl cs b)
+  obtain ⟨ν, rfl⟩ := StandardParabolic.lift_word J hω
+  let bb := (StandardParabolic.system (M := M) J).wordProd ν
+  have hb : StandardParabolic.inclusion cs J bb = b :=
+    (StandardParabolic.inclusion_word cs J ν).trans hprod
+  have hub : BruhatLE cs u b := (bruhat_trans cs x.property.1 x.property.2)
+  obtain ⟨ub, hu⟩ := StandardParabolic.lower_range cs J bb (hb ▸ hub)
+  subst u
+  clear hprod
+  subst b
+  let e := StandardParabolic.intervalIso cs J ub bb
+  let xx := e.symm x
+  let yy := e.symm y
+  have hx : StandardParabolic.inclusion cs J xx.val = x.val :=
+    congrArg Subtype.val (e.apply_symm_apply x)
+  have hy : StandardParabolic.inclusion cs J yy.val = y.val :=
+    congrArg Subtype.val (e.apply_symm_apply y)
+  have hh : BruhatStep (StandardParabolic.system (M := M) J) xx.val yy.val := by
+    apply (StandardParabolic.bruhatStep_inclusion_iff cs J _ _).mp
+    rwa [hx, hy]
+  have hg := (diamondEdges_of_bruhatStep (StandardParabolic.system (M := M) J) hh).map e
+  simpa only [xx, yy, e.apply_symm_apply] using hg
+
+theorem diamondEdges_bruhatStep_general (cs : CoxeterSystem M W)
+    {u b : W} {x y : Interval cs u b} (hxy : DiamondEdges x y) :
+    BruhatStep cs x.val y.val := by
+  classical
+  obtain ⟨J, hJ, hwords⟩ := lower_ideal_finite_generator_support cs b
+  let : Fintype J := hJ.fintype
+  obtain ⟨ω, hω, hprod⟩ := hwords b (bruhat_refl cs b)
+  obtain ⟨ν, rfl⟩ := StandardParabolic.lift_word J hω
+  let bb := (StandardParabolic.system (M := M) J).wordProd ν
+  have hb : StandardParabolic.inclusion cs J bb = b :=
+    (StandardParabolic.inclusion_word cs J ν).trans hprod
+  have hub : BruhatLE cs u b := (bruhat_trans cs x.property.1 x.property.2)
+  obtain ⟨ub, hu⟩ := StandardParabolic.lower_range cs J bb (hb ▸ hub)
+  subst u
+  clear hprod
+  subst b
+  let e := StandardParabolic.intervalIso cs J ub bb
+  let xx := e.symm x
+  let yy := e.symm y
+  have hx : StandardParabolic.inclusion cs J xx.val = x.val :=
+    congrArg Subtype.val (e.apply_symm_apply x)
+  have hy : StandardParabolic.inclusion cs J yy.val = y.val :=
+    congrArg Subtype.val (e.apply_symm_apply y)
+  have hh : DiamondEdges xx yy := hxy.map e.symm
+  have hg := diamondEdges_bruhatStep (StandardParabolic.system (M := M) J) hh
+  have hs := (StandardParabolic.bruhatStep_inclusion_iff cs J xx.val yy.val).mpr hg
+  rwa [hx, hy] at hs
+
+/-- Exact unlabelled directed graph invariance, also for arbitrary infinite
+Coxeter systems. This is not yet KL polynomial invariance. -/
+theorem intervalIso_bruhatStep_iff_general
+    {I' : Type u'} {W' : Type v'} [Group W'] {M' : CoxeterMatrix I'}
+    (cs : CoxeterSystem M W) (cs' : CoxeterSystem M' W')
+    {u b : W} {u' b' : W'} (e : Interval cs u b ≃o Interval cs' u' b')
+    (x y : Interval cs u b) :
+    BruhatStep cs' (e x).val (e y).val ↔ BruhatStep cs x.val y.val := by
+  constructor
+  · intro h
+    exact diamondEdges_bruhatStep_general cs ((DiamondEdges.map_iff e).mp
+      (diamondEdges_of_bruhatStep_general cs' h))
+  · intro h
+    exact diamondEdges_bruhatStep_general cs' ((diamondEdges_of_bruhatStep_general cs h).map e)
+
+end KLInvariance
+
+end
+
+
+section
+
+
+namespace KLInvariance.TitsSpace
+open Module
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+noncomputable section
+
+namespace PlaneBasis
+variable {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+
+theorem graphRoot_mem {a₀ : W} {g h : planeSubgroup M cs P}
+    (he : BruhatStep cs ((g : W) * a₀) ((h : W) * a₀)) :
+    (graphRoot he).val ∈ P := by
+  exact reflection_between_coset_mem M cs (a := a₀) g.property h.property (by group)
+    (graphRoot_represents he)
+
+include B
+
+/-- The two labels on a composable coset path span the whole actual plane. -/
+theorem graphRoot_path_plane {a₀ : W} {g h k : planeSubgroup M cs P}
+    (he : BruhatStep cs ((g : W) * a₀) ((h : W) * a₀))
+    (hf : BruhatStep cs ((h : W) * a₀) ((k : W) * a₀)) :
+    rootPlane (graphRoot he) (graphRoot hf) = P := by
+  exact (rootPlane_eq_of_mem B.left B.right (graphRoot he) (graphRoot hf)
+    B.distinct (B.plane_eq.ge (graphRoot_mem he)) (B.plane_eq.ge (graphRoot_mem hf))
+    (graphRoot_ne_of_path he hf)).trans B.plane_eq
+
+
+theorem twoStep_mem_coset {a₀ : W} {g h k : planeSubgroup M cs P}
+    (he : BruhatStep cs ((g : W) * a₀) ((h : W) * a₀))
+    (hf : BruhatStep cs ((h : W) * a₀) ((k : W) * a₀))
+    {z : W} (hez : BruhatStep cs ((g : W) * a₀) z)
+    (hzf : BruhatStep cs z ((k : W) * a₀)) :
+    z * a₀⁻¹ ∈ planeSubgroup M cs P := by
+  by_cases hz : (h : W) * a₀ = z
+  · rw [← hz, mul_inv_cancel_right]
+    exact h.property
+  have hplane := graphRoot_square_plane he hez hf hzf hz
+  have hroot : (graphRoot hez).val ∈ P := by
+    apply (hplane.symm.trans (B.graphRoot_path_plane he hf)).le
+    exact Submodule.subset_span (Set.mem_insert_of_mem _ (Set.mem_singleton _))
+  have href : z * ((g : W) * a₀)⁻¹ ∈ planeSubgroup M cs P :=
+    (Represents.planeSubgroup_mem_iff M cs (graphRoot_represents hez)).mpr hroot
+  convert (planeSubgroup M cs P).mul_mem href g.property using 1; group
+
+/-- A relative rank-two interval has exactly its two ambient two-edge paths,
+with no assumption that the two ambient and relative orders coincide. -/
+theorem ambient_two_step_rank_two {a₀ : W}
+    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z * a₀))
+    {g k : planeSubgroup M cs P} (hlen : B.system.length k = B.system.length g + 2) :
+    ∃ a b : W, a ≠ b ∧ ∀ z : W,
+      (BruhatStep cs ((g : W) * a₀) z ∧ BruhatStep cs z ((k : W) * a₀)) ↔
+        z = a ∨ z = b := by
+  obtain ⟨a,b,hab,hpaths⟩ := Dihedral.two_step_rank_two B.system hlen
+  have ha := (hpaths a).mpr (Or.inl rfl)
+  have he := (B.coset_step_iff hmin g a).mpr ha.1
+  have hf := (B.coset_step_iff hmin a k).mpr ha.2
+  refine ⟨(a : W)*a₀,(b : W)*a₀,?_,fun z => ?_⟩
+  · intro heq
+    exact hab (Subtype.ext (mul_right_cancel heq))
+  constructor
+  · intro hz
+    let z' : planeSubgroup M cs P := ⟨z*a₀⁻¹,B.twoStep_mem_coset he hf hz.1 hz.2⟩
+    have hz' : (z' : W)*a₀=z := by dsimp [z']; group
+    have hp : BruhatStep B.system g z' ∧ BruhatStep B.system z' k := by
+      constructor
+      · exact (B.coset_step_iff hmin g z').mp (hz' ▸ hz.1)
+      · exact (B.coset_step_iff hmin z' k).mp (hz' ▸ hz.2)
+    rcases (hpaths z').mp hp with hza | hzb
+    · exact Or.inl (hz'.symm.trans (congrArg (fun q : planeSubgroup M cs P => (q:W)*a₀) hza))
+    · exact Or.inr (hz'.symm.trans (congrArg (fun q : planeSubgroup M cs P => (q:W)*a₀) hzb))
+  · rintro (rfl | rfl)
+    · exact ⟨he,hf⟩
+    · have hb := (hpaths b).mpr (Or.inr rfl)
+      exact ⟨(B.coset_step_iff hmin g b).mpr hb.1,
+        (B.coset_step_iff hmin b k).mpr hb.2⟩
+
+end PlaneBasis
+end
+end KLInvariance.TitsSpace
+
+namespace KLInvariance.Dihedral
+variable {G : Type*} [Group G] {N : CoxeterMatrix Bool}
+
+/-- Exactly two intermediate vertices characterize relative rank two as soon
+as one two-edge path exists. Used to rule out skipped ranks after transport. -/
+theorem rank_two_of_two_step_bound (ds : CoxeterSystem N G) {x y a b : G}
+    (ha : BruhatStep ds x a) (hay : BruhatStep ds a y)
+    (hbound : ∀ z, BruhatStep ds x z → BruhatStep ds z y → z = a ∨ z = b) :
+    ds.length y = ds.length x + 2 := by
+  have hxa := (step_iff_odd ds x a).mp ha
+  have hay' := (step_iff_odd ds a y).mp hay
+  rw [Nat.odd_iff] at hxa hay'
+  have heven : (ds.length y - ds.length x) % 2 = 0 := by omega
+  by_contra hlen
+  have hlarge : ds.length x + 3 < ds.length y := by omega
+  obtain ⟨p,q,hpq,hlevel⟩ := two_elements_length ds
+    (by omega : 0 < ds.length x+1) (w := y) (by omega)
+  have hp := (hlevel p).mpr (Or.inl rfl)
+  have hq := (hlevel q).mpr (Or.inr rfl)
+  obtain ⟨r,s,hrs,hlevel'⟩ := two_elements_length ds
+    (by omega : 0 < ds.length x+3) hlarge
+  have hr := (hlevel' r).mpr (Or.inl rfl)
+  have paths : ∀ z, ds.length z = ds.length x+1 ∨ ds.length z = ds.length x+3 →
+      BruhatStep ds x z ∧ BruhatStep ds z y := by
+    intro z hz
+    rw [step_iff_odd,step_iff_odd,Nat.odd_iff,Nat.odd_iff]
+    rcases hz with hz | hz <;> omega
+  have pp := paths p (Or.inl hp)
+  have pq := paths q (Or.inl hq)
+  have pr := paths r (Or.inr hr)
+  have hpr : p ≠ r := by intro he; rw [he] at hp; omega
+  have hqr : q ≠ r := by intro he; rw [he] at hq; omega
+  rcases hbound p pp.1 pp.2 with hp' | hp' <;>
+    rcases hbound q pq.1 pq.2 with hq' | hq' <;>
+    rcases hbound r pr.1 pr.2 with hr' | hr' <;> grind
+
+end KLInvariance.Dihedral
+
+namespace KLInvariance.TitsSpace
+open Module
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+noncomputable section
+
+theorem graphRoot_mem_of_coset {P : Submodule ℝ (I → ℝ)} {a₀ x y : W}
+    (h : BruhatStep cs x y) (hx : x*a₀⁻¹ ∈ planeSubgroup M cs P)
+    (hy : y*a₀⁻¹ ∈ planeSubgroup M cs P) : (graphRoot h).val ∈ P := by
+  apply (Represents.planeSubgroup_mem_iff M cs (graphRoot_represents h)).mp
+  convert (planeSubgroup M cs P).mul_mem hy ((planeSubgroup M cs P).inv_mem hx) using 1
+  group
+
+theorem mem_coset_of_graphRoot {P : Submodule ℝ (I → ℝ)} {a₀ x y : W}
+    (h : BruhatStep cs x y) (hx : x*a₀⁻¹ ∈ planeSubgroup M cs P)
+    (hy : (graphRoot h).val ∈ P) : y*a₀⁻¹ ∈ planeSubgroup M cs P := by
+  have hm := (Represents.planeSubgroup_mem_iff M cs (graphRoot_represents h)).mpr hy
+  convert (planeSubgroup M cs P).mul_mem hm hx using 1
+  group
+
+variable {D : Type*} [Group D] {N : CoxeterMatrix Bool} (ds : CoxeterSystem N D)
+  {l r : D}
+
+/-- The image of a full relative dihedral interval lies in one maximal plane
+coset. This is derived just from the directed graph and distinct vertices. -/
+theorem dihedral_image_in_plane
+    (f : Interval ds l r → W) (hinj : Function.Injective f)
+    (hgraph : ∀ a b : Interval ds l r,
+      BruhatStep ds a.val b.val → BruhatStep cs (f a) (f b))
+    (hlr : BruhatLE ds l r) (hrank : ds.length l + 2 ≤ ds.length r) :
+    ∃ a b : PositiveRoot M cs, a ≠ b ∧
+      ∀ z : Interval ds l r,
+        f z * (f ⟨l,bruhat_refl ds l,hlr⟩)⁻¹ ∈ planeSubgroup M cs (rootPlane a b) := by
+  let lo : Interval ds l r := ⟨l,bruhat_refl ds l,hlr⟩
+  let hi : Interval ds l r := ⟨r,hlr,bruhat_refl ds r⟩
+  obtain ⟨a,b,hab,ha,hb⟩ := Dihedral.two_interval_elements_length ds lo hi
+    (n := ds.length l+1) (by dsimp [lo]; omega) (by dsimp [hi]; omega)
+  have cover : ∀ p q : Interval ds l r, ds.length q.val = ds.length p.val+1 →
+      BruhatStep cs (f p) (f q) := by
+    intro p q hlen
+    apply hgraph
+    rw [Dihedral.step_iff_odd,Nat.odd_iff]
+    omega
+  have hla := cover lo a ha
+  have hlb := cover lo b hb
+  have hrootne := graphRoot_ne_of_same_source hla hlb (fun h => hab (hinj h))
+  let P := rootPlane (graphRoot hla) (graphRoot hlb)
+  refine ⟨graphRoot hla,graphRoot hlb,hrootne,?_⟩
+  change ∀ z : Interval ds l r, f z * (f lo)⁻¹ ∈ planeSubgroup M cs P
+  have hlo : f lo * (f lo)⁻¹ ∈ planeSubgroup M cs P := by simp
+  have hat : f a * (f lo)⁻¹ ∈ planeSubgroup M cs P := by
+    exact mem_coset_of_graphRoot hla hlo (Submodule.subset_span (Set.mem_insert _ _))
+  have hbt : f b * (f lo)⁻¹ ∈ planeSubgroup M cs P := by
+    exact mem_coset_of_graphRoot hlb hlo
+      (Submodule.subset_span (Set.mem_insert_of_mem _ (Set.mem_singleton _)))
+  intro z
+  suffices ∀ n, ∀ z : Interval ds l r, ds.length z.val = n →
+      f z * (f lo)⁻¹ ∈ planeSubgroup M cs P from this _ z rfl
+  intro n
+  induction n using Nat.strong_induction_on with
+  | h n ih =>
+    intro z hz
+    have hzmin := length_le_of_bruhat ds z.property.1
+    by_cases hzlo : z = lo
+    · simpa only [hzlo] using hlo
+    have hzgt : ds.length l < ds.length z.val :=
+      length_lt_of_bruhat_ne ds z.property.1 (fun h => hzlo (Subtype.ext h.symm))
+    by_cases hzone : ds.length z.val = ds.length l+1
+    · obtain ⟨p,q,hpq,hlev⟩ := Dihedral.two_elements_length ds
+        (by omega : 0 < ds.length l+1) (w := r) (by omega)
+      have ap := (hlev a.val).mp ha
+      have bp := (hlev b.val).mp hb
+      have zp := (hlev z.val).mp hzone
+      have habval : a.val ≠ b.val := fun h => hab (Subtype.ext h)
+      have hza : z = a ∨ z = b := by
+        apply (show z.val = a.val ∨ z.val = b.val → z = a ∨ z = b from
+          fun h => h.elim (fun h => Or.inl (Subtype.ext h)) (fun h => Or.inr (Subtype.ext h)))
+        grind
+      rcases hza with rfl | rfl <;> assumption
+    have htwo : ds.length l+2 ≤ ds.length z.val := by omega
+    have hexp : ∃ p : Interval ds l r, ds.length p.val+2=ds.length z.val := by
+      by_cases he : ds.length z.val = ds.length l+2
+      · exact ⟨lo,he.symm⟩
+      · obtain ⟨p,q,_,hp,_⟩ := Dihedral.two_interval_elements_length ds lo z
+          (n := ds.length z.val-2) (by dsimp [lo]; omega) (by omega)
+        exact ⟨p,by omega⟩
+    obtain ⟨p,hp⟩ := hexp
+    obtain ⟨q,s,hqs,hq,hs⟩ := Dihedral.two_interval_elements_length ds lo z
+      (n := ds.length z.val-1) (by dsimp [lo]; omega) (by omega)
+    have hp' := ih (ds.length p.val) (by omega) p rfl
+    have hq' := ih (ds.length q.val) (by omega) q rfl
+    have hs' := ih (ds.length s.val) (by omega) s rfl
+    have hpq := cover p q (by omega)
+    have hps := cover p s (by omega)
+    have hqz := cover q z (by omega)
+    have hsz := cover s z (by omega)
+    have hplane : rootPlane (graphRoot hpq) (graphRoot hps) = P :=
+      rootPlane_eq_of_mem _ _ _ _ hrootne (graphRoot_mem_of_coset hpq hp' hq')
+        (graphRoot_mem_of_coset hps hp' hs')
+        (graphRoot_ne_of_same_source hpq hps (fun h => hqs (hinj h)))
+    have hsquare := graphRoot_square_plane hpq hps hqz hsz (fun h => hqs (hinj h))
+    apply mem_coset_of_graphRoot hqz hq'
+    apply (hsquare.trans hplane).le
+    exact Submodule.subset_span (Set.mem_insert_of_mem _ (Set.mem_singleton _))
+
+end
+end KLInvariance.TitsSpace
+
+namespace KLInvariance.Dihedral
+universe u₁ u₂
+variable {D : Type u₁} [Group D] {N : CoxeterMatrix Bool} (ds : CoxeterSystem N D)
+  {l r : D}
+
+theorem exists_interval_length (hlr : BruhatLE ds l r) {n : ℕ}
+    (hnl : ds.length l ≤ n) (hnr : n ≤ ds.length r) :
+    ∃ z : Interval ds l r, ds.length z.val = n := by
+  by_cases hl : n = ds.length l
+  · exact ⟨⟨l,bruhat_refl ds l,hlr⟩,hl.symm⟩
+  by_cases hr : n = ds.length r
+  · exact ⟨⟨r,hlr,bruhat_refl ds r⟩,hr.symm⟩
+  obtain ⟨a,b,_,ha,_⟩ := two_interval_elements_length ds
+    (⟨l,bruhat_refl ds l,hlr⟩ : Interval ds l r)
+    (⟨r,hlr,bruhat_refl ds r⟩ : Interval ds l r)
+    (n := n) (by dsimp; omega) (by dsimp; omega)
+  exact ⟨a,ha⟩
+
+variable {D' : Type u₂} [Group D'] {N' : CoxeterMatrix Bool}
+  (ds' : CoxeterSystem N' D')
+
+theorem map_cover_length
+    (f : Interval ds l r → D')
+    (hgraph : ∀ x y : Interval ds l r, BruhatStep ds x.val y.val →
+      BruhatStep ds' (f x) (f y))
+    (htwo : ∀ x y : Interval ds l r, ds.length y.val = ds.length x.val+2 →
+      ds'.length (f y) = ds'.length (f x)+2)
+    (hlr : BruhatLE ds l r) (hrank : ds.length l+2 ≤ ds.length r)
+    {x y : Interval ds l r} (hxy : ds.length y.val = ds.length x.val+1) :
+    ds'.length (f y) = ds'.length (f x)+1 := by
+  have hstep : ∀ p q : Interval ds l r, ds.length q.val = ds.length p.val+1 →
+      ds'.length (f p) < ds'.length (f q) := by
+    intro p q hpq
+    apply (hgraph p q ?_).1
+    rw [step_iff_odd,Nat.odd_iff]
+    omega
+  have hxy' := hstep x y hxy
+  have hyr := length_le_of_bruhat ds y.property.2
+  by_cases hy : ds.length y.val < ds.length r
+  · obtain ⟨z,hz⟩ := exists_interval_length ds hlr
+      (n := ds.length y.val+1)
+      (by have := length_le_of_bruhat ds y.property.1; omega) (by omega)
+    have hyz := hstep y z hz
+    have hxz := htwo x z (by omega)
+    omega
+  · obtain ⟨z,hz⟩ := exists_interval_length ds hlr
+      (n := ds.length x.val-1) (by omega) (by omega)
+    have hzx := hstep z x (by omega)
+    have hzy := htwo z y (by omega)
+    omega
+
+/-- A directed dihedral-interval graph injection which does not stretch any
+rank-two interval preserves all relative ranks. -/
+theorem map_length_affine
+    (f : Interval ds l r → D')
+    (hgraph : ∀ x y : Interval ds l r, BruhatStep ds x.val y.val →
+      BruhatStep ds' (f x) (f y))
+    (htwo : ∀ x y : Interval ds l r, ds.length y.val = ds.length x.val+2 →
+      ds'.length (f y) = ds'.length (f x)+2)
+    (hlr : BruhatLE ds l r) (hrank : ds.length l+2 ≤ ds.length r)
+    (z : Interval ds l r) :
+    ds'.length (f z) + ds.length l =
+      ds'.length (f ⟨l,bruhat_refl ds l,hlr⟩) + ds.length z.val := by
+  let lo : Interval ds l r := ⟨l,bruhat_refl ds l,hlr⟩
+  change ds'.length (f z)+ds.length l=ds'.length (f lo)+ds.length z.val
+  suffices ∀ n, ∀ z : Interval ds l r, ds.length z.val=n →
+      ds'.length (f z)+ds.length l=ds'.length (f lo)+ds.length z.val from this _ z rfl
+  intro n
+  induction n using Nat.strong_induction_on with
+  | h n ih =>
+    intro z hz
+    by_cases he : z=lo
+    · simp [he,lo]
+    have hlt : ds.length l < ds.length z.val :=
+      length_lt_of_bruhat_ne ds z.property.1 (fun h => he (Subtype.ext h.symm))
+    obtain ⟨p,hp⟩ := exists_interval_length ds hlr (n := ds.length z.val-1)
+      (by omega) (by have := length_le_of_bruhat ds z.property.2; omega)
+    have ihp := ih (ds.length p.val) (by omega) p rfl
+    have hc := map_cover_length ds ds' f hgraph htwo hlr hrank (x := p) (y := z)
+      (by omega)
+    omega
+
+/-- Once relative covers are recovered, the image exhausts rather than merely
+embeds in the target dihedral interval. -/
+theorem range_eq_interval_of_rank_two
+    (f : Interval ds l r → D') (hinj : Function.Injective f)
+    (hgraph : ∀ x y : Interval ds l r, BruhatStep ds x.val y.val →
+      BruhatStep ds' (f x) (f y))
+    (htwo : ∀ x y : Interval ds l r, ds.length y.val = ds.length x.val+2 →
+      ds'.length (f y) = ds'.length (f x)+2)
+    (hlr : BruhatLE ds l r) (hrank : ds.length l+2 ≤ ds.length r) :
+    Set.range f = {z | BruhatLE ds' (f ⟨l,bruhat_refl ds l,hlr⟩) z ∧
+      BruhatLE ds' z (f ⟨r,hlr,bruhat_refl ds r⟩)} := by
+  let lo : Interval ds l r := ⟨l,bruhat_refl ds l,hlr⟩
+  let hi : Interval ds l r := ⟨r,hlr,bruhat_refl ds r⟩
+  have hlen (z : Interval ds l r) : ds'.length (f z)+ds.length l =
+      ds'.length (f lo)+ds.length z.val :=
+    map_length_affine ds ds' f hgraph htwo hlr hrank z
+  have hhi : ds'.length (f hi)+ds.length l=ds'.length (f lo)+ds.length r := hlen hi
+  have horder (x y : Interval ds l r) :
+      BruhatLE ds' (f x) (f y) ↔ x ≤ y := by
+    change BruhatLE ds' (f x) (f y) ↔ BruhatLE ds x.val y.val
+    rw [bruhat_iff_length_lt_or_eq,bruhat_iff_length_lt_or_eq]
+    have hx := hlen x
+    have hy := hlen y
+    constructor
+    · rintro (h | h)
+      · left; omega
+      · exact Or.inr (congrArg Subtype.val (hinj h))
+    · rintro (h | h)
+      · left; omega
+      · exact Or.inr (congrArg f (Subtype.ext h))
+  apply Set.ext
+  intro z
+  change (∃ x, f x=z) ↔ BruhatLE ds' (f lo) z ∧ BruhatLE ds' z (f hi)
+  constructor
+  · rintro ⟨x,rfl⟩
+    exact ⟨(horder lo x).mpr x.property.1,(horder x hi).mpr x.property.2⟩
+  · intro hz
+    by_cases hlo : z=f lo
+    · exact ⟨lo,hlo.symm⟩
+    by_cases htop : z=f hi
+    · exact ⟨hi,htop.symm⟩
+    have hzlo := length_lt_of_bruhat_ne ds' hz.1 (Ne.symm hlo)
+    have hzhi := length_lt_of_bruhat_ne ds' hz.2 htop
+    have hnlo : ds.length l < ds.length l+(ds'.length z-ds'.length (f lo)) := by omega
+    have hnhi : ds.length l+(ds'.length z-ds'.length (f lo)) < ds.length r := by omega
+    obtain ⟨a,b,hab,ha,hb⟩ := two_interval_elements_length ds lo hi hnlo hnhi
+    have hfa : ds'.length (f a)=ds'.length z := by have := hlen a; omega
+    have hfb : ds'.length (f b)=ds'.length z := by have := hlen b; omega
+    obtain ⟨p,q,hpq,helements⟩ := two_elements_length ds' (by omega : 0<ds'.length z) hzhi
+    have hp := (helements (f a)).mp hfa
+    have hq := (helements (f b)).mp hfb
+    have hr := (helements z).mp rfl
+    have hab' : f a ≠ f b := fun h => hab (hinj h)
+    have he : z=f a ∨ z=f b := by grind
+    rcases he with he | he
+    · exact ⟨a,he.symm⟩
+    · exact ⟨b,he.symm⟩
+
+end KLInvariance.Dihedral
+
+namespace KLInvariance
+universe u v u' v'
+variable {I : Type u} {W : Type v} [Group W] {M : CoxeterMatrix I}
+  {I' : Type u'} {W' : Type v'} [Group W'] {M' : CoxeterMatrix I'}
+  {cs : CoxeterSystem M W} {cs' : CoxeterSystem M' W'}
+  {u b : W} {u' b' : W'}
+
+/-- An interval graph isomorphism preserves the exact two-edge intermediate
+set, including ambient intermediates not specified as interval data. -/
+theorem intervalIso_twoStep_iff
+    (e : Interval cs u b ≃o Interval cs' u' b') (x y a c : Interval cs u b)
+    (hpaths : ∀ z : W, (BruhatStep cs x.val z ∧ BruhatStep cs z y.val) ↔
+      z=a.val ∨ z=c.val) (z : W') :
+    (BruhatStep cs' (e x).val z ∧ BruhatStep cs' z (e y).val) ↔
+      z=(e a).val ∨ z=(e c).val := by
+  constructor
+  · intro hz
+    let z' : Interval cs' u' b' := ⟨z,
+      bruhat_trans cs' (e x).property.1 (Relation.ReflTransGen.single hz.1),
+      bruhat_trans cs' (Relation.ReflTransGen.single hz.2) (e y).property.2⟩
+    have hz₁ : BruhatStep cs x.val (e.symm z').val := by
+      apply (intervalIso_bruhatStep_iff_general cs cs' e x (e.symm z')).mp
+      simpa only [e.apply_symm_apply] using hz.1
+    have hz₂ : BruhatStep cs (e.symm z').val y.val := by
+      apply (intervalIso_bruhatStep_iff_general cs cs' e (e.symm z') y).mp
+      simpa only [e.apply_symm_apply] using hz.2
+    rcases (hpaths (e.symm z').val).mp ⟨hz₁,hz₂⟩ with ha | hc
+    · have he : e.symm z'=a := Subtype.ext ha
+      have he' : z'=e a := by rw [← he,e.apply_symm_apply]
+      exact Or.inl (congrArg Subtype.val he')
+    · have he : e.symm z'=c := Subtype.ext hc
+      have he' : z'=e c := by rw [← he,e.apply_symm_apply]
+      exact Or.inr (congrArg Subtype.val he')
+  · rintro (rfl | rfl)
+    · have h := (hpaths a.val).mpr (Or.inl rfl)
+      exact ⟨(intervalIso_bruhatStep_iff_general cs cs' e x a).mpr h.1,
+        (intervalIso_bruhatStep_iff_general cs cs' e a y).mpr h.2⟩
+    · have h := (hpaths c.val).mpr (Or.inr rfl)
+      exact ⟨(intervalIso_bruhatStep_iff_general cs cs' e x c).mpr h.1,
+        (intervalIso_bruhatStep_iff_general cs cs' e c y).mpr h.2⟩
+
+end KLInvariance
+
+namespace KLInvariance.TitsSpace.PlaneBasis
+open Module
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+noncomputable section
+
+def cosetIntervalMap {a₀ u b : W}
+    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+    {g k : planeSubgroup M cs P}
+    (hlo : BruhatLE cs u ((g : W)*a₀)) (hhi : BruhatLE cs ((k : W)*a₀) b) :
+    Interval B.system g k ↪ Interval cs u b where
+  toFun z := ⟨(z.val : W)*a₀,bruhat_trans cs hlo (B.coset_bruhat hmin z.property.1),
+    bruhat_trans cs (B.coset_bruhat hmin z.property.2) hhi⟩
+  inj' := by
+    intro x y he
+    exact Subtype.ext (Subtype.ext (mul_right_cancel (congrArg Subtype.val he)))
+
+theorem twoStep_interval_iff {a₀ : W}
+    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+    {g k : planeSubgroup M cs P} (x y : Interval B.system g k)
+    (hlen : B.system.length y.val=B.system.length x.val+2) :
+    ∃ a b : Interval B.system g k, a ≠ b ∧ ∀ z : W,
+      (BruhatStep cs ((x.val : W)*a₀) z ∧ BruhatStep cs z ((y.val : W)*a₀)) ↔
+        z=(a.val : W)*a₀ ∨ z=(b.val : W)*a₀ := by
+  obtain ⟨a,b,hab,hpaths⟩ := Dihedral.two_step_rank_two B.system hlen
+  have ha := (hpaths a).mpr (Or.inl rfl)
+  have hb := (hpaths b).mpr (Or.inr rfl)
+  let ai : Interval B.system g k := ⟨a,
+    bruhat_trans B.system x.property.1 (Relation.ReflTransGen.single ha.1),
+    bruhat_trans B.system (Relation.ReflTransGen.single ha.2) y.property.2⟩
+  let bi : Interval B.system g k := ⟨b,
+    bruhat_trans B.system x.property.1 (Relation.ReflTransGen.single hb.1),
+    bruhat_trans B.system (Relation.ReflTransGen.single hb.2) y.property.2⟩
+  have he := (B.coset_step_iff hmin x.val a).mpr ha.1
+  have hf := (B.coset_step_iff hmin a y.val).mpr ha.2
+  refine ⟨ai,bi,fun h => hab (congrArg Subtype.val h),fun z => ?_⟩
+  constructor
+  · intro hz
+    let z' : planeSubgroup M cs P := ⟨z*a₀⁻¹,B.twoStep_mem_coset he hf hz.1 hz.2⟩
+    have hz' : (z' : W)*a₀=z := by dsimp [z']; group
+    have hp : BruhatStep B.system x.val z' ∧ BruhatStep B.system z' y.val := by
+      constructor
+      · exact (B.coset_step_iff hmin x.val z').mp (hz' ▸ hz.1)
+      · exact (B.coset_step_iff hmin z' y.val).mp (hz' ▸ hz.2)
+    rcases (hpaths z').mp hp with hza | hzb
+    · exact Or.inl (hz'.symm.trans (congrArg (fun q : planeSubgroup M cs P => (q:W)*a₀) hza))
+    · exact Or.inr (hz'.symm.trans (congrArg (fun q : planeSubgroup M cs P => (q:W)*a₀) hzb))
+  · rintro (rfl | rfl)
+    · exact ⟨he,hf⟩
+    · exact ⟨(B.coset_step_iff hmin x.val b).mpr hb.1,
+        (B.coset_step_iff hmin b y.val).mpr hb.2⟩
+
+end
+end KLInvariance.TitsSpace.PlaneBasis
+
+namespace KLInvariance.TitsSpace.PlaneBasis
+open Module
+universe u v u' v'
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
+  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
+  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+noncomputable section
+
+
+theorem transport_interval {a₀ u b : W} {u' b' : W'}
+    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+    {g k : planeSubgroup M cs P} (hgk : BruhatLE B.system g k)
+    (hrank : B.system.length g+2 ≤ B.system.length k)
+    (hlo : BruhatLE cs u ((g : W)*a₀)) (hhi : BruhatLE cs ((k : W)*a₀) b)
+    (e : Interval cs u b ≃o Interval cs' u' b') :
+    ∃ (P' : Submodule ℝ (I' → ℝ)) (B' : PlaneBasis (M := M') (cs := cs') P')
+      (a' : W') (F : Interval B.system g k → planeSubgroup M' cs' P'),
+      (∀ z ∈ planeSubgroup M' cs' P', cs'.length a' ≤ cs'.length (z*a')) ∧
+      Function.Injective F ∧
+      (∀ z, (F z : W')*a' = (e (B.cosetIntervalMap hmin hlo hhi z)).val) ∧
+      (∀ x y, BruhatLE B'.system (F x) (F y) ↔ x ≤ y) ∧
+      (∀ x y, BruhatStep B'.system (F x) (F y) ↔ BruhatStep B.system x.val y.val) ∧
+      (∀ z, B'.system.length (F z)+B.system.length g =
+        B'.system.length (F ⟨g,bruhat_refl B.system g,hgk⟩)+B.system.length z.val) ∧
+      Set.range F = {z | BruhatLE B'.system (F ⟨g,bruhat_refl B.system g,hgk⟩) z ∧
+        BruhatLE B'.system z (F ⟨k,hgk,bruhat_refl B.system k⟩)} := by
+  let inc := B.cosetIntervalMap hmin hlo hhi
+  let f : Interval B.system g k → W' := fun z => (e (inc z)).val
+  let lo : Interval B.system g k := ⟨g,bruhat_refl B.system g,hgk⟩
+  have hinj : Function.Injective f := by
+    intro x y h
+    exact inc.injective (e.injective (Subtype.ext h))
+  have hgraph (x y : Interval B.system g k) (h : BruhatStep B.system x.val y.val) :
+      BruhatStep cs' (f x) (f y) := by
+    exact (intervalIso_bruhatStep_iff_general cs cs' e (inc x) (inc y)).mpr
+      ((B.coset_step_iff hmin x.val y.val).mpr h)
+  obtain ⟨α,β,hαβ,hplane⟩ := dihedral_image_in_plane B.system f hinj hgraph hgk hrank
+  let P' := rootPlane α β
+  obtain ⟨B'⟩ := nonempty_planeBasis α β hαβ
+  obtain ⟨a',g₀,hg₀,hg₀a,hmin'⟩ := planeCoset_exists_minimal M' cs' P' (f lo)
+  have hmem (z : Interval B.system g k) : f z*a'⁻¹ ∈ planeSubgroup M' cs' P' := by
+    convert (planeSubgroup M' cs' P').mul_mem (hplane z) hg₀ using 1
+    change f z*a'⁻¹=f z*(f lo)⁻¹*g₀
+    rw [hg₀a]
+    group
+  let F : Interval B.system g k → planeSubgroup M' cs' P' := fun z => ⟨f z*a'⁻¹,hmem z⟩
+  have hF (z : Interval B.system g k) : (F z : W')*a'=f z := by
+    dsimp [F]; group
+  have hFinj : Function.Injective F := by
+    intro x y h
+    apply hinj
+    rw [← hF x,← hF y,h]
+  have hFgraph (x y : Interval B.system g k) (h : BruhatStep B.system x.val y.val) :
+      BruhatStep B'.system (F x) (F y) := by
+    apply (B'.coset_step_iff hmin' (F x) (F y)).mp
+    rw [hF x,hF y]
+    exact hgraph x y h
+  have htwo (x y : Interval B.system g k)
+      (hlen : B.system.length y.val=B.system.length x.val+2) :
+      B'.system.length (F y)=B'.system.length (F x)+2 := by
+    obtain ⟨a,c,hac,hpaths⟩ := B.twoStep_interval_iff hmin x y hlen
+    have hpaths' (z : W') : (BruhatStep cs' (f x) z ∧ BruhatStep cs' z (f y)) ↔
+        z=f a ∨ z=f c := intervalIso_twoStep_iff e (inc x) (inc y) (inc a) (inc c) hpaths z
+    have hpa := (hpaths' (f a)).mpr (Or.inl rfl)
+    have hxa : BruhatStep B'.system (F x) (F a) := by
+      apply (B'.coset_step_iff hmin' (F x) (F a)).mp
+      simpa only [hF] using hpa.1
+    have hay : BruhatStep B'.system (F a) (F y) := by
+      apply (B'.coset_step_iff hmin' (F a) (F y)).mp
+      simpa only [hF] using hpa.2
+    apply Dihedral.rank_two_of_two_step_bound B'.system hxa hay (b := F c)
+    intro z hxz hzy
+    have hp₁ := (B'.coset_step_iff hmin' (F x) z).mpr hxz
+    have hp₂ := (B'.coset_step_iff hmin' z (F y)).mpr hzy
+    rw [hF x] at hp₁
+    rw [hF y] at hp₂
+    rcases (hpaths' ((z : W')*a')).mp ⟨hp₁,hp₂⟩ with he | he
+    · exact Or.inl (Subtype.ext (mul_right_cancel (he.trans (hF a).symm)))
+    · exact Or.inr (Subtype.ext (mul_right_cancel (he.trans (hF c).symm)))
+  have hlen := Dihedral.map_length_affine B.system B'.system F hFgraph htwo hgk hrank
+  have horder (x y : Interval B.system g k) : BruhatLE B'.system (F x) (F y) ↔ x ≤ y := by
+    change BruhatLE B'.system (F x) (F y) ↔ BruhatLE B.system x.val y.val
+    rw [Dihedral.bruhat_iff_length_lt_or_eq,Dihedral.bruhat_iff_length_lt_or_eq]
+    have hx := hlen x
+    have hy := hlen y
+    constructor
+    · rintro (h | h)
+      · left; omega
+      · exact Or.inr (congrArg Subtype.val (hFinj h))
+    · rintro (h | h)
+      · left; omega
+      · exact Or.inr (congrArg F (Subtype.ext h))
+  refine ⟨P',B',a',F,hmin',hFinj,hF,horder,?_,hlen,?_⟩
+  · intro x y
+    rw [Dihedral.step_iff_odd,Dihedral.step_iff_odd,Nat.odd_iff,Nat.odd_iff]
+    have hx := hlen x
+    have hy := hlen y
+    omega
+  · exact Dihedral.range_eq_interval_of_rank_two B.system B'.system F hFinj
+      hFgraph htwo hgk hrank
+
+end
+end KLInvariance.TitsSpace.PlaneBasis
+
+end
+
+
+section
+
+
+namespace KLInvariance.TitsSpace
+open Module
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+noncomputable section
+
+theorem inverted_of_positive_combination (w : W) (a b c : PositiveRoot M cs)
+    {r s : ℝ} (hr : 0 ≤ r) (hs : 0 ≤ s)
+    (hc : c.val = r • a.val + s • b.val)
+    (ha : rootInverted w a) (hb : rootInverted w b) :
+    rootInverted w c := by
+  intro i
+  rw [hc, map_add, map_smul, map_smul]
+  exact add_nonpos (mul_nonpos_of_nonneg_of_nonpos hr (ha i))
+    (mul_nonpos_of_nonneg_of_nonpos hs (hb i))
+
+/-- Incoming graph labels are actual inversions at their upper vertex. -/
+theorem graphRoot_inverted_target {x y : W} (h : BruhatStep cs x y) :
+    rootInverted y (graphRoot h) := by
+  apply ((graphRoot_represents h).inversion_iff (graphRoot h).property.2 y).mpr
+  refine ⟨h.label_reflection, ?_⟩
+  have he : (y*x⁻¹)*y=x := by
+    calc
+      (y*x⁻¹)*y = (y*x⁻¹)*((y*x⁻¹)*x) := by group
+      _ = x := by rw [← mul_assoc,h.label_reflection.mul_self,one_mul]
+  simpa only [he] using h.1
+
+/-- Outgoing graph labels are actual noninversions at their lower vertex. -/
+theorem graphRoot_not_inverted_source {x y : W} (h : BruhatStep cs x y) :
+    ¬ rootInverted x (graphRoot h) := by
+  intro hi
+  have hh := ((graphRoot_represents h).inversion_iff (graphRoot h).property.2 x).mp hi
+  have he : (y*x⁻¹)*x=y := by group
+  have hh' := hh.2
+  rw [he] at hh'
+  exact lt_asymm h.1 hh'
+
+namespace PlaneBasis
+variable {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+
+/-- In a plane with its first simple root inverted, every noninverted root
+is later than every inverted root. This uses convexity, not a signed-index
+formula for a particular realization. -/
+theorem slope_lt_of_inverted_left (w : W) (hl : rootInverted w B.left)
+    (a b : PositiveRoot M cs) (haP : a.val ∈ P) (hbP : b.val ∈ P)
+    (ha : rootInverted w a) (hb : ¬ rootInverted w b) :
+    geometricSlope a < geometricSlope b := by
+  by_contra h
+  have hba : geometricSlope b < geometricSlope a :=
+    lt_of_le_of_ne (le_of_not_gt h) (fun he => hb (geometricSlope_injective he ▸ ha))
+  have hlb : geometricSlope B.left < geometricSlope b :=
+    lt_of_le_of_ne (B.left_min b hbP) (fun he => hb (geometricSlope_injective he ▸ hl))
+  have heq : rootPlane B.left a = P :=
+    rootPlane_eq_of_mem B.left B.right B.left a B.distinct
+      (Submodule.subset_span (Set.mem_insert _ _))
+      (B.plane_eq.ge haP) (fun he => by rw [he] at hlb; exact (lt_asymm hlb hba)) |>.trans B.plane_eq
+  obtain ⟨r,s,hr,hs,hc⟩ := positive_combination_of_slope_between B.left a b
+    (heq.symm ▸ hbP) hlb hba
+  exact hb (inverted_of_positive_combination w B.left a b hr.le hs.le hc hl ha)
+
+/-- The other angular cut, when the last simple root is inverted. -/
+theorem slope_lt_of_inverted_right (w : W) (hr : rootInverted w B.right)
+    (a b : PositiveRoot M cs) (haP : a.val ∈ P) (hbP : b.val ∈ P)
+    (ha : rootInverted w a) (hb : ¬ rootInverted w b) :
+    geometricSlope b < geometricSlope a := by
+  by_contra h
+  have hab : geometricSlope a < geometricSlope b :=
+    lt_of_le_of_ne (le_of_not_gt h) (fun he => hb (geometricSlope_injective he ▸ ha))
+  have hbr : geometricSlope b < geometricSlope B.right :=
+    lt_of_le_of_ne (B.right_max b hbP) (fun he => hb (geometricSlope_injective he ▸ hr))
+  have har : a ≠ B.right := by
+    intro he
+    rw [he] at hab
+    exact lt_asymm hbr hab
+  have heq : rootPlane a B.right = P :=
+    (rootPlane_eq_of_mem B.left B.right a B.right B.distinct
+      (B.plane_eq.ge haP) (Submodule.subset_span (Set.mem_insert_of_mem _ (Set.mem_singleton _)))
+      har).trans B.plane_eq
+  obtain ⟨r,s,hrr,hss,hc⟩ := positive_combination_of_slope_between a B.right b
+    (heq.symm ▸ hbP) hab hbr
+  exact hb (inverted_of_positive_combination w a B.right b hrr.le hss.le hc ha hr)
+
+/-- In the mixed-sign case the angular cut is determined by the first
+canonical simple root. In particular the comparison of consecutive path
+labels depends only on their middle vertex. -/
+theorem slope_lt_iff_inverted_left (w : W)
+    (a b : PositiveRoot M cs) (haP : a.val ∈ P) (hbP : b.val ∈ P)
+    (ha : rootInverted w a) (hb : ¬ rootInverted w b) :
+    geometricSlope a < geometricSlope b ↔ rootInverted w B.left := by
+  constructor
+  · intro hab
+    by_contra hl
+    have hr : rootInverted w B.right := by
+      by_contra hr
+      obtain ⟨r,s,hrr,hss,he⟩ := B.positive_cone a haP
+      exact noninverted_of_positive_combination w B.left B.right a hrr hss he hl hr ha
+    exact lt_asymm hab (B.slope_lt_of_inverted_right w hr a b haP hbP ha hb)
+  · intro hl
+    exact B.slope_lt_of_inverted_left w hl a b haP hbP ha hb
+
+/-- The exact middle-vertex test for two ascending graph arrows with both
+labels in this root plane. -/
+theorem consecutive_slopes_iff {x y z : W}
+    (hxy : BruhatStep cs x y) (hyz : BruhatStep cs y z)
+    (ha : (graphRoot hxy).val ∈ P) (hb : (graphRoot hyz).val ∈ P) :
+    geometricSlope (graphRoot hxy) < geometricSlope (graphRoot hyz) ↔
+      rootInverted y B.left :=
+  B.slope_lt_iff_inverted_left y _ _ ha hb
+    (graphRoot_inverted_target hxy) (graphRoot_not_inverted_source hyz)
+
+end PlaneBasis
+end
+end KLInvariance.TitsSpace
+
+namespace KLInvariance.Dihedral
+variable {W : Type*} [Group W] {M : CoxeterMatrix Bool}
+noncomputable section
+
+/-- The genuine diagram automorphism of an arbitrary rank-two Coxeter system,
+including the infinite one. -/
+def swapHom (cs : CoxeterSystem M W) : W →* W :=
+  cs.lift ⟨fun i => cs.simple (!i), by
+    intro i j
+    have he : M i j = M (!i) (!j) := by
+      cases i <;> cases j
+      · simp
+      · exact M.symmetric _ _
+      · exact M.symmetric _ _
+      · simp
+    rw [he]
+    exact cs.simple_mul_simple_pow _ _⟩
+
+@[simp] theorem swapHom_simple (cs : CoxeterSystem M W) (i : Bool) :
+    swapHom cs (cs.simple i) = cs.simple (!i) := cs.lift_apply_simple _ i
+
+@[simp] theorem swapHom_involutive (cs : CoxeterSystem M W) :
+    Function.Involutive (swapHom cs) := by
+  intro w
+  apply cs.simple_induction w
+  · intro i
+    simp
+  · simp
+  · intro a b ha hb
+    simp only [map_mul,ha,hb]
+
+def swap (cs : CoxeterSystem M W) : W ≃* W where
+  toFun := swapHom cs
+  invFun := swapHom cs
+  left_inv := swapHom_involutive cs
+  right_inv := swapHom_involutive cs
+  map_mul' := map_mul (swapHom cs)
+
+@[simp] theorem swap_simple (cs : CoxeterSystem M W) (i : Bool) :
+    swap cs (cs.simple i) = cs.simple (!i) := swapHom_simple cs i
+
+@[simp] theorem swap_swap (cs : CoxeterSystem M W) (w : W) :
+    swap cs (swap cs w) = w := swapHom_involutive cs w
+
+theorem swap_wordProd (cs : CoxeterSystem M W) (ω : List Bool) :
+    swap cs (cs.wordProd ω) = cs.wordProd (ω.map Bool.not) := by
+  induction ω with
+  | nil => simp
+  | cons i ω ih =>
+    simp only [cs.wordProd_cons, map_mul, swap_simple, List.map_cons, ih]
+
+@[simp] theorem length_swap (cs : CoxeterSystem M W) (w : W) :
+    cs.length (swap cs w) = cs.length w := by
+  have hle (z : W) : cs.length (swap cs z) ≤ cs.length z := by
+    obtain ⟨ω,hω,rfl⟩ := cs.exists_isReduced z
+    rw [swap_wordProd, hω.eq]
+    simpa using cs.length_wordProd_le (ω.map Bool.not)
+  exact le_antisymm (hle w) (by simpa using hle (swap cs w))
+
+@[simp] theorem descent_swap (cs : CoxeterSystem M W) (w : W) (i : Bool) :
+    cs.IsLeftDescent (swap cs w) i ↔ cs.IsLeftDescent w (!i) := by
+  unfold CoxeterSystem.IsLeftDescent
+  rw [length_swap, ← Bool.not_not i, ← swap_simple, ← map_mul, length_swap]
+  simp
+
+/-- A nonzero length below an existing greater length has exactly one left
+simple descent. This is not asserted for a finite dihedral longest element. -/
+theorem unique_leftDescent (cs : CoxeterSystem M W) {w z : W}
+    (hw : 0 < cs.length w) (hz : cs.length w < cs.length z) :
+    ∃! i : Bool, cs.IsLeftDescent w i := by
+  obtain ⟨i,hi⟩ := cs.exists_leftDescent_of_ne_one (w := w) (by
+    intro he
+    rw [he,cs.length_one] at hw
+    omega)
+  refine ⟨i,hi,?_⟩
+  intro j hj
+  by_contra hji
+  have hj' : j = !i := by cases i <;> cases j <;> simp_all
+  have he : cs.wordProd (alt i (cs.length w)) =
+      cs.wordProd (alt (!i) (cs.length w)) := by
+    rw [← eq_alt_of_descent cs i hi, ← hj', ← eq_alt_of_descent cs j hj]
+  obtain ⟨a,b,hab,hab'⟩ := two_elements_length cs hw hz
+  have all_eq (v : W) (hv : cs.length v = cs.length w) : v=w := by
+    obtain ⟨k,hk⟩ := exists_alt cs v
+    rw [hv] at hk
+    have hki : k=i ∨ k=(!i) := by cases k <;> cases i <;> simp
+    rcases hki with hki | hki
+    · rw [hki] at hk
+      exact hk.trans (eq_alt_of_descent cs i hi).symm
+    · rw [hki] at hk
+      exact hk.trans (he.symm.trans (eq_alt_of_descent cs i hi).symm)
+  exact hab ((all_eq a ((hab' a).mpr (.inl rfl))).trans
+    (all_eq b ((hab' b).mpr (.inr rfl))).symm)
+
+/-- Internally, the diagram involution interchanges the two unique-descent
+vertices of each rank. -/
+theorem descent_swap_false_iff_not (cs : CoxeterSystem M W) {w z : W}
+    (hw : 0 < cs.length w) (hz : cs.length w < cs.length z) :
+    cs.IsLeftDescent (swap cs w) false ↔ ¬ cs.IsLeftDescent w false := by
+  rw [descent_swap]
+  obtain ⟨i,hi,hu⟩ := unique_leftDescent cs hw hz
+  cases i
+  · constructor
+    · intro h
+      have hh := hu true h
+      contradiction
+    · exact fun h => (h hi).elim
+  · constructor
+    · intro _ h
+      have hh := hu false h
+      contradiction
+    · exact fun _ => hi
+
+end
+end KLInvariance.Dihedral
+
+namespace KLInvariance.TitsSpace
+open Module
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+noncomputable section
+namespace PlaneBasis
+variable {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+
+/-- The ambient sign of a canonical plane root in a minimal coset is the
+intrinsic simple-descent sign, with the original left labels. -/
+theorem simpleRoot_inverted_coset_iff {a₀ : W}
+    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+    (w : planeSubgroup M cs P) (i : Bool) :
+    rootInverted ((w : W)*a₀) (B.simpleRoot i) ↔ B.system.IsLeftDescent w i := by
+  change (∀ k, action M cs ((w : W)*a₀)⁻¹ (B.simpleRoot i).val k ≤ 0) ↔ _
+  rw [root_inversion_coset_iff hmin w.property (B.simpleRoot i).property.1
+    (B.simpleRoot_mem i)]
+  have hr : Represents M cs (B.system.simple i : W) (B.simpleRoot i).val := by
+    rw [B.system_simple]
+    exact rootReflection_represents M cs _
+  rw [B.inversion_iff hr (B.simpleRoot_mem i) (B.simpleRoot i).property.2 w]
+  exact B.system.isLeftInversion_simple_iff_isLeftDescent w i
+
+/-- Geometric label of an intrinsic graph edge, taken in the actual ambient
+root plane rather than an abstract alternate representation. -/
+def edgeRoot {x y : planeSubgroup M cs P} (h : BruhatStep B.system x y) :
+    PositiveRoot M cs := graphRoot ((B.bruhatStep_iff x y).mpr h)
+
+theorem edgeRoot_plane {x y : planeSubgroup M cs P} (h : BruhatStep B.system x y) :
+    (B.edgeRoot h).val ∈ P := by
+  apply (Represents.planeSubgroup_mem_iff M cs (graphRoot_represents
+    ((B.bruhatStep_iff x y).mpr h))).mp
+  exact (planeSubgroup M cs P).mul_mem y.property
+    ((planeSubgroup M cs P).inv_mem x.property)
+
+/-- In the intrinsic system, the middle-vertex test uses its actual left
+simple generator; it therefore applies to any induced dihedral realization. -/
+theorem edgeRoot_slopes_iff {x y z : planeSubgroup M cs P}
+    (hxy : BruhatStep B.system x y) (hyz : BruhatStep B.system y z) :
+    geometricSlope (B.edgeRoot hxy) < geometricSlope (B.edgeRoot hyz) ↔
+      B.system.IsLeftDescent y false := by
+  unfold edgeRoot
+  rw [B.consecutive_slopes_iff ((B.bruhatStep_iff x y).mpr hxy)
+    ((B.bruhatStep_iff y z).mpr hyz) (B.edgeRoot_plane hxy) (B.edgeRoot_plane hyz)]
+  have hmin : ∀ z ∈ planeSubgroup M cs P, cs.length (1:W) ≤ cs.length (z*1) :=
+    fun z _ => by simp
+  simpa only [mul_one, simpleRoot, Bool.false_eq_true, ↓reduceIte] using
+    B.simpleRoot_inverted_coset_iff hmin y false
+
+end PlaneBasis
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Geometric orders adapted to a fixed upper vertex. These are honest
+ratios of linear functionals, with a strictly positive denominator. They
+will be used only as a permissible choice of the source's schedule. -/
+namespace KLInvariance.TitsSpace
+open Module
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+def simplePositive (i : I) : PositiveRoot M cs :=
+  ⟨root i, ⟨1,i,by simp⟩, root_nonneg i⟩
+
+@[simp] theorem simplePositive_val (i : I) :
+    (simplePositive (M := M) (cs := cs) i).val = root i := rfl
+
+@[simp] theorem rootRef_simplePositive (i : I) :
+    rootRef (simplePositive (M := M) (cs := cs) i) = cs.simple i := by
+  exact (rootReflection_represents M cs _).reflection_unique ⟨1,i,by simp,by simp⟩
+
+theorem simplePositive_injective : Function.Injective (simplePositive (M := M) (cs := cs)) := by
+  intro i j hij
+  have he := congrArg (fun a : PositiveRoot M cs => a.val i) hij
+  by_contra hne
+  simp [simplePositive, root, hne] at he
+
+theorem coordinate_sum (a : I → ℝ) : ∑ i, a i • root i = a := by
+  ext j
+  simp [root, Pi.single_apply, Finset.sum_apply]
+
+theorem dual_coordinate_sum (h : Module.Dual ℝ (I → ℝ)) (a : I → ℝ) :
+    h a = ∑ i, a i * h (root i) := by
+  conv_lhs => rw [← coordinate_sum a]
+  simp only [map_sum,map_smul,smul_eq_mul]
+
+def PositiveFunctional (h : Module.Dual ℝ (I → ℝ)) : Prop := ∀ i, 0 < h (root i)
+
+theorem PositiveFunctional.root_pos {h : Module.Dual ℝ (I → ℝ)}
+    (hh : PositiveFunctional h) (a : PositiveRoot M cs) : 0 < h a.val := by
+  rw [dual_coordinate_sum]
+  obtain ⟨i,hi⟩ : ∃ i, a.val i ≠ 0 := by
+    by_contra hn
+    push Not at hn
+    exact a.property.1.ne_zero (funext hn)
+  exact Finset.sum_pos' (fun j _ => mul_nonneg (a.property.2 j) (hh j).le)
+    ⟨i,Finset.mem_univ _,mul_pos (lt_of_le_of_ne' (a.property.2 i) hi) (hh i)⟩
+
+theorem height_positive : PositiveFunctional (height (I := I)) := by
+  intro i
+  simp [height,root]
+
+/-- Both functionals are actual dual vectors; the root order is not a
+label-carrying addition to the interval isomorphism. -/
+structure RatioOrder where
+  denominator : Module.Dual ℝ (I → ℝ)
+  numerator : Module.Dual ℝ (I → ℝ)
+  positive : PositiveFunctional denominator
+  generic : Function.Injective (fun a : PositiveRoot M cs => numerator a.val / denominator a.val)
+
+namespace RatioOrder
+variable (O : RatioOrder (M := M) (cs := cs))
+
+def slope (a : PositiveRoot M cs) : ℝ := O.numerator a.val / O.denominator a.val
+
+theorem denominator_pos (a : PositiveRoot M cs) : 0 < O.denominator a.val :=
+  O.positive.root_pos a
+
+theorem slope_injective : Function.Injective O.slope := O.generic
+
+/-- The cut at zero is the genuine inversion set of the upper vertex. -/
+def Adapted (y : W) : Prop := ∀ a : PositiveRoot M cs,
+  (O.slope a < 0 ↔ rootInverted y a) ∧ O.slope a ≠ 0
+
+theorem simple_lower_bound (i : I)
+    (hi : ∀ j, O.slope (simplePositive i) ≤ O.slope (simplePositive j))
+    (a : PositiveRoot M cs) : O.slope (simplePositive i) ≤ O.slope a := by
+  change O.slope (simplePositive i) ≤ O.numerator a.val / O.denominator a.val
+  rw [le_div_iff₀ (O.denominator_pos a)]
+  rw [dual_coordinate_sum O.numerator a.val, dual_coordinate_sum O.denominator a.val,
+    Finset.mul_sum]
+  apply Finset.sum_le_sum
+  intro j _
+  have hj := (le_div_iff₀ (O.positive j)).mp (hi j)
+  change O.slope (simplePositive i) * O.denominator (root j) ≤ O.numerator (root j) at hj
+  nlinarith [mul_nonneg (a.property.2 j) (sub_nonneg.mpr hj)]
+
+theorem exists_least_simple [Nonempty I] : ∃ i : I,
+    ∀ a : PositiveRoot M cs, O.slope (simplePositive i) ≤ O.slope a := by
+  obtain ⟨i,_,hi⟩ := Finset.exists_min_image (Finset.univ : Finset I)
+    (fun i => O.slope (simplePositive i)) Finset.univ_nonempty
+  exact ⟨i, O.simple_lower_bound i (fun j => hi j (Finset.mem_univ j))⟩
+
+theorem least_simple_descent {y : W} (hy : y ≠ 1) (ha : O.Adapted y) :
+    ∃ i : I, cs.IsLeftDescent y i ∧
+      O.slope (simplePositive i) < 0 ∧
+      ∀ a : PositiveRoot M cs, O.slope (simplePositive i) ≤ O.slope a := by
+  obtain ⟨j,hj⟩ := cs.exists_leftDescent_of_ne_one hy
+  let : Nonempty I := ⟨j⟩
+  obtain ⟨i,hi⟩ := O.exists_least_simple
+  have hjr : rootInverted y (simplePositive (M := M) (cs := cs) j) := by
+    apply ((rootReflection_represents M cs _).inversion_iff (simplePositive j).property.2 y).mpr
+    simpa only [rootRef_simplePositive] using (show cs.IsLeftInversion y (cs.simple j) from
+      ⟨cs.isReflection_simple j,hj⟩)
+  have hi0 := lt_of_le_of_lt (hi (simplePositive j)) ((ha _).1.mpr hjr)
+  refine ⟨i,?_,hi0,hi⟩
+  have hir := ((rootReflection_represents M cs _).inversion_iff
+    (simplePositive i).property.2 y).mp ((ha _).1.mp hi0)
+  simpa only [rootRef_simplePositive, CoxeterSystem.IsLeftDescent] using hir.2
+
+
+/-- A functional in the chamber indexed by y has exactly Inv(y) as its
+negative roots. This is stronger data used only to construct the order. -/
+def Chamber (y : W) : Prop := ∀ i, 0 < O.numerator (action M cs y (root i))
+
+theorem adapted_of_chamber {y : W} (hc : O.Chamber y) : O.Adapted y := by
+  let f : Module.Dual ℝ (I → ℝ) := O.numerator.comp (action M cs y).toLinearMap
+  have hf : PositiveFunctional f := hc
+  intro a
+  have he (v : I → ℝ) : f (action M cs y⁻¹ v) = O.numerator v := by
+    change O.numerator (action M cs y (action M cs y⁻¹ v)) = _
+    rw [← LinearEquiv.mul_apply, ← map_mul, mul_inv_cancel, map_one]
+    rfl
+  by_cases ha : rootInverted y a
+  · have hv := hf.root_pos (positiveAction y⁻¹ a)
+    rw [positiveAction_eq_of_nonpos _ _ ha, map_neg, he] at hv
+    have hn : O.slope a < 0 := div_neg_of_neg_of_pos (by linarith) (O.denominator_pos a)
+    exact ⟨iff_of_true hn ha, hn.ne⟩
+  · have hp := (not_rootInverted_iff y a).mp ha
+    have hv := hf.root_pos (positiveAction y⁻¹ a)
+    rw [positiveAction_eq_of_nonneg _ _ hp, he] at hv
+    have hn : 0 < O.slope a := div_pos hv (O.denominator_pos a)
+    exact ⟨iff_of_false (not_lt.mpr hn.le) ha, hn.ne'⟩
+
+end RatioOrder
+
+/-- The generic functional can be chosen in any prescribed Coxeter chamber,
+not just outside the countably many equality hyperplanes. The explicit
+polynomial perturbation makes both requirements simultaneous. -/
+theorem exists_ratioOrder_chamber_height (y : W) :
+    ∃ O : RatioOrder (M := M) (cs := cs), O.Chamber y ∧ O.denominator=height := by
+  classical
+  let := countable_Positiveroot M cs
+  let f₀ : Module.Dual ℝ (I → ℝ) := height.comp (action M cs y⁻¹).toLinearMap
+  let f : ℝ → Module.Dual ℝ (I → ℝ) := fun t =>
+    f₀ + t • ReflectionSchedules.momentFunctional t
+  have hf (t : ℝ) (v : I → ℝ) : f t v = f₀ v +
+      t * (ReflectionSchedules.coordinatePolynomial v).eval t := by
+    simp [f,ReflectionSchedules.coordinatePolynomial_eval]
+  let U : Set ℝ := {t | ∀ i, 0 < f t (action M cs y (root i))}
+  have hu : IsOpen U := by
+    change IsOpen {t | ∀ i, 0 < f t (action M cs y (root i))}
+    rw [Set.ofPred_forall]
+    apply isOpen_iInter_of_finite
+    intro i
+    apply isOpen_lt continuous_const
+    simp only [hf]
+    exact continuous_const.add (continuous_id.mul (Polynomial.continuous _))
+  have h0 : 0 ∈ U := by
+    intro i
+    simp only [f,zero_smul,add_zero]
+    change 0 < height (action M cs y⁻¹ (action M cs y (root i)))
+    rw [← LinearEquiv.mul_apply,← map_mul,inv_mul_cancel,map_one]
+    exact height_positive i
+  let P := {p : PositiveRoot M cs × PositiveRoot M cs // p.1 ≠ p.2}
+  let v : P → (I → ℝ) := fun p => normalizedRoot p.val.1 - normalizedRoot p.val.2
+  let d : P → Polynomial ℝ := fun p => Polynomial.C (f₀ (v p)) +
+    Polynomial.X * ReflectionSchedules.coordinatePolynomial (v p)
+  have hd : ∀ p, d p ≠ 0 := by
+    intro p hp
+    have hz := congrArg (fun q : Polynomial ℝ => q.eval 0) hp
+    have hz' : f₀ (v p) = 0 := by simpa [d] using hz
+    have hq : ReflectionSchedules.coordinatePolynomial (v p) = 0 := by
+      simpa only [d,hz',Polynomial.C_0,zero_add,mul_eq_zero,Polynomial.X_ne_zero,false_or] using hp
+    have hv : v p = 0 := ReflectionSchedules.coordinatePolynomial_injective (by simpa [ReflectionSchedules.coordinatePolynomial] using hq)
+    exact p.property (normalizedRoot_injective (sub_eq_zero.mp hv))
+  let bad : Set ℝ := ⋃ p : P, {t | (d p).IsRoot t}
+  have hbad : bad.Countable := Set.countable_iUnion
+    (fun p => (Polynomial.finite_setOfPred_isRoot (hd p)).countable)
+  obtain ⟨t,htU,ht⟩ := (hbad.dense_compl ℝ).inter_open_nonempty U hu ⟨0,h0⟩
+  have hinj : Function.Injective (fun a : PositiveRoot M cs => f t a.val / height a.val) := by
+    intro a b heq
+    by_contra hab
+    apply ht
+    apply Set.mem_iUnion.mpr
+    refine ⟨⟨(a,b),hab⟩,?_⟩
+    change (d ⟨(a,b),hab⟩).eval t = 0
+    dsimp only [d]
+    rw [Polynomial.eval_add,Polynomial.eval_C,Polynomial.eval_mul,Polynomial.eval_X,← hf]
+    change f t (normalizedRoot a - normalizedRoot b) = 0
+    rw [map_sub]
+    apply sub_eq_zero.mpr
+    simpa only [normalizedRoot,map_smul,smul_eq_mul,div_eq_mul_inv,mul_comm] using heq
+  exact ⟨⟨height,f t,height_positive,hinj⟩,htU,rfl⟩
+
+theorem exists_ratioOrder_chamber (y : W) :
+    ∃ O : RatioOrder (M := M) (cs := cs), O.Chamber y := by
+  obtain ⟨O,hO,_⟩ := exists_ratioOrder_chamber_height (M := M) (cs := cs) y
+  exact ⟨O,hO⟩
+
+@[simp] theorem positiveAction_simplePositive (i : I) :
+    positiveAction (cs.simple i) (simplePositive (M := M) (cs := cs) i) = simplePositive i := by
+  apply Subtype.ext
+  rw [positiveAction_eq_of_nonpos]
+  · simp only [simplePositive_val,action_simple_root,neg_neg]
+  · simpa only [simplePositive_val,action_simple_root,Pi.neg_apply,neg_nonpos] using root_nonneg i
+
+theorem simple_action_nonneg (i : I) (a : PositiveRoot M cs)
+    (ha : a ≠ simplePositive i) : ∀ k, 0 ≤ action M cs (cs.simple i) a.val k := by
+  apply a.property.1.act (cs.simple i) |>.sign.resolve_right
+  intro hn
+  exact ha (Subtype.ext (a.property.1.eq_simple_of_nonpos a.property.2 i hn))
+
+theorem simple_positiveAction_val (i : I) (a : PositiveRoot M cs)
+    (ha : a ≠ simplePositive i) :
+    (positiveAction (cs.simple i) a).val = action M cs (cs.simple i) a.val :=
+  positiveAction_eq_of_nonneg _ _ (simple_action_nonneg i a ha)
+
+theorem simple_positiveAction_ne (i : I) (a : PositiveRoot M cs)
+    (ha : a ≠ simplePositive i) : positiveAction (cs.simple i) a ≠ simplePositive i := by
+  intro he
+  apply ha
+  exact (positiveActionEquiv (cs.simple i)).injective (he.trans (positiveAction_simplePositive i).symm)
+
+theorem exists_uniform_pos_lt (v : I → ℝ) (hv : ∀ i, 0 < v i) :
+    ∃ ε : ℝ, 0 < ε ∧ ∀ i, ε < v i := by
+  have aux (s : Finset I) : ∃ ε : ℝ, 0 < ε ∧ ∀ i ∈ s, ε < v i := by
+    induction s using Finset.induction_on with
+    | empty => exact ⟨1,by norm_num,by simp⟩
+    | @insert a s ha ih =>
+      obtain ⟨ε,hε,hlt⟩ := ih
+      refine ⟨min ε (v a / 2),lt_min hε (half_pos (hv a)),?_⟩
+      intro i hi
+      rcases Finset.mem_insert.mp hi with rfl|hi
+      · exact lt_of_le_of_lt (min_le_right _ _) (half_lt_self (hv i))
+      · exact lt_of_le_of_lt (min_le_left _ _) (hlt i hi)
+  obtain ⟨ε,hε,hlt⟩ := aux Finset.univ
+  exact ⟨ε,hε,fun i => hlt i (Finset.mem_univ i)⟩
+
+namespace RatioOrder
+variable (O : RatioOrder (M := M) (cs := cs))
+
+def turnNumerator (i : I) : Module.Dual ℝ (I → ℝ) :=
+  O.numerator.comp (action M cs (cs.simple i)).toLinearMap
+
+def turnDenominator (i : I) (k : ℝ) : Module.Dual ℝ (I → ℝ) :=
+  (O.numerator - k • O.denominator).comp (action M cs (cs.simple i)).toLinearMap
+
+theorem exists_turn_parameter (i : I) (hi0 : O.slope (simplePositive i) < 0)
+    (hi : ∀ a : PositiveRoot M cs, O.slope (simplePositive i) ≤ O.slope a) :
+    ∃ k : ℝ, k < 0 ∧ O.slope (simplePositive i) < k ∧
+      PositiveFunctional (O.turnDenominator i k) := by
+  let c := O.slope (simplePositive i)
+  let v : I → ℝ := fun j => if j=i then -c else
+    O.slope (positiveAction (cs.simple i) (simplePositive j)) - c
+  have hv : ∀ j, 0 < v j := by
+    intro j
+    by_cases hj : j=i
+    · simp only [v,ite_eq_left hj]
+      exact neg_pos.mpr hi0
+    · simp only [v,ite_eq_right hj]
+      apply sub_pos.mpr
+      apply lt_of_le_of_ne (hi _)
+      intro he
+      exact simple_positiveAction_ne i (simplePositive j)
+        (fun hh => hj (simplePositive_injective hh)) (O.slope_injective he.symm)
+  obtain ⟨ε,hε,hεv⟩ := exists_uniform_pos_lt v hv
+  have hεc : ε < -c := by simpa only [v,ite_eq_left rfl] using hεv i
+  refine ⟨c+ε,by linarith,by dsimp [c]; linarith,?_⟩
+  intro j
+  change 0 < O.numerator (action M cs (cs.simple i) (root j)) -
+    (c+ε)*O.denominator (action M cs (cs.simple i) (root j))
+  by_cases hj : j=i
+  · subst j
+    rw [action_simple_root,map_neg,map_neg]
+    have hc : c * O.denominator (root i) = O.numerator (root i) := by
+      exact div_mul_cancel₀ _ (O.positive i).ne'
+    nlinarith [mul_pos hε (O.positive i)]
+  · let a := positiveAction (cs.simple i) (simplePositive (M := M) (cs := cs) j)
+    have ha : a.val = action M cs (cs.simple i) (root j) :=
+      simple_positiveAction_val i (simplePositive j) (fun hh => hj (simplePositive_injective hh))
+    rw [← ha]
+    have hvj : ε < O.slope a - c := by simpa only [v,ite_eq_right hj] using hεv j
+    have hlt : c+ε < O.slope a := by linarith
+    exact sub_pos.mpr ((lt_div_iff₀ (O.denominator_pos a)).mp hlt)
+
+theorem turn_ratio_injective (i : I) (k : ℝ) (hk : k ≠ 0)
+    (hp : PositiveFunctional (O.turnDenominator i k)) :
+    Function.Injective (fun a : PositiveRoot M cs =>
+      O.turnNumerator i a.val / O.turnDenominator i k a.val) := by
+  intro a b hab
+  have hh (a : PositiveRoot M cs) : O.denominator (action M cs (cs.simple i) a.val) ≠ 0 := by
+    rcases (a.property.1.act (cs.simple i)).sign with hpos|hneg
+    · have h := O.denominator_pos (positiveAction (cs.simple i) a)
+      rw [positiveAction_eq_of_nonneg _ _ hpos] at h
+      exact h.ne'
+    · have h := O.denominator_pos (positiveAction (cs.simple i) a)
+      rw [positiveAction_eq_of_nonpos _ _ hneg,map_neg] at h
+      exact (neg_pos.mp h).ne
+  have hc := (div_eq_div_iff (hp.root_pos a).ne' (hp.root_pos b).ne').mp hab
+  change O.numerator (action M cs (cs.simple i) a.val) *
+      (O.numerator (action M cs (cs.simple i) b.val) -
+        k*O.denominator (action M cs (cs.simple i) b.val)) =
+    O.numerator (action M cs (cs.simple i) b.val) *
+      (O.numerator (action M cs (cs.simple i) a.val) -
+        k*O.denominator (action M cs (cs.simple i) a.val)) at hc
+  have hcross : O.numerator (action M cs (cs.simple i) a.val) *
+      O.denominator (action M cs (cs.simple i) b.val) =
+    O.numerator (action M cs (cs.simple i) b.val) *
+      O.denominator (action M cs (cs.simple i) a.val) := by
+    apply sub_eq_zero.mp
+    apply (mul_eq_zero.mp (show k * (O.numerator (action M cs (cs.simple i) a.val) *
+      O.denominator (action M cs (cs.simple i) b.val) -
+        O.numerator (action M cs (cs.simple i) b.val) *
+      O.denominator (action M cs (cs.simple i) a.val)) = 0 by nlinarith [hc])).resolve_left hk
+  have he := (div_eq_div_iff (hh a) (hh b)).mpr hcross
+  have hval (a : PositiveRoot M cs) :
+      O.numerator (action M cs (cs.simple i) a.val) /
+          O.denominator (action M cs (cs.simple i) a.val) =
+        O.slope (positiveAction (cs.simple i) a) := by
+    unfold slope
+    rcases (a.property.1.act (cs.simple i)).sign with hpos|hneg
+    · rw [positiveAction_eq_of_nonneg _ _ hpos]
+    · rw [positiveAction_eq_of_nonpos _ _ hneg,map_neg,map_neg,neg_div_neg_eq]
+  rw [hval a,hval b] at he
+  exact (positiveActionEquiv (cs.simple i)).injective (O.slope_injective he)
+
+def turn (i : I) (k : ℝ) (hk : k ≠ 0)
+    (hp : PositiveFunctional (O.turnDenominator i k)) : RatioOrder (M := M) (cs := cs) :=
+  ⟨O.turnDenominator i k,O.turnNumerator i,hp,O.turn_ratio_injective i k hk hp⟩
+
+
+theorem turn_chamber (i : I) (k : ℝ) (hk : k ≠ 0)
+    (hp : PositiveFunctional (O.turnDenominator i k)) {y : W} (hy : O.Chamber y) :
+    (O.turn i k hk hp).Chamber (cs.simple i * y) := by
+  intro j
+  change 0 < O.numerator (action M cs (cs.simple i)
+    (action M cs (cs.simple i * y) (root j)))
+  rw [← LinearEquiv.mul_apply,← map_mul,cs.simple_mul_simple_cancel_left]
+  exact hy j
+
+theorem turn_simple_greatest (i : I) (k : ℝ) (hk : k < 0)
+    (hp : PositiveFunctional (O.turnDenominator i k)) (a : PositiveRoot M cs)
+    (ha : a ≠ simplePositive i) :
+    (O.turn i k hk.ne hp).slope a < (O.turn i k hk.ne hp).slope (simplePositive i) := by
+  have hlt : (O.turn i k hk.ne hp).slope a < 1 := by
+    apply (div_lt_iff₀ (hp.root_pos a)).mpr
+    change O.numerator (action M cs (cs.simple i) a.val) <
+      1 * (O.numerator (action M cs (cs.simple i) a.val) -
+        k * O.denominator (action M cs (cs.simple i) a.val))
+    have hap := O.denominator_pos (positiveAction (cs.simple i) a)
+    rw [simple_positiveAction_val i a ha] at hap
+    nlinarith [mul_neg_of_neg_of_pos hk hap]
+  have hgt : 1 < (O.turn i k hk.ne hp).slope (simplePositive i) := by
+    apply (lt_div_iff₀ (hp.root_pos (simplePositive i))).mpr
+    change 1 * (O.numerator (action M cs (cs.simple i) (root i)) -
+       k * O.denominator (action M cs (cs.simple i) (root i))) <
+      O.numerator (action M cs (cs.simple i) (root i))
+    rw [action_simple_root,map_neg,map_neg]
+    nlinarith [mul_neg_of_neg_of_pos hk (O.positive i)]
+  exact hlt.trans hgt
+
+theorem turn_slope_lt_iff (i : I) (k : ℝ) (hk : k < 0)
+    (hp : PositiveFunctional (O.turnDenominator i k))
+    (a b : PositiveRoot M cs) (ha : a ≠ simplePositive i) (hb : b ≠ simplePositive i) :
+    (O.turn i k hk.ne hp).slope (positiveAction (cs.simple i) a) <
+      (O.turn i k hk.ne hp).slope (positiveAction (cs.simple i) b) ↔ O.slope a < O.slope b := by
+  have hact (a : PositiveRoot M cs) (ha : a ≠ simplePositive i) :
+      action M cs (cs.simple i) (positiveAction (cs.simple i) a).val = a.val := by
+    rw [simple_positiveAction_val i a ha,← LinearEquiv.mul_apply,← map_mul,
+      (cs.isReflection_simple i).mul_self,map_one]
+    rfl
+  have haD := hp.root_pos (positiveAction (cs.simple i) a)
+  have hbD := hp.root_pos (positiveAction (cs.simple i) b)
+  change 0 < O.numerator (action M cs (cs.simple i) (positiveAction (cs.simple i) a).val) -
+    k * O.denominator (action M cs (cs.simple i) (positiveAction (cs.simple i) a).val) at haD
+  change 0 < O.numerator (action M cs (cs.simple i) (positiveAction (cs.simple i) b).val) -
+    k * O.denominator (action M cs (cs.simple i) (positiveAction (cs.simple i) b).val) at hbD
+  rw [hact a ha] at haD
+  rw [hact b hb] at hbD
+  change O.numerator (action M cs (cs.simple i) (positiveAction (cs.simple i) a).val) /
+    (O.numerator (action M cs (cs.simple i) (positiveAction (cs.simple i) a).val) -
+      k * O.denominator (action M cs (cs.simple i) (positiveAction (cs.simple i) a).val)) <
+    O.numerator (action M cs (cs.simple i) (positiveAction (cs.simple i) b).val) /
+    (O.numerator (action M cs (cs.simple i) (positiveAction (cs.simple i) b).val) -
+      k * O.denominator (action M cs (cs.simple i) (positiveAction (cs.simple i) b).val)) ↔
+    O.numerator a.val / O.denominator a.val < O.numerator b.val / O.denominator b.val
+  rw [hact a ha,hact b hb,div_lt_div_iff₀ haD hbD,
+    div_lt_div_iff₀ (O.denominator_pos a) (O.denominator_pos b)]
+  constructor
+  · intro h
+    by_contra hn
+    have := mul_nonpos_of_nonpos_of_nonneg hk.le (sub_nonneg.mpr (le_of_not_gt hn))
+    nlinarith
+  · intro h
+    have := mul_neg_of_neg_of_pos hk (sub_pos.mpr h)
+    nlinarith
+
+/-- The actual geometric rotation needed for the Hecke recursion. It moves
+the least simple root to last and preserves all other root comparisons. -/
+theorem exists_rotation {y : W} (hy : y ≠ 1) (hc : O.Chamber y) :
+    ∃ (i : I) (O' : RatioOrder (M := M) (cs := cs)),
+      cs.IsLeftDescent y i ∧ O'.Chamber (cs.simple i * y) ∧
+      (∀ a : PositiveRoot M cs, a ≠ simplePositive i →
+        O.slope (simplePositive i) < O.slope a ∧ O'.slope a < O'.slope (simplePositive i)) ∧
+      (∀ a b : PositiveRoot M cs, a ≠ simplePositive i → b ≠ simplePositive i →
+        (O'.slope (positiveAction (cs.simple i) a) < O'.slope (positiveAction (cs.simple i) b) ↔
+          O.slope a < O.slope b)) := by
+  obtain ⟨i,hi,hi0,hmin⟩ := O.least_simple_descent hy (O.adapted_of_chamber hc)
+  obtain ⟨k,hk,_,hpos⟩ := O.exists_turn_parameter i hi0 hmin
+  refine ⟨i,O.turn i k hk.ne hpos,hi,O.turn_chamber i k hk.ne hpos hc,?_,?_⟩
+  · intro a ha
+    exact ⟨lt_of_le_of_ne (hmin a) (fun he => ha (O.slope_injective he).symm),
+      O.turn_simple_greatest i k hk hpos a ha⟩
+  · exact O.turn_slope_lt_iff i k hk hpos
+
+end RatioOrder
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Rank-two path reversal in the actual induced root plane, for every finite
+full interval. This is a required Dyer wall-crossing relation, not the Dyer
+identification itself. -/
+namespace KLInvariance.Dihedral
+variable {W : Type*} [Group W] {M : CoxeterMatrix Bool}
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+/-- Switch the two vertices at every strictly internal rank, leaving all
+other ranks fixed. The diagram involution supplies the intrinsic switch. -/
+def rankFlip (cs : CoxeterSystem M W) (L H : ℕ) (w : W) : W :=
+  if L < cs.length w ∧ cs.length w < H then swap cs w else w
+
+@[simp] theorem length_rankFlip (cs : CoxeterSystem M W) (L H : ℕ) (w : W) :
+    cs.length (rankFlip cs L H w) = cs.length w := by
+  unfold rankFlip
+  split <;> simp
+
+@[simp] theorem rankFlip_involutive (cs : CoxeterSystem M W) (L H : ℕ) :
+    Function.Involutive (rankFlip cs L H) := by
+  intro w
+  unfold rankFlip
+  split <;> simp_all
+
+@[simp] theorem rankFlip_lower (cs : CoxeterSystem M W) (x y : W) :
+    rankFlip cs (cs.length x) (cs.length y) x = x := by simp [rankFlip]
+
+@[simp] theorem rankFlip_upper (cs : CoxeterSystem M W) (x y : W) :
+    rankFlip cs (cs.length x) (cs.length y) y = y := by simp [rankFlip]
+
+@[simp] theorem rankFlip_bruhat_iff (cs : CoxeterSystem M W) (L H : ℕ) (x y : W) :
+    BruhatLE cs (rankFlip cs L H x) (rankFlip cs L H y) ↔ BruhatLE cs x y := by
+  simp only [bruhat_iff_length_lt_or_eq, length_rankFlip,
+    (rankFlip_involutive cs L H).injective.eq_iff]
+
+@[simp] theorem rankFlip_step_iff (cs : CoxeterSystem M W) (L H : ℕ) (x y : W) :
+    BruhatStep cs (rankFlip cs L H x) (rankFlip cs L H y) ↔ BruhatStep cs x y := by
+  simp only [step_iff_odd, length_rankFlip]
+
+/-- An automorphism of the unlabelled full dihedral interval, fixing its ends.
+It need not arise from a Coxeter-system automorphism at those ends. -/
+def intervalFlip (cs : CoxeterSystem M W) (lo hi : W) :
+    Interval cs lo hi ≃o Interval cs lo hi where
+  toFun x := ⟨rankFlip cs (cs.length lo) (cs.length hi) x.val, by
+    constructor
+    · simpa only [rankFlip_lower] using
+        (rankFlip_bruhat_iff cs (cs.length lo) (cs.length hi) lo x.val).mpr x.property.1
+    · simpa only [rankFlip_upper] using
+        (rankFlip_bruhat_iff cs (cs.length lo) (cs.length hi) x.val hi).mpr x.property.2⟩
+  invFun x := ⟨rankFlip cs (cs.length lo) (cs.length hi) x.val, by
+    constructor
+    · simpa only [rankFlip_lower] using
+        (rankFlip_bruhat_iff cs (cs.length lo) (cs.length hi) lo x.val).mpr x.property.1
+    · simpa only [rankFlip_upper] using
+        (rankFlip_bruhat_iff cs (cs.length lo) (cs.length hi) x.val hi).mpr x.property.2⟩
+  left_inv x := Subtype.ext (rankFlip_involutive cs _ _ x.val)
+  right_inv x := Subtype.ext (rankFlip_involutive cs _ _ x.val)
+  map_rel_iff' := rankFlip_bruhat_iff cs _ _ _ _
+
+@[simp] theorem intervalFlip_val (cs : CoxeterSystem M W) (lo hi : W)
+    (x : Interval cs lo hi) :
+    (intervalFlip cs lo hi x).val = rankFlip cs (cs.length lo) (cs.length hi) x.val := rfl
+
+@[simp] theorem intervalFlip_involutive (cs : CoxeterSystem M W) (lo hi : W) :
+    Function.Involutive (intervalFlip cs lo hi) :=
+  fun x => Subtype.ext (rankFlip_involutive cs _ _ x.val)
+
+def edgeFlip (cs : CoxeterSystem M W) (lo hi : W) :
+    BruhatGraph.Edge cs lo hi ≃ BruhatGraph.Edge cs lo hi where
+  toFun e := ⟨(intervalFlip cs lo hi e.val.1, intervalFlip cs lo hi e.val.2),
+    (rankFlip_step_iff cs _ _ _ _).mpr e.property⟩
+  invFun e := ⟨(intervalFlip cs lo hi e.val.1, intervalFlip cs lo hi e.val.2),
+    (rankFlip_step_iff cs _ _ _ _).mpr e.property⟩
+  left_inv e := Subtype.ext (Prod.ext
+    (intervalFlip_involutive cs lo hi e.val.1)
+    (intervalFlip_involutive cs lo hi e.val.2))
+  right_inv e := Subtype.ext (Prod.ext
+    (intervalFlip_involutive cs lo hi e.val.1)
+    (intervalFlip_involutive cs lo hi e.val.2))
+
+@[simp] theorem edgeFlip_source (cs : CoxeterSystem M W) (lo hi : W)
+    (e : BruhatGraph.Edge cs lo hi) :
+    BruhatGraph.source cs lo hi (edgeFlip cs lo hi e) =
+      intervalFlip cs lo hi (BruhatGraph.source cs lo hi e) := rfl
+
+@[simp] theorem edgeFlip_target (cs : CoxeterSystem M W) (lo hi : W)
+    (e : BruhatGraph.Edge cs lo hi) :
+    BruhatGraph.target cs lo hi (edgeFlip cs lo hi e) =
+      intervalFlip cs lo hi (BruhatGraph.target cs lo hi e) := rfl
+
+@[simp] theorem edgeFlip_involutive (cs : CoxeterSystem M W) (lo hi : W) :
+    Function.Involutive (edgeFlip cs lo hi) := by
+  intro e
+  apply Subtype.ext
+  change (intervalFlip cs lo hi (intervalFlip cs lo hi e.val.1),
+    intervalFlip cs lo hi (intervalFlip cs lo hi e.val.2)) = e.val
+  exact Prod.ext (intervalFlip_involutive cs lo hi e.val.1)
+    (intervalFlip_involutive cs lo hi e.val.2)
+
+end
+end KLInvariance.Dihedral
+
+namespace KLInvariance.SchedulePaths
+variable {V E V' E' : Type*} (source target : E → V) (source' target' : E' → V')
+
+/-- Relabelling graph vertices and edges preserves the actual path predicate. -/
+theorem IsPath.map {x y : V} {p : List E}
+    (hp : IsPath source target x y p) (f : V → V') (g : E → E')
+    (hs : ∀ e, source' (g e) = f (source e))
+    (ht : ∀ e, target' (g e) = f (target e)) :
+    IsPath source' target' (f x) (f y) (p.map g) := by
+  induction p generalizing x with
+  | nil => exact congrArg f hp
+  | cons e p ih =>
+    exact ⟨(hs e).trans (congrArg f hp.1), ht e ▸ ih hp.2⟩
+
+/-- Every adjacent pair in a genuine path is composable. -/
+theorem IsPath.isChain {x y : V} {p : List E} (hp : IsPath source target x y p) :
+    p.IsChain (fun e f => target e = source f) := by
+  induction p generalizing x with
+  | nil => exact .nil
+  | cons e p ih =>
+    cases p with
+    | nil => exact .singleton e
+    | cons f p =>
+      apply List.isChain_cons_cons.mpr
+      exact ⟨hp.2.1.symm, ih hp.2⟩
+
+/-- A path's adjacent comparisons suffice for a transitive label order. The
+statement allows the comparison to change under an edge map, provided the
+local relation is checked only on composable edges. -/
+theorem IsPath.pairwise_map_of_local {Λ Λ' : Type*} [LinearOrder Λ] [LinearOrder Λ']
+    {x y : V} {p : List E} (hp : IsPath source target x y p)
+    (label : E → Λ) (label' : E' → Λ') (g : E → E')
+    (hlocal : ∀ e f, target e = source f →
+      label e < label f → label' (g e) < label' (g f))
+    (horder : p.Pairwise (fun e f => label e < label f)) :
+    (p.map g).Pairwise (fun e f => label' e < label' f) := by
+  rw [← List.isChain_iff_pairwise] at horder ⊢
+  rw [List.isChain_map]
+  induction p generalizing x with
+  | nil => exact .nil
+  | cons e p ih =>
+    cases p with
+    | nil => exact .singleton e
+    | cons f p =>
+      rw [List.isChain_cons_cons] at horder ⊢
+      exact ⟨hlocal e f hp.2.1.symm horder.1, ih hp.2 horder.2⟩
+
+end KLInvariance.SchedulePaths
+
+namespace KLInvariance.TitsSpace
+open Module
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+noncomputable section
+namespace PlaneBasis
+variable {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+open Dihedral BruhatGraph
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+theorem edgeRoot_ne_of_path {x y z : planeSubgroup M cs P}
+    (hxy : BruhatStep B.system x y) (hyz : BruhatStep B.system y z) :
+    B.edgeRoot hxy ≠ B.edgeRoot hyz :=
+  graphRoot_ne_of_path ((B.bruhatStep_iff x y).mpr hxy) ((B.bruhatStep_iff y z).mpr hyz)
+
+/-- At every internal vertex, the cut switches side under the path involution.
+This is the local relation behind rank-two path-product reversal. -/
+theorem edgeRoot_slopes_rankFlip {x y z top : planeSubgroup M cs P} {L : ℕ}
+    (hxy : BruhatStep B.system x y) (hyz : BruhatStep B.system y z)
+    (hL : L < B.system.length y) (hH : B.system.length y < B.system.length top) :
+    let hx := (rankFlip_step_iff B.system L (B.system.length top) x y).mpr hxy
+    let hz := (rankFlip_step_iff B.system L (B.system.length top) y z).mpr hyz
+    geometricSlope (B.edgeRoot hxy) < geometricSlope (B.edgeRoot hyz) ↔
+      geometricSlope (B.edgeRoot hz) < geometricSlope (B.edgeRoot hx) := by
+  dsimp only
+  let hx := (rankFlip_step_iff B.system L (B.system.length top) x y).mpr hxy
+  let hz := (rankFlip_step_iff B.system L (B.system.length top) y z).mpr hyz
+  have hsign : B.system.IsLeftDescent
+      (rankFlip B.system L (B.system.length top) y) false ↔
+      ¬ B.system.IsLeftDescent y false := by
+    rw [rankFlip,ite_eq_left ⟨hL,hH⟩]
+    exact descent_swap_false_iff_not B.system (by omega) hH
+  have hne : geometricSlope (B.edgeRoot hx) ≠ geometricSlope (B.edgeRoot hz) :=
+    fun he => B.edgeRoot_ne_of_path hx hz (geometricSlope_injective he)
+  rw [B.edgeRoot_slopes_iff]
+  constructor
+  · intro hi
+    have hn : ¬ geometricSlope (B.edgeRoot hx) < geometricSlope (B.edgeRoot hz) := by
+      rw [B.edgeRoot_slopes_iff,hsign]
+      exact not_not.mpr hi
+    exact lt_of_le_of_ne (le_of_not_gt hn) hne.symm
+  · intro hi
+    by_contra hn
+    have hlt := (B.edgeRoot_slopes_iff hx hz).mpr (hsign.mpr hn)
+    exact lt_asymm hlt hi
+
+/-- Restriction of the actual geometric root slope to a full plane interval. -/
+def intervalEdgeSlope (lo hi : planeSubgroup M cs P) (e : Edge B.system lo hi) : ℝ :=
+  geometricSlope (B.edgeRoot e.property)
+
+/-- The global edge map reverses the comparison for every composable pair.
+No statement is asserted for disjoint edge labels, which can share ties. -/
+theorem intervalEdgeSlope_flip {lo hi : planeSubgroup M cs P}
+    (e f : Edge B.system lo hi)
+    (hjoin : target B.system lo hi e = source B.system lo hi f) :
+    B.intervalEdgeSlope lo hi e < B.intervalEdgeSlope lo hi f ↔
+      B.intervalEdgeSlope lo hi (edgeFlip B.system lo hi f) <
+        B.intervalEdgeSlope lo hi (edgeFlip B.system lo hi e) := by
+  rcases e with ⟨⟨ex,ey⟩,he⟩
+  rcases f with ⟨⟨fx,fy⟩,hf⟩
+  change ey = fx at hjoin
+  subst fx
+  have hl : B.system.length lo < B.system.length ey.val :=
+    lt_of_le_of_lt (length_le_of_bruhat B.system ex.property.1) he.1
+  have hh : B.system.length ey.val < B.system.length hi :=
+    lt_of_lt_of_le hf.1 (length_le_of_bruhat B.system fy.property.2)
+  exact B.edgeRoot_slopes_rankFlip he hf hl hh
+
+end PlaneBasis
+end
+end KLInvariance.TitsSpace
+
+namespace KLInvariance.SchedulePaths
+open scoped BigOperators
+variable {V E R : Type*} (source target : E → V)
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+/-- The finite set of genuine directed paths selected by an edge schedule. -/
+def subpaths (schedule : List E) (x y : V) : Finset (List E) :=
+  schedule.sublists'.toFinset.filter (IsPath source target x y)
+
+@[simp] theorem mem_subpaths (schedule : List E) (x y : V) (p : List E) :
+    p ∈ subpaths source target schedule x y ↔
+      p.Sublist schedule ∧ IsPath source target x y p := by
+  simp [subpaths, List.mem_sublists']
+
+variable [Semiring R]
+
+theorem dispatch_eq_subpaths_sum (T : R) (schedule : List E) (hnd : schedule.Nodup)
+    (x y : V) :
+    dispatch source target T (fun z => if z=y then 1 else 0) schedule x =
+      ∑ p ∈ subpaths source target schedule x y, T ^ p.length := by
+  rw [entry_eq_weighted_paths, subpaths, Finset.sum_filter,
+    List.sum_toFinset _ (List.nodup_sublists'.mpr hnd)]
+
+/-- A length-preserving involution on the actual schedule-path sets proves
+an operator-entry relation, with no Hecke polynomial used in the assumptions. -/
+theorem dispatch_eq_of_path_involution (T : R) (schedule schedule' : List E)
+    (hnd : schedule.Nodup) (hnd' : schedule'.Nodup) (x y : V) (g : E → E)
+    (hg : Function.Involutive g)
+    (hforward : ∀ p ∈ subpaths source target schedule x y,
+      p.map g ∈ subpaths source target schedule' x y)
+    (hbackward : ∀ p ∈ subpaths source target schedule' x y,
+      p.map g ∈ subpaths source target schedule x y) :
+    dispatch source target T (fun z => if z=y then 1 else 0) schedule x =
+      dispatch source target T (fun z => if z=y then 1 else 0) schedule' x := by
+  rw [dispatch_eq_subpaths_sum source target T schedule hnd,
+    dispatch_eq_subpaths_sum source target T schedule' hnd']
+  exact Finset.sum_nbij' (List.map g) (List.map g) hforward hbackward
+    (fun p _ => List.map_involutive_iff.mpr hg p)
+    (fun p _ => List.map_involutive_iff.mpr hg p) (fun p _ => by simp)
+
+end
+end KLInvariance.SchedulePaths
+
+namespace KLInvariance.SchedulePaths
+variable {V E : Type*} (source target : E → V)
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+/-- Sorted full schedules select precisely their strictly monotone paths;
+an actual graph map satisfying the adjacent comparison test transports them. -/
+theorem subpaths_map_of_sorted (label label' : E → ℝ)
+    (schedule schedule' : List E) (x y : V)
+    (hs : schedule.Pairwise (fun e f => label f ≤ label e))
+    (hs' : schedule'.Pairwise (fun e f => label' f ≤ label' e))
+    (hfull' : ∀ e, e ∈ schedule')
+    (hties : ∀ e f, label e = label f → target e ≠ source f)
+    (f : V → V) (g : E → E)
+    (hsource : ∀ e, source (g e) = f (source e))
+    (htarget : ∀ e, target (g e) = f (target e))
+    (hx : f x=x) (hy : f y=y)
+    (hlocal : ∀ e f, target e=source f → label f<label e → label' (g f)<label' (g e))
+    (p : List E) (hp : p ∈ subpaths source target schedule x y) :
+    p.map g ∈ subpaths source target schedule' x y := by
+  obtain ⟨hsub,hp⟩ := (mem_subpaths source target schedule x y p).mp hp
+  have hstrict := decreasing_of_schedule_path source target label hties hp (hs.sublist hsub)
+  have horder := hp.pairwise_map_of_local source target
+    (Λ := OrderDual ℝ) (Λ' := OrderDual ℝ)
+    (fun e => (label e : OrderDual ℝ)) (fun e => (label' e : OrderDual ℝ)) g hlocal hstrict
+  have hp' := hp.map source target source target f g hsource htarget
+  rw [hx,hy] at hp'
+  exact (mem_subpaths source target schedule' x y (p.map g)).mpr
+    ⟨sublist_of_decreasing_subset label' horder hs' (fun _ _ => hfull' _), hp'⟩
+
+end
+end KLInvariance.SchedulePaths
+
+namespace KLInvariance.TitsSpace
+open Module
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+noncomputable section
+namespace PlaneBasis
+variable {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+open Dihedral BruhatGraph SchedulePaths
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+theorem intervalEdgeSlope_ties {lo hi : planeSubgroup M cs P}
+    (e f : Edge B.system lo hi) (heq : B.intervalEdgeSlope lo hi e = B.intervalEdgeSlope lo hi f) :
+    target B.system lo hi e ≠ source B.system lo hi f := by
+  intro hjoin
+  rcases e with ⟨⟨ex,ey⟩,he⟩
+  rcases f with ⟨⟨fx,fy⟩,hf⟩
+  change ey = fx at hjoin
+  subst fx
+  exact B.edgeRoot_ne_of_path he hf (geometricSlope_injective heq)
+
+theorem intervalEdgeSlope_flip_reverse {lo hi : planeSubgroup M cs P}
+    (e f : Edge B.system lo hi)
+    (hjoin : target B.system lo hi e = source B.system lo hi f) :
+    B.intervalEdgeSlope lo hi f < B.intervalEdgeSlope lo hi e ↔
+      B.intervalEdgeSlope lo hi (edgeFlip B.system lo hi e) <
+        B.intervalEdgeSlope lo hi (edgeFlip B.system lo hi f) := by
+  have hj : target B.system lo hi (edgeFlip B.system lo hi e) =
+      source B.system lo hi (edgeFlip B.system lo hi f) := by
+    simpa only [edgeFlip_source,edgeFlip_target] using congrArg (intervalFlip B.system lo hi) hjoin
+  simpa only [edgeFlip_involutive B.system lo hi e, edgeFlip_involutive B.system lo hi f] using
+    (B.intervalEdgeSlope_flip (edgeFlip B.system lo hi e) (edgeFlip B.system lo hi f) hj).symm
+
+/-- Rank-two reversal of the genuine path-product entry between the two
+ends of any full interval. The order is the actual induced angular order. -/
+theorem dispatch_reverse_endpoints {R : Type*} [Semiring R]
+    {lo hi : planeSubgroup M cs P} (hlohi : BruhatLE B.system lo hi)
+    (schedule : List (Edge B.system lo hi)) (hnd : schedule.Nodup)
+    (hfull : ∀ e, e ∈ schedule)
+    (hs : schedule.Pairwise (fun e f => B.intervalEdgeSlope lo hi f ≤ B.intervalEdgeSlope lo hi e))
+    (T : R) :
+    let x : Interval B.system lo hi := ⟨lo,bruhat_refl B.system lo,hlohi⟩
+    let y : Interval B.system lo hi := ⟨hi,hlohi,bruhat_refl B.system hi⟩
+    dispatch (source B.system lo hi) (target B.system lo hi) T
+      (fun z => if z=y then 1 else 0) schedule x =
+    dispatch (source B.system lo hi) (target B.system lo hi) T
+      (fun z => if z=y then 1 else 0) schedule.reverse x := by
+  dsimp only
+  let x : Interval B.system lo hi := ⟨lo,bruhat_refl B.system lo,hlohi⟩
+  let y : Interval B.system lo hi := ⟨hi,hlohi,bruhat_refl B.system hi⟩
+  have hx : intervalFlip B.system lo hi x=x := Subtype.ext (rankFlip_lower B.system lo hi)
+  have hy : intervalFlip B.system lo hi y=y := Subtype.ext (rankFlip_upper B.system lo hi)
+  have hrs : schedule.reverse.Pairwise
+      (fun e f => -B.intervalEdgeSlope lo hi f ≤ -B.intervalEdgeSlope lo hi e) := by
+    rw [List.pairwise_reverse]
+    exact hs.imp (fun h => neg_le_neg h)
+  have hrfull : ∀ e, e ∈ schedule.reverse := fun e => List.mem_reverse.mpr (hfull e)
+  apply dispatch_eq_of_path_involution _ _ T schedule schedule.reverse hnd (List.nodup_reverse.mpr hnd) x y
+    (edgeFlip B.system lo hi) (edgeFlip_involutive B.system lo hi)
+  · apply subpaths_map_of_sorted _ _ (B.intervalEdgeSlope lo hi)
+      (fun e => -B.intervalEdgeSlope lo hi e) schedule schedule.reverse x y hs hrs hrfull
+      (B.intervalEdgeSlope_ties) (intervalFlip B.system lo hi) (edgeFlip B.system lo hi)
+      (edgeFlip_source B.system lo hi) (edgeFlip_target B.system lo hi) hx hy
+    intro e f hj hlt
+    exact neg_lt_neg ((B.intervalEdgeSlope_flip_reverse e f hj).mp hlt)
+  · apply subpaths_map_of_sorted _ _ (fun e => -B.intervalEdgeSlope lo hi e)
+      (B.intervalEdgeSlope lo hi) schedule.reverse schedule x y hrs hs hfull
+      (fun e f he => B.intervalEdgeSlope_ties e f (neg_injective he))
+      (intervalFlip B.system lo hi) (edgeFlip B.system lo hi)
+      (edgeFlip_source B.system lo hi) (edgeFlip_target B.system lo hi) hx hy
+    intro e f hj hlt
+    exact (B.intervalEdgeSlope_flip e f hj).mp (neg_lt_neg_iff.mp hlt)
+
+end PlaneBasis
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Genuine Dyer identification for geometric orders adapted to a fixed top.
+The path values below are defined from actual reflection edges; equality with
+`Hecke.rTilde` will be proved by its genuine simple-descent recurrence. -/
+namespace KLInvariance.AdaptedDyer
+open TitsSpace SchedulePaths
+open scoped BigOperators
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+abbrev Edge := {e : W × W // BruhatStep cs e.1 e.2}
+def source (e : Edge cs) : W := e.val.1
+def target (e : Edge cs) : W := e.val.2
+def rootLabel (e : Edge cs) : PositiveRoot M cs := graphRoot e.property
+
+@[simp] theorem rootRef_rootLabel (e : Edge cs) :
+    rootRef (rootLabel cs e) = target cs e * (source cs e)⁻¹ := by
+  exact congrArg Subtype.val ((positiveRootEquiv M cs).left_inv
+    ⟨e.val.2 * e.val.1⁻¹,e.property.label_reflection⟩)
+
+theorem target_eq_root (e : Edge cs) :
+    target cs e = rootRef (rootLabel cs e) * source cs e := by
+  rw [rootRef_rootLabel]
+  group
+
+theorem rootLabel_ne_of_join (e f : Edge cs) (h : target cs e = source cs f) :
+    rootLabel cs e ≠ rootLabel cs f := by
+  rcases e with ⟨⟨x,y⟩,he⟩
+  rcases f with ⟨⟨z,w⟩,hf⟩
+  change y=z at h
+  subst z
+  exact graphRoot_ne_of_path he hf
+
+variable {cs}
+def score (O : RatioOrder (M := M) (cs := cs)) (e : Edge cs) : ℝ :=
+  O.slope (rootLabel cs e)
+
+theorem score_ties (O : RatioOrder (M := M) (cs := cs)) (e f : Edge cs)
+    (h : score O e = score O f) : target cs e ≠ source cs f :=
+  fun hj => rootLabel_ne_of_join cs e f hj (O.slope_injective h)
+
+variable (cs)
+def edgeFinset (b : W) : Finset (Edge cs) :=
+  let P := ((lowerFinset cs b).product (lowerFinset cs b)).filter
+    (fun e => BruhatStep cs e.1 e.2)
+  P.attach.image (fun e => (⟨e.val,by
+    exact (Finset.mem_filter.mp e.property).2⟩ : Edge cs))
+
+omit [Fintype I] in
+@[simp] theorem mem_edgeFinset (b : W) (e : Edge cs) :
+    e ∈ edgeFinset cs b ↔ BruhatLE cs (target cs e) b := by
+  unfold edgeFinset
+  constructor
+  · intro h
+    obtain ⟨p,_,he⟩ := Finset.mem_image.mp h
+    subst e
+    exact (mem_lowerFinset cs b p.val.2).mp
+      (Finset.mem_product.mp (Finset.mem_filter.mp p.property).1).2
+  · intro h
+    apply Finset.mem_image.mpr
+    have hx : BruhatLE cs (source cs e) b := (show BruhatLE cs (source cs e) (target cs e) from .single e.property).trans h
+    refine ⟨⟨e.val,Finset.mem_filter.mpr ⟨Finset.mem_product.mpr
+      ⟨(mem_lowerFinset cs b _).mpr hx,(mem_lowerFinset cs b _).mpr h⟩,e.property⟩⟩,
+      Finset.mem_attach _ _,?_⟩
+    rfl
+
+variable {cs}
+def schedule (O : RatioOrder (M := M) (cs := cs)) (b : W) : List (Edge cs) :=
+  (edgeFinset cs b).toList.mergeSort (fun e f => score O e ≤ score O f)
+
+@[simp] theorem mem_schedule (O : RatioOrder (M := M) (cs := cs)) (b : W) (e : Edge cs) :
+    e ∈ schedule O b ↔ BruhatLE cs (target cs e) b := by
+  simp [schedule]
+
+theorem schedule_nodup (O : RatioOrder (M := M) (cs := cs)) (b : W) :
+    (schedule O b).Nodup :=
+  (List.mergeSort_perm _ _).nodup_iff.mpr (Finset.nodup_toList _)
+
+theorem schedule_increasing (O : RatioOrder (M := M) (cs := cs)) (b : W) :
+    (schedule O b).Pairwise (fun e f => score O e ≤ score O f) := by
+  unfold schedule
+  apply List.pairwise_mergeSort'
+
+variable (cs)
+omit [Fintype I] in
+theorem path_le {x y : W} {p : List (Edge cs)}
+    (hp : IsPath (source cs) (target cs) x y p) : BruhatLE cs x y := by
+  induction p generalizing x with
+  | nil => exact hp ▸ .refl
+  | cons e p ih =>
+    exact hp.1 ▸ (show BruhatLE cs (source cs e) (target cs e) from .single e.property).trans (ih hp.2)
+
+omit [Fintype I] in
+theorem path_targets_le {x y : W} {p : List (Edge cs)}
+    (hp : IsPath (source cs) (target cs) x y p) :
+    ∀ e ∈ p, BruhatLE cs (target cs e) y := by
+  induction p generalizing x with
+  | nil => simp
+  | cons e p ih =>
+    intro f hf
+    rcases List.mem_cons.mp hf with rfl | hf
+    · exact path_le cs hp.2
+    · exact ih hp.2 f hf
+
+variable {cs}
+def paths (O : RatioOrder (M := M) (cs := cs)) (x y : W) : Finset (List (Edge cs)) :=
+  subpaths (source cs) (target cs) (schedule O y) x y
+
+@[simp] theorem mem_paths (O : RatioOrder (M := M) (cs := cs)) (x y : W) (p : List (Edge cs)) :
+    p ∈ paths O x y ↔ IsPath (source cs) (target cs) x y p ∧
+      p.Pairwise (fun e f => score O e < score O f) := by
+  rw [paths,mem_subpaths]
+  constructor
+  · rintro ⟨hsub,hp⟩
+    refine ⟨hp,?_⟩
+    exact decreasing_of_schedule_path (source cs) (target cs)
+      (Λ := OrderDual ℝ) (fun e => (score O e : OrderDual ℝ))
+      (fun e f h => score_ties O e f h) hp ((schedule_increasing O y).sublist hsub)
+  · rintro ⟨hp,hord⟩
+    refine ⟨?_,hp⟩
+    apply sublist_of_decreasing_subset (Λ := OrderDual ℝ)
+      (fun e => (score O e : OrderDual ℝ)) hord (schedule_increasing O y)
+    intro e he
+    exact (mem_schedule O y e).mpr (path_targets_le cs hp e he)
+
+def polynomial (O : RatioOrder (M := M) (cs := cs)) (x y : W) : Polynomial ℤ :=
+  ∑ p ∈ paths O x y, Polynomial.X ^ p.length
+
+theorem polynomial_dispatch (O : RatioOrder (M := M) (cs := cs)) (x y : W) :
+    polynomial O x y = dispatch (source cs) (target cs) Polynomial.X
+      (fun z => if z=y then 1 else 0) (schedule O y) x :=
+  (dispatch_eq_subpaths_sum _ _ _ _ (schedule_nodup O y) x y).symm
+
+
+variable (cs)
+theorem rootRef_injective : Function.Injective (rootRef (M := M) (cs := cs)) := by
+  intro a b h
+  apply (positiveRootEquiv M cs).symm.injective
+  exact Subtype.ext h
+
+theorem rootLabel_simple_iff (i : I) (e : Edge cs) :
+    rootLabel cs e = simplePositive i ↔ target cs e = cs.simple i * source cs e := by
+  constructor
+  · intro h
+    rw [target_eq_root cs e,h,rootRef_simplePositive]
+  · intro h
+    apply rootRef_injective cs
+    rw [rootRef_rootLabel,rootRef_simplePositive,h]
+    group
+
+/-- On a non-simple-labelled edge this is actual left multiplication by s.
+Simple-labelled edges are fixed merely to make the extension a total involution;
+all path uses below exclude them before applying this map. -/
+def rotateEdge (i : I) (e : Edge cs) : Edge cs :=
+  if h : target cs e = cs.simple i * source cs e then e else
+    ⟨(cs.simple i * source cs e,cs.simple i * target cs e),
+      bruhatStep_simple_mul cs e.property i h⟩
+
+theorem rotateEdge_val (i : I) (e : Edge cs) (h : rootLabel cs e ≠ simplePositive i) :
+    (rotateEdge cs i e).val = (cs.simple i * source cs e,cs.simple i * target cs e) := by
+  rw [rotateEdge,dite_eq_right (fun hh => h ((rootLabel_simple_iff cs i e).mpr hh))]
+
+theorem rotateEdge_root (i : I) (e : Edge cs) (h : rootLabel cs e ≠ simplePositive i) :
+    rootLabel cs (rotateEdge cs i e) = positiveAction (cs.simple i) (rootLabel cs e) := by
+  apply rootRef_injective cs
+  rw [rootRef_rootLabel]
+  change (rotateEdge cs i e).val.2 * (rotateEdge cs i e).val.1⁻¹ =
+    rootRef (positiveAction (cs.simple i) (rootLabel cs e))
+  have hc : rootRef (positiveAction (cs.simple i) (rootLabel cs e)) =
+      cs.simple i * rootRef (rootLabel cs e) * (cs.simple i)⁻¹ :=
+    rootReflection_positiveAction _ _
+  rw [rotateEdge_val cs i e h,hc,rootRef_rootLabel]
+  group
+
+theorem rotateEdge_nonsimple (i : I) (e : Edge cs) (h : rootLabel cs e ≠ simplePositive i) :
+    rootLabel cs (rotateEdge cs i e) ≠ simplePositive i := by
+  rw [rotateEdge_root cs i e h]
+  exact simple_positiveAction_ne i _ h
+
+@[simp] theorem rotateEdge_involutive (i : I) : Function.Involutive (rotateEdge cs i) := by
+  intro e
+  by_cases h : rootLabel cs e = simplePositive i
+  · have he : rotateEdge cs i e = e := by
+      rw [rotateEdge,dite_eq_left ((rootLabel_simple_iff cs i e).mp h)]
+    rw [he,he]
+  · apply Subtype.ext
+    rw [rotateEdge_val cs i _ (rotateEdge_nonsimple cs i e h)]
+    change (cs.simple i * (rotateEdge cs i e).val.1,
+      cs.simple i * (rotateEdge cs i e).val.2) = e.val
+    rw [rotateEdge_val cs i e h]
+    simp only [cs.simple_mul_simple_cancel_left]
+    rfl
+
+omit [Fintype I] in
+theorem path_map_of_mem {E E' V V' : Type*} (src tgt : E → V) (src' tgt' : E' → V')
+    {p : List E} {x y : V} (hp : IsPath src tgt x y p) (f : V → V') (g : E → E')
+    (hs : ∀ e ∈ p, src' (g e) = f (src e)) (ht : ∀ e ∈ p, tgt' (g e) = f (tgt e)) :
+    IsPath src' tgt' (f x) (f y) (p.map g) := by
+  induction p generalizing x with
+  | nil => exact congrArg f hp
+  | cons e p ih =>
+    refine ⟨(hs e List.mem_cons_self).trans (congrArg f hp.1),?_⟩
+    rw [ht e List.mem_cons_self]
+    exact ih hp.2 (fun a ha => hs a (List.mem_cons_of_mem e ha))
+      (fun a ha => ht a (List.mem_cons_of_mem e ha))
+
+theorem path_rotate {x y : W} {p : List (Edge cs)} (i : I)
+    (hp : IsPath (source cs) (target cs) x y p)
+    (hn : ∀ e ∈ p, rootLabel cs e ≠ simplePositive i) :
+    IsPath (source cs) (target cs) (cs.simple i*x) (cs.simple i*y) (p.map (rotateEdge cs i)) := by
+  apply path_map_of_mem (source cs) (target cs) (source cs) (target cs)
+    hp (fun z => cs.simple i*z) (rotateEdge cs i)
+  · intro e he
+    exact congrArg Prod.fst (rotateEdge_val cs i e (hn e he))
+  · intro e he
+    exact congrArg Prod.snd (rotateEdge_val cs i e (hn e he))
+
+
+@[simp] theorem rotateEdge_twice (i : I) (e : Edge cs) :
+    rotateEdge cs i (rotateEdge cs i e) = e := rotateEdge_involutive cs i e
+
+variable {cs}
+def Nonsimple (i : I) (p : List (Edge cs)) : Prop :=
+  ∀ e ∈ p, rootLabel cs e ≠ simplePositive i
+
+@[simp] theorem nonsimple_map (i : I) {p : List (Edge cs)} (hn : Nonsimple i p) :
+    Nonsimple i (p.map (rotateEdge cs i)) := by
+  intro e he
+  obtain ⟨f,hf,rfl⟩ := List.mem_map.mp he
+  exact rotateEdge_nonsimple cs i f (hn f hf)
+
+@[simp] theorem path_map_twice (i : I) (p : List (Edge cs)) :
+    (p.map (rotateEdge cs i)).map (rotateEdge cs i) = p := by
+  simp only [List.map_map,Function.comp_def,rotateEdge_twice]
+  exact List.map_id p
+
+theorem path_rotate_iff (i : I) {x y : W} {p : List (Edge cs)} (hn : Nonsimple i p) :
+    IsPath (source cs) (target cs) (cs.simple i*x) (cs.simple i*y)
+      (p.map (rotateEdge cs i)) ↔ IsPath (source cs) (target cs) x y p := by
+  constructor
+  · intro hp
+    simpa only [cs.simple_mul_simple_cancel_left,path_map_twice] using
+      path_rotate cs i hp (nonsimple_map i hn)
+  · intro hp
+    exact path_rotate cs i hp hn
+
+theorem paths_rotate_iff (O O' : RatioOrder (M := M) (cs := cs)) (i : I)
+    (hcmp : ∀ a b : PositiveRoot M cs, a ≠ simplePositive i → b ≠ simplePositive i →
+      (O'.slope (positiveAction (cs.simple i) a) < O'.slope (positiveAction (cs.simple i) b) ↔
+        O.slope a < O.slope b))
+    {x y : W} {p : List (Edge cs)} (hn : Nonsimple i p) :
+    p.map (rotateEdge cs i) ∈ paths O' (cs.simple i*x) (cs.simple i*y) ↔ p ∈ paths O x y := by
+  rw [mem_paths,mem_paths,path_rotate_iff i hn,List.pairwise_map]
+  apply and_congr_right
+  intro _
+  apply List.Pairwise.iff_of_mem
+  intro e f he hf
+  simpa only [score,rotateEdge_root cs i e (hn e he),rotateEdge_root cs i f (hn f hf)] using
+    hcmp (rootLabel cs e) (rootLabel cs f) (hn e he) (hn f hf)
+
+theorem nonsimple_of_greatest (O : RatioOrder (M := M) (cs := cs)) (i : I)
+    (hmax : ∀ a : PositiveRoot M cs, a ≠ simplePositive i →
+      O.slope a < O.slope (simplePositive i))
+    {x y : W} (hy : ¬cs.length (cs.simple i*y) < cs.length y)
+    {p : List (Edge cs)} (hp : p ∈ paths O x y) : Nonsimple i p := by
+  rw [mem_paths] at hp
+  rcases hp with ⟨hp,ho⟩
+  induction p generalizing x with
+  | nil => simp [Nonsimple]
+  | cons e p ih =>
+    have he : rootLabel cs e ≠ simplePositive i := by
+      intro he
+      cases p with
+      | nil =>
+        have hy' : target cs e = y := hp.2
+        have hr := (rootLabel_simple_iff cs i e).mp he
+        have hx' : source cs e = cs.simple i*y := by
+          rw [← hy',hr,cs.simple_mul_simple_cancel_left]
+        have hl := e.property.1
+        change cs.length (source cs e) < cs.length (target cs e) at hl
+        rw [hy',hx'] at hl
+        exact hy hl
+      | cons f p =>
+        have hlt := (List.pairwise_cons.mp ho).1 f List.mem_cons_self
+        have hle : score O f ≤ O.slope (simplePositive i) := by
+          by_cases hf : rootLabel cs f = simplePositive i
+          · exact le_of_eq (congrArg O.slope hf)
+          · exact (hmax _ hf).le
+        have hes : score O e = O.slope (simplePositive i) := congrArg O.slope he
+        exact (not_lt_of_ge hle) (hes ▸ hlt)
+    intro f hf
+    rcases List.mem_cons.mp hf with rfl|hf
+    · exact he
+    · exact ih hp.2 ho.tail f hf
+
+theorem nonsimple_tail_of_least (O : RatioOrder (M := M) (cs := cs)) (i : I)
+    (hmin : ∀ a : PositiveRoot M cs, a ≠ simplePositive i →
+      O.slope (simplePositive i) < O.slope a)
+    {e : Edge cs} {p : List (Edge cs)}
+    (ho : (e::p).Pairwise (fun e f => score O e < score O f)) : Nonsimple i p := by
+  intro f hf heq
+  have hef := (List.pairwise_cons.mp ho).1 f hf
+  have he : O.slope (simplePositive i) ≤ score O e := by
+    by_cases he : rootLabel cs e = simplePositive i
+    · exact le_of_eq (congrArg O.slope he).symm
+    · exact (hmin _ he).le
+  have hf' : score O f = O.slope (simplePositive i) := congrArg O.slope heq
+  exact (not_lt_of_ge he) (hf' ▸ hef)
+
+theorem nonsimple_of_least_descent (O : RatioOrder (M := M) (cs := cs)) (i : I)
+    (hmin : ∀ a : PositiveRoot M cs, a ≠ simplePositive i →
+      O.slope (simplePositive i) < O.slope a)
+    {x y : W} (hx : cs.length (cs.simple i*x) < cs.length x)
+    {p : List (Edge cs)} (hp : p ∈ paths O x y) : Nonsimple i p := by
+  obtain ⟨hp,ho⟩ := (mem_paths O x y p).mp hp
+  cases p with
+  | nil => simp [Nonsimple]
+  | cons e p =>
+    have he : rootLabel cs e ≠ simplePositive i := by
+      intro he
+      have hr := (rootLabel_simple_iff cs i e).mp he
+      have hl := e.property.1
+      change cs.length (source cs e) < cs.length (target cs e) at hl
+      rw [hr,hp.1] at hl
+      exact (lt_asymm hx hl)
+    intro f hf
+    rcases List.mem_cons.mp hf with rfl|hf
+    · exact he
+    · exact nonsimple_tail_of_least O i hmin ho f hf
+
+def nonsimplePaths (O : RatioOrder (M := M) (cs := cs)) (i : I) (x y : W) :
+    Finset (List (Edge cs)) := (paths O x y).filter (Nonsimple i)
+
+@[simp] theorem mem_nonsimplePaths (O : RatioOrder (M := M) (cs := cs)) (i : I)
+    (x y : W) (p : List (Edge cs)) :
+    p ∈ nonsimplePaths O i x y ↔ p ∈ paths O x y ∧ Nonsimple i p := by
+  simp [nonsimplePaths]
+
+theorem sum_nonsimple_rotate (O O' : RatioOrder (M := M) (cs := cs)) (i : I)
+    (hmax : ∀ a : PositiveRoot M cs, a ≠ simplePositive i →
+      O'.slope a < O'.slope (simplePositive i))
+    (hcmp : ∀ a b : PositiveRoot M cs, a ≠ simplePositive i → b ≠ simplePositive i →
+      (O'.slope (positiveAction (cs.simple i) a) < O'.slope (positiveAction (cs.simple i) b) ↔
+        O.slope a < O.slope b))
+    (x y : W) (hy : cs.length (cs.simple i*y) < cs.length y) :
+    ∑ p ∈ nonsimplePaths O i x y, (Polynomial.X : Polynomial ℤ)^p.length =
+      polynomial O' (cs.simple i*x) (cs.simple i*y) := by
+  unfold polynomial
+  have hasc : ¬cs.length (cs.simple i*(cs.simple i*y)) < cs.length (cs.simple i*y) := by
+    rw [cs.simple_mul_simple_cancel_left]
+    omega
+  apply Finset.sum_bij (fun p _ => p.map (rotateEdge cs i))
+  · intro p hp
+    obtain ⟨hp,hn⟩ := (mem_nonsimplePaths O i x y p).mp hp
+    exact (paths_rotate_iff O O' i hcmp hn).mpr hp
+  · intro p hp q hq he
+    have := congrArg (List.map (rotateEdge cs i)) he
+    simpa only [path_map_twice] using this
+  · intro q hq
+    have hn := nonsimple_of_greatest O' i hmax hasc hq
+    refine ⟨q.map (rotateEdge cs i),?_,path_map_twice i q⟩
+    have hn' := nonsimple_map i hn
+    apply (mem_nonsimplePaths O i x y _).mpr
+    refine ⟨?_,hn'⟩
+    apply (paths_rotate_iff O O' i hcmp hn').mp
+    simpa only [path_map_twice] using hq
+  · intro p hp
+    simp only [List.length_map]
+
+
+variable (cs)
+def simpleEdge (i : I) (x : W) (hx : cs.length x < cs.length (cs.simple i*x)) : Edge cs :=
+  ⟨(x,cs.simple i*x),hx,cs.simple i,cs.isReflection_simple i,rfl⟩
+
+omit [Fintype I] in
+@[simp] theorem simpleEdge_source (i : I) (x : W) (hx : cs.length x < cs.length (cs.simple i*x)) :
+    source cs (simpleEdge cs i x hx) = x := rfl
+
+omit [Fintype I] in
+@[simp] theorem simpleEdge_target (i : I) (x : W) (hx : cs.length x < cs.length (cs.simple i*x)) :
+    target cs (simpleEdge cs i x hx) = cs.simple i*x := rfl
+
+@[simp] theorem simpleEdge_root (i : I) (x : W) (hx : cs.length x < cs.length (cs.simple i*x)) :
+    rootLabel cs (simpleEdge cs i x hx) = simplePositive i :=
+  (rootLabel_simple_iff cs i _).mpr rfl
+
+variable {cs}
+theorem cons_simple_mem (O : RatioOrder (M := M) (cs := cs)) (i : I)
+    (hmin : ∀ a : PositiveRoot M cs, a ≠ simplePositive i →
+      O.slope (simplePositive i) < O.slope a)
+    {x y : W} (hx : cs.length x < cs.length (cs.simple i*x))
+    {p : List (Edge cs)} (hp : p ∈ nonsimplePaths O i (cs.simple i*x) y) :
+    simpleEdge cs i x hx :: p ∈ paths O x y := by
+  obtain ⟨hp,hn⟩ := (mem_nonsimplePaths O i _ _ _).mp hp
+  obtain ⟨hp,ho⟩ := (mem_paths O _ _ _).mp hp
+  apply (mem_paths O _ _ _).mpr
+  refine ⟨⟨rfl,hp⟩,List.pairwise_cons.mpr ⟨?_,ho⟩⟩
+  intro e he
+  change O.slope (rootLabel cs (simpleEdge cs i x hx)) < O.slope (rootLabel cs e)
+  rw [simpleEdge_root]
+  exact hmin _ (hn e he)
+
+theorem not_nonsimple_iff_cons (O : RatioOrder (M := M) (cs := cs)) (i : I)
+    (hmin : ∀ a : PositiveRoot M cs, a ≠ simplePositive i →
+      O.slope (simplePositive i) < O.slope a)
+    {x y : W} (hx : cs.length x < cs.length (cs.simple i*x)) (p : List (Edge cs)) :
+    p ∈ (paths O x y).filter (fun p => ¬Nonsimple i p) ↔
+      ∃ q ∈ nonsimplePaths O i (cs.simple i*x) y, p = simpleEdge cs i x hx :: q := by
+  constructor
+  · intro hp
+    obtain ⟨hp,hn⟩ := Finset.mem_filter.mp hp
+    obtain ⟨hp,ho⟩ := (mem_paths O _ _ _).mp hp
+    cases p with
+    | nil => exact (hn (by simp [Nonsimple])).elim
+    | cons e p =>
+      have hnp : Nonsimple i p := nonsimple_tail_of_least O i hmin ho
+      have he : rootLabel cs e = simplePositive i := by
+        by_contra he
+        apply hn
+        intro f hf
+        rcases List.mem_cons.mp hf with rfl|hf
+        · exact he
+        · exact hnp f hf
+      have heq : e = simpleEdge cs i x hx := by
+        apply Subtype.ext
+        apply Prod.ext
+        · exact hp.1
+        · change target cs e = cs.simple i*x
+          rw [(rootLabel_simple_iff cs i e).mp he,hp.1]
+      subst e
+      refine ⟨p,(mem_nonsimplePaths O i _ _ _).mpr ⟨?_,hnp⟩,rfl⟩
+      exact (mem_paths O _ _ _).mpr ⟨hp.2,ho.tail⟩
+  · rintro ⟨q,hq,rfl⟩
+    apply Finset.mem_filter.mpr
+    refine ⟨cons_simple_mem O i hmin hx hq,?_⟩
+    intro hn
+    exact hn _ List.mem_cons_self (simpleEdge_root cs i x hx)
+
+theorem polynomial_split (O : RatioOrder (M := M) (cs := cs)) (i : I)
+    (hmin : ∀ a : PositiveRoot M cs, a ≠ simplePositive i →
+      O.slope (simplePositive i) < O.slope a)
+    {x y : W} (hx : cs.length x < cs.length (cs.simple i*x)) :
+    polynomial O x y =
+      (∑ p ∈ nonsimplePaths O i x y, (Polynomial.X : Polynomial ℤ)^p.length) +
+      Polynomial.X * (∑ p ∈ nonsimplePaths O i (cs.simple i*x) y,
+        (Polynomial.X : Polynomial ℤ)^p.length) := by
+  unfold polynomial
+  rw [← Finset.sum_filter_add_sum_filter_not (paths O x y) (Nonsimple i)]
+  congr 1
+  rw [Finset.mul_sum]
+  symm
+  apply Finset.sum_bij (fun p _ => simpleEdge cs i x hx :: p)
+  · intro p hp
+    exact (not_nonsimple_iff_cons O i hmin hx _).mpr ⟨p,hp,rfl⟩
+  · intro p hp q hq he
+    exact List.cons.inj he |>.2
+  · intro q hq
+    obtain ⟨p,hp,rfl⟩ := (not_nonsimple_iff_cons O i hmin hx q).mp hq
+    exact ⟨p,hp,rfl⟩
+  · intro p hp
+    simp only [List.length_cons,pow_succ']
+
+/-- The path expansion satisfies exactly the genuine normalized Hecke
+recursion, because the chosen geometric order can be rotated at a descent. -/
+theorem polynomial_recursion (O O' : RatioOrder (M := M) (cs := cs)) (i : I)
+    (hmin : ∀ a : PositiveRoot M cs, a ≠ simplePositive i →
+      O.slope (simplePositive i) < O.slope a)
+    (hmax : ∀ a : PositiveRoot M cs, a ≠ simplePositive i →
+      O'.slope a < O'.slope (simplePositive i))
+    (hcmp : ∀ a b : PositiveRoot M cs, a ≠ simplePositive i → b ≠ simplePositive i →
+      (O'.slope (positiveAction (cs.simple i) a) < O'.slope (positiveAction (cs.simple i) b) ↔
+        O.slope a < O.slope b))
+    (x y : W) (hy : cs.length (cs.simple i*y) < cs.length y) :
+    polynomial O x y = if cs.length (cs.simple i*x) < cs.length x then
+      polynomial O' (cs.simple i*x) (cs.simple i*y) else
+      polynomial O' (cs.simple i*x) (cs.simple i*y) +
+        Polynomial.X * polynomial O' x (cs.simple i*y) := by
+  split_ifs with hx
+  · have heq : nonsimplePaths O i x y = paths O x y := by
+      ext p
+      rw [mem_nonsimplePaths]
+      exact and_iff_left_of_imp (nonsimple_of_least_descent O i hmin hx)
+    rw [← sum_nonsimple_rotate O O' i hmax hcmp x y hy,heq]
+    rfl
+  · have ha : cs.length x < cs.length (cs.simple i*x) := by
+      have := cs.length_simple_mul x i
+      omega
+    rw [polynomial_split O i hmin ha,
+      sum_nonsimple_rotate O O' i hmax hcmp x y hy,
+      sum_nonsimple_rotate O O' i hmax hcmp (cs.simple i*x) y hy,
+      cs.simple_mul_simple_cancel_left]
+
+
+@[simp] theorem paths_diagonal (O : RatioOrder (M := M) (cs := cs)) (x : W) :
+    paths O x x = {[]} := by
+  ext p
+  rw [mem_paths,Finset.mem_singleton]
+  cases p with
+  | nil => simp [IsPath]
+  | cons e p =>
+    constructor
+    · rintro ⟨hp,_⟩
+      have hl := length_le_of_bruhat cs (path_le cs hp.2)
+      have he := e.property.1
+      change cs.length (source cs e) < cs.length (target cs e) at he
+      rw [hp.1] at he
+      omega
+    · simp
+
+@[simp] theorem polynomial_diagonal (O : RatioOrder (M := M) (cs := cs)) (x : W) :
+    polynomial O x x = 1 := by
+  simp [polynomial]
+
+theorem polynomial_zero (O : RatioOrder (M := M) (cs := cs)) (x y : W)
+    (h : ¬BruhatLE cs x y) : polynomial O x y = 0 := by
+  have hp : paths O x y = ∅ := by
+    apply Finset.eq_empty_iff_forall_notMem.mpr
+    intro p hp
+    exact h (path_le cs ((mem_paths O x y p).mp hp).1)
+  simp [polynomial,hp]
+
+/-- Dyer's increasing-path formula for a genuine generic geometric order
+whose numerator is in the top chamber. The target is the Hecke-defined
+normalized R-polynomial, not a renamed path polynomial. -/
+theorem polynomial_eq_rTilde (O : RatioOrder (M := M) (cs := cs)) (x y : W)
+    (hc : O.Chamber y) : polynomial O x y = Hecke.rTilde cs x y := by
+  generalize hn : cs.length y = n
+  induction n using Nat.strong_induction_on generalizing O x y with
+  | h n ih =>
+    by_cases hy : y=1
+    · subst y
+      by_cases hx : x=1
+      · subst x
+        rw [polynomial_diagonal,Hecke.rTilde_diagonal]
+      · have hxy : ¬ BruhatLE cs x 1 := by
+          intro h
+          have hl := length_le_of_bruhat cs h
+          rw [cs.length_one] at hl
+          exact hx (cs.length_eq_zero_iff.mp (Nat.eq_zero_of_le_zero hl))
+        rw [polynomial_zero O x 1 hxy,Hecke.rTilde_zero cs x 1 hxy]
+    · obtain ⟨i,O',hi,hc',hends,hcmp⟩ := O.exists_rotation hy hc
+      have hi' : cs.length (cs.simple i*y) < n := hn ▸ hi
+      have hsx := ih _ hi' O' (cs.simple i*x) (cs.simple i*y) hc' rfl
+      have hx := ih _ hi' O' x (cs.simple i*y) hc' rfl
+      rw [polynomial_recursion O O' i (fun a ha => (hends a ha).1)
+        (fun a ha => (hends a ha).2) hcmp x y hi,
+        Hecke.rTilde_recursion cs x y i hi,hsx,hx]
+
+/-- An adapted geometric order exists for every top, without adding labels
+or geometric data to the given interval isomorphism. -/
+theorem exists_dyer_schedule (y : W) :
+    ∃ O : RatioOrder (M := M) (cs := cs), O.Chamber y ∧
+      ∀ x : W, polynomial O x y = Hecke.rTilde cs x y := by
+  obtain ⟨O,hO⟩ := exists_ratioOrder_chamber (M := M) (cs := cs) y
+  exact ⟨O,hO,fun x => polynomial_eq_rTilde O x y hO⟩
+
+end
+end KLInvariance.AdaptedDyer
+
+end
+
+
+section
+
+/-! Finite edge-product algebra needed for geometric wall crossing. Every
+operator is the actual `1 + T E_xy`; no R/KL identity is assumed here. -/
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/KazhdanLusztig/EdgeLocalization.lean b/lean/OAI/RepresentationTheory/KazhdanLusztig/EdgeLocalization.lean
index b37025b0fea07ffe20656104c11ed2cf2e454585..3e4b8e4cf2cb856b76f4513ab02a562ecd758eb9 100644
--- a/lean/OAI/RepresentationTheory/KazhdanLusztig/EdgeLocalization.lean
+++ b/lean/OAI/RepresentationTheory/KazhdanLusztig/EdgeLocalization.lean
@@ -1,1129 +1,1129 @@
-import OAI.RepresentationTheory.KazhdanLusztig.PlaneLocalization
-
-namespace OAI
-
-/-!
-Bruhat sheaf localization, graded transport, local edge-image bounds and support.
--/
-
-section
-
-namespace KLInvariance.BruhatGraph
-open TitsSpace SchedulePaths
-universe u v u' v'
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
-  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
-  {u b : W} {u' b' : W'}
-  (φ : Interval cs u b ≃o Interval cs' u' b')
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
- def transportedEdge (e : Edge cs u b) : AdaptedDyer.Edge cs' :=
-  ⟨((φ (source cs u b e)).val,(φ (target cs u b e)).val),
-    (intervalIso_bruhatStep_iff_general cs cs' φ _ _).mpr e.property⟩
-
-omit [Fintype I] [Fintype I'] in
- theorem transportedEdge_injective : Function.Injective (transportedEdge φ) := by
-  intro e f h
-  apply Subtype.ext
-  have hh := congrArg Subtype.val h
-  exact Prod.ext (φ.injective (Subtype.ext (congrArg Prod.fst hh)))
-    (φ.injective (Subtype.ext (congrArg Prod.snd hh)))
-
- def transportedScore (e : Edge cs u b) : ℝ ×ₗ ℝ :=
-  lexEdgeScore (geometricFunctional (M := M') (cs := cs')) (transportedEdge φ e)
-
- /-- Decreasing list, in the convention for the outermost-first matrix
- product. Its reverse is chronological increasing reflection order. -/
- def transportedSchedule : List (Edge cs u b) := by
-  letI := Fintype.ofFinite (Edge cs u b)
-  exact sortedSchedule Finset.univ (transportedScore φ)
-
-omit [Fintype I] in
- theorem transportedSchedule_nodup : (transportedSchedule φ).Nodup := by
-  unfold transportedSchedule
-  exact sortedSchedule_nodup _ _
-
-omit [Fintype I] in
- theorem transportedSchedule_full (e : Edge cs u b) : e ∈ transportedSchedule φ := by
-  let := Fintype.ofFinite (Edge cs u b)
-  unfold transportedSchedule
-  exact (mem_sortedSchedule _ _ _).mpr (Finset.mem_univ _)
-
-omit [Fintype I] in
- theorem transportedSchedule_pairwise : (transportedSchedule φ).Pairwise
-     (fun e f => transportedScore φ f ≤ transportedScore φ e) := by
-  unfold transportedSchedule
-  exact sortedSchedule_pairwise _ _
-
-omit [Fintype I] [Fintype I'] in
- theorem transportedEdge_lift (e : AdaptedDyer.Edge cs')
-    (hu : BruhatLE cs' u' (AdaptedDyer.source cs' e))
-    (hb : BruhatLE cs' (AdaptedDyer.target cs' e) b') : ∃ f, transportedEdge φ f=e := by
-  let x : Interval cs' u' b' := ⟨AdaptedDyer.source cs' e,hu,
-    (show BruhatLE cs' (AdaptedDyer.source cs' e) (AdaptedDyer.target cs' e) from
-      .single e.property).trans hb⟩
-  let y : Interval cs' u' b' := ⟨AdaptedDyer.target cs' e,
-    hu.trans (.single e.property),hb⟩
-  let f : Edge cs u b := ⟨(φ.symm x,φ.symm y),
-    (intervalIso_bruhatStep_iff_general cs' cs φ.symm x y).mpr e.property⟩
-  refine ⟨f,?_⟩
-  apply Subtype.ext
-  change ((φ (φ.symm x)).val,(φ (φ.symm y)).val)=e.val
-  rw [φ.apply_symm_apply,φ.apply_symm_apply]
-  rfl
-
-omit [Fintype I] in
- theorem transported_dispatch_entry {R : Type*} [Semiring R] (T : R)
-    (x y : Interval cs u b) :
-    dispatch (source cs u b) (target cs u b) T (fun z => if z=y then 1 else 0)
-      (transportedSchedule φ) x =
-    WallDyer.entry (cs := cs') T (geometricFunctional (M := M') (cs := cs'))
-      b' (φ x).val (φ y).val := by
-  have hsource {z w : W'} {p : List (AdaptedDyer.Edge cs')}
-      (hp : IsPath (AdaptedDyer.source cs') (AdaptedDyer.target cs') z w p) :
-      ∀ e ∈ p, BruhatLE cs' z (AdaptedDyer.source cs' e) := by
-    induction p generalizing z with
-    | nil => simp
-    | cons e p ih =>
-      intro f hf
-      rcases List.mem_cons.mp hf with rfl | hf
-      · exact hp.1 ▸ bruhat_refl cs' z
-      · rw [← hp.1]
-        exact (show BruhatLE cs' (AdaptedDyer.source cs' e)
-          (AdaptedDyer.target cs' e) from .single e.property).trans (ih hp.2 f hf)
-  symm
-  apply dispatch_entry_graph_embedding (AdaptedDyer.source cs') (AdaptedDyer.target cs')
-    (source cs u b) (target cs u b) T
-    (lexEdgeScore (geometricFunctional (M := M') (cs := cs')))
-    (lexEdgeScore_ties _) (fun z : Interval cs u b => (φ z).val)
-    (fun _ _ h => φ.injective (Subtype.ext h)) (transportedEdge φ)
-    (transportedEdge_injective φ) (fun _ => rfl) (fun _ => rfl)
-    (WallDyer.schedule (geometricFunctional (M := M') (cs := cs')) b')
-    (transportedSchedule φ) (sortedSchedule_nodup _ _)
-    (transportedSchedule_nodup φ) (sortedSchedule_pairwise _ _)
-    (transportedSchedule_pairwise φ) (transportedSchedule_full φ)
-  · intro e
-    exact (mem_sortedSchedule _ _ _).mpr ((AdaptedDyer.mem_edgeFinset cs' b' _).mpr
-      (φ (target cs u b e)).property.2)
-  · intro r hr e he
-    have hp := ((mem_subpaths _ _ _ _ _ r).mp hr).2
-    exact transportedEdge_lift φ e ((φ x).property.1.trans (hsource hp e he))
-      ((AdaptedDyer.path_targets_le cs' hp e he).trans (φ y).property.2)
-
-omit [Fintype I] in
-
- theorem transported_entry_eq_rTilde (x y : Interval cs u b) :
-    dispatch (source cs u b) (target cs u b) (Polynomial.X : Polynomial ℤ)
-      (fun z => if z=y then 1 else 0) (transportedSchedule φ) x =
-      Hecke.rTilde cs' (φ x).val (φ y).val := by
-  rw [transported_dispatch_entry]
-  exact WallDyer.geometric_entry_eq_rTilde b' (φ x).val (φ y).val
-    (φ x).property.2 (φ y).property.2
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-/-! Actual transported numerical R-tilde matrix and its genuine KL reciprocal
-vector. No sheaf or desired invariance conclusion is assumed. -/
-namespace KLInvariance.SchedulePaths
-universe u v w w'
-variable {V : Type u} {E : Type v} {R : Type w} {S : Type w'}
-  [Semiring R] [Semiring S] (source target : E → V)
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
- theorem dispatch_map (f : R →+* S) (T : R) (v : V → R) (p : List E) (x : V) :
-    f (dispatch source target T v p x) =
-      dispatch source target (f T) (fun z => f (v z)) p x := by
-  induction p generalizing x with
-  | nil => rfl
-  | cons e p ih =>
-    simp only [dispatch,step,map_add]
-    split_ifs <;> simp only [map_mul,map_zero,ih]
-
-end
-end KLInvariance.SchedulePaths
-
-namespace KLInvariance
-universe u v
-variable {I : Type u} {M : CoxeterMatrix I} {W : Type v} [Group W]
-  (cs : CoxeterSystem M W)
-noncomputable section
-
- theorem scaledKL_reciprocity_upper {C : Type*} [Field C]
-    (t : C) (ht : t ≠ 0) (u b : W) (x : Interval cs u b) :
-    scaledKL cs t x.val b = ∑ᶠ z : Interval cs u b,
-      Polynomial.eval₂ (Int.castRingHom C) (t⁻¹-t) (Hecke.rTilde cs x.val z.val) *
-        scaledKL cs t⁻¹ z.val b := by
-  classical
-  let := interval_finite cs u b
-  let := Fintype.ofFinite (Interval cs u b)
-  let := interval_finite cs x.val b
-  let := Fintype.ofFinite (Interval cs x.val b)
-  rw [scaledKL_reciprocity cs t ht x.val b x.property.2,
-    finsum_eq_sum_of_fintype,finsum_eq_sum_of_fintype]
-  let f : W → C := fun z => Polynomial.eval₂ (Int.castRingHom C) (t⁻¹-t)
-    (Hecke.rTilde cs x.val z) * scaledKL cs t⁻¹ z b
-  have hs (a : W) [Fintype (Interval cs a b)] :
-      (∑ z : Interval cs a b, f z.val) =
-        ∑ z ∈ (lowerFinset cs b).filter (fun z => BruhatLE cs a z), f z := by
-    exact (Finset.sum_subtype (F := (inferInstance : Fintype (Interval cs a b)))
-      ((lowerFinset cs b).filter (fun z => BruhatLE cs a z))
-      (fun z => by simp only [Finset.mem_filter,mem_lowerFinset]; exact and_comm) f).symm
-  change (∑ z : Interval cs x.val b, f z.val) = ∑ z : Interval cs u b, f z.val
-  rw [hs,hs]
-  apply Finset.sum_subset
-  · intro z hz
-    rcases Finset.mem_filter.mp hz with ⟨hz,hxz⟩
-    exact Finset.mem_filter.mpr ⟨hz,x.property.1.trans hxz⟩
-  · intro z hz hn
-    have hxz : ¬ BruhatLE cs x.val z := by
-      intro h
-      exact hn (Finset.mem_filter.mpr ⟨(Finset.mem_filter.mp hz).1,h⟩)
-    dsimp [f]
-    rw [Hecke.rTilde_zero cs _ _ hxz,Polynomial.eval₂_zero,zero_mul]
-
-end
-end KLInvariance
-
-namespace KLInvariance.BruhatGraph
-open TitsSpace SchedulePaths
-universe u v u' v'
-variable {I : Type u} {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
-  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
-  {u b : W} {u' b' : W'}
-  (φ : Interval cs u b ≃o Interval cs' u' b')
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
- theorem transported_entry_eval {C : Type*} [CommSemiring C] (T : C)
-    (x y : Interval cs u b) (f : ℤ →+* C) :
-    dispatch (source cs u b) (target cs u b) T
-      (fun z => if z=y then 1 else 0) (transportedSchedule φ) x =
-      Polynomial.eval₂ f T (Hecke.rTilde cs' (φ x).val (φ y).val) := by
-  rw [← transported_entry_eq_rTilde φ x y]
-  have h := dispatch_map (source cs u b) (target cs u b) (Polynomial.eval₂RingHom f T)
-    Polynomial.X (fun z : Interval cs u b => if z=y then 1 else 0) (transportedSchedule φ) x
-  simpa using h.symm
-
- theorem transported_dispatch_matrix {C : Type*} [CommSemiring C] (T : C)
-    (f : ℤ →+* C) (v : Interval cs u b → C) (x : Interval cs u b) :
-    dispatch (source cs u b) (target cs u b) T v (transportedSchedule φ) x =
-      ∑ᶠ y : Interval cs u b,
-        Polynomial.eval₂ f T (Hecke.rTilde cs' (φ x).val (φ y).val) * v y := by
-  let := interval_finite cs u b
-  let := Fintype.ofFinite (Interval cs u b)
-  rw [dispatch_eq_sum_entries,finsum_eq_sum_of_fintype]
-  exact Finset.sum_congr rfl (fun y _ => congrArg (fun a => a*v y)
-    (transported_entry_eval φ T x y f))
-
- theorem transported_reciprocal_vector {C : Type*} [Field C]
-    (t : C) (ht : t ≠ 0) (x : Interval cs u b) :
-    scaledKL cs' t (φ x).val b' =
-      dispatch (source cs u b) (target cs u b) (t⁻¹-t)
-        (fun y => scaledKL cs' t⁻¹ (φ y).val b') (transportedSchedule φ) x := by
-  let := interval_finite cs u b
-  let := Fintype.ofFinite (Interval cs u b)
-  let := interval_finite cs' u' b'
-  let := Fintype.ofFinite (Interval cs' u' b')
-  rw [transported_dispatch_matrix φ _ (Int.castRingHom C),
-    scaledKL_reciprocity_upper cs' t ht u' b' (φ x),
-    finsum_eq_sum_of_fintype,finsum_eq_sum_of_fintype]
-  exact (φ.toEquiv.sum_comp _).symm
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  (hub : BruhatLE cs u b)
-  (C : BoundarySheaf (polynomialGrading_negative ℝ σ)
-    (graph cs u b) (polynomialLabel cs u b basis) ⟨b,hub,bruhat_refl cs b⟩)
-noncomputable section
-
- theorem boundarySheaf_gradedIso :
-    Nonempty (GradedSheaf.Iso (boundarySheaf cs u b basis hub).sheaf C.sheaf) := by
-  classical
-  let : Finite (Interval cs u b) := interval_finite cs u b
-  let : Fintype (Interval cs u b) := Fintype.ofFinite _
-  let : Fintype (Edge cs u b) := Fintype.ofFinite _
-  exact BoundarySheaf.exists_gradedIso (polynomialGrading_zero ℝ σ)
-    (boundarySheaf cs u b basis hub) C (fun x => x.property.2)
-
- theorem stalkCharacter_eq_character (x : Interval cs u b) :
-    let _ := C.free x
-    stalkCharacter cs u b basis hub x =
-      Graded.character (σ := σ) (C.sheaf.vertex x).piece (C.sheaf.vertex x).nonneg := by
-  let _ := C.free x
-  let B := boundarySheaf cs u b basis hub
-  let _ := B.free x
-  obtain ⟨j⟩ := boundarySheaf_gradedIso cs u b basis hub C
-  exact ModuleData.character_eq_of_gradedEquiv (j.vertex x) (j.vertex_graded x)
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-/-! Faithful upper-interval restriction of the actual minimal boundary sheaf.
-The bottom is arbitrary. No roots or labels are supplied by the interval iso. -/
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
-noncomputable section
-
- def upperInclusion : GraphMap (graph cs u b) (graph cs 1 b) where
-  vertex x := ⟨x.val,one_bruhat cs x.val,x.property.2⟩
-  edge e := ⟨(⟨(source cs u b e).val,one_bruhat cs _,(source cs u b e).property.2⟩,
-    ⟨(target cs u b e).val,one_bruhat cs _,(target cs u b e).property.2⟩),e.property⟩
-  source _ := rfl
-  target _ := rfl
-
-omit [Fintype I] in
- theorem upperInclusion_injective : Function.Injective (upperInclusion cs u b).vertex := by
-  intro x y h
-  exact Subtype.ext (congrArg (fun z : Interval cs 1 b => z.val) h)
-
-omit [Fintype I] in
- theorem upperInclusion_outgoing (e : Edge cs 1 b) (x : Interval cs u b)
-    (hx : (upperInclusion cs u b).vertex x = source cs 1 b e) :
-    ∃ a, (upperInclusion cs u b).edge a=e ∧ source cs u b a=x := by
-  have hxval := congrArg Subtype.val hx
-  let y : Interval cs u b := ⟨(target cs 1 b e).val,
-    bruhat_trans cs x.property.1 (hxval.symm ▸ Relation.ReflTransGen.single e.property),
-    (target cs 1 b e).property.2⟩
-  let a : Edge cs u b := ⟨(x,y),hxval.symm ▸ e.property⟩
-  refine ⟨a,?_,rfl⟩
-  apply Subtype.ext
-  apply Prod.ext
-  · exact hx
-  · rfl
-
-variable {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  (hub : BruhatLE cs u b)
-  (B : BoundarySheaf (polynomialGrading_negative ℝ σ)
-    (graph cs 1 b) (polynomialLabel cs 1 b basis) ⟨b,one_bruhat cs b,bruhat_refl cs b⟩)
-
- def restrictBoundary : BoundarySheaf (polynomialGrading_negative ℝ σ)
-    (graph cs u b) (polynomialLabel cs u b basis) ⟨b,hub,bruhat_refl cs b⟩ :=
-  B.pullback (upperInclusion cs u b) (upperInclusion_injective cs u b)
-    (upperInclusion_outgoing cs u b) ⟨b,hub,bruhat_refl cs b⟩ rfl
-
- theorem stalkCharacter_upperRestriction (x : Interval cs u b) :
-    let _ := B.free ((upperInclusion cs u b).vertex x)
-    stalkCharacter cs u b basis hub x =
-      Graded.character (σ := σ) (B.sheaf.vertex ((upperInclusion cs u b).vertex x)).piece
-        (B.sheaf.vertex ((upperInclusion cs u b).vertex x)).nonneg :=
-  stalkCharacter_eq_character cs u b basis hub (restrictBoundary cs u b basis hub B) x
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph Polynomial SchedulePaths
-universe u
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W) (b : W)
-  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-noncomputable section
-local instance scaledInterval : Fintype (Interval cs 1 b) := by
-  let := interval_finite cs 1 b
-  exact Fintype.ofFinite _
-
- theorem stalkCharacter_natDegree_le (x : Interval cs 1 b) :
-    (stalkCharacter cs 1 b basis (one_bruhat cs b) x).natDegree≤rankDifference cs x.val b := by
-  classical
-  let B := boundarySheaf cs 1 b basis (one_bruhat cs b)
-  let := B.free x
-  let c := (exists_nonnegativeBasis (σ := σ) (B.sheaf.vertex x).piece
-    (B.sheaf.vertex x).nonneg).some
-  rw [stalkCharacter_eq_character cs 1 b basis (one_bruhat cs b) B x,
-    character_eq (B.sheaf.vertex x).piece (B.sheaf.vertex x).nonneg c]
-  apply Polynomial.natDegree_sum_le_of_forall_le
-  intro i _
-  simpa only [Polynomial.natDegree_X_pow] using stalkBasis_degree_le cs b basis x c i
-
- def scaledStalk {C : Type*} [Field C] (t : C) (x : Interval cs 1 b) : C :=
-  t^((cs.length x.val : ℤ)-cs.length b)*
-    eval₂ (Int.castRingHom C) (t^2) (stalkCharacter cs 1 b basis (one_bruhat cs b) x)
-
- theorem scaledStalk_reciprocity_inverse {C : Type*} [Field C]
-    (v : C) (hv : v≠0) (x : Interval cs 1 b) :
-    scaledStalk cs b basis v⁻¹ x=∑ z : Interval cs 1 b,
-      eval₂ (Int.castRingHom C) (v-v⁻¹) (Hecke.rTilde cs x.val z.val)*
-        scaledStalk cs b basis v z := by
-  classical
-  have hrec := stalkCharacter_reciprocity cs basis b x
-  let q : Cˣ := Units.mk0 (v^2) (pow_ne_zero 2 hv)
-  have heval := congrArg (eval₂ (Int.castRingHom C) (q : C)) hrec
-  rw [eval₂_reflect_unit q _ _ (stalkCharacter_natDegree_le cs b basis x)] at heval
-  simp only [eval₂_finsetSum,eval₂_mul] at heval
-  have hq : (q : C)=v^2 := rfl
-  have hqi : (↑q⁻¹ : C)=(v⁻¹)^2 := by simp [q,inv_pow]
-  rw [hq,hqi] at heval
-  have heq := congrArg (fun a : C => v^((cs.length x.val : ℤ)-cs.length b)*a) heval
-  have hd : (rankDifference cs x.val b : ℤ)=(cs.length b : ℤ)-cs.length x.val := by
-    simp [rankDifference,Int.ofNat_sub (length_le_of_bruhat cs x.property.2)]
-  have hscale : v^((cs.length x.val : ℤ)-cs.length b)*(v^2)^(rankDifference cs x.val b)=
-      (v⁻¹)^((cs.length x.val : ℤ)-cs.length b) := by
-    rw [←pow_mul,←zpow_natCast,←zpow_add₀ hv,inv_zpow,←zpow_neg]
-    congr 1
-    push_cast
-    omega
-  rw [←mul_assoc,hscale,Finset.mul_sum] at heq
-  change (v⁻¹)^((cs.length x.val : ℤ)-cs.length b)*
-    eval₂ (Int.castRingHom C) ((v⁻¹)^2) (stalkCharacter cs 1 b basis (one_bruhat cs b) x)=_
-  rw [heq]
-  apply Finset.sum_congr rfl
-  intro z _
-  have hr : OrdinaryR.polynomial cs x.val z.val=rPolynomial cs x.val z.val :=
-    (rPolynomial_eq_constructed cs x.val z.val).symm
-  rw [hr,rPolynomial_normalization cs v hv]
-  unfold scaledStalk
-  calc
-    _ = (v^((cs.length x.val : ℤ)-cs.length b)*v^((cs.length z.val : ℤ)-cs.length x.val))*
-        eval₂ (Int.castRingHom C) (v-v⁻¹) (Hecke.rTilde cs x.val z.val)*
-        eval₂ (Int.castRingHom C) (v^2) (stalkCharacter cs 1 b basis (one_bruhat cs b) z) := by ring
-    _ = _ := by
-      rw [←zpow_add₀ hv,show (cs.length x.val : ℤ)-cs.length b+
-        ((cs.length z.val : ℤ)-cs.length x.val)=(cs.length z.val : ℤ)-cs.length b by ring]
-      ring
-
- theorem scaledStalk_reciprocity {C : Type*} [Field C]
-    (t : C) (ht : t≠0) (x : Interval cs 1 b) :
-    scaledStalk cs b basis t x=∑ z : Interval cs 1 b,
-      eval₂ (Int.castRingHom C) (t⁻¹-t) (Hecke.rTilde cs x.val z.val)*
-        scaledStalk cs b basis t⁻¹ z := by
-  simpa only [inv_inv] using scaledStalk_reciprocity_inverse cs b basis t⁻¹ (inv_ne_zero ht) x
-
- theorem scaledStalk_reciprocity_upper {C : Type*} [Field C]
-    (t : C) (ht : t≠0) (a : W) (x : Interval cs a b) :
-    scaledStalk cs b basis t ((upperInclusion cs a b).vertex x)=
-      ∑ᶠ z : Interval cs a b,
-        eval₂ (Int.castRingHom C) (t⁻¹-t) (Hecke.rTilde cs x.val z.val)*
-          scaledStalk cs b basis t⁻¹ ((upperInclusion cs a b).vertex z) := by
-  classical
-  let := interval_finite cs a b
-  let := Fintype.ofFinite (Interval cs a b)
-  rw [scaledStalk_reciprocity cs b basis t ht, finsum_eq_sum_of_fintype]
-  let e : {z : Interval cs 1 b // BruhatLE cs a z.val} ≃ Interval cs a b :=
-    ⟨fun z => ⟨z.val.val,z.property,z.val.property.2⟩,
-      fun z => ⟨(upperInclusion cs a b).vertex z,z.property.1⟩,fun _ => rfl,fun _ => rfl⟩
-  let g (z : Interval cs 1 b) :=
-    eval₂ (Int.castRingHom C) (t⁻¹-t) (Hecke.rTilde cs x.val z.val)*scaledStalk cs b basis t⁻¹ z
-  change (∑ z : Interval cs 1 b, g z)=_
-  calc
-    _ = ∑ z ∈ Finset.univ.filter (fun z : Interval cs 1 b => BruhatLE cs a z.val),g z := by
-      symm
-      apply Finset.sum_subset (Finset.filter_subset _ _)
-      intro z _ hz
-      have hnot : ¬BruhatLE cs x.val z.val := by
-        intro hxz
-        exact hz (Finset.mem_filter.mpr ⟨Finset.mem_univ _,bruhat_trans cs x.property.1 hxz⟩)
-      simp [g,Hecke.rTilde_zero cs x.val z.val hnot]
-    _ = ∑ z : {z : Interval cs 1 b // BruhatLE cs a z.val}, g z.val :=
-      Finset.sum_subtype _ (by simp) g
-    _ = _ := Equiv.sum_comp e (fun z =>
-      eval₂ (Int.castRingHom C) (t⁻¹-t) (Hecke.rTilde cs x.val z.val)*
-        scaledStalk cs b basis t⁻¹ ((upperInclusion cs a b).vertex z))
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-/-! A specified morphism between honest minimal free covers is an isomorphism
-if it covers the identity. This will recognize the canonical restricted word
-form, not merely an arbitrary choice of self-duality. -/
-namespace KLInvariance.Graded.ModuleData
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-  [IsDomain A] [IsNoetherianRing A]
-  (hneg : ∀ n < 0, 𝓐 n = ⊥)
-  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
-  {T : ModuleData.{uk,ua,um} 𝓐}
-
-include hneg hzero in
- theorem MinimalCover.hom_bijective (C D : MinimalCover T)
-    (f : Hom C.source D.source) (hf : D.hom.map.comp f.map=C.hom.map) :
-    Function.Bijective f.map := by
-  let := C.free
-  let := D.free
-  obtain ⟨g,hg⟩ := exists_hom_lift hneg hzero C.hom C.surjective D.hom
-  have hfa (v : C.source) : D.hom.map (f.map v)=C.hom.map v := LinearMap.congr_fun hf v
-  have hga (v : D.source) : C.hom.map (g.map v)=D.hom.map v := LinearMap.congr_fun hg v
-  have hgf : Function.Bijective (g.comp f).map := by
-    apply Hom.bijective_of_mod_positive
-    intro m
-    apply C.minimal
-    change C.hom.map (m-g.map (f.map m))=0
-    rw [map_sub,hga,hfa,sub_self]
-  have hfg : Function.Bijective (f.comp g).map := by
-    apply Hom.bijective_of_mod_positive
-    intro m
-    apply D.minimal
-    change D.hom.map (m-f.map (g.map m))=0
-    rw [map_sub,hfa,hga,sub_self]
-  exact ⟨Function.Injective.of_comp hgf.1,Function.Surjective.of_comp hfg.2⟩
-
-end KLInvariance.Graded.ModuleData
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.Sheaf.AboveIso
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-  {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
-  [Fintype E] [IsNoetherianRing A] [IsDomain A]
-  (hneg : ∀ n < 0, 𝓐 n = ⊥)
-  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
-  {B C : GradedSheaf (𝓐 := 𝓐) G} {x : V}
-  (j : AboveIso B.forget C.forget x)
-
-include hneg hzero in
- theorem stalk_hom_bijective
-    (hj : ∀ e (h : x < G.target e) n v, v ∈ (B.edge e).piece n →
-      j.edge e h v ∈ (C.edge e).piece n)
-    (hBfree : Module.Free A (B.vertex x)) (hCfree : Module.Free A (C.vertex x))
-    (hBf : B.forget.Flabby) (hBg : B.forget.Generated)
-    (hCf : C.forget.Flabby) (hCg : C.forget.Generated)
-    (hBmin : B.KernelMinimal x) (hCmin : C.KernelMinimal x)
-    (f₀ : ModuleData.Hom (B.vertex x) (C.vertex x))
-    (hf₀ : ∀ v, j.boundary (B.forget.stalkMap x v) =
-      C.forget.stalkMap x (f₀.map v)) : Function.Bijective f₀.map := by
-  let T := ModuleData.range (C.stalkHom x)
-  let f : ModuleData.Hom (B.vertex x) T :=
-    { map := LinearMap.codRestrict (LinearMap.range (C.forget.stalkMap x))
-        (j.boundary.toLinearMap.comp (B.forget.stalkMap x)) (by
-          intro v
-          rw [← j.map_stalk_image hBf hBg hCf hCg]
-          exact ⟨B.forget.stalkMap x v,⟨v,rfl⟩,rfl⟩)
-      graded := by
-        intro n v hv e _
-        exact hj e.val _ n _ ((B.stalkHom x).graded n v hv e (Set.mem_univ e)) }
-  let g : ModuleData.Hom (C.vertex x) T :=
-    { map := (C.forget.stalkMap x).rangeRestrict
-      graded := fun n v hv => (C.stalkHom x).graded n v hv }
-  have hfs : Function.Surjective f.map := by
-    intro v
-    have hv : v.val ∈ (LinearMap.range (B.forget.stalkMap x)).map j.boundary.toLinearMap := by
-      rw [j.map_stalk_image hBf hBg hCf hCg]
-      exact v.property
-    obtain ⟨w,⟨z,rfl⟩,hz⟩ := hv
-    exact ⟨z,Subtype.ext hz⟩
-  have hgs : Function.Surjective g.map := by
-    intro v
-    obtain ⟨w,hw⟩ := v.property
-    exact ⟨w,Subtype.ext hw⟩
-  have hfm : LinearMap.ker f.map ≤ positiveIdeal 𝓐 • (⊤ : Submodule A (B.vertex x)) := by
-    intro v hv
-    apply (B.kernelMinimal_iff x).mp hBmin
-    apply LinearMap.mem_ker.mpr
-    apply j.boundary.injective
-    rw [map_zero]
-    have hh : f.map v = 0 := LinearMap.mem_ker.mp hv
-    exact congrArg (fun z : T => z.val) hh
-  have hgm : LinearMap.ker g.map ≤ positiveIdeal 𝓐 • (⊤ : Submodule A (C.vertex x)) := by
-    intro v hv
-    apply (C.kernelMinimal_iff x).mp hCmin
-    have hh : g.map v = 0 := LinearMap.mem_ker.mp hv
-    exact congrArg (fun z : T => z.val) hh
-  let F : ModuleData.MinimalCover T := ⟨B.vertex x,hBfree,f,hfs,hfm⟩
-  let D : ModuleData.MinimalCover T := ⟨C.vertex x,hCfree,g,hgs,hgm⟩
-  apply F.hom_bijective hneg hzero D f₀
-  apply LinearMap.ext
-  intro v
-  exact Subtype.ext (hf₀ v).symm
-
-end KLInvariance.MomentGraph.Sheaf.AboveIso
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
-  {G : OrderedGraph V E} {B C : GradedSheaf (𝓐 := 𝓐) G}
-noncomputable section
-
- theorem Hom.edge_bijective_of_vertex (f : Hom B C) (α : E → A)
-    (hB : B.UpperQuotient α) (hC : C.UpperQuotient α) (e : E)
-    (hv : Function.Bijective (f.vertex (G.target e)).map) :
-    Function.Bijective (f.edge e).map := by
-  let v := LinearEquiv.ofBijective (f.vertex (G.target e)).map hv
-  let j := QuotientEdge.iso (B.upper e).map (C.upper e).map
-    (hB e).1 (hC e).1 v (QuotientEdge.map_kernel_scalar _ _ (α e) (hB e).2 (hC e).2 v)
-  have he : (f.edge e).map=j.toLinearMap := by
-    apply LinearMap.ext
-    intro m
-    obtain ⟨m,rfl⟩ := (hB e).1 m
-    rw [f.upper]
-    exact (QuotientEdge.iso_apply (B.upper e).map (C.upper e).map
-      (hB e).1 (hC e).1 v
-      (QuotientEdge.map_kernel_scalar _ _ (α e) (hB e).2 (hC e).2 v) m).symm
-  rw [he]
-  exact j.bijective
-
- def Hom.isoOfBijective (f : Hom B C)
-    (hv : ∀ x, Function.Bijective (f.vertex x).map)
-    (he : ∀ e, Function.Bijective (f.edge e).map) : Iso B C where
-  vertex x := LinearEquiv.ofBijective (f.vertex x).map (hv x)
-  edge e := LinearEquiv.ofBijective (f.edge e).map (he e)
-  lower := f.lower
-  upper := f.upper
-  vertex_graded x := (f.vertex x).graded
-  edge_graded e := (f.edge e).graded
-
- def Hom.aboveIso (f : Hom B C) (x : V)
-    (hv : ∀ y, x<y → Function.Bijective (f.vertex y).map)
-    (he : ∀ e, x<G.target e → Function.Bijective (f.edge e).map) :
-    Sheaf.AboveIso B.forget C.forget x where
-  vertex y hy := LinearEquiv.ofBijective (f.vertex y).map (hv y hy)
-  edge e he' := LinearEquiv.ofBijective (f.edge e).map (he e he')
-  lower e _ := f.lower e
-  upper e _ := f.upper e
-
-end
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-/-! Invertibility detected at the top of the actual normalized minimal-boundary
-sheaf. This is stronger than uniqueness of its isomorphism class. -/
-namespace KLInvariance.MomentGraph.BoundarySheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-  [IsDomain A] [IsNoetherianRing A]
-  {hneg : ∀ n < 0, 𝓐 n = ⊥}
-  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
-  {V E : Type um} [PartialOrder V] [Fintype V] [Fintype E]
-  {G : OrderedGraph V E} {α : E → A} {b : V}
-  (B C : BoundarySheaf hneg G α b)
-noncomputable section
-
- include hzero in
- theorem hom_vertex_bijective (f : GradedSheaf.Hom B.sheaf C.sheaf)
-    (hb : Function.Bijective (f.vertex b).map) (x : V) :
-    Function.Bijective (f.vertex x).map := by
-  classical
-  refine wellFounded_gt.induction (C := fun y => Function.Bijective (f.vertex y).map) x ?_
-  intro x ih
-  by_cases hx : x=b
-  · subst x
-    exact hb
-  have he : ∀ e, x<G.target e → Function.Bijective (f.edge e).map :=
-    fun e h => f.edge_bijective_of_vertex α B.quotient C.quotient e (ih _ h)
-  let j := f.aboveIso x ih he
-  apply j.stalk_hom_bijective hneg hzero
-    (fun e h n v hv => (f.edge e).graded n v hv)
-    (B.free x) (C.free x) B.flabby.flabby B.generated C.flabby.flabby C.generated
-    (B.minimal x hx) (C.minimal x hx) (f.vertex x)
-  intro v
-  funext e
-  change (f.edge e.val).map (B.sheaf.forget.stalkMap x v e)=
-    C.sheaf.forget.stalkMap x ((f.vertex x).map v) e
-  rcases e with ⟨e,he⟩
-  subst x
-  exact f.lower e v
-
- def homIso (f : GradedSheaf.Hom B.sheaf C.sheaf)
-    (hb : Function.Bijective (f.vertex b).map) : GradedSheaf.Iso B.sheaf C.sheaf :=
-  f.isoOfBijective (hom_vertex_bijective hzero B C f hb)
-    (fun e => f.edge_bijective_of_vertex α B.quotient C.quotient e
-      (hom_vertex_bijective hzero B C f hb _))
-
-end
-end KLInvariance.MomentGraph.BoundarySheaf
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
-  {G : OrderedGraph V E} {B C : GradedSheaf (𝓐 := 𝓐) G}
-noncomputable section
-
- def Iso.hom (j : Iso B C) : Hom B C where
-  vertex x := ⟨(j.vertex x).toLinearMap,j.vertex_graded x⟩
-  edge e := ⟨(j.edge e).toLinearMap,j.edge_graded e⟩
-  lower := j.lower
-  upper := j.upper
-
- def Iso.invHom (j : Iso B C) : Hom C B where
-  vertex x := ⟨(j.vertex x).symm.toLinearMap,fun n v hv =>
-    (j.hom.vertex x).mem_of_map_mem (j.vertex x).injective n _
-      (by change j.vertex x ((j.vertex x).symm v) ∈ _
-          exact ((j.vertex x).apply_symm_apply v).symm ▸ hv)⟩
-  edge e := ⟨(j.edge e).symm.toLinearMap,fun n v hv =>
-    (j.hom.edge e).mem_of_map_mem (j.edge e).injective n _
-      (by change j.edge e ((j.edge e).symm v) ∈ _
-          exact ((j.edge e).apply_symm_apply v).symm ▸ hv)⟩
-  lower e v := by
-    apply (j.edge e).injective
-    exact ((j.edge e).apply_symm_apply _).trans
-      ((j.lower e ((j.vertex (G.source e)).symm v)).trans
-        (congrArg (C.forget.lower e) ((j.vertex (G.source e)).apply_symm_apply v))).symm
-  upper e v := by
-    apply (j.edge e).injective
-    exact ((j.edge e).apply_symm_apply _).trans
-      ((j.upper e ((j.vertex (G.target e)).symm v)).trans
-        (congrArg (C.forget.upper e) ((j.vertex (G.target e)).apply_symm_apply v))).symm
-
-end
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.BoundarySheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-  [IsDomain A] [IsNoetherianRing A]
-  {hneg : ∀ n < 0, 𝓐 n = ⊥}
-  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
-  {V E : Type um} [PartialOrder V] [Fintype V] [Fintype E]
-  {G : OrderedGraph V E} {α : E → A} {b : V}
-  (B : BoundarySheaf hneg G α b) {C : GradedSheaf (𝓐 := 𝓐) G}
-noncomputable section
-
- include hzero in
- theorem hom_vertex_bijective_of_iso (j : GradedSheaf.Iso B.sheaf C)
-    (f : GradedSheaf.Hom B.sheaf C) (hb : Function.Bijective (f.vertex b).map)
-    (x : V) : Function.Bijective (f.vertex x).map := by
-  let g := j.invHom.comp f
-  have hgb : Function.Bijective (g.vertex b).map := (j.vertex b).symm.bijective.comp hb
-  have hg := hom_vertex_bijective hzero B B g hgb x
-  constructor
-  · intro v w h
-    apply hg.1
-    exact congrArg (j.vertex x).symm h
-  · intro w
-    obtain ⟨v,hv⟩ := hg.2 ((j.vertex x).symm w)
-    exact ⟨v,(j.vertex x).symm.injective hv⟩
-
- def homIsoOfIso (j : GradedSheaf.Iso B.sheaf C) (hq : C.UpperQuotient α)
-    (f : GradedSheaf.Hom B.sheaf C) (hb : Function.Bijective (f.vertex b).map) :
-    GradedSheaf.Iso B.sheaf C :=
-  f.isoOfBijective (hom_vertex_bijective_of_iso hzero B j f hb)
-    (fun e => f.edge_bijective_of_vertex α B.quotient hq e
-      (hom_vertex_bijective_of_iso hzero B j f hb _))
-
-end
-end KLInvariance.MomentGraph.BoundarySheaf
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
-  {G : OrderedGraph V E} {B C D : GradedSheaf (𝓐 := 𝓐) G}
-  (s : SplitIn B D) (t : SplitIn C D)
-
- theorem SplitIn.comparison_bijective (x : V)
-    (hs : ∀ v, (s.projector.vertex x).map v=v)
-    (ht : ∀ v, (t.projector.vertex x).map v=v) :
-    Function.Bijective ((t.projection.comp s.inclusion).vertex x).map := by
-  let f := ((t.projection.comp s.inclusion).vertex x).map
-  let g := ((s.projection.comp t.inclusion).vertex x).map
-  have hgf (v : B.vertex x) : g (f v)=v := by
-    exact (congrArg (s.projection.vertex x).map (ht ((s.inclusion.vertex x).map v))).trans
-      (s.vertex_split x v)
-  have hfg (v : C.vertex x) : f (g v)=v := by
-    exact (congrArg (t.projection.vertex x).map (hs ((t.inclusion.vertex x).map v))).trans
-      (t.vertex_split x v)
-  exact ⟨Function.LeftInverse.injective hgf,Function.RightInverse.surjective hfg⟩
-
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-/-! The CANONICAL map from a normalized BMP summand to its adjoint image is
-invertible. Consequently the ambient WORD form restricts perfectly on this
-particular summand; no such assertion is made for an arbitrary summand. -/
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
-universe u us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  (l : List I) (hl : cs.IsReduced l)
-  (B : BoundarySheaf (symmetricGrading_negative basis)
-    (graph cs 1 (cs.wordProd l)) (symmetricLabel cs 1 (cs.wordProd l))
-    ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩)
-  (s : GradedSheaf.SplitIn B.sheaf (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
-noncomputable section
-
- def bottCanonicalAdjointHom : GradedSheaf.Hom B.sheaf
-    (bottSplitAdjointSheaf cs basis l hl B.sheaf s) :=
-  (bottSplitAdjointSplit cs basis l hl B s).projection.comp s.inclusion
-
- theorem bottCanonicalAdjointHom_top_bijective : Function.Bijective
-    ((bottCanonicalAdjointHom cs basis l hl B s).vertex
-      ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩).map := by
-  apply s.comparison_bijective (bottSplitAdjointSplit cs basis l hl B s)
-  · exact bottBoundarySplit_top_fixed cs basis l hl _ B s
-  · exact bottBoundarySplitAdjoint_top_fixed cs basis l hl _ B s
-
- def bottCanonicalAdjointIso : GradedSheaf.Iso B.sheaf
-    (bottSplitAdjointSheaf cs basis l hl B.sheaf s) := by
-  classical
-  let := symmetricCoefficientNoetherian basis
-  let := interval_finite cs 1 (cs.wordProd l)
-  let := Fintype.ofFinite (Interval cs 1 (cs.wordProd l))
-  let := Fintype.ofFinite (Edge cs 1 (cs.wordProd l))
-  let j := (bottBoundarySheaf_iso_adjoint cs basis l hl B s).some
-  exact B.homIsoOfIso (symmetricGrading_zero basis) j
-    ((bottSplitAdjointSplit cs basis l hl B s).upperQuotient
-      (bottGradedProjectionSheaf_upperQuotient cs basis l 1 (cs.wordProd l)))
-    (bottCanonicalAdjointHom cs basis l hl B s)
-    (bottCanonicalAdjointHom_top_bijective cs basis l hl B s)
-
- theorem bottCanonicalAdjointIso_vertex (x : Interval cs 1 (cs.wordProd l))
-    (v : B.sheaf.vertex x) :
-    ((bottCanonicalAdjointIso cs basis l hl B s).vertex x v).val=
-      ((bottAdjointSheaf cs basis l hl s.projector).vertex x).map
-        ((s.inclusion.vertex x).map v) := rfl
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! On the normalized BMP summand the canonical adjoint recognition proves,
-rather than assumes, perfection of the restricted symmetric word form. -/
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
-universe u us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  (l : List I) (hl : cs.IsReduced l)
-  (B : BoundarySheaf (symmetricGrading_negative basis)
-    (graph cs 1 (cs.wordProd l)) (symmetricLabel cs 1 (cs.wordProd l))
-    ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩)
-  (s : GradedSheaf.SplitIn B.sheaf (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
-noncomputable section
-
- theorem bottSplitRightWord_fixed (n : B.sheaf.forget.sections Set.univ) :
-    bottEndOfSheaf cs basis l hl s.projector (bottSplitRightWord cs basis l hl B.sheaf s n)=
-      bottSplitRightWord cs basis l hl B.sheaf s n := by
-  apply bottVertexProjection_injective cs l hl
-  funext x
-  change bottStalkProjection cs l x.val _=bottStalkProjection cs l x.val _
-  rw [bottEndOfSheaf_coordinates,bottSplitRightWord_coordinates]
-  change (s.inclusion.vertex x).map
-      ((s.projection.vertex x).map ((s.inclusion.vertex x).map (n.val x)))=
-    (s.inclusion.vertex x).map (n.val x)
-  exact congrArg (s.inclusion.vertex x).map (s.vertex_split x (n.val x))
-
- theorem bottCanonicalAdjointWord (m : B.sheaf.forget.sections Set.univ) :
-    bottSplitLeftWord cs basis l hl B.sheaf s
-      ((bottCanonicalAdjointIso cs basis l hl B s).toIso.sections Set.univ m)=
-    bottAdjoint cs l (bottEndOfSheaf cs basis l hl s.projector)
-      (bottSplitRightWord cs basis l hl B.sheaf s m) := by
-  apply bottVertexProjection_injective cs l hl
-  funext x
-  change bottStalkProjection cs l x.val _=bottStalkProjection cs l x.val _
-  rw [bottSplitLeftWord_coordinates]
-  exact (bottCanonicalAdjointIso_vertex cs basis l hl B s x (m.val x)).trans
-    ((congrArg ((bottAdjointSheaf cs basis l hl s.projector).vertex x).map
-      (bottSplitRightWord_coordinates cs basis l hl B.sheaf s m x).symm).trans
-      (bottAdjointSheaf_vertex cs basis l hl s.projector x _))
-
- def bottCanonicalPerfectPairing : GradedSheaf.PerfectPairing B.sheaf (l.length : ℤ) :=
-  bottRecognizedPerfectPairing cs basis l hl B.sheaf s
-    (bottCanonicalAdjointIso cs basis l hl B s)
-
- theorem bottCanonicalPerfectPairing_apply (m n : B.sheaf.forget.sections Set.univ) :
-    (bottCanonicalPerfectPairing cs basis l hl B s).pairing m n=
-      bottSamelsonDuality cs l (bottSplitRightWord cs basis l hl B.sheaf s m)
-        (bottSplitRightWord cs basis l hl B.sheaf s n) := by
-  change bottSplitAdjointEquivDual cs basis l hl B.sheaf s
-    ((bottCanonicalAdjointIso cs basis l hl B s).toIso.sections Set.univ m) n=_
-  rw [bottSplitAdjointWord_apply,bottCanonicalAdjointWord]
-  change bottSamelsonDuality cs l
-    (PerfectAdjoint.adjoint (bottSamelsonDuality cs l)
-      (bottEndOfSheaf cs basis l hl s.projector) _) _=_
-  rw [PerfectAdjoint.pairing,bottSplitRightWord_fixed]
-
- theorem bottCanonicalPerfectPairing_symmetric (m n : B.sheaf.forget.sections Set.univ) :
-    (bottCanonicalPerfectPairing cs basis l hl B s).pairing m n=
-      (bottCanonicalPerfectPairing cs basis l hl B s).pairing n m := by
-  exact (bottCanonicalPerfectPairing_apply cs basis l hl B s m n).trans
-    ((bottSamelson_symmetric cs l _ _).trans
-      (bottCanonicalPerfectPairing_apply cs basis l hl B s n m).symm)
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded Module
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
-  {G : OrderedGraph V E}
-noncomputable section
-
- structure SymmetricPerfectPairing (S : GradedSheaf (𝓐 := 𝓐) G) (d : ℤ)
-    extends PerfectPairing S d where
-  symmetric : ∀ m n, pairing m n=pairing n m
-
- def SymmetricPerfectPairing.pullback {S T : GradedSheaf (𝓐 := 𝓐) G} {d : ℤ}
-    (p : SymmetricPerfectPairing T d) (j : Iso S T) : SymmetricPerfectPairing S d where
-  toPerfectPairing := p.toPerfectPairing.pullback j
-  symmetric _m _n := p.symmetric _ _
-
- variable {B : Type ua} [CommRing B] [Algebra k B] {𝓑 : ℤ → Submodule k B}
-  (e : A ≃ₐ[k] B) (he : ∀ n a, a∈𝓐 n ↔ e a∈𝓑 n)
-
- def SymmetricPerfectPairing.coefficientChange {S : GradedSheaf (𝓐 := 𝓐) G} {d : ℤ}
-    (p : SymmetricPerfectPairing S d) : SymmetricPerfectPairing (coefficientChange e he S) d where
-  toPerfectPairing := p.toPerfectPairing.coefficientChange e he
-  symmetric _m _n := congrArg e (p.symmetric _ _)
-
-end
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-/-! Unconditional symmetric, integral, support-orthogonal graded perfection
-for the actual minimal-boundary sheaves; symmetry is inherited from the
-proved canonical-adjoint recognition, not an arbitrary choice of a dual. -/
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
-universe u us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- theorem symmetricBoundarySheaf_symmetricPairing_word (l : List I) (hl : cs.IsReduced l)
-    (B : BoundarySheaf (symmetricGrading_negative basis)
-      (graph cs 1 (cs.wordProd l)) (symmetricLabel cs 1 (cs.wordProd l))
-      ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩) :
-    Nonempty (GradedSheaf.SymmetricPerfectPairing B.sheaf (cs.length (cs.wordProd l) : ℤ)) := by
-  obtain ⟨s⟩ := bottBoundarySheaf_splitIn cs basis l hl B
-  have hd : cs.length (cs.wordProd l)=l.length := hl
-  rw [hd]
-  exact ⟨{ toPerfectPairing := bottCanonicalPerfectPairing cs basis l hl B s
-           symmetric := bottCanonicalPerfectPairing_symmetric cs basis l hl B s }⟩
-
- theorem symmetricBoundarySheaf_symmetricPairing (b : W) :
-    Nonempty (GradedSheaf.SymmetricPerfectPairing
-      (symmetricBoundarySheaf cs basis b).sheaf (cs.length b : ℤ)) := by
-  obtain ⟨l,hl,he⟩ := cs.exists_isReduced b
-  subst b
-  exact symmetricBoundarySheaf_symmetricPairing_word cs basis l hl _
-
-end
-end KLInvariance.TitsSpace
-
-namespace KLInvariance.BruhatGraph
-open Module _root_.OAI.KLInvariance.Graded MomentGraph TitsSpace
-universe u
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W) (b : W)
-  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- theorem boundarySheaf_symmetricPairing :
-    Nonempty (GradedSheaf.SymmetricPerfectPairing
-      (boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf (cs.length b : ℤ)) := by
-  classical
-  let := interval_finite cs 1 b
-  let := Fintype.ofFinite (Interval cs 1 b)
-  let := Fintype.ofFinite (Edge cs 1 b)
-  let S := symmetricBoundarySheaf cs basis b
-  let T := boundarySheaf cs 1 b basis (one_bruhat cs b)
-  let e := SymmetricAlgebra.equivMvPolynomial basis
-  have he : ∀ n a, a∈symmetricGrading basis n ↔ e a∈polynomialGrading ℝ σ n :=
-    mem_symmetricGrading basis
-  obtain ⟨p⟩ := symmetricBoundarySheaf_symmetricPairing cs basis b
-  obtain ⟨j⟩ := BoundarySheaf.iso_coefficientChange e he S T
-    (polynomialGrading_zero ℝ σ) (fun x => x.property.2)
-  exact ⟨(p.coefficientChange e he).pullback j⟩
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-/-! The exact degree mechanism needed for the character theorem. Minimality
-puts every local intersection coordinate in the positive ideal. A nonzero
-homogeneous diagonal then forces the strict half-degree inequality. This
-lemma DOES NOT assert the local Hodge/nonzero-diagonal input. -/
-namespace KLInvariance.Graded
-open Module
-universe uk ua um ui
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  (𝓐 : ℤ → Submodule k A) [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-  (hneg : ∀ n < 0, 𝓐 n = ⊥)
-
- include hneg in
- theorem homogeneous_positive_nonzero {n : ℤ} {a : A}
-    (ha : a∈𝓐 n) (hp : a∈positiveIdeal 𝓐) (hn : a≠0) : 0<n := by
-  by_contra hh
-  have hnle : n≤0 := le_of_not_gt hh
-  by_cases hz : n=0
-  · subst n
-    rw [positiveIdeal_eq_augmentation_ker 𝓐 hneg] at hp
-    have hc : component 𝓐 0 a=0 := hp
-    have he : component 𝓐 0 a=a := DirectSum.decompose_of_mem_same 𝓐 ha
-    exact hn (he.symm.trans hc)
-  · have hlt : n<0 := lt_of_le_of_ne hnle hz
-    rw [hneg n hlt,Submodule.mem_bot] at ha
-    exact hn ha
-
-variable {M : Type um} [AddCommGroup M] [Module A M]
-
-omit [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐] in
- theorem dual_mem_ideal (J : Ideal A) (f : Dual A M) {m : M}
-    (hm : m∈J • (⊤ : Submodule A M)) : f m∈J := by
-  refine Submodule.smul_induction_on hm ?_ ?_
-  · intro a ha v _
-    rw [map_smul]
-    exact J.mul_mem_right (f v) ha
-  · intro u v hu hv
-    rw [map_add]
-    exact J.add_mem hu hv
-
-variable [Module k M] [IsScalarTower k A M]
-  (𝓜 : ℤ → Submodule k M) [DirectSum.Decomposition 𝓜] [SetLike.GradedSMul 𝓐 𝓜]
-  {ι : Type ui} [Fintype ι] [DecidableEq ι] (b : Basis ι A M) (d : ι → ℤ)
-  (hb : ∀ i, b i∈𝓜 (d i)) (D : ℤ) (Q : Dual A M →ₗ[A] M)
-
- include hneg hb in
- omit [IsScalarTower k A M] in
- theorem local_diagonal_strict_degree
-    (hq : ∀ i, Q (b.dualBasis i)∈𝓜 (D-d i))
-    (hmin : LinearMap.range Q ≤ positiveIdeal 𝓐 • (⊤ : Submodule A M))
-    (i : ι) (hne : b.dualBasis i (Q (b.dualBasis i))≠0) : 2*d i<D := by
-  have hc : b.dualBasis i (Q (b.dualBasis i))∈𝓐 (D-d i+(-d i)) :=
-    dualBasis_homogeneous 𝓐 𝓜 b d hb i _ _ (hq i)
-  have hp : b.dualBasis i (Q (b.dualBasis i))∈positiveIdeal 𝓐 :=
-    dual_mem_ideal (positiveIdeal 𝓐) (b.dualBasis i) (hmin ⟨b.dualBasis i,rfl⟩)
-  have hh := homogeneous_positive_nonzero 𝓐 hneg hc hp hne
-  omega
-
-end KLInvariance.Graded
-
-end
-
-
-section
-
-/-! Integral evaluation compatibility of the actual support/dual equivalence. -/
-namespace KLInvariance.IntegralSupport
-universe ur uv um
-variable {R : Type ur} [CommRing R] {V : Type uv} [DecidableEq V]
-  {A : V → Type um} [∀ x, AddCommGroup (A x)] [∀ x, Module R (A x)]
-  (M : Submodule R (∀ x, A x)) (p : R)
-  (hcut : ∀ x (m : M), (p • Pi.single x (m.val x)) ∈ M)
-  [IsDomain R] [∀ x, Module.IsTorsionFree R (A x)] (hp : p≠0)
-  (d : M ≃ₗ[R] Module.Dual R M)
-  (hbalanced : ∀ x m n, d (cut M p hcut x m) n=d m (cut M p hcut x n))
-  (x : V) (hπ : Function.Surjective (projection M x))
-noncomputable section
-
- theorem supportEquivDual_pairing (m : supported M x) (n : M) :
-    supportEquivDual M p hcut hp d hbalanced x hπ m (projection M x n)=d m.val n := by
-  let e := supportEquivDual M p hcut hp d hbalanced x hπ
-  have he := congrArg (fun s : supported M x => d s.val n) (e.symm_apply_apply m)
-  change d (d.symm ((projection M x).dualMap (e m))) n=d m.val n at he
-  rw [LinearEquiv.apply_symm_apply] at he
-  exact he
-
-end
-end KLInvariance.IntegralSupport
-
-end
-
-
-section
-
-/-! Evaluation and symmetry of the actual integral local intersection map.
-The support factor is explicitly cancelled only after its nonvanishing is proved. -/
-
-end
-
-end OAI
+import OAI.RepresentationTheory.KazhdanLusztig.PlaneLocalization
+
+namespace OAI
+
+/-!
+Bruhat sheaf localization, graded transport, local edge-image bounds and support.
+-/
+
+section
+
+namespace KLInvariance.BruhatGraph
+open TitsSpace SchedulePaths
+universe u v u' v'
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
+  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
+  {u b : W} {u' b' : W'}
+  (φ : Interval cs u b ≃o Interval cs' u' b')
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+ def transportedEdge (e : Edge cs u b) : AdaptedDyer.Edge cs' :=
+  ⟨((φ (source cs u b e)).val,(φ (target cs u b e)).val),
+    (intervalIso_bruhatStep_iff_general cs cs' φ _ _).mpr e.property⟩
+
+omit [Fintype I] [Fintype I'] in
+ theorem transportedEdge_injective : Function.Injective (transportedEdge φ) := by
+  intro e f h
+  apply Subtype.ext
+  have hh := congrArg Subtype.val h
+  exact Prod.ext (φ.injective (Subtype.ext (congrArg Prod.fst hh)))
+    (φ.injective (Subtype.ext (congrArg Prod.snd hh)))
+
+ def transportedScore (e : Edge cs u b) : ℝ ×ₗ ℝ :=
+  lexEdgeScore (geometricFunctional (M := M') (cs := cs')) (transportedEdge φ e)
+
+ /-- Decreasing list, in the convention for the outermost-first matrix
+ product. Its reverse is chronological increasing reflection order. -/
+ def transportedSchedule : List (Edge cs u b) := by
+  letI := Fintype.ofFinite (Edge cs u b)
+  exact sortedSchedule Finset.univ (transportedScore φ)
+
+omit [Fintype I] in
+ theorem transportedSchedule_nodup : (transportedSchedule φ).Nodup := by
+  unfold transportedSchedule
+  exact sortedSchedule_nodup _ _
+
+omit [Fintype I] in
+ theorem transportedSchedule_full (e : Edge cs u b) : e ∈ transportedSchedule φ := by
+  let := Fintype.ofFinite (Edge cs u b)
+  unfold transportedSchedule
+  exact (mem_sortedSchedule _ _ _).mpr (Finset.mem_univ _)
+
+omit [Fintype I] in
+ theorem transportedSchedule_pairwise : (transportedSchedule φ).Pairwise
+     (fun e f => transportedScore φ f ≤ transportedScore φ e) := by
+  unfold transportedSchedule
+  exact sortedSchedule_pairwise _ _
+
+omit [Fintype I] [Fintype I'] in
+ theorem transportedEdge_lift (e : AdaptedDyer.Edge cs')
+    (hu : BruhatLE cs' u' (AdaptedDyer.source cs' e))
+    (hb : BruhatLE cs' (AdaptedDyer.target cs' e) b') : ∃ f, transportedEdge φ f=e := by
+  let x : Interval cs' u' b' := ⟨AdaptedDyer.source cs' e,hu,
+    (show BruhatLE cs' (AdaptedDyer.source cs' e) (AdaptedDyer.target cs' e) from
+      .single e.property).trans hb⟩
+  let y : Interval cs' u' b' := ⟨AdaptedDyer.target cs' e,
+    hu.trans (.single e.property),hb⟩
+  let f : Edge cs u b := ⟨(φ.symm x,φ.symm y),
+    (intervalIso_bruhatStep_iff_general cs' cs φ.symm x y).mpr e.property⟩
+  refine ⟨f,?_⟩
+  apply Subtype.ext
+  change ((φ (φ.symm x)).val,(φ (φ.symm y)).val)=e.val
+  rw [φ.apply_symm_apply,φ.apply_symm_apply]
+  rfl
+
+omit [Fintype I] in
+ theorem transported_dispatch_entry {R : Type*} [Semiring R] (T : R)
+    (x y : Interval cs u b) :
+    dispatch (source cs u b) (target cs u b) T (fun z => if z=y then 1 else 0)
+      (transportedSchedule φ) x =
+    WallDyer.entry (cs := cs') T (geometricFunctional (M := M') (cs := cs'))
+      b' (φ x).val (φ y).val := by
+  have hsource {z w : W'} {p : List (AdaptedDyer.Edge cs')}
+      (hp : IsPath (AdaptedDyer.source cs') (AdaptedDyer.target cs') z w p) :
+      ∀ e ∈ p, BruhatLE cs' z (AdaptedDyer.source cs' e) := by
+    induction p generalizing z with
+    | nil => simp
+    | cons e p ih =>
+      intro f hf
+      rcases List.mem_cons.mp hf with rfl | hf
+      · exact hp.1 ▸ bruhat_refl cs' z
+      · rw [← hp.1]
+        exact (show BruhatLE cs' (AdaptedDyer.source cs' e)
+          (AdaptedDyer.target cs' e) from .single e.property).trans (ih hp.2 f hf)
+  symm
+  apply dispatch_entry_graph_embedding (AdaptedDyer.source cs') (AdaptedDyer.target cs')
+    (source cs u b) (target cs u b) T
+    (lexEdgeScore (geometricFunctional (M := M') (cs := cs')))
+    (lexEdgeScore_ties _) (fun z : Interval cs u b => (φ z).val)
+    (fun _ _ h => φ.injective (Subtype.ext h)) (transportedEdge φ)
+    (transportedEdge_injective φ) (fun _ => rfl) (fun _ => rfl)
+    (WallDyer.schedule (geometricFunctional (M := M') (cs := cs')) b')
+    (transportedSchedule φ) (sortedSchedule_nodup _ _)
+    (transportedSchedule_nodup φ) (sortedSchedule_pairwise _ _)
+    (transportedSchedule_pairwise φ) (transportedSchedule_full φ)
+  · intro e
+    exact (mem_sortedSchedule _ _ _).mpr ((AdaptedDyer.mem_edgeFinset cs' b' _).mpr
+      (φ (target cs u b e)).property.2)
+  · intro r hr e he
+    have hp := ((mem_subpaths _ _ _ _ _ r).mp hr).2
+    exact transportedEdge_lift φ e ((φ x).property.1.trans (hsource hp e he))
+      ((AdaptedDyer.path_targets_le cs' hp e he).trans (φ y).property.2)
+
+omit [Fintype I] in
+
+ theorem transported_entry_eq_rTilde (x y : Interval cs u b) :
+    dispatch (source cs u b) (target cs u b) (Polynomial.X : Polynomial ℤ)
+      (fun z => if z=y then 1 else 0) (transportedSchedule φ) x =
+      Hecke.rTilde cs' (φ x).val (φ y).val := by
+  rw [transported_dispatch_entry]
+  exact WallDyer.geometric_entry_eq_rTilde b' (φ x).val (φ y).val
+    (φ x).property.2 (φ y).property.2
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+/-! Actual transported numerical R-tilde matrix and its genuine KL reciprocal
+vector. No sheaf or desired invariance conclusion is assumed. -/
+namespace KLInvariance.SchedulePaths
+universe u v w w'
+variable {V : Type u} {E : Type v} {R : Type w} {S : Type w'}
+  [Semiring R] [Semiring S] (source target : E → V)
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+ theorem dispatch_map (f : R →+* S) (T : R) (v : V → R) (p : List E) (x : V) :
+    f (dispatch source target T v p x) =
+      dispatch source target (f T) (fun z => f (v z)) p x := by
+  induction p generalizing x with
+  | nil => rfl
+  | cons e p ih =>
+    simp only [dispatch,step,map_add]
+    split_ifs <;> simp only [map_mul,map_zero,ih]
+
+end
+end KLInvariance.SchedulePaths
+
+namespace KLInvariance
+universe u v
+variable {I : Type u} {M : CoxeterMatrix I} {W : Type v} [Group W]
+  (cs : CoxeterSystem M W)
+noncomputable section
+
+ theorem scaledKL_reciprocity_upper {C : Type*} [Field C]
+    (t : C) (ht : t ≠ 0) (u b : W) (x : Interval cs u b) :
+    scaledKL cs t x.val b = ∑ᶠ z : Interval cs u b,
+      Polynomial.eval₂ (Int.castRingHom C) (t⁻¹-t) (Hecke.rTilde cs x.val z.val) *
+        scaledKL cs t⁻¹ z.val b := by
+  classical
+  let := interval_finite cs u b
+  let := Fintype.ofFinite (Interval cs u b)
+  let := interval_finite cs x.val b
+  let := Fintype.ofFinite (Interval cs x.val b)
+  rw [scaledKL_reciprocity cs t ht x.val b x.property.2,
+    finsum_eq_sum_of_fintype,finsum_eq_sum_of_fintype]
+  let f : W → C := fun z => Polynomial.eval₂ (Int.castRingHom C) (t⁻¹-t)
+    (Hecke.rTilde cs x.val z) * scaledKL cs t⁻¹ z b
+  have hs (a : W) [Fintype (Interval cs a b)] :
+      (∑ z : Interval cs a b, f z.val) =
+        ∑ z ∈ (lowerFinset cs b).filter (fun z => BruhatLE cs a z), f z := by
+    exact (Finset.sum_subtype (F := (inferInstance : Fintype (Interval cs a b)))
+      ((lowerFinset cs b).filter (fun z => BruhatLE cs a z))
+      (fun z => by simp only [Finset.mem_filter,mem_lowerFinset]; exact and_comm) f).symm
+  change (∑ z : Interval cs x.val b, f z.val) = ∑ z : Interval cs u b, f z.val
+  rw [hs,hs]
+  apply Finset.sum_subset
+  · intro z hz
+    rcases Finset.mem_filter.mp hz with ⟨hz,hxz⟩
+    exact Finset.mem_filter.mpr ⟨hz,x.property.1.trans hxz⟩
+  · intro z hz hn
+    have hxz : ¬ BruhatLE cs x.val z := by
+      intro h
+      exact hn (Finset.mem_filter.mpr ⟨(Finset.mem_filter.mp hz).1,h⟩)
+    dsimp [f]
+    rw [Hecke.rTilde_zero cs _ _ hxz,Polynomial.eval₂_zero,zero_mul]
+
+end
+end KLInvariance
+
+namespace KLInvariance.BruhatGraph
+open TitsSpace SchedulePaths
+universe u v u' v'
+variable {I : Type u} {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
+  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
+  {u b : W} {u' b' : W'}
+  (φ : Interval cs u b ≃o Interval cs' u' b')
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+ theorem transported_entry_eval {C : Type*} [CommSemiring C] (T : C)
+    (x y : Interval cs u b) (f : ℤ →+* C) :
+    dispatch (source cs u b) (target cs u b) T
+      (fun z => if z=y then 1 else 0) (transportedSchedule φ) x =
+      Polynomial.eval₂ f T (Hecke.rTilde cs' (φ x).val (φ y).val) := by
+  rw [← transported_entry_eq_rTilde φ x y]
+  have h := dispatch_map (source cs u b) (target cs u b) (Polynomial.eval₂RingHom f T)
+    Polynomial.X (fun z : Interval cs u b => if z=y then 1 else 0) (transportedSchedule φ) x
+  simpa using h.symm
+
+ theorem transported_dispatch_matrix {C : Type*} [CommSemiring C] (T : C)
+    (f : ℤ →+* C) (v : Interval cs u b → C) (x : Interval cs u b) :
+    dispatch (source cs u b) (target cs u b) T v (transportedSchedule φ) x =
+      ∑ᶠ y : Interval cs u b,
+        Polynomial.eval₂ f T (Hecke.rTilde cs' (φ x).val (φ y).val) * v y := by
+  let := interval_finite cs u b
+  let := Fintype.ofFinite (Interval cs u b)
+  rw [dispatch_eq_sum_entries,finsum_eq_sum_of_fintype]
+  exact Finset.sum_congr rfl (fun y _ => congrArg (fun a => a*v y)
+    (transported_entry_eval φ T x y f))
+
+ theorem transported_reciprocal_vector {C : Type*} [Field C]
+    (t : C) (ht : t ≠ 0) (x : Interval cs u b) :
+    scaledKL cs' t (φ x).val b' =
+      dispatch (source cs u b) (target cs u b) (t⁻¹-t)
+        (fun y => scaledKL cs' t⁻¹ (φ y).val b') (transportedSchedule φ) x := by
+  let := interval_finite cs u b
+  let := Fintype.ofFinite (Interval cs u b)
+  let := interval_finite cs' u' b'
+  let := Fintype.ofFinite (Interval cs' u' b')
+  rw [transported_dispatch_matrix φ _ (Int.castRingHom C),
+    scaledKL_reciprocity_upper cs' t ht u' b' (φ x),
+    finsum_eq_sum_of_fintype,finsum_eq_sum_of_fintype]
+  exact (φ.toEquiv.sum_comp _).symm
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  (hub : BruhatLE cs u b)
+  (C : BoundarySheaf (polynomialGrading_negative ℝ σ)
+    (graph cs u b) (polynomialLabel cs u b basis) ⟨b,hub,bruhat_refl cs b⟩)
+noncomputable section
+
+ theorem boundarySheaf_gradedIso :
+    Nonempty (GradedSheaf.Iso (boundarySheaf cs u b basis hub).sheaf C.sheaf) := by
+  classical
+  let : Finite (Interval cs u b) := interval_finite cs u b
+  let : Fintype (Interval cs u b) := Fintype.ofFinite _
+  let : Fintype (Edge cs u b) := Fintype.ofFinite _
+  exact BoundarySheaf.exists_gradedIso (polynomialGrading_zero ℝ σ)
+    (boundarySheaf cs u b basis hub) C (fun x => x.property.2)
+
+ theorem stalkCharacter_eq_character (x : Interval cs u b) :
+    let _ := C.free x
+    stalkCharacter cs u b basis hub x =
+      Graded.character (σ := σ) (C.sheaf.vertex x).piece (C.sheaf.vertex x).nonneg := by
+  let _ := C.free x
+  let B := boundarySheaf cs u b basis hub
+  let _ := B.free x
+  obtain ⟨j⟩ := boundarySheaf_gradedIso cs u b basis hub C
+  exact ModuleData.character_eq_of_gradedEquiv (j.vertex x) (j.vertex_graded x)
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+/-! Faithful upper-interval restriction of the actual minimal boundary sheaf.
+The bottom is arbitrary. No roots or labels are supplied by the interval iso. -/
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
+noncomputable section
+
+ def upperInclusion : GraphMap (graph cs u b) (graph cs 1 b) where
+  vertex x := ⟨x.val,one_bruhat cs x.val,x.property.2⟩
+  edge e := ⟨(⟨(source cs u b e).val,one_bruhat cs _,(source cs u b e).property.2⟩,
+    ⟨(target cs u b e).val,one_bruhat cs _,(target cs u b e).property.2⟩),e.property⟩
+  source _ := rfl
+  target _ := rfl
+
+omit [Fintype I] in
+ theorem upperInclusion_injective : Function.Injective (upperInclusion cs u b).vertex := by
+  intro x y h
+  exact Subtype.ext (congrArg (fun z : Interval cs 1 b => z.val) h)
+
+omit [Fintype I] in
+ theorem upperInclusion_outgoing (e : Edge cs 1 b) (x : Interval cs u b)
+    (hx : (upperInclusion cs u b).vertex x = source cs 1 b e) :
+    ∃ a, (upperInclusion cs u b).edge a=e ∧ source cs u b a=x := by
+  have hxval := congrArg Subtype.val hx
+  let y : Interval cs u b := ⟨(target cs 1 b e).val,
+    bruhat_trans cs x.property.1 (hxval.symm ▸ Relation.ReflTransGen.single e.property),
+    (target cs 1 b e).property.2⟩
+  let a : Edge cs u b := ⟨(x,y),hxval.symm ▸ e.property⟩
+  refine ⟨a,?_,rfl⟩
+  apply Subtype.ext
+  apply Prod.ext
+  · exact hx
+  · rfl
+
+variable {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  (hub : BruhatLE cs u b)
+  (B : BoundarySheaf (polynomialGrading_negative ℝ σ)
+    (graph cs 1 b) (polynomialLabel cs 1 b basis) ⟨b,one_bruhat cs b,bruhat_refl cs b⟩)
+
+ def restrictBoundary : BoundarySheaf (polynomialGrading_negative ℝ σ)
+    (graph cs u b) (polynomialLabel cs u b basis) ⟨b,hub,bruhat_refl cs b⟩ :=
+  B.pullback (upperInclusion cs u b) (upperInclusion_injective cs u b)
+    (upperInclusion_outgoing cs u b) ⟨b,hub,bruhat_refl cs b⟩ rfl
+
+ theorem stalkCharacter_upperRestriction (x : Interval cs u b) :
+    let _ := B.free ((upperInclusion cs u b).vertex x)
+    stalkCharacter cs u b basis hub x =
+      Graded.character (σ := σ) (B.sheaf.vertex ((upperInclusion cs u b).vertex x)).piece
+        (B.sheaf.vertex ((upperInclusion cs u b).vertex x)).nonneg :=
+  stalkCharacter_eq_character cs u b basis hub (restrictBoundary cs u b basis hub B) x
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph Polynomial SchedulePaths
+universe u
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W) (b : W)
+  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+noncomputable section
+local instance scaledInterval : Fintype (Interval cs 1 b) := by
+  let := interval_finite cs 1 b
+  exact Fintype.ofFinite _
+
+ theorem stalkCharacter_natDegree_le (x : Interval cs 1 b) :
+    (stalkCharacter cs 1 b basis (one_bruhat cs b) x).natDegree≤rankDifference cs x.val b := by
+  classical
+  let B := boundarySheaf cs 1 b basis (one_bruhat cs b)
+  let := B.free x
+  let c := (exists_nonnegativeBasis (σ := σ) (B.sheaf.vertex x).piece
+    (B.sheaf.vertex x).nonneg).some
+  rw [stalkCharacter_eq_character cs 1 b basis (one_bruhat cs b) B x,
+    character_eq (B.sheaf.vertex x).piece (B.sheaf.vertex x).nonneg c]
+  apply Polynomial.natDegree_sum_le_of_forall_le
+  intro i _
+  simpa only [Polynomial.natDegree_X_pow] using stalkBasis_degree_le cs b basis x c i
+
+ def scaledStalk {C : Type*} [Field C] (t : C) (x : Interval cs 1 b) : C :=
+  t^((cs.length x.val : ℤ)-cs.length b)*
+    eval₂ (Int.castRingHom C) (t^2) (stalkCharacter cs 1 b basis (one_bruhat cs b) x)
+
+ theorem scaledStalk_reciprocity_inverse {C : Type*} [Field C]
+    (v : C) (hv : v≠0) (x : Interval cs 1 b) :
+    scaledStalk cs b basis v⁻¹ x=∑ z : Interval cs 1 b,
+      eval₂ (Int.castRingHom C) (v-v⁻¹) (Hecke.rTilde cs x.val z.val)*
+        scaledStalk cs b basis v z := by
+  classical
+  have hrec := stalkCharacter_reciprocity cs basis b x
+  let q : Cˣ := Units.mk0 (v^2) (pow_ne_zero 2 hv)
+  have heval := congrArg (eval₂ (Int.castRingHom C) (q : C)) hrec
+  rw [eval₂_reflect_unit q _ _ (stalkCharacter_natDegree_le cs b basis x)] at heval
+  simp only [eval₂_finsetSum,eval₂_mul] at heval
+  have hq : (q : C)=v^2 := rfl
+  have hqi : (↑q⁻¹ : C)=(v⁻¹)^2 := by simp [q,inv_pow]
+  rw [hq,hqi] at heval
+  have heq := congrArg (fun a : C => v^((cs.length x.val : ℤ)-cs.length b)*a) heval
+  have hd : (rankDifference cs x.val b : ℤ)=(cs.length b : ℤ)-cs.length x.val := by
+    simp [rankDifference,Int.ofNat_sub (length_le_of_bruhat cs x.property.2)]
+  have hscale : v^((cs.length x.val : ℤ)-cs.length b)*(v^2)^(rankDifference cs x.val b)=
+      (v⁻¹)^((cs.length x.val : ℤ)-cs.length b) := by
+    rw [←pow_mul,←zpow_natCast,←zpow_add₀ hv,inv_zpow,←zpow_neg]
+    congr 1
+    push_cast
+    omega
+  rw [←mul_assoc,hscale,Finset.mul_sum] at heq
+  change (v⁻¹)^((cs.length x.val : ℤ)-cs.length b)*
+    eval₂ (Int.castRingHom C) ((v⁻¹)^2) (stalkCharacter cs 1 b basis (one_bruhat cs b) x)=_
+  rw [heq]
+  apply Finset.sum_congr rfl
+  intro z _
+  have hr : OrdinaryR.polynomial cs x.val z.val=rPolynomial cs x.val z.val :=
+    (rPolynomial_eq_constructed cs x.val z.val).symm
+  rw [hr,rPolynomial_normalization cs v hv]
+  unfold scaledStalk
+  calc
+    _ = (v^((cs.length x.val : ℤ)-cs.length b)*v^((cs.length z.val : ℤ)-cs.length x.val))*
+        eval₂ (Int.castRingHom C) (v-v⁻¹) (Hecke.rTilde cs x.val z.val)*
+        eval₂ (Int.castRingHom C) (v^2) (stalkCharacter cs 1 b basis (one_bruhat cs b) z) := by ring
+    _ = _ := by
+      rw [←zpow_add₀ hv,show (cs.length x.val : ℤ)-cs.length b+
+        ((cs.length z.val : ℤ)-cs.length x.val)=(cs.length z.val : ℤ)-cs.length b by ring]
+      ring
+
+ theorem scaledStalk_reciprocity {C : Type*} [Field C]
+    (t : C) (ht : t≠0) (x : Interval cs 1 b) :
+    scaledStalk cs b basis t x=∑ z : Interval cs 1 b,
+      eval₂ (Int.castRingHom C) (t⁻¹-t) (Hecke.rTilde cs x.val z.val)*
+        scaledStalk cs b basis t⁻¹ z := by
+  simpa only [inv_inv] using scaledStalk_reciprocity_inverse cs b basis t⁻¹ (inv_ne_zero ht) x
+
+ theorem scaledStalk_reciprocity_upper {C : Type*} [Field C]
+    (t : C) (ht : t≠0) (a : W) (x : Interval cs a b) :
+    scaledStalk cs b basis t ((upperInclusion cs a b).vertex x)=
+      ∑ᶠ z : Interval cs a b,
+        eval₂ (Int.castRingHom C) (t⁻¹-t) (Hecke.rTilde cs x.val z.val)*
+          scaledStalk cs b basis t⁻¹ ((upperInclusion cs a b).vertex z) := by
+  classical
+  let := interval_finite cs a b
+  let := Fintype.ofFinite (Interval cs a b)
+  rw [scaledStalk_reciprocity cs b basis t ht, finsum_eq_sum_of_fintype]
+  let e : {z : Interval cs 1 b // BruhatLE cs a z.val} ≃ Interval cs a b :=
+    ⟨fun z => ⟨z.val.val,z.property,z.val.property.2⟩,
+      fun z => ⟨(upperInclusion cs a b).vertex z,z.property.1⟩,fun _ => rfl,fun _ => rfl⟩
+  let g (z : Interval cs 1 b) :=
+    eval₂ (Int.castRingHom C) (t⁻¹-t) (Hecke.rTilde cs x.val z.val)*scaledStalk cs b basis t⁻¹ z
+  change (∑ z : Interval cs 1 b, g z)=_
+  calc
+    _ = ∑ z ∈ Finset.univ.filter (fun z : Interval cs 1 b => BruhatLE cs a z.val),g z := by
+      symm
+      apply Finset.sum_subset (Finset.filter_subset _ _)
+      intro z _ hz
+      have hnot : ¬BruhatLE cs x.val z.val := by
+        intro hxz
+        exact hz (Finset.mem_filter.mpr ⟨Finset.mem_univ _,bruhat_trans cs x.property.1 hxz⟩)
+      simp [g,Hecke.rTilde_zero cs x.val z.val hnot]
+    _ = ∑ z : {z : Interval cs 1 b // BruhatLE cs a z.val}, g z.val :=
+      Finset.sum_subtype _ (by simp) g
+    _ = _ := Equiv.sum_comp e (fun z =>
+      eval₂ (Int.castRingHom C) (t⁻¹-t) (Hecke.rTilde cs x.val z.val)*
+        scaledStalk cs b basis t⁻¹ ((upperInclusion cs a b).vertex z))
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+/-! A specified morphism between honest minimal free covers is an isomorphism
+if it covers the identity. This will recognize the canonical restricted word
+form, not merely an arbitrary choice of self-duality. -/
+namespace KLInvariance.Graded.ModuleData
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+  [IsDomain A] [IsNoetherianRing A]
+  (hneg : ∀ n < 0, 𝓐 n = ⊥)
+  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
+  {T : ModuleData.{uk,ua,um} 𝓐}
+
+include hneg hzero in
+ theorem MinimalCover.hom_bijective (C D : MinimalCover T)
+    (f : Hom C.source D.source) (hf : D.hom.map.comp f.map=C.hom.map) :
+    Function.Bijective f.map := by
+  let := C.free
+  let := D.free
+  obtain ⟨g,hg⟩ := exists_hom_lift hneg hzero C.hom C.surjective D.hom
+  have hfa (v : C.source) : D.hom.map (f.map v)=C.hom.map v := LinearMap.congr_fun hf v
+  have hga (v : D.source) : C.hom.map (g.map v)=D.hom.map v := LinearMap.congr_fun hg v
+  have hgf : Function.Bijective (g.comp f).map := by
+    apply Hom.bijective_of_mod_positive
+    intro m
+    apply C.minimal
+    change C.hom.map (m-g.map (f.map m))=0
+    rw [map_sub,hga,hfa,sub_self]
+  have hfg : Function.Bijective (f.comp g).map := by
+    apply Hom.bijective_of_mod_positive
+    intro m
+    apply D.minimal
+    change D.hom.map (m-f.map (g.map m))=0
+    rw [map_sub,hfa,hga,sub_self]
+  exact ⟨Function.Injective.of_comp hgf.1,Function.Surjective.of_comp hfg.2⟩
+
+end KLInvariance.Graded.ModuleData
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.Sheaf.AboveIso
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+  {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
+  [Fintype E] [IsNoetherianRing A] [IsDomain A]
+  (hneg : ∀ n < 0, 𝓐 n = ⊥)
+  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
+  {B C : GradedSheaf (𝓐 := 𝓐) G} {x : V}
+  (j : AboveIso B.forget C.forget x)
+
+include hneg hzero in
+ theorem stalk_hom_bijective
+    (hj : ∀ e (h : x < G.target e) n v, v ∈ (B.edge e).piece n →
+      j.edge e h v ∈ (C.edge e).piece n)
+    (hBfree : Module.Free A (B.vertex x)) (hCfree : Module.Free A (C.vertex x))
+    (hBf : B.forget.Flabby) (hBg : B.forget.Generated)
+    (hCf : C.forget.Flabby) (hCg : C.forget.Generated)
+    (hBmin : B.KernelMinimal x) (hCmin : C.KernelMinimal x)
+    (f₀ : ModuleData.Hom (B.vertex x) (C.vertex x))
+    (hf₀ : ∀ v, j.boundary (B.forget.stalkMap x v) =
+      C.forget.stalkMap x (f₀.map v)) : Function.Bijective f₀.map := by
+  let T := ModuleData.range (C.stalkHom x)
+  let f : ModuleData.Hom (B.vertex x) T :=
+    { map := LinearMap.codRestrict (LinearMap.range (C.forget.stalkMap x))
+        (j.boundary.toLinearMap.comp (B.forget.stalkMap x)) (by
+          intro v
+          rw [← j.map_stalk_image hBf hBg hCf hCg]
+          exact ⟨B.forget.stalkMap x v,⟨v,rfl⟩,rfl⟩)
+      graded := by
+        intro n v hv e _
+        exact hj e.val _ n _ ((B.stalkHom x).graded n v hv e (Set.mem_univ e)) }
+  let g : ModuleData.Hom (C.vertex x) T :=
+    { map := (C.forget.stalkMap x).rangeRestrict
+      graded := fun n v hv => (C.stalkHom x).graded n v hv }
+  have hfs : Function.Surjective f.map := by
+    intro v
+    have hv : v.val ∈ (LinearMap.range (B.forget.stalkMap x)).map j.boundary.toLinearMap := by
+      rw [j.map_stalk_image hBf hBg hCf hCg]
+      exact v.property
+    obtain ⟨w,⟨z,rfl⟩,hz⟩ := hv
+    exact ⟨z,Subtype.ext hz⟩
+  have hgs : Function.Surjective g.map := by
+    intro v
+    obtain ⟨w,hw⟩ := v.property
+    exact ⟨w,Subtype.ext hw⟩
+  have hfm : LinearMap.ker f.map ≤ positiveIdeal 𝓐 • (⊤ : Submodule A (B.vertex x)) := by
+    intro v hv
+    apply (B.kernelMinimal_iff x).mp hBmin
+    apply LinearMap.mem_ker.mpr
+    apply j.boundary.injective
+    rw [map_zero]
+    have hh : f.map v = 0 := LinearMap.mem_ker.mp hv
+    exact congrArg (fun z : T => z.val) hh
+  have hgm : LinearMap.ker g.map ≤ positiveIdeal 𝓐 • (⊤ : Submodule A (C.vertex x)) := by
+    intro v hv
+    apply (C.kernelMinimal_iff x).mp hCmin
+    have hh : g.map v = 0 := LinearMap.mem_ker.mp hv
+    exact congrArg (fun z : T => z.val) hh
+  let F : ModuleData.MinimalCover T := ⟨B.vertex x,hBfree,f,hfs,hfm⟩
+  let D : ModuleData.MinimalCover T := ⟨C.vertex x,hCfree,g,hgs,hgm⟩
+  apply F.hom_bijective hneg hzero D f₀
+  apply LinearMap.ext
+  intro v
+  exact Subtype.ext (hf₀ v).symm
+
+end KLInvariance.MomentGraph.Sheaf.AboveIso
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
+  {G : OrderedGraph V E} {B C : GradedSheaf (𝓐 := 𝓐) G}
+noncomputable section
+
+ theorem Hom.edge_bijective_of_vertex (f : Hom B C) (α : E → A)
+    (hB : B.UpperQuotient α) (hC : C.UpperQuotient α) (e : E)
+    (hv : Function.Bijective (f.vertex (G.target e)).map) :
+    Function.Bijective (f.edge e).map := by
+  let v := LinearEquiv.ofBijective (f.vertex (G.target e)).map hv
+  let j := QuotientEdge.iso (B.upper e).map (C.upper e).map
+    (hB e).1 (hC e).1 v (QuotientEdge.map_kernel_scalar _ _ (α e) (hB e).2 (hC e).2 v)
+  have he : (f.edge e).map=j.toLinearMap := by
+    apply LinearMap.ext
+    intro m
+    obtain ⟨m,rfl⟩ := (hB e).1 m
+    rw [f.upper]
+    exact (QuotientEdge.iso_apply (B.upper e).map (C.upper e).map
+      (hB e).1 (hC e).1 v
+      (QuotientEdge.map_kernel_scalar _ _ (α e) (hB e).2 (hC e).2 v) m).symm
+  rw [he]
+  exact j.bijective
+
+ def Hom.isoOfBijective (f : Hom B C)
+    (hv : ∀ x, Function.Bijective (f.vertex x).map)
+    (he : ∀ e, Function.Bijective (f.edge e).map) : Iso B C where
+  vertex x := LinearEquiv.ofBijective (f.vertex x).map (hv x)
+  edge e := LinearEquiv.ofBijective (f.edge e).map (he e)
+  lower := f.lower
+  upper := f.upper
+  vertex_graded x := (f.vertex x).graded
+  edge_graded e := (f.edge e).graded
+
+ def Hom.aboveIso (f : Hom B C) (x : V)
+    (hv : ∀ y, x<y → Function.Bijective (f.vertex y).map)
+    (he : ∀ e, x<G.target e → Function.Bijective (f.edge e).map) :
+    Sheaf.AboveIso B.forget C.forget x where
+  vertex y hy := LinearEquiv.ofBijective (f.vertex y).map (hv y hy)
+  edge e he' := LinearEquiv.ofBijective (f.edge e).map (he e he')
+  lower e _ := f.lower e
+  upper e _ := f.upper e
+
+end
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+/-! Invertibility detected at the top of the actual normalized minimal-boundary
+sheaf. This is stronger than uniqueness of its isomorphism class. -/
+namespace KLInvariance.MomentGraph.BoundarySheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+  [IsDomain A] [IsNoetherianRing A]
+  {hneg : ∀ n < 0, 𝓐 n = ⊥}
+  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
+  {V E : Type um} [PartialOrder V] [Fintype V] [Fintype E]
+  {G : OrderedGraph V E} {α : E → A} {b : V}
+  (B C : BoundarySheaf hneg G α b)
+noncomputable section
+
+ include hzero in
+ theorem hom_vertex_bijective (f : GradedSheaf.Hom B.sheaf C.sheaf)
+    (hb : Function.Bijective (f.vertex b).map) (x : V) :
+    Function.Bijective (f.vertex x).map := by
+  classical
+  refine wellFounded_gt.induction (C := fun y => Function.Bijective (f.vertex y).map) x ?_
+  intro x ih
+  by_cases hx : x=b
+  · subst x
+    exact hb
+  have he : ∀ e, x<G.target e → Function.Bijective (f.edge e).map :=
+    fun e h => f.edge_bijective_of_vertex α B.quotient C.quotient e (ih _ h)
+  let j := f.aboveIso x ih he
+  apply j.stalk_hom_bijective hneg hzero
+    (fun e h n v hv => (f.edge e).graded n v hv)
+    (B.free x) (C.free x) B.flabby.flabby B.generated C.flabby.flabby C.generated
+    (B.minimal x hx) (C.minimal x hx) (f.vertex x)
+  intro v
+  funext e
+  change (f.edge e.val).map (B.sheaf.forget.stalkMap x v e)=
+    C.sheaf.forget.stalkMap x ((f.vertex x).map v) e
+  rcases e with ⟨e,he⟩
+  subst x
+  exact f.lower e v
+
+ def homIso (f : GradedSheaf.Hom B.sheaf C.sheaf)
+    (hb : Function.Bijective (f.vertex b).map) : GradedSheaf.Iso B.sheaf C.sheaf :=
+  f.isoOfBijective (hom_vertex_bijective hzero B C f hb)
+    (fun e => f.edge_bijective_of_vertex α B.quotient C.quotient e
+      (hom_vertex_bijective hzero B C f hb _))
+
+end
+end KLInvariance.MomentGraph.BoundarySheaf
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
+  {G : OrderedGraph V E} {B C : GradedSheaf (𝓐 := 𝓐) G}
+noncomputable section
+
+ def Iso.hom (j : Iso B C) : Hom B C where
+  vertex x := ⟨(j.vertex x).toLinearMap,j.vertex_graded x⟩
+  edge e := ⟨(j.edge e).toLinearMap,j.edge_graded e⟩
+  lower := j.lower
+  upper := j.upper
+
+ def Iso.invHom (j : Iso B C) : Hom C B where
+  vertex x := ⟨(j.vertex x).symm.toLinearMap,fun n v hv =>
+    (j.hom.vertex x).mem_of_map_mem (j.vertex x).injective n _
+      (by change j.vertex x ((j.vertex x).symm v) ∈ _
+          exact ((j.vertex x).apply_symm_apply v).symm ▸ hv)⟩
+  edge e := ⟨(j.edge e).symm.toLinearMap,fun n v hv =>
+    (j.hom.edge e).mem_of_map_mem (j.edge e).injective n _
+      (by change j.edge e ((j.edge e).symm v) ∈ _
+          exact ((j.edge e).apply_symm_apply v).symm ▸ hv)⟩
+  lower e v := by
+    apply (j.edge e).injective
+    exact ((j.edge e).apply_symm_apply _).trans
+      ((j.lower e ((j.vertex (G.source e)).symm v)).trans
+        (congrArg (C.forget.lower e) ((j.vertex (G.source e)).apply_symm_apply v))).symm
+  upper e v := by
+    apply (j.edge e).injective
+    exact ((j.edge e).apply_symm_apply _).trans
+      ((j.upper e ((j.vertex (G.target e)).symm v)).trans
+        (congrArg (C.forget.upper e) ((j.vertex (G.target e)).apply_symm_apply v))).symm
+
+end
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.BoundarySheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+  [IsDomain A] [IsNoetherianRing A]
+  {hneg : ∀ n < 0, 𝓐 n = ⊥}
+  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
+  {V E : Type um} [PartialOrder V] [Fintype V] [Fintype E]
+  {G : OrderedGraph V E} {α : E → A} {b : V}
+  (B : BoundarySheaf hneg G α b) {C : GradedSheaf (𝓐 := 𝓐) G}
+noncomputable section
+
+ include hzero in
+ theorem hom_vertex_bijective_of_iso (j : GradedSheaf.Iso B.sheaf C)
+    (f : GradedSheaf.Hom B.sheaf C) (hb : Function.Bijective (f.vertex b).map)
+    (x : V) : Function.Bijective (f.vertex x).map := by
+  let g := j.invHom.comp f
+  have hgb : Function.Bijective (g.vertex b).map := (j.vertex b).symm.bijective.comp hb
+  have hg := hom_vertex_bijective hzero B B g hgb x
+  constructor
+  · intro v w h
+    apply hg.1
+    exact congrArg (j.vertex x).symm h
+  · intro w
+    obtain ⟨v,hv⟩ := hg.2 ((j.vertex x).symm w)
+    exact ⟨v,(j.vertex x).symm.injective hv⟩
+
+ def homIsoOfIso (j : GradedSheaf.Iso B.sheaf C) (hq : C.UpperQuotient α)
+    (f : GradedSheaf.Hom B.sheaf C) (hb : Function.Bijective (f.vertex b).map) :
+    GradedSheaf.Iso B.sheaf C :=
+  f.isoOfBijective (hom_vertex_bijective_of_iso hzero B j f hb)
+    (fun e => f.edge_bijective_of_vertex α B.quotient hq e
+      (hom_vertex_bijective_of_iso hzero B j f hb _))
+
+end
+end KLInvariance.MomentGraph.BoundarySheaf
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
+  {G : OrderedGraph V E} {B C D : GradedSheaf (𝓐 := 𝓐) G}
+  (s : SplitIn B D) (t : SplitIn C D)
+
+ theorem SplitIn.comparison_bijective (x : V)
+    (hs : ∀ v, (s.projector.vertex x).map v=v)
+    (ht : ∀ v, (t.projector.vertex x).map v=v) :
+    Function.Bijective ((t.projection.comp s.inclusion).vertex x).map := by
+  let f := ((t.projection.comp s.inclusion).vertex x).map
+  let g := ((s.projection.comp t.inclusion).vertex x).map
+  have hgf (v : B.vertex x) : g (f v)=v := by
+    exact (congrArg (s.projection.vertex x).map (ht ((s.inclusion.vertex x).map v))).trans
+      (s.vertex_split x v)
+  have hfg (v : C.vertex x) : f (g v)=v := by
+    exact (congrArg (t.projection.vertex x).map (hs ((t.inclusion.vertex x).map v))).trans
+      (t.vertex_split x v)
+  exact ⟨Function.LeftInverse.injective hgf,Function.RightInverse.surjective hfg⟩
+
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+/-! The CANONICAL map from a normalized BMP summand to its adjoint image is
+invertible. Consequently the ambient WORD form restricts perfectly on this
+particular summand; no such assertion is made for an arbitrary summand. -/
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
+universe u us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  (l : List I) (hl : cs.IsReduced l)
+  (B : BoundarySheaf (symmetricGrading_negative basis)
+    (graph cs 1 (cs.wordProd l)) (symmetricLabel cs 1 (cs.wordProd l))
+    ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩)
+  (s : GradedSheaf.SplitIn B.sheaf (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
+noncomputable section
+
+ def bottCanonicalAdjointHom : GradedSheaf.Hom B.sheaf
+    (bottSplitAdjointSheaf cs basis l hl B.sheaf s) :=
+  (bottSplitAdjointSplit cs basis l hl B s).projection.comp s.inclusion
+
+ theorem bottCanonicalAdjointHom_top_bijective : Function.Bijective
+    ((bottCanonicalAdjointHom cs basis l hl B s).vertex
+      ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩).map := by
+  apply s.comparison_bijective (bottSplitAdjointSplit cs basis l hl B s)
+  · exact bottBoundarySplit_top_fixed cs basis l hl _ B s
+  · exact bottBoundarySplitAdjoint_top_fixed cs basis l hl _ B s
+
+ def bottCanonicalAdjointIso : GradedSheaf.Iso B.sheaf
+    (bottSplitAdjointSheaf cs basis l hl B.sheaf s) := by
+  classical
+  let := symmetricCoefficientNoetherian basis
+  let := interval_finite cs 1 (cs.wordProd l)
+  let := Fintype.ofFinite (Interval cs 1 (cs.wordProd l))
+  let := Fintype.ofFinite (Edge cs 1 (cs.wordProd l))
+  let j := (bottBoundarySheaf_iso_adjoint cs basis l hl B s).some
+  exact B.homIsoOfIso (symmetricGrading_zero basis) j
+    ((bottSplitAdjointSplit cs basis l hl B s).upperQuotient
+      (bottGradedProjectionSheaf_upperQuotient cs basis l 1 (cs.wordProd l)))
+    (bottCanonicalAdjointHom cs basis l hl B s)
+    (bottCanonicalAdjointHom_top_bijective cs basis l hl B s)
+
+ theorem bottCanonicalAdjointIso_vertex (x : Interval cs 1 (cs.wordProd l))
+    (v : B.sheaf.vertex x) :
+    ((bottCanonicalAdjointIso cs basis l hl B s).vertex x v).val=
+      ((bottAdjointSheaf cs basis l hl s.projector).vertex x).map
+        ((s.inclusion.vertex x).map v) := rfl
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! On the normalized BMP summand the canonical adjoint recognition proves,
+rather than assumes, perfection of the restricted symmetric word form. -/
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
+universe u us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  (l : List I) (hl : cs.IsReduced l)
+  (B : BoundarySheaf (symmetricGrading_negative basis)
+    (graph cs 1 (cs.wordProd l)) (symmetricLabel cs 1 (cs.wordProd l))
+    ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩)
+  (s : GradedSheaf.SplitIn B.sheaf (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
+noncomputable section
+
+ theorem bottSplitRightWord_fixed (n : B.sheaf.forget.sections Set.univ) :
+    bottEndOfSheaf cs basis l hl s.projector (bottSplitRightWord cs basis l hl B.sheaf s n)=
+      bottSplitRightWord cs basis l hl B.sheaf s n := by
+  apply bottVertexProjection_injective cs l hl
+  funext x
+  change bottStalkProjection cs l x.val _=bottStalkProjection cs l x.val _
+  rw [bottEndOfSheaf_coordinates,bottSplitRightWord_coordinates]
+  change (s.inclusion.vertex x).map
+      ((s.projection.vertex x).map ((s.inclusion.vertex x).map (n.val x)))=
+    (s.inclusion.vertex x).map (n.val x)
+  exact congrArg (s.inclusion.vertex x).map (s.vertex_split x (n.val x))
+
+ theorem bottCanonicalAdjointWord (m : B.sheaf.forget.sections Set.univ) :
+    bottSplitLeftWord cs basis l hl B.sheaf s
+      ((bottCanonicalAdjointIso cs basis l hl B s).toIso.sections Set.univ m)=
+    bottAdjoint cs l (bottEndOfSheaf cs basis l hl s.projector)
+      (bottSplitRightWord cs basis l hl B.sheaf s m) := by
+  apply bottVertexProjection_injective cs l hl
+  funext x
+  change bottStalkProjection cs l x.val _=bottStalkProjection cs l x.val _
+  rw [bottSplitLeftWord_coordinates]
+  exact (bottCanonicalAdjointIso_vertex cs basis l hl B s x (m.val x)).trans
+    ((congrArg ((bottAdjointSheaf cs basis l hl s.projector).vertex x).map
+      (bottSplitRightWord_coordinates cs basis l hl B.sheaf s m x).symm).trans
+      (bottAdjointSheaf_vertex cs basis l hl s.projector x _))
+
+ def bottCanonicalPerfectPairing : GradedSheaf.PerfectPairing B.sheaf (l.length : ℤ) :=
+  bottRecognizedPerfectPairing cs basis l hl B.sheaf s
+    (bottCanonicalAdjointIso cs basis l hl B s)
+
+ theorem bottCanonicalPerfectPairing_apply (m n : B.sheaf.forget.sections Set.univ) :
+    (bottCanonicalPerfectPairing cs basis l hl B s).pairing m n=
+      bottSamelsonDuality cs l (bottSplitRightWord cs basis l hl B.sheaf s m)
+        (bottSplitRightWord cs basis l hl B.sheaf s n) := by
+  change bottSplitAdjointEquivDual cs basis l hl B.sheaf s
+    ((bottCanonicalAdjointIso cs basis l hl B s).toIso.sections Set.univ m) n=_
+  rw [bottSplitAdjointWord_apply,bottCanonicalAdjointWord]
+  change bottSamelsonDuality cs l
+    (PerfectAdjoint.adjoint (bottSamelsonDuality cs l)
+      (bottEndOfSheaf cs basis l hl s.projector) _) _=_
+  rw [PerfectAdjoint.pairing,bottSplitRightWord_fixed]
+
+ theorem bottCanonicalPerfectPairing_symmetric (m n : B.sheaf.forget.sections Set.univ) :
+    (bottCanonicalPerfectPairing cs basis l hl B s).pairing m n=
+      (bottCanonicalPerfectPairing cs basis l hl B s).pairing n m := by
+  exact (bottCanonicalPerfectPairing_apply cs basis l hl B s m n).trans
+    ((bottSamelson_symmetric cs l _ _).trans
+      (bottCanonicalPerfectPairing_apply cs basis l hl B s n m).symm)
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded Module
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
+  {G : OrderedGraph V E}
+noncomputable section
+
+ structure SymmetricPerfectPairing (S : GradedSheaf (𝓐 := 𝓐) G) (d : ℤ)
+    extends PerfectPairing S d where
+  symmetric : ∀ m n, pairing m n=pairing n m
+
+ def SymmetricPerfectPairing.pullback {S T : GradedSheaf (𝓐 := 𝓐) G} {d : ℤ}
+    (p : SymmetricPerfectPairing T d) (j : Iso S T) : SymmetricPerfectPairing S d where
+  toPerfectPairing := p.toPerfectPairing.pullback j
+  symmetric _m _n := p.symmetric _ _
+
+ variable {B : Type ua} [CommRing B] [Algebra k B] {𝓑 : ℤ → Submodule k B}
+  (e : A ≃ₐ[k] B) (he : ∀ n a, a∈𝓐 n ↔ e a∈𝓑 n)
+
+ def SymmetricPerfectPairing.coefficientChange {S : GradedSheaf (𝓐 := 𝓐) G} {d : ℤ}
+    (p : SymmetricPerfectPairing S d) : SymmetricPerfectPairing (coefficientChange e he S) d where
+  toPerfectPairing := p.toPerfectPairing.coefficientChange e he
+  symmetric _m _n := congrArg e (p.symmetric _ _)
+
+end
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+/-! Unconditional symmetric, integral, support-orthogonal graded perfection
+for the actual minimal-boundary sheaves; symmetry is inherited from the
+proved canonical-adjoint recognition, not an arbitrary choice of a dual. -/
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
+universe u us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ theorem symmetricBoundarySheaf_symmetricPairing_word (l : List I) (hl : cs.IsReduced l)
+    (B : BoundarySheaf (symmetricGrading_negative basis)
+      (graph cs 1 (cs.wordProd l)) (symmetricLabel cs 1 (cs.wordProd l))
+      ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩) :
+    Nonempty (GradedSheaf.SymmetricPerfectPairing B.sheaf (cs.length (cs.wordProd l) : ℤ)) := by
+  obtain ⟨s⟩ := bottBoundarySheaf_splitIn cs basis l hl B
+  have hd : cs.length (cs.wordProd l)=l.length := hl
+  rw [hd]
+  exact ⟨{ toPerfectPairing := bottCanonicalPerfectPairing cs basis l hl B s
+           symmetric := bottCanonicalPerfectPairing_symmetric cs basis l hl B s }⟩
+
+ theorem symmetricBoundarySheaf_symmetricPairing (b : W) :
+    Nonempty (GradedSheaf.SymmetricPerfectPairing
+      (symmetricBoundarySheaf cs basis b).sheaf (cs.length b : ℤ)) := by
+  obtain ⟨l,hl,he⟩ := cs.exists_isReduced b
+  subst b
+  exact symmetricBoundarySheaf_symmetricPairing_word cs basis l hl _
+
+end
+end KLInvariance.TitsSpace
+
+namespace KLInvariance.BruhatGraph
+open Module _root_.OAI.KLInvariance.Graded MomentGraph TitsSpace
+universe u
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W) (b : W)
+  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ theorem boundarySheaf_symmetricPairing :
+    Nonempty (GradedSheaf.SymmetricPerfectPairing
+      (boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf (cs.length b : ℤ)) := by
+  classical
+  let := interval_finite cs 1 b
+  let := Fintype.ofFinite (Interval cs 1 b)
+  let := Fintype.ofFinite (Edge cs 1 b)
+  let S := symmetricBoundarySheaf cs basis b
+  let T := boundarySheaf cs 1 b basis (one_bruhat cs b)
+  let e := SymmetricAlgebra.equivMvPolynomial basis
+  have he : ∀ n a, a∈symmetricGrading basis n ↔ e a∈polynomialGrading ℝ σ n :=
+    mem_symmetricGrading basis
+  obtain ⟨p⟩ := symmetricBoundarySheaf_symmetricPairing cs basis b
+  obtain ⟨j⟩ := BoundarySheaf.iso_coefficientChange e he S T
+    (polynomialGrading_zero ℝ σ) (fun x => x.property.2)
+  exact ⟨(p.coefficientChange e he).pullback j⟩
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+/-! The exact degree mechanism needed for the character theorem. Minimality
+puts every local intersection coordinate in the positive ideal. A nonzero
+homogeneous diagonal then forces the strict half-degree inequality. This
+lemma DOES NOT assert the local Hodge/nonzero-diagonal input. -/
+namespace KLInvariance.Graded
+open Module
+universe uk ua um ui
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  (𝓐 : ℤ → Submodule k A) [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+  (hneg : ∀ n < 0, 𝓐 n = ⊥)
+
+ include hneg in
+ theorem homogeneous_positive_nonzero {n : ℤ} {a : A}
+    (ha : a∈𝓐 n) (hp : a∈positiveIdeal 𝓐) (hn : a≠0) : 0<n := by
+  by_contra hh
+  have hnle : n≤0 := le_of_not_gt hh
+  by_cases hz : n=0
+  · subst n
+    rw [positiveIdeal_eq_augmentation_ker 𝓐 hneg] at hp
+    have hc : component 𝓐 0 a=0 := hp
+    have he : component 𝓐 0 a=a := DirectSum.decompose_of_mem_same 𝓐 ha
+    exact hn (he.symm.trans hc)
+  · have hlt : n<0 := lt_of_le_of_ne hnle hz
+    rw [hneg n hlt,Submodule.mem_bot] at ha
+    exact hn ha
+
+variable {M : Type um} [AddCommGroup M] [Module A M]
+
+omit [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐] in
+ theorem dual_mem_ideal (J : Ideal A) (f : Dual A M) {m : M}
+    (hm : m∈J • (⊤ : Submodule A M)) : f m∈J := by
+  refine Submodule.smul_induction_on hm ?_ ?_
+  · intro a ha v _
+    rw [map_smul]
+    exact J.mul_mem_right (f v) ha
+  · intro u v hu hv
+    rw [map_add]
+    exact J.add_mem hu hv
+
+variable [Module k M] [IsScalarTower k A M]
+  (𝓜 : ℤ → Submodule k M) [DirectSum.Decomposition 𝓜] [SetLike.GradedSMul 𝓐 𝓜]
+  {ι : Type ui} [Fintype ι] [DecidableEq ι] (b : Basis ι A M) (d : ι → ℤ)
+  (hb : ∀ i, b i∈𝓜 (d i)) (D : ℤ) (Q : Dual A M →ₗ[A] M)
+
+ include hneg hb in
+ omit [IsScalarTower k A M] in
+ theorem local_diagonal_strict_degree
+    (hq : ∀ i, Q (b.dualBasis i)∈𝓜 (D-d i))
+    (hmin : LinearMap.range Q ≤ positiveIdeal 𝓐 • (⊤ : Submodule A M))
+    (i : ι) (hne : b.dualBasis i (Q (b.dualBasis i))≠0) : 2*d i<D := by
+  have hc : b.dualBasis i (Q (b.dualBasis i))∈𝓐 (D-d i+(-d i)) :=
+    dualBasis_homogeneous 𝓐 𝓜 b d hb i _ _ (hq i)
+  have hp : b.dualBasis i (Q (b.dualBasis i))∈positiveIdeal 𝓐 :=
+    dual_mem_ideal (positiveIdeal 𝓐) (b.dualBasis i) (hmin ⟨b.dualBasis i,rfl⟩)
+  have hh := homogeneous_positive_nonzero 𝓐 hneg hc hp hne
+  omega
+
+end KLInvariance.Graded
+
+end
+
+
+section
+
+/-! Integral evaluation compatibility of the actual support/dual equivalence. -/
+namespace KLInvariance.IntegralSupport
+universe ur uv um
+variable {R : Type ur} [CommRing R] {V : Type uv} [DecidableEq V]
+  {A : V → Type um} [∀ x, AddCommGroup (A x)] [∀ x, Module R (A x)]
+  (M : Submodule R (∀ x, A x)) (p : R)
+  (hcut : ∀ x (m : M), (p • Pi.single x (m.val x)) ∈ M)
+  [IsDomain R] [∀ x, Module.IsTorsionFree R (A x)] (hp : p≠0)
+  (d : M ≃ₗ[R] Module.Dual R M)
+  (hbalanced : ∀ x m n, d (cut M p hcut x m) n=d m (cut M p hcut x n))
+  (x : V) (hπ : Function.Surjective (projection M x))
+noncomputable section
+
+ theorem supportEquivDual_pairing (m : supported M x) (n : M) :
+    supportEquivDual M p hcut hp d hbalanced x hπ m (projection M x n)=d m.val n := by
+  let e := supportEquivDual M p hcut hp d hbalanced x hπ
+  have he := congrArg (fun s : supported M x => d s.val n) (e.symm_apply_apply m)
+  change d (d.symm ((projection M x).dualMap (e m))) n=d m.val n at he
+  rw [LinearEquiv.apply_symm_apply] at he
+  exact he
+
+end
+end KLInvariance.IntegralSupport
+
+end
+
+
+section
+
+/-! Evaluation and symmetry of the actual integral local intersection map.
+The support factor is explicitly cancelled only after its nonvanishing is proved. -/
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/KazhdanLusztig/Frobenius.lean b/lean/OAI/RepresentationTheory/KazhdanLusztig/Frobenius.lean
index 8c9be22293431bd9c5a7c8cbcc5f6fec41253f9f..d204229cbdbbb7be7cac1271e1222fb9cc039b0d 100644
--- a/lean/OAI/RepresentationTheory/KazhdanLusztig/Frobenius.lean
+++ b/lean/OAI/RepresentationTheory/KazhdanLusztig/Frobenius.lean
@@ -1,2227 +1,2227 @@
-import OAI.RepresentationTheory.KazhdanLusztig.Decomposition
-
-namespace OAI
-
-/-!
-Coxeter Frobenius extensions, tensor duality and translation across reflection pairs.
--/
-
-section
-
-namespace KLInvariance.TitsSpace
-open Module ReflectionFrobenius
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
- abbrev SymmetricCoefficient := SymmetricAlgebra ℝ (Extended M)
-
- def coefficientAction (w : W) : SymmetricCoefficient (M := M) ≃ₐ[ℝ]
-    SymmetricCoefficient (M := M) := symmetricAction (extendedAction M cs w)
-
-@[simp] theorem coefficientAction_ι (w : W) (v : Extended M) :
-    coefficientAction cs w (SymmetricAlgebra.ι ℝ (Extended M) v) =
-      SymmetricAlgebra.ι ℝ (Extended M) (extendedAction M cs w v) :=
-  symmetricAction_ι _ _
-
- theorem coefficientAction_simple (i : I) : coefficientAction cs (cs.simple i) =
-    reflectionAction (extendedRoot M i) (extendedForm M (extendedRoot M i))
-      (extendedForm_root_self M i) := by
-  rw [coefficientAction,extendedAction_simple_eq]
-
- theorem coefficientAction_mul (w z : W) (a : SymmetricCoefficient (M := M)) :
-    coefficientAction cs (w*z) a = coefficientAction cs w (coefficientAction cs z a) := by
-  have he : (coefficientAction cs (w*z)).toAlgHom =
-      (coefficientAction cs w).toAlgHom.comp (coefficientAction cs z).toAlgHom := by
-    apply SymmetricAlgebra.algHom_ext
-    apply LinearMap.ext
-    intro x
-    change coefficientAction cs (w*z) (SymmetricAlgebra.ι ℝ (Extended M) x) =
-      coefficientAction cs w (coefficientAction cs z (SymmetricAlgebra.ι ℝ (Extended M) x))
-    simp only [coefficientAction_ι,map_mul,LinearEquiv.mul_apply]
-  exact congrArg (fun f : SymmetricCoefficient (M := M) →ₐ[ℝ] SymmetricCoefficient (M := M) => f a) he
-
- def simpleFixed (i : I) : Subalgebra ℝ (SymmetricCoefficient (M := M)) :=
-  fixed (coefficientAction cs (cs.simple i))
-
- abbrev simpleRootPolynomial (i : I) : SymmetricCoefficient (M := M) :=
-  SymmetricAlgebra.ι ℝ (Extended M) (extendedRoot M i)
-
- theorem simple_involutive (i : I) : Function.Involutive (coefficientAction cs (cs.simple i)) := by
-  rw [coefficientAction_simple]
-  exact reflectionAction_involutive _ _ _
-
- theorem simpleRootPolynomial_ne_zero (i : I) : simpleRootPolynomial (M := M) i ≠ 0 :=
-  reflectionRoot_ne_zero _ _ (extendedForm_root_self M i)
-
- theorem simple_anti (i : I) : coefficientAction cs (cs.simple i)
-    (simpleRootPolynomial (M := M) i) = -simpleRootPolynomial (M := M) i := by
-  rw [coefficientAction_simple]
-  exact reflectionAction_anti _ _ _
-
- theorem simple_divisible (i : I) (a : SymmetricCoefficient (M := M)) :
-    simpleRootPolynomial (M := M) i ∣ a-coefficientAction cs (cs.simple i) a := by
-  rw [coefficientAction_simple]
-  exact reflectionAction_divisible _ _ _ a
-
- def simpleCoordinates (i : I) : SymmetricCoefficient (M := M) ≃ₗ[simpleFixed cs i]
-    (simpleFixed cs i × simpleFixed cs i) :=
-  coordinates _ _ (simple_involutive cs i) (simpleRootPolynomial_ne_zero i)
-    (simple_anti cs i) (simple_divisible cs i)
-
- instance simpleFixed_free (i : I) : Module.Free (simpleFixed cs i) (SymmetricCoefficient (M := M)) :=
-  Module.Free.of_equiv (simpleCoordinates cs i).symm
-
- instance simpleFixed_finite (i : I) : Module.Finite (simpleFixed cs i) (SymmetricCoefficient (M := M)) :=
-  Module.Finite.equiv (simpleCoordinates cs i).symm
-
- def simpleTrace (i : I) : SymmetricCoefficient (M := M) →ₗ[simpleFixed cs i] simpleFixed cs i :=
-  trace _ _ (simple_involutive cs i) (simpleRootPolynomial_ne_zero i)
-    (simple_anti cs i) (simple_divisible cs i)
-
- def simpleDuality (i : I) : SymmetricCoefficient (M := M) ≃ₗ[simpleFixed cs i]
-    Dual (simpleFixed cs i) (SymmetricCoefficient (M := M)) :=
-  duality _ _ (simple_involutive cs i) (simpleRootPolynomial_ne_zero i)
-    (simple_anti cs i) (simple_divisible cs i)
-
- theorem simpleDuality_apply (i : I) (a b : SymmetricCoefficient (M := M)) :
-    simpleDuality cs i a b = simpleTrace cs i (a*b) :=
-  duality_apply _ _ _ _ _ _ a b
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Integral restriction and extension of perfect pairings for a genuine
-Frobenius extension. This supplies the duality operation used by
-Bott--Samelson translation, not an assumed duality on a BMP sheaf. -/
-namespace KLInvariance.FrobeniusDuality
-open Module TensorProduct
-universe ur ua um
-variable {R : Type ur} [CommRing R] {A : Type ua} [CommRing A] [Algebra R A]
-  (tr : A →ₗ[R] R) (d : A ≃ₗ[R] Dual R A)
-  (hd : ∀ a b, d a b = tr (a*b))
-  (M : Type um) [AddCommGroup M] [Module A M] [Module R M] [IsScalarTower R A M]
-noncomputable section
-
- def restrictDual : Dual A M →ₗ[R] Dual R M where
-  toFun f := tr.comp (f.restrictScalars R)
-  map_add' _ _ := by ext; simp
-  map_smul' _ _ := by ext; simp
-
- def fiber (f : Dual R M) (m : M) : Dual R A :=
-  f.comp ((LinearMap.toSpanSingleton A M m).restrictScalars R)
-
-@[simp] theorem fiber_apply (f : Dual R M) (m : M) (a : A) :
-    fiber M f m a = f (a • m) := rfl
-
-include hd in
- theorem trace_recover_fiber (f : Dual R M) (m : M) (a : A) :
-    tr (d.symm (fiber M f m) * a) = f (a • m) := by
-  rw [← hd,d.apply_symm_apply,fiber_apply]
-
-include hd in
- theorem recover_fiber_smul (f : Dual R M) (m : M) (a : A) :
-    d.symm (fiber M f (a • m)) = a * d.symm (fiber M f m) := by
-  apply d.injective
-  apply LinearMap.ext
-  intro b
-  rw [d.apply_symm_apply,hd,fiber_apply]
-  have he : a * d.symm (fiber M f m) * b = d.symm (fiber M f m) * (b*a) := by ring
-  rw [he,trace_recover_fiber tr d hd]
-  simp only [mul_smul]
-
- def recoverDual (f : Dual R M) : Dual A M where
-  toFun m := d.symm (fiber M f m)
-  map_add' m n := by
-    apply d.injective
-    rw [map_add,d.apply_symm_apply,d.apply_symm_apply,d.apply_symm_apply]
-    ext a
-    simp [smul_add]
-  map_smul' a m := by
-    simpa only [RingHom.id_apply,smul_eq_mul] using recover_fiber_smul tr d hd M f m a
-
-@[simp] theorem recoverDual_apply (f : Dual R M) (m : M) :
-    recoverDual tr d hd M f m = d.symm (fiber M f m) := rfl
-
- theorem restrict_recover (f : Dual R M) :
-    restrictDual tr M (recoverDual tr d hd M f) = f := by
-  ext m
-  change tr (d.symm (fiber M f m)) = f m
-  simpa only [mul_one,one_smul] using trace_recover_fiber tr d hd M f m 1
-
- theorem recover_restrict (f : Dual A M) :
-    recoverDual tr d hd M (restrictDual tr M f) = f := by
-  ext m
-  apply d.injective
-  change d (d.symm _) = d (f m)
-  rw [d.apply_symm_apply]
-  ext a
-  change tr (f (a • m)) = d (f m) a
-  rw [map_smul,smul_eq_mul,hd,mul_comm]
-
- def restrictionEquiv : Dual A M ≃ₗ[R] Dual R M :=
-  LinearEquiv.ofBijective (restrictDual tr M)
-    ⟨fun f g h => by
-      have hh := congrArg (recoverDual tr d hd M) h
-      simpa only [recover_restrict] using hh,
-    fun f => ⟨recoverDual tr d hd M f,restrict_recover tr d hd M f⟩⟩
-
- theorem restrictionEquiv_apply (f : Dual A M) (m : M) :
-    restrictionEquiv tr d hd M f m = tr (f m) := rfl
-
-/-- Restricting an integral perfect pairing along a Frobenius trace is again
-integral and perfect; no localization or fraction-field argument is used. -/
- def restrictPairing (p : M ≃ₗ[A] Dual A M) : M ≃ₗ[R] Dual R M :=
-  (p.restrictScalars R).trans (restrictionEquiv tr d hd M)
-
- theorem restrictPairing_apply (p : M ≃ₗ[A] Dual A M) (m n : M) :
-    restrictPairing tr d hd M p m n = tr (p m n) := rfl
-
-end
-end KLInvariance.FrobeniusDuality
-
-namespace KLInvariance.FrobeniusDuality
-open Module TensorProduct
-universe ur ua um
-variable {R : Type ur} [CommRing R] (A : Type ua) [CommRing A] [Algebra R A]
-  {M : Type um} [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M]
-noncomputable section
-
- def extendPairing (p : M ≃ₗ[R] Dual R M) :
-    (A ⊗[R] M) ≃ₗ[A] Dual A (A ⊗[R] M) :=
-  (p.baseChange R A M (Dual R M)).trans
-    (TensorProduct.isBaseChange (R := R) (S := A) (M := M)).toDualBaseChange
-
- theorem extendPairing_tmul (p : M ≃ₗ[R] Dual R M) (a b : A) (m n : M) :
-    extendPairing A p (a ⊗ₜ[R] m) (b ⊗ₜ[R] n) =
-      a * b * algebraMap R A (p m n) := by
-  have hn : b ⊗ₜ[R] n = b • (1 ⊗ₜ[R] n) := by simp only [smul_tmul',smul_eq_mul,mul_one]
-  rw [hn,map_smul]
-  change b * ((TensorProduct.isBaseChange (R := R) (S := A) (M := M)).toDualBaseChange
-    (a ⊗ₜ[R] p m) ((TensorProduct.mk R A M 1) n)) = _
-  rw [IsBaseChange.toDualBaseChange_tmul]
-  ring
-
-end
-end KLInvariance.FrobeniusDuality
-
-end
-
-
-section
-
-/-! Tensor induction of integral self-dual bimodules across a Frobenius
-extension. All pairings refer to actual integral modules. -/
-namespace KLInvariance.FrobeniusDuality
-open Module TensorProduct
-universe ur ua um
-noncomputable section
-
-structure PairedModule (A : Type ua) [CommRing A] where
-  carrier : Type um
-  [add : AddCommGroup carrier]
-  [module : Module A carrier]
-  free : Module.Free A carrier
-  finite : Module.Finite A carrier
-  pairing : carrier ≃ₗ[A] Dual A carrier
-
-attribute [instance] PairedModule.add PairedModule.module PairedModule.free PairedModule.finite
-instance {A : Type ua} [CommRing A] : CoeSort (PairedModule.{ua,um} A) (Type um) := ⟨PairedModule.carrier⟩
-
- def unitPairing (A : Type ua) [CommRing A] : A ≃ₗ[A] Dual A A where
-  toFun a := LinearMap.mul A A a
-  invFun f := f 1
-  left_inv a := mul_one a
-  right_inv f := by
-    apply LinearMap.ext
-    intro a
-    change f 1 * a = f a
-    rw [mul_comm,← smul_eq_mul,← map_smul,smul_eq_mul,mul_one]
-  map_add' _ _ := by ext; simp
-  map_smul' _ _ := by ext; simp
-
- def unitModule (A : Type ua) [CommRing A] : PairedModule A where
-  carrier := A
-  free := inferInstance
-  finite := inferInstance
-  pairing := unitPairing A
-
-variable {R : Type ur} [CommRing R] {A : Type ua} [CommRing A] [Algebra R A]
-  [Module.Free R A] [Module.Finite R A]
-  (tr : A →ₗ[R] R) (d : A ≃ₗ[R] Dual R A) (hd : ∀ a b, d a b = tr (a*b))
-  (N : PairedModule.{ua,um} A)
-
-local instance restrictedModule : Module R N := Module.compHom N (algebraMap R A)
-local instance restrictedTower : IsScalarTower R A N := IsScalarTower.of_algebraMap_smul fun _ _ => rfl
-
- def restrictionBasis : Basis
-    (Module.Free.ChooseBasisIndex R A × Module.Free.ChooseBasisIndex A N) R N :=
-  (Module.Free.chooseBasis R A).smulTower (Module.Free.chooseBasis A N)
-
-local instance restrictedFree : Module.Free R N := Module.Free.of_basis (restrictionBasis N)
-local instance restrictedFinite : Module.Finite R N := Module.Finite.trans A N
-
- def inducedModule : PairedModule A where
-  carrier := A ⊗[R] N
-  free := inferInstance
-  finite := inferInstance
-  pairing := extendPairing A (restrictPairing tr d hd N N.pairing)
-
- theorem inducedModule_pairing_tmul (a b : A) (m n : N) :
-    (inducedModule tr d hd N).pairing (a ⊗ₜ[R] m) (b ⊗ₜ[R] n) =
-      a*b*algebraMap R A (tr (N.pairing m n)) :=
-  extendPairing_tmul A _ a b m n
-
-end
-end KLInvariance.FrobeniusDuality
-
-end
-
-
-section
-
-/-! Integral balanced duality under Bott--Samelson tensor induction. -/
-namespace KLInvariance.FrobeniusDuality
-open Module TensorProduct
-universe ur ua um
-noncomputable section
-
-structure PairedBimodule (A : Type ua) [CommRing A] where
-  obj : PairedModule.{ua,um} A
-  right : A →+* End A obj
-  balanced : ∀ a m n, obj.pairing (right a m) n = obj.pairing m (right a n)
-  symmetric : ∀ m n, obj.pairing m n = obj.pairing n m
-
- def unitBimodule (A : Type ua) [CommRing A] : PairedBimodule A where
-  obj := unitModule A
-  right := Algebra.lmul A A |>.toRingHom
-  balanced a m n := by
-    change A at m n
-    change (a*m)*n = m*(a*n)
-    ring
-  symmetric m n := by
-    change A at m n
-    exact mul_comm m n
-
-variable {R : Type ur} [CommRing R] {A : Type ua} [CommRing A] [Algebra R A]
-  [Module.Free R A] [Module.Finite R A]
-  (tr : A →ₗ[R] R) (d : A ≃ₗ[R] Dual R A) (hd : ∀ a b, d a b = tr (a*b))
-  (N : PairedModule.{ua,um} A)
-
-attribute [local instance] restrictedModule restrictedTower restrictedFree restrictedFinite
-
- def restrictedRight (right : A →+* End A N) : A →+* End R N where
-  toFun a := (right a).restrictScalars R
-  map_one' := by ext m; change (right 1) m = m; rw [map_one]; rfl
-  map_mul' a b := by
-    ext m
-    change (right (a*b)) m = (right a) ((right b) m)
-    rw [map_mul]
-    rfl
-  map_zero' := by ext m; change (right 0) m = 0; rw [map_zero]; rfl
-  map_add' a b := by ext m; change (right (a+b)) m = (right a) m + (right b) m; rw [map_add]; rfl
-
- def inducedRight (right : A →+* End A N) : A →+* End A (inducedModule tr d hd N) :=
-  (Module.End.baseChangeHom R A N).toRingHom.comp (restrictedRight N right)
-
-@[simp] theorem inducedRight_tmul (right : A →+* End A N) (r a : A) (m : N) :
-    inducedRight tr d hd N right r (a ⊗ₜ[R] m) = a ⊗ₜ[R] right r m := rfl
-
- theorem inducedPairing_balanced (right : A →+* End A N)
-    (hb : ∀ a m n, N.pairing (right a m) n = N.pairing m (right a n))
-    (r : A) (m n : inducedModule tr d hd N) :
-    (inducedModule tr d hd N).pairing (inducedRight tr d hd N right r m) n =
-      (inducedModule tr d hd N).pairing m (inducedRight tr d hd N right r n) := by
-  change A ⊗[R] N at m n
-  change extendPairing A (restrictPairing tr d hd N N.pairing)
-    (((right r).restrictScalars R).baseChange A m) n =
-    extendPairing A (restrictPairing tr d hd N N.pairing) m
-    (((right r).restrictScalars R).baseChange A n)
-  induction m using TensorProduct.inductionOn with
-  | add m₁ m₂ h₁ h₂ => simp only [map_add,LinearMap.add_apply,h₁,h₂]
-  | tmul a m =>
-    induction n using TensorProduct.inductionOn with
-    | add n₁ n₂ h₁ h₂ => simp only [map_add,h₁,h₂]
-    | tmul b n =>
-      simp only [LinearMap.baseChange_tmul,extendPairing_tmul,restrictPairing_apply,
-        LinearMap.restrictScalars_apply,hb]
-
- theorem inducedPairing_symmetric (hs : ∀ m n, N.pairing m n = N.pairing n m)
-    (m n : inducedModule tr d hd N) :
-    (inducedModule tr d hd N).pairing m n = (inducedModule tr d hd N).pairing n m := by
-  change A ⊗[R] N at m n
-  change extendPairing A (restrictPairing tr d hd N N.pairing) m n =
-    extendPairing A (restrictPairing tr d hd N N.pairing) n m
-  induction m using TensorProduct.inductionOn with
-  | add m₁ m₂ h₁ h₂ => simp only [map_add,LinearMap.add_apply,h₁,h₂]
-  | tmul a m =>
-    induction n using TensorProduct.inductionOn with
-    | add n₁ n₂ h₁ h₂ => simp only [map_add,LinearMap.add_apply,h₁,h₂]
-    | tmul b n =>
-      rw [extendPairing_tmul,extendPairing_tmul,restrictPairing_apply,restrictPairing_apply,hs,mul_comm a b]
-
- def inducedBimodule (N : PairedBimodule.{ua,um} A) : PairedBimodule A where
-  obj := inducedModule tr d hd N.obj
-  right := inducedRight tr d hd N.obj N.right
-  balanced := inducedPairing_balanced tr d hd N.obj N.right N.balanced
-  symmetric := inducedPairing_symmetric tr d hd N.obj N.symmetric
-
-end
-end KLInvariance.FrobeniusDuality
-
-end
-
-
-section
-
-/-! Genuine tensor-defined Bott--Samelson bimodules for the extended real
-Coxeter realization. The perfect pairing is integral and balanced. This
-construction alone does not identify indecomposable BMP summands or prove
-the Elias--Williamson character theorem. -/
-namespace KLInvariance.TitsSpace
-open Module TensorProduct FrobeniusDuality
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
- def bottSamelson : List I → PairedBimodule.{u,u} (SymmetricCoefficient (M := M))
-  | [] => unitBimodule _
-  | i :: l => inducedBimodule (simpleTrace cs i) (simpleDuality cs i)
-    (simpleDuality_apply cs i) (bottSamelson l)
-
- abbrev BottSamelsonModule (l : List I) := (bottSamelson cs l).obj
-
- def bottSamelsonDuality (l : List I) : BottSamelsonModule cs l ≃ₗ[SymmetricCoefficient (M := M)]
-    Dual (SymmetricCoefficient (M := M)) (BottSamelsonModule cs l) :=
-  (bottSamelson cs l).obj.pairing
-
- def bottSamelsonRight (l : List I) : SymmetricCoefficient (M := M) →+*
-    End (SymmetricCoefficient (M := M)) (BottSamelsonModule cs l) :=
-  (bottSamelson cs l).right
-
- theorem bottSamelson_balanced (l : List I) (a : SymmetricCoefficient (M := M))
-    (m n : BottSamelsonModule cs l) :
-    bottSamelsonDuality cs l (bottSamelsonRight cs l a m) n =
-      bottSamelsonDuality cs l m (bottSamelsonRight cs l a n) :=
-  (bottSamelson cs l).balanced a m n
-
- theorem bottSamelson_symmetric (l : List I) (m n : BottSamelsonModule cs l) :
-    bottSamelsonDuality cs l m n = bottSamelsonDuality cs l n m :=
-  (bottSamelson cs l).symmetric m n
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! The integral reflection trace lowers the actual polynomial grading by one.
-This is the graded rank-two Frobenius input, not a BMP character assertion. -/
-namespace KLInvariance.Graded
-universe uk ua
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [IsDomain A]
-  [Algebra k A] (𝓐 : ℤ → Submodule k A) [DirectSum.Decomposition 𝓐]
-  [SetLike.GradedSMul 𝓐 𝓐]
-
- theorem homogeneous_of_mul {a b : A} {d n : ℤ}
-    (ha : a ∈ 𝓐 d) (hne : a ≠ 0) (hab : a*b ∈ 𝓐 n) : b ∈ 𝓐 (n-d) := by
-  have hc := component_homogeneous_smul 𝓐 𝓐 d a ha n b
-  have he : component 𝓐 n (a*b)=a*b :=
-    DirectSum.decompose_of_mem_same 𝓐 hab
-  rw [smul_eq_mul,he,smul_eq_mul] at hc
-  have hb : b=component 𝓐 (n-d) b := mul_left_cancel₀ hne hc
-  rw [hb]
-  exact (DirectSum.decompose 𝓐 b (n-d)).property
-
-end KLInvariance.Graded
-
-namespace KLInvariance.ReflectionFrobenius
-open KLInvariance.Graded
-universe uk ua
-variable {k : Type uk} [Field k] [CharZero k]
-  {A : Type ua} [CommRing A] [IsDomain A] [Algebra k A]
-  (τ : A ≃ₐ[k] A) (α : A) (hinv : Function.Involutive τ) (hα : α ≠ 0)
-  (hanti : τ α = -α) (hdiv : ∀ a : A, α ∣ a-τ a)
-noncomputable section
-
- theorem trace_difference (a : A) :
-    2*α*(trace τ α hinv hα hanti hdiv a : A) = a-τ a := by
-  have h := expand_coordinates τ α hinv hα hanti hdiv a
-  have ht := congrArg τ h
-  simp only [expand_apply,map_add,map_mul,fixed_val,hanti] at h ht
-  change 2*α*((coordinates τ α hinv hα hanti hdiv a).2 : A) = a-τ a
-  linear_combination h-ht
-
- theorem trace_homogeneous (𝓐 : ℤ → Submodule k A)
-    [DirectSum.Decomposition 𝓐] [SetLike.GradedSMul 𝓐 𝓐]
-    (hroot : α ∈ 𝓐 1) (hτ : ∀ n a, a ∈ 𝓐 n → τ a ∈ 𝓐 n)
-    {n : ℤ} {a : A} (ha : a ∈ 𝓐 n) :
-    (trace τ α hinv hα hanti hdiv a : A) ∈ 𝓐 (n-1) := by
-  apply homogeneous_of_mul 𝓐 hroot hα
-  have hd := (𝓐 n).sub_mem ha (hτ n a ha)
-  have hal := (𝓐 n).smul_mem ((2:k)⁻¹) hd
-  have ht := trace_difference τ α hinv hα hanti hdiv a
-  have htwo : (2:A)*algebraMap k A ((2:k)⁻¹) = 1 := by
-    rw [show (2:A)=algebraMap k A (2:k) by exact (map_ofNat (algebraMap k A) 2).symm, ← map_mul]
-    simp
-  have he : α*(trace τ α hinv hα hanti hdiv a : A) =
-      (2:k)⁻¹ • (a-τ a) := by
-    rw [Algebra.smul_def]
-    linear_combination (algebraMap k A ((2:k)⁻¹))*ht -
-      (α*(trace τ α hinv hα hanti hdiv a : A))*htwo
-  exact he ▸ hal
-
- theorem coordinates_homogeneous (𝓐 : ℤ → Submodule k A)
-    [DirectSum.Decomposition 𝓐] [SetLike.GradedSMul 𝓐 𝓐]
-    (hroot : α ∈ 𝓐 1) (hτ : ∀ n a, a ∈ 𝓐 n → τ a ∈ 𝓐 n)
-    {n : ℤ} {a : A} (ha : a ∈ 𝓐 n) :
-    ((coordinates τ α hinv hα hanti hdiv a).1 : A) ∈ 𝓐 n ∧
-    ((coordinates τ α hinv hα hanti hdiv a).2 : A) ∈ 𝓐 (n-1) := by
-  have ht := trace_homogeneous τ α hinv hα hanti hdiv 𝓐 hroot hτ ha
-  constructor
-  · have hp : α*(trace τ α hinv hα hanti hdiv a : A) ∈ 𝓐 n := by
-      simpa only [smul_eq_mul,vadd_eq_add,show (1:ℤ)+(n-1)=n by omega] using
-        SetLike.GradedSMul.smul_mem hroot ht
-    have he : ((coordinates τ α hinv hα hanti hdiv a).1 : A) =
-        a-α*(trace τ α hinv hα hanti hdiv a : A) := by
-      have h := expand_coordinates τ α hinv hα hanti hdiv a
-      change ((coordinates τ α hinv hα hanti hdiv a).1 : A) +
-        α*(trace τ α hinv hα hanti hdiv a : A)=a at h
-      linear_combination h
-    exact he ▸ (𝓐 n).sub_mem ha hp
-  · exact ht
-
-end
-end KLInvariance.ReflectionFrobenius
-
-end
-
-
-section
-
-/-! The actual two-element homogeneous basis of a reflection Frobenius
-extension. This basis is used in the tensor-defined Bott--Samelson modules. -/
-namespace KLInvariance.ReflectionFrobenius
-open Module
-universe uk ua
-variable {k : Type uk} [Field k] [CharZero k]
-  {A : Type ua} [CommRing A] [IsDomain A] [Algebra k A]
-  (τ : A ≃ₐ[k] A) (α : A) (hinv : Function.Involutive τ) (hα : α ≠ 0)
-  (hanti : τ α = -α) (hdiv : ∀ a : A, α ∣ a-τ a)
-noncomputable section
-
- def reflectionBasis : Basis (Fin 2) (fixed τ) A :=
-  (Basis.finTwoProd (fixed τ)).map (coordinates τ α hinv hα hanti hdiv).symm
-
- theorem reflectionBasis_zero : reflectionBasis τ α hinv hα hanti hdiv 0 = 1 := by
-  change expand τ α (Basis.finTwoProd (fixed τ) 0) = 1
-  simp [expand_apply]
-
- theorem reflectionBasis_one : reflectionBasis τ α hinv hα hanti hdiv 1 = α := by
-  change expand τ α (Basis.finTwoProd (fixed τ) 1) = α
-  simp [expand_apply]
-
- theorem reflectionBasis_homogeneous (𝓐 : ℤ → Submodule k A)
-    [SetLike.GradedOne 𝓐] (hroot : α ∈ 𝓐 1) (i : Fin 2) :
-    reflectionBasis τ α hinv hα hanti hdiv i ∈ 𝓐 (i.val : ℤ) := by
-  fin_cases i
-  · change reflectionBasis τ α hinv hα hanti hdiv (0 : Fin 2) ∈ 𝓐 0
-    rw [reflectionBasis_zero]
-    exact SetLike.one_mem_graded 𝓐
-  · change reflectionBasis τ α hinv hα hanti hdiv (1 : Fin 2) ∈ 𝓐 1
-    rw [reflectionBasis_one]
-    exact hroot
-
-end
-end KLInvariance.ReflectionFrobenius
-
-end
-
-
-section
-
-/-! Homogeneous integral Bott--Samelson pairing at the genuine tensor basis.
-No indecomposable summand or character identity is postulated here. -/
-namespace KLInvariance.FrobeniusDuality
-open Module TensorProduct ReflectionFrobenius
-universe uk ua um ui
-variable {k : Type uk} [Field k] [CharZero k]
-  {A : Type ua} [CommRing A] [IsDomain A] [Algebra k A]
-  (τ : A ≃ₐ[k] A) (α : A) (hinv : Function.Involutive τ) (hα : α ≠ 0)
-  (hanti : τ α = -α) (hdiv : ∀ a : A, α ∣ a-τ a)
-noncomputable section
-
-variable [Module.Free (fixed τ) A] [Module.Finite (fixed τ) A]
-
-variable (N : PairedModule.{ua,um} A)
-  {ι : Type ui} (b : Basis ι A N)
-attribute [local instance] restrictedModule restrictedTower restrictedFree restrictedFinite
-
- def inducedBasis : Basis (Fin 2 × ι) A
-    (inducedModule (trace τ α hinv hα hanti hdiv)
-      (duality τ α hinv hα hanti hdiv) (duality_apply τ α hinv hα hanti hdiv) N) :=
-  ((reflectionBasis τ α hinv hα hanti hdiv).smulTower b).baseChange A
-
- theorem inducedBasis_apply (i : Fin 2) (j : ι) :
-    inducedBasis τ α hinv hα hanti hdiv N b (i,j) =
-      (1:A) ⊗ₜ[fixed τ] (reflectionBasis τ α hinv hα hanti hdiv i • b j) := by
-  change (((reflectionBasis τ α hinv hα hanti hdiv).smulTower b).baseChange A) (i,j) = _
-  rw [Basis.baseChange_apply,Basis.smulTower_apply]
-
- theorem inducedPairing_basis (i i' : Fin 2) (j j' : ι) :
-    (inducedModule (trace τ α hinv hα hanti hdiv)
-      (duality τ α hinv hα hanti hdiv) (duality_apply τ α hinv hα hanti hdiv) N).pairing
-      (inducedBasis τ α hinv hα hanti hdiv N b (i,j))
-      (inducedBasis τ α hinv hα hanti hdiv N b (i',j')) =
-      (trace τ α hinv hα hanti hdiv
-        (reflectionBasis τ α hinv hα hanti hdiv i *
-          reflectionBasis τ α hinv hα hanti hdiv i' * N.pairing (b j) (b j')) : A) := by
-  rw [inducedBasis_apply,inducedBasis_apply]
-  let : Module.Free (fixed τ) N := Module.Free.of_basis
-    ((reflectionBasis τ α hinv hα hanti hdiv).smulTower b)
-  let : Module.Finite (fixed τ) N := Module.Finite.trans A N
-  change extendPairing A (restrictPairing (trace τ α hinv hα hanti hdiv)
-    (duality τ α hinv hα hanti hdiv) (duality_apply τ α hinv hα hanti hdiv) N N.pairing) _ _ = _
-  rw [extendPairing_tmul,restrictPairing_apply]
-  simp only [one_mul]
-  change ((trace τ α hinv hα hanti hdiv _) : A) = _
-  apply congrArg Subtype.val
-  apply congrArg (trace τ α hinv hα hanti hdiv)
-  simp only [map_smul,LinearMap.smul_apply,smul_eq_mul]
-  ring
-
- theorem inducedPairing_basis_homogeneous (𝓐 : ℤ → Submodule k A)
-    [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-    (hroot : α ∈ 𝓐 1) (hτ : ∀ n a, a ∈ 𝓐 n → τ a ∈ 𝓐 n)
-    (d : ι → ℤ) (l : ℤ)
-    (hb : ∀ j j', N.pairing (b j) (b j') ∈ 𝓐 (d j+d j'-l))
-    (i i' : Fin 2) (j j' : ι) :
-    (inducedModule (trace τ α hinv hα hanti hdiv)
-      (duality τ α hinv hα hanti hdiv) (duality_apply τ α hinv hα hanti hdiv) N).pairing
-      (inducedBasis τ α hinv hα hanti hdiv N b (i,j))
-      (inducedBasis τ α hinv hα hanti hdiv N b (i',j')) ∈
-      𝓐 ((i.val+d j)+(i'.val+d j')-(l+1)) := by
-  rw [inducedPairing_basis]
-  have h := trace_homogeneous τ α hinv hα hanti hdiv 𝓐 hroot hτ
-    (SetLike.GradedMul.mul_mem
-      (SetLike.GradedMul.mul_mem
-        (reflectionBasis_homogeneous τ α hinv hα hanti hdiv 𝓐 hroot i)
-        (reflectionBasis_homogeneous τ α hinv hα hanti hdiv 𝓐 hroot i'))
-      (hb j j'))
-  convert h using 1
-  congr 1
-  omega
-
-end
-end KLInvariance.FrobeniusDuality
-
-end
-
-
-section
-
-namespace KLInvariance.Graded
-universe uk us
-variable {k : Type uk} [Field k] {σ : Type us}
-
- theorem algHom_polynomialGrading (f : MvPolynomial σ k →ₐ[k] MvPolynomial σ k)
-    (hX : ∀ i, f (MvPolynomial.X i) ∈ polynomialGrading k σ 1)
-    {n : ℤ} {p : MvPolynomial σ k} (hp : p ∈ polynomialGrading k σ n) :
-    f p ∈ polynomialGrading k σ n := by
-  classical
-  induction hp using MvPolynomial.IsWeightedHomogeneous.induction_on with
-  | zero => simpa only [map_zero] using (polynomialGrading k σ n).zero_mem
-  | add p q _ _ hp hq => simpa only [map_add] using (polynomialGrading k σ n).add_mem hp hq
-  | monomial d c hd =>
-    rw [MvPolynomial.monomial_eq,map_mul]
-    have hc : f (MvPolynomial.C c)=MvPolynomial.C c := f.commutes c
-    rw [hc]
-    apply MvPolynomial.IsWeightedHomogeneous.C_mul
-    simp only [Finsupp.prod,map_prod,map_pow]
-    have hh := SetLike.prod_pow_mem_graded (polynomialGrading k σ)
-      (fun _ : σ => (1:ℤ)) (fun i => f (MvPolynomial.X i)) d (F := d.support)
-      (fun i _ => hX i)
-    simpa only [Finsupp.weight_apply,Finsupp.sum] using hd ▸ hh
-
-end KLInvariance.Graded
-
-namespace KLInvariance.ReflectionFrobenius
-open Module KLInvariance.Graded
-universe uk uv us
-variable {k : Type uk} [Field k] [CharZero k]
-  {V : Type uv} [AddCommGroup V] [Module k V]
-  {σ : Type us} [Fintype σ] (basis : Basis σ k V)
-noncomputable section
-
- def polynomialLinear (v : V) : MvPolynomial σ k :=
-  SymmetricAlgebra.equivMvPolynomial basis (SymmetricAlgebra.ι k V v)
-
-omit [CharZero k] in
- theorem polynomialLinear_homogeneous (v : V) :
-    polynomialLinear basis v ∈ polynomialGrading k σ 1 := by
-  classical
-  unfold polynomialLinear
-  rw [← basis.sum_repr v]
-  simp only [map_sum,map_smul,SymmetricAlgebra.equivMvPolynomial_ι_apply]
-  apply Submodule.sum_mem
-  intro i _
-  exact (polynomialGrading k σ 1).smul_mem _
-    (MvPolynomial.isWeightedHomogeneous_X k (fun _ : σ => (1:ℤ)) i)
-
- def polynomialAction (e : V ≃ₗ[k] V) : MvPolynomial σ k ≃ₐ[k] MvPolynomial σ k :=
-  ((SymmetricAlgebra.equivMvPolynomial basis).symm.trans (symmetricAction e)).trans
-    (SymmetricAlgebra.equivMvPolynomial basis)
-
-omit [CharZero k] [Fintype σ] in
-@[simp] theorem polynomialAction_linear (e : V ≃ₗ[k] V) (v : V) :
-    polynomialAction basis e (polynomialLinear basis v) = polynomialLinear basis (e v) := by
-  simp [polynomialAction,polynomialLinear]
-
-omit [CharZero k] in
- theorem polynomialAction_homogeneous (e : V ≃ₗ[k] V) {n : ℤ}
-    {p : MvPolynomial σ k} (hp : p ∈ polynomialGrading k σ n) :
-    polynomialAction basis e p ∈ polynomialGrading k σ n := by
-  apply algHom_polynomialGrading (polynomialAction basis e).toAlgHom _ hp
-  intro i
-  change polynomialAction basis e (MvPolynomial.X i) ∈ _
-  have hx : MvPolynomial.X i = polynomialLinear basis (basis i) := by
-    simp [polynomialLinear]
-  rw [hx,polynomialAction_linear]
-  exact polynomialLinear_homogeneous basis _
-
-variable (root : V) (coroot : Dual k V) (hroot : coroot root=2)
-
- abbrev polynomialReflection : MvPolynomial σ k ≃ₐ[k] MvPolynomial σ k :=
-  polynomialAction basis (Module.reflection hroot)
-
-omit [CharZero k] [Fintype σ] in
- theorem polynomialReflection_involutive : Function.Involutive
-    (polynomialReflection basis root coroot hroot) := by
-  intro p
-  change SymmetricAlgebra.equivMvPolynomial basis
-    (reflectionAction root coroot hroot
-      ((SymmetricAlgebra.equivMvPolynomial basis).symm
-        (SymmetricAlgebra.equivMvPolynomial basis
-          (reflectionAction root coroot hroot
-            ((SymmetricAlgebra.equivMvPolynomial basis).symm p)))))=p
-  rw [AlgEquiv.symm_apply_apply,reflectionAction_involutive,AlgEquiv.apply_symm_apply]
-
-omit [CharZero k] [Fintype σ] in
- theorem polynomialReflection_anti : polynomialReflection basis root coroot hroot
-    (polynomialLinear basis root) = -polynomialLinear basis root := by
-  change SymmetricAlgebra.equivMvPolynomial basis
-    (reflectionAction root coroot hroot
-      ((SymmetricAlgebra.equivMvPolynomial basis).symm
-        (SymmetricAlgebra.equivMvPolynomial basis (SymmetricAlgebra.ι k V root)))) = _
-  rw [AlgEquiv.symm_apply_apply,reflectionAction_anti,map_neg]
-  rfl
-
-omit [Fintype σ] in
-include coroot hroot in
- theorem polynomialLinear_ne_zero : polynomialLinear basis root ≠ 0 := by
-  intro h
-  apply reflectionRoot_ne_zero root coroot hroot
-  exact (SymmetricAlgebra.equivMvPolynomial basis).injective (by simpa [polynomialLinear] using h)
-
-omit [CharZero k] [Fintype σ] in
- theorem polynomialReflection_divisible (p : MvPolynomial σ k) :
-    polynomialLinear basis root ∣ p-polynomialReflection basis root coroot hroot p := by
-  obtain ⟨d,hd⟩ := reflectionAction_divisible root coroot hroot
-    ((SymmetricAlgebra.equivMvPolynomial basis).symm p)
-  refine ⟨SymmetricAlgebra.equivMvPolynomial basis d,?_⟩
-  simpa [polynomialLinear,polynomialReflection,polynomialAction] using
-    congrArg (SymmetricAlgebra.equivMvPolynomial basis) hd
-
- def polynomialTrace : MvPolynomial σ k →ₗ[fixed (polynomialReflection basis root coroot hroot)]
-    fixed (polynomialReflection basis root coroot hroot) :=
-  trace _ _ (polynomialReflection_involutive basis root coroot hroot)
-    (polynomialLinear_ne_zero basis root coroot hroot)
-    (polynomialReflection_anti basis root coroot hroot)
-    (polynomialReflection_divisible basis root coroot hroot)
-
- theorem polynomialTrace_homogeneous {n : ℤ} {p : MvPolynomial σ k}
-    (hp : p ∈ polynomialGrading k σ n) :
-    (polynomialTrace basis root coroot hroot p : MvPolynomial σ k) ∈ polynomialGrading k σ (n-1) :=
-  trace_homogeneous _ _ _ _ _ _ (polynomialGrading k σ)
-    (polynomialLinear_homogeneous basis root)
-    (fun _ _ h => polynomialAction_homogeneous basis (Module.reflection hroot) h) hp
-
-end
-end KLInvariance.ReflectionFrobenius
-
-end
-
-
-section
-
-/-! The genuine symmetric-algebra grading of the extended real realization,
-transported along a specified realization basis only to access polynomial
-homogeneous components. The algebra and action are unchanged. -/
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded ReflectionFrobenius
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- def symmetricGrading : ℤ → Submodule ℝ (SymmetricCoefficient (M := M)) :=
-  fun n => (polynomialGrading ℝ σ n).map
-    (SymmetricAlgebra.equivMvPolynomial basis).symm.toLinearMap
-
-omit [Fintype σ] in
- theorem mem_symmetricGrading (n : ℤ) (a : SymmetricCoefficient (M := M)) :
-    a ∈ symmetricGrading basis n ↔
-      SymmetricAlgebra.equivMvPolynomial basis a ∈ polynomialGrading ℝ σ n := by
-  constructor
-  · rintro ⟨p,hp,hpa⟩
-    rw [← hpa]
-    simpa using hp
-  · intro h
-    exact ⟨_,h,(SymmetricAlgebra.equivMvPolynomial basis).symm_apply_apply a⟩
-
- instance symmetricGrading_decomposition : DirectSum.Decomposition (symmetricGrading basis) := by
-  apply DirectSum.IsInternal.chooseDecomposition
-  apply image_isInternal (polynomialGrading ℝ σ)
-    (SymmetricAlgebra.equivMvPolynomial basis).symm.toLinearMap
-    (SymmetricAlgebra.equivMvPolynomial basis).symm.surjective
-  intro n p hp
-  have hz : p=0 := (SymmetricAlgebra.equivMvPolynomial basis).symm.injective
-    (by simpa using hp)
-  subst p
-  simp
-
- instance symmetricGrading_graded : SetLike.GradedMonoid (symmetricGrading basis) where
-  one_mem := (mem_symmetricGrading basis 0 1).mpr (by
-    rw [map_one]
-    exact SetLike.one_mem_graded (polynomialGrading ℝ σ))
-  mul_mem := by
-    intro n m a b ha hb
-    apply (mem_symmetricGrading basis (n+m) (a*b)).mpr
-    rw [map_mul]
-    exact SetLike.GradedMul.mul_mem ((mem_symmetricGrading basis n a).mp ha)
-      ((mem_symmetricGrading basis m b).mp hb)
-
-omit [Fintype σ] in
- theorem symmetricGrading_negative (n : ℤ) (hn : n<0) :
-    symmetricGrading basis n = ⊥ := by
-  simp [symmetricGrading,polynomialGrading_negative ℝ σ n hn]
-
- theorem simpleRoot_homogeneous (i : I) :
-    simpleRootPolynomial (M := M) i ∈ symmetricGrading basis 1 := by
-  apply (mem_symmetricGrading basis 1 _).mpr
-  exact polynomialLinear_homogeneous basis _
-
- theorem coefficientAction_homogeneous (w : W) (n : ℤ)
-    (a : SymmetricCoefficient (M := M)) (ha : a ∈ symmetricGrading basis n) :
-    coefficientAction cs w a ∈ symmetricGrading basis n := by
-  apply (mem_symmetricGrading basis n _).mpr
-  have h := polynomialAction_homogeneous basis (extendedAction M cs w)
-    ((mem_symmetricGrading basis n a).mp ha)
-  simpa only [polynomialAction,AlgEquiv.trans_apply,AlgEquiv.symm_apply_apply,
-    coefficientAction] using h
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! A grading defined by an actual homogeneous free basis, with an explicit
-coefficient test. Used to put the genuine grading on tensor Bott--Samelsons. -/
-namespace KLInvariance.Graded
-open Module
-universe uk ua um ui
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  (𝓐 : ℤ → Submodule k A) [DirectSum.Decomposition 𝓐]
-  [SetLike.GradedMonoid 𝓐]
-  {N : Type um} [AddCommGroup N] [Module A N] [Module k N] [IsScalarTower k A N]
-  {ι : Type ui} [Fintype ι] (b : Basis ι A N) (d : ι → ℤ)
-noncomputable section
-
- def basisPiece (n : ℤ) : Submodule k N :=
-  (shiftedPiece 𝓐 d n).comap (b.equivFun.restrictScalars k).toLinearMap
-
-omit [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐] in
- theorem mem_basisPiece (n : ℤ) (m : N) :
-    m ∈ basisPiece 𝓐 b d n ↔ ∀ i, b.repr m i ∈ 𝓐 (n-d i) := Iff.rfl
-
- instance basisPiece_decomposition : DirectSum.Decomposition (basisPiece 𝓐 b d) := by
-  let _ : DirectSum.Decomposition (shiftedPiece 𝓐 d) := (shifted_isInternal 𝓐 d).chooseDecomposition
-  apply DirectSum.IsInternal.chooseDecomposition
-  have he : basisPiece 𝓐 b d = fun n => (shiftedPiece 𝓐 d n).map
-      (b.equivFun.restrictScalars k).symm.toLinearMap := by
-    funext n
-    exact Submodule.comap_equiv_eq_map_symm _ _
-  rw [he]
-  apply image_isInternal (shiftedPiece 𝓐 d)
-    (b.equivFun.restrictScalars k).symm.toLinearMap
-    (b.equivFun.restrictScalars k).symm.surjective
-  intro n m hm
-  have hz : m=0 := (b.equivFun.restrictScalars k).symm.injective (by simpa using hm)
-  subst m
-  simp
-
- instance basisPiece_graded : SetLike.GradedSMul 𝓐 (basisPiece 𝓐 b d) where
-  smul_mem := by
-    intro n m a x ha hx i
-    change b.repr (a • x) i ∈ 𝓐 ((n+m)-d i)
-    rw [map_smul,Finsupp.smul_apply,smul_eq_mul]
-    change ∀ i, b.repr x i ∈ 𝓐 (m-d i) at hx
-    have h := SetLike.GradedMul.mul_mem ha (hx i)
-    simpa only [add_sub_assoc] using h
-
-omit [DirectSum.Decomposition 𝓐] in
- theorem basis_mem_piece (i : ι) : b i ∈ basisPiece 𝓐 b d (d i) := by
-  classical
-  intro j
-  change b.repr (b i) j ∈ 𝓐 (d i-d j)
-  by_cases hij : i=j
-  · subst j
-    simp only [Basis.repr_self,Finsupp.single_eq_same,sub_self]
-    exact SetLike.one_mem_graded 𝓐
-  · rw [Basis.repr_self,Finsupp.single_eq_of_ne (Ne.symm hij)]
-    exact Submodule.zero_mem _
-
-omit [DirectSum.Decomposition 𝓐] in
- theorem pairing_homogeneous (p : N →ₗ[A] Dual A N) (l : ℤ)
-    (hp : ∀ i j, p (b i) (b j) ∈ 𝓐 (d i+d j-l))
-    {n m : ℤ} {x y : N} (hx : x ∈ basisPiece 𝓐 b d n)
-    (hy : y ∈ basisPiece 𝓐 b d m) : p x y ∈ 𝓐 (n+m-l) := by
-  classical
-  have hexp : p x y = ∑ i, ∑ j,
-      b.repr x i * b.repr y j * p (b i) (b j) := by
-    conv_lhs => rw [← b.sum_repr x,← b.sum_repr y]
-    simp only [map_sum,LinearMap.sum_apply,map_smul,LinearMap.smul_apply,smul_eq_mul,
-      Finset.mul_sum,mul_assoc]
-    rw [Finset.sum_comm]
-    apply Finset.sum_congr rfl
-    intro i _
-    apply Finset.sum_congr rfl
-    intro j _
-    ring
-  rw [hexp]
-  apply Submodule.sum_mem
-  intro i _
-  apply Submodule.sum_mem
-  intro j _
-  change ∀ i, b.repr x i ∈ 𝓐 (n-d i) at hx
-  change ∀ i, b.repr y i ∈ 𝓐 (m-d i) at hy
-  have h := SetLike.GradedMul.mul_mem
-    (SetLike.GradedMul.mul_mem (hx i) (hy j)) (hp i j)
-  convert h using 1
-  congr 1
-  omega
-
-end
-end KLInvariance.Graded
-
-end
-
-
-section
-
-/-! The actual tensor-defined Bott--Samelson module has the word grading and
-its integral duality lowers degree by word length (degree-one roots).
-This does not identify an indecomposable summand with the BMP sheaf. -/
-namespace KLInvariance.TitsSpace
-open Module TensorProduct FrobeniusDuality ReflectionFrobenius
-open _root_.OAI.KLInvariance.Graded
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- def BottIndex : List I → Type u
-  | [] => PUnit
-  | _ :: l => Fin 2 × BottIndex l
-
- instance bottIndex_fintype : (l : List I) → Fintype (BottIndex l)
-  | [] => inferInstanceAs (Fintype PUnit)
-  | _ :: l => by
-      let _ := bottIndex_fintype l
-      exact inferInstanceAs (Fintype (Fin 2 × BottIndex l))
-
- def bottDegree : (l : List I) → BottIndex l → ℤ
-  | [], _ => 0
-  | _ :: l, e => e.1.val + bottDegree l e.2
-
-omit [Fintype I] in
- theorem bottDegree_nonneg (l : List I) (e : BottIndex l) : 0 ≤ bottDegree l e := by
-  induction l with
-  | nil => exact le_rfl
-  | cons i l ih => exact add_nonneg (Int.natCast_nonneg _) (ih e.2)
-
- def bottBasis : (l : List I) → Basis (BottIndex l) (SymmetricCoefficient (M := M))
-    (BottSamelsonModule cs l)
-  | [] => Basis.singleton PUnit _
-  | i :: l => by
-      let : Module.Free (fixed (coefficientAction cs (cs.simple i))) (SymmetricCoefficient (M := M)) :=
-        simpleFixed_free cs i
-      let : Module.Finite (fixed (coefficientAction cs (cs.simple i))) (SymmetricCoefficient (M := M)) :=
-        simpleFixed_finite cs i
-      exact inducedBasis (coefficientAction cs (cs.simple i))
-        (simpleRootPolynomial i) (simple_involutive cs i) (simpleRootPolynomial_ne_zero i)
-        (simple_anti cs i) (simple_divisible cs i) (bottSamelson cs l).obj (bottBasis l)
-
- theorem bottBasis_pairing_homogeneous (l : List I) (e f : BottIndex l) :
-    bottSamelsonDuality cs l (bottBasis cs l e) (bottBasis cs l f) ∈
-      symmetricGrading basis (bottDegree l e+bottDegree l f-l.length) := by
-  induction l with
-  | nil =>
-    change PUnit at e f
-    change (Basis.singleton PUnit (SymmetricCoefficient (M := M)) e) *
-      (Basis.singleton PUnit (SymmetricCoefficient (M := M)) f) ∈ symmetricGrading basis 0
-    simp only [Basis.singleton_apply, one_mul]
-    exact SetLike.one_mem_graded _
-  | cons i l ih =>
-    change (Fin 2 × BottIndex l) at e f
-    rcases e with ⟨ei, ej⟩
-    rcases f with ⟨fi, fj⟩
-    let : Module.Free (fixed (coefficientAction cs (cs.simple i))) (SymmetricCoefficient (M := M)) :=
-      simpleFixed_free cs i
-    let : Module.Finite (fixed (coefficientAction cs (cs.simple i))) (SymmetricCoefficient (M := M)) :=
-      simpleFixed_finite cs i
-    have h := inducedPairing_basis_homogeneous
-      (coefficientAction cs (cs.simple i)) (simpleRootPolynomial i)
-      (simple_involutive cs i) (simpleRootPolynomial_ne_zero i)
-      (simple_anti cs i) (simple_divisible cs i) (bottSamelson cs l).obj
-      (bottBasis cs l) (symmetricGrading basis) (simpleRoot_homogeneous basis i)
-      (coefficientAction_homogeneous cs basis (cs.simple i))
-      (bottDegree l) l.length ih ei fi ej fj
-    simp only [List.length_cons,Nat.cast_add,Nat.cast_one,bottDegree]
-    convert h using 1; rfl
-
- instance bottRealModule (l : List I) : Module ℝ (BottSamelsonModule cs l) :=
-  Module.compHom _ (algebraMap ℝ (SymmetricCoefficient (M := M)))
-
- instance bottRealTower (l : List I) :
-    IsScalarTower ℝ (SymmetricCoefficient (M := M)) (BottSamelsonModule cs l) :=
-  IsScalarTower.of_algebraMap_smul fun _ _ => rfl
-
- def bottPiece (l : List I) : ℤ → Submodule ℝ (BottSamelsonModule cs l) :=
-  basisPiece (symmetricGrading basis) (bottBasis cs l) (bottDegree l)
-
- instance bottPiece_decomposition (l : List I) : DirectSum.Decomposition (bottPiece cs basis l) :=
-  basisPiece_decomposition _ _ _
-
- instance bottPiece_graded (l : List I) :
-    SetLike.GradedSMul (symmetricGrading basis) (bottPiece cs basis l) :=
-  basisPiece_graded _ _ _
-
- theorem bottDuality_homogeneous (l : List I) {n m : ℤ}
-    {x y : BottSamelsonModule cs l} (hx : x ∈ bottPiece cs basis l n)
-    (hy : y ∈ bottPiece cs basis l m) :
-    bottSamelsonDuality cs l x y ∈ symmetricGrading basis (n+m-l.length) := by
-  exact pairing_homogeneous (symmetricGrading basis) (bottBasis cs l) (bottDegree l)
-    (bottSamelsonDuality cs l).toLinearMap l.length
-    (bottBasis_pairing_homogeneous cs basis l) hx hy
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Actual subexpression-weight coordinates of the tensor Bott--Samelson
-module, and the genuine right-action eigenvalue. These are the localization
-maps required for moment-sheaf realization; no character theorem is assumed. -/
-namespace KLInvariance.TitsSpace
-open Module TensorProduct FrobeniusDuality ReflectionFrobenius
-universe u v un
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
- theorem coefficientAction_one (a : SymmetricCoefficient (M := M)) :
-    coefficientAction cs 1 a = a := by
-  have h := coefficientAction_mul cs 1 1 a
-  rw [one_mul] at h
-  exact (coefficientAction cs 1).injective h.symm
-
- def bottWeight : (l : List I) → BottIndex l → W
-  | [], _ => 1
-  | i::l, e => if e.1=0 then bottWeight l e.2 else cs.simple i*bottWeight l e.2
-
-attribute [local instance] restrictedModule restrictedTower restrictedFree restrictedFinite
-
- def weightFunctional (i : I) (N : PairedModule.{u,un} (SymmetricCoefficient (M := M)))
-    (f : N →ₗ[SymmetricCoefficient (M := M)] SymmetricCoefficient (M := M))
-    (ε : Fin 2) : N →ₗ[simpleFixed cs i] SymmetricCoefficient (M := M) where
-  toFun m := if ε=0 then f m else coefficientAction cs (cs.simple i) (f m)
-  map_add' m n := by split_ifs <;> simp only [map_add]
-  map_smul' c m := by
-    change (if ε=0 then f ((c:SymmetricCoefficient (M:=M)) • m) else
-      coefficientAction cs (cs.simple i) (f ((c:SymmetricCoefficient (M:=M)) • m))) = _
-    split_ifs
-    · simp only [map_smul,Subalgebra.smul_def,RingHom.id_apply]
-    · rw [map_smul]
-      change coefficientAction cs (cs.simple i)
-        ((c:SymmetricCoefficient (M:=M))*f m) =
-        (c:SymmetricCoefficient (M:=M))*coefficientAction cs (cs.simple i) (f m)
-      rw [map_mul, show coefficientAction cs (cs.simple i) (c:SymmetricCoefficient (M:=M)) = c from c.property]
-
- def inducedFunctional (i : I) (N : PairedModule.{u,un} (SymmetricCoefficient (M := M)))
-    (f : N →ₗ[simpleFixed cs i] SymmetricCoefficient (M := M)) :
-    (SymmetricCoefficient (M := M) ⊗[simpleFixed cs i] N) →ₗ[SymmetricCoefficient (M := M)]
-      SymmetricCoefficient (M := M) :=
-  AlgebraTensorModule.lift ((LinearMap.id : SymmetricCoefficient (M := M) →ₗ[
-    SymmetricCoefficient (M := M)] SymmetricCoefficient (M := M)).smulRight f)
-
- def bottEvaluation : (l : List I) → BottIndex l →
-    BottSamelsonModule cs l →ₗ[SymmetricCoefficient (M := M)] SymmetricCoefficient (M := M)
-  | [], _ => LinearMap.id
-  | i::l, e => inducedFunctional cs i (bottSamelson cs l).obj
-      (weightFunctional cs i (bottSamelson cs l).obj (bottEvaluation l e.2) e.1)
-
- theorem bottEvaluation_tmul (i : I) (l : List I) (e : BottIndex (i::l))
-    (a : SymmetricCoefficient (M := M)) (m : BottSamelsonModule cs l) :
-    bottEvaluation cs (i::l) e (a ⊗ₜ[simpleFixed cs i] m) =
-      a*(if e.1=0 then bottEvaluation cs l e.2 m else
-        coefficientAction cs (cs.simple i) (bottEvaluation cs l e.2 m)) := rfl
-
- theorem bottEvaluation_right (l : List I) (e : BottIndex l)
-    (a : SymmetricCoefficient (M := M)) (m : BottSamelsonModule cs l) :
-    bottEvaluation cs l e (bottSamelsonRight cs l a m) =
-      coefficientAction cs (bottWeight cs l e) a * bottEvaluation cs l e m := by
-  induction l with
-  | nil =>
-    change SymmetricCoefficient (M := M) at m
-    change a*m = coefficientAction cs 1 a * m
-    rw [coefficientAction_one]
-  | cons i l ih =>
-    change SymmetricCoefficient (M := M) ⊗[simpleFixed cs i] BottSamelsonModule cs l at m
-    induction m using TensorProduct.inductionOn with
-    | add m n hm hn => erw [map_add,map_add,map_add,hm,hn,mul_add]
-    | tmul b m =>
-      change bottEvaluation cs (i::l) e (b ⊗ₜ[simpleFixed cs i] bottSamelsonRight cs l a m) = _
-      erw [bottEvaluation_tmul,bottEvaluation_tmul,ih]
-      change b*(if e.1=0 then _ else _) = coefficientAction cs
-        (if e.1=0 then bottWeight cs l e.2 else cs.simple i*bottWeight cs l e.2) a * _
-      split_ifs with h
-      · ring
-      · rw [map_mul,coefficientAction_mul]
-        ring
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Localization coordinates for tensor induction over a reflection wall.
-This is an integral injectivity argument, not character normalization. -/
-namespace KLInvariance.ReflectionFrobenius
-open scoped Matrix
-universe ua ui
-variable {A : Type ua} [CommRing A] [IsDomain A] [CharZero A]
-  {ι : Type ui} [Fintype ι] (τ : A ≃+* A) (α : A)
-noncomputable section
-
- def wallCoordinate (E : Matrix ι ι A) (v : Fin 2 × ι → A) : Fin 2 × ι → A := fun e =>
-  if e.1=0 then (E *ᵥ (fun j => v (0,j)+α*v (1,j))) e.2
-  else ((E.map τ) *ᵥ (fun j => v (0,j)-α*v (1,j))) e.2
-
- def wallMatrix (E : Matrix ι ι A) : Matrix (Fin 2 × ι) (Fin 2 × ι) A := fun e f =>
-  if e.1=0 then (if f.1=0 then E e.2 f.2 else α*E e.2 f.2)
-  else if f.1=0 then τ (E e.2 f.2) else -α*τ (E e.2 f.2)
-
-omit [IsDomain A] [CharZero A] in
- theorem wallMatrix_mulVec (E : Matrix ι ι A) (v : Fin 2 × ι → A) :
-    wallMatrix τ α E *ᵥ v = wallCoordinate τ α E v := by
-  classical
-  funext ⟨r,j⟩
-  change (∑ c : Fin 2 × ι, wallMatrix τ α E (r,j) c * v c) = _
-  rw [Fintype.sum_prod_type,Fin.sum_univ_two]
-  fin_cases r <;>
-    simp [wallMatrix,wallCoordinate,Matrix.mulVec,dotProduct,
-      Finset.sum_add_distrib,mul_add,
-      mul_left_comm,mul_assoc,sub_eq_add_neg]
-
-omit [IsDomain A] [CharZero A] in
- theorem matrixTwist_injective (E : Matrix ι ι A)
-    (hE : Function.Injective (Matrix.mulVec E)) : Function.Injective (Matrix.mulVec (E.map τ)) := by
-  intro v w hv
-  have hh : E *ᵥ (fun j => τ.symm (v j)) = E *ᵥ (fun j => τ.symm (w j)) := by
-    funext i
-    have h := congrArg τ.symm (congrFun hv i)
-    simpa only [Matrix.mulVec,dotProduct,map_sum,map_mul,Matrix.map_apply,
-      RingEquiv.symm_apply_apply] using h
-  have hh' := hE hh
-  funext i
-  exact τ.symm.injective (congrFun hh' i)
-
- theorem wallCoordinate_injective (hα : α ≠ 0) (E : Matrix ι ι A)
-    (hE : Function.Injective (Matrix.mulVec E)) : Function.Injective (wallCoordinate τ α E) := by
-  intro v w hv
-  have h₀ : (fun j => v (0,j)+α*v (1,j)) = (fun j => w (0,j)+α*w (1,j)) := by
-    apply hE
-    funext j
-    simpa only [wallCoordinate,ite_eq_left rfl,ite_true] using congrFun hv (0,j)
-  have h₁ : (fun j => v (0,j)-α*v (1,j)) = (fun j => w (0,j)-α*w (1,j)) := by
-    apply matrixTwist_injective τ E hE
-    funext j
-    simpa only [wallCoordinate,show (1:Fin 2) ≠ 0 by decide,ite_false] using congrFun hv (1,j)
-  funext ⟨ε,j⟩
-  have h0 := congrFun h₀ j
-  have h1 := congrFun h₁ j
-  have hi1 : v (1,j)=w (1,j) := by
-    have hh : (2:A)*α*(v (1,j)-w (1,j))=0 := by linear_combination h0-h1
-    exact sub_eq_zero.mp ((mul_eq_zero.mp hh).resolve_left (mul_ne_zero two_ne_zero hα))
-  have hi0 : v (0,j)=w (0,j) := by simpa only [hi1,add_left_inj] using h0
-  fin_cases ε
-  · exact hi0
-  · exact hi1
-
-end
-end KLInvariance.ReflectionFrobenius
-
-end
-
-
-section
-
-/-! The actual tensor Bott--Samelson module embeds integrally in its individual
-subexpression coordinates. Together with the checked weight eigenvalue this
-identifies its generic support. It does not assert the integral moment-sheaf
-recognition or the Elias--Williamson character theorem. -/
-namespace KLInvariance.TitsSpace
-open Module TensorProduct FrobeniusDuality ReflectionFrobenius
-open scoped Matrix
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-attribute [local instance] restrictedModule restrictedTower restrictedFree restrictedFinite
-
- def bottEvaluationMatrix (l : List I) : Matrix (BottIndex l) (BottIndex l)
-    (SymmetricCoefficient (M := M)) := fun e j => bottEvaluation cs l e (bottBasis cs l j)
-
- theorem bottEvaluation_expand (l : List I) (e : BottIndex l) (m : BottSamelsonModule cs l) :
-    bottEvaluation cs l e m =
-      (bottEvaluationMatrix cs l *ᵥ (bottBasis cs l).equivFun m) e := by
-  classical
-  conv_lhs => rw [← (bottBasis cs l).sum_repr m]
-  simp only [map_sum,map_smul,smul_eq_mul,Matrix.mulVec,dotProduct,
-    bottEvaluationMatrix,Basis.equivFun_apply]
-  exact Finset.sum_congr rfl (fun _ _ => mul_comm _ _)
-
- theorem bottEvaluationMatrix_cons (i : I) (l : List I)
-    (r c : Fin 2) (j k : BottIndex l) :
-    bottEvaluationMatrix cs (i::l) (r,j) (c,k) =
-      if r=0 then (if c=0 then bottEvaluationMatrix cs l j k else
-        simpleRootPolynomial i*bottEvaluationMatrix cs l j k)
-      else if c=0 then coefficientAction cs (cs.simple i) (bottEvaluationMatrix cs l j k)
-      else -simpleRootPolynomial i*coefficientAction cs (cs.simple i) (bottEvaluationMatrix cs l j k) := by
-  let : Module.Free (fixed (coefficientAction cs (cs.simple i))) (SymmetricCoefficient (M := M)) :=
-    simpleFixed_free cs i
-  let : Module.Finite (fixed (coefficientAction cs (cs.simple i))) (SymmetricCoefficient (M := M)) :=
-    simpleFixed_finite cs i
-  change bottEvaluation cs (i::l) (r,j)
-    (inducedBasis (coefficientAction cs (cs.simple i)) (simpleRootPolynomial i)
-      (simple_involutive cs i) (simpleRootPolynomial_ne_zero i) (simple_anti cs i)
-      (simple_divisible cs i) (bottSamelson cs l).obj (bottBasis cs l) (c,k)) = _
-  rw [inducedBasis_apply]
-  erw [bottEvaluation_tmul]
-  fin_cases r <;> fin_cases c <;>
-    simp only [Fin.zero_eta,Fin.isValue,Fin.mk_one,ite_true,
-      show (1:Fin 2) ≠ 0 by decide,ite_false,reflectionBasis_zero,reflectionBasis_one,
-      one_smul,one_mul,map_smul,smul_eq_mul,map_mul,simple_anti,bottEvaluationMatrix]
-
- theorem bottEvaluationMatrix_mulVec_cons (i : I) (l : List I)
-    (v : BottIndex (i::l) → SymmetricCoefficient (M := M)) :
-    bottEvaluationMatrix cs (i::l) *ᵥ v =
-      wallCoordinate (coefficientAction cs (cs.simple i)).toRingEquiv
-        (simpleRootPolynomial i) (bottEvaluationMatrix cs l) v := by
-  have heq : bottEvaluationMatrix cs (i::l) =
-      wallMatrix (coefficientAction cs (cs.simple i)).toRingEquiv
-        (simpleRootPolynomial i) (bottEvaluationMatrix cs l) := by
-    funext e f
-    rcases e with ⟨r,j⟩
-    rcases f with ⟨c,k⟩
-    exact bottEvaluationMatrix_cons cs i l r c j k
-  rw [heq]
-  exact wallMatrix_mulVec (coefficientAction cs (cs.simple i)).toRingEquiv
-    (simpleRootPolynomial i) (bottEvaluationMatrix cs l) v
-
- theorem bottEvaluationMatrix_injective (l : List I) :
-    Function.Injective (Matrix.mulVec (bottEvaluationMatrix cs l)) := by
-  classical
-  let : CharZero (SymmetricCoefficient (M := M)) := Algebra.charZero_of_charZero ℝ _
-  induction l with
-  | nil =>
-    intro v w h
-    funext j
-    change PUnit at j
-    cases j
-    have hh := congrFun h PUnit.unit
-    let : Unique (BottIndex ([] : List I)) := inferInstanceAs (Unique PUnit)
-    have hcol (k : BottIndex ([] : List I)) :
-        bottEvaluationMatrix cs [] PUnit.unit k = 1 := by
-      change PUnit at k
-      change (Basis.singleton PUnit (SymmetricCoefficient (M := M)) k) = 1
-      simp only [Basis.singleton_apply]
-    simp only [Matrix.mulVec,dotProduct,hcol,one_mul,Finset.univ_unique,
-      Finset.sum_singleton] at hh
-    simpa only [Subsingleton.elim (default : BottIndex ([] : List I)) PUnit.unit] using hh
-  | cons i l ih =>
-    intro v w h
-    apply wallCoordinate_injective (coefficientAction cs (cs.simple i)).toRingEquiv
-      (simpleRootPolynomial i) (simpleRootPolynomial_ne_zero i) (bottEvaluationMatrix cs l) ih
-    rw [← bottEvaluationMatrix_mulVec_cons,← bottEvaluationMatrix_mulVec_cons]
-    exact h
-
- theorem bottEvaluation_injective (l : List I) :
-    Function.Injective (fun m : BottSamelsonModule cs l => fun e => bottEvaluation cs l e m) := by
-  intro m n h
-  apply (bottBasis cs l).equivFun.injective
-  apply bottEvaluationMatrix_injective cs l
-  funext e
-  rw [← bottEvaluation_expand,← bottEvaluation_expand]
-  exact congrFun h e
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Integral denominators exhibit the full generic subexpression support of
-an actual tensor word. No moment-sheaf character assertion is made. -/
-namespace KLInvariance.TitsSpace
-open Module
-open scoped Matrix
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
- def bottCoordinateDenominator (l : List I) : SymmetricCoefficient (M := M) := by
-  classical
-  exact (bottEvaluationMatrix cs l).det
-
- theorem bottCoordinateDenominator_ne_zero (l : List I) :
-    bottCoordinateDenominator cs l ≠ 0 := by
-  classical
-  apply Matrix.nonsingular_iff_det_ne_zero.mp
-  apply Matrix.Nonsingular.of_linearIndependent_col
-  exact Matrix.mulVec_injective_iff.mp (bottEvaluationMatrix_injective cs l)
-
- theorem bottEvaluation_contains_scaled_coordinates (l : List I)
-    (v : BottIndex l → SymmetricCoefficient (M := M)) :
-    ∃ m : BottSamelsonModule cs l, ∀ e,
-      bottEvaluation cs l e m = bottCoordinateDenominator cs l * v e := by
-  classical
-  refine ⟨(bottBasis cs l).equivFun.symm ((bottEvaluationMatrix cs l).adjugate *ᵥ v),?_⟩
-  intro e
-  rw [bottEvaluation_expand]
-  simp only [LinearEquiv.apply_symm_apply,Matrix.mulVec_mulVec,Matrix.mul_adjugate,
-    Matrix.smul_mulVec,Matrix.one_mulVec,Pi.smul_apply,smul_eq_mul,bottCoordinateDenominator]
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Faithfulness of the actual polynomial action; distinct Coxeter weights
-have distinct integral polynomial characters. -/
-namespace KLInvariance.TitsSpace
-open Module
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
- theorem symmetricCoefficient_ι_injective :
-    Function.Injective (SymmetricAlgebra.ι ℝ (Extended M)) := by
-  intro x y h
-  apply sub_eq_zero.mp
-  apply (Module.forall_dual_apply_eq_zero_iff ℝ (x-y)).mp
-  intro f
-  rw [map_sub]
-  apply sub_eq_zero.mpr
-  have he := congrArg (SymmetricAlgebra.lift f) h
-  simpa only [SymmetricAlgebra.lift_ι_apply] using he
-
- theorem coefficientAction_injective : Function.Injective (coefficientAction cs) := by
-  intro x y h
-  apply extendedAction_injective M cs
-  apply LinearEquiv.ext
-  intro v
-  apply symmetricCoefficient_ι_injective (M := M)
-  have he := congrArg (fun g : SymmetricCoefficient (M := M) ≃ₐ[ℝ]
-    SymmetricCoefficient (M := M) => g (SymmetricAlgebra.ι ℝ (Extended M) v)) h
-  simpa only [coefficientAction_ι] using he
-
- theorem coefficientAction_separates {x y : W} (hxy : x ≠ y) :
-    ∃ a : SymmetricCoefficient (M := M), coefficientAction cs x a ≠ coefficientAction cs y a := by
-  by_contra! h
-  apply hxy
-  apply coefficientAction_injective cs
-  exact AlgEquiv.ext h
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Integral weight support and orthogonality for the actual tensor module.
-These are the vertex support statements behind the section pairing, not an
-identification of the minimal BMP sheaf or its character. -/
-namespace KLInvariance.TitsSpace
-open Module
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-local instance (l : List I) : DecidableEq (BottIndex l) := Classical.decEq _
-
- def bottWeightSupported (l : List I) (x : W) : Submodule (SymmetricCoefficient (M := M))
-    (BottSamelsonModule cs l) where
-  carrier := {m | ∀ e, bottWeight cs l e ≠ x → bottEvaluation cs l e m = 0}
-  zero_mem' := by intro e _; exact map_zero _
-  add_mem' := by intro m n hm hn e he; rw [map_add,hm e he,hn e he,zero_add]
-  smul_mem' := by intro a m hm e he; rw [map_smul,hm e he,smul_zero]
-
- theorem bott_weight_eigen (l : List I) (x : W) {m : BottSamelsonModule cs l}
-    (hm : m ∈ bottWeightSupported cs l x) (a : SymmetricCoefficient (M := M)) :
-    bottSamelsonRight cs l a m = coefficientAction cs x a • m := by
-  apply bottEvaluation_injective cs l
-  funext e
-  change bottEvaluation cs l e (bottSamelsonRight cs l a m) =
-    bottEvaluation cs l e (coefficientAction cs x a • m)
-  erw [bottEvaluation_right,map_smul]
-  change _ = coefficientAction cs x a * bottEvaluation cs l e m
-  by_cases he : bottWeight cs l e = x
-  · rw [he]
-  · rw [hm e he,mul_zero,mul_zero]
-
- theorem bott_eigen_supported (l : List I) (x : W) {m : BottSamelsonModule cs l}
-    (hm : ∀ a : SymmetricCoefficient (M := M),
-      bottSamelsonRight cs l a m = coefficientAction cs x a • m) :
-    m ∈ bottWeightSupported cs l x := by
-  intro e he
-  obtain ⟨a,ha⟩ := coefficientAction_separates cs he
-  have h := congrArg (bottEvaluation cs l e) (hm a)
-  erw [bottEvaluation_right,map_smul] at h
-  change coefficientAction cs (bottWeight cs l e) a * bottEvaluation cs l e m =
-    coefficientAction cs x a * bottEvaluation cs l e m at h
-  have hh : (coefficientAction cs (bottWeight cs l e) a-coefficientAction cs x a) *
-      bottEvaluation cs l e m = 0 := by rw [sub_mul,h,sub_self]
-  exact (mul_eq_zero.mp hh).resolve_left (sub_ne_zero.mpr ha)
-
- theorem bott_distinct_weight_orthogonal (l : List I) {x y : W} (hxy : x ≠ y)
-    {m n : BottSamelsonModule cs l} (hm : m ∈ bottWeightSupported cs l x)
-    (hn : n ∈ bottWeightSupported cs l y) : bottSamelsonDuality cs l m n = 0 := by
-  obtain ⟨a,ha⟩ := coefficientAction_separates cs hxy
-  have h := bottSamelson_balanced cs l a m n
-  rw [bott_weight_eigen cs l x hm,bott_weight_eigen cs l y hn] at h
-  simp only [map_smul,LinearMap.smul_apply,smul_eq_mul] at h
-  have he : (coefficientAction cs x a-coefficientAction cs y a) *
-      bottSamelsonDuality cs l m n = 0 := by rw [sub_mul,h,sub_self]
-  exact (mul_eq_zero.mp he).resolve_left (sub_ne_zero.mpr ha)
-
- def bottCoordinateLift (l : List I) (e : BottIndex l) : BottSamelsonModule cs l := by
-  classical
-  exact (bottEvaluation_contains_scaled_coordinates cs l (Pi.single e 1)).choose
-
- theorem bottCoordinateLift_eval (l : List I) (e j : BottIndex l) :
-    bottEvaluation cs l j (bottCoordinateLift cs l e) =
-      bottCoordinateDenominator cs l * (Pi.single e 1 : BottIndex l → SymmetricCoefficient (M := M)) j := by
-  classical
-  exact (bottEvaluation_contains_scaled_coordinates cs l (Pi.single e 1)).choose_spec j
-
- theorem bottCoordinateLift_supported (l : List I) (e : BottIndex l) :
-    bottCoordinateLift cs l e ∈ bottWeightSupported cs l (bottWeight cs l e) := by
-  intro j hj
-  rw [bottCoordinateLift_eval]
-  have hje : j ≠ e := fun h => hj (congrArg (bottWeight cs l) h)
-  rw [Pi.single_eq_of_ne hje, mul_zero]
-
- theorem bottCoordinateExpansion (l : List I) (m : BottSamelsonModule cs l) :
-    bottCoordinateDenominator cs l • m =
-      ∑ e : BottIndex l, bottEvaluation cs l e m • bottCoordinateLift cs l e := by
-  classical
-  apply bottEvaluation_injective cs l
-  funext j
-  simp only [map_smul,map_sum,bottCoordinateLift_eval,smul_eq_mul]
-  rw [Finset.sum_eq_single j]
-  · simp only [Pi.single_eq_same,mul_one,mul_comm]
-  · intro e _ hej
-    rw [Pi.single_eq_of_ne (Ne.symm hej),mul_zero,mul_zero]
-  · simp
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! The actual tensor word's integral weight-support lattice is the dual of
-its integral projected stalk lattice. This is not merely a fraction-field
-isomorphism. It still does NOT identify BMP sections with an indecomposable
-summand of the tensor word, or prove the character theorem. -/
-namespace KLInvariance.TitsSpace
-open Module
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
- def bottWeightProjection (l : List I) (x : W) : BottSamelsonModule cs l →ₗ[
-    SymmetricCoefficient (M := M)]
-    ({e : BottIndex l // bottWeight cs l e=x} → SymmetricCoefficient (M := M)) :=
-  LinearMap.pi (fun e => bottEvaluation cs l e.val)
-
- theorem bottWeightProjection_eq_zero (l : List I) (x : W) (m : BottSamelsonModule cs l) :
-    bottWeightProjection cs l x m = 0 ↔ ∀ e, bottWeight cs l e=x → bottEvaluation cs l e m=0 := by
-  constructor
-  · intro h e he
-    exact congrFun h ⟨e,he⟩
-  · intro h
-    funext e
-    exact h e.val e.property
-
- theorem bott_supported_annihilates (l : List I) (x : W) {m : BottSamelsonModule cs l}
-    (hm : m ∈ bottWeightSupported cs l x) (n : BottSamelsonModule cs l)
-    (hn : bottWeightProjection cs l x n=0) : bottSamelsonDuality cs l m n=0 := by
-  classical
-  have hn' := (bottWeightProjection_eq_zero cs l x n).mp hn
-  have hsum : ∑ e : BottIndex l, bottEvaluation cs l e n *
-      bottSamelsonDuality cs l m (bottCoordinateLift cs l e) = 0 := by
-    apply Finset.sum_eq_zero
-    intro e _
-    by_cases he : bottWeight cs l e=x
-    · rw [hn' e he,zero_mul]
-    · rw [bott_distinct_weight_orthogonal cs l (Ne.symm he) hm
-        (bottCoordinateLift_supported cs l e),mul_zero]
-  have hh := congrArg (bottSamelsonDuality cs l m) (bottCoordinateExpansion cs l n)
-  simp only [map_smul,map_sum,smul_eq_mul] at hh
-  rw [hsum] at hh
-  exact (mul_eq_zero.mp hh).resolve_left (bottCoordinateDenominator_ne_zero cs l)
-
- theorem bott_annihilates_supported (l : List I) (x : W) {m : BottSamelsonModule cs l}
-    (hm : ∀ n : BottSamelsonModule cs l, bottWeightProjection cs l x n=0 →
-      bottSamelsonDuality cs l m n=0) : m ∈ bottWeightSupported cs l x := by
-  apply bott_eigen_supported cs l x
-  intro a
-  apply (bottSamelsonDuality cs l).injective
-  apply LinearMap.ext
-  intro n
-  have hn : bottWeightProjection cs l x
-      (bottSamelsonRight cs l a n - coefficientAction cs x a • n)=0 := by
-    apply (bottWeightProjection_eq_zero cs l x _).mpr
-    intro e he
-    erw [map_sub,bottEvaluation_right,map_smul]
-    change coefficientAction cs (bottWeight cs l e) a * bottEvaluation cs l e n -
-      coefficientAction cs x a * bottEvaluation cs l e n=0
-    rw [he,sub_self]
-  have hh := hm _ hn
-  simp only [map_sub,map_smul,smul_eq_mul] at hh
-  rw [bottSamelson_balanced]
-  change bottSamelsonDuality cs l m (bottSamelsonRight cs l a n) =
-    (bottSamelsonDuality cs l (coefficientAction cs x a • m)) n
-  rw [map_smul,LinearMap.smul_apply,smul_eq_mul]
-  exact sub_eq_zero.mp hh
-
- theorem bott_supported_iff_annihilates (l : List I) (x : W) (m : BottSamelsonModule cs l) :
-    m ∈ bottWeightSupported cs l x ↔ ∀ n : BottSamelsonModule cs l,
-      bottWeightProjection cs l x n=0 → bottSamelsonDuality cs l m n=0 :=
-  ⟨fun h => bott_supported_annihilates cs l x h,fun h => bott_annihilates_supported cs l x h⟩
-
- def bottStalk (l : List I) (x : W) : Submodule (SymmetricCoefficient (M := M))
-    ({e : BottIndex l // bottWeight cs l e=x} → SymmetricCoefficient (M := M)) :=
-  LinearMap.range (bottWeightProjection cs l x)
-
- def bottStalkProjection (l : List I) (x : W) : BottSamelsonModule cs l →ₗ[
-    SymmetricCoefficient (M := M)] bottStalk cs l x :=
-  (bottWeightProjection cs l x).rangeRestrict
-
- theorem bottStalkProjection_surjective (l : List I) (x : W) :
-    Function.Surjective (bottStalkProjection cs l x) := by
-  rintro ⟨v,m,hm⟩
-  exact ⟨m,Subtype.ext hm⟩
-
- theorem bottStalkProjection_eq_zero (l : List I) (x : W) (m : BottSamelsonModule cs l) :
-    bottStalkProjection cs l x m=0 ↔ bottWeightProjection cs l x m=0 := by
-  exact Subtype.ext_iff
-
- def bottStalkDualToSupport (l : List I) (x : W) :
-    Dual (SymmetricCoefficient (M := M)) (bottStalk cs l x) →ₗ[SymmetricCoefficient (M := M)]
-      bottWeightSupported cs l x :=
-  ((bottSamelsonDuality cs l).symm.toLinearMap.comp
-    (bottStalkProjection cs l x).dualMap).codRestrict (bottWeightSupported cs l x) (by
-    intro lam
-    apply bott_annihilates_supported cs l x
-    intro n hn
-    simp only [LinearMap.comp_apply,LinearEquiv.coe_coe,LinearEquiv.apply_symm_apply,
-      LinearMap.dualMap_apply]
-    change lam (bottStalkProjection cs l x n)=0
-    rw [(bottStalkProjection_eq_zero cs l x n).mpr hn,map_zero])
-
- theorem bottStalkDualToSupport_bijective (l : List I) (x : W) :
-    Function.Bijective (bottStalkDualToSupport cs l x) := by
-  constructor
-  · intro lam mu he
-    apply LinearMap.dualMap_injective_of_surjective (bottStalkProjection_surjective cs l x)
-    apply (bottSamelsonDuality cs l).symm.injective
-    exact congrArg Subtype.val he
-  · intro m
-    have hm : bottSamelsonDuality cs l m.val ∈
-        (LinearMap.ker (bottStalkProjection cs l x)).dualAnnihilator := by
-      rw [Submodule.mem_dualAnnihilator]
-      intro n hn
-      exact bott_supported_annihilates cs l x m.property n
-        ((bottStalkProjection_eq_zero cs l x n).mp hn)
-    rw [← LinearMap.range_dualMap_eq_dualAnnihilator_ker_of_surjective _
-      (bottStalkProjection_surjective cs l x)] at hm
-    obtain ⟨lam,hlam⟩ := hm
-    refine ⟨lam,?_⟩
-    apply Subtype.ext
-    change (bottSamelsonDuality cs l).symm ((bottStalkProjection cs l x).dualMap lam)=m.val
-    rw [hlam,LinearEquiv.symm_apply_apply]
-
- def bottStalkDuality (l : List I) (x : W) : bottWeightSupported cs l x ≃ₗ[
-    SymmetricCoefficient (M := M)] Dual (SymmetricCoefficient (M := M)) (bottStalk cs l x) :=
-  (LinearEquiv.ofBijective (bottStalkDualToSupport cs l x)
-    (bottStalkDualToSupport_bijective cs l x)).symm
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Full reflection labels, rather than just simple-wall labels, in the
-actual tensor bimodule coefficient action. -/
-namespace KLInvariance.TitsSpace
-open Module ReflectionFrobenius
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
- theorem coefficientAction_reflection {t : W} {a : I → ℝ} (ha : Represents M cs t a) :
-    coefficientAction cs t = reflectionAction (embed M a) (extendedForm M (embed M a))
-      (show extendedForm M (embed M a) (embed M a) = 2 by
-        rw [(extendedForm_spec M).2.2]; exact ha.isRoot.norm) := by
-  rw [coefficientAction,ha.extendedAction_eq]
-
- theorem reflectionCoefficient_anti {t : W} {a : I → ℝ} (ha : Represents M cs t a) :
-    coefficientAction cs t (SymmetricAlgebra.ι ℝ (Extended M) (embed M a)) =
-      -SymmetricAlgebra.ι ℝ (Extended M) (embed M a) := by
-  rw [coefficientAction_reflection cs ha]
-  exact reflectionAction_anti _ _ _
-
- theorem reflectionCoefficient_divisible {t : W} {a : I → ℝ} (ha : Represents M cs t a)
-    (p : SymmetricCoefficient (M := M)) :
-    SymmetricAlgebra.ι ℝ (Extended M) (embed M a) ∣ p-coefficientAction cs t p := by
-  rw [coefficientAction_reflection cs ha]
-  exact reflectionAction_divisible _ _ _ p
-
- theorem reflectionCoefficient_difference {t : W} {a : I → ℝ} (ha : Represents M cs t a)
-    (x : W) : ∃ p : SymmetricCoefficient (M := M),
-    coefficientAction cs x p-coefficientAction cs (t*x) p =
-      SymmetricAlgebra.ι ℝ (Extended M) (embed M a) := by
-  let α := SymmetricAlgebra.ι ℝ (Extended M) (embed M a)
-  let c : SymmetricCoefficient (M := M) := (1/2 : ℝ) • α
-  refine ⟨(coefficientAction cs x).symm c,?_⟩
-  rw [coefficientAction_mul,AlgEquiv.apply_symm_apply]
-  change (1/2 : ℝ) • α - coefficientAction cs t ((1/2 : ℝ) • α) = α
-  rw [map_smul,reflectionCoefficient_anti cs ha]
-  module
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Exact Bruhat support of the actual tensor word's subexpression weights.
-The ambient Coxeter system is not assumed crystallographic. -/
-namespace KLInvariance.TitsSpace
-universe u v
-variable {I : Type u} {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
- def bottSubword : (l : List I) → BottIndex l → List I
-  | [], _ => []
-  | i::l, e => if e.1=0 then bottSubword l e.2 else i::bottSubword l e.2
-
- theorem bottSubword_sublist (l : List I) (e : BottIndex l) :
-    (bottSubword l e).Sublist l := by
-  induction l with
-  | nil => exact List.Sublist.refl []
-  | cons i l ih =>
-    by_cases h : e.1=0
-    · rw [bottSubword,ite_eq_left h]
-      exact List.Sublist.cons i (ih e.2)
-    · rw [bottSubword,ite_eq_right h]
-      exact List.Sublist.cons_cons i (ih e.2)
-
- theorem bottSubword_wordProd (l : List I) (e : BottIndex l) :
-    cs.wordProd (bottSubword l e) = bottWeight cs l e := by
-  induction l with
-  | nil => exact cs.wordProd_nil
-  | cons i l ih =>
-    by_cases h : e.1=0
-    · simp only [bottSubword,bottWeight,ite_eq_left h,ih]
-    · simp only [bottSubword,bottWeight,ite_eq_right h,cs.wordProd_cons,ih]
-
- theorem exists_bottSubword {ν l : List I} (h : ν.Sublist l) :
-    ∃ e : BottIndex l, bottSubword l e = ν := by
-  induction h with
-  | slnil => exact ⟨PUnit.unit,rfl⟩
-  | cons i h ih =>
-    obtain ⟨e,he⟩ := ih
-    exact ⟨(0,e),by simpa only [bottSubword,ite_true] using he⟩
-  | cons_cons i h ih =>
-    obtain ⟨e,he⟩ := ih
-    exact ⟨(1,e),by simp only [bottSubword,show (1:Fin 2) ≠ 0 by decide,ite_false,he]⟩
-
- theorem bottWeight_bruhat (l : List I) (hl : cs.IsReduced l) (e : BottIndex l) :
-    BruhatLE cs (bottWeight cs l e) (cs.wordProd l) := by
-  rw [← bottSubword_wordProd]
-  exact bruhat_of_subword cs (bottSubword_sublist l e) hl
-
- theorem exists_bottWeight (l : List I) (hl : cs.IsReduced l) {x : W}
-    (hx : BruhatLE cs x (cs.wordProd l)) : ∃ e : BottIndex l, bottWeight cs l e=x := by
-  obtain ⟨ν,hν,_,he⟩ := (bruhat_iff_reduced_subword cs hl).mp hx
-  obtain ⟨e,he'⟩ := exists_bottSubword hν
-  exact ⟨e,by rw [← bottSubword_wordProd,he',he]⟩
-
- theorem bottWeight_range (l : List I) (hl : cs.IsReduced l) :
-    Set.range (bottWeight cs l) = {x | BruhatLE cs x (cs.wordProd l)} := by
-  ext x
-  constructor
-  · rintro ⟨e,rfl⟩
-    exact bottWeight_bruhat cs l hl e
-  · exact exists_bottWeight cs l hl
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! The actual integral two-vertex projection sheaf of a tensor
-Bott--Samelson module. This is the localization-functor construction, not
-an assumed BMP sheaf. Surjectivity onto ALL sections, flabbiness, free stalks,
-and the indecomposable normalization remain separate obligations. -/
-namespace KLInvariance.TitsSpace
-open Module MomentGraph BruhatGraph
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
- theorem bottStalkProjection_right (l : List I) (x : W)
-    (a : SymmetricCoefficient (M := M)) (m : BottSamelsonModule cs l) :
-    bottStalkProjection cs l x (bottSamelsonRight cs l a m) =
-      coefficientAction cs x a • bottStalkProjection cs l x m := by
-  apply Subtype.ext
-  funext e
-  change bottEvaluation cs l e.val (bottSamelsonRight cs l a m) =
-    coefficientAction cs x a * bottEvaluation cs l e.val m
-  rw [bottEvaluation_right,e.property]
-
- def bottProjectionSheaf (l : List I) (u b : W) : Sheaf (R := SymmetricCoefficient (M := M))
-    (graph cs u b) where
-  vertex x := ModuleCat.of _ (bottStalk cs l x.val)
-  edge e := ModuleCat.of _ (PairProjection.Edge
-    (bottStalkProjection cs l (source cs u b e).val)
-    (bottStalkProjection cs l (target cs u b e).val))
-  lower e := by
-    change (bottStalk cs l (source cs u b e).val) →ₗ[SymmetricCoefficient (M := M)] _
-    exact PairProjection.lower (bottStalkProjection cs l (source cs u b e).val)
-      (bottStalkProjection cs l (target cs u b e).val)
-  upper e := by
-    change (bottStalk cs l (target cs u b e).val) →ₗ[SymmetricCoefficient (M := M)] _
-    exact PairProjection.upper (bottStalkProjection cs l (source cs u b e).val)
-      (bottStalkProjection cs l (target cs u b e).val)
-
- def bottSection (l : List I) (u b : W) : BottSamelsonModule cs l →ₗ[
-    SymmetricCoefficient (M := M)] (bottProjectionSheaf cs l u b).sections Set.univ where
-  toFun m := ⟨fun x => bottStalkProjection cs l x.val m,by
-    intro e _ _
-    exact PairProjection.compatible _ _ m⟩
-  map_add' := by
-    intro m n
-    apply Subtype.ext
-    funext x
-    exact map_add (bottStalkProjection cs l x.val) m n
-  map_smul' := by
-    intro r m
-    apply Subtype.ext
-    funext x
-    exact map_smul (bottStalkProjection cs l x.val) r m
-
- theorem bottProjectionSheaf_generated (l : List I) (u b : W) :
-    (bottProjectionSheaf cs l u b).Generated := by
-  intro x v
-  obtain ⟨m,hm⟩ := bottStalkProjection_surjective cs l x.val v
-  exact ⟨(bottSection cs l u b m).val,(bottSection cs l u b m).property,hm⟩
-
- theorem bottProjectionSheaf_lower_surjective (l : List I) (u b : W) (e : Edge cs u b) :
-    Function.Surjective ((bottProjectionSheaf cs l u b).lower e) :=
-  PairProjection.lower_surjective _ _ (bottStalkProjection_surjective cs l _)
-
- theorem bottProjectionSheaf_upper_surjective (l : List I) (u b : W) (e : Edge cs u b) :
-    Function.Surjective ((bottProjectionSheaf cs l u b).upper e) :=
-  PairProjection.upper_surjective _ _ (bottStalkProjection_surjective cs l _)
-
-/-- An actual Bruhat reflection root annihilates the integral edge quotient. -/
- theorem bottProjectionSheaf_annihilated (l : List I) (u b : W) (e : Edge cs u b)
-    {a : I → ℝ} (ha : Represents M cs (label cs u b e) a)
-    (z : (bottProjectionSheaf cs l u b).edge e) :
-    SymmetricAlgebra.ι ℝ (Extended M) (embed M a) • z = 0 := by
-  obtain ⟨p,hp⟩ := reflectionCoefficient_difference cs ha (source cs u b e).val
-  rw [label_mul_source] at hp
-  rw [← hp]
-  exact PairProjection.difference_annihilates _ _
-    (bottStalkProjection_surjective cs l _) (bottStalkProjection_surjective cs l _)
-    (bottSamelsonRight cs l p) _ _
-    (bottStalkProjection_right cs l _ p) (bottStalkProjection_right cs l _ p) z
-
- theorem bottSection_injective (l : List I) (hl : cs.IsReduced l) :
-    Function.Injective (bottSection cs l 1 (cs.wordProd l)) := by
-  intro m n h
-  apply bottEvaluation_injective cs l
-  funext e
-  let x : Interval cs 1 (cs.wordProd l) :=
-    ⟨bottWeight cs l e,one_bruhat cs _,bottWeight_bruhat cs l hl e⟩
-  have hh := congrArg (fun z : (bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ =>
-    z.val x) h
-  exact congrFun (congrArg Subtype.val hh) ⟨e,rfl⟩
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Exact kernels of the actual pair-projection quotient, in a form that
-separates integral divisibility from annihilation of the edge. -/
-namespace KLInvariance.PairProjection
-universe u v w z
-variable {R : Type u} [CommRing R]
-  {M : Type v} [AddCommGroup M] [Module R M]
-  {X : Type w} [AddCommGroup X] [Module R X]
-  {Y : Type z} [AddCommGroup Y] [Module R Y]
-  (f : M →ₗ[R] X) (g : M →ₗ[R] Y)
-
- theorem kernel_upper (y : Y) : upper f g y=0 ↔ ∃ m : M, f m=0 ∧ g m=y := by
-  have he : upper f g y=0 ↔ lower f g 0=upper f g y := by
-    rw [map_zero, eq_comm]
-  rw [he, compatible_iff]
-
- theorem upper_kernel_of_divisibility (α : R)
-    (hker : ∀ m : M, f m=0 → ∃ n : Y, g m=α • n)
-    (hann : ∀ z : Edge f g, α • z=0) (y : Y) :
-    upper f g y=0 ↔ ∃ n : Y, y=α • n := by
-  constructor
-  · intro h
-    obtain ⟨m,hm,rfl⟩ := (kernel_upper f g y).mp h
-    exact hker m hm
-  · rintro ⟨n,rfl⟩
-    rw [map_smul]
-    exact hann _
-
-end KLInvariance.PairProjection
-
-end
-
-
-section
-
-/-! The exact simple-wall kernel for the actual integral tensor word module.
-This is a step toward word-sheaf recognition, not a character theorem or a
-proof of combinatorial invariance. No stalk freeness is assumed. -/
-namespace KLInvariance.TitsSpace
-open Module TensorProduct FrobeniusDuality ReflectionFrobenius
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
-attribute [local instance] restrictedModule restrictedTower restrictedFree restrictedFinite
-
-/-- The reflection is linear over its actual invariant subalgebra. -/
-def simpleFixedAction (i : I) : SymmetricCoefficient (M := M) →ₗ[simpleFixed cs i]
-    SymmetricCoefficient (M := M) where
-  toFun := coefficientAction cs (cs.simple i)
-  map_add' := map_add _
-  map_smul' c a := by
-    change coefficientAction cs (cs.simple i) ((c : SymmetricCoefficient (M := M))*a) =
-      (c : SymmetricCoefficient (M := M))*coefficientAction cs (cs.simple i) a
-    rw [map_mul, show coefficientAction cs (cs.simple i)
-      (c : SymmetricCoefficient (M := M)) = c from c.property]
-
-/-- Reflect the first tensor factor. It is not asserted to be left A-linear. -/
-def bottSimpleFlip (i : I) (l : List I) : BottSamelsonModule cs (i::l) →ₗ[simpleFixed cs i]
-    BottSamelsonModule cs (i::l) :=
-  TensorProduct.map (simpleFixedAction cs i)
-    (LinearMap.id : BottSamelsonModule cs l →ₗ[simpleFixed cs i] BottSamelsonModule cs l)
-
-@[simp] theorem bottSimpleFlip_tmul (i : I) (l : List I)
-    (a : SymmetricCoefficient (M := M)) (m : BottSamelsonModule cs l) :
-    bottSimpleFlip cs i l (a ⊗ₜ[simpleFixed cs i] m) =
-      coefficientAction cs (cs.simple i) a ⊗ₜ[simpleFixed cs i] m := rfl
-
-/-- The integral divisibility statement, before any localization. -/
-theorem bottSimpleFlip_divisible (i : I) (l : List I) (m : BottSamelsonModule cs (i::l)) :
-    ∃ n : BottSamelsonModule cs (i::l),
-      m-bottSimpleFlip cs i l m = simpleRootPolynomial (M := M) i • n := by
-  change SymmetricCoefficient (M := M) ⊗[simpleFixed cs i] BottSamelsonModule cs l at m
-  induction m using TensorProduct.inductionOn with
-  | add m n hm hn =>
-    obtain ⟨m',hm'⟩ := hm
-    obtain ⟨n',hn'⟩ := hn
-    refine ⟨m'+n', ?_⟩
-    erw [map_add, add_sub_add_comm, hm', hn', smul_add]
-  | tmul a m =>
-    obtain ⟨b,hb⟩ := simple_divisible cs i a
-    refine ⟨b ⊗ₜ[simpleFixed cs i] m, ?_⟩
-    erw [bottSimpleFlip_tmul, ← sub_tmul, hb]
-    rfl
-
-
-/-- Exchange the two branches of the first subexpression slot. -/
-def bottIndexFlip (i : I) (l : List I) (e : BottIndex (i::l)) : BottIndex (i::l) :=
-  (1-e.1,e.2)
-
-omit [Fintype I] in
-theorem bottWeight_flip (i : I) (l : List I) (e : BottIndex (i::l)) :
-    bottWeight cs (i::l) (bottIndexFlip i l e) = cs.simple i * bottWeight cs (i::l) e := by
-  rcases e with ⟨ε,e⟩
-  fin_cases ε <;> simp [bottIndexFlip, bottWeight, cs.simple_mul_simple_cancel_left]
-
-theorem bottEvaluation_flip (i : I) (l : List I) (e : BottIndex (i::l))
-    (m : BottSamelsonModule cs (i::l)) :
-    bottEvaluation cs (i::l) e (bottSimpleFlip cs i l m) =
-      coefficientAction cs (cs.simple i)
-        (bottEvaluation cs (i::l) (bottIndexFlip i l e) m) := by
-  change SymmetricCoefficient (M := M) ⊗[simpleFixed cs i] BottSamelsonModule cs l at m
-  induction m using TensorProduct.inductionOn with
-  | add m n hm hn => erw [map_add, map_add, map_add, map_add, hm, hn]
-  | tmul a m =>
-    rcases e with ⟨ε,e⟩
-    fin_cases ε
-    · change coefficientAction cs (cs.simple i) a * bottEvaluation cs l e m =
-        coefficientAction cs (cs.simple i)
-          (a * coefficientAction cs (cs.simple i) (bottEvaluation cs l e m))
-      rw [map_mul, simple_involutive cs i]
-    · change coefficientAction cs (cs.simple i) a *
-        coefficientAction cs (cs.simple i) (bottEvaluation cs l e m) =
-        coefficientAction cs (cs.simple i) (a * bottEvaluation cs l e m)
-      rw [map_mul]
-
-theorem bottStalkProjection_flip_zero (i : I) (l : List I) (x : W)
-    {m : BottSamelsonModule cs (i::l)} (hm : bottStalkProjection cs (i::l) x m=0) :
-    bottStalkProjection cs (i::l) (cs.simple i*x) (bottSimpleFlip cs i l m)=0 := by
-  apply (bottStalkProjection_eq_zero cs _ _ _).mpr
-  apply (bottWeightProjection_eq_zero cs _ _ _).mpr
-  intro e he
-  rw [bottEvaluation_flip]
-  have hh := (bottWeightProjection_eq_zero cs _ _ _).mp
-    ((bottStalkProjection_eq_zero cs _ _ _).mp hm)
-  rw [hh (bottIndexFlip i l e) (by
-    rw [bottWeight_flip, he, cs.simple_mul_simple_cancel_left]), map_zero]
-
-/-- The reverse inclusion to edge annihilation for the simple wall at the
-first tensor letter, expressed directly on kernel projections. -/
-theorem bottSimpleWall_kernel_projection (i : I) (l : List I) (x : W)
-    {m : BottSamelsonModule cs (i::l)} (hm : bottStalkProjection cs (i::l) x m=0) :
-    ∃ n : bottStalk cs (i::l) (cs.simple i*x),
-      bottStalkProjection cs (i::l) (cs.simple i*x) m = simpleRootPolynomial (M := M) i • n := by
-  obtain ⟨n,hn⟩ := bottSimpleFlip_divisible cs i l m
-  refine ⟨bottStalkProjection cs (i::l) (cs.simple i*x) n, ?_⟩
-  have he := congrArg (bottStalkProjection cs (i::l) (cs.simple i*x)) hn
-  rw [map_sub, map_smul, bottStalkProjection_flip_zero cs i l x hm, sub_zero] at he
-  exact he
-
-theorem simpleCoefficient_difference (i : I) (x : W) :
-    ∃ p : SymmetricCoefficient (M := M),
-      coefficientAction cs x p - coefficientAction cs (cs.simple i*x) p =
-        simpleRootPolynomial (M := M) i := by
-  let α := simpleRootPolynomial (M := M) i
-  refine ⟨(coefficientAction cs x).symm ((1/2 : ℝ) • α), ?_⟩
-  rw [coefficientAction_mul, AlgEquiv.apply_symm_apply]
-  change (1/2 : ℝ) • α - coefficientAction cs (cs.simple i) ((1/2 : ℝ) • α) = α
-  rw [map_smul, simple_anti]
-  module
-
-
-/-- Any old word's simple-edge quotient is killed by the actual label. -/
-theorem bottSimpleEdge_annihilated (i : I) (l : List I) (x : W)
-    (z : PairProjection.Edge (bottStalkProjection cs l x)
-      (bottStalkProjection cs l (cs.simple i*x))) :
-    simpleRootPolynomial (M := M) i • z = 0 := by
-  obtain ⟨p,hp⟩ := simpleCoefficient_difference cs i x
-  rw [← hp]
-  exact PairProjection.difference_annihilates _ _
-    (bottStalkProjection_surjective cs _ _) (bottStalkProjection_surjective cs _ _)
-    (bottSamelsonRight cs l p) _ _
-    (bottStalkProjection_right cs _ _ p) (bottStalkProjection_right cs _ _ p) z
-
-theorem bottSimpleEdge_kernel_inclusion (i : I) (l : List I) (x : W)
-    (y : bottStalk cs l (cs.simple i*x)) :
-    ∃ n : BottSamelsonModule cs l, bottStalkProjection cs l x n=0 ∧
-      bottStalkProjection cs l (cs.simple i*x) n=simpleRootPolynomial (M := M) i • y := by
-  apply (PairProjection.kernel_upper _ _ _).mp
-  rw [map_smul]
-  exact bottSimpleEdge_annihilated cs i l x _
-
-/-- The integral edge quotient is killed by its actual simple-root label. -/
-theorem bottSimpleWall_annihilated (i : I) (l : List I) (x : W)
-    (z : PairProjection.Edge (bottStalkProjection cs (i::l) x)
-      (bottStalkProjection cs (i::l) (cs.simple i*x))) :
-    simpleRootPolynomial (M := M) i • z = 0 :=
-  bottSimpleEdge_annihilated cs i (i::l) x z
-
-/-- The exact integral quotient kernel, without a free-stalk hypothesis. -/
-theorem bottSimpleWall_upper_kernel (i : I) (l : List I) (x : W)
-    (z : bottStalk cs (i::l) (cs.simple i*x)) :
-    PairProjection.upper (bottStalkProjection cs (i::l) x)
-      (bottStalkProjection cs (i::l) (cs.simple i*x)) z = 0 ↔
-    ∃ n : bottStalk cs (i::l) (cs.simple i*x), z = simpleRootPolynomial (M := M) i • n :=
-  PairProjection.upper_kernel_of_divisibility _ _ _
-    (fun _ hm => bottSimpleWall_kernel_projection cs i l x hm)
-    (bottSimpleWall_annihilated cs i l x) z
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Integral simple-wall translation of a two-stalk projection lattice.
-The hypotheses are explicit old-word edge properties, not character claims.
-No freeness of a submodule is inferred merely from finite generation. -/
-namespace KLInvariance.ReflectionPairTranslation
-universe u v w z
-variable {R : Type u} [CommRing R]
-
-attribute [local instance] RingHomInvPair.of_ringEquiv
-
-/-- Restriction of the scalar action along a ring automorphism. -/
-@[ext] structure Twist (_τ : R ≃+* R) (Y : Type z) where
-  val : Y
-namespace Twist
-variable (τ : R ≃+* R) (Y : Type z) [AddCommGroup Y] [Module R Y]
-local instance : RingHomInvPair τ.symm.toRingHom τ.toRingHom :=
-  RingHomInvPair.of_ringEquiv_symm τ
-local instance : RingHomInvPair τ.toRingHom τ.symm.toRingHom :=
-  RingHomInvPair.of_ringEquiv τ
-def equivVal : Twist τ Y ≃ Y where
-  toFun := val
-  invFun := mk
-  left_inv _ := rfl
-  right_inv _ := rfl
-instance : AddCommGroup (Twist τ Y) := (equivVal τ Y).addCommGroup
-instance : Module R (Twist τ Y) := by
-  letI : Module R Y := Module.compHom Y τ.toRingHom
-  exact ({ equivVal τ Y with map_add' := fun _ _ => rfl } : Twist τ Y ≃+ Y).module R
-
-def equiv : Y ≃ₛₗ[τ.symm.toRingHom] Twist τ Y where
-  toFun := mk
-  invFun := val
-  left_inv _ := rfl
-  right_inv _ := rfl
-  map_add' _ _ := rfl
-  map_smul' r y := by
-    change mk (r • y) = mk (τ (τ.symm r) • y)
-    rw [τ.apply_symm_apply]
-
-instance [Module.Free R Y] : Module.Free R (Twist τ Y) :=
-  Module.Free.of_equiv (equiv τ Y)
-end Twist
-
-variable {N : Type v} [AddCommGroup N] [Module R N]
-  {X : Type w} [AddCommGroup X] [Module R X]
-  {Y : Type z} [AddCommGroup Y] [Module R Y]
-  (f : N →ₗ[R] X) (g : N →ₗ[R] Y) (τ : R ≃+* R) (α : R)
-  (hdiv : ∀ r : R, α ∣ τ r-r)
-  (hinc : ∀ y : Y, ∃ n : N, f n=0 ∧ g n=α • y)
-
-/-- The old pair relation with the scalar action on its second factor twisted.
-Alpha-annihilation of the old edge is what makes this an R-submodule. -/
-def lattice : Submodule R (X × Twist τ Y) where
-  carrier := {p | ∃ n : N, f n=p.1 ∧ g n=p.2.val}
-  zero_mem' := ⟨0,map_zero _,map_zero _⟩
-  add_mem' := by
-    rintro _ _ ⟨m,hm,hm'⟩ ⟨n,hn,hn'⟩
-    exact ⟨m+n,by rw [map_add,hm,hn]; rfl,by rw [map_add,hm',hn']; rfl⟩
-  smul_mem' := by
-    rintro r ⟨x,y⟩ ⟨n,hn,hn'⟩
-    obtain ⟨c,hc⟩ := hdiv r
-    obtain ⟨m,hm,hm'⟩ := hinc (c • y.val)
-    have hm'' : g m=(τ r-r) • y.val := by rw [hm',hc,mul_smul]
-    refine ⟨r • n+m,?_,?_⟩
-    · change f (r • n+m)=r • x
-      rw [map_add,map_smul,hn,hm,add_zero]
-    · change g (r • n+m)=τ r • y.val
-      rw [map_add,map_smul,hn',hm'',sub_smul]
-      abel
-
-/-- Saturation away from the simple wall. Regularity on the old edge module,
-not an unjustified reflexivity assertion, removes a non-wall factor. -/
-theorem lattice_saturated (β : R)
-    (hann : ∀ e : PairProjection.Edge f g, α • e=0)
-    (hreg : Function.Injective (fun e : PairProjection.Edge f g => β • e))
-    (p : X × Twist τ Y) :
-    β • p ∈ lattice f g τ α hdiv hinc ↔ p ∈ lattice f g τ α hdiv hinc := by
-  constructor
-  · rintro ⟨n,hn,hn'⟩
-    apply (PairProjection.compatible_iff f g p.1 p.2.val).mp
-    apply hreg
-    have hn'' : g n=τ β • p.2.val := hn'
-    have h := PairProjection.compatible f g n
-    have hn1 : f n=β • p.1 := hn
-    rw [hn1,hn'',map_smul,map_smul] at h
-    obtain ⟨c,hc⟩ := hdiv β
-    have he : (τ β-β) • PairProjection.upper f g p.2.val=0 := by
-      rw [hc,mul_smul,smul_comm,hann,smul_zero]
-    rw [sub_smul,sub_eq_zero] at he
-    exact h.trans he
-  · exact (lattice f g τ α hdiv hinc).smul_mem β
-
-/-- Surjective projection to the untwisted (lower) stalk. -/
-def left : lattice f g τ α hdiv hinc →ₗ[R] X :=
-  (LinearMap.fst R X (Twist τ Y)).comp (lattice f g τ α hdiv hinc).subtype
-
- theorem left_surjective (hf : Function.Surjective f) :
-    Function.Surjective (left f g τ α hdiv hinc) := by
-  intro x
-  obtain ⟨n,rfl⟩ := hf x
-  exact ⟨⟨(f n,⟨g n⟩),n,rfl,rfl⟩,rfl⟩
-
-/-- Its kernel is the alpha-multiple of the twisted upper stalk. -/
-def kernelMap : Twist τ Y →ₗ[R] LinearMap.ker (left f g τ α hdiv hinc) where
-  toFun y := ⟨⟨(0,⟨α • y.val⟩),hinc y.val⟩,rfl⟩
-  map_add' y z := by
-    apply Subtype.ext
-    apply Subtype.ext
-    apply Prod.ext
-    · change (0 : X)=0+0; simp
-    · change Twist.mk (α • (y.val+z.val)) = Twist.mk (α • y.val+α • z.val)
-      rw [smul_add]
-  map_smul' r y := by
-    apply Subtype.ext
-    apply Subtype.ext
-    change (0,Twist.mk (α • (τ r • y.val)))=(r • (0 : X),Twist.mk (τ r • (α • y.val)))
-    rw [smul_zero,smul_comm α (τ r)]
-
- variable (hker : ∀ y : Y, (∃ n : N, f n=0 ∧ g n=y) ↔ ∃ z : Y, y=α • z)
-
- include hker in
- theorem kernelMap_surjective : Function.Surjective (kernelMap f g τ α hdiv hinc) := by
-  rintro ⟨⟨⟨x,y⟩,n,hn,hn'⟩,hx⟩
-  change x=0 at hx
-  subst x
-  obtain ⟨z,hz⟩ := (hker y.val).mp ⟨n,hn,hn'⟩
-  refine ⟨⟨z⟩,?_⟩
-  apply Subtype.ext
-  apply Subtype.ext
-  exact Prod.ext rfl (congrArg Twist.mk hz.symm)
-
- theorem kernelMap_injective (hinj : Function.Injective (fun y : Y => α • y)) :
-    Function.Injective (kernelMap f g τ α hdiv hinc) := by
-  intro y z h
-  apply Twist.ext
-  exact hinj (congrArg (fun p => p.val.val.2.val) h)
-
-noncomputable def kernelEquiv (hinj : Function.Injective (fun y : Y => α • y)) :
-    Twist τ Y ≃ₗ[R] LinearMap.ker (left f g τ α hdiv hinc) :=
-  LinearEquiv.ofBijective (kernelMap f g τ α hdiv hinc)
-    ⟨kernelMap_injective f g τ α hdiv hinc hinj,kernelMap_surjective f g τ α hdiv hinc hker⟩
-
-/-- A split extension of free modules is free. This does not assert the
-false statement that every submodule of a free polynomial module is free. -/
-theorem free_of_surjective_of_ker_free {P : Type*} [AddCommGroup P] [Module R P]
-    {Q : Type*} [AddCommGroup Q] [Module R Q]
-    (π : P →ₗ[R] Q) (hπ : Function.Surjective π)
-    [Module.Free R Q] [Module.Free R (LinearMap.ker π)] : Module.Free R P := by
-  obtain ⟨s,hs⟩ := π.exists_rightInverse_of_surjective (LinearMap.range_eq_top.mpr hπ)
-  let e : (LinearMap.ker π × Q) →ₗ[R] P :=
-    (LinearMap.ker π).subtype.coprod s
-  have hps (q : Q) : π (s q)=q := DFunLike.congr_fun hs q
-  have he : Function.Bijective e := by
-    constructor
-    · rintro ⟨k,q⟩ ⟨k',q'⟩ h
-      have hq : q=q' := by
-        have hh := congrArg π h
-        simpa only [e,LinearMap.coprod_apply,Submodule.subtype_apply,map_add,LinearMap.mem_ker.mp k.property,
-          LinearMap.mem_ker.mp k'.property,hps,zero_add] using hh
-      subst q'
-      refine Prod.ext ?_ rfl
-      apply Subtype.ext
-      exact add_right_cancel h
-    · intro p
-      refine ⟨(⟨p-s (π p),?_⟩,π p),?_⟩
-      · change π (p-s (π p))=0
-        rw [map_sub,hps,sub_self]
-      · change p-s (π p)+s (π p)=p
-        abel
-  exact Module.Free.of_equiv (LinearEquiv.ofBijective e he)
-
- include hker in
- theorem lattice_free (hf : Function.Surjective f)
-    (hinj : Function.Injective (fun y : Y => α • y))
-    [Module.Free R X] [Module.Free R Y] :
-    Module.Free R (lattice f g τ α hdiv hinc) := by
-  let : Module.Free R (LinearMap.ker (left f g τ α hdiv hinc)) :=
-    Module.Free.of_equiv (kernelEquiv f g τ α hdiv hinc hker hinj)
-  exact free_of_surjective_of_ker_free (left f g τ α hdiv hinc)
-    (left_surjective f g τ α hdiv hinc hf)
-
-end KLInvariance.ReflectionPairTranslation
-
-end
-
-
-section
-
-/-! A generic tensor-generation lemma, kept separate so application to the
-actual recursively paired word carrier does not unfold that carrier's data. -/
-
-end
-
-end OAI
+import OAI.RepresentationTheory.KazhdanLusztig.Decomposition
+
+namespace OAI
+
+/-!
+Coxeter Frobenius extensions, tensor duality and translation across reflection pairs.
+-/
+
+section
+
+namespace KLInvariance.TitsSpace
+open Module ReflectionFrobenius
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+ abbrev SymmetricCoefficient := SymmetricAlgebra ℝ (Extended M)
+
+ def coefficientAction (w : W) : SymmetricCoefficient (M := M) ≃ₐ[ℝ]
+    SymmetricCoefficient (M := M) := symmetricAction (extendedAction M cs w)
+
+@[simp] theorem coefficientAction_ι (w : W) (v : Extended M) :
+    coefficientAction cs w (SymmetricAlgebra.ι ℝ (Extended M) v) =
+      SymmetricAlgebra.ι ℝ (Extended M) (extendedAction M cs w v) :=
+  symmetricAction_ι _ _
+
+ theorem coefficientAction_simple (i : I) : coefficientAction cs (cs.simple i) =
+    reflectionAction (extendedRoot M i) (extendedForm M (extendedRoot M i))
+      (extendedForm_root_self M i) := by
+  rw [coefficientAction,extendedAction_simple_eq]
+
+ theorem coefficientAction_mul (w z : W) (a : SymmetricCoefficient (M := M)) :
+    coefficientAction cs (w*z) a = coefficientAction cs w (coefficientAction cs z a) := by
+  have he : (coefficientAction cs (w*z)).toAlgHom =
+      (coefficientAction cs w).toAlgHom.comp (coefficientAction cs z).toAlgHom := by
+    apply SymmetricAlgebra.algHom_ext
+    apply LinearMap.ext
+    intro x
+    change coefficientAction cs (w*z) (SymmetricAlgebra.ι ℝ (Extended M) x) =
+      coefficientAction cs w (coefficientAction cs z (SymmetricAlgebra.ι ℝ (Extended M) x))
+    simp only [coefficientAction_ι,map_mul,LinearEquiv.mul_apply]
+  exact congrArg (fun f : SymmetricCoefficient (M := M) →ₐ[ℝ] SymmetricCoefficient (M := M) => f a) he
+
+ def simpleFixed (i : I) : Subalgebra ℝ (SymmetricCoefficient (M := M)) :=
+  fixed (coefficientAction cs (cs.simple i))
+
+ abbrev simpleRootPolynomial (i : I) : SymmetricCoefficient (M := M) :=
+  SymmetricAlgebra.ι ℝ (Extended M) (extendedRoot M i)
+
+ theorem simple_involutive (i : I) : Function.Involutive (coefficientAction cs (cs.simple i)) := by
+  rw [coefficientAction_simple]
+  exact reflectionAction_involutive _ _ _
+
+ theorem simpleRootPolynomial_ne_zero (i : I) : simpleRootPolynomial (M := M) i ≠ 0 :=
+  reflectionRoot_ne_zero _ _ (extendedForm_root_self M i)
+
+ theorem simple_anti (i : I) : coefficientAction cs (cs.simple i)
+    (simpleRootPolynomial (M := M) i) = -simpleRootPolynomial (M := M) i := by
+  rw [coefficientAction_simple]
+  exact reflectionAction_anti _ _ _
+
+ theorem simple_divisible (i : I) (a : SymmetricCoefficient (M := M)) :
+    simpleRootPolynomial (M := M) i ∣ a-coefficientAction cs (cs.simple i) a := by
+  rw [coefficientAction_simple]
+  exact reflectionAction_divisible _ _ _ a
+
+ def simpleCoordinates (i : I) : SymmetricCoefficient (M := M) ≃ₗ[simpleFixed cs i]
+    (simpleFixed cs i × simpleFixed cs i) :=
+  coordinates _ _ (simple_involutive cs i) (simpleRootPolynomial_ne_zero i)
+    (simple_anti cs i) (simple_divisible cs i)
+
+ instance simpleFixed_free (i : I) : Module.Free (simpleFixed cs i) (SymmetricCoefficient (M := M)) :=
+  Module.Free.of_equiv (simpleCoordinates cs i).symm
+
+ instance simpleFixed_finite (i : I) : Module.Finite (simpleFixed cs i) (SymmetricCoefficient (M := M)) :=
+  Module.Finite.equiv (simpleCoordinates cs i).symm
+
+ def simpleTrace (i : I) : SymmetricCoefficient (M := M) →ₗ[simpleFixed cs i] simpleFixed cs i :=
+  trace _ _ (simple_involutive cs i) (simpleRootPolynomial_ne_zero i)
+    (simple_anti cs i) (simple_divisible cs i)
+
+ def simpleDuality (i : I) : SymmetricCoefficient (M := M) ≃ₗ[simpleFixed cs i]
+    Dual (simpleFixed cs i) (SymmetricCoefficient (M := M)) :=
+  duality _ _ (simple_involutive cs i) (simpleRootPolynomial_ne_zero i)
+    (simple_anti cs i) (simple_divisible cs i)
+
+ theorem simpleDuality_apply (i : I) (a b : SymmetricCoefficient (M := M)) :
+    simpleDuality cs i a b = simpleTrace cs i (a*b) :=
+  duality_apply _ _ _ _ _ _ a b
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Integral restriction and extension of perfect pairings for a genuine
+Frobenius extension. This supplies the duality operation used by
+Bott--Samelson translation, not an assumed duality on a BMP sheaf. -/
+namespace KLInvariance.FrobeniusDuality
+open Module TensorProduct
+universe ur ua um
+variable {R : Type ur} [CommRing R] {A : Type ua} [CommRing A] [Algebra R A]
+  (tr : A →ₗ[R] R) (d : A ≃ₗ[R] Dual R A)
+  (hd : ∀ a b, d a b = tr (a*b))
+  (M : Type um) [AddCommGroup M] [Module A M] [Module R M] [IsScalarTower R A M]
+noncomputable section
+
+ def restrictDual : Dual A M →ₗ[R] Dual R M where
+  toFun f := tr.comp (f.restrictScalars R)
+  map_add' _ _ := by ext; simp
+  map_smul' _ _ := by ext; simp
+
+ def fiber (f : Dual R M) (m : M) : Dual R A :=
+  f.comp ((LinearMap.toSpanSingleton A M m).restrictScalars R)
+
+@[simp] theorem fiber_apply (f : Dual R M) (m : M) (a : A) :
+    fiber M f m a = f (a • m) := rfl
+
+include hd in
+ theorem trace_recover_fiber (f : Dual R M) (m : M) (a : A) :
+    tr (d.symm (fiber M f m) * a) = f (a • m) := by
+  rw [← hd,d.apply_symm_apply,fiber_apply]
+
+include hd in
+ theorem recover_fiber_smul (f : Dual R M) (m : M) (a : A) :
+    d.symm (fiber M f (a • m)) = a * d.symm (fiber M f m) := by
+  apply d.injective
+  apply LinearMap.ext
+  intro b
+  rw [d.apply_symm_apply,hd,fiber_apply]
+  have he : a * d.symm (fiber M f m) * b = d.symm (fiber M f m) * (b*a) := by ring
+  rw [he,trace_recover_fiber tr d hd]
+  simp only [mul_smul]
+
+ def recoverDual (f : Dual R M) : Dual A M where
+  toFun m := d.symm (fiber M f m)
+  map_add' m n := by
+    apply d.injective
+    rw [map_add,d.apply_symm_apply,d.apply_symm_apply,d.apply_symm_apply]
+    ext a
+    simp [smul_add]
+  map_smul' a m := by
+    simpa only [RingHom.id_apply,smul_eq_mul] using recover_fiber_smul tr d hd M f m a
+
+@[simp] theorem recoverDual_apply (f : Dual R M) (m : M) :
+    recoverDual tr d hd M f m = d.symm (fiber M f m) := rfl
+
+ theorem restrict_recover (f : Dual R M) :
+    restrictDual tr M (recoverDual tr d hd M f) = f := by
+  ext m
+  change tr (d.symm (fiber M f m)) = f m
+  simpa only [mul_one,one_smul] using trace_recover_fiber tr d hd M f m 1
+
+ theorem recover_restrict (f : Dual A M) :
+    recoverDual tr d hd M (restrictDual tr M f) = f := by
+  ext m
+  apply d.injective
+  change d (d.symm _) = d (f m)
+  rw [d.apply_symm_apply]
+  ext a
+  change tr (f (a • m)) = d (f m) a
+  rw [map_smul,smul_eq_mul,hd,mul_comm]
+
+ def restrictionEquiv : Dual A M ≃ₗ[R] Dual R M :=
+  LinearEquiv.ofBijective (restrictDual tr M)
+    ⟨fun f g h => by
+      have hh := congrArg (recoverDual tr d hd M) h
+      simpa only [recover_restrict] using hh,
+    fun f => ⟨recoverDual tr d hd M f,restrict_recover tr d hd M f⟩⟩
+
+ theorem restrictionEquiv_apply (f : Dual A M) (m : M) :
+    restrictionEquiv tr d hd M f m = tr (f m) := rfl
+
+/-- Restricting an integral perfect pairing along a Frobenius trace is again
+integral and perfect; no localization or fraction-field argument is used. -/
+ def restrictPairing (p : M ≃ₗ[A] Dual A M) : M ≃ₗ[R] Dual R M :=
+  (p.restrictScalars R).trans (restrictionEquiv tr d hd M)
+
+ theorem restrictPairing_apply (p : M ≃ₗ[A] Dual A M) (m n : M) :
+    restrictPairing tr d hd M p m n = tr (p m n) := rfl
+
+end
+end KLInvariance.FrobeniusDuality
+
+namespace KLInvariance.FrobeniusDuality
+open Module TensorProduct
+universe ur ua um
+variable {R : Type ur} [CommRing R] (A : Type ua) [CommRing A] [Algebra R A]
+  {M : Type um} [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M]
+noncomputable section
+
+ def extendPairing (p : M ≃ₗ[R] Dual R M) :
+    (A ⊗[R] M) ≃ₗ[A] Dual A (A ⊗[R] M) :=
+  (p.baseChange R A M (Dual R M)).trans
+    (TensorProduct.isBaseChange (R := R) (S := A) (M := M)).toDualBaseChange
+
+ theorem extendPairing_tmul (p : M ≃ₗ[R] Dual R M) (a b : A) (m n : M) :
+    extendPairing A p (a ⊗ₜ[R] m) (b ⊗ₜ[R] n) =
+      a * b * algebraMap R A (p m n) := by
+  have hn : b ⊗ₜ[R] n = b • (1 ⊗ₜ[R] n) := by simp only [smul_tmul',smul_eq_mul,mul_one]
+  rw [hn,map_smul]
+  change b * ((TensorProduct.isBaseChange (R := R) (S := A) (M := M)).toDualBaseChange
+    (a ⊗ₜ[R] p m) ((TensorProduct.mk R A M 1) n)) = _
+  rw [IsBaseChange.toDualBaseChange_tmul]
+  ring
+
+end
+end KLInvariance.FrobeniusDuality
+
+end
+
+
+section
+
+/-! Tensor induction of integral self-dual bimodules across a Frobenius
+extension. All pairings refer to actual integral modules. -/
+namespace KLInvariance.FrobeniusDuality
+open Module TensorProduct
+universe ur ua um
+noncomputable section
+
+structure PairedModule (A : Type ua) [CommRing A] where
+  carrier : Type um
+  [add : AddCommGroup carrier]
+  [module : Module A carrier]
+  free : Module.Free A carrier
+  finite : Module.Finite A carrier
+  pairing : carrier ≃ₗ[A] Dual A carrier
+
+attribute [instance] PairedModule.add PairedModule.module PairedModule.free PairedModule.finite
+instance {A : Type ua} [CommRing A] : CoeSort (PairedModule.{ua,um} A) (Type um) := ⟨PairedModule.carrier⟩
+
+ def unitPairing (A : Type ua) [CommRing A] : A ≃ₗ[A] Dual A A where
+  toFun a := LinearMap.mul A A a
+  invFun f := f 1
+  left_inv a := mul_one a
+  right_inv f := by
+    apply LinearMap.ext
+    intro a
+    change f 1 * a = f a
+    rw [mul_comm,← smul_eq_mul,← map_smul,smul_eq_mul,mul_one]
+  map_add' _ _ := by ext; simp
+  map_smul' _ _ := by ext; simp
+
+ def unitModule (A : Type ua) [CommRing A] : PairedModule A where
+  carrier := A
+  free := inferInstance
+  finite := inferInstance
+  pairing := unitPairing A
+
+variable {R : Type ur} [CommRing R] {A : Type ua} [CommRing A] [Algebra R A]
+  [Module.Free R A] [Module.Finite R A]
+  (tr : A →ₗ[R] R) (d : A ≃ₗ[R] Dual R A) (hd : ∀ a b, d a b = tr (a*b))
+  (N : PairedModule.{ua,um} A)
+
+local instance restrictedModule : Module R N := Module.compHom N (algebraMap R A)
+local instance restrictedTower : IsScalarTower R A N := IsScalarTower.of_algebraMap_smul fun _ _ => rfl
+
+ def restrictionBasis : Basis
+    (Module.Free.ChooseBasisIndex R A × Module.Free.ChooseBasisIndex A N) R N :=
+  (Module.Free.chooseBasis R A).smulTower (Module.Free.chooseBasis A N)
+
+local instance restrictedFree : Module.Free R N := Module.Free.of_basis (restrictionBasis N)
+local instance restrictedFinite : Module.Finite R N := Module.Finite.trans A N
+
+ def inducedModule : PairedModule A where
+  carrier := A ⊗[R] N
+  free := inferInstance
+  finite := inferInstance
+  pairing := extendPairing A (restrictPairing tr d hd N N.pairing)
+
+ theorem inducedModule_pairing_tmul (a b : A) (m n : N) :
+    (inducedModule tr d hd N).pairing (a ⊗ₜ[R] m) (b ⊗ₜ[R] n) =
+      a*b*algebraMap R A (tr (N.pairing m n)) :=
+  extendPairing_tmul A _ a b m n
+
+end
+end KLInvariance.FrobeniusDuality
+
+end
+
+
+section
+
+/-! Integral balanced duality under Bott--Samelson tensor induction. -/
+namespace KLInvariance.FrobeniusDuality
+open Module TensorProduct
+universe ur ua um
+noncomputable section
+
+structure PairedBimodule (A : Type ua) [CommRing A] where
+  obj : PairedModule.{ua,um} A
+  right : A →+* End A obj
+  balanced : ∀ a m n, obj.pairing (right a m) n = obj.pairing m (right a n)
+  symmetric : ∀ m n, obj.pairing m n = obj.pairing n m
+
+ def unitBimodule (A : Type ua) [CommRing A] : PairedBimodule A where
+  obj := unitModule A
+  right := Algebra.lmul A A |>.toRingHom
+  balanced a m n := by
+    change A at m n
+    change (a*m)*n = m*(a*n)
+    ring
+  symmetric m n := by
+    change A at m n
+    exact mul_comm m n
+
+variable {R : Type ur} [CommRing R] {A : Type ua} [CommRing A] [Algebra R A]
+  [Module.Free R A] [Module.Finite R A]
+  (tr : A →ₗ[R] R) (d : A ≃ₗ[R] Dual R A) (hd : ∀ a b, d a b = tr (a*b))
+  (N : PairedModule.{ua,um} A)
+
+attribute [local instance] restrictedModule restrictedTower restrictedFree restrictedFinite
+
+ def restrictedRight (right : A →+* End A N) : A →+* End R N where
+  toFun a := (right a).restrictScalars R
+  map_one' := by ext m; change (right 1) m = m; rw [map_one]; rfl
+  map_mul' a b := by
+    ext m
+    change (right (a*b)) m = (right a) ((right b) m)
+    rw [map_mul]
+    rfl
+  map_zero' := by ext m; change (right 0) m = 0; rw [map_zero]; rfl
+  map_add' a b := by ext m; change (right (a+b)) m = (right a) m + (right b) m; rw [map_add]; rfl
+
+ def inducedRight (right : A →+* End A N) : A →+* End A (inducedModule tr d hd N) :=
+  (Module.End.baseChangeHom R A N).toRingHom.comp (restrictedRight N right)
+
+@[simp] theorem inducedRight_tmul (right : A →+* End A N) (r a : A) (m : N) :
+    inducedRight tr d hd N right r (a ⊗ₜ[R] m) = a ⊗ₜ[R] right r m := rfl
+
+ theorem inducedPairing_balanced (right : A →+* End A N)
+    (hb : ∀ a m n, N.pairing (right a m) n = N.pairing m (right a n))
+    (r : A) (m n : inducedModule tr d hd N) :
+    (inducedModule tr d hd N).pairing (inducedRight tr d hd N right r m) n =
+      (inducedModule tr d hd N).pairing m (inducedRight tr d hd N right r n) := by
+  change A ⊗[R] N at m n
+  change extendPairing A (restrictPairing tr d hd N N.pairing)
+    (((right r).restrictScalars R).baseChange A m) n =
+    extendPairing A (restrictPairing tr d hd N N.pairing) m
+    (((right r).restrictScalars R).baseChange A n)
+  induction m using TensorProduct.inductionOn with
+  | add m₁ m₂ h₁ h₂ => simp only [map_add,LinearMap.add_apply,h₁,h₂]
+  | tmul a m =>
+    induction n using TensorProduct.inductionOn with
+    | add n₁ n₂ h₁ h₂ => simp only [map_add,h₁,h₂]
+    | tmul b n =>
+      simp only [LinearMap.baseChange_tmul,extendPairing_tmul,restrictPairing_apply,
+        LinearMap.restrictScalars_apply,hb]
+
+ theorem inducedPairing_symmetric (hs : ∀ m n, N.pairing m n = N.pairing n m)
+    (m n : inducedModule tr d hd N) :
+    (inducedModule tr d hd N).pairing m n = (inducedModule tr d hd N).pairing n m := by
+  change A ⊗[R] N at m n
+  change extendPairing A (restrictPairing tr d hd N N.pairing) m n =
+    extendPairing A (restrictPairing tr d hd N N.pairing) n m
+  induction m using TensorProduct.inductionOn with
+  | add m₁ m₂ h₁ h₂ => simp only [map_add,LinearMap.add_apply,h₁,h₂]
+  | tmul a m =>
+    induction n using TensorProduct.inductionOn with
+    | add n₁ n₂ h₁ h₂ => simp only [map_add,LinearMap.add_apply,h₁,h₂]
+    | tmul b n =>
+      rw [extendPairing_tmul,extendPairing_tmul,restrictPairing_apply,restrictPairing_apply,hs,mul_comm a b]
+
+ def inducedBimodule (N : PairedBimodule.{ua,um} A) : PairedBimodule A where
+  obj := inducedModule tr d hd N.obj
+  right := inducedRight tr d hd N.obj N.right
+  balanced := inducedPairing_balanced tr d hd N.obj N.right N.balanced
+  symmetric := inducedPairing_symmetric tr d hd N.obj N.symmetric
+
+end
+end KLInvariance.FrobeniusDuality
+
+end
+
+
+section
+
+/-! Genuine tensor-defined Bott--Samelson bimodules for the extended real
+Coxeter realization. The perfect pairing is integral and balanced. This
+construction alone does not identify indecomposable BMP summands or prove
+the Elias--Williamson character theorem. -/
+namespace KLInvariance.TitsSpace
+open Module TensorProduct FrobeniusDuality
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+ def bottSamelson : List I → PairedBimodule.{u,u} (SymmetricCoefficient (M := M))
+  | [] => unitBimodule _
+  | i :: l => inducedBimodule (simpleTrace cs i) (simpleDuality cs i)
+    (simpleDuality_apply cs i) (bottSamelson l)
+
+ abbrev BottSamelsonModule (l : List I) := (bottSamelson cs l).obj
+
+ def bottSamelsonDuality (l : List I) : BottSamelsonModule cs l ≃ₗ[SymmetricCoefficient (M := M)]
+    Dual (SymmetricCoefficient (M := M)) (BottSamelsonModule cs l) :=
+  (bottSamelson cs l).obj.pairing
+
+ def bottSamelsonRight (l : List I) : SymmetricCoefficient (M := M) →+*
+    End (SymmetricCoefficient (M := M)) (BottSamelsonModule cs l) :=
+  (bottSamelson cs l).right
+
+ theorem bottSamelson_balanced (l : List I) (a : SymmetricCoefficient (M := M))
+    (m n : BottSamelsonModule cs l) :
+    bottSamelsonDuality cs l (bottSamelsonRight cs l a m) n =
+      bottSamelsonDuality cs l m (bottSamelsonRight cs l a n) :=
+  (bottSamelson cs l).balanced a m n
+
+ theorem bottSamelson_symmetric (l : List I) (m n : BottSamelsonModule cs l) :
+    bottSamelsonDuality cs l m n = bottSamelsonDuality cs l n m :=
+  (bottSamelson cs l).symmetric m n
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! The integral reflection trace lowers the actual polynomial grading by one.
+This is the graded rank-two Frobenius input, not a BMP character assertion. -/
+namespace KLInvariance.Graded
+universe uk ua
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [IsDomain A]
+  [Algebra k A] (𝓐 : ℤ → Submodule k A) [DirectSum.Decomposition 𝓐]
+  [SetLike.GradedSMul 𝓐 𝓐]
+
+ theorem homogeneous_of_mul {a b : A} {d n : ℤ}
+    (ha : a ∈ 𝓐 d) (hne : a ≠ 0) (hab : a*b ∈ 𝓐 n) : b ∈ 𝓐 (n-d) := by
+  have hc := component_homogeneous_smul 𝓐 𝓐 d a ha n b
+  have he : component 𝓐 n (a*b)=a*b :=
+    DirectSum.decompose_of_mem_same 𝓐 hab
+  rw [smul_eq_mul,he,smul_eq_mul] at hc
+  have hb : b=component 𝓐 (n-d) b := mul_left_cancel₀ hne hc
+  rw [hb]
+  exact (DirectSum.decompose 𝓐 b (n-d)).property
+
+end KLInvariance.Graded
+
+namespace KLInvariance.ReflectionFrobenius
+open KLInvariance.Graded
+universe uk ua
+variable {k : Type uk} [Field k] [CharZero k]
+  {A : Type ua} [CommRing A] [IsDomain A] [Algebra k A]
+  (τ : A ≃ₐ[k] A) (α : A) (hinv : Function.Involutive τ) (hα : α ≠ 0)
+  (hanti : τ α = -α) (hdiv : ∀ a : A, α ∣ a-τ a)
+noncomputable section
+
+ theorem trace_difference (a : A) :
+    2*α*(trace τ α hinv hα hanti hdiv a : A) = a-τ a := by
+  have h := expand_coordinates τ α hinv hα hanti hdiv a
+  have ht := congrArg τ h
+  simp only [expand_apply,map_add,map_mul,fixed_val,hanti] at h ht
+  change 2*α*((coordinates τ α hinv hα hanti hdiv a).2 : A) = a-τ a
+  linear_combination h-ht
+
+ theorem trace_homogeneous (𝓐 : ℤ → Submodule k A)
+    [DirectSum.Decomposition 𝓐] [SetLike.GradedSMul 𝓐 𝓐]
+    (hroot : α ∈ 𝓐 1) (hτ : ∀ n a, a ∈ 𝓐 n → τ a ∈ 𝓐 n)
+    {n : ℤ} {a : A} (ha : a ∈ 𝓐 n) :
+    (trace τ α hinv hα hanti hdiv a : A) ∈ 𝓐 (n-1) := by
+  apply homogeneous_of_mul 𝓐 hroot hα
+  have hd := (𝓐 n).sub_mem ha (hτ n a ha)
+  have hal := (𝓐 n).smul_mem ((2:k)⁻¹) hd
+  have ht := trace_difference τ α hinv hα hanti hdiv a
+  have htwo : (2:A)*algebraMap k A ((2:k)⁻¹) = 1 := by
+    rw [show (2:A)=algebraMap k A (2:k) by exact (map_ofNat (algebraMap k A) 2).symm, ← map_mul]
+    simp
+  have he : α*(trace τ α hinv hα hanti hdiv a : A) =
+      (2:k)⁻¹ • (a-τ a) := by
+    rw [Algebra.smul_def]
+    linear_combination (algebraMap k A ((2:k)⁻¹))*ht -
+      (α*(trace τ α hinv hα hanti hdiv a : A))*htwo
+  exact he ▸ hal
+
+ theorem coordinates_homogeneous (𝓐 : ℤ → Submodule k A)
+    [DirectSum.Decomposition 𝓐] [SetLike.GradedSMul 𝓐 𝓐]
+    (hroot : α ∈ 𝓐 1) (hτ : ∀ n a, a ∈ 𝓐 n → τ a ∈ 𝓐 n)
+    {n : ℤ} {a : A} (ha : a ∈ 𝓐 n) :
+    ((coordinates τ α hinv hα hanti hdiv a).1 : A) ∈ 𝓐 n ∧
+    ((coordinates τ α hinv hα hanti hdiv a).2 : A) ∈ 𝓐 (n-1) := by
+  have ht := trace_homogeneous τ α hinv hα hanti hdiv 𝓐 hroot hτ ha
+  constructor
+  · have hp : α*(trace τ α hinv hα hanti hdiv a : A) ∈ 𝓐 n := by
+      simpa only [smul_eq_mul,vadd_eq_add,show (1:ℤ)+(n-1)=n by omega] using
+        SetLike.GradedSMul.smul_mem hroot ht
+    have he : ((coordinates τ α hinv hα hanti hdiv a).1 : A) =
+        a-α*(trace τ α hinv hα hanti hdiv a : A) := by
+      have h := expand_coordinates τ α hinv hα hanti hdiv a
+      change ((coordinates τ α hinv hα hanti hdiv a).1 : A) +
+        α*(trace τ α hinv hα hanti hdiv a : A)=a at h
+      linear_combination h
+    exact he ▸ (𝓐 n).sub_mem ha hp
+  · exact ht
+
+end
+end KLInvariance.ReflectionFrobenius
+
+end
+
+
+section
+
+/-! The actual two-element homogeneous basis of a reflection Frobenius
+extension. This basis is used in the tensor-defined Bott--Samelson modules. -/
+namespace KLInvariance.ReflectionFrobenius
+open Module
+universe uk ua
+variable {k : Type uk} [Field k] [CharZero k]
+  {A : Type ua} [CommRing A] [IsDomain A] [Algebra k A]
+  (τ : A ≃ₐ[k] A) (α : A) (hinv : Function.Involutive τ) (hα : α ≠ 0)
+  (hanti : τ α = -α) (hdiv : ∀ a : A, α ∣ a-τ a)
+noncomputable section
+
+ def reflectionBasis : Basis (Fin 2) (fixed τ) A :=
+  (Basis.finTwoProd (fixed τ)).map (coordinates τ α hinv hα hanti hdiv).symm
+
+ theorem reflectionBasis_zero : reflectionBasis τ α hinv hα hanti hdiv 0 = 1 := by
+  change expand τ α (Basis.finTwoProd (fixed τ) 0) = 1
+  simp [expand_apply]
+
+ theorem reflectionBasis_one : reflectionBasis τ α hinv hα hanti hdiv 1 = α := by
+  change expand τ α (Basis.finTwoProd (fixed τ) 1) = α
+  simp [expand_apply]
+
+ theorem reflectionBasis_homogeneous (𝓐 : ℤ → Submodule k A)
+    [SetLike.GradedOne 𝓐] (hroot : α ∈ 𝓐 1) (i : Fin 2) :
+    reflectionBasis τ α hinv hα hanti hdiv i ∈ 𝓐 (i.val : ℤ) := by
+  fin_cases i
+  · change reflectionBasis τ α hinv hα hanti hdiv (0 : Fin 2) ∈ 𝓐 0
+    rw [reflectionBasis_zero]
+    exact SetLike.one_mem_graded 𝓐
+  · change reflectionBasis τ α hinv hα hanti hdiv (1 : Fin 2) ∈ 𝓐 1
+    rw [reflectionBasis_one]
+    exact hroot
+
+end
+end KLInvariance.ReflectionFrobenius
+
+end
+
+
+section
+
+/-! Homogeneous integral Bott--Samelson pairing at the genuine tensor basis.
+No indecomposable summand or character identity is postulated here. -/
+namespace KLInvariance.FrobeniusDuality
+open Module TensorProduct ReflectionFrobenius
+universe uk ua um ui
+variable {k : Type uk} [Field k] [CharZero k]
+  {A : Type ua} [CommRing A] [IsDomain A] [Algebra k A]
+  (τ : A ≃ₐ[k] A) (α : A) (hinv : Function.Involutive τ) (hα : α ≠ 0)
+  (hanti : τ α = -α) (hdiv : ∀ a : A, α ∣ a-τ a)
+noncomputable section
+
+variable [Module.Free (fixed τ) A] [Module.Finite (fixed τ) A]
+
+variable (N : PairedModule.{ua,um} A)
+  {ι : Type ui} (b : Basis ι A N)
+attribute [local instance] restrictedModule restrictedTower restrictedFree restrictedFinite
+
+ def inducedBasis : Basis (Fin 2 × ι) A
+    (inducedModule (trace τ α hinv hα hanti hdiv)
+      (duality τ α hinv hα hanti hdiv) (duality_apply τ α hinv hα hanti hdiv) N) :=
+  ((reflectionBasis τ α hinv hα hanti hdiv).smulTower b).baseChange A
+
+ theorem inducedBasis_apply (i : Fin 2) (j : ι) :
+    inducedBasis τ α hinv hα hanti hdiv N b (i,j) =
+      (1:A) ⊗ₜ[fixed τ] (reflectionBasis τ α hinv hα hanti hdiv i • b j) := by
+  change (((reflectionBasis τ α hinv hα hanti hdiv).smulTower b).baseChange A) (i,j) = _
+  rw [Basis.baseChange_apply,Basis.smulTower_apply]
+
+ theorem inducedPairing_basis (i i' : Fin 2) (j j' : ι) :
+    (inducedModule (trace τ α hinv hα hanti hdiv)
+      (duality τ α hinv hα hanti hdiv) (duality_apply τ α hinv hα hanti hdiv) N).pairing
+      (inducedBasis τ α hinv hα hanti hdiv N b (i,j))
+      (inducedBasis τ α hinv hα hanti hdiv N b (i',j')) =
+      (trace τ α hinv hα hanti hdiv
+        (reflectionBasis τ α hinv hα hanti hdiv i *
+          reflectionBasis τ α hinv hα hanti hdiv i' * N.pairing (b j) (b j')) : A) := by
+  rw [inducedBasis_apply,inducedBasis_apply]
+  let : Module.Free (fixed τ) N := Module.Free.of_basis
+    ((reflectionBasis τ α hinv hα hanti hdiv).smulTower b)
+  let : Module.Finite (fixed τ) N := Module.Finite.trans A N
+  change extendPairing A (restrictPairing (trace τ α hinv hα hanti hdiv)
+    (duality τ α hinv hα hanti hdiv) (duality_apply τ α hinv hα hanti hdiv) N N.pairing) _ _ = _
+  rw [extendPairing_tmul,restrictPairing_apply]
+  simp only [one_mul]
+  change ((trace τ α hinv hα hanti hdiv _) : A) = _
+  apply congrArg Subtype.val
+  apply congrArg (trace τ α hinv hα hanti hdiv)
+  simp only [map_smul,LinearMap.smul_apply,smul_eq_mul]
+  ring
+
+ theorem inducedPairing_basis_homogeneous (𝓐 : ℤ → Submodule k A)
+    [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+    (hroot : α ∈ 𝓐 1) (hτ : ∀ n a, a ∈ 𝓐 n → τ a ∈ 𝓐 n)
+    (d : ι → ℤ) (l : ℤ)
+    (hb : ∀ j j', N.pairing (b j) (b j') ∈ 𝓐 (d j+d j'-l))
+    (i i' : Fin 2) (j j' : ι) :
+    (inducedModule (trace τ α hinv hα hanti hdiv)
+      (duality τ α hinv hα hanti hdiv) (duality_apply τ α hinv hα hanti hdiv) N).pairing
+      (inducedBasis τ α hinv hα hanti hdiv N b (i,j))
+      (inducedBasis τ α hinv hα hanti hdiv N b (i',j')) ∈
+      𝓐 ((i.val+d j)+(i'.val+d j')-(l+1)) := by
+  rw [inducedPairing_basis]
+  have h := trace_homogeneous τ α hinv hα hanti hdiv 𝓐 hroot hτ
+    (SetLike.GradedMul.mul_mem
+      (SetLike.GradedMul.mul_mem
+        (reflectionBasis_homogeneous τ α hinv hα hanti hdiv 𝓐 hroot i)
+        (reflectionBasis_homogeneous τ α hinv hα hanti hdiv 𝓐 hroot i'))
+      (hb j j'))
+  convert h using 1
+  congr 1
+  omega
+
+end
+end KLInvariance.FrobeniusDuality
+
+end
+
+
+section
+
+namespace KLInvariance.Graded
+universe uk us
+variable {k : Type uk} [Field k] {σ : Type us}
+
+ theorem algHom_polynomialGrading (f : MvPolynomial σ k →ₐ[k] MvPolynomial σ k)
+    (hX : ∀ i, f (MvPolynomial.X i) ∈ polynomialGrading k σ 1)
+    {n : ℤ} {p : MvPolynomial σ k} (hp : p ∈ polynomialGrading k σ n) :
+    f p ∈ polynomialGrading k σ n := by
+  classical
+  induction hp using MvPolynomial.IsWeightedHomogeneous.induction_on with
+  | zero => simpa only [map_zero] using (polynomialGrading k σ n).zero_mem
+  | add p q _ _ hp hq => simpa only [map_add] using (polynomialGrading k σ n).add_mem hp hq
+  | monomial d c hd =>
+    rw [MvPolynomial.monomial_eq,map_mul]
+    have hc : f (MvPolynomial.C c)=MvPolynomial.C c := f.commutes c
+    rw [hc]
+    apply MvPolynomial.IsWeightedHomogeneous.C_mul
+    simp only [Finsupp.prod,map_prod,map_pow]
+    have hh := SetLike.prod_pow_mem_graded (polynomialGrading k σ)
+      (fun _ : σ => (1:ℤ)) (fun i => f (MvPolynomial.X i)) d (F := d.support)
+      (fun i _ => hX i)
+    simpa only [Finsupp.weight_apply,Finsupp.sum] using hd ▸ hh
+
+end KLInvariance.Graded
+
+namespace KLInvariance.ReflectionFrobenius
+open Module KLInvariance.Graded
+universe uk uv us
+variable {k : Type uk} [Field k] [CharZero k]
+  {V : Type uv} [AddCommGroup V] [Module k V]
+  {σ : Type us} [Fintype σ] (basis : Basis σ k V)
+noncomputable section
+
+ def polynomialLinear (v : V) : MvPolynomial σ k :=
+  SymmetricAlgebra.equivMvPolynomial basis (SymmetricAlgebra.ι k V v)
+
+omit [CharZero k] in
+ theorem polynomialLinear_homogeneous (v : V) :
+    polynomialLinear basis v ∈ polynomialGrading k σ 1 := by
+  classical
+  unfold polynomialLinear
+  rw [← basis.sum_repr v]
+  simp only [map_sum,map_smul,SymmetricAlgebra.equivMvPolynomial_ι_apply]
+  apply Submodule.sum_mem
+  intro i _
+  exact (polynomialGrading k σ 1).smul_mem _
+    (MvPolynomial.isWeightedHomogeneous_X k (fun _ : σ => (1:ℤ)) i)
+
+ def polynomialAction (e : V ≃ₗ[k] V) : MvPolynomial σ k ≃ₐ[k] MvPolynomial σ k :=
+  ((SymmetricAlgebra.equivMvPolynomial basis).symm.trans (symmetricAction e)).trans
+    (SymmetricAlgebra.equivMvPolynomial basis)
+
+omit [CharZero k] [Fintype σ] in
+@[simp] theorem polynomialAction_linear (e : V ≃ₗ[k] V) (v : V) :
+    polynomialAction basis e (polynomialLinear basis v) = polynomialLinear basis (e v) := by
+  simp [polynomialAction,polynomialLinear]
+
+omit [CharZero k] in
+ theorem polynomialAction_homogeneous (e : V ≃ₗ[k] V) {n : ℤ}
+    {p : MvPolynomial σ k} (hp : p ∈ polynomialGrading k σ n) :
+    polynomialAction basis e p ∈ polynomialGrading k σ n := by
+  apply algHom_polynomialGrading (polynomialAction basis e).toAlgHom _ hp
+  intro i
+  change polynomialAction basis e (MvPolynomial.X i) ∈ _
+  have hx : MvPolynomial.X i = polynomialLinear basis (basis i) := by
+    simp [polynomialLinear]
+  rw [hx,polynomialAction_linear]
+  exact polynomialLinear_homogeneous basis _
+
+variable (root : V) (coroot : Dual k V) (hroot : coroot root=2)
+
+ abbrev polynomialReflection : MvPolynomial σ k ≃ₐ[k] MvPolynomial σ k :=
+  polynomialAction basis (Module.reflection hroot)
+
+omit [CharZero k] [Fintype σ] in
+ theorem polynomialReflection_involutive : Function.Involutive
+    (polynomialReflection basis root coroot hroot) := by
+  intro p
+  change SymmetricAlgebra.equivMvPolynomial basis
+    (reflectionAction root coroot hroot
+      ((SymmetricAlgebra.equivMvPolynomial basis).symm
+        (SymmetricAlgebra.equivMvPolynomial basis
+          (reflectionAction root coroot hroot
+            ((SymmetricAlgebra.equivMvPolynomial basis).symm p)))))=p
+  rw [AlgEquiv.symm_apply_apply,reflectionAction_involutive,AlgEquiv.apply_symm_apply]
+
+omit [CharZero k] [Fintype σ] in
+ theorem polynomialReflection_anti : polynomialReflection basis root coroot hroot
+    (polynomialLinear basis root) = -polynomialLinear basis root := by
+  change SymmetricAlgebra.equivMvPolynomial basis
+    (reflectionAction root coroot hroot
+      ((SymmetricAlgebra.equivMvPolynomial basis).symm
+        (SymmetricAlgebra.equivMvPolynomial basis (SymmetricAlgebra.ι k V root)))) = _
+  rw [AlgEquiv.symm_apply_apply,reflectionAction_anti,map_neg]
+  rfl
+
+omit [Fintype σ] in
+include coroot hroot in
+ theorem polynomialLinear_ne_zero : polynomialLinear basis root ≠ 0 := by
+  intro h
+  apply reflectionRoot_ne_zero root coroot hroot
+  exact (SymmetricAlgebra.equivMvPolynomial basis).injective (by simpa [polynomialLinear] using h)
+
+omit [CharZero k] [Fintype σ] in
+ theorem polynomialReflection_divisible (p : MvPolynomial σ k) :
+    polynomialLinear basis root ∣ p-polynomialReflection basis root coroot hroot p := by
+  obtain ⟨d,hd⟩ := reflectionAction_divisible root coroot hroot
+    ((SymmetricAlgebra.equivMvPolynomial basis).symm p)
+  refine ⟨SymmetricAlgebra.equivMvPolynomial basis d,?_⟩
+  simpa [polynomialLinear,polynomialReflection,polynomialAction] using
+    congrArg (SymmetricAlgebra.equivMvPolynomial basis) hd
+
+ def polynomialTrace : MvPolynomial σ k →ₗ[fixed (polynomialReflection basis root coroot hroot)]
+    fixed (polynomialReflection basis root coroot hroot) :=
+  trace _ _ (polynomialReflection_involutive basis root coroot hroot)
+    (polynomialLinear_ne_zero basis root coroot hroot)
+    (polynomialReflection_anti basis root coroot hroot)
+    (polynomialReflection_divisible basis root coroot hroot)
+
+ theorem polynomialTrace_homogeneous {n : ℤ} {p : MvPolynomial σ k}
+    (hp : p ∈ polynomialGrading k σ n) :
+    (polynomialTrace basis root coroot hroot p : MvPolynomial σ k) ∈ polynomialGrading k σ (n-1) :=
+  trace_homogeneous _ _ _ _ _ _ (polynomialGrading k σ)
+    (polynomialLinear_homogeneous basis root)
+    (fun _ _ h => polynomialAction_homogeneous basis (Module.reflection hroot) h) hp
+
+end
+end KLInvariance.ReflectionFrobenius
+
+end
+
+
+section
+
+/-! The genuine symmetric-algebra grading of the extended real realization,
+transported along a specified realization basis only to access polynomial
+homogeneous components. The algebra and action are unchanged. -/
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded ReflectionFrobenius
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ def symmetricGrading : ℤ → Submodule ℝ (SymmetricCoefficient (M := M)) :=
+  fun n => (polynomialGrading ℝ σ n).map
+    (SymmetricAlgebra.equivMvPolynomial basis).symm.toLinearMap
+
+omit [Fintype σ] in
+ theorem mem_symmetricGrading (n : ℤ) (a : SymmetricCoefficient (M := M)) :
+    a ∈ symmetricGrading basis n ↔
+      SymmetricAlgebra.equivMvPolynomial basis a ∈ polynomialGrading ℝ σ n := by
+  constructor
+  · rintro ⟨p,hp,hpa⟩
+    rw [← hpa]
+    simpa using hp
+  · intro h
+    exact ⟨_,h,(SymmetricAlgebra.equivMvPolynomial basis).symm_apply_apply a⟩
+
+ instance symmetricGrading_decomposition : DirectSum.Decomposition (symmetricGrading basis) := by
+  apply DirectSum.IsInternal.chooseDecomposition
+  apply image_isInternal (polynomialGrading ℝ σ)
+    (SymmetricAlgebra.equivMvPolynomial basis).symm.toLinearMap
+    (SymmetricAlgebra.equivMvPolynomial basis).symm.surjective
+  intro n p hp
+  have hz : p=0 := (SymmetricAlgebra.equivMvPolynomial basis).symm.injective
+    (by simpa using hp)
+  subst p
+  simp
+
+ instance symmetricGrading_graded : SetLike.GradedMonoid (symmetricGrading basis) where
+  one_mem := (mem_symmetricGrading basis 0 1).mpr (by
+    rw [map_one]
+    exact SetLike.one_mem_graded (polynomialGrading ℝ σ))
+  mul_mem := by
+    intro n m a b ha hb
+    apply (mem_symmetricGrading basis (n+m) (a*b)).mpr
+    rw [map_mul]
+    exact SetLike.GradedMul.mul_mem ((mem_symmetricGrading basis n a).mp ha)
+      ((mem_symmetricGrading basis m b).mp hb)
+
+omit [Fintype σ] in
+ theorem symmetricGrading_negative (n : ℤ) (hn : n<0) :
+    symmetricGrading basis n = ⊥ := by
+  simp [symmetricGrading,polynomialGrading_negative ℝ σ n hn]
+
+ theorem simpleRoot_homogeneous (i : I) :
+    simpleRootPolynomial (M := M) i ∈ symmetricGrading basis 1 := by
+  apply (mem_symmetricGrading basis 1 _).mpr
+  exact polynomialLinear_homogeneous basis _
+
+ theorem coefficientAction_homogeneous (w : W) (n : ℤ)
+    (a : SymmetricCoefficient (M := M)) (ha : a ∈ symmetricGrading basis n) :
+    coefficientAction cs w a ∈ symmetricGrading basis n := by
+  apply (mem_symmetricGrading basis n _).mpr
+  have h := polynomialAction_homogeneous basis (extendedAction M cs w)
+    ((mem_symmetricGrading basis n a).mp ha)
+  simpa only [polynomialAction,AlgEquiv.trans_apply,AlgEquiv.symm_apply_apply,
+    coefficientAction] using h
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! A grading defined by an actual homogeneous free basis, with an explicit
+coefficient test. Used to put the genuine grading on tensor Bott--Samelsons. -/
+namespace KLInvariance.Graded
+open Module
+universe uk ua um ui
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  (𝓐 : ℤ → Submodule k A) [DirectSum.Decomposition 𝓐]
+  [SetLike.GradedMonoid 𝓐]
+  {N : Type um} [AddCommGroup N] [Module A N] [Module k N] [IsScalarTower k A N]
+  {ι : Type ui} [Fintype ι] (b : Basis ι A N) (d : ι → ℤ)
+noncomputable section
+
+ def basisPiece (n : ℤ) : Submodule k N :=
+  (shiftedPiece 𝓐 d n).comap (b.equivFun.restrictScalars k).toLinearMap
+
+omit [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐] in
+ theorem mem_basisPiece (n : ℤ) (m : N) :
+    m ∈ basisPiece 𝓐 b d n ↔ ∀ i, b.repr m i ∈ 𝓐 (n-d i) := Iff.rfl
+
+ instance basisPiece_decomposition : DirectSum.Decomposition (basisPiece 𝓐 b d) := by
+  let _ : DirectSum.Decomposition (shiftedPiece 𝓐 d) := (shifted_isInternal 𝓐 d).chooseDecomposition
+  apply DirectSum.IsInternal.chooseDecomposition
+  have he : basisPiece 𝓐 b d = fun n => (shiftedPiece 𝓐 d n).map
+      (b.equivFun.restrictScalars k).symm.toLinearMap := by
+    funext n
+    exact Submodule.comap_equiv_eq_map_symm _ _
+  rw [he]
+  apply image_isInternal (shiftedPiece 𝓐 d)
+    (b.equivFun.restrictScalars k).symm.toLinearMap
+    (b.equivFun.restrictScalars k).symm.surjective
+  intro n m hm
+  have hz : m=0 := (b.equivFun.restrictScalars k).symm.injective (by simpa using hm)
+  subst m
+  simp
+
+ instance basisPiece_graded : SetLike.GradedSMul 𝓐 (basisPiece 𝓐 b d) where
+  smul_mem := by
+    intro n m a x ha hx i
+    change b.repr (a • x) i ∈ 𝓐 ((n+m)-d i)
+    rw [map_smul,Finsupp.smul_apply,smul_eq_mul]
+    change ∀ i, b.repr x i ∈ 𝓐 (m-d i) at hx
+    have h := SetLike.GradedMul.mul_mem ha (hx i)
+    simpa only [add_sub_assoc] using h
+
+omit [DirectSum.Decomposition 𝓐] in
+ theorem basis_mem_piece (i : ι) : b i ∈ basisPiece 𝓐 b d (d i) := by
+  classical
+  intro j
+  change b.repr (b i) j ∈ 𝓐 (d i-d j)
+  by_cases hij : i=j
+  · subst j
+    simp only [Basis.repr_self,Finsupp.single_eq_same,sub_self]
+    exact SetLike.one_mem_graded 𝓐
+  · rw [Basis.repr_self,Finsupp.single_eq_of_ne (Ne.symm hij)]
+    exact Submodule.zero_mem _
+
+omit [DirectSum.Decomposition 𝓐] in
+ theorem pairing_homogeneous (p : N →ₗ[A] Dual A N) (l : ℤ)
+    (hp : ∀ i j, p (b i) (b j) ∈ 𝓐 (d i+d j-l))
+    {n m : ℤ} {x y : N} (hx : x ∈ basisPiece 𝓐 b d n)
+    (hy : y ∈ basisPiece 𝓐 b d m) : p x y ∈ 𝓐 (n+m-l) := by
+  classical
+  have hexp : p x y = ∑ i, ∑ j,
+      b.repr x i * b.repr y j * p (b i) (b j) := by
+    conv_lhs => rw [← b.sum_repr x,← b.sum_repr y]
+    simp only [map_sum,LinearMap.sum_apply,map_smul,LinearMap.smul_apply,smul_eq_mul,
+      Finset.mul_sum,mul_assoc]
+    rw [Finset.sum_comm]
+    apply Finset.sum_congr rfl
+    intro i _
+    apply Finset.sum_congr rfl
+    intro j _
+    ring
+  rw [hexp]
+  apply Submodule.sum_mem
+  intro i _
+  apply Submodule.sum_mem
+  intro j _
+  change ∀ i, b.repr x i ∈ 𝓐 (n-d i) at hx
+  change ∀ i, b.repr y i ∈ 𝓐 (m-d i) at hy
+  have h := SetLike.GradedMul.mul_mem
+    (SetLike.GradedMul.mul_mem (hx i) (hy j)) (hp i j)
+  convert h using 1
+  congr 1
+  omega
+
+end
+end KLInvariance.Graded
+
+end
+
+
+section
+
+/-! The actual tensor-defined Bott--Samelson module has the word grading and
+its integral duality lowers degree by word length (degree-one roots).
+This does not identify an indecomposable summand with the BMP sheaf. -/
+namespace KLInvariance.TitsSpace
+open Module TensorProduct FrobeniusDuality ReflectionFrobenius
+open _root_.OAI.KLInvariance.Graded
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ def BottIndex : List I → Type u
+  | [] => PUnit
+  | _ :: l => Fin 2 × BottIndex l
+
+ instance bottIndex_fintype : (l : List I) → Fintype (BottIndex l)
+  | [] => inferInstanceAs (Fintype PUnit)
+  | _ :: l => by
+      let _ := bottIndex_fintype l
+      exact inferInstanceAs (Fintype (Fin 2 × BottIndex l))
+
+ def bottDegree : (l : List I) → BottIndex l → ℤ
+  | [], _ => 0
+  | _ :: l, e => e.1.val + bottDegree l e.2
+
+omit [Fintype I] in
+ theorem bottDegree_nonneg (l : List I) (e : BottIndex l) : 0 ≤ bottDegree l e := by
+  induction l with
+  | nil => exact le_rfl
+  | cons i l ih => exact add_nonneg (Int.natCast_nonneg _) (ih e.2)
+
+ def bottBasis : (l : List I) → Basis (BottIndex l) (SymmetricCoefficient (M := M))
+    (BottSamelsonModule cs l)
+  | [] => Basis.singleton PUnit _
+  | i :: l => by
+      let : Module.Free (fixed (coefficientAction cs (cs.simple i))) (SymmetricCoefficient (M := M)) :=
+        simpleFixed_free cs i
+      let : Module.Finite (fixed (coefficientAction cs (cs.simple i))) (SymmetricCoefficient (M := M)) :=
+        simpleFixed_finite cs i
+      exact inducedBasis (coefficientAction cs (cs.simple i))
+        (simpleRootPolynomial i) (simple_involutive cs i) (simpleRootPolynomial_ne_zero i)
+        (simple_anti cs i) (simple_divisible cs i) (bottSamelson cs l).obj (bottBasis l)
+
+ theorem bottBasis_pairing_homogeneous (l : List I) (e f : BottIndex l) :
+    bottSamelsonDuality cs l (bottBasis cs l e) (bottBasis cs l f) ∈
+      symmetricGrading basis (bottDegree l e+bottDegree l f-l.length) := by
+  induction l with
+  | nil =>
+    change PUnit at e f
+    change (Basis.singleton PUnit (SymmetricCoefficient (M := M)) e) *
+      (Basis.singleton PUnit (SymmetricCoefficient (M := M)) f) ∈ symmetricGrading basis 0
+    simp only [Basis.singleton_apply, one_mul]
+    exact SetLike.one_mem_graded _
+  | cons i l ih =>
+    change (Fin 2 × BottIndex l) at e f
+    rcases e with ⟨ei, ej⟩
+    rcases f with ⟨fi, fj⟩
+    let : Module.Free (fixed (coefficientAction cs (cs.simple i))) (SymmetricCoefficient (M := M)) :=
+      simpleFixed_free cs i
+    let : Module.Finite (fixed (coefficientAction cs (cs.simple i))) (SymmetricCoefficient (M := M)) :=
+      simpleFixed_finite cs i
+    have h := inducedPairing_basis_homogeneous
+      (coefficientAction cs (cs.simple i)) (simpleRootPolynomial i)
+      (simple_involutive cs i) (simpleRootPolynomial_ne_zero i)
+      (simple_anti cs i) (simple_divisible cs i) (bottSamelson cs l).obj
+      (bottBasis cs l) (symmetricGrading basis) (simpleRoot_homogeneous basis i)
+      (coefficientAction_homogeneous cs basis (cs.simple i))
+      (bottDegree l) l.length ih ei fi ej fj
+    simp only [List.length_cons,Nat.cast_add,Nat.cast_one,bottDegree]
+    convert h using 1; rfl
+
+ instance bottRealModule (l : List I) : Module ℝ (BottSamelsonModule cs l) :=
+  Module.compHom _ (algebraMap ℝ (SymmetricCoefficient (M := M)))
+
+ instance bottRealTower (l : List I) :
+    IsScalarTower ℝ (SymmetricCoefficient (M := M)) (BottSamelsonModule cs l) :=
+  IsScalarTower.of_algebraMap_smul fun _ _ => rfl
+
+ def bottPiece (l : List I) : ℤ → Submodule ℝ (BottSamelsonModule cs l) :=
+  basisPiece (symmetricGrading basis) (bottBasis cs l) (bottDegree l)
+
+ instance bottPiece_decomposition (l : List I) : DirectSum.Decomposition (bottPiece cs basis l) :=
+  basisPiece_decomposition _ _ _
+
+ instance bottPiece_graded (l : List I) :
+    SetLike.GradedSMul (symmetricGrading basis) (bottPiece cs basis l) :=
+  basisPiece_graded _ _ _
+
+ theorem bottDuality_homogeneous (l : List I) {n m : ℤ}
+    {x y : BottSamelsonModule cs l} (hx : x ∈ bottPiece cs basis l n)
+    (hy : y ∈ bottPiece cs basis l m) :
+    bottSamelsonDuality cs l x y ∈ symmetricGrading basis (n+m-l.length) := by
+  exact pairing_homogeneous (symmetricGrading basis) (bottBasis cs l) (bottDegree l)
+    (bottSamelsonDuality cs l).toLinearMap l.length
+    (bottBasis_pairing_homogeneous cs basis l) hx hy
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Actual subexpression-weight coordinates of the tensor Bott--Samelson
+module, and the genuine right-action eigenvalue. These are the localization
+maps required for moment-sheaf realization; no character theorem is assumed. -/
+namespace KLInvariance.TitsSpace
+open Module TensorProduct FrobeniusDuality ReflectionFrobenius
+universe u v un
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+ theorem coefficientAction_one (a : SymmetricCoefficient (M := M)) :
+    coefficientAction cs 1 a = a := by
+  have h := coefficientAction_mul cs 1 1 a
+  rw [one_mul] at h
+  exact (coefficientAction cs 1).injective h.symm
+
+ def bottWeight : (l : List I) → BottIndex l → W
+  | [], _ => 1
+  | i::l, e => if e.1=0 then bottWeight l e.2 else cs.simple i*bottWeight l e.2
+
+attribute [local instance] restrictedModule restrictedTower restrictedFree restrictedFinite
+
+ def weightFunctional (i : I) (N : PairedModule.{u,un} (SymmetricCoefficient (M := M)))
+    (f : N →ₗ[SymmetricCoefficient (M := M)] SymmetricCoefficient (M := M))
+    (ε : Fin 2) : N →ₗ[simpleFixed cs i] SymmetricCoefficient (M := M) where
+  toFun m := if ε=0 then f m else coefficientAction cs (cs.simple i) (f m)
+  map_add' m n := by split_ifs <;> simp only [map_add]
+  map_smul' c m := by
+    change (if ε=0 then f ((c:SymmetricCoefficient (M:=M)) • m) else
+      coefficientAction cs (cs.simple i) (f ((c:SymmetricCoefficient (M:=M)) • m))) = _
+    split_ifs
+    · simp only [map_smul,Subalgebra.smul_def,RingHom.id_apply]
+    · rw [map_smul]
+      change coefficientAction cs (cs.simple i)
+        ((c:SymmetricCoefficient (M:=M))*f m) =
+        (c:SymmetricCoefficient (M:=M))*coefficientAction cs (cs.simple i) (f m)
+      rw [map_mul, show coefficientAction cs (cs.simple i) (c:SymmetricCoefficient (M:=M)) = c from c.property]
+
+ def inducedFunctional (i : I) (N : PairedModule.{u,un} (SymmetricCoefficient (M := M)))
+    (f : N →ₗ[simpleFixed cs i] SymmetricCoefficient (M := M)) :
+    (SymmetricCoefficient (M := M) ⊗[simpleFixed cs i] N) →ₗ[SymmetricCoefficient (M := M)]
+      SymmetricCoefficient (M := M) :=
+  AlgebraTensorModule.lift ((LinearMap.id : SymmetricCoefficient (M := M) →ₗ[
+    SymmetricCoefficient (M := M)] SymmetricCoefficient (M := M)).smulRight f)
+
+ def bottEvaluation : (l : List I) → BottIndex l →
+    BottSamelsonModule cs l →ₗ[SymmetricCoefficient (M := M)] SymmetricCoefficient (M := M)
+  | [], _ => LinearMap.id
+  | i::l, e => inducedFunctional cs i (bottSamelson cs l).obj
+      (weightFunctional cs i (bottSamelson cs l).obj (bottEvaluation l e.2) e.1)
+
+ theorem bottEvaluation_tmul (i : I) (l : List I) (e : BottIndex (i::l))
+    (a : SymmetricCoefficient (M := M)) (m : BottSamelsonModule cs l) :
+    bottEvaluation cs (i::l) e (a ⊗ₜ[simpleFixed cs i] m) =
+      a*(if e.1=0 then bottEvaluation cs l e.2 m else
+        coefficientAction cs (cs.simple i) (bottEvaluation cs l e.2 m)) := rfl
+
+ theorem bottEvaluation_right (l : List I) (e : BottIndex l)
+    (a : SymmetricCoefficient (M := M)) (m : BottSamelsonModule cs l) :
+    bottEvaluation cs l e (bottSamelsonRight cs l a m) =
+      coefficientAction cs (bottWeight cs l e) a * bottEvaluation cs l e m := by
+  induction l with
+  | nil =>
+    change SymmetricCoefficient (M := M) at m
+    change a*m = coefficientAction cs 1 a * m
+    rw [coefficientAction_one]
+  | cons i l ih =>
+    change SymmetricCoefficient (M := M) ⊗[simpleFixed cs i] BottSamelsonModule cs l at m
+    induction m using TensorProduct.inductionOn with
+    | add m n hm hn => erw [map_add,map_add,map_add,hm,hn,mul_add]
+    | tmul b m =>
+      change bottEvaluation cs (i::l) e (b ⊗ₜ[simpleFixed cs i] bottSamelsonRight cs l a m) = _
+      erw [bottEvaluation_tmul,bottEvaluation_tmul,ih]
+      change b*(if e.1=0 then _ else _) = coefficientAction cs
+        (if e.1=0 then bottWeight cs l e.2 else cs.simple i*bottWeight cs l e.2) a * _
+      split_ifs with h
+      · ring
+      · rw [map_mul,coefficientAction_mul]
+        ring
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Localization coordinates for tensor induction over a reflection wall.
+This is an integral injectivity argument, not character normalization. -/
+namespace KLInvariance.ReflectionFrobenius
+open scoped Matrix
+universe ua ui
+variable {A : Type ua} [CommRing A] [IsDomain A] [CharZero A]
+  {ι : Type ui} [Fintype ι] (τ : A ≃+* A) (α : A)
+noncomputable section
+
+ def wallCoordinate (E : Matrix ι ι A) (v : Fin 2 × ι → A) : Fin 2 × ι → A := fun e =>
+  if e.1=0 then (E *ᵥ (fun j => v (0,j)+α*v (1,j))) e.2
+  else ((E.map τ) *ᵥ (fun j => v (0,j)-α*v (1,j))) e.2
+
+ def wallMatrix (E : Matrix ι ι A) : Matrix (Fin 2 × ι) (Fin 2 × ι) A := fun e f =>
+  if e.1=0 then (if f.1=0 then E e.2 f.2 else α*E e.2 f.2)
+  else if f.1=0 then τ (E e.2 f.2) else -α*τ (E e.2 f.2)
+
+omit [IsDomain A] [CharZero A] in
+ theorem wallMatrix_mulVec (E : Matrix ι ι A) (v : Fin 2 × ι → A) :
+    wallMatrix τ α E *ᵥ v = wallCoordinate τ α E v := by
+  classical
+  funext ⟨r,j⟩
+  change (∑ c : Fin 2 × ι, wallMatrix τ α E (r,j) c * v c) = _
+  rw [Fintype.sum_prod_type,Fin.sum_univ_two]
+  fin_cases r <;>
+    simp [wallMatrix,wallCoordinate,Matrix.mulVec,dotProduct,
+      Finset.sum_add_distrib,mul_add,
+      mul_left_comm,mul_assoc,sub_eq_add_neg]
+
+omit [IsDomain A] [CharZero A] in
+ theorem matrixTwist_injective (E : Matrix ι ι A)
+    (hE : Function.Injective (Matrix.mulVec E)) : Function.Injective (Matrix.mulVec (E.map τ)) := by
+  intro v w hv
+  have hh : E *ᵥ (fun j => τ.symm (v j)) = E *ᵥ (fun j => τ.symm (w j)) := by
+    funext i
+    have h := congrArg τ.symm (congrFun hv i)
+    simpa only [Matrix.mulVec,dotProduct,map_sum,map_mul,Matrix.map_apply,
+      RingEquiv.symm_apply_apply] using h
+  have hh' := hE hh
+  funext i
+  exact τ.symm.injective (congrFun hh' i)
+
+ theorem wallCoordinate_injective (hα : α ≠ 0) (E : Matrix ι ι A)
+    (hE : Function.Injective (Matrix.mulVec E)) : Function.Injective (wallCoordinate τ α E) := by
+  intro v w hv
+  have h₀ : (fun j => v (0,j)+α*v (1,j)) = (fun j => w (0,j)+α*w (1,j)) := by
+    apply hE
+    funext j
+    simpa only [wallCoordinate,ite_eq_left rfl,ite_true] using congrFun hv (0,j)
+  have h₁ : (fun j => v (0,j)-α*v (1,j)) = (fun j => w (0,j)-α*w (1,j)) := by
+    apply matrixTwist_injective τ E hE
+    funext j
+    simpa only [wallCoordinate,show (1:Fin 2) ≠ 0 by decide,ite_false] using congrFun hv (1,j)
+  funext ⟨ε,j⟩
+  have h0 := congrFun h₀ j
+  have h1 := congrFun h₁ j
+  have hi1 : v (1,j)=w (1,j) := by
+    have hh : (2:A)*α*(v (1,j)-w (1,j))=0 := by linear_combination h0-h1
+    exact sub_eq_zero.mp ((mul_eq_zero.mp hh).resolve_left (mul_ne_zero two_ne_zero hα))
+  have hi0 : v (0,j)=w (0,j) := by simpa only [hi1,add_left_inj] using h0
+  fin_cases ε
+  · exact hi0
+  · exact hi1
+
+end
+end KLInvariance.ReflectionFrobenius
+
+end
+
+
+section
+
+/-! The actual tensor Bott--Samelson module embeds integrally in its individual
+subexpression coordinates. Together with the checked weight eigenvalue this
+identifies its generic support. It does not assert the integral moment-sheaf
+recognition or the Elias--Williamson character theorem. -/
+namespace KLInvariance.TitsSpace
+open Module TensorProduct FrobeniusDuality ReflectionFrobenius
+open scoped Matrix
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+attribute [local instance] restrictedModule restrictedTower restrictedFree restrictedFinite
+
+ def bottEvaluationMatrix (l : List I) : Matrix (BottIndex l) (BottIndex l)
+    (SymmetricCoefficient (M := M)) := fun e j => bottEvaluation cs l e (bottBasis cs l j)
+
+ theorem bottEvaluation_expand (l : List I) (e : BottIndex l) (m : BottSamelsonModule cs l) :
+    bottEvaluation cs l e m =
+      (bottEvaluationMatrix cs l *ᵥ (bottBasis cs l).equivFun m) e := by
+  classical
+  conv_lhs => rw [← (bottBasis cs l).sum_repr m]
+  simp only [map_sum,map_smul,smul_eq_mul,Matrix.mulVec,dotProduct,
+    bottEvaluationMatrix,Basis.equivFun_apply]
+  exact Finset.sum_congr rfl (fun _ _ => mul_comm _ _)
+
+ theorem bottEvaluationMatrix_cons (i : I) (l : List I)
+    (r c : Fin 2) (j k : BottIndex l) :
+    bottEvaluationMatrix cs (i::l) (r,j) (c,k) =
+      if r=0 then (if c=0 then bottEvaluationMatrix cs l j k else
+        simpleRootPolynomial i*bottEvaluationMatrix cs l j k)
+      else if c=0 then coefficientAction cs (cs.simple i) (bottEvaluationMatrix cs l j k)
+      else -simpleRootPolynomial i*coefficientAction cs (cs.simple i) (bottEvaluationMatrix cs l j k) := by
+  let : Module.Free (fixed (coefficientAction cs (cs.simple i))) (SymmetricCoefficient (M := M)) :=
+    simpleFixed_free cs i
+  let : Module.Finite (fixed (coefficientAction cs (cs.simple i))) (SymmetricCoefficient (M := M)) :=
+    simpleFixed_finite cs i
+  change bottEvaluation cs (i::l) (r,j)
+    (inducedBasis (coefficientAction cs (cs.simple i)) (simpleRootPolynomial i)
+      (simple_involutive cs i) (simpleRootPolynomial_ne_zero i) (simple_anti cs i)
+      (simple_divisible cs i) (bottSamelson cs l).obj (bottBasis cs l) (c,k)) = _
+  rw [inducedBasis_apply]
+  erw [bottEvaluation_tmul]
+  fin_cases r <;> fin_cases c <;>
+    simp only [Fin.zero_eta,Fin.isValue,Fin.mk_one,ite_true,
+      show (1:Fin 2) ≠ 0 by decide,ite_false,reflectionBasis_zero,reflectionBasis_one,
+      one_smul,one_mul,map_smul,smul_eq_mul,map_mul,simple_anti,bottEvaluationMatrix]
+
+ theorem bottEvaluationMatrix_mulVec_cons (i : I) (l : List I)
+    (v : BottIndex (i::l) → SymmetricCoefficient (M := M)) :
+    bottEvaluationMatrix cs (i::l) *ᵥ v =
+      wallCoordinate (coefficientAction cs (cs.simple i)).toRingEquiv
+        (simpleRootPolynomial i) (bottEvaluationMatrix cs l) v := by
+  have heq : bottEvaluationMatrix cs (i::l) =
+      wallMatrix (coefficientAction cs (cs.simple i)).toRingEquiv
+        (simpleRootPolynomial i) (bottEvaluationMatrix cs l) := by
+    funext e f
+    rcases e with ⟨r,j⟩
+    rcases f with ⟨c,k⟩
+    exact bottEvaluationMatrix_cons cs i l r c j k
+  rw [heq]
+  exact wallMatrix_mulVec (coefficientAction cs (cs.simple i)).toRingEquiv
+    (simpleRootPolynomial i) (bottEvaluationMatrix cs l) v
+
+ theorem bottEvaluationMatrix_injective (l : List I) :
+    Function.Injective (Matrix.mulVec (bottEvaluationMatrix cs l)) := by
+  classical
+  let : CharZero (SymmetricCoefficient (M := M)) := Algebra.charZero_of_charZero ℝ _
+  induction l with
+  | nil =>
+    intro v w h
+    funext j
+    change PUnit at j
+    cases j
+    have hh := congrFun h PUnit.unit
+    let : Unique (BottIndex ([] : List I)) := inferInstanceAs (Unique PUnit)
+    have hcol (k : BottIndex ([] : List I)) :
+        bottEvaluationMatrix cs [] PUnit.unit k = 1 := by
+      change PUnit at k
+      change (Basis.singleton PUnit (SymmetricCoefficient (M := M)) k) = 1
+      simp only [Basis.singleton_apply]
+    simp only [Matrix.mulVec,dotProduct,hcol,one_mul,Finset.univ_unique,
+      Finset.sum_singleton] at hh
+    simpa only [Subsingleton.elim (default : BottIndex ([] : List I)) PUnit.unit] using hh
+  | cons i l ih =>
+    intro v w h
+    apply wallCoordinate_injective (coefficientAction cs (cs.simple i)).toRingEquiv
+      (simpleRootPolynomial i) (simpleRootPolynomial_ne_zero i) (bottEvaluationMatrix cs l) ih
+    rw [← bottEvaluationMatrix_mulVec_cons,← bottEvaluationMatrix_mulVec_cons]
+    exact h
+
+ theorem bottEvaluation_injective (l : List I) :
+    Function.Injective (fun m : BottSamelsonModule cs l => fun e => bottEvaluation cs l e m) := by
+  intro m n h
+  apply (bottBasis cs l).equivFun.injective
+  apply bottEvaluationMatrix_injective cs l
+  funext e
+  rw [← bottEvaluation_expand,← bottEvaluation_expand]
+  exact congrFun h e
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Integral denominators exhibit the full generic subexpression support of
+an actual tensor word. No moment-sheaf character assertion is made. -/
+namespace KLInvariance.TitsSpace
+open Module
+open scoped Matrix
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+ def bottCoordinateDenominator (l : List I) : SymmetricCoefficient (M := M) := by
+  classical
+  exact (bottEvaluationMatrix cs l).det
+
+ theorem bottCoordinateDenominator_ne_zero (l : List I) :
+    bottCoordinateDenominator cs l ≠ 0 := by
+  classical
+  apply Matrix.nonsingular_iff_det_ne_zero.mp
+  apply Matrix.Nonsingular.of_linearIndependent_col
+  exact Matrix.mulVec_injective_iff.mp (bottEvaluationMatrix_injective cs l)
+
+ theorem bottEvaluation_contains_scaled_coordinates (l : List I)
+    (v : BottIndex l → SymmetricCoefficient (M := M)) :
+    ∃ m : BottSamelsonModule cs l, ∀ e,
+      bottEvaluation cs l e m = bottCoordinateDenominator cs l * v e := by
+  classical
+  refine ⟨(bottBasis cs l).equivFun.symm ((bottEvaluationMatrix cs l).adjugate *ᵥ v),?_⟩
+  intro e
+  rw [bottEvaluation_expand]
+  simp only [LinearEquiv.apply_symm_apply,Matrix.mulVec_mulVec,Matrix.mul_adjugate,
+    Matrix.smul_mulVec,Matrix.one_mulVec,Pi.smul_apply,smul_eq_mul,bottCoordinateDenominator]
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Faithfulness of the actual polynomial action; distinct Coxeter weights
+have distinct integral polynomial characters. -/
+namespace KLInvariance.TitsSpace
+open Module
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+ theorem symmetricCoefficient_ι_injective :
+    Function.Injective (SymmetricAlgebra.ι ℝ (Extended M)) := by
+  intro x y h
+  apply sub_eq_zero.mp
+  apply (Module.forall_dual_apply_eq_zero_iff ℝ (x-y)).mp
+  intro f
+  rw [map_sub]
+  apply sub_eq_zero.mpr
+  have he := congrArg (SymmetricAlgebra.lift f) h
+  simpa only [SymmetricAlgebra.lift_ι_apply] using he
+
+ theorem coefficientAction_injective : Function.Injective (coefficientAction cs) := by
+  intro x y h
+  apply extendedAction_injective M cs
+  apply LinearEquiv.ext
+  intro v
+  apply symmetricCoefficient_ι_injective (M := M)
+  have he := congrArg (fun g : SymmetricCoefficient (M := M) ≃ₐ[ℝ]
+    SymmetricCoefficient (M := M) => g (SymmetricAlgebra.ι ℝ (Extended M) v)) h
+  simpa only [coefficientAction_ι] using he
+
+ theorem coefficientAction_separates {x y : W} (hxy : x ≠ y) :
+    ∃ a : SymmetricCoefficient (M := M), coefficientAction cs x a ≠ coefficientAction cs y a := by
+  by_contra! h
+  apply hxy
+  apply coefficientAction_injective cs
+  exact AlgEquiv.ext h
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Integral weight support and orthogonality for the actual tensor module.
+These are the vertex support statements behind the section pairing, not an
+identification of the minimal BMP sheaf or its character. -/
+namespace KLInvariance.TitsSpace
+open Module
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+local instance (l : List I) : DecidableEq (BottIndex l) := Classical.decEq _
+
+ def bottWeightSupported (l : List I) (x : W) : Submodule (SymmetricCoefficient (M := M))
+    (BottSamelsonModule cs l) where
+  carrier := {m | ∀ e, bottWeight cs l e ≠ x → bottEvaluation cs l e m = 0}
+  zero_mem' := by intro e _; exact map_zero _
+  add_mem' := by intro m n hm hn e he; rw [map_add,hm e he,hn e he,zero_add]
+  smul_mem' := by intro a m hm e he; rw [map_smul,hm e he,smul_zero]
+
+ theorem bott_weight_eigen (l : List I) (x : W) {m : BottSamelsonModule cs l}
+    (hm : m ∈ bottWeightSupported cs l x) (a : SymmetricCoefficient (M := M)) :
+    bottSamelsonRight cs l a m = coefficientAction cs x a • m := by
+  apply bottEvaluation_injective cs l
+  funext e
+  change bottEvaluation cs l e (bottSamelsonRight cs l a m) =
+    bottEvaluation cs l e (coefficientAction cs x a • m)
+  erw [bottEvaluation_right,map_smul]
+  change _ = coefficientAction cs x a * bottEvaluation cs l e m
+  by_cases he : bottWeight cs l e = x
+  · rw [he]
+  · rw [hm e he,mul_zero,mul_zero]
+
+ theorem bott_eigen_supported (l : List I) (x : W) {m : BottSamelsonModule cs l}
+    (hm : ∀ a : SymmetricCoefficient (M := M),
+      bottSamelsonRight cs l a m = coefficientAction cs x a • m) :
+    m ∈ bottWeightSupported cs l x := by
+  intro e he
+  obtain ⟨a,ha⟩ := coefficientAction_separates cs he
+  have h := congrArg (bottEvaluation cs l e) (hm a)
+  erw [bottEvaluation_right,map_smul] at h
+  change coefficientAction cs (bottWeight cs l e) a * bottEvaluation cs l e m =
+    coefficientAction cs x a * bottEvaluation cs l e m at h
+  have hh : (coefficientAction cs (bottWeight cs l e) a-coefficientAction cs x a) *
+      bottEvaluation cs l e m = 0 := by rw [sub_mul,h,sub_self]
+  exact (mul_eq_zero.mp hh).resolve_left (sub_ne_zero.mpr ha)
+
+ theorem bott_distinct_weight_orthogonal (l : List I) {x y : W} (hxy : x ≠ y)
+    {m n : BottSamelsonModule cs l} (hm : m ∈ bottWeightSupported cs l x)
+    (hn : n ∈ bottWeightSupported cs l y) : bottSamelsonDuality cs l m n = 0 := by
+  obtain ⟨a,ha⟩ := coefficientAction_separates cs hxy
+  have h := bottSamelson_balanced cs l a m n
+  rw [bott_weight_eigen cs l x hm,bott_weight_eigen cs l y hn] at h
+  simp only [map_smul,LinearMap.smul_apply,smul_eq_mul] at h
+  have he : (coefficientAction cs x a-coefficientAction cs y a) *
+      bottSamelsonDuality cs l m n = 0 := by rw [sub_mul,h,sub_self]
+  exact (mul_eq_zero.mp he).resolve_left (sub_ne_zero.mpr ha)
+
+ def bottCoordinateLift (l : List I) (e : BottIndex l) : BottSamelsonModule cs l := by
+  classical
+  exact (bottEvaluation_contains_scaled_coordinates cs l (Pi.single e 1)).choose
+
+ theorem bottCoordinateLift_eval (l : List I) (e j : BottIndex l) :
+    bottEvaluation cs l j (bottCoordinateLift cs l e) =
+      bottCoordinateDenominator cs l * (Pi.single e 1 : BottIndex l → SymmetricCoefficient (M := M)) j := by
+  classical
+  exact (bottEvaluation_contains_scaled_coordinates cs l (Pi.single e 1)).choose_spec j
+
+ theorem bottCoordinateLift_supported (l : List I) (e : BottIndex l) :
+    bottCoordinateLift cs l e ∈ bottWeightSupported cs l (bottWeight cs l e) := by
+  intro j hj
+  rw [bottCoordinateLift_eval]
+  have hje : j ≠ e := fun h => hj (congrArg (bottWeight cs l) h)
+  rw [Pi.single_eq_of_ne hje, mul_zero]
+
+ theorem bottCoordinateExpansion (l : List I) (m : BottSamelsonModule cs l) :
+    bottCoordinateDenominator cs l • m =
+      ∑ e : BottIndex l, bottEvaluation cs l e m • bottCoordinateLift cs l e := by
+  classical
+  apply bottEvaluation_injective cs l
+  funext j
+  simp only [map_smul,map_sum,bottCoordinateLift_eval,smul_eq_mul]
+  rw [Finset.sum_eq_single j]
+  · simp only [Pi.single_eq_same,mul_one,mul_comm]
+  · intro e _ hej
+    rw [Pi.single_eq_of_ne (Ne.symm hej),mul_zero,mul_zero]
+  · simp
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! The actual tensor word's integral weight-support lattice is the dual of
+its integral projected stalk lattice. This is not merely a fraction-field
+isomorphism. It still does NOT identify BMP sections with an indecomposable
+summand of the tensor word, or prove the character theorem. -/
+namespace KLInvariance.TitsSpace
+open Module
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+ def bottWeightProjection (l : List I) (x : W) : BottSamelsonModule cs l →ₗ[
+    SymmetricCoefficient (M := M)]
+    ({e : BottIndex l // bottWeight cs l e=x} → SymmetricCoefficient (M := M)) :=
+  LinearMap.pi (fun e => bottEvaluation cs l e.val)
+
+ theorem bottWeightProjection_eq_zero (l : List I) (x : W) (m : BottSamelsonModule cs l) :
+    bottWeightProjection cs l x m = 0 ↔ ∀ e, bottWeight cs l e=x → bottEvaluation cs l e m=0 := by
+  constructor
+  · intro h e he
+    exact congrFun h ⟨e,he⟩
+  · intro h
+    funext e
+    exact h e.val e.property
+
+ theorem bott_supported_annihilates (l : List I) (x : W) {m : BottSamelsonModule cs l}
+    (hm : m ∈ bottWeightSupported cs l x) (n : BottSamelsonModule cs l)
+    (hn : bottWeightProjection cs l x n=0) : bottSamelsonDuality cs l m n=0 := by
+  classical
+  have hn' := (bottWeightProjection_eq_zero cs l x n).mp hn
+  have hsum : ∑ e : BottIndex l, bottEvaluation cs l e n *
+      bottSamelsonDuality cs l m (bottCoordinateLift cs l e) = 0 := by
+    apply Finset.sum_eq_zero
+    intro e _
+    by_cases he : bottWeight cs l e=x
+    · rw [hn' e he,zero_mul]
+    · rw [bott_distinct_weight_orthogonal cs l (Ne.symm he) hm
+        (bottCoordinateLift_supported cs l e),mul_zero]
+  have hh := congrArg (bottSamelsonDuality cs l m) (bottCoordinateExpansion cs l n)
+  simp only [map_smul,map_sum,smul_eq_mul] at hh
+  rw [hsum] at hh
+  exact (mul_eq_zero.mp hh).resolve_left (bottCoordinateDenominator_ne_zero cs l)
+
+ theorem bott_annihilates_supported (l : List I) (x : W) {m : BottSamelsonModule cs l}
+    (hm : ∀ n : BottSamelsonModule cs l, bottWeightProjection cs l x n=0 →
+      bottSamelsonDuality cs l m n=0) : m ∈ bottWeightSupported cs l x := by
+  apply bott_eigen_supported cs l x
+  intro a
+  apply (bottSamelsonDuality cs l).injective
+  apply LinearMap.ext
+  intro n
+  have hn : bottWeightProjection cs l x
+      (bottSamelsonRight cs l a n - coefficientAction cs x a • n)=0 := by
+    apply (bottWeightProjection_eq_zero cs l x _).mpr
+    intro e he
+    erw [map_sub,bottEvaluation_right,map_smul]
+    change coefficientAction cs (bottWeight cs l e) a * bottEvaluation cs l e n -
+      coefficientAction cs x a * bottEvaluation cs l e n=0
+    rw [he,sub_self]
+  have hh := hm _ hn
+  simp only [map_sub,map_smul,smul_eq_mul] at hh
+  rw [bottSamelson_balanced]
+  change bottSamelsonDuality cs l m (bottSamelsonRight cs l a n) =
+    (bottSamelsonDuality cs l (coefficientAction cs x a • m)) n
+  rw [map_smul,LinearMap.smul_apply,smul_eq_mul]
+  exact sub_eq_zero.mp hh
+
+ theorem bott_supported_iff_annihilates (l : List I) (x : W) (m : BottSamelsonModule cs l) :
+    m ∈ bottWeightSupported cs l x ↔ ∀ n : BottSamelsonModule cs l,
+      bottWeightProjection cs l x n=0 → bottSamelsonDuality cs l m n=0 :=
+  ⟨fun h => bott_supported_annihilates cs l x h,fun h => bott_annihilates_supported cs l x h⟩
+
+ def bottStalk (l : List I) (x : W) : Submodule (SymmetricCoefficient (M := M))
+    ({e : BottIndex l // bottWeight cs l e=x} → SymmetricCoefficient (M := M)) :=
+  LinearMap.range (bottWeightProjection cs l x)
+
+ def bottStalkProjection (l : List I) (x : W) : BottSamelsonModule cs l →ₗ[
+    SymmetricCoefficient (M := M)] bottStalk cs l x :=
+  (bottWeightProjection cs l x).rangeRestrict
+
+ theorem bottStalkProjection_surjective (l : List I) (x : W) :
+    Function.Surjective (bottStalkProjection cs l x) := by
+  rintro ⟨v,m,hm⟩
+  exact ⟨m,Subtype.ext hm⟩
+
+ theorem bottStalkProjection_eq_zero (l : List I) (x : W) (m : BottSamelsonModule cs l) :
+    bottStalkProjection cs l x m=0 ↔ bottWeightProjection cs l x m=0 := by
+  exact Subtype.ext_iff
+
+ def bottStalkDualToSupport (l : List I) (x : W) :
+    Dual (SymmetricCoefficient (M := M)) (bottStalk cs l x) →ₗ[SymmetricCoefficient (M := M)]
+      bottWeightSupported cs l x :=
+  ((bottSamelsonDuality cs l).symm.toLinearMap.comp
+    (bottStalkProjection cs l x).dualMap).codRestrict (bottWeightSupported cs l x) (by
+    intro lam
+    apply bott_annihilates_supported cs l x
+    intro n hn
+    simp only [LinearMap.comp_apply,LinearEquiv.coe_coe,LinearEquiv.apply_symm_apply,
+      LinearMap.dualMap_apply]
+    change lam (bottStalkProjection cs l x n)=0
+    rw [(bottStalkProjection_eq_zero cs l x n).mpr hn,map_zero])
+
+ theorem bottStalkDualToSupport_bijective (l : List I) (x : W) :
+    Function.Bijective (bottStalkDualToSupport cs l x) := by
+  constructor
+  · intro lam mu he
+    apply LinearMap.dualMap_injective_of_surjective (bottStalkProjection_surjective cs l x)
+    apply (bottSamelsonDuality cs l).symm.injective
+    exact congrArg Subtype.val he
+  · intro m
+    have hm : bottSamelsonDuality cs l m.val ∈
+        (LinearMap.ker (bottStalkProjection cs l x)).dualAnnihilator := by
+      rw [Submodule.mem_dualAnnihilator]
+      intro n hn
+      exact bott_supported_annihilates cs l x m.property n
+        ((bottStalkProjection_eq_zero cs l x n).mp hn)
+    rw [← LinearMap.range_dualMap_eq_dualAnnihilator_ker_of_surjective _
+      (bottStalkProjection_surjective cs l x)] at hm
+    obtain ⟨lam,hlam⟩ := hm
+    refine ⟨lam,?_⟩
+    apply Subtype.ext
+    change (bottSamelsonDuality cs l).symm ((bottStalkProjection cs l x).dualMap lam)=m.val
+    rw [hlam,LinearEquiv.symm_apply_apply]
+
+ def bottStalkDuality (l : List I) (x : W) : bottWeightSupported cs l x ≃ₗ[
+    SymmetricCoefficient (M := M)] Dual (SymmetricCoefficient (M := M)) (bottStalk cs l x) :=
+  (LinearEquiv.ofBijective (bottStalkDualToSupport cs l x)
+    (bottStalkDualToSupport_bijective cs l x)).symm
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Full reflection labels, rather than just simple-wall labels, in the
+actual tensor bimodule coefficient action. -/
+namespace KLInvariance.TitsSpace
+open Module ReflectionFrobenius
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+ theorem coefficientAction_reflection {t : W} {a : I → ℝ} (ha : Represents M cs t a) :
+    coefficientAction cs t = reflectionAction (embed M a) (extendedForm M (embed M a))
+      (show extendedForm M (embed M a) (embed M a) = 2 by
+        rw [(extendedForm_spec M).2.2]; exact ha.isRoot.norm) := by
+  rw [coefficientAction,ha.extendedAction_eq]
+
+ theorem reflectionCoefficient_anti {t : W} {a : I → ℝ} (ha : Represents M cs t a) :
+    coefficientAction cs t (SymmetricAlgebra.ι ℝ (Extended M) (embed M a)) =
+      -SymmetricAlgebra.ι ℝ (Extended M) (embed M a) := by
+  rw [coefficientAction_reflection cs ha]
+  exact reflectionAction_anti _ _ _
+
+ theorem reflectionCoefficient_divisible {t : W} {a : I → ℝ} (ha : Represents M cs t a)
+    (p : SymmetricCoefficient (M := M)) :
+    SymmetricAlgebra.ι ℝ (Extended M) (embed M a) ∣ p-coefficientAction cs t p := by
+  rw [coefficientAction_reflection cs ha]
+  exact reflectionAction_divisible _ _ _ p
+
+ theorem reflectionCoefficient_difference {t : W} {a : I → ℝ} (ha : Represents M cs t a)
+    (x : W) : ∃ p : SymmetricCoefficient (M := M),
+    coefficientAction cs x p-coefficientAction cs (t*x) p =
+      SymmetricAlgebra.ι ℝ (Extended M) (embed M a) := by
+  let α := SymmetricAlgebra.ι ℝ (Extended M) (embed M a)
+  let c : SymmetricCoefficient (M := M) := (1/2 : ℝ) • α
+  refine ⟨(coefficientAction cs x).symm c,?_⟩
+  rw [coefficientAction_mul,AlgEquiv.apply_symm_apply]
+  change (1/2 : ℝ) • α - coefficientAction cs t ((1/2 : ℝ) • α) = α
+  rw [map_smul,reflectionCoefficient_anti cs ha]
+  module
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Exact Bruhat support of the actual tensor word's subexpression weights.
+The ambient Coxeter system is not assumed crystallographic. -/
+namespace KLInvariance.TitsSpace
+universe u v
+variable {I : Type u} {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+ def bottSubword : (l : List I) → BottIndex l → List I
+  | [], _ => []
+  | i::l, e => if e.1=0 then bottSubword l e.2 else i::bottSubword l e.2
+
+ theorem bottSubword_sublist (l : List I) (e : BottIndex l) :
+    (bottSubword l e).Sublist l := by
+  induction l with
+  | nil => exact List.Sublist.refl []
+  | cons i l ih =>
+    by_cases h : e.1=0
+    · rw [bottSubword,ite_eq_left h]
+      exact List.Sublist.cons i (ih e.2)
+    · rw [bottSubword,ite_eq_right h]
+      exact List.Sublist.cons_cons i (ih e.2)
+
+ theorem bottSubword_wordProd (l : List I) (e : BottIndex l) :
+    cs.wordProd (bottSubword l e) = bottWeight cs l e := by
+  induction l with
+  | nil => exact cs.wordProd_nil
+  | cons i l ih =>
+    by_cases h : e.1=0
+    · simp only [bottSubword,bottWeight,ite_eq_left h,ih]
+    · simp only [bottSubword,bottWeight,ite_eq_right h,cs.wordProd_cons,ih]
+
+ theorem exists_bottSubword {ν l : List I} (h : ν.Sublist l) :
+    ∃ e : BottIndex l, bottSubword l e = ν := by
+  induction h with
+  | slnil => exact ⟨PUnit.unit,rfl⟩
+  | cons i h ih =>
+    obtain ⟨e,he⟩ := ih
+    exact ⟨(0,e),by simpa only [bottSubword,ite_true] using he⟩
+  | cons_cons i h ih =>
+    obtain ⟨e,he⟩ := ih
+    exact ⟨(1,e),by simp only [bottSubword,show (1:Fin 2) ≠ 0 by decide,ite_false,he]⟩
+
+ theorem bottWeight_bruhat (l : List I) (hl : cs.IsReduced l) (e : BottIndex l) :
+    BruhatLE cs (bottWeight cs l e) (cs.wordProd l) := by
+  rw [← bottSubword_wordProd]
+  exact bruhat_of_subword cs (bottSubword_sublist l e) hl
+
+ theorem exists_bottWeight (l : List I) (hl : cs.IsReduced l) {x : W}
+    (hx : BruhatLE cs x (cs.wordProd l)) : ∃ e : BottIndex l, bottWeight cs l e=x := by
+  obtain ⟨ν,hν,_,he⟩ := (bruhat_iff_reduced_subword cs hl).mp hx
+  obtain ⟨e,he'⟩ := exists_bottSubword hν
+  exact ⟨e,by rw [← bottSubword_wordProd,he',he]⟩
+
+ theorem bottWeight_range (l : List I) (hl : cs.IsReduced l) :
+    Set.range (bottWeight cs l) = {x | BruhatLE cs x (cs.wordProd l)} := by
+  ext x
+  constructor
+  · rintro ⟨e,rfl⟩
+    exact bottWeight_bruhat cs l hl e
+  · exact exists_bottWeight cs l hl
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! The actual integral two-vertex projection sheaf of a tensor
+Bott--Samelson module. This is the localization-functor construction, not
+an assumed BMP sheaf. Surjectivity onto ALL sections, flabbiness, free stalks,
+and the indecomposable normalization remain separate obligations. -/
+namespace KLInvariance.TitsSpace
+open Module MomentGraph BruhatGraph
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+ theorem bottStalkProjection_right (l : List I) (x : W)
+    (a : SymmetricCoefficient (M := M)) (m : BottSamelsonModule cs l) :
+    bottStalkProjection cs l x (bottSamelsonRight cs l a m) =
+      coefficientAction cs x a • bottStalkProjection cs l x m := by
+  apply Subtype.ext
+  funext e
+  change bottEvaluation cs l e.val (bottSamelsonRight cs l a m) =
+    coefficientAction cs x a * bottEvaluation cs l e.val m
+  rw [bottEvaluation_right,e.property]
+
+ def bottProjectionSheaf (l : List I) (u b : W) : Sheaf (R := SymmetricCoefficient (M := M))
+    (graph cs u b) where
+  vertex x := ModuleCat.of _ (bottStalk cs l x.val)
+  edge e := ModuleCat.of _ (PairProjection.Edge
+    (bottStalkProjection cs l (source cs u b e).val)
+    (bottStalkProjection cs l (target cs u b e).val))
+  lower e := by
+    change (bottStalk cs l (source cs u b e).val) →ₗ[SymmetricCoefficient (M := M)] _
+    exact PairProjection.lower (bottStalkProjection cs l (source cs u b e).val)
+      (bottStalkProjection cs l (target cs u b e).val)
+  upper e := by
+    change (bottStalk cs l (target cs u b e).val) →ₗ[SymmetricCoefficient (M := M)] _
+    exact PairProjection.upper (bottStalkProjection cs l (source cs u b e).val)
+      (bottStalkProjection cs l (target cs u b e).val)
+
+ def bottSection (l : List I) (u b : W) : BottSamelsonModule cs l →ₗ[
+    SymmetricCoefficient (M := M)] (bottProjectionSheaf cs l u b).sections Set.univ where
+  toFun m := ⟨fun x => bottStalkProjection cs l x.val m,by
+    intro e _ _
+    exact PairProjection.compatible _ _ m⟩
+  map_add' := by
+    intro m n
+    apply Subtype.ext
+    funext x
+    exact map_add (bottStalkProjection cs l x.val) m n
+  map_smul' := by
+    intro r m
+    apply Subtype.ext
+    funext x
+    exact map_smul (bottStalkProjection cs l x.val) r m
+
+ theorem bottProjectionSheaf_generated (l : List I) (u b : W) :
+    (bottProjectionSheaf cs l u b).Generated := by
+  intro x v
+  obtain ⟨m,hm⟩ := bottStalkProjection_surjective cs l x.val v
+  exact ⟨(bottSection cs l u b m).val,(bottSection cs l u b m).property,hm⟩
+
+ theorem bottProjectionSheaf_lower_surjective (l : List I) (u b : W) (e : Edge cs u b) :
+    Function.Surjective ((bottProjectionSheaf cs l u b).lower e) :=
+  PairProjection.lower_surjective _ _ (bottStalkProjection_surjective cs l _)
+
+ theorem bottProjectionSheaf_upper_surjective (l : List I) (u b : W) (e : Edge cs u b) :
+    Function.Surjective ((bottProjectionSheaf cs l u b).upper e) :=
+  PairProjection.upper_surjective _ _ (bottStalkProjection_surjective cs l _)
+
+/-- An actual Bruhat reflection root annihilates the integral edge quotient. -/
+ theorem bottProjectionSheaf_annihilated (l : List I) (u b : W) (e : Edge cs u b)
+    {a : I → ℝ} (ha : Represents M cs (label cs u b e) a)
+    (z : (bottProjectionSheaf cs l u b).edge e) :
+    SymmetricAlgebra.ι ℝ (Extended M) (embed M a) • z = 0 := by
+  obtain ⟨p,hp⟩ := reflectionCoefficient_difference cs ha (source cs u b e).val
+  rw [label_mul_source] at hp
+  rw [← hp]
+  exact PairProjection.difference_annihilates _ _
+    (bottStalkProjection_surjective cs l _) (bottStalkProjection_surjective cs l _)
+    (bottSamelsonRight cs l p) _ _
+    (bottStalkProjection_right cs l _ p) (bottStalkProjection_right cs l _ p) z
+
+ theorem bottSection_injective (l : List I) (hl : cs.IsReduced l) :
+    Function.Injective (bottSection cs l 1 (cs.wordProd l)) := by
+  intro m n h
+  apply bottEvaluation_injective cs l
+  funext e
+  let x : Interval cs 1 (cs.wordProd l) :=
+    ⟨bottWeight cs l e,one_bruhat cs _,bottWeight_bruhat cs l hl e⟩
+  have hh := congrArg (fun z : (bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ =>
+    z.val x) h
+  exact congrFun (congrArg Subtype.val hh) ⟨e,rfl⟩
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Exact kernels of the actual pair-projection quotient, in a form that
+separates integral divisibility from annihilation of the edge. -/
+namespace KLInvariance.PairProjection
+universe u v w z
+variable {R : Type u} [CommRing R]
+  {M : Type v} [AddCommGroup M] [Module R M]
+  {X : Type w} [AddCommGroup X] [Module R X]
+  {Y : Type z} [AddCommGroup Y] [Module R Y]
+  (f : M →ₗ[R] X) (g : M →ₗ[R] Y)
+
+ theorem kernel_upper (y : Y) : upper f g y=0 ↔ ∃ m : M, f m=0 ∧ g m=y := by
+  have he : upper f g y=0 ↔ lower f g 0=upper f g y := by
+    rw [map_zero, eq_comm]
+  rw [he, compatible_iff]
+
+ theorem upper_kernel_of_divisibility (α : R)
+    (hker : ∀ m : M, f m=0 → ∃ n : Y, g m=α • n)
+    (hann : ∀ z : Edge f g, α • z=0) (y : Y) :
+    upper f g y=0 ↔ ∃ n : Y, y=α • n := by
+  constructor
+  · intro h
+    obtain ⟨m,hm,rfl⟩ := (kernel_upper f g y).mp h
+    exact hker m hm
+  · rintro ⟨n,rfl⟩
+    rw [map_smul]
+    exact hann _
+
+end KLInvariance.PairProjection
+
+end
+
+
+section
+
+/-! The exact simple-wall kernel for the actual integral tensor word module.
+This is a step toward word-sheaf recognition, not a character theorem or a
+proof of combinatorial invariance. No stalk freeness is assumed. -/
+namespace KLInvariance.TitsSpace
+open Module TensorProduct FrobeniusDuality ReflectionFrobenius
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+attribute [local instance] restrictedModule restrictedTower restrictedFree restrictedFinite
+
+/-- The reflection is linear over its actual invariant subalgebra. -/
+def simpleFixedAction (i : I) : SymmetricCoefficient (M := M) →ₗ[simpleFixed cs i]
+    SymmetricCoefficient (M := M) where
+  toFun := coefficientAction cs (cs.simple i)
+  map_add' := map_add _
+  map_smul' c a := by
+    change coefficientAction cs (cs.simple i) ((c : SymmetricCoefficient (M := M))*a) =
+      (c : SymmetricCoefficient (M := M))*coefficientAction cs (cs.simple i) a
+    rw [map_mul, show coefficientAction cs (cs.simple i)
+      (c : SymmetricCoefficient (M := M)) = c from c.property]
+
+/-- Reflect the first tensor factor. It is not asserted to be left A-linear. -/
+def bottSimpleFlip (i : I) (l : List I) : BottSamelsonModule cs (i::l) →ₗ[simpleFixed cs i]
+    BottSamelsonModule cs (i::l) :=
+  TensorProduct.map (simpleFixedAction cs i)
+    (LinearMap.id : BottSamelsonModule cs l →ₗ[simpleFixed cs i] BottSamelsonModule cs l)
+
+@[simp] theorem bottSimpleFlip_tmul (i : I) (l : List I)
+    (a : SymmetricCoefficient (M := M)) (m : BottSamelsonModule cs l) :
+    bottSimpleFlip cs i l (a ⊗ₜ[simpleFixed cs i] m) =
+      coefficientAction cs (cs.simple i) a ⊗ₜ[simpleFixed cs i] m := rfl
+
+/-- The integral divisibility statement, before any localization. -/
+theorem bottSimpleFlip_divisible (i : I) (l : List I) (m : BottSamelsonModule cs (i::l)) :
+    ∃ n : BottSamelsonModule cs (i::l),
+      m-bottSimpleFlip cs i l m = simpleRootPolynomial (M := M) i • n := by
+  change SymmetricCoefficient (M := M) ⊗[simpleFixed cs i] BottSamelsonModule cs l at m
+  induction m using TensorProduct.inductionOn with
+  | add m n hm hn =>
+    obtain ⟨m',hm'⟩ := hm
+    obtain ⟨n',hn'⟩ := hn
+    refine ⟨m'+n', ?_⟩
+    erw [map_add, add_sub_add_comm, hm', hn', smul_add]
+  | tmul a m =>
+    obtain ⟨b,hb⟩ := simple_divisible cs i a
+    refine ⟨b ⊗ₜ[simpleFixed cs i] m, ?_⟩
+    erw [bottSimpleFlip_tmul, ← sub_tmul, hb]
+    rfl
+
+
+/-- Exchange the two branches of the first subexpression slot. -/
+def bottIndexFlip (i : I) (l : List I) (e : BottIndex (i::l)) : BottIndex (i::l) :=
+  (1-e.1,e.2)
+
+omit [Fintype I] in
+theorem bottWeight_flip (i : I) (l : List I) (e : BottIndex (i::l)) :
+    bottWeight cs (i::l) (bottIndexFlip i l e) = cs.simple i * bottWeight cs (i::l) e := by
+  rcases e with ⟨ε,e⟩
+  fin_cases ε <;> simp [bottIndexFlip, bottWeight, cs.simple_mul_simple_cancel_left]
+
+theorem bottEvaluation_flip (i : I) (l : List I) (e : BottIndex (i::l))
+    (m : BottSamelsonModule cs (i::l)) :
+    bottEvaluation cs (i::l) e (bottSimpleFlip cs i l m) =
+      coefficientAction cs (cs.simple i)
+        (bottEvaluation cs (i::l) (bottIndexFlip i l e) m) := by
+  change SymmetricCoefficient (M := M) ⊗[simpleFixed cs i] BottSamelsonModule cs l at m
+  induction m using TensorProduct.inductionOn with
+  | add m n hm hn => erw [map_add, map_add, map_add, map_add, hm, hn]
+  | tmul a m =>
+    rcases e with ⟨ε,e⟩
+    fin_cases ε
+    · change coefficientAction cs (cs.simple i) a * bottEvaluation cs l e m =
+        coefficientAction cs (cs.simple i)
+          (a * coefficientAction cs (cs.simple i) (bottEvaluation cs l e m))
+      rw [map_mul, simple_involutive cs i]
+    · change coefficientAction cs (cs.simple i) a *
+        coefficientAction cs (cs.simple i) (bottEvaluation cs l e m) =
+        coefficientAction cs (cs.simple i) (a * bottEvaluation cs l e m)
+      rw [map_mul]
+
+theorem bottStalkProjection_flip_zero (i : I) (l : List I) (x : W)
+    {m : BottSamelsonModule cs (i::l)} (hm : bottStalkProjection cs (i::l) x m=0) :
+    bottStalkProjection cs (i::l) (cs.simple i*x) (bottSimpleFlip cs i l m)=0 := by
+  apply (bottStalkProjection_eq_zero cs _ _ _).mpr
+  apply (bottWeightProjection_eq_zero cs _ _ _).mpr
+  intro e he
+  rw [bottEvaluation_flip]
+  have hh := (bottWeightProjection_eq_zero cs _ _ _).mp
+    ((bottStalkProjection_eq_zero cs _ _ _).mp hm)
+  rw [hh (bottIndexFlip i l e) (by
+    rw [bottWeight_flip, he, cs.simple_mul_simple_cancel_left]), map_zero]
+
+/-- The reverse inclusion to edge annihilation for the simple wall at the
+first tensor letter, expressed directly on kernel projections. -/
+theorem bottSimpleWall_kernel_projection (i : I) (l : List I) (x : W)
+    {m : BottSamelsonModule cs (i::l)} (hm : bottStalkProjection cs (i::l) x m=0) :
+    ∃ n : bottStalk cs (i::l) (cs.simple i*x),
+      bottStalkProjection cs (i::l) (cs.simple i*x) m = simpleRootPolynomial (M := M) i • n := by
+  obtain ⟨n,hn⟩ := bottSimpleFlip_divisible cs i l m
+  refine ⟨bottStalkProjection cs (i::l) (cs.simple i*x) n, ?_⟩
+  have he := congrArg (bottStalkProjection cs (i::l) (cs.simple i*x)) hn
+  rw [map_sub, map_smul, bottStalkProjection_flip_zero cs i l x hm, sub_zero] at he
+  exact he
+
+theorem simpleCoefficient_difference (i : I) (x : W) :
+    ∃ p : SymmetricCoefficient (M := M),
+      coefficientAction cs x p - coefficientAction cs (cs.simple i*x) p =
+        simpleRootPolynomial (M := M) i := by
+  let α := simpleRootPolynomial (M := M) i
+  refine ⟨(coefficientAction cs x).symm ((1/2 : ℝ) • α), ?_⟩
+  rw [coefficientAction_mul, AlgEquiv.apply_symm_apply]
+  change (1/2 : ℝ) • α - coefficientAction cs (cs.simple i) ((1/2 : ℝ) • α) = α
+  rw [map_smul, simple_anti]
+  module
+
+
+/-- Any old word's simple-edge quotient is killed by the actual label. -/
+theorem bottSimpleEdge_annihilated (i : I) (l : List I) (x : W)
+    (z : PairProjection.Edge (bottStalkProjection cs l x)
+      (bottStalkProjection cs l (cs.simple i*x))) :
+    simpleRootPolynomial (M := M) i • z = 0 := by
+  obtain ⟨p,hp⟩ := simpleCoefficient_difference cs i x
+  rw [← hp]
+  exact PairProjection.difference_annihilates _ _
+    (bottStalkProjection_surjective cs _ _) (bottStalkProjection_surjective cs _ _)
+    (bottSamelsonRight cs l p) _ _
+    (bottStalkProjection_right cs _ _ p) (bottStalkProjection_right cs _ _ p) z
+
+theorem bottSimpleEdge_kernel_inclusion (i : I) (l : List I) (x : W)
+    (y : bottStalk cs l (cs.simple i*x)) :
+    ∃ n : BottSamelsonModule cs l, bottStalkProjection cs l x n=0 ∧
+      bottStalkProjection cs l (cs.simple i*x) n=simpleRootPolynomial (M := M) i • y := by
+  apply (PairProjection.kernel_upper _ _ _).mp
+  rw [map_smul]
+  exact bottSimpleEdge_annihilated cs i l x _
+
+/-- The integral edge quotient is killed by its actual simple-root label. -/
+theorem bottSimpleWall_annihilated (i : I) (l : List I) (x : W)
+    (z : PairProjection.Edge (bottStalkProjection cs (i::l) x)
+      (bottStalkProjection cs (i::l) (cs.simple i*x))) :
+    simpleRootPolynomial (M := M) i • z = 0 :=
+  bottSimpleEdge_annihilated cs i (i::l) x z
+
+/-- The exact integral quotient kernel, without a free-stalk hypothesis. -/
+theorem bottSimpleWall_upper_kernel (i : I) (l : List I) (x : W)
+    (z : bottStalk cs (i::l) (cs.simple i*x)) :
+    PairProjection.upper (bottStalkProjection cs (i::l) x)
+      (bottStalkProjection cs (i::l) (cs.simple i*x)) z = 0 ↔
+    ∃ n : bottStalk cs (i::l) (cs.simple i*x), z = simpleRootPolynomial (M := M) i • n :=
+  PairProjection.upper_kernel_of_divisibility _ _ _
+    (fun _ hm => bottSimpleWall_kernel_projection cs i l x hm)
+    (bottSimpleWall_annihilated cs i l x) z
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Integral simple-wall translation of a two-stalk projection lattice.
+The hypotheses are explicit old-word edge properties, not character claims.
+No freeness of a submodule is inferred merely from finite generation. -/
+namespace KLInvariance.ReflectionPairTranslation
+universe u v w z
+variable {R : Type u} [CommRing R]
+
+attribute [local instance] RingHomInvPair.of_ringEquiv
+
+/-- Restriction of the scalar action along a ring automorphism. -/
+@[ext] structure Twist (_τ : R ≃+* R) (Y : Type z) where
+  val : Y
+namespace Twist
+variable (τ : R ≃+* R) (Y : Type z) [AddCommGroup Y] [Module R Y]
+local instance : RingHomInvPair τ.symm.toRingHom τ.toRingHom :=
+  RingHomInvPair.of_ringEquiv_symm τ
+local instance : RingHomInvPair τ.toRingHom τ.symm.toRingHom :=
+  RingHomInvPair.of_ringEquiv τ
+def equivVal : Twist τ Y ≃ Y where
+  toFun := val
+  invFun := mk
+  left_inv _ := rfl
+  right_inv _ := rfl
+instance : AddCommGroup (Twist τ Y) := (equivVal τ Y).addCommGroup
+instance : Module R (Twist τ Y) := by
+  letI : Module R Y := Module.compHom Y τ.toRingHom
+  exact ({ equivVal τ Y with map_add' := fun _ _ => rfl } : Twist τ Y ≃+ Y).module R
+
+def equiv : Y ≃ₛₗ[τ.symm.toRingHom] Twist τ Y where
+  toFun := mk
+  invFun := val
+  left_inv _ := rfl
+  right_inv _ := rfl
+  map_add' _ _ := rfl
+  map_smul' r y := by
+    change mk (r • y) = mk (τ (τ.symm r) • y)
+    rw [τ.apply_symm_apply]
+
+instance [Module.Free R Y] : Module.Free R (Twist τ Y) :=
+  Module.Free.of_equiv (equiv τ Y)
+end Twist
+
+variable {N : Type v} [AddCommGroup N] [Module R N]
+  {X : Type w} [AddCommGroup X] [Module R X]
+  {Y : Type z} [AddCommGroup Y] [Module R Y]
+  (f : N →ₗ[R] X) (g : N →ₗ[R] Y) (τ : R ≃+* R) (α : R)
+  (hdiv : ∀ r : R, α ∣ τ r-r)
+  (hinc : ∀ y : Y, ∃ n : N, f n=0 ∧ g n=α • y)
+
+/-- The old pair relation with the scalar action on its second factor twisted.
+Alpha-annihilation of the old edge is what makes this an R-submodule. -/
+def lattice : Submodule R (X × Twist τ Y) where
+  carrier := {p | ∃ n : N, f n=p.1 ∧ g n=p.2.val}
+  zero_mem' := ⟨0,map_zero _,map_zero _⟩
+  add_mem' := by
+    rintro _ _ ⟨m,hm,hm'⟩ ⟨n,hn,hn'⟩
+    exact ⟨m+n,by rw [map_add,hm,hn]; rfl,by rw [map_add,hm',hn']; rfl⟩
+  smul_mem' := by
+    rintro r ⟨x,y⟩ ⟨n,hn,hn'⟩
+    obtain ⟨c,hc⟩ := hdiv r
+    obtain ⟨m,hm,hm'⟩ := hinc (c • y.val)
+    have hm'' : g m=(τ r-r) • y.val := by rw [hm',hc,mul_smul]
+    refine ⟨r • n+m,?_,?_⟩
+    · change f (r • n+m)=r • x
+      rw [map_add,map_smul,hn,hm,add_zero]
+    · change g (r • n+m)=τ r • y.val
+      rw [map_add,map_smul,hn',hm'',sub_smul]
+      abel
+
+/-- Saturation away from the simple wall. Regularity on the old edge module,
+not an unjustified reflexivity assertion, removes a non-wall factor. -/
+theorem lattice_saturated (β : R)
+    (hann : ∀ e : PairProjection.Edge f g, α • e=0)
+    (hreg : Function.Injective (fun e : PairProjection.Edge f g => β • e))
+    (p : X × Twist τ Y) :
+    β • p ∈ lattice f g τ α hdiv hinc ↔ p ∈ lattice f g τ α hdiv hinc := by
+  constructor
+  · rintro ⟨n,hn,hn'⟩
+    apply (PairProjection.compatible_iff f g p.1 p.2.val).mp
+    apply hreg
+    have hn'' : g n=τ β • p.2.val := hn'
+    have h := PairProjection.compatible f g n
+    have hn1 : f n=β • p.1 := hn
+    rw [hn1,hn'',map_smul,map_smul] at h
+    obtain ⟨c,hc⟩ := hdiv β
+    have he : (τ β-β) • PairProjection.upper f g p.2.val=0 := by
+      rw [hc,mul_smul,smul_comm,hann,smul_zero]
+    rw [sub_smul,sub_eq_zero] at he
+    exact h.trans he
+  · exact (lattice f g τ α hdiv hinc).smul_mem β
+
+/-- Surjective projection to the untwisted (lower) stalk. -/
+def left : lattice f g τ α hdiv hinc →ₗ[R] X :=
+  (LinearMap.fst R X (Twist τ Y)).comp (lattice f g τ α hdiv hinc).subtype
+
+ theorem left_surjective (hf : Function.Surjective f) :
+    Function.Surjective (left f g τ α hdiv hinc) := by
+  intro x
+  obtain ⟨n,rfl⟩ := hf x
+  exact ⟨⟨(f n,⟨g n⟩),n,rfl,rfl⟩,rfl⟩
+
+/-- Its kernel is the alpha-multiple of the twisted upper stalk. -/
+def kernelMap : Twist τ Y →ₗ[R] LinearMap.ker (left f g τ α hdiv hinc) where
+  toFun y := ⟨⟨(0,⟨α • y.val⟩),hinc y.val⟩,rfl⟩
+  map_add' y z := by
+    apply Subtype.ext
+    apply Subtype.ext
+    apply Prod.ext
+    · change (0 : X)=0+0; simp
+    · change Twist.mk (α • (y.val+z.val)) = Twist.mk (α • y.val+α • z.val)
+      rw [smul_add]
+  map_smul' r y := by
+    apply Subtype.ext
+    apply Subtype.ext
+    change (0,Twist.mk (α • (τ r • y.val)))=(r • (0 : X),Twist.mk (τ r • (α • y.val)))
+    rw [smul_zero,smul_comm α (τ r)]
+
+ variable (hker : ∀ y : Y, (∃ n : N, f n=0 ∧ g n=y) ↔ ∃ z : Y, y=α • z)
+
+ include hker in
+ theorem kernelMap_surjective : Function.Surjective (kernelMap f g τ α hdiv hinc) := by
+  rintro ⟨⟨⟨x,y⟩,n,hn,hn'⟩,hx⟩
+  change x=0 at hx
+  subst x
+  obtain ⟨z,hz⟩ := (hker y.val).mp ⟨n,hn,hn'⟩
+  refine ⟨⟨z⟩,?_⟩
+  apply Subtype.ext
+  apply Subtype.ext
+  exact Prod.ext rfl (congrArg Twist.mk hz.symm)
+
+ theorem kernelMap_injective (hinj : Function.Injective (fun y : Y => α • y)) :
+    Function.Injective (kernelMap f g τ α hdiv hinc) := by
+  intro y z h
+  apply Twist.ext
+  exact hinj (congrArg (fun p => p.val.val.2.val) h)
+
+noncomputable def kernelEquiv (hinj : Function.Injective (fun y : Y => α • y)) :
+    Twist τ Y ≃ₗ[R] LinearMap.ker (left f g τ α hdiv hinc) :=
+  LinearEquiv.ofBijective (kernelMap f g τ α hdiv hinc)
+    ⟨kernelMap_injective f g τ α hdiv hinc hinj,kernelMap_surjective f g τ α hdiv hinc hker⟩
+
+/-- A split extension of free modules is free. This does not assert the
+false statement that every submodule of a free polynomial module is free. -/
+theorem free_of_surjective_of_ker_free {P : Type*} [AddCommGroup P] [Module R P]
+    {Q : Type*} [AddCommGroup Q] [Module R Q]
+    (π : P →ₗ[R] Q) (hπ : Function.Surjective π)
+    [Module.Free R Q] [Module.Free R (LinearMap.ker π)] : Module.Free R P := by
+  obtain ⟨s,hs⟩ := π.exists_rightInverse_of_surjective (LinearMap.range_eq_top.mpr hπ)
+  let e : (LinearMap.ker π × Q) →ₗ[R] P :=
+    (LinearMap.ker π).subtype.coprod s
+  have hps (q : Q) : π (s q)=q := DFunLike.congr_fun hs q
+  have he : Function.Bijective e := by
+    constructor
+    · rintro ⟨k,q⟩ ⟨k',q'⟩ h
+      have hq : q=q' := by
+        have hh := congrArg π h
+        simpa only [e,LinearMap.coprod_apply,Submodule.subtype_apply,map_add,LinearMap.mem_ker.mp k.property,
+          LinearMap.mem_ker.mp k'.property,hps,zero_add] using hh
+      subst q'
+      refine Prod.ext ?_ rfl
+      apply Subtype.ext
+      exact add_right_cancel h
+    · intro p
+      refine ⟨(⟨p-s (π p),?_⟩,π p),?_⟩
+      · change π (p-s (π p))=0
+        rw [map_sub,hps,sub_self]
+      · change p-s (π p)+s (π p)=p
+        abel
+  exact Module.Free.of_equiv (LinearEquiv.ofBijective e he)
+
+ include hker in
+ theorem lattice_free (hf : Function.Surjective f)
+    (hinj : Function.Injective (fun y : Y => α • y))
+    [Module.Free R X] [Module.Free R Y] :
+    Module.Free R (lattice f g τ α hdiv hinc) := by
+  let : Module.Free R (LinearMap.ker (left f g τ α hdiv hinc)) :=
+    Module.Free.of_equiv (kernelEquiv f g τ α hdiv hinc hker hinj)
+  exact free_of_surjective_of_ker_free (left f g τ α hdiv hinc)
+    (left_surjective f g τ α hdiv hinc hf)
+
+end KLInvariance.ReflectionPairTranslation
+
+end
+
+
+section
+
+/-! A generic tensor-generation lemma, kept separate so application to the
+actual recursively paired word carrier does not unfold that carrier's data. -/
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/KazhdanLusztig/GradedModules.lean b/lean/OAI/RepresentationTheory/KazhdanLusztig/GradedModules.lean
index ef876a6a703a44266b6268b7ed43e3816491734e..864a5c1db1fbfaa3e3f054b032b34bec012bd339 100644
--- a/lean/OAI/RepresentationTheory/KazhdanLusztig/GradedModules.lean
+++ b/lean/OAI/RepresentationTheory/KazhdanLusztig/GradedModules.lean
@@ -1,1458 +1,1458 @@
-import OAI.RepresentationTheory.KazhdanLusztig.ModuleAlgebra
-
-namespace OAI
-
-/-!
-Graded polynomial modules, projective presentations, finite homogeneous generators and uniqueness of minimal covers.
--/
-
-section
-
-namespace KLInvariance.Graded
-open scoped Pointwise
-universe uk ua um
-variable {k : Type uk} {A : Type ua} {M : Type um} [Field k] [CommRing A] [klPreservedInstance9 : Algebra k A]
-  [AddCommGroup M] [Module k M] [Module A M] [klPreservedInstance8 : IsScalarTower k A M]
-
-include klPreservedInstance8 in
-/-- Graded Nakayama in its generation form. The submodule is closed under
-actual homogeneous components, rather than assumed to be the whole module. -/
-theorem eq_top_of_positive_smul_sup
-    (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M)
-    [DirectSum.Decomposition 𝓜] [SetLike.GradedSMul 𝓐 𝓜]
-    (d₀ : ℤ) (hbound : ∀ j < d₀, 𝓜 j = ⊥)
-    (N : Submodule A M) (hN : ∀ n m, m ∈ N → (DirectSum.decompose 𝓜 m n : M) ∈ N)
-    (hgen : N ⊔ positiveIdeal 𝓐 • (⊤ : Submodule A M) = ⊤) : N = ⊤ := by
-  have _ := klPreservedInstance8
-  classical
-  have hlow : ∀ j < d₀, ∀ m ∈ 𝓜 j, m ∈ N := by
-    intro j hj m hm
-    have hm0 : m = 0 := by simpa [hbound j hj] using hm
-    simp [hm0]
-  have hind : ∀ n : ℕ, ∀ m ∈ 𝓜 (d₀+n), m ∈ N := by
-    intro n
-    induction n using Nat.strong_induction_on with
-    | h n ih =>
-      let p := component 𝓜 (d₀+n)
-      have hsmul (i : ℤ) (hi : 0 < i) (a : A) (ha : a ∈ 𝓐 i) :
-          ∀ m : M, p (a • m) ∈ N := by
-        intro m
-        refine DirectSum.Decomposition.inductionOn 𝓜 ?_ ?_ ?_ m
-        · simp
-        · intro j m
-          by_cases hj : j < d₀+n
-          · have hm : (m : M) ∈ N := by
-              by_cases hj₀ : j < d₀
-              · exact hlow j hj₀ m m.property
-              · have heq : d₀ + (j-d₀).toNat = j := by omega
-                exact ih (j-d₀).toNat (by omega) m (heq.symm ▸ m.property)
-            exact hN _ _ (N.smul_mem a hm)
-          · have hdeg : i+j ≠ d₀+n := by omega
-            have ham : a • (m : M) ∈ 𝓜 (i+j) :=
-              SetLike.GradedSMul.smul_mem ha m.property
-            have heq : p (a • (m : M)) = 0 :=
-              DirectSum.decompose_of_mem_ne 𝓜 ham hdeg
-            simp [heq]
-        · intro x y hx hy
-          simpa only [smul_add, map_add] using N.add_mem hx hy
-      let J : Ideal A :=
-        { carrier := {a | ∀ m : M, p (a • m) ∈ N}
-          zero_mem' := by simp [N.zero_mem]
-          add_mem' := by
-            intro a b ha hb m
-            simpa only [add_smul, map_add] using N.add_mem (ha m) (hb m)
-          smul_mem' := by
-            intro r a ha m
-            change p ((r*a) • m) ∈ N
-            rw [mul_comm r a, mul_smul]
-            exact ha (r • m) }
-      have hIJ : positiveIdeal 𝓐 ≤ J := by
-        apply Ideal.span_le.mpr
-        rintro a ⟨i,hi,ha⟩
-        exact hsmul i hi a ha
-      have hp : ∀ m : M, p m ∈ N := by
-        intro m
-        have hm : m ∈ N ⊔ positiveIdeal 𝓐 • (⊤ : Submodule A M) := by
-          rw [hgen]; trivial
-        obtain ⟨x,hx,y,hy,rfl⟩ := Submodule.mem_sup.mp hm
-        rw [map_add]
-        apply N.add_mem (hN _ _ hx)
-        refine Submodule.smul_induction_on hy ?_ ?_
-        · intro a ha z _; exact hIJ ha z
-        · intro z w hz hw; simpa only [map_add] using N.add_mem hz hw
-      intro m hm
-      have hcomp : p m=m := DirectSum.decompose_of_mem_same 𝓜 hm
-      exact hcomp ▸ hp m
-  have homogeneous_mem (j : ℤ) (m : M) (hm : m ∈ 𝓜 j) : m ∈ N := by
-    by_cases hj : j < d₀
-    · exact hlow j hj m hm
-    · have heq : d₀ + (j-d₀).toNat = j := by omega
-      exact hind (j-d₀).toNat m (heq.symm ▸ hm)
-  apply top_unique
-  intro m _
-  exact DirectSum.Decomposition.inductionOn 𝓜 N.zero_mem
-    (fun m => homogeneous_mem _ m m.property)
-    (fun _ _ hx hy => N.add_mem hx hy) m
-
-include klPreservedInstance8 in
-/-- Projection of a scalar multiple of a homogeneous vector. -/
-theorem component_smul_homogeneous
-    (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M)
-    [DirectSum.Decomposition 𝓐] [DirectSum.Decomposition 𝓜]
-    [SetLike.GradedSMul 𝓐 𝓜] (a : A) (m : M) (d n : ℤ) (hm : m ∈ 𝓜 d) :
-    component 𝓜 n (a • m) = component 𝓐 (n-d) a • m := by
-  have _ := klPreservedInstance8
-  classical
-  refine DirectSum.Decomposition.inductionOn 𝓐 ?_ ?_ ?_ a
-  · simp
-  · intro i a
-    by_cases hi : i = n-d
-    · have hsum : i+d = n := by omega
-      have ham : (a : A) • m ∈ 𝓜 n := hsum ▸
-        SetLike.GradedSMul.smul_mem a.property hm
-      rw [show component 𝓜 n ((a : A) • m) = (a : A) • m from
-        DirectSum.decompose_of_mem_same 𝓜 ham]
-      rw [show component 𝓐 (n-d) (a : A) = (a : A) from
-        DirectSum.decompose_of_mem_same 𝓐 (hi ▸ a.property)]
-    · have hsum : i+d ≠ n := by omega
-      rw [show component 𝓜 n ((a : A) • m) = 0 from
-        DirectSum.decompose_of_mem_ne 𝓜
-          (SetLike.GradedSMul.smul_mem a.property hm) hsum,
-        show component 𝓐 (n-d) (a : A) = 0 from
-        DirectSum.decompose_of_mem_ne 𝓐 a.property hi, zero_smul]
-  · intro a b ha hb
-    simp only [add_smul, map_add, ha, hb]
-
-include klPreservedInstance9 klPreservedInstance8 in
-/-- Finite generation can be made homogeneous using the actual finite
-support decomposition of each generator. -/
-theorem exists_finite_homogeneous_generators
-    (𝓜 : ℤ → Submodule k M) [DirectSum.Decomposition 𝓜] [Module.Finite A M] :
-    ∃ t : Finset M, Submodule.span A (t : Set M) = ⊤ ∧
-      ∀ m ∈ t, ∃ d : ℤ, m ∈ 𝓜 d := by
-  have _ := klPreservedInstance9
-  have _ := klPreservedInstance8
-  classical
-  obtain ⟨s,hs⟩ := Module.Finite.fg_top (R := A) (M := M)
-  let t : Finset M := s.biUnion fun m =>
-    (DirectSum.decompose 𝓜 m).support.image fun j => (DirectSum.decompose 𝓜 m j : M)
-  refine ⟨t, ?_, ?_⟩
-  · apply top_unique
-    rw [← hs]
-    apply Submodule.span_le.mpr
-    intro m hm
-    rw [← DirectSum.sum_support_decompose 𝓜 m]
-    apply Submodule.sum_mem
-    intro j hj
-    apply Submodule.subset_span
-    exact Finset.mem_biUnion.mpr ⟨m,hm,Finset.mem_image.mpr ⟨j,hj,rfl⟩⟩
-  · intro m hm
-    obtain ⟨a,ha,hma⟩ := Finset.mem_biUnion.mp hm
-    obtain ⟨j,hj,rfl⟩ := Finset.mem_image.mp hma
-    exact ⟨j,(DirectSum.decompose 𝓜 a j).property⟩
-
-/-- The span of finitely many homogeneous vectors is closed under each
-component, as required for graded Nakayama. -/
-theorem homogeneous_span_components {ι : Type*} [Fintype ι]
-    (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M)
-    [DirectSum.Decomposition 𝓐] [DirectSum.Decomposition 𝓜]
-    [SetLike.GradedSMul 𝓐 𝓜] (v : ι → M) (d : ι → ℤ)
-    (hv : ∀ i, v i ∈ 𝓜 (d i)) (n : ℤ) (m : M)
-    (hm : m ∈ Submodule.span A (Set.range v)) :
-    component 𝓜 n m ∈ Submodule.span A (Set.range v) := by
-  classical
-  obtain ⟨c,rfl⟩ := (Submodule.mem_span_range_iff_exists_fun A).mp hm
-  rw [map_sum]
-  apply Submodule.sum_mem
-  intro i _
-  rw [component_smul_homogeneous 𝓐 𝓜 _ _ (d i) n (hv i)]
-  exact Submodule.smul_mem _ _ (Submodule.subset_span (Set.mem_range_self i))
-
-/-- Reduction of a scalar modulo the augmentation ideal gives precisely
-its scalar-field action on the reduced module. -/
-theorem quotient_smul_of_scalar_mod (I : Ideal A) (a : A) (c : k)
-    (hac : a-algebraMap k A c ∈ I) (m : M) :
-    (I • (⊤ : Submodule A M)).mkQ (a • m) =
-      c • (I • (⊤ : Submodule A M)).mkQ m := by
-  let J := I • (⊤ : Submodule A M)
-  have hz : J.mkQ ((a-algebraMap k A c) • m) = 0 :=
-    (Submodule.Quotient.mk_eq_zero J).mpr
-      (Submodule.smul_mem_smul hac (Submodule.mem_top : m ∈ (⊤ : Submodule A M)))
-  rw [sub_smul, map_sub] at hz
-  have heq := sub_eq_zero.mp hz
-  calc
-    _ = J.mkQ (algebraMap k A c • m) := heq
-    _ = algebraMap k A c • J.mkQ m := map_smul J.mkQ _ _
-    _ = c • J.mkQ m := IsScalarTower.algebraMap_smul A c (J.mkQ m)
-
-/-- A finite free cover of a projective module, minimal at any proper
-augmentation ideal of a noetherian domain, is injective. The determinant
-form of Nakayama removes the complementary torsion-free kernel. -/
-theorem linearIndependent_of_projective_cover {ι : Type*} [Fintype ι]
-    [IsDomain A] [IsNoetherianRing A] [Module.Projective A M]
-    (I : Ideal A) (hI : I ≠ ⊤)
-    (hscalar : ∀ a : A, ∃ c : k, a-algebraMap k A c ∈ I)
-    (v : ι → M) (hspan : Submodule.span A (Set.range v) = ⊤)
-    (hind : LinearIndependent k
-      (fun i => (I • (⊤ : Submodule A M)).mkQ (v i))) :
-    LinearIndependent A v := by
-  classical
-  let J := I • (⊤ : Submodule A M)
-  let f := Fintype.linearCombination A v
-  have hsurj : Function.Surjective f := by
-    intro m
-    exact (Submodule.mem_span_range_iff_exists_fun A).mp (hspan ▸ Submodule.mem_top)
-  obtain ⟨s,hs⟩ := Module.projective_lifting_property f (LinearMap.id : M →ₗ[A] M) hsurj
-  have hmin : LinearMap.ker f ≤ I • (⊤ : Submodule A (ι → A)) := by
-    intro a ha
-    choose c hc using fun i => hscalar (a i)
-    have heq : ∑ i, c i • J.mkQ (v i) = 0 := by
-      calc
-        _ = J.mkQ (∑ i, a i • v i) := by
-          rw [map_sum]
-          apply Finset.sum_congr rfl
-          intro i _
-          exact (quotient_smul_of_scalar_mod I (a i) (c i) (hc i) (v i)).symm
-        _ = 0 := congrArg J.mkQ (LinearMap.mem_ker.mp ha) |>.trans (map_zero _)
-    have hc0 : ∀ i, c i=0 := (Fintype.linearIndependent_iff.mp hind) c heq
-    have hai (i : ι) : a i ∈ I := by simpa [hc0 i] using hc i
-    have hrebuild : a = ∑ i, a i • Pi.single i (1 : A) := by
-      ext j
-      simp [Finset.sum_apply, Pi.smul_apply, Pi.single_apply]
-    rw [hrebuild]
-    apply Submodule.sum_mem
-    intro i _
-    exact Submodule.smul_mem_smul (hai i) Submodule.mem_top
-  have htop := positive_smul_kernel_of_minimal_split I f s hs hmin
-  have hfg : (⊤ : Submodule A (LinearMap.ker f)).FG := IsNoetherian.noetherian _
-  obtain ⟨r,hr,hkill⟩ := Submodule.exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul
-    I (⊤ : Submodule A (LinearMap.ker f)) hfg (by rw [htop])
-  have hr0 : r ≠ 0 := by
-    intro heq
-    have hone : (1 : A) ∈ I := by simpa [heq] using I.neg_mem hr
-    exact hI (Ideal.eq_top_of_isUnit_mem I hone isUnit_one)
-  have hker : LinearMap.ker f = ⊥ := by
-    apply le_antisymm _ bot_le
-    intro a ha
-    have hz : r • (⟨a,ha⟩ : LinearMap.ker f) = 0 := hkill _ Submodule.mem_top
-    have ha0 : (⟨a,ha⟩ : LinearMap.ker f) = 0 := (smul_eq_zero.mp hz).resolve_left hr0
-    exact congrArg Subtype.val ha0
-  exact linearIndependent_iff_injective_fintypeLinearCombination.mpr
-    (LinearMap.ker_eq_bot.mp hker)
-
-/-- Images of any finite generating family span the augmentation quotient
-as a vector space over the degree-zero field. -/
-theorem quotient_span_of_scalar_mod {ι : Type*} [Fintype ι]
-    (I : Ideal A) (hscalar : ∀ a : A, ∃ c : k, a-algebraMap k A c ∈ I)
-    (v : ι → M) (hv : Submodule.span A (Set.range v) = ⊤) :
-    Submodule.span k (Set.range (fun i => (I • (⊤ : Submodule A M)).mkQ (v i))) = ⊤ := by
-  classical
-  let J := I • (⊤ : Submodule A M)
-  apply top_unique
-  intro z _
-  obtain ⟨m,rfl⟩ := J.mkQ_surjective z
-  obtain ⟨a,ha⟩ := (Submodule.mem_span_range_iff_exists_fun A).mp (hv ▸ Submodule.mem_top :
-    m ∈ Submodule.span A (Set.range v))
-  choose c hc using fun i => hscalar (a i)
-  apply (Submodule.mem_span_range_iff_exists_fun k).mpr
-  refine ⟨c, ?_⟩
-  calc
-    _ = J.mkQ (∑ i, a i • v i) := by
-      rw [map_sum]
-      apply Finset.sum_congr rfl
-      intro i _
-      exact (quotient_smul_of_scalar_mod I (a i) (c i) (hc i) (v i)).symm
-    _ = J.mkQ m := congrArg J.mkQ ha
-
-
-theorem exists_homogeneous_basis_of_scalar_mod
-    [IsDomain A] [IsNoetherianRing A] [Module.Projective A M] [Module.Finite A M]
-    (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M)
-    [DirectSum.Decomposition 𝓐] [DirectSum.Decomposition 𝓜]
-    [SetLike.GradedSMul 𝓐 𝓜]
-    (d₀ : ℤ) (hbound : ∀ j < d₀, 𝓜 j = ⊥)
-    (hproper : positiveIdeal 𝓐 ≠ ⊤)
-    (hscalar : ∀ a : A, ∃ c : k, a-algebraMap k A c ∈ positiveIdeal 𝓐) :
-    ∃ (ι : Type um), Finite ι ∧ ∃ b : Module.Basis ι A M,
-      ∀ i, ∃ d : ℤ, b i ∈ 𝓜 d := by
-  classical
-  obtain ⟨t,ht,hth⟩ := exists_finite_homogeneous_generators (A := A) 𝓜
-  let v : t → M := Subtype.val
-  have hv : Submodule.span A (Set.range v) = ⊤ := by
-    simpa [v] using ht
-  let I := positiveIdeal 𝓐
-  let J := I • (⊤ : Submodule A M)
-  have hq := quotient_span_of_scalar_mod I hscalar v hv
-  obtain ⟨ι,a,ha,hsp,hli⟩ := exists_linearIndependent' k (fun i => J.mkQ (v i))
-  let : Finite ι := Finite.of_injective a ha
-  let : Fintype ι := Fintype.ofFinite ι
-  let w : ι → M := v ∘ a
-  have hw : ∀ i, ∃ d : ℤ, w i ∈ 𝓜 d := fun i => hth _ (a i).property
-  choose d hd using hw
-  have hqw : Submodule.span k (Set.range (fun i => J.mkQ (w i))) = ⊤ := hsp.trans hq
-  let N := Submodule.span A (Set.range w)
-  have hN : ∀ n m, m ∈ N → (DirectSum.decompose 𝓜 m n : M) ∈ N :=
-    homogeneous_span_components 𝓐 𝓜 w d hd
-  have hgen : N ⊔ I • (⊤ : Submodule A M) = ⊤ := by
-    apply top_unique
-    intro m _
-    obtain ⟨c,hc⟩ := (Submodule.mem_span_range_iff_exists_fun k).mp
-      (hqw ▸ Submodule.mem_top : J.mkQ m ∈
-        Submodule.span k (Set.range (fun i => J.mkQ (w i))))
-    let z : M := ∑ i, algebraMap k A (c i) • w i
-    have hz : z ∈ N := by
-      apply Submodule.sum_mem
-      intro i _
-      exact N.smul_mem _ (Submodule.subset_span (Set.mem_range_self i))
-    have hqz : J.mkQ z = J.mkQ m := by
-      change J.mkQ (∑ i, algebraMap k A (c i) • w i) = J.mkQ m
-      rw [map_sum]
-      convert hc using 1
-      apply Finset.sum_congr rfl
-      intro i _
-      exact (map_smul J.mkQ _ _).trans (IsScalarTower.algebraMap_smul A (c i) _)
-    have hdiff : m-z ∈ J := by
-      apply (Submodule.Quotient.mk_eq_zero J).mp
-      change J.mkQ (m-z)=0
-      rw [map_sub,hqz,sub_self]
-    exact Submodule.mem_sup.mpr ⟨z,hz,m-z,hdiff,by abel⟩
-  have hspan : N=⊤ := eq_top_of_positive_smul_sup 𝓐 𝓜 d₀ hbound N hN hgen
-  have hliA : LinearIndependent A w :=
-    linearIndependent_of_projective_cover I hproper hscalar w hspan hli
-  refine ⟨ι,inferInstance,Module.Basis.mk hliA hspan.ge,?_⟩
-  intro i
-  exact ⟨d i, by simpa only [Module.Basis.mk_apply] using hd i⟩
-
-/-- The augmentation for a nonnegatively supported integer grading is
-actual degree-zero projection. -/
-noncomputable def nonnegativeAugmentation
-    (𝓐 : ℤ → Submodule k A) [DirectSum.Decomposition 𝓐]
-    [SetLike.GradedMonoid 𝓐] (hneg : ∀ j < 0, 𝓐 j = ⊥) : A →+* A where
-  toFun := component 𝓐 0
-  map_zero' := map_zero _
-  map_one' := DirectSum.decompose_of_mem_same 𝓐 SetLike.GradedOne.one_mem
-  map_add' := map_add _
-  map_mul' := by
-    intro a b
-    refine DirectSum.Decomposition.inductionOn 𝓐 ?_ ?_ ?_ a
-    · simp
-    · intro i a
-      refine DirectSum.Decomposition.inductionOn 𝓐 ?_ ?_ ?_ b
-      · simp
-      · intro j b
-        by_cases hi : i < 0
-        · have ha : (a : A) = 0 := by simpa [hneg i hi] using a.property
-          simp [ha]
-        by_cases hj : j < 0
-        · have hb : (b : A) = 0 := by simpa [hneg j hj] using b.property
-          simp [hb]
-        by_cases hij : i+j=0
-        · have hi0 : i=0 := by omega
-          have hj0 : j=0 := by omega
-          have hab := SetLike.GradedMul.mul_mem a.property b.property
-          change component 𝓐 0 ((a:A)*(b:A)) = component 𝓐 0 (a:A) * component 𝓐 0 (b:A)
-          rw [show component 𝓐 0 ((a:A)*(b:A)) = (a:A)*(b:A) from
-            DirectSum.decompose_of_mem_same 𝓐 (hij ▸ hab),
-            show component 𝓐 0 (a:A) = a from
-              DirectSum.decompose_of_mem_same 𝓐 (hi0 ▸ a.property),
-            show component 𝓐 0 (b:A) = b from
-              DirectSum.decompose_of_mem_same 𝓐 (hj0 ▸ b.property)]
-        · change component 𝓐 0 ((a:A)*(b:A)) = component 𝓐 0 (a:A) * component 𝓐 0 (b:A)
-          rw [show component 𝓐 0 ((a:A)*(b:A))=0 from
-            DirectSum.decompose_of_mem_ne 𝓐
-              (SetLike.GradedMul.mul_mem a.property b.property) hij]
-          by_cases hi0 : i=0
-          · have hj0 : j≠0 := by omega
-            rw [show component 𝓐 0 (b:A)=0 from
-              DirectSum.decompose_of_mem_ne 𝓐 b.property hj0,mul_zero]
-          · rw [show component 𝓐 0 (a:A)=0 from
-              DirectSum.decompose_of_mem_ne 𝓐 a.property hi0,zero_mul]
-      · intro b c hb hc
-        simp only [mul_add,map_add,hb,hc]
-    · intro a c ha hc
-      simp only [add_mul,map_add,ha,hc]
-
-/-- The positive ideal is exactly the augmentation kernel. -/
-theorem positiveIdeal_eq_augmentation_ker
-    (𝓐 : ℤ → Submodule k A) [DirectSum.Decomposition 𝓐]
-    [SetLike.GradedMonoid 𝓐] (hneg : ∀ j < 0, 𝓐 j = ⊥) :
-    positiveIdeal 𝓐 = RingHom.ker (nonnegativeAugmentation 𝓐 hneg) := by
-  classical
-  apply le_antisymm
-  · apply Ideal.span_le.mpr
-    rintro a ⟨i,hi,ha⟩
-    exact DirectSum.decompose_of_mem_ne 𝓐 ha (by omega)
-  · intro a ha
-    rw [← DirectSum.sum_support_decompose 𝓐 a]
-    apply Ideal.sum_mem
-    intro j hj
-    by_cases hjpos : 0 < j
-    · exact Ideal.subset_span ⟨j,hjpos,(DirectSum.decompose 𝓐 a j).property⟩
-    have hz : (DirectSum.decompose 𝓐 a j : A)=0 := by
-      by_cases hjzero : j=0
-      · subst j; exact ha
-      · have hjneg : j<0 := by omega
-        simpa [hneg j hjneg] using (DirectSum.decompose 𝓐 a j).property
-    rw [hz]
-    exact Ideal.zero_mem _
-
-
-theorem exists_homogeneous_basis_connected
-    [IsDomain A] [IsNoetherianRing A] [Module.Projective A M] [Module.Finite A M]
-    (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M)
-    [DirectSum.Decomposition 𝓐] [DirectSum.Decomposition 𝓜]
-    [SetLike.GradedMonoid 𝓐] [SetLike.GradedSMul 𝓐 𝓜]
-    (hneg : ∀ j < 0, 𝓐 j = ⊥)
-    (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
-    (d₀ : ℤ) (hbound : ∀ j < d₀, 𝓜 j = ⊥) :
-    ∃ (ι : Type um), Finite ι ∧ ∃ b : Module.Basis ι A M,
-      ∀ i, ∃ d : ℤ, b i ∈ 𝓜 d := by
-  apply exists_homogeneous_basis_of_scalar_mod 𝓐 𝓜 d₀ hbound
-  · rw [positiveIdeal_eq_augmentation_ker 𝓐 hneg]
-    exact RingHom.ker_ne_top _
-  · intro a
-    obtain ⟨c,hc⟩ := hzero _ (DirectSum.decompose 𝓐 a 0).property
-    refine ⟨c,?_⟩
-    rw [positiveIdeal_eq_augmentation_ker 𝓐 hneg]
-    change component 𝓐 0 (a-algebraMap k A c)=0
-    rw [map_sub,hc]
-    rw [show component 𝓐 0 (DirectSum.decompose 𝓐 a 0 : A) =
-      (DirectSum.decompose 𝓐 a 0 : A) from
-      DirectSum.decompose_of_mem_same 𝓐 (DirectSum.decompose 𝓐 a 0).property]
-    exact sub_self _
-
-/-- The actual polynomial grading, with each variable in degree one, on
-integer degrees as used for shifted stalks. -/
-abbrev polynomialGrading (k : Type*) [Field k] (σ : Type*) :=
-  MvPolynomial.weightedHomogeneousSubmodule k (fun _ : σ => (1 : ℤ))
-
-noncomputable instance polynomialGrading_decomposition (k : Type*) [Field k] (σ : Type*) :
-    DirectSum.Decomposition (polynomialGrading k σ) :=
-  MvPolynomial.weightedDecomposition k (fun _ : σ => (1 : ℤ))
-
-instance polynomialGrading_monoid (k : Type*) [Field k] (σ : Type*) :
-    SetLike.GradedMonoid (polynomialGrading k σ) :=
-  MvPolynomial.WeightedHomogeneousSubmodule.gradedMonoid
-
-theorem polynomialGrading_negative (k : Type*) [Field k] (σ : Type*)
-    (j : ℤ) (hj : j < 0) : polynomialGrading k σ j = ⊥ := by
-  classical
-  apply eq_bot_iff.mpr
-  intro p hp
-  change p=0
-  apply MvPolynomial.ext
-  intro d
-  by_contra hd
-  have heq := hp (show p.coeff d≠0 by simpa using hd)
-  have hn : 0 ≤ Finsupp.weight (fun _ : σ => (1 : ℤ)) d := by
-    simp only [Finsupp.weight_apply,Finsupp.sum]
-    exact Finset.sum_nonneg fun i _ => nsmul_nonneg (by norm_num) _
-  omega
-
-theorem polynomialGrading_zero (k : Type*) [Field k] (σ : Type*)
-    (p : MvPolynomial σ k) (hp : p ∈ polynomialGrading k σ 0) :
-    ∃ c : k, algebraMap k (MvPolynomial σ k) c=p := by
-  classical
-  refine ⟨p.coeff 0,?_⟩
-  apply MvPolynomial.ext
-  intro d
-  by_cases hd : d=0
-  · subst d; simp
-  · have hc : p.coeff d=0 := by
-      by_contra hc
-      have heq := hp hc
-      obtain ⟨i,hi⟩ := Finsupp.ne_iff.mp hd
-      have hb := Finsupp.le_weight_of_ne_zero (M := ℤ)
-        (w := fun _ : σ => (1 : ℤ)) (fun _ => by norm_num)
-        (show d i≠0 by simpa using hi)
-      omega
-    simp [MvPolynomial.coeff_C,Ne.symm hd,hc]
-
-
-theorem exists_homogeneous_basis_polynomial {σ : Type*} [Finite σ]
-    [Module (MvPolynomial σ k) M] [IsScalarTower k (MvPolynomial σ k) M]
-    [Module.Projective (MvPolynomial σ k) M] [Module.Finite (MvPolynomial σ k) M]
-    (𝓜 : ℤ → Submodule k M) [DirectSum.Decomposition 𝓜]
-    [SetLike.GradedSMul (polynomialGrading k σ) 𝓜]
-    (d₀ : ℤ) (hbound : ∀ j < d₀, 𝓜 j = ⊥) :
-    ∃ (ι : Type um), Finite ι ∧ ∃ b : Module.Basis ι (MvPolynomial σ k) M,
-      ∀ i, ∃ d : ℤ, b i ∈ 𝓜 d := by
-  exact exists_homogeneous_basis_connected (polynomialGrading k σ) 𝓜
-    (polynomialGrading_negative k σ) (polynomialGrading_zero k σ) d₀ hbound
-
-end KLInvariance.Graded
-
-
-namespace KLInvariance.ReverseFiltration
-universe ua up ui
-variable {A : Type ua} [CommRing A] [klPreservedInstance10 : IsDomain A]
-noncomputable section
-
-def scalarQuotientMap (a : A) (ι : Type ui) :
-    (ι → A) →ₗ[A] (ι → A ⧸ Ideal.span {a}) where
-  toFun v i := Ideal.Quotient.mk _ (v i)
-  map_add' _ _ := by ext i; simp
-  map_smul' _ _ := by ext i; simp [Algebra.smul_def]
-
-include klPreservedInstance10 in
-theorem scalarQuotientMap_surjective (a : A) (ι : Type ui) :
-    Function.Surjective (scalarQuotientMap a ι) := by
-  have _ := klPreservedInstance10
-  intro v
-  choose w hw using fun i => Ideal.Quotient.mk_surjective (v i)
-  exact ⟨w,funext hw⟩
-
-/-- Multiplication by the actual nonzero label identifies the kernel of
-coordinate reduction with a free module. -/
-def scalarKernelMap (a : A) (ι : Type ui) :
-    (ι → A) →ₗ[A] LinearMap.ker (scalarQuotientMap a ι) where
-  toFun v := ⟨fun i => a*v i, by
-    ext i
-    change Ideal.Quotient.mk (Ideal.span {a}) (a*v i)=0
-    apply Ideal.Quotient.eq_zero_iff_mem.mpr
-    exact Ideal.mem_span_singleton.mpr (dvd_mul_right a (v i))⟩
-  map_add' v w := by apply Subtype.ext; ext i; exact mul_add _ _ _
-  map_smul' r v := by apply Subtype.ext; ext i; change a*(r*v i)=r*(a*v i); ring
-
-theorem scalarKernelMap_bijective (a : A) (ha : a≠0) (ι : Type ui) :
-    Function.Bijective (scalarKernelMap a ι) := by
-  constructor
-  · intro v w hvw
-    funext i
-    exact mul_left_cancel₀ ha (congrFun (congrArg Subtype.val hvw) i)
-  · intro v
-    have hv (i : ι) : a ∣ v.val i := by
-      apply Ideal.mem_span_singleton.mp
-      apply Ideal.Quotient.eq_zero_iff_mem.mp
-      exact congrFun v.property i
-    choose w hw using hv
-    refine ⟨w,Subtype.ext (funext fun i => (hw i).symm)⟩
-
-theorem scalarKernel_projective (a : A) (ha : a≠0) (ι : Type ui) [Finite ι] :
-    Module.Projective A (LinearMap.ker (scalarQuotientMap a ι)) := by
-  exact Module.Projective.of_equiv
-    (LinearEquiv.ofBijective (scalarKernelMap a ι) (scalarKernelMap_bijective a ha ι))
-
-/-- Every finite free module over A/(a) has an actual free A-presentation
-with projective kernel; no projective-dimension hypothesis is inserted. -/
-theorem exists_scalar_presentation (a : A) (ha : a≠0)
-    (P : Type up) [AddCommGroup P] [Module A P]
-    [Module (A ⧸ Ideal.span {a}) P] [IsScalarTower A (A ⧸ Ideal.span {a}) P]
-    [Module.Free (A ⧸ Ideal.span {a}) P] [Module.Finite (A ⧸ Ideal.span {a}) P] :
-    ∃ (ι : Type up), Finite ι ∧ ∃ g : (ι → A) →ₗ[A] P,
-      Function.Surjective g ∧ Module.Projective A (LinearMap.ker g) := by
-  classical
-  let C := A ⧸ Ideal.span {a}
-  let b := Module.Free.chooseBasis C P
-  let ι := Module.Free.ChooseBasisIndex C P
-  let : Fintype ι := Fintype.ofFinite ι
-  let e : P ≃ₗ[A] (ι → C) := b.equivFun.restrictScalars A
-  let g : (ι → A) →ₗ[A] P := e.symm.toLinearMap.comp (scalarQuotientMap a ι)
-  refine ⟨ι,inferInstance,g,e.symm.surjective.comp (scalarQuotientMap_surjective a ι),?_⟩
-  have hker : LinearMap.ker g = LinearMap.ker (scalarQuotientMap a ι) := by
-    ext v
-    change e.symm (scalarQuotientMap a ι v)=0 ↔ scalarQuotientMap a ι v=0
-    exact e.symm.map_eq_zero_iff
-  rw [hker]
-  exact scalarKernel_projective a ha ι
-
-/-- The reverse-filtration kernel is projective when the edge image is
-actually finite free over the hypersurface quotient. -/
-theorem kernel_projective_of_free_scalar_quotient
-    (a : A) (ha : a≠0) {F : Type*} [AddCommGroup F] [Module A F]
-    [Module.Projective A F]
-    {P : Type up} [AddCommGroup P] [Module A P]
-    [Module (A ⧸ Ideal.span {a}) P] [IsScalarTower A (A ⧸ Ideal.span {a}) P]
-    [Module.Free (A ⧸ Ideal.span {a}) P] [Module.Finite (A ⧸ Ideal.span {a}) P]
-    (f : F →ₗ[A] P) (hf : Function.Surjective f) :
-    Module.Projective A (LinearMap.ker f) := by
-  obtain ⟨ι,hι,g,hg,hgker⟩ := exists_scalar_presentation a ha P
-  let : Finite ι := hι
-  let := hgker
-  exact kernel_projective_of_presentation f g hf hg
-
-end
-end KLInvariance.ReverseFiltration
-
-
-end
-
-section
-
-namespace KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {M : Type um} [AddCommGroup M] [Module k M] [Module A M] [IsScalarTower k A M]
-
-theorem exists_homogeneous_minimal_generators
-    [Module.Finite A M]
-    (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M)
-    [DirectSum.Decomposition 𝓐] [DirectSum.Decomposition 𝓜]
-    [SetLike.GradedSMul 𝓐 𝓜]
-    (d₀ : ℤ) (hbound : ∀ j < d₀, 𝓜 j = ⊥)
-    (hscalar : ∀ a : A, ∃ c : k, a-algebraMap k A c ∈ positiveIdeal 𝓐) :
-    ∃ (ι : Type um), Finite ι ∧ ∃ (v : ι → M) (d : ι → ℤ),
-      (∀ i, v i ∈ 𝓜 (d i)) ∧ Submodule.span A (Set.range v) = ⊤ ∧
-      LinearIndependent k (fun i => (positiveIdeal 𝓐 • (⊤ : Submodule A M)).mkQ (v i)) := by
-  classical
-  obtain ⟨t,ht,hth⟩ := exists_finite_homogeneous_generators (A := A) 𝓜
-  let v : t → M := Subtype.val
-  have hv : Submodule.span A (Set.range v) = ⊤ := by
-    simpa [v] using ht
-  let I := positiveIdeal 𝓐
-  let J := I • (⊤ : Submodule A M)
-  have hq := quotient_span_of_scalar_mod I hscalar v hv
-  obtain ⟨ι,a,ha,hsp,hli⟩ := exists_linearIndependent' k (fun i => J.mkQ (v i))
-  let : Finite ι := Finite.of_injective a ha
-  let : Fintype ι := Fintype.ofFinite ι
-  let w : ι → M := v ∘ a
-  have hw : ∀ i, ∃ d : ℤ, w i ∈ 𝓜 d := fun i => hth _ (a i).property
-  choose d hd using hw
-  have hqw : Submodule.span k (Set.range (fun i => J.mkQ (w i))) = ⊤ := hsp.trans hq
-  let N := Submodule.span A (Set.range w)
-  have hN : ∀ n m, m ∈ N → (DirectSum.decompose 𝓜 m n : M) ∈ N :=
-    homogeneous_span_components 𝓐 𝓜 w d hd
-  have hgen : N ⊔ I • (⊤ : Submodule A M) = ⊤ := by
-    apply top_unique
-    intro m _
-    obtain ⟨c,hc⟩ := (Submodule.mem_span_range_iff_exists_fun k).mp
-      (hqw ▸ Submodule.mem_top : J.mkQ m ∈
-        Submodule.span k (Set.range (fun i => J.mkQ (w i))))
-    let z : M := ∑ i, algebraMap k A (c i) • w i
-    have hz : z ∈ N := by
-      apply Submodule.sum_mem
-      intro i _
-      exact N.smul_mem _ (Submodule.subset_span (Set.mem_range_self i))
-    have hqz : J.mkQ z = J.mkQ m := by
-      change J.mkQ (∑ i, algebraMap k A (c i) • w i) = J.mkQ m
-      rw [map_sum]
-      convert hc using 1
-      apply Finset.sum_congr rfl
-      intro i _
-      exact (map_smul J.mkQ _ _).trans (IsScalarTower.algebraMap_smul A (c i) _)
-    have hdiff : m-z ∈ J := by
-      apply (Submodule.Quotient.mk_eq_zero J).mp
-      change J.mkQ (m-z)=0
-      rw [map_sub,hqz,sub_self]
-    exact Submodule.mem_sup.mpr ⟨z,hz,m-z,hdiff,by abel⟩
-  have hspan : N=⊤ := eq_top_of_positive_smul_sup 𝓐 𝓜 d₀ hbound N hN hgen
-  exact ⟨ι,inferInstance,w,d,hd,hspan,hli⟩
-
- theorem kernel_minimal_cover (I : Ideal A)
-    (hscalar : ∀ a : A, ∃ c : k, a-algebraMap k A c ∈ I)
-    {ι : Type*} [Fintype ι] (v : ι → M)
-    (hind : LinearIndependent k (fun i => (I • (⊤ : Submodule A M)).mkQ (v i))) :
-    LinearMap.ker (Fintype.linearCombination A v) ≤ I • (⊤ : Submodule A (ι → A)) := by
-  classical
-  intro f hf
-  let J := I • (⊤ : Submodule A M)
-  choose c hc using fun i => hscalar (f i)
-  have hz : ∑ i, c i • J.mkQ (v i) = 0 := by
-    have hm : ∑ i, f i • v i = 0 := LinearMap.mem_ker.mp hf
-    have hq := congrArg J.mkQ hm
-    rw [map_sum,map_zero] at hq
-    convert hq using 1
-    apply Finset.sum_congr rfl
-    intro i _
-    exact (quotient_smul_of_scalar_mod I (f i) (c i) (hc i) (v i)).symm
-  have hc0 : ∀ i, c i = 0 := (Fintype.linearIndependent_iff.mp hind) c hz
-  have hfi (i : ι) : f i ∈ I := by simpa [hc0 i] using hc i
-  have he : f = ∑ i, f i • Pi.single i (1 : A) := by
-    ext i
-    simp [Pi.single_apply]
-  rw [he]
-  apply Submodule.sum_mem
-  intro i _
-  exact Submodule.smul_mem_smul (hfi i) Submodule.mem_top
-
-end KLInvariance.Graded
-
-end
-
-
-section
-
-namespace KLInvariance.Graded
-universe uk ua um un
-variable {k : Type uk} [Field k]
-  {M : Type um} [AddCommGroup M] [Module k M]
-  {N : Type un} [AddCommGroup N] [Module k N]
-
- theorem internal_of_components
-    (𝓜 : ℤ → Submodule k M) (p : ℤ → M →ₗ[k] M)
-    (hmem : ∀ i m, p i m ∈ 𝓜 i)
-    (hsame : ∀ i m, m ∈ 𝓜 i → p i m = m)
-    (hne : ∀ i j m, i ≠ j → m ∈ 𝓜 j → p i m = 0)
-    (hsum : ∀ m, ∃ s : Finset ℤ, ∑ i ∈ s, p i m = m) :
-    DirectSum.IsInternal 𝓜 := by
-  classical
-  apply DirectSum.isInternal_submodule_of_iSupIndep_of_iSup_eq_top
-  · apply (iSupIndep_iff_finsetSum_eq_zero_imp_eq_zero _).mpr
-    intro s v hv hz i hi
-    have hp := congrArg (p i) hz
-    rw [map_sum, map_zero, Finset.sum_eq_single_of_mem i hi] at hp
-    · simpa only [hsame i _ (hv i hi)] using hp
-    · intro j hj hji
-      exact hne i j _ (Ne.symm hji) (hv j hj)
-  · apply top_unique
-    intro m _
-    obtain ⟨s,hs⟩ := hsum m
-    rw [← hs]
-    apply Submodule.sum_mem
-    intro i _
-    exact (le_iSup 𝓜 i) (hmem i m)
-
- theorem image_isInternal
-    (𝓜 : ℤ → Submodule k M) [DirectSum.Decomposition 𝓜]
-    (f : M →ₗ[k] N) (hf : Function.Surjective f)
-    (hker : ∀ i m, f m = 0 → f (component 𝓜 i m) = 0) :
-    DirectSum.IsInternal (fun i => (𝓜 i).map f) := by
-  classical
-  apply DirectSum.isInternal_submodule_of_iSupIndep_of_iSup_eq_top
-  · apply (iSupIndep_iff_finsetSum_eq_zero_imp_eq_zero _).mpr
-    intro s v hv hz i hi
-    choose w hw hfw using fun j : s => (Submodule.mem_map.mp (hv j.val j.property))
-    let w' : ℤ → M := fun j => if hj : j ∈ s then w ⟨j,hj⟩ else 0
-    have hw' (j) : w' j ∈ 𝓜 j := by
-      dsimp [w']; split_ifs with hj
-      · exact hw ⟨j,hj⟩
-      · exact Submodule.zero_mem _
-    have hfw' (j) (hj : j ∈ s) : f (w' j) = v j := by simpa [w',hj] using hfw ⟨j,hj⟩
-    have hsum : f (∑ j ∈ s, w' j) = 0 := by
-      rw [map_sum]
-      exact (Finset.sum_congr rfl hfw').trans hz
-    have hc := hker i _ hsum
-    rw [map_sum, Finset.sum_eq_single_of_mem i hi] at hc
-    · change f (DirectSum.decompose 𝓜 (w' i) i : M) = 0 at hc
-      rw [DirectSum.decompose_of_mem_same 𝓜 (hw' i), hfw' i hi] at hc
-      exact hc
-    · intro j _ hji
-      exact DirectSum.decompose_of_mem_ne 𝓜 (hw' j) hji
-  · apply top_unique
-    intro n _
-    obtain ⟨m,rfl⟩ := hf n
-    rw [← DirectSum.sum_support_decompose 𝓜 m, map_sum]
-    apply Submodule.sum_mem
-    intro i _
-    exact (le_iSup (fun i => (𝓜 i).map f) i)
-      (Submodule.mem_map.mpr ⟨_,(DirectSum.decompose 𝓜 m i).property,rfl⟩)
-
-variable {A : Type ua} [CommRing A] [Algebra k A]
-  [Module A M] [IsScalarTower k A M]
-
- def subPiece (𝓜 : ℤ → Submodule k M) (S : Submodule A M) : ℤ → Submodule k S :=
-  fun i => (𝓜 i).comap (S.subtype.restrictScalars k)
-
- theorem sub_isInternal
-    (𝓜 : ℤ → Submodule k M) [DirectSum.Decomposition 𝓜]
-    (S : Submodule A M)
-    (hS : ∀ i m, m ∈ S → component 𝓜 i m ∈ S) :
-    DirectSum.IsInternal (subPiece 𝓜 S) := by
-  classical
-  let p : ℤ → S →ₗ[k] S := fun i =>
-    LinearMap.codRestrict (S.restrictScalars k)
-      ((component 𝓜 i).comp (S.subtype.restrictScalars k)) (fun m => hS i m m.property)
-  apply internal_of_components (subPiece 𝓜 S) p
-  · intro i m
-    exact (DirectSum.decompose 𝓜 (m : M) i).property
-  · intro i m hm
-    apply Subtype.ext
-    exact DirectSum.decompose_of_mem_same 𝓜 hm
-  · intro i j m hij hm
-    apply Subtype.ext
-    exact DirectSum.decompose_of_mem_ne 𝓜 hm hij.symm
-  · intro m
-    refine ⟨(DirectSum.decompose 𝓜 (m : M)).support, ?_⟩
-    apply Subtype.ext
-    change (∑ i ∈ (DirectSum.decompose 𝓜 (m : M)).support, p i m : S).val = (m : M)
-    rw [Submodule.coe_sum]
-    exact DirectSum.sum_support_decompose 𝓜 (m : M)
-
- def quotientPiece (𝓜 : ℤ → Submodule k M) (S : Submodule A M) :
-    ℤ → Submodule k (M ⧸ S) := fun i => (𝓜 i).map (S.mkQ.restrictScalars k)
-
- theorem quotient_isInternal
-    (𝓜 : ℤ → Submodule k M) [DirectSum.Decomposition 𝓜]
-    (S : Submodule A M)
-    (hS : ∀ i m, m ∈ S → component 𝓜 i m ∈ S) :
-    DirectSum.IsInternal (quotientPiece 𝓜 S) := by
-  apply image_isInternal 𝓜 (S.mkQ.restrictScalars k) S.mkQ_surjective
-  intro i m hm
-  apply (Submodule.Quotient.mk_eq_zero S).mpr
-  exact hS i m ((Submodule.Quotient.mk_eq_zero S).mp hm)
-
- instance sub_gradedSMul
-    (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M)
-    [SetLike.GradedSMul 𝓐 𝓜] (S : Submodule A M) :
-    SetLike.GradedSMul 𝓐 (subPiece 𝓜 S) where
-  smul_mem := by
-    intro i j a m ha hm
-    change a • (m : M) ∈ 𝓜 (i+j)
-    exact SetLike.GradedSMul.smul_mem ha (show (m : M) ∈ 𝓜 j from hm)
-
- instance quotient_gradedSMul
-    (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M)
-    [SetLike.GradedSMul 𝓐 𝓜] (S : Submodule A M) :
-    SetLike.GradedSMul 𝓐 (quotientPiece 𝓜 S) where
-  smul_mem := by
-    intro i j a m ha hm
-    obtain ⟨v,hv,rfl⟩ := Submodule.mem_map.mp hm
-    exact Submodule.mem_map.mpr ⟨a • v,SetLike.GradedSMul.smul_mem ha hv,map_smul S.mkQ a v⟩
-
- theorem subPiece_eq_bot (𝓜 : ℤ → Submodule k M) (S : Submodule A M)
-    (i : ℤ) (h : 𝓜 i = ⊥) : subPiece 𝓜 S i = ⊥ := by
-  apply le_antisymm _ bot_le
-  intro m hm
-  apply Subtype.ext
-  exact (Submodule.mem_bot k).mp (h ▸ hm)
-
- theorem quotientPiece_eq_bot (𝓜 : ℤ → Submodule k M) (S : Submodule A M)
-    (i : ℤ) (h : 𝓜 i = ⊥) : quotientPiece 𝓜 S i = ⊥ := by
-  simp [quotientPiece,h]
-
- def shiftedPiece {ι : Type*} (𝓐 : ℤ → Submodule k A) (d : ι → ℤ) :
-    ℤ → Submodule k (ι → A) := fun n =>
-  { carrier := {v | ∀ i, v i ∈ 𝓐 (n-d i)}
-    zero_mem' := fun _ => Submodule.zero_mem _
-    add_mem' := fun hx hy i => Submodule.add_mem _ (hx i) (hy i)
-    smul_mem' := fun c _ hx i => Submodule.smul_mem _ c (hx i) }
-
- theorem shifted_isInternal {ι : Type*} [Fintype ι]
-    (𝓐 : ℤ → Submodule k A) [DirectSum.Decomposition 𝓐] (d : ι → ℤ) :
-    DirectSum.IsInternal (shiftedPiece 𝓐 d) := by
-  classical
-  let p : ℤ → (ι → A) →ₗ[k] (ι → A) := fun n =>
-    LinearMap.pi fun i => (component 𝓐 (n-d i)).comp (LinearMap.proj i)
-  apply internal_of_components (shiftedPiece 𝓐 d) p
-  · intro n v i
-    exact (DirectSum.decompose 𝓐 (v i) (n-d i)).property
-  · intro n v hv
-    funext i
-    exact DirectSum.decompose_of_mem_same 𝓐 (hv i)
-  · intro n j v hnj hv
-    funext i
-    apply DirectSum.decompose_of_mem_ne 𝓐 (hv i)
-    omega
-  · intro v
-    let s : ι → Finset ℤ := fun i =>
-      (DirectSum.decompose 𝓐 (v i)).support.image (fun n => n+d i)
-    refine ⟨Finset.univ.biUnion s, ?_⟩
-    funext i
-    change (∑ n ∈ Finset.univ.biUnion s, p n v) i = v i
-    rw [Finset.sum_apply]
-    have hsub : s i ⊆ Finset.univ.biUnion s :=
-      Finset.subset_biUnion_of_mem s (Finset.mem_univ i)
-    have hout (n : ℤ) (_ : n ∈ Finset.univ.biUnion s) (hn : n ∉ s i) :
-        p n v i = 0 := by
-      have hni : n-d i ∉ (DirectSum.decompose 𝓐 (v i)).support := by
-        intro hni
-        exact hn (Finset.mem_image.mpr ⟨n-d i,hni,by omega⟩)
-      have hz := DFinsupp.notMem_support_iff.mp hni
-      exact congrArg Subtype.val hz
-    rw [← Finset.sum_subset hsub hout]
-    change ∑ n ∈ (DirectSum.decompose 𝓐 (v i)).support.image (fun n => n+d i),
-      component 𝓐 (n-d i) (v i) = v i
-    rw [Finset.sum_image (fun a _ b _ h => by omega)]
-    simp only [add_sub_cancel_right]
-    exact DirectSum.sum_support_decompose 𝓐 (v i)
-
- instance shifted_gradedSMul {ι : Type*}
-    (𝓐 : ℤ → Submodule k A) [SetLike.GradedMonoid 𝓐] (d : ι → ℤ) :
-    SetLike.GradedSMul 𝓐 (shiftedPiece 𝓐 d) where
-  smul_mem := by
-    intro i j a m ha hm t
-    have h := SetLike.GradedMul.mul_mem ha (hm t)
-    simpa [sub_eq_add_neg,add_assoc] using h
-
- theorem shiftedPiece_eq_bot {ι : Type*}
-    (𝓐 : ℤ → Submodule k A) (hneg : ∀ i < 0, 𝓐 i = ⊥) (d : ι → ℤ)
-    (n : ℤ) (h : ∀ i, n < d i) : shiftedPiece 𝓐 d n = ⊥ := by
-  apply le_antisymm _ bot_le
-  intro m hm
-  funext i
-  have hi := hm i
-  rw [hneg _ (by have := h i; omega)] at hi
-  exact hi
-
-end KLInvariance.Graded
-
-end
-
-
-section
-
-namespace KLInvariance.Graded
-universe uk ua um un ui
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {M : Type um} [AddCommGroup M] [Module k M] [Module A M] [IsScalarTower k A M]
-  {N : Type un} [AddCommGroup N] [Module k N] [Module A N] [IsScalarTower k A N]
-
-omit [Algebra k A] [IsScalarTower k A M] [IsScalarTower k A N] in
- theorem map_component
-    (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
-    [DirectSum.Decomposition 𝓜] [DirectSum.Decomposition 𝓝]
-    (f : M →ₗ[A] N) (d : ℤ)
-    (hf : ∀ i m, m ∈ 𝓜 i → f m ∈ 𝓝 (i+d)) (n : ℤ) (m : M) :
-    component 𝓝 n (f m) = f (component 𝓜 (n-d) m) := by
-  classical
-  refine DirectSum.Decomposition.inductionOn 𝓜 ?_ ?_ ?_ m
-  · simp
-  · intro j v
-    by_cases h : j+d=n
-    · have h' : n-d=j := by omega
-      change (DirectSum.decompose 𝓝 (f (v : M)) n : N) =
-        f (DirectSum.decompose 𝓜 (v : M) (n-d) : M)
-      rw [← h, DirectSum.decompose_of_mem_same 𝓝 (hf j v v.property),
-        show j+d-d=j by omega, DirectSum.decompose_of_mem_same 𝓜 v.property]
-    · have h' : j ≠ n-d := by omega
-      change (DirectSum.decompose 𝓝 (f (v : M)) n : N) =
-        f (DirectSum.decompose 𝓜 (v : M) (n-d) : M)
-      rw [DirectSum.decompose_of_mem_ne 𝓝 (hf j v v.property) h,
-        DirectSum.decompose_of_mem_ne 𝓜 v.property h', map_zero]
-  · intro x y hx hy
-    simp only [map_add,hx,hy]
-
-omit [Algebra k A] [IsScalarTower k A M] [IsScalarTower k A N] in
- theorem kernel_homogeneous
-    (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
-    [DirectSum.Decomposition 𝓜] [DirectSum.Decomposition 𝓝]
-    (f : M →ₗ[A] N) (d : ℤ)
-    (hf : ∀ i m, m ∈ 𝓜 i → f m ∈ 𝓝 (i+d)) :
-    ∀ n m, m ∈ LinearMap.ker f → component 𝓜 n m ∈ LinearMap.ker f := by
-  intro n m hm
-  change f (component 𝓜 n m)=0
-  have h := map_component 𝓜 𝓝 f d hf (n+d) m
-  simpa [LinearMap.mem_ker.mp hm] using h.symm
-
-omit [Algebra k A] [IsScalarTower k A M] [IsScalarTower k A N] in
- theorem range_homogeneous
-    (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
-    [DirectSum.Decomposition 𝓜] [DirectSum.Decomposition 𝓝]
-    (f : M →ₗ[A] N) (d : ℤ)
-    (hf : ∀ i m, m ∈ 𝓜 i → f m ∈ 𝓝 (i+d)) :
-    ∀ n m, m ∈ LinearMap.range f → component 𝓝 n m ∈ LinearMap.range f := by
-  rintro n _ ⟨m,rfl⟩
-  exact ⟨component 𝓜 (n-d) m,(map_component 𝓜 𝓝 f d hf n m).symm⟩
-
-omit [IsScalarTower k A M] in
- theorem component_homogeneous_smul
-    (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M)
-    [DirectSum.Decomposition 𝓜] [SetLike.GradedSMul 𝓐 𝓜]
-    (d : ℤ) (a : A) (ha : a ∈ 𝓐 d) (n : ℤ) (m : M) :
-    component 𝓜 n (a • m) = a • component 𝓜 (n-d) m := by
-  apply map_component 𝓜 𝓜 (LinearMap.lsmul A M a) d
-  intro i v hv
-  simpa [add_comm] using SetLike.GradedSMul.smul_mem ha hv
-
-omit [IsScalarTower k A M] in
- theorem scalar_range_homogeneous
-    (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M)
-    [DirectSum.Decomposition 𝓜] [SetLike.GradedSMul 𝓐 𝓜]
-    (d : ℤ) (a : A) (ha : a ∈ 𝓐 d) :
-    ∀ n m, m ∈ LinearMap.range (LinearMap.lsmul A M a) →
-      component 𝓜 n m ∈ LinearMap.range (LinearMap.lsmul A M a) := by
-  apply range_homogeneous 𝓜 𝓜 (LinearMap.lsmul A M a) d
-  intro i v hv
-  simpa [add_comm] using SetLike.GradedSMul.smul_mem ha hv
-
- theorem exists_graded_minimal_free_cover
-    [Module.Finite A M]
-    (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M)
-    [DirectSum.Decomposition 𝓐] [DirectSum.Decomposition 𝓜]
-    [SetLike.GradedSMul 𝓐 𝓜]
-    (d₀ : ℤ) (hbound : ∀ j < d₀, 𝓜 j = ⊥)
-    (hscalar : ∀ a : A, ∃ c : k, a-algebraMap k A c ∈ positiveIdeal 𝓐) :
-    ∃ (ι : Type um), Finite ι ∧ ∃ (d : ι → ℤ) (f : (ι → A) →ₗ[A] M),
-      Function.Surjective f ∧
-      (∀ n v, v ∈ shiftedPiece 𝓐 d n → f v ∈ 𝓜 n) ∧
-      LinearMap.ker f ≤ positiveIdeal 𝓐 • (⊤ : Submodule A (ι → A)) := by
-  classical
-  obtain ⟨ι,hι,v,d,hd,hspan,hind⟩ :=
-    exists_homogeneous_minimal_generators 𝓐 𝓜 d₀ hbound hscalar
-  let : Finite ι := hι
-  let : Fintype ι := Fintype.ofFinite ι
-  refine ⟨ι,hι,d,Fintype.linearCombination A v,?_,?_,?_⟩
-  · intro m
-    exact (Submodule.mem_span_range_iff_exists_fun A).mp (hspan ▸ Submodule.mem_top)
-  · intro n c hc
-    apply Submodule.sum_mem
-    intro i _
-    simpa using SetLike.GradedSMul.smul_mem (hc i) (hd i)
-  · exact kernel_minimal_cover _ hscalar v hind
-
-section Pi
-variable {ι : Type ui} {P : ι → Type um}
-  [∀ i, AddCommGroup (P i)] [∀ i, Module k (P i)]
-
- def piPiece (𝓟 : ∀ i, ℤ → Submodule k (P i)) : ℤ → Submodule k (∀ i, P i) :=
-  fun n => Submodule.pi Set.univ (fun i => 𝓟 i n)
-
- theorem pi_isInternal [Fintype ι]
-    (𝓟 : ∀ i, ℤ → Submodule k (P i)) [∀ i, DirectSum.Decomposition (𝓟 i)] :
-    DirectSum.IsInternal (piPiece 𝓟) := by
-  classical
-  let p : ℤ → (∀ i, P i) →ₗ[k] (∀ i, P i) := fun n =>
-    LinearMap.pi fun i => (component (𝓟 i) n).comp (LinearMap.proj i)
-  apply internal_of_components (piPiece 𝓟) p
-  · intro n v i _
-    exact (DirectSum.decompose (𝓟 i) (v i) n).property
-  · intro n v hv
-    funext i
-    exact DirectSum.decompose_of_mem_same (𝓟 i) (hv i (Set.mem_univ i))
-  · intro n j v hnj hv
-    funext i
-    exact DirectSum.decompose_of_mem_ne (𝓟 i) (hv i (Set.mem_univ i)) hnj.symm
-  · intro v
-    let s : ι → Finset ℤ := fun i => (DirectSum.decompose (𝓟 i) (v i)).support
-    refine ⟨Finset.univ.biUnion s, ?_⟩
-    funext i
-    rw [Finset.sum_apply]
-    have hsub : s i ⊆ Finset.univ.biUnion s :=
-      Finset.subset_biUnion_of_mem s (Finset.mem_univ i)
-    have hout (n : ℤ) (_ : n ∈ Finset.univ.biUnion s) (hn : n ∉ s i) :
-        p n v i = 0 := by
-      change n ∉ (DirectSum.decompose (𝓟 i) (v i)).support at hn
-      have hz := DFinsupp.notMem_support_iff.mp hn
-      exact congrArg Subtype.val hz
-    rw [← Finset.sum_subset hsub hout]
-    exact DirectSum.sum_support_decompose (𝓟 i) (v i)
-
- instance pi_gradedSMul [∀ i, Module A (P i)] [∀ i, IsScalarTower k A (P i)]
-    (𝓐 : ℤ → Submodule k A) (𝓟 : ∀ i, ℤ → Submodule k (P i))
-    [∀ i, SetLike.GradedSMul 𝓐 (𝓟 i)] : SetLike.GradedSMul 𝓐 (piPiece 𝓟) where
-  smul_mem := by
-    intro n j a v ha hv i hi
-    exact SetLike.GradedSMul.smul_mem ha (hv i hi)
-
- theorem piPiece_eq_bot
-    (𝓟 : ∀ i, ℤ → Submodule k (P i)) (n : ℤ) (h : ∀ i, 𝓟 i n = ⊥) :
-    piPiece 𝓟 n = ⊥ := by
-  apply le_antisymm _ bot_le
-  intro v hv
-  funext i
-  have hi : v i ∈ 𝓟 i n := hv i (Set.mem_univ i)
-  rw [h i] at hi
-  exact hi
-
-end Pi
-end KLInvariance.Graded
-
-end
-
-
-section
-
-namespace KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-
-/-- Actual finite nonnegatively graded modules; no character or duality property
-is folded into this data. -/
-structure ModuleData (𝓐 : ℤ → Submodule k A) where
-  obj : ModuleCat.{max ua um} A
-  kModule : Module k obj
-  tower : IsScalarTower k A obj
-  piece : ℤ → Submodule k obj
-  decomposition : DirectSum.Decomposition piece
-  graded : SetLike.GradedSMul 𝓐 piece
-  nonneg : ∀ n < 0, piece n = ⊥
-  finite : Module.Finite A obj
-
-attribute [instance] ModuleData.kModule ModuleData.tower ModuleData.decomposition
-  ModuleData.graded ModuleData.finite
-
-instance {𝓐 : ℤ → Submodule k A} : CoeSort (ModuleData.{uk,ua,um} 𝓐) (Type (max ua um)) :=
-  ⟨fun M => M.obj⟩
-
-namespace ModuleData
-variable {𝓐 : ℤ → Submodule k A}
-
-instance (M : ModuleData.{uk,ua,um} 𝓐) : AddCommGroup M := inferInstanceAs (AddCommGroup M.obj)
-instance (M : ModuleData.{uk,ua,um} 𝓐) : Module A M := inferInstanceAs (Module A M.obj)
-
-structure Hom (M N : ModuleData.{uk,ua,um} 𝓐) where
-  map : M →ₗ[A] N
-  graded : ∀ n m, m ∈ M.piece n → map m ∈ N.piece n
-
-noncomputable def sub [IsNoetherianRing A] (M : ModuleData.{uk,ua,um} 𝓐)
-    (S : Submodule A M) (hS : ∀ n m, m ∈ S → component M.piece n m ∈ S) :
-    ModuleData.{uk,ua,um} 𝓐 where
-  obj := ModuleCat.of A S
-  kModule := inferInstance
-  tower := inferInstance
-  piece := subPiece M.piece S
-  decomposition := (sub_isInternal M.piece S hS).chooseDecomposition
-  graded := inferInstance
-  nonneg := fun n hn => subPiece_eq_bot M.piece S n (M.nonneg n hn)
-  finite := inferInstance
-
-noncomputable def quotient (M : ModuleData.{uk,ua,um} 𝓐)
-    (S : Submodule A M) (hS : ∀ n m, m ∈ S → component M.piece n m ∈ S) :
-    ModuleData.{uk,ua,um} 𝓐 where
-  obj := ModuleCat.of A (M ⧸ S)
-  kModule := inferInstance
-  tower := inferInstance
-  piece := quotientPiece M.piece S
-  decomposition := (quotient_isInternal M.piece S hS).chooseDecomposition
-  graded := inferInstance
-  nonneg := fun n hn => quotientPiece_eq_bot M.piece S n (M.nonneg n hn)
-  finite := inferInstance
-
-noncomputable def pi {ι : Type um} [Fintype ι] (M : ι → ModuleData.{uk,ua,um} 𝓐) :
-    ModuleData.{uk,ua,um} 𝓐 where
-  obj := ModuleCat.of A (∀ i, M i)
-  kModule := inferInstance
-  tower := inferInstance
-  piece := piPiece fun i => (M i).piece
-  decomposition := (pi_isInternal (fun i => (M i).piece)).chooseDecomposition
-  graded := inferInstance
-  nonneg := fun n hn => piPiece_eq_bot (fun i => (M i).piece) n (fun i => (M i).nonneg n hn)
-  finite := inferInstance
-
-noncomputable def kernel [IsNoetherianRing A] {M N : ModuleData.{uk,ua,um} 𝓐}
-    (f : Hom M N) : ModuleData.{uk,ua,um} 𝓐 :=
-  sub M f.map.ker (kernel_homogeneous M.piece N.piece f.map 0
-    (by simpa using f.graded))
-
-noncomputable def range [IsNoetherianRing A] {M N : ModuleData.{uk,ua,um} 𝓐}
-    (f : Hom M N) : ModuleData.{uk,ua,um} 𝓐 :=
-  sub N f.map.range (range_homogeneous M.piece N.piece f.map 0
-    (by simpa using f.graded))
-
-noncomputable def scalarQuotient (M : ModuleData.{uk,ua,um} 𝓐)
-    (a : A) (d : ℤ) (ha : a ∈ 𝓐 d) : ModuleData.{uk,ua,um} 𝓐 :=
-  quotient M (LinearMap.range (LinearMap.lsmul A M a))
-    (scalar_range_homogeneous 𝓐 M.piece d a ha)
-
- def projection {ι : Type um} [Fintype ι] (M : ι → ModuleData.{uk,ua,um} 𝓐) (i : ι) :
-    Hom (pi M) (M i) where
-  map := LinearMap.proj i
-  graded := fun _ _ hm => hm i (Set.mem_univ i)
-
- def quotientMap (M : ModuleData.{uk,ua,um} 𝓐)
-    (S : Submodule A M) (hS : ∀ n m, m ∈ S → component M.piece n m ∈ S) :
-    Hom M (quotient M S hS) where
-  map := S.mkQ
-  graded := by
-    intro n m hm
-    change S.mkQ m ∈ quotientPiece M.piece S n
-    exact Submodule.mem_map.mpr ⟨m,hm,rfl⟩
-
- def subtype [IsNoetherianRing A] (M : ModuleData.{uk,ua,um} 𝓐)
-    (S : Submodule A M) (hS : ∀ n m, m ∈ S → component M.piece n m ∈ S) :
-    Hom (sub M S hS) M where
-  map := S.subtype
-  graded := fun _ _ hm => hm
-
-
- def Hom.comp {M N P : ModuleData.{uk,ua,um} 𝓐} (g : Hom N P) (f : Hom M N) :
-    Hom M P where
-  map := g.map.comp f.map
-  graded := fun n m hm => g.graded n _ (f.graded n m hm)
-
- def Hom.id (M : ModuleData.{uk,ua,um} 𝓐) : Hom M M where
-  map := LinearMap.id
-  graded := fun _ _ h => h
-
- theorem Hom.component {M N : ModuleData.{uk,ua,um} 𝓐} (f : Hom M N) (n : ℤ) (m : M) :
-    component N.piece n (f.map m) = f.map (component M.piece n m) := by
-  simpa using map_component M.piece N.piece f.map 0 (by simpa using f.graded) n m
-
-variable [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-
-noncomputable def free {ι : Type (max ua um)} [Fintype ι]
-    (hneg : ∀ n < 0, 𝓐 n = ⊥) (d : ι → ℤ) (hd : ∀ i, 0 ≤ d i) :
-    ModuleData.{uk,ua,um} 𝓐 where
-  obj := ModuleCat.of A (ι → A)
-  kModule := inferInstance
-  tower := inferInstance
-  piece := shiftedPiece 𝓐 d
-  decomposition := (shifted_isInternal 𝓐 d).chooseDecomposition
-  graded := inferInstance
-  nonneg := fun n hn => shiftedPiece_eq_bot 𝓐 hneg d n (fun i => lt_of_lt_of_le hn (hd i))
-  finite := inferInstance
-
- structure MinimalCover (M : ModuleData.{uk,ua,um} 𝓐) where
-  source : ModuleData.{uk,ua,um} 𝓐
-  free : Module.Free A source
-  hom : Hom source M
-  surjective : Function.Surjective hom.map
-  minimal : LinearMap.ker hom.map ≤ positiveIdeal 𝓐 • (⊤ : Submodule A source)
-
- theorem exists_minimalCover
-    (hneg : ∀ n < 0, 𝓐 n = ⊥)
-    (hscalar : ∀ a : A, ∃ c : k, a-algebraMap k A c ∈ positiveIdeal 𝓐)
-    (M : ModuleData.{uk,ua,um} 𝓐) : Nonempty (MinimalCover M) := by
-  classical
-  obtain ⟨ι,hι,v,d,hd,hspan,hind⟩ :=
-    exists_homogeneous_minimal_generators 𝓐 M.piece 0 M.nonneg hscalar
-  let : Finite ι := hι
-  let : Fintype ι := Fintype.ofFinite ι
-  have hd0 (i : ι) : 0 ≤ d i := by
-    by_contra h
-    have hbot := M.nonneg (d i) (lt_of_not_ge h)
-    have hv0 : v i = 0 := by simpa only [hbot, Submodule.mem_bot] using hd i
-    have hz := hind.ne_zero i
-    exact hz (by rw [hv0,map_zero])
-  let F := free hneg d hd0
-  let f : Hom F M :=
-    { map := Fintype.linearCombination A v
-      graded := by
-        intro n c hc
-        apply Submodule.sum_mem
-        intro i _
-        simpa using SetLike.GradedSMul.smul_mem (hc i) (hd i) }
-  have hfree : Module.Free A F := inferInstanceAs (Module.Free A (ι → A))
-  refine ⟨⟨F,hfree,f,?_,?_⟩⟩
-  · intro m
-    exact (Submodule.mem_span_range_iff_exists_fun A).mp (hspan ▸ Submodule.mem_top)
-  · exact kernel_minimal_cover _ hscalar v hind
-
-end ModuleData
-end KLInvariance.Graded
-
-end
-
-
-section
-
-/-! Honest nonnegative grading shifts; the underlying integral module is unchanged. -/
-namespace KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A}
-noncomputable section
-
- theorem shift_isInternal (M : ModuleData.{uk,ua,um} 𝓐) (d : ℤ) :
-    DirectSum.IsInternal (fun n => M.piece (n-d)) := by
-  classical
-  apply internal_of_components _ (fun n => component M.piece (n-d))
-  · intro n m
-    exact (DirectSum.decompose M.piece m (n-d)).property
-  · intro n m hm
-    exact DirectSum.decompose_of_mem_same M.piece hm
-  · intro n j m hnj hm
-    apply DirectSum.decompose_of_mem_ne M.piece hm
-    omega
-  · intro m
-    refine ⟨(DirectSum.decompose M.piece m).support.image (fun n => n+d),?_⟩
-    rw [Finset.sum_image (fun a _ b _ h => by omega)]
-    simp only [add_sub_cancel_right]
-    exact DirectSum.sum_support_decompose M.piece m
-
-namespace ModuleData
- def shift (M : ModuleData.{uk,ua,um} 𝓐) (d : ℕ) : ModuleData.{uk,ua,um} 𝓐 where
-  obj := M.obj
-  kModule := M.kModule
-  tower := M.tower
-  piece n := M.piece (n-d)
-  decomposition := (shift_isInternal M d).chooseDecomposition
-  graded := ⟨by
-    intro i j a m ha hm
-    change a • m ∈ M.piece (i+j-d)
-    have hh : a • m ∈ M.piece (i+(j-(d:ℤ))) := SetLike.GradedSMul.smul_mem ha hm
-    convert hh using 1
-    congr 1
-    omega⟩
-  nonneg n hn := M.nonneg (n-d) (by omega)
-  finite := M.finite
-
- def Hom.shift {M N : ModuleData.{uk,ua,um} 𝓐} (f : Hom M N) (d : ℕ) :
-    Hom (M.shift d) (N.shift d) where
-  map := f.map
-  graded n m hm := f.graded (n-d) m hm
-
- def prod (M N : ModuleData.{uk,ua,um} 𝓐) : ModuleData.{uk,ua,um} 𝓐 where
-  obj := ModuleCat.of A (M × N)
-  kModule := inferInstance
-  tower := inferInstance
-  piece n := (M.piece n).prod (N.piece n)
-  decomposition := (show DirectSum.IsInternal (fun n => (M.piece n).prod (N.piece n)) from by
-    apply internal_of_components _ (fun n =>
-      (component M.piece n).prodMap (component N.piece n))
-    · intro n m
-      exact ⟨(DirectSum.decompose M.piece m.1 n).property,
-        (DirectSum.decompose N.piece m.2 n).property⟩
-    · intro n m hm
-      exact Prod.ext (DirectSum.decompose_of_mem_same M.piece hm.1)
-        (DirectSum.decompose_of_mem_same N.piece hm.2)
-    · intro i j m hij hm
-      exact Prod.ext (DirectSum.decompose_of_mem_ne M.piece hm.1 (Ne.symm hij))
-        (DirectSum.decompose_of_mem_ne N.piece hm.2 (Ne.symm hij))
-    · intro m
-      classical
-      refine ⟨(DirectSum.decompose M.piece m.1).support ∪ (DirectSum.decompose N.piece m.2).support,?_⟩
-      apply Prod.ext
-      · change (∑ n ∈ _, (component M.piece n).prodMap (component N.piece n) m).1=m.1
-        rw [Prod.fst_sum]
-        rw [← Finset.sum_subset Finset.subset_union_left]
-        · exact DirectSum.sum_support_decompose M.piece m.1
-        · intro n _ hn
-          have hh : (DirectSum.decompose M.piece m.1) n=0 := DFinsupp.notMem_support_iff.mp hn
-          exact congrArg (fun z : M.piece n => (z : M)) hh
-      · change (∑ n ∈ _, (component M.piece n).prodMap (component N.piece n) m).2=m.2
-        rw [Prod.snd_sum]
-        rw [← Finset.sum_subset Finset.subset_union_right]
-        · exact DirectSum.sum_support_decompose N.piece m.2
-        · intro n _ hn
-          have hh : (DirectSum.decompose N.piece m.2) n=0 := DFinsupp.notMem_support_iff.mp hn
-          exact congrArg (fun z : N.piece n => (z : N)) hh).chooseDecomposition
-  graded := ⟨by
-    intro i j a m ha hm
-    exact ⟨SetLike.GradedSMul.smul_mem ha hm.1,SetLike.GradedSMul.smul_mem ha hm.2⟩⟩
-  nonneg n hn := by simp [M.nonneg n hn,N.nonneg n hn]
-  finite := inferInstance
-
- def Hom.prodMap {M N P Q : ModuleData.{uk,ua,um} 𝓐} (f : Hom M P) (g : Hom N Q) :
-    Hom (M.prod N) (P.prod Q) where
-  map := f.map.prodMap g.map
-  graded n _ hm := ⟨f.graded n _ hm.1,g.graded n _ hm.2⟩
-
-end ModuleData
-end
-end KLInvariance.Graded
-
-end
-
-
-section
-
-/-! Homogeneous projectivity and uniqueness of the actual minimal graded
-covers. These are recognition dependencies, not assumptions of the main. -/
-namespace KLInvariance.Graded.ModuleData
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A}
-
- theorem Hom.mem_of_map_mem {M N : ModuleData.{uk,ua,um} 𝓐} (f : Hom M N)
-    (hi : Function.Injective f.map) (n : ℤ) (m : M) (hm : f.map m ∈ N.piece n) :
-    m ∈ M.piece n := by
-  have h : f.map (Graded.component M.piece n m) = f.map m := by
-    rw [← f.component]
-    exact DirectSum.decompose_of_mem_same N.piece hm
-  rw [← hi h]
-  exact (DirectSum.decompose M.piece m n).property
-
- theorem Hom.homogeneous_preimage {M N : ModuleData.{uk,ua,um} 𝓐} (f : Hom M N)
-    (hs : Function.Surjective f.map) (n : ℤ) (v : N) (hv : v ∈ N.piece n) :
-    ∃ m : M, m ∈ M.piece n ∧ f.map m = v := by
-  obtain ⟨m,rfl⟩ := hs v
-  refine ⟨Graded.component M.piece n m,(DirectSum.decompose M.piece m n).property,?_⟩
-  rw [← f.component]
-  exact DirectSum.decompose_of_mem_same N.piece hv
-
-variable [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-  (hneg : ∀ n < 0, 𝓐 n = ⊥)
-
-include hneg in
- theorem exists_hom_lift_of_basis [Nontrivial A] {F N P : ModuleData.{uk,ua,um} 𝓐}
-    {ι : Type (max ua um)} [Fintype ι] (b : Module.Basis ι A F)
-    (d : ι → ℤ) (hd : ∀ i, b i ∈ F.piece (d i))
-    (f : Hom N P) (hs : Function.Surjective f.map) (g : Hom F P) :
-    ∃ l : Hom F N, f.map.comp l.map = g.map := by
-  classical
-  have hd0 (i : ι) : 0 ≤ d i := by
-    by_contra h
-    have hz : b i = 0 := by simpa [F.nonneg (d i) (lt_of_not_ge h)] using hd i
-    exact b.ne_zero i hz
-  let H := free.{uk,ua,um} hneg d hd0
-  let e : Hom H F :=
-    { map := b.equivFun.symm.toLinearMap
-      graded := by
-        intro n v hv
-        change ι → A at v
-        change v ∈ shiftedPiece 𝓐 d n at hv
-        change b.equivFun.symm v ∈ F.piece n
-        rw [b.equivFun_symm_apply]
-        apply Submodule.sum_mem
-        intro i _
-        simpa using SetLike.GradedSMul.smul_mem (hv i) (hd i) }
-  have he : Function.Injective e.map := b.equivFun.symm.injective
-  have hcoeff (n : ℤ) (v : F) (hv : v ∈ F.piece n) :
-      b.equivFun v ∈ shiftedPiece 𝓐 d n := by
-    apply e.mem_of_map_mem he n
-    change b.equivFun.symm (b.equivFun v) ∈ F.piece n
-    simpa only [LinearEquiv.symm_apply_apply] using hv
-  have hlifts (i : ι) : ∃ v : N, v ∈ N.piece (d i) ∧ f.map v = g.map (b i) :=
-    f.homogeneous_preimage hs (d i) _ (g.graded (d i) _ (hd i))
-  choose v hvg hvf using hlifts
-  let l : Hom F N :=
-    { map := b.constr A v
-      graded := by
-        intro n m hm
-        rw [b.constr_apply]
-        apply Submodule.sum_mem
-        intro i _
-        simpa using SetLike.GradedSMul.smul_mem (hcoeff n m hm i) (hvg i) }
-  refine ⟨l,?_⟩
-  apply b.ext
-  intro i
-  change f.map ((b.constr A v) (b i)) = g.map (b i)
-  rw [b.constr_basis,hvf]
-
-variable [IsDomain A] [IsNoetherianRing A]
-  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
-
-include hneg hzero in
- theorem exists_hom_lift {F N P : ModuleData.{uk,ua,um} 𝓐}
-    [Module.Free A F] (f : Hom N P) (hs : Function.Surjective f.map) (g : Hom F P) :
-    ∃ l : Hom F N, f.map.comp l.map = g.map := by
-  classical
-  obtain ⟨ι,hι,b,hd⟩ := exists_homogeneous_basis_connected 𝓐 F.piece hneg hzero 0 F.nonneg
-  let : Finite ι := hι
-  let : Fintype ι := Fintype.ofFinite ι
-  choose d hd using hd
-  exact exists_hom_lift_of_basis hneg b d hd f hs g
-
-omit [IsDomain A] [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐] hneg hzero in
- theorem Hom.bijective_of_mod_positive {M : ModuleData.{uk,ua,um} 𝓐} (f : Hom M M)
-    (hm : ∀ m, m-f.map m ∈ positiveIdeal 𝓐 • (⊤ : Submodule A M)) :
-    Function.Bijective f.map := by
-  have hr : LinearMap.range f.map = ⊤ :=
-    eq_top_of_positive_smul_sup 𝓐 M.piece 0 M.nonneg (LinearMap.range f.map)
-      (range_homogeneous M.piece M.piece f.map 0 (by simpa using f.graded))
-      (by
-        apply top_unique
-        intro m _
-        exact Submodule.mem_sup.mpr ⟨f.map m,⟨m,rfl⟩,m-f.map m,hm m,add_sub_cancel _ _⟩)
-  have hs := LinearMap.range_eq_top.mp hr
-  exact ⟨IsNoetherian.injective_of_surjective_endomorphism f.map hs,hs⟩
-
-include hneg hzero in
- theorem MinimalCover.equiv {M : ModuleData.{uk,ua,um} 𝓐} (C D : MinimalCover M) :
-    ∃ e : C.source ≃ₗ[A] D.source,
-      (∀ n v, v ∈ C.source.piece n → e v ∈ D.source.piece n) ∧
-      D.hom.map.comp e.toLinearMap = C.hom.map := by
-  let := C.free
-  let := D.free
-  obtain ⟨f,hf⟩ := exists_hom_lift hneg hzero D.hom D.surjective C.hom
-  obtain ⟨g,hg⟩ := exists_hom_lift hneg hzero C.hom C.surjective D.hom
-  have hfa (v : C.source) : D.hom.map (f.map v)=C.hom.map v := LinearMap.congr_fun hf v
-  have hga (v : D.source) : C.hom.map (g.map v)=D.hom.map v := LinearMap.congr_fun hg v
-  have hgc : Function.Bijective (g.comp f).map := by
-    apply Hom.bijective_of_mod_positive
-    intro m
-    apply C.minimal
-    change C.hom.map (m-g.map (f.map m))=0
-    rw [map_sub,hga,hfa,sub_self]
-  have hfd : Function.Bijective (f.comp g).map := by
-    apply Hom.bijective_of_mod_positive
-    intro m
-    apply D.minimal
-    change D.hom.map (m-f.map (g.map m))=0
-    rw [map_sub,hfa,hga,sub_self]
-  have hfb : Function.Bijective f.map :=
-    ⟨Function.Injective.of_comp hgc.1,Function.Surjective.of_comp hfd.2⟩
-  exact ⟨LinearEquiv.ofBijective f.map hfb,f.graded,hf⟩
-
-end KLInvariance.Graded.ModuleData
-
-end
-
-
-section
-
-
-end
-
-end OAI
+import OAI.RepresentationTheory.KazhdanLusztig.ModuleAlgebra
+
+namespace OAI
+
+/-!
+Graded polynomial modules, projective presentations, finite homogeneous generators and uniqueness of minimal covers.
+-/
+
+section
+
+namespace KLInvariance.Graded
+open scoped Pointwise
+universe uk ua um
+variable {k : Type uk} {A : Type ua} {M : Type um} [Field k] [CommRing A] [klPreservedInstance9 : Algebra k A]
+  [AddCommGroup M] [Module k M] [Module A M] [klPreservedInstance8 : IsScalarTower k A M]
+
+include klPreservedInstance8 in
+/-- Graded Nakayama in its generation form. The submodule is closed under
+actual homogeneous components, rather than assumed to be the whole module. -/
+theorem eq_top_of_positive_smul_sup
+    (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M)
+    [DirectSum.Decomposition 𝓜] [SetLike.GradedSMul 𝓐 𝓜]
+    (d₀ : ℤ) (hbound : ∀ j < d₀, 𝓜 j = ⊥)
+    (N : Submodule A M) (hN : ∀ n m, m ∈ N → (DirectSum.decompose 𝓜 m n : M) ∈ N)
+    (hgen : N ⊔ positiveIdeal 𝓐 • (⊤ : Submodule A M) = ⊤) : N = ⊤ := by
+  have _ := klPreservedInstance8
+  classical
+  have hlow : ∀ j < d₀, ∀ m ∈ 𝓜 j, m ∈ N := by
+    intro j hj m hm
+    have hm0 : m = 0 := by simpa [hbound j hj] using hm
+    simp [hm0]
+  have hind : ∀ n : ℕ, ∀ m ∈ 𝓜 (d₀+n), m ∈ N := by
+    intro n
+    induction n using Nat.strong_induction_on with
+    | h n ih =>
+      let p := component 𝓜 (d₀+n)
+      have hsmul (i : ℤ) (hi : 0 < i) (a : A) (ha : a ∈ 𝓐 i) :
+          ∀ m : M, p (a • m) ∈ N := by
+        intro m
+        refine DirectSum.Decomposition.inductionOn 𝓜 ?_ ?_ ?_ m
+        · simp
+        · intro j m
+          by_cases hj : j < d₀+n
+          · have hm : (m : M) ∈ N := by
+              by_cases hj₀ : j < d₀
+              · exact hlow j hj₀ m m.property
+              · have heq : d₀ + (j-d₀).toNat = j := by omega
+                exact ih (j-d₀).toNat (by omega) m (heq.symm ▸ m.property)
+            exact hN _ _ (N.smul_mem a hm)
+          · have hdeg : i+j ≠ d₀+n := by omega
+            have ham : a • (m : M) ∈ 𝓜 (i+j) :=
+              SetLike.GradedSMul.smul_mem ha m.property
+            have heq : p (a • (m : M)) = 0 :=
+              DirectSum.decompose_of_mem_ne 𝓜 ham hdeg
+            simp [heq]
+        · intro x y hx hy
+          simpa only [smul_add, map_add] using N.add_mem hx hy
+      let J : Ideal A :=
+        { carrier := {a | ∀ m : M, p (a • m) ∈ N}
+          zero_mem' := by simp [N.zero_mem]
+          add_mem' := by
+            intro a b ha hb m
+            simpa only [add_smul, map_add] using N.add_mem (ha m) (hb m)
+          smul_mem' := by
+            intro r a ha m
+            change p ((r*a) • m) ∈ N
+            rw [mul_comm r a, mul_smul]
+            exact ha (r • m) }
+      have hIJ : positiveIdeal 𝓐 ≤ J := by
+        apply Ideal.span_le.mpr
+        rintro a ⟨i,hi,ha⟩
+        exact hsmul i hi a ha
+      have hp : ∀ m : M, p m ∈ N := by
+        intro m
+        have hm : m ∈ N ⊔ positiveIdeal 𝓐 • (⊤ : Submodule A M) := by
+          rw [hgen]; trivial
+        obtain ⟨x,hx,y,hy,rfl⟩ := Submodule.mem_sup.mp hm
+        rw [map_add]
+        apply N.add_mem (hN _ _ hx)
+        refine Submodule.smul_induction_on hy ?_ ?_
+        · intro a ha z _; exact hIJ ha z
+        · intro z w hz hw; simpa only [map_add] using N.add_mem hz hw
+      intro m hm
+      have hcomp : p m=m := DirectSum.decompose_of_mem_same 𝓜 hm
+      exact hcomp ▸ hp m
+  have homogeneous_mem (j : ℤ) (m : M) (hm : m ∈ 𝓜 j) : m ∈ N := by
+    by_cases hj : j < d₀
+    · exact hlow j hj m hm
+    · have heq : d₀ + (j-d₀).toNat = j := by omega
+      exact hind (j-d₀).toNat m (heq.symm ▸ hm)
+  apply top_unique
+  intro m _
+  exact DirectSum.Decomposition.inductionOn 𝓜 N.zero_mem
+    (fun m => homogeneous_mem _ m m.property)
+    (fun _ _ hx hy => N.add_mem hx hy) m
+
+include klPreservedInstance8 in
+/-- Projection of a scalar multiple of a homogeneous vector. -/
+theorem component_smul_homogeneous
+    (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M)
+    [DirectSum.Decomposition 𝓐] [DirectSum.Decomposition 𝓜]
+    [SetLike.GradedSMul 𝓐 𝓜] (a : A) (m : M) (d n : ℤ) (hm : m ∈ 𝓜 d) :
+    component 𝓜 n (a • m) = component 𝓐 (n-d) a • m := by
+  have _ := klPreservedInstance8
+  classical
+  refine DirectSum.Decomposition.inductionOn 𝓐 ?_ ?_ ?_ a
+  · simp
+  · intro i a
+    by_cases hi : i = n-d
+    · have hsum : i+d = n := by omega
+      have ham : (a : A) • m ∈ 𝓜 n := hsum ▸
+        SetLike.GradedSMul.smul_mem a.property hm
+      rw [show component 𝓜 n ((a : A) • m) = (a : A) • m from
+        DirectSum.decompose_of_mem_same 𝓜 ham]
+      rw [show component 𝓐 (n-d) (a : A) = (a : A) from
+        DirectSum.decompose_of_mem_same 𝓐 (hi ▸ a.property)]
+    · have hsum : i+d ≠ n := by omega
+      rw [show component 𝓜 n ((a : A) • m) = 0 from
+        DirectSum.decompose_of_mem_ne 𝓜
+          (SetLike.GradedSMul.smul_mem a.property hm) hsum,
+        show component 𝓐 (n-d) (a : A) = 0 from
+        DirectSum.decompose_of_mem_ne 𝓐 a.property hi, zero_smul]
+  · intro a b ha hb
+    simp only [add_smul, map_add, ha, hb]
+
+include klPreservedInstance9 klPreservedInstance8 in
+/-- Finite generation can be made homogeneous using the actual finite
+support decomposition of each generator. -/
+theorem exists_finite_homogeneous_generators
+    (𝓜 : ℤ → Submodule k M) [DirectSum.Decomposition 𝓜] [Module.Finite A M] :
+    ∃ t : Finset M, Submodule.span A (t : Set M) = ⊤ ∧
+      ∀ m ∈ t, ∃ d : ℤ, m ∈ 𝓜 d := by
+  have _ := klPreservedInstance9
+  have _ := klPreservedInstance8
+  classical
+  obtain ⟨s,hs⟩ := Module.Finite.fg_top (R := A) (M := M)
+  let t : Finset M := s.biUnion fun m =>
+    (DirectSum.decompose 𝓜 m).support.image fun j => (DirectSum.decompose 𝓜 m j : M)
+  refine ⟨t, ?_, ?_⟩
+  · apply top_unique
+    rw [← hs]
+    apply Submodule.span_le.mpr
+    intro m hm
+    rw [← DirectSum.sum_support_decompose 𝓜 m]
+    apply Submodule.sum_mem
+    intro j hj
+    apply Submodule.subset_span
+    exact Finset.mem_biUnion.mpr ⟨m,hm,Finset.mem_image.mpr ⟨j,hj,rfl⟩⟩
+  · intro m hm
+    obtain ⟨a,ha,hma⟩ := Finset.mem_biUnion.mp hm
+    obtain ⟨j,hj,rfl⟩ := Finset.mem_image.mp hma
+    exact ⟨j,(DirectSum.decompose 𝓜 a j).property⟩
+
+/-- The span of finitely many homogeneous vectors is closed under each
+component, as required for graded Nakayama. -/
+theorem homogeneous_span_components {ι : Type*} [Fintype ι]
+    (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M)
+    [DirectSum.Decomposition 𝓐] [DirectSum.Decomposition 𝓜]
+    [SetLike.GradedSMul 𝓐 𝓜] (v : ι → M) (d : ι → ℤ)
+    (hv : ∀ i, v i ∈ 𝓜 (d i)) (n : ℤ) (m : M)
+    (hm : m ∈ Submodule.span A (Set.range v)) :
+    component 𝓜 n m ∈ Submodule.span A (Set.range v) := by
+  classical
+  obtain ⟨c,rfl⟩ := (Submodule.mem_span_range_iff_exists_fun A).mp hm
+  rw [map_sum]
+  apply Submodule.sum_mem
+  intro i _
+  rw [component_smul_homogeneous 𝓐 𝓜 _ _ (d i) n (hv i)]
+  exact Submodule.smul_mem _ _ (Submodule.subset_span (Set.mem_range_self i))
+
+/-- Reduction of a scalar modulo the augmentation ideal gives precisely
+its scalar-field action on the reduced module. -/
+theorem quotient_smul_of_scalar_mod (I : Ideal A) (a : A) (c : k)
+    (hac : a-algebraMap k A c ∈ I) (m : M) :
+    (I • (⊤ : Submodule A M)).mkQ (a • m) =
+      c • (I • (⊤ : Submodule A M)).mkQ m := by
+  let J := I • (⊤ : Submodule A M)
+  have hz : J.mkQ ((a-algebraMap k A c) • m) = 0 :=
+    (Submodule.Quotient.mk_eq_zero J).mpr
+      (Submodule.smul_mem_smul hac (Submodule.mem_top : m ∈ (⊤ : Submodule A M)))
+  rw [sub_smul, map_sub] at hz
+  have heq := sub_eq_zero.mp hz
+  calc
+    _ = J.mkQ (algebraMap k A c • m) := heq
+    _ = algebraMap k A c • J.mkQ m := map_smul J.mkQ _ _
+    _ = c • J.mkQ m := IsScalarTower.algebraMap_smul A c (J.mkQ m)
+
+/-- A finite free cover of a projective module, minimal at any proper
+augmentation ideal of a noetherian domain, is injective. The determinant
+form of Nakayama removes the complementary torsion-free kernel. -/
+theorem linearIndependent_of_projective_cover {ι : Type*} [Fintype ι]
+    [IsDomain A] [IsNoetherianRing A] [Module.Projective A M]
+    (I : Ideal A) (hI : I ≠ ⊤)
+    (hscalar : ∀ a : A, ∃ c : k, a-algebraMap k A c ∈ I)
+    (v : ι → M) (hspan : Submodule.span A (Set.range v) = ⊤)
+    (hind : LinearIndependent k
+      (fun i => (I • (⊤ : Submodule A M)).mkQ (v i))) :
+    LinearIndependent A v := by
+  classical
+  let J := I • (⊤ : Submodule A M)
+  let f := Fintype.linearCombination A v
+  have hsurj : Function.Surjective f := by
+    intro m
+    exact (Submodule.mem_span_range_iff_exists_fun A).mp (hspan ▸ Submodule.mem_top)
+  obtain ⟨s,hs⟩ := Module.projective_lifting_property f (LinearMap.id : M →ₗ[A] M) hsurj
+  have hmin : LinearMap.ker f ≤ I • (⊤ : Submodule A (ι → A)) := by
+    intro a ha
+    choose c hc using fun i => hscalar (a i)
+    have heq : ∑ i, c i • J.mkQ (v i) = 0 := by
+      calc
+        _ = J.mkQ (∑ i, a i • v i) := by
+          rw [map_sum]
+          apply Finset.sum_congr rfl
+          intro i _
+          exact (quotient_smul_of_scalar_mod I (a i) (c i) (hc i) (v i)).symm
+        _ = 0 := congrArg J.mkQ (LinearMap.mem_ker.mp ha) |>.trans (map_zero _)
+    have hc0 : ∀ i, c i=0 := (Fintype.linearIndependent_iff.mp hind) c heq
+    have hai (i : ι) : a i ∈ I := by simpa [hc0 i] using hc i
+    have hrebuild : a = ∑ i, a i • Pi.single i (1 : A) := by
+      ext j
+      simp [Finset.sum_apply, Pi.smul_apply, Pi.single_apply]
+    rw [hrebuild]
+    apply Submodule.sum_mem
+    intro i _
+    exact Submodule.smul_mem_smul (hai i) Submodule.mem_top
+  have htop := positive_smul_kernel_of_minimal_split I f s hs hmin
+  have hfg : (⊤ : Submodule A (LinearMap.ker f)).FG := IsNoetherian.noetherian _
+  obtain ⟨r,hr,hkill⟩ := Submodule.exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul
+    I (⊤ : Submodule A (LinearMap.ker f)) hfg (by rw [htop])
+  have hr0 : r ≠ 0 := by
+    intro heq
+    have hone : (1 : A) ∈ I := by simpa [heq] using I.neg_mem hr
+    exact hI (Ideal.eq_top_of_isUnit_mem I hone isUnit_one)
+  have hker : LinearMap.ker f = ⊥ := by
+    apply le_antisymm _ bot_le
+    intro a ha
+    have hz : r • (⟨a,ha⟩ : LinearMap.ker f) = 0 := hkill _ Submodule.mem_top
+    have ha0 : (⟨a,ha⟩ : LinearMap.ker f) = 0 := (smul_eq_zero.mp hz).resolve_left hr0
+    exact congrArg Subtype.val ha0
+  exact linearIndependent_iff_injective_fintypeLinearCombination.mpr
+    (LinearMap.ker_eq_bot.mp hker)
+
+/-- Images of any finite generating family span the augmentation quotient
+as a vector space over the degree-zero field. -/
+theorem quotient_span_of_scalar_mod {ι : Type*} [Fintype ι]
+    (I : Ideal A) (hscalar : ∀ a : A, ∃ c : k, a-algebraMap k A c ∈ I)
+    (v : ι → M) (hv : Submodule.span A (Set.range v) = ⊤) :
+    Submodule.span k (Set.range (fun i => (I • (⊤ : Submodule A M)).mkQ (v i))) = ⊤ := by
+  classical
+  let J := I • (⊤ : Submodule A M)
+  apply top_unique
+  intro z _
+  obtain ⟨m,rfl⟩ := J.mkQ_surjective z
+  obtain ⟨a,ha⟩ := (Submodule.mem_span_range_iff_exists_fun A).mp (hv ▸ Submodule.mem_top :
+    m ∈ Submodule.span A (Set.range v))
+  choose c hc using fun i => hscalar (a i)
+  apply (Submodule.mem_span_range_iff_exists_fun k).mpr
+  refine ⟨c, ?_⟩
+  calc
+    _ = J.mkQ (∑ i, a i • v i) := by
+      rw [map_sum]
+      apply Finset.sum_congr rfl
+      intro i _
+      exact (quotient_smul_of_scalar_mod I (a i) (c i) (hc i) (v i)).symm
+    _ = J.mkQ m := congrArg J.mkQ ha
+
+
+theorem exists_homogeneous_basis_of_scalar_mod
+    [IsDomain A] [IsNoetherianRing A] [Module.Projective A M] [Module.Finite A M]
+    (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M)
+    [DirectSum.Decomposition 𝓐] [DirectSum.Decomposition 𝓜]
+    [SetLike.GradedSMul 𝓐 𝓜]
+    (d₀ : ℤ) (hbound : ∀ j < d₀, 𝓜 j = ⊥)
+    (hproper : positiveIdeal 𝓐 ≠ ⊤)
+    (hscalar : ∀ a : A, ∃ c : k, a-algebraMap k A c ∈ positiveIdeal 𝓐) :
+    ∃ (ι : Type um), Finite ι ∧ ∃ b : Module.Basis ι A M,
+      ∀ i, ∃ d : ℤ, b i ∈ 𝓜 d := by
+  classical
+  obtain ⟨t,ht,hth⟩ := exists_finite_homogeneous_generators (A := A) 𝓜
+  let v : t → M := Subtype.val
+  have hv : Submodule.span A (Set.range v) = ⊤ := by
+    simpa [v] using ht
+  let I := positiveIdeal 𝓐
+  let J := I • (⊤ : Submodule A M)
+  have hq := quotient_span_of_scalar_mod I hscalar v hv
+  obtain ⟨ι,a,ha,hsp,hli⟩ := exists_linearIndependent' k (fun i => J.mkQ (v i))
+  let : Finite ι := Finite.of_injective a ha
+  let : Fintype ι := Fintype.ofFinite ι
+  let w : ι → M := v ∘ a
+  have hw : ∀ i, ∃ d : ℤ, w i ∈ 𝓜 d := fun i => hth _ (a i).property
+  choose d hd using hw
+  have hqw : Submodule.span k (Set.range (fun i => J.mkQ (w i))) = ⊤ := hsp.trans hq
+  let N := Submodule.span A (Set.range w)
+  have hN : ∀ n m, m ∈ N → (DirectSum.decompose 𝓜 m n : M) ∈ N :=
+    homogeneous_span_components 𝓐 𝓜 w d hd
+  have hgen : N ⊔ I • (⊤ : Submodule A M) = ⊤ := by
+    apply top_unique
+    intro m _
+    obtain ⟨c,hc⟩ := (Submodule.mem_span_range_iff_exists_fun k).mp
+      (hqw ▸ Submodule.mem_top : J.mkQ m ∈
+        Submodule.span k (Set.range (fun i => J.mkQ (w i))))
+    let z : M := ∑ i, algebraMap k A (c i) • w i
+    have hz : z ∈ N := by
+      apply Submodule.sum_mem
+      intro i _
+      exact N.smul_mem _ (Submodule.subset_span (Set.mem_range_self i))
+    have hqz : J.mkQ z = J.mkQ m := by
+      change J.mkQ (∑ i, algebraMap k A (c i) • w i) = J.mkQ m
+      rw [map_sum]
+      convert hc using 1
+      apply Finset.sum_congr rfl
+      intro i _
+      exact (map_smul J.mkQ _ _).trans (IsScalarTower.algebraMap_smul A (c i) _)
+    have hdiff : m-z ∈ J := by
+      apply (Submodule.Quotient.mk_eq_zero J).mp
+      change J.mkQ (m-z)=0
+      rw [map_sub,hqz,sub_self]
+    exact Submodule.mem_sup.mpr ⟨z,hz,m-z,hdiff,by abel⟩
+  have hspan : N=⊤ := eq_top_of_positive_smul_sup 𝓐 𝓜 d₀ hbound N hN hgen
+  have hliA : LinearIndependent A w :=
+    linearIndependent_of_projective_cover I hproper hscalar w hspan hli
+  refine ⟨ι,inferInstance,Module.Basis.mk hliA hspan.ge,?_⟩
+  intro i
+  exact ⟨d i, by simpa only [Module.Basis.mk_apply] using hd i⟩
+
+/-- The augmentation for a nonnegatively supported integer grading is
+actual degree-zero projection. -/
+noncomputable def nonnegativeAugmentation
+    (𝓐 : ℤ → Submodule k A) [DirectSum.Decomposition 𝓐]
+    [SetLike.GradedMonoid 𝓐] (hneg : ∀ j < 0, 𝓐 j = ⊥) : A →+* A where
+  toFun := component 𝓐 0
+  map_zero' := map_zero _
+  map_one' := DirectSum.decompose_of_mem_same 𝓐 SetLike.GradedOne.one_mem
+  map_add' := map_add _
+  map_mul' := by
+    intro a b
+    refine DirectSum.Decomposition.inductionOn 𝓐 ?_ ?_ ?_ a
+    · simp
+    · intro i a
+      refine DirectSum.Decomposition.inductionOn 𝓐 ?_ ?_ ?_ b
+      · simp
+      · intro j b
+        by_cases hi : i < 0
+        · have ha : (a : A) = 0 := by simpa [hneg i hi] using a.property
+          simp [ha]
+        by_cases hj : j < 0
+        · have hb : (b : A) = 0 := by simpa [hneg j hj] using b.property
+          simp [hb]
+        by_cases hij : i+j=0
+        · have hi0 : i=0 := by omega
+          have hj0 : j=0 := by omega
+          have hab := SetLike.GradedMul.mul_mem a.property b.property
+          change component 𝓐 0 ((a:A)*(b:A)) = component 𝓐 0 (a:A) * component 𝓐 0 (b:A)
+          rw [show component 𝓐 0 ((a:A)*(b:A)) = (a:A)*(b:A) from
+            DirectSum.decompose_of_mem_same 𝓐 (hij ▸ hab),
+            show component 𝓐 0 (a:A) = a from
+              DirectSum.decompose_of_mem_same 𝓐 (hi0 ▸ a.property),
+            show component 𝓐 0 (b:A) = b from
+              DirectSum.decompose_of_mem_same 𝓐 (hj0 ▸ b.property)]
+        · change component 𝓐 0 ((a:A)*(b:A)) = component 𝓐 0 (a:A) * component 𝓐 0 (b:A)
+          rw [show component 𝓐 0 ((a:A)*(b:A))=0 from
+            DirectSum.decompose_of_mem_ne 𝓐
+              (SetLike.GradedMul.mul_mem a.property b.property) hij]
+          by_cases hi0 : i=0
+          · have hj0 : j≠0 := by omega
+            rw [show component 𝓐 0 (b:A)=0 from
+              DirectSum.decompose_of_mem_ne 𝓐 b.property hj0,mul_zero]
+          · rw [show component 𝓐 0 (a:A)=0 from
+              DirectSum.decompose_of_mem_ne 𝓐 a.property hi0,zero_mul]
+      · intro b c hb hc
+        simp only [mul_add,map_add,hb,hc]
+    · intro a c ha hc
+      simp only [add_mul,map_add,ha,hc]
+
+/-- The positive ideal is exactly the augmentation kernel. -/
+theorem positiveIdeal_eq_augmentation_ker
+    (𝓐 : ℤ → Submodule k A) [DirectSum.Decomposition 𝓐]
+    [SetLike.GradedMonoid 𝓐] (hneg : ∀ j < 0, 𝓐 j = ⊥) :
+    positiveIdeal 𝓐 = RingHom.ker (nonnegativeAugmentation 𝓐 hneg) := by
+  classical
+  apply le_antisymm
+  · apply Ideal.span_le.mpr
+    rintro a ⟨i,hi,ha⟩
+    exact DirectSum.decompose_of_mem_ne 𝓐 ha (by omega)
+  · intro a ha
+    rw [← DirectSum.sum_support_decompose 𝓐 a]
+    apply Ideal.sum_mem
+    intro j hj
+    by_cases hjpos : 0 < j
+    · exact Ideal.subset_span ⟨j,hjpos,(DirectSum.decompose 𝓐 a j).property⟩
+    have hz : (DirectSum.decompose 𝓐 a j : A)=0 := by
+      by_cases hjzero : j=0
+      · subst j; exact ha
+      · have hjneg : j<0 := by omega
+        simpa [hneg j hjneg] using (DirectSum.decompose 𝓐 a j).property
+    rw [hz]
+    exact Ideal.zero_mem _
+
+
+theorem exists_homogeneous_basis_connected
+    [IsDomain A] [IsNoetherianRing A] [Module.Projective A M] [Module.Finite A M]
+    (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M)
+    [DirectSum.Decomposition 𝓐] [DirectSum.Decomposition 𝓜]
+    [SetLike.GradedMonoid 𝓐] [SetLike.GradedSMul 𝓐 𝓜]
+    (hneg : ∀ j < 0, 𝓐 j = ⊥)
+    (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
+    (d₀ : ℤ) (hbound : ∀ j < d₀, 𝓜 j = ⊥) :
+    ∃ (ι : Type um), Finite ι ∧ ∃ b : Module.Basis ι A M,
+      ∀ i, ∃ d : ℤ, b i ∈ 𝓜 d := by
+  apply exists_homogeneous_basis_of_scalar_mod 𝓐 𝓜 d₀ hbound
+  · rw [positiveIdeal_eq_augmentation_ker 𝓐 hneg]
+    exact RingHom.ker_ne_top _
+  · intro a
+    obtain ⟨c,hc⟩ := hzero _ (DirectSum.decompose 𝓐 a 0).property
+    refine ⟨c,?_⟩
+    rw [positiveIdeal_eq_augmentation_ker 𝓐 hneg]
+    change component 𝓐 0 (a-algebraMap k A c)=0
+    rw [map_sub,hc]
+    rw [show component 𝓐 0 (DirectSum.decompose 𝓐 a 0 : A) =
+      (DirectSum.decompose 𝓐 a 0 : A) from
+      DirectSum.decompose_of_mem_same 𝓐 (DirectSum.decompose 𝓐 a 0).property]
+    exact sub_self _
+
+/-- The actual polynomial grading, with each variable in degree one, on
+integer degrees as used for shifted stalks. -/
+abbrev polynomialGrading (k : Type*) [Field k] (σ : Type*) :=
+  MvPolynomial.weightedHomogeneousSubmodule k (fun _ : σ => (1 : ℤ))
+
+noncomputable instance polynomialGrading_decomposition (k : Type*) [Field k] (σ : Type*) :
+    DirectSum.Decomposition (polynomialGrading k σ) :=
+  MvPolynomial.weightedDecomposition k (fun _ : σ => (1 : ℤ))
+
+instance polynomialGrading_monoid (k : Type*) [Field k] (σ : Type*) :
+    SetLike.GradedMonoid (polynomialGrading k σ) :=
+  MvPolynomial.WeightedHomogeneousSubmodule.gradedMonoid
+
+theorem polynomialGrading_negative (k : Type*) [Field k] (σ : Type*)
+    (j : ℤ) (hj : j < 0) : polynomialGrading k σ j = ⊥ := by
+  classical
+  apply eq_bot_iff.mpr
+  intro p hp
+  change p=0
+  apply MvPolynomial.ext
+  intro d
+  by_contra hd
+  have heq := hp (show p.coeff d≠0 by simpa using hd)
+  have hn : 0 ≤ Finsupp.weight (fun _ : σ => (1 : ℤ)) d := by
+    simp only [Finsupp.weight_apply,Finsupp.sum]
+    exact Finset.sum_nonneg fun i _ => nsmul_nonneg (by norm_num) _
+  omega
+
+theorem polynomialGrading_zero (k : Type*) [Field k] (σ : Type*)
+    (p : MvPolynomial σ k) (hp : p ∈ polynomialGrading k σ 0) :
+    ∃ c : k, algebraMap k (MvPolynomial σ k) c=p := by
+  classical
+  refine ⟨p.coeff 0,?_⟩
+  apply MvPolynomial.ext
+  intro d
+  by_cases hd : d=0
+  · subst d; simp
+  · have hc : p.coeff d=0 := by
+      by_contra hc
+      have heq := hp hc
+      obtain ⟨i,hi⟩ := Finsupp.ne_iff.mp hd
+      have hb := Finsupp.le_weight_of_ne_zero (M := ℤ)
+        (w := fun _ : σ => (1 : ℤ)) (fun _ => by norm_num)
+        (show d i≠0 by simpa using hi)
+      omega
+    simp [MvPolynomial.coeff_C,Ne.symm hd,hc]
+
+
+theorem exists_homogeneous_basis_polynomial {σ : Type*} [Finite σ]
+    [Module (MvPolynomial σ k) M] [IsScalarTower k (MvPolynomial σ k) M]
+    [Module.Projective (MvPolynomial σ k) M] [Module.Finite (MvPolynomial σ k) M]
+    (𝓜 : ℤ → Submodule k M) [DirectSum.Decomposition 𝓜]
+    [SetLike.GradedSMul (polynomialGrading k σ) 𝓜]
+    (d₀ : ℤ) (hbound : ∀ j < d₀, 𝓜 j = ⊥) :
+    ∃ (ι : Type um), Finite ι ∧ ∃ b : Module.Basis ι (MvPolynomial σ k) M,
+      ∀ i, ∃ d : ℤ, b i ∈ 𝓜 d := by
+  exact exists_homogeneous_basis_connected (polynomialGrading k σ) 𝓜
+    (polynomialGrading_negative k σ) (polynomialGrading_zero k σ) d₀ hbound
+
+end KLInvariance.Graded
+
+
+namespace KLInvariance.ReverseFiltration
+universe ua up ui
+variable {A : Type ua} [CommRing A] [klPreservedInstance10 : IsDomain A]
+noncomputable section
+
+def scalarQuotientMap (a : A) (ι : Type ui) :
+    (ι → A) →ₗ[A] (ι → A ⧸ Ideal.span {a}) where
+  toFun v i := Ideal.Quotient.mk _ (v i)
+  map_add' _ _ := by ext i; simp
+  map_smul' _ _ := by ext i; simp [Algebra.smul_def]
+
+include klPreservedInstance10 in
+theorem scalarQuotientMap_surjective (a : A) (ι : Type ui) :
+    Function.Surjective (scalarQuotientMap a ι) := by
+  have _ := klPreservedInstance10
+  intro v
+  choose w hw using fun i => Ideal.Quotient.mk_surjective (v i)
+  exact ⟨w,funext hw⟩
+
+/-- Multiplication by the actual nonzero label identifies the kernel of
+coordinate reduction with a free module. -/
+def scalarKernelMap (a : A) (ι : Type ui) :
+    (ι → A) →ₗ[A] LinearMap.ker (scalarQuotientMap a ι) where
+  toFun v := ⟨fun i => a*v i, by
+    ext i
+    change Ideal.Quotient.mk (Ideal.span {a}) (a*v i)=0
+    apply Ideal.Quotient.eq_zero_iff_mem.mpr
+    exact Ideal.mem_span_singleton.mpr (dvd_mul_right a (v i))⟩
+  map_add' v w := by apply Subtype.ext; ext i; exact mul_add _ _ _
+  map_smul' r v := by apply Subtype.ext; ext i; change a*(r*v i)=r*(a*v i); ring
+
+theorem scalarKernelMap_bijective (a : A) (ha : a≠0) (ι : Type ui) :
+    Function.Bijective (scalarKernelMap a ι) := by
+  constructor
+  · intro v w hvw
+    funext i
+    exact mul_left_cancel₀ ha (congrFun (congrArg Subtype.val hvw) i)
+  · intro v
+    have hv (i : ι) : a ∣ v.val i := by
+      apply Ideal.mem_span_singleton.mp
+      apply Ideal.Quotient.eq_zero_iff_mem.mp
+      exact congrFun v.property i
+    choose w hw using hv
+    refine ⟨w,Subtype.ext (funext fun i => (hw i).symm)⟩
+
+theorem scalarKernel_projective (a : A) (ha : a≠0) (ι : Type ui) [Finite ι] :
+    Module.Projective A (LinearMap.ker (scalarQuotientMap a ι)) := by
+  exact Module.Projective.of_equiv
+    (LinearEquiv.ofBijective (scalarKernelMap a ι) (scalarKernelMap_bijective a ha ι))
+
+/-- Every finite free module over A/(a) has an actual free A-presentation
+with projective kernel; no projective-dimension hypothesis is inserted. -/
+theorem exists_scalar_presentation (a : A) (ha : a≠0)
+    (P : Type up) [AddCommGroup P] [Module A P]
+    [Module (A ⧸ Ideal.span {a}) P] [IsScalarTower A (A ⧸ Ideal.span {a}) P]
+    [Module.Free (A ⧸ Ideal.span {a}) P] [Module.Finite (A ⧸ Ideal.span {a}) P] :
+    ∃ (ι : Type up), Finite ι ∧ ∃ g : (ι → A) →ₗ[A] P,
+      Function.Surjective g ∧ Module.Projective A (LinearMap.ker g) := by
+  classical
+  let C := A ⧸ Ideal.span {a}
+  let b := Module.Free.chooseBasis C P
+  let ι := Module.Free.ChooseBasisIndex C P
+  let : Fintype ι := Fintype.ofFinite ι
+  let e : P ≃ₗ[A] (ι → C) := b.equivFun.restrictScalars A
+  let g : (ι → A) →ₗ[A] P := e.symm.toLinearMap.comp (scalarQuotientMap a ι)
+  refine ⟨ι,inferInstance,g,e.symm.surjective.comp (scalarQuotientMap_surjective a ι),?_⟩
+  have hker : LinearMap.ker g = LinearMap.ker (scalarQuotientMap a ι) := by
+    ext v
+    change e.symm (scalarQuotientMap a ι v)=0 ↔ scalarQuotientMap a ι v=0
+    exact e.symm.map_eq_zero_iff
+  rw [hker]
+  exact scalarKernel_projective a ha ι
+
+/-- The reverse-filtration kernel is projective when the edge image is
+actually finite free over the hypersurface quotient. -/
+theorem kernel_projective_of_free_scalar_quotient
+    (a : A) (ha : a≠0) {F : Type*} [AddCommGroup F] [Module A F]
+    [Module.Projective A F]
+    {P : Type up} [AddCommGroup P] [Module A P]
+    [Module (A ⧸ Ideal.span {a}) P] [IsScalarTower A (A ⧸ Ideal.span {a}) P]
+    [Module.Free (A ⧸ Ideal.span {a}) P] [Module.Finite (A ⧸ Ideal.span {a}) P]
+    (f : F →ₗ[A] P) (hf : Function.Surjective f) :
+    Module.Projective A (LinearMap.ker f) := by
+  obtain ⟨ι,hι,g,hg,hgker⟩ := exists_scalar_presentation a ha P
+  let : Finite ι := hι
+  let := hgker
+  exact kernel_projective_of_presentation f g hf hg
+
+end
+end KLInvariance.ReverseFiltration
+
+
+end
+
+section
+
+namespace KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {M : Type um} [AddCommGroup M] [Module k M] [Module A M] [IsScalarTower k A M]
+
+theorem exists_homogeneous_minimal_generators
+    [Module.Finite A M]
+    (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M)
+    [DirectSum.Decomposition 𝓐] [DirectSum.Decomposition 𝓜]
+    [SetLike.GradedSMul 𝓐 𝓜]
+    (d₀ : ℤ) (hbound : ∀ j < d₀, 𝓜 j = ⊥)
+    (hscalar : ∀ a : A, ∃ c : k, a-algebraMap k A c ∈ positiveIdeal 𝓐) :
+    ∃ (ι : Type um), Finite ι ∧ ∃ (v : ι → M) (d : ι → ℤ),
+      (∀ i, v i ∈ 𝓜 (d i)) ∧ Submodule.span A (Set.range v) = ⊤ ∧
+      LinearIndependent k (fun i => (positiveIdeal 𝓐 • (⊤ : Submodule A M)).mkQ (v i)) := by
+  classical
+  obtain ⟨t,ht,hth⟩ := exists_finite_homogeneous_generators (A := A) 𝓜
+  let v : t → M := Subtype.val
+  have hv : Submodule.span A (Set.range v) = ⊤ := by
+    simpa [v] using ht
+  let I := positiveIdeal 𝓐
+  let J := I • (⊤ : Submodule A M)
+  have hq := quotient_span_of_scalar_mod I hscalar v hv
+  obtain ⟨ι,a,ha,hsp,hli⟩ := exists_linearIndependent' k (fun i => J.mkQ (v i))
+  let : Finite ι := Finite.of_injective a ha
+  let : Fintype ι := Fintype.ofFinite ι
+  let w : ι → M := v ∘ a
+  have hw : ∀ i, ∃ d : ℤ, w i ∈ 𝓜 d := fun i => hth _ (a i).property
+  choose d hd using hw
+  have hqw : Submodule.span k (Set.range (fun i => J.mkQ (w i))) = ⊤ := hsp.trans hq
+  let N := Submodule.span A (Set.range w)
+  have hN : ∀ n m, m ∈ N → (DirectSum.decompose 𝓜 m n : M) ∈ N :=
+    homogeneous_span_components 𝓐 𝓜 w d hd
+  have hgen : N ⊔ I • (⊤ : Submodule A M) = ⊤ := by
+    apply top_unique
+    intro m _
+    obtain ⟨c,hc⟩ := (Submodule.mem_span_range_iff_exists_fun k).mp
+      (hqw ▸ Submodule.mem_top : J.mkQ m ∈
+        Submodule.span k (Set.range (fun i => J.mkQ (w i))))
+    let z : M := ∑ i, algebraMap k A (c i) • w i
+    have hz : z ∈ N := by
+      apply Submodule.sum_mem
+      intro i _
+      exact N.smul_mem _ (Submodule.subset_span (Set.mem_range_self i))
+    have hqz : J.mkQ z = J.mkQ m := by
+      change J.mkQ (∑ i, algebraMap k A (c i) • w i) = J.mkQ m
+      rw [map_sum]
+      convert hc using 1
+      apply Finset.sum_congr rfl
+      intro i _
+      exact (map_smul J.mkQ _ _).trans (IsScalarTower.algebraMap_smul A (c i) _)
+    have hdiff : m-z ∈ J := by
+      apply (Submodule.Quotient.mk_eq_zero J).mp
+      change J.mkQ (m-z)=0
+      rw [map_sub,hqz,sub_self]
+    exact Submodule.mem_sup.mpr ⟨z,hz,m-z,hdiff,by abel⟩
+  have hspan : N=⊤ := eq_top_of_positive_smul_sup 𝓐 𝓜 d₀ hbound N hN hgen
+  exact ⟨ι,inferInstance,w,d,hd,hspan,hli⟩
+
+ theorem kernel_minimal_cover (I : Ideal A)
+    (hscalar : ∀ a : A, ∃ c : k, a-algebraMap k A c ∈ I)
+    {ι : Type*} [Fintype ι] (v : ι → M)
+    (hind : LinearIndependent k (fun i => (I • (⊤ : Submodule A M)).mkQ (v i))) :
+    LinearMap.ker (Fintype.linearCombination A v) ≤ I • (⊤ : Submodule A (ι → A)) := by
+  classical
+  intro f hf
+  let J := I • (⊤ : Submodule A M)
+  choose c hc using fun i => hscalar (f i)
+  have hz : ∑ i, c i • J.mkQ (v i) = 0 := by
+    have hm : ∑ i, f i • v i = 0 := LinearMap.mem_ker.mp hf
+    have hq := congrArg J.mkQ hm
+    rw [map_sum,map_zero] at hq
+    convert hq using 1
+    apply Finset.sum_congr rfl
+    intro i _
+    exact (quotient_smul_of_scalar_mod I (f i) (c i) (hc i) (v i)).symm
+  have hc0 : ∀ i, c i = 0 := (Fintype.linearIndependent_iff.mp hind) c hz
+  have hfi (i : ι) : f i ∈ I := by simpa [hc0 i] using hc i
+  have he : f = ∑ i, f i • Pi.single i (1 : A) := by
+    ext i
+    simp [Pi.single_apply]
+  rw [he]
+  apply Submodule.sum_mem
+  intro i _
+  exact Submodule.smul_mem_smul (hfi i) Submodule.mem_top
+
+end KLInvariance.Graded
+
+end
+
+
+section
+
+namespace KLInvariance.Graded
+universe uk ua um un
+variable {k : Type uk} [Field k]
+  {M : Type um} [AddCommGroup M] [Module k M]
+  {N : Type un} [AddCommGroup N] [Module k N]
+
+ theorem internal_of_components
+    (𝓜 : ℤ → Submodule k M) (p : ℤ → M →ₗ[k] M)
+    (hmem : ∀ i m, p i m ∈ 𝓜 i)
+    (hsame : ∀ i m, m ∈ 𝓜 i → p i m = m)
+    (hne : ∀ i j m, i ≠ j → m ∈ 𝓜 j → p i m = 0)
+    (hsum : ∀ m, ∃ s : Finset ℤ, ∑ i ∈ s, p i m = m) :
+    DirectSum.IsInternal 𝓜 := by
+  classical
+  apply DirectSum.isInternal_submodule_of_iSupIndep_of_iSup_eq_top
+  · apply (iSupIndep_iff_finsetSum_eq_zero_imp_eq_zero _).mpr
+    intro s v hv hz i hi
+    have hp := congrArg (p i) hz
+    rw [map_sum, map_zero, Finset.sum_eq_single_of_mem i hi] at hp
+    · simpa only [hsame i _ (hv i hi)] using hp
+    · intro j hj hji
+      exact hne i j _ (Ne.symm hji) (hv j hj)
+  · apply top_unique
+    intro m _
+    obtain ⟨s,hs⟩ := hsum m
+    rw [← hs]
+    apply Submodule.sum_mem
+    intro i _
+    exact (le_iSup 𝓜 i) (hmem i m)
+
+ theorem image_isInternal
+    (𝓜 : ℤ → Submodule k M) [DirectSum.Decomposition 𝓜]
+    (f : M →ₗ[k] N) (hf : Function.Surjective f)
+    (hker : ∀ i m, f m = 0 → f (component 𝓜 i m) = 0) :
+    DirectSum.IsInternal (fun i => (𝓜 i).map f) := by
+  classical
+  apply DirectSum.isInternal_submodule_of_iSupIndep_of_iSup_eq_top
+  · apply (iSupIndep_iff_finsetSum_eq_zero_imp_eq_zero _).mpr
+    intro s v hv hz i hi
+    choose w hw hfw using fun j : s => (Submodule.mem_map.mp (hv j.val j.property))
+    let w' : ℤ → M := fun j => if hj : j ∈ s then w ⟨j,hj⟩ else 0
+    have hw' (j) : w' j ∈ 𝓜 j := by
+      dsimp [w']; split_ifs with hj
+      · exact hw ⟨j,hj⟩
+      · exact Submodule.zero_mem _
+    have hfw' (j) (hj : j ∈ s) : f (w' j) = v j := by simpa [w',hj] using hfw ⟨j,hj⟩
+    have hsum : f (∑ j ∈ s, w' j) = 0 := by
+      rw [map_sum]
+      exact (Finset.sum_congr rfl hfw').trans hz
+    have hc := hker i _ hsum
+    rw [map_sum, Finset.sum_eq_single_of_mem i hi] at hc
+    · change f (DirectSum.decompose 𝓜 (w' i) i : M) = 0 at hc
+      rw [DirectSum.decompose_of_mem_same 𝓜 (hw' i), hfw' i hi] at hc
+      exact hc
+    · intro j _ hji
+      exact DirectSum.decompose_of_mem_ne 𝓜 (hw' j) hji
+  · apply top_unique
+    intro n _
+    obtain ⟨m,rfl⟩ := hf n
+    rw [← DirectSum.sum_support_decompose 𝓜 m, map_sum]
+    apply Submodule.sum_mem
+    intro i _
+    exact (le_iSup (fun i => (𝓜 i).map f) i)
+      (Submodule.mem_map.mpr ⟨_,(DirectSum.decompose 𝓜 m i).property,rfl⟩)
+
+variable {A : Type ua} [CommRing A] [Algebra k A]
+  [Module A M] [IsScalarTower k A M]
+
+ def subPiece (𝓜 : ℤ → Submodule k M) (S : Submodule A M) : ℤ → Submodule k S :=
+  fun i => (𝓜 i).comap (S.subtype.restrictScalars k)
+
+ theorem sub_isInternal
+    (𝓜 : ℤ → Submodule k M) [DirectSum.Decomposition 𝓜]
+    (S : Submodule A M)
+    (hS : ∀ i m, m ∈ S → component 𝓜 i m ∈ S) :
+    DirectSum.IsInternal (subPiece 𝓜 S) := by
+  classical
+  let p : ℤ → S →ₗ[k] S := fun i =>
+    LinearMap.codRestrict (S.restrictScalars k)
+      ((component 𝓜 i).comp (S.subtype.restrictScalars k)) (fun m => hS i m m.property)
+  apply internal_of_components (subPiece 𝓜 S) p
+  · intro i m
+    exact (DirectSum.decompose 𝓜 (m : M) i).property
+  · intro i m hm
+    apply Subtype.ext
+    exact DirectSum.decompose_of_mem_same 𝓜 hm
+  · intro i j m hij hm
+    apply Subtype.ext
+    exact DirectSum.decompose_of_mem_ne 𝓜 hm hij.symm
+  · intro m
+    refine ⟨(DirectSum.decompose 𝓜 (m : M)).support, ?_⟩
+    apply Subtype.ext
+    change (∑ i ∈ (DirectSum.decompose 𝓜 (m : M)).support, p i m : S).val = (m : M)
+    rw [Submodule.coe_sum]
+    exact DirectSum.sum_support_decompose 𝓜 (m : M)
+
+ def quotientPiece (𝓜 : ℤ → Submodule k M) (S : Submodule A M) :
+    ℤ → Submodule k (M ⧸ S) := fun i => (𝓜 i).map (S.mkQ.restrictScalars k)
+
+ theorem quotient_isInternal
+    (𝓜 : ℤ → Submodule k M) [DirectSum.Decomposition 𝓜]
+    (S : Submodule A M)
+    (hS : ∀ i m, m ∈ S → component 𝓜 i m ∈ S) :
+    DirectSum.IsInternal (quotientPiece 𝓜 S) := by
+  apply image_isInternal 𝓜 (S.mkQ.restrictScalars k) S.mkQ_surjective
+  intro i m hm
+  apply (Submodule.Quotient.mk_eq_zero S).mpr
+  exact hS i m ((Submodule.Quotient.mk_eq_zero S).mp hm)
+
+ instance sub_gradedSMul
+    (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M)
+    [SetLike.GradedSMul 𝓐 𝓜] (S : Submodule A M) :
+    SetLike.GradedSMul 𝓐 (subPiece 𝓜 S) where
+  smul_mem := by
+    intro i j a m ha hm
+    change a • (m : M) ∈ 𝓜 (i+j)
+    exact SetLike.GradedSMul.smul_mem ha (show (m : M) ∈ 𝓜 j from hm)
+
+ instance quotient_gradedSMul
+    (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M)
+    [SetLike.GradedSMul 𝓐 𝓜] (S : Submodule A M) :
+    SetLike.GradedSMul 𝓐 (quotientPiece 𝓜 S) where
+  smul_mem := by
+    intro i j a m ha hm
+    obtain ⟨v,hv,rfl⟩ := Submodule.mem_map.mp hm
+    exact Submodule.mem_map.mpr ⟨a • v,SetLike.GradedSMul.smul_mem ha hv,map_smul S.mkQ a v⟩
+
+ theorem subPiece_eq_bot (𝓜 : ℤ → Submodule k M) (S : Submodule A M)
+    (i : ℤ) (h : 𝓜 i = ⊥) : subPiece 𝓜 S i = ⊥ := by
+  apply le_antisymm _ bot_le
+  intro m hm
+  apply Subtype.ext
+  exact (Submodule.mem_bot k).mp (h ▸ hm)
+
+ theorem quotientPiece_eq_bot (𝓜 : ℤ → Submodule k M) (S : Submodule A M)
+    (i : ℤ) (h : 𝓜 i = ⊥) : quotientPiece 𝓜 S i = ⊥ := by
+  simp [quotientPiece,h]
+
+ def shiftedPiece {ι : Type*} (𝓐 : ℤ → Submodule k A) (d : ι → ℤ) :
+    ℤ → Submodule k (ι → A) := fun n =>
+  { carrier := {v | ∀ i, v i ∈ 𝓐 (n-d i)}
+    zero_mem' := fun _ => Submodule.zero_mem _
+    add_mem' := fun hx hy i => Submodule.add_mem _ (hx i) (hy i)
+    smul_mem' := fun c _ hx i => Submodule.smul_mem _ c (hx i) }
+
+ theorem shifted_isInternal {ι : Type*} [Fintype ι]
+    (𝓐 : ℤ → Submodule k A) [DirectSum.Decomposition 𝓐] (d : ι → ℤ) :
+    DirectSum.IsInternal (shiftedPiece 𝓐 d) := by
+  classical
+  let p : ℤ → (ι → A) →ₗ[k] (ι → A) := fun n =>
+    LinearMap.pi fun i => (component 𝓐 (n-d i)).comp (LinearMap.proj i)
+  apply internal_of_components (shiftedPiece 𝓐 d) p
+  · intro n v i
+    exact (DirectSum.decompose 𝓐 (v i) (n-d i)).property
+  · intro n v hv
+    funext i
+    exact DirectSum.decompose_of_mem_same 𝓐 (hv i)
+  · intro n j v hnj hv
+    funext i
+    apply DirectSum.decompose_of_mem_ne 𝓐 (hv i)
+    omega
+  · intro v
+    let s : ι → Finset ℤ := fun i =>
+      (DirectSum.decompose 𝓐 (v i)).support.image (fun n => n+d i)
+    refine ⟨Finset.univ.biUnion s, ?_⟩
+    funext i
+    change (∑ n ∈ Finset.univ.biUnion s, p n v) i = v i
+    rw [Finset.sum_apply]
+    have hsub : s i ⊆ Finset.univ.biUnion s :=
+      Finset.subset_biUnion_of_mem s (Finset.mem_univ i)
+    have hout (n : ℤ) (_ : n ∈ Finset.univ.biUnion s) (hn : n ∉ s i) :
+        p n v i = 0 := by
+      have hni : n-d i ∉ (DirectSum.decompose 𝓐 (v i)).support := by
+        intro hni
+        exact hn (Finset.mem_image.mpr ⟨n-d i,hni,by omega⟩)
+      have hz := DFinsupp.notMem_support_iff.mp hni
+      exact congrArg Subtype.val hz
+    rw [← Finset.sum_subset hsub hout]
+    change ∑ n ∈ (DirectSum.decompose 𝓐 (v i)).support.image (fun n => n+d i),
+      component 𝓐 (n-d i) (v i) = v i
+    rw [Finset.sum_image (fun a _ b _ h => by omega)]
+    simp only [add_sub_cancel_right]
+    exact DirectSum.sum_support_decompose 𝓐 (v i)
+
+ instance shifted_gradedSMul {ι : Type*}
+    (𝓐 : ℤ → Submodule k A) [SetLike.GradedMonoid 𝓐] (d : ι → ℤ) :
+    SetLike.GradedSMul 𝓐 (shiftedPiece 𝓐 d) where
+  smul_mem := by
+    intro i j a m ha hm t
+    have h := SetLike.GradedMul.mul_mem ha (hm t)
+    simpa [sub_eq_add_neg,add_assoc] using h
+
+ theorem shiftedPiece_eq_bot {ι : Type*}
+    (𝓐 : ℤ → Submodule k A) (hneg : ∀ i < 0, 𝓐 i = ⊥) (d : ι → ℤ)
+    (n : ℤ) (h : ∀ i, n < d i) : shiftedPiece 𝓐 d n = ⊥ := by
+  apply le_antisymm _ bot_le
+  intro m hm
+  funext i
+  have hi := hm i
+  rw [hneg _ (by have := h i; omega)] at hi
+  exact hi
+
+end KLInvariance.Graded
+
+end
+
+
+section
+
+namespace KLInvariance.Graded
+universe uk ua um un ui
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {M : Type um} [AddCommGroup M] [Module k M] [Module A M] [IsScalarTower k A M]
+  {N : Type un} [AddCommGroup N] [Module k N] [Module A N] [IsScalarTower k A N]
+
+omit [Algebra k A] [IsScalarTower k A M] [IsScalarTower k A N] in
+ theorem map_component
+    (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
+    [DirectSum.Decomposition 𝓜] [DirectSum.Decomposition 𝓝]
+    (f : M →ₗ[A] N) (d : ℤ)
+    (hf : ∀ i m, m ∈ 𝓜 i → f m ∈ 𝓝 (i+d)) (n : ℤ) (m : M) :
+    component 𝓝 n (f m) = f (component 𝓜 (n-d) m) := by
+  classical
+  refine DirectSum.Decomposition.inductionOn 𝓜 ?_ ?_ ?_ m
+  · simp
+  · intro j v
+    by_cases h : j+d=n
+    · have h' : n-d=j := by omega
+      change (DirectSum.decompose 𝓝 (f (v : M)) n : N) =
+        f (DirectSum.decompose 𝓜 (v : M) (n-d) : M)
+      rw [← h, DirectSum.decompose_of_mem_same 𝓝 (hf j v v.property),
+        show j+d-d=j by omega, DirectSum.decompose_of_mem_same 𝓜 v.property]
+    · have h' : j ≠ n-d := by omega
+      change (DirectSum.decompose 𝓝 (f (v : M)) n : N) =
+        f (DirectSum.decompose 𝓜 (v : M) (n-d) : M)
+      rw [DirectSum.decompose_of_mem_ne 𝓝 (hf j v v.property) h,
+        DirectSum.decompose_of_mem_ne 𝓜 v.property h', map_zero]
+  · intro x y hx hy
+    simp only [map_add,hx,hy]
+
+omit [Algebra k A] [IsScalarTower k A M] [IsScalarTower k A N] in
+ theorem kernel_homogeneous
+    (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
+    [DirectSum.Decomposition 𝓜] [DirectSum.Decomposition 𝓝]
+    (f : M →ₗ[A] N) (d : ℤ)
+    (hf : ∀ i m, m ∈ 𝓜 i → f m ∈ 𝓝 (i+d)) :
+    ∀ n m, m ∈ LinearMap.ker f → component 𝓜 n m ∈ LinearMap.ker f := by
+  intro n m hm
+  change f (component 𝓜 n m)=0
+  have h := map_component 𝓜 𝓝 f d hf (n+d) m
+  simpa [LinearMap.mem_ker.mp hm] using h.symm
+
+omit [Algebra k A] [IsScalarTower k A M] [IsScalarTower k A N] in
+ theorem range_homogeneous
+    (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
+    [DirectSum.Decomposition 𝓜] [DirectSum.Decomposition 𝓝]
+    (f : M →ₗ[A] N) (d : ℤ)
+    (hf : ∀ i m, m ∈ 𝓜 i → f m ∈ 𝓝 (i+d)) :
+    ∀ n m, m ∈ LinearMap.range f → component 𝓝 n m ∈ LinearMap.range f := by
+  rintro n _ ⟨m,rfl⟩
+  exact ⟨component 𝓜 (n-d) m,(map_component 𝓜 𝓝 f d hf n m).symm⟩
+
+omit [IsScalarTower k A M] in
+ theorem component_homogeneous_smul
+    (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M)
+    [DirectSum.Decomposition 𝓜] [SetLike.GradedSMul 𝓐 𝓜]
+    (d : ℤ) (a : A) (ha : a ∈ 𝓐 d) (n : ℤ) (m : M) :
+    component 𝓜 n (a • m) = a • component 𝓜 (n-d) m := by
+  apply map_component 𝓜 𝓜 (LinearMap.lsmul A M a) d
+  intro i v hv
+  simpa [add_comm] using SetLike.GradedSMul.smul_mem ha hv
+
+omit [IsScalarTower k A M] in
+ theorem scalar_range_homogeneous
+    (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M)
+    [DirectSum.Decomposition 𝓜] [SetLike.GradedSMul 𝓐 𝓜]
+    (d : ℤ) (a : A) (ha : a ∈ 𝓐 d) :
+    ∀ n m, m ∈ LinearMap.range (LinearMap.lsmul A M a) →
+      component 𝓜 n m ∈ LinearMap.range (LinearMap.lsmul A M a) := by
+  apply range_homogeneous 𝓜 𝓜 (LinearMap.lsmul A M a) d
+  intro i v hv
+  simpa [add_comm] using SetLike.GradedSMul.smul_mem ha hv
+
+ theorem exists_graded_minimal_free_cover
+    [Module.Finite A M]
+    (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M)
+    [DirectSum.Decomposition 𝓐] [DirectSum.Decomposition 𝓜]
+    [SetLike.GradedSMul 𝓐 𝓜]
+    (d₀ : ℤ) (hbound : ∀ j < d₀, 𝓜 j = ⊥)
+    (hscalar : ∀ a : A, ∃ c : k, a-algebraMap k A c ∈ positiveIdeal 𝓐) :
+    ∃ (ι : Type um), Finite ι ∧ ∃ (d : ι → ℤ) (f : (ι → A) →ₗ[A] M),
+      Function.Surjective f ∧
+      (∀ n v, v ∈ shiftedPiece 𝓐 d n → f v ∈ 𝓜 n) ∧
+      LinearMap.ker f ≤ positiveIdeal 𝓐 • (⊤ : Submodule A (ι → A)) := by
+  classical
+  obtain ⟨ι,hι,v,d,hd,hspan,hind⟩ :=
+    exists_homogeneous_minimal_generators 𝓐 𝓜 d₀ hbound hscalar
+  let : Finite ι := hι
+  let : Fintype ι := Fintype.ofFinite ι
+  refine ⟨ι,hι,d,Fintype.linearCombination A v,?_,?_,?_⟩
+  · intro m
+    exact (Submodule.mem_span_range_iff_exists_fun A).mp (hspan ▸ Submodule.mem_top)
+  · intro n c hc
+    apply Submodule.sum_mem
+    intro i _
+    simpa using SetLike.GradedSMul.smul_mem (hc i) (hd i)
+  · exact kernel_minimal_cover _ hscalar v hind
+
+section Pi
+variable {ι : Type ui} {P : ι → Type um}
+  [∀ i, AddCommGroup (P i)] [∀ i, Module k (P i)]
+
+ def piPiece (𝓟 : ∀ i, ℤ → Submodule k (P i)) : ℤ → Submodule k (∀ i, P i) :=
+  fun n => Submodule.pi Set.univ (fun i => 𝓟 i n)
+
+ theorem pi_isInternal [Fintype ι]
+    (𝓟 : ∀ i, ℤ → Submodule k (P i)) [∀ i, DirectSum.Decomposition (𝓟 i)] :
+    DirectSum.IsInternal (piPiece 𝓟) := by
+  classical
+  let p : ℤ → (∀ i, P i) →ₗ[k] (∀ i, P i) := fun n =>
+    LinearMap.pi fun i => (component (𝓟 i) n).comp (LinearMap.proj i)
+  apply internal_of_components (piPiece 𝓟) p
+  · intro n v i _
+    exact (DirectSum.decompose (𝓟 i) (v i) n).property
+  · intro n v hv
+    funext i
+    exact DirectSum.decompose_of_mem_same (𝓟 i) (hv i (Set.mem_univ i))
+  · intro n j v hnj hv
+    funext i
+    exact DirectSum.decompose_of_mem_ne (𝓟 i) (hv i (Set.mem_univ i)) hnj.symm
+  · intro v
+    let s : ι → Finset ℤ := fun i => (DirectSum.decompose (𝓟 i) (v i)).support
+    refine ⟨Finset.univ.biUnion s, ?_⟩
+    funext i
+    rw [Finset.sum_apply]
+    have hsub : s i ⊆ Finset.univ.biUnion s :=
+      Finset.subset_biUnion_of_mem s (Finset.mem_univ i)
+    have hout (n : ℤ) (_ : n ∈ Finset.univ.biUnion s) (hn : n ∉ s i) :
+        p n v i = 0 := by
+      change n ∉ (DirectSum.decompose (𝓟 i) (v i)).support at hn
+      have hz := DFinsupp.notMem_support_iff.mp hn
+      exact congrArg Subtype.val hz
+    rw [← Finset.sum_subset hsub hout]
+    exact DirectSum.sum_support_decompose (𝓟 i) (v i)
+
+ instance pi_gradedSMul [∀ i, Module A (P i)] [∀ i, IsScalarTower k A (P i)]
+    (𝓐 : ℤ → Submodule k A) (𝓟 : ∀ i, ℤ → Submodule k (P i))
+    [∀ i, SetLike.GradedSMul 𝓐 (𝓟 i)] : SetLike.GradedSMul 𝓐 (piPiece 𝓟) where
+  smul_mem := by
+    intro n j a v ha hv i hi
+    exact SetLike.GradedSMul.smul_mem ha (hv i hi)
+
+ theorem piPiece_eq_bot
+    (𝓟 : ∀ i, ℤ → Submodule k (P i)) (n : ℤ) (h : ∀ i, 𝓟 i n = ⊥) :
+    piPiece 𝓟 n = ⊥ := by
+  apply le_antisymm _ bot_le
+  intro v hv
+  funext i
+  have hi : v i ∈ 𝓟 i n := hv i (Set.mem_univ i)
+  rw [h i] at hi
+  exact hi
+
+end Pi
+end KLInvariance.Graded
+
+end
+
+
+section
+
+namespace KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+
+/-- Actual finite nonnegatively graded modules; no character or duality property
+is folded into this data. -/
+structure ModuleData (𝓐 : ℤ → Submodule k A) where
+  obj : ModuleCat.{max ua um} A
+  kModule : Module k obj
+  tower : IsScalarTower k A obj
+  piece : ℤ → Submodule k obj
+  decomposition : DirectSum.Decomposition piece
+  graded : SetLike.GradedSMul 𝓐 piece
+  nonneg : ∀ n < 0, piece n = ⊥
+  finite : Module.Finite A obj
+
+attribute [instance] ModuleData.kModule ModuleData.tower ModuleData.decomposition
+  ModuleData.graded ModuleData.finite
+
+instance {𝓐 : ℤ → Submodule k A} : CoeSort (ModuleData.{uk,ua,um} 𝓐) (Type (max ua um)) :=
+  ⟨fun M => M.obj⟩
+
+namespace ModuleData
+variable {𝓐 : ℤ → Submodule k A}
+
+instance (M : ModuleData.{uk,ua,um} 𝓐) : AddCommGroup M := inferInstanceAs (AddCommGroup M.obj)
+instance (M : ModuleData.{uk,ua,um} 𝓐) : Module A M := inferInstanceAs (Module A M.obj)
+
+structure Hom (M N : ModuleData.{uk,ua,um} 𝓐) where
+  map : M →ₗ[A] N
+  graded : ∀ n m, m ∈ M.piece n → map m ∈ N.piece n
+
+noncomputable def sub [IsNoetherianRing A] (M : ModuleData.{uk,ua,um} 𝓐)
+    (S : Submodule A M) (hS : ∀ n m, m ∈ S → component M.piece n m ∈ S) :
+    ModuleData.{uk,ua,um} 𝓐 where
+  obj := ModuleCat.of A S
+  kModule := inferInstance
+  tower := inferInstance
+  piece := subPiece M.piece S
+  decomposition := (sub_isInternal M.piece S hS).chooseDecomposition
+  graded := inferInstance
+  nonneg := fun n hn => subPiece_eq_bot M.piece S n (M.nonneg n hn)
+  finite := inferInstance
+
+noncomputable def quotient (M : ModuleData.{uk,ua,um} 𝓐)
+    (S : Submodule A M) (hS : ∀ n m, m ∈ S → component M.piece n m ∈ S) :
+    ModuleData.{uk,ua,um} 𝓐 where
+  obj := ModuleCat.of A (M ⧸ S)
+  kModule := inferInstance
+  tower := inferInstance
+  piece := quotientPiece M.piece S
+  decomposition := (quotient_isInternal M.piece S hS).chooseDecomposition
+  graded := inferInstance
+  nonneg := fun n hn => quotientPiece_eq_bot M.piece S n (M.nonneg n hn)
+  finite := inferInstance
+
+noncomputable def pi {ι : Type um} [Fintype ι] (M : ι → ModuleData.{uk,ua,um} 𝓐) :
+    ModuleData.{uk,ua,um} 𝓐 where
+  obj := ModuleCat.of A (∀ i, M i)
+  kModule := inferInstance
+  tower := inferInstance
+  piece := piPiece fun i => (M i).piece
+  decomposition := (pi_isInternal (fun i => (M i).piece)).chooseDecomposition
+  graded := inferInstance
+  nonneg := fun n hn => piPiece_eq_bot (fun i => (M i).piece) n (fun i => (M i).nonneg n hn)
+  finite := inferInstance
+
+noncomputable def kernel [IsNoetherianRing A] {M N : ModuleData.{uk,ua,um} 𝓐}
+    (f : Hom M N) : ModuleData.{uk,ua,um} 𝓐 :=
+  sub M f.map.ker (kernel_homogeneous M.piece N.piece f.map 0
+    (by simpa using f.graded))
+
+noncomputable def range [IsNoetherianRing A] {M N : ModuleData.{uk,ua,um} 𝓐}
+    (f : Hom M N) : ModuleData.{uk,ua,um} 𝓐 :=
+  sub N f.map.range (range_homogeneous M.piece N.piece f.map 0
+    (by simpa using f.graded))
+
+noncomputable def scalarQuotient (M : ModuleData.{uk,ua,um} 𝓐)
+    (a : A) (d : ℤ) (ha : a ∈ 𝓐 d) : ModuleData.{uk,ua,um} 𝓐 :=
+  quotient M (LinearMap.range (LinearMap.lsmul A M a))
+    (scalar_range_homogeneous 𝓐 M.piece d a ha)
+
+ def projection {ι : Type um} [Fintype ι] (M : ι → ModuleData.{uk,ua,um} 𝓐) (i : ι) :
+    Hom (pi M) (M i) where
+  map := LinearMap.proj i
+  graded := fun _ _ hm => hm i (Set.mem_univ i)
+
+ def quotientMap (M : ModuleData.{uk,ua,um} 𝓐)
+    (S : Submodule A M) (hS : ∀ n m, m ∈ S → component M.piece n m ∈ S) :
+    Hom M (quotient M S hS) where
+  map := S.mkQ
+  graded := by
+    intro n m hm
+    change S.mkQ m ∈ quotientPiece M.piece S n
+    exact Submodule.mem_map.mpr ⟨m,hm,rfl⟩
+
+ def subtype [IsNoetherianRing A] (M : ModuleData.{uk,ua,um} 𝓐)
+    (S : Submodule A M) (hS : ∀ n m, m ∈ S → component M.piece n m ∈ S) :
+    Hom (sub M S hS) M where
+  map := S.subtype
+  graded := fun _ _ hm => hm
+
+
+ def Hom.comp {M N P : ModuleData.{uk,ua,um} 𝓐} (g : Hom N P) (f : Hom M N) :
+    Hom M P where
+  map := g.map.comp f.map
+  graded := fun n m hm => g.graded n _ (f.graded n m hm)
+
+ def Hom.id (M : ModuleData.{uk,ua,um} 𝓐) : Hom M M where
+  map := LinearMap.id
+  graded := fun _ _ h => h
+
+ theorem Hom.component {M N : ModuleData.{uk,ua,um} 𝓐} (f : Hom M N) (n : ℤ) (m : M) :
+    component N.piece n (f.map m) = f.map (component M.piece n m) := by
+  simpa using map_component M.piece N.piece f.map 0 (by simpa using f.graded) n m
+
+variable [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+
+noncomputable def free {ι : Type (max ua um)} [Fintype ι]
+    (hneg : ∀ n < 0, 𝓐 n = ⊥) (d : ι → ℤ) (hd : ∀ i, 0 ≤ d i) :
+    ModuleData.{uk,ua,um} 𝓐 where
+  obj := ModuleCat.of A (ι → A)
+  kModule := inferInstance
+  tower := inferInstance
+  piece := shiftedPiece 𝓐 d
+  decomposition := (shifted_isInternal 𝓐 d).chooseDecomposition
+  graded := inferInstance
+  nonneg := fun n hn => shiftedPiece_eq_bot 𝓐 hneg d n (fun i => lt_of_lt_of_le hn (hd i))
+  finite := inferInstance
+
+ structure MinimalCover (M : ModuleData.{uk,ua,um} 𝓐) where
+  source : ModuleData.{uk,ua,um} 𝓐
+  free : Module.Free A source
+  hom : Hom source M
+  surjective : Function.Surjective hom.map
+  minimal : LinearMap.ker hom.map ≤ positiveIdeal 𝓐 • (⊤ : Submodule A source)
+
+ theorem exists_minimalCover
+    (hneg : ∀ n < 0, 𝓐 n = ⊥)
+    (hscalar : ∀ a : A, ∃ c : k, a-algebraMap k A c ∈ positiveIdeal 𝓐)
+    (M : ModuleData.{uk,ua,um} 𝓐) : Nonempty (MinimalCover M) := by
+  classical
+  obtain ⟨ι,hι,v,d,hd,hspan,hind⟩ :=
+    exists_homogeneous_minimal_generators 𝓐 M.piece 0 M.nonneg hscalar
+  let : Finite ι := hι
+  let : Fintype ι := Fintype.ofFinite ι
+  have hd0 (i : ι) : 0 ≤ d i := by
+    by_contra h
+    have hbot := M.nonneg (d i) (lt_of_not_ge h)
+    have hv0 : v i = 0 := by simpa only [hbot, Submodule.mem_bot] using hd i
+    have hz := hind.ne_zero i
+    exact hz (by rw [hv0,map_zero])
+  let F := free hneg d hd0
+  let f : Hom F M :=
+    { map := Fintype.linearCombination A v
+      graded := by
+        intro n c hc
+        apply Submodule.sum_mem
+        intro i _
+        simpa using SetLike.GradedSMul.smul_mem (hc i) (hd i) }
+  have hfree : Module.Free A F := inferInstanceAs (Module.Free A (ι → A))
+  refine ⟨⟨F,hfree,f,?_,?_⟩⟩
+  · intro m
+    exact (Submodule.mem_span_range_iff_exists_fun A).mp (hspan ▸ Submodule.mem_top)
+  · exact kernel_minimal_cover _ hscalar v hind
+
+end ModuleData
+end KLInvariance.Graded
+
+end
+
+
+section
+
+/-! Honest nonnegative grading shifts; the underlying integral module is unchanged. -/
+namespace KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A}
+noncomputable section
+
+ theorem shift_isInternal (M : ModuleData.{uk,ua,um} 𝓐) (d : ℤ) :
+    DirectSum.IsInternal (fun n => M.piece (n-d)) := by
+  classical
+  apply internal_of_components _ (fun n => component M.piece (n-d))
+  · intro n m
+    exact (DirectSum.decompose M.piece m (n-d)).property
+  · intro n m hm
+    exact DirectSum.decompose_of_mem_same M.piece hm
+  · intro n j m hnj hm
+    apply DirectSum.decompose_of_mem_ne M.piece hm
+    omega
+  · intro m
+    refine ⟨(DirectSum.decompose M.piece m).support.image (fun n => n+d),?_⟩
+    rw [Finset.sum_image (fun a _ b _ h => by omega)]
+    simp only [add_sub_cancel_right]
+    exact DirectSum.sum_support_decompose M.piece m
+
+namespace ModuleData
+ def shift (M : ModuleData.{uk,ua,um} 𝓐) (d : ℕ) : ModuleData.{uk,ua,um} 𝓐 where
+  obj := M.obj
+  kModule := M.kModule
+  tower := M.tower
+  piece n := M.piece (n-d)
+  decomposition := (shift_isInternal M d).chooseDecomposition
+  graded := ⟨by
+    intro i j a m ha hm
+    change a • m ∈ M.piece (i+j-d)
+    have hh : a • m ∈ M.piece (i+(j-(d:ℤ))) := SetLike.GradedSMul.smul_mem ha hm
+    convert hh using 1
+    congr 1
+    omega⟩
+  nonneg n hn := M.nonneg (n-d) (by omega)
+  finite := M.finite
+
+ def Hom.shift {M N : ModuleData.{uk,ua,um} 𝓐} (f : Hom M N) (d : ℕ) :
+    Hom (M.shift d) (N.shift d) where
+  map := f.map
+  graded n m hm := f.graded (n-d) m hm
+
+ def prod (M N : ModuleData.{uk,ua,um} 𝓐) : ModuleData.{uk,ua,um} 𝓐 where
+  obj := ModuleCat.of A (M × N)
+  kModule := inferInstance
+  tower := inferInstance
+  piece n := (M.piece n).prod (N.piece n)
+  decomposition := (show DirectSum.IsInternal (fun n => (M.piece n).prod (N.piece n)) from by
+    apply internal_of_components _ (fun n =>
+      (component M.piece n).prodMap (component N.piece n))
+    · intro n m
+      exact ⟨(DirectSum.decompose M.piece m.1 n).property,
+        (DirectSum.decompose N.piece m.2 n).property⟩
+    · intro n m hm
+      exact Prod.ext (DirectSum.decompose_of_mem_same M.piece hm.1)
+        (DirectSum.decompose_of_mem_same N.piece hm.2)
+    · intro i j m hij hm
+      exact Prod.ext (DirectSum.decompose_of_mem_ne M.piece hm.1 (Ne.symm hij))
+        (DirectSum.decompose_of_mem_ne N.piece hm.2 (Ne.symm hij))
+    · intro m
+      classical
+      refine ⟨(DirectSum.decompose M.piece m.1).support ∪ (DirectSum.decompose N.piece m.2).support,?_⟩
+      apply Prod.ext
+      · change (∑ n ∈ _, (component M.piece n).prodMap (component N.piece n) m).1=m.1
+        rw [Prod.fst_sum]
+        rw [← Finset.sum_subset Finset.subset_union_left]
+        · exact DirectSum.sum_support_decompose M.piece m.1
+        · intro n _ hn
+          have hh : (DirectSum.decompose M.piece m.1) n=0 := DFinsupp.notMem_support_iff.mp hn
+          exact congrArg (fun z : M.piece n => (z : M)) hh
+      · change (∑ n ∈ _, (component M.piece n).prodMap (component N.piece n) m).2=m.2
+        rw [Prod.snd_sum]
+        rw [← Finset.sum_subset Finset.subset_union_right]
+        · exact DirectSum.sum_support_decompose N.piece m.2
+        · intro n _ hn
+          have hh : (DirectSum.decompose N.piece m.2) n=0 := DFinsupp.notMem_support_iff.mp hn
+          exact congrArg (fun z : N.piece n => (z : N)) hh).chooseDecomposition
+  graded := ⟨by
+    intro i j a m ha hm
+    exact ⟨SetLike.GradedSMul.smul_mem ha hm.1,SetLike.GradedSMul.smul_mem ha hm.2⟩⟩
+  nonneg n hn := by simp [M.nonneg n hn,N.nonneg n hn]
+  finite := inferInstance
+
+ def Hom.prodMap {M N P Q : ModuleData.{uk,ua,um} 𝓐} (f : Hom M P) (g : Hom N Q) :
+    Hom (M.prod N) (P.prod Q) where
+  map := f.map.prodMap g.map
+  graded n _ hm := ⟨f.graded n _ hm.1,g.graded n _ hm.2⟩
+
+end ModuleData
+end
+end KLInvariance.Graded
+
+end
+
+
+section
+
+/-! Homogeneous projectivity and uniqueness of the actual minimal graded
+covers. These are recognition dependencies, not assumptions of the main. -/
+namespace KLInvariance.Graded.ModuleData
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A}
+
+ theorem Hom.mem_of_map_mem {M N : ModuleData.{uk,ua,um} 𝓐} (f : Hom M N)
+    (hi : Function.Injective f.map) (n : ℤ) (m : M) (hm : f.map m ∈ N.piece n) :
+    m ∈ M.piece n := by
+  have h : f.map (Graded.component M.piece n m) = f.map m := by
+    rw [← f.component]
+    exact DirectSum.decompose_of_mem_same N.piece hm
+  rw [← hi h]
+  exact (DirectSum.decompose M.piece m n).property
+
+ theorem Hom.homogeneous_preimage {M N : ModuleData.{uk,ua,um} 𝓐} (f : Hom M N)
+    (hs : Function.Surjective f.map) (n : ℤ) (v : N) (hv : v ∈ N.piece n) :
+    ∃ m : M, m ∈ M.piece n ∧ f.map m = v := by
+  obtain ⟨m,rfl⟩ := hs v
+  refine ⟨Graded.component M.piece n m,(DirectSum.decompose M.piece m n).property,?_⟩
+  rw [← f.component]
+  exact DirectSum.decompose_of_mem_same N.piece hv
+
+variable [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+  (hneg : ∀ n < 0, 𝓐 n = ⊥)
+
+include hneg in
+ theorem exists_hom_lift_of_basis [Nontrivial A] {F N P : ModuleData.{uk,ua,um} 𝓐}
+    {ι : Type (max ua um)} [Fintype ι] (b : Module.Basis ι A F)
+    (d : ι → ℤ) (hd : ∀ i, b i ∈ F.piece (d i))
+    (f : Hom N P) (hs : Function.Surjective f.map) (g : Hom F P) :
+    ∃ l : Hom F N, f.map.comp l.map = g.map := by
+  classical
+  have hd0 (i : ι) : 0 ≤ d i := by
+    by_contra h
+    have hz : b i = 0 := by simpa [F.nonneg (d i) (lt_of_not_ge h)] using hd i
+    exact b.ne_zero i hz
+  let H := free.{uk,ua,um} hneg d hd0
+  let e : Hom H F :=
+    { map := b.equivFun.symm.toLinearMap
+      graded := by
+        intro n v hv
+        change ι → A at v
+        change v ∈ shiftedPiece 𝓐 d n at hv
+        change b.equivFun.symm v ∈ F.piece n
+        rw [b.equivFun_symm_apply]
+        apply Submodule.sum_mem
+        intro i _
+        simpa using SetLike.GradedSMul.smul_mem (hv i) (hd i) }
+  have he : Function.Injective e.map := b.equivFun.symm.injective
+  have hcoeff (n : ℤ) (v : F) (hv : v ∈ F.piece n) :
+      b.equivFun v ∈ shiftedPiece 𝓐 d n := by
+    apply e.mem_of_map_mem he n
+    change b.equivFun.symm (b.equivFun v) ∈ F.piece n
+    simpa only [LinearEquiv.symm_apply_apply] using hv
+  have hlifts (i : ι) : ∃ v : N, v ∈ N.piece (d i) ∧ f.map v = g.map (b i) :=
+    f.homogeneous_preimage hs (d i) _ (g.graded (d i) _ (hd i))
+  choose v hvg hvf using hlifts
+  let l : Hom F N :=
+    { map := b.constr A v
+      graded := by
+        intro n m hm
+        rw [b.constr_apply]
+        apply Submodule.sum_mem
+        intro i _
+        simpa using SetLike.GradedSMul.smul_mem (hcoeff n m hm i) (hvg i) }
+  refine ⟨l,?_⟩
+  apply b.ext
+  intro i
+  change f.map ((b.constr A v) (b i)) = g.map (b i)
+  rw [b.constr_basis,hvf]
+
+variable [IsDomain A] [IsNoetherianRing A]
+  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
+
+include hneg hzero in
+ theorem exists_hom_lift {F N P : ModuleData.{uk,ua,um} 𝓐}
+    [Module.Free A F] (f : Hom N P) (hs : Function.Surjective f.map) (g : Hom F P) :
+    ∃ l : Hom F N, f.map.comp l.map = g.map := by
+  classical
+  obtain ⟨ι,hι,b,hd⟩ := exists_homogeneous_basis_connected 𝓐 F.piece hneg hzero 0 F.nonneg
+  let : Finite ι := hι
+  let : Fintype ι := Fintype.ofFinite ι
+  choose d hd using hd
+  exact exists_hom_lift_of_basis hneg b d hd f hs g
+
+omit [IsDomain A] [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐] hneg hzero in
+ theorem Hom.bijective_of_mod_positive {M : ModuleData.{uk,ua,um} 𝓐} (f : Hom M M)
+    (hm : ∀ m, m-f.map m ∈ positiveIdeal 𝓐 • (⊤ : Submodule A M)) :
+    Function.Bijective f.map := by
+  have hr : LinearMap.range f.map = ⊤ :=
+    eq_top_of_positive_smul_sup 𝓐 M.piece 0 M.nonneg (LinearMap.range f.map)
+      (range_homogeneous M.piece M.piece f.map 0 (by simpa using f.graded))
+      (by
+        apply top_unique
+        intro m _
+        exact Submodule.mem_sup.mpr ⟨f.map m,⟨m,rfl⟩,m-f.map m,hm m,add_sub_cancel _ _⟩)
+  have hs := LinearMap.range_eq_top.mp hr
+  exact ⟨IsNoetherian.injective_of_surjective_endomorphism f.map hs,hs⟩
+
+include hneg hzero in
+ theorem MinimalCover.equiv {M : ModuleData.{uk,ua,um} 𝓐} (C D : MinimalCover M) :
+    ∃ e : C.source ≃ₗ[A] D.source,
+      (∀ n v, v ∈ C.source.piece n → e v ∈ D.source.piece n) ∧
+      D.hom.map.comp e.toLinearMap = C.hom.map := by
+  let := C.free
+  let := D.free
+  obtain ⟨f,hf⟩ := exists_hom_lift hneg hzero D.hom D.surjective C.hom
+  obtain ⟨g,hg⟩ := exists_hom_lift hneg hzero C.hom C.surjective D.hom
+  have hfa (v : C.source) : D.hom.map (f.map v)=C.hom.map v := LinearMap.congr_fun hf v
+  have hga (v : D.source) : C.hom.map (g.map v)=D.hom.map v := LinearMap.congr_fun hg v
+  have hgc : Function.Bijective (g.comp f).map := by
+    apply Hom.bijective_of_mod_positive
+    intro m
+    apply C.minimal
+    change C.hom.map (m-g.map (f.map m))=0
+    rw [map_sub,hga,hfa,sub_self]
+  have hfd : Function.Bijective (f.comp g).map := by
+    apply Hom.bijective_of_mod_positive
+    intro m
+    apply D.minimal
+    change D.hom.map (m-f.map (g.map m))=0
+    rw [map_sub,hfa,hga,sub_self]
+  have hfb : Function.Bijective f.map :=
+    ⟨Function.Injective.of_comp hgc.1,Function.Surjective.of_comp hfd.2⟩
+  exact ⟨LinearEquiv.ofBijective f.map hfb,f.graded,hf⟩
+
+end KLInvariance.Graded.ModuleData
+
+end
+
+
+section
+
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/KazhdanLusztig/GradedSheaves.lean b/lean/OAI/RepresentationTheory/KazhdanLusztig/GradedSheaves.lean
index a7a33c5971bac24545ed0744a6b8093beb986515..f760f67863a4b95b3f6311441223078ac587d1ec 100644
--- a/lean/OAI/RepresentationTheory/KazhdanLusztig/GradedSheaves.lean
+++ b/lean/OAI/RepresentationTheory/KazhdanLusztig/GradedSheaves.lean
@@ -1,1252 +1,1252 @@
-import OAI.RepresentationTheory.KazhdanLusztig.Sheaves
-import OAI.RepresentationTheory.KazhdanLusztig.GradedModules
-import OAI.RepresentationTheory.KazhdanLusztig.ReflectionSchedules
-
-namespace OAI
-
-/-!
-Graded moment-graph sheaves, boundary sections, graded covers and morphisms.
--/
-
-section
-
-namespace KLInvariance.MomentGraph
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A}
-  {V E : Type um} [PartialOrder V] (G : OrderedGraph V E)
-
-structure GradedSheaf where
-  vertex : V → ModuleData.{uk,ua,um} 𝓐
-  edge : E → ModuleData.{uk,ua,um} 𝓐
-  lower : ∀ e, ModuleData.Hom (vertex (G.source e)) (edge e)
-  upper : ∀ e, ModuleData.Hom (vertex (G.target e)) (edge e)
-
-namespace GradedSheaf
-variable {G} (B : GradedSheaf (𝓐 := 𝓐) G)
-
-def forget : Sheaf (R := A) G where
-  vertex := fun x => (B.vertex x).obj
-  edge := fun e => (B.edge e).obj
-  lower := fun e => (B.lower e).map
-  upper := fun e => (B.upper e).map
-
-/-- Extension for sets closed under actual edges. It implies the source's
-ordinary flabbiness, and is convenient for the descending construction. -/
-def EdgeFlabby : Prop :=
-  ∀ Ω : Set V, B.forget.ClosedUnder Set.univ Ω →
-    ∀ f : B.forget.Assignment, B.forget.CompatibleOn Ω f →
-    ∃ g : B.forget.Assignment, B.forget.IsGlobal g ∧ ∀ x ∈ Ω, g x = f x
-
- theorem EdgeFlabby.flabby (h : B.EdgeFlabby) : B.forget.Flabby := by
-  intro Ω hΩ f hf
-  exact h Ω (fun e _ he => hΩ (G.increasing e).le he) f hf
-
-variable [Fintype V] [Fintype E]
-
-noncomputable def assignments : ModuleData.{uk,ua,um} 𝓐 := ModuleData.pi B.vertex
-
-noncomputable def lowerGlobal (e : E) : ModuleData.Hom B.assignments (B.edge e) :=
-  (B.lower e).comp (ModuleData.projection B.vertex (G.source e))
-
-noncomputable def upperGlobal (e : E) : ModuleData.Hom B.assignments (B.edge e) :=
-  (B.upper e).comp (ModuleData.projection B.vertex (G.target e))
-
-omit [Fintype E] in
- theorem sections_homogeneous (Ω : Set V) :
-    ∀ n f, f ∈ B.forget.sections Ω → component B.assignments.piece n f ∈ B.forget.sections Ω := by
-  intro n f hf e hs ht
-  change (B.lowerGlobal e).map (component B.assignments.piece n f) =
-    (B.upperGlobal e).map (component B.assignments.piece n f)
-  rw [← (B.lowerGlobal e).component n f, ← (B.upperGlobal e).component n f]
-  exact congrArg (component (B.edge e).piece n) (hf e hs ht)
-
-noncomputable def sections [IsNoetherianRing A] (Ω : Set V) : ModuleData.{uk,ua,um} 𝓐 :=
-  ModuleData.sub B.assignments (B.forget.sections Ω) (B.sections_homogeneous Ω)
-
-noncomputable def boundary (x : V) : ModuleData.{uk,ua,um} 𝓐 :=
-by
-  classical
-  letI : Fintype (Sheaf.Outgoing (G := G) x) := inferInstanceAs (Fintype {e : E // G.source e = x})
-  exact ModuleData.pi (fun e : Sheaf.Outgoing (G := G) x => B.edge e.val)
-
-noncomputable def boundaryHom [IsNoetherianRing A] (x : V) :
-    ModuleData.Hom (B.sections (Set.Ioi x)) (B.boundary x) where
-  map := B.forget.boundaryMap x
-  graded := by
-    intro n f hf e _
-    exact (B.upper e.val).graded n (f.val (G.target e.val)) (hf (G.target e.val) (Set.mem_univ _))
-
-noncomputable def boundaryImage [IsNoetherianRing A] (x : V) : ModuleData.{uk,ua,um} 𝓐 :=
-  ModuleData.range (B.boundaryHom x)
-
- theorem exists_boundaryCover [IsNoetherianRing A]
-    [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-    (hneg : ∀ n < 0, 𝓐 n = ⊥)
-    (hscalar : ∀ a : A, ∃ c : k, a-algebraMap k A c ∈ positiveIdeal 𝓐) (x : V) :
-    Nonempty (ModuleData.MinimalCover (B.boundaryImage x)) :=
-  ModuleData.exists_minimalCover hneg hscalar (B.boundaryImage x)
-
-end GradedSheaf
-end KLInvariance.MomentGraph
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A}
-  {V E D : Type um} [PartialOrder V] (G : OrderedGraph V E)
-
-/-- Adjoin a vertex below the old vertices, with specified outgoing edges.
-Only the edge orientations, not the extra comparabilities, enter sections. -/
-def bottomGraph (τ : D → V) : OrderedGraph (WithBot V) (E ⊕ D) where
-  source := Sum.elim (fun e => (G.source e : WithBot V)) (fun _ => ⊥)
-  target := Sum.elim (fun e => (G.target e : WithBot V)) (fun d => (τ d : WithBot V))
-  increasing := by
-    intro e
-    cases e with
-    | inl e => exact WithBot.coe_lt_coe.mpr (G.increasing e)
-    | inr d => exact WithBot.bot_lt_coe (τ d)
-
-namespace GradedSheaf
-variable {G} (B : GradedSheaf (𝓐 := 𝓐) G)
-  [Fintype V] [Fintype D] [IsNoetherianRing A]
-  (τ : D → V) (α : D → A) (hα : ∀ d, α d ∈ 𝓐 1)
-
-noncomputable def attachEdges (d : D) : ModuleData.{uk,ua,um} 𝓐 :=
-  (B.vertex (τ d)).scalarQuotient (α d) 1 (hα d)
-
-noncomputable def attachQuotient (d : D) :
-    ModuleData.Hom (B.vertex (τ d)) (B.attachEdges τ α hα d) :=
-  ModuleData.quotientMap _ _ (scalar_range_homogeneous 𝓐 (B.vertex (τ d)).piece 1 (α d) (hα d))
-
-noncomputable def attachBoundary : ModuleData.{uk,ua,um} 𝓐 :=
-  ModuleData.pi (B.attachEdges τ α hα)
-
-noncomputable def attachBoundaryMap :
-    ModuleData.Hom (B.sections Set.univ) (B.attachBoundary τ α hα) where
-  map := LinearMap.pi fun d => (B.attachQuotient τ α hα d).map.comp
-    ((LinearMap.proj (τ d)).comp (B.forget.sections Set.univ).subtype)
-  graded := by
-    intro n f hf d _
-    exact (B.attachQuotient τ α hα d).graded n _ (hf (τ d) (Set.mem_univ _))
-
-noncomputable def attachImage : ModuleData.{uk,ua,um} 𝓐 :=
-  ModuleData.range (B.attachBoundaryMap τ α hα)
-
-variable (C : ModuleData.MinimalCover (B.attachImage τ α hα))
-
-noncomputable def attachStalkMap : ModuleData.Hom C.source (B.attachBoundary τ α hα) :=
-  (ModuleData.subtype (B.attachBoundary τ α hα) _
-    (range_homogeneous _ _ (B.attachBoundaryMap τ α hα).map 0
-      (by simpa using (B.attachBoundaryMap τ α hα).graded))).comp C.hom
-
-noncomputable def extendBottom : GradedSheaf (𝓐 := 𝓐) (bottomGraph G τ) where
-  vertex := WithBot.recBotCoe C.source B.vertex
-  edge := Sum.elim B.edge (B.attachEdges τ α hα)
-  lower := by
-    intro e
-    cases e with
-    | inl e => exact B.lower e
-    | inr d =>
-      exact (ModuleData.projection (B.attachEdges τ α hα) d).comp
-        (B.attachStalkMap τ α hα C)
-  upper := by
-    intro e
-    cases e with
-    | inl e => exact B.upper e
-    | inr d => exact B.attachQuotient τ α hα d
-
- def attachAssignment (z : C.source) (f : B.forget.Assignment) :
-    (B.extendBottom τ α hα C).forget.Assignment := WithBot.recBotCoe z f
-
- theorem attachAssignment_global (z : C.source) (f : B.forget.Assignment)
-    (hf : B.forget.IsGlobal f)
-    (hz : ∀ d, (B.attachStalkMap τ α hα C).map z d =
-      (B.attachQuotient τ α hα d).map (f (τ d))) :
-    (B.extendBottom τ α hα C).forget.IsGlobal (B.attachAssignment τ α hα C z f) := by
-  intro e _ _
-  cases e with
-  | inl e => exact hf e (Set.mem_univ _) (Set.mem_univ _)
-  | inr d => exact hz d
-
- theorem lift_global (f : B.forget.Assignment) (hf : B.forget.IsGlobal f) :
-    ∃ z : C.source, (B.extendBottom τ α hα C).forget.IsGlobal
-      (B.attachAssignment τ α hα C z f) := by
-  let s : B.sections Set.univ := ⟨f,hf⟩
-  let v : B.attachImage τ α hα :=
-    ⟨(B.attachBoundaryMap τ α hα).map s, ⟨s,rfl⟩⟩
-  obtain ⟨z,hz⟩ := C.surjective v
-  refine ⟨z,B.attachAssignment_global τ α hα C z f hf ?_⟩
-  intro d
-  exact congrArg (fun t : B.attachImage τ α hα => t.val d) hz
-
- theorem extendBottom_generated (hg : B.forget.Generated) :
-    (B.extendBottom τ α hα C).forget.Generated := by
-  intro x
-  induction x using WithBot.recBotCoe with
-  | bot =>
-    intro z
-    obtain ⟨s,hs⟩ := (C.hom.map z).property
-    refine ⟨B.attachAssignment τ α hα C z s.val,
-      B.attachAssignment_global τ α hα C z s.val s.property ?_,rfl⟩
-    intro d
-    exact congrFun hs d |>.symm
-  | coe x =>
-    intro v
-    obtain ⟨f,hf,hfx⟩ := hg x v
-    obtain ⟨z,hz⟩ := B.lift_global τ α hα C f hf
-    exact ⟨B.attachAssignment τ α hα C z f,hz,hfx⟩
-
- theorem extendBottom_edgeFlabby (hf : B.EdgeFlabby) :
-    (B.extendBottom τ α hα C).EdgeFlabby := by
-  intro Ω hΩ f hcompat
-  let Ω' : Set V := {x | (x : WithBot V) ∈ Ω}
-  let f' : B.forget.Assignment := fun x => f (x : WithBot V)
-  have hc : B.forget.ClosedUnder Set.univ Ω' := by
-    intro e _ hs
-    exact hΩ (Sum.inl e) (Set.mem_univ _) hs
-  have hcf : B.forget.CompatibleOn Ω' f' := by
-    intro e hs ht
-    exact hcompat (Sum.inl e) hs ht
-  obtain ⟨g,hg,hgf⟩ := hf Ω' hc f' hcf
-  classical
-  by_cases hb : (⊥ : WithBot V) ∈ Ω
-  · refine ⟨B.attachAssignment τ α hα C (f ⊥) g,
-      B.attachAssignment_global τ α hα C (f ⊥) g hg ?_,?_⟩
-    · intro d
-      have hd : (τ d : WithBot V) ∈ Ω := hΩ (Sum.inr d) (Set.mem_univ _) hb
-      rw [hgf (τ d) hd]
-      exact hcompat (Sum.inr d) hb hd
-    · intro x hx
-      induction x using WithBot.recBotCoe with
-      | bot => rfl
-      | coe x => exact hgf x hx
-  · obtain ⟨z,hz⟩ := B.lift_global τ α hα C g hg
-    refine ⟨B.attachAssignment τ α hα C z g,hz,?_⟩
-    intro x hx
-    induction x using WithBot.recBotCoe with
-    | bot => exact (hb hx).elim
-    | coe x => exact hgf x hx
-
-end GradedSheaf
-end KLInvariance.MomentGraph
-
-end
-
-
-section
-
-namespace KLInvariance.Graded.ModuleData
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A}
-
-def Hom.ofEq {M N : ModuleData.{uk,ua,um} 𝓐} (h : M = N) : Hom M N := h ▸ Hom.id M
-
-@[simp] theorem Hom.ofEq_rfl (M : ModuleData.{uk,ua,um} 𝓐) : Hom.ofEq (rfl : M=M) = Hom.id M := rfl
-
-theorem Hom.ofEq_apply {ι : Type*} (M : ι → ModuleData.{uk,ua,um} 𝓐)
-    {x y : ι} (h : x = y) (v : ∀ z, M z) :
-    (Hom.ofEq (congrArg M h)).map (v x) = v y := by subst y; rfl
-
-end KLInvariance.Graded.ModuleData
-
-namespace KLInvariance.MomentGraph
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A}
-  {V E W F : Type um} [PartialOrder V] [PartialOrder W]
-  {G : OrderedGraph V E} {H : OrderedGraph W F}
-
-/-- An isomorphism of directed graphs, with no order-reflection requirement. -/
-structure GraphEquiv (G : OrderedGraph V E) (H : OrderedGraph W F) where
-  vertex : V ≃ W
-  edge : E ≃ F
-  source : ∀ e, vertex (G.source e) = H.source (edge e)
-  target : ∀ e, vertex (G.target e) = H.target (edge e)
-
-namespace GradedSheaf
-variable (B : GradedSheaf (𝓐 := 𝓐) H) (e : GraphEquiv G H)
-
-def reindex : GradedSheaf (𝓐 := 𝓐) G where
-  vertex := fun x => B.vertex (e.vertex x)
-  edge := fun a => B.edge (e.edge a)
-  lower := fun a => (B.lower (e.edge a)).comp (ModuleData.Hom.ofEq (congrArg B.vertex (e.source a)))
-  upper := fun a => (B.upper (e.edge a)).comp (ModuleData.Hom.ofEq (congrArg B.vertex (e.target a)))
-
-def reindexAssignment (f : B.forget.Assignment) : (B.reindex e).forget.Assignment :=
-  fun x => f (e.vertex x)
-
-@[simp] theorem reindex_lower (a : E) (f : B.forget.Assignment) :
-    (B.reindex e).forget.lower a (B.reindexAssignment e f (G.source a)) =
-      B.forget.lower (e.edge a) (f (H.source (e.edge a))) := by
-  change (B.lower (e.edge a)).map ((ModuleData.Hom.ofEq (congrArg B.vertex (e.source a))).map
-    (f (e.vertex (G.source a)))) = _
-  exact congrArg (B.lower (e.edge a)).map
-    (ModuleData.Hom.ofEq_apply B.vertex (e.source a) f)
-
-@[simp] theorem reindex_upper (a : E) (f : B.forget.Assignment) :
-    (B.reindex e).forget.upper a (B.reindexAssignment e f (G.target a)) =
-      B.forget.upper (e.edge a) (f (H.target (e.edge a))) := by
-  change (B.upper (e.edge a)).map ((ModuleData.Hom.ofEq (congrArg B.vertex (e.target a))).map
-    (f (e.vertex (G.target a)))) = _
-  exact congrArg (B.upper (e.edge a)).map
-    (ModuleData.Hom.ofEq_apply B.vertex (e.target a) f)
-
-def unreindexAssignment (f : (B.reindex e).forget.Assignment) : B.forget.Assignment :=
-  fun y => (ModuleData.Hom.ofEq (congrArg B.vertex (e.vertex.apply_symm_apply y))).map
-    (f (e.vertex.symm y))
-
-@[simp] theorem unreindex_reindexAssignment (f : B.forget.Assignment) :
-    B.unreindexAssignment e (B.reindexAssignment e f) = f := by
-  funext y
-  exact ModuleData.Hom.ofEq_apply B.vertex (e.vertex.apply_symm_apply y) f
-
-@[simp] theorem reindex_unreindexAssignment (f : (B.reindex e).forget.Assignment) :
-    B.reindexAssignment e (B.unreindexAssignment e f) = f := by
-  funext x
-  change (ModuleData.Hom.ofEq (congrArg B.vertex (e.vertex.apply_symm_apply (e.vertex x)))).map
-    (f (e.vertex.symm (e.vertex x))) = f x
-  have h : e.vertex.symm (e.vertex x) = x := e.vertex.symm_apply_apply x
-  have hh : (ModuleData.Hom.ofEq
-      (congrArg (fun t => B.vertex (e.vertex t)) h)).map
-      (f (e.vertex.symm (e.vertex x))) = f x :=
-    ModuleData.Hom.ofEq_apply (fun t => B.vertex (e.vertex t)) h f
-  exact hh
-
- theorem reindex_global (f : B.forget.Assignment) (hf : B.forget.IsGlobal f) :
-    (B.reindex e).forget.IsGlobal (B.reindexAssignment e f) := by
-  intro a _ _
-  rw [B.reindex_lower, B.reindex_upper]
-  exact hf (e.edge a) (Set.mem_univ _) (Set.mem_univ _)
-
- theorem reindex_generated (hg : B.forget.Generated) : (B.reindex e).forget.Generated := by
-  intro x v
-  obtain ⟨f,hf,hfx⟩ := hg (e.vertex x) v
-  exact ⟨B.reindexAssignment e f,B.reindex_global e f hf,hfx⟩
-
- theorem reindex_edgeFlabby (hb : B.EdgeFlabby) : (B.reindex e).EdgeFlabby := by
-  intro Ω hΩ f hf
-  let Ω' := e.vertex '' Ω
-  have hc : B.forget.ClosedUnder Set.univ Ω' := by
-    intro a _ hs
-    obtain ⟨a,rfl⟩ := e.edge.surjective a
-    obtain ⟨x,hx,hxs⟩ := hs
-    have hx' : x = G.source a := e.vertex.injective (hxs.trans (e.source a).symm)
-    exact ⟨G.target a,hΩ a (Set.mem_univ _) (hx' ▸ hx),e.target a⟩
-  have hcf : B.forget.CompatibleOn Ω' (B.unreindexAssignment e f) := by
-    intro a hs ht
-    obtain ⟨a,rfl⟩ := e.edge.surjective a
-    have hh (x : V) (h : e.vertex x ∈ Ω') : x ∈ Ω := by
-      obtain ⟨y,hy,hyx⟩ := h
-      exact (e.vertex.injective hyx) ▸ hy
-    have hfa := hf a (hh _ ((e.source a).symm ▸ hs)) (hh _ ((e.target a).symm ▸ ht))
-    rw [← B.reindex_unreindexAssignment e f] at hfa
-    exact (B.reindex_lower e a (B.unreindexAssignment e f)).symm.trans
-      (hfa.trans (B.reindex_upper e a (B.unreindexAssignment e f)))
-  obtain ⟨g,hg,hgf⟩ := hb Ω' hc (B.unreindexAssignment e f) hcf
-  refine ⟨B.reindexAssignment e g,B.reindex_global e g hg,?_⟩
-  intro x hx
-  have hh := hgf (e.vertex x) ⟨x,hx,rfl⟩
-  change g (e.vertex x) = f x
-  rw [hh]
-  exact congrFun (B.reindex_unreindexAssignment e f) x
-
-end GradedSheaf
-end KLInvariance.MomentGraph
-
-end
-
-
-section
-
-namespace KLInvariance.Graded.ModuleData
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A}
-
-theorem Hom.ofEq_bijective {M N : ModuleData.{uk,ua,um} 𝓐} (h : M=N) :
-    Function.Bijective (Hom.ofEq h).map := by subst N; exact Function.bijective_id
-
-noncomputable def unit [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-    (hneg : ∀ n < 0, 𝓐 n = ⊥) : ModuleData.{uk,ua,um} 𝓐 :=
-  free (ι := PUnit.{max ua um + 1}) hneg (fun _ => 0) (fun _ => le_rfl)
-
-instance unit_free [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-    (hneg : ∀ n < 0, 𝓐 n = ⊥) : Module.Free A (unit.{uk,ua,um} hneg) :=
-  inferInstanceAs (Module.Free A (PUnit.{max ua um + 1} → A))
-
-end KLInvariance.Graded.ModuleData
-
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A}
-  {V E W F : Type um} [PartialOrder V] [PartialOrder W]
-  {G : OrderedGraph V E} {H : OrderedGraph W F}
-
- def FreeStalks (B : GradedSheaf (𝓐 := 𝓐) G) : Prop := ∀ x, Module.Free A (B.vertex x)
-
-/-- The genuine upper-end quotient rule, expressed without choosing a quotient
-presentation of the edge object. -/
- def UpperQuotient (B : GradedSheaf (𝓐 := 𝓐) G) (α : E → A) : Prop :=
-  ∀ e, Function.Surjective (B.upper e).map ∧
-    ∀ v, (B.upper e).map v = 0 ↔ ∃ w, v = α e • w
-
-/-- Minimality of the full outgoing boundary cover. The assignment formulation
-avoids dependent casts and is equivalent to the usual kernel containment. -/
- def KernelMinimal (B : GradedSheaf (𝓐 := 𝓐) G) (x : V) : Prop :=
-  ∀ f : B.forget.Assignment,
-    (∀ e, G.source e = x → B.forget.lower e (f (G.source e)) = 0) →
-    f x ∈ positiveIdeal 𝓐 • (⊤ : Submodule A (B.vertex x))
-
- theorem kernelMinimal_iff (B : GradedSheaf (𝓐 := 𝓐) G) (x : V) :
-    B.KernelMinimal x ↔ LinearMap.ker (B.forget.stalkMap x) ≤
-      positiveIdeal 𝓐 • (⊤ : Submodule A (B.vertex x)) := by
-  classical
-  constructor
-  · intro h v hv
-    let f : B.forget.Assignment := Function.update 0 x v
-    have hf : f x = v := by simp [f]
-    rw [← hf]
-    apply h f
-    intro e he
-    have hh := congrFun (LinearMap.mem_ker.mp hv) (⟨e,he⟩ : Sheaf.Outgoing x)
-    subst x
-    simpa [Sheaf.stalkMap,f] using hh
-  · intro h f hf
-    apply h
-    apply LinearMap.mem_ker.mpr
-    funext e
-    rcases e with ⟨e,he⟩
-    subst x
-    exact hf e rfl
-
- theorem reindex_freeStalks (B : GradedSheaf (𝓐 := 𝓐) H) (e : GraphEquiv G H)
-    (h : B.FreeStalks) : (B.reindex e).FreeStalks := fun x => h (e.vertex x)
-
- theorem reindex_upperQuotient (B : GradedSheaf (𝓐 := 𝓐) H) (e : GraphEquiv G H)
-    {α : F → A} (h : B.UpperQuotient α) : (B.reindex e).UpperQuotient (α ∘ e.edge) := by
-  intro a
-  let j := ModuleData.Hom.ofEq (congrArg B.vertex (e.target a))
-  have hj := ModuleData.Hom.ofEq_bijective (congrArg B.vertex (e.target a))
-  refine ⟨(h (e.edge a)).1.comp hj.2,?_⟩
-  intro v
-  change (B.upper (e.edge a)).map (j.map v) = 0 ↔ ∃ w, v = α (e.edge a) • w
-  rw [(h (e.edge a)).2]
-  constructor
-  · rintro ⟨w,hw⟩
-    obtain ⟨z,rfl⟩ := hj.2 w
-    exact ⟨z,hj.1 (hw.trans (map_smul j.map (α (e.edge a)) z).symm)⟩
-  · rintro ⟨w,rfl⟩
-    exact ⟨j.map w,map_smul _ _ _⟩
-
- theorem reindex_kernelMinimal (B : GradedSheaf (𝓐 := 𝓐) H) (e : GraphEquiv G H)
-    (x : V) (h : B.KernelMinimal (e.vertex x)) : (B.reindex e).KernelMinimal x := by
-  intro f hf
-  have hh : ∀ a, H.source a = e.vertex x →
-      B.forget.lower a (B.unreindexAssignment e f (H.source a)) = 0 := by
-    intro a ha
-    obtain ⟨a,rfl⟩ := e.edge.surjective a
-    have hs : G.source a = x := e.vertex.injective ((e.source a).trans ha)
-    have hv := hf a hs
-    rw [← B.reindex_unreindexAssignment e f] at hv
-    exact (B.reindex_lower e a (B.unreindexAssignment e f)).symm.trans hv
-  have hm := h (B.unreindexAssignment e f) hh
-  have he := congrFun (B.reindex_unreindexAssignment e f) x
-  exact he ▸ hm
-
-variable {D : Type um} [Fintype V] [Fintype D] [IsNoetherianRing A]
-  (B : GradedSheaf (𝓐 := 𝓐) G) (τ : D → V) (α : D → A)
-  (hα : ∀ d, α d ∈ 𝓐 1) (C : ModuleData.MinimalCover (B.attachImage τ α hα))
-
- theorem extendBottom_freeStalks (hb : B.FreeStalks) :
-    (B.extendBottom τ α hα C).FreeStalks := by
-  intro x
-  induction x using WithBot.recBotCoe with
-  | bot => exact C.free
-  | coe x => exact hb x
-
- theorem extendBottom_upperQuotient {αold : E → A} (hb : B.UpperQuotient αold) :
-    (B.extendBottom τ α hα C).UpperQuotient (Sum.elim αold α) := by
-  intro e
-  cases e with
-  | inl e => exact hb e
-  | inr d =>
-    let S := LinearMap.range (LinearMap.lsmul A (B.vertex (τ d)) (α d))
-    refine ⟨S.mkQ_surjective,?_⟩
-    intro v
-    change S.mkQ v = 0 ↔ ∃ w, v = α d • w
-    refine (Submodule.Quotient.mk_eq_zero S).trans ?_
-    change (∃ w, α d • w = v) ↔ ∃ w, v = α d • w
-    exact ⟨fun ⟨w,hw⟩ => ⟨w,hw.symm⟩,fun ⟨w,hw⟩ => ⟨w,hw.symm⟩⟩
-
- theorem extendBottom_kernelMinimal_bot : (B.extendBottom τ α hα C).KernelMinimal ⊥ := by
-  intro f hf
-  apply C.minimal
-  apply LinearMap.mem_ker.mpr
-  apply Subtype.ext
-  funext d
-  exact hf (Sum.inr d) rfl
-
- theorem extendBottom_kernelMinimal_coe (x : V) (h : B.KernelMinimal x) :
-    (B.extendBottom τ α hα C).KernelMinimal (x : WithBot V) := by
-  intro f hf
-  apply h (fun y => f (y : WithBot V))
-  intro a ha
-  exact hf (Sum.inl a) (congrArg (fun y : V => (y : WithBot V)) ha)
-
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-
-namespace KLInvariance.MomentGraph.Sheaf
-universe u v w z z'
-variable {R : Type u} [CommRing R] {V : Type v} [PartialOrder V]
-  {E : Type w} {G : OrderedGraph V E}
-
-structure Iso (B : Sheaf.{u,v,w,z} (R := R) G) (C : Sheaf.{u,v,w,z'} (R := R) G) where
-  vertex : ∀ x, B.vertex x ≃ₗ[R] C.vertex x
-  edge : ∀ e, B.edge e ≃ₗ[R] C.edge e
-  lower : ∀ e v, edge e (B.lower e v) = C.lower e (vertex (G.source e) v)
-  upper : ∀ e v, edge e (B.upper e v) = C.upper e (vertex (G.target e) v)
-
-namespace Iso
-variable {B : Sheaf.{u,v,w,z} (R := R) G} {C : Sheaf.{u,v,w,z'} (R := R) G}
-  (j : Iso B C)
-
-def symm : Iso C B where
-  vertex x := (j.vertex x).symm
-  edge e := (j.edge e).symm
-  lower e v := by
-    apply (j.edge e).injective
-    simp only [LinearEquiv.apply_symm_apply, j.lower]
-  upper e v := by
-    apply (j.edge e).injective
-    simp only [LinearEquiv.apply_symm_apply, j.upper]
-
-noncomputable def assignments : B.Assignment ≃ₗ[R] C.Assignment :=
-  LinearEquiv.piCongrRight j.vertex
-
-theorem compatible_iff (Ω : Set V) (f : B.Assignment) :
-    C.CompatibleOn Ω (j.assignments f) ↔ B.CompatibleOn Ω f := by
-  constructor <;> intro h e hs ht
-  · apply (j.edge e).injective
-    rw [j.lower,j.upper]
-    exact h e hs ht
-  · change C.lower e (j.vertex _ (f _)) = C.upper e (j.vertex _ (f _))
-    rw [← j.lower,← j.upper,h e hs ht]
-
-theorem map_sections (Ω : Set V) :
-    (B.sections Ω).map j.assignments.toLinearMap = C.sections Ω := by
-  ext g
-  constructor
-  · rintro ⟨f,hf,rfl⟩
-    exact (j.compatible_iff Ω f).mpr hf
-  · intro hg
-    refine ⟨j.assignments.symm g,?_,j.assignments.apply_symm_apply g⟩
-    apply (j.compatible_iff Ω _).mp
-    change C.CompatibleOn Ω g at hg
-    simpa using hg
-
-noncomputable def sections (Ω : Set V) : B.sections Ω ≃ₗ[R] C.sections Ω :=
-  j.assignments.ofSubmodules _ _ (j.map_sections Ω)
-
-noncomputable def boundary (x : V) : B.Boundary x ≃ₗ[R] C.Boundary x :=
-  LinearEquiv.piCongrRight (fun e => j.edge e.val)
-
-theorem stalkMap (x : V) (v : B.vertex x) :
-    j.boundary x (B.stalkMap x v) = C.stalkMap x (j.vertex x v) := by
-  funext e
-  rcases e with ⟨e,he⟩
-  subst x
-  exact j.lower e v
-
-theorem boundaryMap (x : V) (f : B.sections (Set.Ioi x)) :
-    j.boundary x (B.boundaryMap x f) = C.boundaryMap x (j.sections _ f) := by
-  funext e
-  exact j.upper e.val (f.val (G.target e.val))
-
-theorem map_boundaryRange (x : V) :
-    (LinearMap.range (B.boundaryMap x)).map (j.boundary x).toLinearMap =
-      LinearMap.range (C.boundaryMap x) := by
-  ext v
-  constructor
-  · rintro ⟨w,⟨f,rfl⟩,rfl⟩
-    exact ⟨j.sections _ f,(j.boundaryMap x f).symm⟩
-  · rintro ⟨f,rfl⟩
-    refine ⟨B.boundaryMap x ((j.sections _).symm f),⟨_,rfl⟩,?_⟩
-    change j.boundary x (B.boundaryMap x ((j.sections _).symm f)) = _
-    rw [j.boundaryMap,LinearEquiv.apply_symm_apply]
-
-end Iso
-
-variable (B : Sheaf.{u,v,w,z} (R := R) G)
-
-def upwardKernel (x : V) (F : Set (Outgoing (G := G) x)) : Submodule R (B.vertex x) where
-  carrier := {v | ∀ e ∈ F, B.stalkMap x v e = 0}
-  zero_mem' := by intro e _; simp
-  add_mem' := by intro v w hv hw e he; simp [hv e he,hw e he]
-  smul_mem' := by intro r v hv e he; simp [hv e he]
-
-namespace Iso
-variable {B : Sheaf.{u,v,w,z} (R := R) G} {C : Sheaf.{u,v,w,z'} (R := R) G}
-  (j : Iso B C)
-
-theorem upwardKernel_iff (x : V) (F : Set (Outgoing (G := G) x)) (v : B.vertex x) :
-    j.vertex x v ∈ C.upwardKernel x F ↔ v ∈ B.upwardKernel x F := by
-  constructor <;> intro h e he
-  · apply (j.edge e.val).injective
-    rw [map_zero]
-    exact (congrFun (j.stalkMap x v) e).trans (h e he)
-  · have hh := congrFun (j.stalkMap x v) e
-    change j.edge e.val (B.stalkMap x v e) = _ at hh
-    rw [h e he,map_zero] at hh
-    exact hh.symm
-
-theorem map_upwardKernel (x : V) (F : Set (Outgoing (G := G) x)) :
-    (B.upwardKernel x F).map (j.vertex x).toLinearMap = C.upwardKernel x F := by
-  ext v
-  constructor
-  · rintro ⟨w,hw,rfl⟩
-    exact (j.upwardKernel_iff x F w).mpr hw
-  · intro hv
-    refine ⟨(j.vertex x).symm v,?_,(j.vertex x).apply_symm_apply v⟩
-    apply (j.upwardKernel_iff x F _).mp
-    simpa using hv
-
-theorem map_lowerImage (e : E) (F : Set (Outgoing (G := G) (G.source e))) :
-    ((B.upwardKernel (G.source e) F).map (B.lower e)).map (j.edge e).toLinearMap =
-      (C.upwardKernel (G.source e) F).map (C.lower e) := by
-  rw [← Submodule.map_comp]
-  have h : (j.edge e).toLinearMap.comp (B.lower e) =
-      (C.lower e).comp (j.vertex (G.source e)).toLinearMap := LinearMap.ext (j.lower e)
-  rw [h,Submodule.map_comp,j.map_upwardKernel]
-
-theorem map_upperImage (e : E) (F : Set (Outgoing (G := G) (G.target e))) :
-    ((B.upwardKernel (G.target e) F).map (B.upper e)).map (j.edge e).toLinearMap =
-      (C.upwardKernel (G.target e) F).map (C.upper e) := by
-  rw [← Submodule.map_comp]
-  have h : (j.edge e).toLinearMap.comp (B.upper e) =
-      (C.upper e).comp (j.vertex (G.target e)).toLinearMap := LinearMap.ext (j.upper e)
-  rw [h,Submodule.map_comp,j.map_upwardKernel]
-
-end Iso
-end KLInvariance.MomentGraph.Sheaf
-
-namespace KLInvariance.EmbeddedModule
-variable {R M N : Type*} [CommRing R] [AddCommGroup M] [Module R M]
-  [AddCommGroup N] [Module R N]
-
-def scaled (r : R) (K : Submodule R M) : Submodule R M :=
-  K.map (LinearMap.lsmul R M r)
-
-theorem map_scaled (j : M ≃ₗ[R] N) (r : R) (K : Submodule R M) :
-    (scaled r K).map j.toLinearMap = scaled r (K.map j.toLinearMap) := by
-  simp only [scaled,← Submodule.map_comp]
-  congr 1
-  ext v
-  exact j.map_smul r v
-
-/-- Scalar comparison is transported by one and the same edge isomorphism. -/
-theorem map_eq_scaled_iff (j : M ≃ₗ[R] N) (r : R) (I J : Submodule R M) :
-    I.map j.toLinearMap = scaled r (J.map j.toLinearMap) ↔ I = scaled r J := by
-  rw [← map_scaled]
-  constructor
-  · intro h
-    have h' := congrArg (Submodule.map j.symm.toLinearMap) h
-    simpa only [← Submodule.map_comp,LinearEquiv.symm_comp,
-      Submodule.map_id] using h'
-  · exact congrArg (Submodule.map j.toLinearMap)
-
-end KLInvariance.EmbeddedModule
-
-end
-
-
-section
-
-
-namespace KLInvariance.MomentGraph.Sheaf
-universe u v w z z'
-variable {R : Type u} [CommRing R] {V : Type v} [PartialOrder V]
-  {E : Type w} {G : OrderedGraph V E}
-  (B : Sheaf.{u,v,w,z} (R := R) G) (C : Sheaf.{u,v,w,z'} (R := R) G) (x : V)
-
-/-- An isomorphism already constructed strictly above `x`, including the edge
-modules whose upper endpoints are above `x`. The latter are canonically supplied
-by the upper quotient rule in the local decomposition. -/
-structure AboveIso where
-  vertex : ∀ y, x < y → B.vertex y ≃ₗ[R] C.vertex y
-  edge : ∀ e, x < G.target e → B.edge e ≃ₗ[R] C.edge e
-  lower : ∀ e (h : x < G.source e) v,
-    edge e (h.trans (G.increasing e)) (B.lower e v) =
-      C.lower e (vertex (G.source e) h v)
-  upper : ∀ e (h : x < G.target e) v,
-    edge e h (B.upper e v) = C.upper e (vertex (G.target e) h v)
-
-namespace AboveIso
-variable {B C x} (j : AboveIso B C x)
-
-def symm : AboveIso C B x where
-  vertex y h := (j.vertex y h).symm
-  edge e h := (j.edge e h).symm
-  lower e h v := by
-    apply (j.edge e (h.trans (G.increasing e))).injective
-    rw [LinearEquiv.apply_symm_apply,j.lower e h,LinearEquiv.apply_symm_apply]
-  upper e h v := by
-    apply (j.edge e h).injective
-    simp only [LinearEquiv.apply_symm_apply,j.upper]
-
-noncomputable def mapAssignment (f : B.Assignment) : C.Assignment :=
-by
-  classical
-  exact fun y => if h : x < y then j.vertex y h (f y) else 0
-
-lemma mapAssignment_at (f : B.Assignment) (y : V) (h : x < y) :
-    j.mapAssignment f y = j.vertex y h (f y) := by
-  classical
-  simp [mapAssignment,h]
-
-lemma compatible (f : B.Assignment) (hf : B.CompatibleOn (Set.Ioi x) f) :
-    C.CompatibleOn (Set.Ioi x) (j.mapAssignment f) := by
-  intro e hs ht
-  change x < G.source e at hs
-  change x < G.target e at ht
-  rw [j.mapAssignment_at f _ hs,j.mapAssignment_at f _ ht,← j.lower e hs,← j.upper e ht]
-  exact congrArg (j.edge e ht) (hf e hs ht)
-
-noncomputable def boundary : B.Boundary x ≃ₗ[R] C.Boundary x :=
-  LinearEquiv.piCongrRight (fun e => j.edge e.val (lt_of_eq_of_lt e.property.symm (G.increasing e.val)))
-
-lemma boundaryMap (f : B.sections (Set.Ioi x)) :
-    j.boundary (B.boundaryMap x f) =
-      C.boundaryMap x ⟨j.mapAssignment f.val,j.compatible f.val f.property⟩ := by
-  funext e
-  change j.edge e.val _ (B.upper e.val (f.val (G.target e.val))) =
-    C.upper e.val (j.mapAssignment f.val (G.target e.val))
-  rw [j.mapAssignment_at _ _ (lt_of_eq_of_lt e.property.symm (G.increasing e.val)),j.upper]
-
-lemma map_boundaryImage : (B.boundaryImage x).map j.boundary.toLinearMap = C.boundaryImage x := by
-  apply le_antisymm
-  · rintro v ⟨w,⟨f,rfl⟩,rfl⟩
-    exact ⟨⟨j.mapAssignment f.val,j.compatible f.val f.property⟩,(j.boundaryMap f).symm⟩
-  · rintro v ⟨f,rfl⟩
-    refine ⟨B.boundaryMap x ⟨j.symm.mapAssignment f.val,j.symm.compatible f.val f.property⟩,
-      ⟨_,rfl⟩,?_⟩
-    funext e
-    change j.edge e.val _ (B.upper e.val (j.symm.mapAssignment f.val (G.target e.val))) =
-      C.upper e.val (f.val (G.target e.val))
-    rw [j.symm.mapAssignment_at _ _ (lt_of_eq_of_lt e.property.symm (G.increasing e.val)),j.upper]
-    exact congrArg (C.upper e.val) ((j.vertex _ _).apply_symm_apply _)
-
-lemma map_stalk_image (hBf : B.Flabby) (hBg : B.Generated)
-    (hCf : C.Flabby) (hCg : C.Generated) :
-    (LinearMap.range (B.stalkMap x)).map j.boundary.toLinearMap =
-      LinearMap.range (C.stalkMap x) := by
-  rw [B.range_stalkMap_eq_boundaryImage hBf hBg,C.range_stalkMap_eq_boundaryImage hCf hCg]
-  exact j.map_boundaryImage
-
-/-- One genuine descending splitting: the finite-free new top summand is
-proved from the local-ring minimum cover, and all outgoing restriction maps
-commute with the selected edge isomorphisms. -/
-theorem extend_stalk [IsLocalRing R]
-    [Module.Free R (B.vertex x)] [Module.Finite R (B.vertex x)]
-    [Module.Free R (C.vertex x)] [Module.Finite R (C.vertex x)]
-    (hBf : B.Flabby) (hBg : B.Generated) (hCf : C.Flabby) (hCg : C.Generated)
-    (hmin : LinearMap.ker (C.stalkMap x) ≤
-      IsLocalRing.maximalIdeal R • (⊤ : Submodule R (C.vertex x))) :
-    ∃ (g : B.vertex x →ₗ[R] C.vertex x)
-      (jv : B.vertex x ≃ₗ[R] C.vertex x × LinearMap.ker g),
-      Module.Free R (LinearMap.ker g) ∧ Module.Finite R (LinearMap.ker g) ∧
-      ∀ (b : B.vertex x) (e : Outgoing (G := G) x),
-        j.edge e.val (lt_of_eq_of_lt e.property.symm (G.increasing e.val)) (B.stalkMap x b e) =
-          C.stalkMap x (jv b).1 e := by
-  let N := LinearMap.range (C.stalkMap x)
-  let π : C.vertex x →ₗ[R] N := (C.stalkMap x).rangeRestrict
-  have him := j.map_stalk_image hBf hBg hCf hCg
-  let φ : B.vertex x →ₗ[R] N := LinearMap.codRestrict N
-    (j.boundary.toLinearMap.comp (B.stalkMap x)) (by
-      intro b
-      change j.boundary (B.stalkMap x b) ∈ LinearMap.range (C.stalkMap x)
-      rw [← him]
-      exact ⟨B.stalkMap x b,⟨b,rfl⟩,rfl⟩)
-  have hφ : Function.Surjective φ := by
-    intro n
-    have hn : n.val ∈ (LinearMap.range (B.stalkMap x)).map j.boundary.toLinearMap := by
-      rw [him]
-      exact n.property
-    rcases hn with ⟨w,⟨b,rfl⟩,hb⟩
-    exact ⟨b,Subtype.ext hb⟩
-  have hπmin : LinearMap.ker π ≤ IsLocalRing.maximalIdeal R • (⊤ : Submodule R (C.vertex x)) := by
-    intro c hc
-    apply hmin
-    exact congrArg Subtype.val (LinearMap.mem_ker.mp hc)
-  obtain ⟨g,jv,_,hg,hfree,hfinite,hjv⟩ :=
-    LocalCover.exists_split_minimal_cover π φ (by intro n; obtain ⟨c,hc⟩ := n.property; exact ⟨c,Subtype.ext hc⟩) hφ hπmin
-  refine ⟨g,jv,hfree,hfinite,?_⟩
-  intro b e
-  have h := congrArg (fun n : N => n.val e) (hjv b).2
-  exact h.symm
-
-end AboveIso
-end KLInvariance.MomentGraph.Sheaf
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.Sheaf
-universe u v w z i
-variable {R : Type u} [CommRing R] {V : Type v} [PartialOrder V]
-  {E : Type w} {G : OrderedGraph V E} {ι : Type i}
-
-/-- Finite direct sums represented by products, as used in the plane localization. -/
-def pi (B : ι → Sheaf.{u,v,w,z} (R := R) G) : Sheaf (R := R) G where
-  vertex x := ModuleCat.of R (∀ i, (B i).vertex x)
-  edge e := ModuleCat.of R (∀ i, (B i).edge e)
-  lower e := LinearMap.pi (fun i => ((B i).lower e).comp (LinearMap.proj i))
-  upper e := LinearMap.pi (fun i => ((B i).upper e).comp (LinearMap.proj i))
-
-lemma pi_compatible (B : ι → Sheaf.{u,v,w,z} (R := R) G) (Ω : Set V)
-    (f : (pi B).Assignment) :
-    (pi B).CompatibleOn Ω f ↔ ∀ i, (B i).CompatibleOn Ω (fun x => f x i) := by
-  constructor
-  · intro h i e hs ht
-    exact congrFun (h e hs ht) i
-  · intro h e hs ht
-    funext i
-    exact h i e hs ht
-
-lemma pi_generated (B : ι → Sheaf.{u,v,w,z} (R := R) G)
-    (h : ∀ i, (B i).Generated) : (pi B).Generated := by
-  intro x v
-  classical
-  choose f hf hfx using fun i => h i x (v i)
-  exact ⟨fun y i => f i y,(pi_compatible B Set.univ _).mpr hf,funext hfx⟩
-
-lemma pi_flabby (B : ι → Sheaf.{u,v,w,z} (R := R) G)
-    (h : ∀ i, (B i).Flabby) : (pi B).Flabby := by
-  intro Ω hΩ f hf
-  classical
-  choose g hg hgf using fun i => h i Ω hΩ (fun x => f x i) ((pi_compatible B Ω f).mp hf i)
-  exact ⟨fun y i => g i y,(pi_compatible B Set.univ _).mpr hg,fun x hx => funext (fun i => hgf i x hx)⟩
-
-lemma pi_kernel_iff (B : ι → Sheaf.{u,v,w,z} (R := R) G) (x : V)
-    (f : (pi B).vertex x) :
-    f ∈ LinearMap.ker ((pi B).stalkMap x) ↔ ∀ i, f i ∈ LinearMap.ker ((B i).stalkMap x) := by
-  constructor
-  · intro h i
-    apply LinearMap.mem_ker.mpr
-    funext e
-    rcases e with ⟨e,he⟩
-    subst x
-    exact congrFun (congrFun (LinearMap.mem_ker.mp h) ⟨e,rfl⟩) i
-  · intro h
-    apply LinearMap.mem_ker.mpr
-    funext e i
-    rcases e with ⟨e,he⟩
-    subst x
-    exact congrFun (LinearMap.mem_ker.mp (h i)) ⟨e,rfl⟩
-
-end KLInvariance.MomentGraph.Sheaf
-
-namespace KLInvariance.LocalCover
-universe u v w
-variable {R : Type u} [CommRing R] {ι : Type v} [Fintype ι]
-  (M : ι → Type w) [∀ i, AddCommGroup (M i)] [∀ i, Module R (M i)]
-
-/-- Coordinatewise membership in the ideal multiple of a finite product. -/
-lemma pi_mem_ideal_smul (I : Ideal R) (f : ∀ i, M i)
-    (hf : ∀ i, f i ∈ I • (⊤ : Submodule R (M i))) :
-    f ∈ I • (⊤ : Submodule R (∀ i, M i)) := by
-  classical
-  have hs (i : ι) : Pi.single i (f i) ∈ I • (⊤ : Submodule R (∀ i, M i)) := by
-    have hmap : (I • (⊤ : Submodule R (M i))).map (LinearMap.single R M i) ≤
-        I • (⊤ : Submodule R (∀ i, M i)) := by
-      rw [Submodule.map_smul'']
-      exact Submodule.smul_mono le_rfl le_top
-    exact hmap ⟨f i,hf i,rfl⟩
-  simpa only [Finset.univ_sum_single] using Submodule.sum_mem (I • (⊤ : Submodule R (∀ i, M i))) (t := Finset.univ) (fun i _ => hs i)
-
-end KLInvariance.LocalCover
-
-namespace KLInvariance.MomentGraph.Sheaf
-universe u v w z i
-variable {R : Type u} [CommRing R] {V : Type v} [PartialOrder V]
-  {E : Type w} {G : OrderedGraph V E} {ι : Type i} [Fintype ι]
-
-lemma pi_minimal (B : ι → Sheaf.{u,v,w,z} (R := R) G) (x : V) (I : Ideal R)
-    (h : ∀ i, LinearMap.ker ((B i).stalkMap x) ≤ I • (⊤ : Submodule R ((B i).vertex x))) :
-    LinearMap.ker ((pi B).stalkMap x) ≤ I • (⊤ : Submodule R ((pi B).vertex x)) := by
-  intro f hf
-  exact LocalCover.pi_mem_ideal_smul (fun i => (B i).vertex x) I f
-    (fun i => h i ((pi_kernel_iff B x f).mp hf i))
-
-end KLInvariance.MomentGraph.Sheaf
-
-end
-
-
-section
-
-namespace KLInvariance.QuotientEdge
-variable {R M N E F : Type*} [CommRing R]
-  [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N]
-  [AddCommGroup E] [Module R E] [AddCommGroup F] [Module R F]
-
-/-- The edge isomorphism forced by an upper stalk isomorphism and exact quotient maps. -/
-noncomputable def iso (f : M →ₗ[R] E) (g : N →ₗ[R] F)
-    (hf : Function.Surjective f) (hg : Function.Surjective g)
-    (h : M ≃ₗ[R] N) (hk : (LinearMap.ker f).map h.toLinearMap = LinearMap.ker g) : E ≃ₗ[R] F :=
-  (f.quotKerEquivOfSurjective hf).symm.trans
-    ((Submodule.Quotient.equiv _ _ h hk).trans (g.quotKerEquivOfSurjective hg))
-
-@[simp] lemma iso_apply (f : M →ₗ[R] E) (g : N →ₗ[R] F)
-    (hf : Function.Surjective f) (hg : Function.Surjective g)
-    (h : M ≃ₗ[R] N) (hk : (LinearMap.ker f).map h.toLinearMap = LinearMap.ker g) (v : M) :
-    iso f g hf hg h hk (f v) = g (h v) := by
-  simp [iso,Submodule.Quotient.equiv]
-
-lemma map_kernel_scalar (f : M →ₗ[R] E) (g : N →ₗ[R] F) (r : R)
-    (hf : ∀ v, f v = 0 ↔ ∃ w, v = r • w)
-    (hg : ∀ v, g v = 0 ↔ ∃ w, v = r • w) (h : M ≃ₗ[R] N) :
-    (LinearMap.ker f).map h.toLinearMap = LinearMap.ker g := by
-  ext v
-  constructor
-  · rintro ⟨w,hw,rfl⟩
-    obtain ⟨u,rfl⟩ := (hf w).mp hw
-    exact (hg _).mpr ⟨h u,h.map_smul r u⟩
-  · intro hv
-    obtain ⟨w,rfl⟩ := (hg v).mp hv
-    refine ⟨r • h.symm w,(hf _).mpr ⟨h.symm w,rfl⟩,?_⟩
-    simp
-
-/-- No separate choice of edge data or scalar compatibility hypothesis is needed. -/
-theorem exists_scalar_iso (f : M →ₗ[R] E) (g : N →ₗ[R] F) (r : R)
-    (hfs : Function.Surjective f) (hgs : Function.Surjective g)
-    (hf : ∀ v, f v = 0 ↔ ∃ w, v = r • w)
-    (hg : ∀ v, g v = 0 ↔ ∃ w, v = r • w) (h : M ≃ₗ[R] N) :
-    ∃ j : E ≃ₗ[R] F, ∀ v, j (f v) = g (h v) :=
-  ⟨iso f g hfs hgs h (map_kernel_scalar f g r hf hg h),iso_apply f g hfs hgs h _⟩
-
-end KLInvariance.QuotientEdge
-
-end
-
-
-section
-
-/-! Partial sheaf isomorphisms on upper sets, and their actual descending
-extension. In particular the incoming edge component is induced by the upper
-scalar quotient, not independently assumed. -/
-namespace KLInvariance.MomentGraph.Sheaf
-universe u v w z z'
-variable {R : Type u} [CommRing R] {V : Type v} [PartialOrder V]
-  {E : Type w} {G : OrderedGraph V E}
-  (B : Sheaf.{u,v,w,z} (R := R) G) (C : Sheaf.{u,v,w,z'} (R := R) G)
-
- def UpperQuotient (α : E → R) : Prop :=
-  ∀ e, Function.Surjective (B.upper e) ∧
-    ∀ v, B.upper e v = 0 ↔ ∃ w, v = α e • w
-
-def castStalkIso {x y : V} (h : x=y) (j : B.vertex x ≃ₗ[R] C.vertex x) :
-    B.vertex y ≃ₗ[R] C.vertex y := h ▸ j
-
-structure OnIso (U : Set V) where
-  vertex : ∀ x, x ∈ U → B.vertex x ≃ₗ[R] C.vertex x
-  edge : ∀ e, G.target e ∈ U → B.edge e ≃ₗ[R] C.edge e
-  lower : ∀ e (hs : G.source e ∈ U) (ht : G.target e ∈ U) v,
-    edge e ht (B.lower e v) = C.lower e (vertex (G.source e) hs v)
-  upper : ∀ e (ht : G.target e ∈ U) v,
-    edge e ht (B.upper e v) = C.upper e (vertex (G.target e) ht v)
-
-namespace OnIso
-variable {B C} {U : Set V} (j : OnIso B C U)
-
- def above (x : V) (hx : Set.Ioi x ⊆ U) : AboveIso B C x where
-  vertex y h := j.vertex y (hx h)
-  edge e h := j.edge e (hx h)
-  lower e h v := j.lower e (hx h) (hx (h.trans (G.increasing e))) v
-  upper e h v := j.upper e (hx h) v
-
- def ofUniv (j : OnIso B C Set.univ) : Iso B C where
-  vertex x := j.vertex x (Set.mem_univ x)
-  edge e := j.edge e (Set.mem_univ _)
-  lower e v := j.lower e (Set.mem_univ _) (Set.mem_univ _) v
-  upper e v := j.upper e (Set.mem_univ _) v
-
-/-- Given the new stalk and its outgoing commuting squares, all new incoming
-edges are forced by the scalar quotient rule. -/
-noncomputable def extend (hU : IsUpperSet U) (x : V) (hx : x ∉ U)
-    (hmax : Set.Ioi x ⊆ U) (α : E → R)
-    (hBq : B.UpperQuotient α) (hCq : C.UpperQuotient α)
-    (jv : B.vertex x ≃ₗ[R] C.vertex x)
-    (hout : ∀ e (he : G.source e = x) v,
-      j.edge e (hmax (lt_of_eq_of_lt he.symm (G.increasing e)))
-        (B.lower e v) = C.lower e (castStalkIso B C he.symm jv v)) :
-    OnIso B C (insert x U) := by
-  classical
-  let vj : ∀ y, y ∈ insert x U → B.vertex y ≃ₗ[R] C.vertex y := fun y hy =>
-    if h : y = x then castStalkIso B C h.symm jv
-    else j.vertex y ((Set.mem_insert_iff.mp hy).resolve_left h)
-  have vj_old (y : V) (hy : y ∈ U) : vj y (Set.mem_insert_of_mem x hy) = j.vertex y hy := by
-    simp only [vj,dite_eq_right (show y ≠ x from fun h => hx (h ▸ hy))]
-  have vj_new : vj x (Set.mem_insert x U) = jv := by simp [vj,castStalkIso]
-  let ej : ∀ e, G.target e ∈ insert x U → B.edge e ≃ₗ[R] C.edge e := fun e he =>
-    if h : G.target e = x then
-      QuotientEdge.iso (B.upper e) (C.upper e) (hBq e).1 (hCq e).1
-        (vj (G.target e) he)
-        (QuotientEdge.map_kernel_scalar (B.upper e) (C.upper e) (α e)
-          (hBq e).2 (hCq e).2 (vj (G.target e) he))
-    else j.edge e ((Set.mem_insert_iff.mp he).resolve_left h)
-  have ej_old (e : E) (he : G.target e ∈ U) :
-      ej e (Set.mem_insert_of_mem x he) = j.edge e he := by
-    simp only [ej,dite_eq_right (show G.target e ≠ x from fun h => hx (h ▸ he))]
-  refine ⟨vj,ej,?_,?_⟩
-  · intro e hs ht v
-    have htU : G.target e ∈ U := by
-      rcases Set.mem_insert_iff.mp hs with hs | hs
-      · exact hmax (lt_of_eq_of_lt hs.symm (G.increasing e))
-      · exact hU (G.increasing e).le hs
-    rw [ej_old e htU]
-    by_cases he : G.source e = x
-    · subst x
-      rw [vj_new]
-      exact hout e rfl v
-    · have hsU : G.source e ∈ U := (Set.mem_insert_iff.mp hs).resolve_left he
-      rw [vj_old _ hsU]
-      exact j.lower e hsU htU v
-  · intro e ht v
-    by_cases he : G.target e = x
-    · simp only [ej,dite_eq_left he]
-      exact QuotientEdge.iso_apply _ _ _ _ _ _ v
-    · have htU : G.target e ∈ U := (Set.mem_insert_iff.mp ht).resolve_left he
-      rw [ej_old e htU,vj_old _ htU]
-      exact j.upper e htU v
-
-end OnIso
-end KLInvariance.MomentGraph.Sheaf
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph
-universe um
-variable {V E : Type um} [PartialOrder V] (G : OrderedGraph V E)
-
- def removeGraph (x : V) (ht : ∀ e, G.target e ≠ x) :
-    OrderedGraph {y : V // y ≠ x} {e : E // G.source e ≠ x} where
-  source e := ⟨G.source e.val,e.property⟩
-  target e := ⟨G.target e.val,ht e.val⟩
-  increasing e := G.increasing e.val
-
- def removeTarget (x : V) (ht : ∀ e, G.target e ≠ x) :
-    {e : E // G.source e = x} → {y : V // y ≠ x} := fun e => ⟨G.target e.val,ht e.val⟩
-
-noncomputable def splitVertex (x : V) : V ≃ WithBot {y : V // y ≠ x} := by
-  classical
-  exact
-    { toFun := fun y => if h : y=x then ⊥ else (⟨y,h⟩ : {y : V // y ≠ x})
-      invFun := WithBot.recBotCoe x Subtype.val
-      left_inv := by intro y; by_cases h : y=x <;> simp [h]
-      right_inv := by
-        intro y
-        induction y using WithBot.recBotCoe with
-        | bot => simp
-        | coe y => simp [y.property] }
-
-noncomputable def splitEdge (x : V) : E ≃ {e : E // G.source e ≠ x} ⊕ {e : E // G.source e = x} := by
-  classical
-  exact
-    { toFun := fun e => if h : G.source e=x then Sum.inr ⟨e,h⟩ else Sum.inl ⟨e,h⟩
-      invFun := Sum.elim Subtype.val Subtype.val
-      left_inv := by intro e; by_cases h : G.source e=x <;> simp [h]
-      right_inv := by
-        intro e
-        cases e with
-        | inl e => simp [e.property]
-        | inr e => simp [e.property] }
-
-omit [PartialOrder V] in
-@[simp] theorem splitVertex_self (x : V) : splitVertex x x = ⊥ := by
-  classical
-  simp [splitVertex]
-
-omit [PartialOrder V] in
- theorem splitVertex_ne (x y : V) (hy : y ≠ x) :
-    splitVertex x y = (⟨y,hy⟩ : {y : V // y ≠ x}) := by
-  classical
-  simp [splitVertex,hy]
-
-noncomputable def splitGraph (x : V) (ht : ∀ e, G.target e ≠ x) :
-    GraphEquiv G (bottomGraph (removeGraph G x ht) (removeTarget G x ht)) where
-  vertex := splitVertex x
-  edge := splitEdge G x
-  source := by
-    classical
-    intro e
-    by_cases h : G.source e=x
-    · simp [splitEdge,splitVertex,h,bottomGraph]
-    · simp [splitEdge,splitVertex,h,bottomGraph,removeGraph]
-  target := by
-    classical
-    intro e
-    by_cases h : G.source e=x
-    · simp [splitEdge,splitVertex,h,ht e,bottomGraph,removeTarget]
-    · simp [splitEdge,splitVertex,h,ht e,bottomGraph,removeGraph]
-
- theorem splitEdge_labels {A : Type*} (α : E → A) (x : V) :
-    (Sum.elim (fun e : {e : E // G.source e ≠ x} => α e.val)
-      (fun e : {e : E // G.source e = x} => α e.val)) ∘ (splitEdge G x) = α := by
-  classical
-  funext e
-  by_cases h : G.source e=x <;> simp [splitEdge,h]
-
-end KLInvariance.MomentGraph
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-  (hneg : ∀ n < 0, 𝓐 n = ⊥)
-
-/-- The outcome of the actual minimal boundary construction. Full upward-kernel
-freeness, integral duality and KL characters are NOT part of this structure. -/
-structure BoundarySheaf {V E : Type um} [PartialOrder V] (G : OrderedGraph V E)
-    (α : E → A) (b : V) where
-  sheaf : GradedSheaf (𝓐 := 𝓐) G
-  free : sheaf.FreeStalks
-  quotient : sheaf.UpperQuotient α
-  flabby : sheaf.EdgeFlabby
-  generated : sheaf.forget.Generated
-  top : sheaf.vertex b = ModuleData.unit hneg
-  minimal : ∀ x, x ≠ b → sheaf.KernelMinimal x
-
-variable {V E : Type um} [PartialOrder V] (G : OrderedGraph V E)
-
-noncomputable def singletonSheaf [IsEmpty E] : GradedSheaf (𝓐 := 𝓐) G where
-  vertex := fun _ => ModuleData.unit hneg
-  edge := isEmptyElim
-  lower := isEmptyElim
-  upper := isEmptyElim
-
- theorem singleton_boundarySheaf (b : V) (hV : ∀ x, x=b) (α : E → A) :
-    Nonempty (BoundarySheaf hneg G α b) := by
-  have hempty : IsEmpty E := ⟨fun e => (G.increasing e).ne ((hV _).trans (hV _).symm)⟩
-  let := hempty
-  let B := singletonSheaf hneg G
-  refine ⟨⟨B,?_,?_,?_,?_,rfl,?_⟩⟩
-  · intro x
-    exact ModuleData.unit_free hneg
-  · intro e
-    exact isEmptyElim e
-  · intro Ω _ f _
-    exact ⟨f,fun e => isEmptyElim e,fun _ _ => rfl⟩
-  · intro x v
-    exact ⟨fun _ => v,fun e => isEmptyElim e,rfl⟩
-  · intro x hx
-    exact (hx (hV x)).elim
-
-variable [IsNoetherianRing A]
-  (hscalar : ∀ a : A, ∃ c : k, a-algebraMap k A c ∈ positiveIdeal 𝓐)
-
-include hscalar in
- theorem exists_boundarySheaf_card (n : ℕ)
-    [Fintype V] [Fintype E] (b : V) (hb : ∀ x, x ≤ b)
-    (α : E → A) (hα : ∀ e, α e ∈ 𝓐 1) (hn : Fintype.card V = n) :
-    Nonempty (BoundarySheaf hneg G α b) := by
-  classical
-  induction n using Nat.strong_induction_on generalizing V E with
-  | h n ih =>
-    by_cases hV : ∀ x : V, x=b
-    · exact singleton_boundarySheaf hneg G b hV α
-    obtain ⟨y,hy⟩ := not_forall.mp hV
-    obtain ⟨x,hx⟩ := exists_minimal_of_wellFoundedLT (fun _ : V => True) ⟨b,trivial⟩
-    have hxb : x ≠ b := by
-      intro he
-      have hlt : y < x := he ▸ lt_of_le_of_ne (hb y) hy
-      exact hx.not_lt trivial hlt
-    have ht (e : E) : G.target e ≠ x := by
-      intro he
-      exact hx.not_lt trivial (he ▸ G.increasing e)
-    let V' := {y : V // y ≠ x}
-    let E' := {e : E // G.source e ≠ x}
-    let D := {e : E // G.source e = x}
-    let G' := removeGraph G x ht
-    let τ := removeTarget G x ht
-    let b' : V' := ⟨b,hxb.symm⟩
-    let α' : E' → A := fun e => α e.val
-    let β : D → A := fun e => α e.val
-    have hβ : ∀ e : D, β e ∈ 𝓐 1 := fun e => hα e.val
-    have hcard : Fintype.card V' < n := by
-      rw [← hn]
-      exact Fintype.card_subtype_lt (x := x) (not_not.mpr rfl)
-    obtain ⟨B⟩ := ih (Fintype.card V') hcard G' b' (fun y => hb y.val)
-      α' (fun e => hα e.val) rfl
-    obtain ⟨C⟩ := ModuleData.exists_minimalCover hneg hscalar (B.sheaf.attachImage τ β hβ)
-    let H := B.sheaf.extendBottom τ β hβ C
-    let e := splitGraph G x ht
-    refine ⟨⟨H.reindex e,?_,?_,?_,?_,?_,?_⟩⟩
-    · exact H.reindex_freeStalks e (B.sheaf.extendBottom_freeStalks τ β hβ C B.free)
-    · have hh := H.reindex_upperQuotient e
-        (B.sheaf.extendBottom_upperQuotient τ β hβ C B.quotient)
-      have he : (Sum.elim α' β) ∘ e.edge = α := splitEdge_labels G α x
-      exact he ▸ hh
-    · exact H.reindex_edgeFlabby e (B.sheaf.extendBottom_edgeFlabby τ β hβ C B.flabby)
-    · exact H.reindex_generated e (B.sheaf.extendBottom_generated τ β hβ C B.generated)
-    · change H.vertex (splitVertex x b) = _
-      rw [splitVertex_ne x b hxb.symm]
-      exact B.top
-    · intro z hz
-      apply H.reindex_kernelMinimal e z
-      by_cases hzx : z=x
-      · subst z
-        change H.KernelMinimal (splitVertex x x)
-        rw [splitVertex_self]
-        exact B.sheaf.extendBottom_kernelMinimal_bot τ β hβ C
-      · change H.KernelMinimal (splitVertex x z)
-        rw [splitVertex_ne x z hzx]
-        apply B.sheaf.extendBottom_kernelMinimal_coe τ β hβ C ⟨z,hzx⟩
-        apply B.minimal
-        intro hzz
-        exact hz (congrArg Subtype.val hzz)
-
-include hscalar in
- theorem exists_boundarySheaf [Fintype V] [Fintype E] (b : V) (hb : ∀ x, x ≤ b)
-    (α : E → A) (hα : ∀ e, α e ∈ 𝓐 1) : Nonempty (BoundarySheaf hneg G α b) :=
-  exists_boundarySheaf_card hneg G hscalar (Fintype.card V) b hb α hα rfl
-
-end KLInvariance.MomentGraph
-
-end
-
-
-section
-
-
-end
-
-end OAI
+import OAI.RepresentationTheory.KazhdanLusztig.Sheaves
+import OAI.RepresentationTheory.KazhdanLusztig.GradedModules
+import OAI.RepresentationTheory.KazhdanLusztig.ReflectionSchedules
+
+namespace OAI
+
+/-!
+Graded moment-graph sheaves, boundary sections, graded covers and morphisms.
+-/
+
+section
+
+namespace KLInvariance.MomentGraph
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A}
+  {V E : Type um} [PartialOrder V] (G : OrderedGraph V E)
+
+structure GradedSheaf where
+  vertex : V → ModuleData.{uk,ua,um} 𝓐
+  edge : E → ModuleData.{uk,ua,um} 𝓐
+  lower : ∀ e, ModuleData.Hom (vertex (G.source e)) (edge e)
+  upper : ∀ e, ModuleData.Hom (vertex (G.target e)) (edge e)
+
+namespace GradedSheaf
+variable {G} (B : GradedSheaf (𝓐 := 𝓐) G)
+
+def forget : Sheaf (R := A) G where
+  vertex := fun x => (B.vertex x).obj
+  edge := fun e => (B.edge e).obj
+  lower := fun e => (B.lower e).map
+  upper := fun e => (B.upper e).map
+
+/-- Extension for sets closed under actual edges. It implies the source's
+ordinary flabbiness, and is convenient for the descending construction. -/
+def EdgeFlabby : Prop :=
+  ∀ Ω : Set V, B.forget.ClosedUnder Set.univ Ω →
+    ∀ f : B.forget.Assignment, B.forget.CompatibleOn Ω f →
+    ∃ g : B.forget.Assignment, B.forget.IsGlobal g ∧ ∀ x ∈ Ω, g x = f x
+
+ theorem EdgeFlabby.flabby (h : B.EdgeFlabby) : B.forget.Flabby := by
+  intro Ω hΩ f hf
+  exact h Ω (fun e _ he => hΩ (G.increasing e).le he) f hf
+
+variable [Fintype V] [Fintype E]
+
+noncomputable def assignments : ModuleData.{uk,ua,um} 𝓐 := ModuleData.pi B.vertex
+
+noncomputable def lowerGlobal (e : E) : ModuleData.Hom B.assignments (B.edge e) :=
+  (B.lower e).comp (ModuleData.projection B.vertex (G.source e))
+
+noncomputable def upperGlobal (e : E) : ModuleData.Hom B.assignments (B.edge e) :=
+  (B.upper e).comp (ModuleData.projection B.vertex (G.target e))
+
+omit [Fintype E] in
+ theorem sections_homogeneous (Ω : Set V) :
+    ∀ n f, f ∈ B.forget.sections Ω → component B.assignments.piece n f ∈ B.forget.sections Ω := by
+  intro n f hf e hs ht
+  change (B.lowerGlobal e).map (component B.assignments.piece n f) =
+    (B.upperGlobal e).map (component B.assignments.piece n f)
+  rw [← (B.lowerGlobal e).component n f, ← (B.upperGlobal e).component n f]
+  exact congrArg (component (B.edge e).piece n) (hf e hs ht)
+
+noncomputable def sections [IsNoetherianRing A] (Ω : Set V) : ModuleData.{uk,ua,um} 𝓐 :=
+  ModuleData.sub B.assignments (B.forget.sections Ω) (B.sections_homogeneous Ω)
+
+noncomputable def boundary (x : V) : ModuleData.{uk,ua,um} 𝓐 :=
+by
+  classical
+  letI : Fintype (Sheaf.Outgoing (G := G) x) := inferInstanceAs (Fintype {e : E // G.source e = x})
+  exact ModuleData.pi (fun e : Sheaf.Outgoing (G := G) x => B.edge e.val)
+
+noncomputable def boundaryHom [IsNoetherianRing A] (x : V) :
+    ModuleData.Hom (B.sections (Set.Ioi x)) (B.boundary x) where
+  map := B.forget.boundaryMap x
+  graded := by
+    intro n f hf e _
+    exact (B.upper e.val).graded n (f.val (G.target e.val)) (hf (G.target e.val) (Set.mem_univ _))
+
+noncomputable def boundaryImage [IsNoetherianRing A] (x : V) : ModuleData.{uk,ua,um} 𝓐 :=
+  ModuleData.range (B.boundaryHom x)
+
+ theorem exists_boundaryCover [IsNoetherianRing A]
+    [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+    (hneg : ∀ n < 0, 𝓐 n = ⊥)
+    (hscalar : ∀ a : A, ∃ c : k, a-algebraMap k A c ∈ positiveIdeal 𝓐) (x : V) :
+    Nonempty (ModuleData.MinimalCover (B.boundaryImage x)) :=
+  ModuleData.exists_minimalCover hneg hscalar (B.boundaryImage x)
+
+end GradedSheaf
+end KLInvariance.MomentGraph
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A}
+  {V E D : Type um} [PartialOrder V] (G : OrderedGraph V E)
+
+/-- Adjoin a vertex below the old vertices, with specified outgoing edges.
+Only the edge orientations, not the extra comparabilities, enter sections. -/
+def bottomGraph (τ : D → V) : OrderedGraph (WithBot V) (E ⊕ D) where
+  source := Sum.elim (fun e => (G.source e : WithBot V)) (fun _ => ⊥)
+  target := Sum.elim (fun e => (G.target e : WithBot V)) (fun d => (τ d : WithBot V))
+  increasing := by
+    intro e
+    cases e with
+    | inl e => exact WithBot.coe_lt_coe.mpr (G.increasing e)
+    | inr d => exact WithBot.bot_lt_coe (τ d)
+
+namespace GradedSheaf
+variable {G} (B : GradedSheaf (𝓐 := 𝓐) G)
+  [Fintype V] [Fintype D] [IsNoetherianRing A]
+  (τ : D → V) (α : D → A) (hα : ∀ d, α d ∈ 𝓐 1)
+
+noncomputable def attachEdges (d : D) : ModuleData.{uk,ua,um} 𝓐 :=
+  (B.vertex (τ d)).scalarQuotient (α d) 1 (hα d)
+
+noncomputable def attachQuotient (d : D) :
+    ModuleData.Hom (B.vertex (τ d)) (B.attachEdges τ α hα d) :=
+  ModuleData.quotientMap _ _ (scalar_range_homogeneous 𝓐 (B.vertex (τ d)).piece 1 (α d) (hα d))
+
+noncomputable def attachBoundary : ModuleData.{uk,ua,um} 𝓐 :=
+  ModuleData.pi (B.attachEdges τ α hα)
+
+noncomputable def attachBoundaryMap :
+    ModuleData.Hom (B.sections Set.univ) (B.attachBoundary τ α hα) where
+  map := LinearMap.pi fun d => (B.attachQuotient τ α hα d).map.comp
+    ((LinearMap.proj (τ d)).comp (B.forget.sections Set.univ).subtype)
+  graded := by
+    intro n f hf d _
+    exact (B.attachQuotient τ α hα d).graded n _ (hf (τ d) (Set.mem_univ _))
+
+noncomputable def attachImage : ModuleData.{uk,ua,um} 𝓐 :=
+  ModuleData.range (B.attachBoundaryMap τ α hα)
+
+variable (C : ModuleData.MinimalCover (B.attachImage τ α hα))
+
+noncomputable def attachStalkMap : ModuleData.Hom C.source (B.attachBoundary τ α hα) :=
+  (ModuleData.subtype (B.attachBoundary τ α hα) _
+    (range_homogeneous _ _ (B.attachBoundaryMap τ α hα).map 0
+      (by simpa using (B.attachBoundaryMap τ α hα).graded))).comp C.hom
+
+noncomputable def extendBottom : GradedSheaf (𝓐 := 𝓐) (bottomGraph G τ) where
+  vertex := WithBot.recBotCoe C.source B.vertex
+  edge := Sum.elim B.edge (B.attachEdges τ α hα)
+  lower := by
+    intro e
+    cases e with
+    | inl e => exact B.lower e
+    | inr d =>
+      exact (ModuleData.projection (B.attachEdges τ α hα) d).comp
+        (B.attachStalkMap τ α hα C)
+  upper := by
+    intro e
+    cases e with
+    | inl e => exact B.upper e
+    | inr d => exact B.attachQuotient τ α hα d
+
+ def attachAssignment (z : C.source) (f : B.forget.Assignment) :
+    (B.extendBottom τ α hα C).forget.Assignment := WithBot.recBotCoe z f
+
+ theorem attachAssignment_global (z : C.source) (f : B.forget.Assignment)
+    (hf : B.forget.IsGlobal f)
+    (hz : ∀ d, (B.attachStalkMap τ α hα C).map z d =
+      (B.attachQuotient τ α hα d).map (f (τ d))) :
+    (B.extendBottom τ α hα C).forget.IsGlobal (B.attachAssignment τ α hα C z f) := by
+  intro e _ _
+  cases e with
+  | inl e => exact hf e (Set.mem_univ _) (Set.mem_univ _)
+  | inr d => exact hz d
+
+ theorem lift_global (f : B.forget.Assignment) (hf : B.forget.IsGlobal f) :
+    ∃ z : C.source, (B.extendBottom τ α hα C).forget.IsGlobal
+      (B.attachAssignment τ α hα C z f) := by
+  let s : B.sections Set.univ := ⟨f,hf⟩
+  let v : B.attachImage τ α hα :=
+    ⟨(B.attachBoundaryMap τ α hα).map s, ⟨s,rfl⟩⟩
+  obtain ⟨z,hz⟩ := C.surjective v
+  refine ⟨z,B.attachAssignment_global τ α hα C z f hf ?_⟩
+  intro d
+  exact congrArg (fun t : B.attachImage τ α hα => t.val d) hz
+
+ theorem extendBottom_generated (hg : B.forget.Generated) :
+    (B.extendBottom τ α hα C).forget.Generated := by
+  intro x
+  induction x using WithBot.recBotCoe with
+  | bot =>
+    intro z
+    obtain ⟨s,hs⟩ := (C.hom.map z).property
+    refine ⟨B.attachAssignment τ α hα C z s.val,
+      B.attachAssignment_global τ α hα C z s.val s.property ?_,rfl⟩
+    intro d
+    exact congrFun hs d |>.symm
+  | coe x =>
+    intro v
+    obtain ⟨f,hf,hfx⟩ := hg x v
+    obtain ⟨z,hz⟩ := B.lift_global τ α hα C f hf
+    exact ⟨B.attachAssignment τ α hα C z f,hz,hfx⟩
+
+ theorem extendBottom_edgeFlabby (hf : B.EdgeFlabby) :
+    (B.extendBottom τ α hα C).EdgeFlabby := by
+  intro Ω hΩ f hcompat
+  let Ω' : Set V := {x | (x : WithBot V) ∈ Ω}
+  let f' : B.forget.Assignment := fun x => f (x : WithBot V)
+  have hc : B.forget.ClosedUnder Set.univ Ω' := by
+    intro e _ hs
+    exact hΩ (Sum.inl e) (Set.mem_univ _) hs
+  have hcf : B.forget.CompatibleOn Ω' f' := by
+    intro e hs ht
+    exact hcompat (Sum.inl e) hs ht
+  obtain ⟨g,hg,hgf⟩ := hf Ω' hc f' hcf
+  classical
+  by_cases hb : (⊥ : WithBot V) ∈ Ω
+  · refine ⟨B.attachAssignment τ α hα C (f ⊥) g,
+      B.attachAssignment_global τ α hα C (f ⊥) g hg ?_,?_⟩
+    · intro d
+      have hd : (τ d : WithBot V) ∈ Ω := hΩ (Sum.inr d) (Set.mem_univ _) hb
+      rw [hgf (τ d) hd]
+      exact hcompat (Sum.inr d) hb hd
+    · intro x hx
+      induction x using WithBot.recBotCoe with
+      | bot => rfl
+      | coe x => exact hgf x hx
+  · obtain ⟨z,hz⟩ := B.lift_global τ α hα C g hg
+    refine ⟨B.attachAssignment τ α hα C z g,hz,?_⟩
+    intro x hx
+    induction x using WithBot.recBotCoe with
+    | bot => exact (hb hx).elim
+    | coe x => exact hgf x hx
+
+end GradedSheaf
+end KLInvariance.MomentGraph
+
+end
+
+
+section
+
+namespace KLInvariance.Graded.ModuleData
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A}
+
+def Hom.ofEq {M N : ModuleData.{uk,ua,um} 𝓐} (h : M = N) : Hom M N := h ▸ Hom.id M
+
+@[simp] theorem Hom.ofEq_rfl (M : ModuleData.{uk,ua,um} 𝓐) : Hom.ofEq (rfl : M=M) = Hom.id M := rfl
+
+theorem Hom.ofEq_apply {ι : Type*} (M : ι → ModuleData.{uk,ua,um} 𝓐)
+    {x y : ι} (h : x = y) (v : ∀ z, M z) :
+    (Hom.ofEq (congrArg M h)).map (v x) = v y := by subst y; rfl
+
+end KLInvariance.Graded.ModuleData
+
+namespace KLInvariance.MomentGraph
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A}
+  {V E W F : Type um} [PartialOrder V] [PartialOrder W]
+  {G : OrderedGraph V E} {H : OrderedGraph W F}
+
+/-- An isomorphism of directed graphs, with no order-reflection requirement. -/
+structure GraphEquiv (G : OrderedGraph V E) (H : OrderedGraph W F) where
+  vertex : V ≃ W
+  edge : E ≃ F
+  source : ∀ e, vertex (G.source e) = H.source (edge e)
+  target : ∀ e, vertex (G.target e) = H.target (edge e)
+
+namespace GradedSheaf
+variable (B : GradedSheaf (𝓐 := 𝓐) H) (e : GraphEquiv G H)
+
+def reindex : GradedSheaf (𝓐 := 𝓐) G where
+  vertex := fun x => B.vertex (e.vertex x)
+  edge := fun a => B.edge (e.edge a)
+  lower := fun a => (B.lower (e.edge a)).comp (ModuleData.Hom.ofEq (congrArg B.vertex (e.source a)))
+  upper := fun a => (B.upper (e.edge a)).comp (ModuleData.Hom.ofEq (congrArg B.vertex (e.target a)))
+
+def reindexAssignment (f : B.forget.Assignment) : (B.reindex e).forget.Assignment :=
+  fun x => f (e.vertex x)
+
+@[simp] theorem reindex_lower (a : E) (f : B.forget.Assignment) :
+    (B.reindex e).forget.lower a (B.reindexAssignment e f (G.source a)) =
+      B.forget.lower (e.edge a) (f (H.source (e.edge a))) := by
+  change (B.lower (e.edge a)).map ((ModuleData.Hom.ofEq (congrArg B.vertex (e.source a))).map
+    (f (e.vertex (G.source a)))) = _
+  exact congrArg (B.lower (e.edge a)).map
+    (ModuleData.Hom.ofEq_apply B.vertex (e.source a) f)
+
+@[simp] theorem reindex_upper (a : E) (f : B.forget.Assignment) :
+    (B.reindex e).forget.upper a (B.reindexAssignment e f (G.target a)) =
+      B.forget.upper (e.edge a) (f (H.target (e.edge a))) := by
+  change (B.upper (e.edge a)).map ((ModuleData.Hom.ofEq (congrArg B.vertex (e.target a))).map
+    (f (e.vertex (G.target a)))) = _
+  exact congrArg (B.upper (e.edge a)).map
+    (ModuleData.Hom.ofEq_apply B.vertex (e.target a) f)
+
+def unreindexAssignment (f : (B.reindex e).forget.Assignment) : B.forget.Assignment :=
+  fun y => (ModuleData.Hom.ofEq (congrArg B.vertex (e.vertex.apply_symm_apply y))).map
+    (f (e.vertex.symm y))
+
+@[simp] theorem unreindex_reindexAssignment (f : B.forget.Assignment) :
+    B.unreindexAssignment e (B.reindexAssignment e f) = f := by
+  funext y
+  exact ModuleData.Hom.ofEq_apply B.vertex (e.vertex.apply_symm_apply y) f
+
+@[simp] theorem reindex_unreindexAssignment (f : (B.reindex e).forget.Assignment) :
+    B.reindexAssignment e (B.unreindexAssignment e f) = f := by
+  funext x
+  change (ModuleData.Hom.ofEq (congrArg B.vertex (e.vertex.apply_symm_apply (e.vertex x)))).map
+    (f (e.vertex.symm (e.vertex x))) = f x
+  have h : e.vertex.symm (e.vertex x) = x := e.vertex.symm_apply_apply x
+  have hh : (ModuleData.Hom.ofEq
+      (congrArg (fun t => B.vertex (e.vertex t)) h)).map
+      (f (e.vertex.symm (e.vertex x))) = f x :=
+    ModuleData.Hom.ofEq_apply (fun t => B.vertex (e.vertex t)) h f
+  exact hh
+
+ theorem reindex_global (f : B.forget.Assignment) (hf : B.forget.IsGlobal f) :
+    (B.reindex e).forget.IsGlobal (B.reindexAssignment e f) := by
+  intro a _ _
+  rw [B.reindex_lower, B.reindex_upper]
+  exact hf (e.edge a) (Set.mem_univ _) (Set.mem_univ _)
+
+ theorem reindex_generated (hg : B.forget.Generated) : (B.reindex e).forget.Generated := by
+  intro x v
+  obtain ⟨f,hf,hfx⟩ := hg (e.vertex x) v
+  exact ⟨B.reindexAssignment e f,B.reindex_global e f hf,hfx⟩
+
+ theorem reindex_edgeFlabby (hb : B.EdgeFlabby) : (B.reindex e).EdgeFlabby := by
+  intro Ω hΩ f hf
+  let Ω' := e.vertex '' Ω
+  have hc : B.forget.ClosedUnder Set.univ Ω' := by
+    intro a _ hs
+    obtain ⟨a,rfl⟩ := e.edge.surjective a
+    obtain ⟨x,hx,hxs⟩ := hs
+    have hx' : x = G.source a := e.vertex.injective (hxs.trans (e.source a).symm)
+    exact ⟨G.target a,hΩ a (Set.mem_univ _) (hx' ▸ hx),e.target a⟩
+  have hcf : B.forget.CompatibleOn Ω' (B.unreindexAssignment e f) := by
+    intro a hs ht
+    obtain ⟨a,rfl⟩ := e.edge.surjective a
+    have hh (x : V) (h : e.vertex x ∈ Ω') : x ∈ Ω := by
+      obtain ⟨y,hy,hyx⟩ := h
+      exact (e.vertex.injective hyx) ▸ hy
+    have hfa := hf a (hh _ ((e.source a).symm ▸ hs)) (hh _ ((e.target a).symm ▸ ht))
+    rw [← B.reindex_unreindexAssignment e f] at hfa
+    exact (B.reindex_lower e a (B.unreindexAssignment e f)).symm.trans
+      (hfa.trans (B.reindex_upper e a (B.unreindexAssignment e f)))
+  obtain ⟨g,hg,hgf⟩ := hb Ω' hc (B.unreindexAssignment e f) hcf
+  refine ⟨B.reindexAssignment e g,B.reindex_global e g hg,?_⟩
+  intro x hx
+  have hh := hgf (e.vertex x) ⟨x,hx,rfl⟩
+  change g (e.vertex x) = f x
+  rw [hh]
+  exact congrFun (B.reindex_unreindexAssignment e f) x
+
+end GradedSheaf
+end KLInvariance.MomentGraph
+
+end
+
+
+section
+
+namespace KLInvariance.Graded.ModuleData
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A}
+
+theorem Hom.ofEq_bijective {M N : ModuleData.{uk,ua,um} 𝓐} (h : M=N) :
+    Function.Bijective (Hom.ofEq h).map := by subst N; exact Function.bijective_id
+
+noncomputable def unit [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+    (hneg : ∀ n < 0, 𝓐 n = ⊥) : ModuleData.{uk,ua,um} 𝓐 :=
+  free (ι := PUnit.{max ua um + 1}) hneg (fun _ => 0) (fun _ => le_rfl)
+
+instance unit_free [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+    (hneg : ∀ n < 0, 𝓐 n = ⊥) : Module.Free A (unit.{uk,ua,um} hneg) :=
+  inferInstanceAs (Module.Free A (PUnit.{max ua um + 1} → A))
+
+end KLInvariance.Graded.ModuleData
+
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A}
+  {V E W F : Type um} [PartialOrder V] [PartialOrder W]
+  {G : OrderedGraph V E} {H : OrderedGraph W F}
+
+ def FreeStalks (B : GradedSheaf (𝓐 := 𝓐) G) : Prop := ∀ x, Module.Free A (B.vertex x)
+
+/-- The genuine upper-end quotient rule, expressed without choosing a quotient
+presentation of the edge object. -/
+ def UpperQuotient (B : GradedSheaf (𝓐 := 𝓐) G) (α : E → A) : Prop :=
+  ∀ e, Function.Surjective (B.upper e).map ∧
+    ∀ v, (B.upper e).map v = 0 ↔ ∃ w, v = α e • w
+
+/-- Minimality of the full outgoing boundary cover. The assignment formulation
+avoids dependent casts and is equivalent to the usual kernel containment. -/
+ def KernelMinimal (B : GradedSheaf (𝓐 := 𝓐) G) (x : V) : Prop :=
+  ∀ f : B.forget.Assignment,
+    (∀ e, G.source e = x → B.forget.lower e (f (G.source e)) = 0) →
+    f x ∈ positiveIdeal 𝓐 • (⊤ : Submodule A (B.vertex x))
+
+ theorem kernelMinimal_iff (B : GradedSheaf (𝓐 := 𝓐) G) (x : V) :
+    B.KernelMinimal x ↔ LinearMap.ker (B.forget.stalkMap x) ≤
+      positiveIdeal 𝓐 • (⊤ : Submodule A (B.vertex x)) := by
+  classical
+  constructor
+  · intro h v hv
+    let f : B.forget.Assignment := Function.update 0 x v
+    have hf : f x = v := by simp [f]
+    rw [← hf]
+    apply h f
+    intro e he
+    have hh := congrFun (LinearMap.mem_ker.mp hv) (⟨e,he⟩ : Sheaf.Outgoing x)
+    subst x
+    simpa [Sheaf.stalkMap,f] using hh
+  · intro h f hf
+    apply h
+    apply LinearMap.mem_ker.mpr
+    funext e
+    rcases e with ⟨e,he⟩
+    subst x
+    exact hf e rfl
+
+ theorem reindex_freeStalks (B : GradedSheaf (𝓐 := 𝓐) H) (e : GraphEquiv G H)
+    (h : B.FreeStalks) : (B.reindex e).FreeStalks := fun x => h (e.vertex x)
+
+ theorem reindex_upperQuotient (B : GradedSheaf (𝓐 := 𝓐) H) (e : GraphEquiv G H)
+    {α : F → A} (h : B.UpperQuotient α) : (B.reindex e).UpperQuotient (α ∘ e.edge) := by
+  intro a
+  let j := ModuleData.Hom.ofEq (congrArg B.vertex (e.target a))
+  have hj := ModuleData.Hom.ofEq_bijective (congrArg B.vertex (e.target a))
+  refine ⟨(h (e.edge a)).1.comp hj.2,?_⟩
+  intro v
+  change (B.upper (e.edge a)).map (j.map v) = 0 ↔ ∃ w, v = α (e.edge a) • w
+  rw [(h (e.edge a)).2]
+  constructor
+  · rintro ⟨w,hw⟩
+    obtain ⟨z,rfl⟩ := hj.2 w
+    exact ⟨z,hj.1 (hw.trans (map_smul j.map (α (e.edge a)) z).symm)⟩
+  · rintro ⟨w,rfl⟩
+    exact ⟨j.map w,map_smul _ _ _⟩
+
+ theorem reindex_kernelMinimal (B : GradedSheaf (𝓐 := 𝓐) H) (e : GraphEquiv G H)
+    (x : V) (h : B.KernelMinimal (e.vertex x)) : (B.reindex e).KernelMinimal x := by
+  intro f hf
+  have hh : ∀ a, H.source a = e.vertex x →
+      B.forget.lower a (B.unreindexAssignment e f (H.source a)) = 0 := by
+    intro a ha
+    obtain ⟨a,rfl⟩ := e.edge.surjective a
+    have hs : G.source a = x := e.vertex.injective ((e.source a).trans ha)
+    have hv := hf a hs
+    rw [← B.reindex_unreindexAssignment e f] at hv
+    exact (B.reindex_lower e a (B.unreindexAssignment e f)).symm.trans hv
+  have hm := h (B.unreindexAssignment e f) hh
+  have he := congrFun (B.reindex_unreindexAssignment e f) x
+  exact he ▸ hm
+
+variable {D : Type um} [Fintype V] [Fintype D] [IsNoetherianRing A]
+  (B : GradedSheaf (𝓐 := 𝓐) G) (τ : D → V) (α : D → A)
+  (hα : ∀ d, α d ∈ 𝓐 1) (C : ModuleData.MinimalCover (B.attachImage τ α hα))
+
+ theorem extendBottom_freeStalks (hb : B.FreeStalks) :
+    (B.extendBottom τ α hα C).FreeStalks := by
+  intro x
+  induction x using WithBot.recBotCoe with
+  | bot => exact C.free
+  | coe x => exact hb x
+
+ theorem extendBottom_upperQuotient {αold : E → A} (hb : B.UpperQuotient αold) :
+    (B.extendBottom τ α hα C).UpperQuotient (Sum.elim αold α) := by
+  intro e
+  cases e with
+  | inl e => exact hb e
+  | inr d =>
+    let S := LinearMap.range (LinearMap.lsmul A (B.vertex (τ d)) (α d))
+    refine ⟨S.mkQ_surjective,?_⟩
+    intro v
+    change S.mkQ v = 0 ↔ ∃ w, v = α d • w
+    refine (Submodule.Quotient.mk_eq_zero S).trans ?_
+    change (∃ w, α d • w = v) ↔ ∃ w, v = α d • w
+    exact ⟨fun ⟨w,hw⟩ => ⟨w,hw.symm⟩,fun ⟨w,hw⟩ => ⟨w,hw.symm⟩⟩
+
+ theorem extendBottom_kernelMinimal_bot : (B.extendBottom τ α hα C).KernelMinimal ⊥ := by
+  intro f hf
+  apply C.minimal
+  apply LinearMap.mem_ker.mpr
+  apply Subtype.ext
+  funext d
+  exact hf (Sum.inr d) rfl
+
+ theorem extendBottom_kernelMinimal_coe (x : V) (h : B.KernelMinimal x) :
+    (B.extendBottom τ α hα C).KernelMinimal (x : WithBot V) := by
+  intro f hf
+  apply h (fun y => f (y : WithBot V))
+  intro a ha
+  exact hf (Sum.inl a) (congrArg (fun y : V => (y : WithBot V)) ha)
+
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+
+namespace KLInvariance.MomentGraph.Sheaf
+universe u v w z z'
+variable {R : Type u} [CommRing R] {V : Type v} [PartialOrder V]
+  {E : Type w} {G : OrderedGraph V E}
+
+structure Iso (B : Sheaf.{u,v,w,z} (R := R) G) (C : Sheaf.{u,v,w,z'} (R := R) G) where
+  vertex : ∀ x, B.vertex x ≃ₗ[R] C.vertex x
+  edge : ∀ e, B.edge e ≃ₗ[R] C.edge e
+  lower : ∀ e v, edge e (B.lower e v) = C.lower e (vertex (G.source e) v)
+  upper : ∀ e v, edge e (B.upper e v) = C.upper e (vertex (G.target e) v)
+
+namespace Iso
+variable {B : Sheaf.{u,v,w,z} (R := R) G} {C : Sheaf.{u,v,w,z'} (R := R) G}
+  (j : Iso B C)
+
+def symm : Iso C B where
+  vertex x := (j.vertex x).symm
+  edge e := (j.edge e).symm
+  lower e v := by
+    apply (j.edge e).injective
+    simp only [LinearEquiv.apply_symm_apply, j.lower]
+  upper e v := by
+    apply (j.edge e).injective
+    simp only [LinearEquiv.apply_symm_apply, j.upper]
+
+noncomputable def assignments : B.Assignment ≃ₗ[R] C.Assignment :=
+  LinearEquiv.piCongrRight j.vertex
+
+theorem compatible_iff (Ω : Set V) (f : B.Assignment) :
+    C.CompatibleOn Ω (j.assignments f) ↔ B.CompatibleOn Ω f := by
+  constructor <;> intro h e hs ht
+  · apply (j.edge e).injective
+    rw [j.lower,j.upper]
+    exact h e hs ht
+  · change C.lower e (j.vertex _ (f _)) = C.upper e (j.vertex _ (f _))
+    rw [← j.lower,← j.upper,h e hs ht]
+
+theorem map_sections (Ω : Set V) :
+    (B.sections Ω).map j.assignments.toLinearMap = C.sections Ω := by
+  ext g
+  constructor
+  · rintro ⟨f,hf,rfl⟩
+    exact (j.compatible_iff Ω f).mpr hf
+  · intro hg
+    refine ⟨j.assignments.symm g,?_,j.assignments.apply_symm_apply g⟩
+    apply (j.compatible_iff Ω _).mp
+    change C.CompatibleOn Ω g at hg
+    simpa using hg
+
+noncomputable def sections (Ω : Set V) : B.sections Ω ≃ₗ[R] C.sections Ω :=
+  j.assignments.ofSubmodules _ _ (j.map_sections Ω)
+
+noncomputable def boundary (x : V) : B.Boundary x ≃ₗ[R] C.Boundary x :=
+  LinearEquiv.piCongrRight (fun e => j.edge e.val)
+
+theorem stalkMap (x : V) (v : B.vertex x) :
+    j.boundary x (B.stalkMap x v) = C.stalkMap x (j.vertex x v) := by
+  funext e
+  rcases e with ⟨e,he⟩
+  subst x
+  exact j.lower e v
+
+theorem boundaryMap (x : V) (f : B.sections (Set.Ioi x)) :
+    j.boundary x (B.boundaryMap x f) = C.boundaryMap x (j.sections _ f) := by
+  funext e
+  exact j.upper e.val (f.val (G.target e.val))
+
+theorem map_boundaryRange (x : V) :
+    (LinearMap.range (B.boundaryMap x)).map (j.boundary x).toLinearMap =
+      LinearMap.range (C.boundaryMap x) := by
+  ext v
+  constructor
+  · rintro ⟨w,⟨f,rfl⟩,rfl⟩
+    exact ⟨j.sections _ f,(j.boundaryMap x f).symm⟩
+  · rintro ⟨f,rfl⟩
+    refine ⟨B.boundaryMap x ((j.sections _).symm f),⟨_,rfl⟩,?_⟩
+    change j.boundary x (B.boundaryMap x ((j.sections _).symm f)) = _
+    rw [j.boundaryMap,LinearEquiv.apply_symm_apply]
+
+end Iso
+
+variable (B : Sheaf.{u,v,w,z} (R := R) G)
+
+def upwardKernel (x : V) (F : Set (Outgoing (G := G) x)) : Submodule R (B.vertex x) where
+  carrier := {v | ∀ e ∈ F, B.stalkMap x v e = 0}
+  zero_mem' := by intro e _; simp
+  add_mem' := by intro v w hv hw e he; simp [hv e he,hw e he]
+  smul_mem' := by intro r v hv e he; simp [hv e he]
+
+namespace Iso
+variable {B : Sheaf.{u,v,w,z} (R := R) G} {C : Sheaf.{u,v,w,z'} (R := R) G}
+  (j : Iso B C)
+
+theorem upwardKernel_iff (x : V) (F : Set (Outgoing (G := G) x)) (v : B.vertex x) :
+    j.vertex x v ∈ C.upwardKernel x F ↔ v ∈ B.upwardKernel x F := by
+  constructor <;> intro h e he
+  · apply (j.edge e.val).injective
+    rw [map_zero]
+    exact (congrFun (j.stalkMap x v) e).trans (h e he)
+  · have hh := congrFun (j.stalkMap x v) e
+    change j.edge e.val (B.stalkMap x v e) = _ at hh
+    rw [h e he,map_zero] at hh
+    exact hh.symm
+
+theorem map_upwardKernel (x : V) (F : Set (Outgoing (G := G) x)) :
+    (B.upwardKernel x F).map (j.vertex x).toLinearMap = C.upwardKernel x F := by
+  ext v
+  constructor
+  · rintro ⟨w,hw,rfl⟩
+    exact (j.upwardKernel_iff x F w).mpr hw
+  · intro hv
+    refine ⟨(j.vertex x).symm v,?_,(j.vertex x).apply_symm_apply v⟩
+    apply (j.upwardKernel_iff x F _).mp
+    simpa using hv
+
+theorem map_lowerImage (e : E) (F : Set (Outgoing (G := G) (G.source e))) :
+    ((B.upwardKernel (G.source e) F).map (B.lower e)).map (j.edge e).toLinearMap =
+      (C.upwardKernel (G.source e) F).map (C.lower e) := by
+  rw [← Submodule.map_comp]
+  have h : (j.edge e).toLinearMap.comp (B.lower e) =
+      (C.lower e).comp (j.vertex (G.source e)).toLinearMap := LinearMap.ext (j.lower e)
+  rw [h,Submodule.map_comp,j.map_upwardKernel]
+
+theorem map_upperImage (e : E) (F : Set (Outgoing (G := G) (G.target e))) :
+    ((B.upwardKernel (G.target e) F).map (B.upper e)).map (j.edge e).toLinearMap =
+      (C.upwardKernel (G.target e) F).map (C.upper e) := by
+  rw [← Submodule.map_comp]
+  have h : (j.edge e).toLinearMap.comp (B.upper e) =
+      (C.upper e).comp (j.vertex (G.target e)).toLinearMap := LinearMap.ext (j.upper e)
+  rw [h,Submodule.map_comp,j.map_upwardKernel]
+
+end Iso
+end KLInvariance.MomentGraph.Sheaf
+
+namespace KLInvariance.EmbeddedModule
+variable {R M N : Type*} [CommRing R] [AddCommGroup M] [Module R M]
+  [AddCommGroup N] [Module R N]
+
+def scaled (r : R) (K : Submodule R M) : Submodule R M :=
+  K.map (LinearMap.lsmul R M r)
+
+theorem map_scaled (j : M ≃ₗ[R] N) (r : R) (K : Submodule R M) :
+    (scaled r K).map j.toLinearMap = scaled r (K.map j.toLinearMap) := by
+  simp only [scaled,← Submodule.map_comp]
+  congr 1
+  ext v
+  exact j.map_smul r v
+
+/-- Scalar comparison is transported by one and the same edge isomorphism. -/
+theorem map_eq_scaled_iff (j : M ≃ₗ[R] N) (r : R) (I J : Submodule R M) :
+    I.map j.toLinearMap = scaled r (J.map j.toLinearMap) ↔ I = scaled r J := by
+  rw [← map_scaled]
+  constructor
+  · intro h
+    have h' := congrArg (Submodule.map j.symm.toLinearMap) h
+    simpa only [← Submodule.map_comp,LinearEquiv.symm_comp,
+      Submodule.map_id] using h'
+  · exact congrArg (Submodule.map j.toLinearMap)
+
+end KLInvariance.EmbeddedModule
+
+end
+
+
+section
+
+
+namespace KLInvariance.MomentGraph.Sheaf
+universe u v w z z'
+variable {R : Type u} [CommRing R] {V : Type v} [PartialOrder V]
+  {E : Type w} {G : OrderedGraph V E}
+  (B : Sheaf.{u,v,w,z} (R := R) G) (C : Sheaf.{u,v,w,z'} (R := R) G) (x : V)
+
+/-- An isomorphism already constructed strictly above `x`, including the edge
+modules whose upper endpoints are above `x`. The latter are canonically supplied
+by the upper quotient rule in the local decomposition. -/
+structure AboveIso where
+  vertex : ∀ y, x < y → B.vertex y ≃ₗ[R] C.vertex y
+  edge : ∀ e, x < G.target e → B.edge e ≃ₗ[R] C.edge e
+  lower : ∀ e (h : x < G.source e) v,
+    edge e (h.trans (G.increasing e)) (B.lower e v) =
+      C.lower e (vertex (G.source e) h v)
+  upper : ∀ e (h : x < G.target e) v,
+    edge e h (B.upper e v) = C.upper e (vertex (G.target e) h v)
+
+namespace AboveIso
+variable {B C x} (j : AboveIso B C x)
+
+def symm : AboveIso C B x where
+  vertex y h := (j.vertex y h).symm
+  edge e h := (j.edge e h).symm
+  lower e h v := by
+    apply (j.edge e (h.trans (G.increasing e))).injective
+    rw [LinearEquiv.apply_symm_apply,j.lower e h,LinearEquiv.apply_symm_apply]
+  upper e h v := by
+    apply (j.edge e h).injective
+    simp only [LinearEquiv.apply_symm_apply,j.upper]
+
+noncomputable def mapAssignment (f : B.Assignment) : C.Assignment :=
+by
+  classical
+  exact fun y => if h : x < y then j.vertex y h (f y) else 0
+
+lemma mapAssignment_at (f : B.Assignment) (y : V) (h : x < y) :
+    j.mapAssignment f y = j.vertex y h (f y) := by
+  classical
+  simp [mapAssignment,h]
+
+lemma compatible (f : B.Assignment) (hf : B.CompatibleOn (Set.Ioi x) f) :
+    C.CompatibleOn (Set.Ioi x) (j.mapAssignment f) := by
+  intro e hs ht
+  change x < G.source e at hs
+  change x < G.target e at ht
+  rw [j.mapAssignment_at f _ hs,j.mapAssignment_at f _ ht,← j.lower e hs,← j.upper e ht]
+  exact congrArg (j.edge e ht) (hf e hs ht)
+
+noncomputable def boundary : B.Boundary x ≃ₗ[R] C.Boundary x :=
+  LinearEquiv.piCongrRight (fun e => j.edge e.val (lt_of_eq_of_lt e.property.symm (G.increasing e.val)))
+
+lemma boundaryMap (f : B.sections (Set.Ioi x)) :
+    j.boundary (B.boundaryMap x f) =
+      C.boundaryMap x ⟨j.mapAssignment f.val,j.compatible f.val f.property⟩ := by
+  funext e
+  change j.edge e.val _ (B.upper e.val (f.val (G.target e.val))) =
+    C.upper e.val (j.mapAssignment f.val (G.target e.val))
+  rw [j.mapAssignment_at _ _ (lt_of_eq_of_lt e.property.symm (G.increasing e.val)),j.upper]
+
+lemma map_boundaryImage : (B.boundaryImage x).map j.boundary.toLinearMap = C.boundaryImage x := by
+  apply le_antisymm
+  · rintro v ⟨w,⟨f,rfl⟩,rfl⟩
+    exact ⟨⟨j.mapAssignment f.val,j.compatible f.val f.property⟩,(j.boundaryMap f).symm⟩
+  · rintro v ⟨f,rfl⟩
+    refine ⟨B.boundaryMap x ⟨j.symm.mapAssignment f.val,j.symm.compatible f.val f.property⟩,
+      ⟨_,rfl⟩,?_⟩
+    funext e
+    change j.edge e.val _ (B.upper e.val (j.symm.mapAssignment f.val (G.target e.val))) =
+      C.upper e.val (f.val (G.target e.val))
+    rw [j.symm.mapAssignment_at _ _ (lt_of_eq_of_lt e.property.symm (G.increasing e.val)),j.upper]
+    exact congrArg (C.upper e.val) ((j.vertex _ _).apply_symm_apply _)
+
+lemma map_stalk_image (hBf : B.Flabby) (hBg : B.Generated)
+    (hCf : C.Flabby) (hCg : C.Generated) :
+    (LinearMap.range (B.stalkMap x)).map j.boundary.toLinearMap =
+      LinearMap.range (C.stalkMap x) := by
+  rw [B.range_stalkMap_eq_boundaryImage hBf hBg,C.range_stalkMap_eq_boundaryImage hCf hCg]
+  exact j.map_boundaryImage
+
+/-- One genuine descending splitting: the finite-free new top summand is
+proved from the local-ring minimum cover, and all outgoing restriction maps
+commute with the selected edge isomorphisms. -/
+theorem extend_stalk [IsLocalRing R]
+    [Module.Free R (B.vertex x)] [Module.Finite R (B.vertex x)]
+    [Module.Free R (C.vertex x)] [Module.Finite R (C.vertex x)]
+    (hBf : B.Flabby) (hBg : B.Generated) (hCf : C.Flabby) (hCg : C.Generated)
+    (hmin : LinearMap.ker (C.stalkMap x) ≤
+      IsLocalRing.maximalIdeal R • (⊤ : Submodule R (C.vertex x))) :
+    ∃ (g : B.vertex x →ₗ[R] C.vertex x)
+      (jv : B.vertex x ≃ₗ[R] C.vertex x × LinearMap.ker g),
+      Module.Free R (LinearMap.ker g) ∧ Module.Finite R (LinearMap.ker g) ∧
+      ∀ (b : B.vertex x) (e : Outgoing (G := G) x),
+        j.edge e.val (lt_of_eq_of_lt e.property.symm (G.increasing e.val)) (B.stalkMap x b e) =
+          C.stalkMap x (jv b).1 e := by
+  let N := LinearMap.range (C.stalkMap x)
+  let π : C.vertex x →ₗ[R] N := (C.stalkMap x).rangeRestrict
+  have him := j.map_stalk_image hBf hBg hCf hCg
+  let φ : B.vertex x →ₗ[R] N := LinearMap.codRestrict N
+    (j.boundary.toLinearMap.comp (B.stalkMap x)) (by
+      intro b
+      change j.boundary (B.stalkMap x b) ∈ LinearMap.range (C.stalkMap x)
+      rw [← him]
+      exact ⟨B.stalkMap x b,⟨b,rfl⟩,rfl⟩)
+  have hφ : Function.Surjective φ := by
+    intro n
+    have hn : n.val ∈ (LinearMap.range (B.stalkMap x)).map j.boundary.toLinearMap := by
+      rw [him]
+      exact n.property
+    rcases hn with ⟨w,⟨b,rfl⟩,hb⟩
+    exact ⟨b,Subtype.ext hb⟩
+  have hπmin : LinearMap.ker π ≤ IsLocalRing.maximalIdeal R • (⊤ : Submodule R (C.vertex x)) := by
+    intro c hc
+    apply hmin
+    exact congrArg Subtype.val (LinearMap.mem_ker.mp hc)
+  obtain ⟨g,jv,_,hg,hfree,hfinite,hjv⟩ :=
+    LocalCover.exists_split_minimal_cover π φ (by intro n; obtain ⟨c,hc⟩ := n.property; exact ⟨c,Subtype.ext hc⟩) hφ hπmin
+  refine ⟨g,jv,hfree,hfinite,?_⟩
+  intro b e
+  have h := congrArg (fun n : N => n.val e) (hjv b).2
+  exact h.symm
+
+end AboveIso
+end KLInvariance.MomentGraph.Sheaf
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.Sheaf
+universe u v w z i
+variable {R : Type u} [CommRing R] {V : Type v} [PartialOrder V]
+  {E : Type w} {G : OrderedGraph V E} {ι : Type i}
+
+/-- Finite direct sums represented by products, as used in the plane localization. -/
+def pi (B : ι → Sheaf.{u,v,w,z} (R := R) G) : Sheaf (R := R) G where
+  vertex x := ModuleCat.of R (∀ i, (B i).vertex x)
+  edge e := ModuleCat.of R (∀ i, (B i).edge e)
+  lower e := LinearMap.pi (fun i => ((B i).lower e).comp (LinearMap.proj i))
+  upper e := LinearMap.pi (fun i => ((B i).upper e).comp (LinearMap.proj i))
+
+lemma pi_compatible (B : ι → Sheaf.{u,v,w,z} (R := R) G) (Ω : Set V)
+    (f : (pi B).Assignment) :
+    (pi B).CompatibleOn Ω f ↔ ∀ i, (B i).CompatibleOn Ω (fun x => f x i) := by
+  constructor
+  · intro h i e hs ht
+    exact congrFun (h e hs ht) i
+  · intro h e hs ht
+    funext i
+    exact h i e hs ht
+
+lemma pi_generated (B : ι → Sheaf.{u,v,w,z} (R := R) G)
+    (h : ∀ i, (B i).Generated) : (pi B).Generated := by
+  intro x v
+  classical
+  choose f hf hfx using fun i => h i x (v i)
+  exact ⟨fun y i => f i y,(pi_compatible B Set.univ _).mpr hf,funext hfx⟩
+
+lemma pi_flabby (B : ι → Sheaf.{u,v,w,z} (R := R) G)
+    (h : ∀ i, (B i).Flabby) : (pi B).Flabby := by
+  intro Ω hΩ f hf
+  classical
+  choose g hg hgf using fun i => h i Ω hΩ (fun x => f x i) ((pi_compatible B Ω f).mp hf i)
+  exact ⟨fun y i => g i y,(pi_compatible B Set.univ _).mpr hg,fun x hx => funext (fun i => hgf i x hx)⟩
+
+lemma pi_kernel_iff (B : ι → Sheaf.{u,v,w,z} (R := R) G) (x : V)
+    (f : (pi B).vertex x) :
+    f ∈ LinearMap.ker ((pi B).stalkMap x) ↔ ∀ i, f i ∈ LinearMap.ker ((B i).stalkMap x) := by
+  constructor
+  · intro h i
+    apply LinearMap.mem_ker.mpr
+    funext e
+    rcases e with ⟨e,he⟩
+    subst x
+    exact congrFun (congrFun (LinearMap.mem_ker.mp h) ⟨e,rfl⟩) i
+  · intro h
+    apply LinearMap.mem_ker.mpr
+    funext e i
+    rcases e with ⟨e,he⟩
+    subst x
+    exact congrFun (LinearMap.mem_ker.mp (h i)) ⟨e,rfl⟩
+
+end KLInvariance.MomentGraph.Sheaf
+
+namespace KLInvariance.LocalCover
+universe u v w
+variable {R : Type u} [CommRing R] {ι : Type v} [Fintype ι]
+  (M : ι → Type w) [∀ i, AddCommGroup (M i)] [∀ i, Module R (M i)]
+
+/-- Coordinatewise membership in the ideal multiple of a finite product. -/
+lemma pi_mem_ideal_smul (I : Ideal R) (f : ∀ i, M i)
+    (hf : ∀ i, f i ∈ I • (⊤ : Submodule R (M i))) :
+    f ∈ I • (⊤ : Submodule R (∀ i, M i)) := by
+  classical
+  have hs (i : ι) : Pi.single i (f i) ∈ I • (⊤ : Submodule R (∀ i, M i)) := by
+    have hmap : (I • (⊤ : Submodule R (M i))).map (LinearMap.single R M i) ≤
+        I • (⊤ : Submodule R (∀ i, M i)) := by
+      rw [Submodule.map_smul'']
+      exact Submodule.smul_mono le_rfl le_top
+    exact hmap ⟨f i,hf i,rfl⟩
+  simpa only [Finset.univ_sum_single] using Submodule.sum_mem (I • (⊤ : Submodule R (∀ i, M i))) (t := Finset.univ) (fun i _ => hs i)
+
+end KLInvariance.LocalCover
+
+namespace KLInvariance.MomentGraph.Sheaf
+universe u v w z i
+variable {R : Type u} [CommRing R] {V : Type v} [PartialOrder V]
+  {E : Type w} {G : OrderedGraph V E} {ι : Type i} [Fintype ι]
+
+lemma pi_minimal (B : ι → Sheaf.{u,v,w,z} (R := R) G) (x : V) (I : Ideal R)
+    (h : ∀ i, LinearMap.ker ((B i).stalkMap x) ≤ I • (⊤ : Submodule R ((B i).vertex x))) :
+    LinearMap.ker ((pi B).stalkMap x) ≤ I • (⊤ : Submodule R ((pi B).vertex x)) := by
+  intro f hf
+  exact LocalCover.pi_mem_ideal_smul (fun i => (B i).vertex x) I f
+    (fun i => h i ((pi_kernel_iff B x f).mp hf i))
+
+end KLInvariance.MomentGraph.Sheaf
+
+end
+
+
+section
+
+namespace KLInvariance.QuotientEdge
+variable {R M N E F : Type*} [CommRing R]
+  [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N]
+  [AddCommGroup E] [Module R E] [AddCommGroup F] [Module R F]
+
+/-- The edge isomorphism forced by an upper stalk isomorphism and exact quotient maps. -/
+noncomputable def iso (f : M →ₗ[R] E) (g : N →ₗ[R] F)
+    (hf : Function.Surjective f) (hg : Function.Surjective g)
+    (h : M ≃ₗ[R] N) (hk : (LinearMap.ker f).map h.toLinearMap = LinearMap.ker g) : E ≃ₗ[R] F :=
+  (f.quotKerEquivOfSurjective hf).symm.trans
+    ((Submodule.Quotient.equiv _ _ h hk).trans (g.quotKerEquivOfSurjective hg))
+
+@[simp] lemma iso_apply (f : M →ₗ[R] E) (g : N →ₗ[R] F)
+    (hf : Function.Surjective f) (hg : Function.Surjective g)
+    (h : M ≃ₗ[R] N) (hk : (LinearMap.ker f).map h.toLinearMap = LinearMap.ker g) (v : M) :
+    iso f g hf hg h hk (f v) = g (h v) := by
+  simp [iso,Submodule.Quotient.equiv]
+
+lemma map_kernel_scalar (f : M →ₗ[R] E) (g : N →ₗ[R] F) (r : R)
+    (hf : ∀ v, f v = 0 ↔ ∃ w, v = r • w)
+    (hg : ∀ v, g v = 0 ↔ ∃ w, v = r • w) (h : M ≃ₗ[R] N) :
+    (LinearMap.ker f).map h.toLinearMap = LinearMap.ker g := by
+  ext v
+  constructor
+  · rintro ⟨w,hw,rfl⟩
+    obtain ⟨u,rfl⟩ := (hf w).mp hw
+    exact (hg _).mpr ⟨h u,h.map_smul r u⟩
+  · intro hv
+    obtain ⟨w,rfl⟩ := (hg v).mp hv
+    refine ⟨r • h.symm w,(hf _).mpr ⟨h.symm w,rfl⟩,?_⟩
+    simp
+
+/-- No separate choice of edge data or scalar compatibility hypothesis is needed. -/
+theorem exists_scalar_iso (f : M →ₗ[R] E) (g : N →ₗ[R] F) (r : R)
+    (hfs : Function.Surjective f) (hgs : Function.Surjective g)
+    (hf : ∀ v, f v = 0 ↔ ∃ w, v = r • w)
+    (hg : ∀ v, g v = 0 ↔ ∃ w, v = r • w) (h : M ≃ₗ[R] N) :
+    ∃ j : E ≃ₗ[R] F, ∀ v, j (f v) = g (h v) :=
+  ⟨iso f g hfs hgs h (map_kernel_scalar f g r hf hg h),iso_apply f g hfs hgs h _⟩
+
+end KLInvariance.QuotientEdge
+
+end
+
+
+section
+
+/-! Partial sheaf isomorphisms on upper sets, and their actual descending
+extension. In particular the incoming edge component is induced by the upper
+scalar quotient, not independently assumed. -/
+namespace KLInvariance.MomentGraph.Sheaf
+universe u v w z z'
+variable {R : Type u} [CommRing R] {V : Type v} [PartialOrder V]
+  {E : Type w} {G : OrderedGraph V E}
+  (B : Sheaf.{u,v,w,z} (R := R) G) (C : Sheaf.{u,v,w,z'} (R := R) G)
+
+ def UpperQuotient (α : E → R) : Prop :=
+  ∀ e, Function.Surjective (B.upper e) ∧
+    ∀ v, B.upper e v = 0 ↔ ∃ w, v = α e • w
+
+def castStalkIso {x y : V} (h : x=y) (j : B.vertex x ≃ₗ[R] C.vertex x) :
+    B.vertex y ≃ₗ[R] C.vertex y := h ▸ j
+
+structure OnIso (U : Set V) where
+  vertex : ∀ x, x ∈ U → B.vertex x ≃ₗ[R] C.vertex x
+  edge : ∀ e, G.target e ∈ U → B.edge e ≃ₗ[R] C.edge e
+  lower : ∀ e (hs : G.source e ∈ U) (ht : G.target e ∈ U) v,
+    edge e ht (B.lower e v) = C.lower e (vertex (G.source e) hs v)
+  upper : ∀ e (ht : G.target e ∈ U) v,
+    edge e ht (B.upper e v) = C.upper e (vertex (G.target e) ht v)
+
+namespace OnIso
+variable {B C} {U : Set V} (j : OnIso B C U)
+
+ def above (x : V) (hx : Set.Ioi x ⊆ U) : AboveIso B C x where
+  vertex y h := j.vertex y (hx h)
+  edge e h := j.edge e (hx h)
+  lower e h v := j.lower e (hx h) (hx (h.trans (G.increasing e))) v
+  upper e h v := j.upper e (hx h) v
+
+ def ofUniv (j : OnIso B C Set.univ) : Iso B C where
+  vertex x := j.vertex x (Set.mem_univ x)
+  edge e := j.edge e (Set.mem_univ _)
+  lower e v := j.lower e (Set.mem_univ _) (Set.mem_univ _) v
+  upper e v := j.upper e (Set.mem_univ _) v
+
+/-- Given the new stalk and its outgoing commuting squares, all new incoming
+edges are forced by the scalar quotient rule. -/
+noncomputable def extend (hU : IsUpperSet U) (x : V) (hx : x ∉ U)
+    (hmax : Set.Ioi x ⊆ U) (α : E → R)
+    (hBq : B.UpperQuotient α) (hCq : C.UpperQuotient α)
+    (jv : B.vertex x ≃ₗ[R] C.vertex x)
+    (hout : ∀ e (he : G.source e = x) v,
+      j.edge e (hmax (lt_of_eq_of_lt he.symm (G.increasing e)))
+        (B.lower e v) = C.lower e (castStalkIso B C he.symm jv v)) :
+    OnIso B C (insert x U) := by
+  classical
+  let vj : ∀ y, y ∈ insert x U → B.vertex y ≃ₗ[R] C.vertex y := fun y hy =>
+    if h : y = x then castStalkIso B C h.symm jv
+    else j.vertex y ((Set.mem_insert_iff.mp hy).resolve_left h)
+  have vj_old (y : V) (hy : y ∈ U) : vj y (Set.mem_insert_of_mem x hy) = j.vertex y hy := by
+    simp only [vj,dite_eq_right (show y ≠ x from fun h => hx (h ▸ hy))]
+  have vj_new : vj x (Set.mem_insert x U) = jv := by simp [vj,castStalkIso]
+  let ej : ∀ e, G.target e ∈ insert x U → B.edge e ≃ₗ[R] C.edge e := fun e he =>
+    if h : G.target e = x then
+      QuotientEdge.iso (B.upper e) (C.upper e) (hBq e).1 (hCq e).1
+        (vj (G.target e) he)
+        (QuotientEdge.map_kernel_scalar (B.upper e) (C.upper e) (α e)
+          (hBq e).2 (hCq e).2 (vj (G.target e) he))
+    else j.edge e ((Set.mem_insert_iff.mp he).resolve_left h)
+  have ej_old (e : E) (he : G.target e ∈ U) :
+      ej e (Set.mem_insert_of_mem x he) = j.edge e he := by
+    simp only [ej,dite_eq_right (show G.target e ≠ x from fun h => hx (h ▸ he))]
+  refine ⟨vj,ej,?_,?_⟩
+  · intro e hs ht v
+    have htU : G.target e ∈ U := by
+      rcases Set.mem_insert_iff.mp hs with hs | hs
+      · exact hmax (lt_of_eq_of_lt hs.symm (G.increasing e))
+      · exact hU (G.increasing e).le hs
+    rw [ej_old e htU]
+    by_cases he : G.source e = x
+    · subst x
+      rw [vj_new]
+      exact hout e rfl v
+    · have hsU : G.source e ∈ U := (Set.mem_insert_iff.mp hs).resolve_left he
+      rw [vj_old _ hsU]
+      exact j.lower e hsU htU v
+  · intro e ht v
+    by_cases he : G.target e = x
+    · simp only [ej,dite_eq_left he]
+      exact QuotientEdge.iso_apply _ _ _ _ _ _ v
+    · have htU : G.target e ∈ U := (Set.mem_insert_iff.mp ht).resolve_left he
+      rw [ej_old e htU,vj_old _ htU]
+      exact j.upper e htU v
+
+end OnIso
+end KLInvariance.MomentGraph.Sheaf
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph
+universe um
+variable {V E : Type um} [PartialOrder V] (G : OrderedGraph V E)
+
+ def removeGraph (x : V) (ht : ∀ e, G.target e ≠ x) :
+    OrderedGraph {y : V // y ≠ x} {e : E // G.source e ≠ x} where
+  source e := ⟨G.source e.val,e.property⟩
+  target e := ⟨G.target e.val,ht e.val⟩
+  increasing e := G.increasing e.val
+
+ def removeTarget (x : V) (ht : ∀ e, G.target e ≠ x) :
+    {e : E // G.source e = x} → {y : V // y ≠ x} := fun e => ⟨G.target e.val,ht e.val⟩
+
+noncomputable def splitVertex (x : V) : V ≃ WithBot {y : V // y ≠ x} := by
+  classical
+  exact
+    { toFun := fun y => if h : y=x then ⊥ else (⟨y,h⟩ : {y : V // y ≠ x})
+      invFun := WithBot.recBotCoe x Subtype.val
+      left_inv := by intro y; by_cases h : y=x <;> simp [h]
+      right_inv := by
+        intro y
+        induction y using WithBot.recBotCoe with
+        | bot => simp
+        | coe y => simp [y.property] }
+
+noncomputable def splitEdge (x : V) : E ≃ {e : E // G.source e ≠ x} ⊕ {e : E // G.source e = x} := by
+  classical
+  exact
+    { toFun := fun e => if h : G.source e=x then Sum.inr ⟨e,h⟩ else Sum.inl ⟨e,h⟩
+      invFun := Sum.elim Subtype.val Subtype.val
+      left_inv := by intro e; by_cases h : G.source e=x <;> simp [h]
+      right_inv := by
+        intro e
+        cases e with
+        | inl e => simp [e.property]
+        | inr e => simp [e.property] }
+
+omit [PartialOrder V] in
+@[simp] theorem splitVertex_self (x : V) : splitVertex x x = ⊥ := by
+  classical
+  simp [splitVertex]
+
+omit [PartialOrder V] in
+ theorem splitVertex_ne (x y : V) (hy : y ≠ x) :
+    splitVertex x y = (⟨y,hy⟩ : {y : V // y ≠ x}) := by
+  classical
+  simp [splitVertex,hy]
+
+noncomputable def splitGraph (x : V) (ht : ∀ e, G.target e ≠ x) :
+    GraphEquiv G (bottomGraph (removeGraph G x ht) (removeTarget G x ht)) where
+  vertex := splitVertex x
+  edge := splitEdge G x
+  source := by
+    classical
+    intro e
+    by_cases h : G.source e=x
+    · simp [splitEdge,splitVertex,h,bottomGraph]
+    · simp [splitEdge,splitVertex,h,bottomGraph,removeGraph]
+  target := by
+    classical
+    intro e
+    by_cases h : G.source e=x
+    · simp [splitEdge,splitVertex,h,ht e,bottomGraph,removeTarget]
+    · simp [splitEdge,splitVertex,h,ht e,bottomGraph,removeGraph]
+
+ theorem splitEdge_labels {A : Type*} (α : E → A) (x : V) :
+    (Sum.elim (fun e : {e : E // G.source e ≠ x} => α e.val)
+      (fun e : {e : E // G.source e = x} => α e.val)) ∘ (splitEdge G x) = α := by
+  classical
+  funext e
+  by_cases h : G.source e=x <;> simp [splitEdge,h]
+
+end KLInvariance.MomentGraph
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+  (hneg : ∀ n < 0, 𝓐 n = ⊥)
+
+/-- The outcome of the actual minimal boundary construction. Full upward-kernel
+freeness, integral duality and KL characters are NOT part of this structure. -/
+structure BoundarySheaf {V E : Type um} [PartialOrder V] (G : OrderedGraph V E)
+    (α : E → A) (b : V) where
+  sheaf : GradedSheaf (𝓐 := 𝓐) G
+  free : sheaf.FreeStalks
+  quotient : sheaf.UpperQuotient α
+  flabby : sheaf.EdgeFlabby
+  generated : sheaf.forget.Generated
+  top : sheaf.vertex b = ModuleData.unit hneg
+  minimal : ∀ x, x ≠ b → sheaf.KernelMinimal x
+
+variable {V E : Type um} [PartialOrder V] (G : OrderedGraph V E)
+
+noncomputable def singletonSheaf [IsEmpty E] : GradedSheaf (𝓐 := 𝓐) G where
+  vertex := fun _ => ModuleData.unit hneg
+  edge := isEmptyElim
+  lower := isEmptyElim
+  upper := isEmptyElim
+
+ theorem singleton_boundarySheaf (b : V) (hV : ∀ x, x=b) (α : E → A) :
+    Nonempty (BoundarySheaf hneg G α b) := by
+  have hempty : IsEmpty E := ⟨fun e => (G.increasing e).ne ((hV _).trans (hV _).symm)⟩
+  let := hempty
+  let B := singletonSheaf hneg G
+  refine ⟨⟨B,?_,?_,?_,?_,rfl,?_⟩⟩
+  · intro x
+    exact ModuleData.unit_free hneg
+  · intro e
+    exact isEmptyElim e
+  · intro Ω _ f _
+    exact ⟨f,fun e => isEmptyElim e,fun _ _ => rfl⟩
+  · intro x v
+    exact ⟨fun _ => v,fun e => isEmptyElim e,rfl⟩
+  · intro x hx
+    exact (hx (hV x)).elim
+
+variable [IsNoetherianRing A]
+  (hscalar : ∀ a : A, ∃ c : k, a-algebraMap k A c ∈ positiveIdeal 𝓐)
+
+include hscalar in
+ theorem exists_boundarySheaf_card (n : ℕ)
+    [Fintype V] [Fintype E] (b : V) (hb : ∀ x, x ≤ b)
+    (α : E → A) (hα : ∀ e, α e ∈ 𝓐 1) (hn : Fintype.card V = n) :
+    Nonempty (BoundarySheaf hneg G α b) := by
+  classical
+  induction n using Nat.strong_induction_on generalizing V E with
+  | h n ih =>
+    by_cases hV : ∀ x : V, x=b
+    · exact singleton_boundarySheaf hneg G b hV α
+    obtain ⟨y,hy⟩ := not_forall.mp hV
+    obtain ⟨x,hx⟩ := exists_minimal_of_wellFoundedLT (fun _ : V => True) ⟨b,trivial⟩
+    have hxb : x ≠ b := by
+      intro he
+      have hlt : y < x := he ▸ lt_of_le_of_ne (hb y) hy
+      exact hx.not_lt trivial hlt
+    have ht (e : E) : G.target e ≠ x := by
+      intro he
+      exact hx.not_lt trivial (he ▸ G.increasing e)
+    let V' := {y : V // y ≠ x}
+    let E' := {e : E // G.source e ≠ x}
+    let D := {e : E // G.source e = x}
+    let G' := removeGraph G x ht
+    let τ := removeTarget G x ht
+    let b' : V' := ⟨b,hxb.symm⟩
+    let α' : E' → A := fun e => α e.val
+    let β : D → A := fun e => α e.val
+    have hβ : ∀ e : D, β e ∈ 𝓐 1 := fun e => hα e.val
+    have hcard : Fintype.card V' < n := by
+      rw [← hn]
+      exact Fintype.card_subtype_lt (x := x) (not_not.mpr rfl)
+    obtain ⟨B⟩ := ih (Fintype.card V') hcard G' b' (fun y => hb y.val)
+      α' (fun e => hα e.val) rfl
+    obtain ⟨C⟩ := ModuleData.exists_minimalCover hneg hscalar (B.sheaf.attachImage τ β hβ)
+    let H := B.sheaf.extendBottom τ β hβ C
+    let e := splitGraph G x ht
+    refine ⟨⟨H.reindex e,?_,?_,?_,?_,?_,?_⟩⟩
+    · exact H.reindex_freeStalks e (B.sheaf.extendBottom_freeStalks τ β hβ C B.free)
+    · have hh := H.reindex_upperQuotient e
+        (B.sheaf.extendBottom_upperQuotient τ β hβ C B.quotient)
+      have he : (Sum.elim α' β) ∘ e.edge = α := splitEdge_labels G α x
+      exact he ▸ hh
+    · exact H.reindex_edgeFlabby e (B.sheaf.extendBottom_edgeFlabby τ β hβ C B.flabby)
+    · exact H.reindex_generated e (B.sheaf.extendBottom_generated τ β hβ C B.generated)
+    · change H.vertex (splitVertex x b) = _
+      rw [splitVertex_ne x b hxb.symm]
+      exact B.top
+    · intro z hz
+      apply H.reindex_kernelMinimal e z
+      by_cases hzx : z=x
+      · subst z
+        change H.KernelMinimal (splitVertex x x)
+        rw [splitVertex_self]
+        exact B.sheaf.extendBottom_kernelMinimal_bot τ β hβ C
+      · change H.KernelMinimal (splitVertex x z)
+        rw [splitVertex_ne x z hzx]
+        apply B.sheaf.extendBottom_kernelMinimal_coe τ β hβ C ⟨z,hzx⟩
+        apply B.minimal
+        intro hzz
+        exact hz (congrArg Subtype.val hzz)
+
+include hscalar in
+ theorem exists_boundarySheaf [Fintype V] [Fintype E] (b : V) (hb : ∀ x, x ≤ b)
+    (α : E → A) (hα : ∀ e, α e ∈ 𝓐 1) : Nonempty (BoundarySheaf hneg G α b) :=
+  exists_boundarySheaf_card hneg G hscalar (Fintype.card V) b hb α hα rfl
+
+end KLInvariance.MomentGraph
+
+end
+
+
+section
+
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/KazhdanLusztig/Hecke.lean b/lean/OAI/RepresentationTheory/KazhdanLusztig/Hecke.lean
index 151d40e55995df5397c6aae30ff4caf93d28d79e..c3449bd23b79103c78f6a87e7cb1cfd1fbba0309 100644
--- a/lean/OAI/RepresentationTheory/KazhdanLusztig/Hecke.lean
+++ b/lean/OAI/RepresentationTheory/KazhdanLusztig/Hecke.lean
@@ -1,1118 +1,1118 @@
-import OAI.RepresentationTheory.KazhdanLusztig.Bruhat
-
-namespace OAI
-
-/-!
-Hecke and Laurent actions, braid independence, bar involution, triangular support and inverse coefficients.
--/
-
-section
-namespace KLInvariance
-
-open Polynomial
-
-universe u v u' v'
-
-variable {B : Type u} {W : Type v} [klPreservedInstance1 : Group W] {M : CoxeterMatrix B}
-
-
-namespace Hecke
-
-variable {C : Type*} [CommRing C]
-
-/-- The standard free module of the actual Coxeter group. -/
-abbrev Space (W C : Type*) [Zero C] := W →₀ C
-
-noncomputable def basis (w : W) : Space W C := Finsupp.single w 1
-
-include klPreservedInstance1 in
-theorem end_ext {f g : Module.End C (Space W C)}
-    (h : ∀ w, f (basis w) = g (basis w)) : f = g := by
-  have _ := klPreservedInstance1
-  apply Finsupp.lhom_ext
-  intro w c
-  have hc : Finsupp.single w c = c • basis w := by
-    simp [basis, Finsupp.smul_single]
-  rw [hc, map_smul, map_smul, h]
-
-/-- Normalized Hecke multiplication: H_s² = 1 + a H_s. -/
-noncomputable def left (cs : CoxeterSystem M W) (a : C) (i : B) :
-    Module.End C (Space W C) :=
-  Finsupp.linearCombination C (fun w => basis (cs.simple i * w) +
-    if cs.length (cs.simple i * w) < cs.length w then a • basis w else 0)
-
-noncomputable def right (cs : CoxeterSystem M W) (a : C) (i : B) :
-    Module.End C (Space W C) :=
-  Finsupp.linearCombination C (fun w => basis (w * cs.simple i) +
-    if cs.length (w * cs.simple i) < cs.length w then a • basis w else 0)
-
-@[simp] theorem left_basis (cs : CoxeterSystem M W) (a : C) (i : B) (w : W) :
-    left cs a i (basis w) = basis (cs.simple i * w) +
-      if cs.length (cs.simple i * w) < cs.length w then a • basis w else 0 := by
-  simp [left, basis, Finsupp.linearCombination_single]
-
-@[simp] theorem right_basis (cs : CoxeterSystem M W) (a : C) (i : B) (w : W) :
-    right cs a i (basis w) = basis (w * cs.simple i) +
-      if cs.length (w * cs.simple i) < cs.length w then a • basis w else 0 := by
-  simp [right, basis, Finsupp.linearCombination_single]
-
-theorem left_right_commute (cs : CoxeterSystem M W) (a : C) (i j : B) :
-    Commute (left cs a i) (right cs a j) := by
-  apply end_ext
-  intro w
-  simp only [Module.End.mul_apply, right_basis, left_basis, map_add]
-  by_cases heq : cs.simple i * w = w * cs.simple j
-  · have heq' : cs.simple i * (w * cs.simple j) = w := by rw [← heq]; simp
-    have heq'' : cs.simple i * w * cs.simple j = w := by simpa [mul_assoc] using heq'
-    rw [heq', heq'']
-    by_cases h : cs.length (cs.simple i * w) < cs.length w
-    · have h' : cs.length (w * cs.simple j) < cs.length w := heq ▸ h
-      simp only [map_smul, right_basis, left_basis,
-        heq, ite_eq_left h']
-    · have h' : ¬cs.length (w * cs.simple j) < cs.length w := heq ▸ h
-      simp only [ite_eq_right h', map_zero, add_zero, heq]
-  · have hl := left_descent_mul_simple_iff cs i j w heq
-    have hr := right_descent_simple_mul_iff cs i j w heq
-    simp only [hl, hr]
-    by_cases hi : cs.length (cs.simple i * w) < cs.length w <;>
-      by_cases hj : cs.length (w * cs.simple j) < cs.length w <;>
-      simp only [hi, hj, ite_true, ite_false, map_zero, add_zero, map_smul,
-        left_basis, right_basis, smul_add, smul_smul, mul_assoc] ;
-      abel
-
-theorem left_quadratic (cs : CoxeterSystem M W) (a : C) (i : B) :
-    left cs a i * left cs a i = 1 + a • left cs a i := by
-  apply end_ext
-  intro w
-  simp only [Module.End.mul_apply, left_basis, map_add,
-    cs.simple_mul_simple_cancel_left, LinearMap.add_apply, LinearMap.smul_apply,
-    Module.End.one_apply]
-  rcases cs.length_simple_mul w i with h | h
-  · have hasc : ¬cs.length (cs.simple i * w) < cs.length w := by omega
-    have hdesc : cs.length w < cs.length (cs.simple i * w) := by omega
-    simp [hasc, hdesc]
-  · have hdesc : cs.length (cs.simple i * w) < cs.length w := by omega
-    have hasc : ¬cs.length w < cs.length (cs.simple i * w) := by omega
-    simp [hasc, hdesc, smul_add, smul_smul]
-
-/-- Inverse normalized Hecke generator, available without inverting any scalar. -/
-noncomputable def leftInv (cs : CoxeterSystem M W) (a : C) (i : B) :
-    Module.End C (Space W C) := left cs a i - a • 1
-
-theorem left_mul_leftInv (cs : CoxeterSystem M W) (a : C) (i : B) :
-    left cs a i * leftInv cs a i = 1 := by
-  apply end_ext
-  intro w
-  have h := congrArg (fun f : Module.End C (Space W C) => f (basis w))
-    (left_quadratic cs a i)
-  simp only [Module.End.mul_apply, LinearMap.add_apply, LinearMap.smul_apply,
-    Module.End.one_apply] at h
-  simp only [Module.End.mul_apply, leftInv, LinearMap.sub_apply, LinearMap.smul_apply,
-    Module.End.one_apply, map_sub, map_smul]
-  rw [h]
-  abel
-
-theorem leftInv_mul_left (cs : CoxeterSystem M W) (a : C) (i : B) :
-    leftInv cs a i * left cs a i = 1 := by
-  apply end_ext
-  intro w
-  have h := congrArg (fun f : Module.End C (Space W C) => f (basis w))
-    (left_quadratic cs a i)
-  simp only [Module.End.mul_apply, LinearMap.add_apply, LinearMap.smul_apply,
-    Module.End.one_apply] at h
-  simp only [Module.End.mul_apply, leftInv, LinearMap.sub_apply, LinearMap.smul_apply,
-    Module.End.one_apply]
-  rw [h]
-  abel
-
-noncomputable def leftWord (cs : CoxeterSystem M W) (a : C) (ω : List B) :
-    Module.End C (Space W C) := (ω.map (left cs a)).prod
-
-@[simp] theorem leftWord_nil (cs : CoxeterSystem M W) (a : C) :
-    leftWord cs a [] = 1 := rfl
-
-@[simp] theorem leftWord_cons (cs : CoxeterSystem M W) (a : C) (i : B) (ω : List B) :
-    leftWord cs a (i :: ω) = left cs a i * leftWord cs a ω := rfl
-
-/-- Acting on the identity along a reduced word gives exactly its standard
-basis element. -/
-theorem leftWord_one (cs : CoxeterSystem M W) (a : C) {ω : List B}
-    (hω : cs.IsReduced ω) :
-    leftWord cs a ω (basis 1) = basis (cs.wordProd ω) := by
-  induction ω with
-  | nil => simp
-  | cons i ω ih =>
-    have hω' : cs.IsReduced ω := by simpa using hω.drop 1
-    have hasc : ¬cs.length (cs.simple i * cs.wordProd ω) < cs.length (cs.wordProd ω) := by
-      have := hω.eq
-      rw [cs.wordProd_cons, List.length_cons, ← hω'.eq] at this
-      omega
-    simp [Module.End.mul_apply, ih hω', hasc, cs.wordProd_cons]
-
-noncomputable def rightWord (cs : CoxeterSystem M W) (a : C) (ω : List B) :
-    Module.End C (Space W C) := (ω.map (right cs a)).prod
-
-@[simp] theorem rightWord_nil (cs : CoxeterSystem M W) (a : C) :
-    rightWord cs a [] = 1 := rfl
-
-@[simp] theorem rightWord_cons (cs : CoxeterSystem M W) (a : C) (i : B) (ω : List B) :
-    rightWord cs a (i :: ω) = right cs a i * rightWord cs a ω := rfl
-
-theorem rightWord_one (cs : CoxeterSystem M W) (a : C) {ω : List B}
-    (hω : cs.IsReduced ω) :
-    rightWord cs a ω (basis 1) = basis (cs.wordProd ω)⁻¹ := by
-  induction ω with
-  | nil => simp
-  | cons i ω ih =>
-    have hω' : cs.IsReduced ω := by simpa using hω.drop 1
-    have hlen : cs.length ((cs.wordProd ω)⁻¹ * cs.simple i) =
-        cs.length (cs.simple i * cs.wordProd ω) := by
-      rw [← cs.length_inv (cs.simple i * cs.wordProd ω)]
-      simp [mul_inv_rev]
-    have hasc : ¬cs.length ((cs.wordProd ω)⁻¹ * cs.simple i) <
-        cs.length (cs.wordProd ω)⁻¹ := by
-      rw [hlen, cs.length_inv]
-      have := hω.eq
-      rw [cs.wordProd_cons, List.length_cons, ← hω'.eq] at this
-      omega
-    have hasc' : ¬cs.length ((cs.wordProd ω)⁻¹ * cs.simple i) <
-        cs.length (cs.wordProd ω) := by simpa using hasc
-    simp [Module.End.mul_apply, ih hω', hasc', cs.wordProd_cons, mul_inv_rev]
-
-theorem commute_rightWord (cs : CoxeterSystem M W) (a : C)
-    {f : Module.End C (Space W C)} (h : ∀ i, Commute f (right cs a i)) (ω : List B) :
-    Commute f (rightWord cs a ω) := by
-  apply Commute.list_prod_right
-  intro r hr
-  obtain ⟨i, _, rfl⟩ := List.mem_map.mp hr
-  exact h i
-
-/-- The regular right action is cyclic, with cyclic vector the identity. -/
-theorem commuting_ext (cs : CoxeterSystem M W) (a : C)
-    {f g : Module.End C (Space W C)}
-    (hf : ∀ i, Commute f (right cs a i))
-    (hg : ∀ i, Commute g (right cs a i))
-    (h : f (basis 1) = g (basis 1)) : f = g := by
-  apply end_ext
-  intro w
-  obtain ⟨ω, hω, heq⟩ := cs.exists_isReduced w⁻¹
-  have hw : rightWord cs a ω (basis 1) = basis w := by
-    rw [rightWord_one cs a hω, ← heq, inv_inv]
-  rw [← hw]
-  have hf' := congrArg (fun t : Module.End C (Space W C) => t (basis 1))
-    (commute_rightWord cs a hf ω).eq
-  have hg' := congrArg (fun t : Module.End C (Space W C) => t (basis 1))
-    (commute_rightWord cs a hg ω).eq
-  simp only [Module.End.mul_apply] at hf' hg'
-  rw [h] at hf'
-  exact hf'.trans hg'.symm
-
-theorem leftWord_commute_right (cs : CoxeterSystem M W) (a : C) (ω : List B) (i : B) :
-    Commute (leftWord cs a ω) (right cs a i) := by
-  apply Commute.list_prod_left
-  intro r hr
-  obtain ⟨j, _, rfl⟩ := List.mem_map.mp hr
-  exact left_right_commute cs a j i
-
-
-theorem leftWord_independent (cs : CoxeterSystem M W) (a : C) {ω η : List B}
-    (hω : cs.IsReduced ω) (hη : cs.IsReduced η) (heq : cs.wordProd ω = cs.wordProd η) :
-    leftWord cs a ω = leftWord cs a η := by
-  apply commuting_ext cs a (leftWord_commute_right cs a ω)
-    (leftWord_commute_right cs a η)
-  rw [leftWord_one cs a hω, leftWord_one cs a hη, heq]
-
-noncomputable def unit (cs : CoxeterSystem M W) (a : C) (i : B) :
-    (Module.End C (Space W C))ˣ where
-  val := left cs a i
-  inv := leftInv cs a i
-  val_inv := left_mul_leftInv cs a i
-  inv_val := leftInv_mul_left cs a i
-
-noncomputable def wordUnit (cs : CoxeterSystem M W) (a : C) (ω : List B) :
-    (Module.End C (Space W C))ˣ := (ω.map (unit cs a)).prod
-
-@[simp] theorem wordUnit_val (cs : CoxeterSystem M W) (a : C) (ω : List B) :
-    (wordUnit cs a ω).val = leftWord cs a ω := by
-  induction ω with
-  | nil => rfl
-  | cons i ω ih =>
-    change (unit cs a i).val * (wordUnit cs a ω).val = left cs a i * leftWord cs a ω
-    rw [ih]
-    rfl
-
-theorem wordUnit_independent (cs : CoxeterSystem M W) (a : C) {ω η : List B}
-    (hω : cs.IsReduced ω) (hη : cs.IsReduced η) (heq : cs.wordProd ω = cs.wordProd η) :
-    wordUnit cs a ω = wordUnit cs a η := by
-  apply Units.ext
-  simpa using leftWord_independent cs a hω hη heq
-
-/-- The genuine normalized standard-basis action, a unit in the regular
-endomorphism algebra. No presentation relations are assumed. -/
-noncomputable def standardUnit (cs : CoxeterSystem M W) (a : C) (w : W) :
-    (Module.End C (Space W C))ˣ :=
-  wordUnit cs a (Classical.choose (cs.exists_isReduced w))
-
-theorem standardUnit_word (cs : CoxeterSystem M W) (a : C) {ω : List B}
-    (hω : cs.IsReduced ω) :
-    standardUnit cs a (cs.wordProd ω) = wordUnit cs a ω := by
-  apply wordUnit_independent cs a (Classical.choose_spec (cs.exists_isReduced _)).1 hω
-  exact (Classical.choose_spec (cs.exists_isReduced _)).2.symm
-
-@[simp] theorem standardUnit_one (cs : CoxeterSystem M W) (a : C) :
-    standardUnit cs a 1 = 1 := by
-  have h := standardUnit_word cs a (ω := []) (by simp [CoxeterSystem.IsReduced])
-  simpa [wordUnit] using h
-
-@[simp] theorem standardUnit_basis_one (cs : CoxeterSystem M W) (a : C) (w : W) :
-    (standardUnit cs a w).val (basis 1) = basis w := by
-  obtain ⟨ω, hω, rfl⟩ := cs.exists_isReduced w
-  rw [standardUnit_word cs a hω, wordUnit_val, leftWord_one cs a hω]
-
-theorem standardUnit_simple_mul (cs : CoxeterSystem M W) (a : C) (i : B) (w : W)
-    (hasc : cs.length w < cs.length (cs.simple i * w)) :
-    standardUnit cs a (cs.simple i * w) = unit cs a i * standardUnit cs a w := by
-  obtain ⟨ω, hω, rfl⟩ := cs.exists_isReduced w
-  have hred : cs.IsReduced (i :: ω) := by
-    unfold CoxeterSystem.IsReduced
-    have := cs.length_simple_mul (cs.wordProd ω) i
-    simp only [cs.wordProd_cons, List.length_cons]
-    rw [← hω.eq]
-    omega
-  rw [← cs.wordProd_cons, standardUnit_word cs a hred, standardUnit_word cs a hω]
-  simp [wordUnit]
-
-
-theorem standardUnit_mul_simple (cs : CoxeterSystem M W) (a : C) (i : B) (w : W)
-    (hasc : cs.length w < cs.length (w * cs.simple i)) :
-    standardUnit cs a (w * cs.simple i) = standardUnit cs a w * unit cs a i := by
-  obtain ⟨ω, hω, rfl⟩ := cs.exists_isReduced w
-  have hred : cs.IsReduced (ω ++ [i]) := by
-    unfold CoxeterSystem.IsReduced
-    have := cs.length_mul_simple (cs.wordProd ω) i
-    simp only [cs.wordProd_append, cs.wordProd_singleton, List.length_append,
-      List.length_singleton]
-    rw [← hω.eq]
-    omega
-  rw [← cs.wordProd_singleton i, ← cs.wordProd_append, standardUnit_word cs a hred,
-    standardUnit_word cs a hω]
-  simp [wordUnit, List.map_append, List.prod_append]
-
-/-- The bar image of an actual standard basis vector; inverse units give
-expression independence before any polynomial coefficients are extracted. -/
-noncomputable def barBasis (cs : CoxeterSystem M W) (a : C) (w : W) : Space W C :=
-  ((standardUnit cs a w⁻¹)⁻¹).val (basis 1)
-
-@[simp] theorem barBasis_one (cs : CoxeterSystem M W) (a : C) :
-    barBasis cs a 1 = basis 1 := by simp [barBasis]
-
-theorem barBasis_simple_mul_ascent (cs : CoxeterSystem M W) (a : C) (i : B) (w : W)
-    (hasc : cs.length w < cs.length (cs.simple i * w)) :
-    barBasis cs a (cs.simple i * w) = leftInv cs a i (barBasis cs a w) := by
-  have hasc' : cs.length w⁻¹ < cs.length (w⁻¹ * cs.simple i) := by
-    have hlen : cs.length (w⁻¹ * cs.simple i) = cs.length (cs.simple i * w) := by
-      rw [← cs.length_inv (cs.simple i * w)]
-      simp [mul_inv_rev]
-    simpa [hlen] using hasc
-  simp only [barBasis, mul_inv_rev, cs.inv_simple,
-    standardUnit_mul_simple cs a i w⁻¹ hasc', Units.val_mul,
-    Module.End.mul_apply]
-  rfl
-
-theorem leftInv_barBasis (cs : CoxeterSystem M W) (a : C) (i : B) (w : W) :
-    leftInv cs a i (barBasis cs a w) = barBasis cs a (cs.simple i * w) -
-      if cs.length (cs.simple i * w) < cs.length w then a • barBasis cs a w else 0 := by
-  by_cases h : cs.length (cs.simple i * w) < cs.length w
-  · have hasc : cs.length (cs.simple i * w) <
-        cs.length (cs.simple i * (cs.simple i * w)) := by simpa using h
-    have hb := barBasis_simple_mul_ascent cs a i (cs.simple i * w) hasc
-    simp only [cs.simple_mul_simple_cancel_left] at hb
-    have hl : left cs a i (barBasis cs a w) = barBasis cs a (cs.simple i * w) := by
-      rw [hb, ← Module.End.mul_apply, left_mul_leftInv, Module.End.one_apply]
-    simp only [ite_eq_left h, leftInv, LinearMap.sub_apply, LinearMap.smul_apply,
-      Module.End.one_apply, hl]
-  · have hasc : cs.length w < cs.length (cs.simple i * w) := by
-      have := cs.length_simple_mul w i
-      omega
-    rw [ite_eq_right h, sub_zero, barBasis_simple_mul_ascent cs a i w hasc]
-
-noncomputable def barLinear (cs : CoxeterSystem M W) (a : C) : Module.End C (Space W C) :=
-  Finsupp.linearCombination C (barBasis cs a)
-
-@[simp] theorem barLinear_basis (cs : CoxeterSystem M W) (a : C) (w : W) :
-    barLinear cs a (basis w) = barBasis cs a w := by
-  simp [barLinear, basis, Finsupp.linearCombination_single]
-
-/-- Intertwining with the opposite parameter. In Laurent coefficients this
-is the semilinear Hecke bar relation. -/
-theorem barLinear_left (cs : CoxeterSystem M W) (a : C) (i : B) :
-    barLinear cs a * left cs (-a) i = leftInv cs a i * barLinear cs a := by
-  apply end_ext
-  intro w
-  simp only [Module.End.mul_apply, left_basis, map_add, barLinear_basis,
-    leftInv_barBasis]
-  by_cases h : cs.length (cs.simple i * w) < cs.length w
-  · simp [h, sub_eq_add_neg]
-  · simp [h]
-
-theorem barLinear_leftInv (cs : CoxeterSystem M W) (a : C) (i : B) (v : Space W C) :
-    barLinear cs a (leftInv cs (-a) i v) = left cs a i (barLinear cs a v) := by
-  have h := congrArg (fun f : Module.End C (Space W C) => f v) (barLinear_left cs a i)
-  simp only [Module.End.mul_apply, leftInv, LinearMap.sub_apply,
-    LinearMap.smul_apply, Module.End.one_apply] at h ⊢
-  rw [map_sub, map_smul, h]
-  simp
-
-/-- The inverse-parameter bar transforms are mutually inverse. -/
-theorem barLinear_barBasis (cs : CoxeterSystem M W) (a : C) (w : W) :
-    barLinear cs a (barBasis cs (-a) w) = basis w := by
-  obtain ⟨ω, hω, rfl⟩ := cs.exists_isReduced w
-  induction ω with
-  | nil => simp
-  | cons i ω ih =>
-    have hω' : cs.IsReduced ω := by simpa using hω.drop 1
-    have hasc : cs.length (cs.wordProd ω) < cs.length (cs.simple i * cs.wordProd ω) := by
-      have := hω.eq
-      simp only [cs.wordProd_cons, List.length_cons, ← hω'.eq] at this
-      omega
-    rw [cs.wordProd_cons, barBasis_simple_mul_ascent cs (-a) i _ hasc,
-      barLinear_leftInv, ih hω', left_basis, ite_eq_right (by omega), add_zero]
-
-theorem barLinear_inverse (cs : CoxeterSystem M W) (a : C) :
-    barLinear cs a * barLinear cs (-a) = 1 := by
-  apply end_ext
-  intro w
-  simp [Module.End.mul_apply, barLinear_barBasis]
-
-
-@[simp] theorem leftInv_basis (cs : CoxeterSystem M W) (a : C) (i : B) (w : W) :
-    leftInv cs a i (basis w) = basis (cs.simple i * w) -
-      if cs.length (cs.simple i * w) < cs.length w then 0 else a • basis w := by
-  simp only [leftInv, LinearMap.sub_apply, LinearMap.smul_apply,
-    Module.End.one_apply, left_basis]
-  split_ifs <;> abel
-
-theorem left_coeff (cs : CoxeterSystem M W) (a : C) (i : B) (v : Space W C) (x : W) :
-    (left cs a i v) x = v (cs.simple i * x) +
-      if cs.length (cs.simple i * x) < cs.length x then a * v x else 0 := by
-  classical
-  induction v using Finsupp.induction_linear with
-  | zero => simp
-  | add v v' hv hv' =>
-    simp only [map_add, Finsupp.add_apply, hv, hv', mul_add]
-    split_ifs <;> ring
-  | single w c =>
-    have hc : Finsupp.single w c = c • (basis w : Space W C) := by
-      simp [basis, Finsupp.smul_single]
-    rw [hc, map_smul, left_basis]
-    have heq : cs.simple i * w = x ↔ w = cs.simple i * x := by
-      constructor <;> intro h
-      · rw [← h]; simp
-      · rw [h]; simp
-    simp only [Finsupp.smul_apply, smul_eq_mul, Finsupp.add_apply, basis,
-      Finsupp.single_apply, heq]
-    by_cases h : w = x
-    · subst w
-      split_ifs <;> try contradiction
-      all_goals simp only [Finsupp.single_eq_same, Finsupp.zero_apply,
-        Finsupp.smul_apply, smul_eq_mul, mul_one, mul_zero] ; ring
-    · simp only [h, ite_false, mul_zero]
-      by_cases hd : cs.length (cs.simple i * w) < cs.length w <;>
-        by_cases hd' : cs.length (cs.simple i * x) < cs.length x <;>
-        simp [hd, hd', h]
-
-theorem leftInv_coeff (cs : CoxeterSystem M W) (a : C) (i : B) (v : Space W C) (x : W) :
-    (leftInv cs a i v) x = v (cs.simple i * x) -
-      if cs.length (cs.simple i * x) < cs.length x then 0 else a * v x := by
-  simp only [leftInv, LinearMap.sub_apply, LinearMap.smul_apply,
-    Module.End.one_apply, Finsupp.sub_apply, Finsupp.smul_apply, smul_eq_mul, left_coeff]
-  split_ifs <;> ring
-
-theorem simple_mul_le_ascent (cs : CoxeterSystem M W) (i : B) {x y : W}
-    (hxy : BruhatLE cs x y) (hy : cs.length y < cs.length (cs.simple i * y)) :
-    BruhatLE cs (cs.simple i * x) (cs.simple i * y) := by
-  have hx := le_upperSimple cs i (cs.simple i * x)
-  rw [upperSimple_mul] at hx
-  have hm := upperSimple_mono cs i hxy
-  simp only [upperSimple, ite_eq_right (show ¬cs.length (cs.simple i * y) < cs.length y by omega)] at hm
-  exact hx.trans hm
-
-/-- Triangularity for the genuine reflection-generated order. -/
-theorem barBasis_zero (cs : CoxeterSystem M W) (a : C) (x y : W)
-    (hxy : ¬BruhatLE cs x y) : barBasis cs a y x = 0 := by
-  obtain ⟨ω, hω, rfl⟩ := cs.exists_isReduced y
-  induction ω generalizing x with
-  | nil =>
-    have hx : x ≠ 1 := fun h => hxy (h ▸ bruhat_refl cs 1)
-    simp [basis, Ne.symm hx]
-  | cons i ω ih =>
-    have hω' : cs.IsReduced ω := by simpa using hω.drop 1
-    have hasc : cs.length (cs.wordProd ω) < cs.length (cs.simple i * cs.wordProd ω) := by
-      have := hω.eq
-      simp only [cs.wordProd_cons, List.length_cons, ← hω'.eq] at this
-      omega
-    have hle : BruhatLE cs (cs.wordProd ω) (cs.simple i * cs.wordProd ω) :=
-      Relation.ReflTransGen.single ⟨hasc, cs.simple i, cs.isReflection_simple i, rfl⟩
-    have hn : ¬BruhatLE cs x (cs.wordProd ω) := fun h => hxy (h.trans hle)
-    have hn' : ¬BruhatLE cs (cs.simple i * x) (cs.wordProd ω) := by
-      intro h
-      have := simple_mul_le_ascent cs i h hasc
-      simp only [cs.simple_mul_simple_cancel_left] at this
-      exact hxy this
-    rw [cs.wordProd_cons, barBasis_simple_mul_ascent cs a i _ hasc, leftInv_coeff,
-      ih x hω' hn, ih (cs.simple i * x) hω' hn']
-    split_ifs <;> simp
-
-@[simp] theorem barBasis_diagonal (cs : CoxeterSystem M W) (a : C) (y : W) :
-    barBasis cs a y y = 1 := by
-  obtain ⟨ω, hω, rfl⟩ := cs.exists_isReduced y
-  induction ω with
-  | nil => simp [basis]
-  | cons i ω ih =>
-    have hω' : cs.IsReduced ω := by simpa using hω.drop 1
-    have hasc : cs.length (cs.wordProd ω) < cs.length (cs.simple i * cs.wordProd ω) := by
-      have := hω.eq
-      simp only [cs.wordProd_cons, List.length_cons, ← hω'.eq] at this
-      omega
-    rw [cs.wordProd_cons, barBasis_simple_mul_ascent cs a i _ hasc, leftInv_coeff]
-    simpa [hasc] using ih hω'
-
-/-- Normalized R-polynomials, obtained from the bar action rather than
-postulated with an inconsistent choice of descent. -/
-noncomputable def rTilde (cs : CoxeterSystem M W) (x y : W) : ℤ[X] :=
-  barBasis cs (-X : ℤ[X]) y x
-
-@[simp] theorem rTilde_diagonal (cs : CoxeterSystem M W) (x : W) :
-    rTilde cs x x = 1 := barBasis_diagonal cs (-X) x
-
-theorem rTilde_zero (cs : CoxeterSystem M W) (x y : W) (h : ¬BruhatLE cs x y) :
-    rTilde cs x y = 0 := barBasis_zero cs (-X) x y h
-
-theorem rTilde_recursion (cs : CoxeterSystem M W) (x y : W) (i : B)
-    (hy : cs.length (cs.simple i * y) < cs.length y) :
-    rTilde cs x y = if cs.length (cs.simple i * x) < cs.length x then
-      rTilde cs (cs.simple i * x) (cs.simple i * y)
-      else rTilde cs (cs.simple i * x) (cs.simple i * y) + X * rTilde cs x (cs.simple i * y) := by
-  have hasc : cs.length (cs.simple i * y) <
-      cs.length (cs.simple i * (cs.simple i * y)) := by simpa using hy
-  have hb := barBasis_simple_mul_ascent cs (-X : ℤ[X]) i (cs.simple i * y) hasc
-  simp only [cs.simple_mul_simple_cancel_left] at hb
-  unfold rTilde
-  rw [hb, leftInv_coeff]
-  split_ifs <;> ring
-
-
-end Hecke
-
-
-namespace HeckeQ
-
-open Hecke (Space basis end_ext)
-variable {C : Type*} [CommRing C]
-
-/-- Unnormalized equal-parameter Hecke multiplication, T_s²=q+(q−1)T_s. -/
-noncomputable def left (cs : CoxeterSystem M W) (q : C) (i : B) :
-    Module.End C (Space W C) :=
-  Finsupp.linearCombination C (fun w =>
-    if cs.length (cs.simple i * w) < cs.length w then
-      q • basis (cs.simple i * w) + (q - 1) • basis w
-    else basis (cs.simple i * w))
-
-noncomputable def right (cs : CoxeterSystem M W) (q : C) (i : B) :
-    Module.End C (Space W C) :=
-  Finsupp.linearCombination C (fun w =>
-    if cs.length (w * cs.simple i) < cs.length w then
-      q • basis (w * cs.simple i) + (q - 1) • basis w
-    else basis (w * cs.simple i))
-
-@[simp] theorem left_basis (cs : CoxeterSystem M W) (q : C) (i : B) (w : W) :
-    left cs q i (basis w) = if cs.length (cs.simple i * w) < cs.length w then
-      q • basis (cs.simple i * w) + (q - 1) • basis w
-    else basis (cs.simple i * w) := by
-  simp [left, basis, Finsupp.linearCombination_single]
-
-@[simp] theorem right_basis (cs : CoxeterSystem M W) (q : C) (i : B) (w : W) :
-    right cs q i (basis w) = if cs.length (w * cs.simple i) < cs.length w then
-      q • basis (w * cs.simple i) + (q - 1) • basis w
-    else basis (w * cs.simple i) := by
-  simp [right, basis, Finsupp.linearCombination_single]
-
-theorem left_right_commute (cs : CoxeterSystem M W) (q : C) (i j : B) :
-    Commute (left cs q i) (right cs q j) := by
-  apply end_ext
-  intro w
-  simp only [Module.End.mul_apply, right_basis, left_basis]
-  by_cases heq : cs.simple i * w = w * cs.simple j
-  · have heq' : cs.simple i * (w * cs.simple j) = w := by rw [← heq]; simp
-    have heq'' : cs.simple i * w * cs.simple j = w := by simpa [mul_assoc] using heq'
-    by_cases h : cs.length (cs.simple i * w) < cs.length w
-    · have h' : cs.length (w * cs.simple j) < cs.length w := heq ▸ h
-      have hn : ¬cs.length w < cs.length (cs.simple i * w) := by omega
-      have hn' : ¬cs.length w < cs.length (w * cs.simple j) := heq ▸ hn
-      simp [h', map_add, map_smul, heq', hn', heq]
-    · have h' : ¬cs.length (w * cs.simple j) < cs.length w := heq ▸ h
-      have hl : cs.length w < cs.length (cs.simple i * w) := by
-        have := cs.length_simple_mul w i; omega
-      have hl' : cs.length w < cs.length (w * cs.simple j) := heq ▸ hl
-      simp [h', heq', hl', heq]
-  · have hl := left_descent_mul_simple_iff cs i j w heq
-    have hr := right_descent_simple_mul_iff cs i j w heq
-    have hr' : cs.length (cs.simple i * (w * cs.simple j)) <
-        cs.length (cs.simple i * w) ↔ cs.length (w * cs.simple j) < cs.length w := by
-      simpa only [mul_assoc] using hr
-    by_cases hi : cs.length (cs.simple i * w) < cs.length w <;>
-      by_cases hj : cs.length (w * cs.simple j) < cs.length w <;>
-      simp only [hi, hj, ite_true, ite_false, map_add, map_smul,
-        left_basis, right_basis, hl, hr', mul_assoc, smul_add, smul_smul] ;
-      module
-
-theorem left_quadratic (cs : CoxeterSystem M W) (q : C) (i : B) :
-    left cs q i * left cs q i = q • 1 + (q - 1) • left cs q i := by
-  apply end_ext
-  intro w
-  simp only [Module.End.mul_apply, left_basis, LinearMap.add_apply, LinearMap.smul_apply,
-    Module.End.one_apply]
-  rcases cs.length_simple_mul w i with h | h
-  · have hasc : ¬cs.length (cs.simple i * w) < cs.length w := by omega
-    have hdesc : cs.length w < cs.length (cs.simple i * w) := by omega
-    simp [hasc, hdesc]
-  · have hdesc : cs.length (cs.simple i * w) < cs.length w := by omega
-    have hasc : ¬cs.length w < cs.length (cs.simple i * w) := by omega
-    simp [hasc, hdesc, smul_add, smul_smul]
-
-noncomputable def leftInv (cs : CoxeterSystem M W) (q : Cˣ) (i : B) :
-    Module.End C (Space W C) := (↑q⁻¹ : C) • (left cs (q : C) i - ((q : C) - 1) • 1)
-
-theorem leftInv_apply (cs : CoxeterSystem M W) (q : Cˣ) (i : B) (v : Space W C) :
-    leftInv cs q i v = (↑q⁻¹ : C) • (left cs (q : C) i v - ((q : C) - 1) • v) := rfl
-
-theorem left_mul_leftInv (cs : CoxeterSystem M W) (q : Cˣ) (i : B) :
-    left cs (q : C) i * leftInv cs q i = 1 := by
-  apply end_ext
-  intro w
-  have h := congrArg (fun f : Module.End C (Space W C) => f (basis w))
-    (left_quadratic cs (q:C) i)
-  simp only [Module.End.mul_apply, LinearMap.add_apply, LinearMap.smul_apply,
-    Module.End.one_apply] at h
-  simp only [Module.End.mul_apply, leftInv_apply, Module.End.one_apply,
-    map_smul, map_sub]
-  rw [h]
-  simp [smul_smul]
-
-theorem leftInv_mul_left (cs : CoxeterSystem M W) (q : Cˣ) (i : B) :
-    leftInv cs q i * left cs (q : C) i = 1 := by
-  apply end_ext
-  intro w
-  have h := congrArg (fun f : Module.End C (Space W C) => f (basis w))
-    (left_quadratic cs (q:C) i)
-  simp only [Module.End.mul_apply, LinearMap.add_apply, LinearMap.smul_apply,
-    Module.End.one_apply] at h
-  simp only [Module.End.mul_apply, leftInv_apply, Module.End.one_apply]
-  rw [h]
-  simp [smul_smul]
-
-
-noncomputable def leftWord (cs : CoxeterSystem M W) (q : C) (ω : List B) :
-    Module.End C (Space W C) := (ω.map (left cs q)).prod
-
-@[simp] theorem leftWord_nil (cs : CoxeterSystem M W) (q : C) :
-    leftWord cs q [] = 1 := rfl
-
-@[simp] theorem leftWord_cons (cs : CoxeterSystem M W) (q : C) (i : B) (ω : List B) :
-    leftWord cs q (i :: ω) = left cs q i * leftWord cs q ω := rfl
-
-/-- Acting on the identity along q reduced word gives exactly its standard
-basis element. -/
-theorem leftWord_one (cs : CoxeterSystem M W) (q : C) {ω : List B}
-    (hω : cs.IsReduced ω) :
-    leftWord cs q ω (basis 1) = basis (cs.wordProd ω) := by
-  induction ω with
-  | nil => simp
-  | cons i ω ih =>
-    have hω' : cs.IsReduced ω := by simpa using hω.drop 1
-    have hasc : ¬cs.length (cs.simple i * cs.wordProd ω) < cs.length (cs.wordProd ω) := by
-      have := hω.eq
-      rw [cs.wordProd_cons, List.length_cons, ← hω'.eq] at this
-      omega
-    simp [Module.End.mul_apply, ih hω', hasc, cs.wordProd_cons]
-
-noncomputable def rightWord (cs : CoxeterSystem M W) (q : C) (ω : List B) :
-    Module.End C (Space W C) := (ω.map (right cs q)).prod
-
-@[simp] theorem rightWord_nil (cs : CoxeterSystem M W) (q : C) :
-    rightWord cs q [] = 1 := rfl
-
-@[simp] theorem rightWord_cons (cs : CoxeterSystem M W) (q : C) (i : B) (ω : List B) :
-    rightWord cs q (i :: ω) = right cs q i * rightWord cs q ω := rfl
-
-theorem rightWord_one (cs : CoxeterSystem M W) (q : C) {ω : List B}
-    (hω : cs.IsReduced ω) :
-    rightWord cs q ω (basis 1) = basis (cs.wordProd ω)⁻¹ := by
-  induction ω with
-  | nil => simp
-  | cons i ω ih =>
-    have hω' : cs.IsReduced ω := by simpa using hω.drop 1
-    have hlen : cs.length ((cs.wordProd ω)⁻¹ * cs.simple i) =
-        cs.length (cs.simple i * cs.wordProd ω) := by
-      rw [← cs.length_inv (cs.simple i * cs.wordProd ω)]
-      simp [mul_inv_rev]
-    have hasc : ¬cs.length ((cs.wordProd ω)⁻¹ * cs.simple i) <
-        cs.length (cs.wordProd ω)⁻¹ := by
-      rw [hlen, cs.length_inv]
-      have := hω.eq
-      rw [cs.wordProd_cons, List.length_cons, ← hω'.eq] at this
-      omega
-    have hasc' : ¬cs.length ((cs.wordProd ω)⁻¹ * cs.simple i) <
-        cs.length (cs.wordProd ω) := by simpa using hasc
-    simp [Module.End.mul_apply, ih hω', hasc', cs.wordProd_cons, mul_inv_rev]
-
-theorem commute_rightWord (cs : CoxeterSystem M W) (q : C)
-    {f : Module.End C (Space W C)} (h : ∀ i, Commute f (right cs q i)) (ω : List B) :
-    Commute f (rightWord cs q ω) := by
-  apply Commute.list_prod_right
-  intro r hr
-  obtain ⟨i, _, rfl⟩ := List.mem_map.mp hr
-  exact h i
-
-/-- The regular right action is cyclic, with cyclic vector the identity. -/
-theorem commuting_ext (cs : CoxeterSystem M W) (q : C)
-    {f g : Module.End C (Space W C)}
-    (hf : ∀ i, Commute f (right cs q i))
-    (hg : ∀ i, Commute g (right cs q i))
-    (h : f (basis 1) = g (basis 1)) : f = g := by
-  apply end_ext
-  intro w
-  obtain ⟨ω, hω, heq⟩ := cs.exists_isReduced w⁻¹
-  have hw : rightWord cs q ω (basis 1) = basis w := by
-    rw [rightWord_one cs q hω, ← heq, inv_inv]
-  rw [← hw]
-  have hf' := congrArg (fun t : Module.End C (Space W C) => t (basis 1))
-    (commute_rightWord cs q hf ω).eq
-  have hg' := congrArg (fun t : Module.End C (Space W C) => t (basis 1))
-    (commute_rightWord cs q hg ω).eq
-  simp only [Module.End.mul_apply] at hf' hg'
-  rw [h] at hf'
-  exact hf'.trans hg'.symm
-
-theorem leftWord_commute_right (cs : CoxeterSystem M W) (q : C) (ω : List B) (i : B) :
-    Commute (leftWord cs q ω) (right cs q i) := by
-  apply Commute.list_prod_left
-  intro r hr
-  obtain ⟨j, _, rfl⟩ := List.mem_map.mp hr
-  exact left_right_commute cs q j i
-
-
-theorem leftWord_independent (cs : CoxeterSystem M W) (q : C) {ω η : List B}
-    (hω : cs.IsReduced ω) (hη : cs.IsReduced η) (heq : cs.wordProd ω = cs.wordProd η) :
-    leftWord cs q ω = leftWord cs q η := by
-  apply commuting_ext cs q (leftWord_commute_right cs q ω)
-    (leftWord_commute_right cs q η)
-  rw [leftWord_one cs q hω, leftWord_one cs q hη, heq]
-
-noncomputable def unit (cs : CoxeterSystem M W) (q : Cˣ) (i : B) :
-    (Module.End C (Space W C))ˣ where
-  val := left cs (q : C) i
-  inv := leftInv cs q i
-  val_inv := left_mul_leftInv cs q i
-  inv_val := leftInv_mul_left cs q i
-
-noncomputable def wordUnit (cs : CoxeterSystem M W) (q : Cˣ) (ω : List B) :
-    (Module.End C (Space W C))ˣ := (ω.map (unit cs q)).prod
-
-@[simp] theorem wordUnit_val (cs : CoxeterSystem M W) (q : Cˣ) (ω : List B) :
-    (wordUnit cs q ω).val = leftWord cs (q : C) ω := by
-  induction ω with
-  | nil => rfl
-  | cons i ω ih =>
-    change (unit cs q i).val * (wordUnit cs q ω).val = left cs (q : C) i * leftWord cs (q : C) ω
-    rw [ih]
-    rfl
-
-theorem wordUnit_independent (cs : CoxeterSystem M W) (q : Cˣ) {ω η : List B}
-    (hω : cs.IsReduced ω) (hη : cs.IsReduced η) (heq : cs.wordProd ω = cs.wordProd η) :
-    wordUnit cs q ω = wordUnit cs q η := by
-  apply Units.ext
-  simpa using leftWord_independent cs (q : C) hω hη heq
-
-/-- The genuine normalized standard-basis action, a unit in the regular
-endomorphism algebra. No presentation relations are assumed. -/
-noncomputable def standardUnit (cs : CoxeterSystem M W) (q : Cˣ) (w : W) :
-    (Module.End C (Space W C))ˣ :=
-  wordUnit cs q (Classical.choose (cs.exists_isReduced w))
-
-theorem standardUnit_word (cs : CoxeterSystem M W) (q : Cˣ) {ω : List B}
-    (hω : cs.IsReduced ω) :
-    standardUnit cs q (cs.wordProd ω) = wordUnit cs q ω := by
-  apply wordUnit_independent cs q (Classical.choose_spec (cs.exists_isReduced _)).1 hω
-  exact (Classical.choose_spec (cs.exists_isReduced _)).2.symm
-
-@[simp] theorem standardUnit_one (cs : CoxeterSystem M W) (q : Cˣ) :
-    standardUnit cs q 1 = 1 := by
-  have h := standardUnit_word cs q (ω := []) (by simp [CoxeterSystem.IsReduced])
-  simpa [wordUnit] using h
-
-@[simp] theorem standardUnit_basis_one (cs : CoxeterSystem M W) (q : Cˣ) (w : W) :
-    (standardUnit cs q w).val (basis 1) = basis w := by
-  obtain ⟨ω, hω, rfl⟩ := cs.exists_isReduced w
-  rw [standardUnit_word cs q hω, wordUnit_val, leftWord_one cs (q : C) hω]
-
-theorem standardUnit_simple_mul (cs : CoxeterSystem M W) (q : Cˣ) (i : B) (w : W)
-    (hasc : cs.length w < cs.length (cs.simple i * w)) :
-    standardUnit cs q (cs.simple i * w) = unit cs q i * standardUnit cs q w := by
-  obtain ⟨ω, hω, rfl⟩ := cs.exists_isReduced w
-  have hred : cs.IsReduced (i :: ω) := by
-    unfold CoxeterSystem.IsReduced
-    have := cs.length_simple_mul (cs.wordProd ω) i
-    simp only [cs.wordProd_cons, List.length_cons]
-    rw [← hω.eq]
-    omega
-  rw [← cs.wordProd_cons, standardUnit_word cs q hred, standardUnit_word cs q hω]
-  simp [wordUnit]
-
-
-theorem standardUnit_mul_simple (cs : CoxeterSystem M W) (q : Cˣ) (i : B) (w : W)
-    (hasc : cs.length w < cs.length (w * cs.simple i)) :
-    standardUnit cs q (w * cs.simple i) = standardUnit cs q w * unit cs q i := by
-  obtain ⟨ω, hω, rfl⟩ := cs.exists_isReduced w
-  have hred : cs.IsReduced (ω ++ [i]) := by
-    unfold CoxeterSystem.IsReduced
-    have := cs.length_mul_simple (cs.wordProd ω) i
-    simp only [cs.wordProd_append, cs.wordProd_singleton, List.length_append,
-      List.length_singleton]
-    rw [← hω.eq]
-    omega
-  rw [← cs.wordProd_singleton i, ← cs.wordProd_append, standardUnit_word cs q hred,
-    standardUnit_word cs q hω]
-  simp [wordUnit, List.map_append, List.prod_append]
-
-/-- The bar image of an actual standard basis vector; inverse units give
-expression independence before any polynomial coefficients are extracted. -/
-noncomputable def barBasis (cs : CoxeterSystem M W) (q : Cˣ) (w : W) : Space W C :=
-  ((standardUnit cs q w⁻¹)⁻¹).val (basis 1)
-
-@[simp] theorem barBasis_one (cs : CoxeterSystem M W) (q : Cˣ) :
-    barBasis cs q 1 = basis 1 := by simp [barBasis]
-
-theorem barBasis_simple_mul_ascent (cs : CoxeterSystem M W) (q : Cˣ) (i : B) (w : W)
-    (hasc : cs.length w < cs.length (cs.simple i * w)) :
-    barBasis cs q (cs.simple i * w) = leftInv cs q i (barBasis cs q w) := by
-  have hasc' : cs.length w⁻¹ < cs.length (w⁻¹ * cs.simple i) := by
-    have hlen : cs.length (w⁻¹ * cs.simple i) = cs.length (cs.simple i * w) := by
-      rw [← cs.length_inv (cs.simple i * w)]
-      simp [mul_inv_rev]
-    simpa [hlen] using hasc
-  simp only [barBasis, mul_inv_rev, cs.inv_simple,
-    standardUnit_mul_simple cs q i w⁻¹ hasc', Units.val_mul,
-    Module.End.mul_apply]
-  rfl
-
-
-theorem leftInv_barBasis (cs : CoxeterSystem M W) (q : Cˣ) (i : B) (w : W) :
-    leftInv cs q i (barBasis cs q w) =
-      if cs.length (cs.simple i * w) < cs.length w then
-        (↑q⁻¹ : C) • barBasis cs q (cs.simple i * w) +
-          ((↑q⁻¹ : C) - 1) • barBasis cs q w
-      else barBasis cs q (cs.simple i * w) := by
-  by_cases h : cs.length (cs.simple i * w) < cs.length w
-  · have hasc : cs.length (cs.simple i * w) <
-        cs.length (cs.simple i * (cs.simple i * w)) := by simpa using h
-    have hb := barBasis_simple_mul_ascent cs q i (cs.simple i * w) hasc
-    simp only [cs.simple_mul_simple_cancel_left] at hb
-    have hl : left cs (q : C) i (barBasis cs q w) = barBasis cs q (cs.simple i * w) := by
-      rw [hb, ← Module.End.mul_apply, left_mul_leftInv, Module.End.one_apply]
-    rw [ite_eq_left h, leftInv_apply, hl, smul_sub, smul_smul]
-    have hscalar : (↑q⁻¹ : C) * ((q : C) - 1) = 1 - (↑q⁻¹ : C) := by
-      rw [mul_sub, q.inv_mul, mul_one]
-    rw [hscalar, sub_smul, one_smul]
-    module
-  · have hasc : cs.length w < cs.length (cs.simple i * w) := by
-      have := cs.length_simple_mul w i
-      omega
-    rw [ite_eq_right h, barBasis_simple_mul_ascent cs q i w hasc]
-
-noncomputable def barLinear (cs : CoxeterSystem M W) (q : Cˣ) : Module.End C (Space W C) :=
-  Finsupp.linearCombination C (barBasis cs q)
-
-@[simp] theorem barLinear_basis (cs : CoxeterSystem M W) (q : Cˣ) (w : W) :
-    barLinear cs q (basis w) = barBasis cs q w := by
-  simp [barLinear, basis, Finsupp.linearCombination_single]
-
-/-- Linear intertwining, before specializing coefficients to a Laurent ring. -/
-theorem barLinear_left (cs : CoxeterSystem M W) (q : Cˣ) (i : B) :
-    barLinear cs q * left cs (↑q⁻¹ : C) i = leftInv cs q i * barLinear cs q := by
-  apply end_ext
-  intro w
-  simp only [Module.End.mul_apply, left_basis, leftInv_barBasis, barLinear_basis]
-  split_ifs <;> simp
-
-theorem barLinear_leftInv (cs : CoxeterSystem M W) (q : Cˣ) (i : B) (v : Space W C) :
-    barLinear cs q (leftInv cs q⁻¹ i v) = left cs (q : C) i (barLinear cs q v) := by
-  have h := congrArg (fun f : Module.End C (Space W C) => f (leftInv cs q⁻¹ i v))
-    (barLinear_left cs q i)
-  simp only [Module.End.mul_apply] at h
-  have hv : left cs (↑q⁻¹ : C) i (leftInv cs q⁻¹ i v) = v := by
-    rw [← Module.End.mul_apply, left_mul_leftInv, Module.End.one_apply]
-  rw [hv] at h
-  have h' := congrArg (left cs (q : C) i) h
-  have hl (t : Space W C) : left cs (q : C) i (leftInv cs q i t) = t := by
-    rw [← Module.End.mul_apply, left_mul_leftInv, Module.End.one_apply]
-  simpa only [hl] using h'.symm
-
-/-- Applying coefficient inversion and the Hecke bar twice is the identity. -/
-theorem barLinear_barBasis (cs : CoxeterSystem M W) (q : Cˣ) (w : W) :
-    barLinear cs q (barBasis cs q⁻¹ w) = basis w := by
-  obtain ⟨ω, hω, rfl⟩ := cs.exists_isReduced w
-  induction ω with
-  | nil => simp
-  | cons i ω ih =>
-    have hω' : cs.IsReduced ω := by simpa using hω.drop 1
-    have hasc : cs.length (cs.wordProd ω) < cs.length (cs.simple i * cs.wordProd ω) := by
-      have := hω.eq
-      simp only [cs.wordProd_cons, List.length_cons, ← hω'.eq] at this
-      omega
-    rw [cs.wordProd_cons, barBasis_simple_mul_ascent cs q⁻¹ i _ hasc,
-      barLinear_leftInv, ih hω', left_basis, ite_eq_right (by omega)]
-
-theorem barLinear_inverse (cs : CoxeterSystem M W) (q : Cˣ) :
-    barLinear cs q * barLinear cs q⁻¹ = 1 := by
-  apply end_ext
-  intro w
-  simp [Module.End.mul_apply, barLinear_barBasis]
-
-@[simp] theorem leftInv_basis (cs : CoxeterSystem M W) (q : Cˣ) (i : B) (w : W) :
-    leftInv cs q i (basis w) =
-      if cs.length (cs.simple i * w) < cs.length w then basis (cs.simple i * w)
-      else (↑q⁻¹ : C) • basis (cs.simple i * w) + ((↑q⁻¹ : C) - 1) • basis w := by
-  rw [leftInv_apply, left_basis]
-  split_ifs
-  · simp [smul_smul]
-  · rw [smul_sub, smul_smul]
-    have hscalar : (↑q⁻¹ : C) * ((q : C) - 1) = 1 - (↑q⁻¹ : C) := by
-      rw [mul_sub, q.inv_mul, mul_one]
-    rw [hscalar, sub_smul, one_smul]
-    module
-
-theorem leftInv_coeff (cs : CoxeterSystem M W) (q : Cˣ) (i : B) (v : Space W C) (x : W) :
-    leftInv cs q i v x =
-      if cs.length (cs.simple i * x) < cs.length x then (↑q⁻¹ : C) * v (cs.simple i * x)
-      else v (cs.simple i * x) + ((↑q⁻¹ : C) - 1) * v x := by
-  classical
-  induction v using Finsupp.induction_linear with
-  | zero => simp
-  | add v v' hv hv' =>
-    simp only [map_add, Finsupp.add_apply, hv, hv']
-    split_ifs <;> ring
-  | single w c =>
-    have hc : Finsupp.single w c = c • (basis w : Space W C) := by
-      simp [basis, Finsupp.smul_single]
-    rw [hc, map_smul, leftInv_basis]
-    by_cases h : w = cs.simple i * x
-    · subst w
-      have hd : cs.length (cs.simple i * (cs.simple i * x)) < cs.length (cs.simple i * x) ↔
-          ¬ cs.length (cs.simple i * x) < cs.length x := by
-        simp only [cs.simple_mul_simple_cancel_left]
-        have := cs.length_simple_mul x i
-        omega
-      have hn : cs.simple i * x ≠ x := by
-        intro he
-        have := cs.length_simple_mul x i
-        rw [he] at this
-        omega
-      simp only [Finsupp.smul_apply, smul_eq_mul, basis, Finsupp.single_apply,
-        cs.simple_mul_simple_cancel_left, hn, ite_false]
-      have hlength := cs.length_simple_mul x i
-      split_ifs <;> try omega
-      all_goals simp [Ne.symm hn] <;> try ring
-    · have hn : cs.simple i * w ≠ x := by
-        intro he
-        apply h
-        rw [← he]
-        simp
-      by_cases hw : w = x
-      · subst w
-        simp only [Finsupp.smul_apply, smul_eq_mul, basis, Finsupp.single_apply,
-          Ne.symm hn, ite_false, ]
-        split_ifs <;> simp [Ne.symm hn] ; try ring
-      · simp only [Finsupp.smul_apply, smul_eq_mul, basis, Finsupp.single_apply,
-          h, hw, ite_false]
-        split_ifs <;> simp [hn, hw]
-
-/-- Triangularity in the actual Bruhat order. -/
-theorem barBasis_zero (cs : CoxeterSystem M W) (q : Cˣ) (x y : W)
-    (hxy : ¬BruhatLE cs x y) : barBasis cs q y x = 0 := by
-  obtain ⟨ω, hω, rfl⟩ := cs.exists_isReduced y
-  induction ω generalizing x with
-  | nil =>
-    have hx : x ≠ 1 := fun h => hxy (h ▸ bruhat_refl cs 1)
-    simp [basis, Ne.symm hx]
-  | cons i ω ih =>
-    have hω' : cs.IsReduced ω := by simpa using hω.drop 1
-    have hasc : cs.length (cs.wordProd ω) < cs.length (cs.simple i * cs.wordProd ω) := by
-      have := hω.eq
-      simp only [cs.wordProd_cons, List.length_cons, ← hω'.eq] at this
-      omega
-    have hle : BruhatLE cs (cs.wordProd ω) (cs.simple i * cs.wordProd ω) :=
-      Relation.ReflTransGen.single ⟨hasc, cs.simple i, cs.isReflection_simple i, rfl⟩
-    have hn : ¬BruhatLE cs x (cs.wordProd ω) := fun h => hxy (h.trans hle)
-    have hn' : ¬BruhatLE cs (cs.simple i * x) (cs.wordProd ω) := by
-      intro h
-      have := Hecke.simple_mul_le_ascent cs i h hasc
-      simp only [cs.simple_mul_simple_cancel_left] at this
-      exact hxy this
-    rw [cs.wordProd_cons, barBasis_simple_mul_ascent cs q i _ hasc, leftInv_coeff,
-      ih x hω' hn, ih (cs.simple i * x) hω' hn']
-    split_ifs <;> simp
-
-@[simp] theorem barBasis_diagonal (cs : CoxeterSystem M W) (q : Cˣ) (y : W) :
-    barBasis cs q y y = (↑q⁻¹ : C) ^ cs.length y := by
-  obtain ⟨ω, hω, rfl⟩ := cs.exists_isReduced y
-  induction ω with
-  | nil => simp [basis]
-  | cons i ω ih =>
-    have hω' : cs.IsReduced ω := by simpa using hω.drop 1
-    have hasc : cs.length (cs.wordProd ω) < cs.length (cs.simple i * cs.wordProd ω) := by
-      have := hω.eq
-      simp only [cs.wordProd_cons, List.length_cons, ← hω'.eq] at this
-      omega
-    have hlen : cs.length (cs.simple i * cs.wordProd ω) = cs.length (cs.wordProd ω) + 1 := by
-      have := cs.length_simple_mul (cs.wordProd ω) i
-      omega
-    rw [cs.wordProd_cons, barBasis_simple_mul_ascent cs q i _ hasc, leftInv_coeff]
-    rw [cs.simple_mul_simple_cancel_left, ite_eq_left hasc, ih hω', hlen, pow_succ]
-    ring
-
-/-- R-value extracted from the genuine Hecke bar, with the classical sign
-and leading q-power. No recurrence is imposed in this definition. -/
-noncomputable def rValue (cs : CoxeterSystem M W) (q : Cˣ) (x y : W) : C :=
-  (-1 : C) ^ cs.length x * (-(q : C)) ^ cs.length y * barBasis cs q y x
-
-@[simp] theorem rValue_diagonal (cs : CoxeterSystem M W) (q : Cˣ) (x : W) :
-    rValue cs q x x = 1 := by
-  rw [rValue, barBasis_diagonal, ← mul_pow, ← mul_pow]
-  simp
-
-theorem rValue_zero (cs : CoxeterSystem M W) (q : Cˣ) (x y : W)
-    (hxy : ¬BruhatLE cs x y) : rValue cs q x y = 0 := by
-  rw [rValue, barBasis_zero cs q x y hxy, mul_zero]
-
-/-- The classical R recursion holds for *every* left descent, by Hecke
-operator independence. This is the consistency obligation for actual R. -/
-theorem rValue_recursion (cs : CoxeterSystem M W) (q : Cˣ) (x y : W) (i : B)
-    (hy : cs.length (cs.simple i * y) < cs.length y) :
-    rValue cs q x y =
-      if cs.length (cs.simple i * x) < cs.length x then
-        rValue cs q (cs.simple i * x) (cs.simple i * y)
-      else ((q : C) - 1) * rValue cs q x (cs.simple i * y) +
-        (q : C) * rValue cs q (cs.simple i * x) (cs.simple i * y) := by
-  have hasc : cs.length (cs.simple i * y) <
-      cs.length (cs.simple i * (cs.simple i * y)) := by simpa using hy
-  have hb := barBasis_simple_mul_ascent cs q i (cs.simple i * y) hasc
-  simp only [cs.simple_mul_simple_cancel_left] at hb
-  have hylen : cs.length y = cs.length (cs.simple i * y) + 1 := by
-    have := cs.length_simple_mul y i
-    omega
-  have hu : (q : C) * (↑q⁻¹ : C) = 1 := q.val_inv
-  by_cases hx : cs.length (cs.simple i * x) < cs.length x
-  · have hxlen : cs.length x = cs.length (cs.simple i * x) + 1 := by
-      have := cs.length_simple_mul x i
-      omega
-    rw [ite_eq_left hx]
-    simp only [rValue, hb, leftInv_coeff, ite_eq_left hx]
-    rw [hxlen, hylen]
-    simp only [pow_succ]
-    linear_combination ((-1 : C) ^ cs.length (cs.simple i * x) *
-      (-(q : C)) ^ cs.length (cs.simple i * y) *
-      barBasis cs q (cs.simple i * y) (cs.simple i * x)) * hu
-  · have hxlen : cs.length (cs.simple i * x) = cs.length x + 1 := by
-      have := cs.length_simple_mul x i
-      omega
-    rw [ite_eq_right hx]
-    simp only [rValue, hb, leftInv_coeff, ite_eq_right hx]
-    rw [hxlen, hylen]
-    simp only [pow_succ]
-    linear_combination -((-1 : C) ^ cs.length x *
-      (-(q : C)) ^ cs.length (cs.simple i * y) *
-      barBasis cs q (cs.simple i * y) x) * hu
-
-end HeckeQ
-
-noncomputable local instance : DecidableEq W := Classical.decEq W
-
-/-- A concrete finite lower ideal of the genuine Bruhat order. -/
-noncomputable def lowerFinset (cs : CoxeterSystem M W) (y : W) : Finset W :=
-  (finite_bruhat_lower cs y).toFinset
-
-@[simp] theorem mem_lowerFinset (cs : CoxeterSystem M W) (y x : W) :
-    x ∈ lowerFinset cs y ↔ BruhatLE cs x y := by
-  classical
-  exact Set.Finite.mem_toFinset _
-
-namespace HeckeQ
-
-open Hecke (Space basis)
-variable {C : Type*} [CommRing C]
-
-theorem bar_coeff_inverse (cs : CoxeterSystem M W) (q : Cˣ) (x y : W) :
-    ∑ z ∈ lowerFinset cs y, barBasis cs q z x * barBasis cs q⁻¹ y z =
-      if x = y then 1 else 0 := by
-  classical
-  have h := congrArg (fun v : Space W C => v x) (barLinear_barBasis cs q y)
-  rw [barLinear, Finsupp.linearCombination_apply, Finsupp.sum_apply] at h
-  have hs : (barBasis cs q⁻¹ y).support ⊆ lowerFinset cs y := by
-    intro z hz
-    apply (mem_lowerFinset cs y z).mpr
-    by_contra hn
-    exact Finsupp.mem_support_iff.mp hz (barBasis_zero cs _ _ _ hn)
-  rw [Finsupp.sum_of_support_subset _ hs _ (by simp), basis, Finsupp.single_apply] at h
-  simpa only [Finsupp.smul_apply, smul_eq_mul, mul_comm, eq_comm] using h
-
-theorem rValue_inverse_factor (cs : CoxeterSystem M W) (q : Cˣ) (x z y : W) :
-    rValue cs q x z * (q : C) ^ cs.length y * (↑q⁻¹ : C) ^ cs.length z *
-        rValue cs q⁻¹ z y =
-      ((-1 : C) ^ cs.length x * (-1 : C) ^ cs.length y) *
-        (barBasis cs q z x * barBasis cs q⁻¹ y z) := by
-  simp only [rValue]
-  rw [neg_pow (q : C), neg_pow (↑q⁻¹ : C)]
-  calc
-    _ = ((-1 : C) ^ cs.length x * (-1 : C) ^ cs.length y) *
-        (((-1 : C) * (-1)) ^ cs.length z *
-          ((q : C) * (↑q⁻¹ : C)) ^ cs.length z *
-          ((q : C) * (↑q⁻¹ : C)) ^ cs.length y) *
-        (barBasis cs q z x * barBasis cs q⁻¹ y z) := by simp only [mul_pow]; ring
-    _ = _ := by simp
-
-/-- The exact R-kernel identity, before specialization to polynomials. -/
-theorem rValue_inverse (cs : CoxeterSystem M W) (q : Cˣ) (x y : W) :
-    ∑ z ∈ lowerFinset cs y,
-      rValue cs q x z * (q : C) ^ cs.length y * (↑q⁻¹ : C) ^ cs.length z *
-        rValue cs q⁻¹ z y = if x = y then 1 else 0 := by
-  classical
-  simp_rw [rValue_inverse_factor]
-  rw [← Finset.mul_sum, bar_coeff_inverse]
-  split_ifs with h
-  · subst y
-    rw [← mul_pow]
-    simp
-  · simp
-
-end HeckeQ
-
-/-- Reflection of a bounded-degree polynomial evaluated at an actual unit. -/
-theorem eval₂_reflect_unit {C : Type*} [CommRing C] (q : Cˣ)
-    (p : ℤ[X]) (d : ℕ) (hp : p.natDegree ≤ d) :
-    Polynomial.eval₂ (Int.castRingHom C) (q : C) (reflect d p) =
-      (q : C) ^ d * Polynomial.eval₂ (Int.castRingHom C) (↑q⁻¹ : C) p := by
-  let := q⁻¹.invertible
-  have h := Polynomial.eval₂_reflect_mul_pow (Int.castRingHom C) (↑q⁻¹ : C) d p hp
-  simp only [invOf_units, inv_inv] at h
-  have h' := congrArg (fun a : C => a * (q : C) ^ d) h
-  calc
-    _ = Polynomial.eval₂ (Int.castRingHom C) (↑q⁻¹ : C) p * (q : C) ^ d := by
-      simpa only [mul_assoc, ← mul_pow, q.inv_mul, one_pow, mul_one] using h'
-    _ = _ := mul_comm _ _
-
-
-end KLInvariance
-end
-
-end OAI
+import OAI.RepresentationTheory.KazhdanLusztig.Bruhat
+
+namespace OAI
+
+/-!
+Hecke and Laurent actions, braid independence, bar involution, triangular support and inverse coefficients.
+-/
+
+section
+namespace KLInvariance
+
+open Polynomial
+
+universe u v u' v'
+
+variable {B : Type u} {W : Type v} [klPreservedInstance1 : Group W] {M : CoxeterMatrix B}
+
+
+namespace Hecke
+
+variable {C : Type*} [CommRing C]
+
+/-- The standard free module of the actual Coxeter group. -/
+abbrev Space (W C : Type*) [Zero C] := W →₀ C
+
+noncomputable def basis (w : W) : Space W C := Finsupp.single w 1
+
+include klPreservedInstance1 in
+theorem end_ext {f g : Module.End C (Space W C)}
+    (h : ∀ w, f (basis w) = g (basis w)) : f = g := by
+  have _ := klPreservedInstance1
+  apply Finsupp.lhom_ext
+  intro w c
+  have hc : Finsupp.single w c = c • basis w := by
+    simp [basis, Finsupp.smul_single]
+  rw [hc, map_smul, map_smul, h]
+
+/-- Normalized Hecke multiplication: H_s² = 1 + a H_s. -/
+noncomputable def left (cs : CoxeterSystem M W) (a : C) (i : B) :
+    Module.End C (Space W C) :=
+  Finsupp.linearCombination C (fun w => basis (cs.simple i * w) +
+    if cs.length (cs.simple i * w) < cs.length w then a • basis w else 0)
+
+noncomputable def right (cs : CoxeterSystem M W) (a : C) (i : B) :
+    Module.End C (Space W C) :=
+  Finsupp.linearCombination C (fun w => basis (w * cs.simple i) +
+    if cs.length (w * cs.simple i) < cs.length w then a • basis w else 0)
+
+@[simp] theorem left_basis (cs : CoxeterSystem M W) (a : C) (i : B) (w : W) :
+    left cs a i (basis w) = basis (cs.simple i * w) +
+      if cs.length (cs.simple i * w) < cs.length w then a • basis w else 0 := by
+  simp [left, basis, Finsupp.linearCombination_single]
+
+@[simp] theorem right_basis (cs : CoxeterSystem M W) (a : C) (i : B) (w : W) :
+    right cs a i (basis w) = basis (w * cs.simple i) +
+      if cs.length (w * cs.simple i) < cs.length w then a • basis w else 0 := by
+  simp [right, basis, Finsupp.linearCombination_single]
+
+theorem left_right_commute (cs : CoxeterSystem M W) (a : C) (i j : B) :
+    Commute (left cs a i) (right cs a j) := by
+  apply end_ext
+  intro w
+  simp only [Module.End.mul_apply, right_basis, left_basis, map_add]
+  by_cases heq : cs.simple i * w = w * cs.simple j
+  · have heq' : cs.simple i * (w * cs.simple j) = w := by rw [← heq]; simp
+    have heq'' : cs.simple i * w * cs.simple j = w := by simpa [mul_assoc] using heq'
+    rw [heq', heq'']
+    by_cases h : cs.length (cs.simple i * w) < cs.length w
+    · have h' : cs.length (w * cs.simple j) < cs.length w := heq ▸ h
+      simp only [map_smul, right_basis, left_basis,
+        heq, ite_eq_left h']
+    · have h' : ¬cs.length (w * cs.simple j) < cs.length w := heq ▸ h
+      simp only [ite_eq_right h', map_zero, add_zero, heq]
+  · have hl := left_descent_mul_simple_iff cs i j w heq
+    have hr := right_descent_simple_mul_iff cs i j w heq
+    simp only [hl, hr]
+    by_cases hi : cs.length (cs.simple i * w) < cs.length w <;>
+      by_cases hj : cs.length (w * cs.simple j) < cs.length w <;>
+      simp only [hi, hj, ite_true, ite_false, map_zero, add_zero, map_smul,
+        left_basis, right_basis, smul_add, smul_smul, mul_assoc] ;
+      abel
+
+theorem left_quadratic (cs : CoxeterSystem M W) (a : C) (i : B) :
+    left cs a i * left cs a i = 1 + a • left cs a i := by
+  apply end_ext
+  intro w
+  simp only [Module.End.mul_apply, left_basis, map_add,
+    cs.simple_mul_simple_cancel_left, LinearMap.add_apply, LinearMap.smul_apply,
+    Module.End.one_apply]
+  rcases cs.length_simple_mul w i with h | h
+  · have hasc : ¬cs.length (cs.simple i * w) < cs.length w := by omega
+    have hdesc : cs.length w < cs.length (cs.simple i * w) := by omega
+    simp [hasc, hdesc]
+  · have hdesc : cs.length (cs.simple i * w) < cs.length w := by omega
+    have hasc : ¬cs.length w < cs.length (cs.simple i * w) := by omega
+    simp [hasc, hdesc, smul_add, smul_smul]
+
+/-- Inverse normalized Hecke generator, available without inverting any scalar. -/
+noncomputable def leftInv (cs : CoxeterSystem M W) (a : C) (i : B) :
+    Module.End C (Space W C) := left cs a i - a • 1
+
+theorem left_mul_leftInv (cs : CoxeterSystem M W) (a : C) (i : B) :
+    left cs a i * leftInv cs a i = 1 := by
+  apply end_ext
+  intro w
+  have h := congrArg (fun f : Module.End C (Space W C) => f (basis w))
+    (left_quadratic cs a i)
+  simp only [Module.End.mul_apply, LinearMap.add_apply, LinearMap.smul_apply,
+    Module.End.one_apply] at h
+  simp only [Module.End.mul_apply, leftInv, LinearMap.sub_apply, LinearMap.smul_apply,
+    Module.End.one_apply, map_sub, map_smul]
+  rw [h]
+  abel
+
+theorem leftInv_mul_left (cs : CoxeterSystem M W) (a : C) (i : B) :
+    leftInv cs a i * left cs a i = 1 := by
+  apply end_ext
+  intro w
+  have h := congrArg (fun f : Module.End C (Space W C) => f (basis w))
+    (left_quadratic cs a i)
+  simp only [Module.End.mul_apply, LinearMap.add_apply, LinearMap.smul_apply,
+    Module.End.one_apply] at h
+  simp only [Module.End.mul_apply, leftInv, LinearMap.sub_apply, LinearMap.smul_apply,
+    Module.End.one_apply]
+  rw [h]
+  abel
+
+noncomputable def leftWord (cs : CoxeterSystem M W) (a : C) (ω : List B) :
+    Module.End C (Space W C) := (ω.map (left cs a)).prod
+
+@[simp] theorem leftWord_nil (cs : CoxeterSystem M W) (a : C) :
+    leftWord cs a [] = 1 := rfl
+
+@[simp] theorem leftWord_cons (cs : CoxeterSystem M W) (a : C) (i : B) (ω : List B) :
+    leftWord cs a (i :: ω) = left cs a i * leftWord cs a ω := rfl
+
+/-- Acting on the identity along a reduced word gives exactly its standard
+basis element. -/
+theorem leftWord_one (cs : CoxeterSystem M W) (a : C) {ω : List B}
+    (hω : cs.IsReduced ω) :
+    leftWord cs a ω (basis 1) = basis (cs.wordProd ω) := by
+  induction ω with
+  | nil => simp
+  | cons i ω ih =>
+    have hω' : cs.IsReduced ω := by simpa using hω.drop 1
+    have hasc : ¬cs.length (cs.simple i * cs.wordProd ω) < cs.length (cs.wordProd ω) := by
+      have := hω.eq
+      rw [cs.wordProd_cons, List.length_cons, ← hω'.eq] at this
+      omega
+    simp [Module.End.mul_apply, ih hω', hasc, cs.wordProd_cons]
+
+noncomputable def rightWord (cs : CoxeterSystem M W) (a : C) (ω : List B) :
+    Module.End C (Space W C) := (ω.map (right cs a)).prod
+
+@[simp] theorem rightWord_nil (cs : CoxeterSystem M W) (a : C) :
+    rightWord cs a [] = 1 := rfl
+
+@[simp] theorem rightWord_cons (cs : CoxeterSystem M W) (a : C) (i : B) (ω : List B) :
+    rightWord cs a (i :: ω) = right cs a i * rightWord cs a ω := rfl
+
+theorem rightWord_one (cs : CoxeterSystem M W) (a : C) {ω : List B}
+    (hω : cs.IsReduced ω) :
+    rightWord cs a ω (basis 1) = basis (cs.wordProd ω)⁻¹ := by
+  induction ω with
+  | nil => simp
+  | cons i ω ih =>
+    have hω' : cs.IsReduced ω := by simpa using hω.drop 1
+    have hlen : cs.length ((cs.wordProd ω)⁻¹ * cs.simple i) =
+        cs.length (cs.simple i * cs.wordProd ω) := by
+      rw [← cs.length_inv (cs.simple i * cs.wordProd ω)]
+      simp [mul_inv_rev]
+    have hasc : ¬cs.length ((cs.wordProd ω)⁻¹ * cs.simple i) <
+        cs.length (cs.wordProd ω)⁻¹ := by
+      rw [hlen, cs.length_inv]
+      have := hω.eq
+      rw [cs.wordProd_cons, List.length_cons, ← hω'.eq] at this
+      omega
+    have hasc' : ¬cs.length ((cs.wordProd ω)⁻¹ * cs.simple i) <
+        cs.length (cs.wordProd ω) := by simpa using hasc
+    simp [Module.End.mul_apply, ih hω', hasc', cs.wordProd_cons, mul_inv_rev]
+
+theorem commute_rightWord (cs : CoxeterSystem M W) (a : C)
+    {f : Module.End C (Space W C)} (h : ∀ i, Commute f (right cs a i)) (ω : List B) :
+    Commute f (rightWord cs a ω) := by
+  apply Commute.list_prod_right
+  intro r hr
+  obtain ⟨i, _, rfl⟩ := List.mem_map.mp hr
+  exact h i
+
+/-- The regular right action is cyclic, with cyclic vector the identity. -/
+theorem commuting_ext (cs : CoxeterSystem M W) (a : C)
+    {f g : Module.End C (Space W C)}
+    (hf : ∀ i, Commute f (right cs a i))
+    (hg : ∀ i, Commute g (right cs a i))
+    (h : f (basis 1) = g (basis 1)) : f = g := by
+  apply end_ext
+  intro w
+  obtain ⟨ω, hω, heq⟩ := cs.exists_isReduced w⁻¹
+  have hw : rightWord cs a ω (basis 1) = basis w := by
+    rw [rightWord_one cs a hω, ← heq, inv_inv]
+  rw [← hw]
+  have hf' := congrArg (fun t : Module.End C (Space W C) => t (basis 1))
+    (commute_rightWord cs a hf ω).eq
+  have hg' := congrArg (fun t : Module.End C (Space W C) => t (basis 1))
+    (commute_rightWord cs a hg ω).eq
+  simp only [Module.End.mul_apply] at hf' hg'
+  rw [h] at hf'
+  exact hf'.trans hg'.symm
+
+theorem leftWord_commute_right (cs : CoxeterSystem M W) (a : C) (ω : List B) (i : B) :
+    Commute (leftWord cs a ω) (right cs a i) := by
+  apply Commute.list_prod_left
+  intro r hr
+  obtain ⟨j, _, rfl⟩ := List.mem_map.mp hr
+  exact left_right_commute cs a j i
+
+
+theorem leftWord_independent (cs : CoxeterSystem M W) (a : C) {ω η : List B}
+    (hω : cs.IsReduced ω) (hη : cs.IsReduced η) (heq : cs.wordProd ω = cs.wordProd η) :
+    leftWord cs a ω = leftWord cs a η := by
+  apply commuting_ext cs a (leftWord_commute_right cs a ω)
+    (leftWord_commute_right cs a η)
+  rw [leftWord_one cs a hω, leftWord_one cs a hη, heq]
+
+noncomputable def unit (cs : CoxeterSystem M W) (a : C) (i : B) :
+    (Module.End C (Space W C))ˣ where
+  val := left cs a i
+  inv := leftInv cs a i
+  val_inv := left_mul_leftInv cs a i
+  inv_val := leftInv_mul_left cs a i
+
+noncomputable def wordUnit (cs : CoxeterSystem M W) (a : C) (ω : List B) :
+    (Module.End C (Space W C))ˣ := (ω.map (unit cs a)).prod
+
+@[simp] theorem wordUnit_val (cs : CoxeterSystem M W) (a : C) (ω : List B) :
+    (wordUnit cs a ω).val = leftWord cs a ω := by
+  induction ω with
+  | nil => rfl
+  | cons i ω ih =>
+    change (unit cs a i).val * (wordUnit cs a ω).val = left cs a i * leftWord cs a ω
+    rw [ih]
+    rfl
+
+theorem wordUnit_independent (cs : CoxeterSystem M W) (a : C) {ω η : List B}
+    (hω : cs.IsReduced ω) (hη : cs.IsReduced η) (heq : cs.wordProd ω = cs.wordProd η) :
+    wordUnit cs a ω = wordUnit cs a η := by
+  apply Units.ext
+  simpa using leftWord_independent cs a hω hη heq
+
+/-- The genuine normalized standard-basis action, a unit in the regular
+endomorphism algebra. No presentation relations are assumed. -/
+noncomputable def standardUnit (cs : CoxeterSystem M W) (a : C) (w : W) :
+    (Module.End C (Space W C))ˣ :=
+  wordUnit cs a (Classical.choose (cs.exists_isReduced w))
+
+theorem standardUnit_word (cs : CoxeterSystem M W) (a : C) {ω : List B}
+    (hω : cs.IsReduced ω) :
+    standardUnit cs a (cs.wordProd ω) = wordUnit cs a ω := by
+  apply wordUnit_independent cs a (Classical.choose_spec (cs.exists_isReduced _)).1 hω
+  exact (Classical.choose_spec (cs.exists_isReduced _)).2.symm
+
+@[simp] theorem standardUnit_one (cs : CoxeterSystem M W) (a : C) :
+    standardUnit cs a 1 = 1 := by
+  have h := standardUnit_word cs a (ω := []) (by simp [CoxeterSystem.IsReduced])
+  simpa [wordUnit] using h
+
+@[simp] theorem standardUnit_basis_one (cs : CoxeterSystem M W) (a : C) (w : W) :
+    (standardUnit cs a w).val (basis 1) = basis w := by
+  obtain ⟨ω, hω, rfl⟩ := cs.exists_isReduced w
+  rw [standardUnit_word cs a hω, wordUnit_val, leftWord_one cs a hω]
+
+theorem standardUnit_simple_mul (cs : CoxeterSystem M W) (a : C) (i : B) (w : W)
+    (hasc : cs.length w < cs.length (cs.simple i * w)) :
+    standardUnit cs a (cs.simple i * w) = unit cs a i * standardUnit cs a w := by
+  obtain ⟨ω, hω, rfl⟩ := cs.exists_isReduced w
+  have hred : cs.IsReduced (i :: ω) := by
+    unfold CoxeterSystem.IsReduced
+    have := cs.length_simple_mul (cs.wordProd ω) i
+    simp only [cs.wordProd_cons, List.length_cons]
+    rw [← hω.eq]
+    omega
+  rw [← cs.wordProd_cons, standardUnit_word cs a hred, standardUnit_word cs a hω]
+  simp [wordUnit]
+
+
+theorem standardUnit_mul_simple (cs : CoxeterSystem M W) (a : C) (i : B) (w : W)
+    (hasc : cs.length w < cs.length (w * cs.simple i)) :
+    standardUnit cs a (w * cs.simple i) = standardUnit cs a w * unit cs a i := by
+  obtain ⟨ω, hω, rfl⟩ := cs.exists_isReduced w
+  have hred : cs.IsReduced (ω ++ [i]) := by
+    unfold CoxeterSystem.IsReduced
+    have := cs.length_mul_simple (cs.wordProd ω) i
+    simp only [cs.wordProd_append, cs.wordProd_singleton, List.length_append,
+      List.length_singleton]
+    rw [← hω.eq]
+    omega
+  rw [← cs.wordProd_singleton i, ← cs.wordProd_append, standardUnit_word cs a hred,
+    standardUnit_word cs a hω]
+  simp [wordUnit, List.map_append, List.prod_append]
+
+/-- The bar image of an actual standard basis vector; inverse units give
+expression independence before any polynomial coefficients are extracted. -/
+noncomputable def barBasis (cs : CoxeterSystem M W) (a : C) (w : W) : Space W C :=
+  ((standardUnit cs a w⁻¹)⁻¹).val (basis 1)
+
+@[simp] theorem barBasis_one (cs : CoxeterSystem M W) (a : C) :
+    barBasis cs a 1 = basis 1 := by simp [barBasis]
+
+theorem barBasis_simple_mul_ascent (cs : CoxeterSystem M W) (a : C) (i : B) (w : W)
+    (hasc : cs.length w < cs.length (cs.simple i * w)) :
+    barBasis cs a (cs.simple i * w) = leftInv cs a i (barBasis cs a w) := by
+  have hasc' : cs.length w⁻¹ < cs.length (w⁻¹ * cs.simple i) := by
+    have hlen : cs.length (w⁻¹ * cs.simple i) = cs.length (cs.simple i * w) := by
+      rw [← cs.length_inv (cs.simple i * w)]
+      simp [mul_inv_rev]
+    simpa [hlen] using hasc
+  simp only [barBasis, mul_inv_rev, cs.inv_simple,
+    standardUnit_mul_simple cs a i w⁻¹ hasc', Units.val_mul,
+    Module.End.mul_apply]
+  rfl
+
+theorem leftInv_barBasis (cs : CoxeterSystem M W) (a : C) (i : B) (w : W) :
+    leftInv cs a i (barBasis cs a w) = barBasis cs a (cs.simple i * w) -
+      if cs.length (cs.simple i * w) < cs.length w then a • barBasis cs a w else 0 := by
+  by_cases h : cs.length (cs.simple i * w) < cs.length w
+  · have hasc : cs.length (cs.simple i * w) <
+        cs.length (cs.simple i * (cs.simple i * w)) := by simpa using h
+    have hb := barBasis_simple_mul_ascent cs a i (cs.simple i * w) hasc
+    simp only [cs.simple_mul_simple_cancel_left] at hb
+    have hl : left cs a i (barBasis cs a w) = barBasis cs a (cs.simple i * w) := by
+      rw [hb, ← Module.End.mul_apply, left_mul_leftInv, Module.End.one_apply]
+    simp only [ite_eq_left h, leftInv, LinearMap.sub_apply, LinearMap.smul_apply,
+      Module.End.one_apply, hl]
+  · have hasc : cs.length w < cs.length (cs.simple i * w) := by
+      have := cs.length_simple_mul w i
+      omega
+    rw [ite_eq_right h, sub_zero, barBasis_simple_mul_ascent cs a i w hasc]
+
+noncomputable def barLinear (cs : CoxeterSystem M W) (a : C) : Module.End C (Space W C) :=
+  Finsupp.linearCombination C (barBasis cs a)
+
+@[simp] theorem barLinear_basis (cs : CoxeterSystem M W) (a : C) (w : W) :
+    barLinear cs a (basis w) = barBasis cs a w := by
+  simp [barLinear, basis, Finsupp.linearCombination_single]
+
+/-- Intertwining with the opposite parameter. In Laurent coefficients this
+is the semilinear Hecke bar relation. -/
+theorem barLinear_left (cs : CoxeterSystem M W) (a : C) (i : B) :
+    barLinear cs a * left cs (-a) i = leftInv cs a i * barLinear cs a := by
+  apply end_ext
+  intro w
+  simp only [Module.End.mul_apply, left_basis, map_add, barLinear_basis,
+    leftInv_barBasis]
+  by_cases h : cs.length (cs.simple i * w) < cs.length w
+  · simp [h, sub_eq_add_neg]
+  · simp [h]
+
+theorem barLinear_leftInv (cs : CoxeterSystem M W) (a : C) (i : B) (v : Space W C) :
+    barLinear cs a (leftInv cs (-a) i v) = left cs a i (barLinear cs a v) := by
+  have h := congrArg (fun f : Module.End C (Space W C) => f v) (barLinear_left cs a i)
+  simp only [Module.End.mul_apply, leftInv, LinearMap.sub_apply,
+    LinearMap.smul_apply, Module.End.one_apply] at h ⊢
+  rw [map_sub, map_smul, h]
+  simp
+
+/-- The inverse-parameter bar transforms are mutually inverse. -/
+theorem barLinear_barBasis (cs : CoxeterSystem M W) (a : C) (w : W) :
+    barLinear cs a (barBasis cs (-a) w) = basis w := by
+  obtain ⟨ω, hω, rfl⟩ := cs.exists_isReduced w
+  induction ω with
+  | nil => simp
+  | cons i ω ih =>
+    have hω' : cs.IsReduced ω := by simpa using hω.drop 1
+    have hasc : cs.length (cs.wordProd ω) < cs.length (cs.simple i * cs.wordProd ω) := by
+      have := hω.eq
+      simp only [cs.wordProd_cons, List.length_cons, ← hω'.eq] at this
+      omega
+    rw [cs.wordProd_cons, barBasis_simple_mul_ascent cs (-a) i _ hasc,
+      barLinear_leftInv, ih hω', left_basis, ite_eq_right (by omega), add_zero]
+
+theorem barLinear_inverse (cs : CoxeterSystem M W) (a : C) :
+    barLinear cs a * barLinear cs (-a) = 1 := by
+  apply end_ext
+  intro w
+  simp [Module.End.mul_apply, barLinear_barBasis]
+
+
+@[simp] theorem leftInv_basis (cs : CoxeterSystem M W) (a : C) (i : B) (w : W) :
+    leftInv cs a i (basis w) = basis (cs.simple i * w) -
+      if cs.length (cs.simple i * w) < cs.length w then 0 else a • basis w := by
+  simp only [leftInv, LinearMap.sub_apply, LinearMap.smul_apply,
+    Module.End.one_apply, left_basis]
+  split_ifs <;> abel
+
+theorem left_coeff (cs : CoxeterSystem M W) (a : C) (i : B) (v : Space W C) (x : W) :
+    (left cs a i v) x = v (cs.simple i * x) +
+      if cs.length (cs.simple i * x) < cs.length x then a * v x else 0 := by
+  classical
+  induction v using Finsupp.induction_linear with
+  | zero => simp
+  | add v v' hv hv' =>
+    simp only [map_add, Finsupp.add_apply, hv, hv', mul_add]
+    split_ifs <;> ring
+  | single w c =>
+    have hc : Finsupp.single w c = c • (basis w : Space W C) := by
+      simp [basis, Finsupp.smul_single]
+    rw [hc, map_smul, left_basis]
+    have heq : cs.simple i * w = x ↔ w = cs.simple i * x := by
+      constructor <;> intro h
+      · rw [← h]; simp
+      · rw [h]; simp
+    simp only [Finsupp.smul_apply, smul_eq_mul, Finsupp.add_apply, basis,
+      Finsupp.single_apply, heq]
+    by_cases h : w = x
+    · subst w
+      split_ifs <;> try contradiction
+      all_goals simp only [Finsupp.single_eq_same, Finsupp.zero_apply,
+        Finsupp.smul_apply, smul_eq_mul, mul_one, mul_zero] ; ring
+    · simp only [h, ite_false, mul_zero]
+      by_cases hd : cs.length (cs.simple i * w) < cs.length w <;>
+        by_cases hd' : cs.length (cs.simple i * x) < cs.length x <;>
+        simp [hd, hd', h]
+
+theorem leftInv_coeff (cs : CoxeterSystem M W) (a : C) (i : B) (v : Space W C) (x : W) :
+    (leftInv cs a i v) x = v (cs.simple i * x) -
+      if cs.length (cs.simple i * x) < cs.length x then 0 else a * v x := by
+  simp only [leftInv, LinearMap.sub_apply, LinearMap.smul_apply,
+    Module.End.one_apply, Finsupp.sub_apply, Finsupp.smul_apply, smul_eq_mul, left_coeff]
+  split_ifs <;> ring
+
+theorem simple_mul_le_ascent (cs : CoxeterSystem M W) (i : B) {x y : W}
+    (hxy : BruhatLE cs x y) (hy : cs.length y < cs.length (cs.simple i * y)) :
+    BruhatLE cs (cs.simple i * x) (cs.simple i * y) := by
+  have hx := le_upperSimple cs i (cs.simple i * x)
+  rw [upperSimple_mul] at hx
+  have hm := upperSimple_mono cs i hxy
+  simp only [upperSimple, ite_eq_right (show ¬cs.length (cs.simple i * y) < cs.length y by omega)] at hm
+  exact hx.trans hm
+
+/-- Triangularity for the genuine reflection-generated order. -/
+theorem barBasis_zero (cs : CoxeterSystem M W) (a : C) (x y : W)
+    (hxy : ¬BruhatLE cs x y) : barBasis cs a y x = 0 := by
+  obtain ⟨ω, hω, rfl⟩ := cs.exists_isReduced y
+  induction ω generalizing x with
+  | nil =>
+    have hx : x ≠ 1 := fun h => hxy (h ▸ bruhat_refl cs 1)
+    simp [basis, Ne.symm hx]
+  | cons i ω ih =>
+    have hω' : cs.IsReduced ω := by simpa using hω.drop 1
+    have hasc : cs.length (cs.wordProd ω) < cs.length (cs.simple i * cs.wordProd ω) := by
+      have := hω.eq
+      simp only [cs.wordProd_cons, List.length_cons, ← hω'.eq] at this
+      omega
+    have hle : BruhatLE cs (cs.wordProd ω) (cs.simple i * cs.wordProd ω) :=
+      Relation.ReflTransGen.single ⟨hasc, cs.simple i, cs.isReflection_simple i, rfl⟩
+    have hn : ¬BruhatLE cs x (cs.wordProd ω) := fun h => hxy (h.trans hle)
+    have hn' : ¬BruhatLE cs (cs.simple i * x) (cs.wordProd ω) := by
+      intro h
+      have := simple_mul_le_ascent cs i h hasc
+      simp only [cs.simple_mul_simple_cancel_left] at this
+      exact hxy this
+    rw [cs.wordProd_cons, barBasis_simple_mul_ascent cs a i _ hasc, leftInv_coeff,
+      ih x hω' hn, ih (cs.simple i * x) hω' hn']
+    split_ifs <;> simp
+
+@[simp] theorem barBasis_diagonal (cs : CoxeterSystem M W) (a : C) (y : W) :
+    barBasis cs a y y = 1 := by
+  obtain ⟨ω, hω, rfl⟩ := cs.exists_isReduced y
+  induction ω with
+  | nil => simp [basis]
+  | cons i ω ih =>
+    have hω' : cs.IsReduced ω := by simpa using hω.drop 1
+    have hasc : cs.length (cs.wordProd ω) < cs.length (cs.simple i * cs.wordProd ω) := by
+      have := hω.eq
+      simp only [cs.wordProd_cons, List.length_cons, ← hω'.eq] at this
+      omega
+    rw [cs.wordProd_cons, barBasis_simple_mul_ascent cs a i _ hasc, leftInv_coeff]
+    simpa [hasc] using ih hω'
+
+/-- Normalized R-polynomials, obtained from the bar action rather than
+postulated with an inconsistent choice of descent. -/
+noncomputable def rTilde (cs : CoxeterSystem M W) (x y : W) : ℤ[X] :=
+  barBasis cs (-X : ℤ[X]) y x
+
+@[simp] theorem rTilde_diagonal (cs : CoxeterSystem M W) (x : W) :
+    rTilde cs x x = 1 := barBasis_diagonal cs (-X) x
+
+theorem rTilde_zero (cs : CoxeterSystem M W) (x y : W) (h : ¬BruhatLE cs x y) :
+    rTilde cs x y = 0 := barBasis_zero cs (-X) x y h
+
+theorem rTilde_recursion (cs : CoxeterSystem M W) (x y : W) (i : B)
+    (hy : cs.length (cs.simple i * y) < cs.length y) :
+    rTilde cs x y = if cs.length (cs.simple i * x) < cs.length x then
+      rTilde cs (cs.simple i * x) (cs.simple i * y)
+      else rTilde cs (cs.simple i * x) (cs.simple i * y) + X * rTilde cs x (cs.simple i * y) := by
+  have hasc : cs.length (cs.simple i * y) <
+      cs.length (cs.simple i * (cs.simple i * y)) := by simpa using hy
+  have hb := barBasis_simple_mul_ascent cs (-X : ℤ[X]) i (cs.simple i * y) hasc
+  simp only [cs.simple_mul_simple_cancel_left] at hb
+  unfold rTilde
+  rw [hb, leftInv_coeff]
+  split_ifs <;> ring
+
+
+end Hecke
+
+
+namespace HeckeQ
+
+open Hecke (Space basis end_ext)
+variable {C : Type*} [CommRing C]
+
+/-- Unnormalized equal-parameter Hecke multiplication, T_s²=q+(q−1)T_s. -/
+noncomputable def left (cs : CoxeterSystem M W) (q : C) (i : B) :
+    Module.End C (Space W C) :=
+  Finsupp.linearCombination C (fun w =>
+    if cs.length (cs.simple i * w) < cs.length w then
+      q • basis (cs.simple i * w) + (q - 1) • basis w
+    else basis (cs.simple i * w))
+
+noncomputable def right (cs : CoxeterSystem M W) (q : C) (i : B) :
+    Module.End C (Space W C) :=
+  Finsupp.linearCombination C (fun w =>
+    if cs.length (w * cs.simple i) < cs.length w then
+      q • basis (w * cs.simple i) + (q - 1) • basis w
+    else basis (w * cs.simple i))
+
+@[simp] theorem left_basis (cs : CoxeterSystem M W) (q : C) (i : B) (w : W) :
+    left cs q i (basis w) = if cs.length (cs.simple i * w) < cs.length w then
+      q • basis (cs.simple i * w) + (q - 1) • basis w
+    else basis (cs.simple i * w) := by
+  simp [left, basis, Finsupp.linearCombination_single]
+
+@[simp] theorem right_basis (cs : CoxeterSystem M W) (q : C) (i : B) (w : W) :
+    right cs q i (basis w) = if cs.length (w * cs.simple i) < cs.length w then
+      q • basis (w * cs.simple i) + (q - 1) • basis w
+    else basis (w * cs.simple i) := by
+  simp [right, basis, Finsupp.linearCombination_single]
+
+theorem left_right_commute (cs : CoxeterSystem M W) (q : C) (i j : B) :
+    Commute (left cs q i) (right cs q j) := by
+  apply end_ext
+  intro w
+  simp only [Module.End.mul_apply, right_basis, left_basis]
+  by_cases heq : cs.simple i * w = w * cs.simple j
+  · have heq' : cs.simple i * (w * cs.simple j) = w := by rw [← heq]; simp
+    have heq'' : cs.simple i * w * cs.simple j = w := by simpa [mul_assoc] using heq'
+    by_cases h : cs.length (cs.simple i * w) < cs.length w
+    · have h' : cs.length (w * cs.simple j) < cs.length w := heq ▸ h
+      have hn : ¬cs.length w < cs.length (cs.simple i * w) := by omega
+      have hn' : ¬cs.length w < cs.length (w * cs.simple j) := heq ▸ hn
+      simp [h', map_add, map_smul, heq', hn', heq]
+    · have h' : ¬cs.length (w * cs.simple j) < cs.length w := heq ▸ h
+      have hl : cs.length w < cs.length (cs.simple i * w) := by
+        have := cs.length_simple_mul w i; omega
+      have hl' : cs.length w < cs.length (w * cs.simple j) := heq ▸ hl
+      simp [h', heq', hl', heq]
+  · have hl := left_descent_mul_simple_iff cs i j w heq
+    have hr := right_descent_simple_mul_iff cs i j w heq
+    have hr' : cs.length (cs.simple i * (w * cs.simple j)) <
+        cs.length (cs.simple i * w) ↔ cs.length (w * cs.simple j) < cs.length w := by
+      simpa only [mul_assoc] using hr
+    by_cases hi : cs.length (cs.simple i * w) < cs.length w <;>
+      by_cases hj : cs.length (w * cs.simple j) < cs.length w <;>
+      simp only [hi, hj, ite_true, ite_false, map_add, map_smul,
+        left_basis, right_basis, hl, hr', mul_assoc, smul_add, smul_smul] ;
+      module
+
+theorem left_quadratic (cs : CoxeterSystem M W) (q : C) (i : B) :
+    left cs q i * left cs q i = q • 1 + (q - 1) • left cs q i := by
+  apply end_ext
+  intro w
+  simp only [Module.End.mul_apply, left_basis, LinearMap.add_apply, LinearMap.smul_apply,
+    Module.End.one_apply]
+  rcases cs.length_simple_mul w i with h | h
+  · have hasc : ¬cs.length (cs.simple i * w) < cs.length w := by omega
+    have hdesc : cs.length w < cs.length (cs.simple i * w) := by omega
+    simp [hasc, hdesc]
+  · have hdesc : cs.length (cs.simple i * w) < cs.length w := by omega
+    have hasc : ¬cs.length w < cs.length (cs.simple i * w) := by omega
+    simp [hasc, hdesc, smul_add, smul_smul]
+
+noncomputable def leftInv (cs : CoxeterSystem M W) (q : Cˣ) (i : B) :
+    Module.End C (Space W C) := (↑q⁻¹ : C) • (left cs (q : C) i - ((q : C) - 1) • 1)
+
+theorem leftInv_apply (cs : CoxeterSystem M W) (q : Cˣ) (i : B) (v : Space W C) :
+    leftInv cs q i v = (↑q⁻¹ : C) • (left cs (q : C) i v - ((q : C) - 1) • v) := rfl
+
+theorem left_mul_leftInv (cs : CoxeterSystem M W) (q : Cˣ) (i : B) :
+    left cs (q : C) i * leftInv cs q i = 1 := by
+  apply end_ext
+  intro w
+  have h := congrArg (fun f : Module.End C (Space W C) => f (basis w))
+    (left_quadratic cs (q:C) i)
+  simp only [Module.End.mul_apply, LinearMap.add_apply, LinearMap.smul_apply,
+    Module.End.one_apply] at h
+  simp only [Module.End.mul_apply, leftInv_apply, Module.End.one_apply,
+    map_smul, map_sub]
+  rw [h]
+  simp [smul_smul]
+
+theorem leftInv_mul_left (cs : CoxeterSystem M W) (q : Cˣ) (i : B) :
+    leftInv cs q i * left cs (q : C) i = 1 := by
+  apply end_ext
+  intro w
+  have h := congrArg (fun f : Module.End C (Space W C) => f (basis w))
+    (left_quadratic cs (q:C) i)
+  simp only [Module.End.mul_apply, LinearMap.add_apply, LinearMap.smul_apply,
+    Module.End.one_apply] at h
+  simp only [Module.End.mul_apply, leftInv_apply, Module.End.one_apply]
+  rw [h]
+  simp [smul_smul]
+
+
+noncomputable def leftWord (cs : CoxeterSystem M W) (q : C) (ω : List B) :
+    Module.End C (Space W C) := (ω.map (left cs q)).prod
+
+@[simp] theorem leftWord_nil (cs : CoxeterSystem M W) (q : C) :
+    leftWord cs q [] = 1 := rfl
+
+@[simp] theorem leftWord_cons (cs : CoxeterSystem M W) (q : C) (i : B) (ω : List B) :
+    leftWord cs q (i :: ω) = left cs q i * leftWord cs q ω := rfl
+
+/-- Acting on the identity along q reduced word gives exactly its standard
+basis element. -/
+theorem leftWord_one (cs : CoxeterSystem M W) (q : C) {ω : List B}
+    (hω : cs.IsReduced ω) :
+    leftWord cs q ω (basis 1) = basis (cs.wordProd ω) := by
+  induction ω with
+  | nil => simp
+  | cons i ω ih =>
+    have hω' : cs.IsReduced ω := by simpa using hω.drop 1
+    have hasc : ¬cs.length (cs.simple i * cs.wordProd ω) < cs.length (cs.wordProd ω) := by
+      have := hω.eq
+      rw [cs.wordProd_cons, List.length_cons, ← hω'.eq] at this
+      omega
+    simp [Module.End.mul_apply, ih hω', hasc, cs.wordProd_cons]
+
+noncomputable def rightWord (cs : CoxeterSystem M W) (q : C) (ω : List B) :
+    Module.End C (Space W C) := (ω.map (right cs q)).prod
+
+@[simp] theorem rightWord_nil (cs : CoxeterSystem M W) (q : C) :
+    rightWord cs q [] = 1 := rfl
+
+@[simp] theorem rightWord_cons (cs : CoxeterSystem M W) (q : C) (i : B) (ω : List B) :
+    rightWord cs q (i :: ω) = right cs q i * rightWord cs q ω := rfl
+
+theorem rightWord_one (cs : CoxeterSystem M W) (q : C) {ω : List B}
+    (hω : cs.IsReduced ω) :
+    rightWord cs q ω (basis 1) = basis (cs.wordProd ω)⁻¹ := by
+  induction ω with
+  | nil => simp
+  | cons i ω ih =>
+    have hω' : cs.IsReduced ω := by simpa using hω.drop 1
+    have hlen : cs.length ((cs.wordProd ω)⁻¹ * cs.simple i) =
+        cs.length (cs.simple i * cs.wordProd ω) := by
+      rw [← cs.length_inv (cs.simple i * cs.wordProd ω)]
+      simp [mul_inv_rev]
+    have hasc : ¬cs.length ((cs.wordProd ω)⁻¹ * cs.simple i) <
+        cs.length (cs.wordProd ω)⁻¹ := by
+      rw [hlen, cs.length_inv]
+      have := hω.eq
+      rw [cs.wordProd_cons, List.length_cons, ← hω'.eq] at this
+      omega
+    have hasc' : ¬cs.length ((cs.wordProd ω)⁻¹ * cs.simple i) <
+        cs.length (cs.wordProd ω) := by simpa using hasc
+    simp [Module.End.mul_apply, ih hω', hasc', cs.wordProd_cons, mul_inv_rev]
+
+theorem commute_rightWord (cs : CoxeterSystem M W) (q : C)
+    {f : Module.End C (Space W C)} (h : ∀ i, Commute f (right cs q i)) (ω : List B) :
+    Commute f (rightWord cs q ω) := by
+  apply Commute.list_prod_right
+  intro r hr
+  obtain ⟨i, _, rfl⟩ := List.mem_map.mp hr
+  exact h i
+
+/-- The regular right action is cyclic, with cyclic vector the identity. -/
+theorem commuting_ext (cs : CoxeterSystem M W) (q : C)
+    {f g : Module.End C (Space W C)}
+    (hf : ∀ i, Commute f (right cs q i))
+    (hg : ∀ i, Commute g (right cs q i))
+    (h : f (basis 1) = g (basis 1)) : f = g := by
+  apply end_ext
+  intro w
+  obtain ⟨ω, hω, heq⟩ := cs.exists_isReduced w⁻¹
+  have hw : rightWord cs q ω (basis 1) = basis w := by
+    rw [rightWord_one cs q hω, ← heq, inv_inv]
+  rw [← hw]
+  have hf' := congrArg (fun t : Module.End C (Space W C) => t (basis 1))
+    (commute_rightWord cs q hf ω).eq
+  have hg' := congrArg (fun t : Module.End C (Space W C) => t (basis 1))
+    (commute_rightWord cs q hg ω).eq
+  simp only [Module.End.mul_apply] at hf' hg'
+  rw [h] at hf'
+  exact hf'.trans hg'.symm
+
+theorem leftWord_commute_right (cs : CoxeterSystem M W) (q : C) (ω : List B) (i : B) :
+    Commute (leftWord cs q ω) (right cs q i) := by
+  apply Commute.list_prod_left
+  intro r hr
+  obtain ⟨j, _, rfl⟩ := List.mem_map.mp hr
+  exact left_right_commute cs q j i
+
+
+theorem leftWord_independent (cs : CoxeterSystem M W) (q : C) {ω η : List B}
+    (hω : cs.IsReduced ω) (hη : cs.IsReduced η) (heq : cs.wordProd ω = cs.wordProd η) :
+    leftWord cs q ω = leftWord cs q η := by
+  apply commuting_ext cs q (leftWord_commute_right cs q ω)
+    (leftWord_commute_right cs q η)
+  rw [leftWord_one cs q hω, leftWord_one cs q hη, heq]
+
+noncomputable def unit (cs : CoxeterSystem M W) (q : Cˣ) (i : B) :
+    (Module.End C (Space W C))ˣ where
+  val := left cs (q : C) i
+  inv := leftInv cs q i
+  val_inv := left_mul_leftInv cs q i
+  inv_val := leftInv_mul_left cs q i
+
+noncomputable def wordUnit (cs : CoxeterSystem M W) (q : Cˣ) (ω : List B) :
+    (Module.End C (Space W C))ˣ := (ω.map (unit cs q)).prod
+
+@[simp] theorem wordUnit_val (cs : CoxeterSystem M W) (q : Cˣ) (ω : List B) :
+    (wordUnit cs q ω).val = leftWord cs (q : C) ω := by
+  induction ω with
+  | nil => rfl
+  | cons i ω ih =>
+    change (unit cs q i).val * (wordUnit cs q ω).val = left cs (q : C) i * leftWord cs (q : C) ω
+    rw [ih]
+    rfl
+
+theorem wordUnit_independent (cs : CoxeterSystem M W) (q : Cˣ) {ω η : List B}
+    (hω : cs.IsReduced ω) (hη : cs.IsReduced η) (heq : cs.wordProd ω = cs.wordProd η) :
+    wordUnit cs q ω = wordUnit cs q η := by
+  apply Units.ext
+  simpa using leftWord_independent cs (q : C) hω hη heq
+
+/-- The genuine normalized standard-basis action, a unit in the regular
+endomorphism algebra. No presentation relations are assumed. -/
+noncomputable def standardUnit (cs : CoxeterSystem M W) (q : Cˣ) (w : W) :
+    (Module.End C (Space W C))ˣ :=
+  wordUnit cs q (Classical.choose (cs.exists_isReduced w))
+
+theorem standardUnit_word (cs : CoxeterSystem M W) (q : Cˣ) {ω : List B}
+    (hω : cs.IsReduced ω) :
+    standardUnit cs q (cs.wordProd ω) = wordUnit cs q ω := by
+  apply wordUnit_independent cs q (Classical.choose_spec (cs.exists_isReduced _)).1 hω
+  exact (Classical.choose_spec (cs.exists_isReduced _)).2.symm
+
+@[simp] theorem standardUnit_one (cs : CoxeterSystem M W) (q : Cˣ) :
+    standardUnit cs q 1 = 1 := by
+  have h := standardUnit_word cs q (ω := []) (by simp [CoxeterSystem.IsReduced])
+  simpa [wordUnit] using h
+
+@[simp] theorem standardUnit_basis_one (cs : CoxeterSystem M W) (q : Cˣ) (w : W) :
+    (standardUnit cs q w).val (basis 1) = basis w := by
+  obtain ⟨ω, hω, rfl⟩ := cs.exists_isReduced w
+  rw [standardUnit_word cs q hω, wordUnit_val, leftWord_one cs (q : C) hω]
+
+theorem standardUnit_simple_mul (cs : CoxeterSystem M W) (q : Cˣ) (i : B) (w : W)
+    (hasc : cs.length w < cs.length (cs.simple i * w)) :
+    standardUnit cs q (cs.simple i * w) = unit cs q i * standardUnit cs q w := by
+  obtain ⟨ω, hω, rfl⟩ := cs.exists_isReduced w
+  have hred : cs.IsReduced (i :: ω) := by
+    unfold CoxeterSystem.IsReduced
+    have := cs.length_simple_mul (cs.wordProd ω) i
+    simp only [cs.wordProd_cons, List.length_cons]
+    rw [← hω.eq]
+    omega
+  rw [← cs.wordProd_cons, standardUnit_word cs q hred, standardUnit_word cs q hω]
+  simp [wordUnit]
+
+
+theorem standardUnit_mul_simple (cs : CoxeterSystem M W) (q : Cˣ) (i : B) (w : W)
+    (hasc : cs.length w < cs.length (w * cs.simple i)) :
+    standardUnit cs q (w * cs.simple i) = standardUnit cs q w * unit cs q i := by
+  obtain ⟨ω, hω, rfl⟩ := cs.exists_isReduced w
+  have hred : cs.IsReduced (ω ++ [i]) := by
+    unfold CoxeterSystem.IsReduced
+    have := cs.length_mul_simple (cs.wordProd ω) i
+    simp only [cs.wordProd_append, cs.wordProd_singleton, List.length_append,
+      List.length_singleton]
+    rw [← hω.eq]
+    omega
+  rw [← cs.wordProd_singleton i, ← cs.wordProd_append, standardUnit_word cs q hred,
+    standardUnit_word cs q hω]
+  simp [wordUnit, List.map_append, List.prod_append]
+
+/-- The bar image of an actual standard basis vector; inverse units give
+expression independence before any polynomial coefficients are extracted. -/
+noncomputable def barBasis (cs : CoxeterSystem M W) (q : Cˣ) (w : W) : Space W C :=
+  ((standardUnit cs q w⁻¹)⁻¹).val (basis 1)
+
+@[simp] theorem barBasis_one (cs : CoxeterSystem M W) (q : Cˣ) :
+    barBasis cs q 1 = basis 1 := by simp [barBasis]
+
+theorem barBasis_simple_mul_ascent (cs : CoxeterSystem M W) (q : Cˣ) (i : B) (w : W)
+    (hasc : cs.length w < cs.length (cs.simple i * w)) :
+    barBasis cs q (cs.simple i * w) = leftInv cs q i (barBasis cs q w) := by
+  have hasc' : cs.length w⁻¹ < cs.length (w⁻¹ * cs.simple i) := by
+    have hlen : cs.length (w⁻¹ * cs.simple i) = cs.length (cs.simple i * w) := by
+      rw [← cs.length_inv (cs.simple i * w)]
+      simp [mul_inv_rev]
+    simpa [hlen] using hasc
+  simp only [barBasis, mul_inv_rev, cs.inv_simple,
+    standardUnit_mul_simple cs q i w⁻¹ hasc', Units.val_mul,
+    Module.End.mul_apply]
+  rfl
+
+
+theorem leftInv_barBasis (cs : CoxeterSystem M W) (q : Cˣ) (i : B) (w : W) :
+    leftInv cs q i (barBasis cs q w) =
+      if cs.length (cs.simple i * w) < cs.length w then
+        (↑q⁻¹ : C) • barBasis cs q (cs.simple i * w) +
+          ((↑q⁻¹ : C) - 1) • barBasis cs q w
+      else barBasis cs q (cs.simple i * w) := by
+  by_cases h : cs.length (cs.simple i * w) < cs.length w
+  · have hasc : cs.length (cs.simple i * w) <
+        cs.length (cs.simple i * (cs.simple i * w)) := by simpa using h
+    have hb := barBasis_simple_mul_ascent cs q i (cs.simple i * w) hasc
+    simp only [cs.simple_mul_simple_cancel_left] at hb
+    have hl : left cs (q : C) i (barBasis cs q w) = barBasis cs q (cs.simple i * w) := by
+      rw [hb, ← Module.End.mul_apply, left_mul_leftInv, Module.End.one_apply]
+    rw [ite_eq_left h, leftInv_apply, hl, smul_sub, smul_smul]
+    have hscalar : (↑q⁻¹ : C) * ((q : C) - 1) = 1 - (↑q⁻¹ : C) := by
+      rw [mul_sub, q.inv_mul, mul_one]
+    rw [hscalar, sub_smul, one_smul]
+    module
+  · have hasc : cs.length w < cs.length (cs.simple i * w) := by
+      have := cs.length_simple_mul w i
+      omega
+    rw [ite_eq_right h, barBasis_simple_mul_ascent cs q i w hasc]
+
+noncomputable def barLinear (cs : CoxeterSystem M W) (q : Cˣ) : Module.End C (Space W C) :=
+  Finsupp.linearCombination C (barBasis cs q)
+
+@[simp] theorem barLinear_basis (cs : CoxeterSystem M W) (q : Cˣ) (w : W) :
+    barLinear cs q (basis w) = barBasis cs q w := by
+  simp [barLinear, basis, Finsupp.linearCombination_single]
+
+/-- Linear intertwining, before specializing coefficients to a Laurent ring. -/
+theorem barLinear_left (cs : CoxeterSystem M W) (q : Cˣ) (i : B) :
+    barLinear cs q * left cs (↑q⁻¹ : C) i = leftInv cs q i * barLinear cs q := by
+  apply end_ext
+  intro w
+  simp only [Module.End.mul_apply, left_basis, leftInv_barBasis, barLinear_basis]
+  split_ifs <;> simp
+
+theorem barLinear_leftInv (cs : CoxeterSystem M W) (q : Cˣ) (i : B) (v : Space W C) :
+    barLinear cs q (leftInv cs q⁻¹ i v) = left cs (q : C) i (barLinear cs q v) := by
+  have h := congrArg (fun f : Module.End C (Space W C) => f (leftInv cs q⁻¹ i v))
+    (barLinear_left cs q i)
+  simp only [Module.End.mul_apply] at h
+  have hv : left cs (↑q⁻¹ : C) i (leftInv cs q⁻¹ i v) = v := by
+    rw [← Module.End.mul_apply, left_mul_leftInv, Module.End.one_apply]
+  rw [hv] at h
+  have h' := congrArg (left cs (q : C) i) h
+  have hl (t : Space W C) : left cs (q : C) i (leftInv cs q i t) = t := by
+    rw [← Module.End.mul_apply, left_mul_leftInv, Module.End.one_apply]
+  simpa only [hl] using h'.symm
+
+/-- Applying coefficient inversion and the Hecke bar twice is the identity. -/
+theorem barLinear_barBasis (cs : CoxeterSystem M W) (q : Cˣ) (w : W) :
+    barLinear cs q (barBasis cs q⁻¹ w) = basis w := by
+  obtain ⟨ω, hω, rfl⟩ := cs.exists_isReduced w
+  induction ω with
+  | nil => simp
+  | cons i ω ih =>
+    have hω' : cs.IsReduced ω := by simpa using hω.drop 1
+    have hasc : cs.length (cs.wordProd ω) < cs.length (cs.simple i * cs.wordProd ω) := by
+      have := hω.eq
+      simp only [cs.wordProd_cons, List.length_cons, ← hω'.eq] at this
+      omega
+    rw [cs.wordProd_cons, barBasis_simple_mul_ascent cs q⁻¹ i _ hasc,
+      barLinear_leftInv, ih hω', left_basis, ite_eq_right (by omega)]
+
+theorem barLinear_inverse (cs : CoxeterSystem M W) (q : Cˣ) :
+    barLinear cs q * barLinear cs q⁻¹ = 1 := by
+  apply end_ext
+  intro w
+  simp [Module.End.mul_apply, barLinear_barBasis]
+
+@[simp] theorem leftInv_basis (cs : CoxeterSystem M W) (q : Cˣ) (i : B) (w : W) :
+    leftInv cs q i (basis w) =
+      if cs.length (cs.simple i * w) < cs.length w then basis (cs.simple i * w)
+      else (↑q⁻¹ : C) • basis (cs.simple i * w) + ((↑q⁻¹ : C) - 1) • basis w := by
+  rw [leftInv_apply, left_basis]
+  split_ifs
+  · simp [smul_smul]
+  · rw [smul_sub, smul_smul]
+    have hscalar : (↑q⁻¹ : C) * ((q : C) - 1) = 1 - (↑q⁻¹ : C) := by
+      rw [mul_sub, q.inv_mul, mul_one]
+    rw [hscalar, sub_smul, one_smul]
+    module
+
+theorem leftInv_coeff (cs : CoxeterSystem M W) (q : Cˣ) (i : B) (v : Space W C) (x : W) :
+    leftInv cs q i v x =
+      if cs.length (cs.simple i * x) < cs.length x then (↑q⁻¹ : C) * v (cs.simple i * x)
+      else v (cs.simple i * x) + ((↑q⁻¹ : C) - 1) * v x := by
+  classical
+  induction v using Finsupp.induction_linear with
+  | zero => simp
+  | add v v' hv hv' =>
+    simp only [map_add, Finsupp.add_apply, hv, hv']
+    split_ifs <;> ring
+  | single w c =>
+    have hc : Finsupp.single w c = c • (basis w : Space W C) := by
+      simp [basis, Finsupp.smul_single]
+    rw [hc, map_smul, leftInv_basis]
+    by_cases h : w = cs.simple i * x
+    · subst w
+      have hd : cs.length (cs.simple i * (cs.simple i * x)) < cs.length (cs.simple i * x) ↔
+          ¬ cs.length (cs.simple i * x) < cs.length x := by
+        simp only [cs.simple_mul_simple_cancel_left]
+        have := cs.length_simple_mul x i
+        omega
+      have hn : cs.simple i * x ≠ x := by
+        intro he
+        have := cs.length_simple_mul x i
+        rw [he] at this
+        omega
+      simp only [Finsupp.smul_apply, smul_eq_mul, basis, Finsupp.single_apply,
+        cs.simple_mul_simple_cancel_left, hn, ite_false]
+      have hlength := cs.length_simple_mul x i
+      split_ifs <;> try omega
+      all_goals simp [Ne.symm hn] <;> try ring
+    · have hn : cs.simple i * w ≠ x := by
+        intro he
+        apply h
+        rw [← he]
+        simp
+      by_cases hw : w = x
+      · subst w
+        simp only [Finsupp.smul_apply, smul_eq_mul, basis, Finsupp.single_apply,
+          Ne.symm hn, ite_false, ]
+        split_ifs <;> simp [Ne.symm hn] ; try ring
+      · simp only [Finsupp.smul_apply, smul_eq_mul, basis, Finsupp.single_apply,
+          h, hw, ite_false]
+        split_ifs <;> simp [hn, hw]
+
+/-- Triangularity in the actual Bruhat order. -/
+theorem barBasis_zero (cs : CoxeterSystem M W) (q : Cˣ) (x y : W)
+    (hxy : ¬BruhatLE cs x y) : barBasis cs q y x = 0 := by
+  obtain ⟨ω, hω, rfl⟩ := cs.exists_isReduced y
+  induction ω generalizing x with
+  | nil =>
+    have hx : x ≠ 1 := fun h => hxy (h ▸ bruhat_refl cs 1)
+    simp [basis, Ne.symm hx]
+  | cons i ω ih =>
+    have hω' : cs.IsReduced ω := by simpa using hω.drop 1
+    have hasc : cs.length (cs.wordProd ω) < cs.length (cs.simple i * cs.wordProd ω) := by
+      have := hω.eq
+      simp only [cs.wordProd_cons, List.length_cons, ← hω'.eq] at this
+      omega
+    have hle : BruhatLE cs (cs.wordProd ω) (cs.simple i * cs.wordProd ω) :=
+      Relation.ReflTransGen.single ⟨hasc, cs.simple i, cs.isReflection_simple i, rfl⟩
+    have hn : ¬BruhatLE cs x (cs.wordProd ω) := fun h => hxy (h.trans hle)
+    have hn' : ¬BruhatLE cs (cs.simple i * x) (cs.wordProd ω) := by
+      intro h
+      have := Hecke.simple_mul_le_ascent cs i h hasc
+      simp only [cs.simple_mul_simple_cancel_left] at this
+      exact hxy this
+    rw [cs.wordProd_cons, barBasis_simple_mul_ascent cs q i _ hasc, leftInv_coeff,
+      ih x hω' hn, ih (cs.simple i * x) hω' hn']
+    split_ifs <;> simp
+
+@[simp] theorem barBasis_diagonal (cs : CoxeterSystem M W) (q : Cˣ) (y : W) :
+    barBasis cs q y y = (↑q⁻¹ : C) ^ cs.length y := by
+  obtain ⟨ω, hω, rfl⟩ := cs.exists_isReduced y
+  induction ω with
+  | nil => simp [basis]
+  | cons i ω ih =>
+    have hω' : cs.IsReduced ω := by simpa using hω.drop 1
+    have hasc : cs.length (cs.wordProd ω) < cs.length (cs.simple i * cs.wordProd ω) := by
+      have := hω.eq
+      simp only [cs.wordProd_cons, List.length_cons, ← hω'.eq] at this
+      omega
+    have hlen : cs.length (cs.simple i * cs.wordProd ω) = cs.length (cs.wordProd ω) + 1 := by
+      have := cs.length_simple_mul (cs.wordProd ω) i
+      omega
+    rw [cs.wordProd_cons, barBasis_simple_mul_ascent cs q i _ hasc, leftInv_coeff]
+    rw [cs.simple_mul_simple_cancel_left, ite_eq_left hasc, ih hω', hlen, pow_succ]
+    ring
+
+/-- R-value extracted from the genuine Hecke bar, with the classical sign
+and leading q-power. No recurrence is imposed in this definition. -/
+noncomputable def rValue (cs : CoxeterSystem M W) (q : Cˣ) (x y : W) : C :=
+  (-1 : C) ^ cs.length x * (-(q : C)) ^ cs.length y * barBasis cs q y x
+
+@[simp] theorem rValue_diagonal (cs : CoxeterSystem M W) (q : Cˣ) (x : W) :
+    rValue cs q x x = 1 := by
+  rw [rValue, barBasis_diagonal, ← mul_pow, ← mul_pow]
+  simp
+
+theorem rValue_zero (cs : CoxeterSystem M W) (q : Cˣ) (x y : W)
+    (hxy : ¬BruhatLE cs x y) : rValue cs q x y = 0 := by
+  rw [rValue, barBasis_zero cs q x y hxy, mul_zero]
+
+/-- The classical R recursion holds for *every* left descent, by Hecke
+operator independence. This is the consistency obligation for actual R. -/
+theorem rValue_recursion (cs : CoxeterSystem M W) (q : Cˣ) (x y : W) (i : B)
+    (hy : cs.length (cs.simple i * y) < cs.length y) :
+    rValue cs q x y =
+      if cs.length (cs.simple i * x) < cs.length x then
+        rValue cs q (cs.simple i * x) (cs.simple i * y)
+      else ((q : C) - 1) * rValue cs q x (cs.simple i * y) +
+        (q : C) * rValue cs q (cs.simple i * x) (cs.simple i * y) := by
+  have hasc : cs.length (cs.simple i * y) <
+      cs.length (cs.simple i * (cs.simple i * y)) := by simpa using hy
+  have hb := barBasis_simple_mul_ascent cs q i (cs.simple i * y) hasc
+  simp only [cs.simple_mul_simple_cancel_left] at hb
+  have hylen : cs.length y = cs.length (cs.simple i * y) + 1 := by
+    have := cs.length_simple_mul y i
+    omega
+  have hu : (q : C) * (↑q⁻¹ : C) = 1 := q.val_inv
+  by_cases hx : cs.length (cs.simple i * x) < cs.length x
+  · have hxlen : cs.length x = cs.length (cs.simple i * x) + 1 := by
+      have := cs.length_simple_mul x i
+      omega
+    rw [ite_eq_left hx]
+    simp only [rValue, hb, leftInv_coeff, ite_eq_left hx]
+    rw [hxlen, hylen]
+    simp only [pow_succ]
+    linear_combination ((-1 : C) ^ cs.length (cs.simple i * x) *
+      (-(q : C)) ^ cs.length (cs.simple i * y) *
+      barBasis cs q (cs.simple i * y) (cs.simple i * x)) * hu
+  · have hxlen : cs.length (cs.simple i * x) = cs.length x + 1 := by
+      have := cs.length_simple_mul x i
+      omega
+    rw [ite_eq_right hx]
+    simp only [rValue, hb, leftInv_coeff, ite_eq_right hx]
+    rw [hxlen, hylen]
+    simp only [pow_succ]
+    linear_combination -((-1 : C) ^ cs.length x *
+      (-(q : C)) ^ cs.length (cs.simple i * y) *
+      barBasis cs q (cs.simple i * y) x) * hu
+
+end HeckeQ
+
+noncomputable local instance : DecidableEq W := Classical.decEq W
+
+/-- A concrete finite lower ideal of the genuine Bruhat order. -/
+noncomputable def lowerFinset (cs : CoxeterSystem M W) (y : W) : Finset W :=
+  (finite_bruhat_lower cs y).toFinset
+
+@[simp] theorem mem_lowerFinset (cs : CoxeterSystem M W) (y x : W) :
+    x ∈ lowerFinset cs y ↔ BruhatLE cs x y := by
+  classical
+  exact Set.Finite.mem_toFinset _
+
+namespace HeckeQ
+
+open Hecke (Space basis)
+variable {C : Type*} [CommRing C]
+
+theorem bar_coeff_inverse (cs : CoxeterSystem M W) (q : Cˣ) (x y : W) :
+    ∑ z ∈ lowerFinset cs y, barBasis cs q z x * barBasis cs q⁻¹ y z =
+      if x = y then 1 else 0 := by
+  classical
+  have h := congrArg (fun v : Space W C => v x) (barLinear_barBasis cs q y)
+  rw [barLinear, Finsupp.linearCombination_apply, Finsupp.sum_apply] at h
+  have hs : (barBasis cs q⁻¹ y).support ⊆ lowerFinset cs y := by
+    intro z hz
+    apply (mem_lowerFinset cs y z).mpr
+    by_contra hn
+    exact Finsupp.mem_support_iff.mp hz (barBasis_zero cs _ _ _ hn)
+  rw [Finsupp.sum_of_support_subset _ hs _ (by simp), basis, Finsupp.single_apply] at h
+  simpa only [Finsupp.smul_apply, smul_eq_mul, mul_comm, eq_comm] using h
+
+theorem rValue_inverse_factor (cs : CoxeterSystem M W) (q : Cˣ) (x z y : W) :
+    rValue cs q x z * (q : C) ^ cs.length y * (↑q⁻¹ : C) ^ cs.length z *
+        rValue cs q⁻¹ z y =
+      ((-1 : C) ^ cs.length x * (-1 : C) ^ cs.length y) *
+        (barBasis cs q z x * barBasis cs q⁻¹ y z) := by
+  simp only [rValue]
+  rw [neg_pow (q : C), neg_pow (↑q⁻¹ : C)]
+  calc
+    _ = ((-1 : C) ^ cs.length x * (-1 : C) ^ cs.length y) *
+        (((-1 : C) * (-1)) ^ cs.length z *
+          ((q : C) * (↑q⁻¹ : C)) ^ cs.length z *
+          ((q : C) * (↑q⁻¹ : C)) ^ cs.length y) *
+        (barBasis cs q z x * barBasis cs q⁻¹ y z) := by simp only [mul_pow]; ring
+    _ = _ := by simp
+
+/-- The exact R-kernel identity, before specialization to polynomials. -/
+theorem rValue_inverse (cs : CoxeterSystem M W) (q : Cˣ) (x y : W) :
+    ∑ z ∈ lowerFinset cs y,
+      rValue cs q x z * (q : C) ^ cs.length y * (↑q⁻¹ : C) ^ cs.length z *
+        rValue cs q⁻¹ z y = if x = y then 1 else 0 := by
+  classical
+  simp_rw [rValue_inverse_factor]
+  rw [← Finset.mul_sum, bar_coeff_inverse]
+  split_ifs with h
+  · subst y
+    rw [← mul_pow]
+    simp
+  · simp
+
+end HeckeQ
+
+/-- Reflection of a bounded-degree polynomial evaluated at an actual unit. -/
+theorem eval₂_reflect_unit {C : Type*} [CommRing C] (q : Cˣ)
+    (p : ℤ[X]) (d : ℕ) (hp : p.natDegree ≤ d) :
+    Polynomial.eval₂ (Int.castRingHom C) (q : C) (reflect d p) =
+      (q : C) ^ d * Polynomial.eval₂ (Int.castRingHom C) (↑q⁻¹ : C) p := by
+  let := q⁻¹.invertible
+  have h := Polynomial.eval₂_reflect_mul_pow (Int.castRingHom C) (↑q⁻¹ : C) d p hp
+  simp only [invOf_units, inv_inv] at h
+  have h' := congrArg (fun a : C => a * (q : C) ^ d) h
+  calc
+    _ = Polynomial.eval₂ (Int.castRingHom C) (↑q⁻¹ : C) p * (q : C) ^ d := by
+      simpa only [mul_assoc, ← mul_pow, q.inv_mul, one_pow, mul_one] using h'
+    _ = _ := mul_comm _ _
+
+
+end KLInvariance
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/KazhdanLusztig/HilbertSeries.lean b/lean/OAI/RepresentationTheory/KazhdanLusztig/HilbertSeries.lean
index 207c05c8f0a0833430f51e44657085a5ac9effc4..fec32f6f2cf1d39a3ea209d0ec4671e8079c164b 100644
--- a/lean/OAI/RepresentationTheory/KazhdanLusztig/HilbertSeries.lean
+++ b/lean/OAI/RepresentationTheory/KazhdanLusztig/HilbertSeries.lean
@@ -1,553 +1,553 @@
-import OAI.RepresentationTheory.KazhdanLusztig.GradedSheaves
-
-namespace OAI
-
-/-!
-Hilbert series of graded modules and boundary sheaves, positivity and strict detection.
--/
-
-section
-
-namespace KLInvariance.Hilbert
-
-/-- Sum over all monomials in finitely many variables, before grouping by degree. -/
-theorem hasSum_monomials (σ : Type*) [Fintype σ] {q : ℝ} (hq : 0 ≤ q) (hq1 : q < 1) :
-    HasSum (fun d : σ → ℕ => q ^ (∑ i, d i)) ((1-q)⁻¹ ^ Fintype.card σ) := by
-  classical
-  apply Fintype.induction_empty_option (P := fun σ _ =>
-    HasSum (fun d : σ → ℕ => q ^ (∑ i, d i)) ((1-q)⁻¹ ^ Fintype.card σ))
-  · intro α β _ e ih
-    let : Fintype α := Fintype.ofEquiv β e.symm
-    rw [← Fintype.card_congr e]
-    apply (Equiv.piCongrLeft (fun _ : β => ℕ) e).hasSum_iff.mp
-    convert ih using 1
-    ext d
-    simp only [Function.comp_apply, Equiv.piCongrLeft_apply]
-    congr 1
-    simpa using e.symm.sum_comp d
-  · simp
-  · intro α _ ih
-    have hg := hasSum_geometric_of_lt_one hq hq1
-    have hs := hg.summable.mul_of_nonneg ih.summable
-      (fun _ => pow_nonneg hq _) (fun _ => pow_nonneg hq _)
-    have hh := hg.mul ih hs
-    have he := (Equiv.piOptionEquivProd (β := fun _ : Option α => ℕ)).hasSum_iff.mpr hh
-    simpa [Function.comp_def, Fintype.sum_option, pow_add, pow_succ, mul_comm] using he
-
-
-lemma intWeight_eq_natWeight {σ : Type*} (d : σ →₀ ℕ) :
-    Finsupp.weight (fun _ : σ => (1 : ℤ)) d =
-      (Finsupp.weight (fun _ : σ => (1 : ℕ)) d : ℕ) := by
-  simp [Finsupp.weight_apply, Finsupp.sum, Nat.cast_sum]
-
-lemma polynomialGrading_nat (k : Type*) [Field k] (σ : Type*) (n : ℕ) :
-    Graded.polynomialGrading k σ n =
-      MvPolynomial.weightedHomogeneousSubmodule k (fun _ : σ => (1 : ℕ)) n := by
-  ext p
-  change (∀ d, p.coeff d ≠ 0 →
-    Finsupp.weight (fun _ : σ => (1 : ℤ)) d = (n : ℤ)) ↔ _
-  simp only [intWeight_eq_natWeight, Nat.cast_inj]
-  rfl
-
-instance polynomialPiece_finite (k : Type*) [Field k] (σ : Type*) [Finite σ] (n : ℤ) :
-    Module.Finite k (Graded.polynomialGrading k σ n) := by
-  by_cases hn : 0 ≤ n
-  · obtain ⟨m, rfl⟩ := Int.eq_ofNat_of_zero_le hn
-    rw [polynomialGrading_nat]
-    exact Module.Finite.iff_fg.mpr
-      (MvPolynomial.weightedHomogeneousSubmodule_fg k (fun _ : σ => 1) (by simp) m)
-  · rw [Graded.polynomialGrading_negative k σ n (lt_of_not_ge hn)]
-    infer_instance
-
-lemma polynomial_finrank_nat (k : Type*) [Field k] (σ : Type*) (n : ℕ) :
-    Module.finrank k (Graded.polynomialGrading k σ n) =
-      Nat.card {d : σ →₀ ℕ // Finsupp.weight (fun _ => (1 : ℕ)) d = n} := by
-  rw [polynomialGrading_nat, MvPolynomial.weightedHomogeneousSubmodule_eq_finsupp_supported]
-  exact Module.finrank_eq_nat_card_basis (MvPolynomial.basisRestrictSupport k _)
-
-lemma hasSum_finsupp_monomials (σ : Type*) [Fintype σ] {q : ℝ}
-    (hq : 0 ≤ q) (hq1 : q < 1) :
-    HasSum (fun d : σ →₀ ℕ => q ^ Finsupp.weight (fun _ => (1 : ℕ)) d)
-      ((1-q)⁻¹ ^ Fintype.card σ) := by
-  have h := (Finsupp.equivFunOnFinite (α := σ) (M := ℕ)).hasSum_iff.mpr
-    (hasSum_monomials σ hq hq1)
-  convert h using 1
-  ext d
-  congr 1
-  simp only [Finsupp.weight_apply, smul_eq_mul, mul_one]
-  exact Finsupp.sum_fintype d _ (by simp)
-
-/-- The actual finite polynomial ring has Hilbert series `(1-q)⁻ᴺ`. -/
-theorem hasSum_polynomial_nat (k : Type*) [Field k] (σ : Type*) [Fintype σ]
-    {q : ℝ} (hq : 0 ≤ q) (hq1 : q < 1) :
-    HasSum (fun n : ℕ => (Module.finrank k (Graded.polynomialGrading k σ n) : ℝ) * q^n)
-      ((1-q)⁻¹ ^ Fintype.card σ) := by
-  have h := (hasSum_finsupp_monomials σ hq hq1).tsum_fiberwise
-    (Finsupp.weight (fun _ : σ => (1 : ℕ)))
-  apply h.congr_fun
-  intro n
-  rw [polynomial_finrank_nat]
-  have hf : (fun d : ↑((Finsupp.weight (R := ℕ) (fun _ : σ => (1 : ℕ))) ⁻¹' {n}) =>
-      q ^ (Finsupp.weight (fun _ : σ => (1 : ℕ))) d.val) = fun _ => q^n := by
-    funext d
-    rw [show Finsupp.weight (fun _ : σ => (1 : ℕ)) d.val = n from d.property]
-  rw [hf, tsum_const, nsmul_eq_mul]
-  simp only [Set.preimage, Set.mem_singleton_iff]
-  rfl
-
-
-/-- Integer-indexed grading, including its zero negative pieces. -/
-theorem hasSum_polynomial (k : Type*) [Field k] (σ : Type*) [Fintype σ]
-    {q : ℝ} (hq : 0 ≤ q) (hq1 : q < 1) :
-    HasSum (fun n : ℤ => (Module.finrank k (Graded.polynomialGrading k σ n) : ℝ) * q^n)
-      ((1-q)⁻¹ ^ Fintype.card σ) := by
-  apply ((Nat.cast_injective : Function.Injective ((↑) : ℕ → ℤ)).hasSum_iff ?_).mp
-  · simpa only [Function.comp_def, zpow_natCast] using hasSum_polynomial_nat k σ hq hq1
-  · intro n hn
-    have hn' : n < 0 := by
-      by_contra h
-      obtain ⟨m, rfl⟩ := Int.eq_ofNat_of_zero_le (le_of_not_gt h)
-      exact hn ⟨m, rfl⟩
-    rw [Graded.polynomialGrading_negative k σ n hn']
-    simp
-
-end KLInvariance.Hilbert
-
-
-end
-
-
-section
-
-namespace KLInvariance.Hilbert
-universe uk ua um ui
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-
-/-- Degree pieces of a shifted finite free module, as a genuine linear equivalence. -/
-def shiftedPieceEquiv {ι : Type ui} (𝓐 : ℤ → Submodule k A) (d : ι → ℤ) (n : ℤ) :
-    Graded.shiftedPiece 𝓐 d n ≃ₗ[k] (∀ i, 𝓐 (n-d i)) where
-  toFun v i := ⟨v.val i, v.property i⟩
-  invFun v := ⟨fun i => v i, fun i => (v i).property⟩
-  left_inv _ := rfl
-  right_inv _ := rfl
-  map_add' _ _ := rfl
-  map_smul' _ _ := rfl
-
-instance shiftedPiece_finite {ι : Type ui} [Finite ι]
-    (𝓐 : ℤ → Submodule k A) [∀ n, Module.Finite k (𝓐 n)] (d : ι → ℤ) (n : ℤ) :
-    Module.Finite k (Graded.shiftedPiece 𝓐 d n) :=
-  Module.Finite.of_surjective (shiftedPieceEquiv 𝓐 d n).symm.toLinearMap
-    (shiftedPieceEquiv 𝓐 d n).symm.surjective
-
-lemma shiftedPiece_finrank {ι : Type ui} [Fintype ι]
-    (𝓐 : ℤ → Submodule k A) [∀ n, Module.Finite k (𝓐 n)] (d : ι → ℤ) (n : ℤ) :
-    Module.finrank k (Graded.shiftedPiece 𝓐 d n) = ∑ i, Module.finrank k (𝓐 (n-d i)) := by
-  rw [(shiftedPieceEquiv 𝓐 d n).finrank_eq, Module.finrank_pi_fintype]
-
-lemma hasSum_shifted_polynomial (k : Type*) [Field k] (σ : Type*) [Fintype σ]
-    (d : ℤ) {q : ℝ} (hq : 0 < q) (hq1 : q < 1) :
-    HasSum (fun n : ℤ => (Module.finrank k (Graded.polynomialGrading k σ (n-d)) : ℝ) * q^n)
-      (q^d * (1-q)⁻¹ ^ Fintype.card σ) := by
-  apply (Equiv.addRight d).hasSum_iff.mp
-  have h := (hasSum_polynomial k σ hq.le hq1).mul_left (q^d)
-  apply h.congr_fun
-  intro n
-  change (Module.finrank k (Graded.polynomialGrading k σ (n+d-d)) : ℝ) * q^(n+d) = _
-  rw [add_sub_cancel_right, zpow_add₀ hq.ne']
-  ring
-
-/-- Convergent Hilbert series and its normalization for the actual shifted free module. -/
-theorem hasSum_shiftedFree (k : Type*) [Field k] (σ : Type*) [Fintype σ]
-    {ι : Type*} [Fintype ι] (d : ι → ℤ) {q : ℝ} (hq : 0 < q) (hq1 : q < 1) :
-    HasSum (fun n : ℤ =>
-      (Module.finrank k (Graded.shiftedPiece (Graded.polynomialGrading k σ) d n) : ℝ) * q^n)
-      ((∑ i, q^(d i)) * (1-q)⁻¹ ^ Fintype.card σ) := by
-  have h := hasSum_sum (s := Finset.univ)
-    (fun i _ => hasSum_shifted_polynomial k σ (d i) hq hq1)
-  rw [← Finset.sum_mul] at h
-  apply h.congr_fun
-  intro n
-  rw [shiftedPiece_finrank, Nat.cast_sum, Finset.sum_mul]
-
-
-section FiniteModule
-variable {M : Type um} [AddCommGroup M] [Module k M] [Module A M] [IsScalarTower k A M]
-  {N : Type*} [AddCommGroup N] [Module k N] [Module A N] [IsScalarTower k A N]
-
-/-- Any finite graded module admits a finite homogeneous free cover. No projectivity is assumed. -/
-theorem exists_finite_graded_cover [Module.Finite A M]
-    (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M) [DirectSum.Decomposition 𝓜]
-    [SetLike.GradedSMul 𝓐 𝓜] :
-    ∃ (ι : Type um), Finite ι ∧ ∃ (d : ι → ℤ) (f : (ι → A) →ₗ[A] M),
-      Function.Surjective f ∧ ∀ n v, v ∈ Graded.shiftedPiece 𝓐 d n → f v ∈ 𝓜 n := by
-  classical
-  obtain ⟨t,ht,hd⟩ := Graded.exists_finite_homogeneous_generators (A := A) 𝓜
-  choose d hd using fun i : t => hd i.val i.property
-  have hspan : Submodule.span A (Set.range fun i : t => i.val) = ⊤ := by simpa using ht
-  refine ⟨t, inferInstance, d, Fintype.linearCombination A (fun i : t => i.val), ?_, ?_⟩
-  · intro m
-    exact (Submodule.mem_span_range_iff_exists_fun A).mp (hspan ▸ Submodule.mem_top)
-  · intro n v hv
-    change (∑ i, v i • (i : M)) ∈ 𝓜 n
-    apply Submodule.sum_mem
-    intro i _
-    simpa using SetLike.GradedSMul.smul_mem (hv i) (hd i)
-
-def degreeMap (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
-    (f : M →ₗ[A] N) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 n) (n : ℤ) :
-    𝓜 n →ₗ[k] 𝓝 n :=
-  LinearMap.codRestrict (𝓝 n) ((f.restrictScalars k).comp (𝓜 n).subtype)
-    (fun m => hf n m m.property)
-
-lemma degreeMap_surjective
-    (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
-    [DirectSum.Decomposition 𝓜] [DirectSum.Decomposition 𝓝]
-    (f : M →ₗ[A] N) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 n)
-    (hs : Function.Surjective f) (n : ℤ) : Function.Surjective (degreeMap 𝓜 𝓝 f hf n) := by
-  intro m
-  obtain ⟨v,hv⟩ := hs m.val
-  refine ⟨DirectSum.decompose 𝓜 v n, ?_⟩
-  apply Subtype.ext
-  change f (Graded.component 𝓜 n v) = m.val
-  have h := Graded.map_component 𝓜 𝓝 f 0 (by simpa using hf) n v
-  rw [sub_zero, hv] at h
-  exact h.symm.trans (DirectSum.decompose_of_mem_same 𝓝 m.property)
-end FiniteModule
-
-lemma term_top {M : Type um} [AddCommGroup M] [Module k M]
-    (𝓜 : ℤ → Submodule k M) (q : ℝ) (n : ℤ) :
-    term 𝓜 ⊤ q n = (Module.finrank k (𝓜 n) : ℝ)*q^n := by
-  rw [term, piece, Submodule.comap_top, finrank_top]
-
-section PolynomialModule
-variable (k : Type uk) [Field k] (σ : Type ua) [Fintype σ]
-  {M : Type um} [AddCommGroup M] [Module k M] [Module (MvPolynomial σ k) M]
-  [IsScalarTower k (MvPolynomial σ k) M] [Module.Finite (MvPolynomial σ k) M]
-  (𝓜 : ℤ → Submodule k M) [DirectSum.Decomposition 𝓜]
-  [SetLike.GradedSMul (Graded.polynomialGrading k σ) 𝓜]
-
-include σ
-
-/-- Finiteness of every genuine homogeneous piece, not an extra assumption on the Hilbert series. -/
-theorem polynomialModule_piece_finite (n : ℤ) : Module.Finite k (𝓜 n) := by
-  classical
-  obtain ⟨ι,hι,d,f,hs,hf⟩ := exists_finite_graded_cover (Graded.polynomialGrading k σ) 𝓜
-  let : Finite ι := hι
-  let : Fintype ι := Fintype.ofFinite ι
-  let : DirectSum.Decomposition (Graded.shiftedPiece (Graded.polynomialGrading k σ) d) :=
-    (Graded.shifted_isInternal (Graded.polynomialGrading k σ) d).chooseDecomposition
-  exact Module.Finite.of_surjective
-    (degreeMap (Graded.shiftedPiece (Graded.polynomialGrading k σ) d) 𝓜 f hf n)
-    (degreeMap_surjective _ _ f hf hs n)
-
-/-- Every finite graded module over the actual finite polynomial ring has a convergent
-Hilbert series at every `0<q<1`, even when it is not free. -/
-theorem summable_polynomialModule {q : ℝ} (hq : 0 < q) (hq1 : q < 1) :
-    Summable (fun n : ℤ => (Module.finrank k (𝓜 n) : ℝ)*q^n) := by
-  classical
-  obtain ⟨ι,hι,d,f,hs,hf⟩ := exists_finite_graded_cover (Graded.polynomialGrading k σ) 𝓜
-  let : Finite ι := hι
-  let : Fintype ι := Fintype.ofFinite ι
-  let : DirectSum.Decomposition (Graded.shiftedPiece (Graded.polynomialGrading k σ) d) :=
-    (Graded.shifted_isInternal (Graded.polynomialGrading k σ) d).chooseDecomposition
-  apply (hasSum_shiftedFree k σ d hq hq1).summable.of_nonneg_of_le
-  · intro n
-    exact mul_nonneg (Nat.cast_nonneg _) (zpow_nonneg hq.le _)
-  · intro n
-    apply mul_le_mul_of_nonneg_right _ (zpow_nonneg hq.le _)
-    exact_mod_cast LinearMap.finrank_le_finrank_of_surjective
-      (degreeMap_surjective (Graded.shiftedPiece (Graded.polynomialGrading k σ) d) 𝓜 f hf hs n)
-
-/-- Convergence for every submodule in the fixed finite graded ambient module. -/
-theorem summable_term (S : Submodule k M) {q : ℝ} (hq : 0 < q) (hq1 : q < 1) :
-    Summable (term 𝓜 S q) := by
-  let : ∀ n, Module.Finite k (𝓜 n) := polynomialModule_piece_finite k σ 𝓜
-  have heq : term 𝓜 ⊤ q = fun n : ℤ => (Module.finrank k (𝓜 n) : ℝ)*q^n :=
-    funext (term_top 𝓜 q)
-  have htop : Summable (term 𝓜 ⊤ q) := by
-    rw [heq]
-    exact summable_polynomialModule k σ 𝓜 hq hq1
-  exact summable_of_le 𝓜 le_top hq htop
-
-/-- Global detection, now with all the manuscript's finite-polynomial-ring hypotheses supplied. -/
-theorem polynomial_eq_of_value_eq {S T : Submodule k M}
-    (h : S ≤ T) (hhom : ∀ n m, m ∈ T → Graded.component 𝓜 n m ∈ T)
-    {q : ℝ} (hq : 0 < q) (hq1 : q < 1)
-    (hv : value 𝓜 S q = value 𝓜 T q) : S = T := by
-  let : ∀ n, Module.Finite k (𝓜 n) := polynomialModule_piece_finite k σ 𝓜
-  exact eq_of_value_eq 𝓜 h hhom hq (summable_term k σ 𝓜 T hq hq1) hv
-end PolynomialModule
-
-section HomogeneousBasis
-variable {σ : Type ua} [Fintype σ]
-  {M : Type um} [AddCommGroup M] [Module k M] [Module (MvPolynomial σ k) M]
-  [IsScalarTower k (MvPolynomial σ k) M]
-  (𝓜 : ℤ → Submodule k M) [DirectSum.Decomposition 𝓜]
-  [SetLike.GradedSMul (Graded.polynomialGrading k σ) 𝓜]
-  {ι : Type ui} [Fintype ι] (b : Module.Basis ι (MvPolynomial σ k) M)
-  (d : ι → ℤ) (hd : ∀ i, b i ∈ 𝓜 (d i))
-
-include hd
-
-omit [Fintype σ] [IsScalarTower k (MvPolynomial σ k) M] [DirectSum.Decomposition 𝓜] in
-lemma basis_equivFun_symm_graded (n : ℤ) (v : ι → MvPolynomial σ k)
-    (hv : v ∈ Graded.shiftedPiece (Graded.polynomialGrading k σ) d n) :
-    b.equivFun.symm v ∈ 𝓜 n := by
-  rw [b.equivFun_symm_apply]
-  apply Submodule.sum_mem
-  intro i _
-  simpa using SetLike.GradedSMul.smul_mem (hv i) (hd i)
-
-lemma finrank_homogeneous_basis (n : ℤ) :
-    Module.finrank k (𝓜 n) =
-      ∑ i, Module.finrank k (Graded.polynomialGrading k σ (n-d i)) := by
-  let : DirectSum.Decomposition (Graded.shiftedPiece (Graded.polynomialGrading k σ) d) :=
-    (Graded.shifted_isInternal (Graded.polynomialGrading k σ) d).chooseDecomposition
-  let f := degreeMap (Graded.shiftedPiece (Graded.polynomialGrading k σ) d) 𝓜
-    b.equivFun.symm.toLinearMap (basis_equivFun_symm_graded 𝓜 b d hd) n
-  have hf : Function.Bijective f := by
-    constructor
-    · intro x y h
-      apply Subtype.ext
-      apply b.equivFun.symm.injective
-      exact congrArg Subtype.val h
-    · exact degreeMap_surjective _ _ b.equivFun.symm.toLinearMap
-        (basis_equivFun_symm_graded 𝓜 b d hd) b.equivFun.symm.surjective n
-  rw [← (LinearEquiv.ofBijective f hf).finrank_eq]
-  exact shiftedPiece_finrank _ d n
-
-/-- The precise Hilbert normalization for a finite homogeneous basis, with the source's
-convention that a generator in degree `d` contributes `q^d`. -/
-theorem hasSum_homogeneous_basis {q : ℝ} (hq : 0 < q) (hq1 : q < 1) :
-    HasSum (fun n : ℤ => (Module.finrank k (𝓜 n) : ℝ)*q^n)
-      ((∑ i, q^(d i)) * (1-q)⁻¹ ^ Fintype.card σ) := by
-  apply (hasSum_shiftedFree k σ d hq hq1).congr_fun
-  intro n
-  rw [finrank_homogeneous_basis 𝓜 b d hd, shiftedPiece_finrank]
-end HomogeneousBasis
-
-
-end KLInvariance.Hilbert
-
-
-end
-
-
-section
-
-namespace KLInvariance.Hilbert
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um un
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {M : Type um} [AddCommGroup M] [Module k M] [Module A M] [IsScalarTower k A M]
-  {N : Type un} [AddCommGroup N] [Module k N] [Module A N] [IsScalarTower k A N]
-
-/-- The actual degreewise kernel, not merely an equality of dimensions. -/
-def degreeKernelEquiv (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
-    (f : M →ₗ[A] N) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 n) (n : ℤ) :
-    subPiece 𝓜 (LinearMap.ker f) n ≃ₗ[k] LinearMap.ker (degreeMap 𝓜 𝓝 f hf n) where
-  toFun m := ⟨⟨m.val.val,m.property⟩, by
-    apply Subtype.ext
-    exact m.val.property⟩
-  invFun m := ⟨⟨m.val.val, by
-    have h := congrArg Subtype.val m.property
-    exact h⟩,m.val.property⟩
-  left_inv _ := rfl
-  right_inv _ := rfl
-  map_add' _ _ := rfl
-  map_smul' _ _ := rfl
-
- theorem finrank_exact
-    (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
-    [DirectSum.Decomposition 𝓜] [DirectSum.Decomposition 𝓝]
-    [∀ n, Module.Finite k (𝓜 n)]
-    (f : M →ₗ[A] N) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 n)
-    (hs : Function.Surjective f) (n : ℤ) :
-    Module.finrank k (𝓜 n) = Module.finrank k (subPiece 𝓜 (LinearMap.ker f) n) +
-      Module.finrank k (𝓝 n) := by
-  have h := LinearMap.finrank_range_add_finrank_ker (degreeMap 𝓜 𝓝 f hf n)
-  rw [LinearMap.range_eq_top.mpr (degreeMap_surjective 𝓜 𝓝 f hf hs n),finrank_top,
-    ← (degreeKernelEquiv 𝓜 𝓝 f hf n).finrank_eq] at h
-  omega
-
-
- theorem value_exact
-    (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
-    [DirectSum.Decomposition 𝓜] [DirectSum.Decomposition 𝓝]
-    [∀ n, Module.Finite k (𝓜 n)]
-    (f : M →ₗ[A] N) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 n)
-    (hs : Function.Surjective f) (q : ℝ)
-    (hK : Summable (term (subPiece 𝓜 (LinearMap.ker f)) ⊤ q))
-    (hN : Summable (term 𝓝 ⊤ q)) :
-    value 𝓜 ⊤ q = value (subPiece 𝓜 (LinearMap.ker f)) ⊤ q + value 𝓝 ⊤ q := by
-  have ht : term 𝓜 ⊤ q = fun n => term (subPiece 𝓜 (LinearMap.ker f)) ⊤ q n +
-      term 𝓝 ⊤ q n := by
-    funext n
-    rw [term_top,term_top,term_top,finrank_exact 𝓜 𝓝 f hf hs n,Nat.cast_add,add_mul]
-  rw [value,ht,hK.tsum_add hN]
-  rfl
-
-/-- Finite degreewise exactness supplies convergence of both terms; no
-free-module hypothesis is used. -/
- theorem value_exact_of_summable
-    (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
-    [DirectSum.Decomposition 𝓜] [DirectSum.Decomposition 𝓝]
-    [∀ n, Module.Finite k (𝓜 n)]
-    (f : M →ₗ[A] N) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 n)
-    (hs : Function.Surjective f) {q : ℝ} (hq : 0 < q)
-    (hM : Summable (term 𝓜 ⊤ q)) :
-    value 𝓜 ⊤ q = value (subPiece 𝓜 (LinearMap.ker f)) ⊤ q + value 𝓝 ⊤ q := by
-  apply value_exact 𝓜 𝓝 f hf hs q
-  · apply hM.of_nonneg_of_le
-    · intro n
-      rw [term_top]
-      positivity
-    · intro n
-      rw [term_top,term_top]
-      apply mul_le_mul_of_nonneg_right _ (zpow_nonneg hq.le _)
-      exact_mod_cast (show Module.finrank k (subPiece 𝓜 (LinearMap.ker f) n) ≤
-        Module.finrank k (𝓜 n) by rw [finrank_exact 𝓜 𝓝 f hf hs n]; omega)
-  · apply hM.of_nonneg_of_le
-    · intro n
-      rw [term_top]
-      positivity
-    · intro n
-      rw [term_top,term_top]
-      apply mul_le_mul_of_nonneg_right _ (zpow_nonneg hq.le _)
-      exact_mod_cast (show Module.finrank k (𝓝 n) ≤ Module.finrank k (𝓜 n) by
-        rw [finrank_exact 𝓜 𝓝 f hf hs n]; omega)
-
-end KLInvariance.Hilbert
-
-end
-
-
-section
-
-namespace KLInvariance.Hilbert
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um un
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {M : Type um} [AddCommGroup M] [Module k M] [Module A M] [IsScalarTower k A M]
-  {N : Type un} [AddCommGroup N] [Module k N] [Module A N] [IsScalarTower k A N]
-
- def subPieceEquiv (𝓜 : ℤ → Submodule k M) (S : Submodule A M) (n : ℤ) :
-    subPiece 𝓜 S n ≃ₗ[k] piece 𝓜 (S.restrictScalars k) n where
-  toFun v := ⟨⟨v.val.val,v.property⟩,v.val.property⟩
-  invFun v := ⟨⟨v.val.val,v.property⟩,v.val.property⟩
-  left_inv _ := rfl
-  right_inv _ := rfl
-  map_add' _ _ := rfl
-  map_smul' _ _ := rfl
-
- theorem term_subPiece (𝓜 : ℤ → Submodule k M) (S : Submodule A M) (q : ℝ) (n : ℤ) :
-    term (subPiece 𝓜 S) ⊤ q n = term 𝓜 (S.restrictScalars k) q n := by
-  rw [term_top,term,(subPieceEquiv 𝓜 S n).finrank_eq]
-
- theorem value_subPiece (𝓜 : ℤ → Submodule k M) (S : Submodule A M) (q : ℝ) :
-    value (subPiece 𝓜 S) ⊤ q = value 𝓜 (S.restrictScalars k) q := by
-  exact tsum_congr (term_subPiece 𝓜 S q)
-
- def imagePieceMap (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
-    (S : Submodule A M) (f : M →ₗ[A] N) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 n)
-    (n : ℤ) : subPiece 𝓜 S n →ₗ[k] subPiece 𝓝 (S.map f) n where
-  toFun v := ⟨⟨f v.val.val,Submodule.mem_map.mpr ⟨_,v.val.property,rfl⟩⟩,
-    hf n v.val.val v.property⟩
-  map_add' _ _ := by ext; exact map_add f _ _
-  map_smul' c v := by ext; exact (f.restrictScalars k).map_smul c v.val.val
-
- theorem imagePieceMap_surjective
-    (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
-    [DirectSum.Decomposition 𝓜] [DirectSum.Decomposition 𝓝]
-    (S : Submodule A M) (hS : ∀ n m, m ∈ S → component 𝓜 n m ∈ S)
-    (f : M →ₗ[A] N) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 n) (n : ℤ) :
-    Function.Surjective (imagePieceMap 𝓜 𝓝 S f hf n) := by
-  rintro ⟨⟨m,hm⟩,hgrad⟩
-  obtain ⟨v,hv,rfl⟩ := Submodule.mem_map.mp hm
-  refine ⟨⟨⟨component 𝓜 n v,hS n v hv⟩,(DirectSum.decompose 𝓜 v n).property⟩,?_⟩
-  apply Subtype.ext
-  apply Subtype.ext
-  change f (component 𝓜 n v) = f v
-  have hh := map_component 𝓜 𝓝 f 0 (by simpa using hf) n v
-  rw [sub_zero] at hh
-  rw [← hh]
-  exact DirectSum.decompose_of_mem_same 𝓝 hgrad
-
- def imageKernelEquiv (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
-    (S : Submodule A M) (f : M →ₗ[A] N) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 n)
-    (n : ℤ) : subPiece 𝓜 (S ⊓ LinearMap.ker f) n ≃ₗ[k]
-      LinearMap.ker (imagePieceMap 𝓜 𝓝 S f hf n) where
-  toFun v := ⟨⟨⟨v.val.val,v.val.property.1⟩,v.property⟩,by
-    apply Subtype.ext
-    apply Subtype.ext
-    exact v.val.property.2⟩
-  invFun v := ⟨⟨v.val.val.val,⟨v.val.val.property,by
-    have h := congrArg (fun z => z.val.val) v.property
-    exact h⟩⟩,v.val.property⟩
-  left_inv _ := rfl
-  right_inv _ := rfl
-  map_add' _ _ := rfl
-  map_smul' _ _ := rfl
-
- theorem finrank_image_exact
-    (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
-    [DirectSum.Decomposition 𝓜] [DirectSum.Decomposition 𝓝]
-    [∀ n, Module.Finite k (𝓜 n)]
-    (S : Submodule A M) (hS : ∀ n m, m ∈ S → component 𝓜 n m ∈ S)
-    (f : M →ₗ[A] N) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 n) (n : ℤ) :
-    Module.finrank k (piece 𝓜 (S.restrictScalars k) n) =
-      Module.finrank k (piece 𝓜 ((S ⊓ LinearMap.ker f).restrictScalars k) n) +
-      Module.finrank k (piece 𝓝 ((S.map f).restrictScalars k) n) := by
-  let : Module.Finite k (subPiece 𝓜 S n) := Module.Finite.of_surjective
-    (subPieceEquiv 𝓜 S n).symm.toLinearMap (subPieceEquiv 𝓜 S n).symm.surjective
-  have h := LinearMap.finrank_range_add_finrank_ker (imagePieceMap 𝓜 𝓝 S f hf n)
-  rw [LinearMap.range_eq_top.mpr (imagePieceMap_surjective 𝓜 𝓝 S hS f hf n),finrank_top,
-    ← (imageKernelEquiv 𝓜 𝓝 S f hf n).finrank_eq,
-    (subPieceEquiv 𝓜 S n).finrank_eq,
-    (subPieceEquiv 𝓜 (S ⊓ LinearMap.ker f) n).finrank_eq,
-    (subPieceEquiv 𝓝 (S.map f) n).finrank_eq] at h
-  omega
-
-/-- Removing a kernel condition adds exactly the Hilbert value of its
-embedded restriction image. No freeness of the source kernel is used. -/
- theorem value_image_exact
-    (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
-    [DirectSum.Decomposition 𝓜] [DirectSum.Decomposition 𝓝]
-    [∀ n, Module.Finite k (𝓜 n)]
-    (S : Submodule A M) (hS : ∀ n m, m ∈ S → component 𝓜 n m ∈ S)
-    (f : M →ₗ[A] N) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 n)
-    {q : ℝ} (hq : 0 < q) (hs : Summable (term 𝓜 (S.restrictScalars k) q)) :
-    value 𝓜 (S.restrictScalars k) q =
-      value 𝓜 ((S ⊓ LinearMap.ker f).restrictScalars k) q +
-      value 𝓝 ((S.map f).restrictScalars k) q := by
-  have ht (n : ℤ) : term 𝓜 (S.restrictScalars k) q n =
-      term 𝓜 ((S ⊓ LinearMap.ker f).restrictScalars k) q n +
-      term 𝓝 ((S.map f).restrictScalars k) q n := by
-    unfold term
-    rw [finrank_image_exact 𝓜 𝓝 S hS f hf n,Nat.cast_add,add_mul]
-  have hk : Summable (term 𝓜 ((S ⊓ LinearMap.ker f).restrictScalars k) q) := by
-    apply hs.of_nonneg_of_le
-    · exact fun n => term_nonneg _ _ hq n
-    · intro n
-      rw [ht n]
-      exact le_add_of_nonneg_right (term_nonneg _ _ hq n)
-  have hi : Summable (term 𝓝 ((S.map f).restrictScalars k) q) := by
-    apply hs.of_nonneg_of_le
-    · exact fun n => term_nonneg _ _ hq n
-    · intro n
-      rw [ht n]
-      exact le_add_of_nonneg_left (term_nonneg _ _ hq n)
-  unfold value
-  rw [show term 𝓜 (S.restrictScalars k) q = _ from funext ht,hk.tsum_add hi]
-
-end KLInvariance.Hilbert
-
-end
-
-
-section
-
-
-end
-
-end OAI
+import OAI.RepresentationTheory.KazhdanLusztig.GradedSheaves
+
+namespace OAI
+
+/-!
+Hilbert series of graded modules and boundary sheaves, positivity and strict detection.
+-/
+
+section
+
+namespace KLInvariance.Hilbert
+
+/-- Sum over all monomials in finitely many variables, before grouping by degree. -/
+theorem hasSum_monomials (σ : Type*) [Fintype σ] {q : ℝ} (hq : 0 ≤ q) (hq1 : q < 1) :
+    HasSum (fun d : σ → ℕ => q ^ (∑ i, d i)) ((1-q)⁻¹ ^ Fintype.card σ) := by
+  classical
+  apply Fintype.induction_empty_option (P := fun σ _ =>
+    HasSum (fun d : σ → ℕ => q ^ (∑ i, d i)) ((1-q)⁻¹ ^ Fintype.card σ))
+  · intro α β _ e ih
+    let : Fintype α := Fintype.ofEquiv β e.symm
+    rw [← Fintype.card_congr e]
+    apply (Equiv.piCongrLeft (fun _ : β => ℕ) e).hasSum_iff.mp
+    convert ih using 1
+    ext d
+    simp only [Function.comp_apply, Equiv.piCongrLeft_apply]
+    congr 1
+    simpa using e.symm.sum_comp d
+  · simp
+  · intro α _ ih
+    have hg := hasSum_geometric_of_lt_one hq hq1
+    have hs := hg.summable.mul_of_nonneg ih.summable
+      (fun _ => pow_nonneg hq _) (fun _ => pow_nonneg hq _)
+    have hh := hg.mul ih hs
+    have he := (Equiv.piOptionEquivProd (β := fun _ : Option α => ℕ)).hasSum_iff.mpr hh
+    simpa [Function.comp_def, Fintype.sum_option, pow_add, pow_succ, mul_comm] using he
+
+
+lemma intWeight_eq_natWeight {σ : Type*} (d : σ →₀ ℕ) :
+    Finsupp.weight (fun _ : σ => (1 : ℤ)) d =
+      (Finsupp.weight (fun _ : σ => (1 : ℕ)) d : ℕ) := by
+  simp [Finsupp.weight_apply, Finsupp.sum, Nat.cast_sum]
+
+lemma polynomialGrading_nat (k : Type*) [Field k] (σ : Type*) (n : ℕ) :
+    Graded.polynomialGrading k σ n =
+      MvPolynomial.weightedHomogeneousSubmodule k (fun _ : σ => (1 : ℕ)) n := by
+  ext p
+  change (∀ d, p.coeff d ≠ 0 →
+    Finsupp.weight (fun _ : σ => (1 : ℤ)) d = (n : ℤ)) ↔ _
+  simp only [intWeight_eq_natWeight, Nat.cast_inj]
+  rfl
+
+instance polynomialPiece_finite (k : Type*) [Field k] (σ : Type*) [Finite σ] (n : ℤ) :
+    Module.Finite k (Graded.polynomialGrading k σ n) := by
+  by_cases hn : 0 ≤ n
+  · obtain ⟨m, rfl⟩ := Int.eq_ofNat_of_zero_le hn
+    rw [polynomialGrading_nat]
+    exact Module.Finite.iff_fg.mpr
+      (MvPolynomial.weightedHomogeneousSubmodule_fg k (fun _ : σ => 1) (by simp) m)
+  · rw [Graded.polynomialGrading_negative k σ n (lt_of_not_ge hn)]
+    infer_instance
+
+lemma polynomial_finrank_nat (k : Type*) [Field k] (σ : Type*) (n : ℕ) :
+    Module.finrank k (Graded.polynomialGrading k σ n) =
+      Nat.card {d : σ →₀ ℕ // Finsupp.weight (fun _ => (1 : ℕ)) d = n} := by
+  rw [polynomialGrading_nat, MvPolynomial.weightedHomogeneousSubmodule_eq_finsupp_supported]
+  exact Module.finrank_eq_nat_card_basis (MvPolynomial.basisRestrictSupport k _)
+
+lemma hasSum_finsupp_monomials (σ : Type*) [Fintype σ] {q : ℝ}
+    (hq : 0 ≤ q) (hq1 : q < 1) :
+    HasSum (fun d : σ →₀ ℕ => q ^ Finsupp.weight (fun _ => (1 : ℕ)) d)
+      ((1-q)⁻¹ ^ Fintype.card σ) := by
+  have h := (Finsupp.equivFunOnFinite (α := σ) (M := ℕ)).hasSum_iff.mpr
+    (hasSum_monomials σ hq hq1)
+  convert h using 1
+  ext d
+  congr 1
+  simp only [Finsupp.weight_apply, smul_eq_mul, mul_one]
+  exact Finsupp.sum_fintype d _ (by simp)
+
+/-- The actual finite polynomial ring has Hilbert series `(1-q)⁻ᴺ`. -/
+theorem hasSum_polynomial_nat (k : Type*) [Field k] (σ : Type*) [Fintype σ]
+    {q : ℝ} (hq : 0 ≤ q) (hq1 : q < 1) :
+    HasSum (fun n : ℕ => (Module.finrank k (Graded.polynomialGrading k σ n) : ℝ) * q^n)
+      ((1-q)⁻¹ ^ Fintype.card σ) := by
+  have h := (hasSum_finsupp_monomials σ hq hq1).tsum_fiberwise
+    (Finsupp.weight (fun _ : σ => (1 : ℕ)))
+  apply h.congr_fun
+  intro n
+  rw [polynomial_finrank_nat]
+  have hf : (fun d : ↑((Finsupp.weight (R := ℕ) (fun _ : σ => (1 : ℕ))) ⁻¹' {n}) =>
+      q ^ (Finsupp.weight (fun _ : σ => (1 : ℕ))) d.val) = fun _ => q^n := by
+    funext d
+    rw [show Finsupp.weight (fun _ : σ => (1 : ℕ)) d.val = n from d.property]
+  rw [hf, tsum_const, nsmul_eq_mul]
+  simp only [Set.preimage, Set.mem_singleton_iff]
+  rfl
+
+
+/-- Integer-indexed grading, including its zero negative pieces. -/
+theorem hasSum_polynomial (k : Type*) [Field k] (σ : Type*) [Fintype σ]
+    {q : ℝ} (hq : 0 ≤ q) (hq1 : q < 1) :
+    HasSum (fun n : ℤ => (Module.finrank k (Graded.polynomialGrading k σ n) : ℝ) * q^n)
+      ((1-q)⁻¹ ^ Fintype.card σ) := by
+  apply ((Nat.cast_injective : Function.Injective ((↑) : ℕ → ℤ)).hasSum_iff ?_).mp
+  · simpa only [Function.comp_def, zpow_natCast] using hasSum_polynomial_nat k σ hq hq1
+  · intro n hn
+    have hn' : n < 0 := by
+      by_contra h
+      obtain ⟨m, rfl⟩ := Int.eq_ofNat_of_zero_le (le_of_not_gt h)
+      exact hn ⟨m, rfl⟩
+    rw [Graded.polynomialGrading_negative k σ n hn']
+    simp
+
+end KLInvariance.Hilbert
+
+
+end
+
+
+section
+
+namespace KLInvariance.Hilbert
+universe uk ua um ui
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+
+/-- Degree pieces of a shifted finite free module, as a genuine linear equivalence. -/
+def shiftedPieceEquiv {ι : Type ui} (𝓐 : ℤ → Submodule k A) (d : ι → ℤ) (n : ℤ) :
+    Graded.shiftedPiece 𝓐 d n ≃ₗ[k] (∀ i, 𝓐 (n-d i)) where
+  toFun v i := ⟨v.val i, v.property i⟩
+  invFun v := ⟨fun i => v i, fun i => (v i).property⟩
+  left_inv _ := rfl
+  right_inv _ := rfl
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+
+instance shiftedPiece_finite {ι : Type ui} [Finite ι]
+    (𝓐 : ℤ → Submodule k A) [∀ n, Module.Finite k (𝓐 n)] (d : ι → ℤ) (n : ℤ) :
+    Module.Finite k (Graded.shiftedPiece 𝓐 d n) :=
+  Module.Finite.of_surjective (shiftedPieceEquiv 𝓐 d n).symm.toLinearMap
+    (shiftedPieceEquiv 𝓐 d n).symm.surjective
+
+lemma shiftedPiece_finrank {ι : Type ui} [Fintype ι]
+    (𝓐 : ℤ → Submodule k A) [∀ n, Module.Finite k (𝓐 n)] (d : ι → ℤ) (n : ℤ) :
+    Module.finrank k (Graded.shiftedPiece 𝓐 d n) = ∑ i, Module.finrank k (𝓐 (n-d i)) := by
+  rw [(shiftedPieceEquiv 𝓐 d n).finrank_eq, Module.finrank_pi_fintype]
+
+lemma hasSum_shifted_polynomial (k : Type*) [Field k] (σ : Type*) [Fintype σ]
+    (d : ℤ) {q : ℝ} (hq : 0 < q) (hq1 : q < 1) :
+    HasSum (fun n : ℤ => (Module.finrank k (Graded.polynomialGrading k σ (n-d)) : ℝ) * q^n)
+      (q^d * (1-q)⁻¹ ^ Fintype.card σ) := by
+  apply (Equiv.addRight d).hasSum_iff.mp
+  have h := (hasSum_polynomial k σ hq.le hq1).mul_left (q^d)
+  apply h.congr_fun
+  intro n
+  change (Module.finrank k (Graded.polynomialGrading k σ (n+d-d)) : ℝ) * q^(n+d) = _
+  rw [add_sub_cancel_right, zpow_add₀ hq.ne']
+  ring
+
+/-- Convergent Hilbert series and its normalization for the actual shifted free module. -/
+theorem hasSum_shiftedFree (k : Type*) [Field k] (σ : Type*) [Fintype σ]
+    {ι : Type*} [Fintype ι] (d : ι → ℤ) {q : ℝ} (hq : 0 < q) (hq1 : q < 1) :
+    HasSum (fun n : ℤ =>
+      (Module.finrank k (Graded.shiftedPiece (Graded.polynomialGrading k σ) d n) : ℝ) * q^n)
+      ((∑ i, q^(d i)) * (1-q)⁻¹ ^ Fintype.card σ) := by
+  have h := hasSum_sum (s := Finset.univ)
+    (fun i _ => hasSum_shifted_polynomial k σ (d i) hq hq1)
+  rw [← Finset.sum_mul] at h
+  apply h.congr_fun
+  intro n
+  rw [shiftedPiece_finrank, Nat.cast_sum, Finset.sum_mul]
+
+
+section FiniteModule
+variable {M : Type um} [AddCommGroup M] [Module k M] [Module A M] [IsScalarTower k A M]
+  {N : Type*} [AddCommGroup N] [Module k N] [Module A N] [IsScalarTower k A N]
+
+/-- Any finite graded module admits a finite homogeneous free cover. No projectivity is assumed. -/
+theorem exists_finite_graded_cover [Module.Finite A M]
+    (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M) [DirectSum.Decomposition 𝓜]
+    [SetLike.GradedSMul 𝓐 𝓜] :
+    ∃ (ι : Type um), Finite ι ∧ ∃ (d : ι → ℤ) (f : (ι → A) →ₗ[A] M),
+      Function.Surjective f ∧ ∀ n v, v ∈ Graded.shiftedPiece 𝓐 d n → f v ∈ 𝓜 n := by
+  classical
+  obtain ⟨t,ht,hd⟩ := Graded.exists_finite_homogeneous_generators (A := A) 𝓜
+  choose d hd using fun i : t => hd i.val i.property
+  have hspan : Submodule.span A (Set.range fun i : t => i.val) = ⊤ := by simpa using ht
+  refine ⟨t, inferInstance, d, Fintype.linearCombination A (fun i : t => i.val), ?_, ?_⟩
+  · intro m
+    exact (Submodule.mem_span_range_iff_exists_fun A).mp (hspan ▸ Submodule.mem_top)
+  · intro n v hv
+    change (∑ i, v i • (i : M)) ∈ 𝓜 n
+    apply Submodule.sum_mem
+    intro i _
+    simpa using SetLike.GradedSMul.smul_mem (hv i) (hd i)
+
+def degreeMap (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
+    (f : M →ₗ[A] N) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 n) (n : ℤ) :
+    𝓜 n →ₗ[k] 𝓝 n :=
+  LinearMap.codRestrict (𝓝 n) ((f.restrictScalars k).comp (𝓜 n).subtype)
+    (fun m => hf n m m.property)
+
+lemma degreeMap_surjective
+    (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
+    [DirectSum.Decomposition 𝓜] [DirectSum.Decomposition 𝓝]
+    (f : M →ₗ[A] N) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 n)
+    (hs : Function.Surjective f) (n : ℤ) : Function.Surjective (degreeMap 𝓜 𝓝 f hf n) := by
+  intro m
+  obtain ⟨v,hv⟩ := hs m.val
+  refine ⟨DirectSum.decompose 𝓜 v n, ?_⟩
+  apply Subtype.ext
+  change f (Graded.component 𝓜 n v) = m.val
+  have h := Graded.map_component 𝓜 𝓝 f 0 (by simpa using hf) n v
+  rw [sub_zero, hv] at h
+  exact h.symm.trans (DirectSum.decompose_of_mem_same 𝓝 m.property)
+end FiniteModule
+
+lemma term_top {M : Type um} [AddCommGroup M] [Module k M]
+    (𝓜 : ℤ → Submodule k M) (q : ℝ) (n : ℤ) :
+    term 𝓜 ⊤ q n = (Module.finrank k (𝓜 n) : ℝ)*q^n := by
+  rw [term, piece, Submodule.comap_top, finrank_top]
+
+section PolynomialModule
+variable (k : Type uk) [Field k] (σ : Type ua) [Fintype σ]
+  {M : Type um} [AddCommGroup M] [Module k M] [Module (MvPolynomial σ k) M]
+  [IsScalarTower k (MvPolynomial σ k) M] [Module.Finite (MvPolynomial σ k) M]
+  (𝓜 : ℤ → Submodule k M) [DirectSum.Decomposition 𝓜]
+  [SetLike.GradedSMul (Graded.polynomialGrading k σ) 𝓜]
+
+include σ
+
+/-- Finiteness of every genuine homogeneous piece, not an extra assumption on the Hilbert series. -/
+theorem polynomialModule_piece_finite (n : ℤ) : Module.Finite k (𝓜 n) := by
+  classical
+  obtain ⟨ι,hι,d,f,hs,hf⟩ := exists_finite_graded_cover (Graded.polynomialGrading k σ) 𝓜
+  let : Finite ι := hι
+  let : Fintype ι := Fintype.ofFinite ι
+  let : DirectSum.Decomposition (Graded.shiftedPiece (Graded.polynomialGrading k σ) d) :=
+    (Graded.shifted_isInternal (Graded.polynomialGrading k σ) d).chooseDecomposition
+  exact Module.Finite.of_surjective
+    (degreeMap (Graded.shiftedPiece (Graded.polynomialGrading k σ) d) 𝓜 f hf n)
+    (degreeMap_surjective _ _ f hf hs n)
+
+/-- Every finite graded module over the actual finite polynomial ring has a convergent
+Hilbert series at every `0<q<1`, even when it is not free. -/
+theorem summable_polynomialModule {q : ℝ} (hq : 0 < q) (hq1 : q < 1) :
+    Summable (fun n : ℤ => (Module.finrank k (𝓜 n) : ℝ)*q^n) := by
+  classical
+  obtain ⟨ι,hι,d,f,hs,hf⟩ := exists_finite_graded_cover (Graded.polynomialGrading k σ) 𝓜
+  let : Finite ι := hι
+  let : Fintype ι := Fintype.ofFinite ι
+  let : DirectSum.Decomposition (Graded.shiftedPiece (Graded.polynomialGrading k σ) d) :=
+    (Graded.shifted_isInternal (Graded.polynomialGrading k σ) d).chooseDecomposition
+  apply (hasSum_shiftedFree k σ d hq hq1).summable.of_nonneg_of_le
+  · intro n
+    exact mul_nonneg (Nat.cast_nonneg _) (zpow_nonneg hq.le _)
+  · intro n
+    apply mul_le_mul_of_nonneg_right _ (zpow_nonneg hq.le _)
+    exact_mod_cast LinearMap.finrank_le_finrank_of_surjective
+      (degreeMap_surjective (Graded.shiftedPiece (Graded.polynomialGrading k σ) d) 𝓜 f hf hs n)
+
+/-- Convergence for every submodule in the fixed finite graded ambient module. -/
+theorem summable_term (S : Submodule k M) {q : ℝ} (hq : 0 < q) (hq1 : q < 1) :
+    Summable (term 𝓜 S q) := by
+  let : ∀ n, Module.Finite k (𝓜 n) := polynomialModule_piece_finite k σ 𝓜
+  have heq : term 𝓜 ⊤ q = fun n : ℤ => (Module.finrank k (𝓜 n) : ℝ)*q^n :=
+    funext (term_top 𝓜 q)
+  have htop : Summable (term 𝓜 ⊤ q) := by
+    rw [heq]
+    exact summable_polynomialModule k σ 𝓜 hq hq1
+  exact summable_of_le 𝓜 le_top hq htop
+
+/-- Global detection, now with all the manuscript's finite-polynomial-ring hypotheses supplied. -/
+theorem polynomial_eq_of_value_eq {S T : Submodule k M}
+    (h : S ≤ T) (hhom : ∀ n m, m ∈ T → Graded.component 𝓜 n m ∈ T)
+    {q : ℝ} (hq : 0 < q) (hq1 : q < 1)
+    (hv : value 𝓜 S q = value 𝓜 T q) : S = T := by
+  let : ∀ n, Module.Finite k (𝓜 n) := polynomialModule_piece_finite k σ 𝓜
+  exact eq_of_value_eq 𝓜 h hhom hq (summable_term k σ 𝓜 T hq hq1) hv
+end PolynomialModule
+
+section HomogeneousBasis
+variable {σ : Type ua} [Fintype σ]
+  {M : Type um} [AddCommGroup M] [Module k M] [Module (MvPolynomial σ k) M]
+  [IsScalarTower k (MvPolynomial σ k) M]
+  (𝓜 : ℤ → Submodule k M) [DirectSum.Decomposition 𝓜]
+  [SetLike.GradedSMul (Graded.polynomialGrading k σ) 𝓜]
+  {ι : Type ui} [Fintype ι] (b : Module.Basis ι (MvPolynomial σ k) M)
+  (d : ι → ℤ) (hd : ∀ i, b i ∈ 𝓜 (d i))
+
+include hd
+
+omit [Fintype σ] [IsScalarTower k (MvPolynomial σ k) M] [DirectSum.Decomposition 𝓜] in
+lemma basis_equivFun_symm_graded (n : ℤ) (v : ι → MvPolynomial σ k)
+    (hv : v ∈ Graded.shiftedPiece (Graded.polynomialGrading k σ) d n) :
+    b.equivFun.symm v ∈ 𝓜 n := by
+  rw [b.equivFun_symm_apply]
+  apply Submodule.sum_mem
+  intro i _
+  simpa using SetLike.GradedSMul.smul_mem (hv i) (hd i)
+
+lemma finrank_homogeneous_basis (n : ℤ) :
+    Module.finrank k (𝓜 n) =
+      ∑ i, Module.finrank k (Graded.polynomialGrading k σ (n-d i)) := by
+  let : DirectSum.Decomposition (Graded.shiftedPiece (Graded.polynomialGrading k σ) d) :=
+    (Graded.shifted_isInternal (Graded.polynomialGrading k σ) d).chooseDecomposition
+  let f := degreeMap (Graded.shiftedPiece (Graded.polynomialGrading k σ) d) 𝓜
+    b.equivFun.symm.toLinearMap (basis_equivFun_symm_graded 𝓜 b d hd) n
+  have hf : Function.Bijective f := by
+    constructor
+    · intro x y h
+      apply Subtype.ext
+      apply b.equivFun.symm.injective
+      exact congrArg Subtype.val h
+    · exact degreeMap_surjective _ _ b.equivFun.symm.toLinearMap
+        (basis_equivFun_symm_graded 𝓜 b d hd) b.equivFun.symm.surjective n
+  rw [← (LinearEquiv.ofBijective f hf).finrank_eq]
+  exact shiftedPiece_finrank _ d n
+
+/-- The precise Hilbert normalization for a finite homogeneous basis, with the source's
+convention that a generator in degree `d` contributes `q^d`. -/
+theorem hasSum_homogeneous_basis {q : ℝ} (hq : 0 < q) (hq1 : q < 1) :
+    HasSum (fun n : ℤ => (Module.finrank k (𝓜 n) : ℝ)*q^n)
+      ((∑ i, q^(d i)) * (1-q)⁻¹ ^ Fintype.card σ) := by
+  apply (hasSum_shiftedFree k σ d hq hq1).congr_fun
+  intro n
+  rw [finrank_homogeneous_basis 𝓜 b d hd, shiftedPiece_finrank]
+end HomogeneousBasis
+
+
+end KLInvariance.Hilbert
+
+
+end
+
+
+section
+
+namespace KLInvariance.Hilbert
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um un
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {M : Type um} [AddCommGroup M] [Module k M] [Module A M] [IsScalarTower k A M]
+  {N : Type un} [AddCommGroup N] [Module k N] [Module A N] [IsScalarTower k A N]
+
+/-- The actual degreewise kernel, not merely an equality of dimensions. -/
+def degreeKernelEquiv (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
+    (f : M →ₗ[A] N) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 n) (n : ℤ) :
+    subPiece 𝓜 (LinearMap.ker f) n ≃ₗ[k] LinearMap.ker (degreeMap 𝓜 𝓝 f hf n) where
+  toFun m := ⟨⟨m.val.val,m.property⟩, by
+    apply Subtype.ext
+    exact m.val.property⟩
+  invFun m := ⟨⟨m.val.val, by
+    have h := congrArg Subtype.val m.property
+    exact h⟩,m.val.property⟩
+  left_inv _ := rfl
+  right_inv _ := rfl
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+
+ theorem finrank_exact
+    (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
+    [DirectSum.Decomposition 𝓜] [DirectSum.Decomposition 𝓝]
+    [∀ n, Module.Finite k (𝓜 n)]
+    (f : M →ₗ[A] N) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 n)
+    (hs : Function.Surjective f) (n : ℤ) :
+    Module.finrank k (𝓜 n) = Module.finrank k (subPiece 𝓜 (LinearMap.ker f) n) +
+      Module.finrank k (𝓝 n) := by
+  have h := LinearMap.finrank_range_add_finrank_ker (degreeMap 𝓜 𝓝 f hf n)
+  rw [LinearMap.range_eq_top.mpr (degreeMap_surjective 𝓜 𝓝 f hf hs n),finrank_top,
+    ← (degreeKernelEquiv 𝓜 𝓝 f hf n).finrank_eq] at h
+  omega
+
+
+ theorem value_exact
+    (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
+    [DirectSum.Decomposition 𝓜] [DirectSum.Decomposition 𝓝]
+    [∀ n, Module.Finite k (𝓜 n)]
+    (f : M →ₗ[A] N) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 n)
+    (hs : Function.Surjective f) (q : ℝ)
+    (hK : Summable (term (subPiece 𝓜 (LinearMap.ker f)) ⊤ q))
+    (hN : Summable (term 𝓝 ⊤ q)) :
+    value 𝓜 ⊤ q = value (subPiece 𝓜 (LinearMap.ker f)) ⊤ q + value 𝓝 ⊤ q := by
+  have ht : term 𝓜 ⊤ q = fun n => term (subPiece 𝓜 (LinearMap.ker f)) ⊤ q n +
+      term 𝓝 ⊤ q n := by
+    funext n
+    rw [term_top,term_top,term_top,finrank_exact 𝓜 𝓝 f hf hs n,Nat.cast_add,add_mul]
+  rw [value,ht,hK.tsum_add hN]
+  rfl
+
+/-- Finite degreewise exactness supplies convergence of both terms; no
+free-module hypothesis is used. -/
+ theorem value_exact_of_summable
+    (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
+    [DirectSum.Decomposition 𝓜] [DirectSum.Decomposition 𝓝]
+    [∀ n, Module.Finite k (𝓜 n)]
+    (f : M →ₗ[A] N) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 n)
+    (hs : Function.Surjective f) {q : ℝ} (hq : 0 < q)
+    (hM : Summable (term 𝓜 ⊤ q)) :
+    value 𝓜 ⊤ q = value (subPiece 𝓜 (LinearMap.ker f)) ⊤ q + value 𝓝 ⊤ q := by
+  apply value_exact 𝓜 𝓝 f hf hs q
+  · apply hM.of_nonneg_of_le
+    · intro n
+      rw [term_top]
+      positivity
+    · intro n
+      rw [term_top,term_top]
+      apply mul_le_mul_of_nonneg_right _ (zpow_nonneg hq.le _)
+      exact_mod_cast (show Module.finrank k (subPiece 𝓜 (LinearMap.ker f) n) ≤
+        Module.finrank k (𝓜 n) by rw [finrank_exact 𝓜 𝓝 f hf hs n]; omega)
+  · apply hM.of_nonneg_of_le
+    · intro n
+      rw [term_top]
+      positivity
+    · intro n
+      rw [term_top,term_top]
+      apply mul_le_mul_of_nonneg_right _ (zpow_nonneg hq.le _)
+      exact_mod_cast (show Module.finrank k (𝓝 n) ≤ Module.finrank k (𝓜 n) by
+        rw [finrank_exact 𝓜 𝓝 f hf hs n]; omega)
+
+end KLInvariance.Hilbert
+
+end
+
+
+section
+
+namespace KLInvariance.Hilbert
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um un
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {M : Type um} [AddCommGroup M] [Module k M] [Module A M] [IsScalarTower k A M]
+  {N : Type un} [AddCommGroup N] [Module k N] [Module A N] [IsScalarTower k A N]
+
+ def subPieceEquiv (𝓜 : ℤ → Submodule k M) (S : Submodule A M) (n : ℤ) :
+    subPiece 𝓜 S n ≃ₗ[k] piece 𝓜 (S.restrictScalars k) n where
+  toFun v := ⟨⟨v.val.val,v.property⟩,v.val.property⟩
+  invFun v := ⟨⟨v.val.val,v.property⟩,v.val.property⟩
+  left_inv _ := rfl
+  right_inv _ := rfl
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+
+ theorem term_subPiece (𝓜 : ℤ → Submodule k M) (S : Submodule A M) (q : ℝ) (n : ℤ) :
+    term (subPiece 𝓜 S) ⊤ q n = term 𝓜 (S.restrictScalars k) q n := by
+  rw [term_top,term,(subPieceEquiv 𝓜 S n).finrank_eq]
+
+ theorem value_subPiece (𝓜 : ℤ → Submodule k M) (S : Submodule A M) (q : ℝ) :
+    value (subPiece 𝓜 S) ⊤ q = value 𝓜 (S.restrictScalars k) q := by
+  exact tsum_congr (term_subPiece 𝓜 S q)
+
+ def imagePieceMap (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
+    (S : Submodule A M) (f : M →ₗ[A] N) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 n)
+    (n : ℤ) : subPiece 𝓜 S n →ₗ[k] subPiece 𝓝 (S.map f) n where
+  toFun v := ⟨⟨f v.val.val,Submodule.mem_map.mpr ⟨_,v.val.property,rfl⟩⟩,
+    hf n v.val.val v.property⟩
+  map_add' _ _ := by ext; exact map_add f _ _
+  map_smul' c v := by ext; exact (f.restrictScalars k).map_smul c v.val.val
+
+ theorem imagePieceMap_surjective
+    (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
+    [DirectSum.Decomposition 𝓜] [DirectSum.Decomposition 𝓝]
+    (S : Submodule A M) (hS : ∀ n m, m ∈ S → component 𝓜 n m ∈ S)
+    (f : M →ₗ[A] N) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 n) (n : ℤ) :
+    Function.Surjective (imagePieceMap 𝓜 𝓝 S f hf n) := by
+  rintro ⟨⟨m,hm⟩,hgrad⟩
+  obtain ⟨v,hv,rfl⟩ := Submodule.mem_map.mp hm
+  refine ⟨⟨⟨component 𝓜 n v,hS n v hv⟩,(DirectSum.decompose 𝓜 v n).property⟩,?_⟩
+  apply Subtype.ext
+  apply Subtype.ext
+  change f (component 𝓜 n v) = f v
+  have hh := map_component 𝓜 𝓝 f 0 (by simpa using hf) n v
+  rw [sub_zero] at hh
+  rw [← hh]
+  exact DirectSum.decompose_of_mem_same 𝓝 hgrad
+
+ def imageKernelEquiv (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
+    (S : Submodule A M) (f : M →ₗ[A] N) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 n)
+    (n : ℤ) : subPiece 𝓜 (S ⊓ LinearMap.ker f) n ≃ₗ[k]
+      LinearMap.ker (imagePieceMap 𝓜 𝓝 S f hf n) where
+  toFun v := ⟨⟨⟨v.val.val,v.val.property.1⟩,v.property⟩,by
+    apply Subtype.ext
+    apply Subtype.ext
+    exact v.val.property.2⟩
+  invFun v := ⟨⟨v.val.val.val,⟨v.val.val.property,by
+    have h := congrArg (fun z => z.val.val) v.property
+    exact h⟩⟩,v.val.property⟩
+  left_inv _ := rfl
+  right_inv _ := rfl
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+
+ theorem finrank_image_exact
+    (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
+    [DirectSum.Decomposition 𝓜] [DirectSum.Decomposition 𝓝]
+    [∀ n, Module.Finite k (𝓜 n)]
+    (S : Submodule A M) (hS : ∀ n m, m ∈ S → component 𝓜 n m ∈ S)
+    (f : M →ₗ[A] N) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 n) (n : ℤ) :
+    Module.finrank k (piece 𝓜 (S.restrictScalars k) n) =
+      Module.finrank k (piece 𝓜 ((S ⊓ LinearMap.ker f).restrictScalars k) n) +
+      Module.finrank k (piece 𝓝 ((S.map f).restrictScalars k) n) := by
+  let : Module.Finite k (subPiece 𝓜 S n) := Module.Finite.of_surjective
+    (subPieceEquiv 𝓜 S n).symm.toLinearMap (subPieceEquiv 𝓜 S n).symm.surjective
+  have h := LinearMap.finrank_range_add_finrank_ker (imagePieceMap 𝓜 𝓝 S f hf n)
+  rw [LinearMap.range_eq_top.mpr (imagePieceMap_surjective 𝓜 𝓝 S hS f hf n),finrank_top,
+    ← (imageKernelEquiv 𝓜 𝓝 S f hf n).finrank_eq,
+    (subPieceEquiv 𝓜 S n).finrank_eq,
+    (subPieceEquiv 𝓜 (S ⊓ LinearMap.ker f) n).finrank_eq,
+    (subPieceEquiv 𝓝 (S.map f) n).finrank_eq] at h
+  omega
+
+/-- Removing a kernel condition adds exactly the Hilbert value of its
+embedded restriction image. No freeness of the source kernel is used. -/
+ theorem value_image_exact
+    (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
+    [DirectSum.Decomposition 𝓜] [DirectSum.Decomposition 𝓝]
+    [∀ n, Module.Finite k (𝓜 n)]
+    (S : Submodule A M) (hS : ∀ n m, m ∈ S → component 𝓜 n m ∈ S)
+    (f : M →ₗ[A] N) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 n)
+    {q : ℝ} (hq : 0 < q) (hs : Summable (term 𝓜 (S.restrictScalars k) q)) :
+    value 𝓜 (S.restrictScalars k) q =
+      value 𝓜 ((S ⊓ LinearMap.ker f).restrictScalars k) q +
+      value 𝓝 ((S.map f).restrictScalars k) q := by
+  have ht (n : ℤ) : term 𝓜 (S.restrictScalars k) q n =
+      term 𝓜 ((S ⊓ LinearMap.ker f).restrictScalars k) q n +
+      term 𝓝 ((S.map f).restrictScalars k) q n := by
+    unfold term
+    rw [finrank_image_exact 𝓜 𝓝 S hS f hf n,Nat.cast_add,add_mul]
+  have hk : Summable (term 𝓜 ((S ⊓ LinearMap.ker f).restrictScalars k) q) := by
+    apply hs.of_nonneg_of_le
+    · exact fun n => term_nonneg _ _ hq n
+    · intro n
+      rw [ht n]
+      exact le_add_of_nonneg_right (term_nonneg _ _ hq n)
+  have hi : Summable (term 𝓝 ((S.map f).restrictScalars k) q) := by
+    apply hs.of_nonneg_of_le
+    · exact fun n => term_nonneg _ _ hq n
+    · intro n
+      rw [ht n]
+      exact le_add_of_nonneg_left (term_nonneg _ _ hq n)
+  unfold value
+  rw [show term 𝓜 (S.restrictScalars k) q = _ from funext ht,hk.tsum_add hi]
+
+end KLInvariance.Hilbert
+
+end
+
+
+section
+
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/KazhdanLusztig/IntegralPairings.lean b/lean/OAI/RepresentationTheory/KazhdanLusztig/IntegralPairings.lean
index 6d4dc0afadbcf92f364c818da1a867a86201ed1b..93c3faf61492e72339bb1c6a578c58280640e5f3 100644
--- a/lean/OAI/RepresentationTheory/KazhdanLusztig/IntegralPairings.lean
+++ b/lean/OAI/RepresentationTheory/KazhdanLusztig/IntegralPairings.lean
@@ -1,2366 +1,2366 @@
-import OAI.RepresentationTheory.KazhdanLusztig.BottSamelson
-
-namespace OAI
-
-/-!
-Hilbert recurrences, projective section ranges, integral lattices and graded perfect adjoints.
--/
-
-section
-
-namespace KLInvariance.Hilbert
-open Polynomial
-
- theorem hasSum_shift_coeff {f : ℤ → ℝ} {q c : ℝ} (hq : q≠0)
-    (hf : HasSum (fun n : ℤ => f n*q^n) c) (d : ℤ) :
-    HasSum (fun n : ℤ => f (n-d)*q^n) (q^d*c) := by
-  apply (Equiv.addRight d).hasSum_iff.mp
-  apply (hf.mul_left (q^d)).congr_fun
-  intro n
-  change f (n+d-d)*q^(n+d)=q^d*(f n*q^n)
-  rw [add_sub_cancel_right,zpow_add₀ hq]
-  ring
-
- theorem numerator_unique (f : ℤ → ℕ) (F G : ℤ[X]) (D : ℝ → ℝ)
-    (hD : ∀ q, 0<q → q<1 → D q≠0)
-    (hF : ∀ q, 0<q → q<1 → HasSum (fun n : ℤ => (f n : ℝ)*q^n)
-      (eval₂ (Int.castRingHom ℝ) q F*D q))
-    (hG : ∀ q, 0<q → q<1 → HasSum (fun n : ℤ => (f n : ℝ)*q^n)
-      (eval₂ (Int.castRingHom ℝ) q G*D q)) : F=G := by
-  apply Polynomial.map_injective (Int.castRingHom ℝ) Int.cast_injective
-  apply Polynomial.eq_of_infinite_eval_eq
-  apply Set.Infinite.mono (s := Set.Ioo (0 : ℝ) 1) ?_ (Set.Ioo_infinite (by norm_num))
-  intro q hq
-  have he := (hF q hq.1 hq.2).unique (hG q hq.1 hq.2)
-  change eval q (F.map (Int.castRingHom ℝ))=eval q (G.map (Int.castRingHom ℝ))
-  simpa only [eval_map] using (mul_right_cancel₀ (hD q hq.1 hq.2) he)
-
- theorem numerator_add_shift (f g h : ℤ → ℕ) (F G H : ℤ[X]) (D : ℝ → ℝ)
-    (hD : ∀ q, 0<q → q<1 → D q≠0)
-    (hF : ∀ q, 0<q → q<1 → HasSum (fun n : ℤ => (f n : ℝ)*q^n)
-      (eval₂ (Int.castRingHom ℝ) q F*D q))
-    (hG : ∀ q, 0<q → q<1 → HasSum (fun n : ℤ => (g n : ℝ)*q^n)
-      (eval₂ (Int.castRingHom ℝ) q G*D q))
-    (hH : ∀ q, 0<q → q<1 → HasSum (fun n : ℤ => (h n : ℝ)*q^n)
-      (eval₂ (Int.castRingHom ℝ) q H*D q))
-    (he : ∀ n, f n=g n+h (n-1)) : F=G+X*H := by
-  apply Polynomial.map_injective (Int.castRingHom ℝ) Int.cast_injective
-  apply Polynomial.eq_of_infinite_eval_eq
-  apply Set.Infinite.mono (s := Set.Ioo (0 : ℝ) 1) ?_ (Set.Ioo_infinite (by norm_num))
-  intro q hq
-  have hh := hasSum_shift_coeff hq.1.ne' (hH q hq.1 hq.2) 1
-  have hs := (hG q hq.1 hq.2).add hh
-  have hf := hF q hq.1 hq.2
-  have heq : eval₂ (Int.castRingHom ℝ) q F*D q=
-      eval₂ (Int.castRingHom ℝ) q G*D q+q*(eval₂ (Int.castRingHom ℝ) q H*D q) := by
-    rw [zpow_one] at hs
-    apply hf.unique
-    apply hs.congr_fun
-    intro n
-    rw [he,Nat.cast_add,add_mul]
-  change eval q (F.map (Int.castRingHom ℝ))=eval q ((G+X*H).map (Int.castRingHom ℝ))
-  simp only [eval_map,eval₂_add,eval₂_mul,eval₂_X]
-  apply mul_right_cancel₀ (hD q hq.1 hq.2)
-  calc
-    _ = eval₂ (Int.castRingHom ℝ) q G*D q+q*(eval₂ (Int.castRingHom ℝ) q H*D q) := heq
-    _ = _ := by ring
-
-end KLInvariance.Hilbert
-
-end
-
-
-section
-
-/-! Exact polynomial character recurrences for the actual integral word stalks.
-These are the unnormalized Hecke coefficients for multiplication by T_s+1.
-They are proved from the projection lattices, not from the KL conjecture. -/
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded
-universe u
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- theorem bottStalk_finrank_descent (i : I) (l : List I) (x : W)
-    (h : cs.length (cs.simple i*x)<cs.length x) (n : ℤ) :
-    Module.finrank ℝ (bottStalkPiece cs basis (i::l) x n)=
-      Module.finrank ℝ (bottStalkPiece cs basis l (cs.simple i*x) n)+
-      Module.finrank ℝ (bottStalkPiece cs basis l x (n-1)) := by
-  have hdim := Hilbert.finrank_exact (A := ℝ) (bottStalkPiece cs basis (i::l) x)
-    (bottStalkPiece cs basis l (cs.simple i*x)) (bottHeadRight cs i l x)
-    (bottHeadRight_homogeneous cs basis i l x) (bottHeadRight_surjective cs i l x) n
-  have hk := finrank_homogeneousEquiv (k := ℝ)
-    (N := LinearMap.ker (bottHeadRight cs i l x)) (bottStalkPiece cs basis l x)
-    (bottHeadRightKernelPiece cs basis i l x) (bottHeadRightKernelEquiv cs i l x h) 1
-    (bottHeadRightKernelEquiv_homogeneous cs basis i l x h) n
-  change Module.finrank ℝ (bottStalkPiece cs basis (i::l) x n)=
-    Module.finrank ℝ (bottHeadRightKernelPiece cs basis i l x n)+
-      Module.finrank ℝ (bottStalkPiece cs basis l (cs.simple i*x) n) at hdim
-  rw [hk] at hdim
-  omega
-
- theorem bottStalkCharacter_ascent (i : I) (l : List I) (x : W)
-    (h : cs.length x<cs.length (cs.simple i*x)) :
-    bottStalkCharacter cs basis (i::l) x=bottStalkCharacter cs basis l x+
-      Polynomial.X*bottStalkCharacter cs basis l (cs.simple i*x) := by
-  apply Hilbert.numerator_add_shift
-    (fun n => Module.finrank ℝ (bottStalkPiece cs basis (i::l) x n))
-    (fun n => Module.finrank ℝ (bottStalkPiece cs basis l x n))
-    (fun n => Module.finrank ℝ (bottStalkPiece cs basis l (cs.simple i*x) n))
-    _ _ _ (fun q => (1-q)⁻¹ ^ Fintype.card σ)
-  · intro q _ hq
-    exact pow_ne_zero _ (inv_ne_zero (sub_ne_zero.mpr (ne_of_gt hq)))
-  · exact fun _ => bottStalkCharacter_hasSum cs basis (i::l) x
-  · exact fun _ => bottStalkCharacter_hasSum cs basis l x
-  · exact fun _ => bottStalkCharacter_hasSum cs basis l (cs.simple i*x)
-  · exact bottStalk_finrank_ascent cs basis i l x h
-
- theorem bottStalkCharacter_descent (i : I) (l : List I) (x : W)
-    (h : cs.length (cs.simple i*x)<cs.length x) :
-    bottStalkCharacter cs basis (i::l) x=bottStalkCharacter cs basis l (cs.simple i*x)+
-      Polynomial.X*bottStalkCharacter cs basis l x := by
-  apply Hilbert.numerator_add_shift
-    (fun n => Module.finrank ℝ (bottStalkPiece cs basis (i::l) x n))
-    (fun n => Module.finrank ℝ (bottStalkPiece cs basis l (cs.simple i*x) n))
-    (fun n => Module.finrank ℝ (bottStalkPiece cs basis l x n))
-    _ _ _ (fun q => (1-q)⁻¹ ^ Fintype.card σ)
-  · intro q _ hq
-    exact pow_ne_zero _ (inv_ne_zero (sub_ne_zero.mpr (ne_of_gt hq)))
-  · exact fun _ => bottStalkCharacter_hasSum cs basis (i::l) x
-  · exact fun _ => bottStalkCharacter_hasSum cs basis l (cs.simple i*x)
-  · exact fun _ => bottStalkCharacter_hasSum cs basis l x
-  · exact bottStalk_finrank_descent cs basis i l x h
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! The honest unnormalized Hecke word (T_s+1)...(T_t+1), including exact
-coefficient recurrence over a general commutative ring, without invertibility
-of q or any KL/BMP character assertion. -/
-namespace KLInvariance.HeckeQ
-open Hecke (Space basis)
-variable {I W C : Type*} [Group W] {M : CoxeterMatrix I} [CommRing C]
-noncomputable section
-
- theorem left_coeff (cs : CoxeterSystem M W) (q : C) (i : I) (v : Space W C) (x : W) :
-    left cs q i v x =
-      if cs.length (cs.simple i*x)<cs.length x then v (cs.simple i*x)+(q-1)*v x
-      else q*v (cs.simple i*x) := by
-  classical
-  induction v using Finsupp.induction_linear with
-  | zero => simp
-  | add v v' hv hv' =>
-    simp only [map_add,Finsupp.add_apply,hv,hv']
-    split_ifs <;> ring
-  | single w c =>
-    have hc : Finsupp.single w c=c • (basis w : Space W C) := by
-      simp [basis,Finsupp.smul_single]
-    rw [hc,map_smul,left_basis]
-    by_cases h : w=cs.simple i*x
-    · subst w
-      have hd : cs.length (cs.simple i*(cs.simple i*x))<cs.length (cs.simple i*x) ↔
-          ¬cs.length (cs.simple i*x)<cs.length x := by
-        simp only [cs.simple_mul_simple_cancel_left]
-        have := cs.length_simple_mul x i
-        omega
-      have hn : cs.simple i*x≠x := by
-        intro he
-        have := cs.length_simple_mul x i
-        rw [he] at this
-        omega
-      simp only [Finsupp.smul_apply,smul_eq_mul,basis,Finsupp.single_apply,
-        cs.simple_mul_simple_cancel_left,hn,ite_false]
-      have hlength := cs.length_simple_mul x i
-      split_ifs <;> try omega
-      all_goals simp [Ne.symm hn] <;> try ring
-    · have hn : cs.simple i*w≠x := by
-        intro he
-        apply h
-        rw [←he]
-        simp
-      by_cases hw : w=x
-      · subst w
-        simp only [Finsupp.smul_apply,smul_eq_mul,basis,Finsupp.single_apply,
-          Ne.symm hn,ite_false]
-        split_ifs <;> simp [Ne.symm hn] ; try ring
-      · simp only [Finsupp.smul_apply,smul_eq_mul,basis,Finsupp.single_apply,
-          h,hw,ite_false]
-        split_ifs <;> simp [hn,hw]
-
- theorem left_plus_coeff (cs : CoxeterSystem M W) (q : C) (i : I)
-    (v : Space W C) (x : W) :
-    (v+left cs q i v) x = if cs.length (cs.simple i*x)<cs.length x then
-      v (cs.simple i*x)+q*v x else v x+q*v (cs.simple i*x) := by
-  rw [Finsupp.add_apply,left_coeff]
-  split_ifs <;> ring
-
- def positiveWord (cs : CoxeterSystem M W) (q : C) : List I → Space W C
-  | [] => basis 1
-  | i::l => positiveWord cs q l+left cs q i (positiveWord cs q l)
-
- @[simp] theorem positiveWord_nil (cs : CoxeterSystem M W) (q : C) :
-    positiveWord cs q []=basis 1 := rfl
-
- theorem positiveWord_cons_coeff (cs : CoxeterSystem M W) (q : C) (i : I)
-    (l : List I) (x : W) :
-    positiveWord cs q (i::l) x = if cs.length (cs.simple i*x)<cs.length x then
-      positiveWord cs q l (cs.simple i*x)+q*positiveWord cs q l x
-      else positiveWord cs q l x+q*positiveWord cs q l (cs.simple i*x) :=
-  left_plus_coeff cs q i (positiveWord cs q l) x
-
-end
-end KLInvariance.HeckeQ
-
-end
-
-
-section
-
-/-! Genuine word stalk normalization and identification with the honest
-Hecke word. This does not identify the indecomposable BMP character with KL. -/
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded
-universe u
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
- theorem bottStalkCharacter_zero (l : List I) (x : W)
-    [Subsingleton (bottStalk cs l x)] : bottStalkCharacter cs basis l x=0 := by
-  apply Hilbert.numerator_unique
-    (fun n => Module.finrank ℝ (bottStalkPiece cs basis l x n)) _ _
-    (fun q => (1-q)⁻¹ ^ Fintype.card σ)
-  · intro q _ hq
-    exact pow_ne_zero _ (inv_ne_zero (sub_ne_zero.mpr (ne_of_gt hq)))
-  · exact fun _ => bottStalkCharacter_hasSum cs basis l x
-  · intro q _ _
-    simpa only [Module.finrank_zero_of_subsingleton,Nat.cast_zero,zero_mul,Polynomial.eval₂_zero] using (hasSum_zero : HasSum (fun _ : ℤ => (0 : ℝ)) 0)
-
- def bottTopPolynomialEquiv (l : List I) (hl : cs.IsReduced l) :
-    bottStalk cs l (cs.wordProd l) ≃ₗ[ℝ] MvPolynomial σ ℝ :=
-  ((bottTopStalkEquiv cs l hl).restrictScalars ℝ).trans
-    (SymmetricAlgebra.equivMvPolynomial basis).toLinearEquiv
-
-omit [Fintype σ] in
- theorem bottTopPolynomialEquiv_homogeneous (l : List I) (hl : cs.IsReduced l)
-    (n : ℤ) (s : bottStalk cs l (cs.wordProd l)) (hs : s∈bottStalkPiece cs basis l (cs.wordProd l) n) :
-    bottTopPolynomialEquiv cs basis l hl s∈polynomialGrading ℝ σ n := by
-  apply (mem_symmetricGrading basis n _).mp
-  exact (bottTopStalkEquiv_homogeneous cs basis l hl n s).mpr hs
-
- theorem bottStalkCharacter_top (l : List I) (hl : cs.IsReduced l) :
-    bottStalkCharacter cs basis l (cs.wordProd l)=1 := by
-  have hd (n : ℤ) : Module.finrank ℝ (bottStalkPiece cs basis l (cs.wordProd l) n)=
-      Module.finrank ℝ (polynomialGrading ℝ σ n) := by
-    have hh := finrank_homogeneousEquiv (bottStalkPiece cs basis l (cs.wordProd l))
-      (polynomialGrading ℝ σ) (bottTopPolynomialEquiv cs basis l hl) 0
-      (fun n s hs => by simpa only [add_zero] using bottTopPolynomialEquiv_homogeneous cs basis l hl n s hs) n
-    have hz : n-(0:ℤ)=n := Int.sub_zero n
-    rw [hz] at hh
-    exact hh.symm
-  apply Hilbert.numerator_unique
-    (fun n => Module.finrank ℝ (bottStalkPiece cs basis l (cs.wordProd l) n)) _ _
-    (fun q => (1-q)⁻¹ ^ Fintype.card σ)
-  · intro q _ hq
-    exact pow_ne_zero _ (inv_ne_zero (sub_ne_zero.mpr (ne_of_gt hq)))
-  · exact fun _ => bottStalkCharacter_hasSum cs basis l (cs.wordProd l)
-  · intro q hq hq1
-    simpa only [hd,Polynomial.eval₂_one,one_mul] using Hilbert.hasSum_polynomial ℝ σ hq.le hq1
-
- theorem bottStalkCharacter_nil (x : W) :
-    bottStalkCharacter cs basis [] x=if x=1 then 1 else 0 := by
-  classical
-  by_cases hx : x=1
-  · subst x
-    simpa using bottStalkCharacter_top cs basis [] (by simp [CoxeterSystem.IsReduced])
-  · let := bottStalk_nil_subsingleton cs x hx
-    rw [ite_eq_right hx]
-    exact bottStalkCharacter_zero cs basis [] x
-
- theorem bottStalkCharacter_eq_heckeWord (l : List I) (x : W) :
-    bottStalkCharacter cs basis l x=HeckeQ.positiveWord cs (Polynomial.X : Polynomial ℤ) l x := by
-  classical
-  induction l generalizing x with
-  | nil => simp only [bottStalkCharacter_nil,HeckeQ.positiveWord_nil,Hecke.basis,
-      Finsupp.single_apply]; split_ifs <;> simp_all
-  | cons i l ih =>
-    rw [HeckeQ.positiveWord_cons_coeff]
-    by_cases hd : cs.length (cs.simple i*x)<cs.length x
-    · rw [ite_eq_left hd,bottStalkCharacter_descent cs basis i l x hd,ih,ih]
-    · have ha : cs.length x<cs.length (cs.simple i*x) := by
-        have := cs.length_simple_mul x i
-        omega
-      rw [ite_eq_right hd,bottStalkCharacter_ascent cs basis i l x ha,ih,ih]
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Range-valued projections to sets of coordinates and their natural
-surjective restriction maps. -/
-namespace KLInvariance.CoordinateProjection
-universe ur un uv ux
-variable {R : Type ur} [CommRing R] {N : Type un} [AddCommGroup N] [Module R N]
-  {V : Type uv} {X : V → Type ux} [∀ x, AddCommGroup (X x)] [∀ x, Module R (X x)]
-  (f : ∀ x, N →ₗ[R] X x)
-noncomputable section
-
- def eval (U : Set V) : N →ₗ[R] (∀ x : U, X x.val) := LinearMap.pi (fun x => f x.val)
- def image (U : Set V) : Submodule R (∀ x : U, X x.val) := LinearMap.range (eval f U)
- def project (U : Set V) : N →ₗ[R] image f U := (eval f U).rangeRestrict
-
- theorem project_surjective (U : Set V) : Function.Surjective (project f U) :=
-by
-  rintro ⟨z,n,hn⟩
-  exact ⟨n,Subtype.ext hn⟩
-
- @[simp] theorem project_apply (U : Set V) (n : N) (x : U) : (project f U n).val x=f x.val n := rfl
-
- theorem project_zero_iff (U : Set V) (n : N) : project f U n=0 ↔ ∀ x∈U, f x n=0 := by
-  constructor
-  · intro h x hx
-    exact congrArg (fun z : image f U => z.val ⟨x,hx⟩) h
-  · intro h
-    apply Subtype.ext
-    funext x
-    exact h x.val x.property
-
- def restrict {U K : Set V} (h : U⊆K) : image f K →ₗ[R] image f U where
-  toFun z := ⟨fun x => z.val ⟨x.val,h x.property⟩,by
-    obtain ⟨n,hn⟩ := z.property
-    refine ⟨n,?_⟩
-    funext x
-    exact congrFun hn ⟨x.val,h x.property⟩⟩
-  map_add' _ _ := rfl
-  map_smul' _ _ := rfl
-
- @[simp] theorem restrict_project {U K : Set V} (h : U⊆K) (n : N) :
-    restrict f h (project f K n)=project f U n := rfl
-
- theorem restrict_surjective {U K : Set V} (h : U⊆K) : Function.Surjective (restrict f h) := by
-  intro z
-  obtain ⟨n,rfl⟩ := project_surjective f U z
-  exact ⟨project f K n,rfl⟩
-
- theorem restrict_zero_iff {U K : Set V} (h : U⊆K) (z : image f K) :
-    restrict f h z=0 ↔ ∀ x hx, x∈U → z.val ⟨x,hx⟩=0 := by
-  constructor
-  · intro hz x hx hu
-    exact congrArg (fun q : image f U => q.val ⟨x,hu⟩) hz
-  · intro hz
-    apply Subtype.ext
-    funext x
-    exact hz x.val (h x.property) x.property
-
-end
-end KLInvariance.CoordinateProjection
-
-end
-
-
-section
-
-/-! Upper sets and actual finite-word projection predicates. -/
-namespace KLInvariance.TitsSpace
-open Module CoordinateProjection
-universe u v
-variable {I : Type u} [klPreservedInstance11 : Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
- def BruhatUpper (U : Set W) : Prop :=
-  ∀ {x y}, BruhatLE cs x y → x∈U → y∈U
-
- def simpleUnion (i : I) (U : Set W) : Set W := {x | x∈U ∨ cs.simple i*x∈U}
- def simpleInter (i : I) (U : Set W) : Set W := {x | x∈U ∧ cs.simple i*x∈U}
-
- omit [Fintype I] in
- theorem BruhatUpper.mem_simpleUnion (U : Set W) (hU : BruhatUpper cs U) (i : I) (x : W) :
-    x∈simpleUnion cs i U ↔ upperSimple cs i x∈U := by
-  constructor
-  · rintro (hx|hx)
-    · exact hU (le_upperSimple cs i x) hx
-    · have hh := hU (le_upperSimple cs i (cs.simple i*x)) hx
-      simpa only [upperSimple_mul] using hh
-  · intro hx
-    classical
-    unfold upperSimple at hx
-    split_ifs at hx with h
-    · exact Or.inl hx
-    · exact Or.inr hx
-
- omit [Fintype I] in
- theorem BruhatUpper.mem_simpleInter (U : Set W) (hU : BruhatUpper cs U) (i : I) (x : W) :
-    x∈simpleInter cs i U ↔ lowerSimple cs i x∈U := by
-  constructor
-  · rintro ⟨hx,hsx⟩
-    classical
-    unfold lowerSimple
-    split_ifs <;> assumption
-  · intro hx
-    refine ⟨hU (lowerSimple_le cs i x) hx,?_⟩
-    have he : lowerSimple cs i (cs.simple i*x)∈U := by simpa only [lowerSimple_mul] using hx
-    exact hU (lowerSimple_le cs i (cs.simple i*x)) he
-
-include klPreservedInstance11 in
- theorem BruhatUpper.simpleUnion (U : Set W) (hU : BruhatUpper cs U) (i : I) :
-    BruhatUpper cs (simpleUnion cs i U) := by
-  have _ := klPreservedInstance11
-  intro x y hxy hx
-  exact (hU.mem_simpleUnion cs U i y).mpr
-    (hU (upperSimple_mono cs i hxy) ((hU.mem_simpleUnion cs U i x).mp hx))
-
-include klPreservedInstance11 in
- theorem BruhatUpper.simpleInter (U : Set W) (hU : BruhatUpper cs U) (i : I) :
-    BruhatUpper cs (simpleInter cs i U) := by
-  have _ := klPreservedInstance11
-  intro x y hxy hx
-  exact (hU.mem_simpleInter cs U i y).mpr
-    (hU (lowerSimple_mono cs i hxy) ((hU.mem_simpleInter cs U i x).mp hx))
-
- omit [Fintype I] in
- theorem simpleUnion_stable (i : I) (U : Set W) (x : W) :
-    cs.simple i*x∈simpleUnion cs i U ↔ x∈simpleUnion cs i U := by
-  change (cs.simple i*x∈U ∨ cs.simple i*(cs.simple i*x)∈U) ↔ (x∈U ∨ cs.simple i*x∈U)
-  rw [cs.simple_mul_simple_cancel_left,or_comm]
-
- omit [Fintype I] in
- theorem simpleInter_stable (i : I) (U : Set W) (x : W) :
-    cs.simple i*x∈simpleInter cs i U ↔ x∈simpleInter cs i U := by
-  change (cs.simple i*x∈U ∧ cs.simple i*(cs.simple i*x)∈U) ↔ (x∈U ∧ cs.simple i*x∈U)
-  rw [cs.simple_mul_simple_cancel_left,and_comm]
-
- omit [Fintype I] in
- theorem cross_edge_is_simple (U : Set W) (hU : BruhatUpper cs U) (i : I)
-    {x y : W} (hxy : BruhatStep cs x y) (hsx : cs.simple i*x∈U)
-    (hsy : cs.simple i*y∉U) : y=cs.simple i*x := by
-  by_contra he
-  exact hsy (hU (.single (bruhatStep_simple_mul cs hxy i he)) hsx)
-
- def bottSetImage (l : List I) (U : Set W) := image (bottStalkProjection cs l) U
- instance bottSetImageAddCommGroup (l : List I) (U : Set W) : AddCommGroup (bottSetImage cs l U) :=
-  Submodule.addCommGroup _
- instance bottSetImageModule (l : List I) (U : Set W) : Module (SymmetricCoefficient (M := M)) (bottSetImage cs l U) :=
-  Submodule.module _
-
- def bottSetProjection (l : List I) (U : Set W) : BottSamelsonModule cs l →ₗ[
-    SymmetricCoefficient (M := M)] bottSetImage cs l U := project (bottStalkProjection cs l) U
-
- def BottCompatible (l : List I) (U : Set W) (z : ∀ x : U, bottStalk cs l x.val) : Prop :=
-  ∀ (x y : W) (hx : x∈U) (hy : y∈U), BruhatStep cs x y →
-    ∃ m : BottSamelsonModule cs l,
-      bottStalkProjection cs l x m=z ⟨x,hx⟩ ∧ bottStalkProjection cs l y m=z ⟨y,hy⟩
-
- def BottFlabby (l : List I) : Prop :=
-  ∀ U, BruhatUpper cs U → ∀ z, BottCompatible cs l U z → z∈bottSetImage cs l U
-
- def BottProjectiveUpper (l : List I) : Prop :=
-  ∀ U, BruhatUpper cs U → Module.Projective (SymmetricCoefficient (M := M)) (bottSetImage cs l U)
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Explicit linear descent and split exact sequences for projected modules. -/
-namespace KLInvariance.ProjectiveRange
-universe ur um up uq
-variable {R : Type ur} [CommRing R]
-  {M : Type um} [AddCommGroup M] [Module R M]
-  {P : Type up} [AddCommGroup P] [Module R P]
-  {Q : Type uq} [AddCommGroup Q] [Module R Q]
-noncomputable section
-
- def descend (f : M →ₗ[R] P) (hf : Function.Surjective f) (g : M →ₗ[R] Q)
-    (hg : ∀ m, f m=0 → g m=0) : P →ₗ[R] Q :=
-  ((LinearMap.ker f).liftQ g hg).comp (f.quotKerEquivOfSurjective hf).symm.toLinearMap
-
- theorem descend_apply (f : M →ₗ[R] P) (hf : Function.Surjective f) (g : M →ₗ[R] Q)
-    (hg : ∀ m, f m=0 → g m=0) (m : M) : descend f hf g hg (f m)=g m := by
-  change (LinearMap.ker f).liftQ g hg ((f.quotKerEquivOfSurjective hf).symm (f m))=g m
-  rw [LinearMap.quotKerEquivOfSurjective_symm_apply]
-  rfl
-
- theorem projective_of_end (f : M →ₗ[R] P) (hf : Function.Surjective f)
-    [Module.Projective R M] (e : Module.End R M)
-    (he : ∀ m, f (e m)=f m) (hz : ∀ m, f m=0 → e m=0) : Module.Projective R P := by
-  apply Module.Projective.of_split (descend f hf e hz) f
-  apply LinearMap.ext
-  intro p
-  obtain ⟨m,rfl⟩ := hf p
-  rw [LinearMap.comp_apply,descend_apply,he,LinearMap.id_apply]
-
- def kernelRetraction (f : M →ₗ[R] P) (s : P →ₗ[R] M)
-    (hs : ∀ p, f (s p)=p) : M →ₗ[R] LinearMap.ker f :=
-  (LinearMap.id-s.comp f).codRestrict (LinearMap.ker f) (fun m => by
-    change f (m-s (f m))=0
-    rw [map_sub,hs,sub_self])
-
- theorem kernel_projective (f : M →ₗ[R] P) (hf : Function.Surjective f)
-    [Module.Projective R M] [Module.Projective R P] : Module.Projective R (LinearMap.ker f) := by
-  obtain ⟨s,hs⟩ := Module.projective_lifting_property f (LinearMap.id : P →ₗ[R] P) hf
-  have hs' : ∀ p, f (s p)=p := LinearMap.congr_fun hs
-  apply Module.Projective.of_split (LinearMap.ker f).subtype (kernelRetraction f s hs')
-  apply LinearMap.ext
-  intro m
-  apply Subtype.ext
-  change (m : M)-s (f m)=m
-  rw [m.property,map_zero,sub_zero]
-
- def splitEquiv (f : M →ₗ[R] P) (s : P →ₗ[R] M) (hs : ∀ p, f (s p)=p) :
-    M ≃ₗ[R] (LinearMap.ker f × P) :=
-  { (kernelRetraction f s hs).prod f with
-    invFun := fun z => z.1.val+s z.2
-    left_inv := fun m => sub_add_cancel m (s (f m))
-    right_inv := by
-      intro z
-      apply Prod.ext
-      · apply Subtype.ext
-        change z.1.val+s z.2-s (f (z.1.val+s z.2))=z.1.val
-        rw [map_add,z.1.property,zero_add,hs]
-        exact add_sub_cancel_right _ _
-      · change f (z.1.val+s z.2)=z.2
-        rw [map_add,z.1.property,zero_add,hs] }
-
- theorem projective_of_kernel (f : M →ₗ[R] P) (hf : Function.Surjective f)
-    [Module.Projective R P] [Module.Projective R (LinearMap.ker f)] : Module.Projective R M := by
-  obtain ⟨s,hs⟩ := Module.projective_lifting_property f (LinearMap.id : P →ₗ[R] P) hf
-  exact Module.Projective.of_equiv (splitEquiv f s (LinearMap.congr_fun hs)).symm
-
-end
-end KLInvariance.ProjectiveRange
-
-end
-
-
-section
-
-namespace KLInvariance.ReflectionPairTranslation
-open TensorProduct
-universe uc ua un
-variable {C : Type uc} [CommRing C] {A : Type ua} [CommRing A] [Algebra C A]
-  {N : Type un} [AddCommGroup N] [Module A N] [Module C N] [IsScalarTower C A N]
-  (τ : A ≃+* A) (hfix : ∀ c : C, τ (algebraMap C A c)=algebraMap C A c)
-noncomputable section
-
- theorem tensorFirst_map (e : Module.End A N) (m : A ⊗[C] N) :
-    tensorFirst (C := C) (AlgebraTensorModule.map (LinearMap.id : A →ₗ[A] A) (e.restrictScalars C) m)=
-      e (tensorFirst (C := C) m) := by
-  induction m using TensorProduct.inductionOn with
-  | add m n hm hn => simp only [map_add,hm,hn]
-  | tmul a n => simp only [AlgebraTensorModule.map_tmul,LinearMap.id_apply,
-      LinearMap.restrictScalars_apply,tensorFirst_tmul,map_smul]
-
- theorem tensorSecond_map (e : Module.End A N) (m : A ⊗[C] N) :
-    tensorSecond τ hfix (AlgebraTensorModule.map (LinearMap.id : A →ₗ[A] A) (e.restrictScalars C) m)=
-      twistMap τ e (tensorSecond τ hfix m) := by
-  induction m using TensorProduct.inductionOn with
-  | add m n hm hn => simp only [map_add,hm,hn]
-  | tmul a n =>
-    simp only [AlgebraTensorModule.map_tmul,LinearMap.id_apply,
-      LinearMap.restrictScalars_apply,tensorSecond_tmul]
-    apply Twist.ext
-    exact (map_smul e (τ a) n).symm
-
-end
-end KLInvariance.ReflectionPairTranslation
-
-end
-
-
-section
-
-namespace KLInvariance
-universe ur un up uq
-variable {R : Type ur} [CommRing R]
-  {N : Type un} [klPreservedInstance12 : AddCommGroup N] [Module R N]
-  {P : Type up} [klPreservedInstance13 : AddCommGroup P] [Module R P]
-  {Q : Type uq} [klPreservedInstance14 : AddCommGroup Q] [Module R Q]
- theorem injective_of_zero_lift (f : N →ₗ[R] P) (hf : Function.Surjective f) (g : P →ₗ[R] Q)
-    (h : ∀ n, g (f n)=0 → f n=0) : Function.Injective g := by
-  apply LinearMap.ker_eq_bot.mp
-  rw [LinearMap.ker_eq_bot']
-  intro p hp
-  obtain ⟨n,rfl⟩ := hf p
-  exact h n hp
- theorem maps_eq_iff_of_zero_iff (f : N →ₗ[R] P) (g : N →ₗ[R] Q)
-    (h : ∀ n, f n=0 ↔ g n=0) (m n : N) : f m=f n ↔ g m=g n := by
-  rw [← sub_eq_zero,← map_sub,h,map_sub,sub_eq_zero]
-include klPreservedInstance12 klPreservedInstance13 klPreservedInstance14 in
- theorem functions_eq_of_surjective (f : N → P) (hf : Function.Surjective f)
-    (g h : P → Q) (he : ∀ n, g (f n)=h (f n)) (p : P) : g p=h p := by
-  have _ := klPreservedInstance12
-  have _ := klPreservedInstance13
-  have _ := klPreservedInstance14
-  obtain ⟨n,rfl⟩ := hf p
-  exact he n
-end KLInvariance
-
-end
-
-
-section
-
-/-! Genuine first/reflected-tail maps on projected upper-set images. -/
-namespace KLInvariance.TitsSpace
-open Module TensorProduct FrobeniusDuality ReflectionFrobenius ReflectionPairTranslation
-open CoordinateProjection ProjectiveRange
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-attribute [local instance] restrictedModule restrictedTower restrictedFree restrictedFinite
-
- def simplePull (i : I) (U : Set W) : Set W := {x | cs.simple i*x∈U}
-
- theorem bottSetProjection_zero_iff (l : List I) (U : Set W) (m : BottSamelsonModule cs l) :
-    bottSetProjection cs l U m=0 ↔ ∀ x∈U, bottStalkProjection cs l x m=0 :=
-  project_zero_iff _ _ _
-
- theorem bottSetProjection_surjective (l : List I) (U : Set W) :
-    Function.Surjective (bottSetProjection cs l U) := project_surjective _ _
-
- theorem bottStalkProjection_zero_of_eq (l : List I) {x y : W} (h : x=y)
-    (m : BottSamelsonModule cs l) (hm : bottStalkProjection cs l x m=0) :
-    bottStalkProjection cs l y m=0 := by
-  subst y
-  exact hm
-
- theorem bottHeadSet_zero_iff (i : I) (l : List I) (U : Set W)
-    (m : BottSamelsonModule cs (i::l)) :
-    bottSetProjection cs (i::l) U m=0 ↔
-      bottSetProjection cs l U (bottHeadTail cs i l m)=0 ∧
-      bottSetProjection cs l (simplePull cs i U) (bottHeadReflectedTail cs i l m).val=0 := by
-  constructor
-  · intro hm
-    have h := (bottSetProjection_zero_iff cs (i::l) U m).mp hm
-    refine ⟨?_,?_⟩
-    · apply (bottSetProjection_zero_iff cs l U _).mpr
-      intro x hx
-      exact congrArg Prod.fst ((bottHeadLocalPair_zero_iff cs i l x m).mp (h x hx))
-    · apply (bottSetProjection_zero_iff cs l (simplePull cs i U) _).mpr
-      intro x hx
-      have hh : bottStalkProjection cs l (cs.simple i*(cs.simple i*x))
-          (bottHeadReflectedTail cs i l m).val=0 := congrArg (fun z => z.2.val)
-        ((bottHeadLocalPair_zero_iff cs i l (cs.simple i*x) m).mp (h _ hx))
-      exact bottStalkProjection_zero_of_eq cs l (cs.simple_mul_simple_cancel_left i) _ hh
-  · rintro ⟨hp,hq⟩
-    apply (bottSetProjection_zero_iff cs (i::l) U m).mpr
-    intro x hx
-    apply (bottHeadLocalPair_zero_iff cs i l x m).mpr
-    apply Prod.ext
-    · exact (bottSetProjection_zero_iff cs l U _).mp hp x hx
-    · apply Twist.ext
-      exact (bottSetProjection_zero_iff cs l (simplePull cs i U) _).mp hq (cs.simple i*x)
-        (by simpa only [simplePull,Set.mem_ofPred_eq,cs.simple_mul_simple_cancel_left] using hx)
-
- def bottSetFirst (i : I) (l : List I) (U : Set W) : bottSetImage cs (i::l) U →ₗ[
-    SymmetricCoefficient (M := M)] bottSetImage cs l U :=
-  descend (bottSetProjection cs (i::l) U) (bottSetProjection_surjective cs (i::l) U)
-    ((bottSetProjection cs l U).comp (bottHeadTail cs i l))
-    (fun m hm => ((bottHeadSet_zero_iff cs i l U m).mp hm).1)
-
- def bottSetSecond (i : I) (l : List I) (U : Set W) : bottSetImage cs (i::l) U →ₗ[
-    SymmetricCoefficient (M := M)]
-    Twist (coefficientAction cs (cs.simple i)).toRingEquiv (bottSetImage cs l (simplePull cs i U)) :=
-  descend (bottSetProjection cs (i::l) U) (bottSetProjection_surjective cs (i::l) U)
-    ((twistMap (coefficientAction cs (cs.simple i)).toRingEquiv
-      (bottSetProjection cs l (simplePull cs i U))).comp (bottHeadReflectedTail cs i l))
-    (fun m hm => Twist.ext (((bottHeadSet_zero_iff cs i l U m).mp hm).2))
-
- theorem bottSetFirst_project (i : I) (l : List I) (U : Set W) (m : BottSamelsonModule cs (i::l)) :
-    bottSetFirst cs i l U (bottSetProjection cs (i::l) U m)=
-      bottSetProjection cs l U (bottHeadTail cs i l m) := descend_apply _ _ _ _ m
-
- theorem bottSetSecond_project (i : I) (l : List I) (U : Set W) (m : BottSamelsonModule cs (i::l)) :
-    bottSetSecond cs i l U (bottSetProjection cs (i::l) U m)=
-      Twist.mk (bottSetProjection cs l (simplePull cs i U) (bottHeadReflectedTail cs i l m).val) :=
-  descend_apply _ _ _ _ m
-
- theorem bottSetComponents_injective (i : I) (l : List I) (U : Set W) :
-    Function.Injective ((bottSetFirst cs i l U).prod (bottSetSecond cs i l U)) := by
-  apply injective_of_zero_lift (bottSetProjection cs (i::l) U)
-    (bottSetProjection_surjective cs (i::l) U)
-  intro m hz
-  apply (bottHeadSet_zero_iff cs i l U m).mpr
-  have hp := congrArg Prod.fst hz
-  have hq := congrArg (fun z => z.2.val) hz
-  exact ⟨(bottSetFirst_project cs i l U m).symm.trans hp,
-    (congrArg Twist.val (bottSetSecond_project cs i l U m)).symm.trans hq⟩
-
- def bottTensorEnd (i : I) (l : List I)
-    (e : Module.End (SymmetricCoefficient (M := M)) (BottSamelsonModule cs l)) :
-    Module.End (SymmetricCoefficient (M := M)) (BottSamelsonModule cs (i::l)) :=
-  AlgebraTensorModule.map (LinearMap.id : SymmetricCoefficient (M := M) →ₗ[
-    SymmetricCoefficient (M := M)] SymmetricCoefficient (M := M)) (e.restrictScalars (simpleFixed cs i))
-
- theorem bottTensorEnd_seed (i : I) (l : List I)
-    (e : Module.End (SymmetricCoefficient (M := M)) (BottSamelsonModule cs l))
-    (n : BottSamelsonModule cs l) :
-    bottTensorEnd cs i l e ((1 : SymmetricCoefficient (M := M)) ⊗ₜ[simpleFixed cs i] n)=
-      (1 : SymmetricCoefficient (M := M)) ⊗ₜ[simpleFixed cs i] e n := rfl
-
- theorem bottTensorEnd_first (i : I) (l : List I)
-    (e : Module.End (SymmetricCoefficient (M := M)) (BottSamelsonModule cs l))
-    (m : BottSamelsonModule cs (i::l)) :
-    bottHeadTail cs i l (bottTensorEnd cs i l e m)=e (bottHeadTail cs i l m) :=
-  tensorFirst_map e m
-
- theorem bottTensorEnd_second (i : I) (l : List I)
-    (e : Module.End (SymmetricCoefficient (M := M)) (BottSamelsonModule cs l))
-    (m : BottSamelsonModule cs (i::l)) :
-    bottHeadReflectedTail cs i l (bottTensorEnd cs i l e m)=
-      twistMap (coefficientAction cs (cs.simple i)).toRingEquiv e (bottHeadReflectedTail cs i l m) :=
-  tensorSecond_map (coefficientAction cs (cs.simple i)).toRingEquiv (fun c => c.property) e m
-
- theorem bottHead_components_injective (i : I) (l : List I)
-    (m : BottSamelsonModule cs (i::l))
-    (hp : bottHeadTail cs i l m=0) (hq : bottHeadReflectedTail cs i l m=0) : m=0 := by
-  apply bottEvaluation_injective cs (i::l)
-  funext e
-  have hz : bottStalkProjection cs (i::l) (bottWeight cs (i::l) e) m=0 := by
-    apply (bottHeadLocalPair_zero_iff cs i l _ m).mpr
-    rw [bottHeadLocalPair_value,hp,hq]
-    apply Prod.ext
-    · exact map_zero _
-    · apply Twist.ext
-      exact map_zero _
-  exact congrArg (fun z => z.val ⟨e,rfl⟩) hz
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-namespace KLInvariance.ReflectionPairTranslation
-open TensorProduct
-universe uc ua un
-variable {C : Type uc} [CommRing C] {A : Type ua} [CommRing A] [Algebra C A]
-  {N : Type un} [AddCommGroup N] [Module A N] [Module C N] [IsScalarTower C A N]
-  (τ : A ≃+* A) (hfix : ∀ c : C, τ (algebraMap C A c)=algebraMap C A c)
-noncomputable section
- theorem tensor_difference (α : A) (hdiv : ∀ a : A, α ∣ a-τ a)
-    (m : A ⊗[C] N) :
-    ∃ n : N, tensorFirst (C := C) m-(tensorSecond τ hfix m).val=α • n := by
-  induction m using TensorProduct.inductionOn with
-  | add m n hm hn =>
-    obtain ⟨p,hp⟩ := hm
-    obtain ⟨q,hq⟩ := hn
-    refine ⟨p+q,?_⟩
-    simp only [map_add]
-    change (tensorFirst (C := C) m+tensorFirst (C := C) n)-
-      ((tensorSecond τ hfix m).val+(tensorSecond τ hfix n).val)=α • (p+q)
-    rw [add_sub_add_comm,hp,hq,smul_add]
-  | tmul a n =>
-    obtain ⟨b,hb⟩ := hdiv a
-    refine ⟨b • n,?_⟩
-    change a • n-τ a • n=α • (b • n)
-    rw [← sub_smul,hb,mul_smul]
-end
-end KLInvariance.ReflectionPairTranslation
-
-end
-
-
-section
-
-namespace KLInvariance.PairProjection
-universe u v w z
-variable {R : Type u} [CommRing R]
-  {N : Type v} [AddCommGroup N] [Module R N]
-  {X : Type w} [AddCommGroup X] [Module R X]
-  {Y : Type z} [AddCommGroup Y] [Module R Y]
-  (f : N →ₗ[R] X) (g : N →ₗ[R] Y)
-
- theorem cross_lift (α : R) (hann : ∀ z : Edge f g, α • z=0)
-    (p q : N) (h : ∃ n : N, p-q=α • n) :
-    ∃ n : N, f n=f q ∧ g n=g p := by
-  apply (compatible_iff f g _ _).mp
-  obtain ⟨n,hn⟩ := h
-  have hd := congrArg ((lower f g).comp f) hn
-  simp only [LinearMap.comp_apply,map_sub,map_smul,hann] at hd
-  exact (sub_eq_zero.mp hd).symm.trans (compatible f g p)
-
-end KLInvariance.PairProjection
-
-end
-
-
-section
-
-/-! Honest head/tail gluing. An exclusive-part crossing can only be the
-simple head edge; alpha-divisibility handles precisely that crossing. -/
-namespace KLInvariance.TitsSpace
-open Module CoordinateProjection ReflectionPairTranslation
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-open FrobeniusDuality ReflectionFrobenius
-attribute [local instance] restrictedModule restrictedTower restrictedFree restrictedFinite
-
- theorem bottHead_difference (i : I) (l : List I) (m : BottSamelsonModule cs (i::l)) :
-    ∃ n : BottSamelsonModule cs l,
-      bottHeadTail cs i l m-(bottHeadReflectedTail cs i l m).val=simpleRootPolynomial (M := M) i • n :=
-  tensor_difference (coefficientAction cs (cs.simple i)).toRingEquiv (fun c => c.property)
-    (simpleRootPolynomial (M := M) i) (simple_divisible cs i) m
-
- def bottHeadPatch (i : I) (l : List I) (U : Set W) (m : BottSamelsonModule cs (i::l)) :
-    ∀ x : simpleUnion cs i U, bottStalk cs l x.val := by
-  classical
-  exact fun x => if x.val∈U then bottStalkProjection cs l x.val (bottHeadTail cs i l m)
-    else bottStalkProjection cs l x.val (bottHeadReflectedTail cs i l m).val
-
- theorem bottHeadPatch_compatible (i : I) (l : List I) (U : Set W) (hU : BruhatUpper cs U)
-    (m : BottSamelsonModule cs (i::l))
-    (hJ : ∀ x∈simpleInter cs i U,
-      bottStalkProjection cs l x (bottHeadTail cs i l m)=
-        bottStalkProjection cs l x (bottHeadReflectedTail cs i l m).val) :
-    BottCompatible cs l (simpleUnion cs i U) (bottHeadPatch cs i l U m) := by
-  classical
-  intro x y hx hy hxy
-  by_cases hxU : x∈U
-  · have hyU := hU (.single hxy) hxU
-    simp only [bottHeadPatch,ite_eq_left hxU,ite_eq_left hyU]
-    exact ⟨bottHeadTail cs i l m,rfl,rfl⟩
-  · by_cases hyU : y∈U
-    · simp only [bottHeadPatch,ite_eq_right hxU,ite_eq_left hyU]
-      by_cases hsy : cs.simple i*y∈U
-      · exact ⟨(bottHeadReflectedTail cs i l m).val,rfl,(hJ y ⟨hyU,hsy⟩).symm⟩
-      · have he := cross_edge_is_simple cs U hU i hxy (hx.resolve_left hxU) hsy
-        subst y
-        exact PairProjection.cross_lift _ _ (simpleRootPolynomial (M := M) i)
-          (bottReflectionEdge_annihilated cs l x (cs.simple i) (simple_represents cs i))
-          (bottHeadTail cs i l m) (bottHeadReflectedTail cs i l m).val (bottHead_difference cs i l m)
-    · simp only [bottHeadPatch,ite_eq_right hxU,ite_eq_right hyU]
-      exact ⟨(bottHeadReflectedTail cs i l m).val,rfl,rfl⟩
-
- theorem bottHeadPatch_lift (i : I) (l : List I) (H : BottFlabby cs l)
-    (U : Set W) (hU : BruhatUpper cs U) (m : BottSamelsonModule cs (i::l))
-    (hJ : ∀ x∈simpleInter cs i U,
-      bottStalkProjection cs l x (bottHeadTail cs i l m)=
-        bottStalkProjection cs l x (bottHeadReflectedTail cs i l m).val) :
-    ∃ n : BottSamelsonModule cs l,
-      (∀ x∈U, bottStalkProjection cs l x n=bottStalkProjection cs l x (bottHeadTail cs i l m)) ∧
-      (∀ x∈simplePull cs i U,
-        bottStalkProjection cs l x n=bottStalkProjection cs l x (bottHeadReflectedTail cs i l m).val) := by
-  classical
-  obtain ⟨n,hn⟩ := H (simpleUnion cs i U) (hU.simpleUnion cs U i)
-    (bottHeadPatch cs i l U m) (bottHeadPatch_compatible cs i l U hU m hJ)
-  refine ⟨n,?_,?_⟩
-  · intro x hx
-    have hh := congrFun hn ⟨x,Or.inl hx⟩
-    simpa only [CoordinateProjection.eval,LinearMap.pi_apply,bottHeadPatch,ite_eq_left hx] using hh
-  · intro x hx
-    have hh := congrFun hn ⟨x,Or.inr hx⟩
-    by_cases hxU : x∈U
-    · have hh' : bottStalkProjection cs l x n=bottStalkProjection cs l x (bottHeadTail cs i l m) := by
-        simpa only [CoordinateProjection.eval,LinearMap.pi_apply,bottHeadPatch,ite_eq_left hxU] using hh
-      exact hh'.trans (hJ x ⟨hxU,hx⟩)
-    · simpa only [CoordinateProjection.eval,LinearMap.pi_apply,bottHeadPatch,ite_eq_right hxU] using hh
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-namespace KLInvariance.TitsSpace
-open Module TensorProduct CoordinateProjection ReflectionPairTranslation
-open FrobeniusDuality ReflectionFrobenius
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-attribute [local instance] restrictedModule restrictedTower restrictedFree restrictedFinite
-
- def bottSetRestrict (l : List I) {U K : Set W} (h : U⊆K) :
-    bottSetImage cs l K →ₗ[SymmetricCoefficient (M := M)] bottSetImage cs l U :=
-  restrict (bottStalkProjection cs l) h
-
- theorem bottSetRestrict_project (l : List I) {U K : Set W} (h : U⊆K)
-    (n : BottSamelsonModule cs l) :
-    bottSetRestrict cs l h (bottSetProjection cs l K n)=bottSetProjection cs l U n := rfl
-
- theorem bottSetRestrict_surjective (l : List I) {U K : Set W} (h : U⊆K) :
-    Function.Surjective (bottSetRestrict cs l h) := restrict_surjective _ h
-
- theorem bottSetRestrict_zero_iff (l : List I) {U K : Set W} (h : U⊆K)
-    (z : bottSetImage cs l K) : bottSetRestrict cs l h z=0 ↔
-      ∀ x hx, x∈U → z.val ⟨x,hx⟩=0 := restrict_zero_iff (bottStalkProjection cs l) h z
-
- def bottSeed (i : I) (l : List I) (n : BottSamelsonModule cs l) :
-    BottSamelsonModule cs (i::l) := (1 : SymmetricCoefficient (M := M)) ⊗ₜ[simpleFixed cs i] n
-
- theorem bottSeed_first (i : I) (l : List I) (n : BottSamelsonModule cs l) :
-    bottHeadTail cs i l (bottSeed cs i l n)=n := bottHeadTail_seed cs i l n
-
- theorem bottSeed_second (i : I) (l : List I) (n : BottSamelsonModule cs l) :
-    bottHeadReflectedTail cs i l (bottSeed cs i l n)=Twist.mk n := bottHeadReflectedTail_seed cs i l n
-
- theorem bottSetFirst_seed (i : I) (l : List I) (U : Set W) (n : BottSamelsonModule cs l) :
-    bottSetFirst cs i l U (bottSetProjection cs (i::l) U (bottSeed cs i l n))=
-      bottSetProjection cs l U n :=
-  (bottSetFirst_project cs i l U _).trans
-    (congrArg (bottSetProjection cs l U) (bottSeed_first cs i l n))
-
- theorem bottSetSecond_seed (i : I) (l : List I) (U : Set W) (n : BottSamelsonModule cs l) :
-    bottSetSecond cs i l U (bottSetProjection cs (i::l) U (bottSeed cs i l n))=
-      Twist.mk (bottSetProjection cs l (simplePull cs i U) n) :=
-  (bottSetSecond_project cs i l U _).trans
-    (congrArg (fun t => Twist.mk (bottSetProjection cs l (simplePull cs i U) t.val))
-      (bottSeed_second cs i l n))
-
- theorem bottSeed_zero_stable (i : I) (l : List I) (U : Set W)
-    (hU : ∀ x, cs.simple i*x∈U ↔ x∈U)
-    (n : BottSamelsonModule cs l) (hn : bottSetProjection cs l U n=0) :
-    bottSetProjection cs (i::l) U (bottSeed cs i l n)=0 := by
-  apply (bottHeadSet_zero_iff cs i l U _).mpr
-  constructor
-  · exact (congrArg (bottSetProjection cs l U) (bottSeed_first cs i l n)).trans hn
-  · apply ((congrArg (fun t => bottSetProjection cs l (simplePull cs i U) t.val)
-      (bottSeed_second cs i l n)).trans ·)
-    apply (bottSetProjection_zero_iff cs l (simplePull cs i U) n).mpr
-    intro x hx
-    exact (bottSetProjection_zero_iff cs l U n).mp hn x ((hU x).mp hx)
-
- theorem bottSetFirst_restrict (i : I) (l : List I) {U K : Set W} (h : U⊆K)
-    (z : bottSetImage cs (i::l) K) :
-    bottSetFirst cs i l U (bottSetRestrict cs (i::l) h z)=
-      bottSetRestrict cs l h (bottSetFirst cs i l K z) := by
-  refine functions_eq_of_surjective (bottSetProjection cs (i::l) K)
-    (bottSetProjection_surjective cs (i::l) K)
-    (fun z => bottSetFirst cs i l U (bottSetRestrict cs (i::l) h z))
-    (fun z => bottSetRestrict cs l h (bottSetFirst cs i l K z)) ?_ z
-  intro m
-  exact (bottSetFirst_project cs i l U m).trans
-    (congrArg (bottSetRestrict cs l h) (bottSetFirst_project cs i l K m)).symm
-
- theorem bottSeed_zero_inter (i : I) (l : List I) (U : Set W)
-    (n : BottSamelsonModule cs l) (hn : bottSetProjection cs l U n=0) :
-    bottSetProjection cs (i::l) (simpleInter cs i U) (bottSeed cs i l n)=0 := by
-  apply (bottHeadSet_zero_iff cs i l (simpleInter cs i U) _).mpr
-  have h := (bottSetProjection_zero_iff cs l U n).mp hn
-  constructor
-  · apply ((congrArg (bottSetProjection cs l (simpleInter cs i U))
-      (bottSeed_first cs i l n)).trans ·)
-    apply (bottSetProjection_zero_iff cs l (simpleInter cs i U) n).mpr
-    intro x hx
-    exact h x hx.1
-  · apply ((congrArg (fun t => bottSetProjection cs l (simplePull cs i (simpleInter cs i U)) t.val)
-      (bottSeed_second cs i l n)).trans ·)
-    apply (bottSetProjection_zero_iff cs l (simplePull cs i (simpleInter cs i U)) n).mpr
-    intro x hx
-    exact h x (by simpa only [cs.simple_mul_simple_cancel_left] using hx.2)
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-namespace KLInvariance
-universe ur um un up
-variable {R : Type ur} [CommRing R]
-  {M : Type um} [AddCommGroup M] [Module R M]
-  {N : Type un} [AddCommGroup N] [Module R N]
-  {P : Type up} [AddCommGroup P] [Module R P]
-noncomputable section
- theorem projective_of_same_range (f : M →ₗ[R] P) (hf : Function.Injective f)
-    (g : N →ₗ[R] P) (hg : Function.Injective g) (h : LinearMap.range f=LinearMap.range g)
-    [Module.Projective R N] : Module.Projective R M :=
-  Module.Projective.of_equiv (((LinearEquiv.ofInjective g hg).trans
-    (LinearEquiv.ofEq _ _ h.symm)).trans (LinearEquiv.ofInjective f hf).symm)
-
-attribute [local instance] RingHomInvPair.of_ringEquiv RingHomInvPair.of_ringEquiv_symm
-
-namespace ReflectionPairTranslation.Twist
- theorem projective (τ : R ≃+* R) [Module.Projective R N] : Module.Projective R (Twist τ N) := by
-  let : RingHomInvPair τ.symm.toRingHom τ.toRingHom := RingHomInvPair.of_ringEquiv_symm τ
-  let : RingHomInvPair τ.toRingHom τ.symm.toRingHom := RingHomInvPair.of_ringEquiv τ
-  exact Module.Projective.of_equiv (σ' := τ.toRingHom) (equiv τ N)
-end ReflectionPairTranslation.Twist
-end
-end KLInvariance
-
-end
-
-
-section
-
-namespace KLInvariance
-universe ur um uh up us
-variable {R : Type ur} [CommRing R]
-  {M : Type um} [AddCommGroup M] [Module R M]
-  {H : Type uh} [AddCommGroup H] [Module R H]
-  {P : Type up} [AddCommGroup P] [Module R P]
-  {S : Type us} [AddCommGroup S] [Module R S]
-
- theorem kernel_coordinate_injective (f : M →ₗ[R] P) (g : M →ₗ[R] S)
-    (h : Function.Injective (f.prod g)) :
-    Function.Injective (g.comp (LinearMap.ker f).subtype) := by
-  intro m n he
-  apply Subtype.ext
-  apply h
-  exact Prod.ext (m.property.trans n.property.symm) he
-
- theorem common_kernel_coordinate_injective (r : M →ₗ[R] H) (f : M →ₗ[R] P) (g : M →ₗ[R] S)
-    (h : Function.Injective (f.prod g)) :
-    Function.Injective (g.comp (LinearMap.ker r ⊓ LinearMap.ker f).subtype) := by
-  intro m n he
-  apply Subtype.ext
-  apply h
-  exact Prod.ext (m.property.2.trans n.property.2.symm) he
-
- theorem projective_of_coordinates (f : M →ₗ[R] P) (hf : Function.Injective f)
-    (g : H →ₗ[R] P) (hg : Function.Injective g)
-    (hfg : ∀ m, ∃ n, f m=g n) (hgf : ∀ n, ∃ m, f m=g n)
-    [Module.Projective R H] : Module.Projective R M := by
-  apply projective_of_same_range f hf g hg
-  ext p
-  constructor
-  · rintro ⟨m,rfl⟩
-    obtain ⟨n,hn⟩ := hfg m
-    exact ⟨n,hn.symm⟩
-  · rintro ⟨n,rfl⟩
-    exact hgf n
-
-namespace ReflectionPairTranslation
- theorem twistMap_injective (τ : R ≃+* R) (f : M →ₗ[R] P) (hf : Function.Injective f) :
-    Function.Injective (twistMap τ f) := by
-  intro m n h
-  exact Twist.ext (hf (congrArg Twist.val h))
-end ReflectionPairTranslation
-
-namespace CoordinateProjection
-universe uv ux
-variable {V : Type uv} {X : V → Type ux} [∀ x, AddCommGroup (X x)] [∀ x, Module R (X x)]
-  (f : ∀ x, M →ₗ[R] X x)
- theorem restrict_joint_injective {U V' K : Set V} (hU : U⊆K) (hV : V'⊆K)
-    (hc : ∀ x∈K, x∈U ∨ x∈V') :
-    Function.Injective ((restrict f hU).prod (restrict f hV)) := by
-  intro p q h
-  apply Subtype.ext
-  funext x
-  rcases hc x.val x.property with hx|hx
-  · exact congrArg (fun z : image f U => z.val ⟨x.val,hx⟩) (congrArg Prod.fst h)
-  · exact congrArg (fun z : image f V' => z.val ⟨x.val,hx⟩) (congrArg Prod.snd h)
-end CoordinateProjection
-end KLInvariance
-
-end
-
-
-section
-
-namespace KLInvariance.TitsSpace
-open Module CoordinateProjection ReflectionPairTranslation
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
- def bottInterRestriction (i : I) (l : List I) (U : Set W) :
-    bottSetImage cs l U →ₗ[SymmetricCoefficient (M := M)] bottSetImage cs l (simpleInter cs i U) :=
-  bottSetRestrict cs l (fun _ h => h.1)
-
- def bottUnionRestriction (i : I) (l : List I) (U : Set W) :
-    bottSetImage cs l (simpleUnion cs i U) →ₗ[SymmetricCoefficient (M := M)] bottSetImage cs l U :=
-  bottSetRestrict cs l (fun _ h => Or.inl h)
-
- def bottUnionPullRestriction (i : I) (l : List I) (U : Set W) :
-    bottSetImage cs l (simpleUnion cs i U) →ₗ[SymmetricCoefficient (M := M)]
-      bottSetImage cs l (simplePull cs i U) :=
-  bottSetRestrict cs l (fun _ h => Or.inr h)
-
- def bottCommonKernel (i : I) (l : List I) (U : Set W) :
-    Submodule (SymmetricCoefficient (M := M)) (bottSetImage cs (i::l) U) :=
-  LinearMap.ker (bottInterRestriction cs i (i::l) U) ⊓ LinearMap.ker (bottSetFirst cs i l U)
-
- def bottCommonSecond (i : I) (l : List I) (U : Set W) :
-    bottCommonKernel cs i l U →ₗ[SymmetricCoefficient (M := M)]
-      Twist (coefficientAction cs (cs.simple i)).toRingEquiv (bottSetImage cs l (simplePull cs i U)) :=
-  (bottSetSecond cs i l U).comp (bottCommonKernel cs i l U).subtype
-
- theorem bottCommonSecond_injective (i : I) (l : List I) (U : Set W) :
-    Function.Injective (bottCommonSecond cs i l U) :=
-  common_kernel_coordinate_injective _ _ _ (bottSetComponents_injective cs i l U)
-
- def bottUnionKernelSecond (i : I) (l : List I) (U : Set W) :
-    LinearMap.ker (bottUnionRestriction cs i l U) →ₗ[SymmetricCoefficient (M := M)]
-      bottSetImage cs l (simplePull cs i U) :=
-  (bottUnionPullRestriction cs i l U).comp (LinearMap.ker (bottUnionRestriction cs i l U)).subtype
-
- theorem bottUnionKernelSecond_injective (i : I) (l : List I) (U : Set W) :
-    Function.Injective (bottUnionKernelSecond cs i l U) := by
-  apply kernel_coordinate_injective
-  exact restrict_joint_injective (bottStalkProjection cs l)
-    (fun _ h => Or.inl h) (fun _ h => Or.inr h) (fun _ h => h)
-
- theorem bottHeadPatch_lift_of_zero_inter (i : I) (l : List I) (H : BottFlabby cs l)
-    (U : Set W) (hU : BruhatUpper cs U) (m : BottSamelsonModule cs (i::l))
-    (hm : bottSetProjection cs (i::l) (simpleInter cs i U) m=0) :
-    ∃ n : BottSamelsonModule cs l,
-      (∀ x∈U, bottStalkProjection cs l x n=bottStalkProjection cs l x (bottHeadTail cs i l m)) ∧
-      (∀ x∈simplePull cs i U,
-        bottStalkProjection cs l x n=bottStalkProjection cs l x (bottHeadReflectedTail cs i l m).val) := by
-  apply bottHeadPatch_lift cs i l H U hU m
-  have hz := (bottHeadSet_zero_iff cs i l (simpleInter cs i U) m).mp hm
-  intro x hx
-  apply ((bottSetProjection_zero_iff cs l (simpleInter cs i U) _).mp hz.1 x hx).trans
-  exact ((bottSetProjection_zero_iff cs l (simplePull cs i (simpleInter cs i U)) _).mp hz.2 x
-    ((simpleInter_stable cs i U x).mpr hx)).symm
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-namespace KLInvariance.TitsSpace
-open Module CoordinateProjection ReflectionPairTranslation
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
- theorem bottCommon_lift (i : I) (l : List I) (U : Set W)
-    (z : bottCommonKernel cs i l U) :
-    ∃ m : BottSamelsonModule cs (i::l),
-      bottSetProjection cs (i::l) U m=z.val ∧
-      bottSetProjection cs (i::l) (simpleInter cs i U) m=0 ∧
-      bottSetProjection cs l U (bottHeadTail cs i l m)=0 := by
-  let m : BottSamelsonModule cs (i::l) :=
-    (bottSetProjection_surjective cs (i::l) U z.val).choose
-  have hm : bottSetProjection cs (i::l) U m=z.val :=
-    (bottSetProjection_surjective cs (i::l) U z.val).choose_spec
-  refine ⟨m,hm,?_,?_⟩
-  · exact (congrArg (bottInterRestriction cs i (i::l) U) hm).trans z.property.1
-  · exact (bottSetFirst_project cs i l U m).symm.trans
-      ((congrArg (bottSetFirst cs i l U) hm).trans z.property.2)
-
- theorem bottCommon_patch_lift (i : I) (l : List I) (H : BottFlabby cs l)
-    (U : Set W) (hU : BruhatUpper cs U) (m : BottSamelsonModule cs (i::l))
-    (hmJ : bottSetProjection cs (i::l) (simpleInter cs i U) m=0)
-    (hmp : bottSetProjection cs l U (bottHeadTail cs i l m)=0) :
-    ∃ n : BottSamelsonModule cs l, bottSetProjection cs l U n=0 ∧
-      bottSetProjection cs l (simplePull cs i U) n=
-      bottSetProjection cs l (simplePull cs i U) (bottHeadReflectedTail cs i l m).val := by
-  let n : BottSamelsonModule cs l := (bottHeadPatch_lift_of_zero_inter cs i l H U hU m hmJ).choose
-  have hnp := (bottHeadPatch_lift_of_zero_inter cs i l H U hU m hmJ).choose_spec.1
-  have hnq := (bottHeadPatch_lift_of_zero_inter cs i l H U hU m hmJ).choose_spec.2
-  refine ⟨n,?_,Subtype.ext (funext fun x => hnq x.val x.property)⟩
-  apply (bottSetProjection_zero_iff cs l U n).mpr
-  intro x hx
-  exact (hnp x hx).trans ((bottSetProjection_zero_iff cs l U _).mp hmp x hx)
-
- theorem bottCommon_range_of_patch (i : I) (l : List I) (U : Set W)
-    (z : bottCommonKernel cs i l U) (m : BottSamelsonModule cs (i::l))
-    (hm : bottSetProjection cs (i::l) U m=z.val) (n : BottSamelsonModule cs l)
-    (hnU : bottSetProjection cs l U n=0)
-    (hnq : bottSetProjection cs l (simplePull cs i U) n=
-      bottSetProjection cs l (simplePull cs i U) (bottHeadReflectedTail cs i l m).val) :
-    ∃ d : Twist (coefficientAction cs (cs.simple i)).toRingEquiv
-        (LinearMap.ker (bottUnionRestriction cs i l U)),
-      bottCommonSecond cs i l U z=
-        twistMap (coefficientAction cs (cs.simple i)).toRingEquiv (bottUnionKernelSecond cs i l U) d := by
-  let d : LinearMap.ker (bottUnionRestriction cs i l U) :=
-    ⟨bottSetProjection cs l (simpleUnion cs i U) n,hnU⟩
-  refine ⟨Twist.mk d,?_⟩
-  change bottSetSecond cs i l U z.val=Twist.mk (bottSetProjection cs l (simplePull cs i U) n)
-  exact ((congrArg (bottSetSecond cs i l U) hm).symm.trans
-    (bottSetSecond_project cs i l U m)).trans (congrArg Twist.mk hnq.symm)
-
- theorem bottCommon_range_of_lift (i : I) (l : List I) (H : BottFlabby cs l)
-    (U : Set W) (hU : BruhatUpper cs U) (z : bottCommonKernel cs i l U)
-    (m : BottSamelsonModule cs (i::l)) (hm : bottSetProjection cs (i::l) U m=z.val)
-    (hmJ : bottSetProjection cs (i::l) (simpleInter cs i U) m=0)
-    (hmp : bottSetProjection cs l U (bottHeadTail cs i l m)=0) :
-    ∃ d : Twist (coefficientAction cs (cs.simple i)).toRingEquiv
-        (LinearMap.ker (bottUnionRestriction cs i l U)),
-      bottCommonSecond cs i l U z=
-        twistMap (coefficientAction cs (cs.simple i)).toRingEquiv (bottUnionKernelSecond cs i l U) d := by
-  exact bottCommon_range_of_patch cs i l U z m hm _
-    (bottCommon_patch_lift cs i l H U hU m hmJ hmp).choose_spec.1
-    (bottCommon_patch_lift cs i l H U hU m hmJ hmp).choose_spec.2
-
- theorem bottCommon_range_forward (i : I) (l : List I) (H : BottFlabby cs l)
-    (U : Set W) (hU : BruhatUpper cs U) (z : bottCommonKernel cs i l U) :
-    ∃ d : Twist (coefficientAction cs (cs.simple i)).toRingEquiv
-        (LinearMap.ker (bottUnionRestriction cs i l U)),
-      bottCommonSecond cs i l U z=
-        twistMap (coefficientAction cs (cs.simple i)).toRingEquiv (bottUnionKernelSecond cs i l U) d :=
-  bottCommon_range_of_lift cs i l H U hU z _ (bottCommon_lift cs i l U z).choose_spec.1
-    (bottCommon_lift cs i l U z).choose_spec.2.1
-    (bottCommon_lift cs i l U z).choose_spec.2.2
-
- theorem bottCommon_range_backward (i : I) (l : List I) (U : Set W)
-    (d : Twist (coefficientAction cs (cs.simple i)).toRingEquiv
-      (LinearMap.ker (bottUnionRestriction cs i l U))) :
-    ∃ z : bottCommonKernel cs i l U,
-      bottCommonSecond cs i l U z=
-        twistMap (coefficientAction cs (cs.simple i)).toRingEquiv (bottUnionKernelSecond cs i l U) d := by
-  let n : BottSamelsonModule cs l :=
-    (bottSetProjection_surjective cs l (simpleUnion cs i U) d.val.val).choose
-  have hn : bottSetProjection cs l (simpleUnion cs i U) n=d.val.val :=
-    (bottSetProjection_surjective cs l (simpleUnion cs i U) d.val.val).choose_spec
-  have hnU : bottSetProjection cs l U n=0 :=
-    (congrArg (bottUnionRestriction cs i l U) hn).trans d.val.property
-  let z : bottCommonKernel cs i l U :=
-    ⟨bottSetProjection cs (i::l) U (bottSeed cs i l n),
-      bottSeed_zero_inter cs i l U n hnU,(bottSetFirst_seed cs i l U n).trans hnU⟩
-  refine ⟨z,?_⟩
-  exact (bottSetSecond_seed cs i l U n).trans
-    (congrArg Twist.mk (congrArg (bottUnionPullRestriction cs i l U) hn))
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Projectivity on reflection-stable upper sets, via an ACTUAL tensor
-splitting rather than an asserted reflexivity of a projected lattice. -/
-namespace KLInvariance.TitsSpace
-open Module CoordinateProjection ProjectiveRange ReflectionPairTranslation
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
- theorem bottHeadLocalPair_eq_iff (i : I) (l : List I) (x : W)
-    (m n : BottSamelsonModule cs (i::l)) :
-    bottStalkProjection cs (i::l) x m=bottStalkProjection cs (i::l) x n ↔
-      bottHeadLocalPair cs i l x m=bottHeadLocalPair cs i l x n :=
-  maps_eq_iff_of_zero_iff _ _ (bottHeadLocalPair_zero_iff cs i l x) m n
-
- theorem bottTensorEnd_set_preserve (i : I) (l : List I) (U : Set W)
-    (hU : ∀ x, cs.simple i*x∈U ↔ x∈U)
-    (e : Module.End (SymmetricCoefficient (M := M)) (BottSamelsonModule cs l))
-    (he : ∀ n x, x∈U → bottStalkProjection cs l x (e n)=bottStalkProjection cs l x n)
-    (m : BottSamelsonModule cs (i::l)) :
-    bottSetProjection cs (i::l) U (bottTensorEnd cs i l e m)=bottSetProjection cs (i::l) U m := by
-  apply Subtype.ext
-  funext x
-  apply (bottHeadLocalPair_eq_iff cs i l x.val _ _).mpr
-  apply (bottHeadLocalPair_value cs i l x.val _).trans
-  apply Eq.trans ?_ (bottHeadLocalPair_value cs i l x.val m).symm
-  apply Prod.ext
-  · exact (congrArg (bottStalkProjection cs l x.val) (bottTensorEnd_first cs i l e m)).trans
-      (he _ x.val x.property)
-  · apply Twist.ext
-    exact (congrArg (fun t => bottStalkProjection cs l (cs.simple i*x.val) t.val)
-      (bottTensorEnd_second cs i l e m)).trans
-      (he _ (cs.simple i*x.val) ((hU x.val).mpr x.property))
-
- theorem bottTensorEnd_set_kills (i : I) (l : List I) (U : Set W)
-    (hU : ∀ x, cs.simple i*x∈U ↔ x∈U)
-    (e : Module.End (SymmetricCoefficient (M := M)) (BottSamelsonModule cs l))
-    (hz : ∀ n, bottSetProjection cs l U n=0 → e n=0)
-    (m : BottSamelsonModule cs (i::l)) (hm : bottSetProjection cs (i::l) U m=0) :
-    bottTensorEnd cs i l e m=0 := by
-  have hmpq := (bottHeadSet_zero_iff cs i l U m).mp hm
-  have hqzero : bottSetProjection cs l U (bottHeadReflectedTail cs i l m).val=0 := by
-    apply (bottSetProjection_zero_iff cs l U _).mpr
-    intro x hx
-    exact (bottSetProjection_zero_iff cs l (simplePull cs i U) _).mp hmpq.2 x ((hU x).mpr hx)
-  apply bottHead_components_injective cs i l
-  · exact (bottTensorEnd_first cs i l e m).trans (hz _ hmpq.1)
-  · apply Twist.ext
-    exact (congrArg Twist.val (bottTensorEnd_second cs i l e m)).trans (hz _ hqzero)
-
- theorem bottStableUpper_projective (i : I) (l : List I) (U : Set W)
-    (hU : ∀ x, cs.simple i*x∈U ↔ x∈U)
-    [Module.Projective (SymmetricCoefficient (M := M)) (bottSetImage cs l U)] :
-    Module.Projective (SymmetricCoefficient (M := M)) (bottSetImage cs (i::l) U) := by
-  obtain ⟨s,hs⟩ := Module.projective_lifting_property (bottSetProjection cs l U)
-    (LinearMap.id : bottSetImage cs l U →ₗ[SymmetricCoefficient (M := M)] bottSetImage cs l U)
-    (bottSetProjection_surjective cs l U)
-  have hs' : ∀ p, bottSetProjection cs l U (s p)=p := LinearMap.congr_fun hs
-  let e := s.comp (bottSetProjection cs l U)
-  have he : ∀ n x, x∈U → bottStalkProjection cs l x (e n)=bottStalkProjection cs l x n := by
-    intro n x hx
-    exact congrArg (fun z : bottSetImage cs l U => z.val ⟨x,hx⟩) (hs' (bottSetProjection cs l U n))
-  have hz : ∀ n, bottSetProjection cs l U n=0 → e n=0 := by
-    intro n hn
-    exact (congrArg s hn).trans s.map_zero
-  exact projective_of_end (bottSetProjection cs (i::l) U) (bottSetProjection_surjective cs (i::l) U)
-    (bottTensorEnd cs i l e) (bottTensorEnd_set_preserve cs i l U hU e he)
-    (bottTensorEnd_set_kills cs i l U hU e hz)
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-namespace KLInvariance.ProjectiveRange
-universe ur um uh up us
-variable {R : Type ur} [CommRing R]
-  {M : Type um} [AddCommGroup M] [Module R M]
-  {H : Type uh} [AddCommGroup H] [Module R H]
-  {P : Type up} [AddCommGroup P] [Module R P]
-  {S : Type us} [AddCommGroup S] [Module R S]
-  (r : M →ₗ[R] H) (f : M →ₗ[R] P) (k : P →ₗ[R] S)
-  (h : ∀ m, r m=0 → k (f m)=0)
-noncomputable section
-
- def onKernel : LinearMap.ker r →ₗ[R] LinearMap.ker k :=
-  (f.comp (LinearMap.ker r).subtype).codRestrict (LinearMap.ker k) (fun m => h m.val m.property)
-
- def kernelOnKernelEquiv : ↥(LinearMap.ker r ⊓ LinearMap.ker f) ≃ₗ[R]
-    LinearMap.ker (onKernel r f k h) where
-  toFun z := ⟨⟨z.val,z.property.1⟩,Subtype.ext z.property.2⟩
-  invFun z := ⟨z.val.val,z.val.property,congrArg Subtype.val z.property⟩
-  left_inv _ := rfl
-  right_inv _ := rfl
-  map_add' _ _ := rfl
-  map_smul' _ _ := rfl
-
- theorem projective_of_two_stage (hr : Function.Surjective r)
-    (hf : Function.Surjective (onKernel r f k h))
-    [Module.Projective R H] [Module.Projective R (LinearMap.ker k)]
-    [Module.Projective R ↥(LinearMap.ker r ⊓ LinearMap.ker f)] :
-    Module.Projective R M := by
-  let : Module.Projective R (LinearMap.ker (onKernel r f k h)) :=
-    Module.Projective.of_equiv (kernelOnKernelEquiv r f k h)
-  let : Module.Projective R (LinearMap.ker r) := projective_of_kernel (onKernel r f k h) hf
-  exact projective_of_kernel r hr
-end
-end KLInvariance.ProjectiveRange
-
-end
-
-
-section
-
-namespace KLInvariance.TitsSpace
-open Module CoordinateProjection ReflectionPairTranslation ProjectiveRange
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
- theorem bottInter_first_zero (i : I) (l : List I) (U : Set W)
-    (z : bottSetImage cs (i::l) U) (hz : bottInterRestriction cs i (i::l) U z=0) :
-    bottInterRestriction cs i l U (bottSetFirst cs i l U z)=0 :=
-  (bottSetFirst_restrict cs i l (fun _ h => h.1) z).symm.trans
-    ((congrArg (bottSetFirst cs i l (simpleInter cs i U)) hz).trans (map_zero _))
-
- theorem bottInterKernelFirst_surjective (i : I) (l : List I) (U : Set W) :
-    Function.Surjective (onKernel (bottInterRestriction cs i (i::l) U) (bottSetFirst cs i l U)
-      (bottInterRestriction cs i l U) (bottInter_first_zero cs i l U)) := by
-  intro z
-  let n : BottSamelsonModule cs l := (bottSetProjection_surjective cs l U z.val).choose
-  have hn : bottSetProjection cs l U n=z.val := (bottSetProjection_surjective cs l U z.val).choose_spec
-  have hnJ : bottSetProjection cs l (simpleInter cs i U) n=0 :=
-    (congrArg (bottInterRestriction cs i l U) hn).trans z.property
-  let w : LinearMap.ker (bottInterRestriction cs i (i::l) U) :=
-    ⟨bottSetProjection cs (i::l) U (bottSeed cs i l n),
-      bottSeed_zero_stable cs i l (simpleInter cs i U) (simpleInter_stable cs i U) n hnJ⟩
-  exact ⟨w,Subtype.ext ((bottSetFirst_seed cs i l U n).trans hn)⟩
-
- theorem bottCommon_projective (i : I) (l : List I) (HP : BottProjectiveUpper cs l)
-    (HF : BottFlabby cs l) (U : Set W) (hU : BruhatUpper cs U) :
-    Module.Projective (SymmetricCoefficient (M := M)) (bottCommonKernel cs i l U) := by
-  let : Module.Projective (SymmetricCoefficient (M := M)) (bottSetImage cs l U) := HP U hU
-  let : Module.Projective (SymmetricCoefficient (M := M)) (bottSetImage cs l (simpleUnion cs i U)) :=
-    HP (simpleUnion cs i U) (hU.simpleUnion cs U i)
-  let : Module.Projective (SymmetricCoefficient (M := M)) (LinearMap.ker (bottUnionRestriction cs i l U)) :=
-    kernel_projective (bottUnionRestriction cs i l U) (bottSetRestrict_surjective cs l (fun _ h => Or.inl h))
-  let : Module.Projective (SymmetricCoefficient (M := M))
-      (Twist (coefficientAction cs (cs.simple i)).toRingEquiv (LinearMap.ker (bottUnionRestriction cs i l U))) :=
-    Twist.projective _
-  exact projective_of_coordinates (bottCommonSecond cs i l U) (bottCommonSecond_injective cs i l U)
-    (twistMap (coefficientAction cs (cs.simple i)).toRingEquiv (bottUnionKernelSecond cs i l U))
-    (twistMap_injective _ _ (bottUnionKernelSecond_injective cs i l U))
-    (bottCommon_range_forward cs i l HF U hU) (bottCommon_range_backward cs i l U)
-
- theorem bottProjectiveUpper_cons (i : I) (l : List I) (HP : BottProjectiveUpper cs l)
-    (HF : BottFlabby cs l) : BottProjectiveUpper cs (i::l) := by
-  intro U hU
-  let : Module.Projective (SymmetricCoefficient (M := M)) (bottSetImage cs l U) := HP U hU
-  let : Module.Projective (SymmetricCoefficient (M := M)) (bottSetImage cs l (simpleInter cs i U)) :=
-    HP (simpleInter cs i U) (hU.simpleInter cs U i)
-  let : Module.Projective (SymmetricCoefficient (M := M)) (bottSetImage cs (i::l) (simpleInter cs i U)) :=
-    bottStableUpper_projective cs i l (simpleInter cs i U) (simpleInter_stable cs i U)
-  let : Module.Projective (SymmetricCoefficient (M := M)) (LinearMap.ker (bottInterRestriction cs i l U)) :=
-    kernel_projective (bottInterRestriction cs i l U) (bottSetRestrict_surjective cs l (fun _ h => h.1))
-  let : Module.Projective (SymmetricCoefficient (M := M))
-      ↥(LinearMap.ker (bottInterRestriction cs i (i::l) U) ⊓ LinearMap.ker (bottSetFirst cs i l U)) :=
-    bottCommon_projective cs i l HP HF U hU
-  exact projective_of_two_stage (bottInterRestriction cs i (i::l) U) (bottSetFirst cs i l U)
-    (bottInterRestriction cs i l U) (bottInter_first_zero cs i l U)
-    (bottSetRestrict_surjective cs (i::l) (fun _ h => h.1)) (bottInterKernelFirst_surjective cs i l U)
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! The codimension-one two-character distinction needed for integral
-Bott--Samelson section realization. No BMP character or pairing is assumed. -/
-namespace KLInvariance.Realization
-open Module
-variable {V : Type*} [AddCommGroup V] [Module ℝ V]
-
- theorem orthogonal_eq_id_or_rootReflection
-    (B : LinearMap.BilinForm ℝ V) (a : V) (ha : B a a=2)
-    (g : V ≃ₗ[ℝ] V) (hg : ∀ v w, B (g v) (g w)=B v w)
-    (hline : ∀ v, g v-v ∈ Submodule.span ℝ {a}) :
-    g.toLinearMap=LinearMap.id ∨ ∀ v, g v=Module.preReflection a (B a) v := by
-  obtain ⟨r,hr⟩ := Submodule.mem_span_singleton.mp (hline a)
-  have hga : g a=(1+r) • a := by rw [add_smul,one_smul,hr]; module
-  have hn := hg a a
-  rw [hga] at hn
-  simp only [map_smul,LinearMap.smul_apply,smul_eq_mul,ha] at hn
-  have hr0 : r*(r+2)=0 := by nlinarith
-  rcases mul_eq_zero.mp hr0 with hr | hr
-  · left
-    have hga' : g a=a := by simpa only [hr,add_zero,one_smul] using hga
-    ext v
-    obtain ⟨c,hc⟩ := Submodule.mem_span_singleton.mp (hline v)
-    have hgv : g v=v+c • a := by rw [hc]; module
-    have h := hg a v
-    rw [hga',hgv] at h
-    simp only [map_add,map_smul,ha,smul_eq_mul] at h
-    have hc0 : c=0 := by linarith
-    change g v=v
-    simpa only [hc0,zero_smul,add_zero] using hgv
-  · right
-    have hr' : r= -2 := by linarith
-    have hga' : g a= -a := by simpa only [hr',show (1:ℝ)+ -2= -1 by norm_num,neg_one_smul] using hga
-    intro v
-    obtain ⟨c,hc⟩ := Submodule.mem_span_singleton.mp (hline v)
-    have hgv : g v=v+c • a := by rw [hc]; module
-    have h := hg a v
-    rw [hga',hgv] at h
-    simp only [map_neg,LinearMap.neg_apply,map_add,map_smul,ha,smul_eq_mul] at h
-    have hc' : c= -(B a v) := by linarith
-    rw [hgv,hc',Module.preReflection_apply,neg_smul,sub_eq_add_neg]
-
-end KLInvariance.Realization
-
-end
-
-
-section
-
-/-! Exact root-prime character classes of the actual Coxeter polynomial
-coefficient action. These are the two-vertex blocks needed in integral
-section realization; they are not BMP or KL-character assumptions. -/
-namespace KLInvariance.TitsSpace
-open Module
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
- theorem reflection_character_congruence_iff {t : W} {a : I → ℝ}
-    (ha : Represents M cs t a) (x y : W) :
-    (∀ p : SymmetricCoefficient (M := M),
-      SymmetricAlgebra.ι ℝ (Extended M) (embed M a) ∣
-        coefficientAction cs x p-coefficientAction cs y p) ↔
-      x=y ∨ y=t*x := by
-  constructor
-  · intro hp
-    let w := y*x⁻¹
-    have hline (v : Extended M) :
-        extendedAction M cs w v-v ∈ Submodule.span ℝ {embed M a} := by
-      let p := (coefficientAction cs x).symm (SymmetricAlgebra.ι ℝ (Extended M) v)
-      have hx : coefficientAction cs x p=SymmetricAlgebra.ι ℝ (Extended M) v :=
-        AlgEquiv.apply_symm_apply _ _
-      have hy : coefficientAction cs y p =
-          coefficientAction cs w (SymmetricAlgebra.ι ℝ (Extended M) v) := by
-        have h := coefficientAction_mul cs w x p
-        rw [show w*x=y by dsimp [w]; group,hx] at h
-        exact h
-      have h := hp p
-      rw [hx,hy,coefficientAction_ι,← map_sub] at h
-      have hs := (EdgeAlgebra.symmetric_ι_dvd_iff (embed M a)
-        (v-extendedAction M cs w v)).mp h
-      simpa only [neg_sub,EdgeAlgebra.rootLine] using (EdgeAlgebra.rootLine (k := ℝ) (embed M a)).neg_mem hs
-    have hnorm : extendedForm M (embed M a) (embed M a)=2 := by
-      rw [(extendedForm_spec M).2.2]
-      exact ha.isRoot.norm
-    rcases Realization.orthogonal_eq_id_or_rootReflection
-      (extendedForm M) (embed M a) hnorm (extendedAction M cs w)
-      (extendedAction_preserves M cs w) hline with hid | href
-    · left
-      have hw : w=1 := by
-        apply extendedAction_injective M cs
-        apply LinearEquiv.ext
-        intro v
-        change extendedAction M cs w v=extendedAction M cs 1 v
-        rw [map_one]
-        change extendedAction M cs w v=v
-        exact LinearMap.congr_fun hid v
-      exact (mul_inv_eq_one.mp hw).symm
-    · right
-      have hw : w=t := by
-        apply extendedAction_injective M cs
-        apply LinearEquiv.ext
-        intro v
-        rw [ha.extendedAction_eq]
-        exact href v
-      exact mul_inv_eq_iff_eq_mul.mp hw
-  · rintro (rfl | rfl)
-    · intro p
-      rw [sub_self]
-      exact dvd_zero _
-    · intro p
-      rw [coefficientAction_mul]
-      exact reflectionCoefficient_divisible cs ha (coefficientAction cs x p)
-
- theorem reflection_character_separates {t : W} {a : I → ℝ}
-    (ha : Represents M cs t a) {x y : W} (hxy : x ≠ y) (ht : y ≠ t*x) :
-    ∃ p : SymmetricCoefficient (M := M),
-      ¬ SymmetricAlgebra.ι ℝ (Extended M) (embed M a) ∣
-        coefficientAction cs x p-coefficientAction cs y p := by
-  by_contra! h
-  exact (reflection_character_congruence_iff cs ha x y).mp h |>.elim hxy ht
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Every nontrivial prime-local congruence class of actual Coxeter
-polynomial characters is a two-vertex reflection block. -/
-namespace KLInvariance.TitsSpace
-open Module
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
- theorem isReflection_of_prime_character_congruence
-    (p : SymmetricCoefficient (M := M)) (hp : Prime p)
-    {x y : W} (hxy : x ≠ y)
-    (hcon : ∀ a, p ∣ coefficientAction cs x a-coefficientAction cs y a) :
-    cs.IsReflection (y*x⁻¹) := by
-  let w := y*x⁻¹
-  have hw : w ≠ 1 := by
-    intro h
-    exact hxy (mul_inv_eq_one.mp h).symm
-  have hex : ∃ v : Extended M, extendedAction M cs w v-v ≠ 0 := by
-    by_contra! h
-    apply hw
-    apply extendedAction_injective M cs
-    apply LinearEquiv.ext
-    intro v
-    rw [map_one]
-    exact sub_eq_zero.mp (h v)
-  obtain ⟨v,hv⟩ := hex
-  let d := extendedAction M cs w v-v
-  have hd : d ≠ 0 := hv
-  have hdiv (z : Extended M) : p ∣ SymmetricAlgebra.ι ℝ (Extended M)
-      (extendedAction M cs w z-z) := by
-    let a := (coefficientAction cs x).symm (SymmetricAlgebra.ι ℝ (Extended M) z)
-    have hx : coefficientAction cs x a=SymmetricAlgebra.ι ℝ (Extended M) z :=
-      AlgEquiv.apply_symm_apply _ _
-    have hy : coefficientAction cs y a =
-        coefficientAction cs w (SymmetricAlgebra.ι ℝ (Extended M) z) := by
-      have h := coefficientAction_mul cs w x a
-      rw [show w*x=y by dsimp [w]; group,hx] at h
-      exact h
-    have h := hcon a
-    rw [hx,hy,coefficientAction_ι,← map_sub] at h
-    simpa only [← map_neg,neg_sub] using dvd_neg.mpr h
-  have hassoc := hp.associated_of_dvd (EdgeAlgebra.symmetric_ι_prime d hd) (hdiv v)
-  have heq : LinearMap.range ((extendedAction M cs w).toLinearMap-LinearMap.id) =
-      Submodule.span ℝ {d} := by
-    apply le_antisymm
-    · rintro z ⟨a,rfl⟩
-      change extendedAction M cs w a-a ∈ Submodule.span ℝ {d}
-      exact (EdgeAlgebra.symmetric_ι_dvd_iff d _).mp
-        (dvd_trans hassoc.symm.dvd (hdiv a))
-    · apply Submodule.span_le.mpr
-      intro z hz
-      rcases Set.mem_singleton_iff.mp hz with rfl
-      exact ⟨v,rfl⟩
-  apply isReflection_of_moving_rank_one M cs w
-  rw [heq,finrank_span_singleton hd]
-
- theorem prime_character_class (p : SymmetricCoefficient (M := M)) (hp : Prime p)
-    {x y : W} (hxy : x ≠ y)
-    (hcon : ∀ a, p ∣ coefficientAction cs x a-coefficientAction cs y a) :
-    ∃ t : W, cs.IsReflection t ∧ y=t*x ∧
-      ∀ z, (∀ a, p ∣ coefficientAction cs x a-coefficientAction cs z a) ↔
-        x=z ∨ z=t*x := by
-  let t := y*x⁻¹
-  have ht := isReflection_of_prime_character_congruence cs p hp hxy hcon
-  let a := (positiveRoot M cs ⟨t,ht⟩).val
-  have ha : Represents M cs t a := positiveRoot_represents M cs ⟨t,ht⟩
-  have he : t*x=y := by dsimp [t]; group
-  obtain ⟨c,hc⟩ := reflectionCoefficient_difference cs ha x
-  rw [he] at hc
-  have hpa : p ∣ SymmetricAlgebra.ι ℝ (Extended M) (embed M a) := hc ▸ hcon c
-  have hane : embed M a ≠ 0 := by
-    intro h
-    have hn : extendedForm M (embed M a) (embed M a)=2 := by
-      rw [(extendedForm_spec M).2.2]
-      exact ha.isRoot.norm
-    rw [h,map_zero] at hn
-    norm_num at hn
-  have hassoc := hp.associated_of_dvd (EdgeAlgebra.symmetric_ι_prime (embed M a) hane) hpa
-  refine ⟨t,ht,he.symm,?_⟩
-  intro z
-  rw [← reflection_character_congruence_iff cs ha x z]
-  exact forall_congr' fun c => ⟨fun h => dvd_trans hassoc.symm.dvd h,
-    fun h => dvd_trans hassoc.dvd h⟩
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Integral local projectors for finite character lattices. This lemma does
-not assert any BMP character theorem or identify a chosen summand with BMP. -/
-namespace KLInvariance.FiniteCharacterLocalization
-universe u v w
-variable {R : Type u} [CommRing R] {X : Type v}
-  {E : X → Type w} [∀ x, AddCommGroup (E x)] [∀ x, Module R (E x)]
-noncomputable section
-
- def diagonal (f : X → R) : (∀ x, E x) →ₗ[R] (∀ x, E x) where
-  toFun m x := f x • m x
-  map_add' := by intro m n; funext x; exact smul_add _ _ _
-  map_smul' := by intro r m; funext x; exact smul_comm _ _ _
-
- def stabilizer (L : Submodule R (∀ x, E x)) : Subalgebra R (X → R) where
-  carrier := {f | ∀ m ∈ L, diagonal f m ∈ L}
-  mul_mem' := by
-    intro f g hf hg m hm
-    have h := hf _ (hg m hm)
-    simpa only [diagonal,LinearMap.coe_mk,AddHom.coe_mk,Pi.mul_apply,mul_smul] using h
-  one_mem' := by intro m hm; simpa [diagonal] using hm
-  add_mem' := by
-    intro f g hf hg m hm
-    have he : diagonal (f+g) m = diagonal f m+diagonal g m := by
-      funext x
-      exact add_smul _ _ _
-    rw [he]
-    exact L.add_mem (hf m hm) (hg m hm)
-  zero_mem' := by
-    intro m _
-    have he : diagonal (0 : X → R) m=0 := by
-      funext x
-      exact zero_smul _ _
-    rw [he]
-    exact L.zero_mem
-  algebraMap_mem' := by
-    intro r m hm
-    exact L.smul_mem r hm
-
- theorem diagonal_mem {L : Submodule R (∀ x, E x)} {f : X → R}
-    (hf : f ∈ stabilizer L) {m : ∀ x, E x} (hm : m ∈ L) :
-    diagonal f m ∈ L := hf m hm
-
-/-- A polynomial in stable diagonal weights which kills the complement of
-`C` and is a common scalar on `C`. Its scalar is a product, not a quotient. -/
- theorem uniform_projector (L : Submodule R (∀ x, E x)) (C : Finset X)
-    (f : X → R) (hf : f ∈ stabilizer L) (hz : ∀ x ∉ C, f x=0) :
-    ∃ g ∈ stabilizer L,
-      (∀ x ∈ C, g x=∏ y ∈ C, f y) ∧ (∀ x ∉ C, g x=0) := by
-  classical
-  let d : R := ∏ y ∈ C, f y
-  let h : X → R := ∏ y ∈ C, (algebraMap R (X → R) (f y)-f)
-  have hh : h ∈ stabilizer L := by
-    exact (stabilizer L).prod_mem fun y _ =>
-      (stabilizer L).sub_mem ((stabilizer L).algebraMap_mem (f y)) hf
-  refine ⟨algebraMap R (X → R) d-h,
-    (stabilizer L).sub_mem ((stabilizer L).algebraMap_mem d) hh,?_,?_⟩
-  · intro x hx
-    change d-h x=d
-    suffices h x=0 by rw [this,sub_zero]
-    dsimp [h]
-    rw [Finset.prod_apply]
-    apply Finset.prod_eq_zero hx
-    simp
-  · intro x hx
-    change d-h x=0
-    suffices h x=d by rw [this,sub_self]
-    dsimp [h,d]
-    rw [Finset.prod_apply]
-    apply Finset.prod_congr rfl
-    intro y _
-    simp [hz x hx]
-
-/-- Separate one residue-character class from its complement without
-inverting any element of the coefficient domain. -/
- theorem class_projector [Fintype X] (L : Submodule R (∀ x, E x)) (p : R) (hp : Prime p)
-    (C : Finset X) (x : X)
-    (test : X → X → R) (htest : ∀ y, test y ∈ stabilizer L)
-    (hzero : ∀ y ∉ C, test y y=0)
-    (hunit : ∀ y ∉ C, ¬ p ∣ test y x)
-    (hcongr : ∀ z ∈ C, ∀ y, p ∣ test y z-test y x) :
-    ∃ d : R, ¬ p ∣ d ∧ ∃ g ∈ stabilizer L,
-      (∀ z ∈ C, g z=d) ∧ (∀ z ∉ C, g z=0) := by
-  classical
-  let f : X → R := ∏ y ∈ Finset.univ \ C, test y
-  have hf : f ∈ stabilizer L :=
-    (stabilizer L).prod_mem fun y _ => htest y
-  have hfu (z : X) (hz : z ∈ C) : ¬ p ∣ f z := by
-    dsimp [f]
-    rw [Finset.prod_apply]
-    apply hp.not_dvd_finsetProd
-    intro y hy
-    have hyC := (Finset.mem_sdiff.mp hy).2
-    intro h
-    apply hunit y hyC
-    simpa only [sub_sub_cancel] using dvd_sub h (hcongr z hz y)
-  have hfz (z : X) (hz : z ∉ C) : f z=0 := by
-    dsimp [f]
-    rw [Finset.prod_apply]
-    exact Finset.prod_eq_zero (Finset.mem_sdiff.mpr ⟨Finset.mem_univ _,hz⟩) (hzero z hz)
-  obtain ⟨g,hg,hgc,hgz⟩ := uniform_projector L C f hf hfz
-  refine ⟨∏ z ∈ C, f z,?_,g,hg,hgc,hgz⟩
-  exact hp.not_dvd_finsetProd (fun z hz => hfu z hz)
-
-/-- Local lifts on each finite character block assemble after a single
-scalar denominator away from the prime. -/
- theorem partition_local_denominator {Q : Type*} [Fintype Q]
-    (L : Submodule R (∀ x, E x)) (p : R) (hp : Prime p) (q : X → Q)
-    (z : ∀ x, E x) (d : Q → R) (hd : ∀ j, ¬ p ∣ d j)
-    (g : Q → X → R) (hg : ∀ j, g j ∈ stabilizer L)
-    (hgin : ∀ j x, q x=j → g j x=d j)
-    (hgout : ∀ j x, q x≠j → g j x=0)
-    (m : Q → ∀ x, E x) (hm : ∀ j, m j ∈ L)
-    (hmatch : ∀ j x, q x=j → m j x=z x) :
-    ∃ a : R, ¬ p ∣ a ∧ a • z ∈ L := by
-  classical
-  refine ⟨∏ j, d j,hp.not_dvd_finsetProd (fun j _ => hd j),?_⟩
-  have hh : (∑ j : Q, (∏ k ∈ Finset.univ.erase j, d k) •
-      diagonal (g j) (m j)) ∈ L := by
-    exact L.sum_mem fun j _ => L.smul_mem _ (hg j _ (hm j))
-  convert hh using 1
-  funext x
-  simp only [Pi.smul_apply,Finset.sum_apply]
-  rw [Finset.sum_eq_single (q x)]
-  · change (∏ j, d j) • z x = (∏ k ∈ Finset.univ.erase (q x),d k) •
-      (g (q x) x • m (q x) x)
-    rw [hgin _ _ rfl,hmatch _ _ rfl,← mul_smul,
-      Finset.prod_erase_mul _ _ (Finset.mem_univ _)]
-  · intro j _ hj
-    change (∏ k ∈ Finset.univ.erase j,d k) • (g j x • m j x)=0
-    rw [hgout j x (Ne.symm hj),zero_smul,smul_zero]
-  · simp
-
- def residueSetoid {A : Type*} (p : R) (χ : A → X → R) : Setoid X where
-  r x y := ∀ a, p ∣ χ a x-χ a y
-  iseqv := ⟨fun x a => by rw [sub_self]; exact dvd_zero _,
-    fun h a => by simpa only [neg_sub] using dvd_neg.mpr (h a),
-    fun h h' a => by simpa only [sub_add_sub_cancel] using dvd_add (h a) (h' a)⟩
-
- theorem character_class_projector [Fintype X] {A : Type*}
-    (L : Submodule R (∀ x, E x)) (p : R) (hp : Prime p)
-    (χ : A → X → R) (hχ : ∀ a, χ a ∈ stabilizer L) (x : X) :
-    ∃ d : R, ¬ p ∣ d ∧ ∃ g ∈ stabilizer L,
-      (∀ z, (residueSetoid p χ).r x z → g z=d) ∧
-      (∀ z, ¬ (residueSetoid p χ).r x z → g z=0) := by
-  classical
-  let C : Finset X := Finset.univ.filter ((residueSetoid p χ).r x)
-  have hc (z : X) : z ∈ C ↔ (residueSetoid p χ).r x z := by simp [C]
-  have ha : ∀ y : {y : X // y ∉ C}, ∃ a, ¬ p ∣ χ a x-χ a y.val := by
-    intro y
-    have hh : ¬ ∀ a, p ∣ χ a x-χ a y.val := fun h => y.property ((hc _).mpr h)
-    exact not_forall.mp hh
-  choose a ha using ha
-  let test (y : X) : X → R := if hy : y ∉ C then
-    χ (a ⟨y,hy⟩)-algebraMap R (X → R) (χ (a ⟨y,hy⟩) y) else 0
-  have ht (y : X) : test y ∈ stabilizer L := by
-    by_cases hy : y ∉ C
-    · simp only [test,dite_eq_left hy]
-      exact (stabilizer L).sub_mem (hχ _) ((stabilizer L).algebraMap_mem _)
-    · simp only [test,dite_eq_right hy]
-      exact (stabilizer L).zero_mem
-  have hz (y : X) (hy : y ∉ C) : test y y=0 := by simp [test,hy]
-  have hu (y : X) (hy : y ∉ C) : ¬ p ∣ test y x := by simpa [test,hy] using ha ⟨y,hy⟩
-  have hcon (z : X) (hz : z ∈ C) (y : X) : p ∣ test y z-test y x := by
-    by_cases hy : y ∉ C
-    · simp only [test,dite_eq_left hy]
-      change p ∣ (χ (a ⟨y,hy⟩) z-χ (a ⟨y,hy⟩) y)-
-        (χ (a ⟨y,hy⟩) x-χ (a ⟨y,hy⟩) y)
-      rw [sub_sub_sub_cancel_right]
-      exact (residueSetoid p χ).symm ((hc z).mp hz) _
-    · simp only [test,dite_eq_right hy,Pi.zero_apply,sub_self]
-      exact dvd_zero _
-  obtain ⟨d,hd,g,hg,hgc,hgz⟩ := class_projector L p hp C x test ht hz hu hcon
-  exact ⟨d,hd,g,hg,fun z h => hgc z ((hc z).mpr h),
-    fun z h => hgz z (fun hz => h ((hc z).mp hz))⟩
-
-/-- Pair/blockwise liftability at a prime suffices for integral local
-membership in a finite character-stable lattice. -/
- theorem character_local_denominator [Fintype X] {A : Type*}
-    (L : Submodule R (∀ x, E x)) (p : R) (hp : Prime p)
-    (χ : A → X → R) (hχ : ∀ a, χ a ∈ stabilizer L) (z : ∀ x, E x)
-    (hlift : ∀ x, ∃ m ∈ L, ∀ y, (residueSetoid p χ).r x y → m y=z y) :
-    ∃ a : R, ¬ p ∣ a ∧ a • z ∈ L := by
-  classical
-  let Q := Quotient (residueSetoid p χ)
-  let : Fintype Q := Fintype.ofFinite Q
-  let q : X → Q := Quotient.mk''
-  have hq (j : Q) (y : X) : q y=j ↔ (residueSetoid p χ).r j.out y := by
-    constructor
-    · intro h
-      exact (residueSetoid p χ).symm (Quotient.exact (h.trans (Quotient.out_eq j).symm))
-    · intro h
-      exact (Quotient.sound ((residueSetoid p χ).symm h)).trans (Quotient.out_eq j)
-  have hproj (j : Q) := character_class_projector L p hp χ hχ j.out
-  choose d hd g hg hgin hgout using hproj
-  have hm (j : Q) := hlift j.out
-  choose m hm hmatch using hm
-  exact partition_local_denominator L p hp q z d hd g hg
-    (fun j y h => hgin j y ((hq j y).mp h))
-    (fun j y h => hgout j y (fun hh => h ((hq j y).mpr hh)))
-    m hm (fun j y h => hmatch j y ((hq j y).mp h))
-
-end
-end KLInvariance.FiniteCharacterLocalization
-
-end
-
-
-section
-
-
-namespace KLInvariance.ProjectiveLattice
-variable {R : Type*} [CommRing R] [klPreservedInstance15 : IsDomain R] [klPreservedInstance16 : UniqueFactorizationMonoid R]
-
- omit [IsDomain R] [UniqueFactorizationMonoid R] in
- theorem finsupp_divide {ι : Type*} (r : R) (f : ι →₀ R)
-    (h : ∀ i, r ∣ f i) : ∃ g : ι →₀ R, r • g = f := by
-  classical
-  let g : ι →₀ R := ∑ i ∈ f.support, Finsupp.single i (Classical.choose (h i))
-  refine ⟨g, ?_⟩
-  dsimp [g]
-  rw [Finset.smul_sum]
-  have heq : (∑ i ∈ f.support, r • Finsupp.single i (Classical.choose (h i))) =
-      ∑ i ∈ f.support, Finsupp.single i (f i) := by
-    apply Finset.sum_congr rfl
-    intro i hi
-    rw [Finsupp.smul_single, smul_eq_mul, ← Classical.choose_spec (h i)]
-  rw [heq]
-  exact Finsupp.sum_single f
-
-include klPreservedInstance15 klPreservedInstance16 in
- theorem prime_divide_of_relation {P : Type*} [AddCommGroup P] [Module R P]
-    [Module.Projective R P] (p a c : R) (hp : Prime p) (ha : ¬p ∣ a)
-    (m n : P) (h : a • m=(p*c) • n) : ∃ z : P, p • z=m := by
-  have _ := klPreservedInstance15
-  have _ := klPreservedInstance16
-  obtain ⟨s,hs⟩ := Module.Projective.out (R := R) (P := P)
-  have hdvd : ∀ i, p ∣ s m i := by
-    intro i
-    have he := congrArg (fun z : P => s z i) h
-    simp only [map_smul, Finsupp.smul_apply, smul_eq_mul] at he
-    have hm : p ∣ a*s m i := by
-      rw [he]
-      exact dvd_mul_of_dvd_left (dvd_mul_right p c) _
-    exact (hp.dvd_mul.mp hm).resolve_left ha
-  obtain ⟨z,hz⟩ := finsupp_divide p (s m) hdvd
-  refine ⟨Finsupp.linearCombination R id z,?_⟩
-  rw [← map_smul,hz,hs]
-
- theorem mem_of_prime_local_membership
-    {E : Type*} [AddCommGroup E] [Module R E] [Module.IsTorsionFree R E]
-    (M : Submodule R E) [Module.Projective R M] (x : E)
-    (hgeneric : ∃ c : R, c ≠ 0 ∧ c • x ∈ M)
-    (hlocal : ∀ p : R, Prime p → ∃ a : R, ¬p ∣ a ∧ a • x ∈ M) : x ∈ M := by
-  obtain ⟨c,hcz,hcx⟩ := hgeneric
-  suffices ∀ c : R, c≠0 → c • x ∈ M → x ∈ M from this c hcz hcx
-  intro c
-  induction c using UniqueFactorizationMonoid.induction_on_prime with
-  | h₁ => simp
-  | h₂ c hc =>
-    intro _ hcx
-    obtain ⟨u,rfl⟩ := hc
-    simpa using M.smul_mem (↑u⁻¹ : R) hcx
-  | h₃ c p hcz hp ih =>
-    intro _ hpcx
-    obtain ⟨a,hpa,hax⟩ := hlocal p hp
-    let m : M := ⟨(p*c) • x,hpcx⟩
-    let n : M := ⟨a • x,hax⟩
-    have he : a • m=(p*c) • n := by
-      apply Subtype.ext
-      exact smul_comm a (p*c) x
-    obtain ⟨z,hz⟩ := prime_divide_of_relation p a c hp hpa m n he
-    have hcx : c • x=(z : E) := by
-      apply smul_right_injective E hp.ne_zero
-      have hv := congrArg Subtype.val hz
-      change p • (z : E)=(p*c) • x at hv
-      simpa only [mul_smul] using hv.symm
-    exact ih hcz (hcx ▸ z.property)
-
-end KLInvariance.ProjectiveLattice
-
-end
-
-
-section
-
-/-! Prime-local realization on arbitrary upper coordinate sets. Projectivity
-of the actual image is kept explicit here and discharged by word induction. -/
-namespace KLInvariance.TitsSpace
-open Module CoordinateProjection FiniteCharacterLocalization
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
- theorem bottSet_scaled (l : List I) (U : Set W)
-    (z : ∀ x : U, bottStalk cs l x.val) :
-    bottCoordinateDenominator cs l • z ∈ bottSetImage cs l U := by
-  classical
-  let c (e : BottIndex l) : SymmetricCoefficient (M := M) :=
-    if h : bottWeight cs l e∈U then (z ⟨bottWeight cs l e,h⟩).val ⟨e,rfl⟩ else 0
-  obtain ⟨m,hm⟩ := bottEvaluation_contains_scaled_coordinates cs l c
-  refine ⟨m,?_⟩
-  funext x
-  apply Subtype.ext
-  funext e
-  rcases x with ⟨x,hx⟩
-  rcases e with ⟨e,he⟩
-  change bottWeight cs l e=x at he
-  subst x
-  change bottEvaluation cs l e m=bottCoordinateDenominator cs l * (z ⟨bottWeight cs l e,hx⟩).val ⟨e,rfl⟩
-  simpa only [c,dite_eq_left hx] using hm e
-
- theorem BottCompatible.reflection_lift {l : List I} {U : Set W}
-    {z : ∀ x : U, bottStalk cs l x.val} (hz : BottCompatible cs l U z)
-    (x y : U) (t : W) (ht : cs.IsReflection t) (he : t*x.val=y.val) :
-    ∃ m : BottSamelsonModule cs l,
-      bottStalkProjection cs l x.val m=z x ∧ bottStalkProjection cs l y.val m=z y := by
-  have he' : t*y.val=x.val := by rw [← he,← mul_assoc,ht.mul_self,one_mul]
-  have hne : cs.length y.val ≠ cs.length x.val := by
-    rw [← he]
-    exact ht.length_mul_right_ne x.val
-  rcases lt_or_gt_of_ne hne with hlt | hlt
-  · obtain ⟨m,hm,hn⟩ := hz y.val x.val y.property x.property ⟨hlt,t,ht,he'.symm⟩
-    exact ⟨m,hn,hm⟩
-  · exact hz x.val y.val x.property y.property ⟨hlt,t,ht,he.symm⟩
-
- theorem BottCompatible.prime_lift {l : List I} {U : Set W}
-    {z : ∀ x : U, bottStalk cs l x.val} (hz : BottCompatible cs l U z)
-    (p : SymmetricCoefficient (M := M)) (hp : Prime p) (x : U) :
-    ∃ m : BottSamelsonModule cs l, ∀ y : U,
-      (∀ a, p ∣ coefficientAction cs x.val a-coefficientAction cs y.val a) →
-      bottStalkProjection cs l y.val m=z y := by
-  classical
-  by_cases hx : ∃ y : U, x≠y ∧
-      ∀ a, p ∣ coefficientAction cs x.val a-coefficientAction cs y.val a
-  · obtain ⟨y,hxy,hcon⟩ := hx
-    obtain ⟨t,ht,he,hclass⟩ := prime_character_class cs p hp
-      (fun h => hxy (Subtype.ext h)) hcon
-    obtain ⟨m,hm,hn⟩ := hz.reflection_lift cs x y t ht he.symm
-    refine ⟨m,?_⟩
-    intro w hw
-    rcases (hclass w.val).mp hw with hxw | hwy
-    · have h : w=x := Subtype.ext hxw.symm
-      subst w
-      exact hm
-    · have h : w=y := Subtype.ext (hwy.trans he.symm)
-      subst w
-      exact hn
-  · obtain ⟨m,hm⟩ := bottStalkProjection_surjective cs l x.val (z x)
-    refine ⟨m,?_⟩
-    intro y hy
-    have h : y=x := by
-      by_contra h
-      exact hx ⟨y,Ne.symm h,hy⟩
-    subst y
-    exact hm
-
- def BottUpperIndex (l : List I) (U : Set W) := {e : BottIndex l // bottWeight cs l e∈U}
-
- def bottIndexSetEval (l : List I) (U : Set W) : BottSamelsonModule cs l →ₗ[
-    SymmetricCoefficient (M := M)]
-    (∀ e : BottUpperIndex cs l U, bottStalk cs l (bottWeight cs l e.val)) :=
-  LinearMap.pi (fun e => bottStalkProjection cs l (bottWeight cs l e.val))
-
- theorem bottIndexSetEval_stable (l : List I) (U : Set W)
-    (a : SymmetricCoefficient (M := M)) :
-    (fun e : BottUpperIndex cs l U => coefficientAction cs (bottWeight cs l e.val) a) ∈
-      stabilizer (LinearMap.range (bottIndexSetEval cs l U)) := by
-  rintro z ⟨m,rfl⟩
-  refine ⟨bottSamelsonRight cs l a m,?_⟩
-  funext e
-  exact bottStalkProjection_right cs l _ a m
-
- theorem BottCompatible.local_denominator {l : List I} {U : Set W}
-    {z : ∀ x : U, bottStalk cs l x.val} (hz : BottCompatible cs l U z)
-    (p : SymmetricCoefficient (M := M)) (hp : Prime p) :
-    ∃ a : SymmetricCoefficient (M := M), ¬p ∣ a ∧ a • z∈bottSetImage cs l U := by
-  classical
-  let : Fintype (BottUpperIndex cs l U) := inferInstanceAs (Fintype {e : BottIndex l // bottWeight cs l e∈U})
-  let zz : ∀ e : BottUpperIndex cs l U, bottStalk cs l (bottWeight cs l e.val) :=
-    fun e => z ⟨bottWeight cs l e.val,e.property⟩
-  have hlift : ∀ e : BottUpperIndex cs l U, ∃ m ∈ LinearMap.range (bottIndexSetEval cs l U),
-      ∀ f, (residueSetoid p (fun a e => coefficientAction cs (bottWeight cs l e.val) a)).r e f → m f=zz f := by
-    intro e
-    obtain ⟨m,hm⟩ := hz.prime_lift cs p hp ⟨bottWeight cs l e.val,e.property⟩
-    exact ⟨bottIndexSetEval cs l U m,⟨m,rfl⟩,fun f hf => hm ⟨bottWeight cs l f.val,f.property⟩ hf⟩
-  obtain ⟨a,ha,m,hm⟩ := character_local_denominator
-    (LinearMap.range (bottIndexSetEval cs l U)) p hp
-    (fun a e => coefficientAction cs (bottWeight cs l e.val) a)
-    (bottIndexSetEval_stable cs l U) zz hlift
-  refine ⟨a,ha,m,?_⟩
-  funext x
-  apply Subtype.ext
-  funext e
-  rcases x with ⟨x,hx⟩
-  rcases e with ⟨e,he⟩
-  change bottWeight cs l e=x at he
-  subst x
-  have hh := congrArg (fun q => (q ⟨e,hx⟩).val ⟨e,rfl⟩) hm
-  exact hh
-
- theorem bottFlabby_of_projective (l : List I) (H : BottProjectiveUpper cs l) :
-    BottFlabby cs l := by
-  intro U hU z hz
-  let : Module.Projective (SymmetricCoefficient (M := M)) (bottSetImage cs l U) := H U hU
-  let : UniqueFactorizationMonoid (SymmetricCoefficient (M := M)) :=
-    (SymmetricAlgebra.equivMvPolynomial (Module.Free.chooseBasis ℝ (Extended M))).symm.toRingEquiv.toMulEquiv.uniqueFactorizationMonoid inferInstance
-  exact ProjectiveLattice.mem_of_prime_local_membership (bottSetImage cs l U) z
-    ⟨bottCoordinateDenominator cs l,bottCoordinateDenominator_ne_zero cs l,bottSet_scaled cs l U z⟩
-    (hz.local_denominator cs)
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Unconditional flabbiness of the actual integral tensor-word projection
-sheaf, obtained from projectivity of every upper-coordinate image. -/
-namespace KLInvariance.TitsSpace
-open Module CoordinateProjection
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
- theorem bottSetProjection_nil_injective (U : Set W) (hU : (1 : W)∈U) :
-    Function.Injective (bottSetProjection cs [] U) := by
-  intro m n h
-  apply bottEvaluation_injective cs []
-  funext e
-  exact congrArg (fun z : bottSetImage cs [] U => (z.val ⟨1,hU⟩).val ⟨e,rfl⟩) h
-
- theorem bottSetImage_nil_subsingleton (U : Set W) (hU : (1 : W)∉U) :
-    Subsingleton (bottSetImage cs [] U) := by
-  constructor
-  intro p q
-  apply Subtype.ext
-  funext x
-  have hx : x.val≠1 := fun h => hU (h ▸ x.property)
-  let := bottStalk_nil_subsingleton cs x.val hx
-  exact Subsingleton.elim _ _
-
- theorem bottSetImage_nil_projective (U : Set W) :
-    Module.Projective (SymmetricCoefficient (M := M)) (bottSetImage cs [] U) := by
-  by_cases hU : (1 : W)∈U
-  · exact Module.Projective.of_equiv (LinearEquiv.ofBijective (bottSetProjection cs [] U)
-      ⟨bottSetProjection_nil_injective cs U hU,bottSetProjection_surjective cs [] U⟩)
-  · let := bottSetImage_nil_subsingleton cs U hU
-    infer_instance
-
- theorem bottProjectiveUpper (l : List I) : BottProjectiveUpper cs l := by
-  induction l with
-  | nil => exact fun U _ => bottSetImage_nil_projective cs U
-  | cons i l ih => exact bottProjectiveUpper_cons cs i l ih (bottFlabby_of_projective cs l ih)
-
- theorem bottFlabby (l : List I) : BottFlabby cs l :=
-  bottFlabby_of_projective cs l (bottProjectiveUpper cs l)
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Flabbiness of the genuine interval word sheaf; the extension outside the
-finite support is zero, not an extra assumption on the source interval. -/
-namespace KLInvariance.TitsSpace
-open Module MomentGraph BruhatGraph
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
- theorem bottStalk_subsingleton_of_not_bruhat (l : List I) (hl : cs.IsReduced l)
-    (x : W) (hx : ¬BruhatLE cs x (cs.wordProd l)) : Subsingleton (bottStalk cs l x) := by
-  constructor
-  intro p q
-  apply Subtype.ext
-  funext e
-  exact (hx (e.property ▸ bottWeight_bruhat cs l hl e.val)).elim
-
- def intervalUpperClosure (b : W) (U : Set (Interval cs 1 b)) : Set W :=
-  {x | ∃ a : Interval cs 1 b, a∈U ∧ BruhatLE cs a.val x}
-
- omit [Fintype I] in
- theorem intervalUpperClosure_upper (b : W) (U : Set (Interval cs 1 b)) :
-    BruhatUpper cs (intervalUpperClosure cs b U) := by
-  rintro x y hxy ⟨a,ha,hax⟩
-  exact ⟨a,ha,bruhat_trans cs hax hxy⟩
-
- omit [Fintype I] in
- theorem intervalUpperClosure_member (b : W) (U : Set (Interval cs 1 b)) (hU : IsUpperSet U)
-    (x : W) (hx : x∈intervalUpperClosure cs b U) (hxb : BruhatLE cs x b) :
-    (⟨x,one_bruhat cs x,hxb⟩ : Interval cs 1 b)∈U := by
-  obtain ⟨a,ha,hax⟩ := hx
-  exact hU hax ha
-
- def bottIntervalExtension (l : List I) (U : Set (Interval cs 1 (cs.wordProd l)))
-    (f : (bottProjectionSheaf cs l 1 (cs.wordProd l)).Assignment)
-    (x : intervalUpperClosure cs (cs.wordProd l) U) : bottStalk cs l x.val := by
-  classical
-  exact if h : BruhatLE cs x.val (cs.wordProd l) then f ⟨x.val,one_bruhat cs x.val,h⟩ else 0
-
- theorem bottIntervalExtension_apply (l : List I) (U : Set (Interval cs 1 (cs.wordProd l)))
-    (f : (bottProjectionSheaf cs l 1 (cs.wordProd l)).Assignment)
-    (x : Interval cs 1 (cs.wordProd l)) (hx : x.val∈intervalUpperClosure cs (cs.wordProd l) U) :
-    bottIntervalExtension cs l U f ⟨x.val,hx⟩=f x := by
-  classical
-  dsimp only [bottIntervalExtension]
-  exact dite_eq_left x.property.2
-
- theorem bottIntervalExtension_compatible (l : List I) (hl : cs.IsReduced l)
-    (U : Set (Interval cs 1 (cs.wordProd l))) (hU : IsUpperSet U)
-    (f : (bottProjectionSheaf cs l 1 (cs.wordProd l)).Assignment)
-    (hf : (bottProjectionSheaf cs l 1 (cs.wordProd l)).CompatibleOn U f) :
-    BottCompatible cs l (intervalUpperClosure cs (cs.wordProd l) U) (bottIntervalExtension cs l U f) := by
-  intro x y hx hy hxy
-  by_cases hyb : BruhatLE cs y (cs.wordProd l)
-  · have hxb : BruhatLE cs x (cs.wordProd l) := bruhat_trans cs (.single hxy) hyb
-    let xx : Interval cs 1 (cs.wordProd l) := ⟨x,one_bruhat cs x,hxb⟩
-    let yy : Interval cs 1 (cs.wordProd l) := ⟨y,one_bruhat cs y,hyb⟩
-    let e : Edge cs 1 (cs.wordProd l) := ⟨(xx,yy),hxy⟩
-    have hh := hf e (intervalUpperClosure_member cs _ U hU x hx hxb)
-      (intervalUpperClosure_member cs _ U hU y hy hyb)
-    obtain ⟨m,hm,hn⟩ := (PairProjection.compatible_iff _ _ _ _).mp hh
-    exact ⟨m,hm.trans (bottIntervalExtension_apply cs l U f xx hx).symm,
-      hn.trans (bottIntervalExtension_apply cs l U f yy hy).symm⟩
-  · let := bottStalk_subsingleton_of_not_bruhat cs l hl y hyb
-    obtain ⟨m,hm⟩ := bottStalkProjection_surjective cs l x (bottIntervalExtension cs l U f ⟨x,hx⟩)
-    exact ⟨m,hm,Subsingleton.elim _ _⟩
-
- theorem bottProjectionSheaf_flabby (l : List I) (hl : cs.IsReduced l) :
-    (bottProjectionSheaf cs l 1 (cs.wordProd l)).Flabby := by
-  intro U hU f hf
-  obtain ⟨m,hm⟩ := bottFlabby cs l _ (intervalUpperClosure_upper cs _ U)
-    (bottIntervalExtension cs l U f) (bottIntervalExtension_compatible cs l hl U hU f hf)
-  refine ⟨(bottSection cs l 1 (cs.wordProd l) m).val,(bottSection cs l 1 (cs.wordProd l) m).property,?_⟩
-  intro x hx
-  exact (congrFun hm ⟨x.val,x,hx,bruhat_refl cs x.val⟩).trans
-    (bottIntervalExtension_apply cs l U f x ⟨x,hx,bruhat_refl cs x.val⟩)
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Integral adjoints, not restrictions of a form to arbitrary summands.
-This is used to transport a genuine BMP projector through the word duality. -/
-namespace PerfectAdjoint
-open Module
-universe ur um
-variable {R : Type ur} [CommRing R]
-  {M : Type um} [AddCommGroup M] [Module R M]
-  (d : M ≃ₗ[R] Dual R M)
-noncomputable section
-
- def adjoint (f : End R M) : End R M :=
-  d.symm.toLinearMap.comp (f.dualMap.comp d.toLinearMap)
-
- theorem pairing (f : End R M) (m n : M) : d (adjoint d f m) n=d m (f n) := by
-  change d (d.symm (f.dualMap (d m))) n=d m (f n)
-  rw [LinearEquiv.apply_symm_apply]
-  rfl
-
- theorem adjoint_comp (f g : End R M) : adjoint d (f.comp g)=(adjoint d g).comp (adjoint d f) := by
-  apply LinearMap.ext
-  intro m
-  apply d.injective
-  apply LinearMap.ext
-  intro n
-  rw [pairing]
-  change d m (f (g n))=d (adjoint d g (adjoint d f m)) n
-  rw [pairing,pairing]
-
- theorem adjoint_idempotent (f : End R M) (hf : ∀ m, f (f m)=f m) (m : M) :
-    adjoint d f (adjoint d f m)=adjoint d f m := by
-  apply d.injective
-  apply LinearMap.ext
-  intro n
-  rw [pairing,pairing,hf,pairing]
-
- theorem commutes_of_balanced (f t : End R M)
-    (ht : ∀ m n, d (t m) n=d m (t n)) (hf : ∀ m, f (t m)=t (f m)) (m : M) :
-    adjoint d f (t m)=t (adjoint d f m) := by
-  apply d.injective
-  apply LinearMap.ext
-  intro n
-  rw [pairing,ht,ht,pairing,hf]
-
- theorem symmetric_pairing (hsym : ∀ m n, d m n=d n m)
-    (f : End R M) (m n : M) : d m (adjoint d f n)=d (f m) n := by
-  rw [hsym,pairing,hsym]
-
- theorem adjoint_adjoint (hsym : ∀ m n, d m n=d n m) (f : End R M) :
-    adjoint d (adjoint d f)=f := by
-  apply LinearMap.ext
-  intro m
-  apply d.injective
-  apply LinearMap.ext
-  intro n
-  rw [pairing,symmetric_pairing d hsym]
-
-end
-end PerfectAdjoint
-
-end
-
-
-section
-
-
-namespace KLInvariance.Graded
-open Module
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {M : Type um} [AddCommGroup M] [Module k M] [Module A M] [IsScalarTower k A M]
-  (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M)
-  [DirectSum.Decomposition 𝓐] [DirectSum.Decomposition 𝓜]
-noncomputable section
-
-omit [IsScalarTower k A M] in
- theorem perfect_inverse_of_homogeneous_functional
-    (p : M ≃ₗ[A] Dual A M) (l : ℤ)
-    (hp : ∀ i j (m n : M), m ∈ 𝓜 i → n ∈ 𝓜 j → p m n ∈ 𝓐 (i+j-l))
-    (r : ℤ) (lam : Dual A M)
-    (hlam : ∀ j (n : M), n ∈ 𝓜 j → lam n ∈ 𝓐 (j+r)) :
-    p.symm lam ∈ 𝓜 (r+l) := by
-  have hc : component 𝓜 (r+l) (p.symm lam) = p.symm lam := by
-    apply p.injective
-    apply LinearMap.ext
-    intro y
-    refine DirectSum.Decomposition.inductionOn 𝓜 ?_ ?_ ?_ y
-    · simp
-    · intro j v
-      let f : M →ₗ[A] A :=
-        { toFun := fun w => p w (v : M)
-          map_add' := by intros; simp
-          map_smul' := by intros; simp }
-      have hh : ∀ i w, w ∈ 𝓜 i → f w ∈ 𝓐 (i+(j-l)) := by
-        intro i w hw
-        simpa only [f,LinearMap.coe_mk,AddHom.coe_mk,add_sub_assoc] using hp i j w v hw v.property
-      have he := map_component 𝓜 𝓐 f (j-l) hh (j+r) (p.symm lam)
-      have hi : j+r-(j-l)=r+l := by omega
-      rw [hi] at he
-      have hf : f (p.symm lam)=lam v := by
-        dsimp only [f,LinearMap.coe_mk,AddHom.coe_mk]
-        rw [LinearEquiv.apply_symm_apply]
-      rw [hf] at he
-      have hsame : component 𝓐 (j+r) (lam v)=lam v :=
-        DirectSum.decompose_of_mem_same 𝓐 (hlam j v v.property)
-      rw [hsame] at he
-      change f (component 𝓜 (r+l) (p.symm lam)) = p (p.symm lam) v
-      rw [LinearEquiv.apply_symm_apply]
-      exact he.symm
-    · intro x y hx hy
-      simp only [map_add,hx,hy]
-  rw [← hc]
-  exact (DirectSum.decompose 𝓜 (p.symm lam) (r+l)).property
-
-end
-end KLInvariance.Graded
-
-end
-
-
-section
-
-namespace KLInvariance.Graded
-open Module
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {M : Type um} [AddCommGroup M] [Module k M] [Module A M]
-  (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M)
-  [DirectSum.Decomposition 𝓐] [DirectSum.Decomposition 𝓜]
-noncomputable section
-
- theorem adjoint_homogeneous
-    (p : M ≃ₗ[A] Dual A M) (l : ℤ)
-    (hp : ∀ i j (m n : M), m ∈ 𝓜 i → n ∈ 𝓜 j → p m n ∈ 𝓐 (i+j-l))
-    (f : End A M) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓜 n)
-    (n : ℤ) (m : M) (hm : m ∈ 𝓜 n) : PerfectAdjoint.adjoint p f m ∈ 𝓜 n := by
-  have h := perfect_inverse_of_homogeneous_functional 𝓐 𝓜 p l hp (n-l)
-    (f.dualMap (p m)) (by
-      intro j v hv
-      have hv' := hp n j m (f v) hm (hf j v hv)
-      have he : n+j-l=j+(n-l) := by omega
-      change p m (f v) ∈ 𝓐 (j+(n-l))
-      rw [← he]
-      exact hv')
-  simpa only [sub_add_cancel,PerfectAdjoint.adjoint,LinearMap.comp_apply,LinearEquiv.coe_coe] using h
-
-end
-end KLInvariance.Graded
-
-end
-
-
-section
-
-
-end
-
-end OAI
+import OAI.RepresentationTheory.KazhdanLusztig.BottSamelson
+
+namespace OAI
+
+/-!
+Hilbert recurrences, projective section ranges, integral lattices and graded perfect adjoints.
+-/
+
+section
+
+namespace KLInvariance.Hilbert
+open Polynomial
+
+ theorem hasSum_shift_coeff {f : ℤ → ℝ} {q c : ℝ} (hq : q≠0)
+    (hf : HasSum (fun n : ℤ => f n*q^n) c) (d : ℤ) :
+    HasSum (fun n : ℤ => f (n-d)*q^n) (q^d*c) := by
+  apply (Equiv.addRight d).hasSum_iff.mp
+  apply (hf.mul_left (q^d)).congr_fun
+  intro n
+  change f (n+d-d)*q^(n+d)=q^d*(f n*q^n)
+  rw [add_sub_cancel_right,zpow_add₀ hq]
+  ring
+
+ theorem numerator_unique (f : ℤ → ℕ) (F G : ℤ[X]) (D : ℝ → ℝ)
+    (hD : ∀ q, 0<q → q<1 → D q≠0)
+    (hF : ∀ q, 0<q → q<1 → HasSum (fun n : ℤ => (f n : ℝ)*q^n)
+      (eval₂ (Int.castRingHom ℝ) q F*D q))
+    (hG : ∀ q, 0<q → q<1 → HasSum (fun n : ℤ => (f n : ℝ)*q^n)
+      (eval₂ (Int.castRingHom ℝ) q G*D q)) : F=G := by
+  apply Polynomial.map_injective (Int.castRingHom ℝ) Int.cast_injective
+  apply Polynomial.eq_of_infinite_eval_eq
+  apply Set.Infinite.mono (s := Set.Ioo (0 : ℝ) 1) ?_ (Set.Ioo_infinite (by norm_num))
+  intro q hq
+  have he := (hF q hq.1 hq.2).unique (hG q hq.1 hq.2)
+  change eval q (F.map (Int.castRingHom ℝ))=eval q (G.map (Int.castRingHom ℝ))
+  simpa only [eval_map] using (mul_right_cancel₀ (hD q hq.1 hq.2) he)
+
+ theorem numerator_add_shift (f g h : ℤ → ℕ) (F G H : ℤ[X]) (D : ℝ → ℝ)
+    (hD : ∀ q, 0<q → q<1 → D q≠0)
+    (hF : ∀ q, 0<q → q<1 → HasSum (fun n : ℤ => (f n : ℝ)*q^n)
+      (eval₂ (Int.castRingHom ℝ) q F*D q))
+    (hG : ∀ q, 0<q → q<1 → HasSum (fun n : ℤ => (g n : ℝ)*q^n)
+      (eval₂ (Int.castRingHom ℝ) q G*D q))
+    (hH : ∀ q, 0<q → q<1 → HasSum (fun n : ℤ => (h n : ℝ)*q^n)
+      (eval₂ (Int.castRingHom ℝ) q H*D q))
+    (he : ∀ n, f n=g n+h (n-1)) : F=G+X*H := by
+  apply Polynomial.map_injective (Int.castRingHom ℝ) Int.cast_injective
+  apply Polynomial.eq_of_infinite_eval_eq
+  apply Set.Infinite.mono (s := Set.Ioo (0 : ℝ) 1) ?_ (Set.Ioo_infinite (by norm_num))
+  intro q hq
+  have hh := hasSum_shift_coeff hq.1.ne' (hH q hq.1 hq.2) 1
+  have hs := (hG q hq.1 hq.2).add hh
+  have hf := hF q hq.1 hq.2
+  have heq : eval₂ (Int.castRingHom ℝ) q F*D q=
+      eval₂ (Int.castRingHom ℝ) q G*D q+q*(eval₂ (Int.castRingHom ℝ) q H*D q) := by
+    rw [zpow_one] at hs
+    apply hf.unique
+    apply hs.congr_fun
+    intro n
+    rw [he,Nat.cast_add,add_mul]
+  change eval q (F.map (Int.castRingHom ℝ))=eval q ((G+X*H).map (Int.castRingHom ℝ))
+  simp only [eval_map,eval₂_add,eval₂_mul,eval₂_X]
+  apply mul_right_cancel₀ (hD q hq.1 hq.2)
+  calc
+    _ = eval₂ (Int.castRingHom ℝ) q G*D q+q*(eval₂ (Int.castRingHom ℝ) q H*D q) := heq
+    _ = _ := by ring
+
+end KLInvariance.Hilbert
+
+end
+
+
+section
+
+/-! Exact polynomial character recurrences for the actual integral word stalks.
+These are the unnormalized Hecke coefficients for multiplication by T_s+1.
+They are proved from the projection lattices, not from the KL conjecture. -/
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded
+universe u
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ theorem bottStalk_finrank_descent (i : I) (l : List I) (x : W)
+    (h : cs.length (cs.simple i*x)<cs.length x) (n : ℤ) :
+    Module.finrank ℝ (bottStalkPiece cs basis (i::l) x n)=
+      Module.finrank ℝ (bottStalkPiece cs basis l (cs.simple i*x) n)+
+      Module.finrank ℝ (bottStalkPiece cs basis l x (n-1)) := by
+  have hdim := Hilbert.finrank_exact (A := ℝ) (bottStalkPiece cs basis (i::l) x)
+    (bottStalkPiece cs basis l (cs.simple i*x)) (bottHeadRight cs i l x)
+    (bottHeadRight_homogeneous cs basis i l x) (bottHeadRight_surjective cs i l x) n
+  have hk := finrank_homogeneousEquiv (k := ℝ)
+    (N := LinearMap.ker (bottHeadRight cs i l x)) (bottStalkPiece cs basis l x)
+    (bottHeadRightKernelPiece cs basis i l x) (bottHeadRightKernelEquiv cs i l x h) 1
+    (bottHeadRightKernelEquiv_homogeneous cs basis i l x h) n
+  change Module.finrank ℝ (bottStalkPiece cs basis (i::l) x n)=
+    Module.finrank ℝ (bottHeadRightKernelPiece cs basis i l x n)+
+      Module.finrank ℝ (bottStalkPiece cs basis l (cs.simple i*x) n) at hdim
+  rw [hk] at hdim
+  omega
+
+ theorem bottStalkCharacter_ascent (i : I) (l : List I) (x : W)
+    (h : cs.length x<cs.length (cs.simple i*x)) :
+    bottStalkCharacter cs basis (i::l) x=bottStalkCharacter cs basis l x+
+      Polynomial.X*bottStalkCharacter cs basis l (cs.simple i*x) := by
+  apply Hilbert.numerator_add_shift
+    (fun n => Module.finrank ℝ (bottStalkPiece cs basis (i::l) x n))
+    (fun n => Module.finrank ℝ (bottStalkPiece cs basis l x n))
+    (fun n => Module.finrank ℝ (bottStalkPiece cs basis l (cs.simple i*x) n))
+    _ _ _ (fun q => (1-q)⁻¹ ^ Fintype.card σ)
+  · intro q _ hq
+    exact pow_ne_zero _ (inv_ne_zero (sub_ne_zero.mpr (ne_of_gt hq)))
+  · exact fun _ => bottStalkCharacter_hasSum cs basis (i::l) x
+  · exact fun _ => bottStalkCharacter_hasSum cs basis l x
+  · exact fun _ => bottStalkCharacter_hasSum cs basis l (cs.simple i*x)
+  · exact bottStalk_finrank_ascent cs basis i l x h
+
+ theorem bottStalkCharacter_descent (i : I) (l : List I) (x : W)
+    (h : cs.length (cs.simple i*x)<cs.length x) :
+    bottStalkCharacter cs basis (i::l) x=bottStalkCharacter cs basis l (cs.simple i*x)+
+      Polynomial.X*bottStalkCharacter cs basis l x := by
+  apply Hilbert.numerator_add_shift
+    (fun n => Module.finrank ℝ (bottStalkPiece cs basis (i::l) x n))
+    (fun n => Module.finrank ℝ (bottStalkPiece cs basis l (cs.simple i*x) n))
+    (fun n => Module.finrank ℝ (bottStalkPiece cs basis l x n))
+    _ _ _ (fun q => (1-q)⁻¹ ^ Fintype.card σ)
+  · intro q _ hq
+    exact pow_ne_zero _ (inv_ne_zero (sub_ne_zero.mpr (ne_of_gt hq)))
+  · exact fun _ => bottStalkCharacter_hasSum cs basis (i::l) x
+  · exact fun _ => bottStalkCharacter_hasSum cs basis l (cs.simple i*x)
+  · exact fun _ => bottStalkCharacter_hasSum cs basis l x
+  · exact bottStalk_finrank_descent cs basis i l x h
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! The honest unnormalized Hecke word (T_s+1)...(T_t+1), including exact
+coefficient recurrence over a general commutative ring, without invertibility
+of q or any KL/BMP character assertion. -/
+namespace KLInvariance.HeckeQ
+open Hecke (Space basis)
+variable {I W C : Type*} [Group W] {M : CoxeterMatrix I} [CommRing C]
+noncomputable section
+
+ theorem left_coeff (cs : CoxeterSystem M W) (q : C) (i : I) (v : Space W C) (x : W) :
+    left cs q i v x =
+      if cs.length (cs.simple i*x)<cs.length x then v (cs.simple i*x)+(q-1)*v x
+      else q*v (cs.simple i*x) := by
+  classical
+  induction v using Finsupp.induction_linear with
+  | zero => simp
+  | add v v' hv hv' =>
+    simp only [map_add,Finsupp.add_apply,hv,hv']
+    split_ifs <;> ring
+  | single w c =>
+    have hc : Finsupp.single w c=c • (basis w : Space W C) := by
+      simp [basis,Finsupp.smul_single]
+    rw [hc,map_smul,left_basis]
+    by_cases h : w=cs.simple i*x
+    · subst w
+      have hd : cs.length (cs.simple i*(cs.simple i*x))<cs.length (cs.simple i*x) ↔
+          ¬cs.length (cs.simple i*x)<cs.length x := by
+        simp only [cs.simple_mul_simple_cancel_left]
+        have := cs.length_simple_mul x i
+        omega
+      have hn : cs.simple i*x≠x := by
+        intro he
+        have := cs.length_simple_mul x i
+        rw [he] at this
+        omega
+      simp only [Finsupp.smul_apply,smul_eq_mul,basis,Finsupp.single_apply,
+        cs.simple_mul_simple_cancel_left,hn,ite_false]
+      have hlength := cs.length_simple_mul x i
+      split_ifs <;> try omega
+      all_goals simp [Ne.symm hn] <;> try ring
+    · have hn : cs.simple i*w≠x := by
+        intro he
+        apply h
+        rw [←he]
+        simp
+      by_cases hw : w=x
+      · subst w
+        simp only [Finsupp.smul_apply,smul_eq_mul,basis,Finsupp.single_apply,
+          Ne.symm hn,ite_false]
+        split_ifs <;> simp [Ne.symm hn] ; try ring
+      · simp only [Finsupp.smul_apply,smul_eq_mul,basis,Finsupp.single_apply,
+          h,hw,ite_false]
+        split_ifs <;> simp [hn,hw]
+
+ theorem left_plus_coeff (cs : CoxeterSystem M W) (q : C) (i : I)
+    (v : Space W C) (x : W) :
+    (v+left cs q i v) x = if cs.length (cs.simple i*x)<cs.length x then
+      v (cs.simple i*x)+q*v x else v x+q*v (cs.simple i*x) := by
+  rw [Finsupp.add_apply,left_coeff]
+  split_ifs <;> ring
+
+ def positiveWord (cs : CoxeterSystem M W) (q : C) : List I → Space W C
+  | [] => basis 1
+  | i::l => positiveWord cs q l+left cs q i (positiveWord cs q l)
+
+ @[simp] theorem positiveWord_nil (cs : CoxeterSystem M W) (q : C) :
+    positiveWord cs q []=basis 1 := rfl
+
+ theorem positiveWord_cons_coeff (cs : CoxeterSystem M W) (q : C) (i : I)
+    (l : List I) (x : W) :
+    positiveWord cs q (i::l) x = if cs.length (cs.simple i*x)<cs.length x then
+      positiveWord cs q l (cs.simple i*x)+q*positiveWord cs q l x
+      else positiveWord cs q l x+q*positiveWord cs q l (cs.simple i*x) :=
+  left_plus_coeff cs q i (positiveWord cs q l) x
+
+end
+end KLInvariance.HeckeQ
+
+end
+
+
+section
+
+/-! Genuine word stalk normalization and identification with the honest
+Hecke word. This does not identify the indecomposable BMP character with KL. -/
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded
+universe u
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+ theorem bottStalkCharacter_zero (l : List I) (x : W)
+    [Subsingleton (bottStalk cs l x)] : bottStalkCharacter cs basis l x=0 := by
+  apply Hilbert.numerator_unique
+    (fun n => Module.finrank ℝ (bottStalkPiece cs basis l x n)) _ _
+    (fun q => (1-q)⁻¹ ^ Fintype.card σ)
+  · intro q _ hq
+    exact pow_ne_zero _ (inv_ne_zero (sub_ne_zero.mpr (ne_of_gt hq)))
+  · exact fun _ => bottStalkCharacter_hasSum cs basis l x
+  · intro q _ _
+    simpa only [Module.finrank_zero_of_subsingleton,Nat.cast_zero,zero_mul,Polynomial.eval₂_zero] using (hasSum_zero : HasSum (fun _ : ℤ => (0 : ℝ)) 0)
+
+ def bottTopPolynomialEquiv (l : List I) (hl : cs.IsReduced l) :
+    bottStalk cs l (cs.wordProd l) ≃ₗ[ℝ] MvPolynomial σ ℝ :=
+  ((bottTopStalkEquiv cs l hl).restrictScalars ℝ).trans
+    (SymmetricAlgebra.equivMvPolynomial basis).toLinearEquiv
+
+omit [Fintype σ] in
+ theorem bottTopPolynomialEquiv_homogeneous (l : List I) (hl : cs.IsReduced l)
+    (n : ℤ) (s : bottStalk cs l (cs.wordProd l)) (hs : s∈bottStalkPiece cs basis l (cs.wordProd l) n) :
+    bottTopPolynomialEquiv cs basis l hl s∈polynomialGrading ℝ σ n := by
+  apply (mem_symmetricGrading basis n _).mp
+  exact (bottTopStalkEquiv_homogeneous cs basis l hl n s).mpr hs
+
+ theorem bottStalkCharacter_top (l : List I) (hl : cs.IsReduced l) :
+    bottStalkCharacter cs basis l (cs.wordProd l)=1 := by
+  have hd (n : ℤ) : Module.finrank ℝ (bottStalkPiece cs basis l (cs.wordProd l) n)=
+      Module.finrank ℝ (polynomialGrading ℝ σ n) := by
+    have hh := finrank_homogeneousEquiv (bottStalkPiece cs basis l (cs.wordProd l))
+      (polynomialGrading ℝ σ) (bottTopPolynomialEquiv cs basis l hl) 0
+      (fun n s hs => by simpa only [add_zero] using bottTopPolynomialEquiv_homogeneous cs basis l hl n s hs) n
+    have hz : n-(0:ℤ)=n := Int.sub_zero n
+    rw [hz] at hh
+    exact hh.symm
+  apply Hilbert.numerator_unique
+    (fun n => Module.finrank ℝ (bottStalkPiece cs basis l (cs.wordProd l) n)) _ _
+    (fun q => (1-q)⁻¹ ^ Fintype.card σ)
+  · intro q _ hq
+    exact pow_ne_zero _ (inv_ne_zero (sub_ne_zero.mpr (ne_of_gt hq)))
+  · exact fun _ => bottStalkCharacter_hasSum cs basis l (cs.wordProd l)
+  · intro q hq hq1
+    simpa only [hd,Polynomial.eval₂_one,one_mul] using Hilbert.hasSum_polynomial ℝ σ hq.le hq1
+
+ theorem bottStalkCharacter_nil (x : W) :
+    bottStalkCharacter cs basis [] x=if x=1 then 1 else 0 := by
+  classical
+  by_cases hx : x=1
+  · subst x
+    simpa using bottStalkCharacter_top cs basis [] (by simp [CoxeterSystem.IsReduced])
+  · let := bottStalk_nil_subsingleton cs x hx
+    rw [ite_eq_right hx]
+    exact bottStalkCharacter_zero cs basis [] x
+
+ theorem bottStalkCharacter_eq_heckeWord (l : List I) (x : W) :
+    bottStalkCharacter cs basis l x=HeckeQ.positiveWord cs (Polynomial.X : Polynomial ℤ) l x := by
+  classical
+  induction l generalizing x with
+  | nil => simp only [bottStalkCharacter_nil,HeckeQ.positiveWord_nil,Hecke.basis,
+      Finsupp.single_apply]; split_ifs <;> simp_all
+  | cons i l ih =>
+    rw [HeckeQ.positiveWord_cons_coeff]
+    by_cases hd : cs.length (cs.simple i*x)<cs.length x
+    · rw [ite_eq_left hd,bottStalkCharacter_descent cs basis i l x hd,ih,ih]
+    · have ha : cs.length x<cs.length (cs.simple i*x) := by
+        have := cs.length_simple_mul x i
+        omega
+      rw [ite_eq_right hd,bottStalkCharacter_ascent cs basis i l x ha,ih,ih]
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Range-valued projections to sets of coordinates and their natural
+surjective restriction maps. -/
+namespace KLInvariance.CoordinateProjection
+universe ur un uv ux
+variable {R : Type ur} [CommRing R] {N : Type un} [AddCommGroup N] [Module R N]
+  {V : Type uv} {X : V → Type ux} [∀ x, AddCommGroup (X x)] [∀ x, Module R (X x)]
+  (f : ∀ x, N →ₗ[R] X x)
+noncomputable section
+
+ def eval (U : Set V) : N →ₗ[R] (∀ x : U, X x.val) := LinearMap.pi (fun x => f x.val)
+ def image (U : Set V) : Submodule R (∀ x : U, X x.val) := LinearMap.range (eval f U)
+ def project (U : Set V) : N →ₗ[R] image f U := (eval f U).rangeRestrict
+
+ theorem project_surjective (U : Set V) : Function.Surjective (project f U) :=
+by
+  rintro ⟨z,n,hn⟩
+  exact ⟨n,Subtype.ext hn⟩
+
+ @[simp] theorem project_apply (U : Set V) (n : N) (x : U) : (project f U n).val x=f x.val n := rfl
+
+ theorem project_zero_iff (U : Set V) (n : N) : project f U n=0 ↔ ∀ x∈U, f x n=0 := by
+  constructor
+  · intro h x hx
+    exact congrArg (fun z : image f U => z.val ⟨x,hx⟩) h
+  · intro h
+    apply Subtype.ext
+    funext x
+    exact h x.val x.property
+
+ def restrict {U K : Set V} (h : U⊆K) : image f K →ₗ[R] image f U where
+  toFun z := ⟨fun x => z.val ⟨x.val,h x.property⟩,by
+    obtain ⟨n,hn⟩ := z.property
+    refine ⟨n,?_⟩
+    funext x
+    exact congrFun hn ⟨x.val,h x.property⟩⟩
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+
+ @[simp] theorem restrict_project {U K : Set V} (h : U⊆K) (n : N) :
+    restrict f h (project f K n)=project f U n := rfl
+
+ theorem restrict_surjective {U K : Set V} (h : U⊆K) : Function.Surjective (restrict f h) := by
+  intro z
+  obtain ⟨n,rfl⟩ := project_surjective f U z
+  exact ⟨project f K n,rfl⟩
+
+ theorem restrict_zero_iff {U K : Set V} (h : U⊆K) (z : image f K) :
+    restrict f h z=0 ↔ ∀ x hx, x∈U → z.val ⟨x,hx⟩=0 := by
+  constructor
+  · intro hz x hx hu
+    exact congrArg (fun q : image f U => q.val ⟨x,hu⟩) hz
+  · intro hz
+    apply Subtype.ext
+    funext x
+    exact hz x.val (h x.property) x.property
+
+end
+end KLInvariance.CoordinateProjection
+
+end
+
+
+section
+
+/-! Upper sets and actual finite-word projection predicates. -/
+namespace KLInvariance.TitsSpace
+open Module CoordinateProjection
+universe u v
+variable {I : Type u} [klPreservedInstance11 : Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+ def BruhatUpper (U : Set W) : Prop :=
+  ∀ {x y}, BruhatLE cs x y → x∈U → y∈U
+
+ def simpleUnion (i : I) (U : Set W) : Set W := {x | x∈U ∨ cs.simple i*x∈U}
+ def simpleInter (i : I) (U : Set W) : Set W := {x | x∈U ∧ cs.simple i*x∈U}
+
+ omit [Fintype I] in
+ theorem BruhatUpper.mem_simpleUnion (U : Set W) (hU : BruhatUpper cs U) (i : I) (x : W) :
+    x∈simpleUnion cs i U ↔ upperSimple cs i x∈U := by
+  constructor
+  · rintro (hx|hx)
+    · exact hU (le_upperSimple cs i x) hx
+    · have hh := hU (le_upperSimple cs i (cs.simple i*x)) hx
+      simpa only [upperSimple_mul] using hh
+  · intro hx
+    classical
+    unfold upperSimple at hx
+    split_ifs at hx with h
+    · exact Or.inl hx
+    · exact Or.inr hx
+
+ omit [Fintype I] in
+ theorem BruhatUpper.mem_simpleInter (U : Set W) (hU : BruhatUpper cs U) (i : I) (x : W) :
+    x∈simpleInter cs i U ↔ lowerSimple cs i x∈U := by
+  constructor
+  · rintro ⟨hx,hsx⟩
+    classical
+    unfold lowerSimple
+    split_ifs <;> assumption
+  · intro hx
+    refine ⟨hU (lowerSimple_le cs i x) hx,?_⟩
+    have he : lowerSimple cs i (cs.simple i*x)∈U := by simpa only [lowerSimple_mul] using hx
+    exact hU (lowerSimple_le cs i (cs.simple i*x)) he
+
+include klPreservedInstance11 in
+ theorem BruhatUpper.simpleUnion (U : Set W) (hU : BruhatUpper cs U) (i : I) :
+    BruhatUpper cs (simpleUnion cs i U) := by
+  have _ := klPreservedInstance11
+  intro x y hxy hx
+  exact (hU.mem_simpleUnion cs U i y).mpr
+    (hU (upperSimple_mono cs i hxy) ((hU.mem_simpleUnion cs U i x).mp hx))
+
+include klPreservedInstance11 in
+ theorem BruhatUpper.simpleInter (U : Set W) (hU : BruhatUpper cs U) (i : I) :
+    BruhatUpper cs (simpleInter cs i U) := by
+  have _ := klPreservedInstance11
+  intro x y hxy hx
+  exact (hU.mem_simpleInter cs U i y).mpr
+    (hU (lowerSimple_mono cs i hxy) ((hU.mem_simpleInter cs U i x).mp hx))
+
+ omit [Fintype I] in
+ theorem simpleUnion_stable (i : I) (U : Set W) (x : W) :
+    cs.simple i*x∈simpleUnion cs i U ↔ x∈simpleUnion cs i U := by
+  change (cs.simple i*x∈U ∨ cs.simple i*(cs.simple i*x)∈U) ↔ (x∈U ∨ cs.simple i*x∈U)
+  rw [cs.simple_mul_simple_cancel_left,or_comm]
+
+ omit [Fintype I] in
+ theorem simpleInter_stable (i : I) (U : Set W) (x : W) :
+    cs.simple i*x∈simpleInter cs i U ↔ x∈simpleInter cs i U := by
+  change (cs.simple i*x∈U ∧ cs.simple i*(cs.simple i*x)∈U) ↔ (x∈U ∧ cs.simple i*x∈U)
+  rw [cs.simple_mul_simple_cancel_left,and_comm]
+
+ omit [Fintype I] in
+ theorem cross_edge_is_simple (U : Set W) (hU : BruhatUpper cs U) (i : I)
+    {x y : W} (hxy : BruhatStep cs x y) (hsx : cs.simple i*x∈U)
+    (hsy : cs.simple i*y∉U) : y=cs.simple i*x := by
+  by_contra he
+  exact hsy (hU (.single (bruhatStep_simple_mul cs hxy i he)) hsx)
+
+ def bottSetImage (l : List I) (U : Set W) := image (bottStalkProjection cs l) U
+ instance bottSetImageAddCommGroup (l : List I) (U : Set W) : AddCommGroup (bottSetImage cs l U) :=
+  Submodule.addCommGroup _
+ instance bottSetImageModule (l : List I) (U : Set W) : Module (SymmetricCoefficient (M := M)) (bottSetImage cs l U) :=
+  Submodule.module _
+
+ def bottSetProjection (l : List I) (U : Set W) : BottSamelsonModule cs l →ₗ[
+    SymmetricCoefficient (M := M)] bottSetImage cs l U := project (bottStalkProjection cs l) U
+
+ def BottCompatible (l : List I) (U : Set W) (z : ∀ x : U, bottStalk cs l x.val) : Prop :=
+  ∀ (x y : W) (hx : x∈U) (hy : y∈U), BruhatStep cs x y →
+    ∃ m : BottSamelsonModule cs l,
+      bottStalkProjection cs l x m=z ⟨x,hx⟩ ∧ bottStalkProjection cs l y m=z ⟨y,hy⟩
+
+ def BottFlabby (l : List I) : Prop :=
+  ∀ U, BruhatUpper cs U → ∀ z, BottCompatible cs l U z → z∈bottSetImage cs l U
+
+ def BottProjectiveUpper (l : List I) : Prop :=
+  ∀ U, BruhatUpper cs U → Module.Projective (SymmetricCoefficient (M := M)) (bottSetImage cs l U)
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Explicit linear descent and split exact sequences for projected modules. -/
+namespace KLInvariance.ProjectiveRange
+universe ur um up uq
+variable {R : Type ur} [CommRing R]
+  {M : Type um} [AddCommGroup M] [Module R M]
+  {P : Type up} [AddCommGroup P] [Module R P]
+  {Q : Type uq} [AddCommGroup Q] [Module R Q]
+noncomputable section
+
+ def descend (f : M →ₗ[R] P) (hf : Function.Surjective f) (g : M →ₗ[R] Q)
+    (hg : ∀ m, f m=0 → g m=0) : P →ₗ[R] Q :=
+  ((LinearMap.ker f).liftQ g hg).comp (f.quotKerEquivOfSurjective hf).symm.toLinearMap
+
+ theorem descend_apply (f : M →ₗ[R] P) (hf : Function.Surjective f) (g : M →ₗ[R] Q)
+    (hg : ∀ m, f m=0 → g m=0) (m : M) : descend f hf g hg (f m)=g m := by
+  change (LinearMap.ker f).liftQ g hg ((f.quotKerEquivOfSurjective hf).symm (f m))=g m
+  rw [LinearMap.quotKerEquivOfSurjective_symm_apply]
+  rfl
+
+ theorem projective_of_end (f : M →ₗ[R] P) (hf : Function.Surjective f)
+    [Module.Projective R M] (e : Module.End R M)
+    (he : ∀ m, f (e m)=f m) (hz : ∀ m, f m=0 → e m=0) : Module.Projective R P := by
+  apply Module.Projective.of_split (descend f hf e hz) f
+  apply LinearMap.ext
+  intro p
+  obtain ⟨m,rfl⟩ := hf p
+  rw [LinearMap.comp_apply,descend_apply,he,LinearMap.id_apply]
+
+ def kernelRetraction (f : M →ₗ[R] P) (s : P →ₗ[R] M)
+    (hs : ∀ p, f (s p)=p) : M →ₗ[R] LinearMap.ker f :=
+  (LinearMap.id-s.comp f).codRestrict (LinearMap.ker f) (fun m => by
+    change f (m-s (f m))=0
+    rw [map_sub,hs,sub_self])
+
+ theorem kernel_projective (f : M →ₗ[R] P) (hf : Function.Surjective f)
+    [Module.Projective R M] [Module.Projective R P] : Module.Projective R (LinearMap.ker f) := by
+  obtain ⟨s,hs⟩ := Module.projective_lifting_property f (LinearMap.id : P →ₗ[R] P) hf
+  have hs' : ∀ p, f (s p)=p := LinearMap.congr_fun hs
+  apply Module.Projective.of_split (LinearMap.ker f).subtype (kernelRetraction f s hs')
+  apply LinearMap.ext
+  intro m
+  apply Subtype.ext
+  change (m : M)-s (f m)=m
+  rw [m.property,map_zero,sub_zero]
+
+ def splitEquiv (f : M →ₗ[R] P) (s : P →ₗ[R] M) (hs : ∀ p, f (s p)=p) :
+    M ≃ₗ[R] (LinearMap.ker f × P) :=
+  { (kernelRetraction f s hs).prod f with
+    invFun := fun z => z.1.val+s z.2
+    left_inv := fun m => sub_add_cancel m (s (f m))
+    right_inv := by
+      intro z
+      apply Prod.ext
+      · apply Subtype.ext
+        change z.1.val+s z.2-s (f (z.1.val+s z.2))=z.1.val
+        rw [map_add,z.1.property,zero_add,hs]
+        exact add_sub_cancel_right _ _
+      · change f (z.1.val+s z.2)=z.2
+        rw [map_add,z.1.property,zero_add,hs] }
+
+ theorem projective_of_kernel (f : M →ₗ[R] P) (hf : Function.Surjective f)
+    [Module.Projective R P] [Module.Projective R (LinearMap.ker f)] : Module.Projective R M := by
+  obtain ⟨s,hs⟩ := Module.projective_lifting_property f (LinearMap.id : P →ₗ[R] P) hf
+  exact Module.Projective.of_equiv (splitEquiv f s (LinearMap.congr_fun hs)).symm
+
+end
+end KLInvariance.ProjectiveRange
+
+end
+
+
+section
+
+namespace KLInvariance.ReflectionPairTranslation
+open TensorProduct
+universe uc ua un
+variable {C : Type uc} [CommRing C] {A : Type ua} [CommRing A] [Algebra C A]
+  {N : Type un} [AddCommGroup N] [Module A N] [Module C N] [IsScalarTower C A N]
+  (τ : A ≃+* A) (hfix : ∀ c : C, τ (algebraMap C A c)=algebraMap C A c)
+noncomputable section
+
+ theorem tensorFirst_map (e : Module.End A N) (m : A ⊗[C] N) :
+    tensorFirst (C := C) (AlgebraTensorModule.map (LinearMap.id : A →ₗ[A] A) (e.restrictScalars C) m)=
+      e (tensorFirst (C := C) m) := by
+  induction m using TensorProduct.inductionOn with
+  | add m n hm hn => simp only [map_add,hm,hn]
+  | tmul a n => simp only [AlgebraTensorModule.map_tmul,LinearMap.id_apply,
+      LinearMap.restrictScalars_apply,tensorFirst_tmul,map_smul]
+
+ theorem tensorSecond_map (e : Module.End A N) (m : A ⊗[C] N) :
+    tensorSecond τ hfix (AlgebraTensorModule.map (LinearMap.id : A →ₗ[A] A) (e.restrictScalars C) m)=
+      twistMap τ e (tensorSecond τ hfix m) := by
+  induction m using TensorProduct.inductionOn with
+  | add m n hm hn => simp only [map_add,hm,hn]
+  | tmul a n =>
+    simp only [AlgebraTensorModule.map_tmul,LinearMap.id_apply,
+      LinearMap.restrictScalars_apply,tensorSecond_tmul]
+    apply Twist.ext
+    exact (map_smul e (τ a) n).symm
+
+end
+end KLInvariance.ReflectionPairTranslation
+
+end
+
+
+section
+
+namespace KLInvariance
+universe ur un up uq
+variable {R : Type ur} [CommRing R]
+  {N : Type un} [klPreservedInstance12 : AddCommGroup N] [Module R N]
+  {P : Type up} [klPreservedInstance13 : AddCommGroup P] [Module R P]
+  {Q : Type uq} [klPreservedInstance14 : AddCommGroup Q] [Module R Q]
+ theorem injective_of_zero_lift (f : N →ₗ[R] P) (hf : Function.Surjective f) (g : P →ₗ[R] Q)
+    (h : ∀ n, g (f n)=0 → f n=0) : Function.Injective g := by
+  apply LinearMap.ker_eq_bot.mp
+  rw [LinearMap.ker_eq_bot']
+  intro p hp
+  obtain ⟨n,rfl⟩ := hf p
+  exact h n hp
+ theorem maps_eq_iff_of_zero_iff (f : N →ₗ[R] P) (g : N →ₗ[R] Q)
+    (h : ∀ n, f n=0 ↔ g n=0) (m n : N) : f m=f n ↔ g m=g n := by
+  rw [← sub_eq_zero,← map_sub,h,map_sub,sub_eq_zero]
+include klPreservedInstance12 klPreservedInstance13 klPreservedInstance14 in
+ theorem functions_eq_of_surjective (f : N → P) (hf : Function.Surjective f)
+    (g h : P → Q) (he : ∀ n, g (f n)=h (f n)) (p : P) : g p=h p := by
+  have _ := klPreservedInstance12
+  have _ := klPreservedInstance13
+  have _ := klPreservedInstance14
+  obtain ⟨n,rfl⟩ := hf p
+  exact he n
+end KLInvariance
+
+end
+
+
+section
+
+/-! Genuine first/reflected-tail maps on projected upper-set images. -/
+namespace KLInvariance.TitsSpace
+open Module TensorProduct FrobeniusDuality ReflectionFrobenius ReflectionPairTranslation
+open CoordinateProjection ProjectiveRange
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+attribute [local instance] restrictedModule restrictedTower restrictedFree restrictedFinite
+
+ def simplePull (i : I) (U : Set W) : Set W := {x | cs.simple i*x∈U}
+
+ theorem bottSetProjection_zero_iff (l : List I) (U : Set W) (m : BottSamelsonModule cs l) :
+    bottSetProjection cs l U m=0 ↔ ∀ x∈U, bottStalkProjection cs l x m=0 :=
+  project_zero_iff _ _ _
+
+ theorem bottSetProjection_surjective (l : List I) (U : Set W) :
+    Function.Surjective (bottSetProjection cs l U) := project_surjective _ _
+
+ theorem bottStalkProjection_zero_of_eq (l : List I) {x y : W} (h : x=y)
+    (m : BottSamelsonModule cs l) (hm : bottStalkProjection cs l x m=0) :
+    bottStalkProjection cs l y m=0 := by
+  subst y
+  exact hm
+
+ theorem bottHeadSet_zero_iff (i : I) (l : List I) (U : Set W)
+    (m : BottSamelsonModule cs (i::l)) :
+    bottSetProjection cs (i::l) U m=0 ↔
+      bottSetProjection cs l U (bottHeadTail cs i l m)=0 ∧
+      bottSetProjection cs l (simplePull cs i U) (bottHeadReflectedTail cs i l m).val=0 := by
+  constructor
+  · intro hm
+    have h := (bottSetProjection_zero_iff cs (i::l) U m).mp hm
+    refine ⟨?_,?_⟩
+    · apply (bottSetProjection_zero_iff cs l U _).mpr
+      intro x hx
+      exact congrArg Prod.fst ((bottHeadLocalPair_zero_iff cs i l x m).mp (h x hx))
+    · apply (bottSetProjection_zero_iff cs l (simplePull cs i U) _).mpr
+      intro x hx
+      have hh : bottStalkProjection cs l (cs.simple i*(cs.simple i*x))
+          (bottHeadReflectedTail cs i l m).val=0 := congrArg (fun z => z.2.val)
+        ((bottHeadLocalPair_zero_iff cs i l (cs.simple i*x) m).mp (h _ hx))
+      exact bottStalkProjection_zero_of_eq cs l (cs.simple_mul_simple_cancel_left i) _ hh
+  · rintro ⟨hp,hq⟩
+    apply (bottSetProjection_zero_iff cs (i::l) U m).mpr
+    intro x hx
+    apply (bottHeadLocalPair_zero_iff cs i l x m).mpr
+    apply Prod.ext
+    · exact (bottSetProjection_zero_iff cs l U _).mp hp x hx
+    · apply Twist.ext
+      exact (bottSetProjection_zero_iff cs l (simplePull cs i U) _).mp hq (cs.simple i*x)
+        (by simpa only [simplePull,Set.mem_ofPred_eq,cs.simple_mul_simple_cancel_left] using hx)
+
+ def bottSetFirst (i : I) (l : List I) (U : Set W) : bottSetImage cs (i::l) U →ₗ[
+    SymmetricCoefficient (M := M)] bottSetImage cs l U :=
+  descend (bottSetProjection cs (i::l) U) (bottSetProjection_surjective cs (i::l) U)
+    ((bottSetProjection cs l U).comp (bottHeadTail cs i l))
+    (fun m hm => ((bottHeadSet_zero_iff cs i l U m).mp hm).1)
+
+ def bottSetSecond (i : I) (l : List I) (U : Set W) : bottSetImage cs (i::l) U →ₗ[
+    SymmetricCoefficient (M := M)]
+    Twist (coefficientAction cs (cs.simple i)).toRingEquiv (bottSetImage cs l (simplePull cs i U)) :=
+  descend (bottSetProjection cs (i::l) U) (bottSetProjection_surjective cs (i::l) U)
+    ((twistMap (coefficientAction cs (cs.simple i)).toRingEquiv
+      (bottSetProjection cs l (simplePull cs i U))).comp (bottHeadReflectedTail cs i l))
+    (fun m hm => Twist.ext (((bottHeadSet_zero_iff cs i l U m).mp hm).2))
+
+ theorem bottSetFirst_project (i : I) (l : List I) (U : Set W) (m : BottSamelsonModule cs (i::l)) :
+    bottSetFirst cs i l U (bottSetProjection cs (i::l) U m)=
+      bottSetProjection cs l U (bottHeadTail cs i l m) := descend_apply _ _ _ _ m
+
+ theorem bottSetSecond_project (i : I) (l : List I) (U : Set W) (m : BottSamelsonModule cs (i::l)) :
+    bottSetSecond cs i l U (bottSetProjection cs (i::l) U m)=
+      Twist.mk (bottSetProjection cs l (simplePull cs i U) (bottHeadReflectedTail cs i l m).val) :=
+  descend_apply _ _ _ _ m
+
+ theorem bottSetComponents_injective (i : I) (l : List I) (U : Set W) :
+    Function.Injective ((bottSetFirst cs i l U).prod (bottSetSecond cs i l U)) := by
+  apply injective_of_zero_lift (bottSetProjection cs (i::l) U)
+    (bottSetProjection_surjective cs (i::l) U)
+  intro m hz
+  apply (bottHeadSet_zero_iff cs i l U m).mpr
+  have hp := congrArg Prod.fst hz
+  have hq := congrArg (fun z => z.2.val) hz
+  exact ⟨(bottSetFirst_project cs i l U m).symm.trans hp,
+    (congrArg Twist.val (bottSetSecond_project cs i l U m)).symm.trans hq⟩
+
+ def bottTensorEnd (i : I) (l : List I)
+    (e : Module.End (SymmetricCoefficient (M := M)) (BottSamelsonModule cs l)) :
+    Module.End (SymmetricCoefficient (M := M)) (BottSamelsonModule cs (i::l)) :=
+  AlgebraTensorModule.map (LinearMap.id : SymmetricCoefficient (M := M) →ₗ[
+    SymmetricCoefficient (M := M)] SymmetricCoefficient (M := M)) (e.restrictScalars (simpleFixed cs i))
+
+ theorem bottTensorEnd_seed (i : I) (l : List I)
+    (e : Module.End (SymmetricCoefficient (M := M)) (BottSamelsonModule cs l))
+    (n : BottSamelsonModule cs l) :
+    bottTensorEnd cs i l e ((1 : SymmetricCoefficient (M := M)) ⊗ₜ[simpleFixed cs i] n)=
+      (1 : SymmetricCoefficient (M := M)) ⊗ₜ[simpleFixed cs i] e n := rfl
+
+ theorem bottTensorEnd_first (i : I) (l : List I)
+    (e : Module.End (SymmetricCoefficient (M := M)) (BottSamelsonModule cs l))
+    (m : BottSamelsonModule cs (i::l)) :
+    bottHeadTail cs i l (bottTensorEnd cs i l e m)=e (bottHeadTail cs i l m) :=
+  tensorFirst_map e m
+
+ theorem bottTensorEnd_second (i : I) (l : List I)
+    (e : Module.End (SymmetricCoefficient (M := M)) (BottSamelsonModule cs l))
+    (m : BottSamelsonModule cs (i::l)) :
+    bottHeadReflectedTail cs i l (bottTensorEnd cs i l e m)=
+      twistMap (coefficientAction cs (cs.simple i)).toRingEquiv e (bottHeadReflectedTail cs i l m) :=
+  tensorSecond_map (coefficientAction cs (cs.simple i)).toRingEquiv (fun c => c.property) e m
+
+ theorem bottHead_components_injective (i : I) (l : List I)
+    (m : BottSamelsonModule cs (i::l))
+    (hp : bottHeadTail cs i l m=0) (hq : bottHeadReflectedTail cs i l m=0) : m=0 := by
+  apply bottEvaluation_injective cs (i::l)
+  funext e
+  have hz : bottStalkProjection cs (i::l) (bottWeight cs (i::l) e) m=0 := by
+    apply (bottHeadLocalPair_zero_iff cs i l _ m).mpr
+    rw [bottHeadLocalPair_value,hp,hq]
+    apply Prod.ext
+    · exact map_zero _
+    · apply Twist.ext
+      exact map_zero _
+  exact congrArg (fun z => z.val ⟨e,rfl⟩) hz
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+namespace KLInvariance.ReflectionPairTranslation
+open TensorProduct
+universe uc ua un
+variable {C : Type uc} [CommRing C] {A : Type ua} [CommRing A] [Algebra C A]
+  {N : Type un} [AddCommGroup N] [Module A N] [Module C N] [IsScalarTower C A N]
+  (τ : A ≃+* A) (hfix : ∀ c : C, τ (algebraMap C A c)=algebraMap C A c)
+noncomputable section
+ theorem tensor_difference (α : A) (hdiv : ∀ a : A, α ∣ a-τ a)
+    (m : A ⊗[C] N) :
+    ∃ n : N, tensorFirst (C := C) m-(tensorSecond τ hfix m).val=α • n := by
+  induction m using TensorProduct.inductionOn with
+  | add m n hm hn =>
+    obtain ⟨p,hp⟩ := hm
+    obtain ⟨q,hq⟩ := hn
+    refine ⟨p+q,?_⟩
+    simp only [map_add]
+    change (tensorFirst (C := C) m+tensorFirst (C := C) n)-
+      ((tensorSecond τ hfix m).val+(tensorSecond τ hfix n).val)=α • (p+q)
+    rw [add_sub_add_comm,hp,hq,smul_add]
+  | tmul a n =>
+    obtain ⟨b,hb⟩ := hdiv a
+    refine ⟨b • n,?_⟩
+    change a • n-τ a • n=α • (b • n)
+    rw [← sub_smul,hb,mul_smul]
+end
+end KLInvariance.ReflectionPairTranslation
+
+end
+
+
+section
+
+namespace KLInvariance.PairProjection
+universe u v w z
+variable {R : Type u} [CommRing R]
+  {N : Type v} [AddCommGroup N] [Module R N]
+  {X : Type w} [AddCommGroup X] [Module R X]
+  {Y : Type z} [AddCommGroup Y] [Module R Y]
+  (f : N →ₗ[R] X) (g : N →ₗ[R] Y)
+
+ theorem cross_lift (α : R) (hann : ∀ z : Edge f g, α • z=0)
+    (p q : N) (h : ∃ n : N, p-q=α • n) :
+    ∃ n : N, f n=f q ∧ g n=g p := by
+  apply (compatible_iff f g _ _).mp
+  obtain ⟨n,hn⟩ := h
+  have hd := congrArg ((lower f g).comp f) hn
+  simp only [LinearMap.comp_apply,map_sub,map_smul,hann] at hd
+  exact (sub_eq_zero.mp hd).symm.trans (compatible f g p)
+
+end KLInvariance.PairProjection
+
+end
+
+
+section
+
+/-! Honest head/tail gluing. An exclusive-part crossing can only be the
+simple head edge; alpha-divisibility handles precisely that crossing. -/
+namespace KLInvariance.TitsSpace
+open Module CoordinateProjection ReflectionPairTranslation
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+open FrobeniusDuality ReflectionFrobenius
+attribute [local instance] restrictedModule restrictedTower restrictedFree restrictedFinite
+
+ theorem bottHead_difference (i : I) (l : List I) (m : BottSamelsonModule cs (i::l)) :
+    ∃ n : BottSamelsonModule cs l,
+      bottHeadTail cs i l m-(bottHeadReflectedTail cs i l m).val=simpleRootPolynomial (M := M) i • n :=
+  tensor_difference (coefficientAction cs (cs.simple i)).toRingEquiv (fun c => c.property)
+    (simpleRootPolynomial (M := M) i) (simple_divisible cs i) m
+
+ def bottHeadPatch (i : I) (l : List I) (U : Set W) (m : BottSamelsonModule cs (i::l)) :
+    ∀ x : simpleUnion cs i U, bottStalk cs l x.val := by
+  classical
+  exact fun x => if x.val∈U then bottStalkProjection cs l x.val (bottHeadTail cs i l m)
+    else bottStalkProjection cs l x.val (bottHeadReflectedTail cs i l m).val
+
+ theorem bottHeadPatch_compatible (i : I) (l : List I) (U : Set W) (hU : BruhatUpper cs U)
+    (m : BottSamelsonModule cs (i::l))
+    (hJ : ∀ x∈simpleInter cs i U,
+      bottStalkProjection cs l x (bottHeadTail cs i l m)=
+        bottStalkProjection cs l x (bottHeadReflectedTail cs i l m).val) :
+    BottCompatible cs l (simpleUnion cs i U) (bottHeadPatch cs i l U m) := by
+  classical
+  intro x y hx hy hxy
+  by_cases hxU : x∈U
+  · have hyU := hU (.single hxy) hxU
+    simp only [bottHeadPatch,ite_eq_left hxU,ite_eq_left hyU]
+    exact ⟨bottHeadTail cs i l m,rfl,rfl⟩
+  · by_cases hyU : y∈U
+    · simp only [bottHeadPatch,ite_eq_right hxU,ite_eq_left hyU]
+      by_cases hsy : cs.simple i*y∈U
+      · exact ⟨(bottHeadReflectedTail cs i l m).val,rfl,(hJ y ⟨hyU,hsy⟩).symm⟩
+      · have he := cross_edge_is_simple cs U hU i hxy (hx.resolve_left hxU) hsy
+        subst y
+        exact PairProjection.cross_lift _ _ (simpleRootPolynomial (M := M) i)
+          (bottReflectionEdge_annihilated cs l x (cs.simple i) (simple_represents cs i))
+          (bottHeadTail cs i l m) (bottHeadReflectedTail cs i l m).val (bottHead_difference cs i l m)
+    · simp only [bottHeadPatch,ite_eq_right hxU,ite_eq_right hyU]
+      exact ⟨(bottHeadReflectedTail cs i l m).val,rfl,rfl⟩
+
+ theorem bottHeadPatch_lift (i : I) (l : List I) (H : BottFlabby cs l)
+    (U : Set W) (hU : BruhatUpper cs U) (m : BottSamelsonModule cs (i::l))
+    (hJ : ∀ x∈simpleInter cs i U,
+      bottStalkProjection cs l x (bottHeadTail cs i l m)=
+        bottStalkProjection cs l x (bottHeadReflectedTail cs i l m).val) :
+    ∃ n : BottSamelsonModule cs l,
+      (∀ x∈U, bottStalkProjection cs l x n=bottStalkProjection cs l x (bottHeadTail cs i l m)) ∧
+      (∀ x∈simplePull cs i U,
+        bottStalkProjection cs l x n=bottStalkProjection cs l x (bottHeadReflectedTail cs i l m).val) := by
+  classical
+  obtain ⟨n,hn⟩ := H (simpleUnion cs i U) (hU.simpleUnion cs U i)
+    (bottHeadPatch cs i l U m) (bottHeadPatch_compatible cs i l U hU m hJ)
+  refine ⟨n,?_,?_⟩
+  · intro x hx
+    have hh := congrFun hn ⟨x,Or.inl hx⟩
+    simpa only [CoordinateProjection.eval,LinearMap.pi_apply,bottHeadPatch,ite_eq_left hx] using hh
+  · intro x hx
+    have hh := congrFun hn ⟨x,Or.inr hx⟩
+    by_cases hxU : x∈U
+    · have hh' : bottStalkProjection cs l x n=bottStalkProjection cs l x (bottHeadTail cs i l m) := by
+        simpa only [CoordinateProjection.eval,LinearMap.pi_apply,bottHeadPatch,ite_eq_left hxU] using hh
+      exact hh'.trans (hJ x ⟨hxU,hx⟩)
+    · simpa only [CoordinateProjection.eval,LinearMap.pi_apply,bottHeadPatch,ite_eq_right hxU] using hh
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+namespace KLInvariance.TitsSpace
+open Module TensorProduct CoordinateProjection ReflectionPairTranslation
+open FrobeniusDuality ReflectionFrobenius
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+attribute [local instance] restrictedModule restrictedTower restrictedFree restrictedFinite
+
+ def bottSetRestrict (l : List I) {U K : Set W} (h : U⊆K) :
+    bottSetImage cs l K →ₗ[SymmetricCoefficient (M := M)] bottSetImage cs l U :=
+  restrict (bottStalkProjection cs l) h
+
+ theorem bottSetRestrict_project (l : List I) {U K : Set W} (h : U⊆K)
+    (n : BottSamelsonModule cs l) :
+    bottSetRestrict cs l h (bottSetProjection cs l K n)=bottSetProjection cs l U n := rfl
+
+ theorem bottSetRestrict_surjective (l : List I) {U K : Set W} (h : U⊆K) :
+    Function.Surjective (bottSetRestrict cs l h) := restrict_surjective _ h
+
+ theorem bottSetRestrict_zero_iff (l : List I) {U K : Set W} (h : U⊆K)
+    (z : bottSetImage cs l K) : bottSetRestrict cs l h z=0 ↔
+      ∀ x hx, x∈U → z.val ⟨x,hx⟩=0 := restrict_zero_iff (bottStalkProjection cs l) h z
+
+ def bottSeed (i : I) (l : List I) (n : BottSamelsonModule cs l) :
+    BottSamelsonModule cs (i::l) := (1 : SymmetricCoefficient (M := M)) ⊗ₜ[simpleFixed cs i] n
+
+ theorem bottSeed_first (i : I) (l : List I) (n : BottSamelsonModule cs l) :
+    bottHeadTail cs i l (bottSeed cs i l n)=n := bottHeadTail_seed cs i l n
+
+ theorem bottSeed_second (i : I) (l : List I) (n : BottSamelsonModule cs l) :
+    bottHeadReflectedTail cs i l (bottSeed cs i l n)=Twist.mk n := bottHeadReflectedTail_seed cs i l n
+
+ theorem bottSetFirst_seed (i : I) (l : List I) (U : Set W) (n : BottSamelsonModule cs l) :
+    bottSetFirst cs i l U (bottSetProjection cs (i::l) U (bottSeed cs i l n))=
+      bottSetProjection cs l U n :=
+  (bottSetFirst_project cs i l U _).trans
+    (congrArg (bottSetProjection cs l U) (bottSeed_first cs i l n))
+
+ theorem bottSetSecond_seed (i : I) (l : List I) (U : Set W) (n : BottSamelsonModule cs l) :
+    bottSetSecond cs i l U (bottSetProjection cs (i::l) U (bottSeed cs i l n))=
+      Twist.mk (bottSetProjection cs l (simplePull cs i U) n) :=
+  (bottSetSecond_project cs i l U _).trans
+    (congrArg (fun t => Twist.mk (bottSetProjection cs l (simplePull cs i U) t.val))
+      (bottSeed_second cs i l n))
+
+ theorem bottSeed_zero_stable (i : I) (l : List I) (U : Set W)
+    (hU : ∀ x, cs.simple i*x∈U ↔ x∈U)
+    (n : BottSamelsonModule cs l) (hn : bottSetProjection cs l U n=0) :
+    bottSetProjection cs (i::l) U (bottSeed cs i l n)=0 := by
+  apply (bottHeadSet_zero_iff cs i l U _).mpr
+  constructor
+  · exact (congrArg (bottSetProjection cs l U) (bottSeed_first cs i l n)).trans hn
+  · apply ((congrArg (fun t => bottSetProjection cs l (simplePull cs i U) t.val)
+      (bottSeed_second cs i l n)).trans ·)
+    apply (bottSetProjection_zero_iff cs l (simplePull cs i U) n).mpr
+    intro x hx
+    exact (bottSetProjection_zero_iff cs l U n).mp hn x ((hU x).mp hx)
+
+ theorem bottSetFirst_restrict (i : I) (l : List I) {U K : Set W} (h : U⊆K)
+    (z : bottSetImage cs (i::l) K) :
+    bottSetFirst cs i l U (bottSetRestrict cs (i::l) h z)=
+      bottSetRestrict cs l h (bottSetFirst cs i l K z) := by
+  refine functions_eq_of_surjective (bottSetProjection cs (i::l) K)
+    (bottSetProjection_surjective cs (i::l) K)
+    (fun z => bottSetFirst cs i l U (bottSetRestrict cs (i::l) h z))
+    (fun z => bottSetRestrict cs l h (bottSetFirst cs i l K z)) ?_ z
+  intro m
+  exact (bottSetFirst_project cs i l U m).trans
+    (congrArg (bottSetRestrict cs l h) (bottSetFirst_project cs i l K m)).symm
+
+ theorem bottSeed_zero_inter (i : I) (l : List I) (U : Set W)
+    (n : BottSamelsonModule cs l) (hn : bottSetProjection cs l U n=0) :
+    bottSetProjection cs (i::l) (simpleInter cs i U) (bottSeed cs i l n)=0 := by
+  apply (bottHeadSet_zero_iff cs i l (simpleInter cs i U) _).mpr
+  have h := (bottSetProjection_zero_iff cs l U n).mp hn
+  constructor
+  · apply ((congrArg (bottSetProjection cs l (simpleInter cs i U))
+      (bottSeed_first cs i l n)).trans ·)
+    apply (bottSetProjection_zero_iff cs l (simpleInter cs i U) n).mpr
+    intro x hx
+    exact h x hx.1
+  · apply ((congrArg (fun t => bottSetProjection cs l (simplePull cs i (simpleInter cs i U)) t.val)
+      (bottSeed_second cs i l n)).trans ·)
+    apply (bottSetProjection_zero_iff cs l (simplePull cs i (simpleInter cs i U)) n).mpr
+    intro x hx
+    exact h x (by simpa only [cs.simple_mul_simple_cancel_left] using hx.2)
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+namespace KLInvariance
+universe ur um un up
+variable {R : Type ur} [CommRing R]
+  {M : Type um} [AddCommGroup M] [Module R M]
+  {N : Type un} [AddCommGroup N] [Module R N]
+  {P : Type up} [AddCommGroup P] [Module R P]
+noncomputable section
+ theorem projective_of_same_range (f : M →ₗ[R] P) (hf : Function.Injective f)
+    (g : N →ₗ[R] P) (hg : Function.Injective g) (h : LinearMap.range f=LinearMap.range g)
+    [Module.Projective R N] : Module.Projective R M :=
+  Module.Projective.of_equiv (((LinearEquiv.ofInjective g hg).trans
+    (LinearEquiv.ofEq _ _ h.symm)).trans (LinearEquiv.ofInjective f hf).symm)
+
+attribute [local instance] RingHomInvPair.of_ringEquiv RingHomInvPair.of_ringEquiv_symm
+
+namespace ReflectionPairTranslation.Twist
+ theorem projective (τ : R ≃+* R) [Module.Projective R N] : Module.Projective R (Twist τ N) := by
+  let : RingHomInvPair τ.symm.toRingHom τ.toRingHom := RingHomInvPair.of_ringEquiv_symm τ
+  let : RingHomInvPair τ.toRingHom τ.symm.toRingHom := RingHomInvPair.of_ringEquiv τ
+  exact Module.Projective.of_equiv (σ' := τ.toRingHom) (equiv τ N)
+end ReflectionPairTranslation.Twist
+end
+end KLInvariance
+
+end
+
+
+section
+
+namespace KLInvariance
+universe ur um uh up us
+variable {R : Type ur} [CommRing R]
+  {M : Type um} [AddCommGroup M] [Module R M]
+  {H : Type uh} [AddCommGroup H] [Module R H]
+  {P : Type up} [AddCommGroup P] [Module R P]
+  {S : Type us} [AddCommGroup S] [Module R S]
+
+ theorem kernel_coordinate_injective (f : M →ₗ[R] P) (g : M →ₗ[R] S)
+    (h : Function.Injective (f.prod g)) :
+    Function.Injective (g.comp (LinearMap.ker f).subtype) := by
+  intro m n he
+  apply Subtype.ext
+  apply h
+  exact Prod.ext (m.property.trans n.property.symm) he
+
+ theorem common_kernel_coordinate_injective (r : M →ₗ[R] H) (f : M →ₗ[R] P) (g : M →ₗ[R] S)
+    (h : Function.Injective (f.prod g)) :
+    Function.Injective (g.comp (LinearMap.ker r ⊓ LinearMap.ker f).subtype) := by
+  intro m n he
+  apply Subtype.ext
+  apply h
+  exact Prod.ext (m.property.2.trans n.property.2.symm) he
+
+ theorem projective_of_coordinates (f : M →ₗ[R] P) (hf : Function.Injective f)
+    (g : H →ₗ[R] P) (hg : Function.Injective g)
+    (hfg : ∀ m, ∃ n, f m=g n) (hgf : ∀ n, ∃ m, f m=g n)
+    [Module.Projective R H] : Module.Projective R M := by
+  apply projective_of_same_range f hf g hg
+  ext p
+  constructor
+  · rintro ⟨m,rfl⟩
+    obtain ⟨n,hn⟩ := hfg m
+    exact ⟨n,hn.symm⟩
+  · rintro ⟨n,rfl⟩
+    exact hgf n
+
+namespace ReflectionPairTranslation
+ theorem twistMap_injective (τ : R ≃+* R) (f : M →ₗ[R] P) (hf : Function.Injective f) :
+    Function.Injective (twistMap τ f) := by
+  intro m n h
+  exact Twist.ext (hf (congrArg Twist.val h))
+end ReflectionPairTranslation
+
+namespace CoordinateProjection
+universe uv ux
+variable {V : Type uv} {X : V → Type ux} [∀ x, AddCommGroup (X x)] [∀ x, Module R (X x)]
+  (f : ∀ x, M →ₗ[R] X x)
+ theorem restrict_joint_injective {U V' K : Set V} (hU : U⊆K) (hV : V'⊆K)
+    (hc : ∀ x∈K, x∈U ∨ x∈V') :
+    Function.Injective ((restrict f hU).prod (restrict f hV)) := by
+  intro p q h
+  apply Subtype.ext
+  funext x
+  rcases hc x.val x.property with hx|hx
+  · exact congrArg (fun z : image f U => z.val ⟨x.val,hx⟩) (congrArg Prod.fst h)
+  · exact congrArg (fun z : image f V' => z.val ⟨x.val,hx⟩) (congrArg Prod.snd h)
+end CoordinateProjection
+end KLInvariance
+
+end
+
+
+section
+
+namespace KLInvariance.TitsSpace
+open Module CoordinateProjection ReflectionPairTranslation
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+ def bottInterRestriction (i : I) (l : List I) (U : Set W) :
+    bottSetImage cs l U →ₗ[SymmetricCoefficient (M := M)] bottSetImage cs l (simpleInter cs i U) :=
+  bottSetRestrict cs l (fun _ h => h.1)
+
+ def bottUnionRestriction (i : I) (l : List I) (U : Set W) :
+    bottSetImage cs l (simpleUnion cs i U) →ₗ[SymmetricCoefficient (M := M)] bottSetImage cs l U :=
+  bottSetRestrict cs l (fun _ h => Or.inl h)
+
+ def bottUnionPullRestriction (i : I) (l : List I) (U : Set W) :
+    bottSetImage cs l (simpleUnion cs i U) →ₗ[SymmetricCoefficient (M := M)]
+      bottSetImage cs l (simplePull cs i U) :=
+  bottSetRestrict cs l (fun _ h => Or.inr h)
+
+ def bottCommonKernel (i : I) (l : List I) (U : Set W) :
+    Submodule (SymmetricCoefficient (M := M)) (bottSetImage cs (i::l) U) :=
+  LinearMap.ker (bottInterRestriction cs i (i::l) U) ⊓ LinearMap.ker (bottSetFirst cs i l U)
+
+ def bottCommonSecond (i : I) (l : List I) (U : Set W) :
+    bottCommonKernel cs i l U →ₗ[SymmetricCoefficient (M := M)]
+      Twist (coefficientAction cs (cs.simple i)).toRingEquiv (bottSetImage cs l (simplePull cs i U)) :=
+  (bottSetSecond cs i l U).comp (bottCommonKernel cs i l U).subtype
+
+ theorem bottCommonSecond_injective (i : I) (l : List I) (U : Set W) :
+    Function.Injective (bottCommonSecond cs i l U) :=
+  common_kernel_coordinate_injective _ _ _ (bottSetComponents_injective cs i l U)
+
+ def bottUnionKernelSecond (i : I) (l : List I) (U : Set W) :
+    LinearMap.ker (bottUnionRestriction cs i l U) →ₗ[SymmetricCoefficient (M := M)]
+      bottSetImage cs l (simplePull cs i U) :=
+  (bottUnionPullRestriction cs i l U).comp (LinearMap.ker (bottUnionRestriction cs i l U)).subtype
+
+ theorem bottUnionKernelSecond_injective (i : I) (l : List I) (U : Set W) :
+    Function.Injective (bottUnionKernelSecond cs i l U) := by
+  apply kernel_coordinate_injective
+  exact restrict_joint_injective (bottStalkProjection cs l)
+    (fun _ h => Or.inl h) (fun _ h => Or.inr h) (fun _ h => h)
+
+ theorem bottHeadPatch_lift_of_zero_inter (i : I) (l : List I) (H : BottFlabby cs l)
+    (U : Set W) (hU : BruhatUpper cs U) (m : BottSamelsonModule cs (i::l))
+    (hm : bottSetProjection cs (i::l) (simpleInter cs i U) m=0) :
+    ∃ n : BottSamelsonModule cs l,
+      (∀ x∈U, bottStalkProjection cs l x n=bottStalkProjection cs l x (bottHeadTail cs i l m)) ∧
+      (∀ x∈simplePull cs i U,
+        bottStalkProjection cs l x n=bottStalkProjection cs l x (bottHeadReflectedTail cs i l m).val) := by
+  apply bottHeadPatch_lift cs i l H U hU m
+  have hz := (bottHeadSet_zero_iff cs i l (simpleInter cs i U) m).mp hm
+  intro x hx
+  apply ((bottSetProjection_zero_iff cs l (simpleInter cs i U) _).mp hz.1 x hx).trans
+  exact ((bottSetProjection_zero_iff cs l (simplePull cs i (simpleInter cs i U)) _).mp hz.2 x
+    ((simpleInter_stable cs i U x).mpr hx)).symm
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+namespace KLInvariance.TitsSpace
+open Module CoordinateProjection ReflectionPairTranslation
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+ theorem bottCommon_lift (i : I) (l : List I) (U : Set W)
+    (z : bottCommonKernel cs i l U) :
+    ∃ m : BottSamelsonModule cs (i::l),
+      bottSetProjection cs (i::l) U m=z.val ∧
+      bottSetProjection cs (i::l) (simpleInter cs i U) m=0 ∧
+      bottSetProjection cs l U (bottHeadTail cs i l m)=0 := by
+  let m : BottSamelsonModule cs (i::l) :=
+    (bottSetProjection_surjective cs (i::l) U z.val).choose
+  have hm : bottSetProjection cs (i::l) U m=z.val :=
+    (bottSetProjection_surjective cs (i::l) U z.val).choose_spec
+  refine ⟨m,hm,?_,?_⟩
+  · exact (congrArg (bottInterRestriction cs i (i::l) U) hm).trans z.property.1
+  · exact (bottSetFirst_project cs i l U m).symm.trans
+      ((congrArg (bottSetFirst cs i l U) hm).trans z.property.2)
+
+ theorem bottCommon_patch_lift (i : I) (l : List I) (H : BottFlabby cs l)
+    (U : Set W) (hU : BruhatUpper cs U) (m : BottSamelsonModule cs (i::l))
+    (hmJ : bottSetProjection cs (i::l) (simpleInter cs i U) m=0)
+    (hmp : bottSetProjection cs l U (bottHeadTail cs i l m)=0) :
+    ∃ n : BottSamelsonModule cs l, bottSetProjection cs l U n=0 ∧
+      bottSetProjection cs l (simplePull cs i U) n=
+      bottSetProjection cs l (simplePull cs i U) (bottHeadReflectedTail cs i l m).val := by
+  let n : BottSamelsonModule cs l := (bottHeadPatch_lift_of_zero_inter cs i l H U hU m hmJ).choose
+  have hnp := (bottHeadPatch_lift_of_zero_inter cs i l H U hU m hmJ).choose_spec.1
+  have hnq := (bottHeadPatch_lift_of_zero_inter cs i l H U hU m hmJ).choose_spec.2
+  refine ⟨n,?_,Subtype.ext (funext fun x => hnq x.val x.property)⟩
+  apply (bottSetProjection_zero_iff cs l U n).mpr
+  intro x hx
+  exact (hnp x hx).trans ((bottSetProjection_zero_iff cs l U _).mp hmp x hx)
+
+ theorem bottCommon_range_of_patch (i : I) (l : List I) (U : Set W)
+    (z : bottCommonKernel cs i l U) (m : BottSamelsonModule cs (i::l))
+    (hm : bottSetProjection cs (i::l) U m=z.val) (n : BottSamelsonModule cs l)
+    (hnU : bottSetProjection cs l U n=0)
+    (hnq : bottSetProjection cs l (simplePull cs i U) n=
+      bottSetProjection cs l (simplePull cs i U) (bottHeadReflectedTail cs i l m).val) :
+    ∃ d : Twist (coefficientAction cs (cs.simple i)).toRingEquiv
+        (LinearMap.ker (bottUnionRestriction cs i l U)),
+      bottCommonSecond cs i l U z=
+        twistMap (coefficientAction cs (cs.simple i)).toRingEquiv (bottUnionKernelSecond cs i l U) d := by
+  let d : LinearMap.ker (bottUnionRestriction cs i l U) :=
+    ⟨bottSetProjection cs l (simpleUnion cs i U) n,hnU⟩
+  refine ⟨Twist.mk d,?_⟩
+  change bottSetSecond cs i l U z.val=Twist.mk (bottSetProjection cs l (simplePull cs i U) n)
+  exact ((congrArg (bottSetSecond cs i l U) hm).symm.trans
+    (bottSetSecond_project cs i l U m)).trans (congrArg Twist.mk hnq.symm)
+
+ theorem bottCommon_range_of_lift (i : I) (l : List I) (H : BottFlabby cs l)
+    (U : Set W) (hU : BruhatUpper cs U) (z : bottCommonKernel cs i l U)
+    (m : BottSamelsonModule cs (i::l)) (hm : bottSetProjection cs (i::l) U m=z.val)
+    (hmJ : bottSetProjection cs (i::l) (simpleInter cs i U) m=0)
+    (hmp : bottSetProjection cs l U (bottHeadTail cs i l m)=0) :
+    ∃ d : Twist (coefficientAction cs (cs.simple i)).toRingEquiv
+        (LinearMap.ker (bottUnionRestriction cs i l U)),
+      bottCommonSecond cs i l U z=
+        twistMap (coefficientAction cs (cs.simple i)).toRingEquiv (bottUnionKernelSecond cs i l U) d := by
+  exact bottCommon_range_of_patch cs i l U z m hm _
+    (bottCommon_patch_lift cs i l H U hU m hmJ hmp).choose_spec.1
+    (bottCommon_patch_lift cs i l H U hU m hmJ hmp).choose_spec.2
+
+ theorem bottCommon_range_forward (i : I) (l : List I) (H : BottFlabby cs l)
+    (U : Set W) (hU : BruhatUpper cs U) (z : bottCommonKernel cs i l U) :
+    ∃ d : Twist (coefficientAction cs (cs.simple i)).toRingEquiv
+        (LinearMap.ker (bottUnionRestriction cs i l U)),
+      bottCommonSecond cs i l U z=
+        twistMap (coefficientAction cs (cs.simple i)).toRingEquiv (bottUnionKernelSecond cs i l U) d :=
+  bottCommon_range_of_lift cs i l H U hU z _ (bottCommon_lift cs i l U z).choose_spec.1
+    (bottCommon_lift cs i l U z).choose_spec.2.1
+    (bottCommon_lift cs i l U z).choose_spec.2.2
+
+ theorem bottCommon_range_backward (i : I) (l : List I) (U : Set W)
+    (d : Twist (coefficientAction cs (cs.simple i)).toRingEquiv
+      (LinearMap.ker (bottUnionRestriction cs i l U))) :
+    ∃ z : bottCommonKernel cs i l U,
+      bottCommonSecond cs i l U z=
+        twistMap (coefficientAction cs (cs.simple i)).toRingEquiv (bottUnionKernelSecond cs i l U) d := by
+  let n : BottSamelsonModule cs l :=
+    (bottSetProjection_surjective cs l (simpleUnion cs i U) d.val.val).choose
+  have hn : bottSetProjection cs l (simpleUnion cs i U) n=d.val.val :=
+    (bottSetProjection_surjective cs l (simpleUnion cs i U) d.val.val).choose_spec
+  have hnU : bottSetProjection cs l U n=0 :=
+    (congrArg (bottUnionRestriction cs i l U) hn).trans d.val.property
+  let z : bottCommonKernel cs i l U :=
+    ⟨bottSetProjection cs (i::l) U (bottSeed cs i l n),
+      bottSeed_zero_inter cs i l U n hnU,(bottSetFirst_seed cs i l U n).trans hnU⟩
+  refine ⟨z,?_⟩
+  exact (bottSetSecond_seed cs i l U n).trans
+    (congrArg Twist.mk (congrArg (bottUnionPullRestriction cs i l U) hn))
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Projectivity on reflection-stable upper sets, via an ACTUAL tensor
+splitting rather than an asserted reflexivity of a projected lattice. -/
+namespace KLInvariance.TitsSpace
+open Module CoordinateProjection ProjectiveRange ReflectionPairTranslation
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+ theorem bottHeadLocalPair_eq_iff (i : I) (l : List I) (x : W)
+    (m n : BottSamelsonModule cs (i::l)) :
+    bottStalkProjection cs (i::l) x m=bottStalkProjection cs (i::l) x n ↔
+      bottHeadLocalPair cs i l x m=bottHeadLocalPair cs i l x n :=
+  maps_eq_iff_of_zero_iff _ _ (bottHeadLocalPair_zero_iff cs i l x) m n
+
+ theorem bottTensorEnd_set_preserve (i : I) (l : List I) (U : Set W)
+    (hU : ∀ x, cs.simple i*x∈U ↔ x∈U)
+    (e : Module.End (SymmetricCoefficient (M := M)) (BottSamelsonModule cs l))
+    (he : ∀ n x, x∈U → bottStalkProjection cs l x (e n)=bottStalkProjection cs l x n)
+    (m : BottSamelsonModule cs (i::l)) :
+    bottSetProjection cs (i::l) U (bottTensorEnd cs i l e m)=bottSetProjection cs (i::l) U m := by
+  apply Subtype.ext
+  funext x
+  apply (bottHeadLocalPair_eq_iff cs i l x.val _ _).mpr
+  apply (bottHeadLocalPair_value cs i l x.val _).trans
+  apply Eq.trans ?_ (bottHeadLocalPair_value cs i l x.val m).symm
+  apply Prod.ext
+  · exact (congrArg (bottStalkProjection cs l x.val) (bottTensorEnd_first cs i l e m)).trans
+      (he _ x.val x.property)
+  · apply Twist.ext
+    exact (congrArg (fun t => bottStalkProjection cs l (cs.simple i*x.val) t.val)
+      (bottTensorEnd_second cs i l e m)).trans
+      (he _ (cs.simple i*x.val) ((hU x.val).mpr x.property))
+
+ theorem bottTensorEnd_set_kills (i : I) (l : List I) (U : Set W)
+    (hU : ∀ x, cs.simple i*x∈U ↔ x∈U)
+    (e : Module.End (SymmetricCoefficient (M := M)) (BottSamelsonModule cs l))
+    (hz : ∀ n, bottSetProjection cs l U n=0 → e n=0)
+    (m : BottSamelsonModule cs (i::l)) (hm : bottSetProjection cs (i::l) U m=0) :
+    bottTensorEnd cs i l e m=0 := by
+  have hmpq := (bottHeadSet_zero_iff cs i l U m).mp hm
+  have hqzero : bottSetProjection cs l U (bottHeadReflectedTail cs i l m).val=0 := by
+    apply (bottSetProjection_zero_iff cs l U _).mpr
+    intro x hx
+    exact (bottSetProjection_zero_iff cs l (simplePull cs i U) _).mp hmpq.2 x ((hU x).mpr hx)
+  apply bottHead_components_injective cs i l
+  · exact (bottTensorEnd_first cs i l e m).trans (hz _ hmpq.1)
+  · apply Twist.ext
+    exact (congrArg Twist.val (bottTensorEnd_second cs i l e m)).trans (hz _ hqzero)
+
+ theorem bottStableUpper_projective (i : I) (l : List I) (U : Set W)
+    (hU : ∀ x, cs.simple i*x∈U ↔ x∈U)
+    [Module.Projective (SymmetricCoefficient (M := M)) (bottSetImage cs l U)] :
+    Module.Projective (SymmetricCoefficient (M := M)) (bottSetImage cs (i::l) U) := by
+  obtain ⟨s,hs⟩ := Module.projective_lifting_property (bottSetProjection cs l U)
+    (LinearMap.id : bottSetImage cs l U →ₗ[SymmetricCoefficient (M := M)] bottSetImage cs l U)
+    (bottSetProjection_surjective cs l U)
+  have hs' : ∀ p, bottSetProjection cs l U (s p)=p := LinearMap.congr_fun hs
+  let e := s.comp (bottSetProjection cs l U)
+  have he : ∀ n x, x∈U → bottStalkProjection cs l x (e n)=bottStalkProjection cs l x n := by
+    intro n x hx
+    exact congrArg (fun z : bottSetImage cs l U => z.val ⟨x,hx⟩) (hs' (bottSetProjection cs l U n))
+  have hz : ∀ n, bottSetProjection cs l U n=0 → e n=0 := by
+    intro n hn
+    exact (congrArg s hn).trans s.map_zero
+  exact projective_of_end (bottSetProjection cs (i::l) U) (bottSetProjection_surjective cs (i::l) U)
+    (bottTensorEnd cs i l e) (bottTensorEnd_set_preserve cs i l U hU e he)
+    (bottTensorEnd_set_kills cs i l U hU e hz)
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+namespace KLInvariance.ProjectiveRange
+universe ur um uh up us
+variable {R : Type ur} [CommRing R]
+  {M : Type um} [AddCommGroup M] [Module R M]
+  {H : Type uh} [AddCommGroup H] [Module R H]
+  {P : Type up} [AddCommGroup P] [Module R P]
+  {S : Type us} [AddCommGroup S] [Module R S]
+  (r : M →ₗ[R] H) (f : M →ₗ[R] P) (k : P →ₗ[R] S)
+  (h : ∀ m, r m=0 → k (f m)=0)
+noncomputable section
+
+ def onKernel : LinearMap.ker r →ₗ[R] LinearMap.ker k :=
+  (f.comp (LinearMap.ker r).subtype).codRestrict (LinearMap.ker k) (fun m => h m.val m.property)
+
+ def kernelOnKernelEquiv : ↥(LinearMap.ker r ⊓ LinearMap.ker f) ≃ₗ[R]
+    LinearMap.ker (onKernel r f k h) where
+  toFun z := ⟨⟨z.val,z.property.1⟩,Subtype.ext z.property.2⟩
+  invFun z := ⟨z.val.val,z.val.property,congrArg Subtype.val z.property⟩
+  left_inv _ := rfl
+  right_inv _ := rfl
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+
+ theorem projective_of_two_stage (hr : Function.Surjective r)
+    (hf : Function.Surjective (onKernel r f k h))
+    [Module.Projective R H] [Module.Projective R (LinearMap.ker k)]
+    [Module.Projective R ↥(LinearMap.ker r ⊓ LinearMap.ker f)] :
+    Module.Projective R M := by
+  let : Module.Projective R (LinearMap.ker (onKernel r f k h)) :=
+    Module.Projective.of_equiv (kernelOnKernelEquiv r f k h)
+  let : Module.Projective R (LinearMap.ker r) := projective_of_kernel (onKernel r f k h) hf
+  exact projective_of_kernel r hr
+end
+end KLInvariance.ProjectiveRange
+
+end
+
+
+section
+
+namespace KLInvariance.TitsSpace
+open Module CoordinateProjection ReflectionPairTranslation ProjectiveRange
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+ theorem bottInter_first_zero (i : I) (l : List I) (U : Set W)
+    (z : bottSetImage cs (i::l) U) (hz : bottInterRestriction cs i (i::l) U z=0) :
+    bottInterRestriction cs i l U (bottSetFirst cs i l U z)=0 :=
+  (bottSetFirst_restrict cs i l (fun _ h => h.1) z).symm.trans
+    ((congrArg (bottSetFirst cs i l (simpleInter cs i U)) hz).trans (map_zero _))
+
+ theorem bottInterKernelFirst_surjective (i : I) (l : List I) (U : Set W) :
+    Function.Surjective (onKernel (bottInterRestriction cs i (i::l) U) (bottSetFirst cs i l U)
+      (bottInterRestriction cs i l U) (bottInter_first_zero cs i l U)) := by
+  intro z
+  let n : BottSamelsonModule cs l := (bottSetProjection_surjective cs l U z.val).choose
+  have hn : bottSetProjection cs l U n=z.val := (bottSetProjection_surjective cs l U z.val).choose_spec
+  have hnJ : bottSetProjection cs l (simpleInter cs i U) n=0 :=
+    (congrArg (bottInterRestriction cs i l U) hn).trans z.property
+  let w : LinearMap.ker (bottInterRestriction cs i (i::l) U) :=
+    ⟨bottSetProjection cs (i::l) U (bottSeed cs i l n),
+      bottSeed_zero_stable cs i l (simpleInter cs i U) (simpleInter_stable cs i U) n hnJ⟩
+  exact ⟨w,Subtype.ext ((bottSetFirst_seed cs i l U n).trans hn)⟩
+
+ theorem bottCommon_projective (i : I) (l : List I) (HP : BottProjectiveUpper cs l)
+    (HF : BottFlabby cs l) (U : Set W) (hU : BruhatUpper cs U) :
+    Module.Projective (SymmetricCoefficient (M := M)) (bottCommonKernel cs i l U) := by
+  let : Module.Projective (SymmetricCoefficient (M := M)) (bottSetImage cs l U) := HP U hU
+  let : Module.Projective (SymmetricCoefficient (M := M)) (bottSetImage cs l (simpleUnion cs i U)) :=
+    HP (simpleUnion cs i U) (hU.simpleUnion cs U i)
+  let : Module.Projective (SymmetricCoefficient (M := M)) (LinearMap.ker (bottUnionRestriction cs i l U)) :=
+    kernel_projective (bottUnionRestriction cs i l U) (bottSetRestrict_surjective cs l (fun _ h => Or.inl h))
+  let : Module.Projective (SymmetricCoefficient (M := M))
+      (Twist (coefficientAction cs (cs.simple i)).toRingEquiv (LinearMap.ker (bottUnionRestriction cs i l U))) :=
+    Twist.projective _
+  exact projective_of_coordinates (bottCommonSecond cs i l U) (bottCommonSecond_injective cs i l U)
+    (twistMap (coefficientAction cs (cs.simple i)).toRingEquiv (bottUnionKernelSecond cs i l U))
+    (twistMap_injective _ _ (bottUnionKernelSecond_injective cs i l U))
+    (bottCommon_range_forward cs i l HF U hU) (bottCommon_range_backward cs i l U)
+
+ theorem bottProjectiveUpper_cons (i : I) (l : List I) (HP : BottProjectiveUpper cs l)
+    (HF : BottFlabby cs l) : BottProjectiveUpper cs (i::l) := by
+  intro U hU
+  let : Module.Projective (SymmetricCoefficient (M := M)) (bottSetImage cs l U) := HP U hU
+  let : Module.Projective (SymmetricCoefficient (M := M)) (bottSetImage cs l (simpleInter cs i U)) :=
+    HP (simpleInter cs i U) (hU.simpleInter cs U i)
+  let : Module.Projective (SymmetricCoefficient (M := M)) (bottSetImage cs (i::l) (simpleInter cs i U)) :=
+    bottStableUpper_projective cs i l (simpleInter cs i U) (simpleInter_stable cs i U)
+  let : Module.Projective (SymmetricCoefficient (M := M)) (LinearMap.ker (bottInterRestriction cs i l U)) :=
+    kernel_projective (bottInterRestriction cs i l U) (bottSetRestrict_surjective cs l (fun _ h => h.1))
+  let : Module.Projective (SymmetricCoefficient (M := M))
+      ↥(LinearMap.ker (bottInterRestriction cs i (i::l) U) ⊓ LinearMap.ker (bottSetFirst cs i l U)) :=
+    bottCommon_projective cs i l HP HF U hU
+  exact projective_of_two_stage (bottInterRestriction cs i (i::l) U) (bottSetFirst cs i l U)
+    (bottInterRestriction cs i l U) (bottInter_first_zero cs i l U)
+    (bottSetRestrict_surjective cs (i::l) (fun _ h => h.1)) (bottInterKernelFirst_surjective cs i l U)
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! The codimension-one two-character distinction needed for integral
+Bott--Samelson section realization. No BMP character or pairing is assumed. -/
+namespace KLInvariance.Realization
+open Module
+variable {V : Type*} [AddCommGroup V] [Module ℝ V]
+
+ theorem orthogonal_eq_id_or_rootReflection
+    (B : LinearMap.BilinForm ℝ V) (a : V) (ha : B a a=2)
+    (g : V ≃ₗ[ℝ] V) (hg : ∀ v w, B (g v) (g w)=B v w)
+    (hline : ∀ v, g v-v ∈ Submodule.span ℝ {a}) :
+    g.toLinearMap=LinearMap.id ∨ ∀ v, g v=Module.preReflection a (B a) v := by
+  obtain ⟨r,hr⟩ := Submodule.mem_span_singleton.mp (hline a)
+  have hga : g a=(1+r) • a := by rw [add_smul,one_smul,hr]; module
+  have hn := hg a a
+  rw [hga] at hn
+  simp only [map_smul,LinearMap.smul_apply,smul_eq_mul,ha] at hn
+  have hr0 : r*(r+2)=0 := by nlinarith
+  rcases mul_eq_zero.mp hr0 with hr | hr
+  · left
+    have hga' : g a=a := by simpa only [hr,add_zero,one_smul] using hga
+    ext v
+    obtain ⟨c,hc⟩ := Submodule.mem_span_singleton.mp (hline v)
+    have hgv : g v=v+c • a := by rw [hc]; module
+    have h := hg a v
+    rw [hga',hgv] at h
+    simp only [map_add,map_smul,ha,smul_eq_mul] at h
+    have hc0 : c=0 := by linarith
+    change g v=v
+    simpa only [hc0,zero_smul,add_zero] using hgv
+  · right
+    have hr' : r= -2 := by linarith
+    have hga' : g a= -a := by simpa only [hr',show (1:ℝ)+ -2= -1 by norm_num,neg_one_smul] using hga
+    intro v
+    obtain ⟨c,hc⟩ := Submodule.mem_span_singleton.mp (hline v)
+    have hgv : g v=v+c • a := by rw [hc]; module
+    have h := hg a v
+    rw [hga',hgv] at h
+    simp only [map_neg,LinearMap.neg_apply,map_add,map_smul,ha,smul_eq_mul] at h
+    have hc' : c= -(B a v) := by linarith
+    rw [hgv,hc',Module.preReflection_apply,neg_smul,sub_eq_add_neg]
+
+end KLInvariance.Realization
+
+end
+
+
+section
+
+/-! Exact root-prime character classes of the actual Coxeter polynomial
+coefficient action. These are the two-vertex blocks needed in integral
+section realization; they are not BMP or KL-character assumptions. -/
+namespace KLInvariance.TitsSpace
+open Module
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+ theorem reflection_character_congruence_iff {t : W} {a : I → ℝ}
+    (ha : Represents M cs t a) (x y : W) :
+    (∀ p : SymmetricCoefficient (M := M),
+      SymmetricAlgebra.ι ℝ (Extended M) (embed M a) ∣
+        coefficientAction cs x p-coefficientAction cs y p) ↔
+      x=y ∨ y=t*x := by
+  constructor
+  · intro hp
+    let w := y*x⁻¹
+    have hline (v : Extended M) :
+        extendedAction M cs w v-v ∈ Submodule.span ℝ {embed M a} := by
+      let p := (coefficientAction cs x).symm (SymmetricAlgebra.ι ℝ (Extended M) v)
+      have hx : coefficientAction cs x p=SymmetricAlgebra.ι ℝ (Extended M) v :=
+        AlgEquiv.apply_symm_apply _ _
+      have hy : coefficientAction cs y p =
+          coefficientAction cs w (SymmetricAlgebra.ι ℝ (Extended M) v) := by
+        have h := coefficientAction_mul cs w x p
+        rw [show w*x=y by dsimp [w]; group,hx] at h
+        exact h
+      have h := hp p
+      rw [hx,hy,coefficientAction_ι,← map_sub] at h
+      have hs := (EdgeAlgebra.symmetric_ι_dvd_iff (embed M a)
+        (v-extendedAction M cs w v)).mp h
+      simpa only [neg_sub,EdgeAlgebra.rootLine] using (EdgeAlgebra.rootLine (k := ℝ) (embed M a)).neg_mem hs
+    have hnorm : extendedForm M (embed M a) (embed M a)=2 := by
+      rw [(extendedForm_spec M).2.2]
+      exact ha.isRoot.norm
+    rcases Realization.orthogonal_eq_id_or_rootReflection
+      (extendedForm M) (embed M a) hnorm (extendedAction M cs w)
+      (extendedAction_preserves M cs w) hline with hid | href
+    · left
+      have hw : w=1 := by
+        apply extendedAction_injective M cs
+        apply LinearEquiv.ext
+        intro v
+        change extendedAction M cs w v=extendedAction M cs 1 v
+        rw [map_one]
+        change extendedAction M cs w v=v
+        exact LinearMap.congr_fun hid v
+      exact (mul_inv_eq_one.mp hw).symm
+    · right
+      have hw : w=t := by
+        apply extendedAction_injective M cs
+        apply LinearEquiv.ext
+        intro v
+        rw [ha.extendedAction_eq]
+        exact href v
+      exact mul_inv_eq_iff_eq_mul.mp hw
+  · rintro (rfl | rfl)
+    · intro p
+      rw [sub_self]
+      exact dvd_zero _
+    · intro p
+      rw [coefficientAction_mul]
+      exact reflectionCoefficient_divisible cs ha (coefficientAction cs x p)
+
+ theorem reflection_character_separates {t : W} {a : I → ℝ}
+    (ha : Represents M cs t a) {x y : W} (hxy : x ≠ y) (ht : y ≠ t*x) :
+    ∃ p : SymmetricCoefficient (M := M),
+      ¬ SymmetricAlgebra.ι ℝ (Extended M) (embed M a) ∣
+        coefficientAction cs x p-coefficientAction cs y p := by
+  by_contra! h
+  exact (reflection_character_congruence_iff cs ha x y).mp h |>.elim hxy ht
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Every nontrivial prime-local congruence class of actual Coxeter
+polynomial characters is a two-vertex reflection block. -/
+namespace KLInvariance.TitsSpace
+open Module
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+ theorem isReflection_of_prime_character_congruence
+    (p : SymmetricCoefficient (M := M)) (hp : Prime p)
+    {x y : W} (hxy : x ≠ y)
+    (hcon : ∀ a, p ∣ coefficientAction cs x a-coefficientAction cs y a) :
+    cs.IsReflection (y*x⁻¹) := by
+  let w := y*x⁻¹
+  have hw : w ≠ 1 := by
+    intro h
+    exact hxy (mul_inv_eq_one.mp h).symm
+  have hex : ∃ v : Extended M, extendedAction M cs w v-v ≠ 0 := by
+    by_contra! h
+    apply hw
+    apply extendedAction_injective M cs
+    apply LinearEquiv.ext
+    intro v
+    rw [map_one]
+    exact sub_eq_zero.mp (h v)
+  obtain ⟨v,hv⟩ := hex
+  let d := extendedAction M cs w v-v
+  have hd : d ≠ 0 := hv
+  have hdiv (z : Extended M) : p ∣ SymmetricAlgebra.ι ℝ (Extended M)
+      (extendedAction M cs w z-z) := by
+    let a := (coefficientAction cs x).symm (SymmetricAlgebra.ι ℝ (Extended M) z)
+    have hx : coefficientAction cs x a=SymmetricAlgebra.ι ℝ (Extended M) z :=
+      AlgEquiv.apply_symm_apply _ _
+    have hy : coefficientAction cs y a =
+        coefficientAction cs w (SymmetricAlgebra.ι ℝ (Extended M) z) := by
+      have h := coefficientAction_mul cs w x a
+      rw [show w*x=y by dsimp [w]; group,hx] at h
+      exact h
+    have h := hcon a
+    rw [hx,hy,coefficientAction_ι,← map_sub] at h
+    simpa only [← map_neg,neg_sub] using dvd_neg.mpr h
+  have hassoc := hp.associated_of_dvd (EdgeAlgebra.symmetric_ι_prime d hd) (hdiv v)
+  have heq : LinearMap.range ((extendedAction M cs w).toLinearMap-LinearMap.id) =
+      Submodule.span ℝ {d} := by
+    apply le_antisymm
+    · rintro z ⟨a,rfl⟩
+      change extendedAction M cs w a-a ∈ Submodule.span ℝ {d}
+      exact (EdgeAlgebra.symmetric_ι_dvd_iff d _).mp
+        (dvd_trans hassoc.symm.dvd (hdiv a))
+    · apply Submodule.span_le.mpr
+      intro z hz
+      rcases Set.mem_singleton_iff.mp hz with rfl
+      exact ⟨v,rfl⟩
+  apply isReflection_of_moving_rank_one M cs w
+  rw [heq,finrank_span_singleton hd]
+
+ theorem prime_character_class (p : SymmetricCoefficient (M := M)) (hp : Prime p)
+    {x y : W} (hxy : x ≠ y)
+    (hcon : ∀ a, p ∣ coefficientAction cs x a-coefficientAction cs y a) :
+    ∃ t : W, cs.IsReflection t ∧ y=t*x ∧
+      ∀ z, (∀ a, p ∣ coefficientAction cs x a-coefficientAction cs z a) ↔
+        x=z ∨ z=t*x := by
+  let t := y*x⁻¹
+  have ht := isReflection_of_prime_character_congruence cs p hp hxy hcon
+  let a := (positiveRoot M cs ⟨t,ht⟩).val
+  have ha : Represents M cs t a := positiveRoot_represents M cs ⟨t,ht⟩
+  have he : t*x=y := by dsimp [t]; group
+  obtain ⟨c,hc⟩ := reflectionCoefficient_difference cs ha x
+  rw [he] at hc
+  have hpa : p ∣ SymmetricAlgebra.ι ℝ (Extended M) (embed M a) := hc ▸ hcon c
+  have hane : embed M a ≠ 0 := by
+    intro h
+    have hn : extendedForm M (embed M a) (embed M a)=2 := by
+      rw [(extendedForm_spec M).2.2]
+      exact ha.isRoot.norm
+    rw [h,map_zero] at hn
+    norm_num at hn
+  have hassoc := hp.associated_of_dvd (EdgeAlgebra.symmetric_ι_prime (embed M a) hane) hpa
+  refine ⟨t,ht,he.symm,?_⟩
+  intro z
+  rw [← reflection_character_congruence_iff cs ha x z]
+  exact forall_congr' fun c => ⟨fun h => dvd_trans hassoc.symm.dvd h,
+    fun h => dvd_trans hassoc.dvd h⟩
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Integral local projectors for finite character lattices. This lemma does
+not assert any BMP character theorem or identify a chosen summand with BMP. -/
+namespace KLInvariance.FiniteCharacterLocalization
+universe u v w
+variable {R : Type u} [CommRing R] {X : Type v}
+  {E : X → Type w} [∀ x, AddCommGroup (E x)] [∀ x, Module R (E x)]
+noncomputable section
+
+ def diagonal (f : X → R) : (∀ x, E x) →ₗ[R] (∀ x, E x) where
+  toFun m x := f x • m x
+  map_add' := by intro m n; funext x; exact smul_add _ _ _
+  map_smul' := by intro r m; funext x; exact smul_comm _ _ _
+
+ def stabilizer (L : Submodule R (∀ x, E x)) : Subalgebra R (X → R) where
+  carrier := {f | ∀ m ∈ L, diagonal f m ∈ L}
+  mul_mem' := by
+    intro f g hf hg m hm
+    have h := hf _ (hg m hm)
+    simpa only [diagonal,LinearMap.coe_mk,AddHom.coe_mk,Pi.mul_apply,mul_smul] using h
+  one_mem' := by intro m hm; simpa [diagonal] using hm
+  add_mem' := by
+    intro f g hf hg m hm
+    have he : diagonal (f+g) m = diagonal f m+diagonal g m := by
+      funext x
+      exact add_smul _ _ _
+    rw [he]
+    exact L.add_mem (hf m hm) (hg m hm)
+  zero_mem' := by
+    intro m _
+    have he : diagonal (0 : X → R) m=0 := by
+      funext x
+      exact zero_smul _ _
+    rw [he]
+    exact L.zero_mem
+  algebraMap_mem' := by
+    intro r m hm
+    exact L.smul_mem r hm
+
+ theorem diagonal_mem {L : Submodule R (∀ x, E x)} {f : X → R}
+    (hf : f ∈ stabilizer L) {m : ∀ x, E x} (hm : m ∈ L) :
+    diagonal f m ∈ L := hf m hm
+
+/-- A polynomial in stable diagonal weights which kills the complement of
+`C` and is a common scalar on `C`. Its scalar is a product, not a quotient. -/
+ theorem uniform_projector (L : Submodule R (∀ x, E x)) (C : Finset X)
+    (f : X → R) (hf : f ∈ stabilizer L) (hz : ∀ x ∉ C, f x=0) :
+    ∃ g ∈ stabilizer L,
+      (∀ x ∈ C, g x=∏ y ∈ C, f y) ∧ (∀ x ∉ C, g x=0) := by
+  classical
+  let d : R := ∏ y ∈ C, f y
+  let h : X → R := ∏ y ∈ C, (algebraMap R (X → R) (f y)-f)
+  have hh : h ∈ stabilizer L := by
+    exact (stabilizer L).prod_mem fun y _ =>
+      (stabilizer L).sub_mem ((stabilizer L).algebraMap_mem (f y)) hf
+  refine ⟨algebraMap R (X → R) d-h,
+    (stabilizer L).sub_mem ((stabilizer L).algebraMap_mem d) hh,?_,?_⟩
+  · intro x hx
+    change d-h x=d
+    suffices h x=0 by rw [this,sub_zero]
+    dsimp [h]
+    rw [Finset.prod_apply]
+    apply Finset.prod_eq_zero hx
+    simp
+  · intro x hx
+    change d-h x=0
+    suffices h x=d by rw [this,sub_self]
+    dsimp [h,d]
+    rw [Finset.prod_apply]
+    apply Finset.prod_congr rfl
+    intro y _
+    simp [hz x hx]
+
+/-- Separate one residue-character class from its complement without
+inverting any element of the coefficient domain. -/
+ theorem class_projector [Fintype X] (L : Submodule R (∀ x, E x)) (p : R) (hp : Prime p)
+    (C : Finset X) (x : X)
+    (test : X → X → R) (htest : ∀ y, test y ∈ stabilizer L)
+    (hzero : ∀ y ∉ C, test y y=0)
+    (hunit : ∀ y ∉ C, ¬ p ∣ test y x)
+    (hcongr : ∀ z ∈ C, ∀ y, p ∣ test y z-test y x) :
+    ∃ d : R, ¬ p ∣ d ∧ ∃ g ∈ stabilizer L,
+      (∀ z ∈ C, g z=d) ∧ (∀ z ∉ C, g z=0) := by
+  classical
+  let f : X → R := ∏ y ∈ Finset.univ \ C, test y
+  have hf : f ∈ stabilizer L :=
+    (stabilizer L).prod_mem fun y _ => htest y
+  have hfu (z : X) (hz : z ∈ C) : ¬ p ∣ f z := by
+    dsimp [f]
+    rw [Finset.prod_apply]
+    apply hp.not_dvd_finsetProd
+    intro y hy
+    have hyC := (Finset.mem_sdiff.mp hy).2
+    intro h
+    apply hunit y hyC
+    simpa only [sub_sub_cancel] using dvd_sub h (hcongr z hz y)
+  have hfz (z : X) (hz : z ∉ C) : f z=0 := by
+    dsimp [f]
+    rw [Finset.prod_apply]
+    exact Finset.prod_eq_zero (Finset.mem_sdiff.mpr ⟨Finset.mem_univ _,hz⟩) (hzero z hz)
+  obtain ⟨g,hg,hgc,hgz⟩ := uniform_projector L C f hf hfz
+  refine ⟨∏ z ∈ C, f z,?_,g,hg,hgc,hgz⟩
+  exact hp.not_dvd_finsetProd (fun z hz => hfu z hz)
+
+/-- Local lifts on each finite character block assemble after a single
+scalar denominator away from the prime. -/
+ theorem partition_local_denominator {Q : Type*} [Fintype Q]
+    (L : Submodule R (∀ x, E x)) (p : R) (hp : Prime p) (q : X → Q)
+    (z : ∀ x, E x) (d : Q → R) (hd : ∀ j, ¬ p ∣ d j)
+    (g : Q → X → R) (hg : ∀ j, g j ∈ stabilizer L)
+    (hgin : ∀ j x, q x=j → g j x=d j)
+    (hgout : ∀ j x, q x≠j → g j x=0)
+    (m : Q → ∀ x, E x) (hm : ∀ j, m j ∈ L)
+    (hmatch : ∀ j x, q x=j → m j x=z x) :
+    ∃ a : R, ¬ p ∣ a ∧ a • z ∈ L := by
+  classical
+  refine ⟨∏ j, d j,hp.not_dvd_finsetProd (fun j _ => hd j),?_⟩
+  have hh : (∑ j : Q, (∏ k ∈ Finset.univ.erase j, d k) •
+      diagonal (g j) (m j)) ∈ L := by
+    exact L.sum_mem fun j _ => L.smul_mem _ (hg j _ (hm j))
+  convert hh using 1
+  funext x
+  simp only [Pi.smul_apply,Finset.sum_apply]
+  rw [Finset.sum_eq_single (q x)]
+  · change (∏ j, d j) • z x = (∏ k ∈ Finset.univ.erase (q x),d k) •
+      (g (q x) x • m (q x) x)
+    rw [hgin _ _ rfl,hmatch _ _ rfl,← mul_smul,
+      Finset.prod_erase_mul _ _ (Finset.mem_univ _)]
+  · intro j _ hj
+    change (∏ k ∈ Finset.univ.erase j,d k) • (g j x • m j x)=0
+    rw [hgout j x (Ne.symm hj),zero_smul,smul_zero]
+  · simp
+
+ def residueSetoid {A : Type*} (p : R) (χ : A → X → R) : Setoid X where
+  r x y := ∀ a, p ∣ χ a x-χ a y
+  iseqv := ⟨fun x a => by rw [sub_self]; exact dvd_zero _,
+    fun h a => by simpa only [neg_sub] using dvd_neg.mpr (h a),
+    fun h h' a => by simpa only [sub_add_sub_cancel] using dvd_add (h a) (h' a)⟩
+
+ theorem character_class_projector [Fintype X] {A : Type*}
+    (L : Submodule R (∀ x, E x)) (p : R) (hp : Prime p)
+    (χ : A → X → R) (hχ : ∀ a, χ a ∈ stabilizer L) (x : X) :
+    ∃ d : R, ¬ p ∣ d ∧ ∃ g ∈ stabilizer L,
+      (∀ z, (residueSetoid p χ).r x z → g z=d) ∧
+      (∀ z, ¬ (residueSetoid p χ).r x z → g z=0) := by
+  classical
+  let C : Finset X := Finset.univ.filter ((residueSetoid p χ).r x)
+  have hc (z : X) : z ∈ C ↔ (residueSetoid p χ).r x z := by simp [C]
+  have ha : ∀ y : {y : X // y ∉ C}, ∃ a, ¬ p ∣ χ a x-χ a y.val := by
+    intro y
+    have hh : ¬ ∀ a, p ∣ χ a x-χ a y.val := fun h => y.property ((hc _).mpr h)
+    exact not_forall.mp hh
+  choose a ha using ha
+  let test (y : X) : X → R := if hy : y ∉ C then
+    χ (a ⟨y,hy⟩)-algebraMap R (X → R) (χ (a ⟨y,hy⟩) y) else 0
+  have ht (y : X) : test y ∈ stabilizer L := by
+    by_cases hy : y ∉ C
+    · simp only [test,dite_eq_left hy]
+      exact (stabilizer L).sub_mem (hχ _) ((stabilizer L).algebraMap_mem _)
+    · simp only [test,dite_eq_right hy]
+      exact (stabilizer L).zero_mem
+  have hz (y : X) (hy : y ∉ C) : test y y=0 := by simp [test,hy]
+  have hu (y : X) (hy : y ∉ C) : ¬ p ∣ test y x := by simpa [test,hy] using ha ⟨y,hy⟩
+  have hcon (z : X) (hz : z ∈ C) (y : X) : p ∣ test y z-test y x := by
+    by_cases hy : y ∉ C
+    · simp only [test,dite_eq_left hy]
+      change p ∣ (χ (a ⟨y,hy⟩) z-χ (a ⟨y,hy⟩) y)-
+        (χ (a ⟨y,hy⟩) x-χ (a ⟨y,hy⟩) y)
+      rw [sub_sub_sub_cancel_right]
+      exact (residueSetoid p χ).symm ((hc z).mp hz) _
+    · simp only [test,dite_eq_right hy,Pi.zero_apply,sub_self]
+      exact dvd_zero _
+  obtain ⟨d,hd,g,hg,hgc,hgz⟩ := class_projector L p hp C x test ht hz hu hcon
+  exact ⟨d,hd,g,hg,fun z h => hgc z ((hc z).mpr h),
+    fun z h => hgz z (fun hz => h ((hc z).mp hz))⟩
+
+/-- Pair/blockwise liftability at a prime suffices for integral local
+membership in a finite character-stable lattice. -/
+ theorem character_local_denominator [Fintype X] {A : Type*}
+    (L : Submodule R (∀ x, E x)) (p : R) (hp : Prime p)
+    (χ : A → X → R) (hχ : ∀ a, χ a ∈ stabilizer L) (z : ∀ x, E x)
+    (hlift : ∀ x, ∃ m ∈ L, ∀ y, (residueSetoid p χ).r x y → m y=z y) :
+    ∃ a : R, ¬ p ∣ a ∧ a • z ∈ L := by
+  classical
+  let Q := Quotient (residueSetoid p χ)
+  let : Fintype Q := Fintype.ofFinite Q
+  let q : X → Q := Quotient.mk''
+  have hq (j : Q) (y : X) : q y=j ↔ (residueSetoid p χ).r j.out y := by
+    constructor
+    · intro h
+      exact (residueSetoid p χ).symm (Quotient.exact (h.trans (Quotient.out_eq j).symm))
+    · intro h
+      exact (Quotient.sound ((residueSetoid p χ).symm h)).trans (Quotient.out_eq j)
+  have hproj (j : Q) := character_class_projector L p hp χ hχ j.out
+  choose d hd g hg hgin hgout using hproj
+  have hm (j : Q) := hlift j.out
+  choose m hm hmatch using hm
+  exact partition_local_denominator L p hp q z d hd g hg
+    (fun j y h => hgin j y ((hq j y).mp h))
+    (fun j y h => hgout j y (fun hh => h ((hq j y).mpr hh)))
+    m hm (fun j y h => hmatch j y ((hq j y).mp h))
+
+end
+end KLInvariance.FiniteCharacterLocalization
+
+end
+
+
+section
+
+
+namespace KLInvariance.ProjectiveLattice
+variable {R : Type*} [CommRing R] [klPreservedInstance15 : IsDomain R] [klPreservedInstance16 : UniqueFactorizationMonoid R]
+
+ omit [IsDomain R] [UniqueFactorizationMonoid R] in
+ theorem finsupp_divide {ι : Type*} (r : R) (f : ι →₀ R)
+    (h : ∀ i, r ∣ f i) : ∃ g : ι →₀ R, r • g = f := by
+  classical
+  let g : ι →₀ R := ∑ i ∈ f.support, Finsupp.single i (Classical.choose (h i))
+  refine ⟨g, ?_⟩
+  dsimp [g]
+  rw [Finset.smul_sum]
+  have heq : (∑ i ∈ f.support, r • Finsupp.single i (Classical.choose (h i))) =
+      ∑ i ∈ f.support, Finsupp.single i (f i) := by
+    apply Finset.sum_congr rfl
+    intro i hi
+    rw [Finsupp.smul_single, smul_eq_mul, ← Classical.choose_spec (h i)]
+  rw [heq]
+  exact Finsupp.sum_single f
+
+include klPreservedInstance15 klPreservedInstance16 in
+ theorem prime_divide_of_relation {P : Type*} [AddCommGroup P] [Module R P]
+    [Module.Projective R P] (p a c : R) (hp : Prime p) (ha : ¬p ∣ a)
+    (m n : P) (h : a • m=(p*c) • n) : ∃ z : P, p • z=m := by
+  have _ := klPreservedInstance15
+  have _ := klPreservedInstance16
+  obtain ⟨s,hs⟩ := Module.Projective.out (R := R) (P := P)
+  have hdvd : ∀ i, p ∣ s m i := by
+    intro i
+    have he := congrArg (fun z : P => s z i) h
+    simp only [map_smul, Finsupp.smul_apply, smul_eq_mul] at he
+    have hm : p ∣ a*s m i := by
+      rw [he]
+      exact dvd_mul_of_dvd_left (dvd_mul_right p c) _
+    exact (hp.dvd_mul.mp hm).resolve_left ha
+  obtain ⟨z,hz⟩ := finsupp_divide p (s m) hdvd
+  refine ⟨Finsupp.linearCombination R id z,?_⟩
+  rw [← map_smul,hz,hs]
+
+ theorem mem_of_prime_local_membership
+    {E : Type*} [AddCommGroup E] [Module R E] [Module.IsTorsionFree R E]
+    (M : Submodule R E) [Module.Projective R M] (x : E)
+    (hgeneric : ∃ c : R, c ≠ 0 ∧ c • x ∈ M)
+    (hlocal : ∀ p : R, Prime p → ∃ a : R, ¬p ∣ a ∧ a • x ∈ M) : x ∈ M := by
+  obtain ⟨c,hcz,hcx⟩ := hgeneric
+  suffices ∀ c : R, c≠0 → c • x ∈ M → x ∈ M from this c hcz hcx
+  intro c
+  induction c using UniqueFactorizationMonoid.induction_on_prime with
+  | h₁ => simp
+  | h₂ c hc =>
+    intro _ hcx
+    obtain ⟨u,rfl⟩ := hc
+    simpa using M.smul_mem (↑u⁻¹ : R) hcx
+  | h₃ c p hcz hp ih =>
+    intro _ hpcx
+    obtain ⟨a,hpa,hax⟩ := hlocal p hp
+    let m : M := ⟨(p*c) • x,hpcx⟩
+    let n : M := ⟨a • x,hax⟩
+    have he : a • m=(p*c) • n := by
+      apply Subtype.ext
+      exact smul_comm a (p*c) x
+    obtain ⟨z,hz⟩ := prime_divide_of_relation p a c hp hpa m n he
+    have hcx : c • x=(z : E) := by
+      apply smul_right_injective E hp.ne_zero
+      have hv := congrArg Subtype.val hz
+      change p • (z : E)=(p*c) • x at hv
+      simpa only [mul_smul] using hv.symm
+    exact ih hcz (hcx ▸ z.property)
+
+end KLInvariance.ProjectiveLattice
+
+end
+
+
+section
+
+/-! Prime-local realization on arbitrary upper coordinate sets. Projectivity
+of the actual image is kept explicit here and discharged by word induction. -/
+namespace KLInvariance.TitsSpace
+open Module CoordinateProjection FiniteCharacterLocalization
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+ theorem bottSet_scaled (l : List I) (U : Set W)
+    (z : ∀ x : U, bottStalk cs l x.val) :
+    bottCoordinateDenominator cs l • z ∈ bottSetImage cs l U := by
+  classical
+  let c (e : BottIndex l) : SymmetricCoefficient (M := M) :=
+    if h : bottWeight cs l e∈U then (z ⟨bottWeight cs l e,h⟩).val ⟨e,rfl⟩ else 0
+  obtain ⟨m,hm⟩ := bottEvaluation_contains_scaled_coordinates cs l c
+  refine ⟨m,?_⟩
+  funext x
+  apply Subtype.ext
+  funext e
+  rcases x with ⟨x,hx⟩
+  rcases e with ⟨e,he⟩
+  change bottWeight cs l e=x at he
+  subst x
+  change bottEvaluation cs l e m=bottCoordinateDenominator cs l * (z ⟨bottWeight cs l e,hx⟩).val ⟨e,rfl⟩
+  simpa only [c,dite_eq_left hx] using hm e
+
+ theorem BottCompatible.reflection_lift {l : List I} {U : Set W}
+    {z : ∀ x : U, bottStalk cs l x.val} (hz : BottCompatible cs l U z)
+    (x y : U) (t : W) (ht : cs.IsReflection t) (he : t*x.val=y.val) :
+    ∃ m : BottSamelsonModule cs l,
+      bottStalkProjection cs l x.val m=z x ∧ bottStalkProjection cs l y.val m=z y := by
+  have he' : t*y.val=x.val := by rw [← he,← mul_assoc,ht.mul_self,one_mul]
+  have hne : cs.length y.val ≠ cs.length x.val := by
+    rw [← he]
+    exact ht.length_mul_right_ne x.val
+  rcases lt_or_gt_of_ne hne with hlt | hlt
+  · obtain ⟨m,hm,hn⟩ := hz y.val x.val y.property x.property ⟨hlt,t,ht,he'.symm⟩
+    exact ⟨m,hn,hm⟩
+  · exact hz x.val y.val x.property y.property ⟨hlt,t,ht,he.symm⟩
+
+ theorem BottCompatible.prime_lift {l : List I} {U : Set W}
+    {z : ∀ x : U, bottStalk cs l x.val} (hz : BottCompatible cs l U z)
+    (p : SymmetricCoefficient (M := M)) (hp : Prime p) (x : U) :
+    ∃ m : BottSamelsonModule cs l, ∀ y : U,
+      (∀ a, p ∣ coefficientAction cs x.val a-coefficientAction cs y.val a) →
+      bottStalkProjection cs l y.val m=z y := by
+  classical
+  by_cases hx : ∃ y : U, x≠y ∧
+      ∀ a, p ∣ coefficientAction cs x.val a-coefficientAction cs y.val a
+  · obtain ⟨y,hxy,hcon⟩ := hx
+    obtain ⟨t,ht,he,hclass⟩ := prime_character_class cs p hp
+      (fun h => hxy (Subtype.ext h)) hcon
+    obtain ⟨m,hm,hn⟩ := hz.reflection_lift cs x y t ht he.symm
+    refine ⟨m,?_⟩
+    intro w hw
+    rcases (hclass w.val).mp hw with hxw | hwy
+    · have h : w=x := Subtype.ext hxw.symm
+      subst w
+      exact hm
+    · have h : w=y := Subtype.ext (hwy.trans he.symm)
+      subst w
+      exact hn
+  · obtain ⟨m,hm⟩ := bottStalkProjection_surjective cs l x.val (z x)
+    refine ⟨m,?_⟩
+    intro y hy
+    have h : y=x := by
+      by_contra h
+      exact hx ⟨y,Ne.symm h,hy⟩
+    subst y
+    exact hm
+
+ def BottUpperIndex (l : List I) (U : Set W) := {e : BottIndex l // bottWeight cs l e∈U}
+
+ def bottIndexSetEval (l : List I) (U : Set W) : BottSamelsonModule cs l →ₗ[
+    SymmetricCoefficient (M := M)]
+    (∀ e : BottUpperIndex cs l U, bottStalk cs l (bottWeight cs l e.val)) :=
+  LinearMap.pi (fun e => bottStalkProjection cs l (bottWeight cs l e.val))
+
+ theorem bottIndexSetEval_stable (l : List I) (U : Set W)
+    (a : SymmetricCoefficient (M := M)) :
+    (fun e : BottUpperIndex cs l U => coefficientAction cs (bottWeight cs l e.val) a) ∈
+      stabilizer (LinearMap.range (bottIndexSetEval cs l U)) := by
+  rintro z ⟨m,rfl⟩
+  refine ⟨bottSamelsonRight cs l a m,?_⟩
+  funext e
+  exact bottStalkProjection_right cs l _ a m
+
+ theorem BottCompatible.local_denominator {l : List I} {U : Set W}
+    {z : ∀ x : U, bottStalk cs l x.val} (hz : BottCompatible cs l U z)
+    (p : SymmetricCoefficient (M := M)) (hp : Prime p) :
+    ∃ a : SymmetricCoefficient (M := M), ¬p ∣ a ∧ a • z∈bottSetImage cs l U := by
+  classical
+  let : Fintype (BottUpperIndex cs l U) := inferInstanceAs (Fintype {e : BottIndex l // bottWeight cs l e∈U})
+  let zz : ∀ e : BottUpperIndex cs l U, bottStalk cs l (bottWeight cs l e.val) :=
+    fun e => z ⟨bottWeight cs l e.val,e.property⟩
+  have hlift : ∀ e : BottUpperIndex cs l U, ∃ m ∈ LinearMap.range (bottIndexSetEval cs l U),
+      ∀ f, (residueSetoid p (fun a e => coefficientAction cs (bottWeight cs l e.val) a)).r e f → m f=zz f := by
+    intro e
+    obtain ⟨m,hm⟩ := hz.prime_lift cs p hp ⟨bottWeight cs l e.val,e.property⟩
+    exact ⟨bottIndexSetEval cs l U m,⟨m,rfl⟩,fun f hf => hm ⟨bottWeight cs l f.val,f.property⟩ hf⟩
+  obtain ⟨a,ha,m,hm⟩ := character_local_denominator
+    (LinearMap.range (bottIndexSetEval cs l U)) p hp
+    (fun a e => coefficientAction cs (bottWeight cs l e.val) a)
+    (bottIndexSetEval_stable cs l U) zz hlift
+  refine ⟨a,ha,m,?_⟩
+  funext x
+  apply Subtype.ext
+  funext e
+  rcases x with ⟨x,hx⟩
+  rcases e with ⟨e,he⟩
+  change bottWeight cs l e=x at he
+  subst x
+  have hh := congrArg (fun q => (q ⟨e,hx⟩).val ⟨e,rfl⟩) hm
+  exact hh
+
+ theorem bottFlabby_of_projective (l : List I) (H : BottProjectiveUpper cs l) :
+    BottFlabby cs l := by
+  intro U hU z hz
+  let : Module.Projective (SymmetricCoefficient (M := M)) (bottSetImage cs l U) := H U hU
+  let : UniqueFactorizationMonoid (SymmetricCoefficient (M := M)) :=
+    (SymmetricAlgebra.equivMvPolynomial (Module.Free.chooseBasis ℝ (Extended M))).symm.toRingEquiv.toMulEquiv.uniqueFactorizationMonoid inferInstance
+  exact ProjectiveLattice.mem_of_prime_local_membership (bottSetImage cs l U) z
+    ⟨bottCoordinateDenominator cs l,bottCoordinateDenominator_ne_zero cs l,bottSet_scaled cs l U z⟩
+    (hz.local_denominator cs)
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Unconditional flabbiness of the actual integral tensor-word projection
+sheaf, obtained from projectivity of every upper-coordinate image. -/
+namespace KLInvariance.TitsSpace
+open Module CoordinateProjection
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+ theorem bottSetProjection_nil_injective (U : Set W) (hU : (1 : W)∈U) :
+    Function.Injective (bottSetProjection cs [] U) := by
+  intro m n h
+  apply bottEvaluation_injective cs []
+  funext e
+  exact congrArg (fun z : bottSetImage cs [] U => (z.val ⟨1,hU⟩).val ⟨e,rfl⟩) h
+
+ theorem bottSetImage_nil_subsingleton (U : Set W) (hU : (1 : W)∉U) :
+    Subsingleton (bottSetImage cs [] U) := by
+  constructor
+  intro p q
+  apply Subtype.ext
+  funext x
+  have hx : x.val≠1 := fun h => hU (h ▸ x.property)
+  let := bottStalk_nil_subsingleton cs x.val hx
+  exact Subsingleton.elim _ _
+
+ theorem bottSetImage_nil_projective (U : Set W) :
+    Module.Projective (SymmetricCoefficient (M := M)) (bottSetImage cs [] U) := by
+  by_cases hU : (1 : W)∈U
+  · exact Module.Projective.of_equiv (LinearEquiv.ofBijective (bottSetProjection cs [] U)
+      ⟨bottSetProjection_nil_injective cs U hU,bottSetProjection_surjective cs [] U⟩)
+  · let := bottSetImage_nil_subsingleton cs U hU
+    infer_instance
+
+ theorem bottProjectiveUpper (l : List I) : BottProjectiveUpper cs l := by
+  induction l with
+  | nil => exact fun U _ => bottSetImage_nil_projective cs U
+  | cons i l ih => exact bottProjectiveUpper_cons cs i l ih (bottFlabby_of_projective cs l ih)
+
+ theorem bottFlabby (l : List I) : BottFlabby cs l :=
+  bottFlabby_of_projective cs l (bottProjectiveUpper cs l)
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Flabbiness of the genuine interval word sheaf; the extension outside the
+finite support is zero, not an extra assumption on the source interval. -/
+namespace KLInvariance.TitsSpace
+open Module MomentGraph BruhatGraph
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+ theorem bottStalk_subsingleton_of_not_bruhat (l : List I) (hl : cs.IsReduced l)
+    (x : W) (hx : ¬BruhatLE cs x (cs.wordProd l)) : Subsingleton (bottStalk cs l x) := by
+  constructor
+  intro p q
+  apply Subtype.ext
+  funext e
+  exact (hx (e.property ▸ bottWeight_bruhat cs l hl e.val)).elim
+
+ def intervalUpperClosure (b : W) (U : Set (Interval cs 1 b)) : Set W :=
+  {x | ∃ a : Interval cs 1 b, a∈U ∧ BruhatLE cs a.val x}
+
+ omit [Fintype I] in
+ theorem intervalUpperClosure_upper (b : W) (U : Set (Interval cs 1 b)) :
+    BruhatUpper cs (intervalUpperClosure cs b U) := by
+  rintro x y hxy ⟨a,ha,hax⟩
+  exact ⟨a,ha,bruhat_trans cs hax hxy⟩
+
+ omit [Fintype I] in
+ theorem intervalUpperClosure_member (b : W) (U : Set (Interval cs 1 b)) (hU : IsUpperSet U)
+    (x : W) (hx : x∈intervalUpperClosure cs b U) (hxb : BruhatLE cs x b) :
+    (⟨x,one_bruhat cs x,hxb⟩ : Interval cs 1 b)∈U := by
+  obtain ⟨a,ha,hax⟩ := hx
+  exact hU hax ha
+
+ def bottIntervalExtension (l : List I) (U : Set (Interval cs 1 (cs.wordProd l)))
+    (f : (bottProjectionSheaf cs l 1 (cs.wordProd l)).Assignment)
+    (x : intervalUpperClosure cs (cs.wordProd l) U) : bottStalk cs l x.val := by
+  classical
+  exact if h : BruhatLE cs x.val (cs.wordProd l) then f ⟨x.val,one_bruhat cs x.val,h⟩ else 0
+
+ theorem bottIntervalExtension_apply (l : List I) (U : Set (Interval cs 1 (cs.wordProd l)))
+    (f : (bottProjectionSheaf cs l 1 (cs.wordProd l)).Assignment)
+    (x : Interval cs 1 (cs.wordProd l)) (hx : x.val∈intervalUpperClosure cs (cs.wordProd l) U) :
+    bottIntervalExtension cs l U f ⟨x.val,hx⟩=f x := by
+  classical
+  dsimp only [bottIntervalExtension]
+  exact dite_eq_left x.property.2
+
+ theorem bottIntervalExtension_compatible (l : List I) (hl : cs.IsReduced l)
+    (U : Set (Interval cs 1 (cs.wordProd l))) (hU : IsUpperSet U)
+    (f : (bottProjectionSheaf cs l 1 (cs.wordProd l)).Assignment)
+    (hf : (bottProjectionSheaf cs l 1 (cs.wordProd l)).CompatibleOn U f) :
+    BottCompatible cs l (intervalUpperClosure cs (cs.wordProd l) U) (bottIntervalExtension cs l U f) := by
+  intro x y hx hy hxy
+  by_cases hyb : BruhatLE cs y (cs.wordProd l)
+  · have hxb : BruhatLE cs x (cs.wordProd l) := bruhat_trans cs (.single hxy) hyb
+    let xx : Interval cs 1 (cs.wordProd l) := ⟨x,one_bruhat cs x,hxb⟩
+    let yy : Interval cs 1 (cs.wordProd l) := ⟨y,one_bruhat cs y,hyb⟩
+    let e : Edge cs 1 (cs.wordProd l) := ⟨(xx,yy),hxy⟩
+    have hh := hf e (intervalUpperClosure_member cs _ U hU x hx hxb)
+      (intervalUpperClosure_member cs _ U hU y hy hyb)
+    obtain ⟨m,hm,hn⟩ := (PairProjection.compatible_iff _ _ _ _).mp hh
+    exact ⟨m,hm.trans (bottIntervalExtension_apply cs l U f xx hx).symm,
+      hn.trans (bottIntervalExtension_apply cs l U f yy hy).symm⟩
+  · let := bottStalk_subsingleton_of_not_bruhat cs l hl y hyb
+    obtain ⟨m,hm⟩ := bottStalkProjection_surjective cs l x (bottIntervalExtension cs l U f ⟨x,hx⟩)
+    exact ⟨m,hm,Subsingleton.elim _ _⟩
+
+ theorem bottProjectionSheaf_flabby (l : List I) (hl : cs.IsReduced l) :
+    (bottProjectionSheaf cs l 1 (cs.wordProd l)).Flabby := by
+  intro U hU f hf
+  obtain ⟨m,hm⟩ := bottFlabby cs l _ (intervalUpperClosure_upper cs _ U)
+    (bottIntervalExtension cs l U f) (bottIntervalExtension_compatible cs l hl U hU f hf)
+  refine ⟨(bottSection cs l 1 (cs.wordProd l) m).val,(bottSection cs l 1 (cs.wordProd l) m).property,?_⟩
+  intro x hx
+  exact (congrFun hm ⟨x.val,x,hx,bruhat_refl cs x.val⟩).trans
+    (bottIntervalExtension_apply cs l U f x ⟨x,hx,bruhat_refl cs x.val⟩)
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Integral adjoints, not restrictions of a form to arbitrary summands.
+This is used to transport a genuine BMP projector through the word duality. -/
+namespace PerfectAdjoint
+open Module
+universe ur um
+variable {R : Type ur} [CommRing R]
+  {M : Type um} [AddCommGroup M] [Module R M]
+  (d : M ≃ₗ[R] Dual R M)
+noncomputable section
+
+ def adjoint (f : End R M) : End R M :=
+  d.symm.toLinearMap.comp (f.dualMap.comp d.toLinearMap)
+
+ theorem pairing (f : End R M) (m n : M) : d (adjoint d f m) n=d m (f n) := by
+  change d (d.symm (f.dualMap (d m))) n=d m (f n)
+  rw [LinearEquiv.apply_symm_apply]
+  rfl
+
+ theorem adjoint_comp (f g : End R M) : adjoint d (f.comp g)=(adjoint d g).comp (adjoint d f) := by
+  apply LinearMap.ext
+  intro m
+  apply d.injective
+  apply LinearMap.ext
+  intro n
+  rw [pairing]
+  change d m (f (g n))=d (adjoint d g (adjoint d f m)) n
+  rw [pairing,pairing]
+
+ theorem adjoint_idempotent (f : End R M) (hf : ∀ m, f (f m)=f m) (m : M) :
+    adjoint d f (adjoint d f m)=adjoint d f m := by
+  apply d.injective
+  apply LinearMap.ext
+  intro n
+  rw [pairing,pairing,hf,pairing]
+
+ theorem commutes_of_balanced (f t : End R M)
+    (ht : ∀ m n, d (t m) n=d m (t n)) (hf : ∀ m, f (t m)=t (f m)) (m : M) :
+    adjoint d f (t m)=t (adjoint d f m) := by
+  apply d.injective
+  apply LinearMap.ext
+  intro n
+  rw [pairing,ht,ht,pairing,hf]
+
+ theorem symmetric_pairing (hsym : ∀ m n, d m n=d n m)
+    (f : End R M) (m n : M) : d m (adjoint d f n)=d (f m) n := by
+  rw [hsym,pairing,hsym]
+
+ theorem adjoint_adjoint (hsym : ∀ m n, d m n=d n m) (f : End R M) :
+    adjoint d (adjoint d f)=f := by
+  apply LinearMap.ext
+  intro m
+  apply d.injective
+  apply LinearMap.ext
+  intro n
+  rw [pairing,symmetric_pairing d hsym]
+
+end
+end PerfectAdjoint
+
+end
+
+
+section
+
+
+namespace KLInvariance.Graded
+open Module
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {M : Type um} [AddCommGroup M] [Module k M] [Module A M] [IsScalarTower k A M]
+  (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M)
+  [DirectSum.Decomposition 𝓐] [DirectSum.Decomposition 𝓜]
+noncomputable section
+
+omit [IsScalarTower k A M] in
+ theorem perfect_inverse_of_homogeneous_functional
+    (p : M ≃ₗ[A] Dual A M) (l : ℤ)
+    (hp : ∀ i j (m n : M), m ∈ 𝓜 i → n ∈ 𝓜 j → p m n ∈ 𝓐 (i+j-l))
+    (r : ℤ) (lam : Dual A M)
+    (hlam : ∀ j (n : M), n ∈ 𝓜 j → lam n ∈ 𝓐 (j+r)) :
+    p.symm lam ∈ 𝓜 (r+l) := by
+  have hc : component 𝓜 (r+l) (p.symm lam) = p.symm lam := by
+    apply p.injective
+    apply LinearMap.ext
+    intro y
+    refine DirectSum.Decomposition.inductionOn 𝓜 ?_ ?_ ?_ y
+    · simp
+    · intro j v
+      let f : M →ₗ[A] A :=
+        { toFun := fun w => p w (v : M)
+          map_add' := by intros; simp
+          map_smul' := by intros; simp }
+      have hh : ∀ i w, w ∈ 𝓜 i → f w ∈ 𝓐 (i+(j-l)) := by
+        intro i w hw
+        simpa only [f,LinearMap.coe_mk,AddHom.coe_mk,add_sub_assoc] using hp i j w v hw v.property
+      have he := map_component 𝓜 𝓐 f (j-l) hh (j+r) (p.symm lam)
+      have hi : j+r-(j-l)=r+l := by omega
+      rw [hi] at he
+      have hf : f (p.symm lam)=lam v := by
+        dsimp only [f,LinearMap.coe_mk,AddHom.coe_mk]
+        rw [LinearEquiv.apply_symm_apply]
+      rw [hf] at he
+      have hsame : component 𝓐 (j+r) (lam v)=lam v :=
+        DirectSum.decompose_of_mem_same 𝓐 (hlam j v v.property)
+      rw [hsame] at he
+      change f (component 𝓜 (r+l) (p.symm lam)) = p (p.symm lam) v
+      rw [LinearEquiv.apply_symm_apply]
+      exact he.symm
+    · intro x y hx hy
+      simp only [map_add,hx,hy]
+  rw [← hc]
+  exact (DirectSum.decompose 𝓜 (p.symm lam) (r+l)).property
+
+end
+end KLInvariance.Graded
+
+end
+
+
+section
+
+namespace KLInvariance.Graded
+open Module
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {M : Type um} [AddCommGroup M] [Module k M] [Module A M]
+  (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M)
+  [DirectSum.Decomposition 𝓐] [DirectSum.Decomposition 𝓜]
+noncomputable section
+
+ theorem adjoint_homogeneous
+    (p : M ≃ₗ[A] Dual A M) (l : ℤ)
+    (hp : ∀ i j (m n : M), m ∈ 𝓜 i → n ∈ 𝓜 j → p m n ∈ 𝓐 (i+j-l))
+    (f : End A M) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓜 n)
+    (n : ℤ) (m : M) (hm : m ∈ 𝓜 n) : PerfectAdjoint.adjoint p f m ∈ 𝓜 n := by
+  have h := perfect_inverse_of_homogeneous_functional 𝓐 𝓜 p l hp (n-l)
+    (f.dualMap (p m)) (by
+      intro j v hv
+      have hv' := hp n j m (f v) hm (hf j v hv)
+      have he : n+j-l=j+(n-l) := by omega
+      change p m (f v) ∈ 𝓐 (j+(n-l))
+      rw [← he]
+      exact hv')
+  simpa only [sub_add_cancel,PerfectAdjoint.adjoint,LinearMap.comp_apply,LinearEquiv.coe_coe] using h
+
+end
+end KLInvariance.Graded
+
+end
+
+
+section
+
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/KazhdanLusztig/Invariance.lean b/lean/OAI/RepresentationTheory/KazhdanLusztig/Invariance.lean
index c02b168107c44a264031083cd43bacfd22483def..2464d455ca2ae383fd64c2c27909c242886fd362 100644
--- a/lean/OAI/RepresentationTheory/KazhdanLusztig/Invariance.lean
+++ b/lean/OAI/RepresentationTheory/KazhdanLusztig/Invariance.lean
@@ -1,572 +1,572 @@
-import OAI.RepresentationTheory.KazhdanLusztig.SimultaneousInvariance
-
-namespace OAI
-
-/-!
-Genuine KL invariance, finite-parabolic transport and unrestricted finite interval models; the main theorem for two arbitrary Coxeter systems in four independent universes.
--/
-
-section
-
-namespace KLInvariance
-universe u v u' v'
-variable {I : Type u} {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {I' : Type u'} {M' : CoxeterMatrix I'}
-  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
-  {a b : W} {a' b' : W'}
-noncomputable section
-
- theorem rankDifference_subinterval_lt (x z : Interval cs a b) (hxz : x≤z)
-    (hne : x.val≠a ∨ z.val≠b) :
-    rankDifference cs x.val z.val<rankDifference cs a b := by
-  have ha := length_le_of_bruhat cs x.property.1
-  have hb := length_le_of_bruhat cs z.property.2
-  have hm := length_le_of_bruhat cs hxz
-  dsimp only [rankDifference]
-  rcases hne with h|h
-  · have := length_lt_of_bruhat_ne cs x.property.1 h.symm
-    omega
-  · have := length_lt_of_bruhat_ne cs z.property.2 h
-    omega
-
- theorem rTilde_transport_all (hab : BruhatLE cs a b) (hab' : BruhatLE cs' a' b')
-    (φ : Interval cs a b ≃o Interval cs' a' b')
-    (hr : Hecke.rTilde cs a b=Hecke.rTilde cs' a' b')
-    (hsmall : ∀ (x z : Interval cs a b), x≤z →
-      rankDifference cs x.val z.val<rankDifference cs a b →
-      Hecke.rTilde cs x.val z.val=Hecke.rTilde cs' (φ x).val (φ z).val) :
-    ∀ x z : Interval cs a b, Hecke.rTilde cs x.val z.val=
-      Hecke.rTilde cs' (φ x).val (φ z).val := by
-  intro x z
-  by_cases hxz : x≤z
-  · by_cases hxa : x.val=a
-    · by_cases hzb : z.val=b
-      · have hx : x=intervalBottom cs hab := Subtype.ext hxa
-        have hz : z=intervalTop cs hab := Subtype.ext hzb
-        rw [hx,hz,intervalIso_bottom cs cs' hab hab' φ,intervalIso_top cs cs' hab hab' φ]
-        exact hr
-      · exact hsmall x z hxz (rankDifference_subinterval_lt x z hxz (Or.inr hzb))
-    · exact hsmall x z hxz (rankDifference_subinterval_lt x z hxz (Or.inl hxa))
-  · have hnot : ¬BruhatLE cs' (φ x).val (φ z).val := fun h => hxz (φ.le_iff_le.mp h)
-    rw [Hecke.rTilde_zero cs x.val z.val hxz,Hecke.rTilde_zero cs' _ _ hnot]
-
-end
-end KLInvariance
-
-end
-
-
-section
-
-/-! One rank-induction step for genuine R invariance. All inputs concern strictly
-smaller intervals in the eventual induction; no KL character theorem is assumed. -/
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph Polynomial SchedulePaths
-universe u u'
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] {cs : CoxeterSystem M W}
-  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
-  {W' : Type u'} [Group W'] {cs' : CoxeterSystem M' W'}
-  {σ' : Type u'} [Fintype σ'] (basis' : Basis σ' ℝ (Extended M'))
-  {a b : W} {a' b' : W'} (hab : BruhatLE cs a b) (hab' : BruhatLE cs' a' b')
-  (φ : Interval cs a b ≃o Interval cs' a' b')
-noncomputable section
-
- theorem actual_target_freeness
-    (hupper : ∀ y : Interval cs a b, y.val≠a →
-      FreeSuffixes basis y.property.2 (upperIntervalIso φ y)) :
-    ∀ e q, e::q <:+ chronologicalSchedule φ →
-      Module.Free (Coefficient (σ := σ))
-        ((intervalBoundary basis hab).sheaf.upwardSubmodule (target cs a b e)
-          (remainingConditions cs a b q (target cs a b e))) := by
-  intro e q hq
-  apply target_free_of_upper_suffixes basis hab φ hq
-  apply hupper
-  have hle := length_le_of_bruhat cs (source cs a b e).property.1
-  have hlt := e.property.1
-  change cs.length (source cs a b e).val<cs.length (target cs a b e).val at hlt
-  intro heq
-  rw [heq] at hlt
-  omega
-
-include hab hab' in
- theorem rTilde_eq_of_smaller_data
-    (hrow : ∀ z : Interval cs a b, z.val≠b →
-      Hecke.rTilde cs a z.val=Hecke.rTilde cs' a' (φ z).val)
-    (hrow' : ∀ z : Interval cs' a' b', z.val≠b' →
-      Hecke.rTilde cs' a' z.val=Hecke.rTilde cs a (φ.symm z).val)
-    (hupper : ∀ y : Interval cs a b, y.val≠a →
-      FreeSuffixes basis y.property.2 (upperIntervalIso φ y))
-    (hupper' : ∀ y : Interval cs' a' b', y.val≠a' →
-      FreeSuffixes basis' y.property.2 (upperIntervalIso φ.symm y)) :
-    Hecke.rTilde cs a b=Hecke.rTilde cs' a' b' := by
-  apply Comparison.polynomial_eq_of_transfer_eval_eq
-  intro t ht ht1
-  apply le_antisymm
-  · exact rTilde_eval_le_of_upper_freeness basis hab hab' φ hrow
-      (actual_target_freeness basis hab φ hupper) ht ht1
-  · exact rTilde_eval_le_of_upper_freeness basis' hab' hab φ.symm hrow'
-      (actual_target_freeness basis' hab' φ.symm hupper') ht ht1
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-/-! Unconditional genuine R-tilde invariance and transported terminal freeness.
-Induction is over arbitrary interval rank, with both endpoints varying. Actual
-BMP characters are only test vectors for the edge inequalities. Their genuine
-R-reciprocity and top1 are enough; strict halfdegree is NOT assumed. -/
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph Polynomial SchedulePaths
-universe u u'
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] {cs : CoxeterSystem M W}
-  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
-  {W' : Type u'} [Group W'] {cs' : CoxeterSystem M' W'}
-  {σ' : Type u'} [Fintype σ'] (basis' : Basis σ' ℝ (Extended M'))
-noncomputable section
-
- theorem rTilde_freeness_of_below
-    (a b : W) (a' b' : W') (hab : BruhatLE cs a b) (hab' : BruhatLE cs' a' b')
-    (φ : Interval cs a b ≃o Interval cs' a' b') (n : ℕ) (hn : rankDifference cs a b=n)
-    (hbelow : ∀ (c d : W) (c' d' : W') (hcd : BruhatLE cs c d) (hcd' : BruhatLE cs' c' d')
-      (ψ : Interval cs c d ≃o Interval cs' c' d'), rankDifference cs c d<n →
-      Hecke.rTilde cs c d=Hecke.rTilde cs' c' d' ∧
-        FreeSuffixes basis hcd ψ ∧ FreeSuffixes basis' hcd' ψ.symm) :
-    Hecke.rTilde cs a b=Hecke.rTilde cs' a' b' ∧
-      FreeSuffixes basis hab φ ∧ FreeSuffixes basis' hab' φ.symm := by
-  have hn' : rankDifference cs' a' b'=n :=
-    (intervalIso_full_rankDifference cs cs' hab hab' φ).symm.trans hn
-  have hsmall (x z : Interval cs a b) (hxz : x≤z)
-      (hlt : rankDifference cs x.val z.val<rankDifference cs a b) :
-      Hecke.rTilde cs x.val z.val=Hecke.rTilde cs' (φ x).val (φ z).val :=
-    (hbelow x.val z.val (φ x).val (φ z).val hxz (φ.monotone hxz)
-      (intervalIso_restrict cs cs' φ x z) (hn ▸ hlt)).1
-  have hrow (z : Interval cs a b) (hz : z.val≠b) :
-      Hecke.rTilde cs a z.val=Hecke.rTilde cs' a' (φ z).val := by
-    have h := hsmall (intervalBottom cs hab) z z.property.1
-      (rankDifference_subinterval_lt (intervalBottom cs hab) z z.property.1 (Or.inr hz))
-    rw [intervalIso_bottom cs cs' hab hab' φ] at h
-    exact h
-  have hrow' (z : Interval cs' a' b') (hz : z.val≠b') :
-      Hecke.rTilde cs' a' z.val=Hecke.rTilde cs a (φ.symm z).val := by
-    let x := intervalBottom cs' hab'
-    have hxz : x≤z := z.property.1
-    have hlt : rankDifference cs (φ.symm x).val (φ.symm z).val<n := by
-      rw [←intervalIso_rankDifference cs' cs φ.symm hxz]
-      exact hn' ▸ rankDifference_subinterval_lt x z hxz (Or.inr hz)
-    have h := (hbelow (φ.symm x).val (φ.symm z).val x.val z.val
-      (φ.symm.monotone hxz) hxz (intervalIso_restrict cs' cs φ.symm x z).symm hlt).1.symm
-    dsimp only [x] at h
-    rw [intervalIso_bottom cs' cs hab' hab φ.symm] at h
-    exact h
-  have hupper (y : Interval cs a b) (hy : y.val≠a) :
-      FreeSuffixes basis y.property.2 (upperIntervalIso φ y) := by
-    exact (hbelow y.val b (φ y).val b' y.property.2 (φ y).property.2
-      (upperIntervalIso φ y)
-      (hn ▸ rankDifference_subinterval_lt y (intervalTop cs hab) y.property.2 (Or.inl hy))).2.1
-  have hupper' (y : Interval cs' a' b') (hy : y.val≠a') :
-      FreeSuffixes basis' y.property.2 (upperIntervalIso φ.symm y) := by
-    let ψ := (upperIntervalIso φ.symm y).symm
-    have hlt : rankDifference cs (φ.symm y).val b<n := by
-      rw [intervalIso_full_rankDifference cs cs' (φ.symm y).property.2 y.property.2 ψ]
-      exact hn' ▸ rankDifference_subinterval_lt y (intervalTop cs' hab') y.property.2 (Or.inl hy)
-    exact (hbelow (φ.symm y).val b y.val b' (φ.symm y).property.2 y.property.2 ψ hlt).2.2
-  have hr := rTilde_eq_of_smaller_data basis basis' hab hab' φ hrow hrow' hupper hupper'
-  have hrall := rTilde_transport_all hab hab' φ hr hsmall
-  refine ⟨hr,actual_freeSuffixes_of_rTilde_eq basis hab φ hrall
-    (actual_target_freeness basis hab φ hupper),?_⟩
-  apply actual_freeSuffixes_of_rTilde_eq basis' hab' φ.symm
-    (fun x z => by simpa only [φ.apply_symm_apply] using (hrall (φ.symm x) (φ.symm z)).symm)
-    (actual_target_freeness basis' hab' φ.symm hupper')
-
- theorem rTilde_invariance_and_freeness
-    (a b : W) (a' b' : W') (hab : BruhatLE cs a b) (hab' : BruhatLE cs' a' b')
-    (φ : Interval cs a b ≃o Interval cs' a' b') :
-    Hecke.rTilde cs a b=Hecke.rTilde cs' a' b' ∧
-      FreeSuffixes basis hab φ ∧ FreeSuffixes basis' hab' φ.symm := by
-  have hp : ∀ n, ∀ (c d : W) (c' d' : W') (hcd : BruhatLE cs c d) (hcd' : BruhatLE cs' c' d')
-      (ψ : Interval cs c d ≃o Interval cs' c' d'), rankDifference cs c d=n →
-      Hecke.rTilde cs c d=Hecke.rTilde cs' c' d' ∧
-        FreeSuffixes basis hcd ψ ∧ FreeSuffixes basis' hcd' ψ.symm := by
-    intro n
-    induction n using Nat.strong_induction_on with
-    | h n ih =>
-      intro c d c' d' hcd hcd' ψ hn
-      apply rTilde_freeness_of_below basis basis' c d c' d' hcd hcd' ψ n hn
-      intro e f e' f' hef hef' θ hlt
-      exact ih _ hlt e f e' f' hef hef' θ rfl
-  exact hp _ a b a' b' hab hab' φ rfl
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-
-namespace KLInvariance
-open Polynomial
-universe u v u' v'
-variable {I : Type u} {M : CoxeterMatrix I} {W : Type v} [Group W]
-  (cs : CoxeterSystem M W)
-noncomputable section
-
- theorem scaledKL_eq_scaledPolynomial (t : ℝ) {x b : W} (hxb : BruhatLE cs x b) :
-    scaledKL cs t x b = Comparison.scaledPolynomial t (rankDifference cs x b)
-      (klPolynomial cs x b) := by
-  have hh := length_le_of_bruhat cs hxb
-  have he : (cs.length x : ℤ)-cs.length b = -(rankDifference cs x b : ℤ) := by
-    unfold rankDifference
-    omega
-  simp only [scaledKL,Comparison.scaledPolynomial,he]
-
-variable {I' : Type u'} {M' : CoxeterMatrix I'} {W' : Type v'} [Group W']
-  (cs' : CoxeterSystem M' W') {u b : W} {u' b' : W'}
-  (hub : BruhatLE cs u b) (hub' : BruhatLE cs' u' b')
-  (φ : Interval cs u b ≃o Interval cs' u' b')
-
-include hub hub'
-
- theorem intervalIso_upper_rankDifference (x : Interval cs u b) :
-    rankDifference cs x.val b = rankDifference cs' (φ x).val b' := by
-  have h := intervalIso_rankDifference cs cs' φ
-    (show x ≤ intervalTop cs hub from x.property.2)
-  rw [intervalIso_top cs cs' hub hub' φ] at h
-  exact h
-
- theorem scaledKL_transport (t : ℝ) (x : Interval cs u b)
-    (hx : klPolynomial cs x.val b=klPolynomial cs' (φ x).val b') :
-    scaledKL cs t x.val b=scaledKL cs' t (φ x).val b' := by
-  rw [scaledKL_eq_scaledPolynomial cs t x.property.2,
-    scaledKL_eq_scaledPolynomial cs' t (φ x).property.2,
-    intervalIso_upper_rankDifference cs cs' hub hub' φ x,hx]
-
-end
-end KLInvariance
-
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph SchedulePaths
-universe u v u' v' us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {σ : Type (max us v)} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
-  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
-  {u b : W} {u' b' : W'} (hub : BruhatLE cs u b)
-  (φ : Interval cs u b ≃o Interval cs' u' b')
-noncomputable section
-
- theorem comparison_of_source02
-    [Fintype (Edge cs 1 b)] [Fintype (Interval cs 1 b)] [DecidableEq (Interval cs 1 b)]
-    (d : SectionDuality cs 1 b basis (one_bruhat cs b)
-      (boundarySheaf cs 1 b basis (one_bruhat cs b)))
-    (hd : ∀ i j (m n : (boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf.sections Set.univ),
-      m ∈ ((boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf.sections Set.univ).piece i →
-      n ∈ ((boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf.sections Set.univ).piece j →
-      d.pairing m n ∈ polynomialGrading ℝ σ (i+j-(cs.length b : ℤ)))
-    (hc : StalkCharacterObligation cs 1 b basis (one_bruhat cs b))
-    (hfree : ∀ e q, e::q <:+ chronologicalSchedule φ →
-      Module.Free (Coefficient (σ := σ))
-        ((intervalBoundary basis hub).sheaf.upwardSubmodule (target cs u b e)
-          (remainingConditions cs u b q (target cs u b e))))
-    {t : ℝ} (ht : 0 < t) (ht1 : t < 1) :
-    (fun x : Interval cs u b => scaledKL cs t x.val b) ≤
-      dispatch (source cs u b) (target cs u b) (t⁻¹-t)
-        (fun x => scaledKL cs t⁻¹ x.val b) (transportedSchedule φ) := by
-  classical
-  have h := schedule_comparison_of_upper_freeness basis hub φ hfree ht ht1
-  have hf : scheduleState cs u b basis hub (intervalBoundary basis hub) t [] =
-      fun x => scaledKL cs t x.val b :=
-    funext (schedule_final_of_character basis hub hc ht ht1)
-  have hi : scheduleState cs u b basis hub (intervalBoundary basis hub) t (chronologicalSchedule φ) =
-      fun x => scaledKL cs t⁻¹ x.val b :=
-    funext (schedule_initial_of_duality_character basis hub φ d hd hc ht ht1)
-  rwa [hf,hi] at h
-
-omit [Fintype I] in
-include hub in
- theorem comparison_difference_le (hub' : BruhatLE cs' u' b')
-    (x : Interval cs u b)
-    (hother : ∀ y : Interval cs u b, y ≠ x →
-      klPolynomial cs y.val b=klPolynomial cs' (φ y).val b')
-    {t : ℝ} (ht : t ≠ 0)
-    (hcomp : scaledKL cs t x.val b ≤
-      dispatch (source cs u b) (target cs u b) (t⁻¹-t)
-        (fun y => scaledKL cs t⁻¹ y.val b) (transportedSchedule φ) x) :
-    scaledKL cs t x.val b-scaledKL cs' t (φ x).val b' ≤
-      scaledKL cs t⁻¹ x.val b-scaledKL cs' t⁻¹ (φ x).val b' := by
-  classical
-  let := interval_finite cs u b
-  let := Fintype.ofFinite (Interval cs u b)
-  have hp' := transported_reciprocal_vector φ t ht x
-  rw [transported_dispatch_matrix φ _ (Int.castRingHom ℝ),finsum_eq_sum_of_fintype] at hcomp hp'
-  exact Comparison.matrix_difference_le _ _ _ x _ _ (by simp)
-    (fun y hy => scaledKL_transport cs cs' hub hub' φ t⁻¹ y (hother y hy)) hcomp hp'
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-
-namespace KLInvariance.BruhatGraph
-open TitsSpace SchedulePaths
-universe u v u' v'
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
-  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
-  {u b : W} {u' b' : W'} (hub : BruhatLE cs u b) (hub' : BruhatLE cs' u' b')
-  (φ : Interval cs u b ≃o Interval cs' u' b')
-noncomputable section
-
- theorem bottom_eq_of_comparisons (hne : u ≠ b)
-    (hupper : ∀ y : Interval cs u b, y.val ≠ u →
-      klPolynomial cs y.val b=klPolynomial cs' (φ y).val b')
-    (hforward : ∀ t : ℝ, 0<t → t<1 →
-      scaledKL cs t u b ≤ dispatch (source cs u b) (target cs u b) (t⁻¹-t)
-        (fun y => scaledKL cs t⁻¹ y.val b) (transportedSchedule φ) (intervalBottom cs hub))
-    (hbackward : ∀ t : ℝ, 0<t → t<1 →
-      scaledKL cs' t u' b' ≤ dispatch (source cs' u' b') (target cs' u' b') (t⁻¹-t)
-        (fun y => scaledKL cs' t⁻¹ y.val b') (transportedSchedule φ.symm)
-          (intervalBottom cs' hub')) : klPolynomial cs u b=klPolynomial cs' u' b' := by
-  have hr := intervalIso_full_rankDifference cs cs' hub hub' φ
-  have hne' : u' ≠ b' := by
-    intro h
-    have hp := length_lt_of_bruhat_ne cs hub hne
-    unfold rankDifference at hr
-    simp only [h,Nat.sub_self] at hr
-    omega
-  apply Comparison.eq_of_scaled_difference_reciprocal
-    (klPolynomial cs u b) (klPolynomial cs' u' b') (rankDifference cs u b)
-    ((klFamilies_spec cs).P_degree u b hub hne)
-  · rw [hr]
-    exact (klFamilies_spec cs').P_degree u' b' hub' hne'
-  intro t ht ht1
-  have hf := comparison_difference_le hub φ hub' (intervalBottom cs hub)
-    (fun y hy => hupper y (fun he => hy (Subtype.ext he))) ht.ne' (hforward t ht ht1)
-  have hb := comparison_difference_le hub' φ.symm hub (intervalBottom cs' hub')
-    (fun y hy => by
-      have hyy : (φ.symm y).val ≠ u := by
-        intro he
-        have he' : φ.symm y=intervalBottom cs hub := Subtype.ext he
-        apply hy
-        have hh := congrArg φ he'
-        rw [φ.apply_symm_apply,intervalIso_bottom cs cs' hub hub' φ] at hh
-        exact hh
-      simpa only [φ.apply_symm_apply] using (hupper (φ.symm y) hyy).symm)
-    ht.ne' (hbackward t ht ht1)
-  rw [intervalIso_bottom cs cs' hub hub' φ] at hf
-  rw [intervalIso_bottom cs' cs hub' hub φ.symm] at hb
-  change scaledKL cs t u b-scaledKL cs' t u' b' ≤
-    scaledKL cs t⁻¹ u b-scaledKL cs' t⁻¹ u' b' at hf
-  change scaledKL cs' t u' b'-scaledKL cs t u b ≤
-    scaledKL cs' t⁻¹ u' b'-scaledKL cs t⁻¹ u b at hb
-  have he : scaledKL cs t u b-scaledKL cs' t u' b' =
-      scaledKL cs t⁻¹ u b-scaledKL cs' t⁻¹ u' b' := by linarith
-  simpa only [scaledKL_eq_scaledPolynomial cs _ hub,
-    scaledKL_eq_scaledPolynomial cs' _ hub',hr] using he
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-/-! Closing the main theorem in finite generator rank. The actual BMP sheaves
-are used only to prove the genuine R-kernel invariance. Canonical KL degree
-bounds and reciprocity now identify the genuine KL polynomials; no BMP stalk
-character identification or strict halfdegree assumption is used. -/
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace SchedulePaths
-universe u u'
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] {cs : CoxeterSystem M W}
-  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
-  {W' : Type u'} [Group W'] {cs' : CoxeterSystem M' W'}
-  {σ' : Type u'} [Fintype σ'] (basis' : Basis σ' ℝ (Extended M'))
-noncomputable section
-include basis basis'
-
- theorem rTilde_invariance_all_entries {a b : W} {a' b' : W'}
-    (φ : Interval cs a b ≃o Interval cs' a' b') (x z : Interval cs a b) :
-    Hecke.rTilde cs x.val z.val=Hecke.rTilde cs' (φ x).val (φ z).val := by
-  by_cases h : x≤z
-  · exact (rTilde_invariance_and_freeness basis basis' x.val z.val (φ x).val (φ z).val
-      h (φ.monotone h) (intervalIso_restrict cs cs' φ x z)).1
-  · rw [Hecke.rTilde_zero cs _ _ h, Hecke.rTilde_zero cs' _ _ (fun h' => h (φ.le_iff_le.mp h'))]
-
- theorem scaledKL_eq_transported {a b : W} {a' b' : W'}
-    (φ : Interval cs a b ≃o Interval cs' a' b')
-    {C : Type*} [Field C] (t : C) (ht : t≠0) (x : Interval cs a b) :
-    scaledKL cs t x.val b =
-      dispatch (source cs a b) (target cs a b) (t⁻¹-t)
-        (fun y => scaledKL cs t⁻¹ y.val b) (transportedSchedule φ) x := by
-  rw [transported_dispatch_matrix φ _ (Int.castRingHom C),
-    scaledKL_reciprocity_upper cs t ht a b x]
-  apply finsum_congr
-  intro z
-  rw [rTilde_invariance_all_entries basis basis' φ x z]
-
- theorem klPolynomial_invariance_finite (a b : W) (a' b' : W')
-    (hab : BruhatLE cs a b) (hab' : BruhatLE cs' a' b')
-    (φ : Interval cs a b ≃o Interval cs' a' b') :
-    klPolynomial cs a b=klPolynomial cs' a' b' := by
-  have hp : ∀ n, ∀ (c : W) (c' : W')
-      (hcb : BruhatLE cs c b) (hcb' : BruhatLE cs' c' b')
-      (ψ : Interval cs c b ≃o Interval cs' c' b'), rankDifference cs c b=n →
-      klPolynomial cs c b=klPolynomial cs' c' b' := by
-    intro n
-    induction n using Nat.strong_induction_on with
-    | h n ih =>
-      intro c c' hcb hcb' ψ hn
-      by_cases hne : c=b
-      · have hne' : c'=b' := by
-          by_contra h
-          have hp := length_lt_of_bruhat_ne cs' hcb' h
-          have hr := intervalIso_full_rankDifference cs cs' hcb hcb' ψ
-          unfold rankDifference at hr
-          rw [hne,Nat.sub_self] at hr
-          omega
-        simp only [hne,hne',klPolynomial,(klFamilies_spec cs).P_diagonal,(klFamilies_spec cs').P_diagonal]
-      · apply bottom_eq_of_comparisons hcb hcb' ψ hne
-        · intro y hy
-          apply ih (rankDifference cs y.val b)
-            (hn ▸ rankDifference_subinterval_lt y (intervalTop cs hcb) y.property.2 (Or.inl hy))
-            y.val (ψ y).val y.property.2 (ψ y).property.2 (upperIntervalIso ψ y) rfl
-        · intro t ht _ht1
-          exact (scaledKL_eq_transported basis basis' ψ t ht.ne' (intervalBottom cs hcb)).le
-        · intro t ht _ht1
-          exact (scaledKL_eq_transported basis' basis ψ.symm t ht.ne' (intervalBottom cs' hcb')).le
-  exact hp _ a a' hab hab' φ rfl
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-/-! The finite-rank reduction of the exact main target.  The model's interval
-and genuine KL polynomial are identified with those of the unrestricted
-ambient Coxeter system, not with an order-theoretically defined invariant. -/
-namespace KLInvariance
-universe u v
-variable {I : Type u} {M : CoxeterMatrix I} {W : Type v} [Group W]
-  (cs : CoxeterSystem M W) (u b : W)
-noncomputable section
-
-structure FiniteIntervalModel where
-  indices : Set I
-  finite : indices.Finite
-  lower : (StandardParabolic.matrix (M := M) indices).Group
-  upper : (StandardParabolic.matrix (M := M) indices).Group
-  lower_eq : StandardParabolic.inclusion cs indices lower = u
-  upper_eq : StandardParabolic.inclusion cs indices upper = b
-
- theorem exists_finiteIntervalModel (hub : BruhatLE cs u b) :
-    Nonempty (FiniteIntervalModel cs u b) := by
-  obtain ⟨J,hJ,hJgen⟩ := lower_ideal_finite_generator_support cs b
-  obtain ⟨v,hv,hvu⟩ := hJgen u hub
-  obtain ⟨w,hw,hwb⟩ := hJgen b (bruhat_refl cs b)
-  obtain ⟨v',hv'⟩ := StandardParabolic.lift_word J hv
-  obtain ⟨w',hw'⟩ := StandardParabolic.lift_word J hw
-  refine ⟨⟨J,hJ,(StandardParabolic.system (M := M) J).wordProd v',
-    (StandardParabolic.system (M := M) J).wordProd w',?_,?_⟩⟩
-  · rw [StandardParabolic.inclusion_word,hv',hvu]
-  · rw [StandardParabolic.inclusion_word,hw',hwb]
-
-def intervalEndpointCongr {a b a' b' : W} (ha : a = a') (hb : b = b') :
-    Interval cs a b ≃o Interval cs a' b' := by
-  subst a'
-  subst b'
-  exact OrderIso.refl _
-
-namespace FiniteIntervalModel
-variable {cs u b} (F : FiniteIntervalModel cs u b)
-
- def orderIso : Interval (StandardParabolic.system (M := M) F.indices) F.lower F.upper ≃o
-    Interval cs u b := by
-  let _ : Fintype F.indices := F.finite.fintype
-  have h := StandardParabolic.intervalIso cs F.indices F.lower F.upper
-  exact h.trans (intervalEndpointCongr cs F.lower_eq F.upper_eq)
-
- theorem bruhat (hub : BruhatLE cs u b) :
-    BruhatLE (StandardParabolic.system (M := M) F.indices) F.lower F.upper := by
-  let _ : Fintype F.indices := F.finite.fintype
-  apply (StandardParabolic.bruhat_inclusion_iff cs F.indices F.lower F.upper).mp
-  simpa only [F.lower_eq,F.upper_eq] using hub
-
- theorem kl_eq : klPolynomial (StandardParabolic.system (M := M) F.indices) F.lower F.upper =
-    klPolynomial cs u b := by
-  let _ : Fintype F.indices := F.finite.fintype
-  simpa only [F.lower_eq,F.upper_eq] using
-    (StandardParabolic.klPolynomial_inclusion cs F.indices F.lower F.upper).symm
-
- theorem rank_eq : rankDifference (StandardParabolic.system (M := M) F.indices) F.lower F.upper =
-    rankDifference cs u b := by
-  let _ : Fintype F.indices := F.finite.fintype
-  simpa only [F.lower_eq,F.upper_eq] using
-    (StandardParabolic.rankDifference_inclusion cs F.indices F.lower F.upper).symm
-
-end FiniteIntervalModel
-end
-end KLInvariance
-
-end
-
-
-section
-
-/-! Exact unrestricted main theorem. Finite parabolic restriction preserves
-actual intervals and genuine equal-parameter KL polynomials. Bases used in
-the temporary finite realization are selected here, not added as hypotheses. -/
-namespace KLInvariance
-open Module TitsSpace BruhatGraph
-universe u v u' v'
-variable {B : Type u} {W : Type v} [Group W] {M : CoxeterMatrix B}
-noncomputable section
-
- theorem combinatorial_invariance
-    {B' : Type u'} {W' : Type v'} [Group W'] {M' : CoxeterMatrix B'}
-    (cs : CoxeterSystem M W) (cs' : CoxeterSystem M' W')
-    {u b : W} {u' b' : W'}
-    (hub : BruhatLE cs u b) (hub' : BruhatLE cs' u' b')
-    (ι : Interval cs u b ≃o Interval cs' u' b') :
-    klPolynomial cs u b = klPolynomial cs' u' b' := by
-  obtain ⟨F⟩ := exists_finiteIntervalModel cs u b hub
-  obtain ⟨F'⟩ := exists_finiteIntervalModel cs' u' b' hub'
-  let : Fintype F.indices := F.finite.fintype
-  let : Fintype F'.indices := F'.finite.fintype
-  let basis := Module.Free.chooseBasis ℝ (Extended (StandardParabolic.matrix (M := M) F.indices))
-  let basis' := Module.Free.chooseBasis ℝ (Extended (StandardParabolic.matrix (M := M') F'.indices))
-  rw [←F.kl_eq,←F'.kl_eq]
-  exact klPolynomial_invariance_finite basis basis' F.lower F.upper F'.lower F'.upper
-    (F.bruhat hub) (F'.bruhat hub') ((F.orderIso.trans ι).trans F'.orderIso.symm)
-
-end
-end KLInvariance
-
-end
-
-
--- Recursive standard-axiom closure of the exact unrestricted exported main.
-
-end OAI
+import OAI.RepresentationTheory.KazhdanLusztig.SimultaneousInvariance
+
+namespace OAI
+
+/-!
+Genuine KL invariance, finite-parabolic transport and unrestricted finite interval models; the main theorem for two arbitrary Coxeter systems in four independent universes.
+-/
+
+section
+
+namespace KLInvariance
+universe u v u' v'
+variable {I : Type u} {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {I' : Type u'} {M' : CoxeterMatrix I'}
+  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
+  {a b : W} {a' b' : W'}
+noncomputable section
+
+ theorem rankDifference_subinterval_lt (x z : Interval cs a b) (hxz : x≤z)
+    (hne : x.val≠a ∨ z.val≠b) :
+    rankDifference cs x.val z.val<rankDifference cs a b := by
+  have ha := length_le_of_bruhat cs x.property.1
+  have hb := length_le_of_bruhat cs z.property.2
+  have hm := length_le_of_bruhat cs hxz
+  dsimp only [rankDifference]
+  rcases hne with h|h
+  · have := length_lt_of_bruhat_ne cs x.property.1 h.symm
+    omega
+  · have := length_lt_of_bruhat_ne cs z.property.2 h
+    omega
+
+ theorem rTilde_transport_all (hab : BruhatLE cs a b) (hab' : BruhatLE cs' a' b')
+    (φ : Interval cs a b ≃o Interval cs' a' b')
+    (hr : Hecke.rTilde cs a b=Hecke.rTilde cs' a' b')
+    (hsmall : ∀ (x z : Interval cs a b), x≤z →
+      rankDifference cs x.val z.val<rankDifference cs a b →
+      Hecke.rTilde cs x.val z.val=Hecke.rTilde cs' (φ x).val (φ z).val) :
+    ∀ x z : Interval cs a b, Hecke.rTilde cs x.val z.val=
+      Hecke.rTilde cs' (φ x).val (φ z).val := by
+  intro x z
+  by_cases hxz : x≤z
+  · by_cases hxa : x.val=a
+    · by_cases hzb : z.val=b
+      · have hx : x=intervalBottom cs hab := Subtype.ext hxa
+        have hz : z=intervalTop cs hab := Subtype.ext hzb
+        rw [hx,hz,intervalIso_bottom cs cs' hab hab' φ,intervalIso_top cs cs' hab hab' φ]
+        exact hr
+      · exact hsmall x z hxz (rankDifference_subinterval_lt x z hxz (Or.inr hzb))
+    · exact hsmall x z hxz (rankDifference_subinterval_lt x z hxz (Or.inl hxa))
+  · have hnot : ¬BruhatLE cs' (φ x).val (φ z).val := fun h => hxz (φ.le_iff_le.mp h)
+    rw [Hecke.rTilde_zero cs x.val z.val hxz,Hecke.rTilde_zero cs' _ _ hnot]
+
+end
+end KLInvariance
+
+end
+
+
+section
+
+/-! One rank-induction step for genuine R invariance. All inputs concern strictly
+smaller intervals in the eventual induction; no KL character theorem is assumed. -/
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph Polynomial SchedulePaths
+universe u u'
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] {cs : CoxeterSystem M W}
+  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
+  {W' : Type u'} [Group W'] {cs' : CoxeterSystem M' W'}
+  {σ' : Type u'} [Fintype σ'] (basis' : Basis σ' ℝ (Extended M'))
+  {a b : W} {a' b' : W'} (hab : BruhatLE cs a b) (hab' : BruhatLE cs' a' b')
+  (φ : Interval cs a b ≃o Interval cs' a' b')
+noncomputable section
+
+ theorem actual_target_freeness
+    (hupper : ∀ y : Interval cs a b, y.val≠a →
+      FreeSuffixes basis y.property.2 (upperIntervalIso φ y)) :
+    ∀ e q, e::q <:+ chronologicalSchedule φ →
+      Module.Free (Coefficient (σ := σ))
+        ((intervalBoundary basis hab).sheaf.upwardSubmodule (target cs a b e)
+          (remainingConditions cs a b q (target cs a b e))) := by
+  intro e q hq
+  apply target_free_of_upper_suffixes basis hab φ hq
+  apply hupper
+  have hle := length_le_of_bruhat cs (source cs a b e).property.1
+  have hlt := e.property.1
+  change cs.length (source cs a b e).val<cs.length (target cs a b e).val at hlt
+  intro heq
+  rw [heq] at hlt
+  omega
+
+include hab hab' in
+ theorem rTilde_eq_of_smaller_data
+    (hrow : ∀ z : Interval cs a b, z.val≠b →
+      Hecke.rTilde cs a z.val=Hecke.rTilde cs' a' (φ z).val)
+    (hrow' : ∀ z : Interval cs' a' b', z.val≠b' →
+      Hecke.rTilde cs' a' z.val=Hecke.rTilde cs a (φ.symm z).val)
+    (hupper : ∀ y : Interval cs a b, y.val≠a →
+      FreeSuffixes basis y.property.2 (upperIntervalIso φ y))
+    (hupper' : ∀ y : Interval cs' a' b', y.val≠a' →
+      FreeSuffixes basis' y.property.2 (upperIntervalIso φ.symm y)) :
+    Hecke.rTilde cs a b=Hecke.rTilde cs' a' b' := by
+  apply Comparison.polynomial_eq_of_transfer_eval_eq
+  intro t ht ht1
+  apply le_antisymm
+  · exact rTilde_eval_le_of_upper_freeness basis hab hab' φ hrow
+      (actual_target_freeness basis hab φ hupper) ht ht1
+  · exact rTilde_eval_le_of_upper_freeness basis' hab' hab φ.symm hrow'
+      (actual_target_freeness basis' hab' φ.symm hupper') ht ht1
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+/-! Unconditional genuine R-tilde invariance and transported terminal freeness.
+Induction is over arbitrary interval rank, with both endpoints varying. Actual
+BMP characters are only test vectors for the edge inequalities. Their genuine
+R-reciprocity and top1 are enough; strict halfdegree is NOT assumed. -/
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph Polynomial SchedulePaths
+universe u u'
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] {cs : CoxeterSystem M W}
+  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
+  {W' : Type u'} [Group W'] {cs' : CoxeterSystem M' W'}
+  {σ' : Type u'} [Fintype σ'] (basis' : Basis σ' ℝ (Extended M'))
+noncomputable section
+
+ theorem rTilde_freeness_of_below
+    (a b : W) (a' b' : W') (hab : BruhatLE cs a b) (hab' : BruhatLE cs' a' b')
+    (φ : Interval cs a b ≃o Interval cs' a' b') (n : ℕ) (hn : rankDifference cs a b=n)
+    (hbelow : ∀ (c d : W) (c' d' : W') (hcd : BruhatLE cs c d) (hcd' : BruhatLE cs' c' d')
+      (ψ : Interval cs c d ≃o Interval cs' c' d'), rankDifference cs c d<n →
+      Hecke.rTilde cs c d=Hecke.rTilde cs' c' d' ∧
+        FreeSuffixes basis hcd ψ ∧ FreeSuffixes basis' hcd' ψ.symm) :
+    Hecke.rTilde cs a b=Hecke.rTilde cs' a' b' ∧
+      FreeSuffixes basis hab φ ∧ FreeSuffixes basis' hab' φ.symm := by
+  have hn' : rankDifference cs' a' b'=n :=
+    (intervalIso_full_rankDifference cs cs' hab hab' φ).symm.trans hn
+  have hsmall (x z : Interval cs a b) (hxz : x≤z)
+      (hlt : rankDifference cs x.val z.val<rankDifference cs a b) :
+      Hecke.rTilde cs x.val z.val=Hecke.rTilde cs' (φ x).val (φ z).val :=
+    (hbelow x.val z.val (φ x).val (φ z).val hxz (φ.monotone hxz)
+      (intervalIso_restrict cs cs' φ x z) (hn ▸ hlt)).1
+  have hrow (z : Interval cs a b) (hz : z.val≠b) :
+      Hecke.rTilde cs a z.val=Hecke.rTilde cs' a' (φ z).val := by
+    have h := hsmall (intervalBottom cs hab) z z.property.1
+      (rankDifference_subinterval_lt (intervalBottom cs hab) z z.property.1 (Or.inr hz))
+    rw [intervalIso_bottom cs cs' hab hab' φ] at h
+    exact h
+  have hrow' (z : Interval cs' a' b') (hz : z.val≠b') :
+      Hecke.rTilde cs' a' z.val=Hecke.rTilde cs a (φ.symm z).val := by
+    let x := intervalBottom cs' hab'
+    have hxz : x≤z := z.property.1
+    have hlt : rankDifference cs (φ.symm x).val (φ.symm z).val<n := by
+      rw [←intervalIso_rankDifference cs' cs φ.symm hxz]
+      exact hn' ▸ rankDifference_subinterval_lt x z hxz (Or.inr hz)
+    have h := (hbelow (φ.symm x).val (φ.symm z).val x.val z.val
+      (φ.symm.monotone hxz) hxz (intervalIso_restrict cs' cs φ.symm x z).symm hlt).1.symm
+    dsimp only [x] at h
+    rw [intervalIso_bottom cs' cs hab' hab φ.symm] at h
+    exact h
+  have hupper (y : Interval cs a b) (hy : y.val≠a) :
+      FreeSuffixes basis y.property.2 (upperIntervalIso φ y) := by
+    exact (hbelow y.val b (φ y).val b' y.property.2 (φ y).property.2
+      (upperIntervalIso φ y)
+      (hn ▸ rankDifference_subinterval_lt y (intervalTop cs hab) y.property.2 (Or.inl hy))).2.1
+  have hupper' (y : Interval cs' a' b') (hy : y.val≠a') :
+      FreeSuffixes basis' y.property.2 (upperIntervalIso φ.symm y) := by
+    let ψ := (upperIntervalIso φ.symm y).symm
+    have hlt : rankDifference cs (φ.symm y).val b<n := by
+      rw [intervalIso_full_rankDifference cs cs' (φ.symm y).property.2 y.property.2 ψ]
+      exact hn' ▸ rankDifference_subinterval_lt y (intervalTop cs' hab') y.property.2 (Or.inl hy)
+    exact (hbelow (φ.symm y).val b y.val b' (φ.symm y).property.2 y.property.2 ψ hlt).2.2
+  have hr := rTilde_eq_of_smaller_data basis basis' hab hab' φ hrow hrow' hupper hupper'
+  have hrall := rTilde_transport_all hab hab' φ hr hsmall
+  refine ⟨hr,actual_freeSuffixes_of_rTilde_eq basis hab φ hrall
+    (actual_target_freeness basis hab φ hupper),?_⟩
+  apply actual_freeSuffixes_of_rTilde_eq basis' hab' φ.symm
+    (fun x z => by simpa only [φ.apply_symm_apply] using (hrall (φ.symm x) (φ.symm z)).symm)
+    (actual_target_freeness basis' hab' φ.symm hupper')
+
+ theorem rTilde_invariance_and_freeness
+    (a b : W) (a' b' : W') (hab : BruhatLE cs a b) (hab' : BruhatLE cs' a' b')
+    (φ : Interval cs a b ≃o Interval cs' a' b') :
+    Hecke.rTilde cs a b=Hecke.rTilde cs' a' b' ∧
+      FreeSuffixes basis hab φ ∧ FreeSuffixes basis' hab' φ.symm := by
+  have hp : ∀ n, ∀ (c d : W) (c' d' : W') (hcd : BruhatLE cs c d) (hcd' : BruhatLE cs' c' d')
+      (ψ : Interval cs c d ≃o Interval cs' c' d'), rankDifference cs c d=n →
+      Hecke.rTilde cs c d=Hecke.rTilde cs' c' d' ∧
+        FreeSuffixes basis hcd ψ ∧ FreeSuffixes basis' hcd' ψ.symm := by
+    intro n
+    induction n using Nat.strong_induction_on with
+    | h n ih =>
+      intro c d c' d' hcd hcd' ψ hn
+      apply rTilde_freeness_of_below basis basis' c d c' d' hcd hcd' ψ n hn
+      intro e f e' f' hef hef' θ hlt
+      exact ih _ hlt e f e' f' hef hef' θ rfl
+  exact hp _ a b a' b' hab hab' φ rfl
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+
+namespace KLInvariance
+open Polynomial
+universe u v u' v'
+variable {I : Type u} {M : CoxeterMatrix I} {W : Type v} [Group W]
+  (cs : CoxeterSystem M W)
+noncomputable section
+
+ theorem scaledKL_eq_scaledPolynomial (t : ℝ) {x b : W} (hxb : BruhatLE cs x b) :
+    scaledKL cs t x b = Comparison.scaledPolynomial t (rankDifference cs x b)
+      (klPolynomial cs x b) := by
+  have hh := length_le_of_bruhat cs hxb
+  have he : (cs.length x : ℤ)-cs.length b = -(rankDifference cs x b : ℤ) := by
+    unfold rankDifference
+    omega
+  simp only [scaledKL,Comparison.scaledPolynomial,he]
+
+variable {I' : Type u'} {M' : CoxeterMatrix I'} {W' : Type v'} [Group W']
+  (cs' : CoxeterSystem M' W') {u b : W} {u' b' : W'}
+  (hub : BruhatLE cs u b) (hub' : BruhatLE cs' u' b')
+  (φ : Interval cs u b ≃o Interval cs' u' b')
+
+include hub hub'
+
+ theorem intervalIso_upper_rankDifference (x : Interval cs u b) :
+    rankDifference cs x.val b = rankDifference cs' (φ x).val b' := by
+  have h := intervalIso_rankDifference cs cs' φ
+    (show x ≤ intervalTop cs hub from x.property.2)
+  rw [intervalIso_top cs cs' hub hub' φ] at h
+  exact h
+
+ theorem scaledKL_transport (t : ℝ) (x : Interval cs u b)
+    (hx : klPolynomial cs x.val b=klPolynomial cs' (φ x).val b') :
+    scaledKL cs t x.val b=scaledKL cs' t (φ x).val b' := by
+  rw [scaledKL_eq_scaledPolynomial cs t x.property.2,
+    scaledKL_eq_scaledPolynomial cs' t (φ x).property.2,
+    intervalIso_upper_rankDifference cs cs' hub hub' φ x,hx]
+
+end
+end KLInvariance
+
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph SchedulePaths
+universe u v u' v' us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {σ : Type (max us v)} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
+  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
+  {u b : W} {u' b' : W'} (hub : BruhatLE cs u b)
+  (φ : Interval cs u b ≃o Interval cs' u' b')
+noncomputable section
+
+ theorem comparison_of_source02
+    [Fintype (Edge cs 1 b)] [Fintype (Interval cs 1 b)] [DecidableEq (Interval cs 1 b)]
+    (d : SectionDuality cs 1 b basis (one_bruhat cs b)
+      (boundarySheaf cs 1 b basis (one_bruhat cs b)))
+    (hd : ∀ i j (m n : (boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf.sections Set.univ),
+      m ∈ ((boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf.sections Set.univ).piece i →
+      n ∈ ((boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf.sections Set.univ).piece j →
+      d.pairing m n ∈ polynomialGrading ℝ σ (i+j-(cs.length b : ℤ)))
+    (hc : StalkCharacterObligation cs 1 b basis (one_bruhat cs b))
+    (hfree : ∀ e q, e::q <:+ chronologicalSchedule φ →
+      Module.Free (Coefficient (σ := σ))
+        ((intervalBoundary basis hub).sheaf.upwardSubmodule (target cs u b e)
+          (remainingConditions cs u b q (target cs u b e))))
+    {t : ℝ} (ht : 0 < t) (ht1 : t < 1) :
+    (fun x : Interval cs u b => scaledKL cs t x.val b) ≤
+      dispatch (source cs u b) (target cs u b) (t⁻¹-t)
+        (fun x => scaledKL cs t⁻¹ x.val b) (transportedSchedule φ) := by
+  classical
+  have h := schedule_comparison_of_upper_freeness basis hub φ hfree ht ht1
+  have hf : scheduleState cs u b basis hub (intervalBoundary basis hub) t [] =
+      fun x => scaledKL cs t x.val b :=
+    funext (schedule_final_of_character basis hub hc ht ht1)
+  have hi : scheduleState cs u b basis hub (intervalBoundary basis hub) t (chronologicalSchedule φ) =
+      fun x => scaledKL cs t⁻¹ x.val b :=
+    funext (schedule_initial_of_duality_character basis hub φ d hd hc ht ht1)
+  rwa [hf,hi] at h
+
+omit [Fintype I] in
+include hub in
+ theorem comparison_difference_le (hub' : BruhatLE cs' u' b')
+    (x : Interval cs u b)
+    (hother : ∀ y : Interval cs u b, y ≠ x →
+      klPolynomial cs y.val b=klPolynomial cs' (φ y).val b')
+    {t : ℝ} (ht : t ≠ 0)
+    (hcomp : scaledKL cs t x.val b ≤
+      dispatch (source cs u b) (target cs u b) (t⁻¹-t)
+        (fun y => scaledKL cs t⁻¹ y.val b) (transportedSchedule φ) x) :
+    scaledKL cs t x.val b-scaledKL cs' t (φ x).val b' ≤
+      scaledKL cs t⁻¹ x.val b-scaledKL cs' t⁻¹ (φ x).val b' := by
+  classical
+  let := interval_finite cs u b
+  let := Fintype.ofFinite (Interval cs u b)
+  have hp' := transported_reciprocal_vector φ t ht x
+  rw [transported_dispatch_matrix φ _ (Int.castRingHom ℝ),finsum_eq_sum_of_fintype] at hcomp hp'
+  exact Comparison.matrix_difference_le _ _ _ x _ _ (by simp)
+    (fun y hy => scaledKL_transport cs cs' hub hub' φ t⁻¹ y (hother y hy)) hcomp hp'
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+
+namespace KLInvariance.BruhatGraph
+open TitsSpace SchedulePaths
+universe u v u' v'
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
+  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
+  {u b : W} {u' b' : W'} (hub : BruhatLE cs u b) (hub' : BruhatLE cs' u' b')
+  (φ : Interval cs u b ≃o Interval cs' u' b')
+noncomputable section
+
+ theorem bottom_eq_of_comparisons (hne : u ≠ b)
+    (hupper : ∀ y : Interval cs u b, y.val ≠ u →
+      klPolynomial cs y.val b=klPolynomial cs' (φ y).val b')
+    (hforward : ∀ t : ℝ, 0<t → t<1 →
+      scaledKL cs t u b ≤ dispatch (source cs u b) (target cs u b) (t⁻¹-t)
+        (fun y => scaledKL cs t⁻¹ y.val b) (transportedSchedule φ) (intervalBottom cs hub))
+    (hbackward : ∀ t : ℝ, 0<t → t<1 →
+      scaledKL cs' t u' b' ≤ dispatch (source cs' u' b') (target cs' u' b') (t⁻¹-t)
+        (fun y => scaledKL cs' t⁻¹ y.val b') (transportedSchedule φ.symm)
+          (intervalBottom cs' hub')) : klPolynomial cs u b=klPolynomial cs' u' b' := by
+  have hr := intervalIso_full_rankDifference cs cs' hub hub' φ
+  have hne' : u' ≠ b' := by
+    intro h
+    have hp := length_lt_of_bruhat_ne cs hub hne
+    unfold rankDifference at hr
+    simp only [h,Nat.sub_self] at hr
+    omega
+  apply Comparison.eq_of_scaled_difference_reciprocal
+    (klPolynomial cs u b) (klPolynomial cs' u' b') (rankDifference cs u b)
+    ((klFamilies_spec cs).P_degree u b hub hne)
+  · rw [hr]
+    exact (klFamilies_spec cs').P_degree u' b' hub' hne'
+  intro t ht ht1
+  have hf := comparison_difference_le hub φ hub' (intervalBottom cs hub)
+    (fun y hy => hupper y (fun he => hy (Subtype.ext he))) ht.ne' (hforward t ht ht1)
+  have hb := comparison_difference_le hub' φ.symm hub (intervalBottom cs' hub')
+    (fun y hy => by
+      have hyy : (φ.symm y).val ≠ u := by
+        intro he
+        have he' : φ.symm y=intervalBottom cs hub := Subtype.ext he
+        apply hy
+        have hh := congrArg φ he'
+        rw [φ.apply_symm_apply,intervalIso_bottom cs cs' hub hub' φ] at hh
+        exact hh
+      simpa only [φ.apply_symm_apply] using (hupper (φ.symm y) hyy).symm)
+    ht.ne' (hbackward t ht ht1)
+  rw [intervalIso_bottom cs cs' hub hub' φ] at hf
+  rw [intervalIso_bottom cs' cs hub' hub φ.symm] at hb
+  change scaledKL cs t u b-scaledKL cs' t u' b' ≤
+    scaledKL cs t⁻¹ u b-scaledKL cs' t⁻¹ u' b' at hf
+  change scaledKL cs' t u' b'-scaledKL cs t u b ≤
+    scaledKL cs' t⁻¹ u' b'-scaledKL cs t⁻¹ u b at hb
+  have he : scaledKL cs t u b-scaledKL cs' t u' b' =
+      scaledKL cs t⁻¹ u b-scaledKL cs' t⁻¹ u' b' := by linarith
+  simpa only [scaledKL_eq_scaledPolynomial cs _ hub,
+    scaledKL_eq_scaledPolynomial cs' _ hub',hr] using he
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+/-! Closing the main theorem in finite generator rank. The actual BMP sheaves
+are used only to prove the genuine R-kernel invariance. Canonical KL degree
+bounds and reciprocity now identify the genuine KL polynomials; no BMP stalk
+character identification or strict halfdegree assumption is used. -/
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace SchedulePaths
+universe u u'
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] {cs : CoxeterSystem M W}
+  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
+  {W' : Type u'} [Group W'] {cs' : CoxeterSystem M' W'}
+  {σ' : Type u'} [Fintype σ'] (basis' : Basis σ' ℝ (Extended M'))
+noncomputable section
+include basis basis'
+
+ theorem rTilde_invariance_all_entries {a b : W} {a' b' : W'}
+    (φ : Interval cs a b ≃o Interval cs' a' b') (x z : Interval cs a b) :
+    Hecke.rTilde cs x.val z.val=Hecke.rTilde cs' (φ x).val (φ z).val := by
+  by_cases h : x≤z
+  · exact (rTilde_invariance_and_freeness basis basis' x.val z.val (φ x).val (φ z).val
+      h (φ.monotone h) (intervalIso_restrict cs cs' φ x z)).1
+  · rw [Hecke.rTilde_zero cs _ _ h, Hecke.rTilde_zero cs' _ _ (fun h' => h (φ.le_iff_le.mp h'))]
+
+ theorem scaledKL_eq_transported {a b : W} {a' b' : W'}
+    (φ : Interval cs a b ≃o Interval cs' a' b')
+    {C : Type*} [Field C] (t : C) (ht : t≠0) (x : Interval cs a b) :
+    scaledKL cs t x.val b =
+      dispatch (source cs a b) (target cs a b) (t⁻¹-t)
+        (fun y => scaledKL cs t⁻¹ y.val b) (transportedSchedule φ) x := by
+  rw [transported_dispatch_matrix φ _ (Int.castRingHom C),
+    scaledKL_reciprocity_upper cs t ht a b x]
+  apply finsum_congr
+  intro z
+  rw [rTilde_invariance_all_entries basis basis' φ x z]
+
+ theorem klPolynomial_invariance_finite (a b : W) (a' b' : W')
+    (hab : BruhatLE cs a b) (hab' : BruhatLE cs' a' b')
+    (φ : Interval cs a b ≃o Interval cs' a' b') :
+    klPolynomial cs a b=klPolynomial cs' a' b' := by
+  have hp : ∀ n, ∀ (c : W) (c' : W')
+      (hcb : BruhatLE cs c b) (hcb' : BruhatLE cs' c' b')
+      (ψ : Interval cs c b ≃o Interval cs' c' b'), rankDifference cs c b=n →
+      klPolynomial cs c b=klPolynomial cs' c' b' := by
+    intro n
+    induction n using Nat.strong_induction_on with
+    | h n ih =>
+      intro c c' hcb hcb' ψ hn
+      by_cases hne : c=b
+      · have hne' : c'=b' := by
+          by_contra h
+          have hp := length_lt_of_bruhat_ne cs' hcb' h
+          have hr := intervalIso_full_rankDifference cs cs' hcb hcb' ψ
+          unfold rankDifference at hr
+          rw [hne,Nat.sub_self] at hr
+          omega
+        simp only [hne,hne',klPolynomial,(klFamilies_spec cs).P_diagonal,(klFamilies_spec cs').P_diagonal]
+      · apply bottom_eq_of_comparisons hcb hcb' ψ hne
+        · intro y hy
+          apply ih (rankDifference cs y.val b)
+            (hn ▸ rankDifference_subinterval_lt y (intervalTop cs hcb) y.property.2 (Or.inl hy))
+            y.val (ψ y).val y.property.2 (ψ y).property.2 (upperIntervalIso ψ y) rfl
+        · intro t ht _ht1
+          exact (scaledKL_eq_transported basis basis' ψ t ht.ne' (intervalBottom cs hcb)).le
+        · intro t ht _ht1
+          exact (scaledKL_eq_transported basis' basis ψ.symm t ht.ne' (intervalBottom cs' hcb')).le
+  exact hp _ a a' hab hab' φ rfl
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+/-! The finite-rank reduction of the exact main target.  The model's interval
+and genuine KL polynomial are identified with those of the unrestricted
+ambient Coxeter system, not with an order-theoretically defined invariant. -/
+namespace KLInvariance
+universe u v
+variable {I : Type u} {M : CoxeterMatrix I} {W : Type v} [Group W]
+  (cs : CoxeterSystem M W) (u b : W)
+noncomputable section
+
+structure FiniteIntervalModel where
+  indices : Set I
+  finite : indices.Finite
+  lower : (StandardParabolic.matrix (M := M) indices).Group
+  upper : (StandardParabolic.matrix (M := M) indices).Group
+  lower_eq : StandardParabolic.inclusion cs indices lower = u
+  upper_eq : StandardParabolic.inclusion cs indices upper = b
+
+ theorem exists_finiteIntervalModel (hub : BruhatLE cs u b) :
+    Nonempty (FiniteIntervalModel cs u b) := by
+  obtain ⟨J,hJ,hJgen⟩ := lower_ideal_finite_generator_support cs b
+  obtain ⟨v,hv,hvu⟩ := hJgen u hub
+  obtain ⟨w,hw,hwb⟩ := hJgen b (bruhat_refl cs b)
+  obtain ⟨v',hv'⟩ := StandardParabolic.lift_word J hv
+  obtain ⟨w',hw'⟩ := StandardParabolic.lift_word J hw
+  refine ⟨⟨J,hJ,(StandardParabolic.system (M := M) J).wordProd v',
+    (StandardParabolic.system (M := M) J).wordProd w',?_,?_⟩⟩
+  · rw [StandardParabolic.inclusion_word,hv',hvu]
+  · rw [StandardParabolic.inclusion_word,hw',hwb]
+
+def intervalEndpointCongr {a b a' b' : W} (ha : a = a') (hb : b = b') :
+    Interval cs a b ≃o Interval cs a' b' := by
+  subst a'
+  subst b'
+  exact OrderIso.refl _
+
+namespace FiniteIntervalModel
+variable {cs u b} (F : FiniteIntervalModel cs u b)
+
+ def orderIso : Interval (StandardParabolic.system (M := M) F.indices) F.lower F.upper ≃o
+    Interval cs u b := by
+  let _ : Fintype F.indices := F.finite.fintype
+  have h := StandardParabolic.intervalIso cs F.indices F.lower F.upper
+  exact h.trans (intervalEndpointCongr cs F.lower_eq F.upper_eq)
+
+ theorem bruhat (hub : BruhatLE cs u b) :
+    BruhatLE (StandardParabolic.system (M := M) F.indices) F.lower F.upper := by
+  let _ : Fintype F.indices := F.finite.fintype
+  apply (StandardParabolic.bruhat_inclusion_iff cs F.indices F.lower F.upper).mp
+  simpa only [F.lower_eq,F.upper_eq] using hub
+
+ theorem kl_eq : klPolynomial (StandardParabolic.system (M := M) F.indices) F.lower F.upper =
+    klPolynomial cs u b := by
+  let _ : Fintype F.indices := F.finite.fintype
+  simpa only [F.lower_eq,F.upper_eq] using
+    (StandardParabolic.klPolynomial_inclusion cs F.indices F.lower F.upper).symm
+
+ theorem rank_eq : rankDifference (StandardParabolic.system (M := M) F.indices) F.lower F.upper =
+    rankDifference cs u b := by
+  let _ : Fintype F.indices := F.finite.fintype
+  simpa only [F.lower_eq,F.upper_eq] using
+    (StandardParabolic.rankDifference_inclusion cs F.indices F.lower F.upper).symm
+
+end FiniteIntervalModel
+end
+end KLInvariance
+
+end
+
+
+section
+
+/-! Exact unrestricted main theorem. Finite parabolic restriction preserves
+actual intervals and genuine equal-parameter KL polynomials. Bases used in
+the temporary finite realization are selected here, not added as hypotheses. -/
+namespace KLInvariance
+open Module TitsSpace BruhatGraph
+universe u v u' v'
+variable {B : Type u} {W : Type v} [Group W] {M : CoxeterMatrix B}
+noncomputable section
+
+ theorem combinatorial_invariance
+    {B' : Type u'} {W' : Type v'} [Group W'] {M' : CoxeterMatrix B'}
+    (cs : CoxeterSystem M W) (cs' : CoxeterSystem M' W')
+    {u b : W} {u' b' : W'}
+    (hub : BruhatLE cs u b) (hub' : BruhatLE cs' u' b')
+    (ι : Interval cs u b ≃o Interval cs' u' b') :
+    klPolynomial cs u b = klPolynomial cs' u' b' := by
+  obtain ⟨F⟩ := exists_finiteIntervalModel cs u b hub
+  obtain ⟨F'⟩ := exists_finiteIntervalModel cs' u' b' hub'
+  let : Fintype F.indices := F.finite.fintype
+  let : Fintype F'.indices := F'.finite.fintype
+  let basis := Module.Free.chooseBasis ℝ (Extended (StandardParabolic.matrix (M := M) F.indices))
+  let basis' := Module.Free.chooseBasis ℝ (Extended (StandardParabolic.matrix (M := M') F'.indices))
+  rw [←F.kl_eq,←F'.kl_eq]
+  exact klPolynomial_invariance_finite basis basis' F.lower F.upper F'.lower F'.upper
+    (F.bruhat hub) (F'.bruhat hub') ((F.orderIso.trans ι).trans F'.orderIso.symm)
+
+end
+end KLInvariance
+
+end
+
+
+-- Recursive standard-axiom closure of the exact unrestricted exported main.
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/KazhdanLusztig/MinimalExtensions.lean b/lean/OAI/RepresentationTheory/KazhdanLusztig/MinimalExtensions.lean
index 0283628d5680f524d4b47e8426986db7e29cf080..a2c7533a7d6d6ce015b834671f37c1b8189d216d 100644
--- a/lean/OAI/RepresentationTheory/KazhdanLusztig/MinimalExtensions.lean
+++ b/lean/OAI/RepresentationTheory/KazhdanLusztig/MinimalExtensions.lean
@@ -1,1739 +1,1739 @@
-import OAI.RepresentationTheory.KazhdanLusztig.HilbertSeries
-import OAI.RepresentationTheory.KazhdanLusztig.Comparison
-
-namespace OAI
-
-/-!
-Minimal-boundary extension sheaves, graded splittings and sheaf decomposition.
--/
-
-section
-
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded _root_.OAI.KLInvariance.Hilbert
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
-  {G : OrderedGraph V E} (B : GradedSheaf (𝓐 := 𝓐) G)
-
-/-- The actual lower restriction, with its source explicitly identified. -/
-def lowerAt (x : V) (e : Sheaf.Outgoing (G := G) x) :
-    ModuleData.Hom (B.vertex x) (B.edge e.val) := by
-  rcases e with ⟨e,rfl⟩
-  exact B.lower e
-
- theorem lowerAt_map (x : V) (e : Sheaf.Outgoing (G := G) x) (v : B.vertex x) :
-    (B.lowerAt x e).map v = B.forget.stalkMap x v e := by
-  rcases e with ⟨e,rfl⟩
-  rfl
-
-/-- The same kernel, typed in the graded stalk so base-field instances are explicit. -/
-def upwardSubmodule (x : V) (F : Set (Sheaf.Outgoing (G := G) x)) :
-    Submodule A (B.vertex x) := B.forget.upwardKernel x F
-
- theorem upwardKernel_homogeneous (x : V) (F : Set (Sheaf.Outgoing (G := G) x)) :
-    ∀ n v, v ∈ B.upwardSubmodule x F →
-      component (B.vertex x).piece n v ∈ B.upwardSubmodule x F := by
-  intro n v hv e he
-  rw [← B.lowerAt_map]
-  rw [← (B.lowerAt x e).component]
-  rw [B.lowerAt_map,hv e he]
-  exact map_zero (component (B.edge e.val).piece n)
-
-noncomputable def upwardKernel [IsNoetherianRing A]
-    (x : V) (F : Set (Sheaf.Outgoing (G := G) x)) : ModuleData.{uk,ua,um} 𝓐 :=
-  ModuleData.sub (B.vertex x) (B.upwardSubmodule x F) (B.upwardKernel_homogeneous x F)
-
- theorem upwardKernel_insert (x : V) (F : Set (Sheaf.Outgoing (G := G) x))
-    (e : Sheaf.Outgoing (G := G) x) :
-    B.upwardSubmodule x (insert e F) =
-      B.upwardSubmodule x F ⊓ LinearMap.ker (B.lowerAt x e).map := by
-  ext v
-  change (∀ a ∈ insert e F, B.forget.stalkMap x v a = 0) ↔
-    (∀ a ∈ F, B.forget.stalkMap x v a = 0) ∧ (B.lowerAt x e).map v = 0
-  rw [B.lowerAt_map]
-  simp only [Set.mem_insert_iff,forall_eq_or_imp]
-  exact and_comm
-
-
- theorem hilbert_edge_increment
-    (x : V) (F : Set (Sheaf.Outgoing (G := G) x))
-    (e : Sheaf.Outgoing (G := G) x)
-    [∀ n, Module.Finite k ((B.vertex x).piece n)]
-    {q : ℝ} (hq : 0 < q)
-    (hs : Summable (term (B.vertex x).piece
-      ((B.upwardSubmodule x F).restrictScalars k) q)) :
-    value (B.vertex x).piece ((B.upwardSubmodule x F).restrictScalars k) q =
-      value (B.vertex x).piece
-        ((B.upwardSubmodule x (insert e F)).restrictScalars k) q +
-      value (B.edge e.val).piece
-        (((B.upwardSubmodule x F).map (B.lowerAt x e).map).restrictScalars k) q := by
-  rw [B.upwardKernel_insert]
-  exact value_image_exact (B.vertex x).piece (B.edge e.val).piece
-    (B.upwardSubmodule x F) (B.upwardKernel_homogeneous x F)
-    (B.lowerAt x e).map (B.lowerAt x e).graded hq hs
-
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-/-! Graded uniqueness/recognition for the genuinely constructed minimal
-boundary covers. Character and self-duality are not included in this data. -/
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
-  {G : OrderedGraph V E} [Fintype V] [Fintype E]
-  (B : GradedSheaf (𝓐 := 𝓐) G)
-
-noncomputable def stalkHom (x : V) : ModuleData.Hom (B.vertex x) (B.boundary x) where
-  map := B.forget.stalkMap x
-  graded := by
-    intro n v hv e _
-    change B.forget.stalkMap x v e ∈ (B.edge e.val).piece n
-    rw [← B.lowerAt_map]
-    exact (B.lowerAt x e).graded n v hv
-
-end KLInvariance.MomentGraph.GradedSheaf
-
-namespace KLInvariance.MomentGraph.Sheaf.AboveIso
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-  {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
-  [Fintype E] [IsNoetherianRing A] [IsDomain A]
-  (hneg : ∀ n < 0, 𝓐 n = ⊥)
-  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
-  {B C : GradedSheaf (𝓐 := 𝓐) G} {x : V}
-  (j : AboveIso B.forget C.forget x)
-
-include hneg hzero in
- theorem extend_stalk_graded
-    (hj : ∀ e (h : x < G.target e) n v, v ∈ (B.edge e).piece n →
-      j.edge e h v ∈ (C.edge e).piece n)
-    (hBfree : Module.Free A (B.vertex x)) (hCfree : Module.Free A (C.vertex x))
-    (hBf : B.forget.Flabby) (hBg : B.forget.Generated)
-    (hCf : C.forget.Flabby) (hCg : C.forget.Generated)
-    (hBmin : B.KernelMinimal x) (hCmin : C.KernelMinimal x) :
-    ∃ e : B.vertex x ≃ₗ[A] C.vertex x,
-      (∀ n v, v ∈ (B.vertex x).piece n → e v ∈ (C.vertex x).piece n) ∧
-      ∀ v, j.boundary (B.forget.stalkMap x v) = C.forget.stalkMap x (e v) := by
-  let T := ModuleData.range (C.stalkHom x)
-  let f : ModuleData.Hom (B.vertex x) T :=
-    { map := LinearMap.codRestrict (LinearMap.range (C.forget.stalkMap x))
-        (j.boundary.toLinearMap.comp (B.forget.stalkMap x)) (by
-          intro v
-          rw [← j.map_stalk_image hBf hBg hCf hCg]
-          exact ⟨B.forget.stalkMap x v,⟨v,rfl⟩,rfl⟩)
-      graded := by
-        intro n v hv e _
-        exact hj e.val _ n _ ((B.stalkHom x).graded n v hv e (Set.mem_univ e)) }
-  let g : ModuleData.Hom (C.vertex x) T :=
-    { map := (C.forget.stalkMap x).rangeRestrict
-      graded := fun n v hv => (C.stalkHom x).graded n v hv }
-  have hfs : Function.Surjective f.map := by
-    intro v
-    have hv : v.val ∈ (LinearMap.range (B.forget.stalkMap x)).map j.boundary.toLinearMap := by
-      rw [j.map_stalk_image hBf hBg hCf hCg]
-      exact v.property
-    obtain ⟨w,⟨z,rfl⟩,hz⟩ := hv
-    exact ⟨z,Subtype.ext hz⟩
-  have hgs : Function.Surjective g.map := by
-    intro v
-    obtain ⟨w,hw⟩ := v.property
-    exact ⟨w,Subtype.ext hw⟩
-  have hfm : LinearMap.ker f.map ≤ positiveIdeal 𝓐 • (⊤ : Submodule A (B.vertex x)) := by
-    intro v hv
-    apply (B.kernelMinimal_iff x).mp hBmin
-    apply LinearMap.mem_ker.mpr
-    apply j.boundary.injective
-    rw [map_zero]
-    have hh : f.map v = 0 := LinearMap.mem_ker.mp hv
-    exact congrArg (fun z : T => z.val) hh
-  have hgm : LinearMap.ker g.map ≤ positiveIdeal 𝓐 • (⊤ : Submodule A (C.vertex x)) := by
-    intro v hv
-    apply (C.kernelMinimal_iff x).mp hCmin
-    have hh : g.map v = 0 := LinearMap.mem_ker.mp hv
-    exact congrArg (fun z : T => z.val) hh
-  let F : ModuleData.MinimalCover T := ⟨B.vertex x,hBfree,f,hfs,hfm⟩
-  let D : ModuleData.MinimalCover T := ⟨C.vertex x,hCfree,g,hgs,hgm⟩
-  obtain ⟨e,he,hef⟩ := ModuleData.MinimalCover.equiv hneg hzero F D
-  refine ⟨e,he,?_⟩
-  intro v
-  exact (congrArg Subtype.val (LinearMap.congr_fun hef v)).symm
-
-end KLInvariance.MomentGraph.Sheaf.AboveIso
-
-end
-
-
-section
-
-namespace KLInvariance.Graded.ModuleData
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A}
-
-def equivOfEq {M N : ModuleData.{uk,ua,um} 𝓐} (h : M=N) : M ≃ₗ[A] N :=
-  h ▸ LinearEquiv.refl A M
-
- theorem equivOfEq_graded {M N : ModuleData.{uk,ua,um} 𝓐} (h : M=N)
-    (n : ℤ) (v : M) (hv : v ∈ M.piece n) : equivOfEq h v ∈ N.piece n := by
-  subst N
-  exact hv
-
- theorem quotient_iso_graded {M N E F : ModuleData.{uk,ua,um} 𝓐}
-    (f : Hom M E) (g : Hom N F) (hf : Function.Surjective f.map)
-    (hg : Function.Surjective g.map) (e : M ≃ₗ[A] N)
-    (he : ∀ n v, v ∈ M.piece n → e v ∈ N.piece n)
-    (hk : (LinearMap.ker f.map).map e.toLinearMap = LinearMap.ker g.map)
-    (n : ℤ) (v : E) (hv : v ∈ E.piece n) :
-    QuotientEdge.iso f.map g.map hf hg e hk v ∈ F.piece n := by
-  obtain ⟨w,hw,rfl⟩ := f.homogeneous_preimage hf n v hv
-  rw [QuotientEdge.iso_apply]
-  exact g.graded n _ (he n w hw)
-
-end KLInvariance.Graded.ModuleData
-
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
-  {G : OrderedGraph V E}
-
-structure Iso (B C : GradedSheaf (𝓐 := 𝓐) G) extends Sheaf.Iso B.forget C.forget where
-  vertex_graded : ∀ x n v, v ∈ (B.vertex x).piece n → vertex x v ∈ (C.vertex x).piece n
-  edge_graded : ∀ e n v, v ∈ (B.edge e).piece n → edge e v ∈ (C.edge e).piece n
-
-structure OnIso (B C : GradedSheaf (𝓐 := 𝓐) G) (U : Set V)
-    extends Sheaf.OnIso B.forget C.forget U where
-  vertex_graded : ∀ x (hx : x ∈ U) n v,
-    v ∈ (B.vertex x).piece n → vertex x hx v ∈ (C.vertex x).piece n
-  edge_graded : ∀ e (he : G.target e ∈ U) n v,
-    v ∈ (B.edge e).piece n → edge e he v ∈ (C.edge e).piece n
-
-namespace OnIso
-variable {B C : GradedSheaf (𝓐 := 𝓐) G} {U : Set V} (j : OnIso B C U)
-
- def ofUniv (j : OnIso B C Set.univ) : Iso B C where
-  toIso := j.toOnIso.ofUniv
-  vertex_graded x := j.vertex_graded x (Set.mem_univ x)
-  edge_graded e := j.edge_graded e (Set.mem_univ _)
-
-noncomputable def extend (hU : IsUpperSet U) (x : V) (hx : x ∉ U)
-    (hmax : Set.Ioi x ⊆ U) (α : E → A)
-    (hBq : B.UpperQuotient α) (hCq : C.UpperQuotient α)
-    (jv : B.vertex x ≃ₗ[A] C.vertex x)
-    (hjv : ∀ n v, v ∈ (B.vertex x).piece n → jv v ∈ (C.vertex x).piece n)
-    (hout : ∀ e (he : G.source e = x) v,
-      j.edge e (hmax (lt_of_eq_of_lt he.symm (G.increasing e)))
-        (B.forget.lower e v) = C.forget.lower e (Sheaf.castStalkIso B.forget C.forget he.symm jv v)) :
-    OnIso B C (insert x U) := by
-  classical
-  let p := j.toOnIso.extend hU x hx hmax α hBq hCq jv hout
-  have hvp : ∀ y (hy : y ∈ insert x U) n v,
-      v ∈ (B.vertex y).piece n → p.vertex y hy v ∈ (C.vertex y).piece n := by
-    intro y hy n v hv
-    dsimp only [p,Sheaf.OnIso.extend]
-    split_ifs with h
-    · subst y
-      exact hjv n v hv
-    · exact j.vertex_graded y _ n v hv
-  refine ⟨p,hvp,?_⟩
-  intro e he n v hv
-  dsimp only [p,Sheaf.OnIso.extend]
-  split_ifs with h
-  · apply ModuleData.quotient_iso_graded (B.upper e) (C.upper e) (hBq e).1 (hCq e).1
-    · subst x
-      exact hjv
-    · exact hv
-  · exact j.edge_graded e _ n v hv
-
-end OnIso
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.Sheaf
-universe u v w z
-variable {R : Type u} [CommRing R] {V : Type v} [PartialOrder V]
-  {E : Type w} {G : OrderedGraph V E}
-
- def zero : Sheaf.{u,v,w,z} (R := R) G where
-  vertex _ := ModuleCat.of R PUnit.{z+1}
-  edge _ := ModuleCat.of R PUnit.{z+1}
-  lower _ := 0
-  upper _ := 0
-
- def prod (B C : Sheaf.{u,v,w,z} (R := R) G) : Sheaf.{u,v,w,z} (R := R) G where
-  vertex x := ModuleCat.of R (B.vertex x × C.vertex x)
-  edge e := ModuleCat.of R (B.edge e × C.edge e)
-  lower e := (B.lower e).prodMap (C.lower e)
-  upper e := (B.upper e).prodMap (C.upper e)
-
-lemma prod_generated (B C : Sheaf.{u,v,w,z} (R := R) G)
-    (hB : B.Generated) (hC : C.Generated) : (prod B C).Generated := by
-  intro x v
-  obtain ⟨f,hf,hfx⟩ := hB x v.1
-  obtain ⟨g,hg,hgx⟩ := hC x v.2
-  exact ⟨fun y => (f y,g y),fun e hs ht => Prod.ext (hf e hs ht) (hg e hs ht),Prod.ext hfx hgx⟩
-
-lemma prod_flabby (B C : Sheaf.{u,v,w,z} (R := R) G)
-    (hB : B.Flabby) (hC : C.Flabby) : (prod B C).Flabby := by
-  intro U hU f hf
-  obtain ⟨g,hg,hgf⟩ := hB U hU (fun y => (f y).1) (fun e hs ht => congrArg Prod.fst (hf e hs ht))
-  obtain ⟨t,ht,htf⟩ := hC U hU (fun y => (f y).2) (fun e hs htt => congrArg Prod.snd (hf e hs htt))
-  exact ⟨fun y => (g y,t y),fun e hs htt => Prod.ext (hg e hs htt) (ht e hs htt),
-    fun y hy => Prod.ext (hgf y hy) (htf y hy)⟩
-
-lemma prod_upperQuotient (B C : Sheaf.{u,v,w,z} (R := R) G) (α : E → R)
-    (hB : B.UpperQuotient α) (hC : C.UpperQuotient α) : (prod B C).UpperQuotient α := by
-  intro e
-  refine ⟨?_,?_⟩
-  · intro v
-    obtain ⟨b,hb⟩ := (hB e).1 v.1
-    obtain ⟨c,hc⟩ := (hC e).1 v.2
-    exact ⟨(b,c),Prod.ext hb hc⟩
-  · intro v
-    constructor
-    · intro h
-      obtain ⟨b,hb⟩ := (hB e).2 v.1 |>.mp (congrArg Prod.fst h)
-      obtain ⟨c,hc⟩ := (hC e).2 v.2 |>.mp (congrArg Prod.snd h)
-      exact ⟨(b,c),Prod.ext hb hc⟩
-    · rintro ⟨⟨b,c⟩,rfl⟩
-      exact Prod.ext ((hB e).2 _ |>.mpr ⟨b,rfl⟩) ((hC e).2 _ |>.mpr ⟨c,rfl⟩)
-
-lemma pi_upperQuotient {ι : Type*} (B : ι → Sheaf.{u,v,w,z} (R := R) G) (α : E → R)
-    (hB : ∀ i, (B i).UpperQuotient α) : (pi B).UpperQuotient α := by
-  classical
-  intro e
-  refine ⟨?_,?_⟩
-  · intro v
-    choose b hb using fun i => (hB i e).1 (v i)
-    exact ⟨b,funext hb⟩
-  · intro v
-    constructor
-    · intro h
-      choose b hb using fun i => (hB i e).2 (v i) |>.mp (congrFun h i)
-      exact ⟨b,funext hb⟩
-    · rintro ⟨b,rfl⟩
-      funext i
-      exact (hB i e).2 _ |>.mpr ⟨b i,rfl⟩
-
-lemma prod_kernel_iff (B C : Sheaf.{u,v,w,z} (R := R) G) (x : V) (v : (prod B C).vertex x) :
-    v ∈ LinearMap.ker ((prod B C).stalkMap x) ↔
-      v.1 ∈ LinearMap.ker (B.stalkMap x) ∧ v.2 ∈ LinearMap.ker (C.stalkMap x) := by
-  constructor
-  · intro h
-    constructor <;> apply LinearMap.mem_ker.mpr <;> funext e
-    · rcases e with ⟨e,he⟩
-      subst x
-      exact congrArg Prod.fst (congrFun (LinearMap.mem_ker.mp h) ⟨e,rfl⟩)
-    · rcases e with ⟨e,he⟩
-      subst x
-      exact congrArg Prod.snd (congrFun (LinearMap.mem_ker.mp h) ⟨e,rfl⟩)
-  · rintro ⟨hB,hC⟩
-    apply LinearMap.mem_ker.mpr
-    funext e
-    rcases e with ⟨e,he⟩
-    subst x
-    exact Prod.ext (congrFun (LinearMap.mem_ker.mp hB) ⟨e,rfl⟩)
-      (congrFun (LinearMap.mem_ker.mp hC) ⟨e,rfl⟩)
-
-lemma prod_minimal (B C : Sheaf.{u,v,w,z} (R := R) G) (x : V) (I : Ideal R)
-    (hB : LinearMap.ker (B.stalkMap x) ≤ I • (⊤ : Submodule R (B.vertex x)))
-    (hC : LinearMap.ker (C.stalkMap x) ≤ I • (⊤ : Submodule R (C.vertex x))) :
-    LinearMap.ker ((prod B C).stalkMap x) ≤ I • (⊤ : Submodule R ((prod B C).vertex x)) := by
-  intro v hv
-  obtain ⟨hvB,hvC⟩ := (prod_kernel_iff B C x v).mp hv
-  have h (M N : Type z) [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N]
-      (f : M →ₗ[R] N) : (I • (⊤ : Submodule R M)).map f ≤ I • (⊤ : Submodule R N) := by
-    rw [Submodule.map_smul'']
-    exact Submodule.smul_mono le_rfl le_top
-  have hb := h _ _ (LinearMap.inl R (B.vertex x) (C.vertex x)) ⟨v.1,hB hvB,rfl⟩
-  have hc := h _ _ (LinearMap.inr R (B.vertex x) (C.vertex x)) ⟨v.2,hC hvC,rfl⟩
-  have heq : (v.1,0) + (0,v.2) = v := Prod.ext (add_zero _) (zero_add _)
-  have hh := (I • (⊤ : Submodule R ((prod B C).vertex x))).add_mem hb hc
-  change (v.1,0) + (0,v.2) ∈ I • (⊤ : Submodule R ((prod B C).vertex x)) at hh
-  exact heq ▸ hh
-
-end KLInvariance.MomentGraph.Sheaf
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.Sheaf
-universe u v w z
-variable {R : Type u} [CommRing R] {V : Type v} [PartialOrder V]
-  {E : Type w} {G : OrderedGraph V E}
-  {B C : Sheaf.{u,v,w,z} (R := R) G} (D : Sheaf.{u,v,w,z} (R := R) G)
-
-noncomputable def addZeroRight (M N : Type z) [AddCommGroup M] [Module R M]
-    [AddCommGroup N] [Module R N] [Subsingleton N] : M ≃ₗ[R] M × N where
-  toFun m := (m,0)
-  invFun := Prod.fst
-  left_inv _ := rfl
-  right_inv _ := Prod.ext rfl (Subsingleton.elim _ _)
-  map_add' _ _ := Prod.ext rfl (zero_add _).symm
-  map_smul' _ _ := Prod.ext rfl (smul_zero _).symm
-
-@[simp] lemma addZeroRight_apply (M N : Type z) [AddCommGroup M] [Module R M]
-    [AddCommGroup N] [Module R N] [Subsingleton N] (m : M) :
-    addZeroRight (R := R) M N m = (m,0) := rfl
-
-noncomputable def OnIso.prodZero {U : Set V} (j : OnIso B C U)
-    (hV : ∀ x ∈ U, Subsingleton (D.vertex x))
-    (hE : ∀ e, G.target e ∈ U → Subsingleton (D.edge e)) : OnIso B (prod C D) U where
-  vertex x hx := by
-    letI := hV x hx
-    exact (j.vertex x hx).trans (addZeroRight _ _)
-  edge e he := by
-    letI := hE e he
-    exact (j.edge e he).trans (addZeroRight _ _)
-  lower e hs ht v := by
-    apply Prod.ext
-    · exact j.lower e hs ht v
-    · exact (D.lower e).map_zero.symm
-  upper e ht v := by
-    apply Prod.ext
-    · exact j.upper e ht v
-    · exact (D.upper e).map_zero.symm
-
-lemma pi_subsingleton_vertex {ι : Type*} (B : ι → Sheaf.{u,v,w,z} (R := R) G)
-    (x : V) (h : ∀ i, Subsingleton ((B i).vertex x)) : Subsingleton ((pi B).vertex x) := by
-  exact ⟨fun f g => funext (fun i => @Subsingleton.elim _ (h i) (f i) (g i))⟩
-
-lemma pi_subsingleton_edge {ι : Type*} (B : ι → Sheaf.{u,v,w,z} (R := R) G)
-    (e : E) (h : ∀ i, Subsingleton ((B i).edge e)) : Subsingleton ((pi B).edge e) := by
-  exact ⟨fun f g => funext (fun i => @Subsingleton.elim _ (h i) (f i) (g i))⟩
-
-end KLInvariance.MomentGraph.Sheaf
-
-end
-
-
-section
-
-namespace KLInvariance.Graded.ModuleData
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A}
-noncomputable section
- def zero : ModuleData.{uk,ua,um} 𝓐 where
-  obj := ModuleCat.of A PUnit.{max ua um+1}
-  kModule := inferInstance
-  tower := inferInstance
-  piece _ := ⊥
-  decomposition := (show DirectSum.IsInternal (fun _ : ℤ => (⊥ : Submodule k PUnit.{max ua um+1})) from by
-    apply internal_of_components _ (fun _ => 0)
-    · intro _ _; exact Submodule.zero_mem _
-    · intro _ _ _; exact Subsingleton.elim _ _
-    · intro _ _ _ _ _; rfl
-    · intro m; exact ⟨∅,Subsingleton.elim _ _⟩).chooseDecomposition
-  graded := ⟨by intro _ _ _ _ _ _; exact Subsingleton.elim _ _⟩
-  nonneg _ _ := rfl
-  finite := inferInstance
-
- def Hom.zero (M N : ModuleData.{uk,ua,um} 𝓐) : Hom M N :=
-  ⟨0,fun _ _ _ => Submodule.zero_mem _⟩
-
- def Hom.pi {ι : Type um} [Fintype ι] {M N : ι → ModuleData.{uk,ua,um} 𝓐}
-    (f : ∀ i, Hom (M i) (N i)) : Hom (ModuleData.pi M) (ModuleData.pi N) where
-  map := LinearMap.pi fun i => (f i).map.comp (LinearMap.proj i)
-  graded n _ hm i hi := (f i).graded n _ (hm i hi)
-end
-end KLInvariance.Graded.ModuleData
-
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
-  {G : OrderedGraph V E}
-noncomputable section
- def shift (B : GradedSheaf (𝓐 := 𝓐) G) (d : ℕ) : GradedSheaf (𝓐 := 𝓐) G where
-  vertex x := (B.vertex x).shift d
-  edge e := (B.edge e).shift d
-  lower e := (B.lower e).shift d
-  upper e := (B.upper e).shift d
-
- def zero : GradedSheaf (𝓐 := 𝓐) G where
-  vertex _ := ModuleData.zero
-  edge _ := ModuleData.zero
-  lower _ := ModuleData.Hom.zero _ _
-  upper _ := ModuleData.Hom.zero _ _
-
- def prod (B C : GradedSheaf (𝓐 := 𝓐) G) : GradedSheaf (𝓐 := 𝓐) G where
-  vertex x := (B.vertex x).prod (C.vertex x)
-  edge e := (B.edge e).prod (C.edge e)
-  lower e := (B.lower e).prodMap (C.lower e)
-  upper e := (B.upper e).prodMap (C.upper e)
-
- def pi {ι : Type um} [Fintype ι] (B : ι → GradedSheaf (𝓐 := 𝓐) G) :
-    GradedSheaf (𝓐 := 𝓐) G where
-  vertex x := ModuleData.pi (fun i => (B i).vertex x)
-  edge e := ModuleData.pi (fun i => (B i).edge e)
-  lower e := ModuleData.Hom.pi (fun i => (B i).lower e)
-  upper e := ModuleData.Hom.pi (fun i => (B i).upper e)
-
- def OnIso.prodZero {B C : GradedSheaf (𝓐 := 𝓐) G} {U : Set V} (j : OnIso B C U)
-    (D : GradedSheaf (𝓐 := 𝓐) G)
-    (hV : ∀ x ∈ U, Subsingleton (D.vertex x))
-    (hE : ∀ e, G.target e ∈ U → Subsingleton (D.edge e)) : OnIso B (prod C D) U where
-  toOnIso := j.toOnIso.prodZero D.forget hV hE
-  vertex_graded x hx n v hv := ⟨j.vertex_graded x hx n v hv,Submodule.zero_mem _⟩
-  edge_graded e he n v hv := ⟨j.edge_graded e he n v hv,Submodule.zero_mem _⟩
-
- theorem shift_forget (B : GradedSheaf (𝓐 := 𝓐) G) (d : ℕ) :
-    (B.shift d).forget=B.forget := rfl
- theorem zero_forget : (zero (𝓐 := 𝓐) (G := G)).forget=Sheaf.zero := rfl
- theorem prod_forget (B C : GradedSheaf (𝓐 := 𝓐) G) :
-    (prod B C).forget=Sheaf.prod B.forget C.forget := rfl
- theorem pi_forget {ι : Type um} [Fintype ι] (B : ι → GradedSheaf (𝓐 := 𝓐) G) :
-    (pi B).forget=Sheaf.pi (fun i => (B i).forget) := rfl
-end
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-/-! Extension by zero from a full lower ideal. Proof-indexed finite products
-avoid informal identifications of the integral stalks at boundary vertices. -/
-namespace KLInvariance.MomentGraph
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
-  (G : OrderedGraph V E) (K : Set V) (hK : IsLowerSet K)
-
- def lowerGraph : OrderedGraph K {e : E // G.target e ∈ K} where
-  source e := ⟨G.source e.val,hK (G.increasing e.val).le e.property⟩
-  target e := ⟨G.target e.val,e.property⟩
-  increasing e := G.increasing e.val
-
-namespace GradedSheaf
-noncomputable section
-attribute [local instance] Classical.propDecidable
-
-abbrev Witness (p : Prop) := ULift.{um} (PLift p)
-
-variable {G K hK} (B : GradedSheaf (𝓐 := 𝓐) (lowerGraph G K hK))
-
- def lowerExtend : GradedSheaf (𝓐 := 𝓐) G where
-  vertex x := ModuleData.pi (fun h : Witness.{um} (x ∈ K) => B.vertex ⟨x,h.down.down⟩)
-  edge e := ModuleData.pi (fun h : Witness.{um} (G.target e ∈ K) => B.edge ⟨e,h.down.down⟩)
-  lower e :=
-    { map := LinearMap.pi (fun h => (B.lower ⟨e,h.down.down⟩).map.comp
-        (LinearMap.proj ⟨⟨hK (G.increasing e).le h.down.down⟩⟩))
-      graded n _v hv h _ := (B.lower ⟨e,h.down.down⟩).graded n _ (hv _ (Set.mem_univ _)) }
-  upper e :=
-    { map := LinearMap.pi (fun h => (B.upper ⟨e,h.down.down⟩).map.comp (LinearMap.proj h))
-      graded n _v hv h _ := (B.upper ⟨e,h.down.down⟩).graded n _ (hv _ (Set.mem_univ _)) }
-
- def lowerExtendVertex (x : K) : (B.lowerExtend.vertex x.val) ≃ₗ[A] B.vertex x where
-  toFun f := f ⟨⟨x.property⟩⟩
-  invFun v _ := v
-  left_inv f := by
-    funext h
-    rfl
-  right_inv _ := rfl
-  map_add' _ _ := rfl
-  map_smul' _ _ := rfl
-
- theorem lowerExtendVertex_graded (x : K) (n : ℤ) (v : B.lowerExtend.vertex x.val)
-    (hv : v ∈ (B.lowerExtend.vertex x.val).piece n) :
-    B.lowerExtendVertex x v ∈ (B.vertex x).piece n := hv _ (Set.mem_univ _)
-
- theorem lowerExtend_support (x : V) (hx : x ∉ K) : Subsingleton (B.lowerExtend.vertex x) :=
-  ⟨fun _f _g => funext (fun h => (hx h.down.down).elim)⟩
-
- theorem lowerExtend_free (hb : ∀ x, Module.Free A (B.vertex x)) (x : V) :
-    Module.Free A (B.lowerExtend.vertex x) := by
-  let : ∀ h : Witness.{um} (x ∈ K), Module.Free A (B.vertex ⟨x,h.down.down⟩) := fun h => hb _
-  change Module.Free A (∀ h : Witness.{um} (x ∈ K), B.vertex ⟨x,h.down.down⟩)
-  infer_instance
-
- def liftLowerAssignment (f : B.forget.Assignment) : B.lowerExtend.forget.Assignment :=
-  fun x h => f ⟨x,h.down.down⟩
-
- def projectLowerAssignment (f : B.lowerExtend.forget.Assignment) : B.forget.Assignment :=
-  fun x => f x.val ⟨⟨x.property⟩⟩
-
- theorem projectLower_compatible {U : Set V} {f : B.lowerExtend.forget.Assignment}
-    (hf : B.lowerExtend.forget.CompatibleOn U f) :
-    B.forget.CompatibleOn (Subtype.val ⁻¹' U) (B.projectLowerAssignment f) := by
-  intro e hs ht
-  exact congrFun (hf e.val hs ht) ⟨⟨e.property⟩⟩
-
- theorem liftLower_compatible {U : Set V} {f : B.forget.Assignment}
-    (hf : B.forget.CompatibleOn (Subtype.val ⁻¹' U) f) :
-    B.lowerExtend.forget.CompatibleOn U (B.liftLowerAssignment f) := by
-  intro e hs ht
-  funext h
-  exact hf ⟨e,h.down.down⟩ hs ht
-
- theorem lowerExtend_generated (hb : B.forget.Generated) : B.lowerExtend.forget.Generated := by
-  intro x v
-  by_cases hx : x ∈ K
-  · obtain ⟨f,hf,hfx⟩ := hb ⟨x,hx⟩ (v ⟨⟨hx⟩⟩)
-    refine ⟨B.liftLowerAssignment f,B.liftLower_compatible hf,?_⟩
-    funext h
-    exact hfx
-  · let : Subsingleton (B.lowerExtend.forget.vertex x) := B.lowerExtend_support x hx
-    exact ⟨0,fun _ _ _ => by simp,Subsingleton.elim _ _⟩
-
- theorem lowerExtend_flabby (hb : B.forget.Flabby) : B.lowerExtend.forget.Flabby := by
-  intro U hU f hf
-  have hu : IsUpperSet (Subtype.val ⁻¹' U : Set K) := fun _ _ hab ha => hU hab ha
-  obtain ⟨g,hg,hgf⟩ := hb (Subtype.val ⁻¹' U) hu (B.projectLowerAssignment f) (B.projectLower_compatible hf)
-  refine ⟨B.liftLowerAssignment g,B.liftLower_compatible hg,?_⟩
-  intro x hx
-  funext h
-  exact hgf ⟨x,h.down.down⟩ hx
-
- theorem lowerExtend_quotient (α : E → A)
-    (hb : B.UpperQuotient (fun e => α e.val)) : B.lowerExtend.UpperQuotient α := by
-  intro e
-  constructor
-  · intro v
-    have hh : ∀ h : Witness.{um} (G.target e ∈ K), ∃ w, (B.upper ⟨e,h.down.down⟩).map w=v h :=
-      fun h => (hb ⟨e,h.down.down⟩).1 (v h)
-    choose w hw using hh
-    exact ⟨w,funext hw⟩
-  · intro v
-    constructor
-    · intro hv
-      have hh : ∀ h : Witness.{um} (G.target e ∈ K), ∃ w, v h=α e • w :=
-        fun h => (hb ⟨e,h.down.down⟩).2 (v h) |>.mp (congrFun hv h)
-      choose w hw using hh
-      exact ⟨w,funext hw⟩
-    · rintro ⟨w,rfl⟩
-      funext h
-      exact (hb ⟨e,h.down.down⟩).2 _ |>.mpr ⟨w h,rfl⟩
-
- theorem lowerExtend_minimal (x : V)
-    (hb : ∀ hx : x ∈ K, B.KernelMinimal ⟨x,hx⟩) : B.lowerExtend.KernelMinimal x := by
-  intro f hf
-  apply LocalCover.pi_mem_ideal_smul
-  intro h
-  apply hb h.down.down (B.projectLowerAssignment f)
-  intro e he
-  have hes : G.source e.val=x := congrArg Subtype.val he
-  exact congrFun (hf e.val hes) ⟨⟨e.property⟩⟩
-
-end
-end GradedSheaf
-end KLInvariance.MomentGraph
-
-end
-
-
-section
-
-/-! Finite sums of honest shifted normalized boundary models. A model family
-is data for the decomposition lemma, not an assumption of the KL main. -/
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-  (hneg : ∀ n < 0, 𝓐 n = ⊥)
-  {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
-
-instance zeroVertex_subsingleton (x : V) : Subsingleton ((zero (𝓐 := 𝓐) (G := G)).vertex x) :=
-  inferInstanceAs (Subsingleton PUnit.{max ua um+1})
-instance zeroEdge_subsingleton (e : E) : Subsingleton ((zero (𝓐 := 𝓐) (G := G)).edge e) :=
-  inferInstanceAs (Subsingleton PUnit.{max ua um+1})
-
-structure BoundaryModels (α : E → A) where
-  sheaf : V → GradedSheaf (𝓐 := 𝓐) G
-  free : ∀ c x, Module.Free A ((sheaf c).vertex x)
-  quotient : ∀ c, (sheaf c).UpperQuotient α
-  generated : ∀ c, (sheaf c).forget.Generated
-  flabby : ∀ c, (sheaf c).forget.Flabby
-  support : ∀ c x, ¬x ≤ c → Subsingleton ((sheaf c).vertex x)
-  top : ∀ c, (sheaf c).vertex c ≃ₗ[A] ModuleData.unit.{uk,ua,um} hneg
-  top_graded : ∀ c n v, v ∈ ((sheaf c).vertex c).piece n → top c v ∈ (ModuleData.unit.{uk,ua,um} hneg).piece n
-  minimal : ∀ c x, x ≠ c → (sheaf c).KernelMinimal x
-
-inductive ModelSum (O : V → GradedSheaf (𝓐 := 𝓐) G) (U : Set V) :
-    GradedSheaf (𝓐 := 𝓐) G → Prop where
-  | zero : ModelSum O U zero
-  | add {B} : ModelSum O U B → ∀ c ∈ U, ∀ (n : ℕ) (d : ULift.{um} (Fin n) → ℕ),
-      ModelSum O U (prod B (pi (fun i => (O c).shift (d i))))
-
-namespace ModelSum
-variable {hneg} {α : E → A} (O : BoundaryModels hneg (G := G) α)
-  {U : Set V} {B : GradedSheaf (𝓐 := 𝓐) G}
-
-omit [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐] in
-lemma mono {O : V → GradedSheaf (𝓐 := 𝓐) G} (h : ModelSum O U B)
-    {T : Set V} (hUT : U ⊆ T) : ModelSum O T B := by
-  induction h with
-  | zero => exact .zero
-  | add _ c hc n d ih => exact .add ih c (hUT hc) n d
-
-lemma free (h : ModelSum O.sheaf U B) (x : V) : Module.Free A (B.vertex x) := by
-  induction h with
-  | zero => exact inferInstanceAs (Module.Free A PUnit.{max ua um+1})
-  | @add C _ c _ n d ih =>
-    let _ : Module.Free A (C.vertex x) := ih
-    let _ := O.free c x
-    change Module.Free A (C.vertex x × (ULift.{um} (Fin n) → (O.sheaf c).vertex x))
-    infer_instance
-
-lemma quotient (h : ModelSum O.sheaf U B) : B.UpperQuotient α := by
-  induction h with
-  | zero =>
-    intro e
-    exact ⟨fun v => ⟨0,Subsingleton.elim _ _⟩,fun v => ⟨fun _ => ⟨0,Subsingleton.elim _ _⟩,fun _ => Subsingleton.elim _ _⟩⟩
-  | @add C _ c _ n d ih =>
-    exact Sheaf.prod_upperQuotient C.forget (Sheaf.pi (fun _ : ULift.{um} (Fin n) => (O.sheaf c).forget)) α ih
-      (Sheaf.pi_upperQuotient (fun _ : ULift.{um} (Fin n) => (O.sheaf c).forget) α (fun _ => O.quotient c))
-
-lemma generated (h : ModelSum O.sheaf U B) : B.forget.Generated := by
-  induction h with
-  | zero =>
-    intro x v
-    let : Subsingleton ((GradedSheaf.zero (𝓐 := 𝓐) (G := G)).forget.vertex x) := zeroVertex_subsingleton (𝓐 := 𝓐) (G := G) x
-    exact ⟨0,fun _ _ _ => rfl,Subsingleton.elim _ _⟩
-  | add _ c _ n d ih => exact Sheaf.prod_generated _ _ ih (Sheaf.pi_generated _ (fun _ => O.generated c))
-
-lemma flabby (h : ModelSum O.sheaf U B) : B.forget.Flabby := by
-  induction h with
-  | zero => exact fun U _ f _ => ⟨f,fun _ _ _ => rfl,fun _ _ => rfl⟩
-  | add _ c _ n d ih => exact Sheaf.prod_flabby _ _ ih (Sheaf.pi_flabby _ (fun _ => O.flabby c))
-
-lemma minimal (h : ModelSum O.sheaf U B) (x : V) (hx : x ∉ U) : B.KernelMinimal x := by
-  induction h with
-  | zero =>
-    apply (GradedSheaf.zero.kernelMinimal_iff x).mpr
-    intro v _
-    let : Subsingleton ((GradedSheaf.zero (𝓐 := 𝓐) (G := G)).forget.vertex x) := zeroVertex_subsingleton (𝓐 := 𝓐) (G := G) x
-    rw [Subsingleton.elim v 0]
-    exact Submodule.zero_mem _
-  | @add C _ c hc n d ih =>
-    apply ((prod C (pi (fun i => (O.sheaf c).shift (d i)))).kernelMinimal_iff x).mpr
-    exact Sheaf.prod_minimal C.forget (Sheaf.pi (fun _ : ULift.{um} (Fin n) => (O.sheaf c).forget)) x _ ((C.kernelMinimal_iff x).mp ih)
-      (Sheaf.pi_minimal (fun _ : ULift.{um} (Fin n) => (O.sheaf c).forget) x _ (fun _ =>
-        ((O.sheaf c).kernelMinimal_iff x).mp (O.minimal c x (fun he => hx (he ▸ hc)))))
-
-end ModelSum
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-/-! Actual normalized BMP models at every vertex of a finite ordered graph,
-constructed on its lower ideal and extended by zero. -/
-namespace KLInvariance.MomentGraph
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-  [IsNoetherianRing A]
-  (hneg : ∀ n < 0, 𝓐 n = ⊥)
-  (hscalar : ∀ a : A, ∃ c : k, a-algebraMap k A c ∈ positiveIdeal 𝓐)
-  {V E : Type um} [PartialOrder V] [Fintype V] [Fintype E]
-  (G : OrderedGraph V E) (α : E → A) (hα : ∀ e, α e ∈ 𝓐 1)
-noncomputable section
-attribute [local instance] Classical.propDecidable
-
- def lowerBoundary (c : V) : BoundarySheaf hneg (lowerGraph G (Set.Iic c) (isLowerSet_Iic c))
-    (fun e => α e.val) ⟨c,le_rfl⟩ :=
-  (exists_boundarySheaf hneg (lowerGraph G (Set.Iic c) (isLowerSet_Iic c)) hscalar
-    ⟨c,le_rfl⟩ (fun x => x.property) (fun e => α e.val) (fun e => hα e.val)).some
-
- def actualBoundaryModels : GradedSheaf.BoundaryModels hneg (G := G) α where
-  sheaf c := (lowerBoundary hneg hscalar G α hα c).sheaf.lowerExtend
-  free c := (lowerBoundary hneg hscalar G α hα c).sheaf.lowerExtend_free
-    (lowerBoundary hneg hscalar G α hα c).free
-  quotient c := (lowerBoundary hneg hscalar G α hα c).sheaf.lowerExtend_quotient α
-    (lowerBoundary hneg hscalar G α hα c).quotient
-  generated c := (lowerBoundary hneg hscalar G α hα c).sheaf.lowerExtend_generated
-    (lowerBoundary hneg hscalar G α hα c).generated
-  flabby c := (lowerBoundary hneg hscalar G α hα c).sheaf.lowerExtend_flabby
-    (lowerBoundary hneg hscalar G α hα c).flabby.flabby
-  support c x hx := (lowerBoundary hneg hscalar G α hα c).sheaf.lowerExtend_support x hx
-  top c := ((lowerBoundary hneg hscalar G α hα c).sheaf.lowerExtendVertex ⟨c,le_rfl⟩).trans
-    (ModuleData.equivOfEq (lowerBoundary hneg hscalar G α hα c).top)
-  top_graded c n v hv := ModuleData.equivOfEq_graded
-    (lowerBoundary hneg hscalar G α hα c).top n _
-    ((lowerBoundary hneg hscalar G α hα c).sheaf.lowerExtendVertex_graded ⟨c,le_rfl⟩ n v hv)
-  minimal c x hx := (lowerBoundary hneg hscalar G α hα c).sheaf.lowerExtend_minimal x
-    (fun hxc => (lowerBoundary hneg hscalar G α hα c).minimal ⟨x,hxc⟩
-      (fun hh => hx (congrArg Subtype.val hh)))
-
-end
-end KLInvariance.MomentGraph
-
-end
-
-
-section
-
-/-! Recognition of the actual finite minimal boundary sheaf. This proves
-choice-independence of the constructed graded object, not its KL character. -/
-namespace KLInvariance.MomentGraph.BoundarySheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-  [IsDomain A] [IsNoetherianRing A]
-  {hneg : ∀ n < 0, 𝓐 n = ⊥}
-  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
-  {V E : Type um} [PartialOrder V] [Fintype V] [Fintype E]
-  {G : OrderedGraph V E} {α : E → A} {b : V}
-  (B C : BoundarySheaf hneg G α b) (hb : ∀ x, x ≤ b)
-
-omit [Fintype V] in
-include hzero hb in
- theorem extend_iso {U : Set V} (hU : IsUpperSet U)
-    (j : GradedSheaf.OnIso B.sheaf C.sheaf U)
-    (x : V) (hx : x ∉ U) (hmax : Set.Ioi x ⊆ U) :
-    Nonempty (GradedSheaf.OnIso B.sheaf C.sheaf (insert x U)) := by
-  classical
-  by_cases hxb : x=b
-  · subst x
-    let eqv := ModuleData.equivOfEq (B.top.trans C.top.symm)
-    have heqv := ModuleData.equivOfEq_graded (B.top.trans C.top.symm)
-    refine ⟨j.extend hU b hx hmax α B.quotient C.quotient eqv heqv ?_⟩
-    intro e he
-    exact False.elim ((not_lt_of_ge (hb (G.target e))) (he ▸ G.increasing e))
-  · let ja := j.toOnIso.above x hmax
-    obtain ⟨e,he,hcomm⟩ := ja.extend_stalk_graded hneg hzero
-      (fun e ht => j.edge_graded e (hmax ht)) (B.free x) (C.free x)
-      B.flabby.flabby B.generated C.flabby.flabby C.generated
-      (B.minimal x hxb) (C.minimal x hxb)
-    refine ⟨j.extend hU x hx hmax α B.quotient C.quotient e he ?_⟩
-    intro f hf v
-    subst x
-    exact congrFun (hcomm v) ⟨f,rfl⟩
-
-include hzero hb in
- theorem exists_gradedIso : Nonempty (GradedSheaf.Iso B.sheaf C.sheaf) := by
-  classical
-  have h : ∀ s : Finset V, IsUpperSet ((s : Set V)ᶜ) →
-      GradedSheaf.OnIso B.sheaf C.sheaf ((s : Set V)ᶜ) →
-      Nonempty (GradedSheaf.Iso B.sheaf C.sheaf) := by
-    intro s
-    induction s using Finset.strongInductionOn with
-    | _ s ih =>
-      intro hU j
-      by_cases hs : s=∅
-      · subst s
-        have heq : ((∅ : Finset V) : Set V)ᶜ=Set.univ := by simp
-        exact ⟨GradedSheaf.OnIso.ofUniv (heq ▸ j)⟩
-      · obtain ⟨x,hxs,hxmax⟩ := s.exists_maximal (Finset.nonempty_iff_ne_empty.mpr hs)
-        have hx : x ∉ ((s : Set V)ᶜ) := by simpa using hxs
-        have hmax : Set.Ioi x ⊆ ((s : Set V)ᶜ) := by
-          intro y hy
-          change y ∉ s
-          intro hys
-          exact (not_le_of_gt hy) (hxmax hys hy.le)
-        obtain ⟨jj⟩ := B.extend_iso hzero C hb hU j x hx hmax
-        have heq : insert x ((s : Set V)ᶜ) = ((s.erase x : Finset V) : Set V)ᶜ := by
-          ext y
-          simp only [Set.mem_insert_iff,Set.mem_compl_iff,Finset.mem_coe,Finset.mem_erase]
-          tauto
-        have hU' : IsUpperSet (insert x ((s : Set V)ᶜ)) := by
-          intro y t hyt hy
-          rcases Set.mem_insert_iff.mp hy with hy | hy
-          · subst y
-            by_cases htx : t=x
-            · exact Set.mem_insert_iff.mpr (Or.inl htx)
-            · exact Set.mem_insert_of_mem x (hmax (lt_of_le_of_ne hyt (Ne.symm htx)))
-          · exact Set.mem_insert_of_mem x (hU hyt hy)
-        exact ih (s.erase x) (Finset.erase_ssubset hxs) (heq ▸ hU') (heq ▸ jj)
-  have hset : ((Finset.univ : Finset V) : Set V)ᶜ = ∅ := by simp
-  have hj : GradedSheaf.OnIso B.sheaf C.sheaf (∅ : Set V) := by
-    refine { toOnIso := ?_, vertex_graded := ?_, edge_graded := ?_ }
-    · exact ⟨fun x hx => False.elim hx, fun e he => False.elim he,
-        fun e hs => False.elim hs, fun e ht => False.elim ht⟩
-    · intro x hx; exact False.elim hx
-    · intro e he; exact False.elim he
-  have hh := h Finset.univ
-  rw [hset] at hh
-  exact hh isUpperSet_empty hj
-
-end KLInvariance.MomentGraph.BoundarySheaf
-
-end
-
-
-section
-
-
-namespace KLInvariance.MomentGraph.Sheaf
-universe ur uv ue
-variable {R : Type ur} [CommRing R] {V : Type uv} [PartialOrder V]
-  {E : Type ue} (G : OrderedGraph V E) (α : E → R) (c : V)
-noncomputable section
-
-private def quotientLinear (e : E) : R →ₗ[R] R ⧸ Ideal.span {α e} :=
-  (Ideal.Quotient.mkₐ R (Ideal.span {α e})).toLinearMap
-
- def structureSheaf : Sheaf (R := R) G where
-  vertex x := ModuleCat.of R (PLift (x ≤ c) → R)
-  edge e := ModuleCat.of R (PLift (G.target e ≤ c) → R ⧸ Ideal.span {α e})
-  upper e := LinearMap.pi (fun h => (quotientLinear α e).comp (LinearMap.proj h))
-  lower e := LinearMap.pi (fun h => (quotientLinear α e).comp
-    (LinearMap.proj ⟨(G.increasing e).le.trans h.down⟩))
-
- theorem structureSheaf_free (x : V) : Module.Free R ((structureSheaf G α c).vertex x) :=
-  inferInstanceAs (Module.Free R (PLift (x ≤ c) → R))
-
- theorem structureSheaf_finite (x : V) : Module.Finite R ((structureSheaf G α c).vertex x) :=
-  inferInstanceAs (Module.Finite R (PLift (x ≤ c) → R))
-
- theorem structureSheaf_support (x : V) (hx : ¬ x ≤ c) :
-    Subsingleton ((structureSheaf G α c).vertex x) := by
-  refine ⟨fun f g => funext fun h => False.elim (hx h.down)⟩
-
- def structureVertexIso (x : V) (hx : x ≤ c) :
-    (structureSheaf G α c).vertex x ≃ₗ[R] R where
-  toFun f := f ⟨hx⟩
-  invFun r := fun _ => r
-  left_inv _ := rfl
-  right_inv _ := rfl
-  map_add' _ _ := rfl
-  map_smul' _ _ := rfl
-
- theorem structureSheaf_generated : (structureSheaf G α c).Generated := by
-  classical
-  intro x v
-  by_cases hx : x ≤ c
-  · refine ⟨fun _ _ => v ⟨hx⟩, fun _ _ _ => rfl, rfl⟩
-  · let _ := structureSheaf_support G α c x hx
-    exact ⟨0,fun _ _ _ => rfl,Subsingleton.elim _ _⟩
-
- theorem structureSheaf_upperQuotient : (structureSheaf G α c).UpperQuotient α := by
-  classical
-  intro e
-  refine ⟨?_,?_⟩
-  · intro v
-    choose r hr using fun h : PLift (G.target e ≤ c) => Ideal.Quotient.mk_surjective (v h)
-    exact ⟨r,funext hr⟩
-  · intro v
-    constructor
-    · intro hv
-      have hh (h : PLift (G.target e ≤ c)) :
-          v h ∈ Ideal.span {α e} := Ideal.Quotient.eq_zero_iff_mem.mp (congrFun hv h)
-      choose r hr using fun h => Ideal.mem_span_singleton.mp (hh h)
-      exact ⟨r,funext hr⟩
-    · rintro ⟨w,rfl⟩
-      funext h
-      exact Ideal.Quotient.eq_zero_iff_mem.mpr
-        (Ideal.mem_span_singleton.mpr ⟨w h,rfl⟩)
-
- theorem structureSheaf_minimal [IsLocalRing R]
-    (hα : ∀ e, α e ∈ IsLocalRing.maximalIdeal R)
-    (hreach : ∀ x, x < c → ∃ e, G.source e = x ∧ G.target e ≤ c)
-    (x : V) (hxc : x ≠ c) :
-    LinearMap.ker ((structureSheaf G α c).stalkMap x) ≤
-      IsLocalRing.maximalIdeal R • (⊤ : Submodule R ((structureSheaf G α c).vertex x)) := by
-  classical
-  intro v hv
-  by_cases hx : x ≤ c
-  · obtain ⟨e,he,het⟩ := hreach x (lt_of_le_of_ne hx hxc)
-    subst x
-    have heq := congrFun (LinearMap.mem_ker.mp hv) ⟨e,rfl⟩
-    have hvα : v ⟨hx⟩ ∈ Ideal.span {α e} :=
-      Ideal.Quotient.eq_zero_iff_mem.mp (congrFun heq ⟨het⟩)
-    have hm : v ⟨hx⟩ ∈ IsLocalRing.maximalIdeal R :=
-      (Ideal.span_le.mpr (Set.singleton_subset_iff.mpr (hα e))) hvα
-    let one : (structureSheaf G α c).vertex (G.source e) := fun _ => 1
-    have hs : v ⟨hx⟩ • one ∈ IsLocalRing.maximalIdeal R •
-        (⊤ : Submodule R ((structureSheaf G α c).vertex (G.source e))) :=
-      Submodule.smul_mem_smul hm (show one ∈
-        (⊤ : Submodule R ((structureSheaf G α c).vertex (G.source e))) from Submodule.mem_top)
-    have heqv : v = v ⟨hx⟩ • one := by
-      funext h
-      exact (mul_one _).symm
-    rw [heqv]
-    exact hs
-  · let _ := structureSheaf_support G α c x hx
-    rw [Subsingleton.elim v 0]
-    exact Submodule.zero_mem _
-
-end
-end KLInvariance.MomentGraph.Sheaf
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.Sheaf
-open TensorProduct
-universe ur us uv ue um
-variable {R : Type ur} [CommRing R] (S : Type us) [CommRing S] [Algebra R S]
-  {V : Type uv} [PartialOrder V] {E : Type ue} {G : OrderedGraph V E}
-  (B : Sheaf.{ur,uv,ue,um} (R := R) G)
-
-noncomputable def baseChange : Sheaf (R := S) G where
-  vertex x := ModuleCat.of S (S ⊗[R] B.vertex x)
-  edge e := ModuleCat.of S (S ⊗[R] B.edge e)
-  lower e := AlgebraTensorModule.lTensor S S (B.lower e)
-  upper e := AlgebraTensorModule.lTensor S S (B.upper e)
-
-noncomputable def tensorAssignments : S ⊗[R] B.Assignment →ₗ[S] (B.baseChange S).Assignment :=
-  TensorProduct.piRightHom R S S (fun x => B.vertex x)
-
-@[simp] theorem tensorAssignments_tmul (s : S) (f : B.Assignment) (x : V) :
-    B.tensorAssignments S (s ⊗ₜ f) x = s ⊗ₜ f x := rfl
-
-lemma baseChange_tmul_compatible (Ω : Set V) (s : S) (f : B.Assignment)
-    (hf : B.CompatibleOn Ω f) : (B.baseChange S).CompatibleOn Ω (fun x => s ⊗ₜ f x) := by
-  intro e hs ht
-  change s ⊗ₜ[R] B.lower e (f (G.source e)) = s ⊗ₜ[R] B.upper e (f (G.target e))
-  rw [hf e hs ht]
-
-noncomputable def tensorSections (Ω : Set V) :
-    S ⊗[R] B.sections Ω →ₗ[S] (B.baseChange S).sections Ω :=
-  ((B.tensorAssignments S).comp (AlgebraTensorModule.lTensor S S (B.sections Ω).subtype)).codRestrict
-    ((B.baseChange S).sections Ω) (by
-      intro z
-      induction z using TensorProduct.inductionOn with
-      | tmul s f => exact B.baseChange_tmul_compatible S Ω s f.val f.property
-      | add z t hz ht => simpa only [map_add] using Submodule.add_mem _ hz ht)
-
-@[simp] theorem tensorSections_tmul (Ω : Set V) (s : S) (f : B.sections Ω) (x : V) :
-    (B.tensorSections S Ω (s ⊗ₜ f)).val x = s ⊗ₜ f.val x := rfl
-
-theorem baseChange_generated (h : B.Generated) : (B.baseChange S).Generated := by
-  intro x v
-  induction v using TensorProduct.inductionOn with
-  | tmul s v =>
-    obtain ⟨f,hf,hx⟩ := h x v
-    exact ⟨fun y => s ⊗ₜ f y,B.baseChange_tmul_compatible S Set.univ s f hf,congrArg (fun v => s ⊗ₜ[R] v) hx⟩
-  | add v w hv hw =>
-    obtain ⟨f,hf,hfv⟩ := hv
-    obtain ⟨g,hg,hgw⟩ := hw
-    exact ⟨f+g,Submodule.add_mem ((B.baseChange S).sections Set.univ) hf hg,congrArg₂ (·+·) hfv hgw⟩
-
-/-- The finite section constraint whose kernel is the actual section module. -/
-noncomputable def constraint (Ω : Set V) : B.Assignment →ₗ[R]
-    (∀ e : {e : E // G.source e ∈ Ω ∧ G.target e ∈ Ω}, B.edge e.val) :=
-  LinearMap.pi (fun e => (B.lower e.val).comp (LinearMap.proj (G.source e.val)) -
-    (B.upper e.val).comp (LinearMap.proj (G.target e.val)))
-
-lemma ker_constraint (Ω : Set V) : LinearMap.ker (B.constraint Ω) = B.sections Ω := by
-  ext f
-  constructor
-  · intro h e hs ht
-    have hh := congrFun (LinearMap.mem_ker.mp h) ⟨e,hs,ht⟩
-    exact sub_eq_zero.mp hh
-  · intro h
-    apply LinearMap.mem_ker.mpr
-    funext e
-    exact sub_eq_zero.mpr (h e.val e.property.1 e.property.2)
-
-end KLInvariance.MomentGraph.Sheaf
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.Sheaf
-open TensorProduct
-universe ur us uv ue um
-variable {R : Type ur} [CommRing R] (S : Type us) [CommRing S] [Algebra R S]
-  {V : Type uv} [PartialOrder V] {E : Type ue} {G : OrderedGraph V E}
-  (B : Sheaf.{ur,uv,ue,um} (R := R) G)
-
- theorem baseChange_vertex_free (x : V) [Module.Free R (B.vertex x)] :
-    Module.Free S ((B.baseChange S).vertex x) :=
-  inferInstanceAs (Module.Free S (S ⊗[R] B.vertex x))
-
- theorem baseChange_vertex_finite (x : V) [Module.Finite R (B.vertex x)] :
-    Module.Finite S ((B.baseChange S).vertex x) :=
-  inferInstanceAs (Module.Finite S (S ⊗[R] B.vertex x))
-
- theorem baseChange_upperQuotient [Module.Flat R S] (α : E → R)
-    (h : B.UpperQuotient α) : (B.baseChange S).UpperQuotient (fun e => algebraMap R S (α e)) := by
-  intro e
-  refine ⟨LinearMap.lTensor_surjective S (h e).1, ?_⟩
-  have hs : Function.Exact (DistribSMul.toLinearMap R (B.vertex (G.target e)) (α e))
-      (B.upper e) := by
-    intro v
-    exact ((h e).2 v).trans (by simp only [Set.mem_range, DistribSMul.toLinearMap_apply,
-      eq_comm])
-  have ht := Module.Flat.lTensor_exact S hs
-  intro v
-  change (B.upper e).lTensor S v = 0 ↔ ∃ w, v = algebraMap R S (α e) • w
-  rw [ht v, LinearMap.lTensor_smul_action]
-  change (∃ w : S ⊗[R] B.vertex (G.target e), α e • w = v) ↔
-    ∃ w : S ⊗[R] B.vertex (G.target e), v = algebraMap R S (α e) • w
-  simp only [IsScalarTower.algebraMap_smul]
-  constructor <;> rintro ⟨w,hw⟩ <;> exact ⟨w,hw.symm⟩
-
-end KLInvariance.MomentGraph.Sheaf
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.Sheaf
-open TensorProduct
-universe ur us uv ue um
-variable {R : Type ur} [CommRing R] (S : Type us) [CommRing S] [Algebra R S]
-  {V : Type uv} [PartialOrder V] {E : Type ue} {G : OrderedGraph V E}
-  (B : Sheaf.{ur,uv,ue,um} (R := R) G)
-noncomputable section
-
- theorem tensorAssignments_bijective [Finite V] : Function.Bijective (B.tensorAssignments S) := by
-  classical
-  let := Fintype.ofFinite V
-  exact (TensorProduct.piRight R S S (fun x => B.vertex x)).bijective
-
- theorem constraint_tensorAssignments (Ω : Set V) (z : S ⊗[R] B.Assignment) :
-    (B.baseChange S).constraint Ω (B.tensorAssignments S z) =
-      TensorProduct.piRightHom R S S (fun e : {e : E // G.source e ∈ Ω ∧ G.target e ∈ Ω} =>
-        B.edge e.val) (AlgebraTensorModule.lTensor S S (B.constraint Ω) z) := by
-  induction z using TensorProduct.inductionOn with
-  | tmul s f =>
-    ext e
-    change s ⊗ₜ[R] B.lower e.val (f (G.source e.val)) -
-      s ⊗ₜ[R] B.upper e.val (f (G.target e.val)) =
-      s ⊗ₜ[R] (B.lower e.val (f (G.source e.val)) -
-        B.upper e.val (f (G.target e.val)))
-    exact (tmul_sub _ _ _).symm
-  | add z w hz hw => simp only [map_add,hz,hw]; rfl
-
- theorem tensorSections_surjective [Finite V] [Finite E] [Module.Flat R S] (Ω : Set V) :
-    Function.Surjective (B.tensorSections S Ω) := by
-  classical
-  let := Fintype.ofFinite V
-  let := Fintype.ofFinite E
-  intro f
-  obtain ⟨z,hz⟩ := (B.tensorAssignments_bijective S).2 f.val
-  have hc : (B.baseChange S).constraint Ω (B.tensorAssignments S z) = 0 := by
-    rw [hz]
-    apply LinearMap.mem_ker.mp
-    rw [(B.baseChange S).ker_constraint Ω]
-    exact f.property
-  have he : AlgebraTensorModule.lTensor S S (B.constraint Ω) z = 0 := by
-    apply (TensorProduct.piRight R S S
-      (fun e : {e : E // G.source e ∈ Ω ∧ G.target e ∈ Ω} => B.edge e.val)).injective
-    rw [map_zero]
-    exact (B.constraint_tensorAssignments S Ω z).symm.trans hc
-  have hex : Function.Exact (B.sections Ω).subtype (B.constraint Ω) := by
-    rw [← B.ker_constraint Ω]
-    exact (B.constraint Ω).exact_subtype_ker_map
-  obtain ⟨w,hw⟩ := (Module.Flat.lTensor_exact S hex z).mp he
-  refine ⟨w,?_⟩
-  apply Subtype.ext
-  change B.tensorAssignments S (AlgebraTensorModule.lTensor S S (B.sections Ω).subtype w) = _
-  exact (congrArg (B.tensorAssignments S) hw).trans hz
-
- theorem tensorSections_injective [Finite V] [Module.Flat R S] (Ω : Set V) :
-    Function.Injective (B.tensorSections S Ω) := by
-  intro x y h
-  apply Module.Flat.lTensor_preserves_injective_linearMap
-    (B.sections Ω).subtype (B.sections Ω).injective_subtype
-  apply (B.tensorAssignments_bijective S).1
-  exact congrArg Subtype.val h
-
-/-- Sections over every vertex subset commute with flat extension of scalars
-for a finite moment graph. No freeness or BMP character input is used. -/
- def tensorSectionsEquiv [Finite V] [Finite E] [Module.Flat R S] (Ω : Set V) :
-    S ⊗[R] B.sections Ω ≃ₗ[S] (B.baseChange S).sections Ω :=
-  LinearEquiv.ofBijective (B.tensorSections S Ω)
-    ⟨B.tensorSections_injective S Ω,B.tensorSections_surjective S Ω⟩
-
- theorem baseChange_flabby [Finite V] [Finite E] [Module.Flat R S]
-    (hb : B.Flabby) : (B.baseChange S).Flabby := by
-  intro Ω hΩ f hf
-  obtain ⟨z,hz⟩ := B.tensorSections_surjective S Ω ⟨f,hf⟩
-  suffices ∃ g : (B.baseChange S).Assignment, (B.baseChange S).IsGlobal g ∧
-      ∀ x ∈ Ω, g x = (B.tensorSections S Ω z).val x by
-    simpa only [hz] using this
-  clear hz f hf
-  induction z using TensorProduct.inductionOn with
-  | tmul s f =>
-    obtain ⟨g,hg,hgf⟩ := hb Ω hΩ f.val f.property
-    refine ⟨fun x => s ⊗ₜ g x,B.baseChange_tmul_compatible S Set.univ s g hg,?_⟩
-    intro x hx
-    change s ⊗ₜ[R] g x = s ⊗ₜ[R] f.val x
-    rw [hgf x hx]
-  | add z w hz hw =>
-    obtain ⟨f,hf,hfz⟩ := hz
-    obtain ⟨g,hg,hgw⟩ := hw
-    refine ⟨f+g,Submodule.add_mem ((B.baseChange S).sections Set.univ) hf hg,?_⟩
-    intro x hx
-    simp only [map_add,Submodule.coe_add,Pi.add_apply]
-    rw [hfz x hx,hgw x hx]
-
-end
-end KLInvariance.MomentGraph.Sheaf
-
-end
-
-
-section
-
-
-namespace KLInvariance.MomentGraph.Sheaf
-open TensorProduct
-universe ur us uv ue um un
-variable {R : Type ur} [CommRing R] {V : Type uv} [PartialOrder V]
-  {E : Type ue} {G : OrderedGraph V E}
-noncomputable section
-
-namespace Iso
-variable {B : Sheaf.{ur,uv,ue,um} (R := R) G}
-  {C : Sheaf.{ur,uv,ue,un} (R := R) G} (j : Iso B C)
-include j
-
- theorem flabby (hb : B.Flabby) : C.Flabby := by
-  intro Ω hΩ f hf
-  have hsf : B.CompatibleOn Ω (j.symm.assignments f) :=
-    (j.symm.compatible_iff Ω f).mpr hf
-  obtain ⟨g,hg,hgf⟩ := hb Ω hΩ (j.symm.assignments f) hsf
-  refine ⟨j.assignments g,(j.compatible_iff Set.univ g).mpr hg,?_⟩
-  intro x hx
-  change j.vertex x (g x) = f x
-  rw [hgf x hx]
-  exact (j.vertex x).apply_symm_apply _
-
- theorem generated (hb : B.Generated) : C.Generated := by
-  intro x v
-  obtain ⟨f,hf,hfv⟩ := hb x ((j.vertex x).symm v)
-  refine ⟨j.assignments f,(j.compatible_iff Set.univ f).mpr hf,?_⟩
-  change j.vertex x (f x) = v
-  rw [hfv,LinearEquiv.apply_symm_apply]
-
-end Iso
-
-variable (S : Type us) [CommRing S] [Algebra R S]
-
-def baseChangeScalar : S ⊗[R] R ≃ₗ[S] S := AlgebraTensorModule.rid R S S
-
-@[simp] theorem baseChangeScalar_tmul (s : S) (r : R) :
-    baseChangeScalar S (s ⊗ₜ[R] r) = s * algebraMap R S r := by
-  change r • s = s * algebraMap R S r
-  rw [Algebra.smul_def,mul_comm]
-
- theorem span_map (a : R) :
-    (Ideal.span {a}).map (algebraMap R S) = Ideal.span {algebraMap R S a} := by
-  rw [Ideal.map_span,Set.image_singleton]
-
- def baseChangeQuotient (a : R) :
-    S ⊗[R] (R ⧸ Ideal.span {a}) ≃ₗ[S] S ⧸ Ideal.span {algebraMap R S a} :=
-  (Algebra.TensorProduct.quotIdealMapEquivTensorQuot S (Ideal.span {a})).symm.toLinearEquiv.trans
-    (Ideal.quotientEquivAlgOfEq S (span_map S a)).toLinearEquiv
-
-@[simp] theorem baseChangeQuotient_tmul (a r : R) (s : S) :
-    baseChangeQuotient S a (s ⊗ₜ[R] Ideal.Quotient.mk _ r) =
-      Ideal.Quotient.mk _ (s * algebraMap R S r) := by
-  simp only [baseChangeQuotient,LinearEquiv.trans_apply,AlgEquiv.toLinearEquiv_apply,
-    Algebra.TensorProduct.quotIdealMapEquivTensorQuot_symm_tmul]
-  change Ideal.Quotient.mk _ (r • s) = Ideal.Quotient.mk _ (s * algebraMap R S r)
-  rw [Algebra.smul_def,mul_comm]
-
-
- def baseChangePiScalar (p : Prop) :
-    S ⊗[R] (PLift p → R) ≃ₗ[S] (PLift p → S) := by
-  classical
-  exact (TensorProduct.piRight R S S (fun _ : PLift p => R)).trans
-    (LinearEquiv.piCongrRight (fun _ => baseChangeScalar S))
-
- def baseChangePiQuotient (a : R) (p : Prop) :
-    S ⊗[R] (PLift p → R ⧸ Ideal.span {a}) ≃ₗ[S]
-      (PLift p → S ⧸ Ideal.span {algebraMap R S a}) := by
-  classical
-  exact (TensorProduct.piRight R S S
-    (fun _ : PLift p => R ⧸ Ideal.span {a})).trans
-      (LinearEquiv.piCongrRight (fun _ => baseChangeQuotient S a))
-
- theorem baseChangePiScalar_tmul (p : Prop) (s : S) (v : PLift p → R) (h : PLift p) :
-    baseChangePiScalar S p (s ⊗ₜ v) h = s * algebraMap R S (v h) :=
-  baseChangeScalar_tmul S s (v h)
-
- theorem baseChangePiQuotient_tmul (a : R) (p : Prop)
-    (s : S) (v : PLift p → R ⧸ Ideal.span {a}) (h : PLift p) :
-    baseChangePiQuotient S a p (s ⊗ₜ v) h = baseChangeQuotient S a (s ⊗ₜ v h) := rfl
-
- def quotientPullback (a : R) (p q : Prop) (h : q → p) :
-    (PLift p → R) →ₗ[R] (PLift q → R ⧸ Ideal.span {a}) :=
-  LinearMap.pi fun t => (Ideal.Quotient.mkₐ R (Ideal.span {a})).toLinearMap.comp
-    (LinearMap.proj ⟨h t.down⟩)
-
- theorem baseChangePi_natural (a : R) (p q : Prop) (h : q → p)
-    (v : S ⊗[R] (PLift p → R)) :
-    baseChangePiQuotient S a q
-      (AlgebraTensorModule.lTensor S S (quotientPullback a p q h) v) =
-      quotientPullback (algebraMap R S a) p q h (baseChangePiScalar S p v) := by
-  induction v using TensorProduct.inductionOn with
-  | tmul s v =>
-    funext hh
-    change baseChangePiQuotient S a q (s ⊗ₜ quotientPullback a p q h v) hh = _
-    rw [baseChangePiQuotient_tmul]
-    change baseChangeQuotient S a (s ⊗ₜ Ideal.Quotient.mk _ (v ⟨h hh.down⟩)) = _
-    rw [baseChangeQuotient_tmul]
-    exact congrArg (Ideal.Quotient.mk _) (baseChangePiScalar_tmul S p s v _).symm
-  | add v w hv hw => simp only [map_add,hv,hw]
-
- def structureBaseChangeIso (G : OrderedGraph V E) (α : E → R) (c : V) :
-    Iso ((structureSheaf G α c).baseChange S)
-      (structureSheaf G (fun e => algebraMap R S (α e)) c) where
-  vertex x := baseChangePiScalar S (x ≤ c)
-  edge e := baseChangePiQuotient S (α e) (G.target e ≤ c)
-  lower e v := baseChangePi_natural S (α e) (G.source e ≤ c) (G.target e ≤ c)
-    ((G.increasing e).le.trans ·) v
-  upper e v := baseChangePi_natural S (α e) (G.target e ≤ c) (G.target e ≤ c) id v
-
- theorem structureSheaf_flat_flabby [Finite V] [Finite E] [Module.Flat R S]
-    (G : OrderedGraph V E) (α : E → R) (c : V) (hf : (structureSheaf G α c).Flabby) :
-    (structureSheaf G (fun e => algebraMap R S (α e)) c).Flabby :=
-  (structureBaseChangeIso S G α c).flabby ((structureSheaf G α c).baseChange_flabby S hf)
-
-end
-end KLInvariance.MomentGraph.Sheaf
-
-end
-
-
-section
-
-/-! Restriction to a full vertex subgraph, including the relative-flabbiness
-criterion used after plane localization. -/
-namespace KLInvariance.MomentGraph
-universe ur uv ue um
-variable {V : Type uv} [PartialOrder V] {E : Type ue} (G : OrderedGraph V E)
-
- def inducedGraph (K : Set V) : OrderedGraph K {e : E // G.source e ∈ K ∧ G.target e ∈ K} where
-  source e := ⟨G.source e.val,e.property.1⟩
-  target e := ⟨G.target e.val,e.property.2⟩
-  increasing e := G.increasing e.val
-
-namespace Sheaf
-variable {G} {R : Type ur} [CommRing R] (B : Sheaf.{ur,uv,ue,um} (R := R) G)
-  (K : Set V)
-noncomputable section
-
- def restrict : Sheaf (R := R) (inducedGraph G K) where
-  vertex x := B.vertex x.val
-  edge e := B.edge e.val
-  lower e := B.lower e.val
-  upper e := B.upper e.val
-
- def restrictAssignment (f : B.Assignment) : (B.restrict K).Assignment := fun x => f x.val
-
- def extendAssignment (f : (B.restrict K).Assignment) : B.Assignment := by
-  classical
-  exact fun x => if h : x ∈ K then f ⟨x,h⟩ else 0
-
-@[simp] theorem extendAssignment_mem (f : (B.restrict K).Assignment) (x : K) :
-    B.extendAssignment K f x.val = f x := by
-  classical
-  change (if h : x.val ∈ K then f ⟨x.val,h⟩ else 0) = f x
-  split_ifs with h
-  · rfl
-  · exact (h x.property).elim
-
- theorem restrict_compatible {Ω : Set V} {f : B.Assignment} (hf : B.CompatibleOn Ω f) :
-    (B.restrict K).CompatibleOn (Subtype.val ⁻¹' Ω) (B.restrictAssignment K f) := by
-  intro e hs ht
-  exact hf e.val hs ht
-
- theorem restrict_global (f : B.Assignment) (hf : B.IsGlobal f) :
-    (B.restrict K).IsGlobal (B.restrictAssignment K f) := by
-  intro e _ _
-  exact hf e.val (Set.mem_univ _) (Set.mem_univ _)
-
- theorem restrict_generated (hg : B.Generated) : (B.restrict K).Generated := by
-  intro x v
-  obtain ⟨f,hf,hfx⟩ := hg x.val v
-  exact ⟨B.restrictAssignment K f,B.restrict_global K f hf,hfx⟩
-
- theorem extend_compatible (Ω : Set K) (f : (B.restrict K).Assignment)
-    (hf : (B.restrict K).CompatibleOn Ω f) :
-    B.CompatibleOn (Subtype.val '' Ω) (B.extendAssignment K f) := by
-  intro e hs ht
-  obtain ⟨x,hx,hxe⟩ := hs
-  obtain ⟨y,hy,hye⟩ := ht
-  have hsk : G.source e ∈ K := hxe ▸ x.property
-  have htk : G.target e ∈ K := hye ▸ y.property
-  let ee : {e : E // G.source e ∈ K ∧ G.target e ∈ K} := ⟨e,hsk,htk⟩
-  have hsx : (inducedGraph G K).source ee = x := Subtype.ext hxe.symm
-  have hty : (inducedGraph G K).target ee = y := Subtype.ext hye.symm
-  have hh := hf ee (hsx ▸ hx) (hty ▸ hy)
-  change B.lower e (f ⟨G.source e,hsk⟩) = B.upper e (f ⟨G.target e,htk⟩) at hh
-  simpa only [← B.extendAssignment_mem K f] using hh
-
-/-- The induced subgraph may use upper sets which are not ambient upper
-sets. All nonzero exiting edges must stay in the subgraph. -/
- theorem restrict_flabby_live [Finite V] (hb : B.Flabby) (live : Set E)
-    (hdead : ∀ e, e ∉ live → Subsingleton (B.edge e))
-    (hclosed : ∀ e ∈ live, G.source e ∈ K → G.target e ∈ K) :
-    (B.restrict K).Flabby := by
-  intro Ω hΩ f hf
-  have hc : B.ClosedUnder live (Subtype.val '' Ω) := by
-    intro e he hs
-    obtain ⟨x,hx,hxe⟩ := hs
-    have hsk : G.source e ∈ K := hxe ▸ x.property
-    have htk := hclosed e he hsk
-    refine ⟨⟨G.target e,htk⟩,?_,rfl⟩
-    apply hΩ _ hx
-    change x.val ≤ G.target e
-    rw [hxe]
-    exact (G.increasing e).le
-  obtain ⟨g,hg,hgf⟩ := B.extend_closedUnder hb live hdead (Subtype.val '' Ω) hc
-    (B.extendAssignment K f) (B.extend_compatible K Ω f hf)
-  refine ⟨B.restrictAssignment K g,B.restrict_global K g hg,?_⟩
-  intro x hx
-  exact (hgf x.val ⟨x,hx,rfl⟩).trans (B.extendAssignment_mem K f x)
-
- theorem restrict_quotient (α : E → R) (hB : B.UpperQuotient α) :
-    (B.restrict K).UpperQuotient (fun e => α e.val) := fun e => hB e.val
-
-end
-end Sheaf
-end KLInvariance.MomentGraph
-
-namespace KLInvariance.MomentGraph.Sheaf
-universe ur uv ue
-variable {R : Type ur} [CommRing R] {V : Type uv} [PartialOrder V]
-  {E : Type ue} (G : OrderedGraph V E) (α : E → R) (K : Set V)
-noncomputable section
-
- def structureRestrictIso (c : K) :
-    Iso ((structureSheaf G α c.val).restrict K)
-      (structureSheaf (inducedGraph G K) (fun e => α e.val) c) where
-  vertex _ := LinearEquiv.refl R _
-  edge _ := LinearEquiv.refl R _
-  lower _ _ := rfl
-  upper _ _ := rfl
-
- theorem structureSheaf_edge_subsingleton (c : V) (e : E) (he : ¬ G.target e ≤ c) :
-    Subsingleton ((structureSheaf G α c).edge e) :=
-  ⟨fun _ _ => funext (fun h => False.elim (he h.down))⟩
-
- theorem structureSheaf_lowerIdeal_flabby [Finite V] (hK : IsLowerSet K) (c : K)
-    (hf : (structureSheaf G α c.val).Flabby) :
-    (structureSheaf (inducedGraph G K) (fun e => α e.val) c).Flabby := by
-  apply (structureRestrictIso G α K c).flabby
-  apply (structureSheaf G α c.val).restrict_flabby_live K hf {e | G.target e ∈ K}
-  · intro e he
-    exact structureSheaf_edge_subsingleton G α c.val e (fun hc => he (hK hc c.property))
-  · intro e he _
-    exact he
-
-end
-end KLInvariance.MomentGraph.Sheaf
-
-end
-
-
-section
-
-/-! Pullback along a directed graph map. Relative-order flabbiness requires
-only outgoing completeness for the surviving edges, not equality of the two
-orders on a coset. -/
-namespace KLInvariance.MomentGraph
-universe ur uv ue uw uf um
-variable {V : Type uv} [PartialOrder V] {E : Type ue}
-  {W : Type uw} [PartialOrder W] {F : Type uf}
-
-structure GraphMap (G : OrderedGraph V E) (H : OrderedGraph W F) where
-  vertex : V → W
-  edge : E → F
-  source : ∀ e, vertex (G.source e) = H.source (edge e)
-  target : ∀ e, vertex (G.target e) = H.target (edge e)
-
-namespace Sheaf
-variable {R : Type ur} [CommRing R] {G : OrderedGraph V E} {H : OrderedGraph W F}
-  (B : Sheaf.{ur,uw,uf,um} (R := R) H) (f : GraphMap G H)
-noncomputable section
-
- def moduleEq {M N : ModuleCat.{um} R} (h : M=N) : M ≃ₗ[R] N := h ▸ LinearEquiv.refl R M
-
- theorem moduleEq_apply {X : Type*} (M : X → ModuleCat.{um} R) {x y : X} (h : x=y)
-    (s : ∀ z, M z) : moduleEq (congrArg M h) (s x) = s y := by subst y; rfl
-
- def pullback : Sheaf (R := R) G where
-  vertex x := B.vertex (f.vertex x)
-  edge e := B.edge (f.edge e)
-  lower e := (B.lower (f.edge e)).comp (moduleEq (congrArg B.vertex (f.source e))).toLinearMap
-  upper e := (B.upper (f.edge e)).comp (moduleEq (congrArg B.vertex (f.target e))).toLinearMap
-
- def pullAssignment (s : B.Assignment) : (B.pullback f).Assignment := fun x => s (f.vertex x)
-
-@[simp] theorem pull_lower (e : E) (s : B.Assignment) :
-    (B.pullback f).lower e (B.pullAssignment f s (G.source e)) =
-      B.lower (f.edge e) (s (H.source (f.edge e))) := by
-  exact congrArg (B.lower (f.edge e)) (moduleEq_apply B.vertex (f.source e) s)
-
-@[simp] theorem pull_upper (e : E) (s : B.Assignment) :
-    (B.pullback f).upper e (B.pullAssignment f s (G.target e)) =
-      B.upper (f.edge e) (s (H.target (f.edge e))) := by
-  exact congrArg (B.upper (f.edge e)) (moduleEq_apply B.vertex (f.target e) s)
-
- theorem pull_global (s : B.Assignment) (hs : B.IsGlobal s) :
-    (B.pullback f).IsGlobal (B.pullAssignment f s) := by
-  intro e _ _
-  rw [pull_lower,pull_upper]
-  exact hs (f.edge e) (Set.mem_univ _) (Set.mem_univ _)
-
- theorem pullback_generated (hg : B.Generated) : (B.pullback f).Generated := by
-  intro x v
-  obtain ⟨s,hs,hx⟩ := hg (f.vertex x) v
-  exact ⟨B.pullAssignment f s,B.pull_global f s hs,hx⟩
-
- def pushAssignment (s : (B.pullback f).Assignment) : B.Assignment := by
-  classical
-  exact fun y => if h : ∃ x, f.vertex x=y then
-    moduleEq (congrArg B.vertex h.choose_spec) (s h.choose) else 0
-
- theorem pull_push (hi : Function.Injective f.vertex) (s : (B.pullback f).Assignment) :
-    B.pullAssignment f (B.pushAssignment f s) = s := by
-  classical
-  change (∀ x, B.vertex (f.vertex x)) at s
-  funext x
-  have h : ∃ z, f.vertex z=f.vertex x := ⟨x,rfl⟩
-  change @Eq (B.vertex (f.vertex x))
-    (if hh : ∃ z, f.vertex z=f.vertex x then
-      moduleEq (congrArg B.vertex hh.choose_spec) (s hh.choose) else 0) (s x)
-  rw [dite_eq_left h]
-  have hx : h.choose=x := hi h.choose_spec
-  exact moduleEq_apply (fun z => B.vertex (f.vertex z)) hx s
-
- theorem pullback_flabby_live [Finite W] (hb : B.Flabby)
-    (hi : Function.Injective f.vertex) (live : Set F)
-    (hdead : ∀ e, e ∉ live → Subsingleton (B.edge e))
-    (hlift : ∀ e ∈ live, ∀ x, f.vertex x = H.source e →
-      ∃ a, f.edge a=e ∧ G.source a=x) : (B.pullback f).Flabby := by
-  intro Ω hΩ s hs
-  have hc : B.ClosedUnder live (f.vertex '' Ω) := by
-    intro e he hsource
-    obtain ⟨x,hx,hxe⟩ := hsource
-    obtain ⟨a,rfl,rfl⟩ := hlift e he x hxe
-    exact ⟨G.target a,hΩ (G.increasing a).le hx,f.target a⟩
-  have hcompat : B.CompatibleOn (f.vertex '' Ω) (B.pushAssignment f s) := by
-    intro e hsource htarget
-    by_cases he : e ∈ live
-    · obtain ⟨x,hx,hxe⟩ := hsource
-      obtain ⟨a,rfl,rfl⟩ := hlift e he x hxe
-      obtain ⟨y,hy,hye⟩ := htarget
-      have hya : y=G.target a := hi (hye.trans (f.target a).symm)
-      subst y
-      have hh := hs a hx hy
-      rw [← B.pull_push f hi s] at hh
-      change (B.pullback f).lower a (B.pullAssignment f (B.pushAssignment f s) (G.source a)) =
-        (B.pullback f).upper a (B.pullAssignment f (B.pushAssignment f s) (G.target a)) at hh
-      rw [pull_lower,pull_upper] at hh
-      exact hh
-    · exact @Subsingleton.elim _ (hdead e he) _ _
-  obtain ⟨t,ht,hts⟩ := B.extend_closedUnder hb live hdead (f.vertex '' Ω) hc
-    (B.pushAssignment f s) hcompat
-  refine ⟨B.pullAssignment f t,B.pull_global f t ht,?_⟩
-  intro x hx
-  change t (f.vertex x) = s x
-  rw [hts (f.vertex x) ⟨x,hx,rfl⟩]
-  exact congrFun (B.pull_push f hi s) x
-
- theorem pullback_quotient (α : F → R) (hq : B.UpperQuotient α) :
-    (B.pullback f).UpperQuotient (fun e => α (f.edge e)) := by
-  intro e
-  let j := moduleEq (congrArg B.vertex (f.target e))
-  refine ⟨?_,?_⟩
-  · exact (hq (f.edge e)).1.comp j.surjective
-  · intro v
-    change B.upper (f.edge e) (j v) = 0 ↔ _
-    rw [(hq (f.edge e)).2]
-    constructor
-    · rintro ⟨w,hw⟩
-      refine ⟨j.symm w,?_⟩
-      apply j.injective
-      change j v = j (α (f.edge e) • j.symm w)
-      rw [map_smul,LinearEquiv.apply_symm_apply]
-      exact hw
-    · rintro ⟨w,rfl⟩
-      exact ⟨j w,j.map_smul _ _⟩
-
-end
-end Sheaf
-end KLInvariance.MomentGraph
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.Sheaf
-universe ur uv ue uw uf um
-variable {R : Type ur} [CommRing R]
-  {V : Type uv} [PartialOrder V] {E : Type ue} {G : OrderedGraph V E}
-  {W : Type uw} [PartialOrder W] {F : Type uf} {H : OrderedGraph W F}
-  (B : Sheaf.{ur,uw,uf,um} (R := R) H) (f : GraphMap G H)
-noncomputable section
-
- theorem stalkMap_assignment (s : B.Assignment) (x : W) (e : Outgoing (G := H) x) :
-    B.stalkMap x (s x) e = B.lower e.val (s (H.source e.val)) := by
-  rcases e with ⟨e,he⟩
-  subst x
-  rfl
-
- theorem pullback_kernel (x : V) (T : Set F)
-    (live : Set F) (hdead : ∀ e, e ∉ live → Subsingleton (B.edge e))
-    (hlift : ∀ e ∈ live, f.vertex x = H.source e →
-      ∃ a, f.edge a=e ∧ G.source a=x) :
-    (B.pullback f).upwardKernel x {e | f.edge e.val ∈ T} =
-      B.upwardKernel (f.vertex x) {e | e.val ∈ T} := by
-  classical
-  ext v
-  change B.vertex (f.vertex x) at v
-  let s : B.Assignment := Function.update (fun _ => 0) (f.vertex x) v
-  have hs : s (f.vertex x)=v := Function.update_self (β := fun y => B.vertex y) (f.vertex x) v (fun _ => 0)
-  have hxs : B.pullAssignment f s x = v := hs
-  constructor
-  · intro hv e he
-    rw [← hs,B.stalkMap_assignment s]
-    by_cases hl : e.val ∈ live
-    · obtain ⟨a,ha,hax⟩ := hlift e.val hl e.property.symm
-      have hh := hv ⟨a,hax⟩ (show f.edge a ∈ T from ha.symm ▸ he)
-      rw [← hxs,(B.pullback f).stalkMap_assignment (B.pullAssignment f s)] at hh
-      rw [B.pull_lower f] at hh
-      exact (congrArg (fun q : F => B.lower q (s (H.source q)) = 0) ha).mp hh
-    · exact @Subsingleton.elim _ (hdead e.val hl) _ _
-  · intro hv e he
-    have hh := hv ⟨f.edge e.val,(f.source e.val).symm.trans (congrArg f.vertex e.property)⟩ he
-    rw [← hs,B.stalkMap_assignment s] at hh
-    rw [← hxs,(B.pullback f).stalkMap_assignment (B.pullAssignment f s),B.pull_lower f]
-    exact hh
-
-end
-end KLInvariance.MomentGraph.Sheaf
-
-end
-
-
-section
-
-/-! Restriction along an outgoing-complete actual graph embedding.  In
-particular this supports passage from [1,b] to [u,b] without changing stalks. -/
-namespace KLInvariance.MomentGraph
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A}
-  {V E W F : Type um} [PartialOrder V] [PartialOrder W]
-  {G : OrderedGraph V E} {H : OrderedGraph W F}
-namespace GradedSheaf
-variable (B : GradedSheaf (𝓐 := 𝓐) H) (f : GraphMap G H)
-
- theorem moduleEq_graded {x y : W} (h : x=y) (n : ℤ)
-    (v : B.vertex x) (hv : v ∈ (B.vertex x).piece n) :
-    Sheaf.moduleEq (congrArg B.forget.vertex h) v ∈ (B.vertex y).piece n := by
-  subst y
-  exact hv
-
- def pullback : GradedSheaf (𝓐 := 𝓐) G where
-  vertex x := B.vertex (f.vertex x)
-  edge e := B.edge (f.edge e)
-  lower e := ⟨(B.forget.pullback f).lower e,fun n v hv =>
-    (B.lower (f.edge e)).graded n _ (B.moduleEq_graded (f.source e) n v hv)⟩
-  upper e := ⟨(B.forget.pullback f).upper e,fun n v hv =>
-    (B.upper (f.edge e)).graded n _ (B.moduleEq_graded (f.target e) n v hv)⟩
-
-@[simp] theorem forget_pullback : (B.pullback f).forget = B.forget.pullback f := rfl
-
- theorem pullback_edgeFlabby (hb : B.EdgeFlabby)
-    (hi : Function.Injective f.vertex)
-    (hlift : ∀ e x, f.vertex x = H.source e → ∃ a, f.edge a=e ∧ G.source a=x) :
-    (B.pullback f).EdgeFlabby := by
-  intro Ω hΩ s hs
-  have hc : B.forget.ClosedUnder Set.univ (f.vertex '' Ω) := by
-    intro e _ hsource
-    obtain ⟨x,hx,hxe⟩ := hsource
-    obtain ⟨a,rfl,rfl⟩ := hlift e x hxe
-    exact ⟨G.target a,hΩ a (Set.mem_univ a) hx,f.target a⟩
-  have hcompat : B.forget.CompatibleOn (f.vertex '' Ω) (B.forget.pushAssignment f s) := by
-    intro e hsource htarget
-    obtain ⟨x,hx,hxe⟩ := hsource
-    obtain ⟨a,rfl,rfl⟩ := hlift e x hxe
-    obtain ⟨y,hy,hye⟩ := htarget
-    have hya : y=G.target a := hi (hye.trans (f.target a).symm)
-    subst y
-    have hh := hs a hx hy
-    change (B.forget.pullback f).lower a (s (G.source a)) =
-      (B.forget.pullback f).upper a (s (G.target a)) at hh
-    rw [← B.forget.pull_push f hi s] at hh
-    rw [Sheaf.pull_lower,Sheaf.pull_upper] at hh
-    exact hh
-  obtain ⟨t,ht,hts⟩ := hb (f.vertex '' Ω) hc (B.forget.pushAssignment f s) hcompat
-  refine ⟨B.forget.pullAssignment f t,B.forget.pull_global f t ht,?_⟩
-  intro x hx
-  change t (f.vertex x) = s x
-  rw [hts (f.vertex x) ⟨x,hx,rfl⟩]
-  exact congrFun (B.forget.pull_push f hi s) x
-
- theorem pullback_kernelMinimal (hi : Function.Injective f.vertex)
-    (hlift : ∀ e x, f.vertex x = H.source e → ∃ a, f.edge a=e ∧ G.source a=x)
-    (x : V) (hx : B.KernelMinimal (f.vertex x)) :
-    (B.pullback f).KernelMinimal x := by
-  intro s hs
-  have h := hx (B.forget.pushAssignment f s) (by
-    intro e he
-    obtain ⟨a,rfl,hax⟩ := hlift e x he.symm
-    have hh := hs a hax
-    change (B.forget.pullback f).lower a (s (G.source a)) = 0 at hh
-    rw [← B.forget.pull_push f hi s,Sheaf.pull_lower] at hh
-    exact hh)
-  have he : B.forget.pushAssignment f s (f.vertex x)=s x :=
-    congrFun (B.forget.pull_push f hi s) x
-  rwa [he] at h
-
-end GradedSheaf
-
-namespace BoundarySheaf
-variable [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-  {hneg : ∀ n < 0, 𝓐 n = ⊥} {α : F → A} {b : W}
-  (B : BoundarySheaf hneg H α b) (f : GraphMap G H)
-  (hi : Function.Injective f.vertex)
-  (hlift : ∀ e x, f.vertex x = H.source e → ∃ a, f.edge a=e ∧ G.source a=x)
-  (c : V) (hc : f.vertex c=b)
-
- def pullback : BoundarySheaf hneg G (fun e => α (f.edge e)) c where
-  sheaf := B.sheaf.pullback f
-  free x := B.free (f.vertex x)
-  quotient := B.sheaf.forget.pullback_quotient f α B.quotient
-  flabby := B.sheaf.pullback_edgeFlabby f B.flabby hi hlift
-  generated := B.sheaf.forget.pullback_generated f B.generated
-  top := by change B.sheaf.vertex (f.vertex c)=_; rw [hc,B.top]
-  minimal x hx := B.sheaf.pullback_kernelMinimal f hi hlift x
-    (B.minimal (f.vertex x) (fun h => hx (hi (h.trans hc.symm))))
-
-end BoundarySheaf
-end KLInvariance.MomentGraph
-
-end
-
-
-section
-
-/-! The actual polynomial character of a nonnegative finite graded free module.
-This is not a definition of KL polynomials, and includes no character identity. -/
-
-end
-
-end OAI
+import OAI.RepresentationTheory.KazhdanLusztig.HilbertSeries
+import OAI.RepresentationTheory.KazhdanLusztig.Comparison
+
+namespace OAI
+
+/-!
+Minimal-boundary extension sheaves, graded splittings and sheaf decomposition.
+-/
+
+section
+
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded _root_.OAI.KLInvariance.Hilbert
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
+  {G : OrderedGraph V E} (B : GradedSheaf (𝓐 := 𝓐) G)
+
+/-- The actual lower restriction, with its source explicitly identified. -/
+def lowerAt (x : V) (e : Sheaf.Outgoing (G := G) x) :
+    ModuleData.Hom (B.vertex x) (B.edge e.val) := by
+  rcases e with ⟨e,rfl⟩
+  exact B.lower e
+
+ theorem lowerAt_map (x : V) (e : Sheaf.Outgoing (G := G) x) (v : B.vertex x) :
+    (B.lowerAt x e).map v = B.forget.stalkMap x v e := by
+  rcases e with ⟨e,rfl⟩
+  rfl
+
+/-- The same kernel, typed in the graded stalk so base-field instances are explicit. -/
+def upwardSubmodule (x : V) (F : Set (Sheaf.Outgoing (G := G) x)) :
+    Submodule A (B.vertex x) := B.forget.upwardKernel x F
+
+ theorem upwardKernel_homogeneous (x : V) (F : Set (Sheaf.Outgoing (G := G) x)) :
+    ∀ n v, v ∈ B.upwardSubmodule x F →
+      component (B.vertex x).piece n v ∈ B.upwardSubmodule x F := by
+  intro n v hv e he
+  rw [← B.lowerAt_map]
+  rw [← (B.lowerAt x e).component]
+  rw [B.lowerAt_map,hv e he]
+  exact map_zero (component (B.edge e.val).piece n)
+
+noncomputable def upwardKernel [IsNoetherianRing A]
+    (x : V) (F : Set (Sheaf.Outgoing (G := G) x)) : ModuleData.{uk,ua,um} 𝓐 :=
+  ModuleData.sub (B.vertex x) (B.upwardSubmodule x F) (B.upwardKernel_homogeneous x F)
+
+ theorem upwardKernel_insert (x : V) (F : Set (Sheaf.Outgoing (G := G) x))
+    (e : Sheaf.Outgoing (G := G) x) :
+    B.upwardSubmodule x (insert e F) =
+      B.upwardSubmodule x F ⊓ LinearMap.ker (B.lowerAt x e).map := by
+  ext v
+  change (∀ a ∈ insert e F, B.forget.stalkMap x v a = 0) ↔
+    (∀ a ∈ F, B.forget.stalkMap x v a = 0) ∧ (B.lowerAt x e).map v = 0
+  rw [B.lowerAt_map]
+  simp only [Set.mem_insert_iff,forall_eq_or_imp]
+  exact and_comm
+
+
+ theorem hilbert_edge_increment
+    (x : V) (F : Set (Sheaf.Outgoing (G := G) x))
+    (e : Sheaf.Outgoing (G := G) x)
+    [∀ n, Module.Finite k ((B.vertex x).piece n)]
+    {q : ℝ} (hq : 0 < q)
+    (hs : Summable (term (B.vertex x).piece
+      ((B.upwardSubmodule x F).restrictScalars k) q)) :
+    value (B.vertex x).piece ((B.upwardSubmodule x F).restrictScalars k) q =
+      value (B.vertex x).piece
+        ((B.upwardSubmodule x (insert e F)).restrictScalars k) q +
+      value (B.edge e.val).piece
+        (((B.upwardSubmodule x F).map (B.lowerAt x e).map).restrictScalars k) q := by
+  rw [B.upwardKernel_insert]
+  exact value_image_exact (B.vertex x).piece (B.edge e.val).piece
+    (B.upwardSubmodule x F) (B.upwardKernel_homogeneous x F)
+    (B.lowerAt x e).map (B.lowerAt x e).graded hq hs
+
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+/-! Graded uniqueness/recognition for the genuinely constructed minimal
+boundary covers. Character and self-duality are not included in this data. -/
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
+  {G : OrderedGraph V E} [Fintype V] [Fintype E]
+  (B : GradedSheaf (𝓐 := 𝓐) G)
+
+noncomputable def stalkHom (x : V) : ModuleData.Hom (B.vertex x) (B.boundary x) where
+  map := B.forget.stalkMap x
+  graded := by
+    intro n v hv e _
+    change B.forget.stalkMap x v e ∈ (B.edge e.val).piece n
+    rw [← B.lowerAt_map]
+    exact (B.lowerAt x e).graded n v hv
+
+end KLInvariance.MomentGraph.GradedSheaf
+
+namespace KLInvariance.MomentGraph.Sheaf.AboveIso
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+  {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
+  [Fintype E] [IsNoetherianRing A] [IsDomain A]
+  (hneg : ∀ n < 0, 𝓐 n = ⊥)
+  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
+  {B C : GradedSheaf (𝓐 := 𝓐) G} {x : V}
+  (j : AboveIso B.forget C.forget x)
+
+include hneg hzero in
+ theorem extend_stalk_graded
+    (hj : ∀ e (h : x < G.target e) n v, v ∈ (B.edge e).piece n →
+      j.edge e h v ∈ (C.edge e).piece n)
+    (hBfree : Module.Free A (B.vertex x)) (hCfree : Module.Free A (C.vertex x))
+    (hBf : B.forget.Flabby) (hBg : B.forget.Generated)
+    (hCf : C.forget.Flabby) (hCg : C.forget.Generated)
+    (hBmin : B.KernelMinimal x) (hCmin : C.KernelMinimal x) :
+    ∃ e : B.vertex x ≃ₗ[A] C.vertex x,
+      (∀ n v, v ∈ (B.vertex x).piece n → e v ∈ (C.vertex x).piece n) ∧
+      ∀ v, j.boundary (B.forget.stalkMap x v) = C.forget.stalkMap x (e v) := by
+  let T := ModuleData.range (C.stalkHom x)
+  let f : ModuleData.Hom (B.vertex x) T :=
+    { map := LinearMap.codRestrict (LinearMap.range (C.forget.stalkMap x))
+        (j.boundary.toLinearMap.comp (B.forget.stalkMap x)) (by
+          intro v
+          rw [← j.map_stalk_image hBf hBg hCf hCg]
+          exact ⟨B.forget.stalkMap x v,⟨v,rfl⟩,rfl⟩)
+      graded := by
+        intro n v hv e _
+        exact hj e.val _ n _ ((B.stalkHom x).graded n v hv e (Set.mem_univ e)) }
+  let g : ModuleData.Hom (C.vertex x) T :=
+    { map := (C.forget.stalkMap x).rangeRestrict
+      graded := fun n v hv => (C.stalkHom x).graded n v hv }
+  have hfs : Function.Surjective f.map := by
+    intro v
+    have hv : v.val ∈ (LinearMap.range (B.forget.stalkMap x)).map j.boundary.toLinearMap := by
+      rw [j.map_stalk_image hBf hBg hCf hCg]
+      exact v.property
+    obtain ⟨w,⟨z,rfl⟩,hz⟩ := hv
+    exact ⟨z,Subtype.ext hz⟩
+  have hgs : Function.Surjective g.map := by
+    intro v
+    obtain ⟨w,hw⟩ := v.property
+    exact ⟨w,Subtype.ext hw⟩
+  have hfm : LinearMap.ker f.map ≤ positiveIdeal 𝓐 • (⊤ : Submodule A (B.vertex x)) := by
+    intro v hv
+    apply (B.kernelMinimal_iff x).mp hBmin
+    apply LinearMap.mem_ker.mpr
+    apply j.boundary.injective
+    rw [map_zero]
+    have hh : f.map v = 0 := LinearMap.mem_ker.mp hv
+    exact congrArg (fun z : T => z.val) hh
+  have hgm : LinearMap.ker g.map ≤ positiveIdeal 𝓐 • (⊤ : Submodule A (C.vertex x)) := by
+    intro v hv
+    apply (C.kernelMinimal_iff x).mp hCmin
+    have hh : g.map v = 0 := LinearMap.mem_ker.mp hv
+    exact congrArg (fun z : T => z.val) hh
+  let F : ModuleData.MinimalCover T := ⟨B.vertex x,hBfree,f,hfs,hfm⟩
+  let D : ModuleData.MinimalCover T := ⟨C.vertex x,hCfree,g,hgs,hgm⟩
+  obtain ⟨e,he,hef⟩ := ModuleData.MinimalCover.equiv hneg hzero F D
+  refine ⟨e,he,?_⟩
+  intro v
+  exact (congrArg Subtype.val (LinearMap.congr_fun hef v)).symm
+
+end KLInvariance.MomentGraph.Sheaf.AboveIso
+
+end
+
+
+section
+
+namespace KLInvariance.Graded.ModuleData
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A}
+
+def equivOfEq {M N : ModuleData.{uk,ua,um} 𝓐} (h : M=N) : M ≃ₗ[A] N :=
+  h ▸ LinearEquiv.refl A M
+
+ theorem equivOfEq_graded {M N : ModuleData.{uk,ua,um} 𝓐} (h : M=N)
+    (n : ℤ) (v : M) (hv : v ∈ M.piece n) : equivOfEq h v ∈ N.piece n := by
+  subst N
+  exact hv
+
+ theorem quotient_iso_graded {M N E F : ModuleData.{uk,ua,um} 𝓐}
+    (f : Hom M E) (g : Hom N F) (hf : Function.Surjective f.map)
+    (hg : Function.Surjective g.map) (e : M ≃ₗ[A] N)
+    (he : ∀ n v, v ∈ M.piece n → e v ∈ N.piece n)
+    (hk : (LinearMap.ker f.map).map e.toLinearMap = LinearMap.ker g.map)
+    (n : ℤ) (v : E) (hv : v ∈ E.piece n) :
+    QuotientEdge.iso f.map g.map hf hg e hk v ∈ F.piece n := by
+  obtain ⟨w,hw,rfl⟩ := f.homogeneous_preimage hf n v hv
+  rw [QuotientEdge.iso_apply]
+  exact g.graded n _ (he n w hw)
+
+end KLInvariance.Graded.ModuleData
+
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
+  {G : OrderedGraph V E}
+
+structure Iso (B C : GradedSheaf (𝓐 := 𝓐) G) extends Sheaf.Iso B.forget C.forget where
+  vertex_graded : ∀ x n v, v ∈ (B.vertex x).piece n → vertex x v ∈ (C.vertex x).piece n
+  edge_graded : ∀ e n v, v ∈ (B.edge e).piece n → edge e v ∈ (C.edge e).piece n
+
+structure OnIso (B C : GradedSheaf (𝓐 := 𝓐) G) (U : Set V)
+    extends Sheaf.OnIso B.forget C.forget U where
+  vertex_graded : ∀ x (hx : x ∈ U) n v,
+    v ∈ (B.vertex x).piece n → vertex x hx v ∈ (C.vertex x).piece n
+  edge_graded : ∀ e (he : G.target e ∈ U) n v,
+    v ∈ (B.edge e).piece n → edge e he v ∈ (C.edge e).piece n
+
+namespace OnIso
+variable {B C : GradedSheaf (𝓐 := 𝓐) G} {U : Set V} (j : OnIso B C U)
+
+ def ofUniv (j : OnIso B C Set.univ) : Iso B C where
+  toIso := j.toOnIso.ofUniv
+  vertex_graded x := j.vertex_graded x (Set.mem_univ x)
+  edge_graded e := j.edge_graded e (Set.mem_univ _)
+
+noncomputable def extend (hU : IsUpperSet U) (x : V) (hx : x ∉ U)
+    (hmax : Set.Ioi x ⊆ U) (α : E → A)
+    (hBq : B.UpperQuotient α) (hCq : C.UpperQuotient α)
+    (jv : B.vertex x ≃ₗ[A] C.vertex x)
+    (hjv : ∀ n v, v ∈ (B.vertex x).piece n → jv v ∈ (C.vertex x).piece n)
+    (hout : ∀ e (he : G.source e = x) v,
+      j.edge e (hmax (lt_of_eq_of_lt he.symm (G.increasing e)))
+        (B.forget.lower e v) = C.forget.lower e (Sheaf.castStalkIso B.forget C.forget he.symm jv v)) :
+    OnIso B C (insert x U) := by
+  classical
+  let p := j.toOnIso.extend hU x hx hmax α hBq hCq jv hout
+  have hvp : ∀ y (hy : y ∈ insert x U) n v,
+      v ∈ (B.vertex y).piece n → p.vertex y hy v ∈ (C.vertex y).piece n := by
+    intro y hy n v hv
+    dsimp only [p,Sheaf.OnIso.extend]
+    split_ifs with h
+    · subst y
+      exact hjv n v hv
+    · exact j.vertex_graded y _ n v hv
+  refine ⟨p,hvp,?_⟩
+  intro e he n v hv
+  dsimp only [p,Sheaf.OnIso.extend]
+  split_ifs with h
+  · apply ModuleData.quotient_iso_graded (B.upper e) (C.upper e) (hBq e).1 (hCq e).1
+    · subst x
+      exact hjv
+    · exact hv
+  · exact j.edge_graded e _ n v hv
+
+end OnIso
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.Sheaf
+universe u v w z
+variable {R : Type u} [CommRing R] {V : Type v} [PartialOrder V]
+  {E : Type w} {G : OrderedGraph V E}
+
+ def zero : Sheaf.{u,v,w,z} (R := R) G where
+  vertex _ := ModuleCat.of R PUnit.{z+1}
+  edge _ := ModuleCat.of R PUnit.{z+1}
+  lower _ := 0
+  upper _ := 0
+
+ def prod (B C : Sheaf.{u,v,w,z} (R := R) G) : Sheaf.{u,v,w,z} (R := R) G where
+  vertex x := ModuleCat.of R (B.vertex x × C.vertex x)
+  edge e := ModuleCat.of R (B.edge e × C.edge e)
+  lower e := (B.lower e).prodMap (C.lower e)
+  upper e := (B.upper e).prodMap (C.upper e)
+
+lemma prod_generated (B C : Sheaf.{u,v,w,z} (R := R) G)
+    (hB : B.Generated) (hC : C.Generated) : (prod B C).Generated := by
+  intro x v
+  obtain ⟨f,hf,hfx⟩ := hB x v.1
+  obtain ⟨g,hg,hgx⟩ := hC x v.2
+  exact ⟨fun y => (f y,g y),fun e hs ht => Prod.ext (hf e hs ht) (hg e hs ht),Prod.ext hfx hgx⟩
+
+lemma prod_flabby (B C : Sheaf.{u,v,w,z} (R := R) G)
+    (hB : B.Flabby) (hC : C.Flabby) : (prod B C).Flabby := by
+  intro U hU f hf
+  obtain ⟨g,hg,hgf⟩ := hB U hU (fun y => (f y).1) (fun e hs ht => congrArg Prod.fst (hf e hs ht))
+  obtain ⟨t,ht,htf⟩ := hC U hU (fun y => (f y).2) (fun e hs htt => congrArg Prod.snd (hf e hs htt))
+  exact ⟨fun y => (g y,t y),fun e hs htt => Prod.ext (hg e hs htt) (ht e hs htt),
+    fun y hy => Prod.ext (hgf y hy) (htf y hy)⟩
+
+lemma prod_upperQuotient (B C : Sheaf.{u,v,w,z} (R := R) G) (α : E → R)
+    (hB : B.UpperQuotient α) (hC : C.UpperQuotient α) : (prod B C).UpperQuotient α := by
+  intro e
+  refine ⟨?_,?_⟩
+  · intro v
+    obtain ⟨b,hb⟩ := (hB e).1 v.1
+    obtain ⟨c,hc⟩ := (hC e).1 v.2
+    exact ⟨(b,c),Prod.ext hb hc⟩
+  · intro v
+    constructor
+    · intro h
+      obtain ⟨b,hb⟩ := (hB e).2 v.1 |>.mp (congrArg Prod.fst h)
+      obtain ⟨c,hc⟩ := (hC e).2 v.2 |>.mp (congrArg Prod.snd h)
+      exact ⟨(b,c),Prod.ext hb hc⟩
+    · rintro ⟨⟨b,c⟩,rfl⟩
+      exact Prod.ext ((hB e).2 _ |>.mpr ⟨b,rfl⟩) ((hC e).2 _ |>.mpr ⟨c,rfl⟩)
+
+lemma pi_upperQuotient {ι : Type*} (B : ι → Sheaf.{u,v,w,z} (R := R) G) (α : E → R)
+    (hB : ∀ i, (B i).UpperQuotient α) : (pi B).UpperQuotient α := by
+  classical
+  intro e
+  refine ⟨?_,?_⟩
+  · intro v
+    choose b hb using fun i => (hB i e).1 (v i)
+    exact ⟨b,funext hb⟩
+  · intro v
+    constructor
+    · intro h
+      choose b hb using fun i => (hB i e).2 (v i) |>.mp (congrFun h i)
+      exact ⟨b,funext hb⟩
+    · rintro ⟨b,rfl⟩
+      funext i
+      exact (hB i e).2 _ |>.mpr ⟨b i,rfl⟩
+
+lemma prod_kernel_iff (B C : Sheaf.{u,v,w,z} (R := R) G) (x : V) (v : (prod B C).vertex x) :
+    v ∈ LinearMap.ker ((prod B C).stalkMap x) ↔
+      v.1 ∈ LinearMap.ker (B.stalkMap x) ∧ v.2 ∈ LinearMap.ker (C.stalkMap x) := by
+  constructor
+  · intro h
+    constructor <;> apply LinearMap.mem_ker.mpr <;> funext e
+    · rcases e with ⟨e,he⟩
+      subst x
+      exact congrArg Prod.fst (congrFun (LinearMap.mem_ker.mp h) ⟨e,rfl⟩)
+    · rcases e with ⟨e,he⟩
+      subst x
+      exact congrArg Prod.snd (congrFun (LinearMap.mem_ker.mp h) ⟨e,rfl⟩)
+  · rintro ⟨hB,hC⟩
+    apply LinearMap.mem_ker.mpr
+    funext e
+    rcases e with ⟨e,he⟩
+    subst x
+    exact Prod.ext (congrFun (LinearMap.mem_ker.mp hB) ⟨e,rfl⟩)
+      (congrFun (LinearMap.mem_ker.mp hC) ⟨e,rfl⟩)
+
+lemma prod_minimal (B C : Sheaf.{u,v,w,z} (R := R) G) (x : V) (I : Ideal R)
+    (hB : LinearMap.ker (B.stalkMap x) ≤ I • (⊤ : Submodule R (B.vertex x)))
+    (hC : LinearMap.ker (C.stalkMap x) ≤ I • (⊤ : Submodule R (C.vertex x))) :
+    LinearMap.ker ((prod B C).stalkMap x) ≤ I • (⊤ : Submodule R ((prod B C).vertex x)) := by
+  intro v hv
+  obtain ⟨hvB,hvC⟩ := (prod_kernel_iff B C x v).mp hv
+  have h (M N : Type z) [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N]
+      (f : M →ₗ[R] N) : (I • (⊤ : Submodule R M)).map f ≤ I • (⊤ : Submodule R N) := by
+    rw [Submodule.map_smul'']
+    exact Submodule.smul_mono le_rfl le_top
+  have hb := h _ _ (LinearMap.inl R (B.vertex x) (C.vertex x)) ⟨v.1,hB hvB,rfl⟩
+  have hc := h _ _ (LinearMap.inr R (B.vertex x) (C.vertex x)) ⟨v.2,hC hvC,rfl⟩
+  have heq : (v.1,0) + (0,v.2) = v := Prod.ext (add_zero _) (zero_add _)
+  have hh := (I • (⊤ : Submodule R ((prod B C).vertex x))).add_mem hb hc
+  change (v.1,0) + (0,v.2) ∈ I • (⊤ : Submodule R ((prod B C).vertex x)) at hh
+  exact heq ▸ hh
+
+end KLInvariance.MomentGraph.Sheaf
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.Sheaf
+universe u v w z
+variable {R : Type u} [CommRing R] {V : Type v} [PartialOrder V]
+  {E : Type w} {G : OrderedGraph V E}
+  {B C : Sheaf.{u,v,w,z} (R := R) G} (D : Sheaf.{u,v,w,z} (R := R) G)
+
+noncomputable def addZeroRight (M N : Type z) [AddCommGroup M] [Module R M]
+    [AddCommGroup N] [Module R N] [Subsingleton N] : M ≃ₗ[R] M × N where
+  toFun m := (m,0)
+  invFun := Prod.fst
+  left_inv _ := rfl
+  right_inv _ := Prod.ext rfl (Subsingleton.elim _ _)
+  map_add' _ _ := Prod.ext rfl (zero_add _).symm
+  map_smul' _ _ := Prod.ext rfl (smul_zero _).symm
+
+@[simp] lemma addZeroRight_apply (M N : Type z) [AddCommGroup M] [Module R M]
+    [AddCommGroup N] [Module R N] [Subsingleton N] (m : M) :
+    addZeroRight (R := R) M N m = (m,0) := rfl
+
+noncomputable def OnIso.prodZero {U : Set V} (j : OnIso B C U)
+    (hV : ∀ x ∈ U, Subsingleton (D.vertex x))
+    (hE : ∀ e, G.target e ∈ U → Subsingleton (D.edge e)) : OnIso B (prod C D) U where
+  vertex x hx := by
+    letI := hV x hx
+    exact (j.vertex x hx).trans (addZeroRight _ _)
+  edge e he := by
+    letI := hE e he
+    exact (j.edge e he).trans (addZeroRight _ _)
+  lower e hs ht v := by
+    apply Prod.ext
+    · exact j.lower e hs ht v
+    · exact (D.lower e).map_zero.symm
+  upper e ht v := by
+    apply Prod.ext
+    · exact j.upper e ht v
+    · exact (D.upper e).map_zero.symm
+
+lemma pi_subsingleton_vertex {ι : Type*} (B : ι → Sheaf.{u,v,w,z} (R := R) G)
+    (x : V) (h : ∀ i, Subsingleton ((B i).vertex x)) : Subsingleton ((pi B).vertex x) := by
+  exact ⟨fun f g => funext (fun i => @Subsingleton.elim _ (h i) (f i) (g i))⟩
+
+lemma pi_subsingleton_edge {ι : Type*} (B : ι → Sheaf.{u,v,w,z} (R := R) G)
+    (e : E) (h : ∀ i, Subsingleton ((B i).edge e)) : Subsingleton ((pi B).edge e) := by
+  exact ⟨fun f g => funext (fun i => @Subsingleton.elim _ (h i) (f i) (g i))⟩
+
+end KLInvariance.MomentGraph.Sheaf
+
+end
+
+
+section
+
+namespace KLInvariance.Graded.ModuleData
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A}
+noncomputable section
+ def zero : ModuleData.{uk,ua,um} 𝓐 where
+  obj := ModuleCat.of A PUnit.{max ua um+1}
+  kModule := inferInstance
+  tower := inferInstance
+  piece _ := ⊥
+  decomposition := (show DirectSum.IsInternal (fun _ : ℤ => (⊥ : Submodule k PUnit.{max ua um+1})) from by
+    apply internal_of_components _ (fun _ => 0)
+    · intro _ _; exact Submodule.zero_mem _
+    · intro _ _ _; exact Subsingleton.elim _ _
+    · intro _ _ _ _ _; rfl
+    · intro m; exact ⟨∅,Subsingleton.elim _ _⟩).chooseDecomposition
+  graded := ⟨by intro _ _ _ _ _ _; exact Subsingleton.elim _ _⟩
+  nonneg _ _ := rfl
+  finite := inferInstance
+
+ def Hom.zero (M N : ModuleData.{uk,ua,um} 𝓐) : Hom M N :=
+  ⟨0,fun _ _ _ => Submodule.zero_mem _⟩
+
+ def Hom.pi {ι : Type um} [Fintype ι] {M N : ι → ModuleData.{uk,ua,um} 𝓐}
+    (f : ∀ i, Hom (M i) (N i)) : Hom (ModuleData.pi M) (ModuleData.pi N) where
+  map := LinearMap.pi fun i => (f i).map.comp (LinearMap.proj i)
+  graded n _ hm i hi := (f i).graded n _ (hm i hi)
+end
+end KLInvariance.Graded.ModuleData
+
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
+  {G : OrderedGraph V E}
+noncomputable section
+ def shift (B : GradedSheaf (𝓐 := 𝓐) G) (d : ℕ) : GradedSheaf (𝓐 := 𝓐) G where
+  vertex x := (B.vertex x).shift d
+  edge e := (B.edge e).shift d
+  lower e := (B.lower e).shift d
+  upper e := (B.upper e).shift d
+
+ def zero : GradedSheaf (𝓐 := 𝓐) G where
+  vertex _ := ModuleData.zero
+  edge _ := ModuleData.zero
+  lower _ := ModuleData.Hom.zero _ _
+  upper _ := ModuleData.Hom.zero _ _
+
+ def prod (B C : GradedSheaf (𝓐 := 𝓐) G) : GradedSheaf (𝓐 := 𝓐) G where
+  vertex x := (B.vertex x).prod (C.vertex x)
+  edge e := (B.edge e).prod (C.edge e)
+  lower e := (B.lower e).prodMap (C.lower e)
+  upper e := (B.upper e).prodMap (C.upper e)
+
+ def pi {ι : Type um} [Fintype ι] (B : ι → GradedSheaf (𝓐 := 𝓐) G) :
+    GradedSheaf (𝓐 := 𝓐) G where
+  vertex x := ModuleData.pi (fun i => (B i).vertex x)
+  edge e := ModuleData.pi (fun i => (B i).edge e)
+  lower e := ModuleData.Hom.pi (fun i => (B i).lower e)
+  upper e := ModuleData.Hom.pi (fun i => (B i).upper e)
+
+ def OnIso.prodZero {B C : GradedSheaf (𝓐 := 𝓐) G} {U : Set V} (j : OnIso B C U)
+    (D : GradedSheaf (𝓐 := 𝓐) G)
+    (hV : ∀ x ∈ U, Subsingleton (D.vertex x))
+    (hE : ∀ e, G.target e ∈ U → Subsingleton (D.edge e)) : OnIso B (prod C D) U where
+  toOnIso := j.toOnIso.prodZero D.forget hV hE
+  vertex_graded x hx n v hv := ⟨j.vertex_graded x hx n v hv,Submodule.zero_mem _⟩
+  edge_graded e he n v hv := ⟨j.edge_graded e he n v hv,Submodule.zero_mem _⟩
+
+ theorem shift_forget (B : GradedSheaf (𝓐 := 𝓐) G) (d : ℕ) :
+    (B.shift d).forget=B.forget := rfl
+ theorem zero_forget : (zero (𝓐 := 𝓐) (G := G)).forget=Sheaf.zero := rfl
+ theorem prod_forget (B C : GradedSheaf (𝓐 := 𝓐) G) :
+    (prod B C).forget=Sheaf.prod B.forget C.forget := rfl
+ theorem pi_forget {ι : Type um} [Fintype ι] (B : ι → GradedSheaf (𝓐 := 𝓐) G) :
+    (pi B).forget=Sheaf.pi (fun i => (B i).forget) := rfl
+end
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+/-! Extension by zero from a full lower ideal. Proof-indexed finite products
+avoid informal identifications of the integral stalks at boundary vertices. -/
+namespace KLInvariance.MomentGraph
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
+  (G : OrderedGraph V E) (K : Set V) (hK : IsLowerSet K)
+
+ def lowerGraph : OrderedGraph K {e : E // G.target e ∈ K} where
+  source e := ⟨G.source e.val,hK (G.increasing e.val).le e.property⟩
+  target e := ⟨G.target e.val,e.property⟩
+  increasing e := G.increasing e.val
+
+namespace GradedSheaf
+noncomputable section
+attribute [local instance] Classical.propDecidable
+
+abbrev Witness (p : Prop) := ULift.{um} (PLift p)
+
+variable {G K hK} (B : GradedSheaf (𝓐 := 𝓐) (lowerGraph G K hK))
+
+ def lowerExtend : GradedSheaf (𝓐 := 𝓐) G where
+  vertex x := ModuleData.pi (fun h : Witness.{um} (x ∈ K) => B.vertex ⟨x,h.down.down⟩)
+  edge e := ModuleData.pi (fun h : Witness.{um} (G.target e ∈ K) => B.edge ⟨e,h.down.down⟩)
+  lower e :=
+    { map := LinearMap.pi (fun h => (B.lower ⟨e,h.down.down⟩).map.comp
+        (LinearMap.proj ⟨⟨hK (G.increasing e).le h.down.down⟩⟩))
+      graded n _v hv h _ := (B.lower ⟨e,h.down.down⟩).graded n _ (hv _ (Set.mem_univ _)) }
+  upper e :=
+    { map := LinearMap.pi (fun h => (B.upper ⟨e,h.down.down⟩).map.comp (LinearMap.proj h))
+      graded n _v hv h _ := (B.upper ⟨e,h.down.down⟩).graded n _ (hv _ (Set.mem_univ _)) }
+
+ def lowerExtendVertex (x : K) : (B.lowerExtend.vertex x.val) ≃ₗ[A] B.vertex x where
+  toFun f := f ⟨⟨x.property⟩⟩
+  invFun v _ := v
+  left_inv f := by
+    funext h
+    rfl
+  right_inv _ := rfl
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+
+ theorem lowerExtendVertex_graded (x : K) (n : ℤ) (v : B.lowerExtend.vertex x.val)
+    (hv : v ∈ (B.lowerExtend.vertex x.val).piece n) :
+    B.lowerExtendVertex x v ∈ (B.vertex x).piece n := hv _ (Set.mem_univ _)
+
+ theorem lowerExtend_support (x : V) (hx : x ∉ K) : Subsingleton (B.lowerExtend.vertex x) :=
+  ⟨fun _f _g => funext (fun h => (hx h.down.down).elim)⟩
+
+ theorem lowerExtend_free (hb : ∀ x, Module.Free A (B.vertex x)) (x : V) :
+    Module.Free A (B.lowerExtend.vertex x) := by
+  let : ∀ h : Witness.{um} (x ∈ K), Module.Free A (B.vertex ⟨x,h.down.down⟩) := fun h => hb _
+  change Module.Free A (∀ h : Witness.{um} (x ∈ K), B.vertex ⟨x,h.down.down⟩)
+  infer_instance
+
+ def liftLowerAssignment (f : B.forget.Assignment) : B.lowerExtend.forget.Assignment :=
+  fun x h => f ⟨x,h.down.down⟩
+
+ def projectLowerAssignment (f : B.lowerExtend.forget.Assignment) : B.forget.Assignment :=
+  fun x => f x.val ⟨⟨x.property⟩⟩
+
+ theorem projectLower_compatible {U : Set V} {f : B.lowerExtend.forget.Assignment}
+    (hf : B.lowerExtend.forget.CompatibleOn U f) :
+    B.forget.CompatibleOn (Subtype.val ⁻¹' U) (B.projectLowerAssignment f) := by
+  intro e hs ht
+  exact congrFun (hf e.val hs ht) ⟨⟨e.property⟩⟩
+
+ theorem liftLower_compatible {U : Set V} {f : B.forget.Assignment}
+    (hf : B.forget.CompatibleOn (Subtype.val ⁻¹' U) f) :
+    B.lowerExtend.forget.CompatibleOn U (B.liftLowerAssignment f) := by
+  intro e hs ht
+  funext h
+  exact hf ⟨e,h.down.down⟩ hs ht
+
+ theorem lowerExtend_generated (hb : B.forget.Generated) : B.lowerExtend.forget.Generated := by
+  intro x v
+  by_cases hx : x ∈ K
+  · obtain ⟨f,hf,hfx⟩ := hb ⟨x,hx⟩ (v ⟨⟨hx⟩⟩)
+    refine ⟨B.liftLowerAssignment f,B.liftLower_compatible hf,?_⟩
+    funext h
+    exact hfx
+  · let : Subsingleton (B.lowerExtend.forget.vertex x) := B.lowerExtend_support x hx
+    exact ⟨0,fun _ _ _ => by simp,Subsingleton.elim _ _⟩
+
+ theorem lowerExtend_flabby (hb : B.forget.Flabby) : B.lowerExtend.forget.Flabby := by
+  intro U hU f hf
+  have hu : IsUpperSet (Subtype.val ⁻¹' U : Set K) := fun _ _ hab ha => hU hab ha
+  obtain ⟨g,hg,hgf⟩ := hb (Subtype.val ⁻¹' U) hu (B.projectLowerAssignment f) (B.projectLower_compatible hf)
+  refine ⟨B.liftLowerAssignment g,B.liftLower_compatible hg,?_⟩
+  intro x hx
+  funext h
+  exact hgf ⟨x,h.down.down⟩ hx
+
+ theorem lowerExtend_quotient (α : E → A)
+    (hb : B.UpperQuotient (fun e => α e.val)) : B.lowerExtend.UpperQuotient α := by
+  intro e
+  constructor
+  · intro v
+    have hh : ∀ h : Witness.{um} (G.target e ∈ K), ∃ w, (B.upper ⟨e,h.down.down⟩).map w=v h :=
+      fun h => (hb ⟨e,h.down.down⟩).1 (v h)
+    choose w hw using hh
+    exact ⟨w,funext hw⟩
+  · intro v
+    constructor
+    · intro hv
+      have hh : ∀ h : Witness.{um} (G.target e ∈ K), ∃ w, v h=α e • w :=
+        fun h => (hb ⟨e,h.down.down⟩).2 (v h) |>.mp (congrFun hv h)
+      choose w hw using hh
+      exact ⟨w,funext hw⟩
+    · rintro ⟨w,rfl⟩
+      funext h
+      exact (hb ⟨e,h.down.down⟩).2 _ |>.mpr ⟨w h,rfl⟩
+
+ theorem lowerExtend_minimal (x : V)
+    (hb : ∀ hx : x ∈ K, B.KernelMinimal ⟨x,hx⟩) : B.lowerExtend.KernelMinimal x := by
+  intro f hf
+  apply LocalCover.pi_mem_ideal_smul
+  intro h
+  apply hb h.down.down (B.projectLowerAssignment f)
+  intro e he
+  have hes : G.source e.val=x := congrArg Subtype.val he
+  exact congrFun (hf e.val hes) ⟨⟨e.property⟩⟩
+
+end
+end GradedSheaf
+end KLInvariance.MomentGraph
+
+end
+
+
+section
+
+/-! Finite sums of honest shifted normalized boundary models. A model family
+is data for the decomposition lemma, not an assumption of the KL main. -/
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+  (hneg : ∀ n < 0, 𝓐 n = ⊥)
+  {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
+
+instance zeroVertex_subsingleton (x : V) : Subsingleton ((zero (𝓐 := 𝓐) (G := G)).vertex x) :=
+  inferInstanceAs (Subsingleton PUnit.{max ua um+1})
+instance zeroEdge_subsingleton (e : E) : Subsingleton ((zero (𝓐 := 𝓐) (G := G)).edge e) :=
+  inferInstanceAs (Subsingleton PUnit.{max ua um+1})
+
+structure BoundaryModels (α : E → A) where
+  sheaf : V → GradedSheaf (𝓐 := 𝓐) G
+  free : ∀ c x, Module.Free A ((sheaf c).vertex x)
+  quotient : ∀ c, (sheaf c).UpperQuotient α
+  generated : ∀ c, (sheaf c).forget.Generated
+  flabby : ∀ c, (sheaf c).forget.Flabby
+  support : ∀ c x, ¬x ≤ c → Subsingleton ((sheaf c).vertex x)
+  top : ∀ c, (sheaf c).vertex c ≃ₗ[A] ModuleData.unit.{uk,ua,um} hneg
+  top_graded : ∀ c n v, v ∈ ((sheaf c).vertex c).piece n → top c v ∈ (ModuleData.unit.{uk,ua,um} hneg).piece n
+  minimal : ∀ c x, x ≠ c → (sheaf c).KernelMinimal x
+
+inductive ModelSum (O : V → GradedSheaf (𝓐 := 𝓐) G) (U : Set V) :
+    GradedSheaf (𝓐 := 𝓐) G → Prop where
+  | zero : ModelSum O U zero
+  | add {B} : ModelSum O U B → ∀ c ∈ U, ∀ (n : ℕ) (d : ULift.{um} (Fin n) → ℕ),
+      ModelSum O U (prod B (pi (fun i => (O c).shift (d i))))
+
+namespace ModelSum
+variable {hneg} {α : E → A} (O : BoundaryModels hneg (G := G) α)
+  {U : Set V} {B : GradedSheaf (𝓐 := 𝓐) G}
+
+omit [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐] in
+lemma mono {O : V → GradedSheaf (𝓐 := 𝓐) G} (h : ModelSum O U B)
+    {T : Set V} (hUT : U ⊆ T) : ModelSum O T B := by
+  induction h with
+  | zero => exact .zero
+  | add _ c hc n d ih => exact .add ih c (hUT hc) n d
+
+lemma free (h : ModelSum O.sheaf U B) (x : V) : Module.Free A (B.vertex x) := by
+  induction h with
+  | zero => exact inferInstanceAs (Module.Free A PUnit.{max ua um+1})
+  | @add C _ c _ n d ih =>
+    let _ : Module.Free A (C.vertex x) := ih
+    let _ := O.free c x
+    change Module.Free A (C.vertex x × (ULift.{um} (Fin n) → (O.sheaf c).vertex x))
+    infer_instance
+
+lemma quotient (h : ModelSum O.sheaf U B) : B.UpperQuotient α := by
+  induction h with
+  | zero =>
+    intro e
+    exact ⟨fun v => ⟨0,Subsingleton.elim _ _⟩,fun v => ⟨fun _ => ⟨0,Subsingleton.elim _ _⟩,fun _ => Subsingleton.elim _ _⟩⟩
+  | @add C _ c _ n d ih =>
+    exact Sheaf.prod_upperQuotient C.forget (Sheaf.pi (fun _ : ULift.{um} (Fin n) => (O.sheaf c).forget)) α ih
+      (Sheaf.pi_upperQuotient (fun _ : ULift.{um} (Fin n) => (O.sheaf c).forget) α (fun _ => O.quotient c))
+
+lemma generated (h : ModelSum O.sheaf U B) : B.forget.Generated := by
+  induction h with
+  | zero =>
+    intro x v
+    let : Subsingleton ((GradedSheaf.zero (𝓐 := 𝓐) (G := G)).forget.vertex x) := zeroVertex_subsingleton (𝓐 := 𝓐) (G := G) x
+    exact ⟨0,fun _ _ _ => rfl,Subsingleton.elim _ _⟩
+  | add _ c _ n d ih => exact Sheaf.prod_generated _ _ ih (Sheaf.pi_generated _ (fun _ => O.generated c))
+
+lemma flabby (h : ModelSum O.sheaf U B) : B.forget.Flabby := by
+  induction h with
+  | zero => exact fun U _ f _ => ⟨f,fun _ _ _ => rfl,fun _ _ => rfl⟩
+  | add _ c _ n d ih => exact Sheaf.prod_flabby _ _ ih (Sheaf.pi_flabby _ (fun _ => O.flabby c))
+
+lemma minimal (h : ModelSum O.sheaf U B) (x : V) (hx : x ∉ U) : B.KernelMinimal x := by
+  induction h with
+  | zero =>
+    apply (GradedSheaf.zero.kernelMinimal_iff x).mpr
+    intro v _
+    let : Subsingleton ((GradedSheaf.zero (𝓐 := 𝓐) (G := G)).forget.vertex x) := zeroVertex_subsingleton (𝓐 := 𝓐) (G := G) x
+    rw [Subsingleton.elim v 0]
+    exact Submodule.zero_mem _
+  | @add C _ c hc n d ih =>
+    apply ((prod C (pi (fun i => (O.sheaf c).shift (d i)))).kernelMinimal_iff x).mpr
+    exact Sheaf.prod_minimal C.forget (Sheaf.pi (fun _ : ULift.{um} (Fin n) => (O.sheaf c).forget)) x _ ((C.kernelMinimal_iff x).mp ih)
+      (Sheaf.pi_minimal (fun _ : ULift.{um} (Fin n) => (O.sheaf c).forget) x _ (fun _ =>
+        ((O.sheaf c).kernelMinimal_iff x).mp (O.minimal c x (fun he => hx (he ▸ hc)))))
+
+end ModelSum
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+/-! Actual normalized BMP models at every vertex of a finite ordered graph,
+constructed on its lower ideal and extended by zero. -/
+namespace KLInvariance.MomentGraph
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+  [IsNoetherianRing A]
+  (hneg : ∀ n < 0, 𝓐 n = ⊥)
+  (hscalar : ∀ a : A, ∃ c : k, a-algebraMap k A c ∈ positiveIdeal 𝓐)
+  {V E : Type um} [PartialOrder V] [Fintype V] [Fintype E]
+  (G : OrderedGraph V E) (α : E → A) (hα : ∀ e, α e ∈ 𝓐 1)
+noncomputable section
+attribute [local instance] Classical.propDecidable
+
+ def lowerBoundary (c : V) : BoundarySheaf hneg (lowerGraph G (Set.Iic c) (isLowerSet_Iic c))
+    (fun e => α e.val) ⟨c,le_rfl⟩ :=
+  (exists_boundarySheaf hneg (lowerGraph G (Set.Iic c) (isLowerSet_Iic c)) hscalar
+    ⟨c,le_rfl⟩ (fun x => x.property) (fun e => α e.val) (fun e => hα e.val)).some
+
+ def actualBoundaryModels : GradedSheaf.BoundaryModels hneg (G := G) α where
+  sheaf c := (lowerBoundary hneg hscalar G α hα c).sheaf.lowerExtend
+  free c := (lowerBoundary hneg hscalar G α hα c).sheaf.lowerExtend_free
+    (lowerBoundary hneg hscalar G α hα c).free
+  quotient c := (lowerBoundary hneg hscalar G α hα c).sheaf.lowerExtend_quotient α
+    (lowerBoundary hneg hscalar G α hα c).quotient
+  generated c := (lowerBoundary hneg hscalar G α hα c).sheaf.lowerExtend_generated
+    (lowerBoundary hneg hscalar G α hα c).generated
+  flabby c := (lowerBoundary hneg hscalar G α hα c).sheaf.lowerExtend_flabby
+    (lowerBoundary hneg hscalar G α hα c).flabby.flabby
+  support c x hx := (lowerBoundary hneg hscalar G α hα c).sheaf.lowerExtend_support x hx
+  top c := ((lowerBoundary hneg hscalar G α hα c).sheaf.lowerExtendVertex ⟨c,le_rfl⟩).trans
+    (ModuleData.equivOfEq (lowerBoundary hneg hscalar G α hα c).top)
+  top_graded c n v hv := ModuleData.equivOfEq_graded
+    (lowerBoundary hneg hscalar G α hα c).top n _
+    ((lowerBoundary hneg hscalar G α hα c).sheaf.lowerExtendVertex_graded ⟨c,le_rfl⟩ n v hv)
+  minimal c x hx := (lowerBoundary hneg hscalar G α hα c).sheaf.lowerExtend_minimal x
+    (fun hxc => (lowerBoundary hneg hscalar G α hα c).minimal ⟨x,hxc⟩
+      (fun hh => hx (congrArg Subtype.val hh)))
+
+end
+end KLInvariance.MomentGraph
+
+end
+
+
+section
+
+/-! Recognition of the actual finite minimal boundary sheaf. This proves
+choice-independence of the constructed graded object, not its KL character. -/
+namespace KLInvariance.MomentGraph.BoundarySheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+  [IsDomain A] [IsNoetherianRing A]
+  {hneg : ∀ n < 0, 𝓐 n = ⊥}
+  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
+  {V E : Type um} [PartialOrder V] [Fintype V] [Fintype E]
+  {G : OrderedGraph V E} {α : E → A} {b : V}
+  (B C : BoundarySheaf hneg G α b) (hb : ∀ x, x ≤ b)
+
+omit [Fintype V] in
+include hzero hb in
+ theorem extend_iso {U : Set V} (hU : IsUpperSet U)
+    (j : GradedSheaf.OnIso B.sheaf C.sheaf U)
+    (x : V) (hx : x ∉ U) (hmax : Set.Ioi x ⊆ U) :
+    Nonempty (GradedSheaf.OnIso B.sheaf C.sheaf (insert x U)) := by
+  classical
+  by_cases hxb : x=b
+  · subst x
+    let eqv := ModuleData.equivOfEq (B.top.trans C.top.symm)
+    have heqv := ModuleData.equivOfEq_graded (B.top.trans C.top.symm)
+    refine ⟨j.extend hU b hx hmax α B.quotient C.quotient eqv heqv ?_⟩
+    intro e he
+    exact False.elim ((not_lt_of_ge (hb (G.target e))) (he ▸ G.increasing e))
+  · let ja := j.toOnIso.above x hmax
+    obtain ⟨e,he,hcomm⟩ := ja.extend_stalk_graded hneg hzero
+      (fun e ht => j.edge_graded e (hmax ht)) (B.free x) (C.free x)
+      B.flabby.flabby B.generated C.flabby.flabby C.generated
+      (B.minimal x hxb) (C.minimal x hxb)
+    refine ⟨j.extend hU x hx hmax α B.quotient C.quotient e he ?_⟩
+    intro f hf v
+    subst x
+    exact congrFun (hcomm v) ⟨f,rfl⟩
+
+include hzero hb in
+ theorem exists_gradedIso : Nonempty (GradedSheaf.Iso B.sheaf C.sheaf) := by
+  classical
+  have h : ∀ s : Finset V, IsUpperSet ((s : Set V)ᶜ) →
+      GradedSheaf.OnIso B.sheaf C.sheaf ((s : Set V)ᶜ) →
+      Nonempty (GradedSheaf.Iso B.sheaf C.sheaf) := by
+    intro s
+    induction s using Finset.strongInductionOn with
+    | _ s ih =>
+      intro hU j
+      by_cases hs : s=∅
+      · subst s
+        have heq : ((∅ : Finset V) : Set V)ᶜ=Set.univ := by simp
+        exact ⟨GradedSheaf.OnIso.ofUniv (heq ▸ j)⟩
+      · obtain ⟨x,hxs,hxmax⟩ := s.exists_maximal (Finset.nonempty_iff_ne_empty.mpr hs)
+        have hx : x ∉ ((s : Set V)ᶜ) := by simpa using hxs
+        have hmax : Set.Ioi x ⊆ ((s : Set V)ᶜ) := by
+          intro y hy
+          change y ∉ s
+          intro hys
+          exact (not_le_of_gt hy) (hxmax hys hy.le)
+        obtain ⟨jj⟩ := B.extend_iso hzero C hb hU j x hx hmax
+        have heq : insert x ((s : Set V)ᶜ) = ((s.erase x : Finset V) : Set V)ᶜ := by
+          ext y
+          simp only [Set.mem_insert_iff,Set.mem_compl_iff,Finset.mem_coe,Finset.mem_erase]
+          tauto
+        have hU' : IsUpperSet (insert x ((s : Set V)ᶜ)) := by
+          intro y t hyt hy
+          rcases Set.mem_insert_iff.mp hy with hy | hy
+          · subst y
+            by_cases htx : t=x
+            · exact Set.mem_insert_iff.mpr (Or.inl htx)
+            · exact Set.mem_insert_of_mem x (hmax (lt_of_le_of_ne hyt (Ne.symm htx)))
+          · exact Set.mem_insert_of_mem x (hU hyt hy)
+        exact ih (s.erase x) (Finset.erase_ssubset hxs) (heq ▸ hU') (heq ▸ jj)
+  have hset : ((Finset.univ : Finset V) : Set V)ᶜ = ∅ := by simp
+  have hj : GradedSheaf.OnIso B.sheaf C.sheaf (∅ : Set V) := by
+    refine { toOnIso := ?_, vertex_graded := ?_, edge_graded := ?_ }
+    · exact ⟨fun x hx => False.elim hx, fun e he => False.elim he,
+        fun e hs => False.elim hs, fun e ht => False.elim ht⟩
+    · intro x hx; exact False.elim hx
+    · intro e he; exact False.elim he
+  have hh := h Finset.univ
+  rw [hset] at hh
+  exact hh isUpperSet_empty hj
+
+end KLInvariance.MomentGraph.BoundarySheaf
+
+end
+
+
+section
+
+
+namespace KLInvariance.MomentGraph.Sheaf
+universe ur uv ue
+variable {R : Type ur} [CommRing R] {V : Type uv} [PartialOrder V]
+  {E : Type ue} (G : OrderedGraph V E) (α : E → R) (c : V)
+noncomputable section
+
+private def quotientLinear (e : E) : R →ₗ[R] R ⧸ Ideal.span {α e} :=
+  (Ideal.Quotient.mkₐ R (Ideal.span {α e})).toLinearMap
+
+ def structureSheaf : Sheaf (R := R) G where
+  vertex x := ModuleCat.of R (PLift (x ≤ c) → R)
+  edge e := ModuleCat.of R (PLift (G.target e ≤ c) → R ⧸ Ideal.span {α e})
+  upper e := LinearMap.pi (fun h => (quotientLinear α e).comp (LinearMap.proj h))
+  lower e := LinearMap.pi (fun h => (quotientLinear α e).comp
+    (LinearMap.proj ⟨(G.increasing e).le.trans h.down⟩))
+
+ theorem structureSheaf_free (x : V) : Module.Free R ((structureSheaf G α c).vertex x) :=
+  inferInstanceAs (Module.Free R (PLift (x ≤ c) → R))
+
+ theorem structureSheaf_finite (x : V) : Module.Finite R ((structureSheaf G α c).vertex x) :=
+  inferInstanceAs (Module.Finite R (PLift (x ≤ c) → R))
+
+ theorem structureSheaf_support (x : V) (hx : ¬ x ≤ c) :
+    Subsingleton ((structureSheaf G α c).vertex x) := by
+  refine ⟨fun f g => funext fun h => False.elim (hx h.down)⟩
+
+ def structureVertexIso (x : V) (hx : x ≤ c) :
+    (structureSheaf G α c).vertex x ≃ₗ[R] R where
+  toFun f := f ⟨hx⟩
+  invFun r := fun _ => r
+  left_inv _ := rfl
+  right_inv _ := rfl
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+
+ theorem structureSheaf_generated : (structureSheaf G α c).Generated := by
+  classical
+  intro x v
+  by_cases hx : x ≤ c
+  · refine ⟨fun _ _ => v ⟨hx⟩, fun _ _ _ => rfl, rfl⟩
+  · let _ := structureSheaf_support G α c x hx
+    exact ⟨0,fun _ _ _ => rfl,Subsingleton.elim _ _⟩
+
+ theorem structureSheaf_upperQuotient : (structureSheaf G α c).UpperQuotient α := by
+  classical
+  intro e
+  refine ⟨?_,?_⟩
+  · intro v
+    choose r hr using fun h : PLift (G.target e ≤ c) => Ideal.Quotient.mk_surjective (v h)
+    exact ⟨r,funext hr⟩
+  · intro v
+    constructor
+    · intro hv
+      have hh (h : PLift (G.target e ≤ c)) :
+          v h ∈ Ideal.span {α e} := Ideal.Quotient.eq_zero_iff_mem.mp (congrFun hv h)
+      choose r hr using fun h => Ideal.mem_span_singleton.mp (hh h)
+      exact ⟨r,funext hr⟩
+    · rintro ⟨w,rfl⟩
+      funext h
+      exact Ideal.Quotient.eq_zero_iff_mem.mpr
+        (Ideal.mem_span_singleton.mpr ⟨w h,rfl⟩)
+
+ theorem structureSheaf_minimal [IsLocalRing R]
+    (hα : ∀ e, α e ∈ IsLocalRing.maximalIdeal R)
+    (hreach : ∀ x, x < c → ∃ e, G.source e = x ∧ G.target e ≤ c)
+    (x : V) (hxc : x ≠ c) :
+    LinearMap.ker ((structureSheaf G α c).stalkMap x) ≤
+      IsLocalRing.maximalIdeal R • (⊤ : Submodule R ((structureSheaf G α c).vertex x)) := by
+  classical
+  intro v hv
+  by_cases hx : x ≤ c
+  · obtain ⟨e,he,het⟩ := hreach x (lt_of_le_of_ne hx hxc)
+    subst x
+    have heq := congrFun (LinearMap.mem_ker.mp hv) ⟨e,rfl⟩
+    have hvα : v ⟨hx⟩ ∈ Ideal.span {α e} :=
+      Ideal.Quotient.eq_zero_iff_mem.mp (congrFun heq ⟨het⟩)
+    have hm : v ⟨hx⟩ ∈ IsLocalRing.maximalIdeal R :=
+      (Ideal.span_le.mpr (Set.singleton_subset_iff.mpr (hα e))) hvα
+    let one : (structureSheaf G α c).vertex (G.source e) := fun _ => 1
+    have hs : v ⟨hx⟩ • one ∈ IsLocalRing.maximalIdeal R •
+        (⊤ : Submodule R ((structureSheaf G α c).vertex (G.source e))) :=
+      Submodule.smul_mem_smul hm (show one ∈
+        (⊤ : Submodule R ((structureSheaf G α c).vertex (G.source e))) from Submodule.mem_top)
+    have heqv : v = v ⟨hx⟩ • one := by
+      funext h
+      exact (mul_one _).symm
+    rw [heqv]
+    exact hs
+  · let _ := structureSheaf_support G α c x hx
+    rw [Subsingleton.elim v 0]
+    exact Submodule.zero_mem _
+
+end
+end KLInvariance.MomentGraph.Sheaf
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.Sheaf
+open TensorProduct
+universe ur us uv ue um
+variable {R : Type ur} [CommRing R] (S : Type us) [CommRing S] [Algebra R S]
+  {V : Type uv} [PartialOrder V] {E : Type ue} {G : OrderedGraph V E}
+  (B : Sheaf.{ur,uv,ue,um} (R := R) G)
+
+noncomputable def baseChange : Sheaf (R := S) G where
+  vertex x := ModuleCat.of S (S ⊗[R] B.vertex x)
+  edge e := ModuleCat.of S (S ⊗[R] B.edge e)
+  lower e := AlgebraTensorModule.lTensor S S (B.lower e)
+  upper e := AlgebraTensorModule.lTensor S S (B.upper e)
+
+noncomputable def tensorAssignments : S ⊗[R] B.Assignment →ₗ[S] (B.baseChange S).Assignment :=
+  TensorProduct.piRightHom R S S (fun x => B.vertex x)
+
+@[simp] theorem tensorAssignments_tmul (s : S) (f : B.Assignment) (x : V) :
+    B.tensorAssignments S (s ⊗ₜ f) x = s ⊗ₜ f x := rfl
+
+lemma baseChange_tmul_compatible (Ω : Set V) (s : S) (f : B.Assignment)
+    (hf : B.CompatibleOn Ω f) : (B.baseChange S).CompatibleOn Ω (fun x => s ⊗ₜ f x) := by
+  intro e hs ht
+  change s ⊗ₜ[R] B.lower e (f (G.source e)) = s ⊗ₜ[R] B.upper e (f (G.target e))
+  rw [hf e hs ht]
+
+noncomputable def tensorSections (Ω : Set V) :
+    S ⊗[R] B.sections Ω →ₗ[S] (B.baseChange S).sections Ω :=
+  ((B.tensorAssignments S).comp (AlgebraTensorModule.lTensor S S (B.sections Ω).subtype)).codRestrict
+    ((B.baseChange S).sections Ω) (by
+      intro z
+      induction z using TensorProduct.inductionOn with
+      | tmul s f => exact B.baseChange_tmul_compatible S Ω s f.val f.property
+      | add z t hz ht => simpa only [map_add] using Submodule.add_mem _ hz ht)
+
+@[simp] theorem tensorSections_tmul (Ω : Set V) (s : S) (f : B.sections Ω) (x : V) :
+    (B.tensorSections S Ω (s ⊗ₜ f)).val x = s ⊗ₜ f.val x := rfl
+
+theorem baseChange_generated (h : B.Generated) : (B.baseChange S).Generated := by
+  intro x v
+  induction v using TensorProduct.inductionOn with
+  | tmul s v =>
+    obtain ⟨f,hf,hx⟩ := h x v
+    exact ⟨fun y => s ⊗ₜ f y,B.baseChange_tmul_compatible S Set.univ s f hf,congrArg (fun v => s ⊗ₜ[R] v) hx⟩
+  | add v w hv hw =>
+    obtain ⟨f,hf,hfv⟩ := hv
+    obtain ⟨g,hg,hgw⟩ := hw
+    exact ⟨f+g,Submodule.add_mem ((B.baseChange S).sections Set.univ) hf hg,congrArg₂ (·+·) hfv hgw⟩
+
+/-- The finite section constraint whose kernel is the actual section module. -/
+noncomputable def constraint (Ω : Set V) : B.Assignment →ₗ[R]
+    (∀ e : {e : E // G.source e ∈ Ω ∧ G.target e ∈ Ω}, B.edge e.val) :=
+  LinearMap.pi (fun e => (B.lower e.val).comp (LinearMap.proj (G.source e.val)) -
+    (B.upper e.val).comp (LinearMap.proj (G.target e.val)))
+
+lemma ker_constraint (Ω : Set V) : LinearMap.ker (B.constraint Ω) = B.sections Ω := by
+  ext f
+  constructor
+  · intro h e hs ht
+    have hh := congrFun (LinearMap.mem_ker.mp h) ⟨e,hs,ht⟩
+    exact sub_eq_zero.mp hh
+  · intro h
+    apply LinearMap.mem_ker.mpr
+    funext e
+    exact sub_eq_zero.mpr (h e.val e.property.1 e.property.2)
+
+end KLInvariance.MomentGraph.Sheaf
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.Sheaf
+open TensorProduct
+universe ur us uv ue um
+variable {R : Type ur} [CommRing R] (S : Type us) [CommRing S] [Algebra R S]
+  {V : Type uv} [PartialOrder V] {E : Type ue} {G : OrderedGraph V E}
+  (B : Sheaf.{ur,uv,ue,um} (R := R) G)
+
+ theorem baseChange_vertex_free (x : V) [Module.Free R (B.vertex x)] :
+    Module.Free S ((B.baseChange S).vertex x) :=
+  inferInstanceAs (Module.Free S (S ⊗[R] B.vertex x))
+
+ theorem baseChange_vertex_finite (x : V) [Module.Finite R (B.vertex x)] :
+    Module.Finite S ((B.baseChange S).vertex x) :=
+  inferInstanceAs (Module.Finite S (S ⊗[R] B.vertex x))
+
+ theorem baseChange_upperQuotient [Module.Flat R S] (α : E → R)
+    (h : B.UpperQuotient α) : (B.baseChange S).UpperQuotient (fun e => algebraMap R S (α e)) := by
+  intro e
+  refine ⟨LinearMap.lTensor_surjective S (h e).1, ?_⟩
+  have hs : Function.Exact (DistribSMul.toLinearMap R (B.vertex (G.target e)) (α e))
+      (B.upper e) := by
+    intro v
+    exact ((h e).2 v).trans (by simp only [Set.mem_range, DistribSMul.toLinearMap_apply,
+      eq_comm])
+  have ht := Module.Flat.lTensor_exact S hs
+  intro v
+  change (B.upper e).lTensor S v = 0 ↔ ∃ w, v = algebraMap R S (α e) • w
+  rw [ht v, LinearMap.lTensor_smul_action]
+  change (∃ w : S ⊗[R] B.vertex (G.target e), α e • w = v) ↔
+    ∃ w : S ⊗[R] B.vertex (G.target e), v = algebraMap R S (α e) • w
+  simp only [IsScalarTower.algebraMap_smul]
+  constructor <;> rintro ⟨w,hw⟩ <;> exact ⟨w,hw.symm⟩
+
+end KLInvariance.MomentGraph.Sheaf
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.Sheaf
+open TensorProduct
+universe ur us uv ue um
+variable {R : Type ur} [CommRing R] (S : Type us) [CommRing S] [Algebra R S]
+  {V : Type uv} [PartialOrder V] {E : Type ue} {G : OrderedGraph V E}
+  (B : Sheaf.{ur,uv,ue,um} (R := R) G)
+noncomputable section
+
+ theorem tensorAssignments_bijective [Finite V] : Function.Bijective (B.tensorAssignments S) := by
+  classical
+  let := Fintype.ofFinite V
+  exact (TensorProduct.piRight R S S (fun x => B.vertex x)).bijective
+
+ theorem constraint_tensorAssignments (Ω : Set V) (z : S ⊗[R] B.Assignment) :
+    (B.baseChange S).constraint Ω (B.tensorAssignments S z) =
+      TensorProduct.piRightHom R S S (fun e : {e : E // G.source e ∈ Ω ∧ G.target e ∈ Ω} =>
+        B.edge e.val) (AlgebraTensorModule.lTensor S S (B.constraint Ω) z) := by
+  induction z using TensorProduct.inductionOn with
+  | tmul s f =>
+    ext e
+    change s ⊗ₜ[R] B.lower e.val (f (G.source e.val)) -
+      s ⊗ₜ[R] B.upper e.val (f (G.target e.val)) =
+      s ⊗ₜ[R] (B.lower e.val (f (G.source e.val)) -
+        B.upper e.val (f (G.target e.val)))
+    exact (tmul_sub _ _ _).symm
+  | add z w hz hw => simp only [map_add,hz,hw]; rfl
+
+ theorem tensorSections_surjective [Finite V] [Finite E] [Module.Flat R S] (Ω : Set V) :
+    Function.Surjective (B.tensorSections S Ω) := by
+  classical
+  let := Fintype.ofFinite V
+  let := Fintype.ofFinite E
+  intro f
+  obtain ⟨z,hz⟩ := (B.tensorAssignments_bijective S).2 f.val
+  have hc : (B.baseChange S).constraint Ω (B.tensorAssignments S z) = 0 := by
+    rw [hz]
+    apply LinearMap.mem_ker.mp
+    rw [(B.baseChange S).ker_constraint Ω]
+    exact f.property
+  have he : AlgebraTensorModule.lTensor S S (B.constraint Ω) z = 0 := by
+    apply (TensorProduct.piRight R S S
+      (fun e : {e : E // G.source e ∈ Ω ∧ G.target e ∈ Ω} => B.edge e.val)).injective
+    rw [map_zero]
+    exact (B.constraint_tensorAssignments S Ω z).symm.trans hc
+  have hex : Function.Exact (B.sections Ω).subtype (B.constraint Ω) := by
+    rw [← B.ker_constraint Ω]
+    exact (B.constraint Ω).exact_subtype_ker_map
+  obtain ⟨w,hw⟩ := (Module.Flat.lTensor_exact S hex z).mp he
+  refine ⟨w,?_⟩
+  apply Subtype.ext
+  change B.tensorAssignments S (AlgebraTensorModule.lTensor S S (B.sections Ω).subtype w) = _
+  exact (congrArg (B.tensorAssignments S) hw).trans hz
+
+ theorem tensorSections_injective [Finite V] [Module.Flat R S] (Ω : Set V) :
+    Function.Injective (B.tensorSections S Ω) := by
+  intro x y h
+  apply Module.Flat.lTensor_preserves_injective_linearMap
+    (B.sections Ω).subtype (B.sections Ω).injective_subtype
+  apply (B.tensorAssignments_bijective S).1
+  exact congrArg Subtype.val h
+
+/-- Sections over every vertex subset commute with flat extension of scalars
+for a finite moment graph. No freeness or BMP character input is used. -/
+ def tensorSectionsEquiv [Finite V] [Finite E] [Module.Flat R S] (Ω : Set V) :
+    S ⊗[R] B.sections Ω ≃ₗ[S] (B.baseChange S).sections Ω :=
+  LinearEquiv.ofBijective (B.tensorSections S Ω)
+    ⟨B.tensorSections_injective S Ω,B.tensorSections_surjective S Ω⟩
+
+ theorem baseChange_flabby [Finite V] [Finite E] [Module.Flat R S]
+    (hb : B.Flabby) : (B.baseChange S).Flabby := by
+  intro Ω hΩ f hf
+  obtain ⟨z,hz⟩ := B.tensorSections_surjective S Ω ⟨f,hf⟩
+  suffices ∃ g : (B.baseChange S).Assignment, (B.baseChange S).IsGlobal g ∧
+      ∀ x ∈ Ω, g x = (B.tensorSections S Ω z).val x by
+    simpa only [hz] using this
+  clear hz f hf
+  induction z using TensorProduct.inductionOn with
+  | tmul s f =>
+    obtain ⟨g,hg,hgf⟩ := hb Ω hΩ f.val f.property
+    refine ⟨fun x => s ⊗ₜ g x,B.baseChange_tmul_compatible S Set.univ s g hg,?_⟩
+    intro x hx
+    change s ⊗ₜ[R] g x = s ⊗ₜ[R] f.val x
+    rw [hgf x hx]
+  | add z w hz hw =>
+    obtain ⟨f,hf,hfz⟩ := hz
+    obtain ⟨g,hg,hgw⟩ := hw
+    refine ⟨f+g,Submodule.add_mem ((B.baseChange S).sections Set.univ) hf hg,?_⟩
+    intro x hx
+    simp only [map_add,Submodule.coe_add,Pi.add_apply]
+    rw [hfz x hx,hgw x hx]
+
+end
+end KLInvariance.MomentGraph.Sheaf
+
+end
+
+
+section
+
+
+namespace KLInvariance.MomentGraph.Sheaf
+open TensorProduct
+universe ur us uv ue um un
+variable {R : Type ur} [CommRing R] {V : Type uv} [PartialOrder V]
+  {E : Type ue} {G : OrderedGraph V E}
+noncomputable section
+
+namespace Iso
+variable {B : Sheaf.{ur,uv,ue,um} (R := R) G}
+  {C : Sheaf.{ur,uv,ue,un} (R := R) G} (j : Iso B C)
+include j
+
+ theorem flabby (hb : B.Flabby) : C.Flabby := by
+  intro Ω hΩ f hf
+  have hsf : B.CompatibleOn Ω (j.symm.assignments f) :=
+    (j.symm.compatible_iff Ω f).mpr hf
+  obtain ⟨g,hg,hgf⟩ := hb Ω hΩ (j.symm.assignments f) hsf
+  refine ⟨j.assignments g,(j.compatible_iff Set.univ g).mpr hg,?_⟩
+  intro x hx
+  change j.vertex x (g x) = f x
+  rw [hgf x hx]
+  exact (j.vertex x).apply_symm_apply _
+
+ theorem generated (hb : B.Generated) : C.Generated := by
+  intro x v
+  obtain ⟨f,hf,hfv⟩ := hb x ((j.vertex x).symm v)
+  refine ⟨j.assignments f,(j.compatible_iff Set.univ f).mpr hf,?_⟩
+  change j.vertex x (f x) = v
+  rw [hfv,LinearEquiv.apply_symm_apply]
+
+end Iso
+
+variable (S : Type us) [CommRing S] [Algebra R S]
+
+def baseChangeScalar : S ⊗[R] R ≃ₗ[S] S := AlgebraTensorModule.rid R S S
+
+@[simp] theorem baseChangeScalar_tmul (s : S) (r : R) :
+    baseChangeScalar S (s ⊗ₜ[R] r) = s * algebraMap R S r := by
+  change r • s = s * algebraMap R S r
+  rw [Algebra.smul_def,mul_comm]
+
+ theorem span_map (a : R) :
+    (Ideal.span {a}).map (algebraMap R S) = Ideal.span {algebraMap R S a} := by
+  rw [Ideal.map_span,Set.image_singleton]
+
+ def baseChangeQuotient (a : R) :
+    S ⊗[R] (R ⧸ Ideal.span {a}) ≃ₗ[S] S ⧸ Ideal.span {algebraMap R S a} :=
+  (Algebra.TensorProduct.quotIdealMapEquivTensorQuot S (Ideal.span {a})).symm.toLinearEquiv.trans
+    (Ideal.quotientEquivAlgOfEq S (span_map S a)).toLinearEquiv
+
+@[simp] theorem baseChangeQuotient_tmul (a r : R) (s : S) :
+    baseChangeQuotient S a (s ⊗ₜ[R] Ideal.Quotient.mk _ r) =
+      Ideal.Quotient.mk _ (s * algebraMap R S r) := by
+  simp only [baseChangeQuotient,LinearEquiv.trans_apply,AlgEquiv.toLinearEquiv_apply,
+    Algebra.TensorProduct.quotIdealMapEquivTensorQuot_symm_tmul]
+  change Ideal.Quotient.mk _ (r • s) = Ideal.Quotient.mk _ (s * algebraMap R S r)
+  rw [Algebra.smul_def,mul_comm]
+
+
+ def baseChangePiScalar (p : Prop) :
+    S ⊗[R] (PLift p → R) ≃ₗ[S] (PLift p → S) := by
+  classical
+  exact (TensorProduct.piRight R S S (fun _ : PLift p => R)).trans
+    (LinearEquiv.piCongrRight (fun _ => baseChangeScalar S))
+
+ def baseChangePiQuotient (a : R) (p : Prop) :
+    S ⊗[R] (PLift p → R ⧸ Ideal.span {a}) ≃ₗ[S]
+      (PLift p → S ⧸ Ideal.span {algebraMap R S a}) := by
+  classical
+  exact (TensorProduct.piRight R S S
+    (fun _ : PLift p => R ⧸ Ideal.span {a})).trans
+      (LinearEquiv.piCongrRight (fun _ => baseChangeQuotient S a))
+
+ theorem baseChangePiScalar_tmul (p : Prop) (s : S) (v : PLift p → R) (h : PLift p) :
+    baseChangePiScalar S p (s ⊗ₜ v) h = s * algebraMap R S (v h) :=
+  baseChangeScalar_tmul S s (v h)
+
+ theorem baseChangePiQuotient_tmul (a : R) (p : Prop)
+    (s : S) (v : PLift p → R ⧸ Ideal.span {a}) (h : PLift p) :
+    baseChangePiQuotient S a p (s ⊗ₜ v) h = baseChangeQuotient S a (s ⊗ₜ v h) := rfl
+
+ def quotientPullback (a : R) (p q : Prop) (h : q → p) :
+    (PLift p → R) →ₗ[R] (PLift q → R ⧸ Ideal.span {a}) :=
+  LinearMap.pi fun t => (Ideal.Quotient.mkₐ R (Ideal.span {a})).toLinearMap.comp
+    (LinearMap.proj ⟨h t.down⟩)
+
+ theorem baseChangePi_natural (a : R) (p q : Prop) (h : q → p)
+    (v : S ⊗[R] (PLift p → R)) :
+    baseChangePiQuotient S a q
+      (AlgebraTensorModule.lTensor S S (quotientPullback a p q h) v) =
+      quotientPullback (algebraMap R S a) p q h (baseChangePiScalar S p v) := by
+  induction v using TensorProduct.inductionOn with
+  | tmul s v =>
+    funext hh
+    change baseChangePiQuotient S a q (s ⊗ₜ quotientPullback a p q h v) hh = _
+    rw [baseChangePiQuotient_tmul]
+    change baseChangeQuotient S a (s ⊗ₜ Ideal.Quotient.mk _ (v ⟨h hh.down⟩)) = _
+    rw [baseChangeQuotient_tmul]
+    exact congrArg (Ideal.Quotient.mk _) (baseChangePiScalar_tmul S p s v _).symm
+  | add v w hv hw => simp only [map_add,hv,hw]
+
+ def structureBaseChangeIso (G : OrderedGraph V E) (α : E → R) (c : V) :
+    Iso ((structureSheaf G α c).baseChange S)
+      (structureSheaf G (fun e => algebraMap R S (α e)) c) where
+  vertex x := baseChangePiScalar S (x ≤ c)
+  edge e := baseChangePiQuotient S (α e) (G.target e ≤ c)
+  lower e v := baseChangePi_natural S (α e) (G.source e ≤ c) (G.target e ≤ c)
+    ((G.increasing e).le.trans ·) v
+  upper e v := baseChangePi_natural S (α e) (G.target e ≤ c) (G.target e ≤ c) id v
+
+ theorem structureSheaf_flat_flabby [Finite V] [Finite E] [Module.Flat R S]
+    (G : OrderedGraph V E) (α : E → R) (c : V) (hf : (structureSheaf G α c).Flabby) :
+    (structureSheaf G (fun e => algebraMap R S (α e)) c).Flabby :=
+  (structureBaseChangeIso S G α c).flabby ((structureSheaf G α c).baseChange_flabby S hf)
+
+end
+end KLInvariance.MomentGraph.Sheaf
+
+end
+
+
+section
+
+/-! Restriction to a full vertex subgraph, including the relative-flabbiness
+criterion used after plane localization. -/
+namespace KLInvariance.MomentGraph
+universe ur uv ue um
+variable {V : Type uv} [PartialOrder V] {E : Type ue} (G : OrderedGraph V E)
+
+ def inducedGraph (K : Set V) : OrderedGraph K {e : E // G.source e ∈ K ∧ G.target e ∈ K} where
+  source e := ⟨G.source e.val,e.property.1⟩
+  target e := ⟨G.target e.val,e.property.2⟩
+  increasing e := G.increasing e.val
+
+namespace Sheaf
+variable {G} {R : Type ur} [CommRing R] (B : Sheaf.{ur,uv,ue,um} (R := R) G)
+  (K : Set V)
+noncomputable section
+
+ def restrict : Sheaf (R := R) (inducedGraph G K) where
+  vertex x := B.vertex x.val
+  edge e := B.edge e.val
+  lower e := B.lower e.val
+  upper e := B.upper e.val
+
+ def restrictAssignment (f : B.Assignment) : (B.restrict K).Assignment := fun x => f x.val
+
+ def extendAssignment (f : (B.restrict K).Assignment) : B.Assignment := by
+  classical
+  exact fun x => if h : x ∈ K then f ⟨x,h⟩ else 0
+
+@[simp] theorem extendAssignment_mem (f : (B.restrict K).Assignment) (x : K) :
+    B.extendAssignment K f x.val = f x := by
+  classical
+  change (if h : x.val ∈ K then f ⟨x.val,h⟩ else 0) = f x
+  split_ifs with h
+  · rfl
+  · exact (h x.property).elim
+
+ theorem restrict_compatible {Ω : Set V} {f : B.Assignment} (hf : B.CompatibleOn Ω f) :
+    (B.restrict K).CompatibleOn (Subtype.val ⁻¹' Ω) (B.restrictAssignment K f) := by
+  intro e hs ht
+  exact hf e.val hs ht
+
+ theorem restrict_global (f : B.Assignment) (hf : B.IsGlobal f) :
+    (B.restrict K).IsGlobal (B.restrictAssignment K f) := by
+  intro e _ _
+  exact hf e.val (Set.mem_univ _) (Set.mem_univ _)
+
+ theorem restrict_generated (hg : B.Generated) : (B.restrict K).Generated := by
+  intro x v
+  obtain ⟨f,hf,hfx⟩ := hg x.val v
+  exact ⟨B.restrictAssignment K f,B.restrict_global K f hf,hfx⟩
+
+ theorem extend_compatible (Ω : Set K) (f : (B.restrict K).Assignment)
+    (hf : (B.restrict K).CompatibleOn Ω f) :
+    B.CompatibleOn (Subtype.val '' Ω) (B.extendAssignment K f) := by
+  intro e hs ht
+  obtain ⟨x,hx,hxe⟩ := hs
+  obtain ⟨y,hy,hye⟩ := ht
+  have hsk : G.source e ∈ K := hxe ▸ x.property
+  have htk : G.target e ∈ K := hye ▸ y.property
+  let ee : {e : E // G.source e ∈ K ∧ G.target e ∈ K} := ⟨e,hsk,htk⟩
+  have hsx : (inducedGraph G K).source ee = x := Subtype.ext hxe.symm
+  have hty : (inducedGraph G K).target ee = y := Subtype.ext hye.symm
+  have hh := hf ee (hsx ▸ hx) (hty ▸ hy)
+  change B.lower e (f ⟨G.source e,hsk⟩) = B.upper e (f ⟨G.target e,htk⟩) at hh
+  simpa only [← B.extendAssignment_mem K f] using hh
+
+/-- The induced subgraph may use upper sets which are not ambient upper
+sets. All nonzero exiting edges must stay in the subgraph. -/
+ theorem restrict_flabby_live [Finite V] (hb : B.Flabby) (live : Set E)
+    (hdead : ∀ e, e ∉ live → Subsingleton (B.edge e))
+    (hclosed : ∀ e ∈ live, G.source e ∈ K → G.target e ∈ K) :
+    (B.restrict K).Flabby := by
+  intro Ω hΩ f hf
+  have hc : B.ClosedUnder live (Subtype.val '' Ω) := by
+    intro e he hs
+    obtain ⟨x,hx,hxe⟩ := hs
+    have hsk : G.source e ∈ K := hxe ▸ x.property
+    have htk := hclosed e he hsk
+    refine ⟨⟨G.target e,htk⟩,?_,rfl⟩
+    apply hΩ _ hx
+    change x.val ≤ G.target e
+    rw [hxe]
+    exact (G.increasing e).le
+  obtain ⟨g,hg,hgf⟩ := B.extend_closedUnder hb live hdead (Subtype.val '' Ω) hc
+    (B.extendAssignment K f) (B.extend_compatible K Ω f hf)
+  refine ⟨B.restrictAssignment K g,B.restrict_global K g hg,?_⟩
+  intro x hx
+  exact (hgf x.val ⟨x,hx,rfl⟩).trans (B.extendAssignment_mem K f x)
+
+ theorem restrict_quotient (α : E → R) (hB : B.UpperQuotient α) :
+    (B.restrict K).UpperQuotient (fun e => α e.val) := fun e => hB e.val
+
+end
+end Sheaf
+end KLInvariance.MomentGraph
+
+namespace KLInvariance.MomentGraph.Sheaf
+universe ur uv ue
+variable {R : Type ur} [CommRing R] {V : Type uv} [PartialOrder V]
+  {E : Type ue} (G : OrderedGraph V E) (α : E → R) (K : Set V)
+noncomputable section
+
+ def structureRestrictIso (c : K) :
+    Iso ((structureSheaf G α c.val).restrict K)
+      (structureSheaf (inducedGraph G K) (fun e => α e.val) c) where
+  vertex _ := LinearEquiv.refl R _
+  edge _ := LinearEquiv.refl R _
+  lower _ _ := rfl
+  upper _ _ := rfl
+
+ theorem structureSheaf_edge_subsingleton (c : V) (e : E) (he : ¬ G.target e ≤ c) :
+    Subsingleton ((structureSheaf G α c).edge e) :=
+  ⟨fun _ _ => funext (fun h => False.elim (he h.down))⟩
+
+ theorem structureSheaf_lowerIdeal_flabby [Finite V] (hK : IsLowerSet K) (c : K)
+    (hf : (structureSheaf G α c.val).Flabby) :
+    (structureSheaf (inducedGraph G K) (fun e => α e.val) c).Flabby := by
+  apply (structureRestrictIso G α K c).flabby
+  apply (structureSheaf G α c.val).restrict_flabby_live K hf {e | G.target e ∈ K}
+  · intro e he
+    exact structureSheaf_edge_subsingleton G α c.val e (fun hc => he (hK hc c.property))
+  · intro e he _
+    exact he
+
+end
+end KLInvariance.MomentGraph.Sheaf
+
+end
+
+
+section
+
+/-! Pullback along a directed graph map. Relative-order flabbiness requires
+only outgoing completeness for the surviving edges, not equality of the two
+orders on a coset. -/
+namespace KLInvariance.MomentGraph
+universe ur uv ue uw uf um
+variable {V : Type uv} [PartialOrder V] {E : Type ue}
+  {W : Type uw} [PartialOrder W] {F : Type uf}
+
+structure GraphMap (G : OrderedGraph V E) (H : OrderedGraph W F) where
+  vertex : V → W
+  edge : E → F
+  source : ∀ e, vertex (G.source e) = H.source (edge e)
+  target : ∀ e, vertex (G.target e) = H.target (edge e)
+
+namespace Sheaf
+variable {R : Type ur} [CommRing R] {G : OrderedGraph V E} {H : OrderedGraph W F}
+  (B : Sheaf.{ur,uw,uf,um} (R := R) H) (f : GraphMap G H)
+noncomputable section
+
+ def moduleEq {M N : ModuleCat.{um} R} (h : M=N) : M ≃ₗ[R] N := h ▸ LinearEquiv.refl R M
+
+ theorem moduleEq_apply {X : Type*} (M : X → ModuleCat.{um} R) {x y : X} (h : x=y)
+    (s : ∀ z, M z) : moduleEq (congrArg M h) (s x) = s y := by subst y; rfl
+
+ def pullback : Sheaf (R := R) G where
+  vertex x := B.vertex (f.vertex x)
+  edge e := B.edge (f.edge e)
+  lower e := (B.lower (f.edge e)).comp (moduleEq (congrArg B.vertex (f.source e))).toLinearMap
+  upper e := (B.upper (f.edge e)).comp (moduleEq (congrArg B.vertex (f.target e))).toLinearMap
+
+ def pullAssignment (s : B.Assignment) : (B.pullback f).Assignment := fun x => s (f.vertex x)
+
+@[simp] theorem pull_lower (e : E) (s : B.Assignment) :
+    (B.pullback f).lower e (B.pullAssignment f s (G.source e)) =
+      B.lower (f.edge e) (s (H.source (f.edge e))) := by
+  exact congrArg (B.lower (f.edge e)) (moduleEq_apply B.vertex (f.source e) s)
+
+@[simp] theorem pull_upper (e : E) (s : B.Assignment) :
+    (B.pullback f).upper e (B.pullAssignment f s (G.target e)) =
+      B.upper (f.edge e) (s (H.target (f.edge e))) := by
+  exact congrArg (B.upper (f.edge e)) (moduleEq_apply B.vertex (f.target e) s)
+
+ theorem pull_global (s : B.Assignment) (hs : B.IsGlobal s) :
+    (B.pullback f).IsGlobal (B.pullAssignment f s) := by
+  intro e _ _
+  rw [pull_lower,pull_upper]
+  exact hs (f.edge e) (Set.mem_univ _) (Set.mem_univ _)
+
+ theorem pullback_generated (hg : B.Generated) : (B.pullback f).Generated := by
+  intro x v
+  obtain ⟨s,hs,hx⟩ := hg (f.vertex x) v
+  exact ⟨B.pullAssignment f s,B.pull_global f s hs,hx⟩
+
+ def pushAssignment (s : (B.pullback f).Assignment) : B.Assignment := by
+  classical
+  exact fun y => if h : ∃ x, f.vertex x=y then
+    moduleEq (congrArg B.vertex h.choose_spec) (s h.choose) else 0
+
+ theorem pull_push (hi : Function.Injective f.vertex) (s : (B.pullback f).Assignment) :
+    B.pullAssignment f (B.pushAssignment f s) = s := by
+  classical
+  change (∀ x, B.vertex (f.vertex x)) at s
+  funext x
+  have h : ∃ z, f.vertex z=f.vertex x := ⟨x,rfl⟩
+  change @Eq (B.vertex (f.vertex x))
+    (if hh : ∃ z, f.vertex z=f.vertex x then
+      moduleEq (congrArg B.vertex hh.choose_spec) (s hh.choose) else 0) (s x)
+  rw [dite_eq_left h]
+  have hx : h.choose=x := hi h.choose_spec
+  exact moduleEq_apply (fun z => B.vertex (f.vertex z)) hx s
+
+ theorem pullback_flabby_live [Finite W] (hb : B.Flabby)
+    (hi : Function.Injective f.vertex) (live : Set F)
+    (hdead : ∀ e, e ∉ live → Subsingleton (B.edge e))
+    (hlift : ∀ e ∈ live, ∀ x, f.vertex x = H.source e →
+      ∃ a, f.edge a=e ∧ G.source a=x) : (B.pullback f).Flabby := by
+  intro Ω hΩ s hs
+  have hc : B.ClosedUnder live (f.vertex '' Ω) := by
+    intro e he hsource
+    obtain ⟨x,hx,hxe⟩ := hsource
+    obtain ⟨a,rfl,rfl⟩ := hlift e he x hxe
+    exact ⟨G.target a,hΩ (G.increasing a).le hx,f.target a⟩
+  have hcompat : B.CompatibleOn (f.vertex '' Ω) (B.pushAssignment f s) := by
+    intro e hsource htarget
+    by_cases he : e ∈ live
+    · obtain ⟨x,hx,hxe⟩ := hsource
+      obtain ⟨a,rfl,rfl⟩ := hlift e he x hxe
+      obtain ⟨y,hy,hye⟩ := htarget
+      have hya : y=G.target a := hi (hye.trans (f.target a).symm)
+      subst y
+      have hh := hs a hx hy
+      rw [← B.pull_push f hi s] at hh
+      change (B.pullback f).lower a (B.pullAssignment f (B.pushAssignment f s) (G.source a)) =
+        (B.pullback f).upper a (B.pullAssignment f (B.pushAssignment f s) (G.target a)) at hh
+      rw [pull_lower,pull_upper] at hh
+      exact hh
+    · exact @Subsingleton.elim _ (hdead e he) _ _
+  obtain ⟨t,ht,hts⟩ := B.extend_closedUnder hb live hdead (f.vertex '' Ω) hc
+    (B.pushAssignment f s) hcompat
+  refine ⟨B.pullAssignment f t,B.pull_global f t ht,?_⟩
+  intro x hx
+  change t (f.vertex x) = s x
+  rw [hts (f.vertex x) ⟨x,hx,rfl⟩]
+  exact congrFun (B.pull_push f hi s) x
+
+ theorem pullback_quotient (α : F → R) (hq : B.UpperQuotient α) :
+    (B.pullback f).UpperQuotient (fun e => α (f.edge e)) := by
+  intro e
+  let j := moduleEq (congrArg B.vertex (f.target e))
+  refine ⟨?_,?_⟩
+  · exact (hq (f.edge e)).1.comp j.surjective
+  · intro v
+    change B.upper (f.edge e) (j v) = 0 ↔ _
+    rw [(hq (f.edge e)).2]
+    constructor
+    · rintro ⟨w,hw⟩
+      refine ⟨j.symm w,?_⟩
+      apply j.injective
+      change j v = j (α (f.edge e) • j.symm w)
+      rw [map_smul,LinearEquiv.apply_symm_apply]
+      exact hw
+    · rintro ⟨w,rfl⟩
+      exact ⟨j w,j.map_smul _ _⟩
+
+end
+end Sheaf
+end KLInvariance.MomentGraph
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.Sheaf
+universe ur uv ue uw uf um
+variable {R : Type ur} [CommRing R]
+  {V : Type uv} [PartialOrder V] {E : Type ue} {G : OrderedGraph V E}
+  {W : Type uw} [PartialOrder W] {F : Type uf} {H : OrderedGraph W F}
+  (B : Sheaf.{ur,uw,uf,um} (R := R) H) (f : GraphMap G H)
+noncomputable section
+
+ theorem stalkMap_assignment (s : B.Assignment) (x : W) (e : Outgoing (G := H) x) :
+    B.stalkMap x (s x) e = B.lower e.val (s (H.source e.val)) := by
+  rcases e with ⟨e,he⟩
+  subst x
+  rfl
+
+ theorem pullback_kernel (x : V) (T : Set F)
+    (live : Set F) (hdead : ∀ e, e ∉ live → Subsingleton (B.edge e))
+    (hlift : ∀ e ∈ live, f.vertex x = H.source e →
+      ∃ a, f.edge a=e ∧ G.source a=x) :
+    (B.pullback f).upwardKernel x {e | f.edge e.val ∈ T} =
+      B.upwardKernel (f.vertex x) {e | e.val ∈ T} := by
+  classical
+  ext v
+  change B.vertex (f.vertex x) at v
+  let s : B.Assignment := Function.update (fun _ => 0) (f.vertex x) v
+  have hs : s (f.vertex x)=v := Function.update_self (β := fun y => B.vertex y) (f.vertex x) v (fun _ => 0)
+  have hxs : B.pullAssignment f s x = v := hs
+  constructor
+  · intro hv e he
+    rw [← hs,B.stalkMap_assignment s]
+    by_cases hl : e.val ∈ live
+    · obtain ⟨a,ha,hax⟩ := hlift e.val hl e.property.symm
+      have hh := hv ⟨a,hax⟩ (show f.edge a ∈ T from ha.symm ▸ he)
+      rw [← hxs,(B.pullback f).stalkMap_assignment (B.pullAssignment f s)] at hh
+      rw [B.pull_lower f] at hh
+      exact (congrArg (fun q : F => B.lower q (s (H.source q)) = 0) ha).mp hh
+    · exact @Subsingleton.elim _ (hdead e.val hl) _ _
+  · intro hv e he
+    have hh := hv ⟨f.edge e.val,(f.source e.val).symm.trans (congrArg f.vertex e.property)⟩ he
+    rw [← hs,B.stalkMap_assignment s] at hh
+    rw [← hxs,(B.pullback f).stalkMap_assignment (B.pullAssignment f s),B.pull_lower f]
+    exact hh
+
+end
+end KLInvariance.MomentGraph.Sheaf
+
+end
+
+
+section
+
+/-! Restriction along an outgoing-complete actual graph embedding.  In
+particular this supports passage from [1,b] to [u,b] without changing stalks. -/
+namespace KLInvariance.MomentGraph
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A}
+  {V E W F : Type um} [PartialOrder V] [PartialOrder W]
+  {G : OrderedGraph V E} {H : OrderedGraph W F}
+namespace GradedSheaf
+variable (B : GradedSheaf (𝓐 := 𝓐) H) (f : GraphMap G H)
+
+ theorem moduleEq_graded {x y : W} (h : x=y) (n : ℤ)
+    (v : B.vertex x) (hv : v ∈ (B.vertex x).piece n) :
+    Sheaf.moduleEq (congrArg B.forget.vertex h) v ∈ (B.vertex y).piece n := by
+  subst y
+  exact hv
+
+ def pullback : GradedSheaf (𝓐 := 𝓐) G where
+  vertex x := B.vertex (f.vertex x)
+  edge e := B.edge (f.edge e)
+  lower e := ⟨(B.forget.pullback f).lower e,fun n v hv =>
+    (B.lower (f.edge e)).graded n _ (B.moduleEq_graded (f.source e) n v hv)⟩
+  upper e := ⟨(B.forget.pullback f).upper e,fun n v hv =>
+    (B.upper (f.edge e)).graded n _ (B.moduleEq_graded (f.target e) n v hv)⟩
+
+@[simp] theorem forget_pullback : (B.pullback f).forget = B.forget.pullback f := rfl
+
+ theorem pullback_edgeFlabby (hb : B.EdgeFlabby)
+    (hi : Function.Injective f.vertex)
+    (hlift : ∀ e x, f.vertex x = H.source e → ∃ a, f.edge a=e ∧ G.source a=x) :
+    (B.pullback f).EdgeFlabby := by
+  intro Ω hΩ s hs
+  have hc : B.forget.ClosedUnder Set.univ (f.vertex '' Ω) := by
+    intro e _ hsource
+    obtain ⟨x,hx,hxe⟩ := hsource
+    obtain ⟨a,rfl,rfl⟩ := hlift e x hxe
+    exact ⟨G.target a,hΩ a (Set.mem_univ a) hx,f.target a⟩
+  have hcompat : B.forget.CompatibleOn (f.vertex '' Ω) (B.forget.pushAssignment f s) := by
+    intro e hsource htarget
+    obtain ⟨x,hx,hxe⟩ := hsource
+    obtain ⟨a,rfl,rfl⟩ := hlift e x hxe
+    obtain ⟨y,hy,hye⟩ := htarget
+    have hya : y=G.target a := hi (hye.trans (f.target a).symm)
+    subst y
+    have hh := hs a hx hy
+    change (B.forget.pullback f).lower a (s (G.source a)) =
+      (B.forget.pullback f).upper a (s (G.target a)) at hh
+    rw [← B.forget.pull_push f hi s] at hh
+    rw [Sheaf.pull_lower,Sheaf.pull_upper] at hh
+    exact hh
+  obtain ⟨t,ht,hts⟩ := hb (f.vertex '' Ω) hc (B.forget.pushAssignment f s) hcompat
+  refine ⟨B.forget.pullAssignment f t,B.forget.pull_global f t ht,?_⟩
+  intro x hx
+  change t (f.vertex x) = s x
+  rw [hts (f.vertex x) ⟨x,hx,rfl⟩]
+  exact congrFun (B.forget.pull_push f hi s) x
+
+ theorem pullback_kernelMinimal (hi : Function.Injective f.vertex)
+    (hlift : ∀ e x, f.vertex x = H.source e → ∃ a, f.edge a=e ∧ G.source a=x)
+    (x : V) (hx : B.KernelMinimal (f.vertex x)) :
+    (B.pullback f).KernelMinimal x := by
+  intro s hs
+  have h := hx (B.forget.pushAssignment f s) (by
+    intro e he
+    obtain ⟨a,rfl,hax⟩ := hlift e x he.symm
+    have hh := hs a hax
+    change (B.forget.pullback f).lower a (s (G.source a)) = 0 at hh
+    rw [← B.forget.pull_push f hi s,Sheaf.pull_lower] at hh
+    exact hh)
+  have he : B.forget.pushAssignment f s (f.vertex x)=s x :=
+    congrFun (B.forget.pull_push f hi s) x
+  rwa [he] at h
+
+end GradedSheaf
+
+namespace BoundarySheaf
+variable [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+  {hneg : ∀ n < 0, 𝓐 n = ⊥} {α : F → A} {b : W}
+  (B : BoundarySheaf hneg H α b) (f : GraphMap G H)
+  (hi : Function.Injective f.vertex)
+  (hlift : ∀ e x, f.vertex x = H.source e → ∃ a, f.edge a=e ∧ G.source a=x)
+  (c : V) (hc : f.vertex c=b)
+
+ def pullback : BoundarySheaf hneg G (fun e => α (f.edge e)) c where
+  sheaf := B.sheaf.pullback f
+  free x := B.free (f.vertex x)
+  quotient := B.sheaf.forget.pullback_quotient f α B.quotient
+  flabby := B.sheaf.pullback_edgeFlabby f B.flabby hi hlift
+  generated := B.sheaf.forget.pullback_generated f B.generated
+  top := by change B.sheaf.vertex (f.vertex c)=_; rw [hc,B.top]
+  minimal x hx := B.sheaf.pullback_kernelMinimal f hi hlift x
+    (B.minimal (f.vertex x) (fun h => hx (hi (h.trans hc.symm))))
+
+end BoundarySheaf
+end KLInvariance.MomentGraph
+
+end
+
+
+section
+
+/-! The actual polynomial character of a nonnegative finite graded free module.
+This is not a definition of KL polynomials, and includes no character identity. -/
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/KazhdanLusztig/Model.lean b/lean/OAI/RepresentationTheory/KazhdanLusztig/Model.lean
index 4241bd3c8bdd140122f256bea6207e36414d976c..3e23e83739c86e3d962295a8bfafffd192f0a184 100644
--- a/lean/OAI/RepresentationTheory/KazhdanLusztig/Model.lean
+++ b/lean/OAI/RepresentationTheory/KazhdanLusztig/Model.lean
@@ -1,82 +1,82 @@
-import Mathlib
-
-namespace OAI
-
-namespace KLInvariance
-
-open Polynomial
-
-universe u v u' v'
-
-variable {B : Type u} {W : Type v} [Group W] {M : CoxeterMatrix B}
-
-/-- An upward edge in the *strong* Bruhat graph, with left reflection labels. -/
-def BruhatStep (cs : CoxeterSystem M W) (x y : W) : Prop :=
-  cs.length x < cs.length y ∧ ∃ t : W, cs.IsReflection t ∧ y = t * x
-
-/-- The actual strong Bruhat order, not the weak order. -/
-def BruhatLE (cs : CoxeterSystem M W) : W → W → Prop :=
-  Relation.ReflTransGen (BruhatStep cs)
-
-/-- The unlabelled interval, including both endpoints. -/
-def Interval (cs : CoxeterSystem M W) (u b : W) :=
-  {x : W // BruhatLE cs u x ∧ BruhatLE cs x b}
-
-instance intervalOrder (cs : CoxeterSystem M W) (u b : W) :
-    PartialOrder (Interval cs u b) where
-  le left right := BruhatLE cs left.val right.val
-  le_refl _ := Relation.ReflTransGen.refl
-  le_trans _ _ _ forward backward := forward.trans backward
-  le_antisymm left right forward backward := by
-    apply Subtype.ext
-    have length_mono : ∀ {lower upper : W}, BruhatLE cs lower upper →
-        cs.length lower ≤ cs.length upper := by
-      intro lower upper relation
-      induction relation using Relation.ReflTransGen.head_induction_on with
-      | refl => exact le_rfl
-      | head step _ rest => exact le_trans step.1.le rest
-    by_contra distinct
-    rcases forward.cases_head with same | ⟨middle, step, rest⟩
-    · exact distinct same
-    · exact (not_lt_of_ge (length_mono backward))
-        (lt_of_lt_of_le step.1 (length_mono rest))
-
-/-- `d(x,y)` in the manuscript, used only for comparable endpoints. -/
-noncomputable def rankDifference (cs : CoxeterSystem M W) (x y : W) : ℕ :=
-  cs.length y - cs.length x
-
-abbrev PolynomialFamilies (W : Type v) := (W → W → ℤ[X]) × (W → W → ℤ[X])
-
-/-- Exact equal-parameter normalization in the introduction.
-The first family is R and the second is P.  `reflect d p` is `q^d p(q⁻¹)`
-when `p.natDegree ≤ d`, which follows here from the degree bound (or the
-diagonal normalization). The sum is over the real Bruhat interval;
-its finiteness is a separate theorem, not a different definition of order. -/
-structure NormalizedKL (cs : CoxeterSystem M W) (RP : PolynomialFamilies W) : Prop where
-  R_diagonal : ∀ x, RP.1 x x = 1
-  R_zero : ∀ x y, ¬ BruhatLE cs x y → RP.1 x y = 0
-  R_recursion : ∀ x y i, cs.length (cs.simple i * y) < cs.length y →
-    RP.1 x y =
-      if cs.length (cs.simple i * x) < cs.length x then
-        RP.1 (cs.simple i * x) (cs.simple i * y)
-      else
-        (X - 1) * RP.1 x (cs.simple i * y) +
-          X * RP.1 (cs.simple i * x) (cs.simple i * y)
-  P_diagonal : ∀ x, RP.2 x x = 1
-  P_zero : ∀ x y, ¬ BruhatLE cs x y → RP.2 x y = 0
-  P_degree : ∀ x y, BruhatLE cs x y → x ≠ y →
-    2 * (RP.2 x y).natDegree < rankDifference cs x y
-  reciprocity : ∀ x y, BruhatLE cs x y →
-    reflect (rankDifference cs x y) (RP.2 x y) =
-      ∑ᶠ z : Interval cs x y, RP.1 x z.val * RP.2 z.val y
-
-/-- The polynomial families selected by their equal-parameter normalization. -/
-noncomputable def klFamilies (cs : CoxeterSystem M W) : PolynomialFamilies W :=
-  Classical.epsilon (NormalizedKL cs)
-
-noncomputable def klPolynomial (cs : CoxeterSystem M W) (x y : W) : ℤ[X] :=
-  (klFamilies cs).2 x y
-
-end KLInvariance
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+namespace KLInvariance
+
+open Polynomial
+
+universe u v u' v'
+
+variable {B : Type u} {W : Type v} [Group W] {M : CoxeterMatrix B}
+
+/-- An upward edge in the *strong* Bruhat graph, with left reflection labels. -/
+def BruhatStep (cs : CoxeterSystem M W) (x y : W) : Prop :=
+  cs.length x < cs.length y ∧ ∃ t : W, cs.IsReflection t ∧ y = t * x
+
+/-- The actual strong Bruhat order, not the weak order. -/
+def BruhatLE (cs : CoxeterSystem M W) : W → W → Prop :=
+  Relation.ReflTransGen (BruhatStep cs)
+
+/-- The unlabelled interval, including both endpoints. -/
+def Interval (cs : CoxeterSystem M W) (u b : W) :=
+  {x : W // BruhatLE cs u x ∧ BruhatLE cs x b}
+
+instance intervalOrder (cs : CoxeterSystem M W) (u b : W) :
+    PartialOrder (Interval cs u b) where
+  le left right := BruhatLE cs left.val right.val
+  le_refl _ := Relation.ReflTransGen.refl
+  le_trans _ _ _ forward backward := forward.trans backward
+  le_antisymm left right forward backward := by
+    apply Subtype.ext
+    have length_mono : ∀ {lower upper : W}, BruhatLE cs lower upper →
+        cs.length lower ≤ cs.length upper := by
+      intro lower upper relation
+      induction relation using Relation.ReflTransGen.head_induction_on with
+      | refl => exact le_rfl
+      | head step _ rest => exact le_trans step.1.le rest
+    by_contra distinct
+    rcases forward.cases_head with same | ⟨middle, step, rest⟩
+    · exact distinct same
+    · exact (not_lt_of_ge (length_mono backward))
+        (lt_of_lt_of_le step.1 (length_mono rest))
+
+/-- `d(x,y)` in the manuscript, used only for comparable endpoints. -/
+noncomputable def rankDifference (cs : CoxeterSystem M W) (x y : W) : ℕ :=
+  cs.length y - cs.length x
+
+abbrev PolynomialFamilies (W : Type v) := (W → W → ℤ[X]) × (W → W → ℤ[X])
+
+/-- Exact equal-parameter normalization in the introduction.
+The first family is R and the second is P.  `reflect d p` is `q^d p(q⁻¹)`
+when `p.natDegree ≤ d`, which follows here from the degree bound (or the
+diagonal normalization). The sum is over the real Bruhat interval;
+its finiteness is a separate theorem, not a different definition of order. -/
+structure NormalizedKL (cs : CoxeterSystem M W) (RP : PolynomialFamilies W) : Prop where
+  R_diagonal : ∀ x, RP.1 x x = 1
+  R_zero : ∀ x y, ¬ BruhatLE cs x y → RP.1 x y = 0
+  R_recursion : ∀ x y i, cs.length (cs.simple i * y) < cs.length y →
+    RP.1 x y =
+      if cs.length (cs.simple i * x) < cs.length x then
+        RP.1 (cs.simple i * x) (cs.simple i * y)
+      else
+        (X - 1) * RP.1 x (cs.simple i * y) +
+          X * RP.1 (cs.simple i * x) (cs.simple i * y)
+  P_diagonal : ∀ x, RP.2 x x = 1
+  P_zero : ∀ x y, ¬ BruhatLE cs x y → RP.2 x y = 0
+  P_degree : ∀ x y, BruhatLE cs x y → x ≠ y →
+    2 * (RP.2 x y).natDegree < rankDifference cs x y
+  reciprocity : ∀ x y, BruhatLE cs x y →
+    reflect (rankDifference cs x y) (RP.2 x y) =
+      ∑ᶠ z : Interval cs x y, RP.1 x z.val * RP.2 z.val y
+
+/-- The polynomial families selected by their equal-parameter normalization. -/
+noncomputable def klFamilies (cs : CoxeterSystem M W) : PolynomialFamilies W :=
+  Classical.epsilon (NormalizedKL cs)
+
+noncomputable def klPolynomial (cs : CoxeterSystem M W) (x y : W) : ℤ[X] :=
+  (klFamilies cs).2 x y
+
+end KLInvariance
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/KazhdanLusztig/ModuleAlgebra.lean b/lean/OAI/RepresentationTheory/KazhdanLusztig/ModuleAlgebra.lean
index fa9dbf946c6f621aa3d18faf585978a0a46589db..041cbc16f61cd627aeba39fabb34036d66d84060 100644
--- a/lean/OAI/RepresentationTheory/KazhdanLusztig/ModuleAlgebra.lean
+++ b/lean/OAI/RepresentationTheory/KazhdanLusztig/ModuleAlgebra.lean
@@ -1,628 +1,628 @@
-import Mathlib
-
-namespace OAI
-
-/-!
-Height-one membership into free targets, minimal covers, graded Nakayama, reverse projectivity, saturated edge kernels and strict Hilbert detection.
--/
-
-section
-
-
-namespace KLInvariance.EdgeLattice
-
-variable {R : Type*} [CommRing R] [klPreservedInstance2 : IsDomain R] [klPreservedInstance3 : UniqueFactorizationMonoid R]
-
-include klPreservedInstance2 klPreservedInstance3 in
-private theorem finsupp_divide {ι : Type*} (r : R) (f : ι →₀ R)
-    (h : ∀ i, r ∣ f i) : ∃ g : ι →₀ R, r • g = f := by
-  have _ := klPreservedInstance2
-  have _ := klPreservedInstance3
-  classical
-  let g : ι →₀ R := ∑ i ∈ f.support, Finsupp.single i (Classical.choose (h i))
-  refine ⟨g, ?_⟩
-  dsimp [g]
-  rw [Finset.smul_sum]
-  have heq : (∑ i ∈ f.support, r • Finsupp.single i (Classical.choose (h i))) =
-      ∑ i ∈ f.support, Finsupp.single i (f i) := by
-    apply Finset.sum_congr rfl
-    intro i hi
-    rw [Finsupp.smul_single, smul_eq_mul, ← Classical.choose_spec (h i)]
-  rw [heq]
-  exact Finsupp.sum_single f
-
-/-- In a free module, divisibility can be checked on a basis. -/
-private theorem basis_divide {E : Type*} [AddCommGroup E] [Module R E]
-    {ι : Type*} (b : Module.Basis ι R E) (p : R) (m : E)
-    (h : ∀ i, p ∣ b.repr m i) : ∃ n : E, p • n = m := by
-  obtain ⟨f, hf⟩ := finsupp_divide p (b.repr m) h
-  refine ⟨b.repr.symm f, b.repr.injective ?_⟩
-  simpa only [map_smul, b.repr.apply_symm_apply] using hf
-
-/-- Scalar-denominator formulation of the height-one intersection property.
-A generic-span witness and local denominators away from each prime element
-force membership in a free target, with no reflexivity hypothesis on a source. -/
-theorem mem_of_prime_local_membership
-    {E : Type*} [AddCommGroup E] [Module R E] [Module.IsTorsionFree R E]
-    (M : Submodule R E) [Module.Free R M] (x : E)
-    (hgeneric : ∃ c : R, c ≠ 0 ∧ c • x ∈ M)
-    (hlocal : ∀ p : R, Prime p → ∃ a : R, ¬p ∣ a ∧ a • x ∈ M) : x ∈ M := by
-  classical
-  obtain ⟨c, hcz, hcx⟩ := hgeneric
-  suffices ∀ c : R, c ≠ 0 → c • x ∈ M → x ∈ M from this c hcz hcx
-  intro c
-  induction c using UniqueFactorizationMonoid.induction_on_prime with
-  | h₁ => simp
-  | h₂ c hc =>
-    intro _ hcx
-    obtain ⟨u, rfl⟩ := hc
-    simpa using M.smul_mem (↑u⁻¹ : R) hcx
-  | h₃ c p hcz hp ih =>
-    intro _ hpcx
-    obtain ⟨a, hpa, hax⟩ := hlocal p hp
-    let m : M := ⟨(p * c) • x, hpcx⟩
-    let n : M := ⟨a • x, hax⟩
-    have heq : a • m = (p * c) • n := by
-      apply Subtype.ext
-      change a • ((p * c) • x) = (p * c) • (a • x)
-      exact smul_comm a (p * c) x
-    let b := Module.Free.chooseBasis R M
-    have hdvd : ∀ i, p ∣ b.repr m i := by
-      intro i
-      have hcoord := congrArg (fun z : M => b.repr z i) heq
-      simp only [map_smul, Finsupp.smul_apply, smul_eq_mul] at hcoord
-      have hmul : p ∣ a * b.repr m i := by
-        rw [hcoord]
-        exact dvd_mul_of_dvd_left (dvd_mul_right p c) _
-      exact (hp.dvd_mul.mp hmul).resolve_left hpa
-    obtain ⟨m', hm'⟩ := basis_divide b p m hdvd
-    have hcx : c • x = (m' : E) := by
-      apply smul_right_injective E hp.ne_zero
-      have hval := congrArg (fun z : M => (z : E)) hm'
-      change p • (m' : E) = (p * c) • x at hval
-      simpa only [mul_smul] using hval.symm
-    exact ih hcz (hcx ▸ m'.property)
-
-include klPreservedInstance2 klPreservedInstance3 in
-/-- A localization membership can be written with a scalar denominator outside
-its prime, without requiring a choice of basis in the ambient module. -/
-theorem exists_denominator_of_localized_mem
-    {E : Type*} [AddCommGroup E] [Module R E]
-    (M : Submodule R E) (P : Ideal R) [P.IsPrime] (x : E)
-    (hx : LocalizedModule.mk x (1 : P.primeCompl) ∈ M.localized P.primeCompl) :
-    ∃ a : R, a ∉ P ∧ a • x ∈ M := by
-  have _ := klPreservedInstance2
-  have _ := klPreservedInstance3
-  obtain ⟨m, hm, s, hs⟩ := hx
-  rw [← IsLocalizedModule.mk_eq_mk'] at hs
-  obtain ⟨t, ht⟩ := LocalizedModule.mk_eq.mp hs
-  refine ⟨(t : R) * s, (t * s).property, ?_⟩
-  have hh := M.smul_mem (t : R) hm
-  simp only [Submonoid.smul_def, one_smul] at ht
-  rw [ht] at hh
-  simpa only [mul_smul] using hh
-
-
-theorem mem_iff_localized_at_height_one
-    [IsNoetherianRing R]
-    {E : Type*} [AddCommGroup E] [Module R E] [Module.IsTorsionFree R E]
-    (M : Submodule R E) [Module.Free R M]
-    (hgeneric : ∀ x : E, ∃ c : R, c ≠ 0 ∧ c • x ∈ M) (x : E) :
-    x ∈ M ↔ ∀ (P : Ideal R) [P.IsPrime], P.height = 1 →
-      LocalizedModule.mk x (1 : P.primeCompl) ∈ M.localized P.primeCompl := by
-  constructor
-  · intro hx P _ _
-    exact ⟨x, hx, 1, by simp⟩
-  · intro hx
-    apply mem_of_prime_local_membership M x (hgeneric x)
-    intro p hp
-    let P := Ideal.span ({p} : Set R)
-    have : P.IsPrime := Ideal.isPrime_span_singleton_of_prime hp
-    have hheight : P.height = 1 :=
-      Ideal.height_span_singleton_eq_one_of_mem_nonZeroDivisors
-        (mem_nonZeroDivisors_iff_ne_zero.mpr hp.ne_zero) hp.not_isUnit
-    obtain ⟨a, ha, hax⟩ := exists_denominator_of_localized_mem M P x (hx P hheight)
-    exact ⟨a, by simpa only [P, Ideal.mem_span_singleton] using ha, hax⟩
-
-/-- Equality with the intersection of preimages of all height-one localizations.
-It includes the field case: an empty height-one intersection is the full module. -/
-theorem eq_iInf_height_one_localizations
-    [IsNoetherianRing R]
-    {E : Type*} [AddCommGroup E] [Module R E] [Module.IsTorsionFree R E]
-    (M : Submodule R E) [Module.Free R M]
-    (hgeneric : ∀ x : E, ∃ c : R, c ≠ 0 ∧ c • x ∈ M) :
-    M = ⨅ (P : PrimeSpectrum R) (_ : P.asIdeal.height = 1),
-      ((M.localized P.asIdeal.primeCompl).restrictScalars R).comap
-        (LocalizedModule.mkLinearMap P.asIdeal.primeCompl E) := by
-  ext x
-  simp only [Submodule.mem_iInf, Submodule.mem_comap, Submodule.restrictScalars_mem,
-    LocalizedModule.mkLinearMap_apply]
-  rw [mem_iff_localized_at_height_one M hgeneric x]
-  constructor
-  · intro h P hp
-    exact h P.asIdeal hp
-  · intro h P hP hp
-    exact h ⟨P, hP⟩ hp
-
-end KLInvariance.EdgeLattice
-
-
-end
-
-section
-
-
-/-! The local algebra step in `loc:decomposition`, source §4.4. The maps
-are actual linear maps; neither a decomposition nor a lift is assumed. -/
-namespace KLInvariance.LocalCover
-
-variable {R B P N : Type*} [CommRing R] [IsLocalRing R]
-  [AddCommGroup B] [Module R B] [AddCommGroup P] [Module R P]
-  [AddCommGroup N] [Module R N]
-
-/-- A lift through a minimal free cover is automatically surjective. This
-is the exact Nakayama step of plane-localized BMP decomposition. -/
-theorem lift_surjective [Module.Finite R P]
-    (π : P →ₗ[R] N) (φ : B →ₗ[R] N) (g : B →ₗ[R] P)
-    (hφ : Function.Surjective φ) (hg : π.comp g = φ)
-    (hmin : LinearMap.ker π ≤ IsLocalRing.maximalIdeal R • (⊤ : Submodule R P)) :
-    Function.Surjective g := by
-  apply LinearMap.surjective_of_surjective_comp_mkQ g (IsLocalRing.maximalIdeal R)
-    (IsLocalRing.maximalIdeal_le_jacobson ⊥)
-  intro q
-  obtain ⟨p, rfl⟩ := Submodule.mkQ_surjective _ q
-  obtain ⟨b, hb⟩ := hφ (π p)
-  refine ⟨b, ?_⟩
-  change Submodule.Quotient.mk (g b) = Submodule.Quotient.mk p
-  apply (Submodule.Quotient.eq _).mpr
-  apply hmin
-  change π (g b - p) = 0
-  rw [map_sub, ← LinearMap.comp_apply, hg, hb, sub_self]
-
-/-- An explicit linear equivalence for a split surjection, with the kernel
-as the second summand. It is used below to produce the local sheaf summand. -/
-noncomputable def splitEquiv (g : B →ₗ[R] P) (s : P →ₗ[R] B)
-    (hs : g.comp s = LinearMap.id) : B ≃ₗ[R] P × LinearMap.ker g := by
-  have hs' (p : P) : g (s p) = p := LinearMap.congr_fun hs p
-  let k : B →ₗ[R] LinearMap.ker g :=
-    { toFun := fun b => ⟨b - s (g b), by simp [LinearMap.mem_ker, hs']⟩
-      map_add' := fun x y => Subtype.ext (by simp; abel)
-      map_smul' := fun r x => Subtype.ext (by simp [smul_sub]) }
-  exact
-    { toFun := fun b => (g b, k b)
-      invFun := fun p => s p.1 + p.2
-      map_add' := fun x y => by simp
-      map_smul' := fun r x => by simp
-      left_inv := fun b => by simp [k]
-      right_inv := fun ⟨p, b⟩ => by
-        apply Prod.ext
-        · simp [hs']
-        · apply Subtype.ext
-          simp [k, hs'] }
-
-/-- The full minimal-cover splitting required at each descending vertex.
-The new kernel is proved finite free over the local ring, not introduced
-as a free summand hypothesis. -/
-theorem exists_split_minimal_cover [Module.Free R B] [Module.Finite R B]
-    [Module.Free R P] [Module.Finite R P]
-    (π : P →ₗ[R] N) (φ : B →ₗ[R] N)
-    (hπ : Function.Surjective π) (hφ : Function.Surjective φ)
-    (hmin : LinearMap.ker π ≤ IsLocalRing.maximalIdeal R • (⊤ : Submodule R P)) :
-    ∃ (g : B →ₗ[R] P) (e : B ≃ₗ[R] P × LinearMap.ker g),
-      Function.Surjective g ∧ π.comp g = φ ∧
-      Module.Free R (LinearMap.ker g) ∧ Module.Finite R (LinearMap.ker g) ∧
-      ∀ b, (e b).1 = g b ∧ π (e b).1 = φ b := by
-  obtain ⟨g, hg⟩ := Module.projective_lifting_property π φ hπ
-  have hsurj := lift_surjective π φ g hφ hg hmin
-  obtain ⟨s, hs⟩ := g.exists_rightInverse_of_surjective (LinearMap.range_eq_top.mpr hsurj)
-  let e := splitEquiv g s hs
-  let k : B →ₗ[R] LinearMap.ker g := (LinearMap.snd R P (LinearMap.ker g)).comp e.toLinearMap
-  have hk : Function.Surjective k := by
-    intro b
-    exact ⟨e.symm (0, b), by simp [k]⟩
-  have hki : k.comp (LinearMap.ker g).subtype = LinearMap.id := by
-    ext b
-    change ((b : B) - s (g b)) = b
-    simp []
-  let : Module.Finite R (LinearMap.ker g) := Module.Finite.of_surjective k hk
-  let : Module.Projective R (LinearMap.ker g) :=
-    Module.Projective.of_split (LinearMap.ker g).subtype k hki
-  let : Module.Free R (LinearMap.ker g) := Module.free_of_flat_of_isLocalRing
-  refine ⟨g, e, hsurj, hg, inferInstance, inferInstance, ?_⟩
-  intro b
-  exact ⟨rfl, LinearMap.congr_fun hg b⟩
-
-end KLInvariance.LocalCover
-
-
-namespace KLInvariance.ReverseFiltration
-
-variable {R F E I : Type*} [CommRing R]
-  [AddCommGroup F] [Module R F] [AddCommGroup E] [Module R E]
-  [AddCommGroup I] [Module R I]
-
-/-- The kernel of a surjection from a projective module is projective if
-the target has an actual length-one projective presentation. -/
-theorem kernel_projective_of_presentation
-    [Module.Projective R F] [Module.Projective R E]
-    (f : F →ₗ[R] I) (g : E →ₗ[R] I)
-    (hf : Function.Surjective f) (hg : Function.Surjective g)
-    [Module.Projective R (LinearMap.ker g)] :
-    Module.Projective R (LinearMap.ker f) := by
-  obtain ⟨a, ha⟩ := Module.projective_lifting_property g f hg
-  obtain ⟨b, hb⟩ := Module.projective_lifting_property f g hf
-  have ha' (x : F) : g (a x) = f x := LinearMap.congr_fun ha x
-  have hb' (x : E) : f (b x) = g x := LinearMap.congr_fun hb x
-  let a₀ : LinearMap.ker f →ₗ[R] LinearMap.ker g :=
-    { toFun := fun x => ⟨a x, by simp [LinearMap.mem_ker, ha']⟩
-      map_add' := fun x y => Subtype.ext (by simp)
-      map_smul' := fun r x => Subtype.ext (by simp) }
-  let i : LinearMap.ker f →ₗ[R] F × LinearMap.ker g :=
-    (LinearMap.ker f).subtype.prod a₀
-  let r : F × LinearMap.ker g →ₗ[R] LinearMap.ker f :=
-    { toFun := fun x => ⟨x.1 - b (a x.1) + b x.2, by
-        simp [LinearMap.mem_ker, hb', ha']⟩
-      map_add' := fun x y => Subtype.ext (by simp; abel)
-      map_smul' := fun r x => Subtype.ext (by simp [smul_add, smul_sub]) }
-  apply Module.Projective.of_split i r
-  ext x
-  change (x : F) - b (a x) + b (a x) = x
-  exact sub_add_cancel _ _
-
-end KLInvariance.ReverseFiltration
-
-
-namespace KLInvariance.Graded
-
-variable {k A M : Type*} [CommRing k] [CommRing A] [Algebra k A]
-  [AddCommGroup M] [Module k M] [Module A M] [klPreservedInstance4 : IsScalarTower k A M]
-
-noncomputable def component (𝓜 : ℤ → Submodule k M)
-    [DirectSum.Decomposition 𝓜] (n : ℤ) : M →ₗ[k] M where
-  toFun m := DirectSum.decompose 𝓜 m n
-  map_add' x y := by simp [DirectSum.decompose_add]
-  map_smul' r m := by simp [DirectSum.decompose_smul, DirectSum.smul_apply]
-
-def positiveIdeal (𝓐 : ℤ → Submodule k A) : Ideal A :=
-  Ideal.span {a | ∃ n : ℤ, 0 < n ∧ a ∈ 𝓐 n}
-
-include klPreservedInstance4 in
-/-- A bounded-below internally graded module which equals its positive-ideal
-multiple is zero.  This is the exact graded Nakayama vanishing step used to
-remove the complementary kernel in the source's free-cover construction. -/
-theorem subsingleton_of_positiveIdeal_smul_eq_top
-    (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M)
-    [DirectSum.Decomposition 𝓜] [SetLike.GradedSMul 𝓐 𝓜]
-    (d₀ : ℤ) (hbound : ∀ j < d₀, 𝓜 j = ⊥)
-    (hgen : positiveIdeal 𝓐 • (⊤ : Submodule A M) = ⊤) :
-    Subsingleton M := by
-  have _ := klPreservedInstance4
-  classical
-  have hlow : ∀ j < d₀, ∀ m ∈ 𝓜 j, m = 0 := by
-    intro j hj m hm
-    simpa [hbound j hj] using hm
-  have hind : ∀ n : ℕ, ∀ m ∈ 𝓜 (d₀ + n), m = 0 := by
-    intro n
-    induction n using Nat.strong_induction_on with
-    | h n ih =>
-      let p := component 𝓜 (d₀ + n)
-      have hpadd (x y : M) : p (x + y) = p x + p y := map_add p x y
-      have hpzero : p 0 = 0 := map_zero p
-      have hsmul (i : ℤ) (hi : 0 < i) (a : A) (ha : a ∈ 𝓐 i) :
-          ∀ m : M, p (a • m) = 0 := by
-        intro m
-        refine DirectSum.Decomposition.inductionOn 𝓜 ?_ ?_ ?_ m
-        · simp
-        · intro j m
-          by_cases hj : j < d₀ + n
-          · have hm : (m : M) = 0 := by
-              by_cases hj₀ : j < d₀
-              · exact hlow j hj₀ m m.property
-              · have heq : d₀ + (j - d₀).toNat = j := by omega
-                exact ih (j - d₀).toNat (by omega) m (heq.symm ▸ m.property)
-            simp [hm]
-          · have hdeg : i + j ≠ d₀ + n := by omega
-            have ham : a • (m : M) ∈ 𝓜 (i + j) :=
-              SetLike.GradedSMul.smul_mem ha m.property
-            exact DirectSum.decompose_of_mem_ne 𝓜 ham hdeg
-        · intro x y hx hy
-          simp only [smul_add, hpadd, hx, hy, add_zero]
-      let K : Ideal A :=
-        { carrier := {a | ∀ m : M, p (a • m) = 0}
-          zero_mem' := by simp
-          add_mem' := by
-            intro a b ha hb m
-            rw [add_smul, hpadd, ha m, hb m, add_zero]
-          smul_mem' := by
-            intro r a ha m
-            change p ((r * a) • m) = 0
-            rw [mul_comm r a, mul_smul]
-            exact ha (r • m) }
-      have hIK : positiveIdeal 𝓐 ≤ K := by
-        apply Ideal.span_le.mpr
-        rintro a ⟨i, hi, ha⟩
-        exact hsmul i hi a ha
-      have hp : ∀ m : M, p m = 0 := by
-        intro m
-        have hm : m ∈ positiveIdeal 𝓐 • (⊤ : Submodule A M) := by
-          rw [hgen]
-          trivial
-        refine Submodule.smul_induction_on hm ?_ ?_
-        · intro a ha x _
-          exact hIK ha x
-        · intro x y hx hy
-          rw [hpadd, hx, hy, add_zero]
-      intro m hm
-      have hcomp : p m = m :=
-        DirectSum.decompose_of_mem_same 𝓜 hm
-      exact hcomp.symm.trans (hp m)
-  have homogeneous_zero (j : ℤ) (m : M) (hm : m ∈ 𝓜 j) : m = 0 := by
-    by_cases hj : j < d₀
-    · exact hlow j hj m hm
-    · have heq : d₀ + (j - d₀).toNat = j := by omega
-      exact hind (j - d₀).toNat m (heq.symm ▸ hm)
-  have all_zero : ∀ m : M, m = 0 :=
-    DirectSum.Decomposition.inductionOn 𝓜 rfl
-      (fun m => homogeneous_zero _ m m.property)
-      (fun x y hx hy => by rw [hx, hy, add_zero])
-  exact ⟨fun x y => (all_zero x).trans (all_zero y).symm⟩
-
-end KLInvariance.Graded
-
-
-namespace KLInvariance.EdgeImage
-open scoped Pointwise
-
-variable {A B D : Type*} [CommRing A]
-  [AddCommGroup B] [Module A B] [AddCommGroup D] [Module A D]
-
-/-- The simultaneous vanishing conditions are saturated with respect to α
-when multiplication by α is injective on their target. -/
-theorem kernel_saturated (f : B →ₗ[A] D) (α : A)
-    (hα : Function.Injective (fun x : D => α • x)) (b : B) :
-    α • b ∈ LinearMap.ker f ↔ b ∈ LinearMap.ker f := by
-  simp only [LinearMap.mem_ker, map_smul]
-  constructor
-  · intro h
-    apply hα
-    simpa using h
-  · intro h
-    simp [h]
-
-/-- Reducing the saturated kernel modulo α introduces no additional
-relations. This is the injectivity behind L/αL ≃ J. -/
-theorem reduced_kernel_injective (f : B →ₗ[A] D) (α : A)
-    (hα : Function.Injective (fun x : D => α • x)) :
-    Function.Injective (QuotSMulTop.map α (LinearMap.ker f).subtype) := by
-  apply (LinearMap.ker_eq_bot).mp
-  apply le_antisymm _ bot_le
-  intro x hx
-  change x = 0
-  obtain ⟨b, rfl⟩ := Submodule.mkQ_surjective _ x
-  have hb : (b : B) ∈ α • (⊤ : Submodule A B) := by
-    apply (Submodule.Quotient.mk_eq_zero _).mp
-    exact hx
-  obtain ⟨c, _, hc⟩ := (Submodule.mem_smul_pointwise_iff_exists _ _ _).mp hb
-  have hfc : c ∈ LinearMap.ker f :=
-    (kernel_saturated f α hα c).mp (hc.symm ▸ b.property)
-  apply (Submodule.Quotient.mk_eq_zero _).mpr
-  apply (Submodule.mem_smul_pointwise_iff_exists _ _ _).mpr
-  exact ⟨⟨c, hfc⟩, trivial, Subtype.ext hc⟩
-
-/-- The inclusion after reduction, viewed over the actual factor ring. -/
-def reducedInclusion (f : B →ₗ[A] D) (α : A) :
-    QuotSMulTop α (LinearMap.ker f) →ₗ[A ⧸ Ideal.span {α}] QuotSMulTop α B :=
-  (QuotSMulTop.map α (LinearMap.ker f).subtype).extendScalarsOfSurjective
-    Ideal.Quotient.mk_surjective
-
-/-- Freeness of the later-edge kernel proves freeness of its actual edge
-image over A/(α), not merely generic freeness. -/
-theorem range_reducedInclusion_free (f : B →ₗ[A] D) (α : A)
-    (hα : Function.Injective (fun x : D => α • x))
-    [Module.Free A (LinearMap.ker f)] :
-    Module.Free (A ⧸ Ideal.span {α}) (LinearMap.range (reducedInclusion f α)) := by
-  let e := LinearEquiv.ofInjective (reducedInclusion f α)
-    (reduced_kernel_injective f α hα)
-  exact Module.Free.of_equiv e
-
-end KLInvariance.EdgeImage
-
-end
-
-section
-
-namespace KLInvariance.Graded
-open scoped Pointwise
-variable {A F P : Type*} [CommRing A]
-  [AddCommGroup F] [Module A F] [AddCommGroup P] [Module A P]
-
-/-- If a split cover is minimal modulo I, its complementary kernel equals
-its own I-multiple. This is the exact input to the graded Nakayama step. -/
-theorem positive_smul_kernel_of_minimal_split
-    (I : Ideal A) (f : F →ₗ[A] P) (s : P →ₗ[A] F)
-    (hs : f.comp s = LinearMap.id)
-    (hmin : LinearMap.ker f ≤ I • (⊤ : Submodule A F)) :
-    I • (⊤ : Submodule A (LinearMap.ker f)) = ⊤ := by
-  have hs' (y : P) : f (s y) = y := LinearMap.congr_fun hs y
-  let k : F →ₗ[A] LinearMap.ker f :=
-    { toFun := fun x => ⟨x - s (f x), by simp [LinearMap.mem_ker, hs']⟩
-      map_add' := fun x y => Subtype.ext (by simp; abel)
-      map_smul' := fun r x => Subtype.ext (by simp [smul_sub]) }
-  apply le_antisymm le_top
-  intro x _
-  have hx : (x : F) ∈ I • (⊤ : Submodule A F) := hmin x.property
-  have hkx : k (x : F) = x := by
-    apply Subtype.ext
-    simp [k]
-  rw [← hkx]
-  refine Submodule.smul_induction_on hx ?_ ?_
-  · intro r hr y _
-    rw [map_smul]
-    exact Submodule.smul_mem_smul hr (Submodule.mem_top : k y ∈ (⊤ : Submodule A _))
-  · intro y z hy hz
-    rw [map_add]
-    exact Submodule.add_mem _ hy hz
-
-/-- Minimality makes a split free cover an isomorphism in the bounded-below
-graded situation. Its kernel is not assumed to vanish. -/
-theorem bijective_of_graded_minimal_split
-    {k : Type*} [CommRing k] [Algebra k A]
-    (𝓐 : ℤ → Submodule k A) (f : F →ₗ[A] P) (s : P →ₗ[A] F)
-    (hs : f.comp s = LinearMap.id)
-    (hmin : LinearMap.ker f ≤ positiveIdeal 𝓐 • (⊤ : Submodule A F))
-    [Module k (LinearMap.ker f)] [IsScalarTower k A (LinearMap.ker f)]
-    (𝓚 : ℤ → Submodule k (LinearMap.ker f))
-    [DirectSum.Decomposition 𝓚] [SetLike.GradedSMul 𝓐 𝓚]
-    (d₀ : ℤ) (hbound : ∀ j < d₀, 𝓚 j = ⊥) : Function.Bijective f := by
-  let := subsingleton_of_positiveIdeal_smul_eq_top 𝓐 𝓚 d₀ hbound
-    (positive_smul_kernel_of_minimal_split (positiveIdeal 𝓐) f s hs hmin)
-  constructor
-  · apply LinearMap.ker_eq_bot.mp
-    refine le_antisymm ?_ bot_le
-    intro x hx
-    exact congrArg Subtype.val (Subsingleton.elim (⟨x, hx⟩ : LinearMap.ker f) 0)
-  · intro y
-    exact ⟨s y, LinearMap.congr_fun hs y⟩
-
-end KLInvariance.Graded
-
-
-end
-
-section
-
-namespace KLInvariance.Hilbert
-universe uk um
-variable {k : Type uk} {M : Type um} [Field k] [AddCommGroup M] [Module k M]
-
-/-- The actual degree-n vector space of a submodule, expressed in the fixed
-ambient homogeneous piece. -/
-def piece (𝓜 : ℤ → Submodule k M) (S : Submodule k M) (n : ℤ) :
-    Submodule k (𝓜 n) := S.comap (𝓜 n).subtype
-
-noncomputable def term (𝓜 : ℤ → Submodule k M) (S : Submodule k M)
-    (q : ℝ) (n : ℤ) : ℝ := (Module.finrank k (piece 𝓜 S n) : ℝ)*q^n
-
-noncomputable def value (𝓜 : ℤ → Submodule k M) (S : Submodule k M)
-    (q : ℝ) : ℝ := ∑' n : ℤ, term 𝓜 S q n
-
-lemma piece_mono (𝓜 : ℤ → Submodule k M) {S T : Submodule k M}
-    (h : S ≤ T) (n : ℤ) : piece 𝓜 S n ≤ piece 𝓜 T n :=
-  Submodule.comap_mono h
-
-lemma term_nonneg (𝓜 : ℤ → Submodule k M) (S : Submodule k M)
-    {q : ℝ} (hq : 0 < q) (n : ℤ) : 0 ≤ term 𝓜 S q n := by
-  exact mul_nonneg (Nat.cast_nonneg _) (le_of_lt (zpow_pos hq _))
-
-lemma term_mono (𝓜 : ℤ → Submodule k M) [∀ n, Module.Finite k (𝓜 n)]
-    {S T : Submodule k M} (h : S ≤ T) {q : ℝ} (hq : 0 < q) (n : ℤ) :
-    term 𝓜 S q n ≤ term 𝓜 T q n := by
-  apply mul_le_mul_of_nonneg_right _ (le_of_lt (zpow_pos hq _))
-  exact_mod_cast Submodule.finrank_mono (piece_mono 𝓜 h n)
-
-lemma summable_of_le (𝓜 : ℤ → Submodule k M) [∀ n, Module.Finite k (𝓜 n)]
-    {S T : Submodule k M} (h : S ≤ T) {q : ℝ} (hq : 0 < q)
-    (hT : Summable (term 𝓜 T q)) : Summable (term 𝓜 S q) :=
-  Summable.of_nonneg_of_le (term_nonneg 𝓜 S hq) (term_mono 𝓜 h hq) hT
-
-/-- Equality at one positive convergent parameter, with actual inclusion,
-forces equality of each homogeneous piece. -/
-theorem pieces_eq_of_value_eq (𝓜 : ℤ → Submodule k M)
-    [∀ n, Module.Finite k (𝓜 n)] {S T : Submodule k M}
-    (h : S ≤ T) {q : ℝ} (hq : 0 < q)
-    (hT : Summable (term 𝓜 T q)) (hv : value 𝓜 S q = value 𝓜 T q) :
-    ∀ n, piece 𝓜 S n = piece 𝓜 T n := by
-  intro n
-  apply Submodule.eq_of_le_of_finrank_eq (piece_mono 𝓜 h n)
-  by_contra hne
-  have hlt : Module.finrank k (piece 𝓜 S n) < Module.finrank k (piece 𝓜 T n) :=
-    lt_of_le_of_ne (Submodule.finrank_mono (piece_mono 𝓜 h n)) hne
-  have hterm : term 𝓜 S q n < term 𝓜 T q n := by
-    apply mul_lt_mul_of_pos_right _ (zpow_pos hq n)
-    exact_mod_cast hlt
-  have hstrict := (summable_of_le 𝓜 h hq hT).tsum_lt_tsum
-    (term_mono 𝓜 h hq) hterm hT
-  exact (ne_of_lt hstrict) hv
-
-/-- No assumption on support codimension is involved. Equality of convergent
-Hilbert values detects equality with a homogeneous target globally. -/
-theorem eq_of_value_eq (𝓜 : ℤ → Submodule k M) [DirectSum.Decomposition 𝓜]
-    [∀ n, Module.Finite k (𝓜 n)] {S T : Submodule k M}
-    (h : S ≤ T) (hhom : ∀ n m, m ∈ T → (DirectSum.decompose 𝓜 m n : M) ∈ T)
-    {q : ℝ} (hq : 0 < q) (hT : Summable (term 𝓜 T q))
-    (hv : value 𝓜 S q = value 𝓜 T q) : S = T := by
-  classical
-  apply le_antisymm h
-  intro m hm
-  rw [← DirectSum.sum_support_decompose 𝓜 m]
-  apply Submodule.sum_mem
-  intro n hn
-  have heq := pieces_eq_of_value_eq 𝓜 h hq hT hv n
-  have hmem : DirectSum.decompose 𝓜 m n ∈ piece 𝓜 T n := hhom n m hm
-  rw [←heq] at hmem
-  exact hmem
-
-lemma value_mono (𝓜 : ℤ → Submodule k M) [∀ n, Module.Finite k (𝓜 n)]
-    {S T : Submodule k M} (h : S ≤ T) {q : ℝ} (hq : 0 < q)
-    (hT : Summable (term 𝓜 T q)) : value 𝓜 S q ≤ value 𝓜 T q :=
-  (summable_of_le 𝓜 h hq hT).tsum_le_tsum (term_mono 𝓜 h hq) hT
-
-/-- The degree-n piece of the actual quotient T/S. -/
-abbrev defectPiece (𝓜 : ℤ → Submodule k M) (S T : Submodule k M) (n : ℤ) :=
-  piece 𝓜 T n ⧸ (piece 𝓜 S n).comap (piece 𝓜 T n).subtype
-
-noncomputable def defectTerm (𝓜 : ℤ → Submodule k M) (S T : Submodule k M)
-    (q : ℝ) (n : ℤ) : ℝ := (Module.finrank k (defectPiece 𝓜 S T n) : ℝ)*q^n
-
-lemma defectTerm_eq_sub (𝓜 : ℤ → Submodule k M)
-    [∀ n, Module.Finite k (𝓜 n)] {S T : Submodule k M} (h : S ≤ T)
-    (q : ℝ) (n : ℤ) : defectTerm 𝓜 S T q n = term 𝓜 T q n-term 𝓜 S q n := by
-  have heq := ((piece 𝓜 S n).comap (piece 𝓜 T n).subtype).finrank_quotient_add_finrank
-  rw [(Submodule.comapSubtypeEquivOfLe (piece_mono 𝓜 h n)).finrank_eq] at heq
-  have hc : (Module.finrank k (defectPiece 𝓜 S T n) : ℝ)+
-      (Module.finrank k (piece 𝓜 S n) : ℝ) =
-        Module.finrank k (piece 𝓜 T n) := by exact_mod_cast heq
-  unfold defectTerm term
-  rw [←hc]
-  ring
-
-/-- Additivity of convergent Hilbert values for the actual degreewise quotient. -/
-theorem defect_value_eq_sub (𝓜 : ℤ → Submodule k M)
-    [∀ n, Module.Finite k (𝓜 n)] {S T : Submodule k M} (h : S ≤ T)
-    {q : ℝ} (hq : 0 < q) (hT : Summable (term 𝓜 T q)) :
-    Summable (defectTerm 𝓜 S T q) ∧
-      ∑' n, defectTerm 𝓜 S T q n = value 𝓜 T q-value 𝓜 S q := by
-  have hf : defectTerm 𝓜 S T q = fun n => term 𝓜 T q n-term 𝓜 S q n :=
-    funext (defectTerm_eq_sub 𝓜 h q)
-  rw [hf]
-  exact ⟨hT.sub (summable_of_le 𝓜 h hq hT),
-    hT.tsum_sub (summable_of_le 𝓜 h hq hT)⟩
-
-/-- All proper homogeneous defects have a strictly positive Hilbert series;
-there is no exception for support in codimension ≥2 or finite length. -/
-theorem defect_value_pos (𝓜 : ℤ → Submodule k M) [DirectSum.Decomposition 𝓜]
-    [∀ n, Module.Finite k (𝓜 n)] {S T : Submodule k M}
-    (h : S ≤ T) (hne : S ≠ T)
-    (hhom : ∀ n m, m ∈ T → (DirectSum.decompose 𝓜 m n : M) ∈ T)
-    {q : ℝ} (hq : 0 < q) (hT : Summable (term 𝓜 T q)) :
-    0 < ∑' n, defectTerm 𝓜 S T q n := by
-  rw [(defect_value_eq_sub 𝓜 h hq hT).2]
-  apply sub_pos.mpr
-  apply lt_of_le_of_ne (value_mono 𝓜 h hq hT)
-  intro heq
-  exact hne (eq_of_value_eq 𝓜 h hhom hq hT heq)
-
-end KLInvariance.Hilbert
-
-end
-
-
-section
-
-/-! Minimal homogeneous free covers, restored after the accidental filename
-collision with GradedCoverUniqueness. The generator argument is the existing
-homogeneous-basis proof before its projectivity-only final step. -/
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-!
+Height-one membership into free targets, minimal covers, graded Nakayama, reverse projectivity, saturated edge kernels and strict Hilbert detection.
+-/
+
+section
+
+
+namespace KLInvariance.EdgeLattice
+
+variable {R : Type*} [CommRing R] [klPreservedInstance2 : IsDomain R] [klPreservedInstance3 : UniqueFactorizationMonoid R]
+
+include klPreservedInstance2 klPreservedInstance3 in
+private theorem finsupp_divide {ι : Type*} (r : R) (f : ι →₀ R)
+    (h : ∀ i, r ∣ f i) : ∃ g : ι →₀ R, r • g = f := by
+  have _ := klPreservedInstance2
+  have _ := klPreservedInstance3
+  classical
+  let g : ι →₀ R := ∑ i ∈ f.support, Finsupp.single i (Classical.choose (h i))
+  refine ⟨g, ?_⟩
+  dsimp [g]
+  rw [Finset.smul_sum]
+  have heq : (∑ i ∈ f.support, r • Finsupp.single i (Classical.choose (h i))) =
+      ∑ i ∈ f.support, Finsupp.single i (f i) := by
+    apply Finset.sum_congr rfl
+    intro i hi
+    rw [Finsupp.smul_single, smul_eq_mul, ← Classical.choose_spec (h i)]
+  rw [heq]
+  exact Finsupp.sum_single f
+
+/-- In a free module, divisibility can be checked on a basis. -/
+private theorem basis_divide {E : Type*} [AddCommGroup E] [Module R E]
+    {ι : Type*} (b : Module.Basis ι R E) (p : R) (m : E)
+    (h : ∀ i, p ∣ b.repr m i) : ∃ n : E, p • n = m := by
+  obtain ⟨f, hf⟩ := finsupp_divide p (b.repr m) h
+  refine ⟨b.repr.symm f, b.repr.injective ?_⟩
+  simpa only [map_smul, b.repr.apply_symm_apply] using hf
+
+/-- Scalar-denominator formulation of the height-one intersection property.
+A generic-span witness and local denominators away from each prime element
+force membership in a free target, with no reflexivity hypothesis on a source. -/
+theorem mem_of_prime_local_membership
+    {E : Type*} [AddCommGroup E] [Module R E] [Module.IsTorsionFree R E]
+    (M : Submodule R E) [Module.Free R M] (x : E)
+    (hgeneric : ∃ c : R, c ≠ 0 ∧ c • x ∈ M)
+    (hlocal : ∀ p : R, Prime p → ∃ a : R, ¬p ∣ a ∧ a • x ∈ M) : x ∈ M := by
+  classical
+  obtain ⟨c, hcz, hcx⟩ := hgeneric
+  suffices ∀ c : R, c ≠ 0 → c • x ∈ M → x ∈ M from this c hcz hcx
+  intro c
+  induction c using UniqueFactorizationMonoid.induction_on_prime with
+  | h₁ => simp
+  | h₂ c hc =>
+    intro _ hcx
+    obtain ⟨u, rfl⟩ := hc
+    simpa using M.smul_mem (↑u⁻¹ : R) hcx
+  | h₃ c p hcz hp ih =>
+    intro _ hpcx
+    obtain ⟨a, hpa, hax⟩ := hlocal p hp
+    let m : M := ⟨(p * c) • x, hpcx⟩
+    let n : M := ⟨a • x, hax⟩
+    have heq : a • m = (p * c) • n := by
+      apply Subtype.ext
+      change a • ((p * c) • x) = (p * c) • (a • x)
+      exact smul_comm a (p * c) x
+    let b := Module.Free.chooseBasis R M
+    have hdvd : ∀ i, p ∣ b.repr m i := by
+      intro i
+      have hcoord := congrArg (fun z : M => b.repr z i) heq
+      simp only [map_smul, Finsupp.smul_apply, smul_eq_mul] at hcoord
+      have hmul : p ∣ a * b.repr m i := by
+        rw [hcoord]
+        exact dvd_mul_of_dvd_left (dvd_mul_right p c) _
+      exact (hp.dvd_mul.mp hmul).resolve_left hpa
+    obtain ⟨m', hm'⟩ := basis_divide b p m hdvd
+    have hcx : c • x = (m' : E) := by
+      apply smul_right_injective E hp.ne_zero
+      have hval := congrArg (fun z : M => (z : E)) hm'
+      change p • (m' : E) = (p * c) • x at hval
+      simpa only [mul_smul] using hval.symm
+    exact ih hcz (hcx ▸ m'.property)
+
+include klPreservedInstance2 klPreservedInstance3 in
+/-- A localization membership can be written with a scalar denominator outside
+its prime, without requiring a choice of basis in the ambient module. -/
+theorem exists_denominator_of_localized_mem
+    {E : Type*} [AddCommGroup E] [Module R E]
+    (M : Submodule R E) (P : Ideal R) [P.IsPrime] (x : E)
+    (hx : LocalizedModule.mk x (1 : P.primeCompl) ∈ M.localized P.primeCompl) :
+    ∃ a : R, a ∉ P ∧ a • x ∈ M := by
+  have _ := klPreservedInstance2
+  have _ := klPreservedInstance3
+  obtain ⟨m, hm, s, hs⟩ := hx
+  rw [← IsLocalizedModule.mk_eq_mk'] at hs
+  obtain ⟨t, ht⟩ := LocalizedModule.mk_eq.mp hs
+  refine ⟨(t : R) * s, (t * s).property, ?_⟩
+  have hh := M.smul_mem (t : R) hm
+  simp only [Submonoid.smul_def, one_smul] at ht
+  rw [ht] at hh
+  simpa only [mul_smul] using hh
+
+
+theorem mem_iff_localized_at_height_one
+    [IsNoetherianRing R]
+    {E : Type*} [AddCommGroup E] [Module R E] [Module.IsTorsionFree R E]
+    (M : Submodule R E) [Module.Free R M]
+    (hgeneric : ∀ x : E, ∃ c : R, c ≠ 0 ∧ c • x ∈ M) (x : E) :
+    x ∈ M ↔ ∀ (P : Ideal R) [P.IsPrime], P.height = 1 →
+      LocalizedModule.mk x (1 : P.primeCompl) ∈ M.localized P.primeCompl := by
+  constructor
+  · intro hx P _ _
+    exact ⟨x, hx, 1, by simp⟩
+  · intro hx
+    apply mem_of_prime_local_membership M x (hgeneric x)
+    intro p hp
+    let P := Ideal.span ({p} : Set R)
+    have : P.IsPrime := Ideal.isPrime_span_singleton_of_prime hp
+    have hheight : P.height = 1 :=
+      Ideal.height_span_singleton_eq_one_of_mem_nonZeroDivisors
+        (mem_nonZeroDivisors_iff_ne_zero.mpr hp.ne_zero) hp.not_isUnit
+    obtain ⟨a, ha, hax⟩ := exists_denominator_of_localized_mem M P x (hx P hheight)
+    exact ⟨a, by simpa only [P, Ideal.mem_span_singleton] using ha, hax⟩
+
+/-- Equality with the intersection of preimages of all height-one localizations.
+It includes the field case: an empty height-one intersection is the full module. -/
+theorem eq_iInf_height_one_localizations
+    [IsNoetherianRing R]
+    {E : Type*} [AddCommGroup E] [Module R E] [Module.IsTorsionFree R E]
+    (M : Submodule R E) [Module.Free R M]
+    (hgeneric : ∀ x : E, ∃ c : R, c ≠ 0 ∧ c • x ∈ M) :
+    M = ⨅ (P : PrimeSpectrum R) (_ : P.asIdeal.height = 1),
+      ((M.localized P.asIdeal.primeCompl).restrictScalars R).comap
+        (LocalizedModule.mkLinearMap P.asIdeal.primeCompl E) := by
+  ext x
+  simp only [Submodule.mem_iInf, Submodule.mem_comap, Submodule.restrictScalars_mem,
+    LocalizedModule.mkLinearMap_apply]
+  rw [mem_iff_localized_at_height_one M hgeneric x]
+  constructor
+  · intro h P hp
+    exact h P.asIdeal hp
+  · intro h P hP hp
+    exact h ⟨P, hP⟩ hp
+
+end KLInvariance.EdgeLattice
+
+
+end
+
+section
+
+
+/-! The local algebra step in `loc:decomposition`, source §4.4. The maps
+are actual linear maps; neither a decomposition nor a lift is assumed. -/
+namespace KLInvariance.LocalCover
+
+variable {R B P N : Type*} [CommRing R] [IsLocalRing R]
+  [AddCommGroup B] [Module R B] [AddCommGroup P] [Module R P]
+  [AddCommGroup N] [Module R N]
+
+/-- A lift through a minimal free cover is automatically surjective. This
+is the exact Nakayama step of plane-localized BMP decomposition. -/
+theorem lift_surjective [Module.Finite R P]
+    (π : P →ₗ[R] N) (φ : B →ₗ[R] N) (g : B →ₗ[R] P)
+    (hφ : Function.Surjective φ) (hg : π.comp g = φ)
+    (hmin : LinearMap.ker π ≤ IsLocalRing.maximalIdeal R • (⊤ : Submodule R P)) :
+    Function.Surjective g := by
+  apply LinearMap.surjective_of_surjective_comp_mkQ g (IsLocalRing.maximalIdeal R)
+    (IsLocalRing.maximalIdeal_le_jacobson ⊥)
+  intro q
+  obtain ⟨p, rfl⟩ := Submodule.mkQ_surjective _ q
+  obtain ⟨b, hb⟩ := hφ (π p)
+  refine ⟨b, ?_⟩
+  change Submodule.Quotient.mk (g b) = Submodule.Quotient.mk p
+  apply (Submodule.Quotient.eq _).mpr
+  apply hmin
+  change π (g b - p) = 0
+  rw [map_sub, ← LinearMap.comp_apply, hg, hb, sub_self]
+
+/-- An explicit linear equivalence for a split surjection, with the kernel
+as the second summand. It is used below to produce the local sheaf summand. -/
+noncomputable def splitEquiv (g : B →ₗ[R] P) (s : P →ₗ[R] B)
+    (hs : g.comp s = LinearMap.id) : B ≃ₗ[R] P × LinearMap.ker g := by
+  have hs' (p : P) : g (s p) = p := LinearMap.congr_fun hs p
+  let k : B →ₗ[R] LinearMap.ker g :=
+    { toFun := fun b => ⟨b - s (g b), by simp [LinearMap.mem_ker, hs']⟩
+      map_add' := fun x y => Subtype.ext (by simp; abel)
+      map_smul' := fun r x => Subtype.ext (by simp [smul_sub]) }
+  exact
+    { toFun := fun b => (g b, k b)
+      invFun := fun p => s p.1 + p.2
+      map_add' := fun x y => by simp
+      map_smul' := fun r x => by simp
+      left_inv := fun b => by simp [k]
+      right_inv := fun ⟨p, b⟩ => by
+        apply Prod.ext
+        · simp [hs']
+        · apply Subtype.ext
+          simp [k, hs'] }
+
+/-- The full minimal-cover splitting required at each descending vertex.
+The new kernel is proved finite free over the local ring, not introduced
+as a free summand hypothesis. -/
+theorem exists_split_minimal_cover [Module.Free R B] [Module.Finite R B]
+    [Module.Free R P] [Module.Finite R P]
+    (π : P →ₗ[R] N) (φ : B →ₗ[R] N)
+    (hπ : Function.Surjective π) (hφ : Function.Surjective φ)
+    (hmin : LinearMap.ker π ≤ IsLocalRing.maximalIdeal R • (⊤ : Submodule R P)) :
+    ∃ (g : B →ₗ[R] P) (e : B ≃ₗ[R] P × LinearMap.ker g),
+      Function.Surjective g ∧ π.comp g = φ ∧
+      Module.Free R (LinearMap.ker g) ∧ Module.Finite R (LinearMap.ker g) ∧
+      ∀ b, (e b).1 = g b ∧ π (e b).1 = φ b := by
+  obtain ⟨g, hg⟩ := Module.projective_lifting_property π φ hπ
+  have hsurj := lift_surjective π φ g hφ hg hmin
+  obtain ⟨s, hs⟩ := g.exists_rightInverse_of_surjective (LinearMap.range_eq_top.mpr hsurj)
+  let e := splitEquiv g s hs
+  let k : B →ₗ[R] LinearMap.ker g := (LinearMap.snd R P (LinearMap.ker g)).comp e.toLinearMap
+  have hk : Function.Surjective k := by
+    intro b
+    exact ⟨e.symm (0, b), by simp [k]⟩
+  have hki : k.comp (LinearMap.ker g).subtype = LinearMap.id := by
+    ext b
+    change ((b : B) - s (g b)) = b
+    simp []
+  let : Module.Finite R (LinearMap.ker g) := Module.Finite.of_surjective k hk
+  let : Module.Projective R (LinearMap.ker g) :=
+    Module.Projective.of_split (LinearMap.ker g).subtype k hki
+  let : Module.Free R (LinearMap.ker g) := Module.free_of_flat_of_isLocalRing
+  refine ⟨g, e, hsurj, hg, inferInstance, inferInstance, ?_⟩
+  intro b
+  exact ⟨rfl, LinearMap.congr_fun hg b⟩
+
+end KLInvariance.LocalCover
+
+
+namespace KLInvariance.ReverseFiltration
+
+variable {R F E I : Type*} [CommRing R]
+  [AddCommGroup F] [Module R F] [AddCommGroup E] [Module R E]
+  [AddCommGroup I] [Module R I]
+
+/-- The kernel of a surjection from a projective module is projective if
+the target has an actual length-one projective presentation. -/
+theorem kernel_projective_of_presentation
+    [Module.Projective R F] [Module.Projective R E]
+    (f : F →ₗ[R] I) (g : E →ₗ[R] I)
+    (hf : Function.Surjective f) (hg : Function.Surjective g)
+    [Module.Projective R (LinearMap.ker g)] :
+    Module.Projective R (LinearMap.ker f) := by
+  obtain ⟨a, ha⟩ := Module.projective_lifting_property g f hg
+  obtain ⟨b, hb⟩ := Module.projective_lifting_property f g hf
+  have ha' (x : F) : g (a x) = f x := LinearMap.congr_fun ha x
+  have hb' (x : E) : f (b x) = g x := LinearMap.congr_fun hb x
+  let a₀ : LinearMap.ker f →ₗ[R] LinearMap.ker g :=
+    { toFun := fun x => ⟨a x, by simp [LinearMap.mem_ker, ha']⟩
+      map_add' := fun x y => Subtype.ext (by simp)
+      map_smul' := fun r x => Subtype.ext (by simp) }
+  let i : LinearMap.ker f →ₗ[R] F × LinearMap.ker g :=
+    (LinearMap.ker f).subtype.prod a₀
+  let r : F × LinearMap.ker g →ₗ[R] LinearMap.ker f :=
+    { toFun := fun x => ⟨x.1 - b (a x.1) + b x.2, by
+        simp [LinearMap.mem_ker, hb', ha']⟩
+      map_add' := fun x y => Subtype.ext (by simp; abel)
+      map_smul' := fun r x => Subtype.ext (by simp [smul_add, smul_sub]) }
+  apply Module.Projective.of_split i r
+  ext x
+  change (x : F) - b (a x) + b (a x) = x
+  exact sub_add_cancel _ _
+
+end KLInvariance.ReverseFiltration
+
+
+namespace KLInvariance.Graded
+
+variable {k A M : Type*} [CommRing k] [CommRing A] [Algebra k A]
+  [AddCommGroup M] [Module k M] [Module A M] [klPreservedInstance4 : IsScalarTower k A M]
+
+noncomputable def component (𝓜 : ℤ → Submodule k M)
+    [DirectSum.Decomposition 𝓜] (n : ℤ) : M →ₗ[k] M where
+  toFun m := DirectSum.decompose 𝓜 m n
+  map_add' x y := by simp [DirectSum.decompose_add]
+  map_smul' r m := by simp [DirectSum.decompose_smul, DirectSum.smul_apply]
+
+def positiveIdeal (𝓐 : ℤ → Submodule k A) : Ideal A :=
+  Ideal.span {a | ∃ n : ℤ, 0 < n ∧ a ∈ 𝓐 n}
+
+include klPreservedInstance4 in
+/-- A bounded-below internally graded module which equals its positive-ideal
+multiple is zero.  This is the exact graded Nakayama vanishing step used to
+remove the complementary kernel in the source's free-cover construction. -/
+theorem subsingleton_of_positiveIdeal_smul_eq_top
+    (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M)
+    [DirectSum.Decomposition 𝓜] [SetLike.GradedSMul 𝓐 𝓜]
+    (d₀ : ℤ) (hbound : ∀ j < d₀, 𝓜 j = ⊥)
+    (hgen : positiveIdeal 𝓐 • (⊤ : Submodule A M) = ⊤) :
+    Subsingleton M := by
+  have _ := klPreservedInstance4
+  classical
+  have hlow : ∀ j < d₀, ∀ m ∈ 𝓜 j, m = 0 := by
+    intro j hj m hm
+    simpa [hbound j hj] using hm
+  have hind : ∀ n : ℕ, ∀ m ∈ 𝓜 (d₀ + n), m = 0 := by
+    intro n
+    induction n using Nat.strong_induction_on with
+    | h n ih =>
+      let p := component 𝓜 (d₀ + n)
+      have hpadd (x y : M) : p (x + y) = p x + p y := map_add p x y
+      have hpzero : p 0 = 0 := map_zero p
+      have hsmul (i : ℤ) (hi : 0 < i) (a : A) (ha : a ∈ 𝓐 i) :
+          ∀ m : M, p (a • m) = 0 := by
+        intro m
+        refine DirectSum.Decomposition.inductionOn 𝓜 ?_ ?_ ?_ m
+        · simp
+        · intro j m
+          by_cases hj : j < d₀ + n
+          · have hm : (m : M) = 0 := by
+              by_cases hj₀ : j < d₀
+              · exact hlow j hj₀ m m.property
+              · have heq : d₀ + (j - d₀).toNat = j := by omega
+                exact ih (j - d₀).toNat (by omega) m (heq.symm ▸ m.property)
+            simp [hm]
+          · have hdeg : i + j ≠ d₀ + n := by omega
+            have ham : a • (m : M) ∈ 𝓜 (i + j) :=
+              SetLike.GradedSMul.smul_mem ha m.property
+            exact DirectSum.decompose_of_mem_ne 𝓜 ham hdeg
+        · intro x y hx hy
+          simp only [smul_add, hpadd, hx, hy, add_zero]
+      let K : Ideal A :=
+        { carrier := {a | ∀ m : M, p (a • m) = 0}
+          zero_mem' := by simp
+          add_mem' := by
+            intro a b ha hb m
+            rw [add_smul, hpadd, ha m, hb m, add_zero]
+          smul_mem' := by
+            intro r a ha m
+            change p ((r * a) • m) = 0
+            rw [mul_comm r a, mul_smul]
+            exact ha (r • m) }
+      have hIK : positiveIdeal 𝓐 ≤ K := by
+        apply Ideal.span_le.mpr
+        rintro a ⟨i, hi, ha⟩
+        exact hsmul i hi a ha
+      have hp : ∀ m : M, p m = 0 := by
+        intro m
+        have hm : m ∈ positiveIdeal 𝓐 • (⊤ : Submodule A M) := by
+          rw [hgen]
+          trivial
+        refine Submodule.smul_induction_on hm ?_ ?_
+        · intro a ha x _
+          exact hIK ha x
+        · intro x y hx hy
+          rw [hpadd, hx, hy, add_zero]
+      intro m hm
+      have hcomp : p m = m :=
+        DirectSum.decompose_of_mem_same 𝓜 hm
+      exact hcomp.symm.trans (hp m)
+  have homogeneous_zero (j : ℤ) (m : M) (hm : m ∈ 𝓜 j) : m = 0 := by
+    by_cases hj : j < d₀
+    · exact hlow j hj m hm
+    · have heq : d₀ + (j - d₀).toNat = j := by omega
+      exact hind (j - d₀).toNat m (heq.symm ▸ hm)
+  have all_zero : ∀ m : M, m = 0 :=
+    DirectSum.Decomposition.inductionOn 𝓜 rfl
+      (fun m => homogeneous_zero _ m m.property)
+      (fun x y hx hy => by rw [hx, hy, add_zero])
+  exact ⟨fun x y => (all_zero x).trans (all_zero y).symm⟩
+
+end KLInvariance.Graded
+
+
+namespace KLInvariance.EdgeImage
+open scoped Pointwise
+
+variable {A B D : Type*} [CommRing A]
+  [AddCommGroup B] [Module A B] [AddCommGroup D] [Module A D]
+
+/-- The simultaneous vanishing conditions are saturated with respect to α
+when multiplication by α is injective on their target. -/
+theorem kernel_saturated (f : B →ₗ[A] D) (α : A)
+    (hα : Function.Injective (fun x : D => α • x)) (b : B) :
+    α • b ∈ LinearMap.ker f ↔ b ∈ LinearMap.ker f := by
+  simp only [LinearMap.mem_ker, map_smul]
+  constructor
+  · intro h
+    apply hα
+    simpa using h
+  · intro h
+    simp [h]
+
+/-- Reducing the saturated kernel modulo α introduces no additional
+relations. This is the injectivity behind L/αL ≃ J. -/
+theorem reduced_kernel_injective (f : B →ₗ[A] D) (α : A)
+    (hα : Function.Injective (fun x : D => α • x)) :
+    Function.Injective (QuotSMulTop.map α (LinearMap.ker f).subtype) := by
+  apply (LinearMap.ker_eq_bot).mp
+  apply le_antisymm _ bot_le
+  intro x hx
+  change x = 0
+  obtain ⟨b, rfl⟩ := Submodule.mkQ_surjective _ x
+  have hb : (b : B) ∈ α • (⊤ : Submodule A B) := by
+    apply (Submodule.Quotient.mk_eq_zero _).mp
+    exact hx
+  obtain ⟨c, _, hc⟩ := (Submodule.mem_smul_pointwise_iff_exists _ _ _).mp hb
+  have hfc : c ∈ LinearMap.ker f :=
+    (kernel_saturated f α hα c).mp (hc.symm ▸ b.property)
+  apply (Submodule.Quotient.mk_eq_zero _).mpr
+  apply (Submodule.mem_smul_pointwise_iff_exists _ _ _).mpr
+  exact ⟨⟨c, hfc⟩, trivial, Subtype.ext hc⟩
+
+/-- The inclusion after reduction, viewed over the actual factor ring. -/
+def reducedInclusion (f : B →ₗ[A] D) (α : A) :
+    QuotSMulTop α (LinearMap.ker f) →ₗ[A ⧸ Ideal.span {α}] QuotSMulTop α B :=
+  (QuotSMulTop.map α (LinearMap.ker f).subtype).extendScalarsOfSurjective
+    Ideal.Quotient.mk_surjective
+
+/-- Freeness of the later-edge kernel proves freeness of its actual edge
+image over A/(α), not merely generic freeness. -/
+theorem range_reducedInclusion_free (f : B →ₗ[A] D) (α : A)
+    (hα : Function.Injective (fun x : D => α • x))
+    [Module.Free A (LinearMap.ker f)] :
+    Module.Free (A ⧸ Ideal.span {α}) (LinearMap.range (reducedInclusion f α)) := by
+  let e := LinearEquiv.ofInjective (reducedInclusion f α)
+    (reduced_kernel_injective f α hα)
+  exact Module.Free.of_equiv e
+
+end KLInvariance.EdgeImage
+
+end
+
+section
+
+namespace KLInvariance.Graded
+open scoped Pointwise
+variable {A F P : Type*} [CommRing A]
+  [AddCommGroup F] [Module A F] [AddCommGroup P] [Module A P]
+
+/-- If a split cover is minimal modulo I, its complementary kernel equals
+its own I-multiple. This is the exact input to the graded Nakayama step. -/
+theorem positive_smul_kernel_of_minimal_split
+    (I : Ideal A) (f : F →ₗ[A] P) (s : P →ₗ[A] F)
+    (hs : f.comp s = LinearMap.id)
+    (hmin : LinearMap.ker f ≤ I • (⊤ : Submodule A F)) :
+    I • (⊤ : Submodule A (LinearMap.ker f)) = ⊤ := by
+  have hs' (y : P) : f (s y) = y := LinearMap.congr_fun hs y
+  let k : F →ₗ[A] LinearMap.ker f :=
+    { toFun := fun x => ⟨x - s (f x), by simp [LinearMap.mem_ker, hs']⟩
+      map_add' := fun x y => Subtype.ext (by simp; abel)
+      map_smul' := fun r x => Subtype.ext (by simp [smul_sub]) }
+  apply le_antisymm le_top
+  intro x _
+  have hx : (x : F) ∈ I • (⊤ : Submodule A F) := hmin x.property
+  have hkx : k (x : F) = x := by
+    apply Subtype.ext
+    simp [k]
+  rw [← hkx]
+  refine Submodule.smul_induction_on hx ?_ ?_
+  · intro r hr y _
+    rw [map_smul]
+    exact Submodule.smul_mem_smul hr (Submodule.mem_top : k y ∈ (⊤ : Submodule A _))
+  · intro y z hy hz
+    rw [map_add]
+    exact Submodule.add_mem _ hy hz
+
+/-- Minimality makes a split free cover an isomorphism in the bounded-below
+graded situation. Its kernel is not assumed to vanish. -/
+theorem bijective_of_graded_minimal_split
+    {k : Type*} [CommRing k] [Algebra k A]
+    (𝓐 : ℤ → Submodule k A) (f : F →ₗ[A] P) (s : P →ₗ[A] F)
+    (hs : f.comp s = LinearMap.id)
+    (hmin : LinearMap.ker f ≤ positiveIdeal 𝓐 • (⊤ : Submodule A F))
+    [Module k (LinearMap.ker f)] [IsScalarTower k A (LinearMap.ker f)]
+    (𝓚 : ℤ → Submodule k (LinearMap.ker f))
+    [DirectSum.Decomposition 𝓚] [SetLike.GradedSMul 𝓐 𝓚]
+    (d₀ : ℤ) (hbound : ∀ j < d₀, 𝓚 j = ⊥) : Function.Bijective f := by
+  let := subsingleton_of_positiveIdeal_smul_eq_top 𝓐 𝓚 d₀ hbound
+    (positive_smul_kernel_of_minimal_split (positiveIdeal 𝓐) f s hs hmin)
+  constructor
+  · apply LinearMap.ker_eq_bot.mp
+    refine le_antisymm ?_ bot_le
+    intro x hx
+    exact congrArg Subtype.val (Subsingleton.elim (⟨x, hx⟩ : LinearMap.ker f) 0)
+  · intro y
+    exact ⟨s y, LinearMap.congr_fun hs y⟩
+
+end KLInvariance.Graded
+
+
+end
+
+section
+
+namespace KLInvariance.Hilbert
+universe uk um
+variable {k : Type uk} {M : Type um} [Field k] [AddCommGroup M] [Module k M]
+
+/-- The actual degree-n vector space of a submodule, expressed in the fixed
+ambient homogeneous piece. -/
+def piece (𝓜 : ℤ → Submodule k M) (S : Submodule k M) (n : ℤ) :
+    Submodule k (𝓜 n) := S.comap (𝓜 n).subtype
+
+noncomputable def term (𝓜 : ℤ → Submodule k M) (S : Submodule k M)
+    (q : ℝ) (n : ℤ) : ℝ := (Module.finrank k (piece 𝓜 S n) : ℝ)*q^n
+
+noncomputable def value (𝓜 : ℤ → Submodule k M) (S : Submodule k M)
+    (q : ℝ) : ℝ := ∑' n : ℤ, term 𝓜 S q n
+
+lemma piece_mono (𝓜 : ℤ → Submodule k M) {S T : Submodule k M}
+    (h : S ≤ T) (n : ℤ) : piece 𝓜 S n ≤ piece 𝓜 T n :=
+  Submodule.comap_mono h
+
+lemma term_nonneg (𝓜 : ℤ → Submodule k M) (S : Submodule k M)
+    {q : ℝ} (hq : 0 < q) (n : ℤ) : 0 ≤ term 𝓜 S q n := by
+  exact mul_nonneg (Nat.cast_nonneg _) (le_of_lt (zpow_pos hq _))
+
+lemma term_mono (𝓜 : ℤ → Submodule k M) [∀ n, Module.Finite k (𝓜 n)]
+    {S T : Submodule k M} (h : S ≤ T) {q : ℝ} (hq : 0 < q) (n : ℤ) :
+    term 𝓜 S q n ≤ term 𝓜 T q n := by
+  apply mul_le_mul_of_nonneg_right _ (le_of_lt (zpow_pos hq _))
+  exact_mod_cast Submodule.finrank_mono (piece_mono 𝓜 h n)
+
+lemma summable_of_le (𝓜 : ℤ → Submodule k M) [∀ n, Module.Finite k (𝓜 n)]
+    {S T : Submodule k M} (h : S ≤ T) {q : ℝ} (hq : 0 < q)
+    (hT : Summable (term 𝓜 T q)) : Summable (term 𝓜 S q) :=
+  Summable.of_nonneg_of_le (term_nonneg 𝓜 S hq) (term_mono 𝓜 h hq) hT
+
+/-- Equality at one positive convergent parameter, with actual inclusion,
+forces equality of each homogeneous piece. -/
+theorem pieces_eq_of_value_eq (𝓜 : ℤ → Submodule k M)
+    [∀ n, Module.Finite k (𝓜 n)] {S T : Submodule k M}
+    (h : S ≤ T) {q : ℝ} (hq : 0 < q)
+    (hT : Summable (term 𝓜 T q)) (hv : value 𝓜 S q = value 𝓜 T q) :
+    ∀ n, piece 𝓜 S n = piece 𝓜 T n := by
+  intro n
+  apply Submodule.eq_of_le_of_finrank_eq (piece_mono 𝓜 h n)
+  by_contra hne
+  have hlt : Module.finrank k (piece 𝓜 S n) < Module.finrank k (piece 𝓜 T n) :=
+    lt_of_le_of_ne (Submodule.finrank_mono (piece_mono 𝓜 h n)) hne
+  have hterm : term 𝓜 S q n < term 𝓜 T q n := by
+    apply mul_lt_mul_of_pos_right _ (zpow_pos hq n)
+    exact_mod_cast hlt
+  have hstrict := (summable_of_le 𝓜 h hq hT).tsum_lt_tsum
+    (term_mono 𝓜 h hq) hterm hT
+  exact (ne_of_lt hstrict) hv
+
+/-- No assumption on support codimension is involved. Equality of convergent
+Hilbert values detects equality with a homogeneous target globally. -/
+theorem eq_of_value_eq (𝓜 : ℤ → Submodule k M) [DirectSum.Decomposition 𝓜]
+    [∀ n, Module.Finite k (𝓜 n)] {S T : Submodule k M}
+    (h : S ≤ T) (hhom : ∀ n m, m ∈ T → (DirectSum.decompose 𝓜 m n : M) ∈ T)
+    {q : ℝ} (hq : 0 < q) (hT : Summable (term 𝓜 T q))
+    (hv : value 𝓜 S q = value 𝓜 T q) : S = T := by
+  classical
+  apply le_antisymm h
+  intro m hm
+  rw [← DirectSum.sum_support_decompose 𝓜 m]
+  apply Submodule.sum_mem
+  intro n hn
+  have heq := pieces_eq_of_value_eq 𝓜 h hq hT hv n
+  have hmem : DirectSum.decompose 𝓜 m n ∈ piece 𝓜 T n := hhom n m hm
+  rw [←heq] at hmem
+  exact hmem
+
+lemma value_mono (𝓜 : ℤ → Submodule k M) [∀ n, Module.Finite k (𝓜 n)]
+    {S T : Submodule k M} (h : S ≤ T) {q : ℝ} (hq : 0 < q)
+    (hT : Summable (term 𝓜 T q)) : value 𝓜 S q ≤ value 𝓜 T q :=
+  (summable_of_le 𝓜 h hq hT).tsum_le_tsum (term_mono 𝓜 h hq) hT
+
+/-- The degree-n piece of the actual quotient T/S. -/
+abbrev defectPiece (𝓜 : ℤ → Submodule k M) (S T : Submodule k M) (n : ℤ) :=
+  piece 𝓜 T n ⧸ (piece 𝓜 S n).comap (piece 𝓜 T n).subtype
+
+noncomputable def defectTerm (𝓜 : ℤ → Submodule k M) (S T : Submodule k M)
+    (q : ℝ) (n : ℤ) : ℝ := (Module.finrank k (defectPiece 𝓜 S T n) : ℝ)*q^n
+
+lemma defectTerm_eq_sub (𝓜 : ℤ → Submodule k M)
+    [∀ n, Module.Finite k (𝓜 n)] {S T : Submodule k M} (h : S ≤ T)
+    (q : ℝ) (n : ℤ) : defectTerm 𝓜 S T q n = term 𝓜 T q n-term 𝓜 S q n := by
+  have heq := ((piece 𝓜 S n).comap (piece 𝓜 T n).subtype).finrank_quotient_add_finrank
+  rw [(Submodule.comapSubtypeEquivOfLe (piece_mono 𝓜 h n)).finrank_eq] at heq
+  have hc : (Module.finrank k (defectPiece 𝓜 S T n) : ℝ)+
+      (Module.finrank k (piece 𝓜 S n) : ℝ) =
+        Module.finrank k (piece 𝓜 T n) := by exact_mod_cast heq
+  unfold defectTerm term
+  rw [←hc]
+  ring
+
+/-- Additivity of convergent Hilbert values for the actual degreewise quotient. -/
+theorem defect_value_eq_sub (𝓜 : ℤ → Submodule k M)
+    [∀ n, Module.Finite k (𝓜 n)] {S T : Submodule k M} (h : S ≤ T)
+    {q : ℝ} (hq : 0 < q) (hT : Summable (term 𝓜 T q)) :
+    Summable (defectTerm 𝓜 S T q) ∧
+      ∑' n, defectTerm 𝓜 S T q n = value 𝓜 T q-value 𝓜 S q := by
+  have hf : defectTerm 𝓜 S T q = fun n => term 𝓜 T q n-term 𝓜 S q n :=
+    funext (defectTerm_eq_sub 𝓜 h q)
+  rw [hf]
+  exact ⟨hT.sub (summable_of_le 𝓜 h hq hT),
+    hT.tsum_sub (summable_of_le 𝓜 h hq hT)⟩
+
+/-- All proper homogeneous defects have a strictly positive Hilbert series;
+there is no exception for support in codimension ≥2 or finite length. -/
+theorem defect_value_pos (𝓜 : ℤ → Submodule k M) [DirectSum.Decomposition 𝓜]
+    [∀ n, Module.Finite k (𝓜 n)] {S T : Submodule k M}
+    (h : S ≤ T) (hne : S ≠ T)
+    (hhom : ∀ n m, m ∈ T → (DirectSum.decompose 𝓜 m n : M) ∈ T)
+    {q : ℝ} (hq : 0 < q) (hT : Summable (term 𝓜 T q)) :
+    0 < ∑' n, defectTerm 𝓜 S T q n := by
+  rw [(defect_value_eq_sub 𝓜 h hq hT).2]
+  apply sub_pos.mpr
+  apply lt_of_le_of_ne (value_mono 𝓜 h hq hT)
+  intro heq
+  exact hne (eq_of_value_eq 𝓜 h hhom hq hT heq)
+
+end KLInvariance.Hilbert
+
+end
+
+
+section
+
+/-! Minimal homogeneous free covers, restored after the accidental filename
+collision with GradedCoverUniqueness. The generator argument is the existing
+homogeneous-basis proof before its projectivity-only final step. -/
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/KazhdanLusztig/OrdinaryR.lean b/lean/OAI/RepresentationTheory/KazhdanLusztig/OrdinaryR.lean
index 494cb52246e448f1faf5391f9d36bd71a5baa0f1..02276d4397e3ee003f0668f5377d13b65d424cf8 100644
--- a/lean/OAI/RepresentationTheory/KazhdanLusztig/OrdinaryR.lean
+++ b/lean/OAI/RepresentationTheory/KazhdanLusztig/OrdinaryR.lean
@@ -1,300 +1,300 @@
-import OAI.RepresentationTheory.KazhdanLusztig.Hecke
-
-namespace OAI
-
-/-!
-Construction of ordinary R-polynomials, left-simple recursion, degree bounds and inverse kernels.
--/
-
-section
-namespace KLInvariance
-
-open Polynomial
-
-universe u v u' v'
-
-variable {B : Type u} {W : Type v} [klPreservedInstance1 : Group W] {M : CoxeterMatrix B}
-
-noncomputable local instance : DecidableEq W := Classical.decEq W
-
-namespace OrdinaryR
-
-open LaurentPolynomial
-
-noncomputable def variableUnit : (LaurentPolynomial ℤ)ˣ :=
-  (LaurentPolynomial.isUnit_T (R := ℤ) 1).unit
-
-@[simp] theorem variableUnit_val : (variableUnit : LaurentPolynomial ℤ) = T 1 :=
-  (LaurentPolynomial.isUnit_T (R := ℤ) 1).unit_spec
-
-/-- Polynomiality of the R-values is proved from the Hecke-derived recursion. -/
-theorem rValue_polynomial (cs : CoxeterSystem M W) (x y : W) :
-    ∃ p : ℤ[X], p.toLaurent = HeckeQ.rValue cs variableUnit x y := by
-  classical
-  suffices ∀ n : ℕ, ∀ y : W, cs.length y = n → ∀ x,
-      ∃ p : ℤ[X], p.toLaurent = HeckeQ.rValue cs variableUnit x y by
-    exact this (cs.length y) y rfl x
-  intro n
-  induction n using Nat.strong_induction_on with
-  | h n ih =>
-    intro y hy x
-    by_cases h1 : y = 1
-    · subst y
-      by_cases hx : x = 1
-      · subst x
-        exact ⟨1, by simp⟩
-      · have hn : ¬ BruhatLE cs x 1 := by
-          intro h
-          have hl := length_le_of_bruhat cs h
-          have hx0 : cs.length x = 0 := by simpa using hl
-          exact hx (cs.length_eq_zero_iff.mp hx0)
-        exact ⟨0, by rw [HeckeQ.rValue_zero cs _ _ _ hn]; simp⟩
-    · obtain ⟨i, hi⟩ := cs.exists_leftDescent_of_ne_one h1
-      have hlt : cs.length (cs.simple i * y) < n := hy ▸ hi
-      obtain ⟨p, hp⟩ := ih _ hlt _ rfl x
-      obtain ⟨r, hr⟩ := ih _ hlt _ rfl (cs.simple i * x)
-      rw [HeckeQ.rValue_recursion cs variableUnit x y i hi]
-      split_ifs
-      · exact ⟨r, hr⟩
-      · refine ⟨(X - 1) * p + X * r, ?_⟩
-        simp [hp, hr]
-
-noncomputable def polynomial (cs : CoxeterSystem M W) (x y : W) : ℤ[X] :=
-  LaurentPolynomial.trunc (HeckeQ.rValue cs variableUnit x y)
-
-@[simp] theorem toLaurent_polynomial (cs : CoxeterSystem M W) (x y : W) :
-    (polynomial cs x y).toLaurent = HeckeQ.rValue cs variableUnit x y := by
-  obtain ⟨p, hp⟩ := rValue_polynomial cs x y
-  rw [polynomial, ← hp, Polynomial.trunc_toLaurent]
-
-@[simp] theorem polynomial_diagonal (cs : CoxeterSystem M W) (x : W) :
-    polynomial cs x x = 1 := by
-  apply Polynomial.toLaurent_injective
-  simp
-
-theorem polynomial_zero (cs : CoxeterSystem M W) (x y : W)
-    (hxy : ¬ BruhatLE cs x y) : polynomial cs x y = 0 := by
-  apply Polynomial.toLaurent_injective
-  simp [HeckeQ.rValue_zero cs _ x y hxy]
-
-theorem polynomial_recursion (cs : CoxeterSystem M W) (x y : W) (i : B)
-    (hy : cs.length (cs.simple i * y) < cs.length y) :
-    polynomial cs x y =
-      if cs.length (cs.simple i * x) < cs.length x then
-        polynomial cs (cs.simple i * x) (cs.simple i * y)
-      else (X - 1) * polynomial cs x (cs.simple i * y) +
-        X * polynomial cs (cs.simple i * x) (cs.simple i * y) := by
-  apply Polynomial.toLaurent_injective
-  rw [toLaurent_polynomial, HeckeQ.rValue_recursion cs variableUnit x y i hy]
-  split_ifs <;> simp
-
-/-- Specializing the honest integral polynomial recovers the Hecke R-value
-at every unit parameter, not just the Laurent indeterminate. -/
-theorem polynomial_eval {C : Type*} [CommRing C] (cs : CoxeterSystem M W)
-    (q : Cˣ) (x y : W) :
-    Polynomial.eval₂ (Int.castRingHom C) (q : C) (polynomial cs x y) =
-      HeckeQ.rValue cs q x y := by
-  classical
-  suffices ∀ n : ℕ, ∀ y : W, cs.length y = n → ∀ x,
-      Polynomial.eval₂ (Int.castRingHom C) (q : C) (polynomial cs x y) =
-        HeckeQ.rValue cs q x y by
-    exact this (cs.length y) y rfl x
-  intro n
-  induction n using Nat.strong_induction_on with
-  | h n ih =>
-    intro y hy x
-    by_cases h1 : y = 1
-    · subst y
-      by_cases hx : x = 1
-      · subst x; simp
-      · have hn : ¬ BruhatLE cs x 1 := by
-          intro h
-          have hl := length_le_of_bruhat cs h
-          have hx0 : cs.length x = 0 := by simpa using hl
-          exact hx (cs.length_eq_zero_iff.mp hx0)
-        rw [polynomial_zero cs _ _ hn, HeckeQ.rValue_zero cs _ _ _ hn]
-        simp
-    · obtain ⟨i, hi⟩ := cs.exists_leftDescent_of_ne_one h1
-      have hlt : cs.length (cs.simple i * y) < n := hy ▸ hi
-      rw [polynomial_recursion cs x y i hi, HeckeQ.rValue_recursion cs q x y i hi]
-      split_ifs
-      · exact ih _ hlt _ rfl (cs.simple i * x)
-      · simp [ih _ hlt _ rfl x, ih _ hlt _ rfl (cs.simple i * x)]
-
-/-- The degree bound needed for the classical polynomial reciprocity. -/
-theorem polynomial_degree (cs : CoxeterSystem M W) (x y : W) :
-    (polynomial cs x y).natDegree ≤ rankDifference cs x y := by
-  classical
-  suffices ∀ n : ℕ, ∀ y : W, cs.length y = n → ∀ x,
-      (polynomial cs x y).natDegree ≤ rankDifference cs x y by
-    exact this (cs.length y) y rfl x
-  intro n
-  induction n using Nat.strong_induction_on with
-  | h n ih =>
-    intro y hy x
-    by_cases hxy : BruhatLE cs x y
-    · by_cases heq : x = y
-      · subst x; simp [rankDifference]
-      · have hlen := length_lt_of_bruhat_ne cs hxy heq
-        have hone : y ≠ 1 := by
-          intro h; subst y; simp at hlen
-        obtain ⟨i, hi⟩ := cs.exists_leftDescent_of_ne_one hone
-        have hlt : cs.length (cs.simple i * y) < n := hy ▸ hi
-        have hylen : cs.length y = cs.length (cs.simple i * y) + 1 := by
-          have := cs.length_simple_mul y i
-          omega
-        rw [polynomial_recursion cs x y i hi]
-        by_cases hx : cs.length (cs.simple i * x) < cs.length x
-        · rw [ite_eq_left hx]
-          have hxlen : cs.length x = cs.length (cs.simple i * x) + 1 := by
-            have := cs.length_simple_mul x i
-            omega
-          have hn := ih _ hlt _ rfl (cs.simple i * x)
-          simpa [rankDifference, hylen, hxlen] using hn
-        · rw [ite_eq_right hx]
-          have hxlen : cs.length (cs.simple i * x) = cs.length x + 1 := by
-            have := cs.length_simple_mul x i
-            omega
-          refine (Polynomial.natDegree_add_le _ _).trans (max_le ?_ ?_)
-          · have hn := ih _ hlt _ rfl x
-            have hp : (X - 1 : ℤ[X]).natDegree ≤ 1 := by
-              exact (Polynomial.natDegree_sub_le _ _).trans (by simp)
-            refine Polynomial.natDegree_mul_le.trans ?_
-            dsimp [rankDifference] at hn ⊢
-            omega
-          · by_cases hsy : BruhatLE cs (cs.simple i * x) (cs.simple i * y)
-            · have hl := length_le_of_bruhat cs hsy
-              have hn := ih _ hlt _ rfl (cs.simple i * x)
-              refine Polynomial.natDegree_mul_le.trans ?_
-              simp only [Polynomial.natDegree_X]
-              dsimp [rankDifference] at hn ⊢
-              omega
-            · rw [polynomial_zero cs _ _ hsy, mul_zero]
-              simp
-    · rw [polynomial_zero cs _ _ hxy]
-      simp
-
-@[simp] theorem eval_variableUnit (p : ℤ[X]) :
-    Polynomial.eval₂ (Int.castRingHom (LaurentPolynomial ℤ))
-      (variableUnit : LaurentPolynomial ℤ) p = p.toLaurent := by
-  have h : Polynomial.eval₂RingHom (Int.castRingHom (LaurentPolynomial ℤ))
-      (variableUnit : LaurentPolynomial ℤ) = Polynomial.toLaurent := by
-    apply Polynomial.ringHom_ext
-    · intro a
-      simp only [Polynomial.coe_eval₂RingHom, Polynomial.eval₂_C, Polynomial.toLaurent_C]
-      exact (map_intCast (LaurentPolynomial.C : ℤ →+* LaurentPolynomial ℤ) a).symm
-    · simp
-  exact congrArg (fun f : ℤ[X] →+* LaurentPolynomial ℤ => f p) h
-
-/-- Polynomial R is a kernel for the length-reflection involution. -/
-theorem polynomial_kernel (cs : CoxeterSystem M W) (x y : W) :
-    ∑ z ∈ lowerFinset cs y,
-      polynomial cs x z * reflect (rankDifference cs z y) (polynomial cs z y) =
-      if x = y then 1 else 0 := by
-  classical
-  apply Polynomial.toLaurent_injective
-  rw [map_sum]
-  have h := HeckeQ.rValue_inverse cs variableUnit x y
-  have heq : ∀ z ∈ lowerFinset cs y,
-      (polynomial cs x z * reflect (rankDifference cs z y) (polynomial cs z y)).toLaurent =
-      HeckeQ.rValue cs variableUnit x z *
-        (variableUnit : LaurentPolynomial ℤ) ^ cs.length y *
-        (↑variableUnit⁻¹ : LaurentPolynomial ℤ) ^ cs.length z *
-        HeckeQ.rValue cs variableUnit⁻¹ z y := by
-    intro z hz
-    have hle := length_le_of_bruhat cs ((mem_lowerFinset cs y z).mp hz)
-    rw [map_mul, toLaurent_polynomial, ← eval_variableUnit,
-      eval₂_reflect_unit _ _ _ (polynomial_degree cs z y), polynomial_eval]
-    have hlen : cs.length y = rankDifference cs z y + cs.length z := by
-      dsimp [rankDifference]; omega
-    rw [hlen, pow_add]
-    simp only [mul_assoc]
-    rw [← mul_assoc ((variableUnit : LaurentPolynomial ℤ) ^ cs.length z),
-      ← mul_pow, Units.mul_inv, one_pow, one_mul]
-  rw [Finset.sum_congr rfl heq, h]
-  split_ifs <;> simp
-
-end OrdinaryR
-
-noncomputable def properIntervalFinset (cs : CoxeterSystem M W) (x y : W) : Finset W := by
-  classical
-  exact (lowerFinset cs y).filter (fun z => BruhatLE cs x z ∧ x ≠ z)
-
-@[simp] theorem mem_properIntervalFinset (cs : CoxeterSystem M W) (x y z : W) :
-    z ∈ properIntervalFinset cs x y ↔ BruhatLE cs z y ∧ BruhatLE cs x z ∧ x ≠ z := by
-  classical
-  simp [properIntervalFinset]
-
-theorem rankDifference_add (cs : CoxeterSystem M W) {x z y : W}
-    (hxz : BruhatLE cs x z) (hzy : BruhatLE cs z y) :
-    rankDifference cs x y = rankDifference cs x z + rankDifference cs z y := by
-  have := length_le_of_bruhat cs hxz
-  have := length_le_of_bruhat cs hzy
-  dsimp [rankDifference]
-  omega
-
-theorem reflect_finset_sum {A : Type*} (s : Finset A) (d : ℕ) (f : A → ℤ[X]) :
-    reflect d (∑ a ∈ s, f a) = ∑ a ∈ s, reflect d (f a) := by
-  classical
-  induction s using Finset.induction_on with
-  | empty => simp
-  | @insert a s ha ih => simp only [Finset.sum_insert ha, reflect_add, ih]
-
-namespace OrdinaryR
-
-theorem polynomial_kernel_reverse (cs : CoxeterSystem M W) (x y : W) :
-    ∑ z ∈ lowerFinset cs y,
-      reflect (rankDifference cs x z) (polynomial cs x z) * polynomial cs z y =
-      if x = y then 1 else 0 := by
-  classical
-  by_cases hxy : BruhatLE cs x y
-  · have h := congrArg (reflect (rankDifference cs x y)) (polynomial_kernel cs x y)
-    rw [reflect_finset_sum] at h
-    have heq : ∀ z ∈ lowerFinset cs y,
-        reflect (rankDifference cs x y)
-          (polynomial cs x z * reflect (rankDifference cs z y) (polynomial cs z y)) =
-        reflect (rankDifference cs x z) (polynomial cs x z) * polynomial cs z y := by
-      intro z hz
-      by_cases hxz : BruhatLE cs x z
-      · rw [rankDifference_add cs hxz ((mem_lowerFinset cs y z).mp hz),
-          reflect_mul _ _ (polynomial_degree cs x z)
-            (Polynomial.natDegree_reflect_le.trans (max_le le_rfl (polynomial_degree cs z y))), reflect_reflect]
-      · rw [polynomial_zero cs x z hxz, zero_mul, reflect_zero, reflect_zero, zero_mul]
-    rw [Finset.sum_congr rfl heq] at h
-    convert h using 1
-    split_ifs with he
-    · subst y
-      simp [rankDifference]
-    · simp
-  · rw [ite_eq_right (by intro he; subst y; exact hxy (bruhat_refl cs x))]
-    apply Finset.sum_eq_zero
-    intro z hz
-    have hxz : ¬BruhatLE cs x z := fun hxz =>
-      hxy (bruhat_trans cs hxz ((mem_lowerFinset cs y z).mp hz))
-    rw [polynomial_zero cs x z hxz, reflect_zero, zero_mul]
-
-end OrdinaryR
-
-theorem sum_proper_eq_sub (cs : CoxeterSystem M W) (x y : W)
-    (hxy : BruhatLE cs x y) (f : W → ℤ[X])
-    (hzero : ∀ z, ¬BruhatLE cs x z → f z = 0) :
-    ∑ z ∈ properIntervalFinset cs x y, f z =
-      (∑ z ∈ lowerFinset cs y, f z) - f x := by
-  classical
-  rw [← Finset.sum_erase_eq_sub ((mem_lowerFinset cs y x).mpr hxy)]
-  apply Finset.sum_subset
-  · intro z hz
-    have hz := (mem_properIntervalFinset cs x y z).mp hz
-    exact Finset.mem_erase.mpr ⟨Ne.symm hz.2.2, (mem_lowerFinset cs y z).mpr hz.1⟩
-  · intro z hz hnot
-    apply hzero
-    intro hxz
-    have hz := Finset.mem_erase.mp hz
-    exact hnot ((mem_properIntervalFinset cs x y z).mpr
-      ⟨(mem_lowerFinset cs y z).mp hz.2, hxz, Ne.symm hz.1⟩)
-
-
-end KLInvariance
-end
-
-end OAI
+import OAI.RepresentationTheory.KazhdanLusztig.Hecke
+
+namespace OAI
+
+/-!
+Construction of ordinary R-polynomials, left-simple recursion, degree bounds and inverse kernels.
+-/
+
+section
+namespace KLInvariance
+
+open Polynomial
+
+universe u v u' v'
+
+variable {B : Type u} {W : Type v} [klPreservedInstance1 : Group W] {M : CoxeterMatrix B}
+
+noncomputable local instance : DecidableEq W := Classical.decEq W
+
+namespace OrdinaryR
+
+open LaurentPolynomial
+
+noncomputable def variableUnit : (LaurentPolynomial ℤ)ˣ :=
+  (LaurentPolynomial.isUnit_T (R := ℤ) 1).unit
+
+@[simp] theorem variableUnit_val : (variableUnit : LaurentPolynomial ℤ) = T 1 :=
+  (LaurentPolynomial.isUnit_T (R := ℤ) 1).unit_spec
+
+/-- Polynomiality of the R-values is proved from the Hecke-derived recursion. -/
+theorem rValue_polynomial (cs : CoxeterSystem M W) (x y : W) :
+    ∃ p : ℤ[X], p.toLaurent = HeckeQ.rValue cs variableUnit x y := by
+  classical
+  suffices ∀ n : ℕ, ∀ y : W, cs.length y = n → ∀ x,
+      ∃ p : ℤ[X], p.toLaurent = HeckeQ.rValue cs variableUnit x y by
+    exact this (cs.length y) y rfl x
+  intro n
+  induction n using Nat.strong_induction_on with
+  | h n ih =>
+    intro y hy x
+    by_cases h1 : y = 1
+    · subst y
+      by_cases hx : x = 1
+      · subst x
+        exact ⟨1, by simp⟩
+      · have hn : ¬ BruhatLE cs x 1 := by
+          intro h
+          have hl := length_le_of_bruhat cs h
+          have hx0 : cs.length x = 0 := by simpa using hl
+          exact hx (cs.length_eq_zero_iff.mp hx0)
+        exact ⟨0, by rw [HeckeQ.rValue_zero cs _ _ _ hn]; simp⟩
+    · obtain ⟨i, hi⟩ := cs.exists_leftDescent_of_ne_one h1
+      have hlt : cs.length (cs.simple i * y) < n := hy ▸ hi
+      obtain ⟨p, hp⟩ := ih _ hlt _ rfl x
+      obtain ⟨r, hr⟩ := ih _ hlt _ rfl (cs.simple i * x)
+      rw [HeckeQ.rValue_recursion cs variableUnit x y i hi]
+      split_ifs
+      · exact ⟨r, hr⟩
+      · refine ⟨(X - 1) * p + X * r, ?_⟩
+        simp [hp, hr]
+
+noncomputable def polynomial (cs : CoxeterSystem M W) (x y : W) : ℤ[X] :=
+  LaurentPolynomial.trunc (HeckeQ.rValue cs variableUnit x y)
+
+@[simp] theorem toLaurent_polynomial (cs : CoxeterSystem M W) (x y : W) :
+    (polynomial cs x y).toLaurent = HeckeQ.rValue cs variableUnit x y := by
+  obtain ⟨p, hp⟩ := rValue_polynomial cs x y
+  rw [polynomial, ← hp, Polynomial.trunc_toLaurent]
+
+@[simp] theorem polynomial_diagonal (cs : CoxeterSystem M W) (x : W) :
+    polynomial cs x x = 1 := by
+  apply Polynomial.toLaurent_injective
+  simp
+
+theorem polynomial_zero (cs : CoxeterSystem M W) (x y : W)
+    (hxy : ¬ BruhatLE cs x y) : polynomial cs x y = 0 := by
+  apply Polynomial.toLaurent_injective
+  simp [HeckeQ.rValue_zero cs _ x y hxy]
+
+theorem polynomial_recursion (cs : CoxeterSystem M W) (x y : W) (i : B)
+    (hy : cs.length (cs.simple i * y) < cs.length y) :
+    polynomial cs x y =
+      if cs.length (cs.simple i * x) < cs.length x then
+        polynomial cs (cs.simple i * x) (cs.simple i * y)
+      else (X - 1) * polynomial cs x (cs.simple i * y) +
+        X * polynomial cs (cs.simple i * x) (cs.simple i * y) := by
+  apply Polynomial.toLaurent_injective
+  rw [toLaurent_polynomial, HeckeQ.rValue_recursion cs variableUnit x y i hy]
+  split_ifs <;> simp
+
+/-- Specializing the honest integral polynomial recovers the Hecke R-value
+at every unit parameter, not just the Laurent indeterminate. -/
+theorem polynomial_eval {C : Type*} [CommRing C] (cs : CoxeterSystem M W)
+    (q : Cˣ) (x y : W) :
+    Polynomial.eval₂ (Int.castRingHom C) (q : C) (polynomial cs x y) =
+      HeckeQ.rValue cs q x y := by
+  classical
+  suffices ∀ n : ℕ, ∀ y : W, cs.length y = n → ∀ x,
+      Polynomial.eval₂ (Int.castRingHom C) (q : C) (polynomial cs x y) =
+        HeckeQ.rValue cs q x y by
+    exact this (cs.length y) y rfl x
+  intro n
+  induction n using Nat.strong_induction_on with
+  | h n ih =>
+    intro y hy x
+    by_cases h1 : y = 1
+    · subst y
+      by_cases hx : x = 1
+      · subst x; simp
+      · have hn : ¬ BruhatLE cs x 1 := by
+          intro h
+          have hl := length_le_of_bruhat cs h
+          have hx0 : cs.length x = 0 := by simpa using hl
+          exact hx (cs.length_eq_zero_iff.mp hx0)
+        rw [polynomial_zero cs _ _ hn, HeckeQ.rValue_zero cs _ _ _ hn]
+        simp
+    · obtain ⟨i, hi⟩ := cs.exists_leftDescent_of_ne_one h1
+      have hlt : cs.length (cs.simple i * y) < n := hy ▸ hi
+      rw [polynomial_recursion cs x y i hi, HeckeQ.rValue_recursion cs q x y i hi]
+      split_ifs
+      · exact ih _ hlt _ rfl (cs.simple i * x)
+      · simp [ih _ hlt _ rfl x, ih _ hlt _ rfl (cs.simple i * x)]
+
+/-- The degree bound needed for the classical polynomial reciprocity. -/
+theorem polynomial_degree (cs : CoxeterSystem M W) (x y : W) :
+    (polynomial cs x y).natDegree ≤ rankDifference cs x y := by
+  classical
+  suffices ∀ n : ℕ, ∀ y : W, cs.length y = n → ∀ x,
+      (polynomial cs x y).natDegree ≤ rankDifference cs x y by
+    exact this (cs.length y) y rfl x
+  intro n
+  induction n using Nat.strong_induction_on with
+  | h n ih =>
+    intro y hy x
+    by_cases hxy : BruhatLE cs x y
+    · by_cases heq : x = y
+      · subst x; simp [rankDifference]
+      · have hlen := length_lt_of_bruhat_ne cs hxy heq
+        have hone : y ≠ 1 := by
+          intro h; subst y; simp at hlen
+        obtain ⟨i, hi⟩ := cs.exists_leftDescent_of_ne_one hone
+        have hlt : cs.length (cs.simple i * y) < n := hy ▸ hi
+        have hylen : cs.length y = cs.length (cs.simple i * y) + 1 := by
+          have := cs.length_simple_mul y i
+          omega
+        rw [polynomial_recursion cs x y i hi]
+        by_cases hx : cs.length (cs.simple i * x) < cs.length x
+        · rw [ite_eq_left hx]
+          have hxlen : cs.length x = cs.length (cs.simple i * x) + 1 := by
+            have := cs.length_simple_mul x i
+            omega
+          have hn := ih _ hlt _ rfl (cs.simple i * x)
+          simpa [rankDifference, hylen, hxlen] using hn
+        · rw [ite_eq_right hx]
+          have hxlen : cs.length (cs.simple i * x) = cs.length x + 1 := by
+            have := cs.length_simple_mul x i
+            omega
+          refine (Polynomial.natDegree_add_le _ _).trans (max_le ?_ ?_)
+          · have hn := ih _ hlt _ rfl x
+            have hp : (X - 1 : ℤ[X]).natDegree ≤ 1 := by
+              exact (Polynomial.natDegree_sub_le _ _).trans (by simp)
+            refine Polynomial.natDegree_mul_le.trans ?_
+            dsimp [rankDifference] at hn ⊢
+            omega
+          · by_cases hsy : BruhatLE cs (cs.simple i * x) (cs.simple i * y)
+            · have hl := length_le_of_bruhat cs hsy
+              have hn := ih _ hlt _ rfl (cs.simple i * x)
+              refine Polynomial.natDegree_mul_le.trans ?_
+              simp only [Polynomial.natDegree_X]
+              dsimp [rankDifference] at hn ⊢
+              omega
+            · rw [polynomial_zero cs _ _ hsy, mul_zero]
+              simp
+    · rw [polynomial_zero cs _ _ hxy]
+      simp
+
+@[simp] theorem eval_variableUnit (p : ℤ[X]) :
+    Polynomial.eval₂ (Int.castRingHom (LaurentPolynomial ℤ))
+      (variableUnit : LaurentPolynomial ℤ) p = p.toLaurent := by
+  have h : Polynomial.eval₂RingHom (Int.castRingHom (LaurentPolynomial ℤ))
+      (variableUnit : LaurentPolynomial ℤ) = Polynomial.toLaurent := by
+    apply Polynomial.ringHom_ext
+    · intro a
+      simp only [Polynomial.coe_eval₂RingHom, Polynomial.eval₂_C, Polynomial.toLaurent_C]
+      exact (map_intCast (LaurentPolynomial.C : ℤ →+* LaurentPolynomial ℤ) a).symm
+    · simp
+  exact congrArg (fun f : ℤ[X] →+* LaurentPolynomial ℤ => f p) h
+
+/-- Polynomial R is a kernel for the length-reflection involution. -/
+theorem polynomial_kernel (cs : CoxeterSystem M W) (x y : W) :
+    ∑ z ∈ lowerFinset cs y,
+      polynomial cs x z * reflect (rankDifference cs z y) (polynomial cs z y) =
+      if x = y then 1 else 0 := by
+  classical
+  apply Polynomial.toLaurent_injective
+  rw [map_sum]
+  have h := HeckeQ.rValue_inverse cs variableUnit x y
+  have heq : ∀ z ∈ lowerFinset cs y,
+      (polynomial cs x z * reflect (rankDifference cs z y) (polynomial cs z y)).toLaurent =
+      HeckeQ.rValue cs variableUnit x z *
+        (variableUnit : LaurentPolynomial ℤ) ^ cs.length y *
+        (↑variableUnit⁻¹ : LaurentPolynomial ℤ) ^ cs.length z *
+        HeckeQ.rValue cs variableUnit⁻¹ z y := by
+    intro z hz
+    have hle := length_le_of_bruhat cs ((mem_lowerFinset cs y z).mp hz)
+    rw [map_mul, toLaurent_polynomial, ← eval_variableUnit,
+      eval₂_reflect_unit _ _ _ (polynomial_degree cs z y), polynomial_eval]
+    have hlen : cs.length y = rankDifference cs z y + cs.length z := by
+      dsimp [rankDifference]; omega
+    rw [hlen, pow_add]
+    simp only [mul_assoc]
+    rw [← mul_assoc ((variableUnit : LaurentPolynomial ℤ) ^ cs.length z),
+      ← mul_pow, Units.mul_inv, one_pow, one_mul]
+  rw [Finset.sum_congr rfl heq, h]
+  split_ifs <;> simp
+
+end OrdinaryR
+
+noncomputable def properIntervalFinset (cs : CoxeterSystem M W) (x y : W) : Finset W := by
+  classical
+  exact (lowerFinset cs y).filter (fun z => BruhatLE cs x z ∧ x ≠ z)
+
+@[simp] theorem mem_properIntervalFinset (cs : CoxeterSystem M W) (x y z : W) :
+    z ∈ properIntervalFinset cs x y ↔ BruhatLE cs z y ∧ BruhatLE cs x z ∧ x ≠ z := by
+  classical
+  simp [properIntervalFinset]
+
+theorem rankDifference_add (cs : CoxeterSystem M W) {x z y : W}
+    (hxz : BruhatLE cs x z) (hzy : BruhatLE cs z y) :
+    rankDifference cs x y = rankDifference cs x z + rankDifference cs z y := by
+  have := length_le_of_bruhat cs hxz
+  have := length_le_of_bruhat cs hzy
+  dsimp [rankDifference]
+  omega
+
+theorem reflect_finset_sum {A : Type*} (s : Finset A) (d : ℕ) (f : A → ℤ[X]) :
+    reflect d (∑ a ∈ s, f a) = ∑ a ∈ s, reflect d (f a) := by
+  classical
+  induction s using Finset.induction_on with
+  | empty => simp
+  | @insert a s ha ih => simp only [Finset.sum_insert ha, reflect_add, ih]
+
+namespace OrdinaryR
+
+theorem polynomial_kernel_reverse (cs : CoxeterSystem M W) (x y : W) :
+    ∑ z ∈ lowerFinset cs y,
+      reflect (rankDifference cs x z) (polynomial cs x z) * polynomial cs z y =
+      if x = y then 1 else 0 := by
+  classical
+  by_cases hxy : BruhatLE cs x y
+  · have h := congrArg (reflect (rankDifference cs x y)) (polynomial_kernel cs x y)
+    rw [reflect_finset_sum] at h
+    have heq : ∀ z ∈ lowerFinset cs y,
+        reflect (rankDifference cs x y)
+          (polynomial cs x z * reflect (rankDifference cs z y) (polynomial cs z y)) =
+        reflect (rankDifference cs x z) (polynomial cs x z) * polynomial cs z y := by
+      intro z hz
+      by_cases hxz : BruhatLE cs x z
+      · rw [rankDifference_add cs hxz ((mem_lowerFinset cs y z).mp hz),
+          reflect_mul _ _ (polynomial_degree cs x z)
+            (Polynomial.natDegree_reflect_le.trans (max_le le_rfl (polynomial_degree cs z y))), reflect_reflect]
+      · rw [polynomial_zero cs x z hxz, zero_mul, reflect_zero, reflect_zero, zero_mul]
+    rw [Finset.sum_congr rfl heq] at h
+    convert h using 1
+    split_ifs with he
+    · subst y
+      simp [rankDifference]
+    · simp
+  · rw [ite_eq_right (by intro he; subst y; exact hxy (bruhat_refl cs x))]
+    apply Finset.sum_eq_zero
+    intro z hz
+    have hxz : ¬BruhatLE cs x z := fun hxz =>
+      hxy (bruhat_trans cs hxz ((mem_lowerFinset cs y z).mp hz))
+    rw [polynomial_zero cs x z hxz, reflect_zero, zero_mul]
+
+end OrdinaryR
+
+theorem sum_proper_eq_sub (cs : CoxeterSystem M W) (x y : W)
+    (hxy : BruhatLE cs x y) (f : W → ℤ[X])
+    (hzero : ∀ z, ¬BruhatLE cs x z → f z = 0) :
+    ∑ z ∈ properIntervalFinset cs x y, f z =
+      (∑ z ∈ lowerFinset cs y, f z) - f x := by
+  classical
+  rw [← Finset.sum_erase_eq_sub ((mem_lowerFinset cs y x).mpr hxy)]
+  apply Finset.sum_subset
+  · intro z hz
+    have hz := (mem_properIntervalFinset cs x y z).mp hz
+    exact Finset.mem_erase.mpr ⟨Ne.symm hz.2.2, (mem_lowerFinset cs y z).mpr hz.1⟩
+  · intro z hz hnot
+    apply hzero
+    intro hxz
+    have hz := Finset.mem_erase.mp hz
+    exact hnot ((mem_properIntervalFinset cs x y z).mpr
+      ⟨(mem_lowerFinset cs y z).mp hz.2, hxz, Ne.symm hz.1⟩)
+
+
+end KLInvariance
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/KazhdanLusztig/PlaneLocalization.lean b/lean/OAI/RepresentationTheory/KazhdanLusztig/PlaneLocalization.lean
index 4fe964b3a89bdf962a2a2d04d17ef4db4d693674..19b044d244799dfcd5816fdc3455b76c0e73b2b3 100644
--- a/lean/OAI/RepresentationTheory/KazhdanLusztig/PlaneLocalization.lean
+++ b/lean/OAI/RepresentationTheory/KazhdanLusztig/PlaneLocalization.lean
@@ -1,1763 +1,1763 @@
-import OAI.RepresentationTheory.KazhdanLusztig.DihedralIntervals
-
-namespace OAI
-
-/-!
-Symmetric plane ideals, sheaf localization and scaled local products along genuine edges.
--/
-
-section
-
-namespace KLInvariance.SymmetricPlaneIdeal
-universe uk uv
-variable {k : Type uk} [Field k] {V : Type uv} [AddCommGroup V] [Module k V]
-noncomputable section
-
- def ideal (P : Submodule k V) : Ideal (SymmetricAlgebra k V) :=
-  Ideal.span ((SymmetricAlgebra.ι k V) '' (P : Set V))
-
- def quotientMap (P : Submodule k V) :
-    SymmetricAlgebra k V →ₐ[k] SymmetricAlgebra k (V ⧸ P) :=
-  SymmetricAlgebra.lift ((SymmetricAlgebra.ι k (V ⧸ P)).comp P.mkQ)
-
-@[simp] theorem quotientMap_ι (P : Submodule k V) (v : V) :
-    quotientMap P (SymmetricAlgebra.ι k V v) = SymmetricAlgebra.ι k (V ⧸ P) (P.mkQ v) :=
-  SymmetricAlgebra.lift_ι_apply _ _
-
- theorem ideal_le_ker (P : Submodule k V) : ideal P ≤ RingHom.ker (quotientMap P) := by
-  apply Ideal.span_le.mpr
-  rintro _ ⟨v,hv,rfl⟩
-  change quotientMap P (SymmetricAlgebra.ι k V v) = 0
-  rw [quotientMap_ι,show P.mkQ v = 0 from (Submodule.Quotient.mk_eq_zero P).mpr hv,map_zero]
-
- def forward (P : Submodule k V) : (SymmetricAlgebra k V ⧸ ideal P) →ₐ[k]
-    SymmetricAlgebra k (V ⧸ P) :=
-  Ideal.Quotient.liftₐ _ (quotientMap P) (fun _ hz => ideal_le_ker P hz)
-
- def linearQuotient (P : Submodule k V) : V ⧸ P →ₗ[k]
-    (SymmetricAlgebra k V ⧸ ideal P) :=
-  P.liftQ ((Ideal.Quotient.mkₐ k (ideal P)).toLinearMap.comp (SymmetricAlgebra.ι k V)) (by
-    intro v hv
-    apply LinearMap.mem_ker.mpr
-    exact Ideal.Quotient.eq_zero_iff_mem.mpr (Ideal.subset_span ⟨v,hv,rfl⟩))
-
- def backward (P : Submodule k V) : SymmetricAlgebra k (V ⧸ P) →ₐ[k]
-    (SymmetricAlgebra k V ⧸ ideal P) := SymmetricAlgebra.lift (linearQuotient P)
-
-@[simp] theorem forward_mk (P : Submodule k V) (z : SymmetricAlgebra k V) :
-    forward P (Ideal.Quotient.mk (ideal P) z) = quotientMap P z := rfl
-
-@[simp] theorem backward_ι_mk (P : Submodule k V) (v : V) :
-    backward P (SymmetricAlgebra.ι k (V ⧸ P) (P.mkQ v)) =
-      Ideal.Quotient.mk (ideal P) (SymmetricAlgebra.ι k V v) := by
-  rw [backward,SymmetricAlgebra.lift_ι_apply]
-  rfl
-
- def quotientEquiv (P : Submodule k V) : (SymmetricAlgebra k V ⧸ ideal P) ≃ₐ[k]
-    SymmetricAlgebra k (V ⧸ P) :=
-  AlgEquiv.ofAlgHom (forward P) (backward P)
-    (by
-      apply SymmetricAlgebra.algHom_ext
-      apply LinearMap.ext
-      intro v
-      obtain ⟨v,rfl⟩ := P.mkQ_surjective v
-      change forward P (backward P (SymmetricAlgebra.ι k _ (P.mkQ v))) = _
-      rw [backward_ι_mk,forward_mk,quotientMap_ι]
-      rfl)
-    (by
-      apply Ideal.Quotient.algHom_ext
-      apply SymmetricAlgebra.algHom_ext
-      apply LinearMap.ext
-      intro v
-      change backward P (forward P (Ideal.Quotient.mk (ideal P) (SymmetricAlgebra.ι k V v))) = _
-      rw [forward_mk,quotientMap_ι,backward_ι_mk]
-      rfl)
-
-@[simp] theorem quotientEquiv_mk_ι (P : Submodule k V) (v : V) :
-    quotientEquiv P (Ideal.Quotient.mk (ideal P) (SymmetricAlgebra.ι k V v)) =
-      SymmetricAlgebra.ι k (V ⧸ P) (P.mkQ v) := quotientMap_ι P v
-
- theorem ι_mem_iff (P : Submodule k V) (v : V) :
-    SymmetricAlgebra.ι k V v ∈ ideal P ↔ v ∈ P := by
-  constructor
-  · intro hv
-    have hh := ideal_le_ker P hv
-    change quotientMap P (SymmetricAlgebra.ι k V v) = 0 at hh
-    rw [quotientMap_ι] at hh
-    apply (Submodule.Quotient.mk_eq_zero P).mp
-    have he : SymmetricAlgebra.ι k (V ⧸ P) (P.mkQ v) =
-        SymmetricAlgebra.ι k (V ⧸ P) 0 := by simpa only [map_zero] using hh
-    by_contra hn
-    obtain ⟨f,hf⟩ := Module.Projective.exists_dual_ne_zero k hn
-    have h := congrArg (SymmetricAlgebra.lift f) he
-    rw [SymmetricAlgebra.lift_ι_apply,SymmetricAlgebra.lift_ι_apply,map_zero] at h
-    exact hf h
-  · intro hv
-    exact Ideal.subset_span ⟨v,hv,rfl⟩
-
- theorem ideal_eq_ker (P : Submodule k V) : ideal P = RingHom.ker (quotientMap P) := by
-  apply le_antisymm (ideal_le_ker P)
-  intro z hz
-  apply Ideal.Quotient.eq_zero_iff_mem.mp
-  apply (quotientEquiv P).injective
-  change quotientMap P z = quotientEquiv P 0
-  rw [map_zero]
-  exact hz
-
- instance ideal_isPrime (P : Submodule k V) : (ideal P).IsPrime := by
-  rw [ideal_eq_ker]
-  exact RingHom.ker_isPrime (quotientMap P)
-
- theorem ideal_span_eq (s : Set V) :
-    ideal (Submodule.span k s) = Ideal.span ((SymmetricAlgebra.ι k V) '' s) := by
-  apply le_antisymm
-  · apply Ideal.span_le.mpr
-    rintro _ ⟨v,hv,rfl⟩
-    induction hv using Submodule.span_induction with
-    | mem x hx => exact Ideal.subset_span ⟨x,hx,rfl⟩
-    | zero =>
-      rw [map_zero]
-      exact (Ideal.span ((SymmetricAlgebra.ι k V) '' s)).zero_mem
-    | add x y _ _ hx hy =>
-      rw [map_add]
-      exact (Ideal.span ((SymmetricAlgebra.ι k V) '' s)).add_mem hx hy
-    | smul r x _ hx =>
-      rw [map_smul,Algebra.smul_def]
-      exact (Ideal.span ((SymmetricAlgebra.ι k V) '' s)).mul_mem_left _ hx
-  · apply Ideal.span_le.mpr
-    rintro _ ⟨v,hv,rfl⟩
-    exact (ι_mem_iff _ _).mpr (Submodule.subset_span hv)
-
-end
-end KLInvariance.SymmetricPlaneIdeal
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.Sheaf
-universe u v w z
-variable {R : Type u} [CommRing R] [IsLocalRing R]
-  {V : Type v} [PartialOrder V] {E : Type w} {G : OrderedGraph V E}
-  {α : E → R} (O : LocalModels.{u,v,w,z} (G := G) α)
-  {B C : Sheaf.{u,v,w,z} (R := R) G}
-
-
-theorem extend_modelSum {U : Set V} (hU : IsUpperSet U)
-    (hC : ModelSum O.sheaf U C) (j : OnIso B C U)
-    (hBq : B.UpperQuotient α) (hBf : B.Flabby) (hBg : B.Generated)
-    (x : V) (hx : x ∉ U) (hmax : Set.Ioi x ⊆ U)
-    [Module.Free R (B.vertex x)] [Module.Finite R (B.vertex x)] :
-    ∃ D : Sheaf.{u,v,w,z} (R := R) G,
-      ModelSum O.sheaf (insert x U) D ∧ Nonempty (OnIso B D (insert x U)) := by
-  classical
-  let := hC.free O x
-  let := hC.finite O x
-  obtain ⟨g,jv,hfree,hfinite,hjv⟩ := (j.above x hmax).extend_stalk
-    hBf hBg (hC.flabby O) (hC.generated O) (hC.minimal O x hx)
-  let := hfree
-  let := hfinite
-  let n := Module.finrank R (LinearMap.ker g)
-  let S : Sheaf.{u,v,w,z} (R := R) G := pi (fun _ : Fin n => O.sheaf x)
-  let jq : LinearMap.ker g ≃ₗ[R] S.vertex x :=
-    (Module.finBasis R (LinearMap.ker g)).equivFun.trans
-      (LinearEquiv.piCongrRight (fun _ : Fin n => O.top x)).symm
-  let jv' : B.vertex x ≃ₗ[R] (prod C S).vertex x :=
-    jv.trans (LinearEquiv.prodCongr (LinearEquiv.refl R (C.vertex x)) jq)
-  have hSV (y : V) (hy : y ∈ U) : Subsingleton (S.vertex y) :=
-    pi_subsingleton_vertex _ y (fun _ => O.support x y (fun hyx => hx (hU hyx hy)))
-  have hSE (e : E) (he : G.target e ∈ U) : Subsingleton (S.edge e) := by
-    apply pi_subsingleton_edge
-    intro i
-    let := O.support x (G.target e) (fun h => hx (hU h he))
-    exact (O.quotient x e).1.subsingleton
-  let jj : OnIso B (prod C S) U := j.prodZero S hSV hSE
-  have hq : (prod C S).UpperQuotient α :=
-    prod_upperQuotient C S α (hC.quotient O) (pi_upperQuotient _ α (fun _ => O.quotient x))
-  have hout : ∀ e (he : G.source e = x) v,
-      jj.edge e (hmax (lt_of_eq_of_lt he.symm (G.increasing e)))
-        (B.lower e v) = (prod C S).lower e (castStalkIso B (prod C S) he.symm jv' v) := by
-    intro e he v
-    subst x
-    apply Prod.ext
-    · exact hjv v ⟨e,rfl⟩
-    · let := hSE e (hmax (G.increasing e))
-      exact Subsingleton.elim _ _
-  refine ⟨prod C S,?_,⟨jj.extend hU x hx hmax α hBq hq jv' hout⟩⟩
-  exact .add (hC.mono (Set.subset_insert x U)) x (Set.mem_insert x U) n
-
-end KLInvariance.MomentGraph.Sheaf
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.Sheaf
-universe u v w z
-variable {R : Type u} [CommRing R] [IsLocalRing R]
-  {V : Type v} [PartialOrder V] {E : Type w} {G : OrderedGraph V E}
-  {α : E → R} (O : LocalModels.{u,v,w,z} (G := G) α)
-  (B : Sheaf.{u,v,w,z} (R := R) G)
-
-/-- Actual finite descending decomposition over a local ring. This does not
-assume that localization preserves indecomposability. Its explicitly stated
-rank-one model criteria must be proved before applying it to a root plane. -/
-theorem exists_modelSumIso [Finite V]
-    (hBfree : ∀ x, Module.Free R (B.vertex x))
-    (hBfinite : ∀ x, Module.Finite R (B.vertex x))
-    (hBq : B.UpperQuotient α) (hBf : B.Flabby) (hBg : B.Generated) :
-    ∃ C : Sheaf.{u,v,w,z} (R := R) G,
-      ModelSum O.sheaf Set.univ C ∧ Nonempty (Iso B C) := by
-  classical
-  let := Fintype.ofFinite V
-  have h : ∀ s : Finset V, IsUpperSet ((s : Set V)ᶜ) →
-      ∀ C : Sheaf.{u,v,w,z} (R := R) G,
-      ModelSum O.sheaf ((s : Set V)ᶜ) C → OnIso B C ((s : Set V)ᶜ) →
-      ∃ D : Sheaf.{u,v,w,z} (R := R) G,
-        ModelSum O.sheaf Set.univ D ∧ Nonempty (Iso B D) := by
-    intro s
-    induction s using Finset.strongInductionOn with
-    | _ s ih =>
-      intro hU C hC j
-      by_cases hs : s = ∅
-      · subst s
-        have hh : ((∅ : Finset V) : Set V)ᶜ = Set.univ := by simp
-        exact ⟨C,hh ▸ hC,⟨OnIso.ofUniv (hh ▸ j)⟩⟩
-      · obtain ⟨x,hxs,hxmax⟩ := s.exists_maximal (Finset.nonempty_iff_ne_empty.mpr hs)
-        have hx : x ∉ ((s : Set V)ᶜ) := by simpa using hxs
-        have hmax : Set.Ioi x ⊆ ((s : Set V)ᶜ) := by
-          intro y hy
-          change y ∉ s
-          intro hys
-          exact (not_le_of_gt hy) (hxmax hys hy.le)
-        let := hBfree x
-        let := hBfinite x
-        obtain ⟨D,hD,⟨jj⟩⟩ := extend_modelSum O hU hC j hBq hBf hBg x hx hmax
-        have heq : insert x ((s : Set V)ᶜ) = ((s.erase x : Finset V) : Set V)ᶜ := by
-          ext y
-          simp only [Set.mem_insert_iff,Set.mem_compl_iff,Finset.mem_coe,Finset.mem_erase]
-          tauto
-        have hU' : IsUpperSet (insert x ((s : Set V)ᶜ)) := by
-          intro y t hyt hy
-          rcases Set.mem_insert_iff.mp hy with hy | hy
-          · subst y
-            by_cases htx : t=x
-            · exact Set.mem_insert_iff.mpr (Or.inl htx)
-            · exact Set.mem_insert_of_mem x (hmax (lt_of_le_of_ne hyt (Ne.symm htx)))
-          · exact Set.mem_insert_of_mem x (hU hyt hy)
-        exact ih (s.erase x) (Finset.erase_ssubset hxs) (heq ▸ hU') D (heq ▸ hD) (heq ▸ jj)
-  have hset : ((Finset.univ : Finset V) : Set V)ᶜ = ∅ := by simp
-  have hj : OnIso B zero (∅ : Set V) :=
-    ⟨fun x hx => False.elim hx,fun e he => False.elim he,
-      fun e hs => False.elim hs,fun e ht => False.elim ht⟩
-  have hinit := h Finset.univ
-  rw [hset] at hinit
-  exact hinit isUpperSet_empty zero .zero hj
-
-end KLInvariance.MomentGraph.Sheaf
-
-end
-
-
-section
-
-/-! The rank-one plane models over the actual flat local coefficient ring.
-The local-ring label condition is later supplied by the plane prime. -/
-namespace KLInvariance.BruhatGraph
-universe u v
-variable {I : Type u} {M : CoxeterMatrix I} {W : Type v} [Group W]
-  (cs : CoxeterSystem M W) (u b : W)
-
- theorem exists_outgoing_below (x c : Interval cs u b) (hxc : x < c) :
-    ∃ e : Edge cs u b, source cs u b e = x ∧ target cs u b e ≤ c := by
-  rcases Relation.ReflTransGen.cases_head hxc.le with heq | ⟨z,hxz,hzc⟩
-  · exact (hxc.ne (Subtype.ext heq)).elim
-  · let y : Interval cs u b := ⟨z,bruhat_trans cs x.property.1 (.single hxz),
-      bruhat_trans cs hzc c.property.2⟩
-    exact ⟨⟨(x,y),hxz⟩,rfl,hzc⟩
-
-end KLInvariance.BruhatGraph
-
-namespace KLInvariance.MomentGraph.Sheaf
-universe ur uv ue
-variable {R : Type ur} [CommRing R] [IsLocalRing R]
-  {V : Type uv} [PartialOrder V] {E : Type ue} (G : OrderedGraph V E) (α : E → R)
-noncomputable section
-
- def structureLocalModels
-    (hα : ∀ e, α e ∈ IsLocalRing.maximalIdeal R)
-    (hreach : ∀ x c, x < c → ∃ e, G.source e = x ∧ G.target e ≤ c)
-    (hfl : ∀ c, (structureSheaf G α c).Flabby) : LocalModels (G := G) α where
-  sheaf := structureSheaf G α
-  free := structureSheaf_free G α
-  finite := structureSheaf_finite G α
-  quotient := structureSheaf_upperQuotient G α
-  generated := structureSheaf_generated G α
-  flabby := hfl
-  support := structureSheaf_support G α
-  top c := structureVertexIso G α c c le_rfl
-  minimal c x hx := structureSheaf_minimal G α c hα (fun x => hreach x c) x hx
-
-end
-end KLInvariance.MomentGraph.Sheaf
-
-namespace KLInvariance.TitsSpace.PlaneBasis
-open Module BruhatGraph MomentGraph MomentGraph.Sheaf
-universe u v us ur
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-  {σ : Type us} (basis : Basis σ ℝ (Extended M))
-  (R : Type ur) [CommRing R] [Algebra (MvPolynomial σ ℝ) R]
-  [Module.Flat (MvPolynomial σ ℝ) R] [IsLocalRing R]
-noncomputable section
-
- def localModels (b : planeSubgroup M cs P)
-    (hα : ∀ e, algebraMap (MvPolynomial σ ℝ) R (B.modelLabel basis b e) ∈
-      IsLocalRing.maximalIdeal R) :
-    LocalModels (G := graph B.system 1 b)
-      (fun e => algebraMap (MvPolynomial σ ℝ) R (B.modelLabel basis b e)) := by
-  classical
-  let := interval_finite B.system 1 b
-  let := Fintype.ofFinite (Edge B.system 1 b)
-  exact structureLocalModels _ _ hα (exists_outgoing_below B.system 1 b)
-    (fun c => structureSheaf_flat_flabby R _ _ c (B.model_flabby basis b c))
-
-end
-end KLInvariance.TitsSpace.PlaneBasis
-
-end
-
-
-section
-
-/-! The actual plane prime and its localization in the coefficient polynomial
-ring of the chosen nondegenerate ambient realization. -/
-namespace KLInvariance.TitsSpace
-open Module
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {σ : Type us} (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- def embedLinear : (I → ℝ) →ₗ[ℝ] Extended M where
-  toFun := embed M
-  map_add' _ _ := by simp [embed]
-  map_smul' _ _ := by simp [embed]
-
- theorem embedLinear_injective : Function.Injective (embedLinear (M := M)) := by
-  intro a b hab
-  exact congrArg Prod.fst hab
-
- def planePrime (P : Submodule ℝ (I → ℝ)) : Ideal (MvPolynomial σ ℝ) :=
-  Ideal.map (SymmetricAlgebra.equivMvPolynomial basis).toRingHom
-    (SymmetricPlaneIdeal.ideal (P.map (embedLinear (M := M))))
-
- instance planePrime_isPrime (P : Submodule ℝ (I → ℝ)) : (planePrime basis P).IsPrime :=
-  Ideal.map_isPrime_of_equiv (SymmetricAlgebra.equivMvPolynomial basis)
-
- theorem linear_mem_planePrime_iff (P : Submodule ℝ (I → ℝ)) (a : I → ℝ) :
-    linearPolynomial basis (embed M a) ∈ planePrime basis P ↔ a ∈ P := by
-  rw [planePrime,Ideal.mem_map_iff_of_surjective (SymmetricAlgebra.equivMvPolynomial basis).toRingHom
-    (SymmetricAlgebra.equivMvPolynomial basis).surjective]
-  change (∃ z, z ∈ SymmetricPlaneIdeal.ideal (P.map (embedLinear (M := M))) ∧
-    SymmetricAlgebra.equivMvPolynomial basis z =
-      SymmetricAlgebra.equivMvPolynomial basis (SymmetricAlgebra.ι ℝ (Extended M) (embed M a))) ↔ _
-  constructor
-  · rintro ⟨z,hz,he⟩
-    have h := (SymmetricAlgebra.equivMvPolynomial basis).injective he
-    subst z
-    have ha := (SymmetricPlaneIdeal.ι_mem_iff _ _).mp hz
-    obtain ⟨b,hb,he⟩ := ha
-    have he' : b = a := embedLinear_injective he
-    exact he' ▸ hb
-  · intro ha
-    refine ⟨_,(SymmetricPlaneIdeal.ι_mem_iff _ _).mpr ?_,rfl⟩
-    exact ⟨a,ha,rfl⟩
-
- abbrev PlaneRing (P : Submodule ℝ (I → ℝ)) := Localization.AtPrime (planePrime basis P)
-
- theorem linear_mem_maximal_iff (P : Submodule ℝ (I → ℝ)) (a : I → ℝ) :
-    algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P) (linearPolynomial basis (embed M a)) ∈
-      IsLocalRing.maximalIdeal (PlaneRing basis P) ↔ a ∈ P := by
-  rw [IsLocalization.AtPrime.to_map_mem_maximal_iff _ (planePrime basis P),
-    linear_mem_planePrime_iff]
-
- theorem linear_isUnit_iff (P : Submodule ℝ (I → ℝ)) (a : I → ℝ) :
-    IsUnit (algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P)
-      (linearPolynomial basis (embed M a))) ↔ a ∉ P := by
-  rw [IsLocalization.AtPrime.isUnit_to_map_iff _ (planePrime basis P)]
-  change _ ∉ planePrime basis P ↔ _
-  rw [linear_mem_planePrime_iff]
-
-end
-end KLInvariance.TitsSpace
-
-namespace KLInvariance.TitsSpace.PlaneBasis
-open Module BruhatGraph MomentGraph MomentGraph.Sheaf
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-  {σ : Type us} (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- theorem modelLabel_mem_maximal (b : planeSubgroup M cs P) (e : Edge B.system 1 b) :
-    algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P) (B.modelLabel basis b e) ∈
-      IsLocalRing.maximalIdeal (PlaneRing basis P) := by
-  apply (linear_mem_maximal_iff basis P _).mpr
-  apply (Represents.planeSubgroup_mem_iff M cs (positiveRoot_represents M cs _)).mp
-  exact (label B.system 1 b e).property
-
- def localizedModels (b : planeSubgroup M cs P) :
-    LocalModels (G := graph B.system 1 b)
-      (fun e => algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P) (B.modelLabel basis b e)) :=
-  B.localModels basis (PlaneRing basis P) b (B.modelLabel_mem_maximal basis b)
-
-end
-end KLInvariance.TitsSpace.PlaneBasis
-
-end
-
-
-section
-
-/-! The actual finite relative lower ideal in a maximal root-plane coset.
-No order-reflection identification with ambient comparability is used. -/
-namespace KLInvariance.TitsSpace.PlaneBasis
-open Module BruhatGraph MomentGraph
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-  (a₀ b : W)
-noncomputable section
-
- def CosetVertex := {d : planeSubgroup M cs P // BruhatLE B.system 1 d ∧
-    BruhatLE cs ((d:W)*a₀) b}
-
- instance cosetOrder : PartialOrder (B.CosetVertex a₀ b) where
-  le x y := BruhatLE B.system x.val y.val
-  le_refl x := bruhat_refl B.system x.val
-  le_trans _ _ _ := bruhat_trans B.system
-  le_antisymm x y hxy hyx := Subtype.ext (bruhat_antisymm B.system hxy hyx)
-
- def cosetAmbient (x : B.CosetVertex a₀ b) : Interval cs 1 b :=
-  ⟨(x.val:W)*a₀,one_bruhat cs _,x.property.2⟩
-
- theorem cosetAmbient_injective : Function.Injective (B.cosetAmbient a₀ b) := by
-  intro x y h
-  apply Subtype.ext
-  apply Subtype.ext
-  exact mul_right_cancel (congrArg Subtype.val h)
-
- instance cosetFinite : Finite (B.CosetVertex a₀ b) := by
-  let := interval_finite cs 1 b
-  exact Finite.of_injective _ (B.cosetAmbient_injective a₀ b)
-
- def CosetEdge := {p : B.CosetVertex a₀ b × B.CosetVertex a₀ b //
-    BruhatStep B.system p.1.val p.2.val}
-
- def cosetGraph : OrderedGraph (B.CosetVertex a₀ b) (B.CosetEdge a₀ b) where
-  source e := e.val.1
-  target e := e.val.2
-  increasing e := ⟨.single e.property,fun h => not_lt_of_ge
-    (length_le_of_bruhat B.system h) e.property.1⟩
-
-variable (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-
- def cosetEdgeAmbient (e : B.CosetEdge a₀ b) : Edge cs 1 b :=
-  ⟨(B.cosetAmbient a₀ b e.val.1,B.cosetAmbient a₀ b e.val.2),
-    (B.coset_step_iff hmin _ _).mpr e.property⟩
-
- def cosetGraphMap : GraphMap (B.cosetGraph a₀ b) (graph cs 1 b) where
-  vertex := B.cosetAmbient a₀ b
-  edge := B.cosetEdgeAmbient a₀ b hmin
-  source _ := rfl
-  target _ := rfl
-
- include hmin in
- theorem coset_lower {d : planeSubgroup M cs P} {x : B.CosetVertex a₀ b}
-    (hx : BruhatLE B.system d x.val) : BruhatLE cs ((d:W)*a₀) b :=
-  bruhat_trans cs (B.coset_bruhat hmin hx) x.property.2
-
- include hmin in
- theorem coset_outgoing_below (x c : B.CosetVertex a₀ b) (hxc : x<c) :
-    ∃ e : B.CosetEdge a₀ b, (B.cosetGraph a₀ b).source e=x ∧
-      (B.cosetGraph a₀ b).target e ≤ c := by
-  rcases Relation.ReflTransGen.cases_head hxc.le with heq | ⟨z,hxz,hzc⟩
-  · exact (hxc.ne (Subtype.ext heq)).elim
-  · let y : B.CosetVertex a₀ b := ⟨z,one_bruhat B.system z,
-      B.coset_lower a₀ b hmin hzc⟩
-    exact ⟨⟨(x,y),hxz⟩,rfl,hzc⟩
-
- theorem coset_lift_live (e : Edge cs 1 b)
-    (he : label cs 1 b e ∈ planeSubgroup M cs P)
-    (x : B.CosetVertex a₀ b) (hx : B.cosetAmbient a₀ b x=source cs 1 b e) :
-    ∃ a : B.CosetEdge a₀ b, B.cosetEdgeAmbient a₀ b hmin a=e ∧
-      (B.cosetGraph a₀ b).source a=x := by
-  let t : planeSubgroup M cs P := ⟨label cs 1 b e,he⟩
-  have hy : ((t*x.val : planeSubgroup M cs P):W)*a₀ = (target cs 1 b e).val := by
-    change (label cs 1 b e*(x.val:W))*a₀ = _
-    rw [mul_assoc,show (x.val:W)*a₀=(source cs 1 b e).val from congrArg Subtype.val hx]
-    exact label_mul_source cs 1 b e
-  let y : B.CosetVertex a₀ b := ⟨t*x.val,one_bruhat B.system _,hy.symm ▸ (target cs 1 b e).property.2⟩
-  have hye : B.cosetAmbient a₀ b y=target cs 1 b e := Subtype.ext hy
-  have hs : BruhatStep B.system x.val y.val := by
-    apply (B.coset_step_iff hmin _ _).mp
-    change BruhatStep cs (B.cosetAmbient a₀ b x).val (B.cosetAmbient a₀ b y).val
-    rw [hx,hye]
-    exact e.property
-  refine ⟨⟨(x,y),hs⟩,?_,rfl⟩
-  apply Subtype.ext
-  exact Prod.ext hx hye
-
-end
-end KLInvariance.TitsSpace.PlaneBasis
-
-end
-
-
-section
-
-/-! Actual plane localization of the boundary-constructed sheaf. Its relative
-coset restriction is free, generated and flabby with upper quotient maps;
-no character or upward-kernel freeness theorem is used. -/
-namespace KLInvariance.MomentGraph.Sheaf
-universe ur uv ue um
-variable {R : Type ur} [CommRing R] {V : Type uv} [PartialOrder V]
-  {E : Type ue} {G : OrderedGraph V E} (B : Sheaf.{ur,uv,ue,um} (R := R) G)
-
- theorem edge_subsingleton_of_unit (α : E → R) (hq : B.UpperQuotient α)
-    (e : E) (he : IsUnit (α e)) : Subsingleton (B.edge e) := by
-  obtain ⟨a,ha⟩ := he
-  have hz : ∀ v : B.edge e, v=0 := by
-    intro v
-    obtain ⟨w,rfl⟩ := (hq e).1 v
-    apply ((hq e).2 w).mpr
-    refine ⟨(↑a⁻¹ : R) • w,?_⟩
-    rw [← ha,smul_smul]
-    simp
-  exact ⟨fun x y => (hz x).trans (hz y).symm⟩
-
-end KLInvariance.MomentGraph.Sheaf
-
-namespace KLInvariance.TitsSpace.PlaneBasis
-open Module BruhatGraph MomentGraph MomentGraph.Sheaf
-universe u v us um
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  (a₀ b : W) (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-noncomputable section
-
- def cosetLabel (e : B.CosetEdge a₀ b) : PlaneRing basis P :=
-  algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P)
-    (polynomialLabel cs 1 b basis (B.cosetEdgeAmbient a₀ b hmin e))
-
- def localizedCoset (C : Sheaf.{us,v,v,um} (R := MvPolynomial σ ℝ) (graph cs 1 b)) :
-    Sheaf (R := PlaneRing basis P) (B.cosetGraph a₀ b) :=
-  (C.baseChange (PlaneRing basis P)).pullback (B.cosetGraphMap a₀ b hmin)
-
- omit [Fintype σ] in
- theorem polynomialLabel_unit_iff (e : Edge cs 1 b) :
-    IsUnit (algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P)
-      (polynomialLabel cs 1 b basis e)) ↔ label cs 1 b e ∉ planeSubgroup M cs P := by
-  change IsUnit (algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P)
-    (linearPolynomial basis (embed M (BruhatGraph.root cs 1 b e).val))) ↔ _
-  rw [linear_isUnit_iff]
-  have hr := positiveRoot_represents M cs ⟨label cs 1 b e,label_isReflection cs 1 b e⟩
-  exact not_congr (Represents.planeSubgroup_mem_iff M cs hr).symm
-
- omit [Fintype σ] in
- theorem localizedCoset_quotient (C : Sheaf.{us,v,v,um} (R := MvPolynomial σ ℝ) (graph cs 1 b))
-    (hq : C.UpperQuotient (polynomialLabel cs 1 b basis)) :
-    (B.localizedCoset basis a₀ b hmin C).UpperQuotient (B.cosetLabel basis a₀ b hmin) :=
-  (C.baseChange (PlaneRing basis P)).pullback_quotient (B.cosetGraphMap a₀ b hmin)
-    (fun e => algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P) (polynomialLabel cs 1 b basis e))
-    (C.baseChange_upperQuotient (PlaneRing basis P) (polynomialLabel cs 1 b basis) hq)
-
- omit [Fintype σ] in
- theorem localizedCoset_generated (C : Sheaf.{us,v,v,um} (R := MvPolynomial σ ℝ) (graph cs 1 b))
-    (hg : C.Generated) : (B.localizedCoset basis a₀ b hmin C).Generated :=
-  (C.baseChange (PlaneRing basis P)).pullback_generated _
-    (C.baseChange_generated (PlaneRing basis P) hg)
-
- omit [Fintype σ] in
- theorem localizedCoset_flabby (C : Sheaf.{us,v,v,um} (R := MvPolynomial σ ℝ) (graph cs 1 b))
-    (hq : C.UpperQuotient (polynomialLabel cs 1 b basis)) (hf : C.Flabby) :
-    (B.localizedCoset basis a₀ b hmin C).Flabby := by
-  let := interval_finite cs 1 b
-  apply (C.baseChange (PlaneRing basis P)).pullback_flabby_live _
-    (C.baseChange_flabby (PlaneRing basis P) hf) (B.cosetAmbient_injective a₀ b)
-    {e | label cs 1 b e ∈ planeSubgroup M cs P}
-  · intro e he
-    exact (C.baseChange (PlaneRing basis P)).edge_subsingleton_of_unit _
-      (C.baseChange_upperQuotient (PlaneRing basis P) _ hq) e
-      ((polynomialLabel_unit_iff (P := P) basis b e).mpr he)
-  · exact B.coset_lift_live a₀ b hmin
-
-end
-end KLInvariance.TitsSpace.PlaneBasis
-
-end
-
-
-section
-
-/-! The relative rank-one structure models are natural under full strictly
-order-compatible graph embeddings, with exactly the pulled-back labels. -/
-namespace KLInvariance.MomentGraph.Sheaf
-universe ur uv ue uw uf
-variable {R : Type ur} [CommRing R] {V : Type uv} [PartialOrder V]
-  {E : Type ue} {G : OrderedGraph V E} {W : Type uw} [PartialOrder W]
-  {F : Type uf} {H : OrderedGraph W F}
-noncomputable section
-
- def propPiEquiv (N : Type*) [AddCommGroup N] [Module R N]
-    {p q : Prop} (h : p ↔ q) : (PLift p → N) ≃ₗ[R] (PLift q → N) where
-  toFun v t := v ⟨h.mpr t.down⟩
-  invFun v t := v ⟨h.mp t.down⟩
-  left_inv _ := rfl
-  right_inv _ := rfl
-  map_add' _ _ := rfl
-  map_smul' _ _ := rfl
-
- theorem moduleEq_pi_apply {X : Type*} (p : X → Prop) {x y : X} (h : x=y)
-    (v : PLift (p x) → R) (t : PLift (p y)) :
-    moduleEq (congrArg (fun z => ModuleCat.of R (PLift (p z) → R)) h) v t =
-      v ⟨h.symm ▸ t.down⟩ := by
-  subst y
-  rfl
-
- def structurePullbackIso (f : GraphMap G H) (α : F → R) (c : V)
-    (ho : ∀ x y, f.vertex x ≤ f.vertex y ↔ x ≤ y) :
-    Iso ((structureSheaf H α (f.vertex c)).pullback f)
-      (structureSheaf G (fun e => α (f.edge e)) c) where
-  vertex x := propPiEquiv R (ho x c)
-  edge e := propPiEquiv (R ⧸ Ideal.span {α (f.edge e)}) (by
-    rw [← f.target e]; exact ho (G.target e) c)
-  lower e v := by
-    change (PLift (f.vertex (G.source e) ≤ f.vertex c) → R) at v
-    funext h
-    change Ideal.Quotient.mk _
-      (moduleEq (congrArg (fun z => ModuleCat.of R (PLift (z ≤ f.vertex c) → R))
-        (f.source e)) v _) = Ideal.Quotient.mk _ (v _)
-    apply congrArg (Ideal.Quotient.mk _)
-    exact moduleEq_pi_apply (fun z => z ≤ f.vertex c) (f.source e) v _
-  upper e v := by
-    change (PLift (f.vertex (G.target e) ≤ f.vertex c) → R) at v
-    funext h
-    change Ideal.Quotient.mk _
-      (moduleEq (congrArg (fun z => ModuleCat.of R (PLift (z ≤ f.vertex c) → R))
-        (f.target e)) v _) = Ideal.Quotient.mk _ (v _)
-    apply congrArg (Ideal.Quotient.mk _)
-    exact moduleEq_pi_apply (fun z => z ≤ f.vertex c) (f.target e) v _
-
-end
-end KLInvariance.MomentGraph.Sheaf
-
-end
-
-
-section
-
-/-! Rank-one models on the actual finite relative lower ideal in an ambient
-plane coset. The ideal need not have a greatest element. -/
-namespace KLInvariance.TitsSpace.PlaneBasis
-open Module BruhatGraph MomentGraph MomentGraph.Sheaf
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-  {σ : Type us} (basis : Basis σ ℝ (Extended M))
-  (a₀ b : W) (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-noncomputable section
-
- def cosetInto (d : planeSubgroup M cs P) (hd : ∀ x : B.CosetVertex a₀ b, BruhatLE B.system x.val d) :
-    B.CosetVertex a₀ b → Interval B.system 1 d := fun x => ⟨x.val,x.property.1,hd x⟩
-
- def cosetIntoEdge (d : planeSubgroup M cs P)
-    (hd : ∀ x : B.CosetVertex a₀ b, BruhatLE B.system x.val d) (e : B.CosetEdge a₀ b) :
-    Edge B.system 1 d := ⟨(B.cosetInto a₀ b d hd e.val.1,B.cosetInto a₀ b d hd e.val.2),e.property⟩
-
- def cosetIntoGraph (d : planeSubgroup M cs P)
-    (hd : ∀ x : B.CosetVertex a₀ b, BruhatLE B.system x.val d) :
-    GraphMap (B.cosetGraph a₀ b) (graph B.system 1 d) where
-  vertex := B.cosetInto a₀ b d hd
-  edge := B.cosetIntoEdge a₀ b d hd
-  source _ := rfl
-  target _ := rfl
-
- theorem cosetInto_injective (d : planeSubgroup M cs P)
-    (hd : ∀ x : B.CosetVertex a₀ b, BruhatLE B.system x.val d) :
-    Function.Injective (B.cosetInto a₀ b d hd) := by
-  intro x y h
-  apply Subtype.ext
-  exact congrArg (fun z : Interval B.system 1 d => z.val) h
-
- theorem coset_label_eq (e : B.CosetEdge a₀ b) :
-    label cs 1 b (B.cosetEdgeAmbient a₀ b hmin e) = (e.val.2.val : W)*(e.val.1.val:W)⁻¹ := by
-  change ((e.val.2.val:W)*a₀)*((e.val.1.val:W)*a₀)⁻¹ = _
-  group
-
- theorem coset_modelLabel_eq (d : planeSubgroup M cs P)
-    (hd : ∀ x : B.CosetVertex a₀ b, BruhatLE B.system x.val d) (e : B.CosetEdge a₀ b) :
-    B.modelLabel basis d (B.cosetIntoEdge a₀ b d hd e) =
-      polynomialLabel cs 1 b basis (B.cosetEdgeAmbient a₀ b hmin e) := by
-  have ht : (⟨(label B.system 1 d (B.cosetIntoEdge a₀ b d hd e)).val,
-      (B.reflection_iff _).mp (label_isReflection B.system 1 d _)⟩ : {t : W // cs.IsReflection t}) =
-      ⟨label cs 1 b (B.cosetEdgeAmbient a₀ b hmin e),label_isReflection cs 1 b _⟩ := by
-    apply Subtype.ext
-    change (label B.system 1 d (B.cosetIntoEdge a₀ b d hd e)).val =
-      label cs 1 b (B.cosetEdgeAmbient a₀ b hmin e)
-    rw [B.coset_label_eq a₀ b hmin e]
-    rfl
-  exact congrArg (fun t : {t : W // cs.IsReflection t} =>
-    linearPolynomial basis (embed M (positiveRoot M cs t).val)) ht
-
- include hmin in
- theorem coset_model_flabby (c : B.CosetVertex a₀ b) :
-    (structureSheaf (B.cosetGraph a₀ b) (B.cosetLabel basis a₀ b hmin) c).Flabby := by
-  classical
-  let := Fintype.ofFinite (B.CosetVertex a₀ b)
-  obtain ⟨d,hd⟩ := Dihedral.exists_common_upper B.system
-    (Finset.univ.image (fun x : B.CosetVertex a₀ b => x.val))
-  have hbnd (x : B.CosetVertex a₀ b) : BruhatLE B.system x.val d :=
-    hd _ (Finset.mem_image.mpr ⟨x,Finset.mem_univ _,rfl⟩)
-  let f := B.cosetIntoGraph a₀ b d hbnd
-  let α : Edge B.system 1 d → PlaneRing basis P := fun e =>
-    algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P) (B.modelLabel basis d e)
-  let C := structureSheaf (graph B.system 1 d) α (f.vertex c)
-  have hfl : C.Flabby := (B.localizedModels basis d).flabby (f.vertex c)
-  let := interval_finite B.system 1 d
-  have hpull : (C.pullback f).Flabby := by
-    apply C.pullback_flabby_live f hfl (B.cosetInto_injective a₀ b d hbnd)
-      {e | target B.system 1 d e ≤ f.vertex c}
-    · intro e he
-      exact structureSheaf_edge_subsingleton _ α (f.vertex c) e he
-    · intro e he x hx
-      have hlow : BruhatLE B.system (target B.system 1 d e).val c.val := he
-      let y : B.CosetVertex a₀ b := ⟨(target B.system 1 d e).val,
-        (target B.system 1 d e).property.1,B.coset_lower a₀ b hmin hlow⟩
-      have hye : f.vertex y=target B.system 1 d e := rfl
-      have hs : BruhatStep B.system x.val y.val := by
-        change BruhatStep B.system (f.vertex x).val (f.vertex y).val
-        rw [hx,hye]
-        exact e.property
-      refine ⟨⟨(x,y),hs⟩,?_,rfl⟩
-      exact Subtype.ext (Prod.ext hx hye)
-  have hh := (structurePullbackIso f α c (fun _ _ => Iff.rfl)).flabby hpull
-  have hα : (fun e => α (f.edge e)) = B.cosetLabel basis a₀ b hmin := by
-    funext e
-    exact congrArg (algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P))
-      (B.coset_modelLabel_eq basis a₀ b hmin d hbnd e)
-  rwa [hα] at hh
-
- def cosetModels : LocalModels (G := B.cosetGraph a₀ b) (B.cosetLabel basis a₀ b hmin) :=
-  structureLocalModels _ _
-    (fun e => (linear_mem_maximal_iff basis P _).mpr (by
-      apply (Represents.planeSubgroup_mem_iff M cs
-        (positiveRoot_represents M cs ⟨label cs 1 b (B.cosetEdgeAmbient a₀ b hmin e),
-          label_isReflection cs 1 b _⟩)).mp
-      change label cs 1 b (B.cosetEdgeAmbient a₀ b hmin e) ∈ planeSubgroup M cs P
-      rw [B.coset_label_eq a₀ b hmin e]
-      exact (planeSubgroup M cs P).mul_mem e.val.2.val.property
-        ((planeSubgroup M cs P).inv_mem e.val.1.val.property)))
-    (B.coset_outgoing_below a₀ b hmin) (B.coset_model_flabby basis a₀ b hmin)
-
-end
-end KLInvariance.TitsSpace.PlaneBasis
-
-end
-
-
-section
-
-/-! Partial upward kernels of the actual rank-one model, with their embedded
-edge images. No character or freeness property of BMP sheaves is assumed. -/
-namespace KLInvariance.MomentGraph.Sheaf
-open scoped Pointwise
-universe ur uv ue
-variable {R : Type ur} [CommRing R] [DecompositionMonoid R]
-  {V : Type uv} [PartialOrder V] {E : Type ue} [Fintype E]
-  (G : OrderedGraph V E) (α : E → R) (c : V)
-noncomputable section
-
- def supportedOutgoing (x : V) (F : Set (Outgoing (G := G) x)) : Finset (Outgoing (G := G) x) := by
-  classical
-  exact Finset.univ.filter (fun e => e ∈ F ∧ G.target e.val ≤ c)
-
- def kernelProduct (x : V) (F : Set (Outgoing (G := G) x)) : R :=
-  ∏ e ∈ supportedOutgoing G c x F, α e.val
-
- theorem structure_kernel_iff_dvd (x : V) (hx : x ≤ c)
-    (F : Set (Outgoing (G := G) x))
-    (hrel : (supportedOutgoing G c x F : Set (Outgoing (G := G) x)).Pairwise
-      (Function.onFun IsRelPrime (fun e => α e.val)))
-    (v : (structureSheaf G α c).vertex x) :
-    v ∈ (structureSheaf G α c).upwardKernel x F ↔ kernelProduct G α c x F ∣ v ⟨hx⟩ := by
-  classical
-  constructor
-  · intro hv
-    apply Finset.prod_dvd_of_isRelPrime hrel
-    intro e he
-    have hh := (Finset.mem_filter.mp he).2
-    have hvv := congrFun (hv e hh.1) ⟨hh.2⟩
-    rcases e with ⟨e,he⟩
-    subst x
-    exact Ideal.mem_span_singleton.mp (Ideal.Quotient.eq_zero_iff_mem.mp hvv)
-  · intro hv e he
-    funext h
-    have hh : e ∈ supportedOutgoing G c x F :=
-      Finset.mem_filter.mpr ⟨Finset.mem_univ _,he,h.down⟩
-    have hd : α e.val ∣ v ⟨hx⟩ := (Finset.dvd_prod_of_mem (fun a => α a.val) hh).trans hv
-    rcases e with ⟨e,heq⟩
-    subst x
-    exact Ideal.Quotient.eq_zero_iff_mem.mpr (Ideal.mem_span_singleton.mpr hd)
-
- theorem structure_kernel_product (x : V) (F : Set (Outgoing (G := G) x))
-    (hrel : (supportedOutgoing G c x F : Set (Outgoing (G := G) x)).Pairwise
-      (Function.onFun IsRelPrime (fun e => α e.val))) :
-    (structureSheaf G α c).upwardKernel x F =
-      kernelProduct G α c x F • (⊤ : Submodule R ((structureSheaf G α c).vertex x)) := by
-  classical
-  ext v
-  rw [Submodule.mem_smul_pointwise_iff_exists]
-  by_cases hx : x ≤ c
-  · rw [structure_kernel_iff_dvd G α c x hx F hrel]
-    constructor
-    · rintro ⟨r,hr⟩
-      refine ⟨fun _ => r,Submodule.mem_top,?_⟩
-      funext h
-      exact hr.symm
-    · rintro ⟨w,_,hw⟩
-      exact ⟨w ⟨hx⟩,(congrFun hw ⟨hx⟩).symm⟩
-  · let _ := structureSheaf_support G α c x hx
-    constructor
-    · intro _
-      exact ⟨0,Submodule.mem_top,Subsingleton.elim _ _⟩
-    · intro _
-      rw [Subsingleton.elim v 0]
-      exact Submodule.zero_mem _
-
-end
-end KLInvariance.MomentGraph.Sheaf
-
-namespace KLInvariance.MomentGraph.Sheaf
-open scoped Pointwise
-universe ur uv ue
-variable {R : Type ur} [CommRing R] {V : Type uv} [PartialOrder V]
-  {E : Type ue} {G : OrderedGraph V E}
-
- theorem lower_surjective_of_generated (B : Sheaf (R := R) G) (hg : B.Generated)
-    (hu : ∀ e, Function.Surjective (B.upper e)) (e : E) : Function.Surjective (B.lower e) := by
-  intro v
-  obtain ⟨w,hw⟩ := hu e v
-  obtain ⟨s,hs,ht⟩ := hg (G.target e) w
-  refine ⟨s (G.source e),?_⟩
-  rw [hs e (Set.mem_univ _) (Set.mem_univ _),ht,hw]
-
-variable [DecompositionMonoid R] [Fintype E] (α : E → R) (c : V)
-
- theorem structure_lower_image (e : E) (F : Set (Outgoing (G := G) (G.source e)))
-    (hrel : (supportedOutgoing G c (G.source e) F : Set (Outgoing (G := G) (G.source e))).Pairwise
-      (Function.onFun IsRelPrime (fun a => α a.val))) :
-    ((structureSheaf G α c).upwardKernel (G.source e) F).map ((structureSheaf G α c).lower e) =
-      kernelProduct G α c (G.source e) F • (⊤ : Submodule R ((structureSheaf G α c).edge e)) := by
-  rw [structure_kernel_product G α c _ F hrel,Submodule.map_pointwise_smul,Submodule.map_top]
-  congr 1
-  apply LinearMap.range_eq_top.mpr
-  exact lower_surjective_of_generated _ (structureSheaf_generated G α c)
-    (fun a => (structureSheaf_upperQuotient G α c a).1) e
-
- theorem structure_upper_image (e : E) (F : Set (Outgoing (G := G) (G.target e)))
-    (hrel : (supportedOutgoing G c (G.target e) F : Set (Outgoing (G := G) (G.target e))).Pairwise
-      (Function.onFun IsRelPrime (fun a => α a.val))) :
-    ((structureSheaf G α c).upwardKernel (G.target e) F).map ((structureSheaf G α c).upper e) =
-      kernelProduct G α c (G.target e) F • (⊤ : Submodule R ((structureSheaf G α c).edge e)) := by
-  rw [structure_kernel_product G α c _ F hrel,Submodule.map_pointwise_smul,Submodule.map_top]
-  congr 1
-  exact LinearMap.range_eq_top.mpr (structureSheaf_upperQuotient G α c e).1
-
-end KLInvariance.MomentGraph.Sheaf
-
-end
-
-
-section
-
-/-! Distinct prime root labels remain relatively prime after genuine prime
-localization. The prime may contain both labels; coprimality here is gcd
-coprimality, not comaximality. -/
-namespace KLInvariance.LocalizationRelativePrime
-variable {A : Type*} [CommRing A] [IsDomain A] (p : Ideal A) [p.IsPrime]
-
-omit [IsDomain A] in
- theorem not_dvd_map {a b : A} (ha : Prime a) (hap : a ∈ p) (hab : ¬a ∣ b) :
-    ¬ algebraMap A (Localization.AtPrime p) a ∣ algebraMap A (Localization.AtPrime p) b := by
-  intro h
-  obtain ⟨s,hs,hdiv⟩ := (IsLocalization.toLocalizationMap p.primeCompl (Localization.AtPrime p)).map_dvd_map.mp h
-  rcases ha.dvd_mul.mp hdiv with has | hab'
-  · exact hs (p.mem_of_dvd has hap)
-  · exact hab hab'
-
- theorem map_prime {a : A} (ha : Prime a) (hap : a ∈ p) :
-    Prime (algebraMap A (Localization.AtPrime p) a) := by
-  apply (IsLocalization.toLocalizationMap p.primeCompl (Localization.AtPrime p)).map_prime ha
-  · change algebraMap A (Localization.AtPrime p) a ≠ 0
-    simpa only [map_zero] using (FaithfulSMul.algebraMap_injective A (Localization.AtPrime p)).ne ha.ne_zero
-  · change ¬IsUnit (algebraMap A (Localization.AtPrime p) a)
-    rw [IsLocalization.AtPrime.isUnit_to_map_iff _ p]
-    exact not_not_intro hap
-
- theorem map_relPrime {a b : A} (ha : Prime a) (hab : ¬a ∣ b) :
-    IsRelPrime (algebraMap A (Localization.AtPrime p) a)
-      (algebraMap A (Localization.AtPrime p) b) := by
-  by_cases hap : a ∈ p
-  · exact ((map_prime p ha hap).irreducible.isRelPrime_iff_not_dvd).mpr (not_dvd_map p ha hap hab)
-  · exact ((IsLocalization.AtPrime.isUnit_to_map_iff (Localization.AtPrime p) p a).mpr hap).isRelPrime_left
-
-end KLInvariance.LocalizationRelativePrime
-
-end
-
-
-section
-
-/-! The local product-kernel lemma on the actual maximal-plane coset.
-All regularity hypotheses come from distinct genuine incident reflections. -/
-namespace KLInvariance.TitsSpace.PlaneBasis
-open Module BruhatGraph MomentGraph MomentGraph.Sheaf
-open scoped Pointwise
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-  {σ : Type us} (basis : Basis σ ℝ (Extended M))
-  (a₀ b : W) (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-noncomputable section
-
- theorem cosetEdgeAmbient_injective : Function.Injective (B.cosetEdgeAmbient a₀ b hmin) := by
-  intro e f h
-  apply Subtype.ext
-  apply Prod.ext
-  · exact B.cosetAmbient_injective a₀ b (congrArg (fun z : Edge cs 1 b => z.val.1) h)
-  · exact B.cosetAmbient_injective a₀ b (congrArg (fun z : Edge cs 1 b => z.val.2) h)
-
- theorem cosetLabel_relPrime {e f : B.CosetEdge a₀ b} (hef : e ≠ f)
-    (hs : (B.cosetGraph a₀ b).source e = (B.cosetGraph a₀ b).source f) :
-    IsRelPrime (B.cosetLabel basis a₀ b hmin e) (B.cosetLabel basis a₀ b hmin f) := by
-  apply LocalizationRelativePrime.map_relPrime (planePrime basis P)
-    (polynomialLabel_prime cs 1 b basis _)
-  intro hd
-  apply hef
-  apply B.cosetEdgeAmbient_injective a₀ b hmin
-  apply eq_of_label_source_eq cs 1 b (label_eq_of_polynomialLabel_dvd cs 1 b basis hd)
-  exact congrArg (B.cosetAmbient a₀ b) hs
-
- theorem coset_supported_pairwise [Fintype (B.CosetEdge a₀ b)]
-    (c x : B.CosetVertex a₀ b)
-    (F : Set (Outgoing (G := B.cosetGraph a₀ b) x)) :
-    (supportedOutgoing (B.cosetGraph a₀ b) c x F :
-      Set (Outgoing (G := B.cosetGraph a₀ b) x)).Pairwise
-      (Function.onFun IsRelPrime (fun e => B.cosetLabel basis a₀ b hmin e.val)) := by
-  intro e _ f _ hef
-  exact B.cosetLabel_relPrime basis a₀ b hmin
-    (fun h => hef (Subtype.ext h)) (e.property.trans f.property.symm)
-
- theorem coset_kernel_product [Fintype (B.CosetEdge a₀ b)]
-    (c x : B.CosetVertex a₀ b) (F : Set (Outgoing (G := B.cosetGraph a₀ b) x)) :
-    (structureSheaf (B.cosetGraph a₀ b) (B.cosetLabel basis a₀ b hmin) c).upwardKernel x F =
-      kernelProduct (B.cosetGraph a₀ b) (B.cosetLabel basis a₀ b hmin) c x F •
-        (⊤ : Submodule (PlaneRing basis P)
-          ((structureSheaf (B.cosetGraph a₀ b) (B.cosetLabel basis a₀ b hmin) c).vertex x)) :=
-  structure_kernel_product _ _ _ _ _ (B.coset_supported_pairwise basis a₀ b hmin c x F)
-
-end
-end KLInvariance.TitsSpace.PlaneBasis
-
-end
-
-
-section
-
-namespace KLInvariance.TitsSpace.PlaneBasis
-open Module BruhatGraph MomentGraph MomentGraph.Sheaf
-universe u v u' v'
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
-  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
-  {a₀ u b : W} {u' b' : W'}
-  (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-noncomputable section
-
- def cosetUpperVertex (x : B.CosetVertex a₀ b) (hx : BruhatLE cs u ((x.val:W)*a₀)) :
-    Interval cs u b := ⟨(x.val:W)*a₀,hx,x.property.2⟩
-
- theorem outgoing_step (x : B.CosetVertex a₀ b)
-    (f : Outgoing (G := B.cosetGraph a₀ b) x) :
-    BruhatStep B.system x.val f.val.val.2.val := by
-  rcases f with ⟨f,hf⟩
-  change f.val.1=x at hf
-  subst x
-  exact f.property
-
- include hmin in
- theorem outgoing_upper (x : B.CosetVertex a₀ b) (hx : BruhatLE cs u ((x.val:W)*a₀))
-    (f : Outgoing (G := B.cosetGraph a₀ b) x) :
-    BruhatLE cs u ((f.val.val.2.val:W)*a₀) :=
-  bruhat_trans cs hx (.single ((B.coset_step_iff hmin _ _).mpr (B.outgoing_step x f)))
-
- def scheduledOutgoing (φ : Interval cs u b ≃o Interval cs' u' b')
-    (α : PositiveRoot M' cs') (x : B.CosetVertex a₀ b)
-    (hx : BruhatLE cs u ((x.val:W)*a₀)) : Set (Outgoing (G := B.cosetGraph a₀ b) x) :=
-  {f | ambientAfterIndicator (φ (B.cosetUpperVertex x hx)).val α
-    (φ (B.cosetUpperVertex f.val.val.2 (B.outgoing_upper hmin x hx f))).val = 1}
-
- theorem outgoing_target_injective (x : B.CosetVertex a₀ b) :
-    Function.Injective (fun f : Outgoing (G := B.cosetGraph a₀ b) x => f.val.val.2) := by
-  intro f g h
-  apply Subtype.ext
-  apply Subtype.ext
-  exact Prod.ext (f.property.trans g.property.symm) h
-
- theorem supported_card_count [Fintype (B.CosetEdge a₀ b)]
-    (φ : Interval cs u b ≃o Interval cs' u' b') (α : PositiveRoot M' cs')
-    (a x c : B.CosetVertex a₀ b) (hax : a ≤ x)
-    (ha : BruhatLE cs u ((a.val:W)*a₀)) :
-    let hx := bruhat_trans cs ha (B.coset_bruhat hmin hax)
-    let inc := B.cosetIntervalMap hmin ha c.property.2
-    (supportedOutgoing (B.cosetGraph a₀ b) c x (B.scheduledOutgoing hmin φ α x hx)).card =
-      ∑ᶠ z : Interval B.system a.val c.val,
-        ambientAfterIndicator (φ (B.cosetUpperVertex x hx)).val α (φ (inc z)).val := by
-  classical
-  let := interval_finite B.system a.val c.val
-  let := Fintype.ofFinite (Interval B.system a.val c.val)
-  let hx := bruhat_trans cs ha (B.coset_bruhat hmin hax)
-  let inc := B.cosetIntervalMap hmin ha c.property.2
-  let F := B.scheduledOutgoing hmin φ α x hx
-  let s := supportedOutgoing (B.cosetGraph a₀ b) c x F
-  change s.card = ∑ᶠ z : Interval B.system a.val c.val,
-    ambientAfterIndicator (φ (B.cosetUpperVertex x hx)).val α (φ (inc z)).val
-  rw [finsum_eq_sum_of_fintype]
-  rw [show s.card=(∑ _ ∈ s,(1:ℕ)) by simp]
-  let idx (f : Outgoing (G := B.cosetGraph a₀ b) x) (hf : f ∈ s) :
-      Interval B.system a.val c.val :=
-    ⟨f.val.val.2.val,bruhat_trans B.system hax (.single (B.outgoing_step x f)),
-      (Finset.mem_filter.mp hf).2.2⟩
-  apply Finset.sum_bij_ne_zero (fun f hf _ => idx f hf)
-  · intro _ _ _
-    exact Finset.mem_univ _
-  · intro f hf _ g hg _ h
-    have hz : f.val.val.2.val=g.val.val.2.val := congrArg (fun z : Interval B.system a.val c.val => z.val) h
-    exact B.outgoing_target_injective x (Subtype.ext hz)
-  · intro z _ hz
-    have hsz : BruhatStep cs' (φ (B.cosetUpperVertex x hx)).val (φ (inc z)).val := by
-      by_contra hn
-      exact hz (by simp only [ambientAfterIndicator,dite_eq_right hn])
-    have hcz := (intervalIso_bruhatStep_iff_general cs cs' φ _ _).mp hsz
-    have hs : BruhatStep B.system x.val z.val := (B.coset_step_iff hmin _ _).mp hcz
-    let y : B.CosetVertex a₀ b := ⟨z.val,one_bruhat B.system z.val,
-      bruhat_trans cs (B.coset_bruhat hmin z.property.2) c.property.2⟩
-    let f : Outgoing (G := B.cosetGraph a₀ b) x := ⟨⟨(x,y),hs⟩,rfl⟩
-    have hfF : f ∈ F := by
-      change ambientAfterIndicator (φ (B.cosetUpperVertex x hx)).val α (φ (inc z)).val=1
-      unfold ambientAfterIndicator at hz ⊢
-      split_ifs at hz ⊢ <;> omega
-    have hfs : f ∈ s := Finset.mem_filter.mpr ⟨Finset.mem_univ _,hfF,z.property.2⟩
-    exact ⟨f,hfs,by omega,rfl⟩
-  · intro f hf _
-    exact ((Finset.mem_filter.mp hf).2.1).symm
-
-end
-end KLInvariance.TitsSpace.PlaneBasis
-
-end
-
-
-section
-
-namespace KLInvariance.TitsSpace.PlaneBasis
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-noncomputable section
-
-/-- The combinatorial factor already constructed from ambient inversion
-pairs is exactly the intrinsic relative dihedral edge degree. -/
- theorem planeWeight_coset_edge {a₀ : W}
-    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-    {x y : planeSubgroup M cs P} (hxy : BruhatStep B.system x y) :
-    (planeWeight ((x:W)*a₀) (B.edgeRoot hxy) P).toNat =
-      (B.system.length y-B.system.length x-1)/2 := by
-  classical
-  let α := B.edgeRoot hxy
-  obtain ⟨β,hαβ,hβ⟩ : ∃ β : PositiveRoot M cs, α ≠ β ∧ β.val ∈ P := by
-    by_cases h : α=B.left
-    · exact ⟨B.right,h ▸ B.distinct,B.right_mem⟩
-    · exact ⟨B.left,h,B.left_mem⟩
-  have hplane : rootPlane α β=P :=
-    (rootPlane_eq_of_mem B.left B.right α β B.distinct
-      (B.plane_eq.ge (B.edgeRoot_plane hxy)) (B.plane_eq.ge hβ) hαβ).trans B.plane_eq
-  have ha : ¬rootInverted ((x:W)*a₀) α := by
-    rw [show α=graphRoot ((B.coset_step_iff hmin x y).mpr hxy) from B.edgeRoot_coset hmin hxy]
-    exact graphRoot_not_inverted_source _
-  have href : rootRef α=(y:W)*(x:W)⁻¹ :=
-    (rootReflection_represents M cs α).reflection_unique
-      (graphRoot_represents ((B.bruhatStep_iff x y).mpr hxy))
-  have hends : rootRef α*((x:W)*a₀)=(y:W)*a₀ := by rw [href]; group
-  have h := planeWeight_relative_degree ((x:W)*a₀) α β hαβ ha
-  rw [hplane,hends,B.coset_length_eq hmin x,B.coset_length_eq hmin y] at h
-  have hn := planeWeight_nonneg ((x:W)*a₀) α P
-  have hlen := hxy.1
-  have hc : ((planeWeight ((x:W)*a₀) α P).toNat:ℤ)=planeWeight ((x:W)*a₀) α P :=
-    Int.toNat_of_nonneg hn
-  change (planeWeight ((x:W)*a₀) α P).toNat = _
-  omega
-
-end
-end KLInvariance.TitsSpace.PlaneBasis
-
-end
-
-
-section
-
-namespace KLInvariance.TitsSpace.PlaneBasis
-open Module BruhatGraph MomentGraph MomentGraph.Sheaf
-universe u v u' v'
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
-  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
-  {a₀ u b : W} {u' b' : W'}
-  (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-  (φ : Interval cs u b ≃o Interval cs' u' b')
-noncomputable section
-
- include hmin in
- theorem cosetEdge_target_upper (e : B.CosetEdge a₀ b)
-    (hx : BruhatLE cs u ((e.val.1.val:W)*a₀)) : BruhatLE cs u ((e.val.2.val:W)*a₀) :=
-  bruhat_trans cs hx (.single ((B.coset_step_iff hmin _ _).mpr e.property))
-
- def mappedCosetRoot (e : B.CosetEdge a₀ b) (hx : BruhatLE cs u ((e.val.1.val:W)*a₀)) :
-    PositiveRoot M' cs' :=
-  graphRoot ((intervalIso_bruhatStep_iff_general cs cs' φ
-    (B.cosetUpperVertex e.val.1 hx)
-    (B.cosetUpperVertex e.val.2 (B.cosetEdge_target_upper hmin e hx))).mpr
-    ((B.coset_step_iff hmin _ _).mpr e.property))
-
- def scheduledLower (e : B.CosetEdge a₀ b) (hx : BruhatLE cs u ((e.val.1.val:W)*a₀)) :
-    Set (Outgoing (G := B.cosetGraph a₀ b) e.val.1) :=
-  B.scheduledOutgoing hmin φ (B.mappedCosetRoot hmin φ e hx) e.val.1 hx
-
- def scheduledUpper (e : B.CosetEdge a₀ b) (hx : BruhatLE cs u ((e.val.1.val:W)*a₀)) :
-    Set (Outgoing (G := B.cosetGraph a₀ b) e.val.2) :=
-  B.scheduledOutgoing hmin φ (B.mappedCosetRoot hmin φ e hx) e.val.2
-    (B.cosetEdge_target_upper hmin e hx)
-
- theorem scheduledLower_self (e : B.CosetEdge a₀ b)
-    (hx : BruhatLE cs u ((e.val.1.val:W)*a₀)) :
-    (⟨e,rfl⟩ : Outgoing (G := B.cosetGraph a₀ b) e.val.1) ∉ B.scheduledLower hmin φ e hx := by
-  change ambientAfterIndicator _ (graphRoot _) _ ≠ 1
-  rw [ambientAfterIndicator_edge_self]
-  omega
-
- theorem supported_count_difference [Fintype (B.CosetEdge a₀ b)]
-    (e : B.CosetEdge a₀ b) (hx : BruhatLE cs u ((e.val.1.val:W)*a₀))
-    (c : B.CosetVertex a₀ b) (hyc : e.val.2 ≤ c) :
-    (supportedOutgoing (B.cosetGraph a₀ b) c e.val.1 (B.scheduledLower hmin φ e hx)).card =
-      (supportedOutgoing (B.cosetGraph a₀ b) c e.val.2 (B.scheduledUpper hmin φ e hx)).card+
-        (planeWeight ((e.val.1.val:W)*a₀) (B.edgeRoot e.property) P).toNat := by
-  let x := e.val.1
-  let y := e.val.2
-  let α := B.mappedCosetRoot hmin φ e hx
-  have hxy : BruhatLE B.system x.val y.val := .single e.property
-  have hxc : BruhatLE B.system x.val c.val := bruhat_trans B.system hxy hyc
-  let yy : Interval B.system x.val c.val := ⟨y.val,hxy,hyc⟩
-  have h := B.transported_count_difference_all hmin hxc hx c.property.2 φ yy e.property
-  rw [B.planeWeight_coset_edge hmin e.property]
-  change (supportedOutgoing (B.cosetGraph a₀ b) c x (B.scheduledOutgoing hmin φ α x hx)).card =
-    (supportedOutgoing (B.cosetGraph a₀ b) c y
-      (B.scheduledOutgoing hmin φ α y (B.cosetEdge_target_upper hmin e hx))).card+_
-  rw [B.supported_card_count hmin φ α x x c (bruhat_refl B.system x.val) hx,
-    B.supported_card_count hmin φ α x y c hxy hx]
-  exact h
-
-end
-end KLInvariance.TitsSpace.PlaneBasis
-
-end
-
-
-section
-
-namespace KLInvariance.BruhatGraph
-open TitsSpace MomentGraph MomentGraph.Sheaf
-universe u v u' v'
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
-  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
-  {u b : W} {u' b' : W'}
-  (φ : Interval cs u b ≃o Interval cs' u' b')
-noncomputable section
-
- def ambientLater (α : PositiveRoot M' cs') : Set (Edge cs 1 b) :=
-  {e | ∃ hx : BruhatLE cs u (source cs 1 b e).val,
-    ambientAfterIndicator (φ ⟨(source cs 1 b e).val,hx,(source cs 1 b e).property.2⟩).val α
-      (φ ⟨(target cs 1 b e).val,bruhat_trans cs hx (.single e.property),
-        (target cs 1 b e).property.2⟩).val = 1}
-
-end
-end KLInvariance.BruhatGraph
-
-namespace KLInvariance.TitsSpace.PlaneBasis
-open Module BruhatGraph MomentGraph MomentGraph.Sheaf
-universe u v u' v'
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
-  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
-  {a₀ u b : W} {u' b' : W'}
-  (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-  (φ : Interval cs u b ≃o Interval cs' u' b')
-noncomputable section
-
- theorem scheduledOutgoing_eq (α : PositiveRoot M' cs')
-    (x : B.CosetVertex a₀ b) (hx : BruhatLE cs u ((x.val:W)*a₀)) :
-    B.scheduledOutgoing hmin φ α x hx =
-      {f | B.cosetEdgeAmbient a₀ b hmin f.val ∈ ambientLater φ α} := by
-  ext f
-  rcases f with ⟨⟨⟨s,t⟩,hst⟩,hs⟩
-  change s=x at hs
-  subst s
-  change (_=1) ↔ ∃ hh : BruhatLE cs u ((x.val:W)*a₀), _=1
-  constructor
-  · intro h
-    exact ⟨hx,h⟩
-  · rintro ⟨_,h⟩
-    exact h
-
-end
-end KLInvariance.TitsSpace.PlaneBasis
-
-end
-
-
-section
-
-/-! Reduction of a root plane along one root is genuinely one-dimensional.
-The scalar comparing two nonparallel reduced roots is a nonzero element of
-the original base field, hence a unit after every coefficient localization. -/
-namespace KLInvariance.PlaneReduction
-universe uk uv ua
-variable {k : Type uk} [Field k] {V : Type uv} [AddCommGroup V] [Module k V]
-  {A : Type ua} [CommRing A] [Algebra k A]
-
- theorem scalar_reduction (l : V →ₗ[k] A) {a b c : V} (ha : l a = 0)
-    (hc : c ∈ Submodule.span k ({a,b} : Set V))
-    (hca : c ∉ Submodule.span k ({a} : Set V)) :
-    ∃ r : k, r ≠ 0 ∧ l c = r • l b := by
-  obtain ⟨s,r,hr⟩ := Submodule.mem_span_pair.mp hc
-  have hr0 : r ≠ 0 := by
-    intro hz
-    apply hca
-    rw [← hr,hz,zero_smul,add_zero]
-    exact Submodule.smul_mem _ s (Submodule.subset_span (Set.mem_singleton a))
-  refine ⟨r,hr0,?_⟩
-  rw [← hr,map_add,map_smul,map_smul,ha,smul_zero,zero_add]
-
- theorem associated_reduction (l : V →ₗ[k] A) {a b c : V} (ha : l a = 0)
-    (hc : c ∈ Submodule.span k ({a,b} : Set V))
-    (hca : c ∉ Submodule.span k ({a} : Set V)) : Associated (l c) (l b) := by
-  obtain ⟨r,hr,he⟩ := scalar_reduction l ha hc hca
-  rw [he,Algebra.smul_def]
-  exact associated_unit_mul_left _ _ ((hr.isUnit).map (algebraMap k A))
-
-end KLInvariance.PlaneReduction
-
-end
-
-
-section
-
-/-! Actual root labels in the localized edge quotient: every nonparallel
-root in a given plane has a reduced label associated to any chosen one. -/
-namespace KLInvariance.TitsSpace
-open Module
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {σ : Type us} (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- def localizedLinear (P : Submodule ℝ (I → ℝ)) : (I → ℝ) →ₗ[ℝ] PlaneRing basis P :=
-  (IsScalarTower.toAlgHom ℝ (MvPolynomial σ ℝ) (PlaneRing basis P)).toLinearMap.comp
-    ((linearPolynomial basis).comp (embedLinear (M := M)))
-
- def localizedEdgeIdeal (P : Submodule ℝ (I → ℝ)) (a : I → ℝ) : Ideal (PlaneRing basis P) :=
-  Ideal.span {localizedLinear basis P a}
-
- abbrev LocalizedEdgeRing (P : Submodule ℝ (I → ℝ)) (a : I → ℝ) :=
-  PlaneRing basis P ⧸ localizedEdgeIdeal basis P a
-
- def localizedReduction (P : Submodule ℝ (I → ℝ)) (a : I → ℝ) :
-    (I → ℝ) →ₗ[ℝ] LocalizedEdgeRing basis P a :=
-  (Ideal.Quotient.mkₐ ℝ (localizedEdgeIdeal basis P a)).toLinearMap.comp
-    (localizedLinear basis P)
-
- theorem localizedReduction_self (P : Submodule ℝ (I → ℝ)) (a : I → ℝ) :
-    localizedReduction basis P a a = 0 :=
-  Ideal.Quotient.eq_zero_iff_mem.mpr (Ideal.subset_span (Set.mem_singleton _))
-
- theorem positiveRoot_not_line {a c : PositiveRoot M cs} (hca : c ≠ a) :
-    c.val ∉ Submodule.span ℝ ({a.val} : Set (I → ℝ)) := by
-  intro h
-  obtain ⟨r,hr⟩ := Submodule.mem_span_singleton.mp h
-  exact (linearIndependent_fin2.mp (positiveRoots_independent c a hca)).2 r hr
-
- theorem localizedReduction_associated (P : Submodule ℝ (I → ℝ))
-    (a b c : PositiveRoot M cs) (hc : c.val ∈ rootPlane a b) (hca : c ≠ a) :
-    Associated (localizedReduction basis P a.val c.val)
-      (localizedReduction basis P a.val b.val) :=
-by
-  exact @PlaneReduction.associated_reduction ℝ _ (I → ℝ) _ _
-    (LocalizedEdgeRing basis P a.val) _ _ (localizedReduction basis P a.val)
-    a.val b.val c.val (localizedReduction_self basis P a.val) hc
-    (positiveRoot_not_line hca)
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- def polynomialRoot (a : PositiveRoot M cs) : Coefficient (σ := σ) :=
-  SymmetricAlgebra.equivMvPolynomial basis
-    (SymmetricAlgebra.ι ℝ (Extended M) (embed M a.val))
-
- theorem polynomialRoot_homogeneous (a : PositiveRoot M cs) :
-    polynomialRoot cs basis a ∈ polynomialGrading ℝ σ 1 := by
-  classical
-  unfold polynomialRoot
-  rw [← basis.sum_repr (embed M a.val)]
-  simp only [map_sum,map_smul,SymmetricAlgebra.equivMvPolynomial_ι_apply]
-  apply Submodule.sum_mem
-  intro i _
-  exact (polynomialGrading ℝ σ 1).smul_mem _
-    (MvPolynomial.isWeightedHomogeneous_X ℝ (fun _ : σ => (1 : ℤ)) i)
-
-omit [Fintype σ] in
- theorem polynomialRoot_prime (a : PositiveRoot M cs) : Prime (polynomialRoot cs basis a) := by
-  apply (MulEquiv.prime_iff (SymmetricAlgebra.equivMvPolynomial basis)).mpr
-  apply EdgeAlgebra.symmetric_ι_prime
-  intro h
-  exact a.property.1.ne_zero (congrArg Prod.fst h)
-
-omit [Fintype σ] in
- theorem polynomialRoot_not_dvd {a r : PositiveRoot M cs} (h : r ≠ a) :
-    ¬ polynomialRoot cs basis a ∣ polynomialRoot cs basis r := by
-  rintro ⟨z,hz⟩
-  apply embed_not_rootLine a r h
-  apply (EdgeAlgebra.symmetric_ι_dvd_iff _ _).mp
-  refine ⟨(SymmetricAlgebra.equivMvPolynomial basis).symm z,?_⟩
-  apply (SymmetricAlgebra.equivMvPolynomial basis).injective
-  simpa only [map_mul,AlgEquiv.apply_symm_apply,polynomialRoot] using hz
-
-
- def edgeFactorLift (x : W) (a : PositiveRoot M cs) : Coefficient (σ := σ) :=
-  ∏ P : (planeWeight x a).support,
-    polynomialRoot cs basis (planeRoot x a P) ^ (planeWeight x a P.val).toNat
-
- theorem edgeFactorLift_homogeneous (x : W) (a : PositiveRoot M cs) :
-    edgeFactorLift cs basis x a ∈ polynomialGrading ℝ σ (edgeFactorDegree x a : ℤ) := by
-  classical
-  have h := SetLike.prod_pow_mem_graded (polynomialGrading ℝ σ)
-    (fun _ : (planeWeight x a).support => (1 : ℤ))
-    (fun P => polynomialRoot cs basis (planeRoot x a P))
-    (fun P => (planeWeight x a P.val).toNat)
-    (F := Finset.univ) (fun P _ => polynomialRoot_homogeneous cs basis (planeRoot x a P))
-  simpa only [edgeFactorLift,edgeFactorDegree,Nat.cast_sum,nsmul_eq_mul,mul_one] using h
-
-omit [Fintype σ] in
- theorem edgeFactorLift_not_dvd (x : W) (a : PositiveRoot M cs) :
-    ¬ polynomialRoot cs basis a ∣ edgeFactorLift cs basis x a := by
-  classical
-  apply (polynomialRoot_prime cs basis a).not_dvd_finsetProd
-  intro P _ h
-  exact polynomialRoot_not_dvd cs basis (planeRoot_ne x a P)
-    ((polynomialRoot_prime cs basis a).dvd_of_dvd_pow h)
-
-omit [Fintype σ] in
- theorem edgeFactorLift_reduction (x : W) (a : PositiveRoot M cs) :
-    Ideal.Quotient.mk (EdgeAlgebra.rootIdeal (k := ℝ) (embed M a.val))
-      ((SymmetricAlgebra.equivMvPolynomial basis).symm (edgeFactorLift cs basis x a)) =
-      edgeFactor x a := by
-  simp only [edgeFactorLift,map_prod,map_pow,polynomialRoot,AlgEquiv.symm_apply_apply,
-    edgeFactor,reducedPlaneRoot]
-
-variable (u b : W) (hub : BruhatLE cs u b)
-  (B : BoundarySheaf (polynomialGrading_negative ℝ σ)
-    (graph cs u b) (polynomialLabel cs u b basis) ⟨b,hub,bruhat_refl cs b⟩)
-
-theorem rootRef_root (e : Edge cs u b) :
-    rootRef (root cs u b e) = label cs u b e :=
-  congrArg Subtype.val ((positiveRootEquiv M cs).left_inv
-    ⟨label cs u b e,label_isReflection cs u b e⟩)
-
- theorem root_not_inverted (e : Edge cs u b) :
-    ¬ rootInverted (source cs u b e).val (root cs u b e) := by
-  intro h
-  have hi := ((rootReflection_represents M cs (root cs u b e)).inversion_iff
-    (root cs u b e).property.2 (source cs u b e).val).mp h
-  have hl := hi.2
-  change cs.length (rootRef (root cs u b e) * (source cs u b e).val) <
-    cs.length (source cs u b e).val at hl
-  rw [rootRef_root, label_mul_source] at hl
-  exact (not_lt_of_ge e.property.1.le) hl
-
- theorem edgeFactorDegree_edge (e : Edge cs u b) :
-    2 * (edgeFactorDegree (source cs u b e).val (root cs u b e) : ℤ) =
-      (cs.length (target cs u b e).val : ℤ) - cs.length (source cs u b e).val - 1 := by
-  rw [edgeFactorDegree_cast]
-  have h := sum_planeWeight (source cs u b e).val (root cs u b e)
-    (root_not_inverted cs u b e)
-  rwa [rootRef_root, label_mul_source] at h
-
-/-- Multiplication by the genuine plane product on the actual edge module. -/
- def edgeFactorMap (e : Edge cs u b) : B.sheaf.edge e →ₗ[Coefficient (σ := σ)] B.sheaf.edge e :=
-  LinearMap.lsmul (Coefficient (σ := σ)) (B.sheaf.edge e)
-    (edgeFactorLift cs basis (source cs u b e).val (root cs u b e))
-
- theorem edgeFactorMap_graded (e : Edge cs u b) (n : ℤ) (v : B.sheaf.edge e)
-    (hv : v ∈ (B.sheaf.edge e).piece n) :
-    edgeFactorMap cs basis u b hub B e v ∈ (B.sheaf.edge e).piece
-      (n + edgeFactorDegree (source cs u b e).val (root cs u b e)) := by
-  simpa only [edgeFactorMap,LinearMap.lsmul_apply,vadd_eq_add,add_comm] using
-    SetLike.GradedSMul.smul_mem (edgeFactorLift_homogeneous cs basis
-      (source cs u b e).val (root cs u b e)) hv
-
-omit [Fintype σ] in
- theorem edgeFactorLift_regular (e : Edge cs u b) :
-    Function.Injective (fun v : B.sheaf.edge e =>
-      edgeFactorLift cs basis (source cs u b e).val (root cs u b e) • v) := by
-  let := B.free ((graph cs u b).target e)
-  exact FiniteFreeScalar.scalar_quotient_regular
-    (Module.Free.chooseBasis (Coefficient (σ := σ)) (B.sheaf.vertex ((graph cs u b).target e)))
-    (B.sheaf.upper e).map (B.quotient e).1
-    (polynomialLabel cs u b basis e)
-    (edgeFactorLift cs basis (source cs u b e).val (root cs u b e))
-    (polynomialLabel_prime cs u b basis e)
-    (edgeFactorLift_not_dvd cs basis (source cs u b e).val (root cs u b e)) (B.quotient e).2
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-namespace KLInvariance.LocalProducts
-open scoped Pointwise
-universe ur us um
-variable {R : Type ur} [CommRing R] {S : Type us} [CommRing S] [Algebra R S]
-  {M : Type um} [AddCommGroup M] [Module R M] [Module S M] [IsScalarTower R S M]
-
- theorem smul_top_le_of_map_dvd {p q : R} (h : algebraMap R S q ∣ algebraMap R S p) :
-    p • (⊤ : Submodule R M) ≤ q • (⊤ : Submodule R M) := by
-  intro z hz
-  obtain ⟨w,_,rfl⟩ := (Submodule.mem_smul_pointwise_iff_exists _ _ _).mp hz
-  obtain ⟨t,ht⟩ := h
-  apply (Submodule.mem_smul_pointwise_iff_exists _ _ _).mpr
-  refine ⟨t • w,Submodule.mem_top,?_⟩
-  rw [← IsScalarTower.algebraMap_smul S q,← IsScalarTower.algebraMap_smul S p,
-    smul_smul,← ht]
-
- theorem smul_top_eq_of_map_associated {p q : R}
-    (h : Associated (algebraMap R S p) (algebraMap R S q)) :
-    p • (⊤ : Submodule R M) = q • (⊤ : Submodule R M) :=
-  le_antisymm (smul_top_le_of_map_dvd h.symm.dvd) (smul_top_le_of_map_dvd h.dvd)
-
-variable {ι κ : Type*}
-
- theorem associated_prod_power (s : Finset ι) (a : ι → S) (p : S)
-    (h : ∀ i ∈ s, Associated (a i) p) : Associated (∏ i ∈ s, a i) (p^s.card) := by
-  simpa only [Finset.prod_const] using Associated.prod s a (fun _ => p) h
-
- theorem associated_prod_scale (s : Finset ι) (t : Finset κ) (a : ι → S) (b : κ → S)
-    (p f : S) (k : ℕ) (ha : ∀ i ∈ s, Associated (a i) p)
-    (hb : ∀ j ∈ t, Associated (b j) p) (hf : Associated f (p^k))
-    (hcount : s.card=t.card+k) :
-    Associated (∏ i ∈ s, a i) (f*∏ j ∈ t, b j) := by
-  have hs := associated_prod_power s a p ha
-  have ht := associated_prod_power t b p hb
-  apply hs.trans
-  have hpow : p^s.card=p^k*p^t.card := by rw [hcount,add_comm,pow_add]
-  rw [hpow]
-  exact (hf.mul_mul ht).symm
-
-/-- The local embedded product images agree whenever the reduced labels are
-associated and the actual later-edge count has the required difference. -/
- theorem product_image_scale (s : Finset ι) (t : Finset κ) (a : ι → R) (b : κ → R)
-    (p : S) (f : R) (k : ℕ)
-    (ha : ∀ i ∈ s, Associated (algebraMap R S (a i)) p)
-    (hb : ∀ j ∈ t, Associated (algebraMap R S (b j)) p)
-    (hf : Associated (algebraMap R S f) (p^k)) (hcount : s.card=t.card+k) :
-    (∏ i ∈ s, a i) • (⊤ : Submodule R M) =
-      f • ((∏ j ∈ t, b j) • (⊤ : Submodule R M)) := by
-  rw [← mul_smul]
-  apply smul_top_eq_of_map_associated (S := S)
-  simpa only [map_prod,map_mul] using associated_prod_scale s t
-    (fun i => algebraMap R S (a i)) (fun j => algebraMap R S (b j)) p
-    (algebraMap R S f) k ha hb hf hcount
-
-end KLInvariance.LocalProducts
-
-end
-
-
-section
-
-namespace KLInvariance.TitsSpace
-open Module BruhatGraph
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {σ : Type us} (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- theorem polynomialRoot_localized (P : Submodule ℝ (I → ℝ)) (a : PositiveRoot M cs) :
-    algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P) (polynomialRoot cs basis a) =
-      localizedLinear basis P a.val := rfl
-
- theorem polynomialRoot_reduced (P : Submodule ℝ (I → ℝ)) (a c : PositiveRoot M cs) :
-    algebraMap (PlaneRing basis P) (LocalizedEdgeRing basis P a.val)
-      (algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P) (polynomialRoot cs basis c)) =
-      localizedReduction basis P a.val c.val := rfl
-
- theorem planeRoot_outside {x : W} {a b : PositiveRoot M cs} (hab : a ≠ b)
-    (Q : (planeWeight x a).support) (hQ : Q.val ≠ rootPlane a b) :
-    (planeRoot x a Q).val ∉ rootPlane a b := by
-  intro h
-  apply hQ
-  rw [← planeRoot_plane x a Q]
-  exact rootPlane_eq_of_mem a b a (planeRoot x a Q) hab
-    (Submodule.subset_span (Set.mem_insert _ _)) h (planeRoot_ne x a Q).symm
-
- theorem reduced_planeRoot_other_isUnit {x : W} {a b : PositiveRoot M cs} (hab : a ≠ b)
-    (Q : (planeWeight x a).support) (hQ : Q.val ≠ rootPlane a b) :
-    IsUnit (localizedReduction basis (rootPlane a b) a.val (planeRoot x a Q).val) := by
-  have h := (linear_isUnit_iff basis (rootPlane a b) (planeRoot x a Q).val).mpr
-    (planeRoot_outside hab Q hQ)
-  exact h.map (algebraMap (PlaneRing basis (rootPlane a b))
-    (LocalizedEdgeRing basis (rootPlane a b) a.val))
-
-/-- The actual integral edge factor reduces, in each actual plane, to precisely
-its plane weight. All other factors are proven units, not discarded formally. -/
- theorem edgeFactor_local_associated (x : W) (a b : PositiveRoot M cs) (hab : a ≠ b) :
-    Associated
-      (algebraMap (PlaneRing basis (rootPlane a b)) (LocalizedEdgeRing basis (rootPlane a b) a.val)
-        (algebraMap (MvPolynomial σ ℝ) (PlaneRing basis (rootPlane a b)) (edgeFactorLift cs basis x a)))
-      ((localizedReduction basis (rootPlane a b) a.val b.val)^((planeWeight x a (rootPlane a b)).toNat)) := by
-  classical
-  let p := localizedReduction basis (rootPlane a b) a.val b.val
-  have hfactor (Q : (planeWeight x a).support) :
-      Associated
-        ((localizedReduction basis (rootPlane a b) a.val (planeRoot x a Q).val)^((planeWeight x a Q.val).toNat))
-        (if Q.val=rootPlane a b then p^((planeWeight x a (rootPlane a b)).toNat) else 1) := by
-    by_cases hQ : Q.val=rootPlane a b
-    · rw [ite_eq_left hQ]
-      have hc : (planeRoot x a Q).val ∈ rootPlane a b := by
-        rw [← hQ,← planeRoot_plane x a Q]
-        exact Submodule.subset_span (Set.mem_insert_of_mem _ (Set.mem_singleton _))
-      have h := localizedReduction_associated basis (rootPlane a b) a b (planeRoot x a Q)
-        hc (planeRoot_ne x a Q)
-      simpa only [hQ] using (Associated.pow_pow (M := LocalizedEdgeRing basis (rootPlane a b) a.val) h (n := (planeWeight x a Q.val).toNat))
-    · rw [ite_eq_right hQ]
-      exact associated_one_iff_isUnit.mpr ((reduced_planeRoot_other_isUnit basis hab Q hQ).pow _)
-  have hprod := Associated.prod Finset.univ
-    (fun Q : (planeWeight x a).support =>
-      (localizedReduction basis (rootPlane a b) a.val (planeRoot x a Q).val)^((planeWeight x a Q.val).toNat))
-    (fun Q => if Q.val=rootPlane a b then p^((planeWeight x a (rootPlane a b)).toNat) else 1)
-    (fun Q _ => hfactor Q)
-  have hp : (∏ Q : (planeWeight x a).support,
-      if Q.val=rootPlane a b then p^((planeWeight x a (rootPlane a b)).toNat) else 1) =
-      p^((planeWeight x a (rootPlane a b)).toNat) := by
-    rw [Fintype.prod_ite_eq_ite_exists _ (fun i j hi hj => Subtype.ext (hi.trans hj.symm))]
-    by_cases hP : rootPlane a b ∈ (planeWeight x a).support
-    · rw [ite_eq_left ⟨⟨rootPlane a b,hP⟩,rfl⟩]
-    · have hz := Finsupp.notMem_support_iff.mp hP
-      rw [hz,Int.toNat_zero,pow_zero]
-      split_ifs <;> rfl
-  rw [hp] at hprod
-  simpa only [edgeFactorLift,map_prod,map_pow,polynomialRoot_reduced] using hprod
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-namespace KLInvariance.TitsSpace.PlaneBasis
-open Module BruhatGraph MomentGraph MomentGraph.Sheaf
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-  {σ : Type us} (basis : Basis σ ℝ (Extended M))
-  (a₀ b : W) (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-noncomputable section
-
- theorem cosetRoot_eq (e : B.CosetEdge a₀ b) :
-    BruhatGraph.root cs 1 b (B.cosetEdgeAmbient a₀ b hmin e)=B.edgeRoot e.property := by
-  rw [B.edgeRoot_coset hmin e.property]
-  rfl
-
- theorem cosetLabel_eq_root (e : B.CosetEdge a₀ b) :
-    B.cosetLabel basis a₀ b hmin e=localizedLinear basis P (B.edgeRoot e.property).val := by
-  change algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P)
-    (polynomialRoot cs basis (BruhatGraph.root cs 1 b (B.cosetEdgeAmbient a₀ b hmin e)))=_
-  rw [B.cosetRoot_eq a₀ b hmin e]
-  rfl
-
- def reducedCosetRoot (e : B.CosetEdge a₀ b) (β : PositiveRoot M cs) :
-    PlaneRing basis P ⧸ Ideal.span {B.cosetLabel basis a₀ b hmin e} :=
-  algebraMap (PlaneRing basis P) _ (localizedLinear basis P β.val)
-
- theorem cosetLabel_reduction_associated (e f : B.CosetEdge a₀ b)
-    (β : PositiveRoot M cs) (hplane : rootPlane (B.edgeRoot e.property) β=P)
-    (hne : B.edgeRoot f.property ≠ B.edgeRoot e.property) :
-    Associated
-      (algebraMap (PlaneRing basis P) (PlaneRing basis P ⧸ Ideal.span {B.cosetLabel basis a₀ b hmin e})
-        (B.cosetLabel basis a₀ b hmin f))
-      (B.reducedCosetRoot basis a₀ b hmin e β) := by
-  unfold reducedCosetRoot
-  rw [B.cosetLabel_eq_root basis a₀ b hmin e,B.cosetLabel_eq_root basis a₀ b hmin f]
-  exact localizedReduction_associated basis P (B.edgeRoot e.property) β (B.edgeRoot f.property)
-    (hplane.symm ▸ B.edgeRoot_plane f.property) hne
-
- theorem cosetFactor_reduction_associated (e : B.CosetEdge a₀ b)
-    (β : PositiveRoot M cs) (hab : B.edgeRoot e.property ≠ β)
-    (hplane : rootPlane (B.edgeRoot e.property) β=P) :
-    Associated
-      (algebraMap (PlaneRing basis P) (PlaneRing basis P ⧸ Ideal.span {B.cosetLabel basis a₀ b hmin e})
-        (algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P)
-          (edgeFactorLift cs basis ((e.val.1.val:W)*a₀) (B.edgeRoot e.property))))
-      ((B.reducedCosetRoot basis a₀ b hmin e β)^((planeWeight ((e.val.1.val:W)*a₀) (B.edgeRoot e.property) P).toNat)) := by
-  unfold reducedCosetRoot
-  rw [B.cosetLabel_eq_root basis a₀ b hmin e]
-  have h := edgeFactor_local_associated basis ((e.val.1.val:W)*a₀) (B.edgeRoot e.property) β hab
-  rw [hplane] at h
-  exact h
-
-end
-end KLInvariance.TitsSpace.PlaneBasis
-
-end
-
-
-section
-
-namespace KLInvariance.BruhatGraph
-open TitsSpace
-universe u v u' v'
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
-  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
-  {u b : W} {u' b' : W'}
-  (φ : Interval cs u b ≃o Interval cs' u' b')
-noncomputable section
-
- def upperSource (e : Edge cs 1 b) (hx : BruhatLE cs u (source cs 1 b e).val) : Interval cs u b :=
-  ⟨(source cs 1 b e).val,hx,(source cs 1 b e).property.2⟩
- def upperTarget (e : Edge cs 1 b) (hx : BruhatLE cs u (source cs 1 b e).val) : Interval cs u b :=
-  ⟨(target cs 1 b e).val,bruhat_trans cs hx (.single e.property),(target cs 1 b e).property.2⟩
- def mappedEdgeRoot (e : Edge cs 1 b) (hx : BruhatLE cs u (source cs 1 b e).val) :
-     PositiveRoot M' cs' :=
-   graphRoot ((intervalIso_bruhatStep_iff_general cs cs' φ (upperSource e hx) (upperTarget e hx)).mpr e.property)
-
-end
-end KLInvariance.BruhatGraph
-
-namespace KLInvariance.TitsSpace.PlaneBasis
-open Module BruhatGraph MomentGraph MomentGraph.Sheaf
-universe u v u' v'
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-noncomputable section
-
- theorem exists_coset_edge {b : W} (e : Edge cs 1 b)
-    (he : (BruhatGraph.root cs 1 b e).val ∈ P) :
-    ∃ a₀, ∃ hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀),
-      ∃ d : B.CosetEdge a₀ b, B.cosetEdgeAmbient a₀ b hmin d=e := by
-  obtain ⟨a₀,g,hg,hx,hmin⟩ := planeCoset_exists_minimal M cs P (source cs 1 b e).val
-  have ht : label cs 1 b e ∈ planeSubgroup M cs P :=
-    (Represents.planeSubgroup_mem_iff M cs (positiveRoot_represents M cs
-      ⟨label cs 1 b e,label_isReflection cs 1 b e⟩)).mpr he
-  let gg : planeSubgroup M cs P := ⟨g,hg⟩
-  let hh : planeSubgroup M cs P := ⟨label cs 1 b e*g,(planeSubgroup M cs P).mul_mem ht hg⟩
-  have hy : (hh:W)*a₀=(target cs 1 b e).val := by
-    change (label cs 1 b e*g)*a₀=_
-    rw [mul_assoc,← hx,label_mul_source]
-  let x : B.CosetVertex a₀ b := ⟨gg,one_bruhat B.system gg,hx ▸ (source cs 1 b e).property.2⟩
-  let y : B.CosetVertex a₀ b := ⟨hh,one_bruhat B.system hh,hy.symm ▸ (target cs 1 b e).property.2⟩
-  have hs : BruhatStep B.system gg hh := by
-    apply (B.coset_step_iff hmin gg hh).mp
-    rw [show (gg:W)*a₀=(source cs 1 b e).val from hx.symm,hy]
-    exact e.property
-  refine ⟨a₀,hmin,⟨(x,y),hs⟩,?_⟩
-  apply Subtype.ext
-  exact Prod.ext (Subtype.ext hx.symm) (Subtype.ext hy)
-
-variable {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
-  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
-  {a₀ u b : W} {u' b' : W'}
-  (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-  (φ : Interval cs u b ≃o Interval cs' u' b')
-
- theorem mappedCosetRoot_eq (e : B.CosetEdge a₀ b)
-    (hx : BruhatLE cs u ((e.val.1.val:W)*a₀)) :
-    B.mappedCosetRoot hmin φ e hx = mappedEdgeRoot φ (B.cosetEdgeAmbient a₀ b hmin e) hx := rfl
-
-end
-end KLInvariance.TitsSpace.PlaneBasis
-
-end
-
-
-section
-
-/-! The actual transported finite-interval elementary matrix is the
-second system's genuine R-tilde matrix. The input is only the unlabelled
-order isomorphism; reflection edges are recovered by the checked general
-Bruhat-graph reconstruction theorem. -/
-
-end
-
-end OAI
+import OAI.RepresentationTheory.KazhdanLusztig.DihedralIntervals
+
+namespace OAI
+
+/-!
+Symmetric plane ideals, sheaf localization and scaled local products along genuine edges.
+-/
+
+section
+
+namespace KLInvariance.SymmetricPlaneIdeal
+universe uk uv
+variable {k : Type uk} [Field k] {V : Type uv} [AddCommGroup V] [Module k V]
+noncomputable section
+
+ def ideal (P : Submodule k V) : Ideal (SymmetricAlgebra k V) :=
+  Ideal.span ((SymmetricAlgebra.ι k V) '' (P : Set V))
+
+ def quotientMap (P : Submodule k V) :
+    SymmetricAlgebra k V →ₐ[k] SymmetricAlgebra k (V ⧸ P) :=
+  SymmetricAlgebra.lift ((SymmetricAlgebra.ι k (V ⧸ P)).comp P.mkQ)
+
+@[simp] theorem quotientMap_ι (P : Submodule k V) (v : V) :
+    quotientMap P (SymmetricAlgebra.ι k V v) = SymmetricAlgebra.ι k (V ⧸ P) (P.mkQ v) :=
+  SymmetricAlgebra.lift_ι_apply _ _
+
+ theorem ideal_le_ker (P : Submodule k V) : ideal P ≤ RingHom.ker (quotientMap P) := by
+  apply Ideal.span_le.mpr
+  rintro _ ⟨v,hv,rfl⟩
+  change quotientMap P (SymmetricAlgebra.ι k V v) = 0
+  rw [quotientMap_ι,show P.mkQ v = 0 from (Submodule.Quotient.mk_eq_zero P).mpr hv,map_zero]
+
+ def forward (P : Submodule k V) : (SymmetricAlgebra k V ⧸ ideal P) →ₐ[k]
+    SymmetricAlgebra k (V ⧸ P) :=
+  Ideal.Quotient.liftₐ _ (quotientMap P) (fun _ hz => ideal_le_ker P hz)
+
+ def linearQuotient (P : Submodule k V) : V ⧸ P →ₗ[k]
+    (SymmetricAlgebra k V ⧸ ideal P) :=
+  P.liftQ ((Ideal.Quotient.mkₐ k (ideal P)).toLinearMap.comp (SymmetricAlgebra.ι k V)) (by
+    intro v hv
+    apply LinearMap.mem_ker.mpr
+    exact Ideal.Quotient.eq_zero_iff_mem.mpr (Ideal.subset_span ⟨v,hv,rfl⟩))
+
+ def backward (P : Submodule k V) : SymmetricAlgebra k (V ⧸ P) →ₐ[k]
+    (SymmetricAlgebra k V ⧸ ideal P) := SymmetricAlgebra.lift (linearQuotient P)
+
+@[simp] theorem forward_mk (P : Submodule k V) (z : SymmetricAlgebra k V) :
+    forward P (Ideal.Quotient.mk (ideal P) z) = quotientMap P z := rfl
+
+@[simp] theorem backward_ι_mk (P : Submodule k V) (v : V) :
+    backward P (SymmetricAlgebra.ι k (V ⧸ P) (P.mkQ v)) =
+      Ideal.Quotient.mk (ideal P) (SymmetricAlgebra.ι k V v) := by
+  rw [backward,SymmetricAlgebra.lift_ι_apply]
+  rfl
+
+ def quotientEquiv (P : Submodule k V) : (SymmetricAlgebra k V ⧸ ideal P) ≃ₐ[k]
+    SymmetricAlgebra k (V ⧸ P) :=
+  AlgEquiv.ofAlgHom (forward P) (backward P)
+    (by
+      apply SymmetricAlgebra.algHom_ext
+      apply LinearMap.ext
+      intro v
+      obtain ⟨v,rfl⟩ := P.mkQ_surjective v
+      change forward P (backward P (SymmetricAlgebra.ι k _ (P.mkQ v))) = _
+      rw [backward_ι_mk,forward_mk,quotientMap_ι]
+      rfl)
+    (by
+      apply Ideal.Quotient.algHom_ext
+      apply SymmetricAlgebra.algHom_ext
+      apply LinearMap.ext
+      intro v
+      change backward P (forward P (Ideal.Quotient.mk (ideal P) (SymmetricAlgebra.ι k V v))) = _
+      rw [forward_mk,quotientMap_ι,backward_ι_mk]
+      rfl)
+
+@[simp] theorem quotientEquiv_mk_ι (P : Submodule k V) (v : V) :
+    quotientEquiv P (Ideal.Quotient.mk (ideal P) (SymmetricAlgebra.ι k V v)) =
+      SymmetricAlgebra.ι k (V ⧸ P) (P.mkQ v) := quotientMap_ι P v
+
+ theorem ι_mem_iff (P : Submodule k V) (v : V) :
+    SymmetricAlgebra.ι k V v ∈ ideal P ↔ v ∈ P := by
+  constructor
+  · intro hv
+    have hh := ideal_le_ker P hv
+    change quotientMap P (SymmetricAlgebra.ι k V v) = 0 at hh
+    rw [quotientMap_ι] at hh
+    apply (Submodule.Quotient.mk_eq_zero P).mp
+    have he : SymmetricAlgebra.ι k (V ⧸ P) (P.mkQ v) =
+        SymmetricAlgebra.ι k (V ⧸ P) 0 := by simpa only [map_zero] using hh
+    by_contra hn
+    obtain ⟨f,hf⟩ := Module.Projective.exists_dual_ne_zero k hn
+    have h := congrArg (SymmetricAlgebra.lift f) he
+    rw [SymmetricAlgebra.lift_ι_apply,SymmetricAlgebra.lift_ι_apply,map_zero] at h
+    exact hf h
+  · intro hv
+    exact Ideal.subset_span ⟨v,hv,rfl⟩
+
+ theorem ideal_eq_ker (P : Submodule k V) : ideal P = RingHom.ker (quotientMap P) := by
+  apply le_antisymm (ideal_le_ker P)
+  intro z hz
+  apply Ideal.Quotient.eq_zero_iff_mem.mp
+  apply (quotientEquiv P).injective
+  change quotientMap P z = quotientEquiv P 0
+  rw [map_zero]
+  exact hz
+
+ instance ideal_isPrime (P : Submodule k V) : (ideal P).IsPrime := by
+  rw [ideal_eq_ker]
+  exact RingHom.ker_isPrime (quotientMap P)
+
+ theorem ideal_span_eq (s : Set V) :
+    ideal (Submodule.span k s) = Ideal.span ((SymmetricAlgebra.ι k V) '' s) := by
+  apply le_antisymm
+  · apply Ideal.span_le.mpr
+    rintro _ ⟨v,hv,rfl⟩
+    induction hv using Submodule.span_induction with
+    | mem x hx => exact Ideal.subset_span ⟨x,hx,rfl⟩
+    | zero =>
+      rw [map_zero]
+      exact (Ideal.span ((SymmetricAlgebra.ι k V) '' s)).zero_mem
+    | add x y _ _ hx hy =>
+      rw [map_add]
+      exact (Ideal.span ((SymmetricAlgebra.ι k V) '' s)).add_mem hx hy
+    | smul r x _ hx =>
+      rw [map_smul,Algebra.smul_def]
+      exact (Ideal.span ((SymmetricAlgebra.ι k V) '' s)).mul_mem_left _ hx
+  · apply Ideal.span_le.mpr
+    rintro _ ⟨v,hv,rfl⟩
+    exact (ι_mem_iff _ _).mpr (Submodule.subset_span hv)
+
+end
+end KLInvariance.SymmetricPlaneIdeal
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.Sheaf
+universe u v w z
+variable {R : Type u} [CommRing R] [IsLocalRing R]
+  {V : Type v} [PartialOrder V] {E : Type w} {G : OrderedGraph V E}
+  {α : E → R} (O : LocalModels.{u,v,w,z} (G := G) α)
+  {B C : Sheaf.{u,v,w,z} (R := R) G}
+
+
+theorem extend_modelSum {U : Set V} (hU : IsUpperSet U)
+    (hC : ModelSum O.sheaf U C) (j : OnIso B C U)
+    (hBq : B.UpperQuotient α) (hBf : B.Flabby) (hBg : B.Generated)
+    (x : V) (hx : x ∉ U) (hmax : Set.Ioi x ⊆ U)
+    [Module.Free R (B.vertex x)] [Module.Finite R (B.vertex x)] :
+    ∃ D : Sheaf.{u,v,w,z} (R := R) G,
+      ModelSum O.sheaf (insert x U) D ∧ Nonempty (OnIso B D (insert x U)) := by
+  classical
+  let := hC.free O x
+  let := hC.finite O x
+  obtain ⟨g,jv,hfree,hfinite,hjv⟩ := (j.above x hmax).extend_stalk
+    hBf hBg (hC.flabby O) (hC.generated O) (hC.minimal O x hx)
+  let := hfree
+  let := hfinite
+  let n := Module.finrank R (LinearMap.ker g)
+  let S : Sheaf.{u,v,w,z} (R := R) G := pi (fun _ : Fin n => O.sheaf x)
+  let jq : LinearMap.ker g ≃ₗ[R] S.vertex x :=
+    (Module.finBasis R (LinearMap.ker g)).equivFun.trans
+      (LinearEquiv.piCongrRight (fun _ : Fin n => O.top x)).symm
+  let jv' : B.vertex x ≃ₗ[R] (prod C S).vertex x :=
+    jv.trans (LinearEquiv.prodCongr (LinearEquiv.refl R (C.vertex x)) jq)
+  have hSV (y : V) (hy : y ∈ U) : Subsingleton (S.vertex y) :=
+    pi_subsingleton_vertex _ y (fun _ => O.support x y (fun hyx => hx (hU hyx hy)))
+  have hSE (e : E) (he : G.target e ∈ U) : Subsingleton (S.edge e) := by
+    apply pi_subsingleton_edge
+    intro i
+    let := O.support x (G.target e) (fun h => hx (hU h he))
+    exact (O.quotient x e).1.subsingleton
+  let jj : OnIso B (prod C S) U := j.prodZero S hSV hSE
+  have hq : (prod C S).UpperQuotient α :=
+    prod_upperQuotient C S α (hC.quotient O) (pi_upperQuotient _ α (fun _ => O.quotient x))
+  have hout : ∀ e (he : G.source e = x) v,
+      jj.edge e (hmax (lt_of_eq_of_lt he.symm (G.increasing e)))
+        (B.lower e v) = (prod C S).lower e (castStalkIso B (prod C S) he.symm jv' v) := by
+    intro e he v
+    subst x
+    apply Prod.ext
+    · exact hjv v ⟨e,rfl⟩
+    · let := hSE e (hmax (G.increasing e))
+      exact Subsingleton.elim _ _
+  refine ⟨prod C S,?_,⟨jj.extend hU x hx hmax α hBq hq jv' hout⟩⟩
+  exact .add (hC.mono (Set.subset_insert x U)) x (Set.mem_insert x U) n
+
+end KLInvariance.MomentGraph.Sheaf
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.Sheaf
+universe u v w z
+variable {R : Type u} [CommRing R] [IsLocalRing R]
+  {V : Type v} [PartialOrder V] {E : Type w} {G : OrderedGraph V E}
+  {α : E → R} (O : LocalModels.{u,v,w,z} (G := G) α)
+  (B : Sheaf.{u,v,w,z} (R := R) G)
+
+/-- Actual finite descending decomposition over a local ring. This does not
+assume that localization preserves indecomposability. Its explicitly stated
+rank-one model criteria must be proved before applying it to a root plane. -/
+theorem exists_modelSumIso [Finite V]
+    (hBfree : ∀ x, Module.Free R (B.vertex x))
+    (hBfinite : ∀ x, Module.Finite R (B.vertex x))
+    (hBq : B.UpperQuotient α) (hBf : B.Flabby) (hBg : B.Generated) :
+    ∃ C : Sheaf.{u,v,w,z} (R := R) G,
+      ModelSum O.sheaf Set.univ C ∧ Nonempty (Iso B C) := by
+  classical
+  let := Fintype.ofFinite V
+  have h : ∀ s : Finset V, IsUpperSet ((s : Set V)ᶜ) →
+      ∀ C : Sheaf.{u,v,w,z} (R := R) G,
+      ModelSum O.sheaf ((s : Set V)ᶜ) C → OnIso B C ((s : Set V)ᶜ) →
+      ∃ D : Sheaf.{u,v,w,z} (R := R) G,
+        ModelSum O.sheaf Set.univ D ∧ Nonempty (Iso B D) := by
+    intro s
+    induction s using Finset.strongInductionOn with
+    | _ s ih =>
+      intro hU C hC j
+      by_cases hs : s = ∅
+      · subst s
+        have hh : ((∅ : Finset V) : Set V)ᶜ = Set.univ := by simp
+        exact ⟨C,hh ▸ hC,⟨OnIso.ofUniv (hh ▸ j)⟩⟩
+      · obtain ⟨x,hxs,hxmax⟩ := s.exists_maximal (Finset.nonempty_iff_ne_empty.mpr hs)
+        have hx : x ∉ ((s : Set V)ᶜ) := by simpa using hxs
+        have hmax : Set.Ioi x ⊆ ((s : Set V)ᶜ) := by
+          intro y hy
+          change y ∉ s
+          intro hys
+          exact (not_le_of_gt hy) (hxmax hys hy.le)
+        let := hBfree x
+        let := hBfinite x
+        obtain ⟨D,hD,⟨jj⟩⟩ := extend_modelSum O hU hC j hBq hBf hBg x hx hmax
+        have heq : insert x ((s : Set V)ᶜ) = ((s.erase x : Finset V) : Set V)ᶜ := by
+          ext y
+          simp only [Set.mem_insert_iff,Set.mem_compl_iff,Finset.mem_coe,Finset.mem_erase]
+          tauto
+        have hU' : IsUpperSet (insert x ((s : Set V)ᶜ)) := by
+          intro y t hyt hy
+          rcases Set.mem_insert_iff.mp hy with hy | hy
+          · subst y
+            by_cases htx : t=x
+            · exact Set.mem_insert_iff.mpr (Or.inl htx)
+            · exact Set.mem_insert_of_mem x (hmax (lt_of_le_of_ne hyt (Ne.symm htx)))
+          · exact Set.mem_insert_of_mem x (hU hyt hy)
+        exact ih (s.erase x) (Finset.erase_ssubset hxs) (heq ▸ hU') D (heq ▸ hD) (heq ▸ jj)
+  have hset : ((Finset.univ : Finset V) : Set V)ᶜ = ∅ := by simp
+  have hj : OnIso B zero (∅ : Set V) :=
+    ⟨fun x hx => False.elim hx,fun e he => False.elim he,
+      fun e hs => False.elim hs,fun e ht => False.elim ht⟩
+  have hinit := h Finset.univ
+  rw [hset] at hinit
+  exact hinit isUpperSet_empty zero .zero hj
+
+end KLInvariance.MomentGraph.Sheaf
+
+end
+
+
+section
+
+/-! The rank-one plane models over the actual flat local coefficient ring.
+The local-ring label condition is later supplied by the plane prime. -/
+namespace KLInvariance.BruhatGraph
+universe u v
+variable {I : Type u} {M : CoxeterMatrix I} {W : Type v} [Group W]
+  (cs : CoxeterSystem M W) (u b : W)
+
+ theorem exists_outgoing_below (x c : Interval cs u b) (hxc : x < c) :
+    ∃ e : Edge cs u b, source cs u b e = x ∧ target cs u b e ≤ c := by
+  rcases Relation.ReflTransGen.cases_head hxc.le with heq | ⟨z,hxz,hzc⟩
+  · exact (hxc.ne (Subtype.ext heq)).elim
+  · let y : Interval cs u b := ⟨z,bruhat_trans cs x.property.1 (.single hxz),
+      bruhat_trans cs hzc c.property.2⟩
+    exact ⟨⟨(x,y),hxz⟩,rfl,hzc⟩
+
+end KLInvariance.BruhatGraph
+
+namespace KLInvariance.MomentGraph.Sheaf
+universe ur uv ue
+variable {R : Type ur} [CommRing R] [IsLocalRing R]
+  {V : Type uv} [PartialOrder V] {E : Type ue} (G : OrderedGraph V E) (α : E → R)
+noncomputable section
+
+ def structureLocalModels
+    (hα : ∀ e, α e ∈ IsLocalRing.maximalIdeal R)
+    (hreach : ∀ x c, x < c → ∃ e, G.source e = x ∧ G.target e ≤ c)
+    (hfl : ∀ c, (structureSheaf G α c).Flabby) : LocalModels (G := G) α where
+  sheaf := structureSheaf G α
+  free := structureSheaf_free G α
+  finite := structureSheaf_finite G α
+  quotient := structureSheaf_upperQuotient G α
+  generated := structureSheaf_generated G α
+  flabby := hfl
+  support := structureSheaf_support G α
+  top c := structureVertexIso G α c c le_rfl
+  minimal c x hx := structureSheaf_minimal G α c hα (fun x => hreach x c) x hx
+
+end
+end KLInvariance.MomentGraph.Sheaf
+
+namespace KLInvariance.TitsSpace.PlaneBasis
+open Module BruhatGraph MomentGraph MomentGraph.Sheaf
+universe u v us ur
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+  {σ : Type us} (basis : Basis σ ℝ (Extended M))
+  (R : Type ur) [CommRing R] [Algebra (MvPolynomial σ ℝ) R]
+  [Module.Flat (MvPolynomial σ ℝ) R] [IsLocalRing R]
+noncomputable section
+
+ def localModels (b : planeSubgroup M cs P)
+    (hα : ∀ e, algebraMap (MvPolynomial σ ℝ) R (B.modelLabel basis b e) ∈
+      IsLocalRing.maximalIdeal R) :
+    LocalModels (G := graph B.system 1 b)
+      (fun e => algebraMap (MvPolynomial σ ℝ) R (B.modelLabel basis b e)) := by
+  classical
+  let := interval_finite B.system 1 b
+  let := Fintype.ofFinite (Edge B.system 1 b)
+  exact structureLocalModels _ _ hα (exists_outgoing_below B.system 1 b)
+    (fun c => structureSheaf_flat_flabby R _ _ c (B.model_flabby basis b c))
+
+end
+end KLInvariance.TitsSpace.PlaneBasis
+
+end
+
+
+section
+
+/-! The actual plane prime and its localization in the coefficient polynomial
+ring of the chosen nondegenerate ambient realization. -/
+namespace KLInvariance.TitsSpace
+open Module
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {σ : Type us} (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ def embedLinear : (I → ℝ) →ₗ[ℝ] Extended M where
+  toFun := embed M
+  map_add' _ _ := by simp [embed]
+  map_smul' _ _ := by simp [embed]
+
+ theorem embedLinear_injective : Function.Injective (embedLinear (M := M)) := by
+  intro a b hab
+  exact congrArg Prod.fst hab
+
+ def planePrime (P : Submodule ℝ (I → ℝ)) : Ideal (MvPolynomial σ ℝ) :=
+  Ideal.map (SymmetricAlgebra.equivMvPolynomial basis).toRingHom
+    (SymmetricPlaneIdeal.ideal (P.map (embedLinear (M := M))))
+
+ instance planePrime_isPrime (P : Submodule ℝ (I → ℝ)) : (planePrime basis P).IsPrime :=
+  Ideal.map_isPrime_of_equiv (SymmetricAlgebra.equivMvPolynomial basis)
+
+ theorem linear_mem_planePrime_iff (P : Submodule ℝ (I → ℝ)) (a : I → ℝ) :
+    linearPolynomial basis (embed M a) ∈ planePrime basis P ↔ a ∈ P := by
+  rw [planePrime,Ideal.mem_map_iff_of_surjective (SymmetricAlgebra.equivMvPolynomial basis).toRingHom
+    (SymmetricAlgebra.equivMvPolynomial basis).surjective]
+  change (∃ z, z ∈ SymmetricPlaneIdeal.ideal (P.map (embedLinear (M := M))) ∧
+    SymmetricAlgebra.equivMvPolynomial basis z =
+      SymmetricAlgebra.equivMvPolynomial basis (SymmetricAlgebra.ι ℝ (Extended M) (embed M a))) ↔ _
+  constructor
+  · rintro ⟨z,hz,he⟩
+    have h := (SymmetricAlgebra.equivMvPolynomial basis).injective he
+    subst z
+    have ha := (SymmetricPlaneIdeal.ι_mem_iff _ _).mp hz
+    obtain ⟨b,hb,he⟩ := ha
+    have he' : b = a := embedLinear_injective he
+    exact he' ▸ hb
+  · intro ha
+    refine ⟨_,(SymmetricPlaneIdeal.ι_mem_iff _ _).mpr ?_,rfl⟩
+    exact ⟨a,ha,rfl⟩
+
+ abbrev PlaneRing (P : Submodule ℝ (I → ℝ)) := Localization.AtPrime (planePrime basis P)
+
+ theorem linear_mem_maximal_iff (P : Submodule ℝ (I → ℝ)) (a : I → ℝ) :
+    algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P) (linearPolynomial basis (embed M a)) ∈
+      IsLocalRing.maximalIdeal (PlaneRing basis P) ↔ a ∈ P := by
+  rw [IsLocalization.AtPrime.to_map_mem_maximal_iff _ (planePrime basis P),
+    linear_mem_planePrime_iff]
+
+ theorem linear_isUnit_iff (P : Submodule ℝ (I → ℝ)) (a : I → ℝ) :
+    IsUnit (algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P)
+      (linearPolynomial basis (embed M a))) ↔ a ∉ P := by
+  rw [IsLocalization.AtPrime.isUnit_to_map_iff _ (planePrime basis P)]
+  change _ ∉ planePrime basis P ↔ _
+  rw [linear_mem_planePrime_iff]
+
+end
+end KLInvariance.TitsSpace
+
+namespace KLInvariance.TitsSpace.PlaneBasis
+open Module BruhatGraph MomentGraph MomentGraph.Sheaf
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+  {σ : Type us} (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ theorem modelLabel_mem_maximal (b : planeSubgroup M cs P) (e : Edge B.system 1 b) :
+    algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P) (B.modelLabel basis b e) ∈
+      IsLocalRing.maximalIdeal (PlaneRing basis P) := by
+  apply (linear_mem_maximal_iff basis P _).mpr
+  apply (Represents.planeSubgroup_mem_iff M cs (positiveRoot_represents M cs _)).mp
+  exact (label B.system 1 b e).property
+
+ def localizedModels (b : planeSubgroup M cs P) :
+    LocalModels (G := graph B.system 1 b)
+      (fun e => algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P) (B.modelLabel basis b e)) :=
+  B.localModels basis (PlaneRing basis P) b (B.modelLabel_mem_maximal basis b)
+
+end
+end KLInvariance.TitsSpace.PlaneBasis
+
+end
+
+
+section
+
+/-! The actual finite relative lower ideal in a maximal root-plane coset.
+No order-reflection identification with ambient comparability is used. -/
+namespace KLInvariance.TitsSpace.PlaneBasis
+open Module BruhatGraph MomentGraph
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+  (a₀ b : W)
+noncomputable section
+
+ def CosetVertex := {d : planeSubgroup M cs P // BruhatLE B.system 1 d ∧
+    BruhatLE cs ((d:W)*a₀) b}
+
+ instance cosetOrder : PartialOrder (B.CosetVertex a₀ b) where
+  le x y := BruhatLE B.system x.val y.val
+  le_refl x := bruhat_refl B.system x.val
+  le_trans _ _ _ := bruhat_trans B.system
+  le_antisymm x y hxy hyx := Subtype.ext (bruhat_antisymm B.system hxy hyx)
+
+ def cosetAmbient (x : B.CosetVertex a₀ b) : Interval cs 1 b :=
+  ⟨(x.val:W)*a₀,one_bruhat cs _,x.property.2⟩
+
+ theorem cosetAmbient_injective : Function.Injective (B.cosetAmbient a₀ b) := by
+  intro x y h
+  apply Subtype.ext
+  apply Subtype.ext
+  exact mul_right_cancel (congrArg Subtype.val h)
+
+ instance cosetFinite : Finite (B.CosetVertex a₀ b) := by
+  let := interval_finite cs 1 b
+  exact Finite.of_injective _ (B.cosetAmbient_injective a₀ b)
+
+ def CosetEdge := {p : B.CosetVertex a₀ b × B.CosetVertex a₀ b //
+    BruhatStep B.system p.1.val p.2.val}
+
+ def cosetGraph : OrderedGraph (B.CosetVertex a₀ b) (B.CosetEdge a₀ b) where
+  source e := e.val.1
+  target e := e.val.2
+  increasing e := ⟨.single e.property,fun h => not_lt_of_ge
+    (length_le_of_bruhat B.system h) e.property.1⟩
+
+variable (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+
+ def cosetEdgeAmbient (e : B.CosetEdge a₀ b) : Edge cs 1 b :=
+  ⟨(B.cosetAmbient a₀ b e.val.1,B.cosetAmbient a₀ b e.val.2),
+    (B.coset_step_iff hmin _ _).mpr e.property⟩
+
+ def cosetGraphMap : GraphMap (B.cosetGraph a₀ b) (graph cs 1 b) where
+  vertex := B.cosetAmbient a₀ b
+  edge := B.cosetEdgeAmbient a₀ b hmin
+  source _ := rfl
+  target _ := rfl
+
+ include hmin in
+ theorem coset_lower {d : planeSubgroup M cs P} {x : B.CosetVertex a₀ b}
+    (hx : BruhatLE B.system d x.val) : BruhatLE cs ((d:W)*a₀) b :=
+  bruhat_trans cs (B.coset_bruhat hmin hx) x.property.2
+
+ include hmin in
+ theorem coset_outgoing_below (x c : B.CosetVertex a₀ b) (hxc : x<c) :
+    ∃ e : B.CosetEdge a₀ b, (B.cosetGraph a₀ b).source e=x ∧
+      (B.cosetGraph a₀ b).target e ≤ c := by
+  rcases Relation.ReflTransGen.cases_head hxc.le with heq | ⟨z,hxz,hzc⟩
+  · exact (hxc.ne (Subtype.ext heq)).elim
+  · let y : B.CosetVertex a₀ b := ⟨z,one_bruhat B.system z,
+      B.coset_lower a₀ b hmin hzc⟩
+    exact ⟨⟨(x,y),hxz⟩,rfl,hzc⟩
+
+ theorem coset_lift_live (e : Edge cs 1 b)
+    (he : label cs 1 b e ∈ planeSubgroup M cs P)
+    (x : B.CosetVertex a₀ b) (hx : B.cosetAmbient a₀ b x=source cs 1 b e) :
+    ∃ a : B.CosetEdge a₀ b, B.cosetEdgeAmbient a₀ b hmin a=e ∧
+      (B.cosetGraph a₀ b).source a=x := by
+  let t : planeSubgroup M cs P := ⟨label cs 1 b e,he⟩
+  have hy : ((t*x.val : planeSubgroup M cs P):W)*a₀ = (target cs 1 b e).val := by
+    change (label cs 1 b e*(x.val:W))*a₀ = _
+    rw [mul_assoc,show (x.val:W)*a₀=(source cs 1 b e).val from congrArg Subtype.val hx]
+    exact label_mul_source cs 1 b e
+  let y : B.CosetVertex a₀ b := ⟨t*x.val,one_bruhat B.system _,hy.symm ▸ (target cs 1 b e).property.2⟩
+  have hye : B.cosetAmbient a₀ b y=target cs 1 b e := Subtype.ext hy
+  have hs : BruhatStep B.system x.val y.val := by
+    apply (B.coset_step_iff hmin _ _).mp
+    change BruhatStep cs (B.cosetAmbient a₀ b x).val (B.cosetAmbient a₀ b y).val
+    rw [hx,hye]
+    exact e.property
+  refine ⟨⟨(x,y),hs⟩,?_,rfl⟩
+  apply Subtype.ext
+  exact Prod.ext hx hye
+
+end
+end KLInvariance.TitsSpace.PlaneBasis
+
+end
+
+
+section
+
+/-! Actual plane localization of the boundary-constructed sheaf. Its relative
+coset restriction is free, generated and flabby with upper quotient maps;
+no character or upward-kernel freeness theorem is used. -/
+namespace KLInvariance.MomentGraph.Sheaf
+universe ur uv ue um
+variable {R : Type ur} [CommRing R] {V : Type uv} [PartialOrder V]
+  {E : Type ue} {G : OrderedGraph V E} (B : Sheaf.{ur,uv,ue,um} (R := R) G)
+
+ theorem edge_subsingleton_of_unit (α : E → R) (hq : B.UpperQuotient α)
+    (e : E) (he : IsUnit (α e)) : Subsingleton (B.edge e) := by
+  obtain ⟨a,ha⟩ := he
+  have hz : ∀ v : B.edge e, v=0 := by
+    intro v
+    obtain ⟨w,rfl⟩ := (hq e).1 v
+    apply ((hq e).2 w).mpr
+    refine ⟨(↑a⁻¹ : R) • w,?_⟩
+    rw [← ha,smul_smul]
+    simp
+  exact ⟨fun x y => (hz x).trans (hz y).symm⟩
+
+end KLInvariance.MomentGraph.Sheaf
+
+namespace KLInvariance.TitsSpace.PlaneBasis
+open Module BruhatGraph MomentGraph MomentGraph.Sheaf
+universe u v us um
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  (a₀ b : W) (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+noncomputable section
+
+ def cosetLabel (e : B.CosetEdge a₀ b) : PlaneRing basis P :=
+  algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P)
+    (polynomialLabel cs 1 b basis (B.cosetEdgeAmbient a₀ b hmin e))
+
+ def localizedCoset (C : Sheaf.{us,v,v,um} (R := MvPolynomial σ ℝ) (graph cs 1 b)) :
+    Sheaf (R := PlaneRing basis P) (B.cosetGraph a₀ b) :=
+  (C.baseChange (PlaneRing basis P)).pullback (B.cosetGraphMap a₀ b hmin)
+
+ omit [Fintype σ] in
+ theorem polynomialLabel_unit_iff (e : Edge cs 1 b) :
+    IsUnit (algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P)
+      (polynomialLabel cs 1 b basis e)) ↔ label cs 1 b e ∉ planeSubgroup M cs P := by
+  change IsUnit (algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P)
+    (linearPolynomial basis (embed M (BruhatGraph.root cs 1 b e).val))) ↔ _
+  rw [linear_isUnit_iff]
+  have hr := positiveRoot_represents M cs ⟨label cs 1 b e,label_isReflection cs 1 b e⟩
+  exact not_congr (Represents.planeSubgroup_mem_iff M cs hr).symm
+
+ omit [Fintype σ] in
+ theorem localizedCoset_quotient (C : Sheaf.{us,v,v,um} (R := MvPolynomial σ ℝ) (graph cs 1 b))
+    (hq : C.UpperQuotient (polynomialLabel cs 1 b basis)) :
+    (B.localizedCoset basis a₀ b hmin C).UpperQuotient (B.cosetLabel basis a₀ b hmin) :=
+  (C.baseChange (PlaneRing basis P)).pullback_quotient (B.cosetGraphMap a₀ b hmin)
+    (fun e => algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P) (polynomialLabel cs 1 b basis e))
+    (C.baseChange_upperQuotient (PlaneRing basis P) (polynomialLabel cs 1 b basis) hq)
+
+ omit [Fintype σ] in
+ theorem localizedCoset_generated (C : Sheaf.{us,v,v,um} (R := MvPolynomial σ ℝ) (graph cs 1 b))
+    (hg : C.Generated) : (B.localizedCoset basis a₀ b hmin C).Generated :=
+  (C.baseChange (PlaneRing basis P)).pullback_generated _
+    (C.baseChange_generated (PlaneRing basis P) hg)
+
+ omit [Fintype σ] in
+ theorem localizedCoset_flabby (C : Sheaf.{us,v,v,um} (R := MvPolynomial σ ℝ) (graph cs 1 b))
+    (hq : C.UpperQuotient (polynomialLabel cs 1 b basis)) (hf : C.Flabby) :
+    (B.localizedCoset basis a₀ b hmin C).Flabby := by
+  let := interval_finite cs 1 b
+  apply (C.baseChange (PlaneRing basis P)).pullback_flabby_live _
+    (C.baseChange_flabby (PlaneRing basis P) hf) (B.cosetAmbient_injective a₀ b)
+    {e | label cs 1 b e ∈ planeSubgroup M cs P}
+  · intro e he
+    exact (C.baseChange (PlaneRing basis P)).edge_subsingleton_of_unit _
+      (C.baseChange_upperQuotient (PlaneRing basis P) _ hq) e
+      ((polynomialLabel_unit_iff (P := P) basis b e).mpr he)
+  · exact B.coset_lift_live a₀ b hmin
+
+end
+end KLInvariance.TitsSpace.PlaneBasis
+
+end
+
+
+section
+
+/-! The relative rank-one structure models are natural under full strictly
+order-compatible graph embeddings, with exactly the pulled-back labels. -/
+namespace KLInvariance.MomentGraph.Sheaf
+universe ur uv ue uw uf
+variable {R : Type ur} [CommRing R] {V : Type uv} [PartialOrder V]
+  {E : Type ue} {G : OrderedGraph V E} {W : Type uw} [PartialOrder W]
+  {F : Type uf} {H : OrderedGraph W F}
+noncomputable section
+
+ def propPiEquiv (N : Type*) [AddCommGroup N] [Module R N]
+    {p q : Prop} (h : p ↔ q) : (PLift p → N) ≃ₗ[R] (PLift q → N) where
+  toFun v t := v ⟨h.mpr t.down⟩
+  invFun v t := v ⟨h.mp t.down⟩
+  left_inv _ := rfl
+  right_inv _ := rfl
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+
+ theorem moduleEq_pi_apply {X : Type*} (p : X → Prop) {x y : X} (h : x=y)
+    (v : PLift (p x) → R) (t : PLift (p y)) :
+    moduleEq (congrArg (fun z => ModuleCat.of R (PLift (p z) → R)) h) v t =
+      v ⟨h.symm ▸ t.down⟩ := by
+  subst y
+  rfl
+
+ def structurePullbackIso (f : GraphMap G H) (α : F → R) (c : V)
+    (ho : ∀ x y, f.vertex x ≤ f.vertex y ↔ x ≤ y) :
+    Iso ((structureSheaf H α (f.vertex c)).pullback f)
+      (structureSheaf G (fun e => α (f.edge e)) c) where
+  vertex x := propPiEquiv R (ho x c)
+  edge e := propPiEquiv (R ⧸ Ideal.span {α (f.edge e)}) (by
+    rw [← f.target e]; exact ho (G.target e) c)
+  lower e v := by
+    change (PLift (f.vertex (G.source e) ≤ f.vertex c) → R) at v
+    funext h
+    change Ideal.Quotient.mk _
+      (moduleEq (congrArg (fun z => ModuleCat.of R (PLift (z ≤ f.vertex c) → R))
+        (f.source e)) v _) = Ideal.Quotient.mk _ (v _)
+    apply congrArg (Ideal.Quotient.mk _)
+    exact moduleEq_pi_apply (fun z => z ≤ f.vertex c) (f.source e) v _
+  upper e v := by
+    change (PLift (f.vertex (G.target e) ≤ f.vertex c) → R) at v
+    funext h
+    change Ideal.Quotient.mk _
+      (moduleEq (congrArg (fun z => ModuleCat.of R (PLift (z ≤ f.vertex c) → R))
+        (f.target e)) v _) = Ideal.Quotient.mk _ (v _)
+    apply congrArg (Ideal.Quotient.mk _)
+    exact moduleEq_pi_apply (fun z => z ≤ f.vertex c) (f.target e) v _
+
+end
+end KLInvariance.MomentGraph.Sheaf
+
+end
+
+
+section
+
+/-! Rank-one models on the actual finite relative lower ideal in an ambient
+plane coset. The ideal need not have a greatest element. -/
+namespace KLInvariance.TitsSpace.PlaneBasis
+open Module BruhatGraph MomentGraph MomentGraph.Sheaf
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+  {σ : Type us} (basis : Basis σ ℝ (Extended M))
+  (a₀ b : W) (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+noncomputable section
+
+ def cosetInto (d : planeSubgroup M cs P) (hd : ∀ x : B.CosetVertex a₀ b, BruhatLE B.system x.val d) :
+    B.CosetVertex a₀ b → Interval B.system 1 d := fun x => ⟨x.val,x.property.1,hd x⟩
+
+ def cosetIntoEdge (d : planeSubgroup M cs P)
+    (hd : ∀ x : B.CosetVertex a₀ b, BruhatLE B.system x.val d) (e : B.CosetEdge a₀ b) :
+    Edge B.system 1 d := ⟨(B.cosetInto a₀ b d hd e.val.1,B.cosetInto a₀ b d hd e.val.2),e.property⟩
+
+ def cosetIntoGraph (d : planeSubgroup M cs P)
+    (hd : ∀ x : B.CosetVertex a₀ b, BruhatLE B.system x.val d) :
+    GraphMap (B.cosetGraph a₀ b) (graph B.system 1 d) where
+  vertex := B.cosetInto a₀ b d hd
+  edge := B.cosetIntoEdge a₀ b d hd
+  source _ := rfl
+  target _ := rfl
+
+ theorem cosetInto_injective (d : planeSubgroup M cs P)
+    (hd : ∀ x : B.CosetVertex a₀ b, BruhatLE B.system x.val d) :
+    Function.Injective (B.cosetInto a₀ b d hd) := by
+  intro x y h
+  apply Subtype.ext
+  exact congrArg (fun z : Interval B.system 1 d => z.val) h
+
+ theorem coset_label_eq (e : B.CosetEdge a₀ b) :
+    label cs 1 b (B.cosetEdgeAmbient a₀ b hmin e) = (e.val.2.val : W)*(e.val.1.val:W)⁻¹ := by
+  change ((e.val.2.val:W)*a₀)*((e.val.1.val:W)*a₀)⁻¹ = _
+  group
+
+ theorem coset_modelLabel_eq (d : planeSubgroup M cs P)
+    (hd : ∀ x : B.CosetVertex a₀ b, BruhatLE B.system x.val d) (e : B.CosetEdge a₀ b) :
+    B.modelLabel basis d (B.cosetIntoEdge a₀ b d hd e) =
+      polynomialLabel cs 1 b basis (B.cosetEdgeAmbient a₀ b hmin e) := by
+  have ht : (⟨(label B.system 1 d (B.cosetIntoEdge a₀ b d hd e)).val,
+      (B.reflection_iff _).mp (label_isReflection B.system 1 d _)⟩ : {t : W // cs.IsReflection t}) =
+      ⟨label cs 1 b (B.cosetEdgeAmbient a₀ b hmin e),label_isReflection cs 1 b _⟩ := by
+    apply Subtype.ext
+    change (label B.system 1 d (B.cosetIntoEdge a₀ b d hd e)).val =
+      label cs 1 b (B.cosetEdgeAmbient a₀ b hmin e)
+    rw [B.coset_label_eq a₀ b hmin e]
+    rfl
+  exact congrArg (fun t : {t : W // cs.IsReflection t} =>
+    linearPolynomial basis (embed M (positiveRoot M cs t).val)) ht
+
+ include hmin in
+ theorem coset_model_flabby (c : B.CosetVertex a₀ b) :
+    (structureSheaf (B.cosetGraph a₀ b) (B.cosetLabel basis a₀ b hmin) c).Flabby := by
+  classical
+  let := Fintype.ofFinite (B.CosetVertex a₀ b)
+  obtain ⟨d,hd⟩ := Dihedral.exists_common_upper B.system
+    (Finset.univ.image (fun x : B.CosetVertex a₀ b => x.val))
+  have hbnd (x : B.CosetVertex a₀ b) : BruhatLE B.system x.val d :=
+    hd _ (Finset.mem_image.mpr ⟨x,Finset.mem_univ _,rfl⟩)
+  let f := B.cosetIntoGraph a₀ b d hbnd
+  let α : Edge B.system 1 d → PlaneRing basis P := fun e =>
+    algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P) (B.modelLabel basis d e)
+  let C := structureSheaf (graph B.system 1 d) α (f.vertex c)
+  have hfl : C.Flabby := (B.localizedModels basis d).flabby (f.vertex c)
+  let := interval_finite B.system 1 d
+  have hpull : (C.pullback f).Flabby := by
+    apply C.pullback_flabby_live f hfl (B.cosetInto_injective a₀ b d hbnd)
+      {e | target B.system 1 d e ≤ f.vertex c}
+    · intro e he
+      exact structureSheaf_edge_subsingleton _ α (f.vertex c) e he
+    · intro e he x hx
+      have hlow : BruhatLE B.system (target B.system 1 d e).val c.val := he
+      let y : B.CosetVertex a₀ b := ⟨(target B.system 1 d e).val,
+        (target B.system 1 d e).property.1,B.coset_lower a₀ b hmin hlow⟩
+      have hye : f.vertex y=target B.system 1 d e := rfl
+      have hs : BruhatStep B.system x.val y.val := by
+        change BruhatStep B.system (f.vertex x).val (f.vertex y).val
+        rw [hx,hye]
+        exact e.property
+      refine ⟨⟨(x,y),hs⟩,?_,rfl⟩
+      exact Subtype.ext (Prod.ext hx hye)
+  have hh := (structurePullbackIso f α c (fun _ _ => Iff.rfl)).flabby hpull
+  have hα : (fun e => α (f.edge e)) = B.cosetLabel basis a₀ b hmin := by
+    funext e
+    exact congrArg (algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P))
+      (B.coset_modelLabel_eq basis a₀ b hmin d hbnd e)
+  rwa [hα] at hh
+
+ def cosetModels : LocalModels (G := B.cosetGraph a₀ b) (B.cosetLabel basis a₀ b hmin) :=
+  structureLocalModels _ _
+    (fun e => (linear_mem_maximal_iff basis P _).mpr (by
+      apply (Represents.planeSubgroup_mem_iff M cs
+        (positiveRoot_represents M cs ⟨label cs 1 b (B.cosetEdgeAmbient a₀ b hmin e),
+          label_isReflection cs 1 b _⟩)).mp
+      change label cs 1 b (B.cosetEdgeAmbient a₀ b hmin e) ∈ planeSubgroup M cs P
+      rw [B.coset_label_eq a₀ b hmin e]
+      exact (planeSubgroup M cs P).mul_mem e.val.2.val.property
+        ((planeSubgroup M cs P).inv_mem e.val.1.val.property)))
+    (B.coset_outgoing_below a₀ b hmin) (B.coset_model_flabby basis a₀ b hmin)
+
+end
+end KLInvariance.TitsSpace.PlaneBasis
+
+end
+
+
+section
+
+/-! Partial upward kernels of the actual rank-one model, with their embedded
+edge images. No character or freeness property of BMP sheaves is assumed. -/
+namespace KLInvariance.MomentGraph.Sheaf
+open scoped Pointwise
+universe ur uv ue
+variable {R : Type ur} [CommRing R] [DecompositionMonoid R]
+  {V : Type uv} [PartialOrder V] {E : Type ue} [Fintype E]
+  (G : OrderedGraph V E) (α : E → R) (c : V)
+noncomputable section
+
+ def supportedOutgoing (x : V) (F : Set (Outgoing (G := G) x)) : Finset (Outgoing (G := G) x) := by
+  classical
+  exact Finset.univ.filter (fun e => e ∈ F ∧ G.target e.val ≤ c)
+
+ def kernelProduct (x : V) (F : Set (Outgoing (G := G) x)) : R :=
+  ∏ e ∈ supportedOutgoing G c x F, α e.val
+
+ theorem structure_kernel_iff_dvd (x : V) (hx : x ≤ c)
+    (F : Set (Outgoing (G := G) x))
+    (hrel : (supportedOutgoing G c x F : Set (Outgoing (G := G) x)).Pairwise
+      (Function.onFun IsRelPrime (fun e => α e.val)))
+    (v : (structureSheaf G α c).vertex x) :
+    v ∈ (structureSheaf G α c).upwardKernel x F ↔ kernelProduct G α c x F ∣ v ⟨hx⟩ := by
+  classical
+  constructor
+  · intro hv
+    apply Finset.prod_dvd_of_isRelPrime hrel
+    intro e he
+    have hh := (Finset.mem_filter.mp he).2
+    have hvv := congrFun (hv e hh.1) ⟨hh.2⟩
+    rcases e with ⟨e,he⟩
+    subst x
+    exact Ideal.mem_span_singleton.mp (Ideal.Quotient.eq_zero_iff_mem.mp hvv)
+  · intro hv e he
+    funext h
+    have hh : e ∈ supportedOutgoing G c x F :=
+      Finset.mem_filter.mpr ⟨Finset.mem_univ _,he,h.down⟩
+    have hd : α e.val ∣ v ⟨hx⟩ := (Finset.dvd_prod_of_mem (fun a => α a.val) hh).trans hv
+    rcases e with ⟨e,heq⟩
+    subst x
+    exact Ideal.Quotient.eq_zero_iff_mem.mpr (Ideal.mem_span_singleton.mpr hd)
+
+ theorem structure_kernel_product (x : V) (F : Set (Outgoing (G := G) x))
+    (hrel : (supportedOutgoing G c x F : Set (Outgoing (G := G) x)).Pairwise
+      (Function.onFun IsRelPrime (fun e => α e.val))) :
+    (structureSheaf G α c).upwardKernel x F =
+      kernelProduct G α c x F • (⊤ : Submodule R ((structureSheaf G α c).vertex x)) := by
+  classical
+  ext v
+  rw [Submodule.mem_smul_pointwise_iff_exists]
+  by_cases hx : x ≤ c
+  · rw [structure_kernel_iff_dvd G α c x hx F hrel]
+    constructor
+    · rintro ⟨r,hr⟩
+      refine ⟨fun _ => r,Submodule.mem_top,?_⟩
+      funext h
+      exact hr.symm
+    · rintro ⟨w,_,hw⟩
+      exact ⟨w ⟨hx⟩,(congrFun hw ⟨hx⟩).symm⟩
+  · let _ := structureSheaf_support G α c x hx
+    constructor
+    · intro _
+      exact ⟨0,Submodule.mem_top,Subsingleton.elim _ _⟩
+    · intro _
+      rw [Subsingleton.elim v 0]
+      exact Submodule.zero_mem _
+
+end
+end KLInvariance.MomentGraph.Sheaf
+
+namespace KLInvariance.MomentGraph.Sheaf
+open scoped Pointwise
+universe ur uv ue
+variable {R : Type ur} [CommRing R] {V : Type uv} [PartialOrder V]
+  {E : Type ue} {G : OrderedGraph V E}
+
+ theorem lower_surjective_of_generated (B : Sheaf (R := R) G) (hg : B.Generated)
+    (hu : ∀ e, Function.Surjective (B.upper e)) (e : E) : Function.Surjective (B.lower e) := by
+  intro v
+  obtain ⟨w,hw⟩ := hu e v
+  obtain ⟨s,hs,ht⟩ := hg (G.target e) w
+  refine ⟨s (G.source e),?_⟩
+  rw [hs e (Set.mem_univ _) (Set.mem_univ _),ht,hw]
+
+variable [DecompositionMonoid R] [Fintype E] (α : E → R) (c : V)
+
+ theorem structure_lower_image (e : E) (F : Set (Outgoing (G := G) (G.source e)))
+    (hrel : (supportedOutgoing G c (G.source e) F : Set (Outgoing (G := G) (G.source e))).Pairwise
+      (Function.onFun IsRelPrime (fun a => α a.val))) :
+    ((structureSheaf G α c).upwardKernel (G.source e) F).map ((structureSheaf G α c).lower e) =
+      kernelProduct G α c (G.source e) F • (⊤ : Submodule R ((structureSheaf G α c).edge e)) := by
+  rw [structure_kernel_product G α c _ F hrel,Submodule.map_pointwise_smul,Submodule.map_top]
+  congr 1
+  apply LinearMap.range_eq_top.mpr
+  exact lower_surjective_of_generated _ (structureSheaf_generated G α c)
+    (fun a => (structureSheaf_upperQuotient G α c a).1) e
+
+ theorem structure_upper_image (e : E) (F : Set (Outgoing (G := G) (G.target e)))
+    (hrel : (supportedOutgoing G c (G.target e) F : Set (Outgoing (G := G) (G.target e))).Pairwise
+      (Function.onFun IsRelPrime (fun a => α a.val))) :
+    ((structureSheaf G α c).upwardKernel (G.target e) F).map ((structureSheaf G α c).upper e) =
+      kernelProduct G α c (G.target e) F • (⊤ : Submodule R ((structureSheaf G α c).edge e)) := by
+  rw [structure_kernel_product G α c _ F hrel,Submodule.map_pointwise_smul,Submodule.map_top]
+  congr 1
+  exact LinearMap.range_eq_top.mpr (structureSheaf_upperQuotient G α c e).1
+
+end KLInvariance.MomentGraph.Sheaf
+
+end
+
+
+section
+
+/-! Distinct prime root labels remain relatively prime after genuine prime
+localization. The prime may contain both labels; coprimality here is gcd
+coprimality, not comaximality. -/
+namespace KLInvariance.LocalizationRelativePrime
+variable {A : Type*} [CommRing A] [IsDomain A] (p : Ideal A) [p.IsPrime]
+
+omit [IsDomain A] in
+ theorem not_dvd_map {a b : A} (ha : Prime a) (hap : a ∈ p) (hab : ¬a ∣ b) :
+    ¬ algebraMap A (Localization.AtPrime p) a ∣ algebraMap A (Localization.AtPrime p) b := by
+  intro h
+  obtain ⟨s,hs,hdiv⟩ := (IsLocalization.toLocalizationMap p.primeCompl (Localization.AtPrime p)).map_dvd_map.mp h
+  rcases ha.dvd_mul.mp hdiv with has | hab'
+  · exact hs (p.mem_of_dvd has hap)
+  · exact hab hab'
+
+ theorem map_prime {a : A} (ha : Prime a) (hap : a ∈ p) :
+    Prime (algebraMap A (Localization.AtPrime p) a) := by
+  apply (IsLocalization.toLocalizationMap p.primeCompl (Localization.AtPrime p)).map_prime ha
+  · change algebraMap A (Localization.AtPrime p) a ≠ 0
+    simpa only [map_zero] using (FaithfulSMul.algebraMap_injective A (Localization.AtPrime p)).ne ha.ne_zero
+  · change ¬IsUnit (algebraMap A (Localization.AtPrime p) a)
+    rw [IsLocalization.AtPrime.isUnit_to_map_iff _ p]
+    exact not_not_intro hap
+
+ theorem map_relPrime {a b : A} (ha : Prime a) (hab : ¬a ∣ b) :
+    IsRelPrime (algebraMap A (Localization.AtPrime p) a)
+      (algebraMap A (Localization.AtPrime p) b) := by
+  by_cases hap : a ∈ p
+  · exact ((map_prime p ha hap).irreducible.isRelPrime_iff_not_dvd).mpr (not_dvd_map p ha hap hab)
+  · exact ((IsLocalization.AtPrime.isUnit_to_map_iff (Localization.AtPrime p) p a).mpr hap).isRelPrime_left
+
+end KLInvariance.LocalizationRelativePrime
+
+end
+
+
+section
+
+/-! The local product-kernel lemma on the actual maximal-plane coset.
+All regularity hypotheses come from distinct genuine incident reflections. -/
+namespace KLInvariance.TitsSpace.PlaneBasis
+open Module BruhatGraph MomentGraph MomentGraph.Sheaf
+open scoped Pointwise
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+  {σ : Type us} (basis : Basis σ ℝ (Extended M))
+  (a₀ b : W) (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+noncomputable section
+
+ theorem cosetEdgeAmbient_injective : Function.Injective (B.cosetEdgeAmbient a₀ b hmin) := by
+  intro e f h
+  apply Subtype.ext
+  apply Prod.ext
+  · exact B.cosetAmbient_injective a₀ b (congrArg (fun z : Edge cs 1 b => z.val.1) h)
+  · exact B.cosetAmbient_injective a₀ b (congrArg (fun z : Edge cs 1 b => z.val.2) h)
+
+ theorem cosetLabel_relPrime {e f : B.CosetEdge a₀ b} (hef : e ≠ f)
+    (hs : (B.cosetGraph a₀ b).source e = (B.cosetGraph a₀ b).source f) :
+    IsRelPrime (B.cosetLabel basis a₀ b hmin e) (B.cosetLabel basis a₀ b hmin f) := by
+  apply LocalizationRelativePrime.map_relPrime (planePrime basis P)
+    (polynomialLabel_prime cs 1 b basis _)
+  intro hd
+  apply hef
+  apply B.cosetEdgeAmbient_injective a₀ b hmin
+  apply eq_of_label_source_eq cs 1 b (label_eq_of_polynomialLabel_dvd cs 1 b basis hd)
+  exact congrArg (B.cosetAmbient a₀ b) hs
+
+ theorem coset_supported_pairwise [Fintype (B.CosetEdge a₀ b)]
+    (c x : B.CosetVertex a₀ b)
+    (F : Set (Outgoing (G := B.cosetGraph a₀ b) x)) :
+    (supportedOutgoing (B.cosetGraph a₀ b) c x F :
+      Set (Outgoing (G := B.cosetGraph a₀ b) x)).Pairwise
+      (Function.onFun IsRelPrime (fun e => B.cosetLabel basis a₀ b hmin e.val)) := by
+  intro e _ f _ hef
+  exact B.cosetLabel_relPrime basis a₀ b hmin
+    (fun h => hef (Subtype.ext h)) (e.property.trans f.property.symm)
+
+ theorem coset_kernel_product [Fintype (B.CosetEdge a₀ b)]
+    (c x : B.CosetVertex a₀ b) (F : Set (Outgoing (G := B.cosetGraph a₀ b) x)) :
+    (structureSheaf (B.cosetGraph a₀ b) (B.cosetLabel basis a₀ b hmin) c).upwardKernel x F =
+      kernelProduct (B.cosetGraph a₀ b) (B.cosetLabel basis a₀ b hmin) c x F •
+        (⊤ : Submodule (PlaneRing basis P)
+          ((structureSheaf (B.cosetGraph a₀ b) (B.cosetLabel basis a₀ b hmin) c).vertex x)) :=
+  structure_kernel_product _ _ _ _ _ (B.coset_supported_pairwise basis a₀ b hmin c x F)
+
+end
+end KLInvariance.TitsSpace.PlaneBasis
+
+end
+
+
+section
+
+namespace KLInvariance.TitsSpace.PlaneBasis
+open Module BruhatGraph MomentGraph MomentGraph.Sheaf
+universe u v u' v'
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
+  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
+  {a₀ u b : W} {u' b' : W'}
+  (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+noncomputable section
+
+ def cosetUpperVertex (x : B.CosetVertex a₀ b) (hx : BruhatLE cs u ((x.val:W)*a₀)) :
+    Interval cs u b := ⟨(x.val:W)*a₀,hx,x.property.2⟩
+
+ theorem outgoing_step (x : B.CosetVertex a₀ b)
+    (f : Outgoing (G := B.cosetGraph a₀ b) x) :
+    BruhatStep B.system x.val f.val.val.2.val := by
+  rcases f with ⟨f,hf⟩
+  change f.val.1=x at hf
+  subst x
+  exact f.property
+
+ include hmin in
+ theorem outgoing_upper (x : B.CosetVertex a₀ b) (hx : BruhatLE cs u ((x.val:W)*a₀))
+    (f : Outgoing (G := B.cosetGraph a₀ b) x) :
+    BruhatLE cs u ((f.val.val.2.val:W)*a₀) :=
+  bruhat_trans cs hx (.single ((B.coset_step_iff hmin _ _).mpr (B.outgoing_step x f)))
+
+ def scheduledOutgoing (φ : Interval cs u b ≃o Interval cs' u' b')
+    (α : PositiveRoot M' cs') (x : B.CosetVertex a₀ b)
+    (hx : BruhatLE cs u ((x.val:W)*a₀)) : Set (Outgoing (G := B.cosetGraph a₀ b) x) :=
+  {f | ambientAfterIndicator (φ (B.cosetUpperVertex x hx)).val α
+    (φ (B.cosetUpperVertex f.val.val.2 (B.outgoing_upper hmin x hx f))).val = 1}
+
+ theorem outgoing_target_injective (x : B.CosetVertex a₀ b) :
+    Function.Injective (fun f : Outgoing (G := B.cosetGraph a₀ b) x => f.val.val.2) := by
+  intro f g h
+  apply Subtype.ext
+  apply Subtype.ext
+  exact Prod.ext (f.property.trans g.property.symm) h
+
+ theorem supported_card_count [Fintype (B.CosetEdge a₀ b)]
+    (φ : Interval cs u b ≃o Interval cs' u' b') (α : PositiveRoot M' cs')
+    (a x c : B.CosetVertex a₀ b) (hax : a ≤ x)
+    (ha : BruhatLE cs u ((a.val:W)*a₀)) :
+    let hx := bruhat_trans cs ha (B.coset_bruhat hmin hax)
+    let inc := B.cosetIntervalMap hmin ha c.property.2
+    (supportedOutgoing (B.cosetGraph a₀ b) c x (B.scheduledOutgoing hmin φ α x hx)).card =
+      ∑ᶠ z : Interval B.system a.val c.val,
+        ambientAfterIndicator (φ (B.cosetUpperVertex x hx)).val α (φ (inc z)).val := by
+  classical
+  let := interval_finite B.system a.val c.val
+  let := Fintype.ofFinite (Interval B.system a.val c.val)
+  let hx := bruhat_trans cs ha (B.coset_bruhat hmin hax)
+  let inc := B.cosetIntervalMap hmin ha c.property.2
+  let F := B.scheduledOutgoing hmin φ α x hx
+  let s := supportedOutgoing (B.cosetGraph a₀ b) c x F
+  change s.card = ∑ᶠ z : Interval B.system a.val c.val,
+    ambientAfterIndicator (φ (B.cosetUpperVertex x hx)).val α (φ (inc z)).val
+  rw [finsum_eq_sum_of_fintype]
+  rw [show s.card=(∑ _ ∈ s,(1:ℕ)) by simp]
+  let idx (f : Outgoing (G := B.cosetGraph a₀ b) x) (hf : f ∈ s) :
+      Interval B.system a.val c.val :=
+    ⟨f.val.val.2.val,bruhat_trans B.system hax (.single (B.outgoing_step x f)),
+      (Finset.mem_filter.mp hf).2.2⟩
+  apply Finset.sum_bij_ne_zero (fun f hf _ => idx f hf)
+  · intro _ _ _
+    exact Finset.mem_univ _
+  · intro f hf _ g hg _ h
+    have hz : f.val.val.2.val=g.val.val.2.val := congrArg (fun z : Interval B.system a.val c.val => z.val) h
+    exact B.outgoing_target_injective x (Subtype.ext hz)
+  · intro z _ hz
+    have hsz : BruhatStep cs' (φ (B.cosetUpperVertex x hx)).val (φ (inc z)).val := by
+      by_contra hn
+      exact hz (by simp only [ambientAfterIndicator,dite_eq_right hn])
+    have hcz := (intervalIso_bruhatStep_iff_general cs cs' φ _ _).mp hsz
+    have hs : BruhatStep B.system x.val z.val := (B.coset_step_iff hmin _ _).mp hcz
+    let y : B.CosetVertex a₀ b := ⟨z.val,one_bruhat B.system z.val,
+      bruhat_trans cs (B.coset_bruhat hmin z.property.2) c.property.2⟩
+    let f : Outgoing (G := B.cosetGraph a₀ b) x := ⟨⟨(x,y),hs⟩,rfl⟩
+    have hfF : f ∈ F := by
+      change ambientAfterIndicator (φ (B.cosetUpperVertex x hx)).val α (φ (inc z)).val=1
+      unfold ambientAfterIndicator at hz ⊢
+      split_ifs at hz ⊢ <;> omega
+    have hfs : f ∈ s := Finset.mem_filter.mpr ⟨Finset.mem_univ _,hfF,z.property.2⟩
+    exact ⟨f,hfs,by omega,rfl⟩
+  · intro f hf _
+    exact ((Finset.mem_filter.mp hf).2.1).symm
+
+end
+end KLInvariance.TitsSpace.PlaneBasis
+
+end
+
+
+section
+
+namespace KLInvariance.TitsSpace.PlaneBasis
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+noncomputable section
+
+/-- The combinatorial factor already constructed from ambient inversion
+pairs is exactly the intrinsic relative dihedral edge degree. -/
+ theorem planeWeight_coset_edge {a₀ : W}
+    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+    {x y : planeSubgroup M cs P} (hxy : BruhatStep B.system x y) :
+    (planeWeight ((x:W)*a₀) (B.edgeRoot hxy) P).toNat =
+      (B.system.length y-B.system.length x-1)/2 := by
+  classical
+  let α := B.edgeRoot hxy
+  obtain ⟨β,hαβ,hβ⟩ : ∃ β : PositiveRoot M cs, α ≠ β ∧ β.val ∈ P := by
+    by_cases h : α=B.left
+    · exact ⟨B.right,h ▸ B.distinct,B.right_mem⟩
+    · exact ⟨B.left,h,B.left_mem⟩
+  have hplane : rootPlane α β=P :=
+    (rootPlane_eq_of_mem B.left B.right α β B.distinct
+      (B.plane_eq.ge (B.edgeRoot_plane hxy)) (B.plane_eq.ge hβ) hαβ).trans B.plane_eq
+  have ha : ¬rootInverted ((x:W)*a₀) α := by
+    rw [show α=graphRoot ((B.coset_step_iff hmin x y).mpr hxy) from B.edgeRoot_coset hmin hxy]
+    exact graphRoot_not_inverted_source _
+  have href : rootRef α=(y:W)*(x:W)⁻¹ :=
+    (rootReflection_represents M cs α).reflection_unique
+      (graphRoot_represents ((B.bruhatStep_iff x y).mpr hxy))
+  have hends : rootRef α*((x:W)*a₀)=(y:W)*a₀ := by rw [href]; group
+  have h := planeWeight_relative_degree ((x:W)*a₀) α β hαβ ha
+  rw [hplane,hends,B.coset_length_eq hmin x,B.coset_length_eq hmin y] at h
+  have hn := planeWeight_nonneg ((x:W)*a₀) α P
+  have hlen := hxy.1
+  have hc : ((planeWeight ((x:W)*a₀) α P).toNat:ℤ)=planeWeight ((x:W)*a₀) α P :=
+    Int.toNat_of_nonneg hn
+  change (planeWeight ((x:W)*a₀) α P).toNat = _
+  omega
+
+end
+end KLInvariance.TitsSpace.PlaneBasis
+
+end
+
+
+section
+
+namespace KLInvariance.TitsSpace.PlaneBasis
+open Module BruhatGraph MomentGraph MomentGraph.Sheaf
+universe u v u' v'
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
+  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
+  {a₀ u b : W} {u' b' : W'}
+  (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+  (φ : Interval cs u b ≃o Interval cs' u' b')
+noncomputable section
+
+ include hmin in
+ theorem cosetEdge_target_upper (e : B.CosetEdge a₀ b)
+    (hx : BruhatLE cs u ((e.val.1.val:W)*a₀)) : BruhatLE cs u ((e.val.2.val:W)*a₀) :=
+  bruhat_trans cs hx (.single ((B.coset_step_iff hmin _ _).mpr e.property))
+
+ def mappedCosetRoot (e : B.CosetEdge a₀ b) (hx : BruhatLE cs u ((e.val.1.val:W)*a₀)) :
+    PositiveRoot M' cs' :=
+  graphRoot ((intervalIso_bruhatStep_iff_general cs cs' φ
+    (B.cosetUpperVertex e.val.1 hx)
+    (B.cosetUpperVertex e.val.2 (B.cosetEdge_target_upper hmin e hx))).mpr
+    ((B.coset_step_iff hmin _ _).mpr e.property))
+
+ def scheduledLower (e : B.CosetEdge a₀ b) (hx : BruhatLE cs u ((e.val.1.val:W)*a₀)) :
+    Set (Outgoing (G := B.cosetGraph a₀ b) e.val.1) :=
+  B.scheduledOutgoing hmin φ (B.mappedCosetRoot hmin φ e hx) e.val.1 hx
+
+ def scheduledUpper (e : B.CosetEdge a₀ b) (hx : BruhatLE cs u ((e.val.1.val:W)*a₀)) :
+    Set (Outgoing (G := B.cosetGraph a₀ b) e.val.2) :=
+  B.scheduledOutgoing hmin φ (B.mappedCosetRoot hmin φ e hx) e.val.2
+    (B.cosetEdge_target_upper hmin e hx)
+
+ theorem scheduledLower_self (e : B.CosetEdge a₀ b)
+    (hx : BruhatLE cs u ((e.val.1.val:W)*a₀)) :
+    (⟨e,rfl⟩ : Outgoing (G := B.cosetGraph a₀ b) e.val.1) ∉ B.scheduledLower hmin φ e hx := by
+  change ambientAfterIndicator _ (graphRoot _) _ ≠ 1
+  rw [ambientAfterIndicator_edge_self]
+  omega
+
+ theorem supported_count_difference [Fintype (B.CosetEdge a₀ b)]
+    (e : B.CosetEdge a₀ b) (hx : BruhatLE cs u ((e.val.1.val:W)*a₀))
+    (c : B.CosetVertex a₀ b) (hyc : e.val.2 ≤ c) :
+    (supportedOutgoing (B.cosetGraph a₀ b) c e.val.1 (B.scheduledLower hmin φ e hx)).card =
+      (supportedOutgoing (B.cosetGraph a₀ b) c e.val.2 (B.scheduledUpper hmin φ e hx)).card+
+        (planeWeight ((e.val.1.val:W)*a₀) (B.edgeRoot e.property) P).toNat := by
+  let x := e.val.1
+  let y := e.val.2
+  let α := B.mappedCosetRoot hmin φ e hx
+  have hxy : BruhatLE B.system x.val y.val := .single e.property
+  have hxc : BruhatLE B.system x.val c.val := bruhat_trans B.system hxy hyc
+  let yy : Interval B.system x.val c.val := ⟨y.val,hxy,hyc⟩
+  have h := B.transported_count_difference_all hmin hxc hx c.property.2 φ yy e.property
+  rw [B.planeWeight_coset_edge hmin e.property]
+  change (supportedOutgoing (B.cosetGraph a₀ b) c x (B.scheduledOutgoing hmin φ α x hx)).card =
+    (supportedOutgoing (B.cosetGraph a₀ b) c y
+      (B.scheduledOutgoing hmin φ α y (B.cosetEdge_target_upper hmin e hx))).card+_
+  rw [B.supported_card_count hmin φ α x x c (bruhat_refl B.system x.val) hx,
+    B.supported_card_count hmin φ α x y c hxy hx]
+  exact h
+
+end
+end KLInvariance.TitsSpace.PlaneBasis
+
+end
+
+
+section
+
+namespace KLInvariance.BruhatGraph
+open TitsSpace MomentGraph MomentGraph.Sheaf
+universe u v u' v'
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
+  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
+  {u b : W} {u' b' : W'}
+  (φ : Interval cs u b ≃o Interval cs' u' b')
+noncomputable section
+
+ def ambientLater (α : PositiveRoot M' cs') : Set (Edge cs 1 b) :=
+  {e | ∃ hx : BruhatLE cs u (source cs 1 b e).val,
+    ambientAfterIndicator (φ ⟨(source cs 1 b e).val,hx,(source cs 1 b e).property.2⟩).val α
+      (φ ⟨(target cs 1 b e).val,bruhat_trans cs hx (.single e.property),
+        (target cs 1 b e).property.2⟩).val = 1}
+
+end
+end KLInvariance.BruhatGraph
+
+namespace KLInvariance.TitsSpace.PlaneBasis
+open Module BruhatGraph MomentGraph MomentGraph.Sheaf
+universe u v u' v'
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
+  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
+  {a₀ u b : W} {u' b' : W'}
+  (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+  (φ : Interval cs u b ≃o Interval cs' u' b')
+noncomputable section
+
+ theorem scheduledOutgoing_eq (α : PositiveRoot M' cs')
+    (x : B.CosetVertex a₀ b) (hx : BruhatLE cs u ((x.val:W)*a₀)) :
+    B.scheduledOutgoing hmin φ α x hx =
+      {f | B.cosetEdgeAmbient a₀ b hmin f.val ∈ ambientLater φ α} := by
+  ext f
+  rcases f with ⟨⟨⟨s,t⟩,hst⟩,hs⟩
+  change s=x at hs
+  subst s
+  change (_=1) ↔ ∃ hh : BruhatLE cs u ((x.val:W)*a₀), _=1
+  constructor
+  · intro h
+    exact ⟨hx,h⟩
+  · rintro ⟨_,h⟩
+    exact h
+
+end
+end KLInvariance.TitsSpace.PlaneBasis
+
+end
+
+
+section
+
+/-! Reduction of a root plane along one root is genuinely one-dimensional.
+The scalar comparing two nonparallel reduced roots is a nonzero element of
+the original base field, hence a unit after every coefficient localization. -/
+namespace KLInvariance.PlaneReduction
+universe uk uv ua
+variable {k : Type uk} [Field k] {V : Type uv} [AddCommGroup V] [Module k V]
+  {A : Type ua} [CommRing A] [Algebra k A]
+
+ theorem scalar_reduction (l : V →ₗ[k] A) {a b c : V} (ha : l a = 0)
+    (hc : c ∈ Submodule.span k ({a,b} : Set V))
+    (hca : c ∉ Submodule.span k ({a} : Set V)) :
+    ∃ r : k, r ≠ 0 ∧ l c = r • l b := by
+  obtain ⟨s,r,hr⟩ := Submodule.mem_span_pair.mp hc
+  have hr0 : r ≠ 0 := by
+    intro hz
+    apply hca
+    rw [← hr,hz,zero_smul,add_zero]
+    exact Submodule.smul_mem _ s (Submodule.subset_span (Set.mem_singleton a))
+  refine ⟨r,hr0,?_⟩
+  rw [← hr,map_add,map_smul,map_smul,ha,smul_zero,zero_add]
+
+ theorem associated_reduction (l : V →ₗ[k] A) {a b c : V} (ha : l a = 0)
+    (hc : c ∈ Submodule.span k ({a,b} : Set V))
+    (hca : c ∉ Submodule.span k ({a} : Set V)) : Associated (l c) (l b) := by
+  obtain ⟨r,hr,he⟩ := scalar_reduction l ha hc hca
+  rw [he,Algebra.smul_def]
+  exact associated_unit_mul_left _ _ ((hr.isUnit).map (algebraMap k A))
+
+end KLInvariance.PlaneReduction
+
+end
+
+
+section
+
+/-! Actual root labels in the localized edge quotient: every nonparallel
+root in a given plane has a reduced label associated to any chosen one. -/
+namespace KLInvariance.TitsSpace
+open Module
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {σ : Type us} (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ def localizedLinear (P : Submodule ℝ (I → ℝ)) : (I → ℝ) →ₗ[ℝ] PlaneRing basis P :=
+  (IsScalarTower.toAlgHom ℝ (MvPolynomial σ ℝ) (PlaneRing basis P)).toLinearMap.comp
+    ((linearPolynomial basis).comp (embedLinear (M := M)))
+
+ def localizedEdgeIdeal (P : Submodule ℝ (I → ℝ)) (a : I → ℝ) : Ideal (PlaneRing basis P) :=
+  Ideal.span {localizedLinear basis P a}
+
+ abbrev LocalizedEdgeRing (P : Submodule ℝ (I → ℝ)) (a : I → ℝ) :=
+  PlaneRing basis P ⧸ localizedEdgeIdeal basis P a
+
+ def localizedReduction (P : Submodule ℝ (I → ℝ)) (a : I → ℝ) :
+    (I → ℝ) →ₗ[ℝ] LocalizedEdgeRing basis P a :=
+  (Ideal.Quotient.mkₐ ℝ (localizedEdgeIdeal basis P a)).toLinearMap.comp
+    (localizedLinear basis P)
+
+ theorem localizedReduction_self (P : Submodule ℝ (I → ℝ)) (a : I → ℝ) :
+    localizedReduction basis P a a = 0 :=
+  Ideal.Quotient.eq_zero_iff_mem.mpr (Ideal.subset_span (Set.mem_singleton _))
+
+ theorem positiveRoot_not_line {a c : PositiveRoot M cs} (hca : c ≠ a) :
+    c.val ∉ Submodule.span ℝ ({a.val} : Set (I → ℝ)) := by
+  intro h
+  obtain ⟨r,hr⟩ := Submodule.mem_span_singleton.mp h
+  exact (linearIndependent_fin2.mp (positiveRoots_independent c a hca)).2 r hr
+
+ theorem localizedReduction_associated (P : Submodule ℝ (I → ℝ))
+    (a b c : PositiveRoot M cs) (hc : c.val ∈ rootPlane a b) (hca : c ≠ a) :
+    Associated (localizedReduction basis P a.val c.val)
+      (localizedReduction basis P a.val b.val) :=
+by
+  exact @PlaneReduction.associated_reduction ℝ _ (I → ℝ) _ _
+    (LocalizedEdgeRing basis P a.val) _ _ (localizedReduction basis P a.val)
+    a.val b.val c.val (localizedReduction_self basis P a.val) hc
+    (positiveRoot_not_line hca)
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ def polynomialRoot (a : PositiveRoot M cs) : Coefficient (σ := σ) :=
+  SymmetricAlgebra.equivMvPolynomial basis
+    (SymmetricAlgebra.ι ℝ (Extended M) (embed M a.val))
+
+ theorem polynomialRoot_homogeneous (a : PositiveRoot M cs) :
+    polynomialRoot cs basis a ∈ polynomialGrading ℝ σ 1 := by
+  classical
+  unfold polynomialRoot
+  rw [← basis.sum_repr (embed M a.val)]
+  simp only [map_sum,map_smul,SymmetricAlgebra.equivMvPolynomial_ι_apply]
+  apply Submodule.sum_mem
+  intro i _
+  exact (polynomialGrading ℝ σ 1).smul_mem _
+    (MvPolynomial.isWeightedHomogeneous_X ℝ (fun _ : σ => (1 : ℤ)) i)
+
+omit [Fintype σ] in
+ theorem polynomialRoot_prime (a : PositiveRoot M cs) : Prime (polynomialRoot cs basis a) := by
+  apply (MulEquiv.prime_iff (SymmetricAlgebra.equivMvPolynomial basis)).mpr
+  apply EdgeAlgebra.symmetric_ι_prime
+  intro h
+  exact a.property.1.ne_zero (congrArg Prod.fst h)
+
+omit [Fintype σ] in
+ theorem polynomialRoot_not_dvd {a r : PositiveRoot M cs} (h : r ≠ a) :
+    ¬ polynomialRoot cs basis a ∣ polynomialRoot cs basis r := by
+  rintro ⟨z,hz⟩
+  apply embed_not_rootLine a r h
+  apply (EdgeAlgebra.symmetric_ι_dvd_iff _ _).mp
+  refine ⟨(SymmetricAlgebra.equivMvPolynomial basis).symm z,?_⟩
+  apply (SymmetricAlgebra.equivMvPolynomial basis).injective
+  simpa only [map_mul,AlgEquiv.apply_symm_apply,polynomialRoot] using hz
+
+
+ def edgeFactorLift (x : W) (a : PositiveRoot M cs) : Coefficient (σ := σ) :=
+  ∏ P : (planeWeight x a).support,
+    polynomialRoot cs basis (planeRoot x a P) ^ (planeWeight x a P.val).toNat
+
+ theorem edgeFactorLift_homogeneous (x : W) (a : PositiveRoot M cs) :
+    edgeFactorLift cs basis x a ∈ polynomialGrading ℝ σ (edgeFactorDegree x a : ℤ) := by
+  classical
+  have h := SetLike.prod_pow_mem_graded (polynomialGrading ℝ σ)
+    (fun _ : (planeWeight x a).support => (1 : ℤ))
+    (fun P => polynomialRoot cs basis (planeRoot x a P))
+    (fun P => (planeWeight x a P.val).toNat)
+    (F := Finset.univ) (fun P _ => polynomialRoot_homogeneous cs basis (planeRoot x a P))
+  simpa only [edgeFactorLift,edgeFactorDegree,Nat.cast_sum,nsmul_eq_mul,mul_one] using h
+
+omit [Fintype σ] in
+ theorem edgeFactorLift_not_dvd (x : W) (a : PositiveRoot M cs) :
+    ¬ polynomialRoot cs basis a ∣ edgeFactorLift cs basis x a := by
+  classical
+  apply (polynomialRoot_prime cs basis a).not_dvd_finsetProd
+  intro P _ h
+  exact polynomialRoot_not_dvd cs basis (planeRoot_ne x a P)
+    ((polynomialRoot_prime cs basis a).dvd_of_dvd_pow h)
+
+omit [Fintype σ] in
+ theorem edgeFactorLift_reduction (x : W) (a : PositiveRoot M cs) :
+    Ideal.Quotient.mk (EdgeAlgebra.rootIdeal (k := ℝ) (embed M a.val))
+      ((SymmetricAlgebra.equivMvPolynomial basis).symm (edgeFactorLift cs basis x a)) =
+      edgeFactor x a := by
+  simp only [edgeFactorLift,map_prod,map_pow,polynomialRoot,AlgEquiv.symm_apply_apply,
+    edgeFactor,reducedPlaneRoot]
+
+variable (u b : W) (hub : BruhatLE cs u b)
+  (B : BoundarySheaf (polynomialGrading_negative ℝ σ)
+    (graph cs u b) (polynomialLabel cs u b basis) ⟨b,hub,bruhat_refl cs b⟩)
+
+theorem rootRef_root (e : Edge cs u b) :
+    rootRef (root cs u b e) = label cs u b e :=
+  congrArg Subtype.val ((positiveRootEquiv M cs).left_inv
+    ⟨label cs u b e,label_isReflection cs u b e⟩)
+
+ theorem root_not_inverted (e : Edge cs u b) :
+    ¬ rootInverted (source cs u b e).val (root cs u b e) := by
+  intro h
+  have hi := ((rootReflection_represents M cs (root cs u b e)).inversion_iff
+    (root cs u b e).property.2 (source cs u b e).val).mp h
+  have hl := hi.2
+  change cs.length (rootRef (root cs u b e) * (source cs u b e).val) <
+    cs.length (source cs u b e).val at hl
+  rw [rootRef_root, label_mul_source] at hl
+  exact (not_lt_of_ge e.property.1.le) hl
+
+ theorem edgeFactorDegree_edge (e : Edge cs u b) :
+    2 * (edgeFactorDegree (source cs u b e).val (root cs u b e) : ℤ) =
+      (cs.length (target cs u b e).val : ℤ) - cs.length (source cs u b e).val - 1 := by
+  rw [edgeFactorDegree_cast]
+  have h := sum_planeWeight (source cs u b e).val (root cs u b e)
+    (root_not_inverted cs u b e)
+  rwa [rootRef_root, label_mul_source] at h
+
+/-- Multiplication by the genuine plane product on the actual edge module. -/
+ def edgeFactorMap (e : Edge cs u b) : B.sheaf.edge e →ₗ[Coefficient (σ := σ)] B.sheaf.edge e :=
+  LinearMap.lsmul (Coefficient (σ := σ)) (B.sheaf.edge e)
+    (edgeFactorLift cs basis (source cs u b e).val (root cs u b e))
+
+ theorem edgeFactorMap_graded (e : Edge cs u b) (n : ℤ) (v : B.sheaf.edge e)
+    (hv : v ∈ (B.sheaf.edge e).piece n) :
+    edgeFactorMap cs basis u b hub B e v ∈ (B.sheaf.edge e).piece
+      (n + edgeFactorDegree (source cs u b e).val (root cs u b e)) := by
+  simpa only [edgeFactorMap,LinearMap.lsmul_apply,vadd_eq_add,add_comm] using
+    SetLike.GradedSMul.smul_mem (edgeFactorLift_homogeneous cs basis
+      (source cs u b e).val (root cs u b e)) hv
+
+omit [Fintype σ] in
+ theorem edgeFactorLift_regular (e : Edge cs u b) :
+    Function.Injective (fun v : B.sheaf.edge e =>
+      edgeFactorLift cs basis (source cs u b e).val (root cs u b e) • v) := by
+  let := B.free ((graph cs u b).target e)
+  exact FiniteFreeScalar.scalar_quotient_regular
+    (Module.Free.chooseBasis (Coefficient (σ := σ)) (B.sheaf.vertex ((graph cs u b).target e)))
+    (B.sheaf.upper e).map (B.quotient e).1
+    (polynomialLabel cs u b basis e)
+    (edgeFactorLift cs basis (source cs u b e).val (root cs u b e))
+    (polynomialLabel_prime cs u b basis e)
+    (edgeFactorLift_not_dvd cs basis (source cs u b e).val (root cs u b e)) (B.quotient e).2
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+namespace KLInvariance.LocalProducts
+open scoped Pointwise
+universe ur us um
+variable {R : Type ur} [CommRing R] {S : Type us} [CommRing S] [Algebra R S]
+  {M : Type um} [AddCommGroup M] [Module R M] [Module S M] [IsScalarTower R S M]
+
+ theorem smul_top_le_of_map_dvd {p q : R} (h : algebraMap R S q ∣ algebraMap R S p) :
+    p • (⊤ : Submodule R M) ≤ q • (⊤ : Submodule R M) := by
+  intro z hz
+  obtain ⟨w,_,rfl⟩ := (Submodule.mem_smul_pointwise_iff_exists _ _ _).mp hz
+  obtain ⟨t,ht⟩ := h
+  apply (Submodule.mem_smul_pointwise_iff_exists _ _ _).mpr
+  refine ⟨t • w,Submodule.mem_top,?_⟩
+  rw [← IsScalarTower.algebraMap_smul S q,← IsScalarTower.algebraMap_smul S p,
+    smul_smul,← ht]
+
+ theorem smul_top_eq_of_map_associated {p q : R}
+    (h : Associated (algebraMap R S p) (algebraMap R S q)) :
+    p • (⊤ : Submodule R M) = q • (⊤ : Submodule R M) :=
+  le_antisymm (smul_top_le_of_map_dvd h.symm.dvd) (smul_top_le_of_map_dvd h.dvd)
+
+variable {ι κ : Type*}
+
+ theorem associated_prod_power (s : Finset ι) (a : ι → S) (p : S)
+    (h : ∀ i ∈ s, Associated (a i) p) : Associated (∏ i ∈ s, a i) (p^s.card) := by
+  simpa only [Finset.prod_const] using Associated.prod s a (fun _ => p) h
+
+ theorem associated_prod_scale (s : Finset ι) (t : Finset κ) (a : ι → S) (b : κ → S)
+    (p f : S) (k : ℕ) (ha : ∀ i ∈ s, Associated (a i) p)
+    (hb : ∀ j ∈ t, Associated (b j) p) (hf : Associated f (p^k))
+    (hcount : s.card=t.card+k) :
+    Associated (∏ i ∈ s, a i) (f*∏ j ∈ t, b j) := by
+  have hs := associated_prod_power s a p ha
+  have ht := associated_prod_power t b p hb
+  apply hs.trans
+  have hpow : p^s.card=p^k*p^t.card := by rw [hcount,add_comm,pow_add]
+  rw [hpow]
+  exact (hf.mul_mul ht).symm
+
+/-- The local embedded product images agree whenever the reduced labels are
+associated and the actual later-edge count has the required difference. -/
+ theorem product_image_scale (s : Finset ι) (t : Finset κ) (a : ι → R) (b : κ → R)
+    (p : S) (f : R) (k : ℕ)
+    (ha : ∀ i ∈ s, Associated (algebraMap R S (a i)) p)
+    (hb : ∀ j ∈ t, Associated (algebraMap R S (b j)) p)
+    (hf : Associated (algebraMap R S f) (p^k)) (hcount : s.card=t.card+k) :
+    (∏ i ∈ s, a i) • (⊤ : Submodule R M) =
+      f • ((∏ j ∈ t, b j) • (⊤ : Submodule R M)) := by
+  rw [← mul_smul]
+  apply smul_top_eq_of_map_associated (S := S)
+  simpa only [map_prod,map_mul] using associated_prod_scale s t
+    (fun i => algebraMap R S (a i)) (fun j => algebraMap R S (b j)) p
+    (algebraMap R S f) k ha hb hf hcount
+
+end KLInvariance.LocalProducts
+
+end
+
+
+section
+
+namespace KLInvariance.TitsSpace
+open Module BruhatGraph
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {σ : Type us} (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ theorem polynomialRoot_localized (P : Submodule ℝ (I → ℝ)) (a : PositiveRoot M cs) :
+    algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P) (polynomialRoot cs basis a) =
+      localizedLinear basis P a.val := rfl
+
+ theorem polynomialRoot_reduced (P : Submodule ℝ (I → ℝ)) (a c : PositiveRoot M cs) :
+    algebraMap (PlaneRing basis P) (LocalizedEdgeRing basis P a.val)
+      (algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P) (polynomialRoot cs basis c)) =
+      localizedReduction basis P a.val c.val := rfl
+
+ theorem planeRoot_outside {x : W} {a b : PositiveRoot M cs} (hab : a ≠ b)
+    (Q : (planeWeight x a).support) (hQ : Q.val ≠ rootPlane a b) :
+    (planeRoot x a Q).val ∉ rootPlane a b := by
+  intro h
+  apply hQ
+  rw [← planeRoot_plane x a Q]
+  exact rootPlane_eq_of_mem a b a (planeRoot x a Q) hab
+    (Submodule.subset_span (Set.mem_insert _ _)) h (planeRoot_ne x a Q).symm
+
+ theorem reduced_planeRoot_other_isUnit {x : W} {a b : PositiveRoot M cs} (hab : a ≠ b)
+    (Q : (planeWeight x a).support) (hQ : Q.val ≠ rootPlane a b) :
+    IsUnit (localizedReduction basis (rootPlane a b) a.val (planeRoot x a Q).val) := by
+  have h := (linear_isUnit_iff basis (rootPlane a b) (planeRoot x a Q).val).mpr
+    (planeRoot_outside hab Q hQ)
+  exact h.map (algebraMap (PlaneRing basis (rootPlane a b))
+    (LocalizedEdgeRing basis (rootPlane a b) a.val))
+
+/-- The actual integral edge factor reduces, in each actual plane, to precisely
+its plane weight. All other factors are proven units, not discarded formally. -/
+ theorem edgeFactor_local_associated (x : W) (a b : PositiveRoot M cs) (hab : a ≠ b) :
+    Associated
+      (algebraMap (PlaneRing basis (rootPlane a b)) (LocalizedEdgeRing basis (rootPlane a b) a.val)
+        (algebraMap (MvPolynomial σ ℝ) (PlaneRing basis (rootPlane a b)) (edgeFactorLift cs basis x a)))
+      ((localizedReduction basis (rootPlane a b) a.val b.val)^((planeWeight x a (rootPlane a b)).toNat)) := by
+  classical
+  let p := localizedReduction basis (rootPlane a b) a.val b.val
+  have hfactor (Q : (planeWeight x a).support) :
+      Associated
+        ((localizedReduction basis (rootPlane a b) a.val (planeRoot x a Q).val)^((planeWeight x a Q.val).toNat))
+        (if Q.val=rootPlane a b then p^((planeWeight x a (rootPlane a b)).toNat) else 1) := by
+    by_cases hQ : Q.val=rootPlane a b
+    · rw [ite_eq_left hQ]
+      have hc : (planeRoot x a Q).val ∈ rootPlane a b := by
+        rw [← hQ,← planeRoot_plane x a Q]
+        exact Submodule.subset_span (Set.mem_insert_of_mem _ (Set.mem_singleton _))
+      have h := localizedReduction_associated basis (rootPlane a b) a b (planeRoot x a Q)
+        hc (planeRoot_ne x a Q)
+      simpa only [hQ] using (Associated.pow_pow (M := LocalizedEdgeRing basis (rootPlane a b) a.val) h (n := (planeWeight x a Q.val).toNat))
+    · rw [ite_eq_right hQ]
+      exact associated_one_iff_isUnit.mpr ((reduced_planeRoot_other_isUnit basis hab Q hQ).pow _)
+  have hprod := Associated.prod Finset.univ
+    (fun Q : (planeWeight x a).support =>
+      (localizedReduction basis (rootPlane a b) a.val (planeRoot x a Q).val)^((planeWeight x a Q.val).toNat))
+    (fun Q => if Q.val=rootPlane a b then p^((planeWeight x a (rootPlane a b)).toNat) else 1)
+    (fun Q _ => hfactor Q)
+  have hp : (∏ Q : (planeWeight x a).support,
+      if Q.val=rootPlane a b then p^((planeWeight x a (rootPlane a b)).toNat) else 1) =
+      p^((planeWeight x a (rootPlane a b)).toNat) := by
+    rw [Fintype.prod_ite_eq_ite_exists _ (fun i j hi hj => Subtype.ext (hi.trans hj.symm))]
+    by_cases hP : rootPlane a b ∈ (planeWeight x a).support
+    · rw [ite_eq_left ⟨⟨rootPlane a b,hP⟩,rfl⟩]
+    · have hz := Finsupp.notMem_support_iff.mp hP
+      rw [hz,Int.toNat_zero,pow_zero]
+      split_ifs <;> rfl
+  rw [hp] at hprod
+  simpa only [edgeFactorLift,map_prod,map_pow,polynomialRoot_reduced] using hprod
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+namespace KLInvariance.TitsSpace.PlaneBasis
+open Module BruhatGraph MomentGraph MomentGraph.Sheaf
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+  {σ : Type us} (basis : Basis σ ℝ (Extended M))
+  (a₀ b : W) (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+noncomputable section
+
+ theorem cosetRoot_eq (e : B.CosetEdge a₀ b) :
+    BruhatGraph.root cs 1 b (B.cosetEdgeAmbient a₀ b hmin e)=B.edgeRoot e.property := by
+  rw [B.edgeRoot_coset hmin e.property]
+  rfl
+
+ theorem cosetLabel_eq_root (e : B.CosetEdge a₀ b) :
+    B.cosetLabel basis a₀ b hmin e=localizedLinear basis P (B.edgeRoot e.property).val := by
+  change algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P)
+    (polynomialRoot cs basis (BruhatGraph.root cs 1 b (B.cosetEdgeAmbient a₀ b hmin e)))=_
+  rw [B.cosetRoot_eq a₀ b hmin e]
+  rfl
+
+ def reducedCosetRoot (e : B.CosetEdge a₀ b) (β : PositiveRoot M cs) :
+    PlaneRing basis P ⧸ Ideal.span {B.cosetLabel basis a₀ b hmin e} :=
+  algebraMap (PlaneRing basis P) _ (localizedLinear basis P β.val)
+
+ theorem cosetLabel_reduction_associated (e f : B.CosetEdge a₀ b)
+    (β : PositiveRoot M cs) (hplane : rootPlane (B.edgeRoot e.property) β=P)
+    (hne : B.edgeRoot f.property ≠ B.edgeRoot e.property) :
+    Associated
+      (algebraMap (PlaneRing basis P) (PlaneRing basis P ⧸ Ideal.span {B.cosetLabel basis a₀ b hmin e})
+        (B.cosetLabel basis a₀ b hmin f))
+      (B.reducedCosetRoot basis a₀ b hmin e β) := by
+  unfold reducedCosetRoot
+  rw [B.cosetLabel_eq_root basis a₀ b hmin e,B.cosetLabel_eq_root basis a₀ b hmin f]
+  exact localizedReduction_associated basis P (B.edgeRoot e.property) β (B.edgeRoot f.property)
+    (hplane.symm ▸ B.edgeRoot_plane f.property) hne
+
+ theorem cosetFactor_reduction_associated (e : B.CosetEdge a₀ b)
+    (β : PositiveRoot M cs) (hab : B.edgeRoot e.property ≠ β)
+    (hplane : rootPlane (B.edgeRoot e.property) β=P) :
+    Associated
+      (algebraMap (PlaneRing basis P) (PlaneRing basis P ⧸ Ideal.span {B.cosetLabel basis a₀ b hmin e})
+        (algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P)
+          (edgeFactorLift cs basis ((e.val.1.val:W)*a₀) (B.edgeRoot e.property))))
+      ((B.reducedCosetRoot basis a₀ b hmin e β)^((planeWeight ((e.val.1.val:W)*a₀) (B.edgeRoot e.property) P).toNat)) := by
+  unfold reducedCosetRoot
+  rw [B.cosetLabel_eq_root basis a₀ b hmin e]
+  have h := edgeFactor_local_associated basis ((e.val.1.val:W)*a₀) (B.edgeRoot e.property) β hab
+  rw [hplane] at h
+  exact h
+
+end
+end KLInvariance.TitsSpace.PlaneBasis
+
+end
+
+
+section
+
+namespace KLInvariance.BruhatGraph
+open TitsSpace
+universe u v u' v'
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
+  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
+  {u b : W} {u' b' : W'}
+  (φ : Interval cs u b ≃o Interval cs' u' b')
+noncomputable section
+
+ def upperSource (e : Edge cs 1 b) (hx : BruhatLE cs u (source cs 1 b e).val) : Interval cs u b :=
+  ⟨(source cs 1 b e).val,hx,(source cs 1 b e).property.2⟩
+ def upperTarget (e : Edge cs 1 b) (hx : BruhatLE cs u (source cs 1 b e).val) : Interval cs u b :=
+  ⟨(target cs 1 b e).val,bruhat_trans cs hx (.single e.property),(target cs 1 b e).property.2⟩
+ def mappedEdgeRoot (e : Edge cs 1 b) (hx : BruhatLE cs u (source cs 1 b e).val) :
+     PositiveRoot M' cs' :=
+   graphRoot ((intervalIso_bruhatStep_iff_general cs cs' φ (upperSource e hx) (upperTarget e hx)).mpr e.property)
+
+end
+end KLInvariance.BruhatGraph
+
+namespace KLInvariance.TitsSpace.PlaneBasis
+open Module BruhatGraph MomentGraph MomentGraph.Sheaf
+universe u v u' v'
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+noncomputable section
+
+ theorem exists_coset_edge {b : W} (e : Edge cs 1 b)
+    (he : (BruhatGraph.root cs 1 b e).val ∈ P) :
+    ∃ a₀, ∃ hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀),
+      ∃ d : B.CosetEdge a₀ b, B.cosetEdgeAmbient a₀ b hmin d=e := by
+  obtain ⟨a₀,g,hg,hx,hmin⟩ := planeCoset_exists_minimal M cs P (source cs 1 b e).val
+  have ht : label cs 1 b e ∈ planeSubgroup M cs P :=
+    (Represents.planeSubgroup_mem_iff M cs (positiveRoot_represents M cs
+      ⟨label cs 1 b e,label_isReflection cs 1 b e⟩)).mpr he
+  let gg : planeSubgroup M cs P := ⟨g,hg⟩
+  let hh : planeSubgroup M cs P := ⟨label cs 1 b e*g,(planeSubgroup M cs P).mul_mem ht hg⟩
+  have hy : (hh:W)*a₀=(target cs 1 b e).val := by
+    change (label cs 1 b e*g)*a₀=_
+    rw [mul_assoc,← hx,label_mul_source]
+  let x : B.CosetVertex a₀ b := ⟨gg,one_bruhat B.system gg,hx ▸ (source cs 1 b e).property.2⟩
+  let y : B.CosetVertex a₀ b := ⟨hh,one_bruhat B.system hh,hy.symm ▸ (target cs 1 b e).property.2⟩
+  have hs : BruhatStep B.system gg hh := by
+    apply (B.coset_step_iff hmin gg hh).mp
+    rw [show (gg:W)*a₀=(source cs 1 b e).val from hx.symm,hy]
+    exact e.property
+  refine ⟨a₀,hmin,⟨(x,y),hs⟩,?_⟩
+  apply Subtype.ext
+  exact Prod.ext (Subtype.ext hx.symm) (Subtype.ext hy)
+
+variable {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
+  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
+  {a₀ u b : W} {u' b' : W'}
+  (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+  (φ : Interval cs u b ≃o Interval cs' u' b')
+
+ theorem mappedCosetRoot_eq (e : B.CosetEdge a₀ b)
+    (hx : BruhatLE cs u ((e.val.1.val:W)*a₀)) :
+    B.mappedCosetRoot hmin φ e hx = mappedEdgeRoot φ (B.cosetEdgeAmbient a₀ b hmin e) hx := rfl
+
+end
+end KLInvariance.TitsSpace.PlaneBasis
+
+end
+
+
+section
+
+/-! The actual transported finite-interval elementary matrix is the
+second system's genuine R-tilde matrix. The input is only the unlabelled
+order isomorphism; reflection edges are recovered by the checked general
+Bruhat-graph reconstruction theorem. -/
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/KazhdanLusztig/PositiveTransfer.lean b/lean/OAI/RepresentationTheory/KazhdanLusztig/PositiveTransfer.lean
index 7767f7b71a785080fdbb3bf779dd49e74abc5b1c..f85e6580ba9ec0695256b3117dcb78544f9baff4 100644
--- a/lean/OAI/RepresentationTheory/KazhdanLusztig/PositiveTransfer.lean
+++ b/lean/OAI/RepresentationTheory/KazhdanLusztig/PositiveTransfer.lean
@@ -1,1334 +1,1334 @@
-import OAI.RepresentationTheory.KazhdanLusztig.EdgeLocalization
-
-namespace OAI
-
-/-!
-Positive transfer inequalities, strict Hilbert-series detection including higher-codimension support, and reverse-filtration projective-to-free arguments.
--/
-
-section
-
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  (hub : BruhatLE cs u b)
-  (B : BoundarySheaf (polynomialGrading_negative ℝ σ)
-    (graph cs u b) (polynomialLabel cs u b basis) ⟨b,hub,bruhat_refl cs b⟩)
-  [Fintype (Edge cs u b)] [DecidableEq (Interval cs u b)]
-noncomputable section
-
-omit [Fintype σ] in
- theorem costalkEquivDual_pairing (d : SectionDuality cs u b basis hub B)
-    (x : Interval cs u b) (k : LinearMap.ker (B.sheaf.forget.stalkMap x))
-    (n : B.sheaf.forget.sections Set.univ) :
-    costalkEquivDual cs u b basis hub B d x k (n.val x)=
-      d.pairing (kernelEquivSingleSupport cs u b basis hub B x k).val n := by
-  let _ (y : Interval cs u b) : Module.Free (Coefficient (σ := σ))
-      (B.sheaf.forget.vertex y) := B.free y
-  let _ (y : Interval cs u b) : Module.IsTorsionFree (Coefficient (σ := σ))
-      (B.sheaf.forget.vertex y) := inferInstance
-  exact IntegralSupport.supportEquivDual_pairing (B.sheaf.forget.sections Set.univ)
-    (Sheaf.labelProduct (polynomialLabel cs u b basis))
-    (B.sheaf.forget.scaled_single_section
-      (B.sheaf.forget.labelProduct_annihilates (polynomialLabel cs u b basis)
-        (GradedSheaf.UpperQuotient.annihilates B.sheaf B.quotient)))
-    (Sheaf.labelProduct_ne_zero _ (polynomialLabel_ne_zero cs u b basis))
-    d.pairing d.balanced x (B.sheaf.forget.projection_surjective B.generated x)
-    (kernelEquivSingleSupport cs u b basis hub B x k) n
-
-omit [Fintype σ] in
- theorem costalkEquivDual_symmetric (d : SectionDuality cs u b basis hub B)
-    (hsym : ∀ m n, d.pairing m n=d.pairing n m)
-    (x : Interval cs u b) (k l : LinearMap.ker (B.sheaf.forget.stalkMap x)) :
-    costalkEquivDual cs u b basis hub B d x k l.val=
-      costalkEquivDual cs u b basis hub B d x l k.val := by
-  let e := costalkEquivDual cs u b basis hub B d x
-  let g := Sheaf.incomingProduct (G := graph cs u b) (polynomialLabel cs u b basis) x
-  let f := kernelEquivSingleSupport cs u b basis hub B x
-  have he (k l : LinearMap.ker (B.sheaf.forget.stalkMap x)) :
-      g * (e k) l.val=d.pairing (f k).val (f l).val := calc
-    g * (e k) l.val = (e k) (g • l.val) := (map_smul (e k) g l.val).symm
-    _ = (e k) ((f l).val.val x) := congrArg (e k)
-      (kernelEquivSingleSupport_apply cs u b basis hub B x l).symm
-    _ = _ := costalkEquivDual_pairing cs u b basis hub B d x k (f l).val
-  exact mul_left_cancel₀ (Sheaf.incomingProduct_ne_zero
-    (polynomialLabel cs u b basis) (polynomialLabel_prime cs u b basis) x)
-    ((he k l).trans ((hsym _ _).trans (he l k).symm))
-
-omit [Fintype σ] in
- theorem costalkEquivDual_inverse_symmetric (d : SectionDuality cs u b basis hub B)
-    (hsym : ∀ m n, d.pairing m n=d.pairing n m)
-    (x : Interval cs u b) (lam mu : Dual (Coefficient (σ := σ)) (B.sheaf.vertex x)) :
-    lam ((costalkEquivDual cs u b basis hub B d x).symm mu).val=
-      mu ((costalkEquivDual cs u b basis hub B d x).symm lam).val := by
-  have h := costalkEquivDual_symmetric cs u b basis hub B d hsym x
-    ((costalkEquivDual cs u b basis hub B d x).symm lam)
-    ((costalkEquivDual cs u b basis hub B d x).symm mu)
-  simpa only [LinearEquiv.apply_symm_apply] using h
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-/-! The genuine BMP local intersection map. All freeness, kernel-minimality and
-inverse grading assertions here are now unconditional consequences of the
-constructed integral duality. The nonvanishing/Hodge input is deliberately
-NOT asserted; the last theorem records precisely how it would force the
-strict KL halfdegree bound on ACTUAL homogeneous stalk generators. -/
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
-universe u
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W) (b : W)
-  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  [Fintype (Edge cs 1 b)] [Fintype (Interval cs 1 b)] [DecidableEq (Interval cs 1 b)]
-noncomputable section
-
- abbrev ActualBoundary := boundarySheaf cs 1 b basis (one_bruhat cs b)
-
- def actualSymmetricPairing : GradedSheaf.SymmetricPerfectPairing
-    (ActualBoundary cs b basis).sheaf (cs.length b : ℤ) :=
-  (boundarySheaf_symmetricPairing cs b basis).some
-
- def actualSectionDuality : SectionDuality cs 1 b basis (one_bruhat cs b)
-    (ActualBoundary cs b basis) := by
-  let p := actualSymmetricPairing cs b basis
-  let B := ActualBoundary cs b basis
-  refine ⟨p.pairing,?_⟩
-  intro x m n
-  exact p.toPerfectPairing.cut_balanced
-    (Sheaf.labelProduct (polynomialLabel cs 1 b basis))
-    (Sheaf.labelProduct_ne_zero _ (polynomialLabel_ne_zero cs 1 b basis))
-    (B.sheaf.forget.scaled_single_section
-      (B.sheaf.forget.labelProduct_annihilates (polynomialLabel cs 1 b basis)
-        (GradedSheaf.UpperQuotient.annihilates B.sheaf B.quotient))) x m n
-
- theorem actualSectionDuality_symmetric
-    (m n : (ActualBoundary cs b basis).sheaf.sections Set.univ) :
-    (actualSectionDuality cs b basis).pairing m n=
-      (actualSectionDuality cs b basis).pairing n m :=
-  (actualSymmetricPairing cs b basis).symmetric m n
-
- theorem actualSectionDuality_homogeneous (i j : ℤ)
-    (m n : (ActualBoundary cs b basis).sheaf.sections Set.univ)
-    (hm : m ∈ ((ActualBoundary cs b basis).sheaf.sections Set.univ).piece i)
-    (hn : n ∈ ((ActualBoundary cs b basis).sheaf.sections Set.univ).piece j) :
-    (actualSectionDuality cs b basis).pairing m n ∈
-      polynomialGrading ℝ σ (i+j-(cs.length b : ℤ)) :=
-  (actualSymmetricPairing cs b basis).homogeneous i j m n
-    (fun x => hm x (Set.mem_univ x)) (fun x => hn x (Set.mem_univ x))
-
-omit [Fintype (Interval cs 1 b)] in
- theorem costalkFreeObligation : CostalkFreeObligation cs 1 b basis (one_bruhat cs b) :=
-  costalk_free_of_duality cs 1 b basis (one_bruhat cs b) _ (actualSectionDuality cs b basis)
-
- def localIntersection (x : Interval cs 1 b) :
-    Dual (Coefficient (σ := σ)) ((ActualBoundary cs b basis).sheaf.vertex x)
-      →ₗ[Coefficient (σ := σ)] (ActualBoundary cs b basis).sheaf.vertex x :=
-  (LinearMap.ker ((ActualBoundary cs b basis).sheaf.forget.stalkMap x)).subtype.comp
-    (costalkEquivDual cs 1 b basis (one_bruhat cs b) _
-      (actualSectionDuality cs b basis) x).symm.toLinearMap
-
-omit [Fintype (Interval cs 1 b)] in
- theorem localIntersection_injective (x : Interval cs 1 b) :
-    Function.Injective (localIntersection cs b basis x) :=
-  Subtype.val_injective.comp (costalkEquivDual cs 1 b basis (one_bruhat cs b) _
-    (actualSectionDuality cs b basis) x).symm.injective
-
-omit [Fintype (Interval cs 1 b)] in
- theorem localIntersection_range (x : Interval cs 1 b) :
-    LinearMap.range (localIntersection cs b basis x)=
-      LinearMap.ker ((ActualBoundary cs b basis).sheaf.forget.stalkMap x) := by
-  apply le_antisymm
-  · rintro v ⟨lam,rfl⟩
-    exact ((costalkEquivDual cs 1 b basis (one_bruhat cs b) _
-      (actualSectionDuality cs b basis) x).symm lam).property
-  · intro v hv
-    refine ⟨costalkEquivDual cs 1 b basis (one_bruhat cs b) _
-      (actualSectionDuality cs b basis) x ⟨v,hv⟩,?_⟩
-    exact congrArg Subtype.val ((costalkEquivDual cs 1 b basis (one_bruhat cs b) _
-      (actualSectionDuality cs b basis) x).symm_apply_apply ⟨v,hv⟩)
-
-omit [Fintype (Interval cs 1 b)] in
- theorem localIntersection_minimal (x : Interval cs 1 b) (hx : x.val≠b) :
-    LinearMap.range (localIntersection cs b basis x) ≤ positiveIdeal (polynomialGrading ℝ σ) •
-      (⊤ : Submodule (Coefficient (σ := σ)) ((ActualBoundary cs b basis).sheaf.vertex x)) := by
-  rw [localIntersection_range]
-  apply ((ActualBoundary cs b basis).sheaf.kernelMinimal_iff x).mp
-  exact (ActualBoundary cs b basis).minimal x (fun h => hx (congrArg Subtype.val h))
-
- theorem localIntersection_homogeneous (x : Interval cs 1 b) (r : ℤ)
-    (lam : Dual (Coefficient (σ := σ)) ((ActualBoundary cs b basis).sheaf.vertex x))
-    (hlam : ∀ j (v : (ActualBoundary cs b basis).sheaf.vertex x),
-      v ∈ ((ActualBoundary cs b basis).sheaf.vertex x).piece j →
-      lam v ∈ polynomialGrading ℝ σ (j+r)) :
-    localIntersection cs b basis x lam ∈ ((ActualBoundary cs b basis).sheaf.vertex x).piece
-      (r+(cs.length b : ℤ)-cs.length x.val) := by
-  have h := costalkEquivDual_inverse_homogeneous cs 1 b basis (one_bruhat cs b) _
-    (actualSectionDuality cs b basis) (cs.length b) (actualSectionDuality_homogeneous cs b basis)
-    x r lam hlam
-  rw [incoming_card] at h
-  exact h
-
- theorem localIntersection_symmetric (x : Interval cs 1 b)
-    (lam mu : Dual (Coefficient (σ := σ)) ((ActualBoundary cs b basis).sheaf.vertex x)) :
-    lam (localIntersection cs b basis x mu)=mu (localIntersection cs b basis x lam) :=
-  costalkEquivDual_inverse_symmetric cs 1 b basis (one_bruhat cs b) _
-    (actualSectionDuality cs b basis) (actualSectionDuality_symmetric cs b basis) x lam mu
-
- theorem localIntersection_diagonal_strict_degree (x : Interval cs 1 b) (hx : x.val≠b)
-    (c : NonnegativeBasis (σ := σ) ((ActualBoundary cs b basis).sheaf.vertex x).piece)
-    [DecidableEq c.index] (i : c.index)
-    (hdiag : c.basis.dualBasis i (localIntersection cs b basis x (c.basis.dualBasis i))≠0) :
-    2*c.degree i<rankDifference cs x.val b := by
-  classical
-  have hg (j : c.index) := localIntersection_homogeneous cs b basis x (-(c.degree j : ℤ))
-    (c.basis.dualBasis j) (dualBasis_homogeneous (polynomialGrading ℝ σ)
-      ((ActualBoundary cs b basis).sheaf.vertex x).piece c.basis
-      (fun i => (c.degree i : ℤ)) c.homogeneous j)
-  have hh := local_diagonal_strict_degree (polynomialGrading ℝ σ)
-    (polynomialGrading_negative ℝ σ) ((ActualBoundary cs b basis).sheaf.vertex x).piece
-    c.basis (fun i => (c.degree i : ℤ)) c.homogeneous
-    ((cs.length b : ℤ)-cs.length x.val) (localIntersection cs b basis x)
-    (fun j => by convert hg j using 1; congr 1; omega)
-    (localIntersection_minimal cs b basis x hx) i hdiag
-  have hlen := length_le_of_bruhat cs x.property.2
-  dsimp [rankDifference]
-  omega
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-/-! Strict later-label membership in a finite nondecreasing schedule,
-with arbitrary positions for equal labels. -/
-namespace KLInvariance.SchedulePaths
-universe u v
-variable {E : Type u} {Λ : Type v} [LinearOrder Λ]
-
- theorem mem_suffix_iff_score_lt (score : E → Λ) {p q : List E} {e f : E}
-    (hs : p.Pairwise (fun a b => score a ≤ score b))
-    (hq : e::q <:+ p) (hf : f ∈ p) (hne : score e ≠ score f) :
-    f ∈ q ↔ score e < score f := by
-  obtain ⟨a,rfl⟩ := hq
-  have h := List.pairwise_append.mp hs
-  constructor
-  · intro hfq
-    exact lt_of_le_of_ne ((List.pairwise_cons.mp h.2.1).1 f hfq) hne
-  · intro hlt
-    rcases List.mem_append.mp hf with hfa | hfr
-    · exact False.elim (not_le_of_gt hlt (h.2.2 f hfa e List.mem_cons_self))
-    · rcases List.mem_cons.mp hfr with rfl | hfr
-      · exact False.elim (hne rfl)
-      · exact hfr
-
- theorem mem_suffix_iff_score_lt_injective (score : E → Λ) {p q : List E} {e f : E}
-    (hs : p.Pairwise (fun a b => score a ≤ score b)) (hn : p.Nodup)
-    (hq : e::q <:+ p) (hf : f ∈ p) (heq : score e = score f → e=f) :
-    f ∈ q ↔ score e < score f := by
-  by_cases hne : score e ≠ score f
-  · exact mem_suffix_iff_score_lt score hs hq hf hne
-  · have hef := heq (not_ne_iff.mp hne)
-    subst f
-    have hx := (List.nodup_cons.mp (hn.sublist hq.sublist)).1
-    simp only [hx,lt_self_iff_false]
-
-end KLInvariance.SchedulePaths
-
-end
-
-
-section
-
-namespace KLInvariance.Hilbert
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um un
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {M : Type um} [AddCommGroup M] [Module k M] [Module A M] [IsScalarTower k A M]
-  {N : Type un} [AddCommGroup N] [Module k N] [Module A N] [IsScalarTower k A N]
-
-omit [Algebra k A] [IsScalarTower k A M] [IsScalarTower k A N] in
- theorem shifted_map_homogeneous
-    (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
-    [DirectSum.Decomposition 𝓜] [DirectSum.Decomposition 𝓝]
-    (S : Submodule A M) (hS : ∀ n m, m ∈ S → component 𝓜 n m ∈ S)
-    (f : M →ₗ[A] N) (d : ℤ) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 (n+d)) :
-    ∀ n v, v ∈ S.map f → component 𝓝 n v ∈ S.map f := by
-  intro n v hv
-  obtain ⟨w,hw,rfl⟩ := Submodule.mem_map.mp hv
-  exact Submodule.mem_map.mpr ⟨component 𝓜 (n-d) w,hS (n-d) w hw,
-    (map_component 𝓜 𝓝 f d hf n w).symm⟩
-
- def shiftedImagePieceMap (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
-    (S : Submodule A M) (f : M →ₗ[A] N) (d : ℤ)
-    (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 (n+d)) (n : ℤ) :
-    subPiece 𝓜 S (n-d) →ₗ[k] subPiece 𝓝 (S.map f) n where
-  toFun v := ⟨⟨f v.val.val,Submodule.mem_map.mpr ⟨_,v.val.property,rfl⟩⟩,
-    by
-      change f v.val.val ∈ 𝓝 n
-      simpa using hf (n-d) v.val.val v.property⟩
-  map_add' _ _ := by ext; exact map_add f _ _
-  map_smul' c v := by ext; exact (f.restrictScalars k).map_smul c v.val.val
-
- theorem shiftedImagePieceMap_surjective
-    (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
-    [DirectSum.Decomposition 𝓜] [DirectSum.Decomposition 𝓝]
-    (S : Submodule A M) (hS : ∀ n m, m ∈ S → component 𝓜 n m ∈ S)
-    (f : M →ₗ[A] N) (d : ℤ) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 (n+d)) (n : ℤ) :
-    Function.Surjective (shiftedImagePieceMap 𝓜 𝓝 S f d hf n) := by
-  rintro ⟨⟨m,hm⟩,hgrad⟩
-  obtain ⟨v,hv,rfl⟩ := Submodule.mem_map.mp hm
-  refine ⟨⟨⟨component 𝓜 (n-d) v,hS (n-d) v hv⟩,
-    (DirectSum.decompose 𝓜 v (n-d)).property⟩,?_⟩
-  apply Subtype.ext
-  apply Subtype.ext
-  change f (component 𝓜 (n-d) v) = f v
-  rw [← map_component 𝓜 𝓝 f d hf n v]
-  exact DirectSum.decompose_of_mem_same 𝓝 hgrad
-
-noncomputable def shiftedImagePieceEquiv
-    (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
-    [DirectSum.Decomposition 𝓜] [DirectSum.Decomposition 𝓝]
-    (S : Submodule A M) (hS : ∀ n m, m ∈ S → component 𝓜 n m ∈ S)
-    (f : M →ₗ[A] N) (d : ℤ) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 (n+d))
-    (hinj : Function.Injective f) (n : ℤ) :
-    subPiece 𝓜 S (n-d) ≃ₗ[k] subPiece 𝓝 (S.map f) n :=
-  LinearEquiv.ofBijective (shiftedImagePieceMap 𝓜 𝓝 S f d hf n) ⟨by
-    intro v w h
-    apply Subtype.ext
-    apply Subtype.ext
-    apply hinj
-    exact congrArg (fun z => z.val.val) h,
-    shiftedImagePieceMap_surjective 𝓜 𝓝 S hS f d hf n⟩
-
-/-- Hilbert series of the actual embedded image of a homogeneous injection. -/
- theorem shifted_image_value
-    (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
-    [DirectSum.Decomposition 𝓜] [DirectSum.Decomposition 𝓝]
-    (S : Submodule A M) (hS : ∀ n m, m ∈ S → component 𝓜 n m ∈ S)
-    (f : M →ₗ[A] N) (d : ℤ) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 (n+d))
-    (hinj : Function.Injective f) {q : ℝ} (hq : q ≠ 0)
-    (hs : Summable (term 𝓜 (S.restrictScalars k) q)) :
-    value 𝓝 ((S.map f).restrictScalars k) q =
-      q^d * value 𝓜 (S.restrictScalars k) q := by
-  have ha := hs.hasSum.mul_left (q^d)
-  have hb : HasSum
-      (fun n : ℤ => (Module.finrank k (piece 𝓜 (S.restrictScalars k) (n-d)) : ℝ)*q^n)
-      (q^d * value 𝓜 (S.restrictScalars k) q) := by
-    apply (Equiv.addRight d).hasSum_iff.mp
-    apply ha.congr_fun
-    intro n
-    change (Module.finrank k (piece 𝓜 (S.restrictScalars k) (n+d-d)) : ℝ)*q^(n+d) =
-      q^d * ((Module.finrank k (piece 𝓜 (S.restrictScalars k) n) : ℝ)*q^n)
-    rw [add_sub_cancel_right,zpow_add₀ hq]
-    ring
-  apply HasSum.tsum_eq
-  apply hb.congr_fun
-  intro n
-  unfold term
-  rw [← (subPieceEquiv 𝓝 (S.map f) n).finrank_eq,
-    ← (shiftedImagePieceEquiv 𝓜 𝓝 S hS f d hf hinj n).finrank_eq,
-    (subPieceEquiv 𝓜 S (n-d)).finrank_eq]
-
-end KLInvariance.Hilbert
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded _root_.OAI.KLInvariance.Hilbert
-universe us um
-variable {σ : Type us} [Fintype σ]
-  {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
-  (B : GradedSheaf (𝓐 := polynomialGrading ℝ σ) G)
-noncomputable section
-
- def kernelHilbert (q : ℝ) (x : V) (F : Set (Sheaf.Outgoing (G := G) x)) : ℝ :=
-  value (B.vertex x).piece ((B.upwardSubmodule x F).restrictScalars ℝ) q
-
- def lowerImage (e : E) (F : Set (Sheaf.Outgoing (G := G) (G.source e))) :
-    Submodule (MvPolynomial σ ℝ) (B.edge e) :=
-  (B.upwardSubmodule (G.source e) F).map (B.lower e).map
-
- def upperImage (e : E) (F : Set (Sheaf.Outgoing (G := G) (G.target e))) :
-    Submodule (MvPolynomial σ ℝ) (B.edge e) :=
-  (B.upwardSubmodule (G.target e) F).map (B.upper e).map
-
-omit [Fintype σ] in
- theorem upperImage_homogeneous (e : E) (H : Set (Sheaf.Outgoing (G := G) (G.target e))) :
-    ∀ n v, v ∈ B.upperImage e H → component (B.edge e).piece n v ∈ B.upperImage e H :=
-  map_homogeneous (B.vertex (G.target e)).piece (B.edge e).piece
-    (B.upwardSubmodule (G.target e) H) (B.upwardKernel_homogeneous (G.target e) H)
-    (B.upper e).map (B.upper e).graded
-
- theorem kernelHilbert_edge (e : E) (F : Set (Sheaf.Outgoing (G := G) (G.source e)))
-    {q : ℝ} (hq : 0 < q) (hq1 : q < 1) :
-    B.kernelHilbert q (G.source e) F -
-      B.kernelHilbert q (G.source e) (insert ⟨e,rfl⟩ F) =
-      value (B.edge e).piece ((B.lowerImage e F).restrictScalars ℝ) q := by
-  let : ∀ n, Module.Finite ℝ ((B.vertex (G.source e)).piece n) :=
-    polynomialModule_piece_finite ℝ σ (B.vertex (G.source e)).piece
-  have h := B.hilbert_edge_increment (G.source e) F ⟨e,rfl⟩ hq
-    (summable_term ℝ σ (B.vertex (G.source e)).piece _ hq hq1)
-  exact sub_eq_iff_eq_add.mpr (h.trans (add_comm _ _))
-
- theorem shifted_upperImage_value (α : E → MvPolynomial σ ℝ) (hquot : B.UpperQuotient α)
-    (e : E) (H : Set (Sheaf.Outgoing (G := G) (G.target e)))
-    (ha : α e ∈ polynomialGrading ℝ σ 1)
-    (hreg : ∀ f ∈ H, Function.Injective (fun v : B.edge f.val => α e • v))
-    (hinj : Function.Injective (fun v : B.vertex (G.target e) => α e • v))
-    (φ : B.edge e →ₗ[MvPolynomial σ ℝ] B.edge e) (d : ℤ)
-    (hφ : ∀ n v, v ∈ (B.edge e).piece n → φ v ∈ (B.edge e).piece (n+d))
-    (hφinj : Function.Injective φ) {q : ℝ} (hq : 0 < q) (hq1 : q < 1) :
-    value (B.edge e).piece (((B.upperImage e H).map φ).restrictScalars ℝ) q =
-      q^d * (1-q) * B.kernelHilbert q (G.target e) H := by
-  let : ∀ n, Module.Finite ℝ ((B.vertex (G.target e)).piece n) :=
-    polynomialModule_piece_finite ℝ σ (B.vertex (G.target e)).piece
-  rw [shifted_image_value (B.edge e).piece (B.edge e).piece (B.upperImage e H)
-    (B.upperImage_homogeneous e H) φ d hφ hφinj hq.ne'
-    (summable_term ℝ σ (B.edge e).piece _ hq hq1)]
-  rw [show value (B.edge e).piece ((B.upperImage e H).restrictScalars ℝ) q =
-    (1-q) * B.kernelHilbert q (G.target e) H from
-      B.upper_image_value α hquot e H ha hreg hinj hq
-        (summable_term ℝ σ (B.vertex (G.target e)).piece _ hq hq1)]
-  exact (mul_assoc _ _ _).symm
-
-
- theorem edge_increment_le (α : E → MvPolynomial σ ℝ) (hquot : B.UpperQuotient α)
-    (e : E) (F : Set (Sheaf.Outgoing (G := G) (G.source e)))
-    (H : Set (Sheaf.Outgoing (G := G) (G.target e)))
-    (ha : α e ∈ polynomialGrading ℝ σ 1)
-    (hreg : ∀ f ∈ H, Function.Injective (fun v : B.edge f.val => α e • v))
-    (hinj : Function.Injective (fun v : B.vertex (G.target e) => α e • v))
-    (φ : B.edge e →ₗ[MvPolynomial σ ℝ] B.edge e) (d : ℤ)
-    (hφ : ∀ n v, v ∈ (B.edge e).piece n → φ v ∈ (B.edge e).piece (n+d))
-    (hφinj : Function.Injective φ) (hinc : B.lowerImage e F ≤ (B.upperImage e H).map φ)
-    {q : ℝ} (hq : 0 < q) (hq1 : q < 1) :
-    B.kernelHilbert q (G.source e) F -
-      B.kernelHilbert q (G.source e) (insert ⟨e,rfl⟩ F) ≤
-      q^d * (1-q) * B.kernelHilbert q (G.target e) H := by
-  let : ∀ n, Module.Finite ℝ ((B.edge e).piece n) :=
-    polynomialModule_piece_finite ℝ σ (B.edge e).piece
-  rw [B.kernelHilbert_edge e F hq hq1,
-    ← B.shifted_upperImage_value α hquot e H ha hreg hinj φ d hφ hφinj hq hq1]
-  exact (summable_term ℝ σ (B.edge e).piece _ hq hq1).tsum_le_tsum
-    (term_mono (B.edge e).piece hinc hq)
-    (summable_term ℝ σ (B.edge e).piece _ hq hq1)
-
-/-- Even a finite-length or higher-codimension embedded defect is detected by
-one positive numerical equality. This does not replace the image inclusion. -/
- theorem edge_increment_eq_iff (α : E → MvPolynomial σ ℝ) (hquot : B.UpperQuotient α)
-    (e : E) (F : Set (Sheaf.Outgoing (G := G) (G.source e)))
-    (H : Set (Sheaf.Outgoing (G := G) (G.target e)))
-    (ha : α e ∈ polynomialGrading ℝ σ 1)
-    (hreg : ∀ f ∈ H, Function.Injective (fun v : B.edge f.val => α e • v))
-    (hinj : Function.Injective (fun v : B.vertex (G.target e) => α e • v))
-    (φ : B.edge e →ₗ[MvPolynomial σ ℝ] B.edge e) (d : ℤ)
-    (hφ : ∀ n v, v ∈ (B.edge e).piece n → φ v ∈ (B.edge e).piece (n+d))
-    (hφinj : Function.Injective φ) (hinc : B.lowerImage e F ≤ (B.upperImage e H).map φ)
-    {q : ℝ} (hq : 0 < q) (hq1 : q < 1) :
-    (B.kernelHilbert q (G.source e) F -
-      B.kernelHilbert q (G.source e) (insert ⟨e,rfl⟩ F) =
-      q^d * (1-q) * B.kernelHilbert q (G.target e) H) ↔
-      B.lowerImage e F = (B.upperImage e H).map φ := by
-  rw [B.kernelHilbert_edge e F hq hq1,
-    ← B.shifted_upperImage_value α hquot e H ha hreg hinj φ d hφ hφinj hq hq1]
-  constructor
-  · intro heq
-    have h := polynomial_eq_of_value_eq ℝ σ (B.edge e).piece hinc
-      (shifted_map_homogeneous (B.edge e).piece (B.edge e).piece (B.upperImage e H)
-        (B.upperImage_homogeneous e H) φ d hφ) hq hq1 heq
-    ext v
-    exact SetLike.ext_iff.mp h v
-  · intro heq
-    rw [heq]
-
-end
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  (hub : BruhatLE cs u b)
-  (B : BoundarySheaf (polynomialGrading_negative ℝ σ)
-    (graph cs u b) (polynomialLabel cs u b basis) ⟨b,hub,bruhat_refl cs b⟩)
-noncomputable section
-
-
- theorem actual_edge_increment_le (e : Edge cs u b)
-    (F : Set (Sheaf.Outgoing (G := graph cs u b) ((graph cs u b).source e)))
-    (H : Set (Sheaf.Outgoing (G := graph cs u b) ((graph cs u b).target e)))
-    (hinc : B.sheaf.lowerImage e F ≤ (B.sheaf.upperImage e H).map
-      (edgeFactorMap cs basis u b hub B e))
-    {q : ℝ} (hq : 0 < q) (hq1 : q < 1) :
-    B.sheaf.kernelHilbert q ((graph cs u b).source e) F -
-      B.sheaf.kernelHilbert q ((graph cs u b).source e) (insert ⟨e,rfl⟩ F) ≤
-      q^(edgeFactorDegree (source cs u b e).val (root cs u b e) : ℤ) * (1-q) *
-        B.sheaf.kernelHilbert q ((graph cs u b).target e) H := by
-  apply B.sheaf.edge_increment_le (polynomialLabel cs u b basis) B.quotient e F H
-    (polynomialLabel_homogeneous cs u b basis e)
-  · intro f _
-    exact incoming_label_regular_outgoing cs u b basis hub B e f.val f.property
-  · exact label_regular_stalk cs u b basis hub B e _
-  · exact edgeFactorMap_graded cs basis u b hub B e
-  · exact edgeFactorLift_regular cs basis u b hub B e
-  · exact hinc
-  · exact hq
-  · exact hq1
-
- theorem actual_edge_increment_eq_iff (e : Edge cs u b)
-    (F : Set (Sheaf.Outgoing (G := graph cs u b) ((graph cs u b).source e)))
-    (H : Set (Sheaf.Outgoing (G := graph cs u b) ((graph cs u b).target e)))
-    (hinc : B.sheaf.lowerImage e F ≤ (B.sheaf.upperImage e H).map
-      (edgeFactorMap cs basis u b hub B e))
-    {q : ℝ} (hq : 0 < q) (hq1 : q < 1) :
-    (B.sheaf.kernelHilbert q ((graph cs u b).source e) F -
-      B.sheaf.kernelHilbert q ((graph cs u b).source e) (insert ⟨e,rfl⟩ F) =
-      q^(edgeFactorDegree (source cs u b e).val (root cs u b e) : ℤ) * (1-q) *
-        B.sheaf.kernelHilbert q ((graph cs u b).target e) H) ↔
-      B.sheaf.lowerImage e F = (B.sheaf.upperImage e H).map
-        (edgeFactorMap cs basis u b hub B e) := by
-  apply B.sheaf.edge_increment_eq_iff (polynomialLabel cs u b basis) B.quotient e F H
-    (polynomialLabel_homogeneous cs u b basis e)
-  · intro f _
-    exact incoming_label_regular_outgoing cs u b basis hub B e f.val f.property
-  · exact label_regular_stalk cs u b basis hub B e _
-  · exact edgeFactorMap_graded cs basis u b hub B e
-  · exact edgeFactorLift_regular cs basis u b hub B e
-  · exact hinc
-  · exact hq
-  · exact hq1
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-namespace KLInvariance.Comparison
-
-/-- Degree-one labels, `q=t²`, and exactly the source degree
-`2*k = length(y)-length(x)-1` give the same coefficient at every edge. -/
- theorem edge_scalar_identity {t : ℝ} (ht : t ≠ 0) (m n k : ℤ)
-    (h : m + 2*k = n-1) :
-    t^m * (t^2)^k * (1-t^2) = (t⁻¹-t) * t^n := by
-  have hp : (t^2)^k = t^(2*k) := by
-    rw [zpow_mul]
-    norm_num
-  rw [hp,← zpow_add₀ ht,h]
-  have hsub : t^(n-1) = t^n*t⁻¹ := by
-    rw [sub_eq_add_neg,zpow_add₀ ht,zpow_neg_one]
-  rw [hsub]
-  have hcancel : t⁻¹*t=1 := inv_mul_cancel₀ ht
-  calc
-    t^n*t⁻¹*(1-t^2) = (t⁻¹-t*(t⁻¹*t))*t^n := by ring
-    _ = (t⁻¹-t)*t^n := by rw [hcancel,mul_one]
-
- theorem transferCoefficient_pos {t : ℝ} (ht : 0 < t) (ht1 : t < 1) :
-    0 < t⁻¹-t := by
-  have hi : 1 < t⁻¹ := (one_lt_inv₀ ht).mpr ht1
-  linarith
-
- theorem stateScale_pos {t : ℝ} (ht : 0 < t) (ht1 : t < 1) (N : ℕ) (m : ℤ) :
-    0 < (1-t^2)^N * t^m := by
-  have hsq : t^2 < 1 := by nlinarith
-  exact mul_pos (pow_pos (sub_pos.mpr hsq) _) (zpow_pos ht _)
-
-/-- Cancellation of a positive state scale converts an additive normalized
-edge identity back to its raw Hilbert-series increment. -/
- theorem scaled_increment_eq_iff (a x y k z : ℝ) (ha : a ≠ 0) :
-    a*x = a*y+(a*k)*z ↔ x-y=k*z := by
-  constructor
-  · intro h
-    have he : a*x=a*(y+k*z) := by linear_combination h
-    have hc := mul_left_cancel₀ ha he
-    linarith
-  · intro h
-    rw [show x=y+k*z by linarith]
-    ring
-
-end KLInvariance.Comparison
-
-end
-
-
-section
-
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  (hub : BruhatLE cs u b)
-  (B : BoundarySheaf (polynomialGrading_negative ℝ σ)
-    (graph cs u b) (polynomialLabel cs u b basis) ⟨b,hub,bruhat_refl cs b⟩)
-noncomputable section
-
- def stateScale (t : ℝ) (x : Interval cs u b) : ℝ :=
-  (1-t^2)^(Fintype.card σ) * t^((cs.length x.val : ℤ)-cs.length b)
-
-omit [Fintype I] in
- theorem stateScale_positive {t : ℝ} (ht : 0 < t) (ht1 : t < 1)
-    (x : Interval cs u b) : 0 < stateScale (σ := σ) cs u b t x :=
-  Comparison.stateScale_pos ht ht1 _ _
-
- def normalizedKernel (t : ℝ) (x : Interval cs u b)
-    (F : Set (Sheaf.Outgoing (G := graph cs u b) x)) : ℝ :=
-  stateScale (σ := σ) cs u b t x * B.sheaf.kernelHilbert (t^2) x F
-
- theorem edge_stateScale (e : Edge cs u b) {t : ℝ} (ht : t ≠ 0) :
-    stateScale (σ := σ) cs u b t ((graph cs u b).source e) *
-      ((t^2)^(edgeFactorDegree (source cs u b e).val (root cs u b e) : ℤ) * (1-t^2)) =
-      (t⁻¹-t) * stateScale (σ := σ) cs u b t ((graph cs u b).target e) := by
-  have hd := edgeFactorDegree_edge cs u b e
-  have h := Comparison.edge_scalar_identity ht
-    ((cs.length (source cs u b e).val : ℤ)-cs.length b)
-    ((cs.length (target cs u b e).val : ℤ)-cs.length b)
-    (edgeFactorDegree (source cs u b e).val (root cs u b e)) (by omega)
-  dsimp only [stateScale,graph]
-  calc
-    _ = (1-t^2)^(Fintype.card σ) *
-        (t^((cs.length (source cs u b e).val : ℤ)-cs.length b) *
-          (t^2)^(edgeFactorDegree (source cs u b e).val (root cs u b e) : ℤ) * (1-t^2)) := by ring
-    _ = _ := by rw [h]; ring
-
-/-- A single removal of a later-edge kernel condition has the exact uniform
-transfer coefficient `t⁻¹-t` used by the Dyer path matrix. -/
- theorem normalized_edge_increment_le (e : Edge cs u b)
-    (F : Set (Sheaf.Outgoing (G := graph cs u b) ((graph cs u b).source e)))
-    (H : Set (Sheaf.Outgoing (G := graph cs u b) ((graph cs u b).target e)))
-    (hinc : B.sheaf.lowerImage e F ≤ (B.sheaf.upperImage e H).map
-      (edgeFactorMap cs basis u b hub B e))
-    {t : ℝ} (ht : 0 < t) (ht1 : t < 1) :
-    normalizedKernel cs u b basis hub B t ((graph cs u b).source e) F ≤
-      normalizedKernel cs u b basis hub B t ((graph cs u b).source e) (insert ⟨e,rfl⟩ F) +
-      (t⁻¹-t) * normalizedKernel cs u b basis hub B t ((graph cs u b).target e) H := by
-  have h := actual_edge_increment_le cs u b basis hub B e F H hinc
-    (sq_pos_of_pos ht) (by nlinarith : t^2 < 1)
-  have h' := mul_le_mul_of_nonneg_left h
-    (stateScale_positive (σ := σ) cs u b ht ht1 ((graph cs u b).source e)).le
-  rw [← mul_assoc,edge_stateScale cs u b e ht.ne',mul_sub] at h'
-  simpa only [normalizedKernel,mul_assoc,add_comm] using sub_le_iff_le_add.mp h'
-
- theorem normalized_edge_increment_eq_iff (e : Edge cs u b)
-    (F : Set (Sheaf.Outgoing (G := graph cs u b) ((graph cs u b).source e)))
-    (H : Set (Sheaf.Outgoing (G := graph cs u b) ((graph cs u b).target e)))
-    (hinc : B.sheaf.lowerImage e F ≤ (B.sheaf.upperImage e H).map
-      (edgeFactorMap cs basis u b hub B e))
-    {t : ℝ} (ht : 0 < t) (ht1 : t < 1) :
-    (normalizedKernel cs u b basis hub B t ((graph cs u b).source e) F =
-      normalizedKernel cs u b basis hub B t ((graph cs u b).source e) (insert ⟨e,rfl⟩ F) +
-      (t⁻¹-t) * normalizedKernel cs u b basis hub B t ((graph cs u b).target e) H) ↔
-      B.sheaf.lowerImage e F = (B.sheaf.upperImage e H).map
-        (edgeFactorMap cs basis u b hub B e) := by
-  rw [← actual_edge_increment_eq_iff cs u b basis hub B e F H hinc
-    (sq_pos_of_pos ht) (by nlinarith : t^2 < 1)]
-  have hs := (stateScale_positive (σ := σ) cs u b ht ht1 ((graph cs u b).source e)).ne'
-  simp only [normalizedKernel]
-  rw [← mul_assoc (t⁻¹-t),← edge_stateScale cs u b e ht.ne']
-  exact Comparison.scaled_increment_eq_iff _ _ _ _ _ hs
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-/-! Exact numerical endpoints for the actual sheaf, conditional on the two
-explicit genuine normalization inputs. These implications are NOT a proof of
-existence of the pairing or the Kazhdan--Lusztig character theorem. -/
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  (hub : BruhatLE cs u b)
-  (B : BoundarySheaf (polynomialGrading_negative ℝ σ)
-    (graph cs u b) (polynomialLabel cs u b basis) ⟨b,hub,bruhat_refl cs b⟩)
-noncomputable section
-
- theorem emptyKernel_hilbert (x : Interval cs u b) {q : ℝ} (hq : 0 < q) (hq1 : q < 1) :
-    B.sheaf.kernelHilbert q x ∅ =
-      Polynomial.eval₂ (Int.castRingHom ℝ) q (stalkCharacter cs u b basis hub x) *
-        (1-q)⁻¹ ^ Fintype.card σ := by
-  let _ := B.free x
-  let c := (exists_nonnegativeBasis (σ := σ) (B.sheaf.vertex x).piece
-    (B.sheaf.vertex x).nonneg).some
-  have he : B.sheaf.upwardSubmodule x ∅ = ⊤ := by
-    ext v
-    change (∀ e, e ∈ (∅ : Set (Sheaf.Outgoing (G := graph cs u b) x)) → _) ↔ True
-    simp
-  rw [GradedSheaf.kernelHilbert,he,Submodule.restrictScalars_top,
-    stalkCharacter_eq_character cs u b basis hub B x,
-    character_eq (B.sheaf.vertex x).piece (B.sheaf.vertex x).nonneg c]
-  simpa only [Hilbert.value,Hilbert.term_top] using (c.hasSum hq hq1).tsum_eq
-
- theorem normalized_emptyKernel_of_character (x : Interval cs u b)
-    (hc : stalkCharacter cs u b basis hub x = klPolynomial cs x.val b)
-    {t : ℝ} (ht : 0 < t) (ht1 : t < 1) :
-    normalizedKernel cs u b basis hub B t x ∅ = scaledKL cs t x.val b := by
-  rw [normalizedKernel,emptyKernel_hilbert cs u b basis hub B x
-    (sq_pos_of_pos ht) (by nlinarith),hc]
-  dsimp only [stateScale,scaledKL]
-  have hne : 1-t^2 ≠ 0 := ne_of_gt (by nlinarith : 0 < 1-t^2)
-  rw [inv_pow]
-  field_simp [hne,inv_pow]
-
-variable [Fintype (Edge cs 1 b)] [Fintype (Interval cs 1 b)] [DecidableEq (Interval cs 1 b)]
-  (C : BoundarySheaf (polynomialGrading_negative ℝ σ)
-    (graph cs 1 b) (polynomialLabel cs 1 b basis) ⟨b,one_bruhat cs b,bruhat_refl cs b⟩)
-
- theorem normalized_fullKernel_of_duality_character
-    (d : SectionDuality cs 1 b basis (one_bruhat cs b) C)
-    (hd : ∀ i j (m n : C.sheaf.sections Set.univ),
-      m ∈ (C.sheaf.sections Set.univ).piece i → n ∈ (C.sheaf.sections Set.univ).piece j →
-      d.pairing m n ∈ polynomialGrading ℝ σ (i+j-(cs.length b : ℤ)))
-    (x : Interval cs 1 b)
-    (hc : stalkCharacter cs 1 b basis (one_bruhat cs b) x = klPolynomial cs x.val b)
-    {t : ℝ} (ht : 0 < t) (ht1 : t < 1) :
-    normalizedKernel cs 1 b basis (one_bruhat cs b) C t x Set.univ =
-      scaledKL cs t⁻¹ x.val b := by
-  let _ := C.free x
-  let c := (exists_nonnegativeBasis (σ := σ) (C.sheaf.vertex x).piece
-    (C.sheaf.vertex x).nonneg).some
-  have hcp : c.polynomial = klPolynomial cs x.val b := by
-    rw [← character_eq (C.sheaf.vertex x).piece (C.sheaf.vertex x).nonneg c,
-      ← stalkCharacter_eq_character cs 1 b basis (one_bruhat cs b) C x]
-    exact hc
-  rw [normalizedKernel,GradedSheaf.kernelHilbert,
-    fullKernel_eq cs 1 b basis (one_bruhat cs b) C x,
-    costalk_hilbert_of_duality cs 1 b basis (one_bruhat cs b) C d (cs.length b) hd x c
-      (sq_pos_of_pos ht) (by nlinarith),hcp,incoming_card cs b x]
-  dsimp only [stateScale,scaledKL]
-  have hscale : t^((cs.length x.val : ℤ)-cs.length b) *
-      (t^2)^((cs.length b : ℤ)-cs.length x.val) =
-      (t⁻¹)^((cs.length x.val : ℤ)-cs.length b) := by
-    rw [show (t^2)^((cs.length b : ℤ)-cs.length x.val) =
-      t^(2*((cs.length b : ℤ)-cs.length x.val)) by rw [zpow_mul]; norm_num,
-      ← zpow_add₀ ht.ne',inv_zpow,← zpow_neg]
-    congr 1
-    omega
-  rw [show (t^2)⁻¹=(t⁻¹)^2 by rw [inv_pow]]
-  have hne : 1-t^2 ≠ 0 := ne_of_gt (by nlinarith : 0 < 1-t^2)
-  calc
-    _ = (t^((cs.length x.val : ℤ)-cs.length b) *
-      (t^2)^((cs.length b : ℤ)-cs.length x.val)) *
-      Polynomial.eval₂ (Int.castRingHom ℝ) ((t⁻¹)^2) (klPolynomial cs x.val b) := by
-        have hcancel : (1-t^2)^(Fintype.card σ) *
-            ((1-t^2)⁻¹)^(Fintype.card σ)=1 := by
-          rw [← mul_pow,mul_inv_cancel₀ hne,one_pow]
-        calc
-          _ = ((1-t^2)^(Fintype.card σ) * ((1-t^2)⁻¹)^(Fintype.card σ)) *
-            ((t^((cs.length x.val : ℤ)-cs.length b) *
-              (t^2)^((cs.length b : ℤ)-cs.length x.val)) *
-              Polynomial.eval₂ (Int.castRingHom ℝ) ((t⁻¹)^2) (klPolynomial cs x.val b)) := by ring
-          _ = _ := by rw [hcancel,one_mul]
-    _ = _ := by rw [hscale]
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  (hub : BruhatLE cs u b)
-  (B : BoundarySheaf (polynomialGrading_negative ℝ σ)
-    (graph cs 1 b) (polynomialLabel cs 1 b basis) ⟨b,one_bruhat cs b,bruhat_refl cs b⟩)
-noncomputable section
-
- omit [Fintype σ] in
- theorem restrictBoundary_kernel (x : Interval cs u b) (T : Set (Edge cs 1 b)) :
-    (restrictBoundary cs u b basis hub B).sheaf.upwardSubmodule x
-      {e | (upperInclusion cs u b).edge e.val ∈ T} =
-    B.sheaf.upwardSubmodule ((upperInclusion cs u b).vertex x) {e | e.val ∈ T} := by
-  exact B.sheaf.forget.pullback_kernel (upperInclusion cs u b) x T Set.univ
-    (fun e he => False.elim (he (Set.mem_univ e)))
-    (fun e _ he => upperInclusion_outgoing cs u b e x he)
-
- theorem normalizedKernel_restrictBoundary (x : Interval cs u b) (T : Set (Edge cs 1 b)) (t : ℝ) :
-    normalizedKernel cs u b basis hub (restrictBoundary cs u b basis hub B) t x
-      {e | (upperInclusion cs u b).edge e.val ∈ T} =
-    normalizedKernel cs 1 b basis (one_bruhat cs b) B t
-      ((upperInclusion cs u b).vertex x) {e | e.val ∈ T} := by
-  unfold normalizedKernel GradedSheaf.kernelHilbert
-  rw [restrictBoundary_kernel cs u b basis hub B x T]
-  rfl
-
- theorem normalized_emptyKernel_restrictBoundary (x : Interval cs u b) (t : ℝ) :
-    normalizedKernel cs u b basis hub (restrictBoundary cs u b basis hub B) t x ∅ =
-    normalizedKernel cs 1 b basis (one_bruhat cs b) B t
-      ((upperInclusion cs u b).vertex x) ∅ := by
-  simpa only [Set.mem_empty_iff_false,Set.ofPred_false] using
-    normalizedKernel_restrictBoundary cs u b basis hub B x ∅ t
-
- theorem normalized_fullKernel_restrictBoundary (x : Interval cs u b) (t : ℝ) :
-    normalizedKernel cs u b basis hub (restrictBoundary cs u b basis hub B) t x Set.univ =
-    normalizedKernel cs 1 b basis (one_bruhat cs b) B t
-      ((upperInclusion cs u b).vertex x) Set.univ := by
-  simpa only [Set.mem_univ,Set.ofPred_true] using
-    normalizedKernel_restrictBoundary cs u b basis hub B x Set.univ t
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-
-namespace KLInvariance.SchedulePaths
-universe u v
-variable {V : Type u} {E : Type v}
-  (source target : E → V)
-noncomputable section
-
- theorem step_monotone (T : ℝ) (hT : 0 ≤ T) (e : E) :
-    Monotone (step source target T e) := by
-  intro v w hv x
-  dsimp [step]
-  split_ifs
-  · exact add_le_add (hv x) (mul_le_mul_of_nonneg_left (hv _) hT)
-  · simpa using hv x
-
- theorem step_injective (T : ℝ) (e : E) (hne : source e ≠ target e) :
-    Function.Injective (step source target T e) := by
-  intro v w he
-  have ht := congrFun he (target e)
-  simp only [step,hne,ite_false,add_zero] at ht
-  funext x
-  have hx := congrFun he x
-  dsimp [step] at hx
-  split_ifs at hx with h
-  · rw [ht] at hx
-    exact add_right_cancel hx
-  · exact add_right_cancel hx
-
- theorem dispatch_monotone (T : ℝ) (hT : 0 ≤ T) (p : List E) :
-    Monotone (fun v => dispatch source target T v p) := by
-  induction p with
-  | nil => exact monotone_id
-  | cons e p ih => exact (step_monotone source target T hT e).comp ih
-
- theorem dispatch_injective (T : ℝ) (p : List E)
-    (hne : ∀ e ∈ p, source e ≠ target e) :
-    Function.Injective (fun v => dispatch source target T v p) := by
-  induction p with
-  | nil => exact Function.injective_id
-  | cons e p ih =>
-    exact (step_injective source target T e (hne e (by simp))).comp
-      (ih (fun a ha => hne a (by simp [ha])))
-
- theorem dispatch_append' (T : ℝ) (v : V → ℝ) (p q : List E) :
-    dispatch source target T v (p++q) =
-      dispatch source target T (dispatch source target T v q) p := by
-  induction p with
-  | nil => rfl
-  | cons e p ih => simp only [List.cons_append,dispatch,ih]
-
-/-- A concrete state indexed by the remaining chronological edge list. -/
- theorem remaining_state_le (T : ℝ) (hT : 0 ≤ T)
-    (D : List E → V → ℝ) (p : List E)
-    (hstep : ∀ e q, e::q <:+ p → D q ≤ step source target T e (D (e::q))) :
-    D [] ≤ dispatch source target T (D p) p.reverse := by
-  induction p with
-  | nil => exact le_rfl
-  | cons e p ih =>
-    have hp := ih (fun a q hq => hstep a q (hq.trans (List.suffix_cons e p)))
-    have he := hstep e p (List.suffix_refl _)
-    rw [List.reverse_cons,dispatch_append']
-    exact hp.trans (dispatch_monotone source target T hT p.reverse he)
-
-/-- If the endpoint bound is equality, every embedded one-edge Hilbert bound
-is equality. Injectivity must be used together with the one-sided bounds. -/
- theorem remaining_steps_eq (T : ℝ) (hT : 0 ≤ T)
-    (D : List E → V → ℝ) (p : List E)
-    (hne : ∀ e ∈ p, source e ≠ target e)
-    (hstep : ∀ e q, e::q <:+ p → D q ≤ step source target T e (D (e::q)))
-    (hend : D [] = dispatch source target T (D p) p.reverse) :
-    ∀ e q, e::q <:+ p → D q = step source target T e (D (e::q)) := by
-  induction p with
-  | nil =>
-    intro e q hq
-    have := hq.length_le
-    simp at this
-  | cons e p ih =>
-    have hps := fun a q (hq : a::q <:+ p) => hstep a q (hq.trans (List.suffix_cons e p))
-    have hp := remaining_state_le source target T hT D p hps
-    have he := hstep e p (List.suffix_refl _)
-    have hm := dispatch_monotone source target T hT p.reverse he
-    have hend' : D [] = dispatch source target T
-        (step source target T e (D (e::p))) p.reverse := by
-      simpa only [List.reverse_cons,dispatch_append',dispatch] using hend
-    have hh : dispatch source target T (D p) p.reverse =
-        dispatch source target T (step source target T e (D (e::p))) p.reverse := by
-      apply le_antisymm hm
-      exact hend'.symm.le.trans hp
-    have hep : D p = step source target T e (D (e::p)) :=
-      dispatch_injective source target T p.reverse
-        (fun a ha => hne a (by simp only [List.mem_reverse] at ha; simp [ha])) hh
-    have hendpoint : D [] = dispatch source target T (D p) p.reverse := by
-      exact hend'.trans hh.symm
-    intro a q hq
-    rcases List.suffix_cons_iff.mp hq with h | h
-    · cases h
-      exact hep
-    · exact ih (fun a ha => hne a (by simp [ha])) hps hendpoint a q h
-
-end
-end KLInvariance.SchedulePaths
-
-end
-
-
-section
-
-namespace KLInvariance.ReverseFiltration
-universe ua um un up
-variable {A : Type ua} [CommRing A]
-  {M : Type um} [AddCommGroup M] [Module A M]
-  {N : Type un} [AddCommGroup N] [Module A N]
-  {P : Type up} [AddCommGroup P] [Module A P]
-
-/-- A kernel which is exactly a regular scalar multiple of its domain is
-integrally isomorphic to that domain, not just to its fraction-field span. -/
-noncomputable def scalarKernelEquiv (g : M →ₗ[A] N) (a : A)
-    (hk : ∀ v, g v = 0 ↔ ∃ w, v = a • w)
-    (ha : Function.Injective (fun v : M => a • v)) :
-    M ≃ₗ[A] LinearMap.ker g := by
-  let f : M →ₗ[A] LinearMap.ker g :=
-    (LinearMap.lsmul A M a).codRestrict (LinearMap.ker g) (fun v =>
-      (hk (a • v)).mpr ⟨v,rfl⟩)
-  apply LinearEquiv.ofBijective f
-  constructor
-  · intro x y h
-    exact ha (congrArg Subtype.val h)
-  · intro v
-    obtain ⟨w,hw⟩ := (hk v.val).mp v.property
-    exact ⟨w,Subtype.ext hw.symm⟩
-
-
- theorem projective_kernel_of_scalar_presentation
-    [Module.Projective A M] [Module.Projective A P]
-    (f : P →ₗ[A] N) (g : M →ₗ[A] N)
-    (hf : Function.Surjective f) (hg : Function.Surjective g)
-    (a : A) (hk : ∀ v, g v = 0 ↔ ∃ w, v = a • w)
-    (ha : Function.Injective (fun v : M => a • v)) :
-    Module.Projective A (LinearMap.ker f) := by
-  let : Module.Projective A (LinearMap.ker g) :=
-    Module.Projective.of_equiv (scalarKernelEquiv g a hk ha)
-  exact kernel_projective_of_presentation f g hf hg
-
- def imageMapKernelEquiv (S : Submodule A M) (f : M →ₗ[A] N) :
-    ↥(S ⊓ LinearMap.ker f) ≃ₗ[A] LinearMap.ker (Hilbert.imageMap S f) where
-  toFun v := ⟨⟨v.val,v.property.1⟩,Subtype.ext v.property.2⟩
-  invFun v := ⟨v.val.val,v.val.property,congrArg Subtype.val v.property⟩
-  left_inv _ := rfl
-  right_inv _ := rfl
-  map_add' _ _ := rfl
-  map_smul' _ _ := rfl
-
-/-- Scaled equality of two *embedded* images constructs a presentation with
-regular scalar kernel. This is the pure module-algebra form of the reverse
-freeness step, allowing arbitrary image codimension. -/
- theorem inf_kernel_projective_of_image_eq
-    (S : Submodule A M) (T : Submodule A P)
-    [Module.Projective A S] [Module.Projective A T]
-    (f : M →ₗ[A] N) (g : P →ₗ[A] N) (φ : N →ₗ[A] N)
-    (hφ : Function.Injective φ) (himage : S.map f = (T.map g).map φ)
-    (a : A) (hk : ∀ v, g v = 0 ↔ ∃ w, v = a • w)
-    (hsat : ∀ v, a • v ∈ T → v ∈ T)
-    (ha : Function.Injective (fun v : P => a • v)) :
-    Module.Projective A ↥(S ⊓ LinearMap.ker f) := by
-  let j : T →ₗ[A] S.map f := ((φ.comp g).comp T.subtype).codRestrict (S.map f) (by
-    intro v
-    rw [himage]
-    exact Submodule.mem_map.mpr ⟨g v.val,Submodule.mem_map.mpr ⟨v.val,v.property,rfl⟩,rfl⟩)
-  have hj : Function.Surjective j := by
-    intro v
-    have hv : v.val ∈ (T.map g).map φ := himage ▸ v.property
-    obtain ⟨w,hw,hwv⟩ := Submodule.mem_map.mp hv
-    obtain ⟨z,hz,hzw⟩ := Submodule.mem_map.mp hw
-    exact ⟨⟨z,hz⟩,Subtype.ext ((congrArg φ hzw).trans hwv)⟩
-  have hjker (v : T) : j v = 0 ↔ ∃ w : T, v = a • w := by
-    constructor
-    · intro hv
-      have hv' : φ (g v.val) = 0 := congrArg Subtype.val hv
-      have hz : g v.val = 0 := hφ (hv'.trans (map_zero φ).symm)
-      obtain ⟨w,hw⟩ := (hk v.val).mp hz
-      refine ⟨⟨w,hsat w (hw ▸ v.property)⟩,Subtype.ext hw⟩
-    · rintro ⟨w,rfl⟩
-      apply Subtype.ext
-      change φ (g (a • w.val)) = 0
-      rw [(hk (a • w.val)).mpr ⟨w.val,rfl⟩,map_zero]
-  have haT : Function.Injective (fun v : T => a • v) := by
-    intro v w hvw
-    exact Subtype.ext (ha (congrArg Subtype.val hvw))
-  let : Module.Projective A (LinearMap.ker (Hilbert.imageMap S f)) :=
-    projective_kernel_of_scalar_presentation (Hilbert.imageMap S f) j
-      (Hilbert.imageMap_surjective S f) hj a hjker haT
-  exact Module.Projective.of_equiv (imageMapKernelEquiv S f).symm
-
-end KLInvariance.ReverseFiltration
-
-end
-
-
-section
-
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  (hub : BruhatLE cs u b)
-  (B : BoundarySheaf (polynomialGrading_negative ℝ σ)
-    (graph cs u b) (polynomialLabel cs u b basis) ⟨b,hub,bruhat_refl cs b⟩)
-noncomputable section
-
-omit [Fintype σ] in
-/-- The reverse-freeness step uses precisely the already free upper kernel
-and the preceding lower kernel, plus the actual embedded image equality. -/
- theorem partial_kernel_projective_of_image_eq (e : Edge cs u b)
-    (F : Set (Sheaf.Outgoing (G := graph cs u b) ((graph cs u b).source e)))
-    (H : Set (Sheaf.Outgoing (G := graph cs u b) ((graph cs u b).target e)))
-    [Module.Projective (Coefficient (σ := σ))
-      (B.sheaf.upwardSubmodule ((graph cs u b).source e) F)]
-    [Module.Projective (Coefficient (σ := σ))
-      (B.sheaf.upwardSubmodule ((graph cs u b).target e) H)]
-    (heq : B.sheaf.lowerImage e F = (B.sheaf.upperImage e H).map
-      (edgeFactorMap cs basis u b hub B e)) :
-    Module.Projective (Coefficient (σ := σ))
-      (B.sheaf.upwardSubmodule ((graph cs u b).source e) (insert ⟨e,rfl⟩ F)) := by
-  rw [B.sheaf.upwardKernel_insert]
-  apply ReverseFiltration.inf_kernel_projective_of_image_eq
-    (B.sheaf.upwardSubmodule ((graph cs u b).source e) F)
-    (B.sheaf.upwardSubmodule ((graph cs u b).target e) H)
-    (B.sheaf.lower e).map (B.sheaf.upper e).map (edgeFactorMap cs basis u b hub B e)
-    (edgeFactorLift_regular cs basis u b hub B e) heq
-    (polynomialLabel cs u b basis e) (B.quotient e).2
-  · apply B.sheaf.upwardSubmodule_saturated
-    intro f _
-    exact incoming_label_regular_outgoing cs u b basis hub B e f.val f.property
-  · exact label_regular_stalk cs u b basis hub B e _
-
- theorem partial_kernel_free_of_image_eq (e : Edge cs u b)
-    (F : Set (Sheaf.Outgoing (G := graph cs u b) ((graph cs u b).source e)))
-    (H : Set (Sheaf.Outgoing (G := graph cs u b) ((graph cs u b).target e)))
-    [Module.Free (Coefficient (σ := σ))
-      (B.sheaf.upwardSubmodule ((graph cs u b).source e) F)]
-    [Module.Free (Coefficient (σ := σ))
-      (B.sheaf.upwardSubmodule ((graph cs u b).target e) H)]
-    (heq : B.sheaf.lowerImage e F = (B.sheaf.upperImage e H).map
-      (edgeFactorMap cs basis u b hub B e)) :
-    Module.Free (Coefficient (σ := σ))
-      (B.sheaf.upwardSubmodule ((graph cs u b).source e) (insert ⟨e,rfl⟩ F)) := by
-  let D := B.sheaf.upwardKernel ((graph cs u b).source e) (insert ⟨e,rfl⟩ F)
-  let : Module.Projective (Coefficient (σ := σ)) D :=
-    partial_kernel_projective_of_image_eq cs u b basis hub B e F H heq
-  obtain ⟨ι,_,bs,_⟩ := exists_homogeneous_basis_polynomial (k := ℝ) (σ := σ)
-    D.piece 0 D.nonneg
-  exact Module.Free.of_basis bs
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph SchedulePaths
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  (hub : BruhatLE cs u b)
-  (B : BoundarySheaf (polynomialGrading_negative ℝ σ)
-    (graph cs u b) (polynomialLabel cs u b basis) ⟨b,hub,bruhat_refl cs b⟩)
-noncomputable section
-
- def remainingConditions (p : List (Edge cs u b)) (x : Interval cs u b) :
-    Set (Sheaf.Outgoing (G := graph cs u b) x) := {f | f.val ∈ p}
-
-omit [Fintype I] in
- theorem remainingConditions_nil (x : Interval cs u b) :
-    remainingConditions cs u b [] x = ∅ := by ext f; simp [remainingConditions]
-
-omit [Fintype I] in
- theorem remainingConditions_cons_source (e : Edge cs u b) (p : List (Edge cs u b)) :
-    remainingConditions cs u b (e::p) (source cs u b e) =
-      insert ⟨e,rfl⟩ (remainingConditions cs u b p (source cs u b e)) := by
-  ext f
-  simp only [remainingConditions,Set.mem_ofPred_eq,List.mem_cons,Set.mem_insert_iff]
-  rw [Subtype.ext_iff]
-
-omit [Fintype I] in
- theorem remainingConditions_cons_other (e : Edge cs u b) (p : List (Edge cs u b))
-    (x : Interval cs u b) (hne : source cs u b e ≠ x) :
-    remainingConditions cs u b (e::p) x = remainingConditions cs u b p x := by
-  ext f
-  have hf : f.val ≠ e := by
-    intro he
-    exact hne (he ▸ f.property)
-  simp [remainingConditions,hf]
-
- def scheduleState (t : ℝ) (p : List (Edge cs u b)) : Interval cs u b → ℝ :=
-  fun x => normalizedKernel cs u b basis hub B t x (remainingConditions cs u b p x)
-
- theorem scheduleState_step (e : Edge cs u b) (p : List (Edge cs u b))
-    (hinc : B.sheaf.lowerImage e (remainingConditions cs u b p ((graph cs u b).source e)) ≤
-      (B.sheaf.upperImage e (remainingConditions cs u b p ((graph cs u b).target e))).map
-        (edgeFactorMap cs basis u b hub B e))
-    {t : ℝ} (ht : 0 < t) (ht1 : t < 1) :
-    scheduleState cs u b basis hub B t p ≤
-      step (source cs u b) (target cs u b) (t⁻¹-t) e
-        (scheduleState cs u b basis hub B t (e::p)) := by
-  classical
-  intro x
-  by_cases hx : (graph cs u b).source e = x
-  · subst x
-    have h := normalized_edge_increment_le cs u b basis hub B e
-      (remainingConditions cs u b p ((graph cs u b).source e))
-      (remainingConditions cs u b p ((graph cs u b).target e)) hinc ht ht1
-    have hne : source cs u b e ≠ target cs u b e :=
-      ne_of_lt ((graph cs u b).increasing e)
-    change normalizedKernel cs u b basis hub B t (source cs u b e)
-        (remainingConditions cs u b p (source cs u b e)) ≤
-      normalizedKernel cs u b basis hub B t (source cs u b e)
-        (remainingConditions cs u b (e::p) (source cs u b e)) +
-      if source cs u b e = source cs u b e then
-        (t⁻¹-t) * normalizedKernel cs u b basis hub B t (target cs u b e)
-          (remainingConditions cs u b (e::p) (target cs u b e)) else 0
-    rw [ite_eq_left rfl,remainingConditions_cons_source,
-      remainingConditions_cons_other cs u b e p (target cs u b e) hne]
-    exact h
-  · simp only [scheduleState,step]
-    have hx' : source cs u b e ≠ x := hx
-    rw [ite_eq_right hx',add_zero,remainingConditions_cons_other cs u b e p x hx]
-
- theorem scheduleState_le (p : List (Edge cs u b))
-    (hinc : ∀ e q, e::q <:+ p →
-      B.sheaf.lowerImage e (remainingConditions cs u b q ((graph cs u b).source e)) ≤
-      (B.sheaf.upperImage e (remainingConditions cs u b q ((graph cs u b).target e))).map
-        (edgeFactorMap cs basis u b hub B e))
-    {t : ℝ} (ht : 0 < t) (ht1 : t < 1) :
-    scheduleState cs u b basis hub B t [] ≤
-      dispatch (source cs u b) (target cs u b) (t⁻¹-t)
-        (scheduleState cs u b basis hub B t p) p.reverse := by
-  exact remaining_state_le _ _ _ (Comparison.transferCoefficient_pos ht ht1).le _ p
-    (fun e q hq => scheduleState_step cs u b basis hub B e q (hinc e q hq) ht ht1)
-
-
- theorem edgeImages_eq_of_scheduleState_eq (p : List (Edge cs u b))
-    (hinc : ∀ e q, e::q <:+ p →
-      B.sheaf.lowerImage e (remainingConditions cs u b q ((graph cs u b).source e)) ≤
-      (B.sheaf.upperImage e (remainingConditions cs u b q ((graph cs u b).target e))).map
-        (edgeFactorMap cs basis u b hub B e))
-    {t : ℝ} (ht : 0 < t) (ht1 : t < 1)
-    (hend : scheduleState cs u b basis hub B t [] =
-      dispatch (source cs u b) (target cs u b) (t⁻¹-t)
-        (scheduleState cs u b basis hub B t p) p.reverse) :
-    ∀ e q, e::q <:+ p →
-      B.sheaf.lowerImage e (remainingConditions cs u b q ((graph cs u b).source e)) =
-      (B.sheaf.upperImage e (remainingConditions cs u b q ((graph cs u b).target e))).map
-        (edgeFactorMap cs basis u b hub B e) := by
-  classical
-  have heq := remaining_steps_eq (source cs u b) (target cs u b) (t⁻¹-t)
-    (Comparison.transferCoefficient_pos ht ht1).le
-    (scheduleState cs u b basis hub B t) p
-    (fun e _ => ne_of_lt ((graph cs u b).increasing e))
-    (fun e q hq => scheduleState_step cs u b basis hub B e q (hinc e q hq) ht ht1) hend
-  intro e q hq
-  apply (normalized_edge_increment_eq_iff cs u b basis hub B e
-    (remainingConditions cs u b q ((graph cs u b).source e))
-    (remainingConditions cs u b q ((graph cs u b).target e)) (hinc e q hq) ht ht1).mp
-  have h := congrFun (heq e q hq) ((graph cs u b).source e)
-  have hne : source cs u b e ≠ target cs u b e :=
-    ne_of_lt ((graph cs u b).increasing e)
-  change normalizedKernel cs u b basis hub B t (source cs u b e)
-      (remainingConditions cs u b q (source cs u b e)) =
-    normalizedKernel cs u b basis hub B t (source cs u b e)
-      (remainingConditions cs u b (e::q) (source cs u b e)) +
-    (if source cs u b e = source cs u b e then
-      (t⁻¹-t) * normalizedKernel cs u b basis hub B t (target cs u b e)
-        (remainingConditions cs u b (e::q) (target cs u b e)) else 0) at h
-  rw [ite_eq_left rfl,remainingConditions_cons_source,
-    remainingConditions_cons_other cs u b e q (target cs u b e) hne] at h
-  exact h
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-/-! Exact identification of chronological suffix conditions with the
-strictly-later root conditions used in the height-one edge-image proof.
-Equal-label edges may be ordered arbitrarily and cause no extra incident
-condition. -/
-namespace KLInvariance.BruhatGraph
-open TitsSpace SchedulePaths MomentGraph
-universe u v u' v'
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
-  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
-  {u b : W} {u' b' : W'}
-  (φ : Interval cs u b ≃o Interval cs' u' b')
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
- def transportedSlope (e : Edge cs u b) : ℝ :=
-  geometricSlope (AdaptedDyer.rootLabel cs' (transportedEdge φ e))
-
- def chronologicalSchedule : List (Edge cs u b) := (transportedSchedule φ).reverse
-
-omit [Fintype I] in
- theorem transportedScore_eq (e : Edge cs u b) : transportedScore φ e =
-    toLex (transportedSlope φ e,transportedSlope φ e) := by
-  unfold transportedScore lexEdgeScore lexRootScore transportedSlope
-  rw [geometricFunctional_normalizedRoot]
-
-omit [Fintype I] in
- theorem chronologicalSchedule_pairwise : (chronologicalSchedule φ).Pairwise
-    (fun e f => transportedSlope φ e ≤ transportedSlope φ f) := by
-  rw [chronologicalSchedule,List.pairwise_reverse]
-  apply (transportedSchedule_pairwise φ).imp
-  intro e f h
-  have hh := Prod.Lex.monotone_fst _ _ h
-  rw [transportedScore_eq,transportedScore_eq] at hh
-  exact hh
-
-omit [Fintype I] in
- theorem chronologicalSchedule_nodup : (chronologicalSchedule φ).Nodup :=
-  List.nodup_reverse.mpr (transportedSchedule_nodup φ)
-
-omit [Fintype I] in
- theorem chronologicalSchedule_full (e : Edge cs u b) : e ∈ chronologicalSchedule φ :=
-  List.mem_reverse.mpr (transportedSchedule_full φ e)
-
-omit [Fintype I] in
- theorem transportedSlope_source_unique (e f : Edge cs u b)
-    (hs : source cs u b e=source cs u b f)
-    (h : transportedSlope φ e=transportedSlope φ f) : e=f := by
-  have hr := congrArg rootRef (geometricSlope_injective h)
-  rw [AdaptedDyer.rootRef_rootLabel,AdaptedDyer.rootRef_rootLabel] at hr
-  change (φ (target cs u b e)).val * (φ (source cs u b e)).val⁻¹ =
-    (φ (target cs u b f)).val * (φ (source cs u b f)).val⁻¹ at hr
-  rw [hs] at hr
-  apply Subtype.ext
-  exact Prod.ext hs (φ.injective (Subtype.ext (mul_right_cancel hr)))
-
-omit [Fintype I] in
- theorem transportedSlope_path_ne (e f : Edge cs u b)
-    (hs : target cs u b e=source cs u b f) :
-    transportedSlope φ e ≠ transportedSlope φ f := by
-  intro h
-  exact AdaptedDyer.rootLabel_ne_of_join cs' (transportedEdge φ e) (transportedEdge φ f)
-    (congrArg (fun x => (φ x).val) hs) (geometricSlope_injective h)
-
-omit [Fintype I] in
- theorem suffix_source_iff {e f : Edge cs u b} {q : List (Edge cs u b)}
-    (hq : e::q <:+ chronologicalSchedule φ) (hf : source cs u b f=source cs u b e) :
-    f ∈ q ↔ transportedSlope φ e < transportedSlope φ f :=
-  mem_suffix_iff_score_lt_injective (transportedSlope φ) (chronologicalSchedule_pairwise φ)
-    (chronologicalSchedule_nodup φ) hq (chronologicalSchedule_full φ f)
-    (transportedSlope_source_unique φ e f hf.symm)
-
-omit [Fintype I] in
- theorem suffix_target_iff {e f : Edge cs u b} {q : List (Edge cs u b)}
-    (hq : e::q <:+ chronologicalSchedule φ) (hf : source cs u b f=target cs u b e) :
-    f ∈ q ↔ transportedSlope φ e < transportedSlope φ f :=
-  mem_suffix_iff_score_lt (transportedSlope φ) (chronologicalSchedule_pairwise φ)
-    hq (chronologicalSchedule_full φ f) (transportedSlope_path_ne φ e f hf.symm)
-
-omit [Fintype I] in
- theorem ambientLater_upperInclusion (e f : Edge cs u b) :
-    (upperInclusion cs u b).edge f ∈ ambientLater φ
-      (AdaptedDyer.rootLabel cs' (transportedEdge φ e)) ↔
-    transportedSlope φ e < transportedSlope φ f := by
-  have hs : BruhatStep cs' (φ (source cs u b f)).val (φ (target cs u b f)).val :=
-    (transportedEdge φ f).property
-  have hiff : ambientAfterIndicator (φ (source cs u b f)).val
-      (AdaptedDyer.rootLabel cs' (transportedEdge φ e))
-      (φ (target cs u b f)).val = 1 ↔ transportedSlope φ e < transportedSlope φ f := by
-    simp only [ambientAfterIndicator,dite_eq_left hs]
-    change (if transportedSlope φ e < transportedSlope φ f then 1 else 0)=1 ↔ _
-    split_ifs <;> simp_all
-  constructor
-  · rintro ⟨_,h⟩
-    exact hiff.mp h
-  · intro h
-    exact ⟨(source cs u b f).property.1,hiff.mpr h⟩
-
-omit [Fintype I] in
- theorem remainingConditions_source_ambient {e : Edge cs u b} {q : List (Edge cs u b)}
-    (hq : e::q <:+ chronologicalSchedule φ) :
-    remainingConditions cs u b q (source cs u b e) =
-      {f | (upperInclusion cs u b).edge f.val ∈ ambientLater φ
-        (AdaptedDyer.rootLabel cs' (transportedEdge φ e))} := by
-  ext f
-  exact (suffix_source_iff φ hq f.property).trans (ambientLater_upperInclusion φ e f.val).symm
-
-omit [Fintype I] in
- theorem remainingConditions_target_ambient {e : Edge cs u b} {q : List (Edge cs u b)}
-    (hq : e::q <:+ chronologicalSchedule φ) :
-    remainingConditions cs u b q (target cs u b e) =
-      {f | (upperInclusion cs u b).edge f.val ∈ ambientLater φ
-        (AdaptedDyer.rootLabel cs' (transportedEdge φ e))} := by
-  ext f
-  exact (suffix_target_iff φ hq f.property).trans (ambientLater_upperInclusion φ e f.val).symm
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-
-end
-
-end OAI
+import OAI.RepresentationTheory.KazhdanLusztig.EdgeLocalization
+
+namespace OAI
+
+/-!
+Positive transfer inequalities, strict Hilbert-series detection including higher-codimension support, and reverse-filtration projective-to-free arguments.
+-/
+
+section
+
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  (hub : BruhatLE cs u b)
+  (B : BoundarySheaf (polynomialGrading_negative ℝ σ)
+    (graph cs u b) (polynomialLabel cs u b basis) ⟨b,hub,bruhat_refl cs b⟩)
+  [Fintype (Edge cs u b)] [DecidableEq (Interval cs u b)]
+noncomputable section
+
+omit [Fintype σ] in
+ theorem costalkEquivDual_pairing (d : SectionDuality cs u b basis hub B)
+    (x : Interval cs u b) (k : LinearMap.ker (B.sheaf.forget.stalkMap x))
+    (n : B.sheaf.forget.sections Set.univ) :
+    costalkEquivDual cs u b basis hub B d x k (n.val x)=
+      d.pairing (kernelEquivSingleSupport cs u b basis hub B x k).val n := by
+  let _ (y : Interval cs u b) : Module.Free (Coefficient (σ := σ))
+      (B.sheaf.forget.vertex y) := B.free y
+  let _ (y : Interval cs u b) : Module.IsTorsionFree (Coefficient (σ := σ))
+      (B.sheaf.forget.vertex y) := inferInstance
+  exact IntegralSupport.supportEquivDual_pairing (B.sheaf.forget.sections Set.univ)
+    (Sheaf.labelProduct (polynomialLabel cs u b basis))
+    (B.sheaf.forget.scaled_single_section
+      (B.sheaf.forget.labelProduct_annihilates (polynomialLabel cs u b basis)
+        (GradedSheaf.UpperQuotient.annihilates B.sheaf B.quotient)))
+    (Sheaf.labelProduct_ne_zero _ (polynomialLabel_ne_zero cs u b basis))
+    d.pairing d.balanced x (B.sheaf.forget.projection_surjective B.generated x)
+    (kernelEquivSingleSupport cs u b basis hub B x k) n
+
+omit [Fintype σ] in
+ theorem costalkEquivDual_symmetric (d : SectionDuality cs u b basis hub B)
+    (hsym : ∀ m n, d.pairing m n=d.pairing n m)
+    (x : Interval cs u b) (k l : LinearMap.ker (B.sheaf.forget.stalkMap x)) :
+    costalkEquivDual cs u b basis hub B d x k l.val=
+      costalkEquivDual cs u b basis hub B d x l k.val := by
+  let e := costalkEquivDual cs u b basis hub B d x
+  let g := Sheaf.incomingProduct (G := graph cs u b) (polynomialLabel cs u b basis) x
+  let f := kernelEquivSingleSupport cs u b basis hub B x
+  have he (k l : LinearMap.ker (B.sheaf.forget.stalkMap x)) :
+      g * (e k) l.val=d.pairing (f k).val (f l).val := calc
+    g * (e k) l.val = (e k) (g • l.val) := (map_smul (e k) g l.val).symm
+    _ = (e k) ((f l).val.val x) := congrArg (e k)
+      (kernelEquivSingleSupport_apply cs u b basis hub B x l).symm
+    _ = _ := costalkEquivDual_pairing cs u b basis hub B d x k (f l).val
+  exact mul_left_cancel₀ (Sheaf.incomingProduct_ne_zero
+    (polynomialLabel cs u b basis) (polynomialLabel_prime cs u b basis) x)
+    ((he k l).trans ((hsym _ _).trans (he l k).symm))
+
+omit [Fintype σ] in
+ theorem costalkEquivDual_inverse_symmetric (d : SectionDuality cs u b basis hub B)
+    (hsym : ∀ m n, d.pairing m n=d.pairing n m)
+    (x : Interval cs u b) (lam mu : Dual (Coefficient (σ := σ)) (B.sheaf.vertex x)) :
+    lam ((costalkEquivDual cs u b basis hub B d x).symm mu).val=
+      mu ((costalkEquivDual cs u b basis hub B d x).symm lam).val := by
+  have h := costalkEquivDual_symmetric cs u b basis hub B d hsym x
+    ((costalkEquivDual cs u b basis hub B d x).symm lam)
+    ((costalkEquivDual cs u b basis hub B d x).symm mu)
+  simpa only [LinearEquiv.apply_symm_apply] using h
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+/-! The genuine BMP local intersection map. All freeness, kernel-minimality and
+inverse grading assertions here are now unconditional consequences of the
+constructed integral duality. The nonvanishing/Hodge input is deliberately
+NOT asserted; the last theorem records precisely how it would force the
+strict KL halfdegree bound on ACTUAL homogeneous stalk generators. -/
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
+universe u
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W) (b : W)
+  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  [Fintype (Edge cs 1 b)] [Fintype (Interval cs 1 b)] [DecidableEq (Interval cs 1 b)]
+noncomputable section
+
+ abbrev ActualBoundary := boundarySheaf cs 1 b basis (one_bruhat cs b)
+
+ def actualSymmetricPairing : GradedSheaf.SymmetricPerfectPairing
+    (ActualBoundary cs b basis).sheaf (cs.length b : ℤ) :=
+  (boundarySheaf_symmetricPairing cs b basis).some
+
+ def actualSectionDuality : SectionDuality cs 1 b basis (one_bruhat cs b)
+    (ActualBoundary cs b basis) := by
+  let p := actualSymmetricPairing cs b basis
+  let B := ActualBoundary cs b basis
+  refine ⟨p.pairing,?_⟩
+  intro x m n
+  exact p.toPerfectPairing.cut_balanced
+    (Sheaf.labelProduct (polynomialLabel cs 1 b basis))
+    (Sheaf.labelProduct_ne_zero _ (polynomialLabel_ne_zero cs 1 b basis))
+    (B.sheaf.forget.scaled_single_section
+      (B.sheaf.forget.labelProduct_annihilates (polynomialLabel cs 1 b basis)
+        (GradedSheaf.UpperQuotient.annihilates B.sheaf B.quotient))) x m n
+
+ theorem actualSectionDuality_symmetric
+    (m n : (ActualBoundary cs b basis).sheaf.sections Set.univ) :
+    (actualSectionDuality cs b basis).pairing m n=
+      (actualSectionDuality cs b basis).pairing n m :=
+  (actualSymmetricPairing cs b basis).symmetric m n
+
+ theorem actualSectionDuality_homogeneous (i j : ℤ)
+    (m n : (ActualBoundary cs b basis).sheaf.sections Set.univ)
+    (hm : m ∈ ((ActualBoundary cs b basis).sheaf.sections Set.univ).piece i)
+    (hn : n ∈ ((ActualBoundary cs b basis).sheaf.sections Set.univ).piece j) :
+    (actualSectionDuality cs b basis).pairing m n ∈
+      polynomialGrading ℝ σ (i+j-(cs.length b : ℤ)) :=
+  (actualSymmetricPairing cs b basis).homogeneous i j m n
+    (fun x => hm x (Set.mem_univ x)) (fun x => hn x (Set.mem_univ x))
+
+omit [Fintype (Interval cs 1 b)] in
+ theorem costalkFreeObligation : CostalkFreeObligation cs 1 b basis (one_bruhat cs b) :=
+  costalk_free_of_duality cs 1 b basis (one_bruhat cs b) _ (actualSectionDuality cs b basis)
+
+ def localIntersection (x : Interval cs 1 b) :
+    Dual (Coefficient (σ := σ)) ((ActualBoundary cs b basis).sheaf.vertex x)
+      →ₗ[Coefficient (σ := σ)] (ActualBoundary cs b basis).sheaf.vertex x :=
+  (LinearMap.ker ((ActualBoundary cs b basis).sheaf.forget.stalkMap x)).subtype.comp
+    (costalkEquivDual cs 1 b basis (one_bruhat cs b) _
+      (actualSectionDuality cs b basis) x).symm.toLinearMap
+
+omit [Fintype (Interval cs 1 b)] in
+ theorem localIntersection_injective (x : Interval cs 1 b) :
+    Function.Injective (localIntersection cs b basis x) :=
+  Subtype.val_injective.comp (costalkEquivDual cs 1 b basis (one_bruhat cs b) _
+    (actualSectionDuality cs b basis) x).symm.injective
+
+omit [Fintype (Interval cs 1 b)] in
+ theorem localIntersection_range (x : Interval cs 1 b) :
+    LinearMap.range (localIntersection cs b basis x)=
+      LinearMap.ker ((ActualBoundary cs b basis).sheaf.forget.stalkMap x) := by
+  apply le_antisymm
+  · rintro v ⟨lam,rfl⟩
+    exact ((costalkEquivDual cs 1 b basis (one_bruhat cs b) _
+      (actualSectionDuality cs b basis) x).symm lam).property
+  · intro v hv
+    refine ⟨costalkEquivDual cs 1 b basis (one_bruhat cs b) _
+      (actualSectionDuality cs b basis) x ⟨v,hv⟩,?_⟩
+    exact congrArg Subtype.val ((costalkEquivDual cs 1 b basis (one_bruhat cs b) _
+      (actualSectionDuality cs b basis) x).symm_apply_apply ⟨v,hv⟩)
+
+omit [Fintype (Interval cs 1 b)] in
+ theorem localIntersection_minimal (x : Interval cs 1 b) (hx : x.val≠b) :
+    LinearMap.range (localIntersection cs b basis x) ≤ positiveIdeal (polynomialGrading ℝ σ) •
+      (⊤ : Submodule (Coefficient (σ := σ)) ((ActualBoundary cs b basis).sheaf.vertex x)) := by
+  rw [localIntersection_range]
+  apply ((ActualBoundary cs b basis).sheaf.kernelMinimal_iff x).mp
+  exact (ActualBoundary cs b basis).minimal x (fun h => hx (congrArg Subtype.val h))
+
+ theorem localIntersection_homogeneous (x : Interval cs 1 b) (r : ℤ)
+    (lam : Dual (Coefficient (σ := σ)) ((ActualBoundary cs b basis).sheaf.vertex x))
+    (hlam : ∀ j (v : (ActualBoundary cs b basis).sheaf.vertex x),
+      v ∈ ((ActualBoundary cs b basis).sheaf.vertex x).piece j →
+      lam v ∈ polynomialGrading ℝ σ (j+r)) :
+    localIntersection cs b basis x lam ∈ ((ActualBoundary cs b basis).sheaf.vertex x).piece
+      (r+(cs.length b : ℤ)-cs.length x.val) := by
+  have h := costalkEquivDual_inverse_homogeneous cs 1 b basis (one_bruhat cs b) _
+    (actualSectionDuality cs b basis) (cs.length b) (actualSectionDuality_homogeneous cs b basis)
+    x r lam hlam
+  rw [incoming_card] at h
+  exact h
+
+ theorem localIntersection_symmetric (x : Interval cs 1 b)
+    (lam mu : Dual (Coefficient (σ := σ)) ((ActualBoundary cs b basis).sheaf.vertex x)) :
+    lam (localIntersection cs b basis x mu)=mu (localIntersection cs b basis x lam) :=
+  costalkEquivDual_inverse_symmetric cs 1 b basis (one_bruhat cs b) _
+    (actualSectionDuality cs b basis) (actualSectionDuality_symmetric cs b basis) x lam mu
+
+ theorem localIntersection_diagonal_strict_degree (x : Interval cs 1 b) (hx : x.val≠b)
+    (c : NonnegativeBasis (σ := σ) ((ActualBoundary cs b basis).sheaf.vertex x).piece)
+    [DecidableEq c.index] (i : c.index)
+    (hdiag : c.basis.dualBasis i (localIntersection cs b basis x (c.basis.dualBasis i))≠0) :
+    2*c.degree i<rankDifference cs x.val b := by
+  classical
+  have hg (j : c.index) := localIntersection_homogeneous cs b basis x (-(c.degree j : ℤ))
+    (c.basis.dualBasis j) (dualBasis_homogeneous (polynomialGrading ℝ σ)
+      ((ActualBoundary cs b basis).sheaf.vertex x).piece c.basis
+      (fun i => (c.degree i : ℤ)) c.homogeneous j)
+  have hh := local_diagonal_strict_degree (polynomialGrading ℝ σ)
+    (polynomialGrading_negative ℝ σ) ((ActualBoundary cs b basis).sheaf.vertex x).piece
+    c.basis (fun i => (c.degree i : ℤ)) c.homogeneous
+    ((cs.length b : ℤ)-cs.length x.val) (localIntersection cs b basis x)
+    (fun j => by convert hg j using 1; congr 1; omega)
+    (localIntersection_minimal cs b basis x hx) i hdiag
+  have hlen := length_le_of_bruhat cs x.property.2
+  dsimp [rankDifference]
+  omega
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+/-! Strict later-label membership in a finite nondecreasing schedule,
+with arbitrary positions for equal labels. -/
+namespace KLInvariance.SchedulePaths
+universe u v
+variable {E : Type u} {Λ : Type v} [LinearOrder Λ]
+
+ theorem mem_suffix_iff_score_lt (score : E → Λ) {p q : List E} {e f : E}
+    (hs : p.Pairwise (fun a b => score a ≤ score b))
+    (hq : e::q <:+ p) (hf : f ∈ p) (hne : score e ≠ score f) :
+    f ∈ q ↔ score e < score f := by
+  obtain ⟨a,rfl⟩ := hq
+  have h := List.pairwise_append.mp hs
+  constructor
+  · intro hfq
+    exact lt_of_le_of_ne ((List.pairwise_cons.mp h.2.1).1 f hfq) hne
+  · intro hlt
+    rcases List.mem_append.mp hf with hfa | hfr
+    · exact False.elim (not_le_of_gt hlt (h.2.2 f hfa e List.mem_cons_self))
+    · rcases List.mem_cons.mp hfr with rfl | hfr
+      · exact False.elim (hne rfl)
+      · exact hfr
+
+ theorem mem_suffix_iff_score_lt_injective (score : E → Λ) {p q : List E} {e f : E}
+    (hs : p.Pairwise (fun a b => score a ≤ score b)) (hn : p.Nodup)
+    (hq : e::q <:+ p) (hf : f ∈ p) (heq : score e = score f → e=f) :
+    f ∈ q ↔ score e < score f := by
+  by_cases hne : score e ≠ score f
+  · exact mem_suffix_iff_score_lt score hs hq hf hne
+  · have hef := heq (not_ne_iff.mp hne)
+    subst f
+    have hx := (List.nodup_cons.mp (hn.sublist hq.sublist)).1
+    simp only [hx,lt_self_iff_false]
+
+end KLInvariance.SchedulePaths
+
+end
+
+
+section
+
+namespace KLInvariance.Hilbert
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um un
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {M : Type um} [AddCommGroup M] [Module k M] [Module A M] [IsScalarTower k A M]
+  {N : Type un} [AddCommGroup N] [Module k N] [Module A N] [IsScalarTower k A N]
+
+omit [Algebra k A] [IsScalarTower k A M] [IsScalarTower k A N] in
+ theorem shifted_map_homogeneous
+    (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
+    [DirectSum.Decomposition 𝓜] [DirectSum.Decomposition 𝓝]
+    (S : Submodule A M) (hS : ∀ n m, m ∈ S → component 𝓜 n m ∈ S)
+    (f : M →ₗ[A] N) (d : ℤ) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 (n+d)) :
+    ∀ n v, v ∈ S.map f → component 𝓝 n v ∈ S.map f := by
+  intro n v hv
+  obtain ⟨w,hw,rfl⟩ := Submodule.mem_map.mp hv
+  exact Submodule.mem_map.mpr ⟨component 𝓜 (n-d) w,hS (n-d) w hw,
+    (map_component 𝓜 𝓝 f d hf n w).symm⟩
+
+ def shiftedImagePieceMap (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
+    (S : Submodule A M) (f : M →ₗ[A] N) (d : ℤ)
+    (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 (n+d)) (n : ℤ) :
+    subPiece 𝓜 S (n-d) →ₗ[k] subPiece 𝓝 (S.map f) n where
+  toFun v := ⟨⟨f v.val.val,Submodule.mem_map.mpr ⟨_,v.val.property,rfl⟩⟩,
+    by
+      change f v.val.val ∈ 𝓝 n
+      simpa using hf (n-d) v.val.val v.property⟩
+  map_add' _ _ := by ext; exact map_add f _ _
+  map_smul' c v := by ext; exact (f.restrictScalars k).map_smul c v.val.val
+
+ theorem shiftedImagePieceMap_surjective
+    (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
+    [DirectSum.Decomposition 𝓜] [DirectSum.Decomposition 𝓝]
+    (S : Submodule A M) (hS : ∀ n m, m ∈ S → component 𝓜 n m ∈ S)
+    (f : M →ₗ[A] N) (d : ℤ) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 (n+d)) (n : ℤ) :
+    Function.Surjective (shiftedImagePieceMap 𝓜 𝓝 S f d hf n) := by
+  rintro ⟨⟨m,hm⟩,hgrad⟩
+  obtain ⟨v,hv,rfl⟩ := Submodule.mem_map.mp hm
+  refine ⟨⟨⟨component 𝓜 (n-d) v,hS (n-d) v hv⟩,
+    (DirectSum.decompose 𝓜 v (n-d)).property⟩,?_⟩
+  apply Subtype.ext
+  apply Subtype.ext
+  change f (component 𝓜 (n-d) v) = f v
+  rw [← map_component 𝓜 𝓝 f d hf n v]
+  exact DirectSum.decompose_of_mem_same 𝓝 hgrad
+
+noncomputable def shiftedImagePieceEquiv
+    (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
+    [DirectSum.Decomposition 𝓜] [DirectSum.Decomposition 𝓝]
+    (S : Submodule A M) (hS : ∀ n m, m ∈ S → component 𝓜 n m ∈ S)
+    (f : M →ₗ[A] N) (d : ℤ) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 (n+d))
+    (hinj : Function.Injective f) (n : ℤ) :
+    subPiece 𝓜 S (n-d) ≃ₗ[k] subPiece 𝓝 (S.map f) n :=
+  LinearEquiv.ofBijective (shiftedImagePieceMap 𝓜 𝓝 S f d hf n) ⟨by
+    intro v w h
+    apply Subtype.ext
+    apply Subtype.ext
+    apply hinj
+    exact congrArg (fun z => z.val.val) h,
+    shiftedImagePieceMap_surjective 𝓜 𝓝 S hS f d hf n⟩
+
+/-- Hilbert series of the actual embedded image of a homogeneous injection. -/
+ theorem shifted_image_value
+    (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
+    [DirectSum.Decomposition 𝓜] [DirectSum.Decomposition 𝓝]
+    (S : Submodule A M) (hS : ∀ n m, m ∈ S → component 𝓜 n m ∈ S)
+    (f : M →ₗ[A] N) (d : ℤ) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 (n+d))
+    (hinj : Function.Injective f) {q : ℝ} (hq : q ≠ 0)
+    (hs : Summable (term 𝓜 (S.restrictScalars k) q)) :
+    value 𝓝 ((S.map f).restrictScalars k) q =
+      q^d * value 𝓜 (S.restrictScalars k) q := by
+  have ha := hs.hasSum.mul_left (q^d)
+  have hb : HasSum
+      (fun n : ℤ => (Module.finrank k (piece 𝓜 (S.restrictScalars k) (n-d)) : ℝ)*q^n)
+      (q^d * value 𝓜 (S.restrictScalars k) q) := by
+    apply (Equiv.addRight d).hasSum_iff.mp
+    apply ha.congr_fun
+    intro n
+    change (Module.finrank k (piece 𝓜 (S.restrictScalars k) (n+d-d)) : ℝ)*q^(n+d) =
+      q^d * ((Module.finrank k (piece 𝓜 (S.restrictScalars k) n) : ℝ)*q^n)
+    rw [add_sub_cancel_right,zpow_add₀ hq]
+    ring
+  apply HasSum.tsum_eq
+  apply hb.congr_fun
+  intro n
+  unfold term
+  rw [← (subPieceEquiv 𝓝 (S.map f) n).finrank_eq,
+    ← (shiftedImagePieceEquiv 𝓜 𝓝 S hS f d hf hinj n).finrank_eq,
+    (subPieceEquiv 𝓜 S (n-d)).finrank_eq]
+
+end KLInvariance.Hilbert
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded _root_.OAI.KLInvariance.Hilbert
+universe us um
+variable {σ : Type us} [Fintype σ]
+  {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
+  (B : GradedSheaf (𝓐 := polynomialGrading ℝ σ) G)
+noncomputable section
+
+ def kernelHilbert (q : ℝ) (x : V) (F : Set (Sheaf.Outgoing (G := G) x)) : ℝ :=
+  value (B.vertex x).piece ((B.upwardSubmodule x F).restrictScalars ℝ) q
+
+ def lowerImage (e : E) (F : Set (Sheaf.Outgoing (G := G) (G.source e))) :
+    Submodule (MvPolynomial σ ℝ) (B.edge e) :=
+  (B.upwardSubmodule (G.source e) F).map (B.lower e).map
+
+ def upperImage (e : E) (F : Set (Sheaf.Outgoing (G := G) (G.target e))) :
+    Submodule (MvPolynomial σ ℝ) (B.edge e) :=
+  (B.upwardSubmodule (G.target e) F).map (B.upper e).map
+
+omit [Fintype σ] in
+ theorem upperImage_homogeneous (e : E) (H : Set (Sheaf.Outgoing (G := G) (G.target e))) :
+    ∀ n v, v ∈ B.upperImage e H → component (B.edge e).piece n v ∈ B.upperImage e H :=
+  map_homogeneous (B.vertex (G.target e)).piece (B.edge e).piece
+    (B.upwardSubmodule (G.target e) H) (B.upwardKernel_homogeneous (G.target e) H)
+    (B.upper e).map (B.upper e).graded
+
+ theorem kernelHilbert_edge (e : E) (F : Set (Sheaf.Outgoing (G := G) (G.source e)))
+    {q : ℝ} (hq : 0 < q) (hq1 : q < 1) :
+    B.kernelHilbert q (G.source e) F -
+      B.kernelHilbert q (G.source e) (insert ⟨e,rfl⟩ F) =
+      value (B.edge e).piece ((B.lowerImage e F).restrictScalars ℝ) q := by
+  let : ∀ n, Module.Finite ℝ ((B.vertex (G.source e)).piece n) :=
+    polynomialModule_piece_finite ℝ σ (B.vertex (G.source e)).piece
+  have h := B.hilbert_edge_increment (G.source e) F ⟨e,rfl⟩ hq
+    (summable_term ℝ σ (B.vertex (G.source e)).piece _ hq hq1)
+  exact sub_eq_iff_eq_add.mpr (h.trans (add_comm _ _))
+
+ theorem shifted_upperImage_value (α : E → MvPolynomial σ ℝ) (hquot : B.UpperQuotient α)
+    (e : E) (H : Set (Sheaf.Outgoing (G := G) (G.target e)))
+    (ha : α e ∈ polynomialGrading ℝ σ 1)
+    (hreg : ∀ f ∈ H, Function.Injective (fun v : B.edge f.val => α e • v))
+    (hinj : Function.Injective (fun v : B.vertex (G.target e) => α e • v))
+    (φ : B.edge e →ₗ[MvPolynomial σ ℝ] B.edge e) (d : ℤ)
+    (hφ : ∀ n v, v ∈ (B.edge e).piece n → φ v ∈ (B.edge e).piece (n+d))
+    (hφinj : Function.Injective φ) {q : ℝ} (hq : 0 < q) (hq1 : q < 1) :
+    value (B.edge e).piece (((B.upperImage e H).map φ).restrictScalars ℝ) q =
+      q^d * (1-q) * B.kernelHilbert q (G.target e) H := by
+  let : ∀ n, Module.Finite ℝ ((B.vertex (G.target e)).piece n) :=
+    polynomialModule_piece_finite ℝ σ (B.vertex (G.target e)).piece
+  rw [shifted_image_value (B.edge e).piece (B.edge e).piece (B.upperImage e H)
+    (B.upperImage_homogeneous e H) φ d hφ hφinj hq.ne'
+    (summable_term ℝ σ (B.edge e).piece _ hq hq1)]
+  rw [show value (B.edge e).piece ((B.upperImage e H).restrictScalars ℝ) q =
+    (1-q) * B.kernelHilbert q (G.target e) H from
+      B.upper_image_value α hquot e H ha hreg hinj hq
+        (summable_term ℝ σ (B.vertex (G.target e)).piece _ hq hq1)]
+  exact (mul_assoc _ _ _).symm
+
+
+ theorem edge_increment_le (α : E → MvPolynomial σ ℝ) (hquot : B.UpperQuotient α)
+    (e : E) (F : Set (Sheaf.Outgoing (G := G) (G.source e)))
+    (H : Set (Sheaf.Outgoing (G := G) (G.target e)))
+    (ha : α e ∈ polynomialGrading ℝ σ 1)
+    (hreg : ∀ f ∈ H, Function.Injective (fun v : B.edge f.val => α e • v))
+    (hinj : Function.Injective (fun v : B.vertex (G.target e) => α e • v))
+    (φ : B.edge e →ₗ[MvPolynomial σ ℝ] B.edge e) (d : ℤ)
+    (hφ : ∀ n v, v ∈ (B.edge e).piece n → φ v ∈ (B.edge e).piece (n+d))
+    (hφinj : Function.Injective φ) (hinc : B.lowerImage e F ≤ (B.upperImage e H).map φ)
+    {q : ℝ} (hq : 0 < q) (hq1 : q < 1) :
+    B.kernelHilbert q (G.source e) F -
+      B.kernelHilbert q (G.source e) (insert ⟨e,rfl⟩ F) ≤
+      q^d * (1-q) * B.kernelHilbert q (G.target e) H := by
+  let : ∀ n, Module.Finite ℝ ((B.edge e).piece n) :=
+    polynomialModule_piece_finite ℝ σ (B.edge e).piece
+  rw [B.kernelHilbert_edge e F hq hq1,
+    ← B.shifted_upperImage_value α hquot e H ha hreg hinj φ d hφ hφinj hq hq1]
+  exact (summable_term ℝ σ (B.edge e).piece _ hq hq1).tsum_le_tsum
+    (term_mono (B.edge e).piece hinc hq)
+    (summable_term ℝ σ (B.edge e).piece _ hq hq1)
+
+/-- Even a finite-length or higher-codimension embedded defect is detected by
+one positive numerical equality. This does not replace the image inclusion. -/
+ theorem edge_increment_eq_iff (α : E → MvPolynomial σ ℝ) (hquot : B.UpperQuotient α)
+    (e : E) (F : Set (Sheaf.Outgoing (G := G) (G.source e)))
+    (H : Set (Sheaf.Outgoing (G := G) (G.target e)))
+    (ha : α e ∈ polynomialGrading ℝ σ 1)
+    (hreg : ∀ f ∈ H, Function.Injective (fun v : B.edge f.val => α e • v))
+    (hinj : Function.Injective (fun v : B.vertex (G.target e) => α e • v))
+    (φ : B.edge e →ₗ[MvPolynomial σ ℝ] B.edge e) (d : ℤ)
+    (hφ : ∀ n v, v ∈ (B.edge e).piece n → φ v ∈ (B.edge e).piece (n+d))
+    (hφinj : Function.Injective φ) (hinc : B.lowerImage e F ≤ (B.upperImage e H).map φ)
+    {q : ℝ} (hq : 0 < q) (hq1 : q < 1) :
+    (B.kernelHilbert q (G.source e) F -
+      B.kernelHilbert q (G.source e) (insert ⟨e,rfl⟩ F) =
+      q^d * (1-q) * B.kernelHilbert q (G.target e) H) ↔
+      B.lowerImage e F = (B.upperImage e H).map φ := by
+  rw [B.kernelHilbert_edge e F hq hq1,
+    ← B.shifted_upperImage_value α hquot e H ha hreg hinj φ d hφ hφinj hq hq1]
+  constructor
+  · intro heq
+    have h := polynomial_eq_of_value_eq ℝ σ (B.edge e).piece hinc
+      (shifted_map_homogeneous (B.edge e).piece (B.edge e).piece (B.upperImage e H)
+        (B.upperImage_homogeneous e H) φ d hφ) hq hq1 heq
+    ext v
+    exact SetLike.ext_iff.mp h v
+  · intro heq
+    rw [heq]
+
+end
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  (hub : BruhatLE cs u b)
+  (B : BoundarySheaf (polynomialGrading_negative ℝ σ)
+    (graph cs u b) (polynomialLabel cs u b basis) ⟨b,hub,bruhat_refl cs b⟩)
+noncomputable section
+
+
+ theorem actual_edge_increment_le (e : Edge cs u b)
+    (F : Set (Sheaf.Outgoing (G := graph cs u b) ((graph cs u b).source e)))
+    (H : Set (Sheaf.Outgoing (G := graph cs u b) ((graph cs u b).target e)))
+    (hinc : B.sheaf.lowerImage e F ≤ (B.sheaf.upperImage e H).map
+      (edgeFactorMap cs basis u b hub B e))
+    {q : ℝ} (hq : 0 < q) (hq1 : q < 1) :
+    B.sheaf.kernelHilbert q ((graph cs u b).source e) F -
+      B.sheaf.kernelHilbert q ((graph cs u b).source e) (insert ⟨e,rfl⟩ F) ≤
+      q^(edgeFactorDegree (source cs u b e).val (root cs u b e) : ℤ) * (1-q) *
+        B.sheaf.kernelHilbert q ((graph cs u b).target e) H := by
+  apply B.sheaf.edge_increment_le (polynomialLabel cs u b basis) B.quotient e F H
+    (polynomialLabel_homogeneous cs u b basis e)
+  · intro f _
+    exact incoming_label_regular_outgoing cs u b basis hub B e f.val f.property
+  · exact label_regular_stalk cs u b basis hub B e _
+  · exact edgeFactorMap_graded cs basis u b hub B e
+  · exact edgeFactorLift_regular cs basis u b hub B e
+  · exact hinc
+  · exact hq
+  · exact hq1
+
+ theorem actual_edge_increment_eq_iff (e : Edge cs u b)
+    (F : Set (Sheaf.Outgoing (G := graph cs u b) ((graph cs u b).source e)))
+    (H : Set (Sheaf.Outgoing (G := graph cs u b) ((graph cs u b).target e)))
+    (hinc : B.sheaf.lowerImage e F ≤ (B.sheaf.upperImage e H).map
+      (edgeFactorMap cs basis u b hub B e))
+    {q : ℝ} (hq : 0 < q) (hq1 : q < 1) :
+    (B.sheaf.kernelHilbert q ((graph cs u b).source e) F -
+      B.sheaf.kernelHilbert q ((graph cs u b).source e) (insert ⟨e,rfl⟩ F) =
+      q^(edgeFactorDegree (source cs u b e).val (root cs u b e) : ℤ) * (1-q) *
+        B.sheaf.kernelHilbert q ((graph cs u b).target e) H) ↔
+      B.sheaf.lowerImage e F = (B.sheaf.upperImage e H).map
+        (edgeFactorMap cs basis u b hub B e) := by
+  apply B.sheaf.edge_increment_eq_iff (polynomialLabel cs u b basis) B.quotient e F H
+    (polynomialLabel_homogeneous cs u b basis e)
+  · intro f _
+    exact incoming_label_regular_outgoing cs u b basis hub B e f.val f.property
+  · exact label_regular_stalk cs u b basis hub B e _
+  · exact edgeFactorMap_graded cs basis u b hub B e
+  · exact edgeFactorLift_regular cs basis u b hub B e
+  · exact hinc
+  · exact hq
+  · exact hq1
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+namespace KLInvariance.Comparison
+
+/-- Degree-one labels, `q=t²`, and exactly the source degree
+`2*k = length(y)-length(x)-1` give the same coefficient at every edge. -/
+ theorem edge_scalar_identity {t : ℝ} (ht : t ≠ 0) (m n k : ℤ)
+    (h : m + 2*k = n-1) :
+    t^m * (t^2)^k * (1-t^2) = (t⁻¹-t) * t^n := by
+  have hp : (t^2)^k = t^(2*k) := by
+    rw [zpow_mul]
+    norm_num
+  rw [hp,← zpow_add₀ ht,h]
+  have hsub : t^(n-1) = t^n*t⁻¹ := by
+    rw [sub_eq_add_neg,zpow_add₀ ht,zpow_neg_one]
+  rw [hsub]
+  have hcancel : t⁻¹*t=1 := inv_mul_cancel₀ ht
+  calc
+    t^n*t⁻¹*(1-t^2) = (t⁻¹-t*(t⁻¹*t))*t^n := by ring
+    _ = (t⁻¹-t)*t^n := by rw [hcancel,mul_one]
+
+ theorem transferCoefficient_pos {t : ℝ} (ht : 0 < t) (ht1 : t < 1) :
+    0 < t⁻¹-t := by
+  have hi : 1 < t⁻¹ := (one_lt_inv₀ ht).mpr ht1
+  linarith
+
+ theorem stateScale_pos {t : ℝ} (ht : 0 < t) (ht1 : t < 1) (N : ℕ) (m : ℤ) :
+    0 < (1-t^2)^N * t^m := by
+  have hsq : t^2 < 1 := by nlinarith
+  exact mul_pos (pow_pos (sub_pos.mpr hsq) _) (zpow_pos ht _)
+
+/-- Cancellation of a positive state scale converts an additive normalized
+edge identity back to its raw Hilbert-series increment. -/
+ theorem scaled_increment_eq_iff (a x y k z : ℝ) (ha : a ≠ 0) :
+    a*x = a*y+(a*k)*z ↔ x-y=k*z := by
+  constructor
+  · intro h
+    have he : a*x=a*(y+k*z) := by linear_combination h
+    have hc := mul_left_cancel₀ ha he
+    linarith
+  · intro h
+    rw [show x=y+k*z by linarith]
+    ring
+
+end KLInvariance.Comparison
+
+end
+
+
+section
+
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  (hub : BruhatLE cs u b)
+  (B : BoundarySheaf (polynomialGrading_negative ℝ σ)
+    (graph cs u b) (polynomialLabel cs u b basis) ⟨b,hub,bruhat_refl cs b⟩)
+noncomputable section
+
+ def stateScale (t : ℝ) (x : Interval cs u b) : ℝ :=
+  (1-t^2)^(Fintype.card σ) * t^((cs.length x.val : ℤ)-cs.length b)
+
+omit [Fintype I] in
+ theorem stateScale_positive {t : ℝ} (ht : 0 < t) (ht1 : t < 1)
+    (x : Interval cs u b) : 0 < stateScale (σ := σ) cs u b t x :=
+  Comparison.stateScale_pos ht ht1 _ _
+
+ def normalizedKernel (t : ℝ) (x : Interval cs u b)
+    (F : Set (Sheaf.Outgoing (G := graph cs u b) x)) : ℝ :=
+  stateScale (σ := σ) cs u b t x * B.sheaf.kernelHilbert (t^2) x F
+
+ theorem edge_stateScale (e : Edge cs u b) {t : ℝ} (ht : t ≠ 0) :
+    stateScale (σ := σ) cs u b t ((graph cs u b).source e) *
+      ((t^2)^(edgeFactorDegree (source cs u b e).val (root cs u b e) : ℤ) * (1-t^2)) =
+      (t⁻¹-t) * stateScale (σ := σ) cs u b t ((graph cs u b).target e) := by
+  have hd := edgeFactorDegree_edge cs u b e
+  have h := Comparison.edge_scalar_identity ht
+    ((cs.length (source cs u b e).val : ℤ)-cs.length b)
+    ((cs.length (target cs u b e).val : ℤ)-cs.length b)
+    (edgeFactorDegree (source cs u b e).val (root cs u b e)) (by omega)
+  dsimp only [stateScale,graph]
+  calc
+    _ = (1-t^2)^(Fintype.card σ) *
+        (t^((cs.length (source cs u b e).val : ℤ)-cs.length b) *
+          (t^2)^(edgeFactorDegree (source cs u b e).val (root cs u b e) : ℤ) * (1-t^2)) := by ring
+    _ = _ := by rw [h]; ring
+
+/-- A single removal of a later-edge kernel condition has the exact uniform
+transfer coefficient `t⁻¹-t` used by the Dyer path matrix. -/
+ theorem normalized_edge_increment_le (e : Edge cs u b)
+    (F : Set (Sheaf.Outgoing (G := graph cs u b) ((graph cs u b).source e)))
+    (H : Set (Sheaf.Outgoing (G := graph cs u b) ((graph cs u b).target e)))
+    (hinc : B.sheaf.lowerImage e F ≤ (B.sheaf.upperImage e H).map
+      (edgeFactorMap cs basis u b hub B e))
+    {t : ℝ} (ht : 0 < t) (ht1 : t < 1) :
+    normalizedKernel cs u b basis hub B t ((graph cs u b).source e) F ≤
+      normalizedKernel cs u b basis hub B t ((graph cs u b).source e) (insert ⟨e,rfl⟩ F) +
+      (t⁻¹-t) * normalizedKernel cs u b basis hub B t ((graph cs u b).target e) H := by
+  have h := actual_edge_increment_le cs u b basis hub B e F H hinc
+    (sq_pos_of_pos ht) (by nlinarith : t^2 < 1)
+  have h' := mul_le_mul_of_nonneg_left h
+    (stateScale_positive (σ := σ) cs u b ht ht1 ((graph cs u b).source e)).le
+  rw [← mul_assoc,edge_stateScale cs u b e ht.ne',mul_sub] at h'
+  simpa only [normalizedKernel,mul_assoc,add_comm] using sub_le_iff_le_add.mp h'
+
+ theorem normalized_edge_increment_eq_iff (e : Edge cs u b)
+    (F : Set (Sheaf.Outgoing (G := graph cs u b) ((graph cs u b).source e)))
+    (H : Set (Sheaf.Outgoing (G := graph cs u b) ((graph cs u b).target e)))
+    (hinc : B.sheaf.lowerImage e F ≤ (B.sheaf.upperImage e H).map
+      (edgeFactorMap cs basis u b hub B e))
+    {t : ℝ} (ht : 0 < t) (ht1 : t < 1) :
+    (normalizedKernel cs u b basis hub B t ((graph cs u b).source e) F =
+      normalizedKernel cs u b basis hub B t ((graph cs u b).source e) (insert ⟨e,rfl⟩ F) +
+      (t⁻¹-t) * normalizedKernel cs u b basis hub B t ((graph cs u b).target e) H) ↔
+      B.sheaf.lowerImage e F = (B.sheaf.upperImage e H).map
+        (edgeFactorMap cs basis u b hub B e) := by
+  rw [← actual_edge_increment_eq_iff cs u b basis hub B e F H hinc
+    (sq_pos_of_pos ht) (by nlinarith : t^2 < 1)]
+  have hs := (stateScale_positive (σ := σ) cs u b ht ht1 ((graph cs u b).source e)).ne'
+  simp only [normalizedKernel]
+  rw [← mul_assoc (t⁻¹-t),← edge_stateScale cs u b e ht.ne']
+  exact Comparison.scaled_increment_eq_iff _ _ _ _ _ hs
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+/-! Exact numerical endpoints for the actual sheaf, conditional on the two
+explicit genuine normalization inputs. These implications are NOT a proof of
+existence of the pairing or the Kazhdan--Lusztig character theorem. -/
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  (hub : BruhatLE cs u b)
+  (B : BoundarySheaf (polynomialGrading_negative ℝ σ)
+    (graph cs u b) (polynomialLabel cs u b basis) ⟨b,hub,bruhat_refl cs b⟩)
+noncomputable section
+
+ theorem emptyKernel_hilbert (x : Interval cs u b) {q : ℝ} (hq : 0 < q) (hq1 : q < 1) :
+    B.sheaf.kernelHilbert q x ∅ =
+      Polynomial.eval₂ (Int.castRingHom ℝ) q (stalkCharacter cs u b basis hub x) *
+        (1-q)⁻¹ ^ Fintype.card σ := by
+  let _ := B.free x
+  let c := (exists_nonnegativeBasis (σ := σ) (B.sheaf.vertex x).piece
+    (B.sheaf.vertex x).nonneg).some
+  have he : B.sheaf.upwardSubmodule x ∅ = ⊤ := by
+    ext v
+    change (∀ e, e ∈ (∅ : Set (Sheaf.Outgoing (G := graph cs u b) x)) → _) ↔ True
+    simp
+  rw [GradedSheaf.kernelHilbert,he,Submodule.restrictScalars_top,
+    stalkCharacter_eq_character cs u b basis hub B x,
+    character_eq (B.sheaf.vertex x).piece (B.sheaf.vertex x).nonneg c]
+  simpa only [Hilbert.value,Hilbert.term_top] using (c.hasSum hq hq1).tsum_eq
+
+ theorem normalized_emptyKernel_of_character (x : Interval cs u b)
+    (hc : stalkCharacter cs u b basis hub x = klPolynomial cs x.val b)
+    {t : ℝ} (ht : 0 < t) (ht1 : t < 1) :
+    normalizedKernel cs u b basis hub B t x ∅ = scaledKL cs t x.val b := by
+  rw [normalizedKernel,emptyKernel_hilbert cs u b basis hub B x
+    (sq_pos_of_pos ht) (by nlinarith),hc]
+  dsimp only [stateScale,scaledKL]
+  have hne : 1-t^2 ≠ 0 := ne_of_gt (by nlinarith : 0 < 1-t^2)
+  rw [inv_pow]
+  field_simp [hne,inv_pow]
+
+variable [Fintype (Edge cs 1 b)] [Fintype (Interval cs 1 b)] [DecidableEq (Interval cs 1 b)]
+  (C : BoundarySheaf (polynomialGrading_negative ℝ σ)
+    (graph cs 1 b) (polynomialLabel cs 1 b basis) ⟨b,one_bruhat cs b,bruhat_refl cs b⟩)
+
+ theorem normalized_fullKernel_of_duality_character
+    (d : SectionDuality cs 1 b basis (one_bruhat cs b) C)
+    (hd : ∀ i j (m n : C.sheaf.sections Set.univ),
+      m ∈ (C.sheaf.sections Set.univ).piece i → n ∈ (C.sheaf.sections Set.univ).piece j →
+      d.pairing m n ∈ polynomialGrading ℝ σ (i+j-(cs.length b : ℤ)))
+    (x : Interval cs 1 b)
+    (hc : stalkCharacter cs 1 b basis (one_bruhat cs b) x = klPolynomial cs x.val b)
+    {t : ℝ} (ht : 0 < t) (ht1 : t < 1) :
+    normalizedKernel cs 1 b basis (one_bruhat cs b) C t x Set.univ =
+      scaledKL cs t⁻¹ x.val b := by
+  let _ := C.free x
+  let c := (exists_nonnegativeBasis (σ := σ) (C.sheaf.vertex x).piece
+    (C.sheaf.vertex x).nonneg).some
+  have hcp : c.polynomial = klPolynomial cs x.val b := by
+    rw [← character_eq (C.sheaf.vertex x).piece (C.sheaf.vertex x).nonneg c,
+      ← stalkCharacter_eq_character cs 1 b basis (one_bruhat cs b) C x]
+    exact hc
+  rw [normalizedKernel,GradedSheaf.kernelHilbert,
+    fullKernel_eq cs 1 b basis (one_bruhat cs b) C x,
+    costalk_hilbert_of_duality cs 1 b basis (one_bruhat cs b) C d (cs.length b) hd x c
+      (sq_pos_of_pos ht) (by nlinarith),hcp,incoming_card cs b x]
+  dsimp only [stateScale,scaledKL]
+  have hscale : t^((cs.length x.val : ℤ)-cs.length b) *
+      (t^2)^((cs.length b : ℤ)-cs.length x.val) =
+      (t⁻¹)^((cs.length x.val : ℤ)-cs.length b) := by
+    rw [show (t^2)^((cs.length b : ℤ)-cs.length x.val) =
+      t^(2*((cs.length b : ℤ)-cs.length x.val)) by rw [zpow_mul]; norm_num,
+      ← zpow_add₀ ht.ne',inv_zpow,← zpow_neg]
+    congr 1
+    omega
+  rw [show (t^2)⁻¹=(t⁻¹)^2 by rw [inv_pow]]
+  have hne : 1-t^2 ≠ 0 := ne_of_gt (by nlinarith : 0 < 1-t^2)
+  calc
+    _ = (t^((cs.length x.val : ℤ)-cs.length b) *
+      (t^2)^((cs.length b : ℤ)-cs.length x.val)) *
+      Polynomial.eval₂ (Int.castRingHom ℝ) ((t⁻¹)^2) (klPolynomial cs x.val b) := by
+        have hcancel : (1-t^2)^(Fintype.card σ) *
+            ((1-t^2)⁻¹)^(Fintype.card σ)=1 := by
+          rw [← mul_pow,mul_inv_cancel₀ hne,one_pow]
+        calc
+          _ = ((1-t^2)^(Fintype.card σ) * ((1-t^2)⁻¹)^(Fintype.card σ)) *
+            ((t^((cs.length x.val : ℤ)-cs.length b) *
+              (t^2)^((cs.length b : ℤ)-cs.length x.val)) *
+              Polynomial.eval₂ (Int.castRingHom ℝ) ((t⁻¹)^2) (klPolynomial cs x.val b)) := by ring
+          _ = _ := by rw [hcancel,one_mul]
+    _ = _ := by rw [hscale]
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  (hub : BruhatLE cs u b)
+  (B : BoundarySheaf (polynomialGrading_negative ℝ σ)
+    (graph cs 1 b) (polynomialLabel cs 1 b basis) ⟨b,one_bruhat cs b,bruhat_refl cs b⟩)
+noncomputable section
+
+ omit [Fintype σ] in
+ theorem restrictBoundary_kernel (x : Interval cs u b) (T : Set (Edge cs 1 b)) :
+    (restrictBoundary cs u b basis hub B).sheaf.upwardSubmodule x
+      {e | (upperInclusion cs u b).edge e.val ∈ T} =
+    B.sheaf.upwardSubmodule ((upperInclusion cs u b).vertex x) {e | e.val ∈ T} := by
+  exact B.sheaf.forget.pullback_kernel (upperInclusion cs u b) x T Set.univ
+    (fun e he => False.elim (he (Set.mem_univ e)))
+    (fun e _ he => upperInclusion_outgoing cs u b e x he)
+
+ theorem normalizedKernel_restrictBoundary (x : Interval cs u b) (T : Set (Edge cs 1 b)) (t : ℝ) :
+    normalizedKernel cs u b basis hub (restrictBoundary cs u b basis hub B) t x
+      {e | (upperInclusion cs u b).edge e.val ∈ T} =
+    normalizedKernel cs 1 b basis (one_bruhat cs b) B t
+      ((upperInclusion cs u b).vertex x) {e | e.val ∈ T} := by
+  unfold normalizedKernel GradedSheaf.kernelHilbert
+  rw [restrictBoundary_kernel cs u b basis hub B x T]
+  rfl
+
+ theorem normalized_emptyKernel_restrictBoundary (x : Interval cs u b) (t : ℝ) :
+    normalizedKernel cs u b basis hub (restrictBoundary cs u b basis hub B) t x ∅ =
+    normalizedKernel cs 1 b basis (one_bruhat cs b) B t
+      ((upperInclusion cs u b).vertex x) ∅ := by
+  simpa only [Set.mem_empty_iff_false,Set.ofPred_false] using
+    normalizedKernel_restrictBoundary cs u b basis hub B x ∅ t
+
+ theorem normalized_fullKernel_restrictBoundary (x : Interval cs u b) (t : ℝ) :
+    normalizedKernel cs u b basis hub (restrictBoundary cs u b basis hub B) t x Set.univ =
+    normalizedKernel cs 1 b basis (one_bruhat cs b) B t
+      ((upperInclusion cs u b).vertex x) Set.univ := by
+  simpa only [Set.mem_univ,Set.ofPred_true] using
+    normalizedKernel_restrictBoundary cs u b basis hub B x Set.univ t
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+
+namespace KLInvariance.SchedulePaths
+universe u v
+variable {V : Type u} {E : Type v}
+  (source target : E → V)
+noncomputable section
+
+ theorem step_monotone (T : ℝ) (hT : 0 ≤ T) (e : E) :
+    Monotone (step source target T e) := by
+  intro v w hv x
+  dsimp [step]
+  split_ifs
+  · exact add_le_add (hv x) (mul_le_mul_of_nonneg_left (hv _) hT)
+  · simpa using hv x
+
+ theorem step_injective (T : ℝ) (e : E) (hne : source e ≠ target e) :
+    Function.Injective (step source target T e) := by
+  intro v w he
+  have ht := congrFun he (target e)
+  simp only [step,hne,ite_false,add_zero] at ht
+  funext x
+  have hx := congrFun he x
+  dsimp [step] at hx
+  split_ifs at hx with h
+  · rw [ht] at hx
+    exact add_right_cancel hx
+  · exact add_right_cancel hx
+
+ theorem dispatch_monotone (T : ℝ) (hT : 0 ≤ T) (p : List E) :
+    Monotone (fun v => dispatch source target T v p) := by
+  induction p with
+  | nil => exact monotone_id
+  | cons e p ih => exact (step_monotone source target T hT e).comp ih
+
+ theorem dispatch_injective (T : ℝ) (p : List E)
+    (hne : ∀ e ∈ p, source e ≠ target e) :
+    Function.Injective (fun v => dispatch source target T v p) := by
+  induction p with
+  | nil => exact Function.injective_id
+  | cons e p ih =>
+    exact (step_injective source target T e (hne e (by simp))).comp
+      (ih (fun a ha => hne a (by simp [ha])))
+
+ theorem dispatch_append' (T : ℝ) (v : V → ℝ) (p q : List E) :
+    dispatch source target T v (p++q) =
+      dispatch source target T (dispatch source target T v q) p := by
+  induction p with
+  | nil => rfl
+  | cons e p ih => simp only [List.cons_append,dispatch,ih]
+
+/-- A concrete state indexed by the remaining chronological edge list. -/
+ theorem remaining_state_le (T : ℝ) (hT : 0 ≤ T)
+    (D : List E → V → ℝ) (p : List E)
+    (hstep : ∀ e q, e::q <:+ p → D q ≤ step source target T e (D (e::q))) :
+    D [] ≤ dispatch source target T (D p) p.reverse := by
+  induction p with
+  | nil => exact le_rfl
+  | cons e p ih =>
+    have hp := ih (fun a q hq => hstep a q (hq.trans (List.suffix_cons e p)))
+    have he := hstep e p (List.suffix_refl _)
+    rw [List.reverse_cons,dispatch_append']
+    exact hp.trans (dispatch_monotone source target T hT p.reverse he)
+
+/-- If the endpoint bound is equality, every embedded one-edge Hilbert bound
+is equality. Injectivity must be used together with the one-sided bounds. -/
+ theorem remaining_steps_eq (T : ℝ) (hT : 0 ≤ T)
+    (D : List E → V → ℝ) (p : List E)
+    (hne : ∀ e ∈ p, source e ≠ target e)
+    (hstep : ∀ e q, e::q <:+ p → D q ≤ step source target T e (D (e::q)))
+    (hend : D [] = dispatch source target T (D p) p.reverse) :
+    ∀ e q, e::q <:+ p → D q = step source target T e (D (e::q)) := by
+  induction p with
+  | nil =>
+    intro e q hq
+    have := hq.length_le
+    simp at this
+  | cons e p ih =>
+    have hps := fun a q (hq : a::q <:+ p) => hstep a q (hq.trans (List.suffix_cons e p))
+    have hp := remaining_state_le source target T hT D p hps
+    have he := hstep e p (List.suffix_refl _)
+    have hm := dispatch_monotone source target T hT p.reverse he
+    have hend' : D [] = dispatch source target T
+        (step source target T e (D (e::p))) p.reverse := by
+      simpa only [List.reverse_cons,dispatch_append',dispatch] using hend
+    have hh : dispatch source target T (D p) p.reverse =
+        dispatch source target T (step source target T e (D (e::p))) p.reverse := by
+      apply le_antisymm hm
+      exact hend'.symm.le.trans hp
+    have hep : D p = step source target T e (D (e::p)) :=
+      dispatch_injective source target T p.reverse
+        (fun a ha => hne a (by simp only [List.mem_reverse] at ha; simp [ha])) hh
+    have hendpoint : D [] = dispatch source target T (D p) p.reverse := by
+      exact hend'.trans hh.symm
+    intro a q hq
+    rcases List.suffix_cons_iff.mp hq with h | h
+    · cases h
+      exact hep
+    · exact ih (fun a ha => hne a (by simp [ha])) hps hendpoint a q h
+
+end
+end KLInvariance.SchedulePaths
+
+end
+
+
+section
+
+namespace KLInvariance.ReverseFiltration
+universe ua um un up
+variable {A : Type ua} [CommRing A]
+  {M : Type um} [AddCommGroup M] [Module A M]
+  {N : Type un} [AddCommGroup N] [Module A N]
+  {P : Type up} [AddCommGroup P] [Module A P]
+
+/-- A kernel which is exactly a regular scalar multiple of its domain is
+integrally isomorphic to that domain, not just to its fraction-field span. -/
+noncomputable def scalarKernelEquiv (g : M →ₗ[A] N) (a : A)
+    (hk : ∀ v, g v = 0 ↔ ∃ w, v = a • w)
+    (ha : Function.Injective (fun v : M => a • v)) :
+    M ≃ₗ[A] LinearMap.ker g := by
+  let f : M →ₗ[A] LinearMap.ker g :=
+    (LinearMap.lsmul A M a).codRestrict (LinearMap.ker g) (fun v =>
+      (hk (a • v)).mpr ⟨v,rfl⟩)
+  apply LinearEquiv.ofBijective f
+  constructor
+  · intro x y h
+    exact ha (congrArg Subtype.val h)
+  · intro v
+    obtain ⟨w,hw⟩ := (hk v.val).mp v.property
+    exact ⟨w,Subtype.ext hw.symm⟩
+
+
+ theorem projective_kernel_of_scalar_presentation
+    [Module.Projective A M] [Module.Projective A P]
+    (f : P →ₗ[A] N) (g : M →ₗ[A] N)
+    (hf : Function.Surjective f) (hg : Function.Surjective g)
+    (a : A) (hk : ∀ v, g v = 0 ↔ ∃ w, v = a • w)
+    (ha : Function.Injective (fun v : M => a • v)) :
+    Module.Projective A (LinearMap.ker f) := by
+  let : Module.Projective A (LinearMap.ker g) :=
+    Module.Projective.of_equiv (scalarKernelEquiv g a hk ha)
+  exact kernel_projective_of_presentation f g hf hg
+
+ def imageMapKernelEquiv (S : Submodule A M) (f : M →ₗ[A] N) :
+    ↥(S ⊓ LinearMap.ker f) ≃ₗ[A] LinearMap.ker (Hilbert.imageMap S f) where
+  toFun v := ⟨⟨v.val,v.property.1⟩,Subtype.ext v.property.2⟩
+  invFun v := ⟨v.val.val,v.val.property,congrArg Subtype.val v.property⟩
+  left_inv _ := rfl
+  right_inv _ := rfl
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+
+/-- Scaled equality of two *embedded* images constructs a presentation with
+regular scalar kernel. This is the pure module-algebra form of the reverse
+freeness step, allowing arbitrary image codimension. -/
+ theorem inf_kernel_projective_of_image_eq
+    (S : Submodule A M) (T : Submodule A P)
+    [Module.Projective A S] [Module.Projective A T]
+    (f : M →ₗ[A] N) (g : P →ₗ[A] N) (φ : N →ₗ[A] N)
+    (hφ : Function.Injective φ) (himage : S.map f = (T.map g).map φ)
+    (a : A) (hk : ∀ v, g v = 0 ↔ ∃ w, v = a • w)
+    (hsat : ∀ v, a • v ∈ T → v ∈ T)
+    (ha : Function.Injective (fun v : P => a • v)) :
+    Module.Projective A ↥(S ⊓ LinearMap.ker f) := by
+  let j : T →ₗ[A] S.map f := ((φ.comp g).comp T.subtype).codRestrict (S.map f) (by
+    intro v
+    rw [himage]
+    exact Submodule.mem_map.mpr ⟨g v.val,Submodule.mem_map.mpr ⟨v.val,v.property,rfl⟩,rfl⟩)
+  have hj : Function.Surjective j := by
+    intro v
+    have hv : v.val ∈ (T.map g).map φ := himage ▸ v.property
+    obtain ⟨w,hw,hwv⟩ := Submodule.mem_map.mp hv
+    obtain ⟨z,hz,hzw⟩ := Submodule.mem_map.mp hw
+    exact ⟨⟨z,hz⟩,Subtype.ext ((congrArg φ hzw).trans hwv)⟩
+  have hjker (v : T) : j v = 0 ↔ ∃ w : T, v = a • w := by
+    constructor
+    · intro hv
+      have hv' : φ (g v.val) = 0 := congrArg Subtype.val hv
+      have hz : g v.val = 0 := hφ (hv'.trans (map_zero φ).symm)
+      obtain ⟨w,hw⟩ := (hk v.val).mp hz
+      refine ⟨⟨w,hsat w (hw ▸ v.property)⟩,Subtype.ext hw⟩
+    · rintro ⟨w,rfl⟩
+      apply Subtype.ext
+      change φ (g (a • w.val)) = 0
+      rw [(hk (a • w.val)).mpr ⟨w.val,rfl⟩,map_zero]
+  have haT : Function.Injective (fun v : T => a • v) := by
+    intro v w hvw
+    exact Subtype.ext (ha (congrArg Subtype.val hvw))
+  let : Module.Projective A (LinearMap.ker (Hilbert.imageMap S f)) :=
+    projective_kernel_of_scalar_presentation (Hilbert.imageMap S f) j
+      (Hilbert.imageMap_surjective S f) hj a hjker haT
+  exact Module.Projective.of_equiv (imageMapKernelEquiv S f).symm
+
+end KLInvariance.ReverseFiltration
+
+end
+
+
+section
+
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  (hub : BruhatLE cs u b)
+  (B : BoundarySheaf (polynomialGrading_negative ℝ σ)
+    (graph cs u b) (polynomialLabel cs u b basis) ⟨b,hub,bruhat_refl cs b⟩)
+noncomputable section
+
+omit [Fintype σ] in
+/-- The reverse-freeness step uses precisely the already free upper kernel
+and the preceding lower kernel, plus the actual embedded image equality. -/
+ theorem partial_kernel_projective_of_image_eq (e : Edge cs u b)
+    (F : Set (Sheaf.Outgoing (G := graph cs u b) ((graph cs u b).source e)))
+    (H : Set (Sheaf.Outgoing (G := graph cs u b) ((graph cs u b).target e)))
+    [Module.Projective (Coefficient (σ := σ))
+      (B.sheaf.upwardSubmodule ((graph cs u b).source e) F)]
+    [Module.Projective (Coefficient (σ := σ))
+      (B.sheaf.upwardSubmodule ((graph cs u b).target e) H)]
+    (heq : B.sheaf.lowerImage e F = (B.sheaf.upperImage e H).map
+      (edgeFactorMap cs basis u b hub B e)) :
+    Module.Projective (Coefficient (σ := σ))
+      (B.sheaf.upwardSubmodule ((graph cs u b).source e) (insert ⟨e,rfl⟩ F)) := by
+  rw [B.sheaf.upwardKernel_insert]
+  apply ReverseFiltration.inf_kernel_projective_of_image_eq
+    (B.sheaf.upwardSubmodule ((graph cs u b).source e) F)
+    (B.sheaf.upwardSubmodule ((graph cs u b).target e) H)
+    (B.sheaf.lower e).map (B.sheaf.upper e).map (edgeFactorMap cs basis u b hub B e)
+    (edgeFactorLift_regular cs basis u b hub B e) heq
+    (polynomialLabel cs u b basis e) (B.quotient e).2
+  · apply B.sheaf.upwardSubmodule_saturated
+    intro f _
+    exact incoming_label_regular_outgoing cs u b basis hub B e f.val f.property
+  · exact label_regular_stalk cs u b basis hub B e _
+
+ theorem partial_kernel_free_of_image_eq (e : Edge cs u b)
+    (F : Set (Sheaf.Outgoing (G := graph cs u b) ((graph cs u b).source e)))
+    (H : Set (Sheaf.Outgoing (G := graph cs u b) ((graph cs u b).target e)))
+    [Module.Free (Coefficient (σ := σ))
+      (B.sheaf.upwardSubmodule ((graph cs u b).source e) F)]
+    [Module.Free (Coefficient (σ := σ))
+      (B.sheaf.upwardSubmodule ((graph cs u b).target e) H)]
+    (heq : B.sheaf.lowerImage e F = (B.sheaf.upperImage e H).map
+      (edgeFactorMap cs basis u b hub B e)) :
+    Module.Free (Coefficient (σ := σ))
+      (B.sheaf.upwardSubmodule ((graph cs u b).source e) (insert ⟨e,rfl⟩ F)) := by
+  let D := B.sheaf.upwardKernel ((graph cs u b).source e) (insert ⟨e,rfl⟩ F)
+  let : Module.Projective (Coefficient (σ := σ)) D :=
+    partial_kernel_projective_of_image_eq cs u b basis hub B e F H heq
+  obtain ⟨ι,_,bs,_⟩ := exists_homogeneous_basis_polynomial (k := ℝ) (σ := σ)
+    D.piece 0 D.nonneg
+  exact Module.Free.of_basis bs
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph SchedulePaths
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  (hub : BruhatLE cs u b)
+  (B : BoundarySheaf (polynomialGrading_negative ℝ σ)
+    (graph cs u b) (polynomialLabel cs u b basis) ⟨b,hub,bruhat_refl cs b⟩)
+noncomputable section
+
+ def remainingConditions (p : List (Edge cs u b)) (x : Interval cs u b) :
+    Set (Sheaf.Outgoing (G := graph cs u b) x) := {f | f.val ∈ p}
+
+omit [Fintype I] in
+ theorem remainingConditions_nil (x : Interval cs u b) :
+    remainingConditions cs u b [] x = ∅ := by ext f; simp [remainingConditions]
+
+omit [Fintype I] in
+ theorem remainingConditions_cons_source (e : Edge cs u b) (p : List (Edge cs u b)) :
+    remainingConditions cs u b (e::p) (source cs u b e) =
+      insert ⟨e,rfl⟩ (remainingConditions cs u b p (source cs u b e)) := by
+  ext f
+  simp only [remainingConditions,Set.mem_ofPred_eq,List.mem_cons,Set.mem_insert_iff]
+  rw [Subtype.ext_iff]
+
+omit [Fintype I] in
+ theorem remainingConditions_cons_other (e : Edge cs u b) (p : List (Edge cs u b))
+    (x : Interval cs u b) (hne : source cs u b e ≠ x) :
+    remainingConditions cs u b (e::p) x = remainingConditions cs u b p x := by
+  ext f
+  have hf : f.val ≠ e := by
+    intro he
+    exact hne (he ▸ f.property)
+  simp [remainingConditions,hf]
+
+ def scheduleState (t : ℝ) (p : List (Edge cs u b)) : Interval cs u b → ℝ :=
+  fun x => normalizedKernel cs u b basis hub B t x (remainingConditions cs u b p x)
+
+ theorem scheduleState_step (e : Edge cs u b) (p : List (Edge cs u b))
+    (hinc : B.sheaf.lowerImage e (remainingConditions cs u b p ((graph cs u b).source e)) ≤
+      (B.sheaf.upperImage e (remainingConditions cs u b p ((graph cs u b).target e))).map
+        (edgeFactorMap cs basis u b hub B e))
+    {t : ℝ} (ht : 0 < t) (ht1 : t < 1) :
+    scheduleState cs u b basis hub B t p ≤
+      step (source cs u b) (target cs u b) (t⁻¹-t) e
+        (scheduleState cs u b basis hub B t (e::p)) := by
+  classical
+  intro x
+  by_cases hx : (graph cs u b).source e = x
+  · subst x
+    have h := normalized_edge_increment_le cs u b basis hub B e
+      (remainingConditions cs u b p ((graph cs u b).source e))
+      (remainingConditions cs u b p ((graph cs u b).target e)) hinc ht ht1
+    have hne : source cs u b e ≠ target cs u b e :=
+      ne_of_lt ((graph cs u b).increasing e)
+    change normalizedKernel cs u b basis hub B t (source cs u b e)
+        (remainingConditions cs u b p (source cs u b e)) ≤
+      normalizedKernel cs u b basis hub B t (source cs u b e)
+        (remainingConditions cs u b (e::p) (source cs u b e)) +
+      if source cs u b e = source cs u b e then
+        (t⁻¹-t) * normalizedKernel cs u b basis hub B t (target cs u b e)
+          (remainingConditions cs u b (e::p) (target cs u b e)) else 0
+    rw [ite_eq_left rfl,remainingConditions_cons_source,
+      remainingConditions_cons_other cs u b e p (target cs u b e) hne]
+    exact h
+  · simp only [scheduleState,step]
+    have hx' : source cs u b e ≠ x := hx
+    rw [ite_eq_right hx',add_zero,remainingConditions_cons_other cs u b e p x hx]
+
+ theorem scheduleState_le (p : List (Edge cs u b))
+    (hinc : ∀ e q, e::q <:+ p →
+      B.sheaf.lowerImage e (remainingConditions cs u b q ((graph cs u b).source e)) ≤
+      (B.sheaf.upperImage e (remainingConditions cs u b q ((graph cs u b).target e))).map
+        (edgeFactorMap cs basis u b hub B e))
+    {t : ℝ} (ht : 0 < t) (ht1 : t < 1) :
+    scheduleState cs u b basis hub B t [] ≤
+      dispatch (source cs u b) (target cs u b) (t⁻¹-t)
+        (scheduleState cs u b basis hub B t p) p.reverse := by
+  exact remaining_state_le _ _ _ (Comparison.transferCoefficient_pos ht ht1).le _ p
+    (fun e q hq => scheduleState_step cs u b basis hub B e q (hinc e q hq) ht ht1)
+
+
+ theorem edgeImages_eq_of_scheduleState_eq (p : List (Edge cs u b))
+    (hinc : ∀ e q, e::q <:+ p →
+      B.sheaf.lowerImage e (remainingConditions cs u b q ((graph cs u b).source e)) ≤
+      (B.sheaf.upperImage e (remainingConditions cs u b q ((graph cs u b).target e))).map
+        (edgeFactorMap cs basis u b hub B e))
+    {t : ℝ} (ht : 0 < t) (ht1 : t < 1)
+    (hend : scheduleState cs u b basis hub B t [] =
+      dispatch (source cs u b) (target cs u b) (t⁻¹-t)
+        (scheduleState cs u b basis hub B t p) p.reverse) :
+    ∀ e q, e::q <:+ p →
+      B.sheaf.lowerImage e (remainingConditions cs u b q ((graph cs u b).source e)) =
+      (B.sheaf.upperImage e (remainingConditions cs u b q ((graph cs u b).target e))).map
+        (edgeFactorMap cs basis u b hub B e) := by
+  classical
+  have heq := remaining_steps_eq (source cs u b) (target cs u b) (t⁻¹-t)
+    (Comparison.transferCoefficient_pos ht ht1).le
+    (scheduleState cs u b basis hub B t) p
+    (fun e _ => ne_of_lt ((graph cs u b).increasing e))
+    (fun e q hq => scheduleState_step cs u b basis hub B e q (hinc e q hq) ht ht1) hend
+  intro e q hq
+  apply (normalized_edge_increment_eq_iff cs u b basis hub B e
+    (remainingConditions cs u b q ((graph cs u b).source e))
+    (remainingConditions cs u b q ((graph cs u b).target e)) (hinc e q hq) ht ht1).mp
+  have h := congrFun (heq e q hq) ((graph cs u b).source e)
+  have hne : source cs u b e ≠ target cs u b e :=
+    ne_of_lt ((graph cs u b).increasing e)
+  change normalizedKernel cs u b basis hub B t (source cs u b e)
+      (remainingConditions cs u b q (source cs u b e)) =
+    normalizedKernel cs u b basis hub B t (source cs u b e)
+      (remainingConditions cs u b (e::q) (source cs u b e)) +
+    (if source cs u b e = source cs u b e then
+      (t⁻¹-t) * normalizedKernel cs u b basis hub B t (target cs u b e)
+        (remainingConditions cs u b (e::q) (target cs u b e)) else 0) at h
+  rw [ite_eq_left rfl,remainingConditions_cons_source,
+    remainingConditions_cons_other cs u b e q (target cs u b e) hne] at h
+  exact h
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+/-! Exact identification of chronological suffix conditions with the
+strictly-later root conditions used in the height-one edge-image proof.
+Equal-label edges may be ordered arbitrarily and cause no extra incident
+condition. -/
+namespace KLInvariance.BruhatGraph
+open TitsSpace SchedulePaths MomentGraph
+universe u v u' v'
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
+  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
+  {u b : W} {u' b' : W'}
+  (φ : Interval cs u b ≃o Interval cs' u' b')
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+ def transportedSlope (e : Edge cs u b) : ℝ :=
+  geometricSlope (AdaptedDyer.rootLabel cs' (transportedEdge φ e))
+
+ def chronologicalSchedule : List (Edge cs u b) := (transportedSchedule φ).reverse
+
+omit [Fintype I] in
+ theorem transportedScore_eq (e : Edge cs u b) : transportedScore φ e =
+    toLex (transportedSlope φ e,transportedSlope φ e) := by
+  unfold transportedScore lexEdgeScore lexRootScore transportedSlope
+  rw [geometricFunctional_normalizedRoot]
+
+omit [Fintype I] in
+ theorem chronologicalSchedule_pairwise : (chronologicalSchedule φ).Pairwise
+    (fun e f => transportedSlope φ e ≤ transportedSlope φ f) := by
+  rw [chronologicalSchedule,List.pairwise_reverse]
+  apply (transportedSchedule_pairwise φ).imp
+  intro e f h
+  have hh := Prod.Lex.monotone_fst _ _ h
+  rw [transportedScore_eq,transportedScore_eq] at hh
+  exact hh
+
+omit [Fintype I] in
+ theorem chronologicalSchedule_nodup : (chronologicalSchedule φ).Nodup :=
+  List.nodup_reverse.mpr (transportedSchedule_nodup φ)
+
+omit [Fintype I] in
+ theorem chronologicalSchedule_full (e : Edge cs u b) : e ∈ chronologicalSchedule φ :=
+  List.mem_reverse.mpr (transportedSchedule_full φ e)
+
+omit [Fintype I] in
+ theorem transportedSlope_source_unique (e f : Edge cs u b)
+    (hs : source cs u b e=source cs u b f)
+    (h : transportedSlope φ e=transportedSlope φ f) : e=f := by
+  have hr := congrArg rootRef (geometricSlope_injective h)
+  rw [AdaptedDyer.rootRef_rootLabel,AdaptedDyer.rootRef_rootLabel] at hr
+  change (φ (target cs u b e)).val * (φ (source cs u b e)).val⁻¹ =
+    (φ (target cs u b f)).val * (φ (source cs u b f)).val⁻¹ at hr
+  rw [hs] at hr
+  apply Subtype.ext
+  exact Prod.ext hs (φ.injective (Subtype.ext (mul_right_cancel hr)))
+
+omit [Fintype I] in
+ theorem transportedSlope_path_ne (e f : Edge cs u b)
+    (hs : target cs u b e=source cs u b f) :
+    transportedSlope φ e ≠ transportedSlope φ f := by
+  intro h
+  exact AdaptedDyer.rootLabel_ne_of_join cs' (transportedEdge φ e) (transportedEdge φ f)
+    (congrArg (fun x => (φ x).val) hs) (geometricSlope_injective h)
+
+omit [Fintype I] in
+ theorem suffix_source_iff {e f : Edge cs u b} {q : List (Edge cs u b)}
+    (hq : e::q <:+ chronologicalSchedule φ) (hf : source cs u b f=source cs u b e) :
+    f ∈ q ↔ transportedSlope φ e < transportedSlope φ f :=
+  mem_suffix_iff_score_lt_injective (transportedSlope φ) (chronologicalSchedule_pairwise φ)
+    (chronologicalSchedule_nodup φ) hq (chronologicalSchedule_full φ f)
+    (transportedSlope_source_unique φ e f hf.symm)
+
+omit [Fintype I] in
+ theorem suffix_target_iff {e f : Edge cs u b} {q : List (Edge cs u b)}
+    (hq : e::q <:+ chronologicalSchedule φ) (hf : source cs u b f=target cs u b e) :
+    f ∈ q ↔ transportedSlope φ e < transportedSlope φ f :=
+  mem_suffix_iff_score_lt (transportedSlope φ) (chronologicalSchedule_pairwise φ)
+    hq (chronologicalSchedule_full φ f) (transportedSlope_path_ne φ e f hf.symm)
+
+omit [Fintype I] in
+ theorem ambientLater_upperInclusion (e f : Edge cs u b) :
+    (upperInclusion cs u b).edge f ∈ ambientLater φ
+      (AdaptedDyer.rootLabel cs' (transportedEdge φ e)) ↔
+    transportedSlope φ e < transportedSlope φ f := by
+  have hs : BruhatStep cs' (φ (source cs u b f)).val (φ (target cs u b f)).val :=
+    (transportedEdge φ f).property
+  have hiff : ambientAfterIndicator (φ (source cs u b f)).val
+      (AdaptedDyer.rootLabel cs' (transportedEdge φ e))
+      (φ (target cs u b f)).val = 1 ↔ transportedSlope φ e < transportedSlope φ f := by
+    simp only [ambientAfterIndicator,dite_eq_left hs]
+    change (if transportedSlope φ e < transportedSlope φ f then 1 else 0)=1 ↔ _
+    split_ifs <;> simp_all
+  constructor
+  · rintro ⟨_,h⟩
+    exact hiff.mp h
+  · intro h
+    exact ⟨(source cs u b f).property.1,hiff.mpr h⟩
+
+omit [Fintype I] in
+ theorem remainingConditions_source_ambient {e : Edge cs u b} {q : List (Edge cs u b)}
+    (hq : e::q <:+ chronologicalSchedule φ) :
+    remainingConditions cs u b q (source cs u b e) =
+      {f | (upperInclusion cs u b).edge f.val ∈ ambientLater φ
+        (AdaptedDyer.rootLabel cs' (transportedEdge φ e))} := by
+  ext f
+  exact (suffix_source_iff φ hq f.property).trans (ambientLater_upperInclusion φ e f.val).symm
+
+omit [Fintype I] in
+ theorem remainingConditions_target_ambient {e : Edge cs u b} {q : List (Edge cs u b)}
+    (hq : e::q <:+ chronologicalSchedule φ) :
+    remainingConditions cs u b q (target cs u b e) =
+      {f | (upperInclusion cs u b).edge f.val ∈ ambientLater φ
+        (AdaptedDyer.rootLabel cs' (transportedEdge φ e))} := by
+  ext f
+  exact (suffix_target_iff φ hq f.property).trans (ambientLater_upperInclusion φ e f.val).symm
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/KazhdanLusztig/ReflectionOrders.lean b/lean/OAI/RepresentationTheory/KazhdanLusztig/ReflectionOrders.lean
index 0b9d2166bf9430e5dfcf34400e3eb096b76e68f9..a925457db408b054eb98ebcc6d43aec213f0f243 100644
--- a/lean/OAI/RepresentationTheory/KazhdanLusztig/ReflectionOrders.lean
+++ b/lean/OAI/RepresentationTheory/KazhdanLusztig/ReflectionOrders.lean
@@ -1,1247 +1,1247 @@
-import OAI.RepresentationTheory.KazhdanLusztig.DyerPaths
-
-namespace OAI
-
-/-!
-Finite sweep algebra, wall orders and the full reflection-order path formula.
--/
-
-section
-
-namespace KLInvariance.SchedulePaths
-open scoped BigOperators
-universe u v w
-variable {V : Type u} {E : Type v} {R : Type w}
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-variable (source target : E → V) [Semiring R]
-
-theorem dispatch_append (T : R) (v : V → R) (p q : List E) :
-    dispatch source target T v (p ++ q) =
-      dispatch source target T (dispatch source target T v q) p := by
-  induction p with
-  | nil => rfl
-  | cons e p ih => simp only [List.cons_append,dispatch,ih]
-
-theorem dispatch_add (T : R) (v w : V → R) (p : List E) :
-    dispatch source target T (v+w) p =
-      dispatch source target T v p + dispatch source target T w p := by
-  induction p with
-  | nil => rfl
-  | cons e p ih =>
-    simp only [dispatch,ih]
-    funext x
-    simp only [step,Pi.add_apply]
-    split_ifs <;> simp [mul_add,add_assoc,add_left_comm,add_comm]
-
-theorem dispatch_mul_right (T c : R) (v : V → R) (p : List E) :
-    dispatch source target T (fun x => v x*c) p =
-      fun x => dispatch source target T v p x*c := by
-  induction p with
-  | nil => rfl
-  | cons e p ih =>
-    simp only [dispatch,ih]
-    funext x
-    simp only [step]
-    split_ifs <;> simp [add_mul,mul_assoc]
-
-theorem dispatch_zero (T : R) (p : List E) : dispatch source target T (0 : V → R) p = 0 := by
-  induction p with
-  | nil => rfl
-  | cons e p ih =>
-    simp only [dispatch,ih]
-    funext x
-    simp [step]
-
-theorem dispatch_sum {A : Type*} (T : R) (F : Finset A) (v : A → V → R) (p : List E) :
-    dispatch source target T (∑ a ∈ F, v a) p =
-      ∑ a ∈ F, dispatch source target T (v a) p := by
-  induction F using Finset.induction_on with
-  | empty => simp [dispatch_zero]
-  | @insert a F ha ih => simp only [Finset.sum_insert ha,dispatch_add,ih]
-
-theorem dispatch_eq_sum_entries [Fintype V] (T : R) (v : V → R) (p : List E) (x : V) :
-    dispatch source target T v p x =
-      ∑ y, dispatch source target T (fun z => if z=y then 1 else 0) p x * v y := by
-  have hv : v = ∑ y, (fun z => (if z=y then 1 else 0)*v y) := by
-    funext z
-    simp [Finset.sum_apply]
-  conv_lhs => rw [hv]
-  rw [dispatch_sum]
-  simp only [Finset.sum_apply,dispatch_mul_right]
-
-theorem dispatch_eq_of_entries [Fintype V] (T : R) (p q : List E)
-    (h : ∀ x y, dispatch source target T (fun z => if z=y then 1 else 0) p x =
-      dispatch source target T (fun z => if z=y then 1 else 0) q x) (v : V → R) :
-    dispatch source target T v p = dispatch source target T v q := by
-  funext x
-  rw [dispatch_eq_sum_entries source target T v p x,
-    dispatch_eq_sum_entries source target T v q x]
-  exact Finset.sum_congr rfl (fun y _ => congrArg (fun a => a*v y) (h x y))
-
-/-- A finite lower region is closed under every target occurring in its
-edge schedule, so values outside that region never enter its product. -/
-theorem dispatch_congr_on (T : R) (S : Set V) (p : List E)
-    (htarget : ∀ e ∈ p, target e ∈ S) (v w : V → R)
-    (hvw : ∀ x ∈ S, v x=w x) :
-    ∀ x ∈ S, dispatch source target T v p x = dispatch source target T w p x := by
-  induction p with
-  | nil => exact hvw
-  | cons e p ih =>
-    have htail := ih (fun f hf => htarget f (List.mem_cons_of_mem _ hf))
-    intro x hx
-    simp only [dispatch,step,htail x hx,htail (target e) (htarget e List.mem_cons_self)]
-
-theorem dispatch_eq_finite_entries (T : R) (F : Finset V) (p : List E)
-    (htarget : ∀ e ∈ p, target e ∈ F) (v : V → R) (x : V) (hx : x ∈ F) :
-    dispatch source target T v p x =
-      ∑ y ∈ F, dispatch source target T (fun z => if z=y then 1 else 0) p x * v y := by
-  let w : V → R := ∑ y ∈ F, (fun z => (if z=y then 1 else 0)*v y)
-  have hvw : ∀ z ∈ (F : Set V), v z=w z := by
-    intro z hz
-    simp [w,Finset.sum_apply,show z ∈ F from hz]
-  rw [dispatch_congr_on source target T (F : Set V) p htarget v w hvw x hx]
-  dsimp only [w]
-  rw [dispatch_sum]
-  simp only [Finset.sum_apply,dispatch_mul_right]
-
-theorem dispatch_eq_of_entries_on (T : R) (F : Finset V) (p q : List E)
-    (hp : ∀ e ∈ p, target e ∈ F) (hq : ∀ e ∈ q, target e ∈ F)
-    (h : ∀ x ∈ F, ∀ y ∈ F,
-      dispatch source target T (fun z => if z=y then 1 else 0) p x =
-      dispatch source target T (fun z => if z=y then 1 else 0) q x)
-    (v : V → R) (x : V) (hx : x ∈ F) :
-    dispatch source target T v p x = dispatch source target T v q x := by
-  rw [dispatch_eq_finite_entries source target T F p hp v x hx,
-    dispatch_eq_finite_entries source target T F q hq v x hx]
-  exact Finset.sum_congr rfl (fun y hy => congrArg (fun a => a*v y) (h x hx y hy))
-
-variable {Λ : Type*} [LinearOrder Λ]
-
-/-- A genuine path in a sorted schedule is determined by its edge set and
-strict label order. Arbitrary permutations of repeated reflection labels
-therefore have no effect. -/
-theorem subpaths_eq_of_sorted (label : E → Λ)
-    (hties : ∀ e f, label e = label f → target e ≠ source f)
-    (p q : List E)
-    (hp : p.Pairwise (fun e f => label f ≤ label e))
-    (hq : q.Pairwise (fun e f => label f ≤ label e))
-    (hmem : ∀ e, e ∈ p ↔ e ∈ q) (x y : V) :
-    subpaths source target p x y = subpaths source target q x y := by
-  ext s
-  simp only [mem_subpaths]
-  constructor
-  · rintro ⟨hs,hpath⟩
-    refine ⟨?_,hpath⟩
-    apply sublist_of_decreasing_subset label
-      (decreasing_of_schedule_path source target label hties hpath (hp.sublist hs)) hq
-    exact fun e he => (hmem e).mp (hs.subset he)
-  · rintro ⟨hs,hpath⟩
-    refine ⟨?_,hpath⟩
-    apply sublist_of_decreasing_subset label
-      (decreasing_of_schedule_path source target label hties hpath (hq.sublist hs)) hp
-    exact fun e he => (hmem e).mpr (hs.subset he)
-
-theorem dispatch_eq_of_sorted [Fintype V] (T : R) (label : E → Λ)
-    (hties : ∀ e f, label e = label f → target e ≠ source f)
-    (p q : List E) (hpnd : p.Nodup) (hqnd : q.Nodup)
-    (hp : p.Pairwise (fun e f => label f ≤ label e))
-    (hq : q.Pairwise (fun e f => label f ≤ label e))
-    (hmem : ∀ e, e ∈ p ↔ e ∈ q) (v : V → R) :
-    dispatch source target T v p = dispatch source target T v q := by
-  apply dispatch_eq_of_entries
-  intro x y
-  rw [dispatch_eq_subpaths_sum source target T p hpnd,
-    dispatch_eq_subpaths_sum source target T q hqnd,
-    subpaths_eq_of_sorted source target label hties p q hp hq hmem x y]
-
-/-- Replacing each block by an equal actual operator replaces the product. -/
-theorem dispatch_flatMap_eq {A : Type*} (T : R) (blocks : List A) (p q : A → List E)
-    (h : ∀ a ∈ blocks, ∀ v, dispatch source target T v (p a) =
-      dispatch source target T v (q a)) (v : V → R) :
-    dispatch source target T v (blocks.flatMap p) =
-      dispatch source target T v (blocks.flatMap q) := by
-  induction blocks with
-  | nil => rfl
-  | cons a l ih =>
-    simp only [List.flatMap_cons,dispatch_append]
-    rw [ih (fun b hb => h b (List.mem_cons_of_mem _ hb)),h a List.mem_cons_self]
-
-
-variable {V' E' : Type*} (source' target' : E' → V')
-
-theorem IsPath.map_iff (j : V' → V) (hj : Function.Injective j) (J : E' → E)
-    (hs : ∀ e, source (J e)=j (source' e))
-    (ht : ∀ e, target (J e)=j (target' e)) {x y : V'} {p : List E'} :
-    IsPath source target (j x) (j y) (p.map J) ↔ IsPath source' target' x y p := by
-  induction p generalizing x with
-  | nil => exact hj.eq_iff
-  | cons e p ih =>
-    change (source (J e)=j x ∧ IsPath source target (target (J e)) (j y) (p.map J)) ↔ _
-    rw [hs,ht,ih,hj.eq_iff]
-    rfl
-
-theorem exists_list_lift (J : E' → E) (p : List E)
-    (h : ∀ e ∈ p, ∃ e', J e'=e) : ∃ p' : List E', p'.map J=p := by
-  induction p with
-  | nil => exact ⟨[],rfl⟩
-  | cons e p ih =>
-    obtain ⟨e',he⟩ := h e List.mem_cons_self
-    obtain ⟨p',hp⟩ := ih (fun f hf => h f (List.mem_cons_of_mem _ hf))
-    exact ⟨e'::p',by simp [he,hp]⟩
-
-/-- If all actual paths between the specified endpoints lie in a full embedded
-subgraph, its full sorted product computes the same entry. -/
-theorem dispatch_entry_graph_embedding (T : R) (label : E → Λ)
-    (hties : ∀ e f, label e=label f → target e ≠ source f)
-    (j : V' → V) (hj : Function.Injective j) (J : E' → E) (hJ : Function.Injective J)
-    (hs : ∀ e, source (J e)=j (source' e))
-    (ht : ∀ e, target (J e)=j (target' e))
-    (p : List E) (q : List E') (hpnd : p.Nodup) (hqnd : q.Nodup)
-    (hp : p.Pairwise (fun e f => label f ≤ label e))
-    (hq : q.Pairwise (fun e f => label (J f) ≤ label (J e)))
-    (hqfull : ∀ e, e ∈ q) (hJmem : ∀ e, J e ∈ p)
-    (x y : V')
-    (hpath : ∀ r ∈ subpaths source target p (j x) (j y), ∀ e ∈ r, ∃ e', J e'=e) :
-    dispatch source target T (fun z => if z=j y then 1 else 0) p (j x) =
-      dispatch source' target' T (fun z => if z=y then 1 else 0) q x := by
-  rw [dispatch_eq_subpaths_sum source target T p hpnd,
-    dispatch_eq_subpaths_sum source' target' T q hqnd]
-  symm
-  have hties' : ∀ e f, label (J e)=label (J f) → target' e ≠ source' f := by
-    intro e f he hf
-    apply hties (J e) (J f) he
-    rw [ht,hs,hf]
-  apply Finset.sum_bij (fun r _ => r.map J)
-  · intro r hr
-    obtain ⟨hr,hpr⟩ := (mem_subpaths source' target' q x y r).mp hr
-    apply (mem_subpaths source target p (j x) (j y) (r.map J)).mpr
-    constructor
-    · apply sublist_of_decreasing_subset label
-      · rw [List.pairwise_map]
-        exact decreasing_of_schedule_path source' target' (label ∘ J) hties' hpr (hq.sublist hr)
-      · exact hp
-      · intro e he
-        obtain ⟨f,_,rfl⟩ := List.mem_map.mp he
-        exact hJmem f
-    · exact hpr.map source' target' source target j J hs ht
-  · intro r _ s _ he
-    exact List.map_injective_iff.mpr hJ he
-  · intro r hr
-    obtain ⟨s,he⟩ := exists_list_lift J r (hpath r hr)
-    refine ⟨s,?_,he⟩
-    obtain ⟨hrsub,hrpath⟩ := (mem_subpaths source target p (j x) (j y) r).mp hr
-    have hspath : IsPath source' target' x y s :=
-      (IsPath.map_iff source target source' target' j hj J hs ht).mp (he ▸ hrpath)
-    apply (mem_subpaths source' target' q x y s).mpr
-    refine ⟨?_,hspath⟩
-    apply sublist_of_decreasing_subset (label ∘ J)
-    · have hstrict := decreasing_of_schedule_path source target label hties hrpath (hp.sublist hrsub)
-      rw [← he,List.pairwise_map] at hstrict
-      exact hstrict
-    · exact hq
-    · exact fun e _ => hqfull e
-  · intro r _
-    simp
-
-end
-end KLInvariance.SchedulePaths
-
-end
-
-
-section
-
-/-! Linear coordinates on the actual normalized root line. These give the
-rank-two geometry of finite wall crossings for genuine root schedules. -/
-namespace KLInvariance.TitsSpace
-open Module
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
-@[simp] theorem normalizedRoot_height (a : PositiveRoot M cs) :
-    height (normalizedRoot a) = 1 := by
-  simp [normalizedRoot,a.height_pos.ne']
-
-@[simp] theorem dual_normalizedRoot (f : Module.Dual ℝ (I → ℝ)) (a : PositiveRoot M cs) :
-    f (normalizedRoot a) = f a.val / height a.val := by
-  simp [normalizedRoot,div_eq_mul_inv,mul_comm]
-
-@[simp] theorem geometricFunctional_normalizedRoot (a : PositiveRoot M cs) :
-    geometricFunctional (M := M) (cs := cs) (normalizedRoot a) = geometricSlope a :=
-  dual_normalizedRoot _ a
-
-theorem normalizedRoot_mem_iff (P : Submodule ℝ (I → ℝ)) (a : PositiveRoot M cs) :
-    normalizedRoot a ∈ P ↔ a.val ∈ P := by
-  constructor
-  · intro h
-    have hh := P.smul_mem (height a.val) h
-    simpa [normalizedRoot,smul_smul,a.height_pos.ne'] using hh
-  · exact fun h => P.smul_mem _ h
-
-theorem rootPlane_eq_normalized_span (a b : PositiveRoot M cs) :
-    rootPlane a b = Submodule.span ℝ {normalizedRoot a,normalizedRoot b} := by
-  apply le_antisymm
-  · apply Submodule.span_le.mpr
-    intro x hx
-    rcases Set.mem_insert_iff.mp hx with rfl | hx
-    · exact (normalizedRoot_mem_iff _ a).mp (Submodule.subset_span (Set.mem_insert _ _))
-    · rw [Set.mem_singleton_iff.mp hx]
-      exact (normalizedRoot_mem_iff _ b).mp (Submodule.subset_span (Set.mem_insert_of_mem _ (Set.mem_singleton _)))
-  · apply Submodule.span_le.mpr
-    intro x hx
-    rcases Set.mem_insert_iff.mp hx with rfl | hx
-    · exact (normalizedRoot_mem_iff _ a).mpr (Submodule.subset_span (Set.mem_insert _ _))
-    · rw [Set.mem_singleton_iff.mp hx]
-      exact (normalizedRoot_mem_iff _ b).mpr (Submodule.subset_span (Set.mem_insert_of_mem _ (Set.mem_singleton _)))
-
-/-- The intersection of a root-plane with the height-one hyperplane is the
-actual affine line through the two normalized roots. -/
-theorem normalizedRoot_affine {a b c : PositiveRoot M cs}
-    (hc : c.val ∈ rootPlane a b) :
-    ∃ t : ℝ, normalizedRoot c = t • normalizedRoot a + (1-t) • normalizedRoot b := by
-  have hcn : normalizedRoot c ∈ Submodule.span ℝ {normalizedRoot a,normalizedRoot b} := by
-    rw [← rootPlane_eq_normalized_span]
-    exact (normalizedRoot_mem_iff _ _).mpr hc
-  obtain ⟨t,s,he⟩ := Submodule.mem_span_pair.mp hcn
-  have hh := congrArg (height (I := I)) he
-  simp only [map_add,map_smul,normalizedRoot_height,smul_eq_mul,mul_one] at hh
-  refine ⟨t,?_⟩
-  have hs : s=1-t := by linarith
-  exact hs ▸ he.symm
-
-/-- On each genuine root plane, every height-normalized linear functional
-is an affine function of the fixed angular slope. -/
-theorem exists_affine_rootPlane (f : Module.Dual ℝ (I → ℝ))
-    (a b : PositiveRoot M cs) (hab : a ≠ b) :
-    ∃ c d : ℝ, ∀ r : PositiveRoot M cs, r.val ∈ rootPlane a b →
-      f (normalizedRoot r) = c * geometricSlope r + d := by
-  let A := geometricSlope a
-  let B := geometricSlope b
-  let fa := f (normalizedRoot a)
-  let fb := f (normalizedRoot b)
-  have hAB : A-B ≠ 0 := sub_ne_zero.mpr (fun h => hab (geometricSlope_injective h))
-  let c := (fa-fb)/(A-B)
-  let d := fb-c*B
-  have ha : fa=c*A+d := by
-    dsimp [d,c]
-    field_simp
-    ring
-  have hb : fb=c*B+d := by dsimp [d]; ring
-  refine ⟨c,d,fun r hr => ?_⟩
-  obtain ⟨t,ht⟩ := normalizedRoot_affine hr
-  have hf : f (normalizedRoot r) = t*fa+(1-t)*fb := by
-    rw [ht]
-    simp only [map_add,map_smul,smul_eq_mul]
-    rfl
-  have hg : geometricSlope r = t*A+(1-t)*B := by
-    rw [← geometricFunctional_normalizedRoot,ht]
-    simp only [map_add,map_smul,geometricFunctional_normalizedRoot,smul_eq_mul]
-    rfl
-  rw [hf,hg,ha,hb]
-  ring
-
-namespace PlaneBasis
-variable {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-
-include B in
-theorem exists_affine (f : Module.Dual ℝ (I → ℝ)) :
-    ∃ c d : ℝ, ∀ r : PositiveRoot M cs, r.val ∈ P →
-      f (normalizedRoot r) = c * geometricSlope r + d := by
-  simpa only [B.plane_eq] using exists_affine_rootPlane f B.left B.right B.distinct
-
-end PlaneBasis
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Localization of genuine directed paths in a maximal root-plane coset.
-The full relative interval is proved to contain every such path. -/
-namespace KLInvariance.TitsSpace.PlaneBasis
-open Module SchedulePaths
-open AdaptedDyer
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
-/-- All edges of a plane-labelled path lie in the full induced relative
-interval between its endpoints, with the original ambient labels. -/
-theorem path_coordinates {a₀ : W}
-    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-    (g : planeSubgroup M cs P) {y : W} {p : List (AdaptedDyer.Edge cs)}
-    (hp : IsPath (source cs) (target cs) ((g:W)*a₀) y p)
-    (hP : ∀ e ∈ p, (rootLabel cs e).val ∈ P) :
-    ∃ k : planeSubgroup M cs P, y=(k:W)*a₀ ∧ BruhatLE B.system g k ∧
-      ∀ e ∈ p, ∃ l r : planeSubgroup M cs P,
-        source cs e=(l:W)*a₀ ∧ target cs e=(r:W)*a₀ ∧
-        BruhatLE B.system g l ∧ BruhatStep B.system l r ∧ BruhatLE B.system r k := by
-  induction p generalizing g with
-  | nil => exact ⟨g,hp.symm,.refl,by simp⟩
-  | cons e p ih =>
-    have hg : source cs e*a₀⁻¹ ∈ planeSubgroup M cs P := by
-      rw [hp.1]
-      simpa only [mul_assoc,mul_inv_cancel,mul_one] using g.property
-    have hh := mem_coset_of_graphRoot e.property hg (hP e List.mem_cons_self)
-    let h : planeSubgroup M cs P := ⟨target cs e*a₀⁻¹,hh⟩
-    have heqh : target cs e=(h:W)*a₀ := by dsimp [h]; group
-    have he : BruhatStep B.system g h := by
-      apply (B.coset_step_iff hmin g h).mp
-      rw [← hp.1,← heqh]
-      exact e.property
-    obtain ⟨k,hky,hk,hbody⟩ := ih h (heqh ▸ hp.2)
-      (fun f hf => hP f (List.mem_cons_of_mem _ hf))
-    refine ⟨k,hky,(show BruhatLE B.system g h from .single he).trans hk,?_⟩
-    intro f hf
-    rcases List.mem_cons.mp hf with rfl | hf
-    · exact ⟨g,h,hp.1,heqh,.refl,he,hk⟩
-    · obtain ⟨l,r,hl,hr,hhl,hlr,hrk⟩ := hbody f hf
-      exact ⟨l,r,hl,hr,(show BruhatLE B.system g h from .single he).trans hhl,hlr,hrk⟩
-
-def intervalToAmbient {a₀ : W} {g k : planeSubgroup M cs P}
-    (z : Interval B.system g k) : W := (z.val:W)*a₀
-
-theorem intervalToAmbient_injective {a₀ : W} {g k : planeSubgroup M cs P} :
-    Function.Injective (B.intervalToAmbient (a₀ := a₀) (g := g) (k := k)) := by
-  intro x y h
-  exact Subtype.ext (Subtype.ext (mul_right_cancel h))
-
-def edgeToAmbient {a₀ : W}
-    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-    {g k : planeSubgroup M cs P} (e : BruhatGraph.Edge B.system g k) : AdaptedDyer.Edge cs :=
-  ⟨(B.intervalToAmbient e.val.1, B.intervalToAmbient e.val.2),
-    (B.coset_step_iff hmin e.val.1.val e.val.2.val).mpr e.property⟩
-
-@[simp] theorem edgeToAmbient_source {a₀ : W}
-    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-    {g k : planeSubgroup M cs P} (e : BruhatGraph.Edge B.system g k) :
-    source cs (B.edgeToAmbient hmin e) = B.intervalToAmbient (a₀ := a₀) (BruhatGraph.source B.system g k e) := rfl
-
-@[simp] theorem edgeToAmbient_target {a₀ : W}
-    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-    {g k : planeSubgroup M cs P} (e : BruhatGraph.Edge B.system g k) :
-    target cs (B.edgeToAmbient hmin e) = B.intervalToAmbient (a₀ := a₀) (BruhatGraph.target B.system g k e) := rfl
-
-theorem edgeToAmbient_injective {a₀ : W}
-    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-    {g k : planeSubgroup M cs P} :
-    Function.Injective (B.edgeToAmbient hmin (g := g) (k := k)) := by
-  intro e f h
-  have hh := congrArg Subtype.val h
-  apply Subtype.ext
-  exact Prod.ext (B.intervalToAmbient_injective (congrArg Prod.fst hh))
-    (B.intervalToAmbient_injective (congrArg Prod.snd hh))
-
-@[simp] theorem edgeToAmbient_root {a₀ : W}
-    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-    {g k : planeSubgroup M cs P} (e : BruhatGraph.Edge B.system g k) :
-    rootLabel cs (B.edgeToAmbient hmin e) = B.edgeRoot e.property := by
-  apply (positiveRootEquiv M cs).symm.injective
-  apply Subtype.ext
-  change rootRef (rootLabel cs (B.edgeToAmbient hmin e)) = rootRef (B.edgeRoot e.property)
-  rw [rootRef_rootLabel]
-  have hr : rootRef (B.edgeRoot e.property) = (e.val.2.val:W)*(e.val.1.val:W)⁻¹ :=
-    (rootReflection_represents M cs _).reflection_unique
-      (graphRoot_represents ((B.bruhatStep_iff _ _).mpr e.property))
-  rw [hr]
-  change ((e.val.2.val:W)*a₀)*((e.val.1.val:W)*a₀)⁻¹ = _
-  group
-
-/-- Edge-by-edge lifting of every plane path into the full relative interval.
-It is obtained from actual path connectivity, not from ambient comparability
-alone (which would not suffice). -/
-theorem path_edges_lift {a₀ : W}
-    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-    {g k : planeSubgroup M cs P} {p : List (AdaptedDyer.Edge cs)}
-    (hp : IsPath (source cs) (target cs) ((g:W)*a₀) ((k:W)*a₀) p)
-    (hP : ∀ e ∈ p, (rootLabel cs e).val ∈ P) :
-    ∀ e ∈ p, ∃ e' : BruhatGraph.Edge B.system g k, B.edgeToAmbient hmin e'=e := by
-  obtain ⟨k',he,_,hbody⟩ := B.path_coordinates hmin g hp hP
-  have hk : k'=k := Subtype.ext (mul_right_cancel he.symm)
-  subst k'
-  intro e he
-  obtain ⟨l,r,hl,hr,hgl,hlr,hrk⟩ := hbody e he
-  let vl : Interval B.system g k := ⟨l,hgl,(show BruhatLE B.system l r from .single hlr).trans hrk⟩
-  let vr : Interval B.system g k := ⟨r,hgl.trans (.single hlr),hrk⟩
-  refine ⟨⟨(vl,vr),hlr⟩,?_⟩
-  apply Subtype.ext
-  exact Prod.ext hl.symm hr.symm
-
-
-include B in
-/-- Reversal of a complete angular block of genuine ambient edges preserves
-all its endpoint entries. Fullness is with respect to the actual root plane,
-not to a rank-two subgroup chosen merely to contain two arbitrary reflections. -/
-theorem dispatch_plane_reverse {R : Type*} [Semiring R] (T : R) (b : W)
-    (p : List (AdaptedDyer.Edge cs)) (hnd : p.Nodup)
-    (hfull : ∀ e, e ∈ p ↔ (rootLabel cs e).val ∈ P ∧ BruhatLE cs (target cs e) b)
-    (hs : p.Pairwise (fun e f => geometricSlope (rootLabel cs f) ≤ geometricSlope (rootLabel cs e)))
-    (x y : W) (hy : BruhatLE cs y b) :
-    dispatch (source cs) (target cs) T (fun z => if z=y then 1 else 0) p x =
-      dispatch (source cs) (target cs) T (fun z => if z=y then 1 else 0) p.reverse x := by
-  obtain ⟨a₀,g,hg,hx,hmin⟩ := planeCoset_exists_minimal M cs P x
-  let g' : planeSubgroup M cs P := ⟨g,hg⟩
-  have hx' : x=(g':W)*a₀ := hx
-  let L : AdaptedDyer.Edge cs → ℝ := fun e => geometricSlope (rootLabel cs e)
-  have hties : ∀ e f, L e=L f → target cs e ≠ source cs f := by
-    intro e f he
-    exact fun hj => rootLabel_ne_of_join cs e f hj (geometricSlope_injective he)
-  by_cases hex : ∃ r : List (AdaptedDyer.Edge cs),
-      IsPath (source cs) (target cs) x y r ∧ ∀ e ∈ r, (rootLabel cs e).val ∈ P
-  · obtain ⟨r,hr,hrP⟩ := hex
-    obtain ⟨k,hyk,hgk,_⟩ := B.path_coordinates hmin g' (hx' ▸ hr) hrP
-    let lo : Interval B.system g' k := ⟨g',.refl,hgk⟩
-    let hi : Interval B.system g' k := ⟨k,hgk,.refl⟩
-    let j := B.intervalToAmbient (a₀ := a₀) (g := g') (k := k)
-    let J := B.edgeToAmbient hmin (g := g') (k := k)
-    have hjlo : j lo=x := hx'.symm
-    have hjhi : j hi=y := hyk.symm
-    let : Fintype (BruhatGraph.Edge B.system g' k) := Fintype.ofFinite _
-    let q : List (BruhatGraph.Edge B.system g' k) :=
-      Finset.univ.toList.mergeSort (fun e f => B.intervalEdgeSlope g' k f ≤ B.intervalEdgeSlope g' k e)
-    have hqnd : q.Nodup := (List.mergeSort_perm _ _).nodup_iff.mpr (Finset.nodup_toList _)
-    have hqfull : ∀ e, e ∈ q := by intro e; simp [q]
-    have hqs : q.Pairwise (fun e f => B.intervalEdgeSlope g' k f ≤ B.intervalEdgeSlope g' k e) :=
-      List.pairwise_mergeSort' _ _
-    have hJroot (e : BruhatGraph.Edge B.system g' k) : L (J e)=B.intervalEdgeSlope g' k e := by
-      dsimp [L,J,intervalEdgeSlope]
-      rw [B.edgeToAmbient_root]
-    have hJmem (e : BruhatGraph.Edge B.system g' k) : J e ∈ p := by
-      apply (hfull _).mpr
-      constructor
-      · dsimp [J]
-        rw [B.edgeToAmbient_root]
-        exact B.edgeRoot_plane e.property
-      · have ht : BruhatLE cs ((e.val.2.val:W)*a₀) ((k:W)*a₀) :=
-          B.coset_bruhat hmin e.val.2.property.2
-        exact ht.trans (hyk ▸ hy)
-    have hlift (s : List (AdaptedDyer.Edge cs))
-        (hst : ∀ e ∈ s, e ∈ p)
-        (hp : IsPath (source cs) (target cs) (j lo) (j hi) s) :
-        ∀ e ∈ s, ∃ e' : BruhatGraph.Edge B.system g' k, J e'=e :=
-      B.path_edges_lift hmin hp (fun e he => ((hfull e).mp (hst e he)).1)
-    have hleft := dispatch_entry_graph_embedding (source cs) (target cs)
-      (BruhatGraph.source B.system g' k) (BruhatGraph.target B.system g' k)
-      T L hties j (B.intervalToAmbient_injective) J (B.edgeToAmbient_injective hmin)
-      (B.edgeToAmbient_source hmin) (B.edgeToAmbient_target hmin)
-      p q hnd hqnd hs (by simpa only [hJroot] using hqs) hqfull hJmem lo hi (by
-        intro s hsp
-        obtain ⟨hsub,hpath⟩ := (mem_subpaths _ _ _ _ _ _).mp hsp
-        exact hlift s (fun e he => hsub.subset he) hpath)
-    have hright := dispatch_entry_graph_embedding (source cs) (target cs)
-      (BruhatGraph.source B.system g' k) (BruhatGraph.target B.system g' k)
-      T (fun e => -L e) (fun e f he => hties e f (neg_injective he))
-      j (B.intervalToAmbient_injective) J (B.edgeToAmbient_injective hmin)
-      (B.edgeToAmbient_source hmin) (B.edgeToAmbient_target hmin)
-      p.reverse q.reverse (List.nodup_reverse.mpr hnd) (List.nodup_reverse.mpr hqnd)
-      (by rw [List.pairwise_reverse]; exact hs.imp (fun he => neg_le_neg he))
-      (by rw [List.pairwise_reverse]; exact hqs.imp (fun he => by simpa only [hJroot] using neg_le_neg he))
-      (fun e => List.mem_reverse.mpr (hqfull e))
-      (fun e => List.mem_reverse.mpr (hJmem e)) lo hi (by
-        intro s hsp
-        obtain ⟨hsub,hpath⟩ := (mem_subpaths _ _ _ _ _ _).mp hsp
-        exact hlift s (fun e he => List.mem_reverse.mp (hsub.subset he)) hpath)
-    rw [hjlo,hjhi] at hleft hright
-    rw [hleft,hright]
-    exact B.dispatch_reverse_endpoints hgk q hqnd hqfull hqs T
-  · have hzero (q : List (AdaptedDyer.Edge cs))
-        (hqp : ∀ e ∈ q, e ∈ p) : subpaths (source cs) (target cs) q x y = ∅ := by
-      apply Finset.eq_empty_iff_forall_notMem.mpr
-      intro r hr
-      obtain ⟨hsub,hpath⟩ := (mem_subpaths _ _ _ _ _ _).mp hr
-      exact hex ⟨r,hpath,fun e he => ((hfull e).mp (hqp e (hsub.subset he))).1⟩
-    rw [dispatch_eq_subpaths_sum _ _ _ _ hnd,
-      dispatch_eq_subpaths_sum _ _ _ _ (List.nodup_reverse.mpr hnd),
-      hzero p (fun _ he => he),hzero p.reverse (fun _ he => List.mem_reverse.mp he)]
-
-end
-end KLInvariance.TitsSpace.PlaneBasis
-
-end
-
-
-section
-
-/-! Algebra and finite ordered-list bookkeeping for wall crossings. -/
-namespace KLInvariance.SchedulePaths
-universe u v w
-variable {E : Type u} {A : Type v} {Λ : Type w} [LinearOrder A] [LinearOrder Λ]
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-local instance : DecidableEq E := Classical.decEq E
-
-def sortedSchedule (F : Finset E) (score : E → Λ) : List E :=
-  F.toList.mergeSort (fun e f => score f ≤ score e)
-
-@[simp] theorem mem_sortedSchedule (F : Finset E) (score : E → Λ) (e : E) :
-    e ∈ sortedSchedule F score ↔ e ∈ F := by simp [sortedSchedule]
-
-theorem sortedSchedule_nodup (F : Finset E) (score : E → Λ) :
-    (sortedSchedule F score).Nodup :=
-  (List.mergeSort_perm _ _).nodup_iff.mpr (Finset.nodup_toList _)
-
-theorem sortedSchedule_pairwise (F : Finset E) (score : E → Λ) :
-    (sortedSchedule F score).Pairwise (fun e f => score f ≤ score e) :=
-  List.pairwise_mergeSort' _ _
-
-def blockValues (F : Finset E) (primary : E → A) : List A :=
-  (F.image primary).sort (fun a b => b ≤ a)
-
-def scheduleBlock (F : Finset E) (primary : E → A) (score : E → Λ) (a : A) : List E :=
-  sortedSchedule (F.filter (fun e => primary e=a)) score
-
-def groupedSchedule (F : Finset E) (primary : E → A) (score : E → Λ) : List E :=
-  (blockValues F primary).flatMap (scheduleBlock F primary score)
-
-@[simp] theorem mem_blockValues (F : Finset E) (primary : E → A) (a : A) :
-    a ∈ blockValues F primary ↔ ∃ e ∈ F, primary e=a := by
-  simp [blockValues]
-
-omit [LinearOrder A] in
-@[simp] theorem mem_scheduleBlock (F : Finset E) (primary : E → A) (score : E → Λ)
-    (a : A) (e : E) : e ∈ scheduleBlock F primary score a ↔ e ∈ F ∧ primary e=a := by
-  simp [scheduleBlock]
-
-@[simp] theorem mem_groupedSchedule (F : Finset E) (primary : E → A) (score : E → Λ)
-    (e : E) : e ∈ groupedSchedule F primary score ↔ e ∈ F := by
-  constructor
-  · intro he
-    obtain ⟨a,_,ha⟩ := List.mem_flatMap.mp he
-    exact ((mem_scheduleBlock F primary score a e).mp ha).1
-  · intro he
-    apply List.mem_flatMap.mpr
-    exact ⟨primary e,(mem_blockValues F primary _).mpr ⟨e,he,rfl⟩,
-      (mem_scheduleBlock F primary score _ _).mpr ⟨he,rfl⟩⟩
-
-theorem blockValues_strict (F : Finset E) (primary : E → A) :
-    (blockValues F primary).Pairwise (fun a b => b < a) := by
-  have hs := Finset.pairwise_sort (F.image primary) (fun a b => b ≤ a)
-  have hn := Finset.sort_nodup (F.image primary) (fun a b => b ≤ a)
-  exact (List.pairwise_and_iff.mpr ⟨hs,hn⟩).imp (fun h => lt_of_le_of_ne h.1 h.2.symm)
-
-theorem groupedSchedule_nodup (F : Finset E) (primary : E → A) (score : E → Λ) :
-    (groupedSchedule F primary score).Nodup := by
-  apply List.nodup_flatMap.mpr
-  constructor
-  · intro a _
-    exact sortedSchedule_nodup _ _
-  · apply (blockValues_strict F primary).imp
-    intro a b hab
-    apply List.disjoint_left.mpr
-    intro e hea heb
-    have ha := ((mem_scheduleBlock F primary score a e).mp hea).2
-    have hb := ((mem_scheduleBlock F primary score b e).mp heb).2
-    exact hab.ne (hb.symm.trans ha)
-
-theorem groupedSchedule_pairwise (F : Finset E) (primary : E → A) (score : E → Λ)
-    (h : ∀ e ∈ F, ∀ f ∈ F, primary f < primary e → score f ≤ score e) :
-    (groupedSchedule F primary score).Pairwise (fun e f => score f ≤ score e) := by
-  apply List.pairwise_flatMap.mpr
-  constructor
-  · intro a _
-    exact sortedSchedule_pairwise _ _
-  · apply (blockValues_strict F primary).imp
-    intro a b hab e he f hf
-    obtain ⟨he,heq⟩ := (mem_scheduleBlock F primary score a e).mp he
-    obtain ⟨hf,hfq⟩ := (mem_scheduleBlock F primary score b f).mp hf
-    exact h e he f hf (by simpa only [heq,hfq] using hab)
-
-variable {V : Type*} {R : Type*} [Semiring R] (source target : E → V)
-
-theorem dispatch_entries_eq_of_sorted (T : R) (label : E → Λ)
-    (hties : ∀ e f, label e=label f → target e ≠ source f)
-    (p q : List E) (hpnd : p.Nodup) (hqnd : q.Nodup)
-    (hp : p.Pairwise (fun e f => label f ≤ label e))
-    (hq : q.Pairwise (fun e f => label f ≤ label e))
-    (hmem : ∀ e, e ∈ p ↔ e ∈ q) (x y : V) :
-    dispatch source target T (fun z => if z=y then 1 else 0) p x =
-      dispatch source target T (fun z => if z=y then 1 else 0) q x := by
-  rw [dispatch_eq_subpaths_sum source target T p hpnd,
-    dispatch_eq_subpaths_sum source target T q hqnd,
-    subpaths_eq_of_sorted source target label hties p q hp hq hmem x y]
-
-omit [LinearOrder A] in
-/-- Replacement on a finite lower region requires no artificial finiteness
-assumption on the ambient Coxeter group. -/
-theorem dispatch_flatMap_eq_on (T : R) (S : Set V) (blocks : List A) (p q : A → List E)
-    (htarget : ∀ a ∈ blocks, ∀ e ∈ p a, target e ∈ S)
-    (h : ∀ a ∈ blocks, ∀ v, ∀ x ∈ S,
-      dispatch source target T v (p a) x = dispatch source target T v (q a) x)
-    (v : V → R) : ∀ x ∈ S,
-    dispatch source target T v (blocks.flatMap p) x =
-      dispatch source target T v (blocks.flatMap q) x := by
-  induction blocks with
-  | nil => exact fun _ _ => rfl
-  | cons a l ih =>
-    intro x hx
-    simp only [List.flatMap_cons,dispatch_append]
-    rw [dispatch_congr_on source target T S (p a) (htarget a List.mem_cons_self)
-      _ _ (ih (fun b hb => htarget b (List.mem_cons_of_mem _ hb))
-      (fun b hb => h b (List.mem_cons_of_mem _ hb))) x hx]
-    exact h a List.mem_cons_self _ x hx
-
-end
-end KLInvariance.SchedulePaths
-
-end
-
-
-section
-
-/-! The actual rank-two edge product is independent of every linear
-functional restricted to that root plane, including a wall tie refinement. -/
-namespace KLInvariance.TitsSpace
-open Module SchedulePaths AdaptedDyer
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
-def lexRootScore (f : Module.Dual ℝ (I → ℝ)) (a : PositiveRoot M cs) : ℝ ×ₗ ℝ :=
-  toLex (f (normalizedRoot a),geometricSlope a)
-
-theorem lexRootScore_injective (f : Module.Dual ℝ (I → ℝ)) :
-    Function.Injective (lexRootScore (M := M) (cs := cs) f) := by
-  intro a b hab
-  apply geometricSlope_injective
-  exact congrArg (fun p : ℝ ×ₗ ℝ => (ofLex p).2) hab
-
-def lexEdgeScore (f : Module.Dual ℝ (I → ℝ)) (e : AdaptedDyer.Edge cs) : ℝ ×ₗ ℝ :=
-  lexRootScore f (rootLabel cs e)
-
-theorem lexEdgeScore_ties (f : Module.Dual ℝ (I → ℝ)) (e e' : AdaptedDyer.Edge cs)
-    (he : lexEdgeScore f e=lexEdgeScore f e') : target cs e ≠ source cs e' := by
-  intro hj
-  exact rootLabel_ne_of_join cs e e' hj (lexRootScore_injective f he)
-
-omit [Fintype I] in
-theorem lex_affine_nonneg {c d x y : ℝ} (hc : 0 ≤ c)
-    (h : (toLex (c*x+d,x) : ℝ ×ₗ ℝ) ≤ toLex (c*y+d,y)) : x ≤ y := by
-  rcases Prod.Lex.toLex_le_toLex.mp h with h | h
-  · by_contra hn
-    have hxy : y < x := lt_of_not_ge hn
-    nlinarith [mul_nonneg hc (sub_nonneg.mpr hxy.le)]
-  · exact h.2
-
-omit [Fintype I] in
-theorem lex_affine_neg {c d x y : ℝ} (hc : c < 0)
-    (h : (toLex (c*x+d,x) : ℝ ×ₗ ℝ) ≤ toLex (c*y+d,y)) : y ≤ x := by
-  rcases Prod.Lex.toLex_le_toLex.mp h with h | h
-  · by_contra hn
-    have hxy : x < y := lt_of_not_ge hn
-    nlinarith [mul_nonpos_of_nonpos_of_nonneg hc.le (sub_nonneg.mpr hxy.le)]
-  · have he : x=y := by nlinarith [h.1]
-    exact he.symm.le
-
-namespace PlaneBasis
-variable {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-
-include B in
-/-- Any linearly sorted full plane block is either angularly sorted or
-reverse-angularly sorted. This includes tied primary labels refined by the
-fixed genuine geometric functional. -/
-theorem lex_block_orientation (f : Module.Dual ℝ (I → ℝ))
-    (p : List (AdaptedDyer.Edge cs))
-    (hP : ∀ e ∈ p, (rootLabel cs e).val ∈ P)
-    (hs : p.Pairwise (fun e e' => lexEdgeScore f e' ≤ lexEdgeScore f e)) :
-    p.Pairwise (fun e e' => geometricSlope (rootLabel cs e') ≤ geometricSlope (rootLabel cs e)) ∨
-    p.Pairwise (fun e e' => geometricSlope (rootLabel cs e) ≤ geometricSlope (rootLabel cs e')) := by
-  obtain ⟨c,d,hf⟩ := B.exists_affine f
-  have hv (e : AdaptedDyer.Edge cs) (he : e ∈ p) :
-      lexEdgeScore f e=toLex (c*geometricSlope (rootLabel cs e)+d,geometricSlope (rootLabel cs e)) := by
-    unfold lexEdgeScore lexRootScore
-    rw [hf _ (hP e he)]
-  by_cases hc : 0 ≤ c
-  · left
-    apply hs.imp_of_mem
-    intro e e' he he' hee'
-    rw [hv e he,hv e' he'] at hee'
-    exact lex_affine_nonneg hc hee'
-  · right
-    apply hs.imp_of_mem
-    intro e e' he he' hee'
-    rw [hv e he,hv e' he'] at hee'
-    exact lex_affine_neg (lt_of_not_ge hc) hee'
-
-include B in
-/-- A complete plane block, sorted by any functional, has precisely the
-same actual path product as its angularly sorted version. -/
-theorem dispatch_plane_functional {R : Type*} [Semiring R] (T : R) (b : W)
-    (f : Module.Dual ℝ (I → ℝ)) (p q : List (AdaptedDyer.Edge cs))
-    (hpnd : p.Nodup) (hqnd : q.Nodup)
-    (hpfull : ∀ e, e ∈ p ↔ (rootLabel cs e).val ∈ P ∧ BruhatLE cs (target cs e) b)
-    (hqfull : ∀ e, e ∈ q ↔ (rootLabel cs e).val ∈ P ∧ BruhatLE cs (target cs e) b)
-    (hp : p.Pairwise (fun e e' => lexEdgeScore f e' ≤ lexEdgeScore f e))
-    (hq : q.Pairwise (fun e e' => geometricSlope (rootLabel cs e') ≤ geometricSlope (rootLabel cs e)))
-    (x y : W) (hy : BruhatLE cs y b) :
-    dispatch (source cs) (target cs) T (fun z => if z=y then 1 else 0) p x =
-      dispatch (source cs) (target cs) T (fun z => if z=y then 1 else 0) q x := by
-  let L : AdaptedDyer.Edge cs → ℝ := fun e => geometricSlope (rootLabel cs e)
-  have hties : ∀ e e', L e=L e' → target cs e ≠ source cs e' := by
-    intro e e' he hj
-    exact rootLabel_ne_of_join cs e e' hj (geometricSlope_injective he)
-  rcases B.lex_block_orientation f p (fun e he => ((hpfull e).mp he).1) hp with hs|hs
-  · exact dispatch_entries_eq_of_sorted (source cs) (target cs) T L hties
-      p q hpnd hqnd hs hq (fun e => (hpfull e).trans (hqfull e).symm) x y
-  · have hentry := dispatch_entries_eq_of_sorted (source cs) (target cs) T
-      (fun e => -L e) (fun e e' he => hties e e' (neg_injective he))
-      p q.reverse hpnd (List.nodup_reverse.mpr hqnd)
-      (hs.imp (fun he => neg_le_neg he))
-      (by rw [List.pairwise_reverse]; exact hq.imp (fun he => neg_le_neg he))
-      (fun e => (hpfull e).trans ((hqfull e).symm.trans List.mem_reverse.symm)) x y
-    exact hentry.trans (B.dispatch_plane_reverse T b q hqnd hqfull hq x y hy).symm
-
-end PlaneBasis
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Generic pencils of genuine linear root-label functionals. At a wall all
-roots of a tied block belong to one full actual root plane. -/
-namespace KLInvariance.ReflectionSchedules
-open Polynomial
-universe u v
-variable {I : Type u} [Fintype I]
-noncomputable section
-
-/-- A countable family of nonzero vectors can be simultaneously avoided. -/
-theorem exists_avoiding_functional {A : Type v} [Countable A] (v : A → (I → ℝ)) :
-    ∃ f : Module.Dual ℝ (I → ℝ), ∀ a, v a ≠ 0 → f (v a) ≠ 0 := by
-  classical
-  let P := {a : A // v a ≠ 0}
-  have hn (a : P) : coordinatePolynomial (v a.val) ≠ 0 := by
-    intro h
-    apply a.property
-    apply coordinatePolynomial_injective
-    simpa [coordinatePolynomial] using h
-  let bad : Set ℝ := ⋃ a : P, {t | (coordinatePolynomial (v a.val)).IsRoot t}
-  have hb : bad.Countable := Set.countable_iUnion
-    (fun a => (Polynomial.finite_setOfPred_isRoot (hn a)).countable)
-  obtain ⟨t,ht⟩ : ∃ t, t ∉ bad := by
-    by_contra h
-    push Not at h
-    have he : bad=Set.univ := Set.eq_univ_of_forall h
-    rw [he] at hb
-    exact Set.not_countable_univ hb
-  refine ⟨momentFunctional t,fun a ha he => ht ?_⟩
-  apply Set.mem_iUnion.mpr
-  refine ⟨⟨a,ha⟩,?_⟩
-  simpa only [Set.mem_ofPred_eq,Polynomial.IsRoot,coordinatePolynomial_eval] using he
-
-end
-end KLInvariance.ReflectionSchedules
-
-namespace KLInvariance.TitsSpace
-open Module
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
-def pencil (f g : Module.Dual ℝ (I → ℝ)) (t : ℝ) : Module.Dual ℝ (I → ℝ) :=
-  (1-t) • f + t • g
-
-omit [Fintype I] in
-@[simp] theorem pencil_zero (f g : Module.Dual ℝ (I → ℝ)) : pencil f g 0=f := by
-  simp [pencil]
-omit [Fintype I] in
-@[simp] theorem pencil_one (f g : Module.Dual ℝ (I → ℝ)) : pencil f g 1=g := by
-  simp [pencil]
-
-omit [Fintype I] in
-theorem pencil_apply (f g : Module.Dual ℝ (I → ℝ)) (t : ℝ) (v : I → ℝ) :
-    pencil f g t v=(1-t)*f v+t*g v := rfl
-
-def rootDifference (a b : PositiveRoot M cs) : I → ℝ := normalizedRoot b-normalizedRoot a
-
-def wallVector (f : Module.Dual ℝ (I → ℝ)) (a b c : PositiveRoot M cs) : I → ℝ :=
-  f (rootDifference a b) • rootDifference a c -
-    f (rootDifference a c) • rootDifference a b
-
-/-- This is an actual genericity condition on linear functionals, not
-an assumed path or Hecke identity. -/
-def Transverse (f g : Module.Dual ℝ (I → ℝ)) : Prop :=
-  ∀ a b c : PositiveRoot M cs, g (wallVector f a b c)=0 → wallVector f a b c=0
-
-theorem pencil_tie_plane (f g : Module.Dual ℝ (I → ℝ))
-    (hf : Function.Injective (fun a : PositiveRoot M cs => f (normalizedRoot a)))
-    (hg : Transverse (M := M) (cs := cs) f g) (t : ℝ)
-    (a b : PositiveRoot M cs) (hab : a ≠ b)
-    (he : pencil f g t (normalizedRoot a)=pencil f g t (normalizedRoot b)) :
-    ∀ c : PositiveRoot M cs,
-      pencil f g t (normalizedRoot c)=pencil f g t (normalizedRoot a) ↔
-        c.val ∈ rootPlane a b := by
-  let v := rootDifference a b
-  have hv : f v ≠ 0 := by
-    change f (normalizedRoot b-normalizedRoot a) ≠ 0
-    rw [map_sub,sub_ne_zero]
-    exact fun h => hab (hf h).symm
-  have hvt : (1-t)*f v+t*g v=0 := by
-    rw [← pencil_apply]
-    change pencil f g t (normalizedRoot b-normalizedRoot a)=0
-    rw [map_sub,he,sub_self]
-  have ht : t ≠ 0 := by
-    intro h
-    subst t
-    simp only [sub_zero,one_mul,zero_mul,add_zero] at hvt
-    exact hv hvt
-  intro c
-  constructor
-  · intro hc
-    let w := rootDifference a c
-    have hwt : (1-t)*f w+t*g w=0 := by
-      rw [← pencil_apply]
-      change pencil f g t (normalizedRoot c-normalizedRoot a)=0
-      rw [map_sub,hc,sub_self]
-    have hz : g (wallVector f a b c)=0 := by
-      change g (f v • w-f w • v)=0
-      rw [map_sub,map_smul,map_smul,smul_eq_mul,smul_eq_mul]
-      apply (mul_eq_zero.mp (show t * (f v*g w-f w*g v)=0 by
-        calc
-          _ = f v*((1-t)*f w+t*g w)-f w*((1-t)*f v+t*g v) := by ring
-          _ = 0 := by rw [hwt,hvt]; ring)).resolve_left ht
-    have heq : f v • w=f w • v := sub_eq_zero.mp (hg a b c hz)
-    let P := rootPlane a b
-    have haP : normalizedRoot a ∈ P := (normalizedRoot_mem_iff _ _).mpr
-      (Submodule.subset_span (Set.mem_insert _ _))
-    have hbP : normalizedRoot b ∈ P := (normalizedRoot_mem_iff _ _).mpr
-      (Submodule.subset_span (Set.mem_insert_of_mem _ (Set.mem_singleton _)))
-    have hvP : v ∈ P := P.sub_mem hbP haP
-    have hwP : w ∈ P := by
-      have hh := P.smul_mem (f v)⁻¹ (heq ▸ P.smul_mem (f w) hvP)
-      simpa only [smul_smul,inv_mul_cancel₀ hv,one_smul] using hh
-    apply (normalizedRoot_mem_iff P c).mp
-    have hh := P.add_mem hwP haP
-    simpa only [w,rootDifference,sub_add_cancel] using hh
-  · intro hc
-    obtain ⟨s,hs⟩ := normalizedRoot_affine hc
-    rw [hs,map_add,map_smul,map_smul,smul_eq_mul,smul_eq_mul,← he]
-    ring
-
-/-- One intermediate functional is simultaneously transverse to any two
-specified generic endpoints, even with infinitely many positive roots. -/
-theorem exists_common_transverse (f h : Module.Dual ℝ (I → ℝ)) :
-    ∃ g : Module.Dual ℝ (I → ℝ),
-      Function.Injective (fun a : PositiveRoot M cs => g (normalizedRoot a)) ∧
-      Transverse (M := M) (cs := cs) f g ∧ Transverse (M := M) (cs := cs) h g := by
-  let := countable_Positiveroot M cs
-  let E := PositiveRoot M cs
-  let A := (E × E) ⊕ (Bool × E × E × E)
-  let v : A → (I → ℝ) := fun x => match x with
-    | .inl (a,b) => rootDifference a b
-    | .inr (i,a,b,c) => wallVector (if i then f else h) a b c
-  obtain ⟨g,hg⟩ := ReflectionSchedules.exists_avoiding_functional v
-  refine ⟨g,?_,?_,?_⟩
-  · intro a b he
-    by_contra hab
-    have hn : rootDifference a b ≠ 0 := by
-      exact fun hz => hab (normalizedRoot_injective (sub_eq_zero.mp hz)).symm
-    apply hg (.inl (a,b)) hn
-    simp only [v,rootDifference,map_sub,he,sub_self]
-  · intro a b c he
-    by_contra hz
-    exact hg (.inr (true,a,b,c)) (by simpa [v] using hz)
-      (by simpa [v] using he)
-  · intro a b c he
-    by_contra hz
-    exact hg (.inr (false,a,b,c)) (by simpa [v] using hz)
-      (by simpa [v] using he)
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! Finite wall crossing for actual reflection-edge path matrices. -/
-namespace KLInvariance.WallDyer
-open Module SchedulePaths TitsSpace AdaptedDyer
-open scoped Topology
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
-def rootValue (f : Module.Dual ℝ (I → ℝ)) (e : Edge cs) : ℝ :=
-  f (normalizedRoot (rootLabel cs e))
-
-def rootPlaneTies (f : Module.Dual ℝ (I → ℝ)) : Prop :=
-  ∀ a b : PositiveRoot M cs, a ≠ b → f (normalizedRoot a)=f (normalizedRoot b) →
-    ∀ c : PositiveRoot M cs, f (normalizedRoot c)=f (normalizedRoot a) ↔
-      c.val ∈ rootPlane a b
-
-def schedule (f : Module.Dual ℝ (I → ℝ)) (b : W) : List (Edge cs) :=
-  sortedSchedule (edgeFinset cs b) (lexEdgeScore f)
-
-def entry {R : Type*} [Semiring R] (T : R) (f : Module.Dual ℝ (I → ℝ))
-    (b x y : W) : R :=
-  dispatch (source cs) (target cs) T (fun z => if z=y then 1 else 0) (schedule f b) x
-
-/-- The complete tied block at any genuine simple wall has an actual
-operator independent of the linear tie-breaking functional. -/
-theorem block_entry_independent {R : Type*} [Semiring R] (T : R) (b : W)
-    (f : Module.Dual ℝ (I → ℝ)) (hf : rootPlaneTies (M := M) (cs := cs) f)
-    (g h : Module.Dual ℝ (I → ℝ)) (a : ℝ) (x y : W) (hy : BruhatLE cs y b) :
-    dispatch (source cs) (target cs) T (fun z => if z=y then 1 else 0)
-      (scheduleBlock (edgeFinset cs b) (rootValue f) (lexEdgeScore g) a) x =
-    dispatch (source cs) (target cs) T (fun z => if z=y then 1 else 0)
-      (scheduleBlock (edgeFinset cs b) (rootValue f) (lexEdgeScore h) a) x := by
-  let F := (edgeFinset cs b).filter (fun e => rootValue f e=a)
-  let p := sortedSchedule F (lexEdgeScore g)
-  let q := sortedSchedule F (lexEdgeScore h)
-  let r := sortedSchedule F (fun e => geometricSlope (rootLabel cs e))
-  change dispatch (source cs) (target cs) T _ p x = dispatch (source cs) (target cs) T _ q x
-  have hmem (k : Edge cs) : k ∈ F ↔ BruhatLE cs (target cs k) b ∧ rootValue f k=a := by
-    simp only [F,Finset.mem_filter,mem_edgeFinset]
-  by_cases hex : ∃ e ∈ F, ∃ e' ∈ F, rootLabel cs e ≠ rootLabel cs e'
-  · obtain ⟨e,he,e',he',hne⟩ := hex
-    obtain ⟨B⟩ := nonempty_planeBasis (rootLabel cs e) (rootLabel cs e') hne
-    have hv := hf (rootLabel cs e) (rootLabel cs e') hne
-      (((hmem e).mp he).2.trans ((hmem e').mp he').2.symm)
-    have hfull (k : Edge cs) : k ∈ F ↔
-        (rootLabel cs k).val ∈ rootPlane (rootLabel cs e) (rootLabel cs e') ∧
-        BruhatLE cs (target cs k) b := by
-      rw [hmem,← hv,show f (normalizedRoot (rootLabel cs e))=a from ((hmem e).mp he).2]
-      exact and_comm
-    have hp := B.dispatch_plane_functional T b g p r
-      (sortedSchedule_nodup _ _) (sortedSchedule_nodup _ _)
-      (fun k => (mem_sortedSchedule _ _ k).trans (hfull k))
-      (fun k => (mem_sortedSchedule _ _ k).trans (hfull k))
-      (sortedSchedule_pairwise _ _) (sortedSchedule_pairwise _ _) x y hy
-    have hq := B.dispatch_plane_functional T b h q r
-      (sortedSchedule_nodup _ _) (sortedSchedule_nodup _ _)
-      (fun k => (mem_sortedSchedule _ _ k).trans (hfull k))
-      (fun k => (mem_sortedSchedule _ _ k).trans (hfull k))
-      (sortedSchedule_pairwise _ _) (sortedSchedule_pairwise _ _) x y hy
-    exact hp.trans hq.symm
-  · have heq (e : Edge cs) (he : e ∈ F) (e' : Edge cs) (he' : e' ∈ F) :
-        rootLabel cs e=rootLabel cs e' := by
-      by_contra hn
-      exact hex ⟨e,he,e',he',hn⟩
-    let L : Edge cs → ℝ := fun e => geometricSlope (rootLabel cs e)
-    have hties : ∀ e e', L e=L e' → target cs e ≠ source cs e' := by
-      intro e e' he hj
-      exact rootLabel_ne_of_join cs e e' hj (geometricSlope_injective he)
-    have hs (s : Module.Dual ℝ (I → ℝ)) :
-        (sortedSchedule F (lexEdgeScore s)).Pairwise (fun e e' => L e' ≤ L e) := by
-      apply (sortedSchedule_pairwise F (lexEdgeScore s)).imp_of_mem
-      intro e e' he he' _
-      have hr := heq e ((mem_sortedSchedule _ _ _).mp he) e' ((mem_sortedSchedule _ _ _).mp he')
-      exact (congrArg geometricSlope hr).symm.le
-    exact dispatch_entries_eq_of_sorted (source cs) (target cs) T L hties p q
-      (sortedSchedule_nodup _ _) (sortedSchedule_nodup _ _) (hs g) (hs h)
-      (fun e => (mem_sortedSchedule _ _ e).trans (mem_sortedSchedule _ _ e).symm) x y
-
-/-- All blocks may be replaced simultaneously on the finite lower region. -/
-theorem grouped_entry_independent {R : Type*} [Semiring R] (T : R) (b : W)
-    (f : Module.Dual ℝ (I → ℝ)) (hf : rootPlaneTies (M := M) (cs := cs) f)
-    (g h : Module.Dual ℝ (I → ℝ)) (x y : W) (hx : BruhatLE cs x b) :
-    dispatch (source cs) (target cs) T (fun z => if z=y then 1 else 0)
-      (groupedSchedule (edgeFinset cs b) (rootValue f) (lexEdgeScore g)) x =
-    dispatch (source cs) (target cs) T (fun z => if z=y then 1 else 0)
-      (groupedSchedule (edgeFinset cs b) (rootValue f) (lexEdgeScore h)) x := by
-  let K := lowerFinset cs b
-  have ht (s : Module.Dual ℝ (I → ℝ)) (a : ℝ) (e : Edge cs)
-      (he : e ∈ scheduleBlock (edgeFinset cs b) (rootValue f) (lexEdgeScore s) a) :
-      target cs e ∈ K := by
-    exact (mem_lowerFinset cs b _).mpr ((mem_edgeFinset cs b e).mp
-      ((mem_scheduleBlock _ _ _ _ _).mp he).1)
-  apply dispatch_flatMap_eq_on (source cs) (target cs) T (K : Set W)
-    (blockValues (edgeFinset cs b) (rootValue f))
-    (scheduleBlock (edgeFinset cs b) (rootValue f) (lexEdgeScore g))
-    (scheduleBlock (edgeFinset cs b) (rootValue f) (lexEdgeScore h))
-    (fun a _ e he => ht g a e he) _ _ x ((mem_lowerFinset cs b x).mpr hx)
-  intro a _ v z hz
-  apply dispatch_eq_of_entries_on (source cs) (target cs) T K _ _ (ht g a) (ht h a) _ v z hz
-  intro l _ k hk
-  exact block_entry_independent T b f hf g h a l k ((mem_lowerFinset cs b k).mp hk)
-
-/-- Stable strict primary comparisons identify the genuine sorted schedule
-with the corresponding product of tied blocks. -/
-theorem entry_eq_grouped {R : Type*} [Semiring R] (T : R) (b : W)
-    (f g : Module.Dual ℝ (I → ℝ))
-    (h : ∀ e ∈ edgeFinset cs b, ∀ e' ∈ edgeFinset cs b,
-      rootValue f e' < rootValue f e → rootValue g e' < rootValue g e)
-    (x y : W) :
-    entry (cs := cs) T g b x y =
-    dispatch (source cs) (target cs) T (fun z => if z=y then 1 else 0)
-      (groupedSchedule (edgeFinset cs b) (rootValue f) (lexEdgeScore g)) x := by
-  apply dispatch_entries_eq_of_sorted (source cs) (target cs) T (lexEdgeScore g)
-    (lexEdgeScore_ties g) _ _ (sortedSchedule_nodup _ _) (groupedSchedule_nodup _ _ _)
-    (sortedSchedule_pairwise _ _)
-  · apply groupedSchedule_pairwise
-    intro e he e' he' hlt
-    exact Prod.Lex.toLex_le_toLex.mpr (Or.inl (h e he e' he' hlt))
-  · intro e
-    simp only [mem_sortedSchedule,mem_groupedSchedule]
-
-theorem pencil_strict_stable (b : W) (f g : Module.Dual ℝ (I → ℝ)) (t₀ : ℝ) :
-    ∀ᶠ t in 𝓝 t₀, ∀ e ∈ edgeFinset cs b, ∀ e' ∈ edgeFinset cs b,
-      rootValue (pencil f g t₀) e' < rootValue (pencil f g t₀) e →
-        rootValue (pencil f g t) e' < rootValue (pencil f g t) e := by
-  apply (Filter.eventually_all_finset _).mpr
-  intro e _
-  apply (Filter.eventually_all_finset _).mpr
-  intro e' _
-  have hc (k : Edge cs) : Continuous (fun t : ℝ => rootValue (pencil f g t) k) := by
-    change Continuous (fun t : ℝ => (1-t)*f (normalizedRoot (rootLabel cs k))+
-      t*g (normalizedRoot (rootLabel cs k)))
-    fun_prop
-  by_cases hlt : rootValue (pencil f g t₀) e' < rootValue (pencil f g t₀) e
-  · filter_upwards [(isOpen_lt (hc e') (hc e)).mem_nhds hlt] with t ht
-    exact fun _ => ht
-  · exact Filter.Eventually.of_forall (fun _ h => (hlt h).elim)
-
-/-- Crossing all actual root-plane walls leaves the full path matrix fixed,
-not merely a single adapted endpoint column. -/
-theorem locally_constant_pencil {R : Type*} [Semiring R] (T : R) (b : W)
-    (f g : Module.Dual ℝ (I → ℝ))
-    (hf : Function.Injective (fun a : PositiveRoot M cs => f (normalizedRoot a)))
-    (hg : Transverse (M := M) (cs := cs) f g) (x y : W) (hx : BruhatLE cs x b) :
-    IsLocallyConstant (fun t : ℝ => entry (cs := cs) T (pencil f g t) b x y) := by
-  apply (IsLocallyConstant.iff_eventually_eq _).mpr
-  intro t₀
-  filter_upwards [pencil_strict_stable b f g t₀] with t ht
-  have hp : rootPlaneTies (M := M) (cs := cs) (pencil f g t₀) := by
-    intro a d had he
-    exact pencil_tie_plane f g hf hg t₀ a d had he
-  rw [entry_eq_grouped T b (pencil f g t₀) (pencil f g t) ht,
-    entry_eq_grouped T b (pencil f g t₀) (pencil f g t₀) (fun _ _ _ _ h => h)]
-  exact grouped_entry_independent T b (pencil f g t₀) hp (pencil f g t) (pencil f g t₀) x y hx
-
-/-- Full matrix order independence for genuine finite geometric orders. -/
-theorem entry_independent {R : Type*} [Semiring R] (T : R) (b : W)
-    (f h : Module.Dual ℝ (I → ℝ))
-    (hf : Function.Injective (fun a : PositiveRoot M cs => f (normalizedRoot a)))
-    (hh : Function.Injective (fun a : PositiveRoot M cs => h (normalizedRoot a)))
-    (x y : W) (hx : BruhatLE cs x b) : entry (cs := cs) T f b x y=entry (cs := cs) T h b x y := by
-  obtain ⟨g,_,hfg,hhg⟩ := exists_common_transverse (M := M) (cs := cs) f h
-  have he := (locally_constant_pencil T b f g hf hfg x y hx).apply_eq_of_preconnectedSpace 0 1
-  have he' := (locally_constant_pencil T b h g hh hhg x y hx).apply_eq_of_preconnectedSpace 0 1
-  simp only [pencil_zero,pencil_one] at he he'
-  exact he.trans he'.symm
-
-end
-end KLInvariance.WallDyer
-
-end
-
-
-section
-
-/-! Dyer's full path-matrix identity, for the same geometric schedule in
-every endpoint column, with the genuine Hecke-defined normalized R. -/
-namespace KLInvariance.WallDyer
-open Module SchedulePaths TitsSpace AdaptedDyer
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
-theorem neg_numerator_normalized (O : RatioOrder (M := M) (cs := cs))
-    (hh : O.denominator=height) (a : PositiveRoot M cs) :
-    (-O.numerator) (normalizedRoot a) = -O.slope a := by
-  simp only [LinearMap.neg_apply,dual_normalizedRoot,RatioOrder.slope,hh]
-
-/-- An upper bound larger than the path endpoint creates no extra paths:
-every actual directed path stays inside its endpoint interval. -/
-theorem subpaths_neg_numerator (O : RatioOrder (M := M) (cs := cs))
-    (hh : O.denominator=height) (b x y : W) (hy : BruhatLE cs y b) :
-    subpaths (source cs) (target cs) (WallDyer.schedule (-O.numerator) b) x y =
-      paths O x y := by
-  have hscore (e : Edge cs) : rootValue (-O.numerator) e = -score O e :=
-    neg_numerator_normalized O hh (rootLabel cs e)
-  ext p
-  rw [mem_subpaths,mem_paths]
-  constructor
-  · rintro ⟨hsub,hpath⟩
-    refine ⟨hpath,?_⟩
-    have hs := (sortedSchedule_pairwise (edgeFinset cs b) (lexEdgeScore (-O.numerator))).sublist hsub
-    have hs' : p.Pairwise (fun e e' => score O e ≤ score O e') := by
-      apply hs.imp
-      intro e e' he
-      have hm := Prod.Lex.monotone_fst _ _ he
-      change rootValue (-O.numerator) e' ≤ rootValue (-O.numerator) e at hm
-      rw [hscore e',hscore e] at hm
-      exact neg_le_neg_iff.mp hm
-    exact decreasing_of_schedule_path (source cs) (target cs)
-      (Λ := OrderDual ℝ) (fun e => (score O e : OrderDual ℝ))
-      (fun e e' he => score_ties O e e' he) hpath hs'
-  · rintro ⟨hpath,hord⟩
-    refine ⟨?_,hpath⟩
-    apply sublist_of_decreasing_subset (lexEdgeScore (-O.numerator))
-    · apply hord.imp
-      intro e e' he
-      apply Prod.Lex.toLex_lt_toLex.mpr
-      left
-      change rootValue (-O.numerator) e' < rootValue (-O.numerator) e
-      rw [hscore e',hscore e]
-      exact neg_lt_neg he
-    · exact sortedSchedule_pairwise _ _
-    · intro e he
-      apply (mem_sortedSchedule _ _ _).mpr
-      exact (mem_edgeFinset cs b e).mpr ((path_targets_le cs hpath e he).trans hy)
-
-theorem entry_neg_numerator (O : RatioOrder (M := M) (cs := cs))
-    (hh : O.denominator=height) (b x y : W) (hy : BruhatLE cs y b) :
-    entry (cs := cs) (Polynomial.X : Polynomial ℤ) (-O.numerator) b x y =
-      polynomial O x y := by
-  unfold entry
-  rw [dispatch_eq_subpaths_sum _ _ _ _ (show (WallDyer.schedule (-O.numerator) b).Nodup from sortedSchedule_nodup _ _),
-    subpaths_neg_numerator O hh b x y hy]
-  rfl
-
-
-theorem entry_eq_rTilde (f : Module.Dual ℝ (I → ℝ))
-    (hf : Function.Injective (fun a : PositiveRoot M cs => f (normalizedRoot a)))
-    (b x y : W) (hx : BruhatLE cs x b) (hy : BruhatLE cs y b) :
-    entry (cs := cs) (Polynomial.X : Polynomial ℤ) f b x y = Hecke.rTilde cs x y := by
-  obtain ⟨O,hO,hh⟩ := exists_ratioOrder_chamber_height (M := M) (cs := cs) y
-  have hn : Function.Injective (fun a : PositiveRoot M cs => (-O.numerator) (normalizedRoot a)) := by
-    intro a b he
-    change (-O.numerator) (normalizedRoot a) = (-O.numerator) (normalizedRoot b) at he
-    rw [neg_numerator_normalized O hh a,neg_numerator_normalized O hh b] at he
-    exact O.slope_injective (neg_injective he)
-  rw [entry_independent (Polynomial.X : Polynomial ℤ) b f (-O.numerator) hf hn x y hx,
-    entry_neg_numerator O hh b x y hy,polynomial_eq_rTilde O x y hO]
-
-/-- The manuscript's fixed geometric functional provides such a genuine
-full Dyer schedule for every finite lower Bruhat region. -/
-theorem geometric_entry_eq_rTilde (b x y : W) (hx : BruhatLE cs x b) (hy : BruhatLE cs y b) :
-    entry (cs := cs) (Polynomial.X : Polynomial ℤ) (geometricFunctional (M := M) (cs := cs)) b x y =
-      Hecke.rTilde cs x y := by
-  apply entry_eq_rTilde _ _ b x y hx hy
-  simpa only [geometricFunctional_normalizedRoot] using
-    (geometricSlope_injective (M := M) (cs := cs))
-
-end
-end KLInvariance.WallDyer
-
-end
-
-
-section
-
-
-end
-
-end OAI
+import OAI.RepresentationTheory.KazhdanLusztig.DyerPaths
+
+namespace OAI
+
+/-!
+Finite sweep algebra, wall orders and the full reflection-order path formula.
+-/
+
+section
+
+namespace KLInvariance.SchedulePaths
+open scoped BigOperators
+universe u v w
+variable {V : Type u} {E : Type v} {R : Type w}
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+variable (source target : E → V) [Semiring R]
+
+theorem dispatch_append (T : R) (v : V → R) (p q : List E) :
+    dispatch source target T v (p ++ q) =
+      dispatch source target T (dispatch source target T v q) p := by
+  induction p with
+  | nil => rfl
+  | cons e p ih => simp only [List.cons_append,dispatch,ih]
+
+theorem dispatch_add (T : R) (v w : V → R) (p : List E) :
+    dispatch source target T (v+w) p =
+      dispatch source target T v p + dispatch source target T w p := by
+  induction p with
+  | nil => rfl
+  | cons e p ih =>
+    simp only [dispatch,ih]
+    funext x
+    simp only [step,Pi.add_apply]
+    split_ifs <;> simp [mul_add,add_assoc,add_left_comm,add_comm]
+
+theorem dispatch_mul_right (T c : R) (v : V → R) (p : List E) :
+    dispatch source target T (fun x => v x*c) p =
+      fun x => dispatch source target T v p x*c := by
+  induction p with
+  | nil => rfl
+  | cons e p ih =>
+    simp only [dispatch,ih]
+    funext x
+    simp only [step]
+    split_ifs <;> simp [add_mul,mul_assoc]
+
+theorem dispatch_zero (T : R) (p : List E) : dispatch source target T (0 : V → R) p = 0 := by
+  induction p with
+  | nil => rfl
+  | cons e p ih =>
+    simp only [dispatch,ih]
+    funext x
+    simp [step]
+
+theorem dispatch_sum {A : Type*} (T : R) (F : Finset A) (v : A → V → R) (p : List E) :
+    dispatch source target T (∑ a ∈ F, v a) p =
+      ∑ a ∈ F, dispatch source target T (v a) p := by
+  induction F using Finset.induction_on with
+  | empty => simp [dispatch_zero]
+  | @insert a F ha ih => simp only [Finset.sum_insert ha,dispatch_add,ih]
+
+theorem dispatch_eq_sum_entries [Fintype V] (T : R) (v : V → R) (p : List E) (x : V) :
+    dispatch source target T v p x =
+      ∑ y, dispatch source target T (fun z => if z=y then 1 else 0) p x * v y := by
+  have hv : v = ∑ y, (fun z => (if z=y then 1 else 0)*v y) := by
+    funext z
+    simp [Finset.sum_apply]
+  conv_lhs => rw [hv]
+  rw [dispatch_sum]
+  simp only [Finset.sum_apply,dispatch_mul_right]
+
+theorem dispatch_eq_of_entries [Fintype V] (T : R) (p q : List E)
+    (h : ∀ x y, dispatch source target T (fun z => if z=y then 1 else 0) p x =
+      dispatch source target T (fun z => if z=y then 1 else 0) q x) (v : V → R) :
+    dispatch source target T v p = dispatch source target T v q := by
+  funext x
+  rw [dispatch_eq_sum_entries source target T v p x,
+    dispatch_eq_sum_entries source target T v q x]
+  exact Finset.sum_congr rfl (fun y _ => congrArg (fun a => a*v y) (h x y))
+
+/-- A finite lower region is closed under every target occurring in its
+edge schedule, so values outside that region never enter its product. -/
+theorem dispatch_congr_on (T : R) (S : Set V) (p : List E)
+    (htarget : ∀ e ∈ p, target e ∈ S) (v w : V → R)
+    (hvw : ∀ x ∈ S, v x=w x) :
+    ∀ x ∈ S, dispatch source target T v p x = dispatch source target T w p x := by
+  induction p with
+  | nil => exact hvw
+  | cons e p ih =>
+    have htail := ih (fun f hf => htarget f (List.mem_cons_of_mem _ hf))
+    intro x hx
+    simp only [dispatch,step,htail x hx,htail (target e) (htarget e List.mem_cons_self)]
+
+theorem dispatch_eq_finite_entries (T : R) (F : Finset V) (p : List E)
+    (htarget : ∀ e ∈ p, target e ∈ F) (v : V → R) (x : V) (hx : x ∈ F) :
+    dispatch source target T v p x =
+      ∑ y ∈ F, dispatch source target T (fun z => if z=y then 1 else 0) p x * v y := by
+  let w : V → R := ∑ y ∈ F, (fun z => (if z=y then 1 else 0)*v y)
+  have hvw : ∀ z ∈ (F : Set V), v z=w z := by
+    intro z hz
+    simp [w,Finset.sum_apply,show z ∈ F from hz]
+  rw [dispatch_congr_on source target T (F : Set V) p htarget v w hvw x hx]
+  dsimp only [w]
+  rw [dispatch_sum]
+  simp only [Finset.sum_apply,dispatch_mul_right]
+
+theorem dispatch_eq_of_entries_on (T : R) (F : Finset V) (p q : List E)
+    (hp : ∀ e ∈ p, target e ∈ F) (hq : ∀ e ∈ q, target e ∈ F)
+    (h : ∀ x ∈ F, ∀ y ∈ F,
+      dispatch source target T (fun z => if z=y then 1 else 0) p x =
+      dispatch source target T (fun z => if z=y then 1 else 0) q x)
+    (v : V → R) (x : V) (hx : x ∈ F) :
+    dispatch source target T v p x = dispatch source target T v q x := by
+  rw [dispatch_eq_finite_entries source target T F p hp v x hx,
+    dispatch_eq_finite_entries source target T F q hq v x hx]
+  exact Finset.sum_congr rfl (fun y hy => congrArg (fun a => a*v y) (h x hx y hy))
+
+variable {Λ : Type*} [LinearOrder Λ]
+
+/-- A genuine path in a sorted schedule is determined by its edge set and
+strict label order. Arbitrary permutations of repeated reflection labels
+therefore have no effect. -/
+theorem subpaths_eq_of_sorted (label : E → Λ)
+    (hties : ∀ e f, label e = label f → target e ≠ source f)
+    (p q : List E)
+    (hp : p.Pairwise (fun e f => label f ≤ label e))
+    (hq : q.Pairwise (fun e f => label f ≤ label e))
+    (hmem : ∀ e, e ∈ p ↔ e ∈ q) (x y : V) :
+    subpaths source target p x y = subpaths source target q x y := by
+  ext s
+  simp only [mem_subpaths]
+  constructor
+  · rintro ⟨hs,hpath⟩
+    refine ⟨?_,hpath⟩
+    apply sublist_of_decreasing_subset label
+      (decreasing_of_schedule_path source target label hties hpath (hp.sublist hs)) hq
+    exact fun e he => (hmem e).mp (hs.subset he)
+  · rintro ⟨hs,hpath⟩
+    refine ⟨?_,hpath⟩
+    apply sublist_of_decreasing_subset label
+      (decreasing_of_schedule_path source target label hties hpath (hq.sublist hs)) hp
+    exact fun e he => (hmem e).mpr (hs.subset he)
+
+theorem dispatch_eq_of_sorted [Fintype V] (T : R) (label : E → Λ)
+    (hties : ∀ e f, label e = label f → target e ≠ source f)
+    (p q : List E) (hpnd : p.Nodup) (hqnd : q.Nodup)
+    (hp : p.Pairwise (fun e f => label f ≤ label e))
+    (hq : q.Pairwise (fun e f => label f ≤ label e))
+    (hmem : ∀ e, e ∈ p ↔ e ∈ q) (v : V → R) :
+    dispatch source target T v p = dispatch source target T v q := by
+  apply dispatch_eq_of_entries
+  intro x y
+  rw [dispatch_eq_subpaths_sum source target T p hpnd,
+    dispatch_eq_subpaths_sum source target T q hqnd,
+    subpaths_eq_of_sorted source target label hties p q hp hq hmem x y]
+
+/-- Replacing each block by an equal actual operator replaces the product. -/
+theorem dispatch_flatMap_eq {A : Type*} (T : R) (blocks : List A) (p q : A → List E)
+    (h : ∀ a ∈ blocks, ∀ v, dispatch source target T v (p a) =
+      dispatch source target T v (q a)) (v : V → R) :
+    dispatch source target T v (blocks.flatMap p) =
+      dispatch source target T v (blocks.flatMap q) := by
+  induction blocks with
+  | nil => rfl
+  | cons a l ih =>
+    simp only [List.flatMap_cons,dispatch_append]
+    rw [ih (fun b hb => h b (List.mem_cons_of_mem _ hb)),h a List.mem_cons_self]
+
+
+variable {V' E' : Type*} (source' target' : E' → V')
+
+theorem IsPath.map_iff (j : V' → V) (hj : Function.Injective j) (J : E' → E)
+    (hs : ∀ e, source (J e)=j (source' e))
+    (ht : ∀ e, target (J e)=j (target' e)) {x y : V'} {p : List E'} :
+    IsPath source target (j x) (j y) (p.map J) ↔ IsPath source' target' x y p := by
+  induction p generalizing x with
+  | nil => exact hj.eq_iff
+  | cons e p ih =>
+    change (source (J e)=j x ∧ IsPath source target (target (J e)) (j y) (p.map J)) ↔ _
+    rw [hs,ht,ih,hj.eq_iff]
+    rfl
+
+theorem exists_list_lift (J : E' → E) (p : List E)
+    (h : ∀ e ∈ p, ∃ e', J e'=e) : ∃ p' : List E', p'.map J=p := by
+  induction p with
+  | nil => exact ⟨[],rfl⟩
+  | cons e p ih =>
+    obtain ⟨e',he⟩ := h e List.mem_cons_self
+    obtain ⟨p',hp⟩ := ih (fun f hf => h f (List.mem_cons_of_mem _ hf))
+    exact ⟨e'::p',by simp [he,hp]⟩
+
+/-- If all actual paths between the specified endpoints lie in a full embedded
+subgraph, its full sorted product computes the same entry. -/
+theorem dispatch_entry_graph_embedding (T : R) (label : E → Λ)
+    (hties : ∀ e f, label e=label f → target e ≠ source f)
+    (j : V' → V) (hj : Function.Injective j) (J : E' → E) (hJ : Function.Injective J)
+    (hs : ∀ e, source (J e)=j (source' e))
+    (ht : ∀ e, target (J e)=j (target' e))
+    (p : List E) (q : List E') (hpnd : p.Nodup) (hqnd : q.Nodup)
+    (hp : p.Pairwise (fun e f => label f ≤ label e))
+    (hq : q.Pairwise (fun e f => label (J f) ≤ label (J e)))
+    (hqfull : ∀ e, e ∈ q) (hJmem : ∀ e, J e ∈ p)
+    (x y : V')
+    (hpath : ∀ r ∈ subpaths source target p (j x) (j y), ∀ e ∈ r, ∃ e', J e'=e) :
+    dispatch source target T (fun z => if z=j y then 1 else 0) p (j x) =
+      dispatch source' target' T (fun z => if z=y then 1 else 0) q x := by
+  rw [dispatch_eq_subpaths_sum source target T p hpnd,
+    dispatch_eq_subpaths_sum source' target' T q hqnd]
+  symm
+  have hties' : ∀ e f, label (J e)=label (J f) → target' e ≠ source' f := by
+    intro e f he hf
+    apply hties (J e) (J f) he
+    rw [ht,hs,hf]
+  apply Finset.sum_bij (fun r _ => r.map J)
+  · intro r hr
+    obtain ⟨hr,hpr⟩ := (mem_subpaths source' target' q x y r).mp hr
+    apply (mem_subpaths source target p (j x) (j y) (r.map J)).mpr
+    constructor
+    · apply sublist_of_decreasing_subset label
+      · rw [List.pairwise_map]
+        exact decreasing_of_schedule_path source' target' (label ∘ J) hties' hpr (hq.sublist hr)
+      · exact hp
+      · intro e he
+        obtain ⟨f,_,rfl⟩ := List.mem_map.mp he
+        exact hJmem f
+    · exact hpr.map source' target' source target j J hs ht
+  · intro r _ s _ he
+    exact List.map_injective_iff.mpr hJ he
+  · intro r hr
+    obtain ⟨s,he⟩ := exists_list_lift J r (hpath r hr)
+    refine ⟨s,?_,he⟩
+    obtain ⟨hrsub,hrpath⟩ := (mem_subpaths source target p (j x) (j y) r).mp hr
+    have hspath : IsPath source' target' x y s :=
+      (IsPath.map_iff source target source' target' j hj J hs ht).mp (he ▸ hrpath)
+    apply (mem_subpaths source' target' q x y s).mpr
+    refine ⟨?_,hspath⟩
+    apply sublist_of_decreasing_subset (label ∘ J)
+    · have hstrict := decreasing_of_schedule_path source target label hties hrpath (hp.sublist hrsub)
+      rw [← he,List.pairwise_map] at hstrict
+      exact hstrict
+    · exact hq
+    · exact fun e _ => hqfull e
+  · intro r _
+    simp
+
+end
+end KLInvariance.SchedulePaths
+
+end
+
+
+section
+
+/-! Linear coordinates on the actual normalized root line. These give the
+rank-two geometry of finite wall crossings for genuine root schedules. -/
+namespace KLInvariance.TitsSpace
+open Module
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+@[simp] theorem normalizedRoot_height (a : PositiveRoot M cs) :
+    height (normalizedRoot a) = 1 := by
+  simp [normalizedRoot,a.height_pos.ne']
+
+@[simp] theorem dual_normalizedRoot (f : Module.Dual ℝ (I → ℝ)) (a : PositiveRoot M cs) :
+    f (normalizedRoot a) = f a.val / height a.val := by
+  simp [normalizedRoot,div_eq_mul_inv,mul_comm]
+
+@[simp] theorem geometricFunctional_normalizedRoot (a : PositiveRoot M cs) :
+    geometricFunctional (M := M) (cs := cs) (normalizedRoot a) = geometricSlope a :=
+  dual_normalizedRoot _ a
+
+theorem normalizedRoot_mem_iff (P : Submodule ℝ (I → ℝ)) (a : PositiveRoot M cs) :
+    normalizedRoot a ∈ P ↔ a.val ∈ P := by
+  constructor
+  · intro h
+    have hh := P.smul_mem (height a.val) h
+    simpa [normalizedRoot,smul_smul,a.height_pos.ne'] using hh
+  · exact fun h => P.smul_mem _ h
+
+theorem rootPlane_eq_normalized_span (a b : PositiveRoot M cs) :
+    rootPlane a b = Submodule.span ℝ {normalizedRoot a,normalizedRoot b} := by
+  apply le_antisymm
+  · apply Submodule.span_le.mpr
+    intro x hx
+    rcases Set.mem_insert_iff.mp hx with rfl | hx
+    · exact (normalizedRoot_mem_iff _ a).mp (Submodule.subset_span (Set.mem_insert _ _))
+    · rw [Set.mem_singleton_iff.mp hx]
+      exact (normalizedRoot_mem_iff _ b).mp (Submodule.subset_span (Set.mem_insert_of_mem _ (Set.mem_singleton _)))
+  · apply Submodule.span_le.mpr
+    intro x hx
+    rcases Set.mem_insert_iff.mp hx with rfl | hx
+    · exact (normalizedRoot_mem_iff _ a).mpr (Submodule.subset_span (Set.mem_insert _ _))
+    · rw [Set.mem_singleton_iff.mp hx]
+      exact (normalizedRoot_mem_iff _ b).mpr (Submodule.subset_span (Set.mem_insert_of_mem _ (Set.mem_singleton _)))
+
+/-- The intersection of a root-plane with the height-one hyperplane is the
+actual affine line through the two normalized roots. -/
+theorem normalizedRoot_affine {a b c : PositiveRoot M cs}
+    (hc : c.val ∈ rootPlane a b) :
+    ∃ t : ℝ, normalizedRoot c = t • normalizedRoot a + (1-t) • normalizedRoot b := by
+  have hcn : normalizedRoot c ∈ Submodule.span ℝ {normalizedRoot a,normalizedRoot b} := by
+    rw [← rootPlane_eq_normalized_span]
+    exact (normalizedRoot_mem_iff _ _).mpr hc
+  obtain ⟨t,s,he⟩ := Submodule.mem_span_pair.mp hcn
+  have hh := congrArg (height (I := I)) he
+  simp only [map_add,map_smul,normalizedRoot_height,smul_eq_mul,mul_one] at hh
+  refine ⟨t,?_⟩
+  have hs : s=1-t := by linarith
+  exact hs ▸ he.symm
+
+/-- On each genuine root plane, every height-normalized linear functional
+is an affine function of the fixed angular slope. -/
+theorem exists_affine_rootPlane (f : Module.Dual ℝ (I → ℝ))
+    (a b : PositiveRoot M cs) (hab : a ≠ b) :
+    ∃ c d : ℝ, ∀ r : PositiveRoot M cs, r.val ∈ rootPlane a b →
+      f (normalizedRoot r) = c * geometricSlope r + d := by
+  let A := geometricSlope a
+  let B := geometricSlope b
+  let fa := f (normalizedRoot a)
+  let fb := f (normalizedRoot b)
+  have hAB : A-B ≠ 0 := sub_ne_zero.mpr (fun h => hab (geometricSlope_injective h))
+  let c := (fa-fb)/(A-B)
+  let d := fb-c*B
+  have ha : fa=c*A+d := by
+    dsimp [d,c]
+    field_simp
+    ring
+  have hb : fb=c*B+d := by dsimp [d]; ring
+  refine ⟨c,d,fun r hr => ?_⟩
+  obtain ⟨t,ht⟩ := normalizedRoot_affine hr
+  have hf : f (normalizedRoot r) = t*fa+(1-t)*fb := by
+    rw [ht]
+    simp only [map_add,map_smul,smul_eq_mul]
+    rfl
+  have hg : geometricSlope r = t*A+(1-t)*B := by
+    rw [← geometricFunctional_normalizedRoot,ht]
+    simp only [map_add,map_smul,geometricFunctional_normalizedRoot,smul_eq_mul]
+    rfl
+  rw [hf,hg,ha,hb]
+  ring
+
+namespace PlaneBasis
+variable {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+
+include B in
+theorem exists_affine (f : Module.Dual ℝ (I → ℝ)) :
+    ∃ c d : ℝ, ∀ r : PositiveRoot M cs, r.val ∈ P →
+      f (normalizedRoot r) = c * geometricSlope r + d := by
+  simpa only [B.plane_eq] using exists_affine_rootPlane f B.left B.right B.distinct
+
+end PlaneBasis
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Localization of genuine directed paths in a maximal root-plane coset.
+The full relative interval is proved to contain every such path. -/
+namespace KLInvariance.TitsSpace.PlaneBasis
+open Module SchedulePaths
+open AdaptedDyer
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+/-- All edges of a plane-labelled path lie in the full induced relative
+interval between its endpoints, with the original ambient labels. -/
+theorem path_coordinates {a₀ : W}
+    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+    (g : planeSubgroup M cs P) {y : W} {p : List (AdaptedDyer.Edge cs)}
+    (hp : IsPath (source cs) (target cs) ((g:W)*a₀) y p)
+    (hP : ∀ e ∈ p, (rootLabel cs e).val ∈ P) :
+    ∃ k : planeSubgroup M cs P, y=(k:W)*a₀ ∧ BruhatLE B.system g k ∧
+      ∀ e ∈ p, ∃ l r : planeSubgroup M cs P,
+        source cs e=(l:W)*a₀ ∧ target cs e=(r:W)*a₀ ∧
+        BruhatLE B.system g l ∧ BruhatStep B.system l r ∧ BruhatLE B.system r k := by
+  induction p generalizing g with
+  | nil => exact ⟨g,hp.symm,.refl,by simp⟩
+  | cons e p ih =>
+    have hg : source cs e*a₀⁻¹ ∈ planeSubgroup M cs P := by
+      rw [hp.1]
+      simpa only [mul_assoc,mul_inv_cancel,mul_one] using g.property
+    have hh := mem_coset_of_graphRoot e.property hg (hP e List.mem_cons_self)
+    let h : planeSubgroup M cs P := ⟨target cs e*a₀⁻¹,hh⟩
+    have heqh : target cs e=(h:W)*a₀ := by dsimp [h]; group
+    have he : BruhatStep B.system g h := by
+      apply (B.coset_step_iff hmin g h).mp
+      rw [← hp.1,← heqh]
+      exact e.property
+    obtain ⟨k,hky,hk,hbody⟩ := ih h (heqh ▸ hp.2)
+      (fun f hf => hP f (List.mem_cons_of_mem _ hf))
+    refine ⟨k,hky,(show BruhatLE B.system g h from .single he).trans hk,?_⟩
+    intro f hf
+    rcases List.mem_cons.mp hf with rfl | hf
+    · exact ⟨g,h,hp.1,heqh,.refl,he,hk⟩
+    · obtain ⟨l,r,hl,hr,hhl,hlr,hrk⟩ := hbody f hf
+      exact ⟨l,r,hl,hr,(show BruhatLE B.system g h from .single he).trans hhl,hlr,hrk⟩
+
+def intervalToAmbient {a₀ : W} {g k : planeSubgroup M cs P}
+    (z : Interval B.system g k) : W := (z.val:W)*a₀
+
+theorem intervalToAmbient_injective {a₀ : W} {g k : planeSubgroup M cs P} :
+    Function.Injective (B.intervalToAmbient (a₀ := a₀) (g := g) (k := k)) := by
+  intro x y h
+  exact Subtype.ext (Subtype.ext (mul_right_cancel h))
+
+def edgeToAmbient {a₀ : W}
+    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+    {g k : planeSubgroup M cs P} (e : BruhatGraph.Edge B.system g k) : AdaptedDyer.Edge cs :=
+  ⟨(B.intervalToAmbient e.val.1, B.intervalToAmbient e.val.2),
+    (B.coset_step_iff hmin e.val.1.val e.val.2.val).mpr e.property⟩
+
+@[simp] theorem edgeToAmbient_source {a₀ : W}
+    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+    {g k : planeSubgroup M cs P} (e : BruhatGraph.Edge B.system g k) :
+    source cs (B.edgeToAmbient hmin e) = B.intervalToAmbient (a₀ := a₀) (BruhatGraph.source B.system g k e) := rfl
+
+@[simp] theorem edgeToAmbient_target {a₀ : W}
+    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+    {g k : planeSubgroup M cs P} (e : BruhatGraph.Edge B.system g k) :
+    target cs (B.edgeToAmbient hmin e) = B.intervalToAmbient (a₀ := a₀) (BruhatGraph.target B.system g k e) := rfl
+
+theorem edgeToAmbient_injective {a₀ : W}
+    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+    {g k : planeSubgroup M cs P} :
+    Function.Injective (B.edgeToAmbient hmin (g := g) (k := k)) := by
+  intro e f h
+  have hh := congrArg Subtype.val h
+  apply Subtype.ext
+  exact Prod.ext (B.intervalToAmbient_injective (congrArg Prod.fst hh))
+    (B.intervalToAmbient_injective (congrArg Prod.snd hh))
+
+@[simp] theorem edgeToAmbient_root {a₀ : W}
+    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+    {g k : planeSubgroup M cs P} (e : BruhatGraph.Edge B.system g k) :
+    rootLabel cs (B.edgeToAmbient hmin e) = B.edgeRoot e.property := by
+  apply (positiveRootEquiv M cs).symm.injective
+  apply Subtype.ext
+  change rootRef (rootLabel cs (B.edgeToAmbient hmin e)) = rootRef (B.edgeRoot e.property)
+  rw [rootRef_rootLabel]
+  have hr : rootRef (B.edgeRoot e.property) = (e.val.2.val:W)*(e.val.1.val:W)⁻¹ :=
+    (rootReflection_represents M cs _).reflection_unique
+      (graphRoot_represents ((B.bruhatStep_iff _ _).mpr e.property))
+  rw [hr]
+  change ((e.val.2.val:W)*a₀)*((e.val.1.val:W)*a₀)⁻¹ = _
+  group
+
+/-- Edge-by-edge lifting of every plane path into the full relative interval.
+It is obtained from actual path connectivity, not from ambient comparability
+alone (which would not suffice). -/
+theorem path_edges_lift {a₀ : W}
+    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+    {g k : planeSubgroup M cs P} {p : List (AdaptedDyer.Edge cs)}
+    (hp : IsPath (source cs) (target cs) ((g:W)*a₀) ((k:W)*a₀) p)
+    (hP : ∀ e ∈ p, (rootLabel cs e).val ∈ P) :
+    ∀ e ∈ p, ∃ e' : BruhatGraph.Edge B.system g k, B.edgeToAmbient hmin e'=e := by
+  obtain ⟨k',he,_,hbody⟩ := B.path_coordinates hmin g hp hP
+  have hk : k'=k := Subtype.ext (mul_right_cancel he.symm)
+  subst k'
+  intro e he
+  obtain ⟨l,r,hl,hr,hgl,hlr,hrk⟩ := hbody e he
+  let vl : Interval B.system g k := ⟨l,hgl,(show BruhatLE B.system l r from .single hlr).trans hrk⟩
+  let vr : Interval B.system g k := ⟨r,hgl.trans (.single hlr),hrk⟩
+  refine ⟨⟨(vl,vr),hlr⟩,?_⟩
+  apply Subtype.ext
+  exact Prod.ext hl.symm hr.symm
+
+
+include B in
+/-- Reversal of a complete angular block of genuine ambient edges preserves
+all its endpoint entries. Fullness is with respect to the actual root plane,
+not to a rank-two subgroup chosen merely to contain two arbitrary reflections. -/
+theorem dispatch_plane_reverse {R : Type*} [Semiring R] (T : R) (b : W)
+    (p : List (AdaptedDyer.Edge cs)) (hnd : p.Nodup)
+    (hfull : ∀ e, e ∈ p ↔ (rootLabel cs e).val ∈ P ∧ BruhatLE cs (target cs e) b)
+    (hs : p.Pairwise (fun e f => geometricSlope (rootLabel cs f) ≤ geometricSlope (rootLabel cs e)))
+    (x y : W) (hy : BruhatLE cs y b) :
+    dispatch (source cs) (target cs) T (fun z => if z=y then 1 else 0) p x =
+      dispatch (source cs) (target cs) T (fun z => if z=y then 1 else 0) p.reverse x := by
+  obtain ⟨a₀,g,hg,hx,hmin⟩ := planeCoset_exists_minimal M cs P x
+  let g' : planeSubgroup M cs P := ⟨g,hg⟩
+  have hx' : x=(g':W)*a₀ := hx
+  let L : AdaptedDyer.Edge cs → ℝ := fun e => geometricSlope (rootLabel cs e)
+  have hties : ∀ e f, L e=L f → target cs e ≠ source cs f := by
+    intro e f he
+    exact fun hj => rootLabel_ne_of_join cs e f hj (geometricSlope_injective he)
+  by_cases hex : ∃ r : List (AdaptedDyer.Edge cs),
+      IsPath (source cs) (target cs) x y r ∧ ∀ e ∈ r, (rootLabel cs e).val ∈ P
+  · obtain ⟨r,hr,hrP⟩ := hex
+    obtain ⟨k,hyk,hgk,_⟩ := B.path_coordinates hmin g' (hx' ▸ hr) hrP
+    let lo : Interval B.system g' k := ⟨g',.refl,hgk⟩
+    let hi : Interval B.system g' k := ⟨k,hgk,.refl⟩
+    let j := B.intervalToAmbient (a₀ := a₀) (g := g') (k := k)
+    let J := B.edgeToAmbient hmin (g := g') (k := k)
+    have hjlo : j lo=x := hx'.symm
+    have hjhi : j hi=y := hyk.symm
+    let : Fintype (BruhatGraph.Edge B.system g' k) := Fintype.ofFinite _
+    let q : List (BruhatGraph.Edge B.system g' k) :=
+      Finset.univ.toList.mergeSort (fun e f => B.intervalEdgeSlope g' k f ≤ B.intervalEdgeSlope g' k e)
+    have hqnd : q.Nodup := (List.mergeSort_perm _ _).nodup_iff.mpr (Finset.nodup_toList _)
+    have hqfull : ∀ e, e ∈ q := by intro e; simp [q]
+    have hqs : q.Pairwise (fun e f => B.intervalEdgeSlope g' k f ≤ B.intervalEdgeSlope g' k e) :=
+      List.pairwise_mergeSort' _ _
+    have hJroot (e : BruhatGraph.Edge B.system g' k) : L (J e)=B.intervalEdgeSlope g' k e := by
+      dsimp [L,J,intervalEdgeSlope]
+      rw [B.edgeToAmbient_root]
+    have hJmem (e : BruhatGraph.Edge B.system g' k) : J e ∈ p := by
+      apply (hfull _).mpr
+      constructor
+      · dsimp [J]
+        rw [B.edgeToAmbient_root]
+        exact B.edgeRoot_plane e.property
+      · have ht : BruhatLE cs ((e.val.2.val:W)*a₀) ((k:W)*a₀) :=
+          B.coset_bruhat hmin e.val.2.property.2
+        exact ht.trans (hyk ▸ hy)
+    have hlift (s : List (AdaptedDyer.Edge cs))
+        (hst : ∀ e ∈ s, e ∈ p)
+        (hp : IsPath (source cs) (target cs) (j lo) (j hi) s) :
+        ∀ e ∈ s, ∃ e' : BruhatGraph.Edge B.system g' k, J e'=e :=
+      B.path_edges_lift hmin hp (fun e he => ((hfull e).mp (hst e he)).1)
+    have hleft := dispatch_entry_graph_embedding (source cs) (target cs)
+      (BruhatGraph.source B.system g' k) (BruhatGraph.target B.system g' k)
+      T L hties j (B.intervalToAmbient_injective) J (B.edgeToAmbient_injective hmin)
+      (B.edgeToAmbient_source hmin) (B.edgeToAmbient_target hmin)
+      p q hnd hqnd hs (by simpa only [hJroot] using hqs) hqfull hJmem lo hi (by
+        intro s hsp
+        obtain ⟨hsub,hpath⟩ := (mem_subpaths _ _ _ _ _ _).mp hsp
+        exact hlift s (fun e he => hsub.subset he) hpath)
+    have hright := dispatch_entry_graph_embedding (source cs) (target cs)
+      (BruhatGraph.source B.system g' k) (BruhatGraph.target B.system g' k)
+      T (fun e => -L e) (fun e f he => hties e f (neg_injective he))
+      j (B.intervalToAmbient_injective) J (B.edgeToAmbient_injective hmin)
+      (B.edgeToAmbient_source hmin) (B.edgeToAmbient_target hmin)
+      p.reverse q.reverse (List.nodup_reverse.mpr hnd) (List.nodup_reverse.mpr hqnd)
+      (by rw [List.pairwise_reverse]; exact hs.imp (fun he => neg_le_neg he))
+      (by rw [List.pairwise_reverse]; exact hqs.imp (fun he => by simpa only [hJroot] using neg_le_neg he))
+      (fun e => List.mem_reverse.mpr (hqfull e))
+      (fun e => List.mem_reverse.mpr (hJmem e)) lo hi (by
+        intro s hsp
+        obtain ⟨hsub,hpath⟩ := (mem_subpaths _ _ _ _ _ _).mp hsp
+        exact hlift s (fun e he => List.mem_reverse.mp (hsub.subset he)) hpath)
+    rw [hjlo,hjhi] at hleft hright
+    rw [hleft,hright]
+    exact B.dispatch_reverse_endpoints hgk q hqnd hqfull hqs T
+  · have hzero (q : List (AdaptedDyer.Edge cs))
+        (hqp : ∀ e ∈ q, e ∈ p) : subpaths (source cs) (target cs) q x y = ∅ := by
+      apply Finset.eq_empty_iff_forall_notMem.mpr
+      intro r hr
+      obtain ⟨hsub,hpath⟩ := (mem_subpaths _ _ _ _ _ _).mp hr
+      exact hex ⟨r,hpath,fun e he => ((hfull e).mp (hqp e (hsub.subset he))).1⟩
+    rw [dispatch_eq_subpaths_sum _ _ _ _ hnd,
+      dispatch_eq_subpaths_sum _ _ _ _ (List.nodup_reverse.mpr hnd),
+      hzero p (fun _ he => he),hzero p.reverse (fun _ he => List.mem_reverse.mp he)]
+
+end
+end KLInvariance.TitsSpace.PlaneBasis
+
+end
+
+
+section
+
+/-! Algebra and finite ordered-list bookkeeping for wall crossings. -/
+namespace KLInvariance.SchedulePaths
+universe u v w
+variable {E : Type u} {A : Type v} {Λ : Type w} [LinearOrder A] [LinearOrder Λ]
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+local instance : DecidableEq E := Classical.decEq E
+
+def sortedSchedule (F : Finset E) (score : E → Λ) : List E :=
+  F.toList.mergeSort (fun e f => score f ≤ score e)
+
+@[simp] theorem mem_sortedSchedule (F : Finset E) (score : E → Λ) (e : E) :
+    e ∈ sortedSchedule F score ↔ e ∈ F := by simp [sortedSchedule]
+
+theorem sortedSchedule_nodup (F : Finset E) (score : E → Λ) :
+    (sortedSchedule F score).Nodup :=
+  (List.mergeSort_perm _ _).nodup_iff.mpr (Finset.nodup_toList _)
+
+theorem sortedSchedule_pairwise (F : Finset E) (score : E → Λ) :
+    (sortedSchedule F score).Pairwise (fun e f => score f ≤ score e) :=
+  List.pairwise_mergeSort' _ _
+
+def blockValues (F : Finset E) (primary : E → A) : List A :=
+  (F.image primary).sort (fun a b => b ≤ a)
+
+def scheduleBlock (F : Finset E) (primary : E → A) (score : E → Λ) (a : A) : List E :=
+  sortedSchedule (F.filter (fun e => primary e=a)) score
+
+def groupedSchedule (F : Finset E) (primary : E → A) (score : E → Λ) : List E :=
+  (blockValues F primary).flatMap (scheduleBlock F primary score)
+
+@[simp] theorem mem_blockValues (F : Finset E) (primary : E → A) (a : A) :
+    a ∈ blockValues F primary ↔ ∃ e ∈ F, primary e=a := by
+  simp [blockValues]
+
+omit [LinearOrder A] in
+@[simp] theorem mem_scheduleBlock (F : Finset E) (primary : E → A) (score : E → Λ)
+    (a : A) (e : E) : e ∈ scheduleBlock F primary score a ↔ e ∈ F ∧ primary e=a := by
+  simp [scheduleBlock]
+
+@[simp] theorem mem_groupedSchedule (F : Finset E) (primary : E → A) (score : E → Λ)
+    (e : E) : e ∈ groupedSchedule F primary score ↔ e ∈ F := by
+  constructor
+  · intro he
+    obtain ⟨a,_,ha⟩ := List.mem_flatMap.mp he
+    exact ((mem_scheduleBlock F primary score a e).mp ha).1
+  · intro he
+    apply List.mem_flatMap.mpr
+    exact ⟨primary e,(mem_blockValues F primary _).mpr ⟨e,he,rfl⟩,
+      (mem_scheduleBlock F primary score _ _).mpr ⟨he,rfl⟩⟩
+
+theorem blockValues_strict (F : Finset E) (primary : E → A) :
+    (blockValues F primary).Pairwise (fun a b => b < a) := by
+  have hs := Finset.pairwise_sort (F.image primary) (fun a b => b ≤ a)
+  have hn := Finset.sort_nodup (F.image primary) (fun a b => b ≤ a)
+  exact (List.pairwise_and_iff.mpr ⟨hs,hn⟩).imp (fun h => lt_of_le_of_ne h.1 h.2.symm)
+
+theorem groupedSchedule_nodup (F : Finset E) (primary : E → A) (score : E → Λ) :
+    (groupedSchedule F primary score).Nodup := by
+  apply List.nodup_flatMap.mpr
+  constructor
+  · intro a _
+    exact sortedSchedule_nodup _ _
+  · apply (blockValues_strict F primary).imp
+    intro a b hab
+    apply List.disjoint_left.mpr
+    intro e hea heb
+    have ha := ((mem_scheduleBlock F primary score a e).mp hea).2
+    have hb := ((mem_scheduleBlock F primary score b e).mp heb).2
+    exact hab.ne (hb.symm.trans ha)
+
+theorem groupedSchedule_pairwise (F : Finset E) (primary : E → A) (score : E → Λ)
+    (h : ∀ e ∈ F, ∀ f ∈ F, primary f < primary e → score f ≤ score e) :
+    (groupedSchedule F primary score).Pairwise (fun e f => score f ≤ score e) := by
+  apply List.pairwise_flatMap.mpr
+  constructor
+  · intro a _
+    exact sortedSchedule_pairwise _ _
+  · apply (blockValues_strict F primary).imp
+    intro a b hab e he f hf
+    obtain ⟨he,heq⟩ := (mem_scheduleBlock F primary score a e).mp he
+    obtain ⟨hf,hfq⟩ := (mem_scheduleBlock F primary score b f).mp hf
+    exact h e he f hf (by simpa only [heq,hfq] using hab)
+
+variable {V : Type*} {R : Type*} [Semiring R] (source target : E → V)
+
+theorem dispatch_entries_eq_of_sorted (T : R) (label : E → Λ)
+    (hties : ∀ e f, label e=label f → target e ≠ source f)
+    (p q : List E) (hpnd : p.Nodup) (hqnd : q.Nodup)
+    (hp : p.Pairwise (fun e f => label f ≤ label e))
+    (hq : q.Pairwise (fun e f => label f ≤ label e))
+    (hmem : ∀ e, e ∈ p ↔ e ∈ q) (x y : V) :
+    dispatch source target T (fun z => if z=y then 1 else 0) p x =
+      dispatch source target T (fun z => if z=y then 1 else 0) q x := by
+  rw [dispatch_eq_subpaths_sum source target T p hpnd,
+    dispatch_eq_subpaths_sum source target T q hqnd,
+    subpaths_eq_of_sorted source target label hties p q hp hq hmem x y]
+
+omit [LinearOrder A] in
+/-- Replacement on a finite lower region requires no artificial finiteness
+assumption on the ambient Coxeter group. -/
+theorem dispatch_flatMap_eq_on (T : R) (S : Set V) (blocks : List A) (p q : A → List E)
+    (htarget : ∀ a ∈ blocks, ∀ e ∈ p a, target e ∈ S)
+    (h : ∀ a ∈ blocks, ∀ v, ∀ x ∈ S,
+      dispatch source target T v (p a) x = dispatch source target T v (q a) x)
+    (v : V → R) : ∀ x ∈ S,
+    dispatch source target T v (blocks.flatMap p) x =
+      dispatch source target T v (blocks.flatMap q) x := by
+  induction blocks with
+  | nil => exact fun _ _ => rfl
+  | cons a l ih =>
+    intro x hx
+    simp only [List.flatMap_cons,dispatch_append]
+    rw [dispatch_congr_on source target T S (p a) (htarget a List.mem_cons_self)
+      _ _ (ih (fun b hb => htarget b (List.mem_cons_of_mem _ hb))
+      (fun b hb => h b (List.mem_cons_of_mem _ hb))) x hx]
+    exact h a List.mem_cons_self _ x hx
+
+end
+end KLInvariance.SchedulePaths
+
+end
+
+
+section
+
+/-! The actual rank-two edge product is independent of every linear
+functional restricted to that root plane, including a wall tie refinement. -/
+namespace KLInvariance.TitsSpace
+open Module SchedulePaths AdaptedDyer
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+def lexRootScore (f : Module.Dual ℝ (I → ℝ)) (a : PositiveRoot M cs) : ℝ ×ₗ ℝ :=
+  toLex (f (normalizedRoot a),geometricSlope a)
+
+theorem lexRootScore_injective (f : Module.Dual ℝ (I → ℝ)) :
+    Function.Injective (lexRootScore (M := M) (cs := cs) f) := by
+  intro a b hab
+  apply geometricSlope_injective
+  exact congrArg (fun p : ℝ ×ₗ ℝ => (ofLex p).2) hab
+
+def lexEdgeScore (f : Module.Dual ℝ (I → ℝ)) (e : AdaptedDyer.Edge cs) : ℝ ×ₗ ℝ :=
+  lexRootScore f (rootLabel cs e)
+
+theorem lexEdgeScore_ties (f : Module.Dual ℝ (I → ℝ)) (e e' : AdaptedDyer.Edge cs)
+    (he : lexEdgeScore f e=lexEdgeScore f e') : target cs e ≠ source cs e' := by
+  intro hj
+  exact rootLabel_ne_of_join cs e e' hj (lexRootScore_injective f he)
+
+omit [Fintype I] in
+theorem lex_affine_nonneg {c d x y : ℝ} (hc : 0 ≤ c)
+    (h : (toLex (c*x+d,x) : ℝ ×ₗ ℝ) ≤ toLex (c*y+d,y)) : x ≤ y := by
+  rcases Prod.Lex.toLex_le_toLex.mp h with h | h
+  · by_contra hn
+    have hxy : y < x := lt_of_not_ge hn
+    nlinarith [mul_nonneg hc (sub_nonneg.mpr hxy.le)]
+  · exact h.2
+
+omit [Fintype I] in
+theorem lex_affine_neg {c d x y : ℝ} (hc : c < 0)
+    (h : (toLex (c*x+d,x) : ℝ ×ₗ ℝ) ≤ toLex (c*y+d,y)) : y ≤ x := by
+  rcases Prod.Lex.toLex_le_toLex.mp h with h | h
+  · by_contra hn
+    have hxy : x < y := lt_of_not_ge hn
+    nlinarith [mul_nonpos_of_nonpos_of_nonneg hc.le (sub_nonneg.mpr hxy.le)]
+  · have he : x=y := by nlinarith [h.1]
+    exact he.symm.le
+
+namespace PlaneBasis
+variable {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+
+include B in
+/-- Any linearly sorted full plane block is either angularly sorted or
+reverse-angularly sorted. This includes tied primary labels refined by the
+fixed genuine geometric functional. -/
+theorem lex_block_orientation (f : Module.Dual ℝ (I → ℝ))
+    (p : List (AdaptedDyer.Edge cs))
+    (hP : ∀ e ∈ p, (rootLabel cs e).val ∈ P)
+    (hs : p.Pairwise (fun e e' => lexEdgeScore f e' ≤ lexEdgeScore f e)) :
+    p.Pairwise (fun e e' => geometricSlope (rootLabel cs e') ≤ geometricSlope (rootLabel cs e)) ∨
+    p.Pairwise (fun e e' => geometricSlope (rootLabel cs e) ≤ geometricSlope (rootLabel cs e')) := by
+  obtain ⟨c,d,hf⟩ := B.exists_affine f
+  have hv (e : AdaptedDyer.Edge cs) (he : e ∈ p) :
+      lexEdgeScore f e=toLex (c*geometricSlope (rootLabel cs e)+d,geometricSlope (rootLabel cs e)) := by
+    unfold lexEdgeScore lexRootScore
+    rw [hf _ (hP e he)]
+  by_cases hc : 0 ≤ c
+  · left
+    apply hs.imp_of_mem
+    intro e e' he he' hee'
+    rw [hv e he,hv e' he'] at hee'
+    exact lex_affine_nonneg hc hee'
+  · right
+    apply hs.imp_of_mem
+    intro e e' he he' hee'
+    rw [hv e he,hv e' he'] at hee'
+    exact lex_affine_neg (lt_of_not_ge hc) hee'
+
+include B in
+/-- A complete plane block, sorted by any functional, has precisely the
+same actual path product as its angularly sorted version. -/
+theorem dispatch_plane_functional {R : Type*} [Semiring R] (T : R) (b : W)
+    (f : Module.Dual ℝ (I → ℝ)) (p q : List (AdaptedDyer.Edge cs))
+    (hpnd : p.Nodup) (hqnd : q.Nodup)
+    (hpfull : ∀ e, e ∈ p ↔ (rootLabel cs e).val ∈ P ∧ BruhatLE cs (target cs e) b)
+    (hqfull : ∀ e, e ∈ q ↔ (rootLabel cs e).val ∈ P ∧ BruhatLE cs (target cs e) b)
+    (hp : p.Pairwise (fun e e' => lexEdgeScore f e' ≤ lexEdgeScore f e))
+    (hq : q.Pairwise (fun e e' => geometricSlope (rootLabel cs e') ≤ geometricSlope (rootLabel cs e)))
+    (x y : W) (hy : BruhatLE cs y b) :
+    dispatch (source cs) (target cs) T (fun z => if z=y then 1 else 0) p x =
+      dispatch (source cs) (target cs) T (fun z => if z=y then 1 else 0) q x := by
+  let L : AdaptedDyer.Edge cs → ℝ := fun e => geometricSlope (rootLabel cs e)
+  have hties : ∀ e e', L e=L e' → target cs e ≠ source cs e' := by
+    intro e e' he hj
+    exact rootLabel_ne_of_join cs e e' hj (geometricSlope_injective he)
+  rcases B.lex_block_orientation f p (fun e he => ((hpfull e).mp he).1) hp with hs|hs
+  · exact dispatch_entries_eq_of_sorted (source cs) (target cs) T L hties
+      p q hpnd hqnd hs hq (fun e => (hpfull e).trans (hqfull e).symm) x y
+  · have hentry := dispatch_entries_eq_of_sorted (source cs) (target cs) T
+      (fun e => -L e) (fun e e' he => hties e e' (neg_injective he))
+      p q.reverse hpnd (List.nodup_reverse.mpr hqnd)
+      (hs.imp (fun he => neg_le_neg he))
+      (by rw [List.pairwise_reverse]; exact hq.imp (fun he => neg_le_neg he))
+      (fun e => (hpfull e).trans ((hqfull e).symm.trans List.mem_reverse.symm)) x y
+    exact hentry.trans (B.dispatch_plane_reverse T b q hqnd hqfull hq x y hy).symm
+
+end PlaneBasis
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Generic pencils of genuine linear root-label functionals. At a wall all
+roots of a tied block belong to one full actual root plane. -/
+namespace KLInvariance.ReflectionSchedules
+open Polynomial
+universe u v
+variable {I : Type u} [Fintype I]
+noncomputable section
+
+/-- A countable family of nonzero vectors can be simultaneously avoided. -/
+theorem exists_avoiding_functional {A : Type v} [Countable A] (v : A → (I → ℝ)) :
+    ∃ f : Module.Dual ℝ (I → ℝ), ∀ a, v a ≠ 0 → f (v a) ≠ 0 := by
+  classical
+  let P := {a : A // v a ≠ 0}
+  have hn (a : P) : coordinatePolynomial (v a.val) ≠ 0 := by
+    intro h
+    apply a.property
+    apply coordinatePolynomial_injective
+    simpa [coordinatePolynomial] using h
+  let bad : Set ℝ := ⋃ a : P, {t | (coordinatePolynomial (v a.val)).IsRoot t}
+  have hb : bad.Countable := Set.countable_iUnion
+    (fun a => (Polynomial.finite_setOfPred_isRoot (hn a)).countable)
+  obtain ⟨t,ht⟩ : ∃ t, t ∉ bad := by
+    by_contra h
+    push Not at h
+    have he : bad=Set.univ := Set.eq_univ_of_forall h
+    rw [he] at hb
+    exact Set.not_countable_univ hb
+  refine ⟨momentFunctional t,fun a ha he => ht ?_⟩
+  apply Set.mem_iUnion.mpr
+  refine ⟨⟨a,ha⟩,?_⟩
+  simpa only [Set.mem_ofPred_eq,Polynomial.IsRoot,coordinatePolynomial_eval] using he
+
+end
+end KLInvariance.ReflectionSchedules
+
+namespace KLInvariance.TitsSpace
+open Module
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+def pencil (f g : Module.Dual ℝ (I → ℝ)) (t : ℝ) : Module.Dual ℝ (I → ℝ) :=
+  (1-t) • f + t • g
+
+omit [Fintype I] in
+@[simp] theorem pencil_zero (f g : Module.Dual ℝ (I → ℝ)) : pencil f g 0=f := by
+  simp [pencil]
+omit [Fintype I] in
+@[simp] theorem pencil_one (f g : Module.Dual ℝ (I → ℝ)) : pencil f g 1=g := by
+  simp [pencil]
+
+omit [Fintype I] in
+theorem pencil_apply (f g : Module.Dual ℝ (I → ℝ)) (t : ℝ) (v : I → ℝ) :
+    pencil f g t v=(1-t)*f v+t*g v := rfl
+
+def rootDifference (a b : PositiveRoot M cs) : I → ℝ := normalizedRoot b-normalizedRoot a
+
+def wallVector (f : Module.Dual ℝ (I → ℝ)) (a b c : PositiveRoot M cs) : I → ℝ :=
+  f (rootDifference a b) • rootDifference a c -
+    f (rootDifference a c) • rootDifference a b
+
+/-- This is an actual genericity condition on linear functionals, not
+an assumed path or Hecke identity. -/
+def Transverse (f g : Module.Dual ℝ (I → ℝ)) : Prop :=
+  ∀ a b c : PositiveRoot M cs, g (wallVector f a b c)=0 → wallVector f a b c=0
+
+theorem pencil_tie_plane (f g : Module.Dual ℝ (I → ℝ))
+    (hf : Function.Injective (fun a : PositiveRoot M cs => f (normalizedRoot a)))
+    (hg : Transverse (M := M) (cs := cs) f g) (t : ℝ)
+    (a b : PositiveRoot M cs) (hab : a ≠ b)
+    (he : pencil f g t (normalizedRoot a)=pencil f g t (normalizedRoot b)) :
+    ∀ c : PositiveRoot M cs,
+      pencil f g t (normalizedRoot c)=pencil f g t (normalizedRoot a) ↔
+        c.val ∈ rootPlane a b := by
+  let v := rootDifference a b
+  have hv : f v ≠ 0 := by
+    change f (normalizedRoot b-normalizedRoot a) ≠ 0
+    rw [map_sub,sub_ne_zero]
+    exact fun h => hab (hf h).symm
+  have hvt : (1-t)*f v+t*g v=0 := by
+    rw [← pencil_apply]
+    change pencil f g t (normalizedRoot b-normalizedRoot a)=0
+    rw [map_sub,he,sub_self]
+  have ht : t ≠ 0 := by
+    intro h
+    subst t
+    simp only [sub_zero,one_mul,zero_mul,add_zero] at hvt
+    exact hv hvt
+  intro c
+  constructor
+  · intro hc
+    let w := rootDifference a c
+    have hwt : (1-t)*f w+t*g w=0 := by
+      rw [← pencil_apply]
+      change pencil f g t (normalizedRoot c-normalizedRoot a)=0
+      rw [map_sub,hc,sub_self]
+    have hz : g (wallVector f a b c)=0 := by
+      change g (f v • w-f w • v)=0
+      rw [map_sub,map_smul,map_smul,smul_eq_mul,smul_eq_mul]
+      apply (mul_eq_zero.mp (show t * (f v*g w-f w*g v)=0 by
+        calc
+          _ = f v*((1-t)*f w+t*g w)-f w*((1-t)*f v+t*g v) := by ring
+          _ = 0 := by rw [hwt,hvt]; ring)).resolve_left ht
+    have heq : f v • w=f w • v := sub_eq_zero.mp (hg a b c hz)
+    let P := rootPlane a b
+    have haP : normalizedRoot a ∈ P := (normalizedRoot_mem_iff _ _).mpr
+      (Submodule.subset_span (Set.mem_insert _ _))
+    have hbP : normalizedRoot b ∈ P := (normalizedRoot_mem_iff _ _).mpr
+      (Submodule.subset_span (Set.mem_insert_of_mem _ (Set.mem_singleton _)))
+    have hvP : v ∈ P := P.sub_mem hbP haP
+    have hwP : w ∈ P := by
+      have hh := P.smul_mem (f v)⁻¹ (heq ▸ P.smul_mem (f w) hvP)
+      simpa only [smul_smul,inv_mul_cancel₀ hv,one_smul] using hh
+    apply (normalizedRoot_mem_iff P c).mp
+    have hh := P.add_mem hwP haP
+    simpa only [w,rootDifference,sub_add_cancel] using hh
+  · intro hc
+    obtain ⟨s,hs⟩ := normalizedRoot_affine hc
+    rw [hs,map_add,map_smul,map_smul,smul_eq_mul,smul_eq_mul,← he]
+    ring
+
+/-- One intermediate functional is simultaneously transverse to any two
+specified generic endpoints, even with infinitely many positive roots. -/
+theorem exists_common_transverse (f h : Module.Dual ℝ (I → ℝ)) :
+    ∃ g : Module.Dual ℝ (I → ℝ),
+      Function.Injective (fun a : PositiveRoot M cs => g (normalizedRoot a)) ∧
+      Transverse (M := M) (cs := cs) f g ∧ Transverse (M := M) (cs := cs) h g := by
+  let := countable_Positiveroot M cs
+  let E := PositiveRoot M cs
+  let A := (E × E) ⊕ (Bool × E × E × E)
+  let v : A → (I → ℝ) := fun x => match x with
+    | .inl (a,b) => rootDifference a b
+    | .inr (i,a,b,c) => wallVector (if i then f else h) a b c
+  obtain ⟨g,hg⟩ := ReflectionSchedules.exists_avoiding_functional v
+  refine ⟨g,?_,?_,?_⟩
+  · intro a b he
+    by_contra hab
+    have hn : rootDifference a b ≠ 0 := by
+      exact fun hz => hab (normalizedRoot_injective (sub_eq_zero.mp hz)).symm
+    apply hg (.inl (a,b)) hn
+    simp only [v,rootDifference,map_sub,he,sub_self]
+  · intro a b c he
+    by_contra hz
+    exact hg (.inr (true,a,b,c)) (by simpa [v] using hz)
+      (by simpa [v] using he)
+  · intro a b c he
+    by_contra hz
+    exact hg (.inr (false,a,b,c)) (by simpa [v] using hz)
+      (by simpa [v] using he)
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! Finite wall crossing for actual reflection-edge path matrices. -/
+namespace KLInvariance.WallDyer
+open Module SchedulePaths TitsSpace AdaptedDyer
+open scoped Topology
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+def rootValue (f : Module.Dual ℝ (I → ℝ)) (e : Edge cs) : ℝ :=
+  f (normalizedRoot (rootLabel cs e))
+
+def rootPlaneTies (f : Module.Dual ℝ (I → ℝ)) : Prop :=
+  ∀ a b : PositiveRoot M cs, a ≠ b → f (normalizedRoot a)=f (normalizedRoot b) →
+    ∀ c : PositiveRoot M cs, f (normalizedRoot c)=f (normalizedRoot a) ↔
+      c.val ∈ rootPlane a b
+
+def schedule (f : Module.Dual ℝ (I → ℝ)) (b : W) : List (Edge cs) :=
+  sortedSchedule (edgeFinset cs b) (lexEdgeScore f)
+
+def entry {R : Type*} [Semiring R] (T : R) (f : Module.Dual ℝ (I → ℝ))
+    (b x y : W) : R :=
+  dispatch (source cs) (target cs) T (fun z => if z=y then 1 else 0) (schedule f b) x
+
+/-- The complete tied block at any genuine simple wall has an actual
+operator independent of the linear tie-breaking functional. -/
+theorem block_entry_independent {R : Type*} [Semiring R] (T : R) (b : W)
+    (f : Module.Dual ℝ (I → ℝ)) (hf : rootPlaneTies (M := M) (cs := cs) f)
+    (g h : Module.Dual ℝ (I → ℝ)) (a : ℝ) (x y : W) (hy : BruhatLE cs y b) :
+    dispatch (source cs) (target cs) T (fun z => if z=y then 1 else 0)
+      (scheduleBlock (edgeFinset cs b) (rootValue f) (lexEdgeScore g) a) x =
+    dispatch (source cs) (target cs) T (fun z => if z=y then 1 else 0)
+      (scheduleBlock (edgeFinset cs b) (rootValue f) (lexEdgeScore h) a) x := by
+  let F := (edgeFinset cs b).filter (fun e => rootValue f e=a)
+  let p := sortedSchedule F (lexEdgeScore g)
+  let q := sortedSchedule F (lexEdgeScore h)
+  let r := sortedSchedule F (fun e => geometricSlope (rootLabel cs e))
+  change dispatch (source cs) (target cs) T _ p x = dispatch (source cs) (target cs) T _ q x
+  have hmem (k : Edge cs) : k ∈ F ↔ BruhatLE cs (target cs k) b ∧ rootValue f k=a := by
+    simp only [F,Finset.mem_filter,mem_edgeFinset]
+  by_cases hex : ∃ e ∈ F, ∃ e' ∈ F, rootLabel cs e ≠ rootLabel cs e'
+  · obtain ⟨e,he,e',he',hne⟩ := hex
+    obtain ⟨B⟩ := nonempty_planeBasis (rootLabel cs e) (rootLabel cs e') hne
+    have hv := hf (rootLabel cs e) (rootLabel cs e') hne
+      (((hmem e).mp he).2.trans ((hmem e').mp he').2.symm)
+    have hfull (k : Edge cs) : k ∈ F ↔
+        (rootLabel cs k).val ∈ rootPlane (rootLabel cs e) (rootLabel cs e') ∧
+        BruhatLE cs (target cs k) b := by
+      rw [hmem,← hv,show f (normalizedRoot (rootLabel cs e))=a from ((hmem e).mp he).2]
+      exact and_comm
+    have hp := B.dispatch_plane_functional T b g p r
+      (sortedSchedule_nodup _ _) (sortedSchedule_nodup _ _)
+      (fun k => (mem_sortedSchedule _ _ k).trans (hfull k))
+      (fun k => (mem_sortedSchedule _ _ k).trans (hfull k))
+      (sortedSchedule_pairwise _ _) (sortedSchedule_pairwise _ _) x y hy
+    have hq := B.dispatch_plane_functional T b h q r
+      (sortedSchedule_nodup _ _) (sortedSchedule_nodup _ _)
+      (fun k => (mem_sortedSchedule _ _ k).trans (hfull k))
+      (fun k => (mem_sortedSchedule _ _ k).trans (hfull k))
+      (sortedSchedule_pairwise _ _) (sortedSchedule_pairwise _ _) x y hy
+    exact hp.trans hq.symm
+  · have heq (e : Edge cs) (he : e ∈ F) (e' : Edge cs) (he' : e' ∈ F) :
+        rootLabel cs e=rootLabel cs e' := by
+      by_contra hn
+      exact hex ⟨e,he,e',he',hn⟩
+    let L : Edge cs → ℝ := fun e => geometricSlope (rootLabel cs e)
+    have hties : ∀ e e', L e=L e' → target cs e ≠ source cs e' := by
+      intro e e' he hj
+      exact rootLabel_ne_of_join cs e e' hj (geometricSlope_injective he)
+    have hs (s : Module.Dual ℝ (I → ℝ)) :
+        (sortedSchedule F (lexEdgeScore s)).Pairwise (fun e e' => L e' ≤ L e) := by
+      apply (sortedSchedule_pairwise F (lexEdgeScore s)).imp_of_mem
+      intro e e' he he' _
+      have hr := heq e ((mem_sortedSchedule _ _ _).mp he) e' ((mem_sortedSchedule _ _ _).mp he')
+      exact (congrArg geometricSlope hr).symm.le
+    exact dispatch_entries_eq_of_sorted (source cs) (target cs) T L hties p q
+      (sortedSchedule_nodup _ _) (sortedSchedule_nodup _ _) (hs g) (hs h)
+      (fun e => (mem_sortedSchedule _ _ e).trans (mem_sortedSchedule _ _ e).symm) x y
+
+/-- All blocks may be replaced simultaneously on the finite lower region. -/
+theorem grouped_entry_independent {R : Type*} [Semiring R] (T : R) (b : W)
+    (f : Module.Dual ℝ (I → ℝ)) (hf : rootPlaneTies (M := M) (cs := cs) f)
+    (g h : Module.Dual ℝ (I → ℝ)) (x y : W) (hx : BruhatLE cs x b) :
+    dispatch (source cs) (target cs) T (fun z => if z=y then 1 else 0)
+      (groupedSchedule (edgeFinset cs b) (rootValue f) (lexEdgeScore g)) x =
+    dispatch (source cs) (target cs) T (fun z => if z=y then 1 else 0)
+      (groupedSchedule (edgeFinset cs b) (rootValue f) (lexEdgeScore h)) x := by
+  let K := lowerFinset cs b
+  have ht (s : Module.Dual ℝ (I → ℝ)) (a : ℝ) (e : Edge cs)
+      (he : e ∈ scheduleBlock (edgeFinset cs b) (rootValue f) (lexEdgeScore s) a) :
+      target cs e ∈ K := by
+    exact (mem_lowerFinset cs b _).mpr ((mem_edgeFinset cs b e).mp
+      ((mem_scheduleBlock _ _ _ _ _).mp he).1)
+  apply dispatch_flatMap_eq_on (source cs) (target cs) T (K : Set W)
+    (blockValues (edgeFinset cs b) (rootValue f))
+    (scheduleBlock (edgeFinset cs b) (rootValue f) (lexEdgeScore g))
+    (scheduleBlock (edgeFinset cs b) (rootValue f) (lexEdgeScore h))
+    (fun a _ e he => ht g a e he) _ _ x ((mem_lowerFinset cs b x).mpr hx)
+  intro a _ v z hz
+  apply dispatch_eq_of_entries_on (source cs) (target cs) T K _ _ (ht g a) (ht h a) _ v z hz
+  intro l _ k hk
+  exact block_entry_independent T b f hf g h a l k ((mem_lowerFinset cs b k).mp hk)
+
+/-- Stable strict primary comparisons identify the genuine sorted schedule
+with the corresponding product of tied blocks. -/
+theorem entry_eq_grouped {R : Type*} [Semiring R] (T : R) (b : W)
+    (f g : Module.Dual ℝ (I → ℝ))
+    (h : ∀ e ∈ edgeFinset cs b, ∀ e' ∈ edgeFinset cs b,
+      rootValue f e' < rootValue f e → rootValue g e' < rootValue g e)
+    (x y : W) :
+    entry (cs := cs) T g b x y =
+    dispatch (source cs) (target cs) T (fun z => if z=y then 1 else 0)
+      (groupedSchedule (edgeFinset cs b) (rootValue f) (lexEdgeScore g)) x := by
+  apply dispatch_entries_eq_of_sorted (source cs) (target cs) T (lexEdgeScore g)
+    (lexEdgeScore_ties g) _ _ (sortedSchedule_nodup _ _) (groupedSchedule_nodup _ _ _)
+    (sortedSchedule_pairwise _ _)
+  · apply groupedSchedule_pairwise
+    intro e he e' he' hlt
+    exact Prod.Lex.toLex_le_toLex.mpr (Or.inl (h e he e' he' hlt))
+  · intro e
+    simp only [mem_sortedSchedule,mem_groupedSchedule]
+
+theorem pencil_strict_stable (b : W) (f g : Module.Dual ℝ (I → ℝ)) (t₀ : ℝ) :
+    ∀ᶠ t in 𝓝 t₀, ∀ e ∈ edgeFinset cs b, ∀ e' ∈ edgeFinset cs b,
+      rootValue (pencil f g t₀) e' < rootValue (pencil f g t₀) e →
+        rootValue (pencil f g t) e' < rootValue (pencil f g t) e := by
+  apply (Filter.eventually_all_finset _).mpr
+  intro e _
+  apply (Filter.eventually_all_finset _).mpr
+  intro e' _
+  have hc (k : Edge cs) : Continuous (fun t : ℝ => rootValue (pencil f g t) k) := by
+    change Continuous (fun t : ℝ => (1-t)*f (normalizedRoot (rootLabel cs k))+
+      t*g (normalizedRoot (rootLabel cs k)))
+    fun_prop
+  by_cases hlt : rootValue (pencil f g t₀) e' < rootValue (pencil f g t₀) e
+  · filter_upwards [(isOpen_lt (hc e') (hc e)).mem_nhds hlt] with t ht
+    exact fun _ => ht
+  · exact Filter.Eventually.of_forall (fun _ h => (hlt h).elim)
+
+/-- Crossing all actual root-plane walls leaves the full path matrix fixed,
+not merely a single adapted endpoint column. -/
+theorem locally_constant_pencil {R : Type*} [Semiring R] (T : R) (b : W)
+    (f g : Module.Dual ℝ (I → ℝ))
+    (hf : Function.Injective (fun a : PositiveRoot M cs => f (normalizedRoot a)))
+    (hg : Transverse (M := M) (cs := cs) f g) (x y : W) (hx : BruhatLE cs x b) :
+    IsLocallyConstant (fun t : ℝ => entry (cs := cs) T (pencil f g t) b x y) := by
+  apply (IsLocallyConstant.iff_eventually_eq _).mpr
+  intro t₀
+  filter_upwards [pencil_strict_stable b f g t₀] with t ht
+  have hp : rootPlaneTies (M := M) (cs := cs) (pencil f g t₀) := by
+    intro a d had he
+    exact pencil_tie_plane f g hf hg t₀ a d had he
+  rw [entry_eq_grouped T b (pencil f g t₀) (pencil f g t) ht,
+    entry_eq_grouped T b (pencil f g t₀) (pencil f g t₀) (fun _ _ _ _ h => h)]
+  exact grouped_entry_independent T b (pencil f g t₀) hp (pencil f g t) (pencil f g t₀) x y hx
+
+/-- Full matrix order independence for genuine finite geometric orders. -/
+theorem entry_independent {R : Type*} [Semiring R] (T : R) (b : W)
+    (f h : Module.Dual ℝ (I → ℝ))
+    (hf : Function.Injective (fun a : PositiveRoot M cs => f (normalizedRoot a)))
+    (hh : Function.Injective (fun a : PositiveRoot M cs => h (normalizedRoot a)))
+    (x y : W) (hx : BruhatLE cs x b) : entry (cs := cs) T f b x y=entry (cs := cs) T h b x y := by
+  obtain ⟨g,_,hfg,hhg⟩ := exists_common_transverse (M := M) (cs := cs) f h
+  have he := (locally_constant_pencil T b f g hf hfg x y hx).apply_eq_of_preconnectedSpace 0 1
+  have he' := (locally_constant_pencil T b h g hh hhg x y hx).apply_eq_of_preconnectedSpace 0 1
+  simp only [pencil_zero,pencil_one] at he he'
+  exact he.trans he'.symm
+
+end
+end KLInvariance.WallDyer
+
+end
+
+
+section
+
+/-! Dyer's full path-matrix identity, for the same geometric schedule in
+every endpoint column, with the genuine Hecke-defined normalized R. -/
+namespace KLInvariance.WallDyer
+open Module SchedulePaths TitsSpace AdaptedDyer
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+theorem neg_numerator_normalized (O : RatioOrder (M := M) (cs := cs))
+    (hh : O.denominator=height) (a : PositiveRoot M cs) :
+    (-O.numerator) (normalizedRoot a) = -O.slope a := by
+  simp only [LinearMap.neg_apply,dual_normalizedRoot,RatioOrder.slope,hh]
+
+/-- An upper bound larger than the path endpoint creates no extra paths:
+every actual directed path stays inside its endpoint interval. -/
+theorem subpaths_neg_numerator (O : RatioOrder (M := M) (cs := cs))
+    (hh : O.denominator=height) (b x y : W) (hy : BruhatLE cs y b) :
+    subpaths (source cs) (target cs) (WallDyer.schedule (-O.numerator) b) x y =
+      paths O x y := by
+  have hscore (e : Edge cs) : rootValue (-O.numerator) e = -score O e :=
+    neg_numerator_normalized O hh (rootLabel cs e)
+  ext p
+  rw [mem_subpaths,mem_paths]
+  constructor
+  · rintro ⟨hsub,hpath⟩
+    refine ⟨hpath,?_⟩
+    have hs := (sortedSchedule_pairwise (edgeFinset cs b) (lexEdgeScore (-O.numerator))).sublist hsub
+    have hs' : p.Pairwise (fun e e' => score O e ≤ score O e') := by
+      apply hs.imp
+      intro e e' he
+      have hm := Prod.Lex.monotone_fst _ _ he
+      change rootValue (-O.numerator) e' ≤ rootValue (-O.numerator) e at hm
+      rw [hscore e',hscore e] at hm
+      exact neg_le_neg_iff.mp hm
+    exact decreasing_of_schedule_path (source cs) (target cs)
+      (Λ := OrderDual ℝ) (fun e => (score O e : OrderDual ℝ))
+      (fun e e' he => score_ties O e e' he) hpath hs'
+  · rintro ⟨hpath,hord⟩
+    refine ⟨?_,hpath⟩
+    apply sublist_of_decreasing_subset (lexEdgeScore (-O.numerator))
+    · apply hord.imp
+      intro e e' he
+      apply Prod.Lex.toLex_lt_toLex.mpr
+      left
+      change rootValue (-O.numerator) e' < rootValue (-O.numerator) e
+      rw [hscore e',hscore e]
+      exact neg_lt_neg he
+    · exact sortedSchedule_pairwise _ _
+    · intro e he
+      apply (mem_sortedSchedule _ _ _).mpr
+      exact (mem_edgeFinset cs b e).mpr ((path_targets_le cs hpath e he).trans hy)
+
+theorem entry_neg_numerator (O : RatioOrder (M := M) (cs := cs))
+    (hh : O.denominator=height) (b x y : W) (hy : BruhatLE cs y b) :
+    entry (cs := cs) (Polynomial.X : Polynomial ℤ) (-O.numerator) b x y =
+      polynomial O x y := by
+  unfold entry
+  rw [dispatch_eq_subpaths_sum _ _ _ _ (show (WallDyer.schedule (-O.numerator) b).Nodup from sortedSchedule_nodup _ _),
+    subpaths_neg_numerator O hh b x y hy]
+  rfl
+
+
+theorem entry_eq_rTilde (f : Module.Dual ℝ (I → ℝ))
+    (hf : Function.Injective (fun a : PositiveRoot M cs => f (normalizedRoot a)))
+    (b x y : W) (hx : BruhatLE cs x b) (hy : BruhatLE cs y b) :
+    entry (cs := cs) (Polynomial.X : Polynomial ℤ) f b x y = Hecke.rTilde cs x y := by
+  obtain ⟨O,hO,hh⟩ := exists_ratioOrder_chamber_height (M := M) (cs := cs) y
+  have hn : Function.Injective (fun a : PositiveRoot M cs => (-O.numerator) (normalizedRoot a)) := by
+    intro a b he
+    change (-O.numerator) (normalizedRoot a) = (-O.numerator) (normalizedRoot b) at he
+    rw [neg_numerator_normalized O hh a,neg_numerator_normalized O hh b] at he
+    exact O.slope_injective (neg_injective he)
+  rw [entry_independent (Polynomial.X : Polynomial ℤ) b f (-O.numerator) hf hn x y hx,
+    entry_neg_numerator O hh b x y hy,polynomial_eq_rTilde O x y hO]
+
+/-- The manuscript's fixed geometric functional provides such a genuine
+full Dyer schedule for every finite lower Bruhat region. -/
+theorem geometric_entry_eq_rTilde (b x y : W) (hx : BruhatLE cs x b) (hy : BruhatLE cs y b) :
+    entry (cs := cs) (Polynomial.X : Polynomial ℤ) (geometricFunctional (M := M) (cs := cs)) b x y =
+      Hecke.rTilde cs x y := by
+  apply entry_eq_rTilde _ _ b x y hx hy
+  simpa only [geometricFunctional_normalizedRoot] using
+    (geometricSlope_injective (M := M) (cs := cs))
+
+end
+end KLInvariance.WallDyer
+
+end
+
+
+section
+
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/KazhdanLusztig/ReflectionPlane.lean b/lean/OAI/RepresentationTheory/KazhdanLusztig/ReflectionPlane.lean
index 6140e0b0e04c5d48bc27c1e8ef4f1f9d6777f347..9202213ace07bc1eda9516f0accb6db523a74236 100644
--- a/lean/OAI/RepresentationTheory/KazhdanLusztig/ReflectionPlane.lean
+++ b/lean/OAI/RepresentationTheory/KazhdanLusztig/ReflectionPlane.lean
@@ -1,100 +1,100 @@
-import Mathlib
-
-namespace OAI
-
-/-!
-Linear reflection products and common root planes for nondegenerate ambient forms.
--/
-
-section
-
-
-namespace KLInvariance.ReflectionPlane
-variable {k V : Type*} [Field k] [AddCommGroup V] [Module k V]
-
-/-- Independent linear forms give a surjection to the coordinate plane. -/
-theorem formPair_surjective (f g : Module.Dual k V)
-    (hfg : LinearIndependent k ![f, g]) : Function.Surjective (f.prod g) := by
-  apply LinearMap.dualMap_injective_iff.mp
-  apply LinearMap.ker_eq_bot.mp
-  apply le_antisymm _ bot_le
-  intro l hl
-  change l = 0
-  have hl' (v : V) : l (f v, g v) = 0 := LinearMap.congr_fun hl v
-  have heval (c d : k) : l (c, d) = c * l (1, 0) + d * l (0, 1) := by
-    have heq : (c, d) = c • ((1, 0) : k × k) + d • ((0, 1) : k × k) := by
-      ext <;> simp
-    rw [heq, map_add, map_smul, map_smul]
-    rfl
-  have hrel : l (1, 0) • f + l (0, 1) • g = 0 := by
-    ext v
-    change l (1, 0) * f v + l (0, 1) * g v = 0
-    have hv := hl' v
-    rw [heval] at hv
-    simpa only [mul_comm] using hv
-  obtain ⟨h₁, h₂⟩ := LinearIndependent.pair_iff.mp hfg _ _ hrel
-  apply LinearMap.ext
-  intro x
-  obtain ⟨c, d⟩ := x
-  change l (c, d) = 0
-  rw [heval, h₁, h₂, mul_zero, mul_zero, add_zero]
-
-/-- The range of the difference of a two-reflection product from the
-identity is exactly the span of the two roots. -/
-theorem range_preReflection_product_sub_id (α β : V) (f g : Module.Dual k V)
-    (hfg : LinearIndependent k ![f, g]) :
-    LinearMap.range ((Module.preReflection α f).comp (Module.preReflection β g) -
-      LinearMap.id) = Submodule.span k {α, β} := by
-  have hformula (v : V) :
-      ((Module.preReflection α f).comp (Module.preReflection β g) -
-        (LinearMap.id : V →ₗ[k] V)) v = (-f v + f β * g v) • α + (-g v) • β := by
-    simp only [LinearMap.sub_apply, LinearMap.comp_apply, Module.preReflection_apply,
-      LinearMap.id_apply, map_sub, map_smul]
-    module
-  apply le_antisymm
-  · rintro z ⟨v, rfl⟩
-    rw [hformula]
-    exact Submodule.mem_span_pair.mpr ⟨_, _, rfl⟩
-  · intro z hz
-    obtain ⟨a, b, rfl⟩ := Submodule.mem_span_pair.mp hz
-    obtain ⟨v, hv⟩ := formPair_surjective f g hfg (-a - f β * b, -b)
-    have hf : f v = -a - f β * b := congrArg Prod.fst hv
-    have hg : g v = -b := congrArg Prod.snd hv
-    refine ⟨v, ?_⟩
-    rw [hformula, hf, hg]
-    module
-
-
-theorem range_reflection_product_sub_id
-    (B : LinearMap.BilinForm k V) (hB : B.Nondegenerate)
-    (α β : V) (hα : B α α = 2) (hβ : B β β = 2)
-    (hαβ : LinearIndependent k ![α, β]) :
-    LinearMap.range (((Module.reflection hα).toLinearMap).comp
-      (Module.reflection hβ).toLinearMap - LinearMap.id) = Submodule.span k {α, β} := by
-  apply range_preReflection_product_sub_id
-  have h := hαβ.map' B hB.ker_eq_bot
-  have heq : (⇑B ∘ ![α, β]) = ![B α, B β] := by
-    funext i
-    fin_cases i <;> rfl
-  rw [heq] at h
-  exact h
-
-/-- Equal products of two normalized reflections recover equal root
-planes, the exact local geometric input to `dih:transport`. -/
-theorem plane_eq_of_reflection_product_eq
-    (B : LinearMap.BilinForm k V) (hB : B.Nondegenerate)
-    (α β γ δ : V)
-    (hα : B α α = 2) (hβ : B β β = 2) (hγ : B γ γ = 2) (hδ : B δ δ = 2)
-    (hαβ : LinearIndependent k ![α, β]) (hγδ : LinearIndependent k ![γ, δ])
-    (heq : (Module.reflection hα).toLinearMap.comp (Module.reflection hβ).toLinearMap =
-      (Module.reflection hγ).toLinearMap.comp (Module.reflection hδ).toLinearMap) :
-    Submodule.span k {α, β} = Submodule.span k {γ, δ} := by
-  rw [← range_reflection_product_sub_id B hB α β hα hβ hαβ,
-    ← range_reflection_product_sub_id B hB γ δ hγ hδ hγδ, heq]
-
-end KLInvariance.ReflectionPlane
-
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-!
+Linear reflection products and common root planes for nondegenerate ambient forms.
+-/
+
+section
+
+
+namespace KLInvariance.ReflectionPlane
+variable {k V : Type*} [Field k] [AddCommGroup V] [Module k V]
+
+/-- Independent linear forms give a surjection to the coordinate plane. -/
+theorem formPair_surjective (f g : Module.Dual k V)
+    (hfg : LinearIndependent k ![f, g]) : Function.Surjective (f.prod g) := by
+  apply LinearMap.dualMap_injective_iff.mp
+  apply LinearMap.ker_eq_bot.mp
+  apply le_antisymm _ bot_le
+  intro l hl
+  change l = 0
+  have hl' (v : V) : l (f v, g v) = 0 := LinearMap.congr_fun hl v
+  have heval (c d : k) : l (c, d) = c * l (1, 0) + d * l (0, 1) := by
+    have heq : (c, d) = c • ((1, 0) : k × k) + d • ((0, 1) : k × k) := by
+      ext <;> simp
+    rw [heq, map_add, map_smul, map_smul]
+    rfl
+  have hrel : l (1, 0) • f + l (0, 1) • g = 0 := by
+    ext v
+    change l (1, 0) * f v + l (0, 1) * g v = 0
+    have hv := hl' v
+    rw [heval] at hv
+    simpa only [mul_comm] using hv
+  obtain ⟨h₁, h₂⟩ := LinearIndependent.pair_iff.mp hfg _ _ hrel
+  apply LinearMap.ext
+  intro x
+  obtain ⟨c, d⟩ := x
+  change l (c, d) = 0
+  rw [heval, h₁, h₂, mul_zero, mul_zero, add_zero]
+
+/-- The range of the difference of a two-reflection product from the
+identity is exactly the span of the two roots. -/
+theorem range_preReflection_product_sub_id (α β : V) (f g : Module.Dual k V)
+    (hfg : LinearIndependent k ![f, g]) :
+    LinearMap.range ((Module.preReflection α f).comp (Module.preReflection β g) -
+      LinearMap.id) = Submodule.span k {α, β} := by
+  have hformula (v : V) :
+      ((Module.preReflection α f).comp (Module.preReflection β g) -
+        (LinearMap.id : V →ₗ[k] V)) v = (-f v + f β * g v) • α + (-g v) • β := by
+    simp only [LinearMap.sub_apply, LinearMap.comp_apply, Module.preReflection_apply,
+      LinearMap.id_apply, map_sub, map_smul]
+    module
+  apply le_antisymm
+  · rintro z ⟨v, rfl⟩
+    rw [hformula]
+    exact Submodule.mem_span_pair.mpr ⟨_, _, rfl⟩
+  · intro z hz
+    obtain ⟨a, b, rfl⟩ := Submodule.mem_span_pair.mp hz
+    obtain ⟨v, hv⟩ := formPair_surjective f g hfg (-a - f β * b, -b)
+    have hf : f v = -a - f β * b := congrArg Prod.fst hv
+    have hg : g v = -b := congrArg Prod.snd hv
+    refine ⟨v, ?_⟩
+    rw [hformula, hf, hg]
+    module
+
+
+theorem range_reflection_product_sub_id
+    (B : LinearMap.BilinForm k V) (hB : B.Nondegenerate)
+    (α β : V) (hα : B α α = 2) (hβ : B β β = 2)
+    (hαβ : LinearIndependent k ![α, β]) :
+    LinearMap.range (((Module.reflection hα).toLinearMap).comp
+      (Module.reflection hβ).toLinearMap - LinearMap.id) = Submodule.span k {α, β} := by
+  apply range_preReflection_product_sub_id
+  have h := hαβ.map' B hB.ker_eq_bot
+  have heq : (⇑B ∘ ![α, β]) = ![B α, B β] := by
+    funext i
+    fin_cases i <;> rfl
+  rw [heq] at h
+  exact h
+
+/-- Equal products of two normalized reflections recover equal root
+planes, the exact local geometric input to `dih:transport`. -/
+theorem plane_eq_of_reflection_product_eq
+    (B : LinearMap.BilinForm k V) (hB : B.Nondegenerate)
+    (α β γ δ : V)
+    (hα : B α α = 2) (hβ : B β β = 2) (hγ : B γ γ = 2) (hδ : B δ δ = 2)
+    (hαβ : LinearIndependent k ![α, β]) (hγδ : LinearIndependent k ![γ, δ])
+    (heq : (Module.reflection hα).toLinearMap.comp (Module.reflection hβ).toLinearMap =
+      (Module.reflection hγ).toLinearMap.comp (Module.reflection hδ).toLinearMap) :
+    Submodule.span k {α, β} = Submodule.span k {γ, δ} := by
+  rw [← range_reflection_product_sub_id B hB α β hα hβ hαβ,
+    ← range_reflection_product_sub_id B hB γ δ hγ hδ hγδ, heq]
+
+end KLInvariance.ReflectionPlane
+
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/KazhdanLusztig/ReflectionSchedules.lean b/lean/OAI/RepresentationTheory/KazhdanLusztig/ReflectionSchedules.lean
index 2052ffe56ee4de0a5bb7354ff4d95abdfce07e2b..21acc292c882dc2ea1470a8c14637cf2c25cd0f8 100644
--- a/lean/OAI/RepresentationTheory/KazhdanLusztig/ReflectionSchedules.lean
+++ b/lean/OAI/RepresentationTheory/KazhdanLusztig/ReflectionSchedules.lean
@@ -1,1776 +1,1776 @@
-import OAI.RepresentationTheory.KazhdanLusztig.Tits
-import OAI.RepresentationTheory.KazhdanLusztig.ModuleAlgebra
-
-namespace OAI
-
-/-!
-Reflection schedules, decreasing paths, root-space and edge algebra, and dihedral path recursions.
--/
-
-section
-
-namespace KLInvariance.ReflectionSchedules
-open Polynomial
-universe u v
-variable {I : Type u} [Fintype I]
-noncomputable section
-
-def coordinatePolynomial (a : I → ℝ) : ℝ[X] :=
-  ∑ i, Polynomial.monomial (Fintype.equivFin I i).val (a i)
-
-theorem coordinatePolynomial_coeff (a : I → ℝ) (j : I) :
-    (coordinatePolynomial a).coeff (Fintype.equivFin I j).val = a j := by
-  classical
-  rw [coordinatePolynomial, Polynomial.finsetSum_coeff, Finset.sum_eq_single j]
-  · simp
-  · intro i _ hij
-    have hne : (Fintype.equivFin I i).val ≠ (Fintype.equivFin I j).val := by
-      intro h
-      exact hij ((Fintype.equivFin I).injective (Fin.ext h))
-    simp [Polynomial.coeff_monomial,hne]
-  · simp
-
-theorem coordinatePolynomial_injective : Function.Injective (coordinatePolynomial (I := I)) := by
-  intro a b h
-  funext i
-  simpa only [coordinatePolynomial_coeff] using congrArg (fun p : ℝ[X] => p.coeff
-    (Fintype.equivFin I i).val) h
-
-def momentFunctional (t : ℝ) : Module.Dual ℝ (I → ℝ) where
-  toFun a := ∑ i, a i * t^(Fintype.equivFin I i).val
-  map_add' a b := by simp [add_mul,Finset.sum_add_distrib]
-  map_smul' r a := by simp [Finset.mul_sum,mul_assoc]
-
-theorem coordinatePolynomial_eval (a : I → ℝ) (t : ℝ) :
-    (coordinatePolynomial a).eval t = momentFunctional t a := by
-  simp [coordinatePolynomial,momentFunctional,Polynomial.eval_finsetSum]
-
-
-theorem exists_generic_functional {A : Type v} [Countable A]
-    (v : A → (I → ℝ)) (hv : Function.Injective v) :
-    ∃ f : Module.Dual ℝ (I → ℝ), Function.Injective (fun a => f (v a)) := by
-  classical
-  let P := {p : A × A // p.1 ≠ p.2}
-  let d : P → ℝ[X] := fun p => coordinatePolynomial (v p.val.1) - coordinatePolynomial (v p.val.2)
-  have hd : ∀ p, d p ≠ 0 := by
-    intro p h
-    exact p.property (hv (coordinatePolynomial_injective (sub_eq_zero.mp h)))
-  let bad : Set ℝ := ⋃ p : P, {t | (d p).IsRoot t}
-  have hbad : bad.Countable := Set.countable_iUnion (fun p => (Polynomial.finite_setOfPred_isRoot (hd p)).countable)
-  have hex : ∃ t, t ∉ bad := by
-    by_contra h
-    push Not at h
-    have heq : bad = Set.univ := Set.eq_univ_of_forall h
-    rw [heq] at hbad
-    exact Set.not_countable_univ hbad
-  obtain ⟨t,ht⟩ := hex
-  refine ⟨momentFunctional t,fun a b hab => ?_⟩
-  change momentFunctional t (v a) = momentFunctional t (v b) at hab
-  by_contra hne
-  apply ht
-  apply Set.mem_iUnion.mpr
-  refine ⟨⟨(a,b),hne⟩,?_⟩
-  change (coordinatePolynomial (v a) - coordinatePolynomial (v b)).eval t = 0
-  rw [Polynomial.eval_sub,coordinatePolynomial_eval,coordinatePolynomial_eval,hab,sub_self]
-
-end
-end KLInvariance.ReflectionSchedules
-
-namespace KLInvariance.TitsSpace
-open Module
-universe u v
-variable {I : Type u} [Fintype I] (M : CoxeterMatrix I)
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
-def height : Module.Dual ℝ (I → ℝ) where
-  toFun a := ∑ i, a i
-  map_add' a b := by simp [Finset.sum_add_distrib]
-  map_smul' r a := by simp [Finset.mul_sum]
-
-variable {M cs}
-
-theorem PositiveRoot.height_pos (a : PositiveRoot M cs) : 0 < height a.val := by
-  classical
-  have hex : ∃ i, a.val i ≠ 0 := by
-    by_contra h
-    push Not at h
-    exact a.property.1.ne_zero (funext h)
-  obtain ⟨i,hi⟩ := hex
-  exact Finset.sum_pos' (fun j _ => a.property.2 j)
-    ⟨i,Finset.mem_univ _,lt_of_le_of_ne' (a.property.2 i) hi⟩
-
-def normalizedRoot (a : PositiveRoot M cs) : I → ℝ := (height a.val)⁻¹ • a.val
-
-theorem normalizedRoot_injective : Function.Injective (normalizedRoot (M := M) (cs := cs)) := by
-  intro a b hab
-  have ha : 0 < (height a.val)⁻¹ := inv_pos.mpr a.height_pos
-  have hb : 0 < (height b.val)⁻¹ := inv_pos.mpr b.height_pos
-  have hn := congrArg (fun v => form M v v) hab
-  change form M ((height a.val)⁻¹ • a.val) ((height a.val)⁻¹ • a.val) =
-    form M ((height b.val)⁻¹ • b.val) ((height b.val)⁻¹ • b.val) at hn
-  simp only [map_smul,LinearMap.smul_apply,smul_eq_mul,a.property.1.norm,b.property.1.norm] at hn
-  have hh : (height a.val)⁻¹ = (height b.val)⁻¹ := by nlinarith
-  apply Subtype.ext
-  change (height a.val)⁻¹ • a.val = (height b.val)⁻¹ • b.val at hab
-  rw [← hh] at hab
-  exact (smul_right_injective _ ha.ne').eq_iff.mp hab
-
-variable (M cs)
-
-theorem countable_Positiveroot : Countable (PositiveRoot M cs) := by
-  let : Countable W := cs.wordProd_surjective.countable
-  exact Countable.of_equiv {t : W // cs.IsReflection t} (positiveRootEquiv M cs)
-
-
-theorem exists_geometric_functional : ∃ f : Module.Dual ℝ (I → ℝ),
-    Function.Injective (fun a : PositiveRoot M cs => f a.val / height a.val) := by
-  let := countable_Positiveroot M cs
-  obtain ⟨f,hf⟩ := ReflectionSchedules.exists_generic_functional
-    (normalizedRoot (M := M) (cs := cs)) normalizedRoot_injective
-  refine ⟨f,?_⟩
-  convert hf using 1
-  funext a
-  simp [normalizedRoot,div_eq_mul_inv,mul_comm]
-
-/-- Betweenness of all positive combinations is the angular-order condition
-on every root plane. This is the genuine geometric reflection-order property. -/
-theorem functionalRatio_between (f : Module.Dual ℝ (I → ℝ))
-    (a b c : PositiveRoot M cs) {r s : ℝ} (hr : 0 < r) (hs : 0 < s)
-    (hc : c.val = r • a.val + s • b.val)
-    (hab : f a.val / height a.val < f b.val / height b.val) :
-    f a.val / height a.val < f c.val / height c.val ∧
-      f c.val / height c.val < f b.val / height b.val := by
-  have ha := a.height_pos
-  have hb := b.height_pos
-  have hh := c.height_pos
-  have hfc : f c.val = r*f a.val+s*f b.val := by rw [hc]; simp
-  have hhc : height c.val = r*height a.val+s*height b.val := by rw [hc]; simp
-  rw [div_lt_div_iff₀ ha hb] at hab
-  constructor
-  · rw [div_lt_div_iff₀ ha hh,hfc,hhc]
-    nlinarith [mul_pos hs (sub_pos.mpr hab)]
-  · rw [div_lt_div_iff₀ hh hb,hfc,hhc]
-    nlinarith [mul_pos hr (sub_pos.mpr hab)]
-
-end
-end KLInvariance.TitsSpace
-
-namespace KLInvariance.TitsSpace
-open Module
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-noncomputable section
-
-/-- Conjugation on the actual root lines, represented by positive roots. -/
-def positiveAction (w : W) (a : PositiveRoot M cs) : PositiveRoot M cs :=
-  positiveRoot M cs ⟨w * (rootReflection M cs a).val * w⁻¹,
-    (rootReflection M cs a).property.conj w⟩
-
-theorem positiveAction_represents (w : W) (a : PositiveRoot M cs) :
-    Represents M cs (w * (rootReflection M cs a).val * w⁻¹) (positiveAction w a).val :=
-  positiveRoot_represents M cs ⟨w * (rootReflection M cs a).val * w⁻¹,
-    (rootReflection M cs a).property.conj w⟩
-
-theorem positiveAction_eq_of_nonneg (w : W) (a : PositiveRoot M cs)
-    (hp : ∀ i, 0 ≤ action M cs w a.val i) :
-    (positiveAction w a).val = action M cs w a.val :=
-  (positiveAction_represents w a).unique_positive
-    ((rootReflection_represents M cs a).act w) (positiveAction w a).property.2 hp
-
-theorem positiveAction_eq_of_nonpos (w : W) (a : PositiveRoot M cs)
-    (hn : ∀ i, action M cs w a.val i ≤ 0) :
-    (positiveAction w a).val = -action M cs w a.val :=
-  (positiveAction_represents w a).unique_positive
-    ((rootReflection_represents M cs a).act w).neg (positiveAction w a).property.2
-    (fun i => neg_nonneg.mpr (hn i))
-
-@[simp] theorem rootReflection_positiveAction (w : W) (a : PositiveRoot M cs) :
-    (rootReflection M cs (positiveAction w a)).val = w*(rootReflection M cs a).val*w⁻¹ :=
-  (rootReflection_represents M cs _).reflection_unique (positiveAction_represents w a)
-
-@[simp] theorem positiveAction_one (a : PositiveRoot M cs) : positiveAction 1 a = a := by
-  apply (positiveRootEquiv M cs).symm.injective
-  apply Subtype.ext
-  change (rootReflection M cs (positiveAction 1 a)).val = (rootReflection M cs a).val
-  simp
-
-theorem positiveAction_mul (w z : W) (a : PositiveRoot M cs) :
-    positiveAction (w*z) a = positiveAction w (positiveAction z a) := by
-  apply (positiveRootEquiv M cs).symm.injective
-  apply Subtype.ext
-  change (rootReflection M cs (positiveAction (w*z) a)).val =
-    (rootReflection M cs (positiveAction w (positiveAction z a))).val
-  simp only [rootReflection_positiveAction]
-  group
-
-def positiveActionEquiv (w : W) : Equiv.Perm (PositiveRoot M cs) where
-  toFun := positiveAction w
-  invFun := positiveAction w⁻¹
-  left_inv a := by rw [← positiveAction_mul, inv_mul_cancel, positiveAction_one]
-  right_inv a := by rw [← positiveAction_mul, mul_inv_cancel, positiveAction_one]
-
-abbrev rootRef (a : PositiveRoot M cs) : W := (rootReflection M cs a).val
-
-theorem positiveAction_involutive (a : PositiveRoot M cs) :
-    Function.Involutive (positiveAction (M := M) (cs := cs) (rootRef a)) := by
-  intro b
-  rw [← positiveAction_mul, (rootReflection M cs a).property.mul_self, positiveAction_one]
-
-def rootInverted (w : W) (a : PositiveRoot M cs) : Prop :=
-  ∀ i, action M cs w⁻¹ a.val i ≤ 0
-
-theorem not_rootInverted_iff (w : W) (a : PositiveRoot M cs) :
-    ¬ rootInverted w a ↔ ∀ i, 0 ≤ action M cs w⁻¹ a.val i := by
-  constructor
-  · intro hn
-    exact (a.property.1.act w⁻¹).sign.resolve_right hn
-  · intro hp hn
-    exact (a.property.1.act w⁻¹).not_both hp hn
-
-theorem rootInverted_mul_of_nonneg (x : W) (a b : PositiveRoot M cs)
-    (hp : ∀ i, 0 ≤ action M cs (rootRef a) b.val i) :
-    rootInverted (rootRef a*x) b ↔ rootInverted x (positiveAction (rootRef a) b) := by
-  unfold rootInverted
-  rw [positiveAction_eq_of_nonneg _ _ hp, mul_inv_rev, (rootReflection M cs a).property.inv,
-    map_mul]
-  rfl
-
-theorem rootInverted_mul_of_nonpos (x : W) (a b : PositiveRoot M cs)
-    (hn : ∀ i, action M cs (rootRef a) b.val i ≤ 0) :
-    rootInverted (rootRef a*x) b ↔ ¬ rootInverted x (positiveAction (rootRef a) b) := by
-  rw [not_rootInverted_iff]
-  unfold rootInverted
-  rw [positiveAction_eq_of_nonpos _ _ hn, mul_inv_rev, (rootReflection M cs a).property.inv,
-    map_mul, map_neg]
-  simp only [LinearEquiv.mul_apply, Pi.neg_apply, neg_nonneg]
-
-theorem negative_pair_sum (a b : PositiveRoot M cs)
-    (hn : ∀ i, action M cs (rootRef a) b.val i ≤ 0) :
-    b.val + (positiveAction (rootRef a) b).val = form M a.val b.val • a.val := by
-  rw [positiveAction_eq_of_nonpos _ _ hn, (rootReflection_represents M cs a).action_formula]
-  module
-
-theorem negative_pair_coefficient_pos (a b : PositiveRoot M cs)
-    (hn : ∀ i, action M cs (rootRef a) b.val i ≤ 0) : 0 < form M a.val b.val := by
-  have h := congrArg (fun z => height z) (negative_pair_sum a b hn)
-  simp only [map_add,map_smul,smul_eq_mul] at h
-  have hp := b.height_pos
-  have hq := (positiveAction (rootRef a) b).height_pos
-  have ha := a.height_pos
-  nlinarith
-
-/-- The crucial direct positivity argument: over an ascent in alpha, the
-negative root-pair for its reflection cannot both already be inversions.
-This works in degenerate planes and does not assume a dihedral presentation. -/
-theorem negative_pair_not_both (x : W) (a b : PositiveRoot M cs)
-    (hx : ¬ rootInverted x a)
-    (hn : ∀ i, action M cs (rootRef a) b.val i ≤ 0) :
-    ¬ (rootInverted x b ∧ rootInverted x (positiveAction (rootRef a) b)) := by
-  rintro ⟨hb,hc⟩
-  have hp := (not_rootInverted_iff x a).mp hx
-  have hs := congrArg (action M cs x⁻¹) (negative_pair_sum a b hn)
-  simp only [map_add,map_smul] at hs
-  have hcoef := negative_pair_coefficient_pos a b hn
-  have hna : ∀ i, action M cs x⁻¹ a.val i ≤ 0 := by
-    intro i
-    have hi := congrFun hs i
-    change action M cs x⁻¹ b.val i +
-      action M cs x⁻¹ (positiveAction (rootRef a) b).val i =
-        form M a.val b.val * action M cs x⁻¹ a.val i at hi
-    have hb' := hb i
-    have hc' := hc i
-    nlinarith
-  exact (a.property.1.act x⁻¹).not_both hp hna
-
-end
-end KLInvariance.TitsSpace
-
-namespace KLInvariance.TitsSpace
-open Module
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
-theorem finite_rootInverted (w : W) : {a : PositiveRoot M cs | rootInverted w a}.Finite := by
-  exact Set.finite_coe_iff.mp (rootInversions_finite M cs w)
-
-def inversionVector (w : W) : PositiveRoot M cs →₀ ℤ := by
-  classical
-  refine Finsupp.ofSupportFinite (fun a => if rootInverted w a then 1 else 0) ?_
-  simpa only [Function.support, ne_eq, ite_eq_right_iff, one_ne_zero, imp_false,
-    not_not] using finite_rootInverted (M := M) (cs := cs) w
-
-@[simp] theorem inversionVector_apply (w : W) (a : PositiveRoot M cs) :
-    inversionVector w a = if rootInverted w a then 1 else 0 := by
-  classical
-  rfl
-
-theorem inversionVector_sum (w : W) : (inversionVector (M := M) (cs := cs) w).sum
-    (fun _ n => n) = (cs.length w : ℤ) := by
-  classical
-  let f := inversionVector (M := M) (cs := cs) w
-  have hh : f.support.card = cs.length w := by
-    have heq : (f.support : Set (PositiveRoot M cs)) = {a | rootInverted w a} := by
-      ext a
-      simp [f]
-    rw [← Set.ncard_coe_finset, heq, ← Nat.card_coe_set_eq]
-    exact card_rootInversions M cs w
-  rw [Finsupp.sum]
-  calc
-    ∑ a ∈ f.support, f a = ∑ _a ∈ f.support, (1:ℤ) := by
-      apply Finset.sum_congr rfl
-      intro a ha
-      have hm := Finsupp.mem_support_iff.mp ha
-      by_cases hi : rootInverted w a
-      · simp [f,hi]
-      · simp [f,hi] at hm
-    _ = _ := by simp [hh]
-
-def rootChange (x y : W) : PositiveRoot M cs →₀ ℤ := inversionVector y - inversionVector x
-
-def rootPlane (a b : PositiveRoot M cs) : Submodule ℝ (I → ℝ) :=
-  Submodule.span ℝ {a.val,b.val}
-
-/-- The finitely supported change in root inversions, with the distinguished
-root removed, grouped by actual root planes. -/
-def planeDefect (x y : W) (a : PositiveRoot M cs) : Submodule ℝ (I → ℝ) →₀ ℤ :=
-  Finsupp.mapDomain (rootPlane a) ((rootChange x y).erase a)
-
-theorem rootInverted_mul_self (x : W) (a : PositiveRoot M cs) :
-    rootInverted (rootRef a*x) a ↔ ¬ rootInverted x a := by
-  have h := (rootReflection M cs a).property.isLeftInversion_mul_right_iff (w := x)
-  simpa only [rootInverted, ← (rootReflection_represents M cs a).inversion_iff a.property.2] using h
-
-theorem rootChange_distinguished (x : W) (a : PositiveRoot M cs)
-    (hx : ¬ rootInverted x a) : rootChange x (rootRef a*x) a = 1 := by
-  classical
-  simp only [rootChange, Finsupp.sub_apply, inversionVector_apply,
-    (rootInverted_mul_self x a).mpr hx, hx, ite_true, ite_false, sub_zero]
-
-
-theorem planeDefect_sum (x : W) (a : PositiveRoot M cs) (hx : ¬ rootInverted x a) :
-    (planeDefect x (rootRef a*x) a).sum (fun _ n => n) =
-      (cs.length (rootRef a*x) : ℤ) - cs.length x - 1 := by
-  classical
-  rw [planeDefect, Finsupp.sum_mapDomain_index (fun _ => rfl) (fun _ _ _ => rfl),
-    Finsupp.erase_eq_sub_single, Finsupp.sum_sub_index (fun _ _ _ => rfl),
-    Finsupp.sum_single_index rfl, rootChange_distinguished x a hx,
-    rootChange, Finsupp.sum_sub_index (fun _ _ _ => rfl),
-    inversionVector_sum, inversionVector_sum]
-
-end
-end KLInvariance.TitsSpace
-
-namespace KLInvariance.TitsSpace
-open Module
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
-abbrev partner (a b : PositiveRoot M cs) := positiveAction (rootRef a) b
-
-theorem partner_self (a : PositiveRoot M cs) : partner a a = a := by
-  apply (positiveRootEquiv M cs).symm.injective
-  apply Subtype.ext
-  change (rootReflection M cs (positiveAction (rootRef a) a)).val = rootRef a
-  rw [rootReflection_positiveAction, (rootReflection M cs a).property.mul_self, one_mul,
-    (rootReflection M cs a).property.inv]
-
-theorem partner_involutive (a : PositiveRoot M cs) : Function.Involutive (partner a) :=
-  positiveAction_involutive a
-
-theorem partner_eq_self_of_eq (a b : PositiveRoot M cs) : partner a b = a ↔ b = a := by
-  constructor
-  · intro h
-    have hh := congrArg (partner a) h
-    rwa [partner_involutive a b,partner_self] at hh
-  · rintro rfl
-    exact partner_self _
-
-abbrev flipped (a b : PositiveRoot M cs) : Prop :=
-  ∀ i, action M cs (rootRef a) b.val i ≤ 0
-
-theorem flipped_partner_iff (a b : PositiveRoot M cs) : flipped a (partner a b) ↔ flipped a b := by
-  have hact : action M cs (rootRef a) * action M cs (rootRef a) = 1 := by
-    rw [← map_mul, (rootReflection M cs a).property.mul_self, map_one]
-  by_cases hb : flipped a b
-  · have heq := positiveAction_eq_of_nonpos (rootRef a) b hb
-    have hp : flipped a (partner a b) := by
-      intro i
-      change action M cs (rootRef a) (positiveAction (rootRef a) b).val i ≤ 0
-      rw [heq,map_neg, ← LinearEquiv.mul_apply,hact]
-      exact neg_nonpos.mpr (b.property.2 i)
-    exact iff_of_true hp hb
-  · have hpos := (b.property.1.act (rootRef a)).sign.resolve_right hb
-    have heq := positiveAction_eq_of_nonneg (rootRef a) b hpos
-    have hp : ¬ flipped a (partner a b) := by
-      intro hh
-      have hneg : ∀ i, b.val i ≤ 0 := by
-        intro i
-        have hi := hh i
-        change action M cs (rootRef a) (positiveAction (rootRef a) b).val i ≤ 0 at hi
-        rwa [heq, ← LinearEquiv.mul_apply, hact] at hi
-      exact b.property.1.not_both b.property.2 hneg
-    exact iff_of_false hp hb
-
-theorem partner_mem {P : Submodule ℝ (I → ℝ)} (a b : PositiveRoot M cs)
-    (ha : a.val ∈ P) (hb : b.val ∈ P) : (partner a b).val ∈ P := by
-  have hm : action M cs (rootRef a) b.val ∈ P := by
-    rw [(rootReflection_represents M cs a).action_formula]
-    exact P.sub_mem hb (P.smul_mem _ ha)
-  rcases (b.property.1.act (rootRef a)).sign with hp|hn
-  · exact (positiveAction_eq_of_nonneg (rootRef a) b hp).symm ▸ hm
-  · exact (positiveAction_eq_of_nonpos (rootRef a) b hn).symm ▸ P.neg_mem hm
-
-theorem rootPlane_partner (a b : PositiveRoot M cs) : rootPlane a (partner a b) = rootPlane a b := by
-  have hle (b : PositiveRoot M cs) : rootPlane a (partner a b) ≤ rootPlane a b := by
-    apply Submodule.span_le.mpr
-    intro z hz
-    rcases Set.mem_insert_iff.mp hz with rfl|hz
-    · exact Submodule.subset_span (Set.mem_insert _ _)
-    · rcases Set.mem_singleton_iff.mp hz with rfl
-      exact partner_mem a b (Submodule.subset_span (Set.mem_insert _ _))
-        (Submodule.subset_span (Set.mem_insert_of_mem _ rfl))
-  exact le_antisymm (hle b) (by simpa only [partner_involutive a b] using hle (partner a b))
-
-theorem negative_partner_fixed (a b : PositiveRoot M cs) (hn : flipped a b)
-    (heq : partner a b = b) : b = a := by
-  have hs := negative_pair_sum a b hn
-  change b.val + (partner a b).val = form M a.val b.val • a.val at hs
-  rw [heq] at hs
-  have hc := negative_pair_coefficient_pos a b hn
-  have hb : b.val = (form M a.val b.val / 2) • a.val := by
-    calc
-      b.val = (1/2:ℝ) • (b.val+b.val) := by module
-      _ = _ := by rw [hs]; module
-  have hnorm := b.property.1.norm
-  rw [hb] at hnorm
-  simp only [map_smul,LinearMap.smul_apply,smul_eq_mul,a.property.1.norm] at hnorm
-  have hc2 : form M a.val b.val / 2 = 1 := by nlinarith
-  apply Subtype.ext
-  simpa only [hc2,one_smul] using hb
-
-/-- Roots in a negative pair which are both positive before the ascent. -/
-def goodPairRoot (x : W) (a b : PositiveRoot M cs) : Prop :=
-  flipped a b ∧ ¬ rootInverted x b ∧ ¬ rootInverted x (partner a b)
-
-theorem goodPairRoot_partner (x : W) (a b : PositiveRoot M cs) :
-    goodPairRoot x a (partner a b) ↔ goodPairRoot x a b := by
-  simp only [goodPairRoot,flipped_partner_iff,partner_involutive a b]
-  tauto
-
-/-- The pairwise change is either zero or two, with no negative case. -/
-theorem rootChange_pair (x : W) (a b : PositiveRoot M cs) (hx : ¬ rootInverted x a) :
-    rootChange x (rootRef a*x) b + rootChange x (rootRef a*x) (partner a b) =
-      if goodPairRoot x a b then 2 else 0 := by
-  by_cases hn : flipped a b
-  · have hn' := (flipped_partner_iff a b).mpr hn
-    have hi := rootInverted_mul_of_nonpos x a b hn
-    have hi' := rootInverted_mul_of_nonpos x a (partner a b) hn'
-    have hpair : positiveAction (rootRef a) (partner a b) = b := partner_involutive a b
-    rw [hpair] at hi'
-    have hnnb := negative_pair_not_both x a b hx hn
-    simp only [rootChange,Finsupp.sub_apply,inversionVector_apply,hi,hi',goodPairRoot,]
-    by_cases hb : rootInverted x b <;> by_cases hc : rootInverted x (partner a b) <;>
-      simp_all
-  · have hp := (b.property.1.act (rootRef a)).sign.resolve_right hn
-    have hn' := mt (flipped_partner_iff a b).mp hn
-    have hp' := ((partner a b).property.1.act (rootRef a)).sign.resolve_right hn'
-    have hi := rootInverted_mul_of_nonneg x a b hp
-    have hi' := rootInverted_mul_of_nonneg x a (partner a b) hp'
-    have hpair : positiveAction (rootRef a) (partner a b) = b := partner_involutive a b
-    rw [hpair] at hi'
-    simp only [rootChange,Finsupp.sub_apply,inversionVector_apply,hi,hi',goodPairRoot,hn,false_and,ite_false]
-    ring
-
-end
-end KLInvariance.TitsSpace
-
-namespace KLInvariance.TitsSpace
-open Module
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
-def geometricFunctional : Module.Dual ℝ (I → ℝ) := (exists_geometric_functional M cs).choose
-
-def geometricSlope (a : PositiveRoot M cs) : ℝ := geometricFunctional (M := M) (cs := cs) a.val / height a.val
-
-theorem geometricSlope_injective : Function.Injective (geometricSlope (M := M) (cs := cs)) :=
-  (exists_geometric_functional M cs).choose_spec
-
-@[instance_reducible]
-def geometricOrder : LinearOrder (PositiveRoot M cs) := LinearOrder.lift' geometricSlope geometricSlope_injective
-
-theorem geometricSlope_between (a b c : PositiveRoot M cs) {r s : ℝ}
-    (hr : 0 < r) (hs : 0 < s) (hc : c.val = r • a.val+s • b.val)
-    (hab : geometricSlope a < geometricSlope b) :
-    geometricSlope a < geometricSlope c ∧ geometricSlope c < geometricSlope b :=
-  functionalRatio_between M cs (geometricFunctional (M := M) (cs := cs)) a b c hr hs hc hab
-
-def partnerEquiv (a : PositiveRoot M cs) : Equiv.Perm (PositiveRoot M cs) :=
-  Function.Involutive.toPerm (partner a) (partner_involutive a)
-
-@[simp] theorem partnerEquiv_apply (a b : PositiveRoot M cs) : partnerEquiv a b = partner a b := rfl
-@[simp] theorem partnerEquiv_symm_apply (a b : PositiveRoot M cs) : (partnerEquiv a).symm b = partner a b := rfl
-
-def extraRoot (x : W) (a b : PositiveRoot M cs) : Prop := b ≠ a ∧ goodPairRoot x a b
-
-theorem extraRoot_partner (x : W) (a b : PositiveRoot M cs) :
-    extraRoot x a (partner a b) ↔ extraRoot x a b := by
-  simp only [extraRoot,ne_eq,partner_eq_self_of_eq,goodPairRoot_partner]
-
-theorem extraRoot_finite (x : W) (a : PositiveRoot M cs) : {b | extraRoot x a b}.Finite := by
-  apply (finite_rootInverted (M := M) (cs := cs) (rootRef a*x)).subset
-  intro b hb
-  exact (rootInverted_mul_of_nonpos x a b hb.2.1).mpr hb.2.2.2
-
-def halfRoot (x : W) (a b : PositiveRoot M cs) : Prop :=
-  extraRoot x a b ∧ geometricSlope b < geometricSlope (partner a b)
-
-theorem halfRoot_finite (x : W) (a : PositiveRoot M cs) : {b | halfRoot x a b}.Finite :=
-  (extraRoot_finite x a).subset (fun _ h => h.1)
-
-def rootIndicator (p : PositiveRoot M cs → Prop) (hp : {a | p a}.Finite) : PositiveRoot M cs →₀ ℤ :=
-  Finsupp.ofSupportFinite (fun a => if p a then 1 else 0) (by
-    simpa only [Function.support,ne_eq,ite_eq_right_iff,one_ne_zero,imp_false,not_not] using hp)
-
-@[simp] theorem rootIndicator_apply (p : PositiveRoot M cs → Prop) (hp : {a | p a}.Finite)
-    (a : PositiveRoot M cs) : rootIndicator p hp a = if p a then 1 else 0 := rfl
-
-def extraVector (x : W) (a : PositiveRoot M cs) := rootIndicator (extraRoot x a) (extraRoot_finite x a)
-def halfVector (x : W) (a : PositiveRoot M cs) := rootIndicator (halfRoot x a) (halfRoot_finite x a)
-
-/-- A choice of one root in each actual two-element negative pair. -/
-theorem extraVector_eq_half (x : W) (a : PositiveRoot M cs) :
-    extraVector x a = halfVector x a + Finsupp.mapDomain (partnerEquiv a) (halfVector x a) := by
-  ext b
-  simp only [Finsupp.add_apply,Finsupp.mapDomain_equiv_apply,partnerEquiv_symm_apply,
-    extraVector,halfVector,rootIndicator_apply,halfRoot,extraRoot_partner,partner_involutive a b]
-  by_cases hb : extraRoot x a b
-  · have hne : geometricSlope b ≠ geometricSlope (partner a b) := by
-      intro h
-      exact hb.1 (negative_partner_fixed a b hb.2.1 (geometricSlope_injective h).symm)
-    rcases lt_or_gt_of_ne hne with h|h
-    · simp [hb,h,not_lt_of_gt h]
-    · simp [hb,h,not_lt_of_gt h]
-  · simp [hb]
-
-/-- All apparent negative contributions cancel under the actual reflection on
-root lines; the remaining pairs are precisely `extraVector`. -/
-theorem erasedChange_pair (x : W) (a : PositiveRoot M cs) (hx : ¬ rootInverted x a) :
-    (rootChange x (rootRef a*x)).erase a +
-      Finsupp.mapDomain (partnerEquiv a) ((rootChange x (rootRef a*x)).erase a) =
-        extraVector x a + extraVector x a := by
-  classical
-  ext b
-  simp only [Finsupp.add_apply,Finsupp.mapDomain_equiv_apply,partnerEquiv_symm_apply]
-  by_cases hb : b=a
-  · subst b
-    simp [partner_self,extraVector,extraRoot]
-  · have hpb : partner a b ≠ a := mt (partner_eq_self_of_eq a b).mp hb
-    rw [Finsupp.erase_ne hb,Finsupp.erase_ne hpb,rootChange_pair x a b hx]
-    simp only [extraVector,rootIndicator_apply,extraRoot,]
-    by_cases hg : goodPairRoot x a b <;> simp [hg,hb]
-
-theorem mapDomain_plane_partner (a : PositiveRoot M cs) (f : PositiveRoot M cs →₀ ℤ) :
-    Finsupp.mapDomain (rootPlane a) (Finsupp.mapDomain (partnerEquiv a) f) =
-      Finsupp.mapDomain (rootPlane a) f := by
-  rw [← Finsupp.mapDomain_comp]
-  apply Finsupp.mapDomain_congr
-  intro b _
-  exact rootPlane_partner a b
-
-def planeWeight (x : W) (a : PositiveRoot M cs) : Submodule ℝ (I → ℝ) →₀ ℤ :=
-  Finsupp.mapDomain (rootPlane a) (halfVector x a)
-
-
-theorem planeDefect_eq_double_weight (x : W) (a : PositiveRoot M cs) (hx : ¬ rootInverted x a) :
-    planeDefect x (rootRef a*x) a = planeWeight x a + planeWeight x a := by
-  have h := congrArg (Finsupp.mapDomain (rootPlane a)) (erasedChange_pair x a hx)
-  rw [Finsupp.mapDomain_add,mapDomain_plane_partner,Finsupp.mapDomain_add] at h
-  have heq : planeDefect x (rootRef a*x) a = Finsupp.mapDomain (rootPlane a) (extraVector x a) := by
-    ext P
-    have hh := congrArg (fun f : Submodule ℝ (I → ℝ) →₀ ℤ => f P) h
-    simp only [Finsupp.add_apply] at hh
-    change planeDefect x (rootRef a*x) a P = _
-    dsimp [planeDefect]
-    omega
-  rw [heq,extraVector_eq_half,Finsupp.mapDomain_add,mapDomain_plane_partner]
-  rfl
-
-theorem planeWeight_nonneg (x : W) (a : PositiveRoot M cs) (P : Submodule ℝ (I → ℝ)) :
-    0 ≤ planeWeight x a P := by
-  classical
-  rw [planeWeight,Finsupp.mapDomain,Finsupp.sum_apply]
-  apply Finset.sum_nonneg
-  intro b _
-  simp only [Finsupp.single_apply,halfVector,rootIndicator_apply]
-  split_ifs <;> norm_num
-
-/-- Exact ambient degree formula, with actual roots, finite support and
-nonnegative integral planar weights; no labeled interval isomorphism enters. -/
-theorem sum_planeWeight (x : W) (a : PositiveRoot M cs) (hx : ¬ rootInverted x a) :
-    2 * (planeWeight x a).sum (fun _ n => n) =
-      (cs.length (rootRef a*x) : ℤ) - cs.length x - 1 := by
-  have h := planeDefect_sum x a hx
-  rw [planeDefect_eq_double_weight x a hx,
-    Finsupp.sum_add_index' (fun _ => rfl) (fun _ _ _ => rfl)] at h
-  linarith
-
-end
-end KLInvariance.TitsSpace
-
-namespace KLInvariance.TitsSpace
-open Module
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
-theorem positiveRoots_independent (a b : PositiveRoot M cs) (hab : a ≠ b) :
-    LinearIndependent ℝ ![a.val,b.val] := by
-  apply (rootReflection_represents M cs a).independent_of_ne (rootReflection_represents M cs b)
-  intro h
-  apply hab
-  apply (positiveRootEquiv M cs).symm.injective
-  exact Subtype.ext h
-
-theorem rootPlane_finrank (a b : PositiveRoot M cs) (hab : a ≠ b) :
-    Module.finrank ℝ (rootPlane a b) = 2 := by
-  have h := finrank_span_eq_card (positiveRoots_independent a b hab)
-  have hrange : Set.range ![a.val,b.val] = {a.val,b.val} := by
-    ext z
-    simp only [Set.mem_range,Set.mem_insert_iff,Set.mem_singleton_iff]
-    constructor
-    · rintro ⟨i,rfl⟩
-      fin_cases i <;> simp
-    · rintro (rfl|rfl)
-      · exact ⟨0,rfl⟩
-      · exact ⟨1,rfl⟩
-  rw [hrange] at h
-  exact h
-
-theorem rootPlane_self_finrank (a : PositiveRoot M cs) :
-    Module.finrank ℝ (rootPlane a a) = 1 := by
-  have hset : ({a.val,a.val} : Set (I → ℝ)) = {a.val} := by ext z; simp
-  rw [rootPlane,hset]
-  exact finrank_span_singleton (K := ℝ) (a.property.1.ne_zero)
-
-theorem rootPlane_eq_iff (a b c : PositiveRoot M cs) (hab : a ≠ b) :
-    rootPlane a c = rootPlane a b ↔ c ≠ a ∧ c.val ∈ rootPlane a b := by
-  constructor
-  · intro h
-    constructor
-    · intro hc
-      subst c
-      have h' := congrArg (fun P : Submodule ℝ (I → ℝ) => Module.finrank ℝ P) h
-      rw [rootPlane_self_finrank,rootPlane_finrank a b hab] at h'
-      norm_num at h'
-    · rw [← h]
-      exact Submodule.subset_span (Set.mem_insert_of_mem _ (Set.mem_singleton _))
-  · rintro ⟨hc,hcP⟩
-    apply Submodule.eq_of_le_of_finrank_eq
-    · apply Submodule.span_le.mpr
-      intro z hz
-      rcases (Set.mem_insert_iff.mp hz) with rfl|hz
-      · exact Submodule.subset_span (Set.mem_insert _ _)
-      · have hzc := Set.mem_singleton_iff.mp hz
-        subst z
-        exact hcP
-    · rw [rootPlane_finrank a c (Ne.symm hc),rootPlane_finrank a b hab]
-
-end
-end KLInvariance.TitsSpace
-
-namespace KLInvariance.TitsSpace
-open Module
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
-/-- The actual finite count of ambient inversions belonging to a root plane.
-No Coxeter presentation of the plane subgroup is assumed here. -/
-def planeLength (P : Submodule ℝ (I → ℝ)) (w : W) : ℤ :=
-  (inversionVector (M := M) (cs := cs) w).sum (fun a n => if a.val ∈ P then n else 0)
-
-theorem rootPlane_grouped_erase (a b : PositiveRoot M cs) (hab : a ≠ b)
-    (f : PositiveRoot M cs →₀ ℤ) :
-    Finsupp.mapDomain (rootPlane a) (f.erase a) (rootPlane a b) =
-      f.sum (fun c n => if c.val ∈ rootPlane a b then n else 0) - f a := by
-  classical
-  have haP : a.val ∈ rootPlane a b := Submodule.subset_span (Set.mem_insert _ _)
-  have heq : Finsupp.mapDomain (rootPlane a) (f.erase a) (rootPlane a b) =
-      (f.erase a).sum (fun c n => if c.val ∈ rootPlane a b then n else 0) := by
-    rw [Finsupp.mapDomain,Finsupp.sum_apply]
-    apply Finset.sum_congr rfl
-    intro c hc
-    have hca : c ≠ a := by
-      intro h
-      subst c
-      simp only [Finsupp.mem_support_iff,Finsupp.erase_same,ne_eq,not_true_eq_false] at hc
-    change (Finsupp.single (rootPlane a c) ((f.erase a) c)) (rootPlane a b) =
-      (if c.val ∈ rootPlane a b then (f.erase a) c else 0)
-    rw [Finsupp.single_apply]
-    by_cases hm : c.val ∈ rootPlane a b
-    · rw [ite_eq_left ((rootPlane_eq_iff a b c hab).mpr ⟨hca,hm⟩),ite_eq_left hm]
-    · rw [ite_eq_right (fun heq => hm ((rootPlane_eq_iff a b c hab).mp heq).2),ite_eq_right hm]
-  rw [heq,Finsupp.erase_eq_sub_single,
-    Finsupp.sum_sub_index (fun c m n => by split_ifs <;> simp),
-    Finsupp.sum_single_index (by simp),ite_eq_left haP]
-
-theorem planeDefect_eq_relative_difference (x : W) (a b : PositiveRoot M cs)
-    (hab : a ≠ b) (hx : ¬ rootInverted x a) :
-    planeDefect x (rootRef a*x) a (rootPlane a b) =
-      planeLength (M := M) (cs := cs) (rootPlane a b) (rootRef a*x) - planeLength (M := M) (cs := cs) (rootPlane a b) x - 1 := by
-  rw [planeDefect,rootPlane_grouped_erase a b hab,rootChange_distinguished x a hx,
-    rootChange,Finsupp.sum_sub_index (fun c m n => by split_ifs <;> simp)]
-  rfl
-
-
-theorem planeWeight_relative_degree (x : W) (a b : PositiveRoot M cs)
-    (hab : a ≠ b) (hx : ¬ rootInverted x a) :
-    2 * planeWeight x a (rootPlane a b) =
-      planeLength (M := M) (cs := cs) (rootPlane a b) (rootRef a*x) - planeLength (M := M) (cs := cs) (rootPlane a b) x - 1 := by
-  have h := planeDefect_eq_relative_difference x a b hab hx
-  rw [planeDefect_eq_double_weight x a hx,Finsupp.add_apply] at h
-  linarith
-
-end
-end KLInvariance.TitsSpace
-
-namespace KLInvariance.EdgeAlgebra
-open Module
-universe u v
-variable {k : Type u} [Field k] {V : Type v} [AddCommGroup V] [Module k V]
-noncomputable section
-
-theorem symmetric_ι_injective : Function.Injective (SymmetricAlgebra.ι k V) := by
-  apply LinearMap.ker_eq_bot.mp
-  apply eq_bot_iff.mpr
-  intro v hv
-  change v = 0
-  by_contra hn
-  obtain ⟨f,hf⟩ := Module.Projective.exists_dual_ne_zero k hn
-  have h := congrArg (SymmetricAlgebra.lift f) (LinearMap.mem_ker.mp hv)
-  rw [SymmetricAlgebra.lift_ι_apply,map_zero] at h
-  exact hf h
-
-
-def rootIdeal (a : V) : Ideal (SymmetricAlgebra k V) :=
-  Ideal.span {SymmetricAlgebra.ι k V a}
-
-def rootLine (a : V) : Submodule k V := Submodule.span k {a}
-
-def symQuotientMap (a : V) : SymmetricAlgebra k V →ₐ[k] SymmetricAlgebra k (V ⧸ rootLine (k := k) a) :=
-  SymmetricAlgebra.lift ((SymmetricAlgebra.ι k (V ⧸ rootLine (k := k) a)).comp (rootLine (k := k) a).mkQ)
-
-@[simp] theorem symQuotientMap_ι (a v : V) :
-    symQuotientMap a (SymmetricAlgebra.ι k V v) =
-      SymmetricAlgebra.ι k (V ⧸ rootLine (k := k) a) ((rootLine (k := k) a).mkQ v) :=
-  SymmetricAlgebra.lift_ι_apply _ _
-
-theorem rootIdeal_le_ker (a : V) : rootIdeal (k := k) a ≤ RingHom.ker (symQuotientMap a) := by
-  apply Ideal.span_le.mpr
-  intro z hz
-  have hz' := Set.mem_singleton_iff.mp hz
-  subst z
-  change symQuotientMap a (SymmetricAlgebra.ι k V a) = 0
-  rw [symQuotientMap_ι]
-  have ha : (rootLine (k := k) a).mkQ a = 0 := by
-    exact (Submodule.Quotient.mk_eq_zero (rootLine (k := k) a)).mpr (Submodule.subset_span (Set.mem_singleton a))
-  rw [ha,map_zero]
-
-def quotientForward (a : V) : (SymmetricAlgebra k V ⧸ rootIdeal (k := k) a) →ₐ[k]
-    SymmetricAlgebra k (V ⧸ rootLine (k := k) a) :=
-  Ideal.Quotient.liftₐ _ (symQuotientMap a) (fun _ hz => rootIdeal_le_ker a hz)
-
-def quotientRootMap (a : V) : V ⧸ rootLine (k := k) a →ₗ[k]
-    (SymmetricAlgebra k V ⧸ rootIdeal (k := k) a) :=
-  (rootLine (k := k) a).liftQ ((Ideal.Quotient.mkₐ k (rootIdeal (k := k) a)).toLinearMap.comp (SymmetricAlgebra.ι k V)) (by
-    apply Submodule.span_le.mpr
-    intro v hv
-    have hv' := Set.mem_singleton_iff.mp hv
-    subst v
-    apply LinearMap.mem_ker.mpr
-    change Ideal.Quotient.mk (rootIdeal (k := k) a) (SymmetricAlgebra.ι k V a) = 0
-    exact Ideal.Quotient.eq_zero_iff_mem.mpr (Ideal.subset_span (Set.mem_singleton _)))
-
-def quotientBackward (a : V) : SymmetricAlgebra k (V ⧸ rootLine (k := k) a) →ₐ[k]
-    (SymmetricAlgebra k V ⧸ rootIdeal (k := k) a) := SymmetricAlgebra.lift (quotientRootMap a)
-
-@[simp] theorem quotientForward_mk (a : V) (z : SymmetricAlgebra k V) :
-    quotientForward a (Ideal.Quotient.mk (rootIdeal (k := k) a) z) = symQuotientMap a z := rfl
-
-@[simp] theorem quotientBackward_ι_mk (a v : V) :
-    quotientBackward a (SymmetricAlgebra.ι k (V ⧸ rootLine (k := k) a) ((rootLine (k := k) a).mkQ v)) =
-      Ideal.Quotient.mk (rootIdeal (k := k) a) (SymmetricAlgebra.ι k V v) := by
-  rw [quotientBackward,SymmetricAlgebra.lift_ι_apply]
-  rfl
-
-/-- Exact coefficient-ring identification used for the manuscript's C=A/(alpha).
-This is an isomorphism of the actual quotient, not a replacement invariant. -/
-def rootQuotientEquiv (a : V) : (SymmetricAlgebra k V ⧸ rootIdeal (k := k) a) ≃ₐ[k]
-    SymmetricAlgebra k (V ⧸ rootLine (k := k) a) :=
-  AlgEquiv.ofAlgHom (quotientForward a) (quotientBackward a)
-    (by
-      apply SymmetricAlgebra.algHom_ext
-      apply LinearMap.ext
-      intro v
-      obtain ⟨v,rfl⟩ := (rootLine (k := k) a).mkQ_surjective v
-      change quotientForward a (quotientBackward a (SymmetricAlgebra.ι k _ ((rootLine (k := k) a).mkQ v))) =
-        SymmetricAlgebra.ι k _ ((rootLine (k := k) a).mkQ v)
-      rw [quotientBackward_ι_mk,quotientForward_mk,symQuotientMap_ι])
-    (by
-      apply Ideal.Quotient.algHom_ext
-      apply SymmetricAlgebra.algHom_ext
-      apply LinearMap.ext
-      intro v
-      change quotientBackward a (quotientForward a (Ideal.Quotient.mk (rootIdeal (k := k) a)
-        (SymmetricAlgebra.ι k V v))) = Ideal.Quotient.mk (rootIdeal (k := k) a) (SymmetricAlgebra.ι k V v)
-      rw [quotientForward_mk,symQuotientMap_ι,quotientBackward_ι_mk])
-
-
-@[simp] theorem rootQuotientEquiv_mk_ι (a v : V) :
-    rootQuotientEquiv a (Ideal.Quotient.mk (rootIdeal (k := k) a) (SymmetricAlgebra.ι k V v)) =
-      SymmetricAlgebra.ι k (V ⧸ rootLine (k := k) a) ((rootLine (k := k) a).mkQ v) := by
-  exact symQuotientMap_ι a v
-
-theorem reducedRoot_ne_zero (a v : V) (hv : v ∉ rootLine (k := k) a) :
-    Ideal.Quotient.mk (rootIdeal (k := k) a) (SymmetricAlgebra.ι k V v) ≠ 0 := by
-  intro h
-  apply hv
-  apply (Submodule.Quotient.mk_eq_zero (rootLine (k := k) a)).mp
-  change (rootLine (k := k) a).mkQ v = 0
-  apply symmetric_ι_injective (k := k)
-  have hh := congrArg (rootQuotientEquiv a) h
-  simpa only [rootQuotientEquiv_mk_ι,map_zero] using hh
-
-variable {κ : Type*}
-theorem linear_homogeneous [Fintype κ] (b : Basis κ k V) (v : V) :
-    (SymmetricAlgebra.equivMvPolynomial b (SymmetricAlgebra.ι k V v)).IsHomogeneous 1 := by
-  classical
-  conv_lhs => rw [← b.sum_repr v]
-  simp only [map_sum,map_smul,SymmetricAlgebra.equivMvPolynomial_ι_apply]
-  apply MvPolynomial.IsHomogeneous.sum
-  intro i _
-  simpa only [Algebra.smul_def,MvPolynomial.algebraMap_eq] using (MvPolynomial.isHomogeneous_X k i).C_mul (b.repr v i)
-
-/-- Polynomial coordinates on the actual symmetric-algebra quotient. -/
-def rootQuotientPolynomial (a : V) (b : Basis κ k (V ⧸ rootLine (k := k) a)) :
-    (SymmetricAlgebra k V ⧸ rootIdeal (k := k) a) ≃ₐ[k] MvPolynomial κ k :=
-  (rootQuotientEquiv a).trans (SymmetricAlgebra.equivMvPolynomial b)
-
-theorem reducedRoot_homogeneous [Fintype κ] (a v : V)
-    (b : Basis κ k (V ⧸ rootLine (k := k) a)) :
-    (rootQuotientPolynomial a b
-      (Ideal.Quotient.mk (rootIdeal (k := k) a) (SymmetricAlgebra.ι k V v))).IsHomogeneous 1 := by
-  change (SymmetricAlgebra.equivMvPolynomial b
-    (rootQuotientEquiv a (Ideal.Quotient.mk (rootIdeal (k := k) a) (SymmetricAlgebra.ι k V v)))).IsHomogeneous 1
-  rw [rootQuotientEquiv_mk_ι]
-  exact linear_homogeneous b _
-
-end
-end KLInvariance.EdgeAlgebra
-
-namespace KLInvariance.TitsSpace
-open Module EdgeAlgebra
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
-theorem planeWeight_support_root (x : W) (a : PositiveRoot M cs)
-    (P : Submodule ℝ (I → ℝ)) (hP : P ∈ (planeWeight x a).support) :
-    ∃ b : PositiveRoot M cs, b ≠ a ∧ rootPlane a b = P := by
-  obtain ⟨b,hb,hbP⟩ := Finset.mem_image.mp (Finsupp.mapDomain_support hP)
-  refine ⟨b,?_,hbP⟩
-  have hb' := Finsupp.mem_support_iff.mp hb
-  change halfVector x a b ≠ 0 at hb'
-  simp only [halfVector,rootIndicator_apply] at hb'
-  have hh : halfRoot x a b := by
-    by_contra hn
-    exact hb' (ite_eq_right hn)
-  exact hh.1.1
-
-def planeRoot (x : W) (a : PositiveRoot M cs)
-    (P : (planeWeight x a).support) : PositiveRoot M cs :=
-  (planeWeight_support_root x a P.val P.property).choose
-
-theorem planeRoot_ne (x : W) (a : PositiveRoot M cs)
-    (P : (planeWeight x a).support) : planeRoot x a P ≠ a :=
-  (planeWeight_support_root x a P.val P.property).choose_spec.1
-
-theorem planeRoot_plane (x : W) (a : PositiveRoot M cs)
-    (P : (planeWeight x a).support) : rootPlane a (planeRoot x a P) = P.val :=
-  (planeWeight_support_root x a P.val P.property).choose_spec.2
-
-theorem embed_not_rootLine (a b : PositiveRoot M cs) (hab : b ≠ a) :
-    embed M b.val ∉ rootLine (k := ℝ) (embed M a.val) := by
-  intro h
-  obtain ⟨c,hc⟩ := Submodule.mem_span_singleton.mp h
-  have hli := (linearIndependent_fin2.mp (positiveRoots_independent b a hab)).2 c
-  change c • a.val ≠ b.val at hli
-  apply hli
-  have hh := congrArg Prod.fst hc
-  simpa only [embed,Prod.smul_fst] using hh
-
-abbrev EdgeCoefficient (a : PositiveRoot M cs) :=
-  SymmetricAlgebra ℝ (Extended M) ⧸ rootIdeal (k := ℝ) (embed M a.val)
-
-def reducedPlaneRoot (x : W) (a : PositiveRoot M cs)
-    (P : (planeWeight x a).support) : EdgeCoefficient a :=
-  Ideal.Quotient.mk (rootIdeal (k := ℝ) (embed M a.val))
-    (SymmetricAlgebra.ι ℝ (Extended M) (embed M (planeRoot x a P).val))
-
-theorem reducedPlaneRoot_ne_zero (x : W) (a : PositiveRoot M cs)
-    (P : (planeWeight x a).support) : reducedPlaneRoot x a P ≠ 0 :=
-  reducedRoot_ne_zero _ _ (embed_not_rootLine a _ (planeRoot_ne x a P))
-
-/-- The manuscript's actual product of reduced root forms, with its proved
-nonnegative integral plane exponents. No abstract invariant is substituted. -/
-def edgeFactor (x : W) (a : PositiveRoot M cs) : EdgeCoefficient a :=
-  ∏ P : (planeWeight x a).support, reducedPlaneRoot x a P ^ (planeWeight x a P.val).toNat
-
-def edgeFactorDegree (x : W) (a : PositiveRoot M cs) : ℕ :=
-  ∑ P : (planeWeight x a).support, (planeWeight x a P.val).toNat
-
-def edgePolynomial (a : PositiveRoot M cs) : EdgeCoefficient a ≃ₐ[ℝ]
-    MvPolynomial (Fin (Module.finrank ℝ (Extended M ⧸ rootLine (k := ℝ) (embed M a.val)))) ℝ :=
-  rootQuotientPolynomial (embed M a.val) (Module.finBasis ℝ _)
-
-theorem edgeFactor_nonzero (x : W) (a : PositiveRoot M cs) : edgeFactor x a ≠ 0 := by
-  intro h
-  have hp := congrArg (edgePolynomial a) h
-  simp only [edgeFactor,map_prod,map_pow,map_zero] at hp
-  apply (Finset.prod_ne_zero_iff.mpr ?_) hp
-  intro P _
-  apply pow_ne_zero
-  intro h0
-  apply reducedPlaneRoot_ne_zero x a P
-  apply (edgePolynomial a).injective
-  simpa only [map_zero] using h0
-
-theorem edgeFactor_homogeneous (x : W) (a : PositiveRoot M cs) :
-    (edgePolynomial a (edgeFactor x a)).IsHomogeneous (edgeFactorDegree x a) := by
-  simp only [edgeFactor,map_prod,map_pow,edgeFactorDegree]
-  apply MvPolynomial.IsHomogeneous.prod
-  intro P _
-  have h : (edgePolynomial a (reducedPlaneRoot x a P)).IsHomogeneous 1 :=
-    reducedRoot_homogeneous _ _ _
-  simpa only [one_mul] using h.pow (planeWeight x a P.val).toNat
-
-theorem edgeFactor_totalDegree (x : W) (a : PositiveRoot M cs) :
-    (edgePolynomial a (edgeFactor x a)).totalDegree = edgeFactorDegree x a :=
-  (edgeFactor_homogeneous x a).totalDegree
-    (by
-      intro h0
-      apply edgeFactor_nonzero x a
-      apply (edgePolynomial a).injective
-      simpa only [map_zero] using h0)
-
-
-theorem edgeFactorDegree_cast (x : W) (a : PositiveRoot M cs) :
-    (edgeFactorDegree x a : ℤ) = (planeWeight x a).sum (fun _ n => n) := by
-  classical
-  simp only [edgeFactorDegree,Nat.cast_sum]
-  simp_rw [Int.toNat_of_nonneg (planeWeight_nonneg x a _)]
-  exact Finset.sum_coe_sort (planeWeight x a).support (fun P => planeWeight x a P)
-
-
-theorem edgeFactor_degree_formula (x : W) (a : PositiveRoot M cs)
-    (hx : ¬ rootInverted x a) :
-    2 * ((edgePolynomial a (edgeFactor x a)).totalDegree : ℤ) =
-      (cs.length (rootRef a*x) : ℤ) - cs.length x - 1 := by
-  rw [edgeFactor_totalDegree,edgeFactorDegree_cast]
-  exact sum_planeWeight x a hx
-
-end
-end KLInvariance.TitsSpace
-
-namespace KLInvariance.EdgeAlgebra
-open Module
-universe u v
-variable {k : Type u} [Field k] {V : Type v} [AddCommGroup V] [Module k V]
-noncomputable section
-
-theorem rootIdeal_eq_ker (a : V) : rootIdeal (k := k) a = RingHom.ker (symQuotientMap (k := k) a) := by
-  apply le_antisymm (rootIdeal_le_ker a)
-  intro z hz
-  apply Ideal.Quotient.eq_zero_iff_mem.mp
-  apply (rootQuotientEquiv (k := k) a).injective
-  change quotientForward a (Ideal.Quotient.mk (rootIdeal (k := k) a) z) = _
-  rw [quotientForward_mk,map_zero]
-  exact hz
-
-theorem rootIdeal_isPrime (a : V) : (rootIdeal (k := k) a).IsPrime := by
-  rw [rootIdeal_eq_ker]
-  exact RingHom.ker_isPrime _
-
-theorem symmetric_ι_prime (a : V) (ha : a ≠ 0) : Prime (SymmetricAlgebra.ι k V a) := by
-  have hn : SymmetricAlgebra.ι k V a ≠ 0 := by
-    intro h
-    apply ha
-    apply symmetric_ι_injective (k := k)
-    simpa only [map_zero] using h
-  exact (Ideal.span_singleton_prime hn).mp (rootIdeal_isPrime a)
-
-theorem reducedRoot_prime (a v : V) (hv : v ∉ rootLine (k := k) a) :
-    Prime (Ideal.Quotient.mk (rootIdeal (k := k) a) (SymmetricAlgebra.ι k V v)) := by
-  apply (MulEquiv.prime_iff (rootQuotientEquiv (k := k) a)).mp
-  rw [rootQuotientEquiv_mk_ι]
-  apply symmetric_ι_prime
-  intro h
-  exact hv ((Submodule.Quotient.mk_eq_zero (rootLine (k := k) a)).mp h)
-
-end
-end KLInvariance.EdgeAlgebra
-
-namespace KLInvariance.EdgeAlgebra
-open Module
-universe u v
-variable {k : Type u} [Field k] {V : Type v} [AddCommGroup V] [Module k V]
-noncomputable section
-
-theorem symmetric_ι_dvd_iff (a v : V) :
-    SymmetricAlgebra.ι k V a ∣ SymmetricAlgebra.ι k V v ↔ v ∈ rootLine (k := k) a := by
-  rw [← Ideal.mem_span_singleton]
-  change SymmetricAlgebra.ι k V v ∈ rootIdeal (k := k) a ↔ _
-  rw [rootIdeal_eq_ker]
-  change symQuotientMap a (SymmetricAlgebra.ι k V v) = 0 ↔ _
-  rw [symQuotientMap_ι]
-  constructor
-  · intro h
-    apply (Submodule.Quotient.mk_eq_zero (rootLine (k := k) a)).mp
-    change (rootLine (k := k) a).mkQ v = 0
-    apply symmetric_ι_injective (k := k)
-    simpa only [map_zero] using h
-  · intro h
-    have h0 : (rootLine (k := k) a).mkQ v = 0 :=
-      (Submodule.Quotient.mk_eq_zero (rootLine (k := k) a)).mpr h
-    rw [h0,map_zero]
-
-theorem mkQ_mem_span_iff (a b v : V) :
-    (rootLine (k := k) a).mkQ v ∈
-      Submodule.span k { (rootLine (k := k) a).mkQ b } ↔
-      v ∈ Submodule.span k {a,b} := by
-  have ha : (rootLine (k := k) a).mkQ a = 0 :=
-    (Submodule.Quotient.mk_eq_zero (rootLine (k := k) a)).mpr
-      (Submodule.subset_span (Set.mem_singleton _))
-  constructor
-  · intro h
-    obtain ⟨c,hc⟩ := Submodule.mem_span_singleton.mp h
-    have h0 : (rootLine (k := k) a).mkQ (v-c • b) = 0 := by
-      rw [map_sub,map_smul,← hc,sub_self]
-    have hm := (Submodule.Quotient.mk_eq_zero (rootLine (k := k) a)).mp h0
-    obtain ⟨d,hd⟩ := Submodule.mem_span_singleton.mp hm
-    apply Submodule.mem_span_pair.mpr
-    exact ⟨d,c,by rw [hd]; module⟩
-  · intro h
-    obtain ⟨d,c,hv⟩ := Submodule.mem_span_pair.mp h
-    apply Submodule.mem_span_singleton.mpr
-    refine ⟨c,?_⟩
-    rw [← hv,map_add,map_smul,map_smul,ha,smul_zero,zero_add]
-
-theorem reducedRoot_dvd_iff (a b v : V) :
-    Ideal.Quotient.mk (rootIdeal (k := k) a) (SymmetricAlgebra.ι k V b) ∣
-      Ideal.Quotient.mk (rootIdeal (k := k) a) (SymmetricAlgebra.ι k V v) ↔
-        v ∈ Submodule.span k {a,b} := by
-  let e := rootQuotientEquiv (k := k) a
-  have hd (z w : SymmetricAlgebra k V ⧸ rootIdeal (k := k) a) :
-      z ∣ w ↔ e z ∣ e w := by
-    constructor
-    · exact map_dvd e
-    · intro h
-      simpa only [AlgEquiv.symm_apply_apply] using map_dvd e.symm h
-  rw [hd]
-  change rootQuotientEquiv a _ ∣ rootQuotientEquiv a _ ↔ _
-  rw [rootQuotientEquiv_mk_ι,rootQuotientEquiv_mk_ι,symmetric_ι_dvd_iff]
-  exact mkQ_mem_span_iff a b v
-
-end
-end KLInvariance.EdgeAlgebra
-
-namespace KLInvariance.TitsSpace
-open Module EdgeAlgebra
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
-theorem embed_mem_span_pair (a b c : I → ℝ) :
-    embed M c ∈ Submodule.span ℝ {embed M a,embed M b} ↔
-      c ∈ Submodule.span ℝ {a,b} := by
-  constructor
-  · intro h
-    obtain ⟨r,s,h⟩ := Submodule.mem_span_pair.mp h
-    refine Submodule.mem_span_pair.mpr ⟨r,s,?_⟩
-    exact congrArg Prod.fst h
-  · intro h
-    obtain ⟨r,s,h⟩ := Submodule.mem_span_pair.mp h
-    refine Submodule.mem_span_pair.mpr ⟨r,s,?_⟩
-    apply Prod.ext
-    · exact h
-    · simp only [embed,Prod.snd_add,Prod.smul_snd,smul_zero,add_zero]
-
-theorem reducedPlaneRoot_prime (x : W) (a : PositiveRoot M cs)
-    (P : (planeWeight x a).support) : Prime (reducedPlaneRoot x a P) :=
-  reducedRoot_prime _ _ (embed_not_rootLine a _ (planeRoot_ne x a P))
-
-
-theorem reducedPlaneRoot_dvd_iff (x : W) (a : PositiveRoot M cs)
-    (P Q : (planeWeight x a).support) :
-    reducedPlaneRoot x a P ∣ reducedPlaneRoot x a Q ↔ P = Q := by
-  rw [reducedPlaneRoot,reducedPlaneRoot,reducedRoot_dvd_iff,embed_mem_span_pair]
-  change (planeRoot x a Q).val ∈ rootPlane a (planeRoot x a P) ↔ P = Q
-  have h := rootPlane_eq_iff a (planeRoot x a P) (planeRoot x a Q)
-    (planeRoot_ne x a P).symm
-  rw [planeRoot_plane,planeRoot_plane] at h
-  rw [planeRoot_plane]
-  constructor
-  · intro hm
-    apply Subtype.ext
-    exact (h.mpr ⟨planeRoot_ne x a Q,hm⟩).symm
-  · intro heq
-    exact (h.mp (congrArg Subtype.val heq).symm).2
-
-end
-end KLInvariance.TitsSpace
-
-
-namespace KLInvariance.SchedulePaths
-universe u v w
-variable {V : Type u} {E : Type v} {R : Type w}
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-variable (source target : E → V)
-
-def IsPath (x y : V) : List E → Prop
-  | [] => x = y
-  | e :: p => source e = x ∧ IsPath (target e) y p
-
-variable [Semiring R]
-def pathTerm (T : R) (v : V → R) : List E → V → R
-  | [], x => v x
-  | e :: p, x => if source e = x then T * pathTerm T v p (target e) else 0
-
-/-- One elementary matrix `1+T E_xy`, acting without choosing an enumeration
-of the finite vertex set. -/
-def step (T : R) (e : E) (v : V → R) : V → R :=
-  fun x => v x + if source e = x then T * v (target e) else 0
-
-/-- Edges are listed in decreasing schedule order, so the chronologically
-last transfer is the outermost one. -/
-def dispatch (T : R) (v : V → R) : List E → V → R
-  | [] => v
-  | e :: p => step source target T e (dispatch T v p)
-
-/-- The full matrix expansion, each sublist contributing exactly one path
-term. Noncomposable sublists contribute zero by the definition of pathTerm. -/
-theorem dispatch_eq_path_sum (T : R) (v : V → R) (p : List E) (x : V) :
-    dispatch source target T v p x =
-      ((p.sublists').map (fun q => pathTerm source target T v q x)).sum := by
-  classical
-  induction p generalizing x with
-  | nil => simp [dispatch, List.sublists', pathTerm]
-  | cons e p ih =>
-    rw [dispatch,step,List.sublists'_cons,List.map_append,List.sum_append,List.map_map]
-    rw [ih x]
-    congr 1
-    by_cases he : source e = x
-    · simp only [Function.comp_def,pathTerm,he,ite_true]
-      rw [List.sum_map_mul_left,ih (target e)]
-    · simp [Function.comp_def,he,pathTerm]
-
-/-- With a delta initial value, a path contributes its length weight, not an
-invariant defined by the desired conclusion. -/
-theorem pathTerm_delta (T : R) (x y : V) (p : List E) :
-    pathTerm source target T (fun z => if z = y then 1 else 0) p x =
-      if IsPath source target x y p then T ^ p.length else 0 := by
-  classical
-  induction p generalizing x with
-  | nil => simp [pathTerm,IsPath]
-  | cons e p ih =>
-    simp only [pathTerm,IsPath,List.length_cons]
-    by_cases he : source e = x
-    · rw [ite_eq_left he,ih]
-      by_cases hp : IsPath source target (target e) y p
-      · simp only [hp,he,and_self,ite_true,pow_succ']
-      · simp [hp,he]
-    · simp [he]
-
-/-- The entry of the elementary-matrix product is exactly the sum of weights
-of directed schedule-subpaths. -/
-theorem entry_eq_weighted_paths (T : R) (p : List E) (x y : V) :
-    dispatch source target T (fun z => if z = y then 1 else 0) p x =
-      ((p.sublists').map (fun q =>
-        if IsPath source target x y q then T ^ q.length else 0)).sum := by
-  classical
-  rw [dispatch_eq_path_sum]
-  congr 1
-  exact List.map_congr_left (fun q _ => pathTerm_delta source target T x y q)
-
-
-section LabelOrder
-variable {Λ : Type*} [LinearOrder Λ] (label : E → Λ)
-
-/-- Ties in a legitimate reflection schedule cannot occur on a composable
-pair. Hence every nonzero term in the schedule expansion has strictly
-rather than merely weakly decreasing labels. -/
-theorem decreasing_of_schedule_path
-    (hties : ∀ e f, label e = label f → target e ≠ source f)
-    {p : List E} {x y : V}
-    (hp : IsPath source target x y p)
-    (hs : p.Pairwise (fun e f => label f ≤ label e)) :
-    p.Pairwise (fun e f => label f < label e) := by
-  induction p generalizing x with
-  | nil => exact List.Pairwise.nil
-  | cons e p ih =>
-    cases p with
-    | nil => simp
-    | cons f p =>
-      have ht : IsPath source target (target e) y (f :: p) := hp.2
-      have hef : label f < label e := by
-        apply lt_of_le_of_ne ((List.pairwise_cons.mp hs).1 f (List.mem_cons_self))
-        intro heq
-        exact hties e f heq.symm ht.1.symm
-      apply List.pairwise_cons.mpr
-      refine ⟨?_,ih ht hs.tail⟩
-      intro g hg
-      rcases List.mem_cons.mp hg with rfl | hg
-      · exact hef
-      · exact lt_of_le_of_lt ((List.pairwise_cons.mp hs.tail).1 g hg) hef
-
-/-- Every strictly decreasing edge list contained in a weakly decreasing
-schedule occurs as a sublist. This also handles arbitrary ordering of ties:
-a strict list can use at most one edge of each label. -/
-theorem sublist_of_decreasing_subset {p schedule : List E}
-    (hp : p.Pairwise (fun e f => label f < label e))
-    (hs : schedule.Pairwise (fun e f => label f ≤ label e))
-    (hmem : p ⊆ schedule) : List.Sublist p schedule := by
-  induction schedule generalizing p with
-  | nil =>
-    have : p = [] := List.eq_nil_iff_forall_not_mem.mpr
-      (fun a ha => List.not_mem_nil (hmem ha))
-    subst p
-    exact List.Sublist.refl []
-  | cons e schedule ih =>
-    cases p with
-    | nil => exact List.nil_sublist _
-    | cons f p =>
-      by_cases hef : e = f
-      · subst f
-        apply List.Sublist.cons_cons
-        apply ih hp.tail hs.tail
-        intro g hg
-        rcases List.mem_cons.mp (hmem (List.mem_cons_of_mem e hg)) with heq | hin
-        · have hlt := (List.pairwise_cons.mp hp).1 g hg
-          rw [heq] at hlt
-          exact (lt_irrefl _ hlt).elim
-        · exact hin
-      · apply List.Sublist.cons
-        apply ih hp hs.tail
-        intro g hg
-        have hfg : f ∈ schedule := (List.mem_cons.mp (hmem (List.mem_cons_self))).resolve_left
-          (Ne.symm hef)
-        have hfe : label f ≤ label e := (List.pairwise_cons.mp hs).1 f hfg
-        rcases List.mem_cons.mp (hmem hg) with heq | hin
-        · subst g
-          rcases List.mem_cons.mp hg with h | h
-          · exact (hef h).elim
-          · exact (not_lt_of_ge hfe ((List.pairwise_cons.mp hp).1 e h)).elim
-        · exact hin
-
-end LabelOrder
-
-end
-end KLInvariance.SchedulePaths
-
-namespace KLInvariance.ReflectionPlane
-variable {V : Type*} [AddCommGroup V] [Module ℝ V]
-
-/-- The product of three normalized reflections in a root plane is again
-an actual orthogonal reflection. The plane itself need not be nondegenerate. -/
-theorem reflection_triple_in_plane
-    (B : LinearMap.BilinForm ℝ V) (hB : B.IsSymm)
-    (a b c : V) (ha : B a a = 2) (hb : B b b = 2) (hc : B c c = 2)
-    (hplane : c ∈ Submodule.span ℝ {a,b}) :
-    ∃ d : V, ∃ hd : B d d = 2,
-      Module.reflection ha * Module.reflection hb * Module.reflection hc =
-        Module.reflection hd := by
-  obtain ⟨x,y,hxy⟩ := Submodule.mem_span_pair.mp hplane
-  subst c
-  let p := B a b
-  let r := p*x+y
-  let d := x • b - r • a
-  have hba : B b a = p := hB.eq b a
-  have hnorm : x^2+p*x*y+y^2=1 := by
-    simp only [map_add,map_smul,LinearMap.add_apply,LinearMap.smul_apply,
-      smul_eq_mul,ha,hb,hba] at hc
-    dsimp [p]
-    nlinarith [hc]
-  have hd : B d d = 2 := by
-    dsimp [d,r]
-    simp only [map_sub,map_smul,LinearMap.sub_apply,LinearMap.smul_apply,
-      smul_eq_mul,ha,hb,hba]
-    change x*(x*2-(p*x+y)*p)-(p*x+y)*(x*p-(p*x+y)*2)=2
-    nlinarith [hnorm]
-  refine ⟨d,hd,?_⟩
-  apply LinearEquiv.ext
-  intro v
-  let A := B a v
-  let C := B b v
-  have h₁ : -A+p*C+(x-p*r)*(x*A+y*C)= r*(x*C-r*A) := by
-    dsimp [r]
-    linear_combination (A-p*C)*hnorm
-  have h₂ : -C+r*(x*A+y*C)= -x*(x*C-r*A) := by
-    dsimp [r]
-    linear_combination C*hnorm
-  calc
-    (Module.reflection ha * Module.reflection hb * Module.reflection hc) v =
-        v + (-A+p*C+(x-p*r)*(x*A+y*C)) • a +
-          (-C+r*(x*A+y*C)) • b := by
-      simp only [LinearEquiv.mul_apply,Module.reflection_apply,map_sub,map_smul,
-        map_add,LinearMap.add_apply,LinearMap.smul_apply,smul_eq_mul,ha,hb,hba]
-      dsimp [A,C,r,p]
-      module
-    _ = v + (r*(x*C-r*A)) • a + (-x*(x*C-r*A)) • b := by rw [h₁,h₂]
-    _ = Module.reflection hd v := by
-      simp only [Module.reflection_apply]
-      dsimp [d]
-      simp only [map_sub,map_smul,LinearMap.sub_apply,LinearMap.smul_apply,smul_eq_mul]
-      dsimp [A,C]
-      module
-
-end KLInvariance.ReflectionPlane
-
-namespace KLInvariance.TitsSpace
-open Module
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-noncomputable section
-
-/-- A product of three genuine Coxeter reflections in one root plane is a
-Coxeter reflection, using the proved reflection faithfulness of the actual
-Tits extension. This is the sufficiency direction of graph diamond closure. -/
-theorem Represents.triple_isReflection {t s r : W} {a b c : I → ℝ}
-    (ha : Represents M cs t a) (hb : Represents M cs s b) (hc : Represents M cs r c)
-    (hplane : c ∈ Submodule.span ℝ {a,b}) : cs.IsReflection (t*s*r) := by
-  have hplane' : embed M c ∈ Submodule.span ℝ {embed M a,embed M b} := by
-    obtain ⟨x,y,hxy⟩ := Submodule.mem_span_pair.mp hplane
-    apply Submodule.mem_span_pair.mpr
-    refine ⟨x,y,?_⟩
-    ext
-    · exact congrFun hxy _
-    · simp [embed]
-  obtain ⟨d,hd,he⟩ := ReflectionPlane.reflection_triple_in_plane (extendedForm M)
-    (extendedForm_spec M).1 (embed M a) (embed M b) (embed M c)
-    (by rw [(extendedForm_spec M).2.2]; exact ha.isRoot.norm)
-    (by rw [(extendedForm_spec M).2.2]; exact hb.isRoot.norm)
-    (by rw [(extendedForm_spec M).2.2]; exact hc.isRoot.norm) hplane'
-  apply isReflection_of_moving_rank_one M cs
-  have heq : extendedAction M cs (t*s*r) = Module.reflection hd := by
-    rw [map_mul,map_mul,ha.extendedAction_eq,hb.extendedAction_eq,hc.extendedAction_eq]
-    exact he
-  rw [heq]
-  exact Realization.finrank_reflection_sub_id _ _ hd
-
-end
-end KLInvariance.TitsSpace
-
-/-! NEW GRAPH CLOSURE -/
-namespace KLInvariance
-universe u v
-variable {I : Type u} [klPreservedInstance7 : Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-noncomputable section
-
-include klPreservedInstance7 in
- theorem BruhatStep.label_reflection {x y : W} (h : BruhatStep cs x y) :
-    cs.IsReflection (y*x⁻¹) := by
-  have _ := klPreservedInstance7
-  obtain ⟨_,t,ht,rfl⟩ := h
-  simpa [mul_assoc] using ht
-
- theorem BruhatStep.composable_labels_ne {x y z : W}
-    (h : BruhatStep cs x y) (h' : BruhatStep cs y z) :
-    y*x⁻¹ ≠ z*y⁻¹ := by
-  intro he
-  have hh : z*x⁻¹=1 := by
-    calc
-      z*x⁻¹ = (z*y⁻¹)*(y*x⁻¹) := by group
-      _ = (y*x⁻¹)*(y*x⁻¹) := by rw [← he]
-      _ = 1 := h.label_reflection.mul_self
-  have hzx : z = x := by simpa using congrArg (fun w => w*x) hh
-  have hlen := lt_trans h.1 h'.1
-  simp [hzx] at hlen
-
-namespace TitsSpace
-open Module
-
- def graphRoot {x y : W} (h : BruhatStep cs x y) : PositiveRoot M cs :=
-   positiveRoot M cs ⟨y*x⁻¹,h.label_reflection⟩
-
- theorem graphRoot_represents {x y : W} (h : BruhatStep cs x y) :
-    Represents M cs (y*x⁻¹) (graphRoot h).val :=
-  positiveRoot_represents M cs ⟨y*x⁻¹,h.label_reflection⟩
-
- theorem graphRoot_ne_of_same_source {x a b : W}
-    (ha : BruhatStep cs x a) (hb : BruhatStep cs x b) (hab : a ≠ b) :
-    graphRoot ha ≠ graphRoot hb := by
-  intro he
-  have hl := (graphRoot_represents ha).reflection_unique
-    (he.symm ▸ graphRoot_represents hb)
-  exact hab ((mul_right_cancel_iff).mp hl)
-
- theorem graphRoot_ne_of_path {x a z : W}
-    (ha : BruhatStep cs x a) (hb : BruhatStep cs a z) :
-    graphRoot ha ≠ graphRoot hb := by
-  intro he
-  exact ha.composable_labels_ne hb ((graphRoot_represents ha).reflection_unique
-    (he.symm ▸ graphRoot_represents hb))
-
- theorem graphRoot_square_plane {x a b z : W}
-    (hxa : BruhatStep cs x a) (hxb : BruhatStep cs x b)
-    (haz : BruhatStep cs a z) (hbz : BruhatStep cs b z) (hab : a ≠ b) :
-    rootPlane (graphRoot hxa) (graphRoot haz) =
-      rootPlane (graphRoot hxa) (graphRoot hxb) := by
-  have hp := (graphRoot_represents haz).plane_eq_of_product_eq
-    (graphRoot_represents hxa) (graphRoot_represents hbz) (graphRoot_represents hxb)
-    (Ne.symm (hxa.composable_labels_ne haz)) (Ne.symm (hxb.composable_labels_ne hbz))
-    (by group : (z*a⁻¹)*(a*x⁻¹)=(z*b⁻¹)*(b*x⁻¹))
-  have hp' := congrArg (Submodule.map
-    (LinearMap.fst ℝ (I → ℝ) (Module.Dual ℝ (LinearMap.ker (form M))))) hp
-  have hp'' : rootPlane (graphRoot haz) (graphRoot hxa) =
-      rootPlane (graphRoot hbz) (graphRoot hxb) := by
-    simpa only [rootPlane,Submodule.map_span,Set.image_insert_eq,
-      Set.image_singleton,LinearMap.fst_apply,embed] using hp'
-  have hm : (graphRoot hxb).val ∈ rootPlane (graphRoot hxa) (graphRoot haz) := by
-    rw [rootPlane,Set.pair_comm]
-    change (graphRoot hxb).val ∈ rootPlane (graphRoot haz) (graphRoot hxa)
-    rw [hp'']
-    exact Submodule.subset_span (Set.mem_insert_of_mem _ (Set.mem_singleton _))
-  exact ((rootPlane_eq_iff (graphRoot hxa) (graphRoot haz) (graphRoot hxb)
-    (graphRoot_ne_of_path hxa haz)).mpr
-    ⟨Ne.symm (graphRoot_ne_of_same_source hxa hxb hab),hm⟩).symm
-
-/-- Soundness of the eight-edge diamond closure rule for the genuine Bruhat
-reflection graph. Vertex ranks need not be consecutive; the proof uses actual
-root planes and reflection faithfulness, not a labeled graph isomorphism. -/
- theorem bruhatStep_diamond_closure {x a b c d z : W}
-    (hxa : BruhatStep cs x a) (hxb : BruhatStep cs x b)
-    (hac : BruhatStep cs a c) (had : BruhatStep cs a d)
-    (hbc : BruhatStep cs b c) (hbd : BruhatStep cs b d)
-    (hcz : BruhatStep cs c z) (hdz : BruhatStep cs d z)
-    (hab : a ≠ b) (hcd : c ≠ d) : BruhatStep cs x z := by
-  let P := rootPlane (graphRoot hxa) (graphRoot hxb)
-  let Q := rootPlane (graphRoot hac) (graphRoot had)
-  have hPc := graphRoot_square_plane hxa hxb hac hbc hab
-  have hPd := graphRoot_square_plane hxa hxb had hbd hab
-  have hacP : (graphRoot hac).val ∈ P := by
-    change (graphRoot hac).val ∈ rootPlane (graphRoot hxa) (graphRoot hxb)
-    rw [← hPc]
-    exact Submodule.subset_span (Set.mem_insert_of_mem _ (Set.mem_singleton _))
-  have hadP : (graphRoot had).val ∈ P := by
-    change (graphRoot had).val ∈ rootPlane (graphRoot hxa) (graphRoot hxb)
-    rw [← hPd]
-    exact Submodule.subset_span (Set.mem_insert_of_mem _ (Set.mem_singleton _))
-  have hQP : Q=P := by
-    apply Submodule.eq_of_le_of_finrank_eq
-    · apply Submodule.span_le.mpr
-      intro v hv
-      rcases Set.mem_insert_iff.mp hv with rfl | hv
-      · exact hacP
-      · have hv' := Set.mem_singleton_iff.mp hv
-        rw [hv']
-        exact hadP
-    · dsimp [Q,P]
-      rw [rootPlane_finrank _ _ (graphRoot_ne_of_same_source hac had hcd),
-        rootPlane_finrank _ _ (graphRoot_ne_of_same_source hxa hxb hab)]
-  have hQc := graphRoot_square_plane hac had hcz hdz hcd
-  have hm : (graphRoot hxa).val ∈
-      Submodule.span ℝ {(graphRoot hcz).val,(graphRoot hac).val} := by
-    rw [Set.pair_comm]
-    change (graphRoot hxa).val ∈ rootPlane (graphRoot hac) (graphRoot hcz)
-    rw [hQc]
-    change (graphRoot hxa).val ∈ Q
-    rw [hQP]
-    exact Submodule.subset_span (Set.mem_insert _ _)
-  have href := (graphRoot_represents hcz).triple_isReflection
-    (graphRoot_represents hac) (graphRoot_represents hxa) hm
-  have heq : (z*c⁻¹)*(c*a⁻¹)*(a*x⁻¹)=z*x⁻¹ := by group
-  rw [heq] at href
-  exact ⟨lt_trans hxa.1 (lt_trans hac.1 hcz.1), z*x⁻¹,href,by group⟩
-
-end TitsSpace
-end
-end KLInvariance
-
-namespace KLInvariance
-universe u
-
-/-- The entirely unlabelled closure used in graph reconstruction. Its
-completeness for Bruhat graphs is a separate obligation; this definition
-neither changes Bruhat order nor defines the R/KL polynomials. -/
-inductive DiamondEdges {X : Type u} [PartialOrder X] : X → X → Prop
-  | cover {x y : X} : x ⋖ y → DiamondEdges x y
-  | diamond {x a b c d z : X} :
-      DiamondEdges x a → DiamondEdges x b →
-      DiamondEdges a c → DiamondEdges a d →
-      DiamondEdges b c → DiamondEdges b d →
-      DiamondEdges c z → DiamondEdges d z →
-      a ≠ b → c ≠ d → DiamondEdges x z
-
- theorem DiamondEdges.map {X Y : Type*} [PartialOrder X] [PartialOrder Y]
-    (e : X ≃o Y) {x y : X} (h : DiamondEdges x y) : DiamondEdges (e x) (e y) := by
-  induction h with
-  | cover h => exact DiamondEdges.cover ((apply_covBy_apply_iff e).mpr h)
-  | diamond _ _ _ _ _ _ _ _ hab hcd hxa hxb hac had hbc hbd hcz hdz =>
-    exact DiamondEdges.diamond hxa hxb hac had hbc hbd hcz hdz
-      (fun he => hab (e.injective he)) (fun he => hcd (e.injective he))
-
- theorem DiamondEdges.map_iff {X Y : Type*} [PartialOrder X] [PartialOrder Y]
-    (e : X ≃o Y) {x y : X} : DiamondEdges (e x) (e y) ↔ DiamondEdges x y := by
-  constructor
-  · intro h
-    simpa only [OrderIso.symm_apply_apply] using h.map e.symm
-  · exact DiamondEdges.map e
-
- theorem bruhatStep_of_interval_covBy
-    {I W : Type*} {M : CoxeterMatrix I} [Group W] (cs : CoxeterSystem M W)
-    {u b : W} {x y : Interval cs u b} (h : x ⋖ y) : BruhatStep cs x.val y.val := by
-  rcases Relation.ReflTransGen.cases_head h.le with heq | ⟨z,hxz,hzy⟩
-  · exact (h.ne (Subtype.ext heq)).elim
-  · have hle : BruhatLE cs x.val z := Relation.ReflTransGen.single hxz
-    let v : Interval cs u b := ⟨z,bruhat_trans cs x.property.1 hle,
-      bruhat_trans cs hzy y.property.2⟩
-    have hv : v = x ∨ v = y := h.eq_or_eq hle hzy
-    rcases hv with hv | hv
-    · have heq : z = x.val := congrArg Subtype.val hv
-      exact (lt_irrefl _ (heq ▸ hxz.1)).elim
-    · have heq : z = y.val := congrArg Subtype.val hv
-      exact heq ▸ hxz
-
-
- theorem diamondEdges_bruhatStep
-    {I W : Type*} [Fintype I] {M : CoxeterMatrix I} [Group W]
-    (cs : CoxeterSystem M W) {u b : W} {x y : Interval cs u b}
-    (h : DiamondEdges x y) : BruhatStep cs x.val y.val := by
-  induction h with
-  | cover h => exact bruhatStep_of_interval_covBy cs h
-  | diamond _ _ _ _ _ _ _ _ hab hcd hxa hxb hac had hbc hbd hcz hdz =>
-    exact TitsSpace.bruhatStep_diamond_closure hxa hxb hac had hbc hbd hcz hdz
-      (fun he => hab (Subtype.ext he)) (fun he => hcd (Subtype.ext he))
-
-end KLInvariance
-
-
-namespace KLInvariance.Dihedral
-variable {W : Type*} [Group W] {M : CoxeterMatrix Bool}
-
-theorem alt_add_two (i : Bool) (n : ℕ) :
-    alt i (n+2) = i :: (!i) :: alt i n := by
-  simp only [alt_succ,Bool.not_not]
-
-theorem alt_odd_sandwich (i : Bool) (n : ℕ) :
-    alt i (2*(n+1)+1) = i :: (alt (!i) (2*n+1) ++ [i]) := by
-  induction n generalizing i with
-  | zero => cases i <;> rfl
-  | succ n ih =>
-    have h₁ : 2*(n+1+1)+1 = (2*(n+1)+1)+2 := by omega
-    have h₂ : 2*(n+1)+1 = (2*n+1)+2 := by omega
-    rw [h₁,alt_add_two,ih,h₂,alt_add_two]
-    simp only [Bool.not_not,List.cons_append]
-
-theorem odd_alt_reflection (cs : CoxeterSystem M W) (i : Bool) (n : ℕ) :
-    cs.IsReflection (cs.wordProd (alt i (2*n+1))) := by
-  induction n generalizing i with
-  | zero => simpa [alt] using cs.isReflection_simple i
-  | succ n ih =>
-    rw [alt_odd_sandwich,cs.wordProd_cons,cs.wordProd_append]
-    obtain ⟨w,j,hw⟩ := ih (!i)
-    refine ⟨cs.simple i*w,j,?_⟩
-    simp only [cs.wordProd_cons,cs.wordProd_nil,mul_one,hw,mul_inv_rev,
-      cs.inv_simple]
-    group
-
-theorem reflection_iff_odd (cs : CoxeterSystem M W) (w : W) :
-    cs.IsReflection w ↔ Odd (cs.length w) := by
-  constructor
-  · exact fun h => h.odd_length
-  · intro h
-    obtain ⟨n,hn⟩ := h
-    obtain ⟨i,hi⟩ := exists_alt cs w
-    rw [hi,hn]
-    convert odd_alt_reflection cs i n using 2
-
-theorem step_iff_odd (cs : CoxeterSystem M W) (x y : W) :
-    BruhatStep cs x y ↔ cs.length x < cs.length y ∧ Odd (cs.length y-cs.length x) := by
-  have hm := cs.length_mul_mod_two y x⁻¹
-  rw [cs.length_inv] at hm
-  constructor
-  · rintro ⟨hlen,t,ht,hprod⟩
-    refine ⟨hlen,?_⟩
-    have hteq : y*x⁻¹=t := by rw [hprod]; group
-    have ho := ht.odd_length
-    rw [← hteq,Nat.odd_iff] at ho
-    rw [Nat.odd_iff]
-    omega
-  · rintro ⟨hlen,ho⟩
-    refine ⟨hlen,y*x⁻¹,?_,by group⟩
-    rw [reflection_iff_odd,Nat.odd_iff]
-    rw [Nat.odd_iff] at ho
-    omega
-
-theorem two_elements_length (cs : CoxeterSystem M W) {n : ℕ} (hn : 0 < n)
-    {w : W} (hw : n < cs.length w) :
-    ∃ a b : W, a ≠ b ∧ ∀ z, cs.length z = n ↔ z = a ∨ z = b := by
-  obtain ⟨i,hi⟩ := exists_alt cs w
-  have hr : cs.IsReduced (alt i (cs.length w)) := by
-    change cs.length (cs.wordProd _) = _
-    rw [← hi,length_alt]
-  have hr₁ := reduced_alt_take cs hr (show n+1 ≤ cs.length w by omega)
-  have hr₀ := reduced_alt_take cs hr (Nat.le_of_lt hw)
-  have hr₂ : cs.IsReduced (alt (!i) n) := by simpa using hr₁.drop 1
-  have hlen₀ : cs.length (cs.wordProd (alt i n)) = n := by simpa using hr₀.eq
-  have hlen₂ : cs.length (cs.wordProd (alt (!i) n)) = n := by simpa using hr₂.eq
-  refine ⟨cs.wordProd (alt i n),cs.wordProd (alt (!i) n),?_,?_⟩
-  · intro heq
-    obtain ⟨k,rfl⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : n ≠ 0)
-    have hp : cs.wordProd (alt i (k+1+1)) = cs.wordProd (alt (!i) k) := by
-      rw [alt_succ,cs.wordProd_cons,← heq,alt_succ,cs.wordProd_cons,
-        cs.simple_mul_simple_cancel_left]
-    have hlen := cs.length_wordProd_le (alt (!i) k)
-    have hred := hr₁.eq
-    rw [hp,length_alt] at hred
-    rw [length_alt] at hlen
-    omega
-  · intro z
-    constructor
-    · intro hz
-      obtain ⟨j,hj⟩ := exists_alt cs z
-      rw [hz] at hj
-      have hj' : j=i ∨ j=(!i) := by cases j <;> cases i <;> simp
-      rcases hj' with rfl|rfl
-      · exact Or.inl hj
-      · exact Or.inr hj
-    · rintro (rfl|rfl) <;> assumption
-
-theorem two_step_rank_two (cs : CoxeterSystem M W) {x y : W}
-    (hlen : cs.length y = cs.length x+2) :
-    ∃ a b : W, a ≠ b ∧
-      ∀ z, (BruhatStep cs x z ∧ BruhatStep cs z y) ↔ z=a ∨ z=b := by
-  obtain ⟨a,b,hab,hab'⟩ := two_elements_length cs (by omega : 0 < cs.length x+1)
-    (w := y) (by omega)
-  refine ⟨a,b,hab,?_⟩
-  intro z
-  rw [← hab',step_iff_odd,step_iff_odd]
-  constructor
-  · rintro ⟨⟨hx,_⟩,⟨hy,_⟩⟩
-    omega
-  · intro hz
-    constructor <;> constructor
-    · omega
-    · rw [Nat.odd_iff]; omega
-    · omega
-    · rw [Nat.odd_iff]; omega
-
-end KLInvariance.Dihedral
-
-namespace KLInvariance
-variable {I W : Type*} [Group W] {M : CoxeterMatrix I}
-
-theorem interval_covBy_of_length_succ (cs : CoxeterSystem M W)
-    {u b : W} {x y : Interval cs u b} (hxy : x ≤ y)
-    (hlen : cs.length y.val = cs.length x.val+1) : x ⋖ y := by
-  apply covBy_iff_lt_and_eq_or_eq.mpr
-  refine ⟨lt_iff_le_and_ne.mpr ⟨hxy,?_⟩,?_⟩
-  · intro heq
-    have := congrArg (fun z : Interval cs u b => cs.length z.val) heq
-    omega
-  · intro z hxz hzy
-    by_cases hz : z=x
-    · exact Or.inl hz
-    · right
-      by_contra hzy'
-      have h₁ := length_lt_of_bruhat_ne cs hxz (fun he => hz (Subtype.ext he.symm))
-      have h₂ := length_lt_of_bruhat_ne cs hzy (fun he => hzy' (Subtype.ext he))
-      omega
-
-namespace Dihedral
-variable {M₂ : CoxeterMatrix Bool}
-
-theorem two_interval_elements_length (cs : CoxeterSystem M₂ W)
-    {u b : W} (x y : Interval cs u b) {n : ℕ}
-    (hn₁ : cs.length x.val < n) (hn₂ : n < cs.length y.val) :
-    ∃ a c : Interval cs u b, a ≠ c ∧ cs.length a.val=n ∧ cs.length c.val=n := by
-  obtain ⟨a,c,hac,hlevel⟩ := two_elements_length cs (by omega : 0<n) hn₂
-  have ha := (hlevel a).mpr (Or.inl rfl)
-  have hc := (hlevel c).mpr (Or.inr rfl)
-  have hmem (z : W) (hz : cs.length z=n) :
-      BruhatLE cs u z ∧ BruhatLE cs z b :=
-    ⟨bruhat_trans cs x.property.1 (shorter_bruhat cs (by omega)),
-      bruhat_trans cs (shorter_bruhat cs (by omega)) y.property.2⟩
-  exact ⟨⟨a,hmem a ha⟩,⟨c,hmem c hc⟩,
-    fun he => hac (congrArg Subtype.val he),ha,hc⟩
-
-/-- Completeness of the actual unlabelled eight-edge closure for genuine
-rank-two Coxeter systems. This is a necessary local case of Dyer's general
-reconstruction, not a replacement main theorem. -/
-theorem diamondEdges_of_bruhatStep (cs : CoxeterSystem M₂ W)
-    {u b : W} {x y : Interval cs u b} (hxy : BruhatStep cs x.val y.val) :
-    DiamondEdges x y := by
-  suffices ∀ n : ℕ, ∀ x y : Interval cs u b,
-      cs.length y.val-cs.length x.val=n → BruhatStep cs x.val y.val →
-      DiamondEdges x y by
-    exact this _ x y rfl hxy
-  intro n
-  induction n using Nat.strong_induction_on with
-  | h n ih =>
-    intro x y hlen hxy
-    have hodd := ((step_iff_odd cs x.val y.val).mp hxy).2
-    rw [hlen,Nat.odd_iff] at hodd
-    by_cases hn : n=1
-    · apply DiamondEdges.cover
-      exact interval_covBy_of_length_succ cs (Relation.ReflTransGen.single hxy) (by omega)
-    · have hn3 : 3 ≤ n := by omega
-      obtain ⟨a,a₂,hab,ha,hb⟩ := two_interval_elements_length cs x y
-        (n := cs.length x.val+1) (by omega) (by have := hxy.1; omega)
-      obtain ⟨c,d,hcd,hc,hd⟩ := two_interval_elements_length cs x y
-        (n := cs.length x.val+2) (by omega) (by have := hxy.1; omega)
-      have hcover {p q : Interval cs u b}
-          (hpq : cs.length q.val=cs.length p.val+1) : DiamondEdges p q :=
-        DiamondEdges.cover (interval_covBy_of_length_succ cs
-          (shorter_bruhat cs (by omega)) hpq)
-      have htop {p : Interval cs u b} (hp : cs.length p.val=cs.length x.val+2) :
-          DiamondEdges p y := by
-        apply ih (cs.length y.val-cs.length p.val) (by have := hxy.1; omega) p y rfl
-        rw [step_iff_odd]
-        constructor
-        · have := hxy.1; omega
-        · rw [Nat.odd_iff]; have := hxy.1; omega
-      exact DiamondEdges.diamond (hcover (by omega)) (hcover (by omega))
-        (hcover (by omega)) (hcover (by omega)) (hcover (by omega)) (hcover (by omega))
-        (htop hc) (htop hd) hab hcd
-
-end Dihedral
-end KLInvariance
-
-
-end
-
-end OAI
+import OAI.RepresentationTheory.KazhdanLusztig.Tits
+import OAI.RepresentationTheory.KazhdanLusztig.ModuleAlgebra
+
+namespace OAI
+
+/-!
+Reflection schedules, decreasing paths, root-space and edge algebra, and dihedral path recursions.
+-/
+
+section
+
+namespace KLInvariance.ReflectionSchedules
+open Polynomial
+universe u v
+variable {I : Type u} [Fintype I]
+noncomputable section
+
+def coordinatePolynomial (a : I → ℝ) : ℝ[X] :=
+  ∑ i, Polynomial.monomial (Fintype.equivFin I i).val (a i)
+
+theorem coordinatePolynomial_coeff (a : I → ℝ) (j : I) :
+    (coordinatePolynomial a).coeff (Fintype.equivFin I j).val = a j := by
+  classical
+  rw [coordinatePolynomial, Polynomial.finsetSum_coeff, Finset.sum_eq_single j]
+  · simp
+  · intro i _ hij
+    have hne : (Fintype.equivFin I i).val ≠ (Fintype.equivFin I j).val := by
+      intro h
+      exact hij ((Fintype.equivFin I).injective (Fin.ext h))
+    simp [Polynomial.coeff_monomial,hne]
+  · simp
+
+theorem coordinatePolynomial_injective : Function.Injective (coordinatePolynomial (I := I)) := by
+  intro a b h
+  funext i
+  simpa only [coordinatePolynomial_coeff] using congrArg (fun p : ℝ[X] => p.coeff
+    (Fintype.equivFin I i).val) h
+
+def momentFunctional (t : ℝ) : Module.Dual ℝ (I → ℝ) where
+  toFun a := ∑ i, a i * t^(Fintype.equivFin I i).val
+  map_add' a b := by simp [add_mul,Finset.sum_add_distrib]
+  map_smul' r a := by simp [Finset.mul_sum,mul_assoc]
+
+theorem coordinatePolynomial_eval (a : I → ℝ) (t : ℝ) :
+    (coordinatePolynomial a).eval t = momentFunctional t a := by
+  simp [coordinatePolynomial,momentFunctional,Polynomial.eval_finsetSum]
+
+
+theorem exists_generic_functional {A : Type v} [Countable A]
+    (v : A → (I → ℝ)) (hv : Function.Injective v) :
+    ∃ f : Module.Dual ℝ (I → ℝ), Function.Injective (fun a => f (v a)) := by
+  classical
+  let P := {p : A × A // p.1 ≠ p.2}
+  let d : P → ℝ[X] := fun p => coordinatePolynomial (v p.val.1) - coordinatePolynomial (v p.val.2)
+  have hd : ∀ p, d p ≠ 0 := by
+    intro p h
+    exact p.property (hv (coordinatePolynomial_injective (sub_eq_zero.mp h)))
+  let bad : Set ℝ := ⋃ p : P, {t | (d p).IsRoot t}
+  have hbad : bad.Countable := Set.countable_iUnion (fun p => (Polynomial.finite_setOfPred_isRoot (hd p)).countable)
+  have hex : ∃ t, t ∉ bad := by
+    by_contra h
+    push Not at h
+    have heq : bad = Set.univ := Set.eq_univ_of_forall h
+    rw [heq] at hbad
+    exact Set.not_countable_univ hbad
+  obtain ⟨t,ht⟩ := hex
+  refine ⟨momentFunctional t,fun a b hab => ?_⟩
+  change momentFunctional t (v a) = momentFunctional t (v b) at hab
+  by_contra hne
+  apply ht
+  apply Set.mem_iUnion.mpr
+  refine ⟨⟨(a,b),hne⟩,?_⟩
+  change (coordinatePolynomial (v a) - coordinatePolynomial (v b)).eval t = 0
+  rw [Polynomial.eval_sub,coordinatePolynomial_eval,coordinatePolynomial_eval,hab,sub_self]
+
+end
+end KLInvariance.ReflectionSchedules
+
+namespace KLInvariance.TitsSpace
+open Module
+universe u v
+variable {I : Type u} [Fintype I] (M : CoxeterMatrix I)
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+def height : Module.Dual ℝ (I → ℝ) where
+  toFun a := ∑ i, a i
+  map_add' a b := by simp [Finset.sum_add_distrib]
+  map_smul' r a := by simp [Finset.mul_sum]
+
+variable {M cs}
+
+theorem PositiveRoot.height_pos (a : PositiveRoot M cs) : 0 < height a.val := by
+  classical
+  have hex : ∃ i, a.val i ≠ 0 := by
+    by_contra h
+    push Not at h
+    exact a.property.1.ne_zero (funext h)
+  obtain ⟨i,hi⟩ := hex
+  exact Finset.sum_pos' (fun j _ => a.property.2 j)
+    ⟨i,Finset.mem_univ _,lt_of_le_of_ne' (a.property.2 i) hi⟩
+
+def normalizedRoot (a : PositiveRoot M cs) : I → ℝ := (height a.val)⁻¹ • a.val
+
+theorem normalizedRoot_injective : Function.Injective (normalizedRoot (M := M) (cs := cs)) := by
+  intro a b hab
+  have ha : 0 < (height a.val)⁻¹ := inv_pos.mpr a.height_pos
+  have hb : 0 < (height b.val)⁻¹ := inv_pos.mpr b.height_pos
+  have hn := congrArg (fun v => form M v v) hab
+  change form M ((height a.val)⁻¹ • a.val) ((height a.val)⁻¹ • a.val) =
+    form M ((height b.val)⁻¹ • b.val) ((height b.val)⁻¹ • b.val) at hn
+  simp only [map_smul,LinearMap.smul_apply,smul_eq_mul,a.property.1.norm,b.property.1.norm] at hn
+  have hh : (height a.val)⁻¹ = (height b.val)⁻¹ := by nlinarith
+  apply Subtype.ext
+  change (height a.val)⁻¹ • a.val = (height b.val)⁻¹ • b.val at hab
+  rw [← hh] at hab
+  exact (smul_right_injective _ ha.ne').eq_iff.mp hab
+
+variable (M cs)
+
+theorem countable_Positiveroot : Countable (PositiveRoot M cs) := by
+  let : Countable W := cs.wordProd_surjective.countable
+  exact Countable.of_equiv {t : W // cs.IsReflection t} (positiveRootEquiv M cs)
+
+
+theorem exists_geometric_functional : ∃ f : Module.Dual ℝ (I → ℝ),
+    Function.Injective (fun a : PositiveRoot M cs => f a.val / height a.val) := by
+  let := countable_Positiveroot M cs
+  obtain ⟨f,hf⟩ := ReflectionSchedules.exists_generic_functional
+    (normalizedRoot (M := M) (cs := cs)) normalizedRoot_injective
+  refine ⟨f,?_⟩
+  convert hf using 1
+  funext a
+  simp [normalizedRoot,div_eq_mul_inv,mul_comm]
+
+/-- Betweenness of all positive combinations is the angular-order condition
+on every root plane. This is the genuine geometric reflection-order property. -/
+theorem functionalRatio_between (f : Module.Dual ℝ (I → ℝ))
+    (a b c : PositiveRoot M cs) {r s : ℝ} (hr : 0 < r) (hs : 0 < s)
+    (hc : c.val = r • a.val + s • b.val)
+    (hab : f a.val / height a.val < f b.val / height b.val) :
+    f a.val / height a.val < f c.val / height c.val ∧
+      f c.val / height c.val < f b.val / height b.val := by
+  have ha := a.height_pos
+  have hb := b.height_pos
+  have hh := c.height_pos
+  have hfc : f c.val = r*f a.val+s*f b.val := by rw [hc]; simp
+  have hhc : height c.val = r*height a.val+s*height b.val := by rw [hc]; simp
+  rw [div_lt_div_iff₀ ha hb] at hab
+  constructor
+  · rw [div_lt_div_iff₀ ha hh,hfc,hhc]
+    nlinarith [mul_pos hs (sub_pos.mpr hab)]
+  · rw [div_lt_div_iff₀ hh hb,hfc,hhc]
+    nlinarith [mul_pos hr (sub_pos.mpr hab)]
+
+end
+end KLInvariance.TitsSpace
+
+namespace KLInvariance.TitsSpace
+open Module
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+noncomputable section
+
+/-- Conjugation on the actual root lines, represented by positive roots. -/
+def positiveAction (w : W) (a : PositiveRoot M cs) : PositiveRoot M cs :=
+  positiveRoot M cs ⟨w * (rootReflection M cs a).val * w⁻¹,
+    (rootReflection M cs a).property.conj w⟩
+
+theorem positiveAction_represents (w : W) (a : PositiveRoot M cs) :
+    Represents M cs (w * (rootReflection M cs a).val * w⁻¹) (positiveAction w a).val :=
+  positiveRoot_represents M cs ⟨w * (rootReflection M cs a).val * w⁻¹,
+    (rootReflection M cs a).property.conj w⟩
+
+theorem positiveAction_eq_of_nonneg (w : W) (a : PositiveRoot M cs)
+    (hp : ∀ i, 0 ≤ action M cs w a.val i) :
+    (positiveAction w a).val = action M cs w a.val :=
+  (positiveAction_represents w a).unique_positive
+    ((rootReflection_represents M cs a).act w) (positiveAction w a).property.2 hp
+
+theorem positiveAction_eq_of_nonpos (w : W) (a : PositiveRoot M cs)
+    (hn : ∀ i, action M cs w a.val i ≤ 0) :
+    (positiveAction w a).val = -action M cs w a.val :=
+  (positiveAction_represents w a).unique_positive
+    ((rootReflection_represents M cs a).act w).neg (positiveAction w a).property.2
+    (fun i => neg_nonneg.mpr (hn i))
+
+@[simp] theorem rootReflection_positiveAction (w : W) (a : PositiveRoot M cs) :
+    (rootReflection M cs (positiveAction w a)).val = w*(rootReflection M cs a).val*w⁻¹ :=
+  (rootReflection_represents M cs _).reflection_unique (positiveAction_represents w a)
+
+@[simp] theorem positiveAction_one (a : PositiveRoot M cs) : positiveAction 1 a = a := by
+  apply (positiveRootEquiv M cs).symm.injective
+  apply Subtype.ext
+  change (rootReflection M cs (positiveAction 1 a)).val = (rootReflection M cs a).val
+  simp
+
+theorem positiveAction_mul (w z : W) (a : PositiveRoot M cs) :
+    positiveAction (w*z) a = positiveAction w (positiveAction z a) := by
+  apply (positiveRootEquiv M cs).symm.injective
+  apply Subtype.ext
+  change (rootReflection M cs (positiveAction (w*z) a)).val =
+    (rootReflection M cs (positiveAction w (positiveAction z a))).val
+  simp only [rootReflection_positiveAction]
+  group
+
+def positiveActionEquiv (w : W) : Equiv.Perm (PositiveRoot M cs) where
+  toFun := positiveAction w
+  invFun := positiveAction w⁻¹
+  left_inv a := by rw [← positiveAction_mul, inv_mul_cancel, positiveAction_one]
+  right_inv a := by rw [← positiveAction_mul, mul_inv_cancel, positiveAction_one]
+
+abbrev rootRef (a : PositiveRoot M cs) : W := (rootReflection M cs a).val
+
+theorem positiveAction_involutive (a : PositiveRoot M cs) :
+    Function.Involutive (positiveAction (M := M) (cs := cs) (rootRef a)) := by
+  intro b
+  rw [← positiveAction_mul, (rootReflection M cs a).property.mul_self, positiveAction_one]
+
+def rootInverted (w : W) (a : PositiveRoot M cs) : Prop :=
+  ∀ i, action M cs w⁻¹ a.val i ≤ 0
+
+theorem not_rootInverted_iff (w : W) (a : PositiveRoot M cs) :
+    ¬ rootInverted w a ↔ ∀ i, 0 ≤ action M cs w⁻¹ a.val i := by
+  constructor
+  · intro hn
+    exact (a.property.1.act w⁻¹).sign.resolve_right hn
+  · intro hp hn
+    exact (a.property.1.act w⁻¹).not_both hp hn
+
+theorem rootInverted_mul_of_nonneg (x : W) (a b : PositiveRoot M cs)
+    (hp : ∀ i, 0 ≤ action M cs (rootRef a) b.val i) :
+    rootInverted (rootRef a*x) b ↔ rootInverted x (positiveAction (rootRef a) b) := by
+  unfold rootInverted
+  rw [positiveAction_eq_of_nonneg _ _ hp, mul_inv_rev, (rootReflection M cs a).property.inv,
+    map_mul]
+  rfl
+
+theorem rootInverted_mul_of_nonpos (x : W) (a b : PositiveRoot M cs)
+    (hn : ∀ i, action M cs (rootRef a) b.val i ≤ 0) :
+    rootInverted (rootRef a*x) b ↔ ¬ rootInverted x (positiveAction (rootRef a) b) := by
+  rw [not_rootInverted_iff]
+  unfold rootInverted
+  rw [positiveAction_eq_of_nonpos _ _ hn, mul_inv_rev, (rootReflection M cs a).property.inv,
+    map_mul, map_neg]
+  simp only [LinearEquiv.mul_apply, Pi.neg_apply, neg_nonneg]
+
+theorem negative_pair_sum (a b : PositiveRoot M cs)
+    (hn : ∀ i, action M cs (rootRef a) b.val i ≤ 0) :
+    b.val + (positiveAction (rootRef a) b).val = form M a.val b.val • a.val := by
+  rw [positiveAction_eq_of_nonpos _ _ hn, (rootReflection_represents M cs a).action_formula]
+  module
+
+theorem negative_pair_coefficient_pos (a b : PositiveRoot M cs)
+    (hn : ∀ i, action M cs (rootRef a) b.val i ≤ 0) : 0 < form M a.val b.val := by
+  have h := congrArg (fun z => height z) (negative_pair_sum a b hn)
+  simp only [map_add,map_smul,smul_eq_mul] at h
+  have hp := b.height_pos
+  have hq := (positiveAction (rootRef a) b).height_pos
+  have ha := a.height_pos
+  nlinarith
+
+/-- The crucial direct positivity argument: over an ascent in alpha, the
+negative root-pair for its reflection cannot both already be inversions.
+This works in degenerate planes and does not assume a dihedral presentation. -/
+theorem negative_pair_not_both (x : W) (a b : PositiveRoot M cs)
+    (hx : ¬ rootInverted x a)
+    (hn : ∀ i, action M cs (rootRef a) b.val i ≤ 0) :
+    ¬ (rootInverted x b ∧ rootInverted x (positiveAction (rootRef a) b)) := by
+  rintro ⟨hb,hc⟩
+  have hp := (not_rootInverted_iff x a).mp hx
+  have hs := congrArg (action M cs x⁻¹) (negative_pair_sum a b hn)
+  simp only [map_add,map_smul] at hs
+  have hcoef := negative_pair_coefficient_pos a b hn
+  have hna : ∀ i, action M cs x⁻¹ a.val i ≤ 0 := by
+    intro i
+    have hi := congrFun hs i
+    change action M cs x⁻¹ b.val i +
+      action M cs x⁻¹ (positiveAction (rootRef a) b).val i =
+        form M a.val b.val * action M cs x⁻¹ a.val i at hi
+    have hb' := hb i
+    have hc' := hc i
+    nlinarith
+  exact (a.property.1.act x⁻¹).not_both hp hna
+
+end
+end KLInvariance.TitsSpace
+
+namespace KLInvariance.TitsSpace
+open Module
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+theorem finite_rootInverted (w : W) : {a : PositiveRoot M cs | rootInverted w a}.Finite := by
+  exact Set.finite_coe_iff.mp (rootInversions_finite M cs w)
+
+def inversionVector (w : W) : PositiveRoot M cs →₀ ℤ := by
+  classical
+  refine Finsupp.ofSupportFinite (fun a => if rootInverted w a then 1 else 0) ?_
+  simpa only [Function.support, ne_eq, ite_eq_right_iff, one_ne_zero, imp_false,
+    not_not] using finite_rootInverted (M := M) (cs := cs) w
+
+@[simp] theorem inversionVector_apply (w : W) (a : PositiveRoot M cs) :
+    inversionVector w a = if rootInverted w a then 1 else 0 := by
+  classical
+  rfl
+
+theorem inversionVector_sum (w : W) : (inversionVector (M := M) (cs := cs) w).sum
+    (fun _ n => n) = (cs.length w : ℤ) := by
+  classical
+  let f := inversionVector (M := M) (cs := cs) w
+  have hh : f.support.card = cs.length w := by
+    have heq : (f.support : Set (PositiveRoot M cs)) = {a | rootInverted w a} := by
+      ext a
+      simp [f]
+    rw [← Set.ncard_coe_finset, heq, ← Nat.card_coe_set_eq]
+    exact card_rootInversions M cs w
+  rw [Finsupp.sum]
+  calc
+    ∑ a ∈ f.support, f a = ∑ _a ∈ f.support, (1:ℤ) := by
+      apply Finset.sum_congr rfl
+      intro a ha
+      have hm := Finsupp.mem_support_iff.mp ha
+      by_cases hi : rootInverted w a
+      · simp [f,hi]
+      · simp [f,hi] at hm
+    _ = _ := by simp [hh]
+
+def rootChange (x y : W) : PositiveRoot M cs →₀ ℤ := inversionVector y - inversionVector x
+
+def rootPlane (a b : PositiveRoot M cs) : Submodule ℝ (I → ℝ) :=
+  Submodule.span ℝ {a.val,b.val}
+
+/-- The finitely supported change in root inversions, with the distinguished
+root removed, grouped by actual root planes. -/
+def planeDefect (x y : W) (a : PositiveRoot M cs) : Submodule ℝ (I → ℝ) →₀ ℤ :=
+  Finsupp.mapDomain (rootPlane a) ((rootChange x y).erase a)
+
+theorem rootInverted_mul_self (x : W) (a : PositiveRoot M cs) :
+    rootInverted (rootRef a*x) a ↔ ¬ rootInverted x a := by
+  have h := (rootReflection M cs a).property.isLeftInversion_mul_right_iff (w := x)
+  simpa only [rootInverted, ← (rootReflection_represents M cs a).inversion_iff a.property.2] using h
+
+theorem rootChange_distinguished (x : W) (a : PositiveRoot M cs)
+    (hx : ¬ rootInverted x a) : rootChange x (rootRef a*x) a = 1 := by
+  classical
+  simp only [rootChange, Finsupp.sub_apply, inversionVector_apply,
+    (rootInverted_mul_self x a).mpr hx, hx, ite_true, ite_false, sub_zero]
+
+
+theorem planeDefect_sum (x : W) (a : PositiveRoot M cs) (hx : ¬ rootInverted x a) :
+    (planeDefect x (rootRef a*x) a).sum (fun _ n => n) =
+      (cs.length (rootRef a*x) : ℤ) - cs.length x - 1 := by
+  classical
+  rw [planeDefect, Finsupp.sum_mapDomain_index (fun _ => rfl) (fun _ _ _ => rfl),
+    Finsupp.erase_eq_sub_single, Finsupp.sum_sub_index (fun _ _ _ => rfl),
+    Finsupp.sum_single_index rfl, rootChange_distinguished x a hx,
+    rootChange, Finsupp.sum_sub_index (fun _ _ _ => rfl),
+    inversionVector_sum, inversionVector_sum]
+
+end
+end KLInvariance.TitsSpace
+
+namespace KLInvariance.TitsSpace
+open Module
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+abbrev partner (a b : PositiveRoot M cs) := positiveAction (rootRef a) b
+
+theorem partner_self (a : PositiveRoot M cs) : partner a a = a := by
+  apply (positiveRootEquiv M cs).symm.injective
+  apply Subtype.ext
+  change (rootReflection M cs (positiveAction (rootRef a) a)).val = rootRef a
+  rw [rootReflection_positiveAction, (rootReflection M cs a).property.mul_self, one_mul,
+    (rootReflection M cs a).property.inv]
+
+theorem partner_involutive (a : PositiveRoot M cs) : Function.Involutive (partner a) :=
+  positiveAction_involutive a
+
+theorem partner_eq_self_of_eq (a b : PositiveRoot M cs) : partner a b = a ↔ b = a := by
+  constructor
+  · intro h
+    have hh := congrArg (partner a) h
+    rwa [partner_involutive a b,partner_self] at hh
+  · rintro rfl
+    exact partner_self _
+
+abbrev flipped (a b : PositiveRoot M cs) : Prop :=
+  ∀ i, action M cs (rootRef a) b.val i ≤ 0
+
+theorem flipped_partner_iff (a b : PositiveRoot M cs) : flipped a (partner a b) ↔ flipped a b := by
+  have hact : action M cs (rootRef a) * action M cs (rootRef a) = 1 := by
+    rw [← map_mul, (rootReflection M cs a).property.mul_self, map_one]
+  by_cases hb : flipped a b
+  · have heq := positiveAction_eq_of_nonpos (rootRef a) b hb
+    have hp : flipped a (partner a b) := by
+      intro i
+      change action M cs (rootRef a) (positiveAction (rootRef a) b).val i ≤ 0
+      rw [heq,map_neg, ← LinearEquiv.mul_apply,hact]
+      exact neg_nonpos.mpr (b.property.2 i)
+    exact iff_of_true hp hb
+  · have hpos := (b.property.1.act (rootRef a)).sign.resolve_right hb
+    have heq := positiveAction_eq_of_nonneg (rootRef a) b hpos
+    have hp : ¬ flipped a (partner a b) := by
+      intro hh
+      have hneg : ∀ i, b.val i ≤ 0 := by
+        intro i
+        have hi := hh i
+        change action M cs (rootRef a) (positiveAction (rootRef a) b).val i ≤ 0 at hi
+        rwa [heq, ← LinearEquiv.mul_apply, hact] at hi
+      exact b.property.1.not_both b.property.2 hneg
+    exact iff_of_false hp hb
+
+theorem partner_mem {P : Submodule ℝ (I → ℝ)} (a b : PositiveRoot M cs)
+    (ha : a.val ∈ P) (hb : b.val ∈ P) : (partner a b).val ∈ P := by
+  have hm : action M cs (rootRef a) b.val ∈ P := by
+    rw [(rootReflection_represents M cs a).action_formula]
+    exact P.sub_mem hb (P.smul_mem _ ha)
+  rcases (b.property.1.act (rootRef a)).sign with hp|hn
+  · exact (positiveAction_eq_of_nonneg (rootRef a) b hp).symm ▸ hm
+  · exact (positiveAction_eq_of_nonpos (rootRef a) b hn).symm ▸ P.neg_mem hm
+
+theorem rootPlane_partner (a b : PositiveRoot M cs) : rootPlane a (partner a b) = rootPlane a b := by
+  have hle (b : PositiveRoot M cs) : rootPlane a (partner a b) ≤ rootPlane a b := by
+    apply Submodule.span_le.mpr
+    intro z hz
+    rcases Set.mem_insert_iff.mp hz with rfl|hz
+    · exact Submodule.subset_span (Set.mem_insert _ _)
+    · rcases Set.mem_singleton_iff.mp hz with rfl
+      exact partner_mem a b (Submodule.subset_span (Set.mem_insert _ _))
+        (Submodule.subset_span (Set.mem_insert_of_mem _ rfl))
+  exact le_antisymm (hle b) (by simpa only [partner_involutive a b] using hle (partner a b))
+
+theorem negative_partner_fixed (a b : PositiveRoot M cs) (hn : flipped a b)
+    (heq : partner a b = b) : b = a := by
+  have hs := negative_pair_sum a b hn
+  change b.val + (partner a b).val = form M a.val b.val • a.val at hs
+  rw [heq] at hs
+  have hc := negative_pair_coefficient_pos a b hn
+  have hb : b.val = (form M a.val b.val / 2) • a.val := by
+    calc
+      b.val = (1/2:ℝ) • (b.val+b.val) := by module
+      _ = _ := by rw [hs]; module
+  have hnorm := b.property.1.norm
+  rw [hb] at hnorm
+  simp only [map_smul,LinearMap.smul_apply,smul_eq_mul,a.property.1.norm] at hnorm
+  have hc2 : form M a.val b.val / 2 = 1 := by nlinarith
+  apply Subtype.ext
+  simpa only [hc2,one_smul] using hb
+
+/-- Roots in a negative pair which are both positive before the ascent. -/
+def goodPairRoot (x : W) (a b : PositiveRoot M cs) : Prop :=
+  flipped a b ∧ ¬ rootInverted x b ∧ ¬ rootInverted x (partner a b)
+
+theorem goodPairRoot_partner (x : W) (a b : PositiveRoot M cs) :
+    goodPairRoot x a (partner a b) ↔ goodPairRoot x a b := by
+  simp only [goodPairRoot,flipped_partner_iff,partner_involutive a b]
+  tauto
+
+/-- The pairwise change is either zero or two, with no negative case. -/
+theorem rootChange_pair (x : W) (a b : PositiveRoot M cs) (hx : ¬ rootInverted x a) :
+    rootChange x (rootRef a*x) b + rootChange x (rootRef a*x) (partner a b) =
+      if goodPairRoot x a b then 2 else 0 := by
+  by_cases hn : flipped a b
+  · have hn' := (flipped_partner_iff a b).mpr hn
+    have hi := rootInverted_mul_of_nonpos x a b hn
+    have hi' := rootInverted_mul_of_nonpos x a (partner a b) hn'
+    have hpair : positiveAction (rootRef a) (partner a b) = b := partner_involutive a b
+    rw [hpair] at hi'
+    have hnnb := negative_pair_not_both x a b hx hn
+    simp only [rootChange,Finsupp.sub_apply,inversionVector_apply,hi,hi',goodPairRoot,]
+    by_cases hb : rootInverted x b <;> by_cases hc : rootInverted x (partner a b) <;>
+      simp_all
+  · have hp := (b.property.1.act (rootRef a)).sign.resolve_right hn
+    have hn' := mt (flipped_partner_iff a b).mp hn
+    have hp' := ((partner a b).property.1.act (rootRef a)).sign.resolve_right hn'
+    have hi := rootInverted_mul_of_nonneg x a b hp
+    have hi' := rootInverted_mul_of_nonneg x a (partner a b) hp'
+    have hpair : positiveAction (rootRef a) (partner a b) = b := partner_involutive a b
+    rw [hpair] at hi'
+    simp only [rootChange,Finsupp.sub_apply,inversionVector_apply,hi,hi',goodPairRoot,hn,false_and,ite_false]
+    ring
+
+end
+end KLInvariance.TitsSpace
+
+namespace KLInvariance.TitsSpace
+open Module
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+def geometricFunctional : Module.Dual ℝ (I → ℝ) := (exists_geometric_functional M cs).choose
+
+def geometricSlope (a : PositiveRoot M cs) : ℝ := geometricFunctional (M := M) (cs := cs) a.val / height a.val
+
+theorem geometricSlope_injective : Function.Injective (geometricSlope (M := M) (cs := cs)) :=
+  (exists_geometric_functional M cs).choose_spec
+
+@[instance_reducible]
+def geometricOrder : LinearOrder (PositiveRoot M cs) := LinearOrder.lift' geometricSlope geometricSlope_injective
+
+theorem geometricSlope_between (a b c : PositiveRoot M cs) {r s : ℝ}
+    (hr : 0 < r) (hs : 0 < s) (hc : c.val = r • a.val+s • b.val)
+    (hab : geometricSlope a < geometricSlope b) :
+    geometricSlope a < geometricSlope c ∧ geometricSlope c < geometricSlope b :=
+  functionalRatio_between M cs (geometricFunctional (M := M) (cs := cs)) a b c hr hs hc hab
+
+def partnerEquiv (a : PositiveRoot M cs) : Equiv.Perm (PositiveRoot M cs) :=
+  Function.Involutive.toPerm (partner a) (partner_involutive a)
+
+@[simp] theorem partnerEquiv_apply (a b : PositiveRoot M cs) : partnerEquiv a b = partner a b := rfl
+@[simp] theorem partnerEquiv_symm_apply (a b : PositiveRoot M cs) : (partnerEquiv a).symm b = partner a b := rfl
+
+def extraRoot (x : W) (a b : PositiveRoot M cs) : Prop := b ≠ a ∧ goodPairRoot x a b
+
+theorem extraRoot_partner (x : W) (a b : PositiveRoot M cs) :
+    extraRoot x a (partner a b) ↔ extraRoot x a b := by
+  simp only [extraRoot,ne_eq,partner_eq_self_of_eq,goodPairRoot_partner]
+
+theorem extraRoot_finite (x : W) (a : PositiveRoot M cs) : {b | extraRoot x a b}.Finite := by
+  apply (finite_rootInverted (M := M) (cs := cs) (rootRef a*x)).subset
+  intro b hb
+  exact (rootInverted_mul_of_nonpos x a b hb.2.1).mpr hb.2.2.2
+
+def halfRoot (x : W) (a b : PositiveRoot M cs) : Prop :=
+  extraRoot x a b ∧ geometricSlope b < geometricSlope (partner a b)
+
+theorem halfRoot_finite (x : W) (a : PositiveRoot M cs) : {b | halfRoot x a b}.Finite :=
+  (extraRoot_finite x a).subset (fun _ h => h.1)
+
+def rootIndicator (p : PositiveRoot M cs → Prop) (hp : {a | p a}.Finite) : PositiveRoot M cs →₀ ℤ :=
+  Finsupp.ofSupportFinite (fun a => if p a then 1 else 0) (by
+    simpa only [Function.support,ne_eq,ite_eq_right_iff,one_ne_zero,imp_false,not_not] using hp)
+
+@[simp] theorem rootIndicator_apply (p : PositiveRoot M cs → Prop) (hp : {a | p a}.Finite)
+    (a : PositiveRoot M cs) : rootIndicator p hp a = if p a then 1 else 0 := rfl
+
+def extraVector (x : W) (a : PositiveRoot M cs) := rootIndicator (extraRoot x a) (extraRoot_finite x a)
+def halfVector (x : W) (a : PositiveRoot M cs) := rootIndicator (halfRoot x a) (halfRoot_finite x a)
+
+/-- A choice of one root in each actual two-element negative pair. -/
+theorem extraVector_eq_half (x : W) (a : PositiveRoot M cs) :
+    extraVector x a = halfVector x a + Finsupp.mapDomain (partnerEquiv a) (halfVector x a) := by
+  ext b
+  simp only [Finsupp.add_apply,Finsupp.mapDomain_equiv_apply,partnerEquiv_symm_apply,
+    extraVector,halfVector,rootIndicator_apply,halfRoot,extraRoot_partner,partner_involutive a b]
+  by_cases hb : extraRoot x a b
+  · have hne : geometricSlope b ≠ geometricSlope (partner a b) := by
+      intro h
+      exact hb.1 (negative_partner_fixed a b hb.2.1 (geometricSlope_injective h).symm)
+    rcases lt_or_gt_of_ne hne with h|h
+    · simp [hb,h,not_lt_of_gt h]
+    · simp [hb,h,not_lt_of_gt h]
+  · simp [hb]
+
+/-- All apparent negative contributions cancel under the actual reflection on
+root lines; the remaining pairs are precisely `extraVector`. -/
+theorem erasedChange_pair (x : W) (a : PositiveRoot M cs) (hx : ¬ rootInverted x a) :
+    (rootChange x (rootRef a*x)).erase a +
+      Finsupp.mapDomain (partnerEquiv a) ((rootChange x (rootRef a*x)).erase a) =
+        extraVector x a + extraVector x a := by
+  classical
+  ext b
+  simp only [Finsupp.add_apply,Finsupp.mapDomain_equiv_apply,partnerEquiv_symm_apply]
+  by_cases hb : b=a
+  · subst b
+    simp [partner_self,extraVector,extraRoot]
+  · have hpb : partner a b ≠ a := mt (partner_eq_self_of_eq a b).mp hb
+    rw [Finsupp.erase_ne hb,Finsupp.erase_ne hpb,rootChange_pair x a b hx]
+    simp only [extraVector,rootIndicator_apply,extraRoot,]
+    by_cases hg : goodPairRoot x a b <;> simp [hg,hb]
+
+theorem mapDomain_plane_partner (a : PositiveRoot M cs) (f : PositiveRoot M cs →₀ ℤ) :
+    Finsupp.mapDomain (rootPlane a) (Finsupp.mapDomain (partnerEquiv a) f) =
+      Finsupp.mapDomain (rootPlane a) f := by
+  rw [← Finsupp.mapDomain_comp]
+  apply Finsupp.mapDomain_congr
+  intro b _
+  exact rootPlane_partner a b
+
+def planeWeight (x : W) (a : PositiveRoot M cs) : Submodule ℝ (I → ℝ) →₀ ℤ :=
+  Finsupp.mapDomain (rootPlane a) (halfVector x a)
+
+
+theorem planeDefect_eq_double_weight (x : W) (a : PositiveRoot M cs) (hx : ¬ rootInverted x a) :
+    planeDefect x (rootRef a*x) a = planeWeight x a + planeWeight x a := by
+  have h := congrArg (Finsupp.mapDomain (rootPlane a)) (erasedChange_pair x a hx)
+  rw [Finsupp.mapDomain_add,mapDomain_plane_partner,Finsupp.mapDomain_add] at h
+  have heq : planeDefect x (rootRef a*x) a = Finsupp.mapDomain (rootPlane a) (extraVector x a) := by
+    ext P
+    have hh := congrArg (fun f : Submodule ℝ (I → ℝ) →₀ ℤ => f P) h
+    simp only [Finsupp.add_apply] at hh
+    change planeDefect x (rootRef a*x) a P = _
+    dsimp [planeDefect]
+    omega
+  rw [heq,extraVector_eq_half,Finsupp.mapDomain_add,mapDomain_plane_partner]
+  rfl
+
+theorem planeWeight_nonneg (x : W) (a : PositiveRoot M cs) (P : Submodule ℝ (I → ℝ)) :
+    0 ≤ planeWeight x a P := by
+  classical
+  rw [planeWeight,Finsupp.mapDomain,Finsupp.sum_apply]
+  apply Finset.sum_nonneg
+  intro b _
+  simp only [Finsupp.single_apply,halfVector,rootIndicator_apply]
+  split_ifs <;> norm_num
+
+/-- Exact ambient degree formula, with actual roots, finite support and
+nonnegative integral planar weights; no labeled interval isomorphism enters. -/
+theorem sum_planeWeight (x : W) (a : PositiveRoot M cs) (hx : ¬ rootInverted x a) :
+    2 * (planeWeight x a).sum (fun _ n => n) =
+      (cs.length (rootRef a*x) : ℤ) - cs.length x - 1 := by
+  have h := planeDefect_sum x a hx
+  rw [planeDefect_eq_double_weight x a hx,
+    Finsupp.sum_add_index' (fun _ => rfl) (fun _ _ _ => rfl)] at h
+  linarith
+
+end
+end KLInvariance.TitsSpace
+
+namespace KLInvariance.TitsSpace
+open Module
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+theorem positiveRoots_independent (a b : PositiveRoot M cs) (hab : a ≠ b) :
+    LinearIndependent ℝ ![a.val,b.val] := by
+  apply (rootReflection_represents M cs a).independent_of_ne (rootReflection_represents M cs b)
+  intro h
+  apply hab
+  apply (positiveRootEquiv M cs).symm.injective
+  exact Subtype.ext h
+
+theorem rootPlane_finrank (a b : PositiveRoot M cs) (hab : a ≠ b) :
+    Module.finrank ℝ (rootPlane a b) = 2 := by
+  have h := finrank_span_eq_card (positiveRoots_independent a b hab)
+  have hrange : Set.range ![a.val,b.val] = {a.val,b.val} := by
+    ext z
+    simp only [Set.mem_range,Set.mem_insert_iff,Set.mem_singleton_iff]
+    constructor
+    · rintro ⟨i,rfl⟩
+      fin_cases i <;> simp
+    · rintro (rfl|rfl)
+      · exact ⟨0,rfl⟩
+      · exact ⟨1,rfl⟩
+  rw [hrange] at h
+  exact h
+
+theorem rootPlane_self_finrank (a : PositiveRoot M cs) :
+    Module.finrank ℝ (rootPlane a a) = 1 := by
+  have hset : ({a.val,a.val} : Set (I → ℝ)) = {a.val} := by ext z; simp
+  rw [rootPlane,hset]
+  exact finrank_span_singleton (K := ℝ) (a.property.1.ne_zero)
+
+theorem rootPlane_eq_iff (a b c : PositiveRoot M cs) (hab : a ≠ b) :
+    rootPlane a c = rootPlane a b ↔ c ≠ a ∧ c.val ∈ rootPlane a b := by
+  constructor
+  · intro h
+    constructor
+    · intro hc
+      subst c
+      have h' := congrArg (fun P : Submodule ℝ (I → ℝ) => Module.finrank ℝ P) h
+      rw [rootPlane_self_finrank,rootPlane_finrank a b hab] at h'
+      norm_num at h'
+    · rw [← h]
+      exact Submodule.subset_span (Set.mem_insert_of_mem _ (Set.mem_singleton _))
+  · rintro ⟨hc,hcP⟩
+    apply Submodule.eq_of_le_of_finrank_eq
+    · apply Submodule.span_le.mpr
+      intro z hz
+      rcases (Set.mem_insert_iff.mp hz) with rfl|hz
+      · exact Submodule.subset_span (Set.mem_insert _ _)
+      · have hzc := Set.mem_singleton_iff.mp hz
+        subst z
+        exact hcP
+    · rw [rootPlane_finrank a c (Ne.symm hc),rootPlane_finrank a b hab]
+
+end
+end KLInvariance.TitsSpace
+
+namespace KLInvariance.TitsSpace
+open Module
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+/-- The actual finite count of ambient inversions belonging to a root plane.
+No Coxeter presentation of the plane subgroup is assumed here. -/
+def planeLength (P : Submodule ℝ (I → ℝ)) (w : W) : ℤ :=
+  (inversionVector (M := M) (cs := cs) w).sum (fun a n => if a.val ∈ P then n else 0)
+
+theorem rootPlane_grouped_erase (a b : PositiveRoot M cs) (hab : a ≠ b)
+    (f : PositiveRoot M cs →₀ ℤ) :
+    Finsupp.mapDomain (rootPlane a) (f.erase a) (rootPlane a b) =
+      f.sum (fun c n => if c.val ∈ rootPlane a b then n else 0) - f a := by
+  classical
+  have haP : a.val ∈ rootPlane a b := Submodule.subset_span (Set.mem_insert _ _)
+  have heq : Finsupp.mapDomain (rootPlane a) (f.erase a) (rootPlane a b) =
+      (f.erase a).sum (fun c n => if c.val ∈ rootPlane a b then n else 0) := by
+    rw [Finsupp.mapDomain,Finsupp.sum_apply]
+    apply Finset.sum_congr rfl
+    intro c hc
+    have hca : c ≠ a := by
+      intro h
+      subst c
+      simp only [Finsupp.mem_support_iff,Finsupp.erase_same,ne_eq,not_true_eq_false] at hc
+    change (Finsupp.single (rootPlane a c) ((f.erase a) c)) (rootPlane a b) =
+      (if c.val ∈ rootPlane a b then (f.erase a) c else 0)
+    rw [Finsupp.single_apply]
+    by_cases hm : c.val ∈ rootPlane a b
+    · rw [ite_eq_left ((rootPlane_eq_iff a b c hab).mpr ⟨hca,hm⟩),ite_eq_left hm]
+    · rw [ite_eq_right (fun heq => hm ((rootPlane_eq_iff a b c hab).mp heq).2),ite_eq_right hm]
+  rw [heq,Finsupp.erase_eq_sub_single,
+    Finsupp.sum_sub_index (fun c m n => by split_ifs <;> simp),
+    Finsupp.sum_single_index (by simp),ite_eq_left haP]
+
+theorem planeDefect_eq_relative_difference (x : W) (a b : PositiveRoot M cs)
+    (hab : a ≠ b) (hx : ¬ rootInverted x a) :
+    planeDefect x (rootRef a*x) a (rootPlane a b) =
+      planeLength (M := M) (cs := cs) (rootPlane a b) (rootRef a*x) - planeLength (M := M) (cs := cs) (rootPlane a b) x - 1 := by
+  rw [planeDefect,rootPlane_grouped_erase a b hab,rootChange_distinguished x a hx,
+    rootChange,Finsupp.sum_sub_index (fun c m n => by split_ifs <;> simp)]
+  rfl
+
+
+theorem planeWeight_relative_degree (x : W) (a b : PositiveRoot M cs)
+    (hab : a ≠ b) (hx : ¬ rootInverted x a) :
+    2 * planeWeight x a (rootPlane a b) =
+      planeLength (M := M) (cs := cs) (rootPlane a b) (rootRef a*x) - planeLength (M := M) (cs := cs) (rootPlane a b) x - 1 := by
+  have h := planeDefect_eq_relative_difference x a b hab hx
+  rw [planeDefect_eq_double_weight x a hx,Finsupp.add_apply] at h
+  linarith
+
+end
+end KLInvariance.TitsSpace
+
+namespace KLInvariance.EdgeAlgebra
+open Module
+universe u v
+variable {k : Type u} [Field k] {V : Type v} [AddCommGroup V] [Module k V]
+noncomputable section
+
+theorem symmetric_ι_injective : Function.Injective (SymmetricAlgebra.ι k V) := by
+  apply LinearMap.ker_eq_bot.mp
+  apply eq_bot_iff.mpr
+  intro v hv
+  change v = 0
+  by_contra hn
+  obtain ⟨f,hf⟩ := Module.Projective.exists_dual_ne_zero k hn
+  have h := congrArg (SymmetricAlgebra.lift f) (LinearMap.mem_ker.mp hv)
+  rw [SymmetricAlgebra.lift_ι_apply,map_zero] at h
+  exact hf h
+
+
+def rootIdeal (a : V) : Ideal (SymmetricAlgebra k V) :=
+  Ideal.span {SymmetricAlgebra.ι k V a}
+
+def rootLine (a : V) : Submodule k V := Submodule.span k {a}
+
+def symQuotientMap (a : V) : SymmetricAlgebra k V →ₐ[k] SymmetricAlgebra k (V ⧸ rootLine (k := k) a) :=
+  SymmetricAlgebra.lift ((SymmetricAlgebra.ι k (V ⧸ rootLine (k := k) a)).comp (rootLine (k := k) a).mkQ)
+
+@[simp] theorem symQuotientMap_ι (a v : V) :
+    symQuotientMap a (SymmetricAlgebra.ι k V v) =
+      SymmetricAlgebra.ι k (V ⧸ rootLine (k := k) a) ((rootLine (k := k) a).mkQ v) :=
+  SymmetricAlgebra.lift_ι_apply _ _
+
+theorem rootIdeal_le_ker (a : V) : rootIdeal (k := k) a ≤ RingHom.ker (symQuotientMap a) := by
+  apply Ideal.span_le.mpr
+  intro z hz
+  have hz' := Set.mem_singleton_iff.mp hz
+  subst z
+  change symQuotientMap a (SymmetricAlgebra.ι k V a) = 0
+  rw [symQuotientMap_ι]
+  have ha : (rootLine (k := k) a).mkQ a = 0 := by
+    exact (Submodule.Quotient.mk_eq_zero (rootLine (k := k) a)).mpr (Submodule.subset_span (Set.mem_singleton a))
+  rw [ha,map_zero]
+
+def quotientForward (a : V) : (SymmetricAlgebra k V ⧸ rootIdeal (k := k) a) →ₐ[k]
+    SymmetricAlgebra k (V ⧸ rootLine (k := k) a) :=
+  Ideal.Quotient.liftₐ _ (symQuotientMap a) (fun _ hz => rootIdeal_le_ker a hz)
+
+def quotientRootMap (a : V) : V ⧸ rootLine (k := k) a →ₗ[k]
+    (SymmetricAlgebra k V ⧸ rootIdeal (k := k) a) :=
+  (rootLine (k := k) a).liftQ ((Ideal.Quotient.mkₐ k (rootIdeal (k := k) a)).toLinearMap.comp (SymmetricAlgebra.ι k V)) (by
+    apply Submodule.span_le.mpr
+    intro v hv
+    have hv' := Set.mem_singleton_iff.mp hv
+    subst v
+    apply LinearMap.mem_ker.mpr
+    change Ideal.Quotient.mk (rootIdeal (k := k) a) (SymmetricAlgebra.ι k V a) = 0
+    exact Ideal.Quotient.eq_zero_iff_mem.mpr (Ideal.subset_span (Set.mem_singleton _)))
+
+def quotientBackward (a : V) : SymmetricAlgebra k (V ⧸ rootLine (k := k) a) →ₐ[k]
+    (SymmetricAlgebra k V ⧸ rootIdeal (k := k) a) := SymmetricAlgebra.lift (quotientRootMap a)
+
+@[simp] theorem quotientForward_mk (a : V) (z : SymmetricAlgebra k V) :
+    quotientForward a (Ideal.Quotient.mk (rootIdeal (k := k) a) z) = symQuotientMap a z := rfl
+
+@[simp] theorem quotientBackward_ι_mk (a v : V) :
+    quotientBackward a (SymmetricAlgebra.ι k (V ⧸ rootLine (k := k) a) ((rootLine (k := k) a).mkQ v)) =
+      Ideal.Quotient.mk (rootIdeal (k := k) a) (SymmetricAlgebra.ι k V v) := by
+  rw [quotientBackward,SymmetricAlgebra.lift_ι_apply]
+  rfl
+
+/-- Exact coefficient-ring identification used for the manuscript's C=A/(alpha).
+This is an isomorphism of the actual quotient, not a replacement invariant. -/
+def rootQuotientEquiv (a : V) : (SymmetricAlgebra k V ⧸ rootIdeal (k := k) a) ≃ₐ[k]
+    SymmetricAlgebra k (V ⧸ rootLine (k := k) a) :=
+  AlgEquiv.ofAlgHom (quotientForward a) (quotientBackward a)
+    (by
+      apply SymmetricAlgebra.algHom_ext
+      apply LinearMap.ext
+      intro v
+      obtain ⟨v,rfl⟩ := (rootLine (k := k) a).mkQ_surjective v
+      change quotientForward a (quotientBackward a (SymmetricAlgebra.ι k _ ((rootLine (k := k) a).mkQ v))) =
+        SymmetricAlgebra.ι k _ ((rootLine (k := k) a).mkQ v)
+      rw [quotientBackward_ι_mk,quotientForward_mk,symQuotientMap_ι])
+    (by
+      apply Ideal.Quotient.algHom_ext
+      apply SymmetricAlgebra.algHom_ext
+      apply LinearMap.ext
+      intro v
+      change quotientBackward a (quotientForward a (Ideal.Quotient.mk (rootIdeal (k := k) a)
+        (SymmetricAlgebra.ι k V v))) = Ideal.Quotient.mk (rootIdeal (k := k) a) (SymmetricAlgebra.ι k V v)
+      rw [quotientForward_mk,symQuotientMap_ι,quotientBackward_ι_mk])
+
+
+@[simp] theorem rootQuotientEquiv_mk_ι (a v : V) :
+    rootQuotientEquiv a (Ideal.Quotient.mk (rootIdeal (k := k) a) (SymmetricAlgebra.ι k V v)) =
+      SymmetricAlgebra.ι k (V ⧸ rootLine (k := k) a) ((rootLine (k := k) a).mkQ v) := by
+  exact symQuotientMap_ι a v
+
+theorem reducedRoot_ne_zero (a v : V) (hv : v ∉ rootLine (k := k) a) :
+    Ideal.Quotient.mk (rootIdeal (k := k) a) (SymmetricAlgebra.ι k V v) ≠ 0 := by
+  intro h
+  apply hv
+  apply (Submodule.Quotient.mk_eq_zero (rootLine (k := k) a)).mp
+  change (rootLine (k := k) a).mkQ v = 0
+  apply symmetric_ι_injective (k := k)
+  have hh := congrArg (rootQuotientEquiv a) h
+  simpa only [rootQuotientEquiv_mk_ι,map_zero] using hh
+
+variable {κ : Type*}
+theorem linear_homogeneous [Fintype κ] (b : Basis κ k V) (v : V) :
+    (SymmetricAlgebra.equivMvPolynomial b (SymmetricAlgebra.ι k V v)).IsHomogeneous 1 := by
+  classical
+  conv_lhs => rw [← b.sum_repr v]
+  simp only [map_sum,map_smul,SymmetricAlgebra.equivMvPolynomial_ι_apply]
+  apply MvPolynomial.IsHomogeneous.sum
+  intro i _
+  simpa only [Algebra.smul_def,MvPolynomial.algebraMap_eq] using (MvPolynomial.isHomogeneous_X k i).C_mul (b.repr v i)
+
+/-- Polynomial coordinates on the actual symmetric-algebra quotient. -/
+def rootQuotientPolynomial (a : V) (b : Basis κ k (V ⧸ rootLine (k := k) a)) :
+    (SymmetricAlgebra k V ⧸ rootIdeal (k := k) a) ≃ₐ[k] MvPolynomial κ k :=
+  (rootQuotientEquiv a).trans (SymmetricAlgebra.equivMvPolynomial b)
+
+theorem reducedRoot_homogeneous [Fintype κ] (a v : V)
+    (b : Basis κ k (V ⧸ rootLine (k := k) a)) :
+    (rootQuotientPolynomial a b
+      (Ideal.Quotient.mk (rootIdeal (k := k) a) (SymmetricAlgebra.ι k V v))).IsHomogeneous 1 := by
+  change (SymmetricAlgebra.equivMvPolynomial b
+    (rootQuotientEquiv a (Ideal.Quotient.mk (rootIdeal (k := k) a) (SymmetricAlgebra.ι k V v)))).IsHomogeneous 1
+  rw [rootQuotientEquiv_mk_ι]
+  exact linear_homogeneous b _
+
+end
+end KLInvariance.EdgeAlgebra
+
+namespace KLInvariance.TitsSpace
+open Module EdgeAlgebra
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+theorem planeWeight_support_root (x : W) (a : PositiveRoot M cs)
+    (P : Submodule ℝ (I → ℝ)) (hP : P ∈ (planeWeight x a).support) :
+    ∃ b : PositiveRoot M cs, b ≠ a ∧ rootPlane a b = P := by
+  obtain ⟨b,hb,hbP⟩ := Finset.mem_image.mp (Finsupp.mapDomain_support hP)
+  refine ⟨b,?_,hbP⟩
+  have hb' := Finsupp.mem_support_iff.mp hb
+  change halfVector x a b ≠ 0 at hb'
+  simp only [halfVector,rootIndicator_apply] at hb'
+  have hh : halfRoot x a b := by
+    by_contra hn
+    exact hb' (ite_eq_right hn)
+  exact hh.1.1
+
+def planeRoot (x : W) (a : PositiveRoot M cs)
+    (P : (planeWeight x a).support) : PositiveRoot M cs :=
+  (planeWeight_support_root x a P.val P.property).choose
+
+theorem planeRoot_ne (x : W) (a : PositiveRoot M cs)
+    (P : (planeWeight x a).support) : planeRoot x a P ≠ a :=
+  (planeWeight_support_root x a P.val P.property).choose_spec.1
+
+theorem planeRoot_plane (x : W) (a : PositiveRoot M cs)
+    (P : (planeWeight x a).support) : rootPlane a (planeRoot x a P) = P.val :=
+  (planeWeight_support_root x a P.val P.property).choose_spec.2
+
+theorem embed_not_rootLine (a b : PositiveRoot M cs) (hab : b ≠ a) :
+    embed M b.val ∉ rootLine (k := ℝ) (embed M a.val) := by
+  intro h
+  obtain ⟨c,hc⟩ := Submodule.mem_span_singleton.mp h
+  have hli := (linearIndependent_fin2.mp (positiveRoots_independent b a hab)).2 c
+  change c • a.val ≠ b.val at hli
+  apply hli
+  have hh := congrArg Prod.fst hc
+  simpa only [embed,Prod.smul_fst] using hh
+
+abbrev EdgeCoefficient (a : PositiveRoot M cs) :=
+  SymmetricAlgebra ℝ (Extended M) ⧸ rootIdeal (k := ℝ) (embed M a.val)
+
+def reducedPlaneRoot (x : W) (a : PositiveRoot M cs)
+    (P : (planeWeight x a).support) : EdgeCoefficient a :=
+  Ideal.Quotient.mk (rootIdeal (k := ℝ) (embed M a.val))
+    (SymmetricAlgebra.ι ℝ (Extended M) (embed M (planeRoot x a P).val))
+
+theorem reducedPlaneRoot_ne_zero (x : W) (a : PositiveRoot M cs)
+    (P : (planeWeight x a).support) : reducedPlaneRoot x a P ≠ 0 :=
+  reducedRoot_ne_zero _ _ (embed_not_rootLine a _ (planeRoot_ne x a P))
+
+/-- The manuscript's actual product of reduced root forms, with its proved
+nonnegative integral plane exponents. No abstract invariant is substituted. -/
+def edgeFactor (x : W) (a : PositiveRoot M cs) : EdgeCoefficient a :=
+  ∏ P : (planeWeight x a).support, reducedPlaneRoot x a P ^ (planeWeight x a P.val).toNat
+
+def edgeFactorDegree (x : W) (a : PositiveRoot M cs) : ℕ :=
+  ∑ P : (planeWeight x a).support, (planeWeight x a P.val).toNat
+
+def edgePolynomial (a : PositiveRoot M cs) : EdgeCoefficient a ≃ₐ[ℝ]
+    MvPolynomial (Fin (Module.finrank ℝ (Extended M ⧸ rootLine (k := ℝ) (embed M a.val)))) ℝ :=
+  rootQuotientPolynomial (embed M a.val) (Module.finBasis ℝ _)
+
+theorem edgeFactor_nonzero (x : W) (a : PositiveRoot M cs) : edgeFactor x a ≠ 0 := by
+  intro h
+  have hp := congrArg (edgePolynomial a) h
+  simp only [edgeFactor,map_prod,map_pow,map_zero] at hp
+  apply (Finset.prod_ne_zero_iff.mpr ?_) hp
+  intro P _
+  apply pow_ne_zero
+  intro h0
+  apply reducedPlaneRoot_ne_zero x a P
+  apply (edgePolynomial a).injective
+  simpa only [map_zero] using h0
+
+theorem edgeFactor_homogeneous (x : W) (a : PositiveRoot M cs) :
+    (edgePolynomial a (edgeFactor x a)).IsHomogeneous (edgeFactorDegree x a) := by
+  simp only [edgeFactor,map_prod,map_pow,edgeFactorDegree]
+  apply MvPolynomial.IsHomogeneous.prod
+  intro P _
+  have h : (edgePolynomial a (reducedPlaneRoot x a P)).IsHomogeneous 1 :=
+    reducedRoot_homogeneous _ _ _
+  simpa only [one_mul] using h.pow (planeWeight x a P.val).toNat
+
+theorem edgeFactor_totalDegree (x : W) (a : PositiveRoot M cs) :
+    (edgePolynomial a (edgeFactor x a)).totalDegree = edgeFactorDegree x a :=
+  (edgeFactor_homogeneous x a).totalDegree
+    (by
+      intro h0
+      apply edgeFactor_nonzero x a
+      apply (edgePolynomial a).injective
+      simpa only [map_zero] using h0)
+
+
+theorem edgeFactorDegree_cast (x : W) (a : PositiveRoot M cs) :
+    (edgeFactorDegree x a : ℤ) = (planeWeight x a).sum (fun _ n => n) := by
+  classical
+  simp only [edgeFactorDegree,Nat.cast_sum]
+  simp_rw [Int.toNat_of_nonneg (planeWeight_nonneg x a _)]
+  exact Finset.sum_coe_sort (planeWeight x a).support (fun P => planeWeight x a P)
+
+
+theorem edgeFactor_degree_formula (x : W) (a : PositiveRoot M cs)
+    (hx : ¬ rootInverted x a) :
+    2 * ((edgePolynomial a (edgeFactor x a)).totalDegree : ℤ) =
+      (cs.length (rootRef a*x) : ℤ) - cs.length x - 1 := by
+  rw [edgeFactor_totalDegree,edgeFactorDegree_cast]
+  exact sum_planeWeight x a hx
+
+end
+end KLInvariance.TitsSpace
+
+namespace KLInvariance.EdgeAlgebra
+open Module
+universe u v
+variable {k : Type u} [Field k] {V : Type v} [AddCommGroup V] [Module k V]
+noncomputable section
+
+theorem rootIdeal_eq_ker (a : V) : rootIdeal (k := k) a = RingHom.ker (symQuotientMap (k := k) a) := by
+  apply le_antisymm (rootIdeal_le_ker a)
+  intro z hz
+  apply Ideal.Quotient.eq_zero_iff_mem.mp
+  apply (rootQuotientEquiv (k := k) a).injective
+  change quotientForward a (Ideal.Quotient.mk (rootIdeal (k := k) a) z) = _
+  rw [quotientForward_mk,map_zero]
+  exact hz
+
+theorem rootIdeal_isPrime (a : V) : (rootIdeal (k := k) a).IsPrime := by
+  rw [rootIdeal_eq_ker]
+  exact RingHom.ker_isPrime _
+
+theorem symmetric_ι_prime (a : V) (ha : a ≠ 0) : Prime (SymmetricAlgebra.ι k V a) := by
+  have hn : SymmetricAlgebra.ι k V a ≠ 0 := by
+    intro h
+    apply ha
+    apply symmetric_ι_injective (k := k)
+    simpa only [map_zero] using h
+  exact (Ideal.span_singleton_prime hn).mp (rootIdeal_isPrime a)
+
+theorem reducedRoot_prime (a v : V) (hv : v ∉ rootLine (k := k) a) :
+    Prime (Ideal.Quotient.mk (rootIdeal (k := k) a) (SymmetricAlgebra.ι k V v)) := by
+  apply (MulEquiv.prime_iff (rootQuotientEquiv (k := k) a)).mp
+  rw [rootQuotientEquiv_mk_ι]
+  apply symmetric_ι_prime
+  intro h
+  exact hv ((Submodule.Quotient.mk_eq_zero (rootLine (k := k) a)).mp h)
+
+end
+end KLInvariance.EdgeAlgebra
+
+namespace KLInvariance.EdgeAlgebra
+open Module
+universe u v
+variable {k : Type u} [Field k] {V : Type v} [AddCommGroup V] [Module k V]
+noncomputable section
+
+theorem symmetric_ι_dvd_iff (a v : V) :
+    SymmetricAlgebra.ι k V a ∣ SymmetricAlgebra.ι k V v ↔ v ∈ rootLine (k := k) a := by
+  rw [← Ideal.mem_span_singleton]
+  change SymmetricAlgebra.ι k V v ∈ rootIdeal (k := k) a ↔ _
+  rw [rootIdeal_eq_ker]
+  change symQuotientMap a (SymmetricAlgebra.ι k V v) = 0 ↔ _
+  rw [symQuotientMap_ι]
+  constructor
+  · intro h
+    apply (Submodule.Quotient.mk_eq_zero (rootLine (k := k) a)).mp
+    change (rootLine (k := k) a).mkQ v = 0
+    apply symmetric_ι_injective (k := k)
+    simpa only [map_zero] using h
+  · intro h
+    have h0 : (rootLine (k := k) a).mkQ v = 0 :=
+      (Submodule.Quotient.mk_eq_zero (rootLine (k := k) a)).mpr h
+    rw [h0,map_zero]
+
+theorem mkQ_mem_span_iff (a b v : V) :
+    (rootLine (k := k) a).mkQ v ∈
+      Submodule.span k { (rootLine (k := k) a).mkQ b } ↔
+      v ∈ Submodule.span k {a,b} := by
+  have ha : (rootLine (k := k) a).mkQ a = 0 :=
+    (Submodule.Quotient.mk_eq_zero (rootLine (k := k) a)).mpr
+      (Submodule.subset_span (Set.mem_singleton _))
+  constructor
+  · intro h
+    obtain ⟨c,hc⟩ := Submodule.mem_span_singleton.mp h
+    have h0 : (rootLine (k := k) a).mkQ (v-c • b) = 0 := by
+      rw [map_sub,map_smul,← hc,sub_self]
+    have hm := (Submodule.Quotient.mk_eq_zero (rootLine (k := k) a)).mp h0
+    obtain ⟨d,hd⟩ := Submodule.mem_span_singleton.mp hm
+    apply Submodule.mem_span_pair.mpr
+    exact ⟨d,c,by rw [hd]; module⟩
+  · intro h
+    obtain ⟨d,c,hv⟩ := Submodule.mem_span_pair.mp h
+    apply Submodule.mem_span_singleton.mpr
+    refine ⟨c,?_⟩
+    rw [← hv,map_add,map_smul,map_smul,ha,smul_zero,zero_add]
+
+theorem reducedRoot_dvd_iff (a b v : V) :
+    Ideal.Quotient.mk (rootIdeal (k := k) a) (SymmetricAlgebra.ι k V b) ∣
+      Ideal.Quotient.mk (rootIdeal (k := k) a) (SymmetricAlgebra.ι k V v) ↔
+        v ∈ Submodule.span k {a,b} := by
+  let e := rootQuotientEquiv (k := k) a
+  have hd (z w : SymmetricAlgebra k V ⧸ rootIdeal (k := k) a) :
+      z ∣ w ↔ e z ∣ e w := by
+    constructor
+    · exact map_dvd e
+    · intro h
+      simpa only [AlgEquiv.symm_apply_apply] using map_dvd e.symm h
+  rw [hd]
+  change rootQuotientEquiv a _ ∣ rootQuotientEquiv a _ ↔ _
+  rw [rootQuotientEquiv_mk_ι,rootQuotientEquiv_mk_ι,symmetric_ι_dvd_iff]
+  exact mkQ_mem_span_iff a b v
+
+end
+end KLInvariance.EdgeAlgebra
+
+namespace KLInvariance.TitsSpace
+open Module EdgeAlgebra
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+theorem embed_mem_span_pair (a b c : I → ℝ) :
+    embed M c ∈ Submodule.span ℝ {embed M a,embed M b} ↔
+      c ∈ Submodule.span ℝ {a,b} := by
+  constructor
+  · intro h
+    obtain ⟨r,s,h⟩ := Submodule.mem_span_pair.mp h
+    refine Submodule.mem_span_pair.mpr ⟨r,s,?_⟩
+    exact congrArg Prod.fst h
+  · intro h
+    obtain ⟨r,s,h⟩ := Submodule.mem_span_pair.mp h
+    refine Submodule.mem_span_pair.mpr ⟨r,s,?_⟩
+    apply Prod.ext
+    · exact h
+    · simp only [embed,Prod.snd_add,Prod.smul_snd,smul_zero,add_zero]
+
+theorem reducedPlaneRoot_prime (x : W) (a : PositiveRoot M cs)
+    (P : (planeWeight x a).support) : Prime (reducedPlaneRoot x a P) :=
+  reducedRoot_prime _ _ (embed_not_rootLine a _ (planeRoot_ne x a P))
+
+
+theorem reducedPlaneRoot_dvd_iff (x : W) (a : PositiveRoot M cs)
+    (P Q : (planeWeight x a).support) :
+    reducedPlaneRoot x a P ∣ reducedPlaneRoot x a Q ↔ P = Q := by
+  rw [reducedPlaneRoot,reducedPlaneRoot,reducedRoot_dvd_iff,embed_mem_span_pair]
+  change (planeRoot x a Q).val ∈ rootPlane a (planeRoot x a P) ↔ P = Q
+  have h := rootPlane_eq_iff a (planeRoot x a P) (planeRoot x a Q)
+    (planeRoot_ne x a P).symm
+  rw [planeRoot_plane,planeRoot_plane] at h
+  rw [planeRoot_plane]
+  constructor
+  · intro hm
+    apply Subtype.ext
+    exact (h.mpr ⟨planeRoot_ne x a Q,hm⟩).symm
+  · intro heq
+    exact (h.mp (congrArg Subtype.val heq).symm).2
+
+end
+end KLInvariance.TitsSpace
+
+
+namespace KLInvariance.SchedulePaths
+universe u v w
+variable {V : Type u} {E : Type v} {R : Type w}
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+variable (source target : E → V)
+
+def IsPath (x y : V) : List E → Prop
+  | [] => x = y
+  | e :: p => source e = x ∧ IsPath (target e) y p
+
+variable [Semiring R]
+def pathTerm (T : R) (v : V → R) : List E → V → R
+  | [], x => v x
+  | e :: p, x => if source e = x then T * pathTerm T v p (target e) else 0
+
+/-- One elementary matrix `1+T E_xy`, acting without choosing an enumeration
+of the finite vertex set. -/
+def step (T : R) (e : E) (v : V → R) : V → R :=
+  fun x => v x + if source e = x then T * v (target e) else 0
+
+/-- Edges are listed in decreasing schedule order, so the chronologically
+last transfer is the outermost one. -/
+def dispatch (T : R) (v : V → R) : List E → V → R
+  | [] => v
+  | e :: p => step source target T e (dispatch T v p)
+
+/-- The full matrix expansion, each sublist contributing exactly one path
+term. Noncomposable sublists contribute zero by the definition of pathTerm. -/
+theorem dispatch_eq_path_sum (T : R) (v : V → R) (p : List E) (x : V) :
+    dispatch source target T v p x =
+      ((p.sublists').map (fun q => pathTerm source target T v q x)).sum := by
+  classical
+  induction p generalizing x with
+  | nil => simp [dispatch, List.sublists', pathTerm]
+  | cons e p ih =>
+    rw [dispatch,step,List.sublists'_cons,List.map_append,List.sum_append,List.map_map]
+    rw [ih x]
+    congr 1
+    by_cases he : source e = x
+    · simp only [Function.comp_def,pathTerm,he,ite_true]
+      rw [List.sum_map_mul_left,ih (target e)]
+    · simp [Function.comp_def,he,pathTerm]
+
+/-- With a delta initial value, a path contributes its length weight, not an
+invariant defined by the desired conclusion. -/
+theorem pathTerm_delta (T : R) (x y : V) (p : List E) :
+    pathTerm source target T (fun z => if z = y then 1 else 0) p x =
+      if IsPath source target x y p then T ^ p.length else 0 := by
+  classical
+  induction p generalizing x with
+  | nil => simp [pathTerm,IsPath]
+  | cons e p ih =>
+    simp only [pathTerm,IsPath,List.length_cons]
+    by_cases he : source e = x
+    · rw [ite_eq_left he,ih]
+      by_cases hp : IsPath source target (target e) y p
+      · simp only [hp,he,and_self,ite_true,pow_succ']
+      · simp [hp,he]
+    · simp [he]
+
+/-- The entry of the elementary-matrix product is exactly the sum of weights
+of directed schedule-subpaths. -/
+theorem entry_eq_weighted_paths (T : R) (p : List E) (x y : V) :
+    dispatch source target T (fun z => if z = y then 1 else 0) p x =
+      ((p.sublists').map (fun q =>
+        if IsPath source target x y q then T ^ q.length else 0)).sum := by
+  classical
+  rw [dispatch_eq_path_sum]
+  congr 1
+  exact List.map_congr_left (fun q _ => pathTerm_delta source target T x y q)
+
+
+section LabelOrder
+variable {Λ : Type*} [LinearOrder Λ] (label : E → Λ)
+
+/-- Ties in a legitimate reflection schedule cannot occur on a composable
+pair. Hence every nonzero term in the schedule expansion has strictly
+rather than merely weakly decreasing labels. -/
+theorem decreasing_of_schedule_path
+    (hties : ∀ e f, label e = label f → target e ≠ source f)
+    {p : List E} {x y : V}
+    (hp : IsPath source target x y p)
+    (hs : p.Pairwise (fun e f => label f ≤ label e)) :
+    p.Pairwise (fun e f => label f < label e) := by
+  induction p generalizing x with
+  | nil => exact List.Pairwise.nil
+  | cons e p ih =>
+    cases p with
+    | nil => simp
+    | cons f p =>
+      have ht : IsPath source target (target e) y (f :: p) := hp.2
+      have hef : label f < label e := by
+        apply lt_of_le_of_ne ((List.pairwise_cons.mp hs).1 f (List.mem_cons_self))
+        intro heq
+        exact hties e f heq.symm ht.1.symm
+      apply List.pairwise_cons.mpr
+      refine ⟨?_,ih ht hs.tail⟩
+      intro g hg
+      rcases List.mem_cons.mp hg with rfl | hg
+      · exact hef
+      · exact lt_of_le_of_lt ((List.pairwise_cons.mp hs.tail).1 g hg) hef
+
+/-- Every strictly decreasing edge list contained in a weakly decreasing
+schedule occurs as a sublist. This also handles arbitrary ordering of ties:
+a strict list can use at most one edge of each label. -/
+theorem sublist_of_decreasing_subset {p schedule : List E}
+    (hp : p.Pairwise (fun e f => label f < label e))
+    (hs : schedule.Pairwise (fun e f => label f ≤ label e))
+    (hmem : p ⊆ schedule) : List.Sublist p schedule := by
+  induction schedule generalizing p with
+  | nil =>
+    have : p = [] := List.eq_nil_iff_forall_not_mem.mpr
+      (fun a ha => List.not_mem_nil (hmem ha))
+    subst p
+    exact List.Sublist.refl []
+  | cons e schedule ih =>
+    cases p with
+    | nil => exact List.nil_sublist _
+    | cons f p =>
+      by_cases hef : e = f
+      · subst f
+        apply List.Sublist.cons_cons
+        apply ih hp.tail hs.tail
+        intro g hg
+        rcases List.mem_cons.mp (hmem (List.mem_cons_of_mem e hg)) with heq | hin
+        · have hlt := (List.pairwise_cons.mp hp).1 g hg
+          rw [heq] at hlt
+          exact (lt_irrefl _ hlt).elim
+        · exact hin
+      · apply List.Sublist.cons
+        apply ih hp hs.tail
+        intro g hg
+        have hfg : f ∈ schedule := (List.mem_cons.mp (hmem (List.mem_cons_self))).resolve_left
+          (Ne.symm hef)
+        have hfe : label f ≤ label e := (List.pairwise_cons.mp hs).1 f hfg
+        rcases List.mem_cons.mp (hmem hg) with heq | hin
+        · subst g
+          rcases List.mem_cons.mp hg with h | h
+          · exact (hef h).elim
+          · exact (not_lt_of_ge hfe ((List.pairwise_cons.mp hp).1 e h)).elim
+        · exact hin
+
+end LabelOrder
+
+end
+end KLInvariance.SchedulePaths
+
+namespace KLInvariance.ReflectionPlane
+variable {V : Type*} [AddCommGroup V] [Module ℝ V]
+
+/-- The product of three normalized reflections in a root plane is again
+an actual orthogonal reflection. The plane itself need not be nondegenerate. -/
+theorem reflection_triple_in_plane
+    (B : LinearMap.BilinForm ℝ V) (hB : B.IsSymm)
+    (a b c : V) (ha : B a a = 2) (hb : B b b = 2) (hc : B c c = 2)
+    (hplane : c ∈ Submodule.span ℝ {a,b}) :
+    ∃ d : V, ∃ hd : B d d = 2,
+      Module.reflection ha * Module.reflection hb * Module.reflection hc =
+        Module.reflection hd := by
+  obtain ⟨x,y,hxy⟩ := Submodule.mem_span_pair.mp hplane
+  subst c
+  let p := B a b
+  let r := p*x+y
+  let d := x • b - r • a
+  have hba : B b a = p := hB.eq b a
+  have hnorm : x^2+p*x*y+y^2=1 := by
+    simp only [map_add,map_smul,LinearMap.add_apply,LinearMap.smul_apply,
+      smul_eq_mul,ha,hb,hba] at hc
+    dsimp [p]
+    nlinarith [hc]
+  have hd : B d d = 2 := by
+    dsimp [d,r]
+    simp only [map_sub,map_smul,LinearMap.sub_apply,LinearMap.smul_apply,
+      smul_eq_mul,ha,hb,hba]
+    change x*(x*2-(p*x+y)*p)-(p*x+y)*(x*p-(p*x+y)*2)=2
+    nlinarith [hnorm]
+  refine ⟨d,hd,?_⟩
+  apply LinearEquiv.ext
+  intro v
+  let A := B a v
+  let C := B b v
+  have h₁ : -A+p*C+(x-p*r)*(x*A+y*C)= r*(x*C-r*A) := by
+    dsimp [r]
+    linear_combination (A-p*C)*hnorm
+  have h₂ : -C+r*(x*A+y*C)= -x*(x*C-r*A) := by
+    dsimp [r]
+    linear_combination C*hnorm
+  calc
+    (Module.reflection ha * Module.reflection hb * Module.reflection hc) v =
+        v + (-A+p*C+(x-p*r)*(x*A+y*C)) • a +
+          (-C+r*(x*A+y*C)) • b := by
+      simp only [LinearEquiv.mul_apply,Module.reflection_apply,map_sub,map_smul,
+        map_add,LinearMap.add_apply,LinearMap.smul_apply,smul_eq_mul,ha,hb,hba]
+      dsimp [A,C,r,p]
+      module
+    _ = v + (r*(x*C-r*A)) • a + (-x*(x*C-r*A)) • b := by rw [h₁,h₂]
+    _ = Module.reflection hd v := by
+      simp only [Module.reflection_apply]
+      dsimp [d]
+      simp only [map_sub,map_smul,LinearMap.sub_apply,LinearMap.smul_apply,smul_eq_mul]
+      dsimp [A,C]
+      module
+
+end KLInvariance.ReflectionPlane
+
+namespace KLInvariance.TitsSpace
+open Module
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+noncomputable section
+
+/-- A product of three genuine Coxeter reflections in one root plane is a
+Coxeter reflection, using the proved reflection faithfulness of the actual
+Tits extension. This is the sufficiency direction of graph diamond closure. -/
+theorem Represents.triple_isReflection {t s r : W} {a b c : I → ℝ}
+    (ha : Represents M cs t a) (hb : Represents M cs s b) (hc : Represents M cs r c)
+    (hplane : c ∈ Submodule.span ℝ {a,b}) : cs.IsReflection (t*s*r) := by
+  have hplane' : embed M c ∈ Submodule.span ℝ {embed M a,embed M b} := by
+    obtain ⟨x,y,hxy⟩ := Submodule.mem_span_pair.mp hplane
+    apply Submodule.mem_span_pair.mpr
+    refine ⟨x,y,?_⟩
+    ext
+    · exact congrFun hxy _
+    · simp [embed]
+  obtain ⟨d,hd,he⟩ := ReflectionPlane.reflection_triple_in_plane (extendedForm M)
+    (extendedForm_spec M).1 (embed M a) (embed M b) (embed M c)
+    (by rw [(extendedForm_spec M).2.2]; exact ha.isRoot.norm)
+    (by rw [(extendedForm_spec M).2.2]; exact hb.isRoot.norm)
+    (by rw [(extendedForm_spec M).2.2]; exact hc.isRoot.norm) hplane'
+  apply isReflection_of_moving_rank_one M cs
+  have heq : extendedAction M cs (t*s*r) = Module.reflection hd := by
+    rw [map_mul,map_mul,ha.extendedAction_eq,hb.extendedAction_eq,hc.extendedAction_eq]
+    exact he
+  rw [heq]
+  exact Realization.finrank_reflection_sub_id _ _ hd
+
+end
+end KLInvariance.TitsSpace
+
+/-! NEW GRAPH CLOSURE -/
+namespace KLInvariance
+universe u v
+variable {I : Type u} [klPreservedInstance7 : Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+noncomputable section
+
+include klPreservedInstance7 in
+ theorem BruhatStep.label_reflection {x y : W} (h : BruhatStep cs x y) :
+    cs.IsReflection (y*x⁻¹) := by
+  have _ := klPreservedInstance7
+  obtain ⟨_,t,ht,rfl⟩ := h
+  simpa [mul_assoc] using ht
+
+ theorem BruhatStep.composable_labels_ne {x y z : W}
+    (h : BruhatStep cs x y) (h' : BruhatStep cs y z) :
+    y*x⁻¹ ≠ z*y⁻¹ := by
+  intro he
+  have hh : z*x⁻¹=1 := by
+    calc
+      z*x⁻¹ = (z*y⁻¹)*(y*x⁻¹) := by group
+      _ = (y*x⁻¹)*(y*x⁻¹) := by rw [← he]
+      _ = 1 := h.label_reflection.mul_self
+  have hzx : z = x := by simpa using congrArg (fun w => w*x) hh
+  have hlen := lt_trans h.1 h'.1
+  simp [hzx] at hlen
+
+namespace TitsSpace
+open Module
+
+ def graphRoot {x y : W} (h : BruhatStep cs x y) : PositiveRoot M cs :=
+   positiveRoot M cs ⟨y*x⁻¹,h.label_reflection⟩
+
+ theorem graphRoot_represents {x y : W} (h : BruhatStep cs x y) :
+    Represents M cs (y*x⁻¹) (graphRoot h).val :=
+  positiveRoot_represents M cs ⟨y*x⁻¹,h.label_reflection⟩
+
+ theorem graphRoot_ne_of_same_source {x a b : W}
+    (ha : BruhatStep cs x a) (hb : BruhatStep cs x b) (hab : a ≠ b) :
+    graphRoot ha ≠ graphRoot hb := by
+  intro he
+  have hl := (graphRoot_represents ha).reflection_unique
+    (he.symm ▸ graphRoot_represents hb)
+  exact hab ((mul_right_cancel_iff).mp hl)
+
+ theorem graphRoot_ne_of_path {x a z : W}
+    (ha : BruhatStep cs x a) (hb : BruhatStep cs a z) :
+    graphRoot ha ≠ graphRoot hb := by
+  intro he
+  exact ha.composable_labels_ne hb ((graphRoot_represents ha).reflection_unique
+    (he.symm ▸ graphRoot_represents hb))
+
+ theorem graphRoot_square_plane {x a b z : W}
+    (hxa : BruhatStep cs x a) (hxb : BruhatStep cs x b)
+    (haz : BruhatStep cs a z) (hbz : BruhatStep cs b z) (hab : a ≠ b) :
+    rootPlane (graphRoot hxa) (graphRoot haz) =
+      rootPlane (graphRoot hxa) (graphRoot hxb) := by
+  have hp := (graphRoot_represents haz).plane_eq_of_product_eq
+    (graphRoot_represents hxa) (graphRoot_represents hbz) (graphRoot_represents hxb)
+    (Ne.symm (hxa.composable_labels_ne haz)) (Ne.symm (hxb.composable_labels_ne hbz))
+    (by group : (z*a⁻¹)*(a*x⁻¹)=(z*b⁻¹)*(b*x⁻¹))
+  have hp' := congrArg (Submodule.map
+    (LinearMap.fst ℝ (I → ℝ) (Module.Dual ℝ (LinearMap.ker (form M))))) hp
+  have hp'' : rootPlane (graphRoot haz) (graphRoot hxa) =
+      rootPlane (graphRoot hbz) (graphRoot hxb) := by
+    simpa only [rootPlane,Submodule.map_span,Set.image_insert_eq,
+      Set.image_singleton,LinearMap.fst_apply,embed] using hp'
+  have hm : (graphRoot hxb).val ∈ rootPlane (graphRoot hxa) (graphRoot haz) := by
+    rw [rootPlane,Set.pair_comm]
+    change (graphRoot hxb).val ∈ rootPlane (graphRoot haz) (graphRoot hxa)
+    rw [hp'']
+    exact Submodule.subset_span (Set.mem_insert_of_mem _ (Set.mem_singleton _))
+  exact ((rootPlane_eq_iff (graphRoot hxa) (graphRoot haz) (graphRoot hxb)
+    (graphRoot_ne_of_path hxa haz)).mpr
+    ⟨Ne.symm (graphRoot_ne_of_same_source hxa hxb hab),hm⟩).symm
+
+/-- Soundness of the eight-edge diamond closure rule for the genuine Bruhat
+reflection graph. Vertex ranks need not be consecutive; the proof uses actual
+root planes and reflection faithfulness, not a labeled graph isomorphism. -/
+ theorem bruhatStep_diamond_closure {x a b c d z : W}
+    (hxa : BruhatStep cs x a) (hxb : BruhatStep cs x b)
+    (hac : BruhatStep cs a c) (had : BruhatStep cs a d)
+    (hbc : BruhatStep cs b c) (hbd : BruhatStep cs b d)
+    (hcz : BruhatStep cs c z) (hdz : BruhatStep cs d z)
+    (hab : a ≠ b) (hcd : c ≠ d) : BruhatStep cs x z := by
+  let P := rootPlane (graphRoot hxa) (graphRoot hxb)
+  let Q := rootPlane (graphRoot hac) (graphRoot had)
+  have hPc := graphRoot_square_plane hxa hxb hac hbc hab
+  have hPd := graphRoot_square_plane hxa hxb had hbd hab
+  have hacP : (graphRoot hac).val ∈ P := by
+    change (graphRoot hac).val ∈ rootPlane (graphRoot hxa) (graphRoot hxb)
+    rw [← hPc]
+    exact Submodule.subset_span (Set.mem_insert_of_mem _ (Set.mem_singleton _))
+  have hadP : (graphRoot had).val ∈ P := by
+    change (graphRoot had).val ∈ rootPlane (graphRoot hxa) (graphRoot hxb)
+    rw [← hPd]
+    exact Submodule.subset_span (Set.mem_insert_of_mem _ (Set.mem_singleton _))
+  have hQP : Q=P := by
+    apply Submodule.eq_of_le_of_finrank_eq
+    · apply Submodule.span_le.mpr
+      intro v hv
+      rcases Set.mem_insert_iff.mp hv with rfl | hv
+      · exact hacP
+      · have hv' := Set.mem_singleton_iff.mp hv
+        rw [hv']
+        exact hadP
+    · dsimp [Q,P]
+      rw [rootPlane_finrank _ _ (graphRoot_ne_of_same_source hac had hcd),
+        rootPlane_finrank _ _ (graphRoot_ne_of_same_source hxa hxb hab)]
+  have hQc := graphRoot_square_plane hac had hcz hdz hcd
+  have hm : (graphRoot hxa).val ∈
+      Submodule.span ℝ {(graphRoot hcz).val,(graphRoot hac).val} := by
+    rw [Set.pair_comm]
+    change (graphRoot hxa).val ∈ rootPlane (graphRoot hac) (graphRoot hcz)
+    rw [hQc]
+    change (graphRoot hxa).val ∈ Q
+    rw [hQP]
+    exact Submodule.subset_span (Set.mem_insert _ _)
+  have href := (graphRoot_represents hcz).triple_isReflection
+    (graphRoot_represents hac) (graphRoot_represents hxa) hm
+  have heq : (z*c⁻¹)*(c*a⁻¹)*(a*x⁻¹)=z*x⁻¹ := by group
+  rw [heq] at href
+  exact ⟨lt_trans hxa.1 (lt_trans hac.1 hcz.1), z*x⁻¹,href,by group⟩
+
+end TitsSpace
+end
+end KLInvariance
+
+namespace KLInvariance
+universe u
+
+/-- The entirely unlabelled closure used in graph reconstruction. Its
+completeness for Bruhat graphs is a separate obligation; this definition
+neither changes Bruhat order nor defines the R/KL polynomials. -/
+inductive DiamondEdges {X : Type u} [PartialOrder X] : X → X → Prop
+  | cover {x y : X} : x ⋖ y → DiamondEdges x y
+  | diamond {x a b c d z : X} :
+      DiamondEdges x a → DiamondEdges x b →
+      DiamondEdges a c → DiamondEdges a d →
+      DiamondEdges b c → DiamondEdges b d →
+      DiamondEdges c z → DiamondEdges d z →
+      a ≠ b → c ≠ d → DiamondEdges x z
+
+ theorem DiamondEdges.map {X Y : Type*} [PartialOrder X] [PartialOrder Y]
+    (e : X ≃o Y) {x y : X} (h : DiamondEdges x y) : DiamondEdges (e x) (e y) := by
+  induction h with
+  | cover h => exact DiamondEdges.cover ((apply_covBy_apply_iff e).mpr h)
+  | diamond _ _ _ _ _ _ _ _ hab hcd hxa hxb hac had hbc hbd hcz hdz =>
+    exact DiamondEdges.diamond hxa hxb hac had hbc hbd hcz hdz
+      (fun he => hab (e.injective he)) (fun he => hcd (e.injective he))
+
+ theorem DiamondEdges.map_iff {X Y : Type*} [PartialOrder X] [PartialOrder Y]
+    (e : X ≃o Y) {x y : X} : DiamondEdges (e x) (e y) ↔ DiamondEdges x y := by
+  constructor
+  · intro h
+    simpa only [OrderIso.symm_apply_apply] using h.map e.symm
+  · exact DiamondEdges.map e
+
+ theorem bruhatStep_of_interval_covBy
+    {I W : Type*} {M : CoxeterMatrix I} [Group W] (cs : CoxeterSystem M W)
+    {u b : W} {x y : Interval cs u b} (h : x ⋖ y) : BruhatStep cs x.val y.val := by
+  rcases Relation.ReflTransGen.cases_head h.le with heq | ⟨z,hxz,hzy⟩
+  · exact (h.ne (Subtype.ext heq)).elim
+  · have hle : BruhatLE cs x.val z := Relation.ReflTransGen.single hxz
+    let v : Interval cs u b := ⟨z,bruhat_trans cs x.property.1 hle,
+      bruhat_trans cs hzy y.property.2⟩
+    have hv : v = x ∨ v = y := h.eq_or_eq hle hzy
+    rcases hv with hv | hv
+    · have heq : z = x.val := congrArg Subtype.val hv
+      exact (lt_irrefl _ (heq ▸ hxz.1)).elim
+    · have heq : z = y.val := congrArg Subtype.val hv
+      exact heq ▸ hxz
+
+
+ theorem diamondEdges_bruhatStep
+    {I W : Type*} [Fintype I] {M : CoxeterMatrix I} [Group W]
+    (cs : CoxeterSystem M W) {u b : W} {x y : Interval cs u b}
+    (h : DiamondEdges x y) : BruhatStep cs x.val y.val := by
+  induction h with
+  | cover h => exact bruhatStep_of_interval_covBy cs h
+  | diamond _ _ _ _ _ _ _ _ hab hcd hxa hxb hac had hbc hbd hcz hdz =>
+    exact TitsSpace.bruhatStep_diamond_closure hxa hxb hac had hbc hbd hcz hdz
+      (fun he => hab (Subtype.ext he)) (fun he => hcd (Subtype.ext he))
+
+end KLInvariance
+
+
+namespace KLInvariance.Dihedral
+variable {W : Type*} [Group W] {M : CoxeterMatrix Bool}
+
+theorem alt_add_two (i : Bool) (n : ℕ) :
+    alt i (n+2) = i :: (!i) :: alt i n := by
+  simp only [alt_succ,Bool.not_not]
+
+theorem alt_odd_sandwich (i : Bool) (n : ℕ) :
+    alt i (2*(n+1)+1) = i :: (alt (!i) (2*n+1) ++ [i]) := by
+  induction n generalizing i with
+  | zero => cases i <;> rfl
+  | succ n ih =>
+    have h₁ : 2*(n+1+1)+1 = (2*(n+1)+1)+2 := by omega
+    have h₂ : 2*(n+1)+1 = (2*n+1)+2 := by omega
+    rw [h₁,alt_add_two,ih,h₂,alt_add_two]
+    simp only [Bool.not_not,List.cons_append]
+
+theorem odd_alt_reflection (cs : CoxeterSystem M W) (i : Bool) (n : ℕ) :
+    cs.IsReflection (cs.wordProd (alt i (2*n+1))) := by
+  induction n generalizing i with
+  | zero => simpa [alt] using cs.isReflection_simple i
+  | succ n ih =>
+    rw [alt_odd_sandwich,cs.wordProd_cons,cs.wordProd_append]
+    obtain ⟨w,j,hw⟩ := ih (!i)
+    refine ⟨cs.simple i*w,j,?_⟩
+    simp only [cs.wordProd_cons,cs.wordProd_nil,mul_one,hw,mul_inv_rev,
+      cs.inv_simple]
+    group
+
+theorem reflection_iff_odd (cs : CoxeterSystem M W) (w : W) :
+    cs.IsReflection w ↔ Odd (cs.length w) := by
+  constructor
+  · exact fun h => h.odd_length
+  · intro h
+    obtain ⟨n,hn⟩ := h
+    obtain ⟨i,hi⟩ := exists_alt cs w
+    rw [hi,hn]
+    convert odd_alt_reflection cs i n using 2
+
+theorem step_iff_odd (cs : CoxeterSystem M W) (x y : W) :
+    BruhatStep cs x y ↔ cs.length x < cs.length y ∧ Odd (cs.length y-cs.length x) := by
+  have hm := cs.length_mul_mod_two y x⁻¹
+  rw [cs.length_inv] at hm
+  constructor
+  · rintro ⟨hlen,t,ht,hprod⟩
+    refine ⟨hlen,?_⟩
+    have hteq : y*x⁻¹=t := by rw [hprod]; group
+    have ho := ht.odd_length
+    rw [← hteq,Nat.odd_iff] at ho
+    rw [Nat.odd_iff]
+    omega
+  · rintro ⟨hlen,ho⟩
+    refine ⟨hlen,y*x⁻¹,?_,by group⟩
+    rw [reflection_iff_odd,Nat.odd_iff]
+    rw [Nat.odd_iff] at ho
+    omega
+
+theorem two_elements_length (cs : CoxeterSystem M W) {n : ℕ} (hn : 0 < n)
+    {w : W} (hw : n < cs.length w) :
+    ∃ a b : W, a ≠ b ∧ ∀ z, cs.length z = n ↔ z = a ∨ z = b := by
+  obtain ⟨i,hi⟩ := exists_alt cs w
+  have hr : cs.IsReduced (alt i (cs.length w)) := by
+    change cs.length (cs.wordProd _) = _
+    rw [← hi,length_alt]
+  have hr₁ := reduced_alt_take cs hr (show n+1 ≤ cs.length w by omega)
+  have hr₀ := reduced_alt_take cs hr (Nat.le_of_lt hw)
+  have hr₂ : cs.IsReduced (alt (!i) n) := by simpa using hr₁.drop 1
+  have hlen₀ : cs.length (cs.wordProd (alt i n)) = n := by simpa using hr₀.eq
+  have hlen₂ : cs.length (cs.wordProd (alt (!i) n)) = n := by simpa using hr₂.eq
+  refine ⟨cs.wordProd (alt i n),cs.wordProd (alt (!i) n),?_,?_⟩
+  · intro heq
+    obtain ⟨k,rfl⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : n ≠ 0)
+    have hp : cs.wordProd (alt i (k+1+1)) = cs.wordProd (alt (!i) k) := by
+      rw [alt_succ,cs.wordProd_cons,← heq,alt_succ,cs.wordProd_cons,
+        cs.simple_mul_simple_cancel_left]
+    have hlen := cs.length_wordProd_le (alt (!i) k)
+    have hred := hr₁.eq
+    rw [hp,length_alt] at hred
+    rw [length_alt] at hlen
+    omega
+  · intro z
+    constructor
+    · intro hz
+      obtain ⟨j,hj⟩ := exists_alt cs z
+      rw [hz] at hj
+      have hj' : j=i ∨ j=(!i) := by cases j <;> cases i <;> simp
+      rcases hj' with rfl|rfl
+      · exact Or.inl hj
+      · exact Or.inr hj
+    · rintro (rfl|rfl) <;> assumption
+
+theorem two_step_rank_two (cs : CoxeterSystem M W) {x y : W}
+    (hlen : cs.length y = cs.length x+2) :
+    ∃ a b : W, a ≠ b ∧
+      ∀ z, (BruhatStep cs x z ∧ BruhatStep cs z y) ↔ z=a ∨ z=b := by
+  obtain ⟨a,b,hab,hab'⟩ := two_elements_length cs (by omega : 0 < cs.length x+1)
+    (w := y) (by omega)
+  refine ⟨a,b,hab,?_⟩
+  intro z
+  rw [← hab',step_iff_odd,step_iff_odd]
+  constructor
+  · rintro ⟨⟨hx,_⟩,⟨hy,_⟩⟩
+    omega
+  · intro hz
+    constructor <;> constructor
+    · omega
+    · rw [Nat.odd_iff]; omega
+    · omega
+    · rw [Nat.odd_iff]; omega
+
+end KLInvariance.Dihedral
+
+namespace KLInvariance
+variable {I W : Type*} [Group W] {M : CoxeterMatrix I}
+
+theorem interval_covBy_of_length_succ (cs : CoxeterSystem M W)
+    {u b : W} {x y : Interval cs u b} (hxy : x ≤ y)
+    (hlen : cs.length y.val = cs.length x.val+1) : x ⋖ y := by
+  apply covBy_iff_lt_and_eq_or_eq.mpr
+  refine ⟨lt_iff_le_and_ne.mpr ⟨hxy,?_⟩,?_⟩
+  · intro heq
+    have := congrArg (fun z : Interval cs u b => cs.length z.val) heq
+    omega
+  · intro z hxz hzy
+    by_cases hz : z=x
+    · exact Or.inl hz
+    · right
+      by_contra hzy'
+      have h₁ := length_lt_of_bruhat_ne cs hxz (fun he => hz (Subtype.ext he.symm))
+      have h₂ := length_lt_of_bruhat_ne cs hzy (fun he => hzy' (Subtype.ext he))
+      omega
+
+namespace Dihedral
+variable {M₂ : CoxeterMatrix Bool}
+
+theorem two_interval_elements_length (cs : CoxeterSystem M₂ W)
+    {u b : W} (x y : Interval cs u b) {n : ℕ}
+    (hn₁ : cs.length x.val < n) (hn₂ : n < cs.length y.val) :
+    ∃ a c : Interval cs u b, a ≠ c ∧ cs.length a.val=n ∧ cs.length c.val=n := by
+  obtain ⟨a,c,hac,hlevel⟩ := two_elements_length cs (by omega : 0<n) hn₂
+  have ha := (hlevel a).mpr (Or.inl rfl)
+  have hc := (hlevel c).mpr (Or.inr rfl)
+  have hmem (z : W) (hz : cs.length z=n) :
+      BruhatLE cs u z ∧ BruhatLE cs z b :=
+    ⟨bruhat_trans cs x.property.1 (shorter_bruhat cs (by omega)),
+      bruhat_trans cs (shorter_bruhat cs (by omega)) y.property.2⟩
+  exact ⟨⟨a,hmem a ha⟩,⟨c,hmem c hc⟩,
+    fun he => hac (congrArg Subtype.val he),ha,hc⟩
+
+/-- Completeness of the actual unlabelled eight-edge closure for genuine
+rank-two Coxeter systems. This is a necessary local case of Dyer's general
+reconstruction, not a replacement main theorem. -/
+theorem diamondEdges_of_bruhatStep (cs : CoxeterSystem M₂ W)
+    {u b : W} {x y : Interval cs u b} (hxy : BruhatStep cs x.val y.val) :
+    DiamondEdges x y := by
+  suffices ∀ n : ℕ, ∀ x y : Interval cs u b,
+      cs.length y.val-cs.length x.val=n → BruhatStep cs x.val y.val →
+      DiamondEdges x y by
+    exact this _ x y rfl hxy
+  intro n
+  induction n using Nat.strong_induction_on with
+  | h n ih =>
+    intro x y hlen hxy
+    have hodd := ((step_iff_odd cs x.val y.val).mp hxy).2
+    rw [hlen,Nat.odd_iff] at hodd
+    by_cases hn : n=1
+    · apply DiamondEdges.cover
+      exact interval_covBy_of_length_succ cs (Relation.ReflTransGen.single hxy) (by omega)
+    · have hn3 : 3 ≤ n := by omega
+      obtain ⟨a,a₂,hab,ha,hb⟩ := two_interval_elements_length cs x y
+        (n := cs.length x.val+1) (by omega) (by have := hxy.1; omega)
+      obtain ⟨c,d,hcd,hc,hd⟩ := two_interval_elements_length cs x y
+        (n := cs.length x.val+2) (by omega) (by have := hxy.1; omega)
+      have hcover {p q : Interval cs u b}
+          (hpq : cs.length q.val=cs.length p.val+1) : DiamondEdges p q :=
+        DiamondEdges.cover (interval_covBy_of_length_succ cs
+          (shorter_bruhat cs (by omega)) hpq)
+      have htop {p : Interval cs u b} (hp : cs.length p.val=cs.length x.val+2) :
+          DiamondEdges p y := by
+        apply ih (cs.length y.val-cs.length p.val) (by have := hxy.1; omega) p y rfl
+        rw [step_iff_odd]
+        constructor
+        · have := hxy.1; omega
+        · rw [Nat.odd_iff]; have := hxy.1; omega
+      exact DiamondEdges.diamond (hcover (by omega)) (hcover (by omega))
+        (hcover (by omega)) (hcover (by omega)) (hcover (by omega)) (hcover (by omega))
+        (htop hc) (htop hd) hab hcd
+
+end Dihedral
+end KLInvariance
+
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/KazhdanLusztig/SheafMorphisms.lean b/lean/OAI/RepresentationTheory/KazhdanLusztig/SheafMorphisms.lean
index 824de8f4aa231541bbd8a87fa55219f2cd317781..ceeda8ada743b5ee2d5110a2e8fca9aa8ed260fd 100644
--- a/lean/OAI/RepresentationTheory/KazhdanLusztig/SheafMorphisms.lean
+++ b/lean/OAI/RepresentationTheory/KazhdanLusztig/SheafMorphisms.lean
@@ -1,1854 +1,1854 @@
-import OAI.RepresentationTheory.KazhdanLusztig.MinimalExtensions
-
-namespace OAI
-
-/-!
-Graded sheaf morphisms, exact section constructions and character sums.
--/
-
-section
-
-namespace KLInvariance.Graded
-open Polynomial
-universe uk us um ui
-variable {k : Type uk} [Field k] {σ : Type us} [Finite σ]
-  {M : Type um} [AddCommGroup M] [Module k M] [Module (MvPolynomial σ k) M]
-  [IsScalarTower k (MvPolynomial σ k) M]
-  (𝓜 : ℤ → Submodule k M) [DirectSum.Decomposition 𝓜]
-  [SetLike.GradedSMul (polynomialGrading k σ) 𝓜]
-
-structure NonnegativeBasis where
-  index : Type um
-  finite : Fintype index
-  basis : Module.Basis index (MvPolynomial σ k) M
-  degree : index → ℕ
-  homogeneous : ∀ i, basis i ∈ 𝓜 (degree i)
-
-attribute [instance] NonnegativeBasis.finite
-
-theorem exists_nonnegativeBasis [Module.Free (MvPolynomial σ k) M]
-    [Module.Finite (MvPolynomial σ k) M] (hneg : ∀ n < 0, 𝓜 n = ⊥) :
-    Nonempty (NonnegativeBasis (σ := σ) 𝓜) := by
-  classical
-  obtain ⟨ι,hι,b,hb⟩ := exists_homogeneous_basis_polynomial (σ := σ) 𝓜 0 hneg
-  let := hι
-  let : Fintype ι := Fintype.ofFinite ι
-  choose d hd using hb
-  have hd0 (i : ι) : 0 ≤ d i := by
-    by_contra h
-    have hh := hd i
-    rw [hneg _ (lt_of_not_ge h),Submodule.mem_bot] at hh
-    exact b.ne_zero i hh
-  exact ⟨⟨ι,inferInstance,b,fun i => (d i).toNat,fun i => by simpa [Int.toNat_of_nonneg (hd0 i)] using hd i⟩⟩
-
-namespace NonnegativeBasis
-variable {𝓜} (b : NonnegativeBasis (σ := σ) 𝓜)
-
-noncomputable def polynomial : ℤ[X] := ∑ i, X ^ b.degree i
-
-omit [Finite σ] [IsScalarTower k (MvPolynomial σ k) M]
-  [DirectSum.Decomposition 𝓜] [SetLike.GradedSMul (polynomialGrading k σ) 𝓜] in
-lemma coeff_nonneg (n : ℕ) : 0 ≤ b.polynomial.coeff n := by
-  classical
-  simp only [polynomial,Polynomial.finsetSum_coeff,Polynomial.coeff_X_pow]
-  exact Finset.sum_nonneg (fun i _ => by split <;> norm_num)
-
-omit [Finite σ] [IsScalarTower k (MvPolynomial σ k) M]
-  [DirectSum.Decomposition 𝓜] [SetLike.GradedSMul (polynomialGrading k σ) 𝓜] in
-lemma eval₂_eq (q : ℝ) : Polynomial.eval₂ (Int.castRingHom ℝ) q b.polynomial =
-    ∑ i, q ^ b.degree i := by
-  change (Polynomial.eval₂RingHom (Int.castRingHom ℝ) q) (∑ i, X ^ b.degree i) = _
-  simp
-
-omit [Finite σ] in
-lemma hasSum [Fintype σ] {q : ℝ} (hq : 0 < q) (hq1 : q < 1) :
-    HasSum (fun n : ℤ => (Module.finrank k (𝓜 n) : ℝ)*q^n)
-      (Polynomial.eval₂ (Int.castRingHom ℝ) q b.polynomial * (1-q)⁻¹ ^ Fintype.card σ) := by
-  rw [b.eval₂_eq]
-  convert Hilbert.hasSum_homogeneous_basis 𝓜 b.basis (fun i => (b.degree i : ℤ))
-    b.homogeneous hq hq1 using 1
-  simp
-
-lemma polynomial_eq (c : NonnegativeBasis (σ := σ) 𝓜) : b.polynomial = c.polynomial := by
-  classical
-  let : Fintype σ := Fintype.ofFinite σ
-  apply Polynomial.map_injective (Int.castRingHom ℝ) Int.cast_injective
-  apply Polynomial.eq_of_infinite_eval_eq
-  have hsub : Set.Ioo (0 : ℝ) 1 ⊆ {q | Polynomial.eval q (b.polynomial.map (Int.castRingHom ℝ)) =
-      Polynomial.eval q (c.polynomial.map (Int.castRingHom ℝ))} := by
-    intro q hq
-    have h := (b.hasSum hq.1 hq.2).unique (c.hasSum hq.1 hq.2)
-    have hne : (1-q)⁻¹ ^ Fintype.card σ ≠ 0 := pow_ne_zero _ (inv_ne_zero (sub_ne_zero.mpr (ne_of_gt hq.2)))
-    change Polynomial.eval q (b.polynomial.map (Int.castRingHom ℝ)) =
-      Polynomial.eval q (c.polynomial.map (Int.castRingHom ℝ))
-    simpa only [Polynomial.eval_map] using mul_right_cancel₀ hne h
-  exact Set.Infinite.mono hsub (Set.Ioo_infinite (show (0 : ℝ) < 1 by norm_num))
-
-end NonnegativeBasis
-
-noncomputable def character [Module.Free (MvPolynomial σ k) M]
-    [Module.Finite (MvPolynomial σ k) M] (hneg : ∀ n < 0, 𝓜 n = ⊥) : ℤ[X] :=
-  (exists_nonnegativeBasis (σ := σ) 𝓜 hneg).some.polynomial
-
-lemma character_coeff_nonneg [Module.Free (MvPolynomial σ k) M]
-    [Module.Finite (MvPolynomial σ k) M] (hneg : ∀ n < 0, 𝓜 n = ⊥) (n : ℕ) :
-    0 ≤ (character (σ := σ) 𝓜 hneg).coeff n :=
-  (exists_nonnegativeBasis (σ := σ) 𝓜 hneg).some.coeff_nonneg n
-
-lemma character_eq [Module.Free (MvPolynomial σ k) M]
-    [Module.Finite (MvPolynomial σ k) M] (hneg : ∀ n < 0, 𝓜 n = ⊥)
-    (b : NonnegativeBasis (σ := σ) 𝓜) : character (σ := σ) 𝓜 hneg = b.polynomial :=
-  (exists_nonnegativeBasis (σ := σ) 𝓜 hneg).some.polynomial_eq b
-
-end KLInvariance.Graded
-
-end
-
-
-section
-
-namespace KLInvariance.Graded
-universe uk us
-variable (k : Type uk) [Field k] (σ : Type us)
-
- theorem polynomial_scalar_mod_positive (a : MvPolynomial σ k) :
-    ∃ c : k, a-algebraMap k (MvPolynomial σ k) c ∈ positiveIdeal (polynomialGrading k σ) := by
-  obtain ⟨c,hc⟩ := polynomialGrading_zero k σ _
-    (DirectSum.decompose (polynomialGrading k σ) a 0).property
-  refine ⟨c,?_⟩
-  rw [positiveIdeal_eq_augmentation_ker _ (polynomialGrading_negative k σ)]
-  change component (polynomialGrading k σ) 0 (a-algebraMap k (MvPolynomial σ k) c)=0
-  rw [map_sub,hc]
-  rw [show component (polynomialGrading k σ) 0
-    (DirectSum.decompose (polynomialGrading k σ) a 0 : MvPolynomial σ k) =
-    (DirectSum.decompose (polynomialGrading k σ) a 0 : MvPolynomial σ k) from
-    DirectSum.decompose_of_mem_same _ (DirectSum.decompose (polynomialGrading k σ) a 0).property]
-  exact sub_self _
-
-end KLInvariance.Graded
-
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
-
-abbrev Coefficient := MvPolynomial σ ℝ
-
-def root (e : Edge cs u b) : PositiveRoot M cs :=
-  positiveRootEquiv M cs ⟨label cs u b e,label_isReflection cs u b e⟩
-
-def polynomialLabel (e : Edge cs u b) : Coefficient (σ := σ) :=
-  SymmetricAlgebra.equivMvPolynomial basis
-    (SymmetricAlgebra.ι ℝ (Extended M) (embed M (root cs u b e).val))
-
- theorem polynomialLabel_homogeneous (e : Edge cs u b) :
-    polynomialLabel cs u b basis e ∈ polynomialGrading ℝ σ 1 := by
-  classical
-  unfold polynomialLabel
-  rw [← basis.sum_repr (embed M (root cs u b e).val)]
-  simp only [map_sum,map_smul,SymmetricAlgebra.equivMvPolynomial_ι_apply]
-  apply Submodule.sum_mem
-  intro i _
-  exact (polynomialGrading ℝ σ 1).smul_mem _
-    (MvPolynomial.isWeightedHomogeneous_X ℝ (fun _ : σ => (1 : ℤ)) i)
-
-omit [Fintype σ] in
- theorem polynomialLabel_ne_zero (e : Edge cs u b) :
-    polynomialLabel cs u b basis e ≠ 0 := by
-  intro he
-  have h0 : SymmetricAlgebra.ι ℝ (Extended M) (embed M (root cs u b e).val) = 0 :=
-    (SymmetricAlgebra.equivMvPolynomial basis).injective
-      (he.trans (map_zero _).symm)
-  have h1 : embed M (root cs u b e).val = 0 :=
-    EdgeAlgebra.symmetric_ι_injective (k := ℝ) (h0.trans (map_zero _).symm)
-  have h2 := congrArg Prod.fst h1
-  exact (root cs u b e).property.1.ne_zero h2
-
-
- theorem exists_boundarySheaf (hub : BruhatLE cs u b) :
-    Nonempty (BoundarySheaf (polynomialGrading_negative ℝ σ)
-      (graph cs u b) (polynomialLabel cs u b basis) ⟨b,hub,bruhat_refl cs b⟩) := by
-  classical
-  let : Finite (Interval cs u b) := interval_finite cs u b
-  let : Fintype (Interval cs u b) := Fintype.ofFinite _
-  let : Fintype (Edge cs u b) := Fintype.ofFinite _
-  exact MomentGraph.exists_boundarySheaf (polynomialGrading_negative ℝ σ)
-    (graph cs u b) (polynomial_scalar_mod_positive ℝ σ)
-    ⟨b,hub,bruhat_refl cs b⟩ (fun x => x.property.2)
-    (polynomialLabel cs u b basis) (polynomialLabel_homogeneous cs u b basis)
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  (hub : BruhatLE cs u b)
-noncomputable section
-
-def boundarySheaf : BoundarySheaf (polynomialGrading_negative ℝ σ)
-    (graph cs u b) (polynomialLabel cs u b basis) ⟨b,hub,bruhat_refl cs b⟩ :=
-  (exists_boundarySheaf cs u b basis hub).some
-
-def stalkCharacter (x : Interval cs u b) : Polynomial ℤ := by
-  let B := boundarySheaf cs u b basis hub
-  let := B.free x
-  exact Graded.character (σ := σ) (B.sheaf.vertex x).piece (B.sheaf.vertex x).nonneg
-
- theorem stalkCharacter_coeff_nonneg (x : Interval cs u b) (n : ℕ) :
-    0 ≤ (stalkCharacter cs u b basis hub x).coeff n := by
-  let B := boundarySheaf cs u b basis hub
-  let := B.free x
-  exact Graded.character_coeff_nonneg (σ := σ) (B.sheaf.vertex x).piece
-    (B.sheaf.vertex x).nonneg n
-
-
-def StalkCharacterObligation : Prop :=
-  ∀ x : Interval cs u b, stalkCharacter cs u b basis hub x = klPolynomial cs x.val b
-
-
-def CostalkFreeObligation : Prop :=
-  ∀ x : Interval cs u b, Module.Free (Coefficient (σ := σ))
-    (LinearMap.ker ((boundarySheaf cs u b basis hub).sheaf.forget.stalkMap x))
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-/-! Parity matching in finite lower Bruhat intervals, used only to construct
-actual dihedral interpolation sections. -/
-namespace KLInvariance
-universe u v
-variable {I : Type u} {M : CoxeterMatrix I} {W : Type v} [Group W]
-  (cs : CoxeterSystem M W)
-noncomputable section
-
- theorem simple_mul_bruhat_of_descent {c : W} (i : I)
-    (hi : cs.length (cs.simple i*c) < cs.length c) {z : W} (hz : BruhatLE cs z c) :
-    BruhatLE cs (cs.simple i*z) c := by
-  have h := le_upperSimple cs i (cs.simple i*z)
-  rw [upperSimple_mul] at h
-  have htop : upperSimple cs i c = c := by simp only [upperSimple,ite_eq_left hi]
-  rw [← htop]
-  exact bruhat_trans cs h (upperSimple_mono cs i hz)
-
- theorem lower_parity_balance {c : W} (hc : c ≠ 1) {p : ℕ} (hp : p < 2) :
-    ((lowerFinset cs c).filter (fun z => cs.length z % 2 = p)).card =
-      ((lowerFinset cs c).filter (fun z => cs.length z % 2 ≠ p)).card := by
-  classical
-  obtain ⟨i,hi⟩ := cs.exists_leftDescent_of_ne_one hc
-  have hmem {z : W} (hz : z ∈ lowerFinset cs c) : cs.simple i*z ∈ lowerFinset cs c :=
-    (mem_lowerFinset cs c _).mpr (simple_mul_bruhat_of_descent cs i hi
-      ((mem_lowerFinset cs c _).mp hz))
-  refine Finset.card_bij (fun z _ => cs.simple i*z) ?_ ?_ ?_
-  · intro z hz
-    obtain ⟨hz,hp'⟩ := Finset.mem_filter.mp hz
-    refine Finset.mem_filter.mpr ⟨hmem hz,?_⟩
-    have := cs.length_simple_mul z i
-    omega
-  · intro z hz w hw he
-    exact mul_left_cancel he
-  · intro z hz
-    obtain ⟨hz,hp'⟩ := Finset.mem_filter.mp hz
-    refine ⟨cs.simple i*z,Finset.mem_filter.mpr ⟨hmem hz,?_⟩,
-      cs.simple_mul_simple_cancel_left i⟩
-    have := cs.length_simple_mul z i
-    omega
-
- def externalLower (x c : W) : Finset W := by
-  classical
-  exact (lowerFinset cs c) \ (lowerFinset cs x)
-
-@[simp] theorem mem_externalLower (x c z : W) :
-    z ∈ externalLower cs x c ↔ BruhatLE cs z c ∧ ¬ BruhatLE cs z x := by
-  classical
-  simp [externalLower]
-
- theorem one_bruhat (x : W) : BruhatLE cs 1 x := by
-  obtain ⟨ω,hω,hx⟩ := cs.exists_isReduced x
-  simpa only [cs.wordProd_nil,← hx] using bruhat_of_subword cs (List.nil_sublist ω) hω
-
- theorem external_parity_card_le {x c : W} (hxc : BruhatLE cs x c) :
-    ((externalLower cs x c).filter (fun z => cs.length z % 2 = cs.length x % 2)).card ≤
-      ((externalLower cs x c).filter (fun z => cs.length z % 2 ≠ cs.length x % 2)).card := by
-  classical
-  by_cases hc : c = 1
-  · subst c
-    have hx : x = 1 := bruhat_antisymm cs hxc (one_bruhat cs x)
-    subst x
-    simp [externalLower]
-  have hp : cs.length x % 2 < 2 := Nat.mod_lt _ (by omega)
-  have hbal := lower_parity_balance cs hc hp
-  have hsub : lowerFinset cs x ⊆ lowerFinset cs c := by
-    intro z hz
-    exact (mem_lowerFinset cs c z).mpr (((mem_lowerFinset cs x z).mp hz).trans hxc)
-  have hs₀ : (lowerFinset cs x).filter (fun z => cs.length z % 2 = cs.length x % 2) ⊆
-      (lowerFinset cs c).filter (fun z => cs.length z % 2 = cs.length x % 2) :=
-    Finset.filter_subset_filter _ hsub
-  have hs₁ : (lowerFinset cs x).filter (fun z => cs.length z % 2 ≠ cs.length x % 2) ⊆
-      (lowerFinset cs c).filter (fun z => cs.length z % 2 ≠ cs.length x % 2) :=
-    Finset.filter_subset_filter _ hsub
-  have heq (p : W → Prop) [DecidablePred p] :
-      (externalLower cs x c).filter p = (lowerFinset cs c).filter p \ (lowerFinset cs x).filter p := by
-    ext z
-    simp only [externalLower,Finset.mem_filter,Finset.mem_sdiff]
-    tauto
-  rw [heq,heq,Finset.card_sdiff_of_subset hs₀,Finset.card_sdiff_of_subset hs₁]
-  rw [hbal]
-  apply Nat.sub_le_sub_left
-  by_cases hx : x = 1
-  · subst x
-    have hh : lowerFinset cs (1:W) = {1} := by
-      ext z
-      simp only [mem_lowerFinset,Finset.mem_singleton]
-      exact ⟨fun h => bruhat_antisymm cs h (one_bruhat cs z),fun h => h ▸ bruhat_refl cs 1⟩
-    simp [hh,Finset.filter_singleton]
-  · exact (lower_parity_balance cs hx hp).symm.le
-
-end
-end KLInvariance
-
-end
-
-
-section
-
-/-! Every lower-supported model is an actual Bruhat BMP sheaf on [1,c],
-pulled back by the genuine lower-interval graph isomorphism and extended by
-zero. Thus its stalk character is not an abstract model selected for convenience. -/
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
-universe u
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W) (b : W)
-  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-noncomputable section
-local instance (p : Prop) : Decidable p := Classical.propDecidable p
-
- abbrev lowerIdealGraph (c : Interval cs 1 b) :=
-  lowerGraph (graph cs 1 b) (Set.Iic c) (isLowerSet_Iic c)
-
- def lowerIntervalMap (c : Interval cs 1 b) :
-    GraphMap (lowerIdealGraph cs b c) (graph cs 1 c.val) where
-  vertex x := ⟨x.val.val,x.val.property.1,x.property⟩
-  edge e := ⟨(⟨(source cs 1 b e.val).val,(source cs 1 b e.val).property.1,
-      bruhat_trans cs (.single e.val.property) e.property⟩,
-    ⟨(target cs 1 b e.val).val,(target cs 1 b e.val).property.1,e.property⟩),e.val.property⟩
-  source _ := rfl
-  target _ := rfl
-
- omit [Fintype I] in
- theorem lowerIntervalMap_injective (c : Interval cs 1 b) :
-    Function.Injective (lowerIntervalMap cs b c).vertex := by
-  intro x y h
-  exact Subtype.ext (Subtype.ext (show x.val.val=y.val.val from
-    congrArg (fun z : Interval cs 1 c.val => z.val) h))
-
- omit [Fintype I] in
- theorem lowerIntervalMap_outgoing (c : Interval cs 1 b)
-    (e : Edge cs 1 c.val) (x : Set.Iic c)
-    (hx : (lowerIntervalMap cs b c).vertex x=source cs 1 c.val e) :
-    ∃ a, (lowerIntervalMap cs b c).edge a=e ∧ (lowerIdealGraph cs b c).source a=x := by
-  have hxx := congrArg Subtype.val hx
-  let y : Interval cs 1 b := ⟨(target cs 1 c.val e).val,(target cs 1 c.val e).property.1,
-    bruhat_trans cs (target cs 1 c.val e).property.2 c.property.2⟩
-  let a : Edge cs 1 b := ⟨(x.val,y),hxx.symm ▸ e.property⟩
-  refine ⟨⟨a,(target cs 1 c.val e).property.2⟩,?_,?_⟩
-  · apply Subtype.ext
-    exact Prod.ext hx rfl
-  · rfl
-
- def lowerIntervalBoundary (c : Interval cs 1 b) :
-    BoundarySheaf (polynomialGrading_negative ℝ σ) (lowerIdealGraph cs b c)
-      (fun e => polynomialLabel cs 1 b basis e.val) ⟨c,Set.mem_Iic.mpr le_rfl⟩ :=
-  (boundarySheaf cs 1 c.val basis (one_bruhat cs c.val)).pullback
-    (lowerIntervalMap cs b c) (lowerIntervalMap_injective cs b c)
-    (lowerIntervalMap_outgoing cs b c) ⟨c,Set.mem_Iic.mpr le_rfl⟩ rfl
-
- def boundaryModels : GradedSheaf.BoundaryModels (polynomialGrading_negative ℝ σ)
-    (G := graph cs 1 b) (polynomialLabel cs 1 b basis) where
-  sheaf c := (lowerIntervalBoundary cs b basis c).sheaf.lowerExtend
-  free c := (lowerIntervalBoundary cs b basis c).sheaf.lowerExtend_free
-    (lowerIntervalBoundary cs b basis c).free
-  quotient c := (lowerIntervalBoundary cs b basis c).sheaf.lowerExtend_quotient _
-    (lowerIntervalBoundary cs b basis c).quotient
-  generated c := (lowerIntervalBoundary cs b basis c).sheaf.lowerExtend_generated
-    (lowerIntervalBoundary cs b basis c).generated
-  flabby c := (lowerIntervalBoundary cs b basis c).sheaf.lowerExtend_flabby
-    (lowerIntervalBoundary cs b basis c).flabby.flabby
-  support c x hx := (lowerIntervalBoundary cs b basis c).sheaf.lowerExtend_support x hx
-  top c := ((lowerIntervalBoundary cs b basis c).sheaf.lowerExtendVertex ⟨c,Set.mem_Iic.mpr le_rfl⟩).trans
-    (ModuleData.equivOfEq (lowerIntervalBoundary cs b basis c).top)
-  top_graded c n v hv := ModuleData.equivOfEq_graded (lowerIntervalBoundary cs b basis c).top n _
-    ((lowerIntervalBoundary cs b basis c).sheaf.lowerExtendVertex_graded ⟨c,Set.mem_Iic.mpr le_rfl⟩ n v hv)
-  minimal c x hx := (lowerIntervalBoundary cs b basis c).sheaf.lowerExtend_minimal x
-    (fun hxc => (lowerIntervalBoundary cs b basis c).minimal ⟨x,hxc⟩
-      (fun hh => hx (congrArg Subtype.val hh)))
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-namespace KLInvariance.Graded
-universe uk ua um ui
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐]
-  {M : Type um} [AddCommGroup M] [Module A M] [Module k M] [IsScalarTower k A M]
-  (𝓜 : ℤ → Submodule k M) [DirectSum.Decomposition 𝓜] [SetLike.GradedSMul 𝓐 𝓜]
-noncomputable section
-
-omit [IsScalarTower k A M] in
- theorem basis_coordinates_graded {ι : Type ui} [Fintype ι]
-    (b : Module.Basis ι A M) (d : ι → ℤ) (hd : ∀ i, b i ∈ 𝓜 (d i))
-    (n : ℤ) (m : M) (hm : m ∈ 𝓜 n) :
-    ∀ i, b.equivFun m i ∈ 𝓐 (n-d i) := by
-  classical
-  let : DirectSum.Decomposition (shiftedPiece 𝓐 d) := (shifted_isInternal 𝓐 d).chooseDecomposition
-  have he : ∀ n v, v ∈ shiftedPiece 𝓐 d n → b.equivFun.symm v ∈ 𝓜 n := by
-    intro n v hv
-    rw [b.equivFun_symm_apply]
-    apply Submodule.sum_mem
-    intro i _
-    simpa using SetLike.GradedSMul.smul_mem (hv i) (hd i)
-  have hh := map_component (shiftedPiece 𝓐 d) 𝓜 b.equivFun.symm.toLinearMap 0
-    (by simpa using he) n (b.equivFun m)
-  simp only [sub_zero,LinearEquiv.coe_toLinearMap,LinearEquiv.symm_apply_apply] at hh
-  rw [show component 𝓜 n m=m from DirectSum.decompose_of_mem_same 𝓜 hm] at hh
-  have hc : component (shiftedPiece 𝓐 d) n (b.equivFun m)=b.equivFun m := by
-    apply b.equivFun.symm.injective
-    exact hh.symm.trans (b.equivFun.symm_apply_apply m).symm
-  rw [← hc]
-  exact (DirectSum.decompose (shiftedPiece 𝓐 d) (b.equivFun m) n).property
-
-end
-end KLInvariance.Graded
-
-namespace KLInvariance.Graded.ModuleData
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-  [IsDomain A] [IsNoetherianRing A]
-  (hneg : ∀ n < 0, 𝓐 n = ⊥)
-  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
-noncomputable section
-
-include hneg hzero in
- theorem exists_equiv_shiftedUnits (F : ModuleData.{uk,ua,um} 𝓐) [Module.Free A F] :
-    ∃ (n : ℕ) (d : ULift.{um} (Fin n) → ℕ)
-      (e : F ≃ₗ[A] pi (fun i : ULift.{um} (Fin n) => (unit hneg).shift (d i))),
-      ∀ m v, v ∈ F.piece m → e v ∈
-        (pi (fun i : ULift.{um} (Fin n) => (unit hneg).shift (d i))).piece m := by
-  classical
-  obtain ⟨ι,hι,b0,hd0⟩ := exists_homogeneous_basis_connected 𝓐 F.piece hneg hzero 0 F.nonneg
-  let : Finite ι := hι
-  let : Fintype ι := Fintype.ofFinite ι
-  let n := Fintype.card ι
-  let J := ULift.{um} (Fin n)
-  let re : ι ≃ J := (Fintype.equivFin ι).trans Equiv.ulift.symm
-  let b := b0.reindex re
-  choose dz hdz using hd0
-  let d : J → ℕ := fun i => (dz (re.symm i)).toNat
-  have hd (i : J) : b i ∈ F.piece (d i : ℤ) := by
-    have hz : 0 ≤ dz (re.symm i) := by
-      by_contra h
-      have he : b0 (re.symm i)=0 := by
-        simpa only [F.nonneg _ (lt_of_not_ge h),Submodule.mem_bot] using hdz (re.symm i)
-      exact b0.ne_zero (re.symm i) he
-    simpa only [b,Module.Basis.reindex_apply,d,Int.toNat_of_nonneg hz] using hdz (re.symm i)
-  let e : F ≃ₗ[A] pi (fun i : J => (unit hneg).shift (d i)) :=
-    { toFun v i _ := b.equivFun v i
-      invFun v := b.equivFun.symm (fun i => v i PUnit.unit)
-      left_inv v := b.equivFun.symm_apply_apply v
-      right_inv v := by
-        funext i j
-        change b.equivFun (b.equivFun.symm (fun i => v i PUnit.unit)) i=v i j
-        rw [LinearEquiv.apply_symm_apply]
-      map_add' v w := by funext i j; exact congrFun (b.equivFun.map_add v w) i
-      map_smul' a v := by funext i j; exact congrFun (b.equivFun.map_smul a v) i }
-  refine ⟨n,d,e,?_⟩
-  intro m v hv i _ j
-  change b.equivFun v i ∈ 𝓐 (m-(d i : ℤ)-0)
-  rw [sub_zero]
-  exact basis_coordinates_graded F.piece b (fun i => (d i : ℤ)) hd m v hv i
-
-end
-end KLInvariance.Graded.ModuleData
-
-end
-
-
-section
-
-namespace KLInvariance.QuotientEdge
-variable {R M N E F : Type*} [CommRing R]
-  [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N]
-  [AddCommGroup E] [Module R E] [AddCommGroup F] [Module R F]
-noncomputable section
-
- def descend (f : M →ₗ[R] E) (g : N →ₗ[R] F)
-    (hf : Function.Surjective f) (h : M →ₗ[R] N)
-    (hk : LinearMap.ker f ≤ LinearMap.ker (g.comp h)) : E →ₗ[R] F :=
-  ((LinearMap.ker f).liftQ (g.comp h) hk).comp (f.quotKerEquivOfSurjective hf).symm.toLinearMap
-
- theorem descend_apply (f : M →ₗ[R] E) (g : N →ₗ[R] F)
-    (hf : Function.Surjective f) (h : M →ₗ[R] N)
-    (hk : LinearMap.ker f ≤ LinearMap.ker (g.comp h)) (v : M) :
-    descend f g hf h hk (f v) = g (h v) := by
-  simp [descend]
-
- theorem kernel_scalar_le (f : M →ₗ[R] E) (g : N →ₗ[R] F) (r : R)
-    (hf : ∀ v, f v = 0 ↔ ∃ w, v=r • w)
-    (hg : ∀ v, g v = 0 ↔ ∃ w, v=r • w) (h : M →ₗ[R] N) :
-    LinearMap.ker f ≤ LinearMap.ker (g.comp h) := by
-  intro v hv
-  obtain ⟨w,rfl⟩ := (hf v).mp hv
-  change g (h (r • w))=0
-  rw [h.map_smul]
-  exact (hg _).mpr ⟨h w,rfl⟩
-
-end
-end KLInvariance.QuotientEdge
-
-namespace KLInvariance.Graded.ModuleData
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A}
-noncomputable section
-
- def quotientHom {M N E F : ModuleData.{uk,ua,um} 𝓐}
-    (f : Hom M E) (g : Hom N F) (hf : Function.Surjective f.map)
-    (h : Hom M N) (hk : LinearMap.ker f.map ≤ LinearMap.ker (g.map.comp h.map)) : Hom E F where
-  map := QuotientEdge.descend f.map g.map hf h.map hk
-  graded := by
-    intro n v hv
-    obtain ⟨w,hw,rfl⟩ := f.homogeneous_preimage hf n v hv
-    rw [QuotientEdge.descend_apply]
-    exact g.graded n _ (h.graded n w hw)
-
- theorem quotientHom_apply {M N E F : ModuleData.{uk,ua,um} 𝓐}
-    (f : Hom M E) (g : Hom N F) (hf : Function.Surjective f.map)
-    (h : Hom M N) (hk : LinearMap.ker f.map ≤ LinearMap.ker (g.map.comp h.map))
-    (v : M) : (quotientHom f g hf h hk).map (f.map v) = g.map (h.map v) :=
-  QuotientEdge.descend_apply _ _ _ _ _ v
-
-end
-end KLInvariance.Graded.ModuleData
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
-  {G : OrderedGraph V E}
-
-structure Hom (B C : GradedSheaf (𝓐 := 𝓐) G) where
-  vertex : ∀ x, ModuleData.Hom (B.vertex x) (C.vertex x)
-  edge : ∀ e, ModuleData.Hom (B.edge e) (C.edge e)
-  lower : ∀ e v, (edge e).map ((B.lower e).map v) =
-    (C.lower e).map ((vertex (G.source e)).map v)
-  upper : ∀ e v, (edge e).map ((B.upper e).map v) =
-    (C.upper e).map ((vertex (G.target e)).map v)
-
-structure OnHom (B C : GradedSheaf (𝓐 := 𝓐) G) (U : Set V) where
-  vertex : ∀ x, x ∈ U → ModuleData.Hom (B.vertex x) (C.vertex x)
-  edge : ∀ e, G.target e ∈ U → ModuleData.Hom (B.edge e) (C.edge e)
-  lower : ∀ e (hs : G.source e ∈ U) (ht : G.target e ∈ U) v,
-    (edge e ht).map ((B.lower e).map v) =
-      (C.lower e).map ((vertex (G.source e) hs).map v)
-  upper : ∀ e (ht : G.target e ∈ U) v,
-    (edge e ht).map ((B.upper e).map v) =
-      (C.upper e).map ((vertex (G.target e) ht).map v)
-
-namespace Hom
-variable {B C D : GradedSheaf (𝓐 := 𝓐) G}
-
-def comp (g : Hom C D) (f : Hom B C) : Hom B D where
-  vertex x := (g.vertex x).comp (f.vertex x)
-  edge e := (g.edge e).comp (f.edge e)
-  lower e v := by
-    change (g.edge e).map ((f.edge e).map ((B.lower e).map v)) = _
-    rw [f.lower,g.lower]; rfl
-  upper e v := by
-    change (g.edge e).map ((f.edge e).map ((B.upper e).map v)) = _
-    rw [f.upper,g.upper]; rfl
-
-end Hom
-
-namespace OnHom
-variable {B C : GradedSheaf (𝓐 := 𝓐) G} {U : Set V} (j : OnHom B C U)
-noncomputable section
-
- def ofUniv (j : OnHom B C Set.univ) : Hom B C where
-  vertex x := j.vertex x (Set.mem_univ x)
-  edge e := j.edge e (Set.mem_univ _)
-  lower e v := j.lower e (Set.mem_univ _) (Set.mem_univ _) v
-  upper e v := j.upper e (Set.mem_univ _) v
-
- def mapAssignment (x : V) (hx : Set.Ioi x ⊆ U)
-    (f : B.forget.Assignment) : C.forget.Assignment := by
-  classical
-  exact fun y => if h : x < y then (j.vertex y (hx h)).map (f y) else 0
-
- theorem mapAssignment_at (x : V) (hx : Set.Ioi x ⊆ U)
-    (f : B.forget.Assignment) (y : V) (h : x < y) :
-    j.mapAssignment x hx f y = (j.vertex y (hx h)).map (f y) := by
-  classical
-  unfold mapAssignment
-  exact dite_eq_left h
-
- theorem compatible (x : V) (hx : Set.Ioi x ⊆ U)
-    (f : B.forget.Assignment) (hf : B.forget.CompatibleOn (Set.Ioi x) f) :
-    C.forget.CompatibleOn (Set.Ioi x) (j.mapAssignment x hx f) := by
-  intro e hs ht
-  change x < G.source e at hs
-  change x < G.target e at ht
-  rw [j.mapAssignment_at x hx f _ hs,j.mapAssignment_at x hx f _ ht]
-  change (C.lower e).map ((j.vertex _ (hx hs)).map (f _)) =
-    (C.upper e).map ((j.vertex _ (hx ht)).map (f _))
-  exact (j.lower e (hx hs) (hx ht) _).symm.trans
-    ((congrArg (j.edge e (hx ht)).map (hf e hs ht)).trans (j.upper e (hx ht) _))
-
-variable [Fintype E]
-
- def boundary (x : V) (hx : Set.Ioi x ⊆ U) :
-    ModuleData.Hom (B.boundary x) (C.boundary x) where
-  map := LinearMap.pi fun e =>
-    ((j.edge e.val (hx (lt_of_eq_of_lt e.property.symm (G.increasing e.val)))).map).comp
-      (LinearMap.proj e)
-  graded := by
-    intro n v hv e he
-    exact (j.edge e.val _).graded n _ (hv e he)
-
- theorem boundaryMap (x : V) (hx : Set.Ioi x ⊆ U)
-    (f : B.forget.sections (Set.Ioi x)) :
-    (j.boundary x hx).map (B.forget.boundaryMap x f) =
-      C.forget.boundaryMap x ⟨j.mapAssignment x hx f.val,j.compatible x hx f.val f.property⟩ := by
-  funext e
-  change (j.edge e.val _).map ((B.upper e.val).map (f.val (G.target e.val))) =
-    (C.upper e.val).map (j.mapAssignment x hx f.val (G.target e.val))
-  rw [j.mapAssignment_at _ _ _ _ (lt_of_eq_of_lt e.property.symm (G.increasing e.val))]
-  exact j.upper e.val _ _
-
- theorem map_stalk_image_le (x : V) (hx : Set.Ioi x ⊆ U)
-    (hBf : B.forget.Flabby) (hBg : B.forget.Generated)
-    (hCf : C.forget.Flabby) (hCg : C.forget.Generated) :
-    (LinearMap.range (B.forget.stalkMap x)).map (j.boundary x hx).map ≤
-      LinearMap.range (C.forget.stalkMap x) := by
-  rw [B.forget.range_stalkMap_eq_boundaryImage hBf hBg,
-    C.forget.range_stalkMap_eq_boundaryImage hCf hCg]
-  rintro v ⟨w,⟨f,rfl⟩,rfl⟩
-  exact ⟨⟨j.mapAssignment x hx f.val,j.compatible x hx f.val f.property⟩,
-    (j.boundaryMap x hx f).symm⟩
-
-variable [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-  [IsDomain A] [IsNoetherianRing A]
-  (hneg : ∀ n < 0, 𝓐 n = ⊥)
-  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
-
-include hneg hzero in
- theorem extend_stalk (x : V) (hx : Set.Ioi x ⊆ U)
-    [Module.Free A (B.vertex x)]
-    (hBf : B.forget.Flabby) (hBg : B.forget.Generated)
-    (hCf : C.forget.Flabby) (hCg : C.forget.Generated) :
-    ∃ f : ModuleData.Hom (B.vertex x) (C.vertex x), ∀ v,
-      C.forget.stalkMap x (f.map v) = (j.boundary x hx).map (B.forget.stalkMap x v) := by
-  let T := ModuleData.range (C.stalkHom x)
-  let f : ModuleData.Hom (B.vertex x) T :=
-    { map := LinearMap.codRestrict (LinearMap.range (C.forget.stalkMap x))
-        ((j.boundary x hx).map.comp (B.forget.stalkMap x)) (by
-          intro v
-          exact j.map_stalk_image_le x hx hBf hBg hCf hCg ⟨_,⟨v,rfl⟩,rfl⟩)
-      graded := fun n v hv => (j.boundary x hx).graded n _ ((B.stalkHom x).graded n v hv) }
-  let π : ModuleData.Hom (C.vertex x) T :=
-    { map := (C.forget.stalkMap x).rangeRestrict
-      graded := fun n v hv => (C.stalkHom x).graded n v hv }
-  have hπ : Function.Surjective π.map := by
-    intro v
-    obtain ⟨w,hw⟩ := v.property
-    exact ⟨w,Subtype.ext hw⟩
-  obtain ⟨l,hl⟩ := ModuleData.exists_hom_lift hneg hzero π hπ f
-  refine ⟨l,fun v => ?_⟩
-  exact congrArg (fun z : T => z.val) (LinearMap.congr_fun hl v)
-
-end
-end OnHom
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
-  {G : OrderedGraph V E} {B C : GradedSheaf (𝓐 := 𝓐) G}
-
- def castVertexHom {x y : V} (h : x=y)
-    (f : ModuleData.Hom (B.vertex x) (C.vertex x)) :
-    ModuleData.Hom (B.vertex y) (C.vertex y) := h ▸ f
-
-namespace OnHom
-variable {U : Set V} (j : OnHom B C U)
-noncomputable section
-
- def extend (hU : IsUpperSet U) (x : V) (hx : x ∉ U)
-    (hmax : Set.Ioi x ⊆ U) (α : E → A)
-    (hBq : B.UpperQuotient α) (hCq : C.UpperQuotient α)
-    (jv : ModuleData.Hom (B.vertex x) (C.vertex x))
-    (hout : ∀ e (he : G.source e = x) v,
-      (j.edge e (hmax (lt_of_eq_of_lt he.symm (G.increasing e)))).map
-        ((B.lower e).map v) =
-      (C.lower e).map ((castVertexHom he.symm jv).map v)) :
-    OnHom B C (insert x U) := by
-  classical
-  let vj : ∀ y, y ∈ insert x U → ModuleData.Hom (B.vertex y) (C.vertex y) := fun y hy =>
-    if h : y=x then castVertexHom h.symm jv
-    else j.vertex y ((Set.mem_insert_iff.mp hy).resolve_left h)
-  have vj_old (y : V) (hy : y ∈ U) : vj y (Set.mem_insert_of_mem x hy) = j.vertex y hy := by
-    simp only [vj,dite_eq_right (show y ≠ x from fun h => hx (h ▸ hy))]
-  have vj_new : vj x (Set.mem_insert x U) = jv := by simp [vj,castVertexHom]
-  let ej : ∀ e, G.target e ∈ insert x U → ModuleData.Hom (B.edge e) (C.edge e) := fun e he =>
-    if h : G.target e=x then
-      ModuleData.quotientHom (B.upper e) (C.upper e) (hBq e).1 (vj (G.target e) he)
-        (QuotientEdge.kernel_scalar_le (B.upper e).map (C.upper e).map (α e)
-          (hBq e).2 (hCq e).2 (vj (G.target e) he).map)
-    else j.edge e ((Set.mem_insert_iff.mp he).resolve_left h)
-  have ej_old (e : E) (he : G.target e ∈ U) :
-      ej e (Set.mem_insert_of_mem x he) = j.edge e he := by
-    simp only [ej,dite_eq_right (show G.target e ≠ x from fun h => hx (h ▸ he))]
-  refine ⟨vj,ej,?_,?_⟩
-  · intro e hs ht v
-    have htU : G.target e ∈ U := by
-      rcases Set.mem_insert_iff.mp hs with hs | hs
-      · exact hmax (lt_of_eq_of_lt hs.symm (G.increasing e))
-      · exact hU (G.increasing e).le hs
-    rw [ej_old e htU]
-    by_cases he : G.source e=x
-    · subst x
-      rw [vj_new]
-      exact hout e rfl v
-    · have hsU : G.source e ∈ U := (Set.mem_insert_iff.mp hs).resolve_left he
-      rw [vj_old _ hsU]
-      exact j.lower e hsU htU v
-  · intro e ht v
-    by_cases he : G.target e=x
-    · simp only [ej,dite_eq_left he]
-      exact ModuleData.quotientHom_apply _ _ _ _ _ v
-    · have htU : G.target e ∈ U := (Set.mem_insert_iff.mp ht).resolve_left he
-      rw [ej_old e htU,vj_old _ htU]
-      exact j.upper e htU v
-
- theorem extend_vertex_old (hU : IsUpperSet U) (x : V) (hx : x ∉ U)
-    (hmax : Set.Ioi x ⊆ U) (α : E → A)
-    (hBq : B.UpperQuotient α) (hCq : C.UpperQuotient α)
-    (jv : ModuleData.Hom (B.vertex x) (C.vertex x))
-    (hout : ∀ e (he : G.source e = x) v,
-      (j.edge e (hmax (lt_of_eq_of_lt he.symm (G.increasing e)))).map
-        ((B.lower e).map v) =
-      (C.lower e).map ((castVertexHom he.symm jv).map v))
-    (y : V) (hy : y ∈ U) :
-    (j.extend hU x hx hmax α hBq hCq jv hout).vertex y (Set.mem_insert_of_mem x hy) =
-      j.vertex y hy := by
-  classical
-  simp only [extend,dite_eq_right (show y ≠ x from fun h => hx (h ▸ hy))]
-
- theorem extend_vertex_new (hU : IsUpperSet U) (x : V) (hx : x ∉ U)
-    (hmax : Set.Ioi x ⊆ U) (α : E → A)
-    (hBq : B.UpperQuotient α) (hCq : C.UpperQuotient α)
-    (jv : ModuleData.Hom (B.vertex x) (C.vertex x))
-    (hout : ∀ e (he : G.source e = x) v,
-      (j.edge e (hmax (lt_of_eq_of_lt he.symm (G.increasing e)))).map
-        ((B.lower e).map v) =
-      (C.lower e).map ((castVertexHom he.symm jv).map v)) :
-    (j.extend hU x hx hmax α hBq hCq jv hout).vertex x (Set.mem_insert x U) = jv := by
-  classical
-  dsimp only [extend]
-  exact dite_eq_left rfl
-
-end
-end OnHom
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-/-! Integral graded lifting of sheaf morphisms. This is the actual minimal
-sheaf recognition mechanism needed to split it out of a Bott--Samelson sheaf;
-it does not assume or establish the KL character theorem. -/
-namespace KLInvariance.MomentGraph.GradedSheaf.OnHom
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-  [IsDomain A] [IsNoetherianRing A]
-  (hneg : ∀ n < 0, 𝓐 n = ⊥)
-  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
-  {V E : Type um} [PartialOrder V] [Fintype V] [Fintype E]
-  {G : OrderedGraph V E} {B C : GradedSheaf (𝓐 := 𝓐) G}
-  (α : E → A) (hBq : B.UpperQuotient α) (hCq : C.UpperQuotient α)
-  (hBfree : ∀ x, Module.Free A (B.vertex x))
-  (hBf : B.forget.Flabby) (hBg : B.forget.Generated)
-  (hCf : C.forget.Flabby) (hCg : C.forget.Generated)
-noncomputable section
-
-omit [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-  [IsDomain A] [IsNoetherianRing A] [Fintype V] [Fintype E] in
-private theorem cast_vertex {U U' : Set V} (h : U=U') (j : OnHom B C U)
-    (y : V) (hy : y ∈ U') :
-    (h ▸ j).vertex y hy = j.vertex y (h.symm ▸ hy) := by
-  subst U'
-  rfl
-
-include hneg hzero hBq hCq hBfree hBf hBg hCf hCg in
- theorem exists_extension {U : Set V} (hU : IsUpperSet U) (j : OnHom B C U) :
-    ∃ f : Hom B C, ∀ x (hx : x ∈ U), f.vertex x = j.vertex x hx := by
-  classical
-  have h : ∀ s : Finset V, IsUpperSet ((s : Set V)ᶜ) →
-      ∀ j : OnHom B C ((s : Set V)ᶜ),
-      ∃ f : Hom B C, ∀ x (hx : x ∈ ((s : Set V)ᶜ)), f.vertex x = j.vertex x hx := by
-    intro s
-    induction s using Finset.strongInductionOn with
-    | _ s ih =>
-      intro hU j
-      by_cases hs : s=∅
-      · subst s
-        have heq : ((∅ : Finset V) : Set V)ᶜ=Set.univ := by simp
-        let jj : OnHom B C Set.univ := heq ▸ j
-        refine ⟨jj.ofUniv,fun y hy => ?_⟩
-        exact cast_vertex heq j y (Set.mem_univ y)
-      · obtain ⟨x,hxs,hxmax⟩ := s.exists_maximal (Finset.nonempty_iff_ne_empty.mpr hs)
-        have hx : x ∉ ((s : Set V)ᶜ) := by simpa using hxs
-        have hmax : Set.Ioi x ⊆ ((s : Set V)ᶜ) := by
-          intro y hy
-          change y ∉ s
-          intro hys
-          exact (not_le_of_gt hy) (hxmax hys hy.le)
-        let := hBfree x
-        obtain ⟨l,hl⟩ := j.extend_stalk hneg hzero x hmax hBf hBg hCf hCg
-        have hout : ∀ e (he : G.source e=x) v,
-            (j.edge e (hmax (lt_of_eq_of_lt he.symm (G.increasing e)))).map
-              ((B.lower e).map v) =
-            (C.lower e).map ((castVertexHom he.symm l).map v) := by
-          intro e he v
-          subst x
-          exact (congrFun (hl v) ⟨e,rfl⟩).symm
-        let jj := j.extend hU x hx hmax α hBq hCq l hout
-        have heq : insert x ((s : Set V)ᶜ) = ((s.erase x : Finset V) : Set V)ᶜ := by
-          ext y
-          simp only [Set.mem_insert_iff,Set.mem_compl_iff,Finset.mem_coe,Finset.mem_erase]
-          tauto
-        have hU' : IsUpperSet (insert x ((s : Set V)ᶜ)) := by
-          intro y t hyt hy
-          rcases Set.mem_insert_iff.mp hy with hy | hy
-          · subst y
-            by_cases htx : t=x
-            · exact Set.mem_insert_iff.mpr (Or.inl htx)
-            · exact Set.mem_insert_of_mem x (hmax (lt_of_le_of_ne hyt (Ne.symm htx)))
-          · exact Set.mem_insert_of_mem x (hU hyt hy)
-        obtain ⟨f,hf⟩ := ih (s.erase x) (Finset.erase_ssubset hxs) (heq ▸ hU') (heq ▸ jj)
-        refine ⟨f,fun y hy => ?_⟩
-        have hy' : y ∈ ((s.erase x : Finset V) : Set V)ᶜ :=
-          heq ▸ Set.mem_insert_of_mem x hy
-        exact (hf y hy').trans ((cast_vertex heq jj y hy').trans
-          (j.extend_vertex_old hU x hx hmax α hBq hCq l hout y hy))
-  let s := (Uᶜ).toFinset
-  have heq : (s : Set V)ᶜ=U := by simp [s]
-  obtain ⟨f,hf⟩ := h s (heq.symm ▸ hU) (heq.symm ▸ j)
-  refine ⟨f,fun x hx => ?_⟩
-  exact (hf x (heq.symm ▸ hx)).trans (cast_vertex heq.symm j x (heq.symm ▸ hx))
-
-end
-end KLInvariance.MomentGraph.GradedSheaf.OnHom
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-  [IsDomain A] [IsNoetherianRing A]
-  (hneg : ∀ n < 0, 𝓐 n = ⊥)
-  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
-  {V E : Type um} [PartialOrder V] [Fintype V] [Fintype E]
-  {G : OrderedGraph V E} {B C : GradedSheaf (𝓐 := 𝓐) G}
-  (α : E → A) (hBq : B.UpperQuotient α) (hCq : C.UpperQuotient α)
-  (hBfree : ∀ x, Module.Free A (B.vertex x))
-  (hBf : B.forget.Flabby) (hBg : B.forget.Generated)
-  (hCf : C.forget.Flabby) (hCg : C.forget.Generated)
-  (b : V) (hb : ∀ x, x ≤ b)
-noncomputable section
-
-include hneg hzero hBq hCq hBfree hBf hBg hCf hCg hb in
- theorem exists_hom_top (t : ModuleData.Hom (B.vertex b) (C.vertex b)) :
-    ∃ f : Hom B C, f.vertex b=t := by
-  let je : OnHom B C (∅ : Set V) :=
-    { vertex x hx := False.elim hx
-      edge e he := False.elim he
-      lower e hs := False.elim hs
-      upper e ht := False.elim ht }
-  have hmax : Set.Ioi b ⊆ (∅ : Set V) := by
-    intro x hx
-    exact False.elim ((not_lt_of_ge (hb x)) hx)
-  have hout : ∀ e (he : G.source e=b) v,
-      (je.edge e (hmax (lt_of_eq_of_lt he.symm (G.increasing e)))).map
-        ((B.lower e).map v) =
-      (C.lower e).map ((castVertexHom he.symm t).map v) := by
-    intro e he
-    exact False.elim ((not_lt_of_ge (hb (G.target e)))
-      (lt_of_eq_of_lt he.symm (G.increasing e)))
-  let jj := je.extend isUpperSet_empty b (Set.notMem_empty b) hmax α hBq hCq t hout
-  have hU : IsUpperSet (insert b (∅ : Set V)) := by
-    intro x y hxy hx
-    have hxb : x=b := by simpa using hx
-    subst x
-    have hyb : y=b := le_antisymm (hb y) hxy
-    simpa using hyb
-  obtain ⟨f,hf⟩ := jj.exists_extension hneg hzero α hBq hCq hBfree hBf hBg hCf hCg hU
-  exact ⟨f,(hf b (Set.mem_insert b _)).trans
-    (je.extend_vertex_new isUpperSet_empty b (Set.notMem_empty b) hmax α hBq hCq t hout)⟩
-
-end
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-/-! A homogeneous idempotent with image in the positive-ideal multiple of a
-bounded-below graded module is zero. This is the minimal-cover rigidity needed
-to identify the indecomposable in the BMP construction. -/
-namespace KLInvariance.Graded.ModuleData
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} (M : ModuleData.{uk,ua,um} 𝓐)
-
- theorem idempotent_eq_zero (f : Hom M M)
-    (hi : ∀ v, f.map (f.map v) = f.map v)
-    (hm : LinearMap.range f.map ≤ positiveIdeal 𝓐 • (⊤ : Submodule A M)) :
-    f.map = 0 := by
-  have hker : LinearMap.ker f.map = ⊤ :=
-    eq_top_of_positive_smul_sup 𝓐 M.piece 0 M.nonneg (LinearMap.ker f.map)
-      (by
-        intro n m h
-        change f.map (component M.piece n m) = 0
-        rw [← f.component, LinearMap.mem_ker.mp h, map_zero])
-      (by
-        apply top_unique
-        intro m _
-        apply Submodule.mem_sup.mpr
-        refine ⟨m-f.map m,?_,f.map m,hm ⟨m,rfl⟩,sub_add_cancel _ _⟩
-        change f.map (m-f.map m) = 0
-        rw [map_sub,hi,sub_self])
-  exact LinearMap.ker_eq_top.mp hker
-
-end KLInvariance.Graded.ModuleData
-
-end
-
-
-section
-
-/-! The actual minimal boundary sheaf is indecomposable: it has no nontrivial
-homogeneous idempotent. No KL character or costalk freeness is used here. -/
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
-  {G : OrderedGraph V E}
-
-structure Endomorphism (B : GradedSheaf (𝓐 := 𝓐) G) where
-  vertex : ∀ x, ModuleData.Hom (B.vertex x) (B.vertex x)
-  edge : ∀ e, ModuleData.Hom (B.edge e) (B.edge e)
-  lower : ∀ e v, (edge e).map ((B.lower e).map v) =
-    (B.lower e).map ((vertex (G.source e)).map v)
-  upper : ∀ e v, (edge e).map ((B.upper e).map v) =
-    (B.upper e).map ((vertex (G.target e)).map v)
-
-namespace Endomorphism
-variable {B : GradedSheaf (𝓐 := 𝓐) G} (f : Endomorphism B)
-
- def complement : Endomorphism B where
-  vertex x :=
-    { map := LinearMap.id - (f.vertex x).map
-      graded := fun n v hv => (B.vertex x).piece n |>.sub_mem hv ((f.vertex x).graded n v hv) }
-  edge e :=
-    { map := LinearMap.id - (f.edge e).map
-      graded := fun n v hv => (B.edge e).piece n |>.sub_mem hv ((f.edge e).graded n v hv) }
-  lower e v := by
-    change (B.lower e).map v - (f.edge e).map ((B.lower e).map v) =
-      (B.lower e).map (v-(f.vertex (G.source e)).map v)
-    rw [map_sub,f.lower]
-  upper e v := by
-    change (B.upper e).map v - (f.edge e).map ((B.upper e).map v) =
-      (B.upper e).map (v-(f.vertex (G.target e)).map v)
-    rw [map_sub,f.upper]
-
- theorem complement_idempotent (hi : ∀ x v, (f.vertex x).map ((f.vertex x).map v) =
-      (f.vertex x).map v) (x : V) (v : B.vertex x) :
-    (f.complement.vertex x).map ((f.complement.vertex x).map v) =
-      (f.complement.vertex x).map v := by
-  change v-(f.vertex x).map v - (f.vertex x).map (v-(f.vertex x).map v) =
-    v-(f.vertex x).map v
-  rw [map_sub,hi,sub_self,sub_zero]
-
-end Endomorphism
-end KLInvariance.MomentGraph.GradedSheaf
-
-namespace KLInvariance.Graded.ModuleData
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-  (hneg : ∀ n < 0, 𝓐 n = ⊥)
-
- theorem unit_idempotent_trivial [IsDomain A]
-    (f : Hom (unit.{uk,ua,um} hneg) (unit.{uk,ua,um} hneg))
-    (hi : ∀ v, f.map (f.map v) = f.map v) :
-    f.map = 0 ∨ f.map = LinearMap.id := by
-  let e : unit.{uk,ua,um} hneg := fun _ => 1
-  let c : A := f.map e PUnit.unit
-  have expand (v : unit.{uk,ua,um} hneg) : v = v PUnit.unit • e := by
-    funext i
-    cases i
-    exact (mul_one _).symm
-  have hf (v : unit.{uk,ua,um} hneg) (i : PUnit.{max ua um + 1}) :
-      f.map v i = v PUnit.unit * c := by
-    calc
-      f.map v i = f.map (v PUnit.unit • e) i := congrArg (fun v => f.map v i) (expand v)
-      _ = v PUnit.unit * c := by rw [map_smul]; cases i; rfl
-  have hc : c*c=c := by
-    have h := congrFun (hi e) PUnit.unit
-    rw [hf (f.map e) PUnit.unit] at h
-    exact h
-  have hz : c*(c-1)=0 := by rw [mul_sub,mul_one,hc,sub_self]
-  rcases mul_eq_zero.mp hz with hc | hc
-  · left
-    apply LinearMap.ext
-    intro v
-    funext i
-    rw [hf,hc,mul_zero]
-    rfl
-  · right
-    have hc : c=1 := sub_eq_zero.mp hc
-    apply LinearMap.ext
-    intro v
-    funext i
-    rw [hf,hc,mul_one]
-    cases i
-    rfl
-
-end KLInvariance.Graded.ModuleData
-
-namespace KLInvariance.MomentGraph.BoundarySheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-  {hneg : ∀ n < 0, 𝓐 n = ⊥} {V E : Type um} [PartialOrder V]
-  {G : OrderedGraph V E} {α : E → A} {b : V}
-  (B : BoundarySheaf hneg G α b)
-
- theorem idempotent_zero_of_top_zero [Finite V]
-    (f : B.sheaf.Endomorphism)
-    (hi : ∀ x v, (f.vertex x).map ((f.vertex x).map v) = (f.vertex x).map v)
-    (ht : (f.vertex b).map = 0) :
-    (∀ x, (f.vertex x).map = 0) ∧ (∀ e, (f.edge e).map = 0) := by
-  have hv : ∀ x, (f.vertex x).map = 0 := by
-    intro x
-    induction x using WellFoundedGT.induction with
-    | ind x ih =>
-      by_cases hx : x=b
-      · exact hx ▸ ht
-      apply ModuleData.idempotent_eq_zero (B.sheaf.vertex x) (f.vertex x) (hi x)
-      intro v hv
-      obtain ⟨w,rfl⟩ := hv
-      apply (B.sheaf.kernelMinimal_iff x).mp (B.minimal x hx)
-      change B.sheaf.forget.stalkMap x ((f.vertex x).map w) = 0
-      funext e
-      rcases e with ⟨e,he⟩
-      subst x
-      change (B.sheaf.lower e).map ((f.vertex (G.source e)).map w) = 0
-      rw [← f.lower]
-      have hezero : (f.edge e).map = 0 := by
-        ext z
-        obtain ⟨y,rfl⟩ := (B.quotient e).1 z
-        rw [f.upper,ih (G.target e) (G.increasing e)]
-        exact map_zero _
-      rw [hezero]
-      rfl
-  refine ⟨hv,?_⟩
-  intro e
-  ext z
-  obtain ⟨y,rfl⟩ := (B.quotient e).1 z
-  rw [f.upper,hv]
-  exact map_zero _
-
-/-- Actual indecomposability in the idempotent formulation. A direct sum
-splitting supplies such a homogeneous idempotent, hence one summand vanishes. -/
- theorem idempotent_trivial [Finite V] [IsDomain A]
-    (f : B.sheaf.Endomorphism)
-    (hi : ∀ x v, (f.vertex x).map ((f.vertex x).map v) = (f.vertex x).map v) :
-    ((∀ x, (f.vertex x).map = 0) ∧ (∀ e, (f.edge e).map = 0)) ∨
-      ((∀ x, (f.vertex x).map = LinearMap.id) ∧
-        (∀ e, (f.edge e).map = LinearMap.id)) := by
-  have ht : (f.vertex b).map = 0 ∨ (f.vertex b).map = LinearMap.id := by
-    have h := B.top
-    have general (N : ModuleData.{uk,ua,um} 𝓐) (hN : N=ModuleData.unit hneg)
-        (g : ModuleData.Hom N N) (hg : ∀v, g.map (g.map v)=g.map v) :
-        g.map=0 ∨ g.map=LinearMap.id := by
-      subst N
-      exact ModuleData.unit_idempotent_trivial hneg g hg
-    exact general _ h (f.vertex b) (hi b)
-  rcases ht with ht | ht
-  · exact Or.inl (B.idempotent_zero_of_top_zero f hi ht)
-  · right
-    have hh : (f.complement.vertex b).map = 0 := by
-      change LinearMap.id - (f.vertex b).map = 0
-      rw [ht,sub_self]
-    obtain ⟨hv,he⟩ := B.idempotent_zero_of_top_zero f.complement
-      (f.complement_idempotent hi) hh
-    constructor
-    · intro x
-      exact (sub_eq_zero.mp (hv x)).symm
-    · intro e
-      exact (sub_eq_zero.mp (he e)).symm
-
-end KLInvariance.MomentGraph.BoundarySheaf
-
-end
-
-
-section
-
-/-! A genuine homogeneous self-map of a minimal cover is invertible when its
-map on the covered module is invertible. Used for the integral summand bridge. -/
-namespace KLInvariance.Graded.ModuleData
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-  [IsDomain A] [IsNoetherianRing A]
-  (hneg : ∀ n < 0, 𝓐 n = ⊥)
-  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
-noncomputable section
-
-include hneg hzero in
- theorem MinimalCover.bijective_of_boundaryIso {M : ModuleData.{uk,ua,um} 𝓐}
-    (C : MinimalCover M) (f : Hom C.source C.source) (g : M ≃ₗ[A] M)
-    (hg : ∀ n v, v ∈ M.piece n → g v ∈ M.piece n)
-    (hcomm : ∀ v, C.hom.map (f.map v) = g (C.hom.map v)) :
-    Function.Bijective f.map := by
-  let gh : Hom M M := ⟨g.toLinearMap,hg⟩
-  let gi : Hom M M :=
-    { map := g.symm.toLinearMap
-      graded := by
-        intro n v hv
-        apply gh.mem_of_map_mem g.injective n
-        change g (g.symm v) ∈ M.piece n
-        simpa only [LinearEquiv.apply_symm_apply] using hv }
-  let := C.free
-  obtain ⟨l,hl⟩ := exists_hom_lift hneg hzero C.hom C.surjective (gi.comp C.hom)
-  have hla (v : C.source) : C.hom.map (l.map v) = g.symm (C.hom.map v) :=
-    LinearMap.congr_fun hl v
-  have hfl : Function.Bijective (f.comp l).map := by
-    apply Hom.bijective_of_mod_positive
-    intro v
-    apply C.minimal
-    change C.hom.map (v-f.map (l.map v))=0
-    rw [map_sub,hcomm,hla,LinearEquiv.apply_symm_apply,sub_self]
-  have hlf : Function.Bijective (l.comp f).map := by
-    apply Hom.bijective_of_mod_positive
-    intro v
-    apply C.minimal
-    change C.hom.map (v-l.map (f.map v))=0
-    rw [map_sub,hla,hcomm,LinearEquiv.symm_apply_apply,sub_self]
-  exact ⟨Function.Injective.of_comp hlf.1,Function.Surjective.of_comp hfl.2⟩
-
-end
-end KLInvariance.Graded.ModuleData
-
-end
-
-
-section
-
-/-! A graded endomorphism of the actual normalized minimal boundary sheaf
-is an automorphism as soon as its top component is one. -/
-namespace KLInvariance.MomentGraph.GradedSheaf.Endomorphism
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
-  {G : OrderedGraph V E} {B : GradedSheaf (𝓐 := 𝓐) G}
-  (f : Endomorphism B)
-
- theorem edge_bijective_of_vertex {α : E → A}
-    (hBq : B.forget.UpperQuotient α) (e : E)
-    (hv : Function.Bijective (f.vertex (G.target e)).map) :
-    Function.Bijective (f.edge e).map := by
-  obtain ⟨j,hj⟩ := QuotientEdge.exists_scalar_iso
-    (B.upper e).map (B.upper e).map (α e)
-    (hBq e).1 (hBq e).1 (hBq e).2 (hBq e).2
-    (LinearEquiv.ofBijective (f.vertex (G.target e)).map hv)
-  have heq : (f.edge e).map = j.toLinearMap := by
-    apply LinearMap.ext
-    intro v
-    obtain ⟨w,rfl⟩ := (hBq e).1 v
-    exact (f.upper e w).trans (hj w).symm
-  rw [heq]
-  exact j.bijective
-
-end KLInvariance.MomentGraph.GradedSheaf.Endomorphism
-
-namespace KLInvariance.MomentGraph.BoundarySheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-  [IsDomain A] [IsNoetherianRing A]
-  {hneg : ∀ n < 0, 𝓐 n = ⊥}
-  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
-  {V E : Type um} [PartialOrder V] [Fintype V] [Fintype E]
-  {G : OrderedGraph V E} {α : E → A} {b : V}
-  (B : BoundarySheaf hneg G α b)
-noncomputable section
-
-include hzero in
- theorem endomorphism_bijective_of_top (f : B.sheaf.Endomorphism)
-    (ht : Function.Bijective (f.vertex b).map) :
-    (∀ x, Function.Bijective (f.vertex x).map) ∧
-    (∀ e, Function.Bijective (f.edge e).map) := by
-  have hv : ∀ x, Function.Bijective (f.vertex x).map := by
-    intro x
-    induction x using WellFoundedGT.induction with
-    | ind x ih =>
-      by_cases hx : x=b
-      · exact hx ▸ ht
-      let j : Sheaf.AboveIso B.sheaf.forget B.sheaf.forget x :=
-        { vertex y hy := LinearEquiv.ofBijective (f.vertex y).map (ih y hy)
-          edge e he := LinearEquiv.ofBijective (f.edge e).map
-            (f.edge_bijective_of_vertex B.quotient e (ih _ he))
-          lower e he v := f.lower e v
-          upper e he v := f.upper e v }
-      let T := ModuleData.range (B.sheaf.stalkHom x)
-      let g : T ≃ₗ[A] T := LinearEquiv.ofSubmodules j.boundary _ _
-        (j.map_stalk_image B.flabby.flabby B.generated B.flabby.flabby B.generated)
-      have hg : ∀ n v, v ∈ T.piece n → g v ∈ T.piece n := by
-        intro n v hv e he
-        exact (f.edge e.val).graded n _ (hv e he)
-      let π : ModuleData.Hom (B.sheaf.vertex x) T :=
-        { map := (B.sheaf.forget.stalkMap x).rangeRestrict
-          graded := fun n v hv => (B.sheaf.stalkHom x).graded n v hv }
-      have hπs : Function.Surjective π.map := by
-        intro v
-        obtain ⟨w,hw⟩ := v.property
-        exact ⟨w,Subtype.ext hw⟩
-      have hπm : LinearMap.ker π.map ≤
-          positiveIdeal 𝓐 • (⊤ : Submodule A (B.sheaf.vertex x)) := by
-        intro v hv
-        apply (B.sheaf.kernelMinimal_iff x).mp (B.minimal x hx)
-        have hh : π.map v = 0 := LinearMap.mem_ker.mp hv
-        exact congrArg (fun z : T => z.val) hh
-      let C : ModuleData.MinimalCover T := ⟨B.sheaf.vertex x,B.free x,π,hπs,hπm⟩
-      apply C.bijective_of_boundaryIso hneg hzero (f.vertex x) g hg
-      intro v
-      apply Subtype.ext
-      funext e
-      change B.sheaf.forget.stalkMap x ((f.vertex x).map v) e =
-        (f.edge e.val).map (B.sheaf.forget.stalkMap x v e)
-      rcases e with ⟨e,rfl⟩
-      exact (f.lower e v).symm
-  exact ⟨hv,fun e => f.edge_bijective_of_vertex B.quotient e (hv _)⟩
-
-end
-end KLInvariance.MomentGraph.BoundarySheaf
-
-end
-
-
-section
-
-namespace KLInvariance.Graded.ModuleData.Hom
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} {M N : ModuleData.{uk,ua,um} 𝓐}
-noncomputable section
-
- def inverse (f : Hom M N) (hf : Function.Bijective f.map) : Hom N M where
-  map := (LinearEquiv.ofBijective f.map hf).symm.toLinearMap
-  graded := by
-    intro n v hv
-    apply f.mem_of_map_mem hf.1 n
-    change (LinearEquiv.ofBijective f.map hf) ((LinearEquiv.ofBijective f.map hf).symm v) ∈ N.piece n
-    simpa only [LinearEquiv.apply_symm_apply] using hv
-
- theorem map_inverse (f : Hom M N) (hf : Function.Bijective f.map) (v : N) :
-    f.map ((f.inverse hf).map v)=v := (LinearEquiv.ofBijective f.map hf).apply_symm_apply v
-
- theorem inverse_map (f : Hom M N) (hf : Function.Bijective f.map) (v : M) :
-    (f.inverse hf).map (f.map v)=v := (LinearEquiv.ofBijective f.map hf).symm_apply_apply v
-
-end
-end KLInvariance.Graded.ModuleData.Hom
-
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
-  {G : OrderedGraph V E}
-noncomputable section
-
- def Hom.toEndomorphism {B : GradedSheaf (𝓐 := 𝓐) G} (f : Hom B B) : Endomorphism B :=
-  ⟨f.vertex,f.edge,f.lower,f.upper⟩
-
- def Hom.inverse {B C : GradedSheaf (𝓐 := 𝓐) G} (f : Hom B C)
-    (hv : ∀ x, Function.Bijective (f.vertex x).map)
-    (he : ∀ e, Function.Bijective (f.edge e).map) : Hom C B where
-  vertex x := (f.vertex x).inverse (hv x)
-  edge e := (f.edge e).inverse (he e)
-  lower e v := by
-    apply (he e).1
-    rw [(f.edge e).map_inverse]
-    rw [f.lower,(f.vertex _).map_inverse]
-  upper e v := by
-    apply (he e).1
-    rw [(f.edge e).map_inverse]
-    rw [f.upper,(f.vertex _).map_inverse]
-
- structure SplitIn (B C : GradedSheaf (𝓐 := 𝓐) G) where
-  inclusion : Hom B C
-  projection : Hom C B
-  vertex_split : ∀ x v, (projection.vertex x).map ((inclusion.vertex x).map v)=v
-  edge_split : ∀ e v, (projection.edge e).map ((inclusion.edge e).map v)=v
-
-end
-end KLInvariance.MomentGraph.GradedSheaf
-
-namespace KLInvariance.MomentGraph.BoundarySheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-  [IsDomain A] [IsNoetherianRing A]
-  {hneg : ∀ n < 0, 𝓐 n = ⊥}
-  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
-  {V E : Type um} [PartialOrder V] [Fintype V] [Fintype E]
-  {G : OrderedGraph V E} {α : E → A} {b : V}
-  (B : BoundarySheaf hneg G α b) (hb : ∀ x, x ≤ b)
-  (C : GradedSheaf (𝓐 := 𝓐) G)
-  (hCq : C.UpperQuotient α) (hCfree : ∀ x, Module.Free A (C.vertex x))
-  (hCf : C.forget.Flabby) (hCg : C.forget.Generated)
-noncomputable section
-
-include hzero hb hCq hCfree hCf hCg in
- theorem splitIn_of_top_split
-    (i : ModuleData.Hom (B.sheaf.vertex b) (C.vertex b))
-    (p : ModuleData.Hom (C.vertex b) (B.sheaf.vertex b))
-    (hpi : ∀ v, p.map (i.map v)=v) : Nonempty (GradedSheaf.SplitIn B.sheaf C) := by
-  obtain ⟨ii,hii⟩ := GradedSheaf.exists_hom_top hneg hzero α B.quotient hCq B.free
-    B.flabby.flabby B.generated hCf hCg b hb i
-  obtain ⟨pp,hpp⟩ := GradedSheaf.exists_hom_top hneg hzero α hCq B.quotient hCfree
-    hCf hCg B.flabby.flabby B.generated b hb p
-  let f := pp.comp ii
-  have ht : Function.Bijective (f.toEndomorphism.vertex b).map := by
-    have heq : (f.toEndomorphism.vertex b).map = LinearMap.id := by
-      apply LinearMap.ext
-      intro v
-      change (pp.vertex b).map ((ii.vertex b).map v)=v
-      rw [hpp,hii]
-      exact hpi v
-    rw [heq]
-    exact Function.bijective_id
-  obtain ⟨hv,he⟩ := B.endomorphism_bijective_of_top hzero f.toEndomorphism ht
-  let fi := f.inverse hv he
-  refine ⟨⟨ii.comp fi,pp,?_,?_⟩⟩
-  · intro x v
-    exact (f.vertex x).map_inverse (hv x) v
-  · intro e v
-    exact (f.edge e).map_inverse (he e) v
-
-end
-end KLInvariance.MomentGraph.BoundarySheaf
-
-end
-
-
-section
-
-/-! A minimal graded free cover splits out of every finite graded free cover.
-The residual kernel is genuinely graded free over the connected coefficient ring.
-This is the local step for decomposing actual flabby word sheaves. -/
-namespace KLInvariance.Graded.ModuleData
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-  [IsDomain A] [IsNoetherianRing A]
-  (hneg : ∀ n < 0, 𝓐 n = ⊥)
-  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
-noncomputable section
-
-include hneg hzero in
- theorem exists_minimalCover_split {T F : ModuleData.{uk,ua,um} 𝓐}
-    (C : MinimalCover T) [Module.Free A F]
-    (f : Hom F T) (hf : Function.Surjective f.map) :
-    ∃ (g : Hom F C.source) (h : Hom C.source F),
-      C.hom.map.comp g.map = f.map ∧ g.map.comp h.map = LinearMap.id := by
-  let : Module.Free A C.source := C.free
-  obtain ⟨g,hg⟩ := exists_hom_lift hneg hzero C.hom C.surjective f
-  obtain ⟨t,ht⟩ := exists_hom_lift hneg hzero f hf C.hom
-  have hgt : Function.Bijective (g.comp t).map := by
-    apply Hom.bijective_of_mod_positive
-    intro v
-    apply C.minimal
-    change C.hom.map (v-g.map (t.map v))=0
-    rw [map_sub,← LinearMap.comp_apply,hg,← LinearMap.comp_apply,ht,sub_self]
-  refine ⟨g,t.comp ((g.comp t).inverse hgt),hg,?_⟩
-  ext v
-  exact (g.comp t).map_inverse hgt v
-
- def Hom.splitEquiv {F C : ModuleData.{uk,ua,um} 𝓐}
-    (g : Hom F C) (h : Hom C F) (hgh : g.map.comp h.map = LinearMap.id) :
-    F ≃ₗ[A] C × kernel g := LocalCover.splitEquiv g.map h.map hgh
-
-omit [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐] [IsDomain A] in
- theorem Hom.splitEquiv_graded {F C : ModuleData.{uk,ua,um} 𝓐}
-    (g : Hom F C) (h : Hom C F) (hgh : g.map.comp h.map = LinearMap.id)
-    (n : ℤ) (v : F) (hv : v ∈ F.piece n) :
-    g.splitEquiv h hgh v ∈ (prod C (kernel g)).piece n := by
-  constructor
-  · exact g.graded n v hv
-  · exact Submodule.sub_mem _ hv (h.graded n _ (g.graded n v hv))
-
-include hneg hzero in
- theorem Hom.kernel_free_of_split {F C : ModuleData.{uk,ua,um} 𝓐}
-    [Module.Free A F] (g : Hom F C) (h : Hom C F)
-    (hgh : g.map.comp h.map = LinearMap.id) : Module.Free A (kernel g) := by
-  let e := g.splitEquiv h hgh
-  let p : F →ₗ[A] kernel g := (LinearMap.snd A C (kernel g)).comp e.toLinearMap
-  have hp : p.comp g.map.ker.subtype = LinearMap.id := by
-    apply LinearMap.ext
-    intro v
-    apply Subtype.ext
-    change ((v : F) - h.map (g.map v))=v
-    rw [LinearMap.mem_ker.mp v.property,map_zero,sub_zero]
-  let : Module.Projective A (kernel g) := Module.Projective.of_split g.map.ker.subtype p hp
-  obtain ⟨ι,hι,b,_⟩ := exists_homogeneous_basis_connected 𝓐 (kernel g).piece
-    hneg hzero 0 (kernel g).nonneg
-  exact Module.Free.of_basis b
-
-end
-end KLInvariance.Graded.ModuleData
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.Sheaf.AboveIso
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-  {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
-  [Fintype E] [IsNoetherianRing A] [IsDomain A]
-  (hneg : ∀ n < 0, 𝓐 n = ⊥)
-  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
-  {B C : GradedSheaf (𝓐 := 𝓐) G} {x : V}
-  (j : AboveIso B.forget C.forget x)
-
-include hneg hzero in
- theorem split_stalk_graded
-    (hj : ∀ e (h : x < G.target e) n v, v ∈ (B.edge e).piece n →
-      j.edge e h v ∈ (C.edge e).piece n)
-    (hBfree : Module.Free A (B.vertex x)) (hCfree : Module.Free A (C.vertex x))
-    (hBf : B.forget.Flabby) (hBg : B.forget.Generated)
-    (hCf : C.forget.Flabby) (hCg : C.forget.Generated)
-    (hCmin : C.KernelMinimal x) :
-    ∃ (g : ModuleData.Hom (B.vertex x) (C.vertex x))
-      (e : B.vertex x ≃ₗ[A] C.vertex x × ModuleData.kernel g),
-      Module.Free A (ModuleData.kernel g) ∧
-      (∀ n v, v ∈ (B.vertex x).piece n →
-        e v ∈ (ModuleData.prod (C.vertex x) (ModuleData.kernel g)).piece n) ∧
-      ∀ v, j.boundary (B.forget.stalkMap x v) = C.forget.stalkMap x (e v).1 := by
-  let T := ModuleData.range (C.stalkHom x)
-  let f : ModuleData.Hom (B.vertex x) T :=
-    { map := LinearMap.codRestrict (LinearMap.range (C.forget.stalkMap x))
-        (j.boundary.toLinearMap.comp (B.forget.stalkMap x)) (by
-          intro v
-          rw [← j.map_stalk_image hBf hBg hCf hCg]
-          exact ⟨B.forget.stalkMap x v,⟨v,rfl⟩,rfl⟩)
-      graded := by
-        intro n v hv e _
-        exact hj e.val _ n _ ((B.stalkHom x).graded n v hv e (Set.mem_univ e)) }
-  let p : ModuleData.Hom (C.vertex x) T :=
-    { map := (C.forget.stalkMap x).rangeRestrict
-      graded := fun n v hv => (C.stalkHom x).graded n v hv }
-  have hfs : Function.Surjective f.map := by
-    intro v
-    have hv : v.val ∈ (LinearMap.range (B.forget.stalkMap x)).map j.boundary.toLinearMap := by
-      rw [j.map_stalk_image hBf hBg hCf hCg]
-      exact v.property
-    obtain ⟨w,⟨z,rfl⟩,hz⟩ := hv
-    exact ⟨z,Subtype.ext hz⟩
-  have hps : Function.Surjective p.map := by
-    intro v
-    obtain ⟨w,hw⟩ := v.property
-    exact ⟨w,Subtype.ext hw⟩
-  have hpm : LinearMap.ker p.map ≤ positiveIdeal 𝓐 • (⊤ : Submodule A (C.vertex x)) := by
-    intro v hv
-    apply (C.kernelMinimal_iff x).mp hCmin
-    have hh : p.map v = 0 := LinearMap.mem_ker.mp hv
-    exact congrArg (fun z : T => z.val) hh
-  let D : ModuleData.MinimalCover T := ⟨C.vertex x,hCfree,p,hps,hpm⟩
-  let : Module.Free A (B.vertex x) := hBfree
-  obtain ⟨g,h,hg,hgh⟩ := ModuleData.exists_minimalCover_split hneg hzero D f hfs
-  refine ⟨g,g.splitEquiv h hgh,g.kernel_free_of_split hneg hzero h hgh,
-    g.splitEquiv_graded h hgh,?_⟩
-  intro v
-  exact (congrArg Subtype.val (LinearMap.congr_fun hg v)).symm
-
-end KLInvariance.MomentGraph.Sheaf.AboveIso
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-  [IsDomain A] [IsNoetherianRing A]
-  {hneg : ∀ n < 0, 𝓐 n = ⊥}
-  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
-  {V E : Type um} [PartialOrder V] [Fintype E] {G : OrderedGraph V E}
-  {α : E → A} (O : BoundaryModels hneg (G := G) α)
-  {B C : GradedSheaf (𝓐 := 𝓐) G}
-noncomputable section
-
-include hzero in
- theorem extend_modelSum {U : Set V} (hU : IsUpperSet U)
-    (hC : ModelSum O.sheaf U C) (j : OnIso B C U)
-    (hBq : B.UpperQuotient α) (hBf : B.forget.Flabby) (hBg : B.forget.Generated)
-    (x : V) (hx : x ∉ U) (hmax : Set.Ioi x ⊆ U)
-    [Module.Free A (B.vertex x)] :
-    ∃ D : GradedSheaf (𝓐 := 𝓐) G,
-      ModelSum O.sheaf (insert x U) D ∧ Nonempty (OnIso B D (insert x U)) := by
-  classical
-  obtain ⟨g,jv,hfree,hjvgrade,hjv⟩ := (j.toOnIso.above x hmax).split_stalk_graded hneg hzero
-    (fun e he => j.edge_graded e (hmax he)) inferInstance (hC.free O x)
-    hBf hBg (hC.flabby O) (hC.generated O) (hC.minimal O x hx)
-  let := hfree
-  obtain ⟨n,d,jq0,hjq0⟩ := ModuleData.exists_equiv_shiftedUnits hneg hzero (ModuleData.kernel g)
-  let S : GradedSheaf (𝓐 := 𝓐) G := pi (fun i : ULift.{um} (Fin n) => (O.sheaf x).shift (d i))
-  let jq1 : ModuleData.pi (fun i : ULift.{um} (Fin n) => (ModuleData.unit hneg).shift (d i))
-      ≃ₗ[A] S.vertex x := LinearEquiv.piCongrRight (fun _ => (O.top x).symm)
-  have hjq1 (m : ℤ) (v : ModuleData.pi (fun i : ULift.{um} (Fin n) => (ModuleData.unit hneg).shift (d i)))
-      (hv : v ∈ (ModuleData.pi (fun i : ULift.{um} (Fin n) => (ModuleData.unit hneg).shift (d i))).piece m) :
-      jq1 v ∈ (S.vertex x).piece m := by
-    intro i _
-    let t : ModuleData.Hom ((O.sheaf x).vertex x) (ModuleData.unit hneg) :=
-      ⟨(O.top x).toLinearMap,O.top_graded x⟩
-    apply t.mem_of_map_mem (O.top x).injective (m-(d i : ℤ))
-    change O.top x ((O.top x).symm (v i)) ∈ (ModuleData.unit hneg).piece (m-(d i : ℤ))
-    exact ((O.top x).apply_symm_apply (v i)).symm ▸ hv i (Set.mem_univ i)
-  let jq := jq0.trans jq1
-  let jv' : B.vertex x ≃ₗ[A] (prod C S).vertex x :=
-    jv.trans (LinearEquiv.prodCongr (LinearEquiv.refl A (C.vertex x)) jq)
-  have hjv' (m : ℤ) (v : B.vertex x) (hv : v ∈ (B.vertex x).piece m) :
-      jv' v ∈ ((prod C S).vertex x).piece m :=
-    ⟨(hjvgrade m v hv).1,hjq1 m _ (hjq0 m _ (hjvgrade m v hv).2)⟩
-  have hSV (y : V) (hy : y ∈ U) : Subsingleton (S.vertex y) := by
-    let := O.support x y (fun hyx => hx (hU hyx hy))
-    exact inferInstanceAs (Subsingleton (ULift.{um} (Fin n) → (O.sheaf x).vertex y))
-  have hSE (e : E) (he : G.target e ∈ U) : Subsingleton (S.edge e) := by
-    let := O.support x (G.target e) (fun h => hx (hU h he))
-    let : Subsingleton ((O.sheaf x).edge e) := (O.quotient x e).1.subsingleton
-    exact inferInstanceAs (Subsingleton (ULift.{um} (Fin n) → (O.sheaf x).edge e))
-  let jj : OnIso B (prod C S) U := j.prodZero S hSV hSE
-  have hsum : ModelSum O.sheaf (insert x U) (prod C S) :=
-    .add (hC.mono (Set.subset_insert x U)) x (Set.mem_insert x U) n d
-  have hout : ∀ e (he : G.source e = x) v,
-      jj.edge e (hmax (lt_of_eq_of_lt he.symm (G.increasing e)))
-        (B.forget.lower e v) = (prod C S).forget.lower e
-          (Sheaf.castStalkIso B.forget (prod C S).forget he.symm jv' v) := by
-    intro e he v
-    subst x
-    apply Prod.ext
-    · exact congrFun (hjv v) ⟨e,rfl⟩
-    · let := hSE e (hmax (G.increasing e))
-      exact Subsingleton.elim _ _
-  exact ⟨prod C S,hsum,⟨jj.extend hU x hx hmax α hBq (hsum.quotient O) jv' hjv' hout⟩⟩
-
-end
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-  [IsDomain A] [IsNoetherianRing A]
-  {hneg : ∀ n < 0, 𝓐 n = ⊥}
-  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
-  {V E : Type um} [PartialOrder V] [Fintype E] {G : OrderedGraph V E}
-  {α : E → A} (O : BoundaryModels hneg (G := G) α)
-  (B : GradedSheaf (𝓐 := 𝓐) G)
-
-include hzero in
-/-- Finite graded decomposition with actual nonnegative shifts. -/
-theorem exists_modelSumIso [Finite V]
-    (hBfree : ∀ x, Module.Free A (B.vertex x))
-    (hBq : B.UpperQuotient α) (hBf : B.forget.Flabby) (hBg : B.forget.Generated) :
-    ∃ C : GradedSheaf (𝓐 := 𝓐) G,
-      ModelSum O.sheaf Set.univ C ∧ Nonempty (Iso B C) := by
-  classical
-  let := Fintype.ofFinite V
-  have h : ∀ s : Finset V, IsUpperSet ((s : Set V)ᶜ) →
-      ∀ C : GradedSheaf (𝓐 := 𝓐) G,
-      ModelSum O.sheaf ((s : Set V)ᶜ) C → OnIso B C ((s : Set V)ᶜ) →
-      ∃ D : GradedSheaf (𝓐 := 𝓐) G,
-        ModelSum O.sheaf Set.univ D ∧ Nonempty (Iso B D) := by
-    intro s
-    induction s using Finset.strongInductionOn with
-    | _ s ih =>
-      intro hU C hC j
-      by_cases hs : s = ∅
-      · subst s
-        have hh : ((∅ : Finset V) : Set V)ᶜ = Set.univ := by simp
-        exact ⟨C,hh ▸ hC,⟨OnIso.ofUniv (hh ▸ j)⟩⟩
-      · obtain ⟨x,hxs,hxmax⟩ := s.exists_maximal (Finset.nonempty_iff_ne_empty.mpr hs)
-        have hx : x ∉ ((s : Set V)ᶜ) := by simpa using hxs
-        have hmax : Set.Ioi x ⊆ ((s : Set V)ᶜ) := by
-          intro y hy
-          change y ∉ s
-          intro hys
-          exact (not_le_of_gt hy) (hxmax hys hy.le)
-        let := hBfree x
-        obtain ⟨D,hD,⟨jj⟩⟩ := extend_modelSum hzero O hU hC j hBq hBf hBg x hx hmax
-        have heq : insert x ((s : Set V)ᶜ) = ((s.erase x : Finset V) : Set V)ᶜ := by
-          ext y
-          simp only [Set.mem_insert_iff,Set.mem_compl_iff,Finset.mem_coe,Finset.mem_erase]
-          tauto
-        have hU' : IsUpperSet (insert x ((s : Set V)ᶜ)) := by
-          intro y t hyt hy
-          rcases Set.mem_insert_iff.mp hy with hy | hy
-          · subst y
-            by_cases htx : t=x
-            · exact Set.mem_insert_iff.mpr (Or.inl htx)
-            · exact Set.mem_insert_of_mem x (hmax (lt_of_le_of_ne hyt (Ne.symm htx)))
-          · exact Set.mem_insert_of_mem x (hU hyt hy)
-        exact ih (s.erase x) (Finset.erase_ssubset hxs) (heq ▸ hU') D (heq ▸ hD) (heq ▸ jj)
-  have hset : ((Finset.univ : Finset V) : Set V)ᶜ = ∅ := by simp
-  have hj : OnIso B zero (∅ : Set V) :=
-    ⟨⟨fun x hx => False.elim hx,fun e he => False.elim he,
-      fun e hs => False.elim hs,fun e ht => False.elim ht⟩,
-      fun x hx => False.elim hx,fun e he => False.elim he⟩
-  have hinit := h Finset.univ
-  rw [hset] at hinit
-  exact hinit isUpperSet_empty zero .zero hj
-
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-/-! Graded linear equivalence preserves the genuine free-module character.
-This does not assert its identification with any KL polynomial. -/
-namespace KLInvariance.Graded.ModuleData
-open Polynomial
-universe uk us um
-variable {k : Type uk} [Field k] {σ : Type us} [Finite σ]
-  {M N : ModuleData.{uk,max uk us,um} (polynomialGrading k σ)}
-
-noncomputable def mapNonnegativeBasis (e : M ≃ₗ[MvPolynomial σ k] N)
-    (he : ∀ n v, v ∈ M.piece n → e v ∈ N.piece n)
-    (b : NonnegativeBasis (σ := σ) M.piece) : NonnegativeBasis (σ := σ) N.piece where
-  index := b.index
-  finite := b.finite
-  basis := b.basis.map e
-  degree := b.degree
-  homogeneous i := he _ _ (b.homogeneous i)
-
- theorem character_eq_of_gradedEquiv [Module.Free (MvPolynomial σ k) M]
-    [Module.Free (MvPolynomial σ k) N] (e : M ≃ₗ[MvPolynomial σ k] N)
-    (he : ∀ n v, v ∈ M.piece n → e v ∈ N.piece n) :
-    character (σ := σ) M.piece M.nonneg = character (σ := σ) N.piece N.nonneg := by
-  let b := (exists_nonnegativeBasis (σ := σ) M.piece M.nonneg).some
-  rw [character_eq M.piece M.nonneg b,
-    character_eq N.piece N.nonneg (mapNonnegativeBasis e he b)]
-  rfl
-
-end KLInvariance.Graded.ModuleData
-
-end
-
-
-section
-
-/-! Honest polynomial characters commute with actual finite graded sums and
-nonnegative shifts. No Hecke or KL identification enters these lemmas. -/
-namespace KLInvariance.Graded.ModuleData
-open Polynomial Module
-universe uk us um
-variable {k : Type uk} [Field k] {σ : Type us} [Finite σ]
-  (M N : ModuleData.{uk,max uk us,um} (polynomialGrading k σ))
-noncomputable section
-
- abbrev char [Module.Free (MvPolynomial σ k) M] : Polynomial ℤ :=
-  character (σ := σ) M.piece M.nonneg
-
- def shiftedBasis (b : NonnegativeBasis (σ := σ) M.piece) (d : ℕ) :
-    NonnegativeBasis (σ := σ) (M.shift d).piece where
-  index := b.index
-  finite := b.finite
-  basis := b.basis
-  degree i := b.degree i+d
-  homogeneous i := by
-    change b.basis i ∈ M.piece ((b.degree i+d : ℕ)-(d:ℤ))
-    simpa only [Nat.cast_add,add_sub_cancel_right] using b.homogeneous i
-
- instance shift_free [Module.Free (MvPolynomial σ k) M] (d : ℕ) :
-    Module.Free (MvPolynomial σ k) (M.shift d) := ‹Module.Free (MvPolynomial σ k) M›
-
- theorem char_shift [Module.Free (MvPolynomial σ k) M] (d : ℕ) :
-    (M.shift d).char=X^d*M.char := by
-  let b := (exists_nonnegativeBasis (σ := σ) M.piece M.nonneg).some
-  rw [char,character_eq _ _ (M.shiftedBasis b d),char,character_eq _ _ b]
-  change (∑ i : b.index, (X : Polynomial ℤ) ^ (b.degree i+d)) = X^d*∑ i, X^b.degree i
-  simp only [pow_add,mul_comm,Finset.mul_sum]
-
- def productBasis (b : NonnegativeBasis (σ := σ) M.piece)
-    (c : NonnegativeBasis (σ := σ) N.piece) :
-    NonnegativeBasis (σ := σ) (M.prod N).piece where
-  index := b.index ⊕ c.index
-  finite := inferInstance
-  basis := b.basis.prod c.basis
-  degree := Sum.elim b.degree c.degree
-  homogeneous i := by
-    cases i with
-    | inl i =>
-      change ((b.basis.prod c.basis) (Sum.inl i)).1 ∈ M.piece (b.degree i) ∧
-        ((b.basis.prod c.basis) (Sum.inl i)).2 ∈ N.piece (b.degree i)
-      simpa only [Basis.prod_apply_inl_fst,Basis.prod_apply_inl_snd] using
-        And.intro (b.homogeneous i) (Submodule.zero_mem (N.piece (b.degree i)))
-    | inr i =>
-      change ((b.basis.prod c.basis) (Sum.inr i)).1 ∈ M.piece (c.degree i) ∧
-        ((b.basis.prod c.basis) (Sum.inr i)).2 ∈ N.piece (c.degree i)
-      simpa only [Basis.prod_apply_inr_fst,Basis.prod_apply_inr_snd] using
-        And.intro (Submodule.zero_mem (M.piece (c.degree i))) (c.homogeneous i)
-
- instance prod_free [Module.Free (MvPolynomial σ k) M] [Module.Free (MvPolynomial σ k) N] :
-    Module.Free (MvPolynomial σ k) (M.prod N) :=
-  (inferInstance : Module.Free (MvPolynomial σ k) (M × N))
-
- theorem char_prod [Module.Free (MvPolynomial σ k) M] [Module.Free (MvPolynomial σ k) N] :
-    (M.prod N).char=M.char+N.char := by
-  let b := (exists_nonnegativeBasis (σ := σ) M.piece M.nonneg).some
-  let c := (exists_nonnegativeBasis (σ := σ) N.piece N.nonneg).some
-  rw [char,character_eq _ _ (M.productBasis N b c),char,character_eq _ _ b,
-    char,character_eq _ _ c]
-  change (∑ i : b.index ⊕ c.index, (X : Polynomial ℤ)^Sum.elim b.degree c.degree i) =
-    (∑ i : b.index, X^b.degree i)+(∑ i : c.index, X^c.degree i)
-  exact Fintype.sum_sum_type (fun i => (X : Polynomial ℤ)^Sum.elim b.degree c.degree i)
-
- def piBasis {ι : Type um} [Fintype ι]
-    (F : ι → ModuleData.{uk,max uk us,um} (polynomialGrading k σ))
-    (b : ∀ i, NonnegativeBasis (σ := σ) (F i).piece) :
-    NonnegativeBasis (σ := σ) (pi F).piece where
-  index := Σ i, (b i).index
-  finite := inferInstance
-  basis := Pi.basis (fun i => (b i).basis)
-  degree i := (b i.1).degree i.2
-  homogeneous i := by
-    classical
-    intro j _
-    change (Pi.basis (fun i => (b i).basis)) i j ∈ (F j).piece ((b i.1).degree i.2)
-    rw [Pi.basis_apply]
-    by_cases h : j=i.1
-    · subst j
-      simpa only [Pi.single_eq_same] using (b i.1).homogeneous i.2
-    · simp only [Pi.single_eq_of_ne h,Submodule.zero_mem]
-
- instance pi_free {ι : Type um} [Fintype ι]
-    (F : ι → ModuleData.{uk,max uk us,um} (polynomialGrading k σ))
-    [∀ i, Module.Free (MvPolynomial σ k) (F i)] :
-    Module.Free (MvPolynomial σ k) (pi F) :=
-  (inferInstance : Module.Free (MvPolynomial σ k) (∀ i, F i))
-
- theorem char_pi {ι : Type um} [Fintype ι]
-    (F : ι → ModuleData.{uk,max uk us,um} (polynomialGrading k σ))
-    [∀ i, Module.Free (MvPolynomial σ k) (F i)] :
-    (pi F).char=∑ i, (F i).char := by
-  let b := fun i => (exists_nonnegativeBasis (σ := σ) (F i).piece (F i).nonneg).some
-  rw [char,character_eq _ _ (piBasis F b)]
-  change (∑ i : Σ j, (b j).index, (X : Polynomial ℤ)^(b i.1).degree i.2) = _
-  rw [Fintype.sum_sigma]
-  apply Finset.sum_congr rfl
-  intro i _
-  exact (character_eq (F i).piece (F i).nonneg (b i)).symm
-
- theorem char_zero [Subsingleton M] [Module.Free (MvPolynomial σ k) M] : M.char=0 := by
-  let b := (exists_nonnegativeBasis (σ := σ) M.piece M.nonneg).some
-  have : IsEmpty b.index := ⟨fun i => b.basis.ne_zero i (Subsingleton.elim _ _)⟩
-  rw [char,character_eq _ _ b]
-  simp [NonnegativeBasis.polynomial]
-
-end
-end KLInvariance.Graded.ModuleData
-
-end
-
-
-section
-
-/-! The character identity of the actual finite shifted-model decomposition.
-This is an identity of genuine graded stalk ranks, not a KL assertion. -/
-
-end
-
-end OAI
+import OAI.RepresentationTheory.KazhdanLusztig.MinimalExtensions
+
+namespace OAI
+
+/-!
+Graded sheaf morphisms, exact section constructions and character sums.
+-/
+
+section
+
+namespace KLInvariance.Graded
+open Polynomial
+universe uk us um ui
+variable {k : Type uk} [Field k] {σ : Type us} [Finite σ]
+  {M : Type um} [AddCommGroup M] [Module k M] [Module (MvPolynomial σ k) M]
+  [IsScalarTower k (MvPolynomial σ k) M]
+  (𝓜 : ℤ → Submodule k M) [DirectSum.Decomposition 𝓜]
+  [SetLike.GradedSMul (polynomialGrading k σ) 𝓜]
+
+structure NonnegativeBasis where
+  index : Type um
+  finite : Fintype index
+  basis : Module.Basis index (MvPolynomial σ k) M
+  degree : index → ℕ
+  homogeneous : ∀ i, basis i ∈ 𝓜 (degree i)
+
+attribute [instance] NonnegativeBasis.finite
+
+theorem exists_nonnegativeBasis [Module.Free (MvPolynomial σ k) M]
+    [Module.Finite (MvPolynomial σ k) M] (hneg : ∀ n < 0, 𝓜 n = ⊥) :
+    Nonempty (NonnegativeBasis (σ := σ) 𝓜) := by
+  classical
+  obtain ⟨ι,hι,b,hb⟩ := exists_homogeneous_basis_polynomial (σ := σ) 𝓜 0 hneg
+  let := hι
+  let : Fintype ι := Fintype.ofFinite ι
+  choose d hd using hb
+  have hd0 (i : ι) : 0 ≤ d i := by
+    by_contra h
+    have hh := hd i
+    rw [hneg _ (lt_of_not_ge h),Submodule.mem_bot] at hh
+    exact b.ne_zero i hh
+  exact ⟨⟨ι,inferInstance,b,fun i => (d i).toNat,fun i => by simpa [Int.toNat_of_nonneg (hd0 i)] using hd i⟩⟩
+
+namespace NonnegativeBasis
+variable {𝓜} (b : NonnegativeBasis (σ := σ) 𝓜)
+
+noncomputable def polynomial : ℤ[X] := ∑ i, X ^ b.degree i
+
+omit [Finite σ] [IsScalarTower k (MvPolynomial σ k) M]
+  [DirectSum.Decomposition 𝓜] [SetLike.GradedSMul (polynomialGrading k σ) 𝓜] in
+lemma coeff_nonneg (n : ℕ) : 0 ≤ b.polynomial.coeff n := by
+  classical
+  simp only [polynomial,Polynomial.finsetSum_coeff,Polynomial.coeff_X_pow]
+  exact Finset.sum_nonneg (fun i _ => by split <;> norm_num)
+
+omit [Finite σ] [IsScalarTower k (MvPolynomial σ k) M]
+  [DirectSum.Decomposition 𝓜] [SetLike.GradedSMul (polynomialGrading k σ) 𝓜] in
+lemma eval₂_eq (q : ℝ) : Polynomial.eval₂ (Int.castRingHom ℝ) q b.polynomial =
+    ∑ i, q ^ b.degree i := by
+  change (Polynomial.eval₂RingHom (Int.castRingHom ℝ) q) (∑ i, X ^ b.degree i) = _
+  simp
+
+omit [Finite σ] in
+lemma hasSum [Fintype σ] {q : ℝ} (hq : 0 < q) (hq1 : q < 1) :
+    HasSum (fun n : ℤ => (Module.finrank k (𝓜 n) : ℝ)*q^n)
+      (Polynomial.eval₂ (Int.castRingHom ℝ) q b.polynomial * (1-q)⁻¹ ^ Fintype.card σ) := by
+  rw [b.eval₂_eq]
+  convert Hilbert.hasSum_homogeneous_basis 𝓜 b.basis (fun i => (b.degree i : ℤ))
+    b.homogeneous hq hq1 using 1
+  simp
+
+lemma polynomial_eq (c : NonnegativeBasis (σ := σ) 𝓜) : b.polynomial = c.polynomial := by
+  classical
+  let : Fintype σ := Fintype.ofFinite σ
+  apply Polynomial.map_injective (Int.castRingHom ℝ) Int.cast_injective
+  apply Polynomial.eq_of_infinite_eval_eq
+  have hsub : Set.Ioo (0 : ℝ) 1 ⊆ {q | Polynomial.eval q (b.polynomial.map (Int.castRingHom ℝ)) =
+      Polynomial.eval q (c.polynomial.map (Int.castRingHom ℝ))} := by
+    intro q hq
+    have h := (b.hasSum hq.1 hq.2).unique (c.hasSum hq.1 hq.2)
+    have hne : (1-q)⁻¹ ^ Fintype.card σ ≠ 0 := pow_ne_zero _ (inv_ne_zero (sub_ne_zero.mpr (ne_of_gt hq.2)))
+    change Polynomial.eval q (b.polynomial.map (Int.castRingHom ℝ)) =
+      Polynomial.eval q (c.polynomial.map (Int.castRingHom ℝ))
+    simpa only [Polynomial.eval_map] using mul_right_cancel₀ hne h
+  exact Set.Infinite.mono hsub (Set.Ioo_infinite (show (0 : ℝ) < 1 by norm_num))
+
+end NonnegativeBasis
+
+noncomputable def character [Module.Free (MvPolynomial σ k) M]
+    [Module.Finite (MvPolynomial σ k) M] (hneg : ∀ n < 0, 𝓜 n = ⊥) : ℤ[X] :=
+  (exists_nonnegativeBasis (σ := σ) 𝓜 hneg).some.polynomial
+
+lemma character_coeff_nonneg [Module.Free (MvPolynomial σ k) M]
+    [Module.Finite (MvPolynomial σ k) M] (hneg : ∀ n < 0, 𝓜 n = ⊥) (n : ℕ) :
+    0 ≤ (character (σ := σ) 𝓜 hneg).coeff n :=
+  (exists_nonnegativeBasis (σ := σ) 𝓜 hneg).some.coeff_nonneg n
+
+lemma character_eq [Module.Free (MvPolynomial σ k) M]
+    [Module.Finite (MvPolynomial σ k) M] (hneg : ∀ n < 0, 𝓜 n = ⊥)
+    (b : NonnegativeBasis (σ := σ) 𝓜) : character (σ := σ) 𝓜 hneg = b.polynomial :=
+  (exists_nonnegativeBasis (σ := σ) 𝓜 hneg).some.polynomial_eq b
+
+end KLInvariance.Graded
+
+end
+
+
+section
+
+namespace KLInvariance.Graded
+universe uk us
+variable (k : Type uk) [Field k] (σ : Type us)
+
+ theorem polynomial_scalar_mod_positive (a : MvPolynomial σ k) :
+    ∃ c : k, a-algebraMap k (MvPolynomial σ k) c ∈ positiveIdeal (polynomialGrading k σ) := by
+  obtain ⟨c,hc⟩ := polynomialGrading_zero k σ _
+    (DirectSum.decompose (polynomialGrading k σ) a 0).property
+  refine ⟨c,?_⟩
+  rw [positiveIdeal_eq_augmentation_ker _ (polynomialGrading_negative k σ)]
+  change component (polynomialGrading k σ) 0 (a-algebraMap k (MvPolynomial σ k) c)=0
+  rw [map_sub,hc]
+  rw [show component (polynomialGrading k σ) 0
+    (DirectSum.decompose (polynomialGrading k σ) a 0 : MvPolynomial σ k) =
+    (DirectSum.decompose (polynomialGrading k σ) a 0 : MvPolynomial σ k) from
+    DirectSum.decompose_of_mem_same _ (DirectSum.decompose (polynomialGrading k σ) a 0).property]
+  exact sub_self _
+
+end KLInvariance.Graded
+
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+
+abbrev Coefficient := MvPolynomial σ ℝ
+
+def root (e : Edge cs u b) : PositiveRoot M cs :=
+  positiveRootEquiv M cs ⟨label cs u b e,label_isReflection cs u b e⟩
+
+def polynomialLabel (e : Edge cs u b) : Coefficient (σ := σ) :=
+  SymmetricAlgebra.equivMvPolynomial basis
+    (SymmetricAlgebra.ι ℝ (Extended M) (embed M (root cs u b e).val))
+
+ theorem polynomialLabel_homogeneous (e : Edge cs u b) :
+    polynomialLabel cs u b basis e ∈ polynomialGrading ℝ σ 1 := by
+  classical
+  unfold polynomialLabel
+  rw [← basis.sum_repr (embed M (root cs u b e).val)]
+  simp only [map_sum,map_smul,SymmetricAlgebra.equivMvPolynomial_ι_apply]
+  apply Submodule.sum_mem
+  intro i _
+  exact (polynomialGrading ℝ σ 1).smul_mem _
+    (MvPolynomial.isWeightedHomogeneous_X ℝ (fun _ : σ => (1 : ℤ)) i)
+
+omit [Fintype σ] in
+ theorem polynomialLabel_ne_zero (e : Edge cs u b) :
+    polynomialLabel cs u b basis e ≠ 0 := by
+  intro he
+  have h0 : SymmetricAlgebra.ι ℝ (Extended M) (embed M (root cs u b e).val) = 0 :=
+    (SymmetricAlgebra.equivMvPolynomial basis).injective
+      (he.trans (map_zero _).symm)
+  have h1 : embed M (root cs u b e).val = 0 :=
+    EdgeAlgebra.symmetric_ι_injective (k := ℝ) (h0.trans (map_zero _).symm)
+  have h2 := congrArg Prod.fst h1
+  exact (root cs u b e).property.1.ne_zero h2
+
+
+ theorem exists_boundarySheaf (hub : BruhatLE cs u b) :
+    Nonempty (BoundarySheaf (polynomialGrading_negative ℝ σ)
+      (graph cs u b) (polynomialLabel cs u b basis) ⟨b,hub,bruhat_refl cs b⟩) := by
+  classical
+  let : Finite (Interval cs u b) := interval_finite cs u b
+  let : Fintype (Interval cs u b) := Fintype.ofFinite _
+  let : Fintype (Edge cs u b) := Fintype.ofFinite _
+  exact MomentGraph.exists_boundarySheaf (polynomialGrading_negative ℝ σ)
+    (graph cs u b) (polynomial_scalar_mod_positive ℝ σ)
+    ⟨b,hub,bruhat_refl cs b⟩ (fun x => x.property.2)
+    (polynomialLabel cs u b basis) (polynomialLabel_homogeneous cs u b basis)
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  (hub : BruhatLE cs u b)
+noncomputable section
+
+def boundarySheaf : BoundarySheaf (polynomialGrading_negative ℝ σ)
+    (graph cs u b) (polynomialLabel cs u b basis) ⟨b,hub,bruhat_refl cs b⟩ :=
+  (exists_boundarySheaf cs u b basis hub).some
+
+def stalkCharacter (x : Interval cs u b) : Polynomial ℤ := by
+  let B := boundarySheaf cs u b basis hub
+  let := B.free x
+  exact Graded.character (σ := σ) (B.sheaf.vertex x).piece (B.sheaf.vertex x).nonneg
+
+ theorem stalkCharacter_coeff_nonneg (x : Interval cs u b) (n : ℕ) :
+    0 ≤ (stalkCharacter cs u b basis hub x).coeff n := by
+  let B := boundarySheaf cs u b basis hub
+  let := B.free x
+  exact Graded.character_coeff_nonneg (σ := σ) (B.sheaf.vertex x).piece
+    (B.sheaf.vertex x).nonneg n
+
+
+def StalkCharacterObligation : Prop :=
+  ∀ x : Interval cs u b, stalkCharacter cs u b basis hub x = klPolynomial cs x.val b
+
+
+def CostalkFreeObligation : Prop :=
+  ∀ x : Interval cs u b, Module.Free (Coefficient (σ := σ))
+    (LinearMap.ker ((boundarySheaf cs u b basis hub).sheaf.forget.stalkMap x))
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+/-! Parity matching in finite lower Bruhat intervals, used only to construct
+actual dihedral interpolation sections. -/
+namespace KLInvariance
+universe u v
+variable {I : Type u} {M : CoxeterMatrix I} {W : Type v} [Group W]
+  (cs : CoxeterSystem M W)
+noncomputable section
+
+ theorem simple_mul_bruhat_of_descent {c : W} (i : I)
+    (hi : cs.length (cs.simple i*c) < cs.length c) {z : W} (hz : BruhatLE cs z c) :
+    BruhatLE cs (cs.simple i*z) c := by
+  have h := le_upperSimple cs i (cs.simple i*z)
+  rw [upperSimple_mul] at h
+  have htop : upperSimple cs i c = c := by simp only [upperSimple,ite_eq_left hi]
+  rw [← htop]
+  exact bruhat_trans cs h (upperSimple_mono cs i hz)
+
+ theorem lower_parity_balance {c : W} (hc : c ≠ 1) {p : ℕ} (hp : p < 2) :
+    ((lowerFinset cs c).filter (fun z => cs.length z % 2 = p)).card =
+      ((lowerFinset cs c).filter (fun z => cs.length z % 2 ≠ p)).card := by
+  classical
+  obtain ⟨i,hi⟩ := cs.exists_leftDescent_of_ne_one hc
+  have hmem {z : W} (hz : z ∈ lowerFinset cs c) : cs.simple i*z ∈ lowerFinset cs c :=
+    (mem_lowerFinset cs c _).mpr (simple_mul_bruhat_of_descent cs i hi
+      ((mem_lowerFinset cs c _).mp hz))
+  refine Finset.card_bij (fun z _ => cs.simple i*z) ?_ ?_ ?_
+  · intro z hz
+    obtain ⟨hz,hp'⟩ := Finset.mem_filter.mp hz
+    refine Finset.mem_filter.mpr ⟨hmem hz,?_⟩
+    have := cs.length_simple_mul z i
+    omega
+  · intro z hz w hw he
+    exact mul_left_cancel he
+  · intro z hz
+    obtain ⟨hz,hp'⟩ := Finset.mem_filter.mp hz
+    refine ⟨cs.simple i*z,Finset.mem_filter.mpr ⟨hmem hz,?_⟩,
+      cs.simple_mul_simple_cancel_left i⟩
+    have := cs.length_simple_mul z i
+    omega
+
+ def externalLower (x c : W) : Finset W := by
+  classical
+  exact (lowerFinset cs c) \ (lowerFinset cs x)
+
+@[simp] theorem mem_externalLower (x c z : W) :
+    z ∈ externalLower cs x c ↔ BruhatLE cs z c ∧ ¬ BruhatLE cs z x := by
+  classical
+  simp [externalLower]
+
+ theorem one_bruhat (x : W) : BruhatLE cs 1 x := by
+  obtain ⟨ω,hω,hx⟩ := cs.exists_isReduced x
+  simpa only [cs.wordProd_nil,← hx] using bruhat_of_subword cs (List.nil_sublist ω) hω
+
+ theorem external_parity_card_le {x c : W} (hxc : BruhatLE cs x c) :
+    ((externalLower cs x c).filter (fun z => cs.length z % 2 = cs.length x % 2)).card ≤
+      ((externalLower cs x c).filter (fun z => cs.length z % 2 ≠ cs.length x % 2)).card := by
+  classical
+  by_cases hc : c = 1
+  · subst c
+    have hx : x = 1 := bruhat_antisymm cs hxc (one_bruhat cs x)
+    subst x
+    simp [externalLower]
+  have hp : cs.length x % 2 < 2 := Nat.mod_lt _ (by omega)
+  have hbal := lower_parity_balance cs hc hp
+  have hsub : lowerFinset cs x ⊆ lowerFinset cs c := by
+    intro z hz
+    exact (mem_lowerFinset cs c z).mpr (((mem_lowerFinset cs x z).mp hz).trans hxc)
+  have hs₀ : (lowerFinset cs x).filter (fun z => cs.length z % 2 = cs.length x % 2) ⊆
+      (lowerFinset cs c).filter (fun z => cs.length z % 2 = cs.length x % 2) :=
+    Finset.filter_subset_filter _ hsub
+  have hs₁ : (lowerFinset cs x).filter (fun z => cs.length z % 2 ≠ cs.length x % 2) ⊆
+      (lowerFinset cs c).filter (fun z => cs.length z % 2 ≠ cs.length x % 2) :=
+    Finset.filter_subset_filter _ hsub
+  have heq (p : W → Prop) [DecidablePred p] :
+      (externalLower cs x c).filter p = (lowerFinset cs c).filter p \ (lowerFinset cs x).filter p := by
+    ext z
+    simp only [externalLower,Finset.mem_filter,Finset.mem_sdiff]
+    tauto
+  rw [heq,heq,Finset.card_sdiff_of_subset hs₀,Finset.card_sdiff_of_subset hs₁]
+  rw [hbal]
+  apply Nat.sub_le_sub_left
+  by_cases hx : x = 1
+  · subst x
+    have hh : lowerFinset cs (1:W) = {1} := by
+      ext z
+      simp only [mem_lowerFinset,Finset.mem_singleton]
+      exact ⟨fun h => bruhat_antisymm cs h (one_bruhat cs z),fun h => h ▸ bruhat_refl cs 1⟩
+    simp [hh,Finset.filter_singleton]
+  · exact (lower_parity_balance cs hx hp).symm.le
+
+end
+end KLInvariance
+
+end
+
+
+section
+
+/-! Every lower-supported model is an actual Bruhat BMP sheaf on [1,c],
+pulled back by the genuine lower-interval graph isomorphism and extended by
+zero. Thus its stalk character is not an abstract model selected for convenience. -/
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
+universe u
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W) (b : W)
+  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+noncomputable section
+local instance (p : Prop) : Decidable p := Classical.propDecidable p
+
+ abbrev lowerIdealGraph (c : Interval cs 1 b) :=
+  lowerGraph (graph cs 1 b) (Set.Iic c) (isLowerSet_Iic c)
+
+ def lowerIntervalMap (c : Interval cs 1 b) :
+    GraphMap (lowerIdealGraph cs b c) (graph cs 1 c.val) where
+  vertex x := ⟨x.val.val,x.val.property.1,x.property⟩
+  edge e := ⟨(⟨(source cs 1 b e.val).val,(source cs 1 b e.val).property.1,
+      bruhat_trans cs (.single e.val.property) e.property⟩,
+    ⟨(target cs 1 b e.val).val,(target cs 1 b e.val).property.1,e.property⟩),e.val.property⟩
+  source _ := rfl
+  target _ := rfl
+
+ omit [Fintype I] in
+ theorem lowerIntervalMap_injective (c : Interval cs 1 b) :
+    Function.Injective (lowerIntervalMap cs b c).vertex := by
+  intro x y h
+  exact Subtype.ext (Subtype.ext (show x.val.val=y.val.val from
+    congrArg (fun z : Interval cs 1 c.val => z.val) h))
+
+ omit [Fintype I] in
+ theorem lowerIntervalMap_outgoing (c : Interval cs 1 b)
+    (e : Edge cs 1 c.val) (x : Set.Iic c)
+    (hx : (lowerIntervalMap cs b c).vertex x=source cs 1 c.val e) :
+    ∃ a, (lowerIntervalMap cs b c).edge a=e ∧ (lowerIdealGraph cs b c).source a=x := by
+  have hxx := congrArg Subtype.val hx
+  let y : Interval cs 1 b := ⟨(target cs 1 c.val e).val,(target cs 1 c.val e).property.1,
+    bruhat_trans cs (target cs 1 c.val e).property.2 c.property.2⟩
+  let a : Edge cs 1 b := ⟨(x.val,y),hxx.symm ▸ e.property⟩
+  refine ⟨⟨a,(target cs 1 c.val e).property.2⟩,?_,?_⟩
+  · apply Subtype.ext
+    exact Prod.ext hx rfl
+  · rfl
+
+ def lowerIntervalBoundary (c : Interval cs 1 b) :
+    BoundarySheaf (polynomialGrading_negative ℝ σ) (lowerIdealGraph cs b c)
+      (fun e => polynomialLabel cs 1 b basis e.val) ⟨c,Set.mem_Iic.mpr le_rfl⟩ :=
+  (boundarySheaf cs 1 c.val basis (one_bruhat cs c.val)).pullback
+    (lowerIntervalMap cs b c) (lowerIntervalMap_injective cs b c)
+    (lowerIntervalMap_outgoing cs b c) ⟨c,Set.mem_Iic.mpr le_rfl⟩ rfl
+
+ def boundaryModels : GradedSheaf.BoundaryModels (polynomialGrading_negative ℝ σ)
+    (G := graph cs 1 b) (polynomialLabel cs 1 b basis) where
+  sheaf c := (lowerIntervalBoundary cs b basis c).sheaf.lowerExtend
+  free c := (lowerIntervalBoundary cs b basis c).sheaf.lowerExtend_free
+    (lowerIntervalBoundary cs b basis c).free
+  quotient c := (lowerIntervalBoundary cs b basis c).sheaf.lowerExtend_quotient _
+    (lowerIntervalBoundary cs b basis c).quotient
+  generated c := (lowerIntervalBoundary cs b basis c).sheaf.lowerExtend_generated
+    (lowerIntervalBoundary cs b basis c).generated
+  flabby c := (lowerIntervalBoundary cs b basis c).sheaf.lowerExtend_flabby
+    (lowerIntervalBoundary cs b basis c).flabby.flabby
+  support c x hx := (lowerIntervalBoundary cs b basis c).sheaf.lowerExtend_support x hx
+  top c := ((lowerIntervalBoundary cs b basis c).sheaf.lowerExtendVertex ⟨c,Set.mem_Iic.mpr le_rfl⟩).trans
+    (ModuleData.equivOfEq (lowerIntervalBoundary cs b basis c).top)
+  top_graded c n v hv := ModuleData.equivOfEq_graded (lowerIntervalBoundary cs b basis c).top n _
+    ((lowerIntervalBoundary cs b basis c).sheaf.lowerExtendVertex_graded ⟨c,Set.mem_Iic.mpr le_rfl⟩ n v hv)
+  minimal c x hx := (lowerIntervalBoundary cs b basis c).sheaf.lowerExtend_minimal x
+    (fun hxc => (lowerIntervalBoundary cs b basis c).minimal ⟨x,hxc⟩
+      (fun hh => hx (congrArg Subtype.val hh)))
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+namespace KLInvariance.Graded
+universe uk ua um ui
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐]
+  {M : Type um} [AddCommGroup M] [Module A M] [Module k M] [IsScalarTower k A M]
+  (𝓜 : ℤ → Submodule k M) [DirectSum.Decomposition 𝓜] [SetLike.GradedSMul 𝓐 𝓜]
+noncomputable section
+
+omit [IsScalarTower k A M] in
+ theorem basis_coordinates_graded {ι : Type ui} [Fintype ι]
+    (b : Module.Basis ι A M) (d : ι → ℤ) (hd : ∀ i, b i ∈ 𝓜 (d i))
+    (n : ℤ) (m : M) (hm : m ∈ 𝓜 n) :
+    ∀ i, b.equivFun m i ∈ 𝓐 (n-d i) := by
+  classical
+  let : DirectSum.Decomposition (shiftedPiece 𝓐 d) := (shifted_isInternal 𝓐 d).chooseDecomposition
+  have he : ∀ n v, v ∈ shiftedPiece 𝓐 d n → b.equivFun.symm v ∈ 𝓜 n := by
+    intro n v hv
+    rw [b.equivFun_symm_apply]
+    apply Submodule.sum_mem
+    intro i _
+    simpa using SetLike.GradedSMul.smul_mem (hv i) (hd i)
+  have hh := map_component (shiftedPiece 𝓐 d) 𝓜 b.equivFun.symm.toLinearMap 0
+    (by simpa using he) n (b.equivFun m)
+  simp only [sub_zero,LinearEquiv.coe_toLinearMap,LinearEquiv.symm_apply_apply] at hh
+  rw [show component 𝓜 n m=m from DirectSum.decompose_of_mem_same 𝓜 hm] at hh
+  have hc : component (shiftedPiece 𝓐 d) n (b.equivFun m)=b.equivFun m := by
+    apply b.equivFun.symm.injective
+    exact hh.symm.trans (b.equivFun.symm_apply_apply m).symm
+  rw [← hc]
+  exact (DirectSum.decompose (shiftedPiece 𝓐 d) (b.equivFun m) n).property
+
+end
+end KLInvariance.Graded
+
+namespace KLInvariance.Graded.ModuleData
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+  [IsDomain A] [IsNoetherianRing A]
+  (hneg : ∀ n < 0, 𝓐 n = ⊥)
+  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
+noncomputable section
+
+include hneg hzero in
+ theorem exists_equiv_shiftedUnits (F : ModuleData.{uk,ua,um} 𝓐) [Module.Free A F] :
+    ∃ (n : ℕ) (d : ULift.{um} (Fin n) → ℕ)
+      (e : F ≃ₗ[A] pi (fun i : ULift.{um} (Fin n) => (unit hneg).shift (d i))),
+      ∀ m v, v ∈ F.piece m → e v ∈
+        (pi (fun i : ULift.{um} (Fin n) => (unit hneg).shift (d i))).piece m := by
+  classical
+  obtain ⟨ι,hι,b0,hd0⟩ := exists_homogeneous_basis_connected 𝓐 F.piece hneg hzero 0 F.nonneg
+  let : Finite ι := hι
+  let : Fintype ι := Fintype.ofFinite ι
+  let n := Fintype.card ι
+  let J := ULift.{um} (Fin n)
+  let re : ι ≃ J := (Fintype.equivFin ι).trans Equiv.ulift.symm
+  let b := b0.reindex re
+  choose dz hdz using hd0
+  let d : J → ℕ := fun i => (dz (re.symm i)).toNat
+  have hd (i : J) : b i ∈ F.piece (d i : ℤ) := by
+    have hz : 0 ≤ dz (re.symm i) := by
+      by_contra h
+      have he : b0 (re.symm i)=0 := by
+        simpa only [F.nonneg _ (lt_of_not_ge h),Submodule.mem_bot] using hdz (re.symm i)
+      exact b0.ne_zero (re.symm i) he
+    simpa only [b,Module.Basis.reindex_apply,d,Int.toNat_of_nonneg hz] using hdz (re.symm i)
+  let e : F ≃ₗ[A] pi (fun i : J => (unit hneg).shift (d i)) :=
+    { toFun v i _ := b.equivFun v i
+      invFun v := b.equivFun.symm (fun i => v i PUnit.unit)
+      left_inv v := b.equivFun.symm_apply_apply v
+      right_inv v := by
+        funext i j
+        change b.equivFun (b.equivFun.symm (fun i => v i PUnit.unit)) i=v i j
+        rw [LinearEquiv.apply_symm_apply]
+      map_add' v w := by funext i j; exact congrFun (b.equivFun.map_add v w) i
+      map_smul' a v := by funext i j; exact congrFun (b.equivFun.map_smul a v) i }
+  refine ⟨n,d,e,?_⟩
+  intro m v hv i _ j
+  change b.equivFun v i ∈ 𝓐 (m-(d i : ℤ)-0)
+  rw [sub_zero]
+  exact basis_coordinates_graded F.piece b (fun i => (d i : ℤ)) hd m v hv i
+
+end
+end KLInvariance.Graded.ModuleData
+
+end
+
+
+section
+
+namespace KLInvariance.QuotientEdge
+variable {R M N E F : Type*} [CommRing R]
+  [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N]
+  [AddCommGroup E] [Module R E] [AddCommGroup F] [Module R F]
+noncomputable section
+
+ def descend (f : M →ₗ[R] E) (g : N →ₗ[R] F)
+    (hf : Function.Surjective f) (h : M →ₗ[R] N)
+    (hk : LinearMap.ker f ≤ LinearMap.ker (g.comp h)) : E →ₗ[R] F :=
+  ((LinearMap.ker f).liftQ (g.comp h) hk).comp (f.quotKerEquivOfSurjective hf).symm.toLinearMap
+
+ theorem descend_apply (f : M →ₗ[R] E) (g : N →ₗ[R] F)
+    (hf : Function.Surjective f) (h : M →ₗ[R] N)
+    (hk : LinearMap.ker f ≤ LinearMap.ker (g.comp h)) (v : M) :
+    descend f g hf h hk (f v) = g (h v) := by
+  simp [descend]
+
+ theorem kernel_scalar_le (f : M →ₗ[R] E) (g : N →ₗ[R] F) (r : R)
+    (hf : ∀ v, f v = 0 ↔ ∃ w, v=r • w)
+    (hg : ∀ v, g v = 0 ↔ ∃ w, v=r • w) (h : M →ₗ[R] N) :
+    LinearMap.ker f ≤ LinearMap.ker (g.comp h) := by
+  intro v hv
+  obtain ⟨w,rfl⟩ := (hf v).mp hv
+  change g (h (r • w))=0
+  rw [h.map_smul]
+  exact (hg _).mpr ⟨h w,rfl⟩
+
+end
+end KLInvariance.QuotientEdge
+
+namespace KLInvariance.Graded.ModuleData
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A}
+noncomputable section
+
+ def quotientHom {M N E F : ModuleData.{uk,ua,um} 𝓐}
+    (f : Hom M E) (g : Hom N F) (hf : Function.Surjective f.map)
+    (h : Hom M N) (hk : LinearMap.ker f.map ≤ LinearMap.ker (g.map.comp h.map)) : Hom E F where
+  map := QuotientEdge.descend f.map g.map hf h.map hk
+  graded := by
+    intro n v hv
+    obtain ⟨w,hw,rfl⟩ := f.homogeneous_preimage hf n v hv
+    rw [QuotientEdge.descend_apply]
+    exact g.graded n _ (h.graded n w hw)
+
+ theorem quotientHom_apply {M N E F : ModuleData.{uk,ua,um} 𝓐}
+    (f : Hom M E) (g : Hom N F) (hf : Function.Surjective f.map)
+    (h : Hom M N) (hk : LinearMap.ker f.map ≤ LinearMap.ker (g.map.comp h.map))
+    (v : M) : (quotientHom f g hf h hk).map (f.map v) = g.map (h.map v) :=
+  QuotientEdge.descend_apply _ _ _ _ _ v
+
+end
+end KLInvariance.Graded.ModuleData
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
+  {G : OrderedGraph V E}
+
+structure Hom (B C : GradedSheaf (𝓐 := 𝓐) G) where
+  vertex : ∀ x, ModuleData.Hom (B.vertex x) (C.vertex x)
+  edge : ∀ e, ModuleData.Hom (B.edge e) (C.edge e)
+  lower : ∀ e v, (edge e).map ((B.lower e).map v) =
+    (C.lower e).map ((vertex (G.source e)).map v)
+  upper : ∀ e v, (edge e).map ((B.upper e).map v) =
+    (C.upper e).map ((vertex (G.target e)).map v)
+
+structure OnHom (B C : GradedSheaf (𝓐 := 𝓐) G) (U : Set V) where
+  vertex : ∀ x, x ∈ U → ModuleData.Hom (B.vertex x) (C.vertex x)
+  edge : ∀ e, G.target e ∈ U → ModuleData.Hom (B.edge e) (C.edge e)
+  lower : ∀ e (hs : G.source e ∈ U) (ht : G.target e ∈ U) v,
+    (edge e ht).map ((B.lower e).map v) =
+      (C.lower e).map ((vertex (G.source e) hs).map v)
+  upper : ∀ e (ht : G.target e ∈ U) v,
+    (edge e ht).map ((B.upper e).map v) =
+      (C.upper e).map ((vertex (G.target e) ht).map v)
+
+namespace Hom
+variable {B C D : GradedSheaf (𝓐 := 𝓐) G}
+
+def comp (g : Hom C D) (f : Hom B C) : Hom B D where
+  vertex x := (g.vertex x).comp (f.vertex x)
+  edge e := (g.edge e).comp (f.edge e)
+  lower e v := by
+    change (g.edge e).map ((f.edge e).map ((B.lower e).map v)) = _
+    rw [f.lower,g.lower]; rfl
+  upper e v := by
+    change (g.edge e).map ((f.edge e).map ((B.upper e).map v)) = _
+    rw [f.upper,g.upper]; rfl
+
+end Hom
+
+namespace OnHom
+variable {B C : GradedSheaf (𝓐 := 𝓐) G} {U : Set V} (j : OnHom B C U)
+noncomputable section
+
+ def ofUniv (j : OnHom B C Set.univ) : Hom B C where
+  vertex x := j.vertex x (Set.mem_univ x)
+  edge e := j.edge e (Set.mem_univ _)
+  lower e v := j.lower e (Set.mem_univ _) (Set.mem_univ _) v
+  upper e v := j.upper e (Set.mem_univ _) v
+
+ def mapAssignment (x : V) (hx : Set.Ioi x ⊆ U)
+    (f : B.forget.Assignment) : C.forget.Assignment := by
+  classical
+  exact fun y => if h : x < y then (j.vertex y (hx h)).map (f y) else 0
+
+ theorem mapAssignment_at (x : V) (hx : Set.Ioi x ⊆ U)
+    (f : B.forget.Assignment) (y : V) (h : x < y) :
+    j.mapAssignment x hx f y = (j.vertex y (hx h)).map (f y) := by
+  classical
+  unfold mapAssignment
+  exact dite_eq_left h
+
+ theorem compatible (x : V) (hx : Set.Ioi x ⊆ U)
+    (f : B.forget.Assignment) (hf : B.forget.CompatibleOn (Set.Ioi x) f) :
+    C.forget.CompatibleOn (Set.Ioi x) (j.mapAssignment x hx f) := by
+  intro e hs ht
+  change x < G.source e at hs
+  change x < G.target e at ht
+  rw [j.mapAssignment_at x hx f _ hs,j.mapAssignment_at x hx f _ ht]
+  change (C.lower e).map ((j.vertex _ (hx hs)).map (f _)) =
+    (C.upper e).map ((j.vertex _ (hx ht)).map (f _))
+  exact (j.lower e (hx hs) (hx ht) _).symm.trans
+    ((congrArg (j.edge e (hx ht)).map (hf e hs ht)).trans (j.upper e (hx ht) _))
+
+variable [Fintype E]
+
+ def boundary (x : V) (hx : Set.Ioi x ⊆ U) :
+    ModuleData.Hom (B.boundary x) (C.boundary x) where
+  map := LinearMap.pi fun e =>
+    ((j.edge e.val (hx (lt_of_eq_of_lt e.property.symm (G.increasing e.val)))).map).comp
+      (LinearMap.proj e)
+  graded := by
+    intro n v hv e he
+    exact (j.edge e.val _).graded n _ (hv e he)
+
+ theorem boundaryMap (x : V) (hx : Set.Ioi x ⊆ U)
+    (f : B.forget.sections (Set.Ioi x)) :
+    (j.boundary x hx).map (B.forget.boundaryMap x f) =
+      C.forget.boundaryMap x ⟨j.mapAssignment x hx f.val,j.compatible x hx f.val f.property⟩ := by
+  funext e
+  change (j.edge e.val _).map ((B.upper e.val).map (f.val (G.target e.val))) =
+    (C.upper e.val).map (j.mapAssignment x hx f.val (G.target e.val))
+  rw [j.mapAssignment_at _ _ _ _ (lt_of_eq_of_lt e.property.symm (G.increasing e.val))]
+  exact j.upper e.val _ _
+
+ theorem map_stalk_image_le (x : V) (hx : Set.Ioi x ⊆ U)
+    (hBf : B.forget.Flabby) (hBg : B.forget.Generated)
+    (hCf : C.forget.Flabby) (hCg : C.forget.Generated) :
+    (LinearMap.range (B.forget.stalkMap x)).map (j.boundary x hx).map ≤
+      LinearMap.range (C.forget.stalkMap x) := by
+  rw [B.forget.range_stalkMap_eq_boundaryImage hBf hBg,
+    C.forget.range_stalkMap_eq_boundaryImage hCf hCg]
+  rintro v ⟨w,⟨f,rfl⟩,rfl⟩
+  exact ⟨⟨j.mapAssignment x hx f.val,j.compatible x hx f.val f.property⟩,
+    (j.boundaryMap x hx f).symm⟩
+
+variable [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+  [IsDomain A] [IsNoetherianRing A]
+  (hneg : ∀ n < 0, 𝓐 n = ⊥)
+  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
+
+include hneg hzero in
+ theorem extend_stalk (x : V) (hx : Set.Ioi x ⊆ U)
+    [Module.Free A (B.vertex x)]
+    (hBf : B.forget.Flabby) (hBg : B.forget.Generated)
+    (hCf : C.forget.Flabby) (hCg : C.forget.Generated) :
+    ∃ f : ModuleData.Hom (B.vertex x) (C.vertex x), ∀ v,
+      C.forget.stalkMap x (f.map v) = (j.boundary x hx).map (B.forget.stalkMap x v) := by
+  let T := ModuleData.range (C.stalkHom x)
+  let f : ModuleData.Hom (B.vertex x) T :=
+    { map := LinearMap.codRestrict (LinearMap.range (C.forget.stalkMap x))
+        ((j.boundary x hx).map.comp (B.forget.stalkMap x)) (by
+          intro v
+          exact j.map_stalk_image_le x hx hBf hBg hCf hCg ⟨_,⟨v,rfl⟩,rfl⟩)
+      graded := fun n v hv => (j.boundary x hx).graded n _ ((B.stalkHom x).graded n v hv) }
+  let π : ModuleData.Hom (C.vertex x) T :=
+    { map := (C.forget.stalkMap x).rangeRestrict
+      graded := fun n v hv => (C.stalkHom x).graded n v hv }
+  have hπ : Function.Surjective π.map := by
+    intro v
+    obtain ⟨w,hw⟩ := v.property
+    exact ⟨w,Subtype.ext hw⟩
+  obtain ⟨l,hl⟩ := ModuleData.exists_hom_lift hneg hzero π hπ f
+  refine ⟨l,fun v => ?_⟩
+  exact congrArg (fun z : T => z.val) (LinearMap.congr_fun hl v)
+
+end
+end OnHom
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
+  {G : OrderedGraph V E} {B C : GradedSheaf (𝓐 := 𝓐) G}
+
+ def castVertexHom {x y : V} (h : x=y)
+    (f : ModuleData.Hom (B.vertex x) (C.vertex x)) :
+    ModuleData.Hom (B.vertex y) (C.vertex y) := h ▸ f
+
+namespace OnHom
+variable {U : Set V} (j : OnHom B C U)
+noncomputable section
+
+ def extend (hU : IsUpperSet U) (x : V) (hx : x ∉ U)
+    (hmax : Set.Ioi x ⊆ U) (α : E → A)
+    (hBq : B.UpperQuotient α) (hCq : C.UpperQuotient α)
+    (jv : ModuleData.Hom (B.vertex x) (C.vertex x))
+    (hout : ∀ e (he : G.source e = x) v,
+      (j.edge e (hmax (lt_of_eq_of_lt he.symm (G.increasing e)))).map
+        ((B.lower e).map v) =
+      (C.lower e).map ((castVertexHom he.symm jv).map v)) :
+    OnHom B C (insert x U) := by
+  classical
+  let vj : ∀ y, y ∈ insert x U → ModuleData.Hom (B.vertex y) (C.vertex y) := fun y hy =>
+    if h : y=x then castVertexHom h.symm jv
+    else j.vertex y ((Set.mem_insert_iff.mp hy).resolve_left h)
+  have vj_old (y : V) (hy : y ∈ U) : vj y (Set.mem_insert_of_mem x hy) = j.vertex y hy := by
+    simp only [vj,dite_eq_right (show y ≠ x from fun h => hx (h ▸ hy))]
+  have vj_new : vj x (Set.mem_insert x U) = jv := by simp [vj,castVertexHom]
+  let ej : ∀ e, G.target e ∈ insert x U → ModuleData.Hom (B.edge e) (C.edge e) := fun e he =>
+    if h : G.target e=x then
+      ModuleData.quotientHom (B.upper e) (C.upper e) (hBq e).1 (vj (G.target e) he)
+        (QuotientEdge.kernel_scalar_le (B.upper e).map (C.upper e).map (α e)
+          (hBq e).2 (hCq e).2 (vj (G.target e) he).map)
+    else j.edge e ((Set.mem_insert_iff.mp he).resolve_left h)
+  have ej_old (e : E) (he : G.target e ∈ U) :
+      ej e (Set.mem_insert_of_mem x he) = j.edge e he := by
+    simp only [ej,dite_eq_right (show G.target e ≠ x from fun h => hx (h ▸ he))]
+  refine ⟨vj,ej,?_,?_⟩
+  · intro e hs ht v
+    have htU : G.target e ∈ U := by
+      rcases Set.mem_insert_iff.mp hs with hs | hs
+      · exact hmax (lt_of_eq_of_lt hs.symm (G.increasing e))
+      · exact hU (G.increasing e).le hs
+    rw [ej_old e htU]
+    by_cases he : G.source e=x
+    · subst x
+      rw [vj_new]
+      exact hout e rfl v
+    · have hsU : G.source e ∈ U := (Set.mem_insert_iff.mp hs).resolve_left he
+      rw [vj_old _ hsU]
+      exact j.lower e hsU htU v
+  · intro e ht v
+    by_cases he : G.target e=x
+    · simp only [ej,dite_eq_left he]
+      exact ModuleData.quotientHom_apply _ _ _ _ _ v
+    · have htU : G.target e ∈ U := (Set.mem_insert_iff.mp ht).resolve_left he
+      rw [ej_old e htU,vj_old _ htU]
+      exact j.upper e htU v
+
+ theorem extend_vertex_old (hU : IsUpperSet U) (x : V) (hx : x ∉ U)
+    (hmax : Set.Ioi x ⊆ U) (α : E → A)
+    (hBq : B.UpperQuotient α) (hCq : C.UpperQuotient α)
+    (jv : ModuleData.Hom (B.vertex x) (C.vertex x))
+    (hout : ∀ e (he : G.source e = x) v,
+      (j.edge e (hmax (lt_of_eq_of_lt he.symm (G.increasing e)))).map
+        ((B.lower e).map v) =
+      (C.lower e).map ((castVertexHom he.symm jv).map v))
+    (y : V) (hy : y ∈ U) :
+    (j.extend hU x hx hmax α hBq hCq jv hout).vertex y (Set.mem_insert_of_mem x hy) =
+      j.vertex y hy := by
+  classical
+  simp only [extend,dite_eq_right (show y ≠ x from fun h => hx (h ▸ hy))]
+
+ theorem extend_vertex_new (hU : IsUpperSet U) (x : V) (hx : x ∉ U)
+    (hmax : Set.Ioi x ⊆ U) (α : E → A)
+    (hBq : B.UpperQuotient α) (hCq : C.UpperQuotient α)
+    (jv : ModuleData.Hom (B.vertex x) (C.vertex x))
+    (hout : ∀ e (he : G.source e = x) v,
+      (j.edge e (hmax (lt_of_eq_of_lt he.symm (G.increasing e)))).map
+        ((B.lower e).map v) =
+      (C.lower e).map ((castVertexHom he.symm jv).map v)) :
+    (j.extend hU x hx hmax α hBq hCq jv hout).vertex x (Set.mem_insert x U) = jv := by
+  classical
+  dsimp only [extend]
+  exact dite_eq_left rfl
+
+end
+end OnHom
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+/-! Integral graded lifting of sheaf morphisms. This is the actual minimal
+sheaf recognition mechanism needed to split it out of a Bott--Samelson sheaf;
+it does not assume or establish the KL character theorem. -/
+namespace KLInvariance.MomentGraph.GradedSheaf.OnHom
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+  [IsDomain A] [IsNoetherianRing A]
+  (hneg : ∀ n < 0, 𝓐 n = ⊥)
+  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
+  {V E : Type um} [PartialOrder V] [Fintype V] [Fintype E]
+  {G : OrderedGraph V E} {B C : GradedSheaf (𝓐 := 𝓐) G}
+  (α : E → A) (hBq : B.UpperQuotient α) (hCq : C.UpperQuotient α)
+  (hBfree : ∀ x, Module.Free A (B.vertex x))
+  (hBf : B.forget.Flabby) (hBg : B.forget.Generated)
+  (hCf : C.forget.Flabby) (hCg : C.forget.Generated)
+noncomputable section
+
+omit [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+  [IsDomain A] [IsNoetherianRing A] [Fintype V] [Fintype E] in
+private theorem cast_vertex {U U' : Set V} (h : U=U') (j : OnHom B C U)
+    (y : V) (hy : y ∈ U') :
+    (h ▸ j).vertex y hy = j.vertex y (h.symm ▸ hy) := by
+  subst U'
+  rfl
+
+include hneg hzero hBq hCq hBfree hBf hBg hCf hCg in
+ theorem exists_extension {U : Set V} (hU : IsUpperSet U) (j : OnHom B C U) :
+    ∃ f : Hom B C, ∀ x (hx : x ∈ U), f.vertex x = j.vertex x hx := by
+  classical
+  have h : ∀ s : Finset V, IsUpperSet ((s : Set V)ᶜ) →
+      ∀ j : OnHom B C ((s : Set V)ᶜ),
+      ∃ f : Hom B C, ∀ x (hx : x ∈ ((s : Set V)ᶜ)), f.vertex x = j.vertex x hx := by
+    intro s
+    induction s using Finset.strongInductionOn with
+    | _ s ih =>
+      intro hU j
+      by_cases hs : s=∅
+      · subst s
+        have heq : ((∅ : Finset V) : Set V)ᶜ=Set.univ := by simp
+        let jj : OnHom B C Set.univ := heq ▸ j
+        refine ⟨jj.ofUniv,fun y hy => ?_⟩
+        exact cast_vertex heq j y (Set.mem_univ y)
+      · obtain ⟨x,hxs,hxmax⟩ := s.exists_maximal (Finset.nonempty_iff_ne_empty.mpr hs)
+        have hx : x ∉ ((s : Set V)ᶜ) := by simpa using hxs
+        have hmax : Set.Ioi x ⊆ ((s : Set V)ᶜ) := by
+          intro y hy
+          change y ∉ s
+          intro hys
+          exact (not_le_of_gt hy) (hxmax hys hy.le)
+        let := hBfree x
+        obtain ⟨l,hl⟩ := j.extend_stalk hneg hzero x hmax hBf hBg hCf hCg
+        have hout : ∀ e (he : G.source e=x) v,
+            (j.edge e (hmax (lt_of_eq_of_lt he.symm (G.increasing e)))).map
+              ((B.lower e).map v) =
+            (C.lower e).map ((castVertexHom he.symm l).map v) := by
+          intro e he v
+          subst x
+          exact (congrFun (hl v) ⟨e,rfl⟩).symm
+        let jj := j.extend hU x hx hmax α hBq hCq l hout
+        have heq : insert x ((s : Set V)ᶜ) = ((s.erase x : Finset V) : Set V)ᶜ := by
+          ext y
+          simp only [Set.mem_insert_iff,Set.mem_compl_iff,Finset.mem_coe,Finset.mem_erase]
+          tauto
+        have hU' : IsUpperSet (insert x ((s : Set V)ᶜ)) := by
+          intro y t hyt hy
+          rcases Set.mem_insert_iff.mp hy with hy | hy
+          · subst y
+            by_cases htx : t=x
+            · exact Set.mem_insert_iff.mpr (Or.inl htx)
+            · exact Set.mem_insert_of_mem x (hmax (lt_of_le_of_ne hyt (Ne.symm htx)))
+          · exact Set.mem_insert_of_mem x (hU hyt hy)
+        obtain ⟨f,hf⟩ := ih (s.erase x) (Finset.erase_ssubset hxs) (heq ▸ hU') (heq ▸ jj)
+        refine ⟨f,fun y hy => ?_⟩
+        have hy' : y ∈ ((s.erase x : Finset V) : Set V)ᶜ :=
+          heq ▸ Set.mem_insert_of_mem x hy
+        exact (hf y hy').trans ((cast_vertex heq jj y hy').trans
+          (j.extend_vertex_old hU x hx hmax α hBq hCq l hout y hy))
+  let s := (Uᶜ).toFinset
+  have heq : (s : Set V)ᶜ=U := by simp [s]
+  obtain ⟨f,hf⟩ := h s (heq.symm ▸ hU) (heq.symm ▸ j)
+  refine ⟨f,fun x hx => ?_⟩
+  exact (hf x (heq.symm ▸ hx)).trans (cast_vertex heq.symm j x (heq.symm ▸ hx))
+
+end
+end KLInvariance.MomentGraph.GradedSheaf.OnHom
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+  [IsDomain A] [IsNoetherianRing A]
+  (hneg : ∀ n < 0, 𝓐 n = ⊥)
+  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
+  {V E : Type um} [PartialOrder V] [Fintype V] [Fintype E]
+  {G : OrderedGraph V E} {B C : GradedSheaf (𝓐 := 𝓐) G}
+  (α : E → A) (hBq : B.UpperQuotient α) (hCq : C.UpperQuotient α)
+  (hBfree : ∀ x, Module.Free A (B.vertex x))
+  (hBf : B.forget.Flabby) (hBg : B.forget.Generated)
+  (hCf : C.forget.Flabby) (hCg : C.forget.Generated)
+  (b : V) (hb : ∀ x, x ≤ b)
+noncomputable section
+
+include hneg hzero hBq hCq hBfree hBf hBg hCf hCg hb in
+ theorem exists_hom_top (t : ModuleData.Hom (B.vertex b) (C.vertex b)) :
+    ∃ f : Hom B C, f.vertex b=t := by
+  let je : OnHom B C (∅ : Set V) :=
+    { vertex x hx := False.elim hx
+      edge e he := False.elim he
+      lower e hs := False.elim hs
+      upper e ht := False.elim ht }
+  have hmax : Set.Ioi b ⊆ (∅ : Set V) := by
+    intro x hx
+    exact False.elim ((not_lt_of_ge (hb x)) hx)
+  have hout : ∀ e (he : G.source e=b) v,
+      (je.edge e (hmax (lt_of_eq_of_lt he.symm (G.increasing e)))).map
+        ((B.lower e).map v) =
+      (C.lower e).map ((castVertexHom he.symm t).map v) := by
+    intro e he
+    exact False.elim ((not_lt_of_ge (hb (G.target e)))
+      (lt_of_eq_of_lt he.symm (G.increasing e)))
+  let jj := je.extend isUpperSet_empty b (Set.notMem_empty b) hmax α hBq hCq t hout
+  have hU : IsUpperSet (insert b (∅ : Set V)) := by
+    intro x y hxy hx
+    have hxb : x=b := by simpa using hx
+    subst x
+    have hyb : y=b := le_antisymm (hb y) hxy
+    simpa using hyb
+  obtain ⟨f,hf⟩ := jj.exists_extension hneg hzero α hBq hCq hBfree hBf hBg hCf hCg hU
+  exact ⟨f,(hf b (Set.mem_insert b _)).trans
+    (je.extend_vertex_new isUpperSet_empty b (Set.notMem_empty b) hmax α hBq hCq t hout)⟩
+
+end
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+/-! A homogeneous idempotent with image in the positive-ideal multiple of a
+bounded-below graded module is zero. This is the minimal-cover rigidity needed
+to identify the indecomposable in the BMP construction. -/
+namespace KLInvariance.Graded.ModuleData
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} (M : ModuleData.{uk,ua,um} 𝓐)
+
+ theorem idempotent_eq_zero (f : Hom M M)
+    (hi : ∀ v, f.map (f.map v) = f.map v)
+    (hm : LinearMap.range f.map ≤ positiveIdeal 𝓐 • (⊤ : Submodule A M)) :
+    f.map = 0 := by
+  have hker : LinearMap.ker f.map = ⊤ :=
+    eq_top_of_positive_smul_sup 𝓐 M.piece 0 M.nonneg (LinearMap.ker f.map)
+      (by
+        intro n m h
+        change f.map (component M.piece n m) = 0
+        rw [← f.component, LinearMap.mem_ker.mp h, map_zero])
+      (by
+        apply top_unique
+        intro m _
+        apply Submodule.mem_sup.mpr
+        refine ⟨m-f.map m,?_,f.map m,hm ⟨m,rfl⟩,sub_add_cancel _ _⟩
+        change f.map (m-f.map m) = 0
+        rw [map_sub,hi,sub_self])
+  exact LinearMap.ker_eq_top.mp hker
+
+end KLInvariance.Graded.ModuleData
+
+end
+
+
+section
+
+/-! The actual minimal boundary sheaf is indecomposable: it has no nontrivial
+homogeneous idempotent. No KL character or costalk freeness is used here. -/
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
+  {G : OrderedGraph V E}
+
+structure Endomorphism (B : GradedSheaf (𝓐 := 𝓐) G) where
+  vertex : ∀ x, ModuleData.Hom (B.vertex x) (B.vertex x)
+  edge : ∀ e, ModuleData.Hom (B.edge e) (B.edge e)
+  lower : ∀ e v, (edge e).map ((B.lower e).map v) =
+    (B.lower e).map ((vertex (G.source e)).map v)
+  upper : ∀ e v, (edge e).map ((B.upper e).map v) =
+    (B.upper e).map ((vertex (G.target e)).map v)
+
+namespace Endomorphism
+variable {B : GradedSheaf (𝓐 := 𝓐) G} (f : Endomorphism B)
+
+ def complement : Endomorphism B where
+  vertex x :=
+    { map := LinearMap.id - (f.vertex x).map
+      graded := fun n v hv => (B.vertex x).piece n |>.sub_mem hv ((f.vertex x).graded n v hv) }
+  edge e :=
+    { map := LinearMap.id - (f.edge e).map
+      graded := fun n v hv => (B.edge e).piece n |>.sub_mem hv ((f.edge e).graded n v hv) }
+  lower e v := by
+    change (B.lower e).map v - (f.edge e).map ((B.lower e).map v) =
+      (B.lower e).map (v-(f.vertex (G.source e)).map v)
+    rw [map_sub,f.lower]
+  upper e v := by
+    change (B.upper e).map v - (f.edge e).map ((B.upper e).map v) =
+      (B.upper e).map (v-(f.vertex (G.target e)).map v)
+    rw [map_sub,f.upper]
+
+ theorem complement_idempotent (hi : ∀ x v, (f.vertex x).map ((f.vertex x).map v) =
+      (f.vertex x).map v) (x : V) (v : B.vertex x) :
+    (f.complement.vertex x).map ((f.complement.vertex x).map v) =
+      (f.complement.vertex x).map v := by
+  change v-(f.vertex x).map v - (f.vertex x).map (v-(f.vertex x).map v) =
+    v-(f.vertex x).map v
+  rw [map_sub,hi,sub_self,sub_zero]
+
+end Endomorphism
+end KLInvariance.MomentGraph.GradedSheaf
+
+namespace KLInvariance.Graded.ModuleData
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+  (hneg : ∀ n < 0, 𝓐 n = ⊥)
+
+ theorem unit_idempotent_trivial [IsDomain A]
+    (f : Hom (unit.{uk,ua,um} hneg) (unit.{uk,ua,um} hneg))
+    (hi : ∀ v, f.map (f.map v) = f.map v) :
+    f.map = 0 ∨ f.map = LinearMap.id := by
+  let e : unit.{uk,ua,um} hneg := fun _ => 1
+  let c : A := f.map e PUnit.unit
+  have expand (v : unit.{uk,ua,um} hneg) : v = v PUnit.unit • e := by
+    funext i
+    cases i
+    exact (mul_one _).symm
+  have hf (v : unit.{uk,ua,um} hneg) (i : PUnit.{max ua um + 1}) :
+      f.map v i = v PUnit.unit * c := by
+    calc
+      f.map v i = f.map (v PUnit.unit • e) i := congrArg (fun v => f.map v i) (expand v)
+      _ = v PUnit.unit * c := by rw [map_smul]; cases i; rfl
+  have hc : c*c=c := by
+    have h := congrFun (hi e) PUnit.unit
+    rw [hf (f.map e) PUnit.unit] at h
+    exact h
+  have hz : c*(c-1)=0 := by rw [mul_sub,mul_one,hc,sub_self]
+  rcases mul_eq_zero.mp hz with hc | hc
+  · left
+    apply LinearMap.ext
+    intro v
+    funext i
+    rw [hf,hc,mul_zero]
+    rfl
+  · right
+    have hc : c=1 := sub_eq_zero.mp hc
+    apply LinearMap.ext
+    intro v
+    funext i
+    rw [hf,hc,mul_one]
+    cases i
+    rfl
+
+end KLInvariance.Graded.ModuleData
+
+namespace KLInvariance.MomentGraph.BoundarySheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+  {hneg : ∀ n < 0, 𝓐 n = ⊥} {V E : Type um} [PartialOrder V]
+  {G : OrderedGraph V E} {α : E → A} {b : V}
+  (B : BoundarySheaf hneg G α b)
+
+ theorem idempotent_zero_of_top_zero [Finite V]
+    (f : B.sheaf.Endomorphism)
+    (hi : ∀ x v, (f.vertex x).map ((f.vertex x).map v) = (f.vertex x).map v)
+    (ht : (f.vertex b).map = 0) :
+    (∀ x, (f.vertex x).map = 0) ∧ (∀ e, (f.edge e).map = 0) := by
+  have hv : ∀ x, (f.vertex x).map = 0 := by
+    intro x
+    induction x using WellFoundedGT.induction with
+    | ind x ih =>
+      by_cases hx : x=b
+      · exact hx ▸ ht
+      apply ModuleData.idempotent_eq_zero (B.sheaf.vertex x) (f.vertex x) (hi x)
+      intro v hv
+      obtain ⟨w,rfl⟩ := hv
+      apply (B.sheaf.kernelMinimal_iff x).mp (B.minimal x hx)
+      change B.sheaf.forget.stalkMap x ((f.vertex x).map w) = 0
+      funext e
+      rcases e with ⟨e,he⟩
+      subst x
+      change (B.sheaf.lower e).map ((f.vertex (G.source e)).map w) = 0
+      rw [← f.lower]
+      have hezero : (f.edge e).map = 0 := by
+        ext z
+        obtain ⟨y,rfl⟩ := (B.quotient e).1 z
+        rw [f.upper,ih (G.target e) (G.increasing e)]
+        exact map_zero _
+      rw [hezero]
+      rfl
+  refine ⟨hv,?_⟩
+  intro e
+  ext z
+  obtain ⟨y,rfl⟩ := (B.quotient e).1 z
+  rw [f.upper,hv]
+  exact map_zero _
+
+/-- Actual indecomposability in the idempotent formulation. A direct sum
+splitting supplies such a homogeneous idempotent, hence one summand vanishes. -/
+ theorem idempotent_trivial [Finite V] [IsDomain A]
+    (f : B.sheaf.Endomorphism)
+    (hi : ∀ x v, (f.vertex x).map ((f.vertex x).map v) = (f.vertex x).map v) :
+    ((∀ x, (f.vertex x).map = 0) ∧ (∀ e, (f.edge e).map = 0)) ∨
+      ((∀ x, (f.vertex x).map = LinearMap.id) ∧
+        (∀ e, (f.edge e).map = LinearMap.id)) := by
+  have ht : (f.vertex b).map = 0 ∨ (f.vertex b).map = LinearMap.id := by
+    have h := B.top
+    have general (N : ModuleData.{uk,ua,um} 𝓐) (hN : N=ModuleData.unit hneg)
+        (g : ModuleData.Hom N N) (hg : ∀v, g.map (g.map v)=g.map v) :
+        g.map=0 ∨ g.map=LinearMap.id := by
+      subst N
+      exact ModuleData.unit_idempotent_trivial hneg g hg
+    exact general _ h (f.vertex b) (hi b)
+  rcases ht with ht | ht
+  · exact Or.inl (B.idempotent_zero_of_top_zero f hi ht)
+  · right
+    have hh : (f.complement.vertex b).map = 0 := by
+      change LinearMap.id - (f.vertex b).map = 0
+      rw [ht,sub_self]
+    obtain ⟨hv,he⟩ := B.idempotent_zero_of_top_zero f.complement
+      (f.complement_idempotent hi) hh
+    constructor
+    · intro x
+      exact (sub_eq_zero.mp (hv x)).symm
+    · intro e
+      exact (sub_eq_zero.mp (he e)).symm
+
+end KLInvariance.MomentGraph.BoundarySheaf
+
+end
+
+
+section
+
+/-! A genuine homogeneous self-map of a minimal cover is invertible when its
+map on the covered module is invertible. Used for the integral summand bridge. -/
+namespace KLInvariance.Graded.ModuleData
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+  [IsDomain A] [IsNoetherianRing A]
+  (hneg : ∀ n < 0, 𝓐 n = ⊥)
+  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
+noncomputable section
+
+include hneg hzero in
+ theorem MinimalCover.bijective_of_boundaryIso {M : ModuleData.{uk,ua,um} 𝓐}
+    (C : MinimalCover M) (f : Hom C.source C.source) (g : M ≃ₗ[A] M)
+    (hg : ∀ n v, v ∈ M.piece n → g v ∈ M.piece n)
+    (hcomm : ∀ v, C.hom.map (f.map v) = g (C.hom.map v)) :
+    Function.Bijective f.map := by
+  let gh : Hom M M := ⟨g.toLinearMap,hg⟩
+  let gi : Hom M M :=
+    { map := g.symm.toLinearMap
+      graded := by
+        intro n v hv
+        apply gh.mem_of_map_mem g.injective n
+        change g (g.symm v) ∈ M.piece n
+        simpa only [LinearEquiv.apply_symm_apply] using hv }
+  let := C.free
+  obtain ⟨l,hl⟩ := exists_hom_lift hneg hzero C.hom C.surjective (gi.comp C.hom)
+  have hla (v : C.source) : C.hom.map (l.map v) = g.symm (C.hom.map v) :=
+    LinearMap.congr_fun hl v
+  have hfl : Function.Bijective (f.comp l).map := by
+    apply Hom.bijective_of_mod_positive
+    intro v
+    apply C.minimal
+    change C.hom.map (v-f.map (l.map v))=0
+    rw [map_sub,hcomm,hla,LinearEquiv.apply_symm_apply,sub_self]
+  have hlf : Function.Bijective (l.comp f).map := by
+    apply Hom.bijective_of_mod_positive
+    intro v
+    apply C.minimal
+    change C.hom.map (v-l.map (f.map v))=0
+    rw [map_sub,hla,hcomm,LinearEquiv.symm_apply_apply,sub_self]
+  exact ⟨Function.Injective.of_comp hlf.1,Function.Surjective.of_comp hfl.2⟩
+
+end
+end KLInvariance.Graded.ModuleData
+
+end
+
+
+section
+
+/-! A graded endomorphism of the actual normalized minimal boundary sheaf
+is an automorphism as soon as its top component is one. -/
+namespace KLInvariance.MomentGraph.GradedSheaf.Endomorphism
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
+  {G : OrderedGraph V E} {B : GradedSheaf (𝓐 := 𝓐) G}
+  (f : Endomorphism B)
+
+ theorem edge_bijective_of_vertex {α : E → A}
+    (hBq : B.forget.UpperQuotient α) (e : E)
+    (hv : Function.Bijective (f.vertex (G.target e)).map) :
+    Function.Bijective (f.edge e).map := by
+  obtain ⟨j,hj⟩ := QuotientEdge.exists_scalar_iso
+    (B.upper e).map (B.upper e).map (α e)
+    (hBq e).1 (hBq e).1 (hBq e).2 (hBq e).2
+    (LinearEquiv.ofBijective (f.vertex (G.target e)).map hv)
+  have heq : (f.edge e).map = j.toLinearMap := by
+    apply LinearMap.ext
+    intro v
+    obtain ⟨w,rfl⟩ := (hBq e).1 v
+    exact (f.upper e w).trans (hj w).symm
+  rw [heq]
+  exact j.bijective
+
+end KLInvariance.MomentGraph.GradedSheaf.Endomorphism
+
+namespace KLInvariance.MomentGraph.BoundarySheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+  [IsDomain A] [IsNoetherianRing A]
+  {hneg : ∀ n < 0, 𝓐 n = ⊥}
+  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
+  {V E : Type um} [PartialOrder V] [Fintype V] [Fintype E]
+  {G : OrderedGraph V E} {α : E → A} {b : V}
+  (B : BoundarySheaf hneg G α b)
+noncomputable section
+
+include hzero in
+ theorem endomorphism_bijective_of_top (f : B.sheaf.Endomorphism)
+    (ht : Function.Bijective (f.vertex b).map) :
+    (∀ x, Function.Bijective (f.vertex x).map) ∧
+    (∀ e, Function.Bijective (f.edge e).map) := by
+  have hv : ∀ x, Function.Bijective (f.vertex x).map := by
+    intro x
+    induction x using WellFoundedGT.induction with
+    | ind x ih =>
+      by_cases hx : x=b
+      · exact hx ▸ ht
+      let j : Sheaf.AboveIso B.sheaf.forget B.sheaf.forget x :=
+        { vertex y hy := LinearEquiv.ofBijective (f.vertex y).map (ih y hy)
+          edge e he := LinearEquiv.ofBijective (f.edge e).map
+            (f.edge_bijective_of_vertex B.quotient e (ih _ he))
+          lower e he v := f.lower e v
+          upper e he v := f.upper e v }
+      let T := ModuleData.range (B.sheaf.stalkHom x)
+      let g : T ≃ₗ[A] T := LinearEquiv.ofSubmodules j.boundary _ _
+        (j.map_stalk_image B.flabby.flabby B.generated B.flabby.flabby B.generated)
+      have hg : ∀ n v, v ∈ T.piece n → g v ∈ T.piece n := by
+        intro n v hv e he
+        exact (f.edge e.val).graded n _ (hv e he)
+      let π : ModuleData.Hom (B.sheaf.vertex x) T :=
+        { map := (B.sheaf.forget.stalkMap x).rangeRestrict
+          graded := fun n v hv => (B.sheaf.stalkHom x).graded n v hv }
+      have hπs : Function.Surjective π.map := by
+        intro v
+        obtain ⟨w,hw⟩ := v.property
+        exact ⟨w,Subtype.ext hw⟩
+      have hπm : LinearMap.ker π.map ≤
+          positiveIdeal 𝓐 • (⊤ : Submodule A (B.sheaf.vertex x)) := by
+        intro v hv
+        apply (B.sheaf.kernelMinimal_iff x).mp (B.minimal x hx)
+        have hh : π.map v = 0 := LinearMap.mem_ker.mp hv
+        exact congrArg (fun z : T => z.val) hh
+      let C : ModuleData.MinimalCover T := ⟨B.sheaf.vertex x,B.free x,π,hπs,hπm⟩
+      apply C.bijective_of_boundaryIso hneg hzero (f.vertex x) g hg
+      intro v
+      apply Subtype.ext
+      funext e
+      change B.sheaf.forget.stalkMap x ((f.vertex x).map v) e =
+        (f.edge e.val).map (B.sheaf.forget.stalkMap x v e)
+      rcases e with ⟨e,rfl⟩
+      exact (f.lower e v).symm
+  exact ⟨hv,fun e => f.edge_bijective_of_vertex B.quotient e (hv _)⟩
+
+end
+end KLInvariance.MomentGraph.BoundarySheaf
+
+end
+
+
+section
+
+namespace KLInvariance.Graded.ModuleData.Hom
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} {M N : ModuleData.{uk,ua,um} 𝓐}
+noncomputable section
+
+ def inverse (f : Hom M N) (hf : Function.Bijective f.map) : Hom N M where
+  map := (LinearEquiv.ofBijective f.map hf).symm.toLinearMap
+  graded := by
+    intro n v hv
+    apply f.mem_of_map_mem hf.1 n
+    change (LinearEquiv.ofBijective f.map hf) ((LinearEquiv.ofBijective f.map hf).symm v) ∈ N.piece n
+    simpa only [LinearEquiv.apply_symm_apply] using hv
+
+ theorem map_inverse (f : Hom M N) (hf : Function.Bijective f.map) (v : N) :
+    f.map ((f.inverse hf).map v)=v := (LinearEquiv.ofBijective f.map hf).apply_symm_apply v
+
+ theorem inverse_map (f : Hom M N) (hf : Function.Bijective f.map) (v : M) :
+    (f.inverse hf).map (f.map v)=v := (LinearEquiv.ofBijective f.map hf).symm_apply_apply v
+
+end
+end KLInvariance.Graded.ModuleData.Hom
+
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
+  {G : OrderedGraph V E}
+noncomputable section
+
+ def Hom.toEndomorphism {B : GradedSheaf (𝓐 := 𝓐) G} (f : Hom B B) : Endomorphism B :=
+  ⟨f.vertex,f.edge,f.lower,f.upper⟩
+
+ def Hom.inverse {B C : GradedSheaf (𝓐 := 𝓐) G} (f : Hom B C)
+    (hv : ∀ x, Function.Bijective (f.vertex x).map)
+    (he : ∀ e, Function.Bijective (f.edge e).map) : Hom C B where
+  vertex x := (f.vertex x).inverse (hv x)
+  edge e := (f.edge e).inverse (he e)
+  lower e v := by
+    apply (he e).1
+    rw [(f.edge e).map_inverse]
+    rw [f.lower,(f.vertex _).map_inverse]
+  upper e v := by
+    apply (he e).1
+    rw [(f.edge e).map_inverse]
+    rw [f.upper,(f.vertex _).map_inverse]
+
+ structure SplitIn (B C : GradedSheaf (𝓐 := 𝓐) G) where
+  inclusion : Hom B C
+  projection : Hom C B
+  vertex_split : ∀ x v, (projection.vertex x).map ((inclusion.vertex x).map v)=v
+  edge_split : ∀ e v, (projection.edge e).map ((inclusion.edge e).map v)=v
+
+end
+end KLInvariance.MomentGraph.GradedSheaf
+
+namespace KLInvariance.MomentGraph.BoundarySheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+  [IsDomain A] [IsNoetherianRing A]
+  {hneg : ∀ n < 0, 𝓐 n = ⊥}
+  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
+  {V E : Type um} [PartialOrder V] [Fintype V] [Fintype E]
+  {G : OrderedGraph V E} {α : E → A} {b : V}
+  (B : BoundarySheaf hneg G α b) (hb : ∀ x, x ≤ b)
+  (C : GradedSheaf (𝓐 := 𝓐) G)
+  (hCq : C.UpperQuotient α) (hCfree : ∀ x, Module.Free A (C.vertex x))
+  (hCf : C.forget.Flabby) (hCg : C.forget.Generated)
+noncomputable section
+
+include hzero hb hCq hCfree hCf hCg in
+ theorem splitIn_of_top_split
+    (i : ModuleData.Hom (B.sheaf.vertex b) (C.vertex b))
+    (p : ModuleData.Hom (C.vertex b) (B.sheaf.vertex b))
+    (hpi : ∀ v, p.map (i.map v)=v) : Nonempty (GradedSheaf.SplitIn B.sheaf C) := by
+  obtain ⟨ii,hii⟩ := GradedSheaf.exists_hom_top hneg hzero α B.quotient hCq B.free
+    B.flabby.flabby B.generated hCf hCg b hb i
+  obtain ⟨pp,hpp⟩ := GradedSheaf.exists_hom_top hneg hzero α hCq B.quotient hCfree
+    hCf hCg B.flabby.flabby B.generated b hb p
+  let f := pp.comp ii
+  have ht : Function.Bijective (f.toEndomorphism.vertex b).map := by
+    have heq : (f.toEndomorphism.vertex b).map = LinearMap.id := by
+      apply LinearMap.ext
+      intro v
+      change (pp.vertex b).map ((ii.vertex b).map v)=v
+      rw [hpp,hii]
+      exact hpi v
+    rw [heq]
+    exact Function.bijective_id
+  obtain ⟨hv,he⟩ := B.endomorphism_bijective_of_top hzero f.toEndomorphism ht
+  let fi := f.inverse hv he
+  refine ⟨⟨ii.comp fi,pp,?_,?_⟩⟩
+  · intro x v
+    exact (f.vertex x).map_inverse (hv x) v
+  · intro e v
+    exact (f.edge e).map_inverse (he e) v
+
+end
+end KLInvariance.MomentGraph.BoundarySheaf
+
+end
+
+
+section
+
+/-! A minimal graded free cover splits out of every finite graded free cover.
+The residual kernel is genuinely graded free over the connected coefficient ring.
+This is the local step for decomposing actual flabby word sheaves. -/
+namespace KLInvariance.Graded.ModuleData
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+  [IsDomain A] [IsNoetherianRing A]
+  (hneg : ∀ n < 0, 𝓐 n = ⊥)
+  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
+noncomputable section
+
+include hneg hzero in
+ theorem exists_minimalCover_split {T F : ModuleData.{uk,ua,um} 𝓐}
+    (C : MinimalCover T) [Module.Free A F]
+    (f : Hom F T) (hf : Function.Surjective f.map) :
+    ∃ (g : Hom F C.source) (h : Hom C.source F),
+      C.hom.map.comp g.map = f.map ∧ g.map.comp h.map = LinearMap.id := by
+  let : Module.Free A C.source := C.free
+  obtain ⟨g,hg⟩ := exists_hom_lift hneg hzero C.hom C.surjective f
+  obtain ⟨t,ht⟩ := exists_hom_lift hneg hzero f hf C.hom
+  have hgt : Function.Bijective (g.comp t).map := by
+    apply Hom.bijective_of_mod_positive
+    intro v
+    apply C.minimal
+    change C.hom.map (v-g.map (t.map v))=0
+    rw [map_sub,← LinearMap.comp_apply,hg,← LinearMap.comp_apply,ht,sub_self]
+  refine ⟨g,t.comp ((g.comp t).inverse hgt),hg,?_⟩
+  ext v
+  exact (g.comp t).map_inverse hgt v
+
+ def Hom.splitEquiv {F C : ModuleData.{uk,ua,um} 𝓐}
+    (g : Hom F C) (h : Hom C F) (hgh : g.map.comp h.map = LinearMap.id) :
+    F ≃ₗ[A] C × kernel g := LocalCover.splitEquiv g.map h.map hgh
+
+omit [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐] [IsDomain A] in
+ theorem Hom.splitEquiv_graded {F C : ModuleData.{uk,ua,um} 𝓐}
+    (g : Hom F C) (h : Hom C F) (hgh : g.map.comp h.map = LinearMap.id)
+    (n : ℤ) (v : F) (hv : v ∈ F.piece n) :
+    g.splitEquiv h hgh v ∈ (prod C (kernel g)).piece n := by
+  constructor
+  · exact g.graded n v hv
+  · exact Submodule.sub_mem _ hv (h.graded n _ (g.graded n v hv))
+
+include hneg hzero in
+ theorem Hom.kernel_free_of_split {F C : ModuleData.{uk,ua,um} 𝓐}
+    [Module.Free A F] (g : Hom F C) (h : Hom C F)
+    (hgh : g.map.comp h.map = LinearMap.id) : Module.Free A (kernel g) := by
+  let e := g.splitEquiv h hgh
+  let p : F →ₗ[A] kernel g := (LinearMap.snd A C (kernel g)).comp e.toLinearMap
+  have hp : p.comp g.map.ker.subtype = LinearMap.id := by
+    apply LinearMap.ext
+    intro v
+    apply Subtype.ext
+    change ((v : F) - h.map (g.map v))=v
+    rw [LinearMap.mem_ker.mp v.property,map_zero,sub_zero]
+  let : Module.Projective A (kernel g) := Module.Projective.of_split g.map.ker.subtype p hp
+  obtain ⟨ι,hι,b,_⟩ := exists_homogeneous_basis_connected 𝓐 (kernel g).piece
+    hneg hzero 0 (kernel g).nonneg
+  exact Module.Free.of_basis b
+
+end
+end KLInvariance.Graded.ModuleData
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.Sheaf.AboveIso
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+  {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
+  [Fintype E] [IsNoetherianRing A] [IsDomain A]
+  (hneg : ∀ n < 0, 𝓐 n = ⊥)
+  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
+  {B C : GradedSheaf (𝓐 := 𝓐) G} {x : V}
+  (j : AboveIso B.forget C.forget x)
+
+include hneg hzero in
+ theorem split_stalk_graded
+    (hj : ∀ e (h : x < G.target e) n v, v ∈ (B.edge e).piece n →
+      j.edge e h v ∈ (C.edge e).piece n)
+    (hBfree : Module.Free A (B.vertex x)) (hCfree : Module.Free A (C.vertex x))
+    (hBf : B.forget.Flabby) (hBg : B.forget.Generated)
+    (hCf : C.forget.Flabby) (hCg : C.forget.Generated)
+    (hCmin : C.KernelMinimal x) :
+    ∃ (g : ModuleData.Hom (B.vertex x) (C.vertex x))
+      (e : B.vertex x ≃ₗ[A] C.vertex x × ModuleData.kernel g),
+      Module.Free A (ModuleData.kernel g) ∧
+      (∀ n v, v ∈ (B.vertex x).piece n →
+        e v ∈ (ModuleData.prod (C.vertex x) (ModuleData.kernel g)).piece n) ∧
+      ∀ v, j.boundary (B.forget.stalkMap x v) = C.forget.stalkMap x (e v).1 := by
+  let T := ModuleData.range (C.stalkHom x)
+  let f : ModuleData.Hom (B.vertex x) T :=
+    { map := LinearMap.codRestrict (LinearMap.range (C.forget.stalkMap x))
+        (j.boundary.toLinearMap.comp (B.forget.stalkMap x)) (by
+          intro v
+          rw [← j.map_stalk_image hBf hBg hCf hCg]
+          exact ⟨B.forget.stalkMap x v,⟨v,rfl⟩,rfl⟩)
+      graded := by
+        intro n v hv e _
+        exact hj e.val _ n _ ((B.stalkHom x).graded n v hv e (Set.mem_univ e)) }
+  let p : ModuleData.Hom (C.vertex x) T :=
+    { map := (C.forget.stalkMap x).rangeRestrict
+      graded := fun n v hv => (C.stalkHom x).graded n v hv }
+  have hfs : Function.Surjective f.map := by
+    intro v
+    have hv : v.val ∈ (LinearMap.range (B.forget.stalkMap x)).map j.boundary.toLinearMap := by
+      rw [j.map_stalk_image hBf hBg hCf hCg]
+      exact v.property
+    obtain ⟨w,⟨z,rfl⟩,hz⟩ := hv
+    exact ⟨z,Subtype.ext hz⟩
+  have hps : Function.Surjective p.map := by
+    intro v
+    obtain ⟨w,hw⟩ := v.property
+    exact ⟨w,Subtype.ext hw⟩
+  have hpm : LinearMap.ker p.map ≤ positiveIdeal 𝓐 • (⊤ : Submodule A (C.vertex x)) := by
+    intro v hv
+    apply (C.kernelMinimal_iff x).mp hCmin
+    have hh : p.map v = 0 := LinearMap.mem_ker.mp hv
+    exact congrArg (fun z : T => z.val) hh
+  let D : ModuleData.MinimalCover T := ⟨C.vertex x,hCfree,p,hps,hpm⟩
+  let : Module.Free A (B.vertex x) := hBfree
+  obtain ⟨g,h,hg,hgh⟩ := ModuleData.exists_minimalCover_split hneg hzero D f hfs
+  refine ⟨g,g.splitEquiv h hgh,g.kernel_free_of_split hneg hzero h hgh,
+    g.splitEquiv_graded h hgh,?_⟩
+  intro v
+  exact (congrArg Subtype.val (LinearMap.congr_fun hg v)).symm
+
+end KLInvariance.MomentGraph.Sheaf.AboveIso
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+  [IsDomain A] [IsNoetherianRing A]
+  {hneg : ∀ n < 0, 𝓐 n = ⊥}
+  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
+  {V E : Type um} [PartialOrder V] [Fintype E] {G : OrderedGraph V E}
+  {α : E → A} (O : BoundaryModels hneg (G := G) α)
+  {B C : GradedSheaf (𝓐 := 𝓐) G}
+noncomputable section
+
+include hzero in
+ theorem extend_modelSum {U : Set V} (hU : IsUpperSet U)
+    (hC : ModelSum O.sheaf U C) (j : OnIso B C U)
+    (hBq : B.UpperQuotient α) (hBf : B.forget.Flabby) (hBg : B.forget.Generated)
+    (x : V) (hx : x ∉ U) (hmax : Set.Ioi x ⊆ U)
+    [Module.Free A (B.vertex x)] :
+    ∃ D : GradedSheaf (𝓐 := 𝓐) G,
+      ModelSum O.sheaf (insert x U) D ∧ Nonempty (OnIso B D (insert x U)) := by
+  classical
+  obtain ⟨g,jv,hfree,hjvgrade,hjv⟩ := (j.toOnIso.above x hmax).split_stalk_graded hneg hzero
+    (fun e he => j.edge_graded e (hmax he)) inferInstance (hC.free O x)
+    hBf hBg (hC.flabby O) (hC.generated O) (hC.minimal O x hx)
+  let := hfree
+  obtain ⟨n,d,jq0,hjq0⟩ := ModuleData.exists_equiv_shiftedUnits hneg hzero (ModuleData.kernel g)
+  let S : GradedSheaf (𝓐 := 𝓐) G := pi (fun i : ULift.{um} (Fin n) => (O.sheaf x).shift (d i))
+  let jq1 : ModuleData.pi (fun i : ULift.{um} (Fin n) => (ModuleData.unit hneg).shift (d i))
+      ≃ₗ[A] S.vertex x := LinearEquiv.piCongrRight (fun _ => (O.top x).symm)
+  have hjq1 (m : ℤ) (v : ModuleData.pi (fun i : ULift.{um} (Fin n) => (ModuleData.unit hneg).shift (d i)))
+      (hv : v ∈ (ModuleData.pi (fun i : ULift.{um} (Fin n) => (ModuleData.unit hneg).shift (d i))).piece m) :
+      jq1 v ∈ (S.vertex x).piece m := by
+    intro i _
+    let t : ModuleData.Hom ((O.sheaf x).vertex x) (ModuleData.unit hneg) :=
+      ⟨(O.top x).toLinearMap,O.top_graded x⟩
+    apply t.mem_of_map_mem (O.top x).injective (m-(d i : ℤ))
+    change O.top x ((O.top x).symm (v i)) ∈ (ModuleData.unit hneg).piece (m-(d i : ℤ))
+    exact ((O.top x).apply_symm_apply (v i)).symm ▸ hv i (Set.mem_univ i)
+  let jq := jq0.trans jq1
+  let jv' : B.vertex x ≃ₗ[A] (prod C S).vertex x :=
+    jv.trans (LinearEquiv.prodCongr (LinearEquiv.refl A (C.vertex x)) jq)
+  have hjv' (m : ℤ) (v : B.vertex x) (hv : v ∈ (B.vertex x).piece m) :
+      jv' v ∈ ((prod C S).vertex x).piece m :=
+    ⟨(hjvgrade m v hv).1,hjq1 m _ (hjq0 m _ (hjvgrade m v hv).2)⟩
+  have hSV (y : V) (hy : y ∈ U) : Subsingleton (S.vertex y) := by
+    let := O.support x y (fun hyx => hx (hU hyx hy))
+    exact inferInstanceAs (Subsingleton (ULift.{um} (Fin n) → (O.sheaf x).vertex y))
+  have hSE (e : E) (he : G.target e ∈ U) : Subsingleton (S.edge e) := by
+    let := O.support x (G.target e) (fun h => hx (hU h he))
+    let : Subsingleton ((O.sheaf x).edge e) := (O.quotient x e).1.subsingleton
+    exact inferInstanceAs (Subsingleton (ULift.{um} (Fin n) → (O.sheaf x).edge e))
+  let jj : OnIso B (prod C S) U := j.prodZero S hSV hSE
+  have hsum : ModelSum O.sheaf (insert x U) (prod C S) :=
+    .add (hC.mono (Set.subset_insert x U)) x (Set.mem_insert x U) n d
+  have hout : ∀ e (he : G.source e = x) v,
+      jj.edge e (hmax (lt_of_eq_of_lt he.symm (G.increasing e)))
+        (B.forget.lower e v) = (prod C S).forget.lower e
+          (Sheaf.castStalkIso B.forget (prod C S).forget he.symm jv' v) := by
+    intro e he v
+    subst x
+    apply Prod.ext
+    · exact congrFun (hjv v) ⟨e,rfl⟩
+    · let := hSE e (hmax (G.increasing e))
+      exact Subsingleton.elim _ _
+  exact ⟨prod C S,hsum,⟨jj.extend hU x hx hmax α hBq (hsum.quotient O) jv' hjv' hout⟩⟩
+
+end
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+  [IsDomain A] [IsNoetherianRing A]
+  {hneg : ∀ n < 0, 𝓐 n = ⊥}
+  (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c = a)
+  {V E : Type um} [PartialOrder V] [Fintype E] {G : OrderedGraph V E}
+  {α : E → A} (O : BoundaryModels hneg (G := G) α)
+  (B : GradedSheaf (𝓐 := 𝓐) G)
+
+include hzero in
+/-- Finite graded decomposition with actual nonnegative shifts. -/
+theorem exists_modelSumIso [Finite V]
+    (hBfree : ∀ x, Module.Free A (B.vertex x))
+    (hBq : B.UpperQuotient α) (hBf : B.forget.Flabby) (hBg : B.forget.Generated) :
+    ∃ C : GradedSheaf (𝓐 := 𝓐) G,
+      ModelSum O.sheaf Set.univ C ∧ Nonempty (Iso B C) := by
+  classical
+  let := Fintype.ofFinite V
+  have h : ∀ s : Finset V, IsUpperSet ((s : Set V)ᶜ) →
+      ∀ C : GradedSheaf (𝓐 := 𝓐) G,
+      ModelSum O.sheaf ((s : Set V)ᶜ) C → OnIso B C ((s : Set V)ᶜ) →
+      ∃ D : GradedSheaf (𝓐 := 𝓐) G,
+        ModelSum O.sheaf Set.univ D ∧ Nonempty (Iso B D) := by
+    intro s
+    induction s using Finset.strongInductionOn with
+    | _ s ih =>
+      intro hU C hC j
+      by_cases hs : s = ∅
+      · subst s
+        have hh : ((∅ : Finset V) : Set V)ᶜ = Set.univ := by simp
+        exact ⟨C,hh ▸ hC,⟨OnIso.ofUniv (hh ▸ j)⟩⟩
+      · obtain ⟨x,hxs,hxmax⟩ := s.exists_maximal (Finset.nonempty_iff_ne_empty.mpr hs)
+        have hx : x ∉ ((s : Set V)ᶜ) := by simpa using hxs
+        have hmax : Set.Ioi x ⊆ ((s : Set V)ᶜ) := by
+          intro y hy
+          change y ∉ s
+          intro hys
+          exact (not_le_of_gt hy) (hxmax hys hy.le)
+        let := hBfree x
+        obtain ⟨D,hD,⟨jj⟩⟩ := extend_modelSum hzero O hU hC j hBq hBf hBg x hx hmax
+        have heq : insert x ((s : Set V)ᶜ) = ((s.erase x : Finset V) : Set V)ᶜ := by
+          ext y
+          simp only [Set.mem_insert_iff,Set.mem_compl_iff,Finset.mem_coe,Finset.mem_erase]
+          tauto
+        have hU' : IsUpperSet (insert x ((s : Set V)ᶜ)) := by
+          intro y t hyt hy
+          rcases Set.mem_insert_iff.mp hy with hy | hy
+          · subst y
+            by_cases htx : t=x
+            · exact Set.mem_insert_iff.mpr (Or.inl htx)
+            · exact Set.mem_insert_of_mem x (hmax (lt_of_le_of_ne hyt (Ne.symm htx)))
+          · exact Set.mem_insert_of_mem x (hU hyt hy)
+        exact ih (s.erase x) (Finset.erase_ssubset hxs) (heq ▸ hU') D (heq ▸ hD) (heq ▸ jj)
+  have hset : ((Finset.univ : Finset V) : Set V)ᶜ = ∅ := by simp
+  have hj : OnIso B zero (∅ : Set V) :=
+    ⟨⟨fun x hx => False.elim hx,fun e he => False.elim he,
+      fun e hs => False.elim hs,fun e ht => False.elim ht⟩,
+      fun x hx => False.elim hx,fun e he => False.elim he⟩
+  have hinit := h Finset.univ
+  rw [hset] at hinit
+  exact hinit isUpperSet_empty zero .zero hj
+
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+/-! Graded linear equivalence preserves the genuine free-module character.
+This does not assert its identification with any KL polynomial. -/
+namespace KLInvariance.Graded.ModuleData
+open Polynomial
+universe uk us um
+variable {k : Type uk} [Field k] {σ : Type us} [Finite σ]
+  {M N : ModuleData.{uk,max uk us,um} (polynomialGrading k σ)}
+
+noncomputable def mapNonnegativeBasis (e : M ≃ₗ[MvPolynomial σ k] N)
+    (he : ∀ n v, v ∈ M.piece n → e v ∈ N.piece n)
+    (b : NonnegativeBasis (σ := σ) M.piece) : NonnegativeBasis (σ := σ) N.piece where
+  index := b.index
+  finite := b.finite
+  basis := b.basis.map e
+  degree := b.degree
+  homogeneous i := he _ _ (b.homogeneous i)
+
+ theorem character_eq_of_gradedEquiv [Module.Free (MvPolynomial σ k) M]
+    [Module.Free (MvPolynomial σ k) N] (e : M ≃ₗ[MvPolynomial σ k] N)
+    (he : ∀ n v, v ∈ M.piece n → e v ∈ N.piece n) :
+    character (σ := σ) M.piece M.nonneg = character (σ := σ) N.piece N.nonneg := by
+  let b := (exists_nonnegativeBasis (σ := σ) M.piece M.nonneg).some
+  rw [character_eq M.piece M.nonneg b,
+    character_eq N.piece N.nonneg (mapNonnegativeBasis e he b)]
+  rfl
+
+end KLInvariance.Graded.ModuleData
+
+end
+
+
+section
+
+/-! Honest polynomial characters commute with actual finite graded sums and
+nonnegative shifts. No Hecke or KL identification enters these lemmas. -/
+namespace KLInvariance.Graded.ModuleData
+open Polynomial Module
+universe uk us um
+variable {k : Type uk} [Field k] {σ : Type us} [Finite σ]
+  (M N : ModuleData.{uk,max uk us,um} (polynomialGrading k σ))
+noncomputable section
+
+ abbrev char [Module.Free (MvPolynomial σ k) M] : Polynomial ℤ :=
+  character (σ := σ) M.piece M.nonneg
+
+ def shiftedBasis (b : NonnegativeBasis (σ := σ) M.piece) (d : ℕ) :
+    NonnegativeBasis (σ := σ) (M.shift d).piece where
+  index := b.index
+  finite := b.finite
+  basis := b.basis
+  degree i := b.degree i+d
+  homogeneous i := by
+    change b.basis i ∈ M.piece ((b.degree i+d : ℕ)-(d:ℤ))
+    simpa only [Nat.cast_add,add_sub_cancel_right] using b.homogeneous i
+
+ instance shift_free [Module.Free (MvPolynomial σ k) M] (d : ℕ) :
+    Module.Free (MvPolynomial σ k) (M.shift d) := ‹Module.Free (MvPolynomial σ k) M›
+
+ theorem char_shift [Module.Free (MvPolynomial σ k) M] (d : ℕ) :
+    (M.shift d).char=X^d*M.char := by
+  let b := (exists_nonnegativeBasis (σ := σ) M.piece M.nonneg).some
+  rw [char,character_eq _ _ (M.shiftedBasis b d),char,character_eq _ _ b]
+  change (∑ i : b.index, (X : Polynomial ℤ) ^ (b.degree i+d)) = X^d*∑ i, X^b.degree i
+  simp only [pow_add,mul_comm,Finset.mul_sum]
+
+ def productBasis (b : NonnegativeBasis (σ := σ) M.piece)
+    (c : NonnegativeBasis (σ := σ) N.piece) :
+    NonnegativeBasis (σ := σ) (M.prod N).piece where
+  index := b.index ⊕ c.index
+  finite := inferInstance
+  basis := b.basis.prod c.basis
+  degree := Sum.elim b.degree c.degree
+  homogeneous i := by
+    cases i with
+    | inl i =>
+      change ((b.basis.prod c.basis) (Sum.inl i)).1 ∈ M.piece (b.degree i) ∧
+        ((b.basis.prod c.basis) (Sum.inl i)).2 ∈ N.piece (b.degree i)
+      simpa only [Basis.prod_apply_inl_fst,Basis.prod_apply_inl_snd] using
+        And.intro (b.homogeneous i) (Submodule.zero_mem (N.piece (b.degree i)))
+    | inr i =>
+      change ((b.basis.prod c.basis) (Sum.inr i)).1 ∈ M.piece (c.degree i) ∧
+        ((b.basis.prod c.basis) (Sum.inr i)).2 ∈ N.piece (c.degree i)
+      simpa only [Basis.prod_apply_inr_fst,Basis.prod_apply_inr_snd] using
+        And.intro (Submodule.zero_mem (M.piece (c.degree i))) (c.homogeneous i)
+
+ instance prod_free [Module.Free (MvPolynomial σ k) M] [Module.Free (MvPolynomial σ k) N] :
+    Module.Free (MvPolynomial σ k) (M.prod N) :=
+  (inferInstance : Module.Free (MvPolynomial σ k) (M × N))
+
+ theorem char_prod [Module.Free (MvPolynomial σ k) M] [Module.Free (MvPolynomial σ k) N] :
+    (M.prod N).char=M.char+N.char := by
+  let b := (exists_nonnegativeBasis (σ := σ) M.piece M.nonneg).some
+  let c := (exists_nonnegativeBasis (σ := σ) N.piece N.nonneg).some
+  rw [char,character_eq _ _ (M.productBasis N b c),char,character_eq _ _ b,
+    char,character_eq _ _ c]
+  change (∑ i : b.index ⊕ c.index, (X : Polynomial ℤ)^Sum.elim b.degree c.degree i) =
+    (∑ i : b.index, X^b.degree i)+(∑ i : c.index, X^c.degree i)
+  exact Fintype.sum_sum_type (fun i => (X : Polynomial ℤ)^Sum.elim b.degree c.degree i)
+
+ def piBasis {ι : Type um} [Fintype ι]
+    (F : ι → ModuleData.{uk,max uk us,um} (polynomialGrading k σ))
+    (b : ∀ i, NonnegativeBasis (σ := σ) (F i).piece) :
+    NonnegativeBasis (σ := σ) (pi F).piece where
+  index := Σ i, (b i).index
+  finite := inferInstance
+  basis := Pi.basis (fun i => (b i).basis)
+  degree i := (b i.1).degree i.2
+  homogeneous i := by
+    classical
+    intro j _
+    change (Pi.basis (fun i => (b i).basis)) i j ∈ (F j).piece ((b i.1).degree i.2)
+    rw [Pi.basis_apply]
+    by_cases h : j=i.1
+    · subst j
+      simpa only [Pi.single_eq_same] using (b i.1).homogeneous i.2
+    · simp only [Pi.single_eq_of_ne h,Submodule.zero_mem]
+
+ instance pi_free {ι : Type um} [Fintype ι]
+    (F : ι → ModuleData.{uk,max uk us,um} (polynomialGrading k σ))
+    [∀ i, Module.Free (MvPolynomial σ k) (F i)] :
+    Module.Free (MvPolynomial σ k) (pi F) :=
+  (inferInstance : Module.Free (MvPolynomial σ k) (∀ i, F i))
+
+ theorem char_pi {ι : Type um} [Fintype ι]
+    (F : ι → ModuleData.{uk,max uk us,um} (polynomialGrading k σ))
+    [∀ i, Module.Free (MvPolynomial σ k) (F i)] :
+    (pi F).char=∑ i, (F i).char := by
+  let b := fun i => (exists_nonnegativeBasis (σ := σ) (F i).piece (F i).nonneg).some
+  rw [char,character_eq _ _ (piBasis F b)]
+  change (∑ i : Σ j, (b j).index, (X : Polynomial ℤ)^(b i.1).degree i.2) = _
+  rw [Fintype.sum_sigma]
+  apply Finset.sum_congr rfl
+  intro i _
+  exact (character_eq (F i).piece (F i).nonneg (b i)).symm
+
+ theorem char_zero [Subsingleton M] [Module.Free (MvPolynomial σ k) M] : M.char=0 := by
+  let b := (exists_nonnegativeBasis (σ := σ) M.piece M.nonneg).some
+  have : IsEmpty b.index := ⟨fun i => b.basis.ne_zero i (Subsingleton.elim _ _)⟩
+  rw [char,character_eq _ _ b]
+  simp [NonnegativeBasis.polynomial]
+
+end
+end KLInvariance.Graded.ModuleData
+
+end
+
+
+section
+
+/-! The character identity of the actual finite shifted-model decomposition.
+This is an identity of genuine graded stalk ranks, not a KL assertion. -/
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/KazhdanLusztig/Sheaves.lean b/lean/OAI/RepresentationTheory/KazhdanLusztig/Sheaves.lean
index 090f8446cc9e869e6a9b45cf0675ae4339583c6c..db81b8123cf1040162715cecbad29c18179677e7 100644
--- a/lean/OAI/RepresentationTheory/KazhdanLusztig/Sheaves.lean
+++ b/lean/OAI/RepresentationTheory/KazhdanLusztig/Sheaves.lean
@@ -1,249 +1,249 @@
-import Mathlib
-
-namespace OAI
-
-/-!
-Ordered moment graphs, sheaves, flabbiness, sections and boundary images.
--/
-
-section
-
-
-/-! The finite-section extension argument required by source §4.3. No BMP
-character theorem is postulated. These definitions are ordinary module-valued
-sheaves on a directed ordered graph; labels enter later through which edge
-modules vanish after localization. -/
-
-namespace KLInvariance.MomentGraph
-
-universe u v w z
-
-structure OrderedGraph (V : Type v) [PartialOrder V] (E : Type w) where
-  source : E → V
-  target : E → V
-  increasing : ∀ e, source e < target e
-
-variable {R : Type u} [CommRing R] {V : Type v} [PartialOrder V] {E : Type w}
-
-structure Sheaf (G : OrderedGraph V E) where
-  vertex : V → ModuleCat.{z} R
-  edge : E → ModuleCat.{z} R
-  lower : ∀ e, (vertex (G.source e)) →ₗ[R] edge e
-  upper : ∀ e, (vertex (G.target e)) →ₗ[R] edge e
-
-namespace Sheaf
-variable {G : OrderedGraph V E} (B : Sheaf (R := R) G)
-
-abbrev Assignment := ∀ x, B.vertex x
-
-/-- Exactly the usual section compatibility on edges with both endpoints in
-`Ω`; coordinates outside `Ω` are ignored and can be chosen arbitrarily. -/
-def CompatibleOn (Ω : Set V) (f : B.Assignment) : Prop :=
-  ∀ e, G.source e ∈ Ω → G.target e ∈ Ω →
-    B.lower e (f (G.source e)) = B.upper e (f (G.target e))
-
-def IsGlobal (f : B.Assignment) : Prop := B.CompatibleOn Set.univ f
-
-/-- Extension of ordinary sections over ambient upper sets. -/
-def Flabby : Prop :=
-  ∀ Ω : Set V, IsUpperSet Ω → ∀ f : B.Assignment, B.CompatibleOn Ω f →
-    ∃ g : B.Assignment, B.IsGlobal g ∧ ∀ x ∈ Ω, g x = f x
-
-/-- Closure under the edges whose modules survive, strictly weaker than
-being an ambient upper set. -/
-def ClosedUnder (_B : Sheaf (R := R) G) (live : Set E) (Ω : Set V) : Prop :=
-  ∀ e ∈ live, G.source e ∈ Ω → G.target e ∈ Ω
-
-theorem compatible_mono {Ω Ω' : Set V} {f : B.Assignment}
-    (hf : B.CompatibleOn Ω f) (h : Ω' ⊆ Ω) : B.CompatibleOn Ω' f :=
-  fun e hs ht => hf e (h hs) (h ht)
-
-/-- One step of the descending extension in `loc:relative-flabbiness`.
-Previously assigned lower neighbors can only be on zero edge modules. -/
-theorem extend_one (hb : B.Flabby) (live : Set E)
-    (hdead : ∀ e, e ∉ live → Subsingleton (B.edge e))
-    (Ω : Set V) (hΩ : B.ClosedUnder live Ω) (a : V) (ha : a ∉ Ω)
-    (hmax : ∀ y, a < y → y ∈ Ω) (f : B.Assignment) (hf : B.CompatibleOn Ω f) :
-    ∃ g : B.Assignment, B.CompatibleOn (insert a Ω) g ∧
-      (∀ x ∈ Ω, g x = f x) := by
-  classical
-  have hu : IsUpperSet (Set.Ioi a) := fun x y hxy hx => lt_of_lt_of_le hx hxy
-  obtain ⟨t, ht, htf⟩ := hb (Set.Ioi a) hu f
-    (B.compatible_mono hf (fun y hy => hmax y hy))
-  let g : B.Assignment := Function.update f a (t a)
-  have hga : g a = t a := by simp [g]
-  have hgf (x : V) (hx : x ≠ a) : g x = f x := by simp [g, hx]
-  refine ⟨g, ?_, ?_⟩
-  · intro e hs htt
-    by_cases hlive : e ∈ live
-    · by_cases hsa : G.source e = a
-      · have hta : G.target e ≠ a := by simpa [hsa] using (G.increasing e).ne'
-        have hT : G.target e ∈ Ω := hmax _ (hsa ▸ G.increasing e)
-        have hsource : g (G.source e) = t (G.source e) := by subst a; exact hga
-        rw [hsource, hgf _ hta, ← htf _ (hsa ▸ G.increasing e)]
-        exact ht e (Set.mem_univ _) (Set.mem_univ _)
-      · have hS : G.source e ∈ Ω := (Set.mem_insert_iff.mp hs).resolve_left hsa
-        have hT : G.target e ∈ Ω := hΩ e hlive hS
-        have hta : G.target e ≠ a := fun h => ha (h ▸ hT)
-        rw [hgf _ hsa, hgf _ hta]
-        exact hf e hS hT
-    · let := hdead e hlive
-      exact Subsingleton.elim _ _
-  · intro x hx
-    exact hgf x (fun h => ha (h ▸ hx))
-
-theorem closedUnder_insert (live : Set E) (Ω : Set V)
-    (hΩ : B.ClosedUnder live Ω) (a : V) (hmax : ∀ y, a < y → y ∈ Ω) :
-    B.ClosedUnder live (insert a Ω) := by
-  intro e he hs
-  rcases Set.mem_insert_iff.mp hs with hs | hs
-  · exact Set.mem_insert_of_mem _ (hmax _ (hs ▸ G.increasing e))
-  · exact Set.mem_insert_of_mem _ (hΩ e he hs)
-
-/-- A finite flabby graph sheaf extends from every set closed under its
-nonzero edges. This is the source's relative-flabbiness argument in full,
-not merely a restatement of flabbiness for ambient upper sets. -/
-theorem extend_closedUnder [Finite V] (hb : B.Flabby) (live : Set E)
-    (hdead : ∀ e, e ∉ live → Subsingleton (B.edge e))
-    (Ω : Set V) (hΩ : B.ClosedUnder live Ω) (f : B.Assignment)
-    (hf : B.CompatibleOn Ω f) :
-    ∃ g : B.Assignment, B.IsGlobal g ∧ (∀ x ∈ Ω, g x = f x) := by
-  classical
-  let := Fintype.ofFinite V
-  suffices h : ∀ s : Finset V, ∀ f : B.Assignment,
-      B.ClosedUnder live (s : Set V)ᶜ → B.CompatibleOn (s : Set V)ᶜ f →
-      ∃ g : B.Assignment, B.IsGlobal g ∧ ∀ x ∈ (s : Set V)ᶜ, g x = f x by
-    have hs : ((Finset.univ.filter (fun x => x ∉ Ω) : Finset V) : Set V)ᶜ = Ω := by
-      ext x
-      simp
-    have H := h (Finset.univ.filter (fun x => x ∉ Ω)) f
-    rw [hs] at H
-    exact H hΩ hf
-  intro s
-  refine Finset.strongInductionOn s ?_
-  intro s ih f hΩ hf
-  by_cases hs : s = ∅
-  · subst s
-    exact ⟨f, by simpa [IsGlobal] using hf, fun _ _ => rfl⟩
-  obtain ⟨a, ha, hmax⟩ := s.exists_maximal (Finset.nonempty_iff_ne_empty.mpr hs)
-  have ha' : a ∉ (s : Set V)ᶜ := by simpa using ha
-  have hmax' (y : V) (hay : a < y) : y ∈ (s : Set V)ᶜ := by
-    intro hy
-    exact (not_le_of_gt hay) (hmax hy hay.le)
-  obtain ⟨g, hg, hgf⟩ := B.extend_one hb live hdead (s : Set V)ᶜ hΩ a ha' hmax' f hf
-  have hnew : ((s.erase a : Finset V) : Set V)ᶜ = insert a (s : Set V)ᶜ := by
-    ext x
-    simp only [Set.mem_compl_iff, Finset.mem_coe, Finset.mem_erase,
-      Set.mem_insert_iff, not_and_or, not_not]
-  have hsmall : s.erase a ⊂ s := Finset.erase_ssubset ha
-  obtain ⟨t, ht, htg⟩ := ih (s.erase a) hsmall g
-    (by rw [hnew]; exact B.closedUnder_insert live (s : Set V)ᶜ hΩ a hmax')
-      (by rw [hnew]; exact hg)
-  refine ⟨t, ht, ?_⟩
-  intro x hx
-  exact (htg x (by rw [hnew]; exact Set.mem_insert_of_mem _ hx)).trans (hgf x hx)
-
-end Sheaf
-end KLInvariance.MomentGraph
-
-end
-
-section
-
-namespace KLInvariance.MomentGraph.Sheaf
-
-universe u v w z
-variable {R : Type u} [CommRing R] {V : Type v} [PartialOrder V] {E : Type w}
-  {G : OrderedGraph V E} (B : Sheaf.{u,v,w,z} (R := R) G)
-
-/-- The actual module of sections on a vertex set. Coordinates outside the
-set are harmless and ignored by all restriction maps. -/
-def sections (Ω : Set V) : Submodule R B.Assignment where
-  carrier := {f | B.CompatibleOn Ω f}
-  zero_mem' := by intro e hs ht; simp
-  add_mem' := by
-    intro f g hf hg e hs ht
-    change B.lower e (f (G.source e) + g (G.source e)) =
-      B.upper e (f (G.target e) + g (G.target e))
-    rw [map_add, map_add, hf e hs ht, hg e hs ht]
-  smul_mem' := by
-    intro r f hf e hs ht
-    change B.lower e (r • f (G.source e)) = B.upper e (r • f (G.target e))
-    rw [map_smul, map_smul, hf e hs ht]
-
-/-- Generation at a stalk by genuine global sections. -/
-def Generated : Prop :=
-  ∀ x (b : B.vertex x), ∃ f : B.Assignment, B.IsGlobal f ∧ f x = b
-
-abbrev Outgoing (x : V) := {e : E // G.source e = x}
-
-abbrev Boundary (x : V) := ∀ e : Outgoing (G := G) x, B.edge e.val
-
-/-- The lower-end stalk map on all upward edges. -/
-def stalkMap (x : V) : B.vertex x →ₗ[R] B.Boundary x where
-  toFun b e := B.lower e.val (e.property.symm ▸ b)
-  map_add' b c := by ext e; cases e with | mk e he => subst x; simp
-  map_smul' r b := by ext e; cases e with | mk e he => subst x; simp
-
-/-- Boundary values supplied by sections strictly above a vertex. -/
-def boundaryMap (x : V) : B.sections (Set.Ioi x) →ₗ[R] B.Boundary x where
-  toFun f e := B.upper e.val (f.val (G.target e.val))
-  map_add' f g := by ext e; simp
-  map_smul' r f := by ext e; simp
-
-def boundaryImage (x : V) : Submodule R (B.Boundary x) :=
-  LinearMap.range (B.boundaryMap x)
-
-/-- The precise boundary-image identity in `loc:decomposition`.
-Generation supplies one inclusion and flabbiness supplies the other. -/
-theorem range_stalkMap_eq_boundaryImage
-    (hf : B.Flabby) (hg : B.Generated) (x : V) :
-    LinearMap.range (B.stalkMap x) = B.boundaryImage x := by
-  apply le_antisymm
-  · rintro v ⟨b, rfl⟩
-    obtain ⟨f, hglob, hfx⟩ := hg x b
-    refine ⟨⟨f, B.compatible_mono hglob (Set.subset_univ _)⟩, ?_⟩
-    ext e
-    change B.upper e.val (f (G.target e.val)) =
-      B.lower e.val (e.property.symm ▸ b)
-    have hc := hglob e.val (Set.mem_univ _) (Set.mem_univ _)
-    rw [← hc]
-    congr 1
-    cases e with | mk e he => subst x; exact hfx
-  · rintro v ⟨f, rfl⟩
-    have hu : IsUpperSet (Set.Ioi x) := fun a b hab ha => lt_of_lt_of_le ha hab
-    obtain ⟨g, hglob, hgf⟩ := hf (Set.Ioi x) hu f.val f.property
-    refine ⟨g x, ?_⟩
-    ext e
-    change B.lower e.val (e.property.symm ▸ g x) =
-      B.upper e.val (f.val (G.target e.val))
-    have hc := hglob e.val (Set.mem_univ _) (Set.mem_univ _)
-    have ht : G.target e.val ∈ Set.Ioi x := by
-      change x < G.target e.val
-      exact lt_of_eq_of_lt e.property.symm (G.increasing e.val)
-    rw [← hgf _ ht, ← hc]
-    congr 1
-    cases e with | mk e he => subst x; rfl
-
-/-- A sheaf satisfying the two standard section properties gives an actual
-surjection of its stalk onto the boundary module. -/
-def stalkCover (hf : B.Flabby) (hg : B.Generated) (x : V) :
-    B.vertex x →ₗ[R] B.boundaryImage x :=
-  (B.stalkMap x).codRestrict _ (fun b => by
-    rw [← B.range_stalkMap_eq_boundaryImage hf hg x]
-    exact ⟨b, rfl⟩)
-
-theorem stalkCover_surjective (hf : B.Flabby) (hg : B.Generated) (x : V) :
-    Function.Surjective (B.stalkCover hf hg x) := by
-  intro v
-  have hv : (v : B.Boundary x) ∈ LinearMap.range (B.stalkMap x) := by
-    simpa only [B.range_stalkMap_eq_boundaryImage hf hg x] using v.property
-  obtain ⟨b, hb⟩ := hv
-  exact ⟨b, Subtype.ext hb⟩
-
-end KLInvariance.MomentGraph.Sheaf
-
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-!
+Ordered moment graphs, sheaves, flabbiness, sections and boundary images.
+-/
+
+section
+
+
+/-! The finite-section extension argument required by source §4.3. No BMP
+character theorem is postulated. These definitions are ordinary module-valued
+sheaves on a directed ordered graph; labels enter later through which edge
+modules vanish after localization. -/
+
+namespace KLInvariance.MomentGraph
+
+universe u v w z
+
+structure OrderedGraph (V : Type v) [PartialOrder V] (E : Type w) where
+  source : E → V
+  target : E → V
+  increasing : ∀ e, source e < target e
+
+variable {R : Type u} [CommRing R] {V : Type v} [PartialOrder V] {E : Type w}
+
+structure Sheaf (G : OrderedGraph V E) where
+  vertex : V → ModuleCat.{z} R
+  edge : E → ModuleCat.{z} R
+  lower : ∀ e, (vertex (G.source e)) →ₗ[R] edge e
+  upper : ∀ e, (vertex (G.target e)) →ₗ[R] edge e
+
+namespace Sheaf
+variable {G : OrderedGraph V E} (B : Sheaf (R := R) G)
+
+abbrev Assignment := ∀ x, B.vertex x
+
+/-- Exactly the usual section compatibility on edges with both endpoints in
+`Ω`; coordinates outside `Ω` are ignored and can be chosen arbitrarily. -/
+def CompatibleOn (Ω : Set V) (f : B.Assignment) : Prop :=
+  ∀ e, G.source e ∈ Ω → G.target e ∈ Ω →
+    B.lower e (f (G.source e)) = B.upper e (f (G.target e))
+
+def IsGlobal (f : B.Assignment) : Prop := B.CompatibleOn Set.univ f
+
+/-- Extension of ordinary sections over ambient upper sets. -/
+def Flabby : Prop :=
+  ∀ Ω : Set V, IsUpperSet Ω → ∀ f : B.Assignment, B.CompatibleOn Ω f →
+    ∃ g : B.Assignment, B.IsGlobal g ∧ ∀ x ∈ Ω, g x = f x
+
+/-- Closure under the edges whose modules survive, strictly weaker than
+being an ambient upper set. -/
+def ClosedUnder (_B : Sheaf (R := R) G) (live : Set E) (Ω : Set V) : Prop :=
+  ∀ e ∈ live, G.source e ∈ Ω → G.target e ∈ Ω
+
+theorem compatible_mono {Ω Ω' : Set V} {f : B.Assignment}
+    (hf : B.CompatibleOn Ω f) (h : Ω' ⊆ Ω) : B.CompatibleOn Ω' f :=
+  fun e hs ht => hf e (h hs) (h ht)
+
+/-- One step of the descending extension in `loc:relative-flabbiness`.
+Previously assigned lower neighbors can only be on zero edge modules. -/
+theorem extend_one (hb : B.Flabby) (live : Set E)
+    (hdead : ∀ e, e ∉ live → Subsingleton (B.edge e))
+    (Ω : Set V) (hΩ : B.ClosedUnder live Ω) (a : V) (ha : a ∉ Ω)
+    (hmax : ∀ y, a < y → y ∈ Ω) (f : B.Assignment) (hf : B.CompatibleOn Ω f) :
+    ∃ g : B.Assignment, B.CompatibleOn (insert a Ω) g ∧
+      (∀ x ∈ Ω, g x = f x) := by
+  classical
+  have hu : IsUpperSet (Set.Ioi a) := fun x y hxy hx => lt_of_lt_of_le hx hxy
+  obtain ⟨t, ht, htf⟩ := hb (Set.Ioi a) hu f
+    (B.compatible_mono hf (fun y hy => hmax y hy))
+  let g : B.Assignment := Function.update f a (t a)
+  have hga : g a = t a := by simp [g]
+  have hgf (x : V) (hx : x ≠ a) : g x = f x := by simp [g, hx]
+  refine ⟨g, ?_, ?_⟩
+  · intro e hs htt
+    by_cases hlive : e ∈ live
+    · by_cases hsa : G.source e = a
+      · have hta : G.target e ≠ a := by simpa [hsa] using (G.increasing e).ne'
+        have hT : G.target e ∈ Ω := hmax _ (hsa ▸ G.increasing e)
+        have hsource : g (G.source e) = t (G.source e) := by subst a; exact hga
+        rw [hsource, hgf _ hta, ← htf _ (hsa ▸ G.increasing e)]
+        exact ht e (Set.mem_univ _) (Set.mem_univ _)
+      · have hS : G.source e ∈ Ω := (Set.mem_insert_iff.mp hs).resolve_left hsa
+        have hT : G.target e ∈ Ω := hΩ e hlive hS
+        have hta : G.target e ≠ a := fun h => ha (h ▸ hT)
+        rw [hgf _ hsa, hgf _ hta]
+        exact hf e hS hT
+    · let := hdead e hlive
+      exact Subsingleton.elim _ _
+  · intro x hx
+    exact hgf x (fun h => ha (h ▸ hx))
+
+theorem closedUnder_insert (live : Set E) (Ω : Set V)
+    (hΩ : B.ClosedUnder live Ω) (a : V) (hmax : ∀ y, a < y → y ∈ Ω) :
+    B.ClosedUnder live (insert a Ω) := by
+  intro e he hs
+  rcases Set.mem_insert_iff.mp hs with hs | hs
+  · exact Set.mem_insert_of_mem _ (hmax _ (hs ▸ G.increasing e))
+  · exact Set.mem_insert_of_mem _ (hΩ e he hs)
+
+/-- A finite flabby graph sheaf extends from every set closed under its
+nonzero edges. This is the source's relative-flabbiness argument in full,
+not merely a restatement of flabbiness for ambient upper sets. -/
+theorem extend_closedUnder [Finite V] (hb : B.Flabby) (live : Set E)
+    (hdead : ∀ e, e ∉ live → Subsingleton (B.edge e))
+    (Ω : Set V) (hΩ : B.ClosedUnder live Ω) (f : B.Assignment)
+    (hf : B.CompatibleOn Ω f) :
+    ∃ g : B.Assignment, B.IsGlobal g ∧ (∀ x ∈ Ω, g x = f x) := by
+  classical
+  let := Fintype.ofFinite V
+  suffices h : ∀ s : Finset V, ∀ f : B.Assignment,
+      B.ClosedUnder live (s : Set V)ᶜ → B.CompatibleOn (s : Set V)ᶜ f →
+      ∃ g : B.Assignment, B.IsGlobal g ∧ ∀ x ∈ (s : Set V)ᶜ, g x = f x by
+    have hs : ((Finset.univ.filter (fun x => x ∉ Ω) : Finset V) : Set V)ᶜ = Ω := by
+      ext x
+      simp
+    have H := h (Finset.univ.filter (fun x => x ∉ Ω)) f
+    rw [hs] at H
+    exact H hΩ hf
+  intro s
+  refine Finset.strongInductionOn s ?_
+  intro s ih f hΩ hf
+  by_cases hs : s = ∅
+  · subst s
+    exact ⟨f, by simpa [IsGlobal] using hf, fun _ _ => rfl⟩
+  obtain ⟨a, ha, hmax⟩ := s.exists_maximal (Finset.nonempty_iff_ne_empty.mpr hs)
+  have ha' : a ∉ (s : Set V)ᶜ := by simpa using ha
+  have hmax' (y : V) (hay : a < y) : y ∈ (s : Set V)ᶜ := by
+    intro hy
+    exact (not_le_of_gt hay) (hmax hy hay.le)
+  obtain ⟨g, hg, hgf⟩ := B.extend_one hb live hdead (s : Set V)ᶜ hΩ a ha' hmax' f hf
+  have hnew : ((s.erase a : Finset V) : Set V)ᶜ = insert a (s : Set V)ᶜ := by
+    ext x
+    simp only [Set.mem_compl_iff, Finset.mem_coe, Finset.mem_erase,
+      Set.mem_insert_iff, not_and_or, not_not]
+  have hsmall : s.erase a ⊂ s := Finset.erase_ssubset ha
+  obtain ⟨t, ht, htg⟩ := ih (s.erase a) hsmall g
+    (by rw [hnew]; exact B.closedUnder_insert live (s : Set V)ᶜ hΩ a hmax')
+      (by rw [hnew]; exact hg)
+  refine ⟨t, ht, ?_⟩
+  intro x hx
+  exact (htg x (by rw [hnew]; exact Set.mem_insert_of_mem _ hx)).trans (hgf x hx)
+
+end Sheaf
+end KLInvariance.MomentGraph
+
+end
+
+section
+
+namespace KLInvariance.MomentGraph.Sheaf
+
+universe u v w z
+variable {R : Type u} [CommRing R] {V : Type v} [PartialOrder V] {E : Type w}
+  {G : OrderedGraph V E} (B : Sheaf.{u,v,w,z} (R := R) G)
+
+/-- The actual module of sections on a vertex set. Coordinates outside the
+set are harmless and ignored by all restriction maps. -/
+def sections (Ω : Set V) : Submodule R B.Assignment where
+  carrier := {f | B.CompatibleOn Ω f}
+  zero_mem' := by intro e hs ht; simp
+  add_mem' := by
+    intro f g hf hg e hs ht
+    change B.lower e (f (G.source e) + g (G.source e)) =
+      B.upper e (f (G.target e) + g (G.target e))
+    rw [map_add, map_add, hf e hs ht, hg e hs ht]
+  smul_mem' := by
+    intro r f hf e hs ht
+    change B.lower e (r • f (G.source e)) = B.upper e (r • f (G.target e))
+    rw [map_smul, map_smul, hf e hs ht]
+
+/-- Generation at a stalk by genuine global sections. -/
+def Generated : Prop :=
+  ∀ x (b : B.vertex x), ∃ f : B.Assignment, B.IsGlobal f ∧ f x = b
+
+abbrev Outgoing (x : V) := {e : E // G.source e = x}
+
+abbrev Boundary (x : V) := ∀ e : Outgoing (G := G) x, B.edge e.val
+
+/-- The lower-end stalk map on all upward edges. -/
+def stalkMap (x : V) : B.vertex x →ₗ[R] B.Boundary x where
+  toFun b e := B.lower e.val (e.property.symm ▸ b)
+  map_add' b c := by ext e; cases e with | mk e he => subst x; simp
+  map_smul' r b := by ext e; cases e with | mk e he => subst x; simp
+
+/-- Boundary values supplied by sections strictly above a vertex. -/
+def boundaryMap (x : V) : B.sections (Set.Ioi x) →ₗ[R] B.Boundary x where
+  toFun f e := B.upper e.val (f.val (G.target e.val))
+  map_add' f g := by ext e; simp
+  map_smul' r f := by ext e; simp
+
+def boundaryImage (x : V) : Submodule R (B.Boundary x) :=
+  LinearMap.range (B.boundaryMap x)
+
+/-- The precise boundary-image identity in `loc:decomposition`.
+Generation supplies one inclusion and flabbiness supplies the other. -/
+theorem range_stalkMap_eq_boundaryImage
+    (hf : B.Flabby) (hg : B.Generated) (x : V) :
+    LinearMap.range (B.stalkMap x) = B.boundaryImage x := by
+  apply le_antisymm
+  · rintro v ⟨b, rfl⟩
+    obtain ⟨f, hglob, hfx⟩ := hg x b
+    refine ⟨⟨f, B.compatible_mono hglob (Set.subset_univ _)⟩, ?_⟩
+    ext e
+    change B.upper e.val (f (G.target e.val)) =
+      B.lower e.val (e.property.symm ▸ b)
+    have hc := hglob e.val (Set.mem_univ _) (Set.mem_univ _)
+    rw [← hc]
+    congr 1
+    cases e with | mk e he => subst x; exact hfx
+  · rintro v ⟨f, rfl⟩
+    have hu : IsUpperSet (Set.Ioi x) := fun a b hab ha => lt_of_lt_of_le ha hab
+    obtain ⟨g, hglob, hgf⟩ := hf (Set.Ioi x) hu f.val f.property
+    refine ⟨g x, ?_⟩
+    ext e
+    change B.lower e.val (e.property.symm ▸ g x) =
+      B.upper e.val (f.val (G.target e.val))
+    have hc := hglob e.val (Set.mem_univ _) (Set.mem_univ _)
+    have ht : G.target e.val ∈ Set.Ioi x := by
+      change x < G.target e.val
+      exact lt_of_eq_of_lt e.property.symm (G.increasing e.val)
+    rw [← hgf _ ht, ← hc]
+    congr 1
+    cases e with | mk e he => subst x; rfl
+
+/-- A sheaf satisfying the two standard section properties gives an actual
+surjection of its stalk onto the boundary module. -/
+def stalkCover (hf : B.Flabby) (hg : B.Generated) (x : V) :
+    B.vertex x →ₗ[R] B.boundaryImage x :=
+  (B.stalkMap x).codRestrict _ (fun b => by
+    rw [← B.range_stalkMap_eq_boundaryImage hf hg x]
+    exact ⟨b, rfl⟩)
+
+theorem stalkCover_surjective (hf : B.Flabby) (hg : B.Generated) (x : V) :
+    Function.Surjective (B.stalkCover hf hg x) := by
+  intro v
+  have hv : (v : B.Boundary x) ∈ LinearMap.range (B.stalkMap x) := by
+    simpa only [B.range_stalkMap_eq_boundaryImage hf hg x] using v.property
+  obtain ⟨b, hb⟩ := hv
+  exact ⟨b, Subtype.ext hb⟩
+
+end KLInvariance.MomentGraph.Sheaf
+
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/KazhdanLusztig/SimultaneousInvariance.lean b/lean/OAI/RepresentationTheory/KazhdanLusztig/SimultaneousInvariance.lean
index 4e960faf836cc896ac210f8939c08698978ade2b..8496d78b374e9c90ba0d5b5b7d48343860ea598f 100644
--- a/lean/OAI/RepresentationTheory/KazhdanLusztig/SimultaneousInvariance.lean
+++ b/lean/OAI/RepresentationTheory/KazhdanLusztig/SimultaneousInvariance.lean
@@ -1,486 +1,486 @@
-import OAI.RepresentationTheory.KazhdanLusztig.TerminalKernels
-
-namespace OAI
-
-/-!
-R-first simultaneous invariance and transported-schedule terminal-kernel freeness: upper-interval freeness, lower-interval cancellation, reversal and endpoint bottom freeness; all-entry transport and KL degree rigidity.
--/
-
-section
-
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph SchedulePaths
-universe u v u' v' us
-variable {I : Type u} {M : CoxeterMatrix I} {W : Type v} [Group W]
-  {cs : CoxeterSystem M W} {I' : Type u'} {M' : CoxeterMatrix I'}
-  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
-  {u b : W} {u' b' : W'}
-  (φ : Interval cs u b ≃o Interval cs' u' b')
-noncomputable section
-
- def upperIntervalIso (x : Interval cs u b) :
-    Interval cs x.val b ≃o Interval cs' (φ x).val b' where
-  toFun z :=
-    let zI : Interval cs u b := ⟨z.val,x.property.1.trans z.property.1,z.property.2⟩
-    ⟨(φ zI).val,φ.monotone z.property.1,(φ zI).property.2⟩
-  invFun z :=
-    let zI : Interval cs' u' b' := ⟨z.val,(φ x).property.1.trans z.property.1,z.property.2⟩
-    ⟨(φ.symm zI).val,by
-      change x ≤ φ.symm zI
-      simpa only [φ.symm_apply_apply] using φ.symm.monotone
-        (show φ x ≤ zI from z.property.1),(φ.symm zI).property.2⟩
-  left_inv z := by
-    apply Subtype.ext
-    change (φ.symm (φ ⟨z.val,x.property.1.trans z.property.1,z.property.2⟩)).val=z.val
-    exact congrArg (fun a : Interval cs u b => a.val) (φ.symm_apply_apply _)
-  right_inv z := by
-    apply Subtype.ext
-    change (φ (φ.symm ⟨z.val,(φ x).property.1.trans z.property.1,z.property.2⟩)).val=z.val
-    exact congrArg (fun a : Interval cs' u' b' => a.val) (φ.apply_symm_apply _)
-  map_rel_iff' := by
-    intro x y
-    exact φ.map_rel_iff
-
- def upperGraphInclusion (x : Interval cs u b) :
-    GraphMap (graph cs x.val b) (graph cs u b) where
-  vertex z := ⟨z.val,x.property.1.trans z.property.1,z.property.2⟩
-  edge e := ⟨(⟨(source cs x.val b e).val,
-      x.property.1.trans (source cs x.val b e).property.1,(source cs x.val b e).property.2⟩,
-    ⟨(target cs x.val b e).val,
-      x.property.1.trans (target cs x.val b e).property.1,(target cs x.val b e).property.2⟩),e.property⟩
-  source _ := rfl
-  target _ := rfl
-
- theorem upperGraphInclusion_outgoing (x : Interval cs u b) (e : Edge cs u b)
-    (z : Interval cs x.val b)
-    (hz : (upperGraphInclusion x).vertex z=source cs u b e) :
-    ∃ a, (upperGraphInclusion x).edge a=e ∧ source cs x.val b a=z := by
-  have hzval := congrArg Subtype.val hz
-  let y : Interval cs x.val b := ⟨(target cs u b e).val,
-    z.property.1.trans (hzval.symm ▸ Relation.ReflTransGen.single e.property),
-    (target cs u b e).property.2⟩
-  let a : Edge cs x.val b := ⟨(z,y),hzval.symm ▸ e.property⟩
-  refine ⟨a,?_,rfl⟩
-  exact Subtype.ext (Prod.ext hz rfl)
-
- theorem upperIntervalIso_val (x : Interval cs u b) (z : Interval cs x.val b) :
-    (upperIntervalIso φ x z).val=(φ ((upperGraphInclusion x).vertex z)).val := rfl
-
-variable [Fintype I']
-
- theorem upper_transportedSlope (x : Interval cs u b) (e : Edge cs x.val b) :
-    transportedSlope (upperIntervalIso φ x) e =
-      transportedSlope φ ((upperGraphInclusion x).edge e) := rfl
-
- theorem cutoff_is_suffix (s : ℝ) :
-    ∃ q, q <:+ chronologicalSchedule φ ∧
-      ∀ e : Edge cs u b, e ∈ q ↔ s < transportedSlope φ e := by
-  obtain ⟨q,hq,he⟩ := exists_suffix_score_gt (transportedSlope φ) s
-    (chronologicalSchedule φ) (chronologicalSchedule_pairwise φ)
-  exact ⟨q,hq,fun e => by simpa only [chronologicalSchedule_full φ e,true_and] using he e⟩
-
-variable [Fintype I] {σ : Type (max us v)} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  (hub : BruhatLE cs u b)
-
- theorem intervalBoundary_upper_kernel (x : Interval cs u b)
-    (z : Interval cs x.val b) (T : Set (Edge cs u b)) :
-    (intervalBoundary basis x.property.2).sheaf.upwardSubmodule z
-      {e | (upperGraphInclusion x).edge e.val ∈ T} =
-    (intervalBoundary basis hub).sheaf.upwardSubmodule ((upperGraphInclusion x).vertex z)
-      {e | e.val ∈ T} := by
-  exact (intervalBoundary basis hub).sheaf.forget.pullback_kernel
-    (upperGraphInclusion x) z T Set.univ
-    (fun e he => False.elim (he (Set.mem_univ e)))
-    (fun e _ he => upperGraphInclusion_outgoing x e z he)
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-/-! Transport of the actual partial-kernel freeness from a smaller-bottom
-interval. No change of ambient sheaf, grading, or reflection order. -/
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph SchedulePaths
-universe u v u' v' us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {σ : Type (max us v)} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
-  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
-  {u b : W} {u' b' : W'} (hub : BruhatLE cs u b)
-  (φ : Interval cs u b ≃o Interval cs' u' b')
-noncomputable section
-
-def FreeSuffixes : Prop :=
-  ∀ q, q <:+ chronologicalSchedule φ → ∀ x : Interval cs u b,
-    Module.Free (Coefficient (σ := σ))
-      ((intervalBoundary basis hub).sheaf.upwardSubmodule x
-        (remainingConditions cs u b q x))
-
- theorem target_free_of_upper_suffixes {e : Edge cs u b} {q : List (Edge cs u b)}
-    (hq : e::q <:+ chronologicalSchedule φ)
-    (hsmall : FreeSuffixes basis (target cs u b e).property.2
-      (upperIntervalIso φ (target cs u b e))) :
-    Module.Free (Coefficient (σ := σ))
-      ((intervalBoundary basis hub).sheaf.upwardSubmodule (target cs u b e)
-        (remainingConditions cs u b q (target cs u b e))) := by
-  let x := target cs u b e
-  let z := intervalBottom cs x.property.2
-  obtain ⟨p,hp,he⟩ := cutoff_is_suffix (upperIntervalIso φ x) (transportedSlope φ e)
-  have hh := hsmall p hp z
-  let T : Set (Edge cs u b) := {f | transportedSlope φ e < transportedSlope φ f}
-  have hs : remainingConditions cs x.val b p z =
-      {f | (upperGraphInclusion x).edge f.val ∈ T} := by
-    ext f
-    exact he f.val
-  rw [hs,intervalBoundary_upper_kernel basis hub x z T] at hh
-  have hb : remainingConditions cs u b q x = {f | f.val ∈ T} := by
-    ext f
-    exact suffix_target_iff φ hq f.property
-  rw [hb]
-  exact hh
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph SchedulePaths
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  (hub : BruhatLE cs u b)
-  (B : BoundarySheaf (polynomialGrading_negative ℝ σ)
-    (graph cs u b) (polynomialLabel cs u b basis) ⟨b,hub,bruhat_refl cs b⟩)
-noncomputable section
-
-omit [Fintype σ] in
- theorem no_conditions_free (x : Interval cs u b) :
-    Module.Free (Coefficient (σ := σ))
-      (B.sheaf.upwardSubmodule x (remainingConditions cs u b [] x)) := by
-  have he : B.sheaf.upwardSubmodule x (remainingConditions cs u b [] x) = ⊤ := by
-    ext v
-    change (∀ e, e ∈ remainingConditions cs u b [] x → _) ↔ True
-    simp [remainingConditions]
-  rw [he]
-  let := B.free x
-  exact Module.Free.of_equiv (Submodule.topEquiv :
-    (⊤ : Submodule (Coefficient (σ := σ)) (B.sheaf.vertex x)) ≃ₗ[Coefficient] B.sheaf.vertex x).symm
-
-/-- Reverse traversal of the actual schedule, after embedded image equality.
-Both kernels used in Schanuel have already been made free at this point. -/
- theorem schedule_kernels_free_of_images_eq (p : List (Edge cs u b))
-    (heq : ∀ e q, e::q <:+ p →
-      B.sheaf.lowerImage e (remainingConditions cs u b q ((graph cs u b).source e)) =
-      (B.sheaf.upperImage e (remainingConditions cs u b q ((graph cs u b).target e))).map
-        (edgeFactorMap cs basis u b hub B e)) :
-    ∀ q, q <:+ p → ∀ x : Interval cs u b,
-      Module.Free (Coefficient (σ := σ))
-        (B.sheaf.upwardSubmodule x (remainingConditions cs u b q x)) := by
-  induction p with
-  | nil =>
-    intro q hq x
-    have hq' : q=[] := by simpa using hq
-    subst q
-    exact no_conditions_free cs u b basis hub B x
-  | cons e p ih =>
-    have hip := ih (fun a q hq => heq a q (hq.trans (List.suffix_cons e p)))
-    intro q hq x
-    rcases List.suffix_cons_iff.mp hq with h | h
-    · subst q
-      by_cases hx : source cs u b e = x
-      · subst x
-        rw [remainingConditions_cons_source]
-        let hS : Module.Free (Coefficient (σ := σ))
-            (B.sheaf.upwardSubmodule ((graph cs u b).source e)
-              (remainingConditions cs u b p (source cs u b e))) :=
-          hip p (List.suffix_refl _) (source cs u b e)
-        let hT : Module.Free (Coefficient (σ := σ))
-            (B.sheaf.upwardSubmodule ((graph cs u b).target e)
-              (remainingConditions cs u b p ((graph cs u b).target e))) :=
-          hip p (List.suffix_refl _) (target cs u b e)
-        exact partial_kernel_free_of_image_eq cs u b basis hub B e _ _
-          (heq e p (List.suffix_refl _))
-      · rw [remainingConditions_cons_other cs u b e p x hx]
-        exact hip p (List.suffix_refl _) x
-    · exact hip q h x
-
- theorem schedule_kernels_free_of_endpoint_eq (p : List (Edge cs u b))
-    (hinc : ∀ e q, e::q <:+ p →
-      B.sheaf.lowerImage e (remainingConditions cs u b q ((graph cs u b).source e)) ≤
-      (B.sheaf.upperImage e (remainingConditions cs u b q ((graph cs u b).target e))).map
-        (edgeFactorMap cs basis u b hub B e))
-    {t : ℝ} (ht : 0 < t) (ht1 : t < 1)
-    (hend : scheduleState cs u b basis hub B t [] =
-      dispatch (source cs u b) (target cs u b) (t⁻¹-t)
-        (scheduleState cs u b basis hub B t p) p.reverse) :
-    ∀ q, q <:+ p → ∀ x : Interval cs u b,
-      Module.Free (Coefficient (σ := σ))
-        (B.sheaf.upwardSubmodule x (remainingConditions cs u b q x)) := by
-  exact schedule_kernels_free_of_images_eq cs u b basis hub B p
-    (edgeImages_eq_of_scheduleState_eq cs u b basis hub B p hinc ht ht1 hend)
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-/-! Actual-character comparison: no identification with the KL basis is used.
-This permits induction on R kernels rather than on stalk character recognition. -/
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph Polynomial SchedulePaths
-universe u u'
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] {cs : CoxeterSystem M W}
-  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
-  {W' : Type u'} [Group W'] {cs' : CoxeterSystem M' W'}
-  {a b : W} {a' b' : W'} (hab : BruhatLE cs a b)
-  (φ : Interval cs a b ≃o Interval cs' a' b')
-noncomputable section
-local instance (X : Type*) : DecidableEq X := Classical.decEq X
-
- theorem transported_initial_scaledStalk {t : ℝ} (ht : 0<t) (ht1 : t<1)
-    (x : Interval cs a b) :
-    scheduleState cs a b basis hab (intervalBoundary basis hab) t (chronologicalSchedule φ) x=
-      scaledStalk cs b basis t⁻¹ ((upperInclusion cs a b).vertex x) := by
-  rw [scheduleState,schedule_all_conditions φ]
-  unfold intervalBoundary
-  rw [normalized_fullKernel_restrictBoundary]
-  exact normalized_fullKernel_scaledStalk cs b basis _ ht ht1
-
- theorem actual_comparison_of_upper_freeness
-    (hfree : ∀ e q, e::q <:+ chronologicalSchedule φ →
-      Module.Free (Coefficient (σ := σ))
-        ((intervalBoundary basis hab).sheaf.upwardSubmodule (target cs a b e)
-          (remainingConditions cs a b q (target cs a b e))))
-    {t : ℝ} (ht : 0<t) (ht1 : t<1) :
-    (fun x : Interval cs a b => scaledStalk cs b basis t ((upperInclusion cs a b).vertex x)) ≤
-      dispatch (source cs a b) (target cs a b) (t⁻¹-t)
-        (fun x => scaledStalk cs b basis t⁻¹ ((upperInclusion cs a b).vertex x))
-        (transportedSchedule φ) := by
-  have h := schedule_comparison_of_upper_freeness basis hab φ hfree ht ht1
-  rwa [funext (schedule_final_scaledStalk cs b basis a hab ht ht1),
-    funext (transported_initial_scaledStalk basis hab φ ht ht1)] at h
-
- theorem actual_endpoint_of_rTilde_eq
-    (hr : ∀ x z : Interval cs a b, Hecke.rTilde cs x.val z.val=
-      Hecke.rTilde cs' (φ x).val (φ z).val)
-    {t : ℝ} (ht : 0<t) (ht1 : t<1) :
-    scheduleState cs a b basis hab (intervalBoundary basis hab) t []=
-      dispatch (source cs a b) (target cs a b) (t⁻¹-t)
-        (scheduleState cs a b basis hab (intervalBoundary basis hab) t (chronologicalSchedule φ))
-        (chronologicalSchedule φ).reverse := by
-  rw [funext (schedule_final_scaledStalk cs b basis a hab ht ht1),
-    funext (transported_initial_scaledStalk basis hab φ ht ht1),
-    chronologicalSchedule,List.reverse_reverse]
-  funext x
-  rw [transported_dispatch_matrix φ _ (Int.castRingHom ℝ),
-    scaledStalk_reciprocity_upper cs b basis t ht.ne' a x]
-  congr 1
-  funext z
-  rw [hr x z]
-
- theorem actual_freeSuffixes_of_rTilde_eq
-    (hr : ∀ x z : Interval cs a b, Hecke.rTilde cs x.val z.val=
-      Hecke.rTilde cs' (φ x).val (φ z).val)
-    (hfree : ∀ e q, e::q <:+ chronologicalSchedule φ →
-      Module.Free (Coefficient (σ := σ))
-        ((intervalBoundary basis hab).sheaf.upwardSubmodule (target cs a b e)
-          (remainingConditions cs a b q (target cs a b e)))) :
-    FreeSuffixes basis hab φ := by
-  apply schedule_kernels_free_of_endpoint_eq cs a b basis hab (intervalBoundary basis hab)
-    (chronologicalSchedule φ)
-    (fun e q hq => by
-      let := hfree e q hq
-      exact schedule_edge_image_le basis hab φ hq)
-    (t := 1/2) (by norm_num) (by norm_num)
-  exact actual_endpoint_of_rTilde_eq basis hab φ hr (by norm_num) (by norm_num)
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-
-namespace KLInvariance.Comparison
-open Polynomial
-noncomputable section
-
-def scaledPolynomial (t : ℝ) (d : ℕ) (p : ℤ[X]) : ℝ :=
-  t ^ (-(d : ℤ)) * p.eval₂ (Int.castRingHom ℝ) (t^2)
-
- theorem eq_of_scaled_difference_reciprocal (p p' : ℤ[X]) (d : ℕ)
-    (hp : 2*p.natDegree < d) (hp' : 2*p'.natDegree < d)
-    (h : ∀ t : ℝ, 0 < t → t < 1 →
-      scaledPolynomial t d p-scaledPolynomial t d p' =
-        scaledPolynomial t⁻¹ d p-scaledPolynomial t⁻¹ d p') : p=p' := by
-  apply eq_of_eval_difference_reciprocal p p' d hp hp'
-  intro q hq hq1
-  let t := Real.sqrt q
-  have ht : 0 < t := Real.sqrt_pos.2 hq
-  have hsq : t^2=q := Real.sq_sqrt hq.le
-  have ht1 : t < 1 := by nlinarith
-  have hisq : (t⁻¹)^2=q⁻¹ := by rw [inv_pow,hsq]
-  have he := h t ht ht1
-  simp only [scaledPolynomial,zpow_neg,zpow_natCast,inv_pow,inv_inv,hsq,hisq,
-    ← mul_sub] at he
-  have hh := congrArg (fun z : ℝ => t^d*z) he
-  have htd : t^d ≠ 0 := pow_ne_zero d ht.ne'
-  rw [← mul_assoc,mul_inv_cancel₀ htd,one_mul] at hh
-  rw [← mul_assoc,← pow_add] at hh
-  rw [show d+d=2*d by omega,pow_mul,hsq] at hh
-  exact hh
-
- theorem matrix_difference_single {V : Type*} [Fintype V] (a : V → ℝ)
-    (v w : V → ℝ) (x : V) (hdiag : a x=1)
-    (hvw : ∀ y, y ≠ x → v y=w y) :
-    (∑ y, a y*v y)-(∑ y, a y*w y)=v x-w x := by
-  classical
-  rw [← Finset.sum_sub_distrib]
-  calc
-    (∑ y, (a y*v y-a y*w y)) = ∑ y, if y=x then v x-w x else 0 := by
-      apply Finset.sum_congr rfl
-      intro y _
-      by_cases hy : y=x
-      · subst y
-        simp [hdiag]
-      · simp [hy,hvw y hy]
-    _ = v x-w x := by simp
-
- theorem matrix_difference_le {V : Type*} [Fintype V]
-    (a : V → ℝ) (v w : V → ℝ) (x : V) (p p' : ℝ)
-    (hdiag : a x=1) (hvw : ∀ y, y ≠ x → v y=w y)
-    (hp : p ≤ ∑ y, a y*v y) (hp' : p'=∑ y, a y*w y) :
-    p-p' ≤ v x-w x := by
-  rw [← matrix_difference_single a v w x hdiag hvw]
-  exact sub_le_sub hp hp'.ge
-
-end
-end KLInvariance.Comparison
-
-end
-
-
-section
-
-/-! A different closing invariant: genuine R kernels. Comparing actual BMP
-characters, native R-reciprocity cancels all proper-lower-interval entries.
-The top stalk is 1, so the remaining bottom/top R entry is bounded directly.
-No degree or character-identification assertion is required. -/
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph Polynomial SchedulePaths
-universe u u'
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] {cs : CoxeterSystem M W}
-  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
-  {W' : Type u'} [Group W'] {cs' : CoxeterSystem M' W'}
-  {a b : W} {a' b' : W'} (hab : BruhatLE cs a b) (hab' : BruhatLE cs' a' b')
-  (φ : Interval cs a b ≃o Interval cs' a' b')
-noncomputable section
-local instance (X : Type*) : DecidableEq X := Classical.decEq X
-
- theorem scaledStalk_upper_top {C : Type*} [Field C] (t : C) :
-    scaledStalk cs b basis t ((upperInclusion cs a b).vertex (intervalTop cs hab))=1 := by
-  change t^((cs.length b : ℤ)-cs.length b)*
-    eval₂ (Int.castRingHom C) (t^2)
-      (stalkCharacter cs 1 b basis (one_bruhat cs b)
-        ⟨b,one_bruhat cs b,bruhat_refl cs b⟩)=1
-  rw [stalkCharacter_top]
-  simp
-
-include hab hab'
-
- theorem rTilde_eval_le_of_comparison
-    (hrow : ∀ z : Interval cs a b, z.val≠b →
-      Hecke.rTilde cs a z.val=Hecke.rTilde cs' a' (φ z).val)
-    {t : ℝ} (ht : 0<t) (_ht1 : t<1)
-    (hcomp : (fun x : Interval cs a b => scaledStalk cs b basis t ((upperInclusion cs a b).vertex x)) ≤
-      dispatch (source cs a b) (target cs a b) (t⁻¹-t)
-        (fun x => scaledStalk cs b basis t⁻¹ ((upperInclusion cs a b).vertex x))
-        (transportedSchedule φ)) :
-    eval₂ (Int.castRingHom ℝ) (t⁻¹-t) (Hecke.rTilde cs a b) ≤
-      eval₂ (Int.castRingHom ℝ) (t⁻¹-t) (Hecke.rTilde cs' a' b') := by
-  let := interval_finite cs a b
-  let := Fintype.ofFinite (Interval cs a b)
-  have hc := hcomp (intervalBottom cs hab)
-  change scaledStalk cs b basis t ((upperInclusion cs a b).vertex (intervalBottom cs hab)) ≤ _ at hc
-  rw [transported_dispatch_matrix φ _ (Int.castRingHom ℝ),
-    scaledStalk_reciprocity_upper cs b basis t ht.ne' a,
-    intervalIso_bottom cs cs' hab hab' φ,finsum_eq_sum_of_fintype,
-    finsum_eq_sum_of_fintype] at hc
-  have hd := Comparison.matrix_difference_single
-    (fun z : Interval cs a b => scaledStalk cs b basis t⁻¹ ((upperInclusion cs a b).vertex z))
-    (fun z => eval₂ (Int.castRingHom ℝ) (t⁻¹-t) (Hecke.rTilde cs a z.val))
-    (fun z => eval₂ (Int.castRingHom ℝ) (t⁻¹-t) (Hecke.rTilde cs' a' (φ z).val))
-    (intervalTop cs hab) (scaledStalk_upper_top basis hab t⁻¹)
-    (fun z hz => congrArg _ (hrow z (fun h => hz (Subtype.ext h))))
-  rw [intervalIso_top cs cs' hab hab' φ] at hd
-  have hh : (∑ z : Interval cs a b, scaledStalk cs b basis t⁻¹ ((upperInclusion cs a b).vertex z)*
-      eval₂ (Int.castRingHom ℝ) (t⁻¹-t) (Hecke.rTilde cs a z.val))-
-      (∑ z : Interval cs a b, scaledStalk cs b basis t⁻¹ ((upperInclusion cs a b).vertex z)*
-      eval₂ (Int.castRingHom ℝ) (t⁻¹-t) (Hecke.rTilde cs' a' (φ z).val)) ≤ 0 := by
-    apply sub_nonpos.mpr
-    simpa only [mul_comm,intervalBottom] using hc
-  rw [hd] at hh
-  exact sub_nonpos.mp hh
-
- theorem rTilde_eval_le_of_upper_freeness
-    (hrow : ∀ z : Interval cs a b, z.val≠b →
-      Hecke.rTilde cs a z.val=Hecke.rTilde cs' a' (φ z).val)
-    (hfree : ∀ e q, e::q <:+ chronologicalSchedule φ →
-      Module.Free (Coefficient (σ := σ))
-        ((intervalBoundary basis hab).sheaf.upwardSubmodule (target cs a b e)
-          (remainingConditions cs a b q (target cs a b e))))
-    {t : ℝ} (ht : 0<t) (ht1 : t<1) :
-    eval₂ (Int.castRingHom ℝ) (t⁻¹-t) (Hecke.rTilde cs a b) ≤
-      eval₂ (Int.castRingHom ℝ) (t⁻¹-t) (Hecke.rTilde cs' a' b') :=
-  rTilde_eval_le_of_comparison basis hab hab' φ hrow ht ht1
-    (actual_comparison_of_upper_freeness basis hab φ hfree ht ht1)
-
-end
-end KLInvariance.BruhatGraph
-
-namespace KLInvariance.Comparison
-open Polynomial
- theorem polynomial_eq_of_transfer_eval_eq (p q : ℤ[X])
-    (h : ∀ t : ℝ, 0<t → t<1 →
-      eval₂ (Int.castRingHom ℝ) (t⁻¹-t) p=eval₂ (Int.castRingHom ℝ) (t⁻¹-t) q) : p=q := by
-  apply Polynomial.map_injective (f := Int.castRingHom ℝ) Int.cast_injective
-  apply Polynomial.eq_of_infinite_eval_eq
-  have hinj : Set.InjOn (fun t : ℝ => t⁻¹-t) (Set.Ioo 0 1) := by
-    apply StrictAntiOn.injOn
-    intro x hx y _ hxy
-    exact sub_lt_sub (inv_lt_inv₀ (lt_trans hx.1 hxy) hx.1 |>.mpr hxy) hxy
-  apply (Set.Ioo_infinite (show (0:ℝ)<1 by norm_num)).image hinj |>.mono
-  rintro _ ⟨t,ht,rfl⟩
-  simpa only [eval_map,Set.mem_ofPred_eq] using h t ht.1 ht.2
-end KLInvariance.Comparison
-
-end
-
-
-section
-
-/-! Rank bookkeeping for induction over arbitrary subintervals, not merely over
-upper restrictions of a fixed top. This is required for the R-kernel route. -/
-
-end
-
-end OAI
+import OAI.RepresentationTheory.KazhdanLusztig.TerminalKernels
+
+namespace OAI
+
+/-!
+R-first simultaneous invariance and transported-schedule terminal-kernel freeness: upper-interval freeness, lower-interval cancellation, reversal and endpoint bottom freeness; all-entry transport and KL degree rigidity.
+-/
+
+section
+
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph SchedulePaths
+universe u v u' v' us
+variable {I : Type u} {M : CoxeterMatrix I} {W : Type v} [Group W]
+  {cs : CoxeterSystem M W} {I' : Type u'} {M' : CoxeterMatrix I'}
+  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
+  {u b : W} {u' b' : W'}
+  (φ : Interval cs u b ≃o Interval cs' u' b')
+noncomputable section
+
+ def upperIntervalIso (x : Interval cs u b) :
+    Interval cs x.val b ≃o Interval cs' (φ x).val b' where
+  toFun z :=
+    let zI : Interval cs u b := ⟨z.val,x.property.1.trans z.property.1,z.property.2⟩
+    ⟨(φ zI).val,φ.monotone z.property.1,(φ zI).property.2⟩
+  invFun z :=
+    let zI : Interval cs' u' b' := ⟨z.val,(φ x).property.1.trans z.property.1,z.property.2⟩
+    ⟨(φ.symm zI).val,by
+      change x ≤ φ.symm zI
+      simpa only [φ.symm_apply_apply] using φ.symm.monotone
+        (show φ x ≤ zI from z.property.1),(φ.symm zI).property.2⟩
+  left_inv z := by
+    apply Subtype.ext
+    change (φ.symm (φ ⟨z.val,x.property.1.trans z.property.1,z.property.2⟩)).val=z.val
+    exact congrArg (fun a : Interval cs u b => a.val) (φ.symm_apply_apply _)
+  right_inv z := by
+    apply Subtype.ext
+    change (φ (φ.symm ⟨z.val,(φ x).property.1.trans z.property.1,z.property.2⟩)).val=z.val
+    exact congrArg (fun a : Interval cs' u' b' => a.val) (φ.apply_symm_apply _)
+  map_rel_iff' := by
+    intro x y
+    exact φ.map_rel_iff
+
+ def upperGraphInclusion (x : Interval cs u b) :
+    GraphMap (graph cs x.val b) (graph cs u b) where
+  vertex z := ⟨z.val,x.property.1.trans z.property.1,z.property.2⟩
+  edge e := ⟨(⟨(source cs x.val b e).val,
+      x.property.1.trans (source cs x.val b e).property.1,(source cs x.val b e).property.2⟩,
+    ⟨(target cs x.val b e).val,
+      x.property.1.trans (target cs x.val b e).property.1,(target cs x.val b e).property.2⟩),e.property⟩
+  source _ := rfl
+  target _ := rfl
+
+ theorem upperGraphInclusion_outgoing (x : Interval cs u b) (e : Edge cs u b)
+    (z : Interval cs x.val b)
+    (hz : (upperGraphInclusion x).vertex z=source cs u b e) :
+    ∃ a, (upperGraphInclusion x).edge a=e ∧ source cs x.val b a=z := by
+  have hzval := congrArg Subtype.val hz
+  let y : Interval cs x.val b := ⟨(target cs u b e).val,
+    z.property.1.trans (hzval.symm ▸ Relation.ReflTransGen.single e.property),
+    (target cs u b e).property.2⟩
+  let a : Edge cs x.val b := ⟨(z,y),hzval.symm ▸ e.property⟩
+  refine ⟨a,?_,rfl⟩
+  exact Subtype.ext (Prod.ext hz rfl)
+
+ theorem upperIntervalIso_val (x : Interval cs u b) (z : Interval cs x.val b) :
+    (upperIntervalIso φ x z).val=(φ ((upperGraphInclusion x).vertex z)).val := rfl
+
+variable [Fintype I']
+
+ theorem upper_transportedSlope (x : Interval cs u b) (e : Edge cs x.val b) :
+    transportedSlope (upperIntervalIso φ x) e =
+      transportedSlope φ ((upperGraphInclusion x).edge e) := rfl
+
+ theorem cutoff_is_suffix (s : ℝ) :
+    ∃ q, q <:+ chronologicalSchedule φ ∧
+      ∀ e : Edge cs u b, e ∈ q ↔ s < transportedSlope φ e := by
+  obtain ⟨q,hq,he⟩ := exists_suffix_score_gt (transportedSlope φ) s
+    (chronologicalSchedule φ) (chronologicalSchedule_pairwise φ)
+  exact ⟨q,hq,fun e => by simpa only [chronologicalSchedule_full φ e,true_and] using he e⟩
+
+variable [Fintype I] {σ : Type (max us v)} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  (hub : BruhatLE cs u b)
+
+ theorem intervalBoundary_upper_kernel (x : Interval cs u b)
+    (z : Interval cs x.val b) (T : Set (Edge cs u b)) :
+    (intervalBoundary basis x.property.2).sheaf.upwardSubmodule z
+      {e | (upperGraphInclusion x).edge e.val ∈ T} =
+    (intervalBoundary basis hub).sheaf.upwardSubmodule ((upperGraphInclusion x).vertex z)
+      {e | e.val ∈ T} := by
+  exact (intervalBoundary basis hub).sheaf.forget.pullback_kernel
+    (upperGraphInclusion x) z T Set.univ
+    (fun e he => False.elim (he (Set.mem_univ e)))
+    (fun e _ he => upperGraphInclusion_outgoing x e z he)
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+/-! Transport of the actual partial-kernel freeness from a smaller-bottom
+interval. No change of ambient sheaf, grading, or reflection order. -/
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph SchedulePaths
+universe u v u' v' us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {σ : Type (max us v)} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
+  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
+  {u b : W} {u' b' : W'} (hub : BruhatLE cs u b)
+  (φ : Interval cs u b ≃o Interval cs' u' b')
+noncomputable section
+
+def FreeSuffixes : Prop :=
+  ∀ q, q <:+ chronologicalSchedule φ → ∀ x : Interval cs u b,
+    Module.Free (Coefficient (σ := σ))
+      ((intervalBoundary basis hub).sheaf.upwardSubmodule x
+        (remainingConditions cs u b q x))
+
+ theorem target_free_of_upper_suffixes {e : Edge cs u b} {q : List (Edge cs u b)}
+    (hq : e::q <:+ chronologicalSchedule φ)
+    (hsmall : FreeSuffixes basis (target cs u b e).property.2
+      (upperIntervalIso φ (target cs u b e))) :
+    Module.Free (Coefficient (σ := σ))
+      ((intervalBoundary basis hub).sheaf.upwardSubmodule (target cs u b e)
+        (remainingConditions cs u b q (target cs u b e))) := by
+  let x := target cs u b e
+  let z := intervalBottom cs x.property.2
+  obtain ⟨p,hp,he⟩ := cutoff_is_suffix (upperIntervalIso φ x) (transportedSlope φ e)
+  have hh := hsmall p hp z
+  let T : Set (Edge cs u b) := {f | transportedSlope φ e < transportedSlope φ f}
+  have hs : remainingConditions cs x.val b p z =
+      {f | (upperGraphInclusion x).edge f.val ∈ T} := by
+    ext f
+    exact he f.val
+  rw [hs,intervalBoundary_upper_kernel basis hub x z T] at hh
+  have hb : remainingConditions cs u b q x = {f | f.val ∈ T} := by
+    ext f
+    exact suffix_target_iff φ hq f.property
+  rw [hb]
+  exact hh
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph SchedulePaths
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  (hub : BruhatLE cs u b)
+  (B : BoundarySheaf (polynomialGrading_negative ℝ σ)
+    (graph cs u b) (polynomialLabel cs u b basis) ⟨b,hub,bruhat_refl cs b⟩)
+noncomputable section
+
+omit [Fintype σ] in
+ theorem no_conditions_free (x : Interval cs u b) :
+    Module.Free (Coefficient (σ := σ))
+      (B.sheaf.upwardSubmodule x (remainingConditions cs u b [] x)) := by
+  have he : B.sheaf.upwardSubmodule x (remainingConditions cs u b [] x) = ⊤ := by
+    ext v
+    change (∀ e, e ∈ remainingConditions cs u b [] x → _) ↔ True
+    simp [remainingConditions]
+  rw [he]
+  let := B.free x
+  exact Module.Free.of_equiv (Submodule.topEquiv :
+    (⊤ : Submodule (Coefficient (σ := σ)) (B.sheaf.vertex x)) ≃ₗ[Coefficient] B.sheaf.vertex x).symm
+
+/-- Reverse traversal of the actual schedule, after embedded image equality.
+Both kernels used in Schanuel have already been made free at this point. -/
+ theorem schedule_kernels_free_of_images_eq (p : List (Edge cs u b))
+    (heq : ∀ e q, e::q <:+ p →
+      B.sheaf.lowerImage e (remainingConditions cs u b q ((graph cs u b).source e)) =
+      (B.sheaf.upperImage e (remainingConditions cs u b q ((graph cs u b).target e))).map
+        (edgeFactorMap cs basis u b hub B e)) :
+    ∀ q, q <:+ p → ∀ x : Interval cs u b,
+      Module.Free (Coefficient (σ := σ))
+        (B.sheaf.upwardSubmodule x (remainingConditions cs u b q x)) := by
+  induction p with
+  | nil =>
+    intro q hq x
+    have hq' : q=[] := by simpa using hq
+    subst q
+    exact no_conditions_free cs u b basis hub B x
+  | cons e p ih =>
+    have hip := ih (fun a q hq => heq a q (hq.trans (List.suffix_cons e p)))
+    intro q hq x
+    rcases List.suffix_cons_iff.mp hq with h | h
+    · subst q
+      by_cases hx : source cs u b e = x
+      · subst x
+        rw [remainingConditions_cons_source]
+        let hS : Module.Free (Coefficient (σ := σ))
+            (B.sheaf.upwardSubmodule ((graph cs u b).source e)
+              (remainingConditions cs u b p (source cs u b e))) :=
+          hip p (List.suffix_refl _) (source cs u b e)
+        let hT : Module.Free (Coefficient (σ := σ))
+            (B.sheaf.upwardSubmodule ((graph cs u b).target e)
+              (remainingConditions cs u b p ((graph cs u b).target e))) :=
+          hip p (List.suffix_refl _) (target cs u b e)
+        exact partial_kernel_free_of_image_eq cs u b basis hub B e _ _
+          (heq e p (List.suffix_refl _))
+      · rw [remainingConditions_cons_other cs u b e p x hx]
+        exact hip p (List.suffix_refl _) x
+    · exact hip q h x
+
+ theorem schedule_kernels_free_of_endpoint_eq (p : List (Edge cs u b))
+    (hinc : ∀ e q, e::q <:+ p →
+      B.sheaf.lowerImage e (remainingConditions cs u b q ((graph cs u b).source e)) ≤
+      (B.sheaf.upperImage e (remainingConditions cs u b q ((graph cs u b).target e))).map
+        (edgeFactorMap cs basis u b hub B e))
+    {t : ℝ} (ht : 0 < t) (ht1 : t < 1)
+    (hend : scheduleState cs u b basis hub B t [] =
+      dispatch (source cs u b) (target cs u b) (t⁻¹-t)
+        (scheduleState cs u b basis hub B t p) p.reverse) :
+    ∀ q, q <:+ p → ∀ x : Interval cs u b,
+      Module.Free (Coefficient (σ := σ))
+        (B.sheaf.upwardSubmodule x (remainingConditions cs u b q x)) := by
+  exact schedule_kernels_free_of_images_eq cs u b basis hub B p
+    (edgeImages_eq_of_scheduleState_eq cs u b basis hub B p hinc ht ht1 hend)
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+/-! Actual-character comparison: no identification with the KL basis is used.
+This permits induction on R kernels rather than on stalk character recognition. -/
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph Polynomial SchedulePaths
+universe u u'
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] {cs : CoxeterSystem M W}
+  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
+  {W' : Type u'} [Group W'] {cs' : CoxeterSystem M' W'}
+  {a b : W} {a' b' : W'} (hab : BruhatLE cs a b)
+  (φ : Interval cs a b ≃o Interval cs' a' b')
+noncomputable section
+local instance (X : Type*) : DecidableEq X := Classical.decEq X
+
+ theorem transported_initial_scaledStalk {t : ℝ} (ht : 0<t) (ht1 : t<1)
+    (x : Interval cs a b) :
+    scheduleState cs a b basis hab (intervalBoundary basis hab) t (chronologicalSchedule φ) x=
+      scaledStalk cs b basis t⁻¹ ((upperInclusion cs a b).vertex x) := by
+  rw [scheduleState,schedule_all_conditions φ]
+  unfold intervalBoundary
+  rw [normalized_fullKernel_restrictBoundary]
+  exact normalized_fullKernel_scaledStalk cs b basis _ ht ht1
+
+ theorem actual_comparison_of_upper_freeness
+    (hfree : ∀ e q, e::q <:+ chronologicalSchedule φ →
+      Module.Free (Coefficient (σ := σ))
+        ((intervalBoundary basis hab).sheaf.upwardSubmodule (target cs a b e)
+          (remainingConditions cs a b q (target cs a b e))))
+    {t : ℝ} (ht : 0<t) (ht1 : t<1) :
+    (fun x : Interval cs a b => scaledStalk cs b basis t ((upperInclusion cs a b).vertex x)) ≤
+      dispatch (source cs a b) (target cs a b) (t⁻¹-t)
+        (fun x => scaledStalk cs b basis t⁻¹ ((upperInclusion cs a b).vertex x))
+        (transportedSchedule φ) := by
+  have h := schedule_comparison_of_upper_freeness basis hab φ hfree ht ht1
+  rwa [funext (schedule_final_scaledStalk cs b basis a hab ht ht1),
+    funext (transported_initial_scaledStalk basis hab φ ht ht1)] at h
+
+ theorem actual_endpoint_of_rTilde_eq
+    (hr : ∀ x z : Interval cs a b, Hecke.rTilde cs x.val z.val=
+      Hecke.rTilde cs' (φ x).val (φ z).val)
+    {t : ℝ} (ht : 0<t) (ht1 : t<1) :
+    scheduleState cs a b basis hab (intervalBoundary basis hab) t []=
+      dispatch (source cs a b) (target cs a b) (t⁻¹-t)
+        (scheduleState cs a b basis hab (intervalBoundary basis hab) t (chronologicalSchedule φ))
+        (chronologicalSchedule φ).reverse := by
+  rw [funext (schedule_final_scaledStalk cs b basis a hab ht ht1),
+    funext (transported_initial_scaledStalk basis hab φ ht ht1),
+    chronologicalSchedule,List.reverse_reverse]
+  funext x
+  rw [transported_dispatch_matrix φ _ (Int.castRingHom ℝ),
+    scaledStalk_reciprocity_upper cs b basis t ht.ne' a x]
+  congr 1
+  funext z
+  rw [hr x z]
+
+ theorem actual_freeSuffixes_of_rTilde_eq
+    (hr : ∀ x z : Interval cs a b, Hecke.rTilde cs x.val z.val=
+      Hecke.rTilde cs' (φ x).val (φ z).val)
+    (hfree : ∀ e q, e::q <:+ chronologicalSchedule φ →
+      Module.Free (Coefficient (σ := σ))
+        ((intervalBoundary basis hab).sheaf.upwardSubmodule (target cs a b e)
+          (remainingConditions cs a b q (target cs a b e)))) :
+    FreeSuffixes basis hab φ := by
+  apply schedule_kernels_free_of_endpoint_eq cs a b basis hab (intervalBoundary basis hab)
+    (chronologicalSchedule φ)
+    (fun e q hq => by
+      let := hfree e q hq
+      exact schedule_edge_image_le basis hab φ hq)
+    (t := 1/2) (by norm_num) (by norm_num)
+  exact actual_endpoint_of_rTilde_eq basis hab φ hr (by norm_num) (by norm_num)
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+
+namespace KLInvariance.Comparison
+open Polynomial
+noncomputable section
+
+def scaledPolynomial (t : ℝ) (d : ℕ) (p : ℤ[X]) : ℝ :=
+  t ^ (-(d : ℤ)) * p.eval₂ (Int.castRingHom ℝ) (t^2)
+
+ theorem eq_of_scaled_difference_reciprocal (p p' : ℤ[X]) (d : ℕ)
+    (hp : 2*p.natDegree < d) (hp' : 2*p'.natDegree < d)
+    (h : ∀ t : ℝ, 0 < t → t < 1 →
+      scaledPolynomial t d p-scaledPolynomial t d p' =
+        scaledPolynomial t⁻¹ d p-scaledPolynomial t⁻¹ d p') : p=p' := by
+  apply eq_of_eval_difference_reciprocal p p' d hp hp'
+  intro q hq hq1
+  let t := Real.sqrt q
+  have ht : 0 < t := Real.sqrt_pos.2 hq
+  have hsq : t^2=q := Real.sq_sqrt hq.le
+  have ht1 : t < 1 := by nlinarith
+  have hisq : (t⁻¹)^2=q⁻¹ := by rw [inv_pow,hsq]
+  have he := h t ht ht1
+  simp only [scaledPolynomial,zpow_neg,zpow_natCast,inv_pow,inv_inv,hsq,hisq,
+    ← mul_sub] at he
+  have hh := congrArg (fun z : ℝ => t^d*z) he
+  have htd : t^d ≠ 0 := pow_ne_zero d ht.ne'
+  rw [← mul_assoc,mul_inv_cancel₀ htd,one_mul] at hh
+  rw [← mul_assoc,← pow_add] at hh
+  rw [show d+d=2*d by omega,pow_mul,hsq] at hh
+  exact hh
+
+ theorem matrix_difference_single {V : Type*} [Fintype V] (a : V → ℝ)
+    (v w : V → ℝ) (x : V) (hdiag : a x=1)
+    (hvw : ∀ y, y ≠ x → v y=w y) :
+    (∑ y, a y*v y)-(∑ y, a y*w y)=v x-w x := by
+  classical
+  rw [← Finset.sum_sub_distrib]
+  calc
+    (∑ y, (a y*v y-a y*w y)) = ∑ y, if y=x then v x-w x else 0 := by
+      apply Finset.sum_congr rfl
+      intro y _
+      by_cases hy : y=x
+      · subst y
+        simp [hdiag]
+      · simp [hy,hvw y hy]
+    _ = v x-w x := by simp
+
+ theorem matrix_difference_le {V : Type*} [Fintype V]
+    (a : V → ℝ) (v w : V → ℝ) (x : V) (p p' : ℝ)
+    (hdiag : a x=1) (hvw : ∀ y, y ≠ x → v y=w y)
+    (hp : p ≤ ∑ y, a y*v y) (hp' : p'=∑ y, a y*w y) :
+    p-p' ≤ v x-w x := by
+  rw [← matrix_difference_single a v w x hdiag hvw]
+  exact sub_le_sub hp hp'.ge
+
+end
+end KLInvariance.Comparison
+
+end
+
+
+section
+
+/-! A different closing invariant: genuine R kernels. Comparing actual BMP
+characters, native R-reciprocity cancels all proper-lower-interval entries.
+The top stalk is 1, so the remaining bottom/top R entry is bounded directly.
+No degree or character-identification assertion is required. -/
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph Polynomial SchedulePaths
+universe u u'
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] {cs : CoxeterSystem M W}
+  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
+  {W' : Type u'} [Group W'] {cs' : CoxeterSystem M' W'}
+  {a b : W} {a' b' : W'} (hab : BruhatLE cs a b) (hab' : BruhatLE cs' a' b')
+  (φ : Interval cs a b ≃o Interval cs' a' b')
+noncomputable section
+local instance (X : Type*) : DecidableEq X := Classical.decEq X
+
+ theorem scaledStalk_upper_top {C : Type*} [Field C] (t : C) :
+    scaledStalk cs b basis t ((upperInclusion cs a b).vertex (intervalTop cs hab))=1 := by
+  change t^((cs.length b : ℤ)-cs.length b)*
+    eval₂ (Int.castRingHom C) (t^2)
+      (stalkCharacter cs 1 b basis (one_bruhat cs b)
+        ⟨b,one_bruhat cs b,bruhat_refl cs b⟩)=1
+  rw [stalkCharacter_top]
+  simp
+
+include hab hab'
+
+ theorem rTilde_eval_le_of_comparison
+    (hrow : ∀ z : Interval cs a b, z.val≠b →
+      Hecke.rTilde cs a z.val=Hecke.rTilde cs' a' (φ z).val)
+    {t : ℝ} (ht : 0<t) (_ht1 : t<1)
+    (hcomp : (fun x : Interval cs a b => scaledStalk cs b basis t ((upperInclusion cs a b).vertex x)) ≤
+      dispatch (source cs a b) (target cs a b) (t⁻¹-t)
+        (fun x => scaledStalk cs b basis t⁻¹ ((upperInclusion cs a b).vertex x))
+        (transportedSchedule φ)) :
+    eval₂ (Int.castRingHom ℝ) (t⁻¹-t) (Hecke.rTilde cs a b) ≤
+      eval₂ (Int.castRingHom ℝ) (t⁻¹-t) (Hecke.rTilde cs' a' b') := by
+  let := interval_finite cs a b
+  let := Fintype.ofFinite (Interval cs a b)
+  have hc := hcomp (intervalBottom cs hab)
+  change scaledStalk cs b basis t ((upperInclusion cs a b).vertex (intervalBottom cs hab)) ≤ _ at hc
+  rw [transported_dispatch_matrix φ _ (Int.castRingHom ℝ),
+    scaledStalk_reciprocity_upper cs b basis t ht.ne' a,
+    intervalIso_bottom cs cs' hab hab' φ,finsum_eq_sum_of_fintype,
+    finsum_eq_sum_of_fintype] at hc
+  have hd := Comparison.matrix_difference_single
+    (fun z : Interval cs a b => scaledStalk cs b basis t⁻¹ ((upperInclusion cs a b).vertex z))
+    (fun z => eval₂ (Int.castRingHom ℝ) (t⁻¹-t) (Hecke.rTilde cs a z.val))
+    (fun z => eval₂ (Int.castRingHom ℝ) (t⁻¹-t) (Hecke.rTilde cs' a' (φ z).val))
+    (intervalTop cs hab) (scaledStalk_upper_top basis hab t⁻¹)
+    (fun z hz => congrArg _ (hrow z (fun h => hz (Subtype.ext h))))
+  rw [intervalIso_top cs cs' hab hab' φ] at hd
+  have hh : (∑ z : Interval cs a b, scaledStalk cs b basis t⁻¹ ((upperInclusion cs a b).vertex z)*
+      eval₂ (Int.castRingHom ℝ) (t⁻¹-t) (Hecke.rTilde cs a z.val))-
+      (∑ z : Interval cs a b, scaledStalk cs b basis t⁻¹ ((upperInclusion cs a b).vertex z)*
+      eval₂ (Int.castRingHom ℝ) (t⁻¹-t) (Hecke.rTilde cs' a' (φ z).val)) ≤ 0 := by
+    apply sub_nonpos.mpr
+    simpa only [mul_comm,intervalBottom] using hc
+  rw [hd] at hh
+  exact sub_nonpos.mp hh
+
+ theorem rTilde_eval_le_of_upper_freeness
+    (hrow : ∀ z : Interval cs a b, z.val≠b →
+      Hecke.rTilde cs a z.val=Hecke.rTilde cs' a' (φ z).val)
+    (hfree : ∀ e q, e::q <:+ chronologicalSchedule φ →
+      Module.Free (Coefficient (σ := σ))
+        ((intervalBoundary basis hab).sheaf.upwardSubmodule (target cs a b e)
+          (remainingConditions cs a b q (target cs a b e))))
+    {t : ℝ} (ht : 0<t) (ht1 : t<1) :
+    eval₂ (Int.castRingHom ℝ) (t⁻¹-t) (Hecke.rTilde cs a b) ≤
+      eval₂ (Int.castRingHom ℝ) (t⁻¹-t) (Hecke.rTilde cs' a' b') :=
+  rTilde_eval_le_of_comparison basis hab hab' φ hrow ht ht1
+    (actual_comparison_of_upper_freeness basis hab φ hfree ht ht1)
+
+end
+end KLInvariance.BruhatGraph
+
+namespace KLInvariance.Comparison
+open Polynomial
+ theorem polynomial_eq_of_transfer_eval_eq (p q : ℤ[X])
+    (h : ∀ t : ℝ, 0<t → t<1 →
+      eval₂ (Int.castRingHom ℝ) (t⁻¹-t) p=eval₂ (Int.castRingHom ℝ) (t⁻¹-t) q) : p=q := by
+  apply Polynomial.map_injective (f := Int.castRingHom ℝ) Int.cast_injective
+  apply Polynomial.eq_of_infinite_eval_eq
+  have hinj : Set.InjOn (fun t : ℝ => t⁻¹-t) (Set.Ioo 0 1) := by
+    apply StrictAntiOn.injOn
+    intro x hx y _ hxy
+    exact sub_lt_sub (inv_lt_inv₀ (lt_trans hx.1 hxy) hx.1 |>.mpr hxy) hxy
+  apply (Set.Ioo_infinite (show (0:ℝ)<1 by norm_num)).image hinj |>.mono
+  rintro _ ⟨t,ht,rfl⟩
+  simpa only [eval_map,Set.mem_ofPred_eq] using h t ht.1 ht.2
+end KLInvariance.Comparison
+
+end
+
+
+section
+
+/-! Rank bookkeeping for induction over arbitrary subintervals, not merely over
+upper restrictions of a fixed top. This is required for the R-kernel route. -/
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/KazhdanLusztig/Support.lean b/lean/OAI/RepresentationTheory/KazhdanLusztig/Support.lean
index c668aaf1bded0e9ab3853df45e60a48b9c058a4e..aa426f51cdd32459b750fe8c02e9d91fea315747 100644
--- a/lean/OAI/RepresentationTheory/KazhdanLusztig/Support.lean
+++ b/lean/OAI/RepresentationTheory/KazhdanLusztig/Support.lean
@@ -1,1920 +1,1920 @@
-import OAI.RepresentationTheory.KazhdanLusztig.BalancedPairings
-
-namespace OAI
-
-/-!
-Graded unit splittings, integral support and balanced cuts, and boundary-sheaf support and recognition.
--/
-
-section
-
-namespace KLInvariance.Graded.ModuleData
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-  (hneg : ∀ n < 0, 𝓐 n=⊥)
-noncomputable section
-
- theorem unit_endomorphisms_commute (f g : Hom (unit.{uk,ua,um} hneg) (unit hneg))
-    (v : unit hneg) : f.map (g.map v)=g.map (f.map v) := by
-  let z : unit.{uk,ua,um} hneg := fun _ => (1 : A)
-  have he (w : unit.{uk,ua,um} hneg) : w=w PUnit.unit • z := by
-    funext j
-    change w j=w PUnit.unit * 1
-    simp only [mul_one]
-  have hf (w : unit.{uk,ua,um} hneg) : f.map w=w PUnit.unit • f.map z := by
-    conv_lhs => rw [he w]
-    exact map_smul f.map _ z
-  have hg (w : unit.{uk,ua,um} hneg) : g.map w=w PUnit.unit • g.map z := by
-    conv_lhs => rw [he w]
-    exact map_smul g.map _ z
-  rw [hf (g.map v),hg (f.map v),hg v,hf v]
-  funext j
-  change (v PUnit.unit * (g.map z) PUnit.unit) * (f.map z) j =
-    (v PUnit.unit * (f.map z) PUnit.unit) * (g.map z) j
-  have hej : j=PUnit.unit := Subsingleton.elim _ _
-  subst j
-  ring
-
- theorem unit_split_reverse (f g : Hom (unit.{uk,ua,um} hneg) (unit hneg))
-    (h : ∀ v, g.map (f.map v)=v) (v : unit hneg) : f.map (g.map v)=v :=
-  (unit_endomorphisms_commute hneg f g v).trans (h v)
-
-
- theorem split_reverse_of_unit {B N : ModuleData.{uk,ua,um} 𝓐}
-    (hB : B=unit hneg) (o : Hom N (unit hneg)) (o' : Hom (unit hneg) N)
-    (ho : ∀ v, o'.map (o.map v)=v) (i : Hom B N) (p : Hom N B)
-    (hpi : ∀ v, p.map (i.map v)=v) (v : N) : i.map (p.map v)=v := by
-  subst B
-  have hp : ∀ a, (p.comp o').map ((o.comp i).map a)=a := by
-    intro a
-    change p.map (o'.map (o.map (i.map a)))=a
-    rw [ho]
-    exact hpi a
-  have hh := unit_split_reverse hneg (o.comp i) (p.comp o') hp (o.map v)
-  change o.map (i.map (p.map (o'.map (o.map v))))=o.map v at hh
-  rw [ho] at hh
-  exact (ho _).symm.trans ((congrArg o'.map hh).trans (ho v))
-
-end
-end KLInvariance.Graded.ModuleData
-
-end
-
-
-section
-
-/-! The projector of any actual normalized BMP splitting into the reduced
-word, and its integral adjoint, are both the identity at the top. The splitting
-is not asserted to exist; word BMP recognition and character remain separate. -/
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
-universe u us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  (l : List I) (hl : cs.IsReduced l)
-  (α : Edge cs 1 (cs.wordProd l) → SymmetricCoefficient (M := M))
-  (B : BoundarySheaf (symmetricGrading_negative basis)
-    (graph cs 1 (cs.wordProd l)) α
-    ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩)
-  (s : GradedSheaf.SplitIn B.sheaf (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
-noncomputable section
-
-include hl in
- theorem bottBoundarySplit_top_fixed (v : bottStalk cs l (cs.wordProd l)) :
-    (s.projector.vertex ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩).map v=v := by
-  exact ModuleData.split_reverse_of_unit (symmetricGrading_negative basis) B.top
-    (bottTopHom cs basis l hl) (bottTopInvHom cs basis l hl)
-    (bottTopInvHom_top cs basis l hl)
-    (s.inclusion.vertex ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩)
-    (s.projection.vertex ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩)
-    (s.vertex_split ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩) v
-
- theorem bottBoundarySplitAdjoint_top_fixed (v : bottStalk cs l (cs.wordProd l)) :
-    ((bottAdjointSheaf cs basis l hl s.projector).vertex
-      ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩).map v=v :=
-  bottAdjointSheaf_fixed_stalk cs basis l hl s.projector
-    ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩
-    (bottBoundarySplit_top_fixed cs basis l hl α B s) v
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded
-universe u us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {σ : Type us} (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- theorem symmetricGrading_zero (a : SymmetricCoefficient (M := M))
-    (ha : a ∈ symmetricGrading basis 0) :
-    ∃ c : ℝ, algebraMap ℝ (SymmetricCoefficient (M := M)) c=a := by
-  obtain ⟨c,hc⟩ := polynomialGrading_zero ℝ σ
-    (SymmetricAlgebra.equivMvPolynomial basis a) ((mem_symmetricGrading basis 0 a).mp ha)
-  refine ⟨c,?_⟩
-  apply (SymmetricAlgebra.equivMvPolynomial basis).injective
-  exact ((SymmetricAlgebra.equivMvPolynomial basis).commutes c).trans hc
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-/-! The actual BMP axioms inherited by a graded sheaf retract. These are
-implications, not claims that the word sheaf already satisfies the axioms. -/
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
-  {G : OrderedGraph V E} {B C : GradedSheaf (𝓐 := 𝓐) G}
-noncomputable section
-
- theorem SplitIn.flabby (s : SplitIn B C) (hC : C.forget.Flabby) : B.forget.Flabby := by
-  intro Ω hΩ f hf
-  obtain ⟨g,hg,hgf⟩ := hC Ω hΩ (s.inclusion.assignmentMap f)
-    (s.inclusion.assignmentMap_compatible Ω f hf)
-  refine ⟨s.projection.assignmentMap g,
-    s.projection.assignmentMap_compatible Set.univ g hg,?_⟩
-  intro x hx
-  change (s.projection.vertex x).map (g x)=f x
-  rw [hgf x hx]
-  exact s.vertex_split x (f x)
-
- theorem SplitIn.generated (s : SplitIn B C) (hC : C.forget.Generated) : B.forget.Generated := by
-  intro x v
-  obtain ⟨f,hf,hfx⟩ := hC x ((s.inclusion.vertex x).map v)
-  refine ⟨s.projection.assignmentMap f,
-    s.projection.assignmentMap_compatible Set.univ f hf,?_⟩
-  change (s.projection.vertex x).map (f x)=v
-  rw [hfx]
-  exact s.vertex_split x v
-
- theorem SplitIn.upperQuotient (s : SplitIn B C) {α : E → A}
-    (hC : C.UpperQuotient α) : B.UpperQuotient α := by
-  intro e
-  constructor
-  · intro v
-    obtain ⟨c,hc⟩ := (hC e).1 ((s.inclusion.edge e).map v)
-    refine ⟨(s.projection.vertex (G.target e)).map c,?_⟩
-    rw [← s.projection.upper,hc]
-    exact s.edge_split e v
-  · intro v
-    constructor
-    · intro hv
-      have hz : (C.upper e).map ((s.inclusion.vertex _).map v)=0 := by
-        rw [← s.inclusion.upper,hv]
-        exact map_zero _
-      obtain ⟨c,hc⟩ := (hC e).2 _ |>.mp hz
-      refine ⟨(s.projection.vertex _).map c,?_⟩
-      have h := congrArg (s.projection.vertex _).map hc
-      rw [s.vertex_split,map_smul] at h
-      exact h
-    · rintro ⟨w,rfl⟩
-      have hz := ((hC e).2 (α e • (s.inclusion.vertex _).map w)).mpr
-        ⟨(s.inclusion.vertex _).map w,rfl⟩
-      have he : (s.inclusion.edge e).map ((B.upper e).map (α e • w))=0 := by
-        rw [s.inclusion.upper,map_smul]
-        exact hz
-      have h := congrArg (s.projection.edge e).map he
-      rw [s.edge_split,map_zero] at h
-      exact h
-
- theorem SplitIn.vertex_projective (s : SplitIn B C) (x : V)
-    [Module.Projective A (C.vertex x)] : Module.Projective A (B.vertex x) := by
-  apply Module.Projective.of_split (s.inclusion.vertex x).map (s.projection.vertex x).map
-  apply LinearMap.ext
-  exact s.vertex_split x
-
- theorem SplitIn.vertex_free
-    [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
-    [IsDomain A] [IsNoetherianRing A]
-    (hneg : ∀ n < 0, 𝓐 n=⊥)
-    (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c=a)
-    (s : SplitIn B C) (x : V) [Module.Projective A (C.vertex x)] :
-    Module.Free A (B.vertex x) := by
-  let D := B.vertex x
-  let : Module.Projective A D := s.vertex_projective x
-  obtain ⟨ι,_,bs,_⟩ := exists_homogeneous_basis_connected 𝓐 D.piece hneg hzero 0 D.nonneg
-  exact Module.Free.of_basis bs
-
-end
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-namespace KLInvariance.Graded.ModuleData
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A}
-
- theorem equivOfEq_inverse {M N : ModuleData.{uk,ua,um} 𝓐} (h : M=N) (m : M) :
-    equivOfEq h.symm (equivOfEq h m)=m := by
-  subst N
-  rfl
-
- def homOfEq {M N : ModuleData.{uk,ua,um} 𝓐} (h : M=N) : Hom M N :=
-  ⟨(equivOfEq h).toLinearMap,equivOfEq_graded h⟩
-
-end KLInvariance.Graded.ModuleData
-
-end
-
-
-section
-
-/-! The actual minimal-boundary sheaf is a summand of the reduced word.
-No splitting or word-recognition axiom is supplied as a premise. -/
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
-universe u us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- def symmetricLabel (u b : W) (e : Edge cs u b) : SymmetricCoefficient (M := M) :=
-  SymmetricAlgebra.ι ℝ (Extended M) (embed M (BruhatGraph.root cs u b e).val)
-
- theorem symmetricLabel_represents (u b : W) (e : Edge cs u b) :
-    Represents M cs (label cs u b e) (BruhatGraph.root cs u b e).val :=
-  positiveRoot_represents M cs ⟨label cs u b e,label_isReflection cs u b e⟩
-
- theorem bottGradedProjectionSheaf_upperQuotient (l : List I) (u b : W) :
-    (bottGradedProjectionSheaf cs basis l u b).UpperQuotient (symmetricLabel cs u b) := by
-  intro e
-  exact ⟨bottProjectionSheaf_upper_surjective cs l u b e,
-    bottProjectionSheaf_upper_kernel cs l u b e (symmetricLabel_represents cs u b e)⟩
-
- def bottBoundarySheafTopInclusion (l : List I) (hl : cs.IsReduced l)
-    (B : BoundarySheaf (symmetricGrading_negative basis)
-      (graph cs 1 (cs.wordProd l)) (symmetricLabel cs 1 (cs.wordProd l))
-      ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩) :
-    ModuleData.Hom (B.sheaf.vertex ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩)
-      (bottStalkData cs basis l (cs.wordProd l)) :=
-  (bottTopInvHom cs basis l hl).comp
-    ⟨(ModuleData.equivOfEq B.top).toLinearMap,ModuleData.equivOfEq_graded B.top⟩
-
- def bottBoundarySheafTopProjection (l : List I) (hl : cs.IsReduced l)
-    (B : BoundarySheaf (symmetricGrading_negative basis)
-      (graph cs 1 (cs.wordProd l)) (symmetricLabel cs 1 (cs.wordProd l))
-      ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩) :
-    ModuleData.Hom (bottStalkData cs basis l (cs.wordProd l))
-      (B.sheaf.vertex ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩) :=
-  (⟨(ModuleData.equivOfEq B.top.symm).toLinearMap,ModuleData.equivOfEq_graded B.top.symm⟩ :
-    ModuleData.Hom _ _).comp (bottTopHom cs basis l hl)
-
- theorem bottBoundarySheafTop_split (l : List I) (hl : cs.IsReduced l)
-    (B : BoundarySheaf (symmetricGrading_negative basis)
-      (graph cs 1 (cs.wordProd l)) (symmetricLabel cs 1 (cs.wordProd l))
-      ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩)
-    (v : B.sheaf.vertex ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩) :
-    (bottBoundarySheafTopProjection cs basis l hl B).map
-      ((bottBoundarySheafTopInclusion cs basis l hl B).map v)=v := by
-  change ModuleData.equivOfEq B.top.symm ((bottTopHom cs basis l hl).map
-    ((bottTopInvHom cs basis l hl).map (ModuleData.equivOfEq B.top v)))=v
-  rw [bottTopHom_inv]
-  exact ModuleData.equivOfEq_inverse B.top v
-
- theorem bottBoundarySheaf_splitIn (l : List I) (hl : cs.IsReduced l)
-    (B : BoundarySheaf (symmetricGrading_negative basis)
-      (graph cs 1 (cs.wordProd l)) (symmetricLabel cs 1 (cs.wordProd l))
-      ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩) :
-    Nonempty (GradedSheaf.SplitIn B.sheaf (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))) := by
-  classical
-  let := symmetricCoefficientNoetherian basis
-  let := interval_finite cs 1 (cs.wordProd l)
-  let := Fintype.ofFinite (Interval cs 1 (cs.wordProd l))
-  let := Fintype.ofFinite (Edge cs 1 (cs.wordProd l))
-  exact B.splitIn_of_top_split (symmetricGrading_zero basis) (fun x => x.property.2)
-    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
-    (bottGradedProjectionSheaf_upperQuotient cs basis l 1 (cs.wordProd l))
-    (fun x => bottStalkFree cs l x.val) (bottProjectionSheaf_flabby cs l hl)
-    (bottProjectionSheaf_generated cs l 1 (cs.wordProd l))
-    (bottBoundarySheafTopInclusion cs basis l hl B) (bottBoundarySheafTopProjection cs basis l hl B)
-    (bottBoundarySheafTop_split cs basis l hl B)
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A B : Type ua} [CommRing A] [CommRing B]
-  [Algebra k A] [Algebra k B]
-  {𝓐 : ℤ → Submodule k A} {𝓑 : ℤ → Submodule k B}
-  (e : A ≃ₐ[k] B) (he : ∀ n a, a∈𝓐 n ↔ e a∈𝓑 n)
-  {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
-noncomputable section
-
- def coefficientChange (S : GradedSheaf (𝓐 := 𝓐) G) : GradedSheaf (𝓐 := 𝓑) G where
-  vertex x := ModuleData.change e he (S.vertex x)
-  edge f := ModuleData.change e he (S.edge f)
-  lower f := ModuleData.changeHom e he (S.lower f)
-  upper f := ModuleData.changeHom e he (S.upper f)
-
- def Hom.coefficientChange {S T : GradedSheaf (𝓐 := 𝓐) G} (g : Hom S T) :
-    Hom (coefficientChange e he S) (coefficientChange e he T) where
-  vertex x := ModuleData.changeHom e he (g.vertex x)
-  edge f := ModuleData.changeHom e he (g.edge f)
-  lower f m := CoefficientChange.ext (g.lower f m.val)
-  upper f m := CoefficientChange.ext (g.upper f m.val)
-
- theorem coefficientChange_free (S : GradedSheaf (𝓐 := 𝓐) G) (hS : S.FreeStalks) :
-    (coefficientChange e he S).FreeStalks := by
-  intro x
-  let := hS x
-  exact ModuleData.change_free e he (S.vertex x)
-
- theorem coefficientChange_upperQuotient (S : GradedSheaf (𝓐 := 𝓐) G)
-    (α : E → A) (hS : S.UpperQuotient α) :
-    (coefficientChange e he S).UpperQuotient (fun f => e (α f)) := by
-  intro f
-  constructor
-  · exact CoefficientChange.map_surjective e _ (hS f).1
-  · intro m
-    constructor
-    · intro hm
-      have hz : (S.upper f).map m.val=0 := congrArg CoefficientChange.val hm
-      obtain ⟨n,hn⟩ := (hS f).2 m.val |>.mp hz
-      refine ⟨CoefficientChange.mk n,CoefficientChange.ext ?_⟩
-      change m.val=e.symm (e (α f)) • n
-      simpa only [e.symm_apply_apply] using hn
-    · rintro ⟨n,rfl⟩
-      apply CoefficientChange.ext
-      change (S.upper f).map (e.symm (e (α f)) • n.val)=0
-      rw [e.symm_apply_apply]
-      exact ((hS f).2 _).mpr ⟨n.val,rfl⟩
-
- theorem coefficientChange_flabby (S : GradedSheaf (𝓐 := 𝓐) G) (hS : S.forget.Flabby) :
-    (coefficientChange e he S).forget.Flabby := by
-  intro U hU f hf
-  have hf' : S.forget.CompatibleOn U (fun x => (f x).val) := fun q hs ht =>
-    congrArg CoefficientChange.val (hf q hs ht)
-  obtain ⟨g,hg,hgf⟩ := hS U hU (fun x => (f x).val) hf'
-  refine ⟨fun x => CoefficientChange.mk (g x),?_,?_⟩
-  · intro q hs ht
-    exact CoefficientChange.ext (hg q hs ht)
-  · intro x hx
-    exact CoefficientChange.ext (hgf x hx)
-
- theorem coefficientChange_generated (S : GradedSheaf (𝓐 := 𝓐) G) (hS : S.forget.Generated) :
-    (coefficientChange e he S).forget.Generated := by
-  intro x m
-  obtain ⟨g,hg,hgx⟩ := hS x m.val
-  refine ⟨fun x => CoefficientChange.mk (g x),?_,CoefficientChange.ext hgx⟩
-  intro q hs ht
-  exact CoefficientChange.ext (hg q hs ht)
-
- def coefficientChangeSections (S : GradedSheaf (𝓐 := 𝓐) G) (U : Set V) :
-    CoefficientChange e (S.forget.sections U) ≃ₗ[B] (coefficientChange e he S).forget.sections U where
-  toFun m := ⟨fun x => CoefficientChange.mk (m.val.val x),fun q hs ht =>
-    CoefficientChange.ext (m.val.property q hs ht)⟩
-  invFun m := CoefficientChange.mk ⟨fun x => (m.val x).val,fun q hs ht =>
-    congrArg CoefficientChange.val (m.property q hs ht)⟩
-  left_inv _ := rfl
-  right_inv _ := rfl
-  map_add' _ _ := rfl
-  map_smul' _ _ := rfl
-
-end
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-/-! Positive triangular decomposition of the honest word's character into
-actual BMP stalk characters. The KL identification is not assumed. -/
-namespace KLInvariance.TitsSpace
-open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
-universe u
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- def bottPolynomialSheaf (l : List I) (u b : W) :
-    GradedSheaf (𝓐 := polynomialGrading ℝ σ) (graph cs u b) :=
-  GradedSheaf.coefficientChange (SymmetricAlgebra.equivMvPolynomial basis)
-    (mem_symmetricGrading basis) (bottGradedProjectionSheaf cs basis l u b)
-
- theorem bottPolynomialSheaf_free (l : List I) (u b : W) :
-    (bottPolynomialSheaf cs basis l u b).FreeStalks :=
-  GradedSheaf.coefficientChange_free _ _ _ (fun x => bottStalkFree cs l x.val)
-
- theorem bottPolynomialSheaf_quotient (l : List I) (u b : W) :
-    (bottPolynomialSheaf cs basis l u b).UpperQuotient (polynomialLabel cs u b basis) :=
-  GradedSheaf.coefficientChange_upperQuotient _ _ _ _
-    (bottGradedProjectionSheaf_upperQuotient cs basis l u b)
-
- theorem bottPolynomialSheaf_flabby (l : List I) (hl : cs.IsReduced l) :
-    (bottPolynomialSheaf cs basis l 1 (cs.wordProd l)).forget.Flabby :=
-  GradedSheaf.coefficientChange_flabby _ _ _ (bottProjectionSheaf_flabby cs l hl)
-
- theorem bottPolynomialSheaf_generated (l : List I) (u b : W) :
-    (bottPolynomialSheaf cs basis l u b).forget.Generated :=
-  GradedSheaf.coefficientChange_generated _ _ _ (bottProjectionSheaf_generated cs l u b)
-
- theorem bottPolynomialSheaf_character (l : List I) (u b : W) (x : Interval cs u b) :
-    GradedSheaf.freeStalkChar (bottPolynomialSheaf cs basis l u b)
-      (bottPolynomialSheaf_free cs basis l u b) x=bottStalkCharacter cs basis l x.val := rfl
-
- theorem bottStalkCharacter_decomposition (l : List I) (hl : cs.IsReduced l)
-    [Fintype (Interval cs 1 (cs.wordProd l))] :
-    ∃ m : Interval cs 1 (cs.wordProd l) → Polynomial ℤ,
-      (∀ y n, 0≤(m y).coeff n) ∧ ∀ x,
-      bottStalkCharacter cs basis l x.val=
-        ∑ y, m y*lowerCharacter cs (cs.wordProd l) basis y x := by
-  classical
-  let := Fintype.ofFinite (Edge cs 1 (cs.wordProd l))
-  obtain ⟨m,hm,he⟩ := GradedSheaf.exists_stalkCharacter_decomposition
-    (boundaryModels cs (cs.wordProd l) basis)
-    (bottPolynomialSheaf cs basis l 1 (cs.wordProd l))
-    (bottPolynomialSheaf_free cs basis l 1 (cs.wordProd l))
-    (bottPolynomialSheaf_quotient cs basis l 1 (cs.wordProd l))
-    (bottPolynomialSheaf_flabby cs basis l hl)
-    (bottPolynomialSheaf_generated cs basis l 1 (cs.wordProd l))
-  refine ⟨m,hm,fun x => ?_⟩
-  simpa only [bottPolynomialSheaf_character,boundaryModels_character] using he x
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-
-namespace KLInvariance.IntegralSupport
-
-universe ur uv um
-variable {R : Type ur} [CommRing R]
-  {V : Type uv} [DecidableEq V] {A : V → Type um} [∀ x, AddCommGroup (A x)] [∀ x, Module R (A x)]
-  (M : Submodule R (∀ x, A x))
-
-noncomputable def projection (x : V) : M →ₗ[R] A x := (LinearMap.proj x).comp M.subtype
-
-def supported (x : V) : Submodule R M where
-  carrier := {m | ∀ y, y ≠ x → m.val y = 0}
-  zero_mem' := by intro y hy; rfl
-  add_mem' := by intro m n hm hn y hy; exact congrArg₂ (· + ·) (hm y hy) (hn y hy) |>.trans (zero_add _)
-  smul_mem' := by intro r m hm y hy; exact congrArg (r • ·) (hm y hy) |>.trans (smul_zero r)
-
-variable (p : R)
-  (hcut : ∀ x (m : M), (p • Pi.single x (m.val x)) ∈ M)
-
-noncomputable def cut (x : V) : M →ₗ[R] M := by
-  exact ((p • (LinearMap.single R A x).comp (LinearMap.proj x)).domRestrict M).codRestrict M (hcut x)
-
-@[simp] theorem cut_apply (x : V) (m : M) : (cut M p hcut x m).val = p • Pi.single x (m.val x) := rfl
-
-@[simp] theorem cut_self (x : V) (m : M) : (cut M p hcut x m).val x = p • m.val x := by
-  simp [cut_apply]
-
- theorem cut_other (x y : V) (h : y ≠ x) (m : M) : (cut M p hcut x m).val y = 0 := by
-  simp [cut_apply,h]
-
- theorem cut_of_supported (x : V) (m : M) (h : m ∈ supported M x) :
-    cut M p hcut x m = p • m := by
-  apply Subtype.ext
-  funext y
-  by_cases hy : y=x
-  · subst y; exact cut_self M p hcut x m
-  · rw [cut_other M p hcut x y hy]
-    change 0 = p • m.val y
-    rw [h y hy,smul_zero]
-
- theorem cut_eq_zero (x : V) (m : M) (h : projection M x m = 0) : cut M p hcut x m = 0 := by
-  apply Subtype.ext
-  change p • Pi.single x (m.val x) = 0
-  change m.val x = 0 at h
-  simp [h]
-
-variable [IsDomain R] [∀ x, Module.IsTorsionFree R (A x)] (hp : p ≠ 0)
-  (d : M ≃ₗ[R] Module.Dual R M)
-  (hbalanced : ∀ x m n, d (cut M p hcut x m) n = d m (cut M p hcut x n))
-
-include hp hbalanced in
-/-- The exact integral annihilator, not merely its generic vector space. -/
- theorem supported_iff_annihilates (x : V) (m : M) :
-    m ∈ supported M x ↔ ∀ n : M, projection M x n = 0 → d m n = 0 := by
-  constructor
-  · intro hm n hn
-    have hh := hbalanced x m n
-    rw [cut_of_supported M p hcut x m hm,cut_eq_zero M p hcut x n hn] at hh
-    simpa only [map_smul,LinearMap.smul_apply,map_zero,smul_eq_zero_iff_right hp] using hh
-  · intro hm y hy
-    have hc : cut M p hcut y m = 0 := by
-      apply d.injective
-      apply LinearMap.ext
-      intro n
-      rw [hbalanced]
-      simpa only [map_zero,LinearMap.zero_apply] using
-        hm (cut M p hcut y n) (cut_other M p hcut y x (Ne.symm hy) n)
-    have hv := congrArg (fun s : M => s.val y) hc
-    simpa only [cut_self,Submodule.coe_zero,Pi.zero_apply,smul_eq_zero_iff_right hp] using hv
-
-/-- Precomposition with a surjective stalk projection gives the integral
-single-support module. No field extension was used anywhere in this map. -/
-noncomputable def dualToSupport (x : V) : Module.Dual R (A x) →ₗ[R] supported M x :=
-  (d.symm.toLinearMap.comp (projection M x).dualMap).codRestrict (supported M x) (by
-    intro lam
-    apply (supported_iff_annihilates M p hcut hp d hbalanced x _).mpr
-    intro n hn
-    simp only [LinearMap.comp_apply,LinearEquiv.coe_coe,LinearEquiv.apply_symm_apply,
-      LinearMap.dualMap_apply]
-    change lam (projection M x n) = 0
-    rw [hn,map_zero])
-
- theorem dualToSupport_bijective (x : V) (hπ : Function.Surjective (projection M x)) :
-    Function.Bijective (dualToSupport M p hcut hp d hbalanced x) := by
-  constructor
-  · intro lam mu he
-    apply LinearMap.dualMap_injective_of_surjective hπ
-    apply d.symm.injective
-    exact congrArg Subtype.val he
-  · intro m
-    have hm : d m.val ∈ (LinearMap.ker (projection M x)).dualAnnihilator := by
-      rw [Submodule.mem_dualAnnihilator]
-      intro n hn
-      exact (supported_iff_annihilates M p hcut hp d hbalanced x m.val).mp m.property n hn
-    rw [← LinearMap.range_dualMap_eq_dualAnnihilator_ker_of_surjective _ hπ] at hm
-    obtain ⟨lam,hlam⟩ := hm
-    refine ⟨lam,?_⟩
-    apply Subtype.ext
-    change d.symm ((projection M x).dualMap lam) = m.val
-    rw [hlam,LinearEquiv.symm_apply_apply]
-
-noncomputable def supportEquivDual (x : V) (hπ : Function.Surjective (projection M x)) :
-    supported M x ≃ₗ[R] Module.Dual R (A x) :=
-  (LinearEquiv.ofBijective (dualToSupport M p hcut hp d hbalanced x)
-    (dualToSupport_bijective M p hcut hp d hbalanced x hπ)).symm
-
-end KLInvariance.IntegralSupport
-
-end
-
-
-section
-
-
-namespace BalancedCuts
-universe ur um ua
-variable {R : Type ur} [CommRing R] [IsDomain R]
-  {M : Type um} [AddCommGroup M] [Module R M]
-  {A : Type ua} [AddCommGroup A] [Module R A]
-  (d : M →ₗ[R] Module.Dual R M) (hsym : ∀ m n, d m n=d n m)
-  (π : M →ₗ[R] A) (c : M →ₗ[R] M) (p : R) (hp : p ≠ 0)
-  (hπ : ∀ m, π (c m)=p • π m)
-  (horth : ∀ m n, π n=0 → d (c m) n=0)
-
-include hsym hp hπ horth in
- theorem balanced (m n : M) : d (c m) n=d m (c n) := by
-  have hn : π (p • n-c n)=0 := by rw [map_sub,map_smul,hπ,sub_self]
-  have hm : π (p • m-c m)=0 := by rw [map_sub,map_smul,hπ,sub_self]
-  have en := horth m _ hn
-  have em := horth n _ hm
-  simp only [map_sub,map_smul,smul_eq_mul,sub_eq_zero] at en em
-  rw [hsym (c n) m,hsym (c n) (c m)] at em
-  exact mul_left_cancel₀ hp (en.trans em.symm)
-
-end BalancedCuts
-
-end
-
-
-section
-
-
-namespace KLInvariance.TitsSpace
-open Module MomentGraph BruhatGraph
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-local instance (a : Type*) : DecidableEq a := Classical.decEq a
-
- theorem bottSection_supported_annihilates (l : List I) (hl : cs.IsReduced l)
-    (x : Interval cs 1 (cs.wordProd l))
-    (s z : (bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ)
-    (hs : s ∈ IntegralSupport.supported
-      ((bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ) x)
-    (hz : z.val x=0) : bottSectionDuality cs l hl s z=0 := by
-  classical
-  obtain ⟨m,rfl⟩ := bottSection_surjective cs l hl s
-  obtain ⟨n,rfl⟩ := bottSection_surjective cs l hl z
-  rw [bottSectionDuality_apply]
-  apply bott_supported_annihilates cs l x.val
-  · intro e he
-    let y : Interval cs 1 (cs.wordProd l) :=
-      ⟨bottWeight cs l e,one_bruhat cs _,bottWeight_bruhat cs l hl e⟩
-    have hy : y ≠ x := fun h => he (congrArg Subtype.val h)
-    have hh := hs y hy
-    exact congrFun (congrArg Subtype.val hh) ⟨e,rfl⟩
-  · exact congrArg Subtype.val hz
-
- theorem bottSection_projector_balanced (l : List I) (hl : cs.IsReduced l)
-    (p : SymmetricCoefficient (M := M)) (hp : p ≠ 0)
-    (x : Interval cs 1 (cs.wordProd l))
-    (c : End (SymmetricCoefficient (M := M))
-      ((bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ))
-    (hself : ∀ m, (c m).val x=p • m.val x)
-    (hother : ∀ m y, y ≠ x → (c m).val y=0)
-    (s z : (bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ) :
-    bottSectionDuality cs l hl (c s) z = bottSectionDuality cs l hl s (c z) := by
-  classical
-  apply BalancedCuts.balanced (bottSectionDuality cs l hl).toLinearMap
-    (bottSection_symmetric cs l hl)
-    (IntegralSupport.projection (A := fun y : Interval cs 1 (cs.wordProd l) => bottStalk cs l y.val)
-      ((bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ) x)
-    c p hp hself
-  intro m n hn
-  exact bottSection_supported_annihilates cs l hl x (c m) n (hother m) hn
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-namespace KLInvariance.TitsSpace
-open Module MomentGraph BruhatGraph
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
- theorem bottSupported_of_stalkSupport (l : List I) (hl : cs.IsReduced l)
-    (x : Interval cs 1 (cs.wordProd l)) (m : BottSamelsonModule cs l)
-    (hm : ∀ y : Interval cs 1 (cs.wordProd l), y≠x → bottStalkProjection cs l y.val m=0) :
-    m∈bottWeightSupported cs l x.val := by
-  intro e he
-  let y : Interval cs 1 (cs.wordProd l) :=
-    ⟨bottWeight cs l e,one_bruhat cs _,bottWeight_bruhat cs l hl e⟩
-  have hy : y≠x := fun h => he (congrArg Subtype.val h)
-  exact congrFun (congrArg Subtype.val (hm y hy)) ⟨e,rfl⟩
-
- theorem bottStalk_supported_annihilates (l : List I) (hl : cs.IsReduced l)
-    (x : Interval cs 1 (cs.wordProd l)) (m n : BottSamelsonModule cs l)
-    (hm : ∀ y : Interval cs 1 (cs.wordProd l), y≠x → bottStalkProjection cs l y.val m=0)
-    (hn : bottStalkProjection cs l x.val n=0) : bottSamelsonDuality cs l m n=0 :=
-  bott_supported_annihilates cs l x.val (bottSupported_of_stalkSupport cs l hl x m hm) n
-    (congrArg Subtype.val hn)
-
-end
-end KLInvariance.TitsSpace
-
-end
-
-
-section
-
-namespace KLInvariance.Graded
-open Module
-universe uk ua um ui
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {M : Type um} [AddCommGroup M] [Module k M] [Module A M]
-  (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M)
-  [DirectSum.Decomposition 𝓐] [DirectSum.Decomposition 𝓜]
-  [SetLike.GradedSMul 𝓐 𝓜]
-  {ι : Type ui} [Fintype ι] (b : Basis ι A M) (d : ι → ℤ)
-  (hb : ∀ i, b i ∈ 𝓜 (d i))
-noncomputable section
-
-include hb in
- theorem homogeneous_basis_repr (n : ℤ) (v : M) (hv : v ∈ 𝓜 n) (i : ι) :
-    b.repr v i ∈ 𝓐 (n-d i) := by
-  let _ : DirectSum.Decomposition (shiftedPiece 𝓐 d) :=
-    (shifted_isInternal 𝓐 d).chooseDecomposition
-  have hg : ∀ n f, f ∈ shiftedPiece 𝓐 d n → b.equivFun.symm f ∈ 𝓜 (n+0) := by
-    intro n f hf
-    rw [add_zero,b.equivFun_symm_apply]
-    apply Submodule.sum_mem
-    intro j _
-    simpa only [vadd_eq_add,sub_add_cancel] using SetLike.GradedSMul.smul_mem (hf j) (hb j)
-  have h := equiv_inverse_homogeneous (shiftedPiece 𝓐 d) 𝓜 b.equivFun.symm 0 hg n v hv
-  rw [sub_zero] at h
-  exact h i
-
-include hb in
- theorem dualBasis_homogeneous [DecidableEq ι] (i : ι) (n : ℤ) (v : M) (hv : v ∈ 𝓜 n) :
-    b.dualBasis i v ∈ 𝓐 (n+(-d i)) := by
-  simpa only [Basis.coe_dualBasis,Basis.coord_apply,← sub_eq_add_neg] using
-    homogeneous_basis_repr 𝓐 𝓜 b d hb n v hv i
-
-end
-end KLInvariance.Graded
-
-end
-
-
-section
-
-/-! Genuine polynomial reflection from a homogeneous perfect duality. The
-nonnegative lattice supplies a weak degree bound, not the strict KL bound. -/
-namespace KLInvariance.Graded.ModuleData
-open Polynomial Module
-universe uk us um
-variable {k : Type uk} [Field k] {σ : Type us} [Finite σ]
-  (M N : ModuleData.{uk,max uk us,um} (polynomialGrading k σ))
-noncomputable section
-
-omit [Finite σ] in
- theorem degree_le_of_dual_equiv
-    (e : Dual (MvPolynomial σ k) M ≃ₗ[MvPolynomial σ k] N) (l : ℕ)
-    (he : ∀ r lam, (∀ j (v : M), v∈M.piece j →
-      lam v∈polynomialGrading k σ (j+r)) → e lam∈N.piece (r+l))
-    (c : NonnegativeBasis (σ := σ) M.piece) (i : c.index) : c.degree i ≤ l := by
-  classical
-  have h := he (-(c.degree i : ℤ)) (c.basis.dualBasis i)
-    (dualBasis_homogeneous (polynomialGrading k σ) M.piece c.basis
-      (fun i => (c.degree i : ℤ)) c.homogeneous i)
-  by_contra hn
-  have hn' : -(c.degree i : ℤ)+(l:ℤ)<0 := by omega
-  rw [N.nonneg _ hn',Submodule.mem_bot] at h
-  have hz := e.injective (show e (c.basis.dualBasis i)=e 0 from
-    h.trans (map_zero e).symm)
-  exact c.basis.dualBasis.ne_zero i hz
-
- def reflectedBasis
-    (e : Dual (MvPolynomial σ k) M ≃ₗ[MvPolynomial σ k] N) (l : ℕ)
-    (he : ∀ r lam, (∀ j (v : M), v∈M.piece j →
-      lam v∈polynomialGrading k σ (j+r)) → e lam∈N.piece (r+l))
-    (c : NonnegativeBasis (σ := σ) M.piece) : NonnegativeBasis (σ := σ) N.piece := by
-  classical
-  exact {
-    index := c.index
-    finite := c.finite
-    basis := c.basis.dualBasis.map e
-    degree := fun i => l-c.degree i
-    homogeneous := fun i => by
-      have hi := M.degree_le_of_dual_equiv N e l he c i
-      have h := he (-(c.degree i : ℤ)) (c.basis.dualBasis i)
-        (dualBasis_homogeneous (polynomialGrading k σ) M.piece c.basis
-          (fun i => (c.degree i : ℤ)) c.homogeneous i)
-      change e (c.basis.dualBasis i) ∈ N.piece ((l-c.degree i : ℕ):ℤ)
-      have hd : ((l-c.degree i : ℕ):ℤ)=-(c.degree i : ℤ)+(l:ℤ) := by omega
-      rw [hd]
-      exact h }
-
- theorem char_of_dual_equiv [Module.Free (MvPolynomial σ k) M]
-    [Module.Free (MvPolynomial σ k) N]
-    (e : Dual (MvPolynomial σ k) M ≃ₗ[MvPolynomial σ k] N) (l : ℕ)
-    (he : ∀ r lam, (∀ j (v : M), v∈M.piece j →
-      lam v∈polynomialGrading k σ (j+r)) → e lam∈N.piece (r+l)) :
-    N.char=Polynomial.reflect l M.char := by
-  classical
-  let c := (exists_nonnegativeBasis (σ := σ) M.piece M.nonneg).some
-  rw [char,character_eq _ _ (M.reflectedBasis N e l he c),char,character_eq _ _ c]
-  change (∑ i : c.index, (X : Polynomial ℤ)^(l-c.degree i))=
-    Polynomial.reflect l (∑ i : c.index, X^c.degree i)
-  have hs (s : Finset c.index) : Polynomial.reflect l (∑ i∈s, (X : Polynomial ℤ)^c.degree i)=
-      ∑ i∈s, Polynomial.reflect l (X^c.degree i) := by
-    induction s using Finset.induction_on with
-    | empty => simp
-    | @insert a s ha ih => simp [Finset.sum_insert ha,ih]
-  rw [hs]
-  apply Finset.sum_congr rfl
-  intro i _
-  rw [Polynomial.reflect_monomial,Polynomial.revAt_le (M.degree_le_of_dual_equiv N e l he c i)]
-
-end
-end KLInvariance.Graded.ModuleData
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.Sheaf
-universe ur uv ue um
-variable {R : Type ur} [CommRing R] {V : Type uv} [PartialOrder V]
-  {E : Type ue} {G : OrderedGraph V E} (B : Sheaf.{ur,uv,ue,um} (R := R) G)
-
-/-- A common edge annihilator makes all the scaled coordinate idempotents
-integral endomorphisms of the section module. -/
- theorem scaled_single_section [DecidableEq V] {p : R}
-    (hp : ∀ e (v : B.edge e), p • v = 0) (x : V) (m : B.sections Set.univ) :
-    p • Pi.single x (m.val x) ∈ B.sections Set.univ := by
-  intro e _ _
-  simp only [Pi.smul_apply,map_smul,hp]
-
-variable [Fintype E] (α : E → R)
-
-def labelProduct : R := ∏ e, α e
-
- theorem labelProduct_ne_zero [IsDomain R] (hα : ∀ e, α e ≠ 0) : labelProduct α ≠ 0 := by
-  exact Finset.prod_ne_zero_iff.mpr (fun e _ => hα e)
-
- theorem labelProduct_annihilates (hα : ∀ e (v : B.edge e), α e • v = 0)
-    (e : E) (v : B.edge e) : labelProduct α • v = 0 := by
-  obtain ⟨r,hr⟩ := Finset.dvd_prod_of_mem α (Finset.mem_univ e)
-  change (∏ e, α e) • v = 0
-  rw [hr,mul_comm, mul_smul,hα,smul_zero]
-
-omit [Fintype E] in
- theorem projection_surjective (hg : B.Generated) (x : V) :
-    Function.Surjective (IntegralSupport.projection (B.sections Set.univ) x) := by
-  intro a
-  obtain ⟨m,hm,hx⟩ := hg x a
-  exact ⟨⟨m,hm⟩,hx⟩
-
-variable [DecidableEq V] [IsDomain R] [∀ x, Module.IsTorsionFree R (B.vertex x)]
-  (hα : ∀ e, α e ≠ 0) (ha : ∀ e (v : B.edge e), α e • v = 0)
-  (d : B.sections Set.univ ≃ₗ[R] Module.Dual R (B.sections Set.univ))
-  (hbalanced : ∀ x m n,
-    d (IntegralSupport.cut (B.sections Set.univ) (labelProduct α)
-      (B.scaled_single_section (B.labelProduct_annihilates α ha)) x m) n =
-    d m (IntegralSupport.cut (B.sections Set.univ) (labelProduct α)
-      (B.scaled_single_section (B.labelProduct_annihilates α ha)) x n))
-
-
-noncomputable def singleSupportEquivDual (hg : B.Generated) (x : V) :
-    IntegralSupport.supported (B.sections Set.univ) x ≃ₗ[R] Module.Dual R (B.vertex x) :=
-  IntegralSupport.supportEquivDual (B.sections Set.univ) (labelProduct α)
-    (B.scaled_single_section (B.labelProduct_annihilates α ha))
-    (labelProduct_ne_zero α hα) d hbalanced x (B.projection_surjective hg x)
-
-end KLInvariance.MomentGraph.Sheaf
-
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
-  {G : OrderedGraph V E} (B : GradedSheaf (𝓐 := 𝓐) G) {α : E → A}
-
- theorem UpperQuotient.annihilates (h : B.UpperQuotient α) (e : E) (v : B.edge e) :
-    α e • v = 0 := by
-  obtain ⟨w,rfl⟩ := (h e).1 v
-  rw [← map_smul]
-  exact (h e).2 _ |>.mpr ⟨w,rfl⟩
-
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.Sheaf
-universe ur uv ue um ui
-variable {R : Type ur} [CommRing R] {V : Type uv} [PartialOrder V]
-  {E : Type ue} {G : OrderedGraph V E} (B : Sheaf.{ur,uv,ue,um} (R := R) G)
-
- theorem stalkMap_eq_zero_iff (x : V) (v : B.vertex x) :
-    B.stalkMap x v = 0 ↔ ∀ e (h : G.source e = x), B.lower e (h.symm ▸ v) = 0 := by
-  constructor
-  · intro hv e he
-    exact congrFun hv ⟨e,he⟩
-  · intro hv
-    funext e
-    exact hv e.val e.property
-
- theorem single_section_iff [DecidableEq V] (x : V) (v : B.vertex x) :
-    Pi.single x v ∈ B.sections Set.univ ↔
-      B.stalkMap x v = 0 ∧ ∀ e (h : G.target e = x), B.upper e (h.symm ▸ v) = 0 := by
-  constructor
-  · intro hv
-    constructor
-    · apply (B.stalkMap_eq_zero_iff x v).mpr
-      intro e he
-      subst x
-      have hc := hv e (Set.mem_univ _) (Set.mem_univ _)
-      simpa only [Pi.single_eq_same, Pi.single_eq_of_ne (ne_of_gt (G.increasing e)),map_zero] using hc
-    · intro e he
-      subst x
-      have hc := hv e (Set.mem_univ _) (Set.mem_univ _)
-      simpa only [Pi.single_eq_same, Pi.single_eq_of_ne (ne_of_lt (G.increasing e)),map_zero] using hc.symm
-  · rintro ⟨hu,hd⟩ e _ _
-    by_cases hs : G.source e = x
-    · subst x
-      simpa only [Pi.single_eq_same, Pi.single_eq_of_ne (ne_of_gt (G.increasing e)),map_zero]
-        using (B.stalkMap_eq_zero_iff _ _).mp hu e rfl
-    · by_cases ht : G.target e = x
-      · subst x
-        simpa only [Pi.single_eq_same, Pi.single_eq_of_ne (ne_of_lt (G.increasing e)),map_zero]
-          using (hd e rfl).symm
-      · simp only [Pi.single_eq_of_ne hs,Pi.single_eq_of_ne ht,map_zero]
-
-variable [Fintype E] [DecidableEq V]
-
-def incoming (x : V) : Finset E := Finset.univ.filter (fun e => G.target e = x)
-
-@[simp] theorem mem_incoming (x : V) (e : E) : e ∈ incoming (G := G) x ↔ G.target e = x := by
-  simp [incoming]
-
-def incomingProduct (α : E → R) (x : V) : R := ∏ e ∈ incoming (G := G) x, α e
-
-variable [IsDomain R] (α : E → R)
-  (hα : ∀ e, Prime (α e))
-  (hq : ∀ e, Function.Surjective (B.upper e))
-  (hk : ∀ e v, B.upper e v = 0 ↔ ∃ w, v = α e • w)
-  {ι : V → Type ui} [∀ x, Fintype (ι x)] (bs : ∀ x, Module.Basis (ι x) R (B.vertex x))
-
-include hα in
- theorem incomingProduct_ne_zero (x : V) : incomingProduct (G := G) α x ≠ 0 := by
-  exact Finset.prod_ne_zero_iff.mpr (fun e _ => (hα e).ne_zero)
-
-omit [IsDomain R] in
-include hα hq hk bs in
- theorem outgoing_product_regular (x : V)
-    (hsep : ∀ e f, G.source e = x → G.target f = x → ¬ α e ∣ α f)
-    (e : E) (he : G.source e = x) :
-    Function.Injective (fun v : B.edge e => incomingProduct (G := G) α x • v) := by
-  apply FiniteFreeScalar.scalar_quotient_regular (bs (G.target e)) (B.upper e) (hq e)
-    (α e) (incomingProduct (G := G) α x) (hα e) _ (hk e)
-  intro hdiv
-  obtain ⟨f,hf,hfdiv⟩ := ((hα e).dvd_finsetProd_iff α).mp hdiv
-  exact hsep e f he ((mem_incoming (G := G) x f).mp hf) hfdiv
-
-omit [IsDomain R] in
-include hk bs in
-/-- The actual integral single-support conditions. Downward labels impose
-product divisibility, and they are regular at every upward edge. -/
- theorem single_section_iff_prod [DecompositionMonoid R] (x : V)
-    (hrel : (incoming (G := G) x : Set E).Pairwise (Function.onFun IsRelPrime α))
-    (hreg : ∀ e, G.source e = x → Function.Injective
-      (fun v : B.edge e => incomingProduct (G := G) α x • v)) (v : B.vertex x) :
-    Pi.single x v ∈ B.sections Set.univ ↔
-      ∃ w : B.vertex x, v = incomingProduct (G := G) α x • w ∧ B.stalkMap x w = 0 := by
-  constructor
-  · intro hv
-    obtain ⟨hu,hd⟩ := (B.single_section_iff x v).mp hv
-    have hdiv : ∀ e ∈ incoming (G := G) x, ∃ w : B.vertex x, α e • w = v := by
-      intro e he
-      have ht := (mem_incoming (G := G) x e).mp he
-      subst x
-      obtain ⟨w,hw⟩ := (hk e v).mp (hd e rfl)
-      exact ⟨w,hw.symm⟩
-    obtain ⟨w,hw⟩ := FiniteFreeScalar.exists_prod_smul (bs x) (incoming (G := G) x) α hrel v hdiv
-    refine ⟨w,hw.symm,?_⟩
-    apply (B.stalkMap_eq_zero_iff x w).mpr
-    intro e he
-    apply hreg e he
-    change incomingProduct (G := G) α x • _ = incomingProduct (G := G) α x • (0 : B.edge e)
-    rw [smul_zero]
-    have hz := (B.stalkMap_eq_zero_iff x v).mp hu e he
-    subst x
-    rw [← hw,map_smul] at hz
-    exact hz
-  · rintro ⟨w,rfl,hw⟩
-    apply (B.single_section_iff x _).mpr
-    constructor
-    · rw [map_smul,hw,smul_zero]
-    · intro e he
-      subst x
-      obtain ⟨r,hr⟩ := Finset.dvd_prod_of_mem α ((mem_incoming (G := G) _ e).mpr rfl)
-      apply (hk e _).mpr
-      exact ⟨r • w, by rw [incomingProduct,hr,mul_smul]⟩
-
-/-- The single-support part as an actual submodule of its stalk. -/
-def singleStalk (x : V) : Submodule R (B.vertex x) :=
-  (B.sections Set.univ).comap (LinearMap.single R (fun y => B.vertex y) x)
-
-noncomputable def singleStalkEquivSupport (x : V) :
-    B.singleStalk x ≃ₗ[R] IntegralSupport.supported (B.sections Set.univ) x := by
-  let f : B.singleStalk x →ₗ[R] IntegralSupport.supported (B.sections Set.univ) x :=
-    { toFun := fun v => ⟨⟨Pi.single x v.val,v.property⟩,by
-                        intro y hy; exact Pi.single_eq_of_ne hy _⟩
-      map_add' := by intro v w; apply Subtype.ext; apply Subtype.ext; exact Pi.single_add _ _ _
-      map_smul' := by intro r v; apply Subtype.ext; apply Subtype.ext; exact Pi.single_smul _ _ _ }
-  apply LinearEquiv.ofBijective f
-  constructor
-  · intro v w hvw
-    apply Subtype.ext
-    have hv := congrArg (fun s : IntegralSupport.supported (B.sections Set.univ) x => s.val.val x) hvw
-    change (Pi.single x v.val : B.Assignment) x = (Pi.single x w.val : B.Assignment) x at hv
-    simpa only [Pi.single_eq_same] using hv
-  · intro m
-    have he : Pi.single x (m.val.val x) = m.val.val := by
-      funext y
-      by_cases hy : y=x
-      · subst y; simp only [Pi.single_eq_same]
-      · rw [Pi.single_eq_of_ne hy, m.property y hy]
-    refine ⟨⟨m.val.val x,?_⟩,?_⟩
-    · change Pi.single x (m.val.val x) ∈ B.sections Set.univ
-      rw [he]; exact m.val.property
-    · apply Subtype.ext; apply Subtype.ext; exact he
-
-
-noncomputable def kernelEquivSingleStalk [DecompositionMonoid R] (x : V)
-    (hrel : (incoming (G := G) x : Set E).Pairwise (Function.onFun IsRelPrime α))
-    (hreg : ∀ e, G.source e = x → Function.Injective
-      (fun v : B.edge e => incomingProduct (G := G) α x • v)) :
-    LinearMap.ker (B.stalkMap x) ≃ₗ[R] B.singleStalk x := by
-  let p := incomingProduct (G := G) α x
-  let f : LinearMap.ker (B.stalkMap x) →ₗ[R] B.singleStalk x :=
-    (p • (LinearMap.ker (B.stalkMap x)).subtype).codRestrict (B.singleStalk x) (by
-      intro v
-      exact (B.single_section_iff_prod α hk bs x hrel hreg _).mpr ⟨v.val,rfl,v.property⟩)
-  apply LinearEquiv.ofBijective f
-  constructor
-  · intro v w hvw
-    apply Subtype.ext
-    let : Module.Free R (B.vertex x) := Module.Free.of_basis (bs x)
-    exact smul_right_injective (B.vertex x) (incomingProduct_ne_zero α hα x)
-      (congrArg Subtype.val hvw)
-  · intro v
-    obtain ⟨w,hw,hz⟩ := (B.single_section_iff_prod α hk bs x hrel hreg v.val).mp v.property
-    exact ⟨⟨w,hz⟩,Subtype.ext hw.symm⟩
-
-end KLInvariance.MomentGraph.Sheaf
-
-end
-
-
-section
-
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
-  {σ : Type us} (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- theorem root_injective_on_labels {e f : Edge cs u b} (h : root cs u b e = root cs u b f) :
-    label cs u b e = label cs u b f := by
-  exact congrArg Subtype.val ((positiveRootEquiv M cs).injective h)
-
- theorem polynomialLabel_prime (e : Edge cs u b) : Prime (polynomialLabel cs u b basis e) := by
-  apply (MulEquiv.prime_iff (SymmetricAlgebra.equivMvPolynomial basis)).mpr
-  apply EdgeAlgebra.symmetric_ι_prime
-  intro h
-  exact (root cs u b e).property.1.ne_zero (congrArg Prod.fst h)
-
- theorem label_eq_of_polynomialLabel_dvd {e f : Edge cs u b}
-    (h : polynomialLabel cs u b basis e ∣ polynomialLabel cs u b basis f) :
-    label cs u b e = label cs u b f := by
-  apply root_injective_on_labels cs u b
-  by_contra hn
-  have hd : SymmetricAlgebra.ι ℝ (Extended M) (embed M (root cs u b e).val) ∣
-      SymmetricAlgebra.ι ℝ (Extended M) (embed M (root cs u b f).val) := by
-    obtain ⟨z,hz⟩ := h
-    exact ⟨(SymmetricAlgebra.equivMvPolynomial basis).symm z, by
-      apply (SymmetricAlgebra.equivMvPolynomial basis).injective
-      simpa only [map_mul,AlgEquiv.apply_symm_apply,polynomialLabel] using hz⟩
-  have hm := (EdgeAlgebra.symmetric_ι_dvd_iff _ _).mp hd
-  obtain ⟨r,hr⟩ := Submodule.mem_span_singleton.mp hm
-  have he : r • (root cs u b e).val = (1 : ℝ) • (root cs u b f).val := by
-    have hfst := congrArg Prod.fst hr
-    change r • (root cs u b e).val = (root cs u b f).val at hfst
-    simpa only [one_smul] using hfst
-  exact one_ne_zero ((positiveRoots_independent _ _ hn).eq_zero_of_pair' he).2
-
- theorem incoming_pairwise_relPrime [Fintype (Edge cs u b)] [DecidableEq (Interval cs u b)] (x : Interval cs u b) :
-    (Sheaf.incoming (G := graph cs u b) x : Set (Edge cs u b)).Pairwise
-      (Function.onFun IsRelPrime (polynomialLabel cs u b basis)) := by
-  intro e he f hf hne
-  apply ((polynomialLabel_prime cs u b basis e).irreducible.isRelPrime_iff_not_dvd).mpr
-  intro hdiv
-  apply hne
-  apply eq_of_label_target_eq cs u b (label_eq_of_polynomialLabel_dvd cs u b basis hdiv)
-  exact ((Sheaf.mem_incoming x e).mp he).trans ((Sheaf.mem_incoming x f).mp hf).symm
-
- theorem outgoing_incoming_not_dvd (x : Interval cs u b) (e f : Edge cs u b)
-    (he : (graph cs u b).source e = x) (hf : (graph cs u b).target f = x) :
-    ¬ polynomialLabel cs u b basis e ∣ polynomialLabel cs u b basis f := by
-  intro h
-  apply source_ne_target_of_label_eq cs u b (label_eq_of_polynomialLabel_dvd cs u b basis h)
-  exact he.trans hf.symm
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-namespace KLInvariance.Hilbert
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {M : Type um} [AddCommGroup M] [Module k M] [Module A M] [IsScalarTower k A M]
-  (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M)
-  [DirectSum.Decomposition 𝓜] [SetLike.GradedSMul 𝓐 𝓜]
-  (a : A) (d : ℤ) (ha : a ∈ 𝓐 d)
-
- def scalarDegreeMap (n : ℤ) : 𝓜 (n-d) →ₗ[k]
-    subPiece 𝓜 (LinearMap.range (LinearMap.lsmul A M a)) n :=
-  { toFun := fun m => ⟨⟨a • m.val,⟨m.val,rfl⟩⟩,by
-      change a • m.val ∈ 𝓜 n
-      simpa using SetLike.GradedSMul.smul_mem ha m.property⟩
-    map_add' := by intros; ext; exact smul_add _ _ _
-    map_smul' := by intros; ext; exact smul_comm _ _ _ }
-
- theorem scalarDegreeMap_surjective (n : ℤ) :
-    Function.Surjective (scalarDegreeMap 𝓐 𝓜 a d ha n) := by
-  rintro ⟨⟨m,v,hv⟩,hm⟩
-  refine ⟨DirectSum.decompose 𝓜 v (n-d), ?_⟩
-  apply Subtype.ext
-  apply Subtype.ext
-  change a • component 𝓜 (n-d) v = m
-  rw [← component_homogeneous_smul 𝓐 𝓜 d a ha n v]
-  change (DirectSum.decompose 𝓜 (a • v) n : M) = m
-  change a • v = m at hv
-  rw [hv]
-  exact DirectSum.decompose_of_mem_same 𝓜 hm
-
-omit [DirectSum.Decomposition 𝓜] in
- theorem scalarDegreeMap_injective (hinj : Function.Injective (fun m : M => a • m)) (n : ℤ) :
-    Function.Injective (scalarDegreeMap 𝓐 𝓜 a d ha n) := by
-  intro v w h
-  apply Subtype.ext
-  apply hinj
-  exact congrArg (fun t => t.val.val) h
-
-noncomputable def scalarDegreeEquiv
-    (hinj : Function.Injective (fun m : M => a • m)) (n : ℤ) :
-    𝓜 (n-d) ≃ₗ[k] subPiece 𝓜 (LinearMap.range (LinearMap.lsmul A M a)) n :=
-  LinearEquiv.ofBijective (scalarDegreeMap 𝓐 𝓜 a d ha n)
-    ⟨scalarDegreeMap_injective 𝓐 𝓜 a d ha hinj n,
-      scalarDegreeMap_surjective 𝓐 𝓜 a d ha n⟩
-
-include ha in
- theorem scalar_value
-    (hinj : Function.Injective (fun m : M => a • m)) {q : ℝ} (hq : q ≠ 0)
-    (hs : Summable (term 𝓜 ⊤ q)) :
-    value (subPiece 𝓜 (LinearMap.range (LinearMap.lsmul A M a))) ⊤ q =
-      q^d * value 𝓜 ⊤ q := by
-  have h := (hs.hasSum.mul_left (q^d))
-  have hh : HasSum
-      (fun n : ℤ => (Module.finrank k (𝓜 (n-d)) : ℝ)*q^n)
-      (q^d * value 𝓜 ⊤ q) := by
-    apply (Equiv.addRight d).hasSum_iff.mp
-    apply h.congr_fun
-    intro n
-    rw [term_top]
-    change (Module.finrank k (𝓜 (n+d-d)) : ℝ)*q^(n+d) =
-      q^d * ((Module.finrank k (𝓜 n) : ℝ)*q^n)
-    rw [add_sub_cancel_right,zpow_add₀ hq]
-    ring
-  apply HasSum.tsum_eq
-  apply hh.congr_fun
-  intro n
-  rw [term_top,← LinearEquiv.finrank_eq (scalarDegreeEquiv 𝓐 𝓜 a d ha hinj n)]
-
-include ha in
-
-
- theorem scalarQuotient_value [∀ n, Module.Finite k (𝓜 n)]
-    (hinj : Function.Injective (fun m : M => a • m)) {q : ℝ} (hq : 0 < q)
-    (hs : Summable (term 𝓜 ⊤ q)) :
-    value (quotientPiece 𝓜 (LinearMap.range (LinearMap.lsmul A M a))) ⊤ q =
-      (1-q^d) * value 𝓜 ⊤ q := by
-  let S := LinearMap.range (LinearMap.lsmul A M a)
-  let : DirectSum.Decomposition (quotientPiece 𝓜 S) :=
-    (quotient_isInternal 𝓜 S (scalar_range_homogeneous 𝓐 𝓜 d a ha)).chooseDecomposition
-  have h := value_exact_of_summable 𝓜 (quotientPiece 𝓜 S) S.mkQ
-    (fun n m hm => Submodule.mem_map.mpr ⟨m,hm,rfl⟩) S.mkQ_surjective hq hs
-  rw [Submodule.ker_mkQ] at h
-  change value 𝓜 ⊤ q = value (subPiece 𝓜 (LinearMap.range (LinearMap.lsmul A M a))) ⊤ q + _ at h
-  rw [scalar_value 𝓐 𝓜 a d ha hinj hq.ne' hs] at h
-  change value (quotientPiece 𝓜 S) ⊤ q = _
-  linarith
-
-end KLInvariance.Hilbert
-
-end
-
-
-section
-
-namespace KLInvariance.Hilbert
-open _root_.OAI.KLInvariance.Graded
-universe uk ua um un
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {M : Type um} [AddCommGroup M] [Module k M] [Module A M] [IsScalarTower k A M]
-  {N : Type un} [AddCommGroup N] [Module k N] [Module A N] [IsScalarTower k A N]
-
- def imageMap (S : Submodule A M) (f : M →ₗ[A] N) : S →ₗ[A] S.map f where
-  toFun v := ⟨f v.val,Submodule.mem_map.mpr ⟨_,v.property,rfl⟩⟩
-  map_add' _ _ := by ext; exact map_add f _ _
-  map_smul' _ _ := by ext; exact map_smul f _ _
-
- theorem imageMap_surjective (S : Submodule A M) (f : M →ₗ[A] N) :
-    Function.Surjective (imageMap S f) := by
-  rintro ⟨m,hm⟩
-  obtain ⟨v,hv,rfl⟩ := Submodule.mem_map.mp hm
-  exact ⟨⟨v,hv⟩,rfl⟩
-
- theorem imageMap_ker (S : Submodule A M) (f : M →ₗ[A] N) (a : A)
-    (hf : ∀ v, f v = 0 ↔ ∃ w, v = a • w)
-    (hS : ∀ v, a • v ∈ S → v ∈ S) :
-    LinearMap.ker (imageMap S f) = LinearMap.range (LinearMap.lsmul A S a) := by
-  ext v
-  constructor
-  · intro hv
-    have hz : f v.val = 0 := congrArg Subtype.val hv
-    obtain ⟨w,hw⟩ := (hf v.val).mp hz
-    have hwS : w ∈ S := hS w (hw ▸ v.property)
-    exact ⟨⟨w,hwS⟩,Subtype.ext hw.symm⟩
-  · rintro ⟨w,rfl⟩
-    apply Subtype.ext
-    exact (hf (a • w.val)).mpr ⟨w.val,rfl⟩
-
-omit [Algebra k A] [IsScalarTower k A M] [IsScalarTower k A N] in
- theorem map_homogeneous
-    (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
-    [DirectSum.Decomposition 𝓜] [DirectSum.Decomposition 𝓝]
-    (S : Submodule A M) (hS : ∀ n m, m ∈ S → component 𝓜 n m ∈ S)
-    (f : M →ₗ[A] N) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 n) :
-    ∀ n v, v ∈ S.map f → component 𝓝 n v ∈ S.map f := by
-  intro n v hv
-  obtain ⟨w,hw,rfl⟩ := Submodule.mem_map.mp hv
-  refine Submodule.mem_map.mpr ⟨component 𝓜 n w,hS n w hw,?_⟩
-  simpa using (map_component 𝓜 𝓝 f 0 (by simpa using hf) n w).symm
-
-/-- The actual upper image of a saturated graded submodule has precisely
-`(1-q^d)` times its Hilbert series. No choice of edge basis enters this identity. -/
-theorem saturated_image_value
-    (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
-    [DirectSum.Decomposition 𝓜] [DirectSum.Decomposition 𝓝]
-    [SetLike.GradedSMul 𝓐 𝓜] [∀ n, Module.Finite k (𝓜 n)]
-    (S : Submodule A M) (hS : ∀ n m, m ∈ S → component 𝓜 n m ∈ S)
-    (f : M →ₗ[A] N) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 n)
-    (a : A) (d : ℤ) (ha : a ∈ 𝓐 d)
-    (hk : ∀ v, f v = 0 ↔ ∃ w, v = a • w)
-    (hsat : ∀ v, a • v ∈ S → v ∈ S)
-    (hinj : Function.Injective (fun v : M => a • v))
-    {q : ℝ} (hq : 0 < q) (hs : Summable (term 𝓜 (S.restrictScalars k) q)) :
-    value 𝓝 ((S.map f).restrictScalars k) q =
-      (1-q^d) * value 𝓜 (S.restrictScalars k) q := by
-  let : DirectSum.Decomposition (subPiece 𝓜 S) := (sub_isInternal 𝓜 S hS).chooseDecomposition
-  let : DirectSum.Decomposition (subPiece 𝓝 (S.map f)) :=
-    (sub_isInternal 𝓝 (S.map f) (map_homogeneous 𝓜 𝓝 S hS f hf)).chooseDecomposition
-  let (n : ℤ) : Module.Finite k (subPiece 𝓜 S n) := Module.Finite.of_surjective
-    (subPieceEquiv 𝓜 S n).symm.toLinearMap (subPieceEquiv 𝓜 S n).symm.surjective
-  have hs' : Summable (term (subPiece 𝓜 S) ⊤ q) :=
-    hs.congr (fun n => (term_subPiece 𝓜 S q n).symm)
-  have h := value_exact_of_summable (subPiece 𝓜 S) (subPiece 𝓝 (S.map f))
-    (imageMap S f) (fun n v hv => hf n v.val hv) (imageMap_surjective S f) hq hs'
-  rw [imageMap_ker S f a hk hsat] at h
-  have hinj' : Function.Injective (fun v : S => a • v) := by
-    intro v w hv
-    apply Subtype.ext
-    apply hinj
-    exact congrArg Subtype.val hv
-  rw [scalar_value 𝓐 (subPiece 𝓜 S) a d ha hinj' hq.ne' hs',
-    value_subPiece,value_subPiece] at h
-  linarith
-
-end KLInvariance.Hilbert
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded _root_.OAI.KLInvariance.Hilbert
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
-  {G : OrderedGraph V E} (B : GradedSheaf (𝓐 := 𝓐) G)
-
- theorem upwardSubmodule_saturated (x : V) (F : Set (Sheaf.Outgoing (G := G) x))
-    (a : A) (hreg : ∀ e ∈ F, Function.Injective (fun v : B.edge e.val => a • v)) :
-    ∀ v, a • v ∈ B.upwardSubmodule x F → v ∈ B.upwardSubmodule x F := by
-  intro v hv e he
-  have hz : (B.lowerAt x e).map (a • v) = 0 :=
-    (B.lowerAt_map x e (a • v)).trans (hv e he)
-  have hz' : (B.lowerAt x e).map v = 0 := by
-    apply hreg e he
-    change a • (B.lowerAt x e).map v = a • (0 : B.edge e.val)
-    simpa only [map_smul,smul_zero] using hz
-  exact (B.lowerAt_map x e v).symm.trans hz'
-
-
- theorem upper_image_value (α : E → A) (hquot : B.UpperQuotient α)
-    (e : E) (F : Set (Sheaf.Outgoing (G := G) (G.target e)))
-    [∀ n, Module.Finite k ((B.vertex (G.target e)).piece n)]
-    (ha : α e ∈ 𝓐 1)
-    (hreg : ∀ f ∈ F, Function.Injective (fun v : B.edge f.val => α e • v))
-    (hinj : Function.Injective (fun v : B.vertex (G.target e) => α e • v))
-    {q : ℝ} (hq : 0 < q)
-    (hs : Summable (term (B.vertex (G.target e)).piece
-      ((B.upwardSubmodule (G.target e) F).restrictScalars k) q)) :
-    value (B.edge e).piece
-        (((B.upwardSubmodule (G.target e) F).map (B.upper e).map).restrictScalars k) q =
-      (1-q) * value (B.vertex (G.target e)).piece
-        ((B.upwardSubmodule (G.target e) F).restrictScalars k) q := by
-  simpa only [zpow_one] using saturated_image_value 𝓐
-    (B.vertex (G.target e)).piece (B.edge e).piece
-    (B.upwardSubmodule (G.target e) F) (B.upwardKernel_homogeneous (G.target e) F)
-    (B.upper e).map (B.upper e).graded (α e) 1 ha (hquot e).2
-    (B.upwardSubmodule_saturated (G.target e) F (α e) hreg) hinj hq hs
-
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  (hub : BruhatLE cs u b)
-  (B : BoundarySheaf (polynomialGrading_negative ℝ σ)
-    (graph cs u b) (polynomialLabel cs u b basis) ⟨b,hub,bruhat_refl cs b⟩)
-noncomputable section
-
-omit [Fintype σ] in
- theorem label_regular_stalk (e : Edge cs u b) (x : Interval cs u b) :
-    Function.Injective (fun v : B.sheaf.vertex x => polynomialLabel cs u b basis e • v) := by
-  let := B.free x
-  exact smul_right_injective _ (polynomialLabel_ne_zero cs u b basis e)
-
-omit [Fintype σ] in
-
-
- theorem incoming_label_regular_outgoing (e f : Edge cs u b)
-    (h : (graph cs u b).source f = (graph cs u b).target e) :
-    Function.Injective (fun v : B.sheaf.edge f => polynomialLabel cs u b basis e • v) := by
-  let := B.free ((graph cs u b).target f)
-  exact FiniteFreeScalar.scalar_quotient_regular
-    (Module.Free.chooseBasis (Coefficient (σ := σ)) (B.sheaf.vertex ((graph cs u b).target f)))
-    (B.sheaf.upper f).map (B.quotient f).1
-    (polynomialLabel cs u b basis f) (polynomialLabel cs u b basis e)
-    (polynomialLabel_prime cs u b basis f)
-    (outgoing_incoming_not_dvd cs u b basis _ f e h rfl) (B.quotient f).2
-
- theorem upper_image_hilbert (e : Edge cs u b)
-    (F : Set (Sheaf.Outgoing (G := graph cs u b) ((graph cs u b).target e)))
-    [∀ n, Module.Finite ℝ ((B.sheaf.vertex ((graph cs u b).target e)).piece n)]
-    {q : ℝ} (hq : 0 < q)
-    (hs : Summable (Hilbert.term (B.sheaf.vertex ((graph cs u b).target e)).piece
-      ((B.sheaf.upwardSubmodule ((graph cs u b).target e) F).restrictScalars ℝ) q)) :
-    Hilbert.value (B.sheaf.edge e).piece
-      (((B.sheaf.upwardSubmodule ((graph cs u b).target e) F).map
-        (B.sheaf.upper e).map).restrictScalars ℝ) q =
-      (1-q) * Hilbert.value (B.sheaf.vertex ((graph cs u b).target e)).piece
-        ((B.sheaf.upwardSubmodule ((graph cs u b).target e) F).restrictScalars ℝ) q := by
-  apply B.sheaf.upper_image_value (polynomialLabel cs u b basis) B.quotient e F
-    (polynomialLabel_homogeneous cs u b basis e)
-  · intro f _
-    exact incoming_label_regular_outgoing cs u b basis hub B e f.val f.property
-  · exact label_regular_stalk cs u b basis hub B e _
-  · exact hq
-  · exact hs
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
-  {σ : Type us} (basis : Basis σ ℝ (Extended M))
-  (hub : BruhatLE cs u b)
-  (B : BoundarySheaf (polynomialGrading_negative ℝ σ)
-    (graph cs u b) (polynomialLabel cs u b basis) ⟨b,hub,bruhat_refl cs b⟩)
-  [Fintype (Edge cs u b)] [DecidableEq (Interval cs u b)]
-noncomputable section
-
- def kernelEquivSingleSupport (x : Interval cs u b) :
-    LinearMap.ker (B.sheaf.forget.stalkMap x) ≃ₗ[Coefficient (σ := σ)]
-      IntegralSupport.supported (B.sheaf.forget.sections Set.univ) x := by
-  let _ (x : Interval cs u b) : Module.Free (Coefficient (σ := σ)) (B.sheaf.forget.vertex x) := B.free x
-  let _ (x : Interval cs u b) : Module.Finite (Coefficient (σ := σ)) (B.sheaf.forget.vertex x) :=
-    (B.sheaf.vertex x).finite
-  let _ (x : Interval cs u b) : Fintype (Module.Free.ChooseBasisIndex
-      (Coefficient (σ := σ)) (B.sheaf.forget.vertex x)) := Fintype.ofFinite _
-  let bs := fun x : Interval cs u b =>
-    Module.Free.chooseBasis (Coefficient (σ := σ)) (B.sheaf.forget.vertex x)
-  exact (B.sheaf.forget.kernelEquivSingleStalk (polynomialLabel cs u b basis)
-    (polynomialLabel_prime cs u b basis) (fun e => (B.quotient e).2) bs x
-    (incoming_pairwise_relPrime cs u b basis x)
-    (B.sheaf.forget.outgoing_product_regular (polynomialLabel cs u b basis)
-      (polynomialLabel_prime cs u b basis) (fun e => (B.quotient e).1)
-      (fun e => (B.quotient e).2) bs x
-      (outgoing_incoming_not_dvd cs u b basis x))).trans
-        (B.sheaf.forget.singleStalkEquivSupport x)
-
-
-structure SectionDuality where
-  pairing : B.sheaf.forget.sections Set.univ ≃ₗ[Coefficient (σ := σ)]
-    Dual (Coefficient (σ := σ)) (B.sheaf.forget.sections Set.univ)
-  balanced : ∀ x m n,
-    pairing (IntegralSupport.cut (B.sheaf.forget.sections Set.univ)
-      (Sheaf.labelProduct (polynomialLabel cs u b basis))
-      (B.sheaf.forget.scaled_single_section
-        (B.sheaf.forget.labelProduct_annihilates (polynomialLabel cs u b basis)
-          (GradedSheaf.UpperQuotient.annihilates B.sheaf B.quotient))) x m) n =
-    pairing m (IntegralSupport.cut (B.sheaf.forget.sections Set.univ)
-      (Sheaf.labelProduct (polynomialLabel cs u b basis))
-      (B.sheaf.forget.scaled_single_section
-        (B.sheaf.forget.labelProduct_annihilates (polynomialLabel cs u b basis)
-          (GradedSheaf.UpperQuotient.annihilates B.sheaf B.quotient))) x n)
-
- def costalkEquivDual (d : SectionDuality cs u b basis hub B) (x : Interval cs u b) :
-    LinearMap.ker (B.sheaf.forget.stalkMap x) ≃ₗ[Coefficient (σ := σ)]
-      Dual (Coefficient (σ := σ)) (B.sheaf.vertex x) := by
-  let _ (x : Interval cs u b) : Module.Free (Coefficient (σ := σ)) (B.sheaf.forget.vertex x) := B.free x
-  let _ (y : Interval cs u b) : Module.IsTorsionFree (Coefficient (σ := σ))
-      (B.sheaf.forget.vertex y) := inferInstance
-  exact (kernelEquivSingleSupport cs u b basis hub B x).trans
-    (B.sheaf.forget.singleSupportEquivDual (polynomialLabel cs u b basis)
-      (polynomialLabel_ne_zero cs u b basis)
-      (GradedSheaf.UpperQuotient.annihilates B.sheaf B.quotient)
-      d.pairing d.balanced B.generated x)
-
- theorem costalk_free_of_duality (d : SectionDuality cs u b basis hub B)
-    (x : Interval cs u b) :
-    Module.Free (Coefficient (σ := σ)) (LinearMap.ker (B.sheaf.forget.stalkMap x)) := by
-  let _ := B.free x
-  exact Module.Free.of_equiv (costalkEquivDual cs u b basis hub B d x).symm
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-
-namespace KLInvariance.MomentGraph.GradedSheaf
-open Module _root_.OAI.KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐]
-  {V E : Type um} [PartialOrder V] [Fintype V] [Fintype E] [DecidableEq V]
-  {G : OrderedGraph V E} (B : GradedSheaf (𝓐 := 𝓐) G)
-  [IsNoetherianRing A] [IsDomain A]
-  [∀ x, Module.IsTorsionFree A (B.forget.vertex x)]
-  (α : E → A) (hα : ∀ e, α e ≠ 0) (ha : ∀ e (v : B.edge e), α e • v = 0)
-  (p : B.forget.sections Set.univ ≃ₗ[A] Dual A (B.forget.sections Set.univ))
-  (hbalanced : ∀ x m n,
-    p (IntegralSupport.cut (B.forget.sections Set.univ) (Sheaf.labelProduct α)
-      (B.forget.scaled_single_section (B.forget.labelProduct_annihilates α ha)) x m) n =
-    p m (IntegralSupport.cut (B.forget.sections Set.univ) (Sheaf.labelProduct α)
-      (B.forget.scaled_single_section (B.forget.labelProduct_annihilates α ha)) x n))
-noncomputable section
-
- theorem dualToSupport_homogeneous (l : ℤ)
-    (hp : ∀ i j (m n : B.sections Set.univ),
-      m ∈ (B.sections Set.univ).piece i → n ∈ (B.sections Set.univ).piece j →
-      p m n ∈ 𝓐 (i+j-l))
-    (x : V) (r : ℤ) (lam : Dual A (B.vertex x))
-    (hlam : ∀ j (v : B.vertex x), v ∈ (B.vertex x).piece j → lam v ∈ 𝓐 (j+r)) :
-    (IntegralSupport.dualToSupport (B.forget.sections Set.univ) (Sheaf.labelProduct α)
-      (B.forget.scaled_single_section (B.forget.labelProduct_annihilates α ha))
-      (Sheaf.labelProduct_ne_zero α hα) p hbalanced x lam).val ∈
-        (B.sections Set.univ).piece (r+l) := by
-  apply perfect_inverse_of_homogeneous_functional 𝓐 (B.sections Set.univ).piece p l hp r
-  intro j n hn
-  exact hlam j (n.val x) (hn x (Set.mem_univ x))
-
-end
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-namespace KLInvariance.Graded
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {M : Type um} [AddCommGroup M] [Module k M] [Module A M]
-  (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M)
-  [DirectSum.Decomposition 𝓜] [SetLike.GradedSMul 𝓐 𝓜]
-
-/-- Cancellation of an injective homogeneous scalar action includes the
-inverse grading shift. It does not require the scalar to be a unit. -/
- theorem mem_of_homogeneous_smul (d : ℤ) (a : A) (ha : a ∈ 𝓐 d)
-    (hinj : Function.Injective (fun m : M => a • m))
-    (n : ℤ) (m : M) (hm : a • m ∈ 𝓜 n) : m ∈ 𝓜 (n-d) := by
-  have hc : component 𝓜 (n-d) m = m := by
-    apply hinj
-    change a • component 𝓜 (n-d) m = a • m
-    rw [← component_homogeneous_smul 𝓐 𝓜 d a ha]
-    exact DirectSum.decompose_of_mem_same 𝓜 hm
-  rw [← hc]
-  exact (DirectSum.decompose 𝓜 m (n-d)).property
-
-end KLInvariance.Graded
-
-end
-
-
-section
-
-
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  (hub : BruhatLE cs u b)
-  (B : BoundarySheaf (polynomialGrading_negative ℝ σ)
-    (graph cs u b) (polynomialLabel cs u b basis) ⟨b,hub,bruhat_refl cs b⟩)
-  [Fintype (Edge cs u b)] [Fintype (Interval cs u b)] [DecidableEq (Interval cs u b)]
-noncomputable section
-
-omit [Fintype (Interval cs u b)] in
- theorem incomingProduct_homogeneous (x : Interval cs u b) :
-    Sheaf.incomingProduct (G := graph cs u b) (polynomialLabel cs u b basis) x ∈
-      polynomialGrading ℝ σ ((Sheaf.incoming (G := graph cs u b) x).card : ℤ) := by
-  simpa only [Sheaf.incomingProduct,Finset.sum_const,Int.natCast_one,mul_one,nsmul_eq_mul] using
-    SetLike.prod_mem_graded (polynomialGrading ℝ σ) (fun _ : Edge cs u b => (1 : ℤ))
-      (polynomialLabel cs u b basis)
-      (F := Sheaf.incoming (G := graph cs u b) x)
-      (fun e _ => polynomialLabel_homogeneous cs u b basis e)
-
-omit [Fintype σ] [Fintype (Interval cs u b)] in
- theorem kernelEquivSingleSupport_apply (x : Interval cs u b)
-    (v : LinearMap.ker (B.sheaf.forget.stalkMap x)) :
-    ((kernelEquivSingleSupport cs u b basis hub B x) v).val.val x =
-      Sheaf.incomingProduct (G := graph cs u b) (polynomialLabel cs u b basis) x • v.val := by
-  change (Pi.single x
-    (Sheaf.incomingProduct (G := graph cs u b) (polynomialLabel cs u b basis) x • v.val) :
-      B.sheaf.forget.Assignment) x = _
-  exact Pi.single_eq_same _ _
-
- theorem costalkEquivDual_inverse_homogeneous
-    (d : SectionDuality cs u b basis hub B) (l : ℤ)
-    (hd : ∀ i j (m n : B.sheaf.sections Set.univ),
-      m ∈ (B.sheaf.sections Set.univ).piece i → n ∈ (B.sheaf.sections Set.univ).piece j →
-      d.pairing m n ∈ polynomialGrading ℝ σ (i+j-l))
-    (x : Interval cs u b) (r : ℤ) (lam : Dual (Coefficient (σ := σ)) (B.sheaf.vertex x))
-    (hlam : ∀ j (v : B.sheaf.vertex x), v ∈ (B.sheaf.vertex x).piece j →
-      lam v ∈ polynomialGrading ℝ σ (j+r)) :
-    ((costalkEquivDual cs u b basis hub B d x).symm lam).val ∈
-      (B.sheaf.vertex x).piece
-        (r+l-(Sheaf.incoming (G := graph cs u b) x).card) := by
-  let _ (y : Interval cs u b) : Module.Free (Coefficient (σ := σ))
-      (B.sheaf.forget.vertex y) := B.free y
-  let _ (y : Interval cs u b) : Module.IsTorsionFree (Coefficient (σ := σ))
-      (B.sheaf.forget.vertex y) := inferInstance
-  let S := B.sheaf.forget.singleSupportEquivDual (polynomialLabel cs u b basis)
-    (polynomialLabel_ne_zero cs u b basis)
-    (GradedSheaf.UpperQuotient.annihilates B.sheaf B.quotient)
-    d.pairing d.balanced B.generated x
-  have hs : (S.symm lam).val ∈ (B.sheaf.sections Set.univ).piece (r+l) := by
-    exact B.sheaf.dualToSupport_homogeneous (polynomialLabel cs u b basis)
-      (polynomialLabel_ne_zero cs u b basis)
-      (GradedSheaf.UpperQuotient.annihilates B.sheaf B.quotient)
-      d.pairing d.balanced l hd x r lam hlam
-  have hv := hs x (Set.mem_univ x)
-  have he : (S.symm lam).val.val x =
-      Sheaf.incomingProduct (G := graph cs u b) (polynomialLabel cs u b basis) x •
-      ((costalkEquivDual cs u b basis hub B d x).symm lam).val := by
-    rw [← kernelEquivSingleSupport_apply cs u b basis hub B x]
-    change (S.symm lam).val.val x =
-      ((kernelEquivSingleSupport cs u b basis hub B x)
-        ((kernelEquivSingleSupport cs u b basis hub B x).symm (S.symm lam))).val.val x
-    rw [LinearEquiv.apply_symm_apply]
-  change (S.symm lam).val.val x ∈ (B.sheaf.vertex x).piece (r+l) at hv
-  rw [he] at hv
-  apply mem_of_homogeneous_smul (polynomialGrading ℝ σ) (B.sheaf.vertex x).piece
-    ((Sheaf.incoming (G := graph cs u b) x).card : ℤ)
-    (Sheaf.incomingProduct (G := graph cs u b) (polynomialLabel cs u b basis) x)
-    (incomingProduct_homogeneous cs u b basis x) _ (r+l) _ hv
-  let _ := B.free x
-  exact smul_right_injective _ (Sheaf.incomingProduct_ne_zero
-    (polynomialLabel cs u b basis) (polynomialLabel_prime cs u b basis) x)
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-/-! The degree of the actual incoming root product on [1,b].  This is not
-asserted for truncated intervals with arbitrary bottom. -/
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace MomentGraph
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W) (b : W)
-noncomputable section
-
- def incomingEquivInversions (x : Interval cs 1 b) :
-    {e : Edge cs 1 b // target cs 1 b e = x} ≃ {t : W // cs.IsLeftInversion x.val t} where
-  toFun e := ⟨label cs 1 b e.val, label_isReflection cs 1 b e.val, by
-    have hx : (target cs 1 b e.val).val = x.val := congrArg Subtype.val e.property
-    have he := label_mul_target cs 1 b e.val
-    rw [hx] at he
-    rw [he]
-    have h := e.val.property.1
-    change cs.length (source cs 1 b e.val).val < cs.length (target cs 1 b e.val).val at h
-    simpa only [hx] using h⟩
-  invFun t := by
-    have hstep : BruhatStep cs (t.val*x.val) x.val := by
-      refine ⟨t.property.2,t.val,t.property.1,?_⟩
-      rw [← mul_assoc,t.property.1.mul_self,one_mul]
-    let y : Interval cs 1 b := ⟨t.val*x.val,one_bruhat cs _,
-      (Relation.ReflTransGen.single hstep).trans x.property.2⟩
-    exact ⟨⟨(y,x),hstep⟩,rfl⟩
-  left_inv e := by
-    apply Subtype.ext
-    apply eq_of_label_target_eq cs 1 b
-    · change x.val * (label cs 1 b e.val*x.val)⁻¹ = label cs 1 b e.val
-      rw [mul_inv_rev,← mul_assoc,mul_inv_cancel,one_mul,
-        (label_isReflection cs 1 b e.val).inv]
-    · exact e.property.symm
-  right_inv t := by
-    apply Subtype.ext
-    change x.val * (t.val*x.val)⁻¹=t.val
-    rw [mul_inv_rev,← mul_assoc,mul_inv_cancel,one_mul,t.property.1.inv]
-
- theorem incoming_card [Fintype (Edge cs 1 b)] [DecidableEq (Interval cs 1 b)]
-    (x : Interval cs 1 b) : (Sheaf.incoming (G := graph cs 1 b) x).card = cs.length x.val := by
-  classical
-  calc
-    _ = Nat.card {e : Edge cs 1 b // target cs 1 b e = x} := by
-      rw [Nat.card_eq_fintype_card,Fintype.card_subtype]
-      rfl
-    _ = Nat.card {t : W // cs.IsLeftInversion x.val t} := Nat.card_congr (incomingEquivInversions cs b x)
-    _ = Nat.card (RootInversions M cs x.val) := Nat.card_congr (rootInversionEquiv M cs x.val).symm
-    _ = cs.length x.val := card_rootInversions M cs x.val
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  (hub : BruhatLE cs u b)
-  (B : BoundarySheaf (polynomialGrading_negative ℝ σ)
-    (graph cs u b) (polynomialLabel cs u b basis) ⟨b,hub,bruhat_refl cs b⟩)
-  [Fintype (Edge cs u b)] [Fintype (Interval cs u b)] [DecidableEq (Interval cs u b)]
-noncomputable section
-
- def fullSubmodule (x : Interval cs u b) : Submodule (Coefficient (σ := σ)) (B.sheaf.vertex x) :=
-    LinearMap.ker (B.sheaf.forget.stalkMap x)
-
-omit [Fintype σ] [Fintype (Edge cs u b)] [Fintype (Interval cs u b)]
-  [DecidableEq (Interval cs u b)] in
- theorem fullKernel_eq (x : Interval cs u b) :
-    B.sheaf.upwardSubmodule x Set.univ = fullSubmodule cs u b basis hub B x := by
-  ext m
-  change (∀ e ∈ Set.univ, B.sheaf.forget.stalkMap x m e = 0) ↔ B.sheaf.forget.stalkMap x m = 0
-  simp only [Set.mem_univ,forall_const]
-  exact funext_iff.symm
-
- def fullKernel (x : Interval cs u b) : ModuleData (polynomialGrading ℝ σ) :=
-  ModuleData.sub (B.sheaf.vertex x) (fullSubmodule cs u b basis hub B x) (by
-    simpa only [fullKernel_eq cs u b basis hub B x] using
-      B.sheaf.upwardKernel_homogeneous x Set.univ)
-
- theorem costalk_hilbert_of_duality
-    (d : SectionDuality cs u b basis hub B) (l : ℤ)
-    (hd : ∀ i j (m n : B.sheaf.sections Set.univ),
-      m ∈ (B.sheaf.sections Set.univ).piece i → n ∈ (B.sheaf.sections Set.univ).piece j →
-      d.pairing m n ∈ polynomialGrading ℝ σ (i+j-l))
-    (x : Interval cs u b) (c : NonnegativeBasis (σ := σ) (B.sheaf.vertex x).piece)
-    {q : ℝ} (hq : 0 < q) (hq1 : q < 1) :
-    Hilbert.value (B.sheaf.vertex x).piece
-      ((fullSubmodule cs u b basis hub B x).restrictScalars ℝ) q =
-      q^(l-(Sheaf.incoming (G := graph cs u b) x).card) *
-        Polynomial.eval₂ (Int.castRingHom ℝ) q⁻¹ c.polynomial *
-        (1-q)⁻¹ ^ Fintype.card σ := by
-  classical
-  let K := fullKernel cs u b basis hub B x
-  let cb : Basis c.index (Coefficient (σ := σ)) K :=
-    c.basis.dualBasis.map (costalkEquivDual cs u b basis hub B d x).symm
-  have hcb (i : c.index) : cb i ∈ K.piece
-      (l-(Sheaf.incoming (G := graph cs u b) x).card-(c.degree i : ℤ)) := by
-    change ((costalkEquivDual cs u b basis hub B d x).symm (c.basis.dualBasis i)).val ∈
-      (B.sheaf.vertex x).piece _
-    have h := costalkEquivDual_inverse_homogeneous cs u b basis hub B d l hd x
-      (-(c.degree i : ℤ)) (c.basis.dualBasis i)
-      (dualBasis_homogeneous (polynomialGrading ℝ σ) (B.sheaf.vertex x).piece
-        c.basis (fun i => (c.degree i : ℤ)) c.homogeneous i)
-    simpa only [show -(c.degree i : ℤ)+l-(Sheaf.incoming (G := graph cs u b) x).card =
-      l-(Sheaf.incoming (G := graph cs u b) x).card-(c.degree i : ℤ) by omega] using h
-  have hs := Hilbert.hasSum_homogeneous_basis K.piece cb
-    (fun i => l-(Sheaf.incoming (G := graph cs u b) x).card-(c.degree i : ℤ)) hcb hq hq1
-  have hv : Hilbert.value K.piece ⊤ q =
-      (∑ i, q^(l-(Sheaf.incoming (G := graph cs u b) x).card-(c.degree i : ℤ))) *
-        (1-q)⁻¹ ^ Fintype.card σ := by
-    simpa only [Hilbert.value,Hilbert.term_top] using hs.tsum_eq
-  change Hilbert.value (B.sheaf.vertex x).piece
-    ((fullSubmodule cs u b basis hub B x).restrictScalars ℝ) q = _
-  rw [← Hilbert.value_subPiece]
-  change Hilbert.value K.piece ⊤ q = _
-  rw [hv,c.eval₂_eq,Finset.mul_sum]
-  congr 1
-  apply Finset.sum_congr rfl
-  intro i _
-  rw [zpow_sub₀ hq.ne',zpow_natCast,div_eq_mul_inv,inv_pow]
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-namespace KLInvariance.CoefficientChange
-open Module
-universe uk ua ub um un
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {B : Type ub} [CommRing B] [Algebra k B]
-  (e : A ≃ₐ[k] B) {M : Type um} [AddCommGroup M] [Module A M]
-  [Module k M] [IsScalarTower k A M]
-noncomputable section
-
- def mapEquiv {N : Type un} [AddCommGroup N] [Module A N] [Module k N]
-    [IsScalarTower k A N] (f : M ≃ₗ[A] N) : CoefficientChange e M ≃ₗ[B] CoefficientChange e N where
-  toLinearMap := map e f.toLinearMap
-  invFun n := mk (f.symm n.val)
-  left_inv m := CoefficientChange.ext (f.symm_apply_apply m.val)
-  right_inv n := CoefficientChange.ext (f.apply_symm_apply n.val)
-
- def dualTo (f : CoefficientChange e (Dual A M)) : Dual B (CoefficientChange e M) where
-  toFun m := e (f.val m.val)
-  map_add' m n := by change e (f.val (m.val+n.val))=_; rw [map_add,map_add]
-  map_smul' b m := by
-    change e (f.val (e.symm b • m.val))=b * e (f.val m.val)
-    rw [map_smul]
-    change e (e.symm b * f.val m.val)=_
-    rw [map_mul,e.apply_symm_apply]
-
- def dualFrom (f : Dual B (CoefficientChange e M)) : Dual A M where
-  toFun m := e.symm (f (mk m))
-  map_add' m n := by
-    change e.symm (f (mk m+mk n))=_
-    rw [map_add,map_add]
-  map_smul' a m := by
-    change e.symm (f (mk (a • m)))=a * e.symm (f (mk m))
-    rw [show (mk (a • m) : CoefficientChange e M)=e a • mk m from by
-      apply CoefficientChange.ext
-      change a • m=e.symm (e a) • m
-      rw [e.symm_apply_apply],
-      map_smul]
-    change e.symm (e a * f (mk m))=_
-    rw [map_mul,e.symm_apply_apply]
-
- def dualEquiv : CoefficientChange e (Dual A M) ≃ₗ[B] Dual B (CoefficientChange e M) where
-  toFun := dualTo e
-  invFun f := mk (dualFrom e f)
-  left_inv f := by
-    apply CoefficientChange.ext
-    apply LinearMap.ext
-    intro m
-    exact e.symm_apply_apply (f.val m)
-  right_inv f := by
-    apply LinearMap.ext
-    intro m
-    exact e.apply_symm_apply (f m)
-  map_add' f g := by
-    apply LinearMap.ext
-    intro m
-    exact map_add e (f.val m.val) (g.val m.val)
-  map_smul' b f := by
-    apply LinearMap.ext
-    intro m
-    change e (e.symm b * f.val m.val)=b * e (f.val m.val)
-    rw [map_mul,e.apply_symm_apply]
-
- def pairingEquiv (d : M ≃ₗ[A] Dual A M) :
-    CoefficientChange e M ≃ₗ[B] Dual B (CoefficientChange e M) :=
-  (mapEquiv e d).trans (dualEquiv e)
-
- omit [Module k M] [IsScalarTower k A M] in
- theorem pairingEquiv_apply (d : M ≃ₗ[A] Dual A M) (m n : CoefficientChange e M) :
-    pairingEquiv e d m n=e (d m.val n.val) := rfl
-
-end
-end KLInvariance.CoefficientChange
-
-end
-
-
-section
-
-/-! A genuine homogeneous integral perfect pairing with vertex-support
-orthogonality on both sides. Transport does not identify unequal lattices. -/
-namespace KLInvariance.MomentGraph.GradedSheaf
-open _root_.OAI.KLInvariance.Graded Module
-universe uk ua um
-variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
-  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
-  {G : OrderedGraph V E}
-noncomputable section
-
- structure PerfectPairing (S : GradedSheaf (𝓐 := 𝓐) G) (d : ℤ) where
-  pairing : S.forget.sections Set.univ ≃ₗ[A] Dual A (S.forget.sections Set.univ)
-  homogeneous : ∀ i j (m n : S.forget.sections Set.univ),
-    (∀ x, m.val x∈(S.vertex x).piece i) → (∀ x, n.val x∈(S.vertex x).piece j) →
-    pairing m n∈𝓐 (i+j-d)
-  left_orthogonal : ∀ (x : V) (m n : S.forget.sections Set.univ),
-    (∀ y, y≠x → m.val y=0) → n.val x=0 → pairing m n=0
-  right_orthogonal : ∀ (x : V) (m n : S.forget.sections Set.univ),
-    m.val x=0 → (∀ y, y≠x → n.val y=0) → pairing m n=0
-
- def PerfectPairing.pullback {S T : GradedSheaf (𝓐 := 𝓐) G} {d : ℤ}
-    (p : PerfectPairing T d) (j : Iso S T) : PerfectPairing S d where
-  pairing := (j.toIso.sections Set.univ).trans
-    (p.pairing.trans (j.toIso.sections Set.univ).dualMap)
-  homogeneous i q m n hm hn := p.homogeneous i q _ _
-    (fun x => j.vertex_graded x i _ (hm x)) (fun x => j.vertex_graded x q _ (hn x))
-  left_orthogonal x m n hm hn := by
-    apply p.left_orthogonal x
-    · intro y hy
-      change j.vertex y (m.val y)=0
-      rw [hm y hy,map_zero]
-    · change j.vertex x (n.val x)=0
-      rw [hn,map_zero]
-  right_orthogonal x m n hm hn := by
-    apply p.right_orthogonal x
-    · change j.vertex x (m.val x)=0
-      rw [hm,map_zero]
-    · intro y hy
-      change j.vertex y (n.val y)=0
-      rw [hn y hy,map_zero]
-
- variable {B : Type ua} [CommRing B] [Algebra k B] {𝓑 : ℤ → Submodule k B}
-  (e : A ≃ₐ[k] B) (he : ∀ n a, a∈𝓐 n ↔ e a∈𝓑 n)
-
- def PerfectPairing.coefficientChange {S : GradedSheaf (𝓐 := 𝓐) G} {d : ℤ}
-    (p : PerfectPairing S d) : PerfectPairing (coefficientChange e he S) d where
-  pairing := (coefficientChangeSections e he S Set.univ).symm.trans
-    ((CoefficientChange.pairingEquiv e p.pairing).trans
-      (coefficientChangeSections e he S Set.univ).symm.dualMap)
-  homogeneous i j m n hm hn := by
-    apply (he (i+j-d) _).mp
-    exact p.homogeneous i j _ _
-      (fun x => (ModuleData.mem_changePiece e (S.vertex x) i (m.val x)).mp (hm x))
-      (fun x => (ModuleData.mem_changePiece e (S.vertex x) j (n.val x)).mp (hn x))
-  left_orthogonal x m n hm hn := by
-    change e (p.pairing _ _)=0
-    apply (congrArg e ?_).trans (map_zero e)
-    apply p.left_orthogonal x
-    · intro y hy
-      exact congrArg (fun v : CoefficientChange e (S.vertex y) => v.val) (hm y hy)
-    · exact congrArg (fun v : CoefficientChange e (S.vertex x) => v.val) hn
-  right_orthogonal x m n hm hn := by
-    change e (p.pairing _ _)=0
-    apply (congrArg e ?_).trans (map_zero e)
-    apply p.right_orthogonal x
-    · exact congrArg (fun v : CoefficientChange e (S.vertex x) => v.val) hm
-    · intro y hy
-      exact congrArg (fun v : CoefficientChange e (S.vertex y) => v.val) (hn y hy)
-
-end
-end KLInvariance.MomentGraph.GradedSheaf
-
-end
-
-
-section
-
-
-end
-
-end OAI
+import OAI.RepresentationTheory.KazhdanLusztig.BalancedPairings
+
+namespace OAI
+
+/-!
+Graded unit splittings, integral support and balanced cuts, and boundary-sheaf support and recognition.
+-/
+
+section
+
+namespace KLInvariance.Graded.ModuleData
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+  (hneg : ∀ n < 0, 𝓐 n=⊥)
+noncomputable section
+
+ theorem unit_endomorphisms_commute (f g : Hom (unit.{uk,ua,um} hneg) (unit hneg))
+    (v : unit hneg) : f.map (g.map v)=g.map (f.map v) := by
+  let z : unit.{uk,ua,um} hneg := fun _ => (1 : A)
+  have he (w : unit.{uk,ua,um} hneg) : w=w PUnit.unit • z := by
+    funext j
+    change w j=w PUnit.unit * 1
+    simp only [mul_one]
+  have hf (w : unit.{uk,ua,um} hneg) : f.map w=w PUnit.unit • f.map z := by
+    conv_lhs => rw [he w]
+    exact map_smul f.map _ z
+  have hg (w : unit.{uk,ua,um} hneg) : g.map w=w PUnit.unit • g.map z := by
+    conv_lhs => rw [he w]
+    exact map_smul g.map _ z
+  rw [hf (g.map v),hg (f.map v),hg v,hf v]
+  funext j
+  change (v PUnit.unit * (g.map z) PUnit.unit) * (f.map z) j =
+    (v PUnit.unit * (f.map z) PUnit.unit) * (g.map z) j
+  have hej : j=PUnit.unit := Subsingleton.elim _ _
+  subst j
+  ring
+
+ theorem unit_split_reverse (f g : Hom (unit.{uk,ua,um} hneg) (unit hneg))
+    (h : ∀ v, g.map (f.map v)=v) (v : unit hneg) : f.map (g.map v)=v :=
+  (unit_endomorphisms_commute hneg f g v).trans (h v)
+
+
+ theorem split_reverse_of_unit {B N : ModuleData.{uk,ua,um} 𝓐}
+    (hB : B=unit hneg) (o : Hom N (unit hneg)) (o' : Hom (unit hneg) N)
+    (ho : ∀ v, o'.map (o.map v)=v) (i : Hom B N) (p : Hom N B)
+    (hpi : ∀ v, p.map (i.map v)=v) (v : N) : i.map (p.map v)=v := by
+  subst B
+  have hp : ∀ a, (p.comp o').map ((o.comp i).map a)=a := by
+    intro a
+    change p.map (o'.map (o.map (i.map a)))=a
+    rw [ho]
+    exact hpi a
+  have hh := unit_split_reverse hneg (o.comp i) (p.comp o') hp (o.map v)
+  change o.map (i.map (p.map (o'.map (o.map v))))=o.map v at hh
+  rw [ho] at hh
+  exact (ho _).symm.trans ((congrArg o'.map hh).trans (ho v))
+
+end
+end KLInvariance.Graded.ModuleData
+
+end
+
+
+section
+
+/-! The projector of any actual normalized BMP splitting into the reduced
+word, and its integral adjoint, are both the identity at the top. The splitting
+is not asserted to exist; word BMP recognition and character remain separate. -/
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
+universe u us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  (l : List I) (hl : cs.IsReduced l)
+  (α : Edge cs 1 (cs.wordProd l) → SymmetricCoefficient (M := M))
+  (B : BoundarySheaf (symmetricGrading_negative basis)
+    (graph cs 1 (cs.wordProd l)) α
+    ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩)
+  (s : GradedSheaf.SplitIn B.sheaf (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l)))
+noncomputable section
+
+include hl in
+ theorem bottBoundarySplit_top_fixed (v : bottStalk cs l (cs.wordProd l)) :
+    (s.projector.vertex ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩).map v=v := by
+  exact ModuleData.split_reverse_of_unit (symmetricGrading_negative basis) B.top
+    (bottTopHom cs basis l hl) (bottTopInvHom cs basis l hl)
+    (bottTopInvHom_top cs basis l hl)
+    (s.inclusion.vertex ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩)
+    (s.projection.vertex ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩)
+    (s.vertex_split ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩) v
+
+ theorem bottBoundarySplitAdjoint_top_fixed (v : bottStalk cs l (cs.wordProd l)) :
+    ((bottAdjointSheaf cs basis l hl s.projector).vertex
+      ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩).map v=v :=
+  bottAdjointSheaf_fixed_stalk cs basis l hl s.projector
+    ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩
+    (bottBoundarySplit_top_fixed cs basis l hl α B s) v
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded
+universe u us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {σ : Type us} (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ theorem symmetricGrading_zero (a : SymmetricCoefficient (M := M))
+    (ha : a ∈ symmetricGrading basis 0) :
+    ∃ c : ℝ, algebraMap ℝ (SymmetricCoefficient (M := M)) c=a := by
+  obtain ⟨c,hc⟩ := polynomialGrading_zero ℝ σ
+    (SymmetricAlgebra.equivMvPolynomial basis a) ((mem_symmetricGrading basis 0 a).mp ha)
+  refine ⟨c,?_⟩
+  apply (SymmetricAlgebra.equivMvPolynomial basis).injective
+  exact ((SymmetricAlgebra.equivMvPolynomial basis).commutes c).trans hc
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+/-! The actual BMP axioms inherited by a graded sheaf retract. These are
+implications, not claims that the word sheaf already satisfies the axioms. -/
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
+  {G : OrderedGraph V E} {B C : GradedSheaf (𝓐 := 𝓐) G}
+noncomputable section
+
+ theorem SplitIn.flabby (s : SplitIn B C) (hC : C.forget.Flabby) : B.forget.Flabby := by
+  intro Ω hΩ f hf
+  obtain ⟨g,hg,hgf⟩ := hC Ω hΩ (s.inclusion.assignmentMap f)
+    (s.inclusion.assignmentMap_compatible Ω f hf)
+  refine ⟨s.projection.assignmentMap g,
+    s.projection.assignmentMap_compatible Set.univ g hg,?_⟩
+  intro x hx
+  change (s.projection.vertex x).map (g x)=f x
+  rw [hgf x hx]
+  exact s.vertex_split x (f x)
+
+ theorem SplitIn.generated (s : SplitIn B C) (hC : C.forget.Generated) : B.forget.Generated := by
+  intro x v
+  obtain ⟨f,hf,hfx⟩ := hC x ((s.inclusion.vertex x).map v)
+  refine ⟨s.projection.assignmentMap f,
+    s.projection.assignmentMap_compatible Set.univ f hf,?_⟩
+  change (s.projection.vertex x).map (f x)=v
+  rw [hfx]
+  exact s.vertex_split x v
+
+ theorem SplitIn.upperQuotient (s : SplitIn B C) {α : E → A}
+    (hC : C.UpperQuotient α) : B.UpperQuotient α := by
+  intro e
+  constructor
+  · intro v
+    obtain ⟨c,hc⟩ := (hC e).1 ((s.inclusion.edge e).map v)
+    refine ⟨(s.projection.vertex (G.target e)).map c,?_⟩
+    rw [← s.projection.upper,hc]
+    exact s.edge_split e v
+  · intro v
+    constructor
+    · intro hv
+      have hz : (C.upper e).map ((s.inclusion.vertex _).map v)=0 := by
+        rw [← s.inclusion.upper,hv]
+        exact map_zero _
+      obtain ⟨c,hc⟩ := (hC e).2 _ |>.mp hz
+      refine ⟨(s.projection.vertex _).map c,?_⟩
+      have h := congrArg (s.projection.vertex _).map hc
+      rw [s.vertex_split,map_smul] at h
+      exact h
+    · rintro ⟨w,rfl⟩
+      have hz := ((hC e).2 (α e • (s.inclusion.vertex _).map w)).mpr
+        ⟨(s.inclusion.vertex _).map w,rfl⟩
+      have he : (s.inclusion.edge e).map ((B.upper e).map (α e • w))=0 := by
+        rw [s.inclusion.upper,map_smul]
+        exact hz
+      have h := congrArg (s.projection.edge e).map he
+      rw [s.edge_split,map_zero] at h
+      exact h
+
+ theorem SplitIn.vertex_projective (s : SplitIn B C) (x : V)
+    [Module.Projective A (C.vertex x)] : Module.Projective A (B.vertex x) := by
+  apply Module.Projective.of_split (s.inclusion.vertex x).map (s.projection.vertex x).map
+  apply LinearMap.ext
+  exact s.vertex_split x
+
+ theorem SplitIn.vertex_free
+    [DirectSum.Decomposition 𝓐] [SetLike.GradedMonoid 𝓐]
+    [IsDomain A] [IsNoetherianRing A]
+    (hneg : ∀ n < 0, 𝓐 n=⊥)
+    (hzero : ∀ a ∈ 𝓐 0, ∃ c : k, algebraMap k A c=a)
+    (s : SplitIn B C) (x : V) [Module.Projective A (C.vertex x)] :
+    Module.Free A (B.vertex x) := by
+  let D := B.vertex x
+  let : Module.Projective A D := s.vertex_projective x
+  obtain ⟨ι,_,bs,_⟩ := exists_homogeneous_basis_connected 𝓐 D.piece hneg hzero 0 D.nonneg
+  exact Module.Free.of_basis bs
+
+end
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+namespace KLInvariance.Graded.ModuleData
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A}
+
+ theorem equivOfEq_inverse {M N : ModuleData.{uk,ua,um} 𝓐} (h : M=N) (m : M) :
+    equivOfEq h.symm (equivOfEq h m)=m := by
+  subst N
+  rfl
+
+ def homOfEq {M N : ModuleData.{uk,ua,um} 𝓐} (h : M=N) : Hom M N :=
+  ⟨(equivOfEq h).toLinearMap,equivOfEq_graded h⟩
+
+end KLInvariance.Graded.ModuleData
+
+end
+
+
+section
+
+/-! The actual minimal-boundary sheaf is a summand of the reduced word.
+No splitting or word-recognition axiom is supplied as a premise. -/
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
+universe u us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ def symmetricLabel (u b : W) (e : Edge cs u b) : SymmetricCoefficient (M := M) :=
+  SymmetricAlgebra.ι ℝ (Extended M) (embed M (BruhatGraph.root cs u b e).val)
+
+ theorem symmetricLabel_represents (u b : W) (e : Edge cs u b) :
+    Represents M cs (label cs u b e) (BruhatGraph.root cs u b e).val :=
+  positiveRoot_represents M cs ⟨label cs u b e,label_isReflection cs u b e⟩
+
+ theorem bottGradedProjectionSheaf_upperQuotient (l : List I) (u b : W) :
+    (bottGradedProjectionSheaf cs basis l u b).UpperQuotient (symmetricLabel cs u b) := by
+  intro e
+  exact ⟨bottProjectionSheaf_upper_surjective cs l u b e,
+    bottProjectionSheaf_upper_kernel cs l u b e (symmetricLabel_represents cs u b e)⟩
+
+ def bottBoundarySheafTopInclusion (l : List I) (hl : cs.IsReduced l)
+    (B : BoundarySheaf (symmetricGrading_negative basis)
+      (graph cs 1 (cs.wordProd l)) (symmetricLabel cs 1 (cs.wordProd l))
+      ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩) :
+    ModuleData.Hom (B.sheaf.vertex ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩)
+      (bottStalkData cs basis l (cs.wordProd l)) :=
+  (bottTopInvHom cs basis l hl).comp
+    ⟨(ModuleData.equivOfEq B.top).toLinearMap,ModuleData.equivOfEq_graded B.top⟩
+
+ def bottBoundarySheafTopProjection (l : List I) (hl : cs.IsReduced l)
+    (B : BoundarySheaf (symmetricGrading_negative basis)
+      (graph cs 1 (cs.wordProd l)) (symmetricLabel cs 1 (cs.wordProd l))
+      ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩) :
+    ModuleData.Hom (bottStalkData cs basis l (cs.wordProd l))
+      (B.sheaf.vertex ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩) :=
+  (⟨(ModuleData.equivOfEq B.top.symm).toLinearMap,ModuleData.equivOfEq_graded B.top.symm⟩ :
+    ModuleData.Hom _ _).comp (bottTopHom cs basis l hl)
+
+ theorem bottBoundarySheafTop_split (l : List I) (hl : cs.IsReduced l)
+    (B : BoundarySheaf (symmetricGrading_negative basis)
+      (graph cs 1 (cs.wordProd l)) (symmetricLabel cs 1 (cs.wordProd l))
+      ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩)
+    (v : B.sheaf.vertex ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩) :
+    (bottBoundarySheafTopProjection cs basis l hl B).map
+      ((bottBoundarySheafTopInclusion cs basis l hl B).map v)=v := by
+  change ModuleData.equivOfEq B.top.symm ((bottTopHom cs basis l hl).map
+    ((bottTopInvHom cs basis l hl).map (ModuleData.equivOfEq B.top v)))=v
+  rw [bottTopHom_inv]
+  exact ModuleData.equivOfEq_inverse B.top v
+
+ theorem bottBoundarySheaf_splitIn (l : List I) (hl : cs.IsReduced l)
+    (B : BoundarySheaf (symmetricGrading_negative basis)
+      (graph cs 1 (cs.wordProd l)) (symmetricLabel cs 1 (cs.wordProd l))
+      ⟨cs.wordProd l,one_bruhat cs _,bruhat_refl cs _⟩) :
+    Nonempty (GradedSheaf.SplitIn B.sheaf (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))) := by
+  classical
+  let := symmetricCoefficientNoetherian basis
+  let := interval_finite cs 1 (cs.wordProd l)
+  let := Fintype.ofFinite (Interval cs 1 (cs.wordProd l))
+  let := Fintype.ofFinite (Edge cs 1 (cs.wordProd l))
+  exact B.splitIn_of_top_split (symmetricGrading_zero basis) (fun x => x.property.2)
+    (bottGradedProjectionSheaf cs basis l 1 (cs.wordProd l))
+    (bottGradedProjectionSheaf_upperQuotient cs basis l 1 (cs.wordProd l))
+    (fun x => bottStalkFree cs l x.val) (bottProjectionSheaf_flabby cs l hl)
+    (bottProjectionSheaf_generated cs l 1 (cs.wordProd l))
+    (bottBoundarySheafTopInclusion cs basis l hl B) (bottBoundarySheafTopProjection cs basis l hl B)
+    (bottBoundarySheafTop_split cs basis l hl B)
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A B : Type ua} [CommRing A] [CommRing B]
+  [Algebra k A] [Algebra k B]
+  {𝓐 : ℤ → Submodule k A} {𝓑 : ℤ → Submodule k B}
+  (e : A ≃ₐ[k] B) (he : ∀ n a, a∈𝓐 n ↔ e a∈𝓑 n)
+  {V E : Type um} [PartialOrder V] {G : OrderedGraph V E}
+noncomputable section
+
+ def coefficientChange (S : GradedSheaf (𝓐 := 𝓐) G) : GradedSheaf (𝓐 := 𝓑) G where
+  vertex x := ModuleData.change e he (S.vertex x)
+  edge f := ModuleData.change e he (S.edge f)
+  lower f := ModuleData.changeHom e he (S.lower f)
+  upper f := ModuleData.changeHom e he (S.upper f)
+
+ def Hom.coefficientChange {S T : GradedSheaf (𝓐 := 𝓐) G} (g : Hom S T) :
+    Hom (coefficientChange e he S) (coefficientChange e he T) where
+  vertex x := ModuleData.changeHom e he (g.vertex x)
+  edge f := ModuleData.changeHom e he (g.edge f)
+  lower f m := CoefficientChange.ext (g.lower f m.val)
+  upper f m := CoefficientChange.ext (g.upper f m.val)
+
+ theorem coefficientChange_free (S : GradedSheaf (𝓐 := 𝓐) G) (hS : S.FreeStalks) :
+    (coefficientChange e he S).FreeStalks := by
+  intro x
+  let := hS x
+  exact ModuleData.change_free e he (S.vertex x)
+
+ theorem coefficientChange_upperQuotient (S : GradedSheaf (𝓐 := 𝓐) G)
+    (α : E → A) (hS : S.UpperQuotient α) :
+    (coefficientChange e he S).UpperQuotient (fun f => e (α f)) := by
+  intro f
+  constructor
+  · exact CoefficientChange.map_surjective e _ (hS f).1
+  · intro m
+    constructor
+    · intro hm
+      have hz : (S.upper f).map m.val=0 := congrArg CoefficientChange.val hm
+      obtain ⟨n,hn⟩ := (hS f).2 m.val |>.mp hz
+      refine ⟨CoefficientChange.mk n,CoefficientChange.ext ?_⟩
+      change m.val=e.symm (e (α f)) • n
+      simpa only [e.symm_apply_apply] using hn
+    · rintro ⟨n,rfl⟩
+      apply CoefficientChange.ext
+      change (S.upper f).map (e.symm (e (α f)) • n.val)=0
+      rw [e.symm_apply_apply]
+      exact ((hS f).2 _).mpr ⟨n.val,rfl⟩
+
+ theorem coefficientChange_flabby (S : GradedSheaf (𝓐 := 𝓐) G) (hS : S.forget.Flabby) :
+    (coefficientChange e he S).forget.Flabby := by
+  intro U hU f hf
+  have hf' : S.forget.CompatibleOn U (fun x => (f x).val) := fun q hs ht =>
+    congrArg CoefficientChange.val (hf q hs ht)
+  obtain ⟨g,hg,hgf⟩ := hS U hU (fun x => (f x).val) hf'
+  refine ⟨fun x => CoefficientChange.mk (g x),?_,?_⟩
+  · intro q hs ht
+    exact CoefficientChange.ext (hg q hs ht)
+  · intro x hx
+    exact CoefficientChange.ext (hgf x hx)
+
+ theorem coefficientChange_generated (S : GradedSheaf (𝓐 := 𝓐) G) (hS : S.forget.Generated) :
+    (coefficientChange e he S).forget.Generated := by
+  intro x m
+  obtain ⟨g,hg,hgx⟩ := hS x m.val
+  refine ⟨fun x => CoefficientChange.mk (g x),?_,CoefficientChange.ext hgx⟩
+  intro q hs ht
+  exact CoefficientChange.ext (hg q hs ht)
+
+ def coefficientChangeSections (S : GradedSheaf (𝓐 := 𝓐) G) (U : Set V) :
+    CoefficientChange e (S.forget.sections U) ≃ₗ[B] (coefficientChange e he S).forget.sections U where
+  toFun m := ⟨fun x => CoefficientChange.mk (m.val.val x),fun q hs ht =>
+    CoefficientChange.ext (m.val.property q hs ht)⟩
+  invFun m := CoefficientChange.mk ⟨fun x => (m.val x).val,fun q hs ht =>
+    congrArg CoefficientChange.val (m.property q hs ht)⟩
+  left_inv _ := rfl
+  right_inv _ := rfl
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+
+end
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+/-! Positive triangular decomposition of the honest word's character into
+actual BMP stalk characters. The KL identification is not assumed. -/
+namespace KLInvariance.TitsSpace
+open Module _root_.OAI.KLInvariance.Graded MomentGraph BruhatGraph
+universe u
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ def bottPolynomialSheaf (l : List I) (u b : W) :
+    GradedSheaf (𝓐 := polynomialGrading ℝ σ) (graph cs u b) :=
+  GradedSheaf.coefficientChange (SymmetricAlgebra.equivMvPolynomial basis)
+    (mem_symmetricGrading basis) (bottGradedProjectionSheaf cs basis l u b)
+
+ theorem bottPolynomialSheaf_free (l : List I) (u b : W) :
+    (bottPolynomialSheaf cs basis l u b).FreeStalks :=
+  GradedSheaf.coefficientChange_free _ _ _ (fun x => bottStalkFree cs l x.val)
+
+ theorem bottPolynomialSheaf_quotient (l : List I) (u b : W) :
+    (bottPolynomialSheaf cs basis l u b).UpperQuotient (polynomialLabel cs u b basis) :=
+  GradedSheaf.coefficientChange_upperQuotient _ _ _ _
+    (bottGradedProjectionSheaf_upperQuotient cs basis l u b)
+
+ theorem bottPolynomialSheaf_flabby (l : List I) (hl : cs.IsReduced l) :
+    (bottPolynomialSheaf cs basis l 1 (cs.wordProd l)).forget.Flabby :=
+  GradedSheaf.coefficientChange_flabby _ _ _ (bottProjectionSheaf_flabby cs l hl)
+
+ theorem bottPolynomialSheaf_generated (l : List I) (u b : W) :
+    (bottPolynomialSheaf cs basis l u b).forget.Generated :=
+  GradedSheaf.coefficientChange_generated _ _ _ (bottProjectionSheaf_generated cs l u b)
+
+ theorem bottPolynomialSheaf_character (l : List I) (u b : W) (x : Interval cs u b) :
+    GradedSheaf.freeStalkChar (bottPolynomialSheaf cs basis l u b)
+      (bottPolynomialSheaf_free cs basis l u b) x=bottStalkCharacter cs basis l x.val := rfl
+
+ theorem bottStalkCharacter_decomposition (l : List I) (hl : cs.IsReduced l)
+    [Fintype (Interval cs 1 (cs.wordProd l))] :
+    ∃ m : Interval cs 1 (cs.wordProd l) → Polynomial ℤ,
+      (∀ y n, 0≤(m y).coeff n) ∧ ∀ x,
+      bottStalkCharacter cs basis l x.val=
+        ∑ y, m y*lowerCharacter cs (cs.wordProd l) basis y x := by
+  classical
+  let := Fintype.ofFinite (Edge cs 1 (cs.wordProd l))
+  obtain ⟨m,hm,he⟩ := GradedSheaf.exists_stalkCharacter_decomposition
+    (boundaryModels cs (cs.wordProd l) basis)
+    (bottPolynomialSheaf cs basis l 1 (cs.wordProd l))
+    (bottPolynomialSheaf_free cs basis l 1 (cs.wordProd l))
+    (bottPolynomialSheaf_quotient cs basis l 1 (cs.wordProd l))
+    (bottPolynomialSheaf_flabby cs basis l hl)
+    (bottPolynomialSheaf_generated cs basis l 1 (cs.wordProd l))
+  refine ⟨m,hm,fun x => ?_⟩
+  simpa only [bottPolynomialSheaf_character,boundaryModels_character] using he x
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+
+namespace KLInvariance.IntegralSupport
+
+universe ur uv um
+variable {R : Type ur} [CommRing R]
+  {V : Type uv} [DecidableEq V] {A : V → Type um} [∀ x, AddCommGroup (A x)] [∀ x, Module R (A x)]
+  (M : Submodule R (∀ x, A x))
+
+noncomputable def projection (x : V) : M →ₗ[R] A x := (LinearMap.proj x).comp M.subtype
+
+def supported (x : V) : Submodule R M where
+  carrier := {m | ∀ y, y ≠ x → m.val y = 0}
+  zero_mem' := by intro y hy; rfl
+  add_mem' := by intro m n hm hn y hy; exact congrArg₂ (· + ·) (hm y hy) (hn y hy) |>.trans (zero_add _)
+  smul_mem' := by intro r m hm y hy; exact congrArg (r • ·) (hm y hy) |>.trans (smul_zero r)
+
+variable (p : R)
+  (hcut : ∀ x (m : M), (p • Pi.single x (m.val x)) ∈ M)
+
+noncomputable def cut (x : V) : M →ₗ[R] M := by
+  exact ((p • (LinearMap.single R A x).comp (LinearMap.proj x)).domRestrict M).codRestrict M (hcut x)
+
+@[simp] theorem cut_apply (x : V) (m : M) : (cut M p hcut x m).val = p • Pi.single x (m.val x) := rfl
+
+@[simp] theorem cut_self (x : V) (m : M) : (cut M p hcut x m).val x = p • m.val x := by
+  simp [cut_apply]
+
+ theorem cut_other (x y : V) (h : y ≠ x) (m : M) : (cut M p hcut x m).val y = 0 := by
+  simp [cut_apply,h]
+
+ theorem cut_of_supported (x : V) (m : M) (h : m ∈ supported M x) :
+    cut M p hcut x m = p • m := by
+  apply Subtype.ext
+  funext y
+  by_cases hy : y=x
+  · subst y; exact cut_self M p hcut x m
+  · rw [cut_other M p hcut x y hy]
+    change 0 = p • m.val y
+    rw [h y hy,smul_zero]
+
+ theorem cut_eq_zero (x : V) (m : M) (h : projection M x m = 0) : cut M p hcut x m = 0 := by
+  apply Subtype.ext
+  change p • Pi.single x (m.val x) = 0
+  change m.val x = 0 at h
+  simp [h]
+
+variable [IsDomain R] [∀ x, Module.IsTorsionFree R (A x)] (hp : p ≠ 0)
+  (d : M ≃ₗ[R] Module.Dual R M)
+  (hbalanced : ∀ x m n, d (cut M p hcut x m) n = d m (cut M p hcut x n))
+
+include hp hbalanced in
+/-- The exact integral annihilator, not merely its generic vector space. -/
+ theorem supported_iff_annihilates (x : V) (m : M) :
+    m ∈ supported M x ↔ ∀ n : M, projection M x n = 0 → d m n = 0 := by
+  constructor
+  · intro hm n hn
+    have hh := hbalanced x m n
+    rw [cut_of_supported M p hcut x m hm,cut_eq_zero M p hcut x n hn] at hh
+    simpa only [map_smul,LinearMap.smul_apply,map_zero,smul_eq_zero_iff_right hp] using hh
+  · intro hm y hy
+    have hc : cut M p hcut y m = 0 := by
+      apply d.injective
+      apply LinearMap.ext
+      intro n
+      rw [hbalanced]
+      simpa only [map_zero,LinearMap.zero_apply] using
+        hm (cut M p hcut y n) (cut_other M p hcut y x (Ne.symm hy) n)
+    have hv := congrArg (fun s : M => s.val y) hc
+    simpa only [cut_self,Submodule.coe_zero,Pi.zero_apply,smul_eq_zero_iff_right hp] using hv
+
+/-- Precomposition with a surjective stalk projection gives the integral
+single-support module. No field extension was used anywhere in this map. -/
+noncomputable def dualToSupport (x : V) : Module.Dual R (A x) →ₗ[R] supported M x :=
+  (d.symm.toLinearMap.comp (projection M x).dualMap).codRestrict (supported M x) (by
+    intro lam
+    apply (supported_iff_annihilates M p hcut hp d hbalanced x _).mpr
+    intro n hn
+    simp only [LinearMap.comp_apply,LinearEquiv.coe_coe,LinearEquiv.apply_symm_apply,
+      LinearMap.dualMap_apply]
+    change lam (projection M x n) = 0
+    rw [hn,map_zero])
+
+ theorem dualToSupport_bijective (x : V) (hπ : Function.Surjective (projection M x)) :
+    Function.Bijective (dualToSupport M p hcut hp d hbalanced x) := by
+  constructor
+  · intro lam mu he
+    apply LinearMap.dualMap_injective_of_surjective hπ
+    apply d.symm.injective
+    exact congrArg Subtype.val he
+  · intro m
+    have hm : d m.val ∈ (LinearMap.ker (projection M x)).dualAnnihilator := by
+      rw [Submodule.mem_dualAnnihilator]
+      intro n hn
+      exact (supported_iff_annihilates M p hcut hp d hbalanced x m.val).mp m.property n hn
+    rw [← LinearMap.range_dualMap_eq_dualAnnihilator_ker_of_surjective _ hπ] at hm
+    obtain ⟨lam,hlam⟩ := hm
+    refine ⟨lam,?_⟩
+    apply Subtype.ext
+    change d.symm ((projection M x).dualMap lam) = m.val
+    rw [hlam,LinearEquiv.symm_apply_apply]
+
+noncomputable def supportEquivDual (x : V) (hπ : Function.Surjective (projection M x)) :
+    supported M x ≃ₗ[R] Module.Dual R (A x) :=
+  (LinearEquiv.ofBijective (dualToSupport M p hcut hp d hbalanced x)
+    (dualToSupport_bijective M p hcut hp d hbalanced x hπ)).symm
+
+end KLInvariance.IntegralSupport
+
+end
+
+
+section
+
+
+namespace BalancedCuts
+universe ur um ua
+variable {R : Type ur} [CommRing R] [IsDomain R]
+  {M : Type um} [AddCommGroup M] [Module R M]
+  {A : Type ua} [AddCommGroup A] [Module R A]
+  (d : M →ₗ[R] Module.Dual R M) (hsym : ∀ m n, d m n=d n m)
+  (π : M →ₗ[R] A) (c : M →ₗ[R] M) (p : R) (hp : p ≠ 0)
+  (hπ : ∀ m, π (c m)=p • π m)
+  (horth : ∀ m n, π n=0 → d (c m) n=0)
+
+include hsym hp hπ horth in
+ theorem balanced (m n : M) : d (c m) n=d m (c n) := by
+  have hn : π (p • n-c n)=0 := by rw [map_sub,map_smul,hπ,sub_self]
+  have hm : π (p • m-c m)=0 := by rw [map_sub,map_smul,hπ,sub_self]
+  have en := horth m _ hn
+  have em := horth n _ hm
+  simp only [map_sub,map_smul,smul_eq_mul,sub_eq_zero] at en em
+  rw [hsym (c n) m,hsym (c n) (c m)] at em
+  exact mul_left_cancel₀ hp (en.trans em.symm)
+
+end BalancedCuts
+
+end
+
+
+section
+
+
+namespace KLInvariance.TitsSpace
+open Module MomentGraph BruhatGraph
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+local instance (a : Type*) : DecidableEq a := Classical.decEq a
+
+ theorem bottSection_supported_annihilates (l : List I) (hl : cs.IsReduced l)
+    (x : Interval cs 1 (cs.wordProd l))
+    (s z : (bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ)
+    (hs : s ∈ IntegralSupport.supported
+      ((bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ) x)
+    (hz : z.val x=0) : bottSectionDuality cs l hl s z=0 := by
+  classical
+  obtain ⟨m,rfl⟩ := bottSection_surjective cs l hl s
+  obtain ⟨n,rfl⟩ := bottSection_surjective cs l hl z
+  rw [bottSectionDuality_apply]
+  apply bott_supported_annihilates cs l x.val
+  · intro e he
+    let y : Interval cs 1 (cs.wordProd l) :=
+      ⟨bottWeight cs l e,one_bruhat cs _,bottWeight_bruhat cs l hl e⟩
+    have hy : y ≠ x := fun h => he (congrArg Subtype.val h)
+    have hh := hs y hy
+    exact congrFun (congrArg Subtype.val hh) ⟨e,rfl⟩
+  · exact congrArg Subtype.val hz
+
+ theorem bottSection_projector_balanced (l : List I) (hl : cs.IsReduced l)
+    (p : SymmetricCoefficient (M := M)) (hp : p ≠ 0)
+    (x : Interval cs 1 (cs.wordProd l))
+    (c : End (SymmetricCoefficient (M := M))
+      ((bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ))
+    (hself : ∀ m, (c m).val x=p • m.val x)
+    (hother : ∀ m y, y ≠ x → (c m).val y=0)
+    (s z : (bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ) :
+    bottSectionDuality cs l hl (c s) z = bottSectionDuality cs l hl s (c z) := by
+  classical
+  apply BalancedCuts.balanced (bottSectionDuality cs l hl).toLinearMap
+    (bottSection_symmetric cs l hl)
+    (IntegralSupport.projection (A := fun y : Interval cs 1 (cs.wordProd l) => bottStalk cs l y.val)
+      ((bottProjectionSheaf cs l 1 (cs.wordProd l)).sections Set.univ) x)
+    c p hp hself
+  intro m n hn
+  exact bottSection_supported_annihilates cs l hl x (c m) n (hother m) hn
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+namespace KLInvariance.TitsSpace
+open Module MomentGraph BruhatGraph
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+ theorem bottSupported_of_stalkSupport (l : List I) (hl : cs.IsReduced l)
+    (x : Interval cs 1 (cs.wordProd l)) (m : BottSamelsonModule cs l)
+    (hm : ∀ y : Interval cs 1 (cs.wordProd l), y≠x → bottStalkProjection cs l y.val m=0) :
+    m∈bottWeightSupported cs l x.val := by
+  intro e he
+  let y : Interval cs 1 (cs.wordProd l) :=
+    ⟨bottWeight cs l e,one_bruhat cs _,bottWeight_bruhat cs l hl e⟩
+  have hy : y≠x := fun h => he (congrArg Subtype.val h)
+  exact congrFun (congrArg Subtype.val (hm y hy)) ⟨e,rfl⟩
+
+ theorem bottStalk_supported_annihilates (l : List I) (hl : cs.IsReduced l)
+    (x : Interval cs 1 (cs.wordProd l)) (m n : BottSamelsonModule cs l)
+    (hm : ∀ y : Interval cs 1 (cs.wordProd l), y≠x → bottStalkProjection cs l y.val m=0)
+    (hn : bottStalkProjection cs l x.val n=0) : bottSamelsonDuality cs l m n=0 :=
+  bott_supported_annihilates cs l x.val (bottSupported_of_stalkSupport cs l hl x m hm) n
+    (congrArg Subtype.val hn)
+
+end
+end KLInvariance.TitsSpace
+
+end
+
+
+section
+
+namespace KLInvariance.Graded
+open Module
+universe uk ua um ui
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {M : Type um} [AddCommGroup M] [Module k M] [Module A M]
+  (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M)
+  [DirectSum.Decomposition 𝓐] [DirectSum.Decomposition 𝓜]
+  [SetLike.GradedSMul 𝓐 𝓜]
+  {ι : Type ui} [Fintype ι] (b : Basis ι A M) (d : ι → ℤ)
+  (hb : ∀ i, b i ∈ 𝓜 (d i))
+noncomputable section
+
+include hb in
+ theorem homogeneous_basis_repr (n : ℤ) (v : M) (hv : v ∈ 𝓜 n) (i : ι) :
+    b.repr v i ∈ 𝓐 (n-d i) := by
+  let _ : DirectSum.Decomposition (shiftedPiece 𝓐 d) :=
+    (shifted_isInternal 𝓐 d).chooseDecomposition
+  have hg : ∀ n f, f ∈ shiftedPiece 𝓐 d n → b.equivFun.symm f ∈ 𝓜 (n+0) := by
+    intro n f hf
+    rw [add_zero,b.equivFun_symm_apply]
+    apply Submodule.sum_mem
+    intro j _
+    simpa only [vadd_eq_add,sub_add_cancel] using SetLike.GradedSMul.smul_mem (hf j) (hb j)
+  have h := equiv_inverse_homogeneous (shiftedPiece 𝓐 d) 𝓜 b.equivFun.symm 0 hg n v hv
+  rw [sub_zero] at h
+  exact h i
+
+include hb in
+ theorem dualBasis_homogeneous [DecidableEq ι] (i : ι) (n : ℤ) (v : M) (hv : v ∈ 𝓜 n) :
+    b.dualBasis i v ∈ 𝓐 (n+(-d i)) := by
+  simpa only [Basis.coe_dualBasis,Basis.coord_apply,← sub_eq_add_neg] using
+    homogeneous_basis_repr 𝓐 𝓜 b d hb n v hv i
+
+end
+end KLInvariance.Graded
+
+end
+
+
+section
+
+/-! Genuine polynomial reflection from a homogeneous perfect duality. The
+nonnegative lattice supplies a weak degree bound, not the strict KL bound. -/
+namespace KLInvariance.Graded.ModuleData
+open Polynomial Module
+universe uk us um
+variable {k : Type uk} [Field k] {σ : Type us} [Finite σ]
+  (M N : ModuleData.{uk,max uk us,um} (polynomialGrading k σ))
+noncomputable section
+
+omit [Finite σ] in
+ theorem degree_le_of_dual_equiv
+    (e : Dual (MvPolynomial σ k) M ≃ₗ[MvPolynomial σ k] N) (l : ℕ)
+    (he : ∀ r lam, (∀ j (v : M), v∈M.piece j →
+      lam v∈polynomialGrading k σ (j+r)) → e lam∈N.piece (r+l))
+    (c : NonnegativeBasis (σ := σ) M.piece) (i : c.index) : c.degree i ≤ l := by
+  classical
+  have h := he (-(c.degree i : ℤ)) (c.basis.dualBasis i)
+    (dualBasis_homogeneous (polynomialGrading k σ) M.piece c.basis
+      (fun i => (c.degree i : ℤ)) c.homogeneous i)
+  by_contra hn
+  have hn' : -(c.degree i : ℤ)+(l:ℤ)<0 := by omega
+  rw [N.nonneg _ hn',Submodule.mem_bot] at h
+  have hz := e.injective (show e (c.basis.dualBasis i)=e 0 from
+    h.trans (map_zero e).symm)
+  exact c.basis.dualBasis.ne_zero i hz
+
+ def reflectedBasis
+    (e : Dual (MvPolynomial σ k) M ≃ₗ[MvPolynomial σ k] N) (l : ℕ)
+    (he : ∀ r lam, (∀ j (v : M), v∈M.piece j →
+      lam v∈polynomialGrading k σ (j+r)) → e lam∈N.piece (r+l))
+    (c : NonnegativeBasis (σ := σ) M.piece) : NonnegativeBasis (σ := σ) N.piece := by
+  classical
+  exact {
+    index := c.index
+    finite := c.finite
+    basis := c.basis.dualBasis.map e
+    degree := fun i => l-c.degree i
+    homogeneous := fun i => by
+      have hi := M.degree_le_of_dual_equiv N e l he c i
+      have h := he (-(c.degree i : ℤ)) (c.basis.dualBasis i)
+        (dualBasis_homogeneous (polynomialGrading k σ) M.piece c.basis
+          (fun i => (c.degree i : ℤ)) c.homogeneous i)
+      change e (c.basis.dualBasis i) ∈ N.piece ((l-c.degree i : ℕ):ℤ)
+      have hd : ((l-c.degree i : ℕ):ℤ)=-(c.degree i : ℤ)+(l:ℤ) := by omega
+      rw [hd]
+      exact h }
+
+ theorem char_of_dual_equiv [Module.Free (MvPolynomial σ k) M]
+    [Module.Free (MvPolynomial σ k) N]
+    (e : Dual (MvPolynomial σ k) M ≃ₗ[MvPolynomial σ k] N) (l : ℕ)
+    (he : ∀ r lam, (∀ j (v : M), v∈M.piece j →
+      lam v∈polynomialGrading k σ (j+r)) → e lam∈N.piece (r+l)) :
+    N.char=Polynomial.reflect l M.char := by
+  classical
+  let c := (exists_nonnegativeBasis (σ := σ) M.piece M.nonneg).some
+  rw [char,character_eq _ _ (M.reflectedBasis N e l he c),char,character_eq _ _ c]
+  change (∑ i : c.index, (X : Polynomial ℤ)^(l-c.degree i))=
+    Polynomial.reflect l (∑ i : c.index, X^c.degree i)
+  have hs (s : Finset c.index) : Polynomial.reflect l (∑ i∈s, (X : Polynomial ℤ)^c.degree i)=
+      ∑ i∈s, Polynomial.reflect l (X^c.degree i) := by
+    induction s using Finset.induction_on with
+    | empty => simp
+    | @insert a s ha ih => simp [Finset.sum_insert ha,ih]
+  rw [hs]
+  apply Finset.sum_congr rfl
+  intro i _
+  rw [Polynomial.reflect_monomial,Polynomial.revAt_le (M.degree_le_of_dual_equiv N e l he c i)]
+
+end
+end KLInvariance.Graded.ModuleData
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.Sheaf
+universe ur uv ue um
+variable {R : Type ur} [CommRing R] {V : Type uv} [PartialOrder V]
+  {E : Type ue} {G : OrderedGraph V E} (B : Sheaf.{ur,uv,ue,um} (R := R) G)
+
+/-- A common edge annihilator makes all the scaled coordinate idempotents
+integral endomorphisms of the section module. -/
+ theorem scaled_single_section [DecidableEq V] {p : R}
+    (hp : ∀ e (v : B.edge e), p • v = 0) (x : V) (m : B.sections Set.univ) :
+    p • Pi.single x (m.val x) ∈ B.sections Set.univ := by
+  intro e _ _
+  simp only [Pi.smul_apply,map_smul,hp]
+
+variable [Fintype E] (α : E → R)
+
+def labelProduct : R := ∏ e, α e
+
+ theorem labelProduct_ne_zero [IsDomain R] (hα : ∀ e, α e ≠ 0) : labelProduct α ≠ 0 := by
+  exact Finset.prod_ne_zero_iff.mpr (fun e _ => hα e)
+
+ theorem labelProduct_annihilates (hα : ∀ e (v : B.edge e), α e • v = 0)
+    (e : E) (v : B.edge e) : labelProduct α • v = 0 := by
+  obtain ⟨r,hr⟩ := Finset.dvd_prod_of_mem α (Finset.mem_univ e)
+  change (∏ e, α e) • v = 0
+  rw [hr,mul_comm, mul_smul,hα,smul_zero]
+
+omit [Fintype E] in
+ theorem projection_surjective (hg : B.Generated) (x : V) :
+    Function.Surjective (IntegralSupport.projection (B.sections Set.univ) x) := by
+  intro a
+  obtain ⟨m,hm,hx⟩ := hg x a
+  exact ⟨⟨m,hm⟩,hx⟩
+
+variable [DecidableEq V] [IsDomain R] [∀ x, Module.IsTorsionFree R (B.vertex x)]
+  (hα : ∀ e, α e ≠ 0) (ha : ∀ e (v : B.edge e), α e • v = 0)
+  (d : B.sections Set.univ ≃ₗ[R] Module.Dual R (B.sections Set.univ))
+  (hbalanced : ∀ x m n,
+    d (IntegralSupport.cut (B.sections Set.univ) (labelProduct α)
+      (B.scaled_single_section (B.labelProduct_annihilates α ha)) x m) n =
+    d m (IntegralSupport.cut (B.sections Set.univ) (labelProduct α)
+      (B.scaled_single_section (B.labelProduct_annihilates α ha)) x n))
+
+
+noncomputable def singleSupportEquivDual (hg : B.Generated) (x : V) :
+    IntegralSupport.supported (B.sections Set.univ) x ≃ₗ[R] Module.Dual R (B.vertex x) :=
+  IntegralSupport.supportEquivDual (B.sections Set.univ) (labelProduct α)
+    (B.scaled_single_section (B.labelProduct_annihilates α ha))
+    (labelProduct_ne_zero α hα) d hbalanced x (B.projection_surjective hg x)
+
+end KLInvariance.MomentGraph.Sheaf
+
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
+  {G : OrderedGraph V E} (B : GradedSheaf (𝓐 := 𝓐) G) {α : E → A}
+
+ theorem UpperQuotient.annihilates (h : B.UpperQuotient α) (e : E) (v : B.edge e) :
+    α e • v = 0 := by
+  obtain ⟨w,rfl⟩ := (h e).1 v
+  rw [← map_smul]
+  exact (h e).2 _ |>.mpr ⟨w,rfl⟩
+
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.Sheaf
+universe ur uv ue um ui
+variable {R : Type ur} [CommRing R] {V : Type uv} [PartialOrder V]
+  {E : Type ue} {G : OrderedGraph V E} (B : Sheaf.{ur,uv,ue,um} (R := R) G)
+
+ theorem stalkMap_eq_zero_iff (x : V) (v : B.vertex x) :
+    B.stalkMap x v = 0 ↔ ∀ e (h : G.source e = x), B.lower e (h.symm ▸ v) = 0 := by
+  constructor
+  · intro hv e he
+    exact congrFun hv ⟨e,he⟩
+  · intro hv
+    funext e
+    exact hv e.val e.property
+
+ theorem single_section_iff [DecidableEq V] (x : V) (v : B.vertex x) :
+    Pi.single x v ∈ B.sections Set.univ ↔
+      B.stalkMap x v = 0 ∧ ∀ e (h : G.target e = x), B.upper e (h.symm ▸ v) = 0 := by
+  constructor
+  · intro hv
+    constructor
+    · apply (B.stalkMap_eq_zero_iff x v).mpr
+      intro e he
+      subst x
+      have hc := hv e (Set.mem_univ _) (Set.mem_univ _)
+      simpa only [Pi.single_eq_same, Pi.single_eq_of_ne (ne_of_gt (G.increasing e)),map_zero] using hc
+    · intro e he
+      subst x
+      have hc := hv e (Set.mem_univ _) (Set.mem_univ _)
+      simpa only [Pi.single_eq_same, Pi.single_eq_of_ne (ne_of_lt (G.increasing e)),map_zero] using hc.symm
+  · rintro ⟨hu,hd⟩ e _ _
+    by_cases hs : G.source e = x
+    · subst x
+      simpa only [Pi.single_eq_same, Pi.single_eq_of_ne (ne_of_gt (G.increasing e)),map_zero]
+        using (B.stalkMap_eq_zero_iff _ _).mp hu e rfl
+    · by_cases ht : G.target e = x
+      · subst x
+        simpa only [Pi.single_eq_same, Pi.single_eq_of_ne (ne_of_lt (G.increasing e)),map_zero]
+          using (hd e rfl).symm
+      · simp only [Pi.single_eq_of_ne hs,Pi.single_eq_of_ne ht,map_zero]
+
+variable [Fintype E] [DecidableEq V]
+
+def incoming (x : V) : Finset E := Finset.univ.filter (fun e => G.target e = x)
+
+@[simp] theorem mem_incoming (x : V) (e : E) : e ∈ incoming (G := G) x ↔ G.target e = x := by
+  simp [incoming]
+
+def incomingProduct (α : E → R) (x : V) : R := ∏ e ∈ incoming (G := G) x, α e
+
+variable [IsDomain R] (α : E → R)
+  (hα : ∀ e, Prime (α e))
+  (hq : ∀ e, Function.Surjective (B.upper e))
+  (hk : ∀ e v, B.upper e v = 0 ↔ ∃ w, v = α e • w)
+  {ι : V → Type ui} [∀ x, Fintype (ι x)] (bs : ∀ x, Module.Basis (ι x) R (B.vertex x))
+
+include hα in
+ theorem incomingProduct_ne_zero (x : V) : incomingProduct (G := G) α x ≠ 0 := by
+  exact Finset.prod_ne_zero_iff.mpr (fun e _ => (hα e).ne_zero)
+
+omit [IsDomain R] in
+include hα hq hk bs in
+ theorem outgoing_product_regular (x : V)
+    (hsep : ∀ e f, G.source e = x → G.target f = x → ¬ α e ∣ α f)
+    (e : E) (he : G.source e = x) :
+    Function.Injective (fun v : B.edge e => incomingProduct (G := G) α x • v) := by
+  apply FiniteFreeScalar.scalar_quotient_regular (bs (G.target e)) (B.upper e) (hq e)
+    (α e) (incomingProduct (G := G) α x) (hα e) _ (hk e)
+  intro hdiv
+  obtain ⟨f,hf,hfdiv⟩ := ((hα e).dvd_finsetProd_iff α).mp hdiv
+  exact hsep e f he ((mem_incoming (G := G) x f).mp hf) hfdiv
+
+omit [IsDomain R] in
+include hk bs in
+/-- The actual integral single-support conditions. Downward labels impose
+product divisibility, and they are regular at every upward edge. -/
+ theorem single_section_iff_prod [DecompositionMonoid R] (x : V)
+    (hrel : (incoming (G := G) x : Set E).Pairwise (Function.onFun IsRelPrime α))
+    (hreg : ∀ e, G.source e = x → Function.Injective
+      (fun v : B.edge e => incomingProduct (G := G) α x • v)) (v : B.vertex x) :
+    Pi.single x v ∈ B.sections Set.univ ↔
+      ∃ w : B.vertex x, v = incomingProduct (G := G) α x • w ∧ B.stalkMap x w = 0 := by
+  constructor
+  · intro hv
+    obtain ⟨hu,hd⟩ := (B.single_section_iff x v).mp hv
+    have hdiv : ∀ e ∈ incoming (G := G) x, ∃ w : B.vertex x, α e • w = v := by
+      intro e he
+      have ht := (mem_incoming (G := G) x e).mp he
+      subst x
+      obtain ⟨w,hw⟩ := (hk e v).mp (hd e rfl)
+      exact ⟨w,hw.symm⟩
+    obtain ⟨w,hw⟩ := FiniteFreeScalar.exists_prod_smul (bs x) (incoming (G := G) x) α hrel v hdiv
+    refine ⟨w,hw.symm,?_⟩
+    apply (B.stalkMap_eq_zero_iff x w).mpr
+    intro e he
+    apply hreg e he
+    change incomingProduct (G := G) α x • _ = incomingProduct (G := G) α x • (0 : B.edge e)
+    rw [smul_zero]
+    have hz := (B.stalkMap_eq_zero_iff x v).mp hu e he
+    subst x
+    rw [← hw,map_smul] at hz
+    exact hz
+  · rintro ⟨w,rfl,hw⟩
+    apply (B.single_section_iff x _).mpr
+    constructor
+    · rw [map_smul,hw,smul_zero]
+    · intro e he
+      subst x
+      obtain ⟨r,hr⟩ := Finset.dvd_prod_of_mem α ((mem_incoming (G := G) _ e).mpr rfl)
+      apply (hk e _).mpr
+      exact ⟨r • w, by rw [incomingProduct,hr,mul_smul]⟩
+
+/-- The single-support part as an actual submodule of its stalk. -/
+def singleStalk (x : V) : Submodule R (B.vertex x) :=
+  (B.sections Set.univ).comap (LinearMap.single R (fun y => B.vertex y) x)
+
+noncomputable def singleStalkEquivSupport (x : V) :
+    B.singleStalk x ≃ₗ[R] IntegralSupport.supported (B.sections Set.univ) x := by
+  let f : B.singleStalk x →ₗ[R] IntegralSupport.supported (B.sections Set.univ) x :=
+    { toFun := fun v => ⟨⟨Pi.single x v.val,v.property⟩,by
+                        intro y hy; exact Pi.single_eq_of_ne hy _⟩
+      map_add' := by intro v w; apply Subtype.ext; apply Subtype.ext; exact Pi.single_add _ _ _
+      map_smul' := by intro r v; apply Subtype.ext; apply Subtype.ext; exact Pi.single_smul _ _ _ }
+  apply LinearEquiv.ofBijective f
+  constructor
+  · intro v w hvw
+    apply Subtype.ext
+    have hv := congrArg (fun s : IntegralSupport.supported (B.sections Set.univ) x => s.val.val x) hvw
+    change (Pi.single x v.val : B.Assignment) x = (Pi.single x w.val : B.Assignment) x at hv
+    simpa only [Pi.single_eq_same] using hv
+  · intro m
+    have he : Pi.single x (m.val.val x) = m.val.val := by
+      funext y
+      by_cases hy : y=x
+      · subst y; simp only [Pi.single_eq_same]
+      · rw [Pi.single_eq_of_ne hy, m.property y hy]
+    refine ⟨⟨m.val.val x,?_⟩,?_⟩
+    · change Pi.single x (m.val.val x) ∈ B.sections Set.univ
+      rw [he]; exact m.val.property
+    · apply Subtype.ext; apply Subtype.ext; exact he
+
+
+noncomputable def kernelEquivSingleStalk [DecompositionMonoid R] (x : V)
+    (hrel : (incoming (G := G) x : Set E).Pairwise (Function.onFun IsRelPrime α))
+    (hreg : ∀ e, G.source e = x → Function.Injective
+      (fun v : B.edge e => incomingProduct (G := G) α x • v)) :
+    LinearMap.ker (B.stalkMap x) ≃ₗ[R] B.singleStalk x := by
+  let p := incomingProduct (G := G) α x
+  let f : LinearMap.ker (B.stalkMap x) →ₗ[R] B.singleStalk x :=
+    (p • (LinearMap.ker (B.stalkMap x)).subtype).codRestrict (B.singleStalk x) (by
+      intro v
+      exact (B.single_section_iff_prod α hk bs x hrel hreg _).mpr ⟨v.val,rfl,v.property⟩)
+  apply LinearEquiv.ofBijective f
+  constructor
+  · intro v w hvw
+    apply Subtype.ext
+    let : Module.Free R (B.vertex x) := Module.Free.of_basis (bs x)
+    exact smul_right_injective (B.vertex x) (incomingProduct_ne_zero α hα x)
+      (congrArg Subtype.val hvw)
+  · intro v
+    obtain ⟨w,hw,hz⟩ := (B.single_section_iff_prod α hk bs x hrel hreg v.val).mp v.property
+    exact ⟨⟨w,hz⟩,Subtype.ext hw.symm⟩
+
+end KLInvariance.MomentGraph.Sheaf
+
+end
+
+
+section
+
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
+  {σ : Type us} (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ theorem root_injective_on_labels {e f : Edge cs u b} (h : root cs u b e = root cs u b f) :
+    label cs u b e = label cs u b f := by
+  exact congrArg Subtype.val ((positiveRootEquiv M cs).injective h)
+
+ theorem polynomialLabel_prime (e : Edge cs u b) : Prime (polynomialLabel cs u b basis e) := by
+  apply (MulEquiv.prime_iff (SymmetricAlgebra.equivMvPolynomial basis)).mpr
+  apply EdgeAlgebra.symmetric_ι_prime
+  intro h
+  exact (root cs u b e).property.1.ne_zero (congrArg Prod.fst h)
+
+ theorem label_eq_of_polynomialLabel_dvd {e f : Edge cs u b}
+    (h : polynomialLabel cs u b basis e ∣ polynomialLabel cs u b basis f) :
+    label cs u b e = label cs u b f := by
+  apply root_injective_on_labels cs u b
+  by_contra hn
+  have hd : SymmetricAlgebra.ι ℝ (Extended M) (embed M (root cs u b e).val) ∣
+      SymmetricAlgebra.ι ℝ (Extended M) (embed M (root cs u b f).val) := by
+    obtain ⟨z,hz⟩ := h
+    exact ⟨(SymmetricAlgebra.equivMvPolynomial basis).symm z, by
+      apply (SymmetricAlgebra.equivMvPolynomial basis).injective
+      simpa only [map_mul,AlgEquiv.apply_symm_apply,polynomialLabel] using hz⟩
+  have hm := (EdgeAlgebra.symmetric_ι_dvd_iff _ _).mp hd
+  obtain ⟨r,hr⟩ := Submodule.mem_span_singleton.mp hm
+  have he : r • (root cs u b e).val = (1 : ℝ) • (root cs u b f).val := by
+    have hfst := congrArg Prod.fst hr
+    change r • (root cs u b e).val = (root cs u b f).val at hfst
+    simpa only [one_smul] using hfst
+  exact one_ne_zero ((positiveRoots_independent _ _ hn).eq_zero_of_pair' he).2
+
+ theorem incoming_pairwise_relPrime [Fintype (Edge cs u b)] [DecidableEq (Interval cs u b)] (x : Interval cs u b) :
+    (Sheaf.incoming (G := graph cs u b) x : Set (Edge cs u b)).Pairwise
+      (Function.onFun IsRelPrime (polynomialLabel cs u b basis)) := by
+  intro e he f hf hne
+  apply ((polynomialLabel_prime cs u b basis e).irreducible.isRelPrime_iff_not_dvd).mpr
+  intro hdiv
+  apply hne
+  apply eq_of_label_target_eq cs u b (label_eq_of_polynomialLabel_dvd cs u b basis hdiv)
+  exact ((Sheaf.mem_incoming x e).mp he).trans ((Sheaf.mem_incoming x f).mp hf).symm
+
+ theorem outgoing_incoming_not_dvd (x : Interval cs u b) (e f : Edge cs u b)
+    (he : (graph cs u b).source e = x) (hf : (graph cs u b).target f = x) :
+    ¬ polynomialLabel cs u b basis e ∣ polynomialLabel cs u b basis f := by
+  intro h
+  apply source_ne_target_of_label_eq cs u b (label_eq_of_polynomialLabel_dvd cs u b basis h)
+  exact he.trans hf.symm
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+namespace KLInvariance.Hilbert
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {M : Type um} [AddCommGroup M] [Module k M] [Module A M] [IsScalarTower k A M]
+  (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M)
+  [DirectSum.Decomposition 𝓜] [SetLike.GradedSMul 𝓐 𝓜]
+  (a : A) (d : ℤ) (ha : a ∈ 𝓐 d)
+
+ def scalarDegreeMap (n : ℤ) : 𝓜 (n-d) →ₗ[k]
+    subPiece 𝓜 (LinearMap.range (LinearMap.lsmul A M a)) n :=
+  { toFun := fun m => ⟨⟨a • m.val,⟨m.val,rfl⟩⟩,by
+      change a • m.val ∈ 𝓜 n
+      simpa using SetLike.GradedSMul.smul_mem ha m.property⟩
+    map_add' := by intros; ext; exact smul_add _ _ _
+    map_smul' := by intros; ext; exact smul_comm _ _ _ }
+
+ theorem scalarDegreeMap_surjective (n : ℤ) :
+    Function.Surjective (scalarDegreeMap 𝓐 𝓜 a d ha n) := by
+  rintro ⟨⟨m,v,hv⟩,hm⟩
+  refine ⟨DirectSum.decompose 𝓜 v (n-d), ?_⟩
+  apply Subtype.ext
+  apply Subtype.ext
+  change a • component 𝓜 (n-d) v = m
+  rw [← component_homogeneous_smul 𝓐 𝓜 d a ha n v]
+  change (DirectSum.decompose 𝓜 (a • v) n : M) = m
+  change a • v = m at hv
+  rw [hv]
+  exact DirectSum.decompose_of_mem_same 𝓜 hm
+
+omit [DirectSum.Decomposition 𝓜] in
+ theorem scalarDegreeMap_injective (hinj : Function.Injective (fun m : M => a • m)) (n : ℤ) :
+    Function.Injective (scalarDegreeMap 𝓐 𝓜 a d ha n) := by
+  intro v w h
+  apply Subtype.ext
+  apply hinj
+  exact congrArg (fun t => t.val.val) h
+
+noncomputable def scalarDegreeEquiv
+    (hinj : Function.Injective (fun m : M => a • m)) (n : ℤ) :
+    𝓜 (n-d) ≃ₗ[k] subPiece 𝓜 (LinearMap.range (LinearMap.lsmul A M a)) n :=
+  LinearEquiv.ofBijective (scalarDegreeMap 𝓐 𝓜 a d ha n)
+    ⟨scalarDegreeMap_injective 𝓐 𝓜 a d ha hinj n,
+      scalarDegreeMap_surjective 𝓐 𝓜 a d ha n⟩
+
+include ha in
+ theorem scalar_value
+    (hinj : Function.Injective (fun m : M => a • m)) {q : ℝ} (hq : q ≠ 0)
+    (hs : Summable (term 𝓜 ⊤ q)) :
+    value (subPiece 𝓜 (LinearMap.range (LinearMap.lsmul A M a))) ⊤ q =
+      q^d * value 𝓜 ⊤ q := by
+  have h := (hs.hasSum.mul_left (q^d))
+  have hh : HasSum
+      (fun n : ℤ => (Module.finrank k (𝓜 (n-d)) : ℝ)*q^n)
+      (q^d * value 𝓜 ⊤ q) := by
+    apply (Equiv.addRight d).hasSum_iff.mp
+    apply h.congr_fun
+    intro n
+    rw [term_top]
+    change (Module.finrank k (𝓜 (n+d-d)) : ℝ)*q^(n+d) =
+      q^d * ((Module.finrank k (𝓜 n) : ℝ)*q^n)
+    rw [add_sub_cancel_right,zpow_add₀ hq]
+    ring
+  apply HasSum.tsum_eq
+  apply hh.congr_fun
+  intro n
+  rw [term_top,← LinearEquiv.finrank_eq (scalarDegreeEquiv 𝓐 𝓜 a d ha hinj n)]
+
+include ha in
+
+
+ theorem scalarQuotient_value [∀ n, Module.Finite k (𝓜 n)]
+    (hinj : Function.Injective (fun m : M => a • m)) {q : ℝ} (hq : 0 < q)
+    (hs : Summable (term 𝓜 ⊤ q)) :
+    value (quotientPiece 𝓜 (LinearMap.range (LinearMap.lsmul A M a))) ⊤ q =
+      (1-q^d) * value 𝓜 ⊤ q := by
+  let S := LinearMap.range (LinearMap.lsmul A M a)
+  let : DirectSum.Decomposition (quotientPiece 𝓜 S) :=
+    (quotient_isInternal 𝓜 S (scalar_range_homogeneous 𝓐 𝓜 d a ha)).chooseDecomposition
+  have h := value_exact_of_summable 𝓜 (quotientPiece 𝓜 S) S.mkQ
+    (fun n m hm => Submodule.mem_map.mpr ⟨m,hm,rfl⟩) S.mkQ_surjective hq hs
+  rw [Submodule.ker_mkQ] at h
+  change value 𝓜 ⊤ q = value (subPiece 𝓜 (LinearMap.range (LinearMap.lsmul A M a))) ⊤ q + _ at h
+  rw [scalar_value 𝓐 𝓜 a d ha hinj hq.ne' hs] at h
+  change value (quotientPiece 𝓜 S) ⊤ q = _
+  linarith
+
+end KLInvariance.Hilbert
+
+end
+
+
+section
+
+namespace KLInvariance.Hilbert
+open _root_.OAI.KLInvariance.Graded
+universe uk ua um un
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {M : Type um} [AddCommGroup M] [Module k M] [Module A M] [IsScalarTower k A M]
+  {N : Type un} [AddCommGroup N] [Module k N] [Module A N] [IsScalarTower k A N]
+
+ def imageMap (S : Submodule A M) (f : M →ₗ[A] N) : S →ₗ[A] S.map f where
+  toFun v := ⟨f v.val,Submodule.mem_map.mpr ⟨_,v.property,rfl⟩⟩
+  map_add' _ _ := by ext; exact map_add f _ _
+  map_smul' _ _ := by ext; exact map_smul f _ _
+
+ theorem imageMap_surjective (S : Submodule A M) (f : M →ₗ[A] N) :
+    Function.Surjective (imageMap S f) := by
+  rintro ⟨m,hm⟩
+  obtain ⟨v,hv,rfl⟩ := Submodule.mem_map.mp hm
+  exact ⟨⟨v,hv⟩,rfl⟩
+
+ theorem imageMap_ker (S : Submodule A M) (f : M →ₗ[A] N) (a : A)
+    (hf : ∀ v, f v = 0 ↔ ∃ w, v = a • w)
+    (hS : ∀ v, a • v ∈ S → v ∈ S) :
+    LinearMap.ker (imageMap S f) = LinearMap.range (LinearMap.lsmul A S a) := by
+  ext v
+  constructor
+  · intro hv
+    have hz : f v.val = 0 := congrArg Subtype.val hv
+    obtain ⟨w,hw⟩ := (hf v.val).mp hz
+    have hwS : w ∈ S := hS w (hw ▸ v.property)
+    exact ⟨⟨w,hwS⟩,Subtype.ext hw.symm⟩
+  · rintro ⟨w,rfl⟩
+    apply Subtype.ext
+    exact (hf (a • w.val)).mpr ⟨w.val,rfl⟩
+
+omit [Algebra k A] [IsScalarTower k A M] [IsScalarTower k A N] in
+ theorem map_homogeneous
+    (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
+    [DirectSum.Decomposition 𝓜] [DirectSum.Decomposition 𝓝]
+    (S : Submodule A M) (hS : ∀ n m, m ∈ S → component 𝓜 n m ∈ S)
+    (f : M →ₗ[A] N) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 n) :
+    ∀ n v, v ∈ S.map f → component 𝓝 n v ∈ S.map f := by
+  intro n v hv
+  obtain ⟨w,hw,rfl⟩ := Submodule.mem_map.mp hv
+  refine Submodule.mem_map.mpr ⟨component 𝓜 n w,hS n w hw,?_⟩
+  simpa using (map_component 𝓜 𝓝 f 0 (by simpa using hf) n w).symm
+
+/-- The actual upper image of a saturated graded submodule has precisely
+`(1-q^d)` times its Hilbert series. No choice of edge basis enters this identity. -/
+theorem saturated_image_value
+    (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M) (𝓝 : ℤ → Submodule k N)
+    [DirectSum.Decomposition 𝓜] [DirectSum.Decomposition 𝓝]
+    [SetLike.GradedSMul 𝓐 𝓜] [∀ n, Module.Finite k (𝓜 n)]
+    (S : Submodule A M) (hS : ∀ n m, m ∈ S → component 𝓜 n m ∈ S)
+    (f : M →ₗ[A] N) (hf : ∀ n m, m ∈ 𝓜 n → f m ∈ 𝓝 n)
+    (a : A) (d : ℤ) (ha : a ∈ 𝓐 d)
+    (hk : ∀ v, f v = 0 ↔ ∃ w, v = a • w)
+    (hsat : ∀ v, a • v ∈ S → v ∈ S)
+    (hinj : Function.Injective (fun v : M => a • v))
+    {q : ℝ} (hq : 0 < q) (hs : Summable (term 𝓜 (S.restrictScalars k) q)) :
+    value 𝓝 ((S.map f).restrictScalars k) q =
+      (1-q^d) * value 𝓜 (S.restrictScalars k) q := by
+  let : DirectSum.Decomposition (subPiece 𝓜 S) := (sub_isInternal 𝓜 S hS).chooseDecomposition
+  let : DirectSum.Decomposition (subPiece 𝓝 (S.map f)) :=
+    (sub_isInternal 𝓝 (S.map f) (map_homogeneous 𝓜 𝓝 S hS f hf)).chooseDecomposition
+  let (n : ℤ) : Module.Finite k (subPiece 𝓜 S n) := Module.Finite.of_surjective
+    (subPieceEquiv 𝓜 S n).symm.toLinearMap (subPieceEquiv 𝓜 S n).symm.surjective
+  have hs' : Summable (term (subPiece 𝓜 S) ⊤ q) :=
+    hs.congr (fun n => (term_subPiece 𝓜 S q n).symm)
+  have h := value_exact_of_summable (subPiece 𝓜 S) (subPiece 𝓝 (S.map f))
+    (imageMap S f) (fun n v hv => hf n v.val hv) (imageMap_surjective S f) hq hs'
+  rw [imageMap_ker S f a hk hsat] at h
+  have hinj' : Function.Injective (fun v : S => a • v) := by
+    intro v w hv
+    apply Subtype.ext
+    apply hinj
+    exact congrArg Subtype.val hv
+  rw [scalar_value 𝓐 (subPiece 𝓜 S) a d ha hinj' hq.ne' hs',
+    value_subPiece,value_subPiece] at h
+  linarith
+
+end KLInvariance.Hilbert
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded _root_.OAI.KLInvariance.Hilbert
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
+  {G : OrderedGraph V E} (B : GradedSheaf (𝓐 := 𝓐) G)
+
+ theorem upwardSubmodule_saturated (x : V) (F : Set (Sheaf.Outgoing (G := G) x))
+    (a : A) (hreg : ∀ e ∈ F, Function.Injective (fun v : B.edge e.val => a • v)) :
+    ∀ v, a • v ∈ B.upwardSubmodule x F → v ∈ B.upwardSubmodule x F := by
+  intro v hv e he
+  have hz : (B.lowerAt x e).map (a • v) = 0 :=
+    (B.lowerAt_map x e (a • v)).trans (hv e he)
+  have hz' : (B.lowerAt x e).map v = 0 := by
+    apply hreg e he
+    change a • (B.lowerAt x e).map v = a • (0 : B.edge e.val)
+    simpa only [map_smul,smul_zero] using hz
+  exact (B.lowerAt_map x e v).symm.trans hz'
+
+
+ theorem upper_image_value (α : E → A) (hquot : B.UpperQuotient α)
+    (e : E) (F : Set (Sheaf.Outgoing (G := G) (G.target e)))
+    [∀ n, Module.Finite k ((B.vertex (G.target e)).piece n)]
+    (ha : α e ∈ 𝓐 1)
+    (hreg : ∀ f ∈ F, Function.Injective (fun v : B.edge f.val => α e • v))
+    (hinj : Function.Injective (fun v : B.vertex (G.target e) => α e • v))
+    {q : ℝ} (hq : 0 < q)
+    (hs : Summable (term (B.vertex (G.target e)).piece
+      ((B.upwardSubmodule (G.target e) F).restrictScalars k) q)) :
+    value (B.edge e).piece
+        (((B.upwardSubmodule (G.target e) F).map (B.upper e).map).restrictScalars k) q =
+      (1-q) * value (B.vertex (G.target e)).piece
+        ((B.upwardSubmodule (G.target e) F).restrictScalars k) q := by
+  simpa only [zpow_one] using saturated_image_value 𝓐
+    (B.vertex (G.target e)).piece (B.edge e).piece
+    (B.upwardSubmodule (G.target e) F) (B.upwardKernel_homogeneous (G.target e) F)
+    (B.upper e).map (B.upper e).graded (α e) 1 ha (hquot e).2
+    (B.upwardSubmodule_saturated (G.target e) F (α e) hreg) hinj hq hs
+
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  (hub : BruhatLE cs u b)
+  (B : BoundarySheaf (polynomialGrading_negative ℝ σ)
+    (graph cs u b) (polynomialLabel cs u b basis) ⟨b,hub,bruhat_refl cs b⟩)
+noncomputable section
+
+omit [Fintype σ] in
+ theorem label_regular_stalk (e : Edge cs u b) (x : Interval cs u b) :
+    Function.Injective (fun v : B.sheaf.vertex x => polynomialLabel cs u b basis e • v) := by
+  let := B.free x
+  exact smul_right_injective _ (polynomialLabel_ne_zero cs u b basis e)
+
+omit [Fintype σ] in
+
+
+ theorem incoming_label_regular_outgoing (e f : Edge cs u b)
+    (h : (graph cs u b).source f = (graph cs u b).target e) :
+    Function.Injective (fun v : B.sheaf.edge f => polynomialLabel cs u b basis e • v) := by
+  let := B.free ((graph cs u b).target f)
+  exact FiniteFreeScalar.scalar_quotient_regular
+    (Module.Free.chooseBasis (Coefficient (σ := σ)) (B.sheaf.vertex ((graph cs u b).target f)))
+    (B.sheaf.upper f).map (B.quotient f).1
+    (polynomialLabel cs u b basis f) (polynomialLabel cs u b basis e)
+    (polynomialLabel_prime cs u b basis f)
+    (outgoing_incoming_not_dvd cs u b basis _ f e h rfl) (B.quotient f).2
+
+ theorem upper_image_hilbert (e : Edge cs u b)
+    (F : Set (Sheaf.Outgoing (G := graph cs u b) ((graph cs u b).target e)))
+    [∀ n, Module.Finite ℝ ((B.sheaf.vertex ((graph cs u b).target e)).piece n)]
+    {q : ℝ} (hq : 0 < q)
+    (hs : Summable (Hilbert.term (B.sheaf.vertex ((graph cs u b).target e)).piece
+      ((B.sheaf.upwardSubmodule ((graph cs u b).target e) F).restrictScalars ℝ) q)) :
+    Hilbert.value (B.sheaf.edge e).piece
+      (((B.sheaf.upwardSubmodule ((graph cs u b).target e) F).map
+        (B.sheaf.upper e).map).restrictScalars ℝ) q =
+      (1-q) * Hilbert.value (B.sheaf.vertex ((graph cs u b).target e)).piece
+        ((B.sheaf.upwardSubmodule ((graph cs u b).target e) F).restrictScalars ℝ) q := by
+  apply B.sheaf.upper_image_value (polynomialLabel cs u b basis) B.quotient e F
+    (polynomialLabel_homogeneous cs u b basis e)
+  · intro f _
+    exact incoming_label_regular_outgoing cs u b basis hub B e f.val f.property
+  · exact label_regular_stalk cs u b basis hub B e _
+  · exact hq
+  · exact hs
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
+  {σ : Type us} (basis : Basis σ ℝ (Extended M))
+  (hub : BruhatLE cs u b)
+  (B : BoundarySheaf (polynomialGrading_negative ℝ σ)
+    (graph cs u b) (polynomialLabel cs u b basis) ⟨b,hub,bruhat_refl cs b⟩)
+  [Fintype (Edge cs u b)] [DecidableEq (Interval cs u b)]
+noncomputable section
+
+ def kernelEquivSingleSupport (x : Interval cs u b) :
+    LinearMap.ker (B.sheaf.forget.stalkMap x) ≃ₗ[Coefficient (σ := σ)]
+      IntegralSupport.supported (B.sheaf.forget.sections Set.univ) x := by
+  let _ (x : Interval cs u b) : Module.Free (Coefficient (σ := σ)) (B.sheaf.forget.vertex x) := B.free x
+  let _ (x : Interval cs u b) : Module.Finite (Coefficient (σ := σ)) (B.sheaf.forget.vertex x) :=
+    (B.sheaf.vertex x).finite
+  let _ (x : Interval cs u b) : Fintype (Module.Free.ChooseBasisIndex
+      (Coefficient (σ := σ)) (B.sheaf.forget.vertex x)) := Fintype.ofFinite _
+  let bs := fun x : Interval cs u b =>
+    Module.Free.chooseBasis (Coefficient (σ := σ)) (B.sheaf.forget.vertex x)
+  exact (B.sheaf.forget.kernelEquivSingleStalk (polynomialLabel cs u b basis)
+    (polynomialLabel_prime cs u b basis) (fun e => (B.quotient e).2) bs x
+    (incoming_pairwise_relPrime cs u b basis x)
+    (B.sheaf.forget.outgoing_product_regular (polynomialLabel cs u b basis)
+      (polynomialLabel_prime cs u b basis) (fun e => (B.quotient e).1)
+      (fun e => (B.quotient e).2) bs x
+      (outgoing_incoming_not_dvd cs u b basis x))).trans
+        (B.sheaf.forget.singleStalkEquivSupport x)
+
+
+structure SectionDuality where
+  pairing : B.sheaf.forget.sections Set.univ ≃ₗ[Coefficient (σ := σ)]
+    Dual (Coefficient (σ := σ)) (B.sheaf.forget.sections Set.univ)
+  balanced : ∀ x m n,
+    pairing (IntegralSupport.cut (B.sheaf.forget.sections Set.univ)
+      (Sheaf.labelProduct (polynomialLabel cs u b basis))
+      (B.sheaf.forget.scaled_single_section
+        (B.sheaf.forget.labelProduct_annihilates (polynomialLabel cs u b basis)
+          (GradedSheaf.UpperQuotient.annihilates B.sheaf B.quotient))) x m) n =
+    pairing m (IntegralSupport.cut (B.sheaf.forget.sections Set.univ)
+      (Sheaf.labelProduct (polynomialLabel cs u b basis))
+      (B.sheaf.forget.scaled_single_section
+        (B.sheaf.forget.labelProduct_annihilates (polynomialLabel cs u b basis)
+          (GradedSheaf.UpperQuotient.annihilates B.sheaf B.quotient))) x n)
+
+ def costalkEquivDual (d : SectionDuality cs u b basis hub B) (x : Interval cs u b) :
+    LinearMap.ker (B.sheaf.forget.stalkMap x) ≃ₗ[Coefficient (σ := σ)]
+      Dual (Coefficient (σ := σ)) (B.sheaf.vertex x) := by
+  let _ (x : Interval cs u b) : Module.Free (Coefficient (σ := σ)) (B.sheaf.forget.vertex x) := B.free x
+  let _ (y : Interval cs u b) : Module.IsTorsionFree (Coefficient (σ := σ))
+      (B.sheaf.forget.vertex y) := inferInstance
+  exact (kernelEquivSingleSupport cs u b basis hub B x).trans
+    (B.sheaf.forget.singleSupportEquivDual (polynomialLabel cs u b basis)
+      (polynomialLabel_ne_zero cs u b basis)
+      (GradedSheaf.UpperQuotient.annihilates B.sheaf B.quotient)
+      d.pairing d.balanced B.generated x)
+
+ theorem costalk_free_of_duality (d : SectionDuality cs u b basis hub B)
+    (x : Interval cs u b) :
+    Module.Free (Coefficient (σ := σ)) (LinearMap.ker (B.sheaf.forget.stalkMap x)) := by
+  let _ := B.free x
+  exact Module.Free.of_equiv (costalkEquivDual cs u b basis hub B d x).symm
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+
+namespace KLInvariance.MomentGraph.GradedSheaf
+open Module _root_.OAI.KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} [DirectSum.Decomposition 𝓐]
+  {V E : Type um} [PartialOrder V] [Fintype V] [Fintype E] [DecidableEq V]
+  {G : OrderedGraph V E} (B : GradedSheaf (𝓐 := 𝓐) G)
+  [IsNoetherianRing A] [IsDomain A]
+  [∀ x, Module.IsTorsionFree A (B.forget.vertex x)]
+  (α : E → A) (hα : ∀ e, α e ≠ 0) (ha : ∀ e (v : B.edge e), α e • v = 0)
+  (p : B.forget.sections Set.univ ≃ₗ[A] Dual A (B.forget.sections Set.univ))
+  (hbalanced : ∀ x m n,
+    p (IntegralSupport.cut (B.forget.sections Set.univ) (Sheaf.labelProduct α)
+      (B.forget.scaled_single_section (B.forget.labelProduct_annihilates α ha)) x m) n =
+    p m (IntegralSupport.cut (B.forget.sections Set.univ) (Sheaf.labelProduct α)
+      (B.forget.scaled_single_section (B.forget.labelProduct_annihilates α ha)) x n))
+noncomputable section
+
+ theorem dualToSupport_homogeneous (l : ℤ)
+    (hp : ∀ i j (m n : B.sections Set.univ),
+      m ∈ (B.sections Set.univ).piece i → n ∈ (B.sections Set.univ).piece j →
+      p m n ∈ 𝓐 (i+j-l))
+    (x : V) (r : ℤ) (lam : Dual A (B.vertex x))
+    (hlam : ∀ j (v : B.vertex x), v ∈ (B.vertex x).piece j → lam v ∈ 𝓐 (j+r)) :
+    (IntegralSupport.dualToSupport (B.forget.sections Set.univ) (Sheaf.labelProduct α)
+      (B.forget.scaled_single_section (B.forget.labelProduct_annihilates α ha))
+      (Sheaf.labelProduct_ne_zero α hα) p hbalanced x lam).val ∈
+        (B.sections Set.univ).piece (r+l) := by
+  apply perfect_inverse_of_homogeneous_functional 𝓐 (B.sections Set.univ).piece p l hp r
+  intro j n hn
+  exact hlam j (n.val x) (hn x (Set.mem_univ x))
+
+end
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+namespace KLInvariance.Graded
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {M : Type um} [AddCommGroup M] [Module k M] [Module A M]
+  (𝓐 : ℤ → Submodule k A) (𝓜 : ℤ → Submodule k M)
+  [DirectSum.Decomposition 𝓜] [SetLike.GradedSMul 𝓐 𝓜]
+
+/-- Cancellation of an injective homogeneous scalar action includes the
+inverse grading shift. It does not require the scalar to be a unit. -/
+ theorem mem_of_homogeneous_smul (d : ℤ) (a : A) (ha : a ∈ 𝓐 d)
+    (hinj : Function.Injective (fun m : M => a • m))
+    (n : ℤ) (m : M) (hm : a • m ∈ 𝓜 n) : m ∈ 𝓜 (n-d) := by
+  have hc : component 𝓜 (n-d) m = m := by
+    apply hinj
+    change a • component 𝓜 (n-d) m = a • m
+    rw [← component_homogeneous_smul 𝓐 𝓜 d a ha]
+    exact DirectSum.decompose_of_mem_same 𝓜 hm
+  rw [← hc]
+  exact (DirectSum.decompose 𝓜 m (n-d)).property
+
+end KLInvariance.Graded
+
+end
+
+
+section
+
+
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  (hub : BruhatLE cs u b)
+  (B : BoundarySheaf (polynomialGrading_negative ℝ σ)
+    (graph cs u b) (polynomialLabel cs u b basis) ⟨b,hub,bruhat_refl cs b⟩)
+  [Fintype (Edge cs u b)] [Fintype (Interval cs u b)] [DecidableEq (Interval cs u b)]
+noncomputable section
+
+omit [Fintype (Interval cs u b)] in
+ theorem incomingProduct_homogeneous (x : Interval cs u b) :
+    Sheaf.incomingProduct (G := graph cs u b) (polynomialLabel cs u b basis) x ∈
+      polynomialGrading ℝ σ ((Sheaf.incoming (G := graph cs u b) x).card : ℤ) := by
+  simpa only [Sheaf.incomingProduct,Finset.sum_const,Int.natCast_one,mul_one,nsmul_eq_mul] using
+    SetLike.prod_mem_graded (polynomialGrading ℝ σ) (fun _ : Edge cs u b => (1 : ℤ))
+      (polynomialLabel cs u b basis)
+      (F := Sheaf.incoming (G := graph cs u b) x)
+      (fun e _ => polynomialLabel_homogeneous cs u b basis e)
+
+omit [Fintype σ] [Fintype (Interval cs u b)] in
+ theorem kernelEquivSingleSupport_apply (x : Interval cs u b)
+    (v : LinearMap.ker (B.sheaf.forget.stalkMap x)) :
+    ((kernelEquivSingleSupport cs u b basis hub B x) v).val.val x =
+      Sheaf.incomingProduct (G := graph cs u b) (polynomialLabel cs u b basis) x • v.val := by
+  change (Pi.single x
+    (Sheaf.incomingProduct (G := graph cs u b) (polynomialLabel cs u b basis) x • v.val) :
+      B.sheaf.forget.Assignment) x = _
+  exact Pi.single_eq_same _ _
+
+ theorem costalkEquivDual_inverse_homogeneous
+    (d : SectionDuality cs u b basis hub B) (l : ℤ)
+    (hd : ∀ i j (m n : B.sheaf.sections Set.univ),
+      m ∈ (B.sheaf.sections Set.univ).piece i → n ∈ (B.sheaf.sections Set.univ).piece j →
+      d.pairing m n ∈ polynomialGrading ℝ σ (i+j-l))
+    (x : Interval cs u b) (r : ℤ) (lam : Dual (Coefficient (σ := σ)) (B.sheaf.vertex x))
+    (hlam : ∀ j (v : B.sheaf.vertex x), v ∈ (B.sheaf.vertex x).piece j →
+      lam v ∈ polynomialGrading ℝ σ (j+r)) :
+    ((costalkEquivDual cs u b basis hub B d x).symm lam).val ∈
+      (B.sheaf.vertex x).piece
+        (r+l-(Sheaf.incoming (G := graph cs u b) x).card) := by
+  let _ (y : Interval cs u b) : Module.Free (Coefficient (σ := σ))
+      (B.sheaf.forget.vertex y) := B.free y
+  let _ (y : Interval cs u b) : Module.IsTorsionFree (Coefficient (σ := σ))
+      (B.sheaf.forget.vertex y) := inferInstance
+  let S := B.sheaf.forget.singleSupportEquivDual (polynomialLabel cs u b basis)
+    (polynomialLabel_ne_zero cs u b basis)
+    (GradedSheaf.UpperQuotient.annihilates B.sheaf B.quotient)
+    d.pairing d.balanced B.generated x
+  have hs : (S.symm lam).val ∈ (B.sheaf.sections Set.univ).piece (r+l) := by
+    exact B.sheaf.dualToSupport_homogeneous (polynomialLabel cs u b basis)
+      (polynomialLabel_ne_zero cs u b basis)
+      (GradedSheaf.UpperQuotient.annihilates B.sheaf B.quotient)
+      d.pairing d.balanced l hd x r lam hlam
+  have hv := hs x (Set.mem_univ x)
+  have he : (S.symm lam).val.val x =
+      Sheaf.incomingProduct (G := graph cs u b) (polynomialLabel cs u b basis) x •
+      ((costalkEquivDual cs u b basis hub B d x).symm lam).val := by
+    rw [← kernelEquivSingleSupport_apply cs u b basis hub B x]
+    change (S.symm lam).val.val x =
+      ((kernelEquivSingleSupport cs u b basis hub B x)
+        ((kernelEquivSingleSupport cs u b basis hub B x).symm (S.symm lam))).val.val x
+    rw [LinearEquiv.apply_symm_apply]
+  change (S.symm lam).val.val x ∈ (B.sheaf.vertex x).piece (r+l) at hv
+  rw [he] at hv
+  apply mem_of_homogeneous_smul (polynomialGrading ℝ σ) (B.sheaf.vertex x).piece
+    ((Sheaf.incoming (G := graph cs u b) x).card : ℤ)
+    (Sheaf.incomingProduct (G := graph cs u b) (polynomialLabel cs u b basis) x)
+    (incomingProduct_homogeneous cs u b basis x) _ (r+l) _ hv
+  let _ := B.free x
+  exact smul_right_injective _ (Sheaf.incomingProduct_ne_zero
+    (polynomialLabel cs u b basis) (polynomialLabel_prime cs u b basis) x)
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+/-! The degree of the actual incoming root product on [1,b].  This is not
+asserted for truncated intervals with arbitrary bottom. -/
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace MomentGraph
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W) (b : W)
+noncomputable section
+
+ def incomingEquivInversions (x : Interval cs 1 b) :
+    {e : Edge cs 1 b // target cs 1 b e = x} ≃ {t : W // cs.IsLeftInversion x.val t} where
+  toFun e := ⟨label cs 1 b e.val, label_isReflection cs 1 b e.val, by
+    have hx : (target cs 1 b e.val).val = x.val := congrArg Subtype.val e.property
+    have he := label_mul_target cs 1 b e.val
+    rw [hx] at he
+    rw [he]
+    have h := e.val.property.1
+    change cs.length (source cs 1 b e.val).val < cs.length (target cs 1 b e.val).val at h
+    simpa only [hx] using h⟩
+  invFun t := by
+    have hstep : BruhatStep cs (t.val*x.val) x.val := by
+      refine ⟨t.property.2,t.val,t.property.1,?_⟩
+      rw [← mul_assoc,t.property.1.mul_self,one_mul]
+    let y : Interval cs 1 b := ⟨t.val*x.val,one_bruhat cs _,
+      (Relation.ReflTransGen.single hstep).trans x.property.2⟩
+    exact ⟨⟨(y,x),hstep⟩,rfl⟩
+  left_inv e := by
+    apply Subtype.ext
+    apply eq_of_label_target_eq cs 1 b
+    · change x.val * (label cs 1 b e.val*x.val)⁻¹ = label cs 1 b e.val
+      rw [mul_inv_rev,← mul_assoc,mul_inv_cancel,one_mul,
+        (label_isReflection cs 1 b e.val).inv]
+    · exact e.property.symm
+  right_inv t := by
+    apply Subtype.ext
+    change x.val * (t.val*x.val)⁻¹=t.val
+    rw [mul_inv_rev,← mul_assoc,mul_inv_cancel,one_mul,t.property.1.inv]
+
+ theorem incoming_card [Fintype (Edge cs 1 b)] [DecidableEq (Interval cs 1 b)]
+    (x : Interval cs 1 b) : (Sheaf.incoming (G := graph cs 1 b) x).card = cs.length x.val := by
+  classical
+  calc
+    _ = Nat.card {e : Edge cs 1 b // target cs 1 b e = x} := by
+      rw [Nat.card_eq_fintype_card,Fintype.card_subtype]
+      rfl
+    _ = Nat.card {t : W // cs.IsLeftInversion x.val t} := Nat.card_congr (incomingEquivInversions cs b x)
+    _ = Nat.card (RootInversions M cs x.val) := Nat.card_congr (rootInversionEquiv M cs x.val).symm
+    _ = cs.length x.val := card_rootInversions M cs x.val
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  (hub : BruhatLE cs u b)
+  (B : BoundarySheaf (polynomialGrading_negative ℝ σ)
+    (graph cs u b) (polynomialLabel cs u b basis) ⟨b,hub,bruhat_refl cs b⟩)
+  [Fintype (Edge cs u b)] [Fintype (Interval cs u b)] [DecidableEq (Interval cs u b)]
+noncomputable section
+
+ def fullSubmodule (x : Interval cs u b) : Submodule (Coefficient (σ := σ)) (B.sheaf.vertex x) :=
+    LinearMap.ker (B.sheaf.forget.stalkMap x)
+
+omit [Fintype σ] [Fintype (Edge cs u b)] [Fintype (Interval cs u b)]
+  [DecidableEq (Interval cs u b)] in
+ theorem fullKernel_eq (x : Interval cs u b) :
+    B.sheaf.upwardSubmodule x Set.univ = fullSubmodule cs u b basis hub B x := by
+  ext m
+  change (∀ e ∈ Set.univ, B.sheaf.forget.stalkMap x m e = 0) ↔ B.sheaf.forget.stalkMap x m = 0
+  simp only [Set.mem_univ,forall_const]
+  exact funext_iff.symm
+
+ def fullKernel (x : Interval cs u b) : ModuleData (polynomialGrading ℝ σ) :=
+  ModuleData.sub (B.sheaf.vertex x) (fullSubmodule cs u b basis hub B x) (by
+    simpa only [fullKernel_eq cs u b basis hub B x] using
+      B.sheaf.upwardKernel_homogeneous x Set.univ)
+
+ theorem costalk_hilbert_of_duality
+    (d : SectionDuality cs u b basis hub B) (l : ℤ)
+    (hd : ∀ i j (m n : B.sheaf.sections Set.univ),
+      m ∈ (B.sheaf.sections Set.univ).piece i → n ∈ (B.sheaf.sections Set.univ).piece j →
+      d.pairing m n ∈ polynomialGrading ℝ σ (i+j-l))
+    (x : Interval cs u b) (c : NonnegativeBasis (σ := σ) (B.sheaf.vertex x).piece)
+    {q : ℝ} (hq : 0 < q) (hq1 : q < 1) :
+    Hilbert.value (B.sheaf.vertex x).piece
+      ((fullSubmodule cs u b basis hub B x).restrictScalars ℝ) q =
+      q^(l-(Sheaf.incoming (G := graph cs u b) x).card) *
+        Polynomial.eval₂ (Int.castRingHom ℝ) q⁻¹ c.polynomial *
+        (1-q)⁻¹ ^ Fintype.card σ := by
+  classical
+  let K := fullKernel cs u b basis hub B x
+  let cb : Basis c.index (Coefficient (σ := σ)) K :=
+    c.basis.dualBasis.map (costalkEquivDual cs u b basis hub B d x).symm
+  have hcb (i : c.index) : cb i ∈ K.piece
+      (l-(Sheaf.incoming (G := graph cs u b) x).card-(c.degree i : ℤ)) := by
+    change ((costalkEquivDual cs u b basis hub B d x).symm (c.basis.dualBasis i)).val ∈
+      (B.sheaf.vertex x).piece _
+    have h := costalkEquivDual_inverse_homogeneous cs u b basis hub B d l hd x
+      (-(c.degree i : ℤ)) (c.basis.dualBasis i)
+      (dualBasis_homogeneous (polynomialGrading ℝ σ) (B.sheaf.vertex x).piece
+        c.basis (fun i => (c.degree i : ℤ)) c.homogeneous i)
+    simpa only [show -(c.degree i : ℤ)+l-(Sheaf.incoming (G := graph cs u b) x).card =
+      l-(Sheaf.incoming (G := graph cs u b) x).card-(c.degree i : ℤ) by omega] using h
+  have hs := Hilbert.hasSum_homogeneous_basis K.piece cb
+    (fun i => l-(Sheaf.incoming (G := graph cs u b) x).card-(c.degree i : ℤ)) hcb hq hq1
+  have hv : Hilbert.value K.piece ⊤ q =
+      (∑ i, q^(l-(Sheaf.incoming (G := graph cs u b) x).card-(c.degree i : ℤ))) *
+        (1-q)⁻¹ ^ Fintype.card σ := by
+    simpa only [Hilbert.value,Hilbert.term_top] using hs.tsum_eq
+  change Hilbert.value (B.sheaf.vertex x).piece
+    ((fullSubmodule cs u b basis hub B x).restrictScalars ℝ) q = _
+  rw [← Hilbert.value_subPiece]
+  change Hilbert.value K.piece ⊤ q = _
+  rw [hv,c.eval₂_eq,Finset.mul_sum]
+  congr 1
+  apply Finset.sum_congr rfl
+  intro i _
+  rw [zpow_sub₀ hq.ne',zpow_natCast,div_eq_mul_inv,inv_pow]
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+namespace KLInvariance.CoefficientChange
+open Module
+universe uk ua ub um un
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {B : Type ub} [CommRing B] [Algebra k B]
+  (e : A ≃ₐ[k] B) {M : Type um} [AddCommGroup M] [Module A M]
+  [Module k M] [IsScalarTower k A M]
+noncomputable section
+
+ def mapEquiv {N : Type un} [AddCommGroup N] [Module A N] [Module k N]
+    [IsScalarTower k A N] (f : M ≃ₗ[A] N) : CoefficientChange e M ≃ₗ[B] CoefficientChange e N where
+  toLinearMap := map e f.toLinearMap
+  invFun n := mk (f.symm n.val)
+  left_inv m := CoefficientChange.ext (f.symm_apply_apply m.val)
+  right_inv n := CoefficientChange.ext (f.apply_symm_apply n.val)
+
+ def dualTo (f : CoefficientChange e (Dual A M)) : Dual B (CoefficientChange e M) where
+  toFun m := e (f.val m.val)
+  map_add' m n := by change e (f.val (m.val+n.val))=_; rw [map_add,map_add]
+  map_smul' b m := by
+    change e (f.val (e.symm b • m.val))=b * e (f.val m.val)
+    rw [map_smul]
+    change e (e.symm b * f.val m.val)=_
+    rw [map_mul,e.apply_symm_apply]
+
+ def dualFrom (f : Dual B (CoefficientChange e M)) : Dual A M where
+  toFun m := e.symm (f (mk m))
+  map_add' m n := by
+    change e.symm (f (mk m+mk n))=_
+    rw [map_add,map_add]
+  map_smul' a m := by
+    change e.symm (f (mk (a • m)))=a * e.symm (f (mk m))
+    rw [show (mk (a • m) : CoefficientChange e M)=e a • mk m from by
+      apply CoefficientChange.ext
+      change a • m=e.symm (e a) • m
+      rw [e.symm_apply_apply],
+      map_smul]
+    change e.symm (e a * f (mk m))=_
+    rw [map_mul,e.symm_apply_apply]
+
+ def dualEquiv : CoefficientChange e (Dual A M) ≃ₗ[B] Dual B (CoefficientChange e M) where
+  toFun := dualTo e
+  invFun f := mk (dualFrom e f)
+  left_inv f := by
+    apply CoefficientChange.ext
+    apply LinearMap.ext
+    intro m
+    exact e.symm_apply_apply (f.val m)
+  right_inv f := by
+    apply LinearMap.ext
+    intro m
+    exact e.apply_symm_apply (f m)
+  map_add' f g := by
+    apply LinearMap.ext
+    intro m
+    exact map_add e (f.val m.val) (g.val m.val)
+  map_smul' b f := by
+    apply LinearMap.ext
+    intro m
+    change e (e.symm b * f.val m.val)=b * e (f.val m.val)
+    rw [map_mul,e.apply_symm_apply]
+
+ def pairingEquiv (d : M ≃ₗ[A] Dual A M) :
+    CoefficientChange e M ≃ₗ[B] Dual B (CoefficientChange e M) :=
+  (mapEquiv e d).trans (dualEquiv e)
+
+ omit [Module k M] [IsScalarTower k A M] in
+ theorem pairingEquiv_apply (d : M ≃ₗ[A] Dual A M) (m n : CoefficientChange e M) :
+    pairingEquiv e d m n=e (d m.val n.val) := rfl
+
+end
+end KLInvariance.CoefficientChange
+
+end
+
+
+section
+
+/-! A genuine homogeneous integral perfect pairing with vertex-support
+orthogonality on both sides. Transport does not identify unequal lattices. -/
+namespace KLInvariance.MomentGraph.GradedSheaf
+open _root_.OAI.KLInvariance.Graded Module
+universe uk ua um
+variable {k : Type uk} [Field k] {A : Type ua} [CommRing A] [Algebra k A]
+  {𝓐 : ℤ → Submodule k A} {V E : Type um} [PartialOrder V]
+  {G : OrderedGraph V E}
+noncomputable section
+
+ structure PerfectPairing (S : GradedSheaf (𝓐 := 𝓐) G) (d : ℤ) where
+  pairing : S.forget.sections Set.univ ≃ₗ[A] Dual A (S.forget.sections Set.univ)
+  homogeneous : ∀ i j (m n : S.forget.sections Set.univ),
+    (∀ x, m.val x∈(S.vertex x).piece i) → (∀ x, n.val x∈(S.vertex x).piece j) →
+    pairing m n∈𝓐 (i+j-d)
+  left_orthogonal : ∀ (x : V) (m n : S.forget.sections Set.univ),
+    (∀ y, y≠x → m.val y=0) → n.val x=0 → pairing m n=0
+  right_orthogonal : ∀ (x : V) (m n : S.forget.sections Set.univ),
+    m.val x=0 → (∀ y, y≠x → n.val y=0) → pairing m n=0
+
+ def PerfectPairing.pullback {S T : GradedSheaf (𝓐 := 𝓐) G} {d : ℤ}
+    (p : PerfectPairing T d) (j : Iso S T) : PerfectPairing S d where
+  pairing := (j.toIso.sections Set.univ).trans
+    (p.pairing.trans (j.toIso.sections Set.univ).dualMap)
+  homogeneous i q m n hm hn := p.homogeneous i q _ _
+    (fun x => j.vertex_graded x i _ (hm x)) (fun x => j.vertex_graded x q _ (hn x))
+  left_orthogonal x m n hm hn := by
+    apply p.left_orthogonal x
+    · intro y hy
+      change j.vertex y (m.val y)=0
+      rw [hm y hy,map_zero]
+    · change j.vertex x (n.val x)=0
+      rw [hn,map_zero]
+  right_orthogonal x m n hm hn := by
+    apply p.right_orthogonal x
+    · change j.vertex x (m.val x)=0
+      rw [hm,map_zero]
+    · intro y hy
+      change j.vertex y (n.val y)=0
+      rw [hn y hy,map_zero]
+
+ variable {B : Type ua} [CommRing B] [Algebra k B] {𝓑 : ℤ → Submodule k B}
+  (e : A ≃ₐ[k] B) (he : ∀ n a, a∈𝓐 n ↔ e a∈𝓑 n)
+
+ def PerfectPairing.coefficientChange {S : GradedSheaf (𝓐 := 𝓐) G} {d : ℤ}
+    (p : PerfectPairing S d) : PerfectPairing (coefficientChange e he S) d where
+  pairing := (coefficientChangeSections e he S Set.univ).symm.trans
+    ((CoefficientChange.pairingEquiv e p.pairing).trans
+      (coefficientChangeSections e he S Set.univ).symm.dualMap)
+  homogeneous i j m n hm hn := by
+    apply (he (i+j-d) _).mp
+    exact p.homogeneous i j _ _
+      (fun x => (ModuleData.mem_changePiece e (S.vertex x) i (m.val x)).mp (hm x))
+      (fun x => (ModuleData.mem_changePiece e (S.vertex x) j (n.val x)).mp (hn x))
+  left_orthogonal x m n hm hn := by
+    change e (p.pairing _ _)=0
+    apply (congrArg e ?_).trans (map_zero e)
+    apply p.left_orthogonal x
+    · intro y hy
+      exact congrArg (fun v : CoefficientChange e (S.vertex y) => v.val) (hm y hy)
+    · exact congrArg (fun v : CoefficientChange e (S.vertex x) => v.val) hn
+  right_orthogonal x m n hm hn := by
+    change e (p.pairing _ _)=0
+    apply (congrArg e ?_).trans (map_zero e)
+    apply p.right_orthogonal x
+    · exact congrArg (fun v : CoefficientChange e (S.vertex x) => v.val) hm
+    · intro y hy
+      exact congrArg (fun v : CoefficientChange e (S.vertex y) => v.val) (hn y hy)
+
+end
+end KLInvariance.MomentGraph.GradedSheaf
+
+end
+
+
+section
+
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/KazhdanLusztig/TerminalKernels.lean b/lean/OAI/RepresentationTheory/KazhdanLusztig/TerminalKernels.lean
index 183dc49dfe5fd7b77c5d1020655b8abca1cc720f..a7ef89fd9c0c8565fa6f3674915a65671f6eda3c 100644
--- a/lean/OAI/RepresentationTheory/KazhdanLusztig/TerminalKernels.lean
+++ b/lean/OAI/RepresentationTheory/KazhdanLusztig/TerminalKernels.lean
@@ -1,1410 +1,1410 @@
-import OAI.RepresentationTheory.KazhdanLusztig.PositiveTransfer
-
-namespace OAI
-
-/-!
-All-section transport, terminal kernels, scalar quotients, height-one membership in free targets and scaled edge-image inclusion.
--/
-
-section
-
-namespace KLInvariance.MomentGraph.Sheaf.Iso
-universe ur uv ue uz uz'
-variable {R : Type ur} [CommRing R] {V : Type uv} [PartialOrder V]
-  {E : Type ue} {G : OrderedGraph V E}
-  {B : Sheaf.{ur,uv,ue,uz} (R := R) G} {C : Sheaf.{ur,uv,ue,uz'} (R := R) G}
-  (j : Iso B C)
-
-include j in
- theorem scaledImages (e : E) (F : Set (Outgoing (G := G) (G.source e)))
-    (H : Set (Outgoing (G := G) (G.target e))) (r : R)
-    (h : B.ScaledImages e F H r) : C.ScaledImages e F H r := by
-  intro z
-  constructor
-  · rintro ⟨v,hv,he⟩
-    have hv' : (j.vertex (G.source e)).symm v ∈ B.upwardKernel (G.source e) F := by
-      apply (j.upwardKernel_iff _ _ _).mp
-      simpa only [LinearEquiv.apply_symm_apply] using hv
-    have he' : B.lower e ((j.vertex (G.source e)).symm v)=(j.edge e).symm z := by
-      apply (j.edge e).injective
-      rw [j.lower,LinearEquiv.apply_symm_apply,LinearEquiv.apply_symm_apply,he]
-    obtain ⟨w,hw,hew⟩ := (h ((j.edge e).symm z)).mp ⟨_,hv',he'⟩
-    refine ⟨j.vertex (G.target e) w,(j.upwardKernel_iff _ _ _).mpr hw,?_⟩
-    rw [← j.upper,← map_smul,hew,LinearEquiv.apply_symm_apply]
-  · rintro ⟨w,hw,he⟩
-    have hw' : (j.vertex (G.target e)).symm w ∈ B.upwardKernel (G.target e) H := by
-      apply (j.upwardKernel_iff _ _ _).mp
-      simpa only [LinearEquiv.apply_symm_apply] using hw
-    have he' : r • B.upper e ((j.vertex (G.target e)).symm w)=(j.edge e).symm z := by
-      apply (j.edge e).injective
-      rw [map_smul,j.upper,LinearEquiv.apply_symm_apply,LinearEquiv.apply_symm_apply,he]
-    obtain ⟨v,hv,hev⟩ := (h ((j.edge e).symm z)).mpr ⟨_,hw',he'⟩
-    refine ⟨j.vertex (G.source e) v,(j.upwardKernel_iff _ _ _).mpr hv,?_⟩
-    rw [← j.lower,hev,LinearEquiv.apply_symm_apply]
-
-end KLInvariance.MomentGraph.Sheaf.Iso
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.Sheaf
-open scoped Pointwise
-universe ur uv ue
-variable {R : Type ur} [CommRing R] [DecompositionMonoid R]
-  {V : Type uv} [PartialOrder V] {E : Type ue} [Fintype E]
-  (G : OrderedGraph V E) (α : E → R) (c : V)
-noncomputable section
-
-/-- A quotient-product equality gives the actual two embedded model images.
-The common quotient is the edge module's coefficient ring. -/
- theorem structure_scaledImages (e : E)
-    (F : Set (Outgoing (G := G) (G.source e)))
-    (H : Set (Outgoing (G := G) (G.target e))) (r : R)
-    (hF : (supportedOutgoing G c (G.source e) F : Set (Outgoing (G := G) (G.source e))).Pairwise
-      (Function.onFun IsRelPrime (fun a => α a.val)))
-    (hH : (supportedOutgoing G c (G.target e) H : Set (Outgoing (G := G) (G.target e))).Pairwise
-      (Function.onFun IsRelPrime (fun a => α a.val)))
-    (h : Associated
-      (algebraMap R (R ⧸ Ideal.span {α e}) (kernelProduct G α c (G.source e) F))
-      (algebraMap R (R ⧸ Ideal.span {α e}) (r*kernelProduct G α c (G.target e) H))) :
-    (structureSheaf G α c).ScaledImages e F H r := by
-  rw [scaledImages_iff,structure_lower_image (G := G) α c e F hF,
-    structure_upper_image (G := G) α c e H hH,← mul_smul]
-  let S := R ⧸ Ideal.span {α e}
-  let : Module S ((structureSheaf G α c).edge e) :=
-    inferInstanceAs (Module S (PLift (G.target e ≤ c) → S))
-  let : IsScalarTower R S ((structureSheaf G α c).edge e) :=
-    inferInstanceAs (IsScalarTower R S (PLift (G.target e ≤ c) → S))
-  exact LocalProducts.smul_top_eq_of_map_associated (S := S) h
-
- theorem structure_scaledImages_of_count (e : E)
-    (F : Set (Outgoing (G := G) (G.source e)))
-    (H : Set (Outgoing (G := G) (G.target e))) (r : R) (p : R ⧸ Ideal.span {α e}) (k : ℕ)
-    (hF : (supportedOutgoing G c (G.source e) F : Set (Outgoing (G := G) (G.source e))).Pairwise
-      (Function.onFun IsRelPrime (fun a => α a.val)))
-    (hH : (supportedOutgoing G c (G.target e) H : Set (Outgoing (G := G) (G.target e))).Pairwise
-      (Function.onFun IsRelPrime (fun a => α a.val)))
-    (ha : ∀ i ∈ supportedOutgoing G c (G.source e) F,
-      Associated (algebraMap R (R ⧸ Ideal.span {α e}) (α i.val)) p)
-    (hb : ∀ i ∈ supportedOutgoing G c (G.target e) H,
-      Associated (algebraMap R (R ⧸ Ideal.span {α e}) (α i.val)) p)
-    (hr : Associated (algebraMap R (R ⧸ Ideal.span {α e}) r) (p^k))
-    (hc : (supportedOutgoing G c (G.source e) F).card=
-      (supportedOutgoing G c (G.target e) H).card+k) :
-    (structureSheaf G α c).ScaledImages e F H r := by
-  apply structure_scaledImages G α c e F H r hF hH
-  simpa only [kernelProduct,map_prod,map_mul] using
-    LocalProducts.associated_prod_scale _ _ _ _ p _ k ha hb hr hc
-
-end
-end KLInvariance.MomentGraph.Sheaf
-
-end
-
-
-section
-
-namespace KLInvariance.TitsSpace.PlaneBasis
-open Module BruhatGraph MomentGraph MomentGraph.Sheaf
-universe u v us u' v'
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-  {σ : Type us} (basis : Basis σ ℝ (Extended M))
-  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
-  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
-  {a₀ u b : W} {u' b' : W'}
-  (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-  (φ : Interval cs u b ≃o Interval cs' u' b')
-noncomputable section
-
- theorem scheduledLower_root_ne (e : B.CosetEdge a₀ b)
-    (hx : BruhatLE cs u ((e.val.1.val:W)*a₀))
-    (f : Outgoing (G := B.cosetGraph a₀ b) e.val.1)
-    (hf : f ∈ B.scheduledLower hmin φ e hx) :
-    B.edgeRoot f.val.property ≠ B.edgeRoot e.property := by
-  intro he
-  have hroot : BruhatGraph.root cs 1 b (B.cosetEdgeAmbient a₀ b hmin f.val) =
-      BruhatGraph.root cs 1 b (B.cosetEdgeAmbient a₀ b hmin e) := by
-    rwa [B.cosetRoot_eq a₀ b hmin f.val,B.cosetRoot_eq a₀ b hmin e]
-  have hh : f.val=e := B.cosetEdgeAmbient_injective a₀ b hmin
-    (eq_of_label_source_eq cs 1 b (root_injective_on_labels cs 1 b hroot)
-      (congrArg (B.cosetAmbient a₀ b) f.property))
-  have hff : f=(⟨e,rfl⟩ : Outgoing (G := B.cosetGraph a₀ b) e.val.1) := Subtype.ext hh
-  exact B.scheduledLower_self hmin φ e hx (hff ▸ hf)
-
- theorem outgoingUpper_root_ne (e : B.CosetEdge a₀ b)
-    (f : Outgoing (G := B.cosetGraph a₀ b) e.val.2) :
-    B.edgeRoot f.val.property ≠ B.edgeRoot e.property := by
-  rcases f with ⟨⟨⟨s,t⟩,hst⟩,hs⟩
-  change s=e.val.2 at hs
-  subst s
-  exact (B.edgeRoot_ne_of_path e.property hst).symm
-
- theorem coset_structure_scaledImages [Fintype (B.CosetEdge a₀ b)]
-    (e : B.CosetEdge a₀ b) (hx : BruhatLE cs u ((e.val.1.val:W)*a₀))
-    (c : B.CosetVertex a₀ b) :
-    (structureSheaf (B.cosetGraph a₀ b) (B.cosetLabel basis a₀ b hmin) c).ScaledImages
-      e (B.scheduledLower hmin φ e hx) (B.scheduledUpper hmin φ e hx)
-      (algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P)
-        (edgeFactorLift cs basis ((e.val.1.val:W)*a₀) (B.edgeRoot e.property))) := by
-  classical
-  by_cases hyc : e.val.2 ≤ c
-  · let α := B.edgeRoot e.property
-    obtain ⟨β,hαβ,hβ⟩ : ∃ β : PositiveRoot M cs, α ≠ β ∧ β.val ∈ P := by
-      by_cases h : α=B.left
-      · exact ⟨B.right,h ▸ B.distinct,B.right_mem⟩
-      · exact ⟨B.left,h,B.left_mem⟩
-    have hplane : rootPlane α β=P :=
-      (rootPlane_eq_of_mem B.left B.right α β B.distinct
-        (B.plane_eq.ge (B.edgeRoot_plane e.property)) (B.plane_eq.ge hβ) hαβ).trans B.plane_eq
-    apply structure_scaledImages_of_count (B.cosetGraph a₀ b) (B.cosetLabel basis a₀ b hmin) c
-      e _ _ _ (B.reducedCosetRoot basis a₀ b hmin e β)
-      ((planeWeight ((e.val.1.val:W)*a₀) α P).toNat)
-      (B.coset_supported_pairwise basis a₀ b hmin c e.val.1 _)
-      (B.coset_supported_pairwise basis a₀ b hmin c e.val.2 _)
-    · intro f hf
-      exact B.cosetLabel_reduction_associated basis a₀ b hmin e f.val β hplane
-        (B.scheduledLower_root_ne hmin φ e hx f (Finset.mem_filter.mp hf).2.1)
-    · intro f _
-      exact B.cosetLabel_reduction_associated basis a₀ b hmin e f.val β hplane
-        (B.outgoingUpper_root_ne e f)
-    · exact B.cosetFactor_reduction_associated basis a₀ b hmin e β hαβ hplane
-    · exact B.supported_count_difference hmin φ e hx c hyc
-  · let := structureSheaf_edge_subsingleton (B.cosetGraph a₀ b)
-        (B.cosetLabel basis a₀ b hmin) c e hyc
-    intro z
-    constructor <;> intro _ <;> exact ⟨0,Submodule.zero_mem _,Subsingleton.elim _ _⟩
-
-end
-end KLInvariance.TitsSpace.PlaneBasis
-
-end
-
-
-section
-
-
-namespace KLInvariance.TitsSpace.PlaneBasis
-open Module BruhatGraph MomentGraph MomentGraph.Sheaf
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-  {σ : Type (max us v)} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  (a₀ b : W) (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-noncomputable section
-
- theorem localized_boundary_decomposition :
-    ∃ C : Sheaf.{max us v,v,v,max us v} (R := PlaneRing basis P) (B.cosetGraph a₀ b),
-      ModelSum (B.cosetModels basis a₀ b hmin).sheaf Set.univ C ∧
-      Nonempty (Iso (B.localizedCoset basis a₀ b hmin
-        (boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf.forget) C) := by
-  let D := boundarySheaf cs 1 b basis (one_bruhat cs b)
-  apply exists_modelSumIso (B.cosetModels basis a₀ b hmin)
-    (B.localizedCoset basis a₀ b hmin D.sheaf.forget)
-  · intro x
-    let : Module.Free (MvPolynomial σ ℝ)
-        (D.sheaf.forget.vertex ((B.cosetGraphMap a₀ b hmin).vertex x)) :=
-      D.free (B.cosetAmbient a₀ b x)
-    exact D.sheaf.forget.baseChange_vertex_free (PlaneRing basis P) _
-  · intro x
-    let : Module.Finite (MvPolynomial σ ℝ)
-        (D.sheaf.forget.vertex ((B.cosetGraphMap a₀ b hmin).vertex x)) :=
-      (D.sheaf.vertex (B.cosetAmbient a₀ b x)).finite
-    exact D.sheaf.forget.baseChange_vertex_finite (PlaneRing basis P) _
-  · exact B.localizedCoset_quotient basis a₀ b hmin D.sheaf.forget D.quotient
-  · exact B.localizedCoset_flabby basis a₀ b hmin D.sheaf.forget D.quotient D.flabby.flabby
-  · exact B.localizedCoset_generated basis a₀ b hmin D.sheaf.forget D.generated
-
-end
-end KLInvariance.TitsSpace.PlaneBasis
-
-end
-
-
-section
-
-
-namespace KLInvariance.TitsSpace.PlaneBasis
-open Module BruhatGraph MomentGraph MomentGraph.Sheaf
-universe u v us u' v'
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-  {σ : Type (max us v)} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
-  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
-  {a₀ u b : W} {u' b' : W'}
-  (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-  (φ : Interval cs u b ≃o Interval cs' u' b')
-noncomputable section
-
- theorem localized_boundary_scaledImages (e : B.CosetEdge a₀ b)
-    (hx : BruhatLE cs u ((e.val.1.val:W)*a₀)) :
-    (B.localizedCoset basis a₀ b hmin
-      (boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf.forget).ScaledImages
-      e (B.scheduledLower hmin φ e hx) (B.scheduledUpper hmin φ e hx)
-      (algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P)
-        (edgeFactorLift cs basis ((e.val.1.val:W)*a₀) (B.edgeRoot e.property))) := by
-  classical
-  let := Fintype.ofFinite (B.CosetVertex a₀ b)
-  let : Finite (B.CosetEdge a₀ b) := Finite.of_injective
-    (fun e : B.CosetEdge a₀ b => e.val) Subtype.val_injective
-  let := Fintype.ofFinite (B.CosetEdge a₀ b)
-  obtain ⟨C,hC,⟨j⟩⟩ := B.localized_boundary_decomposition basis a₀ b hmin
-  apply j.symm.scaledImages
-  apply hC.scaledImages
-  intro c _
-  exact B.coset_structure_scaledImages basis hmin φ e hx c
-
-end
-end KLInvariance.TitsSpace.PlaneBasis
-
-end
-
-
-section
-
-/-! Flat scalar extension of actual embedded kernel and image modules. -/
-namespace KLInvariance.FlatSubmodule
-open TensorProduct
-universe ur us um un
-variable {R : Type ur} [CommRing R] (S : Type us) [CommRing S] [Algebra R S]
-  {M : Type um} [AddCommGroup M] [Module R M]
-  {N : Type un} [AddCommGroup N] [Module R N]
-
- theorem baseChange_ker [Module.Flat R S] (f : M →ₗ[R] N) :
-    (LinearMap.ker f).baseChange S = LinearMap.ker (f.baseChange S) := by
-  ext z
-  exact (Module.Flat.lTensor_exact S f.exact_subtype_ker_map z).symm
-
- theorem baseChange_map (p : Submodule R M) (f : M →ₗ[R] N) :
-    (p.map f).baseChange S = (p.baseChange S).map (f.baseChange S) := by
-  apply le_antisymm
-  · rintro z ⟨k,rfl⟩
-    induction k using TensorProduct.inductionOn with
-    | tmul s x =>
-      obtain ⟨m,hm,hx⟩ := x.property
-      refine ⟨s ⊗ₜ m,Submodule.tmul_mem_baseChange_of_mem s hm,?_⟩
-      change s ⊗ₜ[R] f m = s ⊗ₜ[R] (x : N)
-      rw [hx]
-    | add x y hx hy => simpa only [map_add] using Submodule.add_mem _ hx hy
-  · rintro z ⟨w,⟨k,hk⟩,rfl⟩
-    subst w
-    induction k using TensorProduct.inductionOn with
-    | tmul s x =>
-      exact Submodule.tmul_mem_baseChange_of_mem s (show f x ∈ p.map f from ⟨x,x.property,rfl⟩)
-    | add x y hx hy => simpa only [map_add] using Submodule.add_mem _ hx hy
-
-end KLInvariance.FlatSubmodule
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.Sheaf
-open TensorProduct
-universe ur us uv ue um
-variable {R : Type ur} [CommRing R] (S : Type us) [CommRing S] [Algebra R S]
-  {V : Type uv} [PartialOrder V] {E : Type ue} {G : OrderedGraph V E}
-  (B : Sheaf.{ur,uv,ue,um} (R := R) G)
-noncomputable section
-
- def kernelConstraint (x : V) (F : Set (Outgoing (G := G) x)) :
-    B.vertex x →ₗ[R] (∀ e : F, B.edge e.val.val) :=
-  LinearMap.pi (fun e => (LinearMap.proj e.val).comp (B.stalkMap x))
-
- theorem ker_kernelConstraint (x : V) (F : Set (Outgoing (G := G) x)) :
-    LinearMap.ker (B.kernelConstraint x F) = B.upwardKernel x F := by
-  ext v
-  constructor
-  · intro h e he
-    exact congrFun (LinearMap.mem_ker.mp h) ⟨e,he⟩
-  · intro h
-    exact LinearMap.mem_ker.mpr (funext fun e => h e.val e.property)
-
- theorem kernelConstraint_tensor (x : V) (F : Set (Outgoing (G := G) x))
-    (z : S ⊗[R] B.vertex x) :
-    (B.baseChange S).kernelConstraint x F z =
-      TensorProduct.piRightHom R S S (fun e : F => B.edge e.val.val)
-        ((B.kernelConstraint x F).baseChange S z) := by
-  let k : (S ⊗[R] B.vertex x) →ₗ[S] (∀ e : F, S ⊗[R] B.edge e.val.val) :=
-    (B.baseChange S).kernelConstraint x F
-  change k z = _
-  induction z using TensorProduct.inductionOn with
-  | tmul s v =>
-    funext e
-    rcases e with ⟨⟨e,he⟩,heF⟩
-    subst x
-    rfl
-  | add z t hz ht =>
-    simp only [map_add,hz,ht]
-
- theorem baseChange_upwardKernel [Finite E] [Module.Flat R S]
-    (x : V) (F : Set (Outgoing (G := G) x)) :
-    (B.upwardKernel x F).baseChange S = (B.baseChange S).upwardKernel x F := by
-  classical
-  let := Fintype.ofFinite E
-  rw [← B.ker_kernelConstraint x F,FlatSubmodule.baseChange_ker,
-    ← (B.baseChange S).ker_kernelConstraint x F]
-  ext z
-  change (B.kernelConstraint x F).baseChange S z = 0 ↔
-    (B.baseChange S).kernelConstraint x F z = 0
-  rw [B.kernelConstraint_tensor S]
-  exact ((TensorProduct.piRight R S S (fun e : F => B.edge e.val.val)).map_eq_zero_iff).symm
-
- theorem baseChange_lowerImage [Finite E] [Module.Flat R S]
-    (e : E) (F : Set (Outgoing (G := G) (G.source e))) :
-    ((B.upwardKernel (G.source e) F).map (B.lower e)).baseChange S =
-      ((B.baseChange S).upwardKernel (G.source e) F).map ((B.baseChange S).lower e) := by
-  rw [FlatSubmodule.baseChange_map,B.baseChange_upwardKernel S]
-  rfl
-
- theorem baseChange_upperImage [Finite E] [Module.Flat R S]
-    (e : E) (F : Set (Outgoing (G := G) (G.target e))) :
-    ((B.upwardKernel (G.target e) F).map (B.upper e)).baseChange S =
-      ((B.baseChange S).upwardKernel (G.target e) F).map ((B.baseChange S).upper e) := by
-  rw [FlatSubmodule.baseChange_map,B.baseChange_upwardKernel S]
-  rfl
-
-end
-end KLInvariance.MomentGraph.Sheaf
-
-end
-
-
-section
-
-namespace KLInvariance.TitsSpace.PlaneBasis
-open Module BruhatGraph MomentGraph MomentGraph.Sheaf
-universe u v us um u' v'
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
-  {σ : Type us} (basis : Basis σ ℝ (Extended M))
-  {a₀ b : W}
-  (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-noncomputable section
-
- theorem localizedCoset_kernel
-    (C : Sheaf.{us,v,v,um} (R := MvPolynomial σ ℝ) (graph cs 1 b))
-    (hq : C.UpperQuotient (polynomialLabel cs 1 b basis))
-    (x : B.CosetVertex a₀ b) (T : Set (Edge cs 1 b)) :
-    (B.localizedCoset basis a₀ b hmin C).upwardKernel x
-      {e | B.cosetEdgeAmbient a₀ b hmin e.val ∈ T} =
-    (C.baseChange (PlaneRing basis P)).upwardKernel (B.cosetAmbient a₀ b x) {e | e.val ∈ T} := by
-  apply (C.baseChange (PlaneRing basis P)).pullback_kernel (B.cosetGraphMap a₀ b hmin) x T
-    {e | label cs 1 b e ∈ planeSubgroup M cs P}
-  · intro e he
-    exact (C.baseChange (PlaneRing basis P)).edge_subsingleton_of_unit _
-      (C.baseChange_upperQuotient (PlaneRing basis P) _ hq) e
-      ((polynomialLabel_unit_iff (P := P) basis b e).mpr he)
-  · intro e he hx
-    exact B.coset_lift_live a₀ b hmin e he x hx
-
-end
-
-variable {σ : Type (max us v)} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
-  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
-  {u : W} {u' b' : W'}
-  (φ : Interval cs u b ≃o Interval cs' u' b')
-noncomputable section
-
- theorem ambient_boundary_scaledImages (e : B.CosetEdge a₀ b)
-    (hx : BruhatLE cs u ((e.val.1.val:W)*a₀)) :
-    let D := (boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf.forget
-    let T := ambientLater φ (B.mappedCosetRoot hmin φ e hx)
-    (D.baseChange (PlaneRing basis P)).ScaledImages (B.cosetEdgeAmbient a₀ b hmin e)
-      {f | f.val ∈ T} {f | f.val ∈ T}
-      (algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P)
-        (edgeFactorLift cs basis ((e.val.1.val:W)*a₀) (B.edgeRoot e.property))) := by
-  let D := boundarySheaf cs 1 b basis (one_bruhat cs b)
-  have h := B.localized_boundary_scaledImages basis hmin φ e hx
-  dsimp only
-  intro z
-  have hh := h z
-  change (∃ v ∈ (B.localizedCoset basis a₀ b hmin D.sheaf.forget).upwardKernel e.val.1
-      (B.scheduledOutgoing hmin φ _ e.val.1 hx), _) ↔
-    ∃ w ∈ (B.localizedCoset basis a₀ b hmin D.sheaf.forget).upwardKernel e.val.2
-      (B.scheduledOutgoing hmin φ _ e.val.2 (B.cosetEdge_target_upper hmin e hx)), _ at hh
-  rw [B.scheduledOutgoing_eq hmin φ,B.scheduledOutgoing_eq hmin φ,
-    B.localizedCoset_kernel basis hmin D.sheaf.forget D.quotient,
-    B.localizedCoset_kernel basis hmin D.sheaf.forget D.quotient] at hh
-  exact hh
-
-end
-end KLInvariance.TitsSpace.PlaneBasis
-
-end
-
-
-section
-
-/-! The local embedded image identity at every actual root-plane prime.
-The edge is arbitrary in [1,b], with source in the comparison interval. -/
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace MomentGraph MomentGraph.Sheaf
-universe u v us u' v'
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {σ : Type (max us v)} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
-  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
-  {u b : W} {u' b' : W'}
-  (φ : Interval cs u b ≃o Interval cs' u' b')
-noncomputable section
-
- theorem boundary_plane_scaledImages (e : Edge cs 1 b)
-    (hx : BruhatLE cs u (source cs 1 b e).val)
-    (β : PositiveRoot M cs) (hβ : root cs 1 b e ≠ β) :
-    let P := rootPlane (root cs 1 b e) β
-    let D := (boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf.forget
-    let T := ambientLater φ (mappedEdgeRoot φ e hx)
-    (D.baseChange (PlaneRing basis P)).ScaledImages e {f | f.val ∈ T} {f | f.val ∈ T}
-      (algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P)
-        (edgeFactorLift cs basis (source cs 1 b e).val (root cs 1 b e))) := by
-  classical
-  let P := rootPlane (root cs 1 b e) β
-  let B := (nonempty_planeBasis (root cs 1 b e) β hβ).some
-  obtain ⟨a₀,hmin,d,hd⟩ := B.exists_coset_edge e (Submodule.subset_span (by simp))
-  have hdx : BruhatLE cs u ((d.val.1.val:W)*a₀) := by
-    change BruhatLE cs u (source cs 1 b (B.cosetEdgeAmbient a₀ b hmin d)).val
-    rwa [hd]
-  have h := B.ambient_boundary_scaledImages hmin basis φ d hdx
-  dsimp only at h ⊢
-  rw [B.mappedCosetRoot_eq hmin φ d hdx,
-    show B.edgeRoot d.property=root cs 1 b (B.cosetEdgeAmbient a₀ b hmin d) from
-      (B.cosetRoot_eq a₀ b hmin d).symm] at h
-  change ((boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf.forget.baseChange
-    (PlaneRing basis P)).ScaledImages (B.cosetEdgeAmbient a₀ b hmin d)
-    {f | f.val ∈ ambientLater φ (mappedEdgeRoot φ (B.cosetEdgeAmbient a₀ b hmin d) hdx)}
-    {f | f.val ∈ ambientLater φ (mappedEdgeRoot φ (B.cosetEdgeAmbient a₀ b hmin d) hdx)}
-    (algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P)
-      (edgeFactorLift cs basis (source cs 1 b (B.cosetEdgeAmbient a₀ b hmin d)).val
-        (root cs 1 b (B.cosetEdgeAmbient a₀ b hmin d)))) at h
-  have h' := h
-  simp only [hd] at h'
-  exact Eq.mp (congrArg (fun z : Edge cs 1 b =>
-    ((boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf.forget.baseChange
-      (PlaneRing basis P)).ScaledImages z
-      {f | f.val ∈ ambientLater φ (mappedEdgeRoot φ e hx)}
-      {f | f.val ∈ ambientLater φ (mappedEdgeRoot φ e hx)}
-      (algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P)
-        (edgeFactorLift cs basis (source cs 1 b e).val (root cs 1 b e)))) hd) h'
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace EdgeAlgebra
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- abbrev EdgeRing (a : PositiveRoot M cs) :=
-   MvPolynomial σ ℝ ⧸ Ideal.span {polynomialRoot cs basis a}
-
- def edgeRingEquiv (a : PositiveRoot M cs) : EdgeRing cs basis a ≃ₐ[ℝ] EdgeCoefficient a :=
-   (Ideal.quotientEquivAlg (rootIdeal (k := ℝ) (embed M a.val))
-     (Ideal.span {polynomialRoot cs basis a}) (SymmetricAlgebra.equivMvPolynomial basis)
-     (by simp only [rootIdeal,Ideal.map_span,Set.image_singleton,polynomialRoot]; rfl)).symm
-
- @[simp] theorem edgeRingEquiv_root (a β : PositiveRoot M cs) :
-    edgeRingEquiv cs basis a (Ideal.Quotient.mk _ (polynomialRoot cs basis β)) =
-      Ideal.Quotient.mk _ (SymmetricAlgebra.ι ℝ (Extended M) (embed M β.val)) := by
-  apply (edgeRingEquiv cs basis a).symm.injective
-  simp only [AlgEquiv.symm_apply_apply]
-  rfl
-
- instance edgeRing_isDomain (a : PositiveRoot M cs) : IsDomain (EdgeRing cs basis a) := by
-   let : (Ideal.span {polynomialRoot cs basis a}).IsPrime :=
-     Ideal.isPrime_span_singleton_of_prime (polynomialRoot_prime cs basis a)
-   infer_instance
-
- instance edgeRing_ufm (a : PositiveRoot M cs) : UniqueFactorizationMonoid (EdgeRing cs basis a) :=
-   ((edgeRingEquiv cs basis a).trans (edgePolynomial a)).toMulEquiv.symm.uniqueFactorizationMonoid inferInstance
-
- theorem reducedPolynomialRoot_prime (a β : PositiveRoot M cs) (hne : a≠β) :
-    Prime (Ideal.Quotient.mk (Ideal.span {polynomialRoot cs basis a})
-      (polynomialRoot cs basis β)) := by
-  apply (MulEquiv.prime_iff (edgeRingEquiv cs basis a)).mp
-  rw [edgeRingEquiv_root]
-  exact reducedRoot_prime _ _ (embed_not_rootLine a β hne.symm)
-
- theorem planePrime_span (a β : PositiveRoot M cs) :
-    planePrime basis (rootPlane a β) =
-      Ideal.span {polynomialRoot cs basis a,polynomialRoot cs basis β} := by
-  simp only [planePrime,rootPlane,Submodule.map_span,Set.image_insert_eq,
-    Set.image_singleton,SymmetricPlaneIdeal.ideal_span_eq,Ideal.map_span,
-    polynomialRoot,embedLinear]
-  rfl
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-/-! The scalar quotient of an actual finite free presentation is free over
-its quotient coefficient ring. There is no quotient-freeness hypothesis. -/
-namespace KLInvariance.ScalarQuotient
-open Module
-universe ur um un ui
-variable {R : Type ur} [CommRing R]
-  {M : Type um} [AddCommGroup M] [Module R M]
-  {N : Type un} [AddCommGroup N] [Module R N]
-  {ι : Type ui} [Fintype ι]
-noncomputable section
-
- def coordinateReduction (b : Basis ι R M) (a : R) :
-    M →ₗ[R] (ι → R ⧸ Ideal.span {a}) :=
-  (LinearMap.pi fun i => (Ideal.span ({a} : Set R)).mkQ.comp
-    ((LinearMap.proj i).comp b.equivFun.toLinearMap))
-
- theorem coordinateReduction_surjective (b : Basis ι R M) (a : R) :
-    Function.Surjective (coordinateReduction b a) := by
-  intro v
-  choose w hw using fun i => Ideal.Quotient.mk_surjective (v i)
-  exact ⟨b.equivFun.symm w,funext fun i => by
-    change Ideal.Quotient.mk (Ideal.span {a}) (b.equivFun (b.equivFun.symm w) i)=v i
-    simpa only [LinearEquiv.apply_symm_apply] using hw i⟩
-
- theorem coordinateReduction_kernel (b : Basis ι R M) (a : R) (v : M) :
-    coordinateReduction b a v=0 ↔ ∃ w, v=a • w := by
-  have h : (∃ w, v=a • w) ↔ ∀ i, a ∣ b.equivFun v i :=
-    (exists_congr fun _ => eq_comm).trans (FiniteFreeScalar.exists_smul_iff b a v)
-  rw [h]
-  simp only [coordinateReduction,LinearMap.pi_apply,LinearMap.comp_apply,
-    LinearMap.proj_apply,LinearEquiv.coe_coe,funext_iff,Pi.zero_apply]
-  exact forall_congr' fun i => (Ideal.Quotient.eq_zero_iff_mem).trans Ideal.mem_span_singleton
-
- theorem free_of_scalar_presentation (a : R)
-    [Module (R ⧸ Ideal.span {a}) N] [IsScalarTower R (R ⧸ Ideal.span {a}) N]
-    (b : Basis ι R M) (q : M →ₗ[R] N) (hq : Function.Surjective q)
-    (hk : ∀ v, q v=0 ↔ ∃ w, v=a • w) :
-    Module.Free (R ⧸ Ideal.span {a}) N := by
-  obtain ⟨j,_⟩ := QuotientEdge.exists_scalar_iso q (coordinateReduction b a) a
-    hq (coordinateReduction_surjective b a) hk (coordinateReduction_kernel b a)
-    (LinearEquiv.refl R M)
-  let k : N ≃ₗ[R ⧸ Ideal.span {a}] (ι → R ⧸ Ideal.span {a}) :=
-    { j.toAddEquiv with
-      map_smul' := by
-        intro c v
-        obtain ⟨r,rfl⟩ := Ideal.Quotient.mk_surjective c
-        change j ((algebraMap R (R ⧸ Ideal.span {a}) r) • v) =
-          (algebraMap R (R ⧸ Ideal.span {a}) r) • j v
-        simpa only [algebraMap_smul] using j.map_smul r v }
-  exact Module.Free.of_equiv k.symm
-
- def submodule (a : R) [Module (R ⧸ Ideal.span {a}) N]
-    [IsScalarTower R (R ⧸ Ideal.span {a}) N] (S : Submodule R N) :
-    Submodule (R ⧸ Ideal.span {a}) N where
-  carrier := S
-  add_mem' := S.add_mem
-  zero_mem' := S.zero_mem
-  smul_mem' c v hv := by
-    obtain ⟨r,rfl⟩ := Ideal.Quotient.mk_surjective c
-    change (algebraMap R (R ⧸ Ideal.span {a}) r) • v ∈ S
-    simpa only [algebraMap_smul] using S.smul_mem r hv
-
- theorem saturated_image_free (a : R)
-    [Module (R ⧸ Ideal.span {a}) N] [IsScalarTower R (R ⧸ Ideal.span {a}) N]
-    (q : M →ₗ[R] N) (hk : ∀ v, q v=0 ↔ ∃ w, v=a • w)
-    (L : Submodule R M) [Module.Free R L] [Module.Finite R L]
-    (hsat : ∀ v, a • v ∈ L → v ∈ L) :
-    Module.Free (R ⧸ Ideal.span {a}) (submodule a (L.map q)) := by
-  let f : L →ₗ[R] submodule a (L.map q) :=
-    (q.comp L.subtype).codRestrict ((submodule a (L.map q)).restrictScalars R)
-      (fun v => ⟨v.val,v.property,rfl⟩)
-  apply free_of_scalar_presentation a (Module.Free.chooseBasis R L) f
-  · rintro ⟨v,hv⟩
-    obtain ⟨w,hw,rfl⟩ := hv
-    exact ⟨⟨w,hw⟩,rfl⟩
-  · intro v
-    constructor
-    · intro hv
-      obtain ⟨w,hw⟩ := (hk v.val).mp (congrArg Subtype.val hv)
-      exact ⟨⟨w,hsat w (hw ▸ v.property)⟩,Subtype.ext hw⟩
-    · rintro ⟨w,rfl⟩
-      exact Subtype.ext ((hk _).mpr ⟨w.val,rfl⟩)
-
-end
-end KLInvariance.ScalarQuotient
-
-end
-
-
-section
-
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
-  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  (hub : BruhatLE cs u b)
-  (B : BoundarySheaf (polynomialGrading_negative ℝ σ)
-    (graph cs u b) (polynomialLabel cs u b basis) ⟨b,hub,bruhat_refl cs b⟩)
-noncomputable section
-
-omit [Fintype σ] in
- theorem edgeTorsion (e : Edge cs u b) :
-    Module.IsTorsionBy (MvPolynomial σ ℝ) (B.sheaf.edge e) (polynomialRoot cs basis (root cs u b e)) :=
-   fun v => GradedSheaf.UpperQuotient.annihilates B.sheaf B.quotient e v
-
- @[instance_reducible] def edgeModule (e : Edge cs u b) :
-    Module (EdgeRing cs basis (root cs u b e)) (B.sheaf.edge e) :=
-   (edgeTorsion cs u b basis hub B e).module
-
-omit [Fintype σ] in
- theorem edgeModule_tower (e : Edge cs u b) :
-    let _ := edgeModule cs u b basis hub B e
-    IsScalarTower (MvPolynomial σ ℝ) (EdgeRing cs basis (root cs u b e)) (B.sheaf.edge e) := by
-   dsimp only [edgeModule,Module.IsTorsionBy.module]
-   exact Module.IsTorsionBySet.isScalarTower _
-
-omit [Fintype σ] in
- theorem edgeModule_free (e : Edge cs u b) :
-    let _ := edgeModule cs u b basis hub B e
-    Module.Free (EdgeRing cs basis (root cs u b e)) (B.sheaf.edge e) := by
-   let _ := edgeModule cs u b basis hub B e
-   let _ := edgeModule_tower cs u b basis hub B e
-   let _ := B.free (target cs u b e)
-   exact ScalarQuotient.free_of_scalar_presentation _
-     (Module.Free.chooseBasis (MvPolynomial σ ℝ) (B.sheaf.vertex (target cs u b e)))
-     (B.sheaf.upper e).map (B.quotient e).1 (B.quotient e).2
-
- theorem upperImage_free (e : Edge cs u b)
-    (F : Set (Sheaf.Outgoing (G := graph cs u b) (target cs u b e)))
-    [Module.Free (MvPolynomial σ ℝ) (B.sheaf.upwardSubmodule (target cs u b e) F)] :
-    let _ := edgeModule cs u b basis hub B e
-    let _ := edgeModule_tower cs u b basis hub B e
-    Module.Free (EdgeRing cs basis (root cs u b e))
-      (ScalarQuotient.submodule (polynomialRoot cs basis (root cs u b e))
-        ((B.sheaf.upwardSubmodule (target cs u b e) F).map (B.sheaf.upper e).map)) := by
-   let _ := edgeModule cs u b basis hub B e
-   let _ := edgeModule_tower cs u b basis hub B e
-   let _ : Module.Finite (MvPolynomial σ ℝ) (B.sheaf.vertex (target cs u b e)) :=
-     (B.sheaf.vertex (target cs u b e)).finite
-   let hfinite : Module.Finite (MvPolynomial σ ℝ) (B.sheaf.upwardSubmodule (target cs u b e) F) :=
-     Module.Finite.of_injective (B.sheaf.upwardSubmodule (target cs u b e) F).subtype Subtype.val_injective
-   dsimp only
-   let hfree := (inferInstance : Module.Free (MvPolynomial σ ℝ) (B.sheaf.upwardSubmodule (target cs u b e) F))
-   exact @ScalarQuotient.saturated_image_free (MvPolynomial σ ℝ) _
-     (B.sheaf.vertex (target cs u b e)) _ _ (B.sheaf.edge e) _ _
-     (polynomialRoot cs basis (root cs u b e)) _ _
-     (B.sheaf.upper e).map (B.quotient e).2 (B.sheaf.upwardSubmodule (target cs u b e) F)
-     hfree hfinite (by
-
-     exact B.sheaf.upwardSubmodule_saturated (target cs u b e) F _
-            (fun f _ => incoming_label_regular_outgoing cs u b basis hub B e f.val f.property))
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace EdgeAlgebra
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- theorem reducedRoot_dvd_iff_mem_plane (a β : PositiveRoot M cs) (z : MvPolynomial σ ℝ) :
-    Ideal.Quotient.mk (Ideal.span {polynomialRoot cs basis a}) (polynomialRoot cs basis β) ∣
-      Ideal.Quotient.mk (Ideal.span {polynomialRoot cs basis a}) z ↔
-    z ∈ planePrime basis (rootPlane a β) := by
-  rw [← Ideal.mem_span_singleton]
-  change z ∈ Ideal.comap (Ideal.Quotient.mk (Ideal.span {polynomialRoot cs basis a}))
-    (Ideal.span {Ideal.Quotient.mk (Ideal.span {polynomialRoot cs basis a})
-      (polynomialRoot cs basis β)}) ↔ _
-  have hi : Ideal.map (Ideal.Quotient.mk (Ideal.span {polynomialRoot cs basis a}))
-      (Ideal.span {polynomialRoot cs basis β}) =
-      Ideal.span {Ideal.Quotient.mk (Ideal.span {polynomialRoot cs basis a})
-        (polynomialRoot cs basis β)} := by rw [Ideal.map_span,Set.image_singleton]
-  rw [← hi,Ideal.comap_map_quotientMk,← Ideal.span_union,Set.singleton_union,planePrime_span]
-
- theorem reducedFactor_ne_zero (x : W) (a : PositiveRoot M cs) :
-    Ideal.Quotient.mk (Ideal.span {polynomialRoot cs basis a})
-      (edgeFactorLift cs basis x a) ≠ 0 := by
-  simpa only [ne_eq,Ideal.Quotient.eq_zero_iff_mem,Ideal.mem_span_singleton] using
-    edgeFactorLift_not_dvd cs basis x a
-
- theorem prime_dvd_reducedFactor (x : W) (a : PositiveRoot M cs)
-    (p : EdgeRing cs basis a) (hp : Prime p)
-    (hd : p ∣ Ideal.Quotient.mk _ (edgeFactorLift cs basis x a)) :
-    ∃ β : PositiveRoot M cs, a ≠ β ∧
-      Associated p (Ideal.Quotient.mk _ (polynomialRoot cs basis β)) := by
-  classical
-  simp only [edgeFactorLift,map_prod,map_pow] at hd
-  obtain ⟨P,_,hP⟩ := (hp.dvd_finsetProd_iff _).mp hd
-  exact ⟨planeRoot x a P,(planeRoot_ne x a P).symm,
-    hp.irreducible.associated_of_dvd
-      (reducedPolynomialRoot_prime cs basis a _ (planeRoot_ne x a P).symm).irreducible
-      (hp.dvd_of_dvd_pow hP)⟩
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.Sheaf
-universe ur uv ue um
-variable {R : Type ur} [CommRing R] {V : Type uv} [PartialOrder V]
-  {E : Type ue} [Finite E] {G : OrderedGraph V E}
-  (B : Sheaf.{ur,uv,ue,um} (R := R) G)
-noncomputable section
-open scoped Classical
-
- def outgoingLabelProduct (α : E → R) (x : V) (F : Set (Outgoing (G := G) x)) : R :=
-   ∏ f ∈ F.toFinite.toFinset, α f.val
-
- theorem outgoingLabelProduct_dvd (α : E → R) (x : V) (F : Set (Outgoing (G := G) x))
-    (f : Outgoing (G := G) x) (hf : f ∈ F) : α f.val ∣ outgoingLabelProduct α x F := by
-   exact Finset.dvd_prod_of_mem _ (F.toFinite.mem_toFinset.mpr hf)
-
- theorem outgoingLabelProduct_smul_mem (α : E → R) (hα : ∀ e (v : B.edge e), α e • v=0)
-    (x : V) (F : Set (Outgoing (G := G) x)) (v : B.vertex x) :
-    outgoingLabelProduct α x F • v ∈ B.upwardKernel x F := by
-   intro f hf
-   obtain ⟨r,hr⟩ := outgoingLabelProduct_dvd α x F f hf
-   rw [map_smul,hr,mul_comm,mul_smul]
-   have h := hα f.val (B.stalkMap x v f)
-   change r • (α f.val • B.stalkMap x v f) = 0
-   rw [h,smul_zero]
-
- theorem outgoingLabelProduct_smul_upperImage (α : E → R) (hα : ∀ e (v : B.edge e), α e • v=0)
-    (e : E) (hq : Function.Surjective (B.upper e))
-    (F : Set (Outgoing (G := G) (G.target e))) (v : B.edge e) :
-    outgoingLabelProduct α (G.target e) F • v ∈
-      (B.upwardKernel (G.target e) F).map (B.upper e) := by
-   obtain ⟨w,rfl⟩ := hq v
-   exact ⟨_,B.outgoingLabelProduct_smul_mem α hα _ F w,(B.upper e).map_smul _ w⟩
-
-end
-end KLInvariance.MomentGraph.Sheaf
-
-end
-
-
-section
-
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace MomentGraph MomentGraph.Sheaf
-universe u v us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
-  {σ : Type us} (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- theorem incoming_root_ne_outgoing (e f : Edge cs u b)
-    (h : source cs u b f=target cs u b e) : root cs u b e ≠ root cs u b f := by
-  intro hr
-  exact source_ne_target_of_label_eq cs u b (root_injective_on_labels cs u b hr.symm) h
-
- theorem reducedKernelProduct_ne_zero (e : Edge cs u b)
-    (F : Set (Outgoing (G := graph cs u b) (target cs u b e))) :
-    Ideal.Quotient.mk (Ideal.span {polynomialRoot cs basis (root cs u b e)})
-      (outgoingLabelProduct (polynomialLabel cs u b basis) (target cs u b e) F) ≠ 0 := by
-  classical
-  rw [outgoingLabelProduct,map_prod]
-  apply Finset.prod_ne_zero_iff.mpr
-  intro f _
-  exact (reducedPolynomialRoot_prime cs basis _ _
-    (incoming_root_ne_outgoing cs u b e f.val f.property)).ne_zero
-
- theorem prime_dvd_reducedKernelProduct (e : Edge cs u b)
-    (F : Set (Outgoing (G := graph cs u b) (target cs u b e)))
-    (p : EdgeRing cs basis (root cs u b e)) (hp : Prime p)
-    (hd : p ∣ Ideal.Quotient.mk _
-      (outgoingLabelProduct (polynomialLabel cs u b basis) (target cs u b e) F)) :
-    ∃ β : PositiveRoot M cs, root cs u b e ≠ β ∧
-      Associated p (Ideal.Quotient.mk _ (polynomialRoot cs basis β)) := by
-  classical
-  rw [outgoingLabelProduct,map_prod] at hd
-  obtain ⟨f,_,hf⟩ := (hp.dvd_finsetProd_iff _).mp hd
-  exact ⟨root cs u b f.val,incoming_root_ne_outgoing cs u b e f.val f.property,
-    hp.irreducible.associated_of_dvd
-      (reducedPolynomialRoot_prime cs basis _ _
-        (incoming_root_ne_outgoing cs u b e f.val f.property)).irreducible hf⟩
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-namespace KLInvariance.ScalarQuotient
-universe ur un
-variable {R : Type ur} [CommRing R]
-  {N : Type un} [AddCommGroup N] [Module R N]
-noncomputable section
-
- theorem scalarImage (a z : R)
-    [Module (R ⧸ Ideal.span {a}) N] [IsScalarTower R (R ⧸ Ideal.span {a}) N]
-    (S : Submodule R N) :
-    submodule a (S.map (LinearMap.lsmul R N z)) =
-      (submodule a S).map
-        (LinearMap.lsmul (R ⧸ Ideal.span {a}) N (Ideal.Quotient.mk _ z)) := by
-  ext v
-  constructor
-  · rintro ⟨w,hw,rfl⟩
-    exact ⟨w,hw,by simp only [LinearMap.lsmul_apply]; exact algebraMap_smul _ _ w⟩
-  · rintro ⟨w,hw,rfl⟩
-    exact ⟨w,hw,by simp only [LinearMap.lsmul_apply]; exact (algebraMap_smul _ _ w).symm⟩
-
- theorem free_scalarImage (a z : R)
-    [Module (R ⧸ Ideal.span {a}) N] [IsScalarTower R (R ⧸ Ideal.span {a}) N]
-    (S : Submodule R N) [Module.Free (R ⧸ Ideal.span {a}) (submodule a S)]
-    (hz : Function.Injective (fun v : N => z • v)) :
-    Module.Free (R ⧸ Ideal.span {a}) (submodule a (S.map (LinearMap.lsmul R N z))) := by
-  rw [scalarImage]
-  apply Module.Free.of_equiv (Submodule.equivMapOfInjective _ ?_ (submodule a S))
-  intro v w h
-  apply hz
-  change (algebraMap R (R ⧸ Ideal.span {a}) z) • v =
-    (algebraMap R (R ⧸ Ideal.span {a}) z) • w at h
-  simpa only [algebraMap_smul] using h
-
-end
-end KLInvariance.ScalarQuotient
-
-end
-
-
-section
-
-namespace KLInvariance.TensorLocalMembership
-open TensorProduct
-universe ur us um
-variable {R : Type ur} [CommRing R] (S : Type us) [CommRing S] [Algebra R S]
-  {M : Type um} [AddCommGroup M] [Module R M]
-  (P : Submonoid R) [IsLocalization P S]
-noncomputable section
-
- theorem baseChange_eq_localized (L : Submodule R M) :
-    L.baseChange S = L.localized' S P (TensorProduct.mk R S M 1) := by
-  rw [Submodule.baseChange_eq_span,Submodule.localized'_eq_span]
-  rfl
-
- theorem denominator (L : Submodule R M) (x : M)
-    (hx : 1 ⊗ₜ[R] x ∈ L.baseChange S) :
-    ∃ a : P, (a:R) • x ∈ L := by
-  rw [baseChange_eq_localized S P L] at hx
-  obtain ⟨m,hm,s,hs⟩ := hx
-  have he := (IsLocalizedModule.mk'_eq_iff).mp hs
-  change (TensorProduct.mk R S M 1) m = (s:R) • (TensorProduct.mk R S M 1) x at he
-  rw [← map_smul] at he
-  obtain ⟨t,ht⟩ := (IsLocalizedModule.eq_iff_exists P (TensorProduct.mk R S M 1)).mp he
-  refine ⟨t*s,?_⟩
-  have hh := L.smul_mem (t:R) hm
-  change (t:R) • m ∈ L at hh
-  change (t:R) • m = (t:R) • ((s:R) • x) at ht
-  simpa only [Submonoid.coe_mul,mul_smul,← ht] using hh
-
-end
-end KLInvariance.TensorLocalMembership
-
-end
-
-
-section
-
-namespace KLInvariance.MomentGraph.Sheaf
-open TensorProduct
-universe ur us uv ue um
-variable {R : Type ur} [CommRing R] (S : Type us) [CommRing S] [Algebra R S]
-  {V : Type uv} [PartialOrder V] {E : Type ue} [Finite E] {G : OrderedGraph V E}
-  (B : Sheaf.{ur,uv,ue,um} (R := R) G)
-  (P : Submonoid R) [IsLocalization P S]
-noncomputable section
-
- theorem scaledImages_denominator
-    (e : E) (F : Set (Outgoing (G := G) (G.source e)))
-    (H : Set (Outgoing (G := G) (G.target e))) (a : R)
-    (h : (B.baseChange S).ScaledImages e F H (algebraMap R S a))
-    (x : B.edge e) (hx : x ∈ (B.upwardKernel (G.source e) F).map (B.lower e)) :
-    ∃ s : P, (s:R) • x ∈ ((B.upwardKernel (G.target e) H).map (B.upper e)).map
-      (LinearMap.lsmul R (B.edge e) a) := by
-  let _ : Module.Flat R S := IsLocalization.flat S P
-  apply TensorLocalMembership.denominator S P _ x
-  rw [FlatSubmodule.baseChange_map,B.baseChange_upperImage S]
-  have hx' : 1 ⊗ₜ[R] x ∈ ((B.baseChange S).upwardKernel (G.source e) F).map
-      ((B.baseChange S).lower e) := by
-    rw [← B.baseChange_lowerImage S]
-    exact Submodule.tmul_mem_baseChange_of_mem 1 hx
-  obtain ⟨v,hv,he⟩ := hx'
-  obtain ⟨w,hw,hwx⟩ := (h (1 ⊗ₜ[R] x)).mp ⟨v,hv,he⟩
-  refine ⟨(B.baseChange S).upper e w,⟨w,hw,rfl⟩,?_⟩
-  change (LinearMap.lsmul R (B.edge e) a).baseChange S ((B.baseChange S).upper e w) = _
-  have hg : (LinearMap.lsmul R (B.edge e) a).baseChange S =
-      LinearMap.lsmul S (S ⊗[R] B.edge e) (algebraMap R S a) := by
-    apply LinearMap.restrictScalars_injective R
-    apply TensorProduct.ext'
-    intro r v
-    change (LinearMap.lsmul R (B.edge e) a).baseChange S (r ⊗ₜ[R] v) =
-      (algebraMap R S a) • (r ⊗ₜ[R] v)
-    simp only [LinearMap.baseChange_tmul,LinearMap.lsmul_apply,
-      algebraMap_smul,TensorProduct.tmul_smul,TensorProduct.smul_tmul']
-  rw [hg]
-  exact hwx
-
-end
-end KLInvariance.MomentGraph.Sheaf
-
-end
-
-
-section
-
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace
-universe u v us un
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-  {σ : Type us} (basis : Basis σ ℝ (Extended M))
-noncomputable section
-
- theorem mem_of_plane_denominators (a : PositiveRoot M cs)
-    {N : Type un} [AddCommGroup N] [Module (MvPolynomial σ ℝ) N]
-    [Module (EdgeRing cs basis a) N] [IsScalarTower (MvPolynomial σ ℝ) (EdgeRing cs basis a) N]
-    [Module.IsTorsionFree (EdgeRing cs basis a) N]
-    (S : Submodule (MvPolynomial σ ℝ) N)
-    [Module.Free (EdgeRing cs basis a) (ScalarQuotient.submodule (polynomialRoot cs basis a) S)]
-    (z : N) (c : MvPolynomial σ ℝ)
-    (hc : Ideal.Quotient.mk (Ideal.span {polynomialRoot cs basis a}) c ≠ 0)
-    (hcz : c • z ∈ S)
-    (hprime : ∀ p : EdgeRing cs basis a, Prime p → p ∣ Ideal.Quotient.mk _ c →
-      ∃ β : PositiveRoot M cs, a≠β ∧ Associated p (Ideal.Quotient.mk _ (polynomialRoot cs basis β)))
-    (hlocal : ∀ β : PositiveRoot M cs, a≠β →
-      ∃ s : MvPolynomial σ ℝ, s ∉ planePrime basis (rootPlane a β) ∧ s • z ∈ S) :
-    z ∈ S := by
-  apply EdgeLattice.mem_of_prime_local_membership
-    (ScalarQuotient.submodule (polynomialRoot cs basis a) S) z
-  · refine ⟨Ideal.Quotient.mk _ c,hc,?_⟩
-    change (algebraMap (MvPolynomial σ ℝ) (EdgeRing cs basis a) c) • z ∈ S
-    simpa only [algebraMap_smul] using hcz
-  · intro p hp
-    by_cases hpc : p ∣ Ideal.Quotient.mk (Ideal.span {polynomialRoot cs basis a}) c
-    · obtain ⟨β,hβ,hpβ⟩ := hprime p hp hpc
-      obtain ⟨s,hs,hsz⟩ := hlocal β hβ
-      refine ⟨Ideal.Quotient.mk _ s,?_,?_⟩
-      · rw [hpβ.dvd_iff_dvd_left,reducedRoot_dvd_iff_mem_plane]
-        exact hs
-      · change (algebraMap (MvPolynomial σ ℝ) (EdgeRing cs basis a) s) • z ∈ S
-        simpa only [algebraMap_smul] using hsz
-    · refine ⟨Ideal.Quotient.mk _ c,hpc,?_⟩
-      change (algebraMap (MvPolynomial σ ℝ) (EdgeRing cs basis a) c) • z ∈ S
-      simpa only [algebraMap_smul] using hcz
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace MomentGraph MomentGraph.Sheaf
-universe u v us u' v'
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {σ : Type (max us v)} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
-  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
-  {u b : W} {u' b' : W'}
-  (φ : Interval cs u b ≃o Interval cs' u' b')
-noncomputable section
-
- theorem boundary_edge_image_le (e : Edge cs 1 b)
-    (hx : BruhatLE cs u (source cs 1 b e).val)
-    [Module.Free (MvPolynomial σ ℝ)
-      ((boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf.upwardSubmodule
-        (target cs 1 b e) {f | f.val ∈ ambientLater φ (mappedEdgeRoot φ e hx)})] :
-    let D := (boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf.forget
-    let T := ambientLater φ (mappedEdgeRoot φ e hx)
-    (D.upwardKernel (source cs 1 b e) {f | f.val ∈ T}).map (D.lower e) ≤
-      ((D.upwardKernel (target cs 1 b e) {f | f.val ∈ T}).map (D.upper e)).map
-        (LinearMap.lsmul (MvPolynomial σ ℝ) (D.edge e)
-          (edgeFactorLift cs basis (source cs 1 b e).val (root cs 1 b e))) := by
-  classical
-  let B := boundarySheaf cs 1 b basis (one_bruhat cs b)
-  let D := B.sheaf.forget
-  let F : Set (Outgoing (G := graph cs 1 b) (source cs 1 b e)) :=
-    {f | f.val ∈ ambientLater φ (mappedEdgeRoot φ e hx)}
-  let G : Set (Outgoing (G := graph cs 1 b) (target cs 1 b e)) :=
-    {f | f.val ∈ ambientLater φ (mappedEdgeRoot φ e hx)}
-  let a := root cs 1 b e
-  let f := edgeFactorLift cs basis (source cs 1 b e).val a
-  let J := (D.upwardKernel (target cs 1 b e) G).map (D.upper e)
-  let S := J.map (LinearMap.lsmul (MvPolynomial σ ℝ) (D.edge e) f)
-  let pG := outgoingLabelProduct (polynomialLabel cs 1 b basis) (target cs 1 b e) G
-  let _ : Module (EdgeRing cs basis a) (D.edge e) :=
-    edgeModule cs 1 b basis (one_bruhat cs b) B e
-  let _ : IsScalarTower (MvPolynomial σ ℝ) (EdgeRing cs basis a) (D.edge e) :=
-    edgeModule_tower cs 1 b basis (one_bruhat cs b) B e
-  let _ : Module.Free (EdgeRing cs basis a) (D.edge e) :=
-    edgeModule_free cs 1 b basis (one_bruhat cs b) B e
-  let _ : Module.Free (EdgeRing cs basis a)
-      (ScalarQuotient.submodule (polynomialRoot cs basis a) J) :=
-    upperImage_free cs 1 b basis (one_bruhat cs b) B e G
-  let _ : Module.Free (EdgeRing cs basis a) (ScalarQuotient.submodule (polynomialRoot cs basis a) S) :=
-    ScalarQuotient.free_scalarImage (polynomialRoot cs basis a) f J
-      (edgeFactorLift_regular cs basis 1 b (one_bruhat cs b) B e)
-  change (D.upwardKernel (source cs 1 b e) F).map (D.lower e) ≤ S
-  intro z hz
-  apply mem_of_plane_denominators cs basis a S z (f*pG)
-  · rw [map_mul]
-    exact mul_ne_zero (reducedFactor_ne_zero cs basis _ a)
-      (reducedKernelProduct_ne_zero cs 1 b basis e G)
-  · refine ⟨pG • z,?_,?_⟩
-    · exact D.outgoingLabelProduct_smul_upperImage (polynomialLabel cs 1 b basis)
-        (GradedSheaf.UpperQuotient.annihilates B.sheaf B.quotient) e (B.quotient e).1 G z
-    · change f • (pG • z) = (f*pG) • z
-      rw [mul_smul]
-  · intro p hp hd
-    rw [map_mul] at hd
-    rcases hp.dvd_mul.mp hd with hf | hG
-    · exact prime_dvd_reducedFactor cs basis _ a p hp hf
-    · exact prime_dvd_reducedKernelProduct cs 1 b basis e G p hp hG
-  · intro β hβ
-    obtain ⟨s,hs⟩ := D.scaledImages_denominator (PlaneRing basis (rootPlane a β))
-      (planePrime basis (rootPlane a β)).primeCompl e F G f
-      (boundary_plane_scaledImages basis φ e hx β hβ) z hz
-    exact ⟨s,s.property,hs⟩
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph SchedulePaths
-universe u v us u' v'
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {σ : Type (max us v)} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
-  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
-  {u b : W} {u' b' : W'} (hub : BruhatLE cs u b)
-  (φ : Interval cs u b ≃o Interval cs' u' b')
-noncomputable section
-
- def intervalBoundary := restrictBoundary cs u b basis hub
-  (boundarySheaf cs 1 b basis (one_bruhat cs b))
-
- theorem schedule_edge_image_le {e : Edge cs u b} {q : List (Edge cs u b)}
-    (hq : e::q <:+ chronologicalSchedule φ)
-    [hF : Module.Free (Coefficient (σ := σ))
-      ((intervalBoundary basis hub).sheaf.upwardSubmodule (target cs u b e)
-        (remainingConditions cs u b q (target cs u b e)))] :
-    (intervalBoundary basis hub).sheaf.lowerImage e
-        (remainingConditions cs u b q (source cs u b e)) ≤
-      ((intervalBoundary basis hub).sheaf.upperImage e
-        (remainingConditions cs u b q (target cs u b e))).map
-        (edgeFactorMap cs basis u b hub (intervalBoundary basis hub) e) := by
-  let B := boundarySheaf cs 1 b basis (one_bruhat cs b)
-  let a := (upperInclusion cs u b).edge e
-  let r := AdaptedDyer.rootLabel cs' (transportedEdge φ e)
-  have hr : mappedEdgeRoot φ a (source cs u b e).property.1 = r := rfl
-  have hk (x : Interval cs u b) :=
-    restrictBoundary_kernel cs u b basis hub B x (ambientLater φ r)
-  have hs : (intervalBoundary basis hub).sheaf.upwardSubmodule (source cs u b e)
-      (remainingConditions cs u b q (source cs u b e)) =
-      B.sheaf.upwardSubmodule (source cs 1 b a) {f | f.val ∈ ambientLater φ r} :=
-    (congrArg ((intervalBoundary basis hub).sheaf.upwardSubmodule (source cs u b e))
-      (remainingConditions_source_ambient φ hq)).trans (hk (source cs u b e))
-  have ht : (intervalBoundary basis hub).sheaf.upwardSubmodule (target cs u b e)
-      (remainingConditions cs u b q (target cs u b e)) =
-      B.sheaf.upwardSubmodule (target cs 1 b a) {f | f.val ∈ ambientLater φ r} :=
-    (congrArg ((intervalBoundary basis hub).sheaf.upwardSubmodule (target cs u b e))
-      (remainingConditions_target_ambient φ hq)).trans (hk (target cs u b e))
-  have hfree : Module.Free (Coefficient (σ := σ))
-      (B.sheaf.upwardSubmodule (target cs 1 b a) {f | f.val ∈ ambientLater φ r}) :=
-    (congrArg (fun N : Submodule (Coefficient (σ := σ)) (B.sheaf.vertex (target cs 1 b a)) =>
-      Module.Free (Coefficient (σ := σ)) N) ht).mp hF
-  let hf : Module.Free (Coefficient (σ := σ))
-      (B.sheaf.upwardSubmodule (target cs 1 b a)
-        {f | f.val ∈ ambientLater φ (mappedEdgeRoot φ a (source cs u b e).property.1)}) := by
-    rw [hr]
-    exact hfree
-  have h := boundary_edge_image_le basis φ a (source cs u b e).property.1
-  dsimp only at h
-  rw [hr] at h
-  have hs' := congrArg (fun N : Submodule (Coefficient (σ := σ)) (B.sheaf.vertex (source cs 1 b a)) =>
-    N.map (B.sheaf.lower a).map) hs
-  have ht' := congrArg (fun N : Submodule (Coefficient (σ := σ)) (B.sheaf.vertex (target cs 1 b a)) =>
-    (N.map (B.sheaf.upper a).map).map
-      (edgeFactorMap cs basis u b hub (intervalBoundary basis hub) e)) ht
-  exact hs'.le.trans (h.trans ht'.symm.le)
-
- theorem schedule_comparison_of_upper_freeness
-    (hfree : ∀ e q, e::q <:+ chronologicalSchedule φ →
-      Module.Free (Coefficient (σ := σ))
-        ((intervalBoundary basis hub).sheaf.upwardSubmodule (target cs u b e)
-          (remainingConditions cs u b q (target cs u b e))))
-    {t : ℝ} (ht : 0 < t) (ht1 : t < 1) :
-    scheduleState cs u b basis hub (intervalBoundary basis hub) t [] ≤
-      dispatch (source cs u b) (target cs u b) (t⁻¹-t)
-        (scheduleState cs u b basis hub (intervalBoundary basis hub) t (chronologicalSchedule φ))
-        (transportedSchedule φ) := by
-  have h := scheduleState_le cs u b basis hub (intervalBoundary basis hub) (chronologicalSchedule φ)
-    (fun e q hq => by
-      let _ := hfree e q hq
-      exact schedule_edge_image_le basis hub φ hq) ht ht1
-  simpa only [chronologicalSchedule,List.reverse_reverse] using h
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph SchedulePaths
-universe u v u' v' us
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-  {σ : Type (max us v)} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
-  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
-  {u b : W} {u' b' : W'} (hub : BruhatLE cs u b)
-  (φ : Interval cs u b ≃o Interval cs' u' b')
-noncomputable section
-
- theorem schedule_final_of_character
-    (hc : StalkCharacterObligation cs 1 b basis (one_bruhat cs b))
-    {t : ℝ} (ht : 0 < t) (ht1 : t < 1) (x : Interval cs u b) :
-    scheduleState cs u b basis hub (intervalBoundary basis hub) t [] x =
-      scaledKL cs t x.val b := by
-  have he : remainingConditions cs u b [] x = ∅ := by
-    ext e
-    simp [remainingConditions]
-  rw [scheduleState,he]
-  unfold intervalBoundary
-  rw [normalized_emptyKernel_restrictBoundary]
-  exact normalized_emptyKernel_of_character cs 1 b basis (one_bruhat cs b)
-    (boundarySheaf cs 1 b basis (one_bruhat cs b))
-    ((upperInclusion cs u b).vertex x) (hc _) ht ht1
-
-omit [Fintype I] [Fintype σ] in
- theorem schedule_all_conditions (x : Interval cs u b) :
-    remainingConditions cs u b (chronologicalSchedule φ) x = Set.univ := by
-  ext e
-  simp only [remainingConditions,Set.mem_ofPred_eq,Set.mem_univ,iff_true]
-  exact chronologicalSchedule_full φ e.val
-
-variable [Fintype (Edge cs 1 b)] [Fintype (Interval cs 1 b)] [DecidableEq (Interval cs 1 b)]
-
- theorem schedule_initial_of_duality_character
-    (d : SectionDuality cs 1 b basis (one_bruhat cs b)
-      (boundarySheaf cs 1 b basis (one_bruhat cs b)))
-    (hd : ∀ i j (m n : (boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf.sections Set.univ),
-      m ∈ ((boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf.sections Set.univ).piece i →
-      n ∈ ((boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf.sections Set.univ).piece j →
-      d.pairing m n ∈ polynomialGrading ℝ σ (i+j-(cs.length b : ℤ)))
-    (hc : StalkCharacterObligation cs 1 b basis (one_bruhat cs b))
-    {t : ℝ} (ht : 0 < t) (ht1 : t < 1) (x : Interval cs u b) :
-    scheduleState cs u b basis hub (intervalBoundary basis hub) t (chronologicalSchedule φ) x =
-      scaledKL cs t⁻¹ x.val b := by
-  rw [scheduleState,schedule_all_conditions φ]
-  unfold intervalBoundary
-  rw [normalized_fullKernel_restrictBoundary]
-  exact normalized_fullKernel_of_duality_character cs b basis
-    (boundarySheaf cs 1 b basis (one_bruhat cs b)) d hd
-    ((upperInclusion cs u b).vertex x) (hc _) ht ht1
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-/-! Native endpoint normalization from the ACTUAL stalk characters. Unlike the
-KL-labelled endpoints, these need neither character recognition nor Hodge. -/
-namespace KLInvariance.BruhatGraph
-open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph Polynomial SchedulePaths
-universe u
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type u} [Group W] (cs : CoxeterSystem M W) (b : W)
-  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
-noncomputable section
-local instance actualInterval : Fintype (Interval cs 1 b) := by
-  let := interval_finite cs 1 b
-  exact Fintype.ofFinite _
-local instance actualEdges : Fintype (Edge cs 1 b) := Fintype.ofFinite _
-local instance (a : Type*) : DecidableEq a := Classical.decEq a
-
- theorem normalized_emptyKernel_scaledStalk (x : Interval cs 1 b)
-    {t : ℝ} (ht : 0<t) (ht1 : t<1) :
-    normalizedKernel cs 1 b basis (one_bruhat cs b) (ActualBoundary cs b basis) t x ∅=
-      scaledStalk cs b basis t x := by
-  rw [normalizedKernel,emptyKernel_hilbert cs 1 b basis (one_bruhat cs b)
-    (ActualBoundary cs b basis) x (sq_pos_of_pos ht) (by nlinarith)]
-  dsimp only [stateScale,scaledStalk]
-  have hne : 1-t^2≠0 := ne_of_gt (by nlinarith : 0<1-t^2)
-  rw [inv_pow]
-  field_simp [hne,inv_pow]
-
- theorem normalized_fullKernel_scaledStalk (x : Interval cs 1 b)
-    {t : ℝ} (ht : 0<t) (ht1 : t<1) :
-    normalizedKernel cs 1 b basis (one_bruhat cs b) (ActualBoundary cs b basis) t x Set.univ=
-      scaledStalk cs b basis t⁻¹ x := by
-  let C := ActualBoundary cs b basis
-  let := C.free x
-  let c := (exists_nonnegativeBasis (σ := σ) (C.sheaf.vertex x).piece
-    (C.sheaf.vertex x).nonneg).some
-  have hcp : c.polynomial=stalkCharacter cs 1 b basis (one_bruhat cs b) x := by
-    rw [stalkCharacter_eq_character cs 1 b basis (one_bruhat cs b) C x,
-      character_eq (C.sheaf.vertex x).piece (C.sheaf.vertex x).nonneg c]
-  rw [normalizedKernel,GradedSheaf.kernelHilbert,
-    fullKernel_eq cs 1 b basis (one_bruhat cs b) C x,
-    costalk_hilbert_of_duality cs 1 b basis (one_bruhat cs b) C (actualSectionDuality cs b basis)
-      (cs.length b) (actualSectionDuality_homogeneous cs b basis) x c
-      (sq_pos_of_pos ht) (by nlinarith),hcp,incoming_card cs b x]
-  dsimp only [stateScale,scaledStalk]
-  have hscale : t^((cs.length x.val : ℤ)-cs.length b)*
-      (t^2)^((cs.length b : ℤ)-cs.length x.val)=
-      (t⁻¹)^((cs.length x.val : ℤ)-cs.length b) := by
-    rw [show (t^2)^((cs.length b : ℤ)-cs.length x.val)=
-      t^(2*((cs.length b : ℤ)-cs.length x.val)) by rw [zpow_mul]; norm_num,
-      ←zpow_add₀ ht.ne',inv_zpow,←zpow_neg]
-    congr 1
-    omega
-  rw [show (t^2)⁻¹=(t⁻¹)^2 by rw [inv_pow]]
-  have hne : 1-t^2≠0 := ne_of_gt (by nlinarith : 0<1-t^2)
-  calc
-    _ = (t^((cs.length x.val : ℤ)-cs.length b)*
-      (t^2)^((cs.length b : ℤ)-cs.length x.val))*
-      eval₂ (Int.castRingHom ℝ) ((t⁻¹)^2) (stalkCharacter cs 1 b basis (one_bruhat cs b) x) := by
-      have hcancel : (1-t^2)^(Fintype.card σ)*((1-t^2)⁻¹)^(Fintype.card σ)=1 := by
-        rw [←mul_pow,mul_inv_cancel₀ hne,one_pow]
-      calc
-        _ = ((1-t^2)^(Fintype.card σ)*((1-t^2)⁻¹)^(Fintype.card σ))*
-            ((t^((cs.length x.val : ℤ)-cs.length b)*(t^2)^((cs.length b : ℤ)-cs.length x.val))*
-            eval₂ (Int.castRingHom ℝ) ((t⁻¹)^2) (stalkCharacter cs 1 b basis (one_bruhat cs b) x)) := by ring
-        _ = _ := by rw [hcancel,one_mul]
-    _ = _ := by rw [hscale]
-
- theorem schedule_final_scaledStalk (a : W) (hab : BruhatLE cs a b)
-    {t : ℝ} (ht : 0<t) (ht1 : t<1) (x : Interval cs a b) :
-    scheduleState cs a b basis hab (intervalBoundary basis hab) t [] x=
-      scaledStalk cs b basis t ((upperInclusion cs a b).vertex x) := by
-  have he : remainingConditions cs a b [] x=∅ := by
-    ext e
-    simp [remainingConditions]
-  rw [scheduleState,he]
-  unfold intervalBoundary
-  rw [normalized_emptyKernel_restrictBoundary]
-  exact normalized_emptyKernel_scaledStalk cs b basis _ ht ht1
-
- theorem schedule_initial_scaledStalk (a : W) (hab : BruhatLE cs a b)
-    {t : ℝ} (ht : 0<t) (ht1 : t<1) (x : Interval cs a b) :
-    scheduleState cs a b basis hab (intervalBoundary basis hab) t
-      (chronologicalSchedule (OrderIso.refl (Interval cs a b))) x=
-      scaledStalk cs b basis t⁻¹ ((upperInclusion cs a b).vertex x) := by
-  rw [scheduleState,schedule_all_conditions (OrderIso.refl (Interval cs a b))]
-  unfold intervalBoundary
-  rw [normalized_fullKernel_restrictBoundary]
-  exact normalized_fullKernel_scaledStalk cs b basis _ ht ht1
-
- theorem native_endpoint_eq (a : W) (hab : BruhatLE cs a b)
-    {t : ℝ} (ht : 0<t) (ht1 : t<1) :
-    scheduleState cs a b basis hab (intervalBoundary basis hab) t []=
-      dispatch (source cs a b) (target cs a b) (t⁻¹-t)
-        (scheduleState cs a b basis hab (intervalBoundary basis hab) t
-           (chronologicalSchedule (OrderIso.refl (Interval cs a b))))
-        (chronologicalSchedule (OrderIso.refl (Interval cs a b))).reverse := by
-  have hf := funext (schedule_final_scaledStalk cs b basis a hab ht ht1)
-  have hi := funext (schedule_initial_scaledStalk cs b basis a hab ht ht1)
-  rw [hf,hi,chronologicalSchedule,List.reverse_reverse]
-  funext x
-  rw [transported_dispatch_matrix (OrderIso.refl (Interval cs a b)) _ (Int.castRingHom ℝ)]
-  exact scaledStalk_reciprocity_upper cs b basis t ht.ne' a x
-
-end
-end KLInvariance.BruhatGraph
-
-end
-
-
-section
-
-namespace KLInvariance.SchedulePaths
-universe u v
-variable {E : Type u} {Λ : Type v} [LinearOrder Λ]
-
- theorem exists_suffix_score_gt (score : E → Λ) (r : Λ) (p : List E)
-    (hp : p.Pairwise (fun a b => score a ≤ score b)) :
-    ∃ q, q <:+ p ∧ ∀ e, e ∈ q ↔ e ∈ p ∧ r < score e := by
-  induction p with
-  | nil => exact ⟨[],List.suffix_refl _,fun e => by simp⟩
-  | cons a p ih =>
-    have hs := List.pairwise_cons.mp hp
-    by_cases ha : r < score a
-    · refine ⟨a::p,List.suffix_refl _,?_⟩
-      intro e
-      constructor
-      · intro he
-        refine ⟨he,?_⟩
-        rcases List.mem_cons.mp he with rfl | he
-        · exact ha
-        · exact lt_of_lt_of_le ha (hs.1 e he)
-      · exact And.left
-    · obtain ⟨q,hq,hmem⟩ := ih hs.2
-      refine ⟨q,hq.trans (List.suffix_cons a p),?_⟩
-      intro e
-      rw [hmem,List.mem_cons]
-      constructor
-      · exact fun h => ⟨Or.inr h.1,h.2⟩
-      · rintro ⟨he,hr⟩
-        rcases he with rfl | he
-        · exact (ha hr).elim
-        · exact ⟨he,hr⟩
-
-end KLInvariance.SchedulePaths
-
-end
-
-
-section
-
-
-end
-
-end OAI
+import OAI.RepresentationTheory.KazhdanLusztig.PositiveTransfer
+
+namespace OAI
+
+/-!
+All-section transport, terminal kernels, scalar quotients, height-one membership in free targets and scaled edge-image inclusion.
+-/
+
+section
+
+namespace KLInvariance.MomentGraph.Sheaf.Iso
+universe ur uv ue uz uz'
+variable {R : Type ur} [CommRing R] {V : Type uv} [PartialOrder V]
+  {E : Type ue} {G : OrderedGraph V E}
+  {B : Sheaf.{ur,uv,ue,uz} (R := R) G} {C : Sheaf.{ur,uv,ue,uz'} (R := R) G}
+  (j : Iso B C)
+
+include j in
+ theorem scaledImages (e : E) (F : Set (Outgoing (G := G) (G.source e)))
+    (H : Set (Outgoing (G := G) (G.target e))) (r : R)
+    (h : B.ScaledImages e F H r) : C.ScaledImages e F H r := by
+  intro z
+  constructor
+  · rintro ⟨v,hv,he⟩
+    have hv' : (j.vertex (G.source e)).symm v ∈ B.upwardKernel (G.source e) F := by
+      apply (j.upwardKernel_iff _ _ _).mp
+      simpa only [LinearEquiv.apply_symm_apply] using hv
+    have he' : B.lower e ((j.vertex (G.source e)).symm v)=(j.edge e).symm z := by
+      apply (j.edge e).injective
+      rw [j.lower,LinearEquiv.apply_symm_apply,LinearEquiv.apply_symm_apply,he]
+    obtain ⟨w,hw,hew⟩ := (h ((j.edge e).symm z)).mp ⟨_,hv',he'⟩
+    refine ⟨j.vertex (G.target e) w,(j.upwardKernel_iff _ _ _).mpr hw,?_⟩
+    rw [← j.upper,← map_smul,hew,LinearEquiv.apply_symm_apply]
+  · rintro ⟨w,hw,he⟩
+    have hw' : (j.vertex (G.target e)).symm w ∈ B.upwardKernel (G.target e) H := by
+      apply (j.upwardKernel_iff _ _ _).mp
+      simpa only [LinearEquiv.apply_symm_apply] using hw
+    have he' : r • B.upper e ((j.vertex (G.target e)).symm w)=(j.edge e).symm z := by
+      apply (j.edge e).injective
+      rw [map_smul,j.upper,LinearEquiv.apply_symm_apply,LinearEquiv.apply_symm_apply,he]
+    obtain ⟨v,hv,hev⟩ := (h ((j.edge e).symm z)).mpr ⟨_,hw',he'⟩
+    refine ⟨j.vertex (G.source e) v,(j.upwardKernel_iff _ _ _).mpr hv,?_⟩
+    rw [← j.lower,hev,LinearEquiv.apply_symm_apply]
+
+end KLInvariance.MomentGraph.Sheaf.Iso
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.Sheaf
+open scoped Pointwise
+universe ur uv ue
+variable {R : Type ur} [CommRing R] [DecompositionMonoid R]
+  {V : Type uv} [PartialOrder V] {E : Type ue} [Fintype E]
+  (G : OrderedGraph V E) (α : E → R) (c : V)
+noncomputable section
+
+/-- A quotient-product equality gives the actual two embedded model images.
+The common quotient is the edge module's coefficient ring. -/
+ theorem structure_scaledImages (e : E)
+    (F : Set (Outgoing (G := G) (G.source e)))
+    (H : Set (Outgoing (G := G) (G.target e))) (r : R)
+    (hF : (supportedOutgoing G c (G.source e) F : Set (Outgoing (G := G) (G.source e))).Pairwise
+      (Function.onFun IsRelPrime (fun a => α a.val)))
+    (hH : (supportedOutgoing G c (G.target e) H : Set (Outgoing (G := G) (G.target e))).Pairwise
+      (Function.onFun IsRelPrime (fun a => α a.val)))
+    (h : Associated
+      (algebraMap R (R ⧸ Ideal.span {α e}) (kernelProduct G α c (G.source e) F))
+      (algebraMap R (R ⧸ Ideal.span {α e}) (r*kernelProduct G α c (G.target e) H))) :
+    (structureSheaf G α c).ScaledImages e F H r := by
+  rw [scaledImages_iff,structure_lower_image (G := G) α c e F hF,
+    structure_upper_image (G := G) α c e H hH,← mul_smul]
+  let S := R ⧸ Ideal.span {α e}
+  let : Module S ((structureSheaf G α c).edge e) :=
+    inferInstanceAs (Module S (PLift (G.target e ≤ c) → S))
+  let : IsScalarTower R S ((structureSheaf G α c).edge e) :=
+    inferInstanceAs (IsScalarTower R S (PLift (G.target e ≤ c) → S))
+  exact LocalProducts.smul_top_eq_of_map_associated (S := S) h
+
+ theorem structure_scaledImages_of_count (e : E)
+    (F : Set (Outgoing (G := G) (G.source e)))
+    (H : Set (Outgoing (G := G) (G.target e))) (r : R) (p : R ⧸ Ideal.span {α e}) (k : ℕ)
+    (hF : (supportedOutgoing G c (G.source e) F : Set (Outgoing (G := G) (G.source e))).Pairwise
+      (Function.onFun IsRelPrime (fun a => α a.val)))
+    (hH : (supportedOutgoing G c (G.target e) H : Set (Outgoing (G := G) (G.target e))).Pairwise
+      (Function.onFun IsRelPrime (fun a => α a.val)))
+    (ha : ∀ i ∈ supportedOutgoing G c (G.source e) F,
+      Associated (algebraMap R (R ⧸ Ideal.span {α e}) (α i.val)) p)
+    (hb : ∀ i ∈ supportedOutgoing G c (G.target e) H,
+      Associated (algebraMap R (R ⧸ Ideal.span {α e}) (α i.val)) p)
+    (hr : Associated (algebraMap R (R ⧸ Ideal.span {α e}) r) (p^k))
+    (hc : (supportedOutgoing G c (G.source e) F).card=
+      (supportedOutgoing G c (G.target e) H).card+k) :
+    (structureSheaf G α c).ScaledImages e F H r := by
+  apply structure_scaledImages G α c e F H r hF hH
+  simpa only [kernelProduct,map_prod,map_mul] using
+    LocalProducts.associated_prod_scale _ _ _ _ p _ k ha hb hr hc
+
+end
+end KLInvariance.MomentGraph.Sheaf
+
+end
+
+
+section
+
+namespace KLInvariance.TitsSpace.PlaneBasis
+open Module BruhatGraph MomentGraph MomentGraph.Sheaf
+universe u v us u' v'
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+  {σ : Type us} (basis : Basis σ ℝ (Extended M))
+  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
+  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
+  {a₀ u b : W} {u' b' : W'}
+  (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+  (φ : Interval cs u b ≃o Interval cs' u' b')
+noncomputable section
+
+ theorem scheduledLower_root_ne (e : B.CosetEdge a₀ b)
+    (hx : BruhatLE cs u ((e.val.1.val:W)*a₀))
+    (f : Outgoing (G := B.cosetGraph a₀ b) e.val.1)
+    (hf : f ∈ B.scheduledLower hmin φ e hx) :
+    B.edgeRoot f.val.property ≠ B.edgeRoot e.property := by
+  intro he
+  have hroot : BruhatGraph.root cs 1 b (B.cosetEdgeAmbient a₀ b hmin f.val) =
+      BruhatGraph.root cs 1 b (B.cosetEdgeAmbient a₀ b hmin e) := by
+    rwa [B.cosetRoot_eq a₀ b hmin f.val,B.cosetRoot_eq a₀ b hmin e]
+  have hh : f.val=e := B.cosetEdgeAmbient_injective a₀ b hmin
+    (eq_of_label_source_eq cs 1 b (root_injective_on_labels cs 1 b hroot)
+      (congrArg (B.cosetAmbient a₀ b) f.property))
+  have hff : f=(⟨e,rfl⟩ : Outgoing (G := B.cosetGraph a₀ b) e.val.1) := Subtype.ext hh
+  exact B.scheduledLower_self hmin φ e hx (hff ▸ hf)
+
+ theorem outgoingUpper_root_ne (e : B.CosetEdge a₀ b)
+    (f : Outgoing (G := B.cosetGraph a₀ b) e.val.2) :
+    B.edgeRoot f.val.property ≠ B.edgeRoot e.property := by
+  rcases f with ⟨⟨⟨s,t⟩,hst⟩,hs⟩
+  change s=e.val.2 at hs
+  subst s
+  exact (B.edgeRoot_ne_of_path e.property hst).symm
+
+ theorem coset_structure_scaledImages [Fintype (B.CosetEdge a₀ b)]
+    (e : B.CosetEdge a₀ b) (hx : BruhatLE cs u ((e.val.1.val:W)*a₀))
+    (c : B.CosetVertex a₀ b) :
+    (structureSheaf (B.cosetGraph a₀ b) (B.cosetLabel basis a₀ b hmin) c).ScaledImages
+      e (B.scheduledLower hmin φ e hx) (B.scheduledUpper hmin φ e hx)
+      (algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P)
+        (edgeFactorLift cs basis ((e.val.1.val:W)*a₀) (B.edgeRoot e.property))) := by
+  classical
+  by_cases hyc : e.val.2 ≤ c
+  · let α := B.edgeRoot e.property
+    obtain ⟨β,hαβ,hβ⟩ : ∃ β : PositiveRoot M cs, α ≠ β ∧ β.val ∈ P := by
+      by_cases h : α=B.left
+      · exact ⟨B.right,h ▸ B.distinct,B.right_mem⟩
+      · exact ⟨B.left,h,B.left_mem⟩
+    have hplane : rootPlane α β=P :=
+      (rootPlane_eq_of_mem B.left B.right α β B.distinct
+        (B.plane_eq.ge (B.edgeRoot_plane e.property)) (B.plane_eq.ge hβ) hαβ).trans B.plane_eq
+    apply structure_scaledImages_of_count (B.cosetGraph a₀ b) (B.cosetLabel basis a₀ b hmin) c
+      e _ _ _ (B.reducedCosetRoot basis a₀ b hmin e β)
+      ((planeWeight ((e.val.1.val:W)*a₀) α P).toNat)
+      (B.coset_supported_pairwise basis a₀ b hmin c e.val.1 _)
+      (B.coset_supported_pairwise basis a₀ b hmin c e.val.2 _)
+    · intro f hf
+      exact B.cosetLabel_reduction_associated basis a₀ b hmin e f.val β hplane
+        (B.scheduledLower_root_ne hmin φ e hx f (Finset.mem_filter.mp hf).2.1)
+    · intro f _
+      exact B.cosetLabel_reduction_associated basis a₀ b hmin e f.val β hplane
+        (B.outgoingUpper_root_ne e f)
+    · exact B.cosetFactor_reduction_associated basis a₀ b hmin e β hαβ hplane
+    · exact B.supported_count_difference hmin φ e hx c hyc
+  · let := structureSheaf_edge_subsingleton (B.cosetGraph a₀ b)
+        (B.cosetLabel basis a₀ b hmin) c e hyc
+    intro z
+    constructor <;> intro _ <;> exact ⟨0,Submodule.zero_mem _,Subsingleton.elim _ _⟩
+
+end
+end KLInvariance.TitsSpace.PlaneBasis
+
+end
+
+
+section
+
+
+namespace KLInvariance.TitsSpace.PlaneBasis
+open Module BruhatGraph MomentGraph MomentGraph.Sheaf
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+  {σ : Type (max us v)} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  (a₀ b : W) (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+noncomputable section
+
+ theorem localized_boundary_decomposition :
+    ∃ C : Sheaf.{max us v,v,v,max us v} (R := PlaneRing basis P) (B.cosetGraph a₀ b),
+      ModelSum (B.cosetModels basis a₀ b hmin).sheaf Set.univ C ∧
+      Nonempty (Iso (B.localizedCoset basis a₀ b hmin
+        (boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf.forget) C) := by
+  let D := boundarySheaf cs 1 b basis (one_bruhat cs b)
+  apply exists_modelSumIso (B.cosetModels basis a₀ b hmin)
+    (B.localizedCoset basis a₀ b hmin D.sheaf.forget)
+  · intro x
+    let : Module.Free (MvPolynomial σ ℝ)
+        (D.sheaf.forget.vertex ((B.cosetGraphMap a₀ b hmin).vertex x)) :=
+      D.free (B.cosetAmbient a₀ b x)
+    exact D.sheaf.forget.baseChange_vertex_free (PlaneRing basis P) _
+  · intro x
+    let : Module.Finite (MvPolynomial σ ℝ)
+        (D.sheaf.forget.vertex ((B.cosetGraphMap a₀ b hmin).vertex x)) :=
+      (D.sheaf.vertex (B.cosetAmbient a₀ b x)).finite
+    exact D.sheaf.forget.baseChange_vertex_finite (PlaneRing basis P) _
+  · exact B.localizedCoset_quotient basis a₀ b hmin D.sheaf.forget D.quotient
+  · exact B.localizedCoset_flabby basis a₀ b hmin D.sheaf.forget D.quotient D.flabby.flabby
+  · exact B.localizedCoset_generated basis a₀ b hmin D.sheaf.forget D.generated
+
+end
+end KLInvariance.TitsSpace.PlaneBasis
+
+end
+
+
+section
+
+
+namespace KLInvariance.TitsSpace.PlaneBasis
+open Module BruhatGraph MomentGraph MomentGraph.Sheaf
+universe u v us u' v'
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+  {σ : Type (max us v)} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
+  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
+  {a₀ u b : W} {u' b' : W'}
+  (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+  (φ : Interval cs u b ≃o Interval cs' u' b')
+noncomputable section
+
+ theorem localized_boundary_scaledImages (e : B.CosetEdge a₀ b)
+    (hx : BruhatLE cs u ((e.val.1.val:W)*a₀)) :
+    (B.localizedCoset basis a₀ b hmin
+      (boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf.forget).ScaledImages
+      e (B.scheduledLower hmin φ e hx) (B.scheduledUpper hmin φ e hx)
+      (algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P)
+        (edgeFactorLift cs basis ((e.val.1.val:W)*a₀) (B.edgeRoot e.property))) := by
+  classical
+  let := Fintype.ofFinite (B.CosetVertex a₀ b)
+  let : Finite (B.CosetEdge a₀ b) := Finite.of_injective
+    (fun e : B.CosetEdge a₀ b => e.val) Subtype.val_injective
+  let := Fintype.ofFinite (B.CosetEdge a₀ b)
+  obtain ⟨C,hC,⟨j⟩⟩ := B.localized_boundary_decomposition basis a₀ b hmin
+  apply j.symm.scaledImages
+  apply hC.scaledImages
+  intro c _
+  exact B.coset_structure_scaledImages basis hmin φ e hx c
+
+end
+end KLInvariance.TitsSpace.PlaneBasis
+
+end
+
+
+section
+
+/-! Flat scalar extension of actual embedded kernel and image modules. -/
+namespace KLInvariance.FlatSubmodule
+open TensorProduct
+universe ur us um un
+variable {R : Type ur} [CommRing R] (S : Type us) [CommRing S] [Algebra R S]
+  {M : Type um} [AddCommGroup M] [Module R M]
+  {N : Type un} [AddCommGroup N] [Module R N]
+
+ theorem baseChange_ker [Module.Flat R S] (f : M →ₗ[R] N) :
+    (LinearMap.ker f).baseChange S = LinearMap.ker (f.baseChange S) := by
+  ext z
+  exact (Module.Flat.lTensor_exact S f.exact_subtype_ker_map z).symm
+
+ theorem baseChange_map (p : Submodule R M) (f : M →ₗ[R] N) :
+    (p.map f).baseChange S = (p.baseChange S).map (f.baseChange S) := by
+  apply le_antisymm
+  · rintro z ⟨k,rfl⟩
+    induction k using TensorProduct.inductionOn with
+    | tmul s x =>
+      obtain ⟨m,hm,hx⟩ := x.property
+      refine ⟨s ⊗ₜ m,Submodule.tmul_mem_baseChange_of_mem s hm,?_⟩
+      change s ⊗ₜ[R] f m = s ⊗ₜ[R] (x : N)
+      rw [hx]
+    | add x y hx hy => simpa only [map_add] using Submodule.add_mem _ hx hy
+  · rintro z ⟨w,⟨k,hk⟩,rfl⟩
+    subst w
+    induction k using TensorProduct.inductionOn with
+    | tmul s x =>
+      exact Submodule.tmul_mem_baseChange_of_mem s (show f x ∈ p.map f from ⟨x,x.property,rfl⟩)
+    | add x y hx hy => simpa only [map_add] using Submodule.add_mem _ hx hy
+
+end KLInvariance.FlatSubmodule
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.Sheaf
+open TensorProduct
+universe ur us uv ue um
+variable {R : Type ur} [CommRing R] (S : Type us) [CommRing S] [Algebra R S]
+  {V : Type uv} [PartialOrder V] {E : Type ue} {G : OrderedGraph V E}
+  (B : Sheaf.{ur,uv,ue,um} (R := R) G)
+noncomputable section
+
+ def kernelConstraint (x : V) (F : Set (Outgoing (G := G) x)) :
+    B.vertex x →ₗ[R] (∀ e : F, B.edge e.val.val) :=
+  LinearMap.pi (fun e => (LinearMap.proj e.val).comp (B.stalkMap x))
+
+ theorem ker_kernelConstraint (x : V) (F : Set (Outgoing (G := G) x)) :
+    LinearMap.ker (B.kernelConstraint x F) = B.upwardKernel x F := by
+  ext v
+  constructor
+  · intro h e he
+    exact congrFun (LinearMap.mem_ker.mp h) ⟨e,he⟩
+  · intro h
+    exact LinearMap.mem_ker.mpr (funext fun e => h e.val e.property)
+
+ theorem kernelConstraint_tensor (x : V) (F : Set (Outgoing (G := G) x))
+    (z : S ⊗[R] B.vertex x) :
+    (B.baseChange S).kernelConstraint x F z =
+      TensorProduct.piRightHom R S S (fun e : F => B.edge e.val.val)
+        ((B.kernelConstraint x F).baseChange S z) := by
+  let k : (S ⊗[R] B.vertex x) →ₗ[S] (∀ e : F, S ⊗[R] B.edge e.val.val) :=
+    (B.baseChange S).kernelConstraint x F
+  change k z = _
+  induction z using TensorProduct.inductionOn with
+  | tmul s v =>
+    funext e
+    rcases e with ⟨⟨e,he⟩,heF⟩
+    subst x
+    rfl
+  | add z t hz ht =>
+    simp only [map_add,hz,ht]
+
+ theorem baseChange_upwardKernel [Finite E] [Module.Flat R S]
+    (x : V) (F : Set (Outgoing (G := G) x)) :
+    (B.upwardKernel x F).baseChange S = (B.baseChange S).upwardKernel x F := by
+  classical
+  let := Fintype.ofFinite E
+  rw [← B.ker_kernelConstraint x F,FlatSubmodule.baseChange_ker,
+    ← (B.baseChange S).ker_kernelConstraint x F]
+  ext z
+  change (B.kernelConstraint x F).baseChange S z = 0 ↔
+    (B.baseChange S).kernelConstraint x F z = 0
+  rw [B.kernelConstraint_tensor S]
+  exact ((TensorProduct.piRight R S S (fun e : F => B.edge e.val.val)).map_eq_zero_iff).symm
+
+ theorem baseChange_lowerImage [Finite E] [Module.Flat R S]
+    (e : E) (F : Set (Outgoing (G := G) (G.source e))) :
+    ((B.upwardKernel (G.source e) F).map (B.lower e)).baseChange S =
+      ((B.baseChange S).upwardKernel (G.source e) F).map ((B.baseChange S).lower e) := by
+  rw [FlatSubmodule.baseChange_map,B.baseChange_upwardKernel S]
+  rfl
+
+ theorem baseChange_upperImage [Finite E] [Module.Flat R S]
+    (e : E) (F : Set (Outgoing (G := G) (G.target e))) :
+    ((B.upwardKernel (G.target e) F).map (B.upper e)).baseChange S =
+      ((B.baseChange S).upwardKernel (G.target e) F).map ((B.baseChange S).upper e) := by
+  rw [FlatSubmodule.baseChange_map,B.baseChange_upwardKernel S]
+  rfl
+
+end
+end KLInvariance.MomentGraph.Sheaf
+
+end
+
+
+section
+
+namespace KLInvariance.TitsSpace.PlaneBasis
+open Module BruhatGraph MomentGraph MomentGraph.Sheaf
+universe u v us um u' v'
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {P : Submodule ℝ (I → ℝ)} (B : PlaneBasis (M := M) (cs := cs) P)
+  {σ : Type us} (basis : Basis σ ℝ (Extended M))
+  {a₀ b : W}
+  (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+noncomputable section
+
+ theorem localizedCoset_kernel
+    (C : Sheaf.{us,v,v,um} (R := MvPolynomial σ ℝ) (graph cs 1 b))
+    (hq : C.UpperQuotient (polynomialLabel cs 1 b basis))
+    (x : B.CosetVertex a₀ b) (T : Set (Edge cs 1 b)) :
+    (B.localizedCoset basis a₀ b hmin C).upwardKernel x
+      {e | B.cosetEdgeAmbient a₀ b hmin e.val ∈ T} =
+    (C.baseChange (PlaneRing basis P)).upwardKernel (B.cosetAmbient a₀ b x) {e | e.val ∈ T} := by
+  apply (C.baseChange (PlaneRing basis P)).pullback_kernel (B.cosetGraphMap a₀ b hmin) x T
+    {e | label cs 1 b e ∈ planeSubgroup M cs P}
+  · intro e he
+    exact (C.baseChange (PlaneRing basis P)).edge_subsingleton_of_unit _
+      (C.baseChange_upperQuotient (PlaneRing basis P) _ hq) e
+      ((polynomialLabel_unit_iff (P := P) basis b e).mpr he)
+  · intro e he hx
+    exact B.coset_lift_live a₀ b hmin e he x hx
+
+end
+
+variable {σ : Type (max us v)} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
+  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
+  {u : W} {u' b' : W'}
+  (φ : Interval cs u b ≃o Interval cs' u' b')
+noncomputable section
+
+ theorem ambient_boundary_scaledImages (e : B.CosetEdge a₀ b)
+    (hx : BruhatLE cs u ((e.val.1.val:W)*a₀)) :
+    let D := (boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf.forget
+    let T := ambientLater φ (B.mappedCosetRoot hmin φ e hx)
+    (D.baseChange (PlaneRing basis P)).ScaledImages (B.cosetEdgeAmbient a₀ b hmin e)
+      {f | f.val ∈ T} {f | f.val ∈ T}
+      (algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P)
+        (edgeFactorLift cs basis ((e.val.1.val:W)*a₀) (B.edgeRoot e.property))) := by
+  let D := boundarySheaf cs 1 b basis (one_bruhat cs b)
+  have h := B.localized_boundary_scaledImages basis hmin φ e hx
+  dsimp only
+  intro z
+  have hh := h z
+  change (∃ v ∈ (B.localizedCoset basis a₀ b hmin D.sheaf.forget).upwardKernel e.val.1
+      (B.scheduledOutgoing hmin φ _ e.val.1 hx), _) ↔
+    ∃ w ∈ (B.localizedCoset basis a₀ b hmin D.sheaf.forget).upwardKernel e.val.2
+      (B.scheduledOutgoing hmin φ _ e.val.2 (B.cosetEdge_target_upper hmin e hx)), _ at hh
+  rw [B.scheduledOutgoing_eq hmin φ,B.scheduledOutgoing_eq hmin φ,
+    B.localizedCoset_kernel basis hmin D.sheaf.forget D.quotient,
+    B.localizedCoset_kernel basis hmin D.sheaf.forget D.quotient] at hh
+  exact hh
+
+end
+end KLInvariance.TitsSpace.PlaneBasis
+
+end
+
+
+section
+
+/-! The local embedded image identity at every actual root-plane prime.
+The edge is arbitrary in [1,b], with source in the comparison interval. -/
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace MomentGraph MomentGraph.Sheaf
+universe u v us u' v'
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {σ : Type (max us v)} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
+  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
+  {u b : W} {u' b' : W'}
+  (φ : Interval cs u b ≃o Interval cs' u' b')
+noncomputable section
+
+ theorem boundary_plane_scaledImages (e : Edge cs 1 b)
+    (hx : BruhatLE cs u (source cs 1 b e).val)
+    (β : PositiveRoot M cs) (hβ : root cs 1 b e ≠ β) :
+    let P := rootPlane (root cs 1 b e) β
+    let D := (boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf.forget
+    let T := ambientLater φ (mappedEdgeRoot φ e hx)
+    (D.baseChange (PlaneRing basis P)).ScaledImages e {f | f.val ∈ T} {f | f.val ∈ T}
+      (algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P)
+        (edgeFactorLift cs basis (source cs 1 b e).val (root cs 1 b e))) := by
+  classical
+  let P := rootPlane (root cs 1 b e) β
+  let B := (nonempty_planeBasis (root cs 1 b e) β hβ).some
+  obtain ⟨a₀,hmin,d,hd⟩ := B.exists_coset_edge e (Submodule.subset_span (by simp))
+  have hdx : BruhatLE cs u ((d.val.1.val:W)*a₀) := by
+    change BruhatLE cs u (source cs 1 b (B.cosetEdgeAmbient a₀ b hmin d)).val
+    rwa [hd]
+  have h := B.ambient_boundary_scaledImages hmin basis φ d hdx
+  dsimp only at h ⊢
+  rw [B.mappedCosetRoot_eq hmin φ d hdx,
+    show B.edgeRoot d.property=root cs 1 b (B.cosetEdgeAmbient a₀ b hmin d) from
+      (B.cosetRoot_eq a₀ b hmin d).symm] at h
+  change ((boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf.forget.baseChange
+    (PlaneRing basis P)).ScaledImages (B.cosetEdgeAmbient a₀ b hmin d)
+    {f | f.val ∈ ambientLater φ (mappedEdgeRoot φ (B.cosetEdgeAmbient a₀ b hmin d) hdx)}
+    {f | f.val ∈ ambientLater φ (mappedEdgeRoot φ (B.cosetEdgeAmbient a₀ b hmin d) hdx)}
+    (algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P)
+      (edgeFactorLift cs basis (source cs 1 b (B.cosetEdgeAmbient a₀ b hmin d)).val
+        (root cs 1 b (B.cosetEdgeAmbient a₀ b hmin d)))) at h
+  have h' := h
+  simp only [hd] at h'
+  exact Eq.mp (congrArg (fun z : Edge cs 1 b =>
+    ((boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf.forget.baseChange
+      (PlaneRing basis P)).ScaledImages z
+      {f | f.val ∈ ambientLater φ (mappedEdgeRoot φ e hx)}
+      {f | f.val ∈ ambientLater φ (mappedEdgeRoot φ e hx)}
+      (algebraMap (MvPolynomial σ ℝ) (PlaneRing basis P)
+        (edgeFactorLift cs basis (source cs 1 b e).val (root cs 1 b e)))) hd) h'
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace EdgeAlgebra
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ abbrev EdgeRing (a : PositiveRoot M cs) :=
+   MvPolynomial σ ℝ ⧸ Ideal.span {polynomialRoot cs basis a}
+
+ def edgeRingEquiv (a : PositiveRoot M cs) : EdgeRing cs basis a ≃ₐ[ℝ] EdgeCoefficient a :=
+   (Ideal.quotientEquivAlg (rootIdeal (k := ℝ) (embed M a.val))
+     (Ideal.span {polynomialRoot cs basis a}) (SymmetricAlgebra.equivMvPolynomial basis)
+     (by simp only [rootIdeal,Ideal.map_span,Set.image_singleton,polynomialRoot]; rfl)).symm
+
+ @[simp] theorem edgeRingEquiv_root (a β : PositiveRoot M cs) :
+    edgeRingEquiv cs basis a (Ideal.Quotient.mk _ (polynomialRoot cs basis β)) =
+      Ideal.Quotient.mk _ (SymmetricAlgebra.ι ℝ (Extended M) (embed M β.val)) := by
+  apply (edgeRingEquiv cs basis a).symm.injective
+  simp only [AlgEquiv.symm_apply_apply]
+  rfl
+
+ instance edgeRing_isDomain (a : PositiveRoot M cs) : IsDomain (EdgeRing cs basis a) := by
+   let : (Ideal.span {polynomialRoot cs basis a}).IsPrime :=
+     Ideal.isPrime_span_singleton_of_prime (polynomialRoot_prime cs basis a)
+   infer_instance
+
+ instance edgeRing_ufm (a : PositiveRoot M cs) : UniqueFactorizationMonoid (EdgeRing cs basis a) :=
+   ((edgeRingEquiv cs basis a).trans (edgePolynomial a)).toMulEquiv.symm.uniqueFactorizationMonoid inferInstance
+
+ theorem reducedPolynomialRoot_prime (a β : PositiveRoot M cs) (hne : a≠β) :
+    Prime (Ideal.Quotient.mk (Ideal.span {polynomialRoot cs basis a})
+      (polynomialRoot cs basis β)) := by
+  apply (MulEquiv.prime_iff (edgeRingEquiv cs basis a)).mp
+  rw [edgeRingEquiv_root]
+  exact reducedRoot_prime _ _ (embed_not_rootLine a β hne.symm)
+
+ theorem planePrime_span (a β : PositiveRoot M cs) :
+    planePrime basis (rootPlane a β) =
+      Ideal.span {polynomialRoot cs basis a,polynomialRoot cs basis β} := by
+  simp only [planePrime,rootPlane,Submodule.map_span,Set.image_insert_eq,
+    Set.image_singleton,SymmetricPlaneIdeal.ideal_span_eq,Ideal.map_span,
+    polynomialRoot,embedLinear]
+  rfl
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+/-! The scalar quotient of an actual finite free presentation is free over
+its quotient coefficient ring. There is no quotient-freeness hypothesis. -/
+namespace KLInvariance.ScalarQuotient
+open Module
+universe ur um un ui
+variable {R : Type ur} [CommRing R]
+  {M : Type um} [AddCommGroup M] [Module R M]
+  {N : Type un} [AddCommGroup N] [Module R N]
+  {ι : Type ui} [Fintype ι]
+noncomputable section
+
+ def coordinateReduction (b : Basis ι R M) (a : R) :
+    M →ₗ[R] (ι → R ⧸ Ideal.span {a}) :=
+  (LinearMap.pi fun i => (Ideal.span ({a} : Set R)).mkQ.comp
+    ((LinearMap.proj i).comp b.equivFun.toLinearMap))
+
+ theorem coordinateReduction_surjective (b : Basis ι R M) (a : R) :
+    Function.Surjective (coordinateReduction b a) := by
+  intro v
+  choose w hw using fun i => Ideal.Quotient.mk_surjective (v i)
+  exact ⟨b.equivFun.symm w,funext fun i => by
+    change Ideal.Quotient.mk (Ideal.span {a}) (b.equivFun (b.equivFun.symm w) i)=v i
+    simpa only [LinearEquiv.apply_symm_apply] using hw i⟩
+
+ theorem coordinateReduction_kernel (b : Basis ι R M) (a : R) (v : M) :
+    coordinateReduction b a v=0 ↔ ∃ w, v=a • w := by
+  have h : (∃ w, v=a • w) ↔ ∀ i, a ∣ b.equivFun v i :=
+    (exists_congr fun _ => eq_comm).trans (FiniteFreeScalar.exists_smul_iff b a v)
+  rw [h]
+  simp only [coordinateReduction,LinearMap.pi_apply,LinearMap.comp_apply,
+    LinearMap.proj_apply,LinearEquiv.coe_coe,funext_iff,Pi.zero_apply]
+  exact forall_congr' fun i => (Ideal.Quotient.eq_zero_iff_mem).trans Ideal.mem_span_singleton
+
+ theorem free_of_scalar_presentation (a : R)
+    [Module (R ⧸ Ideal.span {a}) N] [IsScalarTower R (R ⧸ Ideal.span {a}) N]
+    (b : Basis ι R M) (q : M →ₗ[R] N) (hq : Function.Surjective q)
+    (hk : ∀ v, q v=0 ↔ ∃ w, v=a • w) :
+    Module.Free (R ⧸ Ideal.span {a}) N := by
+  obtain ⟨j,_⟩ := QuotientEdge.exists_scalar_iso q (coordinateReduction b a) a
+    hq (coordinateReduction_surjective b a) hk (coordinateReduction_kernel b a)
+    (LinearEquiv.refl R M)
+  let k : N ≃ₗ[R ⧸ Ideal.span {a}] (ι → R ⧸ Ideal.span {a}) :=
+    { j.toAddEquiv with
+      map_smul' := by
+        intro c v
+        obtain ⟨r,rfl⟩ := Ideal.Quotient.mk_surjective c
+        change j ((algebraMap R (R ⧸ Ideal.span {a}) r) • v) =
+          (algebraMap R (R ⧸ Ideal.span {a}) r) • j v
+        simpa only [algebraMap_smul] using j.map_smul r v }
+  exact Module.Free.of_equiv k.symm
+
+ def submodule (a : R) [Module (R ⧸ Ideal.span {a}) N]
+    [IsScalarTower R (R ⧸ Ideal.span {a}) N] (S : Submodule R N) :
+    Submodule (R ⧸ Ideal.span {a}) N where
+  carrier := S
+  add_mem' := S.add_mem
+  zero_mem' := S.zero_mem
+  smul_mem' c v hv := by
+    obtain ⟨r,rfl⟩ := Ideal.Quotient.mk_surjective c
+    change (algebraMap R (R ⧸ Ideal.span {a}) r) • v ∈ S
+    simpa only [algebraMap_smul] using S.smul_mem r hv
+
+ theorem saturated_image_free (a : R)
+    [Module (R ⧸ Ideal.span {a}) N] [IsScalarTower R (R ⧸ Ideal.span {a}) N]
+    (q : M →ₗ[R] N) (hk : ∀ v, q v=0 ↔ ∃ w, v=a • w)
+    (L : Submodule R M) [Module.Free R L] [Module.Finite R L]
+    (hsat : ∀ v, a • v ∈ L → v ∈ L) :
+    Module.Free (R ⧸ Ideal.span {a}) (submodule a (L.map q)) := by
+  let f : L →ₗ[R] submodule a (L.map q) :=
+    (q.comp L.subtype).codRestrict ((submodule a (L.map q)).restrictScalars R)
+      (fun v => ⟨v.val,v.property,rfl⟩)
+  apply free_of_scalar_presentation a (Module.Free.chooseBasis R L) f
+  · rintro ⟨v,hv⟩
+    obtain ⟨w,hw,rfl⟩ := hv
+    exact ⟨⟨w,hw⟩,rfl⟩
+  · intro v
+    constructor
+    · intro hv
+      obtain ⟨w,hw⟩ := (hk v.val).mp (congrArg Subtype.val hv)
+      exact ⟨⟨w,hsat w (hw ▸ v.property)⟩,Subtype.ext hw⟩
+    · rintro ⟨w,rfl⟩
+      exact Subtype.ext ((hk _).mpr ⟨w.val,rfl⟩)
+
+end
+end KLInvariance.ScalarQuotient
+
+end
+
+
+section
+
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
+  {σ : Type us} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  (hub : BruhatLE cs u b)
+  (B : BoundarySheaf (polynomialGrading_negative ℝ σ)
+    (graph cs u b) (polynomialLabel cs u b basis) ⟨b,hub,bruhat_refl cs b⟩)
+noncomputable section
+
+omit [Fintype σ] in
+ theorem edgeTorsion (e : Edge cs u b) :
+    Module.IsTorsionBy (MvPolynomial σ ℝ) (B.sheaf.edge e) (polynomialRoot cs basis (root cs u b e)) :=
+   fun v => GradedSheaf.UpperQuotient.annihilates B.sheaf B.quotient e v
+
+ @[instance_reducible] def edgeModule (e : Edge cs u b) :
+    Module (EdgeRing cs basis (root cs u b e)) (B.sheaf.edge e) :=
+   (edgeTorsion cs u b basis hub B e).module
+
+omit [Fintype σ] in
+ theorem edgeModule_tower (e : Edge cs u b) :
+    let _ := edgeModule cs u b basis hub B e
+    IsScalarTower (MvPolynomial σ ℝ) (EdgeRing cs basis (root cs u b e)) (B.sheaf.edge e) := by
+   dsimp only [edgeModule,Module.IsTorsionBy.module]
+   exact Module.IsTorsionBySet.isScalarTower _
+
+omit [Fintype σ] in
+ theorem edgeModule_free (e : Edge cs u b) :
+    let _ := edgeModule cs u b basis hub B e
+    Module.Free (EdgeRing cs basis (root cs u b e)) (B.sheaf.edge e) := by
+   let _ := edgeModule cs u b basis hub B e
+   let _ := edgeModule_tower cs u b basis hub B e
+   let _ := B.free (target cs u b e)
+   exact ScalarQuotient.free_of_scalar_presentation _
+     (Module.Free.chooseBasis (MvPolynomial σ ℝ) (B.sheaf.vertex (target cs u b e)))
+     (B.sheaf.upper e).map (B.quotient e).1 (B.quotient e).2
+
+ theorem upperImage_free (e : Edge cs u b)
+    (F : Set (Sheaf.Outgoing (G := graph cs u b) (target cs u b e)))
+    [Module.Free (MvPolynomial σ ℝ) (B.sheaf.upwardSubmodule (target cs u b e) F)] :
+    let _ := edgeModule cs u b basis hub B e
+    let _ := edgeModule_tower cs u b basis hub B e
+    Module.Free (EdgeRing cs basis (root cs u b e))
+      (ScalarQuotient.submodule (polynomialRoot cs basis (root cs u b e))
+        ((B.sheaf.upwardSubmodule (target cs u b e) F).map (B.sheaf.upper e).map)) := by
+   let _ := edgeModule cs u b basis hub B e
+   let _ := edgeModule_tower cs u b basis hub B e
+   let _ : Module.Finite (MvPolynomial σ ℝ) (B.sheaf.vertex (target cs u b e)) :=
+     (B.sheaf.vertex (target cs u b e)).finite
+   let hfinite : Module.Finite (MvPolynomial σ ℝ) (B.sheaf.upwardSubmodule (target cs u b e) F) :=
+     Module.Finite.of_injective (B.sheaf.upwardSubmodule (target cs u b e) F).subtype Subtype.val_injective
+   dsimp only
+   let hfree := (inferInstance : Module.Free (MvPolynomial σ ℝ) (B.sheaf.upwardSubmodule (target cs u b e) F))
+   exact @ScalarQuotient.saturated_image_free (MvPolynomial σ ℝ) _
+     (B.sheaf.vertex (target cs u b e)) _ _ (B.sheaf.edge e) _ _
+     (polynomialRoot cs basis (root cs u b e)) _ _
+     (B.sheaf.upper e).map (B.quotient e).2 (B.sheaf.upwardSubmodule (target cs u b e) F)
+     hfree hfinite (by
+
+     exact B.sheaf.upwardSubmodule_saturated (target cs u b e) F _
+            (fun f _ => incoming_label_regular_outgoing cs u b basis hub B e f.val f.property))
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace EdgeAlgebra
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ theorem reducedRoot_dvd_iff_mem_plane (a β : PositiveRoot M cs) (z : MvPolynomial σ ℝ) :
+    Ideal.Quotient.mk (Ideal.span {polynomialRoot cs basis a}) (polynomialRoot cs basis β) ∣
+      Ideal.Quotient.mk (Ideal.span {polynomialRoot cs basis a}) z ↔
+    z ∈ planePrime basis (rootPlane a β) := by
+  rw [← Ideal.mem_span_singleton]
+  change z ∈ Ideal.comap (Ideal.Quotient.mk (Ideal.span {polynomialRoot cs basis a}))
+    (Ideal.span {Ideal.Quotient.mk (Ideal.span {polynomialRoot cs basis a})
+      (polynomialRoot cs basis β)}) ↔ _
+  have hi : Ideal.map (Ideal.Quotient.mk (Ideal.span {polynomialRoot cs basis a}))
+      (Ideal.span {polynomialRoot cs basis β}) =
+      Ideal.span {Ideal.Quotient.mk (Ideal.span {polynomialRoot cs basis a})
+        (polynomialRoot cs basis β)} := by rw [Ideal.map_span,Set.image_singleton]
+  rw [← hi,Ideal.comap_map_quotientMk,← Ideal.span_union,Set.singleton_union,planePrime_span]
+
+ theorem reducedFactor_ne_zero (x : W) (a : PositiveRoot M cs) :
+    Ideal.Quotient.mk (Ideal.span {polynomialRoot cs basis a})
+      (edgeFactorLift cs basis x a) ≠ 0 := by
+  simpa only [ne_eq,Ideal.Quotient.eq_zero_iff_mem,Ideal.mem_span_singleton] using
+    edgeFactorLift_not_dvd cs basis x a
+
+ theorem prime_dvd_reducedFactor (x : W) (a : PositiveRoot M cs)
+    (p : EdgeRing cs basis a) (hp : Prime p)
+    (hd : p ∣ Ideal.Quotient.mk _ (edgeFactorLift cs basis x a)) :
+    ∃ β : PositiveRoot M cs, a ≠ β ∧
+      Associated p (Ideal.Quotient.mk _ (polynomialRoot cs basis β)) := by
+  classical
+  simp only [edgeFactorLift,map_prod,map_pow] at hd
+  obtain ⟨P,_,hP⟩ := (hp.dvd_finsetProd_iff _).mp hd
+  exact ⟨planeRoot x a P,(planeRoot_ne x a P).symm,
+    hp.irreducible.associated_of_dvd
+      (reducedPolynomialRoot_prime cs basis a _ (planeRoot_ne x a P).symm).irreducible
+      (hp.dvd_of_dvd_pow hP)⟩
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.Sheaf
+universe ur uv ue um
+variable {R : Type ur} [CommRing R] {V : Type uv} [PartialOrder V]
+  {E : Type ue} [Finite E] {G : OrderedGraph V E}
+  (B : Sheaf.{ur,uv,ue,um} (R := R) G)
+noncomputable section
+open scoped Classical
+
+ def outgoingLabelProduct (α : E → R) (x : V) (F : Set (Outgoing (G := G) x)) : R :=
+   ∏ f ∈ F.toFinite.toFinset, α f.val
+
+ theorem outgoingLabelProduct_dvd (α : E → R) (x : V) (F : Set (Outgoing (G := G) x))
+    (f : Outgoing (G := G) x) (hf : f ∈ F) : α f.val ∣ outgoingLabelProduct α x F := by
+   exact Finset.dvd_prod_of_mem _ (F.toFinite.mem_toFinset.mpr hf)
+
+ theorem outgoingLabelProduct_smul_mem (α : E → R) (hα : ∀ e (v : B.edge e), α e • v=0)
+    (x : V) (F : Set (Outgoing (G := G) x)) (v : B.vertex x) :
+    outgoingLabelProduct α x F • v ∈ B.upwardKernel x F := by
+   intro f hf
+   obtain ⟨r,hr⟩ := outgoingLabelProduct_dvd α x F f hf
+   rw [map_smul,hr,mul_comm,mul_smul]
+   have h := hα f.val (B.stalkMap x v f)
+   change r • (α f.val • B.stalkMap x v f) = 0
+   rw [h,smul_zero]
+
+ theorem outgoingLabelProduct_smul_upperImage (α : E → R) (hα : ∀ e (v : B.edge e), α e • v=0)
+    (e : E) (hq : Function.Surjective (B.upper e))
+    (F : Set (Outgoing (G := G) (G.target e))) (v : B.edge e) :
+    outgoingLabelProduct α (G.target e) F • v ∈
+      (B.upwardKernel (G.target e) F).map (B.upper e) := by
+   obtain ⟨w,rfl⟩ := hq v
+   exact ⟨_,B.outgoingLabelProduct_smul_mem α hα _ F w,(B.upper e).map_smul _ w⟩
+
+end
+end KLInvariance.MomentGraph.Sheaf
+
+end
+
+
+section
+
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace MomentGraph MomentGraph.Sheaf
+universe u v us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W) (u b : W)
+  {σ : Type us} (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ theorem incoming_root_ne_outgoing (e f : Edge cs u b)
+    (h : source cs u b f=target cs u b e) : root cs u b e ≠ root cs u b f := by
+  intro hr
+  exact source_ne_target_of_label_eq cs u b (root_injective_on_labels cs u b hr.symm) h
+
+ theorem reducedKernelProduct_ne_zero (e : Edge cs u b)
+    (F : Set (Outgoing (G := graph cs u b) (target cs u b e))) :
+    Ideal.Quotient.mk (Ideal.span {polynomialRoot cs basis (root cs u b e)})
+      (outgoingLabelProduct (polynomialLabel cs u b basis) (target cs u b e) F) ≠ 0 := by
+  classical
+  rw [outgoingLabelProduct,map_prod]
+  apply Finset.prod_ne_zero_iff.mpr
+  intro f _
+  exact (reducedPolynomialRoot_prime cs basis _ _
+    (incoming_root_ne_outgoing cs u b e f.val f.property)).ne_zero
+
+ theorem prime_dvd_reducedKernelProduct (e : Edge cs u b)
+    (F : Set (Outgoing (G := graph cs u b) (target cs u b e)))
+    (p : EdgeRing cs basis (root cs u b e)) (hp : Prime p)
+    (hd : p ∣ Ideal.Quotient.mk _
+      (outgoingLabelProduct (polynomialLabel cs u b basis) (target cs u b e) F)) :
+    ∃ β : PositiveRoot M cs, root cs u b e ≠ β ∧
+      Associated p (Ideal.Quotient.mk _ (polynomialRoot cs basis β)) := by
+  classical
+  rw [outgoingLabelProduct,map_prod] at hd
+  obtain ⟨f,_,hf⟩ := (hp.dvd_finsetProd_iff _).mp hd
+  exact ⟨root cs u b f.val,incoming_root_ne_outgoing cs u b e f.val f.property,
+    hp.irreducible.associated_of_dvd
+      (reducedPolynomialRoot_prime cs basis _ _
+        (incoming_root_ne_outgoing cs u b e f.val f.property)).irreducible hf⟩
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+namespace KLInvariance.ScalarQuotient
+universe ur un
+variable {R : Type ur} [CommRing R]
+  {N : Type un} [AddCommGroup N] [Module R N]
+noncomputable section
+
+ theorem scalarImage (a z : R)
+    [Module (R ⧸ Ideal.span {a}) N] [IsScalarTower R (R ⧸ Ideal.span {a}) N]
+    (S : Submodule R N) :
+    submodule a (S.map (LinearMap.lsmul R N z)) =
+      (submodule a S).map
+        (LinearMap.lsmul (R ⧸ Ideal.span {a}) N (Ideal.Quotient.mk _ z)) := by
+  ext v
+  constructor
+  · rintro ⟨w,hw,rfl⟩
+    exact ⟨w,hw,by simp only [LinearMap.lsmul_apply]; exact algebraMap_smul _ _ w⟩
+  · rintro ⟨w,hw,rfl⟩
+    exact ⟨w,hw,by simp only [LinearMap.lsmul_apply]; exact (algebraMap_smul _ _ w).symm⟩
+
+ theorem free_scalarImage (a z : R)
+    [Module (R ⧸ Ideal.span {a}) N] [IsScalarTower R (R ⧸ Ideal.span {a}) N]
+    (S : Submodule R N) [Module.Free (R ⧸ Ideal.span {a}) (submodule a S)]
+    (hz : Function.Injective (fun v : N => z • v)) :
+    Module.Free (R ⧸ Ideal.span {a}) (submodule a (S.map (LinearMap.lsmul R N z))) := by
+  rw [scalarImage]
+  apply Module.Free.of_equiv (Submodule.equivMapOfInjective _ ?_ (submodule a S))
+  intro v w h
+  apply hz
+  change (algebraMap R (R ⧸ Ideal.span {a}) z) • v =
+    (algebraMap R (R ⧸ Ideal.span {a}) z) • w at h
+  simpa only [algebraMap_smul] using h
+
+end
+end KLInvariance.ScalarQuotient
+
+end
+
+
+section
+
+namespace KLInvariance.TensorLocalMembership
+open TensorProduct
+universe ur us um
+variable {R : Type ur} [CommRing R] (S : Type us) [CommRing S] [Algebra R S]
+  {M : Type um} [AddCommGroup M] [Module R M]
+  (P : Submonoid R) [IsLocalization P S]
+noncomputable section
+
+ theorem baseChange_eq_localized (L : Submodule R M) :
+    L.baseChange S = L.localized' S P (TensorProduct.mk R S M 1) := by
+  rw [Submodule.baseChange_eq_span,Submodule.localized'_eq_span]
+  rfl
+
+ theorem denominator (L : Submodule R M) (x : M)
+    (hx : 1 ⊗ₜ[R] x ∈ L.baseChange S) :
+    ∃ a : P, (a:R) • x ∈ L := by
+  rw [baseChange_eq_localized S P L] at hx
+  obtain ⟨m,hm,s,hs⟩ := hx
+  have he := (IsLocalizedModule.mk'_eq_iff).mp hs
+  change (TensorProduct.mk R S M 1) m = (s:R) • (TensorProduct.mk R S M 1) x at he
+  rw [← map_smul] at he
+  obtain ⟨t,ht⟩ := (IsLocalizedModule.eq_iff_exists P (TensorProduct.mk R S M 1)).mp he
+  refine ⟨t*s,?_⟩
+  have hh := L.smul_mem (t:R) hm
+  change (t:R) • m ∈ L at hh
+  change (t:R) • m = (t:R) • ((s:R) • x) at ht
+  simpa only [Submonoid.coe_mul,mul_smul,← ht] using hh
+
+end
+end KLInvariance.TensorLocalMembership
+
+end
+
+
+section
+
+namespace KLInvariance.MomentGraph.Sheaf
+open TensorProduct
+universe ur us uv ue um
+variable {R : Type ur} [CommRing R] (S : Type us) [CommRing S] [Algebra R S]
+  {V : Type uv} [PartialOrder V] {E : Type ue} [Finite E] {G : OrderedGraph V E}
+  (B : Sheaf.{ur,uv,ue,um} (R := R) G)
+  (P : Submonoid R) [IsLocalization P S]
+noncomputable section
+
+ theorem scaledImages_denominator
+    (e : E) (F : Set (Outgoing (G := G) (G.source e)))
+    (H : Set (Outgoing (G := G) (G.target e))) (a : R)
+    (h : (B.baseChange S).ScaledImages e F H (algebraMap R S a))
+    (x : B.edge e) (hx : x ∈ (B.upwardKernel (G.source e) F).map (B.lower e)) :
+    ∃ s : P, (s:R) • x ∈ ((B.upwardKernel (G.target e) H).map (B.upper e)).map
+      (LinearMap.lsmul R (B.edge e) a) := by
+  let _ : Module.Flat R S := IsLocalization.flat S P
+  apply TensorLocalMembership.denominator S P _ x
+  rw [FlatSubmodule.baseChange_map,B.baseChange_upperImage S]
+  have hx' : 1 ⊗ₜ[R] x ∈ ((B.baseChange S).upwardKernel (G.source e) F).map
+      ((B.baseChange S).lower e) := by
+    rw [← B.baseChange_lowerImage S]
+    exact Submodule.tmul_mem_baseChange_of_mem 1 hx
+  obtain ⟨v,hv,he⟩ := hx'
+  obtain ⟨w,hw,hwx⟩ := (h (1 ⊗ₜ[R] x)).mp ⟨v,hv,he⟩
+  refine ⟨(B.baseChange S).upper e w,⟨w,hw,rfl⟩,?_⟩
+  change (LinearMap.lsmul R (B.edge e) a).baseChange S ((B.baseChange S).upper e w) = _
+  have hg : (LinearMap.lsmul R (B.edge e) a).baseChange S =
+      LinearMap.lsmul S (S ⊗[R] B.edge e) (algebraMap R S a) := by
+    apply LinearMap.restrictScalars_injective R
+    apply TensorProduct.ext'
+    intro r v
+    change (LinearMap.lsmul R (B.edge e) a).baseChange S (r ⊗ₜ[R] v) =
+      (algebraMap R S a) • (r ⊗ₜ[R] v)
+    simp only [LinearMap.baseChange_tmul,LinearMap.lsmul_apply,
+      algebraMap_smul,TensorProduct.tmul_smul,TensorProduct.smul_tmul']
+  rw [hg]
+  exact hwx
+
+end
+end KLInvariance.MomentGraph.Sheaf
+
+end
+
+
+section
+
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace
+universe u v us un
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+  {σ : Type us} (basis : Basis σ ℝ (Extended M))
+noncomputable section
+
+ theorem mem_of_plane_denominators (a : PositiveRoot M cs)
+    {N : Type un} [AddCommGroup N] [Module (MvPolynomial σ ℝ) N]
+    [Module (EdgeRing cs basis a) N] [IsScalarTower (MvPolynomial σ ℝ) (EdgeRing cs basis a) N]
+    [Module.IsTorsionFree (EdgeRing cs basis a) N]
+    (S : Submodule (MvPolynomial σ ℝ) N)
+    [Module.Free (EdgeRing cs basis a) (ScalarQuotient.submodule (polynomialRoot cs basis a) S)]
+    (z : N) (c : MvPolynomial σ ℝ)
+    (hc : Ideal.Quotient.mk (Ideal.span {polynomialRoot cs basis a}) c ≠ 0)
+    (hcz : c • z ∈ S)
+    (hprime : ∀ p : EdgeRing cs basis a, Prime p → p ∣ Ideal.Quotient.mk _ c →
+      ∃ β : PositiveRoot M cs, a≠β ∧ Associated p (Ideal.Quotient.mk _ (polynomialRoot cs basis β)))
+    (hlocal : ∀ β : PositiveRoot M cs, a≠β →
+      ∃ s : MvPolynomial σ ℝ, s ∉ planePrime basis (rootPlane a β) ∧ s • z ∈ S) :
+    z ∈ S := by
+  apply EdgeLattice.mem_of_prime_local_membership
+    (ScalarQuotient.submodule (polynomialRoot cs basis a) S) z
+  · refine ⟨Ideal.Quotient.mk _ c,hc,?_⟩
+    change (algebraMap (MvPolynomial σ ℝ) (EdgeRing cs basis a) c) • z ∈ S
+    simpa only [algebraMap_smul] using hcz
+  · intro p hp
+    by_cases hpc : p ∣ Ideal.Quotient.mk (Ideal.span {polynomialRoot cs basis a}) c
+    · obtain ⟨β,hβ,hpβ⟩ := hprime p hp hpc
+      obtain ⟨s,hs,hsz⟩ := hlocal β hβ
+      refine ⟨Ideal.Quotient.mk _ s,?_,?_⟩
+      · rw [hpβ.dvd_iff_dvd_left,reducedRoot_dvd_iff_mem_plane]
+        exact hs
+      · change (algebraMap (MvPolynomial σ ℝ) (EdgeRing cs basis a) s) • z ∈ S
+        simpa only [algebraMap_smul] using hsz
+    · refine ⟨Ideal.Quotient.mk _ c,hpc,?_⟩
+      change (algebraMap (MvPolynomial σ ℝ) (EdgeRing cs basis a) c) • z ∈ S
+      simpa only [algebraMap_smul] using hcz
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace MomentGraph MomentGraph.Sheaf
+universe u v us u' v'
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {σ : Type (max us v)} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
+  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
+  {u b : W} {u' b' : W'}
+  (φ : Interval cs u b ≃o Interval cs' u' b')
+noncomputable section
+
+ theorem boundary_edge_image_le (e : Edge cs 1 b)
+    (hx : BruhatLE cs u (source cs 1 b e).val)
+    [Module.Free (MvPolynomial σ ℝ)
+      ((boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf.upwardSubmodule
+        (target cs 1 b e) {f | f.val ∈ ambientLater φ (mappedEdgeRoot φ e hx)})] :
+    let D := (boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf.forget
+    let T := ambientLater φ (mappedEdgeRoot φ e hx)
+    (D.upwardKernel (source cs 1 b e) {f | f.val ∈ T}).map (D.lower e) ≤
+      ((D.upwardKernel (target cs 1 b e) {f | f.val ∈ T}).map (D.upper e)).map
+        (LinearMap.lsmul (MvPolynomial σ ℝ) (D.edge e)
+          (edgeFactorLift cs basis (source cs 1 b e).val (root cs 1 b e))) := by
+  classical
+  let B := boundarySheaf cs 1 b basis (one_bruhat cs b)
+  let D := B.sheaf.forget
+  let F : Set (Outgoing (G := graph cs 1 b) (source cs 1 b e)) :=
+    {f | f.val ∈ ambientLater φ (mappedEdgeRoot φ e hx)}
+  let G : Set (Outgoing (G := graph cs 1 b) (target cs 1 b e)) :=
+    {f | f.val ∈ ambientLater φ (mappedEdgeRoot φ e hx)}
+  let a := root cs 1 b e
+  let f := edgeFactorLift cs basis (source cs 1 b e).val a
+  let J := (D.upwardKernel (target cs 1 b e) G).map (D.upper e)
+  let S := J.map (LinearMap.lsmul (MvPolynomial σ ℝ) (D.edge e) f)
+  let pG := outgoingLabelProduct (polynomialLabel cs 1 b basis) (target cs 1 b e) G
+  let _ : Module (EdgeRing cs basis a) (D.edge e) :=
+    edgeModule cs 1 b basis (one_bruhat cs b) B e
+  let _ : IsScalarTower (MvPolynomial σ ℝ) (EdgeRing cs basis a) (D.edge e) :=
+    edgeModule_tower cs 1 b basis (one_bruhat cs b) B e
+  let _ : Module.Free (EdgeRing cs basis a) (D.edge e) :=
+    edgeModule_free cs 1 b basis (one_bruhat cs b) B e
+  let _ : Module.Free (EdgeRing cs basis a)
+      (ScalarQuotient.submodule (polynomialRoot cs basis a) J) :=
+    upperImage_free cs 1 b basis (one_bruhat cs b) B e G
+  let _ : Module.Free (EdgeRing cs basis a) (ScalarQuotient.submodule (polynomialRoot cs basis a) S) :=
+    ScalarQuotient.free_scalarImage (polynomialRoot cs basis a) f J
+      (edgeFactorLift_regular cs basis 1 b (one_bruhat cs b) B e)
+  change (D.upwardKernel (source cs 1 b e) F).map (D.lower e) ≤ S
+  intro z hz
+  apply mem_of_plane_denominators cs basis a S z (f*pG)
+  · rw [map_mul]
+    exact mul_ne_zero (reducedFactor_ne_zero cs basis _ a)
+      (reducedKernelProduct_ne_zero cs 1 b basis e G)
+  · refine ⟨pG • z,?_,?_⟩
+    · exact D.outgoingLabelProduct_smul_upperImage (polynomialLabel cs 1 b basis)
+        (GradedSheaf.UpperQuotient.annihilates B.sheaf B.quotient) e (B.quotient e).1 G z
+    · change f • (pG • z) = (f*pG) • z
+      rw [mul_smul]
+  · intro p hp hd
+    rw [map_mul] at hd
+    rcases hp.dvd_mul.mp hd with hf | hG
+    · exact prime_dvd_reducedFactor cs basis _ a p hp hf
+    · exact prime_dvd_reducedKernelProduct cs 1 b basis e G p hp hG
+  · intro β hβ
+    obtain ⟨s,hs⟩ := D.scaledImages_denominator (PlaneRing basis (rootPlane a β))
+      (planePrime basis (rootPlane a β)).primeCompl e F G f
+      (boundary_plane_scaledImages basis φ e hx β hβ) z hz
+    exact ⟨s,s.property,hs⟩
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph SchedulePaths
+universe u v us u' v'
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {σ : Type (max us v)} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
+  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
+  {u b : W} {u' b' : W'} (hub : BruhatLE cs u b)
+  (φ : Interval cs u b ≃o Interval cs' u' b')
+noncomputable section
+
+ def intervalBoundary := restrictBoundary cs u b basis hub
+  (boundarySheaf cs 1 b basis (one_bruhat cs b))
+
+ theorem schedule_edge_image_le {e : Edge cs u b} {q : List (Edge cs u b)}
+    (hq : e::q <:+ chronologicalSchedule φ)
+    [hF : Module.Free (Coefficient (σ := σ))
+      ((intervalBoundary basis hub).sheaf.upwardSubmodule (target cs u b e)
+        (remainingConditions cs u b q (target cs u b e)))] :
+    (intervalBoundary basis hub).sheaf.lowerImage e
+        (remainingConditions cs u b q (source cs u b e)) ≤
+      ((intervalBoundary basis hub).sheaf.upperImage e
+        (remainingConditions cs u b q (target cs u b e))).map
+        (edgeFactorMap cs basis u b hub (intervalBoundary basis hub) e) := by
+  let B := boundarySheaf cs 1 b basis (one_bruhat cs b)
+  let a := (upperInclusion cs u b).edge e
+  let r := AdaptedDyer.rootLabel cs' (transportedEdge φ e)
+  have hr : mappedEdgeRoot φ a (source cs u b e).property.1 = r := rfl
+  have hk (x : Interval cs u b) :=
+    restrictBoundary_kernel cs u b basis hub B x (ambientLater φ r)
+  have hs : (intervalBoundary basis hub).sheaf.upwardSubmodule (source cs u b e)
+      (remainingConditions cs u b q (source cs u b e)) =
+      B.sheaf.upwardSubmodule (source cs 1 b a) {f | f.val ∈ ambientLater φ r} :=
+    (congrArg ((intervalBoundary basis hub).sheaf.upwardSubmodule (source cs u b e))
+      (remainingConditions_source_ambient φ hq)).trans (hk (source cs u b e))
+  have ht : (intervalBoundary basis hub).sheaf.upwardSubmodule (target cs u b e)
+      (remainingConditions cs u b q (target cs u b e)) =
+      B.sheaf.upwardSubmodule (target cs 1 b a) {f | f.val ∈ ambientLater φ r} :=
+    (congrArg ((intervalBoundary basis hub).sheaf.upwardSubmodule (target cs u b e))
+      (remainingConditions_target_ambient φ hq)).trans (hk (target cs u b e))
+  have hfree : Module.Free (Coefficient (σ := σ))
+      (B.sheaf.upwardSubmodule (target cs 1 b a) {f | f.val ∈ ambientLater φ r}) :=
+    (congrArg (fun N : Submodule (Coefficient (σ := σ)) (B.sheaf.vertex (target cs 1 b a)) =>
+      Module.Free (Coefficient (σ := σ)) N) ht).mp hF
+  let hf : Module.Free (Coefficient (σ := σ))
+      (B.sheaf.upwardSubmodule (target cs 1 b a)
+        {f | f.val ∈ ambientLater φ (mappedEdgeRoot φ a (source cs u b e).property.1)}) := by
+    rw [hr]
+    exact hfree
+  have h := boundary_edge_image_le basis φ a (source cs u b e).property.1
+  dsimp only at h
+  rw [hr] at h
+  have hs' := congrArg (fun N : Submodule (Coefficient (σ := σ)) (B.sheaf.vertex (source cs 1 b a)) =>
+    N.map (B.sheaf.lower a).map) hs
+  have ht' := congrArg (fun N : Submodule (Coefficient (σ := σ)) (B.sheaf.vertex (target cs 1 b a)) =>
+    (N.map (B.sheaf.upper a).map).map
+      (edgeFactorMap cs basis u b hub (intervalBoundary basis hub) e)) ht
+  exact hs'.le.trans (h.trans ht'.symm.le)
+
+ theorem schedule_comparison_of_upper_freeness
+    (hfree : ∀ e q, e::q <:+ chronologicalSchedule φ →
+      Module.Free (Coefficient (σ := σ))
+        ((intervalBoundary basis hub).sheaf.upwardSubmodule (target cs u b e)
+          (remainingConditions cs u b q (target cs u b e))))
+    {t : ℝ} (ht : 0 < t) (ht1 : t < 1) :
+    scheduleState cs u b basis hub (intervalBoundary basis hub) t [] ≤
+      dispatch (source cs u b) (target cs u b) (t⁻¹-t)
+        (scheduleState cs u b basis hub (intervalBoundary basis hub) t (chronologicalSchedule φ))
+        (transportedSchedule φ) := by
+  have h := scheduleState_le cs u b basis hub (intervalBoundary basis hub) (chronologicalSchedule φ)
+    (fun e q hq => by
+      let _ := hfree e q hq
+      exact schedule_edge_image_le basis hub φ hq) ht ht1
+  simpa only [chronologicalSchedule,List.reverse_reverse] using h
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph SchedulePaths
+universe u v u' v' us
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+  {σ : Type (max us v)} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+  {I' : Type u'} [Fintype I'] {M' : CoxeterMatrix I'}
+  {W' : Type v'} [Group W'] {cs' : CoxeterSystem M' W'}
+  {u b : W} {u' b' : W'} (hub : BruhatLE cs u b)
+  (φ : Interval cs u b ≃o Interval cs' u' b')
+noncomputable section
+
+ theorem schedule_final_of_character
+    (hc : StalkCharacterObligation cs 1 b basis (one_bruhat cs b))
+    {t : ℝ} (ht : 0 < t) (ht1 : t < 1) (x : Interval cs u b) :
+    scheduleState cs u b basis hub (intervalBoundary basis hub) t [] x =
+      scaledKL cs t x.val b := by
+  have he : remainingConditions cs u b [] x = ∅ := by
+    ext e
+    simp [remainingConditions]
+  rw [scheduleState,he]
+  unfold intervalBoundary
+  rw [normalized_emptyKernel_restrictBoundary]
+  exact normalized_emptyKernel_of_character cs 1 b basis (one_bruhat cs b)
+    (boundarySheaf cs 1 b basis (one_bruhat cs b))
+    ((upperInclusion cs u b).vertex x) (hc _) ht ht1
+
+omit [Fintype I] [Fintype σ] in
+ theorem schedule_all_conditions (x : Interval cs u b) :
+    remainingConditions cs u b (chronologicalSchedule φ) x = Set.univ := by
+  ext e
+  simp only [remainingConditions,Set.mem_ofPred_eq,Set.mem_univ,iff_true]
+  exact chronologicalSchedule_full φ e.val
+
+variable [Fintype (Edge cs 1 b)] [Fintype (Interval cs 1 b)] [DecidableEq (Interval cs 1 b)]
+
+ theorem schedule_initial_of_duality_character
+    (d : SectionDuality cs 1 b basis (one_bruhat cs b)
+      (boundarySheaf cs 1 b basis (one_bruhat cs b)))
+    (hd : ∀ i j (m n : (boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf.sections Set.univ),
+      m ∈ ((boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf.sections Set.univ).piece i →
+      n ∈ ((boundarySheaf cs 1 b basis (one_bruhat cs b)).sheaf.sections Set.univ).piece j →
+      d.pairing m n ∈ polynomialGrading ℝ σ (i+j-(cs.length b : ℤ)))
+    (hc : StalkCharacterObligation cs 1 b basis (one_bruhat cs b))
+    {t : ℝ} (ht : 0 < t) (ht1 : t < 1) (x : Interval cs u b) :
+    scheduleState cs u b basis hub (intervalBoundary basis hub) t (chronologicalSchedule φ) x =
+      scaledKL cs t⁻¹ x.val b := by
+  rw [scheduleState,schedule_all_conditions φ]
+  unfold intervalBoundary
+  rw [normalized_fullKernel_restrictBoundary]
+  exact normalized_fullKernel_of_duality_character cs b basis
+    (boundarySheaf cs 1 b basis (one_bruhat cs b)) d hd
+    ((upperInclusion cs u b).vertex x) (hc _) ht ht1
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+/-! Native endpoint normalization from the ACTUAL stalk characters. Unlike the
+KL-labelled endpoints, these need neither character recognition nor Hodge. -/
+namespace KLInvariance.BruhatGraph
+open Module TitsSpace _root_.OAI.KLInvariance.Graded MomentGraph Polynomial SchedulePaths
+universe u
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type u} [Group W] (cs : CoxeterSystem M W) (b : W)
+  {σ : Type u} [Fintype σ] (basis : Basis σ ℝ (Extended M))
+noncomputable section
+local instance actualInterval : Fintype (Interval cs 1 b) := by
+  let := interval_finite cs 1 b
+  exact Fintype.ofFinite _
+local instance actualEdges : Fintype (Edge cs 1 b) := Fintype.ofFinite _
+local instance (a : Type*) : DecidableEq a := Classical.decEq a
+
+ theorem normalized_emptyKernel_scaledStalk (x : Interval cs 1 b)
+    {t : ℝ} (ht : 0<t) (ht1 : t<1) :
+    normalizedKernel cs 1 b basis (one_bruhat cs b) (ActualBoundary cs b basis) t x ∅=
+      scaledStalk cs b basis t x := by
+  rw [normalizedKernel,emptyKernel_hilbert cs 1 b basis (one_bruhat cs b)
+    (ActualBoundary cs b basis) x (sq_pos_of_pos ht) (by nlinarith)]
+  dsimp only [stateScale,scaledStalk]
+  have hne : 1-t^2≠0 := ne_of_gt (by nlinarith : 0<1-t^2)
+  rw [inv_pow]
+  field_simp [hne,inv_pow]
+
+ theorem normalized_fullKernel_scaledStalk (x : Interval cs 1 b)
+    {t : ℝ} (ht : 0<t) (ht1 : t<1) :
+    normalizedKernel cs 1 b basis (one_bruhat cs b) (ActualBoundary cs b basis) t x Set.univ=
+      scaledStalk cs b basis t⁻¹ x := by
+  let C := ActualBoundary cs b basis
+  let := C.free x
+  let c := (exists_nonnegativeBasis (σ := σ) (C.sheaf.vertex x).piece
+    (C.sheaf.vertex x).nonneg).some
+  have hcp : c.polynomial=stalkCharacter cs 1 b basis (one_bruhat cs b) x := by
+    rw [stalkCharacter_eq_character cs 1 b basis (one_bruhat cs b) C x,
+      character_eq (C.sheaf.vertex x).piece (C.sheaf.vertex x).nonneg c]
+  rw [normalizedKernel,GradedSheaf.kernelHilbert,
+    fullKernel_eq cs 1 b basis (one_bruhat cs b) C x,
+    costalk_hilbert_of_duality cs 1 b basis (one_bruhat cs b) C (actualSectionDuality cs b basis)
+      (cs.length b) (actualSectionDuality_homogeneous cs b basis) x c
+      (sq_pos_of_pos ht) (by nlinarith),hcp,incoming_card cs b x]
+  dsimp only [stateScale,scaledStalk]
+  have hscale : t^((cs.length x.val : ℤ)-cs.length b)*
+      (t^2)^((cs.length b : ℤ)-cs.length x.val)=
+      (t⁻¹)^((cs.length x.val : ℤ)-cs.length b) := by
+    rw [show (t^2)^((cs.length b : ℤ)-cs.length x.val)=
+      t^(2*((cs.length b : ℤ)-cs.length x.val)) by rw [zpow_mul]; norm_num,
+      ←zpow_add₀ ht.ne',inv_zpow,←zpow_neg]
+    congr 1
+    omega
+  rw [show (t^2)⁻¹=(t⁻¹)^2 by rw [inv_pow]]
+  have hne : 1-t^2≠0 := ne_of_gt (by nlinarith : 0<1-t^2)
+  calc
+    _ = (t^((cs.length x.val : ℤ)-cs.length b)*
+      (t^2)^((cs.length b : ℤ)-cs.length x.val))*
+      eval₂ (Int.castRingHom ℝ) ((t⁻¹)^2) (stalkCharacter cs 1 b basis (one_bruhat cs b) x) := by
+      have hcancel : (1-t^2)^(Fintype.card σ)*((1-t^2)⁻¹)^(Fintype.card σ)=1 := by
+        rw [←mul_pow,mul_inv_cancel₀ hne,one_pow]
+      calc
+        _ = ((1-t^2)^(Fintype.card σ)*((1-t^2)⁻¹)^(Fintype.card σ))*
+            ((t^((cs.length x.val : ℤ)-cs.length b)*(t^2)^((cs.length b : ℤ)-cs.length x.val))*
+            eval₂ (Int.castRingHom ℝ) ((t⁻¹)^2) (stalkCharacter cs 1 b basis (one_bruhat cs b) x)) := by ring
+        _ = _ := by rw [hcancel,one_mul]
+    _ = _ := by rw [hscale]
+
+ theorem schedule_final_scaledStalk (a : W) (hab : BruhatLE cs a b)
+    {t : ℝ} (ht : 0<t) (ht1 : t<1) (x : Interval cs a b) :
+    scheduleState cs a b basis hab (intervalBoundary basis hab) t [] x=
+      scaledStalk cs b basis t ((upperInclusion cs a b).vertex x) := by
+  have he : remainingConditions cs a b [] x=∅ := by
+    ext e
+    simp [remainingConditions]
+  rw [scheduleState,he]
+  unfold intervalBoundary
+  rw [normalized_emptyKernel_restrictBoundary]
+  exact normalized_emptyKernel_scaledStalk cs b basis _ ht ht1
+
+ theorem schedule_initial_scaledStalk (a : W) (hab : BruhatLE cs a b)
+    {t : ℝ} (ht : 0<t) (ht1 : t<1) (x : Interval cs a b) :
+    scheduleState cs a b basis hab (intervalBoundary basis hab) t
+      (chronologicalSchedule (OrderIso.refl (Interval cs a b))) x=
+      scaledStalk cs b basis t⁻¹ ((upperInclusion cs a b).vertex x) := by
+  rw [scheduleState,schedule_all_conditions (OrderIso.refl (Interval cs a b))]
+  unfold intervalBoundary
+  rw [normalized_fullKernel_restrictBoundary]
+  exact normalized_fullKernel_scaledStalk cs b basis _ ht ht1
+
+ theorem native_endpoint_eq (a : W) (hab : BruhatLE cs a b)
+    {t : ℝ} (ht : 0<t) (ht1 : t<1) :
+    scheduleState cs a b basis hab (intervalBoundary basis hab) t []=
+      dispatch (source cs a b) (target cs a b) (t⁻¹-t)
+        (scheduleState cs a b basis hab (intervalBoundary basis hab) t
+           (chronologicalSchedule (OrderIso.refl (Interval cs a b))))
+        (chronologicalSchedule (OrderIso.refl (Interval cs a b))).reverse := by
+  have hf := funext (schedule_final_scaledStalk cs b basis a hab ht ht1)
+  have hi := funext (schedule_initial_scaledStalk cs b basis a hab ht ht1)
+  rw [hf,hi,chronologicalSchedule,List.reverse_reverse]
+  funext x
+  rw [transported_dispatch_matrix (OrderIso.refl (Interval cs a b)) _ (Int.castRingHom ℝ)]
+  exact scaledStalk_reciprocity_upper cs b basis t ht.ne' a x
+
+end
+end KLInvariance.BruhatGraph
+
+end
+
+
+section
+
+namespace KLInvariance.SchedulePaths
+universe u v
+variable {E : Type u} {Λ : Type v} [LinearOrder Λ]
+
+ theorem exists_suffix_score_gt (score : E → Λ) (r : Λ) (p : List E)
+    (hp : p.Pairwise (fun a b => score a ≤ score b)) :
+    ∃ q, q <:+ p ∧ ∀ e, e ∈ q ↔ e ∈ p ∧ r < score e := by
+  induction p with
+  | nil => exact ⟨[],List.suffix_refl _,fun e => by simp⟩
+  | cons a p ih =>
+    have hs := List.pairwise_cons.mp hp
+    by_cases ha : r < score a
+    · refine ⟨a::p,List.suffix_refl _,?_⟩
+      intro e
+      constructor
+      · intro he
+        refine ⟨he,?_⟩
+        rcases List.mem_cons.mp he with rfl | he
+        · exact ha
+        · exact lt_of_lt_of_le ha (hs.1 e he)
+      · exact And.left
+    · obtain ⟨q,hq,hmem⟩ := ih hs.2
+      refine ⟨q,hq.trans (List.suffix_cons a p),?_⟩
+      intro e
+      rw [hmem,List.mem_cons]
+      constructor
+      · exact fun h => ⟨Or.inr h.1,h.2⟩
+      · rintro ⟨he,hr⟩
+        rcases he with rfl | he
+        · exact (ha hr).elim
+        · exact ⟨he,hr⟩
+
+end KLInvariance.SchedulePaths
+
+end
+
+
+section
+
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/KazhdanLusztig/Tits.lean b/lean/OAI/RepresentationTheory/KazhdanLusztig/Tits.lean
index bc384e94329ddac0a8124d1a9768964b6a092a72..b9521ed050ed240b44bf5a6d4684d5b0757c0893 100644
--- a/lean/OAI/RepresentationTheory/KazhdanLusztig/Tits.lean
+++ b/lean/OAI/RepresentationTheory/KazhdanLusztig/Tits.lean
@@ -1,2274 +1,2274 @@
-import OAI.RepresentationTheory.KazhdanLusztig.Bruhat
-import OAI.RepresentationTheory.KazhdanLusztig.Dihedral
-import OAI.RepresentationTheory.KazhdanLusztig.ReflectionPlane
-import OAI.RepresentationTheory.KazhdanLusztig.Sheaves
-
-namespace OAI
-
-/-!
-Actual Bruhat graph labels, Tits realizations and extended root spaces, cotangent coordinates and standard parabolic support.
--/
-
-section
-
-namespace KLInvariance
-namespace BruhatGraph
-universe u v
-variable {B : Type u} {W : Type v} [Group W] {M : CoxeterMatrix B}
-  (cs : CoxeterSystem M W) (u b : W)
-
-/-- Actual finite directed moment-graph edges, including the non-cover
-reflection edges. No reconstruction from the abstract poset is assumed. -/
-def Edge := {p : Interval cs u b × Interval cs u b // BruhatStep cs p.1.val p.2.val}
-
-instance : Finite (Edge cs u b) := by
-  let := interval_finite cs u b
-  unfold Edge
-  infer_instance
-
-def source (e : Edge cs u b) : Interval cs u b := e.val.1
-def target (e : Edge cs u b) : Interval cs u b := e.val.2
-
-def graph : MomentGraph.OrderedGraph (Interval cs u b) (Edge cs u b) where
-  source := source cs u b
-  target := target cs u b
-  increasing e := by
-    refine ⟨Relation.ReflTransGen.single e.property, ?_⟩
-    intro h
-    exact (not_lt_of_ge (length_le_of_bruhat cs h)) e.property.1
-
-/-- Genuine left reflection label; uniquely determined by the endpoints. -/
-def label (e : Edge cs u b) : W := (target cs u b e).val * (source cs u b e).val⁻¹
-
-theorem label_isReflection (e : Edge cs u b) : cs.IsReflection (label cs u b e) := by
-  obtain ⟨t, ht, htx⟩ := e.property.2
-  simpa [label, source, target, htx, mul_assoc] using ht
-
-@[simp] theorem label_mul_source (e : Edge cs u b) :
-    label cs u b e * (source cs u b e).val = (target cs u b e).val := by
-  simp [label, mul_assoc]
-
-@[simp] theorem label_mul_target (e : Edge cs u b) :
-    label cs u b e * (target cs u b e).val = (source cs u b e).val := by
-  rw [← label_mul_source cs u b e, ← mul_assoc,
-    (label_isReflection cs u b e).mul_self, one_mul]
-
-theorem eq_of_label_source_eq {e f : Edge cs u b}
-    (hl : label cs u b e = label cs u b f)
-    (hs : source cs u b e = source cs u b f) : e = f := by
-  have ht : target cs u b e = target cs u b f := by
-    apply Subtype.ext
-    rw [← label_mul_source cs u b e, ← label_mul_source cs u b f, hl, hs]
-  apply Subtype.ext
-  exact Prod.ext hs ht
-
-theorem eq_of_label_target_eq {e f : Edge cs u b}
-    (hl : label cs u b e = label cs u b f)
-    (ht : target cs u b e = target cs u b f) : e = f := by
-  have hs : source cs u b e = source cs u b f := by
-    apply Subtype.ext
-    rw [← label_mul_target cs u b e, ← label_mul_target cs u b f, hl, ht]
-  exact eq_of_label_source_eq cs u b hl hs
-
-theorem source_ne_target_of_label_eq {e f : Edge cs u b}
-    (hl : label cs u b e = label cs u b f) :
-    source cs u b e ≠ target cs u b f := by
-  intro h
-  have ht : target cs u b e = source cs u b f := by
-    apply Subtype.ext
-    rw [← label_mul_source cs u b e, hl, h, label_mul_target]
-  have he := (graph cs u b).increasing e
-  have hf := (graph cs u b).increasing f
-  change source cs u b e < target cs u b e at he
-  change source cs u b f < target cs u b f at hf
-  rw [h, ht] at he
-  exact lt_asymm he hf
-
-
-theorem endpoints_disjoint_of_label_eq {e f : Edge cs u b} (hef : e ≠ f)
-    (hl : label cs u b e = label cs u b f) :
-    Disjoint ({source cs u b e, target cs u b e} : Set _)
-      {source cs u b f, target cs u b f} := by
-  rw [Set.disjoint_left]
-  intro x hx hy
-  simp only [Set.mem_insert_iff, Set.mem_singleton_iff] at hx hy
-  rcases hx with rfl | rfl <;> rcases hy with hy | hy
-  · exact hef (eq_of_label_source_eq cs u b hl hy)
-  · exact source_ne_target_of_label_eq cs u b hl hy
-  · exact source_ne_target_of_label_eq cs u b hl.symm hy.symm
-  · exact hef (eq_of_label_target_eq cs u b hl hy)
-
-end BruhatGraph
-end KLInvariance
-
-
-namespace KLInvariance.Realization
-open Module
-variable {V : Type*} [AddCommGroup V] [Module ℝ V]
-
-/-- An actual rank-one perturbation that preserves a nondegenerate symmetric
-form is an orthogonal reflection, not a unipotent transvection. -/
-theorem orthogonal_rank_one_form
-    (B : LinearMap.BilinForm ℝ V) (hs : B.IsSymm) (hn : B.Nondegenerate)
-    (u : V) (hu : u ≠ 0) (f : Module.Dual ℝ V) (hf : f ≠ 0)
-    (hg : ∀ v w, B (v + f v • u) (w + f w • u) = B v w) :
-    f u = -2 ∧ B u u ≠ 0 ∧
-      ∀ v, v + f v • u = Module.preReflection u ((2 / B u u) • B u) v := by
-  obtain ⟨v, hv⟩ : ∃ v, f v = 1 := by
-    have hex : ∃ w, f w ≠ 0 := by
-      by_contra! h
-      exact hf (LinearMap.ext h)
-    obtain ⟨w, hw⟩ := hex
-    exact ⟨(f w)⁻¹ • w, by simp [hw]⟩
-  have hz (z : V) (h : f z = 0) : B u z = 0 := by
-    have h' := hg v z
-    simpa [hv, h, map_add] using h'
-  have hprop (z : V) : B u z = f z * B u v := by
-    have h := hz (z - f z • v) (by simp [hv])
-    exact sub_eq_zero.mp (by simpa [map_sub, map_smul] using h)
-  have hc : B u v ≠ 0 := by
-    intro h
-    apply hu
-    exact hn.1 u (fun z => by rw [hprop z, h, mul_zero])
-  have hself := hg v v
-  simp only [hv, one_smul, map_add, LinearMap.add_apply] at hself
-  rw [hs.eq v u] at hself
-  have hfu : f u = -2 := by
-    rw [hprop u] at hself
-    apply (mul_right_cancel₀ hc)
-    linarith
-  have hne : B u u ≠ 0 := by
-    rw [hprop u, hfu]
-    exact mul_ne_zero (by norm_num) hc
-  refine ⟨hfu, hne, fun z => ?_⟩
-  rw [Module.preReflection_apply]
-  have heq : ((2 / B u u) • B u) z = -f z := by
-    simp only [LinearMap.smul_apply, smul_eq_mul]
-    rw [hprop z, hprop u, hfu]
-    field_simp [hc]
-  rw [heq, neg_smul, sub_neg_eq_add]
-
-/-- This is the rank-one assertion in `bg:realization` stated without a chosen
-rank-one normal form. Its conclusion uses the library's actual reflection. -/
-theorem orthogonal_finrank_one
-    (B : LinearMap.BilinForm ℝ V) (hs : B.IsSymm) (hn : B.Nondegenerate)
-    (g : V ≃ₗ[ℝ] V) (hg : ∀ v w, B (g v) (g w) = B v w)
-    (hr : Module.finrank ℝ (LinearMap.range (g.toLinearMap - LinearMap.id)) = 1) :
-    ∃ u : V, B u u ≠ 0 ∧
-      ∀ v, g v = Module.preReflection u ((2 / B u u) • B u) v := by
-  let d : V →ₗ[ℝ] V := g.toLinearMap - LinearMap.id
-  obtain ⟨e⟩ := nonempty_linearEquiv_of_finrank_eq_one hr
-  let u : V := (e 1).val
-  let f : Module.Dual ℝ V := e.symm.toLinearMap.comp d.rangeRestrict
-  have hu : u ≠ 0 := by
-    intro h
-    have : e 1 = 0 := Subtype.ext h
-    have := e.injective (this.trans e.map_zero.symm)
-    norm_num at this
-  have hform (v : V) : f v • u = d v := by
-    change f v • (e 1).val = d v
-    have h := congrArg Subtype.val (e.apply_symm_apply (d.rangeRestrict v))
-    have he : e (f v) = f v • e 1 := by
-      rw [← map_smul, smul_eq_mul, mul_one]
-    rw [show e.symm (d.rangeRestrict v) = f v from rfl, he] at h
-    exact h
-  have hf : f ≠ 0 := by
-    intro h
-    obtain ⟨v, hv⟩ := (e 1).property
-    have hh := hform v
-    rw [h, LinearMap.zero_apply, zero_smul, hv] at hh
-    exact hu hh.symm
-  have hgform (v : V) : g v = v + f v • u := by
-    rw [hform]
-    change g v = v + (g v - v)
-    abel
-  obtain ⟨_, hnonzero, hreflection⟩ := orthogonal_rank_one_form B hs hn u hu f hf
-    (fun v w => by rw [← hgform, ← hgform]; exact hg v w)
-  exact ⟨u, hnonzero, fun v => (hgform v).trans (hreflection v)⟩
-end KLInvariance.Realization
-
-/-! The finite Coxeter relations for the actual real geometric reflection
-operators required in `bg:realization`. Faithfulness is not assumed or claimed. -/
-namespace KLInvariance.Realization
-open Module Polynomial.Chebyshev
-variable {V : Type*} [AddCommGroup V] [Module ℝ V]
-
-private theorem reflection_pow_even_eq_one
-    {α β : V} {f g : Module.Dual ℝ V} (hf : f α = 2) (hg : g β = 2)
-    (k : ℕ) (hS : (S ℝ ((k : ℤ) - 1)).eval (f β * g α - 2) = 0) :
-    (Module.reflection hf * Module.reflection hg) ^ (2 * k) = 1 := by
-  ext v
-  rw [Module.reflection_mul_reflection_pow_apply hf hg]
-  have h₁ : ((↑(2 * k) : ℤ) - 2) / 2 = (k : ℤ) - 1 := by omega
-  have h₂ : ((↑(2 * k) : ℤ) - 1) / 2 = (k : ℤ) - 1 := by omega
-  rw [h₁, h₂, hS]
-  simp
-
-private theorem reflection_pow_odd_eq_one
-    {α β : V} {f g : Module.Dual ℝ V} (hf : f α = 2) (hg : g β = 2)
-    (k : ℕ)
-    (hS : (S ℝ (k : ℤ)).eval (f β * g α - 2) +
-      (S ℝ ((k : ℤ) - 1)).eval (f β * g α - 2) = 0) :
-    (Module.reflection hf * Module.reflection hg) ^ (2 * k + 1) = 1 := by
-  ext v
-  rw [Module.reflection_mul_reflection_pow_apply hf hg]
-  have h₀ : (↑(2 * k + 1) : ℤ) / 2 = (k : ℤ) := by omega
-  have h₁ : ((↑(2 * k + 1) : ℤ) - 2) / 2 = (k : ℤ) - 1 := by omega
-  have h₂ : ((↑(2 * k + 1) : ℤ) - 1) / 2 = (k : ℤ) := by omega
-  have h₃ : ((↑(2 * k + 1) : ℤ) - 3) / 2 = (k : ℤ) - 1 := by omega
-  rw [h₀, h₁, h₂, h₃, hS]
-  simp
-
-/-- If the two simple roots have the exact Tits pairing, the corresponding
-linear equivalences satisfy their finite Coxeter relation on the entire
-ambient space, not just on the root plane. -/
-theorem geometric_reflection_relation
-    {α β : V} {f g : Module.Dual ℝ V} (hf : f α = 2) (hg : g β = 2)
-    (m : ℕ) (hm : 2 ≤ m)
-    (hfg : f β = -2 * Real.cos (Real.pi / m))
-    (hgf : g α = -2 * Real.cos (Real.pi / m)) :
-    (Module.reflection hf * Module.reflection hg) ^ m = 1 := by
-  by_cases hm2 : m = 2
-  · subst m
-    have hfg0 : f β = 0 := by simpa using hfg
-    have hgf0 : g α = 0 := by simpa using hgf
-    ext v
-    change ((Module.reflection hf * Module.reflection hg) ^ 2) v = v
-    simp only [pow_two, LinearEquiv.mul_apply, Module.reflection_apply,
-      map_sub, map_smul, hf, hg, hfg0, hgf0, ]
-    module
-  have hm3 : 2 < m := by omega
-  have hmR : (2 : ℝ) < m := by exact_mod_cast hm3
-  let θ : ℝ := Real.pi / m
-  have hθ : (m : ℝ) * θ = Real.pi := by dsimp [θ]; field_simp
-  have hθpos : 0 < θ := div_pos Real.pi_pos (by positivity)
-  have hθlt : 2 * θ < Real.pi := by
-    dsimp [θ]
-    rw [← mul_div_assoc]
-    apply (div_lt_iff₀ (show (0 : ℝ) < m by positivity)).2
-    nlinarith [Real.pi_pos]
-  have hsin : Real.sin (2 * θ) ≠ 0 :=
-    ne_of_gt (Real.sin_pos_of_pos_of_lt_pi (by positivity) hθlt)
-  have ht : f β * g α - 2 = 2 * Real.cos (2 * θ) := by
-    rw [hfg, hgf, Real.cos_two_mul]
-    dsimp [θ]
-    ring
-  rcases Nat.even_or_odd m with he | ho
-  · obtain ⟨k, hk⟩ := he
-    have hm' : m = 2 * k := by omega
-    clear hk
-    subst m
-    apply reflection_pow_even_eq_one hf hg k
-    rw [ht]
-    have h := S_two_mul_real_cos (2 * θ) ((k : ℤ) - 1)
-    have ha : (((k : ℤ) - 1 : ℤ) + 1 : ℝ) * (2 * θ) = Real.pi := by
-      push_cast
-      push_cast at hθ
-      nlinarith [hθ]
-    rw [ha, Real.sin_pi] at h
-    exact (mul_eq_zero.mp h).resolve_right hsin
-  · obtain ⟨k, hk⟩ := ho
-    have hm' : m = 2 * k + 1 := by omega
-    clear hk
-    subst m
-    apply reflection_pow_odd_eq_one hf hg k
-    rw [ht]
-    have h₁ := S_two_mul_real_cos (2 * θ) (k : ℤ)
-    have h₂ := S_two_mul_real_cos (2 * θ) ((k : ℤ) - 1)
-    have ha : ((k : ℤ) + 1 : ℝ) * (2 * θ) = Real.pi + θ := by
-      push_cast
-      push_cast at hθ
-      nlinarith [hθ]
-    have hb : (((k : ℤ) - 1 : ℤ) + 1 : ℝ) * (2 * θ) = Real.pi - θ := by
-      push_cast
-      push_cast at hθ
-      nlinarith [hθ]
-    rw [ha] at h₁
-    simp only [Real.sin_add, Real.sin_pi, Real.cos_pi, zero_mul, neg_one_mul,
-      zero_add] at h₁
-    rw [hb, Real.sin_pi_sub] at h₂
-    apply (mul_eq_zero.mp (show
-      ((S ℝ (k : ℤ)).eval (2 * Real.cos (2 * θ)) +
-      (S ℝ ((k : ℤ) - 1)).eval (2 * Real.cos (2 * θ))) *
-      Real.sin (2 * θ) = 0 by nlinarith [h₁, h₂])).resolve_right hsin
-end KLInvariance.Realization
-
-namespace KLInvariance.Realization
-open Module
-universe u v z
-variable {I : Type u} {W : Type v} [Group W] {M : CoxeterMatrix I}
-variable (cs : CoxeterSystem M W)
-variable {V : Type z} [AddCommGroup V] [Module ℝ V]
-variable (B : LinearMap.BilinForm ℝ V) (α : I → V)
-variable (hα : ∀ i j, B (α i) (α j) = -2 * Real.cos (Real.pi / M i j))
-
-include hα
-
-lemma simple_normal_value (i : I) : B (α i) (α i) = 2 := by
-  rw [hα]
-  simp
-
-noncomputable def geometricSimple (i : I) : V ≃ₗ[ℝ] V :=
-  Module.reflection (simple_normal_value B α hα i)
-
-theorem geometricSimple_liftable : M.IsLiftable (geometricSimple B α hα) := by
-  intro i j
-  by_cases hij : i = j
-  · subst j
-    rw [M.diagonal, pow_one]
-    ext v
-    exact Module.involutive_reflection (simple_normal_value B α hα i) v
-  by_cases hm0 : M i j = 0
-  · rw [hm0, pow_zero]
-  have hm2 : 2 ≤ M i j := by
-    have := M.off_diagonal i j hij
-    omega
-  exact geometric_reflection_relation (simple_normal_value B α hα i)
-    (simple_normal_value B α hα j) (M i j) hm2 (hα i j)
-    (by rw [hα j i, M.symmetric j i])
-
-/-- The genuine geometric reflection action. Each relation has been proved;
-this definition does not presuppose a faithful representation. -/
-noncomputable def geometricAction : W →* (V ≃ₗ[ℝ] V) :=
-  cs.lift ⟨geometricSimple B α hα, geometricSimple_liftable B α hα⟩
-
-@[simp] theorem geometricAction_simple (i : I) :
-    geometricAction cs B α hα (cs.simple i) = geometricSimple B α hα i :=
-  cs.lift_apply_simple (geometricSimple_liftable B α hα) i
-
-theorem geometricSimple_preserves (hs : B.IsSymm) (i : I) (v w : V) :
-    B (geometricSimple B α hα i v) (geometricSimple B α hα i w) = B v w := by
-  simp only [geometricSimple, Module.reflection_apply, map_sub, map_smul,
-    LinearMap.sub_apply, LinearMap.smul_apply, smul_eq_mul]
-  rw [hs.eq v (α i), simple_normal_value B α hα i]
-  ring
-
-theorem geometricAction_preserves (hs : B.IsSymm) (w : W) (v z : V) :
-    B (geometricAction cs B α hα w v) (geometricAction cs B α hα w z) = B v z := by
-  obtain ⟨ω, rfl⟩ := cs.wordProd_surjective w
-  induction ω with
-  | nil => simp
-  | cons i ω ih =>
-    rw [cs.wordProd_cons, map_mul]
-    simp only [LinearEquiv.mul_apply, geometricAction_simple]
-    rw [geometricSimple_preserves B α hα hs, ih]
-
-end KLInvariance.Realization
-
-namespace KLInvariance.TitsSpace
-open Module
-universe u v
-variable {I : Type u} [Fintype I] (M : CoxeterMatrix I)
-noncomputable section
-
-
-def gram (i j : I) : ℝ := -2 * Real.cos (Real.pi / M i j)
-
-def root (i : I) : I → ℝ := by
-  classical
-  exact Pi.single i 1
-
-def normal (i : I) : Module.Dual ℝ (I → ℝ) :=
-  ∑ j, gram M i j • LinearMap.proj j
-
-def form : LinearMap.BilinForm ℝ (I → ℝ) :=
-  ∑ i, (LinearMap.proj i).smulRight (normal M i)
-
-@[simp] theorem normal_root (i j : I) : normal M i (root j) = gram M i j := by
-  classical
-  simp only [normal, root, LinearMap.sum_apply, LinearMap.smul_apply,
-    LinearMap.proj_apply, smul_eq_mul]
-  rw [Finset.sum_eq_single j]
-  · simp
-  · intro b _ hbj
-    simp [hbj]
-  · simp
-
-@[simp] theorem form_root (i : I) (v : I → ℝ) : form M (root i) v = normal M i v := by
-  classical
-  simp [form, root, LinearMap.sum_apply]
-
-theorem form_symm : (form M).IsSymm := by
-  constructor
-  intro v w
-  simp only [form, normal, LinearMap.sum_apply, LinearMap.smulRight_apply,
-    LinearMap.smul_apply, LinearMap.proj_apply, smul_eq_mul]
-  simp_rw [Finset.mul_sum]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro i hi
-  apply Finset.sum_congr rfl
-  intro j hj
-  rw [show gram M j i = gram M i j by simp only [gram, M.symmetric j i]]
-  ring
-
-theorem form_roots (i j : I) : form M (root i) (root j) =
-    -2 * Real.cos (Real.pi / M i j) := by simp [gram]
-
-theorem root_linearIndependent : LinearIndependent ℝ (root (I := I)) := by
-  classical
-  have heq : (root (I := I)) = (Pi.basisFun ℝ I : I → I → ℝ) := by
-    funext i
-    simp [root, Pi.basisFun_apply]
-  rw [heq]
-  exact (Pi.basisFun ℝ I).linearIndependent
-
-/-- A concrete action on the ordinary geometric root space over ℝ. Its
-faithfulness and the root-sign theorem are still required for the main proof. -/
-def action {W : Type v} [Group W] (cs : CoxeterSystem M W) :
-    W →* ((I → ℝ) ≃ₗ[ℝ] (I → ℝ)) :=
-  Realization.geometricAction cs (form M) root (form_roots M)
-
-@[simp] theorem action_simple {W : Type v} [Group W] (cs : CoxeterSystem M W)
-    (i : I) (v : I → ℝ) :
-    action M cs (cs.simple i) v = v - normal M i v • root i := by
-  simp [action, Realization.geometricAction_simple, Realization.geometricSimple,
-    Module.reflection_apply]
-
-theorem action_preserves {W : Type v} [Group W] (cs : CoxeterSystem M W)
-    (w : W) (v z : I → ℝ) : form M (action M cs w v) (action M cs w z) = form M v z :=
-  Realization.geometricAction_preserves cs (form M) root (form_roots M) (form_symm M) w v z
-end
-end KLInvariance.TitsSpace
-
-namespace KLInvariance.Realization
-universe u
-variable {V : Type u} [AddCommGroup V] [Module ℝ V]
-
-
-abbrev Extension (B : LinearMap.BilinForm ℝ V) := V × Module.Dual ℝ (LinearMap.ker B)
-
-noncomputable def extensionForm (B : LinearMap.BilinForm ℝ V)
-    (p : V →ₗ[ℝ] LinearMap.ker B) : LinearMap.BilinForm ℝ (Extension B) where
-  toFun x :=
-    { toFun := fun y => B x.1 y.1 + x.2 (p y.1) + y.2 (p x.1)
-      map_add' := by intro y z; simp [map_add]; ring
-      map_smul' := by intro t y; simp [map_smul]; ring }
-  map_add' := by
-    intro x y
-    apply LinearMap.ext
-    intro z
-    change B (x.1 + y.1) z.1 + (x.2 + y.2) (p z.1) + z.2 (p (x.1 + y.1)) = _
-    simp [map_add]
-    ring
-  map_smul' := by
-    intro t x
-    apply LinearMap.ext
-    intro z
-    change B (t • x.1) z.1 + (t • x.2) (p z.1) + z.2 (p (t • x.1)) = _
-    simp [map_smul]
-    ring
-
-@[simp] theorem extensionForm_apply (B : LinearMap.BilinForm ℝ V)
-    (p : V →ₗ[ℝ] LinearMap.ker B) (x y : Extension B) :
-    extensionForm B p x y = B x.1 y.1 + x.2 (p y.1) + y.2 (p x.1) := rfl
-
-@[simp] theorem extensionForm_inl (B : LinearMap.BilinForm ℝ V)
-    (p : V →ₗ[ℝ] LinearMap.ker B) (v w : V) :
-    extensionForm B p (v, 0) (w, 0) = B v w := by simp
-
-theorem extensionForm_symm (B : LinearMap.BilinForm ℝ V) (hB : B.IsSymm)
-    (p : V →ₗ[ℝ] LinearMap.ker B) : (extensionForm B p).IsSymm := by
-  constructor
-  intro x y
-  simp only [extensionForm_apply]
-  rw [hB.eq x.1 y.1]
-  ring
-
-theorem extensionForm_nondegenerate (B : LinearMap.BilinForm ℝ V)
-    (hB : B.IsSymm) (p : V →ₗ[ℝ] LinearMap.ker B)
-    (hp : ∀ k : LinearMap.ker B, p k.val = k) : (extensionForm B p).Nondegenerate := by
-  have hf : ∀ x : Extension B, (∀ y, extensionForm B p x y = 0) → x = 0 := by
-    rintro ⟨v, φ⟩ hx
-    have hpv : p v = 0 := by
-      apply (Module.forall_dual_apply_eq_zero_iff ℝ (p v)).mp
-      intro ψ
-      simpa using hx (0, ψ)
-    have hφ : φ = 0 := by
-      ext k
-      have hbk : B k.val = 0 := k.property
-      have hvk : B v k.val = 0 := by
-        rw [hB.eq v k.val, hbk]
-        rfl
-      simpa [hvk, hp k, hpv] using hx (k.val, 0)
-    have hbv : v ∈ LinearMap.ker B := by
-      change B v = 0
-      ext w
-      simpa [hφ, hpv] using hx (w, 0)
-    have hv : v = 0 := by
-      have h := hp ⟨v, hbv⟩
-      rw [hpv] at h
-      exact (congrArg Subtype.val h).symm
-    simp [hv, hφ]
-  exact ⟨hf, fun x hx => hf x (fun y => by
-    rw [(extensionForm_symm B hB p).eq]
-    exact hx y)⟩
-
-theorem exists_minimal_nondegenerate_extension (B : LinearMap.BilinForm ℝ V)
-    (hB : B.IsSymm) :
-    ∃ C : LinearMap.BilinForm ℝ (Extension B), C.IsSymm ∧ C.Nondegenerate ∧
-      ∀ v w, C (v, 0) (w, 0) = B v w := by
-  obtain ⟨Q, hQ⟩ := (LinearMap.ker B).exists_isCompl
-  let p := (LinearMap.ker B).projectionOnto Q hQ
-  exact ⟨extensionForm B p, extensionForm_symm B hB p,
-    extensionForm_nondegenerate B hB p (Submodule.projectionOnto_apply_left hQ),
-    extensionForm_inl B p⟩
-
-theorem extension_finrank [FiniteDimensional ℝ V] (B : LinearMap.BilinForm ℝ V) :
-    Module.finrank ℝ (Extension B) = Module.finrank ℝ V + Module.finrank ℝ (LinearMap.ker B) := by
-  rw [Module.finrank_prod, Subspace.dual_finrank_eq]
-
-
-theorem extension_dimension_minimal [FiniteDimensional ℝ V]
-    {E : Type*} [AddCommGroup E] [Module ℝ E] [FiniteDimensional ℝ E]
-    (B : LinearMap.BilinForm ℝ V) (C : LinearMap.BilinForm ℝ E)
-    (hC : C.Nondegenerate) (i : V →ₗ[ℝ] E) (hi : Function.Injective i)
-    (hform : ∀ v w, C (i v) (i w) = B v w) :
-    Module.finrank ℝ V + Module.finrank ℝ (LinearMap.ker B) ≤ Module.finrank ℝ E := by
-  let f : LinearMap.ker B →ₗ[ℝ] E := i.comp (LinearMap.ker B).subtype
-  have hf : Function.Injective f := hi.comp Subtype.val_injective
-  have hsub : LinearMap.range i ≤ C.orthogonal (LinearMap.range f) := by
-    rintro _ ⟨v, rfl⟩
-    apply LinearMap.BilinForm.mem_orthogonal_iff.mpr
-    rintro _ ⟨k, rfl⟩
-    change C (i k.val) (i v) = 0
-    rw [hform, show B k.val = 0 from k.property]
-    rfl
-  have hd := Submodule.finrank_mono hsub
-  rw [LinearMap.finrank_range_of_inj hi, LinearMap.BilinForm.finrank_orthogonal hC,
-    LinearMap.finrank_range_of_inj hf] at hd
-  have hk : Module.finrank ℝ (LinearMap.ker B) ≤ Module.finrank ℝ E := by
-    rw [← LinearMap.finrank_range_of_inj hf]
-    exact Submodule.finrank_le _
-  omega
-
-end KLInvariance.Realization
-
-namespace KLInvariance.TitsSpace
-open Module
-universe u v
-variable {I : Type u} [Fintype I] (M : CoxeterMatrix I)
-noncomputable section
-
-abbrev Extended := Realization.Extension (form M)
-
-def extendedForm : LinearMap.BilinForm ℝ (Extended M) :=
-  Classical.choose (Realization.exists_minimal_nondegenerate_extension (form M) (form_symm M))
-
-theorem extendedForm_spec : (extendedForm M).IsSymm ∧ (extendedForm M).Nondegenerate ∧
-    ∀ v w, extendedForm M (v, 0) (w, 0) = form M v w :=
-  Classical.choose_spec (Realization.exists_minimal_nondegenerate_extension (form M) (form_symm M))
-
-def extendedRoot (i : I) : Extended M := (root i, 0)
-
-@[simp] theorem extendedForm_roots (i j : I) :
-    extendedForm M (extendedRoot M i) (extendedRoot M j) =
-      -2 * Real.cos (Real.pi / M i j) := by
-  rw [extendedRoot, extendedRoot, (extendedForm_spec M).2.2, form_roots]
-
-theorem extendedRoot_linearIndependent : LinearIndependent ℝ (extendedRoot M) := by
-  exact (root_linearIndependent (I := I)).map'
-    (LinearMap.inl ℝ (I → ℝ) (Module.Dual ℝ (LinearMap.ker (form M))))
-    (LinearMap.ker_eq_bot.mpr LinearMap.inl_injective)
-
-/-- Both roots and coroots are independent in the actual minimal extension. -/
-theorem extendedNormal_linearIndependent :
-    LinearIndependent ℝ (fun i => extendedForm M (extendedRoot M i)) :=
-  (extendedRoot_linearIndependent M).map' (extendedForm M) (extendedForm_spec M).2.1.ker_eq_bot
-
-def extendedAction {W : Type v} [Group W] (cs : CoxeterSystem M W) :
-    W →* (Extended M ≃ₗ[ℝ] Extended M) :=
-  Realization.geometricAction cs (extendedForm M) (extendedRoot M) (extendedForm_roots M)
-
-theorem extendedAction_preserves {W : Type v} [Group W] (cs : CoxeterSystem M W)
-    (w : W) (v z : Extended M) :
-    extendedForm M (extendedAction M cs w v) (extendedAction M cs w z) = extendedForm M v z :=
-  Realization.geometricAction_preserves cs (extendedForm M) (extendedRoot M)
-    (extendedForm_roots M) (extendedForm_spec M).1 w v z
-
-@[simp] theorem extendedAction_simple {W : Type v} [Group W] (cs : CoxeterSystem M W)
-    (i : I) (v : Extended M) :
-    extendedAction M cs (cs.simple i) v =
-      v - extendedForm M (extendedRoot M i) v • extendedRoot M i := by
-  simp only [extendedAction, Realization.geometricAction_simple,
-    Realization.geometricSimple, Module.reflection_apply]
-
-/-- The extended action genuinely restricts to the ordinary geometric action.
-Injectivity of that latter action is an outstanding obligation, not a hypothesis
-hidden in the construction of this extension. -/
-theorem extendedAction_restrict {W : Type v} [Group W] (cs : CoxeterSystem M W)
-    (w : W) (v : I → ℝ) :
-    extendedAction M cs w (v, 0) = (action M cs w v, 0) := by
-  obtain ⟨ω, rfl⟩ := cs.wordProd_surjective w
-  induction ω generalizing v with
-  | nil => simp
-  | cons i ω ih =>
-    rw [cs.wordProd_cons, map_mul, map_mul]
-    simp only [LinearEquiv.mul_apply, ih, extendedAction_simple, action_simple]
-    change (action M cs (cs.wordProd ω) v, 0) -
-      extendedForm M (root i, 0) (action M cs (cs.wordProd ω) v, 0) • (root i, 0) = _
-    rw [(extendedForm_spec M).2.2, form_root]
-    ext <;> simp
-
-theorem extended_finrank : Module.finrank ℝ (Extended M) =
-    Fintype.card I + Module.finrank ℝ (LinearMap.ker (form M)) := by
-  rw [Realization.extension_finrank, Module.finrank_pi]
-
-
-def dominant : Module.Dual ℝ (Extended M) :=
-  (∑ i : I, LinearMap.proj i).comp
-    (LinearMap.fst ℝ (I → ℝ) (Module.Dual ℝ (LinearMap.ker (form M))))
-
-@[simp] theorem dominant_root (i : I) : dominant M (extendedRoot M i) = 1 := by
-  classical
-  simp [dominant, extendedRoot, root]
-
-end
-end KLInvariance.TitsSpace
-
-namespace KLInvariance
-universe u v
-variable {B : Type u} {W : Type v} [Group W] {M : CoxeterMatrix B}
-
-namespace ParabolicSupport
-open CoxeterSystem
-
-variable (cs : CoxeterSystem M W) (J : Set B)
-
-/-- The genuine subgroup generated by a subset of simple reflections. -/
-def subgroup : Subgroup W := Subgroup.closure (cs.simple '' J)
-
-theorem word_mem {ν : List B} (hν : ∀ i ∈ ν, i ∈ J) :
-    cs.wordProd ν ∈ subgroup cs J := by
-  induction ν with
-  | nil => exact (subgroup cs J).one_mem
-  | cons i ν ih =>
-    rw [cs.wordProd_cons]
-    exact (subgroup cs J).mul_mem
-      (Subgroup.subset_closure ⟨i, hν i (by simp), rfl⟩)
-      (ih fun j hj => hν j (by simp [hj]))
-
-theorem exists_word {w : W} (hw : w ∈ subgroup cs J) :
-    ∃ ν : List B, (∀ i ∈ ν, i ∈ J) ∧ cs.wordProd ν = w := by
-  induction hw using Subgroup.closure_induction with
-  | mem w hw =>
-    obtain ⟨i, hi, rfl⟩ := hw
-    exact ⟨[i], by simpa using hi, by simp⟩
-  | one => exact ⟨[], by simp, by simp⟩
-  | mul x y hx hy ihx ihy =>
-    obtain ⟨ω, hω, rfl⟩ := ihx
-    obtain ⟨ν, hν, rfl⟩ := ihy
-    exact ⟨ω ++ ν, fun i hi => (List.mem_append.mp hi).elim (hω i) (hν i),
-      cs.wordProd_append ω ν⟩
-  | inv x hx ih =>
-    obtain ⟨ω, hω, rfl⟩ := ih
-    exact ⟨ω.reverse, by simpa using hω, cs.wordProd_reverse ω⟩
-
-theorem exists_reduced_word {w : W} (hw : w ∈ subgroup cs J) :
-    ∃ ν : List B, (∀ i ∈ ν, i ∈ J) ∧ cs.IsReduced ν ∧ cs.wordProd ν = w := by
-  obtain ⟨ω, hω, hprod⟩ := exists_word cs J hw
-  obtain ⟨ν, hsub, hν, heq⟩ := exists_reduced_subword cs ω
-  exact ⟨ν, fun i hi => hω i (hsub.subset hi), hν, heq.trans hprod⟩
-
-/-- A minimal right-coset representative concatenates with every reduced
-word in the parabolic generators without cancellations. -/
-theorem reduced_append_of_minimal {u : W}
-    (hmin : ∀ v ∈ subgroup cs J, cs.length u ≤ cs.length (u * v))
-    {ω ν : List B} (hω : cs.IsReduced ω) (hu : cs.wordProd ω = u)
-    (hν : cs.IsReduced ν) (hJ : ∀ i ∈ ν, i ∈ J) :
-    cs.IsReduced (ω ++ ν) := by
-  obtain ⟨τ, hsub, hτ, hprod⟩ := exists_reduced_subword cs (ω ++ ν)
-  obtain ⟨ω', ν', rfl, hωsub, hνsub⟩ := List.sublist_append_iff.mp hsub
-  have hprod' : cs.wordProd ω' * cs.wordProd ν' = u * cs.wordProd ν := by
-    simpa [cs.wordProd_append, hu] using hprod
-  have hω' : cs.wordProd ω' = u * (cs.wordProd ν * (cs.wordProd ν')⁻¹) := by
-    rw [← mul_assoc, ← hprod']
-    group
-  have hlen := hmin (cs.wordProd ν * (cs.wordProd ν')⁻¹)
-    ((subgroup cs J).mul_mem (word_mem cs J hJ)
-      ((subgroup cs J).inv_mem (word_mem cs J fun i hi => hJ i (hνsub.subset hi))))
-  rw [← hω'] at hlen
-  have hbound := cs.length_wordProd_le ω'
-  have homega := hω.eq
-  rw [hu] at homega
-  have hle := hωsub.length_le
-  have heqω := hωsub.eq_of_length (by omega)
-  subst ω'
-  have heqν : cs.wordProd ν' = cs.wordProd ν := by
-    rw [hu] at hprod'
-    exact mul_left_cancel hprod'
-  have hboundν := cs.length_wordProd_le ν'
-  rw [heqν, hν.eq] at hboundν
-  have heqν' := hνsub.eq_of_length (by have := hνsub.length_le; omega)
-  simpa [heqν'] using hτ
-
-theorem length_mul_of_minimal {u v : W}
-    (hmin : ∀ z ∈ subgroup cs J, cs.length u ≤ cs.length (u * z))
-    (hv : v ∈ subgroup cs J) : cs.length (u * v) = cs.length u + cs.length v := by
-  obtain ⟨ω, hω, hu⟩ := cs.exists_isReduced u
-  obtain ⟨ν, hJ, hν, hvprod⟩ := exists_reduced_word cs J hv
-  have h := (reduced_append_of_minimal cs J hmin hω hu.symm hν hJ).eq
-  simpa [cs.wordProd_append, hu, ← hvprod, hω.eq, hν.eq] using h
-
-/-- Every right parabolic coset has a minimal representative. -/
-theorem exists_minimal (w : W) :
-    ∃ u v : W, v ∈ subgroup cs J ∧ w = u * v ∧
-      ∀ z ∈ subgroup cs J, cs.length u ≤ cs.length (u * z) := by
-  classical
-  have hex : ∃ n : ℕ, ∃ u v : W, v ∈ subgroup cs J ∧ w = u * v ∧ cs.length u = n :=
-    ⟨cs.length w, w, 1, (subgroup cs J).one_mem, by simp, rfl⟩
-  obtain ⟨u, v, hv, hw, hu⟩ := Nat.find_spec hex
-  refine ⟨u, v, hv, hw, fun z hz => ?_⟩
-  rw [hu]
-  apply Nat.find_min' hex
-  refine ⟨u * z, z⁻¹ * v, (subgroup cs J).mul_mem ((subgroup cs J).inv_mem hz) hv, ?_, rfl⟩
-  simpa [mul_assoc] using hw
-
-theorem minimal_simple_ascent {u : W}
-    (hmin : ∀ z ∈ subgroup cs J, cs.length u ≤ cs.length (u * z))
-    {i : B} (hi : i ∈ J) : cs.length (u * cs.simple i) = cs.length u + 1 := by
-  have h := length_mul_of_minimal cs J hmin (Subgroup.subset_closure ⟨i, hi, rfl⟩)
-  simpa using h
-
-/-- The decomposition is length-additive, including after multiplying by
-any of the chosen simple generators. This is used in the root-sign induction. -/
-theorem exists_length_additive_decomposition (w : W) :
-    ∃ u v : W, v ∈ subgroup cs J ∧ w = u * v ∧
-      cs.length w = cs.length u + cs.length v ∧
-      (∀ i ∈ J, cs.length (u * cs.simple i) = cs.length u + 1) ∧
-      (∀ i ∈ J, cs.length (w * cs.simple i) = cs.length u + cs.length (v * cs.simple i)) := by
-  obtain ⟨u, v, hv, hw, hmin⟩ := exists_minimal cs J w
-  refine ⟨u, v, hv, hw, ?_, (fun i hi => minimal_simple_ascent cs J hmin hi), ?_⟩
-  · rw [hw]
-    exact length_mul_of_minimal cs J hmin hv
-  · intro i hi
-    rw [hw, mul_assoc]
-    exact length_mul_of_minimal cs J hmin
-      ((subgroup cs J).mul_mem hv (Subgroup.subset_closure ⟨i, hi, rfl⟩))
-
-end ParabolicSupport
-
-end KLInvariance
-
-namespace KLInvariance.ParabolicSupport
-open CoxeterSystem
-universe u v
-variable {B : Type u} {W : Type v} [Group W] {M : CoxeterMatrix B}
-variable (cs : CoxeterSystem M W)
-
-theorem reduced_of_append {ω ν : List B} (h : cs.IsReduced (ω ++ ν)) :
-    cs.IsReduced ω := by
-  simpa using h.take ω.length
-
-/-- A reduced two-letter word followed by `i` must alternate and end in `j`. -/
-theorem reduced_pair_eq_alternating {ω : List B} {i j : B} (hij : i ≠ j)
-    (hJ : ∀ k ∈ ω, k = i ∨ k = j) (hred : cs.IsReduced (ω ++ [i])) :
-    ω = alternatingWord i j ω.length := by
-  induction ω using List.reverseRecOn generalizing i j with
-  | nil => rfl
-  | append_singleton ω k ih =>
-    have hk : k = i ∨ k = j := hJ k (by simp)
-    have hki : k ≠ i := by
-      rintro rfl
-      have hlen := hred.eq
-      simp only [cs.wordProd_append, cs.wordProd_singleton, List.length_append,
-        List.length_singleton, mul_assoc, cs.simple_mul_simple_self, mul_one] at hlen
-      have := cs.length_wordProd_le ω
-      omega
-    have hkj : k = j := hk.resolve_left hki
-    subst k
-    have hprefix : cs.IsReduced (ω ++ [j]) := reduced_of_append cs hred
-    rw [List.length_append, List.length_singleton, alternatingWord_succ, List.concat_eq_append]
-    congr 1
-    apply ih hij.symm
-    · intro k hk
-      exact (hJ k (List.mem_append.mpr (Or.inl hk))).symm
-    · exact hprefix
-
-/-- The coefficients in the root formula are controlled by genuine reducedness,
-not by a separate rank-two length assumption. -/
-theorem alternating_length_bound {ω : List B} {i j : B} (hij : i ≠ j)
-    (hJ : ∀ k ∈ ω, k = i ∨ k = j) (hred : cs.IsReduced (ω ++ [i])) :
-    M i j = 0 ∨ ω.length + 1 ≤ M i j := by
-  by_cases hm : M i j = 0
-  · exact Or.inl hm
-  · right
-    have heq := reduced_pair_eq_alternating cs hij hJ hred
-    have hr : cs.IsReduced (alternatingWord j i (ω.length + 1)) := by
-      rw [alternatingWord_succ, ← heq]
-      simpa [List.concat_eq_append] using hred
-    by_contra hn
-    apply cs.not_isReduced_alternatingWord j i (by simpa [M.symmetric] using hm)
-      (by simpa [M.symmetric] using (show M i j < ω.length + 1 by omega)) hr
-
-end KLInvariance.ParabolicSupport
-
-namespace KLInvariance.TitsSpace
-open CoxeterSystem Polynomial Polynomial.Chebyshev
-universe u v
-variable {I : Type u} [klPreservedInstance5 : Fintype I] (M : CoxeterMatrix I)
-variable {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
-def rootCoeff (i j : I) (n : ℤ) : ℝ :=
-  (S ℝ n).eval (2 * Real.cos (Real.pi / M i j))
-
-include klPreservedInstance5 in
-@[simp] theorem rootCoeff_zero (i j : I) : rootCoeff M i j 0 = 1 := by
-  have _ := klPreservedInstance5
-  simp [rootCoeff]
-include klPreservedInstance5 in
-@[simp] theorem rootCoeff_neg_one (i j : I) : rootCoeff M i j (-1) = 0 := by
-  have _ := klPreservedInstance5
-  simp [rootCoeff]
-
-include klPreservedInstance5 in
-theorem rootCoeff_succ (i j : I) (n : ℤ) :
-    rootCoeff M i j (n + 1) =
-      (2 * Real.cos (Real.pi / M i j)) * rootCoeff M i j n - rootCoeff M i j (n-1) := by
-  have _ := klPreservedInstance5
-  simp [rootCoeff, S_add_one]
-
-include klPreservedInstance5 in
-@[simp] theorem gram_diag (i : I) : gram M i i = 2 := by
-  have _ := klPreservedInstance5
-  simp [gram]
-
-/-- Exact rank-two root formula, including the infinite pairing. -/
-theorem alternating_root (i j : I) (n : ℕ) :
-    action M cs (cs.wordProd (alternatingWord i j n)) (root i) =
-      if Even n then
-        rootCoeff M i j n • root i + rootCoeff M i j ((n : ℤ)-1) • root j
-      else rootCoeff M i j ((n : ℤ)-1) • root i + rootCoeff M i j n • root j := by
-  induction n with
-  | zero => simp [alternatingWord]
-  | succ n ih =>
-    rw [alternatingWord_succ', cs.wordProd_cons, map_mul, LinearEquiv.mul_apply, ih]
-    have hrec := rootCoeff_succ M i j (n : ℤ)
-    have hsymm : gram M j i = gram M i j := by simp [gram, M.symmetric j i]
-    by_cases hn : Even n
-    · simp only [hn, ite_true, Nat.even_add_one, not_true_eq_false, ite_false]
-      rw [action_simple]
-      simp only [map_add, map_smul, normal_root, gram_diag, hsymm]
-      push_cast
-      rw [show (n : ℤ) + 1 - 1 = n by omega, hrec]
-      simp only [gram]
-      module
-    · simp only [hn, ite_false, Nat.even_add_one, not_false_eq_true, ite_true]
-      rw [action_simple]
-      simp only [map_add, map_smul, normal_root, gram_diag]
-      push_cast
-      rw [show (n : ℤ) + 1 - 1 = n by omega, hrec]
-      simp only [gram]
-      module
-
-/-- For a reduced alternating word followed by the other generator, both
-rank-two coefficients are nonnegative. -/
-theorem rootCoeff_nonneg (i j : I) {n : ℤ} (hn : -1 ≤ n)
-    (hm : M i j = 0 ∨ n + 1 ≤ (M i j : ℤ)) (hij : i ≠ j) :
-    0 ≤ rootCoeff M i j n := by
-  rcases hm with hm | hm
-  · simp only [rootCoeff, hm, Nat.cast_zero, div_zero, Real.cos_zero, mul_one, S_eval_two]
-    exact_mod_cast (by omega : (0 : ℤ) ≤ n + 1)
-  · by_cases hn0 : n = -1
-    · simp [hn0]
-    have hn1 : 0 < n + 1 := by omega
-    have hMpos : (0 : ℝ) < M i j := by exact_mod_cast (by omega : 0 < M i j)
-    have hM2 : 2 ≤ M i j := by
-      have := M.off_diagonal i j hij
-      omega
-    let θ : ℝ := Real.pi / M i j
-    have hθpos : 0 < θ := div_pos Real.pi_pos hMpos
-    have hθlt : θ < Real.pi := by
-      dsimp [θ]
-      apply (div_lt_iff₀ hMpos).2
-      have : (2 : ℝ) ≤ M i j := by exact_mod_cast hM2
-      nlinarith [Real.pi_pos]
-    have hsin := Real.sin_pos_of_pos_of_lt_pi hθpos hθlt
-    have heq := S_two_mul_real_cos θ n
-    have hargpos : 0 ≤ ((n : ℝ) + 1) * θ := by
-      have : (0 : ℝ) < (n : ℝ) + 1 := by exact_mod_cast hn1
-      positivity
-    have hargle : ((n : ℝ) + 1) * θ ≤ Real.pi := by
-      dsimp [θ]
-      rw [← mul_div_assoc]
-      apply (div_le_iff₀ hMpos).2
-      have : (n : ℝ) + 1 ≤ M i j := by exact_mod_cast hm
-      nlinarith [Real.pi_pos]
-    have hs := Real.sin_nonneg_of_nonneg_of_le_pi hargpos hargle
-    change 0 ≤ (S ℝ n).eval (2 * Real.cos θ)
-    nlinarith [heq]
-end
-end KLInvariance.TitsSpace
-
-namespace KLInvariance.TitsSpace
-open CoxeterSystem
-universe u v
-variable {I : Type u} [klPreservedInstance6 : Fintype I] (M : CoxeterMatrix I)
-variable {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
-include klPreservedInstance6 in
-@[simp] theorem root_nonneg (i k : I) : 0 ≤ root i k := by
-  have _ := klPreservedInstance6
-  classical
-  simp only [root, Pi.single_apply]
-  split_ifs <;> norm_num
-
-/-- Positivity of a rank-two reduced tail after an already positive prefix. -/
-theorem pair_image_nonneg (u : W) {i j : I} (hij : i ≠ j)
-    (hi : ∀ k, 0 ≤ action M cs u (root i) k)
-    (hj : ∀ k, 0 ≤ action M cs u (root j) k)
-    {ω : List I} (hJ : ∀ k ∈ ω, k = i ∨ k = j)
-    (hred : cs.IsReduced (ω ++ [i])) :
-    ∀ k, 0 ≤ action M cs (u * cs.wordProd ω) (root i) k := by
-  have heq := ParabolicSupport.reduced_pair_eq_alternating cs hij hJ hred
-  have hbound := ParabolicSupport.alternating_length_bound cs hij hJ hred
-  have hn : 0 ≤ rootCoeff M i j (ω.length : ℤ) :=
-    rootCoeff_nonneg M i j (by omega) (by rcases hbound with h|h; exact Or.inl h; right; omega) hij
-  have hn' : 0 ≤ rootCoeff M i j ((ω.length : ℤ)-1) :=
-    rootCoeff_nonneg M i j (by omega) (by rcases hbound with h|h; exact Or.inl h; right; omega) hij
-  intro k
-  rw [map_mul, LinearEquiv.mul_apply, heq, alternating_root]
-  split_ifs
-  all_goals
-    simp only [map_add, map_smul, Pi.add_apply, Pi.smul_apply, smul_eq_mul]
-    exact add_nonneg (mul_nonneg (by assumption) (by exact hi k))
-      (mul_nonneg (by assumption) (by first | exact hi k | exact hj k))
-
-/-- The genuine Tits root-sign theorem for ascents, proved from exchange,
-minimal parabolic cosets and the rank-two trigonometric formulas. -/
-theorem action_root_nonneg_of_ascent (w : W) (i : I)
-    (hasc : cs.length (w * cs.simple i) = cs.length w + 1) :
-    ∀ k, 0 ≤ action M cs w (root i) k := by
-  suffices ∀ n : ℕ, ∀ w : W, cs.length w = n → ∀ i : I,
-      cs.length (w * cs.simple i) = cs.length w + 1 →
-      ∀ k, 0 ≤ action M cs w (root i) k by
-    exact this (cs.length w) w rfl i hasc
-  intro n
-  induction n using Nat.strong_induction_on with
-  | h n ih =>
-    intro w hwlen i hasc
-    by_cases hw : w = 1
-    · subst w
-      simp
-    obtain ⟨j, hj⟩ := cs.exists_rightDescent_of_ne_one hw
-    change cs.length (w * cs.simple j) < cs.length w at hj
-    have hij : i ≠ j := by rintro rfl; omega
-    obtain ⟨u, v, hv, huv, hlen, huasc, hmul⟩ :=
-      ParabolicSupport.exists_length_additive_decomposition cs ({i, j} : Set I) w
-    have himem : i ∈ ({i, j} : Set I) := by simp
-    have hjmem : j ∈ ({i, j} : Set I) := by simp
-    have hult : cs.length u < n := by
-      have h := hmul j hjmem
-      omega
-    have hui := ih (cs.length u) hult u rfl i (huasc i himem)
-    have huj := ih (cs.length u) hult u rfl j (huasc j hjmem)
-    obtain ⟨ω, hJ, hω, hprod⟩ := ParabolicSupport.exists_reduced_word cs ({i,j} : Set I) hv
-    have hvi : cs.length (v * cs.simple i) = cs.length v + 1 := by
-      have h := hmul i himem
-      omega
-    have hωi : cs.IsReduced (ω ++ [i]) := by
-      change cs.length (cs.wordProd (ω ++ [i])) = (ω ++ [i]).length
-      simp only [cs.wordProd_append, cs.wordProd_singleton, hprod, List.length_append,
-        List.length_singleton]
-      rw [hvi, ← hprod, hω.eq]
-    have hJ' : ∀ k ∈ ω, k = i ∨ k = j := by simpa using hJ
-    rw [huv, ← hprod]
-    exact pair_image_nonneg M cs u hij hui huj hJ' hωi
-
-@[simp] theorem action_simple_root (i : I) : action M cs (cs.simple i) (root i) = -root i := by
-  rw [action_simple, normal_root, gram_diag]
-  module
-
-/-- Descents carry a simple root into the negative cone. -/
-theorem action_root_nonpos_of_descent (w : W) (i : I)
-    (hdesc : cs.length (w * cs.simple i) < cs.length w) :
-    ∀ k, action M cs w (root i) k ≤ 0 := by
-  have hasc : cs.length ((w * cs.simple i) * cs.simple i) = cs.length (w * cs.simple i) + 1 := by
-    rcases cs.length_mul_simple w i with h|h
-    · omega
-    · simpa [mul_assoc] using h.symm
-  have h := action_root_nonneg_of_ascent M cs (w * cs.simple i) i hasc
-  simpa only [map_mul, LinearEquiv.mul_apply, action_simple_root, map_neg,
-    Pi.neg_apply, neg_nonneg] using h
-
-/-- Faithfulness of the actual ordinary Tits action, not an added axiom. -/
-theorem action_injective : Function.Injective (action M cs) := by
-  apply (MonoidHom.ker_eq_bot_iff _).1
-  apply (Subgroup.eq_bot_iff_forall _).2
-  intro w hw
-  by_contra hne
-  obtain ⟨i, hi⟩ := cs.exists_rightDescent_of_ne_one hne
-  have h := action_root_nonpos_of_descent M cs w i hi i
-  have hw' : action M cs w = 1 := hw
-  rw [hw'] at h
-  norm_num [root] at h
-end
-end KLInvariance.TitsSpace
-
-namespace KLInvariance.Realization
-open Module
-variable {V : Type*} [AddCommGroup V] [Module ℝ V]
-
-/-- Rank of the moving subspace is unchanged by conjugacy. -/
-theorem finrank_moving_conj (g h : V ≃ₗ[ℝ] V) :
-    Module.finrank ℝ (LinearMap.range ((h * g * h⁻¹).toLinearMap - LinearMap.id)) =
-      Module.finrank ℝ (LinearMap.range (g.toLinearMap - LinearMap.id)) := by
-  have heq : (h * g * h⁻¹).toLinearMap - LinearMap.id =
-      h.toLinearMap.comp ((g.toLinearMap - LinearMap.id).comp h.symm.toLinearMap) := by
-    ext v
-    simp
-  rw [heq, LinearMap.range_comp, LinearMap.range_comp_of_range_eq_top _ h.symm.range,
-    h.finrank_map_eq]
-
-/-- An orthogonal rank-one map negating a norm-two vector is its exact reflection. -/
-theorem rank_one_eq_reflection_of_negates [FiniteDimensional ℝ V]
-    (B : LinearMap.BilinForm ℝ V) (g : V ≃ₗ[ℝ] V)
-    (hg : ∀ v w, B (g v) (g w) = B v w)
-    (hr : Module.finrank ℝ (LinearMap.range (g.toLinearMap - LinearMap.id)) = 1)
-    (e : V) (he : B e e = 2) (hge : g e = -e) :
-    ∀ v, g v = Module.preReflection e (B e) v := by
-  let d : V →ₗ[ℝ] V := g.toLinearMap - LinearMap.id
-  have he0 : e ≠ 0 := by intro h; simp [h] at he
-  have hemem : e ∈ LinearMap.range d := by
-    refine ⟨(- (1 / 2 : ℝ)) • e, ?_⟩
-    change g (- (1 / 2 : ℝ) • e) - (- (1 / 2 : ℝ) • e) = e
-    rw [map_smul, hge]
-    module
-  have hspan : Submodule.span ℝ {e} = LinearMap.range d := by
-    apply Submodule.eq_of_le_of_finrank_eq
-    · exact Submodule.span_le.mpr (by simpa using hemem)
-    · rw [finrank_span_singleton he0]
-      exact hr.symm
-  intro v
-  have hv : d v ∈ Submodule.span ℝ {e} := by rw [hspan]; exact LinearMap.mem_range_self _ v
-  obtain ⟨c, hc⟩ := Submodule.mem_span_singleton.mp hv
-  have hgform : g v = v + c • e := by
-    change c • e = g v - v at hc
-    rw [hc]
-    abel
-  have hB := hg e v
-  rw [hge, hgform] at hB
-  simp only [map_neg, LinearMap.neg_apply, map_add, map_smul, he, smul_eq_mul] at hB
-  have hc' : c = - B e v := by linarith
-  rw [hgform, hc', Module.preReflection_apply, neg_smul, sub_eq_add_neg]
-
-/-- Commutation with a reflection makes its negative eigenspace invariant;
-a nonpositive coordinate selects the negative, rather than positive, sign. -/
-theorem negates_of_commutes_nonpos
-    (B : LinearMap.BilinForm ℝ V) (g : V ≃ₗ[ℝ] V)
-    (hg : ∀ v w, B (g v) (g w) = B v w)
-    (e : V) (he : B e e = 2)
-    (hcomm : g * Module.reflection he = Module.reflection he * g)
-    (f : Module.Dual ℝ V) (hfe : f e = 1) (hfg : f (g e) ≤ 0) :
-    g e = -e := by
-  have h := LinearEquiv.congr_fun hcomm e
-  simp only [LinearEquiv.mul_apply, Module.reflection_apply_self, map_neg,
-    Module.reflection_apply] at h
-  let c : ℝ := B e (g e) / 2
-  have hge : g e = c • e := by
-    dsimp [c]
-    linear_combination (norm := module) (- (1 / 2 : ℝ)) • h
-  have hc : c ≤ 0 := by simpa [hge, hfe] using hfg
-  have hnorm := hg e e
-  rw [hge] at hnorm
-  simp only [map_smul, LinearMap.smul_apply, he, smul_eq_mul] at hnorm
-  have hc' : c = -1 := by nlinarith [sq_nonneg (c + 1)]
-  rw [hge, hc', neg_one_smul]
-/-- The moving subspace of a genuine reflection is exactly its root line. -/
-theorem range_reflection_sub_id (e : V) (f : Module.Dual ℝ V) (h : f e = 2) :
-    LinearMap.range ((Module.reflection h).toLinearMap - LinearMap.id) =
-      Submodule.span ℝ {e} := by
-  apply le_antisymm
-  · rintro z ⟨v, rfl⟩
-    apply Submodule.mem_span_singleton.mpr
-    refine ⟨-f v, ?_⟩
-    simp only [LinearMap.sub_apply, LinearEquiv.coe_coe, LinearMap.id_apply,
-      Module.reflection_apply, neg_smul]
-    abel
-  · apply Submodule.span_le.mpr
-    intro z hz
-    have hz' : z = e := Set.mem_singleton_iff.mp hz
-    rw [hz']
-    refine ⟨(- (1 / 2 : ℝ)) • e, ?_⟩
-    simp only [LinearMap.sub_apply, LinearEquiv.coe_coe, LinearMap.id_apply,
-      Module.reflection_apply, map_smul, h]
-    module
-
-theorem finrank_reflection_sub_id (e : V) (f : Module.Dual ℝ V) (h : f e = 2) :
-    Module.finrank ℝ (LinearMap.range
-      ((Module.reflection h).toLinearMap - LinearMap.id)) = 1 := by
-  rw [range_reflection_sub_id]
-  apply finrank_span_singleton
-  intro he
-  simp [he] at h
-
-end KLInvariance.Realization
-
-namespace KLInvariance.TitsSpace
-open Module
-universe u v
-variable {I : Type u} [Fintype I] (M : CoxeterMatrix I)
-variable {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
-/-- Faithfulness persists in the actual nondegenerate extension. -/
-theorem extendedAction_injective : Function.Injective (extendedAction M cs) := by
-  intro w z h
-  apply action_injective M cs
-  ext v i
-  have hv := congrArg (fun g : Extended M ≃ₗ[ℝ] Extended M => (g (v, 0)).1 i) h
-  simpa only [extendedAction_restrict] using hv
-
-@[simp] theorem extendedForm_root_self (i : I) :
-    extendedForm M (extendedRoot M i) (extendedRoot M i) = 2 := by
-  rw [extendedForm_roots, M.diagonal]
-  norm_num
-
-@[simp] theorem extendedAction_simple_eq (i : I) :
-    extendedAction M cs (cs.simple i) = Module.reflection (extendedForm_root_self M i) := by
-  apply LinearEquiv.ext
-  intro v
-  rw [extendedAction_simple, Module.reflection_apply]
-
-/-- A rank-one element of the actual extended action is an involution. -/
-theorem involution_of_moving_rank_one (w : W)
-    (hr : Module.finrank ℝ (LinearMap.range
-      ((extendedAction M cs w).toLinearMap - LinearMap.id)) = 1) : w⁻¹ = w := by
-  obtain ⟨e, he, hform⟩ := Realization.orthogonal_finrank_one
-    (extendedForm M) (extendedForm_spec M).1 (extendedForm_spec M).2.1
-    (extendedAction M cs w) (extendedAction_preserves M cs w) hr
-  have he2 : ((2 / extendedForm M e e) • extendedForm M e) e = 2 := by
-    simp only [LinearMap.smul_apply, smul_eq_mul]
-    field_simp
-  have hw2 : w * w = 1 := by
-    apply extendedAction_injective M cs
-    apply LinearEquiv.ext
-    intro v
-    simp only [map_mul, map_one, LinearEquiv.mul_apply, hform]
-    change _ = v
-    exact Module.involutive_preReflection he2 v
-  exact inv_eq_of_mul_eq_one_left hw2
-
-/-- Reflection faithfulness, the difficult converse, proved by length descent.
-This avoids assuming the finite-subgroup theorem cited in the manuscript. -/
-theorem isReflection_of_moving_rank_one (w : W)
-    (hr : Module.finrank ℝ (LinearMap.range
-      ((extendedAction M cs w).toLinearMap - LinearMap.id)) = 1) :
-    cs.IsReflection w := by
-  suffices ∀ n : ℕ, ∀ w : W, cs.length w = n →
-      Module.finrank ℝ (LinearMap.range
-        ((extendedAction M cs w).toLinearMap - LinearMap.id)) = 1 →
-      cs.IsReflection w from this (cs.length w) w rfl hr
-  intro n
-  induction n using Nat.strong_induction_on with
-  | h n ih =>
-    intro w hlen hr
-    have hwne : w ≠ 1 := by
-      intro h
-      subst w
-      have heq : (extendedAction M cs 1).toLinearMap - LinearMap.id = 0 := by
-        apply LinearMap.ext
-        intro v
-        change (extendedAction M cs 1) v - v = 0
-        rw [map_one]
-        change v - v = 0
-        abel
-      rw [heq, LinearMap.range_zero, finrank_bot] at hr
-      omega
-    have hwinv := involution_of_moving_rank_one M cs w hr
-    obtain ⟨i, hi⟩ := cs.exists_rightDescent_of_ne_one hwne
-    change cs.length (w * cs.simple i) < cs.length w at hi
-    have hleft : cs.length (cs.simple i * w) < cs.length w := by
-      have h := cs.length_inv (w * cs.simple i)
-      simp only [mul_inv_rev, cs.inv_simple, hwinv] at h
-      omega
-    by_cases hcomm : cs.simple i * w = w * cs.simple i
-    · have hcomm' : extendedAction M cs w * Module.reflection (extendedForm_root_self M i) =
-          Module.reflection (extendedForm_root_self M i) * extendedAction M cs w := by
-        rw [← extendedAction_simple_eq M cs i, ← map_mul, ← map_mul, hcomm]
-      let f : Module.Dual ℝ (Extended M) := (LinearMap.proj i).comp
-        (LinearMap.fst ℝ (I → ℝ) (Module.Dual ℝ (LinearMap.ker (form M))))
-      have hf : f (extendedRoot M i) = 1 := by simp [f, extendedRoot, root]
-      have hfg : f (extendedAction M cs w (extendedRoot M i)) ≤ 0 := by
-        have hn := action_root_nonpos_of_descent M cs w i hi i
-        simpa only [f, LinearMap.comp_apply, extendedRoot, extendedAction_restrict,
-          LinearMap.fst_apply, LinearMap.proj_apply] using hn
-      have hneg := Realization.negates_of_commutes_nonpos (extendedForm M)
-        (extendedAction M cs w) (extendedAction_preserves M cs w)
-        (extendedRoot M i) (extendedForm_root_self M i) hcomm' f hf hfg
-      have heq : extendedAction M cs w = extendedAction M cs (cs.simple i) := by
-        apply LinearEquiv.ext
-        intro v
-        rw [extendedAction_simple]
-        exact Realization.rank_one_eq_reflection_of_negates (extendedForm M)
-          (extendedAction M cs w) (extendedAction_preserves M cs w) hr
-          (extendedRoot M i) (extendedForm_root_self M i) hneg v
-      rw [extendedAction_injective M cs heq]
-      exact cs.isReflection_simple i
-    · let z := cs.simple i * w * cs.simple i
-      have hzlt : cs.length z < n := by
-        have hz := (right_descent_simple_mul_iff cs i i w hcomm).2 hi
-        dsimp [z]
-        omega
-      have hzrank : Module.finrank ℝ (LinearMap.range
-          ((extendedAction M cs z).toLinearMap - LinearMap.id)) = 1 := by
-        have hc := Realization.finrank_moving_conj (extendedAction M cs w)
-          (extendedAction M cs (cs.simple i))
-        have hgi : (extendedAction M cs (cs.simple i))⁻¹ = extendedAction M cs (cs.simple i) := by
-          rw [← map_inv, cs.inv_simple]
-        have heq : extendedAction M cs z =
-            extendedAction M cs (cs.simple i) * extendedAction M cs w *
-              (extendedAction M cs (cs.simple i))⁻¹ := by
-          dsimp [z]
-          rw [map_mul, map_mul, hgi]
-        rw [heq]
-        exact hc.trans hr
-      have hz := ih (cs.length z) hzlt z rfl hzrank
-      have hc := hz.conj (w := cs.simple i)
-      simpa [z, mul_assoc] using hc
-
-end
-end KLInvariance.TitsSpace
-
-namespace KLInvariance.TitsSpace
-open Module
-universe u v
-variable {I : Type u} [Fintype I] (M : CoxeterMatrix I)
-variable {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
-/-- The other direction of reflection faithfulness for the actual action. -/
-theorem moving_rank_one_of_isReflection (w : W) (hw : cs.IsReflection w) :
-    Module.finrank ℝ (LinearMap.range
-      ((extendedAction M cs w).toLinearMap - LinearMap.id)) = 1 := by
-  obtain ⟨z, i, rfl⟩ := hw
-  have hc := Realization.finrank_moving_conj (extendedAction M cs (cs.simple i))
-    (extendedAction M cs z)
-  have heq : extendedAction M cs (z * cs.simple i * z⁻¹) =
-      extendedAction M cs z * extendedAction M cs (cs.simple i) *
-        (extendedAction M cs z)⁻¹ := by rw [map_mul, map_mul, map_inv]
-  rw [heq, hc, extendedAction_simple_eq]
-  exact Realization.finrank_reflection_sub_id _ _ _
-
-theorem reflectionFaithful (w : W) :
-    Module.finrank ℝ (LinearMap.range
-      ((extendedAction M cs w).toLinearMap - LinearMap.id)) = 1 ↔ cs.IsReflection w :=
-  ⟨isReflection_of_moving_rank_one M cs w, moving_rank_one_of_isReflection M cs w⟩
-
-@[simp] theorem dominant_action_root (w : W) (i : I) :
-    dominant M (extendedAction M cs w (extendedRoot M i)) =
-      ∑ k, action M cs w (root i) k := by
-  simp [extendedRoot, extendedAction_restrict, dominant]
-
-/-- Strict dominant sign, not merely closed-cone positivity. -/
-theorem dominant_pos_iff_ascent (w : W) (i : I) :
-    0 < dominant M (extendedAction M cs w (extendedRoot M i)) ↔
-      cs.length w < cs.length (w * cs.simple i) := by
-  classical
-  rw [dominant_action_root]
-  rcases cs.length_mul_simple w i with h | h
-  · have ha := action_root_nonneg_of_ascent M cs w i h
-    have hv : action M cs w (root i) ≠ 0 := by
-      apply (action M cs w).map_ne_zero_iff.mpr
-      intro hz
-      have hz' := congrFun hz i
-      simp [root] at hz'
-    have hex : ∃ k, 0 < action M cs w (root i) k := by
-      by_contra! hn
-      apply hv
-      funext k
-      exact le_antisymm (hn k) (ha k)
-    have hs : 0 < ∑ k, action M cs w (root i) k := by
-      apply (Finset.sum_pos_iff_of_nonneg (fun k _ => ha k)).mpr
-      obtain ⟨k, hk⟩ := hex
-      exact ⟨k, Finset.mem_univ _, hk⟩
-    exact iff_of_true hs (by omega)
-  · have hn := action_root_nonpos_of_descent M cs w i (by omega)
-    have hs : ∑ k, action M cs w (root i) k ≤ 0 := Finset.sum_nonpos (fun k _ => hn k)
-    exact iff_of_false (by linarith) (by omega)
-
-/-- The strictly dominant linear form in the convention `(w ρ)(αᵢ)=ρ(w⁻¹αᵢ)`
-has exactly the source manuscript's left-ascent sign. -/
-theorem dominant_dual_pos_iff_left_ascent (w : W) (i : I) :
-    0 < dominant M (extendedAction M cs w⁻¹ (extendedRoot M i)) ↔
-      cs.length w < cs.length (cs.simple i * w) := by
-  rw [dominant_pos_iff_ascent, cs.length_inv]
-  have h : cs.length (w⁻¹ * cs.simple i) = cs.length (cs.simple i * w) := by
-    rw [← cs.length_inv (cs.simple i * w)]
-    simp
-  rw [h]
-
-end
-end KLInvariance.TitsSpace
-
-/-! Coordinate-dual geometric action. This exists in arbitrary rank and is
-used only to prove the finite standard-parabolic reduction of the main goal. -/
-namespace KLInvariance.Cotangent
-open Module
-universe u v
-variable {I : Type u} (M : CoxeterMatrix I)
-noncomputable section
-
-def root (i : I) : I → ℝ := fun j => -2 * Real.cos (Real.pi / M i j)
-
-lemma normal_root (i : I) : (LinearMap.proj i : (I → ℝ) →ₗ[ℝ] ℝ) (root M i) = 2 := by
-  simp [root]
-
-def simple (i : I) : (I → ℝ) ≃ₗ[ℝ] (I → ℝ) := Module.reflection (normal_root M i)
-
-theorem simple_liftable : M.IsLiftable (simple M) := by
-  intro i j
-  by_cases hij : i = j
-  · subst j
-    rw [M.diagonal, pow_one]
-    apply LinearEquiv.ext
-    intro v
-    exact Module.involutive_reflection (normal_root M i) v
-  by_cases hm0 : M i j = 0
-  · rw [hm0, pow_zero]
-  have hm2 : 2 ≤ M i j := by
-    have := M.off_diagonal i j hij
-    omega
-  apply Realization.geometric_reflection_relation (normal_root M i) (normal_root M j)
-    (M i j) hm2
-  · simp [root, M.symmetric j i]
-  · rfl
-
-def action {W : Type v} [Group W] (cs : CoxeterSystem M W) :
-    W →* ((I → ℝ) ≃ₗ[ℝ] (I → ℝ)) := cs.lift ⟨simple M, simple_liftable M⟩
-
-@[simp] theorem action_simple {W : Type v} [Group W] (cs : CoxeterSystem M W)
-    (i : I) (l : I → ℝ) :
-    action M cs (cs.simple i) l = l - l i • root M i := by
-  rw [action, cs.lift_apply_simple]
-  rfl
-
-section Finite
-variable [Fintype I] {W : Type v} [Group W] (cs : CoxeterSystem M W)
-
-lemma dot_root (l : I → ℝ) (i : I) : l ⬝ᵥ TitsSpace.root i = l i := by
-  classical
-  simp [TitsSpace.root, dotProduct, Pi.single_apply]
-
-lemma root_dot (i : I) (v : I → ℝ) :
-    root M i ⬝ᵥ v = TitsSpace.normal M i v := by
-  simp [root, TitsSpace.normal, TitsSpace.gram, dotProduct]
-
-lemma simple_pairing (i : I) (l v : I → ℝ) :
-    (action M cs (cs.simple i) l) ⬝ᵥ (TitsSpace.action M cs (cs.simple i) v) =
-      l ⬝ᵥ v := by
-  rw [action_simple, TitsSpace.action_simple]
-  simp only [sub_dotProduct, dotProduct_sub, smul_dotProduct, dotProduct_smul,
-    dot_root, root_dot]
-  have hd : root M i i = 2 := by simp [root]
-  rw [hd]
-  simp only [smul_eq_mul]
-  ring
-
-theorem pairing (w : W) (l v : I → ℝ) :
-    (action M cs w l) ⬝ᵥ (TitsSpace.action M cs w v) = l ⬝ᵥ v := by
-  obtain ⟨ω, rfl⟩ := cs.wordProd_surjective w
-  induction ω generalizing l v with
-  | nil => simp
-  | cons i ω ih =>
-    rw [cs.wordProd_cons, map_mul, map_mul, LinearEquiv.mul_apply,
-      LinearEquiv.mul_apply, simple_pairing, ih]
-
-theorem action_injective : Function.Injective (action M cs) := by
-  apply (injective_iff_map_eq_one _).2
-  intro w hw
-  apply TitsSpace.action_injective M cs
-  rw [map_one]
-  apply LinearEquiv.ext
-  intro v
-  apply dotProduct_eq
-  intro l
-  have h := pairing M cs w l v
-  rw [hw] at h
-  simpa [dotProduct_comm] using h
-end Finite
-end
-end KLInvariance.Cotangent
-
-namespace KLInvariance.StandardParabolic
-open CoxeterSystem
-universe u v
-variable {I : Type u} {W : Type v} [Group W] {M : CoxeterMatrix I}
-variable (cs : CoxeterSystem M W) (J : Set I)
-noncomputable section
-
-def matrix : CoxeterMatrix J where
-  M i j := M i.val j.val
-  isSymm := M.isSymm.submatrix Subtype.val
-  diagonal i := M.diagonal i.val
-  off_diagonal i j h := M.off_diagonal i.val j.val (fun h' => h (Subtype.ext h'))
-
-abbrev system : CoxeterSystem (matrix (M := M) J) (matrix (M := M) J).Group :=
-  (matrix (M := M) J).toCoxeterSystem
-
-def inclusion : (matrix (M := M) J).Group →* W :=
-  (system (M := M) J).lift ⟨fun i => cs.simple i.val,
-    fun i j => cs.simple_mul_simple_pow i.val j.val⟩
-
-@[simp] theorem inclusion_simple (i : J) :
-    inclusion cs J ((system (M := M) J).simple i) = cs.simple i.val :=
-  (system (M := M) J).lift_apply_simple _ i
-
-lemma inclusion_word (ω : List J) :
-    inclusion cs J ((system (M := M) J).wordProd ω) = cs.wordProd (ω.map Subtype.val) := by
-  induction ω with
-  | nil => simp
-  | cons i ω ih =>
-    rw [wordProd_cons, map_mul, inclusion_simple, List.map_cons, wordProd_cons, ih]
-
-def restrictVec (l : I → ℝ) : J → ℝ := fun j => l j.val
-
-lemma simple_restrict (i : J) (l : I → ℝ) :
-    restrictVec J (Cotangent.action M cs (cs.simple i.val) l) =
-      Cotangent.action (matrix (M := M) J) (system (M := M) J)
-        ((system (M := M) J).simple i) (restrictVec J l) := by
-  rw [Cotangent.action_simple, Cotangent.action_simple]
-  funext j
-  rfl
-
-theorem action_restrict (w : (matrix (M := M) J).Group) (l : I → ℝ) :
-    restrictVec J (Cotangent.action M cs (inclusion cs J w) l) =
-      Cotangent.action (matrix (M := M) J) (system (M := M) J) w (restrictVec J l) := by
-  obtain ⟨ω, rfl⟩ := (system (M := M) J).wordProd_surjective w
-  induction ω generalizing l with
-  | nil => simp
-  | cons i ω ih =>
-    rw [wordProd_cons, map_mul, inclusion_simple, map_mul, map_mul,
-      LinearEquiv.mul_apply, LinearEquiv.mul_apply, simple_restrict, ih]
-
-theorem inclusion_injective [Fintype J] : Function.Injective (inclusion cs J) := by
-  classical
-  apply (injective_iff_map_eq_one _).2
-  intro w hw
-  apply Cotangent.action_injective (matrix (M := M) J) (system (M := M) J)
-  rw [map_one]
-  apply LinearEquiv.ext
-  intro l
-  let l' : I → ℝ := fun i => if hi : i ∈ J then l ⟨i, hi⟩ else 0
-  have hr : restrictVec J l' = l := by ext j; simp [restrictVec, l', j.property]
-  have h := action_restrict cs J w l'
-  rw [hw, map_one] at h
-  change restrictVec J l' = _ at h
-  rw [hr] at h
-  exact h.symm
-
-lemma lift_word {ω : List I} (hω : ∀ i ∈ ω, i ∈ J) :
-    ∃ ν : List J, ν.map Subtype.val = ω := by
-  induction ω with
-  | nil => exact ⟨[], rfl⟩
-  | cons i ω ih =>
-    obtain ⟨ν, rfl⟩ := ih (fun j hj => hω j (by simp [hj]))
-    exact ⟨⟨i, hω i (by simp)⟩ :: ν, rfl⟩
-
-theorem inclusion_mem (w : (matrix (M := M) J).Group) :
-    inclusion cs J w ∈ ParabolicSupport.subgroup cs J := by
-  obtain ⟨ω, rfl⟩ := (system (M := M) J).wordProd_surjective w
-  rw [inclusion_word]
-  apply ParabolicSupport.word_mem
-  intro i hi
-  obtain ⟨j, _, rfl⟩ := List.mem_map.mp hi
-  exact j.property
-
-theorem inclusion_range : (inclusion cs J).range = ParabolicSupport.subgroup cs J := by
-  apply le_antisymm
-  · rintro _ ⟨w, rfl⟩
-    exact inclusion_mem cs J w
-  · intro w hw
-    obtain ⟨ω, hω, rfl⟩ := ParabolicSupport.exists_word cs J hw
-    obtain ⟨ν, rfl⟩ := lift_word J hω
-    exact ⟨(system (M := M) J).wordProd ν, inclusion_word cs J ν⟩
-
-theorem length_inclusion [Fintype J] (w : (matrix (M := M) J).Group) :
-    cs.length (inclusion cs J w) = (system (M := M) J).length w := by
-  apply le_antisymm
-  · obtain ⟨ω, hω, rfl⟩ := (system (M := M) J).exists_isReduced w
-    rw [inclusion_word, hω.eq]
-    simpa using cs.length_wordProd_le (ω.map Subtype.val)
-  · obtain ⟨ω, hω, hred, hprod⟩ := ParabolicSupport.exists_reduced_word cs J
-      (inclusion_mem cs J w)
-    obtain ⟨ν, rfl⟩ := lift_word J hω
-    have heq : (system (M := M) J).wordProd ν = w :=
-      inclusion_injective cs J ((inclusion_word cs J ν).trans hprod)
-    have hb := (system (M := M) J).length_wordProd_le ν
-    rw [heq] at hb
-    have hr := hred.eq
-    rw [hprod, List.length_map] at hr
-    omega
-
-lemma reduced_map_iff [Fintype J] (ω : List J) :
-    cs.IsReduced (ω.map Subtype.val) ↔ (system (M := M) J).IsReduced ω := by
-  simp only [CoxeterSystem.IsReduced, List.length_map, ← inclusion_word, length_inclusion]
-
-end
-end KLInvariance.StandardParabolic
-
-namespace KLInvariance.StandardParabolic
-open CoxeterSystem
-universe u v
-variable {I : Type u} {W : Type v} [Group W] {M : CoxeterMatrix I}
-variable (cs : CoxeterSystem M W) (J : Set I) [Fintype J]
-noncomputable section
-
-theorem bruhat_inclusion_iff (x y : (matrix (M := M) J).Group) :
-    BruhatLE cs (inclusion cs J x) (inclusion cs J y) ↔
-      BruhatLE (system (M := M) J) x y := by
-  obtain ⟨ω, hω, rfl⟩ := (system (M := M) J).exists_isReduced y
-  rw [inclusion_word]
-  constructor
-  · intro h
-    obtain ⟨ν, hsub, _, hprod⟩ := (bruhat_iff_reduced_subword cs
-      ((reduced_map_iff cs J ω).2 hω)).1 h
-    obtain ⟨τ, hτ, rfl⟩ := List.sublist_map_iff.mp hsub
-    have hx : (system (M := M) J).wordProd τ = x :=
-      inclusion_injective cs J ((inclusion_word cs J τ).trans hprod)
-    rw [← hx]
-    exact bruhat_of_subword _ hτ hω
-  · intro h
-    obtain ⟨ν, hsub, _, rfl⟩ := (bruhat_iff_reduced_subword _ hω).1 h
-    rw [inclusion_word]
-    exact bruhat_of_subword cs (hsub.map Subtype.val) ((reduced_map_iff cs J ω).2 hω)
-
-theorem lower_range (y : (matrix (M := M) J).Group) {z : W}
-    (hzy : BruhatLE cs z (inclusion cs J y)) :
-    ∃ x, inclusion cs J x = z := by
-  obtain ⟨ω, hω, rfl⟩ := (system (M := M) J).exists_isReduced y
-  rw [inclusion_word] at hzy
-  obtain ⟨ν, hsub, _, hprod⟩ := (bruhat_iff_reduced_subword cs
-    ((reduced_map_iff cs J ω).2 hω)).1 hzy
-  obtain ⟨τ, _, rfl⟩ := List.sublist_map_iff.mp hsub
-  exact ⟨(system (M := M) J).wordProd τ, (inclusion_word cs J τ).trans hprod⟩
-
-def intervalMap (x y : (matrix (M := M) J).Group) :
-    Interval (system (M := M) J) x y →
-      Interval cs (inclusion cs J x) (inclusion cs J y) :=
-  fun z => ⟨inclusion cs J z.val, (bruhat_inclusion_iff cs J x z.val).2 z.property.1,
-    (bruhat_inclusion_iff cs J z.val y).2 z.property.2⟩
-
-lemma intervalMap_bijective (x y : (matrix (M := M) J).Group) :
-    Function.Bijective (intervalMap cs J x y) := by
-  constructor
-  · intro z t h
-    exact Subtype.ext (inclusion_injective cs J (congrArg Subtype.val h))
-  · intro z
-    obtain ⟨w, hw⟩ := lower_range cs J y z.property.2
-    refine ⟨⟨w, ?_, ?_⟩, Subtype.ext hw⟩
-    · apply (bruhat_inclusion_iff cs J x w).1
-      rw [hw]
-      exact z.property.1
-    · apply (bruhat_inclusion_iff cs J w y).1
-      rw [hw]
-      exact z.property.2
-
-/-- The exact unlabelled interval identification used in the finite-rank
-reduction; it arises from a proved embedding, not an added main hypothesis. -/
-def intervalIso (x y : (matrix (M := M) J).Group) :
-    Interval (system (M := M) J) x y ≃o
-      Interval cs (inclusion cs J x) (inclusion cs J y) where
-  toEquiv := Equiv.ofBijective _ (intervalMap_bijective cs J x y)
-  map_rel_iff' := by
-    intro z t
-    exact bruhat_inclusion_iff cs J z.val t.val
-
-@[simp] theorem intervalIso_val (x y : (matrix (M := M) J).Group)
-    (z : Interval (system (M := M) J) x y) :
-    (intervalIso cs J x y z).val = inclusion cs J z.val := rfl
-
-lemma rankDifference_inclusion (x y : (matrix (M := M) J).Group) :
-    rankDifference cs (inclusion cs J x) (inclusion cs J y) =
-      rankDifference (system (M := M) J) x y := by
-  simp only [rankDifference, length_inclusion]
-
-/-- Restriction of the actual normalized families satisfies their exact
-recurrence, degree bound and interval reciprocity in the standard parabolic. -/
-theorem normalized_restriction {RP : PolynomialFamilies W} (hRP : NormalizedKL cs RP) :
-    NormalizedKL (system (M := M) J)
-      (fun x y => RP.1 (inclusion cs J x) (inclusion cs J y),
-       fun x y => RP.2 (inclusion cs J x) (inclusion cs J y)) where
-  R_diagonal x := hRP.R_diagonal _
-  R_zero x y hxy := hRP.R_zero _ _ (fun h => hxy ((bruhat_inclusion_iff cs J x y).1 h))
-  R_recursion := by
-    intro x y i hy
-    have hs (z : (matrix (M := M) J).Group) :
-        inclusion cs J ((system (M := M) J).simple i * z) =
-          cs.simple i.val * inclusion cs J z := by rw [map_mul, inclusion_simple]
-    have hl (z : (matrix (M := M) J).Group) :
-        cs.length (cs.simple i.val * inclusion cs J z) =
-          (system (M := M) J).length ((system (M := M) J).simple i * z) := by
-      rw [← hs, length_inclusion]
-    have h := hRP.R_recursion (inclusion cs J x) (inclusion cs J y) i.val
-      (by rwa [hl, length_inclusion])
-    simpa only [hs, hl, length_inclusion] using h
-  P_diagonal x := hRP.P_diagonal _
-  P_zero x y hxy := hRP.P_zero _ _ (fun h => hxy ((bruhat_inclusion_iff cs J x y).1 h))
-  P_degree := by
-    intro x y hxy hne
-    have h := hRP.P_degree (inclusion cs J x) (inclusion cs J y)
-      ((bruhat_inclusion_iff cs J x y).2 hxy) (fun h => hne (inclusion_injective cs J h))
-    rwa [rankDifference_inclusion] at h
-  reciprocity := by
-    intro x y hxy
-    have h := hRP.reciprocity (inclusion cs J x) (inclusion cs J y)
-      ((bruhat_inclusion_iff cs J x y).2 hxy)
-    rw [rankDifference_inclusion] at h
-    rw [h]
-    let := interval_finite cs (inclusion cs J x) (inclusion cs J y)
-    let := interval_finite (system (M := M) J) x y
-    let := Fintype.ofFinite (Interval cs (inclusion cs J x) (inclusion cs J y))
-    let := Fintype.ofFinite (Interval (system (M := M) J) x y)
-    simp only [finsum_eq_sum_of_fintype]
-    exact ((intervalIso cs J x y).toEquiv.sum_comp
-      (fun z => RP.1 (inclusion cs J x) z.val * RP.2 z.val (inclusion cs J y))).symm
-
-theorem klFamilies_inclusion :
-    (fun x y => rPolynomial cs (inclusion cs J x) (inclusion cs J y),
-     fun x y => klPolynomial cs (inclusion cs J x) (inclusion cs J y)) =
-      klFamilies (system (M := M) J) :=
-  normalizedKL_unique _ (normalized_restriction cs J (klFamilies_spec cs))
-    (klFamilies_spec _)
-
-theorem klPolynomial_inclusion (x y : (matrix (M := M) J).Group) :
-    klPolynomial cs (inclusion cs J x) (inclusion cs J y) =
-      klPolynomial (system (M := M) J) x y :=
-  congrFun (congrFun (congrArg Prod.snd (klFamilies_inclusion cs J)) x) y
-
-theorem rPolynomial_inclusion (x y : (matrix (M := M) J).Group) :
-    rPolynomial cs (inclusion cs J x) (inclusion cs J y) =
-      rPolynomial (system (M := M) J) x y :=
-  congrFun (congrFun (congrArg Prod.fst (klFamilies_inclusion cs J)) x) y
-
-end
-end KLInvariance.StandardParabolic
-
-namespace KLInvariance.TitsSpace
-open Module
-universe u v
-variable {I : Type u} [Fintype I] (M : CoxeterMatrix I)
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
-/-- A genuine real root, obtained from a simple root by the Tits action. -/
-def IsRoot (a : I → ℝ) : Prop := ∃ w : W, ∃ i : I, a = action M cs w (root i)
-
-/-- A normalized real root representing a genuine Coxeter reflection. -/
-def Represents (t : W) (a : I → ℝ) : Prop :=
-  ∃ w : W, ∃ i : I, t = w * cs.simple i * w⁻¹ ∧ a = action M cs w (root i)
-
-abbrev embed (a : I → ℝ) : Extended M := (a, 0)
-
-@[simp] theorem form_root_self (i : I) : form M (root i) (root i) = 2 := by
-  simp []
-
-variable {M cs}
-
-theorem IsRoot.norm {a : I → ℝ} (ha : IsRoot M cs a) : form M a a = 2 := by
-  obtain ⟨w,i,rfl⟩ := ha
-  rw [action_preserves, form_root_self]
-
-theorem IsRoot.ne_zero {a : I → ℝ} (ha : IsRoot M cs a) : a ≠ 0 := by
-  intro he
-  have hn := ha.norm
-  simp [he] at hn
-
-theorem IsRoot.sign {a : I → ℝ} (ha : IsRoot M cs a) :
-    (∀ i, 0 ≤ a i) ∨ (∀ i, a i ≤ 0) := by
-  obtain ⟨w,i,rfl⟩ := ha
-  rcases lt_or_gt_of_ne (cs.length_mul_simple_ne w i) with h|h
-  · exact Or.inr (action_root_nonpos_of_descent M cs w i h)
-  · exact Or.inl (action_root_nonneg_of_ascent M cs w i (by
-      rcases cs.length_mul_simple w i with h'|h' <;> omega))
-
-theorem IsRoot.neg {a : I → ℝ} (ha : IsRoot M cs a) : IsRoot M cs (-a) := by
-  obtain ⟨w,i,rfl⟩ := ha
-  refine ⟨w * cs.simple i, i, ?_⟩
-  simp only [map_mul, LinearEquiv.mul_apply, action_simple_root, map_neg]
-
-theorem IsRoot.act {a : I → ℝ} (ha : IsRoot M cs a) (w : W) :
-    IsRoot M cs (action M cs w a) := by
-  obtain ⟨z,i,rfl⟩ := ha
-  exact ⟨w*z, i, by simp only [map_mul, LinearEquiv.mul_apply]⟩
-
-theorem Represents.isRoot {t : W} {a : I → ℝ} (ha : Represents M cs t a) :
-    IsRoot M cs a := by
-  obtain ⟨w,i,_,h⟩ := ha
-  exact ⟨w,i,h⟩
-
-theorem Represents.isReflection {t : W} {a : I → ℝ} (ha : Represents M cs t a) :
-    cs.IsReflection t := by
-  obtain ⟨w,i,h,_⟩ := ha
-  exact ⟨w,i,h⟩
-
-theorem Represents.neg {t : W} {a : I → ℝ} (ha : Represents M cs t a) :
-    Represents M cs t (-a) := by
-  obtain ⟨w,i,rfl,rfl⟩ := ha
-  refine ⟨w * cs.simple i, i, ?_, ?_⟩
-  · simp [mul_assoc]
-  · simp only [map_mul, LinearEquiv.mul_apply, action_simple_root, map_neg]
-
-theorem Represents.act {t : W} {a : I → ℝ} (ha : Represents M cs t a) (w : W) :
-    Represents M cs (w*t*w⁻¹) (action M cs w a) := by
-  obtain ⟨z,i,rfl,rfl⟩ := ha
-  refine ⟨w*z,i,?_,?_⟩
-  · group
-  · simp only [map_mul, LinearEquiv.mul_apply]
-
-/-- The actual reflection formula for every conjugate root in the minimal
-nondegenerate extension, not merely for simple roots. -/
-theorem Represents.formula {t : W} {a : I → ℝ} (ha : Represents M cs t a)
-    (v : Extended M) :
-    extendedAction M cs t v = v - extendedForm M (embed M a) v • embed M a := by
-  obtain ⟨w,i,rfl,rfl⟩ := ha
-  have hv : extendedAction M cs w (extendedAction M cs w⁻¹ v) = v := by
-    rw [← LinearEquiv.mul_apply, ← map_mul, mul_inv_cancel, map_one]
-    rfl
-  have hp := extendedAction_preserves M cs w (extendedRoot M i)
-    (extendedAction M cs w⁻¹ v)
-  rw [hv] at hp
-  rw [map_mul, map_mul]
-  simp only [LinearEquiv.mul_apply, extendedAction_simple, map_sub, map_smul]
-  rw [hv, ← hp, extendedRoot, extendedAction_restrict]
-
-theorem Represents.action_formula {t : W} {a : I → ℝ} (ha : Represents M cs t a)
-    (v : I → ℝ) : action M cs t v = v - form M a v • a := by
-  have h := congrArg Prod.fst (ha.formula (embed M v))
-  simpa only [extendedAction_restrict, embed, (extendedForm_spec M).2.2,
-    Prod.fst_sub, Prod.smul_fst] using h
-
-theorem Represents.unique_up_to_sign {t : W} {a b : I → ℝ}
-    (ha : Represents M cs t a) (hb : Represents M cs t b) : a = b ∨ a = -b := by
-  have hn := ha.isRoot.norm
-  have hm := hb.isRoot.norm
-  have hf := ha.action_formula a
-  rw [hb.action_formula a, hn] at hf
-  have hmul : form M b a • b = (2 : ℝ) • a := by
-    have h := congrArg (fun z => a-z) hf
-    simpa only [sub_sub_cancel] using h
-  have hab : a = (form M b a / 2) • b := by
-    calc
-      a = (1/2 : ℝ) • ((2 : ℝ) • a) := by module
-      _ = (1/2 : ℝ) • (form M b a • b) := by rw [hmul]
-      _ = _ := by module
-  have hc : (form M b a / 2) ^ 2 = 1 := by
-    rw [hab] at hn
-    simp only [map_smul, LinearMap.smul_apply, smul_eq_mul, hm] at hn
-    nlinarith
-  rcases sq_eq_one_iff.mp hc with h|h
-  · exact Or.inl (by simpa [h] using hab)
-  · exact Or.inr (by simpa [h] using hab)
-
-theorem Represents.reflection_unique {t s : W} {a : I → ℝ}
-    (ha : Represents M cs t a) (hb : Represents M cs s a) : t = s := by
-  apply extendedAction_injective M cs
-  apply LinearEquiv.ext
-  intro v
-  rw [ha.formula, hb.formula]
-
-theorem Represents.unique_positive {t : W} {a b : I → ℝ}
-    (ha : Represents M cs t a) (hb : Represents M cs t b)
-    (hpa : ∀ i, 0 ≤ a i) (hpb : ∀ i, 0 ≤ b i) : a = b := by
-  rcases ha.unique_up_to_sign hb with h|h
-  · exact h
-  · exfalso
-    apply ha.isRoot.ne_zero
-    ext i
-    have hi := congrFun h i
-    simp only [Pi.neg_apply] at hi
-    have := hpa i
-    have := hpb i
-    simp only [Pi.zero_apply]
-    linarith
-
-variable (M) (cs)
-
-theorem exists_positive_root {t : W} (ht : cs.IsReflection t) :
-    ∃ a : I → ℝ, Represents M cs t a ∧ ∀ i, 0 ≤ a i := by
-  obtain ⟨w,i,rfl⟩ := ht
-  have hr : Represents M cs (w * cs.simple i * w⁻¹) (action M cs w (root i)) :=
-    ⟨w,i,rfl,rfl⟩
-  rcases hr.isRoot.sign with h|h
-  · exact ⟨_,hr,h⟩
-  · exact ⟨_,hr.neg,fun i => by simpa only [Pi.neg_apply, neg_nonneg] using h i⟩
-
-/-- Genuine positive roots, with their actual coordinate sign. -/
-def PositiveRoot := {a : I → ℝ // IsRoot M cs a ∧ ∀ i, 0 ≤ a i}
-
-def positiveRoot (t : {t : W // cs.IsReflection t}) : PositiveRoot M cs :=
-  ⟨(exists_positive_root M cs t.property).choose,
-    (exists_positive_root M cs t.property).choose_spec.1.isRoot,
-    (exists_positive_root M cs t.property).choose_spec.2⟩
-
-theorem positiveRoot_represents (t : {t : W // cs.IsReflection t}) :
-    Represents M cs t.val (positiveRoot M cs t).val :=
-  (exists_positive_root M cs t.property).choose_spec.1
-
-def rootReflection (a : PositiveRoot M cs) : {t : W // cs.IsReflection t} := by
-  let w := a.property.1.choose
-  let i := a.property.1.choose_spec.choose
-  exact ⟨w * cs.simple i * w⁻¹, w,i,rfl⟩
-
-theorem rootReflection_represents (a : PositiveRoot M cs) :
-    Represents M cs (rootReflection M cs a).val a.val := by
-  refine ⟨a.property.1.choose, a.property.1.choose_spec.choose, rfl, ?_⟩
-  exact a.property.1.choose_spec.choose_spec
-
-
-def positiveRootEquiv : {t : W // cs.IsReflection t} ≃ PositiveRoot M cs where
-  toFun := positiveRoot M cs
-  invFun := rootReflection M cs
-  left_inv t := Subtype.ext ((rootReflection_represents M cs _).reflection_unique
-      (positiveRoot_represents M cs t))
-  right_inv a := Subtype.ext ((positiveRoot_represents M cs _).unique_positive
-      (rootReflection_represents M cs a) (positiveRoot M cs _).property.2 a.property.2)
-
-end
-end KLInvariance.TitsSpace
-
-namespace KLInvariance.TitsSpace
-open Module
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-noncomputable section
-
-theorem IsRoot.not_both {a : I → ℝ} (ha : IsRoot M cs a)
-    (hp : ∀ i, 0 ≤ a i) (hn : ∀ i, a i ≤ 0) : False := by
-  apply ha.ne_zero
-  ext i
-  exact le_antisymm (hn i) (hp i)
-
-/-- A simple reflection can turn a positive real root negative only on its
-own root. This uses actual Tits coordinates and normalization. -/
-theorem IsRoot.eq_simple_of_nonpos {a : I → ℝ} (ha : IsRoot M cs a)
-    (hp : ∀ k, 0 ≤ a k) (i : I)
-    (hn : ∀ k, action M cs (cs.simple i) a k ≤ 0) : a = root i := by
-  classical
-  have ha0 (k : I) (hk : k ≠ i) : a k = 0 := by
-    have h := hn k
-    simp only [action_simple, Pi.sub_apply, Pi.smul_apply, root,
-      Pi.single_apply, hk, ↓reduceIte, smul_eq_mul, mul_zero, sub_zero] at h
-    exact le_antisymm h (hp k)
-  have heq : a = a i • root i := by
-    ext k
-    by_cases hki : k = i
-    · subst k; simp [root]
-    · simp [root, hki, ha0 k hki]
-  have hnorm := ha.norm
-  rw [heq] at hnorm
-  simp only [map_smul, LinearMap.smul_apply, smul_eq_mul, form_root_self] at hnorm
-  have hai : a i = 1 := by nlinarith [hp i]
-  simpa only [hai, one_smul] using heq
-
-theorem Represents.simple_sign {t : W} {a : I → ℝ} (ha : Represents M cs t a)
-    (hp : ∀ k, 0 ≤ a k) (i : I) :
-    ((∀ k, action M cs (cs.simple i) a k ≤ 0) ↔ t = cs.simple i) ∧
-    ((∀ k, 0 ≤ action M cs (cs.simple i) a k) ↔ t ≠ cs.simple i) := by
-  have hi : Represents M cs (cs.simple i) (root i) := ⟨1,i,by simp,by simp⟩
-  have hneg : (∀ k, action M cs (cs.simple i) a k ≤ 0) ↔ t = cs.simple i := by
-    constructor
-    · intro h
-      have heq := ha.isRoot.eq_simple_of_nonpos hp i h
-      exact ha.reflection_unique (heq.symm ▸ hi)
-    · intro h
-      have heq := (h ▸ ha).unique_positive hi hp (root_nonneg i)
-      simpa only [heq, action_simple_root, Pi.neg_apply, neg_nonpos] using root_nonneg i
-  refine ⟨hneg, ?_⟩
-  constructor
-  · intro h heq
-    exact (ha.isRoot.act (cs.simple i)).not_both h (hneg.mpr heq)
-  · intro h
-    rcases (ha.isRoot.act (cs.simple i)).sign with hp|hn
-    · exact hp
-    · exact False.elim (h (hneg.mp hn))
-
-/-- The signed combinatorial reflection action is the actual root sign
-under the faithful Tits representation. -/
-def SignedRepresents (p : W × ZMod 2) (a : I → ℝ) : Prop :=
-  Represents M cs p.1 a ∧
-    ((p.2 = 0 ∧ ∀ i, 0 ≤ a i) ∨ (p.2 = 1 ∧ ∀ i, a i ≤ 0))
-
-theorem SignedRepresents.simple {p : W × ZMod 2} {a : I → ℝ}
-    (ha : SignedRepresents (M := M) (cs := cs) p a) (i : I) :
-    SignedRepresents (M := M) (cs := cs) (CoxeterSupport.simpleAction cs i p)
-      (action M cs (cs.simple i) a) := by
-  rcases p with ⟨t,z⟩
-  rw [CoxeterSupport.simpleAction_apply]
-  refine ⟨?_,?_⟩
-  · simpa only [cs.inv_simple] using ha.1.act (cs.simple i)
-  · rcases ha.2 with ⟨rfl,hp⟩|⟨rfl,hn⟩
-    · by_cases heq : t = cs.simple i
-      · exact Or.inr ⟨by simp [heq], (ha.1.simple_sign hp i).1.mpr heq⟩
-      · exact Or.inl ⟨by simp [heq, ZModModule.add_self], (ha.1.simple_sign hp i).2.mpr heq⟩
-    · have hp : ∀ k, 0 ≤ (-a) k := fun k => neg_nonneg.mpr (hn k)
-      by_cases heq : t = cs.simple i
-      · refine Or.inl ⟨by simp [heq, ZModModule.add_self], ?_⟩
-        have h := (ha.1.neg.simple_sign hp i).1.mpr heq
-        simpa only [map_neg, Pi.neg_apply, neg_nonpos] using h
-      · refine Or.inr ⟨by simp [heq], ?_⟩
-        have h := (ha.1.neg.simple_sign hp i).2.mpr heq
-        simpa only [map_neg, Pi.neg_apply, neg_nonneg] using h
-
-theorem SignedRepresents.act {p : W × ZMod 2} {a : I → ℝ}
-    (ha : SignedRepresents (M := M) (cs := cs) p a) (w : W) :
-    SignedRepresents (M := M) (cs := cs) (CoxeterSupport.reflectionAction cs w p)
-      (action M cs w a) := by
-  obtain ⟨ω,rfl⟩ := cs.wordProd_surjective w
-  induction ω with
-  | nil => simpa using ha
-  | cons i ω ih =>
-    simpa only [cs.wordProd_cons, map_mul, Equiv.Perm.mul_apply,
-      CoxeterSupport.reflectionAction_simple, LinearEquiv.mul_apply] using ih.simple i
-
-
-theorem Represents.inversion_iff {t : W} {a : I → ℝ} (ha : Represents M cs t a)
-    (hp : ∀ i, 0 ≤ a i) (w : W) :
-    (∀ i, action M cs w⁻¹ a i ≤ 0) ↔ cs.IsLeftInversion w t := by
-  classical
-  obtain ⟨ω,hω,heq⟩ := cs.exists_isReduced w
-  rw [heq]
-  let c : ZMod 2 := (cs.leftInvSeq ω).count t
-  have hc : c+c=0 := ZModModule.add_self c
-  have hrec := CoxeterSupport.wordAction_record cs ω t c
-  rw [← CoxeterSupport.reflectionAction_wordProd] at hrec
-  have hrec' : CoxeterSupport.reflectionAction cs (cs.wordProd ω)
-      ((cs.wordProd ω)⁻¹ * t * cs.wordProd ω, c) = (t,0) := by
-    change CoxeterSupport.reflectionAction cs (cs.wordProd ω)
-      ((cs.wordProd ω)⁻¹ * t * cs.wordProd ω,c) = (t,c+c) at hrec
-    rwa [hc] at hrec
-  have hinv : CoxeterSupport.reflectionAction cs (cs.wordProd ω)⁻¹ (t,0) =
-      ((cs.wordProd ω)⁻¹ * t * cs.wordProd ω,c) := by
-    rw [map_inv, ← hrec']
-    exact Equiv.symm_apply_apply _ _
-  have hs : SignedRepresents (M := M) (cs := cs) (t,0) a := ⟨ha,Or.inl ⟨rfl,hp⟩⟩
-  have hsi := hs.act (cs.wordProd ω)⁻¹
-  rw [hinv] at hsi
-  have hcount : c = if t ∈ cs.leftInvSeq ω then 1 else 0 := by
-    dsimp [c]
-    by_cases hm : t ∈ cs.leftInvSeq ω
-    · rw [ite_eq_left hm, List.count_eq_one_of_mem hω.nodup_leftInvSeq hm]
-      norm_num
-    · rw [ite_eq_right hm, List.count_eq_zero.mpr hm]
-      rfl
-  have hmem : (∀ i, action M cs (cs.wordProd ω)⁻¹ a i ≤ 0) ↔ t ∈ cs.leftInvSeq ω := by
-    by_cases hm : t ∈ cs.leftInvSeq ω
-    · have hc1 : c=1 := by simpa [hm] using hcount
-      simp only [hc1] at hsi
-      rcases hsi.2 with ⟨hbad,_⟩|⟨_,hn⟩
-      · norm_num at hbad
-      · exact iff_of_true hn hm
-    · have hc0 : c=0 := by simpa [hm] using hcount
-      simp only [hc0] at hsi
-      rcases hsi.2 with ⟨_,hpos⟩|⟨hbad,_⟩
-      · exact iff_of_false (fun hn => (ha.isRoot.act _).not_both hpos hn) hm
-      · norm_num at hbad
-  rw [hmem]
-  exact ⟨cs.isLeftInversion_of_mem_leftInvSeq hω,
-    CoxeterSupport.mem_leftInvSeq_of_inversion cs hω⟩
-
-end
-end KLInvariance.TitsSpace
-
-namespace KLInvariance.TitsSpace
-open Module
-universe u v
-variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
-  {W : Type v} [Group W] {cs : CoxeterSystem M W}
-noncomputable section
-
-theorem Represents.independent_of_ne {t s : W} {a b : I → ℝ}
-    (ha : Represents M cs t a) (hb : Represents M cs s b) (hts : t ≠ s) :
-    LinearIndependent ℝ ![a,b] := by
-  apply (LinearIndependent.pair_iff' ha.isRoot.ne_zero).mpr
-  intro c heq
-  have hn := hb.isRoot.norm
-  rw [← heq] at hn
-  simp only [map_smul, LinearMap.smul_apply, smul_eq_mul, ha.isRoot.norm] at hn
-  have hc : c^2 = 1 := by nlinarith
-  rcases sq_eq_one_iff.mp hc with hc|hc
-  · have hab : a=b := by simpa only [hc,one_smul] using heq
-    exact hts (ha.reflection_unique (hab.symm ▸ hb))
-  · have hab : -a=b := by simpa only [hc,neg_one_smul] using heq
-    exact hts (ha.neg.reflection_unique (hab.symm ▸ hb))
-
-theorem Represents.extended_independent_of_ne {t s : W} {a b : I → ℝ}
-    (ha : Represents M cs t a) (hb : Represents M cs s b) (hts : t ≠ s) :
-    LinearIndependent ℝ ![embed M a,embed M b] := by
-  have h := (ha.independent_of_ne hb hts).map'
-    (LinearMap.inl ℝ (I → ℝ) (Module.Dual ℝ (LinearMap.ker (form M))))
-    (LinearMap.ker_eq_bot.mpr LinearMap.inl_injective)
-  have heq : (⇑(LinearMap.inl ℝ (I → ℝ) (Module.Dual ℝ (LinearMap.ker (form M)))) ∘
-      ![a,b]) = ![embed M a,embed M b] := by
-    funext i
-    fin_cases i <;> rfl
-  rwa [heq] at h
-
-theorem Represents.extendedAction_eq {t : W} {a : I → ℝ} (ha : Represents M cs t a) :
-    extendedAction M cs t = Module.reflection
-      (show extendedForm M (embed M a) (embed M a) = 2 by
-        rw [(extendedForm_spec M).2.2]; exact ha.isRoot.norm) := by
-  apply LinearEquiv.ext
-  intro v
-  exact ha.formula v
-
-
-theorem Represents.plane_eq_of_product_eq {t s r q : W} {a b c d : I → ℝ}
-    (ha : Represents M cs t a) (hb : Represents M cs s b)
-    (hc : Represents M cs r c) (hd : Represents M cs q d)
-    (hts : t ≠ s) (hrq : r ≠ q) (heq : t*s=r*q) :
-    Submodule.span ℝ {embed M a, embed M b} =
-      Submodule.span ℝ {embed M c, embed M d} := by
-  apply ReflectionPlane.plane_eq_of_reflection_product_eq (extendedForm M)
-    (extendedForm_spec M).2.1 (embed M a) (embed M b) (embed M c) (embed M d)
-    (by rw [(extendedForm_spec M).2.2]; exact ha.isRoot.norm)
-    (by rw [(extendedForm_spec M).2.2]; exact hb.isRoot.norm)
-    (by rw [(extendedForm_spec M).2.2]; exact hc.isRoot.norm)
-    (by rw [(extendedForm_spec M).2.2]; exact hd.isRoot.norm)
-    (ha.extended_independent_of_ne hb hts) (hc.extended_independent_of_ne hd hrq)
-  have h := congrArg (extendedAction M cs) heq
-  rw [map_mul, map_mul, ha.extendedAction_eq, hb.extendedAction_eq,
-    hc.extendedAction_eq, hd.extendedAction_eq] at h
-  exact congrArg LinearEquiv.toLinearMap h
-
-variable (M) (cs)
-/-- The actual positive ambient roots inverted by `w`. -/
-def RootInversions (w : W) :=
-  {a : PositiveRoot M cs // ∀ i, action M cs w⁻¹ a.val i ≤ 0}
-
-def rootInversionEquiv (w : W) : RootInversions M cs w ≃ {t : W // cs.IsLeftInversion w t} where
-  toFun a := ⟨(rootReflection M cs a.val).val,
-    ((rootReflection_represents M cs a.val).inversion_iff a.val.property.2 w).mp a.property⟩
-  invFun t := ⟨positiveRoot M cs ⟨t.val,t.property.1⟩,
-    ((positiveRoot_represents M cs ⟨t.val,t.property.1⟩).inversion_iff
-      (positiveRoot M cs ⟨t.val,t.property.1⟩).property.2 w).mpr t.property⟩
-  left_inv a := by
-    apply Subtype.ext
-    exact (positiveRootEquiv M cs).apply_symm_apply a.val
-  right_inv t := by
-    apply Subtype.ext
-    change (rootReflection M cs (positiveRoot M cs ⟨t.val,t.property.1⟩)).val = t.val
-    exact congrArg (fun r : {r : W // cs.IsReflection r} => r.val)
-      ((positiveRootEquiv M cs).symm_apply_apply ⟨t.val,t.property.1⟩)
-
-def leftInversionListEquiv {ω : List I} (hω : cs.IsReduced ω) :
-    {t : W // cs.IsLeftInversion (cs.wordProd ω) t} ≃ {t : W // t ∈ cs.leftInvSeq ω} where
-  toFun t := ⟨t.val,CoxeterSupport.mem_leftInvSeq_of_inversion cs hω t.property⟩
-  invFun t := ⟨t.val,cs.isLeftInversion_of_mem_leftInvSeq hω t.property⟩
-  left_inv _ := rfl
-  right_inv _ := rfl
-
-/-- Full ambient inversion count, not a cardinality assumption. -/
-theorem rootInversions_finite (w : W) : Finite (RootInversions M cs w) := by
-  classical
-  obtain ⟨ω,hω,heq⟩ := cs.exists_isReduced w
-  rw [heq]
-  exact Finite.of_equiv (Fin (cs.leftInvSeq ω).length)
-    ((List.Nodup.getEquiv (cs.leftInvSeq ω) hω.nodup_leftInvSeq).trans
-      ((leftInversionListEquiv M cs hω).symm.trans (rootInversionEquiv M cs _).symm))
-
-theorem card_rootInversions (w : W) : Nat.card (RootInversions M cs w) = cs.length w := by
-  classical
-  obtain ⟨ω,hω,heq⟩ := cs.exists_isReduced w
-  rw [heq, Nat.card_congr (rootInversionEquiv M cs _),
-    Nat.card_congr (leftInversionListEquiv M cs hω),
-    ← Nat.card_congr (List.Nodup.getEquiv (cs.leftInvSeq ω) hω.nodup_leftInvSeq),
-    Nat.card_fin, cs.length_leftInvSeq, hω.eq]
-
-end
-end KLInvariance.TitsSpace
-
-namespace KLInvariance.TitsSpace
-open Module
-universe u v
-variable {I : Type u} [Fintype I] (M : CoxeterMatrix I)
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
-/-- Elements whose actual linear displacement lies in `P`. This is a
-subgroup even when the invariant bilinear form restricted to `P` is singular. -/
-def movingSubgroup (P : Submodule ℝ (I → ℝ)) : Subgroup W where
-  carrier := {w | ∀ v, action M cs w v - v ∈ P}
-  one_mem' := by intro v; simp
-  mul_mem' := by
-    intro w z hw hz v
-    have h := P.add_mem (hw (action M cs z v)) (hz v)
-    simpa only [map_mul, LinearEquiv.mul_apply, sub_add_sub_cancel] using h
-  inv_mem' := by
-    intro w hw v
-    have h := P.neg_mem (hw (action M cs w⁻¹ v))
-    have hv : action M cs w (action M cs w⁻¹ v) = v := by
-      rw [← LinearEquiv.mul_apply, ← map_mul, mul_inv_cancel, map_one]
-      rfl
-    simpa only [hv, neg_sub] using h
-
-/-- The manuscript's maximal plane reflection subgroup, defined using
-actual normalized roots, not by assuming a rank-two presentation. -/
-def planeSubgroup (P : Submodule ℝ (I → ℝ)) : Subgroup W :=
-  Subgroup.closure {t | ∃ a, Represents M cs t a ∧ a ∈ P}
-
-theorem planeSubgroup_le_moving (P : Submodule ℝ (I → ℝ)) :
-    planeSubgroup M cs P ≤ movingSubgroup M cs P := by
-  apply (Subgroup.closure_le (movingSubgroup M cs P)).mpr
-  rintro t ⟨a,ha,haP⟩ v
-  rw [ha.action_formula]
-  have heq : v - form M a v • a - v = (-form M a v) • a := by module
-  rw [heq]
-  exact P.smul_mem _ haP
-
-theorem Represents.mem_of_moving {P : Submodule ℝ (I → ℝ)} {t : W} {a : I → ℝ}
-    (ha : Represents M cs t a) (ht : t ∈ movingSubgroup M cs P) : a ∈ P := by
-  have h := ht a
-  change action M cs t a - a ∈ P at h
-  rw [ha.action_formula,ha.isRoot.norm] at h
-  have heq : a - (2 : ℝ) • a - a = (-2 : ℝ) • a := by module
-  rw [heq] at h
-  have h' := P.smul_mem (-1/2 : ℝ) h
-  have heq' : (-1/2 : ℝ) • ((-2 : ℝ) • a) = a := by module
-  rwa [heq'] at h'
-
-
-theorem Represents.planeSubgroup_mem_iff {P : Submodule ℝ (I → ℝ)} {t : W} {a : I → ℝ}
-    (ha : Represents M cs t a) : t ∈ planeSubgroup M cs P ↔ a ∈ P := by
-  constructor
-  · intro ht
-    exact Represents.mem_of_moving M cs ha (planeSubgroup_le_moving M cs P ht)
-  · intro haP
-    exact Subgroup.subset_closure ⟨a,ha,haP⟩
-
-/-- Two ambient vertices in a left coset of this plane subgroup can have
-no reflection edge except one whose actual root is in the plane. -/
-theorem reflection_between_coset_mem {P : Submodule ℝ (I → ℝ)} {t g h a : W}
-    (hg : g ∈ planeSubgroup M cs P) (hh : h ∈ planeSubgroup M cs P)
-    (heq : h*a=t*(g*a)) {r : I → ℝ} (ht : Represents M cs t r) : r ∈ P := by
-  have htg : t*g=h := by apply mul_right_cancel (b := a); simpa only [mul_assoc] using heq.symm
-  have htG : t ∈ planeSubgroup M cs P := by
-    have heq' : t=h*g⁻¹ := by rw [← htg]; group
-    rw [heq']
-    exact (planeSubgroup M cs P).mul_mem hh ((planeSubgroup M cs P).inv_mem hg)
-  exact (Represents.planeSubgroup_mem_iff M cs ht).mp htG
-
-end
-end KLInvariance.TitsSpace
-
-namespace KLInvariance.TitsSpace
-open Module
-universe u v
-variable {I : Type u} [Fintype I] (M : CoxeterMatrix I)
-  {W : Type v} [Group W] (cs : CoxeterSystem M W)
-noncomputable section
-
-/-- Least ambient length in each actual left plane-subgroup coset. No
-uniqueness or rank-two Coxeter presentation is smuggled into this definition. -/
-theorem planeCoset_exists_minimal (P : Submodule ℝ (I → ℝ)) (a : W) :
-    ∃ a₀ g : W, g ∈ planeSubgroup M cs P ∧ a = g*a₀ ∧
-      ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀) := by
-  classical
-  have hex : ∃ n : ℕ, ∃ a₀ g : W, g ∈ planeSubgroup M cs P ∧
-      a = g*a₀ ∧ cs.length a₀ = n :=
-    ⟨cs.length a,a,1,(planeSubgroup M cs P).one_mem,by simp,rfl⟩
-  obtain ⟨a₀,g,hg,ha,hlen⟩ := Nat.find_spec hex
-  refine ⟨a₀,g,hg,ha,fun z hz => ?_⟩
-  rw [hlen]
-  apply Nat.find_min' hex
-  refine ⟨z*a₀,g*z⁻¹,(planeSubgroup M cs P).mul_mem hg
-    ((planeSubgroup M cs P).inv_mem hz),?_,rfl⟩
-  simpa [mul_assoc] using ha
-
-variable {M cs}
-
-theorem planeSubgroup_preserves {P : Submodule ℝ (I → ℝ)} {g : W}
-    (hg : g ∈ planeSubgroup M cs P) {a : I → ℝ} (ha : a ∈ P) :
-    action M cs g a ∈ P := by
-  have h := (planeSubgroup_le_moving M cs P hg) a
-  simpa only [sub_add_cancel] using P.add_mem h ha
-
-/-- A least coset representative sends every positive plane root to a
-positive ambient root under its inverse, by the proved inversion criterion. -/
-theorem root_nonneg_of_coset_minimal {P : Submodule ℝ (I → ℝ)} {a₀ : W}
-    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-    {a : I → ℝ} (ha : IsRoot M cs a) (hp : ∀ i, 0 ≤ a i) (haP : a ∈ P) :
-    ∀ i, 0 ≤ action M cs a₀⁻¹ a i := by
-  let r : PositiveRoot M cs := ⟨a,ha,hp⟩
-  have hr := rootReflection_represents M cs r
-  have htP := (Represents.planeSubgroup_mem_iff M cs hr).mpr haP
-  rcases (ha.act a₀⁻¹).sign with hpos|hneg
-  · exact hpos
-  · have hi := (hr.inversion_iff hp a₀).mp hneg
-    exact (not_lt_of_ge (hmin _ htP) hi.2).elim
-
-
-theorem root_nonpos_coset_minimal_iff {P : Submodule ℝ (I → ℝ)} {a₀ : W}
-    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-    {a : I → ℝ} (ha : IsRoot M cs a) (haP : a ∈ P) :
-    (∀ i, action M cs a₀⁻¹ a i ≤ 0) ↔ ∀ i, a i ≤ 0 := by
-  rcases ha.sign with hp|hn
-  · have hnew := root_nonneg_of_coset_minimal hmin ha hp haP
-    exact iff_of_false ((ha.act _).not_both hnew) (ha.not_both hp)
-  · have hnp : ∀ i, 0 ≤ (-a) i := fun i => neg_nonneg.mpr (hn i)
-    have hh := root_nonneg_of_coset_minimal hmin ha.neg hnp (P.neg_mem haP)
-    have hnew : ∀ i, action M cs a₀⁻¹ a i ≤ 0 := by
-      intro i
-      simpa only [map_neg, Pi.neg_apply, neg_nonneg] using hh i
-    exact iff_of_true hnew hn
-
-theorem root_inversion_coset_iff {P : Submodule ℝ (I → ℝ)} {a₀ g : W}
-    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-    (hg : g ∈ planeSubgroup M cs P) {a : I → ℝ} (ha : IsRoot M cs a)
-    (haP : a ∈ P) :
-    (∀ i, action M cs (g*a₀)⁻¹ a i ≤ 0) ↔
-      ∀ i, action M cs g⁻¹ a i ≤ 0 := by
-  rw [mul_inv_rev, map_mul]
-  change (∀ i, action M cs a₀⁻¹ (action M cs g⁻¹ a) i ≤ 0) ↔ _
-  exact root_nonpos_coset_minimal_iff hmin (ha.act _)
-    (planeSubgroup_preserves ((planeSubgroup M cs P).inv_mem hg) haP)
-
-theorem leftInversion_coset_iff {P : Submodule ℝ (I → ℝ)} {a₀ g t : W}
-    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-    (hg : g ∈ planeSubgroup M cs P) (ht : t ∈ planeSubgroup M cs P)
-    (hr : cs.IsReflection t) :
-    cs.IsLeftInversion (g*a₀) t ↔ cs.IsLeftInversion g t := by
-  let a := positiveRoot M cs ⟨t,hr⟩
-  have ha := positiveRoot_represents M cs ⟨t,hr⟩
-  rw [← ha.inversion_iff a.property.2, ← ha.inversion_iff a.property.2]
-  exact root_inversion_coset_iff hmin hg a.property.1
-    ((Represents.planeSubgroup_mem_iff M cs ha).mp ht)
-
-/-- The directed ambient reflection graph on any plane-subgroup coset is
-exactly transported from that on the subgroup by a least coset representative.
-Identifying this graph with a *dihedral Coxeter* graph still requires the
-canonical rank-two reflection-subgroup theorem. -/
-theorem bruhatStep_coset_iff {P : Submodule ℝ (I → ℝ)} {a₀ g h : W}
-    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
-    (hg : g ∈ planeSubgroup M cs P) (hh : h ∈ planeSubgroup M cs P) :
-    BruhatStep cs (g*a₀) (h*a₀) ↔ BruhatStep cs g h := by
-  have hn : ∀ {t : W}, h=t*g → cs.IsReflection t →
-      (cs.length (g*a₀) < cs.length (h*a₀) ↔ cs.length g < cs.length h) := by
-    intro t heq ht
-    have htP : t ∈ planeSubgroup M cs P := by
-      have htgh : t=h*g⁻¹ := by rw [heq]; group
-      rw [htgh]
-      exact (planeSubgroup M cs P).mul_mem hh ((planeSubgroup M cs P).inv_mem hg)
-    have hi := leftInversion_coset_iff hmin hh htP ht
-    have hth : t*h=g := by rw [heq,← mul_assoc,ht.mul_self,one_mul]
-    simpa only [CoxeterSystem.IsLeftInversion, ht, true_and, ← mul_assoc, hth] using hi
-  constructor
-  · rintro ⟨hlen,t,ht,heq⟩
-    have htg : h=t*g := by apply mul_right_cancel (b := a₀); simpa only [mul_assoc] using heq
-    exact ⟨(hn htg ht).mp hlen,t,ht,htg⟩
-  · rintro ⟨hlen,t,ht,heq⟩
-    exact ⟨(hn heq ht).mpr hlen,t,ht,by rw [heq,mul_assoc]⟩
-
-end
-end KLInvariance.TitsSpace
-
-/-! Construction of the source's geometric reflection schedule. The moment
-curve avoids every forbidden hyperplane of a countable root set, using only
-finite polynomial root sets and uncountability of the real field. -/
-
-end
-
-end OAI
+import OAI.RepresentationTheory.KazhdanLusztig.Bruhat
+import OAI.RepresentationTheory.KazhdanLusztig.Dihedral
+import OAI.RepresentationTheory.KazhdanLusztig.ReflectionPlane
+import OAI.RepresentationTheory.KazhdanLusztig.Sheaves
+
+namespace OAI
+
+/-!
+Actual Bruhat graph labels, Tits realizations and extended root spaces, cotangent coordinates and standard parabolic support.
+-/
+
+section
+
+namespace KLInvariance
+namespace BruhatGraph
+universe u v
+variable {B : Type u} {W : Type v} [Group W] {M : CoxeterMatrix B}
+  (cs : CoxeterSystem M W) (u b : W)
+
+/-- Actual finite directed moment-graph edges, including the non-cover
+reflection edges. No reconstruction from the abstract poset is assumed. -/
+def Edge := {p : Interval cs u b × Interval cs u b // BruhatStep cs p.1.val p.2.val}
+
+instance : Finite (Edge cs u b) := by
+  let := interval_finite cs u b
+  unfold Edge
+  infer_instance
+
+def source (e : Edge cs u b) : Interval cs u b := e.val.1
+def target (e : Edge cs u b) : Interval cs u b := e.val.2
+
+def graph : MomentGraph.OrderedGraph (Interval cs u b) (Edge cs u b) where
+  source := source cs u b
+  target := target cs u b
+  increasing e := by
+    refine ⟨Relation.ReflTransGen.single e.property, ?_⟩
+    intro h
+    exact (not_lt_of_ge (length_le_of_bruhat cs h)) e.property.1
+
+/-- Genuine left reflection label; uniquely determined by the endpoints. -/
+def label (e : Edge cs u b) : W := (target cs u b e).val * (source cs u b e).val⁻¹
+
+theorem label_isReflection (e : Edge cs u b) : cs.IsReflection (label cs u b e) := by
+  obtain ⟨t, ht, htx⟩ := e.property.2
+  simpa [label, source, target, htx, mul_assoc] using ht
+
+@[simp] theorem label_mul_source (e : Edge cs u b) :
+    label cs u b e * (source cs u b e).val = (target cs u b e).val := by
+  simp [label, mul_assoc]
+
+@[simp] theorem label_mul_target (e : Edge cs u b) :
+    label cs u b e * (target cs u b e).val = (source cs u b e).val := by
+  rw [← label_mul_source cs u b e, ← mul_assoc,
+    (label_isReflection cs u b e).mul_self, one_mul]
+
+theorem eq_of_label_source_eq {e f : Edge cs u b}
+    (hl : label cs u b e = label cs u b f)
+    (hs : source cs u b e = source cs u b f) : e = f := by
+  have ht : target cs u b e = target cs u b f := by
+    apply Subtype.ext
+    rw [← label_mul_source cs u b e, ← label_mul_source cs u b f, hl, hs]
+  apply Subtype.ext
+  exact Prod.ext hs ht
+
+theorem eq_of_label_target_eq {e f : Edge cs u b}
+    (hl : label cs u b e = label cs u b f)
+    (ht : target cs u b e = target cs u b f) : e = f := by
+  have hs : source cs u b e = source cs u b f := by
+    apply Subtype.ext
+    rw [← label_mul_target cs u b e, ← label_mul_target cs u b f, hl, ht]
+  exact eq_of_label_source_eq cs u b hl hs
+
+theorem source_ne_target_of_label_eq {e f : Edge cs u b}
+    (hl : label cs u b e = label cs u b f) :
+    source cs u b e ≠ target cs u b f := by
+  intro h
+  have ht : target cs u b e = source cs u b f := by
+    apply Subtype.ext
+    rw [← label_mul_source cs u b e, hl, h, label_mul_target]
+  have he := (graph cs u b).increasing e
+  have hf := (graph cs u b).increasing f
+  change source cs u b e < target cs u b e at he
+  change source cs u b f < target cs u b f at hf
+  rw [h, ht] at he
+  exact lt_asymm he hf
+
+
+theorem endpoints_disjoint_of_label_eq {e f : Edge cs u b} (hef : e ≠ f)
+    (hl : label cs u b e = label cs u b f) :
+    Disjoint ({source cs u b e, target cs u b e} : Set _)
+      {source cs u b f, target cs u b f} := by
+  rw [Set.disjoint_left]
+  intro x hx hy
+  simp only [Set.mem_insert_iff, Set.mem_singleton_iff] at hx hy
+  rcases hx with rfl | rfl <;> rcases hy with hy | hy
+  · exact hef (eq_of_label_source_eq cs u b hl hy)
+  · exact source_ne_target_of_label_eq cs u b hl hy
+  · exact source_ne_target_of_label_eq cs u b hl.symm hy.symm
+  · exact hef (eq_of_label_target_eq cs u b hl hy)
+
+end BruhatGraph
+end KLInvariance
+
+
+namespace KLInvariance.Realization
+open Module
+variable {V : Type*} [AddCommGroup V] [Module ℝ V]
+
+/-- An actual rank-one perturbation that preserves a nondegenerate symmetric
+form is an orthogonal reflection, not a unipotent transvection. -/
+theorem orthogonal_rank_one_form
+    (B : LinearMap.BilinForm ℝ V) (hs : B.IsSymm) (hn : B.Nondegenerate)
+    (u : V) (hu : u ≠ 0) (f : Module.Dual ℝ V) (hf : f ≠ 0)
+    (hg : ∀ v w, B (v + f v • u) (w + f w • u) = B v w) :
+    f u = -2 ∧ B u u ≠ 0 ∧
+      ∀ v, v + f v • u = Module.preReflection u ((2 / B u u) • B u) v := by
+  obtain ⟨v, hv⟩ : ∃ v, f v = 1 := by
+    have hex : ∃ w, f w ≠ 0 := by
+      by_contra! h
+      exact hf (LinearMap.ext h)
+    obtain ⟨w, hw⟩ := hex
+    exact ⟨(f w)⁻¹ • w, by simp [hw]⟩
+  have hz (z : V) (h : f z = 0) : B u z = 0 := by
+    have h' := hg v z
+    simpa [hv, h, map_add] using h'
+  have hprop (z : V) : B u z = f z * B u v := by
+    have h := hz (z - f z • v) (by simp [hv])
+    exact sub_eq_zero.mp (by simpa [map_sub, map_smul] using h)
+  have hc : B u v ≠ 0 := by
+    intro h
+    apply hu
+    exact hn.1 u (fun z => by rw [hprop z, h, mul_zero])
+  have hself := hg v v
+  simp only [hv, one_smul, map_add, LinearMap.add_apply] at hself
+  rw [hs.eq v u] at hself
+  have hfu : f u = -2 := by
+    rw [hprop u] at hself
+    apply (mul_right_cancel₀ hc)
+    linarith
+  have hne : B u u ≠ 0 := by
+    rw [hprop u, hfu]
+    exact mul_ne_zero (by norm_num) hc
+  refine ⟨hfu, hne, fun z => ?_⟩
+  rw [Module.preReflection_apply]
+  have heq : ((2 / B u u) • B u) z = -f z := by
+    simp only [LinearMap.smul_apply, smul_eq_mul]
+    rw [hprop z, hprop u, hfu]
+    field_simp [hc]
+  rw [heq, neg_smul, sub_neg_eq_add]
+
+/-- This is the rank-one assertion in `bg:realization` stated without a chosen
+rank-one normal form. Its conclusion uses the library's actual reflection. -/
+theorem orthogonal_finrank_one
+    (B : LinearMap.BilinForm ℝ V) (hs : B.IsSymm) (hn : B.Nondegenerate)
+    (g : V ≃ₗ[ℝ] V) (hg : ∀ v w, B (g v) (g w) = B v w)
+    (hr : Module.finrank ℝ (LinearMap.range (g.toLinearMap - LinearMap.id)) = 1) :
+    ∃ u : V, B u u ≠ 0 ∧
+      ∀ v, g v = Module.preReflection u ((2 / B u u) • B u) v := by
+  let d : V →ₗ[ℝ] V := g.toLinearMap - LinearMap.id
+  obtain ⟨e⟩ := nonempty_linearEquiv_of_finrank_eq_one hr
+  let u : V := (e 1).val
+  let f : Module.Dual ℝ V := e.symm.toLinearMap.comp d.rangeRestrict
+  have hu : u ≠ 0 := by
+    intro h
+    have : e 1 = 0 := Subtype.ext h
+    have := e.injective (this.trans e.map_zero.symm)
+    norm_num at this
+  have hform (v : V) : f v • u = d v := by
+    change f v • (e 1).val = d v
+    have h := congrArg Subtype.val (e.apply_symm_apply (d.rangeRestrict v))
+    have he : e (f v) = f v • e 1 := by
+      rw [← map_smul, smul_eq_mul, mul_one]
+    rw [show e.symm (d.rangeRestrict v) = f v from rfl, he] at h
+    exact h
+  have hf : f ≠ 0 := by
+    intro h
+    obtain ⟨v, hv⟩ := (e 1).property
+    have hh := hform v
+    rw [h, LinearMap.zero_apply, zero_smul, hv] at hh
+    exact hu hh.symm
+  have hgform (v : V) : g v = v + f v • u := by
+    rw [hform]
+    change g v = v + (g v - v)
+    abel
+  obtain ⟨_, hnonzero, hreflection⟩ := orthogonal_rank_one_form B hs hn u hu f hf
+    (fun v w => by rw [← hgform, ← hgform]; exact hg v w)
+  exact ⟨u, hnonzero, fun v => (hgform v).trans (hreflection v)⟩
+end KLInvariance.Realization
+
+/-! The finite Coxeter relations for the actual real geometric reflection
+operators required in `bg:realization`. Faithfulness is not assumed or claimed. -/
+namespace KLInvariance.Realization
+open Module Polynomial.Chebyshev
+variable {V : Type*} [AddCommGroup V] [Module ℝ V]
+
+private theorem reflection_pow_even_eq_one
+    {α β : V} {f g : Module.Dual ℝ V} (hf : f α = 2) (hg : g β = 2)
+    (k : ℕ) (hS : (S ℝ ((k : ℤ) - 1)).eval (f β * g α - 2) = 0) :
+    (Module.reflection hf * Module.reflection hg) ^ (2 * k) = 1 := by
+  ext v
+  rw [Module.reflection_mul_reflection_pow_apply hf hg]
+  have h₁ : ((↑(2 * k) : ℤ) - 2) / 2 = (k : ℤ) - 1 := by omega
+  have h₂ : ((↑(2 * k) : ℤ) - 1) / 2 = (k : ℤ) - 1 := by omega
+  rw [h₁, h₂, hS]
+  simp
+
+private theorem reflection_pow_odd_eq_one
+    {α β : V} {f g : Module.Dual ℝ V} (hf : f α = 2) (hg : g β = 2)
+    (k : ℕ)
+    (hS : (S ℝ (k : ℤ)).eval (f β * g α - 2) +
+      (S ℝ ((k : ℤ) - 1)).eval (f β * g α - 2) = 0) :
+    (Module.reflection hf * Module.reflection hg) ^ (2 * k + 1) = 1 := by
+  ext v
+  rw [Module.reflection_mul_reflection_pow_apply hf hg]
+  have h₀ : (↑(2 * k + 1) : ℤ) / 2 = (k : ℤ) := by omega
+  have h₁ : ((↑(2 * k + 1) : ℤ) - 2) / 2 = (k : ℤ) - 1 := by omega
+  have h₂ : ((↑(2 * k + 1) : ℤ) - 1) / 2 = (k : ℤ) := by omega
+  have h₃ : ((↑(2 * k + 1) : ℤ) - 3) / 2 = (k : ℤ) - 1 := by omega
+  rw [h₀, h₁, h₂, h₃, hS]
+  simp
+
+/-- If the two simple roots have the exact Tits pairing, the corresponding
+linear equivalences satisfy their finite Coxeter relation on the entire
+ambient space, not just on the root plane. -/
+theorem geometric_reflection_relation
+    {α β : V} {f g : Module.Dual ℝ V} (hf : f α = 2) (hg : g β = 2)
+    (m : ℕ) (hm : 2 ≤ m)
+    (hfg : f β = -2 * Real.cos (Real.pi / m))
+    (hgf : g α = -2 * Real.cos (Real.pi / m)) :
+    (Module.reflection hf * Module.reflection hg) ^ m = 1 := by
+  by_cases hm2 : m = 2
+  · subst m
+    have hfg0 : f β = 0 := by simpa using hfg
+    have hgf0 : g α = 0 := by simpa using hgf
+    ext v
+    change ((Module.reflection hf * Module.reflection hg) ^ 2) v = v
+    simp only [pow_two, LinearEquiv.mul_apply, Module.reflection_apply,
+      map_sub, map_smul, hf, hg, hfg0, hgf0, ]
+    module
+  have hm3 : 2 < m := by omega
+  have hmR : (2 : ℝ) < m := by exact_mod_cast hm3
+  let θ : ℝ := Real.pi / m
+  have hθ : (m : ℝ) * θ = Real.pi := by dsimp [θ]; field_simp
+  have hθpos : 0 < θ := div_pos Real.pi_pos (by positivity)
+  have hθlt : 2 * θ < Real.pi := by
+    dsimp [θ]
+    rw [← mul_div_assoc]
+    apply (div_lt_iff₀ (show (0 : ℝ) < m by positivity)).2
+    nlinarith [Real.pi_pos]
+  have hsin : Real.sin (2 * θ) ≠ 0 :=
+    ne_of_gt (Real.sin_pos_of_pos_of_lt_pi (by positivity) hθlt)
+  have ht : f β * g α - 2 = 2 * Real.cos (2 * θ) := by
+    rw [hfg, hgf, Real.cos_two_mul]
+    dsimp [θ]
+    ring
+  rcases Nat.even_or_odd m with he | ho
+  · obtain ⟨k, hk⟩ := he
+    have hm' : m = 2 * k := by omega
+    clear hk
+    subst m
+    apply reflection_pow_even_eq_one hf hg k
+    rw [ht]
+    have h := S_two_mul_real_cos (2 * θ) ((k : ℤ) - 1)
+    have ha : (((k : ℤ) - 1 : ℤ) + 1 : ℝ) * (2 * θ) = Real.pi := by
+      push_cast
+      push_cast at hθ
+      nlinarith [hθ]
+    rw [ha, Real.sin_pi] at h
+    exact (mul_eq_zero.mp h).resolve_right hsin
+  · obtain ⟨k, hk⟩ := ho
+    have hm' : m = 2 * k + 1 := by omega
+    clear hk
+    subst m
+    apply reflection_pow_odd_eq_one hf hg k
+    rw [ht]
+    have h₁ := S_two_mul_real_cos (2 * θ) (k : ℤ)
+    have h₂ := S_two_mul_real_cos (2 * θ) ((k : ℤ) - 1)
+    have ha : ((k : ℤ) + 1 : ℝ) * (2 * θ) = Real.pi + θ := by
+      push_cast
+      push_cast at hθ
+      nlinarith [hθ]
+    have hb : (((k : ℤ) - 1 : ℤ) + 1 : ℝ) * (2 * θ) = Real.pi - θ := by
+      push_cast
+      push_cast at hθ
+      nlinarith [hθ]
+    rw [ha] at h₁
+    simp only [Real.sin_add, Real.sin_pi, Real.cos_pi, zero_mul, neg_one_mul,
+      zero_add] at h₁
+    rw [hb, Real.sin_pi_sub] at h₂
+    apply (mul_eq_zero.mp (show
+      ((S ℝ (k : ℤ)).eval (2 * Real.cos (2 * θ)) +
+      (S ℝ ((k : ℤ) - 1)).eval (2 * Real.cos (2 * θ))) *
+      Real.sin (2 * θ) = 0 by nlinarith [h₁, h₂])).resolve_right hsin
+end KLInvariance.Realization
+
+namespace KLInvariance.Realization
+open Module
+universe u v z
+variable {I : Type u} {W : Type v} [Group W] {M : CoxeterMatrix I}
+variable (cs : CoxeterSystem M W)
+variable {V : Type z} [AddCommGroup V] [Module ℝ V]
+variable (B : LinearMap.BilinForm ℝ V) (α : I → V)
+variable (hα : ∀ i j, B (α i) (α j) = -2 * Real.cos (Real.pi / M i j))
+
+include hα
+
+lemma simple_normal_value (i : I) : B (α i) (α i) = 2 := by
+  rw [hα]
+  simp
+
+noncomputable def geometricSimple (i : I) : V ≃ₗ[ℝ] V :=
+  Module.reflection (simple_normal_value B α hα i)
+
+theorem geometricSimple_liftable : M.IsLiftable (geometricSimple B α hα) := by
+  intro i j
+  by_cases hij : i = j
+  · subst j
+    rw [M.diagonal, pow_one]
+    ext v
+    exact Module.involutive_reflection (simple_normal_value B α hα i) v
+  by_cases hm0 : M i j = 0
+  · rw [hm0, pow_zero]
+  have hm2 : 2 ≤ M i j := by
+    have := M.off_diagonal i j hij
+    omega
+  exact geometric_reflection_relation (simple_normal_value B α hα i)
+    (simple_normal_value B α hα j) (M i j) hm2 (hα i j)
+    (by rw [hα j i, M.symmetric j i])
+
+/-- The genuine geometric reflection action. Each relation has been proved;
+this definition does not presuppose a faithful representation. -/
+noncomputable def geometricAction : W →* (V ≃ₗ[ℝ] V) :=
+  cs.lift ⟨geometricSimple B α hα, geometricSimple_liftable B α hα⟩
+
+@[simp] theorem geometricAction_simple (i : I) :
+    geometricAction cs B α hα (cs.simple i) = geometricSimple B α hα i :=
+  cs.lift_apply_simple (geometricSimple_liftable B α hα) i
+
+theorem geometricSimple_preserves (hs : B.IsSymm) (i : I) (v w : V) :
+    B (geometricSimple B α hα i v) (geometricSimple B α hα i w) = B v w := by
+  simp only [geometricSimple, Module.reflection_apply, map_sub, map_smul,
+    LinearMap.sub_apply, LinearMap.smul_apply, smul_eq_mul]
+  rw [hs.eq v (α i), simple_normal_value B α hα i]
+  ring
+
+theorem geometricAction_preserves (hs : B.IsSymm) (w : W) (v z : V) :
+    B (geometricAction cs B α hα w v) (geometricAction cs B α hα w z) = B v z := by
+  obtain ⟨ω, rfl⟩ := cs.wordProd_surjective w
+  induction ω with
+  | nil => simp
+  | cons i ω ih =>
+    rw [cs.wordProd_cons, map_mul]
+    simp only [LinearEquiv.mul_apply, geometricAction_simple]
+    rw [geometricSimple_preserves B α hα hs, ih]
+
+end KLInvariance.Realization
+
+namespace KLInvariance.TitsSpace
+open Module
+universe u v
+variable {I : Type u} [Fintype I] (M : CoxeterMatrix I)
+noncomputable section
+
+
+def gram (i j : I) : ℝ := -2 * Real.cos (Real.pi / M i j)
+
+def root (i : I) : I → ℝ := by
+  classical
+  exact Pi.single i 1
+
+def normal (i : I) : Module.Dual ℝ (I → ℝ) :=
+  ∑ j, gram M i j • LinearMap.proj j
+
+def form : LinearMap.BilinForm ℝ (I → ℝ) :=
+  ∑ i, (LinearMap.proj i).smulRight (normal M i)
+
+@[simp] theorem normal_root (i j : I) : normal M i (root j) = gram M i j := by
+  classical
+  simp only [normal, root, LinearMap.sum_apply, LinearMap.smul_apply,
+    LinearMap.proj_apply, smul_eq_mul]
+  rw [Finset.sum_eq_single j]
+  · simp
+  · intro b _ hbj
+    simp [hbj]
+  · simp
+
+@[simp] theorem form_root (i : I) (v : I → ℝ) : form M (root i) v = normal M i v := by
+  classical
+  simp [form, root, LinearMap.sum_apply]
+
+theorem form_symm : (form M).IsSymm := by
+  constructor
+  intro v w
+  simp only [form, normal, LinearMap.sum_apply, LinearMap.smulRight_apply,
+    LinearMap.smul_apply, LinearMap.proj_apply, smul_eq_mul]
+  simp_rw [Finset.mul_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro i hi
+  apply Finset.sum_congr rfl
+  intro j hj
+  rw [show gram M j i = gram M i j by simp only [gram, M.symmetric j i]]
+  ring
+
+theorem form_roots (i j : I) : form M (root i) (root j) =
+    -2 * Real.cos (Real.pi / M i j) := by simp [gram]
+
+theorem root_linearIndependent : LinearIndependent ℝ (root (I := I)) := by
+  classical
+  have heq : (root (I := I)) = (Pi.basisFun ℝ I : I → I → ℝ) := by
+    funext i
+    simp [root, Pi.basisFun_apply]
+  rw [heq]
+  exact (Pi.basisFun ℝ I).linearIndependent
+
+/-- A concrete action on the ordinary geometric root space over ℝ. Its
+faithfulness and the root-sign theorem are still required for the main proof. -/
+def action {W : Type v} [Group W] (cs : CoxeterSystem M W) :
+    W →* ((I → ℝ) ≃ₗ[ℝ] (I → ℝ)) :=
+  Realization.geometricAction cs (form M) root (form_roots M)
+
+@[simp] theorem action_simple {W : Type v} [Group W] (cs : CoxeterSystem M W)
+    (i : I) (v : I → ℝ) :
+    action M cs (cs.simple i) v = v - normal M i v • root i := by
+  simp [action, Realization.geometricAction_simple, Realization.geometricSimple,
+    Module.reflection_apply]
+
+theorem action_preserves {W : Type v} [Group W] (cs : CoxeterSystem M W)
+    (w : W) (v z : I → ℝ) : form M (action M cs w v) (action M cs w z) = form M v z :=
+  Realization.geometricAction_preserves cs (form M) root (form_roots M) (form_symm M) w v z
+end
+end KLInvariance.TitsSpace
+
+namespace KLInvariance.Realization
+universe u
+variable {V : Type u} [AddCommGroup V] [Module ℝ V]
+
+
+abbrev Extension (B : LinearMap.BilinForm ℝ V) := V × Module.Dual ℝ (LinearMap.ker B)
+
+noncomputable def extensionForm (B : LinearMap.BilinForm ℝ V)
+    (p : V →ₗ[ℝ] LinearMap.ker B) : LinearMap.BilinForm ℝ (Extension B) where
+  toFun x :=
+    { toFun := fun y => B x.1 y.1 + x.2 (p y.1) + y.2 (p x.1)
+      map_add' := by intro y z; simp [map_add]; ring
+      map_smul' := by intro t y; simp [map_smul]; ring }
+  map_add' := by
+    intro x y
+    apply LinearMap.ext
+    intro z
+    change B (x.1 + y.1) z.1 + (x.2 + y.2) (p z.1) + z.2 (p (x.1 + y.1)) = _
+    simp [map_add]
+    ring
+  map_smul' := by
+    intro t x
+    apply LinearMap.ext
+    intro z
+    change B (t • x.1) z.1 + (t • x.2) (p z.1) + z.2 (p (t • x.1)) = _
+    simp [map_smul]
+    ring
+
+@[simp] theorem extensionForm_apply (B : LinearMap.BilinForm ℝ V)
+    (p : V →ₗ[ℝ] LinearMap.ker B) (x y : Extension B) :
+    extensionForm B p x y = B x.1 y.1 + x.2 (p y.1) + y.2 (p x.1) := rfl
+
+@[simp] theorem extensionForm_inl (B : LinearMap.BilinForm ℝ V)
+    (p : V →ₗ[ℝ] LinearMap.ker B) (v w : V) :
+    extensionForm B p (v, 0) (w, 0) = B v w := by simp
+
+theorem extensionForm_symm (B : LinearMap.BilinForm ℝ V) (hB : B.IsSymm)
+    (p : V →ₗ[ℝ] LinearMap.ker B) : (extensionForm B p).IsSymm := by
+  constructor
+  intro x y
+  simp only [extensionForm_apply]
+  rw [hB.eq x.1 y.1]
+  ring
+
+theorem extensionForm_nondegenerate (B : LinearMap.BilinForm ℝ V)
+    (hB : B.IsSymm) (p : V →ₗ[ℝ] LinearMap.ker B)
+    (hp : ∀ k : LinearMap.ker B, p k.val = k) : (extensionForm B p).Nondegenerate := by
+  have hf : ∀ x : Extension B, (∀ y, extensionForm B p x y = 0) → x = 0 := by
+    rintro ⟨v, φ⟩ hx
+    have hpv : p v = 0 := by
+      apply (Module.forall_dual_apply_eq_zero_iff ℝ (p v)).mp
+      intro ψ
+      simpa using hx (0, ψ)
+    have hφ : φ = 0 := by
+      ext k
+      have hbk : B k.val = 0 := k.property
+      have hvk : B v k.val = 0 := by
+        rw [hB.eq v k.val, hbk]
+        rfl
+      simpa [hvk, hp k, hpv] using hx (k.val, 0)
+    have hbv : v ∈ LinearMap.ker B := by
+      change B v = 0
+      ext w
+      simpa [hφ, hpv] using hx (w, 0)
+    have hv : v = 0 := by
+      have h := hp ⟨v, hbv⟩
+      rw [hpv] at h
+      exact (congrArg Subtype.val h).symm
+    simp [hv, hφ]
+  exact ⟨hf, fun x hx => hf x (fun y => by
+    rw [(extensionForm_symm B hB p).eq]
+    exact hx y)⟩
+
+theorem exists_minimal_nondegenerate_extension (B : LinearMap.BilinForm ℝ V)
+    (hB : B.IsSymm) :
+    ∃ C : LinearMap.BilinForm ℝ (Extension B), C.IsSymm ∧ C.Nondegenerate ∧
+      ∀ v w, C (v, 0) (w, 0) = B v w := by
+  obtain ⟨Q, hQ⟩ := (LinearMap.ker B).exists_isCompl
+  let p := (LinearMap.ker B).projectionOnto Q hQ
+  exact ⟨extensionForm B p, extensionForm_symm B hB p,
+    extensionForm_nondegenerate B hB p (Submodule.projectionOnto_apply_left hQ),
+    extensionForm_inl B p⟩
+
+theorem extension_finrank [FiniteDimensional ℝ V] (B : LinearMap.BilinForm ℝ V) :
+    Module.finrank ℝ (Extension B) = Module.finrank ℝ V + Module.finrank ℝ (LinearMap.ker B) := by
+  rw [Module.finrank_prod, Subspace.dual_finrank_eq]
+
+
+theorem extension_dimension_minimal [FiniteDimensional ℝ V]
+    {E : Type*} [AddCommGroup E] [Module ℝ E] [FiniteDimensional ℝ E]
+    (B : LinearMap.BilinForm ℝ V) (C : LinearMap.BilinForm ℝ E)
+    (hC : C.Nondegenerate) (i : V →ₗ[ℝ] E) (hi : Function.Injective i)
+    (hform : ∀ v w, C (i v) (i w) = B v w) :
+    Module.finrank ℝ V + Module.finrank ℝ (LinearMap.ker B) ≤ Module.finrank ℝ E := by
+  let f : LinearMap.ker B →ₗ[ℝ] E := i.comp (LinearMap.ker B).subtype
+  have hf : Function.Injective f := hi.comp Subtype.val_injective
+  have hsub : LinearMap.range i ≤ C.orthogonal (LinearMap.range f) := by
+    rintro _ ⟨v, rfl⟩
+    apply LinearMap.BilinForm.mem_orthogonal_iff.mpr
+    rintro _ ⟨k, rfl⟩
+    change C (i k.val) (i v) = 0
+    rw [hform, show B k.val = 0 from k.property]
+    rfl
+  have hd := Submodule.finrank_mono hsub
+  rw [LinearMap.finrank_range_of_inj hi, LinearMap.BilinForm.finrank_orthogonal hC,
+    LinearMap.finrank_range_of_inj hf] at hd
+  have hk : Module.finrank ℝ (LinearMap.ker B) ≤ Module.finrank ℝ E := by
+    rw [← LinearMap.finrank_range_of_inj hf]
+    exact Submodule.finrank_le _
+  omega
+
+end KLInvariance.Realization
+
+namespace KLInvariance.TitsSpace
+open Module
+universe u v
+variable {I : Type u} [Fintype I] (M : CoxeterMatrix I)
+noncomputable section
+
+abbrev Extended := Realization.Extension (form M)
+
+def extendedForm : LinearMap.BilinForm ℝ (Extended M) :=
+  Classical.choose (Realization.exists_minimal_nondegenerate_extension (form M) (form_symm M))
+
+theorem extendedForm_spec : (extendedForm M).IsSymm ∧ (extendedForm M).Nondegenerate ∧
+    ∀ v w, extendedForm M (v, 0) (w, 0) = form M v w :=
+  Classical.choose_spec (Realization.exists_minimal_nondegenerate_extension (form M) (form_symm M))
+
+def extendedRoot (i : I) : Extended M := (root i, 0)
+
+@[simp] theorem extendedForm_roots (i j : I) :
+    extendedForm M (extendedRoot M i) (extendedRoot M j) =
+      -2 * Real.cos (Real.pi / M i j) := by
+  rw [extendedRoot, extendedRoot, (extendedForm_spec M).2.2, form_roots]
+
+theorem extendedRoot_linearIndependent : LinearIndependent ℝ (extendedRoot M) := by
+  exact (root_linearIndependent (I := I)).map'
+    (LinearMap.inl ℝ (I → ℝ) (Module.Dual ℝ (LinearMap.ker (form M))))
+    (LinearMap.ker_eq_bot.mpr LinearMap.inl_injective)
+
+/-- Both roots and coroots are independent in the actual minimal extension. -/
+theorem extendedNormal_linearIndependent :
+    LinearIndependent ℝ (fun i => extendedForm M (extendedRoot M i)) :=
+  (extendedRoot_linearIndependent M).map' (extendedForm M) (extendedForm_spec M).2.1.ker_eq_bot
+
+def extendedAction {W : Type v} [Group W] (cs : CoxeterSystem M W) :
+    W →* (Extended M ≃ₗ[ℝ] Extended M) :=
+  Realization.geometricAction cs (extendedForm M) (extendedRoot M) (extendedForm_roots M)
+
+theorem extendedAction_preserves {W : Type v} [Group W] (cs : CoxeterSystem M W)
+    (w : W) (v z : Extended M) :
+    extendedForm M (extendedAction M cs w v) (extendedAction M cs w z) = extendedForm M v z :=
+  Realization.geometricAction_preserves cs (extendedForm M) (extendedRoot M)
+    (extendedForm_roots M) (extendedForm_spec M).1 w v z
+
+@[simp] theorem extendedAction_simple {W : Type v} [Group W] (cs : CoxeterSystem M W)
+    (i : I) (v : Extended M) :
+    extendedAction M cs (cs.simple i) v =
+      v - extendedForm M (extendedRoot M i) v • extendedRoot M i := by
+  simp only [extendedAction, Realization.geometricAction_simple,
+    Realization.geometricSimple, Module.reflection_apply]
+
+/-- The extended action genuinely restricts to the ordinary geometric action.
+Injectivity of that latter action is an outstanding obligation, not a hypothesis
+hidden in the construction of this extension. -/
+theorem extendedAction_restrict {W : Type v} [Group W] (cs : CoxeterSystem M W)
+    (w : W) (v : I → ℝ) :
+    extendedAction M cs w (v, 0) = (action M cs w v, 0) := by
+  obtain ⟨ω, rfl⟩ := cs.wordProd_surjective w
+  induction ω generalizing v with
+  | nil => simp
+  | cons i ω ih =>
+    rw [cs.wordProd_cons, map_mul, map_mul]
+    simp only [LinearEquiv.mul_apply, ih, extendedAction_simple, action_simple]
+    change (action M cs (cs.wordProd ω) v, 0) -
+      extendedForm M (root i, 0) (action M cs (cs.wordProd ω) v, 0) • (root i, 0) = _
+    rw [(extendedForm_spec M).2.2, form_root]
+    ext <;> simp
+
+theorem extended_finrank : Module.finrank ℝ (Extended M) =
+    Fintype.card I + Module.finrank ℝ (LinearMap.ker (form M)) := by
+  rw [Realization.extension_finrank, Module.finrank_pi]
+
+
+def dominant : Module.Dual ℝ (Extended M) :=
+  (∑ i : I, LinearMap.proj i).comp
+    (LinearMap.fst ℝ (I → ℝ) (Module.Dual ℝ (LinearMap.ker (form M))))
+
+@[simp] theorem dominant_root (i : I) : dominant M (extendedRoot M i) = 1 := by
+  classical
+  simp [dominant, extendedRoot, root]
+
+end
+end KLInvariance.TitsSpace
+
+namespace KLInvariance
+universe u v
+variable {B : Type u} {W : Type v} [Group W] {M : CoxeterMatrix B}
+
+namespace ParabolicSupport
+open CoxeterSystem
+
+variable (cs : CoxeterSystem M W) (J : Set B)
+
+/-- The genuine subgroup generated by a subset of simple reflections. -/
+def subgroup : Subgroup W := Subgroup.closure (cs.simple '' J)
+
+theorem word_mem {ν : List B} (hν : ∀ i ∈ ν, i ∈ J) :
+    cs.wordProd ν ∈ subgroup cs J := by
+  induction ν with
+  | nil => exact (subgroup cs J).one_mem
+  | cons i ν ih =>
+    rw [cs.wordProd_cons]
+    exact (subgroup cs J).mul_mem
+      (Subgroup.subset_closure ⟨i, hν i (by simp), rfl⟩)
+      (ih fun j hj => hν j (by simp [hj]))
+
+theorem exists_word {w : W} (hw : w ∈ subgroup cs J) :
+    ∃ ν : List B, (∀ i ∈ ν, i ∈ J) ∧ cs.wordProd ν = w := by
+  induction hw using Subgroup.closure_induction with
+  | mem w hw =>
+    obtain ⟨i, hi, rfl⟩ := hw
+    exact ⟨[i], by simpa using hi, by simp⟩
+  | one => exact ⟨[], by simp, by simp⟩
+  | mul x y hx hy ihx ihy =>
+    obtain ⟨ω, hω, rfl⟩ := ihx
+    obtain ⟨ν, hν, rfl⟩ := ihy
+    exact ⟨ω ++ ν, fun i hi => (List.mem_append.mp hi).elim (hω i) (hν i),
+      cs.wordProd_append ω ν⟩
+  | inv x hx ih =>
+    obtain ⟨ω, hω, rfl⟩ := ih
+    exact ⟨ω.reverse, by simpa using hω, cs.wordProd_reverse ω⟩
+
+theorem exists_reduced_word {w : W} (hw : w ∈ subgroup cs J) :
+    ∃ ν : List B, (∀ i ∈ ν, i ∈ J) ∧ cs.IsReduced ν ∧ cs.wordProd ν = w := by
+  obtain ⟨ω, hω, hprod⟩ := exists_word cs J hw
+  obtain ⟨ν, hsub, hν, heq⟩ := exists_reduced_subword cs ω
+  exact ⟨ν, fun i hi => hω i (hsub.subset hi), hν, heq.trans hprod⟩
+
+/-- A minimal right-coset representative concatenates with every reduced
+word in the parabolic generators without cancellations. -/
+theorem reduced_append_of_minimal {u : W}
+    (hmin : ∀ v ∈ subgroup cs J, cs.length u ≤ cs.length (u * v))
+    {ω ν : List B} (hω : cs.IsReduced ω) (hu : cs.wordProd ω = u)
+    (hν : cs.IsReduced ν) (hJ : ∀ i ∈ ν, i ∈ J) :
+    cs.IsReduced (ω ++ ν) := by
+  obtain ⟨τ, hsub, hτ, hprod⟩ := exists_reduced_subword cs (ω ++ ν)
+  obtain ⟨ω', ν', rfl, hωsub, hνsub⟩ := List.sublist_append_iff.mp hsub
+  have hprod' : cs.wordProd ω' * cs.wordProd ν' = u * cs.wordProd ν := by
+    simpa [cs.wordProd_append, hu] using hprod
+  have hω' : cs.wordProd ω' = u * (cs.wordProd ν * (cs.wordProd ν')⁻¹) := by
+    rw [← mul_assoc, ← hprod']
+    group
+  have hlen := hmin (cs.wordProd ν * (cs.wordProd ν')⁻¹)
+    ((subgroup cs J).mul_mem (word_mem cs J hJ)
+      ((subgroup cs J).inv_mem (word_mem cs J fun i hi => hJ i (hνsub.subset hi))))
+  rw [← hω'] at hlen
+  have hbound := cs.length_wordProd_le ω'
+  have homega := hω.eq
+  rw [hu] at homega
+  have hle := hωsub.length_le
+  have heqω := hωsub.eq_of_length (by omega)
+  subst ω'
+  have heqν : cs.wordProd ν' = cs.wordProd ν := by
+    rw [hu] at hprod'
+    exact mul_left_cancel hprod'
+  have hboundν := cs.length_wordProd_le ν'
+  rw [heqν, hν.eq] at hboundν
+  have heqν' := hνsub.eq_of_length (by have := hνsub.length_le; omega)
+  simpa [heqν'] using hτ
+
+theorem length_mul_of_minimal {u v : W}
+    (hmin : ∀ z ∈ subgroup cs J, cs.length u ≤ cs.length (u * z))
+    (hv : v ∈ subgroup cs J) : cs.length (u * v) = cs.length u + cs.length v := by
+  obtain ⟨ω, hω, hu⟩ := cs.exists_isReduced u
+  obtain ⟨ν, hJ, hν, hvprod⟩ := exists_reduced_word cs J hv
+  have h := (reduced_append_of_minimal cs J hmin hω hu.symm hν hJ).eq
+  simpa [cs.wordProd_append, hu, ← hvprod, hω.eq, hν.eq] using h
+
+/-- Every right parabolic coset has a minimal representative. -/
+theorem exists_minimal (w : W) :
+    ∃ u v : W, v ∈ subgroup cs J ∧ w = u * v ∧
+      ∀ z ∈ subgroup cs J, cs.length u ≤ cs.length (u * z) := by
+  classical
+  have hex : ∃ n : ℕ, ∃ u v : W, v ∈ subgroup cs J ∧ w = u * v ∧ cs.length u = n :=
+    ⟨cs.length w, w, 1, (subgroup cs J).one_mem, by simp, rfl⟩
+  obtain ⟨u, v, hv, hw, hu⟩ := Nat.find_spec hex
+  refine ⟨u, v, hv, hw, fun z hz => ?_⟩
+  rw [hu]
+  apply Nat.find_min' hex
+  refine ⟨u * z, z⁻¹ * v, (subgroup cs J).mul_mem ((subgroup cs J).inv_mem hz) hv, ?_, rfl⟩
+  simpa [mul_assoc] using hw
+
+theorem minimal_simple_ascent {u : W}
+    (hmin : ∀ z ∈ subgroup cs J, cs.length u ≤ cs.length (u * z))
+    {i : B} (hi : i ∈ J) : cs.length (u * cs.simple i) = cs.length u + 1 := by
+  have h := length_mul_of_minimal cs J hmin (Subgroup.subset_closure ⟨i, hi, rfl⟩)
+  simpa using h
+
+/-- The decomposition is length-additive, including after multiplying by
+any of the chosen simple generators. This is used in the root-sign induction. -/
+theorem exists_length_additive_decomposition (w : W) :
+    ∃ u v : W, v ∈ subgroup cs J ∧ w = u * v ∧
+      cs.length w = cs.length u + cs.length v ∧
+      (∀ i ∈ J, cs.length (u * cs.simple i) = cs.length u + 1) ∧
+      (∀ i ∈ J, cs.length (w * cs.simple i) = cs.length u + cs.length (v * cs.simple i)) := by
+  obtain ⟨u, v, hv, hw, hmin⟩ := exists_minimal cs J w
+  refine ⟨u, v, hv, hw, ?_, (fun i hi => minimal_simple_ascent cs J hmin hi), ?_⟩
+  · rw [hw]
+    exact length_mul_of_minimal cs J hmin hv
+  · intro i hi
+    rw [hw, mul_assoc]
+    exact length_mul_of_minimal cs J hmin
+      ((subgroup cs J).mul_mem hv (Subgroup.subset_closure ⟨i, hi, rfl⟩))
+
+end ParabolicSupport
+
+end KLInvariance
+
+namespace KLInvariance.ParabolicSupport
+open CoxeterSystem
+universe u v
+variable {B : Type u} {W : Type v} [Group W] {M : CoxeterMatrix B}
+variable (cs : CoxeterSystem M W)
+
+theorem reduced_of_append {ω ν : List B} (h : cs.IsReduced (ω ++ ν)) :
+    cs.IsReduced ω := by
+  simpa using h.take ω.length
+
+/-- A reduced two-letter word followed by `i` must alternate and end in `j`. -/
+theorem reduced_pair_eq_alternating {ω : List B} {i j : B} (hij : i ≠ j)
+    (hJ : ∀ k ∈ ω, k = i ∨ k = j) (hred : cs.IsReduced (ω ++ [i])) :
+    ω = alternatingWord i j ω.length := by
+  induction ω using List.reverseRecOn generalizing i j with
+  | nil => rfl
+  | append_singleton ω k ih =>
+    have hk : k = i ∨ k = j := hJ k (by simp)
+    have hki : k ≠ i := by
+      rintro rfl
+      have hlen := hred.eq
+      simp only [cs.wordProd_append, cs.wordProd_singleton, List.length_append,
+        List.length_singleton, mul_assoc, cs.simple_mul_simple_self, mul_one] at hlen
+      have := cs.length_wordProd_le ω
+      omega
+    have hkj : k = j := hk.resolve_left hki
+    subst k
+    have hprefix : cs.IsReduced (ω ++ [j]) := reduced_of_append cs hred
+    rw [List.length_append, List.length_singleton, alternatingWord_succ, List.concat_eq_append]
+    congr 1
+    apply ih hij.symm
+    · intro k hk
+      exact (hJ k (List.mem_append.mpr (Or.inl hk))).symm
+    · exact hprefix
+
+/-- The coefficients in the root formula are controlled by genuine reducedness,
+not by a separate rank-two length assumption. -/
+theorem alternating_length_bound {ω : List B} {i j : B} (hij : i ≠ j)
+    (hJ : ∀ k ∈ ω, k = i ∨ k = j) (hred : cs.IsReduced (ω ++ [i])) :
+    M i j = 0 ∨ ω.length + 1 ≤ M i j := by
+  by_cases hm : M i j = 0
+  · exact Or.inl hm
+  · right
+    have heq := reduced_pair_eq_alternating cs hij hJ hred
+    have hr : cs.IsReduced (alternatingWord j i (ω.length + 1)) := by
+      rw [alternatingWord_succ, ← heq]
+      simpa [List.concat_eq_append] using hred
+    by_contra hn
+    apply cs.not_isReduced_alternatingWord j i (by simpa [M.symmetric] using hm)
+      (by simpa [M.symmetric] using (show M i j < ω.length + 1 by omega)) hr
+
+end KLInvariance.ParabolicSupport
+
+namespace KLInvariance.TitsSpace
+open CoxeterSystem Polynomial Polynomial.Chebyshev
+universe u v
+variable {I : Type u} [klPreservedInstance5 : Fintype I] (M : CoxeterMatrix I)
+variable {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+def rootCoeff (i j : I) (n : ℤ) : ℝ :=
+  (S ℝ n).eval (2 * Real.cos (Real.pi / M i j))
+
+include klPreservedInstance5 in
+@[simp] theorem rootCoeff_zero (i j : I) : rootCoeff M i j 0 = 1 := by
+  have _ := klPreservedInstance5
+  simp [rootCoeff]
+include klPreservedInstance5 in
+@[simp] theorem rootCoeff_neg_one (i j : I) : rootCoeff M i j (-1) = 0 := by
+  have _ := klPreservedInstance5
+  simp [rootCoeff]
+
+include klPreservedInstance5 in
+theorem rootCoeff_succ (i j : I) (n : ℤ) :
+    rootCoeff M i j (n + 1) =
+      (2 * Real.cos (Real.pi / M i j)) * rootCoeff M i j n - rootCoeff M i j (n-1) := by
+  have _ := klPreservedInstance5
+  simp [rootCoeff, S_add_one]
+
+include klPreservedInstance5 in
+@[simp] theorem gram_diag (i : I) : gram M i i = 2 := by
+  have _ := klPreservedInstance5
+  simp [gram]
+
+/-- Exact rank-two root formula, including the infinite pairing. -/
+theorem alternating_root (i j : I) (n : ℕ) :
+    action M cs (cs.wordProd (alternatingWord i j n)) (root i) =
+      if Even n then
+        rootCoeff M i j n • root i + rootCoeff M i j ((n : ℤ)-1) • root j
+      else rootCoeff M i j ((n : ℤ)-1) • root i + rootCoeff M i j n • root j := by
+  induction n with
+  | zero => simp [alternatingWord]
+  | succ n ih =>
+    rw [alternatingWord_succ', cs.wordProd_cons, map_mul, LinearEquiv.mul_apply, ih]
+    have hrec := rootCoeff_succ M i j (n : ℤ)
+    have hsymm : gram M j i = gram M i j := by simp [gram, M.symmetric j i]
+    by_cases hn : Even n
+    · simp only [hn, ite_true, Nat.even_add_one, not_true_eq_false, ite_false]
+      rw [action_simple]
+      simp only [map_add, map_smul, normal_root, gram_diag, hsymm]
+      push_cast
+      rw [show (n : ℤ) + 1 - 1 = n by omega, hrec]
+      simp only [gram]
+      module
+    · simp only [hn, ite_false, Nat.even_add_one, not_false_eq_true, ite_true]
+      rw [action_simple]
+      simp only [map_add, map_smul, normal_root, gram_diag]
+      push_cast
+      rw [show (n : ℤ) + 1 - 1 = n by omega, hrec]
+      simp only [gram]
+      module
+
+/-- For a reduced alternating word followed by the other generator, both
+rank-two coefficients are nonnegative. -/
+theorem rootCoeff_nonneg (i j : I) {n : ℤ} (hn : -1 ≤ n)
+    (hm : M i j = 0 ∨ n + 1 ≤ (M i j : ℤ)) (hij : i ≠ j) :
+    0 ≤ rootCoeff M i j n := by
+  rcases hm with hm | hm
+  · simp only [rootCoeff, hm, Nat.cast_zero, div_zero, Real.cos_zero, mul_one, S_eval_two]
+    exact_mod_cast (by omega : (0 : ℤ) ≤ n + 1)
+  · by_cases hn0 : n = -1
+    · simp [hn0]
+    have hn1 : 0 < n + 1 := by omega
+    have hMpos : (0 : ℝ) < M i j := by exact_mod_cast (by omega : 0 < M i j)
+    have hM2 : 2 ≤ M i j := by
+      have := M.off_diagonal i j hij
+      omega
+    let θ : ℝ := Real.pi / M i j
+    have hθpos : 0 < θ := div_pos Real.pi_pos hMpos
+    have hθlt : θ < Real.pi := by
+      dsimp [θ]
+      apply (div_lt_iff₀ hMpos).2
+      have : (2 : ℝ) ≤ M i j := by exact_mod_cast hM2
+      nlinarith [Real.pi_pos]
+    have hsin := Real.sin_pos_of_pos_of_lt_pi hθpos hθlt
+    have heq := S_two_mul_real_cos θ n
+    have hargpos : 0 ≤ ((n : ℝ) + 1) * θ := by
+      have : (0 : ℝ) < (n : ℝ) + 1 := by exact_mod_cast hn1
+      positivity
+    have hargle : ((n : ℝ) + 1) * θ ≤ Real.pi := by
+      dsimp [θ]
+      rw [← mul_div_assoc]
+      apply (div_le_iff₀ hMpos).2
+      have : (n : ℝ) + 1 ≤ M i j := by exact_mod_cast hm
+      nlinarith [Real.pi_pos]
+    have hs := Real.sin_nonneg_of_nonneg_of_le_pi hargpos hargle
+    change 0 ≤ (S ℝ n).eval (2 * Real.cos θ)
+    nlinarith [heq]
+end
+end KLInvariance.TitsSpace
+
+namespace KLInvariance.TitsSpace
+open CoxeterSystem
+universe u v
+variable {I : Type u} [klPreservedInstance6 : Fintype I] (M : CoxeterMatrix I)
+variable {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+include klPreservedInstance6 in
+@[simp] theorem root_nonneg (i k : I) : 0 ≤ root i k := by
+  have _ := klPreservedInstance6
+  classical
+  simp only [root, Pi.single_apply]
+  split_ifs <;> norm_num
+
+/-- Positivity of a rank-two reduced tail after an already positive prefix. -/
+theorem pair_image_nonneg (u : W) {i j : I} (hij : i ≠ j)
+    (hi : ∀ k, 0 ≤ action M cs u (root i) k)
+    (hj : ∀ k, 0 ≤ action M cs u (root j) k)
+    {ω : List I} (hJ : ∀ k ∈ ω, k = i ∨ k = j)
+    (hred : cs.IsReduced (ω ++ [i])) :
+    ∀ k, 0 ≤ action M cs (u * cs.wordProd ω) (root i) k := by
+  have heq := ParabolicSupport.reduced_pair_eq_alternating cs hij hJ hred
+  have hbound := ParabolicSupport.alternating_length_bound cs hij hJ hred
+  have hn : 0 ≤ rootCoeff M i j (ω.length : ℤ) :=
+    rootCoeff_nonneg M i j (by omega) (by rcases hbound with h|h; exact Or.inl h; right; omega) hij
+  have hn' : 0 ≤ rootCoeff M i j ((ω.length : ℤ)-1) :=
+    rootCoeff_nonneg M i j (by omega) (by rcases hbound with h|h; exact Or.inl h; right; omega) hij
+  intro k
+  rw [map_mul, LinearEquiv.mul_apply, heq, alternating_root]
+  split_ifs
+  all_goals
+    simp only [map_add, map_smul, Pi.add_apply, Pi.smul_apply, smul_eq_mul]
+    exact add_nonneg (mul_nonneg (by assumption) (by exact hi k))
+      (mul_nonneg (by assumption) (by first | exact hi k | exact hj k))
+
+/-- The genuine Tits root-sign theorem for ascents, proved from exchange,
+minimal parabolic cosets and the rank-two trigonometric formulas. -/
+theorem action_root_nonneg_of_ascent (w : W) (i : I)
+    (hasc : cs.length (w * cs.simple i) = cs.length w + 1) :
+    ∀ k, 0 ≤ action M cs w (root i) k := by
+  suffices ∀ n : ℕ, ∀ w : W, cs.length w = n → ∀ i : I,
+      cs.length (w * cs.simple i) = cs.length w + 1 →
+      ∀ k, 0 ≤ action M cs w (root i) k by
+    exact this (cs.length w) w rfl i hasc
+  intro n
+  induction n using Nat.strong_induction_on with
+  | h n ih =>
+    intro w hwlen i hasc
+    by_cases hw : w = 1
+    · subst w
+      simp
+    obtain ⟨j, hj⟩ := cs.exists_rightDescent_of_ne_one hw
+    change cs.length (w * cs.simple j) < cs.length w at hj
+    have hij : i ≠ j := by rintro rfl; omega
+    obtain ⟨u, v, hv, huv, hlen, huasc, hmul⟩ :=
+      ParabolicSupport.exists_length_additive_decomposition cs ({i, j} : Set I) w
+    have himem : i ∈ ({i, j} : Set I) := by simp
+    have hjmem : j ∈ ({i, j} : Set I) := by simp
+    have hult : cs.length u < n := by
+      have h := hmul j hjmem
+      omega
+    have hui := ih (cs.length u) hult u rfl i (huasc i himem)
+    have huj := ih (cs.length u) hult u rfl j (huasc j hjmem)
+    obtain ⟨ω, hJ, hω, hprod⟩ := ParabolicSupport.exists_reduced_word cs ({i,j} : Set I) hv
+    have hvi : cs.length (v * cs.simple i) = cs.length v + 1 := by
+      have h := hmul i himem
+      omega
+    have hωi : cs.IsReduced (ω ++ [i]) := by
+      change cs.length (cs.wordProd (ω ++ [i])) = (ω ++ [i]).length
+      simp only [cs.wordProd_append, cs.wordProd_singleton, hprod, List.length_append,
+        List.length_singleton]
+      rw [hvi, ← hprod, hω.eq]
+    have hJ' : ∀ k ∈ ω, k = i ∨ k = j := by simpa using hJ
+    rw [huv, ← hprod]
+    exact pair_image_nonneg M cs u hij hui huj hJ' hωi
+
+@[simp] theorem action_simple_root (i : I) : action M cs (cs.simple i) (root i) = -root i := by
+  rw [action_simple, normal_root, gram_diag]
+  module
+
+/-- Descents carry a simple root into the negative cone. -/
+theorem action_root_nonpos_of_descent (w : W) (i : I)
+    (hdesc : cs.length (w * cs.simple i) < cs.length w) :
+    ∀ k, action M cs w (root i) k ≤ 0 := by
+  have hasc : cs.length ((w * cs.simple i) * cs.simple i) = cs.length (w * cs.simple i) + 1 := by
+    rcases cs.length_mul_simple w i with h|h
+    · omega
+    · simpa [mul_assoc] using h.symm
+  have h := action_root_nonneg_of_ascent M cs (w * cs.simple i) i hasc
+  simpa only [map_mul, LinearEquiv.mul_apply, action_simple_root, map_neg,
+    Pi.neg_apply, neg_nonneg] using h
+
+/-- Faithfulness of the actual ordinary Tits action, not an added axiom. -/
+theorem action_injective : Function.Injective (action M cs) := by
+  apply (MonoidHom.ker_eq_bot_iff _).1
+  apply (Subgroup.eq_bot_iff_forall _).2
+  intro w hw
+  by_contra hne
+  obtain ⟨i, hi⟩ := cs.exists_rightDescent_of_ne_one hne
+  have h := action_root_nonpos_of_descent M cs w i hi i
+  have hw' : action M cs w = 1 := hw
+  rw [hw'] at h
+  norm_num [root] at h
+end
+end KLInvariance.TitsSpace
+
+namespace KLInvariance.Realization
+open Module
+variable {V : Type*} [AddCommGroup V] [Module ℝ V]
+
+/-- Rank of the moving subspace is unchanged by conjugacy. -/
+theorem finrank_moving_conj (g h : V ≃ₗ[ℝ] V) :
+    Module.finrank ℝ (LinearMap.range ((h * g * h⁻¹).toLinearMap - LinearMap.id)) =
+      Module.finrank ℝ (LinearMap.range (g.toLinearMap - LinearMap.id)) := by
+  have heq : (h * g * h⁻¹).toLinearMap - LinearMap.id =
+      h.toLinearMap.comp ((g.toLinearMap - LinearMap.id).comp h.symm.toLinearMap) := by
+    ext v
+    simp
+  rw [heq, LinearMap.range_comp, LinearMap.range_comp_of_range_eq_top _ h.symm.range,
+    h.finrank_map_eq]
+
+/-- An orthogonal rank-one map negating a norm-two vector is its exact reflection. -/
+theorem rank_one_eq_reflection_of_negates [FiniteDimensional ℝ V]
+    (B : LinearMap.BilinForm ℝ V) (g : V ≃ₗ[ℝ] V)
+    (hg : ∀ v w, B (g v) (g w) = B v w)
+    (hr : Module.finrank ℝ (LinearMap.range (g.toLinearMap - LinearMap.id)) = 1)
+    (e : V) (he : B e e = 2) (hge : g e = -e) :
+    ∀ v, g v = Module.preReflection e (B e) v := by
+  let d : V →ₗ[ℝ] V := g.toLinearMap - LinearMap.id
+  have he0 : e ≠ 0 := by intro h; simp [h] at he
+  have hemem : e ∈ LinearMap.range d := by
+    refine ⟨(- (1 / 2 : ℝ)) • e, ?_⟩
+    change g (- (1 / 2 : ℝ) • e) - (- (1 / 2 : ℝ) • e) = e
+    rw [map_smul, hge]
+    module
+  have hspan : Submodule.span ℝ {e} = LinearMap.range d := by
+    apply Submodule.eq_of_le_of_finrank_eq
+    · exact Submodule.span_le.mpr (by simpa using hemem)
+    · rw [finrank_span_singleton he0]
+      exact hr.symm
+  intro v
+  have hv : d v ∈ Submodule.span ℝ {e} := by rw [hspan]; exact LinearMap.mem_range_self _ v
+  obtain ⟨c, hc⟩ := Submodule.mem_span_singleton.mp hv
+  have hgform : g v = v + c • e := by
+    change c • e = g v - v at hc
+    rw [hc]
+    abel
+  have hB := hg e v
+  rw [hge, hgform] at hB
+  simp only [map_neg, LinearMap.neg_apply, map_add, map_smul, he, smul_eq_mul] at hB
+  have hc' : c = - B e v := by linarith
+  rw [hgform, hc', Module.preReflection_apply, neg_smul, sub_eq_add_neg]
+
+/-- Commutation with a reflection makes its negative eigenspace invariant;
+a nonpositive coordinate selects the negative, rather than positive, sign. -/
+theorem negates_of_commutes_nonpos
+    (B : LinearMap.BilinForm ℝ V) (g : V ≃ₗ[ℝ] V)
+    (hg : ∀ v w, B (g v) (g w) = B v w)
+    (e : V) (he : B e e = 2)
+    (hcomm : g * Module.reflection he = Module.reflection he * g)
+    (f : Module.Dual ℝ V) (hfe : f e = 1) (hfg : f (g e) ≤ 0) :
+    g e = -e := by
+  have h := LinearEquiv.congr_fun hcomm e
+  simp only [LinearEquiv.mul_apply, Module.reflection_apply_self, map_neg,
+    Module.reflection_apply] at h
+  let c : ℝ := B e (g e) / 2
+  have hge : g e = c • e := by
+    dsimp [c]
+    linear_combination (norm := module) (- (1 / 2 : ℝ)) • h
+  have hc : c ≤ 0 := by simpa [hge, hfe] using hfg
+  have hnorm := hg e e
+  rw [hge] at hnorm
+  simp only [map_smul, LinearMap.smul_apply, he, smul_eq_mul] at hnorm
+  have hc' : c = -1 := by nlinarith [sq_nonneg (c + 1)]
+  rw [hge, hc', neg_one_smul]
+/-- The moving subspace of a genuine reflection is exactly its root line. -/
+theorem range_reflection_sub_id (e : V) (f : Module.Dual ℝ V) (h : f e = 2) :
+    LinearMap.range ((Module.reflection h).toLinearMap - LinearMap.id) =
+      Submodule.span ℝ {e} := by
+  apply le_antisymm
+  · rintro z ⟨v, rfl⟩
+    apply Submodule.mem_span_singleton.mpr
+    refine ⟨-f v, ?_⟩
+    simp only [LinearMap.sub_apply, LinearEquiv.coe_coe, LinearMap.id_apply,
+      Module.reflection_apply, neg_smul]
+    abel
+  · apply Submodule.span_le.mpr
+    intro z hz
+    have hz' : z = e := Set.mem_singleton_iff.mp hz
+    rw [hz']
+    refine ⟨(- (1 / 2 : ℝ)) • e, ?_⟩
+    simp only [LinearMap.sub_apply, LinearEquiv.coe_coe, LinearMap.id_apply,
+      Module.reflection_apply, map_smul, h]
+    module
+
+theorem finrank_reflection_sub_id (e : V) (f : Module.Dual ℝ V) (h : f e = 2) :
+    Module.finrank ℝ (LinearMap.range
+      ((Module.reflection h).toLinearMap - LinearMap.id)) = 1 := by
+  rw [range_reflection_sub_id]
+  apply finrank_span_singleton
+  intro he
+  simp [he] at h
+
+end KLInvariance.Realization
+
+namespace KLInvariance.TitsSpace
+open Module
+universe u v
+variable {I : Type u} [Fintype I] (M : CoxeterMatrix I)
+variable {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+/-- Faithfulness persists in the actual nondegenerate extension. -/
+theorem extendedAction_injective : Function.Injective (extendedAction M cs) := by
+  intro w z h
+  apply action_injective M cs
+  ext v i
+  have hv := congrArg (fun g : Extended M ≃ₗ[ℝ] Extended M => (g (v, 0)).1 i) h
+  simpa only [extendedAction_restrict] using hv
+
+@[simp] theorem extendedForm_root_self (i : I) :
+    extendedForm M (extendedRoot M i) (extendedRoot M i) = 2 := by
+  rw [extendedForm_roots, M.diagonal]
+  norm_num
+
+@[simp] theorem extendedAction_simple_eq (i : I) :
+    extendedAction M cs (cs.simple i) = Module.reflection (extendedForm_root_self M i) := by
+  apply LinearEquiv.ext
+  intro v
+  rw [extendedAction_simple, Module.reflection_apply]
+
+/-- A rank-one element of the actual extended action is an involution. -/
+theorem involution_of_moving_rank_one (w : W)
+    (hr : Module.finrank ℝ (LinearMap.range
+      ((extendedAction M cs w).toLinearMap - LinearMap.id)) = 1) : w⁻¹ = w := by
+  obtain ⟨e, he, hform⟩ := Realization.orthogonal_finrank_one
+    (extendedForm M) (extendedForm_spec M).1 (extendedForm_spec M).2.1
+    (extendedAction M cs w) (extendedAction_preserves M cs w) hr
+  have he2 : ((2 / extendedForm M e e) • extendedForm M e) e = 2 := by
+    simp only [LinearMap.smul_apply, smul_eq_mul]
+    field_simp
+  have hw2 : w * w = 1 := by
+    apply extendedAction_injective M cs
+    apply LinearEquiv.ext
+    intro v
+    simp only [map_mul, map_one, LinearEquiv.mul_apply, hform]
+    change _ = v
+    exact Module.involutive_preReflection he2 v
+  exact inv_eq_of_mul_eq_one_left hw2
+
+/-- Reflection faithfulness, the difficult converse, proved by length descent.
+This avoids assuming the finite-subgroup theorem cited in the manuscript. -/
+theorem isReflection_of_moving_rank_one (w : W)
+    (hr : Module.finrank ℝ (LinearMap.range
+      ((extendedAction M cs w).toLinearMap - LinearMap.id)) = 1) :
+    cs.IsReflection w := by
+  suffices ∀ n : ℕ, ∀ w : W, cs.length w = n →
+      Module.finrank ℝ (LinearMap.range
+        ((extendedAction M cs w).toLinearMap - LinearMap.id)) = 1 →
+      cs.IsReflection w from this (cs.length w) w rfl hr
+  intro n
+  induction n using Nat.strong_induction_on with
+  | h n ih =>
+    intro w hlen hr
+    have hwne : w ≠ 1 := by
+      intro h
+      subst w
+      have heq : (extendedAction M cs 1).toLinearMap - LinearMap.id = 0 := by
+        apply LinearMap.ext
+        intro v
+        change (extendedAction M cs 1) v - v = 0
+        rw [map_one]
+        change v - v = 0
+        abel
+      rw [heq, LinearMap.range_zero, finrank_bot] at hr
+      omega
+    have hwinv := involution_of_moving_rank_one M cs w hr
+    obtain ⟨i, hi⟩ := cs.exists_rightDescent_of_ne_one hwne
+    change cs.length (w * cs.simple i) < cs.length w at hi
+    have hleft : cs.length (cs.simple i * w) < cs.length w := by
+      have h := cs.length_inv (w * cs.simple i)
+      simp only [mul_inv_rev, cs.inv_simple, hwinv] at h
+      omega
+    by_cases hcomm : cs.simple i * w = w * cs.simple i
+    · have hcomm' : extendedAction M cs w * Module.reflection (extendedForm_root_self M i) =
+          Module.reflection (extendedForm_root_self M i) * extendedAction M cs w := by
+        rw [← extendedAction_simple_eq M cs i, ← map_mul, ← map_mul, hcomm]
+      let f : Module.Dual ℝ (Extended M) := (LinearMap.proj i).comp
+        (LinearMap.fst ℝ (I → ℝ) (Module.Dual ℝ (LinearMap.ker (form M))))
+      have hf : f (extendedRoot M i) = 1 := by simp [f, extendedRoot, root]
+      have hfg : f (extendedAction M cs w (extendedRoot M i)) ≤ 0 := by
+        have hn := action_root_nonpos_of_descent M cs w i hi i
+        simpa only [f, LinearMap.comp_apply, extendedRoot, extendedAction_restrict,
+          LinearMap.fst_apply, LinearMap.proj_apply] using hn
+      have hneg := Realization.negates_of_commutes_nonpos (extendedForm M)
+        (extendedAction M cs w) (extendedAction_preserves M cs w)
+        (extendedRoot M i) (extendedForm_root_self M i) hcomm' f hf hfg
+      have heq : extendedAction M cs w = extendedAction M cs (cs.simple i) := by
+        apply LinearEquiv.ext
+        intro v
+        rw [extendedAction_simple]
+        exact Realization.rank_one_eq_reflection_of_negates (extendedForm M)
+          (extendedAction M cs w) (extendedAction_preserves M cs w) hr
+          (extendedRoot M i) (extendedForm_root_self M i) hneg v
+      rw [extendedAction_injective M cs heq]
+      exact cs.isReflection_simple i
+    · let z := cs.simple i * w * cs.simple i
+      have hzlt : cs.length z < n := by
+        have hz := (right_descent_simple_mul_iff cs i i w hcomm).2 hi
+        dsimp [z]
+        omega
+      have hzrank : Module.finrank ℝ (LinearMap.range
+          ((extendedAction M cs z).toLinearMap - LinearMap.id)) = 1 := by
+        have hc := Realization.finrank_moving_conj (extendedAction M cs w)
+          (extendedAction M cs (cs.simple i))
+        have hgi : (extendedAction M cs (cs.simple i))⁻¹ = extendedAction M cs (cs.simple i) := by
+          rw [← map_inv, cs.inv_simple]
+        have heq : extendedAction M cs z =
+            extendedAction M cs (cs.simple i) * extendedAction M cs w *
+              (extendedAction M cs (cs.simple i))⁻¹ := by
+          dsimp [z]
+          rw [map_mul, map_mul, hgi]
+        rw [heq]
+        exact hc.trans hr
+      have hz := ih (cs.length z) hzlt z rfl hzrank
+      have hc := hz.conj (w := cs.simple i)
+      simpa [z, mul_assoc] using hc
+
+end
+end KLInvariance.TitsSpace
+
+namespace KLInvariance.TitsSpace
+open Module
+universe u v
+variable {I : Type u} [Fintype I] (M : CoxeterMatrix I)
+variable {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+/-- The other direction of reflection faithfulness for the actual action. -/
+theorem moving_rank_one_of_isReflection (w : W) (hw : cs.IsReflection w) :
+    Module.finrank ℝ (LinearMap.range
+      ((extendedAction M cs w).toLinearMap - LinearMap.id)) = 1 := by
+  obtain ⟨z, i, rfl⟩ := hw
+  have hc := Realization.finrank_moving_conj (extendedAction M cs (cs.simple i))
+    (extendedAction M cs z)
+  have heq : extendedAction M cs (z * cs.simple i * z⁻¹) =
+      extendedAction M cs z * extendedAction M cs (cs.simple i) *
+        (extendedAction M cs z)⁻¹ := by rw [map_mul, map_mul, map_inv]
+  rw [heq, hc, extendedAction_simple_eq]
+  exact Realization.finrank_reflection_sub_id _ _ _
+
+theorem reflectionFaithful (w : W) :
+    Module.finrank ℝ (LinearMap.range
+      ((extendedAction M cs w).toLinearMap - LinearMap.id)) = 1 ↔ cs.IsReflection w :=
+  ⟨isReflection_of_moving_rank_one M cs w, moving_rank_one_of_isReflection M cs w⟩
+
+@[simp] theorem dominant_action_root (w : W) (i : I) :
+    dominant M (extendedAction M cs w (extendedRoot M i)) =
+      ∑ k, action M cs w (root i) k := by
+  simp [extendedRoot, extendedAction_restrict, dominant]
+
+/-- Strict dominant sign, not merely closed-cone positivity. -/
+theorem dominant_pos_iff_ascent (w : W) (i : I) :
+    0 < dominant M (extendedAction M cs w (extendedRoot M i)) ↔
+      cs.length w < cs.length (w * cs.simple i) := by
+  classical
+  rw [dominant_action_root]
+  rcases cs.length_mul_simple w i with h | h
+  · have ha := action_root_nonneg_of_ascent M cs w i h
+    have hv : action M cs w (root i) ≠ 0 := by
+      apply (action M cs w).map_ne_zero_iff.mpr
+      intro hz
+      have hz' := congrFun hz i
+      simp [root] at hz'
+    have hex : ∃ k, 0 < action M cs w (root i) k := by
+      by_contra! hn
+      apply hv
+      funext k
+      exact le_antisymm (hn k) (ha k)
+    have hs : 0 < ∑ k, action M cs w (root i) k := by
+      apply (Finset.sum_pos_iff_of_nonneg (fun k _ => ha k)).mpr
+      obtain ⟨k, hk⟩ := hex
+      exact ⟨k, Finset.mem_univ _, hk⟩
+    exact iff_of_true hs (by omega)
+  · have hn := action_root_nonpos_of_descent M cs w i (by omega)
+    have hs : ∑ k, action M cs w (root i) k ≤ 0 := Finset.sum_nonpos (fun k _ => hn k)
+    exact iff_of_false (by linarith) (by omega)
+
+/-- The strictly dominant linear form in the convention `(w ρ)(αᵢ)=ρ(w⁻¹αᵢ)`
+has exactly the source manuscript's left-ascent sign. -/
+theorem dominant_dual_pos_iff_left_ascent (w : W) (i : I) :
+    0 < dominant M (extendedAction M cs w⁻¹ (extendedRoot M i)) ↔
+      cs.length w < cs.length (cs.simple i * w) := by
+  rw [dominant_pos_iff_ascent, cs.length_inv]
+  have h : cs.length (w⁻¹ * cs.simple i) = cs.length (cs.simple i * w) := by
+    rw [← cs.length_inv (cs.simple i * w)]
+    simp
+  rw [h]
+
+end
+end KLInvariance.TitsSpace
+
+/-! Coordinate-dual geometric action. This exists in arbitrary rank and is
+used only to prove the finite standard-parabolic reduction of the main goal. -/
+namespace KLInvariance.Cotangent
+open Module
+universe u v
+variable {I : Type u} (M : CoxeterMatrix I)
+noncomputable section
+
+def root (i : I) : I → ℝ := fun j => -2 * Real.cos (Real.pi / M i j)
+
+lemma normal_root (i : I) : (LinearMap.proj i : (I → ℝ) →ₗ[ℝ] ℝ) (root M i) = 2 := by
+  simp [root]
+
+def simple (i : I) : (I → ℝ) ≃ₗ[ℝ] (I → ℝ) := Module.reflection (normal_root M i)
+
+theorem simple_liftable : M.IsLiftable (simple M) := by
+  intro i j
+  by_cases hij : i = j
+  · subst j
+    rw [M.diagonal, pow_one]
+    apply LinearEquiv.ext
+    intro v
+    exact Module.involutive_reflection (normal_root M i) v
+  by_cases hm0 : M i j = 0
+  · rw [hm0, pow_zero]
+  have hm2 : 2 ≤ M i j := by
+    have := M.off_diagonal i j hij
+    omega
+  apply Realization.geometric_reflection_relation (normal_root M i) (normal_root M j)
+    (M i j) hm2
+  · simp [root, M.symmetric j i]
+  · rfl
+
+def action {W : Type v} [Group W] (cs : CoxeterSystem M W) :
+    W →* ((I → ℝ) ≃ₗ[ℝ] (I → ℝ)) := cs.lift ⟨simple M, simple_liftable M⟩
+
+@[simp] theorem action_simple {W : Type v} [Group W] (cs : CoxeterSystem M W)
+    (i : I) (l : I → ℝ) :
+    action M cs (cs.simple i) l = l - l i • root M i := by
+  rw [action, cs.lift_apply_simple]
+  rfl
+
+section Finite
+variable [Fintype I] {W : Type v} [Group W] (cs : CoxeterSystem M W)
+
+lemma dot_root (l : I → ℝ) (i : I) : l ⬝ᵥ TitsSpace.root i = l i := by
+  classical
+  simp [TitsSpace.root, dotProduct, Pi.single_apply]
+
+lemma root_dot (i : I) (v : I → ℝ) :
+    root M i ⬝ᵥ v = TitsSpace.normal M i v := by
+  simp [root, TitsSpace.normal, TitsSpace.gram, dotProduct]
+
+lemma simple_pairing (i : I) (l v : I → ℝ) :
+    (action M cs (cs.simple i) l) ⬝ᵥ (TitsSpace.action M cs (cs.simple i) v) =
+      l ⬝ᵥ v := by
+  rw [action_simple, TitsSpace.action_simple]
+  simp only [sub_dotProduct, dotProduct_sub, smul_dotProduct, dotProduct_smul,
+    dot_root, root_dot]
+  have hd : root M i i = 2 := by simp [root]
+  rw [hd]
+  simp only [smul_eq_mul]
+  ring
+
+theorem pairing (w : W) (l v : I → ℝ) :
+    (action M cs w l) ⬝ᵥ (TitsSpace.action M cs w v) = l ⬝ᵥ v := by
+  obtain ⟨ω, rfl⟩ := cs.wordProd_surjective w
+  induction ω generalizing l v with
+  | nil => simp
+  | cons i ω ih =>
+    rw [cs.wordProd_cons, map_mul, map_mul, LinearEquiv.mul_apply,
+      LinearEquiv.mul_apply, simple_pairing, ih]
+
+theorem action_injective : Function.Injective (action M cs) := by
+  apply (injective_iff_map_eq_one _).2
+  intro w hw
+  apply TitsSpace.action_injective M cs
+  rw [map_one]
+  apply LinearEquiv.ext
+  intro v
+  apply dotProduct_eq
+  intro l
+  have h := pairing M cs w l v
+  rw [hw] at h
+  simpa [dotProduct_comm] using h
+end Finite
+end
+end KLInvariance.Cotangent
+
+namespace KLInvariance.StandardParabolic
+open CoxeterSystem
+universe u v
+variable {I : Type u} {W : Type v} [Group W] {M : CoxeterMatrix I}
+variable (cs : CoxeterSystem M W) (J : Set I)
+noncomputable section
+
+def matrix : CoxeterMatrix J where
+  M i j := M i.val j.val
+  isSymm := M.isSymm.submatrix Subtype.val
+  diagonal i := M.diagonal i.val
+  off_diagonal i j h := M.off_diagonal i.val j.val (fun h' => h (Subtype.ext h'))
+
+abbrev system : CoxeterSystem (matrix (M := M) J) (matrix (M := M) J).Group :=
+  (matrix (M := M) J).toCoxeterSystem
+
+def inclusion : (matrix (M := M) J).Group →* W :=
+  (system (M := M) J).lift ⟨fun i => cs.simple i.val,
+    fun i j => cs.simple_mul_simple_pow i.val j.val⟩
+
+@[simp] theorem inclusion_simple (i : J) :
+    inclusion cs J ((system (M := M) J).simple i) = cs.simple i.val :=
+  (system (M := M) J).lift_apply_simple _ i
+
+lemma inclusion_word (ω : List J) :
+    inclusion cs J ((system (M := M) J).wordProd ω) = cs.wordProd (ω.map Subtype.val) := by
+  induction ω with
+  | nil => simp
+  | cons i ω ih =>
+    rw [wordProd_cons, map_mul, inclusion_simple, List.map_cons, wordProd_cons, ih]
+
+def restrictVec (l : I → ℝ) : J → ℝ := fun j => l j.val
+
+lemma simple_restrict (i : J) (l : I → ℝ) :
+    restrictVec J (Cotangent.action M cs (cs.simple i.val) l) =
+      Cotangent.action (matrix (M := M) J) (system (M := M) J)
+        ((system (M := M) J).simple i) (restrictVec J l) := by
+  rw [Cotangent.action_simple, Cotangent.action_simple]
+  funext j
+  rfl
+
+theorem action_restrict (w : (matrix (M := M) J).Group) (l : I → ℝ) :
+    restrictVec J (Cotangent.action M cs (inclusion cs J w) l) =
+      Cotangent.action (matrix (M := M) J) (system (M := M) J) w (restrictVec J l) := by
+  obtain ⟨ω, rfl⟩ := (system (M := M) J).wordProd_surjective w
+  induction ω generalizing l with
+  | nil => simp
+  | cons i ω ih =>
+    rw [wordProd_cons, map_mul, inclusion_simple, map_mul, map_mul,
+      LinearEquiv.mul_apply, LinearEquiv.mul_apply, simple_restrict, ih]
+
+theorem inclusion_injective [Fintype J] : Function.Injective (inclusion cs J) := by
+  classical
+  apply (injective_iff_map_eq_one _).2
+  intro w hw
+  apply Cotangent.action_injective (matrix (M := M) J) (system (M := M) J)
+  rw [map_one]
+  apply LinearEquiv.ext
+  intro l
+  let l' : I → ℝ := fun i => if hi : i ∈ J then l ⟨i, hi⟩ else 0
+  have hr : restrictVec J l' = l := by ext j; simp [restrictVec, l', j.property]
+  have h := action_restrict cs J w l'
+  rw [hw, map_one] at h
+  change restrictVec J l' = _ at h
+  rw [hr] at h
+  exact h.symm
+
+lemma lift_word {ω : List I} (hω : ∀ i ∈ ω, i ∈ J) :
+    ∃ ν : List J, ν.map Subtype.val = ω := by
+  induction ω with
+  | nil => exact ⟨[], rfl⟩
+  | cons i ω ih =>
+    obtain ⟨ν, rfl⟩ := ih (fun j hj => hω j (by simp [hj]))
+    exact ⟨⟨i, hω i (by simp)⟩ :: ν, rfl⟩
+
+theorem inclusion_mem (w : (matrix (M := M) J).Group) :
+    inclusion cs J w ∈ ParabolicSupport.subgroup cs J := by
+  obtain ⟨ω, rfl⟩ := (system (M := M) J).wordProd_surjective w
+  rw [inclusion_word]
+  apply ParabolicSupport.word_mem
+  intro i hi
+  obtain ⟨j, _, rfl⟩ := List.mem_map.mp hi
+  exact j.property
+
+theorem inclusion_range : (inclusion cs J).range = ParabolicSupport.subgroup cs J := by
+  apply le_antisymm
+  · rintro _ ⟨w, rfl⟩
+    exact inclusion_mem cs J w
+  · intro w hw
+    obtain ⟨ω, hω, rfl⟩ := ParabolicSupport.exists_word cs J hw
+    obtain ⟨ν, rfl⟩ := lift_word J hω
+    exact ⟨(system (M := M) J).wordProd ν, inclusion_word cs J ν⟩
+
+theorem length_inclusion [Fintype J] (w : (matrix (M := M) J).Group) :
+    cs.length (inclusion cs J w) = (system (M := M) J).length w := by
+  apply le_antisymm
+  · obtain ⟨ω, hω, rfl⟩ := (system (M := M) J).exists_isReduced w
+    rw [inclusion_word, hω.eq]
+    simpa using cs.length_wordProd_le (ω.map Subtype.val)
+  · obtain ⟨ω, hω, hred, hprod⟩ := ParabolicSupport.exists_reduced_word cs J
+      (inclusion_mem cs J w)
+    obtain ⟨ν, rfl⟩ := lift_word J hω
+    have heq : (system (M := M) J).wordProd ν = w :=
+      inclusion_injective cs J ((inclusion_word cs J ν).trans hprod)
+    have hb := (system (M := M) J).length_wordProd_le ν
+    rw [heq] at hb
+    have hr := hred.eq
+    rw [hprod, List.length_map] at hr
+    omega
+
+lemma reduced_map_iff [Fintype J] (ω : List J) :
+    cs.IsReduced (ω.map Subtype.val) ↔ (system (M := M) J).IsReduced ω := by
+  simp only [CoxeterSystem.IsReduced, List.length_map, ← inclusion_word, length_inclusion]
+
+end
+end KLInvariance.StandardParabolic
+
+namespace KLInvariance.StandardParabolic
+open CoxeterSystem
+universe u v
+variable {I : Type u} {W : Type v} [Group W] {M : CoxeterMatrix I}
+variable (cs : CoxeterSystem M W) (J : Set I) [Fintype J]
+noncomputable section
+
+theorem bruhat_inclusion_iff (x y : (matrix (M := M) J).Group) :
+    BruhatLE cs (inclusion cs J x) (inclusion cs J y) ↔
+      BruhatLE (system (M := M) J) x y := by
+  obtain ⟨ω, hω, rfl⟩ := (system (M := M) J).exists_isReduced y
+  rw [inclusion_word]
+  constructor
+  · intro h
+    obtain ⟨ν, hsub, _, hprod⟩ := (bruhat_iff_reduced_subword cs
+      ((reduced_map_iff cs J ω).2 hω)).1 h
+    obtain ⟨τ, hτ, rfl⟩ := List.sublist_map_iff.mp hsub
+    have hx : (system (M := M) J).wordProd τ = x :=
+      inclusion_injective cs J ((inclusion_word cs J τ).trans hprod)
+    rw [← hx]
+    exact bruhat_of_subword _ hτ hω
+  · intro h
+    obtain ⟨ν, hsub, _, rfl⟩ := (bruhat_iff_reduced_subword _ hω).1 h
+    rw [inclusion_word]
+    exact bruhat_of_subword cs (hsub.map Subtype.val) ((reduced_map_iff cs J ω).2 hω)
+
+theorem lower_range (y : (matrix (M := M) J).Group) {z : W}
+    (hzy : BruhatLE cs z (inclusion cs J y)) :
+    ∃ x, inclusion cs J x = z := by
+  obtain ⟨ω, hω, rfl⟩ := (system (M := M) J).exists_isReduced y
+  rw [inclusion_word] at hzy
+  obtain ⟨ν, hsub, _, hprod⟩ := (bruhat_iff_reduced_subword cs
+    ((reduced_map_iff cs J ω).2 hω)).1 hzy
+  obtain ⟨τ, _, rfl⟩ := List.sublist_map_iff.mp hsub
+  exact ⟨(system (M := M) J).wordProd τ, (inclusion_word cs J τ).trans hprod⟩
+
+def intervalMap (x y : (matrix (M := M) J).Group) :
+    Interval (system (M := M) J) x y →
+      Interval cs (inclusion cs J x) (inclusion cs J y) :=
+  fun z => ⟨inclusion cs J z.val, (bruhat_inclusion_iff cs J x z.val).2 z.property.1,
+    (bruhat_inclusion_iff cs J z.val y).2 z.property.2⟩
+
+lemma intervalMap_bijective (x y : (matrix (M := M) J).Group) :
+    Function.Bijective (intervalMap cs J x y) := by
+  constructor
+  · intro z t h
+    exact Subtype.ext (inclusion_injective cs J (congrArg Subtype.val h))
+  · intro z
+    obtain ⟨w, hw⟩ := lower_range cs J y z.property.2
+    refine ⟨⟨w, ?_, ?_⟩, Subtype.ext hw⟩
+    · apply (bruhat_inclusion_iff cs J x w).1
+      rw [hw]
+      exact z.property.1
+    · apply (bruhat_inclusion_iff cs J w y).1
+      rw [hw]
+      exact z.property.2
+
+/-- The exact unlabelled interval identification used in the finite-rank
+reduction; it arises from a proved embedding, not an added main hypothesis. -/
+def intervalIso (x y : (matrix (M := M) J).Group) :
+    Interval (system (M := M) J) x y ≃o
+      Interval cs (inclusion cs J x) (inclusion cs J y) where
+  toEquiv := Equiv.ofBijective _ (intervalMap_bijective cs J x y)
+  map_rel_iff' := by
+    intro z t
+    exact bruhat_inclusion_iff cs J z.val t.val
+
+@[simp] theorem intervalIso_val (x y : (matrix (M := M) J).Group)
+    (z : Interval (system (M := M) J) x y) :
+    (intervalIso cs J x y z).val = inclusion cs J z.val := rfl
+
+lemma rankDifference_inclusion (x y : (matrix (M := M) J).Group) :
+    rankDifference cs (inclusion cs J x) (inclusion cs J y) =
+      rankDifference (system (M := M) J) x y := by
+  simp only [rankDifference, length_inclusion]
+
+/-- Restriction of the actual normalized families satisfies their exact
+recurrence, degree bound and interval reciprocity in the standard parabolic. -/
+theorem normalized_restriction {RP : PolynomialFamilies W} (hRP : NormalizedKL cs RP) :
+    NormalizedKL (system (M := M) J)
+      (fun x y => RP.1 (inclusion cs J x) (inclusion cs J y),
+       fun x y => RP.2 (inclusion cs J x) (inclusion cs J y)) where
+  R_diagonal x := hRP.R_diagonal _
+  R_zero x y hxy := hRP.R_zero _ _ (fun h => hxy ((bruhat_inclusion_iff cs J x y).1 h))
+  R_recursion := by
+    intro x y i hy
+    have hs (z : (matrix (M := M) J).Group) :
+        inclusion cs J ((system (M := M) J).simple i * z) =
+          cs.simple i.val * inclusion cs J z := by rw [map_mul, inclusion_simple]
+    have hl (z : (matrix (M := M) J).Group) :
+        cs.length (cs.simple i.val * inclusion cs J z) =
+          (system (M := M) J).length ((system (M := M) J).simple i * z) := by
+      rw [← hs, length_inclusion]
+    have h := hRP.R_recursion (inclusion cs J x) (inclusion cs J y) i.val
+      (by rwa [hl, length_inclusion])
+    simpa only [hs, hl, length_inclusion] using h
+  P_diagonal x := hRP.P_diagonal _
+  P_zero x y hxy := hRP.P_zero _ _ (fun h => hxy ((bruhat_inclusion_iff cs J x y).1 h))
+  P_degree := by
+    intro x y hxy hne
+    have h := hRP.P_degree (inclusion cs J x) (inclusion cs J y)
+      ((bruhat_inclusion_iff cs J x y).2 hxy) (fun h => hne (inclusion_injective cs J h))
+    rwa [rankDifference_inclusion] at h
+  reciprocity := by
+    intro x y hxy
+    have h := hRP.reciprocity (inclusion cs J x) (inclusion cs J y)
+      ((bruhat_inclusion_iff cs J x y).2 hxy)
+    rw [rankDifference_inclusion] at h
+    rw [h]
+    let := interval_finite cs (inclusion cs J x) (inclusion cs J y)
+    let := interval_finite (system (M := M) J) x y
+    let := Fintype.ofFinite (Interval cs (inclusion cs J x) (inclusion cs J y))
+    let := Fintype.ofFinite (Interval (system (M := M) J) x y)
+    simp only [finsum_eq_sum_of_fintype]
+    exact ((intervalIso cs J x y).toEquiv.sum_comp
+      (fun z => RP.1 (inclusion cs J x) z.val * RP.2 z.val (inclusion cs J y))).symm
+
+theorem klFamilies_inclusion :
+    (fun x y => rPolynomial cs (inclusion cs J x) (inclusion cs J y),
+     fun x y => klPolynomial cs (inclusion cs J x) (inclusion cs J y)) =
+      klFamilies (system (M := M) J) :=
+  normalizedKL_unique _ (normalized_restriction cs J (klFamilies_spec cs))
+    (klFamilies_spec _)
+
+theorem klPolynomial_inclusion (x y : (matrix (M := M) J).Group) :
+    klPolynomial cs (inclusion cs J x) (inclusion cs J y) =
+      klPolynomial (system (M := M) J) x y :=
+  congrFun (congrFun (congrArg Prod.snd (klFamilies_inclusion cs J)) x) y
+
+theorem rPolynomial_inclusion (x y : (matrix (M := M) J).Group) :
+    rPolynomial cs (inclusion cs J x) (inclusion cs J y) =
+      rPolynomial (system (M := M) J) x y :=
+  congrFun (congrFun (congrArg Prod.fst (klFamilies_inclusion cs J)) x) y
+
+end
+end KLInvariance.StandardParabolic
+
+namespace KLInvariance.TitsSpace
+open Module
+universe u v
+variable {I : Type u} [Fintype I] (M : CoxeterMatrix I)
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+/-- A genuine real root, obtained from a simple root by the Tits action. -/
+def IsRoot (a : I → ℝ) : Prop := ∃ w : W, ∃ i : I, a = action M cs w (root i)
+
+/-- A normalized real root representing a genuine Coxeter reflection. -/
+def Represents (t : W) (a : I → ℝ) : Prop :=
+  ∃ w : W, ∃ i : I, t = w * cs.simple i * w⁻¹ ∧ a = action M cs w (root i)
+
+abbrev embed (a : I → ℝ) : Extended M := (a, 0)
+
+@[simp] theorem form_root_self (i : I) : form M (root i) (root i) = 2 := by
+  simp []
+
+variable {M cs}
+
+theorem IsRoot.norm {a : I → ℝ} (ha : IsRoot M cs a) : form M a a = 2 := by
+  obtain ⟨w,i,rfl⟩ := ha
+  rw [action_preserves, form_root_self]
+
+theorem IsRoot.ne_zero {a : I → ℝ} (ha : IsRoot M cs a) : a ≠ 0 := by
+  intro he
+  have hn := ha.norm
+  simp [he] at hn
+
+theorem IsRoot.sign {a : I → ℝ} (ha : IsRoot M cs a) :
+    (∀ i, 0 ≤ a i) ∨ (∀ i, a i ≤ 0) := by
+  obtain ⟨w,i,rfl⟩ := ha
+  rcases lt_or_gt_of_ne (cs.length_mul_simple_ne w i) with h|h
+  · exact Or.inr (action_root_nonpos_of_descent M cs w i h)
+  · exact Or.inl (action_root_nonneg_of_ascent M cs w i (by
+      rcases cs.length_mul_simple w i with h'|h' <;> omega))
+
+theorem IsRoot.neg {a : I → ℝ} (ha : IsRoot M cs a) : IsRoot M cs (-a) := by
+  obtain ⟨w,i,rfl⟩ := ha
+  refine ⟨w * cs.simple i, i, ?_⟩
+  simp only [map_mul, LinearEquiv.mul_apply, action_simple_root, map_neg]
+
+theorem IsRoot.act {a : I → ℝ} (ha : IsRoot M cs a) (w : W) :
+    IsRoot M cs (action M cs w a) := by
+  obtain ⟨z,i,rfl⟩ := ha
+  exact ⟨w*z, i, by simp only [map_mul, LinearEquiv.mul_apply]⟩
+
+theorem Represents.isRoot {t : W} {a : I → ℝ} (ha : Represents M cs t a) :
+    IsRoot M cs a := by
+  obtain ⟨w,i,_,h⟩ := ha
+  exact ⟨w,i,h⟩
+
+theorem Represents.isReflection {t : W} {a : I → ℝ} (ha : Represents M cs t a) :
+    cs.IsReflection t := by
+  obtain ⟨w,i,h,_⟩ := ha
+  exact ⟨w,i,h⟩
+
+theorem Represents.neg {t : W} {a : I → ℝ} (ha : Represents M cs t a) :
+    Represents M cs t (-a) := by
+  obtain ⟨w,i,rfl,rfl⟩ := ha
+  refine ⟨w * cs.simple i, i, ?_, ?_⟩
+  · simp [mul_assoc]
+  · simp only [map_mul, LinearEquiv.mul_apply, action_simple_root, map_neg]
+
+theorem Represents.act {t : W} {a : I → ℝ} (ha : Represents M cs t a) (w : W) :
+    Represents M cs (w*t*w⁻¹) (action M cs w a) := by
+  obtain ⟨z,i,rfl,rfl⟩ := ha
+  refine ⟨w*z,i,?_,?_⟩
+  · group
+  · simp only [map_mul, LinearEquiv.mul_apply]
+
+/-- The actual reflection formula for every conjugate root in the minimal
+nondegenerate extension, not merely for simple roots. -/
+theorem Represents.formula {t : W} {a : I → ℝ} (ha : Represents M cs t a)
+    (v : Extended M) :
+    extendedAction M cs t v = v - extendedForm M (embed M a) v • embed M a := by
+  obtain ⟨w,i,rfl,rfl⟩ := ha
+  have hv : extendedAction M cs w (extendedAction M cs w⁻¹ v) = v := by
+    rw [← LinearEquiv.mul_apply, ← map_mul, mul_inv_cancel, map_one]
+    rfl
+  have hp := extendedAction_preserves M cs w (extendedRoot M i)
+    (extendedAction M cs w⁻¹ v)
+  rw [hv] at hp
+  rw [map_mul, map_mul]
+  simp only [LinearEquiv.mul_apply, extendedAction_simple, map_sub, map_smul]
+  rw [hv, ← hp, extendedRoot, extendedAction_restrict]
+
+theorem Represents.action_formula {t : W} {a : I → ℝ} (ha : Represents M cs t a)
+    (v : I → ℝ) : action M cs t v = v - form M a v • a := by
+  have h := congrArg Prod.fst (ha.formula (embed M v))
+  simpa only [extendedAction_restrict, embed, (extendedForm_spec M).2.2,
+    Prod.fst_sub, Prod.smul_fst] using h
+
+theorem Represents.unique_up_to_sign {t : W} {a b : I → ℝ}
+    (ha : Represents M cs t a) (hb : Represents M cs t b) : a = b ∨ a = -b := by
+  have hn := ha.isRoot.norm
+  have hm := hb.isRoot.norm
+  have hf := ha.action_formula a
+  rw [hb.action_formula a, hn] at hf
+  have hmul : form M b a • b = (2 : ℝ) • a := by
+    have h := congrArg (fun z => a-z) hf
+    simpa only [sub_sub_cancel] using h
+  have hab : a = (form M b a / 2) • b := by
+    calc
+      a = (1/2 : ℝ) • ((2 : ℝ) • a) := by module
+      _ = (1/2 : ℝ) • (form M b a • b) := by rw [hmul]
+      _ = _ := by module
+  have hc : (form M b a / 2) ^ 2 = 1 := by
+    rw [hab] at hn
+    simp only [map_smul, LinearMap.smul_apply, smul_eq_mul, hm] at hn
+    nlinarith
+  rcases sq_eq_one_iff.mp hc with h|h
+  · exact Or.inl (by simpa [h] using hab)
+  · exact Or.inr (by simpa [h] using hab)
+
+theorem Represents.reflection_unique {t s : W} {a : I → ℝ}
+    (ha : Represents M cs t a) (hb : Represents M cs s a) : t = s := by
+  apply extendedAction_injective M cs
+  apply LinearEquiv.ext
+  intro v
+  rw [ha.formula, hb.formula]
+
+theorem Represents.unique_positive {t : W} {a b : I → ℝ}
+    (ha : Represents M cs t a) (hb : Represents M cs t b)
+    (hpa : ∀ i, 0 ≤ a i) (hpb : ∀ i, 0 ≤ b i) : a = b := by
+  rcases ha.unique_up_to_sign hb with h|h
+  · exact h
+  · exfalso
+    apply ha.isRoot.ne_zero
+    ext i
+    have hi := congrFun h i
+    simp only [Pi.neg_apply] at hi
+    have := hpa i
+    have := hpb i
+    simp only [Pi.zero_apply]
+    linarith
+
+variable (M) (cs)
+
+theorem exists_positive_root {t : W} (ht : cs.IsReflection t) :
+    ∃ a : I → ℝ, Represents M cs t a ∧ ∀ i, 0 ≤ a i := by
+  obtain ⟨w,i,rfl⟩ := ht
+  have hr : Represents M cs (w * cs.simple i * w⁻¹) (action M cs w (root i)) :=
+    ⟨w,i,rfl,rfl⟩
+  rcases hr.isRoot.sign with h|h
+  · exact ⟨_,hr,h⟩
+  · exact ⟨_,hr.neg,fun i => by simpa only [Pi.neg_apply, neg_nonneg] using h i⟩
+
+/-- Genuine positive roots, with their actual coordinate sign. -/
+def PositiveRoot := {a : I → ℝ // IsRoot M cs a ∧ ∀ i, 0 ≤ a i}
+
+def positiveRoot (t : {t : W // cs.IsReflection t}) : PositiveRoot M cs :=
+  ⟨(exists_positive_root M cs t.property).choose,
+    (exists_positive_root M cs t.property).choose_spec.1.isRoot,
+    (exists_positive_root M cs t.property).choose_spec.2⟩
+
+theorem positiveRoot_represents (t : {t : W // cs.IsReflection t}) :
+    Represents M cs t.val (positiveRoot M cs t).val :=
+  (exists_positive_root M cs t.property).choose_spec.1
+
+def rootReflection (a : PositiveRoot M cs) : {t : W // cs.IsReflection t} := by
+  let w := a.property.1.choose
+  let i := a.property.1.choose_spec.choose
+  exact ⟨w * cs.simple i * w⁻¹, w,i,rfl⟩
+
+theorem rootReflection_represents (a : PositiveRoot M cs) :
+    Represents M cs (rootReflection M cs a).val a.val := by
+  refine ⟨a.property.1.choose, a.property.1.choose_spec.choose, rfl, ?_⟩
+  exact a.property.1.choose_spec.choose_spec
+
+
+def positiveRootEquiv : {t : W // cs.IsReflection t} ≃ PositiveRoot M cs where
+  toFun := positiveRoot M cs
+  invFun := rootReflection M cs
+  left_inv t := Subtype.ext ((rootReflection_represents M cs _).reflection_unique
+      (positiveRoot_represents M cs t))
+  right_inv a := Subtype.ext ((positiveRoot_represents M cs _).unique_positive
+      (rootReflection_represents M cs a) (positiveRoot M cs _).property.2 a.property.2)
+
+end
+end KLInvariance.TitsSpace
+
+namespace KLInvariance.TitsSpace
+open Module
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+noncomputable section
+
+theorem IsRoot.not_both {a : I → ℝ} (ha : IsRoot M cs a)
+    (hp : ∀ i, 0 ≤ a i) (hn : ∀ i, a i ≤ 0) : False := by
+  apply ha.ne_zero
+  ext i
+  exact le_antisymm (hn i) (hp i)
+
+/-- A simple reflection can turn a positive real root negative only on its
+own root. This uses actual Tits coordinates and normalization. -/
+theorem IsRoot.eq_simple_of_nonpos {a : I → ℝ} (ha : IsRoot M cs a)
+    (hp : ∀ k, 0 ≤ a k) (i : I)
+    (hn : ∀ k, action M cs (cs.simple i) a k ≤ 0) : a = root i := by
+  classical
+  have ha0 (k : I) (hk : k ≠ i) : a k = 0 := by
+    have h := hn k
+    simp only [action_simple, Pi.sub_apply, Pi.smul_apply, root,
+      Pi.single_apply, hk, ↓reduceIte, smul_eq_mul, mul_zero, sub_zero] at h
+    exact le_antisymm h (hp k)
+  have heq : a = a i • root i := by
+    ext k
+    by_cases hki : k = i
+    · subst k; simp [root]
+    · simp [root, hki, ha0 k hki]
+  have hnorm := ha.norm
+  rw [heq] at hnorm
+  simp only [map_smul, LinearMap.smul_apply, smul_eq_mul, form_root_self] at hnorm
+  have hai : a i = 1 := by nlinarith [hp i]
+  simpa only [hai, one_smul] using heq
+
+theorem Represents.simple_sign {t : W} {a : I → ℝ} (ha : Represents M cs t a)
+    (hp : ∀ k, 0 ≤ a k) (i : I) :
+    ((∀ k, action M cs (cs.simple i) a k ≤ 0) ↔ t = cs.simple i) ∧
+    ((∀ k, 0 ≤ action M cs (cs.simple i) a k) ↔ t ≠ cs.simple i) := by
+  have hi : Represents M cs (cs.simple i) (root i) := ⟨1,i,by simp,by simp⟩
+  have hneg : (∀ k, action M cs (cs.simple i) a k ≤ 0) ↔ t = cs.simple i := by
+    constructor
+    · intro h
+      have heq := ha.isRoot.eq_simple_of_nonpos hp i h
+      exact ha.reflection_unique (heq.symm ▸ hi)
+    · intro h
+      have heq := (h ▸ ha).unique_positive hi hp (root_nonneg i)
+      simpa only [heq, action_simple_root, Pi.neg_apply, neg_nonpos] using root_nonneg i
+  refine ⟨hneg, ?_⟩
+  constructor
+  · intro h heq
+    exact (ha.isRoot.act (cs.simple i)).not_both h (hneg.mpr heq)
+  · intro h
+    rcases (ha.isRoot.act (cs.simple i)).sign with hp|hn
+    · exact hp
+    · exact False.elim (h (hneg.mp hn))
+
+/-- The signed combinatorial reflection action is the actual root sign
+under the faithful Tits representation. -/
+def SignedRepresents (p : W × ZMod 2) (a : I → ℝ) : Prop :=
+  Represents M cs p.1 a ∧
+    ((p.2 = 0 ∧ ∀ i, 0 ≤ a i) ∨ (p.2 = 1 ∧ ∀ i, a i ≤ 0))
+
+theorem SignedRepresents.simple {p : W × ZMod 2} {a : I → ℝ}
+    (ha : SignedRepresents (M := M) (cs := cs) p a) (i : I) :
+    SignedRepresents (M := M) (cs := cs) (CoxeterSupport.simpleAction cs i p)
+      (action M cs (cs.simple i) a) := by
+  rcases p with ⟨t,z⟩
+  rw [CoxeterSupport.simpleAction_apply]
+  refine ⟨?_,?_⟩
+  · simpa only [cs.inv_simple] using ha.1.act (cs.simple i)
+  · rcases ha.2 with ⟨rfl,hp⟩|⟨rfl,hn⟩
+    · by_cases heq : t = cs.simple i
+      · exact Or.inr ⟨by simp [heq], (ha.1.simple_sign hp i).1.mpr heq⟩
+      · exact Or.inl ⟨by simp [heq, ZModModule.add_self], (ha.1.simple_sign hp i).2.mpr heq⟩
+    · have hp : ∀ k, 0 ≤ (-a) k := fun k => neg_nonneg.mpr (hn k)
+      by_cases heq : t = cs.simple i
+      · refine Or.inl ⟨by simp [heq, ZModModule.add_self], ?_⟩
+        have h := (ha.1.neg.simple_sign hp i).1.mpr heq
+        simpa only [map_neg, Pi.neg_apply, neg_nonpos] using h
+      · refine Or.inr ⟨by simp [heq], ?_⟩
+        have h := (ha.1.neg.simple_sign hp i).2.mpr heq
+        simpa only [map_neg, Pi.neg_apply, neg_nonneg] using h
+
+theorem SignedRepresents.act {p : W × ZMod 2} {a : I → ℝ}
+    (ha : SignedRepresents (M := M) (cs := cs) p a) (w : W) :
+    SignedRepresents (M := M) (cs := cs) (CoxeterSupport.reflectionAction cs w p)
+      (action M cs w a) := by
+  obtain ⟨ω,rfl⟩ := cs.wordProd_surjective w
+  induction ω with
+  | nil => simpa using ha
+  | cons i ω ih =>
+    simpa only [cs.wordProd_cons, map_mul, Equiv.Perm.mul_apply,
+      CoxeterSupport.reflectionAction_simple, LinearEquiv.mul_apply] using ih.simple i
+
+
+theorem Represents.inversion_iff {t : W} {a : I → ℝ} (ha : Represents M cs t a)
+    (hp : ∀ i, 0 ≤ a i) (w : W) :
+    (∀ i, action M cs w⁻¹ a i ≤ 0) ↔ cs.IsLeftInversion w t := by
+  classical
+  obtain ⟨ω,hω,heq⟩ := cs.exists_isReduced w
+  rw [heq]
+  let c : ZMod 2 := (cs.leftInvSeq ω).count t
+  have hc : c+c=0 := ZModModule.add_self c
+  have hrec := CoxeterSupport.wordAction_record cs ω t c
+  rw [← CoxeterSupport.reflectionAction_wordProd] at hrec
+  have hrec' : CoxeterSupport.reflectionAction cs (cs.wordProd ω)
+      ((cs.wordProd ω)⁻¹ * t * cs.wordProd ω, c) = (t,0) := by
+    change CoxeterSupport.reflectionAction cs (cs.wordProd ω)
+      ((cs.wordProd ω)⁻¹ * t * cs.wordProd ω,c) = (t,c+c) at hrec
+    rwa [hc] at hrec
+  have hinv : CoxeterSupport.reflectionAction cs (cs.wordProd ω)⁻¹ (t,0) =
+      ((cs.wordProd ω)⁻¹ * t * cs.wordProd ω,c) := by
+    rw [map_inv, ← hrec']
+    exact Equiv.symm_apply_apply _ _
+  have hs : SignedRepresents (M := M) (cs := cs) (t,0) a := ⟨ha,Or.inl ⟨rfl,hp⟩⟩
+  have hsi := hs.act (cs.wordProd ω)⁻¹
+  rw [hinv] at hsi
+  have hcount : c = if t ∈ cs.leftInvSeq ω then 1 else 0 := by
+    dsimp [c]
+    by_cases hm : t ∈ cs.leftInvSeq ω
+    · rw [ite_eq_left hm, List.count_eq_one_of_mem hω.nodup_leftInvSeq hm]
+      norm_num
+    · rw [ite_eq_right hm, List.count_eq_zero.mpr hm]
+      rfl
+  have hmem : (∀ i, action M cs (cs.wordProd ω)⁻¹ a i ≤ 0) ↔ t ∈ cs.leftInvSeq ω := by
+    by_cases hm : t ∈ cs.leftInvSeq ω
+    · have hc1 : c=1 := by simpa [hm] using hcount
+      simp only [hc1] at hsi
+      rcases hsi.2 with ⟨hbad,_⟩|⟨_,hn⟩
+      · norm_num at hbad
+      · exact iff_of_true hn hm
+    · have hc0 : c=0 := by simpa [hm] using hcount
+      simp only [hc0] at hsi
+      rcases hsi.2 with ⟨_,hpos⟩|⟨hbad,_⟩
+      · exact iff_of_false (fun hn => (ha.isRoot.act _).not_both hpos hn) hm
+      · norm_num at hbad
+  rw [hmem]
+  exact ⟨cs.isLeftInversion_of_mem_leftInvSeq hω,
+    CoxeterSupport.mem_leftInvSeq_of_inversion cs hω⟩
+
+end
+end KLInvariance.TitsSpace
+
+namespace KLInvariance.TitsSpace
+open Module
+universe u v
+variable {I : Type u} [Fintype I] {M : CoxeterMatrix I}
+  {W : Type v} [Group W] {cs : CoxeterSystem M W}
+noncomputable section
+
+theorem Represents.independent_of_ne {t s : W} {a b : I → ℝ}
+    (ha : Represents M cs t a) (hb : Represents M cs s b) (hts : t ≠ s) :
+    LinearIndependent ℝ ![a,b] := by
+  apply (LinearIndependent.pair_iff' ha.isRoot.ne_zero).mpr
+  intro c heq
+  have hn := hb.isRoot.norm
+  rw [← heq] at hn
+  simp only [map_smul, LinearMap.smul_apply, smul_eq_mul, ha.isRoot.norm] at hn
+  have hc : c^2 = 1 := by nlinarith
+  rcases sq_eq_one_iff.mp hc with hc|hc
+  · have hab : a=b := by simpa only [hc,one_smul] using heq
+    exact hts (ha.reflection_unique (hab.symm ▸ hb))
+  · have hab : -a=b := by simpa only [hc,neg_one_smul] using heq
+    exact hts (ha.neg.reflection_unique (hab.symm ▸ hb))
+
+theorem Represents.extended_independent_of_ne {t s : W} {a b : I → ℝ}
+    (ha : Represents M cs t a) (hb : Represents M cs s b) (hts : t ≠ s) :
+    LinearIndependent ℝ ![embed M a,embed M b] := by
+  have h := (ha.independent_of_ne hb hts).map'
+    (LinearMap.inl ℝ (I → ℝ) (Module.Dual ℝ (LinearMap.ker (form M))))
+    (LinearMap.ker_eq_bot.mpr LinearMap.inl_injective)
+  have heq : (⇑(LinearMap.inl ℝ (I → ℝ) (Module.Dual ℝ (LinearMap.ker (form M)))) ∘
+      ![a,b]) = ![embed M a,embed M b] := by
+    funext i
+    fin_cases i <;> rfl
+  rwa [heq] at h
+
+theorem Represents.extendedAction_eq {t : W} {a : I → ℝ} (ha : Represents M cs t a) :
+    extendedAction M cs t = Module.reflection
+      (show extendedForm M (embed M a) (embed M a) = 2 by
+        rw [(extendedForm_spec M).2.2]; exact ha.isRoot.norm) := by
+  apply LinearEquiv.ext
+  intro v
+  exact ha.formula v
+
+
+theorem Represents.plane_eq_of_product_eq {t s r q : W} {a b c d : I → ℝ}
+    (ha : Represents M cs t a) (hb : Represents M cs s b)
+    (hc : Represents M cs r c) (hd : Represents M cs q d)
+    (hts : t ≠ s) (hrq : r ≠ q) (heq : t*s=r*q) :
+    Submodule.span ℝ {embed M a, embed M b} =
+      Submodule.span ℝ {embed M c, embed M d} := by
+  apply ReflectionPlane.plane_eq_of_reflection_product_eq (extendedForm M)
+    (extendedForm_spec M).2.1 (embed M a) (embed M b) (embed M c) (embed M d)
+    (by rw [(extendedForm_spec M).2.2]; exact ha.isRoot.norm)
+    (by rw [(extendedForm_spec M).2.2]; exact hb.isRoot.norm)
+    (by rw [(extendedForm_spec M).2.2]; exact hc.isRoot.norm)
+    (by rw [(extendedForm_spec M).2.2]; exact hd.isRoot.norm)
+    (ha.extended_independent_of_ne hb hts) (hc.extended_independent_of_ne hd hrq)
+  have h := congrArg (extendedAction M cs) heq
+  rw [map_mul, map_mul, ha.extendedAction_eq, hb.extendedAction_eq,
+    hc.extendedAction_eq, hd.extendedAction_eq] at h
+  exact congrArg LinearEquiv.toLinearMap h
+
+variable (M) (cs)
+/-- The actual positive ambient roots inverted by `w`. -/
+def RootInversions (w : W) :=
+  {a : PositiveRoot M cs // ∀ i, action M cs w⁻¹ a.val i ≤ 0}
+
+def rootInversionEquiv (w : W) : RootInversions M cs w ≃ {t : W // cs.IsLeftInversion w t} where
+  toFun a := ⟨(rootReflection M cs a.val).val,
+    ((rootReflection_represents M cs a.val).inversion_iff a.val.property.2 w).mp a.property⟩
+  invFun t := ⟨positiveRoot M cs ⟨t.val,t.property.1⟩,
+    ((positiveRoot_represents M cs ⟨t.val,t.property.1⟩).inversion_iff
+      (positiveRoot M cs ⟨t.val,t.property.1⟩).property.2 w).mpr t.property⟩
+  left_inv a := by
+    apply Subtype.ext
+    exact (positiveRootEquiv M cs).apply_symm_apply a.val
+  right_inv t := by
+    apply Subtype.ext
+    change (rootReflection M cs (positiveRoot M cs ⟨t.val,t.property.1⟩)).val = t.val
+    exact congrArg (fun r : {r : W // cs.IsReflection r} => r.val)
+      ((positiveRootEquiv M cs).symm_apply_apply ⟨t.val,t.property.1⟩)
+
+def leftInversionListEquiv {ω : List I} (hω : cs.IsReduced ω) :
+    {t : W // cs.IsLeftInversion (cs.wordProd ω) t} ≃ {t : W // t ∈ cs.leftInvSeq ω} where
+  toFun t := ⟨t.val,CoxeterSupport.mem_leftInvSeq_of_inversion cs hω t.property⟩
+  invFun t := ⟨t.val,cs.isLeftInversion_of_mem_leftInvSeq hω t.property⟩
+  left_inv _ := rfl
+  right_inv _ := rfl
+
+/-- Full ambient inversion count, not a cardinality assumption. -/
+theorem rootInversions_finite (w : W) : Finite (RootInversions M cs w) := by
+  classical
+  obtain ⟨ω,hω,heq⟩ := cs.exists_isReduced w
+  rw [heq]
+  exact Finite.of_equiv (Fin (cs.leftInvSeq ω).length)
+    ((List.Nodup.getEquiv (cs.leftInvSeq ω) hω.nodup_leftInvSeq).trans
+      ((leftInversionListEquiv M cs hω).symm.trans (rootInversionEquiv M cs _).symm))
+
+theorem card_rootInversions (w : W) : Nat.card (RootInversions M cs w) = cs.length w := by
+  classical
+  obtain ⟨ω,hω,heq⟩ := cs.exists_isReduced w
+  rw [heq, Nat.card_congr (rootInversionEquiv M cs _),
+    Nat.card_congr (leftInversionListEquiv M cs hω),
+    ← Nat.card_congr (List.Nodup.getEquiv (cs.leftInvSeq ω) hω.nodup_leftInvSeq),
+    Nat.card_fin, cs.length_leftInvSeq, hω.eq]
+
+end
+end KLInvariance.TitsSpace
+
+namespace KLInvariance.TitsSpace
+open Module
+universe u v
+variable {I : Type u} [Fintype I] (M : CoxeterMatrix I)
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+/-- Elements whose actual linear displacement lies in `P`. This is a
+subgroup even when the invariant bilinear form restricted to `P` is singular. -/
+def movingSubgroup (P : Submodule ℝ (I → ℝ)) : Subgroup W where
+  carrier := {w | ∀ v, action M cs w v - v ∈ P}
+  one_mem' := by intro v; simp
+  mul_mem' := by
+    intro w z hw hz v
+    have h := P.add_mem (hw (action M cs z v)) (hz v)
+    simpa only [map_mul, LinearEquiv.mul_apply, sub_add_sub_cancel] using h
+  inv_mem' := by
+    intro w hw v
+    have h := P.neg_mem (hw (action M cs w⁻¹ v))
+    have hv : action M cs w (action M cs w⁻¹ v) = v := by
+      rw [← LinearEquiv.mul_apply, ← map_mul, mul_inv_cancel, map_one]
+      rfl
+    simpa only [hv, neg_sub] using h
+
+/-- The manuscript's maximal plane reflection subgroup, defined using
+actual normalized roots, not by assuming a rank-two presentation. -/
+def planeSubgroup (P : Submodule ℝ (I → ℝ)) : Subgroup W :=
+  Subgroup.closure {t | ∃ a, Represents M cs t a ∧ a ∈ P}
+
+theorem planeSubgroup_le_moving (P : Submodule ℝ (I → ℝ)) :
+    planeSubgroup M cs P ≤ movingSubgroup M cs P := by
+  apply (Subgroup.closure_le (movingSubgroup M cs P)).mpr
+  rintro t ⟨a,ha,haP⟩ v
+  rw [ha.action_formula]
+  have heq : v - form M a v • a - v = (-form M a v) • a := by module
+  rw [heq]
+  exact P.smul_mem _ haP
+
+theorem Represents.mem_of_moving {P : Submodule ℝ (I → ℝ)} {t : W} {a : I → ℝ}
+    (ha : Represents M cs t a) (ht : t ∈ movingSubgroup M cs P) : a ∈ P := by
+  have h := ht a
+  change action M cs t a - a ∈ P at h
+  rw [ha.action_formula,ha.isRoot.norm] at h
+  have heq : a - (2 : ℝ) • a - a = (-2 : ℝ) • a := by module
+  rw [heq] at h
+  have h' := P.smul_mem (-1/2 : ℝ) h
+  have heq' : (-1/2 : ℝ) • ((-2 : ℝ) • a) = a := by module
+  rwa [heq'] at h'
+
+
+theorem Represents.planeSubgroup_mem_iff {P : Submodule ℝ (I → ℝ)} {t : W} {a : I → ℝ}
+    (ha : Represents M cs t a) : t ∈ planeSubgroup M cs P ↔ a ∈ P := by
+  constructor
+  · intro ht
+    exact Represents.mem_of_moving M cs ha (planeSubgroup_le_moving M cs P ht)
+  · intro haP
+    exact Subgroup.subset_closure ⟨a,ha,haP⟩
+
+/-- Two ambient vertices in a left coset of this plane subgroup can have
+no reflection edge except one whose actual root is in the plane. -/
+theorem reflection_between_coset_mem {P : Submodule ℝ (I → ℝ)} {t g h a : W}
+    (hg : g ∈ planeSubgroup M cs P) (hh : h ∈ planeSubgroup M cs P)
+    (heq : h*a=t*(g*a)) {r : I → ℝ} (ht : Represents M cs t r) : r ∈ P := by
+  have htg : t*g=h := by apply mul_right_cancel (b := a); simpa only [mul_assoc] using heq.symm
+  have htG : t ∈ planeSubgroup M cs P := by
+    have heq' : t=h*g⁻¹ := by rw [← htg]; group
+    rw [heq']
+    exact (planeSubgroup M cs P).mul_mem hh ((planeSubgroup M cs P).inv_mem hg)
+  exact (Represents.planeSubgroup_mem_iff M cs ht).mp htG
+
+end
+end KLInvariance.TitsSpace
+
+namespace KLInvariance.TitsSpace
+open Module
+universe u v
+variable {I : Type u} [Fintype I] (M : CoxeterMatrix I)
+  {W : Type v} [Group W] (cs : CoxeterSystem M W)
+noncomputable section
+
+/-- Least ambient length in each actual left plane-subgroup coset. No
+uniqueness or rank-two Coxeter presentation is smuggled into this definition. -/
+theorem planeCoset_exists_minimal (P : Submodule ℝ (I → ℝ)) (a : W) :
+    ∃ a₀ g : W, g ∈ planeSubgroup M cs P ∧ a = g*a₀ ∧
+      ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀) := by
+  classical
+  have hex : ∃ n : ℕ, ∃ a₀ g : W, g ∈ planeSubgroup M cs P ∧
+      a = g*a₀ ∧ cs.length a₀ = n :=
+    ⟨cs.length a,a,1,(planeSubgroup M cs P).one_mem,by simp,rfl⟩
+  obtain ⟨a₀,g,hg,ha,hlen⟩ := Nat.find_spec hex
+  refine ⟨a₀,g,hg,ha,fun z hz => ?_⟩
+  rw [hlen]
+  apply Nat.find_min' hex
+  refine ⟨z*a₀,g*z⁻¹,(planeSubgroup M cs P).mul_mem hg
+    ((planeSubgroup M cs P).inv_mem hz),?_,rfl⟩
+  simpa [mul_assoc] using ha
+
+variable {M cs}
+
+theorem planeSubgroup_preserves {P : Submodule ℝ (I → ℝ)} {g : W}
+    (hg : g ∈ planeSubgroup M cs P) {a : I → ℝ} (ha : a ∈ P) :
+    action M cs g a ∈ P := by
+  have h := (planeSubgroup_le_moving M cs P hg) a
+  simpa only [sub_add_cancel] using P.add_mem h ha
+
+/-- A least coset representative sends every positive plane root to a
+positive ambient root under its inverse, by the proved inversion criterion. -/
+theorem root_nonneg_of_coset_minimal {P : Submodule ℝ (I → ℝ)} {a₀ : W}
+    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+    {a : I → ℝ} (ha : IsRoot M cs a) (hp : ∀ i, 0 ≤ a i) (haP : a ∈ P) :
+    ∀ i, 0 ≤ action M cs a₀⁻¹ a i := by
+  let r : PositiveRoot M cs := ⟨a,ha,hp⟩
+  have hr := rootReflection_represents M cs r
+  have htP := (Represents.planeSubgroup_mem_iff M cs hr).mpr haP
+  rcases (ha.act a₀⁻¹).sign with hpos|hneg
+  · exact hpos
+  · have hi := (hr.inversion_iff hp a₀).mp hneg
+    exact (not_lt_of_ge (hmin _ htP) hi.2).elim
+
+
+theorem root_nonpos_coset_minimal_iff {P : Submodule ℝ (I → ℝ)} {a₀ : W}
+    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+    {a : I → ℝ} (ha : IsRoot M cs a) (haP : a ∈ P) :
+    (∀ i, action M cs a₀⁻¹ a i ≤ 0) ↔ ∀ i, a i ≤ 0 := by
+  rcases ha.sign with hp|hn
+  · have hnew := root_nonneg_of_coset_minimal hmin ha hp haP
+    exact iff_of_false ((ha.act _).not_both hnew) (ha.not_both hp)
+  · have hnp : ∀ i, 0 ≤ (-a) i := fun i => neg_nonneg.mpr (hn i)
+    have hh := root_nonneg_of_coset_minimal hmin ha.neg hnp (P.neg_mem haP)
+    have hnew : ∀ i, action M cs a₀⁻¹ a i ≤ 0 := by
+      intro i
+      simpa only [map_neg, Pi.neg_apply, neg_nonneg] using hh i
+    exact iff_of_true hnew hn
+
+theorem root_inversion_coset_iff {P : Submodule ℝ (I → ℝ)} {a₀ g : W}
+    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+    (hg : g ∈ planeSubgroup M cs P) {a : I → ℝ} (ha : IsRoot M cs a)
+    (haP : a ∈ P) :
+    (∀ i, action M cs (g*a₀)⁻¹ a i ≤ 0) ↔
+      ∀ i, action M cs g⁻¹ a i ≤ 0 := by
+  rw [mul_inv_rev, map_mul]
+  change (∀ i, action M cs a₀⁻¹ (action M cs g⁻¹ a) i ≤ 0) ↔ _
+  exact root_nonpos_coset_minimal_iff hmin (ha.act _)
+    (planeSubgroup_preserves ((planeSubgroup M cs P).inv_mem hg) haP)
+
+theorem leftInversion_coset_iff {P : Submodule ℝ (I → ℝ)} {a₀ g t : W}
+    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+    (hg : g ∈ planeSubgroup M cs P) (ht : t ∈ planeSubgroup M cs P)
+    (hr : cs.IsReflection t) :
+    cs.IsLeftInversion (g*a₀) t ↔ cs.IsLeftInversion g t := by
+  let a := positiveRoot M cs ⟨t,hr⟩
+  have ha := positiveRoot_represents M cs ⟨t,hr⟩
+  rw [← ha.inversion_iff a.property.2, ← ha.inversion_iff a.property.2]
+  exact root_inversion_coset_iff hmin hg a.property.1
+    ((Represents.planeSubgroup_mem_iff M cs ha).mp ht)
+
+/-- The directed ambient reflection graph on any plane-subgroup coset is
+exactly transported from that on the subgroup by a least coset representative.
+Identifying this graph with a *dihedral Coxeter* graph still requires the
+canonical rank-two reflection-subgroup theorem. -/
+theorem bruhatStep_coset_iff {P : Submodule ℝ (I → ℝ)} {a₀ g h : W}
+    (hmin : ∀ z ∈ planeSubgroup M cs P, cs.length a₀ ≤ cs.length (z*a₀))
+    (hg : g ∈ planeSubgroup M cs P) (hh : h ∈ planeSubgroup M cs P) :
+    BruhatStep cs (g*a₀) (h*a₀) ↔ BruhatStep cs g h := by
+  have hn : ∀ {t : W}, h=t*g → cs.IsReflection t →
+      (cs.length (g*a₀) < cs.length (h*a₀) ↔ cs.length g < cs.length h) := by
+    intro t heq ht
+    have htP : t ∈ planeSubgroup M cs P := by
+      have htgh : t=h*g⁻¹ := by rw [heq]; group
+      rw [htgh]
+      exact (planeSubgroup M cs P).mul_mem hh ((planeSubgroup M cs P).inv_mem hg)
+    have hi := leftInversion_coset_iff hmin hh htP ht
+    have hth : t*h=g := by rw [heq,← mul_assoc,ht.mul_self,one_mul]
+    simpa only [CoxeterSystem.IsLeftInversion, ht, true_and, ← mul_assoc, hth] using hi
+  constructor
+  · rintro ⟨hlen,t,ht,heq⟩
+    have htg : h=t*g := by apply mul_right_cancel (b := a₀); simpa only [mul_assoc] using heq
+    exact ⟨(hn htg ht).mp hlen,t,ht,htg⟩
+  · rintro ⟨hlen,t,ht,heq⟩
+    exact ⟨(hn heq ht).mpr hlen,t,ht,by rw [heq,mul_assoc]⟩
+
+end
+end KLInvariance.TitsSpace
+
+/-! Construction of the source's geometric reflection schedule. The moment
+curve avoids every forbidden hyperplane of a countable root set, using only
+finite polynomial root sets and uncountability of the real field. -/
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/AlternatingTabloids.lean b/lean/OAI/RepresentationTheory/PartialPermutation/AlternatingTabloids.lean
index c48d529bb5dd408a7cbea607f17ee1153dc3451f..c85a416ae4346c974fd10f00b8498867cfe41b35 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/AlternatingTabloids.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/AlternatingTabloids.lean
@@ -1,94 +1,94 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.ColumnSorting
-
-namespace OAI
-
-section
-namespace PartialPermutation
-namespace YoungTabloid
-noncomputable section
-open Finset
-open scoped Classical
-
-def signC {α : Type*} [Fintype α] [DecidableEq α] (g : Equiv.Perm α) : ℂ :=
-  (Equiv.Perm.sign g : ℤ)
-
-@[simp] lemma signC_one {α : Type*} [Fintype α] [DecidableEq α] : signC (1 : Equiv.Perm α)=1 := by
-  simp [signC]
-@[simp] lemma signC_mul {α : Type*} [Fintype α] [DecidableEq α] (g h : Equiv.Perm α) :
-    signC (g*h)=signC g*signC h := by simp [signC]
-@[simp] lemma signC_inv {α : Type*} [Fintype α] [DecidableEq α] (g : Equiv.Perm α) :
-    signC g⁻¹=signC g := by simp [signC]
-@[simp] lemma signC_sq {α : Type*} [Fintype α] [DecidableEq α] (g : Equiv.Perm α) :
-    signC g*signC g=1 := by
-  simpa using (signC_mul g g⁻¹).symm
-@[simp] lemma signC_swap {α : Type*} [Fintype α] [DecidableEq α] (x y : α) (h : x≠y) :
-    signC (Equiv.swap x y) = -1 := by simp [signC,Equiv.Perm.sign_swap h]
-
-lemma swap_mem_column (μ : YoungDiagram) (x y : μ.cells) (h : col μ x=col μ y) :
-    Equiv.swap x y ∈ columnGroup μ := by
-  intro z
-  by_cases hx : z=x
-  · subst z; simpa using h.symm
-  by_cases hy : z=y
-  · subst z; simpa using h
-  simp [Equiv.swap_apply_of_ne_of_ne hx hy]
-
-lemma relabel_swap_fix (μ : YoungDiagram) (r : Tabloid μ) (x y : μ.cells)
-    (h : r.1 x=r.1 y) : relabel μ (Equiv.swap x y) r=r := by
-  apply Subtype.ext
-  funext z
-  change r.1 ((Equiv.swap x y)⁻¹ z)=r.1 z
-  rw [Equiv.swap_inv]
-  by_cases hx : z=x
-  · subst z; simpa using h.symm
-  by_cases hy : z=y
-  · subst z; simpa using h
-  simp [Equiv.swap_apply_of_ne_of_ne hx hy]
-
-def Alternating (μ : YoungDiagram) (f : Tabloid μ → ℂ) : Prop :=
-  ∀ (c : columnGroup μ) r, f (relabel μ c r)=signC c.1*f r
-
-lemma alternating_zero_nontransversal (μ : YoungDiagram) (f : Tabloid μ → ℂ)
-    (hf : Alternating μ f) (r : Tabloid μ) (hr : ¬Transversal μ r) : f r=0 := by
-  simp only [Transversal,not_forall] at hr
-  obtain ⟨x,y,h⟩ := hr
-  obtain ⟨hc,he,hne⟩ := h
-  have hz := hf ⟨Equiv.swap x y,swap_mem_column μ x y hc⟩ r
-  change f (relabel μ (Equiv.swap x y) r)=signC (Equiv.swap x y)*f r at hz
-  rw [relabel_swap_fix μ r x y he] at hz
-  erw [signC_swap x y hne] at hz
-  linear_combination hz / 2
-
-lemma relabel_canonical (μ : YoungDiagram) (c : columnGroup μ) :
-    (relabel μ c (canonical μ)).1 = fun x => row μ (c.1⁻¹ x) := rfl
-
-lemma column_canonical_injective (μ : YoungDiagram) :
-    Function.Injective (fun c : columnGroup μ => relabel μ c (canonical μ)) := by
-  intro c d h
-  have h' : c⁻¹=d⁻¹ := column_row_injective μ (congrArg Subtype.val h)
-  exact inv_injective h'
-
-lemma transversal_column_orbit (μ : YoungDiagram) (r : Tabloid μ) (hr : Transversal μ r) :
-    ∃ c : columnGroup μ, r=relabel μ c (canonical μ) := by
-  obtain ⟨c,hc⟩ := transversal_eq_column μ r hr
-  refine ⟨c⁻¹,Subtype.ext ?_⟩
-  change r.1 = fun x => row μ ((c.1⁻¹)⁻¹ x)
-  simpa only [inv_inv] using hc
-
-lemma alternating_eq_of_canonical_eq (μ : YoungDiagram) (f g : Tabloid μ → ℂ)
-    (hf : Alternating μ f) (hg : Alternating μ g) (h : f (canonical μ)=g (canonical μ)) :
-    f=g := by
-  funext r
-  by_cases hr : Transversal μ r
-  · obtain ⟨c,rfl⟩ := transversal_column_orbit μ r hr
-    rw [hf,hg,h]
-  · rw [alternating_zero_nontransversal μ f hf r hr,
-      alternating_zero_nontransversal μ g hg r hr]
-
-end
-end YoungTabloid
-end PartialPermutation
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.ColumnSorting
+
+namespace OAI
+
+section
+namespace PartialPermutation
+namespace YoungTabloid
+noncomputable section
+open Finset
+open scoped Classical
+
+def signC {α : Type*} [Fintype α] [DecidableEq α] (g : Equiv.Perm α) : ℂ :=
+  (Equiv.Perm.sign g : ℤ)
+
+@[simp] lemma signC_one {α : Type*} [Fintype α] [DecidableEq α] : signC (1 : Equiv.Perm α)=1 := by
+  simp [signC]
+@[simp] lemma signC_mul {α : Type*} [Fintype α] [DecidableEq α] (g h : Equiv.Perm α) :
+    signC (g*h)=signC g*signC h := by simp [signC]
+@[simp] lemma signC_inv {α : Type*} [Fintype α] [DecidableEq α] (g : Equiv.Perm α) :
+    signC g⁻¹=signC g := by simp [signC]
+@[simp] lemma signC_sq {α : Type*} [Fintype α] [DecidableEq α] (g : Equiv.Perm α) :
+    signC g*signC g=1 := by
+  simpa using (signC_mul g g⁻¹).symm
+@[simp] lemma signC_swap {α : Type*} [Fintype α] [DecidableEq α] (x y : α) (h : x≠y) :
+    signC (Equiv.swap x y) = -1 := by simp [signC,Equiv.Perm.sign_swap h]
+
+lemma swap_mem_column (μ : YoungDiagram) (x y : μ.cells) (h : col μ x=col μ y) :
+    Equiv.swap x y ∈ columnGroup μ := by
+  intro z
+  by_cases hx : z=x
+  · subst z; simpa using h.symm
+  by_cases hy : z=y
+  · subst z; simpa using h
+  simp [Equiv.swap_apply_of_ne_of_ne hx hy]
+
+lemma relabel_swap_fix (μ : YoungDiagram) (r : Tabloid μ) (x y : μ.cells)
+    (h : r.1 x=r.1 y) : relabel μ (Equiv.swap x y) r=r := by
+  apply Subtype.ext
+  funext z
+  change r.1 ((Equiv.swap x y)⁻¹ z)=r.1 z
+  rw [Equiv.swap_inv]
+  by_cases hx : z=x
+  · subst z; simpa using h.symm
+  by_cases hy : z=y
+  · subst z; simpa using h
+  simp [Equiv.swap_apply_of_ne_of_ne hx hy]
+
+def Alternating (μ : YoungDiagram) (f : Tabloid μ → ℂ) : Prop :=
+  ∀ (c : columnGroup μ) r, f (relabel μ c r)=signC c.1*f r
+
+lemma alternating_zero_nontransversal (μ : YoungDiagram) (f : Tabloid μ → ℂ)
+    (hf : Alternating μ f) (r : Tabloid μ) (hr : ¬Transversal μ r) : f r=0 := by
+  simp only [Transversal,not_forall] at hr
+  obtain ⟨x,y,h⟩ := hr
+  obtain ⟨hc,he,hne⟩ := h
+  have hz := hf ⟨Equiv.swap x y,swap_mem_column μ x y hc⟩ r
+  change f (relabel μ (Equiv.swap x y) r)=signC (Equiv.swap x y)*f r at hz
+  rw [relabel_swap_fix μ r x y he] at hz
+  erw [signC_swap x y hne] at hz
+  linear_combination hz / 2
+
+lemma relabel_canonical (μ : YoungDiagram) (c : columnGroup μ) :
+    (relabel μ c (canonical μ)).1 = fun x => row μ (c.1⁻¹ x) := rfl
+
+lemma column_canonical_injective (μ : YoungDiagram) :
+    Function.Injective (fun c : columnGroup μ => relabel μ c (canonical μ)) := by
+  intro c d h
+  have h' : c⁻¹=d⁻¹ := column_row_injective μ (congrArg Subtype.val h)
+  exact inv_injective h'
+
+lemma transversal_column_orbit (μ : YoungDiagram) (r : Tabloid μ) (hr : Transversal μ r) :
+    ∃ c : columnGroup μ, r=relabel μ c (canonical μ) := by
+  obtain ⟨c,hc⟩ := transversal_eq_column μ r hr
+  refine ⟨c⁻¹,Subtype.ext ?_⟩
+  change r.1 = fun x => row μ ((c.1⁻¹)⁻¹ x)
+  simpa only [inv_inv] using hc
+
+lemma alternating_eq_of_canonical_eq (μ : YoungDiagram) (f g : Tabloid μ → ℂ)
+    (hf : Alternating μ f) (hg : Alternating μ g) (h : f (canonical μ)=g (canonical μ)) :
+    f=g := by
+  funext r
+  by_cases hr : Transversal μ r
+  · obtain ⟨c,rfl⟩ := transversal_column_orbit μ r hr
+    rw [hf,hg,h]
+  · rw [alternating_zero_nontransversal μ f hf r hr,
+      alternating_zero_nontransversal μ g hg r hr]
+
+end
+end YoungTabloid
+end PartialPermutation
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/BlockSubgroups.lean b/lean/OAI/RepresentationTheory/PartialPermutation/BlockSubgroups.lean
index 36d1051be7417d4c0bf2134469a54b912779847a..991a9cf58de651d4400baf1cb9990a4f734f5e34 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/BlockSubgroups.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/BlockSubgroups.lean
@@ -1,48 +1,48 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.Model
-
-namespace OAI
-
-section
-open scoped Classical
-open scoped BigOperators ComplexConjugate MonoidAlgebra
-open scoped BigOperators ComplexConjugate
-open scoped MonoidAlgebra BigOperators
-open scoped BigOperators MonoidAlgebra Classical
-
-attribute [local instance] Classical.propDecidable
-open scoped MonoidAlgebra BigOperators
-open scoped BigOperators
-
-namespace PartialPermutation
-
-lemma blockSubgroup_commute {X : Type*} {M N : Set X} (h : Disjoint M N)
-    {g k : Equiv.Perm X} (hg : g ∈ blockSubgroup M) (hk : k ∈ blockSubgroup N) :
-    Commute g k := by
-  apply Equiv.Perm.Disjoint.commute
-  intro x
-  by_cases hx : x ∈ M
-  · right
-    exact hk x (fun hn => Set.disjoint_left.mp h hx hn)
-  · exact Or.inl (hg x hx)
-
-lemma jointFactors_generate {G : Type*} [Group G] {b : ℕ} (H : Fin b → Subgroup G) :
-    ⨆ i, (H i).subgroupOf (⨆ j, H j) = ⊤ := by
-  apply Subgroup.map_subtype_inj.mp
-  rw [Subgroup.map_iSup]
-  simp only [Subgroup.map_subgroupOf_eq_of_le (le_iSup H _),
-    ← MonoidHom.range_eq_map, Subgroup.range_subtype]
-
-lemma jointFactors_commute {G : Type*} [Group G] {b : ℕ} (H : Fin b → Subgroup G)
-    (h : ∀ i j, i ≠ j → ∀ g ∈ H i, ∀ k ∈ H j, Commute g k) :
-    ∀ i j, i ≠ j → ∀ g ∈ (H i).subgroupOf (⨆ j, H j),
-      ∀ k ∈ (H j).subgroupOf (⨆ j, H j), Commute g k := by
-  intro i j hij g hg k hk
-  apply Subtype.ext
-  exact h i j hij g hg k hk
-
-end PartialPermutation
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.Model
+
+namespace OAI
+
+section
+open scoped Classical
+open scoped BigOperators ComplexConjugate MonoidAlgebra
+open scoped BigOperators ComplexConjugate
+open scoped MonoidAlgebra BigOperators
+open scoped BigOperators MonoidAlgebra Classical
+
+attribute [local instance] Classical.propDecidable
+open scoped MonoidAlgebra BigOperators
+open scoped BigOperators
+
+namespace PartialPermutation
+
+lemma blockSubgroup_commute {X : Type*} {M N : Set X} (h : Disjoint M N)
+    {g k : Equiv.Perm X} (hg : g ∈ blockSubgroup M) (hk : k ∈ blockSubgroup N) :
+    Commute g k := by
+  apply Equiv.Perm.Disjoint.commute
+  intro x
+  by_cases hx : x ∈ M
+  · right
+    exact hk x (fun hn => Set.disjoint_left.mp h hx hn)
+  · exact Or.inl (hg x hx)
+
+lemma jointFactors_generate {G : Type*} [Group G] {b : ℕ} (H : Fin b → Subgroup G) :
+    ⨆ i, (H i).subgroupOf (⨆ j, H j) = ⊤ := by
+  apply Subgroup.map_subtype_inj.mp
+  rw [Subgroup.map_iSup]
+  simp only [Subgroup.map_subgroupOf_eq_of_le (le_iSup H _),
+    ← MonoidHom.range_eq_map, Subgroup.range_subtype]
+
+lemma jointFactors_commute {G : Type*} [Group G] {b : ℕ} (H : Fin b → Subgroup G)
+    (h : ∀ i j, i ≠ j → ∀ g ∈ H i, ∀ k ∈ H j, Commute g k) :
+    ∀ i j, i ≠ j → ∀ g ∈ (H i).subgroupOf (⨆ j, H j),
+      ∀ k ∈ (H j).subgroupOf (⨆ j, H j), Commute g k := by
+  intro i j hij g hg k hk
+  apply Subtype.ext
+  exact h i j hij g hg k hk
+
+end PartialPermutation
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/CentralKernels.lean b/lean/OAI/RepresentationTheory/PartialPermutation/CentralKernels.lean
index a9c2062f01c298bb6f7de531fe47440e5ab3530f..f001515c402a2103ca3aa9c9d79cb32d96a95efb 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/CentralKernels.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/CentralKernels.lean
@@ -1,113 +1,113 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.Model
-
-namespace OAI
-
-section
-open scoped Classical
-open scoped BigOperators ComplexConjugate MonoidAlgebra
-open scoped BigOperators ComplexConjugate
-open scoped MonoidAlgebra BigOperators
-
-namespace PartialPermutation
-noncomputable section
-open scoped MonoidAlgebra BigOperators
-
-section CentralFourier
-variable {G V : Type*} [Group G] [Fintype G] [AddCommGroup V] [Module ℂ V]
-    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V)
-
-omit [FiniteDimensional ℂ V] in
-lemma centralFourier_commute (a : G → ℂ)
-    (ha : ∀ g h, a (g * h) = a (h * g)) (g : G) :
-    complexFourier ρ a * ρ g = ρ g * complexFourier ρ a := by
-  classical
-  simp only [complexFourier, Finset.sum_mul, Finset.mul_sum, smul_mul_assoc,
-    mul_smul_comm, ← map_mul]
-  calc
-    _ = ∑ h : G, a (g * h * g⁻¹) • ρ (g * h * g⁻¹ * g) := by
-      exact (Fintype.sum_equiv (Equiv.mulLeft g |>.trans (Equiv.mulRight g⁻¹)) _ _
-        (fun h => rfl)).symm
-    _ = _ := by
-      apply Finset.sum_congr rfl
-      intro h _
-      rw [ha (g * h) g⁻¹]
-      simp [mul_assoc]
-
-lemma centralFourier_scalar [Representation.IsIrreducible ρ] (a : G → ℂ)
-    (ha : ∀ g h, a (g * h) = a (h * g)) :
-    ∃ c : ℂ, complexFourier ρ a = c • (1 : Module.End ℂ V) := by
-  let T : Representation.IntertwiningMap ρ ρ :=
-    { toLinearMap := complexFourier ρ a
-      isIntertwining' := fun g => by
-        simpa only [Module.End.mul_eq_comp] using centralFourier_commute ρ a ha g }
-  obtain ⟨c, hc⟩ := Representation.IsIrreducible.algebraMap_intertwiningMap_bijective_of_isAlgClosed
-    (ρ := ρ) |>.surjective T
-  refine ⟨c, ?_⟩
-  rw [Representation.IntertwiningMap.algebraMap_apply] at hc
-  exact (congrArg Representation.IntertwiningMap.toLinearMap hc).symm
-
-end CentralFourier
-
-def centralCharacterKernel {G V : Type*} [Group G] [Fintype G]
-    [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
-    (ρ : Representation ℂ G V) (g : G) : ℂ :=
-  (Module.finrank ℂ V : ℂ) / Fintype.card G * ρ.character g⁻¹
-
-lemma centralCharacterKernel_comm {G V : Type*} [Group G] [Fintype G]
-    [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
-    (ρ : Representation ℂ G V) (g h : G) :
-    centralCharacterKernel ρ (g * h) = centralCharacterKernel ρ (h * g) := by
-  simp only [centralCharacterKernel, mul_inv_rev]
-  rw [Representation.char_mul_comm]
-
-lemma centralCharacterKernel_fourier {G V W : Type*} [Group G] [Fintype G]
-    [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
-    [AddCommGroup W] [Module ℂ W] [FiniteDimensional ℂ W]
-    (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
-    [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ] :
-    complexFourier σ (centralCharacterKernel ρ) =
-      if Nonempty (Representation.Equiv ρ σ) then 1 else 0 := by
-  classical
-  obtain ⟨c, hc⟩ := centralFourier_scalar σ (centralCharacterKernel ρ)
-    (centralCharacterKernel_comm ρ)
-  have : Nontrivial W := IsSimpleModule.nontrivial ℂ[G] σ.asModule
-  have hD : (Module.finrank ℂ W : ℂ) ≠ 0 := by
-    exact_mod_cast (Module.finrank_pos (R := ℂ) (M := W)).ne'
-  let : Invertible (Nat.card G : ℂ) := invertibleOfNonzero (by
-    exact_mod_cast (Nat.card_pos (α := G)).ne')
-  have ht := congrArg (LinearMap.trace ℂ W) hc
-  have ho := Representation.char_orthonormal σ ρ
-  have htrace : LinearMap.trace ℂ W (complexFourier σ (centralCharacterKernel ρ)) =
-      (Module.finrank ℂ V : ℂ) *
-        (if Nonempty (Representation.Equiv ρ σ) then 1 else 0) := by
-    simp only [complexFourier, map_sum, map_smul, smul_eq_mul, centralCharacterKernel]
-    change (∑ g, (Module.finrank ℂ V : ℂ) / Fintype.card G *
-      ρ.character g⁻¹ * σ.character g) = _
-    calc
-      _ = (Module.finrank ℂ V : ℂ) * ((Fintype.card G : ℂ)⁻¹ *
-          ∑ g, σ.character g * ρ.character g⁻¹) := by
-        simp only [Finset.mul_sum]
-        apply Finset.sum_congr rfl
-        intro g _
-        ring
-      _ = _ := by rw [Fintype.card_eq_nat_card, ho]
-  rw [htrace, map_smul, LinearMap.trace_one, smul_eq_mul] at ht
-  split_ifs with he
-  · obtain ⟨e⟩ := he
-    have hd := e.toLinearEquiv.finrank_eq
-    have hcd : c = 1 := by
-      rw [ite_eq_left ⟨e⟩, mul_one, hd] at ht
-      apply (mul_right_cancel₀ hD)
-      simpa using ht.symm
-    simpa [hcd] using hc
-  · have hcd : c = 0 := by
-      rw [ite_eq_right he, mul_zero] at ht
-      exact (mul_eq_zero.mp ht.symm).resolve_right hD
-    simpa [hcd] using hc
-
-end
-end PartialPermutation
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.Model
+
+namespace OAI
+
+section
+open scoped Classical
+open scoped BigOperators ComplexConjugate MonoidAlgebra
+open scoped BigOperators ComplexConjugate
+open scoped MonoidAlgebra BigOperators
+
+namespace PartialPermutation
+noncomputable section
+open scoped MonoidAlgebra BigOperators
+
+section CentralFourier
+variable {G V : Type*} [Group G] [Fintype G] [AddCommGroup V] [Module ℂ V]
+    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V)
+
+omit [FiniteDimensional ℂ V] in
+lemma centralFourier_commute (a : G → ℂ)
+    (ha : ∀ g h, a (g * h) = a (h * g)) (g : G) :
+    complexFourier ρ a * ρ g = ρ g * complexFourier ρ a := by
+  classical
+  simp only [complexFourier, Finset.sum_mul, Finset.mul_sum, smul_mul_assoc,
+    mul_smul_comm, ← map_mul]
+  calc
+    _ = ∑ h : G, a (g * h * g⁻¹) • ρ (g * h * g⁻¹ * g) := by
+      exact (Fintype.sum_equiv (Equiv.mulLeft g |>.trans (Equiv.mulRight g⁻¹)) _ _
+        (fun h => rfl)).symm
+    _ = _ := by
+      apply Finset.sum_congr rfl
+      intro h _
+      rw [ha (g * h) g⁻¹]
+      simp [mul_assoc]
+
+lemma centralFourier_scalar [Representation.IsIrreducible ρ] (a : G → ℂ)
+    (ha : ∀ g h, a (g * h) = a (h * g)) :
+    ∃ c : ℂ, complexFourier ρ a = c • (1 : Module.End ℂ V) := by
+  let T : Representation.IntertwiningMap ρ ρ :=
+    { toLinearMap := complexFourier ρ a
+      isIntertwining' := fun g => by
+        simpa only [Module.End.mul_eq_comp] using centralFourier_commute ρ a ha g }
+  obtain ⟨c, hc⟩ := Representation.IsIrreducible.algebraMap_intertwiningMap_bijective_of_isAlgClosed
+    (ρ := ρ) |>.surjective T
+  refine ⟨c, ?_⟩
+  rw [Representation.IntertwiningMap.algebraMap_apply] at hc
+  exact (congrArg Representation.IntertwiningMap.toLinearMap hc).symm
+
+end CentralFourier
+
+def centralCharacterKernel {G V : Type*} [Group G] [Fintype G]
+    [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
+    (ρ : Representation ℂ G V) (g : G) : ℂ :=
+  (Module.finrank ℂ V : ℂ) / Fintype.card G * ρ.character g⁻¹
+
+lemma centralCharacterKernel_comm {G V : Type*} [Group G] [Fintype G]
+    [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
+    (ρ : Representation ℂ G V) (g h : G) :
+    centralCharacterKernel ρ (g * h) = centralCharacterKernel ρ (h * g) := by
+  simp only [centralCharacterKernel, mul_inv_rev]
+  rw [Representation.char_mul_comm]
+
+lemma centralCharacterKernel_fourier {G V W : Type*} [Group G] [Fintype G]
+    [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
+    [AddCommGroup W] [Module ℂ W] [FiniteDimensional ℂ W]
+    (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
+    [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ] :
+    complexFourier σ (centralCharacterKernel ρ) =
+      if Nonempty (Representation.Equiv ρ σ) then 1 else 0 := by
+  classical
+  obtain ⟨c, hc⟩ := centralFourier_scalar σ (centralCharacterKernel ρ)
+    (centralCharacterKernel_comm ρ)
+  have : Nontrivial W := IsSimpleModule.nontrivial ℂ[G] σ.asModule
+  have hD : (Module.finrank ℂ W : ℂ) ≠ 0 := by
+    exact_mod_cast (Module.finrank_pos (R := ℂ) (M := W)).ne'
+  let : Invertible (Nat.card G : ℂ) := invertibleOfNonzero (by
+    exact_mod_cast (Nat.card_pos (α := G)).ne')
+  have ht := congrArg (LinearMap.trace ℂ W) hc
+  have ho := Representation.char_orthonormal σ ρ
+  have htrace : LinearMap.trace ℂ W (complexFourier σ (centralCharacterKernel ρ)) =
+      (Module.finrank ℂ V : ℂ) *
+        (if Nonempty (Representation.Equiv ρ σ) then 1 else 0) := by
+    simp only [complexFourier, map_sum, map_smul, smul_eq_mul, centralCharacterKernel]
+    change (∑ g, (Module.finrank ℂ V : ℂ) / Fintype.card G *
+      ρ.character g⁻¹ * σ.character g) = _
+    calc
+      _ = (Module.finrank ℂ V : ℂ) * ((Fintype.card G : ℂ)⁻¹ *
+          ∑ g, σ.character g * ρ.character g⁻¹) := by
+        simp only [Finset.mul_sum]
+        apply Finset.sum_congr rfl
+        intro g _
+        ring
+      _ = _ := by rw [Fintype.card_eq_nat_card, ho]
+  rw [htrace, map_smul, LinearMap.trace_one, smul_eq_mul] at ht
+  split_ifs with he
+  · obtain ⟨e⟩ := he
+    have hd := e.toLinearEquiv.finrank_eq
+    have hcd : c = 1 := by
+      rw [ite_eq_left ⟨e⟩, mul_one, hd] at ht
+      apply (mul_right_cancel₀ hD)
+      simpa using ht.symm
+    simpa [hcd] using hc
+  · have hcd : c = 0 := by
+      rw [ite_eq_right he, mul_zero] at ht
+      exact (mul_eq_zero.mp ht.symm).resolve_right hD
+    simpa [hcd] using hc
+
+end
+end PartialPermutation
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/Characters.lean b/lean/OAI/RepresentationTheory/PartialPermutation/Characters.lean
index aa03318bf08ba3fc3a13a12a7e50c73bccc051f0..b77f58aa4a33245b7d788cc16a0db894bf9428ef 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/Characters.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/Characters.lean
@@ -1,83 +1,83 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.MatrixCoefficients
-
-namespace OAI
-
-section
-open scoped Classical
-open scoped BigOperators ComplexConjugate MonoidAlgebra
-open scoped BigOperators ComplexConjugate
-
-namespace PartialPermutation
-open scoped BigOperators ComplexConjugate MonoidAlgebra
-
-noncomputable section
-variable {G V : Type*} [Group G] [Fintype G]
-    [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-    (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
-include hρ
-
-omit [Fintype G] in
-lemma unitary_adjoint (g : G) : LinearMap.adjoint (ρ g) = ρ g⁻¹ := by
-  ext v
-  apply ext_inner_left ℂ
-  intro u
-  rw [LinearMap.adjoint_inner_right]
-  have h := hρ g u (ρ g⁻¹ v)
-  simpa using h
-
-omit [Fintype G] in
-lemma unitary_character_inv (g : G) : ρ.character g⁻¹ = conj (ρ.character g) := by
-  let e := stdOrthonormalBasis ℂ V
-  rw [Representation.character, Representation.character, ← unitary_adjoint ρ hρ,
-    LinearMap.trace_eq_sum_inner _ e, LinearMap.trace_eq_sum_inner _ e, map_sum]
-  apply Finset.sum_congr rfl
-  intro i _
-  rw [LinearMap.adjoint_inner_right, inner_conj_symm]
-
-variable [Representation.IsIrreducible ρ]
-
-lemma character_norm_sq_sum : ∑ g, ‖ρ.character g‖ ^ 2 = (Fintype.card G : ℝ) := by
-  let := invertibleOfNonzero (show (Nat.card G : ℂ) ≠ 0 by simp [Nat.card_eq_fintype_card])
-  have h := Representation.char_orthonormal ρ ρ
-  have he : Nonempty (Representation.Equiv ρ ρ) := ⟨Representation.Equiv.refl ρ⟩
-  simp only [ite_eq_left he, unitary_character_inv ρ hρ] at h
-  have hN : (Nat.card G : ℂ) ≠ 0 := by simp [Nat.card_eq_fintype_card]
-  have hsum : (∑ g, ρ.character g * conj (ρ.character g)) = (Nat.card G : ℂ) := by
-    simpa [hN, mul_assoc] using congrArg (fun z : ℂ => (Nat.card G : ℂ) * z) h
-  have ht := congrArg Complex.re hsum
-  simpa only [Complex.re_sum, Complex.mul_conj, Complex.ofReal_re,
-    Complex.normSq_eq_norm_sq, Complex.natCast_re, Nat.card_eq_fintype_card] using ht
-
-lemma character_kernel_identity :
-    complexFourier ρ (fun g => ((Module.finrank ℂ V : ℂ) / Fintype.card G) *
-      conj (ρ.character g)) = 1 := by
-  classical
-  let e := stdOrthonormalBasis ℂ V
-  have : Nontrivial V := IsSimpleModule.nontrivial ℂ[G] ρ.asModule
-  have hD : (Module.finrank ℂ V : ℂ) ≠ 0 := by
-    exact_mod_cast (Module.finrank_pos (R := ℂ) (M := V)).ne'
-  have hN : (Fintype.card G : ℂ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
-  ext y
-  apply ext_inner_left ℂ
-  intro x
-  simp only [complexFourier, LinearMap.sum_apply, LinearMap.smul_apply, Module.End.one_apply]
-  simp only [inner_sum, inner_smul_right, mul_assoc, ← Finset.mul_sum]
-  have hs : (Fintype.card G : ℂ)⁻¹ * ∑ g, conj (ρ.character g) * inner ℂ x (ρ g y) =
-      (Module.finrank ℂ V : ℂ)⁻¹ * inner ℂ x y := by
-    simp_rw [Representation.character, LinearMap.trace_eq_sum_inner _ e, map_sum,
-      Finset.sum_mul]
-    rw [Finset.sum_comm, Finset.mul_sum]
-    have hc (i : Fin (Module.finrank ℂ V)) :=
-      matrixCoefficient_orthogonality ρ hρ x (e i) y (e i)
-    simp_rw [mul_comm (conj _), hc, div_eq_mul_inv]
-    simpa [Finset.mul_sum, mul_assoc, mul_left_comm, mul_comm] using
-      congrArg (fun z : ℂ => (Module.finrank ℂ V : ℂ)⁻¹ * z) (e.sum_inner_mul_inner x y)
-  rw [div_eq_mul_inv, mul_assoc, hs, ← mul_assoc, mul_inv_cancel₀ hD, one_mul]
-
-end
-end PartialPermutation
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.MatrixCoefficients
+
+namespace OAI
+
+section
+open scoped Classical
+open scoped BigOperators ComplexConjugate MonoidAlgebra
+open scoped BigOperators ComplexConjugate
+
+namespace PartialPermutation
+open scoped BigOperators ComplexConjugate MonoidAlgebra
+
+noncomputable section
+variable {G V : Type*} [Group G] [Fintype G]
+    [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+    (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
+include hρ
+
+omit [Fintype G] in
+lemma unitary_adjoint (g : G) : LinearMap.adjoint (ρ g) = ρ g⁻¹ := by
+  ext v
+  apply ext_inner_left ℂ
+  intro u
+  rw [LinearMap.adjoint_inner_right]
+  have h := hρ g u (ρ g⁻¹ v)
+  simpa using h
+
+omit [Fintype G] in
+lemma unitary_character_inv (g : G) : ρ.character g⁻¹ = conj (ρ.character g) := by
+  let e := stdOrthonormalBasis ℂ V
+  rw [Representation.character, Representation.character, ← unitary_adjoint ρ hρ,
+    LinearMap.trace_eq_sum_inner _ e, LinearMap.trace_eq_sum_inner _ e, map_sum]
+  apply Finset.sum_congr rfl
+  intro i _
+  rw [LinearMap.adjoint_inner_right, inner_conj_symm]
+
+variable [Representation.IsIrreducible ρ]
+
+lemma character_norm_sq_sum : ∑ g, ‖ρ.character g‖ ^ 2 = (Fintype.card G : ℝ) := by
+  let := invertibleOfNonzero (show (Nat.card G : ℂ) ≠ 0 by simp [Nat.card_eq_fintype_card])
+  have h := Representation.char_orthonormal ρ ρ
+  have he : Nonempty (Representation.Equiv ρ ρ) := ⟨Representation.Equiv.refl ρ⟩
+  simp only [ite_eq_left he, unitary_character_inv ρ hρ] at h
+  have hN : (Nat.card G : ℂ) ≠ 0 := by simp [Nat.card_eq_fintype_card]
+  have hsum : (∑ g, ρ.character g * conj (ρ.character g)) = (Nat.card G : ℂ) := by
+    simpa [hN, mul_assoc] using congrArg (fun z : ℂ => (Nat.card G : ℂ) * z) h
+  have ht := congrArg Complex.re hsum
+  simpa only [Complex.re_sum, Complex.mul_conj, Complex.ofReal_re,
+    Complex.normSq_eq_norm_sq, Complex.natCast_re, Nat.card_eq_fintype_card] using ht
+
+lemma character_kernel_identity :
+    complexFourier ρ (fun g => ((Module.finrank ℂ V : ℂ) / Fintype.card G) *
+      conj (ρ.character g)) = 1 := by
+  classical
+  let e := stdOrthonormalBasis ℂ V
+  have : Nontrivial V := IsSimpleModule.nontrivial ℂ[G] ρ.asModule
+  have hD : (Module.finrank ℂ V : ℂ) ≠ 0 := by
+    exact_mod_cast (Module.finrank_pos (R := ℂ) (M := V)).ne'
+  have hN : (Fintype.card G : ℂ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
+  ext y
+  apply ext_inner_left ℂ
+  intro x
+  simp only [complexFourier, LinearMap.sum_apply, LinearMap.smul_apply, Module.End.one_apply]
+  simp only [inner_sum, inner_smul_right, mul_assoc, ← Finset.mul_sum]
+  have hs : (Fintype.card G : ℂ)⁻¹ * ∑ g, conj (ρ.character g) * inner ℂ x (ρ g y) =
+      (Module.finrank ℂ V : ℂ)⁻¹ * inner ℂ x y := by
+    simp_rw [Representation.character, LinearMap.trace_eq_sum_inner _ e, map_sum,
+      Finset.sum_mul]
+    rw [Finset.sum_comm, Finset.mul_sum]
+    have hc (i : Fin (Module.finrank ℂ V)) :=
+      matrixCoefficient_orthogonality ρ hρ x (e i) y (e i)
+    simp_rw [mul_comm (conj _), hc, div_eq_mul_inv]
+    simpa [Finset.mul_sum, mul_assoc, mul_left_comm, mul_comm] using
+      congrArg (fun z : ℂ => (Module.finrank ℂ V : ℂ)⁻¹ * z) (e.sum_inner_mul_inner x y)
+  rw [div_eq_mul_inv, mul_assoc, hs, ← mul_assoc, mul_inv_cancel₀ hD, one_mul]
+
+end
+end PartialPermutation
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/ClassCharacters.lean b/lean/OAI/RepresentationTheory/PartialPermutation/ClassCharacters.lean
index 9d08c33bff44f810b987f72061fc42280bf4e6c0..b9e803554256793c18e50c26d35af0fcf619452e 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/ClassCharacters.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/ClassCharacters.lean
@@ -1,69 +1,69 @@
-import Mathlib
-
-namespace OAI
-
-section
-namespace PartialPermutation
-noncomputable section
-open scoped Classical
-
-variable {G : Type*} [Group G] [Fintype G]
-
-omit [Fintype G] in
-lemma character_eq_of_isConj {V : Type*} [AddCommGroup V] [Module ℂ V]
-    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V) {g h : G} (H : IsConj g h) :
-    ρ.character g=ρ.character h := by
-  obtain ⟨k,rfl⟩ := isConj_iff.mp H
-  exact (ρ.char_conj _ _).symm
-
-def classCharacter {V : Type*} [AddCommGroup V] [Module ℂ V]
-    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V) : ConjClasses G → ℂ :=
-  Quotient.lift ρ.character (fun _ _ h => character_eq_of_isConj ρ h)
-
-omit [Fintype G] in
-lemma classCharacter_mk {V : Type*} [AddCommGroup V] [Module ℂ V]
-    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V) (g : G) :
-    classCharacter ρ (ConjClasses.mk g)=ρ.character g := rfl
-
-lemma independent_classCharacters {ι : Type*} [Fintype ι]
-    (V : ι → Type*) [∀ i, AddCommGroup (V i)] [∀ i, Module ℂ (V i)]
-    [∀ i, FiniteDimensional ℂ (V i)] (ρ : ∀ i, Representation ℂ G (V i))
-    [∀ i, Representation.IsIrreducible (ρ i)]
-    (hd : ∀ i j, Nonempty (Representation.Equiv (ρ i) (ρ j)) ↔ i=j) :
-    LinearIndependent ℂ (fun i => classCharacter (ρ i)) := by
-  let : Invertible (Nat.card G : ℂ) := invertibleOfNonzero (by
-    exact_mod_cast (Nat.card_pos (α := G)).ne')
-  apply Fintype.linearIndependent_iff.mpr
-  intro a ha j
-  have he (g : G) : (∑ i, a i*(ρ i).character g)=0 := by
-    have h := congrFun ha (ConjClasses.mk g)
-    simpa only [Finset.sum_apply,Pi.smul_apply,smul_eq_mul,classCharacter_mk,Pi.zero_apply] using h
-  calc
-    a j = ∑ i, a i * ((Nat.card G : ℂ)⁻¹ * ∑ g, (ρ i).character g * (ρ j).character g⁻¹) := by
-      simp only [Representation.char_orthonormal,hd]
-      simp
-    _ = (Nat.card G : ℂ)⁻¹ * ∑ g, (∑ i, a i*(ρ i).character g) * (ρ j).character g⁻¹ := by
-      simp only [Finset.mul_sum,Finset.sum_mul]
-      rw [Finset.sum_comm]
-      apply Finset.sum_congr rfl
-      intro g _
-      apply Finset.sum_congr rfl
-      intro i _
-      ring
-    _ = 0 := by simp [he]
-
-lemma irreducible_family_card_le_conjClasses {ι : Type*} [Fintype ι]
-    (V : ι → Type*) [∀ i, AddCommGroup (V i)] [∀ i, Module ℂ (V i)]
-    [∀ i, FiniteDimensional ℂ (V i)] (ρ : ∀ i, Representation ℂ G (V i))
-    [∀ i, Representation.IsIrreducible (ρ i)]
-    (hd : ∀ i j, Nonempty (Representation.Equiv (ρ i) (ρ j)) ↔ i=j) :
-    Fintype.card ι ≤ Nat.card (ConjClasses G) := by
-  let : Fintype (ConjClasses G) := Fintype.ofFinite _
-  simpa using (independent_classCharacters V ρ hd).fintype_card_le_finrank
-
-end
-end PartialPermutation
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+section
+namespace PartialPermutation
+noncomputable section
+open scoped Classical
+
+variable {G : Type*} [Group G] [Fintype G]
+
+omit [Fintype G] in
+lemma character_eq_of_isConj {V : Type*} [AddCommGroup V] [Module ℂ V]
+    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V) {g h : G} (H : IsConj g h) :
+    ρ.character g=ρ.character h := by
+  obtain ⟨k,rfl⟩ := isConj_iff.mp H
+  exact (ρ.char_conj _ _).symm
+
+def classCharacter {V : Type*} [AddCommGroup V] [Module ℂ V]
+    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V) : ConjClasses G → ℂ :=
+  Quotient.lift ρ.character (fun _ _ h => character_eq_of_isConj ρ h)
+
+omit [Fintype G] in
+lemma classCharacter_mk {V : Type*} [AddCommGroup V] [Module ℂ V]
+    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V) (g : G) :
+    classCharacter ρ (ConjClasses.mk g)=ρ.character g := rfl
+
+lemma independent_classCharacters {ι : Type*} [Fintype ι]
+    (V : ι → Type*) [∀ i, AddCommGroup (V i)] [∀ i, Module ℂ (V i)]
+    [∀ i, FiniteDimensional ℂ (V i)] (ρ : ∀ i, Representation ℂ G (V i))
+    [∀ i, Representation.IsIrreducible (ρ i)]
+    (hd : ∀ i j, Nonempty (Representation.Equiv (ρ i) (ρ j)) ↔ i=j) :
+    LinearIndependent ℂ (fun i => classCharacter (ρ i)) := by
+  let : Invertible (Nat.card G : ℂ) := invertibleOfNonzero (by
+    exact_mod_cast (Nat.card_pos (α := G)).ne')
+  apply Fintype.linearIndependent_iff.mpr
+  intro a ha j
+  have he (g : G) : (∑ i, a i*(ρ i).character g)=0 := by
+    have h := congrFun ha (ConjClasses.mk g)
+    simpa only [Finset.sum_apply,Pi.smul_apply,smul_eq_mul,classCharacter_mk,Pi.zero_apply] using h
+  calc
+    a j = ∑ i, a i * ((Nat.card G : ℂ)⁻¹ * ∑ g, (ρ i).character g * (ρ j).character g⁻¹) := by
+      simp only [Representation.char_orthonormal,hd]
+      simp
+    _ = (Nat.card G : ℂ)⁻¹ * ∑ g, (∑ i, a i*(ρ i).character g) * (ρ j).character g⁻¹ := by
+      simp only [Finset.mul_sum,Finset.sum_mul]
+      rw [Finset.sum_comm]
+      apply Finset.sum_congr rfl
+      intro g _
+      apply Finset.sum_congr rfl
+      intro i _
+      ring
+    _ = 0 := by simp [he]
+
+lemma irreducible_family_card_le_conjClasses {ι : Type*} [Fintype ι]
+    (V : ι → Type*) [∀ i, AddCommGroup (V i)] [∀ i, Module ℂ (V i)]
+    [∀ i, FiniteDimensional ℂ (V i)] (ρ : ∀ i, Representation ℂ G (V i))
+    [∀ i, Representation.IsIrreducible (ρ i)]
+    (hd : ∀ i j, Nonempty (Representation.Equiv (ρ i) (ρ j)) ↔ i=j) :
+    Fintype.card ι ≤ Nat.card (ConjClasses G) := by
+  let : Fintype (ConjClasses G) := Fintype.ofFinite _
+  simpa using (independent_classCharacters V ρ hd).fintype_card_le_finrank
+
+end
+end PartialPermutation
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/ColumnRearrangement.lean b/lean/OAI/RepresentationTheory/PartialPermutation/ColumnRearrangement.lean
index b3684928a3972f43d746aec19b06752a6c12eb6f..712283fe74eddd125c44fcf9831cea23d3dab445 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/ColumnRearrangement.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/ColumnRearrangement.lean
@@ -1,118 +1,118 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.ColumnSorting
-
-namespace OAI
-
-section
-namespace PartialPermutation.YoungTabloid
-noncomputable section
-open Finset
-open scoped Classical
-
-lemma cell_eq_of_row_col (μ : YoungDiagram) {x y : μ.cells}
-    (hr : row μ x=row μ y) (hc : col μ x=col μ y) : x=y :=
-  Subtype.ext (Prod.ext hr hc)
-
-lemma cell_lt_of_column (μ : YoungDiagram) {x y : μ.cells}
-    (hc : col μ x=col μ y) (hr : row μ x<row μ y) : x<y := by
-  apply lt_of_le_of_ne
-  · exact ⟨hr.le,hc.le⟩
-  · intro h
-    exact hr.ne (congrArg (row μ) h)
-
-def columnRestrict (μ : YoungDiagram) (c : columnGroup μ) (j : ℕ) : Equiv.Perm (Col μ j) where
-  toFun x := ⟨c.1 x.1,(c.2 x.1).trans x.2⟩
-  invFun x := ⟨c.1⁻¹ x.1,((c⁻¹).2 x.1).trans x.2⟩
-  left_inv x := by apply Subtype.ext; exact c.1.symm_apply_apply _
-  right_inv x := by apply Subtype.ext; exact c.1.apply_symm_apply _
-
-lemma column_monovary (μ : YoungDiagram) (f : μ.cells → ℕ) (hf : StrictMono f) (j : ℕ) :
-    Monovary (fun x : Col μ j => f x.1) (fun x => row μ x.1) := by
-  intro x y h
-  exact (hf (cell_lt_of_column μ (x.2.trans y.2.symm) h)).le
-
-lemma column_rearrangement (μ : YoungDiagram) (f : μ.cells → ℕ) (hf : StrictMono f)
-    (c : columnGroup μ) (j : ℕ) :
-    (∑ x : Col μ j, f x.1 * row μ (c.1 x.1)) ≤ ∑ x : Col μ j, f x.1 * row μ x.1 :=
-  (column_monovary μ f hf j).sum_mul_comp_perm_le_sum_mul (σ := columnRestrict μ c j)
-
-lemma fin_perm_mono_eq (n : ℕ) (p : Equiv.Perm (Fin n)) (hp : StrictMono p) : p=1 := by
-  have hle (i : Fin n) : i.val≤(p i).val := strictMono_fin_nat_le _
-    (fun i j h => hp h) i
-  have heq : (∑ i : Fin n, i.val)=∑ i : Fin n, (p i).val := (Equiv.sum_comp p Fin.val).symm
-  have hh := (Finset.sum_eq_sum_iff_of_le (fun i _ => hle i)).mp heq
-  apply Equiv.ext
-  intro i
-  exact Fin.ext (hh i (mem_univ _)).symm
-
-lemma column_rearrangement_eq (μ : YoungDiagram) (f : μ.cells → ℕ) (hf : StrictMono f)
-    (c : columnGroup μ) (j : ℕ)
-    (he : (∑ x : Col μ j, f x.1 * row μ (c.1 x.1)) = ∑ x : Col μ j, f x.1 * row μ x.1) :
-    ∀ x : Col μ j, c.1 x.1=x.1 := by
-  have hm := (column_monovary μ f hf j).sum_mul_comp_perm_eq_sum_mul_iff
-    (σ := columnRestrict μ c j) |>.mp he
-  let e := columnEquiv μ j
-  let p := e.symm.trans ((columnRestrict μ c j).trans e)
-  have hp : StrictMono p := by
-    intro i k hik
-    let x := e.symm i
-    let y := e.symm k
-    have hxy : f x.1 < f y.1 := hf (cell_lt_of_column μ (x.2.trans y.2.symm) hik)
-    change row μ (c.1 x.1) < row μ (c.1 y.1)
-    by_contra h
-    have hr : row μ (c.1 y.1) < row μ (c.1 x.1) := by
-      apply lt_of_le_of_ne (Nat.le_of_not_gt h)
-      intro hh
-      have heq := c.1.injective (cell_eq_of_row_col μ hh
-        ((c.2 y.1).trans ((y.2.trans x.2.symm).trans (c.2 x.1).symm)))
-      exact hxy.ne (congrArg f heq.symm)
-    exact (not_le_of_gt hxy) (hm hr)
-  have hp1 := fin_perm_mono_eq _ p hp
-  intro x
-  have hx := Equiv.congr_fun hp1 (e x)
-  change e (columnRestrict μ c j (e.symm (e x)))=e x at hx
-  erw [Equiv.symm_apply_apply] at hx
-  exact congrArg Subtype.val (e.injective hx)
-
-def colIndex (μ : YoungDiagram) (x : μ.cells) : Fin (μ.rowLen 0) :=
-  ⟨col μ x,(μ.mem_iff_lt_rowLen.mp x.2).trans_le
-    (μ.rowLen_anti 0 x.1.1 (Nat.zero_le _))⟩
-
-def columnDecomp (μ : YoungDiagram) : μ.cells ≃ Σ j : Fin (μ.rowLen 0), Col μ j where
-  toFun x := ⟨colIndex μ x,⟨x,rfl⟩⟩
-  invFun x := x.2.1
-  left_inv _ := rfl
-  right_inv x := by
-    obtain ⟨j,x,hx⟩ := x
-    have hj : colIndex μ x=j := Fin.ext hx
-    cases hj
-    rfl
-
-lemma sum_columns (μ : YoungDiagram) (f : μ.cells → ℕ) :
-    (∑ x, f x)=∑ j : Fin (μ.rowLen 0), ∑ x : Col μ j, f x.1 := by
-  exact (Fintype.sum_equiv (columnDecomp μ) _
-    (fun x : Σ j : Fin (μ.rowLen 0), Col μ j => f x.2.1) (fun _ => rfl)).trans
-    (Fintype.sum_sigma _)
-
-lemma full_rearrangement (μ : YoungDiagram) (f : μ.cells → ℕ) (hf : StrictMono f)
-    (c : columnGroup μ) :
-    (∑ x, f x * row μ (c.1 x)) ≤ ∑ x, f x * row μ x := by
-  rw [sum_columns, sum_columns]
-  exact Finset.sum_le_sum (fun j _ => column_rearrangement μ f hf c j)
-
-lemma full_rearrangement_eq (μ : YoungDiagram) (f : μ.cells → ℕ) (hf : StrictMono f)
-    (c : columnGroup μ) (he : (∑ x, f x * row μ (c.1 x)) = ∑ x, f x * row μ x) : c=1 := by
-  rw [sum_columns, sum_columns] at he
-  have hh := (Finset.sum_eq_sum_iff_of_le
-    (fun (j : Fin (μ.rowLen 0)) _ => column_rearrangement μ f hf c j)).mp he
-  apply Subtype.ext
-  apply Equiv.ext
-  intro x
-  exact column_rearrangement_eq μ f hf c (colIndex μ x)
-    (hh (colIndex μ x) (mem_univ _)) ⟨x,rfl⟩
-
-end
-end PartialPermutation.YoungTabloid
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.ColumnSorting
+
+namespace OAI
+
+section
+namespace PartialPermutation.YoungTabloid
+noncomputable section
+open Finset
+open scoped Classical
+
+lemma cell_eq_of_row_col (μ : YoungDiagram) {x y : μ.cells}
+    (hr : row μ x=row μ y) (hc : col μ x=col μ y) : x=y :=
+  Subtype.ext (Prod.ext hr hc)
+
+lemma cell_lt_of_column (μ : YoungDiagram) {x y : μ.cells}
+    (hc : col μ x=col μ y) (hr : row μ x<row μ y) : x<y := by
+  apply lt_of_le_of_ne
+  · exact ⟨hr.le,hc.le⟩
+  · intro h
+    exact hr.ne (congrArg (row μ) h)
+
+def columnRestrict (μ : YoungDiagram) (c : columnGroup μ) (j : ℕ) : Equiv.Perm (Col μ j) where
+  toFun x := ⟨c.1 x.1,(c.2 x.1).trans x.2⟩
+  invFun x := ⟨c.1⁻¹ x.1,((c⁻¹).2 x.1).trans x.2⟩
+  left_inv x := by apply Subtype.ext; exact c.1.symm_apply_apply _
+  right_inv x := by apply Subtype.ext; exact c.1.apply_symm_apply _
+
+lemma column_monovary (μ : YoungDiagram) (f : μ.cells → ℕ) (hf : StrictMono f) (j : ℕ) :
+    Monovary (fun x : Col μ j => f x.1) (fun x => row μ x.1) := by
+  intro x y h
+  exact (hf (cell_lt_of_column μ (x.2.trans y.2.symm) h)).le
+
+lemma column_rearrangement (μ : YoungDiagram) (f : μ.cells → ℕ) (hf : StrictMono f)
+    (c : columnGroup μ) (j : ℕ) :
+    (∑ x : Col μ j, f x.1 * row μ (c.1 x.1)) ≤ ∑ x : Col μ j, f x.1 * row μ x.1 :=
+  (column_monovary μ f hf j).sum_mul_comp_perm_le_sum_mul (σ := columnRestrict μ c j)
+
+lemma fin_perm_mono_eq (n : ℕ) (p : Equiv.Perm (Fin n)) (hp : StrictMono p) : p=1 := by
+  have hle (i : Fin n) : i.val≤(p i).val := strictMono_fin_nat_le _
+    (fun i j h => hp h) i
+  have heq : (∑ i : Fin n, i.val)=∑ i : Fin n, (p i).val := (Equiv.sum_comp p Fin.val).symm
+  have hh := (Finset.sum_eq_sum_iff_of_le (fun i _ => hle i)).mp heq
+  apply Equiv.ext
+  intro i
+  exact Fin.ext (hh i (mem_univ _)).symm
+
+lemma column_rearrangement_eq (μ : YoungDiagram) (f : μ.cells → ℕ) (hf : StrictMono f)
+    (c : columnGroup μ) (j : ℕ)
+    (he : (∑ x : Col μ j, f x.1 * row μ (c.1 x.1)) = ∑ x : Col μ j, f x.1 * row μ x.1) :
+    ∀ x : Col μ j, c.1 x.1=x.1 := by
+  have hm := (column_monovary μ f hf j).sum_mul_comp_perm_eq_sum_mul_iff
+    (σ := columnRestrict μ c j) |>.mp he
+  let e := columnEquiv μ j
+  let p := e.symm.trans ((columnRestrict μ c j).trans e)
+  have hp : StrictMono p := by
+    intro i k hik
+    let x := e.symm i
+    let y := e.symm k
+    have hxy : f x.1 < f y.1 := hf (cell_lt_of_column μ (x.2.trans y.2.symm) hik)
+    change row μ (c.1 x.1) < row μ (c.1 y.1)
+    by_contra h
+    have hr : row μ (c.1 y.1) < row μ (c.1 x.1) := by
+      apply lt_of_le_of_ne (Nat.le_of_not_gt h)
+      intro hh
+      have heq := c.1.injective (cell_eq_of_row_col μ hh
+        ((c.2 y.1).trans ((y.2.trans x.2.symm).trans (c.2 x.1).symm)))
+      exact hxy.ne (congrArg f heq.symm)
+    exact (not_le_of_gt hxy) (hm hr)
+  have hp1 := fin_perm_mono_eq _ p hp
+  intro x
+  have hx := Equiv.congr_fun hp1 (e x)
+  change e (columnRestrict μ c j (e.symm (e x)))=e x at hx
+  erw [Equiv.symm_apply_apply] at hx
+  exact congrArg Subtype.val (e.injective hx)
+
+def colIndex (μ : YoungDiagram) (x : μ.cells) : Fin (μ.rowLen 0) :=
+  ⟨col μ x,(μ.mem_iff_lt_rowLen.mp x.2).trans_le
+    (μ.rowLen_anti 0 x.1.1 (Nat.zero_le _))⟩
+
+def columnDecomp (μ : YoungDiagram) : μ.cells ≃ Σ j : Fin (μ.rowLen 0), Col μ j where
+  toFun x := ⟨colIndex μ x,⟨x,rfl⟩⟩
+  invFun x := x.2.1
+  left_inv _ := rfl
+  right_inv x := by
+    obtain ⟨j,x,hx⟩ := x
+    have hj : colIndex μ x=j := Fin.ext hx
+    cases hj
+    rfl
+
+lemma sum_columns (μ : YoungDiagram) (f : μ.cells → ℕ) :
+    (∑ x, f x)=∑ j : Fin (μ.rowLen 0), ∑ x : Col μ j, f x.1 := by
+  exact (Fintype.sum_equiv (columnDecomp μ) _
+    (fun x : Σ j : Fin (μ.rowLen 0), Col μ j => f x.2.1) (fun _ => rfl)).trans
+    (Fintype.sum_sigma _)
+
+lemma full_rearrangement (μ : YoungDiagram) (f : μ.cells → ℕ) (hf : StrictMono f)
+    (c : columnGroup μ) :
+    (∑ x, f x * row μ (c.1 x)) ≤ ∑ x, f x * row μ x := by
+  rw [sum_columns, sum_columns]
+  exact Finset.sum_le_sum (fun j _ => column_rearrangement μ f hf c j)
+
+lemma full_rearrangement_eq (μ : YoungDiagram) (f : μ.cells → ℕ) (hf : StrictMono f)
+    (c : columnGroup μ) (he : (∑ x, f x * row μ (c.1 x)) = ∑ x, f x * row μ x) : c=1 := by
+  rw [sum_columns, sum_columns] at he
+  have hh := (Finset.sum_eq_sum_iff_of_le
+    (fun (j : Fin (μ.rowLen 0)) _ => column_rearrangement μ f hf c j)).mp he
+  apply Subtype.ext
+  apply Equiv.ext
+  intro x
+  exact column_rearrangement_eq μ f hf c (colIndex μ x)
+    (hh (colIndex μ x) (mem_univ _)) ⟨x,rfl⟩
+
+end
+end PartialPermutation.YoungTabloid
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/ColumnSorting.lean b/lean/OAI/RepresentationTheory/PartialPermutation/ColumnSorting.lean
index bb0fbb36d114c2f602d72fd062268322345a35bb..a68c61143bcc4b540e6c46ed3b48ef9ed33756b2 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/ColumnSorting.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/ColumnSorting.lean
@@ -1,110 +1,110 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.Tabloids
-
-namespace OAI
-
-section
-namespace PartialPermutation
-namespace YoungTabloid
-noncomputable section
-open Finset
-open scoped Classical
-
-lemma exists_sortingEquiv (α : Type*) [Fintype α] (r : α → ℕ) (hr : Function.Injective r)
-    (n : ℕ) (hn : Fintype.card α=n) :
-    ∃ e : α ≃ Fin n, StrictMono (fun i => r (e.symm i)) := by
-  let : LinearOrder α := LinearOrder.lift' r hr
-  let e := Fintype.orderIsoFinOfCardEq α hn
-  exact ⟨e.toEquiv.symm, e.strictMono⟩
-
-lemma strictMono_fin_nat_le {n : ℕ} (f : Fin n → ℕ) (hf : StrictMono f) (i : Fin n) :
-    i.val ≤ f i := by
-  obtain ⟨i,hi⟩ := i
-  induction i with
-  | zero => exact Nat.zero_le _
-  | succ i ih =>
-    have hi' : i<n := by omega
-    have h := hf (show (⟨i,hi'⟩:Fin n)<⟨i+1,hi⟩ by simp)
-    have ih' := ih hi'
-    change i ≤ f ⟨i,hi'⟩ at ih'
-    change i+1 ≤ f ⟨i+1,hi⟩
-    omega
-
-def Transversal (μ : YoungDiagram) (r : Tabloid μ) : Prop :=
-  ∀ x y : μ.cells, col μ x=col μ y → r.1 x=r.1 y → x=y
-
-lemma transversal_col_injective (μ : YoungDiagram) (r : Tabloid μ) (hr : Transversal μ r)
-    (j : ℕ) : Function.Injective (fun x : Col μ j => r.1 x.1) := by
-  intro x y h
-  apply Subtype.ext
-  exact hr x.1 y.1 (x.2.trans y.2.symm) h
-
-def sortedColumn (μ : YoungDiagram) (r : Tabloid μ) (hr : Transversal μ r) (j : ℕ) :
-    Col μ j ≃ Fin (μ.colLen j) :=
-  (exists_sortingEquiv (Col μ j) (fun x => r.1 x.1)
-    (transversal_col_injective μ r hr j) _ (col_card μ j)).choose
-
-lemma sortedColumn_mono (μ : YoungDiagram) (r : Tabloid μ) (hr : Transversal μ r) (j : ℕ) :
-    StrictMono (fun i => r.1 ((sortedColumn μ r hr j).symm i).1) :=
-  (exists_sortingEquiv (Col μ j) (fun x => r.1 x.1)
-    (transversal_col_injective μ r hr j) _ (col_card μ j)).choose_spec
-
-def fiberPermutation {α ι : Type*} (r : α → ι)
-    (p : ∀ i, Equiv.Perm {a // r a=i}) : Equiv.Perm α :=
-  (Equiv.sigmaFiberEquiv r).symm.trans
-    ((Equiv.sigmaCongrRight p).trans (Equiv.sigmaFiberEquiv r))
-
-lemma fiberPermutation_apply {α ι : Type*} (r : α → ι)
-    (p : ∀ i, Equiv.Perm {a // r a=i}) (a : α) :
-    fiberPermutation r p a = (p (r a) ⟨a,rfl⟩).1 := rfl
-
-lemma fiberPermutation_rank {α ι : Type*} (r : α → ι)
-    (p : ∀ i, Equiv.Perm {a // r a=i}) (a : α) :
-    r (fiberPermutation r p a)=r a := (p (r a) ⟨a,rfl⟩).2
-
-def columnSorting (μ : YoungDiagram) (r : Tabloid μ) (hr : Transversal μ r) :
-    Equiv.Perm μ.cells :=
-  fiberPermutation (col μ) (fun j => (sortedColumn μ r hr j).trans (columnEquiv μ j).symm)
-
-lemma columnSorting_mem (μ : YoungDiagram) (r : Tabloid μ) (hr : Transversal μ r) :
-    columnSorting μ r hr ∈ columnGroup μ :=
-  fiberPermutation_rank (col μ) _
-
-lemma columnSorting_row (μ : YoungDiagram) (r : Tabloid μ) (hr : Transversal μ r)
-    (x : μ.cells) : row μ (columnSorting μ r hr x) ≤ r.1 x := by
-  let e := sortedColumn μ r hr (col μ x)
-  have h := strictMono_fin_nat_le _ (sortedColumn_mono μ r hr (col μ x)) (e ⟨x,rfl⟩)
-  change (e ⟨x,rfl⟩).val ≤ r.1 (e.symm (e ⟨x,rfl⟩)).val at h
-  erw [Equiv.symm_apply_apply] at h
-  exact h
-
-lemma transversal_eq_column (μ : YoungDiagram) (r : Tabloid μ) (hr : Transversal μ r) :
-    ∃ c : columnGroup μ, r.1=fun x => row μ (c.1 x) := by
-  let c := columnSorting μ r hr
-  have hle := columnSorting_row μ r hr
-  have heq : (∑ x, row μ (c x))=∑ x, r.1 x := by
-    rw [rowSum]
-    exact Fintype.sum_equiv c _ _ (fun _ => rfl)
-  refine ⟨⟨c,columnSorting_mem μ r hr⟩,funext (fun x => ?_)⟩
-  exact (Finset.sum_eq_sum_iff_of_le (fun x _ => hle x)).mp heq x (Finset.mem_univ x) |>.symm
-
-lemma column_row_injective (μ : YoungDiagram) :
-    Function.Injective (fun c : columnGroup μ => fun x => row μ (c.1 x)) := by
-  intro c d h
-  apply Subtype.ext
-  apply Equiv.ext
-  intro x
-  apply Subtype.ext
-  exact Prod.ext (congrFun h x) ((c.2 x).trans (d.2 x).symm)
-
-lemma unique_column_of_transversal (μ : YoungDiagram) (r : Tabloid μ) (hr : Transversal μ r) :
-    ∃! c : columnGroup μ, r.1=fun x => row μ (c.1 x) := by
-  obtain ⟨c,hc⟩ := transversal_eq_column μ r hr
-  exact ⟨c,hc,fun d hd => column_row_injective μ (hd.symm.trans hc)⟩
-
-end
-end YoungTabloid
-end PartialPermutation
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.Tabloids
+
+namespace OAI
+
+section
+namespace PartialPermutation
+namespace YoungTabloid
+noncomputable section
+open Finset
+open scoped Classical
+
+lemma exists_sortingEquiv (α : Type*) [Fintype α] (r : α → ℕ) (hr : Function.Injective r)
+    (n : ℕ) (hn : Fintype.card α=n) :
+    ∃ e : α ≃ Fin n, StrictMono (fun i => r (e.symm i)) := by
+  let : LinearOrder α := LinearOrder.lift' r hr
+  let e := Fintype.orderIsoFinOfCardEq α hn
+  exact ⟨e.toEquiv.symm, e.strictMono⟩
+
+lemma strictMono_fin_nat_le {n : ℕ} (f : Fin n → ℕ) (hf : StrictMono f) (i : Fin n) :
+    i.val ≤ f i := by
+  obtain ⟨i,hi⟩ := i
+  induction i with
+  | zero => exact Nat.zero_le _
+  | succ i ih =>
+    have hi' : i<n := by omega
+    have h := hf (show (⟨i,hi'⟩:Fin n)<⟨i+1,hi⟩ by simp)
+    have ih' := ih hi'
+    change i ≤ f ⟨i,hi'⟩ at ih'
+    change i+1 ≤ f ⟨i+1,hi⟩
+    omega
+
+def Transversal (μ : YoungDiagram) (r : Tabloid μ) : Prop :=
+  ∀ x y : μ.cells, col μ x=col μ y → r.1 x=r.1 y → x=y
+
+lemma transversal_col_injective (μ : YoungDiagram) (r : Tabloid μ) (hr : Transversal μ r)
+    (j : ℕ) : Function.Injective (fun x : Col μ j => r.1 x.1) := by
+  intro x y h
+  apply Subtype.ext
+  exact hr x.1 y.1 (x.2.trans y.2.symm) h
+
+def sortedColumn (μ : YoungDiagram) (r : Tabloid μ) (hr : Transversal μ r) (j : ℕ) :
+    Col μ j ≃ Fin (μ.colLen j) :=
+  (exists_sortingEquiv (Col μ j) (fun x => r.1 x.1)
+    (transversal_col_injective μ r hr j) _ (col_card μ j)).choose
+
+lemma sortedColumn_mono (μ : YoungDiagram) (r : Tabloid μ) (hr : Transversal μ r) (j : ℕ) :
+    StrictMono (fun i => r.1 ((sortedColumn μ r hr j).symm i).1) :=
+  (exists_sortingEquiv (Col μ j) (fun x => r.1 x.1)
+    (transversal_col_injective μ r hr j) _ (col_card μ j)).choose_spec
+
+def fiberPermutation {α ι : Type*} (r : α → ι)
+    (p : ∀ i, Equiv.Perm {a // r a=i}) : Equiv.Perm α :=
+  (Equiv.sigmaFiberEquiv r).symm.trans
+    ((Equiv.sigmaCongrRight p).trans (Equiv.sigmaFiberEquiv r))
+
+lemma fiberPermutation_apply {α ι : Type*} (r : α → ι)
+    (p : ∀ i, Equiv.Perm {a // r a=i}) (a : α) :
+    fiberPermutation r p a = (p (r a) ⟨a,rfl⟩).1 := rfl
+
+lemma fiberPermutation_rank {α ι : Type*} (r : α → ι)
+    (p : ∀ i, Equiv.Perm {a // r a=i}) (a : α) :
+    r (fiberPermutation r p a)=r a := (p (r a) ⟨a,rfl⟩).2
+
+def columnSorting (μ : YoungDiagram) (r : Tabloid μ) (hr : Transversal μ r) :
+    Equiv.Perm μ.cells :=
+  fiberPermutation (col μ) (fun j => (sortedColumn μ r hr j).trans (columnEquiv μ j).symm)
+
+lemma columnSorting_mem (μ : YoungDiagram) (r : Tabloid μ) (hr : Transversal μ r) :
+    columnSorting μ r hr ∈ columnGroup μ :=
+  fiberPermutation_rank (col μ) _
+
+lemma columnSorting_row (μ : YoungDiagram) (r : Tabloid μ) (hr : Transversal μ r)
+    (x : μ.cells) : row μ (columnSorting μ r hr x) ≤ r.1 x := by
+  let e := sortedColumn μ r hr (col μ x)
+  have h := strictMono_fin_nat_le _ (sortedColumn_mono μ r hr (col μ x)) (e ⟨x,rfl⟩)
+  change (e ⟨x,rfl⟩).val ≤ r.1 (e.symm (e ⟨x,rfl⟩)).val at h
+  erw [Equiv.symm_apply_apply] at h
+  exact h
+
+lemma transversal_eq_column (μ : YoungDiagram) (r : Tabloid μ) (hr : Transversal μ r) :
+    ∃ c : columnGroup μ, r.1=fun x => row μ (c.1 x) := by
+  let c := columnSorting μ r hr
+  have hle := columnSorting_row μ r hr
+  have heq : (∑ x, row μ (c x))=∑ x, r.1 x := by
+    rw [rowSum]
+    exact Fintype.sum_equiv c _ _ (fun _ => rfl)
+  refine ⟨⟨c,columnSorting_mem μ r hr⟩,funext (fun x => ?_)⟩
+  exact (Finset.sum_eq_sum_iff_of_le (fun x _ => hle x)).mp heq x (Finset.mem_univ x) |>.symm
+
+lemma column_row_injective (μ : YoungDiagram) :
+    Function.Injective (fun c : columnGroup μ => fun x => row μ (c.1 x)) := by
+  intro c d h
+  apply Subtype.ext
+  apply Equiv.ext
+  intro x
+  apply Subtype.ext
+  exact Prod.ext (congrFun h x) ((c.2 x).trans (d.2 x).symm)
+
+lemma unique_column_of_transversal (μ : YoungDiagram) (r : Tabloid μ) (hr : Transversal μ r) :
+    ∃! c : columnGroup μ, r.1=fun x => row μ (c.1 x) := by
+  obtain ⟨c,hc⟩ := transversal_eq_column μ r hr
+  exact ⟨c,hc,fun d hd => column_row_injective μ (hd.symm.trans hc)⟩
+
+end
+end YoungTabloid
+end PartialPermutation
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/CompleteKernels.lean b/lean/OAI/RepresentationTheory/PartialPermutation/CompleteKernels.lean
index 19f6026f389583eaeeb1043c4011253646188941..ff9a3b099d6ddaeac61c737a4b245950de98cb34 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/CompleteKernels.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/CompleteKernels.lean
@@ -1,101 +1,101 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.Components
-import OAI.RepresentationTheory.PartialPermutation.CentralKernels
-
-namespace OAI
-
-section
-open scoped Classical
-open scoped BigOperators ComplexConjugate MonoidAlgebra
-open scoped BigOperators ComplexConjugate
-open scoped MonoidAlgebra BigOperators
-
-namespace PartialPermutation
-noncomputable section
-open scoped MonoidAlgebra BigOperators
-
-section CompleteKernels
-variable {G V : Type*} [Group G] [Fintype G] [AddCommGroup V] [Module ℂ V]
-    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V)
-
-def componentRep (c : isotypicComponents ℂ[G] ρ.asModule) :=
-  (Subrepresentation.ofSubmodule' (componentSimple ρ c)).toRepresentation
-
-instance componentRep_irreducible (c : isotypicComponents ℂ[G] ρ.asModule) :
-    Representation.IsIrreducible (componentRep ρ c) :=
-  (subrep_irreducible_iff ρ _).mpr (componentSimple_simple ρ c)
-
-lemma componentRep_equiv_iff (c d : isotypicComponents ℂ[G] ρ.asModule) :
-    Nonempty (Representation.Equiv (componentRep ρ c) (componentRep ρ d)) ↔ c = d := by
-  constructor
-  · rintro ⟨e⟩
-    apply componentCharacter_injective ρ
-    exact Subtype.ext (Representation.char_iso e)
-  · rintro rfl
-    exact ⟨Representation.Equiv.refl _⟩
-
-omit [FiniteDimensional ℂ V] in
-lemma complexFourier_subrep (σ : Subrepresentation ρ) (a : G → ℂ) (x : σ) :
-    ((complexFourier σ.toRepresentation a x : σ) : V) = complexFourier ρ a x := by
-  simp only [complexFourier, LinearMap.sum_apply, LinearMap.smul_apply,
-    Submodule.coe_sum, Submodule.coe_smul]
-  rfl
-
-omit [FiniteDimensional ℂ V] in
-lemma complexFourier_sum {ι : Type*} [Fintype ι] (a : ι → G → ℂ) :
-    complexFourier ρ (fun g => ∑ i, a i g) = ∑ i, complexFourier ρ (a i) := by
-  simp only [complexFourier, Finset.sum_smul]
-  exact Finset.sum_comm
-
-lemma sum_character_kernels_identity
-    [Fintype (isotypicComponents ℂ[G] ρ.asModule)] :
-    complexFourier ρ (fun g => ∑ c, centralCharacterKernel (componentRep ρ c) g) = 1 := by
-  classical
-  let K : G → ℂ := fun g => ∑ c, centralCharacterKernel (componentRep ρ c) g
-  have hs (σ : Subrepresentation ρ) [Representation.IsIrreducible σ.toRepresentation] :
-      complexFourier σ.toRepresentation K = 1 := by
-    let χ : OccurringTypes ρ := ⟨σ.toRepresentation.character, σ, inferInstance, rfl⟩
-    obtain ⟨c, hc⟩ := componentCharacter_surjective ρ χ
-    have he : Nonempty (Representation.Equiv (componentRep ρ c) σ.toRepresentation) := by
-      exact irreducible_equiv_of_character_eq σ.toRepresentation (componentRep ρ c)
-        (congrArg Subtype.val hc).symm
-    rw [complexFourier_sum]
-    rw [Finset.sum_eq_single c]
-    · rw [centralCharacterKernel_fourier, ite_eq_left he]
-    · intro d _ hdc
-      rw [centralCharacterKernel_fourier, ite_eq_right]
-      rintro ⟨e⟩
-      obtain ⟨ec⟩ := he
-      exact hdc ((componentRep_equiv_iff ρ d c).mp ⟨e.trans ec.symm⟩)
-    · simp
-  let E (S : {S : Submodule ℂ[G] ρ.asModule // IsSimpleModule ℂ[G] S}) : Submodule ℂ V :=
-    S.val.restrictScalars ℂ
-  have htop : (⨆ S, E S) = ⊤ := by
-    change (⨆ S : {S : Submodule ℂ[G] ρ.asModule // IsSimpleModule ℂ[G] S},
-      S.val.restrictScalars ℂ) = ⊤
-    rw [← Submodule.restrictScalars_iSup]
-    have ht : (⨆ S : {S : Submodule ℂ[G] ρ.asModule // IsSimpleModule ℂ[G] S}, S.val) = ⊤ :=
-      (sSup_eq_iSup' {S : Submodule ℂ[G] ρ.asModule | IsSimpleModule ℂ[G] S}).symm.trans
-        (IsSemisimpleModule.sSup_simples_eq_top ℂ[G] ρ.asModule)
-    rw [ht, Submodule.restrictScalars_top]
-  have hker : (⊤ : Submodule ℂ V) ≤ LinearMap.ker (complexFourier ρ K - 1) := by
-    rw [← htop]
-    refine iSup_le fun S => ?_
-    intro x hx
-    let σ := Subrepresentation.ofSubmodule' S.val
-    have : Representation.IsIrreducible σ.toRepresentation :=
-      (subrep_irreducible_iff ρ σ).mpr S.property
-    have ht := congrArg (fun A : Module.End ℂ σ.toSubmodule => ((A ⟨x, hx⟩ : σ.toSubmodule) : V)) (hs σ)
-    rw [complexFourier_subrep] at ht
-    simpa using sub_eq_zero.mpr ht
-  ext x
-  have hx := hker (Submodule.mem_top : x ∈ (⊤ : Submodule ℂ V))
-  simpa [LinearMap.mem_ker, sub_eq_zero] using hx
-
-end CompleteKernels
-end
-end PartialPermutation
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.Components
+import OAI.RepresentationTheory.PartialPermutation.CentralKernels
+
+namespace OAI
+
+section
+open scoped Classical
+open scoped BigOperators ComplexConjugate MonoidAlgebra
+open scoped BigOperators ComplexConjugate
+open scoped MonoidAlgebra BigOperators
+
+namespace PartialPermutation
+noncomputable section
+open scoped MonoidAlgebra BigOperators
+
+section CompleteKernels
+variable {G V : Type*} [Group G] [Fintype G] [AddCommGroup V] [Module ℂ V]
+    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V)
+
+def componentRep (c : isotypicComponents ℂ[G] ρ.asModule) :=
+  (Subrepresentation.ofSubmodule' (componentSimple ρ c)).toRepresentation
+
+instance componentRep_irreducible (c : isotypicComponents ℂ[G] ρ.asModule) :
+    Representation.IsIrreducible (componentRep ρ c) :=
+  (subrep_irreducible_iff ρ _).mpr (componentSimple_simple ρ c)
+
+lemma componentRep_equiv_iff (c d : isotypicComponents ℂ[G] ρ.asModule) :
+    Nonempty (Representation.Equiv (componentRep ρ c) (componentRep ρ d)) ↔ c = d := by
+  constructor
+  · rintro ⟨e⟩
+    apply componentCharacter_injective ρ
+    exact Subtype.ext (Representation.char_iso e)
+  · rintro rfl
+    exact ⟨Representation.Equiv.refl _⟩
+
+omit [FiniteDimensional ℂ V] in
+lemma complexFourier_subrep (σ : Subrepresentation ρ) (a : G → ℂ) (x : σ) :
+    ((complexFourier σ.toRepresentation a x : σ) : V) = complexFourier ρ a x := by
+  simp only [complexFourier, LinearMap.sum_apply, LinearMap.smul_apply,
+    Submodule.coe_sum, Submodule.coe_smul]
+  rfl
+
+omit [FiniteDimensional ℂ V] in
+lemma complexFourier_sum {ι : Type*} [Fintype ι] (a : ι → G → ℂ) :
+    complexFourier ρ (fun g => ∑ i, a i g) = ∑ i, complexFourier ρ (a i) := by
+  simp only [complexFourier, Finset.sum_smul]
+  exact Finset.sum_comm
+
+lemma sum_character_kernels_identity
+    [Fintype (isotypicComponents ℂ[G] ρ.asModule)] :
+    complexFourier ρ (fun g => ∑ c, centralCharacterKernel (componentRep ρ c) g) = 1 := by
+  classical
+  let K : G → ℂ := fun g => ∑ c, centralCharacterKernel (componentRep ρ c) g
+  have hs (σ : Subrepresentation ρ) [Representation.IsIrreducible σ.toRepresentation] :
+      complexFourier σ.toRepresentation K = 1 := by
+    let χ : OccurringTypes ρ := ⟨σ.toRepresentation.character, σ, inferInstance, rfl⟩
+    obtain ⟨c, hc⟩ := componentCharacter_surjective ρ χ
+    have he : Nonempty (Representation.Equiv (componentRep ρ c) σ.toRepresentation) := by
+      exact irreducible_equiv_of_character_eq σ.toRepresentation (componentRep ρ c)
+        (congrArg Subtype.val hc).symm
+    rw [complexFourier_sum]
+    rw [Finset.sum_eq_single c]
+    · rw [centralCharacterKernel_fourier, ite_eq_left he]
+    · intro d _ hdc
+      rw [centralCharacterKernel_fourier, ite_eq_right]
+      rintro ⟨e⟩
+      obtain ⟨ec⟩ := he
+      exact hdc ((componentRep_equiv_iff ρ d c).mp ⟨e.trans ec.symm⟩)
+    · simp
+  let E (S : {S : Submodule ℂ[G] ρ.asModule // IsSimpleModule ℂ[G] S}) : Submodule ℂ V :=
+    S.val.restrictScalars ℂ
+  have htop : (⨆ S, E S) = ⊤ := by
+    change (⨆ S : {S : Submodule ℂ[G] ρ.asModule // IsSimpleModule ℂ[G] S},
+      S.val.restrictScalars ℂ) = ⊤
+    rw [← Submodule.restrictScalars_iSup]
+    have ht : (⨆ S : {S : Submodule ℂ[G] ρ.asModule // IsSimpleModule ℂ[G] S}, S.val) = ⊤ :=
+      (sSup_eq_iSup' {S : Submodule ℂ[G] ρ.asModule | IsSimpleModule ℂ[G] S}).symm.trans
+        (IsSemisimpleModule.sSup_simples_eq_top ℂ[G] ρ.asModule)
+    rw [ht, Submodule.restrictScalars_top]
+  have hker : (⊤ : Submodule ℂ V) ≤ LinearMap.ker (complexFourier ρ K - 1) := by
+    rw [← htop]
+    refine iSup_le fun S => ?_
+    intro x hx
+    let σ := Subrepresentation.ofSubmodule' S.val
+    have : Representation.IsIrreducible σ.toRepresentation :=
+      (subrep_irreducible_iff ρ σ).mpr S.property
+    have ht := congrArg (fun A : Module.End ℂ σ.toSubmodule => ((A ⟨x, hx⟩ : σ.toSubmodule) : V)) (hs σ)
+    rw [complexFourier_subrep] at ht
+    simpa using sub_eq_zero.mpr ht
+  ext x
+  have hx := hker (Submodule.mem_top : x ∈ (⊤ : Submodule ℂ V))
+  simpa [LinearMap.mem_ker, sub_eq_zero] using hx
+
+end CompleteKernels
+end
+end PartialPermutation
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/ComponentGeometry.lean b/lean/OAI/RepresentationTheory/PartialPermutation/ComponentGeometry.lean
index 5895ca4f2474290df42ea62d72a167a95fccf483..af8e99f2a1d969758c125616f0df182a31374965 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/ComponentGeometry.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/ComponentGeometry.lean
@@ -1,96 +1,96 @@
-import Mathlib
-
-namespace OAI
-
-section
-open scoped Classical
-open scoped BigOperators ComplexConjugate MonoidAlgebra
-open scoped BigOperators ComplexConjugate
-open scoped MonoidAlgebra BigOperators
-open scoped BigOperators MonoidAlgebra Classical
-
-namespace PartialPermutation
-noncomputable section
-section Geometry
-variable {G V : Type*} [Group G] [Fintype G]
-    [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-    (ρ : Representation ℂ G V)
-
-def componentSpace (c : isotypicComponents ℂ[G] ρ.asModule) : Submodule ℂ V :=
-  c.val.restrictScalars ℂ
-
-omit [Fintype G] [FiniteDimensional ℂ V] in
-lemma componentSpace_invariant (c : isotypicComponents ℂ[G] ρ.asModule)
-    (g : G) (v : V) (hv : v ∈ componentSpace ρ c) : ρ g v ∈ componentSpace ρ c := by
-  have h := c.val.smul_mem (MonoidAlgebra.single g (1 : ℂ)) hv
-  change ρ.asAlgebraHom (MonoidAlgebra.single g 1) v ∈ c.val at h
-  change ρ g v ∈ c.val
-  simpa using h
-
-omit [FiniteDimensional ℂ V] in
-lemma iSup_componentSpace : ⨆ c, componentSpace ρ c = ⊤ := by
-  change (⨆ c : isotypicComponents ℂ[G] ρ.asModule, c.val.restrictScalars ℂ) = ⊤
-  rw [← Submodule.restrictScalars_iSup, ← sSup_eq_iSup', sSup_isotypicComponents,
-    Submodule.restrictScalars_top]
-
-omit [Fintype G] in
-lemma componentSpace_projection_commute
-    (hρ : ∀ g x y, inner ℂ (ρ g x) (ρ g y) = inner ℂ x y)
-    (c : isotypicComponents ℂ[G] ρ.asModule) (g : G) (v : V) :
-    ρ g ((componentSpace ρ c).starProjection v) =
-      (componentSpace ρ c).starProjection (ρ g v) := by
-  have he : (componentSpace ρ c).map (ρ g) = componentSpace ρ c := by
-    apply le_antisymm
-    · rintro x ⟨v, hv, rfl⟩
-      exact componentSpace_invariant ρ c g v hv
-    · intro v hv
-      exact ⟨ρ g⁻¹ v, componentSpace_invariant ρ c g⁻¹ v hv, by simp⟩
-  have hm := ((ρ g).isometryOfInner (hρ g)).map_starProjection (componentSpace ρ c) v
-  simpa only [LinearMap.coe_isometryOfInner, LinearMap.isometryOfInner_toLinearMap, he] using hm
-
-omit [Fintype G] in
-lemma componentSpace_orthogonal
-    (hρ : ∀ g x y, inner ℂ (ρ g x) (ρ g y) = inner ℂ x y) :
-    OrthogonalFamily ℂ (fun c => componentSpace ρ c) (fun c => (componentSpace ρ c).subtypeₗᵢ) := by
-  apply OrthogonalFamily.of_pairwise
-  intro c d hcd x hx
-  apply (Submodule.starProjection_apply_eq_zero_iff (K := componentSpace ρ d)).mp
-  let p : ρ.IntertwiningMap ρ :=
-    { toLinearMap := (componentSpace ρ d).starProjection.toLinearMap
-      isIntertwining' g := by
-        ext v
-        exact (componentSpace_projection_commute ρ hρ d g v).symm }
-  let f := Representation.IntertwiningMap.equivLinearMapAsModule ρ ρ p
-  have hi : c.val.IsFullyInvariant := .of_mem_isotypicComponents c.property
-  have hmem : f x ∈ c.val := hi f hx
-  have hmem' : f x ∈ d.val := (componentSpace ρ d).starProjection_apply_mem x
-  have hd : Disjoint c.val d.val :=
-    (sSupIndep_isotypicComponents ℂ[G] ρ.asModule).pairwiseDisjoint c.property d.property
-      (Subtype.coe_ne_coe.mpr hcd)
-  exact Submodule.disjoint_def.mp hd (f x) hmem hmem'
-
-lemma sum_component_projections
-    (hρ : ∀ g x y, inner ℂ (ρ g x) (ρ g y) = inner ℂ x y)
-    [Fintype (isotypicComponents ℂ[G] ρ.asModule)] (v : V) :
-    ∑ c, (componentSpace ρ c).starProjection v = v :=
-  (componentSpace_orthogonal ρ hρ).sum_projection_of_mem_iSup v
-    (by rw [iSup_componentSpace]; trivial)
-
-lemma sum_component_norm_sq
-    (hρ : ∀ g x y, inner ℂ (ρ g x) (ρ g y) = inner ℂ x y)
-    [Fintype (isotypicComponents ℂ[G] ρ.asModule)] (v : V) :
-    ∑ c, ‖(componentSpace ρ c).starProjection v‖ ^ 2 = ‖v‖ ^ 2 := by
-  have h := (componentSpace_orthogonal ρ hρ).norm_sum
-    (fun c => (componentSpace ρ c).orthogonalProjectionOnto v) Finset.univ
-  change ‖∑ c, (componentSpace ρ c).starProjection v‖ ^ 2 =
-    ∑ c, ‖(componentSpace ρ c).starProjection v‖ ^ 2 at h
-  rw [sum_component_projections ρ hρ] at h
-  exact h.symm
-
-end Geometry
-end
-end PartialPermutation
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+section
+open scoped Classical
+open scoped BigOperators ComplexConjugate MonoidAlgebra
+open scoped BigOperators ComplexConjugate
+open scoped MonoidAlgebra BigOperators
+open scoped BigOperators MonoidAlgebra Classical
+
+namespace PartialPermutation
+noncomputable section
+section Geometry
+variable {G V : Type*} [Group G] [Fintype G]
+    [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+    (ρ : Representation ℂ G V)
+
+def componentSpace (c : isotypicComponents ℂ[G] ρ.asModule) : Submodule ℂ V :=
+  c.val.restrictScalars ℂ
+
+omit [Fintype G] [FiniteDimensional ℂ V] in
+lemma componentSpace_invariant (c : isotypicComponents ℂ[G] ρ.asModule)
+    (g : G) (v : V) (hv : v ∈ componentSpace ρ c) : ρ g v ∈ componentSpace ρ c := by
+  have h := c.val.smul_mem (MonoidAlgebra.single g (1 : ℂ)) hv
+  change ρ.asAlgebraHom (MonoidAlgebra.single g 1) v ∈ c.val at h
+  change ρ g v ∈ c.val
+  simpa using h
+
+omit [FiniteDimensional ℂ V] in
+lemma iSup_componentSpace : ⨆ c, componentSpace ρ c = ⊤ := by
+  change (⨆ c : isotypicComponents ℂ[G] ρ.asModule, c.val.restrictScalars ℂ) = ⊤
+  rw [← Submodule.restrictScalars_iSup, ← sSup_eq_iSup', sSup_isotypicComponents,
+    Submodule.restrictScalars_top]
+
+omit [Fintype G] in
+lemma componentSpace_projection_commute
+    (hρ : ∀ g x y, inner ℂ (ρ g x) (ρ g y) = inner ℂ x y)
+    (c : isotypicComponents ℂ[G] ρ.asModule) (g : G) (v : V) :
+    ρ g ((componentSpace ρ c).starProjection v) =
+      (componentSpace ρ c).starProjection (ρ g v) := by
+  have he : (componentSpace ρ c).map (ρ g) = componentSpace ρ c := by
+    apply le_antisymm
+    · rintro x ⟨v, hv, rfl⟩
+      exact componentSpace_invariant ρ c g v hv
+    · intro v hv
+      exact ⟨ρ g⁻¹ v, componentSpace_invariant ρ c g⁻¹ v hv, by simp⟩
+  have hm := ((ρ g).isometryOfInner (hρ g)).map_starProjection (componentSpace ρ c) v
+  simpa only [LinearMap.coe_isometryOfInner, LinearMap.isometryOfInner_toLinearMap, he] using hm
+
+omit [Fintype G] in
+lemma componentSpace_orthogonal
+    (hρ : ∀ g x y, inner ℂ (ρ g x) (ρ g y) = inner ℂ x y) :
+    OrthogonalFamily ℂ (fun c => componentSpace ρ c) (fun c => (componentSpace ρ c).subtypeₗᵢ) := by
+  apply OrthogonalFamily.of_pairwise
+  intro c d hcd x hx
+  apply (Submodule.starProjection_apply_eq_zero_iff (K := componentSpace ρ d)).mp
+  let p : ρ.IntertwiningMap ρ :=
+    { toLinearMap := (componentSpace ρ d).starProjection.toLinearMap
+      isIntertwining' g := by
+        ext v
+        exact (componentSpace_projection_commute ρ hρ d g v).symm }
+  let f := Representation.IntertwiningMap.equivLinearMapAsModule ρ ρ p
+  have hi : c.val.IsFullyInvariant := .of_mem_isotypicComponents c.property
+  have hmem : f x ∈ c.val := hi f hx
+  have hmem' : f x ∈ d.val := (componentSpace ρ d).starProjection_apply_mem x
+  have hd : Disjoint c.val d.val :=
+    (sSupIndep_isotypicComponents ℂ[G] ρ.asModule).pairwiseDisjoint c.property d.property
+      (Subtype.coe_ne_coe.mpr hcd)
+  exact Submodule.disjoint_def.mp hd (f x) hmem hmem'
+
+lemma sum_component_projections
+    (hρ : ∀ g x y, inner ℂ (ρ g x) (ρ g y) = inner ℂ x y)
+    [Fintype (isotypicComponents ℂ[G] ρ.asModule)] (v : V) :
+    ∑ c, (componentSpace ρ c).starProjection v = v :=
+  (componentSpace_orthogonal ρ hρ).sum_projection_of_mem_iSup v
+    (by rw [iSup_componentSpace]; trivial)
+
+lemma sum_component_norm_sq
+    (hρ : ∀ g x y, inner ℂ (ρ g x) (ρ g y) = inner ℂ x y)
+    [Fintype (isotypicComponents ℂ[G] ρ.asModule)] (v : V) :
+    ∑ c, ‖(componentSpace ρ c).starProjection v‖ ^ 2 = ‖v‖ ^ 2 := by
+  have h := (componentSpace_orthogonal ρ hρ).norm_sum
+    (fun c => (componentSpace ρ c).orthogonalProjectionOnto v) Finset.univ
+  change ‖∑ c, (componentSpace ρ c).starProjection v‖ ^ 2 =
+    ∑ c, ‖(componentSpace ρ c).starProjection v‖ ^ 2 at h
+  rw [sum_component_projections ρ hρ] at h
+  exact h.symm
+
+end Geometry
+end
+end PartialPermutation
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/Components.lean b/lean/OAI/RepresentationTheory/PartialPermutation/Components.lean
index c6a7df6e5691af7b2f028d4eb09824da56b2ba58..990618f27d339e76d43fb85ebb8a5e44209e5b8b 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/Components.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/Components.lean
@@ -1,85 +1,85 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.Model
-import OAI.RepresentationTheory.PartialPermutation.ModuleTransport
-
-namespace OAI
-
-section
-open scoped Classical
-open scoped BigOperators ComplexConjugate MonoidAlgebra
-open scoped BigOperators ComplexConjugate
-open scoped MonoidAlgebra BigOperators
-
-namespace PartialPermutation
-noncomputable section
-section Components
-variable {G V : Type*} [Group G] [Fintype G] [AddCommGroup V] [Module ℂ V]
-    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V)
-
-omit [Fintype G] [FiniteDimensional ℂ V] in
-lemma componentSimple_simple (c : isotypicComponents ℂ[G] ρ.asModule) :
-    IsSimpleModule ℂ[G] (componentSimple ρ c) := c.property.choose_spec.1
-
-omit [Fintype G] [FiniteDimensional ℂ V] in
-lemma componentSimple_component (c : isotypicComponents ℂ[G] ρ.asModule) :
-    c.val = isotypicComponent ℂ[G] ρ.asModule (componentSimple ρ c) :=
-  c.property.choose_spec.2
-
-noncomputable def componentCharacter (c : isotypicComponents ℂ[G] ρ.asModule) : OccurringTypes ρ :=
-  ⟨(Subrepresentation.ofSubmodule' (componentSimple ρ c)).toRepresentation.character,
-    Subrepresentation.ofSubmodule' (componentSimple ρ c),
-    (subrep_irreducible_iff ρ _).mpr (componentSimple_simple ρ c), rfl⟩
-
-lemma componentCharacter_injective : Function.Injective (componentCharacter ρ) := by
-  intro c d h
-  let σ := Subrepresentation.ofSubmodule' (componentSimple ρ c)
-  let τ := Subrepresentation.ofSubmodule' (componentSimple ρ d)
-  have : Representation.IsIrreducible σ.toRepresentation :=
-    (subrep_irreducible_iff ρ σ).mpr (componentSimple_simple ρ c)
-  have : Representation.IsIrreducible τ.toRepresentation :=
-    (subrep_irreducible_iff ρ τ).mpr (componentSimple_simple ρ d)
-  obtain ⟨e⟩ := irreducible_equiv_of_character_eq σ.toRepresentation τ.toRepresentation
-    (congrArg Subtype.val h)
-  apply Subtype.ext
-  rw [componentSimple_component ρ c, componentSimple_component ρ d]
-  exact (moduleEquivOfSubrepEquiv ρ τ σ e).symm.isotypicComponent_eq
-
-omit [Fintype G] in
-lemma componentCharacter_surjective : Function.Surjective (componentCharacter ρ) := by
-  intro χ
-  obtain ⟨σ, hirr, hχ⟩ := χ.property
-  have := hirr
-  have : IsSimpleModule ℂ[G] σ.asSubmodule := (subrep_irreducible_iff ρ σ).mp hirr
-  let c : isotypicComponents ℂ[G] ρ.asModule :=
-    ⟨isotypicComponent ℂ[G] ρ.asModule σ.asSubmodule, σ.asSubmodule, inferInstance, rfl⟩
-  have := componentSimple_simple ρ c
-  have hs : σ.asSubmodule ≤ c.val := σ.asSubmodule.le_isotypicComponent
-  rw [componentSimple_component ρ c] at hs
-  obtain ⟨e⟩ := isIsotypicOfType_submodule_iff.mp
-    (IsIsotypicOfType.isotypicComponent ℂ[G] ρ.asModule (componentSimple ρ c))
-    σ.asSubmodule hs
-  refine ⟨c, Subtype.ext ?_⟩
-  change (Subrepresentation.ofSubmodule' (componentSimple ρ c)).toRepresentation.character = χ.val
-  rw [hχ]
-  exact Representation.char_iso
-    (subrepEquivOfModuleEquiv ρ (Subrepresentation.ofSubmodule' (componentSimple ρ c)) σ e.symm)
-
-noncomputable def componentTypeEquiv : isotypicComponents ℂ[G] ρ.asModule ≃ OccurringTypes ρ :=
-  Equiv.ofBijective (componentCharacter ρ)
-    ⟨componentCharacter_injective ρ, componentCharacter_surjective ρ⟩
-
-instance occurringTypes_finite : Finite (OccurringTypes ρ) :=
-  Finite.of_surjective (componentCharacter ρ) (componentCharacter_surjective ρ)
-
-lemma card_components_eq_types :
-    Nat.card (isotypicComponents ℂ[G] ρ.asModule) = Nat.card (OccurringTypes ρ) :=
-  Nat.card_congr (componentTypeEquiv ρ)
-
-end Components
-
-end
-end PartialPermutation
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.Model
+import OAI.RepresentationTheory.PartialPermutation.ModuleTransport
+
+namespace OAI
+
+section
+open scoped Classical
+open scoped BigOperators ComplexConjugate MonoidAlgebra
+open scoped BigOperators ComplexConjugate
+open scoped MonoidAlgebra BigOperators
+
+namespace PartialPermutation
+noncomputable section
+section Components
+variable {G V : Type*} [Group G] [Fintype G] [AddCommGroup V] [Module ℂ V]
+    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V)
+
+omit [Fintype G] [FiniteDimensional ℂ V] in
+lemma componentSimple_simple (c : isotypicComponents ℂ[G] ρ.asModule) :
+    IsSimpleModule ℂ[G] (componentSimple ρ c) := c.property.choose_spec.1
+
+omit [Fintype G] [FiniteDimensional ℂ V] in
+lemma componentSimple_component (c : isotypicComponents ℂ[G] ρ.asModule) :
+    c.val = isotypicComponent ℂ[G] ρ.asModule (componentSimple ρ c) :=
+  c.property.choose_spec.2
+
+noncomputable def componentCharacter (c : isotypicComponents ℂ[G] ρ.asModule) : OccurringTypes ρ :=
+  ⟨(Subrepresentation.ofSubmodule' (componentSimple ρ c)).toRepresentation.character,
+    Subrepresentation.ofSubmodule' (componentSimple ρ c),
+    (subrep_irreducible_iff ρ _).mpr (componentSimple_simple ρ c), rfl⟩
+
+lemma componentCharacter_injective : Function.Injective (componentCharacter ρ) := by
+  intro c d h
+  let σ := Subrepresentation.ofSubmodule' (componentSimple ρ c)
+  let τ := Subrepresentation.ofSubmodule' (componentSimple ρ d)
+  have : Representation.IsIrreducible σ.toRepresentation :=
+    (subrep_irreducible_iff ρ σ).mpr (componentSimple_simple ρ c)
+  have : Representation.IsIrreducible τ.toRepresentation :=
+    (subrep_irreducible_iff ρ τ).mpr (componentSimple_simple ρ d)
+  obtain ⟨e⟩ := irreducible_equiv_of_character_eq σ.toRepresentation τ.toRepresentation
+    (congrArg Subtype.val h)
+  apply Subtype.ext
+  rw [componentSimple_component ρ c, componentSimple_component ρ d]
+  exact (moduleEquivOfSubrepEquiv ρ τ σ e).symm.isotypicComponent_eq
+
+omit [Fintype G] in
+lemma componentCharacter_surjective : Function.Surjective (componentCharacter ρ) := by
+  intro χ
+  obtain ⟨σ, hirr, hχ⟩ := χ.property
+  have := hirr
+  have : IsSimpleModule ℂ[G] σ.asSubmodule := (subrep_irreducible_iff ρ σ).mp hirr
+  let c : isotypicComponents ℂ[G] ρ.asModule :=
+    ⟨isotypicComponent ℂ[G] ρ.asModule σ.asSubmodule, σ.asSubmodule, inferInstance, rfl⟩
+  have := componentSimple_simple ρ c
+  have hs : σ.asSubmodule ≤ c.val := σ.asSubmodule.le_isotypicComponent
+  rw [componentSimple_component ρ c] at hs
+  obtain ⟨e⟩ := isIsotypicOfType_submodule_iff.mp
+    (IsIsotypicOfType.isotypicComponent ℂ[G] ρ.asModule (componentSimple ρ c))
+    σ.asSubmodule hs
+  refine ⟨c, Subtype.ext ?_⟩
+  change (Subrepresentation.ofSubmodule' (componentSimple ρ c)).toRepresentation.character = χ.val
+  rw [hχ]
+  exact Representation.char_iso
+    (subrepEquivOfModuleEquiv ρ (Subrepresentation.ofSubmodule' (componentSimple ρ c)) σ e.symm)
+
+noncomputable def componentTypeEquiv : isotypicComponents ℂ[G] ρ.asModule ≃ OccurringTypes ρ :=
+  Equiv.ofBijective (componentCharacter ρ)
+    ⟨componentCharacter_injective ρ, componentCharacter_surjective ρ⟩
+
+instance occurringTypes_finite : Finite (OccurringTypes ρ) :=
+  Finite.of_surjective (componentCharacter ρ) (componentCharacter_surjective ρ)
+
+lemma card_components_eq_types :
+    Nat.card (isotypicComponents ℂ[G] ρ.asModule) = Nat.card (OccurringTypes ρ) :=
+  Nat.card_congr (componentTypeEquiv ρ)
+
+end Components
+
+end
+end PartialPermutation
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/ConjugacyCounts.lean b/lean/OAI/RepresentationTheory/PartialPermutation/ConjugacyCounts.lean
index 144cb9662fa95da8ad1d332108deb2922cc0cacb..fbddb25bcdd116e0115a55599fbcbf5bfc89ec36 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/ConjugacyCounts.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/ConjugacyCounts.lean
@@ -1,73 +1,73 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.DiagramCounts
-
-namespace OAI
-
-section
-
-namespace PartialPermutation.DiagramCounting
-noncomputable section
-open Finset
-open scoped Classical
-
-lemma cellsOfRowLens_card (w : List ℕ) : (YoungDiagram.cellsOfRowLens w).card=w.sum := by
-  induction w with
-  | nil => simp [YoungDiagram.cellsOfRowLens]
-  | cons a w ih =>
-    have hd : Disjoint (({0} : Finset ℕ) ×ˢ range a)
-        ((YoungDiagram.cellsOfRowLens w).map
-          (Function.Embedding.prodMap ⟨Nat.succ,Nat.succ_injective⟩ (Function.Embedding.refl ℕ))) := by
-      apply Finset.disjoint_left.mpr
-      intro x hx hy
-      obtain ⟨z,hz,he⟩ := mem_map.mp hy
-      have hx0 : x.1=0 := by simpa using (mem_product.mp hx).1
-      have hh := congrArg Prod.fst he
-      change z.1+1=x.1 at hh
-      omega
-    simp only [YoungDiagram.cellsOfRowLens,card_union_of_disjoint hd,card_product,
-      card_singleton,card_range,one_mul,card_map,ih,List.sum_cons]
-
-def partitionShape {n : ℕ} (p : n.Partition) : Shape n :=
-  ⟨YoungDiagram.ofRowLens (p.parts.sort (· ≥ ·))
-    (List.sortedGE_iff_pairwise.mpr (Multiset.pairwise_sort _ _)),by
-    rw [YoungDiagram.ofRowLens,cellsOfRowLens_card]
-    have hs := congrArg Multiset.sum (Multiset.sort_eq p.parts (· ≥ ·))
-    exact hs.trans p.parts_sum⟩
-
-lemma partitionShape_injective {n : ℕ} : Function.Injective (@partitionShape n) := by
-  intro p q h
-  have hp : (partitionShape p).1.rowLens=p.parts.sort (· ≥ ·) :=
-    YoungDiagram.rowLens_ofRowLens_eq_self (hw := List.sortedGE_iff_pairwise.mpr (Multiset.pairwise_sort _ _)) (fun x hx => p.parts_pos ((Multiset.mem_sort _).mp hx))
-  have hq : (partitionShape q).1.rowLens=q.parts.sort (· ≥ ·) :=
-    YoungDiagram.rowLens_ofRowLens_eq_self (hw := List.sortedGE_iff_pairwise.mpr (Multiset.pairwise_sort _ _)) (fun x hx => q.parts_pos ((Multiset.mem_sort _).mp hx))
-  apply Nat.Partition.ext
-  have he := congrArg (fun μ : Shape n => μ.1.rowLens) h
-  rw [hp,hq] at he
-  exact (Multiset.sort_eq p.parts (· ≥ ·)).symm.trans
-    ((congrArg (fun l : List ℕ => (l : Multiset ℕ)) he).trans (Multiset.sort_eq q.parts (· ≥ ·)))
-
-def conjugacyPartition {α : Type*} [Fintype α] [DecidableEq α] :
-    ConjClasses (Equiv.Perm α) → (Fintype.card α).Partition :=
-  Quotient.lift Equiv.Perm.partition (fun _ _ h => Equiv.Perm.partition_eq_of_isConj.mp h)
-
-lemma conjugacyPartition_injective {α : Type*} [Fintype α] [DecidableEq α] :
-    Function.Injective (@conjugacyPartition α _ _) := by
-  intro a b
-  induction a using Quotient.inductionOn with | _ g =>
-    induction b using Quotient.inductionOn with | _ h =>
-      intro he
-      exact Quotient.sound (Equiv.Perm.partition_eq_of_isConj.mpr he)
-
-lemma symmetric_conjClasses_card_le (n : ℕ) :
-    Nat.card (ConjClasses (Equiv.Perm (Fin n))) ≤ Fintype.card (Shape n) := by
-  let : Fintype (ConjClasses (Equiv.Perm (Fin n))) := Fintype.ofFinite _
-  have h := Fintype.card_le_of_injective _
-    (partitionShape_injective.comp (@conjugacyPartition_injective (Fin n) _ _))
-  simpa only [Nat.card_eq_fintype_card,Fintype.card_fin] using h
-
-end
-end PartialPermutation.DiagramCounting
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.DiagramCounts
+
+namespace OAI
+
+section
+
+namespace PartialPermutation.DiagramCounting
+noncomputable section
+open Finset
+open scoped Classical
+
+lemma cellsOfRowLens_card (w : List ℕ) : (YoungDiagram.cellsOfRowLens w).card=w.sum := by
+  induction w with
+  | nil => simp [YoungDiagram.cellsOfRowLens]
+  | cons a w ih =>
+    have hd : Disjoint (({0} : Finset ℕ) ×ˢ range a)
+        ((YoungDiagram.cellsOfRowLens w).map
+          (Function.Embedding.prodMap ⟨Nat.succ,Nat.succ_injective⟩ (Function.Embedding.refl ℕ))) := by
+      apply Finset.disjoint_left.mpr
+      intro x hx hy
+      obtain ⟨z,hz,he⟩ := mem_map.mp hy
+      have hx0 : x.1=0 := by simpa using (mem_product.mp hx).1
+      have hh := congrArg Prod.fst he
+      change z.1+1=x.1 at hh
+      omega
+    simp only [YoungDiagram.cellsOfRowLens,card_union_of_disjoint hd,card_product,
+      card_singleton,card_range,one_mul,card_map,ih,List.sum_cons]
+
+def partitionShape {n : ℕ} (p : n.Partition) : Shape n :=
+  ⟨YoungDiagram.ofRowLens (p.parts.sort (· ≥ ·))
+    (List.sortedGE_iff_pairwise.mpr (Multiset.pairwise_sort _ _)),by
+    rw [YoungDiagram.ofRowLens,cellsOfRowLens_card]
+    have hs := congrArg Multiset.sum (Multiset.sort_eq p.parts (· ≥ ·))
+    exact hs.trans p.parts_sum⟩
+
+lemma partitionShape_injective {n : ℕ} : Function.Injective (@partitionShape n) := by
+  intro p q h
+  have hp : (partitionShape p).1.rowLens=p.parts.sort (· ≥ ·) :=
+    YoungDiagram.rowLens_ofRowLens_eq_self (hw := List.sortedGE_iff_pairwise.mpr (Multiset.pairwise_sort _ _)) (fun x hx => p.parts_pos ((Multiset.mem_sort _).mp hx))
+  have hq : (partitionShape q).1.rowLens=q.parts.sort (· ≥ ·) :=
+    YoungDiagram.rowLens_ofRowLens_eq_self (hw := List.sortedGE_iff_pairwise.mpr (Multiset.pairwise_sort _ _)) (fun x hx => q.parts_pos ((Multiset.mem_sort _).mp hx))
+  apply Nat.Partition.ext
+  have he := congrArg (fun μ : Shape n => μ.1.rowLens) h
+  rw [hp,hq] at he
+  exact (Multiset.sort_eq p.parts (· ≥ ·)).symm.trans
+    ((congrArg (fun l : List ℕ => (l : Multiset ℕ)) he).trans (Multiset.sort_eq q.parts (· ≥ ·)))
+
+def conjugacyPartition {α : Type*} [Fintype α] [DecidableEq α] :
+    ConjClasses (Equiv.Perm α) → (Fintype.card α).Partition :=
+  Quotient.lift Equiv.Perm.partition (fun _ _ h => Equiv.Perm.partition_eq_of_isConj.mp h)
+
+lemma conjugacyPartition_injective {α : Type*} [Fintype α] [DecidableEq α] :
+    Function.Injective (@conjugacyPartition α _ _) := by
+  intro a b
+  induction a using Quotient.inductionOn with | _ g =>
+    induction b using Quotient.inductionOn with | _ h =>
+      intro he
+      exact Quotient.sound (Equiv.Perm.partition_eq_of_isConj.mpr he)
+
+lemma symmetric_conjClasses_card_le (n : ℕ) :
+    Nat.card (ConjClasses (Equiv.Perm (Fin n))) ≤ Fintype.card (Shape n) := by
+  let : Fintype (ConjClasses (Equiv.Perm (Fin n))) := Fintype.ofFinite _
+  have h := Fintype.card_le_of_injective _
+    (partitionShape_injective.comp (@conjugacyPartition_injective (Fin n) _ _))
+  simpa only [Nat.card_eq_fintype_card,Fintype.card_fin] using h
+
+end
+end PartialPermutation.DiagramCounting
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/ConvolutionKernels.lean b/lean/OAI/RepresentationTheory/PartialPermutation/ConvolutionKernels.lean
index 77b57e223013c3341239917cf8dd57ff6a7931f0..10096d4152ef4f390ee12d5138c3fc50cc960358 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/ConvolutionKernels.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/ConvolutionKernels.lean
@@ -1,179 +1,179 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.FourierBessel
-import OAI.RepresentationTheory.PartialPermutation.Characters
-import OAI.RepresentationTheory.PartialPermutation.CosetCaps
-
-namespace OAI
-
-section
-open scoped Classical
-open scoped BigOperators ComplexConjugate MonoidAlgebra
-open scoped BigOperators ComplexConjugate
-open scoped MonoidAlgebra BigOperators
-open scoped BigOperators MonoidAlgebra Classical
-
-attribute [local instance] Classical.propDecidable
-namespace PartialPermutation
-open scoped BigOperators ComplexConjugate
-
-noncomputable section
-
-def RightCosetCap {G : Type*} [Group G] [Fintype G]
-    (f : G → ℝ) (H : Subgroup G) (B : ℝ) : Prop :=
-  by classical exact ∀ g, ∑ h : H, f (h * g) ≤ B / H.index
-
-section
-variable {G : Type*} [Group G] [Fintype G]
-
-def leftKernel (H : Subgroup G) (k : H → ℂ) (f : G → ℝ) (g : G) : ℂ :=
-  by classical exact ∑ h : H, k h * (f ((h : G)⁻¹ * g) : ℂ)
-
-lemma leftKernel_l2 (H : Subgroup G) (k : H → ℂ) (f : G → ℝ) (B : ℝ)
-    (hf : ∀ g, 0 ≤ f g) (hcap : RightCosetCap f H B) :
-    ∑ g, ‖leftKernel H k f g‖ ^ 2 ≤
-      B / H.index * mass f * ∑ h : H, ‖k h‖ ^ 2 := by
-  classical
-  have hp (g : G) : ∑ h : H, f ((h : G)⁻¹ * g) ≤ B / H.index := by
-    have he : ∑ h : H, f ((h : G)⁻¹ * g) = ∑ h : H, f ((h : G) * g) := by
-      exact Fintype.sum_equiv (Equiv.inv H) _ _ (fun h => rfl)
-    rw [he]
-    exact hcap g
-  have hpt (g : G) : ‖leftKernel H k f g‖ ^ 2 ≤
-      B / H.index * ∑ h : H, f ((h : G)⁻¹ * g) * ‖k h‖ ^ 2 := by
-    have h := norm_sum_weighted_sq_le (fun h : H => f ((h : G)⁻¹ * g))
-      (fun h => hf _) k
-    change ‖∑ h : H, k h * (f ((h : G)⁻¹ * g) : ℂ)‖ ^ 2 ≤ _
-    simp_rw [mul_comm (k _)]
-    exact h.trans (mul_le_mul_of_nonneg_right (hp g)
-      (Finset.sum_nonneg fun h _ => mul_nonneg (hf _) (sq_nonneg _)))
-  calc
-    _ ≤ ∑ g, B / H.index * ∑ h : H, f ((h : G)⁻¹ * g) * ‖k h‖ ^ 2 :=
-      Finset.sum_le_sum fun g _ => hpt g
-    _ = _ := by
-      rw [← Finset.mul_sum, Finset.sum_comm]
-      have he (h : H) : ∑ g, f ((h : G)⁻¹ * g) = mass f := by
-        exact Fintype.sum_equiv (Equiv.mulLeft (h : G)⁻¹) _ _ (fun g => rfl)
-      simp_rw [← Finset.sum_mul, he]
-      rw [← Finset.mul_sum, mul_assoc]
-
-lemma complexFourier_leftKernel {V : Type*} [AddCommGroup V] [Module ℂ V]
-    (ρ : Representation ℂ G V) (H : Subgroup G) (k : H → ℂ) (f : G → ℝ) :
-    complexFourier ρ (leftKernel H k f) =
-      complexFourier (ρ.comp H.subtype) k * complexFourier ρ (fun g => (f g : ℂ)) := by
-  classical
-  unfold complexFourier leftKernel
-  simp_rw [Finset.sum_smul, Finset.sum_mul, Finset.mul_sum]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro h _
-  apply Fintype.sum_equiv (Equiv.mulLeft (h : G)⁻¹)
-  intro g
-  simp only [smul_mul_smul_comm, ← map_mul, MonoidHom.comp_apply, Equiv.coe_mulLeft,
-    Subgroup.subtype_apply, mul_inv_cancel_left]
-
-lemma kernel_hs_bound {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
-    [Representation.IsIrreducible ρ] (H : Subgroup G) (k : H → ℂ)
-    (d : ℕ) (hk : ∑ h : H, ‖k h‖ ^ 2 = (d : ℝ)^2 / Fintype.card H)
-    (f : G → ℝ) (B : ℝ) (hf : ∀ g, 0 ≤ f g) (hcap : RightCosetCap f H B) :
-    (Module.finrank ℂ V : ℝ) *
-      hsNormSq (complexFourier (ρ.comp H.subtype) k * complexFourier ρ (fun g => f g)) ≤
-      B * mass f * (d : ℝ)^2 := by
-  classical
-  have hc : (H.index : ℝ) * (Fintype.card H : ℝ) = (Fintype.card G : ℝ) := by
-    exact_mod_cast (by simpa only [Nat.card_eq_fintype_card] using H.index_mul_card)
-  have hi : (H.index : ℝ) ≠ 0 := by
-    exact_mod_cast H.index_ne_zero_of_finite
-  have hh : (Fintype.card H : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
-  calc
-    _ ≤ (Fintype.card G : ℝ) * ∑ g, ‖leftKernel H k f g‖ ^ 2 := by
-      rw [← complexFourier_leftKernel]
-      exact fourier_bessel ρ hρ _
-    _ ≤ (Fintype.card G : ℝ) * (B / H.index * mass f * ∑ h : H, ‖k h‖ ^ 2) :=
-      mul_le_mul_of_nonneg_left (leftKernel_l2 H k f B hf hcap) (by positivity)
-    _ = _ := by rw [hk, ← hc]; field_simp
-
-lemma inverse_rightCap (H : Subgroup G) (f : G → ℝ) (B : ℝ)
-    (hcap : LeftCosetCap f H B) : RightCosetCap (fun g => f g⁻¹) H B := by
-  intro g
-  have he : ∑ h : H, f (((h : G) * g)⁻¹) = ∑ h : H, f (g⁻¹ * h) := by
-    apply Fintype.sum_equiv (Equiv.inv H)
-    intro h
-    simp
-  rw [he]
-  exact hcap g⁻¹
-
-lemma mass_inverse (f : G → ℝ) : mass (fun g => f g⁻¹) = mass f := by
-  exact Fintype.sum_equiv (Equiv.inv G) _ _ (fun g => rfl)
-
-lemma complexFourier_adjoint {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (f : G → ℂ) :
-    LinearMap.adjoint (complexFourier ρ f) = complexFourier ρ (fun g => conj (f g⁻¹)) := by
-  unfold complexFourier
-  simp only [map_sum, map_smulₛₗ, unitary_adjoint ρ hρ]
-  apply Fintype.sum_equiv (Equiv.inv G)
-  intro g
-  simp
-
-lemma complexFourier_selfAdjoint {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
-    (f : G → ℂ) (hf : ∀ g, f g⁻¹ = conj (f g)) :
-    LinearMap.adjoint (complexFourier ρ f) = complexFourier ρ f := by
-  rw [complexFourier_adjoint ρ hρ]
-  simp [hf]
-
-lemma kernel_hs_bound_right {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
-    [Representation.IsIrreducible ρ] (H : Subgroup G) (k : H → ℂ)
-    (d : ℕ) (hk : ∑ h : H, ‖k h‖ ^ 2 = (d : ℝ)^2 / Fintype.card H)
-    (hki : ∀ h, k h⁻¹ = conj (k h))
-    (f : G → ℝ) (B : ℝ) (hf : ∀ g, 0 ≤ f g) (hcap : LeftCosetCap f H B) :
-    (Module.finrank ℂ V : ℝ) *
-      hsNormSq (complexFourier ρ (fun g => f g) * complexFourier (ρ.comp H.subtype) k) ≤
-      B * mass f * (d : ℝ)^2 := by
-  have hi := kernel_hs_bound ρ hρ H k d hk (fun g => f g⁻¹) B
-    (fun g => hf _) (inverse_rightCap H f B hcap)
-  rw [mass_inverse] at hi
-  have hP := complexFourier_selfAdjoint (ρ.comp H.subtype) (fun g => hρ g) k hki
-  have hA : LinearMap.adjoint (complexFourier ρ (fun g => (f g⁻¹ : ℂ))) =
-      complexFourier ρ (fun g => (f g : ℂ)) := by
-    rw [complexFourier_adjoint ρ hρ]
-    simp
-  rw [← hsNormSq_adjoint (_ * _), adjoint_mul_eq, hA, hP] at hi
-  exact hi
-
-lemma projection_kernel_mul {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-    [FiniteDimensional ℂ V] (W : Submodule ℂ V) (P : Module.End ℂ V)
-    (hP : LinearMap.adjoint P = P) (hact : ∀ x ∈ W, P x = x) :
-    W.starProjection.toLinearMap * P = W.starProjection.toLinearMap := by
-  have he : P * W.starProjection.toLinearMap = W.starProjection.toLinearMap := by
-    ext x
-    exact hact _ (W.starProjection_apply_mem x)
-  have he' := congrArg LinearMap.adjoint he
-  simpa only [adjoint_mul_eq, adjoint_projection_eq, hP] using he'
-
-lemma hsNormSq_projection_kernel_left {V : Type*} [NormedAddCommGroup V]
-    [InnerProductSpace ℂ V] [FiniteDimensional ℂ V] (W : Submodule ℂ V)
-    (P A : Module.End ℂ V) (hP : LinearMap.adjoint P = P)
-    (hact : ∀ x ∈ W, P x = x) :
-    hsNormSq (W.starProjection.toLinearMap * A) ≤ hsNormSq (P * A) := by
-  have h := hsNormSq_projection_left W (P * A)
-  rw [← mul_assoc, projection_kernel_mul W P hP hact] at h
-  exact h
-
-lemma hsNormSq_projection_kernel_right {V : Type*} [NormedAddCommGroup V]
-    [InnerProductSpace ℂ V] [FiniteDimensional ℂ V] (W : Submodule ℂ V)
-    (P A : Module.End ℂ V) (hP : LinearMap.adjoint P = P)
-    (hact : ∀ x ∈ W, P x = x) :
-    hsNormSq (A * W.starProjection.toLinearMap) ≤ hsNormSq (A * P) := by
-  rw [← hsNormSq_adjoint (A * _), adjoint_mul_eq, adjoint_projection_eq,
-    ← hsNormSq_adjoint (A * P), adjoint_mul_eq, hP]
-  exact hsNormSq_projection_kernel_left W P (LinearMap.adjoint A) hP hact
-
-end
-end
-end PartialPermutation
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.FourierBessel
+import OAI.RepresentationTheory.PartialPermutation.Characters
+import OAI.RepresentationTheory.PartialPermutation.CosetCaps
+
+namespace OAI
+
+section
+open scoped Classical
+open scoped BigOperators ComplexConjugate MonoidAlgebra
+open scoped BigOperators ComplexConjugate
+open scoped MonoidAlgebra BigOperators
+open scoped BigOperators MonoidAlgebra Classical
+
+attribute [local instance] Classical.propDecidable
+namespace PartialPermutation
+open scoped BigOperators ComplexConjugate
+
+noncomputable section
+
+def RightCosetCap {G : Type*} [Group G] [Fintype G]
+    (f : G → ℝ) (H : Subgroup G) (B : ℝ) : Prop :=
+  by classical exact ∀ g, ∑ h : H, f (h * g) ≤ B / H.index
+
+section
+variable {G : Type*} [Group G] [Fintype G]
+
+def leftKernel (H : Subgroup G) (k : H → ℂ) (f : G → ℝ) (g : G) : ℂ :=
+  by classical exact ∑ h : H, k h * (f ((h : G)⁻¹ * g) : ℂ)
+
+lemma leftKernel_l2 (H : Subgroup G) (k : H → ℂ) (f : G → ℝ) (B : ℝ)
+    (hf : ∀ g, 0 ≤ f g) (hcap : RightCosetCap f H B) :
+    ∑ g, ‖leftKernel H k f g‖ ^ 2 ≤
+      B / H.index * mass f * ∑ h : H, ‖k h‖ ^ 2 := by
+  classical
+  have hp (g : G) : ∑ h : H, f ((h : G)⁻¹ * g) ≤ B / H.index := by
+    have he : ∑ h : H, f ((h : G)⁻¹ * g) = ∑ h : H, f ((h : G) * g) := by
+      exact Fintype.sum_equiv (Equiv.inv H) _ _ (fun h => rfl)
+    rw [he]
+    exact hcap g
+  have hpt (g : G) : ‖leftKernel H k f g‖ ^ 2 ≤
+      B / H.index * ∑ h : H, f ((h : G)⁻¹ * g) * ‖k h‖ ^ 2 := by
+    have h := norm_sum_weighted_sq_le (fun h : H => f ((h : G)⁻¹ * g))
+      (fun h => hf _) k
+    change ‖∑ h : H, k h * (f ((h : G)⁻¹ * g) : ℂ)‖ ^ 2 ≤ _
+    simp_rw [mul_comm (k _)]
+    exact h.trans (mul_le_mul_of_nonneg_right (hp g)
+      (Finset.sum_nonneg fun h _ => mul_nonneg (hf _) (sq_nonneg _)))
+  calc
+    _ ≤ ∑ g, B / H.index * ∑ h : H, f ((h : G)⁻¹ * g) * ‖k h‖ ^ 2 :=
+      Finset.sum_le_sum fun g _ => hpt g
+    _ = _ := by
+      rw [← Finset.mul_sum, Finset.sum_comm]
+      have he (h : H) : ∑ g, f ((h : G)⁻¹ * g) = mass f := by
+        exact Fintype.sum_equiv (Equiv.mulLeft (h : G)⁻¹) _ _ (fun g => rfl)
+      simp_rw [← Finset.sum_mul, he]
+      rw [← Finset.mul_sum, mul_assoc]
+
+lemma complexFourier_leftKernel {V : Type*} [AddCommGroup V] [Module ℂ V]
+    (ρ : Representation ℂ G V) (H : Subgroup G) (k : H → ℂ) (f : G → ℝ) :
+    complexFourier ρ (leftKernel H k f) =
+      complexFourier (ρ.comp H.subtype) k * complexFourier ρ (fun g => (f g : ℂ)) := by
+  classical
+  unfold complexFourier leftKernel
+  simp_rw [Finset.sum_smul, Finset.sum_mul, Finset.mul_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro h _
+  apply Fintype.sum_equiv (Equiv.mulLeft (h : G)⁻¹)
+  intro g
+  simp only [smul_mul_smul_comm, ← map_mul, MonoidHom.comp_apply, Equiv.coe_mulLeft,
+    Subgroup.subtype_apply, mul_inv_cancel_left]
+
+lemma kernel_hs_bound {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
+    [Representation.IsIrreducible ρ] (H : Subgroup G) (k : H → ℂ)
+    (d : ℕ) (hk : ∑ h : H, ‖k h‖ ^ 2 = (d : ℝ)^2 / Fintype.card H)
+    (f : G → ℝ) (B : ℝ) (hf : ∀ g, 0 ≤ f g) (hcap : RightCosetCap f H B) :
+    (Module.finrank ℂ V : ℝ) *
+      hsNormSq (complexFourier (ρ.comp H.subtype) k * complexFourier ρ (fun g => f g)) ≤
+      B * mass f * (d : ℝ)^2 := by
+  classical
+  have hc : (H.index : ℝ) * (Fintype.card H : ℝ) = (Fintype.card G : ℝ) := by
+    exact_mod_cast (by simpa only [Nat.card_eq_fintype_card] using H.index_mul_card)
+  have hi : (H.index : ℝ) ≠ 0 := by
+    exact_mod_cast H.index_ne_zero_of_finite
+  have hh : (Fintype.card H : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
+  calc
+    _ ≤ (Fintype.card G : ℝ) * ∑ g, ‖leftKernel H k f g‖ ^ 2 := by
+      rw [← complexFourier_leftKernel]
+      exact fourier_bessel ρ hρ _
+    _ ≤ (Fintype.card G : ℝ) * (B / H.index * mass f * ∑ h : H, ‖k h‖ ^ 2) :=
+      mul_le_mul_of_nonneg_left (leftKernel_l2 H k f B hf hcap) (by positivity)
+    _ = _ := by rw [hk, ← hc]; field_simp
+
+lemma inverse_rightCap (H : Subgroup G) (f : G → ℝ) (B : ℝ)
+    (hcap : LeftCosetCap f H B) : RightCosetCap (fun g => f g⁻¹) H B := by
+  intro g
+  have he : ∑ h : H, f (((h : G) * g)⁻¹) = ∑ h : H, f (g⁻¹ * h) := by
+    apply Fintype.sum_equiv (Equiv.inv H)
+    intro h
+    simp
+  rw [he]
+  exact hcap g⁻¹
+
+lemma mass_inverse (f : G → ℝ) : mass (fun g => f g⁻¹) = mass f := by
+  exact Fintype.sum_equiv (Equiv.inv G) _ _ (fun g => rfl)
+
+lemma complexFourier_adjoint {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (f : G → ℂ) :
+    LinearMap.adjoint (complexFourier ρ f) = complexFourier ρ (fun g => conj (f g⁻¹)) := by
+  unfold complexFourier
+  simp only [map_sum, map_smulₛₗ, unitary_adjoint ρ hρ]
+  apply Fintype.sum_equiv (Equiv.inv G)
+  intro g
+  simp
+
+lemma complexFourier_selfAdjoint {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
+    (f : G → ℂ) (hf : ∀ g, f g⁻¹ = conj (f g)) :
+    LinearMap.adjoint (complexFourier ρ f) = complexFourier ρ f := by
+  rw [complexFourier_adjoint ρ hρ]
+  simp [hf]
+
+lemma kernel_hs_bound_right {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
+    [Representation.IsIrreducible ρ] (H : Subgroup G) (k : H → ℂ)
+    (d : ℕ) (hk : ∑ h : H, ‖k h‖ ^ 2 = (d : ℝ)^2 / Fintype.card H)
+    (hki : ∀ h, k h⁻¹ = conj (k h))
+    (f : G → ℝ) (B : ℝ) (hf : ∀ g, 0 ≤ f g) (hcap : LeftCosetCap f H B) :
+    (Module.finrank ℂ V : ℝ) *
+      hsNormSq (complexFourier ρ (fun g => f g) * complexFourier (ρ.comp H.subtype) k) ≤
+      B * mass f * (d : ℝ)^2 := by
+  have hi := kernel_hs_bound ρ hρ H k d hk (fun g => f g⁻¹) B
+    (fun g => hf _) (inverse_rightCap H f B hcap)
+  rw [mass_inverse] at hi
+  have hP := complexFourier_selfAdjoint (ρ.comp H.subtype) (fun g => hρ g) k hki
+  have hA : LinearMap.adjoint (complexFourier ρ (fun g => (f g⁻¹ : ℂ))) =
+      complexFourier ρ (fun g => (f g : ℂ)) := by
+    rw [complexFourier_adjoint ρ hρ]
+    simp
+  rw [← hsNormSq_adjoint (_ * _), adjoint_mul_eq, hA, hP] at hi
+  exact hi
+
+lemma projection_kernel_mul {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+    [FiniteDimensional ℂ V] (W : Submodule ℂ V) (P : Module.End ℂ V)
+    (hP : LinearMap.adjoint P = P) (hact : ∀ x ∈ W, P x = x) :
+    W.starProjection.toLinearMap * P = W.starProjection.toLinearMap := by
+  have he : P * W.starProjection.toLinearMap = W.starProjection.toLinearMap := by
+    ext x
+    exact hact _ (W.starProjection_apply_mem x)
+  have he' := congrArg LinearMap.adjoint he
+  simpa only [adjoint_mul_eq, adjoint_projection_eq, hP] using he'
+
+lemma hsNormSq_projection_kernel_left {V : Type*} [NormedAddCommGroup V]
+    [InnerProductSpace ℂ V] [FiniteDimensional ℂ V] (W : Submodule ℂ V)
+    (P A : Module.End ℂ V) (hP : LinearMap.adjoint P = P)
+    (hact : ∀ x ∈ W, P x = x) :
+    hsNormSq (W.starProjection.toLinearMap * A) ≤ hsNormSq (P * A) := by
+  have h := hsNormSq_projection_left W (P * A)
+  rw [← mul_assoc, projection_kernel_mul W P hP hact] at h
+  exact h
+
+lemma hsNormSq_projection_kernel_right {V : Type*} [NormedAddCommGroup V]
+    [InnerProductSpace ℂ V] [FiniteDimensional ℂ V] (W : Submodule ℂ V)
+    (P A : Module.End ℂ V) (hP : LinearMap.adjoint P = P)
+    (hact : ∀ x ∈ W, P x = x) :
+    hsNormSq (A * W.starProjection.toLinearMap) ≤ hsNormSq (A * P) := by
+  rw [← hsNormSq_adjoint (A * _), adjoint_mul_eq, adjoint_projection_eq,
+    ← hsNormSq_adjoint (A * P), adjoint_mul_eq, hP]
+  exact hsNormSq_projection_kernel_left W P (LinearMap.adjoint A) hP hact
+
+end
+end
+end PartialPermutation
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/CosetCaps.lean b/lean/OAI/RepresentationTheory/PartialPermutation/CosetCaps.lean
index a7ad773421b9fb468a28fd63cbfbd4585b3e042d..65eb03682e2a0951321c0890b9d383a6680141c4 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/CosetCaps.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/CosetCaps.lean
@@ -1,73 +1,73 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.Model
-
-namespace OAI
-
-section
-open scoped Classical
-open scoped BigOperators ComplexConjugate MonoidAlgebra
-open scoped BigOperators ComplexConjugate
-open scoped MonoidAlgebra BigOperators
-open scoped BigOperators MonoidAlgebra Classical
-
-namespace PartialPermutation
-
-section CosetCap
-variable {G : Type*} [Group G] [Fintype G]
-
-lemma sum_mul_le_of_leftCosetCap (f F : G → ℝ) (H : Subgroup G) (B : ℝ)
-    (hcap : LeftCosetCap f H B) (hF : ∀ g, 0 ≤ F g)
-    (hinv : ∀ (g : G) (h : H), F (g * h) = F g) :
-    ∑ g, f g * F g ≤ B / Fintype.card G * ∑ g, F g := by
-  classical
-  have hsum (h : H) : ∑ g, f (g * h) * F g = ∑ g, f g * F g := by
-    apply Fintype.sum_equiv (Equiv.mulRight (h : G))
-    intro g
-    change f (g * h) * F g = f (g * h) * F (g * h)
-    rw [hinv]
-  have havg : (Fintype.card H : ℝ) * (∑ g, f g * F g) =
-      ∑ g, (∑ h : H, f (g * h)) * F g := by
-    simp_rw [Finset.sum_mul]
-    rw [Finset.sum_comm]
-    simp [hsum]
-  have hc : (H.index : ℝ) * (Fintype.card H : ℝ) = (Fintype.card G : ℝ) := by
-    exact_mod_cast (by simpa only [Nat.card_eq_fintype_card] using H.index_mul_card)
-  have hh : 0 < (Fintype.card H : ℝ) := by positivity
-  have hg : 0 < (Fintype.card G : ℝ) := by positivity
-  have hi : 0 < (H.index : ℝ) := by
-    have := Nat.cast_nonneg (α := ℝ) H.index
-    nlinarith
-  have h : (Fintype.card H : ℝ) * (∑ g, f g * F g) ≤
-      (B / H.index) * ∑ g, F g := by
-    rw [havg, Finset.mul_sum]
-    exact Finset.sum_le_sum fun g _ => mul_le_mul_of_nonneg_right (hcap g) (hF g)
-  calc
-    ∑ g, f g * F g ≤ ((B / H.index) * ∑ g, F g) / Fintype.card H := by
-      apply (le_div_iff₀ hh).mpr
-      simpa only [mul_comm] using h
-    _ = B / Fintype.card G * ∑ g, F g := by
-      rw [← hc]
-      field_simp
-
-lemma norm_sum_weighted_sq_le {ι : Type*} [Fintype ι]
-    (p : ι → ℝ) (hp : ∀ i, 0 ≤ p i) (c : ι → ℂ) :
-    ‖∑ i, (p i : ℂ) * c i‖ ^ 2 ≤ (∑ i, p i) * ∑ i, p i * ‖c i‖ ^ 2 := by
-  have hn : ‖∑ i, (p i : ℂ) * c i‖ ≤ ∑ i, p i * ‖c i‖ := by
-    calc
-      _ ≤ ∑ i, ‖(p i : ℂ) * c i‖ := norm_sum_le _ _
-      _ = _ := by simp [Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg (hp _)]
-  have hnonneg : 0 ≤ ∑ i, p i * ‖c i‖ := Finset.sum_nonneg fun i _ => mul_nonneg (hp i) (norm_nonneg _)
-  have hsq : (∑ i, p i * ‖c i‖) ^ 2 ≤ (∑ i, p i) * ∑ i, p i * ‖c i‖ ^ 2 := by
-    apply Finset.sum_sq_le_sum_mul_sum_of_sq_le_mul
-    · exact fun i _ => hp i
-    · exact fun i _ => mul_nonneg (hp i) (sq_nonneg _)
-    · intro i hi
-      nlinarith
-  exact (pow_le_pow_left₀ (norm_nonneg _) hn 2).trans hsq
-
-end CosetCap
-end PartialPermutation
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.Model
+
+namespace OAI
+
+section
+open scoped Classical
+open scoped BigOperators ComplexConjugate MonoidAlgebra
+open scoped BigOperators ComplexConjugate
+open scoped MonoidAlgebra BigOperators
+open scoped BigOperators MonoidAlgebra Classical
+
+namespace PartialPermutation
+
+section CosetCap
+variable {G : Type*} [Group G] [Fintype G]
+
+lemma sum_mul_le_of_leftCosetCap (f F : G → ℝ) (H : Subgroup G) (B : ℝ)
+    (hcap : LeftCosetCap f H B) (hF : ∀ g, 0 ≤ F g)
+    (hinv : ∀ (g : G) (h : H), F (g * h) = F g) :
+    ∑ g, f g * F g ≤ B / Fintype.card G * ∑ g, F g := by
+  classical
+  have hsum (h : H) : ∑ g, f (g * h) * F g = ∑ g, f g * F g := by
+    apply Fintype.sum_equiv (Equiv.mulRight (h : G))
+    intro g
+    change f (g * h) * F g = f (g * h) * F (g * h)
+    rw [hinv]
+  have havg : (Fintype.card H : ℝ) * (∑ g, f g * F g) =
+      ∑ g, (∑ h : H, f (g * h)) * F g := by
+    simp_rw [Finset.sum_mul]
+    rw [Finset.sum_comm]
+    simp [hsum]
+  have hc : (H.index : ℝ) * (Fintype.card H : ℝ) = (Fintype.card G : ℝ) := by
+    exact_mod_cast (by simpa only [Nat.card_eq_fintype_card] using H.index_mul_card)
+  have hh : 0 < (Fintype.card H : ℝ) := by positivity
+  have hg : 0 < (Fintype.card G : ℝ) := by positivity
+  have hi : 0 < (H.index : ℝ) := by
+    have := Nat.cast_nonneg (α := ℝ) H.index
+    nlinarith
+  have h : (Fintype.card H : ℝ) * (∑ g, f g * F g) ≤
+      (B / H.index) * ∑ g, F g := by
+    rw [havg, Finset.mul_sum]
+    exact Finset.sum_le_sum fun g _ => mul_le_mul_of_nonneg_right (hcap g) (hF g)
+  calc
+    ∑ g, f g * F g ≤ ((B / H.index) * ∑ g, F g) / Fintype.card H := by
+      apply (le_div_iff₀ hh).mpr
+      simpa only [mul_comm] using h
+    _ = B / Fintype.card G * ∑ g, F g := by
+      rw [← hc]
+      field_simp
+
+lemma norm_sum_weighted_sq_le {ι : Type*} [Fintype ι]
+    (p : ι → ℝ) (hp : ∀ i, 0 ≤ p i) (c : ι → ℂ) :
+    ‖∑ i, (p i : ℂ) * c i‖ ^ 2 ≤ (∑ i, p i) * ∑ i, p i * ‖c i‖ ^ 2 := by
+  have hn : ‖∑ i, (p i : ℂ) * c i‖ ≤ ∑ i, p i * ‖c i‖ := by
+    calc
+      _ ≤ ∑ i, ‖(p i : ℂ) * c i‖ := norm_sum_le _ _
+      _ = _ := by simp [Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg (hp _)]
+  have hnonneg : 0 ≤ ∑ i, p i * ‖c i‖ := Finset.sum_nonneg fun i _ => mul_nonneg (hp i) (norm_nonneg _)
+  have hsq : (∑ i, p i * ‖c i‖) ^ 2 ≤ (∑ i, p i) * ∑ i, p i * ‖c i‖ ^ 2 := by
+    apply Finset.sum_sq_le_sum_mul_sum_of_sq_le_mul
+    · exact fun i _ => hp i
+    · exact fun i _ => mul_nonneg (hp i) (sq_nonneg _)
+    · intro i hi
+      nlinarith
+  exact (pow_le_pow_left₀ (norm_nonneg _) hn 2).trans hsq
+
+end CosetCap
+end PartialPermutation
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/CrossTableaux.lean b/lean/OAI/RepresentationTheory/PartialPermutation/CrossTableaux.lean
index 1155041aa06252eb04c91471834263197da651d5..130c0ccce9d7edb7ce562e887c61c702b8d3c61b 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/CrossTableaux.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/CrossTableaux.lean
@@ -1,89 +1,89 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.TableauVectors
-
-namespace OAI
-
-section
-namespace PartialPermutation.YoungTabloid
-noncomputable section
-open Finset
-open scoped Classical
-
-def rowMoment (μ : YoungDiagram) : ℕ := ∑ x, row μ x
-
-def CrossTransversal (μ : YoungDiagram) (r : μ.cells → ℕ) : Prop :=
-  ∀ x y, col μ x=col μ y → r x=r y → x=y
-
-def crossSortedColumn (μ : YoungDiagram) (r : μ.cells → ℕ)
-    (hr : CrossTransversal μ r) (j : ℕ) : Col μ j ≃ Fin (μ.colLen j) :=
-  (exists_sortingEquiv (Col μ j) (fun x => r x.1)
-    (fun x y h => Subtype.ext (hr x.1 y.1 (x.2.trans y.2.symm) h))
-    _ (col_card μ j)).choose
-
-lemma crossSortedColumn_mono (μ : YoungDiagram) (r : μ.cells → ℕ)
-    (hr : CrossTransversal μ r) (j : ℕ) :
-    StrictMono (fun i => r ((crossSortedColumn μ r hr j).symm i).1) :=
-  (exists_sortingEquiv (Col μ j) (fun x => r x.1)
-    (fun x y h => Subtype.ext (hr x.1 y.1 (x.2.trans y.2.symm) h))
-    _ (col_card μ j)).choose_spec
-
-def crossSorting (μ : YoungDiagram) (r : μ.cells → ℕ) (hr : CrossTransversal μ r) :
-    Equiv.Perm μ.cells := fiberPermutation (col μ)
-      (fun j => (crossSortedColumn μ r hr j).trans (columnEquiv μ j).symm)
-
-lemma crossSorting_row (μ : YoungDiagram) (r : μ.cells → ℕ)
-    (hr : CrossTransversal μ r) (x : μ.cells) : row μ (crossSorting μ r hr x) ≤ r x := by
-  let e := crossSortedColumn μ r hr (col μ x)
-  have h := strictMono_fin_nat_le _ (crossSortedColumn_mono μ r hr (col μ x)) (e ⟨x,rfl⟩)
-  change (e ⟨x,rfl⟩).val ≤ r (e.symm (e ⟨x,rfl⟩)).val at h
-  erw [Equiv.symm_apply_apply] at h
-  exact h
-
-lemma cross_rowMoment_le (μ ν : YoungDiagram) (e : μ.cells ≃ ν.cells) (r : Tabloid ν)
-    (hr : CrossTransversal μ (fun x => r.1 (e x))) : rowMoment μ ≤ rowMoment ν := by
-  calc
-    rowMoment μ = ∑ x, row μ (crossSorting μ _ hr x) :=
-      (Equiv.sum_comp (crossSorting μ _ hr) (row μ)).symm
-    _ ≤ ∑ x, r.1 (e x) := Finset.sum_le_sum (fun x _ => crossSorting_row μ _ hr x)
-    _ = ∑ x, r.1 x := Equiv.sum_comp e r.1
-    _ = rowMoment ν := rowSum ν r
-
-def rowFiberEquiv (μ : YoungDiagram) (i : ℕ) : {x : μ.cells // row μ x=i} ≃ Fin (μ.rowLen i) where
-  toFun x := ⟨col μ x.1,by
-    have h := μ.mem_iff_lt_rowLen.mp x.1.2
-    have hi : x.1.1.1=i := x.2
-    simpa only [col,hi] using h⟩
-  invFun j := ⟨rowCell μ i j,rfl⟩
-  left_inv x := by apply Subtype.ext; apply Subtype.ext; exact Prod.ext x.2.symm rfl
-  right_inv _ := rfl
-
-lemma shape_eq_of_row_equiv (μ ν : YoungDiagram) (e : μ.cells ≃ ν.cells)
-    (h : ∀ x, row ν (e x)=row μ x) : μ=ν := by
-  have hr (i : ℕ) : μ.rowLen i=ν.rowLen i := by
-    let er : {x : μ.cells // row μ x=i} ≃ {x : ν.cells // row ν x=i} :=
-      e.subtypeEquiv (fun x => by rw [h])
-    simpa only [Fintype.card_fin] using (Fintype.card_congr (rowFiberEquiv μ i)).symm.trans
-      ((Fintype.card_congr er).trans (Fintype.card_congr (rowFiberEquiv ν i)))
-  apply YoungDiagram.ext
-  ext x
-  exact (μ.mem_iff_lt_rowLen).trans ((hr x.1) ▸ ν.mem_iff_lt_rowLen.symm)
-
-lemma cross_equal_moment_shape_eq (μ ν : YoungDiagram) (e : μ.cells ≃ ν.cells)
-    (r : Tabloid ν) (hr : CrossTransversal μ (fun x => r.1 (e x)))
-    (hm : rowMoment μ=rowMoment ν) : μ=ν := by
-  let p := crossSorting μ _ hr
-  have hs : (∑ x, row μ (p x))=∑ x, r.1 (e x) := by
-    rw [Equiv.sum_comp p,Equiv.sum_comp e,rowSum]
-    exact hm
-  have he : ∀ x, row μ (p x)=r.1 (e x) :=
-    fun x => (Finset.sum_eq_sum_iff_of_le (fun x _ => crossSorting_row μ _ hr x)).mp hs x (mem_univ x)
-  obtain ⟨q,hq⟩ := r.2
-  apply shape_eq_of_row_equiv μ ν ((p.symm.trans e).trans q)
-  intro x
-  simpa [hq] using (he (p.symm x)).symm
-
-end
-end PartialPermutation.YoungTabloid
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.TableauVectors
+
+namespace OAI
+
+section
+namespace PartialPermutation.YoungTabloid
+noncomputable section
+open Finset
+open scoped Classical
+
+def rowMoment (μ : YoungDiagram) : ℕ := ∑ x, row μ x
+
+def CrossTransversal (μ : YoungDiagram) (r : μ.cells → ℕ) : Prop :=
+  ∀ x y, col μ x=col μ y → r x=r y → x=y
+
+def crossSortedColumn (μ : YoungDiagram) (r : μ.cells → ℕ)
+    (hr : CrossTransversal μ r) (j : ℕ) : Col μ j ≃ Fin (μ.colLen j) :=
+  (exists_sortingEquiv (Col μ j) (fun x => r x.1)
+    (fun x y h => Subtype.ext (hr x.1 y.1 (x.2.trans y.2.symm) h))
+    _ (col_card μ j)).choose
+
+lemma crossSortedColumn_mono (μ : YoungDiagram) (r : μ.cells → ℕ)
+    (hr : CrossTransversal μ r) (j : ℕ) :
+    StrictMono (fun i => r ((crossSortedColumn μ r hr j).symm i).1) :=
+  (exists_sortingEquiv (Col μ j) (fun x => r x.1)
+    (fun x y h => Subtype.ext (hr x.1 y.1 (x.2.trans y.2.symm) h))
+    _ (col_card μ j)).choose_spec
+
+def crossSorting (μ : YoungDiagram) (r : μ.cells → ℕ) (hr : CrossTransversal μ r) :
+    Equiv.Perm μ.cells := fiberPermutation (col μ)
+      (fun j => (crossSortedColumn μ r hr j).trans (columnEquiv μ j).symm)
+
+lemma crossSorting_row (μ : YoungDiagram) (r : μ.cells → ℕ)
+    (hr : CrossTransversal μ r) (x : μ.cells) : row μ (crossSorting μ r hr x) ≤ r x := by
+  let e := crossSortedColumn μ r hr (col μ x)
+  have h := strictMono_fin_nat_le _ (crossSortedColumn_mono μ r hr (col μ x)) (e ⟨x,rfl⟩)
+  change (e ⟨x,rfl⟩).val ≤ r (e.symm (e ⟨x,rfl⟩)).val at h
+  erw [Equiv.symm_apply_apply] at h
+  exact h
+
+lemma cross_rowMoment_le (μ ν : YoungDiagram) (e : μ.cells ≃ ν.cells) (r : Tabloid ν)
+    (hr : CrossTransversal μ (fun x => r.1 (e x))) : rowMoment μ ≤ rowMoment ν := by
+  calc
+    rowMoment μ = ∑ x, row μ (crossSorting μ _ hr x) :=
+      (Equiv.sum_comp (crossSorting μ _ hr) (row μ)).symm
+    _ ≤ ∑ x, r.1 (e x) := Finset.sum_le_sum (fun x _ => crossSorting_row μ _ hr x)
+    _ = ∑ x, r.1 x := Equiv.sum_comp e r.1
+    _ = rowMoment ν := rowSum ν r
+
+def rowFiberEquiv (μ : YoungDiagram) (i : ℕ) : {x : μ.cells // row μ x=i} ≃ Fin (μ.rowLen i) where
+  toFun x := ⟨col μ x.1,by
+    have h := μ.mem_iff_lt_rowLen.mp x.1.2
+    have hi : x.1.1.1=i := x.2
+    simpa only [col,hi] using h⟩
+  invFun j := ⟨rowCell μ i j,rfl⟩
+  left_inv x := by apply Subtype.ext; apply Subtype.ext; exact Prod.ext x.2.symm rfl
+  right_inv _ := rfl
+
+lemma shape_eq_of_row_equiv (μ ν : YoungDiagram) (e : μ.cells ≃ ν.cells)
+    (h : ∀ x, row ν (e x)=row μ x) : μ=ν := by
+  have hr (i : ℕ) : μ.rowLen i=ν.rowLen i := by
+    let er : {x : μ.cells // row μ x=i} ≃ {x : ν.cells // row ν x=i} :=
+      e.subtypeEquiv (fun x => by rw [h])
+    simpa only [Fintype.card_fin] using (Fintype.card_congr (rowFiberEquiv μ i)).symm.trans
+      ((Fintype.card_congr er).trans (Fintype.card_congr (rowFiberEquiv ν i)))
+  apply YoungDiagram.ext
+  ext x
+  exact (μ.mem_iff_lt_rowLen).trans ((hr x.1) ▸ ν.mem_iff_lt_rowLen.symm)
+
+lemma cross_equal_moment_shape_eq (μ ν : YoungDiagram) (e : μ.cells ≃ ν.cells)
+    (r : Tabloid ν) (hr : CrossTransversal μ (fun x => r.1 (e x)))
+    (hm : rowMoment μ=rowMoment ν) : μ=ν := by
+  let p := crossSorting μ _ hr
+  have hs : (∑ x, row μ (p x))=∑ x, r.1 (e x) := by
+    rw [Equiv.sum_comp p,Equiv.sum_comp e,rowSum]
+    exact hm
+  have he : ∀ x, row μ (p x)=r.1 (e x) :=
+    fun x => (Finset.sum_eq_sum_iff_of_le (fun x _ => crossSorting_row μ _ hr x)).mp hs x (mem_univ x)
+  obtain ⟨q,hq⟩ := r.2
+  apply shape_eq_of_row_equiv μ ν ((p.symm.trans e).trans q)
+  intro x
+  simpa [hq] using (he (p.symm x)).symm
+
+end
+end PartialPermutation.YoungTabloid
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/DeficitCounts.lean b/lean/OAI/RepresentationTheory/PartialPermutation/DeficitCounts.lean
index 0fef29a878023192ec029b8d2efb6cc7fadd1d86..6e30176124d00494cbc7c82e44b93ec1bfd6bfb3 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/DeficitCounts.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/DeficitCounts.lean
@@ -1,133 +1,133 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.DiagramCounts
-
-namespace OAI
-
-section
-namespace PartialPermutation
-namespace DiagramCounting
-noncomputable section
-open Finset
-
-lemma shift_injective : Function.Injective (fun x : ℕ × ℕ => (x.1+1,x.2)) := by
-  rintro ⟨i,j⟩ ⟨k,l⟩ h
-  simp only [Prod.mk.injEq] at h
-  exact Prod.ext (Nat.add_right_cancel h.1) h.2
-
-def dropTop (μ : YoungDiagram) : YoungDiagram where
-  cells := μ.cells.preimage (fun x : ℕ × ℕ => (x.1+1,x.2)) shift_injective.injOn
-  isLowerSet := by
-    intro y x hxy hy
-    simp only [Finset.mem_coe,Finset.mem_preimage] at hy ⊢
-    have hh : (x.1+1,x.2) ≤ (y.1+1,y.2) := ⟨Nat.add_le_add_right hxy.1 1,hxy.2⟩
-    exact μ.isLowerSet hh hy
-
-lemma mem_dropTop (μ : YoungDiagram) (i j : ℕ) :
-    (i,j) ∈ dropTop μ ↔ (i+1,j) ∈ μ := by
-  change (i,j) ∈ μ.cells.preimage (fun x : ℕ × ℕ => (x.1+1,x.2)) shift_injective.injOn ↔ _
-  exact Finset.mem_preimage
-
-def lowerEquiv (μ : YoungDiagram) : (dropTop μ).cells ≃ {x : μ.cells // x.1.1≠0} where
-  toFun x := ⟨⟨(x.1.1+1,x.1.2), (mem_dropTop μ _ _).mp x.2⟩,Nat.succ_ne_zero _⟩
-  invFun x := ⟨(x.1.1.1-1,x.1.1.2), by
-    apply (mem_dropTop μ _ _).mpr
-    have he : x.1.1.1-1+1=x.1.1.1 := by have h:=x.2; omega
-    rw [he]; exact x.1.2⟩
-  left_inv x := by apply Subtype.ext; simp
-  right_inv x := by
-    apply Subtype.ext; apply Subtype.ext; apply Prod.ext
-    · have h:=x.2; dsimp; omega
-    · rfl
-
-def rowEquiv (μ : YoungDiagram) : {x : μ.cells // x.1.1=0} ≃ Fin (μ.rowLen 0) where
-  toFun x := ⟨x.1.1.2, by have h:=μ.mem_iff_lt_rowLen.mp x.1.2; simpa [x.2] using h⟩
-  invFun i := ⟨⟨(0,i), μ.mem_iff_lt_rowLen.mpr i.2⟩,rfl⟩
-  left_inv x := by apply Subtype.ext; apply Subtype.ext; exact Prod.ext x.2.symm rfl
-  right_inv _ := rfl
-
-lemma dropTop_card (μ : YoungDiagram) :
-    (dropTop μ).cells.card = μ.cells.card-μ.rowLen 0 := by
-  have h := Fintype.card_congr (lowerEquiv μ)
-  rw [Fintype.card_subtype_compl] at h
-  have he := Fintype.card_congr (rowEquiv μ)
-  simpa only [Fintype.card_coe,Fintype.card_fin,he] using h
-
-lemma eq_of_dropTop_row (μ ν : YoungDiagram) (hd : dropTop μ=dropTop ν)
-    (hr : μ.rowLen 0=ν.rowLen 0) : μ=ν := by
-  apply YoungDiagram.ext
-  apply Finset.ext
-  rintro ⟨i,j⟩
-  cases i with
-  | zero => exact μ.mem_iff_lt_rowLen.trans (hr ▸ ν.mem_iff_lt_rowLen.symm)
-  | succ i =>
-    change (i+1,j) ∈ μ ↔ (i+1,j) ∈ ν
-    rw [← mem_dropTop μ, ← mem_dropTop ν,hd]
-
-lemma transpose_card (μ : YoungDiagram) : μ.transpose.cells.card=μ.cells.card := by
-  have e : μ.cells ≃ μ.transpose.cells := {
-    toFun := fun x => ⟨x.1.swap, by simpa only [YoungDiagram.mem_cells,
-      YoungDiagram.mem_transpose,Prod.swap_swap] using x.2⟩
-    invFun := fun x => ⟨x.1.swap,YoungDiagram.mem_transpose.mp x.2⟩
-    left_inv := fun _ => rfl
-    right_inv := fun _ => rfl }
-  simpa only [Fintype.card_coe] using (Fintype.card_congr e).symm
-
-def ShapeDeficit (n k : ℕ) :=
-  {μ : Shape n // μ.1.cells.card-max (μ.1.rowLen 0) (μ.1.colLen 0)=k}
-
-instance (n k : ℕ) : Fintype (ShapeDeficit n k) := by
-  classical
-  unfold ShapeDeficit
-  infer_instance
-
-def deficitCode (n k : ℕ) (μ : ShapeDeficit n k) : Bool × Shape k :=
-  if h : μ.1.1.colLen 0 ≤ μ.1.1.rowLen 0 then
-    (true,⟨dropTop μ.1.1, by rw [dropTop_card]; simpa only [max_eq_left h] using μ.2⟩)
-  else
-    (false,⟨dropTop μ.1.1.transpose, by
-      rw [dropTop_card,transpose_card,YoungDiagram.rowLen_transpose]
-      simpa only [max_eq_right (le_of_not_ge h)] using μ.2⟩)
-
-lemma deficitCode_injective (n k : ℕ) : Function.Injective (deficitCode n k) := by
-  intro μ ν he
-  unfold deficitCode at he
-  split_ifs at he with hμ hν hν
-  · have hd : dropTop μ.1.1=dropTop ν.1.1 := congrArg (fun t : Bool × Shape k => t.2.1) he
-    have hcμ := μ.2
-    have hcν := ν.2
-    rw [μ.1.2,max_eq_left hμ] at hcμ
-    rw [ν.1.2,max_eq_left hν] at hcν
-    have hleμ := rowLen_le_card μ.1.1 0
-    have hleν := rowLen_le_card ν.1.1 0
-    rw [μ.1.2] at hleμ
-    rw [ν.1.2] at hleν
-    apply Subtype.ext; apply Subtype.ext
-    exact eq_of_dropTop_row _ _ hd (by omega)
-  · have hh := congrArg Prod.fst he; cases hh
-  · have hh := congrArg Prod.fst he; cases hh
-  · have hd : dropTop μ.1.1.transpose=dropTop ν.1.1.transpose :=
-      congrArg (fun t : Bool × Shape k => t.2.1) he
-    have hcμ := μ.2
-    have hcν := ν.2
-    rw [μ.1.2,max_eq_right (le_of_not_ge hμ)] at hcμ
-    rw [ν.1.2,max_eq_right (le_of_not_ge hν)] at hcν
-    have hleμ := colLen_le_card μ.1.1 0
-    have hleν := colLen_le_card ν.1.1 0
-    rw [μ.1.2] at hleμ
-    rw [ν.1.2] at hleν
-    have htr := eq_of_dropTop_row _ _ hd (by
-      simp only [YoungDiagram.rowLen_transpose]; omega)
-    apply Subtype.ext; apply Subtype.ext
-    have hh := congrArg YoungDiagram.transpose htr
-    simpa only [YoungDiagram.transpose_transpose] using hh
-
-lemma deficit_count_le (n k : ℕ) : Fintype.card (ShapeDeficit n k) ≤ 2*partitionCount k := by
-  simpa only [Fintype.card_prod,Fintype.card_bool,partitionCount] using
-    Fintype.card_le_of_injective (deficitCode n k) (deficitCode_injective n k)
-
-end
-end DiagramCounting
-end PartialPermutation
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.DiagramCounts
+
+namespace OAI
+
+section
+namespace PartialPermutation
+namespace DiagramCounting
+noncomputable section
+open Finset
+
+lemma shift_injective : Function.Injective (fun x : ℕ × ℕ => (x.1+1,x.2)) := by
+  rintro ⟨i,j⟩ ⟨k,l⟩ h
+  simp only [Prod.mk.injEq] at h
+  exact Prod.ext (Nat.add_right_cancel h.1) h.2
+
+def dropTop (μ : YoungDiagram) : YoungDiagram where
+  cells := μ.cells.preimage (fun x : ℕ × ℕ => (x.1+1,x.2)) shift_injective.injOn
+  isLowerSet := by
+    intro y x hxy hy
+    simp only [Finset.mem_coe,Finset.mem_preimage] at hy ⊢
+    have hh : (x.1+1,x.2) ≤ (y.1+1,y.2) := ⟨Nat.add_le_add_right hxy.1 1,hxy.2⟩
+    exact μ.isLowerSet hh hy
+
+lemma mem_dropTop (μ : YoungDiagram) (i j : ℕ) :
+    (i,j) ∈ dropTop μ ↔ (i+1,j) ∈ μ := by
+  change (i,j) ∈ μ.cells.preimage (fun x : ℕ × ℕ => (x.1+1,x.2)) shift_injective.injOn ↔ _
+  exact Finset.mem_preimage
+
+def lowerEquiv (μ : YoungDiagram) : (dropTop μ).cells ≃ {x : μ.cells // x.1.1≠0} where
+  toFun x := ⟨⟨(x.1.1+1,x.1.2), (mem_dropTop μ _ _).mp x.2⟩,Nat.succ_ne_zero _⟩
+  invFun x := ⟨(x.1.1.1-1,x.1.1.2), by
+    apply (mem_dropTop μ _ _).mpr
+    have he : x.1.1.1-1+1=x.1.1.1 := by have h:=x.2; omega
+    rw [he]; exact x.1.2⟩
+  left_inv x := by apply Subtype.ext; simp
+  right_inv x := by
+    apply Subtype.ext; apply Subtype.ext; apply Prod.ext
+    · have h:=x.2; dsimp; omega
+    · rfl
+
+def rowEquiv (μ : YoungDiagram) : {x : μ.cells // x.1.1=0} ≃ Fin (μ.rowLen 0) where
+  toFun x := ⟨x.1.1.2, by have h:=μ.mem_iff_lt_rowLen.mp x.1.2; simpa [x.2] using h⟩
+  invFun i := ⟨⟨(0,i), μ.mem_iff_lt_rowLen.mpr i.2⟩,rfl⟩
+  left_inv x := by apply Subtype.ext; apply Subtype.ext; exact Prod.ext x.2.symm rfl
+  right_inv _ := rfl
+
+lemma dropTop_card (μ : YoungDiagram) :
+    (dropTop μ).cells.card = μ.cells.card-μ.rowLen 0 := by
+  have h := Fintype.card_congr (lowerEquiv μ)
+  rw [Fintype.card_subtype_compl] at h
+  have he := Fintype.card_congr (rowEquiv μ)
+  simpa only [Fintype.card_coe,Fintype.card_fin,he] using h
+
+lemma eq_of_dropTop_row (μ ν : YoungDiagram) (hd : dropTop μ=dropTop ν)
+    (hr : μ.rowLen 0=ν.rowLen 0) : μ=ν := by
+  apply YoungDiagram.ext
+  apply Finset.ext
+  rintro ⟨i,j⟩
+  cases i with
+  | zero => exact μ.mem_iff_lt_rowLen.trans (hr ▸ ν.mem_iff_lt_rowLen.symm)
+  | succ i =>
+    change (i+1,j) ∈ μ ↔ (i+1,j) ∈ ν
+    rw [← mem_dropTop μ, ← mem_dropTop ν,hd]
+
+lemma transpose_card (μ : YoungDiagram) : μ.transpose.cells.card=μ.cells.card := by
+  have e : μ.cells ≃ μ.transpose.cells := {
+    toFun := fun x => ⟨x.1.swap, by simpa only [YoungDiagram.mem_cells,
+      YoungDiagram.mem_transpose,Prod.swap_swap] using x.2⟩
+    invFun := fun x => ⟨x.1.swap,YoungDiagram.mem_transpose.mp x.2⟩
+    left_inv := fun _ => rfl
+    right_inv := fun _ => rfl }
+  simpa only [Fintype.card_coe] using (Fintype.card_congr e).symm
+
+def ShapeDeficit (n k : ℕ) :=
+  {μ : Shape n // μ.1.cells.card-max (μ.1.rowLen 0) (μ.1.colLen 0)=k}
+
+instance (n k : ℕ) : Fintype (ShapeDeficit n k) := by
+  classical
+  unfold ShapeDeficit
+  infer_instance
+
+def deficitCode (n k : ℕ) (μ : ShapeDeficit n k) : Bool × Shape k :=
+  if h : μ.1.1.colLen 0 ≤ μ.1.1.rowLen 0 then
+    (true,⟨dropTop μ.1.1, by rw [dropTop_card]; simpa only [max_eq_left h] using μ.2⟩)
+  else
+    (false,⟨dropTop μ.1.1.transpose, by
+      rw [dropTop_card,transpose_card,YoungDiagram.rowLen_transpose]
+      simpa only [max_eq_right (le_of_not_ge h)] using μ.2⟩)
+
+lemma deficitCode_injective (n k : ℕ) : Function.Injective (deficitCode n k) := by
+  intro μ ν he
+  unfold deficitCode at he
+  split_ifs at he with hμ hν hν
+  · have hd : dropTop μ.1.1=dropTop ν.1.1 := congrArg (fun t : Bool × Shape k => t.2.1) he
+    have hcμ := μ.2
+    have hcν := ν.2
+    rw [μ.1.2,max_eq_left hμ] at hcμ
+    rw [ν.1.2,max_eq_left hν] at hcν
+    have hleμ := rowLen_le_card μ.1.1 0
+    have hleν := rowLen_le_card ν.1.1 0
+    rw [μ.1.2] at hleμ
+    rw [ν.1.2] at hleν
+    apply Subtype.ext; apply Subtype.ext
+    exact eq_of_dropTop_row _ _ hd (by omega)
+  · have hh := congrArg Prod.fst he; cases hh
+  · have hh := congrArg Prod.fst he; cases hh
+  · have hd : dropTop μ.1.1.transpose=dropTop ν.1.1.transpose :=
+      congrArg (fun t : Bool × Shape k => t.2.1) he
+    have hcμ := μ.2
+    have hcν := ν.2
+    rw [μ.1.2,max_eq_right (le_of_not_ge hμ)] at hcμ
+    rw [ν.1.2,max_eq_right (le_of_not_ge hν)] at hcν
+    have hleμ := colLen_le_card μ.1.1 0
+    have hleν := colLen_le_card ν.1.1 0
+    rw [μ.1.2] at hleμ
+    rw [ν.1.2] at hleν
+    have htr := eq_of_dropTop_row _ _ hd (by
+      simp only [YoungDiagram.rowLen_transpose]; omega)
+    apply Subtype.ext; apply Subtype.ext
+    have hh := congrArg YoungDiagram.transpose htr
+    simpa only [YoungDiagram.transpose_transpose] using hh
+
+lemma deficit_count_le (n k : ℕ) : Fintype.card (ShapeDeficit n k) ≤ 2*partitionCount k := by
+  simpa only [Fintype.card_prod,Fintype.card_bool,partitionCount] using
+    Fintype.card_le_of_injective (deficitCode n k) (deficitCode_injective n k)
+
+end
+end DiagramCounting
+end PartialPermutation
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/DeficitPlancherel.lean b/lean/OAI/RepresentationTheory/PartialPermutation/DeficitPlancherel.lean
index 15732f302bc398895623c05a005417ca8b74846b..05081e14001bf6e593226c9555ceb3fdf15c3470 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/DeficitPlancherel.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/DeficitPlancherel.lean
@@ -1,94 +1,94 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.TrivialRepresentation
-
-namespace OAI
-
-section
-open scoped Classical
-open scoped BigOperators ComplexConjugate MonoidAlgebra
-open scoped BigOperators ComplexConjugate
-open scoped MonoidAlgebra BigOperators
-open scoped BigOperators MonoidAlgebra Classical
-
-namespace PartialPermutation
-noncomputable section
-open scoped BigOperators MonoidAlgebra Classical
-
-variable {G : Type*} [Group G] [Fintype G]
-def nontrivialEnergy (f : G → ℝ) : ℝ :=
-  ∑ c ∈ Finset.univ.erase (trivialIndex G), (irreducibleDegree c : ℝ) *
-    hsNormSq (V := irreducibleSpace c)
-      (complexFourier (V := irreducibleSpace c) (irreducibleRep c) (fun g => (f g : ℂ)))
-
-theorem centered_plancherel (f : G → ℝ) :
-    (Fintype.card G : ℝ) * ∑ g, (f g - (∑ h, f h) / Fintype.card G)^2 =
-      nontrivialEnergy f := by
-  have h := finite_group_plancherel
-    (fun g => (f g : ℂ) - (∑ h, (f h : ℂ)) / Fintype.card G)
-  have hz : complexFourier (irreducibleRep (trivialIndex G))
-      (fun g => (f g : ℂ) - (∑ h, (f h : ℂ)) / Fintype.card G) = 0 := by
-    rw [complexFourier_centered, ite_eq_left rfl, complexFourier_trivial, sub_self]
-  have hj (c : IrreducibleIndex G) (hc : c ≠ trivialIndex G) :
-      complexFourier (irreducibleRep c)
-        (fun g => (f g : ℂ) - (∑ h, (f h : ℂ)) / Fintype.card G) =
-      complexFourier (irreducibleRep c) (fun g => (f g : ℂ)) := by
-    rw [complexFourier_centered, ite_eq_right hc, smul_zero, sub_zero]
-  rw [← Finset.sum_erase_add _ _ (Finset.mem_univ (trivialIndex G)), hz] at h
-  simp only [hsNormSq, map_zero, Complex.zero_re, mul_zero, add_zero] at h
-  have hn (g : G) :
-      ‖(f g : ℂ) - (∑ h, (f h : ℂ)) / Fintype.card G‖^2 =
-        (f g - (∑ h, f h) / Fintype.card G)^2 := by
-    rw [← Complex.ofReal_sum, ← Complex.ofReal_natCast, ← Complex.ofReal_div,
-      ← Complex.ofReal_sub, Complex.norm_real, Real.norm_eq_abs, sq_abs]
-  simp only [hn] at h
-  rw [h]
-  apply Finset.sum_congr rfl
-  intro c hc
-  rw [hj c (Finset.mem_erase.mp hc).1]
-  rfl
-
-lemma uniform_square_decomposition (f : G → ℝ) :
-    (Fintype.card G : ℝ) * ∑ g, (f g - 1 / Fintype.card G)^2 =
-      (1 - ∑ g, f g)^2 +
-        (Fintype.card G : ℝ) * ∑ g, (f g - (∑ h, f h) / Fintype.card G)^2 := by
-  have hN : (Fintype.card G : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
-  simp only [sub_sq, Finset.sum_sub_distrib, Finset.sum_add_distrib,
-    ← Finset.sum_mul, ← Finset.mul_sum, Finset.sum_const, Finset.card_univ, nsmul_eq_mul]
-  field_simp
-  ring
-
-theorem deficit_plancherel (f : G → ℝ) :
-    (∑ g, |f g - 1 / Fintype.card G|)^2 ≤ (1 - ∑ g, f g)^2 +
-      nontrivialEnergy f := by
-  have hcs := Finset.sum_mul_sq_le_sq_mul_sq (Finset.univ : Finset G)
-    (fun _ => (1 : ℝ)) (fun g => |f g - 1 / Fintype.card G|)
-  simp only [one_mul, one_pow, Finset.sum_const, Finset.card_univ,
-    nsmul_eq_mul, mul_one, sq_abs] at hcs
-  rw [uniform_square_decomposition, centered_plancherel] at hcs
-  exact hcs
-
-theorem normalization_tv_le {X : Type*} [Fintype X] (f μ : X → ℝ)
-    (hf : ∀ x, 0 ≤ f x) (hdom : ∀ x, f x ≤ μ x) (hm : ∑ x, μ x = 1)
-    (hz : 0 < ∑ x, f x) :
-    (∑ x, |f x / (∑ y, f y) - μ x|) / 2 ≤ 1 - ∑ x, f x := by
-  have hz1 : ∑ x, f x ≤ 1 := (Finset.sum_le_sum (fun x _ => hdom x)).trans_eq hm
-  have hnorm (x : X) : |f x / (∑ y, f y) - f x| =
-      f x / (∑ y, f y) - f x := by
-    apply abs_of_nonneg
-    exact sub_nonneg.mpr ((le_div_iff₀ hz).mpr (mul_le_of_le_one_right (hf x) hz1))
-  have hdiff (x : X) : |f x - μ x| = μ x - f x := abs_sub_comm _ _ |>.trans
-    (abs_of_nonneg (sub_nonneg.mpr (hdom x)))
-  have htri (x : X) : |f x / (∑ y, f y) - μ x| ≤
-      (f x / (∑ y, f y) - f x) + (μ x - f x) := by
-    simpa only [hnorm, hdiff] using abs_sub_le (f x / (∑ y, f y)) (f x) (μ x)
-  have hs := Finset.sum_le_sum (s := (Finset.univ : Finset X)) (fun x _ => htri x)
-  simp only [Finset.sum_add_distrib, Finset.sum_sub_distrib, ← Finset.sum_div, hm,
-    div_self hz.ne'] at hs
-  linarith
-
-end
-end PartialPermutation
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.TrivialRepresentation
+
+namespace OAI
+
+section
+open scoped Classical
+open scoped BigOperators ComplexConjugate MonoidAlgebra
+open scoped BigOperators ComplexConjugate
+open scoped MonoidAlgebra BigOperators
+open scoped BigOperators MonoidAlgebra Classical
+
+namespace PartialPermutation
+noncomputable section
+open scoped BigOperators MonoidAlgebra Classical
+
+variable {G : Type*} [Group G] [Fintype G]
+def nontrivialEnergy (f : G → ℝ) : ℝ :=
+  ∑ c ∈ Finset.univ.erase (trivialIndex G), (irreducibleDegree c : ℝ) *
+    hsNormSq (V := irreducibleSpace c)
+      (complexFourier (V := irreducibleSpace c) (irreducibleRep c) (fun g => (f g : ℂ)))
+
+theorem centered_plancherel (f : G → ℝ) :
+    (Fintype.card G : ℝ) * ∑ g, (f g - (∑ h, f h) / Fintype.card G)^2 =
+      nontrivialEnergy f := by
+  have h := finite_group_plancherel
+    (fun g => (f g : ℂ) - (∑ h, (f h : ℂ)) / Fintype.card G)
+  have hz : complexFourier (irreducibleRep (trivialIndex G))
+      (fun g => (f g : ℂ) - (∑ h, (f h : ℂ)) / Fintype.card G) = 0 := by
+    rw [complexFourier_centered, ite_eq_left rfl, complexFourier_trivial, sub_self]
+  have hj (c : IrreducibleIndex G) (hc : c ≠ trivialIndex G) :
+      complexFourier (irreducibleRep c)
+        (fun g => (f g : ℂ) - (∑ h, (f h : ℂ)) / Fintype.card G) =
+      complexFourier (irreducibleRep c) (fun g => (f g : ℂ)) := by
+    rw [complexFourier_centered, ite_eq_right hc, smul_zero, sub_zero]
+  rw [← Finset.sum_erase_add _ _ (Finset.mem_univ (trivialIndex G)), hz] at h
+  simp only [hsNormSq, map_zero, Complex.zero_re, mul_zero, add_zero] at h
+  have hn (g : G) :
+      ‖(f g : ℂ) - (∑ h, (f h : ℂ)) / Fintype.card G‖^2 =
+        (f g - (∑ h, f h) / Fintype.card G)^2 := by
+    rw [← Complex.ofReal_sum, ← Complex.ofReal_natCast, ← Complex.ofReal_div,
+      ← Complex.ofReal_sub, Complex.norm_real, Real.norm_eq_abs, sq_abs]
+  simp only [hn] at h
+  rw [h]
+  apply Finset.sum_congr rfl
+  intro c hc
+  rw [hj c (Finset.mem_erase.mp hc).1]
+  rfl
+
+lemma uniform_square_decomposition (f : G → ℝ) :
+    (Fintype.card G : ℝ) * ∑ g, (f g - 1 / Fintype.card G)^2 =
+      (1 - ∑ g, f g)^2 +
+        (Fintype.card G : ℝ) * ∑ g, (f g - (∑ h, f h) / Fintype.card G)^2 := by
+  have hN : (Fintype.card G : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
+  simp only [sub_sq, Finset.sum_sub_distrib, Finset.sum_add_distrib,
+    ← Finset.sum_mul, ← Finset.mul_sum, Finset.sum_const, Finset.card_univ, nsmul_eq_mul]
+  field_simp
+  ring
+
+theorem deficit_plancherel (f : G → ℝ) :
+    (∑ g, |f g - 1 / Fintype.card G|)^2 ≤ (1 - ∑ g, f g)^2 +
+      nontrivialEnergy f := by
+  have hcs := Finset.sum_mul_sq_le_sq_mul_sq (Finset.univ : Finset G)
+    (fun _ => (1 : ℝ)) (fun g => |f g - 1 / Fintype.card G|)
+  simp only [one_mul, one_pow, Finset.sum_const, Finset.card_univ,
+    nsmul_eq_mul, mul_one, sq_abs] at hcs
+  rw [uniform_square_decomposition, centered_plancherel] at hcs
+  exact hcs
+
+theorem normalization_tv_le {X : Type*} [Fintype X] (f μ : X → ℝ)
+    (hf : ∀ x, 0 ≤ f x) (hdom : ∀ x, f x ≤ μ x) (hm : ∑ x, μ x = 1)
+    (hz : 0 < ∑ x, f x) :
+    (∑ x, |f x / (∑ y, f y) - μ x|) / 2 ≤ 1 - ∑ x, f x := by
+  have hz1 : ∑ x, f x ≤ 1 := (Finset.sum_le_sum (fun x _ => hdom x)).trans_eq hm
+  have hnorm (x : X) : |f x / (∑ y, f y) - f x| =
+      f x / (∑ y, f y) - f x := by
+    apply abs_of_nonneg
+    exact sub_nonneg.mpr ((le_div_iff₀ hz).mpr (mul_le_of_le_one_right (hf x) hz1))
+  have hdiff (x : X) : |f x - μ x| = μ x - f x := abs_sub_comm _ _ |>.trans
+    (abs_of_nonneg (sub_nonneg.mpr (hdom x)))
+  have htri (x : X) : |f x / (∑ y, f y) - μ x| ≤
+      (f x / (∑ y, f y) - f x) + (μ x - f x) := by
+    simpa only [hnorm, hdiff] using abs_sub_le (f x / (∑ y, f y)) (f x) (μ x)
+  have hs := Finset.sum_le_sum (s := (Finset.univ : Finset X)) (fun x _ => htri x)
+  simp only [Finset.sum_add_distrib, Finset.sum_sub_distrib, ← Finset.sum_div, hm,
+    div_self hz.ne'] at hs
+  linarith
+
+end
+end PartialPermutation
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/DegreeEstimates.lean b/lean/OAI/RepresentationTheory/PartialPermutation/DegreeEstimates.lean
index 4bbefea1c0ebd1fa5f626a9bd819e894dcceab6d..fb150547eb8f113da041ea2902b681c62815ac65 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/DegreeEstimates.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/DegreeEstimates.lean
@@ -1,58 +1,58 @@
-import Mathlib
-
-namespace OAI
-
-section
-open scoped Classical
-open scoped BigOperators ComplexConjugate MonoidAlgebra
-open scoped BigOperators ComplexConjugate
-open scoped MonoidAlgebra BigOperators
-open scoped BigOperators MonoidAlgebra Classical
-
-attribute [local instance] Classical.propDecidable
-open scoped MonoidAlgebra BigOperators
-open scoped BigOperators
-
-namespace PartialPermutation
-
-lemma exists_degree_le_rpow {b D : ℕ} (hb : 0 < b) (hD : 0 < D)
-    (d : Fin b → ℕ) (hprod : ∏ i, d i ≤ D) :
-    ∃ i, (d i : ℝ) ≤ (D : ℝ) ^ (1 / (b : ℝ)) := by
-  classical
-  by_contra h
-  push Not at h
-  have hb' : (b : ℝ) ≠ 0 := by positivity
-  have hD' : (0 : ℝ) < D := by positivity
-  have hp : ∏ _ : Fin b, (D : ℝ) ^ (1 / (b : ℝ)) < ∏ i, (d i : ℝ) := by
-    apply Finset.prod_lt_prod₀
-    · intro i hi; positivity
-    · intro i hi; exact (h i).le
-    · exact ⟨⟨0, hb⟩, Finset.mem_univ _, h _⟩
-  have he : ∏ _ : Fin b, (D : ℝ) ^ (1 / (b : ℝ)) = D := by
-    simp only [Finset.prod_const, Finset.card_univ, Fintype.card_fin]
-    rw [← Real.rpow_natCast, ← Real.rpow_mul hD'.le]
-    simp [hb']
-  rw [he, ← Nat.cast_prod] at hp
-  exact (not_lt_of_ge (Nat.cast_le.mpr hprod)) hp
-
-lemma degree_sq_div_le {b D d : ℕ} (hD : 0 < D)
-    (hd : (d : ℝ) ≤ (D : ℝ) ^ (1 / (b : ℝ))) :
-    (d : ℝ) ^ 2 / D ≤ ((D : ℝ) ^ (-(1 / 2 : ℝ) + 1 / b)) ^ 2 := by
-  have hD' : (0 : ℝ) < D := by positivity
-  rw [div_le_iff₀ hD']
-  calc
-    (d : ℝ) ^ 2 ≤ ((D : ℝ) ^ (1 / (b : ℝ))) ^ 2 :=
-      pow_le_pow_left₀ (by positivity) hd 2
-    _ = ((D : ℝ) ^ (-(1 / 2 : ℝ) + 1 / b)) ^ 2 * D := by
-      rw [← Real.rpow_two, ← Real.rpow_two, ← Real.rpow_mul hD'.le,
-        ← Real.rpow_mul hD'.le]
-      nth_rw 3 [← Real.rpow_one (D : ℝ)]
-      rw [← Real.rpow_add hD']
-      congr 1
-      ring
-
-end PartialPermutation
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+section
+open scoped Classical
+open scoped BigOperators ComplexConjugate MonoidAlgebra
+open scoped BigOperators ComplexConjugate
+open scoped MonoidAlgebra BigOperators
+open scoped BigOperators MonoidAlgebra Classical
+
+attribute [local instance] Classical.propDecidable
+open scoped MonoidAlgebra BigOperators
+open scoped BigOperators
+
+namespace PartialPermutation
+
+lemma exists_degree_le_rpow {b D : ℕ} (hb : 0 < b) (hD : 0 < D)
+    (d : Fin b → ℕ) (hprod : ∏ i, d i ≤ D) :
+    ∃ i, (d i : ℝ) ≤ (D : ℝ) ^ (1 / (b : ℝ)) := by
+  classical
+  by_contra h
+  push Not at h
+  have hb' : (b : ℝ) ≠ 0 := by positivity
+  have hD' : (0 : ℝ) < D := by positivity
+  have hp : ∏ _ : Fin b, (D : ℝ) ^ (1 / (b : ℝ)) < ∏ i, (d i : ℝ) := by
+    apply Finset.prod_lt_prod₀
+    · intro i hi; positivity
+    · intro i hi; exact (h i).le
+    · exact ⟨⟨0, hb⟩, Finset.mem_univ _, h _⟩
+  have he : ∏ _ : Fin b, (D : ℝ) ^ (1 / (b : ℝ)) = D := by
+    simp only [Finset.prod_const, Finset.card_univ, Fintype.card_fin]
+    rw [← Real.rpow_natCast, ← Real.rpow_mul hD'.le]
+    simp [hb']
+  rw [he, ← Nat.cast_prod] at hp
+  exact (not_lt_of_ge (Nat.cast_le.mpr hprod)) hp
+
+lemma degree_sq_div_le {b D d : ℕ} (hD : 0 < D)
+    (hd : (d : ℝ) ≤ (D : ℝ) ^ (1 / (b : ℝ))) :
+    (d : ℝ) ^ 2 / D ≤ ((D : ℝ) ^ (-(1 / 2 : ℝ) + 1 / b)) ^ 2 := by
+  have hD' : (0 : ℝ) < D := by positivity
+  rw [div_le_iff₀ hD']
+  calc
+    (d : ℝ) ^ 2 ≤ ((D : ℝ) ^ (1 / (b : ℝ))) ^ 2 :=
+      pow_le_pow_left₀ (by positivity) hd 2
+    _ = ((D : ℝ) ^ (-(1 / 2 : ℝ) + 1 / b)) ^ 2 * D := by
+      rw [← Real.rpow_two, ← Real.rpow_two, ← Real.rpow_mul hD'.le,
+        ← Real.rpow_mul hD'.le]
+      nth_rw 3 [← Real.rpow_one (D : ℝ)]
+      rw [← Real.rpow_add hD']
+      congr 1
+      ring
+
+end PartialPermutation
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/DegreeTransfer.lean b/lean/OAI/RepresentationTheory/PartialPermutation/DegreeTransfer.lean
index 36a7f2cbad37a7e00f222e1b9186947d8066696b..743c9eaf5828eaea4c87ee8f339637b89ac1d2c6 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/DegreeTransfer.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/DegreeTransfer.lean
@@ -1,81 +1,81 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.ProjectionCovers
-import OAI.RepresentationTheory.PartialPermutation.JointKernels
-
-namespace OAI
-
-section
-open scoped Classical
-open scoped BigOperators ComplexConjugate MonoidAlgebra
-open scoped BigOperators ComplexConjugate
-open scoped MonoidAlgebra BigOperators
-open scoped BigOperators MonoidAlgebra Classical
-
-attribute [local instance] Classical.propDecidable
-open scoped MonoidAlgebra BigOperators
-open scoped BigOperators
-
-namespace PartialPermutation
-noncomputable section
-variable {G V : Type*} [Group G] [Fintype G]
-    [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-    (ρ : Representation ℂ G V) {b : ℕ} (H : Fin b → Subgroup G)
-
-lemma joint_small_type_cover (hb : 0 < b) (hD : 0 < Module.finrank ℂ V)
-    (hρ : IsUnitary ρ)
-    (hcomm : ∀ i j, i ≠ j → ∀ g ∈ H i, ∀ k ∈ H j, Commute g k)
-    (c : isotypicComponents ℂ[↥(⨆ i, H i)] (Representation.asModule (ρ.comp (⨆ i, H i).subtype))) :
-    ∃ i, ∀ x ∈ componentSpace (ρ.comp (⨆ i, H i).subtype) c,
-      complexFourier (ρ.comp (H i).subtype)
-        (smallTypeKernel (H i) ((Module.finrank ℂ V : ℝ)^(1 / (b : ℝ)))) x = x := by
-  let J := ⨆ i, H i
-  let ρJ : Representation ℂ J V := ρ.comp J.subtype
-  obtain ⟨i, d, k, hd, hk, hki, hka⟩ := component_kernel ρ H hb hD hρ hcomm c
-  let σ : Subrepresentation (ρ.comp (H i).subtype) :=
-    { toSubmodule := componentSpace ρJ c
-      apply_mem_toSubmodule := fun g v hv =>
-        componentSpace_invariant ρJ c ⟨g, le_iSup H i g.property⟩ v hv }
-  have hu : IsUnitary σ.toRepresentation := fun g x y => hρ g x y
-  have ha : complexFourier σ.toRepresentation k = 1 := by
-    ext x
-    rw [complexFourier_subrep]
-    exact hka x.val x.property
-  have hs := smallTypeKernel_identity_of_kernel σ.toRepresentation hu k d
-    ((Module.finrank ℂ V : ℝ)^(1 / (b : ℝ))) hd hk ha
-  refine ⟨i, fun x hx => ?_⟩
-  have ht := congrArg (fun A : Module.End ℂ σ.toSubmodule => A ⟨x, hx⟩) hs
-  have ht' := congrArg (fun y : σ.toSubmodule => (y : V)) ht
-  rw [complexFourier_subrep] at ht'
-  exact ht'
-
-theorem small_degree_sum_transfer [Representation.IsIrreducible ρ]
-    (hb : 0 < b) (hρ : IsUnitary ρ)
-    (hcomm : ∀ i j, i ≠ j → ∀ g ∈ H i, ∀ k ∈ H j, Commute g k)
-    (f : G → ℝ) (B : ℝ) (hf : ∀ g, 0 ≤ f g) (hcap : ∀ i, LeftCosetCap f (H i) B) :
-    (Module.finrank ℂ V : ℝ) * hsNormSq (complexFourier ρ (fun g => f g)) ≤
-      B * mass f * ∑ i, smallDegreeSum (H i) ((Module.finrank ℂ V : ℝ)^(1 / (b : ℝ))) := by
-  have : Nontrivial V := IsSimpleModule.nontrivial ℂ[G] ρ.asModule
-  have hD : 0 < Module.finrank ℂ V := Module.finrank_pos
-  let R := (Module.finrank ℂ V : ℝ)^(1 / (b : ℝ))
-  let P (i : Fin b) := complexFourier (ρ.comp (H i).subtype) (smallTypeKernel (H i) R)
-  have hp (i : Fin b) : LinearMap.adjoint (P i) = P i :=
-    complexFourier_selfAdjoint (G := H i) (ρ.comp (H i).subtype)
-      (fun (g : H i) => hρ g) _ (smallTypeKernel_inv R)
-  have hcover := component_projection_cover_hs (G := ↥(⨆ i, H i))
-    (ρ.comp (⨆ i, H i).subtype) (fun (g : ↥(⨆ i, H i)) => hρ g) P hp
-    (joint_small_type_cover ρ H hb hD hρ hcomm) (complexFourier ρ (fun g => f g))
-  calc
-    _ ≤ (Module.finrank ℂ V : ℝ) * ∑ i, hsNormSq (complexFourier ρ (fun g => f g) * P i) :=
-      mul_le_mul_of_nonneg_left hcover (by positivity)
-    _ = ∑ i, (Module.finrank ℂ V : ℝ) * hsNormSq (complexFourier ρ (fun g => f g) * P i) :=
-      Finset.mul_sum _ _ _
-    _ ≤ ∑ i, B * mass f * smallDegreeSum (H i) R :=
-      Finset.sum_le_sum (fun i _ => small_type_hs_bound ρ hρ (H i) R f B hf (hcap i))
-    _ = _ := by rw [Finset.mul_sum]
-
-end
-end PartialPermutation
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.ProjectionCovers
+import OAI.RepresentationTheory.PartialPermutation.JointKernels
+
+namespace OAI
+
+section
+open scoped Classical
+open scoped BigOperators ComplexConjugate MonoidAlgebra
+open scoped BigOperators ComplexConjugate
+open scoped MonoidAlgebra BigOperators
+open scoped BigOperators MonoidAlgebra Classical
+
+attribute [local instance] Classical.propDecidable
+open scoped MonoidAlgebra BigOperators
+open scoped BigOperators
+
+namespace PartialPermutation
+noncomputable section
+variable {G V : Type*} [Group G] [Fintype G]
+    [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+    (ρ : Representation ℂ G V) {b : ℕ} (H : Fin b → Subgroup G)
+
+lemma joint_small_type_cover (hb : 0 < b) (hD : 0 < Module.finrank ℂ V)
+    (hρ : IsUnitary ρ)
+    (hcomm : ∀ i j, i ≠ j → ∀ g ∈ H i, ∀ k ∈ H j, Commute g k)
+    (c : isotypicComponents ℂ[↥(⨆ i, H i)] (Representation.asModule (ρ.comp (⨆ i, H i).subtype))) :
+    ∃ i, ∀ x ∈ componentSpace (ρ.comp (⨆ i, H i).subtype) c,
+      complexFourier (ρ.comp (H i).subtype)
+        (smallTypeKernel (H i) ((Module.finrank ℂ V : ℝ)^(1 / (b : ℝ)))) x = x := by
+  let J := ⨆ i, H i
+  let ρJ : Representation ℂ J V := ρ.comp J.subtype
+  obtain ⟨i, d, k, hd, hk, hki, hka⟩ := component_kernel ρ H hb hD hρ hcomm c
+  let σ : Subrepresentation (ρ.comp (H i).subtype) :=
+    { toSubmodule := componentSpace ρJ c
+      apply_mem_toSubmodule := fun g v hv =>
+        componentSpace_invariant ρJ c ⟨g, le_iSup H i g.property⟩ v hv }
+  have hu : IsUnitary σ.toRepresentation := fun g x y => hρ g x y
+  have ha : complexFourier σ.toRepresentation k = 1 := by
+    ext x
+    rw [complexFourier_subrep]
+    exact hka x.val x.property
+  have hs := smallTypeKernel_identity_of_kernel σ.toRepresentation hu k d
+    ((Module.finrank ℂ V : ℝ)^(1 / (b : ℝ))) hd hk ha
+  refine ⟨i, fun x hx => ?_⟩
+  have ht := congrArg (fun A : Module.End ℂ σ.toSubmodule => A ⟨x, hx⟩) hs
+  have ht' := congrArg (fun y : σ.toSubmodule => (y : V)) ht
+  rw [complexFourier_subrep] at ht'
+  exact ht'
+
+theorem small_degree_sum_transfer [Representation.IsIrreducible ρ]
+    (hb : 0 < b) (hρ : IsUnitary ρ)
+    (hcomm : ∀ i j, i ≠ j → ∀ g ∈ H i, ∀ k ∈ H j, Commute g k)
+    (f : G → ℝ) (B : ℝ) (hf : ∀ g, 0 ≤ f g) (hcap : ∀ i, LeftCosetCap f (H i) B) :
+    (Module.finrank ℂ V : ℝ) * hsNormSq (complexFourier ρ (fun g => f g)) ≤
+      B * mass f * ∑ i, smallDegreeSum (H i) ((Module.finrank ℂ V : ℝ)^(1 / (b : ℝ))) := by
+  have : Nontrivial V := IsSimpleModule.nontrivial ℂ[G] ρ.asModule
+  have hD : 0 < Module.finrank ℂ V := Module.finrank_pos
+  let R := (Module.finrank ℂ V : ℝ)^(1 / (b : ℝ))
+  let P (i : Fin b) := complexFourier (ρ.comp (H i).subtype) (smallTypeKernel (H i) R)
+  have hp (i : Fin b) : LinearMap.adjoint (P i) = P i :=
+    complexFourier_selfAdjoint (G := H i) (ρ.comp (H i).subtype)
+      (fun (g : H i) => hρ g) _ (smallTypeKernel_inv R)
+  have hcover := component_projection_cover_hs (G := ↥(⨆ i, H i))
+    (ρ.comp (⨆ i, H i).subtype) (fun (g : ↥(⨆ i, H i)) => hρ g) P hp
+    (joint_small_type_cover ρ H hb hD hρ hcomm) (complexFourier ρ (fun g => f g))
+  calc
+    _ ≤ (Module.finrank ℂ V : ℝ) * ∑ i, hsNormSq (complexFourier ρ (fun g => f g) * P i) :=
+      mul_le_mul_of_nonneg_left hcover (by positivity)
+    _ = ∑ i, (Module.finrank ℂ V : ℝ) * hsNormSq (complexFourier ρ (fun g => f g) * P i) :=
+      Finset.mul_sum _ _ _
+    _ ≤ ∑ i, B * mass f * smallDegreeSum (H i) R :=
+      Finset.sum_le_sum (fun i _ => small_type_hs_bound ρ hρ (H i) R f B hf (hcap i))
+    _ = _ := by rw [Finset.mul_sum]
+
+end
+end PartialPermutation
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/DiagonalBounds.lean b/lean/OAI/RepresentationTheory/PartialPermutation/DiagonalBounds.lean
index 73397e42d72db4835f12ccd657f21709f4672e51..01cdd34e512029fabacfaa4483c064f7b1dea67b 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/DiagonalBounds.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/DiagonalBounds.lean
@@ -1,58 +1,58 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.RankFillings
-
-namespace OAI
-
-section
-namespace PartialPermutation
-namespace Tableau
-noncomputable section
-open Finset
-
-def diagonalIndices (μ : YoungDiagram) : Finset ℕ := μ.cells.image (fun x => x.1+x.2)
-
-def diagonalRank (μ : YoungDiagram) (x : μ.cells) : diagonalIndices μ :=
-  ⟨x.1.1+x.1.2, Finset.mem_image.mpr ⟨x.1,x.2,rfl⟩⟩
-
-lemma diagonalRank_strictMono (μ : YoungDiagram) : StrictMono (diagonalRank μ) := by
-  intro x y hxy
-  have h1 : x.1.1 ≤ y.1.1 := hxy.le.1
-  have h2 : x.1.2 ≤ y.1.2 := hxy.le.2
-  have hn : x.1 ≠ y.1 := fun h => (ne_of_lt hxy) (Subtype.ext h)
-  change x.1.1+x.1.2 < y.1.1+y.1.2
-  by_contra h
-  apply hn
-  exact Prod.ext (by omega) (by omega)
-
-lemma diagonal_factorial_lower_bound (μ : YoungDiagram) :
-    (∏ k : diagonalIndices μ,
-      (Fintype.card {x : μ.cells // diagonalRank μ x = k}).factorial) ≤ standardCount μ :=
-  rank_factorial_le_filling_count _ _ (diagonalRank μ) (diagonalRank_strictMono μ)
-
-lemma diagonal_power_lower_bound (μ : YoungDiagram) :
-    2^(μ.cells.card - (diagonalIndices μ).card) ≤ standardCount μ := by
-  simpa only [Fintype.card_coe, standardCount] using
-    rank_power_le_filling_count _ _ (diagonalRank μ) (diagonalRank_strictMono μ)
-
-lemma diagonal_count_le (μ : YoungDiagram) : (diagonalIndices μ).card ≤ 2*maxLine μ-1 := by
-  apply (Finset.card_le_card (t := Finset.range (2*maxLine μ-1)) ?_).trans_eq (Finset.card_range _)
-  intro k hk
-  obtain ⟨⟨i,j⟩,hij,rfl⟩ := Finset.mem_image.mp hk
-  have hj : j < maxLine μ := (μ.mem_iff_lt_rowLen.mp hij).trans_le
-    ((μ.rowLen_anti 0 i (Nat.zero_le _)).trans (le_max_left _ _))
-  have hi : i < maxLine μ := (μ.mem_iff_lt_colLen.mp hij).trans_le
-    ((μ.colLen_anti 0 j (Nat.zero_le _)).trans (le_max_right _ _))
-  simp only [Finset.mem_range]
-  omega
-
-lemma width_height_power_lower_bound (μ : YoungDiagram) :
-    2^(μ.cells.card - (2*maxLine μ-1)) ≤ standardCount μ :=
-  (Nat.pow_le_pow_right (by omega) (Nat.sub_le_sub_left (diagonal_count_le μ) _)).trans
-    (diagonal_power_lower_bound μ)
-
-end
-end Tableau
-end PartialPermutation
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.RankFillings
+
+namespace OAI
+
+section
+namespace PartialPermutation
+namespace Tableau
+noncomputable section
+open Finset
+
+def diagonalIndices (μ : YoungDiagram) : Finset ℕ := μ.cells.image (fun x => x.1+x.2)
+
+def diagonalRank (μ : YoungDiagram) (x : μ.cells) : diagonalIndices μ :=
+  ⟨x.1.1+x.1.2, Finset.mem_image.mpr ⟨x.1,x.2,rfl⟩⟩
+
+lemma diagonalRank_strictMono (μ : YoungDiagram) : StrictMono (diagonalRank μ) := by
+  intro x y hxy
+  have h1 : x.1.1 ≤ y.1.1 := hxy.le.1
+  have h2 : x.1.2 ≤ y.1.2 := hxy.le.2
+  have hn : x.1 ≠ y.1 := fun h => (ne_of_lt hxy) (Subtype.ext h)
+  change x.1.1+x.1.2 < y.1.1+y.1.2
+  by_contra h
+  apply hn
+  exact Prod.ext (by omega) (by omega)
+
+lemma diagonal_factorial_lower_bound (μ : YoungDiagram) :
+    (∏ k : diagonalIndices μ,
+      (Fintype.card {x : μ.cells // diagonalRank μ x = k}).factorial) ≤ standardCount μ :=
+  rank_factorial_le_filling_count _ _ (diagonalRank μ) (diagonalRank_strictMono μ)
+
+lemma diagonal_power_lower_bound (μ : YoungDiagram) :
+    2^(μ.cells.card - (diagonalIndices μ).card) ≤ standardCount μ := by
+  simpa only [Fintype.card_coe, standardCount] using
+    rank_power_le_filling_count _ _ (diagonalRank μ) (diagonalRank_strictMono μ)
+
+lemma diagonal_count_le (μ : YoungDiagram) : (diagonalIndices μ).card ≤ 2*maxLine μ-1 := by
+  apply (Finset.card_le_card (t := Finset.range (2*maxLine μ-1)) ?_).trans_eq (Finset.card_range _)
+  intro k hk
+  obtain ⟨⟨i,j⟩,hij,rfl⟩ := Finset.mem_image.mp hk
+  have hj : j < maxLine μ := (μ.mem_iff_lt_rowLen.mp hij).trans_le
+    ((μ.rowLen_anti 0 i (Nat.zero_le _)).trans (le_max_left _ _))
+  have hi : i < maxLine μ := (μ.mem_iff_lt_colLen.mp hij).trans_le
+    ((μ.colLen_anti 0 j (Nat.zero_le _)).trans (le_max_right _ _))
+  simp only [Finset.mem_range]
+  omega
+
+lemma width_height_power_lower_bound (μ : YoungDiagram) :
+    2^(μ.cells.card - (2*maxLine μ-1)) ≤ standardCount μ :=
+  (Nat.pow_le_pow_right (by omega) (Nat.sub_le_sub_left (diagonal_count_le μ) _)).trans
+    (diagonal_power_lower_bound μ)
+
+end
+end Tableau
+end PartialPermutation
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/DiagramCounts.lean b/lean/OAI/RepresentationTheory/PartialPermutation/DiagramCounts.lean
index 50ea7b1e7a8f056328a6fcb136c0563f4baaf8fa..c9bff35513e698e73154a081481a42fb7829aa1a 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/DiagramCounts.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/DiagramCounts.lean
@@ -1,116 +1,116 @@
-import Mathlib
-
-namespace OAI
-
-section
-namespace PartialPermutation
-namespace DiagramCounting
-noncomputable section
-open Finset
-
-lemma rectangle_card_le (μ : YoungDiagram) (i j : ℕ) (h : (i,j) ∈ μ) :
-    (i+1)*(j+1) ≤ μ.cells.card := by
-  have hs : (Finset.range (i+1)) ×ˢ (Finset.range (j+1)) ⊆ μ.cells := by
-    intro x hx
-    rcases Finset.mem_product.mp hx with ⟨hx,hy⟩
-    exact μ.isLowerSet ⟨by simpa only [Finset.mem_range, Nat.lt_succ_iff] using hx,
-      by simpa only [Finset.mem_range, Nat.lt_succ_iff] using hy⟩ h
-  simpa only [Finset.card_product, Finset.card_range] using Finset.card_le_card hs
-
-lemma cell_bounded (μ : YoungDiagram) (x : μ.cells) :
-    x.1.1 < μ.cells.card ∧ x.1.2 < μ.cells.card := by
-  have h := rectangle_card_le μ x.1.1 x.1.2 x.2
-  constructor <;> nlinarith
-
-lemma rowLen_le_card (μ : YoungDiagram) (i : ℕ) : μ.rowLen i ≤ μ.cells.card := by
-  by_contra h
-  have hx := (μ.mem_iff_lt_rowLen.mpr (show μ.cells.card < μ.rowLen i by omega))
-  have hh := cell_bounded μ ⟨(i,μ.cells.card),hx⟩
-  exact (lt_irrefl _) hh.2
-
-lemma colLen_le_card (μ : YoungDiagram) (j : ℕ) : μ.colLen j ≤ μ.cells.card := by
-  by_contra h
-  have hx := (μ.mem_iff_lt_colLen.mpr (show μ.cells.card < μ.colLen j by omega))
-  have hh := cell_bounded μ ⟨(μ.cells.card,j),hx⟩
-  exact (lt_irrefl _) hh.1
-
-def Shape (n : ℕ) := {μ : YoungDiagram // μ.cells.card=n}
-
-def boxCode (n : ℕ) (μ : Shape n) :
-    ((Finset.range n) ×ˢ (Finset.range n)).powerset := by
-  refine ⟨μ.1.cells,Finset.mem_powerset.mpr ?_⟩
-  intro x hx
-  have hh := cell_bounded μ.1 ⟨x,hx⟩
-  rw [μ.2] at hh
-  simpa only [Finset.mem_product, Finset.mem_range] using hh
-
-lemma boxCode_injective (n : ℕ) : Function.Injective (boxCode n) := by
-  intro μ ν h
-  apply Subtype.ext
-  apply YoungDiagram.ext
-  exact congrArg Subtype.val h
-
-instance (n : ℕ) : Fintype (Shape n) := Fintype.ofInjective (boxCode n) (boxCode_injective n)
-
-def partitionCount (n : ℕ) := Fintype.card (Shape n)
-
-def shortCode (n m : ℕ) (μ : Shape n) :
-    (Fin m → Fin (n+1)) × (Fin m → Fin (n+1)) :=
-  (fun i => ⟨μ.1.rowLen i, by have h:=rowLen_le_card μ.1 i; rw [μ.2] at h; omega⟩,
-   fun j => ⟨μ.1.colLen j, by have h:=colLen_le_card μ.1 j; rw [μ.2] at h; omega⟩)
-
-lemma no_large_corner (n m : ℕ) (hn : n < (m+1)*(m+1)) (μ : Shape n)
-    {i j : ℕ} (h : (i,j) ∈ μ.1) : i < m ∨ j < m := by
-  by_contra hh
-  push Not at hh
-  have hcorner : (m,m) ∈ μ.1 := μ.1.isLowerSet hh h
-  have hc := rectangle_card_le μ.1 m m hcorner
-  rw [μ.2] at hc
-  omega
-
-lemma shortCode_injective (n m : ℕ) (hn : n < (m+1)*(m+1)) :
-    Function.Injective (shortCode n m) := by
-  intro μ ν he
-  have hr : ∀ i : Fin m, μ.1.rowLen i = ν.1.rowLen i := by
-    intro i
-    exact congrArg (fun z : (Fin m → Fin (n+1)) × (Fin m → Fin (n+1)) => (z.1 i).val) he
-  have hc : ∀ j : Fin m, μ.1.colLen j = ν.1.colLen j := by
-    intro j
-    exact congrArg (fun z : (Fin m → Fin (n+1)) × (Fin m → Fin (n+1)) => (z.2 j).val) he
-  apply Subtype.ext
-  apply YoungDiagram.ext
-  apply Finset.ext
-  rintro ⟨i,j⟩
-  constructor
-  · intro h
-    rcases no_large_corner n m hn μ h with hi | hj
-    · apply ν.1.mem_iff_lt_rowLen.mpr
-      rw [← hr ⟨i,hi⟩]
-      exact μ.1.mem_iff_lt_rowLen.mp h
-    · apply ν.1.mem_iff_lt_colLen.mpr
-      rw [← hc ⟨j,hj⟩]
-      exact μ.1.mem_iff_lt_colLen.mp h
-  · intro h
-    rcases no_large_corner n m hn ν h with hi | hj
-    · apply μ.1.mem_iff_lt_rowLen.mpr
-      rw [hr ⟨i,hi⟩]
-      exact ν.1.mem_iff_lt_rowLen.mp h
-    · apply μ.1.mem_iff_lt_colLen.mpr
-      rw [hc ⟨j,hj⟩]
-      exact ν.1.mem_iff_lt_colLen.mp h
-
-lemma partitionCount_le_shortCode (n m : ℕ) (hn : n < (m+1)*(m+1)) :
-    partitionCount n ≤ (n+1)^(2*m) := by
-  have h := Fintype.card_le_of_injective (shortCode n m) (shortCode_injective n m hn)
-  simpa only [Fintype.card_prod, Fintype.card_fun, Fintype.card_fin, ← pow_add,
-    ← two_mul, partitionCount] using h
-
-lemma partitionCount_le_sqrt (n : ℕ) : partitionCount n ≤ (n+1)^(2*n.sqrt) :=
-  partitionCount_le_shortCode n n.sqrt (Nat.lt_succ_sqrt n)
-
-end
-end DiagramCounting
-end PartialPermutation
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+section
+namespace PartialPermutation
+namespace DiagramCounting
+noncomputable section
+open Finset
+
+lemma rectangle_card_le (μ : YoungDiagram) (i j : ℕ) (h : (i,j) ∈ μ) :
+    (i+1)*(j+1) ≤ μ.cells.card := by
+  have hs : (Finset.range (i+1)) ×ˢ (Finset.range (j+1)) ⊆ μ.cells := by
+    intro x hx
+    rcases Finset.mem_product.mp hx with ⟨hx,hy⟩
+    exact μ.isLowerSet ⟨by simpa only [Finset.mem_range, Nat.lt_succ_iff] using hx,
+      by simpa only [Finset.mem_range, Nat.lt_succ_iff] using hy⟩ h
+  simpa only [Finset.card_product, Finset.card_range] using Finset.card_le_card hs
+
+lemma cell_bounded (μ : YoungDiagram) (x : μ.cells) :
+    x.1.1 < μ.cells.card ∧ x.1.2 < μ.cells.card := by
+  have h := rectangle_card_le μ x.1.1 x.1.2 x.2
+  constructor <;> nlinarith
+
+lemma rowLen_le_card (μ : YoungDiagram) (i : ℕ) : μ.rowLen i ≤ μ.cells.card := by
+  by_contra h
+  have hx := (μ.mem_iff_lt_rowLen.mpr (show μ.cells.card < μ.rowLen i by omega))
+  have hh := cell_bounded μ ⟨(i,μ.cells.card),hx⟩
+  exact (lt_irrefl _) hh.2
+
+lemma colLen_le_card (μ : YoungDiagram) (j : ℕ) : μ.colLen j ≤ μ.cells.card := by
+  by_contra h
+  have hx := (μ.mem_iff_lt_colLen.mpr (show μ.cells.card < μ.colLen j by omega))
+  have hh := cell_bounded μ ⟨(μ.cells.card,j),hx⟩
+  exact (lt_irrefl _) hh.1
+
+def Shape (n : ℕ) := {μ : YoungDiagram // μ.cells.card=n}
+
+def boxCode (n : ℕ) (μ : Shape n) :
+    ((Finset.range n) ×ˢ (Finset.range n)).powerset := by
+  refine ⟨μ.1.cells,Finset.mem_powerset.mpr ?_⟩
+  intro x hx
+  have hh := cell_bounded μ.1 ⟨x,hx⟩
+  rw [μ.2] at hh
+  simpa only [Finset.mem_product, Finset.mem_range] using hh
+
+lemma boxCode_injective (n : ℕ) : Function.Injective (boxCode n) := by
+  intro μ ν h
+  apply Subtype.ext
+  apply YoungDiagram.ext
+  exact congrArg Subtype.val h
+
+instance (n : ℕ) : Fintype (Shape n) := Fintype.ofInjective (boxCode n) (boxCode_injective n)
+
+def partitionCount (n : ℕ) := Fintype.card (Shape n)
+
+def shortCode (n m : ℕ) (μ : Shape n) :
+    (Fin m → Fin (n+1)) × (Fin m → Fin (n+1)) :=
+  (fun i => ⟨μ.1.rowLen i, by have h:=rowLen_le_card μ.1 i; rw [μ.2] at h; omega⟩,
+   fun j => ⟨μ.1.colLen j, by have h:=colLen_le_card μ.1 j; rw [μ.2] at h; omega⟩)
+
+lemma no_large_corner (n m : ℕ) (hn : n < (m+1)*(m+1)) (μ : Shape n)
+    {i j : ℕ} (h : (i,j) ∈ μ.1) : i < m ∨ j < m := by
+  by_contra hh
+  push Not at hh
+  have hcorner : (m,m) ∈ μ.1 := μ.1.isLowerSet hh h
+  have hc := rectangle_card_le μ.1 m m hcorner
+  rw [μ.2] at hc
+  omega
+
+lemma shortCode_injective (n m : ℕ) (hn : n < (m+1)*(m+1)) :
+    Function.Injective (shortCode n m) := by
+  intro μ ν he
+  have hr : ∀ i : Fin m, μ.1.rowLen i = ν.1.rowLen i := by
+    intro i
+    exact congrArg (fun z : (Fin m → Fin (n+1)) × (Fin m → Fin (n+1)) => (z.1 i).val) he
+  have hc : ∀ j : Fin m, μ.1.colLen j = ν.1.colLen j := by
+    intro j
+    exact congrArg (fun z : (Fin m → Fin (n+1)) × (Fin m → Fin (n+1)) => (z.2 j).val) he
+  apply Subtype.ext
+  apply YoungDiagram.ext
+  apply Finset.ext
+  rintro ⟨i,j⟩
+  constructor
+  · intro h
+    rcases no_large_corner n m hn μ h with hi | hj
+    · apply ν.1.mem_iff_lt_rowLen.mpr
+      rw [← hr ⟨i,hi⟩]
+      exact μ.1.mem_iff_lt_rowLen.mp h
+    · apply ν.1.mem_iff_lt_colLen.mpr
+      rw [← hc ⟨j,hj⟩]
+      exact μ.1.mem_iff_lt_colLen.mp h
+  · intro h
+    rcases no_large_corner n m hn ν h with hi | hj
+    · apply μ.1.mem_iff_lt_rowLen.mpr
+      rw [hr ⟨i,hi⟩]
+      exact ν.1.mem_iff_lt_rowLen.mp h
+    · apply μ.1.mem_iff_lt_colLen.mpr
+      rw [hc ⟨j,hj⟩]
+      exact ν.1.mem_iff_lt_colLen.mp h
+
+lemma partitionCount_le_shortCode (n m : ℕ) (hn : n < (m+1)*(m+1)) :
+    partitionCount n ≤ (n+1)^(2*m) := by
+  have h := Fintype.card_le_of_injective (shortCode n m) (shortCode_injective n m hn)
+  simpa only [Fintype.card_prod, Fintype.card_fun, Fintype.card_fin, ← pow_add,
+    ← two_mul, partitionCount] using h
+
+lemma partitionCount_le_sqrt (n : ℕ) : partitionCount n ≤ (n+1)^(2*n.sqrt) :=
+  partitionCount_le_shortCode n n.sqrt (Nat.lt_succ_sqrt n)
+
+end
+end DiagramCounting
+end PartialPermutation
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/Equivalences.lean b/lean/OAI/RepresentationTheory/PartialPermutation/Equivalences.lean
index ef1f01f09fc2d4919bbde88db23a674d23261488..72e1ceedee9d1913daaaf752ab85d9e183eda227 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/Equivalences.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/Equivalences.lean
@@ -1,106 +1,106 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.Plancherel
-
-namespace OAI
-
-section
-open scoped Classical
-open scoped BigOperators ComplexConjugate MonoidAlgebra
-open scoped BigOperators ComplexConjugate
-open scoped MonoidAlgebra BigOperators
-open scoped BigOperators MonoidAlgebra Classical
-
-attribute [local instance] Classical.propDecidable
-open scoped MonoidAlgebra BigOperators
-open scoped BigOperators
-
-namespace PartialPermutation
-noncomputable section
-variable {G K V W : Type*} [Group G] [Group K]
-    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
-
-def subrepresentationCompEquiv (ρ : Representation ℂ G V) (e : K ≃* G) :
-    Subrepresentation (ρ.comp e.toMonoidHom) ≃o Subrepresentation ρ where
-  toFun S :=
-    { toSubmodule := S.toSubmodule
-      apply_mem_toSubmodule := by
-        intro g v hv
-        simpa using S.apply_mem_toSubmodule (e.symm g) hv }
-  invFun S :=
-    { toSubmodule := S.toSubmodule
-      apply_mem_toSubmodule := fun g v hv => S.apply_mem_toSubmodule (e g) hv }
-  left_inv S := by ext; rfl
-  right_inv S := by ext; rfl
-  map_rel_iff' := Iff.rfl
-
-instance irreducible_comp_mulEquiv (ρ : Representation ℂ G V)
-    [Representation.IsIrreducible ρ] (e : K ≃* G) :
-    Representation.IsIrreducible (ρ.comp e.toMonoidHom) :=
-  (subrepresentationCompEquiv ρ e).isSimpleOrder_iff.mpr inferInstance
-
-def representationEquivComp {ρ σ : Representation ℂ G V}
-    (a : Representation.Equiv ρ σ) (e : K →* G) :
-    Representation.Equiv (ρ.comp e) (σ.comp e) :=
-  Representation.Equiv.mk a.toLinearEquiv (fun g => a.isIntertwining' (e g))
-
-def representationEquivCompGeneral {ρ : Representation ℂ G V} {σ : Representation ℂ G W}
-    (a : Representation.Equiv ρ σ) (e : K →* G) :
-    Representation.Equiv (ρ.comp e) (σ.comp e) :=
-  Representation.Equiv.mk a.toLinearEquiv (fun g => a.isIntertwining' (e g))
-
-variable [Fintype G] [Fintype K]
-
-def irreducibleIndexMap (e : G ≃* K) (c : IrreducibleIndex G) : IrreducibleIndex K :=
-  (irreducibleRep_complete ((irreducibleRep c).comp e.symm.toMonoidHom)).choose
-
-lemma irreducibleIndexMap_spec (e : G ≃* K) (c : IrreducibleIndex G) :
-    Nonempty (Representation.Equiv (irreducibleRep (irreducibleIndexMap e c))
-      ((irreducibleRep c).comp e.symm.toMonoidHom)) :=
-  (irreducibleRep_complete ((irreducibleRep c).comp e.symm.toMonoidHom)).choose_spec.1
-
-lemma irreducibleIndexMap_inverse (e : G ≃* K) (c : IrreducibleIndex G) :
-    irreducibleIndexMap e.symm (irreducibleIndexMap e c) = c := by
-  obtain ⟨a⟩ := irreducibleIndexMap_spec e c
-  obtain ⟨b⟩ := irreducibleIndexMap_spec e.symm (irreducibleIndexMap e c)
-  have h : Nonempty (Representation.Equiv
-      (irreducibleRep (irreducibleIndexMap e.symm (irreducibleIndexMap e c)))
-      (irreducibleRep c)) := by
-    refine ⟨Representation.Equiv.mk (b.toLinearEquiv.trans a.toLinearEquiv) ?_⟩
-    intro g
-    apply LinearMap.ext
-    intro x
-    change a (b (irreducibleRep _ g x)) = irreducibleRep c g (a (b x))
-    calc
-      _ = a (irreducibleRep (irreducibleIndexMap e c) (e g) (b x)) :=
-        congrArg a (congrArg (fun L : _ →ₗ[ℂ] _ => L x) (b.isIntertwining' g))
-      _ = _ := by
-        have ha := congrArg (fun L : _ →ₗ[ℂ] _ => L (b x)) (a.isIntertwining' (e g))
-        change a (irreducibleRep (irreducibleIndexMap e c) (e g) (b x)) =
-          irreducibleRep c (e.symm (e g)) (a (b x)) at ha
-        simpa only [MulEquiv.symm_apply_apply] using ha
-  exact (componentRep_equiv_iff (regularRep G) _ _).mp h
-
-def irreducibleIndexEquiv (e : G ≃* K) : IrreducibleIndex G ≃ IrreducibleIndex K where
-  toFun := irreducibleIndexMap e
-  invFun := irreducibleIndexMap e.symm
-  left_inv := irreducibleIndexMap_inverse e
-  right_inv := irreducibleIndexMap_inverse e.symm
-
-lemma irreducibleDegree_map (e : G ≃* K) (c : IrreducibleIndex G) :
-    irreducibleDegree (irreducibleIndexMap e c) = irreducibleDegree c := by
-  obtain ⟨a⟩ := irreducibleIndexMap_spec e c
-  exact a.toLinearEquiv.finrank_eq
-
-lemma inverseDegreeSum_equiv (e : G ≃* K) (u : ℝ) :
-    inverseDegreeSum G u = inverseDegreeSum K u := by
-  apply Fintype.sum_equiv (irreducibleIndexEquiv e)
-  intro c
-  change (irreducibleDegree c : ℝ)^(-u) = (irreducibleDegree (irreducibleIndexMap e c) : ℝ)^(-u)
-  rw [irreducibleDegree_map]
-
-end
-end PartialPermutation
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.Plancherel
+
+namespace OAI
+
+section
+open scoped Classical
+open scoped BigOperators ComplexConjugate MonoidAlgebra
+open scoped BigOperators ComplexConjugate
+open scoped MonoidAlgebra BigOperators
+open scoped BigOperators MonoidAlgebra Classical
+
+attribute [local instance] Classical.propDecidable
+open scoped MonoidAlgebra BigOperators
+open scoped BigOperators
+
+namespace PartialPermutation
+noncomputable section
+variable {G K V W : Type*} [Group G] [Group K]
+    [AddCommGroup V] [Module ℂ V] [AddCommGroup W] [Module ℂ W]
+
+def subrepresentationCompEquiv (ρ : Representation ℂ G V) (e : K ≃* G) :
+    Subrepresentation (ρ.comp e.toMonoidHom) ≃o Subrepresentation ρ where
+  toFun S :=
+    { toSubmodule := S.toSubmodule
+      apply_mem_toSubmodule := by
+        intro g v hv
+        simpa using S.apply_mem_toSubmodule (e.symm g) hv }
+  invFun S :=
+    { toSubmodule := S.toSubmodule
+      apply_mem_toSubmodule := fun g v hv => S.apply_mem_toSubmodule (e g) hv }
+  left_inv S := by ext; rfl
+  right_inv S := by ext; rfl
+  map_rel_iff' := Iff.rfl
+
+instance irreducible_comp_mulEquiv (ρ : Representation ℂ G V)
+    [Representation.IsIrreducible ρ] (e : K ≃* G) :
+    Representation.IsIrreducible (ρ.comp e.toMonoidHom) :=
+  (subrepresentationCompEquiv ρ e).isSimpleOrder_iff.mpr inferInstance
+
+def representationEquivComp {ρ σ : Representation ℂ G V}
+    (a : Representation.Equiv ρ σ) (e : K →* G) :
+    Representation.Equiv (ρ.comp e) (σ.comp e) :=
+  Representation.Equiv.mk a.toLinearEquiv (fun g => a.isIntertwining' (e g))
+
+def representationEquivCompGeneral {ρ : Representation ℂ G V} {σ : Representation ℂ G W}
+    (a : Representation.Equiv ρ σ) (e : K →* G) :
+    Representation.Equiv (ρ.comp e) (σ.comp e) :=
+  Representation.Equiv.mk a.toLinearEquiv (fun g => a.isIntertwining' (e g))
+
+variable [Fintype G] [Fintype K]
+
+def irreducibleIndexMap (e : G ≃* K) (c : IrreducibleIndex G) : IrreducibleIndex K :=
+  (irreducibleRep_complete ((irreducibleRep c).comp e.symm.toMonoidHom)).choose
+
+lemma irreducibleIndexMap_spec (e : G ≃* K) (c : IrreducibleIndex G) :
+    Nonempty (Representation.Equiv (irreducibleRep (irreducibleIndexMap e c))
+      ((irreducibleRep c).comp e.symm.toMonoidHom)) :=
+  (irreducibleRep_complete ((irreducibleRep c).comp e.symm.toMonoidHom)).choose_spec.1
+
+lemma irreducibleIndexMap_inverse (e : G ≃* K) (c : IrreducibleIndex G) :
+    irreducibleIndexMap e.symm (irreducibleIndexMap e c) = c := by
+  obtain ⟨a⟩ := irreducibleIndexMap_spec e c
+  obtain ⟨b⟩ := irreducibleIndexMap_spec e.symm (irreducibleIndexMap e c)
+  have h : Nonempty (Representation.Equiv
+      (irreducibleRep (irreducibleIndexMap e.symm (irreducibleIndexMap e c)))
+      (irreducibleRep c)) := by
+    refine ⟨Representation.Equiv.mk (b.toLinearEquiv.trans a.toLinearEquiv) ?_⟩
+    intro g
+    apply LinearMap.ext
+    intro x
+    change a (b (irreducibleRep _ g x)) = irreducibleRep c g (a (b x))
+    calc
+      _ = a (irreducibleRep (irreducibleIndexMap e c) (e g) (b x)) :=
+        congrArg a (congrArg (fun L : _ →ₗ[ℂ] _ => L x) (b.isIntertwining' g))
+      _ = _ := by
+        have ha := congrArg (fun L : _ →ₗ[ℂ] _ => L (b x)) (a.isIntertwining' (e g))
+        change a (irreducibleRep (irreducibleIndexMap e c) (e g) (b x)) =
+          irreducibleRep c (e.symm (e g)) (a (b x)) at ha
+        simpa only [MulEquiv.symm_apply_apply] using ha
+  exact (componentRep_equiv_iff (regularRep G) _ _).mp h
+
+def irreducibleIndexEquiv (e : G ≃* K) : IrreducibleIndex G ≃ IrreducibleIndex K where
+  toFun := irreducibleIndexMap e
+  invFun := irreducibleIndexMap e.symm
+  left_inv := irreducibleIndexMap_inverse e
+  right_inv := irreducibleIndexMap_inverse e.symm
+
+lemma irreducibleDegree_map (e : G ≃* K) (c : IrreducibleIndex G) :
+    irreducibleDegree (irreducibleIndexMap e c) = irreducibleDegree c := by
+  obtain ⟨a⟩ := irreducibleIndexMap_spec e c
+  exact a.toLinearEquiv.finrank_eq
+
+lemma inverseDegreeSum_equiv (e : G ≃* K) (u : ℝ) :
+    inverseDegreeSum G u = inverseDegreeSum K u := by
+  apply Fintype.sum_equiv (irreducibleIndexEquiv e)
+  intro c
+  change (irreducibleDegree c : ℝ)^(-u) = (irreducibleDegree (irreducibleIndexMap e c) : ℝ)^(-u)
+  rw [irreducibleDegree_map]
+
+end
+end PartialPermutation
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/FillingExtensions.lean b/lean/OAI/RepresentationTheory/PartialPermutation/FillingExtensions.lean
index d17a0ca9eb7df0da076fc3475db31e3814588644..deeea35e21a6e4523f7f37de3846b1759b27d4dd 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/FillingExtensions.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/FillingExtensions.lean
@@ -1,116 +1,116 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.StandardTableaux
-
-namespace OAI
-
-section
-namespace PartialPermutation
-namespace Tableau
-noncomputable section
-open Finset
-
-section Extension
-variable {α β : Type*} [Fintype α] [PartialOrder α] [Fintype β] [PartialOrder β]
-
-def fillingCongr (e : α ≃o β) : StandardFilling α ≃ StandardFilling β where
-  toFun T := ⟨e.symm.toEquiv.trans (T.1.trans (finCongr (Fintype.card_congr e.toEquiv))),
-    (Fin.cast_strictMono (Fintype.card_congr e.toEquiv)).comp (T.2.comp e.symm.strictMono)⟩
-  invFun T := ⟨e.toEquiv.trans (T.1.trans (finCongr (Fintype.card_congr e.symm.toEquiv))),
-    (Fin.cast_strictMono (Fintype.card_congr e.symm.toEquiv)).comp (T.2.comp e.strictMono)⟩
-  left_inv T := by apply Subtype.ext; apply Equiv.ext; intro a; apply Fin.ext; simp
-  right_inv T := by apply Subtype.ext; apply Equiv.ext; intro a; apply Fin.ext; simp
-
-omit [PartialOrder α] in
-lemma sum_subtype_card (P : α → Prop) [DecidablePred P] :
-    Fintype.card {a // P a} + Fintype.card {a // ¬P a} = Fintype.card α := by
-  simpa only [Fintype.card_sum] using Fintype.card_congr (Equiv.sumCompl P)
-
-def concatEnumeration (P : α → Prop) [DecidablePred P]
-    (T : StandardFilling {a // P a}) (U : StandardFilling {a // ¬P a}) :
-    α ≃ Fin (Fintype.card α) :=
-  (Equiv.sumCompl P).symm.trans ((Equiv.sumCongr T.1 U.1).trans
-    (finSumFinEquiv.trans (finCongr (sum_subtype_card P))))
-
-lemma concatEnumeration_pos (P : α → Prop) [DecidablePred P]
-    (T : StandardFilling {a // P a}) (U : StandardFilling {a // ¬P a})
-    (a : α) (ha : P a) : (concatEnumeration P T U a).val = (T.1 ⟨a,ha⟩).val := by
-  simp [concatEnumeration, Equiv.sumCompl_symm_apply_of_pos ha]
-
-lemma concatEnumeration_neg (P : α → Prop) [DecidablePred P]
-    (T : StandardFilling {a // P a}) (U : StandardFilling {a // ¬P a})
-    (a : α) (ha : ¬P a) : (concatEnumeration P T U a).val =
-      Fintype.card {a // P a} + (U.1 ⟨a,ha⟩).val := by
-  simp [concatEnumeration, Equiv.sumCompl_symm_apply_of_neg ha]
-
-def extendFilling (P : α → Prop) [DecidablePred P] (hP : IsLowerSet {a | P a})
-    (T : StandardFilling {a // P a}) (U : StandardFilling {a // ¬P a}) :
-    StandardFilling α := by
-  refine ⟨concatEnumeration P T U, ?_⟩
-  intro a b hab
-  change (concatEnumeration P T U a).val < (concatEnumeration P T U b).val
-  by_cases ha : P a <;> by_cases hb : P b
-  · rw [concatEnumeration_pos P T U a ha, concatEnumeration_pos P T U b hb]
-    exact T.2 (show (⟨a,ha⟩ : {a // P a}) < ⟨b,hb⟩ from hab)
-  · rw [concatEnumeration_pos P T U a ha, concatEnumeration_neg P T U b hb]
-    have h := (T.1 ⟨a,ha⟩).isLt
-    omega
-  · exact (ha (hP hab.le hb)).elim
-  · rw [concatEnumeration_neg P T U a ha, concatEnumeration_neg P T U b hb]
-    exact Nat.add_lt_add_left (U.2 (show (⟨a,ha⟩ : {a // ¬P a}) < ⟨b,hb⟩ from hab)) _
-
-lemma filling_count_mono_lowerSet (P : α → Prop) [DecidablePred P]
-    (hP : IsLowerSet {a | P a}) :
-    Fintype.card (StandardFilling {a // P a}) ≤ Fintype.card (StandardFilling α) := by
-  let U := chosenFilling {a // ¬P a}
-  apply Fintype.card_le_of_injective (fun T => extendFilling P hP T U)
-  intro T S he
-  apply Subtype.ext
-  apply Equiv.ext
-  intro a
-  apply Fin.ext
-  have h := congrArg (fun T : StandardFilling α => (T.1 a.1).val) he
-  change (concatEnumeration P T U a.1).val = (concatEnumeration P S U a.1).val at h
-  simpa only [concatEnumeration_pos P _ _ _ a.2] using h
-
-lemma filling_count_mono_complement (P : α → Prop) [DecidablePred P]
-    (hP : IsLowerSet {a | P a}) :
-    Fintype.card (StandardFilling {a // ¬P a}) ≤ Fintype.card (StandardFilling α) := by
-  let T := chosenFilling {a // P a}
-  apply Fintype.card_le_of_injective (fun U => extendFilling P hP T U)
-  intro U V he
-  apply Subtype.ext
-  apply Equiv.ext
-  intro a
-  apply Fin.ext
-  have h := congrArg (fun S : StandardFilling α => (S.1 a.1).val) he
-  change (concatEnumeration P T U a.1).val = (concatEnumeration P T V a.1).val at h
-  simp only [concatEnumeration_neg P _ _ _ a.2] at h
-  exact Nat.add_left_cancel h
-
-end Extension
-
-def subdiagramCellsEquiv {μ ν : YoungDiagram} (h : μ ≤ ν) :
-    μ.cells ≃o {x : ν.cells // x.1 ∈ μ.cells} where
-  toFun x := ⟨⟨x.1,h x.2⟩,x.2⟩
-  invFun x := ⟨x.1.1,x.2⟩
-  left_inv _ := rfl
-  right_inv _ := rfl
-  map_rel_iff' := Iff.rfl
-
-lemma standardCount_mono {μ ν : YoungDiagram} (h : μ ≤ ν) :
-    standardCount μ ≤ standardCount ν := by
-  classical
-  have hh : IsLowerSet {x : ν.cells | x.1 ∈ μ.cells} := by
-    intro x y hxy hy
-    exact μ.isLowerSet hxy hy
-  have hcard := Fintype.card_congr (fillingCongr (subdiagramCellsEquiv h))
-  unfold standardCount
-  rw [hcard]
-  exact filling_count_mono_lowerSet _ hh
-
-end
-end Tableau
-end PartialPermutation
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.StandardTableaux
+
+namespace OAI
+
+section
+namespace PartialPermutation
+namespace Tableau
+noncomputable section
+open Finset
+
+section Extension
+variable {α β : Type*} [Fintype α] [PartialOrder α] [Fintype β] [PartialOrder β]
+
+def fillingCongr (e : α ≃o β) : StandardFilling α ≃ StandardFilling β where
+  toFun T := ⟨e.symm.toEquiv.trans (T.1.trans (finCongr (Fintype.card_congr e.toEquiv))),
+    (Fin.cast_strictMono (Fintype.card_congr e.toEquiv)).comp (T.2.comp e.symm.strictMono)⟩
+  invFun T := ⟨e.toEquiv.trans (T.1.trans (finCongr (Fintype.card_congr e.symm.toEquiv))),
+    (Fin.cast_strictMono (Fintype.card_congr e.symm.toEquiv)).comp (T.2.comp e.strictMono)⟩
+  left_inv T := by apply Subtype.ext; apply Equiv.ext; intro a; apply Fin.ext; simp
+  right_inv T := by apply Subtype.ext; apply Equiv.ext; intro a; apply Fin.ext; simp
+
+omit [PartialOrder α] in
+lemma sum_subtype_card (P : α → Prop) [DecidablePred P] :
+    Fintype.card {a // P a} + Fintype.card {a // ¬P a} = Fintype.card α := by
+  simpa only [Fintype.card_sum] using Fintype.card_congr (Equiv.sumCompl P)
+
+def concatEnumeration (P : α → Prop) [DecidablePred P]
+    (T : StandardFilling {a // P a}) (U : StandardFilling {a // ¬P a}) :
+    α ≃ Fin (Fintype.card α) :=
+  (Equiv.sumCompl P).symm.trans ((Equiv.sumCongr T.1 U.1).trans
+    (finSumFinEquiv.trans (finCongr (sum_subtype_card P))))
+
+lemma concatEnumeration_pos (P : α → Prop) [DecidablePred P]
+    (T : StandardFilling {a // P a}) (U : StandardFilling {a // ¬P a})
+    (a : α) (ha : P a) : (concatEnumeration P T U a).val = (T.1 ⟨a,ha⟩).val := by
+  simp [concatEnumeration, Equiv.sumCompl_symm_apply_of_pos ha]
+
+lemma concatEnumeration_neg (P : α → Prop) [DecidablePred P]
+    (T : StandardFilling {a // P a}) (U : StandardFilling {a // ¬P a})
+    (a : α) (ha : ¬P a) : (concatEnumeration P T U a).val =
+      Fintype.card {a // P a} + (U.1 ⟨a,ha⟩).val := by
+  simp [concatEnumeration, Equiv.sumCompl_symm_apply_of_neg ha]
+
+def extendFilling (P : α → Prop) [DecidablePred P] (hP : IsLowerSet {a | P a})
+    (T : StandardFilling {a // P a}) (U : StandardFilling {a // ¬P a}) :
+    StandardFilling α := by
+  refine ⟨concatEnumeration P T U, ?_⟩
+  intro a b hab
+  change (concatEnumeration P T U a).val < (concatEnumeration P T U b).val
+  by_cases ha : P a <;> by_cases hb : P b
+  · rw [concatEnumeration_pos P T U a ha, concatEnumeration_pos P T U b hb]
+    exact T.2 (show (⟨a,ha⟩ : {a // P a}) < ⟨b,hb⟩ from hab)
+  · rw [concatEnumeration_pos P T U a ha, concatEnumeration_neg P T U b hb]
+    have h := (T.1 ⟨a,ha⟩).isLt
+    omega
+  · exact (ha (hP hab.le hb)).elim
+  · rw [concatEnumeration_neg P T U a ha, concatEnumeration_neg P T U b hb]
+    exact Nat.add_lt_add_left (U.2 (show (⟨a,ha⟩ : {a // ¬P a}) < ⟨b,hb⟩ from hab)) _
+
+lemma filling_count_mono_lowerSet (P : α → Prop) [DecidablePred P]
+    (hP : IsLowerSet {a | P a}) :
+    Fintype.card (StandardFilling {a // P a}) ≤ Fintype.card (StandardFilling α) := by
+  let U := chosenFilling {a // ¬P a}
+  apply Fintype.card_le_of_injective (fun T => extendFilling P hP T U)
+  intro T S he
+  apply Subtype.ext
+  apply Equiv.ext
+  intro a
+  apply Fin.ext
+  have h := congrArg (fun T : StandardFilling α => (T.1 a.1).val) he
+  change (concatEnumeration P T U a.1).val = (concatEnumeration P S U a.1).val at h
+  simpa only [concatEnumeration_pos P _ _ _ a.2] using h
+
+lemma filling_count_mono_complement (P : α → Prop) [DecidablePred P]
+    (hP : IsLowerSet {a | P a}) :
+    Fintype.card (StandardFilling {a // ¬P a}) ≤ Fintype.card (StandardFilling α) := by
+  let T := chosenFilling {a // P a}
+  apply Fintype.card_le_of_injective (fun U => extendFilling P hP T U)
+  intro U V he
+  apply Subtype.ext
+  apply Equiv.ext
+  intro a
+  apply Fin.ext
+  have h := congrArg (fun S : StandardFilling α => (S.1 a.1).val) he
+  change (concatEnumeration P T U a.1).val = (concatEnumeration P T V a.1).val at h
+  simp only [concatEnumeration_neg P _ _ _ a.2] at h
+  exact Nat.add_left_cancel h
+
+end Extension
+
+def subdiagramCellsEquiv {μ ν : YoungDiagram} (h : μ ≤ ν) :
+    μ.cells ≃o {x : ν.cells // x.1 ∈ μ.cells} where
+  toFun x := ⟨⟨x.1,h x.2⟩,x.2⟩
+  invFun x := ⟨x.1.1,x.2⟩
+  left_inv _ := rfl
+  right_inv _ := rfl
+  map_rel_iff' := Iff.rfl
+
+lemma standardCount_mono {μ ν : YoungDiagram} (h : μ ≤ ν) :
+    standardCount μ ≤ standardCount ν := by
+  classical
+  have hh : IsLowerSet {x : ν.cells | x.1 ∈ μ.cells} := by
+    intro x y hxy hy
+    exact μ.isLowerSet hxy hy
+  have hcard := Fintype.card_congr (fillingCongr (subdiagramCellsEquiv h))
+  unfold standardCount
+  rw [hcard]
+  exact filling_count_mono_lowerSet _ hh
+
+end
+end Tableau
+end PartialPermutation
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/FourierBessel.lean b/lean/OAI/RepresentationTheory/PartialPermutation/FourierBessel.lean
index ba41e5e70d6eec1e50f1d71eb12c88aead94576a..738cf1f901b6629febe758643f5792f4ba331fd5 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/FourierBessel.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/FourierBessel.lean
@@ -1,58 +1,58 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.MatrixCoefficients
-import OAI.RepresentationTheory.PartialPermutation.HilbertSchmidt
-
-namespace OAI
-
-section
-open scoped Classical
-open scoped BigOperators ComplexConjugate MonoidAlgebra
-open scoped BigOperators ComplexConjugate
-
-namespace PartialPermutation
-open scoped BigOperators ComplexConjugate MonoidAlgebra
-
-noncomputable section
-
-lemma fourier_bessel {G V : Type*} [Group G] [Fintype G]
-    [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-    (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) [Representation.IsIrreducible ρ]
-    (f : G → ℂ) :
-    (Module.finrank ℂ V : ℝ) * hsNormSq (complexFourier ρ f) ≤
-      (Fintype.card G : ℝ) * ∑ g, ‖f g‖ ^ 2 := by
-  classical
-  let e := stdOrthonormalBasis ℂ V
-  let t := Real.sqrt ((Module.finrank ℂ V : ℝ) / Fintype.card G)
-  let v := fun p : Fin (Module.finrank ℂ V) × Fin (Module.finrank ℂ V) =>
-    (t : ℂ) • WithLp.toLp 2 (fun g : G => conj (inner ℂ (e p.1) (ρ g (e p.2))))
-  have hv : Orthonormal ℂ v := normalized_matrixCoefficients_orthonormal ρ hρ e
-  have hN : (0 : ℝ) < Fintype.card G := by exact_mod_cast Fintype.card_pos
-  have ht : t ^ 2 = (Module.finrank ℂ V : ℝ) / Fintype.card G :=
-    Real.sq_sqrt (by positivity)
-  have inner_eq (p : Fin (Module.finrank ℂ V) × Fin (Module.finrank ℂ V)) :
-      inner ℂ (v p) (WithLp.toLp 2 f) =
-        (t : ℂ) * inner ℂ (e p.1) (complexFourier ρ f (e p.2)) := by
-    simp [v, complexFourier, PiLp.inner_apply, RCLike.inner_apply, LinearMap.sum_apply]
-  have h := hv.sum_inner_products_le (WithLp.toLp 2 f) (s := Finset.univ)
-  simp only [PiLp.norm_sq_eq_of_L2] at h
-  simp only [inner_eq, norm_mul, Complex.norm_real, Real.norm_eq_abs,
-    abs_of_nonneg (show 0 ≤ t from Real.sqrt_nonneg _), mul_pow] at h
-  rw [← Finset.mul_sum, ht, Fintype.sum_prod_type] at h
-  have hs : ∑ i, ∑ j, ‖inner ℂ (e i) (complexFourier ρ f (e j))‖ ^ 2 =
-      hsNormSq (complexFourier ρ f) := by
-    rw [Finset.sum_comm, hsNormSq_eq_sum e]
-    exact Finset.sum_congr rfl (fun j _ => e.sum_sq_norm_inner_right _)
-  rw [hs] at h
-  have h' := mul_le_mul_of_nonneg_left h hN.le
-  calc
-    _ = (Fintype.card G : ℝ) *
-        ((Module.finrank ℂ V : ℝ) / Fintype.card G * hsNormSq (complexFourier ρ f)) := by
-      field_simp
-    _ ≤ _ := h'
-
-end
-end PartialPermutation
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.MatrixCoefficients
+import OAI.RepresentationTheory.PartialPermutation.HilbertSchmidt
+
+namespace OAI
+
+section
+open scoped Classical
+open scoped BigOperators ComplexConjugate MonoidAlgebra
+open scoped BigOperators ComplexConjugate
+
+namespace PartialPermutation
+open scoped BigOperators ComplexConjugate MonoidAlgebra
+
+noncomputable section
+
+lemma fourier_bessel {G V : Type*} [Group G] [Fintype G]
+    [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+    (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) [Representation.IsIrreducible ρ]
+    (f : G → ℂ) :
+    (Module.finrank ℂ V : ℝ) * hsNormSq (complexFourier ρ f) ≤
+      (Fintype.card G : ℝ) * ∑ g, ‖f g‖ ^ 2 := by
+  classical
+  let e := stdOrthonormalBasis ℂ V
+  let t := Real.sqrt ((Module.finrank ℂ V : ℝ) / Fintype.card G)
+  let v := fun p : Fin (Module.finrank ℂ V) × Fin (Module.finrank ℂ V) =>
+    (t : ℂ) • WithLp.toLp 2 (fun g : G => conj (inner ℂ (e p.1) (ρ g (e p.2))))
+  have hv : Orthonormal ℂ v := normalized_matrixCoefficients_orthonormal ρ hρ e
+  have hN : (0 : ℝ) < Fintype.card G := by exact_mod_cast Fintype.card_pos
+  have ht : t ^ 2 = (Module.finrank ℂ V : ℝ) / Fintype.card G :=
+    Real.sq_sqrt (by positivity)
+  have inner_eq (p : Fin (Module.finrank ℂ V) × Fin (Module.finrank ℂ V)) :
+      inner ℂ (v p) (WithLp.toLp 2 f) =
+        (t : ℂ) * inner ℂ (e p.1) (complexFourier ρ f (e p.2)) := by
+    simp [v, complexFourier, PiLp.inner_apply, RCLike.inner_apply, LinearMap.sum_apply]
+  have h := hv.sum_inner_products_le (WithLp.toLp 2 f) (s := Finset.univ)
+  simp only [PiLp.norm_sq_eq_of_L2] at h
+  simp only [inner_eq, norm_mul, Complex.norm_real, Real.norm_eq_abs,
+    abs_of_nonneg (show 0 ≤ t from Real.sqrt_nonneg _), mul_pow] at h
+  rw [← Finset.mul_sum, ht, Fintype.sum_prod_type] at h
+  have hs : ∑ i, ∑ j, ‖inner ℂ (e i) (complexFourier ρ f (e j))‖ ^ 2 =
+      hsNormSq (complexFourier ρ f) := by
+    rw [Finset.sum_comm, hsNormSq_eq_sum e]
+    exact Finset.sum_congr rfl (fun j _ => e.sum_sq_norm_inner_right _)
+  rw [hs] at h
+  have h' := mul_le_mul_of_nonneg_left h hN.le
+  calc
+    _ = (Fintype.card G : ℝ) *
+        ((Module.finrank ℂ V : ℝ) / Fintype.card G * hsNormSq (complexFourier ρ f)) := by
+      field_simp
+    _ ≤ _ := h'
+
+end
+end PartialPermutation
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/HilbertSchmidt.lean b/lean/OAI/RepresentationTheory/PartialPermutation/HilbertSchmidt.lean
index fd9c2e69b2774bb0f4a7bf8862ed467b4f6d4abe..b5e24bef9a423db0b4b5e5a957a58f907fb2b79b 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/HilbertSchmidt.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/HilbertSchmidt.lean
@@ -1,104 +1,104 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.Model
-
-namespace OAI
-
-section
-open scoped Classical
-open scoped BigOperators ComplexConjugate MonoidAlgebra
-open scoped BigOperators ComplexConjugate
-
-namespace PartialPermutation
-noncomputable section
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-    [FiniteDimensional ℂ V]
-
-lemma hsNormSq_eq_sum {ι : Type*} [Fintype ι] (e : OrthonormalBasis ι ℂ V)
-    (A : Module.End ℂ V) : hsNormSq A = ∑ i, ‖A (e i)‖ ^ 2 := by
-  rw [hsNormSq, LinearMap.trace_eq_sum_inner _ e, Complex.re_sum]
-  apply Finset.sum_congr rfl
-  intro i _
-  rw [Module.End.mul_apply, LinearMap.adjoint_inner_right]
-  exact inner_self_eq_norm_sq (𝕜 := ℂ) (A (e i))
-
-lemma hsNormSq_nonneg (A : Module.End ℂ V) : 0 ≤ hsNormSq A := by
-  rw [hsNormSq_eq_sum (stdOrthonormalBasis ℂ V)]
-  positivity
-
-open scoped Matrix.Norms.Frobenius in
-
-lemma hsNormSq_eq_matrix_norm_sq {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (e : OrthonormalBasis ι ℂ V) (A : Module.End ℂ V) :
-    hsNormSq A = ‖LinearMap.toMatrix e.toBasis e.toBasis A‖ ^ 2 := by
-  rw [hsNormSq_eq_sum e, Matrix.frobenius_norm_def]
-  simp only [Real.rpow_two]
-  rw [← Real.sqrt_eq_rpow, Real.sq_sqrt (by positivity)]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro i _
-  rw [← e.sum_sq_norm_inner_right (A (e i))]
-  apply Finset.sum_congr rfl
-  intro j _
-  simp [LinearMap.toMatrix_apply, OrthonormalBasis.repr_apply_apply, e.coe_toBasis_repr_apply]
-
-lemma hsNormSq_adjoint (A : Module.End ℂ V) : hsNormSq (LinearMap.adjoint A) = hsNormSq A := by
-  unfold hsNormSq
-  rw [LinearMap.adjoint_adjoint, LinearMap.trace_mul_comm]
-
-lemma hsNormSq_projection_left (W : Submodule ℂ V) (A : Module.End ℂ V) :
-    hsNormSq (W.starProjection.toLinearMap * A) ≤ hsNormSq A := by
-  rw [hsNormSq_eq_sum (stdOrthonormalBasis ℂ V), hsNormSq_eq_sum (stdOrthonormalBasis ℂ V)]
-  exact Finset.sum_le_sum fun i _ =>
-    pow_le_pow_left₀ (norm_nonneg _) (W.norm_starProjection_apply_le _) 2
-
-lemma adjoint_mul_eq (A B : Module.End ℂ V) :
-    LinearMap.adjoint (A * B) = LinearMap.adjoint B * LinearMap.adjoint A :=
-  LinearMap.adjoint_comp A B
-
-lemma adjoint_projection_eq (W : Submodule ℂ V) :
-    LinearMap.adjoint W.starProjection.toLinearMap = W.starProjection.toLinearMap := by
-  ext v
-  apply ext_inner_left ℂ
-  intro u
-  rw [LinearMap.adjoint_inner_right]
-  exact W.inner_starProjection_left_eq_right u v
-
-lemma hsNormSq_projection_right (W : Submodule ℂ V) (A : Module.End ℂ V) :
-    hsNormSq (A * W.starProjection.toLinearMap) ≤ hsNormSq A := by
-  rw [← hsNormSq_adjoint (A * _), adjoint_mul_eq, adjoint_projection_eq,
-    ← hsNormSq_adjoint A]
-  exact hsNormSq_projection_left W _
-
-open scoped Matrix.Norms.Frobenius in
-
-lemma hsNormSq_mul_le (A B : Module.End ℂ V) :
-    hsNormSq (A * B) ≤ hsNormSq A * hsNormSq B := by
-  let e := stdOrthonormalBasis ℂ V
-  rw [hsNormSq_eq_matrix_norm_sq e, hsNormSq_eq_matrix_norm_sq e,
-    hsNormSq_eq_matrix_norm_sq e, LinearMap.toMatrix_mul]
-  simpa only [mul_pow] using pow_le_pow_left₀ (norm_nonneg _)
-    (Matrix.frobenius_norm_mul (LinearMap.toMatrix e.toBasis e.toBasis A)
-      (LinearMap.toMatrix e.toBasis e.toBasis B)) 2
-
-open scoped Matrix.Norms.Frobenius in
-
-lemma hsNormSq_sum_le {ι : Type*} [Fintype ι] (A : ι → Module.End ℂ V) :
-    hsNormSq (∑ i, A i) ≤ (Fintype.card ι : ℝ) * ∑ i, hsNormSq (A i) := by
-  classical
-  let e := stdOrthonormalBasis ℂ V
-  simp_rw [hsNormSq_eq_matrix_norm_sq e]
-  rw [map_sum]
-  refine (pow_le_pow_left₀ (norm_nonneg _) (norm_sum_le _ _) 2).trans ?_
-  have h := Finset.sum_sq_le_sum_mul_sum_of_sq_le_mul (s := Finset.univ)
-    (f := fun _ : ι => (1 : ℝ))
-    (g := fun i => ‖LinearMap.toMatrix e.toBasis e.toBasis (A i)‖ ^ 2)
-    (r := fun i => ‖LinearMap.toMatrix e.toBasis e.toBasis (A i)‖)
-    (fun _ _ => zero_le_one) (fun _ _ => sq_nonneg _) (fun _ _ => by simp)
-  simpa using h
-
-end
-end PartialPermutation
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.Model
+
+namespace OAI
+
+section
+open scoped Classical
+open scoped BigOperators ComplexConjugate MonoidAlgebra
+open scoped BigOperators ComplexConjugate
+
+namespace PartialPermutation
+noncomputable section
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+    [FiniteDimensional ℂ V]
+
+lemma hsNormSq_eq_sum {ι : Type*} [Fintype ι] (e : OrthonormalBasis ι ℂ V)
+    (A : Module.End ℂ V) : hsNormSq A = ∑ i, ‖A (e i)‖ ^ 2 := by
+  rw [hsNormSq, LinearMap.trace_eq_sum_inner _ e, Complex.re_sum]
+  apply Finset.sum_congr rfl
+  intro i _
+  rw [Module.End.mul_apply, LinearMap.adjoint_inner_right]
+  exact inner_self_eq_norm_sq (𝕜 := ℂ) (A (e i))
+
+lemma hsNormSq_nonneg (A : Module.End ℂ V) : 0 ≤ hsNormSq A := by
+  rw [hsNormSq_eq_sum (stdOrthonormalBasis ℂ V)]
+  positivity
+
+open scoped Matrix.Norms.Frobenius in
+
+lemma hsNormSq_eq_matrix_norm_sq {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (e : OrthonormalBasis ι ℂ V) (A : Module.End ℂ V) :
+    hsNormSq A = ‖LinearMap.toMatrix e.toBasis e.toBasis A‖ ^ 2 := by
+  rw [hsNormSq_eq_sum e, Matrix.frobenius_norm_def]
+  simp only [Real.rpow_two]
+  rw [← Real.sqrt_eq_rpow, Real.sq_sqrt (by positivity)]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro i _
+  rw [← e.sum_sq_norm_inner_right (A (e i))]
+  apply Finset.sum_congr rfl
+  intro j _
+  simp [LinearMap.toMatrix_apply, OrthonormalBasis.repr_apply_apply, e.coe_toBasis_repr_apply]
+
+lemma hsNormSq_adjoint (A : Module.End ℂ V) : hsNormSq (LinearMap.adjoint A) = hsNormSq A := by
+  unfold hsNormSq
+  rw [LinearMap.adjoint_adjoint, LinearMap.trace_mul_comm]
+
+lemma hsNormSq_projection_left (W : Submodule ℂ V) (A : Module.End ℂ V) :
+    hsNormSq (W.starProjection.toLinearMap * A) ≤ hsNormSq A := by
+  rw [hsNormSq_eq_sum (stdOrthonormalBasis ℂ V), hsNormSq_eq_sum (stdOrthonormalBasis ℂ V)]
+  exact Finset.sum_le_sum fun i _ =>
+    pow_le_pow_left₀ (norm_nonneg _) (W.norm_starProjection_apply_le _) 2
+
+lemma adjoint_mul_eq (A B : Module.End ℂ V) :
+    LinearMap.adjoint (A * B) = LinearMap.adjoint B * LinearMap.adjoint A :=
+  LinearMap.adjoint_comp A B
+
+lemma adjoint_projection_eq (W : Submodule ℂ V) :
+    LinearMap.adjoint W.starProjection.toLinearMap = W.starProjection.toLinearMap := by
+  ext v
+  apply ext_inner_left ℂ
+  intro u
+  rw [LinearMap.adjoint_inner_right]
+  exact W.inner_starProjection_left_eq_right u v
+
+lemma hsNormSq_projection_right (W : Submodule ℂ V) (A : Module.End ℂ V) :
+    hsNormSq (A * W.starProjection.toLinearMap) ≤ hsNormSq A := by
+  rw [← hsNormSq_adjoint (A * _), adjoint_mul_eq, adjoint_projection_eq,
+    ← hsNormSq_adjoint A]
+  exact hsNormSq_projection_left W _
+
+open scoped Matrix.Norms.Frobenius in
+
+lemma hsNormSq_mul_le (A B : Module.End ℂ V) :
+    hsNormSq (A * B) ≤ hsNormSq A * hsNormSq B := by
+  let e := stdOrthonormalBasis ℂ V
+  rw [hsNormSq_eq_matrix_norm_sq e, hsNormSq_eq_matrix_norm_sq e,
+    hsNormSq_eq_matrix_norm_sq e, LinearMap.toMatrix_mul]
+  simpa only [mul_pow] using pow_le_pow_left₀ (norm_nonneg _)
+    (Matrix.frobenius_norm_mul (LinearMap.toMatrix e.toBasis e.toBasis A)
+      (LinearMap.toMatrix e.toBasis e.toBasis B)) 2
+
+open scoped Matrix.Norms.Frobenius in
+
+lemma hsNormSq_sum_le {ι : Type*} [Fintype ι] (A : ι → Module.End ℂ V) :
+    hsNormSq (∑ i, A i) ≤ (Fintype.card ι : ℝ) * ∑ i, hsNormSq (A i) := by
+  classical
+  let e := stdOrthonormalBasis ℂ V
+  simp_rw [hsNormSq_eq_matrix_norm_sq e]
+  rw [map_sum]
+  refine (pow_le_pow_left₀ (norm_nonneg _) (norm_sum_le _ _) 2).trans ?_
+  have h := Finset.sum_sq_le_sum_mul_sum_of_sq_le_mul (s := Finset.univ)
+    (f := fun _ : ι => (1 : ℝ))
+    (g := fun i => ‖LinearMap.toMatrix e.toBasis e.toBasis (A i)‖ ^ 2)
+    (r := fun i => ‖LinearMap.toMatrix e.toBasis e.toBasis (A i)‖)
+    (fun _ _ => zero_le_one) (fun _ _ => sq_nonneg _) (fun _ _ => by simp)
+  simpa using h
+
+end
+end PartialPermutation
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/Interleavings.lean b/lean/OAI/RepresentationTheory/PartialPermutation/Interleavings.lean
index 4e6e0ca11cba7807a3615f9cf27d986289cf7423..5f900cbe363285727ab211a5307953ebef9c7d07 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/Interleavings.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/Interleavings.lean
@@ -1,98 +1,98 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.StandardTableaux
-
-namespace OAI
-
-section
-namespace PartialPermutation
-namespace Tableau
-noncomputable section
-open Finset
-
-section Interleave
-variable {α : Type*} [Fintype α] [PartialOrder α]
-
-omit [PartialOrder α] in
-lemma interleaving_compl_card (P : α → Prop) [DecidablePred P]
-    (s : {s : Finset (Fin (Fintype.card α)) // s.card=Fintype.card {a // P a}}) :
-    s.1ᶜ.card = Fintype.card {a // ¬P a} := by
-  rw [Finset.card_compl, Fintype.card_fin, s.2, Fintype.card_subtype_compl]
-
-def interleavingEnumeration (P : α → Prop) [DecidablePred P]
-    (T : StandardFilling {a // P a}) (U : StandardFilling {a // ¬P a})
-    (s : {s : Finset (Fin (Fintype.card α)) // s.card=Fintype.card {a // P a}}) :
-    α ≃ Fin (Fintype.card α) :=
-  (Equiv.sumCompl P).symm.trans ((Equiv.sumCongr T.1 U.1).trans
-    (finSumEquivOfFinset s.2 (interleaving_compl_card P s)))
-
-lemma interleavingEnumeration_pos (P : α → Prop) [DecidablePred P]
-    (T : StandardFilling {a // P a}) (U : StandardFilling {a // ¬P a})
-    (s : {s : Finset (Fin (Fintype.card α)) // s.card=Fintype.card {a // P a}})
-    (a : α) (ha : P a) :
-    interleavingEnumeration P T U s a = s.1.orderEmbOfFin s.2 (T.1 ⟨a,ha⟩) := by
-  simp [interleavingEnumeration, Equiv.sumCompl_symm_apply_of_pos ha]
-
-lemma interleavingEnumeration_neg (P : α → Prop) [DecidablePred P]
-    (T : StandardFilling {a // P a}) (U : StandardFilling {a // ¬P a})
-    (s : {s : Finset (Fin (Fintype.card α)) // s.card=Fintype.card {a // P a}})
-    (a : α) (ha : ¬P a) :
-    interleavingEnumeration P T U s a = s.1ᶜ.orderEmbOfFin
-      (interleaving_compl_card P s) (U.1 ⟨a,ha⟩) := by
-  simp [interleavingEnumeration, Equiv.sumCompl_symm_apply_of_neg ha]
-
-def interleaveFilling (P : α → Prop) [DecidablePred P]
-    (hP : ∀ a b, a < b → (P a ↔ P b))
-    (T : StandardFilling {a // P a}) (U : StandardFilling {a // ¬P a})
-    (s : {s : Finset (Fin (Fintype.card α)) // s.card=Fintype.card {a // P a}}) :
-    StandardFilling α := by
-  refine ⟨interleavingEnumeration P T U s, ?_⟩
-  intro a b hab
-  by_cases ha : P a
-  · have hb := (hP a b hab).mp ha
-    rw [interleavingEnumeration_pos P T U s a ha, interleavingEnumeration_pos P T U s b hb]
-    exact (s.1.orderEmbOfFin s.2).strictMono (T.2 hab)
-  · have hb : ¬P b := fun hp => ha ((hP a b hab).mpr hp)
-    rw [interleavingEnumeration_neg P T U s a ha, interleavingEnumeration_neg P T U s b hb]
-    exact (s.1ᶜ.orderEmbOfFin _).strictMono (U.2 hab)
-
-lemma interleaveFilling_injective (P : α → Prop) [DecidablePred P]
-    (hP : ∀ a b, a < b → (P a ↔ P b))
-    (T : StandardFilling {a // P a}) (U : StandardFilling {a // ¬P a}) :
-    Function.Injective (interleaveFilling P hP T U) := by
-  intro s t hst
-  have hval := congrArg Subtype.val hst
-  have hsr : Set.range (fun a : {a // P a} => interleavingEnumeration P T U s a.1) = s.1 := by
-    have heq : (fun a : {a // P a} => interleavingEnumeration P T U s a.1) =
-        (s.1.orderEmbOfFin s.2) ∘ T.1 := by
-      funext a
-      exact interleavingEnumeration_pos P T U s a.1 a.2
-    rw [heq, Set.range_comp, Equiv.range_eq_univ, Set.image_univ, Finset.range_orderEmbOfFin]
-  have htr : Set.range (fun a : {a // P a} => interleavingEnumeration P T U t a.1) = t.1 := by
-    have heq : (fun a : {a // P a} => interleavingEnumeration P T U t a.1) =
-        (t.1.orderEmbOfFin t.2) ∘ T.1 := by
-      funext a
-      exact interleavingEnumeration_pos P T U t a.1 a.2
-    rw [heq, Set.range_comp, Equiv.range_eq_univ, Set.image_univ, Finset.range_orderEmbOfFin]
-  apply Subtype.ext
-  apply Finset.coe_injective
-  rw [← hsr, ← htr]
-  congr 1
-  funext a
-  exact Equiv.congr_fun hval a.1
-
-lemma interleaving_binomial_lower_bound (P : α → Prop) [DecidablePred P]
-    (hP : ∀ a b, a < b → (P a ↔ P b)) :
-    (Fintype.card α).choose (Fintype.card {a // P a}) ≤
-      Fintype.card (StandardFilling α) := by
-  have h := Fintype.card_le_of_injective
-    (interleaveFilling P hP (chosenFilling _) (chosenFilling _))
-    (interleaveFilling_injective P hP (chosenFilling _) (chosenFilling _))
-  simpa only [Fintype.card_finset_len, Fintype.card_fin] using h
-end Interleave
-
-end
-end Tableau
-end PartialPermutation
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.StandardTableaux
+
+namespace OAI
+
+section
+namespace PartialPermutation
+namespace Tableau
+noncomputable section
+open Finset
+
+section Interleave
+variable {α : Type*} [Fintype α] [PartialOrder α]
+
+omit [PartialOrder α] in
+lemma interleaving_compl_card (P : α → Prop) [DecidablePred P]
+    (s : {s : Finset (Fin (Fintype.card α)) // s.card=Fintype.card {a // P a}}) :
+    s.1ᶜ.card = Fintype.card {a // ¬P a} := by
+  rw [Finset.card_compl, Fintype.card_fin, s.2, Fintype.card_subtype_compl]
+
+def interleavingEnumeration (P : α → Prop) [DecidablePred P]
+    (T : StandardFilling {a // P a}) (U : StandardFilling {a // ¬P a})
+    (s : {s : Finset (Fin (Fintype.card α)) // s.card=Fintype.card {a // P a}}) :
+    α ≃ Fin (Fintype.card α) :=
+  (Equiv.sumCompl P).symm.trans ((Equiv.sumCongr T.1 U.1).trans
+    (finSumEquivOfFinset s.2 (interleaving_compl_card P s)))
+
+lemma interleavingEnumeration_pos (P : α → Prop) [DecidablePred P]
+    (T : StandardFilling {a // P a}) (U : StandardFilling {a // ¬P a})
+    (s : {s : Finset (Fin (Fintype.card α)) // s.card=Fintype.card {a // P a}})
+    (a : α) (ha : P a) :
+    interleavingEnumeration P T U s a = s.1.orderEmbOfFin s.2 (T.1 ⟨a,ha⟩) := by
+  simp [interleavingEnumeration, Equiv.sumCompl_symm_apply_of_pos ha]
+
+lemma interleavingEnumeration_neg (P : α → Prop) [DecidablePred P]
+    (T : StandardFilling {a // P a}) (U : StandardFilling {a // ¬P a})
+    (s : {s : Finset (Fin (Fintype.card α)) // s.card=Fintype.card {a // P a}})
+    (a : α) (ha : ¬P a) :
+    interleavingEnumeration P T U s a = s.1ᶜ.orderEmbOfFin
+      (interleaving_compl_card P s) (U.1 ⟨a,ha⟩) := by
+  simp [interleavingEnumeration, Equiv.sumCompl_symm_apply_of_neg ha]
+
+def interleaveFilling (P : α → Prop) [DecidablePred P]
+    (hP : ∀ a b, a < b → (P a ↔ P b))
+    (T : StandardFilling {a // P a}) (U : StandardFilling {a // ¬P a})
+    (s : {s : Finset (Fin (Fintype.card α)) // s.card=Fintype.card {a // P a}}) :
+    StandardFilling α := by
+  refine ⟨interleavingEnumeration P T U s, ?_⟩
+  intro a b hab
+  by_cases ha : P a
+  · have hb := (hP a b hab).mp ha
+    rw [interleavingEnumeration_pos P T U s a ha, interleavingEnumeration_pos P T U s b hb]
+    exact (s.1.orderEmbOfFin s.2).strictMono (T.2 hab)
+  · have hb : ¬P b := fun hp => ha ((hP a b hab).mpr hp)
+    rw [interleavingEnumeration_neg P T U s a ha, interleavingEnumeration_neg P T U s b hb]
+    exact (s.1ᶜ.orderEmbOfFin _).strictMono (U.2 hab)
+
+lemma interleaveFilling_injective (P : α → Prop) [DecidablePred P]
+    (hP : ∀ a b, a < b → (P a ↔ P b))
+    (T : StandardFilling {a // P a}) (U : StandardFilling {a // ¬P a}) :
+    Function.Injective (interleaveFilling P hP T U) := by
+  intro s t hst
+  have hval := congrArg Subtype.val hst
+  have hsr : Set.range (fun a : {a // P a} => interleavingEnumeration P T U s a.1) = s.1 := by
+    have heq : (fun a : {a // P a} => interleavingEnumeration P T U s a.1) =
+        (s.1.orderEmbOfFin s.2) ∘ T.1 := by
+      funext a
+      exact interleavingEnumeration_pos P T U s a.1 a.2
+    rw [heq, Set.range_comp, Equiv.range_eq_univ, Set.image_univ, Finset.range_orderEmbOfFin]
+  have htr : Set.range (fun a : {a // P a} => interleavingEnumeration P T U t a.1) = t.1 := by
+    have heq : (fun a : {a // P a} => interleavingEnumeration P T U t a.1) =
+        (t.1.orderEmbOfFin t.2) ∘ T.1 := by
+      funext a
+      exact interleavingEnumeration_pos P T U t a.1 a.2
+    rw [heq, Set.range_comp, Equiv.range_eq_univ, Set.image_univ, Finset.range_orderEmbOfFin]
+  apply Subtype.ext
+  apply Finset.coe_injective
+  rw [← hsr, ← htr]
+  congr 1
+  funext a
+  exact Equiv.congr_fun hval a.1
+
+lemma interleaving_binomial_lower_bound (P : α → Prop) [DecidablePred P]
+    (hP : ∀ a b, a < b → (P a ↔ P b)) :
+    (Fintype.card α).choose (Fintype.card {a // P a}) ≤
+      Fintype.card (StandardFilling α) := by
+  have h := Fintype.card_le_of_injective
+    (interleaveFilling P hP (chosenFilling _) (chosenFilling _))
+    (interleaveFilling_injective P hP (chosenFilling _) (chosenFilling _))
+  simpa only [Fintype.card_finset_len, Fintype.card_fin] using h
+end Interleave
+
+end
+end Tableau
+end PartialPermutation
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/IsotypicKernels.lean b/lean/OAI/RepresentationTheory/PartialPermutation/IsotypicKernels.lean
index 7da828740b163b61ec8bf3c6a34bd49b2277784f..5838ead2705707a10e53db0b88462643ffcf1652 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/IsotypicKernels.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/IsotypicKernels.lean
@@ -1,120 +1,120 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.Characters
-import OAI.RepresentationTheory.PartialPermutation.IsotypicMatrices
-
-namespace OAI
-
-section
-open scoped Classical
-open scoped BigOperators ComplexConjugate MonoidAlgebra
-open scoped BigOperators ComplexConjugate
-open scoped MonoidAlgebra BigOperators
-open scoped BigOperators MonoidAlgebra Classical
-
-attribute [local instance] Classical.propDecidable
-open scoped BigOperators
-open scoped MonoidAlgebra BigOperators
-
-namespace PartialPermutation
-open scoped MonoidAlgebra BigOperators ComplexConjugate
-
-noncomputable section
-variable {G M : Type*} [Group G]
-    [NormedAddCommGroup M] [InnerProductSpace ℂ M]
-    [Module ℂ[G] M] [IsScalarTower ℂ ℂ[G] M]
-
-lemma ofModule'_algebra : (Representation.ofModule' (k := ℂ) (G := G) M).asAlgebraHom =
-    Algebra.lsmul ℂ ℂ M := by
-  exact (MonoidAlgebra.lift ℂ (Module.End ℂ M) G).apply_symm_apply _
-
-lemma ofModule'_apply (g : G) (x : M) :
-    Representation.ofModule' (k := ℂ) (G := G) M g x =
-      (MonoidAlgebra.single g (1 : ℂ)) • x := by
-  rw [← Representation.asAlgebraHom_single_one, ofModule'_algebra]
-  rfl
-
-noncomputable def ofModule'ModuleEquiv :
-    (Representation.ofModule' (k := ℂ) (G := G) M).asModule ≃ₗ[ℂ[G]] M where
-  toFun x := x
-  invFun x := x
-  left_inv _ := rfl
-  right_inv _ := rfl
-  map_add' _ _ := rfl
-  map_smul' a x := by
-    change (Representation.ofModule' (k := ℂ) (G := G) M).asAlgebraHom a x = a • (show M from x)
-    rw [ofModule'_algebra]
-    rfl
-
-lemma ofModule'_irreducible [IsSimpleModule ℂ[G] M] :
-    Representation.IsIrreducible (Representation.ofModule' (k := ℂ) (G := G) M) := by
-  rw [Representation.irreducible_iff_isSimpleModule_asModule]
-  exact IsSimpleModule.congr (ofModule'ModuleEquiv (G := G) (M := M))
-
-structure IsotypicKernelModel (G M : Type*) [Group G] [Fintype G]
-    [NormedAddCommGroup M] [InnerProductSpace ℂ M]
-    [Module ℂ[G] M] [IsScalarTower ℂ ℂ[G] M] extends IsotypicMatrixModel ℂ[G] M where
-  kernel : G → ℂ
-  kernel_norm : ∑ g, ‖kernel g‖^2 = (degree : ℝ)^2 / Fintype.card G
-  kernel_inv : ∀ g, kernel g⁻¹ = conj (kernel g)
-  kernel_action : ∀ x : M, ∑ g, kernel g • (MonoidAlgebra.single g (1 : ℂ)) • x = x
-
-variable [Fintype G]
-
-lemma exists_isotypicKernelModel [FiniteDimensional ℂ M] [Nontrivial M]
-    (h : IsIsotypic ℂ[G] M)
-    (hu : ∀ (g : G) (x y : M), inner ℂ ((MonoidAlgebra.single g (1 : ℂ)) • x)
-      ((MonoidAlgebra.single g (1 : ℂ)) • y) = inner ℂ x y) :
-    Nonempty (IsotypicKernelModel G M) := by
-  have : Module.Finite ℂ[G] M := Module.Finite.of_restrictScalars_finite ℂ ℂ[G] M
-  obtain ⟨m, hm, S, hS, ⟨e⟩⟩ := h.linearEquiv_fun
-  let : IsSimpleModule ℂ[G] S := hS
-  have : Nontrivial S := IsSimpleModule.nontrivial ℂ[G] S
-  have : FiniteDimensional ℂ S :=
-    Module.Finite.of_injective (S.subtype.restrictScalars ℂ) S.subtype_injective
-  let : NormedAddCommGroup S := inferInstanceAs (NormedAddCommGroup (S.restrictScalars ℂ))
-  let : InnerProductSpace ℂ S := inferInstanceAs (InnerProductSpace ℂ (S.restrictScalars ℂ))
-  let σ := Representation.ofModule' (k := ℂ) (G := G) S
-  have : Representation.IsIrreducible σ := ofModule'_irreducible
-  have hσ : IsUnitary σ := by
-    intro g x y
-    change inner ℂ (Representation.ofModule' (k := ℂ) (G := G) S g x)
-      (Representation.ofModule' (k := ℂ) (G := G) S g y) = _
-    rw [ofModule'_apply, ofModule'_apply]
-    exact hu g x.val y.val
-  let k : G → ℂ := fun g => ((Module.finrank ℂ S : ℂ) / Fintype.card G) * conj (σ.character g)
-  let F := (isotypicLift e).comp (Matrix.toLinAlgEquiv (Module.finBasis ℂ S)).toAlgHom
-  refine ⟨{ degree := Module.finrank ℂ S
-            degree_pos := Module.finrank_pos
-            lift := F
-            lift_image := fun A => isotypicLift_range e _
-            orbit := fun v => ⟨LinearMap.range (isotypicOrbitMap e v),
-              mem_isotypicOrbitMap_range e v, isotypicOrbitMap_range_invariant e v,
-              finrank_isotypicOrbitMap_range e v⟩
-            kernel := k
-            kernel_norm := ?_
-            kernel_inv := ?_
-            kernel_action := ?_ }⟩
-  · have hN : (Fintype.card G : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
-    simp only [k, norm_mul, norm_div, Complex.norm_natCast, Complex.norm_conj,
-      mul_pow, ← Finset.mul_sum, character_norm_sq_sum σ hσ]
-    field_simp
-  · intro g
-    simp [k, unitary_character_inv σ hσ]
-  · have he' := congrArg (isotypicLift e) (character_kernel_identity σ hσ)
-    have ha (g : G) : isotypicLift e (σ g) =
-        Algebra.lsmul ℂ ℂ M (MonoidAlgebra.single g (1 : ℂ)) := by
-      change isotypicLift e (Representation.ofModule' (k := ℂ) (G := G) S g) = _
-      rw [← Representation.asAlgebraHom_single_one, ofModule'_algebra, isotypicLift_action]
-    change isotypicLift e (complexFourier σ k) = isotypicLift e 1 at he'
-    simp only [complexFourier, map_sum, map_smul, map_one, ha] at he'
-    intro x
-    have ht := congrArg (fun A : Module.End ℂ M => A x) he'
-    simpa only [LinearMap.sum_apply, LinearMap.smul_apply, Algebra.lsmul_apply,
-      Module.End.one_apply] using ht
-
-end
-end PartialPermutation
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.Characters
+import OAI.RepresentationTheory.PartialPermutation.IsotypicMatrices
+
+namespace OAI
+
+section
+open scoped Classical
+open scoped BigOperators ComplexConjugate MonoidAlgebra
+open scoped BigOperators ComplexConjugate
+open scoped MonoidAlgebra BigOperators
+open scoped BigOperators MonoidAlgebra Classical
+
+attribute [local instance] Classical.propDecidable
+open scoped BigOperators
+open scoped MonoidAlgebra BigOperators
+
+namespace PartialPermutation
+open scoped MonoidAlgebra BigOperators ComplexConjugate
+
+noncomputable section
+variable {G M : Type*} [Group G]
+    [NormedAddCommGroup M] [InnerProductSpace ℂ M]
+    [Module ℂ[G] M] [IsScalarTower ℂ ℂ[G] M]
+
+lemma ofModule'_algebra : (Representation.ofModule' (k := ℂ) (G := G) M).asAlgebraHom =
+    Algebra.lsmul ℂ ℂ M := by
+  exact (MonoidAlgebra.lift ℂ (Module.End ℂ M) G).apply_symm_apply _
+
+lemma ofModule'_apply (g : G) (x : M) :
+    Representation.ofModule' (k := ℂ) (G := G) M g x =
+      (MonoidAlgebra.single g (1 : ℂ)) • x := by
+  rw [← Representation.asAlgebraHom_single_one, ofModule'_algebra]
+  rfl
+
+noncomputable def ofModule'ModuleEquiv :
+    (Representation.ofModule' (k := ℂ) (G := G) M).asModule ≃ₗ[ℂ[G]] M where
+  toFun x := x
+  invFun x := x
+  left_inv _ := rfl
+  right_inv _ := rfl
+  map_add' _ _ := rfl
+  map_smul' a x := by
+    change (Representation.ofModule' (k := ℂ) (G := G) M).asAlgebraHom a x = a • (show M from x)
+    rw [ofModule'_algebra]
+    rfl
+
+lemma ofModule'_irreducible [IsSimpleModule ℂ[G] M] :
+    Representation.IsIrreducible (Representation.ofModule' (k := ℂ) (G := G) M) := by
+  rw [Representation.irreducible_iff_isSimpleModule_asModule]
+  exact IsSimpleModule.congr (ofModule'ModuleEquiv (G := G) (M := M))
+
+structure IsotypicKernelModel (G M : Type*) [Group G] [Fintype G]
+    [NormedAddCommGroup M] [InnerProductSpace ℂ M]
+    [Module ℂ[G] M] [IsScalarTower ℂ ℂ[G] M] extends IsotypicMatrixModel ℂ[G] M where
+  kernel : G → ℂ
+  kernel_norm : ∑ g, ‖kernel g‖^2 = (degree : ℝ)^2 / Fintype.card G
+  kernel_inv : ∀ g, kernel g⁻¹ = conj (kernel g)
+  kernel_action : ∀ x : M, ∑ g, kernel g • (MonoidAlgebra.single g (1 : ℂ)) • x = x
+
+variable [Fintype G]
+
+lemma exists_isotypicKernelModel [FiniteDimensional ℂ M] [Nontrivial M]
+    (h : IsIsotypic ℂ[G] M)
+    (hu : ∀ (g : G) (x y : M), inner ℂ ((MonoidAlgebra.single g (1 : ℂ)) • x)
+      ((MonoidAlgebra.single g (1 : ℂ)) • y) = inner ℂ x y) :
+    Nonempty (IsotypicKernelModel G M) := by
+  have : Module.Finite ℂ[G] M := Module.Finite.of_restrictScalars_finite ℂ ℂ[G] M
+  obtain ⟨m, hm, S, hS, ⟨e⟩⟩ := h.linearEquiv_fun
+  let : IsSimpleModule ℂ[G] S := hS
+  have : Nontrivial S := IsSimpleModule.nontrivial ℂ[G] S
+  have : FiniteDimensional ℂ S :=
+    Module.Finite.of_injective (S.subtype.restrictScalars ℂ) S.subtype_injective
+  let : NormedAddCommGroup S := inferInstanceAs (NormedAddCommGroup (S.restrictScalars ℂ))
+  let : InnerProductSpace ℂ S := inferInstanceAs (InnerProductSpace ℂ (S.restrictScalars ℂ))
+  let σ := Representation.ofModule' (k := ℂ) (G := G) S
+  have : Representation.IsIrreducible σ := ofModule'_irreducible
+  have hσ : IsUnitary σ := by
+    intro g x y
+    change inner ℂ (Representation.ofModule' (k := ℂ) (G := G) S g x)
+      (Representation.ofModule' (k := ℂ) (G := G) S g y) = _
+    rw [ofModule'_apply, ofModule'_apply]
+    exact hu g x.val y.val
+  let k : G → ℂ := fun g => ((Module.finrank ℂ S : ℂ) / Fintype.card G) * conj (σ.character g)
+  let F := (isotypicLift e).comp (Matrix.toLinAlgEquiv (Module.finBasis ℂ S)).toAlgHom
+  refine ⟨{ degree := Module.finrank ℂ S
+            degree_pos := Module.finrank_pos
+            lift := F
+            lift_image := fun A => isotypicLift_range e _
+            orbit := fun v => ⟨LinearMap.range (isotypicOrbitMap e v),
+              mem_isotypicOrbitMap_range e v, isotypicOrbitMap_range_invariant e v,
+              finrank_isotypicOrbitMap_range e v⟩
+            kernel := k
+            kernel_norm := ?_
+            kernel_inv := ?_
+            kernel_action := ?_ }⟩
+  · have hN : (Fintype.card G : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
+    simp only [k, norm_mul, norm_div, Complex.norm_natCast, Complex.norm_conj,
+      mul_pow, ← Finset.mul_sum, character_norm_sq_sum σ hσ]
+    field_simp
+  · intro g
+    simp [k, unitary_character_inv σ hσ]
+  · have he' := congrArg (isotypicLift e) (character_kernel_identity σ hσ)
+    have ha (g : G) : isotypicLift e (σ g) =
+        Algebra.lsmul ℂ ℂ M (MonoidAlgebra.single g (1 : ℂ)) := by
+      change isotypicLift e (Representation.ofModule' (k := ℂ) (G := G) S g) = _
+      rw [← Representation.asAlgebraHom_single_one, ofModule'_algebra, isotypicLift_action]
+    change isotypicLift e (complexFourier σ k) = isotypicLift e 1 at he'
+    simp only [complexFourier, map_sum, map_smul, map_one, ha] at he'
+    intro x
+    have ht := congrArg (fun A : Module.End ℂ M => A x) he'
+    simpa only [LinearMap.sum_apply, LinearMap.smul_apply, Algebra.lsmul_apply,
+      Module.End.one_apply] using ht
+
+end
+end PartialPermutation
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/IsotypicMatrices.lean b/lean/OAI/RepresentationTheory/PartialPermutation/IsotypicMatrices.lean
index 401e66d4c234574a97523aa2292f8fe44d32acef..d8053d70e1f9091f847a3e7c68d914830217879b 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/IsotypicMatrices.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/IsotypicMatrices.lean
@@ -1,138 +1,138 @@
-import Mathlib
-
-namespace OAI
-
-section
-open scoped Classical
-open scoped BigOperators ComplexConjugate MonoidAlgebra
-open scoped BigOperators ComplexConjugate
-open scoped MonoidAlgebra BigOperators
-open scoped BigOperators MonoidAlgebra Classical
-
-attribute [local instance] Classical.propDecidable
-open scoped BigOperators
-
-namespace PartialPermutation
-
-section Burnside
-variable {R S : Type*} [Ring R] [Algebra ℂ R] [AddCommGroup S]
-    [Module ℂ S] [Module R S] [IsScalarTower ℂ R S]
-    [IsSimpleModule R S] [FiniteDimensional ℂ S]
-
-lemma simple_action_surjective : Function.Surjective (Algebra.lsmul ℂ ℂ S (A := R)) := by
-  intro f
-  let E := Module.End R S
-  let f' : Module.End E S :=
-    { toFun := f
-      map_add' := f.map_add
-      map_smul' := fun c x => by
-        obtain ⟨t, rfl⟩ := (IsSimpleModule.algebraMap_end_bijective_of_isAlgClosed ℂ
-          (A := R) (V := S)).surjective c
-        change f (t • x) = t • f x
-        exact f.map_smul t x }
-  have : Module.Finite E S := Module.Finite.of_restrictScalars_finite ℂ E S
-  obtain ⟨r, hr⟩ := (Module.Finite.toModuleEnd_moduleEnd_surjective (R := R) (M := S)) f'
-  refine ⟨r, ?_⟩
-  ext x
-  exact congrArg (fun A : Module.End E S => A x) hr
-
-end Burnside
-
-def diagonalEnd {S : Type*} [AddCommGroup S] [Module ℂ S] (m : ℕ) :
-    Module.End ℂ S →ₐ[ℂ] Module.End ℂ (Fin m → S) where
-  toFun A := LinearMap.pi fun i => A.comp (LinearMap.proj i)
-  map_zero' := by ext; rfl
-  map_one' := by ext; rfl
-  map_add' := by intros; ext; rfl
-  map_mul' := by intros; ext; rfl
-  commutes' := by intro; ext; rfl
-
-section Isotypic
-variable {R M S : Type*} [Ring R] [Algebra ℂ R]
-    [AddCommGroup M] [Module ℂ M] [Module R M] [IsScalarTower ℂ R M]
-    [AddCommGroup S] [Module ℂ S] [Module R S] [IsScalarTower ℂ R S]
-
-noncomputable def isotypicLift {m : ℕ} (e : M ≃ₗ[R] Fin m → S) :
-    Module.End ℂ S →ₐ[ℂ] Module.End ℂ M :=
-  (e.restrictScalars ℂ).symm.conjAlgEquiv ℂ |>.toAlgHom.comp (diagonalEnd m)
-
-lemma isotypicLift_action {m : ℕ} (e : M ≃ₗ[R] Fin m → S) (r : R) :
-    isotypicLift e (Algebra.lsmul ℂ ℂ S r) = Algebra.lsmul ℂ ℂ M r := by
-  ext x
-  apply (e.restrictScalars ℂ).injective
-  simp [isotypicLift, diagonalEnd, LinearEquiv.conjAlgEquiv_apply, map_smul]
-  ext i
-  rfl
-
-lemma isotypicLift_range [IsSimpleModule R S] [FiniteDimensional ℂ S]
-    {m : ℕ} (e : M ≃ₗ[R] Fin m → S) (A : Module.End ℂ S) :
-    ∃ r : R, isotypicLift e A = Algebra.lsmul ℂ ℂ M r := by
-  obtain ⟨r, rfl⟩ := simple_action_surjective (R := R) A
-  exact ⟨r, isotypicLift_action e r⟩
-
-noncomputable def isotypicOrbitMap {m : ℕ} (e : M ≃ₗ[R] Fin m → S) (v : M) :
-    Module.End ℂ S →ₗ[ℂ] M where
-  toFun A := isotypicLift e A v
-  map_add' A B := by simp
-  map_smul' c A := by simp
-
-lemma mem_isotypicOrbitMap_range {m : ℕ} (e : M ≃ₗ[R] Fin m → S) (v : M) :
-    v ∈ LinearMap.range (isotypicOrbitMap e v) := by
-  refine ⟨1, ?_⟩
-  simp [isotypicOrbitMap]
-
-lemma isotypicOrbitMap_range_invariant {m : ℕ} (e : M ≃ₗ[R] Fin m → S)
-    (v : M) (r : R) (x : M) (hx : x ∈ LinearMap.range (isotypicOrbitMap e v)) :
-    r • x ∈ LinearMap.range (isotypicOrbitMap e v) := by
-  obtain ⟨A, rfl⟩ := hx
-  refine ⟨Algebra.lsmul ℂ ℂ S r * A, ?_⟩
-  change isotypicLift e (_ * A) v = r • (isotypicLift e A v)
-  rw [map_mul, Module.End.mul_apply, isotypicLift_action]
-  rfl
-
-lemma finrank_isotypicOrbitMap_range [FiniteDimensional ℂ S] {m : ℕ}
-    (e : M ≃ₗ[R] Fin m → S) (v : M) :
-    Module.finrank ℂ (LinearMap.range (isotypicOrbitMap e v)) ≤
-      (Module.finrank ℂ S) ^ 2 := by
-  calc
-    _ ≤ Module.finrank ℂ (Module.End ℂ S) := LinearMap.finrank_range_le _
-    _ = _ := by rw [Module.finrank_linearMap]; ring
-
-end Isotypic
-
-structure IsotypicMatrixModel (R M : Type*) [Ring R] [Algebra ℂ R]
-    [AddCommGroup M] [Module ℂ M] [Module R M] [IsScalarTower ℂ R M] where
-  degree : ℕ
-  degree_pos : 0 < degree
-  lift : Matrix (Fin degree) (Fin degree) ℂ →ₐ[ℂ] Module.End ℂ M
-  lift_image : ∀ A, ∃ r : R, lift A = Algebra.lsmul ℂ ℂ M r
-  orbit : ∀ v : M, ∃ W : Submodule ℂ M, v ∈ W ∧
-    (∀ r : R, ∀ x ∈ W, r • x ∈ W) ∧ Module.finrank ℂ W ≤ degree ^ 2
-
-lemma exists_isotypicMatrixModel {R M : Type*} [Ring R] [Algebra ℂ R]
-    [AddCommGroup M] [Module ℂ M] [Module R M] [IsScalarTower ℂ R M]
-    [FiniteDimensional ℂ M] [Nontrivial M] [IsSemisimpleModule R M]
-    (h : IsIsotypic R M) : Nonempty (IsotypicMatrixModel R M) := by
-  have : Module.Finite R M := Module.Finite.of_restrictScalars_finite ℂ R M
-  obtain ⟨m, hm, S, hS, ⟨e⟩⟩ := h.linearEquiv_fun
-  let : IsSimpleModule R S := hS
-  have : Nontrivial S := IsSimpleModule.nontrivial R S
-  have : FiniteDimensional ℂ S :=
-    Module.Finite.of_injective (S.subtype.restrictScalars ℂ) S.subtype_injective
-  let F := (isotypicLift e).comp (Matrix.toLinAlgEquiv (Module.finBasis ℂ S)).toAlgHom
-  refine ⟨{ degree := Module.finrank ℂ S
-            degree_pos := Module.finrank_pos
-            lift := F
-            lift_image := ?_
-            orbit := ?_ }⟩
-  · intro A
-    exact isotypicLift_range e _
-  · intro v
-    exact ⟨LinearMap.range (isotypicOrbitMap e v), mem_isotypicOrbitMap_range e v,
-      isotypicOrbitMap_range_invariant e v, finrank_isotypicOrbitMap_range e v⟩
-
-end PartialPermutation
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+section
+open scoped Classical
+open scoped BigOperators ComplexConjugate MonoidAlgebra
+open scoped BigOperators ComplexConjugate
+open scoped MonoidAlgebra BigOperators
+open scoped BigOperators MonoidAlgebra Classical
+
+attribute [local instance] Classical.propDecidable
+open scoped BigOperators
+
+namespace PartialPermutation
+
+section Burnside
+variable {R S : Type*} [Ring R] [Algebra ℂ R] [AddCommGroup S]
+    [Module ℂ S] [Module R S] [IsScalarTower ℂ R S]
+    [IsSimpleModule R S] [FiniteDimensional ℂ S]
+
+lemma simple_action_surjective : Function.Surjective (Algebra.lsmul ℂ ℂ S (A := R)) := by
+  intro f
+  let E := Module.End R S
+  let f' : Module.End E S :=
+    { toFun := f
+      map_add' := f.map_add
+      map_smul' := fun c x => by
+        obtain ⟨t, rfl⟩ := (IsSimpleModule.algebraMap_end_bijective_of_isAlgClosed ℂ
+          (A := R) (V := S)).surjective c
+        change f (t • x) = t • f x
+        exact f.map_smul t x }
+  have : Module.Finite E S := Module.Finite.of_restrictScalars_finite ℂ E S
+  obtain ⟨r, hr⟩ := (Module.Finite.toModuleEnd_moduleEnd_surjective (R := R) (M := S)) f'
+  refine ⟨r, ?_⟩
+  ext x
+  exact congrArg (fun A : Module.End E S => A x) hr
+
+end Burnside
+
+def diagonalEnd {S : Type*} [AddCommGroup S] [Module ℂ S] (m : ℕ) :
+    Module.End ℂ S →ₐ[ℂ] Module.End ℂ (Fin m → S) where
+  toFun A := LinearMap.pi fun i => A.comp (LinearMap.proj i)
+  map_zero' := by ext; rfl
+  map_one' := by ext; rfl
+  map_add' := by intros; ext; rfl
+  map_mul' := by intros; ext; rfl
+  commutes' := by intro; ext; rfl
+
+section Isotypic
+variable {R M S : Type*} [Ring R] [Algebra ℂ R]
+    [AddCommGroup M] [Module ℂ M] [Module R M] [IsScalarTower ℂ R M]
+    [AddCommGroup S] [Module ℂ S] [Module R S] [IsScalarTower ℂ R S]
+
+noncomputable def isotypicLift {m : ℕ} (e : M ≃ₗ[R] Fin m → S) :
+    Module.End ℂ S →ₐ[ℂ] Module.End ℂ M :=
+  (e.restrictScalars ℂ).symm.conjAlgEquiv ℂ |>.toAlgHom.comp (diagonalEnd m)
+
+lemma isotypicLift_action {m : ℕ} (e : M ≃ₗ[R] Fin m → S) (r : R) :
+    isotypicLift e (Algebra.lsmul ℂ ℂ S r) = Algebra.lsmul ℂ ℂ M r := by
+  ext x
+  apply (e.restrictScalars ℂ).injective
+  simp [isotypicLift, diagonalEnd, LinearEquiv.conjAlgEquiv_apply, map_smul]
+  ext i
+  rfl
+
+lemma isotypicLift_range [IsSimpleModule R S] [FiniteDimensional ℂ S]
+    {m : ℕ} (e : M ≃ₗ[R] Fin m → S) (A : Module.End ℂ S) :
+    ∃ r : R, isotypicLift e A = Algebra.lsmul ℂ ℂ M r := by
+  obtain ⟨r, rfl⟩ := simple_action_surjective (R := R) A
+  exact ⟨r, isotypicLift_action e r⟩
+
+noncomputable def isotypicOrbitMap {m : ℕ} (e : M ≃ₗ[R] Fin m → S) (v : M) :
+    Module.End ℂ S →ₗ[ℂ] M where
+  toFun A := isotypicLift e A v
+  map_add' A B := by simp
+  map_smul' c A := by simp
+
+lemma mem_isotypicOrbitMap_range {m : ℕ} (e : M ≃ₗ[R] Fin m → S) (v : M) :
+    v ∈ LinearMap.range (isotypicOrbitMap e v) := by
+  refine ⟨1, ?_⟩
+  simp [isotypicOrbitMap]
+
+lemma isotypicOrbitMap_range_invariant {m : ℕ} (e : M ≃ₗ[R] Fin m → S)
+    (v : M) (r : R) (x : M) (hx : x ∈ LinearMap.range (isotypicOrbitMap e v)) :
+    r • x ∈ LinearMap.range (isotypicOrbitMap e v) := by
+  obtain ⟨A, rfl⟩ := hx
+  refine ⟨Algebra.lsmul ℂ ℂ S r * A, ?_⟩
+  change isotypicLift e (_ * A) v = r • (isotypicLift e A v)
+  rw [map_mul, Module.End.mul_apply, isotypicLift_action]
+  rfl
+
+lemma finrank_isotypicOrbitMap_range [FiniteDimensional ℂ S] {m : ℕ}
+    (e : M ≃ₗ[R] Fin m → S) (v : M) :
+    Module.finrank ℂ (LinearMap.range (isotypicOrbitMap e v)) ≤
+      (Module.finrank ℂ S) ^ 2 := by
+  calc
+    _ ≤ Module.finrank ℂ (Module.End ℂ S) := LinearMap.finrank_range_le _
+    _ = _ := by rw [Module.finrank_linearMap]; ring
+
+end Isotypic
+
+structure IsotypicMatrixModel (R M : Type*) [Ring R] [Algebra ℂ R]
+    [AddCommGroup M] [Module ℂ M] [Module R M] [IsScalarTower ℂ R M] where
+  degree : ℕ
+  degree_pos : 0 < degree
+  lift : Matrix (Fin degree) (Fin degree) ℂ →ₐ[ℂ] Module.End ℂ M
+  lift_image : ∀ A, ∃ r : R, lift A = Algebra.lsmul ℂ ℂ M r
+  orbit : ∀ v : M, ∃ W : Submodule ℂ M, v ∈ W ∧
+    (∀ r : R, ∀ x ∈ W, r • x ∈ W) ∧ Module.finrank ℂ W ≤ degree ^ 2
+
+lemma exists_isotypicMatrixModel {R M : Type*} [Ring R] [Algebra ℂ R]
+    [AddCommGroup M] [Module ℂ M] [Module R M] [IsScalarTower ℂ R M]
+    [FiniteDimensional ℂ M] [Nontrivial M] [IsSemisimpleModule R M]
+    (h : IsIsotypic R M) : Nonempty (IsotypicMatrixModel R M) := by
+  have : Module.Finite R M := Module.Finite.of_restrictScalars_finite ℂ R M
+  obtain ⟨m, hm, S, hS, ⟨e⟩⟩ := h.linearEquiv_fun
+  let : IsSimpleModule R S := hS
+  have : Nontrivial S := IsSimpleModule.nontrivial R S
+  have : FiniteDimensional ℂ S :=
+    Module.Finite.of_injective (S.subtype.restrictScalars ℂ) S.subtype_injective
+  let F := (isotypicLift e).comp (Matrix.toLinAlgEquiv (Module.finBasis ℂ S)).toAlgHom
+  refine ⟨{ degree := Module.finrank ℂ S
+            degree_pos := Module.finrank_pos
+            lift := F
+            lift_image := ?_
+            orbit := ?_ }⟩
+  · intro A
+    exact isotypicLift_range e _
+  · intro v
+    exact ⟨LinearMap.range (isotypicOrbitMap e v), mem_isotypicOrbitMap_range e v,
+      isotypicOrbitMap_range_invariant e v, finrank_isotypicOrbitMap_range e v⟩
+
+end PartialPermutation
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/IsotypicProjections.lean b/lean/OAI/RepresentationTheory/PartialPermutation/IsotypicProjections.lean
index a336a5447055b6d72f13bf35b046078fd19a9b81..646ab318c0e9e664bbe06c4eb9ed585e2c94a9be 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/IsotypicProjections.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/IsotypicProjections.lean
@@ -1,97 +1,97 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.CompleteKernels
-import OAI.RepresentationTheory.PartialPermutation.ComponentGeometry
-
-namespace OAI
-
-section
-open scoped Classical
-open scoped BigOperators ComplexConjugate MonoidAlgebra
-open scoped BigOperators ComplexConjugate
-open scoped MonoidAlgebra BigOperators
-open scoped BigOperators MonoidAlgebra Classical
-
-attribute [local instance] Classical.propDecidable
-namespace PartialPermutation
-noncomputable section
-section FullProjection
-variable {G V : Type*} [Group G] [Fintype G]
-    [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-    (ρ : Representation ℂ G V)
-
-omit [FiniteDimensional ℂ V] in
-lemma linearMap_eq_on_simples (A B : Module.End ℂ V)
-    (h : ∀ (σ : Subrepresentation ρ) [Representation.IsIrreducible σ.toRepresentation],
-      ∀ x : σ, A x = B x) : A = B := by
-  let E (S : {S : Submodule ℂ[G] ρ.asModule // IsSimpleModule ℂ[G] S}) : Submodule ℂ V :=
-    S.val.restrictScalars ℂ
-  have htop : (⨆ S, E S) = ⊤ := by
-    change (⨆ S : {S : Submodule ℂ[G] ρ.asModule // IsSimpleModule ℂ[G] S},
-      S.val.restrictScalars ℂ) = ⊤
-    rw [← Submodule.restrictScalars_iSup]
-    have ht : (⨆ S : {S : Submodule ℂ[G] ρ.asModule // IsSimpleModule ℂ[G] S}, S.val) = ⊤ :=
-      (sSup_eq_iSup' {S : Submodule ℂ[G] ρ.asModule | IsSimpleModule ℂ[G] S}).symm.trans
-        (IsSemisimpleModule.sSup_simples_eq_top ℂ[G] ρ.asModule)
-    rw [ht, Submodule.restrictScalars_top]
-  have hker : (⊤ : Submodule ℂ V) ≤ LinearMap.ker (A - B) := by
-    rw [← htop]
-    refine iSup_le fun S => ?_
-    intro x hx
-    let σ := Subrepresentation.ofSubmodule' S.val
-    have : Representation.IsIrreducible σ.toRepresentation :=
-      (subrep_irreducible_iff ρ σ).mpr S.property
-    exact sub_eq_zero.mpr (h σ ⟨x, hx⟩)
-  apply LinearMap.ext
-  intro x
-  have hx := hker (Submodule.mem_top : x ∈ (⊤ : Submodule ℂ V))
-  simpa only [LinearMap.mem_ker, LinearMap.sub_apply, sub_eq_zero] using hx
-
-lemma component_kernel_eq_projection (hρ : IsUnitary ρ)
-    (c : isotypicComponents ℂ[G] ρ.asModule) :
-    complexFourier ρ (centralCharacterKernel (componentRep ρ c)) =
-      (componentSpace ρ c).starProjection.toLinearMap := by
-  apply linearMap_eq_on_simples ρ
-  intro σ hσ x
-  have := hσ
-  have : IsSimpleModule ℂ[G] σ.asSubmodule := (subrep_irreducible_iff ρ σ).mp hσ
-  let d : isotypicComponents ℂ[G] ρ.asModule :=
-    ⟨isotypicComponent ℂ[G] ρ.asModule σ.asSubmodule, σ.asSubmodule, inferInstance, rfl⟩
-  have hx : (x : V) ∈ componentSpace ρ d :=
-    σ.asSubmodule.le_isotypicComponent x.property
-  have he : Nonempty (Representation.Equiv (componentRep ρ d) σ.toRepresentation) := by
-    have := componentSimple_simple ρ d
-    have hs : σ.asSubmodule ≤ d.val := σ.asSubmodule.le_isotypicComponent
-    rw [componentSimple_component ρ d] at hs
-    obtain ⟨e⟩ := isIsotypicOfType_submodule_iff.mp
-      (IsIsotypicOfType.isotypicComponent ℂ[G] ρ.asModule (componentSimple ρ d))
-      σ.asSubmodule hs
-    exact ⟨subrepEquivOfModuleEquiv ρ _ σ e.symm⟩
-  have hi : Nonempty (Representation.Equiv (componentRep ρ c) σ.toRepresentation) ↔ c = d := by
-    constructor
-    · rintro ⟨e⟩
-      obtain ⟨ed⟩ := he
-      exact (componentRep_equiv_iff ρ c d).mp ⟨e.trans ed.symm⟩
-    · rintro rfl
-      exact he
-  rw [← complexFourier_subrep ρ σ]
-  have ht := centralCharacterKernel_fourier (componentRep ρ c) σ.toRepresentation
-  by_cases hcd : c = d
-  · subst c
-    rw [ite_eq_left he] at ht
-    rw [ht]
-    exact ((componentSpace ρ d).starProjection_eq_self_iff.mpr hx).symm
-  · rw [ite_eq_right (mt hi.mp hcd)] at ht
-    rw [ht]
-    change 0 = (componentSpace ρ c).starProjection (x : V)
-    symm
-    apply (Submodule.starProjection_apply_eq_zero_iff (K := componentSpace ρ c)).mpr
-    intro y hy
-    exact componentSpace_orthogonal ρ hρ hcd ⟨y, hy⟩ ⟨x, hx⟩
-
-end FullProjection
-end
-end PartialPermutation
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.CompleteKernels
+import OAI.RepresentationTheory.PartialPermutation.ComponentGeometry
+
+namespace OAI
+
+section
+open scoped Classical
+open scoped BigOperators ComplexConjugate MonoidAlgebra
+open scoped BigOperators ComplexConjugate
+open scoped MonoidAlgebra BigOperators
+open scoped BigOperators MonoidAlgebra Classical
+
+attribute [local instance] Classical.propDecidable
+namespace PartialPermutation
+noncomputable section
+section FullProjection
+variable {G V : Type*} [Group G] [Fintype G]
+    [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+    (ρ : Representation ℂ G V)
+
+omit [FiniteDimensional ℂ V] in
+lemma linearMap_eq_on_simples (A B : Module.End ℂ V)
+    (h : ∀ (σ : Subrepresentation ρ) [Representation.IsIrreducible σ.toRepresentation],
+      ∀ x : σ, A x = B x) : A = B := by
+  let E (S : {S : Submodule ℂ[G] ρ.asModule // IsSimpleModule ℂ[G] S}) : Submodule ℂ V :=
+    S.val.restrictScalars ℂ
+  have htop : (⨆ S, E S) = ⊤ := by
+    change (⨆ S : {S : Submodule ℂ[G] ρ.asModule // IsSimpleModule ℂ[G] S},
+      S.val.restrictScalars ℂ) = ⊤
+    rw [← Submodule.restrictScalars_iSup]
+    have ht : (⨆ S : {S : Submodule ℂ[G] ρ.asModule // IsSimpleModule ℂ[G] S}, S.val) = ⊤ :=
+      (sSup_eq_iSup' {S : Submodule ℂ[G] ρ.asModule | IsSimpleModule ℂ[G] S}).symm.trans
+        (IsSemisimpleModule.sSup_simples_eq_top ℂ[G] ρ.asModule)
+    rw [ht, Submodule.restrictScalars_top]
+  have hker : (⊤ : Submodule ℂ V) ≤ LinearMap.ker (A - B) := by
+    rw [← htop]
+    refine iSup_le fun S => ?_
+    intro x hx
+    let σ := Subrepresentation.ofSubmodule' S.val
+    have : Representation.IsIrreducible σ.toRepresentation :=
+      (subrep_irreducible_iff ρ σ).mpr S.property
+    exact sub_eq_zero.mpr (h σ ⟨x, hx⟩)
+  apply LinearMap.ext
+  intro x
+  have hx := hker (Submodule.mem_top : x ∈ (⊤ : Submodule ℂ V))
+  simpa only [LinearMap.mem_ker, LinearMap.sub_apply, sub_eq_zero] using hx
+
+lemma component_kernel_eq_projection (hρ : IsUnitary ρ)
+    (c : isotypicComponents ℂ[G] ρ.asModule) :
+    complexFourier ρ (centralCharacterKernel (componentRep ρ c)) =
+      (componentSpace ρ c).starProjection.toLinearMap := by
+  apply linearMap_eq_on_simples ρ
+  intro σ hσ x
+  have := hσ
+  have : IsSimpleModule ℂ[G] σ.asSubmodule := (subrep_irreducible_iff ρ σ).mp hσ
+  let d : isotypicComponents ℂ[G] ρ.asModule :=
+    ⟨isotypicComponent ℂ[G] ρ.asModule σ.asSubmodule, σ.asSubmodule, inferInstance, rfl⟩
+  have hx : (x : V) ∈ componentSpace ρ d :=
+    σ.asSubmodule.le_isotypicComponent x.property
+  have he : Nonempty (Representation.Equiv (componentRep ρ d) σ.toRepresentation) := by
+    have := componentSimple_simple ρ d
+    have hs : σ.asSubmodule ≤ d.val := σ.asSubmodule.le_isotypicComponent
+    rw [componentSimple_component ρ d] at hs
+    obtain ⟨e⟩ := isIsotypicOfType_submodule_iff.mp
+      (IsIsotypicOfType.isotypicComponent ℂ[G] ρ.asModule (componentSimple ρ d))
+      σ.asSubmodule hs
+    exact ⟨subrepEquivOfModuleEquiv ρ _ σ e.symm⟩
+  have hi : Nonempty (Representation.Equiv (componentRep ρ c) σ.toRepresentation) ↔ c = d := by
+    constructor
+    · rintro ⟨e⟩
+      obtain ⟨ed⟩ := he
+      exact (componentRep_equiv_iff ρ c d).mp ⟨e.trans ed.symm⟩
+    · rintro rfl
+      exact he
+  rw [← complexFourier_subrep ρ σ]
+  have ht := centralCharacterKernel_fourier (componentRep ρ c) σ.toRepresentation
+  by_cases hcd : c = d
+  · subst c
+    rw [ite_eq_left he] at ht
+    rw [ht]
+    exact ((componentSpace ρ d).starProjection_eq_self_iff.mpr hx).symm
+  · rw [ite_eq_right (mt hi.mp hcd)] at ht
+    rw [ht]
+    change 0 = (componentSpace ρ c).starProjection (x : V)
+    symm
+    apply (Submodule.starProjection_apply_eq_zero_iff (K := componentSpace ρ c)).mpr
+    intro y hy
+    exact componentSpace_orthogonal ρ hρ hcd ⟨y, hy⟩ ⟨x, hx⟩
+
+end FullProjection
+end
+end PartialPermutation
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/JointKernels.lean b/lean/OAI/RepresentationTheory/PartialPermutation/JointKernels.lean
index a23cc7824c6f80322623cdb5f989966e1c3b924c..04a6a68147a9297dd2025e70dc3b295f1eaca72f 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/JointKernels.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/JointKernels.lean
@@ -1,145 +1,145 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.ComponentGeometry
-import OAI.RepresentationTheory.PartialPermutation.MatrixDegrees
-import OAI.RepresentationTheory.PartialPermutation.Restriction
-import OAI.RepresentationTheory.PartialPermutation.IsotypicKernels
-import OAI.RepresentationTheory.PartialPermutation.DegreeEstimates
-import OAI.RepresentationTheory.PartialPermutation.BlockSubgroups
-
-namespace OAI
-
-section
-open scoped Classical
-open scoped BigOperators ComplexConjugate MonoidAlgebra
-open scoped BigOperators ComplexConjugate
-open scoped MonoidAlgebra BigOperators
-open scoped BigOperators MonoidAlgebra Classical
-
-attribute [local instance] Classical.propDecidable
-open scoped MonoidAlgebra BigOperators
-open scoped BigOperators
-
-namespace PartialPermutation
-open scoped MonoidAlgebra BigOperators ComplexConjugate
-
-noncomputable section
-
-lemma isotypic_joint_kernels {G M : Type*} [Group G] [Fintype G]
-    [NormedAddCommGroup M] [InnerProductSpace ℂ M]
-    [Module ℂ[G] M] [IsScalarTower ℂ ℂ[G] M]
-    [FiniteDimensional ℂ M] [Nontrivial M]
-    {b : ℕ} (H : Fin b → Subgroup G) (hgen : ⨆ i, H i = ⊤)
-    (hcomm : ∀ i j, i ≠ j → ∀ g ∈ H i, ∀ k ∈ H j, Commute g k)
-    (h : IsIsotypic ℂ[G] M)
-    (hu : ∀ (g : G) (x y : M), inner ℂ ((MonoidAlgebra.single g (1 : ℂ)) • x)
-      ((MonoidAlgebra.single g (1 : ℂ)) • y) = inner ℂ x y) :
-    ∃ d : Fin b → ℕ, (∀ i, 0 < d i) ∧ (∏ i, d i) ≤ Module.finrank ℂ M ∧
-      ∀ i, ∃ k : H i → ℂ,
-        (∑ g, ‖k g‖^2 = (d i : ℝ)^2 / Fintype.card (H i)) ∧
-        (∀ g, k g⁻¹ = conj (k g)) ∧
-        (∀ x : M, ∑ g, k g • (MonoidAlgebra.single (g : G) (1 : ℂ)) • x = x) := by
-  classical
-  let (i : Fin b) : Module ℂ[H i] M := restrictGroupModule (H i)
-  let (i : Fin b) : IsScalarTower ℂ ℂ[H i] M := restrictGroupModule_tower (H i)
-  have hmodels (i : Fin b) : Nonempty (IsotypicKernelModel (H i) M) := by
-    apply exists_isotypicKernelModel (group_isotypic_restrict H hgen hcomm h i)
-    intro g x y
-    change inner ℂ ((MonoidAlgebra.mapDomainAlgHom ℂ ℂ (H i).subtype
-      (MonoidAlgebra.single g (1 : ℂ))) • x)
-      ((MonoidAlgebra.mapDomainAlgHom ℂ ℂ (H i).subtype
-      (MonoidAlgebra.single g (1 : ℂ))) • y) = _
-    simpa [MonoidAlgebra.mapDomainAlgHom] using hu g x y
-  let F (i : Fin b) := Classical.choice (hmodels i)
-  have hAcomm (i j : Fin b) (hij : i ≠ j) (X Y) : Commute ((F i).lift X) ((F j).lift Y) := by
-    obtain ⟨r, hr⟩ := (F i).lift_image X
-    obtain ⟨s, hs⟩ := (F j).lift_image Y
-    rw [hr, hs]
-    have he (i : Fin b) (r : ℂ[H i]) : Algebra.lsmul ℂ ℂ M r =
-        Algebra.lsmul ℂ ℂ M (MonoidAlgebra.mapDomainAlgHom ℂ ℂ (H i).subtype r) := by
-      ext x
-      rfl
-    rw [he i r, he j s]
-    exact (subgroup_algebra_commute (H i) (H j) (hcomm i j hij) r s).map
-      (Algebra.lsmul ℂ ℂ M : ℂ[G] →ₐ[ℂ] Module.End ℂ M)
-  refine ⟨fun i => (F i).degree, fun i => (F i).degree_pos, ?_, ?_⟩
-  · exact commuting_degrees_le_finrank (fun i => (F i).degree)
-      (fun i => (F i).degree_pos) (fun i => (F i).lift) hAcomm Finset.univ
-  · intro i
-    refine ⟨(F i).kernel, (F i).kernel_norm, (F i).kernel_inv, ?_⟩
-    intro x
-    have ht := (F i).kernel_action x
-    change ∑ g, (F i).kernel g • (MonoidAlgebra.mapDomainAlgHom ℂ ℂ (H i).subtype
-      (MonoidAlgebra.single g (1 : ℂ))) • x = x at ht
-    simpa [MonoidAlgebra.mapDomainAlgHom] using ht
-
-section ComponentKernel
-variable {G V : Type*} [Group G] [Fintype G]
-    [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-    (ρ : Representation ℂ G V) {b : ℕ} (H : Fin b → Subgroup G)
-
-lemma component_kernel (hb : 0 < b) (hD : 0 < Module.finrank ℂ V) (hρ : IsUnitary ρ)
-    (hcomm : ∀ i j, i ≠ j → ∀ g ∈ H i, ∀ k ∈ H j, Commute g k)
-    (c : isotypicComponents ℂ[↥(⨆ i, H i)] (Representation.asModule (ρ.comp (⨆ i, H i).subtype))) :
-    ∃ i, ∃ d : ℕ, ∃ k : H i → ℂ,
-      (d : ℝ) ≤ (Module.finrank ℂ V : ℝ) ^ (1 / (b : ℝ)) ∧
-      (∑ g, ‖k g‖^2 = (d : ℝ)^2 / Fintype.card (H i)) ∧
-      (∀ g, k g⁻¹ = conj (k g)) ∧
-      (∀ x ∈ componentSpace (ρ.comp (⨆ i, H i).subtype) c,
-        complexFourier (ρ.comp (H i).subtype) k x = x) := by
-  classical
-  let J := ⨆ i, H i
-  let ρJ : Representation ℂ J V := ρ.comp J.subtype
-  let : NormedAddCommGroup c.val :=
-    inferInstanceAs (NormedAddCommGroup (componentSpace ρJ c))
-  let : InnerProductSpace ℂ c.val :=
-    inferInstanceAs (InnerProductSpace ℂ (componentSpace ρJ c))
-  let : FiniteDimensional ℂ c.val :=
-    inferInstanceAs (FiniteDimensional ℂ (componentSpace ρJ c))
-  let inc : c.val →ₗ[ℂ] V := c.val.subtype.restrictScalars ℂ
-  have hinc (g : J) (x : c.val) :
-      inc ((MonoidAlgebra.single g (1 : ℂ)) • x) = ρ (g : G) (inc x) := by
-    change ρJ.asAlgebraHom (MonoidAlgebra.single g 1) (inc x) = _
-    rw [Representation.asAlgebraHom_single, one_smul]
-    rfl
-  have hu : ∀ (g : J) (x y : c.val),
-      inner ℂ ((MonoidAlgebra.single g (1 : ℂ)) • x)
-        ((MonoidAlgebra.single g (1 : ℂ)) • y) = inner ℂ x y := by
-    intro g x y
-    change inner ℂ (inc ((MonoidAlgebra.single g (1 : ℂ)) • x))
-      (inc ((MonoidAlgebra.single g (1 : ℂ)) • y)) = inner ℂ (inc x) (inc y)
-    rw [hinc, hinc]
-    exact hρ (g : G) (inc x) (inc y)
-  obtain ⟨d, hdpos, hprod, hkernel⟩ := isotypic_joint_kernels
-    (M := c.val) (fun i => (H i).subgroupOf J)
-    (jointFactors_generate H) (jointFactors_commute H hcomm)
-    (IsIsotypic.isotypicComponents c.property) hu
-  have hdim : Module.finrank ℂ c.val ≤ Module.finrank ℂ V :=
-    Submodule.finrank_le (componentSpace ρJ c)
-  obtain ⟨i, hi⟩ := exists_degree_le_rpow hb hD d (hprod.trans hdim)
-  obtain ⟨k, hk, hki, hka⟩ := hkernel i
-  let e := Subgroup.subgroupOfEquivOfLe (le_iSup H i)
-  let k' : H i → ℂ := fun g => k (e.symm g)
-  refine ⟨i, d i, k', hi, ?_, ?_, ?_⟩
-  · have he : ∑ g, ‖k' g‖^2 = ∑ g, ‖k g‖^2 :=
-      Fintype.sum_equiv e.symm.toEquiv _ _ (fun g => rfl)
-    rw [he, hk, Fintype.card_congr e.toEquiv]
-  · intro g
-    simp only [k', map_inv, hki]
-  · intro x hx
-    have ht := congrArg inc (hka ⟨x, hx⟩)
-    have hix : inc ⟨x, hx⟩ = x := rfl
-    have ht' : ∑ g : (H i).subgroupOf J, k g • ρ (g.val.val) x = x := by
-      simpa only [map_sum, map_smul, hinc, hix] using ht
-    rw [complexFourier, LinearMap.sum_apply]
-    calc
-      _ = ∑ g : (H i).subgroupOf J, k g • ρ (g.val.val) x := by
-        exact Fintype.sum_equiv e.symm.toEquiv _ _ (fun g => rfl)
-      _ = x := ht'
-
-end ComponentKernel
-end
-end PartialPermutation
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.ComponentGeometry
+import OAI.RepresentationTheory.PartialPermutation.MatrixDegrees
+import OAI.RepresentationTheory.PartialPermutation.Restriction
+import OAI.RepresentationTheory.PartialPermutation.IsotypicKernels
+import OAI.RepresentationTheory.PartialPermutation.DegreeEstimates
+import OAI.RepresentationTheory.PartialPermutation.BlockSubgroups
+
+namespace OAI
+
+section
+open scoped Classical
+open scoped BigOperators ComplexConjugate MonoidAlgebra
+open scoped BigOperators ComplexConjugate
+open scoped MonoidAlgebra BigOperators
+open scoped BigOperators MonoidAlgebra Classical
+
+attribute [local instance] Classical.propDecidable
+open scoped MonoidAlgebra BigOperators
+open scoped BigOperators
+
+namespace PartialPermutation
+open scoped MonoidAlgebra BigOperators ComplexConjugate
+
+noncomputable section
+
+lemma isotypic_joint_kernels {G M : Type*} [Group G] [Fintype G]
+    [NormedAddCommGroup M] [InnerProductSpace ℂ M]
+    [Module ℂ[G] M] [IsScalarTower ℂ ℂ[G] M]
+    [FiniteDimensional ℂ M] [Nontrivial M]
+    {b : ℕ} (H : Fin b → Subgroup G) (hgen : ⨆ i, H i = ⊤)
+    (hcomm : ∀ i j, i ≠ j → ∀ g ∈ H i, ∀ k ∈ H j, Commute g k)
+    (h : IsIsotypic ℂ[G] M)
+    (hu : ∀ (g : G) (x y : M), inner ℂ ((MonoidAlgebra.single g (1 : ℂ)) • x)
+      ((MonoidAlgebra.single g (1 : ℂ)) • y) = inner ℂ x y) :
+    ∃ d : Fin b → ℕ, (∀ i, 0 < d i) ∧ (∏ i, d i) ≤ Module.finrank ℂ M ∧
+      ∀ i, ∃ k : H i → ℂ,
+        (∑ g, ‖k g‖^2 = (d i : ℝ)^2 / Fintype.card (H i)) ∧
+        (∀ g, k g⁻¹ = conj (k g)) ∧
+        (∀ x : M, ∑ g, k g • (MonoidAlgebra.single (g : G) (1 : ℂ)) • x = x) := by
+  classical
+  let (i : Fin b) : Module ℂ[H i] M := restrictGroupModule (H i)
+  let (i : Fin b) : IsScalarTower ℂ ℂ[H i] M := restrictGroupModule_tower (H i)
+  have hmodels (i : Fin b) : Nonempty (IsotypicKernelModel (H i) M) := by
+    apply exists_isotypicKernelModel (group_isotypic_restrict H hgen hcomm h i)
+    intro g x y
+    change inner ℂ ((MonoidAlgebra.mapDomainAlgHom ℂ ℂ (H i).subtype
+      (MonoidAlgebra.single g (1 : ℂ))) • x)
+      ((MonoidAlgebra.mapDomainAlgHom ℂ ℂ (H i).subtype
+      (MonoidAlgebra.single g (1 : ℂ))) • y) = _
+    simpa [MonoidAlgebra.mapDomainAlgHom] using hu g x y
+  let F (i : Fin b) := Classical.choice (hmodels i)
+  have hAcomm (i j : Fin b) (hij : i ≠ j) (X Y) : Commute ((F i).lift X) ((F j).lift Y) := by
+    obtain ⟨r, hr⟩ := (F i).lift_image X
+    obtain ⟨s, hs⟩ := (F j).lift_image Y
+    rw [hr, hs]
+    have he (i : Fin b) (r : ℂ[H i]) : Algebra.lsmul ℂ ℂ M r =
+        Algebra.lsmul ℂ ℂ M (MonoidAlgebra.mapDomainAlgHom ℂ ℂ (H i).subtype r) := by
+      ext x
+      rfl
+    rw [he i r, he j s]
+    exact (subgroup_algebra_commute (H i) (H j) (hcomm i j hij) r s).map
+      (Algebra.lsmul ℂ ℂ M : ℂ[G] →ₐ[ℂ] Module.End ℂ M)
+  refine ⟨fun i => (F i).degree, fun i => (F i).degree_pos, ?_, ?_⟩
+  · exact commuting_degrees_le_finrank (fun i => (F i).degree)
+      (fun i => (F i).degree_pos) (fun i => (F i).lift) hAcomm Finset.univ
+  · intro i
+    refine ⟨(F i).kernel, (F i).kernel_norm, (F i).kernel_inv, ?_⟩
+    intro x
+    have ht := (F i).kernel_action x
+    change ∑ g, (F i).kernel g • (MonoidAlgebra.mapDomainAlgHom ℂ ℂ (H i).subtype
+      (MonoidAlgebra.single g (1 : ℂ))) • x = x at ht
+    simpa [MonoidAlgebra.mapDomainAlgHom] using ht
+
+section ComponentKernel
+variable {G V : Type*} [Group G] [Fintype G]
+    [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+    (ρ : Representation ℂ G V) {b : ℕ} (H : Fin b → Subgroup G)
+
+lemma component_kernel (hb : 0 < b) (hD : 0 < Module.finrank ℂ V) (hρ : IsUnitary ρ)
+    (hcomm : ∀ i j, i ≠ j → ∀ g ∈ H i, ∀ k ∈ H j, Commute g k)
+    (c : isotypicComponents ℂ[↥(⨆ i, H i)] (Representation.asModule (ρ.comp (⨆ i, H i).subtype))) :
+    ∃ i, ∃ d : ℕ, ∃ k : H i → ℂ,
+      (d : ℝ) ≤ (Module.finrank ℂ V : ℝ) ^ (1 / (b : ℝ)) ∧
+      (∑ g, ‖k g‖^2 = (d : ℝ)^2 / Fintype.card (H i)) ∧
+      (∀ g, k g⁻¹ = conj (k g)) ∧
+      (∀ x ∈ componentSpace (ρ.comp (⨆ i, H i).subtype) c,
+        complexFourier (ρ.comp (H i).subtype) k x = x) := by
+  classical
+  let J := ⨆ i, H i
+  let ρJ : Representation ℂ J V := ρ.comp J.subtype
+  let : NormedAddCommGroup c.val :=
+    inferInstanceAs (NormedAddCommGroup (componentSpace ρJ c))
+  let : InnerProductSpace ℂ c.val :=
+    inferInstanceAs (InnerProductSpace ℂ (componentSpace ρJ c))
+  let : FiniteDimensional ℂ c.val :=
+    inferInstanceAs (FiniteDimensional ℂ (componentSpace ρJ c))
+  let inc : c.val →ₗ[ℂ] V := c.val.subtype.restrictScalars ℂ
+  have hinc (g : J) (x : c.val) :
+      inc ((MonoidAlgebra.single g (1 : ℂ)) • x) = ρ (g : G) (inc x) := by
+    change ρJ.asAlgebraHom (MonoidAlgebra.single g 1) (inc x) = _
+    rw [Representation.asAlgebraHom_single, one_smul]
+    rfl
+  have hu : ∀ (g : J) (x y : c.val),
+      inner ℂ ((MonoidAlgebra.single g (1 : ℂ)) • x)
+        ((MonoidAlgebra.single g (1 : ℂ)) • y) = inner ℂ x y := by
+    intro g x y
+    change inner ℂ (inc ((MonoidAlgebra.single g (1 : ℂ)) • x))
+      (inc ((MonoidAlgebra.single g (1 : ℂ)) • y)) = inner ℂ (inc x) (inc y)
+    rw [hinc, hinc]
+    exact hρ (g : G) (inc x) (inc y)
+  obtain ⟨d, hdpos, hprod, hkernel⟩ := isotypic_joint_kernels
+    (M := c.val) (fun i => (H i).subgroupOf J)
+    (jointFactors_generate H) (jointFactors_commute H hcomm)
+    (IsIsotypic.isotypicComponents c.property) hu
+  have hdim : Module.finrank ℂ c.val ≤ Module.finrank ℂ V :=
+    Submodule.finrank_le (componentSpace ρJ c)
+  obtain ⟨i, hi⟩ := exists_degree_le_rpow hb hD d (hprod.trans hdim)
+  obtain ⟨k, hk, hki, hka⟩ := hkernel i
+  let e := Subgroup.subgroupOfEquivOfLe (le_iSup H i)
+  let k' : H i → ℂ := fun g => k (e.symm g)
+  refine ⟨i, d i, k', hi, ?_, ?_, ?_⟩
+  · have he : ∑ g, ‖k' g‖^2 = ∑ g, ‖k g‖^2 :=
+      Fintype.sum_equiv e.symm.toEquiv _ _ (fun g => rfl)
+    rw [he, hk, Fintype.card_congr e.toEquiv]
+  · intro g
+    simp only [k', map_inv, hki]
+  · intro x hx
+    have ht := congrArg inc (hka ⟨x, hx⟩)
+    have hix : inc ⟨x, hx⟩ = x := rfl
+    have ht' : ∑ g : (H i).subgroupOf J, k g • ρ (g.val.val) x = x := by
+      simpa only [map_sum, map_smul, hinc, hix] using ht
+    rw [complexFourier, LinearMap.sum_apply]
+    calc
+      _ = ∑ g : (H i).subgroupOf J, k g • ρ (g.val.val) x := by
+        exact Fintype.sum_equiv e.symm.toEquiv _ _ (fun g => rfl)
+      _ = x := ht'
+
+end ComponentKernel
+end
+end PartialPermutation
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/Main.lean b/lean/OAI/RepresentationTheory/PartialPermutation/Main.lean
index 3a103f3ff2b12fc4f030878b0bed29e6bd559749..4130b1d267855e1df9d66de0dc97a1a278bc70ea 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/Main.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/Main.lean
@@ -1,56 +1,56 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.DeficitPlancherel
-import OAI.RepresentationTheory.PartialPermutation.NormBounds
-
-namespace OAI
-
-namespace PartialPermutation
-noncomputable section
-open scoped Classical
-
-theorem complementaryCosets_finite : ComplementaryCosetsFiniteStatement := by
-  intro X _ _ b hb M hnonempty hdis _ V _ _ _ ρ hirr hρ f B u hf hm hu hcap
-  let := hirr
-  have hB : 0 ≤ B := by
-    let i : Fin b := ⟨0,hb⟩
-    have hp : (0:ℝ) < (blockSubgroup (M i)).index := by
-      exact_mod_cast Nat.pos_of_ne_zero (blockSubgroup (M i)).index_ne_zero_of_finite
-    have hz : 0 ≤ B / (blockSubgroup (M i)).index :=
-      (Finset.sum_nonneg (fun (g : blockSubgroup (M i)) _ => hf (1*g))).trans (hcap i 1)
-    simpa only [div_mul_cancel₀ B hp.ne'] using mul_nonneg hz hp.le
-  have hsum : (∑ i, symmetricInverseDegreeSum (Fintype.card (M i)) u) ≤
-      (b:ℝ)*symmetricDegreeConstant u := by
-    calc
-      _ ≤ ∑ _i : Fin b, symmetricDegreeConstant u := by
-        apply Finset.sum_le_sum
-        intro i _
-        let : Nonempty (M i) := (hnonempty i).to_subtype
-        exact symmetricInverseDegreeSum_le_constant (by exact Fintype.card_pos) hu
-      _ = _ := by simp
-  have hs : 0 ≤ ∑ i, symmetricInverseDegreeSum (Fintype.card (M i)) u := by
-    exact Finset.sum_nonneg (fun _ _ => inverseDegreeSum_nonneg _ _)
-  have hR : 0 ≤ (Module.finrank ℂ V : ℝ)^(-1+(u+2)/(b:ℝ)) := Real.rpow_nonneg (by positivity) _
-  have hcoef : B * mass f * (Module.finrank ℂ V:ℝ)^(-1+(u+2)/(b:ℝ)) ≤
-      B * (Module.finrank ℂ V:ℝ)^(-1+(u+2)/(b:ℝ)) := by
-    calc
-      _ ≤ B * 1 * (Module.finrank ℂ V:ℝ)^(-1+(u+2)/(b:ℝ)) :=
-        mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left hm hB) hR
-      _ = _ := by ring
-  have hHS : hsNormSq (complexFourier ρ (fun g => f g)) ≤
-      b * B * symmetricDegreeConstant u * (Module.finrank ℂ V:ℝ)^(-1+(u+2)/(b:ℝ)) := by
-    calc
-      _ ≤ _ := by
-        have h := inverse_degree_sum_transfer ρ (fun i => blockSubgroup (M i)) hb hρ
-          (fun i j hij g hg k hk => blockSubgroup_commute (hdis hij) hg hk) f B u hf hu hcap
-        rw [inverseDegreeSum_block_sum M u] at h
-        exact h
-      _ ≤ (B * (Module.finrank ℂ V:ℝ)^(-1+(u+2)/(b:ℝ))) *
-          ((b:ℝ)*symmetricDegreeConstant u) :=
-        mul_le_mul hcoef hsum hs (mul_nonneg hB hR)
-      _ = _ := by ring
-  exact ⟨hHS,(opNorm_sq_le_hsNormSq _).trans hHS⟩
-
-end
-end PartialPermutation
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.DeficitPlancherel
+import OAI.RepresentationTheory.PartialPermutation.NormBounds
+
+namespace OAI
+
+namespace PartialPermutation
+noncomputable section
+open scoped Classical
+
+theorem complementaryCosets_finite : ComplementaryCosetsFiniteStatement := by
+  intro X _ _ b hb M hnonempty hdis _ V _ _ _ ρ hirr hρ f B u hf hm hu hcap
+  let := hirr
+  have hB : 0 ≤ B := by
+    let i : Fin b := ⟨0,hb⟩
+    have hp : (0:ℝ) < (blockSubgroup (M i)).index := by
+      exact_mod_cast Nat.pos_of_ne_zero (blockSubgroup (M i)).index_ne_zero_of_finite
+    have hz : 0 ≤ B / (blockSubgroup (M i)).index :=
+      (Finset.sum_nonneg (fun (g : blockSubgroup (M i)) _ => hf (1*g))).trans (hcap i 1)
+    simpa only [div_mul_cancel₀ B hp.ne'] using mul_nonneg hz hp.le
+  have hsum : (∑ i, symmetricInverseDegreeSum (Fintype.card (M i)) u) ≤
+      (b:ℝ)*symmetricDegreeConstant u := by
+    calc
+      _ ≤ ∑ _i : Fin b, symmetricDegreeConstant u := by
+        apply Finset.sum_le_sum
+        intro i _
+        let : Nonempty (M i) := (hnonempty i).to_subtype
+        exact symmetricInverseDegreeSum_le_constant (by exact Fintype.card_pos) hu
+      _ = _ := by simp
+  have hs : 0 ≤ ∑ i, symmetricInverseDegreeSum (Fintype.card (M i)) u := by
+    exact Finset.sum_nonneg (fun _ _ => inverseDegreeSum_nonneg _ _)
+  have hR : 0 ≤ (Module.finrank ℂ V : ℝ)^(-1+(u+2)/(b:ℝ)) := Real.rpow_nonneg (by positivity) _
+  have hcoef : B * mass f * (Module.finrank ℂ V:ℝ)^(-1+(u+2)/(b:ℝ)) ≤
+      B * (Module.finrank ℂ V:ℝ)^(-1+(u+2)/(b:ℝ)) := by
+    calc
+      _ ≤ B * 1 * (Module.finrank ℂ V:ℝ)^(-1+(u+2)/(b:ℝ)) :=
+        mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left hm hB) hR
+      _ = _ := by ring
+  have hHS : hsNormSq (complexFourier ρ (fun g => f g)) ≤
+      b * B * symmetricDegreeConstant u * (Module.finrank ℂ V:ℝ)^(-1+(u+2)/(b:ℝ)) := by
+    calc
+      _ ≤ _ := by
+        have h := inverse_degree_sum_transfer ρ (fun i => blockSubgroup (M i)) hb hρ
+          (fun i j hij g hg k hk => blockSubgroup_commute (hdis hij) hg hk) f B u hf hu hcap
+        rw [inverseDegreeSum_block_sum M u] at h
+        exact h
+      _ ≤ (B * (Module.finrank ℂ V:ℝ)^(-1+(u+2)/(b:ℝ))) *
+          ((b:ℝ)*symmetricDegreeConstant u) :=
+        mul_le_mul hcoef hsum hs (mul_nonneg hB hR)
+      _ = _ := by ring
+  exact ⟨hHS,(opNorm_sq_le_hsNormSq _).trans hHS⟩
+
+end
+end PartialPermutation
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/MatrixCoefficients.lean b/lean/OAI/RepresentationTheory/PartialPermutation/MatrixCoefficients.lean
index 57fa23e1f32dec013e82bee50e872a63a2dfa39c..1bb346a0c018f3318f7dcf9f2159db2da4e0689e 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/MatrixCoefficients.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/MatrixCoefficients.lean
@@ -1,74 +1,74 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.Model
-import OAI.RepresentationTheory.PartialPermutation.SchurAverage
-
-namespace OAI
-
-section
-open scoped Classical
-open scoped BigOperators ComplexConjugate MonoidAlgebra
-
-namespace PartialPermutation
-
-section MatrixCoefficients
-variable {G V : Type*} [Group G] [Fintype G]
-    [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-    (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) [Representation.IsIrreducible ρ]
-include hρ
-
-lemma matrixCoefficient_orthogonality (x y u v : V) :
-    (Fintype.card G : ℂ)⁻¹ * ∑ g, inner ℂ x (ρ g u) * conj (inner ℂ y (ρ g v)) =
-      (inner ℂ v u / (Module.finrank ℂ V : ℂ)) * inner ℂ x y := by
-  have h := congrArg (fun A : Module.End ℂ V => inner ℂ x (A y))
-    (conjugateAverage_eq_trace ρ (InnerProductSpace.rankOne ℂ u v).toLinearMap)
-  have hu (g : G) : inner ℂ v (ρ g⁻¹ y) = conj (inner ℂ y (ρ g v)) := by
-    rw [← inner_conj_symm]
-    have ht := hρ g v (ρ g⁻¹ y)
-    simpa using ht.symm
-  simpa [conjugateAverage, LinearMap.sum_apply, InnerProductSpace.rankOne_apply,
-    InnerProductSpace.trace_rankOne, inner_smul_right, inner_sum, hu, mul_comm] using h
-
-lemma normalized_matrixCoefficients_orthonormal {ι : Type*} [Fintype ι]
-    (e : OrthonormalBasis ι ℂ V) :
-    Orthonormal ℂ (fun p : ι × ι =>
-      (Real.sqrt ((Module.finrank ℂ V : ℝ) / Fintype.card G) : ℂ) •
-        (WithLp.toLp 2 (fun g : G => conj (inner ℂ (e p.1) (ρ g (e p.2)))))) := by
-  classical
-  have : Nontrivial V := IsSimpleModule.nontrivial ℂ[G] ρ.asModule
-  have hD : (0 : ℝ) < Module.finrank ℂ V := by exact_mod_cast Module.finrank_pos (R := ℂ) (M := V)
-  have hN : (0 : ℝ) < Fintype.card G := by exact_mod_cast Fintype.card_pos
-  have hDc : (Module.finrank ℂ V : ℂ) ≠ 0 := by exact_mod_cast hD.ne'
-  have hNc : (Fintype.card G : ℂ) ≠ 0 := by exact_mod_cast hN.ne'
-  have ht : (Real.sqrt ((Module.finrank ℂ V : ℝ) / Fintype.card G) : ℂ) ^ 2 =
-      (Module.finrank ℂ V : ℂ) / Fintype.card G := by
-    rw [← Complex.ofReal_pow, Real.sq_sqrt (div_nonneg hD.le hN.le)]
-    push_cast
-    rfl
-  rw [orthonormal_iff_ite]
-  intro p q
-  rw [inner_smul_left, inner_smul_right]
-  simp only [Complex.conj_ofReal, PiLp.inner_apply, RCLike.inner_apply,
-    starRingEnd_self_apply]
-  simp_rw [mul_comm (conj (inner ℂ (e q.1) (ρ _ (e q.2))))]
-  have h := matrixCoefficient_orthogonality ρ hρ (e p.1) (e q.1) (e p.2) (e q.2)
-  have hs : ∑ g : G, inner ℂ (e p.1) (ρ g (e p.2)) *
-      conj (inner ℂ (e q.1) (ρ g (e q.2))) =
-      (Fintype.card G : ℂ) * ((inner ℂ (e q.2) (e p.2) /
-        (Module.finrank ℂ V : ℂ)) * inner ℂ (e p.1) (e q.1)) := by
-    rw [← h]
-    field_simp
-  rw [hs, orthonormal_iff_ite.mp e.orthonormal, orthonormal_iff_ite.mp e.orthonormal]
-  by_cases h1 : p.1 = q.1 <;> by_cases h2 : p.2 = q.2
-  · have hpq : p = q := Prod.ext h1 h2
-    simp only [ite_eq_left, hpq, ← mul_assoc, ← pow_two, ht]
-    field_simp
-  · simp [h1, h2, Ne.symm h2, Prod.ext_iff]
-  · simp [h1, h2, Prod.ext_iff]
-  · simp [h1, h2, Prod.ext_iff]
-
-end MatrixCoefficients
-end PartialPermutation
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.Model
+import OAI.RepresentationTheory.PartialPermutation.SchurAverage
+
+namespace OAI
+
+section
+open scoped Classical
+open scoped BigOperators ComplexConjugate MonoidAlgebra
+
+namespace PartialPermutation
+
+section MatrixCoefficients
+variable {G V : Type*} [Group G] [Fintype G]
+    [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+    (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) [Representation.IsIrreducible ρ]
+include hρ
+
+lemma matrixCoefficient_orthogonality (x y u v : V) :
+    (Fintype.card G : ℂ)⁻¹ * ∑ g, inner ℂ x (ρ g u) * conj (inner ℂ y (ρ g v)) =
+      (inner ℂ v u / (Module.finrank ℂ V : ℂ)) * inner ℂ x y := by
+  have h := congrArg (fun A : Module.End ℂ V => inner ℂ x (A y))
+    (conjugateAverage_eq_trace ρ (InnerProductSpace.rankOne ℂ u v).toLinearMap)
+  have hu (g : G) : inner ℂ v (ρ g⁻¹ y) = conj (inner ℂ y (ρ g v)) := by
+    rw [← inner_conj_symm]
+    have ht := hρ g v (ρ g⁻¹ y)
+    simpa using ht.symm
+  simpa [conjugateAverage, LinearMap.sum_apply, InnerProductSpace.rankOne_apply,
+    InnerProductSpace.trace_rankOne, inner_smul_right, inner_sum, hu, mul_comm] using h
+
+lemma normalized_matrixCoefficients_orthonormal {ι : Type*} [Fintype ι]
+    (e : OrthonormalBasis ι ℂ V) :
+    Orthonormal ℂ (fun p : ι × ι =>
+      (Real.sqrt ((Module.finrank ℂ V : ℝ) / Fintype.card G) : ℂ) •
+        (WithLp.toLp 2 (fun g : G => conj (inner ℂ (e p.1) (ρ g (e p.2)))))) := by
+  classical
+  have : Nontrivial V := IsSimpleModule.nontrivial ℂ[G] ρ.asModule
+  have hD : (0 : ℝ) < Module.finrank ℂ V := by exact_mod_cast Module.finrank_pos (R := ℂ) (M := V)
+  have hN : (0 : ℝ) < Fintype.card G := by exact_mod_cast Fintype.card_pos
+  have hDc : (Module.finrank ℂ V : ℂ) ≠ 0 := by exact_mod_cast hD.ne'
+  have hNc : (Fintype.card G : ℂ) ≠ 0 := by exact_mod_cast hN.ne'
+  have ht : (Real.sqrt ((Module.finrank ℂ V : ℝ) / Fintype.card G) : ℂ) ^ 2 =
+      (Module.finrank ℂ V : ℂ) / Fintype.card G := by
+    rw [← Complex.ofReal_pow, Real.sq_sqrt (div_nonneg hD.le hN.le)]
+    push_cast
+    rfl
+  rw [orthonormal_iff_ite]
+  intro p q
+  rw [inner_smul_left, inner_smul_right]
+  simp only [Complex.conj_ofReal, PiLp.inner_apply, RCLike.inner_apply,
+    starRingEnd_self_apply]
+  simp_rw [mul_comm (conj (inner ℂ (e q.1) (ρ _ (e q.2))))]
+  have h := matrixCoefficient_orthogonality ρ hρ (e p.1) (e q.1) (e p.2) (e q.2)
+  have hs : ∑ g : G, inner ℂ (e p.1) (ρ g (e p.2)) *
+      conj (inner ℂ (e q.1) (ρ g (e q.2))) =
+      (Fintype.card G : ℂ) * ((inner ℂ (e q.2) (e p.2) /
+        (Module.finrank ℂ V : ℂ)) * inner ℂ (e p.1) (e q.1)) := by
+    rw [← h]
+    field_simp
+  rw [hs, orthonormal_iff_ite.mp e.orthonormal, orthonormal_iff_ite.mp e.orthonormal]
+  by_cases h1 : p.1 = q.1 <;> by_cases h2 : p.2 = q.2
+  · have hpq : p = q := Prod.ext h1 h2
+    simp only [ite_eq_left, hpq, ← mul_assoc, ← pow_two, ht]
+    field_simp
+  · simp [h1, h2, Ne.symm h2, Prod.ext_iff]
+  · simp [h1, h2, Prod.ext_iff]
+  · simp [h1, h2, Prod.ext_iff]
+
+end MatrixCoefficients
+end PartialPermutation
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/MatrixDegrees.lean b/lean/OAI/RepresentationTheory/PartialPermutation/MatrixDegrees.lean
index edc6c5a0860fa58630938fc6c89eb457c7b1a58c..4019c48b8a5eda3ebd9e2c2d9b83907566df173c 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/MatrixDegrees.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/MatrixDegrees.lean
@@ -1,125 +1,125 @@
-import Mathlib
-
-namespace OAI
-
-section
-open scoped Classical
-open scoped BigOperators ComplexConjugate MonoidAlgebra
-open scoped BigOperators ComplexConjugate
-open scoped MonoidAlgebra BigOperators
-open scoped BigOperators MonoidAlgebra Classical
-
-attribute [local instance] Classical.propDecidable
-open scoped BigOperators
-
-namespace PartialPermutation
-
-section MatrixTraces
-variable {V : Type*} [AddCommGroup V] [Module ℂ V]
-
-lemma matrixUnit_trace_eq {d : ℕ}
-    (A : Matrix (Fin d) (Fin d) ℂ →ₐ[ℂ] Module.End ℂ V)
-    (B : Module.End ℂ V) (hB : ∀ X, Commute (A X) B) (i j : Fin d) :
-    LinearMap.trace ℂ V (A (Matrix.single i i 1) * B) =
-      LinearMap.trace ℂ V (A (Matrix.single j j 1) * B) := by
-  have hu (i j k : Fin d) : A (Matrix.single i j 1) * A (Matrix.single j k 1) =
-      A (Matrix.single i k 1) := by
-    rw [← map_mul, Matrix.single_mul_single_same, one_mul]
-  calc
-    _ = LinearMap.trace ℂ V
-        (A (Matrix.single i j 1) * (A (Matrix.single j i 1) * B)) := by
-      rw [← mul_assoc, hu]
-    _ = LinearMap.trace ℂ V
-        ((A (Matrix.single j i 1) * B) * A (Matrix.single i j 1)) :=
-      LinearMap.trace_mul_comm ℂ _ _
-    _ = LinearMap.trace ℂ V
-        (A (Matrix.single j i 1) * A (Matrix.single i j 1) * B) := by
-      rw [mul_assoc, ← (hB _).eq, ← mul_assoc]
-    _ = _ := by rw [hu]
-
-lemma degree_mul_matrixUnit_trace {d : ℕ}
-    (A : Matrix (Fin d) (Fin d) ℂ →ₐ[ℂ] Module.End ℂ V)
-    (B : Module.End ℂ V) (hB : ∀ X, Commute (A X) B) (t : Fin d) :
-    (d : ℂ) * LinearMap.trace ℂ V (A (Matrix.single t t 1) * B) =
-      LinearMap.trace ℂ V B := by
-  classical
-  calc
-    _ = ∑ j : Fin d, LinearMap.trace ℂ V (A (Matrix.single t t 1) * B) := by simp
-    _ = ∑ j : Fin d, LinearMap.trace ℂ V (A (Matrix.single j j 1) * B) := by
-      apply Finset.sum_congr rfl
-      intro j hj
-      exact matrixUnit_trace_eq A B hB t j
-    _ = LinearMap.trace ℂ V ((∑ j : Fin d, A (Matrix.single j j 1)) * B) := by
-      simp [Finset.sum_mul]
-    _ = _ := by rw [← map_sum, Matrix.sum_single_one, map_one, one_mul]
-
-end MatrixTraces
-
-section CommutingDegrees
-variable {V : Type*} [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
-    {ι : Type*} [DecidableEq ι] (d : ι → ℕ) (hd : ∀ i, 0 < d i)
-    (A : ∀ i, Matrix (Fin (d i)) (Fin (d i)) ℂ →ₐ[ℂ] Module.End ℂ V)
-    (hcomm : ∀ i j, i ≠ j → ∀ X Y, Commute (A i X) (A j Y))
-
-include hd A hcomm
-
-lemma commuting_degrees_dvd_finrank (s : Finset ι) :
-    (∏ i ∈ s, d i) ∣ Module.finrank ℂ V := by
-  classical
-  let p (i : ι) : Module.End ℂ V :=
-    A i (Matrix.single (⟨0, hd i⟩ : Fin (d i)) ⟨0, hd i⟩ 1)
-  have hp (i : ι) : IsIdempotentElem (p i) := by
-    dsimp [IsIdempotentElem, p]
-    rw [← map_mul, Matrix.single_mul_single_same, one_mul]
-  have hc (i j : ι) (hij : i ≠ j) : Commute (p i) (p j) := hcomm i j hij _ _
-  have hcc (s : Finset ι) : (↑s : Set ι).Pairwise (Function.onFun Commute p) := by
-    intro i hi j hj hij
-    exact hc i j hij
-  let P (s : Finset ι) : Module.End ℂ V :=
-    s.noncommProd p (hcc s)
-  have hP (s : Finset ι) (i : ι) (hi : i ∉ s) (X) : Commute (A i X) (P s) := by
-    apply Finset.noncommProd_commute
-    intro j hj
-    exact hcomm i j (fun he => hi (he ▸ hj)) _ _
-  have hid (s : Finset ι) : IsIdempotentElem (P s) := by
-    induction s using Finset.induction_on with
-    | empty => simp [P, Finset.noncommProd_empty, IsIdempotentElem]
-    | @insert i s hi ih =>
-      dsimp only [P]
-      rw [Finset.noncommProd_insert_of_notMem _ _ _ _ hi]
-      change IsIdempotentElem (p i * P s)
-      change (p i * P s) * (p i * P s) = p i * P s
-      rw [(hP s i hi _).symm.mul_mul_mul_comm, hp i, ih]
-  have ht (s : Finset ι) :
-      ((∏ i ∈ s, d i : ℕ) : ℂ) * LinearMap.trace ℂ V (P s) = Module.finrank ℂ V := by
-    induction s using Finset.induction_on with
-    | empty => simp [P, Finset.noncommProd_empty, Module.End.one_eq_id, LinearMap.trace_id]
-    | @insert i s hi ih =>
-      dsimp only [P]
-      rw [Finset.prod_insert hi, Nat.cast_mul,
-        Finset.noncommProd_insert_of_notMem _ _ _ _ hi]
-      change ((d i : ℂ) * ↑(∏ j ∈ s, d j)) *
-          LinearMap.trace ℂ V (p i * P s) = _
-      calc
-        _ = (↑(∏ j ∈ s, d j) : ℂ) *
-            ((d i : ℂ) * LinearMap.trace ℂ V (p i * P s)) := by ring
-        _ = (↑(∏ j ∈ s, d j) : ℂ) * LinearMap.trace ℂ V (P s) := by
-          rw [degree_mul_matrixUnit_trace (A i) (P s) (hP s i hi)]
-        _ = _ := ih
-  refine ⟨Module.finrank ℂ (LinearMap.range (P s)), ?_⟩
-  have he := ht s
-  rw [(LinearMap.IsIdempotentElem.isProj_range (P s) (hid s)).trace] at he
-  exact_mod_cast he.symm
-
-lemma commuting_degrees_le_finrank [Nontrivial V] (s : Finset ι) :
-    (∏ i ∈ s, d i) ≤ Module.finrank ℂ V :=
-  Nat.le_of_dvd (Module.finrank_pos (R := ℂ))
-    (commuting_degrees_dvd_finrank d hd A hcomm s)
-
-end CommutingDegrees
-
-end PartialPermutation
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+section
+open scoped Classical
+open scoped BigOperators ComplexConjugate MonoidAlgebra
+open scoped BigOperators ComplexConjugate
+open scoped MonoidAlgebra BigOperators
+open scoped BigOperators MonoidAlgebra Classical
+
+attribute [local instance] Classical.propDecidable
+open scoped BigOperators
+
+namespace PartialPermutation
+
+section MatrixTraces
+variable {V : Type*} [AddCommGroup V] [Module ℂ V]
+
+lemma matrixUnit_trace_eq {d : ℕ}
+    (A : Matrix (Fin d) (Fin d) ℂ →ₐ[ℂ] Module.End ℂ V)
+    (B : Module.End ℂ V) (hB : ∀ X, Commute (A X) B) (i j : Fin d) :
+    LinearMap.trace ℂ V (A (Matrix.single i i 1) * B) =
+      LinearMap.trace ℂ V (A (Matrix.single j j 1) * B) := by
+  have hu (i j k : Fin d) : A (Matrix.single i j 1) * A (Matrix.single j k 1) =
+      A (Matrix.single i k 1) := by
+    rw [← map_mul, Matrix.single_mul_single_same, one_mul]
+  calc
+    _ = LinearMap.trace ℂ V
+        (A (Matrix.single i j 1) * (A (Matrix.single j i 1) * B)) := by
+      rw [← mul_assoc, hu]
+    _ = LinearMap.trace ℂ V
+        ((A (Matrix.single j i 1) * B) * A (Matrix.single i j 1)) :=
+      LinearMap.trace_mul_comm ℂ _ _
+    _ = LinearMap.trace ℂ V
+        (A (Matrix.single j i 1) * A (Matrix.single i j 1) * B) := by
+      rw [mul_assoc, ← (hB _).eq, ← mul_assoc]
+    _ = _ := by rw [hu]
+
+lemma degree_mul_matrixUnit_trace {d : ℕ}
+    (A : Matrix (Fin d) (Fin d) ℂ →ₐ[ℂ] Module.End ℂ V)
+    (B : Module.End ℂ V) (hB : ∀ X, Commute (A X) B) (t : Fin d) :
+    (d : ℂ) * LinearMap.trace ℂ V (A (Matrix.single t t 1) * B) =
+      LinearMap.trace ℂ V B := by
+  classical
+  calc
+    _ = ∑ j : Fin d, LinearMap.trace ℂ V (A (Matrix.single t t 1) * B) := by simp
+    _ = ∑ j : Fin d, LinearMap.trace ℂ V (A (Matrix.single j j 1) * B) := by
+      apply Finset.sum_congr rfl
+      intro j hj
+      exact matrixUnit_trace_eq A B hB t j
+    _ = LinearMap.trace ℂ V ((∑ j : Fin d, A (Matrix.single j j 1)) * B) := by
+      simp [Finset.sum_mul]
+    _ = _ := by rw [← map_sum, Matrix.sum_single_one, map_one, one_mul]
+
+end MatrixTraces
+
+section CommutingDegrees
+variable {V : Type*} [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
+    {ι : Type*} [DecidableEq ι] (d : ι → ℕ) (hd : ∀ i, 0 < d i)
+    (A : ∀ i, Matrix (Fin (d i)) (Fin (d i)) ℂ →ₐ[ℂ] Module.End ℂ V)
+    (hcomm : ∀ i j, i ≠ j → ∀ X Y, Commute (A i X) (A j Y))
+
+include hd A hcomm
+
+lemma commuting_degrees_dvd_finrank (s : Finset ι) :
+    (∏ i ∈ s, d i) ∣ Module.finrank ℂ V := by
+  classical
+  let p (i : ι) : Module.End ℂ V :=
+    A i (Matrix.single (⟨0, hd i⟩ : Fin (d i)) ⟨0, hd i⟩ 1)
+  have hp (i : ι) : IsIdempotentElem (p i) := by
+    dsimp [IsIdempotentElem, p]
+    rw [← map_mul, Matrix.single_mul_single_same, one_mul]
+  have hc (i j : ι) (hij : i ≠ j) : Commute (p i) (p j) := hcomm i j hij _ _
+  have hcc (s : Finset ι) : (↑s : Set ι).Pairwise (Function.onFun Commute p) := by
+    intro i hi j hj hij
+    exact hc i j hij
+  let P (s : Finset ι) : Module.End ℂ V :=
+    s.noncommProd p (hcc s)
+  have hP (s : Finset ι) (i : ι) (hi : i ∉ s) (X) : Commute (A i X) (P s) := by
+    apply Finset.noncommProd_commute
+    intro j hj
+    exact hcomm i j (fun he => hi (he ▸ hj)) _ _
+  have hid (s : Finset ι) : IsIdempotentElem (P s) := by
+    induction s using Finset.induction_on with
+    | empty => simp [P, Finset.noncommProd_empty, IsIdempotentElem]
+    | @insert i s hi ih =>
+      dsimp only [P]
+      rw [Finset.noncommProd_insert_of_notMem _ _ _ _ hi]
+      change IsIdempotentElem (p i * P s)
+      change (p i * P s) * (p i * P s) = p i * P s
+      rw [(hP s i hi _).symm.mul_mul_mul_comm, hp i, ih]
+  have ht (s : Finset ι) :
+      ((∏ i ∈ s, d i : ℕ) : ℂ) * LinearMap.trace ℂ V (P s) = Module.finrank ℂ V := by
+    induction s using Finset.induction_on with
+    | empty => simp [P, Finset.noncommProd_empty, Module.End.one_eq_id, LinearMap.trace_id]
+    | @insert i s hi ih =>
+      dsimp only [P]
+      rw [Finset.prod_insert hi, Nat.cast_mul,
+        Finset.noncommProd_insert_of_notMem _ _ _ _ hi]
+      change ((d i : ℂ) * ↑(∏ j ∈ s, d j)) *
+          LinearMap.trace ℂ V (p i * P s) = _
+      calc
+        _ = (↑(∏ j ∈ s, d j) : ℂ) *
+            ((d i : ℂ) * LinearMap.trace ℂ V (p i * P s)) := by ring
+        _ = (↑(∏ j ∈ s, d j) : ℂ) * LinearMap.trace ℂ V (P s) := by
+          rw [degree_mul_matrixUnit_trace (A i) (P s) (hP s i hi)]
+        _ = _ := ih
+  refine ⟨Module.finrank ℂ (LinearMap.range (P s)), ?_⟩
+  have he := ht s
+  rw [(LinearMap.IsIdempotentElem.isProj_range (P s) (hid s)).trace] at he
+  exact_mod_cast he.symm
+
+lemma commuting_degrees_le_finrank [Nontrivial V] (s : Finset ι) :
+    (∏ i ∈ s, d i) ≤ Module.finrank ℂ V :=
+  Nat.le_of_dvd (Module.finrank_pos (R := ℂ))
+    (commuting_degrees_dvd_finrank d hd A hcomm s)
+
+end CommutingDegrees
+
+end PartialPermutation
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/Model.lean b/lean/OAI/RepresentationTheory/PartialPermutation/Model.lean
index d015f3354b669ca5e4e0e042801c66cf0ba65309..6dcfb4049e423b6f48a7b77244746eb45198060d 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/Model.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/Model.lean
@@ -1,113 +1,113 @@
-import Mathlib
-
-namespace OAI
-
-open scoped Classical BigOperators ComplexConjugate MonoidAlgebra
-namespace PartialPermutation
-noncomputable section
-
-def IsUnitary {G V : Type*} [Group G] [NormedAddCommGroup V]
-    [InnerProductSpace ℂ V] (ρ : Representation ℂ G V) : Prop :=
-  ∀ g x y, inner ℂ (ρ g x) (ρ g y) = inner ℂ x y
-
-section
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-    [FiniteDimensional ℂ V]
-def hsNormSq (A : Module.End ℂ V) : ℝ :=
-  (LinearMap.trace ℂ V (LinearMap.adjoint A * A)).re
-
-end
-
-def complexFourier {G V : Type*} [Group G] [Fintype G] [AddCommGroup V] [Module ℂ V]
-    (ρ : Representation ℂ G V) (f : G → ℂ) : Module.End ℂ V := ∑ g, f g • ρ g
-
-section
-variable {G V : Type*} [Group G] [Fintype G] [AddCommGroup V] [Module ℂ V]
-    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V)
-noncomputable def componentSimple (c : isotypicComponents ℂ[G] ρ.asModule) :
-    Submodule ℂ[G] ρ.asModule := c.property.choose
-
-instance components_finite : Finite (isotypicComponents ℂ[G] ρ.asModule) := by
-  let : IsNoetherian ℂ[G] ρ.asModule := isNoetherian_of_tower ℂ inferInstance
-  infer_instance
-
-end
-
-section
-variable (G : Type*) [Group G] [Fintype G]
-def regularRep : Representation ℂ G (EuclideanSpace ℂ G) where
-  toFun g :=
-    { toFun := fun f => WithLp.toLp 2 (fun h => f (g⁻¹ * h))
-      map_add' := by intros; rfl
-      map_smul' := by intros; rfl }
-  map_one' := by ext f h; simp
-  map_mul' := by
-    intro x y
-    ext f h
-    change f ((x * y)⁻¹ * h) = f (y⁻¹ * (x⁻¹ * h))
-    simp [mul_assoc]
-
-end
-
-section
-variable {G : Type*} [Group G] [Fintype G]
-abbrev IrreducibleIndex (G : Type*) [Group G] [Fintype G] :=
-  isotypicComponents ℂ[G] (regularRep G).asModule
-
-noncomputable instance irreducibleIndex_fintype : Fintype (IrreducibleIndex G) :=
-  Fintype.ofFinite _
-
-abbrev irreducibleSpace (c : IrreducibleIndex G) :=
-  (Subrepresentation.ofSubmodule' (componentSimple (regularRep G) c)).toSubmodule
-
-abbrev irreducibleDegree (c : IrreducibleIndex G) := Module.finrank ℂ (irreducibleSpace c)
-
-end
-
-def mass {G : Type*} [Fintype G] (f : G → ℝ) : ℝ := ∑ g, f g
-
-def LeftCosetCap {G : Type*} [Group G] [Fintype G]
-    (f : G → ℝ) (H : Subgroup G) (B : ℝ) : Prop :=
-  by classical exact ∀ g, ∑ h : H, f (g * h) ≤ B / H.index
-
-def blockSubgroup {X : Type*} (M : Set X) : Subgroup (Equiv.Perm X) where
-  carrier := {g | ∀ x, x ∉ M → g x = x}
-  one_mem' := by simp
-  mul_mem' := by
-    intro g h hg hh x hx
-    simp only [Set.mem_ofPred_eq] at hg hh ⊢
-    simp [Equiv.Perm.mul_apply, hh x hx, hg x hx]
-  inv_mem' := by
-    intro g hg x hx
-    exact ((Equiv.eq_symm_apply g).mpr (hg x hx)).symm
-
-def inverseDegreeSum (G : Type*) [Group G] [Fintype G] (u : ℝ) : ℝ :=
-  ∑ c : IrreducibleIndex G, (irreducibleDegree c : ℝ)^(-u)
-
-def symmetricInverseDegreeSum (n : ℕ) (u : ℝ) : ℝ :=
-  inverseDegreeSum (Equiv.Perm (Fin n)) u
-
-def symmetricDegreeConstant (u : ℝ) : ℝ :=
-  sSup {x : ℝ | ∃ n : ℕ, 1 ≤ n ∧ x = symmetricInverseDegreeSum n u}
-
-def ComplementaryCosetsFiniteStatement : Prop :=
-  ∀ (X : Type*) [Fintype X] [DecidableEq X] (b : ℕ) (_ : 0 < b) (M : Fin b → Set X),
-    (∀ i, (M i).Nonempty) →
-    Pairwise (fun i j => Disjoint (M i) (M j)) → (⋃ i, M i) = Set.univ →
-    ∀ (V : Type) [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-      [FiniteDimensional ℂ V] (ρ : Representation ℂ (Equiv.Perm X) V),
-      Representation.IsIrreducible ρ → IsUnitary ρ →
-      ∀ (f : Equiv.Perm X → ℝ) (B u : ℝ),
-      (∀ g, 0 ≤ f g) → mass f ≤ 1 → 0 < u →
-      (∀ i, LeftCosetCap f (blockSubgroup (M i)) B) →
-      (hsNormSq (complexFourier ρ (fun g => f g)) ≤
-        b * B * symmetricDegreeConstant u *
-          (Module.finrank ℂ V : ℝ)^(-1+(u+2)/(b : ℝ))) ∧
-      (‖LinearMap.toContinuousLinearMap (complexFourier ρ (fun g => f g))‖^2 ≤
-        b * B * symmetricDegreeConstant u *
-          (Module.finrank ℂ V : ℝ)^(-1+(u+2)/(b : ℝ)))
-
-end
-end PartialPermutation
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+open scoped Classical BigOperators ComplexConjugate MonoidAlgebra
+namespace PartialPermutation
+noncomputable section
+
+def IsUnitary {G V : Type*} [Group G] [NormedAddCommGroup V]
+    [InnerProductSpace ℂ V] (ρ : Representation ℂ G V) : Prop :=
+  ∀ g x y, inner ℂ (ρ g x) (ρ g y) = inner ℂ x y
+
+section
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+    [FiniteDimensional ℂ V]
+def hsNormSq (A : Module.End ℂ V) : ℝ :=
+  (LinearMap.trace ℂ V (LinearMap.adjoint A * A)).re
+
+end
+
+def complexFourier {G V : Type*} [Group G] [Fintype G] [AddCommGroup V] [Module ℂ V]
+    (ρ : Representation ℂ G V) (f : G → ℂ) : Module.End ℂ V := ∑ g, f g • ρ g
+
+section
+variable {G V : Type*} [Group G] [Fintype G] [AddCommGroup V] [Module ℂ V]
+    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V)
+noncomputable def componentSimple (c : isotypicComponents ℂ[G] ρ.asModule) :
+    Submodule ℂ[G] ρ.asModule := c.property.choose
+
+instance components_finite : Finite (isotypicComponents ℂ[G] ρ.asModule) := by
+  let : IsNoetherian ℂ[G] ρ.asModule := isNoetherian_of_tower ℂ inferInstance
+  infer_instance
+
+end
+
+section
+variable (G : Type*) [Group G] [Fintype G]
+def regularRep : Representation ℂ G (EuclideanSpace ℂ G) where
+  toFun g :=
+    { toFun := fun f => WithLp.toLp 2 (fun h => f (g⁻¹ * h))
+      map_add' := by intros; rfl
+      map_smul' := by intros; rfl }
+  map_one' := by ext f h; simp
+  map_mul' := by
+    intro x y
+    ext f h
+    change f ((x * y)⁻¹ * h) = f (y⁻¹ * (x⁻¹ * h))
+    simp [mul_assoc]
+
+end
+
+section
+variable {G : Type*} [Group G] [Fintype G]
+abbrev IrreducibleIndex (G : Type*) [Group G] [Fintype G] :=
+  isotypicComponents ℂ[G] (regularRep G).asModule
+
+noncomputable instance irreducibleIndex_fintype : Fintype (IrreducibleIndex G) :=
+  Fintype.ofFinite _
+
+abbrev irreducibleSpace (c : IrreducibleIndex G) :=
+  (Subrepresentation.ofSubmodule' (componentSimple (regularRep G) c)).toSubmodule
+
+abbrev irreducibleDegree (c : IrreducibleIndex G) := Module.finrank ℂ (irreducibleSpace c)
+
+end
+
+def mass {G : Type*} [Fintype G] (f : G → ℝ) : ℝ := ∑ g, f g
+
+def LeftCosetCap {G : Type*} [Group G] [Fintype G]
+    (f : G → ℝ) (H : Subgroup G) (B : ℝ) : Prop :=
+  by classical exact ∀ g, ∑ h : H, f (g * h) ≤ B / H.index
+
+def blockSubgroup {X : Type*} (M : Set X) : Subgroup (Equiv.Perm X) where
+  carrier := {g | ∀ x, x ∉ M → g x = x}
+  one_mem' := by simp
+  mul_mem' := by
+    intro g h hg hh x hx
+    simp only [Set.mem_ofPred_eq] at hg hh ⊢
+    simp [Equiv.Perm.mul_apply, hh x hx, hg x hx]
+  inv_mem' := by
+    intro g hg x hx
+    exact ((Equiv.eq_symm_apply g).mpr (hg x hx)).symm
+
+def inverseDegreeSum (G : Type*) [Group G] [Fintype G] (u : ℝ) : ℝ :=
+  ∑ c : IrreducibleIndex G, (irreducibleDegree c : ℝ)^(-u)
+
+def symmetricInverseDegreeSum (n : ℕ) (u : ℝ) : ℝ :=
+  inverseDegreeSum (Equiv.Perm (Fin n)) u
+
+def symmetricDegreeConstant (u : ℝ) : ℝ :=
+  sSup {x : ℝ | ∃ n : ℕ, 1 ≤ n ∧ x = symmetricInverseDegreeSum n u}
+
+def ComplementaryCosetsFiniteStatement : Prop :=
+  ∀ (X : Type*) [Fintype X] [DecidableEq X] (b : ℕ) (_ : 0 < b) (M : Fin b → Set X),
+    (∀ i, (M i).Nonempty) →
+    Pairwise (fun i j => Disjoint (M i) (M j)) → (⋃ i, M i) = Set.univ →
+    ∀ (V : Type) [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+      [FiniteDimensional ℂ V] (ρ : Representation ℂ (Equiv.Perm X) V),
+      Representation.IsIrreducible ρ → IsUnitary ρ →
+      ∀ (f : Equiv.Perm X → ℝ) (B u : ℝ),
+      (∀ g, 0 ≤ f g) → mass f ≤ 1 → 0 < u →
+      (∀ i, LeftCosetCap f (blockSubgroup (M i)) B) →
+      (hsNormSq (complexFourier ρ (fun g => f g)) ≤
+        b * B * symmetricDegreeConstant u *
+          (Module.finrank ℂ V : ℝ)^(-1+(u+2)/(b : ℝ))) ∧
+      (‖LinearMap.toContinuousLinearMap (complexFourier ρ (fun g => f g))‖^2 ≤
+        b * B * symmetricDegreeConstant u *
+          (Module.finrank ℂ V : ℝ)^(-1+(u+2)/(b : ℝ)))
+
+end
+end PartialPermutation
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/ModuleTransport.lean b/lean/OAI/RepresentationTheory/PartialPermutation/ModuleTransport.lean
index f1c1d00154070086577fb0491f832f85cb4d197d..bb23e07ccf60ff7c025397f93dd38ea85a563707 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/ModuleTransport.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/ModuleTransport.lean
@@ -1,90 +1,90 @@
-import Mathlib
-
-namespace OAI
-
-section
-open scoped Classical
-open scoped BigOperators ComplexConjugate MonoidAlgebra
-open scoped BigOperators ComplexConjugate
-open scoped MonoidAlgebra BigOperators
-
-namespace PartialPermutation
-noncomputable section
-
-section ModuleTransport
-variable {G V : Type*} [Group G] [AddCommGroup V] [Module ℂ V]
-    (ρ : Representation ℂ G V)
-
-noncomputable def subrepModuleEquiv (σ : Subrepresentation ρ) :
-    σ.toRepresentation.asModule ≃ₗ[ℂ[G]] σ.asSubmodule where
-  toFun x := ⟨x.1, x.2⟩
-  invFun x := ⟨x.1, x.2⟩
-  left_inv _ := rfl
-  right_inv _ := rfl
-  map_add' _ _ := rfl
-  map_smul' a x := by
-    apply Subtype.ext
-    change ((σ.toRepresentation.asAlgebraHom a) x).val =
-      (ρ.asAlgebraHom a) x.val
-    induction a using MonoidAlgebra.induction_linear with
-    | zero => simp only [map_zero]; rfl
-    | add a b ha hb =>
-      simp only [map_add]
-      change ((σ.toRepresentation.asAlgebraHom a) x).val +
-        ((σ.toRepresentation.asAlgebraHom b) x).val = _ + _
-      rw [ha, hb]
-    | single g c =>
-      rw [Representation.asAlgebraHom_single, Representation.asAlgebraHom_single]
-      rfl
-
-lemma subrep_irreducible_iff (σ : Subrepresentation ρ) :
-    Representation.IsIrreducible σ.toRepresentation ↔ IsSimpleModule ℂ[G] σ.asSubmodule := by
-  rw [Representation.irreducible_iff_isSimpleModule_asModule]
-  exact ⟨fun _ => IsSimpleModule.congr (subrepModuleEquiv ρ σ).symm,
-    fun _ => IsSimpleModule.congr (subrepModuleEquiv ρ σ)⟩
-
-noncomputable def subrepEquivOfModuleEquiv (σ τ : Subrepresentation ρ)
-    (e : σ.asSubmodule ≃ₗ[ℂ[G]] τ.asSubmodule) :
-    Representation.Equiv σ.toRepresentation τ.toRepresentation := by
-  let e' := (subrepModuleEquiv ρ σ).trans (e.trans (subrepModuleEquiv ρ τ).symm)
-  exact ((Representation.IntertwiningMap.equivLinearMapAsModule _ _).symm
-    e'.toLinearMap).ofBijective e'.bijective
-
-noncomputable def moduleEquivOfSubrepEquiv (σ τ : Subrepresentation ρ)
-    (e : Representation.Equiv σ.toRepresentation τ.toRepresentation) :
-    σ.asSubmodule ≃ₗ[ℂ[G]] τ.asSubmodule := by
-  let f := Representation.IntertwiningMap.equivLinearMapAsModule _ _ e.toIntertwiningMap
-  let e' := LinearEquiv.ofBijective f e.bijective
-  exact (subrepModuleEquiv ρ σ).symm.trans (e'.trans (subrepModuleEquiv ρ τ))
-
-end ModuleTransport
-
-lemma irreducible_equiv_of_character_eq {G V W : Type*} [Group G] [Fintype G]
-    [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
-    [AddCommGroup W] [Module ℂ W] [FiniteDimensional ℂ W]
-    (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
-    [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ]
-    (h : ρ.character = σ.character) : Nonempty (Representation.Equiv σ ρ) := by
-  classical
-  let : Invertible (Nat.card G : ℂ) := invertibleOfNonzero (by
-    exact_mod_cast (Nat.card_pos (α := G)).ne')
-  have hself := Representation.char_orthonormal ρ ρ
-  have hother := Representation.char_orthonormal ρ σ
-  rw [h] at hself
-  rw [h] at hother
-  have he : Nonempty (ρ.Equiv ρ) := ⟨Representation.Equiv.refl ρ⟩
-  simp only [he, ite_eq_left] at hself
-  by_contra hn
-  simp only [hn] at hother
-  exact one_ne_zero (hself.symm.trans hother)
-
-def OccurringTypes {G V : Type*} [Group G] [AddCommGroup V] [Module ℂ V]
-    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V) :=
-  {χ : G → ℂ // ∃ σ : Subrepresentation ρ,
-    Representation.IsIrreducible σ.toRepresentation ∧ χ = σ.toRepresentation.character}
-
-end
-end PartialPermutation
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+section
+open scoped Classical
+open scoped BigOperators ComplexConjugate MonoidAlgebra
+open scoped BigOperators ComplexConjugate
+open scoped MonoidAlgebra BigOperators
+
+namespace PartialPermutation
+noncomputable section
+
+section ModuleTransport
+variable {G V : Type*} [Group G] [AddCommGroup V] [Module ℂ V]
+    (ρ : Representation ℂ G V)
+
+noncomputable def subrepModuleEquiv (σ : Subrepresentation ρ) :
+    σ.toRepresentation.asModule ≃ₗ[ℂ[G]] σ.asSubmodule where
+  toFun x := ⟨x.1, x.2⟩
+  invFun x := ⟨x.1, x.2⟩
+  left_inv _ := rfl
+  right_inv _ := rfl
+  map_add' _ _ := rfl
+  map_smul' a x := by
+    apply Subtype.ext
+    change ((σ.toRepresentation.asAlgebraHom a) x).val =
+      (ρ.asAlgebraHom a) x.val
+    induction a using MonoidAlgebra.induction_linear with
+    | zero => simp only [map_zero]; rfl
+    | add a b ha hb =>
+      simp only [map_add]
+      change ((σ.toRepresentation.asAlgebraHom a) x).val +
+        ((σ.toRepresentation.asAlgebraHom b) x).val = _ + _
+      rw [ha, hb]
+    | single g c =>
+      rw [Representation.asAlgebraHom_single, Representation.asAlgebraHom_single]
+      rfl
+
+lemma subrep_irreducible_iff (σ : Subrepresentation ρ) :
+    Representation.IsIrreducible σ.toRepresentation ↔ IsSimpleModule ℂ[G] σ.asSubmodule := by
+  rw [Representation.irreducible_iff_isSimpleModule_asModule]
+  exact ⟨fun _ => IsSimpleModule.congr (subrepModuleEquiv ρ σ).symm,
+    fun _ => IsSimpleModule.congr (subrepModuleEquiv ρ σ)⟩
+
+noncomputable def subrepEquivOfModuleEquiv (σ τ : Subrepresentation ρ)
+    (e : σ.asSubmodule ≃ₗ[ℂ[G]] τ.asSubmodule) :
+    Representation.Equiv σ.toRepresentation τ.toRepresentation := by
+  let e' := (subrepModuleEquiv ρ σ).trans (e.trans (subrepModuleEquiv ρ τ).symm)
+  exact ((Representation.IntertwiningMap.equivLinearMapAsModule _ _).symm
+    e'.toLinearMap).ofBijective e'.bijective
+
+noncomputable def moduleEquivOfSubrepEquiv (σ τ : Subrepresentation ρ)
+    (e : Representation.Equiv σ.toRepresentation τ.toRepresentation) :
+    σ.asSubmodule ≃ₗ[ℂ[G]] τ.asSubmodule := by
+  let f := Representation.IntertwiningMap.equivLinearMapAsModule _ _ e.toIntertwiningMap
+  let e' := LinearEquiv.ofBijective f e.bijective
+  exact (subrepModuleEquiv ρ σ).symm.trans (e'.trans (subrepModuleEquiv ρ τ))
+
+end ModuleTransport
+
+lemma irreducible_equiv_of_character_eq {G V W : Type*} [Group G] [Fintype G]
+    [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
+    [AddCommGroup W] [Module ℂ W] [FiniteDimensional ℂ W]
+    (ρ : Representation ℂ G V) (σ : Representation ℂ G W)
+    [Representation.IsIrreducible ρ] [Representation.IsIrreducible σ]
+    (h : ρ.character = σ.character) : Nonempty (Representation.Equiv σ ρ) := by
+  classical
+  let : Invertible (Nat.card G : ℂ) := invertibleOfNonzero (by
+    exact_mod_cast (Nat.card_pos (α := G)).ne')
+  have hself := Representation.char_orthonormal ρ ρ
+  have hother := Representation.char_orthonormal ρ σ
+  rw [h] at hself
+  rw [h] at hother
+  have he : Nonempty (ρ.Equiv ρ) := ⟨Representation.Equiv.refl ρ⟩
+  simp only [he, ite_eq_left] at hself
+  by_contra hn
+  simp only [hn] at hother
+  exact one_ne_zero (hself.symm.trans hother)
+
+def OccurringTypes {G V : Type*} [Group G] [AddCommGroup V] [Module ℂ V]
+    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V) :=
+  {χ : G → ℂ // ∃ σ : Subrepresentation ρ,
+    Representation.IsIrreducible σ.toRepresentation ∧ χ = σ.toRepresentation.character}
+
+end
+end PartialPermutation
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/NormBounds.lean b/lean/OAI/RepresentationTheory/PartialPermutation/NormBounds.lean
index b50e82544225cc86997d24fd7e6de680f2538c5f..78809090027beeb335b74a71aa9805ba83415deb 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/NormBounds.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/NormBounds.lean
@@ -1,52 +1,52 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.ReciprocalBounds
-import OAI.RepresentationTheory.PartialPermutation.OperatorNorm
-
-namespace OAI
-
-namespace PartialPermutation
-noncomputable section
-open scoped Classical
-
-theorem complementaryCosets : ComplementaryCosetsStatement := by
-  intro n b hb M hnonempty hdis hcover V _ _ _ ρ hirr hρ f B u hf hm hu hcap
-  let := hirr
-  have hB : 0 ≤ B := by
-    let i : Fin b := ⟨0,hb⟩
-    have hp : (0:ℝ) < (blockSubgroup (M i)).index := by
-      exact_mod_cast Nat.pos_of_ne_zero (blockSubgroup (M i)).index_ne_zero_of_finite
-    have hz : 0 ≤ B / (blockSubgroup (M i)).index :=
-      (Finset.sum_nonneg (fun (g : blockSubgroup (M i)) _ => hf (1*g))).trans (hcap i 1)
-    simpa only [div_mul_cancel₀ B hp.ne'] using mul_nonneg hz hp.le
-  have hsum : (∑ i, symmetricInverseDegreeSum (Fintype.card (M i)) u) ≤
-      (b:ℝ)*symmetricDegreeConstant u := by
-    calc
-      _ ≤ ∑ _i : Fin b, symmetricDegreeConstant u := by
-        apply Finset.sum_le_sum
-        intro i _
-        let : Nonempty (M i) := (hnonempty i).to_subtype
-        exact symmetricInverseDegreeSum_le_constant (by exact Fintype.card_pos) hu
-      _ = _ := by simp
-  have hs : 0 ≤ ∑ i, symmetricInverseDegreeSum (Fintype.card (M i)) u := by
-    exact Finset.sum_nonneg (fun i _ => inverseDegreeSum_nonneg _ _)
-  have hR : 0 ≤ (Module.finrank ℂ V : ℝ)^(-1+(u+2)/(b:ℝ)) := Real.rpow_nonneg (by positivity) _
-  have hcoef : B * mass f * (Module.finrank ℂ V:ℝ)^(-1+(u+2)/(b:ℝ)) ≤
-      B * (Module.finrank ℂ V:ℝ)^(-1+(u+2)/(b:ℝ)) := by
-    calc
-      _ ≤ B * 1 * (Module.finrank ℂ V:ℝ)^(-1+(u+2)/(b:ℝ)) :=
-        mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left hm hB) hR
-      _ = _ := by ring
-  have hHS : hsNormSq (complexFourier ρ (fun g => f g)) ≤
-      b * B * symmetricDegreeConstant u * (Module.finrank ℂ V:ℝ)^(-1+(u+2)/(b:ℝ)) := by
-    calc
-      _ ≤ _ := symmetric_block_degree_transfer hb M hdis ρ hρ f B u hf hu hcap
-      _ ≤ (B * (Module.finrank ℂ V:ℝ)^(-1+(u+2)/(b:ℝ))) *
-          ((b:ℝ)*symmetricDegreeConstant u) :=
-        mul_le_mul hcoef hsum hs (mul_nonneg hB hR)
-      _ = _ := by ring
-  exact ⟨hHS,(opNorm_sq_le_hsNormSq _).trans hHS⟩
-
-end
-end PartialPermutation
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.ReciprocalBounds
+import OAI.RepresentationTheory.PartialPermutation.OperatorNorm
+
+namespace OAI
+
+namespace PartialPermutation
+noncomputable section
+open scoped Classical
+
+theorem complementaryCosets : ComplementaryCosetsStatement := by
+  intro n b hb M hnonempty hdis hcover V _ _ _ ρ hirr hρ f B u hf hm hu hcap
+  let := hirr
+  have hB : 0 ≤ B := by
+    let i : Fin b := ⟨0,hb⟩
+    have hp : (0:ℝ) < (blockSubgroup (M i)).index := by
+      exact_mod_cast Nat.pos_of_ne_zero (blockSubgroup (M i)).index_ne_zero_of_finite
+    have hz : 0 ≤ B / (blockSubgroup (M i)).index :=
+      (Finset.sum_nonneg (fun (g : blockSubgroup (M i)) _ => hf (1*g))).trans (hcap i 1)
+    simpa only [div_mul_cancel₀ B hp.ne'] using mul_nonneg hz hp.le
+  have hsum : (∑ i, symmetricInverseDegreeSum (Fintype.card (M i)) u) ≤
+      (b:ℝ)*symmetricDegreeConstant u := by
+    calc
+      _ ≤ ∑ _i : Fin b, symmetricDegreeConstant u := by
+        apply Finset.sum_le_sum
+        intro i _
+        let : Nonempty (M i) := (hnonempty i).to_subtype
+        exact symmetricInverseDegreeSum_le_constant (by exact Fintype.card_pos) hu
+      _ = _ := by simp
+  have hs : 0 ≤ ∑ i, symmetricInverseDegreeSum (Fintype.card (M i)) u := by
+    exact Finset.sum_nonneg (fun i _ => inverseDegreeSum_nonneg _ _)
+  have hR : 0 ≤ (Module.finrank ℂ V : ℝ)^(-1+(u+2)/(b:ℝ)) := Real.rpow_nonneg (by positivity) _
+  have hcoef : B * mass f * (Module.finrank ℂ V:ℝ)^(-1+(u+2)/(b:ℝ)) ≤
+      B * (Module.finrank ℂ V:ℝ)^(-1+(u+2)/(b:ℝ)) := by
+    calc
+      _ ≤ B * 1 * (Module.finrank ℂ V:ℝ)^(-1+(u+2)/(b:ℝ)) :=
+        mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left hm hB) hR
+      _ = _ := by ring
+  have hHS : hsNormSq (complexFourier ρ (fun g => f g)) ≤
+      b * B * symmetricDegreeConstant u * (Module.finrank ℂ V:ℝ)^(-1+(u+2)/(b:ℝ)) := by
+    calc
+      _ ≤ _ := symmetric_block_degree_transfer hb M hdis ρ hρ f B u hf hu hcap
+      _ ≤ (B * (Module.finrank ℂ V:ℝ)^(-1+(u+2)/(b:ℝ))) *
+          ((b:ℝ)*symmetricDegreeConstant u) :=
+        mul_le_mul hcoef hsum hs (mul_nonneg hB hR)
+      _ = _ := by ring
+  exact ⟨hHS,(opNorm_sq_le_hsNormSq _).trans hHS⟩
+
+end
+end PartialPermutation
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/OperatorNorm.lean b/lean/OAI/RepresentationTheory/PartialPermutation/OperatorNorm.lean
index 4304b457855706ee97988d4d042f45f561076322..fca7d6921b13a859615aa61b25a350e5265b961c 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/OperatorNorm.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/OperatorNorm.lean
@@ -1,39 +1,39 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.HilbertSchmidt
-
-namespace OAI
-
-namespace PartialPermutation
-noncomputable section
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-    [FiniteDimensional ℂ V]
-
-lemma norm_apply_sq_le_hsNormSq (A : Module.End ℂ V) (x : V) :
-    ‖A x‖^2 ≤ hsNormSq A * ‖x‖^2 := by
-  let e := stdOrthonormalBasis ℂ V
-  calc
-    ‖A x‖^2 = ∑ i, ‖inner ℂ (e i) (A x)‖^2 := (e.sum_sq_norm_inner_right _).symm
-    _ ≤ ∑ i, ‖LinearMap.adjoint A (e i)‖^2 * ‖x‖^2 := by
-      apply Finset.sum_le_sum
-      intro i _
-      rw [← LinearMap.adjoint_inner_left A]
-      simpa only [mul_pow] using pow_le_pow_left₀ (norm_nonneg (inner ℂ (LinearMap.adjoint A (e i)) x))
-        (norm_inner_le_norm (𝕜 := ℂ) (LinearMap.adjoint A (e i)) x) 2
-    _ = hsNormSq A * ‖x‖^2 := by
-      rw [← Finset.sum_mul, ← hsNormSq_eq_sum e, hsNormSq_adjoint]
-
-lemma opNorm_sq_le_hsNormSq (A : Module.End ℂ V) :
-    ‖LinearMap.toContinuousLinearMap A‖^2 ≤ hsNormSq A := by
-  have h : ‖LinearMap.toContinuousLinearMap A‖ ≤ Real.sqrt (hsNormSq A) := by
-    apply ContinuousLinearMap.opNorm_le_bound _ (Real.sqrt_nonneg _)
-    intro x
-    apply (sq_le_sq₀ (norm_nonneg _) (by positivity)).mp
-    rw [mul_pow, Real.sq_sqrt (hsNormSq_nonneg A)]
-    exact norm_apply_sq_le_hsNormSq A x
-  simpa only [Real.sq_sqrt (hsNormSq_nonneg A)] using
-    pow_le_pow_left₀ (norm_nonneg _) h 2
-
-end
-end PartialPermutation
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.HilbertSchmidt
+
+namespace OAI
+
+namespace PartialPermutation
+noncomputable section
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+    [FiniteDimensional ℂ V]
+
+lemma norm_apply_sq_le_hsNormSq (A : Module.End ℂ V) (x : V) :
+    ‖A x‖^2 ≤ hsNormSq A * ‖x‖^2 := by
+  let e := stdOrthonormalBasis ℂ V
+  calc
+    ‖A x‖^2 = ∑ i, ‖inner ℂ (e i) (A x)‖^2 := (e.sum_sq_norm_inner_right _).symm
+    _ ≤ ∑ i, ‖LinearMap.adjoint A (e i)‖^2 * ‖x‖^2 := by
+      apply Finset.sum_le_sum
+      intro i _
+      rw [← LinearMap.adjoint_inner_left A]
+      simpa only [mul_pow] using pow_le_pow_left₀ (norm_nonneg (inner ℂ (LinearMap.adjoint A (e i)) x))
+        (norm_inner_le_norm (𝕜 := ℂ) (LinearMap.adjoint A (e i)) x) 2
+    _ = hsNormSq A * ‖x‖^2 := by
+      rw [← Finset.sum_mul, ← hsNormSq_eq_sum e, hsNormSq_adjoint]
+
+lemma opNorm_sq_le_hsNormSq (A : Module.End ℂ V) :
+    ‖LinearMap.toContinuousLinearMap A‖^2 ≤ hsNormSq A := by
+  have h : ‖LinearMap.toContinuousLinearMap A‖ ≤ Real.sqrt (hsNormSq A) := by
+    apply ContinuousLinearMap.opNorm_le_bound _ (Real.sqrt_nonneg _)
+    intro x
+    apply (sq_le_sq₀ (norm_nonneg _) (by positivity)).mp
+    rw [mul_pow, Real.sq_sqrt (hsNormSq_nonneg A)]
+    exact norm_apply_sq_le_hsNormSq A x
+  simpa only [Real.sq_sqrt (hsNormSq_nonneg A)] using
+    pow_le_pow_left₀ (norm_nonneg _) h 2
+
+end
+end PartialPermutation
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/PartitionAsymptotics.lean b/lean/OAI/RepresentationTheory/PartialPermutation/PartitionAsymptotics.lean
index 6b8a03d50e37126591e1e94e2764ccce3999f78d..a78ab6accf02fd1e1a041dc00cb8120a9ccc4399 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/PartitionAsymptotics.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/PartitionAsymptotics.lean
@@ -1,53 +1,53 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.DiagramCounts
-
-namespace OAI
-
-section
-namespace PartialPermutation
-namespace DiagramCounting
-noncomputable section
-open Finset
-open Filter in
-
-lemma partitionCount_subexponential (δ : ℝ) (hδ : 0 < δ) :
-    ∀ᶠ n : ℕ in atTop, (partitionCount n : ℝ) ≤ Real.exp (δ*n) := by
-  have hl := (isLittleO_log_rpow_atTop (show (0:ℝ)<1/2 by norm_num)).bound
-    (show 0 < δ/4 by positivity)
-  have ht : Tendsto (fun n : ℕ => (n:ℝ)+1) atTop atTop :=
-    tendsto_atTop_mono (fun n => le_add_of_nonneg_right (show (0:ℝ)≤1 by norm_num))
-      tendsto_natCast_atTop_atTop
-  filter_upwards [ht.eventually hl, eventually_ge_atTop 1] with n hn hn1
-  have hn0 : (0:ℝ) ≤ n := Nat.cast_nonneg n
-  have hn1' : (1:ℝ) ≤ n := by exact_mod_cast hn1
-  have hs : 0 ≤ Real.sqrt (n+1) := Real.sqrt_nonneg _
-  have hsq := Real.sq_sqrt (show (0:ℝ)≤n+1 by positivity)
-  have hl0 : 0 ≤ Real.log ((n:ℝ)+1) := Real.log_nonneg (by linarith)
-  rw [Real.norm_eq_abs,Real.norm_eq_abs,abs_of_nonneg hl0,
-    ← Real.sqrt_eq_rpow,abs_of_nonneg hs] at hn
-  have hnrt : (n.sqrt:ℝ) ≤ Real.sqrt (n+1) := by
-    have h := Nat.sqrt_le n
-    have hr : (n.sqrt:ℝ)^2 ≤ n := by simpa only [pow_two] using (show (n.sqrt:ℝ)*(n.sqrt:ℝ)≤n by exact_mod_cast h)
-    nlinarith [show (0:ℝ)≤n.sqrt by positivity]
-  have hexp : (2*(n.sqrt:ℝ))*Real.log ((n:ℝ)+1) ≤ δ*n := by
-    calc
-      _ ≤ (2*Real.sqrt (n+1))*((δ/4)*Real.sqrt (n+1)) :=
-        mul_le_mul (by linarith) hn hl0 (by positivity)
-      _ = δ/2*((n:ℝ)+1) := by nlinarith
-      _ ≤ δ*n := by nlinarith
-  calc
-    (partitionCount n : ℝ) ≤ ((n+1)^(2*n.sqrt):ℕ) := by
-      exact_mod_cast partitionCount_le_sqrt n
-    _ = Real.exp ((2*(n.sqrt:ℝ))*Real.log ((n:ℝ)+1)) := by
-      push_cast
-      rw [show (2:ℝ)*(n.sqrt:ℝ)=((2*n.sqrt:ℕ):ℝ) by push_cast; rfl,
-        Real.exp_nat_mul, Real.exp_log (by positivity)]
-    _ ≤ Real.exp (δ*n) := Real.exp_le_exp.mpr hexp
-
-end
-end DiagramCounting
-end PartialPermutation
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.DiagramCounts
+
+namespace OAI
+
+section
+namespace PartialPermutation
+namespace DiagramCounting
+noncomputable section
+open Finset
+open Filter in
+
+lemma partitionCount_subexponential (δ : ℝ) (hδ : 0 < δ) :
+    ∀ᶠ n : ℕ in atTop, (partitionCount n : ℝ) ≤ Real.exp (δ*n) := by
+  have hl := (isLittleO_log_rpow_atTop (show (0:ℝ)<1/2 by norm_num)).bound
+    (show 0 < δ/4 by positivity)
+  have ht : Tendsto (fun n : ℕ => (n:ℝ)+1) atTop atTop :=
+    tendsto_atTop_mono (fun n => le_add_of_nonneg_right (show (0:ℝ)≤1 by norm_num))
+      tendsto_natCast_atTop_atTop
+  filter_upwards [ht.eventually hl, eventually_ge_atTop 1] with n hn hn1
+  have hn0 : (0:ℝ) ≤ n := Nat.cast_nonneg n
+  have hn1' : (1:ℝ) ≤ n := by exact_mod_cast hn1
+  have hs : 0 ≤ Real.sqrt (n+1) := Real.sqrt_nonneg _
+  have hsq := Real.sq_sqrt (show (0:ℝ)≤n+1 by positivity)
+  have hl0 : 0 ≤ Real.log ((n:ℝ)+1) := Real.log_nonneg (by linarith)
+  rw [Real.norm_eq_abs,Real.norm_eq_abs,abs_of_nonneg hl0,
+    ← Real.sqrt_eq_rpow,abs_of_nonneg hs] at hn
+  have hnrt : (n.sqrt:ℝ) ≤ Real.sqrt (n+1) := by
+    have h := Nat.sqrt_le n
+    have hr : (n.sqrt:ℝ)^2 ≤ n := by simpa only [pow_two] using (show (n.sqrt:ℝ)*(n.sqrt:ℝ)≤n by exact_mod_cast h)
+    nlinarith [show (0:ℝ)≤n.sqrt by positivity]
+  have hexp : (2*(n.sqrt:ℝ))*Real.log ((n:ℝ)+1) ≤ δ*n := by
+    calc
+      _ ≤ (2*Real.sqrt (n+1))*((δ/4)*Real.sqrt (n+1)) :=
+        mul_le_mul (by linarith) hn hl0 (by positivity)
+      _ = δ/2*((n:ℝ)+1) := by nlinarith
+      _ ≤ δ*n := by nlinarith
+  calc
+    (partitionCount n : ℝ) ≤ ((n+1)^(2*n.sqrt):ℕ) := by
+      exact_mod_cast partitionCount_le_sqrt n
+    _ = Real.exp ((2*(n.sqrt:ℝ))*Real.log ((n:ℝ)+1)) := by
+      push_cast
+      rw [show (2:ℝ)*(n.sqrt:ℝ)=((2*n.sqrt:ℕ):ℝ) by push_cast; rfl,
+        Real.exp_nat_mul, Real.exp_log (by positivity)]
+    _ ≤ Real.exp (δ*n) := Real.exp_le_exp.mpr hexp
+
+end
+end DiagramCounting
+end PartialPermutation
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/Plancherel.lean b/lean/OAI/RepresentationTheory/PartialPermutation/Plancherel.lean
index 4d924fa91a757efdfdceb6771041041c584962c3..c634ad41e98befeecaa3164589271a02143e5f0a 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/Plancherel.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/Plancherel.lean
@@ -1,151 +1,151 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.Characters
-import OAI.RepresentationTheory.PartialPermutation.CompleteKernels
-import OAI.RepresentationTheory.PartialPermutation.RegularRepresentation
-
-namespace OAI
-
-section
-open scoped Classical
-open scoped BigOperators ComplexConjugate MonoidAlgebra
-open scoped BigOperators ComplexConjugate
-open scoped MonoidAlgebra BigOperators
-open scoped BigOperators MonoidAlgebra Classical
-
-namespace PartialPermutation
-noncomputable section
-open scoped BigOperators MonoidAlgebra Classical
-
-variable {G : Type*} [Group G] [Fintype G]
-
-omit [Fintype G] in
-lemma componentRep_unitary {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
-    (c : isotypicComponents ℂ[G] ρ.asModule) : IsUnitary (componentRep ρ c) := by
-  intro g x y
-  exact hρ g x.val y.val
-
-lemma complexFourier_regular_delta (a : G → ℂ) (g : G) :
-    complexFourier (regularRep G) a (deltaOne G) g = a g := by
-  simp [complexFourier, deltaOne, inv_mul_eq_one, eq_comm]
-
-abbrev irreducibleRep (c : IrreducibleIndex G) := componentRep (regularRep G) c
-
-lemma irreducibleRep_unitary (c : IrreducibleIndex G) :
-    IsUnitary (G := G) (V := irreducibleSpace c) (irreducibleRep c) :=
-  componentRep_unitary (regularRep G) (regularRep_unitary G) c
-
-lemma regular_character_identity (g : G) :
-    ∑ c : IrreducibleIndex G, (irreducibleDegree c : ℂ) * (irreducibleRep c).character g =
-      if g = 1 then (Fintype.card G : ℂ) else 0 := by
-  have ht := congrArg (fun A : Module.End ℂ (EuclideanSpace ℂ G) => A (deltaOne G) g⁻¹)
-    (sum_character_kernels_identity (regularRep G))
-  rw [complexFourier_regular_delta] at ht
-  change (∑ c : IrreducibleIndex G, centralCharacterKernel (irreducibleRep c) g⁻¹) =
-    (deltaOne G) g⁻¹ at ht
-  simp only [centralCharacterKernel, inv_inv] at ht
-  have hN : (Fintype.card G : ℂ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
-  have ht' := congrArg (fun z : ℂ => (Fintype.card G : ℂ) * z) ht
-  simp only [Finset.mul_sum] at ht'
-  convert ht' using 1
-  · apply Finset.sum_congr rfl
-    intro c _
-    change _ = (Fintype.card G : ℂ) *
-      ((irreducibleDegree c : ℂ) / Fintype.card G * (irreducibleRep c).character g)
-    field_simp
-  · change (if g = 1 then (Fintype.card G : ℂ) else 0) =
-      (Fintype.card G : ℂ) * (if g⁻¹ = 1 then 1 else 0)
-    simp only [inv_eq_one]
-    split_ifs <;> simp
-
-lemma sum_regular_character_kernels (g : G) :
-    ∑ c : IrreducibleIndex G, centralCharacterKernel (irreducibleRep c) g =
-      if g = 1 then 1 else 0 := by
-  have ht := congrArg (fun A : Module.End ℂ (EuclideanSpace ℂ G) => A (deltaOne G) g)
-    (sum_character_kernels_identity (regularRep G))
-  rw [complexFourier_regular_delta] at ht
-  exact ht
-
-lemma irreducibleRep_complete {V : Type*} [AddCommGroup V] [Module ℂ V]
-    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ] :
-    ∃! c : IrreducibleIndex G, Nonempty (Representation.Equiv (irreducibleRep c) ρ) := by
-  have : Nontrivial V := IsSimpleModule.nontrivial ℂ[G] ρ.asModule
-  have hex : ∃ c : IrreducibleIndex G, Nonempty (Representation.Equiv (irreducibleRep c) ρ) := by
-    by_contra hn
-    push Not at hn
-    have hs : complexFourier ρ (fun g => ∑ c : IrreducibleIndex G,
-        centralCharacterKernel (irreducibleRep c) g) = 1 := by
-      simp only [sum_regular_character_kernels]
-      simp [complexFourier]
-    rw [complexFourier_sum] at hs
-    have hz : (∑ c : IrreducibleIndex G, complexFourier ρ
-        (centralCharacterKernel (irreducibleRep c))) = 0 := by
-      apply Finset.sum_eq_zero
-      intro c _
-      rw [centralCharacterKernel_fourier, ite_eq_right (fun ⟨e⟩ => (hn c).false e)]
-    rw [hz] at hs
-    exact zero_ne_one hs
-  obtain ⟨c, hc⟩ := hex
-  refine ⟨c, hc, ?_⟩
-  intro d hd
-  obtain ⟨e⟩ := hc
-  obtain ⟨f⟩ := hd
-  exact (componentRep_equiv_iff (regularRep G) d c).mp ⟨f.trans e.symm⟩
-
-lemma trace_fourier_energy {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (a : G → ℂ) :
-    LinearMap.trace ℂ V (LinearMap.adjoint (complexFourier ρ a) * complexFourier ρ a) =
-      ∑ h, ∑ g, conj (a h) * a g * ρ.character (h⁻¹ * g) := by
-  have hadj : LinearMap.adjoint (complexFourier ρ a) =
-      ∑ h, conj (a h) • ρ h⁻¹ := by
-    simp only [complexFourier, map_sum, map_smulₛₗ, unitary_adjoint ρ hρ]
-  rw [hadj, complexFourier, Finset.sum_mul]
-  simp only [map_sum, Finset.mul_sum, smul_mul_assoc, mul_smul_comm, map_smul,
-    smul_eq_mul, ← map_mul, Representation.character]
-  apply Finset.sum_congr rfl
-  intro h _
-  apply Finset.sum_congr rfl
-  intro g _
-  ring
-
-theorem finite_group_plancherel (a : G → ℂ) :
-    (Fintype.card G : ℝ) * ∑ g, ‖a g‖^2 =
-      ∑ c : IrreducibleIndex G, (irreducibleDegree c : ℝ) *
-        hsNormSq (V := irreducibleSpace c) (complexFourier (V := irreducibleSpace c) (irreducibleRep c) a) := by
-  have he : ∑ c : IrreducibleIndex G, (irreducibleDegree c : ℂ) *
-      LinearMap.trace ℂ (irreducibleSpace c)
-        (LinearMap.adjoint (𝕜 := ℂ) (E := irreducibleSpace c) (F := irreducibleSpace c) (complexFourier (V := irreducibleSpace c) (irreducibleRep c) a) *
-          complexFourier (V := irreducibleSpace c) (irreducibleRep c) a) =
-        (Fintype.card G : ℂ) * ∑ g, conj (a g) * a g := by
-    have htrace (c : IrreducibleIndex G) := trace_fourier_energy
-      (V := irreducibleSpace c) (irreducibleRep c) (irreducibleRep_unitary c) a
-    simp only [htrace, Finset.mul_sum]
-    rw [Finset.sum_comm]
-    calc
-      _ = ∑ h, ∑ g, conj (a h) * a g *
-          (∑ c : IrreducibleIndex G, (irreducibleDegree c : ℂ) *
-            (irreducibleRep c).character (h⁻¹ * g)) := by
-        apply Finset.sum_congr rfl
-        intro h _
-        rw [Finset.sum_comm]
-        apply Finset.sum_congr rfl
-        intro g _
-        rw [Finset.mul_sum]
-        apply Finset.sum_congr rfl
-        intro c _
-        ring
-      _ = _ := by
-        simp only [regular_character_identity, inv_mul_eq_one]
-        simp [mul_comm, mul_left_comm]
-  have hre := congrArg Complex.re he
-  have hnorm (g : G) : (conj (a g) * a g).re = ‖a g‖^2 := by
-    rw [mul_comm, Complex.mul_conj]
-    exact Complex.normSq_eq_norm_sq _
-  simpa only [Complex.re_sum, Complex.mul_re, Complex.natCast_re, Complex.natCast_im,
-    zero_mul, sub_zero, hnorm, hsNormSq] using hre.symm
-
-end
-end PartialPermutation
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.Characters
+import OAI.RepresentationTheory.PartialPermutation.CompleteKernels
+import OAI.RepresentationTheory.PartialPermutation.RegularRepresentation
+
+namespace OAI
+
+section
+open scoped Classical
+open scoped BigOperators ComplexConjugate MonoidAlgebra
+open scoped BigOperators ComplexConjugate
+open scoped MonoidAlgebra BigOperators
+open scoped BigOperators MonoidAlgebra Classical
+
+namespace PartialPermutation
+noncomputable section
+open scoped BigOperators MonoidAlgebra Classical
+
+variable {G : Type*} [Group G] [Fintype G]
+
+omit [Fintype G] in
+lemma componentRep_unitary {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
+    (c : isotypicComponents ℂ[G] ρ.asModule) : IsUnitary (componentRep ρ c) := by
+  intro g x y
+  exact hρ g x.val y.val
+
+lemma complexFourier_regular_delta (a : G → ℂ) (g : G) :
+    complexFourier (regularRep G) a (deltaOne G) g = a g := by
+  simp [complexFourier, deltaOne, inv_mul_eq_one, eq_comm]
+
+abbrev irreducibleRep (c : IrreducibleIndex G) := componentRep (regularRep G) c
+
+lemma irreducibleRep_unitary (c : IrreducibleIndex G) :
+    IsUnitary (G := G) (V := irreducibleSpace c) (irreducibleRep c) :=
+  componentRep_unitary (regularRep G) (regularRep_unitary G) c
+
+lemma regular_character_identity (g : G) :
+    ∑ c : IrreducibleIndex G, (irreducibleDegree c : ℂ) * (irreducibleRep c).character g =
+      if g = 1 then (Fintype.card G : ℂ) else 0 := by
+  have ht := congrArg (fun A : Module.End ℂ (EuclideanSpace ℂ G) => A (deltaOne G) g⁻¹)
+    (sum_character_kernels_identity (regularRep G))
+  rw [complexFourier_regular_delta] at ht
+  change (∑ c : IrreducibleIndex G, centralCharacterKernel (irreducibleRep c) g⁻¹) =
+    (deltaOne G) g⁻¹ at ht
+  simp only [centralCharacterKernel, inv_inv] at ht
+  have hN : (Fintype.card G : ℂ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
+  have ht' := congrArg (fun z : ℂ => (Fintype.card G : ℂ) * z) ht
+  simp only [Finset.mul_sum] at ht'
+  convert ht' using 1
+  · apply Finset.sum_congr rfl
+    intro c _
+    change _ = (Fintype.card G : ℂ) *
+      ((irreducibleDegree c : ℂ) / Fintype.card G * (irreducibleRep c).character g)
+    field_simp
+  · change (if g = 1 then (Fintype.card G : ℂ) else 0) =
+      (Fintype.card G : ℂ) * (if g⁻¹ = 1 then 1 else 0)
+    simp only [inv_eq_one]
+    split_ifs <;> simp
+
+lemma sum_regular_character_kernels (g : G) :
+    ∑ c : IrreducibleIndex G, centralCharacterKernel (irreducibleRep c) g =
+      if g = 1 then 1 else 0 := by
+  have ht := congrArg (fun A : Module.End ℂ (EuclideanSpace ℂ G) => A (deltaOne G) g)
+    (sum_character_kernels_identity (regularRep G))
+  rw [complexFourier_regular_delta] at ht
+  exact ht
+
+lemma irreducibleRep_complete {V : Type*} [AddCommGroup V] [Module ℂ V]
+    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ] :
+    ∃! c : IrreducibleIndex G, Nonempty (Representation.Equiv (irreducibleRep c) ρ) := by
+  have : Nontrivial V := IsSimpleModule.nontrivial ℂ[G] ρ.asModule
+  have hex : ∃ c : IrreducibleIndex G, Nonempty (Representation.Equiv (irreducibleRep c) ρ) := by
+    by_contra hn
+    push Not at hn
+    have hs : complexFourier ρ (fun g => ∑ c : IrreducibleIndex G,
+        centralCharacterKernel (irreducibleRep c) g) = 1 := by
+      simp only [sum_regular_character_kernels]
+      simp [complexFourier]
+    rw [complexFourier_sum] at hs
+    have hz : (∑ c : IrreducibleIndex G, complexFourier ρ
+        (centralCharacterKernel (irreducibleRep c))) = 0 := by
+      apply Finset.sum_eq_zero
+      intro c _
+      rw [centralCharacterKernel_fourier, ite_eq_right (fun ⟨e⟩ => (hn c).false e)]
+    rw [hz] at hs
+    exact zero_ne_one hs
+  obtain ⟨c, hc⟩ := hex
+  refine ⟨c, hc, ?_⟩
+  intro d hd
+  obtain ⟨e⟩ := hc
+  obtain ⟨f⟩ := hd
+  exact (componentRep_equiv_iff (regularRep G) d c).mp ⟨f.trans e.symm⟩
+
+lemma trace_fourier_energy {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (a : G → ℂ) :
+    LinearMap.trace ℂ V (LinearMap.adjoint (complexFourier ρ a) * complexFourier ρ a) =
+      ∑ h, ∑ g, conj (a h) * a g * ρ.character (h⁻¹ * g) := by
+  have hadj : LinearMap.adjoint (complexFourier ρ a) =
+      ∑ h, conj (a h) • ρ h⁻¹ := by
+    simp only [complexFourier, map_sum, map_smulₛₗ, unitary_adjoint ρ hρ]
+  rw [hadj, complexFourier, Finset.sum_mul]
+  simp only [map_sum, Finset.mul_sum, smul_mul_assoc, mul_smul_comm, map_smul,
+    smul_eq_mul, ← map_mul, Representation.character]
+  apply Finset.sum_congr rfl
+  intro h _
+  apply Finset.sum_congr rfl
+  intro g _
+  ring
+
+theorem finite_group_plancherel (a : G → ℂ) :
+    (Fintype.card G : ℝ) * ∑ g, ‖a g‖^2 =
+      ∑ c : IrreducibleIndex G, (irreducibleDegree c : ℝ) *
+        hsNormSq (V := irreducibleSpace c) (complexFourier (V := irreducibleSpace c) (irreducibleRep c) a) := by
+  have he : ∑ c : IrreducibleIndex G, (irreducibleDegree c : ℂ) *
+      LinearMap.trace ℂ (irreducibleSpace c)
+        (LinearMap.adjoint (𝕜 := ℂ) (E := irreducibleSpace c) (F := irreducibleSpace c) (complexFourier (V := irreducibleSpace c) (irreducibleRep c) a) *
+          complexFourier (V := irreducibleSpace c) (irreducibleRep c) a) =
+        (Fintype.card G : ℂ) * ∑ g, conj (a g) * a g := by
+    have htrace (c : IrreducibleIndex G) := trace_fourier_energy
+      (V := irreducibleSpace c) (irreducibleRep c) (irreducibleRep_unitary c) a
+    simp only [htrace, Finset.mul_sum]
+    rw [Finset.sum_comm]
+    calc
+      _ = ∑ h, ∑ g, conj (a h) * a g *
+          (∑ c : IrreducibleIndex G, (irreducibleDegree c : ℂ) *
+            (irreducibleRep c).character (h⁻¹ * g)) := by
+        apply Finset.sum_congr rfl
+        intro h _
+        rw [Finset.sum_comm]
+        apply Finset.sum_congr rfl
+        intro g _
+        rw [Finset.mul_sum]
+        apply Finset.sum_congr rfl
+        intro c _
+        ring
+      _ = _ := by
+        simp only [regular_character_identity, inv_mul_eq_one]
+        simp [mul_comm, mul_left_comm]
+  have hre := congrArg Complex.re he
+  have hnorm (g : G) : (conj (a g) * a g).re = ‖a g‖^2 := by
+    rw [mul_comm, Complex.mul_conj]
+    exact Complex.normSq_eq_norm_sq _
+  simpa only [Complex.re_sum, Complex.mul_re, Complex.natCast_re, Complex.natCast_im,
+    zero_mul, sub_zero, hnorm, hsNormSq] using hre.symm
+
+end
+end PartialPermutation
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/Polytabloids.lean b/lean/OAI/RepresentationTheory/PartialPermutation/Polytabloids.lean
index e5540771e9927b0709abce82ba21ccd92760b903..ab95e408f9fd194d389e4b284e313ef00ee1482d 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/Polytabloids.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/Polytabloids.lean
@@ -1,129 +1,129 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.AlternatingTabloids
-
-namespace OAI
-
-section
-namespace PartialPermutation
-namespace YoungTabloid
-noncomputable section
-open Finset
-open scoped Classical
-
-def tabloidRep (μ : YoungDiagram) :
-    Representation ℂ (Equiv.Perm μ.cells) (EuclideanSpace ℂ (Tabloid μ)) where
-  toFun g :=
-    { toFun := fun f => WithLp.toLp 2 (fun r => f (relabel μ g⁻¹ r))
-      map_add' := by intros; rfl
-      map_smul' := by intros; rfl }
-  map_one' := by ext f r; simp [relabel_one]
-  map_mul' := by
-    intro g h
-    ext f r
-    change f (relabel μ (g*h)⁻¹ r)=f (relabel μ h⁻¹ (relabel μ g⁻¹ r))
-    rw [mul_inv_rev,relabel_mul]
-
-@[simp] lemma tabloidRep_apply (μ : YoungDiagram) (g : Equiv.Perm μ.cells)
-    (f : EuclideanSpace ℂ (Tabloid μ)) (r : Tabloid μ) :
-    tabloidRep μ g f r=f (relabel μ g⁻¹ r) := rfl
-
-lemma tabloidRep_unitary (μ : YoungDiagram) (g : Equiv.Perm μ.cells)
-    (f h : EuclideanSpace ℂ (Tabloid μ)) :
-    inner ℂ (tabloidRep μ g f) (tabloidRep μ g h)=inner ℂ f h := by
-  simp only [PiLp.inner_apply,tabloidRep_apply]
-  exact Fintype.sum_equiv (relabelEquiv μ g⁻¹) _ _ (fun _ => rfl)
-
-def antisym (μ : YoungDiagram) : Module.End ℂ (EuclideanSpace ℂ (Tabloid μ)) :=
-  ∑ c : columnGroup μ, signC c.1 • tabloidRep μ c.1
-
-lemma antisym_apply (μ : YoungDiagram) (f : EuclideanSpace ℂ (Tabloid μ)) (r : Tabloid μ) :
-    antisym μ f r = ∑ c : columnGroup μ, signC c.1 * f (relabel μ c.1⁻¹ r) := by
-  simp [antisym,LinearMap.sum_apply,LinearMap.smul_apply]
-
-lemma antisym_alternating (μ : YoungDiagram) (f : EuclideanSpace ℂ (Tabloid μ)) :
-    Alternating μ (antisym μ f) := by
-  intro d r
-  rw [antisym_apply,antisym_apply,Finset.mul_sum]
-  calc
-    _ = ∑ c : columnGroup μ, signC (d*c).1 *
-        f (relabel μ (d*c).1⁻¹ (relabel μ d.1 r)) :=
-      (Fintype.sum_equiv (Equiv.mulLeft d) _ _ (fun _ => rfl)).symm
-    _ = _ := by
-      apply Finset.sum_congr rfl
-      intro c _
-      simp only [Subgroup.coe_mul,signC_mul,mul_inv_rev,← relabel_mul]
-      simp only [mul_assoc,inv_mul_cancel,mul_one]
-
-def delta (μ : YoungDiagram) (r : Tabloid μ) : EuclideanSpace ℂ (Tabloid μ) :=
-  WithLp.toLp 2 (fun s => if s=r then 1 else 0)
-
-def polytabloid (μ : YoungDiagram) : EuclideanSpace ℂ (Tabloid μ) :=
-  antisym μ (delta μ (canonical μ))
-
-lemma canonical_fix_iff (μ : YoungDiagram) (c : columnGroup μ) :
-    relabel μ c (canonical μ)=canonical μ ↔ c=1 := by
-  constructor
-  · intro h
-    exact column_canonical_injective μ (h.trans (relabel_one μ _).symm)
-  · rintro rfl
-    rfl
-
-lemma polytabloid_canonical (μ : YoungDiagram) : polytabloid μ (canonical μ)=1 := by
-  rw [polytabloid,antisym_apply]
-  change (∑ c : columnGroup μ, signC c.1 *
-    (if relabel μ c.1⁻¹ (canonical μ)=canonical μ then 1 else 0))=1
-  have hh (c : columnGroup μ) : relabel μ c.1⁻¹ (canonical μ)=canonical μ ↔ c=1 := by
-    change relabel μ (c⁻¹:columnGroup μ) (canonical μ)=canonical μ ↔ c=1
-    rw [canonical_fix_iff,inv_eq_one]
-  simp [hh]
-
-lemma polytabloid_ne_zero (μ : YoungDiagram) : polytabloid μ≠0 := by
-  intro h
-  have hc := polytabloid_canonical μ
-  rw [h] at hc
-  exact zero_ne_one hc
-
-lemma polytabloid_alternating (μ : YoungDiagram) : Alternating μ (polytabloid μ) :=
-  antisym_alternating μ _
-
-lemma antisym_rank_one (μ : YoungDiagram) (f : EuclideanSpace ℂ (Tabloid μ)) :
-    antisym μ f = (antisym μ f (canonical μ)) • polytabloid μ := by
-  apply PiLp.ext
-  intro r
-  have ha : Alternating μ (fun s => antisym μ f (canonical μ)*polytabloid μ s) := by
-    intro c s
-    dsimp only
-    rw [polytabloid_alternating μ c s]
-    ring
-  exact congrFun (alternating_eq_of_canonical_eq μ _ _ (antisym_alternating μ f) ha
-    (by rw [polytabloid_canonical,mul_one])) r
-
-@[simp] lemma signC_conj {α : Type*} [Fintype α] [DecidableEq α] (g : Equiv.Perm α) :
-    starRingEnd ℂ (signC g)=signC g := by simp [signC]
-
-lemma antisym_selfadjoint (μ : YoungDiagram) (f h : EuclideanSpace ℂ (Tabloid μ)) :
-    inner ℂ (antisym μ f) h=inner ℂ f (antisym μ h) := by
-  have hu (c : columnGroup μ) :
-      inner ℂ (tabloidRep μ c.1 f) h=inner ℂ f (tabloidRep μ c.1⁻¹ h) := by
-    have hh := tabloidRep_unitary μ c.1 f (tabloidRep μ c.1⁻¹ h)
-    simpa using hh
-  simp only [antisym,LinearMap.sum_apply,LinearMap.smul_apply,inner_sum,sum_inner,
-    inner_smul_left,inner_smul_right,signC_conj,hu]
-  exact Fintype.sum_equiv (Equiv.inv (columnGroup μ)) _ _ (fun c => by
-    simp only [Equiv.inv_apply,Subgroup.coe_inv,signC_inv])
-
-lemma inner_delta (μ : YoungDiagram) (r : Tabloid μ) (f : EuclideanSpace ℂ (Tabloid μ)) :
-    inner ℂ (delta μ r) f=f r := by simp [PiLp.inner_apply,delta]
-
-lemma antisym_eq_inner (μ : YoungDiagram) (f : EuclideanSpace ℂ (Tabloid μ)) :
-    antisym μ f=inner ℂ (polytabloid μ) f • polytabloid μ := by
-  rw [antisym_rank_one]
-  congr 1
-  rw [polytabloid,antisym_selfadjoint,inner_delta]
-
-end
-end YoungTabloid
-end PartialPermutation
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.AlternatingTabloids
+
+namespace OAI
+
+section
+namespace PartialPermutation
+namespace YoungTabloid
+noncomputable section
+open Finset
+open scoped Classical
+
+def tabloidRep (μ : YoungDiagram) :
+    Representation ℂ (Equiv.Perm μ.cells) (EuclideanSpace ℂ (Tabloid μ)) where
+  toFun g :=
+    { toFun := fun f => WithLp.toLp 2 (fun r => f (relabel μ g⁻¹ r))
+      map_add' := by intros; rfl
+      map_smul' := by intros; rfl }
+  map_one' := by ext f r; simp [relabel_one]
+  map_mul' := by
+    intro g h
+    ext f r
+    change f (relabel μ (g*h)⁻¹ r)=f (relabel μ h⁻¹ (relabel μ g⁻¹ r))
+    rw [mul_inv_rev,relabel_mul]
+
+@[simp] lemma tabloidRep_apply (μ : YoungDiagram) (g : Equiv.Perm μ.cells)
+    (f : EuclideanSpace ℂ (Tabloid μ)) (r : Tabloid μ) :
+    tabloidRep μ g f r=f (relabel μ g⁻¹ r) := rfl
+
+lemma tabloidRep_unitary (μ : YoungDiagram) (g : Equiv.Perm μ.cells)
+    (f h : EuclideanSpace ℂ (Tabloid μ)) :
+    inner ℂ (tabloidRep μ g f) (tabloidRep μ g h)=inner ℂ f h := by
+  simp only [PiLp.inner_apply,tabloidRep_apply]
+  exact Fintype.sum_equiv (relabelEquiv μ g⁻¹) _ _ (fun _ => rfl)
+
+def antisym (μ : YoungDiagram) : Module.End ℂ (EuclideanSpace ℂ (Tabloid μ)) :=
+  ∑ c : columnGroup μ, signC c.1 • tabloidRep μ c.1
+
+lemma antisym_apply (μ : YoungDiagram) (f : EuclideanSpace ℂ (Tabloid μ)) (r : Tabloid μ) :
+    antisym μ f r = ∑ c : columnGroup μ, signC c.1 * f (relabel μ c.1⁻¹ r) := by
+  simp [antisym,LinearMap.sum_apply,LinearMap.smul_apply]
+
+lemma antisym_alternating (μ : YoungDiagram) (f : EuclideanSpace ℂ (Tabloid μ)) :
+    Alternating μ (antisym μ f) := by
+  intro d r
+  rw [antisym_apply,antisym_apply,Finset.mul_sum]
+  calc
+    _ = ∑ c : columnGroup μ, signC (d*c).1 *
+        f (relabel μ (d*c).1⁻¹ (relabel μ d.1 r)) :=
+      (Fintype.sum_equiv (Equiv.mulLeft d) _ _ (fun _ => rfl)).symm
+    _ = _ := by
+      apply Finset.sum_congr rfl
+      intro c _
+      simp only [Subgroup.coe_mul,signC_mul,mul_inv_rev,← relabel_mul]
+      simp only [mul_assoc,inv_mul_cancel,mul_one]
+
+def delta (μ : YoungDiagram) (r : Tabloid μ) : EuclideanSpace ℂ (Tabloid μ) :=
+  WithLp.toLp 2 (fun s => if s=r then 1 else 0)
+
+def polytabloid (μ : YoungDiagram) : EuclideanSpace ℂ (Tabloid μ) :=
+  antisym μ (delta μ (canonical μ))
+
+lemma canonical_fix_iff (μ : YoungDiagram) (c : columnGroup μ) :
+    relabel μ c (canonical μ)=canonical μ ↔ c=1 := by
+  constructor
+  · intro h
+    exact column_canonical_injective μ (h.trans (relabel_one μ _).symm)
+  · rintro rfl
+    rfl
+
+lemma polytabloid_canonical (μ : YoungDiagram) : polytabloid μ (canonical μ)=1 := by
+  rw [polytabloid,antisym_apply]
+  change (∑ c : columnGroup μ, signC c.1 *
+    (if relabel μ c.1⁻¹ (canonical μ)=canonical μ then 1 else 0))=1
+  have hh (c : columnGroup μ) : relabel μ c.1⁻¹ (canonical μ)=canonical μ ↔ c=1 := by
+    change relabel μ (c⁻¹:columnGroup μ) (canonical μ)=canonical μ ↔ c=1
+    rw [canonical_fix_iff,inv_eq_one]
+  simp [hh]
+
+lemma polytabloid_ne_zero (μ : YoungDiagram) : polytabloid μ≠0 := by
+  intro h
+  have hc := polytabloid_canonical μ
+  rw [h] at hc
+  exact zero_ne_one hc
+
+lemma polytabloid_alternating (μ : YoungDiagram) : Alternating μ (polytabloid μ) :=
+  antisym_alternating μ _
+
+lemma antisym_rank_one (μ : YoungDiagram) (f : EuclideanSpace ℂ (Tabloid μ)) :
+    antisym μ f = (antisym μ f (canonical μ)) • polytabloid μ := by
+  apply PiLp.ext
+  intro r
+  have ha : Alternating μ (fun s => antisym μ f (canonical μ)*polytabloid μ s) := by
+    intro c s
+    dsimp only
+    rw [polytabloid_alternating μ c s]
+    ring
+  exact congrFun (alternating_eq_of_canonical_eq μ _ _ (antisym_alternating μ f) ha
+    (by rw [polytabloid_canonical,mul_one])) r
+
+@[simp] lemma signC_conj {α : Type*} [Fintype α] [DecidableEq α] (g : Equiv.Perm α) :
+    starRingEnd ℂ (signC g)=signC g := by simp [signC]
+
+lemma antisym_selfadjoint (μ : YoungDiagram) (f h : EuclideanSpace ℂ (Tabloid μ)) :
+    inner ℂ (antisym μ f) h=inner ℂ f (antisym μ h) := by
+  have hu (c : columnGroup μ) :
+      inner ℂ (tabloidRep μ c.1 f) h=inner ℂ f (tabloidRep μ c.1⁻¹ h) := by
+    have hh := tabloidRep_unitary μ c.1 f (tabloidRep μ c.1⁻¹ h)
+    simpa using hh
+  simp only [antisym,LinearMap.sum_apply,LinearMap.smul_apply,inner_sum,sum_inner,
+    inner_smul_left,inner_smul_right,signC_conj,hu]
+  exact Fintype.sum_equiv (Equiv.inv (columnGroup μ)) _ _ (fun c => by
+    simp only [Equiv.inv_apply,Subgroup.coe_inv,signC_inv])
+
+lemma inner_delta (μ : YoungDiagram) (r : Tabloid μ) (f : EuclideanSpace ℂ (Tabloid μ)) :
+    inner ℂ (delta μ r) f=f r := by simp [PiLp.inner_apply,delta]
+
+lemma antisym_eq_inner (μ : YoungDiagram) (f : EuclideanSpace ℂ (Tabloid μ)) :
+    antisym μ f=inner ℂ (polytabloid μ) f • polytabloid μ := by
+  rw [antisym_rank_one]
+  congr 1
+  rw [polytabloid,antisym_selfadjoint,inner_delta]
+
+end
+end YoungTabloid
+end PartialPermutation
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/ProjectionCovers.lean b/lean/OAI/RepresentationTheory/PartialPermutation/ProjectionCovers.lean
index 2ffbd71fd92fbabf57a1d64679e2deaeb665881a..f2b64830924a7789b88a66cf17dfe7c4f90f614a 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/ProjectionCovers.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/ProjectionCovers.lean
@@ -1,126 +1,126 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.SmallKernels
-
-namespace OAI
-
-section
-open scoped Classical
-open scoped BigOperators ComplexConjugate MonoidAlgebra
-open scoped BigOperators ComplexConjugate
-open scoped MonoidAlgebra BigOperators
-open scoped BigOperators MonoidAlgebra Classical
-
-attribute [local instance] Classical.propDecidable
-namespace PartialPermutation
-noncomputable section
-variable {G : Type*} [Group G] [Fintype G]
-
-def smallDegreeSum (G : Type*) [Group G] [Fintype G] (R : ℝ) : ℝ :=
-  ∑ c : IrreducibleIndex G, if (irreducibleDegree c : ℝ) ≤ R then
-    (irreducibleDegree c : ℝ)^2 else 0
-
-lemma smallTypeKernel_energy (R : ℝ) :
-    (Fintype.card G : ℝ) * ∑ g, ‖smallTypeKernel G R g‖^2 = smallDegreeSum G R := by
-  rw [finite_group_plancherel]
-  unfold smallDegreeSum
-  apply Finset.sum_congr rfl
-  intro c _
-  rw [complexFourier_smallTypeKernel]
-  split_ifs <;> simp [hsNormSq, LinearMap.trace_one, pow_two]
-
-lemma kernel_hs_energy {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
-    [Representation.IsIrreducible ρ] (H : Subgroup G) (k : H → ℂ)
-    (f : G → ℝ) (B : ℝ) (hf : ∀ g, 0 ≤ f g) (hcap : RightCosetCap f H B) :
-    (Module.finrank ℂ V : ℝ) *
-      hsNormSq (complexFourier (ρ.comp H.subtype) k * complexFourier ρ (fun g => f g)) ≤
-      B * mass f * ((Fintype.card H : ℝ) * ∑ h, ‖k h‖^2) := by
-  have hc : (H.index : ℝ) * (Fintype.card H : ℝ) = (Fintype.card G : ℝ) := by
-    exact_mod_cast (by simpa only [Nat.card_eq_fintype_card] using H.index_mul_card)
-  have hi : (H.index : ℝ) ≠ 0 := by exact_mod_cast H.index_ne_zero_of_finite
-  calc
-    _ ≤ (Fintype.card G : ℝ) * ∑ g, ‖leftKernel H k f g‖^2 := by
-      rw [← complexFourier_leftKernel]
-      exact fourier_bessel ρ hρ _
-    _ ≤ (Fintype.card G : ℝ) * (B / H.index * mass f * ∑ h, ‖k h‖^2) :=
-      mul_le_mul_of_nonneg_left (leftKernel_l2 H k f B hf hcap) (by positivity)
-    _ = _ := by rw [← hc]; field_simp
-
-lemma small_type_hs_bound {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
-    [Representation.IsIrreducible ρ] (H : Subgroup G) (R : ℝ)
-    (f : G → ℝ) (B : ℝ) (hf : ∀ g, 0 ≤ f g) (hcap : LeftCosetCap f H B) :
-    (Module.finrank ℂ V : ℝ) *
-      hsNormSq (complexFourier ρ (fun g => f g) *
-        complexFourier (ρ.comp H.subtype) (smallTypeKernel H R)) ≤
-      B * mass f * smallDegreeSum H R := by
-  have hi := kernel_hs_energy ρ hρ H (smallTypeKernel H R) (fun g => f g⁻¹) B
-    (fun g => hf _) (inverse_rightCap H f B hcap)
-  rw [mass_inverse, smallTypeKernel_energy] at hi
-  have hP := complexFourier_selfAdjoint (G := H) (ρ.comp H.subtype)
-    (fun (g : H) => hρ g) (smallTypeKernel H R) (smallTypeKernel_inv R)
-  have hA : LinearMap.adjoint (complexFourier ρ (fun g => (f g⁻¹ : ℂ))) =
-      complexFourier ρ (fun g => (f g : ℂ)) := by
-    rw [complexFourier_adjoint ρ hρ]
-    simp
-  rw [← hsNormSq_adjoint (_ * _), adjoint_mul_eq, hA, hP] at hi
-  exact hi
-
-section Cover
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
-    {ι : Type*} [Fintype ι] (P : ι → Module.End ℂ V)
-    (hP : ∀ i, LinearMap.adjoint (P i) = P i)
-    (hcover : ∀ c : isotypicComponents ℂ[G] ρ.asModule,
-      ∃ i, ∀ x ∈ componentSpace ρ c, P i x = x)
-
-include ρ hρ hP hcover
-
-lemma component_projection_cover_norm (v : V) : ‖v‖^2 ≤ ∑ i, ‖P i v‖^2 := by
-  let : Fintype (isotypicComponents ℂ[G] ρ.asModule) := Fintype.ofFinite _
-  calc
-    _ = ∑ c, ‖(componentSpace ρ c).starProjection v‖^2 :=
-      (sum_component_norm_sq ρ hρ v).symm
-    _ ≤ ∑ c, ∑ i, ‖(componentSpace ρ c).starProjection (P i v)‖^2 := by
-      apply Finset.sum_le_sum
-      intro c _
-      obtain ⟨i, hi⟩ := hcover c
-      have he := congrArg (fun A : Module.End ℂ V => A v)
-        (projection_kernel_mul (componentSpace ρ c) (P i) (hP i) hi)
-      change (componentSpace ρ c).starProjection (P i v) =
-        (componentSpace ρ c).starProjection v at he
-      rw [← he]
-      exact Finset.single_le_sum (fun j _ =>
-        sq_nonneg (‖(componentSpace ρ c).starProjection (P j v)‖)) (Finset.mem_univ i)
-    _ = _ := by
-      rw [Finset.sum_comm]
-      apply Finset.sum_congr rfl
-      intro i _
-      exact sum_component_norm_sq ρ hρ (P i v)
-
-lemma component_projection_cover_hs (A : Module.End ℂ V) :
-    hsNormSq A ≤ ∑ i, hsNormSq (A * P i) := by
-  let e := stdOrthonormalBasis ℂ V
-  calc
-    _ = ∑ j, ‖LinearMap.adjoint A (e j)‖^2 := by
-      rw [← hsNormSq_adjoint A, hsNormSq_eq_sum e]
-    _ ≤ ∑ j, ∑ i, ‖P i (LinearMap.adjoint A (e j))‖^2 :=
-      Finset.sum_le_sum (fun j _ => component_projection_cover_norm ρ hρ P hP hcover _)
-    _ = ∑ i, hsNormSq (P i * LinearMap.adjoint A) := by
-      rw [Finset.sum_comm]
-      apply Finset.sum_congr rfl
-      intro i _
-      rw [hsNormSq_eq_sum e]
-      rfl
-    _ = _ := by
-      apply Finset.sum_congr rfl
-      intro i _
-      rw [← hsNormSq_adjoint (A * P i), adjoint_mul_eq, hP]
-
-end Cover
-end
-end PartialPermutation
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.SmallKernels
+
+namespace OAI
+
+section
+open scoped Classical
+open scoped BigOperators ComplexConjugate MonoidAlgebra
+open scoped BigOperators ComplexConjugate
+open scoped MonoidAlgebra BigOperators
+open scoped BigOperators MonoidAlgebra Classical
+
+attribute [local instance] Classical.propDecidable
+namespace PartialPermutation
+noncomputable section
+variable {G : Type*} [Group G] [Fintype G]
+
+def smallDegreeSum (G : Type*) [Group G] [Fintype G] (R : ℝ) : ℝ :=
+  ∑ c : IrreducibleIndex G, if (irreducibleDegree c : ℝ) ≤ R then
+    (irreducibleDegree c : ℝ)^2 else 0
+
+lemma smallTypeKernel_energy (R : ℝ) :
+    (Fintype.card G : ℝ) * ∑ g, ‖smallTypeKernel G R g‖^2 = smallDegreeSum G R := by
+  rw [finite_group_plancherel]
+  unfold smallDegreeSum
+  apply Finset.sum_congr rfl
+  intro c _
+  rw [complexFourier_smallTypeKernel]
+  split_ifs <;> simp [hsNormSq, LinearMap.trace_one, pow_two]
+
+lemma kernel_hs_energy {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
+    [Representation.IsIrreducible ρ] (H : Subgroup G) (k : H → ℂ)
+    (f : G → ℝ) (B : ℝ) (hf : ∀ g, 0 ≤ f g) (hcap : RightCosetCap f H B) :
+    (Module.finrank ℂ V : ℝ) *
+      hsNormSq (complexFourier (ρ.comp H.subtype) k * complexFourier ρ (fun g => f g)) ≤
+      B * mass f * ((Fintype.card H : ℝ) * ∑ h, ‖k h‖^2) := by
+  have hc : (H.index : ℝ) * (Fintype.card H : ℝ) = (Fintype.card G : ℝ) := by
+    exact_mod_cast (by simpa only [Nat.card_eq_fintype_card] using H.index_mul_card)
+  have hi : (H.index : ℝ) ≠ 0 := by exact_mod_cast H.index_ne_zero_of_finite
+  calc
+    _ ≤ (Fintype.card G : ℝ) * ∑ g, ‖leftKernel H k f g‖^2 := by
+      rw [← complexFourier_leftKernel]
+      exact fourier_bessel ρ hρ _
+    _ ≤ (Fintype.card G : ℝ) * (B / H.index * mass f * ∑ h, ‖k h‖^2) :=
+      mul_le_mul_of_nonneg_left (leftKernel_l2 H k f B hf hcap) (by positivity)
+    _ = _ := by rw [← hc]; field_simp
+
+lemma small_type_hs_bound {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
+    [Representation.IsIrreducible ρ] (H : Subgroup G) (R : ℝ)
+    (f : G → ℝ) (B : ℝ) (hf : ∀ g, 0 ≤ f g) (hcap : LeftCosetCap f H B) :
+    (Module.finrank ℂ V : ℝ) *
+      hsNormSq (complexFourier ρ (fun g => f g) *
+        complexFourier (ρ.comp H.subtype) (smallTypeKernel H R)) ≤
+      B * mass f * smallDegreeSum H R := by
+  have hi := kernel_hs_energy ρ hρ H (smallTypeKernel H R) (fun g => f g⁻¹) B
+    (fun g => hf _) (inverse_rightCap H f B hcap)
+  rw [mass_inverse, smallTypeKernel_energy] at hi
+  have hP := complexFourier_selfAdjoint (G := H) (ρ.comp H.subtype)
+    (fun (g : H) => hρ g) (smallTypeKernel H R) (smallTypeKernel_inv R)
+  have hA : LinearMap.adjoint (complexFourier ρ (fun g => (f g⁻¹ : ℂ))) =
+      complexFourier ρ (fun g => (f g : ℂ)) := by
+    rw [complexFourier_adjoint ρ hρ]
+    simp
+  rw [← hsNormSq_adjoint (_ * _), adjoint_mul_eq, hA, hP] at hi
+  exact hi
+
+section Cover
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
+    {ι : Type*} [Fintype ι] (P : ι → Module.End ℂ V)
+    (hP : ∀ i, LinearMap.adjoint (P i) = P i)
+    (hcover : ∀ c : isotypicComponents ℂ[G] ρ.asModule,
+      ∃ i, ∀ x ∈ componentSpace ρ c, P i x = x)
+
+include ρ hρ hP hcover
+
+lemma component_projection_cover_norm (v : V) : ‖v‖^2 ≤ ∑ i, ‖P i v‖^2 := by
+  let : Fintype (isotypicComponents ℂ[G] ρ.asModule) := Fintype.ofFinite _
+  calc
+    _ = ∑ c, ‖(componentSpace ρ c).starProjection v‖^2 :=
+      (sum_component_norm_sq ρ hρ v).symm
+    _ ≤ ∑ c, ∑ i, ‖(componentSpace ρ c).starProjection (P i v)‖^2 := by
+      apply Finset.sum_le_sum
+      intro c _
+      obtain ⟨i, hi⟩ := hcover c
+      have he := congrArg (fun A : Module.End ℂ V => A v)
+        (projection_kernel_mul (componentSpace ρ c) (P i) (hP i) hi)
+      change (componentSpace ρ c).starProjection (P i v) =
+        (componentSpace ρ c).starProjection v at he
+      rw [← he]
+      exact Finset.single_le_sum (fun j _ =>
+        sq_nonneg (‖(componentSpace ρ c).starProjection (P j v)‖)) (Finset.mem_univ i)
+    _ = _ := by
+      rw [Finset.sum_comm]
+      apply Finset.sum_congr rfl
+      intro i _
+      exact sum_component_norm_sq ρ hρ (P i v)
+
+lemma component_projection_cover_hs (A : Module.End ℂ V) :
+    hsNormSq A ≤ ∑ i, hsNormSq (A * P i) := by
+  let e := stdOrthonormalBasis ℂ V
+  calc
+    _ = ∑ j, ‖LinearMap.adjoint A (e j)‖^2 := by
+      rw [← hsNormSq_adjoint A, hsNormSq_eq_sum e]
+    _ ≤ ∑ j, ∑ i, ‖P i (LinearMap.adjoint A (e j))‖^2 :=
+      Finset.sum_le_sum (fun j _ => component_projection_cover_norm ρ hρ P hP hcover _)
+    _ = ∑ i, hsNormSq (P i * LinearMap.adjoint A) := by
+      rw [Finset.sum_comm]
+      apply Finset.sum_congr rfl
+      intro i _
+      rw [hsNormSq_eq_sum e]
+      rfl
+    _ = _ := by
+      apply Finset.sum_congr rfl
+      intro i _
+      rw [← hsNormSq_adjoint (A * P i), adjoint_mul_eq, hP]
+
+end Cover
+end
+end PartialPermutation
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/ProjectionEnergy.lean b/lean/OAI/RepresentationTheory/PartialPermutation/ProjectionEnergy.lean
index d93b5bddd7e299c7eabc4e7f3b493420801ff523..a1e31afec18eabbfabe2b2bbea46a86817b09776 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/ProjectionEnergy.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/ProjectionEnergy.lean
@@ -1,128 +1,128 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.Plancherel
-import OAI.RepresentationTheory.PartialPermutation.ConvolutionKernels
-import OAI.RepresentationTheory.PartialPermutation.IsotypicProjections
-
-namespace OAI
-
-section
-open scoped Classical
-open scoped BigOperators ComplexConjugate MonoidAlgebra
-open scoped BigOperators ComplexConjugate
-open scoped MonoidAlgebra BigOperators
-open scoped BigOperators MonoidAlgebra Classical
-
-attribute [local instance] Classical.propDecidable
-namespace PartialPermutation
-noncomputable section
-variable {G : Type*} [Group G] [Fintype G]
-
-lemma centralCharacterKernel_norm_sq {W : Type*} [NormedAddCommGroup W]
-    [InnerProductSpace ℂ W] [FiniteDimensional ℂ W]
-    (τ : Representation ℂ G W) [Representation.IsIrreducible τ] (hτ : IsUnitary τ) :
-    ∑ g, ‖centralCharacterKernel τ g‖^2 =
-      (Module.finrank ℂ W : ℝ)^2 / Fintype.card G := by
-  have hN : (Fintype.card G : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
-  simp only [centralCharacterKernel, unitary_character_inv τ hτ, norm_mul,
-    norm_div, Complex.norm_natCast, Complex.norm_conj, mul_pow,
-    ← Finset.mul_sum, character_norm_sq_sum τ hτ]
-  field_simp
-
-lemma centralCharacterKernel_inv {W : Type*} [NormedAddCommGroup W]
-    [InnerProductSpace ℂ W] [FiniteDimensional ℂ W]
-    (τ : Representation ℂ G W) (hτ : IsUnitary τ) (g : G) :
-    centralCharacterKernel τ g⁻¹ = conj (centralCharacterKernel τ g) := by
-  simp [centralCharacterKernel, unitary_character_inv τ hτ]
-
-lemma kernel_hs_global (H : Subgroup G) (k : H → ℂ)
-    (d : ℕ) (hk : ∑ h : H, ‖k h‖^2 = (d : ℝ)^2 / Fintype.card H)
-    (f : G → ℝ) (B : ℝ) (hf : ∀ g, 0 ≤ f g) (hcap : RightCosetCap f H B) :
-    ∑ c : IrreducibleIndex G, (irreducibleDegree c : ℝ) *
-      hsNormSq (V := irreducibleSpace c)
-        (complexFourier ((irreducibleRep c).comp H.subtype) k *
-          complexFourier (irreducibleRep c) (fun g => (f g : ℂ))) ≤
-        B * mass f * (d : ℝ)^2 := by
-  have hc : (H.index : ℝ) * (Fintype.card H : ℝ) = (Fintype.card G : ℝ) := by
-    exact_mod_cast (by simpa only [Nat.card_eq_fintype_card] using H.index_mul_card)
-  have hi : (H.index : ℝ) ≠ 0 := by
-    exact_mod_cast H.index_ne_zero_of_finite
-  have hh : (Fintype.card H : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
-  calc
-    _ = (Fintype.card G : ℝ) * ∑ g, ‖leftKernel H k f g‖^2 := by
-      rw [finite_group_plancherel]
-      apply Finset.sum_congr rfl
-      intro c _
-      rw [complexFourier_leftKernel]
-    _ ≤ (Fintype.card G : ℝ) * (B / H.index * mass f * ∑ h : H, ‖k h‖^2) :=
-      mul_le_mul_of_nonneg_left (leftKernel_l2 H k f B hf hcap) (by positivity)
-    _ = _ := by rw [hk, ← hc]; field_simp
-
-lemma kernel_hs_global_right (H : Subgroup G) (k : H → ℂ)
-    (d : ℕ) (hk : ∑ h : H, ‖k h‖^2 = (d : ℝ)^2 / Fintype.card H)
-    (hki : ∀ h, k h⁻¹ = conj (k h))
-    (f : G → ℝ) (B : ℝ) (hf : ∀ g, 0 ≤ f g) (hcap : LeftCosetCap f H B) :
-    ∑ c : IrreducibleIndex G, (irreducibleDegree c : ℝ) *
-      hsNormSq (V := irreducibleSpace c)
-        (complexFourier (irreducibleRep c) (fun g => (f g : ℂ)) *
-          complexFourier ((irreducibleRep c).comp H.subtype) k) ≤
-        B * mass f * (d : ℝ)^2 := by
-  have hl := kernel_hs_global H k d hk (fun g => f g⁻¹) B
-    (fun g => hf _) (inverse_rightCap H f B hcap)
-  rw [mass_inverse] at hl
-  convert hl using 1
-  apply Finset.sum_congr rfl
-  intro c _
-  have hP := complexFourier_selfAdjoint (G := H) (V := irreducibleSpace c)
-    ((irreducibleRep c).comp H.subtype) (fun (g : H) => irreducibleRep_unitary c g) k hki
-  have hA : LinearMap.adjoint (E := irreducibleSpace c) (F := irreducibleSpace c)
-      (complexFourier (irreducibleRep c) (fun g => (f g⁻¹ : ℂ))) =
-        complexFourier (irreducibleRep c) (fun g => (f g : ℂ)) := by
-    rw [complexFourier_adjoint _ (irreducibleRep_unitary c)]
-    simp
-  rw [← hsNormSq_adjoint (V := irreducibleSpace c)
-    (complexFourier ((irreducibleRep c).comp H.subtype) k *
-      complexFourier (irreducibleRep c) (fun g => (f g⁻¹ : ℂ))), adjoint_mul_eq, hP, hA]
-
-def fullIsotypicProjection {V W : Type*} [AddCommGroup V] [Module ℂ V]
-    [AddCommGroup W] [Module ℂ W] [FiniteDimensional ℂ W]
-    (ρ : Representation ℂ G V) (H : Subgroup G) (τ : Representation ℂ H W) :
-    Module.End ℂ V := complexFourier (ρ.comp H.subtype) (centralCharacterKernel τ)
-
-lemma centralCharacterKernel_equiv {V W : Type*} [AddCommGroup V] [Module ℂ V]
-    [FiniteDimensional ℂ V] [AddCommGroup W] [Module ℂ W] [FiniteDimensional ℂ W]
-    {ρ : Representation ℂ G V} {σ : Representation ℂ G W}
-    (e : Representation.Equiv ρ σ) : centralCharacterKernel ρ = centralCharacterKernel σ := by
-  funext g
-  simp only [centralCharacterKernel, Representation.char_iso e, e.toLinearEquiv.finrank_eq]
-
-lemma fullIsotypicProjection_eq_component {V W : Type*} [NormedAddCommGroup V]
-    [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-    [AddCommGroup W] [Module ℂ W] [FiniteDimensional ℂ W]
-    (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (H : Subgroup G)
-    (τ : Representation ℂ H W)
-    (c : isotypicComponents ℂ[H] (Representation.asModule (ρ.comp H.subtype)))
-    (e : Representation.Equiv τ (componentRep (ρ.comp H.subtype) c)) :
-    fullIsotypicProjection ρ H τ =
-      (componentSpace (ρ.comp H.subtype) c).starProjection.toLinearMap := by
-  rw [fullIsotypicProjection, centralCharacterKernel_equiv e]
-  exact component_kernel_eq_projection (G := H) (ρ.comp H.subtype)
-    (fun (g : H) => hρ g) c
-
-theorem central_isotypic_global (H : Subgroup G) {W : Type*} [NormedAddCommGroup W]
-    [InnerProductSpace ℂ W] [FiniteDimensional ℂ W]
-    (τ : Representation ℂ H W) [Representation.IsIrreducible τ] (hτ : IsUnitary τ)
-    (f : G → ℝ) (B : ℝ) (hf : ∀ g, 0 ≤ f g) (hcap : LeftCosetCap f H B) :
-    ∑ c : IrreducibleIndex G, (irreducibleDegree c : ℝ) *
-      hsNormSq (V := irreducibleSpace c)
-        (complexFourier (irreducibleRep c) (fun g => (f g : ℂ)) *
-          fullIsotypicProjection (irreducibleRep c) H τ) ≤
-      B * mass f * (Module.finrank ℂ W : ℝ)^2 :=
-  kernel_hs_global_right H (centralCharacterKernel τ) _
-    (centralCharacterKernel_norm_sq τ hτ) (centralCharacterKernel_inv τ hτ) f B hf hcap
-
-end
-end PartialPermutation
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.Plancherel
+import OAI.RepresentationTheory.PartialPermutation.ConvolutionKernels
+import OAI.RepresentationTheory.PartialPermutation.IsotypicProjections
+
+namespace OAI
+
+section
+open scoped Classical
+open scoped BigOperators ComplexConjugate MonoidAlgebra
+open scoped BigOperators ComplexConjugate
+open scoped MonoidAlgebra BigOperators
+open scoped BigOperators MonoidAlgebra Classical
+
+attribute [local instance] Classical.propDecidable
+namespace PartialPermutation
+noncomputable section
+variable {G : Type*} [Group G] [Fintype G]
+
+lemma centralCharacterKernel_norm_sq {W : Type*} [NormedAddCommGroup W]
+    [InnerProductSpace ℂ W] [FiniteDimensional ℂ W]
+    (τ : Representation ℂ G W) [Representation.IsIrreducible τ] (hτ : IsUnitary τ) :
+    ∑ g, ‖centralCharacterKernel τ g‖^2 =
+      (Module.finrank ℂ W : ℝ)^2 / Fintype.card G := by
+  have hN : (Fintype.card G : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
+  simp only [centralCharacterKernel, unitary_character_inv τ hτ, norm_mul,
+    norm_div, Complex.norm_natCast, Complex.norm_conj, mul_pow,
+    ← Finset.mul_sum, character_norm_sq_sum τ hτ]
+  field_simp
+
+lemma centralCharacterKernel_inv {W : Type*} [NormedAddCommGroup W]
+    [InnerProductSpace ℂ W] [FiniteDimensional ℂ W]
+    (τ : Representation ℂ G W) (hτ : IsUnitary τ) (g : G) :
+    centralCharacterKernel τ g⁻¹ = conj (centralCharacterKernel τ g) := by
+  simp [centralCharacterKernel, unitary_character_inv τ hτ]
+
+lemma kernel_hs_global (H : Subgroup G) (k : H → ℂ)
+    (d : ℕ) (hk : ∑ h : H, ‖k h‖^2 = (d : ℝ)^2 / Fintype.card H)
+    (f : G → ℝ) (B : ℝ) (hf : ∀ g, 0 ≤ f g) (hcap : RightCosetCap f H B) :
+    ∑ c : IrreducibleIndex G, (irreducibleDegree c : ℝ) *
+      hsNormSq (V := irreducibleSpace c)
+        (complexFourier ((irreducibleRep c).comp H.subtype) k *
+          complexFourier (irreducibleRep c) (fun g => (f g : ℂ))) ≤
+        B * mass f * (d : ℝ)^2 := by
+  have hc : (H.index : ℝ) * (Fintype.card H : ℝ) = (Fintype.card G : ℝ) := by
+    exact_mod_cast (by simpa only [Nat.card_eq_fintype_card] using H.index_mul_card)
+  have hi : (H.index : ℝ) ≠ 0 := by
+    exact_mod_cast H.index_ne_zero_of_finite
+  have hh : (Fintype.card H : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
+  calc
+    _ = (Fintype.card G : ℝ) * ∑ g, ‖leftKernel H k f g‖^2 := by
+      rw [finite_group_plancherel]
+      apply Finset.sum_congr rfl
+      intro c _
+      rw [complexFourier_leftKernel]
+    _ ≤ (Fintype.card G : ℝ) * (B / H.index * mass f * ∑ h : H, ‖k h‖^2) :=
+      mul_le_mul_of_nonneg_left (leftKernel_l2 H k f B hf hcap) (by positivity)
+    _ = _ := by rw [hk, ← hc]; field_simp
+
+lemma kernel_hs_global_right (H : Subgroup G) (k : H → ℂ)
+    (d : ℕ) (hk : ∑ h : H, ‖k h‖^2 = (d : ℝ)^2 / Fintype.card H)
+    (hki : ∀ h, k h⁻¹ = conj (k h))
+    (f : G → ℝ) (B : ℝ) (hf : ∀ g, 0 ≤ f g) (hcap : LeftCosetCap f H B) :
+    ∑ c : IrreducibleIndex G, (irreducibleDegree c : ℝ) *
+      hsNormSq (V := irreducibleSpace c)
+        (complexFourier (irreducibleRep c) (fun g => (f g : ℂ)) *
+          complexFourier ((irreducibleRep c).comp H.subtype) k) ≤
+        B * mass f * (d : ℝ)^2 := by
+  have hl := kernel_hs_global H k d hk (fun g => f g⁻¹) B
+    (fun g => hf _) (inverse_rightCap H f B hcap)
+  rw [mass_inverse] at hl
+  convert hl using 1
+  apply Finset.sum_congr rfl
+  intro c _
+  have hP := complexFourier_selfAdjoint (G := H) (V := irreducibleSpace c)
+    ((irreducibleRep c).comp H.subtype) (fun (g : H) => irreducibleRep_unitary c g) k hki
+  have hA : LinearMap.adjoint (E := irreducibleSpace c) (F := irreducibleSpace c)
+      (complexFourier (irreducibleRep c) (fun g => (f g⁻¹ : ℂ))) =
+        complexFourier (irreducibleRep c) (fun g => (f g : ℂ)) := by
+    rw [complexFourier_adjoint _ (irreducibleRep_unitary c)]
+    simp
+  rw [← hsNormSq_adjoint (V := irreducibleSpace c)
+    (complexFourier ((irreducibleRep c).comp H.subtype) k *
+      complexFourier (irreducibleRep c) (fun g => (f g⁻¹ : ℂ))), adjoint_mul_eq, hP, hA]
+
+def fullIsotypicProjection {V W : Type*} [AddCommGroup V] [Module ℂ V]
+    [AddCommGroup W] [Module ℂ W] [FiniteDimensional ℂ W]
+    (ρ : Representation ℂ G V) (H : Subgroup G) (τ : Representation ℂ H W) :
+    Module.End ℂ V := complexFourier (ρ.comp H.subtype) (centralCharacterKernel τ)
+
+lemma centralCharacterKernel_equiv {V W : Type*} [AddCommGroup V] [Module ℂ V]
+    [FiniteDimensional ℂ V] [AddCommGroup W] [Module ℂ W] [FiniteDimensional ℂ W]
+    {ρ : Representation ℂ G V} {σ : Representation ℂ G W}
+    (e : Representation.Equiv ρ σ) : centralCharacterKernel ρ = centralCharacterKernel σ := by
+  funext g
+  simp only [centralCharacterKernel, Representation.char_iso e, e.toLinearEquiv.finrank_eq]
+
+lemma fullIsotypicProjection_eq_component {V W : Type*} [NormedAddCommGroup V]
+    [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+    [AddCommGroup W] [Module ℂ W] [FiniteDimensional ℂ W]
+    (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) (H : Subgroup G)
+    (τ : Representation ℂ H W)
+    (c : isotypicComponents ℂ[H] (Representation.asModule (ρ.comp H.subtype)))
+    (e : Representation.Equiv τ (componentRep (ρ.comp H.subtype) c)) :
+    fullIsotypicProjection ρ H τ =
+      (componentSpace (ρ.comp H.subtype) c).starProjection.toLinearMap := by
+  rw [fullIsotypicProjection, centralCharacterKernel_equiv e]
+  exact component_kernel_eq_projection (G := H) (ρ.comp H.subtype)
+    (fun (g : H) => hρ g) c
+
+theorem central_isotypic_global (H : Subgroup G) {W : Type*} [NormedAddCommGroup W]
+    [InnerProductSpace ℂ W] [FiniteDimensional ℂ W]
+    (τ : Representation ℂ H W) [Representation.IsIrreducible τ] (hτ : IsUnitary τ)
+    (f : G → ℝ) (B : ℝ) (hf : ∀ g, 0 ≤ f g) (hcap : LeftCosetCap f H B) :
+    ∑ c : IrreducibleIndex G, (irreducibleDegree c : ℝ) *
+      hsNormSq (V := irreducibleSpace c)
+        (complexFourier (irreducibleRep c) (fun g => (f g : ℂ)) *
+          fullIsotypicProjection (irreducibleRep c) H τ) ≤
+      B * mass f * (Module.finrank ℂ W : ℝ)^2 :=
+  kernel_hs_global_right H (centralCharacterKernel τ) _
+    (centralCharacterKernel_norm_sq τ hτ) (centralCharacterKernel_inv τ hτ) f B hf hcap
+
+end
+end PartialPermutation
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/RankFillings.lean b/lean/OAI/RepresentationTheory/PartialPermutation/RankFillings.lean
index ea17e96b45cf0ca39b133bab1b9ae8e0654d7823..89e134c9b1611155d613d522f97a0b9b934604a2 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/RankFillings.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/RankFillings.lean
@@ -1,107 +1,107 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.StandardTableaux
-
-namespace OAI
-
-section
-namespace PartialPermutation
-namespace Tableau
-noncomputable section
-open Finset
-
-lemma exists_rankEnumeration (α ι : Type*) [Fintype α] [LinearOrder ι]
-    (r : α → ι) :
-    ∃ e : α ≃ Fin (Fintype.card α), ∀ a b, r a < r b → e a < e b := by
-  classical
-  let e₀ := Fintype.equivFin α
-  let key : α → ι ×ₗ Fin (Fintype.card α) := fun a => toLex (r a, e₀ a)
-  have hk : Function.Injective key := by
-    intro a b h
-    exact e₀.injective (congrArg (fun z : ι ×ₗ Fin (Fintype.card α) => (ofLex z).2) h)
-  let : LinearOrder α := LinearOrder.lift' key hk
-  let e := (Fintype.orderIsoFinOfCardEq α rfl).symm
-  refine ⟨e.toEquiv, ?_⟩
-  intro a b h
-  apply e.strictMono
-  exact Prod.Lex.left _ _ h
-
-def fiberPermutation {α ι : Type*} (r : α → ι)
-    (p : ∀ i, Equiv.Perm {a // r a = i}) : Equiv.Perm α :=
-  (Equiv.sigmaFiberEquiv r).symm.trans
-    ((Equiv.sigmaCongrRight p).trans (Equiv.sigmaFiberEquiv r))
-
-lemma fiberPermutation_apply {α ι : Type*} (r : α → ι)
-    (p : ∀ i, Equiv.Perm {a // r a = i}) (a : α) :
-    fiberPermutation r p a = (p (r a) ⟨a,rfl⟩).1 := rfl
-
-lemma fiberPermutation_rank {α ι : Type*} (r : α → ι)
-    (p : ∀ i, Equiv.Perm {a // r a = i}) (a : α) :
-    r (fiberPermutation r p a) = r a := (p (r a) ⟨a,rfl⟩).2
-
-lemma fiberPermutation_injective {α ι : Type*} (r : α → ι) :
-    Function.Injective (fiberPermutation r) := by
-  intro p q h
-  funext i
-  apply Equiv.ext
-  rintro ⟨a,ha⟩
-  subst i
-  apply Subtype.ext
-  exact Equiv.congr_fun h a
-
-lemma rank_factorial_le_filling_count (α ι : Type*) [Fintype α] [PartialOrder α]
-    [Fintype ι] [LinearOrder ι] (r : α → ι) (hr : StrictMono r) :
-    (∏ i, (Fintype.card {a // r a = i}).factorial) ≤
-      Fintype.card (StandardFilling α) := by
-  classical
-  obtain ⟨e,he⟩ := exists_rankEnumeration α ι r
-  let f : (∀ i, Equiv.Perm {a // r a = i}) → StandardFilling α := fun p =>
-    ⟨(fiberPermutation r p).trans e, fun a b hab => he _ _ (by
-      simpa only [fiberPermutation_rank] using hr hab)⟩
-  have hf : Function.Injective f := by
-    intro p q hpq
-    apply fiberPermutation_injective r
-    apply Equiv.ext
-    intro a
-    apply e.injective
-    exact congrArg (fun t : StandardFilling α => t.1 a) hpq
-  simpa only [Fintype.card_pi, Fintype.card_perm] using Fintype.card_le_of_injective f hf
-
-lemma two_pow_pred_le_factorial (n : ℕ) : 2^(n-1) ≤ n.factorial := by
-  induction n with
-  | zero => simp
-  | succ n ih =>
-    rcases n with _ | n
-    · simp
-    · rw [Nat.factorial_succ]
-      simp only [Nat.add_sub_cancel] at ih ⊢
-      calc
-        2^(n+1) = 2 * 2^n := by rw [pow_succ]; omega
-        _ ≤ (n+2) * (n+1).factorial := Nat.mul_le_mul (by omega) ih
-
-lemma power_count_sub_ranks_le_factorials (ι : Type*) [Fintype ι] (a : ι → ℕ) :
-    2^((∑ i, a i)-Fintype.card ι) ≤ ∏ i, (a i).factorial := by
-  classical
-  have h : ∑ i, a i ≤ (∑ i, (a i-1)) + Fintype.card ι := by
-    calc
-      _ ≤ ∑ i, (a i-1+1) := Finset.sum_le_sum (fun i _ => by omega)
-      _ = _ := by rw [Finset.sum_add_distrib]; simp
-  calc
-    _ ≤ 2^(∑ i, (a i-1)) := Nat.pow_le_pow_right (by omega) (by omega)
-    _ = ∏ i, 2^(a i-1) := by rw [Finset.prod_pow_eq_pow_sum]
-    _ ≤ _ := Finset.prod_le_prod (fun index _ => two_pow_pred_le_factorial (a index))
-
-lemma rank_power_le_filling_count (α ι : Type*) [Fintype α] [PartialOrder α]
-    [Fintype ι] [LinearOrder ι] (r : α → ι) (hr : StrictMono r) :
-    2^(Fintype.card α - Fintype.card ι) ≤ Fintype.card (StandardFilling α) := by
-  have hh := Fintype.card_congr (Equiv.sigmaFiberEquiv r)
-  rw [Fintype.card_sigma] at hh
-  rw [← hh]
-  exact (power_count_sub_ranks_le_factorials ι (fun i => Fintype.card {a // r a=i})).trans
-    (rank_factorial_le_filling_count α ι r hr)
-
-end
-end Tableau
-end PartialPermutation
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.StandardTableaux
+
+namespace OAI
+
+section
+namespace PartialPermutation
+namespace Tableau
+noncomputable section
+open Finset
+
+lemma exists_rankEnumeration (α ι : Type*) [Fintype α] [LinearOrder ι]
+    (r : α → ι) :
+    ∃ e : α ≃ Fin (Fintype.card α), ∀ a b, r a < r b → e a < e b := by
+  classical
+  let e₀ := Fintype.equivFin α
+  let key : α → ι ×ₗ Fin (Fintype.card α) := fun a => toLex (r a, e₀ a)
+  have hk : Function.Injective key := by
+    intro a b h
+    exact e₀.injective (congrArg (fun z : ι ×ₗ Fin (Fintype.card α) => (ofLex z).2) h)
+  let : LinearOrder α := LinearOrder.lift' key hk
+  let e := (Fintype.orderIsoFinOfCardEq α rfl).symm
+  refine ⟨e.toEquiv, ?_⟩
+  intro a b h
+  apply e.strictMono
+  exact Prod.Lex.left _ _ h
+
+def fiberPermutation {α ι : Type*} (r : α → ι)
+    (p : ∀ i, Equiv.Perm {a // r a = i}) : Equiv.Perm α :=
+  (Equiv.sigmaFiberEquiv r).symm.trans
+    ((Equiv.sigmaCongrRight p).trans (Equiv.sigmaFiberEquiv r))
+
+lemma fiberPermutation_apply {α ι : Type*} (r : α → ι)
+    (p : ∀ i, Equiv.Perm {a // r a = i}) (a : α) :
+    fiberPermutation r p a = (p (r a) ⟨a,rfl⟩).1 := rfl
+
+lemma fiberPermutation_rank {α ι : Type*} (r : α → ι)
+    (p : ∀ i, Equiv.Perm {a // r a = i}) (a : α) :
+    r (fiberPermutation r p a) = r a := (p (r a) ⟨a,rfl⟩).2
+
+lemma fiberPermutation_injective {α ι : Type*} (r : α → ι) :
+    Function.Injective (fiberPermutation r) := by
+  intro p q h
+  funext i
+  apply Equiv.ext
+  rintro ⟨a,ha⟩
+  subst i
+  apply Subtype.ext
+  exact Equiv.congr_fun h a
+
+lemma rank_factorial_le_filling_count (α ι : Type*) [Fintype α] [PartialOrder α]
+    [Fintype ι] [LinearOrder ι] (r : α → ι) (hr : StrictMono r) :
+    (∏ i, (Fintype.card {a // r a = i}).factorial) ≤
+      Fintype.card (StandardFilling α) := by
+  classical
+  obtain ⟨e,he⟩ := exists_rankEnumeration α ι r
+  let f : (∀ i, Equiv.Perm {a // r a = i}) → StandardFilling α := fun p =>
+    ⟨(fiberPermutation r p).trans e, fun a b hab => he _ _ (by
+      simpa only [fiberPermutation_rank] using hr hab)⟩
+  have hf : Function.Injective f := by
+    intro p q hpq
+    apply fiberPermutation_injective r
+    apply Equiv.ext
+    intro a
+    apply e.injective
+    exact congrArg (fun t : StandardFilling α => t.1 a) hpq
+  simpa only [Fintype.card_pi, Fintype.card_perm] using Fintype.card_le_of_injective f hf
+
+lemma two_pow_pred_le_factorial (n : ℕ) : 2^(n-1) ≤ n.factorial := by
+  induction n with
+  | zero => simp
+  | succ n ih =>
+    rcases n with _ | n
+    · simp
+    · rw [Nat.factorial_succ]
+      simp only [Nat.add_sub_cancel] at ih ⊢
+      calc
+        2^(n+1) = 2 * 2^n := by rw [pow_succ]; omega
+        _ ≤ (n+2) * (n+1).factorial := Nat.mul_le_mul (by omega) ih
+
+lemma power_count_sub_ranks_le_factorials (ι : Type*) [Fintype ι] (a : ι → ℕ) :
+    2^((∑ i, a i)-Fintype.card ι) ≤ ∏ i, (a i).factorial := by
+  classical
+  have h : ∑ i, a i ≤ (∑ i, (a i-1)) + Fintype.card ι := by
+    calc
+      _ ≤ ∑ i, (a i-1+1) := Finset.sum_le_sum (fun i _ => by omega)
+      _ = _ := by rw [Finset.sum_add_distrib]; simp
+  calc
+    _ ≤ 2^(∑ i, (a i-1)) := Nat.pow_le_pow_right (by omega) (by omega)
+    _ = ∏ i, 2^(a i-1) := by rw [Finset.prod_pow_eq_pow_sum]
+    _ ≤ _ := Finset.prod_le_prod (fun index _ => two_pow_pred_le_factorial (a index))
+
+lemma rank_power_le_filling_count (α ι : Type*) [Fintype α] [PartialOrder α]
+    [Fintype ι] [LinearOrder ι] (r : α → ι) (hr : StrictMono r) :
+    2^(Fintype.card α - Fintype.card ι) ≤ Fintype.card (StandardFilling α) := by
+  have hh := Fintype.card_congr (Equiv.sigmaFiberEquiv r)
+  rw [Fintype.card_sigma] at hh
+  rw [← hh]
+  exact (power_count_sub_ranks_le_factorials ι (fun i => Fintype.card {a // r a=i})).trans
+    (rank_factorial_le_filling_count α ι r hr)
+
+end
+end Tableau
+end PartialPermutation
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/ReciprocalBounds.lean b/lean/OAI/RepresentationTheory/PartialPermutation/ReciprocalBounds.lean
index 8fe49d7bd3e792f541b8ceeded22a9b73ac5e717..bb87bae8ba2997a99807f8eecef3bb7eb927eac1 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/ReciprocalBounds.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/ReciprocalBounds.lean
@@ -1,43 +1,43 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.SymmetricTransfer
-import OAI.RepresentationTheory.PartialPermutation.TableauLimits
-import OAI.RepresentationTheory.PartialPermutation.SpechtClassification
-
-namespace OAI
-
-namespace PartialPermutation
-noncomputable section
-open scoped Classical
-
-lemma symmetricInverseDegreeSum_le_tableaux (n : ℕ) (u : ℝ) (hu : 0 < u) :
-    symmetricInverseDegreeSum n u ≤ TableauZeta.fullSum n u := by
-  have he : symmetricInverseDegreeSum n u =
-      ∑ μ : DiagramCounting.Shape n, (irreducibleDegree (SpechtActual.index μ):ℝ)^(-u) := by
-    exact (Fintype.sum_equiv (SpechtActual.indexEquiv n) _ _ (fun _ => rfl)).symm
-  rw [he]
-  apply Finset.sum_le_sum
-  intro μ _
-  exact Real.rpow_le_rpow_of_nonpos
-    (by exact_mod_cast Tableau.standardCount_pos μ.1)
-    (by exact_mod_cast SpechtActual.standardCount_le_degree μ) (neg_nonpos.mpr hu.le)
-
-theorem symmetricReciprocalDegreeBounded : SymmetricReciprocalDegreeBoundedStatement := by
-  intro u hu
-  obtain ⟨a,ha⟩ := TableauZeta.fullSum_bddAbove u hu
-  refine ⟨a,?_⟩
-  rintro x ⟨n,_,rfl⟩
-  exact (symmetricInverseDegreeSum_le_tableaux n u hu).trans (ha (Set.mem_range_self n))
-
-lemma symmetricInverseDegreeSum_le_constant {n : ℕ} (hn : 1 ≤ n) {u : ℝ} (hu : 0 < u) :
-    symmetricInverseDegreeSum n u ≤ symmetricDegreeConstant u := by
-  exact le_csSup (symmetricReciprocalDegreeBounded u hu) ⟨n,hn,rfl⟩
-
-lemma symmetricDegreeConstant_nonneg {u : ℝ} (hu : 0 < u) :
-    0 ≤ symmetricDegreeConstant u :=
-  (inverseDegreeSum_nonneg (Equiv.Perm (Fin 1)) u).trans
-    (symmetricInverseDegreeSum_le_constant (n := 1) le_rfl hu)
-
-end
-end PartialPermutation
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.SymmetricTransfer
+import OAI.RepresentationTheory.PartialPermutation.TableauLimits
+import OAI.RepresentationTheory.PartialPermutation.SpechtClassification
+
+namespace OAI
+
+namespace PartialPermutation
+noncomputable section
+open scoped Classical
+
+lemma symmetricInverseDegreeSum_le_tableaux (n : ℕ) (u : ℝ) (hu : 0 < u) :
+    symmetricInverseDegreeSum n u ≤ TableauZeta.fullSum n u := by
+  have he : symmetricInverseDegreeSum n u =
+      ∑ μ : DiagramCounting.Shape n, (irreducibleDegree (SpechtActual.index μ):ℝ)^(-u) := by
+    exact (Fintype.sum_equiv (SpechtActual.indexEquiv n) _ _ (fun _ => rfl)).symm
+  rw [he]
+  apply Finset.sum_le_sum
+  intro μ _
+  exact Real.rpow_le_rpow_of_nonpos
+    (by exact_mod_cast Tableau.standardCount_pos μ.1)
+    (by exact_mod_cast SpechtActual.standardCount_le_degree μ) (neg_nonpos.mpr hu.le)
+
+theorem symmetricReciprocalDegreeBounded : SymmetricReciprocalDegreeBoundedStatement := by
+  intro u hu
+  obtain ⟨a,ha⟩ := TableauZeta.fullSum_bddAbove u hu
+  refine ⟨a,?_⟩
+  rintro x ⟨n,_,rfl⟩
+  exact (symmetricInverseDegreeSum_le_tableaux n u hu).trans (ha (Set.mem_range_self n))
+
+lemma symmetricInverseDegreeSum_le_constant {n : ℕ} (hn : 1 ≤ n) {u : ℝ} (hu : 0 < u) :
+    symmetricInverseDegreeSum n u ≤ symmetricDegreeConstant u := by
+  exact le_csSup (symmetricReciprocalDegreeBounded u hu) ⟨n,hn,rfl⟩
+
+lemma symmetricDegreeConstant_nonneg {u : ℝ} (hu : 0 < u) :
+    0 ≤ symmetricDegreeConstant u :=
+  (inverseDegreeSum_nonneg (Equiv.Perm (Fin 1)) u).trans
+    (symmetricInverseDegreeSum_le_constant (n := 1) le_rfl hu)
+
+end
+end PartialPermutation
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/ReciprocalTransfer.lean b/lean/OAI/RepresentationTheory/PartialPermutation/ReciprocalTransfer.lean
index de68cfd6df1e4504dc876cc00f9c1598d6d28bcf..edac54db40a654833de556b1d916fccfbc3fade9 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/ReciprocalTransfer.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/ReciprocalTransfer.lean
@@ -1,90 +1,90 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.DegreeTransfer
-
-namespace OAI
-
-section
-open scoped Classical
-open scoped BigOperators ComplexConjugate MonoidAlgebra
-open scoped BigOperators ComplexConjugate
-open scoped MonoidAlgebra BigOperators
-open scoped BigOperators MonoidAlgebra Classical
-
-attribute [local instance] Classical.propDecidable
-open scoped MonoidAlgebra BigOperators
-open scoped BigOperators
-
-namespace PartialPermutation
-noncomputable section
-
-lemma inverseDegreeSum_nonneg (G : Type*) [Group G] [Fintype G] (u : ℝ) :
-    0 ≤ inverseDegreeSum G u := by
-  exact Finset.sum_nonneg (fun c _ => Real.rpow_nonneg (by positivity) _)
-
-lemma smallDegreeSum_le_inverseDegreeSum (G : Type*) [Group G] [Fintype G]
-    {u R : ℝ} (hu : 0 < u) (hR : 0 < R) :
-    smallDegreeSum G R ≤ R^(u+2) * inverseDegreeSum G u := by
-  rw [smallDegreeSum, inverseDegreeSum, Finset.mul_sum]
-  apply Finset.sum_le_sum
-  intro c _
-  split_ifs with hc
-  · have hd : 0 < (irreducibleDegree c : ℝ) := by
-      have : Nontrivial (irreducibleSpace c) :=
-        IsSimpleModule.nontrivial ℂ[G] (irreducibleRep c).asModule
-      exact_mod_cast (Module.finrank_pos : 0 < irreducibleDegree c)
-    calc
-      (irreducibleDegree c : ℝ)^2 =
-          (irreducibleDegree c : ℝ)^(u+2) * (irreducibleDegree c : ℝ)^(-u) := by
-        rw [← Real.rpow_add hd]
-        ring_nf
-        rw [Real.rpow_two]
-      _ ≤ _ := mul_le_mul_of_nonneg_right
-        (Real.rpow_le_rpow hd.le hc (by linarith)) (Real.rpow_nonneg hd.le _)
-  · positivity
-
-variable {G V : Type*} [Group G] [Fintype G]
-    [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-    (ρ : Representation ℂ G V) {b : ℕ} (H : Fin b → Subgroup G)
-
-theorem inverse_degree_sum_transfer [Representation.IsIrreducible ρ]
-    (hb : 0 < b) (hρ : IsUnitary ρ)
-    (hcomm : ∀ i j, i ≠ j → ∀ g ∈ H i, ∀ k ∈ H j, Commute g k)
-    (f : G → ℝ) (B u : ℝ) (hf : ∀ g, 0 ≤ f g) (hu : 0 < u)
-    (hcap : ∀ i, LeftCosetCap f (H i) B) :
-    hsNormSq (complexFourier ρ (fun g => f g)) ≤
-      B * mass f * (Module.finrank ℂ V : ℝ)^(-1 + (u+2)/(b : ℝ)) *
-        ∑ i, inverseDegreeSum (H i) u := by
-  have : Nontrivial V := IsSimpleModule.nontrivial ℂ[G] ρ.asModule
-  have hD : 0 < (Module.finrank ℂ V : ℝ) := by exact_mod_cast (Module.finrank_pos : 0 < Module.finrank ℂ V)
-  have hB : 0 ≤ B := by
-    let i : Fin b := ⟨0, hb⟩
-    have hp : (0 : ℝ) < (H i).index := by
-      exact_mod_cast Nat.pos_of_ne_zero (H i).index_ne_zero_of_finite
-    have hz : 0 ≤ B / (H i).index :=
-      (Finset.sum_nonneg (fun (g : H i) _ => hf (1*g))).trans (hcap i 1)
-    simpa only [div_mul_cancel₀ B hp.ne'] using mul_nonneg hz hp.le
-  have hm : 0 ≤ mass f := Finset.sum_nonneg (fun g _ => hf g)
-  have ht := small_degree_sum_transfer ρ H hb hρ hcomm f B hf hcap
-  have hs : ∑ i, smallDegreeSum (H i) ((Module.finrank ℂ V : ℝ)^(1 / (b : ℝ))) ≤
-      (Module.finrank ℂ V : ℝ)^((u+2)/(b : ℝ)) * ∑ i, inverseDegreeSum (H i) u := by
-    rw [Finset.mul_sum]
-    apply Finset.sum_le_sum
-    intro i _
-    have ht := smallDegreeSum_le_inverseDegreeSum (H i) hu (Real.rpow_pos_of_pos hD (1/(b : ℝ)))
-    rw [← Real.rpow_mul hD.le] at ht
-    convert ht using 1; congr 2; ring
-  have hn := ht.trans (mul_le_mul_of_nonneg_left hs (mul_nonneg hB hm))
-  apply (mul_le_mul_iff_right₀ hD).mp
-  calc
-    _ ≤ _ := hn
-    _ = _ := by
-      rw [show (u+2)/(b : ℝ) = (-1+(u+2)/(b : ℝ))+1 by ring,
-        Real.rpow_add hD, Real.rpow_one]
-      ring_nf
-
-end
-end PartialPermutation
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.DegreeTransfer
+
+namespace OAI
+
+section
+open scoped Classical
+open scoped BigOperators ComplexConjugate MonoidAlgebra
+open scoped BigOperators ComplexConjugate
+open scoped MonoidAlgebra BigOperators
+open scoped BigOperators MonoidAlgebra Classical
+
+attribute [local instance] Classical.propDecidable
+open scoped MonoidAlgebra BigOperators
+open scoped BigOperators
+
+namespace PartialPermutation
+noncomputable section
+
+lemma inverseDegreeSum_nonneg (G : Type*) [Group G] [Fintype G] (u : ℝ) :
+    0 ≤ inverseDegreeSum G u := by
+  exact Finset.sum_nonneg (fun c _ => Real.rpow_nonneg (by positivity) _)
+
+lemma smallDegreeSum_le_inverseDegreeSum (G : Type*) [Group G] [Fintype G]
+    {u R : ℝ} (hu : 0 < u) (hR : 0 < R) :
+    smallDegreeSum G R ≤ R^(u+2) * inverseDegreeSum G u := by
+  rw [smallDegreeSum, inverseDegreeSum, Finset.mul_sum]
+  apply Finset.sum_le_sum
+  intro c _
+  split_ifs with hc
+  · have hd : 0 < (irreducibleDegree c : ℝ) := by
+      have : Nontrivial (irreducibleSpace c) :=
+        IsSimpleModule.nontrivial ℂ[G] (irreducibleRep c).asModule
+      exact_mod_cast (Module.finrank_pos : 0 < irreducibleDegree c)
+    calc
+      (irreducibleDegree c : ℝ)^2 =
+          (irreducibleDegree c : ℝ)^(u+2) * (irreducibleDegree c : ℝ)^(-u) := by
+        rw [← Real.rpow_add hd]
+        ring_nf
+        rw [Real.rpow_two]
+      _ ≤ _ := mul_le_mul_of_nonneg_right
+        (Real.rpow_le_rpow hd.le hc (by linarith)) (Real.rpow_nonneg hd.le _)
+  · positivity
+
+variable {G V : Type*} [Group G] [Fintype G]
+    [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+    (ρ : Representation ℂ G V) {b : ℕ} (H : Fin b → Subgroup G)
+
+theorem inverse_degree_sum_transfer [Representation.IsIrreducible ρ]
+    (hb : 0 < b) (hρ : IsUnitary ρ)
+    (hcomm : ∀ i j, i ≠ j → ∀ g ∈ H i, ∀ k ∈ H j, Commute g k)
+    (f : G → ℝ) (B u : ℝ) (hf : ∀ g, 0 ≤ f g) (hu : 0 < u)
+    (hcap : ∀ i, LeftCosetCap f (H i) B) :
+    hsNormSq (complexFourier ρ (fun g => f g)) ≤
+      B * mass f * (Module.finrank ℂ V : ℝ)^(-1 + (u+2)/(b : ℝ)) *
+        ∑ i, inverseDegreeSum (H i) u := by
+  have : Nontrivial V := IsSimpleModule.nontrivial ℂ[G] ρ.asModule
+  have hD : 0 < (Module.finrank ℂ V : ℝ) := by exact_mod_cast (Module.finrank_pos : 0 < Module.finrank ℂ V)
+  have hB : 0 ≤ B := by
+    let i : Fin b := ⟨0, hb⟩
+    have hp : (0 : ℝ) < (H i).index := by
+      exact_mod_cast Nat.pos_of_ne_zero (H i).index_ne_zero_of_finite
+    have hz : 0 ≤ B / (H i).index :=
+      (Finset.sum_nonneg (fun (g : H i) _ => hf (1*g))).trans (hcap i 1)
+    simpa only [div_mul_cancel₀ B hp.ne'] using mul_nonneg hz hp.le
+  have hm : 0 ≤ mass f := Finset.sum_nonneg (fun g _ => hf g)
+  have ht := small_degree_sum_transfer ρ H hb hρ hcomm f B hf hcap
+  have hs : ∑ i, smallDegreeSum (H i) ((Module.finrank ℂ V : ℝ)^(1 / (b : ℝ))) ≤
+      (Module.finrank ℂ V : ℝ)^((u+2)/(b : ℝ)) * ∑ i, inverseDegreeSum (H i) u := by
+    rw [Finset.mul_sum]
+    apply Finset.sum_le_sum
+    intro i _
+    have ht := smallDegreeSum_le_inverseDegreeSum (H i) hu (Real.rpow_pos_of_pos hD (1/(b : ℝ)))
+    rw [← Real.rpow_mul hD.le] at ht
+    convert ht using 1; congr 2; ring
+  have hn := ht.trans (mul_le_mul_of_nonneg_left hs (mul_nonneg hB hm))
+  apply (mul_le_mul_iff_right₀ hD).mp
+  calc
+    _ ≤ _ := hn
+    _ = _ := by
+      rw [show (u+2)/(b : ℝ) = (-1+(u+2)/(b : ℝ))+1 by ring,
+        Real.rpow_add hD, Real.rpow_one]
+      ring_nf
+
+end
+end PartialPermutation
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/RegularRepresentation.lean b/lean/OAI/RepresentationTheory/PartialPermutation/RegularRepresentation.lean
index 5735b6a321656fc8532aa8194b62ba35e581f081..0ed84363cde81d2c048eae24dea0690f7a801dbb 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/RegularRepresentation.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/RegularRepresentation.lean
@@ -1,39 +1,39 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.Model
-
-namespace OAI
-
-section
-open scoped Classical
-open scoped BigOperators ComplexConjugate MonoidAlgebra
-open scoped BigOperators ComplexConjugate
-open scoped MonoidAlgebra BigOperators
-open scoped BigOperators MonoidAlgebra Classical
-
-namespace PartialPermutation
-noncomputable section
-variable (G : Type*) [Group G] [Fintype G]
-
-omit [Fintype G] in
-@[simp] lemma regularRep_apply (g h : G) (f : EuclideanSpace ℂ G) :
-    regularRep G g f h = f (g⁻¹ * h) := rfl
-
-lemma regularRep_unitary (g : G) (f k : EuclideanSpace ℂ G) :
-    inner ℂ (regularRep G g f) (regularRep G g k) = inner ℂ f k := by
-  simp only [PiLp.inner_apply, regularRep_apply]
-  exact Fintype.sum_equiv (Equiv.mulLeft g⁻¹) _ _ (fun h => rfl)
-
-def deltaOne : EuclideanSpace ℂ G := WithLp.toLp 2 (fun g => if g = 1 then 1 else 0)
-
-omit [Fintype G] in
-lemma regularRep_delta (g h : G) :
-    regularRep G g (deltaOne G) h = if h = g then 1 else 0 := by
-  change (if g⁻¹ * h = 1 then (1 : ℂ) else 0) = if h = g then 1 else 0
-  simp [inv_mul_eq_one, eq_comm]
-
-end
-end PartialPermutation
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.Model
+
+namespace OAI
+
+section
+open scoped Classical
+open scoped BigOperators ComplexConjugate MonoidAlgebra
+open scoped BigOperators ComplexConjugate
+open scoped MonoidAlgebra BigOperators
+open scoped BigOperators MonoidAlgebra Classical
+
+namespace PartialPermutation
+noncomputable section
+variable (G : Type*) [Group G] [Fintype G]
+
+omit [Fintype G] in
+@[simp] lemma regularRep_apply (g h : G) (f : EuclideanSpace ℂ G) :
+    regularRep G g f h = f (g⁻¹ * h) := rfl
+
+lemma regularRep_unitary (g : G) (f k : EuclideanSpace ℂ G) :
+    inner ℂ (regularRep G g f) (regularRep G g k) = inner ℂ f k := by
+  simp only [PiLp.inner_apply, regularRep_apply]
+  exact Fintype.sum_equiv (Equiv.mulLeft g⁻¹) _ _ (fun h => rfl)
+
+def deltaOne : EuclideanSpace ℂ G := WithLp.toLp 2 (fun g => if g = 1 then 1 else 0)
+
+omit [Fintype G] in
+lemma regularRep_delta (g h : G) :
+    regularRep G g (deltaOne G) h = if h = g then 1 else 0 := by
+  change (if g⁻¹ * h = 1 then (1 : ℂ) else 0) = if h = g then 1 else 0
+  simp [inv_mul_eq_one, eq_comm]
+
+end
+end PartialPermutation
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/Restriction.lean b/lean/OAI/RepresentationTheory/PartialPermutation/Restriction.lean
index caa1a738701fc1235e630fb1821190f33525c188..efd30b578133403ed2718fbfd67320400275b9af 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/Restriction.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/Restriction.lean
@@ -1,148 +1,148 @@
-import Mathlib
-
-namespace OAI
-
-section
-open scoped Classical
-open scoped BigOperators ComplexConjugate MonoidAlgebra
-open scoped BigOperators ComplexConjugate
-open scoped MonoidAlgebra BigOperators
-open scoped BigOperators MonoidAlgebra Classical
-
-attribute [local instance] Classical.propDecidable
-open scoped BigOperators
-open scoped MonoidAlgebra BigOperators
-
-namespace PartialPermutation
-
-lemma isotypic_restrict_of_fullyInvariant {R S M : Type*}
-    [Ring R] [Ring S] [AddCommGroup M] [Module R M] [Module S M]
-    [IsSemisimpleModule R M] [IsSemisimpleModule S M]
-    (φ : S →+* R) (hsmul : ∀ s : S, ∀ x : M, φ s • x = s • x)
-    (h : IsIsotypic R M)
-    (hinv : ∀ N : Submodule S M, N.IsFullyInvariant →
-      ∀ r : R, ∀ x ∈ N, r • x ∈ N) : IsIsotypic S M := by
-  apply isIsotypic_iff_isFullyInvariant_imp_bot_or_top.mpr
-  intro N hN
-  let NR : Submodule R M := { N with smul_mem' := hinv N hN }
-  have hNR : NR.IsFullyInvariant := by
-    intro f x hx
-    let fS : Module.End S M :=
-      { f.toAddMonoidHom with
-        map_smul' := fun s x => by
-          change f (s • x) = s • f x
-          rw [← hsmul, f.map_smul, hsmul] }
-    exact hN fS hx
-  rcases (isIsotypic_iff_isFullyInvariant_imp_bot_or_top.mp h) NR hNR with h0 | h1
-  · left
-    ext x
-    change x ∈ NR ↔ x = 0
-    rw [h0, Submodule.mem_bot]
-  · right
-    ext x
-    change x ∈ NR ↔ x ∈ (⊤ : Submodule S M)
-    simp [h1]
-
-section GroupRestriction
-variable {G M : Type*} [Group G] [AddCommGroup M] [Module ℂ[G] M]
-
-@[instance_reducible] noncomputable def restrictGroupModule (H : Subgroup G) : Module ℂ[H] M :=
-  Module.compHom M (MonoidAlgebra.mapDomainAlgHom ℂ ℂ H.subtype).toRingHom
-
-lemma restrictGroupModule_smul (H : Subgroup G) (s : ℂ[H]) (x : M) :
-    letI := restrictGroupModule (M := M) H
-    s • x = (MonoidAlgebra.mapDomainAlgHom ℂ ℂ H.subtype s) • x := rfl
-
-lemma group_isotypic_restrict [Finite G] {ι : Type*} (H : ι → Subgroup G)
-    (hgen : ⨆ i, H i = ⊤)
-    (hcomm : ∀ i j, i ≠ j → ∀ g ∈ H i, ∀ k ∈ H j, Commute g k)
-    (h : IsIsotypic ℂ[G] M) (i : ι) :
-    letI := restrictGroupModule (M := M) (H i)
-    IsIsotypic ℂ[H i] M := by
-  let := restrictGroupModule (M := M) (H i)
-  let φ : ℂ[H i] →+* ℂ[G] :=
-    (MonoidAlgebra.mapDomainAlgHom ℂ ℂ (H i).subtype).toRingHom
-  apply isotypic_restrict_of_fullyInvariant φ (fun s x => rfl) h
-  intro N hN
-  have hsingle (g : G) : ∀ x ∈ N, (MonoidAlgebra.single g (1 : ℂ)) • x ∈ N := by
-    have hg : g ∈ ⨆ j, H j := by rw [hgen]; exact Subgroup.mem_top g
-    refine Subgroup.iSup_induction H
-      (C := fun g => ∀ x ∈ N, (MonoidAlgebra.single g (1 : ℂ)) • x ∈ N) hg ?_ ?_ ?_
-    · intro j g hg
-      by_cases hji : j = i
-      · subst j
-        intro x hx
-        have hs := N.smul_mem (MonoidAlgebra.single (⟨g, hg⟩ : H i) (1 : ℂ)) hx
-        change φ (MonoidAlgebra.single (⟨g, hg⟩ : H i) (1 : ℂ)) • x ∈ N at hs
-        simpa [φ, MonoidAlgebra.mapDomainAlgHom] using hs
-      · have hφ (a : ℂ[H i]) : Commute (MonoidAlgebra.single g (1 : ℂ)) (φ a) := by
-          induction a using MonoidAlgebra.induction_linear with
-          | zero => simp
-          | add a b ha hb => simpa using ha.add_right hb
-          | single k a =>
-            simpa [φ, MonoidAlgebra.mapDomainAlgHom] using
-              (MonoidAlgebra.single_commute_single (hcomm j i hji g hg k k.property)
-                (Commute.all (1 : ℂ) a))
-        let T : Module.End ℂ[H i] M :=
-          { toFun := fun x => (MonoidAlgebra.single g (1 : ℂ)) • x
-            map_add' := smul_add _
-            map_smul' := fun a x => by
-              change (MonoidAlgebra.single g (1 : ℂ)) • (φ a • x) =
-                φ a • ((MonoidAlgebra.single g (1 : ℂ)) • x)
-              simp only [smul_smul, (hφ a).eq] }
-        exact fun x hx => hN T hx
-    · intro x hx
-      simpa [← MonoidAlgebra.one_def] using hx
-    · intro g k hg hk x hx
-      have he : (MonoidAlgebra.single (g * k) (1 : ℂ) : ℂ[G]) =
-          MonoidAlgebra.single g 1 * MonoidAlgebra.single k 1 := by
-        simp [MonoidAlgebra.single_mul_single]
-      rw [he, mul_smul]
-      exact hg _ (hk _ hx)
-  intro r x hx
-  induction r using MonoidAlgebra.induction_linear with
-  | zero => simp
-  | add a b ha hb => simpa [add_smul] using N.add_mem ha hb
-  | single g a =>
-    have hg := hsingle g x hx
-    have hs := N.smul_mem (MonoidAlgebra.single (1 : H i) a) hg
-    change φ (MonoidAlgebra.single (1 : H i) a) •
-      ((MonoidAlgebra.single g (1 : ℂ)) • x) ∈ N at hs
-    simpa [φ, MonoidAlgebra.mapDomainAlgHom, smul_smul,
-      MonoidAlgebra.single_mul_single] using hs
-
-lemma restrictGroupModule_tower [Module ℂ M] [IsScalarTower ℂ ℂ[G] M]
-    (H : Subgroup G) :
-    letI := restrictGroupModule (M := M) H
-    IsScalarTower ℂ ℂ[H] M := by
-  let := restrictGroupModule (M := M) H
-  apply IsScalarTower.of_algebraMap_smul
-  intro c x
-  change (MonoidAlgebra.mapDomainAlgHom ℂ ℂ H.subtype (algebraMap ℂ ℂ[H] c)) • x = c • x
-  rw [AlgHom.commutes, algebraMap_smul]
-
-lemma subgroup_algebra_commute (H K : Subgroup G)
-    (h : ∀ g ∈ H, ∀ k ∈ K, Commute g k) (a : ℂ[H]) (b : ℂ[K]) :
-    Commute (MonoidAlgebra.mapDomainAlgHom ℂ ℂ H.subtype a)
-      (MonoidAlgebra.mapDomainAlgHom ℂ ℂ K.subtype b) := by
-  induction a using MonoidAlgebra.induction_linear with
-  | zero => simp
-  | add a₁ a₂ ha₁ ha₂ =>
-    simp only [map_add]
-    exact ha₁.add_left ha₂
-  | single g c =>
-    induction b using MonoidAlgebra.induction_linear with
-    | zero => simp
-    | add b₁ b₂ hb₁ hb₂ =>
-      simp only [map_add]
-      exact hb₁.add_right hb₂
-    | single k d =>
-      simpa [MonoidAlgebra.mapDomainAlgHom] using
-        (MonoidAlgebra.single_commute_single (h g g.property k k.property) (Commute.all c d))
-
-end GroupRestriction
-end PartialPermutation
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+section
+open scoped Classical
+open scoped BigOperators ComplexConjugate MonoidAlgebra
+open scoped BigOperators ComplexConjugate
+open scoped MonoidAlgebra BigOperators
+open scoped BigOperators MonoidAlgebra Classical
+
+attribute [local instance] Classical.propDecidable
+open scoped BigOperators
+open scoped MonoidAlgebra BigOperators
+
+namespace PartialPermutation
+
+lemma isotypic_restrict_of_fullyInvariant {R S M : Type*}
+    [Ring R] [Ring S] [AddCommGroup M] [Module R M] [Module S M]
+    [IsSemisimpleModule R M] [IsSemisimpleModule S M]
+    (φ : S →+* R) (hsmul : ∀ s : S, ∀ x : M, φ s • x = s • x)
+    (h : IsIsotypic R M)
+    (hinv : ∀ N : Submodule S M, N.IsFullyInvariant →
+      ∀ r : R, ∀ x ∈ N, r • x ∈ N) : IsIsotypic S M := by
+  apply isIsotypic_iff_isFullyInvariant_imp_bot_or_top.mpr
+  intro N hN
+  let NR : Submodule R M := { N with smul_mem' := hinv N hN }
+  have hNR : NR.IsFullyInvariant := by
+    intro f x hx
+    let fS : Module.End S M :=
+      { f.toAddMonoidHom with
+        map_smul' := fun s x => by
+          change f (s • x) = s • f x
+          rw [← hsmul, f.map_smul, hsmul] }
+    exact hN fS hx
+  rcases (isIsotypic_iff_isFullyInvariant_imp_bot_or_top.mp h) NR hNR with h0 | h1
+  · left
+    ext x
+    change x ∈ NR ↔ x = 0
+    rw [h0, Submodule.mem_bot]
+  · right
+    ext x
+    change x ∈ NR ↔ x ∈ (⊤ : Submodule S M)
+    simp [h1]
+
+section GroupRestriction
+variable {G M : Type*} [Group G] [AddCommGroup M] [Module ℂ[G] M]
+
+@[instance_reducible] noncomputable def restrictGroupModule (H : Subgroup G) : Module ℂ[H] M :=
+  Module.compHom M (MonoidAlgebra.mapDomainAlgHom ℂ ℂ H.subtype).toRingHom
+
+lemma restrictGroupModule_smul (H : Subgroup G) (s : ℂ[H]) (x : M) :
+    letI := restrictGroupModule (M := M) H
+    s • x = (MonoidAlgebra.mapDomainAlgHom ℂ ℂ H.subtype s) • x := rfl
+
+lemma group_isotypic_restrict [Finite G] {ι : Type*} (H : ι → Subgroup G)
+    (hgen : ⨆ i, H i = ⊤)
+    (hcomm : ∀ i j, i ≠ j → ∀ g ∈ H i, ∀ k ∈ H j, Commute g k)
+    (h : IsIsotypic ℂ[G] M) (i : ι) :
+    letI := restrictGroupModule (M := M) (H i)
+    IsIsotypic ℂ[H i] M := by
+  let := restrictGroupModule (M := M) (H i)
+  let φ : ℂ[H i] →+* ℂ[G] :=
+    (MonoidAlgebra.mapDomainAlgHom ℂ ℂ (H i).subtype).toRingHom
+  apply isotypic_restrict_of_fullyInvariant φ (fun s x => rfl) h
+  intro N hN
+  have hsingle (g : G) : ∀ x ∈ N, (MonoidAlgebra.single g (1 : ℂ)) • x ∈ N := by
+    have hg : g ∈ ⨆ j, H j := by rw [hgen]; exact Subgroup.mem_top g
+    refine Subgroup.iSup_induction H
+      (C := fun g => ∀ x ∈ N, (MonoidAlgebra.single g (1 : ℂ)) • x ∈ N) hg ?_ ?_ ?_
+    · intro j g hg
+      by_cases hji : j = i
+      · subst j
+        intro x hx
+        have hs := N.smul_mem (MonoidAlgebra.single (⟨g, hg⟩ : H i) (1 : ℂ)) hx
+        change φ (MonoidAlgebra.single (⟨g, hg⟩ : H i) (1 : ℂ)) • x ∈ N at hs
+        simpa [φ, MonoidAlgebra.mapDomainAlgHom] using hs
+      · have hφ (a : ℂ[H i]) : Commute (MonoidAlgebra.single g (1 : ℂ)) (φ a) := by
+          induction a using MonoidAlgebra.induction_linear with
+          | zero => simp
+          | add a b ha hb => simpa using ha.add_right hb
+          | single k a =>
+            simpa [φ, MonoidAlgebra.mapDomainAlgHom] using
+              (MonoidAlgebra.single_commute_single (hcomm j i hji g hg k k.property)
+                (Commute.all (1 : ℂ) a))
+        let T : Module.End ℂ[H i] M :=
+          { toFun := fun x => (MonoidAlgebra.single g (1 : ℂ)) • x
+            map_add' := smul_add _
+            map_smul' := fun a x => by
+              change (MonoidAlgebra.single g (1 : ℂ)) • (φ a • x) =
+                φ a • ((MonoidAlgebra.single g (1 : ℂ)) • x)
+              simp only [smul_smul, (hφ a).eq] }
+        exact fun x hx => hN T hx
+    · intro x hx
+      simpa [← MonoidAlgebra.one_def] using hx
+    · intro g k hg hk x hx
+      have he : (MonoidAlgebra.single (g * k) (1 : ℂ) : ℂ[G]) =
+          MonoidAlgebra.single g 1 * MonoidAlgebra.single k 1 := by
+        simp [MonoidAlgebra.single_mul_single]
+      rw [he, mul_smul]
+      exact hg _ (hk _ hx)
+  intro r x hx
+  induction r using MonoidAlgebra.induction_linear with
+  | zero => simp
+  | add a b ha hb => simpa [add_smul] using N.add_mem ha hb
+  | single g a =>
+    have hg := hsingle g x hx
+    have hs := N.smul_mem (MonoidAlgebra.single (1 : H i) a) hg
+    change φ (MonoidAlgebra.single (1 : H i) a) •
+      ((MonoidAlgebra.single g (1 : ℂ)) • x) ∈ N at hs
+    simpa [φ, MonoidAlgebra.mapDomainAlgHom, smul_smul,
+      MonoidAlgebra.single_mul_single] using hs
+
+lemma restrictGroupModule_tower [Module ℂ M] [IsScalarTower ℂ ℂ[G] M]
+    (H : Subgroup G) :
+    letI := restrictGroupModule (M := M) H
+    IsScalarTower ℂ ℂ[H] M := by
+  let := restrictGroupModule (M := M) H
+  apply IsScalarTower.of_algebraMap_smul
+  intro c x
+  change (MonoidAlgebra.mapDomainAlgHom ℂ ℂ H.subtype (algebraMap ℂ ℂ[H] c)) • x = c • x
+  rw [AlgHom.commutes, algebraMap_smul]
+
+lemma subgroup_algebra_commute (H K : Subgroup G)
+    (h : ∀ g ∈ H, ∀ k ∈ K, Commute g k) (a : ℂ[H]) (b : ℂ[K]) :
+    Commute (MonoidAlgebra.mapDomainAlgHom ℂ ℂ H.subtype a)
+      (MonoidAlgebra.mapDomainAlgHom ℂ ℂ K.subtype b) := by
+  induction a using MonoidAlgebra.induction_linear with
+  | zero => simp
+  | add a₁ a₂ ha₁ ha₂ =>
+    simp only [map_add]
+    exact ha₁.add_left ha₂
+  | single g c =>
+    induction b using MonoidAlgebra.induction_linear with
+    | zero => simp
+    | add b₁ b₂ hb₁ hb₂ =>
+      simp only [map_add]
+      exact hb₁.add_right hb₂
+    | single k d =>
+      simpa [MonoidAlgebra.mapDomainAlgHom] using
+        (MonoidAlgebra.single_commute_single (h g g.property k k.property) (Commute.all c d))
+
+end GroupRestriction
+end PartialPermutation
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/RowBounds.lean b/lean/OAI/RepresentationTheory/PartialPermutation/RowBounds.lean
index 789fa031e9e6a8408ee76cc7bb72ee857d4e3ac6..2ef8108a2678b7b4ef6b8cdc0284a6151294aba4 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/RowBounds.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/RowBounds.lean
@@ -1,106 +1,106 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.FillingExtensions
-import OAI.RepresentationTheory.PartialPermutation.Interleavings
-
-namespace OAI
-
-section
-namespace PartialPermutation
-namespace Tableau
-noncomputable section
-open Finset
-
-def topPrefix (μ : YoungDiagram) (c : ℕ) (x : μ.cells) : Prop :=
-  x.1.1=0 ∧ x.1.2<c
-
-instance (μ : YoungDiagram) (c : ℕ) : DecidablePred (topPrefix μ c) :=
-  fun _ => inferInstanceAs (Decidable (_ ∧ _))
-
-lemma topPrefix_lower (μ : YoungDiagram) (c : ℕ) :
-    IsLowerSet {x | topPrefix μ c x} := by
-  intro x y hxy hy
-  exact ⟨Nat.eq_zero_of_le_zero (hxy.1.trans_eq hy.1), hxy.2.trans_lt hy.2⟩
-
-def topPrefixEquiv (μ : YoungDiagram) (c : ℕ) (hc : c ≤ μ.rowLen 0) :
-    {x // topPrefix μ c x} ≃ Fin c where
-  toFun x := ⟨x.1.1.2,x.2.2⟩
-  invFun i := ⟨⟨(0,i), μ.mem_iff_lt_rowLen.mpr (i.2.trans_le hc)⟩,rfl,i.2⟩
-  left_inv x := by apply Subtype.ext; apply Subtype.ext; exact Prod.ext x.2.1.symm rfl
-  right_inv _ := rfl
-
-lemma topPrefix_card (μ : YoungDiagram) (c : ℕ) (hc : c ≤ μ.rowLen 0) :
-    Fintype.card {x // topPrefix μ c x} = c := by
-  simpa using Fintype.card_congr (topPrefixEquiv μ c hc)
-
-def topRowEquiv (μ : YoungDiagram) : {x : μ.cells // x.1.1=0} ≃ Fin (μ.rowLen 0) where
-  toFun x := ⟨x.1.1.2, by have h:=μ.mem_iff_lt_rowLen.mp x.1.2; simpa [x.2] using h⟩
-  invFun i := ⟨⟨(0,i), μ.mem_iff_lt_rowLen.mpr i.2⟩,rfl⟩
-  left_inv x := by apply Subtype.ext; apply Subtype.ext; exact Prod.ext x.2.symm rfl
-  right_inv _ := rfl
-
-lemma lower_card (μ : YoungDiagram) :
-    Fintype.card {x : μ.cells // x.1.1 ≠ 0} = μ.cells.card - μ.rowLen 0 := by
-  rw [Fintype.card_subtype_compl]
-  have h := Fintype.card_congr (topRowEquiv μ)
-  rw [Fintype.card_fin] at h
-  rw [h, Fintype.card_coe]
-
-lemma topRow_le_card (μ : YoungDiagram) : μ.rowLen 0 ≤ μ.cells.card := by
-  have h := Fintype.card_le_of_injective (topRowEquiv μ).symm
-    (topRowEquiv μ).symm.injective
-  have h' := Fintype.card_subtype_le (fun x : μ.cells => x.1.1=0)
-  simpa only [Fintype.card_fin, Fintype.card_coe] using h.trans h'
-
-lemma prefix_upper_card (μ : YoungDiagram) (c : ℕ) (hc : c ≤ μ.rowLen 0) :
-    Fintype.card {x // ¬topPrefix μ c x} = μ.cells.card-c := by
-  rw [Fintype.card_subtype_compl, topPrefix_card μ c hc, Fintype.card_coe]
-
-def lowerInUpperEquiv (μ : YoungDiagram) (c : ℕ) :
-    {x : {x // ¬topPrefix μ c x} // x.1.1.1 ≠ 0} ≃ {x : μ.cells // x.1.1 ≠ 0} where
-  toFun x := ⟨x.1.1,x.2⟩
-  invFun x := ⟨⟨x.1,fun h => x.2 h.1⟩,x.2⟩
-  left_inv _ := rfl
-  right_inv _ := rfl
-
-lemma row_interleaving_lower_bound (μ : YoungDiagram) :
-    (μ.cells.card-μ.rowLen 1).choose (μ.cells.card-μ.rowLen 0) ≤ standardCount μ := by
-  let P := topPrefix μ (μ.rowLen 1)
-  let Q : {x // ¬P x} → Prop := fun x => x.1.1.1 ≠ 0
-  have hP := topPrefix_lower μ (μ.rowLen 1)
-  have hQ : ∀ a b : {x // ¬P x}, a < b → (Q a ↔ Q b) := by
-    intro a b hab
-    change (a.1.1.1 ≠ 0 ↔ b.1.1.1 ≠ 0)
-    have hi := hab.le.1
-    have hj := hab.le.2
-    constructor
-    · intro ha hb
-      exact ha (Nat.eq_zero_of_le_zero (hi.trans_eq hb))
-    · intro hb ha
-      have hcol : b.1.1.2 < μ.rowLen 1 :=
-        (μ.mem_iff_lt_rowLen.mp b.1.2).trans_le (μ.rowLen_anti 1 b.1.1.1 (by omega))
-      exact a.2 ⟨ha,hj.trans_lt hcol⟩
-  have h := interleaving_binomial_lower_bound Q hQ
-  have hc : μ.rowLen 1 ≤ μ.rowLen 0 := μ.rowLen_anti 0 1 (by omega)
-  have hcard : Fintype.card {x : {x // ¬P x} // Q x} = μ.cells.card-μ.rowLen 0 :=
-    (Fintype.card_congr (lowerInUpperEquiv μ (μ.rowLen 1))).trans (lower_card μ)
-  rw [hcard] at h
-  have hu : Fintype.card {x // ¬P x} = μ.cells.card-μ.rowLen 1 := prefix_upper_card μ _ hc
-  rw [hu] at h
-  exact h.trans (filling_count_mono_complement P hP)
-
-lemma secondRow_le_lower (μ : YoungDiagram) : μ.rowLen 1 ≤ μ.cells.card-μ.rowLen 0 := by
-  let f : Fin (μ.rowLen 1) → {x : μ.cells // x.1.1 ≠ 0} := fun i =>
-    ⟨⟨(1,i), μ.mem_iff_lt_rowLen.mpr i.2⟩,Nat.one_ne_zero⟩
-  have hf : Function.Injective f := by
-    intro i j hij
-    apply Fin.ext
-    exact congrArg (fun x : {x : μ.cells // x.1.1 ≠ 0} => x.1.1.2) hij
-  have h := Fintype.card_le_of_injective f hf
-  simpa only [Fintype.card_fin, lower_card] using h
-
-end
-end Tableau
-end PartialPermutation
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.FillingExtensions
+import OAI.RepresentationTheory.PartialPermutation.Interleavings
+
+namespace OAI
+
+section
+namespace PartialPermutation
+namespace Tableau
+noncomputable section
+open Finset
+
+def topPrefix (μ : YoungDiagram) (c : ℕ) (x : μ.cells) : Prop :=
+  x.1.1=0 ∧ x.1.2<c
+
+instance (μ : YoungDiagram) (c : ℕ) : DecidablePred (topPrefix μ c) :=
+  fun _ => inferInstanceAs (Decidable (_ ∧ _))
+
+lemma topPrefix_lower (μ : YoungDiagram) (c : ℕ) :
+    IsLowerSet {x | topPrefix μ c x} := by
+  intro x y hxy hy
+  exact ⟨Nat.eq_zero_of_le_zero (hxy.1.trans_eq hy.1), hxy.2.trans_lt hy.2⟩
+
+def topPrefixEquiv (μ : YoungDiagram) (c : ℕ) (hc : c ≤ μ.rowLen 0) :
+    {x // topPrefix μ c x} ≃ Fin c where
+  toFun x := ⟨x.1.1.2,x.2.2⟩
+  invFun i := ⟨⟨(0,i), μ.mem_iff_lt_rowLen.mpr (i.2.trans_le hc)⟩,rfl,i.2⟩
+  left_inv x := by apply Subtype.ext; apply Subtype.ext; exact Prod.ext x.2.1.symm rfl
+  right_inv _ := rfl
+
+lemma topPrefix_card (μ : YoungDiagram) (c : ℕ) (hc : c ≤ μ.rowLen 0) :
+    Fintype.card {x // topPrefix μ c x} = c := by
+  simpa using Fintype.card_congr (topPrefixEquiv μ c hc)
+
+def topRowEquiv (μ : YoungDiagram) : {x : μ.cells // x.1.1=0} ≃ Fin (μ.rowLen 0) where
+  toFun x := ⟨x.1.1.2, by have h:=μ.mem_iff_lt_rowLen.mp x.1.2; simpa [x.2] using h⟩
+  invFun i := ⟨⟨(0,i), μ.mem_iff_lt_rowLen.mpr i.2⟩,rfl⟩
+  left_inv x := by apply Subtype.ext; apply Subtype.ext; exact Prod.ext x.2.symm rfl
+  right_inv _ := rfl
+
+lemma lower_card (μ : YoungDiagram) :
+    Fintype.card {x : μ.cells // x.1.1 ≠ 0} = μ.cells.card - μ.rowLen 0 := by
+  rw [Fintype.card_subtype_compl]
+  have h := Fintype.card_congr (topRowEquiv μ)
+  rw [Fintype.card_fin] at h
+  rw [h, Fintype.card_coe]
+
+lemma topRow_le_card (μ : YoungDiagram) : μ.rowLen 0 ≤ μ.cells.card := by
+  have h := Fintype.card_le_of_injective (topRowEquiv μ).symm
+    (topRowEquiv μ).symm.injective
+  have h' := Fintype.card_subtype_le (fun x : μ.cells => x.1.1=0)
+  simpa only [Fintype.card_fin, Fintype.card_coe] using h.trans h'
+
+lemma prefix_upper_card (μ : YoungDiagram) (c : ℕ) (hc : c ≤ μ.rowLen 0) :
+    Fintype.card {x // ¬topPrefix μ c x} = μ.cells.card-c := by
+  rw [Fintype.card_subtype_compl, topPrefix_card μ c hc, Fintype.card_coe]
+
+def lowerInUpperEquiv (μ : YoungDiagram) (c : ℕ) :
+    {x : {x // ¬topPrefix μ c x} // x.1.1.1 ≠ 0} ≃ {x : μ.cells // x.1.1 ≠ 0} where
+  toFun x := ⟨x.1.1,x.2⟩
+  invFun x := ⟨⟨x.1,fun h => x.2 h.1⟩,x.2⟩
+  left_inv _ := rfl
+  right_inv _ := rfl
+
+lemma row_interleaving_lower_bound (μ : YoungDiagram) :
+    (μ.cells.card-μ.rowLen 1).choose (μ.cells.card-μ.rowLen 0) ≤ standardCount μ := by
+  let P := topPrefix μ (μ.rowLen 1)
+  let Q : {x // ¬P x} → Prop := fun x => x.1.1.1 ≠ 0
+  have hP := topPrefix_lower μ (μ.rowLen 1)
+  have hQ : ∀ a b : {x // ¬P x}, a < b → (Q a ↔ Q b) := by
+    intro a b hab
+    change (a.1.1.1 ≠ 0 ↔ b.1.1.1 ≠ 0)
+    have hi := hab.le.1
+    have hj := hab.le.2
+    constructor
+    · intro ha hb
+      exact ha (Nat.eq_zero_of_le_zero (hi.trans_eq hb))
+    · intro hb ha
+      have hcol : b.1.1.2 < μ.rowLen 1 :=
+        (μ.mem_iff_lt_rowLen.mp b.1.2).trans_le (μ.rowLen_anti 1 b.1.1.1 (by omega))
+      exact a.2 ⟨ha,hj.trans_lt hcol⟩
+  have h := interleaving_binomial_lower_bound Q hQ
+  have hc : μ.rowLen 1 ≤ μ.rowLen 0 := μ.rowLen_anti 0 1 (by omega)
+  have hcard : Fintype.card {x : {x // ¬P x} // Q x} = μ.cells.card-μ.rowLen 0 :=
+    (Fintype.card_congr (lowerInUpperEquiv μ (μ.rowLen 1))).trans (lower_card μ)
+  rw [hcard] at h
+  have hu : Fintype.card {x // ¬P x} = μ.cells.card-μ.rowLen 1 := prefix_upper_card μ _ hc
+  rw [hu] at h
+  exact h.trans (filling_count_mono_complement P hP)
+
+lemma secondRow_le_lower (μ : YoungDiagram) : μ.rowLen 1 ≤ μ.cells.card-μ.rowLen 0 := by
+  let f : Fin (μ.rowLen 1) → {x : μ.cells // x.1.1 ≠ 0} := fun i =>
+    ⟨⟨(1,i), μ.mem_iff_lt_rowLen.mpr i.2⟩,Nat.one_ne_zero⟩
+  have hf : Function.Injective f := by
+    intro i j hij
+    apply Fin.ext
+    exact congrArg (fun x : {x : μ.cells // x.1.1 ≠ 0} => x.1.1.2) hij
+  have h := Fintype.card_le_of_injective f hf
+  simpa only [Fintype.card_fin, lower_card] using h
+
+end
+end Tableau
+end PartialPermutation
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/SchurAverage.lean b/lean/OAI/RepresentationTheory/PartialPermutation/SchurAverage.lean
index 4a8de9d3318540927d2b6de624811b8f710ee4f5..13bccf8df89aaf174d9d7b9b0c6ef3bfb5d507b4 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/SchurAverage.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/SchurAverage.lean
@@ -1,70 +1,70 @@
-import Mathlib
-
-namespace OAI
-
-section
-
-open scoped Classical
-open scoped BigOperators ComplexConjugate MonoidAlgebra
-
-namespace PartialPermutation
-
-section SchurAverage
-variable {G V : Type*} [Group G] [Fintype G]
-    [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
-    (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
-
-noncomputable def conjugateAverage (A : Module.End ℂ V) : Module.End ℂ V :=
-  (Fintype.card G : ℂ)⁻¹ • ∑ g : G, ρ g * A * ρ g⁻¹
-
-omit [FiniteDimensional ℂ V] [Representation.IsIrreducible ρ] in
-lemma conjugateAverage_eq_averageMap (A : Module.End ℂ V) :
-    conjugateAverage ρ A =
-      letI := invertibleOfNonzero (show (Fintype.card G : ℂ) ≠ 0 by exact_mod_cast Fintype.card_ne_zero)
-      (ρ.linHom ρ).averageMap A := by
-  let := invertibleOfNonzero (show (Fintype.card G : ℂ) ≠ 0 by exact_mod_cast Fintype.card_ne_zero)
-  simp [conjugateAverage, Representation.averageMap, GroupAlgebra.average,
-    Representation.linHom_apply, ← Module.End.mul_eq_comp, mul_assoc]
-
-lemma conjugateAverage_scalar (A : Module.End ℂ V) :
-    ∃ c : ℂ, conjugateAverage ρ A = c • (1 : Module.End ℂ V) := by
-  let := invertibleOfNonzero (show (Fintype.card G : ℂ) ≠ 0 by exact_mod_cast Fintype.card_ne_zero)
-  let F : (ρ.linHom ρ).invariants :=
-    ⟨(ρ.linHom ρ).averageMap A, (ρ.linHom ρ).averageMap_invariant A⟩
-  let T := Representation.invariantsEquivIntertwiningMap ρ ρ F
-  obtain ⟨c, hc⟩ := Representation.IsIrreducible.algebraMap_intertwiningMap_bijective_of_isAlgClosed
-    (ρ := ρ) |>.surjective T
-  refine ⟨c, ?_⟩
-  rw [Representation.IntertwiningMap.algebraMap_apply] at hc
-  have h := congrArg Representation.IntertwiningMap.toLinearMap hc
-  change c • (1 : Module.End ℂ V) = (ρ.linHom ρ).averageMap A at h
-  rw [conjugateAverage_eq_averageMap]
-  exact h.symm
-
-omit [FiniteDimensional ℂ V] [Representation.IsIrreducible ρ] in
-lemma trace_conjugateAverage (A : Module.End ℂ V) :
-    LinearMap.trace ℂ V (conjugateAverage ρ A) = LinearMap.trace ℂ V A := by
-  have hc : (Fintype.card G : ℂ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
-  have ht (g : G) : LinearMap.trace ℂ V (ρ g * A * ρ g⁻¹) = LinearMap.trace ℂ V A := by
-    rw [LinearMap.trace_mul_cycle, ← map_mul, inv_mul_cancel, map_one, one_mul]
-  simp [conjugateAverage, ht, hc]
-
-lemma conjugateAverage_eq_trace (A : Module.End ℂ V) :
-    conjugateAverage ρ A =
-      (LinearMap.trace ℂ V A / (Module.finrank ℂ V : ℂ)) • (1 : Module.End ℂ V) := by
-  obtain ⟨c, hc⟩ := conjugateAverage_scalar ρ A
-  have hd : (Module.finrank ℂ V : ℂ) ≠ 0 := by
-    have : Nontrivial V := by
-      have : Nontrivial ρ.asModule := IsSimpleModule.nontrivial (MonoidAlgebra ℂ G) ρ.asModule
-      exact ρ.asModuleEquiv.symm.toEquiv.nontrivial
-    exact_mod_cast (Module.finrank_pos (R := ℂ) (M := V)).ne'
-  have ht := trace_conjugateAverage ρ A
-  rw [hc, map_smul, LinearMap.trace_one, smul_eq_mul] at ht
-  rw [hc, ← ht, mul_div_cancel_right₀ _ hd]
-
-end SchurAverage
-end PartialPermutation
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+section
+
+open scoped Classical
+open scoped BigOperators ComplexConjugate MonoidAlgebra
+
+namespace PartialPermutation
+
+section SchurAverage
+variable {G V : Type*} [Group G] [Fintype G]
+    [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
+    (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
+
+noncomputable def conjugateAverage (A : Module.End ℂ V) : Module.End ℂ V :=
+  (Fintype.card G : ℂ)⁻¹ • ∑ g : G, ρ g * A * ρ g⁻¹
+
+omit [FiniteDimensional ℂ V] [Representation.IsIrreducible ρ] in
+lemma conjugateAverage_eq_averageMap (A : Module.End ℂ V) :
+    conjugateAverage ρ A =
+      letI := invertibleOfNonzero (show (Fintype.card G : ℂ) ≠ 0 by exact_mod_cast Fintype.card_ne_zero)
+      (ρ.linHom ρ).averageMap A := by
+  let := invertibleOfNonzero (show (Fintype.card G : ℂ) ≠ 0 by exact_mod_cast Fintype.card_ne_zero)
+  simp [conjugateAverage, Representation.averageMap, GroupAlgebra.average,
+    Representation.linHom_apply, ← Module.End.mul_eq_comp, mul_assoc]
+
+lemma conjugateAverage_scalar (A : Module.End ℂ V) :
+    ∃ c : ℂ, conjugateAverage ρ A = c • (1 : Module.End ℂ V) := by
+  let := invertibleOfNonzero (show (Fintype.card G : ℂ) ≠ 0 by exact_mod_cast Fintype.card_ne_zero)
+  let F : (ρ.linHom ρ).invariants :=
+    ⟨(ρ.linHom ρ).averageMap A, (ρ.linHom ρ).averageMap_invariant A⟩
+  let T := Representation.invariantsEquivIntertwiningMap ρ ρ F
+  obtain ⟨c, hc⟩ := Representation.IsIrreducible.algebraMap_intertwiningMap_bijective_of_isAlgClosed
+    (ρ := ρ) |>.surjective T
+  refine ⟨c, ?_⟩
+  rw [Representation.IntertwiningMap.algebraMap_apply] at hc
+  have h := congrArg Representation.IntertwiningMap.toLinearMap hc
+  change c • (1 : Module.End ℂ V) = (ρ.linHom ρ).averageMap A at h
+  rw [conjugateAverage_eq_averageMap]
+  exact h.symm
+
+omit [FiniteDimensional ℂ V] [Representation.IsIrreducible ρ] in
+lemma trace_conjugateAverage (A : Module.End ℂ V) :
+    LinearMap.trace ℂ V (conjugateAverage ρ A) = LinearMap.trace ℂ V A := by
+  have hc : (Fintype.card G : ℂ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
+  have ht (g : G) : LinearMap.trace ℂ V (ρ g * A * ρ g⁻¹) = LinearMap.trace ℂ V A := by
+    rw [LinearMap.trace_mul_cycle, ← map_mul, inv_mul_cancel, map_one, one_mul]
+  simp [conjugateAverage, ht, hc]
+
+lemma conjugateAverage_eq_trace (A : Module.End ℂ V) :
+    conjugateAverage ρ A =
+      (LinearMap.trace ℂ V A / (Module.finrank ℂ V : ℂ)) • (1 : Module.End ℂ V) := by
+  obtain ⟨c, hc⟩ := conjugateAverage_scalar ρ A
+  have hd : (Module.finrank ℂ V : ℂ) ≠ 0 := by
+    have : Nontrivial V := by
+      have : Nontrivial ρ.asModule := IsSimpleModule.nontrivial (MonoidAlgebra ℂ G) ρ.asModule
+      exact ρ.asModuleEquiv.symm.toEquiv.nontrivial
+    exact_mod_cast (Module.finrank_pos (R := ℂ) (M := V)).ne'
+  have ht := trace_conjugateAverage ρ A
+  rw [hc, map_smul, LinearMap.trace_one, smul_eq_mul] at ht
+  rw [hc, ← ht, mul_div_cancel_right₀ _ hd]
+
+end SchurAverage
+end PartialPermutation
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/SmallKernels.lean b/lean/OAI/RepresentationTheory/PartialPermutation/SmallKernels.lean
index be53a9849d5c345b9b3c19e48a22d78e3d5381a5..a14066c235012a2110e53a1768749fcea8b413d5 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/SmallKernels.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/SmallKernels.lean
@@ -1,101 +1,101 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.ProjectionEnergy
-
-namespace OAI
-
-section
-open scoped Classical
-open scoped BigOperators ComplexConjugate MonoidAlgebra
-open scoped BigOperators ComplexConjugate
-open scoped MonoidAlgebra BigOperators
-open scoped BigOperators MonoidAlgebra Classical
-
-attribute [local instance] Classical.propDecidable
-namespace PartialPermutation
-noncomputable section
-variable {G : Type*} [Group G] [Fintype G]
-
-def smallTypeKernel (G : Type*) [Group G] [Fintype G] (R : ℝ) : G → ℂ :=
-  fun g => ∑ c : IrreducibleIndex G,
-    if (irreducibleDegree c : ℝ) ≤ R then centralCharacterKernel (irreducibleRep c) g else 0
-
-lemma smallTypeKernel_inv (R : ℝ) (g : G) :
-    smallTypeKernel G R g⁻¹ = conj (smallTypeKernel G R g) := by
-  simp only [smallTypeKernel, map_sum]
-  apply Finset.sum_congr rfl
-  intro c _
-  split_ifs <;> simp [centralCharacterKernel_inv _ (irreducibleRep_unitary c)]
-
-lemma complexFourier_smallTypeKernel {V : Type*} [AddCommGroup V] [Module ℂ V]
-    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
-    (R : ℝ) : complexFourier ρ (smallTypeKernel G R) =
-      if (Module.finrank ℂ V : ℝ) ≤ R then 1 else 0 := by
-  classical
-  obtain ⟨c, ⟨e⟩, hc⟩ := irreducibleRep_complete ρ
-  have hdeg : irreducibleDegree c = Module.finrank ℂ V := e.toLinearEquiv.finrank_eq
-  have he (d : IrreducibleIndex G) :
-      Nonempty (Representation.Equiv (irreducibleRep d) ρ) ↔ d = c := by
-    exact ⟨hc d, fun h => h ▸ ⟨e⟩⟩
-  unfold smallTypeKernel
-  rw [complexFourier_sum]
-  calc
-    _ = ∑ d : IrreducibleIndex G, if d = c then
-        (if (Module.finrank ℂ V : ℝ) ≤ R then (1 : Module.End ℂ V) else 0) else 0 := by
-      apply Finset.sum_congr rfl
-      intro d _
-      by_cases hd : (irreducibleDegree d : ℝ) ≤ R
-      · simp only [hd, ite_true, centralCharacterKernel_fourier]
-        by_cases hdc : d = c
-        · subst d
-          rw [hdeg] at hd
-          simp [he, hd]
-        · simp [he, hdc]
-      · simp only [hd, ite_false, complexFourier]
-        by_cases hdc : d = c
-        · subst d
-          simp [hdeg] at hd
-          simp [hd]
-        · simp [hdc]
-    _ = _ := by simp
-
-@[simp] lemma hsNormSq_one {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-    [FiniteDimensional ℂ V] : hsNormSq (1 : Module.End ℂ V) = Module.finrank ℂ V := by
-  simp [hsNormSq, LinearMap.trace_one]
-
-lemma degree_le_of_kernel_identity {V : Type*} [NormedAddCommGroup V]
-    [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-    (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
-    (k : G → ℂ) (d : ℕ) (hk : ∑ g, ‖k g‖^2 = (d : ℝ)^2 / Fintype.card G)
-    (hact : complexFourier ρ k = 1) : Module.finrank ℂ V ≤ d := by
-  have h := fourier_bessel ρ hρ k
-  have hN : (Fintype.card G : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
-  rw [hact, hsNormSq_one, hk] at h
-  have hd : (Module.finrank ℂ V : ℝ)^2 ≤ (d : ℝ)^2 := by
-    convert h using 1 <;> field_simp
-  exact_mod_cast (sq_le_sq₀ (by positivity) (by positivity)).mp hd
-
-lemma smallTypeKernel_identity_of_kernel {V : Type*} [NormedAddCommGroup V]
-    [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
-    (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
-    (k : G → ℂ) (d : ℕ) (R : ℝ) (hd : (d : ℝ) ≤ R)
-    (hk : ∑ g, ‖k g‖^2 = (d : ℝ)^2 / Fintype.card G)
-    (hact : complexFourier ρ k = 1) : complexFourier ρ (smallTypeKernel G R) = 1 := by
-  apply linearMap_eq_on_simples ρ
-  intro σ hσ x
-  have := hσ
-  have hu : IsUnitary σ.toRepresentation := fun g x y => hρ g x y
-  have hs : complexFourier σ.toRepresentation k = 1 := by
-    ext y
-    rw [complexFourier_subrep, hact]
-    rfl
-  have hdeg := degree_le_of_kernel_identity σ.toRepresentation hu k d hk hs
-  have hdegR : (Module.finrank ℂ σ.toSubmodule : ℝ) ≤ R := (Nat.cast_le.mpr hdeg).trans hd
-  rw [← complexFourier_subrep, complexFourier_smallTypeKernel, ite_eq_left hdegR]
-  rfl
-
-end
-end PartialPermutation
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.ProjectionEnergy
+
+namespace OAI
+
+section
+open scoped Classical
+open scoped BigOperators ComplexConjugate MonoidAlgebra
+open scoped BigOperators ComplexConjugate
+open scoped MonoidAlgebra BigOperators
+open scoped BigOperators MonoidAlgebra Classical
+
+attribute [local instance] Classical.propDecidable
+namespace PartialPermutation
+noncomputable section
+variable {G : Type*} [Group G] [Fintype G]
+
+def smallTypeKernel (G : Type*) [Group G] [Fintype G] (R : ℝ) : G → ℂ :=
+  fun g => ∑ c : IrreducibleIndex G,
+    if (irreducibleDegree c : ℝ) ≤ R then centralCharacterKernel (irreducibleRep c) g else 0
+
+lemma smallTypeKernel_inv (R : ℝ) (g : G) :
+    smallTypeKernel G R g⁻¹ = conj (smallTypeKernel G R g) := by
+  simp only [smallTypeKernel, map_sum]
+  apply Finset.sum_congr rfl
+  intro c _
+  split_ifs <;> simp [centralCharacterKernel_inv _ (irreducibleRep_unitary c)]
+
+lemma complexFourier_smallTypeKernel {V : Type*} [AddCommGroup V] [Module ℂ V]
+    [FiniteDimensional ℂ V] (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ]
+    (R : ℝ) : complexFourier ρ (smallTypeKernel G R) =
+      if (Module.finrank ℂ V : ℝ) ≤ R then 1 else 0 := by
+  classical
+  obtain ⟨c, ⟨e⟩, hc⟩ := irreducibleRep_complete ρ
+  have hdeg : irreducibleDegree c = Module.finrank ℂ V := e.toLinearEquiv.finrank_eq
+  have he (d : IrreducibleIndex G) :
+      Nonempty (Representation.Equiv (irreducibleRep d) ρ) ↔ d = c := by
+    exact ⟨hc d, fun h => h ▸ ⟨e⟩⟩
+  unfold smallTypeKernel
+  rw [complexFourier_sum]
+  calc
+    _ = ∑ d : IrreducibleIndex G, if d = c then
+        (if (Module.finrank ℂ V : ℝ) ≤ R then (1 : Module.End ℂ V) else 0) else 0 := by
+      apply Finset.sum_congr rfl
+      intro d _
+      by_cases hd : (irreducibleDegree d : ℝ) ≤ R
+      · simp only [hd, ite_true, centralCharacterKernel_fourier]
+        by_cases hdc : d = c
+        · subst d
+          rw [hdeg] at hd
+          simp [he, hd]
+        · simp [he, hdc]
+      · simp only [hd, ite_false, complexFourier]
+        by_cases hdc : d = c
+        · subst d
+          simp [hdeg] at hd
+          simp [hd]
+        · simp [hdc]
+    _ = _ := by simp
+
+@[simp] lemma hsNormSq_one {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+    [FiniteDimensional ℂ V] : hsNormSq (1 : Module.End ℂ V) = Module.finrank ℂ V := by
+  simp [hsNormSq, LinearMap.trace_one]
+
+lemma degree_le_of_kernel_identity {V : Type*} [NormedAddCommGroup V]
+    [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+    (ρ : Representation ℂ G V) [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
+    (k : G → ℂ) (d : ℕ) (hk : ∑ g, ‖k g‖^2 = (d : ℝ)^2 / Fintype.card G)
+    (hact : complexFourier ρ k = 1) : Module.finrank ℂ V ≤ d := by
+  have h := fourier_bessel ρ hρ k
+  have hN : (Fintype.card G : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
+  rw [hact, hsNormSq_one, hk] at h
+  have hd : (Module.finrank ℂ V : ℝ)^2 ≤ (d : ℝ)^2 := by
+    convert h using 1 <;> field_simp
+  exact_mod_cast (sq_le_sq₀ (by positivity) (by positivity)).mp hd
+
+lemma smallTypeKernel_identity_of_kernel {V : Type*} [NormedAddCommGroup V]
+    [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
+    (ρ : Representation ℂ G V) (hρ : IsUnitary ρ)
+    (k : G → ℂ) (d : ℕ) (R : ℝ) (hd : (d : ℝ) ≤ R)
+    (hk : ∑ g, ‖k g‖^2 = (d : ℝ)^2 / Fintype.card G)
+    (hact : complexFourier ρ k = 1) : complexFourier ρ (smallTypeKernel G R) = 1 := by
+  apply linearMap_eq_on_simples ρ
+  intro σ hσ x
+  have := hσ
+  have hu : IsUnitary σ.toRepresentation := fun g x y => hρ g x y
+  have hs : complexFourier σ.toRepresentation k = 1 := by
+    ext y
+    rw [complexFourier_subrep, hact]
+    rfl
+  have hdeg := degree_le_of_kernel_identity σ.toRepresentation hu k d hk hs
+  have hdegR : (Module.finrank ℂ σ.toSubmodule : ℝ) ≤ R := (Nat.cast_le.mpr hdeg).trans hd
+  rw [← complexFourier_subrep, complexFourier_smallTypeKernel, ite_eq_left hdegR]
+  rfl
+
+end
+end PartialPermutation
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/SpechtClassification.lean b/lean/OAI/RepresentationTheory/PartialPermutation/SpechtClassification.lean
index cfce2828127bc2ab009217862d91e675117f7e34..4cc10c1c90fa1cc52974e21db496f52199c63b30 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/SpechtClassification.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/SpechtClassification.lean
@@ -1,83 +1,83 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.Plancherel
-import OAI.RepresentationTheory.PartialPermutation.Equivalences
-import OAI.RepresentationTheory.PartialPermutation.SpechtDimension
-import OAI.RepresentationTheory.PartialPermutation.SpechtShapes
-import OAI.RepresentationTheory.PartialPermutation.ConjugacyCounts
-import OAI.RepresentationTheory.PartialPermutation.ClassCharacters
-
-namespace OAI
-
-section
-namespace PartialPermutation
-namespace SpechtActual
-noncomputable section
-open YoungTabloid DiagramCounting
-open scoped Classical
-
-def labels {n : ℕ} (μ : Shape n) : μ.1.cells ≃ Fin n :=
-  (Fintype.equivFin μ.1.cells).trans (finCongr (by simpa using μ.2))
-
-def rep {n : ℕ} (μ : Shape n) :=
-  (spechtRep μ.1).comp (labels μ).symm.permCongrHom.toMonoidHom
-
-instance {n : ℕ} (μ : Shape n) : Representation.IsIrreducible (rep μ) := by
-  unfold rep
-  infer_instance
-
-def index {n : ℕ} (μ : Shape n) : IrreducibleIndex (Equiv.Perm (Fin n)) :=
-  (irreducibleRep_complete (rep μ)).choose
-
-lemma index_spec {n : ℕ} (μ : Shape n) :
-    Nonempty (Representation.Equiv (irreducibleRep (index μ)) (rep μ)) :=
-  (irreducibleRep_complete (rep μ)).choose_spec.1
-
-lemma rep_equiv_shape_eq {n : ℕ} (μ ν : Shape n)
-    (E : Representation.Equiv (rep μ) (rep ν)) : μ=ν := by
-  apply Subtype.ext
-  let e : μ.1.cells ≃ ν.1.cells := (labels μ).trans (labels ν).symm
-  apply specht_equiv_shape_eq μ.1 ν.1 e E.toLinearEquiv
-  intro g x
-  have hh := LinearMap.congr_fun (E.isIntertwining' ((labels μ).permCongrHom g)) x
-  change E (spechtRep μ.1 ((labels μ).symm.permCongrHom ((labels μ).permCongrHom g)) x)=
-    spechtRep ν.1 ((labels ν).symm.permCongrHom ((labels μ).permCongrHom g)) (E x) at hh
-  have he : (labels μ).symm.permCongrHom ((labels μ).permCongrHom g)=g :=
-    (labels μ).permCongrHom.symm_apply_apply g
-  rw [he] at hh
-  exact hh
-
-lemma index_injective (n : ℕ) : Function.Injective (@index n) := by
-  intro μ ν h
-  obtain ⟨e⟩ := index_spec μ
-  obtain ⟨f⟩ := index_spec ν
-  apply rep_equiv_shape_eq μ ν
-  exact e.symm.trans (h ▸ f)
-
-lemma actual_irreducible_card_le (n : ℕ) :
-    Fintype.card (IrreducibleIndex (Equiv.Perm (Fin n))) ≤ Fintype.card (Shape n) := by
-  exact (irreducible_family_card_le_conjClasses (fun c => irreducibleSpace c)
-    (fun c => irreducibleRep c) (componentRep_equiv_iff (regularRep (Equiv.Perm (Fin n))))).trans
-    (symmetric_conjClasses_card_le n)
-
-theorem index_bijective (n : ℕ) : Function.Bijective (@index n) := by
-  have hc := actual_irreducible_card_le n
-  have hi := index_injective n
-  have he := le_antisymm hc (Fintype.card_le_of_injective _ hi)
-  exact ⟨hi,(Fintype.bijective_iff_injective_and_card _).mpr ⟨hi,he.symm⟩ |>.surjective⟩
-
-def indexEquiv (n : ℕ) : Shape n ≃ IrreducibleIndex (Equiv.Perm (Fin n)) :=
-  Equiv.ofBijective index (index_bijective n)
-
-lemma standardCount_le_degree {n : ℕ} (μ : Shape n) :
-    Tableau.standardCount μ.1 ≤ irreducibleDegree (index μ) := by
-  obtain ⟨e⟩ := index_spec μ
-  have hd := e.toLinearEquiv.finrank_eq
-  exact (standardCount_le_specht_degree μ.1).trans_eq hd.symm
-
-end
-end SpechtActual
-end PartialPermutation
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.Plancherel
+import OAI.RepresentationTheory.PartialPermutation.Equivalences
+import OAI.RepresentationTheory.PartialPermutation.SpechtDimension
+import OAI.RepresentationTheory.PartialPermutation.SpechtShapes
+import OAI.RepresentationTheory.PartialPermutation.ConjugacyCounts
+import OAI.RepresentationTheory.PartialPermutation.ClassCharacters
+
+namespace OAI
+
+section
+namespace PartialPermutation
+namespace SpechtActual
+noncomputable section
+open YoungTabloid DiagramCounting
+open scoped Classical
+
+def labels {n : ℕ} (μ : Shape n) : μ.1.cells ≃ Fin n :=
+  (Fintype.equivFin μ.1.cells).trans (finCongr (by simpa using μ.2))
+
+def rep {n : ℕ} (μ : Shape n) :=
+  (spechtRep μ.1).comp (labels μ).symm.permCongrHom.toMonoidHom
+
+instance {n : ℕ} (μ : Shape n) : Representation.IsIrreducible (rep μ) := by
+  unfold rep
+  infer_instance
+
+def index {n : ℕ} (μ : Shape n) : IrreducibleIndex (Equiv.Perm (Fin n)) :=
+  (irreducibleRep_complete (rep μ)).choose
+
+lemma index_spec {n : ℕ} (μ : Shape n) :
+    Nonempty (Representation.Equiv (irreducibleRep (index μ)) (rep μ)) :=
+  (irreducibleRep_complete (rep μ)).choose_spec.1
+
+lemma rep_equiv_shape_eq {n : ℕ} (μ ν : Shape n)
+    (E : Representation.Equiv (rep μ) (rep ν)) : μ=ν := by
+  apply Subtype.ext
+  let e : μ.1.cells ≃ ν.1.cells := (labels μ).trans (labels ν).symm
+  apply specht_equiv_shape_eq μ.1 ν.1 e E.toLinearEquiv
+  intro g x
+  have hh := LinearMap.congr_fun (E.isIntertwining' ((labels μ).permCongrHom g)) x
+  change E (spechtRep μ.1 ((labels μ).symm.permCongrHom ((labels μ).permCongrHom g)) x)=
+    spechtRep ν.1 ((labels ν).symm.permCongrHom ((labels μ).permCongrHom g)) (E x) at hh
+  have he : (labels μ).symm.permCongrHom ((labels μ).permCongrHom g)=g :=
+    (labels μ).permCongrHom.symm_apply_apply g
+  rw [he] at hh
+  exact hh
+
+lemma index_injective (n : ℕ) : Function.Injective (@index n) := by
+  intro μ ν h
+  obtain ⟨e⟩ := index_spec μ
+  obtain ⟨f⟩ := index_spec ν
+  apply rep_equiv_shape_eq μ ν
+  exact e.symm.trans (h ▸ f)
+
+lemma actual_irreducible_card_le (n : ℕ) :
+    Fintype.card (IrreducibleIndex (Equiv.Perm (Fin n))) ≤ Fintype.card (Shape n) := by
+  exact (irreducible_family_card_le_conjClasses (fun c => irreducibleSpace c)
+    (fun c => irreducibleRep c) (componentRep_equiv_iff (regularRep (Equiv.Perm (Fin n))))).trans
+    (symmetric_conjClasses_card_le n)
+
+theorem index_bijective (n : ℕ) : Function.Bijective (@index n) := by
+  have hc := actual_irreducible_card_le n
+  have hi := index_injective n
+  have he := le_antisymm hc (Fintype.card_le_of_injective _ hi)
+  exact ⟨hi,(Fintype.bijective_iff_injective_and_card _).mpr ⟨hi,he.symm⟩ |>.surjective⟩
+
+def indexEquiv (n : ℕ) : Shape n ≃ IrreducibleIndex (Equiv.Perm (Fin n)) :=
+  Equiv.ofBijective index (index_bijective n)
+
+lemma standardCount_le_degree {n : ℕ} (μ : Shape n) :
+    Tableau.standardCount μ.1 ≤ irreducibleDegree (index μ) := by
+  obtain ⟨e⟩ := index_spec μ
+  have hd := e.toLinearEquiv.finrank_eq
+  exact (standardCount_le_specht_degree μ.1).trans_eq hd.symm
+
+end
+end SpechtActual
+end PartialPermutation
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/SpechtDimension.lean b/lean/OAI/RepresentationTheory/PartialPermutation/SpechtDimension.lean
index 851509f95220be39ec6c79d18d6ed02bf90b3146..7367d803cf317b20da1595b9b2818067706863bd 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/SpechtDimension.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/SpechtDimension.lean
@@ -1,87 +1,87 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.SpechtModules
-import OAI.RepresentationTheory.PartialPermutation.TableauVectors
-
-namespace OAI
-
-section
-namespace PartialPermutation.YoungTabloid
-noncomputable section
-open Finset
-open scoped Classical
-
-lemma tableauVector_diag (μ : YoungDiagram) (t : Std μ) :
-    tableauVector μ t (tableauTabloid μ t)=1 := by
-  change polytabloid μ (relabel μ (tableauPerm μ t)⁻¹
-    (relabel μ (tableauPerm μ t) (canonical μ)))=1
-  rw [← relabel_mul,inv_mul_cancel,relabel_one,polytabloid_canonical]
-
-lemma polytabloid_support (μ : YoungDiagram) (r : Tabloid μ) (h : polytabloid μ r≠0) :
-    ∃ c : columnGroup μ, r=relabel μ c (canonical μ) := by
-  apply transversal_column_orbit μ r
-  by_contra hr
-  exact h (alternating_zero_nontransversal μ _ (polytabloid_alternating μ) r hr)
-
-lemma tableauVector_support (μ : YoungDiagram) (t : Std μ) (r : Tabloid μ)
-    (h : tableauVector μ t r≠0) :
-    ∃ c : columnGroup μ, r=relabel μ (tableauPerm μ t) (relabel μ c (canonical μ)) := by
-  obtain ⟨c,hc⟩ := polytabloid_support μ _ h
-  refine ⟨c,?_⟩
-  have hh := congrArg (relabel μ (tableauPerm μ t)) hc
-  simpa only [← relabel_mul,mul_inv_cancel,relabel_one] using hh
-
-lemma tableauVector_triangular (μ : YoungDiagram) (s t : Std μ)
-    (hne : s≠t) (h : tableauVector μ s (tableauTabloid μ t)≠0) :
-    score μ (tableauTabloid μ t)<score μ (tableauTabloid μ s) := by
-  obtain ⟨c,hc⟩ := tableauVector_support μ s _ h
-  have hle := score_column_le μ s c
-  rw [← hc] at hle
-  apply lt_of_le_of_ne hle
-  intro hh
-  have h1 := score_column_eq μ s c (hc ▸ hh)
-  subst c
-  apply hne
-  apply tableauTabloid_injective μ
-  simpa only [Subgroup.coe_one,relabel_one,tableauTabloid] using hc.symm
-
-lemma triangular_linearIndependent {ι α : Type*} [Fintype ι] [Fintype α]
-    (v : ι → EuclideanSpace ℂ α) (r : ι → α) (k : ι → ℕ)
-    (hd : ∀ i, v i (r i)=1) (ht : ∀ i j, i≠j → v i (r j)≠0 → k j<k i) :
-    LinearIndependent ℂ v := by
-  apply Fintype.linearIndependent_iff.mpr
-  intro a ha i
-  by_contra hi
-  let S := univ.filter (fun j => a j≠0)
-  obtain ⟨j,hj,hmax⟩ := S.exists_max_image k ⟨i,by simpa [S] using hi⟩
-  have haj : a j≠0 := (mem_filter.mp hj).2
-  have hv (l : ι) (hl : l≠j) : a l * v l (r j)=0 := by
-    by_cases hla : a l=0
-    · simp [hla]
-    have hzero : v l (r j)=0 := by
-      by_contra hne
-      exact (not_lt_of_ge (hmax l (by simp [S,hla]))) (ht l j hl hne)
-    simp [hzero]
-  have he := congrArg (fun w : EuclideanSpace ℂ α => w (r j)) ha
-  have hs : (∑ l, a l * v l (r j))=a j := by
-    rw [Finset.sum_eq_single j (fun l _ hl => hv l hl) (fun h => (h (mem_univ j)).elim),hd,mul_one]
-  simp at he
-  exact haj (hs.symm.trans he)
-
-lemma tableauVector_independent (μ : YoungDiagram) : LinearIndependent ℂ (tableauVector μ) :=
-  triangular_linearIndependent _ (tableauTabloid μ) (fun t => score μ (tableauTabloid μ t))
-    (tableauVector_diag μ) (tableauVector_triangular μ)
-
-theorem standardCount_le_specht_degree (μ : YoungDiagram) :
-    Tableau.standardCount μ ≤ Module.finrank ℂ (spechtSub μ).toSubmodule := by
-  let v : Std μ → (spechtSub μ).toSubmodule := fun t =>
-    ⟨tableauVector μ t,(spechtSub μ).apply_mem_toSubmodule _ (polytabloid_mem_specht μ)⟩
-  have hv : LinearIndependent ℂ v := LinearIndependent.of_comp
-    (spechtSub μ).toSubmodule.subtype (tableauVector_independent μ)
-  exact hv.fintype_card_le_finrank
-
-end
-end PartialPermutation.YoungTabloid
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.SpechtModules
+import OAI.RepresentationTheory.PartialPermutation.TableauVectors
+
+namespace OAI
+
+section
+namespace PartialPermutation.YoungTabloid
+noncomputable section
+open Finset
+open scoped Classical
+
+lemma tableauVector_diag (μ : YoungDiagram) (t : Std μ) :
+    tableauVector μ t (tableauTabloid μ t)=1 := by
+  change polytabloid μ (relabel μ (tableauPerm μ t)⁻¹
+    (relabel μ (tableauPerm μ t) (canonical μ)))=1
+  rw [← relabel_mul,inv_mul_cancel,relabel_one,polytabloid_canonical]
+
+lemma polytabloid_support (μ : YoungDiagram) (r : Tabloid μ) (h : polytabloid μ r≠0) :
+    ∃ c : columnGroup μ, r=relabel μ c (canonical μ) := by
+  apply transversal_column_orbit μ r
+  by_contra hr
+  exact h (alternating_zero_nontransversal μ _ (polytabloid_alternating μ) r hr)
+
+lemma tableauVector_support (μ : YoungDiagram) (t : Std μ) (r : Tabloid μ)
+    (h : tableauVector μ t r≠0) :
+    ∃ c : columnGroup μ, r=relabel μ (tableauPerm μ t) (relabel μ c (canonical μ)) := by
+  obtain ⟨c,hc⟩ := polytabloid_support μ _ h
+  refine ⟨c,?_⟩
+  have hh := congrArg (relabel μ (tableauPerm μ t)) hc
+  simpa only [← relabel_mul,mul_inv_cancel,relabel_one] using hh
+
+lemma tableauVector_triangular (μ : YoungDiagram) (s t : Std μ)
+    (hne : s≠t) (h : tableauVector μ s (tableauTabloid μ t)≠0) :
+    score μ (tableauTabloid μ t)<score μ (tableauTabloid μ s) := by
+  obtain ⟨c,hc⟩ := tableauVector_support μ s _ h
+  have hle := score_column_le μ s c
+  rw [← hc] at hle
+  apply lt_of_le_of_ne hle
+  intro hh
+  have h1 := score_column_eq μ s c (hc ▸ hh)
+  subst c
+  apply hne
+  apply tableauTabloid_injective μ
+  simpa only [Subgroup.coe_one,relabel_one,tableauTabloid] using hc.symm
+
+lemma triangular_linearIndependent {ι α : Type*} [Fintype ι] [Fintype α]
+    (v : ι → EuclideanSpace ℂ α) (r : ι → α) (k : ι → ℕ)
+    (hd : ∀ i, v i (r i)=1) (ht : ∀ i j, i≠j → v i (r j)≠0 → k j<k i) :
+    LinearIndependent ℂ v := by
+  apply Fintype.linearIndependent_iff.mpr
+  intro a ha i
+  by_contra hi
+  let S := univ.filter (fun j => a j≠0)
+  obtain ⟨j,hj,hmax⟩ := S.exists_max_image k ⟨i,by simpa [S] using hi⟩
+  have haj : a j≠0 := (mem_filter.mp hj).2
+  have hv (l : ι) (hl : l≠j) : a l * v l (r j)=0 := by
+    by_cases hla : a l=0
+    · simp [hla]
+    have hzero : v l (r j)=0 := by
+      by_contra hne
+      exact (not_lt_of_ge (hmax l (by simp [S,hla]))) (ht l j hl hne)
+    simp [hzero]
+  have he := congrArg (fun w : EuclideanSpace ℂ α => w (r j)) ha
+  have hs : (∑ l, a l * v l (r j))=a j := by
+    rw [Finset.sum_eq_single j (fun l _ hl => hv l hl) (fun h => (h (mem_univ j)).elim),hd,mul_one]
+  simp at he
+  exact haj (hs.symm.trans he)
+
+lemma tableauVector_independent (μ : YoungDiagram) : LinearIndependent ℂ (tableauVector μ) :=
+  triangular_linearIndependent _ (tableauTabloid μ) (fun t => score μ (tableauTabloid μ t))
+    (tableauVector_diag μ) (tableauVector_triangular μ)
+
+theorem standardCount_le_specht_degree (μ : YoungDiagram) :
+    Tableau.standardCount μ ≤ Module.finrank ℂ (spechtSub μ).toSubmodule := by
+  let v : Std μ → (spechtSub μ).toSubmodule := fun t =>
+    ⟨tableauVector μ t,(spechtSub μ).apply_mem_toSubmodule _ (polytabloid_mem_specht μ)⟩
+  have hv : LinearIndependent ℂ v := LinearIndependent.of_comp
+    (spechtSub μ).toSubmodule.subtype (tableauVector_independent μ)
+  exact hv.fintype_card_le_finrank
+
+end
+end PartialPermutation.YoungTabloid
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/SpechtModules.lean b/lean/OAI/RepresentationTheory/PartialPermutation/SpechtModules.lean
index 7129d1b13d49c8f7eb20b10ba5f92585886b77ad..b583806c2fc443d1c8d3490b1660e4a20502f62f 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/SpechtModules.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/SpechtModules.lean
@@ -1,119 +1,119 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.Polytabloids
-
-namespace OAI
-
-section
-namespace PartialPermutation
-namespace YoungTabloid
-noncomputable section
-open Finset
-open scoped Classical
-
-def spechtSub (μ : YoungDiagram) : Subrepresentation (tabloidRep μ) where
-  toSubmodule := Submodule.span ℂ (Set.range (fun g => tabloidRep μ g (polytabloid μ)))
-  apply_mem_toSubmodule g v hv := by
-    induction hv using Submodule.span_induction with
-    | mem u hu =>
-      obtain ⟨h,rfl⟩ := hu
-      apply Submodule.subset_span
-      exact ⟨g*h,by simp [Module.End.mul_apply]⟩
-    | zero => simp
-    | add u v hu hv ih ih' => simpa only [map_add] using Submodule.add_mem _ ih ih'
-    | smul a u hu ih => simpa only [map_smul] using Submodule.smul_mem _ a ih
-
-lemma polytabloid_mem_specht (μ : YoungDiagram) : polytabloid μ ∈ spechtSub μ := by
-  apply Submodule.subset_span
-  exact ⟨1,by simp⟩
-
-lemma specht_le_of_polytabloid_mem (μ : YoungDiagram) (W : Subrepresentation (tabloidRep μ))
-    (hw : polytabloid μ ∈ W) : spechtSub μ ≤ W := by
-  apply Submodule.span_le.mpr
-  rintro _ ⟨g,rfl⟩
-  exact W.apply_mem_toSubmodule g hw
-
-lemma antisym_mem_invariant (μ : YoungDiagram) (W : Subrepresentation (tabloidRep μ))
-    {f : EuclideanSpace ℂ (Tabloid μ)} (hf : f∈W) : antisym μ f∈W := by
-  simp only [antisym,LinearMap.sum_apply,LinearMap.smul_apply]
-  apply Submodule.sum_mem
-  intro c _
-  exact W.toSubmodule.smul_mem _ (W.apply_mem_toSubmodule c.1 hf)
-
-lemma specht_detect (μ : YoungDiagram) (f : EuclideanSpace ℂ (Tabloid μ))
-    (hf : f∈spechtSub μ) (hne : f≠0) :
-    ∃ g : Equiv.Perm μ.cells, inner ℂ (polytabloid μ) (tabloidRep μ g f)≠0 := by
-  by_contra hn
-  simp only [not_exists,not_not] at hn
-  have hz (v) (hv : v∈(spechtSub μ).toSubmodule) : inner ℂ v f=0 := by
-    induction hv using Submodule.span_induction with
-    | mem v hv =>
-      obtain ⟨g,rfl⟩ := hv
-      have hh := tabloidRep_unitary μ g (polytabloid μ) (tabloidRep μ g⁻¹ f)
-      simpa [hn] using hh
-    | zero => simp
-    | add v w hv hw ih ih' => simp [ih,ih']
-    | smul a v hv ih => simp [ih]
-  exact hne (inner_self_eq_zero.mp (hz f hf))
-
-lemma specht_minimal (μ : YoungDiagram) (W : Subrepresentation (tabloidRep μ))
-    (hW : W ≤ spechtSub μ) : W=⊥ ∨ W=spechtSub μ := by
-  by_cases hw : W=⊥
-  · exact Or.inl hw
-  right
-  have hne : W.toSubmodule≠⊥ := by
-    intro hh
-    exact hw (Subrepresentation.ext hh)
-  obtain ⟨f,hf,hf0⟩ := W.toSubmodule.ne_bot_iff.mp hne
-  obtain ⟨g,hg⟩ := specht_detect μ f (hW hf) hf0
-  have ha := antisym_mem_invariant μ W (W.apply_mem_toSubmodule g hf)
-  rw [antisym_eq_inner] at ha
-  have he : polytabloid μ∈W := (W.toSubmodule.smul_mem_iff hg).mp ha
-  exact le_antisymm hW (specht_le_of_polytabloid_mem μ W he)
-
-def spechtRep (μ : YoungDiagram) := (spechtSub μ).toRepresentation
-
-instance specht_irreducible (μ : YoungDiagram) : Representation.IsIrreducible (spechtRep μ) := by
-  let S := (spechtSub μ).toSubmodule
-  let e : S := ⟨polytabloid μ,polytabloid_mem_specht μ⟩
-  have he : e≠0 := fun h => polytabloid_ne_zero μ (congrArg Subtype.val h)
-  let : Nontrivial (Subrepresentation (spechtRep μ)) := by
-    refine ⟨⟨⊥,⊤,?_⟩⟩
-    intro h
-    have hh : e ∈ (⊥ : Subrepresentation (spechtRep μ)) := by rw [h]; trivial
-    exact he hh
-  constructor
-  intro σ
-  let W : Subrepresentation (tabloidRep μ) :=
-    { toSubmodule := σ.toSubmodule.map S.subtype
-      apply_mem_toSubmodule := by
-        intro g v hv
-        obtain ⟨u,hu,rfl⟩ := hv
-        exact ⟨spechtRep μ g u,σ.apply_mem_toSubmodule g hu,rfl⟩ }
-  have hW : W ≤ spechtSub μ := by
-    intro v hv
-    obtain ⟨u,hu,rfl⟩ := hv
-    exact u.2
-  rcases specht_minimal μ W hW with hw | hw
-  · left
-    apply Subrepresentation.ext
-    apply σ.toSubmodule.eq_bot_iff.mpr
-    intro u hu
-    have h : u.1 ∈ W := ⟨u,hu,rfl⟩
-    rw [hw] at h
-    exact Subtype.ext h
-  · right
-    apply Subrepresentation.ext
-    apply top_unique
-    intro u _
-    have h : u.1∈W := by rw [hw]; exact u.2
-    obtain ⟨v,hv,heq⟩ := h
-    have huv : v=u := Subtype.ext heq
-    exact huv ▸ hv
-
-end
-end YoungTabloid
-end PartialPermutation
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.Polytabloids
+
+namespace OAI
+
+section
+namespace PartialPermutation
+namespace YoungTabloid
+noncomputable section
+open Finset
+open scoped Classical
+
+def spechtSub (μ : YoungDiagram) : Subrepresentation (tabloidRep μ) where
+  toSubmodule := Submodule.span ℂ (Set.range (fun g => tabloidRep μ g (polytabloid μ)))
+  apply_mem_toSubmodule g v hv := by
+    induction hv using Submodule.span_induction with
+    | mem u hu =>
+      obtain ⟨h,rfl⟩ := hu
+      apply Submodule.subset_span
+      exact ⟨g*h,by simp [Module.End.mul_apply]⟩
+    | zero => simp
+    | add u v hu hv ih ih' => simpa only [map_add] using Submodule.add_mem _ ih ih'
+    | smul a u hu ih => simpa only [map_smul] using Submodule.smul_mem _ a ih
+
+lemma polytabloid_mem_specht (μ : YoungDiagram) : polytabloid μ ∈ spechtSub μ := by
+  apply Submodule.subset_span
+  exact ⟨1,by simp⟩
+
+lemma specht_le_of_polytabloid_mem (μ : YoungDiagram) (W : Subrepresentation (tabloidRep μ))
+    (hw : polytabloid μ ∈ W) : spechtSub μ ≤ W := by
+  apply Submodule.span_le.mpr
+  rintro _ ⟨g,rfl⟩
+  exact W.apply_mem_toSubmodule g hw
+
+lemma antisym_mem_invariant (μ : YoungDiagram) (W : Subrepresentation (tabloidRep μ))
+    {f : EuclideanSpace ℂ (Tabloid μ)} (hf : f∈W) : antisym μ f∈W := by
+  simp only [antisym,LinearMap.sum_apply,LinearMap.smul_apply]
+  apply Submodule.sum_mem
+  intro c _
+  exact W.toSubmodule.smul_mem _ (W.apply_mem_toSubmodule c.1 hf)
+
+lemma specht_detect (μ : YoungDiagram) (f : EuclideanSpace ℂ (Tabloid μ))
+    (hf : f∈spechtSub μ) (hne : f≠0) :
+    ∃ g : Equiv.Perm μ.cells, inner ℂ (polytabloid μ) (tabloidRep μ g f)≠0 := by
+  by_contra hn
+  simp only [not_exists,not_not] at hn
+  have hz (v) (hv : v∈(spechtSub μ).toSubmodule) : inner ℂ v f=0 := by
+    induction hv using Submodule.span_induction with
+    | mem v hv =>
+      obtain ⟨g,rfl⟩ := hv
+      have hh := tabloidRep_unitary μ g (polytabloid μ) (tabloidRep μ g⁻¹ f)
+      simpa [hn] using hh
+    | zero => simp
+    | add v w hv hw ih ih' => simp [ih,ih']
+    | smul a v hv ih => simp [ih]
+  exact hne (inner_self_eq_zero.mp (hz f hf))
+
+lemma specht_minimal (μ : YoungDiagram) (W : Subrepresentation (tabloidRep μ))
+    (hW : W ≤ spechtSub μ) : W=⊥ ∨ W=spechtSub μ := by
+  by_cases hw : W=⊥
+  · exact Or.inl hw
+  right
+  have hne : W.toSubmodule≠⊥ := by
+    intro hh
+    exact hw (Subrepresentation.ext hh)
+  obtain ⟨f,hf,hf0⟩ := W.toSubmodule.ne_bot_iff.mp hne
+  obtain ⟨g,hg⟩ := specht_detect μ f (hW hf) hf0
+  have ha := antisym_mem_invariant μ W (W.apply_mem_toSubmodule g hf)
+  rw [antisym_eq_inner] at ha
+  have he : polytabloid μ∈W := (W.toSubmodule.smul_mem_iff hg).mp ha
+  exact le_antisymm hW (specht_le_of_polytabloid_mem μ W he)
+
+def spechtRep (μ : YoungDiagram) := (spechtSub μ).toRepresentation
+
+instance specht_irreducible (μ : YoungDiagram) : Representation.IsIrreducible (spechtRep μ) := by
+  let S := (spechtSub μ).toSubmodule
+  let e : S := ⟨polytabloid μ,polytabloid_mem_specht μ⟩
+  have he : e≠0 := fun h => polytabloid_ne_zero μ (congrArg Subtype.val h)
+  let : Nontrivial (Subrepresentation (spechtRep μ)) := by
+    refine ⟨⟨⊥,⊤,?_⟩⟩
+    intro h
+    have hh : e ∈ (⊥ : Subrepresentation (spechtRep μ)) := by rw [h]; trivial
+    exact he hh
+  constructor
+  intro σ
+  let W : Subrepresentation (tabloidRep μ) :=
+    { toSubmodule := σ.toSubmodule.map S.subtype
+      apply_mem_toSubmodule := by
+        intro g v hv
+        obtain ⟨u,hu,rfl⟩ := hv
+        exact ⟨spechtRep μ g u,σ.apply_mem_toSubmodule g hu,rfl⟩ }
+  have hW : W ≤ spechtSub μ := by
+    intro v hv
+    obtain ⟨u,hu,rfl⟩ := hv
+    exact u.2
+  rcases specht_minimal μ W hW with hw | hw
+  · left
+    apply Subrepresentation.ext
+    apply σ.toSubmodule.eq_bot_iff.mpr
+    intro u hu
+    have h : u.1 ∈ W := ⟨u,hu,rfl⟩
+    rw [hw] at h
+    exact Subtype.ext h
+  · right
+    apply Subrepresentation.ext
+    apply top_unique
+    intro u _
+    have h : u.1∈W := by rw [hw]; exact u.2
+    obtain ⟨v,hv,heq⟩ := h
+    have huv : v=u := Subtype.ext heq
+    exact huv ▸ hv
+
+end
+end YoungTabloid
+end PartialPermutation
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/SpechtShapes.lean b/lean/OAI/RepresentationTheory/PartialPermutation/SpechtShapes.lean
index c2f76c5259e754186d00f0c122985b534425f085..539c40d84a7b8781240bdf7a7148e939dd4e0db5 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/SpechtShapes.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/SpechtShapes.lean
@@ -1,88 +1,88 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.SpechtModules
-import OAI.RepresentationTheory.PartialPermutation.CrossTableaux
-
-namespace OAI
-
-section
-namespace PartialPermutation.YoungTabloid
-noncomputable section
-open Finset
-open scoped Classical
-
-lemma polytabloid_specht_sign (μ : YoungDiagram) (c : columnGroup μ) :
-    spechtRep μ c.1 ⟨polytabloid μ,polytabloid_mem_specht μ⟩ =
-      signC c.1 • (⟨polytabloid μ,polytabloid_mem_specht μ⟩ : (spechtSub μ).toSubmodule) := by
-  apply Subtype.ext
-  apply PiLp.ext
-  intro r
-  change polytabloid μ (relabel μ c.1⁻¹ r)=signC c.1 * polytabloid μ r
-  simpa using polytabloid_alternating μ c⁻¹ r
-
-lemma cross_transversal_of_specht_map (μ ν : YoungDiagram) (e : μ.cells ≃ ν.cells)
-    (L : (spechtSub μ).toSubmodule →ₗ[ℂ] (spechtSub ν).toSubmodule)
-    (hL : Function.Injective L)
-    (hcomm : ∀ g x, L (spechtRep μ g x)=spechtRep ν (e.permCongrHom g) (L x)) :
-    ∃ r : Tabloid ν, CrossTransversal μ (fun x => r.1 (e x)) := by
-  let v : (spechtSub μ).toSubmodule := ⟨polytabloid μ,polytabloid_mem_specht μ⟩
-  have hv : v≠0 := fun h => polytabloid_ne_zero μ (congrArg Subtype.val h)
-  have hf : (L v).val ≠ 0 := by
-    intro h
-    exact hv (hL (by simpa using (show L v=0 from Subtype.ext h)))
-  obtain ⟨r,hr⟩ : ∃ r : Tabloid ν, (L v).val r≠0 := by
-    by_contra hh
-    push Not at hh
-    exact hf (PiLp.ext hh)
-  refine ⟨r,?_⟩
-  intro x y hc he
-  by_contra hxy
-  let c : columnGroup μ := ⟨Equiv.swap x y,swap_mem_column μ x y hc⟩
-  have hsg : signC c.1 = -1 := signC_swap x y hxy
-  have hswap : e.permCongrHom c.1=Equiv.swap (e x) (e y) := by
-    apply Equiv.ext
-    intro z
-    change e (Equiv.swap x y (e.symm z))=Equiv.swap (e x) (e y) z
-    by_cases hx : e.symm z=x
-    · have hz : z=e x := by simpa using congrArg e hx
-      simp [hz]
-    by_cases hy : e.symm z=y
-    · have hz : z=e y := by simpa using congrArg e hy
-      simp [hz]
-    have hx' : z≠e x := by intro h; exact hx (by simp [h])
-    have hy' : z≠e y := by intro h; exact hy (by simp [h])
-    simp [Equiv.swap_apply_of_ne_of_ne hx hy,Equiv.swap_apply_of_ne_of_ne hx' hy']
-  have h := hcomm c.1 v
-  rw [polytabloid_specht_sign,map_smul,hsg,hswap] at h
-  have hh := congrArg (fun w : (spechtSub ν).toSubmodule => w.val r) h
-  change (-1 : ℂ) * (L v).val r = (L v).val (relabel ν (Equiv.swap (e x) (e y))⁻¹ r) at hh
-  rw [Equiv.swap_inv,relabel_swap_fix ν r (e x) (e y) he] at hh
-  apply hr
-  linear_combination -hh / 2
-
-lemma specht_map_rowMoment_le (μ ν : YoungDiagram) (e : μ.cells ≃ ν.cells)
-    (L : (spechtSub μ).toSubmodule →ₗ[ℂ] (spechtSub ν).toSubmodule)
-    (hL : Function.Injective L)
-    (hcomm : ∀ g x, L (spechtRep μ g x)=spechtRep ν (e.permCongrHom g) (L x)) :
-    rowMoment μ ≤ rowMoment ν := by
-  obtain ⟨r,hr⟩ := cross_transversal_of_specht_map μ ν e L hL hcomm
-  exact cross_rowMoment_le μ ν e r hr
-
-theorem specht_equiv_shape_eq (μ ν : YoungDiagram) (e : μ.cells ≃ ν.cells)
-    (L : (spechtSub μ).toSubmodule ≃ₗ[ℂ] (spechtSub ν).toSubmodule)
-    (hcomm : ∀ g x, L (spechtRep μ g x)=spechtRep ν (e.permCongrHom g) (L x)) : μ=ν := by
-  have hrev : ∀ g x, L.symm (spechtRep ν g x)=spechtRep μ (e.symm.permCongrHom g) (L.symm x) := by
-    intro g x
-    apply L.injective
-    have he : e.permCongrHom (e.symm.permCongrHom g)=g := e.permCongrHom.apply_symm_apply g
-    simpa only [he,L.apply_symm_apply] using (hcomm (e.symm.permCongrHom g) (L.symm x)).symm
-  have hm := le_antisymm (specht_map_rowMoment_le μ ν e L.toLinearMap L.injective hcomm)
-    (specht_map_rowMoment_le ν μ e.symm L.symm.toLinearMap L.symm.injective hrev)
-  obtain ⟨r,hr⟩ := cross_transversal_of_specht_map μ ν e L.toLinearMap L.injective hcomm
-  exact cross_equal_moment_shape_eq μ ν e r hr hm
-
-end
-end PartialPermutation.YoungTabloid
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.SpechtModules
+import OAI.RepresentationTheory.PartialPermutation.CrossTableaux
+
+namespace OAI
+
+section
+namespace PartialPermutation.YoungTabloid
+noncomputable section
+open Finset
+open scoped Classical
+
+lemma polytabloid_specht_sign (μ : YoungDiagram) (c : columnGroup μ) :
+    spechtRep μ c.1 ⟨polytabloid μ,polytabloid_mem_specht μ⟩ =
+      signC c.1 • (⟨polytabloid μ,polytabloid_mem_specht μ⟩ : (spechtSub μ).toSubmodule) := by
+  apply Subtype.ext
+  apply PiLp.ext
+  intro r
+  change polytabloid μ (relabel μ c.1⁻¹ r)=signC c.1 * polytabloid μ r
+  simpa using polytabloid_alternating μ c⁻¹ r
+
+lemma cross_transversal_of_specht_map (μ ν : YoungDiagram) (e : μ.cells ≃ ν.cells)
+    (L : (spechtSub μ).toSubmodule →ₗ[ℂ] (spechtSub ν).toSubmodule)
+    (hL : Function.Injective L)
+    (hcomm : ∀ g x, L (spechtRep μ g x)=spechtRep ν (e.permCongrHom g) (L x)) :
+    ∃ r : Tabloid ν, CrossTransversal μ (fun x => r.1 (e x)) := by
+  let v : (spechtSub μ).toSubmodule := ⟨polytabloid μ,polytabloid_mem_specht μ⟩
+  have hv : v≠0 := fun h => polytabloid_ne_zero μ (congrArg Subtype.val h)
+  have hf : (L v).val ≠ 0 := by
+    intro h
+    exact hv (hL (by simpa using (show L v=0 from Subtype.ext h)))
+  obtain ⟨r,hr⟩ : ∃ r : Tabloid ν, (L v).val r≠0 := by
+    by_contra hh
+    push Not at hh
+    exact hf (PiLp.ext hh)
+  refine ⟨r,?_⟩
+  intro x y hc he
+  by_contra hxy
+  let c : columnGroup μ := ⟨Equiv.swap x y,swap_mem_column μ x y hc⟩
+  have hsg : signC c.1 = -1 := signC_swap x y hxy
+  have hswap : e.permCongrHom c.1=Equiv.swap (e x) (e y) := by
+    apply Equiv.ext
+    intro z
+    change e (Equiv.swap x y (e.symm z))=Equiv.swap (e x) (e y) z
+    by_cases hx : e.symm z=x
+    · have hz : z=e x := by simpa using congrArg e hx
+      simp [hz]
+    by_cases hy : e.symm z=y
+    · have hz : z=e y := by simpa using congrArg e hy
+      simp [hz]
+    have hx' : z≠e x := by intro h; exact hx (by simp [h])
+    have hy' : z≠e y := by intro h; exact hy (by simp [h])
+    simp [Equiv.swap_apply_of_ne_of_ne hx hy,Equiv.swap_apply_of_ne_of_ne hx' hy']
+  have h := hcomm c.1 v
+  rw [polytabloid_specht_sign,map_smul,hsg,hswap] at h
+  have hh := congrArg (fun w : (spechtSub ν).toSubmodule => w.val r) h
+  change (-1 : ℂ) * (L v).val r = (L v).val (relabel ν (Equiv.swap (e x) (e y))⁻¹ r) at hh
+  rw [Equiv.swap_inv,relabel_swap_fix ν r (e x) (e y) he] at hh
+  apply hr
+  linear_combination -hh / 2
+
+lemma specht_map_rowMoment_le (μ ν : YoungDiagram) (e : μ.cells ≃ ν.cells)
+    (L : (spechtSub μ).toSubmodule →ₗ[ℂ] (spechtSub ν).toSubmodule)
+    (hL : Function.Injective L)
+    (hcomm : ∀ g x, L (spechtRep μ g x)=spechtRep ν (e.permCongrHom g) (L x)) :
+    rowMoment μ ≤ rowMoment ν := by
+  obtain ⟨r,hr⟩ := cross_transversal_of_specht_map μ ν e L hL hcomm
+  exact cross_rowMoment_le μ ν e r hr
+
+theorem specht_equiv_shape_eq (μ ν : YoungDiagram) (e : μ.cells ≃ ν.cells)
+    (L : (spechtSub μ).toSubmodule ≃ₗ[ℂ] (spechtSub ν).toSubmodule)
+    (hcomm : ∀ g x, L (spechtRep μ g x)=spechtRep ν (e.permCongrHom g) (L x)) : μ=ν := by
+  have hrev : ∀ g x, L.symm (spechtRep ν g x)=spechtRep μ (e.symm.permCongrHom g) (L.symm x) := by
+    intro g x
+    apply L.injective
+    have he : e.permCongrHom (e.symm.permCongrHom g)=g := e.permCongrHom.apply_symm_apply g
+    simpa only [he,L.apply_symm_apply] using (hcomm (e.symm.permCongrHom g) (L.symm x)).symm
+  have hm := le_antisymm (specht_map_rowMoment_le μ ν e L.toLinearMap L.injective hcomm)
+    (specht_map_rowMoment_le ν μ e.symm L.symm.toLinearMap L.symm.injective hrev)
+  obtain ⟨r,hr⟩ := cross_transversal_of_specht_map μ ν e L.toLinearMap L.injective hcomm
+  exact cross_equal_moment_shape_eq μ ν e r hr hm
+
+end
+end PartialPermutation.YoungTabloid
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/StandardTableaux.lean b/lean/OAI/RepresentationTheory/PartialPermutation/StandardTableaux.lean
index ddfd508779f830ab3b6ad9e207920dbc843e62e4..9520a4e5658106de085c993b2b17e44054463350 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/StandardTableaux.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/StandardTableaux.lean
@@ -1,54 +1,54 @@
-import Mathlib
-
-namespace OAI
-
-section
-namespace PartialPermutation
-namespace Tableau
-
-noncomputable section
-open Finset
-
-def StandardFilling (α : Type*) [Fintype α] [PartialOrder α] :=
-  {e : α ≃ Fin (Fintype.card α) // StrictMono e}
-
-instance (α : Type*) [Fintype α] [PartialOrder α] : Fintype (StandardFilling α) :=
-  by
-    classical
-    unfold StandardFilling
-    infer_instance
-
-def standardCount (μ : YoungDiagram) : ℕ :=
-  Fintype.card (StandardFilling μ.cells)
-
-def maxLine (μ : YoungDiagram) : ℕ := max (μ.rowLen 0) (μ.colLen 0)
-def deficit (μ : YoungDiagram) : ℕ := μ.cells.card - maxLine μ
-
-def chosenFilling (α : Type*) [Fintype α] [PartialOrder α] : StandardFilling α := by
-  letI : Fintype (LinearExtension α) := inferInstanceAs (Fintype α)
-  let e : LinearExtension α ≃o Fin (Fintype.card α) :=
-    (Fintype.orderIsoFinOfCardEq (LinearExtension α) (by rfl)).symm
-  refine ⟨{toFun := fun a => e (toLinearExtension a)
-           invFun := fun i => (e.symm i : α)
-           left_inv := ?_
-           right_inv := ?_}, ?_⟩
-  · intro a
-    exact e.symm_apply_apply a
-  · intro i
-    exact e.apply_symm_apply i
-  · intro a b hab
-    apply e.strictMono
-    apply lt_of_le_of_ne (toLinearExtension.monotone hab.le)
-    intro h
-    exact (ne_of_lt hab) h
-
-lemma standardCount_pos (μ : YoungDiagram) : 0 < standardCount μ := by
-  let : Nonempty (StandardFilling μ.cells) := ⟨chosenFilling _⟩
-  exact Fintype.card_pos
-
-end
-end Tableau
-end PartialPermutation
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+section
+namespace PartialPermutation
+namespace Tableau
+
+noncomputable section
+open Finset
+
+def StandardFilling (α : Type*) [Fintype α] [PartialOrder α] :=
+  {e : α ≃ Fin (Fintype.card α) // StrictMono e}
+
+instance (α : Type*) [Fintype α] [PartialOrder α] : Fintype (StandardFilling α) :=
+  by
+    classical
+    unfold StandardFilling
+    infer_instance
+
+def standardCount (μ : YoungDiagram) : ℕ :=
+  Fintype.card (StandardFilling μ.cells)
+
+def maxLine (μ : YoungDiagram) : ℕ := max (μ.rowLen 0) (μ.colLen 0)
+def deficit (μ : YoungDiagram) : ℕ := μ.cells.card - maxLine μ
+
+def chosenFilling (α : Type*) [Fintype α] [PartialOrder α] : StandardFilling α := by
+  letI : Fintype (LinearExtension α) := inferInstanceAs (Fintype α)
+  let e : LinearExtension α ≃o Fin (Fintype.card α) :=
+    (Fintype.orderIsoFinOfCardEq (LinearExtension α) (by rfl)).symm
+  refine ⟨{toFun := fun a => e (toLinearExtension a)
+           invFun := fun i => (e.symm i : α)
+           left_inv := ?_
+           right_inv := ?_}, ?_⟩
+  · intro a
+    exact e.symm_apply_apply a
+  · intro i
+    exact e.apply_symm_apply i
+  · intro a b hab
+    apply e.strictMono
+    apply lt_of_le_of_ne (toLinearExtension.monotone hab.le)
+    intro h
+    exact (ne_of_lt hab) h
+
+lemma standardCount_pos (μ : YoungDiagram) : 0 < standardCount μ := by
+  let : Nonempty (StandardFilling μ.cells) := ⟨chosenFilling _⟩
+  exact Fintype.card_pos
+
+end
+end Tableau
+end PartialPermutation
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/SymmetricTransfer.lean b/lean/OAI/RepresentationTheory/PartialPermutation/SymmetricTransfer.lean
index 257df1079a6f67b455ed79af9fb33ae98828483d..07d4940c135e8fa797ec38a86b8f562f8baa91ec 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/SymmetricTransfer.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/SymmetricTransfer.lean
@@ -1,85 +1,85 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.ReciprocalTransfer
-import OAI.RepresentationTheory.PartialPermutation.Equivalences
-
-namespace OAI
-
-section
-open scoped Classical
-open scoped BigOperators ComplexConjugate MonoidAlgebra
-open scoped BigOperators ComplexConjugate
-open scoped MonoidAlgebra BigOperators
-open scoped BigOperators MonoidAlgebra Classical
-
-attribute [local instance] Classical.propDecidable
-open scoped MonoidAlgebra BigOperators
-open scoped BigOperators
-
-namespace PartialPermutation
-noncomputable section
-
-def permBlockEquiv {X : Type*} (M : Set X) : Equiv.Perm M ≃* blockSubgroup M where
-  toEquiv := Equiv.Perm.subtypeEquivSubtypePerm (fun x => x ∈ M)
-  map_mul' f g := by
-    apply Subtype.ext
-    exact Equiv.Perm.ofSubtype.map_mul f g
-
-def blockSubgroupSnEquiv {X : Type*} [Fintype X] [DecidableEq X] (M : Set X) :
-    blockSubgroup M ≃* Equiv.Perm (Fin (Fintype.card M)) :=
-  (permBlockEquiv M).symm.trans (Fintype.equivFin M).permCongrHom
-
-lemma inverseDegreeSum_block {X : Type*} [Fintype X] [DecidableEq X] (M : Set X) (u : ℝ) :
-    inverseDegreeSum (blockSubgroup M) u =
-      inverseDegreeSum (Equiv.Perm (Fin (Fintype.card M))) u :=
-  inverseDegreeSum_equiv (blockSubgroupSnEquiv M) u
-
-def SymmetricReciprocalDegreeBoundedStatement : Prop :=
-  ∀ u : ℝ, 0 < u →
-    BddAbove {x : ℝ | ∃ n : ℕ, 1 ≤ n ∧ x = symmetricInverseDegreeSum n u}
-
-def ComplementaryCosetsStatement : Prop :=
-  ∀ (n b : ℕ) (_ : 0 < b) (M : Fin b → Set (Fin n)),
-    (∀ i, (M i).Nonempty) →
-    Pairwise (fun i j => Disjoint (M i) (M j)) → (⋃ i, M i) = Set.univ →
-    ∀ (V : Type) [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-      [FiniteDimensional ℂ V] (ρ : Representation ℂ (Equiv.Perm (Fin n)) V),
-      Representation.IsIrreducible ρ → IsUnitary ρ →
-      ∀ (f : Equiv.Perm (Fin n) → ℝ) (B u : ℝ),
-      (∀ g, 0 ≤ f g) → mass f ≤ 1 → 0 < u →
-      (∀ i, LeftCosetCap f (blockSubgroup (M i)) B) →
-      (hsNormSq (complexFourier ρ (fun g => f g)) ≤
-        b * B * symmetricDegreeConstant u *
-          (Module.finrank ℂ V : ℝ)^(-1+(u+2)/(b : ℝ))) ∧
-      (‖LinearMap.toContinuousLinearMap (complexFourier ρ (fun g => f g))‖^2 ≤
-        b * B * symmetricDegreeConstant u *
-          (Module.finrank ℂ V : ℝ)^(-1+(u+2)/(b : ℝ)))
-
-lemma inverseDegreeSum_block_sum {X I : Type*} [Fintype X] [DecidableEq X] [Fintype I]
-    (M : I → Set X) (u : ℝ) :
-    (∑ i, inverseDegreeSum (blockSubgroup (M i)) u) =
-      ∑ i, symmetricInverseDegreeSum (Fintype.card (M i)) u := by
-  apply Finset.sum_congr rfl
-  intro i _
-  exact inverseDegreeSum_block (M i) u
-
-theorem symmetric_block_degree_transfer {n b : ℕ} (hb : 0 < b)
-    (M : Fin b → Set (Fin n)) (hdis : Pairwise (fun i j => Disjoint (M i) (M j)))
-    {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-    [FiniteDimensional ℂ V] (ρ : Representation ℂ (Equiv.Perm (Fin n)) V)
-    [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
-    (f : Equiv.Perm (Fin n) → ℝ) (B u : ℝ) (hf : ∀ g, 0 ≤ f g) (hu : 0 < u)
-    (hcap : ∀ i, LeftCosetCap f (blockSubgroup (M i)) B) :
-    hsNormSq (complexFourier ρ (fun g => f g)) ≤
-      B * mass f * (Module.finrank ℂ V : ℝ)^(-1 + (u+2)/(b : ℝ)) *
-        ∑ i, symmetricInverseDegreeSum (Fintype.card (M i)) u := by
-  have h := inverse_degree_sum_transfer ρ (fun i => blockSubgroup (M i)) hb hρ
-    (fun i j hij g hg k hk => blockSubgroup_commute (hdis hij) hg hk) f B u hf hu hcap
-  rw [inverseDegreeSum_block_sum M u] at h
-  exact h
-
-end
-end PartialPermutation
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.ReciprocalTransfer
+import OAI.RepresentationTheory.PartialPermutation.Equivalences
+
+namespace OAI
+
+section
+open scoped Classical
+open scoped BigOperators ComplexConjugate MonoidAlgebra
+open scoped BigOperators ComplexConjugate
+open scoped MonoidAlgebra BigOperators
+open scoped BigOperators MonoidAlgebra Classical
+
+attribute [local instance] Classical.propDecidable
+open scoped MonoidAlgebra BigOperators
+open scoped BigOperators
+
+namespace PartialPermutation
+noncomputable section
+
+def permBlockEquiv {X : Type*} (M : Set X) : Equiv.Perm M ≃* blockSubgroup M where
+  toEquiv := Equiv.Perm.subtypeEquivSubtypePerm (fun x => x ∈ M)
+  map_mul' f g := by
+    apply Subtype.ext
+    exact Equiv.Perm.ofSubtype.map_mul f g
+
+def blockSubgroupSnEquiv {X : Type*} [Fintype X] [DecidableEq X] (M : Set X) :
+    blockSubgroup M ≃* Equiv.Perm (Fin (Fintype.card M)) :=
+  (permBlockEquiv M).symm.trans (Fintype.equivFin M).permCongrHom
+
+lemma inverseDegreeSum_block {X : Type*} [Fintype X] [DecidableEq X] (M : Set X) (u : ℝ) :
+    inverseDegreeSum (blockSubgroup M) u =
+      inverseDegreeSum (Equiv.Perm (Fin (Fintype.card M))) u :=
+  inverseDegreeSum_equiv (blockSubgroupSnEquiv M) u
+
+def SymmetricReciprocalDegreeBoundedStatement : Prop :=
+  ∀ u : ℝ, 0 < u →
+    BddAbove {x : ℝ | ∃ n : ℕ, 1 ≤ n ∧ x = symmetricInverseDegreeSum n u}
+
+def ComplementaryCosetsStatement : Prop :=
+  ∀ (n b : ℕ) (_ : 0 < b) (M : Fin b → Set (Fin n)),
+    (∀ i, (M i).Nonempty) →
+    Pairwise (fun i j => Disjoint (M i) (M j)) → (⋃ i, M i) = Set.univ →
+    ∀ (V : Type) [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+      [FiniteDimensional ℂ V] (ρ : Representation ℂ (Equiv.Perm (Fin n)) V),
+      Representation.IsIrreducible ρ → IsUnitary ρ →
+      ∀ (f : Equiv.Perm (Fin n) → ℝ) (B u : ℝ),
+      (∀ g, 0 ≤ f g) → mass f ≤ 1 → 0 < u →
+      (∀ i, LeftCosetCap f (blockSubgroup (M i)) B) →
+      (hsNormSq (complexFourier ρ (fun g => f g)) ≤
+        b * B * symmetricDegreeConstant u *
+          (Module.finrank ℂ V : ℝ)^(-1+(u+2)/(b : ℝ))) ∧
+      (‖LinearMap.toContinuousLinearMap (complexFourier ρ (fun g => f g))‖^2 ≤
+        b * B * symmetricDegreeConstant u *
+          (Module.finrank ℂ V : ℝ)^(-1+(u+2)/(b : ℝ)))
+
+lemma inverseDegreeSum_block_sum {X I : Type*} [Fintype X] [DecidableEq X] [Fintype I]
+    (M : I → Set X) (u : ℝ) :
+    (∑ i, inverseDegreeSum (blockSubgroup (M i)) u) =
+      ∑ i, symmetricInverseDegreeSum (Fintype.card (M i)) u := by
+  apply Finset.sum_congr rfl
+  intro i _
+  exact inverseDegreeSum_block (M i) u
+
+theorem symmetric_block_degree_transfer {n b : ℕ} (hb : 0 < b)
+    (M : Fin b → Set (Fin n)) (hdis : Pairwise (fun i j => Disjoint (M i) (M j)))
+    {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+    [FiniteDimensional ℂ V] (ρ : Representation ℂ (Equiv.Perm (Fin n)) V)
+    [Representation.IsIrreducible ρ] (hρ : IsUnitary ρ)
+    (f : Equiv.Perm (Fin n) → ℝ) (B u : ℝ) (hf : ∀ g, 0 ≤ f g) (hu : 0 < u)
+    (hcap : ∀ i, LeftCosetCap f (blockSubgroup (M i)) B) :
+    hsNormSq (complexFourier ρ (fun g => f g)) ≤
+      B * mass f * (Module.finrank ℂ V : ℝ)^(-1 + (u+2)/(b : ℝ)) *
+        ∑ i, symmetricInverseDegreeSum (Fintype.card (M i)) u := by
+  have h := inverse_degree_sum_transfer ρ (fun i => blockSubgroup (M i)) hb hρ
+    (fun i j hij g hg k hk => blockSubgroup_commute (hdis hij) hg hk) f B u hf hu hcap
+  rw [inverseDegreeSum_block_sum M u] at h
+  exact h
+
+end
+end PartialPermutation
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/TableauLimits.lean b/lean/OAI/RepresentationTheory/PartialPermutation/TableauLimits.lean
index e896c82460e0bc5aa97140c6a3cc34387cda665d..65d11cab573af7c72d0546f69888e775d733d428 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/TableauLimits.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/TableauLimits.lean
@@ -1,110 +1,110 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.TableauWeights
-
-namespace OAI
-
-section
-namespace PartialPermutation
-namespace TableauZeta
-noncomputable section
-open Finset Filter
-open Tableau DiagramCounting
-
-lemma deficitTerm_zero (n : ℕ) (u : ℝ) : deficitTerm n u 0=0 := by
-  apply Finset.sum_eq_zero
-  intro μ _
-  unfold inverseWeight
-  have he : deficit μ.1.1=0 := μ.2
-  simp only [he,ite_true]
-
-lemma deficitTerm_fixed_bound (n : ℕ) (u : ℝ) (hu : 0 < u) (k : ℕ) (hk : 0 < k)
-    (hkn : 2*k≤n) :
-    deficitTerm n u k ≤ 2*(partitionCount k:ℝ)*((n-2*k+1:ℕ):ℝ)^(-u) := by
-  have hcard : (Fintype.card (ShapeDeficit n k):ℝ) ≤ 2*partitionCount k := by
-    exact_mod_cast deficit_count_le n k
-  calc
-    deficitTerm n u k ≤ ∑ μ : ShapeDeficit n k, ((n-2*k+1:ℕ):ℝ)^(-u) := by
-      apply Finset.sum_le_sum
-      intro μ _
-      have hdef : deficit μ.1.1=k := μ.2
-      have hsize : μ.1.1.cells.card=n := μ.1.2
-      have hcount := fixed_deficit_growth μ.1.1 (by simpa only [hdef] using hk) (by simpa only [hdef,hsize] using hkn)
-      rw [hdef,hsize] at hcount
-      unfold inverseWeight
-      rw [ite_eq_right (by omega)]
-      exact Real.rpow_le_rpow_of_nonpos (by positivity)
-        (by exact_mod_cast hcount) (neg_nonpos.mpr hu.le)
-    _ = (Fintype.card (ShapeDeficit n k):ℝ)*((n-2*k+1:ℕ):ℝ)^(-u) := by simp
-    _ ≤ _ := mul_le_mul_of_nonneg_right hcard (Real.rpow_nonneg (by positivity) _)
-
-lemma deficitTerm_tendsto_zero (u : ℝ) (hu : 0 < u) (k : ℕ) :
-    Tendsto (fun n => deficitTerm n u k) atTop (nhds 0) := by
-  by_cases hk : k=0
-  · subst k
-    simpa only [deficitTerm_zero] using tendsto_const_nhds (x := (0:ℝ))
-  · have ht : Tendsto (fun n : ℕ => n-2*k+1) atTop atTop := by
-      apply tendsto_atTop.2
-      intro b
-      filter_upwards [eventually_ge_atTop (b+2*k)] with n hn
-      omega
-    have hr := (tendsto_rpow_neg_atTop hu).comp (tendsto_natCast_atTop_atTop.comp ht)
-    have hr' := hr.const_mul (2*(partitionCount k:ℝ))
-    simp only [mul_zero] at hr'
-    apply squeeze_zero' (Eventually.of_forall (fun n => deficitTerm_nonneg n u k)) _ hr'
-    filter_upwards [eventually_ge_atTop (2*k)] with n hn
-    exact deficitTerm_fixed_bound n u hu k (by omega) hn
-
-lemma tsum_deficitTerm (n : ℕ) (u : ℝ) :
-    (∑' k, deficitTerm n u k) = nontrivialSum n u := by
-  have hh := (hasSum_fintype (fun μ : Shape n => inverseWeight u μ.1)).tsum_fiberwise
-    (fun μ : Shape n => deficit μ.1)
-  change HasSum (fun k => ∑' μ : ShapeDeficit n k, inverseWeight u μ.1.1)
-    (nontrivialSum n u) at hh
-  simpa only [tsum_fintype,deficitTerm] using hh.tsum_eq
-
-theorem nontrivialSum_tendsto_zero (u : ℝ) (hu : 0 < u) :
-    Tendsto (fun n => nontrivialSum n u) atTop (nhds 0) := by
-  have hh := tendsto_tsum_of_dominated_convergence (summable_majorant u hu)
-    (deficitTerm_tendsto_zero u hu)
-    (Eventually.of_forall (fun n k => deficitTerm_bound n u hu k))
-  simpa only [tsum_deficitTerm,tsum_zero] using hh
-
-lemma fullSum_bound (n : ℕ) (u : ℝ) (hu : 0 < u) :
-    fullSum n u ≤ 2*partitionCount 0 + nontrivialSum n u := by
-  have hi (μ : YoungDiagram) : (standardCount μ:ℝ)^(-u) ≤ 1 := by
-    have hc : (1:ℝ) ≤ standardCount μ := by exact_mod_cast standardCount_pos μ
-    simpa only [Real.one_rpow] using
-      Real.rpow_le_rpow_of_nonpos (by norm_num) hc (neg_nonpos.mpr hu.le)
-  have hc : (Fintype.card (ShapeDeficit n 0):ℝ) ≤ 2*partitionCount 0 := by
-    exact_mod_cast deficit_count_le n 0
-  calc
-    fullSum n u ≤ ∑ μ : Shape n,
-        ((if deficit μ.1=0 then (1:ℝ) else 0) + inverseWeight u μ.1) := by
-      apply Finset.sum_le_sum
-      intro μ _
-      unfold inverseWeight
-      split_ifs
-      · simpa only [add_zero] using hi μ.1
-      · simp only [zero_add,le_refl]
-    _ = (Fintype.card (ShapeDeficit n 0):ℝ) + nontrivialSum n u := by
-      rw [Finset.sum_add_distrib]
-      congr 1
-      change (∑ μ : Shape n, if deficit μ.1=0 then (1:ℝ) else 0) =
-        (Fintype.card {μ : Shape n // deficit μ.1=0}:ℝ)
-      rw [Fintype.card_subtype,Finset.sum_boole]
-    _ ≤ _ := add_le_add_left hc _
-
-theorem fullSum_bddAbove (u : ℝ) (hu : 0 < u) :
-    BddAbove (Set.range (fun n => fullSum n u)) := by
-  obtain ⟨a,ha⟩ := (nontrivialSum_tendsto_zero u hu).bddAbove_range
-  refine ⟨2*partitionCount 0+a,?_⟩
-  rintro x ⟨n,rfl⟩
-  exact (fullSum_bound n u hu).trans (add_le_add_right (ha (Set.mem_range_self n)) _)
-
-end
-end TableauZeta
-end PartialPermutation
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.TableauWeights
+
+namespace OAI
+
+section
+namespace PartialPermutation
+namespace TableauZeta
+noncomputable section
+open Finset Filter
+open Tableau DiagramCounting
+
+lemma deficitTerm_zero (n : ℕ) (u : ℝ) : deficitTerm n u 0=0 := by
+  apply Finset.sum_eq_zero
+  intro μ _
+  unfold inverseWeight
+  have he : deficit μ.1.1=0 := μ.2
+  simp only [he,ite_true]
+
+lemma deficitTerm_fixed_bound (n : ℕ) (u : ℝ) (hu : 0 < u) (k : ℕ) (hk : 0 < k)
+    (hkn : 2*k≤n) :
+    deficitTerm n u k ≤ 2*(partitionCount k:ℝ)*((n-2*k+1:ℕ):ℝ)^(-u) := by
+  have hcard : (Fintype.card (ShapeDeficit n k):ℝ) ≤ 2*partitionCount k := by
+    exact_mod_cast deficit_count_le n k
+  calc
+    deficitTerm n u k ≤ ∑ μ : ShapeDeficit n k, ((n-2*k+1:ℕ):ℝ)^(-u) := by
+      apply Finset.sum_le_sum
+      intro μ _
+      have hdef : deficit μ.1.1=k := μ.2
+      have hsize : μ.1.1.cells.card=n := μ.1.2
+      have hcount := fixed_deficit_growth μ.1.1 (by simpa only [hdef] using hk) (by simpa only [hdef,hsize] using hkn)
+      rw [hdef,hsize] at hcount
+      unfold inverseWeight
+      rw [ite_eq_right (by omega)]
+      exact Real.rpow_le_rpow_of_nonpos (by positivity)
+        (by exact_mod_cast hcount) (neg_nonpos.mpr hu.le)
+    _ = (Fintype.card (ShapeDeficit n k):ℝ)*((n-2*k+1:ℕ):ℝ)^(-u) := by simp
+    _ ≤ _ := mul_le_mul_of_nonneg_right hcard (Real.rpow_nonneg (by positivity) _)
+
+lemma deficitTerm_tendsto_zero (u : ℝ) (hu : 0 < u) (k : ℕ) :
+    Tendsto (fun n => deficitTerm n u k) atTop (nhds 0) := by
+  by_cases hk : k=0
+  · subst k
+    simpa only [deficitTerm_zero] using tendsto_const_nhds (x := (0:ℝ))
+  · have ht : Tendsto (fun n : ℕ => n-2*k+1) atTop atTop := by
+      apply tendsto_atTop.2
+      intro b
+      filter_upwards [eventually_ge_atTop (b+2*k)] with n hn
+      omega
+    have hr := (tendsto_rpow_neg_atTop hu).comp (tendsto_natCast_atTop_atTop.comp ht)
+    have hr' := hr.const_mul (2*(partitionCount k:ℝ))
+    simp only [mul_zero] at hr'
+    apply squeeze_zero' (Eventually.of_forall (fun n => deficitTerm_nonneg n u k)) _ hr'
+    filter_upwards [eventually_ge_atTop (2*k)] with n hn
+    exact deficitTerm_fixed_bound n u hu k (by omega) hn
+
+lemma tsum_deficitTerm (n : ℕ) (u : ℝ) :
+    (∑' k, deficitTerm n u k) = nontrivialSum n u := by
+  have hh := (hasSum_fintype (fun μ : Shape n => inverseWeight u μ.1)).tsum_fiberwise
+    (fun μ : Shape n => deficit μ.1)
+  change HasSum (fun k => ∑' μ : ShapeDeficit n k, inverseWeight u μ.1.1)
+    (nontrivialSum n u) at hh
+  simpa only [tsum_fintype,deficitTerm] using hh.tsum_eq
+
+theorem nontrivialSum_tendsto_zero (u : ℝ) (hu : 0 < u) :
+    Tendsto (fun n => nontrivialSum n u) atTop (nhds 0) := by
+  have hh := tendsto_tsum_of_dominated_convergence (summable_majorant u hu)
+    (deficitTerm_tendsto_zero u hu)
+    (Eventually.of_forall (fun n k => deficitTerm_bound n u hu k))
+  simpa only [tsum_deficitTerm,tsum_zero] using hh
+
+lemma fullSum_bound (n : ℕ) (u : ℝ) (hu : 0 < u) :
+    fullSum n u ≤ 2*partitionCount 0 + nontrivialSum n u := by
+  have hi (μ : YoungDiagram) : (standardCount μ:ℝ)^(-u) ≤ 1 := by
+    have hc : (1:ℝ) ≤ standardCount μ := by exact_mod_cast standardCount_pos μ
+    simpa only [Real.one_rpow] using
+      Real.rpow_le_rpow_of_nonpos (by norm_num) hc (neg_nonpos.mpr hu.le)
+  have hc : (Fintype.card (ShapeDeficit n 0):ℝ) ≤ 2*partitionCount 0 := by
+    exact_mod_cast deficit_count_le n 0
+  calc
+    fullSum n u ≤ ∑ μ : Shape n,
+        ((if deficit μ.1=0 then (1:ℝ) else 0) + inverseWeight u μ.1) := by
+      apply Finset.sum_le_sum
+      intro μ _
+      unfold inverseWeight
+      split_ifs
+      · simpa only [add_zero] using hi μ.1
+      · simp only [zero_add,le_refl]
+    _ = (Fintype.card (ShapeDeficit n 0):ℝ) + nontrivialSum n u := by
+      rw [Finset.sum_add_distrib]
+      congr 1
+      change (∑ μ : Shape n, if deficit μ.1=0 then (1:ℝ) else 0) =
+        (Fintype.card {μ : Shape n // deficit μ.1=0}:ℝ)
+      rw [Fintype.card_subtype,Finset.sum_boole]
+    _ ≤ _ := add_le_add_left hc _
+
+theorem fullSum_bddAbove (u : ℝ) (hu : 0 < u) :
+    BddAbove (Set.range (fun n => fullSum n u)) := by
+  obtain ⟨a,ha⟩ := (nontrivialSum_tendsto_zero u hu).bddAbove_range
+  refine ⟨2*partitionCount 0+a,?_⟩
+  rintro x ⟨n,rfl⟩
+  exact (fullSum_bound n u hu).trans (add_le_add_right (ha (Set.mem_range_self n)) _)
+
+end
+end TableauZeta
+end PartialPermutation
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/TableauVectors.lean b/lean/OAI/RepresentationTheory/PartialPermutation/TableauVectors.lean
index 481d5f6971a5db2f0d75c171b06c2a0a93979552..c13831e4450d40140c43d5af194527e960911c8b 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/TableauVectors.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/TableauVectors.lean
@@ -1,114 +1,114 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.StandardTableaux
-import OAI.RepresentationTheory.PartialPermutation.Polytabloids
-import OAI.RepresentationTheory.PartialPermutation.ColumnRearrangement
-
-namespace OAI
-
-section
-namespace PartialPermutation.YoungTabloid
-noncomputable section
-open Finset
-open scoped Classical
-
-abbrev Std (μ : YoungDiagram) := Tableau.StandardFilling μ.cells
-
-def referenceLabels (μ : YoungDiagram) : μ.cells ≃ Fin (Fintype.card μ.cells) := Fintype.equivFin _
-def tableauPerm (μ : YoungDiagram) (t : Std μ) : Equiv.Perm μ.cells :=
-  t.1.trans (referenceLabels μ).symm
-def tableauTabloid (μ : YoungDiagram) (t : Std μ) : Tabloid μ :=
-  relabel μ (tableauPerm μ t) (canonical μ)
-def tableauVector (μ : YoungDiagram) (t : Std μ) : EuclideanSpace ℂ (Tabloid μ) :=
-  tabloidRep μ (tableauPerm μ t) (polytabloid μ)
-
-def rowCell (μ : YoungDiagram) (i : ℕ) (j : Fin (μ.rowLen i)) : μ.cells :=
-  ⟨(i,j),μ.mem_iff_lt_rowLen.mpr j.2⟩
-
-lemma rowCell_strictMono (μ : YoungDiagram) (i : ℕ) : StrictMono (rowCell μ i) := by
-  intro j k h
-  apply lt_of_le_of_ne
-  · exact ⟨le_refl _,h.le⟩
-  · intro he
-    exact h.ne (Fin.ext (congrArg (fun x : μ.cells => x.1.2) he))
-
-lemma standard_eq_of_inverse_rows (μ : YoungDiagram) (s t : Std μ)
-    (h : ∀ k, row μ (s.1.symm k)=row μ (t.1.symm k)) : s=t := by
-  have hh (i : ℕ) : (fun j => s.1 (rowCell μ i j))=(fun j => t.1 (rowCell μ i j)) := by
-    apply ((s.2.comp (rowCell_strictMono μ i)).range_inj_of_wellFoundedLT
-      (t.2.comp (rowCell_strictMono μ i))).mp
-    ext k
-    constructor
-    · rintro ⟨j,rfl⟩
-      let x := t.1.symm (s.1 (rowCell μ i j))
-      have hx : row μ x=i := by simpa [x,rowCell,row] using (h (s.1 (rowCell μ i j))).symm
-      refine ⟨⟨col μ x,?_⟩,?_⟩
-      · have hc := μ.mem_iff_lt_rowLen.mp x.2
-        simpa only [← hx,row,col] using hc
-      · have he : rowCell μ i ⟨col μ x,by simpa only [← hx,row,col] using μ.mem_iff_lt_rowLen.mp x.2⟩=x :=
-          cell_eq_of_row_col μ hx.symm rfl
-        simp only [Function.comp_apply]
-        rw [he]
-        exact t.1.apply_symm_apply _
-    · rintro ⟨j,rfl⟩
-      let x := s.1.symm (t.1 (rowCell μ i j))
-      have hx : row μ x=i := by simpa [x,rowCell,row] using h (t.1 (rowCell μ i j))
-      refine ⟨⟨col μ x,?_⟩,?_⟩
-      · have hc := μ.mem_iff_lt_rowLen.mp x.2
-        simpa only [← hx,row,col] using hc
-      · have he : rowCell μ i ⟨col μ x,by simpa only [← hx,row,col] using μ.mem_iff_lt_rowLen.mp x.2⟩=x :=
-          cell_eq_of_row_col μ hx.symm rfl
-        simp only [Function.comp_apply]
-        rw [he]
-        exact s.1.apply_symm_apply _
-  apply Subtype.ext
-  apply Equiv.ext
-  intro x
-  exact congrFun (hh (row μ x)) ⟨col μ x,μ.mem_iff_lt_rowLen.mp x.2⟩
-
-lemma tableauTabloid_injective (μ : YoungDiagram) : Function.Injective (tableauTabloid μ) := by
-  intro s t h
-  apply standard_eq_of_inverse_rows μ s t
-  intro k
-  have hh := congrFun (congrArg Subtype.val h) ((referenceLabels μ).symm k)
-  simpa [tableauTabloid,tableauPerm,relabel,canonical,ofPerm] using hh
-
-def score (μ : YoungDiagram) (r : Tabloid μ) : ℕ :=
-  ∑ x, ((referenceLabels μ) x).val * r.1 x
-
-lemma score_relabel (μ : YoungDiagram) (g : Equiv.Perm μ.cells) (r : Tabloid μ) :
-    score μ (relabel μ g r) = ∑ x, ((referenceLabels μ) (g x)).val * r.1 x := by
-  unfold score
-  exact (Fintype.sum_equiv g _ _ (fun x => by simp only [relabel_apply,
-    Equiv.Perm.inv_def,Equiv.symm_apply_apply])).symm
-
-lemma score_tableau_column (μ : YoungDiagram) (t : Std μ) (c : columnGroup μ) :
-    score μ (relabel μ (tableauPerm μ t) (relabel μ c (canonical μ))) =
-      ∑ x, (t.1 x).val * row μ (c.1⁻¹ x) := by
-  rw [score_relabel]
-  apply Finset.sum_congr rfl
-  intro x _
-  simp [tableauPerm,relabel,canonical,ofPerm]
-
-lemma score_tableau (μ : YoungDiagram) (t : Std μ) :
-    score μ (tableauTabloid μ t)=∑ x, (t.1 x).val * row μ x := by
-  simpa only [relabel_one,Subgroup.coe_one,inv_one,Equiv.Perm.one_apply,tableauTabloid]
-    using score_tableau_column μ t 1
-
-lemma score_column_le (μ : YoungDiagram) (t : Std μ) (c : columnGroup μ) :
-    score μ (relabel μ (tableauPerm μ t) (relabel μ c (canonical μ))) ≤
-      score μ (tableauTabloid μ t) := by
-  rw [score_tableau_column,score_tableau]
-  exact full_rearrangement μ _ (fun x y h => t.2 h) c⁻¹
-
-lemma score_column_eq (μ : YoungDiagram) (t : Std μ) (c : columnGroup μ)
-    (h : score μ (relabel μ (tableauPerm μ t) (relabel μ c (canonical μ))) =
-      score μ (tableauTabloid μ t)) : c=1 := by
-  rw [score_tableau_column,score_tableau] at h
-  have hh := full_rearrangement_eq μ _ (fun x y h => t.2 h) c⁻¹ h
-  simpa using hh
-
-end
-end PartialPermutation.YoungTabloid
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.StandardTableaux
+import OAI.RepresentationTheory.PartialPermutation.Polytabloids
+import OAI.RepresentationTheory.PartialPermutation.ColumnRearrangement
+
+namespace OAI
+
+section
+namespace PartialPermutation.YoungTabloid
+noncomputable section
+open Finset
+open scoped Classical
+
+abbrev Std (μ : YoungDiagram) := Tableau.StandardFilling μ.cells
+
+def referenceLabels (μ : YoungDiagram) : μ.cells ≃ Fin (Fintype.card μ.cells) := Fintype.equivFin _
+def tableauPerm (μ : YoungDiagram) (t : Std μ) : Equiv.Perm μ.cells :=
+  t.1.trans (referenceLabels μ).symm
+def tableauTabloid (μ : YoungDiagram) (t : Std μ) : Tabloid μ :=
+  relabel μ (tableauPerm μ t) (canonical μ)
+def tableauVector (μ : YoungDiagram) (t : Std μ) : EuclideanSpace ℂ (Tabloid μ) :=
+  tabloidRep μ (tableauPerm μ t) (polytabloid μ)
+
+def rowCell (μ : YoungDiagram) (i : ℕ) (j : Fin (μ.rowLen i)) : μ.cells :=
+  ⟨(i,j),μ.mem_iff_lt_rowLen.mpr j.2⟩
+
+lemma rowCell_strictMono (μ : YoungDiagram) (i : ℕ) : StrictMono (rowCell μ i) := by
+  intro j k h
+  apply lt_of_le_of_ne
+  · exact ⟨le_refl _,h.le⟩
+  · intro he
+    exact h.ne (Fin.ext (congrArg (fun x : μ.cells => x.1.2) he))
+
+lemma standard_eq_of_inverse_rows (μ : YoungDiagram) (s t : Std μ)
+    (h : ∀ k, row μ (s.1.symm k)=row μ (t.1.symm k)) : s=t := by
+  have hh (i : ℕ) : (fun j => s.1 (rowCell μ i j))=(fun j => t.1 (rowCell μ i j)) := by
+    apply ((s.2.comp (rowCell_strictMono μ i)).range_inj_of_wellFoundedLT
+      (t.2.comp (rowCell_strictMono μ i))).mp
+    ext k
+    constructor
+    · rintro ⟨j,rfl⟩
+      let x := t.1.symm (s.1 (rowCell μ i j))
+      have hx : row μ x=i := by simpa [x,rowCell,row] using (h (s.1 (rowCell μ i j))).symm
+      refine ⟨⟨col μ x,?_⟩,?_⟩
+      · have hc := μ.mem_iff_lt_rowLen.mp x.2
+        simpa only [← hx,row,col] using hc
+      · have he : rowCell μ i ⟨col μ x,by simpa only [← hx,row,col] using μ.mem_iff_lt_rowLen.mp x.2⟩=x :=
+          cell_eq_of_row_col μ hx.symm rfl
+        simp only [Function.comp_apply]
+        rw [he]
+        exact t.1.apply_symm_apply _
+    · rintro ⟨j,rfl⟩
+      let x := s.1.symm (t.1 (rowCell μ i j))
+      have hx : row μ x=i := by simpa [x,rowCell,row] using h (t.1 (rowCell μ i j))
+      refine ⟨⟨col μ x,?_⟩,?_⟩
+      · have hc := μ.mem_iff_lt_rowLen.mp x.2
+        simpa only [← hx,row,col] using hc
+      · have he : rowCell μ i ⟨col μ x,by simpa only [← hx,row,col] using μ.mem_iff_lt_rowLen.mp x.2⟩=x :=
+          cell_eq_of_row_col μ hx.symm rfl
+        simp only [Function.comp_apply]
+        rw [he]
+        exact s.1.apply_symm_apply _
+  apply Subtype.ext
+  apply Equiv.ext
+  intro x
+  exact congrFun (hh (row μ x)) ⟨col μ x,μ.mem_iff_lt_rowLen.mp x.2⟩
+
+lemma tableauTabloid_injective (μ : YoungDiagram) : Function.Injective (tableauTabloid μ) := by
+  intro s t h
+  apply standard_eq_of_inverse_rows μ s t
+  intro k
+  have hh := congrFun (congrArg Subtype.val h) ((referenceLabels μ).symm k)
+  simpa [tableauTabloid,tableauPerm,relabel,canonical,ofPerm] using hh
+
+def score (μ : YoungDiagram) (r : Tabloid μ) : ℕ :=
+  ∑ x, ((referenceLabels μ) x).val * r.1 x
+
+lemma score_relabel (μ : YoungDiagram) (g : Equiv.Perm μ.cells) (r : Tabloid μ) :
+    score μ (relabel μ g r) = ∑ x, ((referenceLabels μ) (g x)).val * r.1 x := by
+  unfold score
+  exact (Fintype.sum_equiv g _ _ (fun x => by simp only [relabel_apply,
+    Equiv.Perm.inv_def,Equiv.symm_apply_apply])).symm
+
+lemma score_tableau_column (μ : YoungDiagram) (t : Std μ) (c : columnGroup μ) :
+    score μ (relabel μ (tableauPerm μ t) (relabel μ c (canonical μ))) =
+      ∑ x, (t.1 x).val * row μ (c.1⁻¹ x) := by
+  rw [score_relabel]
+  apply Finset.sum_congr rfl
+  intro x _
+  simp [tableauPerm,relabel,canonical,ofPerm]
+
+lemma score_tableau (μ : YoungDiagram) (t : Std μ) :
+    score μ (tableauTabloid μ t)=∑ x, (t.1 x).val * row μ x := by
+  simpa only [relabel_one,Subgroup.coe_one,inv_one,Equiv.Perm.one_apply,tableauTabloid]
+    using score_tableau_column μ t 1
+
+lemma score_column_le (μ : YoungDiagram) (t : Std μ) (c : columnGroup μ) :
+    score μ (relabel μ (tableauPerm μ t) (relabel μ c (canonical μ))) ≤
+      score μ (tableauTabloid μ t) := by
+  rw [score_tableau_column,score_tableau]
+  exact full_rearrangement μ _ (fun x y h => t.2 h) c⁻¹
+
+lemma score_column_eq (μ : YoungDiagram) (t : Std μ) (c : columnGroup μ)
+    (h : score μ (relabel μ (tableauPerm μ t) (relabel μ c (canonical μ))) =
+      score μ (tableauTabloid μ t)) : c=1 := by
+  rw [score_tableau_column,score_tableau] at h
+  have hh := full_rearrangement_eq μ _ (fun x y h => t.2 h) c⁻¹ h
+  simpa using hh
+
+end
+end PartialPermutation.YoungTabloid
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/TableauWeights.lean b/lean/OAI/RepresentationTheory/PartialPermutation/TableauWeights.lean
index 8aefe06e47270f86f6037f0ff0e3cb3015392c57..bf08e725e6c3b5081a77c6adcc8e9ef7e3112069 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/TableauWeights.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/TableauWeights.lean
@@ -1,96 +1,96 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.TailBounds
-import OAI.RepresentationTheory.PartialPermutation.DeficitCounts
-import OAI.RepresentationTheory.PartialPermutation.PartitionAsymptotics
-
-namespace OAI
-
-section
-namespace PartialPermutation
-namespace TableauZeta
-noncomputable section
-open Finset Filter
-open Tableau DiagramCounting
-
-lemma inverse_tableau_bound (μ : YoungDiagram) (u : ℝ) (hu : 0 < u) :
-    (standardCount μ : ℝ)^(-u) ≤
-      Real.exp (u*Real.log 2 - (u*Real.log 2/8)*(deficit μ : ℝ)) := by
-  have hc : (0:ℝ)<standardCount μ := by exact_mod_cast standardCount_pos μ
-  have hb : (2:ℝ)^(deficit μ/8) ≤ standardCount μ := by
-    exact_mod_cast tail_power_lower_bound μ
-  have hh := Real.rpow_le_rpow_of_nonpos (show (0:ℝ)<2^(deficit μ/8) by positivity)
-    hb (neg_nonpos.mpr hu.le)
-  apply hh.trans
-  rw [Real.rpow_def_of_pos (by positivity),Real.log_pow]
-  apply Real.exp_le_exp.mpr
-  have hk : (deficit μ : ℝ) < 8*(deficit μ/8 : ℕ)+8 := by
-    exact_mod_cast (show deficit μ < 8*(deficit μ/8)+8 by omega)
-  have hl : 0 < Real.log 2 := Real.log_pos (by norm_num)
-  have hm := mul_le_mul_of_nonneg_left hk.le (show 0 ≤ u*Real.log 2/8 by positivity)
-  nlinarith
-
-def majorant (u : ℝ) (k : ℕ) : ℝ :=
-  2*(partitionCount k : ℝ)*Real.exp (u*Real.log 2 - (u*Real.log 2/8)*k)
-
-lemma summable_majorant (u : ℝ) (hu : 0 < u) : Summable (majorant u) := by
-  let a := u*Real.log 2/8
-  have ha : 0 < a := by dsimp [a]; exact div_pos (mul_pos hu (Real.log_pos (by norm_num))) (by norm_num)
-  have hs : Summable (fun k : ℕ => (2*Real.exp (u*Real.log 2))*Real.exp ((k:ℝ)*(-a/2))) :=
-    (Real.summable_exp_nat_mul_iff.mpr (by linarith)).mul_left _
-  apply hs.of_norm_bounded_eventually_nat
-  filter_upwards [partitionCount_subexponential (a/2) (by positivity)] with k hk
-  rw [Real.norm_eq_abs,abs_of_nonneg (show 0 ≤ majorant u k by unfold majorant; positivity)]
-  calc
-    majorant u k ≤ 2*Real.exp ((a/2)*k)*Real.exp (u*Real.log 2-a*k) := by
-      unfold majorant
-      exact mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left hk (by norm_num)) (Real.exp_nonneg _)
-    _ = (2*Real.exp (u*Real.log 2))*Real.exp ((k:ℝ)*(-a/2)) := by
-      rw [mul_assoc,← Real.exp_add,mul_assoc,← Real.exp_add]
-      congr 2
-      ring
-
-def inverseWeight (u : ℝ) (μ : YoungDiagram) : ℝ :=
-  if deficit μ=0 then 0 else (standardCount μ:ℝ)^(-u)
-
-def deficitTerm (n : ℕ) (u : ℝ) (k : ℕ) : ℝ :=
-  ∑ μ : ShapeDeficit n k, inverseWeight u μ.1.1
-
-def nontrivialSum (n : ℕ) (u : ℝ) : ℝ :=
-  ∑ μ : Shape n, inverseWeight u μ.1
-
-def fullSum (n : ℕ) (u : ℝ) : ℝ :=
-  ∑ μ : Shape n, (standardCount μ.1:ℝ)^(-u)
-
-lemma inverseWeight_nonneg (u : ℝ) (μ : YoungDiagram) : 0 ≤ inverseWeight u μ := by
-  unfold inverseWeight
-  split_ifs <;> positivity
-
-lemma deficitTerm_nonneg (n : ℕ) (u : ℝ) (k : ℕ) : 0 ≤ deficitTerm n u k :=
-  Finset.sum_nonneg (fun μ _ => inverseWeight_nonneg u μ.1.1)
-
-lemma deficitTerm_bound (n : ℕ) (u : ℝ) (hu : 0 < u) (k : ℕ) :
-    ‖deficitTerm n u k‖ ≤ majorant u k := by
-  rw [Real.norm_eq_abs,abs_of_nonneg (deficitTerm_nonneg n u k)]
-  have hcard : (Fintype.card (ShapeDeficit n k):ℝ) ≤ 2*partitionCount k := by
-    exact_mod_cast deficit_count_le n k
-  calc
-    deficitTerm n u k ≤ ∑ μ : ShapeDeficit n k,
-        Real.exp (u*Real.log 2 - (u*Real.log 2/8)*k) := by
-      apply Finset.sum_le_sum
-      intro μ _
-      unfold inverseWeight
-      split_ifs
-      · exact Real.exp_nonneg _
-      · have hh := inverse_tableau_bound μ.1.1 u hu
-        have he : deficit μ.1.1=k := μ.2
-        simpa only [he] using hh
-    _ = (Fintype.card (ShapeDeficit n k):ℝ)*
-        Real.exp (u*Real.log 2 - (u*Real.log 2/8)*k) := by simp
-    _ ≤ majorant u k := mul_le_mul_of_nonneg_right hcard (Real.exp_nonneg _)
-
-end
-end TableauZeta
-end PartialPermutation
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.TailBounds
+import OAI.RepresentationTheory.PartialPermutation.DeficitCounts
+import OAI.RepresentationTheory.PartialPermutation.PartitionAsymptotics
+
+namespace OAI
+
+section
+namespace PartialPermutation
+namespace TableauZeta
+noncomputable section
+open Finset Filter
+open Tableau DiagramCounting
+
+lemma inverse_tableau_bound (μ : YoungDiagram) (u : ℝ) (hu : 0 < u) :
+    (standardCount μ : ℝ)^(-u) ≤
+      Real.exp (u*Real.log 2 - (u*Real.log 2/8)*(deficit μ : ℝ)) := by
+  have hc : (0:ℝ)<standardCount μ := by exact_mod_cast standardCount_pos μ
+  have hb : (2:ℝ)^(deficit μ/8) ≤ standardCount μ := by
+    exact_mod_cast tail_power_lower_bound μ
+  have hh := Real.rpow_le_rpow_of_nonpos (show (0:ℝ)<2^(deficit μ/8) by positivity)
+    hb (neg_nonpos.mpr hu.le)
+  apply hh.trans
+  rw [Real.rpow_def_of_pos (by positivity),Real.log_pow]
+  apply Real.exp_le_exp.mpr
+  have hk : (deficit μ : ℝ) < 8*(deficit μ/8 : ℕ)+8 := by
+    exact_mod_cast (show deficit μ < 8*(deficit μ/8)+8 by omega)
+  have hl : 0 < Real.log 2 := Real.log_pos (by norm_num)
+  have hm := mul_le_mul_of_nonneg_left hk.le (show 0 ≤ u*Real.log 2/8 by positivity)
+  nlinarith
+
+def majorant (u : ℝ) (k : ℕ) : ℝ :=
+  2*(partitionCount k : ℝ)*Real.exp (u*Real.log 2 - (u*Real.log 2/8)*k)
+
+lemma summable_majorant (u : ℝ) (hu : 0 < u) : Summable (majorant u) := by
+  let a := u*Real.log 2/8
+  have ha : 0 < a := by dsimp [a]; exact div_pos (mul_pos hu (Real.log_pos (by norm_num))) (by norm_num)
+  have hs : Summable (fun k : ℕ => (2*Real.exp (u*Real.log 2))*Real.exp ((k:ℝ)*(-a/2))) :=
+    (Real.summable_exp_nat_mul_iff.mpr (by linarith)).mul_left _
+  apply hs.of_norm_bounded_eventually_nat
+  filter_upwards [partitionCount_subexponential (a/2) (by positivity)] with k hk
+  rw [Real.norm_eq_abs,abs_of_nonneg (show 0 ≤ majorant u k by unfold majorant; positivity)]
+  calc
+    majorant u k ≤ 2*Real.exp ((a/2)*k)*Real.exp (u*Real.log 2-a*k) := by
+      unfold majorant
+      exact mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left hk (by norm_num)) (Real.exp_nonneg _)
+    _ = (2*Real.exp (u*Real.log 2))*Real.exp ((k:ℝ)*(-a/2)) := by
+      rw [mul_assoc,← Real.exp_add,mul_assoc,← Real.exp_add]
+      congr 2
+      ring
+
+def inverseWeight (u : ℝ) (μ : YoungDiagram) : ℝ :=
+  if deficit μ=0 then 0 else (standardCount μ:ℝ)^(-u)
+
+def deficitTerm (n : ℕ) (u : ℝ) (k : ℕ) : ℝ :=
+  ∑ μ : ShapeDeficit n k, inverseWeight u μ.1.1
+
+def nontrivialSum (n : ℕ) (u : ℝ) : ℝ :=
+  ∑ μ : Shape n, inverseWeight u μ.1
+
+def fullSum (n : ℕ) (u : ℝ) : ℝ :=
+  ∑ μ : Shape n, (standardCount μ.1:ℝ)^(-u)
+
+lemma inverseWeight_nonneg (u : ℝ) (μ : YoungDiagram) : 0 ≤ inverseWeight u μ := by
+  unfold inverseWeight
+  split_ifs <;> positivity
+
+lemma deficitTerm_nonneg (n : ℕ) (u : ℝ) (k : ℕ) : 0 ≤ deficitTerm n u k :=
+  Finset.sum_nonneg (fun μ _ => inverseWeight_nonneg u μ.1.1)
+
+lemma deficitTerm_bound (n : ℕ) (u : ℝ) (hu : 0 < u) (k : ℕ) :
+    ‖deficitTerm n u k‖ ≤ majorant u k := by
+  rw [Real.norm_eq_abs,abs_of_nonneg (deficitTerm_nonneg n u k)]
+  have hcard : (Fintype.card (ShapeDeficit n k):ℝ) ≤ 2*partitionCount k := by
+    exact_mod_cast deficit_count_le n k
+  calc
+    deficitTerm n u k ≤ ∑ μ : ShapeDeficit n k,
+        Real.exp (u*Real.log 2 - (u*Real.log 2/8)*k) := by
+      apply Finset.sum_le_sum
+      intro μ _
+      unfold inverseWeight
+      split_ifs
+      · exact Real.exp_nonneg _
+      · have hh := inverse_tableau_bound μ.1.1 u hu
+        have he : deficit μ.1.1=k := μ.2
+        simpa only [he] using hh
+    _ = (Fintype.card (ShapeDeficit n k):ℝ)*
+        Real.exp (u*Real.log 2 - (u*Real.log 2/8)*k) := by simp
+    _ ≤ majorant u k := mul_le_mul_of_nonneg_right hcard (Real.exp_nonneg _)
+
+end
+end TableauZeta
+end PartialPermutation
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/Tabloids.lean b/lean/OAI/RepresentationTheory/PartialPermutation/Tabloids.lean
index d9792bf8e6298a14f9e85e34b51c47fb01c2038e..4991840db5bc454cbb59d901e6752dfca4a3f7d2 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/Tabloids.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/Tabloids.lean
@@ -1,87 +1,87 @@
-import Mathlib
-
-namespace OAI
-
-section
-namespace PartialPermutation
-namespace YoungTabloid
-noncomputable section
-open Finset
-open scoped Classical
-
-def row (μ : YoungDiagram) (x : μ.cells) : ℕ := x.1.1
-def col (μ : YoungDiagram) (x : μ.cells) : ℕ := x.1.2
-
-def Tabloid (μ : YoungDiagram) :=
-  {r : μ.cells → ℕ // ∃ p : Equiv.Perm μ.cells, r = fun x => row μ (p x)}
-
-def ofPerm (μ : YoungDiagram) (p : Equiv.Perm μ.cells) : Tabloid μ :=
-  ⟨fun x => row μ (p x),p,rfl⟩
-
-lemma ofPerm_surjective (μ : YoungDiagram) : Function.Surjective (ofPerm μ) := by
-  rintro ⟨r,p,rfl⟩
-  exact ⟨p,rfl⟩
-
-instance (μ : YoungDiagram) : Finite (Tabloid μ) :=
-  Finite.of_surjective (ofPerm μ) (ofPerm_surjective μ)
-instance (μ : YoungDiagram) : Fintype (Tabloid μ) := Fintype.ofFinite _
-
-def canonical (μ : YoungDiagram) : Tabloid μ := ofPerm μ 1
-
-def relabel (μ : YoungDiagram) (g : Equiv.Perm μ.cells) (r : Tabloid μ) : Tabloid μ := by
-  refine ⟨fun x => r.1 (g⁻¹ x),?_⟩
-  obtain ⟨p,hp⟩ := r.2
-  exact ⟨g⁻¹.trans p,by rw [hp]; rfl⟩
-
-@[simp] lemma relabel_apply (μ : YoungDiagram) (g : Equiv.Perm μ.cells) (r : Tabloid μ)
-    (x : μ.cells) : (relabel μ g r).1 x=r.1 (g⁻¹ x) := rfl
-
-lemma relabel_one (μ : YoungDiagram) (r : Tabloid μ) : relabel μ 1 r=r := by rfl
-lemma relabel_mul (μ : YoungDiagram) (g h : Equiv.Perm μ.cells) (r : Tabloid μ) :
-    relabel μ (g*h) r=relabel μ g (relabel μ h r) := by
-  apply Subtype.ext
-  funext x
-  simp only [relabel_apply,mul_inv_rev,Equiv.Perm.mul_apply]
-
-def relabelEquiv (μ : YoungDiagram) (g : Equiv.Perm μ.cells) : Equiv.Perm (Tabloid μ) where
-  toFun := relabel μ g
-  invFun := relabel μ g⁻¹
-  left_inv r := by rw [← relabel_mul,inv_mul_cancel,relabel_one]
-  right_inv r := by rw [← relabel_mul,mul_inv_cancel,relabel_one]
-
-lemma rowSum (μ : YoungDiagram) (r : Tabloid μ) :
-    (∑ x, r.1 x) = ∑ x, row μ x := by
-  obtain ⟨p,hp⟩ := r.2
-  rw [hp]
-  exact Fintype.sum_equiv p _ _ (fun _ => rfl)
-
-def columnGroup (μ : YoungDiagram) : Subgroup (Equiv.Perm μ.cells) where
-  carrier := {g | ∀ x, col μ (g x)=col μ x}
-  one_mem' _ := rfl
-  mul_mem' hg hh x := (hg _).trans (hh x)
-  inv_mem' := by
-    intro g hg x
-    simpa only [Equiv.Perm.inv_def, Equiv.apply_symm_apply] using (hg (g⁻¹ x)).symm
-
-def Col (μ : YoungDiagram) (j : ℕ) := {x : μ.cells // col μ x=j}
-
-instance (μ : YoungDiagram) (j : ℕ) : Fintype (Col μ j) := by dsimp [Col]; infer_instance
-
-def columnEquiv (μ : YoungDiagram) (j : ℕ) : Col μ j ≃ Fin (μ.colLen j) where
-  toFun x := ⟨row μ x.1, by
-    have h := μ.mem_iff_lt_colLen.mp x.1.2
-    have hj : x.1.1.2=j := x.2
-    simpa only [row, hj] using h⟩
-  invFun i := ⟨⟨(i,j),μ.mem_iff_lt_colLen.mpr i.2⟩,rfl⟩
-  left_inv x := by apply Subtype.ext; apply Subtype.ext; exact Prod.ext rfl x.2.symm
-  right_inv _ := rfl
-
-lemma col_card (μ : YoungDiagram) (j : ℕ) : Fintype.card (Col μ j)=μ.colLen j := by
-  simpa only [Fintype.card_fin] using Fintype.card_congr (columnEquiv μ j)
-
-end
-end YoungTabloid
-end PartialPermutation
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+section
+namespace PartialPermutation
+namespace YoungTabloid
+noncomputable section
+open Finset
+open scoped Classical
+
+def row (μ : YoungDiagram) (x : μ.cells) : ℕ := x.1.1
+def col (μ : YoungDiagram) (x : μ.cells) : ℕ := x.1.2
+
+def Tabloid (μ : YoungDiagram) :=
+  {r : μ.cells → ℕ // ∃ p : Equiv.Perm μ.cells, r = fun x => row μ (p x)}
+
+def ofPerm (μ : YoungDiagram) (p : Equiv.Perm μ.cells) : Tabloid μ :=
+  ⟨fun x => row μ (p x),p,rfl⟩
+
+lemma ofPerm_surjective (μ : YoungDiagram) : Function.Surjective (ofPerm μ) := by
+  rintro ⟨r,p,rfl⟩
+  exact ⟨p,rfl⟩
+
+instance (μ : YoungDiagram) : Finite (Tabloid μ) :=
+  Finite.of_surjective (ofPerm μ) (ofPerm_surjective μ)
+instance (μ : YoungDiagram) : Fintype (Tabloid μ) := Fintype.ofFinite _
+
+def canonical (μ : YoungDiagram) : Tabloid μ := ofPerm μ 1
+
+def relabel (μ : YoungDiagram) (g : Equiv.Perm μ.cells) (r : Tabloid μ) : Tabloid μ := by
+  refine ⟨fun x => r.1 (g⁻¹ x),?_⟩
+  obtain ⟨p,hp⟩ := r.2
+  exact ⟨g⁻¹.trans p,by rw [hp]; rfl⟩
+
+@[simp] lemma relabel_apply (μ : YoungDiagram) (g : Equiv.Perm μ.cells) (r : Tabloid μ)
+    (x : μ.cells) : (relabel μ g r).1 x=r.1 (g⁻¹ x) := rfl
+
+lemma relabel_one (μ : YoungDiagram) (r : Tabloid μ) : relabel μ 1 r=r := by rfl
+lemma relabel_mul (μ : YoungDiagram) (g h : Equiv.Perm μ.cells) (r : Tabloid μ) :
+    relabel μ (g*h) r=relabel μ g (relabel μ h r) := by
+  apply Subtype.ext
+  funext x
+  simp only [relabel_apply,mul_inv_rev,Equiv.Perm.mul_apply]
+
+def relabelEquiv (μ : YoungDiagram) (g : Equiv.Perm μ.cells) : Equiv.Perm (Tabloid μ) where
+  toFun := relabel μ g
+  invFun := relabel μ g⁻¹
+  left_inv r := by rw [← relabel_mul,inv_mul_cancel,relabel_one]
+  right_inv r := by rw [← relabel_mul,mul_inv_cancel,relabel_one]
+
+lemma rowSum (μ : YoungDiagram) (r : Tabloid μ) :
+    (∑ x, r.1 x) = ∑ x, row μ x := by
+  obtain ⟨p,hp⟩ := r.2
+  rw [hp]
+  exact Fintype.sum_equiv p _ _ (fun _ => rfl)
+
+def columnGroup (μ : YoungDiagram) : Subgroup (Equiv.Perm μ.cells) where
+  carrier := {g | ∀ x, col μ (g x)=col μ x}
+  one_mem' _ := rfl
+  mul_mem' hg hh x := (hg _).trans (hh x)
+  inv_mem' := by
+    intro g hg x
+    simpa only [Equiv.Perm.inv_def, Equiv.apply_symm_apply] using (hg (g⁻¹ x)).symm
+
+def Col (μ : YoungDiagram) (j : ℕ) := {x : μ.cells // col μ x=j}
+
+instance (μ : YoungDiagram) (j : ℕ) : Fintype (Col μ j) := by dsimp [Col]; infer_instance
+
+def columnEquiv (μ : YoungDiagram) (j : ℕ) : Col μ j ≃ Fin (μ.colLen j) where
+  toFun x := ⟨row μ x.1, by
+    have h := μ.mem_iff_lt_colLen.mp x.1.2
+    have hj : x.1.1.2=j := x.2
+    simpa only [row, hj] using h⟩
+  invFun i := ⟨⟨(i,j),μ.mem_iff_lt_colLen.mpr i.2⟩,rfl⟩
+  left_inv x := by apply Subtype.ext; apply Subtype.ext; exact Prod.ext rfl x.2.symm
+  right_inv _ := rfl
+
+lemma col_card (μ : YoungDiagram) (j : ℕ) : Fintype.card (Col μ j)=μ.colLen j := by
+  simpa only [Fintype.card_fin] using Fintype.card_congr (columnEquiv μ j)
+
+end
+end YoungTabloid
+end PartialPermutation
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/TailBounds.lean b/lean/OAI/RepresentationTheory/PartialPermutation/TailBounds.lean
index bb5b72cf889a85ea4412bed0680027626f4ac177..bb08edf6be6480fcc923d17da6d20850bf723c82 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/TailBounds.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/TailBounds.lean
@@ -1,133 +1,133 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.DiagonalBounds
-import OAI.RepresentationTheory.PartialPermutation.Truncation
-
-namespace OAI
-
-section
-namespace PartialPermutation
-namespace Tableau
-noncomputable section
-open Finset
-
-def transposeCellsEquiv (μ : YoungDiagram) : μ.cells ≃o μ.transpose.cells where
-  toFun x := ⟨x.1.swap, by simpa only [YoungDiagram.mem_cells, YoungDiagram.mem_transpose,
-    Prod.swap_swap] using x.2⟩
-  invFun x := ⟨x.1.swap, YoungDiagram.mem_transpose.mp x.2⟩
-  left_inv _ := rfl
-  right_inv _ := rfl
-  map_rel_iff' := by intro x y; exact ⟨fun h => ⟨h.2,h.1⟩,fun h => ⟨h.2,h.1⟩⟩
-
-lemma standardCount_transpose (μ : YoungDiagram) : standardCount μ.transpose = standardCount μ :=
-  (Fintype.card_congr (fillingCongr (transposeCellsEquiv μ))).symm
-
-lemma card_transpose (μ : YoungDiagram) : μ.transpose.cells.card = μ.cells.card := by
-  simpa only [Fintype.card_coe] using (Fintype.card_congr (transposeCellsEquiv μ).toEquiv).symm
-
-lemma maxLine_transpose (μ : YoungDiagram) : maxLine μ.transpose = maxLine μ := by
-  simp only [maxLine, YoungDiagram.rowLen_transpose, YoungDiagram.colLen_transpose, max_comm]
-
-lemma deficit_transpose (μ : YoungDiagram) : deficit μ.transpose = deficit μ := by
-  simp only [deficit, card_transpose, maxLine_transpose]
-
-lemma two_pow_le_centralBinom (b : ℕ) : 2^b ≤ (2*b).choose b := by
-  change 2^b ≤ Nat.centralBinom b
-  induction b with
-  | zero => simp
-  | succ b ih =>
-    have he := Nat.succ_mul_centralBinom_succ b
-    have hstep : 2 * Nat.centralBinom b ≤ Nat.centralBinom (b+1) := by
-      by_contra h
-      have hh := Nat.mul_lt_mul_of_pos_left (show Nat.centralBinom (b+1) < 2*Nat.centralBinom b by omega)
-        (Nat.succ_pos b)
-      nlinarith
-    calc
-      2^(b+1) = 2 * 2^b := by rw [pow_succ']
-      _ ≤ 2 * Nat.centralBinom b := Nat.mul_le_mul_left _ ih
-      _ ≤ _ := hstep
-
-lemma tail_power_lower_bound_oriented (μ : YoungDiagram)
-    (hor : μ.colLen 0 ≤ μ.rowLen 0) : 2^(deficit μ/8) ≤ standardCount μ := by
-  have hL : maxLine μ = μ.rowLen 0 := max_eq_left hor
-  by_cases hsmall : 4*maxLine μ ≤ μ.cells.card
-  · apply le_trans (Nat.pow_le_pow_right (by omega) (show deficit μ/8 ≤
-        μ.cells.card-(2*maxLine μ-1) by unfold deficit; omega))
-      (width_height_power_lower_bound μ)
-  · let b := min (deficit μ) (μ.rowLen 0/2)
-    have hb₁ : b ≤ deficit μ := min_le_left _ _
-    have hb₂ : b ≤ μ.rowLen 0/2 := min_le_right _ _
-    have hba : b ≤ min (μ.rowLen 0) (μ.cells.card-μ.rowLen 0) := by
-      rw [deficit,hL] at hb₁
-      exact le_min (by omega) hb₁
-    have hkb : deficit μ/8 ≤ b := by
-      have hc := topRow_le_card μ
-      dsimp [b]
-      apply le_min
-      · exact Nat.div_le_self _ _
-      · rw [hL] at hsmall
-        unfold deficit
-        rw [hL]
-        omega
-    calc
-      2^(deficit μ/8) ≤ 2^b := Nat.pow_le_pow_right (by omega) hkb
-      _ ≤ (2*b).choose b := two_pow_le_centralBinom b
-      _ ≤ (μ.rowLen 0).choose b := Nat.choose_le_choose b (by omega)
-      _ ≤ standardCount μ := binomial_lower_bound μ b hba
-
-lemma tail_power_lower_bound (μ : YoungDiagram) : 2^(deficit μ/8) ≤ standardCount μ := by
-  by_cases h : μ.colLen 0 ≤ μ.rowLen 0
-  · exact tail_power_lower_bound_oriented μ h
-  · have hh := tail_power_lower_bound_oriented μ.transpose (by
-      simpa only [YoungDiagram.rowLen_transpose, YoungDiagram.colLen_transpose] using le_of_not_ge h)
-    simpa only [deficit_transpose, standardCount_transpose] using hh
-
-lemma longest_binomial_lower_bound (μ : YoungDiagram) (b : ℕ)
-    (hb : b ≤ min (maxLine μ) (deficit μ)) :
-    (maxLine μ).choose b ≤ standardCount μ := by
-  by_cases h : μ.colLen 0 ≤ μ.rowLen 0
-  · have he : maxLine μ=μ.rowLen 0 := max_eq_left h
-    rw [he]
-    exact binomial_lower_bound μ b (by simpa only [deficit,he] using hb)
-  · have hb' : b ≤ min (μ.transpose.rowLen 0)
-        (μ.transpose.cells.card - μ.transpose.rowLen 0) := by
-      simpa only [deficit,maxLine,max_eq_right (le_of_not_ge h),
-        card_transpose,YoungDiagram.rowLen_transpose] using hb
-    simpa only [YoungDiagram.rowLen_transpose,standardCount_transpose,
-      maxLine,max_eq_right (le_of_not_ge h)] using binomial_lower_bound μ.transpose b hb'
-
-lemma linear_le_choose (a k : ℕ) (hk : 0 < k) (hka : k ≤ a) : a-k+1 ≤ a.choose k := by
-  obtain ⟨r,rfl⟩ := Nat.exists_eq_add_of_le hka
-  simp only [Nat.add_sub_cancel_left]
-  induction r with
-  | zero => simp
-  | succ r ih =>
-    have hk' : k-1 ≤ k+r := by omega
-    have hpos := Nat.choose_pos hk'
-    have he : (k+(r+1)).choose k = (k+r).choose (k-1)+(k+r).choose k := by
-      have h := Nat.choose_succ_succ (k+r) (k-1)
-      simpa only [Nat.succ_eq_add_one, Nat.sub_add_cancel (show 1≤k by omega), Nat.add_assoc] using h
-    rw [he]
-    omega
-
-lemma fixed_deficit_growth (μ : YoungDiagram) (hk : 0 < deficit μ)
-    (hlarge : 2*deficit μ ≤ μ.cells.card) :
-    μ.cells.card-2*deficit μ+1 ≤ standardCount μ := by
-  have hLle : maxLine μ ≤ μ.cells.card := by
-    apply max_le
-    · exact topRow_le_card μ
-    · have h:=topRow_le_card μ.transpose
-      simpa only [YoungDiagram.rowLen_transpose,card_transpose] using h
-  have hkL : deficit μ ≤ maxLine μ := by unfold deficit at *; omega
-  have he : μ.cells.card-2*deficit μ=maxLine μ-deficit μ := by
-    unfold deficit at *; omega
-  rw [he]
-  exact (linear_le_choose _ _ hk hkL).trans
-    (longest_binomial_lower_bound μ (deficit μ) (le_min hkL le_rfl))
-
-end
-end Tableau
-end PartialPermutation
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.DiagonalBounds
+import OAI.RepresentationTheory.PartialPermutation.Truncation
+
+namespace OAI
+
+section
+namespace PartialPermutation
+namespace Tableau
+noncomputable section
+open Finset
+
+def transposeCellsEquiv (μ : YoungDiagram) : μ.cells ≃o μ.transpose.cells where
+  toFun x := ⟨x.1.swap, by simpa only [YoungDiagram.mem_cells, YoungDiagram.mem_transpose,
+    Prod.swap_swap] using x.2⟩
+  invFun x := ⟨x.1.swap, YoungDiagram.mem_transpose.mp x.2⟩
+  left_inv _ := rfl
+  right_inv _ := rfl
+  map_rel_iff' := by intro x y; exact ⟨fun h => ⟨h.2,h.1⟩,fun h => ⟨h.2,h.1⟩⟩
+
+lemma standardCount_transpose (μ : YoungDiagram) : standardCount μ.transpose = standardCount μ :=
+  (Fintype.card_congr (fillingCongr (transposeCellsEquiv μ))).symm
+
+lemma card_transpose (μ : YoungDiagram) : μ.transpose.cells.card = μ.cells.card := by
+  simpa only [Fintype.card_coe] using (Fintype.card_congr (transposeCellsEquiv μ).toEquiv).symm
+
+lemma maxLine_transpose (μ : YoungDiagram) : maxLine μ.transpose = maxLine μ := by
+  simp only [maxLine, YoungDiagram.rowLen_transpose, YoungDiagram.colLen_transpose, max_comm]
+
+lemma deficit_transpose (μ : YoungDiagram) : deficit μ.transpose = deficit μ := by
+  simp only [deficit, card_transpose, maxLine_transpose]
+
+lemma two_pow_le_centralBinom (b : ℕ) : 2^b ≤ (2*b).choose b := by
+  change 2^b ≤ Nat.centralBinom b
+  induction b with
+  | zero => simp
+  | succ b ih =>
+    have he := Nat.succ_mul_centralBinom_succ b
+    have hstep : 2 * Nat.centralBinom b ≤ Nat.centralBinom (b+1) := by
+      by_contra h
+      have hh := Nat.mul_lt_mul_of_pos_left (show Nat.centralBinom (b+1) < 2*Nat.centralBinom b by omega)
+        (Nat.succ_pos b)
+      nlinarith
+    calc
+      2^(b+1) = 2 * 2^b := by rw [pow_succ']
+      _ ≤ 2 * Nat.centralBinom b := Nat.mul_le_mul_left _ ih
+      _ ≤ _ := hstep
+
+lemma tail_power_lower_bound_oriented (μ : YoungDiagram)
+    (hor : μ.colLen 0 ≤ μ.rowLen 0) : 2^(deficit μ/8) ≤ standardCount μ := by
+  have hL : maxLine μ = μ.rowLen 0 := max_eq_left hor
+  by_cases hsmall : 4*maxLine μ ≤ μ.cells.card
+  · apply le_trans (Nat.pow_le_pow_right (by omega) (show deficit μ/8 ≤
+        μ.cells.card-(2*maxLine μ-1) by unfold deficit; omega))
+      (width_height_power_lower_bound μ)
+  · let b := min (deficit μ) (μ.rowLen 0/2)
+    have hb₁ : b ≤ deficit μ := min_le_left _ _
+    have hb₂ : b ≤ μ.rowLen 0/2 := min_le_right _ _
+    have hba : b ≤ min (μ.rowLen 0) (μ.cells.card-μ.rowLen 0) := by
+      rw [deficit,hL] at hb₁
+      exact le_min (by omega) hb₁
+    have hkb : deficit μ/8 ≤ b := by
+      have hc := topRow_le_card μ
+      dsimp [b]
+      apply le_min
+      · exact Nat.div_le_self _ _
+      · rw [hL] at hsmall
+        unfold deficit
+        rw [hL]
+        omega
+    calc
+      2^(deficit μ/8) ≤ 2^b := Nat.pow_le_pow_right (by omega) hkb
+      _ ≤ (2*b).choose b := two_pow_le_centralBinom b
+      _ ≤ (μ.rowLen 0).choose b := Nat.choose_le_choose b (by omega)
+      _ ≤ standardCount μ := binomial_lower_bound μ b hba
+
+lemma tail_power_lower_bound (μ : YoungDiagram) : 2^(deficit μ/8) ≤ standardCount μ := by
+  by_cases h : μ.colLen 0 ≤ μ.rowLen 0
+  · exact tail_power_lower_bound_oriented μ h
+  · have hh := tail_power_lower_bound_oriented μ.transpose (by
+      simpa only [YoungDiagram.rowLen_transpose, YoungDiagram.colLen_transpose] using le_of_not_ge h)
+    simpa only [deficit_transpose, standardCount_transpose] using hh
+
+lemma longest_binomial_lower_bound (μ : YoungDiagram) (b : ℕ)
+    (hb : b ≤ min (maxLine μ) (deficit μ)) :
+    (maxLine μ).choose b ≤ standardCount μ := by
+  by_cases h : μ.colLen 0 ≤ μ.rowLen 0
+  · have he : maxLine μ=μ.rowLen 0 := max_eq_left h
+    rw [he]
+    exact binomial_lower_bound μ b (by simpa only [deficit,he] using hb)
+  · have hb' : b ≤ min (μ.transpose.rowLen 0)
+        (μ.transpose.cells.card - μ.transpose.rowLen 0) := by
+      simpa only [deficit,maxLine,max_eq_right (le_of_not_ge h),
+        card_transpose,YoungDiagram.rowLen_transpose] using hb
+    simpa only [YoungDiagram.rowLen_transpose,standardCount_transpose,
+      maxLine,max_eq_right (le_of_not_ge h)] using binomial_lower_bound μ.transpose b hb'
+
+lemma linear_le_choose (a k : ℕ) (hk : 0 < k) (hka : k ≤ a) : a-k+1 ≤ a.choose k := by
+  obtain ⟨r,rfl⟩ := Nat.exists_eq_add_of_le hka
+  simp only [Nat.add_sub_cancel_left]
+  induction r with
+  | zero => simp
+  | succ r ih =>
+    have hk' : k-1 ≤ k+r := by omega
+    have hpos := Nat.choose_pos hk'
+    have he : (k+(r+1)).choose k = (k+r).choose (k-1)+(k+r).choose k := by
+      have h := Nat.choose_succ_succ (k+r) (k-1)
+      simpa only [Nat.succ_eq_add_one, Nat.sub_add_cancel (show 1≤k by omega), Nat.add_assoc] using h
+    rw [he]
+    omega
+
+lemma fixed_deficit_growth (μ : YoungDiagram) (hk : 0 < deficit μ)
+    (hlarge : 2*deficit μ ≤ μ.cells.card) :
+    μ.cells.card-2*deficit μ+1 ≤ standardCount μ := by
+  have hLle : maxLine μ ≤ μ.cells.card := by
+    apply max_le
+    · exact topRow_le_card μ
+    · have h:=topRow_le_card μ.transpose
+      simpa only [YoungDiagram.rowLen_transpose,card_transpose] using h
+  have hkL : deficit μ ≤ maxLine μ := by unfold deficit at *; omega
+  have he : μ.cells.card-2*deficit μ=maxLine μ-deficit μ := by
+    unfold deficit at *; omega
+  rw [he]
+  exact (linear_le_choose _ _ hk hkL).trans
+    (longest_binomial_lower_bound μ (deficit μ) (le_min hkL le_rfl))
+
+end
+end Tableau
+end PartialPermutation
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/TrivialRepresentation.lean b/lean/OAI/RepresentationTheory/PartialPermutation/TrivialRepresentation.lean
index 374188d8aecafe2657073fa8502bb44bedd6ccce..1fab802eec0293617341323b36ad993c3d4089b3 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/TrivialRepresentation.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/TrivialRepresentation.lean
@@ -1,93 +1,93 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.Plancherel
-
-namespace OAI
-
-section
-open scoped Classical
-open scoped BigOperators ComplexConjugate MonoidAlgebra
-open scoped BigOperators ComplexConjugate
-open scoped MonoidAlgebra BigOperators
-open scoped BigOperators MonoidAlgebra Classical
-
-namespace PartialPermutation
-noncomputable section
-open scoped BigOperators MonoidAlgebra Classical
-
-variable {G : Type*} [Group G] [Fintype G]
-def trivialSubmoduleEquiv (G : Type*) [Group G] : Subrepresentation (Representation.trivial ℂ G ℂ) ≃o Submodule ℂ ℂ where
-  toFun := Subrepresentation.toSubmodule
-  invFun S := ⟨S, fun g x hx => by simpa only [Representation.trivial_apply] using hx⟩
-  left_inv S := rfl
-  right_inv S := rfl
-  map_rel_iff' := Iff.rfl
-
-instance trivial_rep_irreducible (G : Type*) [Group G] : Representation.IsIrreducible (Representation.trivial ℂ G ℂ) :=
-  (OrderIso.isSimpleOrder_iff (trivialSubmoduleEquiv G)).mpr inferInstance
-
-omit [Fintype G] in
-@[simp] lemma trivial_character (g : G) : (Representation.trivial ℂ G ℂ).character g = 1 := by
-  simp [Representation.character, Representation.trivial]
-
-def trivialIndex (G : Type*) [Group G] [Fintype G] : IrreducibleIndex G :=
-  (irreducibleRep_complete (Representation.trivial ℂ G ℂ)).choose
-
-lemma trivialIndex_equiv :
-    Nonempty (Representation.Equiv (irreducibleRep (trivialIndex G))
-      (Representation.trivial ℂ G ℂ)) :=
-  (irreducibleRep_complete (Representation.trivial ℂ G ℂ)).choose_spec.1
-
-lemma trivialIndex_unique (c : IrreducibleIndex G) :
-    Nonempty (Representation.Equiv (Representation.trivial ℂ G ℂ) (irreducibleRep c)) ↔
-      c = trivialIndex G := by
-  constructor
-  · rintro ⟨e⟩
-    exact (irreducibleRep_complete (Representation.trivial ℂ G ℂ)).choose_spec.2 c ⟨e.symm⟩
-  · rintro rfl
-    obtain ⟨e⟩ := trivialIndex_equiv (G := G)
-    exact ⟨e.symm⟩
-
-@[simp] lemma irreducibleDegree_trivial : irreducibleDegree (trivialIndex G) = 1 := by
-  obtain ⟨e⟩ := trivialIndex_equiv (G := G)
-  simpa [irreducibleDegree] using e.toLinearEquiv.finrank_eq
-
-@[simp] lemma irreducibleRep_trivial_apply (g : G) :
-    irreducibleRep (trivialIndex G) g = 1 := by
-  obtain ⟨e⟩ := trivialIndex_equiv (G := G)
-  apply LinearMap.ext
-  intro x
-  apply e.toLinearEquiv.injective
-  simpa using LinearMap.congr_fun (e.toIntertwiningMap.isIntertwining' g) x
-
-lemma complexFourier_trivial (a : G → ℂ) :
-    complexFourier (irreducibleRep (trivialIndex G)) a = (∑ g, a g) • 1 := by
-  simp only [complexFourier, irreducibleRep_trivial_apply, Finset.sum_smul]
-
-lemma complexFourier_uniform (c : IrreducibleIndex G) :
-    complexFourier (irreducibleRep c) (fun _ => (Fintype.card G : ℂ)⁻¹) =
-      if c = trivialIndex G then 1 else 0 := by
-  have hk : centralCharacterKernel (Representation.trivial ℂ G ℂ) =
-      fun _ => (Fintype.card G : ℂ)⁻¹ := by
-    funext g
-    simp [centralCharacterKernel, trivial_character]
-  rw [← hk, centralCharacterKernel_fourier, trivialIndex_unique]
-  split_ifs <;> rfl
-
-lemma complexFourier_centered (a : G → ℂ) (z : ℂ) (c : IrreducibleIndex G) :
-    complexFourier (irreducibleRep c) (fun g => a g - z / Fintype.card G) =
-      complexFourier (irreducibleRep c) a -
-        z • (if c = trivialIndex G then 1 else 0) := by
-  rw [← complexFourier_uniform]
-  simp only [complexFourier, Finset.smul_sum,
-    smul_smul, div_eq_mul_inv]
-  rw [← Finset.sum_sub_distrib]
-  apply Finset.sum_congr rfl
-  intro g _
-  exact sub_smul (R := ℂ) (M := Module.End ℂ (irreducibleSpace c))
-    (a g) (z * (Fintype.card G : ℂ)⁻¹) ((irreducibleRep c) g)
-
-end
-end PartialPermutation
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.Plancherel
+
+namespace OAI
+
+section
+open scoped Classical
+open scoped BigOperators ComplexConjugate MonoidAlgebra
+open scoped BigOperators ComplexConjugate
+open scoped MonoidAlgebra BigOperators
+open scoped BigOperators MonoidAlgebra Classical
+
+namespace PartialPermutation
+noncomputable section
+open scoped BigOperators MonoidAlgebra Classical
+
+variable {G : Type*} [Group G] [Fintype G]
+def trivialSubmoduleEquiv (G : Type*) [Group G] : Subrepresentation (Representation.trivial ℂ G ℂ) ≃o Submodule ℂ ℂ where
+  toFun := Subrepresentation.toSubmodule
+  invFun S := ⟨S, fun g x hx => by simpa only [Representation.trivial_apply] using hx⟩
+  left_inv S := rfl
+  right_inv S := rfl
+  map_rel_iff' := Iff.rfl
+
+instance trivial_rep_irreducible (G : Type*) [Group G] : Representation.IsIrreducible (Representation.trivial ℂ G ℂ) :=
+  (OrderIso.isSimpleOrder_iff (trivialSubmoduleEquiv G)).mpr inferInstance
+
+omit [Fintype G] in
+@[simp] lemma trivial_character (g : G) : (Representation.trivial ℂ G ℂ).character g = 1 := by
+  simp [Representation.character, Representation.trivial]
+
+def trivialIndex (G : Type*) [Group G] [Fintype G] : IrreducibleIndex G :=
+  (irreducibleRep_complete (Representation.trivial ℂ G ℂ)).choose
+
+lemma trivialIndex_equiv :
+    Nonempty (Representation.Equiv (irreducibleRep (trivialIndex G))
+      (Representation.trivial ℂ G ℂ)) :=
+  (irreducibleRep_complete (Representation.trivial ℂ G ℂ)).choose_spec.1
+
+lemma trivialIndex_unique (c : IrreducibleIndex G) :
+    Nonempty (Representation.Equiv (Representation.trivial ℂ G ℂ) (irreducibleRep c)) ↔
+      c = trivialIndex G := by
+  constructor
+  · rintro ⟨e⟩
+    exact (irreducibleRep_complete (Representation.trivial ℂ G ℂ)).choose_spec.2 c ⟨e.symm⟩
+  · rintro rfl
+    obtain ⟨e⟩ := trivialIndex_equiv (G := G)
+    exact ⟨e.symm⟩
+
+@[simp] lemma irreducibleDegree_trivial : irreducibleDegree (trivialIndex G) = 1 := by
+  obtain ⟨e⟩ := trivialIndex_equiv (G := G)
+  simpa [irreducibleDegree] using e.toLinearEquiv.finrank_eq
+
+@[simp] lemma irreducibleRep_trivial_apply (g : G) :
+    irreducibleRep (trivialIndex G) g = 1 := by
+  obtain ⟨e⟩ := trivialIndex_equiv (G := G)
+  apply LinearMap.ext
+  intro x
+  apply e.toLinearEquiv.injective
+  simpa using LinearMap.congr_fun (e.toIntertwiningMap.isIntertwining' g) x
+
+lemma complexFourier_trivial (a : G → ℂ) :
+    complexFourier (irreducibleRep (trivialIndex G)) a = (∑ g, a g) • 1 := by
+  simp only [complexFourier, irreducibleRep_trivial_apply, Finset.sum_smul]
+
+lemma complexFourier_uniform (c : IrreducibleIndex G) :
+    complexFourier (irreducibleRep c) (fun _ => (Fintype.card G : ℂ)⁻¹) =
+      if c = trivialIndex G then 1 else 0 := by
+  have hk : centralCharacterKernel (Representation.trivial ℂ G ℂ) =
+      fun _ => (Fintype.card G : ℂ)⁻¹ := by
+    funext g
+    simp [centralCharacterKernel, trivial_character]
+  rw [← hk, centralCharacterKernel_fourier, trivialIndex_unique]
+  split_ifs <;> rfl
+
+lemma complexFourier_centered (a : G → ℂ) (z : ℂ) (c : IrreducibleIndex G) :
+    complexFourier (irreducibleRep c) (fun g => a g - z / Fintype.card G) =
+      complexFourier (irreducibleRep c) a -
+        z • (if c = trivialIndex G then 1 else 0) := by
+  rw [← complexFourier_uniform]
+  simp only [complexFourier, Finset.smul_sum,
+    smul_smul, div_eq_mul_inv]
+  rw [← Finset.sum_sub_distrib]
+  apply Finset.sum_congr rfl
+  intro g _
+  exact sub_smul (R := ℂ) (M := Module.End ℂ (irreducibleSpace c))
+    (a g) (z * (Fintype.card G : ℂ)⁻¹) ((irreducibleRep c) g)
+
+end
+end PartialPermutation
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/PartialPermutation/Truncation.lean b/lean/OAI/RepresentationTheory/PartialPermutation/Truncation.lean
index c730110e146ada5d569d851e24f2e94aa628130a..51847578fda459e091ed58880756297157711c8b 100644
--- a/lean/OAI/RepresentationTheory/PartialPermutation/Truncation.lean
+++ b/lean/OAI/RepresentationTheory/PartialPermutation/Truncation.lean
@@ -1,125 +1,125 @@
-import Mathlib
-import OAI.RepresentationTheory.PartialPermutation.RowBounds
-
-namespace OAI
-
-section
-namespace PartialPermutation
-namespace Tableau
-noncomputable section
-open Finset
-
-def truncateLower (μ : YoungDiagram)
-    (T : StandardFilling {x : μ.cells // x.1.1 ≠ 0}) (b : ℕ) : YoungDiagram := by
-  classical
-  let P : ℕ × ℕ → Prop := fun x => ∃ hx : x ∈ μ.cells,
-    ∀ hn : x.1 ≠ 0, (T.1 ⟨⟨x,hx⟩,hn⟩).val < b
-  refine ⟨μ.cells.filter P, ?_⟩
-  intro y x hxy hy
-  obtain ⟨hy,hpy⟩ := Finset.mem_filter.mp hy
-  obtain ⟨_,hty⟩ := hpy
-  have hx := μ.isLowerSet hxy hy
-  apply Finset.mem_filter.mpr
-  refine ⟨hx,hx,?_⟩
-  intro hn
-  have hny : y.1≠0 := by intro he; exact hn (Nat.eq_zero_of_le_zero (hxy.1.trans_eq he))
-  have hmono : (T.1 ⟨⟨x,hx⟩,hn⟩).val ≤ (T.1 ⟨⟨y,hy⟩,hny⟩).val :=
-    T.2.monotone (show (⟨⟨x,hx⟩,hn⟩ : {x : μ.cells // x.1.1≠0}) ≤
-      ⟨⟨y,hy⟩,hny⟩ from hxy)
-  exact hmono.trans_lt (hty hny)
-
-lemma mem_truncateLower (μ : YoungDiagram)
-    (T : StandardFilling {x : μ.cells // x.1.1 ≠ 0}) (b : ℕ) (x : ℕ × ℕ) :
-    x ∈ (truncateLower μ T b).cells ↔ ∃ hx : x ∈ μ.cells,
-      ∀ hn : x.1 ≠ 0, (T.1 ⟨⟨x,hx⟩,hn⟩).val < b := by
-  classical
-  simp only [truncateLower, Finset.mem_filter]
-  exact ⟨fun h => h.2, fun h => ⟨h.choose,h⟩⟩
-
-lemma truncateLower_le (μ : YoungDiagram)
-    (T : StandardFilling {x : μ.cells // x.1.1 ≠ 0}) (b : ℕ) :
-    truncateLower μ T b ≤ μ := by
-  intro x hx
-  exact (mem_truncateLower μ T b x).mp hx |>.choose
-
-lemma truncateLower_topRow (μ : YoungDiagram)
-    (T : StandardFilling {x : μ.cells // x.1.1 ≠ 0}) (b : ℕ) :
-    (truncateLower μ T b).rowLen 0 = μ.rowLen 0 := by
-  have hm : ∀ j, (0,j) ∈ truncateLower μ T b ↔ (0,j) ∈ μ := by
-    intro j
-    rw [← YoungDiagram.mem_cells, mem_truncateLower]
-    exact ⟨fun h => h.choose, fun h => ⟨h,fun hn => (hn rfl).elim⟩⟩
-  apply le_antisymm
-  · by_contra h
-    have hh := (hm (μ.rowLen 0)).mp ((truncateLower μ T b).mem_iff_lt_rowLen.mpr (by omega))
-    exact (lt_irrefl _) (μ.mem_iff_lt_rowLen.mp hh)
-  · by_contra h
-    have hh := (hm ((truncateLower μ T b).rowLen 0)).mpr (μ.mem_iff_lt_rowLen.mpr (by omega))
-    exact (lt_irrefl _) ((truncateLower μ T b).mem_iff_lt_rowLen.mp hh)
-
-def truncatedLowerEquiv (μ : YoungDiagram)
-    (T : StandardFilling {x : μ.cells // x.1.1 ≠ 0}) (b : ℕ)
-    (hb : b ≤ μ.cells.card - μ.rowLen 0) :
-    {x : (truncateLower μ T b).cells // x.1.1 ≠ 0} ≃ Fin b := by
-  let incl : {x : (truncateLower μ T b).cells // x.1.1 ≠ 0} →
-      {x : μ.cells // x.1.1 ≠ 0} := fun x =>
-    ⟨⟨x.1.1,truncateLower_le μ T b x.1.2⟩,x.2⟩
-  have hb' : b ≤ Fintype.card {x : μ.cells // x.1.1≠0} := by rwa [lower_card]
-  let val : Fin b → {x : μ.cells // x.1.1 ≠ 0} := fun i =>
-    T.1.symm ⟨i.val,i.isLt.trans_le hb'⟩
-  have hv : ∀ i, (val i).1.1 ∈ (truncateLower μ T b).cells := by
-    intro i
-    apply (mem_truncateLower μ T b _).mpr
-    refine ⟨(val i).1.2,?_⟩
-    intro hn
-    have he : (⟨⟨(val i).1.1,(val i).1.2⟩,hn⟩ : {x : μ.cells // x.1.1 ≠ 0}) = val i := rfl
-    rw [he]
-    simp [val]
-  have ht : ∀ x, (T.1 (incl x)).val < b := by
-    intro x
-    exact ((mem_truncateLower μ T b x.1.1).mp x.1.2).choose_spec x.2
-  refine {toFun := fun x => ⟨T.1 (incl x), ht x⟩
-          invFun := fun i => ⟨⟨(val i).1.1,hv i⟩,(val i).2⟩
-          left_inv := ?_
-          right_inv := ?_}
-  · intro x
-    apply Subtype.ext
-    apply Subtype.ext
-    have he : val ⟨T.1 (incl x), ht x⟩ = incl x := by simp [val]
-    exact congrArg (fun y : {x : μ.cells // x.1.1≠0} => y.1.1) he
-  · intro i
-    apply Fin.ext
-    change (T.1 (val i)).val = i.val
-    simp [val]
-
-lemma truncateLower_card (μ : YoungDiagram)
-    (T : StandardFilling {x : μ.cells // x.1.1 ≠ 0}) (b : ℕ)
-    (hb : b ≤ μ.cells.card - μ.rowLen 0) :
-    (truncateLower μ T b).cells.card = μ.rowLen 0 + b := by
-  have h := Fintype.card_congr (truncatedLowerEquiv μ T b hb)
-  rw [Fintype.card_fin, lower_card, truncateLower_topRow] at h
-  have hle := topRow_le_card (truncateLower μ T b)
-  rw [truncateLower_topRow] at hle
-  omega
-
-lemma binomial_lower_bound (μ : YoungDiagram) (b : ℕ)
-    (hb : b ≤ min (μ.rowLen 0) (μ.cells.card-μ.rowLen 0)) :
-    (μ.rowLen 0).choose b ≤ standardCount μ := by
-  let T := chosenFilling {x : μ.cells // x.1.1≠0}
-  let ν := truncateLower μ T b
-  have hb' := hb.trans (min_le_right _ _)
-  have htop : ν.rowLen 0 = μ.rowLen 0 := truncateLower_topRow μ T b
-  have hcard : ν.cells.card = μ.rowLen 0 + b := truncateLower_card μ T b hb'
-  have hsecond := secondRow_le_lower ν
-  rw [htop,hcard,Nat.add_sub_cancel_left] at hsecond
-  have h := row_interleaving_lower_bound ν
-  rw [htop,hcard,Nat.add_sub_cancel_left] at h
-  exact ((Nat.choose_le_choose b (show μ.rowLen 0 ≤ μ.rowLen 0+b-ν.rowLen 1 by omega)).trans h).trans
-    (standardCount_mono (truncateLower_le μ T b))
-
-end
-end Tableau
-end PartialPermutation
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.PartialPermutation.RowBounds
+
+namespace OAI
+
+section
+namespace PartialPermutation
+namespace Tableau
+noncomputable section
+open Finset
+
+def truncateLower (μ : YoungDiagram)
+    (T : StandardFilling {x : μ.cells // x.1.1 ≠ 0}) (b : ℕ) : YoungDiagram := by
+  classical
+  let P : ℕ × ℕ → Prop := fun x => ∃ hx : x ∈ μ.cells,
+    ∀ hn : x.1 ≠ 0, (T.1 ⟨⟨x,hx⟩,hn⟩).val < b
+  refine ⟨μ.cells.filter P, ?_⟩
+  intro y x hxy hy
+  obtain ⟨hy,hpy⟩ := Finset.mem_filter.mp hy
+  obtain ⟨_,hty⟩ := hpy
+  have hx := μ.isLowerSet hxy hy
+  apply Finset.mem_filter.mpr
+  refine ⟨hx,hx,?_⟩
+  intro hn
+  have hny : y.1≠0 := by intro he; exact hn (Nat.eq_zero_of_le_zero (hxy.1.trans_eq he))
+  have hmono : (T.1 ⟨⟨x,hx⟩,hn⟩).val ≤ (T.1 ⟨⟨y,hy⟩,hny⟩).val :=
+    T.2.monotone (show (⟨⟨x,hx⟩,hn⟩ : {x : μ.cells // x.1.1≠0}) ≤
+      ⟨⟨y,hy⟩,hny⟩ from hxy)
+  exact hmono.trans_lt (hty hny)
+
+lemma mem_truncateLower (μ : YoungDiagram)
+    (T : StandardFilling {x : μ.cells // x.1.1 ≠ 0}) (b : ℕ) (x : ℕ × ℕ) :
+    x ∈ (truncateLower μ T b).cells ↔ ∃ hx : x ∈ μ.cells,
+      ∀ hn : x.1 ≠ 0, (T.1 ⟨⟨x,hx⟩,hn⟩).val < b := by
+  classical
+  simp only [truncateLower, Finset.mem_filter]
+  exact ⟨fun h => h.2, fun h => ⟨h.choose,h⟩⟩
+
+lemma truncateLower_le (μ : YoungDiagram)
+    (T : StandardFilling {x : μ.cells // x.1.1 ≠ 0}) (b : ℕ) :
+    truncateLower μ T b ≤ μ := by
+  intro x hx
+  exact (mem_truncateLower μ T b x).mp hx |>.choose
+
+lemma truncateLower_topRow (μ : YoungDiagram)
+    (T : StandardFilling {x : μ.cells // x.1.1 ≠ 0}) (b : ℕ) :
+    (truncateLower μ T b).rowLen 0 = μ.rowLen 0 := by
+  have hm : ∀ j, (0,j) ∈ truncateLower μ T b ↔ (0,j) ∈ μ := by
+    intro j
+    rw [← YoungDiagram.mem_cells, mem_truncateLower]
+    exact ⟨fun h => h.choose, fun h => ⟨h,fun hn => (hn rfl).elim⟩⟩
+  apply le_antisymm
+  · by_contra h
+    have hh := (hm (μ.rowLen 0)).mp ((truncateLower μ T b).mem_iff_lt_rowLen.mpr (by omega))
+    exact (lt_irrefl _) (μ.mem_iff_lt_rowLen.mp hh)
+  · by_contra h
+    have hh := (hm ((truncateLower μ T b).rowLen 0)).mpr (μ.mem_iff_lt_rowLen.mpr (by omega))
+    exact (lt_irrefl _) ((truncateLower μ T b).mem_iff_lt_rowLen.mp hh)
+
+def truncatedLowerEquiv (μ : YoungDiagram)
+    (T : StandardFilling {x : μ.cells // x.1.1 ≠ 0}) (b : ℕ)
+    (hb : b ≤ μ.cells.card - μ.rowLen 0) :
+    {x : (truncateLower μ T b).cells // x.1.1 ≠ 0} ≃ Fin b := by
+  let incl : {x : (truncateLower μ T b).cells // x.1.1 ≠ 0} →
+      {x : μ.cells // x.1.1 ≠ 0} := fun x =>
+    ⟨⟨x.1.1,truncateLower_le μ T b x.1.2⟩,x.2⟩
+  have hb' : b ≤ Fintype.card {x : μ.cells // x.1.1≠0} := by rwa [lower_card]
+  let val : Fin b → {x : μ.cells // x.1.1 ≠ 0} := fun i =>
+    T.1.symm ⟨i.val,i.isLt.trans_le hb'⟩
+  have hv : ∀ i, (val i).1.1 ∈ (truncateLower μ T b).cells := by
+    intro i
+    apply (mem_truncateLower μ T b _).mpr
+    refine ⟨(val i).1.2,?_⟩
+    intro hn
+    have he : (⟨⟨(val i).1.1,(val i).1.2⟩,hn⟩ : {x : μ.cells // x.1.1 ≠ 0}) = val i := rfl
+    rw [he]
+    simp [val]
+  have ht : ∀ x, (T.1 (incl x)).val < b := by
+    intro x
+    exact ((mem_truncateLower μ T b x.1.1).mp x.1.2).choose_spec x.2
+  refine {toFun := fun x => ⟨T.1 (incl x), ht x⟩
+          invFun := fun i => ⟨⟨(val i).1.1,hv i⟩,(val i).2⟩
+          left_inv := ?_
+          right_inv := ?_}
+  · intro x
+    apply Subtype.ext
+    apply Subtype.ext
+    have he : val ⟨T.1 (incl x), ht x⟩ = incl x := by simp [val]
+    exact congrArg (fun y : {x : μ.cells // x.1.1≠0} => y.1.1) he
+  · intro i
+    apply Fin.ext
+    change (T.1 (val i)).val = i.val
+    simp [val]
+
+lemma truncateLower_card (μ : YoungDiagram)
+    (T : StandardFilling {x : μ.cells // x.1.1 ≠ 0}) (b : ℕ)
+    (hb : b ≤ μ.cells.card - μ.rowLen 0) :
+    (truncateLower μ T b).cells.card = μ.rowLen 0 + b := by
+  have h := Fintype.card_congr (truncatedLowerEquiv μ T b hb)
+  rw [Fintype.card_fin, lower_card, truncateLower_topRow] at h
+  have hle := topRow_le_card (truncateLower μ T b)
+  rw [truncateLower_topRow] at hle
+  omega
+
+lemma binomial_lower_bound (μ : YoungDiagram) (b : ℕ)
+    (hb : b ≤ min (μ.rowLen 0) (μ.cells.card-μ.rowLen 0)) :
+    (μ.rowLen 0).choose b ≤ standardCount μ := by
+  let T := chosenFilling {x : μ.cells // x.1.1≠0}
+  let ν := truncateLower μ T b
+  have hb' := hb.trans (min_le_right _ _)
+  have htop : ν.rowLen 0 = μ.rowLen 0 := truncateLower_topRow μ T b
+  have hcard : ν.cells.card = μ.rowLen 0 + b := truncateLower_card μ T b hb'
+  have hsecond := secondRow_le_lower ν
+  rw [htop,hcard,Nat.add_sub_cancel_left] at hsecond
+  have h := row_interleaving_lower_bound ν
+  rw [htop,hcard,Nat.add_sub_cancel_left] at h
+  exact ((Nat.choose_le_choose b (show μ.rowLen 0 ≤ μ.rowLen 0+b-ν.rowLen 1 by omega)).trans h).trans
+    (standardCount_mono (truncateLower_le μ T b))
+
+end
+end Tableau
+end PartialPermutation
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/Placement/Chart.lean b/lean/OAI/RepresentationTheory/Placement/Chart.lean
index e23ba00314d3e582699d926b045e7b618fcbc2f4..f3f46c6314d21027ee5aba18f8e341209caf21d1 100644
--- a/lean/OAI/RepresentationTheory/Placement/Chart.lean
+++ b/lean/OAI/RepresentationTheory/Placement/Chart.lean
@@ -1,489 +1,489 @@
-import Mathlib
-import OAI.RepresentationTheory.Unitary.Basic
-
-namespace OAI
-
-noncomputable section
-open scoped BigOperators Matrix.Norms.L2Operator ComplexOrder
-attribute [local instance] Classical.propDecidable
-noncomputable section
-open scoped BigOperators
-attribute [local instance] Classical.propDecidable
-namespace CoordinateSweeps.PlacementInduction
-variable {G Ω : Type*} [Group G] [Fintype G] [Fintype Ω] [DecidableEq Ω]
-
-/- A transitive action with actual chosen representatives. For the application,
-Ω is the set of ordered placements; the fiber group fixes the base placement. -/
-structure Chart (a : G →* Equiv.Perm Ω) where
-  base : Ω
-  rep : Ω → G
-  reaches : ∀ x, a (rep x) base=x
-
-namespace Chart
-variable {a : G →* Equiv.Perm Ω} (C : Chart a)
-
-def stabilizer : Subgroup G where
-  carrier := {g | a g C.base=C.base}
-  one_mem' := by simp
-  mul_mem' {g h} hg hh := by
-    change a (g*h) C.base=C.base
-    change a g C.base=C.base at hg
-    change a h C.base=C.base at hh
-    rw [map_mul,Equiv.Perm.mul_apply,hh,hg]
-  inv_mem' {g} hg := by
-    change a g⁻¹ C.base=C.base
-    change a g C.base=C.base at hg
-    apply (a g).injective
-    have hi : a g⁻¹=(a g)⁻¹ := map_inv a g
-    rw [hi,Equiv.Perm.coe_inv,Equiv.apply_symm_apply,hg]
-
-def cocycle (g : G) (x : Ω) : C.stabilizer :=
-  ⟨(C.rep (a g x))⁻¹*g*C.rep x,by
-    change a ((C.rep (a g x))⁻¹*g*C.rep x) C.base=C.base
-    simp only [map_mul,Equiv.Perm.mul_apply,C.reaches,map_inv,Equiv.Perm.coe_inv]
-    exact (Equiv.symm_apply_eq _).mpr (C.reaches (a g x)).symm⟩
-
-omit [Fintype G] [Fintype Ω] [DecidableEq Ω] in
-@[simp] lemma cocycle_mul (g h : G) (x : Ω) :
-    C.cocycle (g*h) x=C.cocycle g (a h x)*C.cocycle h x := by
-  apply Subtype.ext
-  change (C.rep (a (g*h) x))⁻¹*(g*h)*C.rep x=
-    ((C.rep (a g (a h x)))⁻¹*g*C.rep (a h x))*((C.rep (a h x))⁻¹*h*C.rep x)
-  rw [map_mul,Equiv.Perm.mul_apply]
-  group
-
-omit [Fintype G] [Fintype Ω] [DecidableEq Ω] in
-@[simp] lemma cocycle_one (x : Ω) : C.cocycle 1 x=1 := by
-  apply Subtype.ext
-  change (C.rep (a 1 x))⁻¹*1*C.rep x=1
-  rw [map_one,Equiv.Perm.one_apply,mul_one,inv_mul_cancel]
-
-omit [Fintype G] [Fintype Ω] [DecidableEq Ω] in
-lemma cocycle_inv (g : G) (x : Ω) :
-    C.cocycle g⁻¹ (a g x)=(C.cocycle g x)⁻¹ := by
-  have hh := C.cocycle_mul g⁻¹ g x
-  rw [inv_mul_cancel,C.cocycle_one] at hh
-  exact eq_inv_of_mul_eq_one_left hh.symm
-
-variable (ρ : UnitaryIrrep C.stabilizer)
-
-/- Literal placement-by-fiber block matrix of the induced unitary action. -/
-def matrix (g : G) : Matrix (Ω × Fin ρ.dimension) (Ω × Fin ρ.dimension) ℂ :=
-  fun y x => if y.1=a g x.1 then ρ.matrix (C.cocycle g x.1) y.2 x.2 else 0
-
-omit [Fintype G] in
-lemma matrix_mul (g h : G) : C.matrix ρ (g*h)=C.matrix ρ g*C.matrix ρ h := by
-  ext y x
-  rcases y with ⟨y,i⟩
-  rcases x with ⟨x,j⟩
-  rw [Matrix.mul_apply,Fintype.sum_prod_type]
-  have hs : (∑ z : Ω, ∑ k : Fin ρ.dimension,
-      C.matrix ρ g (y,i) (z,k)*C.matrix ρ h (z,k) (x,j))=
-    ∑ k : Fin ρ.dimension, C.matrix ρ g (y,i) (a h x,k)*C.matrix ρ h (a h x,k) (x,j) := by
-    apply Finset.sum_eq_single (a h x)
-    · intro z _ hz
-      simp only [matrix,ite_eq_right hz,mul_zero,Finset.sum_const_zero]
-    · simp
-  rw [hs]
-  by_cases he : y=a (g*h) x
-  · have he' : y=a g (a h x) := by simpa only [map_mul,Equiv.Perm.mul_apply] using he
-    simp only [matrix,ite_eq_left he,ite_eq_left he',ite_true]
-    rw [C.cocycle_mul,map_mul,Matrix.mul_apply]
-  · have he' : ¬ y=a g (a h x) := by simpa only [map_mul,Equiv.Perm.mul_apply] using he
-    simp only [matrix,ite_eq_right he,ite_eq_right he',zero_mul,Finset.sum_const_zero]
-
-omit [Fintype G] [Fintype Ω] in
-lemma matrix_one : C.matrix ρ 1=1 := by
-  ext y x
-  rcases y with ⟨y,i⟩
-  rcases x with ⟨x,j⟩
-  simp only [matrix,map_one,Equiv.Perm.one_apply,C.cocycle_one,Matrix.one_apply,Prod.mk.injEq]
-  by_cases hy : y=x <;> by_cases hi : i=j <;> simp [hy,hi]
-
-def action : G →* Matrix (Ω × Fin ρ.dimension) (Ω × Fin ρ.dimension) ℂ where
-  toFun := C.matrix ρ
-  map_one' := C.matrix_one ρ
-  map_mul' := C.matrix_mul ρ
-
-omit [Fintype G] [Fintype Ω] in
-lemma matrix_inv (g : G) : C.matrix ρ g⁻¹=(C.matrix ρ g).conjTranspose := by
-  ext y x
-  rcases y with ⟨y,i⟩
-  rcases x with ⟨x,j⟩
-  by_cases he : x=a g y
-  · have he' : y=a g⁻¹ x := by
-      rw [he,map_inv,Equiv.Perm.coe_inv,Equiv.symm_apply_apply]
-    simp only [Matrix.conjTranspose_apply,matrix,ite_eq_left he,ite_eq_left he']
-    rw [he,C.cocycle_inv,ρ.matrix_inv]
-    rfl
-  · have he' : ¬ y=a g⁻¹ x := by
-      intro hh
-      apply he
-      rw [hh,map_inv,Equiv.Perm.coe_inv,Equiv.apply_symm_apply]
-    simp only [matrix,ite_eq_right he,ite_eq_right he',Matrix.conjTranspose_apply,star_zero]
-
-omit [Fintype G] in
-lemma matrix_unitary (g : G) : (C.matrix ρ g).conjTranspose*C.matrix ρ g=1 := by
-  rw [← C.matrix_inv,← C.matrix_mul,inv_mul_cancel,C.matrix_one]
-
-omit [Fintype G] in
-lemma trace_matrix (g : G) : Matrix.trace (C.matrix ρ g)=
-    ∑ x : Ω, if a g x=x then Matrix.trace (ρ.matrix (C.cocycle g x)) else 0 := by
-  rw [Matrix.trace,Fintype.sum_prod_type]
-  apply Finset.sum_congr rfl
-  intro x _
-  by_cases hx : a g x=x
-  · simp only [Matrix.diag,matrix,ite_eq_left hx.symm,ite_eq_left hx,Matrix.trace]
-  · simp only [Matrix.diag,matrix,ite_eq_right (Ne.symm hx),Finset.sum_const_zero,ite_eq_right hx]
-
-end Chart
-end CoordinateSweeps.PlacementInduction
-
-namespace CoordinateSweeps.PlacementInduction.Chart
-variable {G Ω : Type*} [Group G] [Fintype G] [Fintype Ω] [DecidableEq Ω]
-    {a : G →* Equiv.Perm Ω} (C : Chart a)
-
-omit [Fintype G] [Fintype Ω] [DecidableEq Ω] in
-lemma invrep_reaches (x : Ω) : a (C.rep x)⁻¹ x=C.base := by
-  rw [map_inv,Equiv.Perm.coe_inv]
-  exact (Equiv.symm_apply_eq _).mpr (C.reaches x).symm
-
-/- Actual orbit--stabilizer decomposition, including its fiber cardinal factor. -/
-def decompose : G ≃ Ω × C.stabilizer where
-  toFun g := (a g C.base,⟨(C.rep (a g C.base))⁻¹*g,by
-    change a ((C.rep (a g C.base))⁻¹*g) C.base=C.base
-    rw [map_mul,Equiv.Perm.mul_apply,C.invrep_reaches]⟩)
-  invFun x := C.rep x.1*x.2.val
-  left_inv g := by
-    change C.rep (a g C.base)*((C.rep (a g C.base))⁻¹*g)=g
-    group
-  right_inv x := by
-    rcases x with ⟨x,h⟩
-    have hx : a (C.rep x*h.val) C.base=x := by
-      rw [map_mul,Equiv.Perm.mul_apply,h.property,C.reaches]
-    apply Prod.ext
-    · exact hx
-    · apply Subtype.ext
-      change (C.rep (a (C.rep x*h.val) C.base))⁻¹*(C.rep x*h.val)=h.val
-      rw [hx]
-      group
-
-omit [DecidableEq Ω] in
-lemma card_group : Fintype.card G=Fintype.card Ω*Fintype.card C.stabilizer := by
-  simpa only [Fintype.card_prod] using Fintype.card_congr C.decompose
-
-def fixedEquiv (x : Ω) : C.stabilizer ≃ {g : G // a g x=x} where
-  toFun h := ⟨C.rep x*h.val*(C.rep x)⁻¹,by
-    rw [map_mul,map_mul,Equiv.Perm.mul_apply,Equiv.Perm.mul_apply,C.invrep_reaches,
-      h.property,C.reaches]⟩
-  invFun g := ⟨(C.rep x)⁻¹*g.val*C.rep x,by
-    change a ((C.rep x)⁻¹*g.val*C.rep x) C.base=C.base
-    rw [map_mul,map_mul,Equiv.Perm.mul_apply,Equiv.Perm.mul_apply,C.reaches,g.property,C.invrep_reaches]⟩
-  left_inv h := by
-    apply Subtype.ext
-    change (C.rep x)⁻¹*(C.rep x*h.val*(C.rep x)⁻¹)*C.rep x=h.val
-    group
-  right_inv g := by
-    apply Subtype.ext
-    change C.rep x*((C.rep x)⁻¹*g.val*C.rep x)*(C.rep x)⁻¹=g.val
-    group
-
-omit [Fintype G] [Fintype Ω] [DecidableEq Ω] in
-lemma cocycle_fixed (x : Ω) (h : C.stabilizer) :
-    C.cocycle ((C.fixedEquiv x h).val) x=h := by
-  apply Subtype.ext
-  change (C.rep (a (C.fixedEquiv x h).val x))⁻¹*(C.rep x*h.val*(C.rep x)⁻¹)*C.rep x=h.val
-  rw [(C.fixedEquiv x h).property]
-  group
-
-end CoordinateSweeps.PlacementInduction.Chart
-
-namespace CoordinateSweeps.PlacementInduction.Chart
-open UnitaryIrrep
-variable {G Ω : Type*} [Group G] [Fintype G] [Fintype Ω] [DecidableEq Ω]
-    {a : G →* Equiv.Perm Ω} (C : Chart a)
-
-omit [Fintype Ω] in
-lemma fixed_character_sum (ρ : UnitaryIrrep C.stabilizer) (τ : UnitaryIrrep G) (x : Ω) :
-    (∑ g : G, if a g x=x then Matrix.trace (ρ.matrix (C.cocycle g x))*Matrix.trace (τ.matrix g⁻¹) else 0)=
-      ∑ h : C.stabilizer, Matrix.trace (ρ.matrix h)*Matrix.trace (τ.matrix (h.val)⁻¹) := by
-  rw [← Finset.sum_filter]
-  rw [Finset.sum_subtype (p := fun g : G => a g x=x) _ (by intro g; simp) (fun g =>
-    Matrix.trace (ρ.matrix (C.cocycle g x))*Matrix.trace (τ.matrix g⁻¹))]
-  rw [← (C.fixedEquiv x).sum_comp]
-  apply Finset.sum_congr rfl
-  intro h _
-  rw [C.cocycle_fixed]
-  congr 1
-  change Matrix.trace (τ.matrix (C.rep x*h.val*(C.rep x)⁻¹)⁻¹)=_
-  rw [show (C.rep x*h.val*(C.rep x)⁻¹)⁻¹=C.rep x*h.val⁻¹*(C.rep x)⁻¹ by group]
-  exact trace_conj τ.matrix _ _
-
-lemma induced_character_sum (ρ : UnitaryIrrep C.stabilizer) (τ : UnitaryIrrep G) :
-    (∑ g : G, Matrix.trace (C.action ρ g)*Matrix.trace (τ.matrix g⁻¹))=
-      (Fintype.card Ω : ℂ)*
-        ∑ h : C.stabilizer, Matrix.trace (ρ.matrix h)*Matrix.trace (τ.matrix (h.val)⁻¹) := by
-  change (∑ g : G, Matrix.trace (C.matrix ρ g)*Matrix.trace (τ.matrix g⁻¹))=_
-  simp_rw [C.trace_matrix,Finset.sum_mul]
-  rw [Finset.sum_comm]
-  have hm (g : G) (x : Ω) :
-      (if a g x=x then Matrix.trace (ρ.matrix (C.cocycle g x)) else 0)*Matrix.trace (τ.matrix g⁻¹)=
-        if a g x=x then Matrix.trace (ρ.matrix (C.cocycle g x))*Matrix.trace (τ.matrix g⁻¹) else 0 := by
-    split_ifs <;> simp
-  simp only [hm,C.fixed_character_sum,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
-
-/- Finite Frobenius reciprocity for the literal placement matrices. The RHS
-is the actual Hom space in restriction, not an assumed branching coefficient. -/
-theorem multiplicity_eq (ρ : UnitaryIrrep C.stabilizer) (τ : UnitaryIrrep G) :
-    Module.finrank ℂ (τ.asRepresentation.IntertwiningMap (matrixRepresentation (C.action ρ)))=
-      Module.finrank ℂ (ρ.asRepresentation.IntertwiningMap
-        (matrixRepresentation (τ.matrix.comp C.stabilizer.subtype))) := by
-  have hl := τ.character_sum_multiplicity (C.action ρ)
-  have hr := ρ.character_sum_multiplicity (τ.matrix.comp C.stabilizer.subtype)
-  have he := C.induced_character_sum ρ τ
-  have hs : (∑ h : C.stabilizer, Matrix.trace (ρ.matrix h)*Matrix.trace (τ.matrix (h.val)⁻¹))=
-      (Fintype.card C.stabilizer : ℂ)*Module.finrank ℂ (ρ.asRepresentation.IntertwiningMap
-        (matrixRepresentation (τ.matrix.comp C.stabilizer.subtype))) := by
-    rw [← hr]
-    have hh := Equiv.sum_comp (Equiv.inv C.stabilizer) (fun h : C.stabilizer =>
-      Matrix.trace (τ.matrix h.val)*Matrix.trace (ρ.matrix h⁻¹))
-    simpa only [Equiv.inv_apply,MonoidHom.comp_apply,Subgroup.subtype_apply,Subgroup.coe_inv,inv_inv,mul_comm]
-      using hh
-  rw [hs,hl,C.card_group,Nat.cast_mul] at he
-  have hcard : (Fintype.card Ω : ℂ)*(Fintype.card C.stabilizer : ℂ) ≠ 0 := by
-    rw [← Nat.cast_mul,← C.card_group]
-    exact_mod_cast Fintype.card_ne_zero
-  have he' := mul_left_cancel₀ hcard (he.trans (mul_assoc _ _ _).symm)
-  exact_mod_cast he'
-
-/- The induced moment really dominates each target moment by its branching multiplicity. -/
-theorem induced_moment_bound (ρ : UnitaryIrrep C.stabilizer) (τ : UnitaryIrrep G)
-    {W : Type*} [Fintype W] (w : W → ℂ) (g : W → G) (q : ℕ) :
-    (Module.finrank ℂ (ρ.asRepresentation.IntertwiningMap
-      (matrixRepresentation (τ.matrix.comp C.stabilizer.subtype))) : ℝ)*
-        (Matrix.trace (((averageMatrix τ.matrix w g).conjTranspose*averageMatrix τ.matrix w g)^q)).re ≤
-          (Matrix.trace (((averageMatrix (C.action ρ) w g).conjTranspose*averageMatrix (C.action ρ) w g)^q)).re := by
-  rw [← C.multiplicity_eq ρ τ]
-  exact τ.multiplicity_moment_le (C.action ρ) (fun g => (C.matrix_inv ρ g).symm) w g q
-
-end CoordinateSweeps.PlacementInduction.Chart
-
-namespace CoordinateSweeps.OrderedPlacements
-variable {I X : Type*} [Fintype I] [Fintype X] [DecidableEq X]
-
-def action : Equiv.Perm X →* Equiv.Perm (I ↪ X) where
-  toFun g :=
-    { toFun := fun x => x.trans g.toEmbedding
-      invFun := fun x => x.trans g.symm.toEmbedding
-      left_inv := fun x => by ext i; simp
-      right_inv := fun x => by ext i; simp }
-  map_one' := by ext x i; rfl
-  map_mul' g h := by ext x i; rfl
-
-omit [Fintype I] [Fintype X] [DecidableEq X] in
-@[simp] lemma action_apply (g : Equiv.Perm X) (x : I ↪ X) (i : I) : action g x i=g (x i) := rfl
-
-def rangeEquiv (x0 x : I ↪ X) : Set.range x0 ≃ Set.range x :=
-  (Equiv.ofInjective x0 x0.injective).symm.trans (Equiv.ofInjective x x.injective)
-
-omit [Fintype I] [Fintype X] [DecidableEq X] in
-lemma rangeEquiv_apply (x0 x : I ↪ X) (i : I) :
-    (rangeEquiv x0 x ⟨x0 i,⟨i,rfl⟩⟩).val=x i := by
-  have he : (Equiv.ofInjective x0 x0.injective).symm ⟨x0 i,⟨i,rfl⟩⟩=i :=
-    (Equiv.ofInjective x0 x0.injective).symm_apply_apply i
-  change x ((Equiv.ofInjective x0 x0.injective).symm ⟨x0 i,⟨i,rfl⟩⟩)=x i
-  rw [he]
-
-/- Every ordered placement is reached by an actual permutation extending the
-bijection on its marked positions. No cardinal or transitivity axiom is used. -/
-def chart (x0 : I ↪ X) : PlacementInduction.Chart (action (I := I) (X := X)) where
-  base := x0
-  rep x := Equiv.extendSubtype (rangeEquiv x0 x)
-  reaches x := by
-    ext i
-    change Equiv.extendSubtype (rangeEquiv x0 x) (x0 i)=x i
-    rw [Equiv.extendSubtype_apply_of_mem _ _ (show x0 i ∈ Set.range x0 from ⟨i,rfl⟩)]
-    exact rangeEquiv_apply x0 x i
-
-lemma mem_stabilizer_iff (x0 : I ↪ X) (g : Equiv.Perm X) :
-    g ∈ (chart x0).stabilizer ↔ ∀ i, g (x0 i)=x0 i := by
-  change action g x0=x0 ↔ _
-  constructor
-  · intro h i
-    exact congrArg (fun x : I ↪ X => x i) h
-  · intro h
-    ext i
-    exact h i
-
-omit [DecidableEq X] in
-lemma card_placements : Fintype.card (I ↪ X)=(Fintype.card X).descFactorial (Fintype.card I) :=
-  Fintype.card_embedding_eq
-
-end CoordinateSweeps.OrderedPlacements
-
-namespace CoordinateSweeps.PlacementInduction.Chart
-open UnitaryIrrep
-variable {G Ω W : Type*} [Group G] [Fintype G] [Fintype Ω] [DecidableEq Ω] [Fintype W]
-    {a : G →* Equiv.Perm Ω} (C : Chart a)
-
-def transition (_ : Chart a) (w : W → ℝ) (g : W → G) (y x : Ω) : ℝ :=
-  ∑ ω, if y=a (g ω) x then w ω else 0
-
-def conditionalBlock (ρ : UnitaryIrrep C.stabilizer) (w : W → ℝ) (g : W → G) (y x : Ω) :
-    Matrix (Fin ρ.dimension) (Fin ρ.dimension) ℂ :=
-  (C.transition w g y x : ℂ)⁻¹ •
-    ∑ ω, if y=a (g ω) x then (w ω : ℂ) • ρ.matrix (C.cocycle (g ω) x) else 0
-
-omit [Fintype G] [Fintype Ω] in
-lemma transition_nonneg (w : W → ℝ) (hw : ∀ ω, 0 ≤ w ω) (g : W → G) (y x : Ω) :
-    0 ≤ C.transition w g y x := by
-  apply Finset.sum_nonneg
-  intro ω _
-  split_ifs
-  · exact hw ω
-  · exact le_refl 0
-
-omit [Fintype G] in
-lemma transition_columns (w : W → ℝ) (hs : ∑ ω, w ω=1) (g : W → G) (x : Ω) :
-    ∑ y, C.transition w g y x=1 := by
-  unfold transition
-  rw [Finset.sum_comm]
-  simpa only [Finset.sum_ite_eq',Finset.mem_univ,ite_true] using hs
-
-omit [Fintype G] in
-lemma transition_rows (w : W → ℝ) (hs : ∑ ω, w ω=1) (g : W → G) (y : Ω) :
-    ∑ x, C.transition w g y x=1 := by
-  unfold transition
-  rw [Finset.sum_comm]
-  have he (ω : W) : (∑ x : Ω, if y=a (g ω) x then w ω else 0)=w ω := by
-    have hh := Equiv.sum_comp (a (g ω)) (fun x => if y=x then w ω else 0)
-    simpa only [Finset.sum_ite_eq,Finset.mem_univ,ite_true] using hh
-  simpa only [he] using hs
-
-omit [Fintype G] [Fintype Ω] in
-lemma weight_le_transition (w : W → ℝ) (hw : ∀ ω, 0 ≤ w ω) (g : W → G)
-    (x y : Ω) (ω : W) (hω : y=a (g ω) x) : w ω ≤ C.transition w g y x := by
-  have hh := Finset.single_le_sum (s := (Finset.univ : Finset W))
-    (f := fun ν => if y=a (g ν) x then w ν else 0)
-    (fun ν _ => by split_ifs; exact hw ν; exact le_refl 0) (Finset.mem_univ ω)
-  simpa only [transition,ite_eq_left hω] using hh
-
-omit [Fintype G] [Fintype Ω] in
-lemma zero_transition_sum (ρ : UnitaryIrrep C.stabilizer) (w : W → ℝ)
-    (hw : ∀ ω, 0 ≤ w ω) (g : W → G) (y x : Ω) (hz : C.transition w g y x=0) :
-    (∑ ω, if y=a (g ω) x then (w ω : ℂ) • ρ.matrix (C.cocycle (g ω) x) else 0)=0 := by
-  apply Finset.sum_eq_zero
-  intro ω _
-  by_cases hω : y=a (g ω) x
-  · have hh := C.weight_le_transition w hw g x y ω hω
-    rw [hz] at hh
-    have hwz : w ω=0 := le_antisymm hh (hw ω)
-    simp [hω,hwz]
-  · exact ite_eq_right hω
-
-/- Every induced average block is exactly probability times its actual normalized
-conditional fiber operator, including impossible transitions. -/
-omit [Fintype G] in
-theorem average_block (ρ : UnitaryIrrep C.stabilizer) (w : W → ℝ) (hw : ∀ ω, 0 ≤ w ω)
-    (g : W → G) (y x : Ω) :
-    (averageMatrix (C.action ρ) (fun ω => (w ω : ℂ)) g).submatrix (fun i => (y,i)) (fun j => (x,j))=
-      (C.transition w g y x : ℂ) • C.conditionalBlock ρ w g y x := by
-  have he : (averageMatrix (C.action ρ) (fun ω => (w ω : ℂ)) g).submatrix
-      (fun i => (y,i)) (fun j => (x,j))=
-      ∑ ω, if y=a (g ω) x then (w ω : ℂ) • ρ.matrix (C.cocycle (g ω) x) else 0 := by
-    ext i j
-    simp only [averageMatrix,Matrix.submatrix_apply,Matrix.sum_apply,Matrix.smul_apply,smul_eq_mul]
-    apply Finset.sum_congr rfl
-    intro ω _
-    change (w ω : ℂ)*(if y=a (g ω) x then ρ.matrix (C.cocycle (g ω) x) i j else 0)=
-      (if y=a (g ω) x then (w ω : ℂ) • ρ.matrix (C.cocycle (g ω) x) else 0) i j
-    split_ifs <;> simp
-  rw [he,conditionalBlock,smul_smul]
-  by_cases hz : C.transition w g y x=0
-  · rw [C.zero_transition_sum ρ w hw g y x hz,smul_zero]
-  · rw [mul_inv_cancel₀ (Complex.ofReal_ne_zero.mpr hz),one_smul]
-
-end CoordinateSweeps.PlacementInduction.Chart
-
-namespace CoordinateSweeps.PlacementInduction.Chart
-variable {G Ω W : Type*} [Group G] [Fintype G] [Fintype Ω] [DecidableEq Ω] [Fintype W]
-    {a : G →* Equiv.Perm Ω} (C : Chart a)
-
-def pointFix (_ : Chart a) (x : Ω) : Subgroup G where
-  carrier g := a g x=x
-  one_mem' := by change a 1 x=x; simp
-  mul_mem' {g h} hg hh := by change a (g*h) x=x; rw [map_mul,Equiv.Perm.mul_apply,hh,hg]
-  inv_mem' {g} hg := by
-    change a g⁻¹ x=x
-    rw [map_inv,Equiv.Perm.coe_inv]
-    exact (Equiv.symm_apply_eq _).mpr hg.symm
-
-def pointEquiv (x : Ω) : C.stabilizer ≃* C.pointFix x where
-  toEquiv := C.fixedEquiv x
-  map_mul' h k := by
-    apply Subtype.ext
-    change C.rep x*(h.val*k.val)*(C.rep x)⁻¹=
-      (C.rep x*h.val*(C.rep x)⁻¹)*(C.rep x*k.val*(C.rep x)⁻¹)
-    group
-
-/- The remaining symmetric-group type transported to the actual input placement. -/
-def pointIrrep (ρ : UnitaryIrrep C.stabilizer) (x : Ω) : UnitaryIrrep (C.pointFix x) where
-  dimension := ρ.dimension
-  positive := ρ.positive
-  matrix := ρ.matrix.comp (C.pointEquiv x).symm.toMonoidHom
-  unitary g := ρ.unitary _
-  irreducible S hS := ρ.irreducible S (by
-    intro g v hv
-    have hh := hS (C.pointEquiv x g) v hv
-    change (ρ.matrix ((C.pointEquiv x).symm (C.pointEquiv x g))).mulVec v ∈ S at hh
-    simpa only [MulEquiv.symm_apply_apply] using hh)
-
-def transitionResidual (x y : Ω) (g₀ g : G) (h₀ : y=a g₀ x) (hg : y=a g x) : C.pointFix x :=
-  ⟨g₀⁻¹*g,by
-    change a (g₀⁻¹*g) x=x
-    rw [map_mul,Equiv.Perm.mul_apply,← hg,h₀,map_inv,Equiv.Perm.coe_inv,Equiv.symm_apply_apply]⟩
-
-omit [Fintype G] [Fintype Ω] [DecidableEq Ω] in
-lemma cocycle_residual (x y : Ω) (g₀ g : G) (h₀ : y=a g₀ x) (hg : y=a g x) :
-    C.cocycle g x=C.cocycle g₀ x*(C.pointEquiv x).symm (C.transitionResidual x y g₀ g h₀ hg) := by
-  apply Subtype.ext
-  change (C.rep (a g x))⁻¹*g*C.rep x=
-    ((C.rep (a g₀ x))⁻¹*g₀*C.rep x)*((C.rep x)⁻¹*(g₀⁻¹*g)*C.rep x)
-  rw [← hg,← h₀]
-  group
-
-/- Normalized residual law on the true point stabilizer. The reference is any
-compatible transition; it is a unitary frame, not a new hypothesis on the law. -/
-def transitionAverage (ρ : UnitaryIrrep C.stabilizer) (w : W → ℝ) (g : W → G)
-    (x y : Ω) (ω₀ : W) (h₀ : y=a (g ω₀) x) : Matrix (Fin ρ.dimension) (Fin ρ.dimension) ℂ :=
-  (C.transition w g y x : ℂ)⁻¹ • ∑ ω : {ω : W // y=a (g ω) x},
-    (w ω.val : ℂ) • ρ.matrix
-      ((C.pointEquiv x).symm (C.transitionResidual x y (g ω₀) (g ω.val) h₀ ω.property))
-
-/- Exact moving-frame identity for each normalized block. -/
-omit [Fintype G] [Fintype Ω] in
-theorem conditionalBlock_eq (ρ : UnitaryIrrep C.stabilizer) (w : W → ℝ) (g : W → G)
-    (x y : Ω) (ω₀ : W) (h₀ : y=a (g ω₀) x) :
-    C.conditionalBlock ρ w g y x=ρ.matrix (C.cocycle (g ω₀) x)*C.transitionAverage ρ w g x y ω₀ h₀ := by
-  unfold conditionalBlock transitionAverage
-  rw [Matrix.mul_smul,Matrix.mul_sum]
-  congr 1
-  rw [← Finset.sum_filter]
-  rw [Finset.sum_subtype (p := fun ω => y=a (g ω) x) (Finset.univ.filter (fun ω => y=a (g ω) x))
-    (by intro ω; simp)]
-  apply Finset.sum_congr rfl
-  intro ω _
-  change (w ω.val : ℂ) • ρ.matrix (C.cocycle (g ω.val) x)=
-    ρ.matrix (C.cocycle (g ω₀) x)*((w ω.val : ℂ) •
-      ρ.matrix ((C.pointEquiv x).symm (C.transitionResidual x y (g ω₀) (g ω.val) h₀ ω.property)))
-  rw [Matrix.mul_smul,← map_mul]
-  congr 2
-  exact C.cocycle_residual x y (g ω₀) (g ω.val) h₀ ω.property
-
-end CoordinateSweeps.PlacementInduction.Chart
-
-end
-end
-open scoped Matrix.Norms.L2Operator
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.Unitary.Basic
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators Matrix.Norms.L2Operator ComplexOrder
+attribute [local instance] Classical.propDecidable
+noncomputable section
+open scoped BigOperators
+attribute [local instance] Classical.propDecidable
+namespace CoordinateSweeps.PlacementInduction
+variable {G Ω : Type*} [Group G] [Fintype G] [Fintype Ω] [DecidableEq Ω]
+
+/- A transitive action with actual chosen representatives. For the application,
+Ω is the set of ordered placements; the fiber group fixes the base placement. -/
+structure Chart (a : G →* Equiv.Perm Ω) where
+  base : Ω
+  rep : Ω → G
+  reaches : ∀ x, a (rep x) base=x
+
+namespace Chart
+variable {a : G →* Equiv.Perm Ω} (C : Chart a)
+
+def stabilizer : Subgroup G where
+  carrier := {g | a g C.base=C.base}
+  one_mem' := by simp
+  mul_mem' {g h} hg hh := by
+    change a (g*h) C.base=C.base
+    change a g C.base=C.base at hg
+    change a h C.base=C.base at hh
+    rw [map_mul,Equiv.Perm.mul_apply,hh,hg]
+  inv_mem' {g} hg := by
+    change a g⁻¹ C.base=C.base
+    change a g C.base=C.base at hg
+    apply (a g).injective
+    have hi : a g⁻¹=(a g)⁻¹ := map_inv a g
+    rw [hi,Equiv.Perm.coe_inv,Equiv.apply_symm_apply,hg]
+
+def cocycle (g : G) (x : Ω) : C.stabilizer :=
+  ⟨(C.rep (a g x))⁻¹*g*C.rep x,by
+    change a ((C.rep (a g x))⁻¹*g*C.rep x) C.base=C.base
+    simp only [map_mul,Equiv.Perm.mul_apply,C.reaches,map_inv,Equiv.Perm.coe_inv]
+    exact (Equiv.symm_apply_eq _).mpr (C.reaches (a g x)).symm⟩
+
+omit [Fintype G] [Fintype Ω] [DecidableEq Ω] in
+@[simp] lemma cocycle_mul (g h : G) (x : Ω) :
+    C.cocycle (g*h) x=C.cocycle g (a h x)*C.cocycle h x := by
+  apply Subtype.ext
+  change (C.rep (a (g*h) x))⁻¹*(g*h)*C.rep x=
+    ((C.rep (a g (a h x)))⁻¹*g*C.rep (a h x))*((C.rep (a h x))⁻¹*h*C.rep x)
+  rw [map_mul,Equiv.Perm.mul_apply]
+  group
+
+omit [Fintype G] [Fintype Ω] [DecidableEq Ω] in
+@[simp] lemma cocycle_one (x : Ω) : C.cocycle 1 x=1 := by
+  apply Subtype.ext
+  change (C.rep (a 1 x))⁻¹*1*C.rep x=1
+  rw [map_one,Equiv.Perm.one_apply,mul_one,inv_mul_cancel]
+
+omit [Fintype G] [Fintype Ω] [DecidableEq Ω] in
+lemma cocycle_inv (g : G) (x : Ω) :
+    C.cocycle g⁻¹ (a g x)=(C.cocycle g x)⁻¹ := by
+  have hh := C.cocycle_mul g⁻¹ g x
+  rw [inv_mul_cancel,C.cocycle_one] at hh
+  exact eq_inv_of_mul_eq_one_left hh.symm
+
+variable (ρ : UnitaryIrrep C.stabilizer)
+
+/- Literal placement-by-fiber block matrix of the induced unitary action. -/
+def matrix (g : G) : Matrix (Ω × Fin ρ.dimension) (Ω × Fin ρ.dimension) ℂ :=
+  fun y x => if y.1=a g x.1 then ρ.matrix (C.cocycle g x.1) y.2 x.2 else 0
+
+omit [Fintype G] in
+lemma matrix_mul (g h : G) : C.matrix ρ (g*h)=C.matrix ρ g*C.matrix ρ h := by
+  ext y x
+  rcases y with ⟨y,i⟩
+  rcases x with ⟨x,j⟩
+  rw [Matrix.mul_apply,Fintype.sum_prod_type]
+  have hs : (∑ z : Ω, ∑ k : Fin ρ.dimension,
+      C.matrix ρ g (y,i) (z,k)*C.matrix ρ h (z,k) (x,j))=
+    ∑ k : Fin ρ.dimension, C.matrix ρ g (y,i) (a h x,k)*C.matrix ρ h (a h x,k) (x,j) := by
+    apply Finset.sum_eq_single (a h x)
+    · intro z _ hz
+      simp only [matrix,ite_eq_right hz,mul_zero,Finset.sum_const_zero]
+    · simp
+  rw [hs]
+  by_cases he : y=a (g*h) x
+  · have he' : y=a g (a h x) := by simpa only [map_mul,Equiv.Perm.mul_apply] using he
+    simp only [matrix,ite_eq_left he,ite_eq_left he',ite_true]
+    rw [C.cocycle_mul,map_mul,Matrix.mul_apply]
+  · have he' : ¬ y=a g (a h x) := by simpa only [map_mul,Equiv.Perm.mul_apply] using he
+    simp only [matrix,ite_eq_right he,ite_eq_right he',zero_mul,Finset.sum_const_zero]
+
+omit [Fintype G] [Fintype Ω] in
+lemma matrix_one : C.matrix ρ 1=1 := by
+  ext y x
+  rcases y with ⟨y,i⟩
+  rcases x with ⟨x,j⟩
+  simp only [matrix,map_one,Equiv.Perm.one_apply,C.cocycle_one,Matrix.one_apply,Prod.mk.injEq]
+  by_cases hy : y=x <;> by_cases hi : i=j <;> simp [hy,hi]
+
+def action : G →* Matrix (Ω × Fin ρ.dimension) (Ω × Fin ρ.dimension) ℂ where
+  toFun := C.matrix ρ
+  map_one' := C.matrix_one ρ
+  map_mul' := C.matrix_mul ρ
+
+omit [Fintype G] [Fintype Ω] in
+lemma matrix_inv (g : G) : C.matrix ρ g⁻¹=(C.matrix ρ g).conjTranspose := by
+  ext y x
+  rcases y with ⟨y,i⟩
+  rcases x with ⟨x,j⟩
+  by_cases he : x=a g y
+  · have he' : y=a g⁻¹ x := by
+      rw [he,map_inv,Equiv.Perm.coe_inv,Equiv.symm_apply_apply]
+    simp only [Matrix.conjTranspose_apply,matrix,ite_eq_left he,ite_eq_left he']
+    rw [he,C.cocycle_inv,ρ.matrix_inv]
+    rfl
+  · have he' : ¬ y=a g⁻¹ x := by
+      intro hh
+      apply he
+      rw [hh,map_inv,Equiv.Perm.coe_inv,Equiv.apply_symm_apply]
+    simp only [matrix,ite_eq_right he,ite_eq_right he',Matrix.conjTranspose_apply,star_zero]
+
+omit [Fintype G] in
+lemma matrix_unitary (g : G) : (C.matrix ρ g).conjTranspose*C.matrix ρ g=1 := by
+  rw [← C.matrix_inv,← C.matrix_mul,inv_mul_cancel,C.matrix_one]
+
+omit [Fintype G] in
+lemma trace_matrix (g : G) : Matrix.trace (C.matrix ρ g)=
+    ∑ x : Ω, if a g x=x then Matrix.trace (ρ.matrix (C.cocycle g x)) else 0 := by
+  rw [Matrix.trace,Fintype.sum_prod_type]
+  apply Finset.sum_congr rfl
+  intro x _
+  by_cases hx : a g x=x
+  · simp only [Matrix.diag,matrix,ite_eq_left hx.symm,ite_eq_left hx,Matrix.trace]
+  · simp only [Matrix.diag,matrix,ite_eq_right (Ne.symm hx),Finset.sum_const_zero,ite_eq_right hx]
+
+end Chart
+end CoordinateSweeps.PlacementInduction
+
+namespace CoordinateSweeps.PlacementInduction.Chart
+variable {G Ω : Type*} [Group G] [Fintype G] [Fintype Ω] [DecidableEq Ω]
+    {a : G →* Equiv.Perm Ω} (C : Chart a)
+
+omit [Fintype G] [Fintype Ω] [DecidableEq Ω] in
+lemma invrep_reaches (x : Ω) : a (C.rep x)⁻¹ x=C.base := by
+  rw [map_inv,Equiv.Perm.coe_inv]
+  exact (Equiv.symm_apply_eq _).mpr (C.reaches x).symm
+
+/- Actual orbit--stabilizer decomposition, including its fiber cardinal factor. -/
+def decompose : G ≃ Ω × C.stabilizer where
+  toFun g := (a g C.base,⟨(C.rep (a g C.base))⁻¹*g,by
+    change a ((C.rep (a g C.base))⁻¹*g) C.base=C.base
+    rw [map_mul,Equiv.Perm.mul_apply,C.invrep_reaches]⟩)
+  invFun x := C.rep x.1*x.2.val
+  left_inv g := by
+    change C.rep (a g C.base)*((C.rep (a g C.base))⁻¹*g)=g
+    group
+  right_inv x := by
+    rcases x with ⟨x,h⟩
+    have hx : a (C.rep x*h.val) C.base=x := by
+      rw [map_mul,Equiv.Perm.mul_apply,h.property,C.reaches]
+    apply Prod.ext
+    · exact hx
+    · apply Subtype.ext
+      change (C.rep (a (C.rep x*h.val) C.base))⁻¹*(C.rep x*h.val)=h.val
+      rw [hx]
+      group
+
+omit [DecidableEq Ω] in
+lemma card_group : Fintype.card G=Fintype.card Ω*Fintype.card C.stabilizer := by
+  simpa only [Fintype.card_prod] using Fintype.card_congr C.decompose
+
+def fixedEquiv (x : Ω) : C.stabilizer ≃ {g : G // a g x=x} where
+  toFun h := ⟨C.rep x*h.val*(C.rep x)⁻¹,by
+    rw [map_mul,map_mul,Equiv.Perm.mul_apply,Equiv.Perm.mul_apply,C.invrep_reaches,
+      h.property,C.reaches]⟩
+  invFun g := ⟨(C.rep x)⁻¹*g.val*C.rep x,by
+    change a ((C.rep x)⁻¹*g.val*C.rep x) C.base=C.base
+    rw [map_mul,map_mul,Equiv.Perm.mul_apply,Equiv.Perm.mul_apply,C.reaches,g.property,C.invrep_reaches]⟩
+  left_inv h := by
+    apply Subtype.ext
+    change (C.rep x)⁻¹*(C.rep x*h.val*(C.rep x)⁻¹)*C.rep x=h.val
+    group
+  right_inv g := by
+    apply Subtype.ext
+    change C.rep x*((C.rep x)⁻¹*g.val*C.rep x)*(C.rep x)⁻¹=g.val
+    group
+
+omit [Fintype G] [Fintype Ω] [DecidableEq Ω] in
+lemma cocycle_fixed (x : Ω) (h : C.stabilizer) :
+    C.cocycle ((C.fixedEquiv x h).val) x=h := by
+  apply Subtype.ext
+  change (C.rep (a (C.fixedEquiv x h).val x))⁻¹*(C.rep x*h.val*(C.rep x)⁻¹)*C.rep x=h.val
+  rw [(C.fixedEquiv x h).property]
+  group
+
+end CoordinateSweeps.PlacementInduction.Chart
+
+namespace CoordinateSweeps.PlacementInduction.Chart
+open UnitaryIrrep
+variable {G Ω : Type*} [Group G] [Fintype G] [Fintype Ω] [DecidableEq Ω]
+    {a : G →* Equiv.Perm Ω} (C : Chart a)
+
+omit [Fintype Ω] in
+lemma fixed_character_sum (ρ : UnitaryIrrep C.stabilizer) (τ : UnitaryIrrep G) (x : Ω) :
+    (∑ g : G, if a g x=x then Matrix.trace (ρ.matrix (C.cocycle g x))*Matrix.trace (τ.matrix g⁻¹) else 0)=
+      ∑ h : C.stabilizer, Matrix.trace (ρ.matrix h)*Matrix.trace (τ.matrix (h.val)⁻¹) := by
+  rw [← Finset.sum_filter]
+  rw [Finset.sum_subtype (p := fun g : G => a g x=x) _ (by intro g; simp) (fun g =>
+    Matrix.trace (ρ.matrix (C.cocycle g x))*Matrix.trace (τ.matrix g⁻¹))]
+  rw [← (C.fixedEquiv x).sum_comp]
+  apply Finset.sum_congr rfl
+  intro h _
+  rw [C.cocycle_fixed]
+  congr 1
+  change Matrix.trace (τ.matrix (C.rep x*h.val*(C.rep x)⁻¹)⁻¹)=_
+  rw [show (C.rep x*h.val*(C.rep x)⁻¹)⁻¹=C.rep x*h.val⁻¹*(C.rep x)⁻¹ by group]
+  exact trace_conj τ.matrix _ _
+
+lemma induced_character_sum (ρ : UnitaryIrrep C.stabilizer) (τ : UnitaryIrrep G) :
+    (∑ g : G, Matrix.trace (C.action ρ g)*Matrix.trace (τ.matrix g⁻¹))=
+      (Fintype.card Ω : ℂ)*
+        ∑ h : C.stabilizer, Matrix.trace (ρ.matrix h)*Matrix.trace (τ.matrix (h.val)⁻¹) := by
+  change (∑ g : G, Matrix.trace (C.matrix ρ g)*Matrix.trace (τ.matrix g⁻¹))=_
+  simp_rw [C.trace_matrix,Finset.sum_mul]
+  rw [Finset.sum_comm]
+  have hm (g : G) (x : Ω) :
+      (if a g x=x then Matrix.trace (ρ.matrix (C.cocycle g x)) else 0)*Matrix.trace (τ.matrix g⁻¹)=
+        if a g x=x then Matrix.trace (ρ.matrix (C.cocycle g x))*Matrix.trace (τ.matrix g⁻¹) else 0 := by
+    split_ifs <;> simp
+  simp only [hm,C.fixed_character_sum,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
+
+/- Finite Frobenius reciprocity for the literal placement matrices. The RHS
+is the actual Hom space in restriction, not an assumed branching coefficient. -/
+theorem multiplicity_eq (ρ : UnitaryIrrep C.stabilizer) (τ : UnitaryIrrep G) :
+    Module.finrank ℂ (τ.asRepresentation.IntertwiningMap (matrixRepresentation (C.action ρ)))=
+      Module.finrank ℂ (ρ.asRepresentation.IntertwiningMap
+        (matrixRepresentation (τ.matrix.comp C.stabilizer.subtype))) := by
+  have hl := τ.character_sum_multiplicity (C.action ρ)
+  have hr := ρ.character_sum_multiplicity (τ.matrix.comp C.stabilizer.subtype)
+  have he := C.induced_character_sum ρ τ
+  have hs : (∑ h : C.stabilizer, Matrix.trace (ρ.matrix h)*Matrix.trace (τ.matrix (h.val)⁻¹))=
+      (Fintype.card C.stabilizer : ℂ)*Module.finrank ℂ (ρ.asRepresentation.IntertwiningMap
+        (matrixRepresentation (τ.matrix.comp C.stabilizer.subtype))) := by
+    rw [← hr]
+    have hh := Equiv.sum_comp (Equiv.inv C.stabilizer) (fun h : C.stabilizer =>
+      Matrix.trace (τ.matrix h.val)*Matrix.trace (ρ.matrix h⁻¹))
+    simpa only [Equiv.inv_apply,MonoidHom.comp_apply,Subgroup.subtype_apply,Subgroup.coe_inv,inv_inv,mul_comm]
+      using hh
+  rw [hs,hl,C.card_group,Nat.cast_mul] at he
+  have hcard : (Fintype.card Ω : ℂ)*(Fintype.card C.stabilizer : ℂ) ≠ 0 := by
+    rw [← Nat.cast_mul,← C.card_group]
+    exact_mod_cast Fintype.card_ne_zero
+  have he' := mul_left_cancel₀ hcard (he.trans (mul_assoc _ _ _).symm)
+  exact_mod_cast he'
+
+/- The induced moment really dominates each target moment by its branching multiplicity. -/
+theorem induced_moment_bound (ρ : UnitaryIrrep C.stabilizer) (τ : UnitaryIrrep G)
+    {W : Type*} [Fintype W] (w : W → ℂ) (g : W → G) (q : ℕ) :
+    (Module.finrank ℂ (ρ.asRepresentation.IntertwiningMap
+      (matrixRepresentation (τ.matrix.comp C.stabilizer.subtype))) : ℝ)*
+        (Matrix.trace (((averageMatrix τ.matrix w g).conjTranspose*averageMatrix τ.matrix w g)^q)).re ≤
+          (Matrix.trace (((averageMatrix (C.action ρ) w g).conjTranspose*averageMatrix (C.action ρ) w g)^q)).re := by
+  rw [← C.multiplicity_eq ρ τ]
+  exact τ.multiplicity_moment_le (C.action ρ) (fun g => (C.matrix_inv ρ g).symm) w g q
+
+end CoordinateSweeps.PlacementInduction.Chart
+
+namespace CoordinateSweeps.OrderedPlacements
+variable {I X : Type*} [Fintype I] [Fintype X] [DecidableEq X]
+
+def action : Equiv.Perm X →* Equiv.Perm (I ↪ X) where
+  toFun g :=
+    { toFun := fun x => x.trans g.toEmbedding
+      invFun := fun x => x.trans g.symm.toEmbedding
+      left_inv := fun x => by ext i; simp
+      right_inv := fun x => by ext i; simp }
+  map_one' := by ext x i; rfl
+  map_mul' g h := by ext x i; rfl
+
+omit [Fintype I] [Fintype X] [DecidableEq X] in
+@[simp] lemma action_apply (g : Equiv.Perm X) (x : I ↪ X) (i : I) : action g x i=g (x i) := rfl
+
+def rangeEquiv (x0 x : I ↪ X) : Set.range x0 ≃ Set.range x :=
+  (Equiv.ofInjective x0 x0.injective).symm.trans (Equiv.ofInjective x x.injective)
+
+omit [Fintype I] [Fintype X] [DecidableEq X] in
+lemma rangeEquiv_apply (x0 x : I ↪ X) (i : I) :
+    (rangeEquiv x0 x ⟨x0 i,⟨i,rfl⟩⟩).val=x i := by
+  have he : (Equiv.ofInjective x0 x0.injective).symm ⟨x0 i,⟨i,rfl⟩⟩=i :=
+    (Equiv.ofInjective x0 x0.injective).symm_apply_apply i
+  change x ((Equiv.ofInjective x0 x0.injective).symm ⟨x0 i,⟨i,rfl⟩⟩)=x i
+  rw [he]
+
+/- Every ordered placement is reached by an actual permutation extending the
+bijection on its marked positions. No cardinal or transitivity axiom is used. -/
+def chart (x0 : I ↪ X) : PlacementInduction.Chart (action (I := I) (X := X)) where
+  base := x0
+  rep x := Equiv.extendSubtype (rangeEquiv x0 x)
+  reaches x := by
+    ext i
+    change Equiv.extendSubtype (rangeEquiv x0 x) (x0 i)=x i
+    rw [Equiv.extendSubtype_apply_of_mem _ _ (show x0 i ∈ Set.range x0 from ⟨i,rfl⟩)]
+    exact rangeEquiv_apply x0 x i
+
+lemma mem_stabilizer_iff (x0 : I ↪ X) (g : Equiv.Perm X) :
+    g ∈ (chart x0).stabilizer ↔ ∀ i, g (x0 i)=x0 i := by
+  change action g x0=x0 ↔ _
+  constructor
+  · intro h i
+    exact congrArg (fun x : I ↪ X => x i) h
+  · intro h
+    ext i
+    exact h i
+
+omit [DecidableEq X] in
+lemma card_placements : Fintype.card (I ↪ X)=(Fintype.card X).descFactorial (Fintype.card I) :=
+  Fintype.card_embedding_eq
+
+end CoordinateSweeps.OrderedPlacements
+
+namespace CoordinateSweeps.PlacementInduction.Chart
+open UnitaryIrrep
+variable {G Ω W : Type*} [Group G] [Fintype G] [Fintype Ω] [DecidableEq Ω] [Fintype W]
+    {a : G →* Equiv.Perm Ω} (C : Chart a)
+
+def transition (_ : Chart a) (w : W → ℝ) (g : W → G) (y x : Ω) : ℝ :=
+  ∑ ω, if y=a (g ω) x then w ω else 0
+
+def conditionalBlock (ρ : UnitaryIrrep C.stabilizer) (w : W → ℝ) (g : W → G) (y x : Ω) :
+    Matrix (Fin ρ.dimension) (Fin ρ.dimension) ℂ :=
+  (C.transition w g y x : ℂ)⁻¹ •
+    ∑ ω, if y=a (g ω) x then (w ω : ℂ) • ρ.matrix (C.cocycle (g ω) x) else 0
+
+omit [Fintype G] [Fintype Ω] in
+lemma transition_nonneg (w : W → ℝ) (hw : ∀ ω, 0 ≤ w ω) (g : W → G) (y x : Ω) :
+    0 ≤ C.transition w g y x := by
+  apply Finset.sum_nonneg
+  intro ω _
+  split_ifs
+  · exact hw ω
+  · exact le_refl 0
+
+omit [Fintype G] in
+lemma transition_columns (w : W → ℝ) (hs : ∑ ω, w ω=1) (g : W → G) (x : Ω) :
+    ∑ y, C.transition w g y x=1 := by
+  unfold transition
+  rw [Finset.sum_comm]
+  simpa only [Finset.sum_ite_eq',Finset.mem_univ,ite_true] using hs
+
+omit [Fintype G] in
+lemma transition_rows (w : W → ℝ) (hs : ∑ ω, w ω=1) (g : W → G) (y : Ω) :
+    ∑ x, C.transition w g y x=1 := by
+  unfold transition
+  rw [Finset.sum_comm]
+  have he (ω : W) : (∑ x : Ω, if y=a (g ω) x then w ω else 0)=w ω := by
+    have hh := Equiv.sum_comp (a (g ω)) (fun x => if y=x then w ω else 0)
+    simpa only [Finset.sum_ite_eq,Finset.mem_univ,ite_true] using hh
+  simpa only [he] using hs
+
+omit [Fintype G] [Fintype Ω] in
+lemma weight_le_transition (w : W → ℝ) (hw : ∀ ω, 0 ≤ w ω) (g : W → G)
+    (x y : Ω) (ω : W) (hω : y=a (g ω) x) : w ω ≤ C.transition w g y x := by
+  have hh := Finset.single_le_sum (s := (Finset.univ : Finset W))
+    (f := fun ν => if y=a (g ν) x then w ν else 0)
+    (fun ν _ => by split_ifs; exact hw ν; exact le_refl 0) (Finset.mem_univ ω)
+  simpa only [transition,ite_eq_left hω] using hh
+
+omit [Fintype G] [Fintype Ω] in
+lemma zero_transition_sum (ρ : UnitaryIrrep C.stabilizer) (w : W → ℝ)
+    (hw : ∀ ω, 0 ≤ w ω) (g : W → G) (y x : Ω) (hz : C.transition w g y x=0) :
+    (∑ ω, if y=a (g ω) x then (w ω : ℂ) • ρ.matrix (C.cocycle (g ω) x) else 0)=0 := by
+  apply Finset.sum_eq_zero
+  intro ω _
+  by_cases hω : y=a (g ω) x
+  · have hh := C.weight_le_transition w hw g x y ω hω
+    rw [hz] at hh
+    have hwz : w ω=0 := le_antisymm hh (hw ω)
+    simp [hω,hwz]
+  · exact ite_eq_right hω
+
+/- Every induced average block is exactly probability times its actual normalized
+conditional fiber operator, including impossible transitions. -/
+omit [Fintype G] in
+theorem average_block (ρ : UnitaryIrrep C.stabilizer) (w : W → ℝ) (hw : ∀ ω, 0 ≤ w ω)
+    (g : W → G) (y x : Ω) :
+    (averageMatrix (C.action ρ) (fun ω => (w ω : ℂ)) g).submatrix (fun i => (y,i)) (fun j => (x,j))=
+      (C.transition w g y x : ℂ) • C.conditionalBlock ρ w g y x := by
+  have he : (averageMatrix (C.action ρ) (fun ω => (w ω : ℂ)) g).submatrix
+      (fun i => (y,i)) (fun j => (x,j))=
+      ∑ ω, if y=a (g ω) x then (w ω : ℂ) • ρ.matrix (C.cocycle (g ω) x) else 0 := by
+    ext i j
+    simp only [averageMatrix,Matrix.submatrix_apply,Matrix.sum_apply,Matrix.smul_apply,smul_eq_mul]
+    apply Finset.sum_congr rfl
+    intro ω _
+    change (w ω : ℂ)*(if y=a (g ω) x then ρ.matrix (C.cocycle (g ω) x) i j else 0)=
+      (if y=a (g ω) x then (w ω : ℂ) • ρ.matrix (C.cocycle (g ω) x) else 0) i j
+    split_ifs <;> simp
+  rw [he,conditionalBlock,smul_smul]
+  by_cases hz : C.transition w g y x=0
+  · rw [C.zero_transition_sum ρ w hw g y x hz,smul_zero]
+  · rw [mul_inv_cancel₀ (Complex.ofReal_ne_zero.mpr hz),one_smul]
+
+end CoordinateSweeps.PlacementInduction.Chart
+
+namespace CoordinateSweeps.PlacementInduction.Chart
+variable {G Ω W : Type*} [Group G] [Fintype G] [Fintype Ω] [DecidableEq Ω] [Fintype W]
+    {a : G →* Equiv.Perm Ω} (C : Chart a)
+
+def pointFix (_ : Chart a) (x : Ω) : Subgroup G where
+  carrier g := a g x=x
+  one_mem' := by change a 1 x=x; simp
+  mul_mem' {g h} hg hh := by change a (g*h) x=x; rw [map_mul,Equiv.Perm.mul_apply,hh,hg]
+  inv_mem' {g} hg := by
+    change a g⁻¹ x=x
+    rw [map_inv,Equiv.Perm.coe_inv]
+    exact (Equiv.symm_apply_eq _).mpr hg.symm
+
+def pointEquiv (x : Ω) : C.stabilizer ≃* C.pointFix x where
+  toEquiv := C.fixedEquiv x
+  map_mul' h k := by
+    apply Subtype.ext
+    change C.rep x*(h.val*k.val)*(C.rep x)⁻¹=
+      (C.rep x*h.val*(C.rep x)⁻¹)*(C.rep x*k.val*(C.rep x)⁻¹)
+    group
+
+/- The remaining symmetric-group type transported to the actual input placement. -/
+def pointIrrep (ρ : UnitaryIrrep C.stabilizer) (x : Ω) : UnitaryIrrep (C.pointFix x) where
+  dimension := ρ.dimension
+  positive := ρ.positive
+  matrix := ρ.matrix.comp (C.pointEquiv x).symm.toMonoidHom
+  unitary g := ρ.unitary _
+  irreducible S hS := ρ.irreducible S (by
+    intro g v hv
+    have hh := hS (C.pointEquiv x g) v hv
+    change (ρ.matrix ((C.pointEquiv x).symm (C.pointEquiv x g))).mulVec v ∈ S at hh
+    simpa only [MulEquiv.symm_apply_apply] using hh)
+
+def transitionResidual (x y : Ω) (g₀ g : G) (h₀ : y=a g₀ x) (hg : y=a g x) : C.pointFix x :=
+  ⟨g₀⁻¹*g,by
+    change a (g₀⁻¹*g) x=x
+    rw [map_mul,Equiv.Perm.mul_apply,← hg,h₀,map_inv,Equiv.Perm.coe_inv,Equiv.symm_apply_apply]⟩
+
+omit [Fintype G] [Fintype Ω] [DecidableEq Ω] in
+lemma cocycle_residual (x y : Ω) (g₀ g : G) (h₀ : y=a g₀ x) (hg : y=a g x) :
+    C.cocycle g x=C.cocycle g₀ x*(C.pointEquiv x).symm (C.transitionResidual x y g₀ g h₀ hg) := by
+  apply Subtype.ext
+  change (C.rep (a g x))⁻¹*g*C.rep x=
+    ((C.rep (a g₀ x))⁻¹*g₀*C.rep x)*((C.rep x)⁻¹*(g₀⁻¹*g)*C.rep x)
+  rw [← hg,← h₀]
+  group
+
+/- Normalized residual law on the true point stabilizer. The reference is any
+compatible transition; it is a unitary frame, not a new hypothesis on the law. -/
+def transitionAverage (ρ : UnitaryIrrep C.stabilizer) (w : W → ℝ) (g : W → G)
+    (x y : Ω) (ω₀ : W) (h₀ : y=a (g ω₀) x) : Matrix (Fin ρ.dimension) (Fin ρ.dimension) ℂ :=
+  (C.transition w g y x : ℂ)⁻¹ • ∑ ω : {ω : W // y=a (g ω) x},
+    (w ω.val : ℂ) • ρ.matrix
+      ((C.pointEquiv x).symm (C.transitionResidual x y (g ω₀) (g ω.val) h₀ ω.property))
+
+/- Exact moving-frame identity for each normalized block. -/
+omit [Fintype G] [Fintype Ω] in
+theorem conditionalBlock_eq (ρ : UnitaryIrrep C.stabilizer) (w : W → ℝ) (g : W → G)
+    (x y : Ω) (ω₀ : W) (h₀ : y=a (g ω₀) x) :
+    C.conditionalBlock ρ w g y x=ρ.matrix (C.cocycle (g ω₀) x)*C.transitionAverage ρ w g x y ω₀ h₀ := by
+  unfold conditionalBlock transitionAverage
+  rw [Matrix.mul_smul,Matrix.mul_sum]
+  congr 1
+  rw [← Finset.sum_filter]
+  rw [Finset.sum_subtype (p := fun ω => y=a (g ω) x) (Finset.univ.filter (fun ω => y=a (g ω) x))
+    (by intro ω; simp)]
+  apply Finset.sum_congr rfl
+  intro ω _
+  change (w ω.val : ℂ) • ρ.matrix (C.cocycle (g ω.val) x)=
+    ρ.matrix (C.cocycle (g ω₀) x)*((w ω.val : ℂ) •
+      ρ.matrix ((C.pointEquiv x).symm (C.transitionResidual x y (g ω₀) (g ω.val) h₀ ω.property)))
+  rw [Matrix.mul_smul,← map_mul]
+  congr 2
+  exact C.cocycle_residual x y (g ω₀) (g ω.val) h₀ ω.property
+
+end CoordinateSweeps.PlacementInduction.Chart
+
+end
+end
+open scoped Matrix.Norms.L2Operator
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/Placement/Occurrence.lean b/lean/OAI/RepresentationTheory/Placement/Occurrence.lean
index d4db5c97d2a2497be5e501999670656bf17abaad..1224ac454d15d951ba8ef2dc3cf1afb1b607491c 100644
--- a/lean/OAI/RepresentationTheory/Placement/Occurrence.lean
+++ b/lean/OAI/RepresentationTheory/Placement/Occurrence.lean
@@ -1,1145 +1,1145 @@
-import Mathlib
-import OAI.RepresentationTheory.Placement.Chart
-import OAI.RepresentationTheory.Young.Classification
-import OAI.Analysis.Matrix.TensorMoment
-
-namespace OAI
-
-noncomputable section
-open scoped BigOperators Matrix.Norms.L2Operator ComplexOrder
-attribute [local instance] Classical.propDecidable
-noncomputable section
-open scoped BigOperators
-attribute [local instance] Classical.propDecidable
-noncomputable section
-open scoped BigOperators
-open MvPolynomial
-noncomputable section
-open MeasureTheory ProbabilityTheory Real Set Filter
-open scoped ENNReal NNReal BigOperators
-noncomputable section
-open scoped BigOperators
-noncomputable section
-open scoped BigOperators Matrix.Norms.L2Operator
-attribute [local instance] Classical.propDecidable
-
-namespace CoordinateSweeps.Placement
-variable {I U X Y : Type*} [Fintype I] [DecidableEq I] [Fintype X] [Fintype Y]
-
-/- Dependence only on the listed coordinate values of a placement. -/
-def DependsOn (coords : Y → I → U) (A : Finset I) (f : Y → ℂ) : Prop :=
-  ∀ y y', (∀ i ∈ A, coords y i = coords y' i) → f y = f y'
-
-/- Counting-measure top-level space. Equivalently the orthogonal complement
-of the span of proper-coordinate cylinder functions (conjugation preserves that span). -/
-def topSpace (coords : Y → I → U) : Submodule ℂ (Y → ℂ) where
-  carrier := {v | ∀ A : Finset I, A ≠ Finset.univ → ∀ f : Y → ℂ,
-    DependsOn coords A f → ∑ y, f y*v y = 0}
-  zero_mem' := by simp
-  add_mem' := by
-    intro v w hv hw A hA f hf
-    simp only [Pi.add_apply, mul_add, Finset.sum_add_distrib, hv A hA f hf, hw A hA f hf, zero_add]
-  smul_mem' := by
-    intro c v hv A hA f hf
-    simp only [Pi.smul_apply, smul_eq_mul]
-    simp_rw [← mul_assoc, mul_comm (f _) c, mul_assoc]
-    rw [← Finset.mul_sum, hv A hA f hf, mul_zero]
-
-/- The source's s-normalized alternating kernel, over all coordinate subsets. -/
-def alternatingKernel (s : ℝ) (p : Finset I → X → Y → ℝ) : Matrix X Y ℂ :=
-  fun x y => (s⁻¹^(Fintype.card I) * ∑ A ∈ (Finset.univ : Finset I).powerset,
-    (-1 : ℝ)^(Fintype.card I-A.card) * s^A.card * p A x y : ℝ)
-
-/- All proper-subset marginal terms vanish on the top level. This is the
-nontrivial algebraic bridge between Q and the actual transition operator. -/
-omit [DecidableEq I] [Fintype X] in
-theorem alternatingKernel_mulVec_top (coords : Y → I → U)
-    (s : ℝ) (hs : s ≠ 0) (p : Finset I → X → Y → ℝ)
-    (hp : ∀ A x, DependsOn coords A (fun y => (p A x y : ℂ)))
-    {v : Y → ℂ} (hv : v ∈ topSpace coords) :
-    (alternatingKernel s p).mulVec v = Matrix.mulVec (fun x y => (p Finset.univ x y : ℂ)) v := by
-  funext x
-  simp only [Matrix.mulVec, dotProduct, alternatingKernel, Complex.ofReal_mul,
-    Complex.ofReal_pow, Complex.ofReal_inv, Complex.ofReal_sum]
-  simp only [mul_assoc, Finset.sum_mul]
-  rw [← Finset.mul_sum, Finset.sum_comm]
-  simp only [← Finset.mul_sum]
-  rw [Finset.sum_eq_single (Finset.univ : Finset I)]
-  · simp only [Finset.card_univ, Nat.sub_self, pow_zero, one_mul]
-    rw [← mul_assoc, ← mul_pow, inv_mul_cancel₀ (Complex.ofReal_ne_zero.mpr hs), one_pow, one_mul]
-  · intro A hA hne
-    have hzero := hv A hne (fun y => (p A x y : ℂ)) (hp A x)
-    simp only [hzero, mul_zero]
-  · simp
-
-/- Counting HS norm always dominates the L2 operator norm, rectangular and
-empty dimensions included. No entrywise max norm is used. -/
-theorem matrix_l2_norm_le_hs (A : Matrix X Y ℂ) :
-    ‖A‖ ≤ Real.sqrt (∑ x, ∑ y, ‖A x y‖^2) := by
-  let L : EuclideanSpace ℂ Y →L[ℂ] EuclideanSpace ℂ X :=
-    LinearMap.toContinuousLinearMap (Matrix.toEuclideanLin A)
-  change ‖L‖ ≤ _
-  refine ContinuousLinearMap.opNorm_le_bound L (by positivity) ?_
-  intro v
-  have hrow (x : X) : ‖(A.mulVec v) x‖^2 ≤
-      (∑ y, ‖A x y‖^2)*(∑ y, ‖v y‖^2) := by
-    have hnorm : ‖(A.mulVec v) x‖ ≤ ∑ y, ‖A x y‖*‖v y‖ := by
-      apply (norm_sum_le _ _).trans
-      exact Finset.sum_le_sum (fun y _ => le_of_eq (norm_mul _ _))
-    have hc := Finset.sum_mul_sq_le_sq_mul_sq Finset.univ (fun y => ‖A x y‖) (fun y => ‖v y‖)
-    exact ((sq_le_sq₀ (norm_nonneg _) (Finset.sum_nonneg (fun _ _ => by positivity))).mpr hnorm).trans hc
-  have hall : (∑ x, ‖(A.mulVec v) x‖^2) ≤
-      (∑ x, ∑ y, ‖A x y‖^2)*(∑ y, ‖v y‖^2) := by
-    rw [Finset.sum_mul]
-    exact Finset.sum_le_sum (fun x _ => hrow x)
-  change ‖Matrix.toEuclideanLin A v‖ ≤ _
-  apply (sq_le_sq₀ (norm_nonneg _) (mul_nonneg (Real.sqrt_nonneg _) (norm_nonneg _))).mp
-  rw [mul_pow, Real.sq_sqrt (by positivity), PiLp.norm_sq_eq_of_L2, PiLp.norm_sq_eq_of_L2]
-  exact hall
-
-/- The actual full marginal has the alternating HS bound on the entire top
-placement space. This is used before choosing an irreducible constituent. -/
-omit [DecidableEq I] in
-theorem fullKernel_top_norm_le (coords : Y → I → U)
-    (s : ℝ) (hs : s ≠ 0) (p : Finset I → X → Y → ℝ)
-    (hp : ∀ A x, DependsOn coords A (fun y => (p A x y : ℂ)))
-    (v : EuclideanSpace ℂ Y) (hv : (fun y => v y) ∈ topSpace coords) :
-    ‖(EuclideanSpace.equiv X ℂ).symm
-      (Matrix.mulVec (fun x y => (p Finset.univ x y : ℂ)) v)‖ ≤
-      Real.sqrt (∑ x, ∑ y, ‖alternatingKernel s p x y‖^2)*‖v‖ := by
-  rw [← alternatingKernel_mulVec_top coords s hs p hp hv]
-  exact ((alternatingKernel s p).l2_opNorm_mulVec v).trans
-    (mul_le_mul_of_nonneg_right (matrix_l2_norm_le_hs _) (norm_nonneg _))
-end CoordinateSweeps.Placement
-namespace CoordinateSweeps
-namespace UnitaryIrrep
-variable {Γ : Type*} [Group Γ] (ρ : UnitaryIrrep Γ)
-
-theorem adjoint_intertwines {X : Type*} [Fintype X] [DecidableEq X]
-    (P : Γ →* Matrix X X ℂ) (hP : ∀ g, (P g).conjTranspose * P g = 1)
-    (B : Matrix (Fin ρ.dimension) X ℂ)
-    (hB : ∀ g, B * P g = ρ.matrix g * B) (g : Γ) :
-    P g * B.conjTranspose = B.conjTranspose * ρ.matrix g := by
-  have hPinv : P g⁻¹ = (P g).conjTranspose := by
-    calc
-      P g⁻¹ = 1 * P g⁻¹ := (one_mul _).symm
-      _ = ((P g).conjTranspose * P g) * P g⁻¹ := by rw [hP g]
-      _ = (P g).conjTranspose := by rw [mul_assoc, ← map_mul, mul_inv_cancel, map_one, mul_one]
-  have h := congrArg Matrix.conjTranspose (hB g⁻¹)
-  have hrinv : ρ.matrix g⁻¹ = (ρ.matrix g).conjTranspose := by
-    calc
-      ρ.matrix g⁻¹ = 1 * ρ.matrix g⁻¹ := (one_mul _).symm
-      _ = ((ρ.matrix g).conjTranspose * ρ.matrix g) * ρ.matrix g⁻¹ := by rw [ρ.unitary g]
-      _ = (ρ.matrix g).conjTranspose := by rw [mul_assoc, ← map_mul, mul_inv_cancel, map_one, mul_one]
-  simpa only [Matrix.conjTranspose_mul, hPinv, hrinv,
-    Matrix.conjTranspose_conjTranspose] using h
-
-/- Schur's lemma makes every nonzero intertwiner a scalar coisometry. -/
-theorem intertwiner_gram_scalar {X : Type*} [Fintype X] [DecidableEq X]
-    (P : Γ →* Matrix X X ℂ) (hP : ∀ g, (P g).conjTranspose * P g = 1)
-    (B : Matrix (Fin ρ.dimension) X ℂ)
-    (hB : ∀ g, B * P g = ρ.matrix g * B) :
-    ∃ c : ℂ, B * B.conjTranspose = c • (1 : Matrix _ _ ℂ) := by
-  apply ρ.commuting_scalar
-  intro g
-  rw [Matrix.mul_assoc, ← ρ.adjoint_intertwines P hP B hB g,
-    ← Matrix.mul_assoc, hB g, Matrix.mul_assoc]
-
-/- Exact norm transfer, with no extraneous dimension loss. Q need agree with
-an averaged permutation operator only on the adjoint intertwiner range. -/
-theorem norm_le_compression {X : Type*} [Fintype X] [DecidableEq X]
-    (P : Γ →* Matrix X X ℂ) (hP : ∀ g, (P g).conjTranspose * P g = 1)
-    (B : Matrix (Fin ρ.dimension) X ℂ) (hne : B ≠ 0)
-    (hB : ∀ g, B * P g = ρ.matrix g * B)
-    (K : Matrix (Fin ρ.dimension) (Fin ρ.dimension) ℂ) (Q : Matrix X X ℂ)
-    (hQ : Q * B.conjTranspose = B.conjTranspose * K) : ‖K‖ ≤ ‖Q‖ := by
-  let : NeZero ρ.dimension := ⟨ρ.positive.ne'⟩
-  obtain ⟨c, hc⟩ := ρ.intertwiner_gram_scalar P hP B hB
-  have hn : ‖B‖ * ‖B‖ = ‖c‖ := by
-    have h := Matrix.l2_opNorm_conjTranspose_mul_self B.conjTranspose
-    simpa only [Matrix.conjTranspose_conjTranspose, Matrix.l2_opNorm_conjTranspose,
-      hc, norm_smul, norm_one, mul_one] using h.symm
-  have hcpos : 0 < ‖c‖ := by rw [← hn]; exact mul_pos (norm_pos_iff.mpr hne) (norm_pos_iff.mpr hne)
-  have he : c • K = B * Q * B.conjTranspose := by
-    rw [Matrix.mul_assoc, hQ, ← Matrix.mul_assoc, hc, Matrix.smul_mul, one_mul]
-  have hbound : ‖c‖ * ‖K‖ ≤ ‖c‖ * ‖Q‖ := by
-    rw [← norm_smul, he]
-    calc
-      ‖B * Q * B.conjTranspose‖ ≤ ‖B * Q‖ * ‖B.conjTranspose‖ := Matrix.l2_opNorm_mul _ _
-      _ ≤ (‖B‖ * ‖Q‖) * ‖B.conjTranspose‖ :=
-        mul_le_mul_of_nonneg_right (Matrix.l2_opNorm_mul _ _) (norm_nonneg _)
-      _ = (‖B‖ * ‖B‖) * ‖Q‖ := by rw [Matrix.l2_opNorm_conjTranspose]; ring
-      _ = ‖c‖ * ‖Q‖ := by rw [hn]
-  exact (mul_le_mul_iff_right₀ hcpos).mp hbound
-end UnitaryIrrep
-end CoordinateSweeps
-
-noncomputable section
-universe u v
-open scoped ComplexConjugate
-
-namespace CoordinateSweeps
-/- The pointwise stabilizer of two sites, used in the sparse dimension bridge. -/
-def fixTwo {Ω : Type*} (a b : Ω) : Subgroup (Equiv.Perm Ω) where
-  carrier := {g | g a = a ∧ g b = b}
-  one_mem' := ⟨rfl, rfl⟩
-  mul_mem' hg hk := by simpa using And.intro (congrArg _ hk.1 |>.trans hg.1) (congrArg _ hk.2 |>.trans hg.2)
-  inv_mem' := by
-    intro g hg
-    constructor
-    · apply g.injective
-      simpa using hg.1.symm
-    · apply g.injective
-      simpa using hg.2.symm
-
-namespace SparseDimension
-variable {Ω V : Type*} [Fintype Ω] [DecidableEq Ω]
-  [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
-
-/- Generic Schur commutant lemma, with no classification assumptions. -/
-theorem commuting_scalar {Γ : Type*} [Group Γ]
-    (ρ : Representation ℂ Γ V) [ρ.IsIrreducible]
-    (A : Module.End ℂ V) (hA : ∀ g, A * ρ g = ρ g * A) :
-    ∃ c : ℂ, A = c • (1 : Module.End ℂ V) := by
-  let F : Representation.IntertwiningMap ρ ρ :=
-    { toLinearMap := A
-      isIntertwining' := hA }
-  obtain ⟨c, hc⟩ := Representation.IsIrreducible.algebraMap_intertwiningMap_bijective_of_isAlgClosed
-    (ρ := ρ) |>.2 F
-  refine ⟨c, ?_⟩
-  have hh := congrArg Representation.IntertwiningMap.toLinearMap hc
-  exact hh.symm
-
-/- If all operators are scalar, irreducibility forces one dimension. -/
-omit [FiniteDimensional ℂ V] in
-theorem finrank_eq_one_of_scalars {Γ : Type*} [Group Γ]
-    (ρ : Representation ℂ Γ V) [ρ.IsIrreducible]
-    (hs : ∀ g, ∃ c : ℂ, ρ g = c • (1 : Module.End ℂ V)) :
-    Module.finrank ℂ V = 1 := by
-  have : Nontrivial V := by
-    by_contra hn
-    have : Subsingleton V := not_nontrivial_iff_subsingleton.mp hn
-    have he : (⊥ : Subrepresentation ρ) = ⊤ := by
-      apply Subrepresentation.toSubmodule_injective
-      exact Subsingleton.elim _ _
-    exact bot_ne_top he
-  obtain ⟨v, hv⟩ := exists_ne (0 : V)
-  let S : Subrepresentation ρ :=
-    { toSubmodule := ℂ ∙ v
-      apply_mem_toSubmodule := by
-        intro g x hx
-        obtain ⟨c, hc⟩ := hs g
-        rw [hc]
-        exact Submodule.smul_mem _ c hx }
-  have hne : S ≠ ⊥ := by
-    intro he
-    have hm : v ∈ S := Submodule.mem_span_singleton_self _
-    rw [he] at hm
-    exact hv hm
-  have he : S = ⊤ := (eq_bot_or_eq_top S).resolve_left hne
-  have he' : (ℂ ∙ v : Submodule ℂ V) = ⊤ := congrArg Subrepresentation.toSubmodule he
-  rw [← finrank_top ℂ V, ← he']
-  exact finrank_span_singleton hv
-
-omit [FiniteDimensional ℂ V] in
-@[simp] theorem subrep_bot_toSubmodule {Γ : Type*} [Group Γ]
-    (ρ : Representation ℂ Γ V) : (⊥ : Subrepresentation ρ).toSubmodule = ⊥ := rfl
-omit [FiniteDimensional ℂ V] in
-@[simp] theorem subrep_top_toSubmodule {Γ : Type*} [Group Γ]
-    (ρ : Representation ℂ Γ V) : (⊤ : Subrepresentation ρ).toSubmodule = ⊤ := rfl
-
-/- Flatten a subrepresentation of a subrepresentation without changing its dimension. -/
-def nested {Γ : Type*} [Group Γ] {ρ : Representation ℂ Γ V}
-    (S : Subrepresentation ρ) (T : Subrepresentation S.toRepresentation) : Subrepresentation ρ where
-  toSubmodule := T.toSubmodule.map S.toSubmodule.subtype
-  apply_mem_toSubmodule := by
-    intro g v hv
-    obtain ⟨w, hw, rfl⟩ := hv
-    refine ⟨S.toRepresentation g w, T.apply_mem_toSubmodule g hw, rfl⟩
-
-/- Existence of a least-dimensional constituent, proved rather than assumed. -/
-theorem exists_minimal_constituent {Γ : Type*} [Group Γ]
-    (ρ : Representation ℂ Γ V) [Nontrivial V] :
-    ∃ S : Subrepresentation ρ, 0 < Module.finrank ℂ S.toSubmodule ∧
-      S.toRepresentation.IsIrreducible ∧
-      ∀ T : Subrepresentation ρ, T ≠ ⊥ →
-        Module.finrank ℂ S.toSubmodule ≤ Module.finrank ℂ T.toSubmodule := by
-  classical
-  have hp : ∃ n, ∃ S : Subrepresentation ρ,
-      0 < Module.finrank ℂ S.toSubmodule ∧ Module.finrank ℂ S.toSubmodule = n :=
-    ⟨Module.finrank ℂ V, ⊤, by simpa using (Module.finrank_pos (R := ℂ) (M := V)), by simp⟩
-  obtain ⟨S, hSpos, hSdim⟩ := Nat.find_spec hp
-  have hmin (T : Subrepresentation ρ) (hT : T ≠ ⊥) :
-      Module.finrank ℂ S.toSubmodule ≤ Module.finrank ℂ T.toSubmodule := by
-    rw [hSdim]
-    apply Nat.find_min' hp
-    refine ⟨T, ?_, rfl⟩
-    apply Module.finrank_pos_iff.mpr
-    apply Submodule.nontrivial_iff_ne_bot.mpr
-    exact fun ht => hT (Subrepresentation.toSubmodule_injective ht)
-  refine ⟨S, hSpos, ?_, hmin⟩
-  have : Nontrivial S.toSubmodule := Module.finrank_pos_iff.mp hSpos
-  refine { exists_pair_ne := ⟨⊥, ⊤, ?_⟩, eq_bot_or_eq_top := ?_ }
-  · intro he
-    exact bot_ne_top (congrArg Subrepresentation.toSubmodule he)
-  · intro T
-    by_cases hT : T = ⊥
-    · exact Or.inl hT
-    right
-    by_contra htop
-    have hTp : 0 < Module.finrank ℂ T.toSubmodule := by
-      apply Module.finrank_pos_iff.mpr
-      apply Submodule.nontrivial_iff_ne_bot.mpr
-      exact fun ht => hT (Subrepresentation.toSubmodule_injective ht)
-    have hdim : Module.finrank ℂ (nested S T).toSubmodule = Module.finrank ℂ T.toSubmodule :=
-      S.toSubmodule.finrank_map_subtype_eq T.toSubmodule
-    have hnb : nested S T ≠ ⊥ := by
-      intro he
-      have hz : Module.finrank ℂ (nested S T).toSubmodule = 0 := by rw [he]; simp
-      omega
-    have hlow := hmin (nested S T) hnb
-    rw [hdim] at hlow
-    have hhigh : Module.finrank ℂ T.toSubmodule < Module.finrank ℂ S.toSubmodule :=
-      Submodule.finrank_lt (fun ht => htop (Subrepresentation.toSubmodule_injective ht))
-    omega
-
-/- Every reducible finite-group representation has a constituent of at most
-half its dimension. This uses Maschke over ℂ, not an assumed decomposition. -/
-theorem exists_constituent_le_half {Γ : Type*} [Group Γ] [Fintype Γ]
-    (ρ : Representation ℂ Γ V) [Nontrivial V]
-    (hn : ¬ρ.IsIrreducible) :
-    ∃ S : Subrepresentation ρ, 0 < Module.finrank ℂ S.toSubmodule ∧
-      S.toRepresentation.IsIrreducible ∧
-      2 * Module.finrank ℂ S.toSubmodule ≤ Module.finrank ℂ V := by
-  classical
-  obtain ⟨S, hpos, hirr, hmin⟩ := exists_minimal_constituent ρ
-  have hnon : ∃ T : Subrepresentation ρ, T ≠ ⊥ ∧ T ≠ ⊤ := by
-    by_contra h
-    apply hn
-    refine { exists_pair_ne := ⟨⊥, ⊤, ?_⟩, eq_bot_or_eq_top := ?_ }
-    · intro he
-      exact bot_ne_top (congrArg Subrepresentation.toSubmodule he)
-    · intro T
-      simpa only [not_exists, not_and_or, not_not] using (not_exists.mp h T)
-  obtain ⟨T, hTbot, hTtop⟩ := hnon
-  have hSt : S ≠ ⊤ := by
-    intro he
-    have hd := hmin T hTbot
-    have ht : Module.finrank ℂ T.toSubmodule < Module.finrank ℂ V :=
-      Submodule.finrank_lt (fun ht => hTtop (Subrepresentation.toSubmodule_injective ht))
-    rw [he] at hd
-    simp only [subrep_top_toSubmodule, finrank_top] at hd
-    omega
-  let : NeZero (Nat.card Γ : ℂ) := ⟨by exact_mod_cast Nat.card_pos.ne'⟩
-  obtain ⟨U, hU⟩ := exists_isCompl S
-  have hUb : U ≠ ⊥ := by
-    intro he
-    have h := hU.sup_eq_top
-    rw [he, sup_bot_eq] at h
-    exact hSt h
-  have hle := hmin U hUb
-  have hcompl : IsCompl S.toSubmodule U.toSubmodule := by
-    apply IsCompl.of_eq
-    · exact congrArg Subrepresentation.toSubmodule hU.inf_eq_bot
-    · exact congrArg Subrepresentation.toSubmodule hU.sup_eq_top
-  have hsum := Submodule.finrank_add_eq_of_isCompl hcompl
-  exact ⟨S, hpos, hirr, by omega⟩
-
-/- Restriction to permutations fixing two distinct sites is not irreducible,
-unless the full irreducible is one-dimensional. This gives a factor-two
-dimension descent every two removed cards, needed for sparse types. -/
-theorem finrank_eq_one_of_fixTwo_irreducible
-    (ρ : Representation ℂ (Equiv.Perm Ω) V) [ρ.IsIrreducible]
-    (a b : Ω) (hab : a ≠ b)
-    [hirr : Representation.IsIrreducible (ρ.comp (fixTwo a b).subtype)] :
-    Module.finrank ℂ V = 1 := by
-  have hc : ∀ g : fixTwo a b,
-      ρ (Equiv.swap a b) * ρ g.val = ρ g.val * ρ (Equiv.swap a b) := by
-    intro g
-    rw [← map_mul, ← map_mul]
-    congr 1
-    rw [Equiv.mul_swap_eq_swap_mul, g.property.1, g.property.2]
-  obtain ⟨c, hc⟩ := commuting_scalar (ρ.comp (fixTwo a b).subtype) (ρ (Equiv.swap a b)) hc
-  have hswap (x y : Ω) (hxy : x ≠ y) : ρ (Equiv.swap x y) = c • (1 : Module.End ℂ V) := by
-    obtain ⟨g, hg⟩ := isConj_iff.mp (Equiv.Perm.isConj_swap hab hxy)
-    rw [← hg, map_mul, map_mul, hc]
-    simp only [mul_smul_comm, mul_one, smul_mul_assoc, ← map_mul, mul_inv_cancel, map_one]
-  apply finrank_eq_one_of_scalars ρ
-  intro g
-  induction g using Equiv.Perm.swap_induction_on with
-  | one => exact ⟨1, by simp⟩
-  | swap_mul g x y hxy ih =>
-    obtain ⟨d, hd⟩ := ih
-    refine ⟨c*d, ?_⟩
-    rw [map_mul, hswap x y hxy, hd, smul_mul_smul_comm, one_mul]
-
-/- Restriction along a surjective homomorphism preserves irreducibility. -/
-omit [FiniteDimensional ℂ V] in
-theorem irreducible_comp_surjective {Γ Δ : Type*} [Group Γ] [Group Δ]
-    (ρ : Representation ℂ Γ V) [ρ.IsIrreducible]
-    (f : Δ →* Γ) (hf : Function.Surjective f) :
-    Representation.IsIrreducible (ρ.comp f) := by
-  have he (S : Subrepresentation (ρ.comp f)) :
-      S.toSubmodule = ⊥ ∨ S.toSubmodule = ⊤ := by
-    let T : Subrepresentation ρ :=
-      { toSubmodule := S.toSubmodule
-        apply_mem_toSubmodule := by
-          intro g v hv
-          obtain ⟨x, rfl⟩ := hf g
-          exact S.apply_mem_toSubmodule x hv }
-    exact (eq_bot_or_eq_top T).imp
-      (congrArg Subrepresentation.toSubmodule) (congrArg Subrepresentation.toSubmodule)
-  have : Nontrivial V := by
-    by_contra hn
-    have : Subsingleton V := not_nontrivial_iff_subsingleton.mp hn
-    apply bot_ne_top (α := Subrepresentation ρ)
-    exact Subrepresentation.toSubmodule_injective (Subsingleton.elim _ _)
-  refine { exists_pair_ne := ⟨⊥, ⊤, ?_⟩, eq_bot_or_eq_top := ?_ }
-  · intro h
-    exact bot_ne_top (congrArg Subrepresentation.toSubmodule h)
-  · intro S
-    exact (he S).imp (fun h => Subrepresentation.toSubmodule_injective h)
-      (fun h => Subrepresentation.toSubmodule_injective h)
-
-/- Permutations of the complementary sites, viewed in the two-point stabilizer. -/
-def fixTwoHom (a b : Ω) : Equiv.Perm {x : Ω // x ≠ a ∧ x ≠ b} →* fixTwo a b where
-  toFun g := ⟨Equiv.Perm.ofSubtype g,
-    Equiv.Perm.ofSubtype_apply_of_not_mem g (by simp),
-    Equiv.Perm.ofSubtype_apply_of_not_mem g (by simp)⟩
-  map_one' := Subtype.ext (map_one _)
-  map_mul' g h := Subtype.ext (map_mul _ g h)
-
-omit [Fintype Ω] in
-@[simp] theorem fixTwoHom_val (a b : Ω) (g : Equiv.Perm {x : Ω // x ≠ a ∧ x ≠ b}) :
-    (fixTwoHom a b g).val = Equiv.Perm.ofSubtype g := rfl
-
-omit [Fintype Ω] in
-theorem fixTwoHom_surjective (a b : Ω) : Function.Surjective (fixTwoHom a b) := by
-  intro g
-  have hi (x : Ω) : ((g.val x ≠ a ∧ g.val x ≠ b) ↔ (x ≠ a ∧ x ≠ b)) := by
-    have ha : g.val x = a ↔ x = a :=
-      ⟨fun h => g.val.injective (h.trans g.property.1.symm), fun h => h ▸ g.property.1⟩
-    have hb : g.val x = b ↔ x = b :=
-      ⟨fun h => g.val.injective (h.trans g.property.2.symm), fun h => h ▸ g.property.2⟩
-    simp only [ne_eq, ha, hb]
-  refine ⟨g.val.subtypePerm hi, Subtype.ext ?_⟩
-  apply Equiv.Perm.ofSubtype_subtypePerm hi
-  intro x hx
-  constructor
-  · intro he; subst x; exact hx g.property.1
-  · intro he; subst x; exact hx g.property.2
-
-/- One factor-two dimension descent, with the remaining symmetric group
-realized literally as permutations of the complement. -/
-theorem exists_half_constituent_fixTwo
-    (ρ : Representation ℂ (Equiv.Perm Ω) V) [ρ.IsIrreducible] [Nontrivial V]
-    (hd : Module.finrank ℂ V ≠ 1) (a b : Ω) (hab : a ≠ b) :
-    ∃ S : Subrepresentation (ρ.comp (fixTwo a b).subtype),
-      0 < Module.finrank ℂ S.toSubmodule ∧
-      Representation.IsIrreducible (S.toRepresentation.comp (fixTwoHom a b)) ∧
-      2 * Module.finrank ℂ S.toSubmodule ≤ Module.finrank ℂ V := by
-  classical
-  have hn : ¬Representation.IsIrreducible (ρ.comp (fixTwo a b).subtype) := by
-    intro hirr
-    let := hirr
-    exact hd (finrank_eq_one_of_fixTwo_irreducible ρ a b hab)
-  obtain ⟨S, hpos, hirr, hdim⟩ := exists_constituent_le_half (ρ.comp (fixTwo a b).subtype) hn
-  let := hirr
-  exact ⟨S, hpos, irreducible_comp_surjective S.toRepresentation
-    (fixTwoHom a b) (fixTwoHom_surjective a b), hdim⟩
-
-/- Nonzero state space follows from irreducibility, including degenerate groups. -/
-omit [FiniteDimensional ℂ V] in
-theorem irreducible_nontrivial {Γ : Type*} [Group Γ]
-    (ρ : Representation ℂ Γ V) [ρ.IsIrreducible] : Nontrivial V := by
-  by_contra hn
-  have : Subsingleton V := not_nontrivial_iff_subsingleton.mp hn
-  apply bot_ne_top (α := Subrepresentation ρ)
-  exact Subrepresentation.toSubmodule_injective (Subsingleton.elim _ _)
-
-/- A one-dimensional constituent is already visible after at most twice the
-base-two logarithm of the dimension many fixed cards. The power formulation
-avoids rounding conventions and is valid also for dimension one. -/
-theorem sparse_line_by_dim (D : ℕ) :
-    ∀ (Ω : Type u) [Fintype Ω] [DecidableEq Ω]
-      (V : Type v) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
-      (ρ : Representation ℂ (Equiv.Perm Ω) V) [ρ.IsIrreducible],
-      Module.finrank ℂ V = D →
-      ∃ (t : ℕ) (A : Finset Ω) (W : Submodule ℂ V),
-        Module.finrank ℂ W = 1 ∧ A.card ≤ 2*t ∧ 2^t ≤ D ∧
-        ∀ g : Equiv.Perm Ω, (∀ a ∈ A, g a = a) →
-          ∀ w ∈ W, ρ g w ∈ W := by
-  classical
-  induction D using Nat.strong_induction_on with
-  | h D ih =>
-    intro Ω _ _ V _ _ _ ρ hirr hD
-    have : Nontrivial V := irreducible_nontrivial ρ
-    have hDpos : 0 < D := hD ▸ Module.finrank_pos
-    by_cases hD1 : D = 1
-    · refine ⟨0, ∅, ⊤, ?_, by simp, by simp [hD1], ?_⟩
-      · simpa only [finrank_top] using hD.trans hD1
-      · intro g hg w hw; trivial
-    have hΩ : Nontrivial Ω := by
-      by_contra hn
-      have : Subsingleton Ω := not_nontrivial_iff_subsingleton.mp hn
-      have hscalar : ∀ g : Equiv.Perm Ω, ∃ c : ℂ, ρ g = c • (1 : Module.End ℂ V) := by
-        intro g
-        have he : g = 1 := Subsingleton.elim _ _
-        exact ⟨1, by rw [he, map_one, one_smul]⟩
-      exact hD1 (hD.symm.trans (finrank_eq_one_of_scalars ρ hscalar))
-    let := hΩ
-    obtain ⟨a, b, hab⟩ := exists_pair_ne Ω
-    obtain ⟨S, hSpos, hSirr, hShalf⟩ := exists_half_constituent_fixTwo ρ
-      (by simpa only [hD] using hD1) a b hab
-    let ρ' : Representation ℂ (Equiv.Perm {x : Ω // x ≠ a ∧ x ≠ b}) S.toSubmodule :=
-      S.toRepresentation.comp (fixTwoHom a b)
-    let : ρ'.IsIrreducible := hSirr
-    have hlt : Module.finrank ℂ S.toSubmodule < D := by rw [hD] at hShalf; omega
-    obtain ⟨t, A, W, hWdim, hAcard, ht, hWstable⟩ :=
-      ih (Module.finrank ℂ S.toSubmodule) hlt {x : Ω // x ≠ a ∧ x ≠ b}
-        S.toSubmodule ρ' rfl
-    let B : Finset Ω := insert a (insert b (A.image Subtype.val))
-    refine ⟨t+1, B, W.map S.toSubmodule.subtype, ?_, ?_, ?_, ?_⟩
-    · rw [Submodule.finrank_map_subtype_eq, hWdim]
-    · have hc := Finset.card_image_le (s := A) (f := Subtype.val)
-      have h1 := Finset.card_insert_le a (insert b (A.image Subtype.val))
-      have h2 := Finset.card_insert_le b (A.image Subtype.val)
-      dsimp [B]
-      omega
-    · rw [pow_succ]
-      rw [hD] at hShalf
-      omega
-    · intro g hg w hw
-      obtain ⟨w, hw, rfl⟩ := hw
-      have hga : g a = a := hg a (by simp [B])
-      have hgb : g b = b := hg b (by simp [B])
-      obtain ⟨σ, hσ⟩ := fixTwoHom_surjective a b ⟨g, hga, hgb⟩
-      have hsfix : ∀ x ∈ A, σ x = x := by
-        intro x hx
-        apply Subtype.ext
-        calc
-          (σ x).val = (fixTwoHom a b σ).val x.val :=
-            (Equiv.Perm.ofSubtype_apply_coe σ x).symm
-          _ = g x.val := congrArg (fun f : fixTwo a b => f.val x.val) hσ
-          _ = x.val := hg x.val (by simp [B, Finset.mem_image_of_mem _ hx])
-      refine ⟨ρ' σ w, hWstable σ hsfix w hw, ?_⟩
-      change ρ ((fixTwoHom a b σ).val) w.val = ρ g w.val
-      rw [hσ]
-
-/- The rounded-free quantitative sparse dimension bridge for the actual
-symmetric group, independent of a partition classification. -/
-theorem exists_sparse_line
-    (ρ : Representation ℂ (Equiv.Perm Ω) V) [ρ.IsIrreducible] :
-    ∃ (t : ℕ) (A : Finset Ω) (W : Submodule ℂ V),
-      Module.finrank ℂ W = 1 ∧ A.card ≤ 2*t ∧ 2^t ≤ Module.finrank ℂ V ∧
-      ∀ g : Equiv.Perm Ω, (∀ a ∈ A, g a = a) →
-        ∀ w ∈ W, ρ g w ∈ W :=
-  sparse_line_by_dim _ Ω V ρ rfl
-
-/- Ordinary permutation sign, embedded into the complex scalars. -/
-def signScalar : Equiv.Perm Ω →* ℂ :=
-  (Int.castRingHom ℂ).toMonoidHom.comp ((Units.coeHom ℤ).comp Equiv.Perm.sign)
-
-@[simp] theorem signScalar_apply (g : Equiv.Perm Ω) :
-    signScalar g = ((Equiv.Perm.sign g : ℤ) : ℂ) := rfl
-
-@[simp] theorem signScalar_swap (a b : Ω) (hab : a ≠ b) :
-    signScalar (Equiv.swap a b) = -1 := by
-  simp [signScalar_apply, Equiv.Perm.sign_swap hab]
-
-/- Every one-dimensional representation of a finite symmetric group is
-literally the trivial or sign representation. This also handles Ω of size 0,1. -/
-theorem oneDimensional_trivial_or_sign
-    (ρ : Representation ℂ (Equiv.Perm Ω) V) (hd : Module.finrank ℂ V = 1) :
-    (∀ g, ρ g = 1) ∨ (∀ g, ρ g = signScalar g • (1 : Module.End ℂ V)) := by
-  classical
-  by_cases hΩ : Nontrivial Ω
-  swap
-  · have : Subsingleton Ω := not_nontrivial_iff_subsingleton.mp hΩ
-    left
-    intro g
-    rw [Subsingleton.elim g 1, map_one]
-  let := hΩ
-  obtain ⟨a, b, hab⟩ := exists_pair_ne Ω
-  obtain ⟨c, hc, _⟩ := LinearMap.existsUnique_eq_smul_id_of_finrank_eq_one hd (ρ (Equiv.swap a b))
-  change ρ (Equiv.swap a b) = c • (1 : Module.End ℂ V) at hc
-  have hsame (x y : Ω) (hxy : x ≠ y) : ρ (Equiv.swap x y) = c • (1 : Module.End ℂ V) := by
-    obtain ⟨g, hg⟩ := isConj_iff.mp (Equiv.Perm.isConj_swap hab hxy)
-    rw [← hg, map_mul, map_mul, hc]
-    simp only [mul_smul_comm, mul_one, smul_mul_assoc, ← map_mul, mul_inv_cancel, map_one]
-  have hc2 : c*c = 1 := by
-    apply (LinearEquiv.smul_id_of_finrank_eq_one hd).injective
-    simp only [LinearEquiv.smul_id_of_finrank_eq_one_apply]
-    change (c*c) • (1 : Module.End ℂ V) = 1 • (1 : Module.End ℂ V)
-    rw [one_smul]
-    calc
-      (c*c) • (1 : Module.End ℂ V) = (c • (1 : Module.End ℂ V))*(c • 1) := by
-        simp only [smul_mul_smul_comm, one_mul]
-      _ = 1 := by rw [← hc, ← map_mul, Equiv.swap_mul_self, map_one]
-  rcases mul_self_eq_one_iff.mp hc2 with hc1 | hcn
-  · left
-    intro g
-    induction g using Equiv.Perm.swap_induction_on with
-    | one => exact map_one _
-    | swap_mul g x y hxy ih => rw [map_mul, hsame x y hxy, hc1, one_smul, ih, one_mul]
-  · right
-    intro g
-    induction g using Equiv.Perm.swap_induction_on with
-    | one => simp
-    | swap_mul g x y hxy ih =>
-      rw [map_mul, hsame x y hxy, hcn, ih, map_mul, signScalar_swap x y hxy,
-        smul_mul_smul_comm, one_mul]
-
-/- The cyclic span proof underlying the irreducible dimension lemma,
-on an arbitrary finite-dimensional realization. -/
-omit [FiniteDimensional ℂ V] in
-theorem orbit_span_top {Γ : Type*} [Group Γ]
-    (ρ : Representation ℂ Γ V) [ρ.IsIrreducible] (v : V) (hv : v ≠ 0) :
-    Submodule.span ℂ (Set.range (fun g => ρ g v)) = ⊤ := by
-  let S : Subrepresentation ρ :=
-    { toSubmodule := Submodule.span ℂ (Set.range (fun g => ρ g v))
-      apply_mem_toSubmodule := by
-        intro g w hw
-        induction hw using Submodule.span_induction with
-        | mem w hw =>
-          obtain ⟨k, rfl⟩ := hw
-          rw [← Module.End.mul_apply, ← map_mul]
-          exact Submodule.subset_span (Set.mem_range_self (g*k))
-        | zero => simp
-        | add x y hx hy hix hiy => simpa only [map_add] using Submodule.add_mem _ hix hiy
-        | smul c x hx hi => simpa only [map_smul] using Submodule.smul_mem _ c hi }
-  have hn : S ≠ ⊥ := by
-    intro he
-    have hm : v ∈ S := by
-      change v ∈ Submodule.span ℂ (Set.range (fun g => ρ g v))
-      apply Submodule.subset_span
-      exact ⟨1, by simp⟩
-    rw [he] at hm
-    exact hv hm
-  exact congrArg Subrepresentation.toSubmodule ((eq_bot_or_eq_top S).resolve_left hn)
-
-/- The upper sparse dimension bound: a line invariant under the stabilizer
-of k named sites gives at most n^k dimensions in its irreducible orbit. -/
-theorem dimension_le_pow_of_stable_line
-    (ρ : Representation ℂ (Equiv.Perm Ω) V) [ρ.IsIrreducible]
-    (A : Finset Ω) (W : Submodule ℂ V) (hW : Module.finrank ℂ W = 1)
-    (hstable : ∀ g : Equiv.Perm Ω, (∀ a ∈ A, g a = a) → ∀ w ∈ W, ρ g w ∈ W) :
-    Module.finrank ℂ V ≤ (Fintype.card Ω)^A.card := by
-  classical
-  have : Nontrivial W := (Module.finrank_pos_iff (R := ℂ) (M := W)).mp (by omega)
-  obtain ⟨v, hv⟩ := exists_ne (0 : W)
-  have hv' : v.val ≠ 0 := by intro he; exact hv (Subtype.ext he)
-  let restrict (g : Equiv.Perm Ω) : A → Ω := fun a => g a.val
-  let chooseG (x : A → Ω) : Equiv.Perm Ω :=
-    if hx : ∃ g, restrict g = x then Classical.choose hx else 1
-  have hchoose (g : Equiv.Perm Ω) : restrict (chooseG (restrict g)) = restrict g := by
-    dsimp [chooseG]
-    rw [dite_eq_left (show ∃ h, restrict h = restrict g from ⟨g, rfl⟩)]
-    exact Classical.choose_spec (show ∃ h, restrict h = restrict g from ⟨g, rfl⟩)
-  let u (x : A → Ω) : V := ρ (chooseG x) v.val
-  have hspan : Submodule.span ℂ (Set.range u) = ⊤ := by
-    rw [eq_top_iff, ← orbit_span_top ρ v.val hv']
-    apply Submodule.span_le.mpr
-    rintro _ ⟨g, rfl⟩
-    let r := chooseG (restrict g)
-    have hfix : ∀ a ∈ A, (r⁻¹*g) a = a := by
-      intro a ha
-      have he := congrFun (hchoose g) ⟨a, ha⟩
-      change r a = g a at he
-      simp only [Equiv.Perm.mul_apply]
-      rw [← he]; simp
-    have hw : ρ (r⁻¹*g) v.val ∈ W := hstable _ hfix _ v.property
-    obtain ⟨c, hc⟩ := exists_smul_eq_of_finrank_eq_one hW hv
-      (⟨ρ (r⁻¹*g) v.val, hw⟩ : W)
-    have he : ρ g v.val = c • u (restrict g) := by
-      have hh := congrArg (fun w : W => ρ r w.val) hc
-      change ρ r (c • v.val) = ρ r (ρ (r⁻¹*g) v.val) at hh
-      rw [map_smul, ← Module.End.mul_apply, ← map_mul, mul_inv_cancel_left] at hh
-      exact hh.symm
-    change ρ g v.val ∈ Submodule.span ℂ (Set.range u)
-    rw [he]
-    exact Submodule.smul_mem _ c (Submodule.subset_span (Set.mem_range_self _))
-  have hd := finrank_le_of_span_eq_top hspan
-  simpa only [Fintype.card_fun, Fintype.card_coe] using hd
-
-/- Pointwise stabilizer of a named finite set of sites. -/
-def fixSet (A : Finset Ω) : Subgroup (Equiv.Perm Ω) where
-  carrier := {g | ∀ a ∈ A, g a = a}
-  one_mem' := by simp
-  mul_mem' hg hh a ha := by simp only [Equiv.Perm.mul_apply, hh a ha, hg a ha]
-  inv_mem' {g} hg a ha := by
-    apply g.injective
-    simpa using (hg a ha).symm
-
-/- The complementary symmetric group onto the pointwise stabilizer. -/
-def fixSetHom (A : Finset Ω) : Equiv.Perm {x : Ω // x ∉ A} →* fixSet A where
-  toFun g := ⟨Equiv.Perm.ofSubtype g, fun _ ha =>
-    Equiv.Perm.ofSubtype_apply_of_not_mem g (not_not.mpr ha)⟩
-  map_one' := Subtype.ext (map_one _)
-  map_mul' g h := Subtype.ext (map_mul _ g h)
-
-omit [Fintype Ω] in
-@[simp] theorem fixSetHom_val (A : Finset Ω) (g : Equiv.Perm {x : Ω // x ∉ A}) :
-    (fixSetHom A g).val = Equiv.Perm.ofSubtype g := rfl
-
-omit [Fintype Ω] in
-theorem fixSetHom_surjective (A : Finset Ω) : Function.Surjective (fixSetHom A) := by
-  intro g
-  have hi (x : Ω) : (g.val x ∉ A) ↔ x ∉ A := by
-    constructor
-    · intro h hx
-      apply h
-      rwa [g.property x hx]
-    · intro h hx
-      have he := g.property (g.val x) hx
-      have he' : g.val x = x := g.val.injective he
-      exact h (he' ▸ hx)
-  refine ⟨g.val.subtypePerm hi, Subtype.ext ?_⟩
-  apply Equiv.Perm.ofSubtype_subtypePerm hi
-  intro x hx
-  by_contra hn
-  exact hx (g.property x (by simpa using hn))
-
-/- A stable line under a pointwise stabilizer is a literal trivial or sign
-line. There is no hypothesis about a partition or a pre-existing embedding. -/
-theorem stable_line_trivial_or_sign
-    (ρ : Representation ℂ (Equiv.Perm Ω) V)
-    (A : Finset Ω) (W : Submodule ℂ V) (hW : Module.finrank ℂ W = 1)
-    (hstable : ∀ g : Equiv.Perm Ω, (∀ a ∈ A, g a = a) → ∀ w ∈ W, ρ g w ∈ W) :
-    ∃ u : V, u ≠ 0 ∧
-      ((∀ g : Equiv.Perm Ω, (∀ a ∈ A, g a = a) → ρ g u = u) ∨
-       (∀ g : Equiv.Perm Ω, (∀ a ∈ A, g a = a) → ρ g u = signScalar g • u)) := by
-  classical
-  let S : Subrepresentation (ρ.comp (fixSet A).subtype) :=
-    { toSubmodule := W
-      apply_mem_toSubmodule := fun g w hw => hstable g.val g.property w hw }
-  let ρ' := S.toRepresentation.comp (fixSetHom A)
-  have : Nontrivial W := (Module.finrank_pos_iff (R := ℂ) (M := W)).mp (by omega)
-  obtain ⟨u, hu⟩ := exists_ne (0 : W)
-  refine ⟨u.val, fun he => hu (Subtype.ext he), ?_⟩
-  rcases oneDimensional_trivial_or_sign ρ' hW with ht | hs
-  · left
-    intro g hg
-    obtain ⟨σ, hσ⟩ := fixSetHom_surjective A ⟨g, hg⟩
-    have he := congrArg (fun T : Module.End ℂ W => (T u).val) (ht σ)
-    change ρ ((fixSetHom A σ).val) u.val = u.val at he
-    simpa only [hσ] using he
-  · right
-    intro g hg
-    obtain ⟨σ, hσ⟩ := fixSetHom_surjective A ⟨g, hg⟩
-    have he := congrArg (fun T : Module.End ℂ W => (T u).val) (hs σ)
-    change ρ ((fixSetHom A σ).val) u.val = signScalar σ • u.val at he
-    simp only [signScalar_apply] at he ⊢
-    rw [← Equiv.Perm.sign_ofSubtype σ] at he
-    simpa only [← fixSetHom_val, hσ] using he
-
-/- Minimal signed placement level with both dimension bounds. The last
-clause rules out every earlier line, hence in particular every earlier fixed
-vector of either the representation or its sign twist. -/
-theorem exists_minimal_sparse_line
-    (ρ : Representation ℂ (Equiv.Perm Ω) V) [ρ.IsIrreducible] :
-    ∃ (k t : ℕ) (A : Finset Ω) (u : V),
-      A.card = k ∧ k ≤ 2*t ∧ 2^t ≤ Module.finrank ℂ V ∧
-      Module.finrank ℂ V ≤ (Fintype.card Ω)^k ∧ u ≠ 0 ∧
-      ((∀ g : Equiv.Perm Ω, (∀ a ∈ A, g a = a) → ρ g u = u) ∨
-       (∀ g : Equiv.Perm Ω, (∀ a ∈ A, g a = a) → ρ g u = signScalar g • u)) ∧
-      ∀ B : Finset Ω, B.card < k → ∀ (v : V),
-        (∀ g : Equiv.Perm Ω, (∀ a ∈ B, g a = a) → ∃ c : ℂ, ρ g v = c • v) → v = 0 := by
-  classical
-  let P (k : ℕ) : Prop := ∃ (A : Finset Ω) (W : Submodule ℂ V), A.card = k ∧
-    Module.finrank ℂ W = 1 ∧
-    ∀ g : Equiv.Perm Ω, (∀ a ∈ A, g a = a) → ∀ w ∈ W, ρ g w ∈ W
-  obtain ⟨t, A₀, W₀, hW₀, hA₀, ht, hs₀⟩ := exists_sparse_line ρ
-  have hex : ∃ k, P k := ⟨A₀.card, A₀, W₀, rfl, hW₀, hs₀⟩
-  obtain ⟨A, W, hA, hW, hs⟩ := Nat.find_spec hex
-  have hk : Nat.find hex ≤ A₀.card := Nat.find_min' hex ⟨A₀, W₀, rfl, hW₀, hs₀⟩
-  obtain ⟨u, hu, hchar⟩ := stable_line_trivial_or_sign ρ A W hW hs
-  refine ⟨Nat.find hex, t, A, u, hA, hk.trans hA₀, ht, ?_, hu, hchar, ?_⟩
-  · rw [← hA]
-    exact dimension_le_pow_of_stable_line ρ A W hW hs
-  · intro B hB v hv
-    by_contra hn
-    have hp : P B.card := by
-      refine ⟨B, ℂ ∙ v, rfl, finrank_span_singleton hn, ?_⟩
-      intro g hg w hw
-      obtain ⟨c, rfl⟩ := Submodule.mem_span_singleton.mp hw
-      obtain ⟨r, hr⟩ := hv g hg
-      rw [map_smul, hr, smul_smul]
-      exact Submodule.smul_mem _ _ (Submodule.mem_span_singleton_self v)
-    exact Nat.find_min hex hB hp
-end SparseDimension
-end CoordinateSweeps
-
-noncomputable section
-open scoped BigOperators
-
-namespace CoordinateSweeps.PlacementOccurrence
-variable {Γ X V : Type*} [Group Γ] [Fintype Γ] [Fintype X] [DecidableEq X] [MulAction Γ X]
-    [AddCommGroup V] [Module ℂ V]
-
-/- Sum of orbit vectors against placement coordinates. The repeated
-stabilizer fibers are left in the sum, so no transversal choice is needed. -/
-def orbitSynthesis (ρ : Representation ℂ Γ V) (u : V) (x₀ : X) : (X → ℂ) →ₗ[ℂ] V where
-  toFun f := ∑ g, f (g • x₀) • ρ g u
-  map_add' f h := by simp only [Pi.add_apply, add_smul, Finset.sum_add_distrib]
-  map_smul' c f := by simp only [Pi.smul_apply, smul_eq_mul, mul_smul, Finset.smul_sum, RingHom.id_apply]
-
-/- Exact covariance, with the source's convention that gh applies h first. -/
-omit [Fintype X] [DecidableEq X] in
-theorem orbitSynthesis_covariant (ρ : Representation ℂ Γ V) (u : V) (x₀ : X)
-    (h : Γ) (f : X → ℂ) :
-    orbitSynthesis ρ u x₀ (fun x => f (h⁻¹ • x)) = ρ h (orbitSynthesis ρ u x₀ f) := by
-  change (∑ g, f (h⁻¹ • (g • x₀)) • ρ g u) = ρ h (∑ g, f (g • x₀) • ρ g u)
-  rw [← Equiv.sum_comp (Equiv.mulLeft h) (fun g => f (h⁻¹ • (g • x₀)) • ρ g u)]
-  simp only [Equiv.coe_mulLeft, mul_smul, inv_smul_smul, map_sum, map_smul,
-    map_mul, Module.End.mul_apply]
-
-/- A vector fixed by the point stabilizer produces a nonzero synthesis map. -/
-omit [Fintype X] in
-theorem orbitSynthesis_delta_ne_zero (ρ : Representation ℂ Γ V) (u : V) (hu : u ≠ 0)
-    (x₀ : X) (hfixed : ∀ g : Γ, g • x₀ = x₀ → ρ g u = u) :
-    orbitSynthesis ρ u x₀ (fun x => if x = x₀ then 1 else 0) ≠ 0 := by
-  classical
-  have he : orbitSynthesis ρ u x₀ (fun x => if x = x₀ then 1 else 0) =
-      ((Finset.univ.filter (fun g : Γ => g • x₀ = x₀)).card : ℂ) • u := by
-    change (∑ g, (if g • x₀ = x₀ then (1 : ℂ) else 0) • ρ g u) = _
-    simp only [ite_smul, one_smul, zero_smul]
-    calc
-      _ = ∑ g : Γ, if g • x₀ = x₀ then u else 0 := by
-        apply Finset.sum_congr rfl
-        intro g _
-        split_ifs with hg
-        · exact hfixed g hg
-        · rfl
-      _ = _ := by rw [← Finset.sum_filter]; simp [Nat.cast_smul_eq_nsmul]
-  rw [he]
-  apply smul_ne_zero _ hu
-  norm_cast
-  apply Nat.ne_of_gt
-  apply Finset.card_pos.mpr
-  exact ⟨1, by simp⟩
-
-/- Frobenius occurrence at the precise stabilizer level, proved by an
-explicit intertwiner rather than postulated decomposition. -/
-omit [Fintype X] in
-theorem orbitSynthesis_surjective (ρ : Representation ℂ Γ V) [ρ.IsIrreducible]
-    (u : V) (hu : u ≠ 0) (x₀ : X)
-    (hfixed : ∀ g : Γ, g • x₀ = x₀ → ρ g u = u) :
-    Function.Surjective (orbitSynthesis ρ u x₀) := by
-  classical
-  let S : Subrepresentation ρ :=
-    { toSubmodule := (orbitSynthesis ρ u x₀).range
-      apply_mem_toSubmodule := by
-        intro g v hv
-        obtain ⟨f, rfl⟩ := hv
-        exact ⟨(fun x => f (g⁻¹ • x)), orbitSynthesis_covariant ρ u x₀ g f⟩ }
-  have hne : S ≠ ⊥ := by
-    intro he
-    have hm : orbitSynthesis ρ u x₀ (fun x => if x = x₀ then 1 else 0) ∈ S :=
-      ⟨_, rfl⟩
-    rw [he] at hm
-    exact orbitSynthesis_delta_ne_zero ρ u hu x₀ hfixed hm
-  have he : S = ⊤ := (eq_bot_or_eq_top S).resolve_left hne
-  apply LinearMap.range_eq_top.mp
-  exact congrArg Subrepresentation.toSubmodule he
-
-/- The subgroup average is fixed under that subgroup, with no assumptions on
-interrelations between different coordinate subgroups. -/
-omit [Fintype Γ] in
-theorem subgroup_sum_fixed (ρ : Representation ℂ Γ V) (K : Subgroup Γ) [Fintype K] (u : V)
-    (k : K) : ρ k.val (∑ h : K, ρ h.val u) = ∑ h : K, ρ h.val u := by
-  simp only [map_sum, ← Module.End.mul_apply, ← map_mul]
-  exact Equiv.sum_comp (Equiv.mulLeft k) (fun h : K => ρ h.val u)
-
-/- Averaging a synthesis over a larger stabilizer acts only on the cyclic
-vector whenever the coefficient function forgets those coordinates. -/
-omit [Fintype X] [DecidableEq X] in
-theorem orbitSynthesis_subgroup_average (ρ : Representation ℂ Γ V) (u : V) (x₀ : X)
-    (K : Subgroup Γ) [Fintype K] (f : X → ℂ)
-    (hf : ∀ (g : Γ) (k : K), f ((g*k.val) • x₀) = f (g • x₀)) :
-    orbitSynthesis ρ (∑ k : K, ρ k.val u) x₀ f =
-      (Fintype.card K : ℂ) • orbitSynthesis ρ u x₀ f := by
-  change (∑ g, f (g • x₀) • ρ g (∑ k : K, ρ k.val u)) = _
-  simp only [map_sum, Finset.smul_sum, ← Module.End.mul_apply, ← map_mul]
-  rw [Finset.sum_comm]
-  have hk (k : K) : (∑ g : Γ, f (g • x₀) • ρ (g*k.val) u) = orbitSynthesis ρ u x₀ f := by
-    calc
-      _ = ∑ g : Γ, f ((g*k.val) • x₀) • ρ (g*k.val) u := by
-        simp only [hf]
-      _ = _ := Equiv.sum_comp (Equiv.mulRight k.val) (fun g => f (g • x₀) • ρ g u)
-  simp only [hk, Finset.sum_const, Finset.card_univ, Nat.cast_smul_eq_nsmul]
-
-/- Exact proper-coordinate annihilation used in source04:eq7: it is the
-absence of fixed vectors one level earlier, not a partial cancellation claim. -/
-omit [Fintype X] [DecidableEq X] in
-theorem orbitSynthesis_eq_zero_of_no_fixed (ρ : Representation ℂ Γ V) (u : V) (x₀ : X)
-    (K : Subgroup Γ) [Fintype K]
-    (hnof : ∀ v : V, (∀ k : K, ρ k.val v = v) → v = 0)
-    (f : X → ℂ) (hf : ∀ (g : Γ) (k : K), f ((g*k.val) • x₀) = f (g • x₀)) :
-    orbitSynthesis ρ u x₀ f = 0 := by
-  have hzero : (∑ k : K, ρ k.val u) = 0 := hnof _ (subgroup_sum_fixed ρ K u)
-  have h := orbitSynthesis_subgroup_average ρ u x₀ K f hf
-  rw [hzero] at h
-  have hz : orbitSynthesis ρ (0 : V) x₀ f = 0 := by simp [orbitSynthesis]
-  rw [hz] at h
-  apply (smul_eq_zero.mp h.symm).resolve_left
-  exact_mod_cast Fintype.card_ne_zero
-end CoordinateSweeps.PlacementOccurrence
-
-namespace CoordinateSweeps.SparseDimension
-variable {Ω V : Type*} [Fintype Ω] [DecidableEq Ω]
-  [AddCommGroup V] [Module ℂ V]
-
-@[simp] theorem signScalar_mul_self (g : Equiv.Perm Ω) :
-    signScalar g * signScalar g = (1 : ℂ) := by
-  simp only [signScalar_apply]
-  rcases Int.units_eq_one_or (Equiv.Perm.sign g) with h | h <;> rw [h] <;> norm_num
-
-/- Tensoring with the ordinary sign character, on exactly the same space. -/
-def signTwist (ρ : Representation ℂ (Equiv.Perm Ω) V) :
-    Representation ℂ (Equiv.Perm Ω) V where
-  toFun g := signScalar g • ρ g
-  map_one' := by simp
-  map_mul' g h := by simp only [map_mul, smul_mul_smul_comm]
-
-@[simp] theorem signTwist_apply (ρ : Representation ℂ (Equiv.Perm Ω) V)
-    (g : Equiv.Perm Ω) (v : V) : signTwist ρ g v = signScalar g • ρ g v := rfl
-
-/- The same invariant subspaces for ρ and its sign twist. -/
-def untwistSubrep (ρ : Representation ℂ (Equiv.Perm Ω) V)
-    (S : Subrepresentation (signTwist ρ)) : Subrepresentation ρ where
-  toSubmodule := S.toSubmodule
-  apply_mem_toSubmodule := by
-    intro g v hv
-    have h := S.toSubmodule.smul_mem (signScalar g) (S.apply_mem_toSubmodule g hv)
-    simpa only [signTwist_apply, smul_smul, signScalar_mul_self, one_smul] using h
-
-instance signTwist_irreducible [FiniteDimensional ℂ V] (ρ : Representation ℂ (Equiv.Perm Ω) V)
-    [ρ.IsIrreducible] : (signTwist ρ).IsIrreducible := by
-  have : Nontrivial V := irreducible_nontrivial ρ
-  refine { exists_pair_ne := ⟨⊥, ⊤, ?_⟩, eq_bot_or_eq_top := ?_ }
-  · intro he
-    exact bot_ne_top (congrArg Subrepresentation.toSubmodule he)
-  · intro S
-    rcases eq_bot_or_eq_top (untwistSubrep ρ S) with h | h
-    · left
-      have hh := congrArg Subrepresentation.toSubmodule h
-      exact Subrepresentation.toSubmodule_injective hh
-    · right
-      have hh := congrArg Subrepresentation.toSubmodule h
-      exact Subrepresentation.toSubmodule_injective hh
-
-/- Either the original representation or its sign twist has a first placement
-level k with both quantitative dimension bounds and no earlier fixed vector. -/
-theorem exists_minimal_fixed_vector [FiniteDimensional ℂ V]
-    (ρ : Representation ℂ (Equiv.Perm Ω) V) [ρ.IsIrreducible] :
-    ∃ (ε : Bool) (k t : ℕ) (A : Finset Ω) (u : V),
-      let τ := if ε then signTwist ρ else ρ
-      A.card = k ∧ k ≤ 2*t ∧ 2^t ≤ Module.finrank ℂ V ∧
-      Module.finrank ℂ V ≤ (Fintype.card Ω)^k ∧ u ≠ 0 ∧
-      (∀ g : Equiv.Perm Ω, (∀ a ∈ A, g a = a) → τ g u = u) ∧
-      ∀ B : Finset Ω, B.card < k → ∀ v : V,
-        (∀ g : Equiv.Perm Ω, (∀ a ∈ B, g a = a) → τ g v = v) → v = 0 := by
-  obtain ⟨k,t,A,u,hA,hk,ht,hd,hu,hchar,hmin⟩ := exists_minimal_sparse_line ρ
-  rcases hchar with htriv | hsign
-  · refine ⟨false,k,t,A,u,hA,hk,ht,hd,hu,htriv, ?_⟩
-    intro B hB v hv
-    exact hmin B hB v (fun g hg => ⟨1, by simpa using hv g hg⟩)
-  · refine ⟨true,k,t,A,u,hA,hk,ht,hd,hu, ?_, ?_⟩
-    · intro g hg
-      simp only [↓reduceIte, signTwist_apply, hsign g hg, smul_smul,
-        signScalar_mul_self, one_smul]
-    · intro B hB v hv
-      apply hmin B hB v
-      intro g hg
-      refine ⟨signScalar g, ?_⟩
-      have h := congrArg (fun w : V => signScalar g • w) (hv g hg)
-      simpa only [Bool.true_eq, ↓reduceIte, signTwist_apply, smul_smul,
-        signScalar_mul_self, one_smul] using h
-end CoordinateSweeps.SparseDimension
-
-namespace CoordinateSweeps.PlacementOccurrence
-open SparseDimension
-variable {Ω I V : Type*} [Fintype Ω] [DecidableEq Ω]
-  [Fintype I] [AddCommGroup V] [Module ℂ V]
-
-/- Stabilizer of the small placement coordinates really is the pointwise
-stabilizer of their image, so the exact earlier-level hypothesis applies. -/
-theorem synthesis_kills_proper_coordinates
-    (ρ : Representation ℂ (Equiv.Perm Ω) V) (u : V) (x₀ : I ↪ Ω)
-    (hnof : ∀ B : Finset Ω, B.card < Fintype.card I → ∀ v : V,
-      (∀ g : Equiv.Perm Ω, (∀ a ∈ B, g a = a) → ρ g v = v) → v = 0)
-    (A : Finset I) (hA : A ≠ Finset.univ) (f : (I ↪ Ω) → ℂ)
-    (hf : ∀ x y : I ↪ Ω, (∀ i ∈ A, x i = y i) → f x = f y) :
-    orbitSynthesis ρ u x₀ f = 0 := by
-  classical
-  let B : Finset Ω := A.image x₀
-  have hB : B.card < Fintype.card I := by
-    dsimp [B]
-    rw [Finset.card_image_of_injective _ x₀.injective]
-    exact Finset.card_lt_card (Finset.ssubset_univ_iff.mpr hA)
-  apply orbitSynthesis_eq_zero_of_no_fixed ρ u x₀ (fixSet B)
-  · intro v hv
-    exact hnof B hB v (fun g hg => hv ⟨g,hg⟩)
-  · intro g k
-    apply hf
-    intro i hi
-    change g (k.val (x₀ i)) = g (x₀ i)
-    rw [k.property (x₀ i) (Finset.mem_image_of_mem _ hi)]
-end CoordinateSweeps.PlacementOccurrence
-
-namespace CoordinateSweeps.Placement
-variable {I U X Y : Type*} [Fintype I] [DecidableEq I] [Fintype X] [Fintype Y]
-/- The adjoint of a synthesis map annihilating all cylinder functions lands
-in the exact alternating-kernel top space. Complex conjugation is essential. -/
-omit [DecidableEq I] in
-theorem conjTranspose_mulVec_mem_top {D : Type*} [Fintype D]
-    (coords : Y → I → U) (B : Matrix D Y ℂ)
-    (hB : ∀ A : Finset I, A ≠ Finset.univ → ∀ f : Y → ℂ,
-      DependsOn coords A f → B.mulVec f = 0) (w : D → ℂ) :
-    B.conjTranspose.mulVec w ∈ topSpace coords := by
-  intro A hA f hf
-  have hfstar : DependsOn coords A (fun y => star (f y)) :=
-    fun y y' h => congrArg star (hf y y' h)
-  have hz := hB A hA (fun y => star (f y)) hfstar
-  have hsum (d : D) : ∑ y, f y * star (B d y) = 0 := by
-    have h := congrArg star (congrFun hz d)
-    simpa only [Matrix.mulVec, dotProduct, star_sum, star_mul,
-      star_star, Pi.zero_apply, star_zero, mul_comm] using h
-  simp only [Matrix.mulVec, dotProduct, Matrix.conjTranspose_apply, Finset.mul_sum]
-  rw [Finset.sum_comm]
-  simp_rw [← mul_assoc, ← Finset.sum_mul, hsum, zero_mul]
-  exact Finset.sum_const_zero
-
-/- Matrix form of top-level compression, ready for Schur norm transfer. -/
-omit [DecidableEq I] [Fintype X] in
-theorem alternatingKernel_mul_conjTranspose {D : Type*} [Fintype D]
-    (coords : Y → I → U) (s : ℝ) (hs : s ≠ 0)
-    (p : Finset I → X → Y → ℝ)
-    (hp : ∀ A x, DependsOn coords A (fun y => (p A x y : ℂ)))
-    (B : Matrix D Y ℂ)
-    (hB : ∀ A : Finset I, A ≠ Finset.univ → ∀ f : Y → ℂ,
-      DependsOn coords A f → B.mulVec f = 0) :
-    alternatingKernel s p * B.conjTranspose =
-      (show Matrix X Y ℂ from fun x y => (p Finset.univ x y : ℂ)) * B.conjTranspose := by
-  apply Matrix.ext
-  intro x d
-  have hv : (fun y => B.conjTranspose y d) ∈ topSpace coords := by
-    classical
-    have h := conjTranspose_mulVec_mem_top coords B hB (Pi.single d 1)
-    have he : B.conjTranspose.mulVec (Pi.single d 1) = (fun y => B.conjTranspose y d) := by
-      ext y; simp [Matrix.mulVec, dotProduct, Pi.single_apply]
-    rw [he] at h
-    exact h
-  exact congrFun (alternatingKernel_mulVec_top coords s hs p hp hv) x
-end CoordinateSweeps.Placement
-
-namespace CoordinateSweeps.PlacementOccurrence
-open scoped Matrix.Norms.L2Operator
-variable {Γ X : Type*} [Group Γ] [Fintype Γ] [Fintype X] [DecidableEq X] [MulAction Γ X]
-
-def actionMatrix : Γ →* Matrix X X ℂ :=
-  Matrix.permMatrixHom.comp (MulAction.toPermHom Γ X)
-
-omit [Fintype Γ] in
-lemma actionMatrix_mulVec (g : Γ) (f : X → ℂ) :
-    (actionMatrix g).mulVec f=fun x => f (g⁻¹ • x) := by
-  change (((MulAction.toPermHom Γ X g)⁻¹).permMatrix ℂ).mulVec f = _
-  rw [Matrix.permMatrix_mulVec]
-  rfl
-
-omit [Fintype Γ] in
-lemma actionMatrix_unitary (g : Γ) : (actionMatrix (X := X) g).conjTranspose*actionMatrix g=1 := by
-  change (((MulAction.toPermHom Γ X g)⁻¹).permMatrix ℂ).conjTranspose*
-    ((MulAction.toPermHom Γ X g)⁻¹).permMatrix ℂ=1
-  rw [Matrix.conjTranspose_permMatrix,inv_inv,← Matrix.permMatrix_mul,inv_mul_cancel,Matrix.permMatrix_one]
-
-def synthesisMatrix (ρ : UnitaryIrrep Γ) (u : Fin ρ.dimension → ℂ) (x₀ : X) :
-    Matrix (Fin ρ.dimension) X ℂ := LinearMap.toMatrix' (orbitSynthesis ρ.asRepresentation u x₀)
-
-lemma synthesisMatrix_mulVec (ρ : UnitaryIrrep Γ) (u : Fin ρ.dimension → ℂ) (x₀ : X) (f : X → ℂ) :
-    (synthesisMatrix ρ u x₀).mulVec f=orbitSynthesis ρ.asRepresentation u x₀ f :=
-  LinearMap.toMatrix'_mulVec _ _
-
-lemma synthesisMatrix_covariant (ρ : UnitaryIrrep Γ) (u : Fin ρ.dimension → ℂ) (x₀ : X) (g : Γ) :
-    synthesisMatrix ρ u x₀*actionMatrix g=ρ.matrix g*synthesisMatrix ρ u x₀ := by
-  change (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul (synthesisMatrix ρ u x₀) (actionMatrix g))=
-    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul (ρ.matrix g) (synthesisMatrix ρ u x₀))
-  apply Matrix.mulVec_injective
-  funext f
-  rw [← Matrix.mulVec_mulVec,← Matrix.mulVec_mulVec,actionMatrix_mulVec,
-    synthesisMatrix_mulVec,synthesisMatrix_mulVec]
-  exact orbitSynthesis_covariant ρ.asRepresentation u x₀ g f
-
-lemma synthesisMatrix_ne_zero (ρ : UnitaryIrrep Γ) (u : Fin ρ.dimension → ℂ) (hu : u ≠ 0) (x₀ : X)
-    (hfixed : ∀ g : Γ, g • x₀=x₀ → ρ.asRepresentation g u=u) : synthesisMatrix ρ u x₀ ≠ 0 := by
-  intro h
-  have hh := orbitSynthesis_delta_ne_zero ρ.asRepresentation u hu x₀ hfixed
-  apply hh
-  rw [← synthesisMatrix_mulVec,h,Matrix.zero_mulVec]
-
-end CoordinateSweeps.PlacementOccurrence
-
-namespace CoordinateSweeps.PlacementOccurrence
-open scoped Matrix.Norms.L2Operator
-open Placement
-variable {I Ω : Type*} [Fintype I] [DecidableEq I] [Fintype Ω] [DecidableEq Ω]
-
-/- Source04:eq7 at its exact first placement level, with a constructed
-nonzero synthesis intertwiner and Schur norm transfer. -/
-omit [DecidableEq I] in
-theorem first_level_norm_le (ρ : UnitaryIrrep (Equiv.Perm Ω))
-    (u : Fin ρ.dimension → ℂ) (hu : u ≠ 0) (x₀ : I ↪ Ω)
-    (hfixed : ∀ g : Equiv.Perm Ω, (∀ i, g (x₀ i)=x₀ i) → ρ.asRepresentation g u=u)
-    (hnof : ∀ A : Finset Ω, A.card < Fintype.card I → ∀ v : Fin ρ.dimension → ℂ,
-      (∀ g : Equiv.Perm Ω, (∀ a ∈ A, g a=a) → ρ.asRepresentation g v=v) → v=0)
-    {W : Type*} [Fintype W] (w : W → ℂ) (g : W → Equiv.Perm Ω)
-    (s : ℝ) (hs : s ≠ 0) (p : Finset I → (I ↪ Ω) → (I ↪ Ω) → ℝ)
-    (hp : ∀ A x, DependsOn (fun y : I ↪ Ω => y) A (fun y => (p A x y : ℂ)))
-    (hfull : (show Matrix (I ↪ Ω) (I ↪ Ω) ℂ from fun x y => (p Finset.univ x y : ℂ))=
-      ∑ a : W, w a • actionMatrix (X := I ↪ Ω) (g a)) :
-    ‖∑ a : W, w a • ρ.matrix (g a)‖ ≤
-      ‖alternatingKernel s p‖ := by
-  let B : Matrix (Fin ρ.dimension) (I ↪ Ω) ℂ := synthesisMatrix ρ u x₀
-  have hB : ∀ A : Finset I, A ≠ Finset.univ → ∀ f : (I ↪ Ω) → ℂ,
-      DependsOn (fun y : I ↪ Ω => y) A f → B.mulVec f=0 := by
-    intro A hA f hf
-    rw [synthesisMatrix_mulVec]
-    exact synthesis_kills_proper_coordinates ρ.asRepresentation u x₀ hnof A hA f hf
-  have hne : B ≠ 0 := synthesisMatrix_ne_zero ρ u hu x₀ (by
-    intro σ hσ
-    exact hfixed σ (fun i => congrArg (fun x : I ↪ Ω => x i) hσ))
-  have hnorm : ‖∑ a : W, w a • ρ.matrix (g a)‖ ≤ ‖alternatingKernel s p‖ := by
-    apply ρ.norm_le_compression (actionMatrix (X := I ↪ Ω)) actionMatrix_unitary B hne
-      (synthesisMatrix_covariant ρ u x₀)
-    rw [alternatingKernel_mul_conjTranspose (fun y : I ↪ Ω => y) s hs p hp B hB,hfull,
-      Matrix.sum_mul,Matrix.mul_sum]
-    apply Finset.sum_congr rfl
-    intro a _
-    rw [Matrix.smul_mul,Matrix.mul_smul]
-    congr 1
-    exact ρ.adjoint_intertwines _ actionMatrix_unitary B (synthesisMatrix_covariant ρ u x₀) (g a)
-  exact hnorm
-
-end CoordinateSweeps.PlacementOccurrence
-
-open scoped Matrix.Norms.L2Operator
-
-namespace CoordinateSweeps.PlacementOccurrence
-variable {Γ X : Type*} [Group Γ] [Fintype X] [DecidableEq X] [MulAction Γ X]
-lemma actionMatrix_entry (g : Γ) (x y : X) : actionMatrix (X := X) g x y=if g⁻¹ • x=y then 1 else 0 := by
-  change (((MulAction.toPermHom Γ X g)⁻¹).permMatrix ℂ) x y=_
-  simp [Equiv.Perm.permMatrix, PEquiv.toMatrix]
-end CoordinateSweeps.PlacementOccurrence
-
-end
-end
-end
-end
-end
-end
-end
-end
-open scoped Matrix.Norms.L2Operator
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.Placement.Chart
+import OAI.RepresentationTheory.Young.Classification
+import OAI.Analysis.Matrix.TensorMoment
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators Matrix.Norms.L2Operator ComplexOrder
+attribute [local instance] Classical.propDecidable
+noncomputable section
+open scoped BigOperators
+attribute [local instance] Classical.propDecidable
+noncomputable section
+open scoped BigOperators
+open MvPolynomial
+noncomputable section
+open MeasureTheory ProbabilityTheory Real Set Filter
+open scoped ENNReal NNReal BigOperators
+noncomputable section
+open scoped BigOperators
+noncomputable section
+open scoped BigOperators Matrix.Norms.L2Operator
+attribute [local instance] Classical.propDecidable
+
+namespace CoordinateSweeps.Placement
+variable {I U X Y : Type*} [Fintype I] [DecidableEq I] [Fintype X] [Fintype Y]
+
+/- Dependence only on the listed coordinate values of a placement. -/
+def DependsOn (coords : Y → I → U) (A : Finset I) (f : Y → ℂ) : Prop :=
+  ∀ y y', (∀ i ∈ A, coords y i = coords y' i) → f y = f y'
+
+/- Counting-measure top-level space. Equivalently the orthogonal complement
+of the span of proper-coordinate cylinder functions (conjugation preserves that span). -/
+def topSpace (coords : Y → I → U) : Submodule ℂ (Y → ℂ) where
+  carrier := {v | ∀ A : Finset I, A ≠ Finset.univ → ∀ f : Y → ℂ,
+    DependsOn coords A f → ∑ y, f y*v y = 0}
+  zero_mem' := by simp
+  add_mem' := by
+    intro v w hv hw A hA f hf
+    simp only [Pi.add_apply, mul_add, Finset.sum_add_distrib, hv A hA f hf, hw A hA f hf, zero_add]
+  smul_mem' := by
+    intro c v hv A hA f hf
+    simp only [Pi.smul_apply, smul_eq_mul]
+    simp_rw [← mul_assoc, mul_comm (f _) c, mul_assoc]
+    rw [← Finset.mul_sum, hv A hA f hf, mul_zero]
+
+/- The source's s-normalized alternating kernel, over all coordinate subsets. -/
+def alternatingKernel (s : ℝ) (p : Finset I → X → Y → ℝ) : Matrix X Y ℂ :=
+  fun x y => (s⁻¹^(Fintype.card I) * ∑ A ∈ (Finset.univ : Finset I).powerset,
+    (-1 : ℝ)^(Fintype.card I-A.card) * s^A.card * p A x y : ℝ)
+
+/- All proper-subset marginal terms vanish on the top level. This is the
+nontrivial algebraic bridge between Q and the actual transition operator. -/
+omit [DecidableEq I] [Fintype X] in
+theorem alternatingKernel_mulVec_top (coords : Y → I → U)
+    (s : ℝ) (hs : s ≠ 0) (p : Finset I → X → Y → ℝ)
+    (hp : ∀ A x, DependsOn coords A (fun y => (p A x y : ℂ)))
+    {v : Y → ℂ} (hv : v ∈ topSpace coords) :
+    (alternatingKernel s p).mulVec v = Matrix.mulVec (fun x y => (p Finset.univ x y : ℂ)) v := by
+  funext x
+  simp only [Matrix.mulVec, dotProduct, alternatingKernel, Complex.ofReal_mul,
+    Complex.ofReal_pow, Complex.ofReal_inv, Complex.ofReal_sum]
+  simp only [mul_assoc, Finset.sum_mul]
+  rw [← Finset.mul_sum, Finset.sum_comm]
+  simp only [← Finset.mul_sum]
+  rw [Finset.sum_eq_single (Finset.univ : Finset I)]
+  · simp only [Finset.card_univ, Nat.sub_self, pow_zero, one_mul]
+    rw [← mul_assoc, ← mul_pow, inv_mul_cancel₀ (Complex.ofReal_ne_zero.mpr hs), one_pow, one_mul]
+  · intro A hA hne
+    have hzero := hv A hne (fun y => (p A x y : ℂ)) (hp A x)
+    simp only [hzero, mul_zero]
+  · simp
+
+/- Counting HS norm always dominates the L2 operator norm, rectangular and
+empty dimensions included. No entrywise max norm is used. -/
+theorem matrix_l2_norm_le_hs (A : Matrix X Y ℂ) :
+    ‖A‖ ≤ Real.sqrt (∑ x, ∑ y, ‖A x y‖^2) := by
+  let L : EuclideanSpace ℂ Y →L[ℂ] EuclideanSpace ℂ X :=
+    LinearMap.toContinuousLinearMap (Matrix.toEuclideanLin A)
+  change ‖L‖ ≤ _
+  refine ContinuousLinearMap.opNorm_le_bound L (by positivity) ?_
+  intro v
+  have hrow (x : X) : ‖(A.mulVec v) x‖^2 ≤
+      (∑ y, ‖A x y‖^2)*(∑ y, ‖v y‖^2) := by
+    have hnorm : ‖(A.mulVec v) x‖ ≤ ∑ y, ‖A x y‖*‖v y‖ := by
+      apply (norm_sum_le _ _).trans
+      exact Finset.sum_le_sum (fun y _ => le_of_eq (norm_mul _ _))
+    have hc := Finset.sum_mul_sq_le_sq_mul_sq Finset.univ (fun y => ‖A x y‖) (fun y => ‖v y‖)
+    exact ((sq_le_sq₀ (norm_nonneg _) (Finset.sum_nonneg (fun _ _ => by positivity))).mpr hnorm).trans hc
+  have hall : (∑ x, ‖(A.mulVec v) x‖^2) ≤
+      (∑ x, ∑ y, ‖A x y‖^2)*(∑ y, ‖v y‖^2) := by
+    rw [Finset.sum_mul]
+    exact Finset.sum_le_sum (fun x _ => hrow x)
+  change ‖Matrix.toEuclideanLin A v‖ ≤ _
+  apply (sq_le_sq₀ (norm_nonneg _) (mul_nonneg (Real.sqrt_nonneg _) (norm_nonneg _))).mp
+  rw [mul_pow, Real.sq_sqrt (by positivity), PiLp.norm_sq_eq_of_L2, PiLp.norm_sq_eq_of_L2]
+  exact hall
+
+/- The actual full marginal has the alternating HS bound on the entire top
+placement space. This is used before choosing an irreducible constituent. -/
+omit [DecidableEq I] in
+theorem fullKernel_top_norm_le (coords : Y → I → U)
+    (s : ℝ) (hs : s ≠ 0) (p : Finset I → X → Y → ℝ)
+    (hp : ∀ A x, DependsOn coords A (fun y => (p A x y : ℂ)))
+    (v : EuclideanSpace ℂ Y) (hv : (fun y => v y) ∈ topSpace coords) :
+    ‖(EuclideanSpace.equiv X ℂ).symm
+      (Matrix.mulVec (fun x y => (p Finset.univ x y : ℂ)) v)‖ ≤
+      Real.sqrt (∑ x, ∑ y, ‖alternatingKernel s p x y‖^2)*‖v‖ := by
+  rw [← alternatingKernel_mulVec_top coords s hs p hp hv]
+  exact ((alternatingKernel s p).l2_opNorm_mulVec v).trans
+    (mul_le_mul_of_nonneg_right (matrix_l2_norm_le_hs _) (norm_nonneg _))
+end CoordinateSweeps.Placement
+namespace CoordinateSweeps
+namespace UnitaryIrrep
+variable {Γ : Type*} [Group Γ] (ρ : UnitaryIrrep Γ)
+
+theorem adjoint_intertwines {X : Type*} [Fintype X] [DecidableEq X]
+    (P : Γ →* Matrix X X ℂ) (hP : ∀ g, (P g).conjTranspose * P g = 1)
+    (B : Matrix (Fin ρ.dimension) X ℂ)
+    (hB : ∀ g, B * P g = ρ.matrix g * B) (g : Γ) :
+    P g * B.conjTranspose = B.conjTranspose * ρ.matrix g := by
+  have hPinv : P g⁻¹ = (P g).conjTranspose := by
+    calc
+      P g⁻¹ = 1 * P g⁻¹ := (one_mul _).symm
+      _ = ((P g).conjTranspose * P g) * P g⁻¹ := by rw [hP g]
+      _ = (P g).conjTranspose := by rw [mul_assoc, ← map_mul, mul_inv_cancel, map_one, mul_one]
+  have h := congrArg Matrix.conjTranspose (hB g⁻¹)
+  have hrinv : ρ.matrix g⁻¹ = (ρ.matrix g).conjTranspose := by
+    calc
+      ρ.matrix g⁻¹ = 1 * ρ.matrix g⁻¹ := (one_mul _).symm
+      _ = ((ρ.matrix g).conjTranspose * ρ.matrix g) * ρ.matrix g⁻¹ := by rw [ρ.unitary g]
+      _ = (ρ.matrix g).conjTranspose := by rw [mul_assoc, ← map_mul, mul_inv_cancel, map_one, mul_one]
+  simpa only [Matrix.conjTranspose_mul, hPinv, hrinv,
+    Matrix.conjTranspose_conjTranspose] using h
+
+/- Schur's lemma makes every nonzero intertwiner a scalar coisometry. -/
+theorem intertwiner_gram_scalar {X : Type*} [Fintype X] [DecidableEq X]
+    (P : Γ →* Matrix X X ℂ) (hP : ∀ g, (P g).conjTranspose * P g = 1)
+    (B : Matrix (Fin ρ.dimension) X ℂ)
+    (hB : ∀ g, B * P g = ρ.matrix g * B) :
+    ∃ c : ℂ, B * B.conjTranspose = c • (1 : Matrix _ _ ℂ) := by
+  apply ρ.commuting_scalar
+  intro g
+  rw [Matrix.mul_assoc, ← ρ.adjoint_intertwines P hP B hB g,
+    ← Matrix.mul_assoc, hB g, Matrix.mul_assoc]
+
+/- Exact norm transfer, with no extraneous dimension loss. Q need agree with
+an averaged permutation operator only on the adjoint intertwiner range. -/
+theorem norm_le_compression {X : Type*} [Fintype X] [DecidableEq X]
+    (P : Γ →* Matrix X X ℂ) (hP : ∀ g, (P g).conjTranspose * P g = 1)
+    (B : Matrix (Fin ρ.dimension) X ℂ) (hne : B ≠ 0)
+    (hB : ∀ g, B * P g = ρ.matrix g * B)
+    (K : Matrix (Fin ρ.dimension) (Fin ρ.dimension) ℂ) (Q : Matrix X X ℂ)
+    (hQ : Q * B.conjTranspose = B.conjTranspose * K) : ‖K‖ ≤ ‖Q‖ := by
+  let : NeZero ρ.dimension := ⟨ρ.positive.ne'⟩
+  obtain ⟨c, hc⟩ := ρ.intertwiner_gram_scalar P hP B hB
+  have hn : ‖B‖ * ‖B‖ = ‖c‖ := by
+    have h := Matrix.l2_opNorm_conjTranspose_mul_self B.conjTranspose
+    simpa only [Matrix.conjTranspose_conjTranspose, Matrix.l2_opNorm_conjTranspose,
+      hc, norm_smul, norm_one, mul_one] using h.symm
+  have hcpos : 0 < ‖c‖ := by rw [← hn]; exact mul_pos (norm_pos_iff.mpr hne) (norm_pos_iff.mpr hne)
+  have he : c • K = B * Q * B.conjTranspose := by
+    rw [Matrix.mul_assoc, hQ, ← Matrix.mul_assoc, hc, Matrix.smul_mul, one_mul]
+  have hbound : ‖c‖ * ‖K‖ ≤ ‖c‖ * ‖Q‖ := by
+    rw [← norm_smul, he]
+    calc
+      ‖B * Q * B.conjTranspose‖ ≤ ‖B * Q‖ * ‖B.conjTranspose‖ := Matrix.l2_opNorm_mul _ _
+      _ ≤ (‖B‖ * ‖Q‖) * ‖B.conjTranspose‖ :=
+        mul_le_mul_of_nonneg_right (Matrix.l2_opNorm_mul _ _) (norm_nonneg _)
+      _ = (‖B‖ * ‖B‖) * ‖Q‖ := by rw [Matrix.l2_opNorm_conjTranspose]; ring
+      _ = ‖c‖ * ‖Q‖ := by rw [hn]
+  exact (mul_le_mul_iff_right₀ hcpos).mp hbound
+end UnitaryIrrep
+end CoordinateSweeps
+
+noncomputable section
+universe u v
+open scoped ComplexConjugate
+
+namespace CoordinateSweeps
+/- The pointwise stabilizer of two sites, used in the sparse dimension bridge. -/
+def fixTwo {Ω : Type*} (a b : Ω) : Subgroup (Equiv.Perm Ω) where
+  carrier := {g | g a = a ∧ g b = b}
+  one_mem' := ⟨rfl, rfl⟩
+  mul_mem' hg hk := by simpa using And.intro (congrArg _ hk.1 |>.trans hg.1) (congrArg _ hk.2 |>.trans hg.2)
+  inv_mem' := by
+    intro g hg
+    constructor
+    · apply g.injective
+      simpa using hg.1.symm
+    · apply g.injective
+      simpa using hg.2.symm
+
+namespace SparseDimension
+variable {Ω V : Type*} [Fintype Ω] [DecidableEq Ω]
+  [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
+
+/- Generic Schur commutant lemma, with no classification assumptions. -/
+theorem commuting_scalar {Γ : Type*} [Group Γ]
+    (ρ : Representation ℂ Γ V) [ρ.IsIrreducible]
+    (A : Module.End ℂ V) (hA : ∀ g, A * ρ g = ρ g * A) :
+    ∃ c : ℂ, A = c • (1 : Module.End ℂ V) := by
+  let F : Representation.IntertwiningMap ρ ρ :=
+    { toLinearMap := A
+      isIntertwining' := hA }
+  obtain ⟨c, hc⟩ := Representation.IsIrreducible.algebraMap_intertwiningMap_bijective_of_isAlgClosed
+    (ρ := ρ) |>.2 F
+  refine ⟨c, ?_⟩
+  have hh := congrArg Representation.IntertwiningMap.toLinearMap hc
+  exact hh.symm
+
+/- If all operators are scalar, irreducibility forces one dimension. -/
+omit [FiniteDimensional ℂ V] in
+theorem finrank_eq_one_of_scalars {Γ : Type*} [Group Γ]
+    (ρ : Representation ℂ Γ V) [ρ.IsIrreducible]
+    (hs : ∀ g, ∃ c : ℂ, ρ g = c • (1 : Module.End ℂ V)) :
+    Module.finrank ℂ V = 1 := by
+  have : Nontrivial V := by
+    by_contra hn
+    have : Subsingleton V := not_nontrivial_iff_subsingleton.mp hn
+    have he : (⊥ : Subrepresentation ρ) = ⊤ := by
+      apply Subrepresentation.toSubmodule_injective
+      exact Subsingleton.elim _ _
+    exact bot_ne_top he
+  obtain ⟨v, hv⟩ := exists_ne (0 : V)
+  let S : Subrepresentation ρ :=
+    { toSubmodule := ℂ ∙ v
+      apply_mem_toSubmodule := by
+        intro g x hx
+        obtain ⟨c, hc⟩ := hs g
+        rw [hc]
+        exact Submodule.smul_mem _ c hx }
+  have hne : S ≠ ⊥ := by
+    intro he
+    have hm : v ∈ S := Submodule.mem_span_singleton_self _
+    rw [he] at hm
+    exact hv hm
+  have he : S = ⊤ := (eq_bot_or_eq_top S).resolve_left hne
+  have he' : (ℂ ∙ v : Submodule ℂ V) = ⊤ := congrArg Subrepresentation.toSubmodule he
+  rw [← finrank_top ℂ V, ← he']
+  exact finrank_span_singleton hv
+
+omit [FiniteDimensional ℂ V] in
+@[simp] theorem subrep_bot_toSubmodule {Γ : Type*} [Group Γ]
+    (ρ : Representation ℂ Γ V) : (⊥ : Subrepresentation ρ).toSubmodule = ⊥ := rfl
+omit [FiniteDimensional ℂ V] in
+@[simp] theorem subrep_top_toSubmodule {Γ : Type*} [Group Γ]
+    (ρ : Representation ℂ Γ V) : (⊤ : Subrepresentation ρ).toSubmodule = ⊤ := rfl
+
+/- Flatten a subrepresentation of a subrepresentation without changing its dimension. -/
+def nested {Γ : Type*} [Group Γ] {ρ : Representation ℂ Γ V}
+    (S : Subrepresentation ρ) (T : Subrepresentation S.toRepresentation) : Subrepresentation ρ where
+  toSubmodule := T.toSubmodule.map S.toSubmodule.subtype
+  apply_mem_toSubmodule := by
+    intro g v hv
+    obtain ⟨w, hw, rfl⟩ := hv
+    refine ⟨S.toRepresentation g w, T.apply_mem_toSubmodule g hw, rfl⟩
+
+/- Existence of a least-dimensional constituent, proved rather than assumed. -/
+theorem exists_minimal_constituent {Γ : Type*} [Group Γ]
+    (ρ : Representation ℂ Γ V) [Nontrivial V] :
+    ∃ S : Subrepresentation ρ, 0 < Module.finrank ℂ S.toSubmodule ∧
+      S.toRepresentation.IsIrreducible ∧
+      ∀ T : Subrepresentation ρ, T ≠ ⊥ →
+        Module.finrank ℂ S.toSubmodule ≤ Module.finrank ℂ T.toSubmodule := by
+  classical
+  have hp : ∃ n, ∃ S : Subrepresentation ρ,
+      0 < Module.finrank ℂ S.toSubmodule ∧ Module.finrank ℂ S.toSubmodule = n :=
+    ⟨Module.finrank ℂ V, ⊤, by simpa using (Module.finrank_pos (R := ℂ) (M := V)), by simp⟩
+  obtain ⟨S, hSpos, hSdim⟩ := Nat.find_spec hp
+  have hmin (T : Subrepresentation ρ) (hT : T ≠ ⊥) :
+      Module.finrank ℂ S.toSubmodule ≤ Module.finrank ℂ T.toSubmodule := by
+    rw [hSdim]
+    apply Nat.find_min' hp
+    refine ⟨T, ?_, rfl⟩
+    apply Module.finrank_pos_iff.mpr
+    apply Submodule.nontrivial_iff_ne_bot.mpr
+    exact fun ht => hT (Subrepresentation.toSubmodule_injective ht)
+  refine ⟨S, hSpos, ?_, hmin⟩
+  have : Nontrivial S.toSubmodule := Module.finrank_pos_iff.mp hSpos
+  refine { exists_pair_ne := ⟨⊥, ⊤, ?_⟩, eq_bot_or_eq_top := ?_ }
+  · intro he
+    exact bot_ne_top (congrArg Subrepresentation.toSubmodule he)
+  · intro T
+    by_cases hT : T = ⊥
+    · exact Or.inl hT
+    right
+    by_contra htop
+    have hTp : 0 < Module.finrank ℂ T.toSubmodule := by
+      apply Module.finrank_pos_iff.mpr
+      apply Submodule.nontrivial_iff_ne_bot.mpr
+      exact fun ht => hT (Subrepresentation.toSubmodule_injective ht)
+    have hdim : Module.finrank ℂ (nested S T).toSubmodule = Module.finrank ℂ T.toSubmodule :=
+      S.toSubmodule.finrank_map_subtype_eq T.toSubmodule
+    have hnb : nested S T ≠ ⊥ := by
+      intro he
+      have hz : Module.finrank ℂ (nested S T).toSubmodule = 0 := by rw [he]; simp
+      omega
+    have hlow := hmin (nested S T) hnb
+    rw [hdim] at hlow
+    have hhigh : Module.finrank ℂ T.toSubmodule < Module.finrank ℂ S.toSubmodule :=
+      Submodule.finrank_lt (fun ht => htop (Subrepresentation.toSubmodule_injective ht))
+    omega
+
+/- Every reducible finite-group representation has a constituent of at most
+half its dimension. This uses Maschke over ℂ, not an assumed decomposition. -/
+theorem exists_constituent_le_half {Γ : Type*} [Group Γ] [Fintype Γ]
+    (ρ : Representation ℂ Γ V) [Nontrivial V]
+    (hn : ¬ρ.IsIrreducible) :
+    ∃ S : Subrepresentation ρ, 0 < Module.finrank ℂ S.toSubmodule ∧
+      S.toRepresentation.IsIrreducible ∧
+      2 * Module.finrank ℂ S.toSubmodule ≤ Module.finrank ℂ V := by
+  classical
+  obtain ⟨S, hpos, hirr, hmin⟩ := exists_minimal_constituent ρ
+  have hnon : ∃ T : Subrepresentation ρ, T ≠ ⊥ ∧ T ≠ ⊤ := by
+    by_contra h
+    apply hn
+    refine { exists_pair_ne := ⟨⊥, ⊤, ?_⟩, eq_bot_or_eq_top := ?_ }
+    · intro he
+      exact bot_ne_top (congrArg Subrepresentation.toSubmodule he)
+    · intro T
+      simpa only [not_exists, not_and_or, not_not] using (not_exists.mp h T)
+  obtain ⟨T, hTbot, hTtop⟩ := hnon
+  have hSt : S ≠ ⊤ := by
+    intro he
+    have hd := hmin T hTbot
+    have ht : Module.finrank ℂ T.toSubmodule < Module.finrank ℂ V :=
+      Submodule.finrank_lt (fun ht => hTtop (Subrepresentation.toSubmodule_injective ht))
+    rw [he] at hd
+    simp only [subrep_top_toSubmodule, finrank_top] at hd
+    omega
+  let : NeZero (Nat.card Γ : ℂ) := ⟨by exact_mod_cast Nat.card_pos.ne'⟩
+  obtain ⟨U, hU⟩ := exists_isCompl S
+  have hUb : U ≠ ⊥ := by
+    intro he
+    have h := hU.sup_eq_top
+    rw [he, sup_bot_eq] at h
+    exact hSt h
+  have hle := hmin U hUb
+  have hcompl : IsCompl S.toSubmodule U.toSubmodule := by
+    apply IsCompl.of_eq
+    · exact congrArg Subrepresentation.toSubmodule hU.inf_eq_bot
+    · exact congrArg Subrepresentation.toSubmodule hU.sup_eq_top
+  have hsum := Submodule.finrank_add_eq_of_isCompl hcompl
+  exact ⟨S, hpos, hirr, by omega⟩
+
+/- Restriction to permutations fixing two distinct sites is not irreducible,
+unless the full irreducible is one-dimensional. This gives a factor-two
+dimension descent every two removed cards, needed for sparse types. -/
+theorem finrank_eq_one_of_fixTwo_irreducible
+    (ρ : Representation ℂ (Equiv.Perm Ω) V) [ρ.IsIrreducible]
+    (a b : Ω) (hab : a ≠ b)
+    [hirr : Representation.IsIrreducible (ρ.comp (fixTwo a b).subtype)] :
+    Module.finrank ℂ V = 1 := by
+  have hc : ∀ g : fixTwo a b,
+      ρ (Equiv.swap a b) * ρ g.val = ρ g.val * ρ (Equiv.swap a b) := by
+    intro g
+    rw [← map_mul, ← map_mul]
+    congr 1
+    rw [Equiv.mul_swap_eq_swap_mul, g.property.1, g.property.2]
+  obtain ⟨c, hc⟩ := commuting_scalar (ρ.comp (fixTwo a b).subtype) (ρ (Equiv.swap a b)) hc
+  have hswap (x y : Ω) (hxy : x ≠ y) : ρ (Equiv.swap x y) = c • (1 : Module.End ℂ V) := by
+    obtain ⟨g, hg⟩ := isConj_iff.mp (Equiv.Perm.isConj_swap hab hxy)
+    rw [← hg, map_mul, map_mul, hc]
+    simp only [mul_smul_comm, mul_one, smul_mul_assoc, ← map_mul, mul_inv_cancel, map_one]
+  apply finrank_eq_one_of_scalars ρ
+  intro g
+  induction g using Equiv.Perm.swap_induction_on with
+  | one => exact ⟨1, by simp⟩
+  | swap_mul g x y hxy ih =>
+    obtain ⟨d, hd⟩ := ih
+    refine ⟨c*d, ?_⟩
+    rw [map_mul, hswap x y hxy, hd, smul_mul_smul_comm, one_mul]
+
+/- Restriction along a surjective homomorphism preserves irreducibility. -/
+omit [FiniteDimensional ℂ V] in
+theorem irreducible_comp_surjective {Γ Δ : Type*} [Group Γ] [Group Δ]
+    (ρ : Representation ℂ Γ V) [ρ.IsIrreducible]
+    (f : Δ →* Γ) (hf : Function.Surjective f) :
+    Representation.IsIrreducible (ρ.comp f) := by
+  have he (S : Subrepresentation (ρ.comp f)) :
+      S.toSubmodule = ⊥ ∨ S.toSubmodule = ⊤ := by
+    let T : Subrepresentation ρ :=
+      { toSubmodule := S.toSubmodule
+        apply_mem_toSubmodule := by
+          intro g v hv
+          obtain ⟨x, rfl⟩ := hf g
+          exact S.apply_mem_toSubmodule x hv }
+    exact (eq_bot_or_eq_top T).imp
+      (congrArg Subrepresentation.toSubmodule) (congrArg Subrepresentation.toSubmodule)
+  have : Nontrivial V := by
+    by_contra hn
+    have : Subsingleton V := not_nontrivial_iff_subsingleton.mp hn
+    apply bot_ne_top (α := Subrepresentation ρ)
+    exact Subrepresentation.toSubmodule_injective (Subsingleton.elim _ _)
+  refine { exists_pair_ne := ⟨⊥, ⊤, ?_⟩, eq_bot_or_eq_top := ?_ }
+  · intro h
+    exact bot_ne_top (congrArg Subrepresentation.toSubmodule h)
+  · intro S
+    exact (he S).imp (fun h => Subrepresentation.toSubmodule_injective h)
+      (fun h => Subrepresentation.toSubmodule_injective h)
+
+/- Permutations of the complementary sites, viewed in the two-point stabilizer. -/
+def fixTwoHom (a b : Ω) : Equiv.Perm {x : Ω // x ≠ a ∧ x ≠ b} →* fixTwo a b where
+  toFun g := ⟨Equiv.Perm.ofSubtype g,
+    Equiv.Perm.ofSubtype_apply_of_not_mem g (by simp),
+    Equiv.Perm.ofSubtype_apply_of_not_mem g (by simp)⟩
+  map_one' := Subtype.ext (map_one _)
+  map_mul' g h := Subtype.ext (map_mul _ g h)
+
+omit [Fintype Ω] in
+@[simp] theorem fixTwoHom_val (a b : Ω) (g : Equiv.Perm {x : Ω // x ≠ a ∧ x ≠ b}) :
+    (fixTwoHom a b g).val = Equiv.Perm.ofSubtype g := rfl
+
+omit [Fintype Ω] in
+theorem fixTwoHom_surjective (a b : Ω) : Function.Surjective (fixTwoHom a b) := by
+  intro g
+  have hi (x : Ω) : ((g.val x ≠ a ∧ g.val x ≠ b) ↔ (x ≠ a ∧ x ≠ b)) := by
+    have ha : g.val x = a ↔ x = a :=
+      ⟨fun h => g.val.injective (h.trans g.property.1.symm), fun h => h ▸ g.property.1⟩
+    have hb : g.val x = b ↔ x = b :=
+      ⟨fun h => g.val.injective (h.trans g.property.2.symm), fun h => h ▸ g.property.2⟩
+    simp only [ne_eq, ha, hb]
+  refine ⟨g.val.subtypePerm hi, Subtype.ext ?_⟩
+  apply Equiv.Perm.ofSubtype_subtypePerm hi
+  intro x hx
+  constructor
+  · intro he; subst x; exact hx g.property.1
+  · intro he; subst x; exact hx g.property.2
+
+/- One factor-two dimension descent, with the remaining symmetric group
+realized literally as permutations of the complement. -/
+theorem exists_half_constituent_fixTwo
+    (ρ : Representation ℂ (Equiv.Perm Ω) V) [ρ.IsIrreducible] [Nontrivial V]
+    (hd : Module.finrank ℂ V ≠ 1) (a b : Ω) (hab : a ≠ b) :
+    ∃ S : Subrepresentation (ρ.comp (fixTwo a b).subtype),
+      0 < Module.finrank ℂ S.toSubmodule ∧
+      Representation.IsIrreducible (S.toRepresentation.comp (fixTwoHom a b)) ∧
+      2 * Module.finrank ℂ S.toSubmodule ≤ Module.finrank ℂ V := by
+  classical
+  have hn : ¬Representation.IsIrreducible (ρ.comp (fixTwo a b).subtype) := by
+    intro hirr
+    let := hirr
+    exact hd (finrank_eq_one_of_fixTwo_irreducible ρ a b hab)
+  obtain ⟨S, hpos, hirr, hdim⟩ := exists_constituent_le_half (ρ.comp (fixTwo a b).subtype) hn
+  let := hirr
+  exact ⟨S, hpos, irreducible_comp_surjective S.toRepresentation
+    (fixTwoHom a b) (fixTwoHom_surjective a b), hdim⟩
+
+/- Nonzero state space follows from irreducibility, including degenerate groups. -/
+omit [FiniteDimensional ℂ V] in
+theorem irreducible_nontrivial {Γ : Type*} [Group Γ]
+    (ρ : Representation ℂ Γ V) [ρ.IsIrreducible] : Nontrivial V := by
+  by_contra hn
+  have : Subsingleton V := not_nontrivial_iff_subsingleton.mp hn
+  apply bot_ne_top (α := Subrepresentation ρ)
+  exact Subrepresentation.toSubmodule_injective (Subsingleton.elim _ _)
+
+/- A one-dimensional constituent is already visible after at most twice the
+base-two logarithm of the dimension many fixed cards. The power formulation
+avoids rounding conventions and is valid also for dimension one. -/
+theorem sparse_line_by_dim (D : ℕ) :
+    ∀ (Ω : Type u) [Fintype Ω] [DecidableEq Ω]
+      (V : Type v) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
+      (ρ : Representation ℂ (Equiv.Perm Ω) V) [ρ.IsIrreducible],
+      Module.finrank ℂ V = D →
+      ∃ (t : ℕ) (A : Finset Ω) (W : Submodule ℂ V),
+        Module.finrank ℂ W = 1 ∧ A.card ≤ 2*t ∧ 2^t ≤ D ∧
+        ∀ g : Equiv.Perm Ω, (∀ a ∈ A, g a = a) →
+          ∀ w ∈ W, ρ g w ∈ W := by
+  classical
+  induction D using Nat.strong_induction_on with
+  | h D ih =>
+    intro Ω _ _ V _ _ _ ρ hirr hD
+    have : Nontrivial V := irreducible_nontrivial ρ
+    have hDpos : 0 < D := hD ▸ Module.finrank_pos
+    by_cases hD1 : D = 1
+    · refine ⟨0, ∅, ⊤, ?_, by simp, by simp [hD1], ?_⟩
+      · simpa only [finrank_top] using hD.trans hD1
+      · intro g hg w hw; trivial
+    have hΩ : Nontrivial Ω := by
+      by_contra hn
+      have : Subsingleton Ω := not_nontrivial_iff_subsingleton.mp hn
+      have hscalar : ∀ g : Equiv.Perm Ω, ∃ c : ℂ, ρ g = c • (1 : Module.End ℂ V) := by
+        intro g
+        have he : g = 1 := Subsingleton.elim _ _
+        exact ⟨1, by rw [he, map_one, one_smul]⟩
+      exact hD1 (hD.symm.trans (finrank_eq_one_of_scalars ρ hscalar))
+    let := hΩ
+    obtain ⟨a, b, hab⟩ := exists_pair_ne Ω
+    obtain ⟨S, hSpos, hSirr, hShalf⟩ := exists_half_constituent_fixTwo ρ
+      (by simpa only [hD] using hD1) a b hab
+    let ρ' : Representation ℂ (Equiv.Perm {x : Ω // x ≠ a ∧ x ≠ b}) S.toSubmodule :=
+      S.toRepresentation.comp (fixTwoHom a b)
+    let : ρ'.IsIrreducible := hSirr
+    have hlt : Module.finrank ℂ S.toSubmodule < D := by rw [hD] at hShalf; omega
+    obtain ⟨t, A, W, hWdim, hAcard, ht, hWstable⟩ :=
+      ih (Module.finrank ℂ S.toSubmodule) hlt {x : Ω // x ≠ a ∧ x ≠ b}
+        S.toSubmodule ρ' rfl
+    let B : Finset Ω := insert a (insert b (A.image Subtype.val))
+    refine ⟨t+1, B, W.map S.toSubmodule.subtype, ?_, ?_, ?_, ?_⟩
+    · rw [Submodule.finrank_map_subtype_eq, hWdim]
+    · have hc := Finset.card_image_le (s := A) (f := Subtype.val)
+      have h1 := Finset.card_insert_le a (insert b (A.image Subtype.val))
+      have h2 := Finset.card_insert_le b (A.image Subtype.val)
+      dsimp [B]
+      omega
+    · rw [pow_succ]
+      rw [hD] at hShalf
+      omega
+    · intro g hg w hw
+      obtain ⟨w, hw, rfl⟩ := hw
+      have hga : g a = a := hg a (by simp [B])
+      have hgb : g b = b := hg b (by simp [B])
+      obtain ⟨σ, hσ⟩ := fixTwoHom_surjective a b ⟨g, hga, hgb⟩
+      have hsfix : ∀ x ∈ A, σ x = x := by
+        intro x hx
+        apply Subtype.ext
+        calc
+          (σ x).val = (fixTwoHom a b σ).val x.val :=
+            (Equiv.Perm.ofSubtype_apply_coe σ x).symm
+          _ = g x.val := congrArg (fun f : fixTwo a b => f.val x.val) hσ
+          _ = x.val := hg x.val (by simp [B, Finset.mem_image_of_mem _ hx])
+      refine ⟨ρ' σ w, hWstable σ hsfix w hw, ?_⟩
+      change ρ ((fixTwoHom a b σ).val) w.val = ρ g w.val
+      rw [hσ]
+
+/- The rounded-free quantitative sparse dimension bridge for the actual
+symmetric group, independent of a partition classification. -/
+theorem exists_sparse_line
+    (ρ : Representation ℂ (Equiv.Perm Ω) V) [ρ.IsIrreducible] :
+    ∃ (t : ℕ) (A : Finset Ω) (W : Submodule ℂ V),
+      Module.finrank ℂ W = 1 ∧ A.card ≤ 2*t ∧ 2^t ≤ Module.finrank ℂ V ∧
+      ∀ g : Equiv.Perm Ω, (∀ a ∈ A, g a = a) →
+        ∀ w ∈ W, ρ g w ∈ W :=
+  sparse_line_by_dim _ Ω V ρ rfl
+
+/- Ordinary permutation sign, embedded into the complex scalars. -/
+def signScalar : Equiv.Perm Ω →* ℂ :=
+  (Int.castRingHom ℂ).toMonoidHom.comp ((Units.coeHom ℤ).comp Equiv.Perm.sign)
+
+@[simp] theorem signScalar_apply (g : Equiv.Perm Ω) :
+    signScalar g = ((Equiv.Perm.sign g : ℤ) : ℂ) := rfl
+
+@[simp] theorem signScalar_swap (a b : Ω) (hab : a ≠ b) :
+    signScalar (Equiv.swap a b) = -1 := by
+  simp [signScalar_apply, Equiv.Perm.sign_swap hab]
+
+/- Every one-dimensional representation of a finite symmetric group is
+literally the trivial or sign representation. This also handles Ω of size 0,1. -/
+theorem oneDimensional_trivial_or_sign
+    (ρ : Representation ℂ (Equiv.Perm Ω) V) (hd : Module.finrank ℂ V = 1) :
+    (∀ g, ρ g = 1) ∨ (∀ g, ρ g = signScalar g • (1 : Module.End ℂ V)) := by
+  classical
+  by_cases hΩ : Nontrivial Ω
+  swap
+  · have : Subsingleton Ω := not_nontrivial_iff_subsingleton.mp hΩ
+    left
+    intro g
+    rw [Subsingleton.elim g 1, map_one]
+  let := hΩ
+  obtain ⟨a, b, hab⟩ := exists_pair_ne Ω
+  obtain ⟨c, hc, _⟩ := LinearMap.existsUnique_eq_smul_id_of_finrank_eq_one hd (ρ (Equiv.swap a b))
+  change ρ (Equiv.swap a b) = c • (1 : Module.End ℂ V) at hc
+  have hsame (x y : Ω) (hxy : x ≠ y) : ρ (Equiv.swap x y) = c • (1 : Module.End ℂ V) := by
+    obtain ⟨g, hg⟩ := isConj_iff.mp (Equiv.Perm.isConj_swap hab hxy)
+    rw [← hg, map_mul, map_mul, hc]
+    simp only [mul_smul_comm, mul_one, smul_mul_assoc, ← map_mul, mul_inv_cancel, map_one]
+  have hc2 : c*c = 1 := by
+    apply (LinearEquiv.smul_id_of_finrank_eq_one hd).injective
+    simp only [LinearEquiv.smul_id_of_finrank_eq_one_apply]
+    change (c*c) • (1 : Module.End ℂ V) = 1 • (1 : Module.End ℂ V)
+    rw [one_smul]
+    calc
+      (c*c) • (1 : Module.End ℂ V) = (c • (1 : Module.End ℂ V))*(c • 1) := by
+        simp only [smul_mul_smul_comm, one_mul]
+      _ = 1 := by rw [← hc, ← map_mul, Equiv.swap_mul_self, map_one]
+  rcases mul_self_eq_one_iff.mp hc2 with hc1 | hcn
+  · left
+    intro g
+    induction g using Equiv.Perm.swap_induction_on with
+    | one => exact map_one _
+    | swap_mul g x y hxy ih => rw [map_mul, hsame x y hxy, hc1, one_smul, ih, one_mul]
+  · right
+    intro g
+    induction g using Equiv.Perm.swap_induction_on with
+    | one => simp
+    | swap_mul g x y hxy ih =>
+      rw [map_mul, hsame x y hxy, hcn, ih, map_mul, signScalar_swap x y hxy,
+        smul_mul_smul_comm, one_mul]
+
+/- The cyclic span proof underlying the irreducible dimension lemma,
+on an arbitrary finite-dimensional realization. -/
+omit [FiniteDimensional ℂ V] in
+theorem orbit_span_top {Γ : Type*} [Group Γ]
+    (ρ : Representation ℂ Γ V) [ρ.IsIrreducible] (v : V) (hv : v ≠ 0) :
+    Submodule.span ℂ (Set.range (fun g => ρ g v)) = ⊤ := by
+  let S : Subrepresentation ρ :=
+    { toSubmodule := Submodule.span ℂ (Set.range (fun g => ρ g v))
+      apply_mem_toSubmodule := by
+        intro g w hw
+        induction hw using Submodule.span_induction with
+        | mem w hw =>
+          obtain ⟨k, rfl⟩ := hw
+          rw [← Module.End.mul_apply, ← map_mul]
+          exact Submodule.subset_span (Set.mem_range_self (g*k))
+        | zero => simp
+        | add x y hx hy hix hiy => simpa only [map_add] using Submodule.add_mem _ hix hiy
+        | smul c x hx hi => simpa only [map_smul] using Submodule.smul_mem _ c hi }
+  have hn : S ≠ ⊥ := by
+    intro he
+    have hm : v ∈ S := by
+      change v ∈ Submodule.span ℂ (Set.range (fun g => ρ g v))
+      apply Submodule.subset_span
+      exact ⟨1, by simp⟩
+    rw [he] at hm
+    exact hv hm
+  exact congrArg Subrepresentation.toSubmodule ((eq_bot_or_eq_top S).resolve_left hn)
+
+/- The upper sparse dimension bound: a line invariant under the stabilizer
+of k named sites gives at most n^k dimensions in its irreducible orbit. -/
+theorem dimension_le_pow_of_stable_line
+    (ρ : Representation ℂ (Equiv.Perm Ω) V) [ρ.IsIrreducible]
+    (A : Finset Ω) (W : Submodule ℂ V) (hW : Module.finrank ℂ W = 1)
+    (hstable : ∀ g : Equiv.Perm Ω, (∀ a ∈ A, g a = a) → ∀ w ∈ W, ρ g w ∈ W) :
+    Module.finrank ℂ V ≤ (Fintype.card Ω)^A.card := by
+  classical
+  have : Nontrivial W := (Module.finrank_pos_iff (R := ℂ) (M := W)).mp (by omega)
+  obtain ⟨v, hv⟩ := exists_ne (0 : W)
+  have hv' : v.val ≠ 0 := by intro he; exact hv (Subtype.ext he)
+  let restrict (g : Equiv.Perm Ω) : A → Ω := fun a => g a.val
+  let chooseG (x : A → Ω) : Equiv.Perm Ω :=
+    if hx : ∃ g, restrict g = x then Classical.choose hx else 1
+  have hchoose (g : Equiv.Perm Ω) : restrict (chooseG (restrict g)) = restrict g := by
+    dsimp [chooseG]
+    rw [dite_eq_left (show ∃ h, restrict h = restrict g from ⟨g, rfl⟩)]
+    exact Classical.choose_spec (show ∃ h, restrict h = restrict g from ⟨g, rfl⟩)
+  let u (x : A → Ω) : V := ρ (chooseG x) v.val
+  have hspan : Submodule.span ℂ (Set.range u) = ⊤ := by
+    rw [eq_top_iff, ← orbit_span_top ρ v.val hv']
+    apply Submodule.span_le.mpr
+    rintro _ ⟨g, rfl⟩
+    let r := chooseG (restrict g)
+    have hfix : ∀ a ∈ A, (r⁻¹*g) a = a := by
+      intro a ha
+      have he := congrFun (hchoose g) ⟨a, ha⟩
+      change r a = g a at he
+      simp only [Equiv.Perm.mul_apply]
+      rw [← he]; simp
+    have hw : ρ (r⁻¹*g) v.val ∈ W := hstable _ hfix _ v.property
+    obtain ⟨c, hc⟩ := exists_smul_eq_of_finrank_eq_one hW hv
+      (⟨ρ (r⁻¹*g) v.val, hw⟩ : W)
+    have he : ρ g v.val = c • u (restrict g) := by
+      have hh := congrArg (fun w : W => ρ r w.val) hc
+      change ρ r (c • v.val) = ρ r (ρ (r⁻¹*g) v.val) at hh
+      rw [map_smul, ← Module.End.mul_apply, ← map_mul, mul_inv_cancel_left] at hh
+      exact hh.symm
+    change ρ g v.val ∈ Submodule.span ℂ (Set.range u)
+    rw [he]
+    exact Submodule.smul_mem _ c (Submodule.subset_span (Set.mem_range_self _))
+  have hd := finrank_le_of_span_eq_top hspan
+  simpa only [Fintype.card_fun, Fintype.card_coe] using hd
+
+/- Pointwise stabilizer of a named finite set of sites. -/
+def fixSet (A : Finset Ω) : Subgroup (Equiv.Perm Ω) where
+  carrier := {g | ∀ a ∈ A, g a = a}
+  one_mem' := by simp
+  mul_mem' hg hh a ha := by simp only [Equiv.Perm.mul_apply, hh a ha, hg a ha]
+  inv_mem' {g} hg a ha := by
+    apply g.injective
+    simpa using (hg a ha).symm
+
+/- The complementary symmetric group onto the pointwise stabilizer. -/
+def fixSetHom (A : Finset Ω) : Equiv.Perm {x : Ω // x ∉ A} →* fixSet A where
+  toFun g := ⟨Equiv.Perm.ofSubtype g, fun _ ha =>
+    Equiv.Perm.ofSubtype_apply_of_not_mem g (not_not.mpr ha)⟩
+  map_one' := Subtype.ext (map_one _)
+  map_mul' g h := Subtype.ext (map_mul _ g h)
+
+omit [Fintype Ω] in
+@[simp] theorem fixSetHom_val (A : Finset Ω) (g : Equiv.Perm {x : Ω // x ∉ A}) :
+    (fixSetHom A g).val = Equiv.Perm.ofSubtype g := rfl
+
+omit [Fintype Ω] in
+theorem fixSetHom_surjective (A : Finset Ω) : Function.Surjective (fixSetHom A) := by
+  intro g
+  have hi (x : Ω) : (g.val x ∉ A) ↔ x ∉ A := by
+    constructor
+    · intro h hx
+      apply h
+      rwa [g.property x hx]
+    · intro h hx
+      have he := g.property (g.val x) hx
+      have he' : g.val x = x := g.val.injective he
+      exact h (he' ▸ hx)
+  refine ⟨g.val.subtypePerm hi, Subtype.ext ?_⟩
+  apply Equiv.Perm.ofSubtype_subtypePerm hi
+  intro x hx
+  by_contra hn
+  exact hx (g.property x (by simpa using hn))
+
+/- A stable line under a pointwise stabilizer is a literal trivial or sign
+line. There is no hypothesis about a partition or a pre-existing embedding. -/
+theorem stable_line_trivial_or_sign
+    (ρ : Representation ℂ (Equiv.Perm Ω) V)
+    (A : Finset Ω) (W : Submodule ℂ V) (hW : Module.finrank ℂ W = 1)
+    (hstable : ∀ g : Equiv.Perm Ω, (∀ a ∈ A, g a = a) → ∀ w ∈ W, ρ g w ∈ W) :
+    ∃ u : V, u ≠ 0 ∧
+      ((∀ g : Equiv.Perm Ω, (∀ a ∈ A, g a = a) → ρ g u = u) ∨
+       (∀ g : Equiv.Perm Ω, (∀ a ∈ A, g a = a) → ρ g u = signScalar g • u)) := by
+  classical
+  let S : Subrepresentation (ρ.comp (fixSet A).subtype) :=
+    { toSubmodule := W
+      apply_mem_toSubmodule := fun g w hw => hstable g.val g.property w hw }
+  let ρ' := S.toRepresentation.comp (fixSetHom A)
+  have : Nontrivial W := (Module.finrank_pos_iff (R := ℂ) (M := W)).mp (by omega)
+  obtain ⟨u, hu⟩ := exists_ne (0 : W)
+  refine ⟨u.val, fun he => hu (Subtype.ext he), ?_⟩
+  rcases oneDimensional_trivial_or_sign ρ' hW with ht | hs
+  · left
+    intro g hg
+    obtain ⟨σ, hσ⟩ := fixSetHom_surjective A ⟨g, hg⟩
+    have he := congrArg (fun T : Module.End ℂ W => (T u).val) (ht σ)
+    change ρ ((fixSetHom A σ).val) u.val = u.val at he
+    simpa only [hσ] using he
+  · right
+    intro g hg
+    obtain ⟨σ, hσ⟩ := fixSetHom_surjective A ⟨g, hg⟩
+    have he := congrArg (fun T : Module.End ℂ W => (T u).val) (hs σ)
+    change ρ ((fixSetHom A σ).val) u.val = signScalar σ • u.val at he
+    simp only [signScalar_apply] at he ⊢
+    rw [← Equiv.Perm.sign_ofSubtype σ] at he
+    simpa only [← fixSetHom_val, hσ] using he
+
+/- Minimal signed placement level with both dimension bounds. The last
+clause rules out every earlier line, hence in particular every earlier fixed
+vector of either the representation or its sign twist. -/
+theorem exists_minimal_sparse_line
+    (ρ : Representation ℂ (Equiv.Perm Ω) V) [ρ.IsIrreducible] :
+    ∃ (k t : ℕ) (A : Finset Ω) (u : V),
+      A.card = k ∧ k ≤ 2*t ∧ 2^t ≤ Module.finrank ℂ V ∧
+      Module.finrank ℂ V ≤ (Fintype.card Ω)^k ∧ u ≠ 0 ∧
+      ((∀ g : Equiv.Perm Ω, (∀ a ∈ A, g a = a) → ρ g u = u) ∨
+       (∀ g : Equiv.Perm Ω, (∀ a ∈ A, g a = a) → ρ g u = signScalar g • u)) ∧
+      ∀ B : Finset Ω, B.card < k → ∀ (v : V),
+        (∀ g : Equiv.Perm Ω, (∀ a ∈ B, g a = a) → ∃ c : ℂ, ρ g v = c • v) → v = 0 := by
+  classical
+  let P (k : ℕ) : Prop := ∃ (A : Finset Ω) (W : Submodule ℂ V), A.card = k ∧
+    Module.finrank ℂ W = 1 ∧
+    ∀ g : Equiv.Perm Ω, (∀ a ∈ A, g a = a) → ∀ w ∈ W, ρ g w ∈ W
+  obtain ⟨t, A₀, W₀, hW₀, hA₀, ht, hs₀⟩ := exists_sparse_line ρ
+  have hex : ∃ k, P k := ⟨A₀.card, A₀, W₀, rfl, hW₀, hs₀⟩
+  obtain ⟨A, W, hA, hW, hs⟩ := Nat.find_spec hex
+  have hk : Nat.find hex ≤ A₀.card := Nat.find_min' hex ⟨A₀, W₀, rfl, hW₀, hs₀⟩
+  obtain ⟨u, hu, hchar⟩ := stable_line_trivial_or_sign ρ A W hW hs
+  refine ⟨Nat.find hex, t, A, u, hA, hk.trans hA₀, ht, ?_, hu, hchar, ?_⟩
+  · rw [← hA]
+    exact dimension_le_pow_of_stable_line ρ A W hW hs
+  · intro B hB v hv
+    by_contra hn
+    have hp : P B.card := by
+      refine ⟨B, ℂ ∙ v, rfl, finrank_span_singleton hn, ?_⟩
+      intro g hg w hw
+      obtain ⟨c, rfl⟩ := Submodule.mem_span_singleton.mp hw
+      obtain ⟨r, hr⟩ := hv g hg
+      rw [map_smul, hr, smul_smul]
+      exact Submodule.smul_mem _ _ (Submodule.mem_span_singleton_self v)
+    exact Nat.find_min hex hB hp
+end SparseDimension
+end CoordinateSweeps
+
+noncomputable section
+open scoped BigOperators
+
+namespace CoordinateSweeps.PlacementOccurrence
+variable {Γ X V : Type*} [Group Γ] [Fintype Γ] [Fintype X] [DecidableEq X] [MulAction Γ X]
+    [AddCommGroup V] [Module ℂ V]
+
+/- Sum of orbit vectors against placement coordinates. The repeated
+stabilizer fibers are left in the sum, so no transversal choice is needed. -/
+def orbitSynthesis (ρ : Representation ℂ Γ V) (u : V) (x₀ : X) : (X → ℂ) →ₗ[ℂ] V where
+  toFun f := ∑ g, f (g • x₀) • ρ g u
+  map_add' f h := by simp only [Pi.add_apply, add_smul, Finset.sum_add_distrib]
+  map_smul' c f := by simp only [Pi.smul_apply, smul_eq_mul, mul_smul, Finset.smul_sum, RingHom.id_apply]
+
+/- Exact covariance, with the source's convention that gh applies h first. -/
+omit [Fintype X] [DecidableEq X] in
+theorem orbitSynthesis_covariant (ρ : Representation ℂ Γ V) (u : V) (x₀ : X)
+    (h : Γ) (f : X → ℂ) :
+    orbitSynthesis ρ u x₀ (fun x => f (h⁻¹ • x)) = ρ h (orbitSynthesis ρ u x₀ f) := by
+  change (∑ g, f (h⁻¹ • (g • x₀)) • ρ g u) = ρ h (∑ g, f (g • x₀) • ρ g u)
+  rw [← Equiv.sum_comp (Equiv.mulLeft h) (fun g => f (h⁻¹ • (g • x₀)) • ρ g u)]
+  simp only [Equiv.coe_mulLeft, mul_smul, inv_smul_smul, map_sum, map_smul,
+    map_mul, Module.End.mul_apply]
+
+/- A vector fixed by the point stabilizer produces a nonzero synthesis map. -/
+omit [Fintype X] in
+theorem orbitSynthesis_delta_ne_zero (ρ : Representation ℂ Γ V) (u : V) (hu : u ≠ 0)
+    (x₀ : X) (hfixed : ∀ g : Γ, g • x₀ = x₀ → ρ g u = u) :
+    orbitSynthesis ρ u x₀ (fun x => if x = x₀ then 1 else 0) ≠ 0 := by
+  classical
+  have he : orbitSynthesis ρ u x₀ (fun x => if x = x₀ then 1 else 0) =
+      ((Finset.univ.filter (fun g : Γ => g • x₀ = x₀)).card : ℂ) • u := by
+    change (∑ g, (if g • x₀ = x₀ then (1 : ℂ) else 0) • ρ g u) = _
+    simp only [ite_smul, one_smul, zero_smul]
+    calc
+      _ = ∑ g : Γ, if g • x₀ = x₀ then u else 0 := by
+        apply Finset.sum_congr rfl
+        intro g _
+        split_ifs with hg
+        · exact hfixed g hg
+        · rfl
+      _ = _ := by rw [← Finset.sum_filter]; simp [Nat.cast_smul_eq_nsmul]
+  rw [he]
+  apply smul_ne_zero _ hu
+  norm_cast
+  apply Nat.ne_of_gt
+  apply Finset.card_pos.mpr
+  exact ⟨1, by simp⟩
+
+/- Frobenius occurrence at the precise stabilizer level, proved by an
+explicit intertwiner rather than postulated decomposition. -/
+omit [Fintype X] in
+theorem orbitSynthesis_surjective (ρ : Representation ℂ Γ V) [ρ.IsIrreducible]
+    (u : V) (hu : u ≠ 0) (x₀ : X)
+    (hfixed : ∀ g : Γ, g • x₀ = x₀ → ρ g u = u) :
+    Function.Surjective (orbitSynthesis ρ u x₀) := by
+  classical
+  let S : Subrepresentation ρ :=
+    { toSubmodule := (orbitSynthesis ρ u x₀).range
+      apply_mem_toSubmodule := by
+        intro g v hv
+        obtain ⟨f, rfl⟩ := hv
+        exact ⟨(fun x => f (g⁻¹ • x)), orbitSynthesis_covariant ρ u x₀ g f⟩ }
+  have hne : S ≠ ⊥ := by
+    intro he
+    have hm : orbitSynthesis ρ u x₀ (fun x => if x = x₀ then 1 else 0) ∈ S :=
+      ⟨_, rfl⟩
+    rw [he] at hm
+    exact orbitSynthesis_delta_ne_zero ρ u hu x₀ hfixed hm
+  have he : S = ⊤ := (eq_bot_or_eq_top S).resolve_left hne
+  apply LinearMap.range_eq_top.mp
+  exact congrArg Subrepresentation.toSubmodule he
+
+/- The subgroup average is fixed under that subgroup, with no assumptions on
+interrelations between different coordinate subgroups. -/
+omit [Fintype Γ] in
+theorem subgroup_sum_fixed (ρ : Representation ℂ Γ V) (K : Subgroup Γ) [Fintype K] (u : V)
+    (k : K) : ρ k.val (∑ h : K, ρ h.val u) = ∑ h : K, ρ h.val u := by
+  simp only [map_sum, ← Module.End.mul_apply, ← map_mul]
+  exact Equiv.sum_comp (Equiv.mulLeft k) (fun h : K => ρ h.val u)
+
+/- Averaging a synthesis over a larger stabilizer acts only on the cyclic
+vector whenever the coefficient function forgets those coordinates. -/
+omit [Fintype X] [DecidableEq X] in
+theorem orbitSynthesis_subgroup_average (ρ : Representation ℂ Γ V) (u : V) (x₀ : X)
+    (K : Subgroup Γ) [Fintype K] (f : X → ℂ)
+    (hf : ∀ (g : Γ) (k : K), f ((g*k.val) • x₀) = f (g • x₀)) :
+    orbitSynthesis ρ (∑ k : K, ρ k.val u) x₀ f =
+      (Fintype.card K : ℂ) • orbitSynthesis ρ u x₀ f := by
+  change (∑ g, f (g • x₀) • ρ g (∑ k : K, ρ k.val u)) = _
+  simp only [map_sum, Finset.smul_sum, ← Module.End.mul_apply, ← map_mul]
+  rw [Finset.sum_comm]
+  have hk (k : K) : (∑ g : Γ, f (g • x₀) • ρ (g*k.val) u) = orbitSynthesis ρ u x₀ f := by
+    calc
+      _ = ∑ g : Γ, f ((g*k.val) • x₀) • ρ (g*k.val) u := by
+        simp only [hf]
+      _ = _ := Equiv.sum_comp (Equiv.mulRight k.val) (fun g => f (g • x₀) • ρ g u)
+  simp only [hk, Finset.sum_const, Finset.card_univ, Nat.cast_smul_eq_nsmul]
+
+/- Exact proper-coordinate annihilation used in source04:eq7: it is the
+absence of fixed vectors one level earlier, not a partial cancellation claim. -/
+omit [Fintype X] [DecidableEq X] in
+theorem orbitSynthesis_eq_zero_of_no_fixed (ρ : Representation ℂ Γ V) (u : V) (x₀ : X)
+    (K : Subgroup Γ) [Fintype K]
+    (hnof : ∀ v : V, (∀ k : K, ρ k.val v = v) → v = 0)
+    (f : X → ℂ) (hf : ∀ (g : Γ) (k : K), f ((g*k.val) • x₀) = f (g • x₀)) :
+    orbitSynthesis ρ u x₀ f = 0 := by
+  have hzero : (∑ k : K, ρ k.val u) = 0 := hnof _ (subgroup_sum_fixed ρ K u)
+  have h := orbitSynthesis_subgroup_average ρ u x₀ K f hf
+  rw [hzero] at h
+  have hz : orbitSynthesis ρ (0 : V) x₀ f = 0 := by simp [orbitSynthesis]
+  rw [hz] at h
+  apply (smul_eq_zero.mp h.symm).resolve_left
+  exact_mod_cast Fintype.card_ne_zero
+end CoordinateSweeps.PlacementOccurrence
+
+namespace CoordinateSweeps.SparseDimension
+variable {Ω V : Type*} [Fintype Ω] [DecidableEq Ω]
+  [AddCommGroup V] [Module ℂ V]
+
+@[simp] theorem signScalar_mul_self (g : Equiv.Perm Ω) :
+    signScalar g * signScalar g = (1 : ℂ) := by
+  simp only [signScalar_apply]
+  rcases Int.units_eq_one_or (Equiv.Perm.sign g) with h | h <;> rw [h] <;> norm_num
+
+/- Tensoring with the ordinary sign character, on exactly the same space. -/
+def signTwist (ρ : Representation ℂ (Equiv.Perm Ω) V) :
+    Representation ℂ (Equiv.Perm Ω) V where
+  toFun g := signScalar g • ρ g
+  map_one' := by simp
+  map_mul' g h := by simp only [map_mul, smul_mul_smul_comm]
+
+@[simp] theorem signTwist_apply (ρ : Representation ℂ (Equiv.Perm Ω) V)
+    (g : Equiv.Perm Ω) (v : V) : signTwist ρ g v = signScalar g • ρ g v := rfl
+
+/- The same invariant subspaces for ρ and its sign twist. -/
+def untwistSubrep (ρ : Representation ℂ (Equiv.Perm Ω) V)
+    (S : Subrepresentation (signTwist ρ)) : Subrepresentation ρ where
+  toSubmodule := S.toSubmodule
+  apply_mem_toSubmodule := by
+    intro g v hv
+    have h := S.toSubmodule.smul_mem (signScalar g) (S.apply_mem_toSubmodule g hv)
+    simpa only [signTwist_apply, smul_smul, signScalar_mul_self, one_smul] using h
+
+instance signTwist_irreducible [FiniteDimensional ℂ V] (ρ : Representation ℂ (Equiv.Perm Ω) V)
+    [ρ.IsIrreducible] : (signTwist ρ).IsIrreducible := by
+  have : Nontrivial V := irreducible_nontrivial ρ
+  refine { exists_pair_ne := ⟨⊥, ⊤, ?_⟩, eq_bot_or_eq_top := ?_ }
+  · intro he
+    exact bot_ne_top (congrArg Subrepresentation.toSubmodule he)
+  · intro S
+    rcases eq_bot_or_eq_top (untwistSubrep ρ S) with h | h
+    · left
+      have hh := congrArg Subrepresentation.toSubmodule h
+      exact Subrepresentation.toSubmodule_injective hh
+    · right
+      have hh := congrArg Subrepresentation.toSubmodule h
+      exact Subrepresentation.toSubmodule_injective hh
+
+/- Either the original representation or its sign twist has a first placement
+level k with both quantitative dimension bounds and no earlier fixed vector. -/
+theorem exists_minimal_fixed_vector [FiniteDimensional ℂ V]
+    (ρ : Representation ℂ (Equiv.Perm Ω) V) [ρ.IsIrreducible] :
+    ∃ (ε : Bool) (k t : ℕ) (A : Finset Ω) (u : V),
+      let τ := if ε then signTwist ρ else ρ
+      A.card = k ∧ k ≤ 2*t ∧ 2^t ≤ Module.finrank ℂ V ∧
+      Module.finrank ℂ V ≤ (Fintype.card Ω)^k ∧ u ≠ 0 ∧
+      (∀ g : Equiv.Perm Ω, (∀ a ∈ A, g a = a) → τ g u = u) ∧
+      ∀ B : Finset Ω, B.card < k → ∀ v : V,
+        (∀ g : Equiv.Perm Ω, (∀ a ∈ B, g a = a) → τ g v = v) → v = 0 := by
+  obtain ⟨k,t,A,u,hA,hk,ht,hd,hu,hchar,hmin⟩ := exists_minimal_sparse_line ρ
+  rcases hchar with htriv | hsign
+  · refine ⟨false,k,t,A,u,hA,hk,ht,hd,hu,htriv, ?_⟩
+    intro B hB v hv
+    exact hmin B hB v (fun g hg => ⟨1, by simpa using hv g hg⟩)
+  · refine ⟨true,k,t,A,u,hA,hk,ht,hd,hu, ?_, ?_⟩
+    · intro g hg
+      simp only [↓reduceIte, signTwist_apply, hsign g hg, smul_smul,
+        signScalar_mul_self, one_smul]
+    · intro B hB v hv
+      apply hmin B hB v
+      intro g hg
+      refine ⟨signScalar g, ?_⟩
+      have h := congrArg (fun w : V => signScalar g • w) (hv g hg)
+      simpa only [Bool.true_eq, ↓reduceIte, signTwist_apply, smul_smul,
+        signScalar_mul_self, one_smul] using h
+end CoordinateSweeps.SparseDimension
+
+namespace CoordinateSweeps.PlacementOccurrence
+open SparseDimension
+variable {Ω I V : Type*} [Fintype Ω] [DecidableEq Ω]
+  [Fintype I] [AddCommGroup V] [Module ℂ V]
+
+/- Stabilizer of the small placement coordinates really is the pointwise
+stabilizer of their image, so the exact earlier-level hypothesis applies. -/
+theorem synthesis_kills_proper_coordinates
+    (ρ : Representation ℂ (Equiv.Perm Ω) V) (u : V) (x₀ : I ↪ Ω)
+    (hnof : ∀ B : Finset Ω, B.card < Fintype.card I → ∀ v : V,
+      (∀ g : Equiv.Perm Ω, (∀ a ∈ B, g a = a) → ρ g v = v) → v = 0)
+    (A : Finset I) (hA : A ≠ Finset.univ) (f : (I ↪ Ω) → ℂ)
+    (hf : ∀ x y : I ↪ Ω, (∀ i ∈ A, x i = y i) → f x = f y) :
+    orbitSynthesis ρ u x₀ f = 0 := by
+  classical
+  let B : Finset Ω := A.image x₀
+  have hB : B.card < Fintype.card I := by
+    dsimp [B]
+    rw [Finset.card_image_of_injective _ x₀.injective]
+    exact Finset.card_lt_card (Finset.ssubset_univ_iff.mpr hA)
+  apply orbitSynthesis_eq_zero_of_no_fixed ρ u x₀ (fixSet B)
+  · intro v hv
+    exact hnof B hB v (fun g hg => hv ⟨g,hg⟩)
+  · intro g k
+    apply hf
+    intro i hi
+    change g (k.val (x₀ i)) = g (x₀ i)
+    rw [k.property (x₀ i) (Finset.mem_image_of_mem _ hi)]
+end CoordinateSweeps.PlacementOccurrence
+
+namespace CoordinateSweeps.Placement
+variable {I U X Y : Type*} [Fintype I] [DecidableEq I] [Fintype X] [Fintype Y]
+/- The adjoint of a synthesis map annihilating all cylinder functions lands
+in the exact alternating-kernel top space. Complex conjugation is essential. -/
+omit [DecidableEq I] in
+theorem conjTranspose_mulVec_mem_top {D : Type*} [Fintype D]
+    (coords : Y → I → U) (B : Matrix D Y ℂ)
+    (hB : ∀ A : Finset I, A ≠ Finset.univ → ∀ f : Y → ℂ,
+      DependsOn coords A f → B.mulVec f = 0) (w : D → ℂ) :
+    B.conjTranspose.mulVec w ∈ topSpace coords := by
+  intro A hA f hf
+  have hfstar : DependsOn coords A (fun y => star (f y)) :=
+    fun y y' h => congrArg star (hf y y' h)
+  have hz := hB A hA (fun y => star (f y)) hfstar
+  have hsum (d : D) : ∑ y, f y * star (B d y) = 0 := by
+    have h := congrArg star (congrFun hz d)
+    simpa only [Matrix.mulVec, dotProduct, star_sum, star_mul,
+      star_star, Pi.zero_apply, star_zero, mul_comm] using h
+  simp only [Matrix.mulVec, dotProduct, Matrix.conjTranspose_apply, Finset.mul_sum]
+  rw [Finset.sum_comm]
+  simp_rw [← mul_assoc, ← Finset.sum_mul, hsum, zero_mul]
+  exact Finset.sum_const_zero
+
+/- Matrix form of top-level compression, ready for Schur norm transfer. -/
+omit [DecidableEq I] [Fintype X] in
+theorem alternatingKernel_mul_conjTranspose {D : Type*} [Fintype D]
+    (coords : Y → I → U) (s : ℝ) (hs : s ≠ 0)
+    (p : Finset I → X → Y → ℝ)
+    (hp : ∀ A x, DependsOn coords A (fun y => (p A x y : ℂ)))
+    (B : Matrix D Y ℂ)
+    (hB : ∀ A : Finset I, A ≠ Finset.univ → ∀ f : Y → ℂ,
+      DependsOn coords A f → B.mulVec f = 0) :
+    alternatingKernel s p * B.conjTranspose =
+      (show Matrix X Y ℂ from fun x y => (p Finset.univ x y : ℂ)) * B.conjTranspose := by
+  apply Matrix.ext
+  intro x d
+  have hv : (fun y => B.conjTranspose y d) ∈ topSpace coords := by
+    classical
+    have h := conjTranspose_mulVec_mem_top coords B hB (Pi.single d 1)
+    have he : B.conjTranspose.mulVec (Pi.single d 1) = (fun y => B.conjTranspose y d) := by
+      ext y; simp [Matrix.mulVec, dotProduct, Pi.single_apply]
+    rw [he] at h
+    exact h
+  exact congrFun (alternatingKernel_mulVec_top coords s hs p hp hv) x
+end CoordinateSweeps.Placement
+
+namespace CoordinateSweeps.PlacementOccurrence
+open scoped Matrix.Norms.L2Operator
+variable {Γ X : Type*} [Group Γ] [Fintype Γ] [Fintype X] [DecidableEq X] [MulAction Γ X]
+
+def actionMatrix : Γ →* Matrix X X ℂ :=
+  Matrix.permMatrixHom.comp (MulAction.toPermHom Γ X)
+
+omit [Fintype Γ] in
+lemma actionMatrix_mulVec (g : Γ) (f : X → ℂ) :
+    (actionMatrix g).mulVec f=fun x => f (g⁻¹ • x) := by
+  change (((MulAction.toPermHom Γ X g)⁻¹).permMatrix ℂ).mulVec f = _
+  rw [Matrix.permMatrix_mulVec]
+  rfl
+
+omit [Fintype Γ] in
+lemma actionMatrix_unitary (g : Γ) : (actionMatrix (X := X) g).conjTranspose*actionMatrix g=1 := by
+  change (((MulAction.toPermHom Γ X g)⁻¹).permMatrix ℂ).conjTranspose*
+    ((MulAction.toPermHom Γ X g)⁻¹).permMatrix ℂ=1
+  rw [Matrix.conjTranspose_permMatrix,inv_inv,← Matrix.permMatrix_mul,inv_mul_cancel,Matrix.permMatrix_one]
+
+def synthesisMatrix (ρ : UnitaryIrrep Γ) (u : Fin ρ.dimension → ℂ) (x₀ : X) :
+    Matrix (Fin ρ.dimension) X ℂ := LinearMap.toMatrix' (orbitSynthesis ρ.asRepresentation u x₀)
+
+lemma synthesisMatrix_mulVec (ρ : UnitaryIrrep Γ) (u : Fin ρ.dimension → ℂ) (x₀ : X) (f : X → ℂ) :
+    (synthesisMatrix ρ u x₀).mulVec f=orbitSynthesis ρ.asRepresentation u x₀ f :=
+  LinearMap.toMatrix'_mulVec _ _
+
+lemma synthesisMatrix_covariant (ρ : UnitaryIrrep Γ) (u : Fin ρ.dimension → ℂ) (x₀ : X) (g : Γ) :
+    synthesisMatrix ρ u x₀*actionMatrix g=ρ.matrix g*synthesisMatrix ρ u x₀ := by
+  change (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul (synthesisMatrix ρ u x₀) (actionMatrix g))=
+    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul (ρ.matrix g) (synthesisMatrix ρ u x₀))
+  apply Matrix.mulVec_injective
+  funext f
+  rw [← Matrix.mulVec_mulVec,← Matrix.mulVec_mulVec,actionMatrix_mulVec,
+    synthesisMatrix_mulVec,synthesisMatrix_mulVec]
+  exact orbitSynthesis_covariant ρ.asRepresentation u x₀ g f
+
+lemma synthesisMatrix_ne_zero (ρ : UnitaryIrrep Γ) (u : Fin ρ.dimension → ℂ) (hu : u ≠ 0) (x₀ : X)
+    (hfixed : ∀ g : Γ, g • x₀=x₀ → ρ.asRepresentation g u=u) : synthesisMatrix ρ u x₀ ≠ 0 := by
+  intro h
+  have hh := orbitSynthesis_delta_ne_zero ρ.asRepresentation u hu x₀ hfixed
+  apply hh
+  rw [← synthesisMatrix_mulVec,h,Matrix.zero_mulVec]
+
+end CoordinateSweeps.PlacementOccurrence
+
+namespace CoordinateSweeps.PlacementOccurrence
+open scoped Matrix.Norms.L2Operator
+open Placement
+variable {I Ω : Type*} [Fintype I] [DecidableEq I] [Fintype Ω] [DecidableEq Ω]
+
+/- Source04:eq7 at its exact first placement level, with a constructed
+nonzero synthesis intertwiner and Schur norm transfer. -/
+omit [DecidableEq I] in
+theorem first_level_norm_le (ρ : UnitaryIrrep (Equiv.Perm Ω))
+    (u : Fin ρ.dimension → ℂ) (hu : u ≠ 0) (x₀ : I ↪ Ω)
+    (hfixed : ∀ g : Equiv.Perm Ω, (∀ i, g (x₀ i)=x₀ i) → ρ.asRepresentation g u=u)
+    (hnof : ∀ A : Finset Ω, A.card < Fintype.card I → ∀ v : Fin ρ.dimension → ℂ,
+      (∀ g : Equiv.Perm Ω, (∀ a ∈ A, g a=a) → ρ.asRepresentation g v=v) → v=0)
+    {W : Type*} [Fintype W] (w : W → ℂ) (g : W → Equiv.Perm Ω)
+    (s : ℝ) (hs : s ≠ 0) (p : Finset I → (I ↪ Ω) → (I ↪ Ω) → ℝ)
+    (hp : ∀ A x, DependsOn (fun y : I ↪ Ω => y) A (fun y => (p A x y : ℂ)))
+    (hfull : (show Matrix (I ↪ Ω) (I ↪ Ω) ℂ from fun x y => (p Finset.univ x y : ℂ))=
+      ∑ a : W, w a • actionMatrix (X := I ↪ Ω) (g a)) :
+    ‖∑ a : W, w a • ρ.matrix (g a)‖ ≤
+      ‖alternatingKernel s p‖ := by
+  let B : Matrix (Fin ρ.dimension) (I ↪ Ω) ℂ := synthesisMatrix ρ u x₀
+  have hB : ∀ A : Finset I, A ≠ Finset.univ → ∀ f : (I ↪ Ω) → ℂ,
+      DependsOn (fun y : I ↪ Ω => y) A f → B.mulVec f=0 := by
+    intro A hA f hf
+    rw [synthesisMatrix_mulVec]
+    exact synthesis_kills_proper_coordinates ρ.asRepresentation u x₀ hnof A hA f hf
+  have hne : B ≠ 0 := synthesisMatrix_ne_zero ρ u hu x₀ (by
+    intro σ hσ
+    exact hfixed σ (fun i => congrArg (fun x : I ↪ Ω => x i) hσ))
+  have hnorm : ‖∑ a : W, w a • ρ.matrix (g a)‖ ≤ ‖alternatingKernel s p‖ := by
+    apply ρ.norm_le_compression (actionMatrix (X := I ↪ Ω)) actionMatrix_unitary B hne
+      (synthesisMatrix_covariant ρ u x₀)
+    rw [alternatingKernel_mul_conjTranspose (fun y : I ↪ Ω => y) s hs p hp B hB,hfull,
+      Matrix.sum_mul,Matrix.mul_sum]
+    apply Finset.sum_congr rfl
+    intro a _
+    rw [Matrix.smul_mul,Matrix.mul_smul]
+    congr 1
+    exact ρ.adjoint_intertwines _ actionMatrix_unitary B (synthesisMatrix_covariant ρ u x₀) (g a)
+  exact hnorm
+
+end CoordinateSweeps.PlacementOccurrence
+
+open scoped Matrix.Norms.L2Operator
+
+namespace CoordinateSweeps.PlacementOccurrence
+variable {Γ X : Type*} [Group Γ] [Fintype X] [DecidableEq X] [MulAction Γ X]
+lemma actionMatrix_entry (g : Γ) (x y : X) : actionMatrix (X := X) g x y=if g⁻¹ • x=y then 1 else 0 := by
+  change (((MulAction.toPermHom Γ X g)⁻¹).permMatrix ℂ) x y=_
+  simp [Equiv.Perm.permMatrix, PEquiv.toMatrix]
+end CoordinateSweeps.PlacementOccurrence
+
+end
+end
+end
+end
+end
+end
+end
+end
+open scoped Matrix.Norms.L2Operator
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/ProjectiveInduction/CosetTransport.lean b/lean/OAI/RepresentationTheory/ProjectiveInduction/CosetTransport.lean
index aba90f48e322e1aa25fc1d06db7e7e691b1ae3db..dc61316fcf75ce70561d58ac6b42b013a1e3cbf7 100644
--- a/lean/OAI/RepresentationTheory/ProjectiveInduction/CosetTransport.lean
+++ b/lean/OAI/RepresentationTheory/ProjectiveInduction/CosetTransport.lean
@@ -1,646 +1,646 @@
-import Mathlib
-
-namespace OAI
-
-/-! Projective coset transport and subgroup graph orbit calculations. -/
-
-noncomputable section
-
-namespace TwistedCorrespondences
-
-universe u v w
-
-structure ScalarCocycle (G : Type u) [Group G] where
-  toFun : G → G → Circle
-  one_left' : ∀ g, toFun 1 g = 1
-  one_right' : ∀ g, toFun g 1 = 1
-  cocycle' : ∀ g h k, toFun g h * toFun (g * h) k = toFun h k * toFun g (h * k)
-
-namespace ScalarCocycle
-
-variable {G : Type u} {H : Type v} {K : Type w}
-variable [Group G] [Group H] [Group K]
-
-instance : CoeFun (ScalarCocycle G) (fun _ => G → G → Circle) := ⟨toFun⟩
-
-@[simp] theorem one_left (α : ScalarCocycle G) (g : G) : α 1 g = 1 := α.one_left' g
-@[simp] theorem one_right (α : ScalarCocycle G) (g : G) : α g 1 = 1 := α.one_right' g
-
-theorem cocycle (α : ScalarCocycle G) (g h k : G) :
-    α g h * α (g * h) k = α h k * α g (h * k) := α.cocycle' g h k
-
-@[ext] theorem ext {α β : ScalarCocycle G} (h : ∀ g k, α g k = β g k) : α = β := by
-  cases α
-  cases β
-  congr
-  exact funext fun g => funext (h g)
-
-def trivial (G : Type u) [Group G] : ScalarCocycle G where
-  toFun _ _ := 1
-  one_left' _ := rfl
-  one_right' _ := rfl
-  cocycle' _ _ _ := rfl
-
-@[simp] theorem trivial_apply (g h : G) : trivial G g h = 1 := rfl
-
-def pullback (α : ScalarCocycle H) (f : G →* H) : ScalarCocycle G where
-  toFun g h := α (f g) (f h)
-  one_left' g := by simp
-  one_right' g := by simp
-  cocycle' g h k := by simpa using α.cocycle (f g) (f h) (f k)
-
-@[simp] theorem pullback_apply (α : ScalarCocycle H) (f : G →* H) (g h : G) :
-    α.pullback f g h = α (f g) (f h) := rfl
-
-def mul (α β : ScalarCocycle G) : ScalarCocycle G where
-  toFun g h := α g h * β g h
-  one_left' g := by simp
-  one_right' g := by simp
-  cocycle' g h k := by
-    calc
-      (α g h * β g h) * (α (g * h) k * β (g * h) k) =
-          (α g h * α (g * h) k) * (β g h * β (g * h) k) := by ac_rfl
-      _ = (α h k * α g (h * k)) * (β h k * β g (h * k)) := by
-        rw [α.cocycle, β.cocycle]
-      _ = (α h k * β h k) * (α g (h * k) * β g (h * k)) := by ac_rfl
-
-@[simp] theorem mul_apply (α β : ScalarCocycle G) (g h : G) :
-    α.mul β g h = α g h * β g h := rfl
-
-def inv (α : ScalarCocycle G) : ScalarCocycle G where
-  toFun g h := (α g h)⁻¹
-  one_left' g := by simp
-  one_right' g := by simp
-  cocycle' g h k := by
-    simpa only [mul_inv] using congrArg Inv.inv (α.cocycle g h k)
-
-@[simp] theorem inv_apply (α : ScalarCocycle G) (g h : G) :
-    α.inv g h = (α g h)⁻¹ := rfl
-
-theorem inv_apply_coe (α : ScalarCocycle G) (g h : G) :
-    (α.inv g h : ℂ) = star (α g h : ℂ) := Circle.coe_inv_eq_conj _
-
-def joint (μ : ScalarCocycle G) (ω : ScalarCocycle H) : ScalarCocycle (G × H) :=
-  (μ.pullback (MonoidHom.fst G H)).mul ((ω.pullback (MonoidHom.snd G H)).inv)
-
-@[simp] theorem joint_apply (μ : ScalarCocycle G) (ω : ScalarCocycle H)
-    (g g' : G) (h h' : H) :
-    joint μ ω (g, h) (g', h') = μ g g' * (ω h h')⁻¹ := rfl
-
-theorem joint_apply_coe (μ : ScalarCocycle G) (ω : ScalarCocycle H)
-    (g g' : G) (h h' : H) :
-    (joint μ ω (g, h) (g', h') : ℂ) = (μ g g' : ℂ) * star (ω h h' : ℂ) := by
-  simp only [joint_apply, Circle.coe_mul, Circle.coe_inv_eq_conj, Complex.star_def]
-
-@[simp] theorem joint_mixed (μ : ScalarCocycle G) (ω : ScalarCocycle H)
-    (g : G) (h : H) : joint μ ω (g, 1) (1, h) = 1 := by simp
-
-@[simp] theorem joint_mixed_rev (μ : ScalarCocycle G) (ω : ScalarCocycle H)
-    (g : G) (h : H) : joint μ ω (1, h) (g, 1) = 1 := by simp
-
-def matchedRatio (μ : ScalarCocycle G) (ω : ScalarCocycle H)
-    {A : Subgroup G} {B : Subgroup H} (δ : A ≃* B) : ScalarCocycle A :=
-  (μ.pullback A.subtype).mul ((ω.pullback (B.subtype.comp δ.toMonoidHom)).inv)
-
-@[simp] theorem matchedRatio_apply (μ : ScalarCocycle G) (ω : ScalarCocycle H)
-    {A : Subgroup G} {B : Subgroup H} (δ : A ≃* B) (a b : A) :
-    matchedRatio μ ω δ a b = μ a b / ω (δ a) (δ b) := rfl
-
-end ScalarCocycle
-
-structure ProjectiveUnitaryRepresentation {G : Type u} [Group G]
-    (α : ScalarCocycle G) (V : Type v) [NormedAddCommGroup V] [InnerProductSpace ℂ V] where
-  op : G → V ≃ₗᵢ[ℂ] V
-  op_one : op 1 = LinearIsometryEquiv.refl ℂ V
-  op_mul : ∀ g h ξ, op g (op h ξ) = (α g h : ℂ) • op (g * h) ξ
-
-namespace ProjectiveUnitaryRepresentation
-
-variable {G : Type u} {H : Type v} [Group G] [Group H]
-variable {V : Type w} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-
-def pullback {α : ScalarCocycle H} (π : ProjectiveUnitaryRepresentation α V)
-    (f : G →* H) : ProjectiveUnitaryRepresentation (α.pullback f) V where
-  op g := π.op (f g)
-  op_one := by simpa using π.op_one
-  op_mul g h ξ := by simpa using π.op_mul (f g) (f h) ξ
-
-def joint {μ : ScalarCocycle G} {ω : ScalarCocycle H}
-    (π : ProjectiveUnitaryRepresentation μ V)
-    (ρ : ProjectiveUnitaryRepresentation ω.inv V)
-    (comm : ∀ g h ξ, ρ.op h (π.op g ξ) = π.op g (ρ.op h ξ)) :
-    ProjectiveUnitaryRepresentation (μ.joint ω) V where
-  op s := (ρ.op s.2).trans (π.op s.1)
-  op_one := by
-    ext ξ
-    simp [π.op_one, ρ.op_one]
-  op_mul s t ξ := by
-    change π.op s.1 (ρ.op s.2 (π.op t.1 (ρ.op t.2 ξ))) =
-      ((μ.joint ω) s t : ℂ) • π.op (s.1 * t.1) (ρ.op (s.2 * t.2) ξ)
-    rw [comm, π.op_mul, ρ.op_mul, map_smul, smul_smul]
-    rfl
-
-end ProjectiveUnitaryRepresentation
-
-section Graphs
-
-variable {G : Type u} {H : Type v} [Group G] [Group H]
-
-def graphHom {A : Subgroup G} {B : Subgroup H} (δ : A ≃* B) : A →* G × H :=
-  A.subtype.prod (B.subtype.comp δ.toMonoidHom)
-
-@[simp] theorem graphHom_apply {A : Subgroup G} {B : Subgroup H} (δ : A ≃* B)
-    (a : A) : graphHom δ a = ((a : G), (δ a : H)) := rfl
-
-def graphSubgroup {A : Subgroup G} {B : Subgroup H} (δ : A ≃* B) : Subgroup (G × H) :=
-  (graphHom δ).range
-
-@[simp] theorem mem_graphSubgroup {A : Subgroup G} {B : Subgroup H} (δ : A ≃* B)
-    (s : G × H) : s ∈ graphSubgroup δ ↔ ∃ a : A, ((a : G), (δ a : H)) = s := Iff.rfl
-
-theorem graph_kernel_left {A : Subgroup G} {B : Subgroup H} (δ : A ≃* B)
-    {g : G} (hg : (g, (1 : H)) ∈ graphSubgroup δ) : g = 1 := by
-  obtain ⟨a, ha⟩ := (mem_graphSubgroup δ _).1 hg
-  have hδ : δ a = 1 := Subtype.ext (congrArg Prod.snd ha)
-  have ha₁ : a = 1 := δ.injective (hδ.trans (map_one δ).symm)
-  simpa [ha₁] using (congrArg Prod.fst ha).symm
-
-theorem graph_kernel_right {A : Subgroup G} {B : Subgroup H} (δ : A ≃* B)
-    {h : H} (hh : ((1 : G), h) ∈ graphSubgroup δ) : h = 1 := by
-  obtain ⟨a, ha⟩ := (mem_graphSubgroup δ _).1 hh
-  have ha₁ : a = 1 := Subtype.ext (congrArg Prod.fst ha)
-  simpa [ha₁] using (congrArg Prod.snd ha).symm
-
-theorem graph_map_fst {A : Subgroup G} {B : Subgroup H} (δ : A ≃* B) :
-    (graphSubgroup δ).map (MonoidHom.fst G H) = A := by
-  ext g
-  constructor
-  · rintro ⟨s, hs, rfl⟩
-    obtain ⟨a, rfl⟩ := (mem_graphSubgroup δ _).1 hs
-    exact a.property
-  · intro hg
-    refine ⟨((g : G), (δ ⟨g, hg⟩ : H)), ?_, rfl⟩
-    exact (mem_graphSubgroup δ _).2 ⟨⟨g, hg⟩, rfl⟩
-
-theorem graph_map_snd {A : Subgroup G} {B : Subgroup H} (δ : A ≃* B) :
-    (graphSubgroup δ).map (MonoidHom.snd G H) = B := by
-  ext h
-  constructor
-  · rintro ⟨s, hs, rfl⟩
-    obtain ⟨a, rfl⟩ := (mem_graphSubgroup δ _).1 hs
-    exact (δ a).property
-  · intro hh
-    refine ⟨((δ.symm ⟨h, hh⟩ : G), h), ?_, rfl⟩
-    refine (mem_graphSubgroup δ _).2 ⟨δ.symm ⟨h, hh⟩, ?_⟩
-    simp
-
-theorem joint_pullback_graph (μ : ScalarCocycle G) (ω : ScalarCocycle H)
-    {A : Subgroup G} {B : Subgroup H} (δ : A ≃* B) :
-    (μ.joint ω).pullback (graphHom δ) = μ.matchedRatio ω δ := by
-  ext a b
-  rfl
-
-theorem graph_multiplier {V : Type w} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-    {μ : ScalarCocycle G} {ω : ScalarCocycle H}
-    (π : ProjectiveUnitaryRepresentation μ V)
-    (ρ : ProjectiveUnitaryRepresentation ω.inv V)
-    (comm : ∀ g h ξ, ρ.op h (π.op g ξ) = π.op g (ρ.op h ξ))
-    {A : Subgroup G} {B : Subgroup H} (δ : A ≃* B) (a b : A) (ξ : V) :
-    π.op a (ρ.op (δ a) (π.op b (ρ.op (δ b) ξ))) =
-      ((μ a b : ℂ) / (ω (δ a) (δ b) : ℂ)) •
-        π.op (a * b) (ρ.op (δ (a * b)) ξ) := by
-  simpa [ProjectiveUnitaryRepresentation.joint, div_eq_mul_inv, map_mul] using
-    (π.joint ρ comm).op_mul (graphHom δ a) (graphHom δ b) ξ
-
-theorem exists_graph_of_coordinate_kernels (C : Subgroup (G × H))
-    (hG : ∀ g : G, (g, (1 : H)) ∈ C → g = 1)
-    (hH : ∀ h : H, ((1 : G), h) ∈ C → h = 1) :
-    ∃ δ : C.map (MonoidHom.fst G H) ≃* C.map (MonoidHom.snd G H),
-      C = graphSubgroup δ := by
-  let p := (MonoidHom.fst G H).subgroupMap C
-  let q := (MonoidHom.snd G H).subgroupMap C
-  have hp : Function.Injective p := by
-    apply (injective_iff_map_eq_one p).2
-    intro c hc
-    have hc₁ : (c : G × H).1 = 1 := congrArg Subtype.val hc
-    have hc₂ : (c : G × H).2 = 1 := by
-      apply hH
-      simpa only [← hc₁, Prod.mk.eta] using c.property
-    apply Subtype.ext
-    exact Prod.ext hc₁ hc₂
-  have hq : Function.Injective q := by
-    apply (injective_iff_map_eq_one q).2
-    intro c hc
-    have hc₂ : (c : G × H).2 = 1 := congrArg Subtype.val hc
-    have hc₁ : (c : G × H).1 = 1 := by
-      apply hG
-      simpa only [← hc₂, Prod.mk.eta] using c.property
-    apply Subtype.ext
-    exact Prod.ext hc₁ hc₂
-  let ep : C ≃* C.map (MonoidHom.fst G H) :=
-    MulEquiv.ofBijective p ⟨hp, (MonoidHom.fst G H).subgroupMap_surjective C⟩
-  let eq : C ≃* C.map (MonoidHom.snd G H) :=
-    MulEquiv.ofBijective q ⟨hq, (MonoidHom.snd G H).subgroupMap_surjective C⟩
-  let δ := ep.symm.trans eq
-  refine ⟨δ, ?_⟩
-  ext s
-  rw [mem_graphSubgroup]
-  constructor
-  · intro hs
-    let c : C := ⟨s, hs⟩
-    refine ⟨ep c, ?_⟩
-    have hc : δ (ep c) = eq c := by simp [δ]
-    simp only [hc]
-    rfl
-  · rintro ⟨a, ha⟩
-    have hc : ((ep.symm a : C) : G × H) = ((a : G), (δ a : H)) := by
-      apply Prod.ext
-      · exact congrArg Subtype.val (ep.apply_symm_apply a)
-      · rfl
-    rw [← ha, ← hc]
-    exact (ep.symm a).property
-
-theorem stabilizer_is_graph {X : Type w} [MulAction (G × H) X] (x : X)
-    (hG : ∀ g : G, (g, (1 : H)) • x = x → g = 1)
-    (hH : ∀ h : H, ((1 : G), h) • x = x → h = 1) :
-    ∃ δ : (MulAction.stabilizer (G × H) x).map (MonoidHom.fst G H) ≃*
-        (MulAction.stabilizer (G × H) x).map (MonoidHom.snd G H),
-      MulAction.stabilizer (G × H) x = graphSubgroup δ := by
-  apply exists_graph_of_coordinate_kernels
-  · exact hG
-  · exact hH
-
-end Graphs
-
-section GraphCosets
-
-variable {G : Type u} {H : Type v} [Group G] [Group H]
-
-theorem graph_cosets_free_left {A : Subgroup G} {B : Subgroup H} (δ : A ≃* B)
-    (g : G) (x : (G × H) ⧸ graphSubgroup δ)
-    (hg : (g, (1 : H)) • x = x) : g = 1 := by
-  induction x using Quotient.inductionOn with
-  | h r =>
-    change (((g, (1 : H)) * r : G × H) : (G × H) ⧸ graphSubgroup δ) =
-      (r : (G × H) ⧸ graphSubgroup δ) at hg
-    have hr := QuotientGroup.eq.mp hg
-    have heq : ((g, (1 : H)) * r)⁻¹ * r = (r.1⁻¹ * g⁻¹ * r.1, (1 : H)) := by
-      apply Prod.ext <;> simp [mul_inv_rev, mul_assoc]
-    have hk : (r.1⁻¹ * g⁻¹ * r.1, (1 : H)) ∈ graphSubgroup δ := heq ▸ hr
-    have hk₁ := graph_kernel_left δ hk
-    have hi : g⁻¹ = 1 := by
-      calc
-        g⁻¹ = r.1 * (r.1⁻¹ * g⁻¹ * r.1) * r.1⁻¹ := by group
-        _ = 1 := by rw [hk₁]; group
-    simpa using hi
-
-theorem graph_cosets_free_right {A : Subgroup G} {B : Subgroup H} (δ : A ≃* B)
-    (h : H) (x : (G × H) ⧸ graphSubgroup δ)
-    (hh : ((1 : G), h) • x = x) : h = 1 := by
-  induction x using Quotient.inductionOn with
-  | h r =>
-    change ((((1 : G), h) * r : G × H) : (G × H) ⧸ graphSubgroup δ) =
-      (r : (G × H) ⧸ graphSubgroup δ) at hh
-    have hr := QuotientGroup.eq.mp hh
-    have heq : (((1 : G), h) * r)⁻¹ * r = ((1 : G), r.2⁻¹ * h⁻¹ * r.2) := by
-      apply Prod.ext <;> simp [mul_inv_rev, mul_assoc]
-    have hk : ((1 : G), r.2⁻¹ * h⁻¹ * r.2) ∈ graphSubgroup δ := heq ▸ hr
-    have hk₂ := graph_kernel_right δ hk
-    have hi : h⁻¹ = 1 := by
-      calc
-        h⁻¹ = r.2 * (r.2⁻¹ * h⁻¹ * r.2) * r.2⁻¹ := by group
-        _ = 1 := by rw [hk₂]; group
-    simpa using hi
-
-def leftCosetOrbitRel (C : Subgroup (G × H)) : Setoid ((G × H) ⧸ C) :=
-  letI := MulAction.compHom ((G × H) ⧸ C) (MonoidHom.inl G H)
-  MulAction.orbitRel G ((G × H) ⧸ C)
-
-@[simp] theorem leftCosetOrbitRel_iff (C : Subgroup (G × H)) (x y : (G × H) ⧸ C) :
-    leftCosetOrbitRel C x y ↔ ∃ g : G, (g, (1 : H)) • y = x := Iff.rfl
-
-def sndCosetProjection (C : Subgroup (G × H)) :
-    (G × H) ⧸ C → H ⧸ C.map (MonoidHom.snd G H) :=
-  Quotient.lift (fun r : G × H => (r.2 : H ⧸ C.map (MonoidHom.snd G H))) (by
-    intro r s hrs
-    apply QuotientGroup.eq.mpr
-    exact ⟨r⁻¹ * s, QuotientGroup.leftRel_apply.mp hrs, rfl⟩)
-
-@[simp] theorem sndCosetProjection_mk (C : Subgroup (G × H)) (r : G × H) :
-    sndCosetProjection C (r : (G × H) ⧸ C) =
-      (r.2 : H ⧸ C.map (MonoidHom.snd G H)) := rfl
-
-@[simp] theorem sndCosetProjection_smul (C : Subgroup (G × H)) (g : G)
-    (x : (G × H) ⧸ C) :
-    sndCosetProjection C ((g, (1 : H)) • x) = sndCosetProjection C x := by
-  induction x using Quotient.inductionOn with
-  | h r =>
-    change ((((g, (1 : H)) * r).2 : H) : H ⧸ C.map (MonoidHom.snd G H)) =
-      (r.2 : H ⧸ C.map (MonoidHom.snd G H))
-    simp
-
-theorem sndCosetProjection_eq_iff (C : Subgroup (G × H)) (x y : (G × H) ⧸ C) :
-    sndCosetProjection C x = sndCosetProjection C y ↔ leftCosetOrbitRel C x y := by
-  constructor
-  · induction x using Quotient.inductionOn with
-    | h r =>
-      induction y using Quotient.inductionOn with
-      | h s =>
-        intro hrs
-        obtain ⟨c, hc, hc₂⟩ := QuotientGroup.eq.mp hrs
-        refine ⟨r.1 * c.1 * s.1⁻¹, ?_⟩
-        change ((((r.1 * c.1 * s.1⁻¹, (1 : H)) * s : G × H)) : (G × H) ⧸ C) =
-          (r : (G × H) ⧸ C)
-        apply Eq.symm
-        apply QuotientGroup.eq.mpr
-        have heq : r⁻¹ * ((r.1 * c.1 * s.1⁻¹, (1 : H)) * s) = c := by
-          apply Prod.ext
-          · simp [mul_assoc]
-          · simpa using hc₂.symm
-        rw [heq]
-        exact hc
-  · rintro ⟨g, rfl⟩
-    exact sndCosetProjection_smul C g y
-
-def leftCosetOrbitEquiv (C : Subgroup (G × H)) :
-    Quotient (leftCosetOrbitRel C) ≃ H ⧸ C.map (MonoidHom.snd G H) := by
-  let f : Quotient (leftCosetOrbitRel C) → H ⧸ C.map (MonoidHom.snd G H) :=
-    Quotient.lift (sndCosetProjection C)
-      (fun x y hxy => (sndCosetProjection_eq_iff C x y).mpr hxy)
-  apply Equiv.ofBijective f
-  constructor
-  · intro x y
-    induction x using Quotient.inductionOn with
-    | h r =>
-      induction y using Quotient.inductionOn with
-      | h s =>
-        intro hrs
-        exact Quotient.sound ((sndCosetProjection_eq_iff C r s).mp hrs)
-  · intro y
-    induction y using Quotient.inductionOn with
-    | h h =>
-      exact ⟨Quotient.mk _ (((1 : G), h) : (G × H) ⧸ C), rfl⟩
-
-theorem graph_left_orbit_count {A : Subgroup G} {B : Subgroup H} (δ : A ≃* B) :
-    Nat.card (Quotient (leftCosetOrbitRel (graphSubgroup δ))) = B.index := by
-  calc
-    _ = Nat.card (H ⧸ (graphSubgroup δ).map (MonoidHom.snd G H)) :=
-      Nat.card_congr (leftCosetOrbitEquiv (graphSubgroup δ))
-    _ = B.index := by rw [graph_map_snd]; rfl
-
-theorem snd_finiteIndex_of_finite_orbits (C : Subgroup (G × H))
-    [Finite (Quotient (leftCosetOrbitRel C))] :
-    (C.map (MonoidHom.snd G H)).FiniteIndex := by
-  have : Finite (H ⧸ C.map (MonoidHom.snd G H)) :=
-    Finite.of_equiv (Quotient (leftCosetOrbitRel C)) (leftCosetOrbitEquiv C)
-  exact Subgroup.finiteIndex_of_finite_quotient
-
-def rightCosetOrbitRel (C : Subgroup (G × H)) : Setoid ((G × H) ⧸ C) :=
-  letI := MulAction.compHom ((G × H) ⧸ C) (MonoidHom.inr G H)
-  MulAction.orbitRel H ((G × H) ⧸ C)
-
-@[simp] theorem rightCosetOrbitRel_iff (C : Subgroup (G × H)) (x y : (G × H) ⧸ C) :
-    rightCosetOrbitRel C x y ↔ ∃ h : H, ((1 : G), h) • y = x := Iff.rfl
-
-def fstCosetProjection (C : Subgroup (G × H)) :
-    (G × H) ⧸ C → G ⧸ C.map (MonoidHom.fst G H) :=
-  Quotient.lift (fun r : G × H => (r.1 : G ⧸ C.map (MonoidHom.fst G H))) (by
-    intro r s hrs
-    apply QuotientGroup.eq.mpr
-    exact ⟨r⁻¹ * s, QuotientGroup.leftRel_apply.mp hrs, rfl⟩)
-
-@[simp] theorem fstCosetProjection_mk (C : Subgroup (G × H)) (r : G × H) :
-    fstCosetProjection C (r : (G × H) ⧸ C) =
-      (r.1 : G ⧸ C.map (MonoidHom.fst G H)) := rfl
-
-@[simp] theorem fstCosetProjection_smul (C : Subgroup (G × H)) (h : H)
-    (x : (G × H) ⧸ C) :
-    fstCosetProjection C (((1 : G), h) • x) = fstCosetProjection C x := by
-  induction x using Quotient.inductionOn with
-  | h r =>
-    change (((((1 : G), h) * r).1 : G) : G ⧸ C.map (MonoidHom.fst G H)) =
-      (r.1 : G ⧸ C.map (MonoidHom.fst G H))
-    simp
-
-theorem fstCosetProjection_eq_iff (C : Subgroup (G × H)) (x y : (G × H) ⧸ C) :
-    fstCosetProjection C x = fstCosetProjection C y ↔ rightCosetOrbitRel C x y := by
-  constructor
-  · induction x using Quotient.inductionOn with
-    | h r =>
-      induction y using Quotient.inductionOn with
-      | h s =>
-        intro hrs
-        obtain ⟨c, hc, hc₁⟩ := QuotientGroup.eq.mp hrs
-        refine ⟨r.2 * c.2 * s.2⁻¹, ?_⟩
-        change (((((1 : G), r.2 * c.2 * s.2⁻¹) * s : G × H)) : (G × H) ⧸ C) =
-          (r : (G × H) ⧸ C)
-        apply Eq.symm
-        apply QuotientGroup.eq.mpr
-        have heq : r⁻¹ * (((1 : G), r.2 * c.2 * s.2⁻¹) * s) = c := by
-          apply Prod.ext
-          · simpa using hc₁.symm
-          · simp [mul_assoc]
-        rw [heq]
-        exact hc
-  · rintro ⟨h, rfl⟩
-    exact fstCosetProjection_smul C h y
-
-def rightCosetOrbitEquiv (C : Subgroup (G × H)) :
-    Quotient (rightCosetOrbitRel C) ≃ G ⧸ C.map (MonoidHom.fst G H) := by
-  let f : Quotient (rightCosetOrbitRel C) → G ⧸ C.map (MonoidHom.fst G H) :=
-    Quotient.lift (fstCosetProjection C)
-      (fun x y hxy => (fstCosetProjection_eq_iff C x y).mpr hxy)
-  apply Equiv.ofBijective f
-  constructor
-  · intro x y
-    induction x using Quotient.inductionOn with
-    | h r =>
-      induction y using Quotient.inductionOn with
-      | h s =>
-        intro hrs
-        exact Quotient.sound ((fstCosetProjection_eq_iff C r s).mp hrs)
-  · intro y
-    induction y using Quotient.inductionOn with
-    | h g =>
-      exact ⟨Quotient.mk _ ((g, (1 : H)) : (G × H) ⧸ C), rfl⟩
-
-theorem graph_right_orbit_count {A : Subgroup G} {B : Subgroup H} (δ : A ≃* B) :
-    Nat.card (Quotient (rightCosetOrbitRel (graphSubgroup δ))) = A.index := by
-  calc
-    _ = Nat.card (G ⧸ (graphSubgroup δ).map (MonoidHom.fst G H)) :=
-      Nat.card_congr (rightCosetOrbitEquiv (graphSubgroup δ))
-    _ = A.index := by rw [graph_map_fst]; rfl
-
-theorem fst_finiteIndex_of_finite_orbits (C : Subgroup (G × H))
-    [Finite (Quotient (rightCosetOrbitRel C))] :
-    (C.map (MonoidHom.fst G H)).FiniteIndex := by
-  have : Finite (G ⧸ C.map (MonoidHom.fst G H)) :=
-    Finite.of_equiv (Quotient (rightCosetOrbitRel C)) (rightCosetOrbitEquiv C)
-  exact Subgroup.finiteIndex_of_finite_quotient
-
-theorem finiteIndex_graph_of_coordinate_kernels (C : Subgroup (G × H))
-    (hG : ∀ g : G, (g, (1 : H)) ∈ C → g = 1)
-    (hH : ∀ h : H, ((1 : G), h) ∈ C → h = 1)
-    [Finite (Quotient (leftCosetOrbitRel C))]
-    [Finite (Quotient (rightCosetOrbitRel C))] :
-    ∃ (A : Subgroup G) (B : Subgroup H),
-      A.FiniteIndex ∧ B.FiniteIndex ∧ ∃ δ : A ≃* B, C = graphSubgroup δ := by
-  exact ⟨C.map (MonoidHom.fst G H), C.map (MonoidHom.snd G H),
-    fst_finiteIndex_of_finite_orbits C, snd_finiteIndex_of_finite_orbits C,
-    exists_graph_of_coordinate_kernels C hG hH⟩
-
-end GraphCosets
-
-namespace ScalarCocycle
-variable {D : Type*} [Group D]
-
-theorem induction_scalar (α : ScalarCocycle D) (s t r₀ r₁ r₂ : D) :
-    let a := r₂⁻¹ * s * r₁
-    let b := r₁⁻¹ * t * r₀
-    (α s r₁ / α r₂ a) * (α t r₀ / α r₁ b) * α a b =
-      α s t * (α (s * t) r₀ / α r₂ (a * b)) := by
-  dsimp only
-  let a := r₂⁻¹ * s * r₁
-  let b := r₁⁻¹ * t * r₀
-  change (α s r₁ / α r₂ a) * (α t r₀ / α r₁ b) * α a b =
-    α s t * (α (s * t) r₀ / α r₂ (a * b))
-  have ha : r₂ * a = s * r₁ := by dsimp [a]; group
-  have hb : r₁ * b = t * r₀ := by dsimp [b]; group
-  have h₃ := α.cocycle r₂ a b
-  rw [ha] at h₃
-  have h₂ := α.cocycle s r₁ b
-  rw [hb] at h₂
-  have hx : α a b = (α r₂ a * α (s * r₁) b) / α r₂ (a * b) :=
-    eq_div_iff_mul_eq'.mpr h₃.symm
-  calc
-    _ = α t r₀ * ((α s r₁ * α (s * r₁) b) / α r₁ b) / α r₂ (a * b) := by
-      rw [hx]
-      simp [div_eq_mul_inv, mul_comm, mul_left_comm, mul_assoc]
-    _ = (α t r₀ * α s (t * r₀)) / α r₂ (a * b) := by
-      rw [h₂]
-      simp [div_eq_mul_inv, mul_comm, mul_assoc]
-    _ = (α s t * α (s * t) r₀) / α r₂ (a * b) := by rw [← α.cocycle]
-    _ = _ := by simp only [mul_div_assoc]
-end ScalarCocycle
-
-def circleSmul {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-    (z : Circle) : V ≃ₗᵢ[ℂ] V where
-  toFun ξ := (z : ℂ) • ξ
-  invFun ξ := ((z⁻¹ : Circle) : ℂ) • ξ
-  left_inv ξ := by simp [smul_smul]
-  right_inv ξ := by simp [smul_smul]
-  map_add' ξ η := smul_add _ _ _
-  map_smul' c ξ := by simp only [RingHom.id_apply, smul_smul, mul_comm]
-  norm_map' ξ := by simp [norm_smul, Circle.norm_coe]
-
-@[simp] theorem circleSmul_apply {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-    (z : Circle) (ξ : V) : circleSmul z ξ = (z : ℂ) • ξ := rfl
-
-structure CosetSection {D : Type*} [Group D] (C : Subgroup D) where
-  rep : D ⧸ C → D
-  rep_spec : ∀ x, (rep x : D ⧸ C) = x
-  rep_one : rep ((1 : D) : D ⧸ C) = 1
-
-namespace CosetSection
-variable {D : Type*} [Group D] {C : Subgroup D}
-
-noncomputable def choose (C : Subgroup D) : CosetSection C := by
-  classical
-  exact {
-    rep := fun x => if x = ((1 : D) : D ⧸ C) then 1 else x.out
-    rep_spec := fun x => by
-      split_ifs with h
-      · exact h.symm
-      · exact Quotient.out_eq' x
-    rep_one := by simp }
-
-def correction (r : CosetSection C) (s : D) (x : D ⧸ C) : C := by
-  refine ⟨(r.rep (s • x))⁻¹ * s * r.rep x, ?_⟩
-  have heq : (r.rep (s • x) : D ⧸ C) = (s * r.rep x : D) := by
-    rw [r.rep_spec]
-    change s • x = s • (r.rep x : D ⧸ C)
-    rw [r.rep_spec]
-  simpa only [mul_assoc] using QuotientGroup.eq.mp heq
-
-@[simp] theorem correction_coe (r : CosetSection C) (s : D) (x : D ⧸ C) :
-    (r.correction s x : D) = (r.rep (s • x))⁻¹ * s * r.rep x := rfl
-
-@[simp] theorem correction_one (r : CosetSection C) (x : D ⧸ C) :
-    r.correction 1 x = 1 := by
-  apply Subtype.ext
-  simp
-
-theorem correction_mul (r : CosetSection C) (s t : D) (x : D ⧸ C) :
-    r.correction s (t • x) * r.correction t x = r.correction (s * t) x := by
-  apply Subtype.ext
-  simp only [Subgroup.coe_mul, correction_coe, mul_smul]
-  group
-
-def transportScalar (r : CosetSection C) (α : ScalarCocycle D) (s : D) (x : D ⧸ C) :
-    Circle := α s (r.rep x) / α (r.rep (s • x)) (r.correction s x)
-
-@[simp] theorem transportScalar_one (r : CosetSection C) (α : ScalarCocycle D)
-    (x : D ⧸ C) : r.transportScalar α 1 x = 1 := by
-  simp [transportScalar]
-
-theorem transportScalar_mul (r : CosetSection C) (α : ScalarCocycle D)
-    (s t : D) (x : D ⧸ C) :
-    r.transportScalar α s (t • x) * r.transportScalar α t x *
-        α (r.correction s (t • x)) (r.correction t x) =
-      α s t * r.transportScalar α (s * t) x := by
-  have hc : ((r.correction s (t • x) : D) * r.correction t x) =
-      (r.correction (s * t) x : D) := congrArg Subtype.val (r.correction_mul s t x)
-  have h := α.induction_scalar s t (r.rep x) (r.rep (t • x)) (r.rep (s • t • x))
-  change r.transportScalar α s (t • x) * r.transportScalar α t x *
-      α (r.correction s (t • x)) (r.correction t x) =
-    _ at h
-  change _ = α s t * (α (s * t) (r.rep x) /
-    α (r.rep (s • t • x)) ((r.correction s (t • x) : D) * r.correction t x)) at h
-  rw [hc] at h
-  simpa only [transportScalar, mul_smul] using h
-
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
-
-def transport (r : CosetSection C) {α : ScalarCocycle D}
-    (γ : ProjectiveUnitaryRepresentation (α.pullback C.subtype) V)
-    (s : D) (x : D ⧸ C) : V ≃ₗᵢ[ℂ] V :=
-  (γ.op (r.correction s x)).trans (circleSmul (r.transportScalar α s x))
-
-@[simp] theorem transport_apply (r : CosetSection C) {α : ScalarCocycle D}
-    (γ : ProjectiveUnitaryRepresentation (α.pullback C.subtype) V)
-    (s : D) (x : D ⧸ C) (ξ : V) :
-    r.transport γ s x ξ =
-      (r.transportScalar α s x : ℂ) • (γ.op (r.correction s x) ξ) := rfl
-
-@[simp] theorem transport_one (r : CosetSection C) {α : ScalarCocycle D}
-    (γ : ProjectiveUnitaryRepresentation (α.pullback C.subtype) V) (x : D ⧸ C) :
-    r.transport γ 1 x = LinearIsometryEquiv.refl ℂ V := by
-  ext ξ
-  simp [γ.op_one]
-
-theorem transport_mul (r : CosetSection C) {α : ScalarCocycle D}
-    (γ : ProjectiveUnitaryRepresentation (α.pullback C.subtype) V)
-    (s t : D) (x : D ⧸ C) (ξ : V) :
-    r.transport γ s (t • x) (r.transport γ t x ξ) =
-      (α s t : ℂ) • r.transport γ (s * t) x ξ := by
-  have hscalar : (r.transportScalar α s (t • x) : ℂ) *
-      (r.transportScalar α t x : ℂ) *
-      (α (r.correction s (t • x)) (r.correction t x) : ℂ) =
-      (α s t : ℂ) * (r.transportScalar α (s * t) x : ℂ) := by
-    simpa only [Circle.coe_mul] using
-      congrArg (fun z : Circle => (z : ℂ)) (r.transportScalar_mul α s t x)
-  change (r.transportScalar α s (t • x) : ℂ) •
-      γ.op (r.correction s (t • x))
-        ((r.transportScalar α t x : ℂ) • γ.op (r.correction t x) ξ) =
-    (α s t : ℂ) • ((r.transportScalar α (s * t) x : ℂ) •
-      γ.op (r.correction (s * t) x) ξ)
-  rw [map_smul, γ.op_mul]
-  simp only [ScalarCocycle.pullback_apply, Subgroup.coe_subtype, smul_smul]
-  rw [← mul_assoc, hscalar, r.correction_mul]
-end CosetSection
-
-end TwistedCorrespondences
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-! Projective coset transport and subgroup graph orbit calculations. -/
+
+noncomputable section
+
+namespace TwistedCorrespondences
+
+universe u v w
+
+structure ScalarCocycle (G : Type u) [Group G] where
+  toFun : G → G → Circle
+  one_left' : ∀ g, toFun 1 g = 1
+  one_right' : ∀ g, toFun g 1 = 1
+  cocycle' : ∀ g h k, toFun g h * toFun (g * h) k = toFun h k * toFun g (h * k)
+
+namespace ScalarCocycle
+
+variable {G : Type u} {H : Type v} {K : Type w}
+variable [Group G] [Group H] [Group K]
+
+instance : CoeFun (ScalarCocycle G) (fun _ => G → G → Circle) := ⟨toFun⟩
+
+@[simp] theorem one_left (α : ScalarCocycle G) (g : G) : α 1 g = 1 := α.one_left' g
+@[simp] theorem one_right (α : ScalarCocycle G) (g : G) : α g 1 = 1 := α.one_right' g
+
+theorem cocycle (α : ScalarCocycle G) (g h k : G) :
+    α g h * α (g * h) k = α h k * α g (h * k) := α.cocycle' g h k
+
+@[ext] theorem ext {α β : ScalarCocycle G} (h : ∀ g k, α g k = β g k) : α = β := by
+  cases α
+  cases β
+  congr
+  exact funext fun g => funext (h g)
+
+def trivial (G : Type u) [Group G] : ScalarCocycle G where
+  toFun _ _ := 1
+  one_left' _ := rfl
+  one_right' _ := rfl
+  cocycle' _ _ _ := rfl
+
+@[simp] theorem trivial_apply (g h : G) : trivial G g h = 1 := rfl
+
+def pullback (α : ScalarCocycle H) (f : G →* H) : ScalarCocycle G where
+  toFun g h := α (f g) (f h)
+  one_left' g := by simp
+  one_right' g := by simp
+  cocycle' g h k := by simpa using α.cocycle (f g) (f h) (f k)
+
+@[simp] theorem pullback_apply (α : ScalarCocycle H) (f : G →* H) (g h : G) :
+    α.pullback f g h = α (f g) (f h) := rfl
+
+def mul (α β : ScalarCocycle G) : ScalarCocycle G where
+  toFun g h := α g h * β g h
+  one_left' g := by simp
+  one_right' g := by simp
+  cocycle' g h k := by
+    calc
+      (α g h * β g h) * (α (g * h) k * β (g * h) k) =
+          (α g h * α (g * h) k) * (β g h * β (g * h) k) := by ac_rfl
+      _ = (α h k * α g (h * k)) * (β h k * β g (h * k)) := by
+        rw [α.cocycle, β.cocycle]
+      _ = (α h k * β h k) * (α g (h * k) * β g (h * k)) := by ac_rfl
+
+@[simp] theorem mul_apply (α β : ScalarCocycle G) (g h : G) :
+    α.mul β g h = α g h * β g h := rfl
+
+def inv (α : ScalarCocycle G) : ScalarCocycle G where
+  toFun g h := (α g h)⁻¹
+  one_left' g := by simp
+  one_right' g := by simp
+  cocycle' g h k := by
+    simpa only [mul_inv] using congrArg Inv.inv (α.cocycle g h k)
+
+@[simp] theorem inv_apply (α : ScalarCocycle G) (g h : G) :
+    α.inv g h = (α g h)⁻¹ := rfl
+
+theorem inv_apply_coe (α : ScalarCocycle G) (g h : G) :
+    (α.inv g h : ℂ) = star (α g h : ℂ) := Circle.coe_inv_eq_conj _
+
+def joint (μ : ScalarCocycle G) (ω : ScalarCocycle H) : ScalarCocycle (G × H) :=
+  (μ.pullback (MonoidHom.fst G H)).mul ((ω.pullback (MonoidHom.snd G H)).inv)
+
+@[simp] theorem joint_apply (μ : ScalarCocycle G) (ω : ScalarCocycle H)
+    (g g' : G) (h h' : H) :
+    joint μ ω (g, h) (g', h') = μ g g' * (ω h h')⁻¹ := rfl
+
+theorem joint_apply_coe (μ : ScalarCocycle G) (ω : ScalarCocycle H)
+    (g g' : G) (h h' : H) :
+    (joint μ ω (g, h) (g', h') : ℂ) = (μ g g' : ℂ) * star (ω h h' : ℂ) := by
+  simp only [joint_apply, Circle.coe_mul, Circle.coe_inv_eq_conj, Complex.star_def]
+
+@[simp] theorem joint_mixed (μ : ScalarCocycle G) (ω : ScalarCocycle H)
+    (g : G) (h : H) : joint μ ω (g, 1) (1, h) = 1 := by simp
+
+@[simp] theorem joint_mixed_rev (μ : ScalarCocycle G) (ω : ScalarCocycle H)
+    (g : G) (h : H) : joint μ ω (1, h) (g, 1) = 1 := by simp
+
+def matchedRatio (μ : ScalarCocycle G) (ω : ScalarCocycle H)
+    {A : Subgroup G} {B : Subgroup H} (δ : A ≃* B) : ScalarCocycle A :=
+  (μ.pullback A.subtype).mul ((ω.pullback (B.subtype.comp δ.toMonoidHom)).inv)
+
+@[simp] theorem matchedRatio_apply (μ : ScalarCocycle G) (ω : ScalarCocycle H)
+    {A : Subgroup G} {B : Subgroup H} (δ : A ≃* B) (a b : A) :
+    matchedRatio μ ω δ a b = μ a b / ω (δ a) (δ b) := rfl
+
+end ScalarCocycle
+
+structure ProjectiveUnitaryRepresentation {G : Type u} [Group G]
+    (α : ScalarCocycle G) (V : Type v) [NormedAddCommGroup V] [InnerProductSpace ℂ V] where
+  op : G → V ≃ₗᵢ[ℂ] V
+  op_one : op 1 = LinearIsometryEquiv.refl ℂ V
+  op_mul : ∀ g h ξ, op g (op h ξ) = (α g h : ℂ) • op (g * h) ξ
+
+namespace ProjectiveUnitaryRepresentation
+
+variable {G : Type u} {H : Type v} [Group G] [Group H]
+variable {V : Type w} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+
+def pullback {α : ScalarCocycle H} (π : ProjectiveUnitaryRepresentation α V)
+    (f : G →* H) : ProjectiveUnitaryRepresentation (α.pullback f) V where
+  op g := π.op (f g)
+  op_one := by simpa using π.op_one
+  op_mul g h ξ := by simpa using π.op_mul (f g) (f h) ξ
+
+def joint {μ : ScalarCocycle G} {ω : ScalarCocycle H}
+    (π : ProjectiveUnitaryRepresentation μ V)
+    (ρ : ProjectiveUnitaryRepresentation ω.inv V)
+    (comm : ∀ g h ξ, ρ.op h (π.op g ξ) = π.op g (ρ.op h ξ)) :
+    ProjectiveUnitaryRepresentation (μ.joint ω) V where
+  op s := (ρ.op s.2).trans (π.op s.1)
+  op_one := by
+    ext ξ
+    simp [π.op_one, ρ.op_one]
+  op_mul s t ξ := by
+    change π.op s.1 (ρ.op s.2 (π.op t.1 (ρ.op t.2 ξ))) =
+      ((μ.joint ω) s t : ℂ) • π.op (s.1 * t.1) (ρ.op (s.2 * t.2) ξ)
+    rw [comm, π.op_mul, ρ.op_mul, map_smul, smul_smul]
+    rfl
+
+end ProjectiveUnitaryRepresentation
+
+section Graphs
+
+variable {G : Type u} {H : Type v} [Group G] [Group H]
+
+def graphHom {A : Subgroup G} {B : Subgroup H} (δ : A ≃* B) : A →* G × H :=
+  A.subtype.prod (B.subtype.comp δ.toMonoidHom)
+
+@[simp] theorem graphHom_apply {A : Subgroup G} {B : Subgroup H} (δ : A ≃* B)
+    (a : A) : graphHom δ a = ((a : G), (δ a : H)) := rfl
+
+def graphSubgroup {A : Subgroup G} {B : Subgroup H} (δ : A ≃* B) : Subgroup (G × H) :=
+  (graphHom δ).range
+
+@[simp] theorem mem_graphSubgroup {A : Subgroup G} {B : Subgroup H} (δ : A ≃* B)
+    (s : G × H) : s ∈ graphSubgroup δ ↔ ∃ a : A, ((a : G), (δ a : H)) = s := Iff.rfl
+
+theorem graph_kernel_left {A : Subgroup G} {B : Subgroup H} (δ : A ≃* B)
+    {g : G} (hg : (g, (1 : H)) ∈ graphSubgroup δ) : g = 1 := by
+  obtain ⟨a, ha⟩ := (mem_graphSubgroup δ _).1 hg
+  have hδ : δ a = 1 := Subtype.ext (congrArg Prod.snd ha)
+  have ha₁ : a = 1 := δ.injective (hδ.trans (map_one δ).symm)
+  simpa [ha₁] using (congrArg Prod.fst ha).symm
+
+theorem graph_kernel_right {A : Subgroup G} {B : Subgroup H} (δ : A ≃* B)
+    {h : H} (hh : ((1 : G), h) ∈ graphSubgroup δ) : h = 1 := by
+  obtain ⟨a, ha⟩ := (mem_graphSubgroup δ _).1 hh
+  have ha₁ : a = 1 := Subtype.ext (congrArg Prod.fst ha)
+  simpa [ha₁] using (congrArg Prod.snd ha).symm
+
+theorem graph_map_fst {A : Subgroup G} {B : Subgroup H} (δ : A ≃* B) :
+    (graphSubgroup δ).map (MonoidHom.fst G H) = A := by
+  ext g
+  constructor
+  · rintro ⟨s, hs, rfl⟩
+    obtain ⟨a, rfl⟩ := (mem_graphSubgroup δ _).1 hs
+    exact a.property
+  · intro hg
+    refine ⟨((g : G), (δ ⟨g, hg⟩ : H)), ?_, rfl⟩
+    exact (mem_graphSubgroup δ _).2 ⟨⟨g, hg⟩, rfl⟩
+
+theorem graph_map_snd {A : Subgroup G} {B : Subgroup H} (δ : A ≃* B) :
+    (graphSubgroup δ).map (MonoidHom.snd G H) = B := by
+  ext h
+  constructor
+  · rintro ⟨s, hs, rfl⟩
+    obtain ⟨a, rfl⟩ := (mem_graphSubgroup δ _).1 hs
+    exact (δ a).property
+  · intro hh
+    refine ⟨((δ.symm ⟨h, hh⟩ : G), h), ?_, rfl⟩
+    refine (mem_graphSubgroup δ _).2 ⟨δ.symm ⟨h, hh⟩, ?_⟩
+    simp
+
+theorem joint_pullback_graph (μ : ScalarCocycle G) (ω : ScalarCocycle H)
+    {A : Subgroup G} {B : Subgroup H} (δ : A ≃* B) :
+    (μ.joint ω).pullback (graphHom δ) = μ.matchedRatio ω δ := by
+  ext a b
+  rfl
+
+theorem graph_multiplier {V : Type w} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+    {μ : ScalarCocycle G} {ω : ScalarCocycle H}
+    (π : ProjectiveUnitaryRepresentation μ V)
+    (ρ : ProjectiveUnitaryRepresentation ω.inv V)
+    (comm : ∀ g h ξ, ρ.op h (π.op g ξ) = π.op g (ρ.op h ξ))
+    {A : Subgroup G} {B : Subgroup H} (δ : A ≃* B) (a b : A) (ξ : V) :
+    π.op a (ρ.op (δ a) (π.op b (ρ.op (δ b) ξ))) =
+      ((μ a b : ℂ) / (ω (δ a) (δ b) : ℂ)) •
+        π.op (a * b) (ρ.op (δ (a * b)) ξ) := by
+  simpa [ProjectiveUnitaryRepresentation.joint, div_eq_mul_inv, map_mul] using
+    (π.joint ρ comm).op_mul (graphHom δ a) (graphHom δ b) ξ
+
+theorem exists_graph_of_coordinate_kernels (C : Subgroup (G × H))
+    (hG : ∀ g : G, (g, (1 : H)) ∈ C → g = 1)
+    (hH : ∀ h : H, ((1 : G), h) ∈ C → h = 1) :
+    ∃ δ : C.map (MonoidHom.fst G H) ≃* C.map (MonoidHom.snd G H),
+      C = graphSubgroup δ := by
+  let p := (MonoidHom.fst G H).subgroupMap C
+  let q := (MonoidHom.snd G H).subgroupMap C
+  have hp : Function.Injective p := by
+    apply (injective_iff_map_eq_one p).2
+    intro c hc
+    have hc₁ : (c : G × H).1 = 1 := congrArg Subtype.val hc
+    have hc₂ : (c : G × H).2 = 1 := by
+      apply hH
+      simpa only [← hc₁, Prod.mk.eta] using c.property
+    apply Subtype.ext
+    exact Prod.ext hc₁ hc₂
+  have hq : Function.Injective q := by
+    apply (injective_iff_map_eq_one q).2
+    intro c hc
+    have hc₂ : (c : G × H).2 = 1 := congrArg Subtype.val hc
+    have hc₁ : (c : G × H).1 = 1 := by
+      apply hG
+      simpa only [← hc₂, Prod.mk.eta] using c.property
+    apply Subtype.ext
+    exact Prod.ext hc₁ hc₂
+  let ep : C ≃* C.map (MonoidHom.fst G H) :=
+    MulEquiv.ofBijective p ⟨hp, (MonoidHom.fst G H).subgroupMap_surjective C⟩
+  let eq : C ≃* C.map (MonoidHom.snd G H) :=
+    MulEquiv.ofBijective q ⟨hq, (MonoidHom.snd G H).subgroupMap_surjective C⟩
+  let δ := ep.symm.trans eq
+  refine ⟨δ, ?_⟩
+  ext s
+  rw [mem_graphSubgroup]
+  constructor
+  · intro hs
+    let c : C := ⟨s, hs⟩
+    refine ⟨ep c, ?_⟩
+    have hc : δ (ep c) = eq c := by simp [δ]
+    simp only [hc]
+    rfl
+  · rintro ⟨a, ha⟩
+    have hc : ((ep.symm a : C) : G × H) = ((a : G), (δ a : H)) := by
+      apply Prod.ext
+      · exact congrArg Subtype.val (ep.apply_symm_apply a)
+      · rfl
+    rw [← ha, ← hc]
+    exact (ep.symm a).property
+
+theorem stabilizer_is_graph {X : Type w} [MulAction (G × H) X] (x : X)
+    (hG : ∀ g : G, (g, (1 : H)) • x = x → g = 1)
+    (hH : ∀ h : H, ((1 : G), h) • x = x → h = 1) :
+    ∃ δ : (MulAction.stabilizer (G × H) x).map (MonoidHom.fst G H) ≃*
+        (MulAction.stabilizer (G × H) x).map (MonoidHom.snd G H),
+      MulAction.stabilizer (G × H) x = graphSubgroup δ := by
+  apply exists_graph_of_coordinate_kernels
+  · exact hG
+  · exact hH
+
+end Graphs
+
+section GraphCosets
+
+variable {G : Type u} {H : Type v} [Group G] [Group H]
+
+theorem graph_cosets_free_left {A : Subgroup G} {B : Subgroup H} (δ : A ≃* B)
+    (g : G) (x : (G × H) ⧸ graphSubgroup δ)
+    (hg : (g, (1 : H)) • x = x) : g = 1 := by
+  induction x using Quotient.inductionOn with
+  | h r =>
+    change (((g, (1 : H)) * r : G × H) : (G × H) ⧸ graphSubgroup δ) =
+      (r : (G × H) ⧸ graphSubgroup δ) at hg
+    have hr := QuotientGroup.eq.mp hg
+    have heq : ((g, (1 : H)) * r)⁻¹ * r = (r.1⁻¹ * g⁻¹ * r.1, (1 : H)) := by
+      apply Prod.ext <;> simp [mul_inv_rev, mul_assoc]
+    have hk : (r.1⁻¹ * g⁻¹ * r.1, (1 : H)) ∈ graphSubgroup δ := heq ▸ hr
+    have hk₁ := graph_kernel_left δ hk
+    have hi : g⁻¹ = 1 := by
+      calc
+        g⁻¹ = r.1 * (r.1⁻¹ * g⁻¹ * r.1) * r.1⁻¹ := by group
+        _ = 1 := by rw [hk₁]; group
+    simpa using hi
+
+theorem graph_cosets_free_right {A : Subgroup G} {B : Subgroup H} (δ : A ≃* B)
+    (h : H) (x : (G × H) ⧸ graphSubgroup δ)
+    (hh : ((1 : G), h) • x = x) : h = 1 := by
+  induction x using Quotient.inductionOn with
+  | h r =>
+    change ((((1 : G), h) * r : G × H) : (G × H) ⧸ graphSubgroup δ) =
+      (r : (G × H) ⧸ graphSubgroup δ) at hh
+    have hr := QuotientGroup.eq.mp hh
+    have heq : (((1 : G), h) * r)⁻¹ * r = ((1 : G), r.2⁻¹ * h⁻¹ * r.2) := by
+      apply Prod.ext <;> simp [mul_inv_rev, mul_assoc]
+    have hk : ((1 : G), r.2⁻¹ * h⁻¹ * r.2) ∈ graphSubgroup δ := heq ▸ hr
+    have hk₂ := graph_kernel_right δ hk
+    have hi : h⁻¹ = 1 := by
+      calc
+        h⁻¹ = r.2 * (r.2⁻¹ * h⁻¹ * r.2) * r.2⁻¹ := by group
+        _ = 1 := by rw [hk₂]; group
+    simpa using hi
+
+def leftCosetOrbitRel (C : Subgroup (G × H)) : Setoid ((G × H) ⧸ C) :=
+  letI := MulAction.compHom ((G × H) ⧸ C) (MonoidHom.inl G H)
+  MulAction.orbitRel G ((G × H) ⧸ C)
+
+@[simp] theorem leftCosetOrbitRel_iff (C : Subgroup (G × H)) (x y : (G × H) ⧸ C) :
+    leftCosetOrbitRel C x y ↔ ∃ g : G, (g, (1 : H)) • y = x := Iff.rfl
+
+def sndCosetProjection (C : Subgroup (G × H)) :
+    (G × H) ⧸ C → H ⧸ C.map (MonoidHom.snd G H) :=
+  Quotient.lift (fun r : G × H => (r.2 : H ⧸ C.map (MonoidHom.snd G H))) (by
+    intro r s hrs
+    apply QuotientGroup.eq.mpr
+    exact ⟨r⁻¹ * s, QuotientGroup.leftRel_apply.mp hrs, rfl⟩)
+
+@[simp] theorem sndCosetProjection_mk (C : Subgroup (G × H)) (r : G × H) :
+    sndCosetProjection C (r : (G × H) ⧸ C) =
+      (r.2 : H ⧸ C.map (MonoidHom.snd G H)) := rfl
+
+@[simp] theorem sndCosetProjection_smul (C : Subgroup (G × H)) (g : G)
+    (x : (G × H) ⧸ C) :
+    sndCosetProjection C ((g, (1 : H)) • x) = sndCosetProjection C x := by
+  induction x using Quotient.inductionOn with
+  | h r =>
+    change ((((g, (1 : H)) * r).2 : H) : H ⧸ C.map (MonoidHom.snd G H)) =
+      (r.2 : H ⧸ C.map (MonoidHom.snd G H))
+    simp
+
+theorem sndCosetProjection_eq_iff (C : Subgroup (G × H)) (x y : (G × H) ⧸ C) :
+    sndCosetProjection C x = sndCosetProjection C y ↔ leftCosetOrbitRel C x y := by
+  constructor
+  · induction x using Quotient.inductionOn with
+    | h r =>
+      induction y using Quotient.inductionOn with
+      | h s =>
+        intro hrs
+        obtain ⟨c, hc, hc₂⟩ := QuotientGroup.eq.mp hrs
+        refine ⟨r.1 * c.1 * s.1⁻¹, ?_⟩
+        change ((((r.1 * c.1 * s.1⁻¹, (1 : H)) * s : G × H)) : (G × H) ⧸ C) =
+          (r : (G × H) ⧸ C)
+        apply Eq.symm
+        apply QuotientGroup.eq.mpr
+        have heq : r⁻¹ * ((r.1 * c.1 * s.1⁻¹, (1 : H)) * s) = c := by
+          apply Prod.ext
+          · simp [mul_assoc]
+          · simpa using hc₂.symm
+        rw [heq]
+        exact hc
+  · rintro ⟨g, rfl⟩
+    exact sndCosetProjection_smul C g y
+
+def leftCosetOrbitEquiv (C : Subgroup (G × H)) :
+    Quotient (leftCosetOrbitRel C) ≃ H ⧸ C.map (MonoidHom.snd G H) := by
+  let f : Quotient (leftCosetOrbitRel C) → H ⧸ C.map (MonoidHom.snd G H) :=
+    Quotient.lift (sndCosetProjection C)
+      (fun x y hxy => (sndCosetProjection_eq_iff C x y).mpr hxy)
+  apply Equiv.ofBijective f
+  constructor
+  · intro x y
+    induction x using Quotient.inductionOn with
+    | h r =>
+      induction y using Quotient.inductionOn with
+      | h s =>
+        intro hrs
+        exact Quotient.sound ((sndCosetProjection_eq_iff C r s).mp hrs)
+  · intro y
+    induction y using Quotient.inductionOn with
+    | h h =>
+      exact ⟨Quotient.mk _ (((1 : G), h) : (G × H) ⧸ C), rfl⟩
+
+theorem graph_left_orbit_count {A : Subgroup G} {B : Subgroup H} (δ : A ≃* B) :
+    Nat.card (Quotient (leftCosetOrbitRel (graphSubgroup δ))) = B.index := by
+  calc
+    _ = Nat.card (H ⧸ (graphSubgroup δ).map (MonoidHom.snd G H)) :=
+      Nat.card_congr (leftCosetOrbitEquiv (graphSubgroup δ))
+    _ = B.index := by rw [graph_map_snd]; rfl
+
+theorem snd_finiteIndex_of_finite_orbits (C : Subgroup (G × H))
+    [Finite (Quotient (leftCosetOrbitRel C))] :
+    (C.map (MonoidHom.snd G H)).FiniteIndex := by
+  have : Finite (H ⧸ C.map (MonoidHom.snd G H)) :=
+    Finite.of_equiv (Quotient (leftCosetOrbitRel C)) (leftCosetOrbitEquiv C)
+  exact Subgroup.finiteIndex_of_finite_quotient
+
+def rightCosetOrbitRel (C : Subgroup (G × H)) : Setoid ((G × H) ⧸ C) :=
+  letI := MulAction.compHom ((G × H) ⧸ C) (MonoidHom.inr G H)
+  MulAction.orbitRel H ((G × H) ⧸ C)
+
+@[simp] theorem rightCosetOrbitRel_iff (C : Subgroup (G × H)) (x y : (G × H) ⧸ C) :
+    rightCosetOrbitRel C x y ↔ ∃ h : H, ((1 : G), h) • y = x := Iff.rfl
+
+def fstCosetProjection (C : Subgroup (G × H)) :
+    (G × H) ⧸ C → G ⧸ C.map (MonoidHom.fst G H) :=
+  Quotient.lift (fun r : G × H => (r.1 : G ⧸ C.map (MonoidHom.fst G H))) (by
+    intro r s hrs
+    apply QuotientGroup.eq.mpr
+    exact ⟨r⁻¹ * s, QuotientGroup.leftRel_apply.mp hrs, rfl⟩)
+
+@[simp] theorem fstCosetProjection_mk (C : Subgroup (G × H)) (r : G × H) :
+    fstCosetProjection C (r : (G × H) ⧸ C) =
+      (r.1 : G ⧸ C.map (MonoidHom.fst G H)) := rfl
+
+@[simp] theorem fstCosetProjection_smul (C : Subgroup (G × H)) (h : H)
+    (x : (G × H) ⧸ C) :
+    fstCosetProjection C (((1 : G), h) • x) = fstCosetProjection C x := by
+  induction x using Quotient.inductionOn with
+  | h r =>
+    change (((((1 : G), h) * r).1 : G) : G ⧸ C.map (MonoidHom.fst G H)) =
+      (r.1 : G ⧸ C.map (MonoidHom.fst G H))
+    simp
+
+theorem fstCosetProjection_eq_iff (C : Subgroup (G × H)) (x y : (G × H) ⧸ C) :
+    fstCosetProjection C x = fstCosetProjection C y ↔ rightCosetOrbitRel C x y := by
+  constructor
+  · induction x using Quotient.inductionOn with
+    | h r =>
+      induction y using Quotient.inductionOn with
+      | h s =>
+        intro hrs
+        obtain ⟨c, hc, hc₁⟩ := QuotientGroup.eq.mp hrs
+        refine ⟨r.2 * c.2 * s.2⁻¹, ?_⟩
+        change (((((1 : G), r.2 * c.2 * s.2⁻¹) * s : G × H)) : (G × H) ⧸ C) =
+          (r : (G × H) ⧸ C)
+        apply Eq.symm
+        apply QuotientGroup.eq.mpr
+        have heq : r⁻¹ * (((1 : G), r.2 * c.2 * s.2⁻¹) * s) = c := by
+          apply Prod.ext
+          · simpa using hc₁.symm
+          · simp [mul_assoc]
+        rw [heq]
+        exact hc
+  · rintro ⟨h, rfl⟩
+    exact fstCosetProjection_smul C h y
+
+def rightCosetOrbitEquiv (C : Subgroup (G × H)) :
+    Quotient (rightCosetOrbitRel C) ≃ G ⧸ C.map (MonoidHom.fst G H) := by
+  let f : Quotient (rightCosetOrbitRel C) → G ⧸ C.map (MonoidHom.fst G H) :=
+    Quotient.lift (fstCosetProjection C)
+      (fun x y hxy => (fstCosetProjection_eq_iff C x y).mpr hxy)
+  apply Equiv.ofBijective f
+  constructor
+  · intro x y
+    induction x using Quotient.inductionOn with
+    | h r =>
+      induction y using Quotient.inductionOn with
+      | h s =>
+        intro hrs
+        exact Quotient.sound ((fstCosetProjection_eq_iff C r s).mp hrs)
+  · intro y
+    induction y using Quotient.inductionOn with
+    | h g =>
+      exact ⟨Quotient.mk _ ((g, (1 : H)) : (G × H) ⧸ C), rfl⟩
+
+theorem graph_right_orbit_count {A : Subgroup G} {B : Subgroup H} (δ : A ≃* B) :
+    Nat.card (Quotient (rightCosetOrbitRel (graphSubgroup δ))) = A.index := by
+  calc
+    _ = Nat.card (G ⧸ (graphSubgroup δ).map (MonoidHom.fst G H)) :=
+      Nat.card_congr (rightCosetOrbitEquiv (graphSubgroup δ))
+    _ = A.index := by rw [graph_map_fst]; rfl
+
+theorem fst_finiteIndex_of_finite_orbits (C : Subgroup (G × H))
+    [Finite (Quotient (rightCosetOrbitRel C))] :
+    (C.map (MonoidHom.fst G H)).FiniteIndex := by
+  have : Finite (G ⧸ C.map (MonoidHom.fst G H)) :=
+    Finite.of_equiv (Quotient (rightCosetOrbitRel C)) (rightCosetOrbitEquiv C)
+  exact Subgroup.finiteIndex_of_finite_quotient
+
+theorem finiteIndex_graph_of_coordinate_kernels (C : Subgroup (G × H))
+    (hG : ∀ g : G, (g, (1 : H)) ∈ C → g = 1)
+    (hH : ∀ h : H, ((1 : G), h) ∈ C → h = 1)
+    [Finite (Quotient (leftCosetOrbitRel C))]
+    [Finite (Quotient (rightCosetOrbitRel C))] :
+    ∃ (A : Subgroup G) (B : Subgroup H),
+      A.FiniteIndex ∧ B.FiniteIndex ∧ ∃ δ : A ≃* B, C = graphSubgroup δ := by
+  exact ⟨C.map (MonoidHom.fst G H), C.map (MonoidHom.snd G H),
+    fst_finiteIndex_of_finite_orbits C, snd_finiteIndex_of_finite_orbits C,
+    exists_graph_of_coordinate_kernels C hG hH⟩
+
+end GraphCosets
+
+namespace ScalarCocycle
+variable {D : Type*} [Group D]
+
+theorem induction_scalar (α : ScalarCocycle D) (s t r₀ r₁ r₂ : D) :
+    let a := r₂⁻¹ * s * r₁
+    let b := r₁⁻¹ * t * r₀
+    (α s r₁ / α r₂ a) * (α t r₀ / α r₁ b) * α a b =
+      α s t * (α (s * t) r₀ / α r₂ (a * b)) := by
+  dsimp only
+  let a := r₂⁻¹ * s * r₁
+  let b := r₁⁻¹ * t * r₀
+  change (α s r₁ / α r₂ a) * (α t r₀ / α r₁ b) * α a b =
+    α s t * (α (s * t) r₀ / α r₂ (a * b))
+  have ha : r₂ * a = s * r₁ := by dsimp [a]; group
+  have hb : r₁ * b = t * r₀ := by dsimp [b]; group
+  have h₃ := α.cocycle r₂ a b
+  rw [ha] at h₃
+  have h₂ := α.cocycle s r₁ b
+  rw [hb] at h₂
+  have hx : α a b = (α r₂ a * α (s * r₁) b) / α r₂ (a * b) :=
+    eq_div_iff_mul_eq'.mpr h₃.symm
+  calc
+    _ = α t r₀ * ((α s r₁ * α (s * r₁) b) / α r₁ b) / α r₂ (a * b) := by
+      rw [hx]
+      simp [div_eq_mul_inv, mul_comm, mul_left_comm, mul_assoc]
+    _ = (α t r₀ * α s (t * r₀)) / α r₂ (a * b) := by
+      rw [h₂]
+      simp [div_eq_mul_inv, mul_comm, mul_assoc]
+    _ = (α s t * α (s * t) r₀) / α r₂ (a * b) := by rw [← α.cocycle]
+    _ = _ := by simp only [mul_div_assoc]
+end ScalarCocycle
+
+def circleSmul {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+    (z : Circle) : V ≃ₗᵢ[ℂ] V where
+  toFun ξ := (z : ℂ) • ξ
+  invFun ξ := ((z⁻¹ : Circle) : ℂ) • ξ
+  left_inv ξ := by simp [smul_smul]
+  right_inv ξ := by simp [smul_smul]
+  map_add' ξ η := smul_add _ _ _
+  map_smul' c ξ := by simp only [RingHom.id_apply, smul_smul, mul_comm]
+  norm_map' ξ := by simp [norm_smul, Circle.norm_coe]
+
+@[simp] theorem circleSmul_apply {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+    (z : Circle) (ξ : V) : circleSmul z ξ = (z : ℂ) • ξ := rfl
+
+structure CosetSection {D : Type*} [Group D] (C : Subgroup D) where
+  rep : D ⧸ C → D
+  rep_spec : ∀ x, (rep x : D ⧸ C) = x
+  rep_one : rep ((1 : D) : D ⧸ C) = 1
+
+namespace CosetSection
+variable {D : Type*} [Group D] {C : Subgroup D}
+
+noncomputable def choose (C : Subgroup D) : CosetSection C := by
+  classical
+  exact {
+    rep := fun x => if x = ((1 : D) : D ⧸ C) then 1 else x.out
+    rep_spec := fun x => by
+      split_ifs with h
+      · exact h.symm
+      · exact Quotient.out_eq' x
+    rep_one := by simp }
+
+def correction (r : CosetSection C) (s : D) (x : D ⧸ C) : C := by
+  refine ⟨(r.rep (s • x))⁻¹ * s * r.rep x, ?_⟩
+  have heq : (r.rep (s • x) : D ⧸ C) = (s * r.rep x : D) := by
+    rw [r.rep_spec]
+    change s • x = s • (r.rep x : D ⧸ C)
+    rw [r.rep_spec]
+  simpa only [mul_assoc] using QuotientGroup.eq.mp heq
+
+@[simp] theorem correction_coe (r : CosetSection C) (s : D) (x : D ⧸ C) :
+    (r.correction s x : D) = (r.rep (s • x))⁻¹ * s * r.rep x := rfl
+
+@[simp] theorem correction_one (r : CosetSection C) (x : D ⧸ C) :
+    r.correction 1 x = 1 := by
+  apply Subtype.ext
+  simp
+
+theorem correction_mul (r : CosetSection C) (s t : D) (x : D ⧸ C) :
+    r.correction s (t • x) * r.correction t x = r.correction (s * t) x := by
+  apply Subtype.ext
+  simp only [Subgroup.coe_mul, correction_coe, mul_smul]
+  group
+
+def transportScalar (r : CosetSection C) (α : ScalarCocycle D) (s : D) (x : D ⧸ C) :
+    Circle := α s (r.rep x) / α (r.rep (s • x)) (r.correction s x)
+
+@[simp] theorem transportScalar_one (r : CosetSection C) (α : ScalarCocycle D)
+    (x : D ⧸ C) : r.transportScalar α 1 x = 1 := by
+  simp [transportScalar]
+
+theorem transportScalar_mul (r : CosetSection C) (α : ScalarCocycle D)
+    (s t : D) (x : D ⧸ C) :
+    r.transportScalar α s (t • x) * r.transportScalar α t x *
+        α (r.correction s (t • x)) (r.correction t x) =
+      α s t * r.transportScalar α (s * t) x := by
+  have hc : ((r.correction s (t • x) : D) * r.correction t x) =
+      (r.correction (s * t) x : D) := congrArg Subtype.val (r.correction_mul s t x)
+  have h := α.induction_scalar s t (r.rep x) (r.rep (t • x)) (r.rep (s • t • x))
+  change r.transportScalar α s (t • x) * r.transportScalar α t x *
+      α (r.correction s (t • x)) (r.correction t x) =
+    _ at h
+  change _ = α s t * (α (s * t) (r.rep x) /
+    α (r.rep (s • t • x)) ((r.correction s (t • x) : D) * r.correction t x)) at h
+  rw [hc] at h
+  simpa only [transportScalar, mul_smul] using h
+
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+
+def transport (r : CosetSection C) {α : ScalarCocycle D}
+    (γ : ProjectiveUnitaryRepresentation (α.pullback C.subtype) V)
+    (s : D) (x : D ⧸ C) : V ≃ₗᵢ[ℂ] V :=
+  (γ.op (r.correction s x)).trans (circleSmul (r.transportScalar α s x))
+
+@[simp] theorem transport_apply (r : CosetSection C) {α : ScalarCocycle D}
+    (γ : ProjectiveUnitaryRepresentation (α.pullback C.subtype) V)
+    (s : D) (x : D ⧸ C) (ξ : V) :
+    r.transport γ s x ξ =
+      (r.transportScalar α s x : ℂ) • (γ.op (r.correction s x) ξ) := rfl
+
+@[simp] theorem transport_one (r : CosetSection C) {α : ScalarCocycle D}
+    (γ : ProjectiveUnitaryRepresentation (α.pullback C.subtype) V) (x : D ⧸ C) :
+    r.transport γ 1 x = LinearIsometryEquiv.refl ℂ V := by
+  ext ξ
+  simp [γ.op_one]
+
+theorem transport_mul (r : CosetSection C) {α : ScalarCocycle D}
+    (γ : ProjectiveUnitaryRepresentation (α.pullback C.subtype) V)
+    (s t : D) (x : D ⧸ C) (ξ : V) :
+    r.transport γ s (t • x) (r.transport γ t x ξ) =
+      (α s t : ℂ) • r.transport γ (s * t) x ξ := by
+  have hscalar : (r.transportScalar α s (t • x) : ℂ) *
+      (r.transportScalar α t x : ℂ) *
+      (α (r.correction s (t • x)) (r.correction t x) : ℂ) =
+      (α s t : ℂ) * (r.transportScalar α (s * t) x : ℂ) := by
+    simpa only [Circle.coe_mul] using
+      congrArg (fun z : Circle => (z : ℂ)) (r.transportScalar_mul α s t x)
+  change (r.transportScalar α s (t • x) : ℂ) •
+      γ.op (r.correction s (t • x))
+        ((r.transportScalar α t x : ℂ) • γ.op (r.correction t x) ξ) =
+    (α s t : ℂ) • ((r.transportScalar α (s * t) x : ℂ) •
+      γ.op (r.correction (s * t) x) ξ)
+  rw [map_smul, γ.op_mul]
+  simp only [ScalarCocycle.pullback_apply, Subgroup.coe_subtype, smul_smul]
+  rw [← mul_assoc, hscalar, r.correction_mul]
+end CosetSection
+
+end TwistedCorrespondences
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/RestrictedTilting/ActionDenominators.lean b/lean/OAI/RepresentationTheory/RestrictedTilting/ActionDenominators.lean
index be717936d75265e1ab9a6b704a8e14ce560f636d..04ae4d2c73148f5533712b5f123d7bd41974e863 100644
--- a/lean/OAI/RepresentationTheory/RestrictedTilting/ActionDenominators.lean
+++ b/lean/OAI/RepresentationTheory/RestrictedTilting/ActionDenominators.lean
@@ -1,152 +1,152 @@
-import Mathlib
-import OAI.RepresentationTheory.RestrictedTilting.LatticeGluing
-import OAI.RepresentationTheory.RestrictedTilting.PolynomialActions
-
-namespace OAI
-
-/-! Polynomial action denominators over coordinates and fraction fields. -/
-
-noncomputable section
-open CategoryTheory CategoryTheory.MonoidalCategory CategoryTheory.Limits
-open scoped BigOperators
-
-namespace FiberFunctorProof
-section
-open scoped BigOperators
-open Matrix
-
-section PolynomialDenominator
-variable {k R K : Type*} [CommRing k] [CommRing R] [Field K]
-    [Algebra k R] [Algebra R K] [Algebra k K] [IsScalarTower k R K]
-
- 
-lemma polynomial_clear_denominator (p : Polynomial k) (b : R)
-    (hb : algebraMap R K b ≠ 0) (d : ℕ) (hd : p.natDegree ≤ d) :
-    ∃ a : R, algebraMap R K a =
-      algebraMap R K b ^ d * Polynomial.aeval (-(algebraMap R K b)⁻¹) p := by
-  refine ⟨∑ n ∈ Finset.range (d+1),
-    algebraMap k R (p.coeff n) * (-1 : R)^n * b^(d-n), ?_⟩
-  rw [Polynomial.aeval_eq_sum_range' (Nat.lt_succ_of_le hd), Finset.mul_sum]
-  simp only [map_sum, map_mul, map_pow, map_neg, map_one,
-    ← IsScalarTower.algebraMap_apply]
-  apply Finset.sum_congr rfl
-  intro n hn
-  rw [pow_sub₀ _ hb (Nat.lt_succ_iff.mp (Finset.mem_range.mp hn))]
-  rw [neg_pow ((algebraMap R K b)⁻¹) n, inv_pow]
-  simp only [Algebra.smul_def]
-  ring
-
-end PolynomialDenominator
-
-variable {k ι : Type*} [Field k] [Infinite k] [Fintype ι] [DecidableEq ι]
-variable (P : Matrix ι ι (Polynomial k))
-    (h0 : polynomialMatrixEval P (0 : k) = 1)
-    (hadd : ∀ s t : k, polynomialMatrixEval P (s+t) =
-      polynomialMatrixEval P s * polynomialMatrixEval P t)
-
- 
-noncomputable def polynomialActionEquiv {K : Type*} [Field K] [Algebra k K] (t : K) :
-    (ι → K) ≃ₗ[K] (ι → K) where
-  toLinearMap := Matrix.toLin' (polynomialMatrixEval P t)
-  invFun := Matrix.mulVec (polynomialMatrixEval P (-t))
-  left_inv v := by
-    change (polynomialMatrixEval P (-t)) *ᵥ ((polynomialMatrixEval P t) *ᵥ v) = v
-    rw [Matrix.mulVec_mulVec, ← polynomialMatrixEval_add P hadd, neg_add_cancel,
-      polynomialMatrixEval_zero P h0, Matrix.one_mulVec]
-  right_inv v := by
-    change (polynomialMatrixEval P t) *ᵥ ((polynomialMatrixEval P (-t)) *ᵥ v) = v
-    rw [Matrix.mulVec_mulVec, ← polynomialMatrixEval_add P hadd, add_neg_cancel,
-      polynomialMatrixEval_zero P h0, Matrix.one_mulVec]
-
-variable {R K : Type*} [CommRing R] [Field K] [Algebra k R] [Algebra R K]
-    [Algebra k K] [IsScalarTower k R K]
-
- 
-
-theorem polynomialAction_denominator (b : R) (hb : algebraMap R K b ≠ 0)
-    (d : ℕ) (hd : ∀ i j, (P i j).natDegree ≤ d) :
-    ∀ v ∈ coordinateLattice (R := R) (K := K) (ι := ι),
-      b^d • polynomialActionEquiv P h0 hadd (-(algebraMap R K b)⁻¹) v ∈
-      coordinateLattice (R := R) (K := K) (ι := ι) := by
-  intro v hv
-  apply (mem_coordinateLattice _).mpr
-  intro i
-  choose w hw using (mem_coordinateLattice v).mp hv
-  choose a ha using fun j => polynomial_clear_denominator (P i j) b hb d (hd i j)
-  refine ⟨∑ j, a j * w j, ?_⟩
-  change algebraMap R K (∑ j, a j * w j) =
-    b^d • ((Matrix.toLin' (polynomialMatrixEval P (-(algebraMap R K b)⁻¹))) v) i
-  rw [Algebra.smul_def]
-  rw [Matrix.toLin'_apply]
-  simp only [map_sum, map_mul, ha, hw, map_pow, Matrix.mulVec, dotProduct,
-    polynomialMatrixEval, Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro j _
-  ring
-
-end
-
-open scoped BigOperators
-open Matrix
-
-section FractionDenominator
-variable {k R K : Type*} [CommRing k] [CommRing R] [Field K]
-    [Algebra k R] [Algebra R K] [Algebra k K] [IsScalarTower k R K]
-
-lemma polynomial_clear_fraction_denominator (p : Polynomial k) (a b : R)
-    (hb : algebraMap R K b ≠ 0) (d : ℕ) (hd : p.natDegree ≤ d) :
-    ∃ c : R, algebraMap R K c =
-      algebraMap R K b ^ d * Polynomial.aeval
-        (algebraMap R K a * (algebraMap R K b)⁻¹) p := by
-  refine ⟨∑ n ∈ Finset.range (d+1),
-    algebraMap k R (p.coeff n) * a^n * b^(d-n), ?_⟩
-  rw [Polynomial.aeval_eq_sum_range' (Nat.lt_succ_of_le hd), Finset.mul_sum]
-  simp only [map_sum, map_mul, map_pow, ← IsScalarTower.algebraMap_apply]
-  apply Finset.sum_congr rfl
-  intro n hn
-  rw [pow_sub₀ _ hb (Nat.lt_succ_iff.mp (Finset.mem_range.mp hn)), mul_pow, inv_pow]
-  simp only [Algebra.smul_def]
-  ring
-
-end FractionDenominator
-
-variable {k ι : Type*} [Field k] [Infinite k] [Fintype ι] [DecidableEq ι]
-variable (P : Matrix ι ι (Polynomial k))
-    (h0 : polynomialMatrixEval P (0 : k) = 1)
-    (hadd : ∀ s t : k, polynomialMatrixEval P (s+t) =
-      polynomialMatrixEval P s * polynomialMatrixEval P t)
-variable {R K : Type*} [CommRing R] [Field K] [Algebra k R] [Algebra R K]
-    [Algebra k K] [IsScalarTower k R K]
-
-lemma polynomialActionEquiv_symm (t : K) :
-    (polynomialActionEquiv P h0 hadd t).symm = polynomialActionEquiv P h0 hadd (-t) := by
-  ext v i
-  rfl
-
-theorem polynomialAction_fraction_denominator (a b : R) (hb : algebraMap R K b ≠ 0)
-    (d : ℕ) (hd : ∀ i j, (P i j).natDegree ≤ d) :
-    ∀ v ∈ coordinateLattice (R := R) (K := K) (ι := ι),
-      b^d • polynomialActionEquiv P h0 hadd
-        (algebraMap R K a * (algebraMap R K b)⁻¹) v ∈
-      coordinateLattice (R := R) (K := K) (ι := ι) := by
-  intro v hv
-  apply (mem_coordinateLattice _).mpr
-  intro i
-  choose w hw using (mem_coordinateLattice v).mp hv
-  choose c hc using fun j => polynomial_clear_fraction_denominator (P i j) a b hb d (hd i j)
-  refine ⟨∑ j, c j * w j, ?_⟩
-  change algebraMap R K (∑ j, c j * w j) =
-    b^d • ((Matrix.toLin' (polynomialMatrixEval P
-      (algebraMap R K a * (algebraMap R K b)⁻¹))) v) i
-  rw [Algebra.smul_def, Matrix.toLin'_apply]
-  simp only [map_sum, map_mul, hc, hw, map_pow, Matrix.mulVec, dotProduct,
-    polynomialMatrixEval, Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro j _
-  ring
-
-end FiberFunctorProof
-
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.RestrictedTilting.LatticeGluing
+import OAI.RepresentationTheory.RestrictedTilting.PolynomialActions
+
+namespace OAI
+
+/-! Polynomial action denominators over coordinates and fraction fields. -/
+
+noncomputable section
+open CategoryTheory CategoryTheory.MonoidalCategory CategoryTheory.Limits
+open scoped BigOperators
+
+namespace FiberFunctorProof
+section
+open scoped BigOperators
+open Matrix
+
+section PolynomialDenominator
+variable {k R K : Type*} [CommRing k] [CommRing R] [Field K]
+    [Algebra k R] [Algebra R K] [Algebra k K] [IsScalarTower k R K]
+
+ 
+lemma polynomial_clear_denominator (p : Polynomial k) (b : R)
+    (hb : algebraMap R K b ≠ 0) (d : ℕ) (hd : p.natDegree ≤ d) :
+    ∃ a : R, algebraMap R K a =
+      algebraMap R K b ^ d * Polynomial.aeval (-(algebraMap R K b)⁻¹) p := by
+  refine ⟨∑ n ∈ Finset.range (d+1),
+    algebraMap k R (p.coeff n) * (-1 : R)^n * b^(d-n), ?_⟩
+  rw [Polynomial.aeval_eq_sum_range' (Nat.lt_succ_of_le hd), Finset.mul_sum]
+  simp only [map_sum, map_mul, map_pow, map_neg, map_one,
+    ← IsScalarTower.algebraMap_apply]
+  apply Finset.sum_congr rfl
+  intro n hn
+  rw [pow_sub₀ _ hb (Nat.lt_succ_iff.mp (Finset.mem_range.mp hn))]
+  rw [neg_pow ((algebraMap R K b)⁻¹) n, inv_pow]
+  simp only [Algebra.smul_def]
+  ring
+
+end PolynomialDenominator
+
+variable {k ι : Type*} [Field k] [Infinite k] [Fintype ι] [DecidableEq ι]
+variable (P : Matrix ι ι (Polynomial k))
+    (h0 : polynomialMatrixEval P (0 : k) = 1)
+    (hadd : ∀ s t : k, polynomialMatrixEval P (s+t) =
+      polynomialMatrixEval P s * polynomialMatrixEval P t)
+
+ 
+noncomputable def polynomialActionEquiv {K : Type*} [Field K] [Algebra k K] (t : K) :
+    (ι → K) ≃ₗ[K] (ι → K) where
+  toLinearMap := Matrix.toLin' (polynomialMatrixEval P t)
+  invFun := Matrix.mulVec (polynomialMatrixEval P (-t))
+  left_inv v := by
+    change (polynomialMatrixEval P (-t)) *ᵥ ((polynomialMatrixEval P t) *ᵥ v) = v
+    rw [Matrix.mulVec_mulVec, ← polynomialMatrixEval_add P hadd, neg_add_cancel,
+      polynomialMatrixEval_zero P h0, Matrix.one_mulVec]
+  right_inv v := by
+    change (polynomialMatrixEval P t) *ᵥ ((polynomialMatrixEval P (-t)) *ᵥ v) = v
+    rw [Matrix.mulVec_mulVec, ← polynomialMatrixEval_add P hadd, add_neg_cancel,
+      polynomialMatrixEval_zero P h0, Matrix.one_mulVec]
+
+variable {R K : Type*} [CommRing R] [Field K] [Algebra k R] [Algebra R K]
+    [Algebra k K] [IsScalarTower k R K]
+
+ 
+
+theorem polynomialAction_denominator (b : R) (hb : algebraMap R K b ≠ 0)
+    (d : ℕ) (hd : ∀ i j, (P i j).natDegree ≤ d) :
+    ∀ v ∈ coordinateLattice (R := R) (K := K) (ι := ι),
+      b^d • polynomialActionEquiv P h0 hadd (-(algebraMap R K b)⁻¹) v ∈
+      coordinateLattice (R := R) (K := K) (ι := ι) := by
+  intro v hv
+  apply (mem_coordinateLattice _).mpr
+  intro i
+  choose w hw using (mem_coordinateLattice v).mp hv
+  choose a ha using fun j => polynomial_clear_denominator (P i j) b hb d (hd i j)
+  refine ⟨∑ j, a j * w j, ?_⟩
+  change algebraMap R K (∑ j, a j * w j) =
+    b^d • ((Matrix.toLin' (polynomialMatrixEval P (-(algebraMap R K b)⁻¹))) v) i
+  rw [Algebra.smul_def]
+  rw [Matrix.toLin'_apply]
+  simp only [map_sum, map_mul, ha, hw, map_pow, Matrix.mulVec, dotProduct,
+    polynomialMatrixEval, Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro j _
+  ring
+
+end
+
+open scoped BigOperators
+open Matrix
+
+section FractionDenominator
+variable {k R K : Type*} [CommRing k] [CommRing R] [Field K]
+    [Algebra k R] [Algebra R K] [Algebra k K] [IsScalarTower k R K]
+
+lemma polynomial_clear_fraction_denominator (p : Polynomial k) (a b : R)
+    (hb : algebraMap R K b ≠ 0) (d : ℕ) (hd : p.natDegree ≤ d) :
+    ∃ c : R, algebraMap R K c =
+      algebraMap R K b ^ d * Polynomial.aeval
+        (algebraMap R K a * (algebraMap R K b)⁻¹) p := by
+  refine ⟨∑ n ∈ Finset.range (d+1),
+    algebraMap k R (p.coeff n) * a^n * b^(d-n), ?_⟩
+  rw [Polynomial.aeval_eq_sum_range' (Nat.lt_succ_of_le hd), Finset.mul_sum]
+  simp only [map_sum, map_mul, map_pow, ← IsScalarTower.algebraMap_apply]
+  apply Finset.sum_congr rfl
+  intro n hn
+  rw [pow_sub₀ _ hb (Nat.lt_succ_iff.mp (Finset.mem_range.mp hn)), mul_pow, inv_pow]
+  simp only [Algebra.smul_def]
+  ring
+
+end FractionDenominator
+
+variable {k ι : Type*} [Field k] [Infinite k] [Fintype ι] [DecidableEq ι]
+variable (P : Matrix ι ι (Polynomial k))
+    (h0 : polynomialMatrixEval P (0 : k) = 1)
+    (hadd : ∀ s t : k, polynomialMatrixEval P (s+t) =
+      polynomialMatrixEval P s * polynomialMatrixEval P t)
+variable {R K : Type*} [CommRing R] [Field K] [Algebra k R] [Algebra R K]
+    [Algebra k K] [IsScalarTower k R K]
+
+lemma polynomialActionEquiv_symm (t : K) :
+    (polynomialActionEquiv P h0 hadd t).symm = polynomialActionEquiv P h0 hadd (-t) := by
+  ext v i
+  rfl
+
+theorem polynomialAction_fraction_denominator (a b : R) (hb : algebraMap R K b ≠ 0)
+    (d : ℕ) (hd : ∀ i j, (P i j).natDegree ≤ d) :
+    ∀ v ∈ coordinateLattice (R := R) (K := K) (ι := ι),
+      b^d • polynomialActionEquiv P h0 hadd
+        (algebraMap R K a * (algebraMap R K b)⁻¹) v ∈
+      coordinateLattice (R := R) (K := K) (ι := ι) := by
+  intro v hv
+  apply (mem_coordinateLattice _).mpr
+  intro i
+  choose w hw using (mem_coordinateLattice v).mp hv
+  choose c hc using fun j => polynomial_clear_fraction_denominator (P i j) a b hb d (hd i j)
+  refine ⟨∑ j, c j * w j, ?_⟩
+  change algebraMap R K (∑ j, c j * w j) =
+    b^d • ((Matrix.toLin' (polynomialMatrixEval P
+      (algebraMap R K a * (algebraMap R K b)⁻¹))) v) i
+  rw [Algebra.smul_def, Matrix.toLin'_apply]
+  simp only [map_sum, map_mul, hc, hw, map_pow, Matrix.mulVec, dotProduct,
+    polynomialMatrixEval, Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro j _
+  ring
+
+end FiberFunctorProof
+
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/RestrictedTilting/AdicJets.lean b/lean/OAI/RepresentationTheory/RestrictedTilting/AdicJets.lean
index e7d2be232f610619993ece7c3e52e23fdb9a89a5..b073509d393455453ba0dd26c4134250ea319e34 100644
--- a/lean/OAI/RepresentationTheory/RestrictedTilting/AdicJets.lean
+++ b/lean/OAI/RepresentationTheory/RestrictedTilting/AdicJets.lean
@@ -1,88 +1,88 @@
-import OAI.RepresentationTheory.RestrictedTilting.PolynomialJets
-
-namespace OAI
-
-namespace FiberFunctorProof
-open scoped Pointwise
-section
-variable {R M : Type*} [CommRing R] [AddCommGroup M] [Module R M]
-
-lemma jetRelation_succ (x y : R) (j : ℕ) :
-    jetRelation (M := M) x y (j+1) = x • jetRelation x y j ⊔ y • jetRelation x y j := by
-  apply le_antisymm
-  · rintro m ⟨v,rfl⟩
-    rw [binarySum_succ]
-    apply Submodule.add_mem
-    · apply (show y • jetRelation x y j ≤ x • jetRelation x y j ⊔ y • jetRelation x y j from le_sup_right)
-      rw [pow_succ', mul_smul]
-      apply Submodule.smul_mem_pointwise_smul
-      exact ⟨Pi.single 0 (v 0), binarySum_single_zero x y j (v 0)⟩
-    · apply (show x • jetRelation x y j ≤ x • jetRelation x y j ⊔ y • jetRelation x y j from le_sup_left)
-      apply Submodule.smul_mem_pointwise_smul
-      exact ⟨_,rfl⟩
-  · apply sup_le
-    · intro m hm
-      obtain ⟨w,⟨v,rfl⟩,rfl⟩ := (Submodule.mem_smul_pointwise_iff_exists m x _).mp hm
-      exact ⟨Fin.cons 0 v, binarySum_prepend_zero x y j v⟩
-    · intro m hm
-      obtain ⟨w,⟨v,rfl⟩,rfl⟩ := (Submodule.mem_smul_pointwise_iff_exists m y _).mp hm
-      exact ⟨Fin.snoc v 0, binarySum_append_zero x y j v⟩
-
- 
-
-theorem jetRelation_eq_pow_smul (x y : R) (j : ℕ) :
-    jetRelation (M := M) x y j = (Ideal.span {x,y})^j • (⊤ : Submodule R M) := by
-  induction j with
-  | zero => simp [jetRelation_zero]
-  | succ j ih =>
-    rw [jetRelation_succ, pow_succ', Submodule.mul_smul, ← ih,
-      Ideal.span_insert, Submodule.sup_smul,
-      Submodule.ideal_span_singleton_smul, Submodule.ideal_span_singleton_smul]
-
-end
-
-section
-universe u
-variable (k : Type u) [Field k]
-variable {M : Type u} [AddCommGroup M] [Module k M]
-  [Module (MvPolynomial (Fin 2) k) M] [IsScalarTower k (MvPolynomial (Fin 2) k) M]
-
-lemma planeIdeal_eq : MvPolynomial.idealOfVars (Fin 2) k =
-    Ideal.span {MvPolynomial.X 0, MvPolynomial.X 1} := by
-  unfold MvPolynomial.idealOfVars
-  congr 1
-  ext P
-  simp only [Set.mem_range, Set.mem_insert_iff, Set.mem_singleton_iff]
-  constructor
-  · rintro ⟨i,rfl⟩
-    fin_cases i <;> simp
-  · rintro (rfl|rfl)
-    · exact ⟨0,rfl⟩
-    · exact ⟨1,rfl⟩
-
-lemma planeJetRelation_eq_pow_smul (j : ℕ) :
-    planeJetRelation (M := M) k j =
-      (((MvPolynomial.idealOfVars (Fin 2) k)^j • (⊤ : Submodule (MvPolynomial (Fin 2) k) M)).restrictScalars k) := by
-  rw [planeJetRelation, planeIdeal_eq, jetRelation_eq_pow_smul]
-
-noncomputable def planeJetAdicEquiv (j : ℕ) :
-    (M ⧸ planeJetRelation (M := M) k j) ≃ₗ[k]
-      (M ⧸ ((MvPolynomial.idealOfVars (Fin 2) k)^j • (⊤ : Submodule (MvPolynomial (Fin 2) k) M))) :=
-  (Submodule.quotEquivOfEq _ _ (planeJetRelation_eq_pow_smul k j)).trans
-    (Submodule.Quotient.restrictScalarsEquiv k _)
-
-lemma planeJetAdicEquiv_mk (j : ℕ) (m : M) :
-    planeJetAdicEquiv k j ((planeJetRelation k j).mkQ m) =
-      (((MvPolynomial.idealOfVars (Fin 2) k)^j • (⊤ : Submodule (MvPolynomial (Fin 2) k) M))).mkQ m := by
-  rfl
-
-lemma planeJetAdicEquiv_transition (j : ℕ) (z : M ⧸ planeJetRelation (M := M) k (j+1)) :
-    AdicCompletion.transitionMap (MvPolynomial.idealOfVars (Fin 2) k) M (Nat.le_succ j)
-      (planeJetAdicEquiv k (j+1) z) = planeJetAdicEquiv k j (planeJetTransition k j z) := by
-  obtain ⟨m,rfl⟩ := (planeJetRelation (M := M) k (j+1)).mkQ_surjective z
-  rfl
-
-end
-end FiberFunctorProof
-
-end OAI
+import OAI.RepresentationTheory.RestrictedTilting.PolynomialJets
+
+namespace OAI
+
+namespace FiberFunctorProof
+open scoped Pointwise
+section
+variable {R M : Type*} [CommRing R] [AddCommGroup M] [Module R M]
+
+lemma jetRelation_succ (x y : R) (j : ℕ) :
+    jetRelation (M := M) x y (j+1) = x • jetRelation x y j ⊔ y • jetRelation x y j := by
+  apply le_antisymm
+  · rintro m ⟨v,rfl⟩
+    rw [binarySum_succ]
+    apply Submodule.add_mem
+    · apply (show y • jetRelation x y j ≤ x • jetRelation x y j ⊔ y • jetRelation x y j from le_sup_right)
+      rw [pow_succ', mul_smul]
+      apply Submodule.smul_mem_pointwise_smul
+      exact ⟨Pi.single 0 (v 0), binarySum_single_zero x y j (v 0)⟩
+    · apply (show x • jetRelation x y j ≤ x • jetRelation x y j ⊔ y • jetRelation x y j from le_sup_left)
+      apply Submodule.smul_mem_pointwise_smul
+      exact ⟨_,rfl⟩
+  · apply sup_le
+    · intro m hm
+      obtain ⟨w,⟨v,rfl⟩,rfl⟩ := (Submodule.mem_smul_pointwise_iff_exists m x _).mp hm
+      exact ⟨Fin.cons 0 v, binarySum_prepend_zero x y j v⟩
+    · intro m hm
+      obtain ⟨w,⟨v,rfl⟩,rfl⟩ := (Submodule.mem_smul_pointwise_iff_exists m y _).mp hm
+      exact ⟨Fin.snoc v 0, binarySum_append_zero x y j v⟩
+
+ 
+
+theorem jetRelation_eq_pow_smul (x y : R) (j : ℕ) :
+    jetRelation (M := M) x y j = (Ideal.span {x,y})^j • (⊤ : Submodule R M) := by
+  induction j with
+  | zero => simp [jetRelation_zero]
+  | succ j ih =>
+    rw [jetRelation_succ, pow_succ', Submodule.mul_smul, ← ih,
+      Ideal.span_insert, Submodule.sup_smul,
+      Submodule.ideal_span_singleton_smul, Submodule.ideal_span_singleton_smul]
+
+end
+
+section
+universe u
+variable (k : Type u) [Field k]
+variable {M : Type u} [AddCommGroup M] [Module k M]
+  [Module (MvPolynomial (Fin 2) k) M] [IsScalarTower k (MvPolynomial (Fin 2) k) M]
+
+lemma planeIdeal_eq : MvPolynomial.idealOfVars (Fin 2) k =
+    Ideal.span {MvPolynomial.X 0, MvPolynomial.X 1} := by
+  unfold MvPolynomial.idealOfVars
+  congr 1
+  ext P
+  simp only [Set.mem_range, Set.mem_insert_iff, Set.mem_singleton_iff]
+  constructor
+  · rintro ⟨i,rfl⟩
+    fin_cases i <;> simp
+  · rintro (rfl|rfl)
+    · exact ⟨0,rfl⟩
+    · exact ⟨1,rfl⟩
+
+lemma planeJetRelation_eq_pow_smul (j : ℕ) :
+    planeJetRelation (M := M) k j =
+      (((MvPolynomial.idealOfVars (Fin 2) k)^j • (⊤ : Submodule (MvPolynomial (Fin 2) k) M)).restrictScalars k) := by
+  rw [planeJetRelation, planeIdeal_eq, jetRelation_eq_pow_smul]
+
+noncomputable def planeJetAdicEquiv (j : ℕ) :
+    (M ⧸ planeJetRelation (M := M) k j) ≃ₗ[k]
+      (M ⧸ ((MvPolynomial.idealOfVars (Fin 2) k)^j • (⊤ : Submodule (MvPolynomial (Fin 2) k) M))) :=
+  (Submodule.quotEquivOfEq _ _ (planeJetRelation_eq_pow_smul k j)).trans
+    (Submodule.Quotient.restrictScalarsEquiv k _)
+
+lemma planeJetAdicEquiv_mk (j : ℕ) (m : M) :
+    planeJetAdicEquiv k j ((planeJetRelation k j).mkQ m) =
+      (((MvPolynomial.idealOfVars (Fin 2) k)^j • (⊤ : Submodule (MvPolynomial (Fin 2) k) M))).mkQ m := by
+  rfl
+
+lemma planeJetAdicEquiv_transition (j : ℕ) (z : M ⧸ planeJetRelation (M := M) k (j+1)) :
+    AdicCompletion.transitionMap (MvPolynomial.idealOfVars (Fin 2) k) M (Nat.le_succ j)
+      (planeJetAdicEquiv k (j+1) z) = planeJetAdicEquiv k j (planeJetTransition k j z) := by
+  obtain ⟨m,rfl⟩ := (planeJetRelation (M := M) k (j+1)).mkQ_surjective z
+  rfl
+
+end
+end FiberFunctorProof
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/RestrictedTilting/AssociatedBundle.lean b/lean/OAI/RepresentationTheory/RestrictedTilting/AssociatedBundle.lean
index 7bcc149bc980654d917f714cb1dc3736506211a3..c60150a59ba9dad9557208b82953d2a1a9133ea7 100644
--- a/lean/OAI/RepresentationTheory/RestrictedTilting/AssociatedBundle.lean
+++ b/lean/OAI/RepresentationTheory/RestrictedTilting/AssociatedBundle.lean
@@ -1,50 +1,50 @@
-import OAI.RepresentationTheory.RestrictedTilting.AssociatedProjective
-import OAI.RepresentationTheory.RestrictedTilting.AssociatedPlaneJet
-import OAI.RepresentationTheory.RestrictedTilting.PlaneFibre
-
-namespace OAI
-
-namespace FiberFunctorProof
-open scoped Pointwise
-universe u
-noncomputable section
-variable (k : Type u) [Field k] [Infinite k]
-variable (ι : Type u) [Fintype ι] [DecidableEq ι]
-variable (ρ : Representation k (Multiplicative k) (ι → k))
-variable (P : Matrix ι ι (Polynomial k))
-variable (hP : ∀ t : k, polynomialMatrixEval P t = LinearMap.toMatrix' (ρ (Multiplicative.ofAdd t)))
-variable (d : ℕ) (hd : ∀ i j, (P i j).natDegree ≤ d)
-local notation "N" => associatedSectionsSubmodule k ι ρ
-local notation "A" => MvPolynomial (Fin 2) k
-
- 
-
-def associatedPlaneBundle : EquivariantPlaneBundle k where
-  Sections := N
-  finite := associatedSections_finite k ι ρ P hP d hd
-  projective := associatedSections_projective k ι ρ P hP d hd
-  action := associatedSectionsRepresentation k ι ρ
-  rational := associatedSections_rational k ι ρ
-  compatible := associatedSections_planeCompatible k ι ρ
-
-omit [Infinite k] in
-lemma planeFibreRelations_origin (M : Type u) [AddCommGroup M] [Module k M] [Module A M]
-    [IsScalarTower k A M] :
-    planeFibreRelations M (0 : Fin 2 → k) = planeJetRelation (M := M) k 1 := by
-  change (planePointIdeal 0 • (⊤ : Submodule A M)).restrictScalars k =
-    (jetRelation (MvPolynomial.X 0 : A) (MvPolynomial.X 1) 1).restrictScalars k
-  have he : planePointIdeal (0 : Fin 2 → k) = planeOriginIdeal k := by
-    rw [planePointIdeal, MvPolynomial.eval_zero]; rfl
-  rw [he, planeOriginIdeal_eq_span, Ideal.span_insert, Submodule.sup_smul,
-    Submodule.ideal_span_singleton_smul, Submodule.ideal_span_singleton_smul,
-    jetRelation_one]
-  rfl
-
-lemma associatedPlaneBundle_originRelation :
-    (associatedPlaneBundle k ι ρ P hP d hd).fibreRelation 0 = associatedOriginRelations k ι ρ := by
-  change planeFibreRelations N (0 : Fin 2 → k) = _
-  rw [planeFibreRelations_origin, associated_planeJet_one]
-end
-end FiberFunctorProof
-
-end OAI
+import OAI.RepresentationTheory.RestrictedTilting.AssociatedProjective
+import OAI.RepresentationTheory.RestrictedTilting.AssociatedPlaneJet
+import OAI.RepresentationTheory.RestrictedTilting.PlaneFibre
+
+namespace OAI
+
+namespace FiberFunctorProof
+open scoped Pointwise
+universe u
+noncomputable section
+variable (k : Type u) [Field k] [Infinite k]
+variable (ι : Type u) [Fintype ι] [DecidableEq ι]
+variable (ρ : Representation k (Multiplicative k) (ι → k))
+variable (P : Matrix ι ι (Polynomial k))
+variable (hP : ∀ t : k, polynomialMatrixEval P t = LinearMap.toMatrix' (ρ (Multiplicative.ofAdd t)))
+variable (d : ℕ) (hd : ∀ i j, (P i j).natDegree ≤ d)
+local notation "N" => associatedSectionsSubmodule k ι ρ
+local notation "A" => MvPolynomial (Fin 2) k
+
+ 
+
+def associatedPlaneBundle : EquivariantPlaneBundle k where
+  Sections := N
+  finite := associatedSections_finite k ι ρ P hP d hd
+  projective := associatedSections_projective k ι ρ P hP d hd
+  action := associatedSectionsRepresentation k ι ρ
+  rational := associatedSections_rational k ι ρ
+  compatible := associatedSections_planeCompatible k ι ρ
+
+omit [Infinite k] in
+lemma planeFibreRelations_origin (M : Type u) [AddCommGroup M] [Module k M] [Module A M]
+    [IsScalarTower k A M] :
+    planeFibreRelations M (0 : Fin 2 → k) = planeJetRelation (M := M) k 1 := by
+  change (planePointIdeal 0 • (⊤ : Submodule A M)).restrictScalars k =
+    (jetRelation (MvPolynomial.X 0 : A) (MvPolynomial.X 1) 1).restrictScalars k
+  have he : planePointIdeal (0 : Fin 2 → k) = planeOriginIdeal k := by
+    rw [planePointIdeal, MvPolynomial.eval_zero]; rfl
+  rw [he, planeOriginIdeal_eq_span, Ideal.span_insert, Submodule.sup_smul,
+    Submodule.ideal_span_singleton_smul, Submodule.ideal_span_singleton_smul,
+    jetRelation_one]
+  rfl
+
+lemma associatedPlaneBundle_originRelation :
+    (associatedPlaneBundle k ι ρ P hP d hd).fibreRelation 0 = associatedOriginRelations k ι ρ := by
+  change planeFibreRelations N (0 : Fin 2 → k) = _
+  rw [planeFibreRelations_origin, associated_planeJet_one]
+end
+end FiberFunctorProof
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/RestrictedTilting/AssociatedFibre.lean b/lean/OAI/RepresentationTheory/RestrictedTilting/AssociatedFibre.lean
index 7fb899cb1577021d157f62ec4cf35d63f51c6146..f608078b42add2500d7025fafb2e59a350e2585f 100644
--- a/lean/OAI/RepresentationTheory/RestrictedTilting/AssociatedFibre.lean
+++ b/lean/OAI/RepresentationTheory/RestrictedTilting/AssociatedFibre.lean
@@ -1,63 +1,63 @@
-import OAI.RepresentationTheory.RestrictedTilting.PlaneFibre
-import OAI.RepresentationTheory.RestrictedTilting.AssociatedNumerator
-import OAI.RepresentationTheory.RestrictedTilting.XChartLineEval
-
-namespace OAI
-
-namespace FiberFunctorProof
-universe u
-noncomputable section
-variable (k : Type u) [Field k] [Infinite k]
-variable (ι : Type u) [Fintype ι] [DecidableEq ι]
-variable (ρ : Representation k (Multiplicative k) (ι → k))
-variable (P : Matrix ι ι (Polynomial k))
-variable (hP : ∀ t : k, polynomialMatrixEval P t = LinearMap.toMatrix' (ρ (Multiplicative.ofAdd t)))
-variable (d : ℕ) (hd : ∀ i j, (P i j).natDegree ≤ d)
-local notation "N" => associatedSectionsSubmodule k ι ρ
-local notation "A" => MvPolynomial (Fin 2) k
-
-def associatedE1Equiv : (N ⧸ planeFibreRelations (k := k) N ![1,0]) ≃ₗ[k] (ι → k) :=
-  (planeFibreEquivOfSplit (k := k) N (ι → A) ![1,0]
-    (associatedNumeratorLinear k ι ρ P hP d hd) (associatedDenominatorLinear k ι ρ P hP d hd)
-    ((MvPolynomial.X 0)^(d+d)) (by simp)
-    (associatedDenominator_numerator k ι ρ P hP d hd)
-    (associatedNumerator_denominator k ι ρ P hP d hd)).trans
-      (planeFreeFibreEquiv (k := k) ι ![1,0])
-
-omit [Infinite k] in
-lemma planeLineEval_one (F : A) : planeLineEval k (1 : kˣ) F = MvPolynomial.eval ![1,0] F := by
-  have he : (fun i : Fin 2 => if i = 0 then ((1 : kˣ) : k) else 0) = ![1,0] := by
-    ext i; fin_cases i <;> simp
-  simp only [planeLineEval, he, MvPolynomial.aeval_def]
-  rfl
-
-lemma associatedNumerator_e1 (f : N) :
-    planeVectorEval ι ![1,0] (associatedNumeratorLinear k ι ρ P hP d hd f) =
-      regularVectorEvaluation k ι 1 f.val := by
-  funext i
-  have he := congrArg (chartLineEval k 1) (associatedNumeratorLinear_spec k ι ρ P hP d hd f i)
-  rw [chartLineEval_algebraMap, planeLineEval_one, map_mul, map_pow,
-    chartLineEval_coordinate, ite_eq_left rfl, Units.val_one, one_pow, one_mul] at he
-  change MvPolynomial.eval ![1,0] _ = _
-  rw [he]
-  have hh := congrFun (associatedXChart_line k ι ρ (1 : kˣ) f) i
-  simpa only [map_one] using hh
-
-lemma associatedE1Equiv_mk (f : N) :
-    associatedE1Equiv k ι ρ P hP d hd ((planeFibreRelations (k := k) N ![1,0]).mkQ f) =
-      regularVectorEvaluation k ι 1 f.val := by
-  change planeVectorEval ι ![1,0] (associatedNumeratorLinear k ι ρ P hP d hd f) = _
-  exact associatedNumerator_e1 k ι ρ P hP d hd f
-
-lemma associatedE1Equiv_unipotent (t : k) (f : N) :
-    associatedE1Equiv k ι ρ P hP d hd ((planeFibreRelations (k := k) N ![1,0]).mkQ
-      (associatedSectionsRepresentation k ι ρ (unipotent k t) f)) =
-      ρ (Multiplicative.ofAdd t) (associatedE1Equiv k ι ρ P hP d hd
-        ((planeFibreRelations (k := k) N ![1,0]).mkQ f)) := by
-  rw [associatedE1Equiv_mk, associatedE1Equiv_mk]
-  simpa only [map_one, Units.val_one, one_pow, div_one] using
-    associatedSections_line_unipotent k ι ρ (1 : kˣ) t f
-end
-end FiberFunctorProof
-
-end OAI
+import OAI.RepresentationTheory.RestrictedTilting.PlaneFibre
+import OAI.RepresentationTheory.RestrictedTilting.AssociatedNumerator
+import OAI.RepresentationTheory.RestrictedTilting.XChartLineEval
+
+namespace OAI
+
+namespace FiberFunctorProof
+universe u
+noncomputable section
+variable (k : Type u) [Field k] [Infinite k]
+variable (ι : Type u) [Fintype ι] [DecidableEq ι]
+variable (ρ : Representation k (Multiplicative k) (ι → k))
+variable (P : Matrix ι ι (Polynomial k))
+variable (hP : ∀ t : k, polynomialMatrixEval P t = LinearMap.toMatrix' (ρ (Multiplicative.ofAdd t)))
+variable (d : ℕ) (hd : ∀ i j, (P i j).natDegree ≤ d)
+local notation "N" => associatedSectionsSubmodule k ι ρ
+local notation "A" => MvPolynomial (Fin 2) k
+
+def associatedE1Equiv : (N ⧸ planeFibreRelations (k := k) N ![1,0]) ≃ₗ[k] (ι → k) :=
+  (planeFibreEquivOfSplit (k := k) N (ι → A) ![1,0]
+    (associatedNumeratorLinear k ι ρ P hP d hd) (associatedDenominatorLinear k ι ρ P hP d hd)
+    ((MvPolynomial.X 0)^(d+d)) (by simp)
+    (associatedDenominator_numerator k ι ρ P hP d hd)
+    (associatedNumerator_denominator k ι ρ P hP d hd)).trans
+      (planeFreeFibreEquiv (k := k) ι ![1,0])
+
+omit [Infinite k] in
+lemma planeLineEval_one (F : A) : planeLineEval k (1 : kˣ) F = MvPolynomial.eval ![1,0] F := by
+  have he : (fun i : Fin 2 => if i = 0 then ((1 : kˣ) : k) else 0) = ![1,0] := by
+    ext i; fin_cases i <;> simp
+  simp only [planeLineEval, he, MvPolynomial.aeval_def]
+  rfl
+
+lemma associatedNumerator_e1 (f : N) :
+    planeVectorEval ι ![1,0] (associatedNumeratorLinear k ι ρ P hP d hd f) =
+      regularVectorEvaluation k ι 1 f.val := by
+  funext i
+  have he := congrArg (chartLineEval k 1) (associatedNumeratorLinear_spec k ι ρ P hP d hd f i)
+  rw [chartLineEval_algebraMap, planeLineEval_one, map_mul, map_pow,
+    chartLineEval_coordinate, ite_eq_left rfl, Units.val_one, one_pow, one_mul] at he
+  change MvPolynomial.eval ![1,0] _ = _
+  rw [he]
+  have hh := congrFun (associatedXChart_line k ι ρ (1 : kˣ) f) i
+  simpa only [map_one] using hh
+
+lemma associatedE1Equiv_mk (f : N) :
+    associatedE1Equiv k ι ρ P hP d hd ((planeFibreRelations (k := k) N ![1,0]).mkQ f) =
+      regularVectorEvaluation k ι 1 f.val := by
+  change planeVectorEval ι ![1,0] (associatedNumeratorLinear k ι ρ P hP d hd f) = _
+  exact associatedNumerator_e1 k ι ρ P hP d hd f
+
+lemma associatedE1Equiv_unipotent (t : k) (f : N) :
+    associatedE1Equiv k ι ρ P hP d hd ((planeFibreRelations (k := k) N ![1,0]).mkQ
+      (associatedSectionsRepresentation k ι ρ (unipotent k t) f)) =
+      ρ (Multiplicative.ofAdd t) (associatedE1Equiv k ι ρ P hP d hd
+        ((planeFibreRelations (k := k) N ![1,0]).mkQ f)) := by
+  rw [associatedE1Equiv_mk, associatedE1Equiv_mk]
+  simpa only [map_one, Units.val_one, one_pow, div_one] using
+    associatedSections_line_unipotent k ι ρ (1 : kˣ) t f
+end
+end FiberFunctorProof
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/RestrictedTilting/AssociatedLattices.lean b/lean/OAI/RepresentationTheory/RestrictedTilting/AssociatedLattices.lean
index f69772810ca3e186e0e1c9038153a3b65b4c4190..7cdeca445a1620c8e06f893f3ee4a1f97c87f68b 100644
--- a/lean/OAI/RepresentationTheory/RestrictedTilting/AssociatedLattices.lean
+++ b/lean/OAI/RepresentationTheory/RestrictedTilting/AssociatedLattices.lean
@@ -1,219 +1,219 @@
-import Mathlib
-import OAI.RepresentationTheory.RestrictedTilting.GlobalLattices
-
-namespace OAI
-
-/-! Associated-origin finiteness, lattice equivalence and numerator evaluations. -/
-
-noncomputable section
-open CategoryTheory CategoryTheory.MonoidalCategory CategoryTheory.Limits
-open scoped BigOperators
-
-namespace FiberFunctorProof
-section
-universe u
-variable (k : Type u) [Field k] [Infinite k]
-variable (ι : Type u) [Fintype ι] [DecidableEq ι]
-variable (ρ : Representation k (Multiplicative k) (ι → k))
-
-omit [Infinite k] [Fintype ι] [DecidableEq ι] in
-lemma associatedCoordinate_smul (i : Fin 2) (f : associatedSectionsSubmodule k ι ρ) :
-    (MvPolynomial.X i : MvPolynomial (Fin 2) k) • f = associatedCoordinateMul k ι ρ i f := by
-  apply Subtype.ext
-  funext j
-  apply Subtype.ext
-  funext g
-  change ((MvPolynomial.X i : MvPolynomial (Fin 2) k) • f.val j).val g =
-    (regularFirstColumn k i).val g * (f.val j).val g
-  rw [Algebra.smul_def, regularPlane_algebraMap]
-  change (regularPlane k (MvPolynomial.X i)).val g * (f.val j).val g = _
-  rw [regularPlane_evaluation, MvPolynomial.eval_X, regularFirstColumn_apply]
-
-omit [Infinite k] [Fintype ι] [DecidableEq ι] in
-lemma associatedOrigin_coordinate_mem (i : Fin 2) (f : associatedSectionsSubmodule k ι ρ) :
-    (MvPolynomial.X i : MvPolynomial (Fin 2) k) • f ∈ associatedOriginRelations k ι ρ := by
-  rw [associatedCoordinate_smul]
-  fin_cases i
-  · exact Submodule.mem_sup_left ⟨f,rfl⟩
-  · exact Submodule.mem_sup_right ⟨f,rfl⟩
-
- 
-
-theorem associatedOrigin_finite
-    (P : Matrix ι ι (Polynomial k))
-    (hP : ∀ t : k, polynomialMatrixEval P t = LinearMap.toMatrix' (ρ (Multiplicative.ofAdd t)))
-    (d : ℕ) (hd : ∀ i j, (P i j).natDegree ≤ d) :
-    Module.Finite k (associatedOriginRep k ι ρ).V := by
-  let : Module.Finite (MvPolynomial (Fin 2) k) (associatedSectionsSubmodule k ι ρ) :=
-    associatedSections_finite k ι ρ P hP d hd
-  exact originQuotient_finite (associatedOriginRelations k ι ρ)
-    (associatedOrigin_coordinate_mem k ι ρ)
-
-end
-
-section
-variable (k : Type*) [Field k]
-
-lemma planeChartToFraction_invSelf (a : Fin 2) :
-    planeChartToFraction k a
-      (IsLocalization.Away.invSelf (MvPolynomial.X a : MvPolynomial (Fin 2) k)) =
-      (algebraMap (MvPolynomial (Fin 2) k) (planeFractionField k) (MvPolynomial.X a))⁻¹ := by
-  have h := congrArg (planeChartToFraction k a)
-    (IsLocalization.Away.mul_invSelf (S := planeCoordinateChart k a)
-      (MvPolynomial.X a : MvPolynomial (Fin 2) k))
-  rw [map_mul, map_one, planeChartToFraction_algebraMap] at h
-  apply mul_left_cancel₀ (planeFractionCoordinate_ne_zero k a)
-  rw [mul_inv_cancel₀ (planeFractionCoordinate_ne_zero k a)]
-  exact h
-
-lemma planeChartVector_exists {ι : Type*} (a : Fin 2) (v : ι → planeFractionField k)
-    (hv : v ∈ powerLocalization (MvPolynomial.X a : MvPolynomial (Fin 2) k) coordinateLattice) :
-    ∃ f : ι → planeCoordinateChart k a, (fun i => planeChartToFraction k a (f i)) = v := by
-  obtain ⟨n,hn⟩ := hv
-  choose b hb using (mem_coordinateLattice _).mp hn
-  refine ⟨fun i => algebraMap (MvPolynomial (Fin 2) k) (planeCoordinateChart k a) (b i) *
-    IsLocalization.Away.invSelf (MvPolynomial.X a : MvPolynomial (Fin 2) k)^n, ?_⟩
-  funext i
-  rw [map_mul, map_pow, planeChartToFraction_algebraMap, planeChartToFraction_invSelf, hb]
-  change ((MvPolynomial.X a : MvPolynomial (Fin 2) k)^n • v i) * _ = _
-  rw [Algebra.smul_def, map_pow]
-  rw [inv_pow, mul_assoc, mul_comm (v i), ← mul_assoc,
-    mul_inv_cancel₀ (pow_ne_zero n (planeFractionCoordinate_ne_zero k a)), one_mul]
-
-lemma planeOverlapToFraction_injective : Function.Injective (planeOverlapToFraction k) := by
-  apply (IsLocalization.injective_iff_map_algebraMap_eq
-    (Submonoid.powers (MvPolynomial.X 0 * MvPolynomial.X 1 : MvPolynomial (Fin 2) k))
-    (planeOverlapToFraction k).toRingHom).mpr
-  intro P Q
-  constructor
-  · exact congrArg (planeOverlapToFraction k)
-  · intro h
-    have h' : algebraMap (MvPolynomial (Fin 2) k) (planeFractionField k) P =
-        algebraMap (MvPolynomial (Fin 2) k) (planeFractionField k) Q := by
-      simpa only [AlgHom.toRingHom_eq_coe, RingHom.coe_coe,
-        planeOverlapToFraction_algebraMap] using h
-    exact congrArg (algebraMap (MvPolynomial (Fin 2) k) (planeOverlapRing k))
-      ((FaithfulSMul.algebraMap_injective _ _) h')
-
-variable {ι : Type*} [Fintype ι]
-lemma planeChartPair_evaluation_relation (P : Matrix ι ι (Polynomial k))
-    (f₀ : ι → planeCoordinateChart k 0) (f₁ : ι → planeCoordinateChart k 1)
-    (hf : (fun i => planeChartToFraction k 1 (f₁ i)) =
-      (polynomialMatrixEval P ((algebraMap (MvPolynomial (Fin 2) k) (planeFractionField k)
-        (MvPolynomial.X 0 * MvPolynomial.X 1))⁻¹)).mulVec
-          (fun i => planeChartToFraction k 0 (f₀ i))) (x : planeTorusPoints k) :
-    (fun i => planeChartEvaluation k 1 (f₁ i) x) =
-      (polynomialMatrixEval P ((x.val 0 * x.val 1)⁻¹)).mulVec
-        (fun i => planeChartEvaluation k 0 (f₀ i) x) := by
-  have hov (i : ι) : planeChartToOverlap k 1 (f₁ i) =
-      ∑ j, Polynomial.aeval (planeOverlapInverse k) (P i j) * planeChartToOverlap k 0 (f₀ j) := by
-    apply planeOverlapToFraction_injective k
-    simp only [map_sum, map_mul, map_aeval_polynomial, planeChartToFraction_overlap,
-      planeOverlapInverse_fraction]
-    simpa only [polynomialMatrixEval, Matrix.mulVec, dotProduct, map_mul] using congrFun hf i
-  funext i
-  let ev : planeOverlapRing k →ₐ[k] k :=
-    (Pi.evalAlgHom k _ x).comp (planeOverlapEvaluation k)
-  have h := congrArg ev (hov i)
-  simp only [map_sum, map_mul, map_aeval_polynomial] at h
-  change planeOverlapEvaluation k (planeChartToOverlap k 1 (f₁ i)) x =
-    ∑ j, Polynomial.aeval (planeOverlapEvaluation k (planeOverlapInverse k) x) (P i j) *
-      planeOverlapEvaluation k (planeChartToOverlap k 0 (f₀ j)) x at h
-  simpa only [planeChartToOverlap_evaluation, planeOverlapInverse_evaluation,
-    polynomialMatrixEval, Matrix.mulVec, dotProduct] using h
-
-end
-
-universe u
-variable (k : Type u) [Field k] [Infinite k]
-variable (ι : Type u) [Fintype ι] [DecidableEq ι]
-variable (ρ : Representation k (Multiplicative k) (ι → k))
-variable (P : Matrix ι ι (Polynomial k))
-variable (hP : ∀ t : k, polynomialMatrixEval P t = LinearMap.toMatrix' (ρ (Multiplicative.ofAdd t)))
-
-noncomputable def associatedLatticeMap :
-    (associatedSectionsSubmodule k ι ρ) →ₗ[MvPolynomial (Fin 2) k]
-      planeGlobalLattice k P (associatedPolynomial_zero k ι ρ P hP)
-        (associatedPolynomial_add k ι ρ P hP) :=
-  (associatedFractionChart k ι ρ).codRestrict _ (associatedFractionChart_mem k ι ρ P hP)
-
-lemma associatedLatticeMap_surjective : Function.Surjective (associatedLatticeMap k ι ρ P hP) := by
-  intro v
-  obtain ⟨f₀,h₀⟩ := planeChartVector_exists k 0 v.val v.property.1
-  obtain ⟨w,hw,he⟩ := v.property.2
-  obtain ⟨f₁,h₁⟩ := planeChartVector_exists k 1 w hw
-  have hrel : (fun i => planeChartToFraction k 1 (f₁ i)) =
-      (polynomialMatrixEval P ((algebraMap (MvPolynomial (Fin 2) k) (planeFractionField k)
-        (MvPolynomial.X 0 * MvPolynomial.X 1))⁻¹)).mulVec
-          (fun i => planeChartToFraction k 0 (f₀ i)) := by
-    rw [h₀, h₁]
-    change planeGlobalTransition k P (associatedPolynomial_zero k ι ρ P hP)
-      (associatedPolynomial_add k ι ρ P hP) w = v.val at he
-    have h := congrArg (planeGlobalTransition k P (associatedPolynomial_zero k ι ρ P hP)
-      (associatedPolynomial_add k ι ρ P hP)).symm he
-    rw [LinearEquiv.symm_apply_apply] at h
-    rw [planeGlobalTransition, polynomialActionEquiv_symm, neg_neg] at h
-    simpa only [polynomialActionEquiv, LinearEquiv.coe_mk, Matrix.toLin'_apply] using h
-  have htrans (x : planeTorusPoints k) :
-      (fun i => planeChartEvaluation k 1 (f₁ i) x) =
-        ρ (Multiplicative.ofAdd ((x.val 0*x.val 1)⁻¹))
-          (fun i => planeChartEvaluation k 0 (f₀ i) x) := by
-    have h := planeChartPair_evaluation_relation k P f₀ f₁ hrel x
-    rwa [hP, LinearMap.toMatrix'_mulVec] at h
-  obtain ⟨f,hf,_⟩ := associatedChartPair_glue k ι ρ P hP f₀ f₁ htrans
-  refine ⟨f, Subtype.ext ?_⟩
-  change (fun i => planeChartToFraction k 0 (associatedXChart k ι ρ f i)) = v.val
-  rw [hf]
-  exact h₀
-
- 
-
-noncomputable def associatedLatticeEquiv :
-    (associatedSectionsSubmodule k ι ρ) ≃ₗ[MvPolynomial (Fin 2) k]
-      planeGlobalLattice k P (associatedPolynomial_zero k ι ρ P hP)
-        (associatedPolynomial_add k ι ρ P hP) :=
-  LinearEquiv.ofBijective (associatedLatticeMap k ι ρ P hP)
-    ⟨fun _ _ h => associatedFractionChart_injective k ι ρ (congrArg Subtype.val h),
-      associatedLatticeMap_surjective k ι ρ P hP⟩
-
-
-omit [Fintype ι] [DecidableEq ι] in
-lemma associatedFractionChart_apply (f : associatedSectionsSubmodule k ι ρ) (i : ι) :
-    associatedFractionChart k ι ρ f i =
-      planeChartToFraction k 0 (associatedXChart k ι ρ f i) := rfl
-
-include hP in
-lemma associatedNumerator_exists (d : ℕ) (hd : ∀ i j, (P i j).natDegree ≤ d)
-    (f : associatedSectionsSubmodule k ι ρ) :
-    ∃ F : ι → MvPolynomial (Fin 2) k, ∀ i,
-      algebraMap (MvPolynomial (Fin 2) k) (planeCoordinateChart k 0) (F i) =
-        planeChartCoordinate k 0 0 ^ d * associatedXChart k ι ρ f i := by
-  have hb := gluedLattice_bound (MvPolynomial.X 0 : MvPolynomial (Fin 2) k)
-    (MvPolynomial.X 1) (MvPolynomial.X_ne_zero 0) (planeCoordinates_relPrime k)
-    (planeGlobalTransition k P (associatedPolynomial_zero k ι ρ P hP)
-      (associatedPolynomial_add k ι ρ P hP)) d
-    (planeGlobalTransition_denominator k P (associatedPolynomial_zero k ι ρ P hP)
-      (associatedPolynomial_add k ι ρ P hP) d hd)
-    (associatedFractionChart_mem k ι ρ P hP f)
-  choose F hF using (mem_coordinateLattice _).mp hb
-  refine ⟨F, fun i => planeChartToFraction_injective k 0 ?_⟩
-  rw [planeChartToFraction_algebraMap, map_mul, map_pow,
-    planeChartCoordinate, planeChartToFraction_algebraMap]
-  simpa only [Pi.smul_apply, Algebra.smul_def, map_pow, associatedFractionChart_apply] using hF i
-
-omit [Fintype ι] [DecidableEq ι] in
-lemma associatedNumerator_evaluation (d : ℕ) (f : associatedSectionsSubmodule k ι ρ)
-    (F : ι → MvPolynomial (Fin 2) k)
-    (hF : ∀ i, algebraMap (MvPolynomial (Fin 2) k) (planeCoordinateChart k 0) (F i) =
-      planeChartCoordinate k 0 0 ^ d * associatedXChart k ι ρ f i)
-    (x : planeTorusPoints k) (i : ι) :
-    MvPolynomial.eval x.val (F i) =
-      (x.val 0)^d * planeChartEvaluation k 0 (associatedXChart k ι ρ f i) x := by
-  have h := congrFun (congrArg (planeChartEvaluation k 0) (hF i)) x
-  simpa only [planeChartEvaluation_algebraMap, planeTorusEvaluation_apply,
-    map_mul, map_pow, Pi.mul_apply, Pi.pow_apply, planeChartEvaluation_coordinate] using h
-
-end FiberFunctorProof
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.RestrictedTilting.GlobalLattices
+
+namespace OAI
+
+/-! Associated-origin finiteness, lattice equivalence and numerator evaluations. -/
+
+noncomputable section
+open CategoryTheory CategoryTheory.MonoidalCategory CategoryTheory.Limits
+open scoped BigOperators
+
+namespace FiberFunctorProof
+section
+universe u
+variable (k : Type u) [Field k] [Infinite k]
+variable (ι : Type u) [Fintype ι] [DecidableEq ι]
+variable (ρ : Representation k (Multiplicative k) (ι → k))
+
+omit [Infinite k] [Fintype ι] [DecidableEq ι] in
+lemma associatedCoordinate_smul (i : Fin 2) (f : associatedSectionsSubmodule k ι ρ) :
+    (MvPolynomial.X i : MvPolynomial (Fin 2) k) • f = associatedCoordinateMul k ι ρ i f := by
+  apply Subtype.ext
+  funext j
+  apply Subtype.ext
+  funext g
+  change ((MvPolynomial.X i : MvPolynomial (Fin 2) k) • f.val j).val g =
+    (regularFirstColumn k i).val g * (f.val j).val g
+  rw [Algebra.smul_def, regularPlane_algebraMap]
+  change (regularPlane k (MvPolynomial.X i)).val g * (f.val j).val g = _
+  rw [regularPlane_evaluation, MvPolynomial.eval_X, regularFirstColumn_apply]
+
+omit [Infinite k] [Fintype ι] [DecidableEq ι] in
+lemma associatedOrigin_coordinate_mem (i : Fin 2) (f : associatedSectionsSubmodule k ι ρ) :
+    (MvPolynomial.X i : MvPolynomial (Fin 2) k) • f ∈ associatedOriginRelations k ι ρ := by
+  rw [associatedCoordinate_smul]
+  fin_cases i
+  · exact Submodule.mem_sup_left ⟨f,rfl⟩
+  · exact Submodule.mem_sup_right ⟨f,rfl⟩
+
+ 
+
+theorem associatedOrigin_finite
+    (P : Matrix ι ι (Polynomial k))
+    (hP : ∀ t : k, polynomialMatrixEval P t = LinearMap.toMatrix' (ρ (Multiplicative.ofAdd t)))
+    (d : ℕ) (hd : ∀ i j, (P i j).natDegree ≤ d) :
+    Module.Finite k (associatedOriginRep k ι ρ).V := by
+  let : Module.Finite (MvPolynomial (Fin 2) k) (associatedSectionsSubmodule k ι ρ) :=
+    associatedSections_finite k ι ρ P hP d hd
+  exact originQuotient_finite (associatedOriginRelations k ι ρ)
+    (associatedOrigin_coordinate_mem k ι ρ)
+
+end
+
+section
+variable (k : Type*) [Field k]
+
+lemma planeChartToFraction_invSelf (a : Fin 2) :
+    planeChartToFraction k a
+      (IsLocalization.Away.invSelf (MvPolynomial.X a : MvPolynomial (Fin 2) k)) =
+      (algebraMap (MvPolynomial (Fin 2) k) (planeFractionField k) (MvPolynomial.X a))⁻¹ := by
+  have h := congrArg (planeChartToFraction k a)
+    (IsLocalization.Away.mul_invSelf (S := planeCoordinateChart k a)
+      (MvPolynomial.X a : MvPolynomial (Fin 2) k))
+  rw [map_mul, map_one, planeChartToFraction_algebraMap] at h
+  apply mul_left_cancel₀ (planeFractionCoordinate_ne_zero k a)
+  rw [mul_inv_cancel₀ (planeFractionCoordinate_ne_zero k a)]
+  exact h
+
+lemma planeChartVector_exists {ι : Type*} (a : Fin 2) (v : ι → planeFractionField k)
+    (hv : v ∈ powerLocalization (MvPolynomial.X a : MvPolynomial (Fin 2) k) coordinateLattice) :
+    ∃ f : ι → planeCoordinateChart k a, (fun i => planeChartToFraction k a (f i)) = v := by
+  obtain ⟨n,hn⟩ := hv
+  choose b hb using (mem_coordinateLattice _).mp hn
+  refine ⟨fun i => algebraMap (MvPolynomial (Fin 2) k) (planeCoordinateChart k a) (b i) *
+    IsLocalization.Away.invSelf (MvPolynomial.X a : MvPolynomial (Fin 2) k)^n, ?_⟩
+  funext i
+  rw [map_mul, map_pow, planeChartToFraction_algebraMap, planeChartToFraction_invSelf, hb]
+  change ((MvPolynomial.X a : MvPolynomial (Fin 2) k)^n • v i) * _ = _
+  rw [Algebra.smul_def, map_pow]
+  rw [inv_pow, mul_assoc, mul_comm (v i), ← mul_assoc,
+    mul_inv_cancel₀ (pow_ne_zero n (planeFractionCoordinate_ne_zero k a)), one_mul]
+
+lemma planeOverlapToFraction_injective : Function.Injective (planeOverlapToFraction k) := by
+  apply (IsLocalization.injective_iff_map_algebraMap_eq
+    (Submonoid.powers (MvPolynomial.X 0 * MvPolynomial.X 1 : MvPolynomial (Fin 2) k))
+    (planeOverlapToFraction k).toRingHom).mpr
+  intro P Q
+  constructor
+  · exact congrArg (planeOverlapToFraction k)
+  · intro h
+    have h' : algebraMap (MvPolynomial (Fin 2) k) (planeFractionField k) P =
+        algebraMap (MvPolynomial (Fin 2) k) (planeFractionField k) Q := by
+      simpa only [AlgHom.toRingHom_eq_coe, RingHom.coe_coe,
+        planeOverlapToFraction_algebraMap] using h
+    exact congrArg (algebraMap (MvPolynomial (Fin 2) k) (planeOverlapRing k))
+      ((FaithfulSMul.algebraMap_injective _ _) h')
+
+variable {ι : Type*} [Fintype ι]
+lemma planeChartPair_evaluation_relation (P : Matrix ι ι (Polynomial k))
+    (f₀ : ι → planeCoordinateChart k 0) (f₁ : ι → planeCoordinateChart k 1)
+    (hf : (fun i => planeChartToFraction k 1 (f₁ i)) =
+      (polynomialMatrixEval P ((algebraMap (MvPolynomial (Fin 2) k) (planeFractionField k)
+        (MvPolynomial.X 0 * MvPolynomial.X 1))⁻¹)).mulVec
+          (fun i => planeChartToFraction k 0 (f₀ i))) (x : planeTorusPoints k) :
+    (fun i => planeChartEvaluation k 1 (f₁ i) x) =
+      (polynomialMatrixEval P ((x.val 0 * x.val 1)⁻¹)).mulVec
+        (fun i => planeChartEvaluation k 0 (f₀ i) x) := by
+  have hov (i : ι) : planeChartToOverlap k 1 (f₁ i) =
+      ∑ j, Polynomial.aeval (planeOverlapInverse k) (P i j) * planeChartToOverlap k 0 (f₀ j) := by
+    apply planeOverlapToFraction_injective k
+    simp only [map_sum, map_mul, map_aeval_polynomial, planeChartToFraction_overlap,
+      planeOverlapInverse_fraction]
+    simpa only [polynomialMatrixEval, Matrix.mulVec, dotProduct, map_mul] using congrFun hf i
+  funext i
+  let ev : planeOverlapRing k →ₐ[k] k :=
+    (Pi.evalAlgHom k _ x).comp (planeOverlapEvaluation k)
+  have h := congrArg ev (hov i)
+  simp only [map_sum, map_mul, map_aeval_polynomial] at h
+  change planeOverlapEvaluation k (planeChartToOverlap k 1 (f₁ i)) x =
+    ∑ j, Polynomial.aeval (planeOverlapEvaluation k (planeOverlapInverse k) x) (P i j) *
+      planeOverlapEvaluation k (planeChartToOverlap k 0 (f₀ j)) x at h
+  simpa only [planeChartToOverlap_evaluation, planeOverlapInverse_evaluation,
+    polynomialMatrixEval, Matrix.mulVec, dotProduct] using h
+
+end
+
+universe u
+variable (k : Type u) [Field k] [Infinite k]
+variable (ι : Type u) [Fintype ι] [DecidableEq ι]
+variable (ρ : Representation k (Multiplicative k) (ι → k))
+variable (P : Matrix ι ι (Polynomial k))
+variable (hP : ∀ t : k, polynomialMatrixEval P t = LinearMap.toMatrix' (ρ (Multiplicative.ofAdd t)))
+
+noncomputable def associatedLatticeMap :
+    (associatedSectionsSubmodule k ι ρ) →ₗ[MvPolynomial (Fin 2) k]
+      planeGlobalLattice k P (associatedPolynomial_zero k ι ρ P hP)
+        (associatedPolynomial_add k ι ρ P hP) :=
+  (associatedFractionChart k ι ρ).codRestrict _ (associatedFractionChart_mem k ι ρ P hP)
+
+lemma associatedLatticeMap_surjective : Function.Surjective (associatedLatticeMap k ι ρ P hP) := by
+  intro v
+  obtain ⟨f₀,h₀⟩ := planeChartVector_exists k 0 v.val v.property.1
+  obtain ⟨w,hw,he⟩ := v.property.2
+  obtain ⟨f₁,h₁⟩ := planeChartVector_exists k 1 w hw
+  have hrel : (fun i => planeChartToFraction k 1 (f₁ i)) =
+      (polynomialMatrixEval P ((algebraMap (MvPolynomial (Fin 2) k) (planeFractionField k)
+        (MvPolynomial.X 0 * MvPolynomial.X 1))⁻¹)).mulVec
+          (fun i => planeChartToFraction k 0 (f₀ i)) := by
+    rw [h₀, h₁]
+    change planeGlobalTransition k P (associatedPolynomial_zero k ι ρ P hP)
+      (associatedPolynomial_add k ι ρ P hP) w = v.val at he
+    have h := congrArg (planeGlobalTransition k P (associatedPolynomial_zero k ι ρ P hP)
+      (associatedPolynomial_add k ι ρ P hP)).symm he
+    rw [LinearEquiv.symm_apply_apply] at h
+    rw [planeGlobalTransition, polynomialActionEquiv_symm, neg_neg] at h
+    simpa only [polynomialActionEquiv, LinearEquiv.coe_mk, Matrix.toLin'_apply] using h
+  have htrans (x : planeTorusPoints k) :
+      (fun i => planeChartEvaluation k 1 (f₁ i) x) =
+        ρ (Multiplicative.ofAdd ((x.val 0*x.val 1)⁻¹))
+          (fun i => planeChartEvaluation k 0 (f₀ i) x) := by
+    have h := planeChartPair_evaluation_relation k P f₀ f₁ hrel x
+    rwa [hP, LinearMap.toMatrix'_mulVec] at h
+  obtain ⟨f,hf,_⟩ := associatedChartPair_glue k ι ρ P hP f₀ f₁ htrans
+  refine ⟨f, Subtype.ext ?_⟩
+  change (fun i => planeChartToFraction k 0 (associatedXChart k ι ρ f i)) = v.val
+  rw [hf]
+  exact h₀
+
+ 
+
+noncomputable def associatedLatticeEquiv :
+    (associatedSectionsSubmodule k ι ρ) ≃ₗ[MvPolynomial (Fin 2) k]
+      planeGlobalLattice k P (associatedPolynomial_zero k ι ρ P hP)
+        (associatedPolynomial_add k ι ρ P hP) :=
+  LinearEquiv.ofBijective (associatedLatticeMap k ι ρ P hP)
+    ⟨fun _ _ h => associatedFractionChart_injective k ι ρ (congrArg Subtype.val h),
+      associatedLatticeMap_surjective k ι ρ P hP⟩
+
+
+omit [Fintype ι] [DecidableEq ι] in
+lemma associatedFractionChart_apply (f : associatedSectionsSubmodule k ι ρ) (i : ι) :
+    associatedFractionChart k ι ρ f i =
+      planeChartToFraction k 0 (associatedXChart k ι ρ f i) := rfl
+
+include hP in
+lemma associatedNumerator_exists (d : ℕ) (hd : ∀ i j, (P i j).natDegree ≤ d)
+    (f : associatedSectionsSubmodule k ι ρ) :
+    ∃ F : ι → MvPolynomial (Fin 2) k, ∀ i,
+      algebraMap (MvPolynomial (Fin 2) k) (planeCoordinateChart k 0) (F i) =
+        planeChartCoordinate k 0 0 ^ d * associatedXChart k ι ρ f i := by
+  have hb := gluedLattice_bound (MvPolynomial.X 0 : MvPolynomial (Fin 2) k)
+    (MvPolynomial.X 1) (MvPolynomial.X_ne_zero 0) (planeCoordinates_relPrime k)
+    (planeGlobalTransition k P (associatedPolynomial_zero k ι ρ P hP)
+      (associatedPolynomial_add k ι ρ P hP)) d
+    (planeGlobalTransition_denominator k P (associatedPolynomial_zero k ι ρ P hP)
+      (associatedPolynomial_add k ι ρ P hP) d hd)
+    (associatedFractionChart_mem k ι ρ P hP f)
+  choose F hF using (mem_coordinateLattice _).mp hb
+  refine ⟨F, fun i => planeChartToFraction_injective k 0 ?_⟩
+  rw [planeChartToFraction_algebraMap, map_mul, map_pow,
+    planeChartCoordinate, planeChartToFraction_algebraMap]
+  simpa only [Pi.smul_apply, Algebra.smul_def, map_pow, associatedFractionChart_apply] using hF i
+
+omit [Fintype ι] [DecidableEq ι] in
+lemma associatedNumerator_evaluation (d : ℕ) (f : associatedSectionsSubmodule k ι ρ)
+    (F : ι → MvPolynomial (Fin 2) k)
+    (hF : ∀ i, algebraMap (MvPolynomial (Fin 2) k) (planeCoordinateChart k 0) (F i) =
+      planeChartCoordinate k 0 0 ^ d * associatedXChart k ι ρ f i)
+    (x : planeTorusPoints k) (i : ι) :
+    MvPolynomial.eval x.val (F i) =
+      (x.val 0)^d * planeChartEvaluation k 0 (associatedXChart k ι ρ f i) x := by
+  have h := congrFun (congrArg (planeChartEvaluation k 0) (hF i)) x
+  simpa only [planeChartEvaluation_algebraMap, planeTorusEvaluation_apply,
+    map_mul, map_pow, Pi.mul_apply, Pi.pow_apply, planeChartEvaluation_coordinate] using h
+
+end FiberFunctorProof
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/RestrictedTilting/AssociatedNumerator.lean b/lean/OAI/RepresentationTheory/RestrictedTilting/AssociatedNumerator.lean
index 92ade533ab520cf1d3493fb52221b09b79ed3189..72a2b9388a919a811246a1f5c52f31f3dec77550 100644
--- a/lean/OAI/RepresentationTheory/RestrictedTilting/AssociatedNumerator.lean
+++ b/lean/OAI/RepresentationTheory/RestrictedTilting/AssociatedNumerator.lean
@@ -1,82 +1,82 @@
-import OAI.RepresentationTheory.RestrictedTilting.LatticeNumerator
-
-namespace OAI
-
-namespace FiberFunctorProof
-universe u
-variable (k : Type u) [Field k] [Infinite k]
-variable (ι : Type u) [Fintype ι] [DecidableEq ι]
-variable (ρ : Representation k (Multiplicative k) (ι → k))
-variable (P : Matrix ι ι (Polynomial k))
-variable (hP : ∀ t : k, polynomialMatrixEval P t = LinearMap.toMatrix' (ρ (Multiplicative.ofAdd t)))
-variable (d : ℕ) (hd : ∀ i j, (P i j).natDegree ≤ d)
-local notation "A" => MvPolynomial (Fin 2) k
-local notation "N" => associatedSectionsSubmodule k ι ρ
-local notation "h0" => associatedPolynomial_zero k ι ρ P hP
-local notation "hadd" => associatedPolynomial_add k ι ρ P hP
-local notation "Q" => planeGlobalTransition k P h0 hadd
-
-include hd in
-lemma planeGlobalTransition_inverse_denominator :
-    ∀ v ∈ coordinateLattice (R := A), (MvPolynomial.X 0 * MvPolynomial.X 1 : A)^d •
-      (Q).symm v ∈ coordinateLattice (R := A) := by
-  intro v hv
-  have h := polynomialAction_fraction_denominator P h0 hadd (1 : A)
-    (MvPolynomial.X 0 * MvPolynomial.X 1) (planeFraction_product_ne_zero k) d hd v hv
-  simpa only [map_one, one_mul, planeGlobalTransition, polynomialActionEquiv_symm, neg_neg] using h
-
-noncomputable def associatedNumeratorLinear : N →ₗ[A] (ι → A) :=
-  (latticeNumerator (MvPolynomial.X 0) (MvPolynomial.X 1) (MvPolynomial.X_ne_zero 0)
-    (planeCoordinates_relPrime k) Q d (planeGlobalTransition_denominator k P h0 hadd d hd)) ∘ₗ
-    (associatedLatticeEquiv k ι ρ P hP).toLinearMap
-
-noncomputable def associatedDenominatorLinear : (ι → A) →ₗ[A] N :=
-  (associatedLatticeEquiv k ι ρ P hP).symm.toLinearMap ∘ₗ
-    latticeDenominator (MvPolynomial.X 0) (MvPolynomial.X 1) Q d
-      (planeGlobalTransition_inverse_denominator k ι ρ P hP d hd)
-
-lemma associatedNumerator_denominator (v : ι → A) :
-    associatedNumeratorLinear k ι ρ P hP d hd (associatedDenominatorLinear k ι ρ P hP d hd v) =
-      (MvPolynomial.X 0 : A)^(d+d) • v := by
-  change latticeNumerator (MvPolynomial.X 0) (MvPolynomial.X 1) (MvPolynomial.X_ne_zero 0)
-    (planeCoordinates_relPrime k) Q d (planeGlobalTransition_denominator k P h0 hadd d hd)
-    ((associatedLatticeEquiv k ι ρ P hP)
-      ((associatedLatticeEquiv k ι ρ P hP).symm (latticeDenominator (MvPolynomial.X 0) (MvPolynomial.X 1) Q d
-        (planeGlobalTransition_inverse_denominator k ι ρ P hP d hd) v))) = _
-  erw [LinearEquiv.apply_symm_apply]
-  exact latticeNumerator_denominator (MvPolynomial.X 0) (MvPolynomial.X 1)
-    (MvPolynomial.X_ne_zero 0) (planeCoordinates_relPrime k) Q d d
-    (planeGlobalTransition_denominator k P h0 hadd d hd)
-    (planeGlobalTransition_inverse_denominator k ι ρ P hP d hd) v
-
-lemma associatedDenominator_numerator (f : N) :
-    associatedDenominatorLinear k ι ρ P hP d hd (associatedNumeratorLinear k ι ρ P hP d hd f) =
-      (MvPolynomial.X 0 : A)^(d+d) • f := by
-  change (associatedLatticeEquiv k ι ρ P hP).symm
-    (latticeDenominator (MvPolynomial.X 0) (MvPolynomial.X 1) Q d
-        (planeGlobalTransition_inverse_denominator k ι ρ P hP d hd) (latticeNumerator (MvPolynomial.X 0) (MvPolynomial.X 1) (MvPolynomial.X_ne_zero 0)
-      (planeCoordinates_relPrime k) Q d (planeGlobalTransition_denominator k P h0 hadd d hd)
-      (associatedLatticeEquiv k ι ρ P hP f))) = _
-  erw [latticeDenominator_numerator, map_smul, LinearEquiv.symm_apply_apply]
-
-lemma associatedNumeratorLinear_spec (f : N) (i : ι) :
-    algebraMap A (planeCoordinateChart k 0) (associatedNumeratorLinear k ι ρ P hP d hd f i) =
-      planeChartCoordinate k 0 0 ^ d * associatedXChart k ι ρ f i := by
-  apply planeChartToFraction_injective k 0
-  rw [planeChartToFraction_algebraMap, map_mul, map_pow,
-    planeChartCoordinate, planeChartToFraction_algebraMap]
-  have h := congrFun (latticeNumerator_coordinates (MvPolynomial.X 0) (MvPolynomial.X 1)
-    (MvPolynomial.X_ne_zero 0) (planeCoordinates_relPrime k) Q d
-    (planeGlobalTransition_denominator k P h0 hadd d hd)
-    (associatedLatticeEquiv k ι ρ P hP f)) i
-  change algebraMap A (planeFractionField k)
-    (latticeNumerator (MvPolynomial.X 0) (MvPolynomial.X 1) (MvPolynomial.X_ne_zero 0)
-      (planeCoordinates_relPrime k) Q d (planeGlobalTransition_denominator k P h0 hadd d hd)
-      (associatedLatticeEquiv k ι ρ P hP f) i) = _
-  change algebraMap A (planeFractionField k) _ =
-    (MvPolynomial.X 0 : A)^d • (associatedLatticeEquiv k ι ρ P hP f).val i at h
-  simp only [Algebra.smul_def, map_pow] at h
-  exact h
-end FiberFunctorProof
-
-end OAI
+import OAI.RepresentationTheory.RestrictedTilting.LatticeNumerator
+
+namespace OAI
+
+namespace FiberFunctorProof
+universe u
+variable (k : Type u) [Field k] [Infinite k]
+variable (ι : Type u) [Fintype ι] [DecidableEq ι]
+variable (ρ : Representation k (Multiplicative k) (ι → k))
+variable (P : Matrix ι ι (Polynomial k))
+variable (hP : ∀ t : k, polynomialMatrixEval P t = LinearMap.toMatrix' (ρ (Multiplicative.ofAdd t)))
+variable (d : ℕ) (hd : ∀ i j, (P i j).natDegree ≤ d)
+local notation "A" => MvPolynomial (Fin 2) k
+local notation "N" => associatedSectionsSubmodule k ι ρ
+local notation "h0" => associatedPolynomial_zero k ι ρ P hP
+local notation "hadd" => associatedPolynomial_add k ι ρ P hP
+local notation "Q" => planeGlobalTransition k P h0 hadd
+
+include hd in
+lemma planeGlobalTransition_inverse_denominator :
+    ∀ v ∈ coordinateLattice (R := A), (MvPolynomial.X 0 * MvPolynomial.X 1 : A)^d •
+      (Q).symm v ∈ coordinateLattice (R := A) := by
+  intro v hv
+  have h := polynomialAction_fraction_denominator P h0 hadd (1 : A)
+    (MvPolynomial.X 0 * MvPolynomial.X 1) (planeFraction_product_ne_zero k) d hd v hv
+  simpa only [map_one, one_mul, planeGlobalTransition, polynomialActionEquiv_symm, neg_neg] using h
+
+noncomputable def associatedNumeratorLinear : N →ₗ[A] (ι → A) :=
+  (latticeNumerator (MvPolynomial.X 0) (MvPolynomial.X 1) (MvPolynomial.X_ne_zero 0)
+    (planeCoordinates_relPrime k) Q d (planeGlobalTransition_denominator k P h0 hadd d hd)) ∘ₗ
+    (associatedLatticeEquiv k ι ρ P hP).toLinearMap
+
+noncomputable def associatedDenominatorLinear : (ι → A) →ₗ[A] N :=
+  (associatedLatticeEquiv k ι ρ P hP).symm.toLinearMap ∘ₗ
+    latticeDenominator (MvPolynomial.X 0) (MvPolynomial.X 1) Q d
+      (planeGlobalTransition_inverse_denominator k ι ρ P hP d hd)
+
+lemma associatedNumerator_denominator (v : ι → A) :
+    associatedNumeratorLinear k ι ρ P hP d hd (associatedDenominatorLinear k ι ρ P hP d hd v) =
+      (MvPolynomial.X 0 : A)^(d+d) • v := by
+  change latticeNumerator (MvPolynomial.X 0) (MvPolynomial.X 1) (MvPolynomial.X_ne_zero 0)
+    (planeCoordinates_relPrime k) Q d (planeGlobalTransition_denominator k P h0 hadd d hd)
+    ((associatedLatticeEquiv k ι ρ P hP)
+      ((associatedLatticeEquiv k ι ρ P hP).symm (latticeDenominator (MvPolynomial.X 0) (MvPolynomial.X 1) Q d
+        (planeGlobalTransition_inverse_denominator k ι ρ P hP d hd) v))) = _
+  erw [LinearEquiv.apply_symm_apply]
+  exact latticeNumerator_denominator (MvPolynomial.X 0) (MvPolynomial.X 1)
+    (MvPolynomial.X_ne_zero 0) (planeCoordinates_relPrime k) Q d d
+    (planeGlobalTransition_denominator k P h0 hadd d hd)
+    (planeGlobalTransition_inverse_denominator k ι ρ P hP d hd) v
+
+lemma associatedDenominator_numerator (f : N) :
+    associatedDenominatorLinear k ι ρ P hP d hd (associatedNumeratorLinear k ι ρ P hP d hd f) =
+      (MvPolynomial.X 0 : A)^(d+d) • f := by
+  change (associatedLatticeEquiv k ι ρ P hP).symm
+    (latticeDenominator (MvPolynomial.X 0) (MvPolynomial.X 1) Q d
+        (planeGlobalTransition_inverse_denominator k ι ρ P hP d hd) (latticeNumerator (MvPolynomial.X 0) (MvPolynomial.X 1) (MvPolynomial.X_ne_zero 0)
+      (planeCoordinates_relPrime k) Q d (planeGlobalTransition_denominator k P h0 hadd d hd)
+      (associatedLatticeEquiv k ι ρ P hP f))) = _
+  erw [latticeDenominator_numerator, map_smul, LinearEquiv.symm_apply_apply]
+
+lemma associatedNumeratorLinear_spec (f : N) (i : ι) :
+    algebraMap A (planeCoordinateChart k 0) (associatedNumeratorLinear k ι ρ P hP d hd f i) =
+      planeChartCoordinate k 0 0 ^ d * associatedXChart k ι ρ f i := by
+  apply planeChartToFraction_injective k 0
+  rw [planeChartToFraction_algebraMap, map_mul, map_pow,
+    planeChartCoordinate, planeChartToFraction_algebraMap]
+  have h := congrFun (latticeNumerator_coordinates (MvPolynomial.X 0) (MvPolynomial.X 1)
+    (MvPolynomial.X_ne_zero 0) (planeCoordinates_relPrime k) Q d
+    (planeGlobalTransition_denominator k P h0 hadd d hd)
+    (associatedLatticeEquiv k ι ρ P hP f)) i
+  change algebraMap A (planeFractionField k)
+    (latticeNumerator (MvPolynomial.X 0) (MvPolynomial.X 1) (MvPolynomial.X_ne_zero 0)
+      (planeCoordinates_relPrime k) Q d (planeGlobalTransition_denominator k P h0 hadd d hd)
+      (associatedLatticeEquiv k ι ρ P hP f) i) = _
+  change algebraMap A (planeFractionField k) _ =
+    (MvPolynomial.X 0 : A)^d • (associatedLatticeEquiv k ι ρ P hP f).val i at h
+  simp only [Algebra.smul_def, map_pow] at h
+  exact h
+end FiberFunctorProof
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/RestrictedTilting/AssociatedPlaneJet.lean b/lean/OAI/RepresentationTheory/RestrictedTilting/AssociatedPlaneJet.lean
index b9b902c1210e5c3d10dac3ce7a757af3083c62f9..dccfac3fb6a2c28f9e1204e3a71cf098021cc47c 100644
--- a/lean/OAI/RepresentationTheory/RestrictedTilting/AssociatedPlaneJet.lean
+++ b/lean/OAI/RepresentationTheory/RestrictedTilting/AssociatedPlaneJet.lean
@@ -1,58 +1,58 @@
-import OAI.RepresentationTheory.RestrictedTilting.FormalSections
-import OAI.RepresentationTheory.RestrictedTilting.AssociatedNumerator
-
-namespace OAI
-
-namespace FiberFunctorProof
-open CategoryTheory CategoryTheory.MonoidalCategory
-open scoped TensorProduct Pointwise
-universe u
-variable (k : Type u) [Field k]
-variable (ι : Type u) [Fintype ι] [DecidableEq ι]
-variable (ρ : Representation k (Multiplicative k) (ι → k))
-local notation "N" => associatedSectionsSubmodule k ι ρ
-local notation "A" => MvPolynomial (Fin 2) k
-
-omit [Fintype ι] [DecidableEq ι] in
-lemma associatedSections_planeCompatible : PlaneCompatible k (associatedSectionsRepresentation k ι ρ) := by
-  intro g P v
-  induction P using MvPolynomial.induction_on generalizing v with
-  | C c =>
-    rw [linearChange_C, MvPolynomial.C_eq_algebraMap]
-    simp only [algebraMap_smul, map_smul]
-  | add P Q hP hQ => simp only [add_smul, map_add, hP, hQ]
-  | mul_X P i hP =>
-    rw [mul_smul, hP, associatedCoordinate_smul, associatedCoordinateMul_action,
-      map_mul, mul_smul, linearChange_X]
-    congr 1
-    simp only [Finset.sum_smul, mul_smul, MvPolynomial.C_eq_algebraMap,
-      algebraMap_smul, associatedCoordinate_smul]
-    rfl
-
-omit [Fintype ι] [DecidableEq ι] in
-lemma associated_planeJet_one :
-    planeJetRelation (M := N) k 1 = associatedOriginRelations k ι ρ := by
-  change (jetRelation (M := N) (MvPolynomial.X 0 : A) (MvPolynomial.X 1) 1).restrictScalars k = _
-  rw [jetRelation_one]
-  ext v
-  change v ∈ pairRelation (MvPolynomial.X 0 : A) (MvPolynomial.X 1) ↔ _
-  rw [mem_pairRelation, associatedOriginRelations, Submodule.mem_sup]
-  constructor
-  · rintro ⟨a,b,rfl⟩
-    exact ⟨_,⟨a,(associatedCoordinate_smul k ι ρ 0 a).symm⟩,
-      _,⟨b,(associatedCoordinate_smul k ι ρ 1 b).symm⟩,rfl⟩
-  · rintro ⟨_,⟨a,rfl⟩,_,⟨b,rfl⟩,rfl⟩
-    exact ⟨a,b, by rw [associatedCoordinate_smul, associatedCoordinate_smul]⟩
-
-variable [Infinite k]
-variable (P : Matrix ι ι (Polynomial k))
-variable (hP : ∀ t : k, polynomialMatrixEval P t = LinearMap.toMatrix' (ρ (Multiplicative.ofAdd t)))
-variable (d : ℕ) (hd : ∀ i j, (P i j).natDegree ≤ d)
-
-include hP hd in
-lemma associated_planeJet_one_finite : Module.Finite k (N ⧸ planeJetRelation (M := N) k 1) := by
-  rw [associated_planeJet_one]
-  exact associatedOrigin_finite k ι ρ P hP d hd
-end FiberFunctorProof
-
-end OAI
+import OAI.RepresentationTheory.RestrictedTilting.FormalSections
+import OAI.RepresentationTheory.RestrictedTilting.AssociatedNumerator
+
+namespace OAI
+
+namespace FiberFunctorProof
+open CategoryTheory CategoryTheory.MonoidalCategory
+open scoped TensorProduct Pointwise
+universe u
+variable (k : Type u) [Field k]
+variable (ι : Type u) [Fintype ι] [DecidableEq ι]
+variable (ρ : Representation k (Multiplicative k) (ι → k))
+local notation "N" => associatedSectionsSubmodule k ι ρ
+local notation "A" => MvPolynomial (Fin 2) k
+
+omit [Fintype ι] [DecidableEq ι] in
+lemma associatedSections_planeCompatible : PlaneCompatible k (associatedSectionsRepresentation k ι ρ) := by
+  intro g P v
+  induction P using MvPolynomial.induction_on generalizing v with
+  | C c =>
+    rw [linearChange_C, MvPolynomial.C_eq_algebraMap]
+    simp only [algebraMap_smul, map_smul]
+  | add P Q hP hQ => simp only [add_smul, map_add, hP, hQ]
+  | mul_X P i hP =>
+    rw [mul_smul, hP, associatedCoordinate_smul, associatedCoordinateMul_action,
+      map_mul, mul_smul, linearChange_X]
+    congr 1
+    simp only [Finset.sum_smul, mul_smul, MvPolynomial.C_eq_algebraMap,
+      algebraMap_smul, associatedCoordinate_smul]
+    rfl
+
+omit [Fintype ι] [DecidableEq ι] in
+lemma associated_planeJet_one :
+    planeJetRelation (M := N) k 1 = associatedOriginRelations k ι ρ := by
+  change (jetRelation (M := N) (MvPolynomial.X 0 : A) (MvPolynomial.X 1) 1).restrictScalars k = _
+  rw [jetRelation_one]
+  ext v
+  change v ∈ pairRelation (MvPolynomial.X 0 : A) (MvPolynomial.X 1) ↔ _
+  rw [mem_pairRelation, associatedOriginRelations, Submodule.mem_sup]
+  constructor
+  · rintro ⟨a,b,rfl⟩
+    exact ⟨_,⟨a,(associatedCoordinate_smul k ι ρ 0 a).symm⟩,
+      _,⟨b,(associatedCoordinate_smul k ι ρ 1 b).symm⟩,rfl⟩
+  · rintro ⟨_,⟨a,rfl⟩,_,⟨b,rfl⟩,rfl⟩
+    exact ⟨a,b, by rw [associatedCoordinate_smul, associatedCoordinate_smul]⟩
+
+variable [Infinite k]
+variable (P : Matrix ι ι (Polynomial k))
+variable (hP : ∀ t : k, polynomialMatrixEval P t = LinearMap.toMatrix' (ρ (Multiplicative.ofAdd t)))
+variable (d : ℕ) (hd : ∀ i j, (P i j).natDegree ≤ d)
+
+include hP hd in
+lemma associated_planeJet_one_finite : Module.Finite k (N ⧸ planeJetRelation (M := N) k 1) := by
+  rw [associated_planeJet_one]
+  exact associatedOrigin_finite k ι ρ P hP d hd
+end FiberFunctorProof
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/RestrictedTilting/AssociatedProjective.lean b/lean/OAI/RepresentationTheory/RestrictedTilting/AssociatedProjective.lean
index 976cad5ddf9681a42e6b1e7f3b9700ac2e7b0983..34155f343f88e16ed6b812d1a9227fca1d7f188f 100644
--- a/lean/OAI/RepresentationTheory/RestrictedTilting/AssociatedProjective.lean
+++ b/lean/OAI/RepresentationTheory/RestrictedTilting/AssociatedProjective.lean
@@ -1,78 +1,78 @@
-import OAI.RepresentationTheory.RestrictedTilting.AssociatedNumerator
-import OAI.RepresentationTheory.RestrictedTilting.LatticeLocalProjectivity
-
-namespace OAI
-
-namespace FiberFunctorProof
-open scoped TensorProduct
-universe u
-variable (k : Type u) [Field k] [Infinite k]
-variable (ι : Type u) [Fintype ι] [DecidableEq ι]
-variable (ρ : Representation k (Multiplicative k) (ι → k))
-variable (P : Matrix ι ι (Polynomial k))
-variable (hP : ∀ t : k, polynomialMatrixEval P t = LinearMap.toMatrix' (ρ (Multiplicative.ofAdd t)))
-variable (d : ℕ) (hd : ∀ i j, (P i j).natDegree ≤ d)
-local notation "A" => MvPolynomial (Fin 2) k
-local notation "N" => associatedSectionsSubmodule k ι ρ
-local notation "h0" => associatedPolynomial_zero k ι ρ P hP
-local notation "hadd" => associatedPolynomial_add k ι ρ P hP
-local notation "Q" => planeGlobalTransition k P h0 hadd
-
-include hP hd in
- 
-
-theorem associatedSections_projective : Module.Projective A N := by
-  let : Module.Finite A N := associatedSections_finite k ι ρ P hP d hd
-  let : Module.FinitePresentation A N := Module.finitePresentation_of_finite A N
-  apply Module.projective_of_localization_maximal
-  intro I hI
-  let S := Localization.AtPrime I
-  have localProj : Module.Projective S (S ⊗[A] N) := by
-    by_cases hI0 : I = planeOriginIdeal k
-    · subst I
-      let := associatedStalk_free k ι ρ P hP d hd
-      change Module.Projective (planeOriginRing k) ((planeOriginRing k) ⊗[A] N)
-      exact Module.Projective.of_free
-    · have hc : (MvPolynomial.X 0 : A) ∉ I ∨ (MvPolynomial.X 1 : A) ∉ I := by
-        by_contra hn
-        push Not at hn
-        apply hI0
-        symm
-        apply (planeOriginIdeal_isMaximal k).eq_of_le hI.ne_top
-        rw [planeOriginIdeal_eq_span]
-        exact Ideal.span_le.mpr (by simpa [Set.insert_subset_iff, Set.singleton_subset_iff] using hn)
-      rcases hc with hx | hy
-      · have hu : IsUnit (algebraMap A S (MvPolynomial.X 0)) :=
-          (IsLocalization.AtPrime.isUnit_to_map_iff S I _).mpr hx
-        let : Module.Projective S (S ⊗[A] planeGlobalLattice k P h0 hadd) :=
-          gluedLattice_projective_baseChange (S := S)
-          (MvPolynomial.X 0 : A) (MvPolynomial.X 1) (MvPolynomial.X_ne_zero 0)
-          (planeCoordinates_relPrime k) Q d d
-          (planeGlobalTransition_denominator k P h0 hadd d hd)
-          (planeGlobalTransition_inverse_denominator k ι ρ P hP d hd) hu
-        exact Module.Projective.of_equiv (LinearEquiv.baseChange A S _ _ (associatedLatticeEquiv k ι ρ P hP).symm)
-      · have hu : IsUnit (algebraMap A S (MvPolynomial.X 1)) :=
-          (IsLocalization.AtPrime.isUnit_to_map_iff S I _).mpr hy
-        have ha : ∀ v ∈ (coordinateLattice (R := A)),
-            (MvPolynomial.X 1 * MvPolynomial.X 0 : A)^d • (Q).symm v ∈ coordinateLattice (R := A) := by
-          rw [show (MvPolynomial.X 1 * MvPolynomial.X 0 : A) =
-            MvPolynomial.X 0 * MvPolynomial.X 1 from mul_comm _ _]
-          exact planeGlobalTransition_inverse_denominator k ι ρ P hP d hd
-        have hb : ∀ v ∈ (coordinateLattice (R := A)),
-            (MvPolynomial.X 1 * MvPolynomial.X 0 : A)^d • (Q).symm.symm v ∈ coordinateLattice (R := A) := by
-          rw [show (MvPolynomial.X 1 * MvPolynomial.X 0 : A) =
-            MvPolynomial.X 0 * MvPolynomial.X 1 from mul_comm _ _]
-          exact planeGlobalTransition_denominator k P h0 hadd d hd
-        let := gluedLattice_projective_baseChange (S := S)
-          (MvPolynomial.X 1 : A) (MvPolynomial.X 0) (MvPolynomial.X_ne_zero 1)
-          (planeCoordinates_relPrime k).symm (Q).symm d d ha hb hu
-        exact Module.Projective.of_equiv
-          (LinearEquiv.baseChange A S _ _
-          (((associatedLatticeEquiv k ι ρ P hP).trans
-            (latticeSwap (MvPolynomial.X 0) (MvPolynomial.X 1) Q)).symm))
-  let := localProj
-  exact Module.Projective.of_equiv
-    (IsLocalizedModule.isBaseChange I.primeCompl S (LocalizedModule.mkLinearMap I.primeCompl N)).equiv
-end FiberFunctorProof
-
-end OAI
+import OAI.RepresentationTheory.RestrictedTilting.AssociatedNumerator
+import OAI.RepresentationTheory.RestrictedTilting.LatticeLocalProjectivity
+
+namespace OAI
+
+namespace FiberFunctorProof
+open scoped TensorProduct
+universe u
+variable (k : Type u) [Field k] [Infinite k]
+variable (ι : Type u) [Fintype ι] [DecidableEq ι]
+variable (ρ : Representation k (Multiplicative k) (ι → k))
+variable (P : Matrix ι ι (Polynomial k))
+variable (hP : ∀ t : k, polynomialMatrixEval P t = LinearMap.toMatrix' (ρ (Multiplicative.ofAdd t)))
+variable (d : ℕ) (hd : ∀ i j, (P i j).natDegree ≤ d)
+local notation "A" => MvPolynomial (Fin 2) k
+local notation "N" => associatedSectionsSubmodule k ι ρ
+local notation "h0" => associatedPolynomial_zero k ι ρ P hP
+local notation "hadd" => associatedPolynomial_add k ι ρ P hP
+local notation "Q" => planeGlobalTransition k P h0 hadd
+
+include hP hd in
+ 
+
+theorem associatedSections_projective : Module.Projective A N := by
+  let : Module.Finite A N := associatedSections_finite k ι ρ P hP d hd
+  let : Module.FinitePresentation A N := Module.finitePresentation_of_finite A N
+  apply Module.projective_of_localization_maximal
+  intro I hI
+  let S := Localization.AtPrime I
+  have localProj : Module.Projective S (S ⊗[A] N) := by
+    by_cases hI0 : I = planeOriginIdeal k
+    · subst I
+      let := associatedStalk_free k ι ρ P hP d hd
+      change Module.Projective (planeOriginRing k) ((planeOriginRing k) ⊗[A] N)
+      exact Module.Projective.of_free
+    · have hc : (MvPolynomial.X 0 : A) ∉ I ∨ (MvPolynomial.X 1 : A) ∉ I := by
+        by_contra hn
+        push Not at hn
+        apply hI0
+        symm
+        apply (planeOriginIdeal_isMaximal k).eq_of_le hI.ne_top
+        rw [planeOriginIdeal_eq_span]
+        exact Ideal.span_le.mpr (by simpa [Set.insert_subset_iff, Set.singleton_subset_iff] using hn)
+      rcases hc with hx | hy
+      · have hu : IsUnit (algebraMap A S (MvPolynomial.X 0)) :=
+          (IsLocalization.AtPrime.isUnit_to_map_iff S I _).mpr hx
+        let : Module.Projective S (S ⊗[A] planeGlobalLattice k P h0 hadd) :=
+          gluedLattice_projective_baseChange (S := S)
+          (MvPolynomial.X 0 : A) (MvPolynomial.X 1) (MvPolynomial.X_ne_zero 0)
+          (planeCoordinates_relPrime k) Q d d
+          (planeGlobalTransition_denominator k P h0 hadd d hd)
+          (planeGlobalTransition_inverse_denominator k ι ρ P hP d hd) hu
+        exact Module.Projective.of_equiv (LinearEquiv.baseChange A S _ _ (associatedLatticeEquiv k ι ρ P hP).symm)
+      · have hu : IsUnit (algebraMap A S (MvPolynomial.X 1)) :=
+          (IsLocalization.AtPrime.isUnit_to_map_iff S I _).mpr hy
+        have ha : ∀ v ∈ (coordinateLattice (R := A)),
+            (MvPolynomial.X 1 * MvPolynomial.X 0 : A)^d • (Q).symm v ∈ coordinateLattice (R := A) := by
+          rw [show (MvPolynomial.X 1 * MvPolynomial.X 0 : A) =
+            MvPolynomial.X 0 * MvPolynomial.X 1 from mul_comm _ _]
+          exact planeGlobalTransition_inverse_denominator k ι ρ P hP d hd
+        have hb : ∀ v ∈ (coordinateLattice (R := A)),
+            (MvPolynomial.X 1 * MvPolynomial.X 0 : A)^d • (Q).symm.symm v ∈ coordinateLattice (R := A) := by
+          rw [show (MvPolynomial.X 1 * MvPolynomial.X 0 : A) =
+            MvPolynomial.X 0 * MvPolynomial.X 1 from mul_comm _ _]
+          exact planeGlobalTransition_denominator k P h0 hadd d hd
+        let := gluedLattice_projective_baseChange (S := S)
+          (MvPolynomial.X 1 : A) (MvPolynomial.X 0) (MvPolynomial.X_ne_zero 1)
+          (planeCoordinates_relPrime k).symm (Q).symm d d ha hb hu
+        exact Module.Projective.of_equiv
+          (LinearEquiv.baseChange A S _ _
+          (((associatedLatticeEquiv k ι ρ P hP).trans
+            (latticeSwap (MvPolynomial.X 0) (MvPolynomial.X 1) Q)).symm))
+  let := localProj
+  exact Module.Projective.of_equiv
+    (IsLocalizedModule.isBaseChange I.primeCompl S (LocalizedModule.mkLinearMap I.primeCompl N)).equiv
+end FiberFunctorProof
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/RestrictedTilting/AssociatedWeights.lean b/lean/OAI/RepresentationTheory/RestrictedTilting/AssociatedWeights.lean
index 0d347a539b1b38ad99b02f8ec1035efa933fc362..e2d7abe3cbfd54146d206f943151ce741f9a2a5d 100644
--- a/lean/OAI/RepresentationTheory/RestrictedTilting/AssociatedWeights.lean
+++ b/lean/OAI/RepresentationTheory/RestrictedTilting/AssociatedWeights.lean
@@ -1,467 +1,467 @@
-import Mathlib
-import OAI.RepresentationTheory.RestrictedTilting.AssociatedLattices
-import OAI.RepresentationTheory.RestrictedTilting.TiltingBounds
-
-namespace OAI
-
-/-! Character transfer, numerator weights and associated-origin tilting. -/
-
-noncomputable section
-open CategoryTheory CategoryTheory.MonoidalCategory CategoryTheory.Limits
-open scoped BigOperators
-
-namespace FiberFunctorProof
-section
-universe u v w
-variable {k : Type u} [Field k] {G : Type v} [Monoid G]
-variable {M : Type w} [AddCommGroup M] [Module k M]
-
-noncomputable def characterFiniteSubmodule (ρ : Representation k G M) : Submodule k M where
-  carrier := {v | ∃ c : (G →* k) →₀ M, ∀ g, ρ g v = characterEvaluation g c}
-  zero_mem' := ⟨0, by intro g; simp⟩
-  add_mem' := by
-    rintro v w ⟨c,hc⟩ ⟨d,hd⟩
-    exact ⟨c+d, fun g => by rw [map_add, hc, hd, map_add]⟩
-  smul_mem' := by
-    rintro a v ⟨c,hc⟩
-    exact ⟨a • c, fun g => by rw [map_smul, hc, map_smul]⟩
-
-lemma characterFinite_eigen (ρ : Representation k G M) (χ : G →* k) (v : M)
-    (hv : ∀ g, ρ g v = χ g • v) : v ∈ characterFiniteSubmodule ρ := by
-  classical
-  refine ⟨Finsupp.single χ v, fun g => ?_⟩
-  simpa [characterEvaluation_apply] using hv g
-
-lemma characterFinite_map {N : Type*} [AddCommGroup N] [Module k N]
-    (ρ : Representation k G M) (σ : Representation k G N) (f : M →ₗ[k] N)
-    (hf : ∀ g v, f (ρ g v) = σ g (f v)) {v : M}
-    (hv : v ∈ characterFiniteSubmodule ρ) : f v ∈ characterFiniteSubmodule σ := by
-  classical
-  obtain ⟨c,hc⟩ := hv
-  refine ⟨c.mapRange f f.map_zero, fun g => ?_⟩
-  rw [characterEvaluation_apply, Finsupp.sum_mapRange_index (fun _ => smul_zero _)]
-  rw [← hf, hc, characterEvaluation_apply]
-  simp only [Finsupp.sum, map_sum, map_smul]
-
-lemma HasCharacterExpansions.of_surjective {N : Type*} [AddCommGroup N] [Module k N]
-    (ρ : Representation k G M) (σ : Representation k G N) (f : M →ₗ[k] N)
-    (hf : ∀ g v, f (ρ g v) = σ g (f v)) (hs : Function.Surjective f)
-    (hρ : HasCharacterExpansions ρ) : HasCharacterExpansions σ := by
-  intro w
-  obtain ⟨v,rfl⟩ := hs w
-  exact characterFinite_map ρ σ f hf (hρ v)
-
-lemma HasCharacterExpansions.of_injective {N : Type*} [AddCommGroup N] [Module k N]
-    (ρ : Representation k G M) (σ : Representation k G N) (f : M →ₗ[k] N)
-    (hf : ∀ g v, f (ρ g v) = σ g (f v)) (hi : Function.Injective f)
-    (hσ : HasCharacterExpansions σ) : HasCharacterExpansions ρ := by
-  classical
-  obtain ⟨s,hs⟩ := f.exists_leftInverse_of_injective (LinearMap.ker_eq_bot.mpr hi)
-  have hsf (v : M) : s (f v) = v := LinearMap.congr_fun hs v
-  intro v
-  obtain ⟨c,hc⟩ := hσ (f v)
-  refine ⟨c.mapRange s s.map_zero, fun g => ?_⟩
-  rw [Finsupp.sum_mapRange_index (fun _ => smul_zero _)]
-  calc
-    ρ g v = s (σ g (f v)) := by rw [← hf, hsf]
-    _ = c.sum (fun χ x => χ g • s x) := by
-      rw [hc]
-      simp only [Finsupp.sum, map_sum, map_smul]
-
-end
-
-section
-open scoped BigOperators
-universe u
-variable (k : Type u) [Field k]
-noncomputable def diagonalPolynomialScale {σ : Type*} (c : σ → k) :
-    MvPolynomial σ k →ₐ[k] MvPolynomial σ k :=
-  MvPolynomial.aeval (fun i => MvPolynomial.C (c i) * MvPolynomial.X i)
-
-lemma diagonalPolynomialScale_monomial {σ : Type*} [Fintype σ]
-    (c : σ → k) (d : σ →₀ ℕ) (r : k) :
-    diagonalPolynomialScale k c (MvPolynomial.monomial d r) =
-      MvPolynomial.monomial d (r * ∏ i, c i ^ d i) := by
-  classical
-  change MvPolynomial.eval₂ MvPolynomial.C
-    (fun i => MvPolynomial.C (c i) * MvPolynomial.X i) (MvPolynomial.monomial d r) = _
-  rw [MvPolynomial.eval₂_monomial, Finsupp.prod_fintype]
-  · simp only [mul_pow, Finset.prod_mul_distrib, ← map_pow, ← map_prod]
-    rw [MvPolynomial.monomial_eq, Finsupp.prod_fintype]
-    · simp only [map_mul, mul_assoc]
-    · intro i; simp
-  · intro i
-    simp
-
-lemma diagonalPolynomialScale_coeff {σ : Type*} [Fintype σ]
-    (c : σ → k) (P : MvPolynomial σ k) (d : σ →₀ ℕ) :
-    (diagonalPolynomialScale k c P).coeff d = P.coeff d * ∏ i, c i ^ d i := by
-  classical
-  induction P using MvPolynomial.induction_on' with
-  | monomial e r =>
-    rw [diagonalPolynomialScale_monomial]
-    simp only [MvPolynomial.coeff_monomial]
-    split_ifs with h
-    · subst e
-      rfl
-    · simp
-  | add P Q hP hQ =>
-    simp only [map_add, AddMonoidAlgebra.coeff_add, Finsupp.add_apply, hP, hQ, add_mul]
-
-noncomputable def planeTorusPolynomial (a : kˣ) :
-    MvPolynomial (Fin 2) k →ₐ[k] MvPolynomial (Fin 2) k :=
-  diagonalPolynomialScale k ![(a⁻¹ : kˣ).val, a.val]
-
-lemma planeTorusPolynomial_coeff (a : kˣ) (P : MvPolynomial (Fin 2) k)
-    (d : Fin 2 →₀ ℕ) :
-    (planeTorusPolynomial k a P).coeff d =
-      P.coeff d * integerCharacter k ((d 1 : ℤ) - d 0) a := by
-  rw [planeTorusPolynomial, diagonalPolynomialScale_coeff]
-  congr 1
-  simp [Fin.prod_univ_two, integerCharacter, zpow_sub, mul_comm]
-
- 
-lemma positive_plane_weight_dvd [Infinite k] (P : MvPolynomial (Fin 2) k)
-    (h : ℤ) (hh : 0 < h)
-    (hP : ∀ a : kˣ, planeTorusPolynomial k a P = integerCharacter k h a • P) :
-    MvPolynomial.X 1 ∣ P := by
-  classical
-  conv_rhs => rw [MvPolynomial.as_sum P]
-  apply Finset.dvd_sum
-  intro d hd
-  rw [MvPolynomial.X_dvd_monomial]
-  right
-  intro hd1
-  have hc : P.coeff d ≠ 0 := MvPolynomial.mem_support_iff.mp hd
-  have he : (d 1 : ℤ) - d 0 = h := by
-    apply integerCharacter_injective k
-    ext a
-    have ha := congrArg (fun polynomial : MvPolynomial (Fin 2) k => polynomial.coeff d) (hP a)
-    rw [planeTorusPolynomial_coeff, MvPolynomial.coeff_smul, smul_eq_mul] at ha
-    exact mul_left_cancel₀ hc (ha.trans (mul_comm _ _))
-  rw [hd1] at he
-  omega
-
-end
-
-section
-open scoped BigOperators Matrix Kronecker
-universe u
-variable (k : Type u) [Field k]
-
-def coordinateCharacter (ij : Fin 2 × Fin 2) : kˣ →* k :=
-  if ij.1 = 0 then inverseUnitCharacter k else Units.coeHom k
-
-lemma linearChange_diagonal {σ : Type*} [Fintype σ] [DecidableEq σ] (c : σ → k) :
-    linearChange (Matrix.diagonal c) = diagonalPolynomialScale k c := by
-  ext i : 1
-  simp [linearChange, diagonalPolynomialScale, Matrix.diagonal_apply, apply_ite]
-
-lemma leftCoordinateMatrix_diagonal (a : kˣ) :
-    leftCoordinateMatrix k (diagonalTorus k a) =
-      Matrix.diagonal (fun ij => coordinateCharacter k ij a) := by
-  ext ⟨i,j⟩ ⟨l,m⟩
-  change (diagonalTorus k a)⁻¹ l i * (1 : Matrix (Fin 2) (Fin 2) k) j m = _
-  rw [← map_inv]
-  change !![((↑a⁻¹ : k)), 0; 0, ((↑(a⁻¹)⁻¹ : k))] l i *
-    (1 : Matrix (Fin 2) (Fin 2) k) j m = _
-  fin_cases i <;> fin_cases j <;> fin_cases l <;> fin_cases m <;>
-    simp [coordinateCharacter, inverseUnitCharacter]
-
-lemma coordinatePolynomial_torus_scale (a : kˣ) (P : MvPolynomial (Fin 2 × Fin 2) k) :
-    coordinatePolynomialRepresentation k (diagonalTorus k a) P =
-      diagonalPolynomialScale k (fun ij => coordinateCharacter k ij a) P := by
-  change linearChange (leftCoordinateMatrix k (diagonalTorus k a)) P = _
-  rw [leftCoordinateMatrix_diagonal, linearChange_diagonal]
-
-lemma coordinatePolynomial_characters : HasCharacterExpansions
-    ((coordinatePolynomialRepresentation k).comp (diagonalTorus k)) := by
-  intro P
-  change P ∈ characterFiniteSubmodule _
-  induction P using MvPolynomial.induction_on' with
-  | monomial d r =>
-    apply characterFinite_eigen _ (∏ ij, coordinateCharacter k ij ^ d ij)
-    intro a
-    change coordinatePolynomialRepresentation k (diagonalTorus k a)
-      (MvPolynomial.monomial d r) = _
-    rw [coordinatePolynomial_torus_scale, diagonalPolynomialScale_monomial]
-    simp only [MonoidHom.finsetProd_apply, MonoidHom.pow_apply, MvPolynomial.smul_monomial,
-      smul_eq_mul, mul_comm]
-  | add P Q hP hQ => exact (characterFiniteSubmodule _).add_mem hP hQ
-
-lemma regularSL2_characters : HasCharacterExpansions
-    ((regularSL2Representation k).comp (diagonalTorus k)) := by
-  apply HasCharacterExpansions.of_surjective
-    ((coordinatePolynomialRepresentation k).comp (diagonalTorus k))
-    ((regularSL2Representation k).comp (diagonalTorus k))
-    (coordinateToRegular k) _ (coordinateToRegular_surjective k)
-    (coordinatePolynomial_characters k)
-  intro a P
-  apply Subtype.ext
-  funext h
-  exact coordinatePolynomial_evaluation k (diagonalTorus k a) h P
-
-lemma regularVector_characters (ι : Type u) [Fintype ι] [DecidableEq ι] :
-    HasCharacterExpansions ((regularVectorRepresentation k ι).comp (diagonalTorus k)) := by
-  intro f
-  let ei (i : ι) : regularSL2Functions k →ₗ[k] (ι → regularSL2Functions k) :=
-    LinearMap.single k (fun _ : ι => regularSL2Functions k) i
-  have hsum : ∑ i, ei i (f i) = f := by ext i; simp [ei]
-  change f ∈ characterFiniteSubmodule _
-  rw [← hsum]
-  apply Submodule.sum_mem
-  intro j _
-  apply characterFinite_map ((regularSL2Representation k).comp (diagonalTorus k))
-    _ (ei j) _ (regularSL2_characters k (f j))
-  intro a v
-  funext i
-  change (ei j (regularSL2Representation k (diagonalTorus k a) v)) i =
-    regularSL2Representation k (diagonalTorus k a) (ei j v i)
-  simp only [ei, LinearMap.single_apply, Pi.single_apply]
-  split_ifs <;> simp_all
-
- 
-
-lemma associatedSections_characters (ι : Type u) [Fintype ι] [DecidableEq ι]
-    (ρ : Representation k (Multiplicative k) (ι → k)) :
-    HasCharacterExpansions (k := k) (M := associatedSectionsSubmodule k ι ρ)
-      ((associatedSectionsRepresentation k ι ρ).comp (diagonalTorus k)) := by
-  exact HasCharacterExpansions.of_injective _ _ (associatedSectionsSubmodule k ι ρ).subtype
-    (fun _ _ => rfl) (associatedSectionsSubmodule k ι ρ).injective_subtype
-    (regularVector_characters k ι)
-
-end
-
-section
-universe u
-variable (k : Type u) [Field k] [Infinite k]
-
-noncomputable def planeTorusScale (a : kˣ) (x : planeTorusPoints k) : planeTorusPoints k :=
-  ⟨![(↑a⁻¹ : k) * x.val 0, (a : k) * x.val 1], by
-    intro i
-    fin_cases i
-    · exact mul_ne_zero (Units.ne_zero a⁻¹) (x.property 0)
-    · exact mul_ne_zero (Units.ne_zero a) (x.property 1)⟩
-
-omit [Infinite k] in
-lemma planeTorusScale_coordinate (a : kˣ) (x : planeTorusPoints k) (i : Fin 2) :
-    (planeTorusScale k a x).val i = ![(↑a⁻¹ : k), (a : k)] i * x.val i := by
-  fin_cases i <;> rfl
-
-omit [Infinite k] in
-lemma diagonalPolynomialScale_eval {σ : Type*} (c x : σ → k) (P : MvPolynomial σ k) :
-    MvPolynomial.eval x (diagonalPolynomialScale k c P) =
-      MvPolynomial.eval (fun i => c i * x i) P := by
-  induction P using MvPolynomial.induction_on with
-  | C r => simp [diagonalPolynomialScale]
-  | add P Q hP hQ => simp only [map_add, hP, hQ]
-  | mul_X P i hP =>
-    simp only [map_mul]
-    rw [hP]
-    congr 1
-    simp [diagonalPolynomialScale]
-
-omit [Infinite k] in
-lemma planeTorusPolynomial_eval (a : kˣ) (x : planeTorusPoints k)
-    (P : MvPolynomial (Fin 2) k) :
-    MvPolynomial.eval x.val (planeTorusPolynomial k a P) =
-      MvPolynomial.eval (planeTorusScale k a x).val P := by
-  rw [planeTorusPolynomial, diagonalPolynomialScale_eval]
-  have hx : (fun i => ![(↑a⁻¹ : k), (a : k)] i * x.val i) =
-      (planeTorusScale k a x).val := by
-    funext i
-    exact (planeTorusScale_coordinate k a x i).symm
-  rw [hx]
-
-omit [Infinite k] in
-lemma planeXSection_torus (a : kˣ) (x : planeTorusPoints k) :
-    (diagonalTorus k a)⁻¹ * planeXSection k x = planeXSection k (planeTorusScale k a x) := by
-  rw [← map_inv]
-  apply Subtype.ext
-  change !![(↑a⁻¹ : k), 0; 0, (↑(a⁻¹)⁻¹ : k)] *
-    !![x.val 0, 0; x.val 1, (x.val 0)⁻¹] =
-      !![(↑a⁻¹ : k) * x.val 0, 0; (a : k) * x.val 1, ((↑a⁻¹ : k) * x.val 0)⁻¹]
-  ext i j
-  fin_cases i <;> fin_cases j <;>
-    simp [Matrix.mul_apply, Fin.sum_univ_two, mul_comm]
-
-variable (ι : Type u) [Fintype ι] [DecidableEq ι]
-variable (ρ : Representation k (Multiplicative k) (ι → k))
-
-omit [Fintype ι] [DecidableEq ι] in
-lemma associatedXChart_torus_evaluation (a : kˣ) (f : associatedSectionsSubmodule k ι ρ)
-    (x : planeTorusPoints k) :
-    (fun i => planeChartEvaluation k 0
-      (associatedXChart k ι ρ (associatedSectionsRepresentation k ι ρ (diagonalTorus k a) f) i) x) =
-    (fun i => planeChartEvaluation k 0 (associatedXChart k ι ρ f i) (planeTorusScale k a x)) := by
-  rw [associatedXChart_evaluation, associatedXChart_evaluation]
-  change regularVectorEvaluation k ι ((diagonalTorus k a)⁻¹ * planeXSection k x) f.val = _
-  rw [planeXSection_torus]
-
-omit [Fintype ι] [DecidableEq ι] in
-lemma associatedNumerator_weight (d : ℕ) (f : associatedSectionsSubmodule k ι ρ)
-    (F : ι → MvPolynomial (Fin 2) k)
-    (hF : ∀ i, algebraMap (MvPolynomial (Fin 2) k) (planeCoordinateChart k 0) (F i) =
-      planeChartCoordinate k 0 0 ^ d * associatedXChart k ι ρ f i)
-    (h : ℤ) (hf : ∀ a : kˣ, associatedSectionsRepresentation k ι ρ (diagonalTorus k a) f =
-      integerCharacter k h a • f) (i : ι) (a : kˣ) :
-    planeTorusPolynomial k a (F i) = integerCharacter k (h-d) a • F i := by
-  apply planeTorusEvaluation_injective k
-  funext x
-  simp only [planeTorusEvaluation_apply]
-  rw [planeTorusPolynomial_eval, MvPolynomial.smul_eq_C_mul, map_mul, MvPolynomial.eval_C,
-    associatedNumerator_evaluation k ι ρ d f F hF,
-    associatedNumerator_evaluation k ι ρ d f F hF]
-  have hx := congrFun (associatedXChart_torus_evaluation k ι ρ a f x) i
-  rw [hf, map_smul] at hx
-  simp only [Pi.smul_apply, map_smul] at hx
-  change integerCharacter k h a * planeChartEvaluation k 0 (associatedXChart k ι ρ f i) x =
-    planeChartEvaluation k 0 (associatedXChart k ι ρ f i) (planeTorusScale k a x) at hx
-  rw [← hx]
-  change ((↑a⁻¹ : k) * x.val 0)^d *
-    (integerCharacter k h a * planeChartEvaluation k 0 (associatedXChart k ι ρ f i) x) = _
-  simp only [integerCharacter, MonoidHom.comp_apply, zpowGroupHom_apply,
-    Units.coeHom_apply, zpow_sub, zpow_natCast, Units.val_mul, Units.val_inv_eq_inv_val,
-    Units.val_pow_eq_pow_val, mul_pow, inv_pow]
-  ring
-
-end
-
-section GenericDivision
-variable {R K ι : Type*} [CommRing R] [Field K] [Algebra R K]
-
-lemma gluedLattice_div_y (x y : R) (T : (ι → K) ≃ₗ[K] (ι → K))
-    (hy : algebraMap R K y ≠ 0) {z : ι → K} (hz : z ∈ gluedLattice x y T)
-    (hx : (algebraMap R K y)⁻¹ • z ∈ powerLocalization x (coordinateLattice (R := R) (K := K) (ι := ι))) :
-    (algebraMap R K y)⁻¹ • z ∈ gluedLattice x y T := by
-  refine ⟨hx, ?_⟩
-  change z ∈ powerLocalization x (coordinateLattice (R := R)) ∧
-    z ∈ (powerLocalization y (coordinateLattice (R := R))).map (T.toLinearMap.restrictScalars R) at hz
-  obtain ⟨w,hw,he⟩ := hz.2
-  refine ⟨(algebraMap R K y)⁻¹ • w,
-    powerLocalization_div (K := K) y (coordinateLattice (R := R) (K := K) (ι := ι)) hy hw, ?_⟩
-  change T ((algebraMap R K y)⁻¹ • w) = (algebraMap R K y)⁻¹ • z
-  change T w = z at he
-  rw [map_smul, he]
-
-end GenericDivision
-
-section
-universe u
-variable (k : Type u) [Field k] [Infinite k]
-variable (ι : Type u) [Fintype ι] [DecidableEq ι]
-variable (ρ : Representation k (Multiplicative k) (ι → k))
-variable (P : Matrix ι ι (Polynomial k))
-variable (hP : ∀ t : k, polynomialMatrixEval P t = LinearMap.toMatrix' (ρ (Multiplicative.ofAdd t)))
-
-include hP in
-lemma associated_y_divisible (d : ℕ) (f : associatedSectionsSubmodule k ι ρ)
-    (F : ι → MvPolynomial (Fin 2) k)
-    (hF : ∀ i, algebraMap (MvPolynomial (Fin 2) k) (planeCoordinateChart k 0) (F i) =
-      planeChartCoordinate k 0 0 ^ d * associatedXChart k ι ρ f i)
-    (hdiv : ∀ i, MvPolynomial.X 1 ∣ F i) :
-    ∃ f' : associatedSectionsSubmodule k ι ρ, f = (MvPolynomial.X 1 : MvPolynomial (Fin 2) k) • f' := by
-  let y : planeFractionField k :=
-    algebraMap (MvPolynomial (Fin 2) k) (planeFractionField k) (MvPolynomial.X 1)
-  have hy : y ≠ 0 := planeFractionCoordinate_ne_zero k 1
-  have hx : y⁻¹ • associatedFractionChart k ι ρ f ∈
-      powerLocalization (MvPolynomial.X 0 : MvPolynomial (Fin 2) k) coordinateLattice := by
-    refine ⟨d, (mem_coordinateLattice _).mpr fun i => ?_⟩
-    obtain ⟨Q,hQ⟩ := hdiv i
-    refine ⟨Q, ?_⟩
-    have hi := congrArg (planeChartToFraction k 0) (hF i)
-    rw [planeChartToFraction_algebraMap, hQ, map_mul, map_mul, map_pow,
-      planeChartCoordinate, planeChartToFraction_algebraMap] at hi
-    change y * algebraMap (MvPolynomial (Fin 2) k) (planeFractionField k) Q =
-      algebraMap (MvPolynomial (Fin 2) k) (planeFractionField k) (MvPolynomial.X 0)^d *
-        associatedFractionChart k ι ρ f i at hi
-    simp only [Pi.smul_apply, Algebra.smul_def, map_pow]
-    change algebraMap (MvPolynomial (Fin 2) k) (planeFractionField k) Q =
-      algebraMap (MvPolynomial (Fin 2) k) (planeFractionField k) (MvPolynomial.X 0)^d *
-        (y⁻¹ * associatedFractionChart k ι ρ f i)
-    calc
-      _ = y⁻¹ * (y * algebraMap (MvPolynomial (Fin 2) k) (planeFractionField k) Q) := by
-        rw [← mul_assoc, inv_mul_cancel₀ hy, one_mul]
-      _ = _ := by rw [hi]; ring
-  have hz := gluedLattice_div_y (MvPolynomial.X 0 : MvPolynomial (Fin 2) k)
-    (MvPolynomial.X 1) (planeGlobalTransition k P (associatedPolynomial_zero k ι ρ P hP)
-      (associatedPolynomial_add k ι ρ P hP)) hy
-    (associatedFractionChart_mem k ι ρ P hP f) hx
-  obtain ⟨f',hf'⟩ := associatedLatticeMap_surjective k ι ρ P hP ⟨_,hz⟩
-  have he : associatedFractionChart k ι ρ f' = y⁻¹ • associatedFractionChart k ι ρ f :=
-    congrArg Subtype.val hf'
-  refine ⟨f', associatedFractionChart_injective k ι ρ ?_⟩
-  rw [map_smul, he]
-  rw [← algebraMap_smul (planeFractionField k) (MvPolynomial.X 1)]
-  change associatedFractionChart k ι ρ f = y • (y⁻¹ • associatedFractionChart k ι ρ f)
-  rw [smul_inv_smul₀ hy]
-
-
-include hP in
-lemma associatedOrigin_weight_upper (d : ℕ) (hd : ∀ i j, (P i j).natDegree ≤ d)
-    (n : ℤ) (hn : (d : ℤ) < n) (v : (associatedOriginRep k ι ρ).V)
-    (hv : ∀ a : kˣ, (associatedOriginRep k ι ρ).ρ (diagonalTorus k a) v =
-      integerCharacter k n a • v) : v = 0 := by
-  let Q : (associatedSectionsSubmodule k ι ρ) →ₗ[k] (associatedOriginRep k ι ρ).V :=
-    (associatedOriginRelations k ι ρ).mkQ
-  obtain ⟨f,hfv,hf⟩ := eigenvector_lift
-    ((associatedSectionsRepresentation k ι ρ).comp (diagonalTorus k))
-    ((associatedOriginRep k ι ρ).ρ.comp (diagonalTorus k))
-    (associatedSections_characters k ι ρ) Q (fun _ _ => rfl)
-    (associatedOriginRelations k ι ρ).mkQ_surjective (integerCharacter k n) v hv
-  obtain ⟨F,hF⟩ := associatedNumerator_exists k ι ρ P hP d hd f
-  have hdiv (i : ι) : MvPolynomial.X 1 ∣ F i := by
-    apply positive_plane_weight_dvd k (F i) (n-d) (by omega)
-    exact associatedNumerator_weight k ι ρ d f F hF n hf i
-  obtain ⟨f',hff'⟩ := associated_y_divisible k ι ρ P hP d f F hF hdiv
-  have hmem : f ∈ associatedOriginRelations k ι ρ := by
-    rw [hff']
-    exact associatedOrigin_coordinate_mem k ι ρ 1 f'
-  rw [← hfv]
-  exact (Submodule.Quotient.mk_eq_zero (associatedOriginRelations k ι ρ)).mpr hmem
-
-end
-
-section
-universe u
-variable (k : Type u) [Field k]
-
-variable [Infinite k]
-variable (ι : Type u) [Fintype ι] [DecidableEq ι]
-variable (ρ : Representation k (Multiplicative k) (ι → k))
-
- 
-
-theorem associatedOrigin_weights
-    (P : Matrix ι ι (Polynomial k))
-    (hP : ∀ t : k, polynomialMatrixEval P t = LinearMap.toMatrix' (ρ (Multiplicative.ofAdd t)))
-    (d : ℕ) (hd : ∀ i j, (P i j).natDegree ≤ d) :
-    HasWeightsBetween (associatedOriginRep k ι ρ).ρ (-(d : ℤ)) d :=
-  weights_between_of_upper k _ d (associatedOrigin_weight_upper k ι ρ P hP d hd)
-
-end
-
-universe u
-variable (k : Type u) [Field k] [Infinite k]
-variable (p : ℕ) [Fact p.Prime] [CharP k p]
-variable (ι : Type u) [Fintype ι] [DecidableEq ι]
-variable (ρ : Representation k (Multiplicative k) (ι → k))
-variable (P : Matrix ι ι (Polynomial k))
-variable (hP : ∀ t : k, polynomialMatrixEval P t = LinearMap.toMatrix' (ρ (Multiplicative.ofAdd t)))
-variable (d : ℕ) (hd : ∀ i j, (P i j).natDegree ≤ d)
-
-noncomputable def associatedOriginFD : FDRep k (SL2 k) := by
-  let : Module.Finite k (associatedOriginRep k ι ρ).V := associatedOrigin_finite k ι ρ P hP d hd
-  exact FDRep.of (associatedOriginRep k ι ρ).ρ
-
- 
-
-theorem associatedOrigin_tilting (m : ℕ) (hm : d < p^(m+1)) :
-    IsTilting k (steinberg k p m ⊗ associatedOriginFD k ι ρ P hP d hd) := by
-  apply bounded_weights_tilting k p m d hm
-  · exact associatedOrigin_rational k ι ρ
-  · exact associatedOrigin_weights k ι ρ P hP d hd
-end FiberFunctorProof
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.RestrictedTilting.AssociatedLattices
+import OAI.RepresentationTheory.RestrictedTilting.TiltingBounds
+
+namespace OAI
+
+/-! Character transfer, numerator weights and associated-origin tilting. -/
+
+noncomputable section
+open CategoryTheory CategoryTheory.MonoidalCategory CategoryTheory.Limits
+open scoped BigOperators
+
+namespace FiberFunctorProof
+section
+universe u v w
+variable {k : Type u} [Field k] {G : Type v} [Monoid G]
+variable {M : Type w} [AddCommGroup M] [Module k M]
+
+noncomputable def characterFiniteSubmodule (ρ : Representation k G M) : Submodule k M where
+  carrier := {v | ∃ c : (G →* k) →₀ M, ∀ g, ρ g v = characterEvaluation g c}
+  zero_mem' := ⟨0, by intro g; simp⟩
+  add_mem' := by
+    rintro v w ⟨c,hc⟩ ⟨d,hd⟩
+    exact ⟨c+d, fun g => by rw [map_add, hc, hd, map_add]⟩
+  smul_mem' := by
+    rintro a v ⟨c,hc⟩
+    exact ⟨a • c, fun g => by rw [map_smul, hc, map_smul]⟩
+
+lemma characterFinite_eigen (ρ : Representation k G M) (χ : G →* k) (v : M)
+    (hv : ∀ g, ρ g v = χ g • v) : v ∈ characterFiniteSubmodule ρ := by
+  classical
+  refine ⟨Finsupp.single χ v, fun g => ?_⟩
+  simpa [characterEvaluation_apply] using hv g
+
+lemma characterFinite_map {N : Type*} [AddCommGroup N] [Module k N]
+    (ρ : Representation k G M) (σ : Representation k G N) (f : M →ₗ[k] N)
+    (hf : ∀ g v, f (ρ g v) = σ g (f v)) {v : M}
+    (hv : v ∈ characterFiniteSubmodule ρ) : f v ∈ characterFiniteSubmodule σ := by
+  classical
+  obtain ⟨c,hc⟩ := hv
+  refine ⟨c.mapRange f f.map_zero, fun g => ?_⟩
+  rw [characterEvaluation_apply, Finsupp.sum_mapRange_index (fun _ => smul_zero _)]
+  rw [← hf, hc, characterEvaluation_apply]
+  simp only [Finsupp.sum, map_sum, map_smul]
+
+lemma HasCharacterExpansions.of_surjective {N : Type*} [AddCommGroup N] [Module k N]
+    (ρ : Representation k G M) (σ : Representation k G N) (f : M →ₗ[k] N)
+    (hf : ∀ g v, f (ρ g v) = σ g (f v)) (hs : Function.Surjective f)
+    (hρ : HasCharacterExpansions ρ) : HasCharacterExpansions σ := by
+  intro w
+  obtain ⟨v,rfl⟩ := hs w
+  exact characterFinite_map ρ σ f hf (hρ v)
+
+lemma HasCharacterExpansions.of_injective {N : Type*} [AddCommGroup N] [Module k N]
+    (ρ : Representation k G M) (σ : Representation k G N) (f : M →ₗ[k] N)
+    (hf : ∀ g v, f (ρ g v) = σ g (f v)) (hi : Function.Injective f)
+    (hσ : HasCharacterExpansions σ) : HasCharacterExpansions ρ := by
+  classical
+  obtain ⟨s,hs⟩ := f.exists_leftInverse_of_injective (LinearMap.ker_eq_bot.mpr hi)
+  have hsf (v : M) : s (f v) = v := LinearMap.congr_fun hs v
+  intro v
+  obtain ⟨c,hc⟩ := hσ (f v)
+  refine ⟨c.mapRange s s.map_zero, fun g => ?_⟩
+  rw [Finsupp.sum_mapRange_index (fun _ => smul_zero _)]
+  calc
+    ρ g v = s (σ g (f v)) := by rw [← hf, hsf]
+    _ = c.sum (fun χ x => χ g • s x) := by
+      rw [hc]
+      simp only [Finsupp.sum, map_sum, map_smul]
+
+end
+
+section
+open scoped BigOperators
+universe u
+variable (k : Type u) [Field k]
+noncomputable def diagonalPolynomialScale {σ : Type*} (c : σ → k) :
+    MvPolynomial σ k →ₐ[k] MvPolynomial σ k :=
+  MvPolynomial.aeval (fun i => MvPolynomial.C (c i) * MvPolynomial.X i)
+
+lemma diagonalPolynomialScale_monomial {σ : Type*} [Fintype σ]
+    (c : σ → k) (d : σ →₀ ℕ) (r : k) :
+    diagonalPolynomialScale k c (MvPolynomial.monomial d r) =
+      MvPolynomial.monomial d (r * ∏ i, c i ^ d i) := by
+  classical
+  change MvPolynomial.eval₂ MvPolynomial.C
+    (fun i => MvPolynomial.C (c i) * MvPolynomial.X i) (MvPolynomial.monomial d r) = _
+  rw [MvPolynomial.eval₂_monomial, Finsupp.prod_fintype]
+  · simp only [mul_pow, Finset.prod_mul_distrib, ← map_pow, ← map_prod]
+    rw [MvPolynomial.monomial_eq, Finsupp.prod_fintype]
+    · simp only [map_mul, mul_assoc]
+    · intro i; simp
+  · intro i
+    simp
+
+lemma diagonalPolynomialScale_coeff {σ : Type*} [Fintype σ]
+    (c : σ → k) (P : MvPolynomial σ k) (d : σ →₀ ℕ) :
+    (diagonalPolynomialScale k c P).coeff d = P.coeff d * ∏ i, c i ^ d i := by
+  classical
+  induction P using MvPolynomial.induction_on' with
+  | monomial e r =>
+    rw [diagonalPolynomialScale_monomial]
+    simp only [MvPolynomial.coeff_monomial]
+    split_ifs with h
+    · subst e
+      rfl
+    · simp
+  | add P Q hP hQ =>
+    simp only [map_add, AddMonoidAlgebra.coeff_add, Finsupp.add_apply, hP, hQ, add_mul]
+
+noncomputable def planeTorusPolynomial (a : kˣ) :
+    MvPolynomial (Fin 2) k →ₐ[k] MvPolynomial (Fin 2) k :=
+  diagonalPolynomialScale k ![(a⁻¹ : kˣ).val, a.val]
+
+lemma planeTorusPolynomial_coeff (a : kˣ) (P : MvPolynomial (Fin 2) k)
+    (d : Fin 2 →₀ ℕ) :
+    (planeTorusPolynomial k a P).coeff d =
+      P.coeff d * integerCharacter k ((d 1 : ℤ) - d 0) a := by
+  rw [planeTorusPolynomial, diagonalPolynomialScale_coeff]
+  congr 1
+  simp [Fin.prod_univ_two, integerCharacter, zpow_sub, mul_comm]
+
+ 
+lemma positive_plane_weight_dvd [Infinite k] (P : MvPolynomial (Fin 2) k)
+    (h : ℤ) (hh : 0 < h)
+    (hP : ∀ a : kˣ, planeTorusPolynomial k a P = integerCharacter k h a • P) :
+    MvPolynomial.X 1 ∣ P := by
+  classical
+  conv_rhs => rw [MvPolynomial.as_sum P]
+  apply Finset.dvd_sum
+  intro d hd
+  rw [MvPolynomial.X_dvd_monomial]
+  right
+  intro hd1
+  have hc : P.coeff d ≠ 0 := MvPolynomial.mem_support_iff.mp hd
+  have he : (d 1 : ℤ) - d 0 = h := by
+    apply integerCharacter_injective k
+    ext a
+    have ha := congrArg (fun polynomial : MvPolynomial (Fin 2) k => polynomial.coeff d) (hP a)
+    rw [planeTorusPolynomial_coeff, MvPolynomial.coeff_smul, smul_eq_mul] at ha
+    exact mul_left_cancel₀ hc (ha.trans (mul_comm _ _))
+  rw [hd1] at he
+  omega
+
+end
+
+section
+open scoped BigOperators Matrix Kronecker
+universe u
+variable (k : Type u) [Field k]
+
+def coordinateCharacter (ij : Fin 2 × Fin 2) : kˣ →* k :=
+  if ij.1 = 0 then inverseUnitCharacter k else Units.coeHom k
+
+lemma linearChange_diagonal {σ : Type*} [Fintype σ] [DecidableEq σ] (c : σ → k) :
+    linearChange (Matrix.diagonal c) = diagonalPolynomialScale k c := by
+  ext i : 1
+  simp [linearChange, diagonalPolynomialScale, Matrix.diagonal_apply, apply_ite]
+
+lemma leftCoordinateMatrix_diagonal (a : kˣ) :
+    leftCoordinateMatrix k (diagonalTorus k a) =
+      Matrix.diagonal (fun ij => coordinateCharacter k ij a) := by
+  ext ⟨i,j⟩ ⟨l,m⟩
+  change (diagonalTorus k a)⁻¹ l i * (1 : Matrix (Fin 2) (Fin 2) k) j m = _
+  rw [← map_inv]
+  change !![((↑a⁻¹ : k)), 0; 0, ((↑(a⁻¹)⁻¹ : k))] l i *
+    (1 : Matrix (Fin 2) (Fin 2) k) j m = _
+  fin_cases i <;> fin_cases j <;> fin_cases l <;> fin_cases m <;>
+    simp [coordinateCharacter, inverseUnitCharacter]
+
+lemma coordinatePolynomial_torus_scale (a : kˣ) (P : MvPolynomial (Fin 2 × Fin 2) k) :
+    coordinatePolynomialRepresentation k (diagonalTorus k a) P =
+      diagonalPolynomialScale k (fun ij => coordinateCharacter k ij a) P := by
+  change linearChange (leftCoordinateMatrix k (diagonalTorus k a)) P = _
+  rw [leftCoordinateMatrix_diagonal, linearChange_diagonal]
+
+lemma coordinatePolynomial_characters : HasCharacterExpansions
+    ((coordinatePolynomialRepresentation k).comp (diagonalTorus k)) := by
+  intro P
+  change P ∈ characterFiniteSubmodule _
+  induction P using MvPolynomial.induction_on' with
+  | monomial d r =>
+    apply characterFinite_eigen _ (∏ ij, coordinateCharacter k ij ^ d ij)
+    intro a
+    change coordinatePolynomialRepresentation k (diagonalTorus k a)
+      (MvPolynomial.monomial d r) = _
+    rw [coordinatePolynomial_torus_scale, diagonalPolynomialScale_monomial]
+    simp only [MonoidHom.finsetProd_apply, MonoidHom.pow_apply, MvPolynomial.smul_monomial,
+      smul_eq_mul, mul_comm]
+  | add P Q hP hQ => exact (characterFiniteSubmodule _).add_mem hP hQ
+
+lemma regularSL2_characters : HasCharacterExpansions
+    ((regularSL2Representation k).comp (diagonalTorus k)) := by
+  apply HasCharacterExpansions.of_surjective
+    ((coordinatePolynomialRepresentation k).comp (diagonalTorus k))
+    ((regularSL2Representation k).comp (diagonalTorus k))
+    (coordinateToRegular k) _ (coordinateToRegular_surjective k)
+    (coordinatePolynomial_characters k)
+  intro a P
+  apply Subtype.ext
+  funext h
+  exact coordinatePolynomial_evaluation k (diagonalTorus k a) h P
+
+lemma regularVector_characters (ι : Type u) [Fintype ι] [DecidableEq ι] :
+    HasCharacterExpansions ((regularVectorRepresentation k ι).comp (diagonalTorus k)) := by
+  intro f
+  let ei (i : ι) : regularSL2Functions k →ₗ[k] (ι → regularSL2Functions k) :=
+    LinearMap.single k (fun _ : ι => regularSL2Functions k) i
+  have hsum : ∑ i, ei i (f i) = f := by ext i; simp [ei]
+  change f ∈ characterFiniteSubmodule _
+  rw [← hsum]
+  apply Submodule.sum_mem
+  intro j _
+  apply characterFinite_map ((regularSL2Representation k).comp (diagonalTorus k))
+    _ (ei j) _ (regularSL2_characters k (f j))
+  intro a v
+  funext i
+  change (ei j (regularSL2Representation k (diagonalTorus k a) v)) i =
+    regularSL2Representation k (diagonalTorus k a) (ei j v i)
+  simp only [ei, LinearMap.single_apply, Pi.single_apply]
+  split_ifs <;> simp_all
+
+ 
+
+lemma associatedSections_characters (ι : Type u) [Fintype ι] [DecidableEq ι]
+    (ρ : Representation k (Multiplicative k) (ι → k)) :
+    HasCharacterExpansions (k := k) (M := associatedSectionsSubmodule k ι ρ)
+      ((associatedSectionsRepresentation k ι ρ).comp (diagonalTorus k)) := by
+  exact HasCharacterExpansions.of_injective _ _ (associatedSectionsSubmodule k ι ρ).subtype
+    (fun _ _ => rfl) (associatedSectionsSubmodule k ι ρ).injective_subtype
+    (regularVector_characters k ι)
+
+end
+
+section
+universe u
+variable (k : Type u) [Field k] [Infinite k]
+
+noncomputable def planeTorusScale (a : kˣ) (x : planeTorusPoints k) : planeTorusPoints k :=
+  ⟨![(↑a⁻¹ : k) * x.val 0, (a : k) * x.val 1], by
+    intro i
+    fin_cases i
+    · exact mul_ne_zero (Units.ne_zero a⁻¹) (x.property 0)
+    · exact mul_ne_zero (Units.ne_zero a) (x.property 1)⟩
+
+omit [Infinite k] in
+lemma planeTorusScale_coordinate (a : kˣ) (x : planeTorusPoints k) (i : Fin 2) :
+    (planeTorusScale k a x).val i = ![(↑a⁻¹ : k), (a : k)] i * x.val i := by
+  fin_cases i <;> rfl
+
+omit [Infinite k] in
+lemma diagonalPolynomialScale_eval {σ : Type*} (c x : σ → k) (P : MvPolynomial σ k) :
+    MvPolynomial.eval x (diagonalPolynomialScale k c P) =
+      MvPolynomial.eval (fun i => c i * x i) P := by
+  induction P using MvPolynomial.induction_on with
+  | C r => simp [diagonalPolynomialScale]
+  | add P Q hP hQ => simp only [map_add, hP, hQ]
+  | mul_X P i hP =>
+    simp only [map_mul]
+    rw [hP]
+    congr 1
+    simp [diagonalPolynomialScale]
+
+omit [Infinite k] in
+lemma planeTorusPolynomial_eval (a : kˣ) (x : planeTorusPoints k)
+    (P : MvPolynomial (Fin 2) k) :
+    MvPolynomial.eval x.val (planeTorusPolynomial k a P) =
+      MvPolynomial.eval (planeTorusScale k a x).val P := by
+  rw [planeTorusPolynomial, diagonalPolynomialScale_eval]
+  have hx : (fun i => ![(↑a⁻¹ : k), (a : k)] i * x.val i) =
+      (planeTorusScale k a x).val := by
+    funext i
+    exact (planeTorusScale_coordinate k a x i).symm
+  rw [hx]
+
+omit [Infinite k] in
+lemma planeXSection_torus (a : kˣ) (x : planeTorusPoints k) :
+    (diagonalTorus k a)⁻¹ * planeXSection k x = planeXSection k (planeTorusScale k a x) := by
+  rw [← map_inv]
+  apply Subtype.ext
+  change !![(↑a⁻¹ : k), 0; 0, (↑(a⁻¹)⁻¹ : k)] *
+    !![x.val 0, 0; x.val 1, (x.val 0)⁻¹] =
+      !![(↑a⁻¹ : k) * x.val 0, 0; (a : k) * x.val 1, ((↑a⁻¹ : k) * x.val 0)⁻¹]
+  ext i j
+  fin_cases i <;> fin_cases j <;>
+    simp [Matrix.mul_apply, Fin.sum_univ_two, mul_comm]
+
+variable (ι : Type u) [Fintype ι] [DecidableEq ι]
+variable (ρ : Representation k (Multiplicative k) (ι → k))
+
+omit [Fintype ι] [DecidableEq ι] in
+lemma associatedXChart_torus_evaluation (a : kˣ) (f : associatedSectionsSubmodule k ι ρ)
+    (x : planeTorusPoints k) :
+    (fun i => planeChartEvaluation k 0
+      (associatedXChart k ι ρ (associatedSectionsRepresentation k ι ρ (diagonalTorus k a) f) i) x) =
+    (fun i => planeChartEvaluation k 0 (associatedXChart k ι ρ f i) (planeTorusScale k a x)) := by
+  rw [associatedXChart_evaluation, associatedXChart_evaluation]
+  change regularVectorEvaluation k ι ((diagonalTorus k a)⁻¹ * planeXSection k x) f.val = _
+  rw [planeXSection_torus]
+
+omit [Fintype ι] [DecidableEq ι] in
+lemma associatedNumerator_weight (d : ℕ) (f : associatedSectionsSubmodule k ι ρ)
+    (F : ι → MvPolynomial (Fin 2) k)
+    (hF : ∀ i, algebraMap (MvPolynomial (Fin 2) k) (planeCoordinateChart k 0) (F i) =
+      planeChartCoordinate k 0 0 ^ d * associatedXChart k ι ρ f i)
+    (h : ℤ) (hf : ∀ a : kˣ, associatedSectionsRepresentation k ι ρ (diagonalTorus k a) f =
+      integerCharacter k h a • f) (i : ι) (a : kˣ) :
+    planeTorusPolynomial k a (F i) = integerCharacter k (h-d) a • F i := by
+  apply planeTorusEvaluation_injective k
+  funext x
+  simp only [planeTorusEvaluation_apply]
+  rw [planeTorusPolynomial_eval, MvPolynomial.smul_eq_C_mul, map_mul, MvPolynomial.eval_C,
+    associatedNumerator_evaluation k ι ρ d f F hF,
+    associatedNumerator_evaluation k ι ρ d f F hF]
+  have hx := congrFun (associatedXChart_torus_evaluation k ι ρ a f x) i
+  rw [hf, map_smul] at hx
+  simp only [Pi.smul_apply, map_smul] at hx
+  change integerCharacter k h a * planeChartEvaluation k 0 (associatedXChart k ι ρ f i) x =
+    planeChartEvaluation k 0 (associatedXChart k ι ρ f i) (planeTorusScale k a x) at hx
+  rw [← hx]
+  change ((↑a⁻¹ : k) * x.val 0)^d *
+    (integerCharacter k h a * planeChartEvaluation k 0 (associatedXChart k ι ρ f i) x) = _
+  simp only [integerCharacter, MonoidHom.comp_apply, zpowGroupHom_apply,
+    Units.coeHom_apply, zpow_sub, zpow_natCast, Units.val_mul, Units.val_inv_eq_inv_val,
+    Units.val_pow_eq_pow_val, mul_pow, inv_pow]
+  ring
+
+end
+
+section GenericDivision
+variable {R K ι : Type*} [CommRing R] [Field K] [Algebra R K]
+
+lemma gluedLattice_div_y (x y : R) (T : (ι → K) ≃ₗ[K] (ι → K))
+    (hy : algebraMap R K y ≠ 0) {z : ι → K} (hz : z ∈ gluedLattice x y T)
+    (hx : (algebraMap R K y)⁻¹ • z ∈ powerLocalization x (coordinateLattice (R := R) (K := K) (ι := ι))) :
+    (algebraMap R K y)⁻¹ • z ∈ gluedLattice x y T := by
+  refine ⟨hx, ?_⟩
+  change z ∈ powerLocalization x (coordinateLattice (R := R)) ∧
+    z ∈ (powerLocalization y (coordinateLattice (R := R))).map (T.toLinearMap.restrictScalars R) at hz
+  obtain ⟨w,hw,he⟩ := hz.2
+  refine ⟨(algebraMap R K y)⁻¹ • w,
+    powerLocalization_div (K := K) y (coordinateLattice (R := R) (K := K) (ι := ι)) hy hw, ?_⟩
+  change T ((algebraMap R K y)⁻¹ • w) = (algebraMap R K y)⁻¹ • z
+  change T w = z at he
+  rw [map_smul, he]
+
+end GenericDivision
+
+section
+universe u
+variable (k : Type u) [Field k] [Infinite k]
+variable (ι : Type u) [Fintype ι] [DecidableEq ι]
+variable (ρ : Representation k (Multiplicative k) (ι → k))
+variable (P : Matrix ι ι (Polynomial k))
+variable (hP : ∀ t : k, polynomialMatrixEval P t = LinearMap.toMatrix' (ρ (Multiplicative.ofAdd t)))
+
+include hP in
+lemma associated_y_divisible (d : ℕ) (f : associatedSectionsSubmodule k ι ρ)
+    (F : ι → MvPolynomial (Fin 2) k)
+    (hF : ∀ i, algebraMap (MvPolynomial (Fin 2) k) (planeCoordinateChart k 0) (F i) =
+      planeChartCoordinate k 0 0 ^ d * associatedXChart k ι ρ f i)
+    (hdiv : ∀ i, MvPolynomial.X 1 ∣ F i) :
+    ∃ f' : associatedSectionsSubmodule k ι ρ, f = (MvPolynomial.X 1 : MvPolynomial (Fin 2) k) • f' := by
+  let y : planeFractionField k :=
+    algebraMap (MvPolynomial (Fin 2) k) (planeFractionField k) (MvPolynomial.X 1)
+  have hy : y ≠ 0 := planeFractionCoordinate_ne_zero k 1
+  have hx : y⁻¹ • associatedFractionChart k ι ρ f ∈
+      powerLocalization (MvPolynomial.X 0 : MvPolynomial (Fin 2) k) coordinateLattice := by
+    refine ⟨d, (mem_coordinateLattice _).mpr fun i => ?_⟩
+    obtain ⟨Q,hQ⟩ := hdiv i
+    refine ⟨Q, ?_⟩
+    have hi := congrArg (planeChartToFraction k 0) (hF i)
+    rw [planeChartToFraction_algebraMap, hQ, map_mul, map_mul, map_pow,
+      planeChartCoordinate, planeChartToFraction_algebraMap] at hi
+    change y * algebraMap (MvPolynomial (Fin 2) k) (planeFractionField k) Q =
+      algebraMap (MvPolynomial (Fin 2) k) (planeFractionField k) (MvPolynomial.X 0)^d *
+        associatedFractionChart k ι ρ f i at hi
+    simp only [Pi.smul_apply, Algebra.smul_def, map_pow]
+    change algebraMap (MvPolynomial (Fin 2) k) (planeFractionField k) Q =
+      algebraMap (MvPolynomial (Fin 2) k) (planeFractionField k) (MvPolynomial.X 0)^d *
+        (y⁻¹ * associatedFractionChart k ι ρ f i)
+    calc
+      _ = y⁻¹ * (y * algebraMap (MvPolynomial (Fin 2) k) (planeFractionField k) Q) := by
+        rw [← mul_assoc, inv_mul_cancel₀ hy, one_mul]
+      _ = _ := by rw [hi]; ring
+  have hz := gluedLattice_div_y (MvPolynomial.X 0 : MvPolynomial (Fin 2) k)
+    (MvPolynomial.X 1) (planeGlobalTransition k P (associatedPolynomial_zero k ι ρ P hP)
+      (associatedPolynomial_add k ι ρ P hP)) hy
+    (associatedFractionChart_mem k ι ρ P hP f) hx
+  obtain ⟨f',hf'⟩ := associatedLatticeMap_surjective k ι ρ P hP ⟨_,hz⟩
+  have he : associatedFractionChart k ι ρ f' = y⁻¹ • associatedFractionChart k ι ρ f :=
+    congrArg Subtype.val hf'
+  refine ⟨f', associatedFractionChart_injective k ι ρ ?_⟩
+  rw [map_smul, he]
+  rw [← algebraMap_smul (planeFractionField k) (MvPolynomial.X 1)]
+  change associatedFractionChart k ι ρ f = y • (y⁻¹ • associatedFractionChart k ι ρ f)
+  rw [smul_inv_smul₀ hy]
+
+
+include hP in
+lemma associatedOrigin_weight_upper (d : ℕ) (hd : ∀ i j, (P i j).natDegree ≤ d)
+    (n : ℤ) (hn : (d : ℤ) < n) (v : (associatedOriginRep k ι ρ).V)
+    (hv : ∀ a : kˣ, (associatedOriginRep k ι ρ).ρ (diagonalTorus k a) v =
+      integerCharacter k n a • v) : v = 0 := by
+  let Q : (associatedSectionsSubmodule k ι ρ) →ₗ[k] (associatedOriginRep k ι ρ).V :=
+    (associatedOriginRelations k ι ρ).mkQ
+  obtain ⟨f,hfv,hf⟩ := eigenvector_lift
+    ((associatedSectionsRepresentation k ι ρ).comp (diagonalTorus k))
+    ((associatedOriginRep k ι ρ).ρ.comp (diagonalTorus k))
+    (associatedSections_characters k ι ρ) Q (fun _ _ => rfl)
+    (associatedOriginRelations k ι ρ).mkQ_surjective (integerCharacter k n) v hv
+  obtain ⟨F,hF⟩ := associatedNumerator_exists k ι ρ P hP d hd f
+  have hdiv (i : ι) : MvPolynomial.X 1 ∣ F i := by
+    apply positive_plane_weight_dvd k (F i) (n-d) (by omega)
+    exact associatedNumerator_weight k ι ρ d f F hF n hf i
+  obtain ⟨f',hff'⟩ := associated_y_divisible k ι ρ P hP d f F hF hdiv
+  have hmem : f ∈ associatedOriginRelations k ι ρ := by
+    rw [hff']
+    exact associatedOrigin_coordinate_mem k ι ρ 1 f'
+  rw [← hfv]
+  exact (Submodule.Quotient.mk_eq_zero (associatedOriginRelations k ι ρ)).mpr hmem
+
+end
+
+section
+universe u
+variable (k : Type u) [Field k]
+
+variable [Infinite k]
+variable (ι : Type u) [Fintype ι] [DecidableEq ι]
+variable (ρ : Representation k (Multiplicative k) (ι → k))
+
+ 
+
+theorem associatedOrigin_weights
+    (P : Matrix ι ι (Polynomial k))
+    (hP : ∀ t : k, polynomialMatrixEval P t = LinearMap.toMatrix' (ρ (Multiplicative.ofAdd t)))
+    (d : ℕ) (hd : ∀ i j, (P i j).natDegree ≤ d) :
+    HasWeightsBetween (associatedOriginRep k ι ρ).ρ (-(d : ℤ)) d :=
+  weights_between_of_upper k _ d (associatedOrigin_weight_upper k ι ρ P hP d hd)
+
+end
+
+universe u
+variable (k : Type u) [Field k] [Infinite k]
+variable (p : ℕ) [Fact p.Prime] [CharP k p]
+variable (ι : Type u) [Fintype ι] [DecidableEq ι]
+variable (ρ : Representation k (Multiplicative k) (ι → k))
+variable (P : Matrix ι ι (Polynomial k))
+variable (hP : ∀ t : k, polynomialMatrixEval P t = LinearMap.toMatrix' (ρ (Multiplicative.ofAdd t)))
+variable (d : ℕ) (hd : ∀ i j, (P i j).natDegree ≤ d)
+
+noncomputable def associatedOriginFD : FDRep k (SL2 k) := by
+  let : Module.Finite k (associatedOriginRep k ι ρ).V := associatedOrigin_finite k ι ρ P hP d hd
+  exact FDRep.of (associatedOriginRep k ι ρ).ρ
+
+ 
+
+theorem associatedOrigin_tilting (m : ℕ) (hm : d < p^(m+1)) :
+    IsTilting k (steinberg k p m ⊗ associatedOriginFD k ι ρ P hP d hd) := by
+  apply bounded_weights_tilting k p m d hm
+  · exact associatedOrigin_rational k ι ρ
+  · exact associatedOrigin_weights k ι ρ P hP d hd
+end FiberFunctorProof
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/RestrictedTilting/BaseChangeProjective.lean b/lean/OAI/RepresentationTheory/RestrictedTilting/BaseChangeProjective.lean
index f0e5b52c92aee82acd988a3a5f1fb70e3b540ba5..99bfff786d56ab54aadfef05d20d3ab7fd53493e 100644
--- a/lean/OAI/RepresentationTheory/RestrictedTilting/BaseChangeProjective.lean
+++ b/lean/OAI/RepresentationTheory/RestrictedTilting/BaseChangeProjective.lean
@@ -1,27 +1,27 @@
-import Mathlib
-
-namespace OAI
-
-namespace FiberFunctorProof
-open scoped TensorProduct
-variable {R S M F : Type*} [CommRing R] [CommRing S] [Algebra R S]
-  [AddCommGroup M] [Module R M] [AddCommGroup F] [Module R F]
-
- 
-
-theorem projective_baseChange_of_split_smul [Module.Projective R F]
-    (f : M →ₗ[R] F) (g : F →ₗ[R] M) (a : R)
-    (h : g.comp f = a • LinearMap.id) (ha : IsUnit (algebraMap R S a)) :
-    Module.Projective S (S ⊗[R] M) := by
-  have hb : (g.baseChange S).comp (f.baseChange S) =
-      algebraMap R S a • LinearMap.id := by
-    rw [← LinearMap.baseChange_comp, h, LinearMap.baseChange_smul,
-      LinearMap.baseChange_id, algebraMap_smul]
-  obtain ⟨u,hu⟩ := ha
-  refine Module.Projective.of_split ((↑u⁻¹ : S) • f.baseChange S) (g.baseChange S) ?_
-  rw [LinearMap.comp_smul, hb, ← hu, smul_smul]
-  simp
-
-end FiberFunctorProof
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+namespace FiberFunctorProof
+open scoped TensorProduct
+variable {R S M F : Type*} [CommRing R] [CommRing S] [Algebra R S]
+  [AddCommGroup M] [Module R M] [AddCommGroup F] [Module R F]
+
+ 
+
+theorem projective_baseChange_of_split_smul [Module.Projective R F]
+    (f : M →ₗ[R] F) (g : F →ₗ[R] M) (a : R)
+    (h : g.comp f = a • LinearMap.id) (ha : IsUnit (algebraMap R S a)) :
+    Module.Projective S (S ⊗[R] M) := by
+  have hb : (g.baseChange S).comp (f.baseChange S) =
+      algebraMap R S a • LinearMap.id := by
+    rw [← LinearMap.baseChange_comp, h, LinearMap.baseChange_smul,
+      LinearMap.baseChange_id, algebraMap_smul]
+  obtain ⟨u,hu⟩ := ha
+  refine Module.Projective.of_split ((↑u⁻¹ : S) • f.baseChange S) (g.baseChange S) ?_
+  rw [LinearMap.comp_smul, hb, ← hu, smul_smul]
+  simp
+
+end FiberFunctorProof
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/RestrictedTilting/Basic.lean b/lean/OAI/RepresentationTheory/RestrictedTilting/Basic.lean
index 532d5838f5d7b622594fee366fc74745efec5a16..7346e0835717595babc3b32a196f906cdcd854d2 100644
--- a/lean/OAI/RepresentationTheory/RestrictedTilting/Basic.lean
+++ b/lean/OAI/RepresentationTheory/RestrictedTilting/Basic.lean
@@ -1,4 +1,4 @@
-import OAI.RepresentationTheory.RestrictedTilting.AssociatedWeights
-import OAI.RepresentationTheory.RestrictedTilting.BinarySequences
-import OAI.RepresentationTheory.RestrictedTilting.JetRepresentations
-import OAI.RepresentationTheory.RestrictedTilting.LocalFreeness
+import OAI.RepresentationTheory.RestrictedTilting.AssociatedWeights
+import OAI.RepresentationTheory.RestrictedTilting.BinarySequences
+import OAI.RepresentationTheory.RestrictedTilting.JetRepresentations
+import OAI.RepresentationTheory.RestrictedTilting.LocalFreeness
diff --git a/lean/OAI/RepresentationTheory/RestrictedTilting/BinarySequences.lean b/lean/OAI/RepresentationTheory/RestrictedTilting/BinarySequences.lean
index 910a18ece21033de0873530db91e22097c212636..5208fe74c55b9cc46e6f019c90d711a29f47476e 100644
--- a/lean/OAI/RepresentationTheory/RestrictedTilting/BinarySequences.lean
+++ b/lean/OAI/RepresentationTheory/RestrictedTilting/BinarySequences.lean
@@ -1,311 +1,311 @@
-import Mathlib
-import OAI.RepresentationTheory.RestrictedTilting.ExtensionSplitting
-import OAI.RepresentationTheory.RestrictedTilting.TensorLimits
-
-namespace OAI
-
-/-! Binary multiplication exact sequences and tensor closure of good filtrations. -/
-
-noncomputable section
-open CategoryTheory CategoryTheory.MonoidalCategory CategoryTheory.Limits
-open scoped BigOperators
-
-namespace FiberFunctorProof
-section
-
-universe u
-variable (k : Type u) [Field k]
-open scoped TensorProduct
-
-abbrev BiBinary (a b : ℕ) := Binary k a ⊗[k] Binary k b
-abbrev BiPolynomial := MvPolynomial (Fin 2) k ⊗[k] MvPolynomial (Fin 2) k
-
-noncomputable instance biBinaryAddCommGroup (a b : ℕ) : AddCommGroup (BiBinary k a b) :=
-  TensorProduct.addCommGroup
-
-noncomputable def binaryMulX (n : ℕ) (i : Fin 2) : Binary k n →ₗ[k] Binary k (n + 1) where
-  toFun f := ⟨MvPolynomial.X i * f.val,
-    by simpa [Nat.add_comm] using (MvPolynomial.isHomogeneous_X k i).mul f.property⟩
-  map_add' f g := by apply Subtype.ext; simp [mul_add]
-  map_smul' c f := by apply Subtype.ext; simp
-
-noncomputable def biBinaryEmbed (a b : ℕ) : BiBinary k a b →ₗ[k] BiPolynomial k :=
-  TensorProduct.map (Binary k a).subtype (Binary k b).subtype
-
-lemma biBinaryEmbed_injective (a b : ℕ) : Function.Injective (biBinaryEmbed k a b) :=
-  TensorProduct.map_injective_of_flat_flat _ _
-    (Binary k a).injective_subtype (Binary k b).injective_subtype
-
- 
-noncomputable def determinantMap (a b : ℕ) : BiBinary k a b →ₗ[k] BiBinary k (a + 1) (b + 1) :=
-  TensorProduct.map (binaryMulX k a 0) (binaryMulX k b 1) -
-    TensorProduct.map (binaryMulX k a 1) (binaryMulX k b 0)
-
-noncomputable def biDeterminant : BiPolynomial k :=
-  MvPolynomial.X 0 ⊗ₜ[k] MvPolynomial.X 1 - MvPolynomial.X 1 ⊗ₜ[k] MvPolynomial.X 0
-
-lemma embed_determinantMap (a b : ℕ) (x : BiBinary k a b) :
-    biBinaryEmbed k (a + 1) (b + 1) (determinantMap k a b x) =
-      biDeterminant k * biBinaryEmbed k a b x := by
-  induction x using TensorProduct.inductionOn with
-  | tmul f g =>
-    simp only [biBinaryEmbed, determinantMap, LinearMap.sub_apply, TensorProduct.map_tmul,
-      map_sub, biDeterminant, sub_mul, Algebra.TensorProduct.tmul_mul_tmul]
-    rfl
-  | add x y hx hy => simp [map_add, hx, hy, mul_add]
-
-lemma biDeterminant_ne_zero :
-    MvPolynomial.tensorEquivSum k (Fin 2) (Fin 2) k (biDeterminant k) ≠ 0 := by
-  intro h
-  have hh := congrArg
-    (MvPolynomial.eval (fun i : Fin 2 ⊕ Fin 2 => match i with
-      | .inl i => if i = 0 then (1 : k) else 0
-      | .inr i => if i = 1 then (1 : k) else 0)) h
-  simp [biDeterminant] at hh
-
-lemma determinantMap_injective (a b : ℕ) : Function.Injective (determinantMap k a b) := by
-  intro x y hxy
-  apply biBinaryEmbed_injective k a b
-  apply (MvPolynomial.tensorEquivSum k (Fin 2) (Fin 2) k).injective
-  apply mul_left_cancel₀ (biDeterminant_ne_zero k)
-  rw [← map_mul, ← map_mul, ← embed_determinantMap, ← embed_determinantMap, hxy]
-
-noncomputable def binaryMultiply (a b : ℕ) : BiBinary k a b →ₗ[k] Binary k (a + b) :=
-  TensorProduct.lift
-    { toFun := fun f =>
-        { toFun := fun g => ⟨f.val * g.val, f.property.mul g.property⟩
-          map_add' := by intro g h; apply Subtype.ext; simp [mul_add]
-          map_smul' := by intro c g; apply Subtype.ext; simp }
-      map_add' := by intro f g; ext h : 1; apply Subtype.ext; simp [add_mul]
-      map_smul' := by intro c f; ext g : 1; apply Subtype.ext; simp }
-
-@[simp] lemma binaryMultiply_tmul (a b : ℕ) (f : Binary k a) (g : Binary k b) :
-    (binaryMultiply k a b (f ⊗ₜ[k] g)).val = f.val * g.val := rfl
-
-lemma binaryMultiply_determinantMap (a b : ℕ) :
-    (binaryMultiply k (a + 1) (b + 1)).comp (determinantMap k a b) = 0 := by
-  apply TensorProduct.ext'
-  intro f g
-  apply Subtype.ext
-  simp only [LinearMap.comp_apply, determinantMap, LinearMap.sub_apply, TensorProduct.map_tmul,
-    map_sub, Submodule.coe_sub, binaryMultiply_tmul]
-  change (MvPolynomial.X 0 * f.val) * (MvPolynomial.X 1 * g.val) -
-    (MvPolynomial.X 1 * f.val) * (MvPolynomial.X 0 * g.val) = 0
-  ring
-
-lemma binaryMultiply_surjective (a b : ℕ) : Function.Surjective (binaryMultiply k a b) := by
-  rw [← LinearMap.range_eq_top]
-  apply top_unique
-  rw [← (binaryBasis k (a + b)).span_eq]
-  apply Submodule.span_le.mpr
-  rintro _ ⟨i, rfl⟩
-  let j : Fin (a + 1) := ⟨min i.val a, by omega⟩
-  let l : Fin (b + 1) := ⟨i.val - j.val, by dsimp [j]; omega⟩
-  refine ⟨binaryBasis k a j ⊗ₜ[k] binaryBasis k b l, ?_⟩
-  apply Subtype.ext
-  simp only [binaryMultiply_tmul, binaryBasis_apply]
-  have h₁ : a - j.val + (b - l.val) = a + b - i.val := by dsimp [j, l]; omega
-  have h₂ : j.val + l.val = i.val := by dsimp [j, l]; omega
-  rw [mul_mul_mul_comm, ← pow_add, ← pow_add, h₁, h₂]
-
-lemma binaryMultiply_exact (a b : ℕ) :
-    LinearMap.range (determinantMap k a b) = LinearMap.ker (binaryMultiply k (a + 1) (b + 1)) := by
-  have hin : LinearMap.range (determinantMap k a b) ≤
-      LinearMap.ker (binaryMultiply k (a + 1) (b + 1)) := by
-    rw [LinearMap.range_le_ker_iff]
-    exact binaryMultiply_determinantMap k a b
-  have : FiniteDimensional k (Binary k a) := Module.Finite.of_basis (binaryBasis k a)
-  have : FiniteDimensional k (Binary k b) := Module.Finite.of_basis (binaryBasis k b)
-  have : FiniteDimensional k (Binary k (a + 1)) := Module.Finite.of_basis (binaryBasis k (a + 1))
-  have : FiniteDimensional k (Binary k (b + 1)) := Module.Finite.of_basis (binaryBasis k (b + 1))
-  apply Submodule.eq_of_le_of_finrank_eq hin
-  have hr := LinearMap.finrank_range_of_inj (determinantMap_injective k a b)
-  have hn := LinearMap.finrank_range_add_finrank_ker (binaryMultiply k (a + 1) (b + 1))
-  rw [LinearMap.range_eq_top.mpr (binaryMultiply_surjective k (a + 1) (b + 1)),
-    finrank_top] at hn
-  rw [Module.finrank_tensorProduct, binaryBasis_finrank, binaryBasis_finrank] at hr hn
-  rw [binaryBasis_finrank] at hn
-  nlinarith
-
-
-
-
-noncomputable def biLinearChange (A : Matrix (Fin 2) (Fin 2) k) :
-    BiPolynomial k →ₐ[k] BiPolynomial k :=
-  Algebra.TensorProduct.map (linearChange A) (linearChange A)
-
-lemma biLinearChange_determinant (A : Matrix (Fin 2) (Fin 2) k) :
-    biLinearChange k A (biDeterminant k) = A.det • biDeterminant k := by
-  apply (MvPolynomial.tensorEquivSum k (Fin 2) (Fin 2) k).injective
-  simp only [biLinearChange, biDeterminant, map_sub, Algebra.TensorProduct.map_tmul,
-    linearChange_X, Fin.sum_univ_two, ← MvPolynomial.smul_eq_C_mul,
-    TensorProduct.add_tmul, TensorProduct.tmul_add, TensorProduct.smul_tmul_smul,
-    map_add, map_smul, MvPolynomial.tensorEquivSum_X_tmul_X]
-  simp only [MvPolynomial.smul_eq_C_mul, Matrix.det_fin_two, map_mul, map_sub]
-  ring
-
-noncomputable def biBinaryAction (a b : ℕ) (A : Matrix (Fin 2) (Fin 2) k) :
-    BiBinary k a b →ₗ[k] BiBinary k a b :=
-  TensorProduct.map (homogeneousRepresentation a A) (homogeneousRepresentation b A)
-
-lemma biBinaryEmbed_action (a b : ℕ) (A : Matrix (Fin 2) (Fin 2) k) (x : BiBinary k a b) :
-    biBinaryEmbed k a b (biBinaryAction k a b A x) =
-      biLinearChange k A (biBinaryEmbed k a b x) := by
-  induction x using TensorProduct.inductionOn with
-  | tmul f g => rfl
-  | add x y hx hy => simp [hx, hy]
-
-lemma determinantMap_equivariant (a b : ℕ) (A : Matrix (Fin 2) (Fin 2) k)
-    (hA : A.det = 1) (x : BiBinary k a b) :
-    determinantMap k a b (biBinaryAction k a b A x) =
-      biBinaryAction k (a + 1) (b + 1) A (determinantMap k a b x) := by
-  apply biBinaryEmbed_injective k (a + 1) (b + 1)
-  rw [embed_determinantMap, biBinaryEmbed_action, biBinaryEmbed_action,
-    embed_determinantMap, map_mul, biLinearChange_determinant, hA, one_smul]
-
-lemma binaryMultiply_equivariant (a b : ℕ) (A : Matrix (Fin 2) (Fin 2) k)
-    (x : BiBinary k a b) :
-    binaryMultiply k a b (biBinaryAction k a b A x) =
-      homogeneousRepresentation (a + b) A (binaryMultiply k a b x) := by
-  induction x using TensorProduct.inductionOn with
-  | tmul f g =>
-    apply Subtype.ext
-    change linearChange A f.val * linearChange A g.val = linearChange A (f.val * g.val)
-    exact (map_mul _ _ _).symm
-  | add x y hx hy => simp [hx, hy]
-
-lemma dualMatrix_det (g : SL2 k) : (dualMatrix k g).det = 1 := by
-  change Matrix.det ((g⁻¹ : SL2 k).val.transpose) = 1
-  rw [Matrix.det_transpose]
-  exact (g⁻¹ : SL2 k).property
-
-end
-
-universe u
-variable (k : Type u) [Field k]
-noncomputable def costandardDeterminant (a b : ℕ) :
-    costandard k a ⊗ costandard k b ⟶ costandard k (a + 1) ⊗ costandard k (b + 1) :=
-  fdrepHom k (determinantMap k a b) (fun g x =>
-    determinantMap_equivariant k a b (dualMatrix k g) (dualMatrix_det k g) x)
-
- 
-noncomputable def costandardMultiply (a b : ℕ) :
-    costandard k a ⊗ costandard k b ⟶ costandard k (a + b) :=
-  fdrepHom k (binaryMultiply k a b) (fun g x =>
-    binaryMultiply_equivariant k a b (dualMatrix k g) x)
-
-lemma costandardDeterminant_multiply (a b : ℕ) :
-    costandardDeterminant k a b ≫ costandardMultiply k (a + 1) (b + 1) = 0 := by
-  apply Action.hom_ext
-  apply FGModuleCat.hom_ext
-  exact binaryMultiply_determinantMap k a b
-
- 
-noncomputable def costandardTensorSequence (a b : ℕ) : ShortComplex (FDRep k (SL2 k)) :=
-  ShortComplex.mk (costandardDeterminant k a b) (costandardMultiply k (a + 1) (b + 1))
-    (costandardDeterminant_multiply k a b)
-
-lemma costandardTensorSequence_shortExact (a b : ℕ) :
-    (costandardTensorSequence k a b).ShortExact :=
-  fdrep_shortExact k _ (determinantMap_injective k a b)
-    (binaryMultiply_surjective k (a + 1) (b + 1)) (binaryMultiply_exact k a b)
-
-
-lemma costandardMultiply_isIso_of_zero (a b : ℕ) (h : a = 0 ∨ b = 0) :
-    IsIso (costandardMultiply k a b) := by
-  let F := fdrepForget k (SL2 k)
-  have hr : Module.finrank k (BiBinary k a b) = Module.finrank k (Binary k (a + b)) := by
-    rw [Module.finrank_tensorProduct, binaryBasis_finrank, binaryBasis_finrank,
-      binaryBasis_finrank]
-    rcases h with rfl | rfl <;> simp
-  have hi : Function.Injective (binaryMultiply k a b) :=
-    (LinearMap.injective_iff_surjective_of_finrank_eq_finrank hr).mpr
-      (binaryMultiply_surjective k a b)
-  have : Mono (F.map (costandardMultiply k a b)) :=
-    (ModuleCat.mono_iff_injective _).mpr hi
-  have : Epi (F.map (costandardMultiply k a b)) :=
-    (ModuleCat.epi_iff_surjective _).mpr (binaryMultiply_surjective k a b)
-  have : Mono (costandardMultiply k a b) := F.mono_of_mono_map (by infer_instance)
-  have : Epi (costandardMultiply k a b) := F.epi_of_epi_map (by infer_instance)
-  exact isIso_of_mono_of_epi (costandardMultiply k a b)
-
- 
-
-theorem costandard_tensor_hasGoodFiltration (a b : ℕ) :
-    HasGoodFiltration k (costandard k a ⊗ costandard k b) := by
-  induction a generalizing b with
-  | zero =>
-    have := costandardMultiply_isIso_of_zero k 0 b (Or.inl rfl)
-    exact (costandard_hasGoodFiltration k (0 + b)).of_iso k
-      (asIso (costandardMultiply k 0 b)).symm
-  | succ a ih =>
-    cases b with
-    | zero =>
-      have := costandardMultiply_isIso_of_zero k (a + 1) 0 (Or.inr rfl)
-      exact (costandard_hasGoodFiltration k (a + 1 + 0)).of_iso k
-        (asIso (costandardMultiply k (a + 1) 0)).symm
-    | succ b =>
-      exact .step _ (costandardDeterminant k a b)
-        (costandardMultiply k (a + 1) (b + 1)) (costandardDeterminant_multiply k a b)
-        (costandardTensorSequence_shortExact k a b) (ih b)
-
-
-lemma HasGoodFiltration.tensor_costandard {A : FDRep k (SL2 k)}
-    (hA : HasGoodFiltration k A) (n : ℕ) :
-    HasGoodFiltration k (A ⊗ costandard k n) := by
-  induction hA with
-  | zero A hA => exact .zero _ ((tensorRight (costandard k n)).map_isZero hA)
-  | @step A B m f g w hS hA ih =>
-    let := tensorRight_preservesFiniteLimits (costandard k n)
-    let := tensorRight_preservesFiniteColimits (costandard k n)
-    have hT := hS.map_of_exact (tensorRight (costandard k n))
-    exact ih.extension k (costandard_tensor_hasGoodFiltration k m n)
-      _ _ _ hT
-
- 
-theorem HasGoodFiltration.tensor {A B : FDRep k (SL2 k)}
-    (hA : HasGoodFiltration k A) (hB : HasGoodFiltration k B) :
-    HasGoodFiltration k (A ⊗ B) := by
-  induction hB with
-  | zero B hB => exact .zero _ ((tensorLeft A).map_isZero hB)
-  | @step B C n f g w hS hB ih =>
-    let := tensorLeft_preservesFiniteLimits A
-    let := tensorLeft_preservesFiniteColimits A
-    have hT := hS.map_of_exact (tensorLeft A)
-    exact ih.extension k (hA.tensor_costandard k n) _ _ _ hT
-
-noncomputable def fdrepBidual (A : FDRep k (SL2 k)) : A ≅ (Aᘁ)ᘁ :=
-  (rightDualIso
-    (@HasRightDual.exact _ _ _ (Aᘁ) (RightRigidCategory.rightDual (Aᘁ)))
-    (BraidedCategory.exactPairing_swap A (Aᘁ))).symm
-
-lemma IsRational.of_iso {A B : FDRep k (SL2 k)}
-    (h : IsRational k A.ρ) (e : A ≅ B) : IsRational k B.ρ :=
-  h.of_surjective k A.ρ B.ρ (fdrepLinear k e.hom) (fdrep_surjective k e.hom)
-    (fdrepLinear_comm k e.hom)
-
-lemma IsTilting.of_iso {A B : FDRep k (SL2 k)}
-    (h : IsTilting k A) (e : A ≅ B) : IsTilting k B := by
-  refine ⟨h.1.of_iso k e, h.2.1.of_iso k e, ?_⟩
-  let eD : Aᘁ ≅ Bᘁ :=
-    { hom := e.invᘁ
-      inv := e.homᘁ
-      hom_inv_id := by rw [← comp_rightAdjointMate, e.hom_inv_id, rightAdjointMate_id]
-      inv_hom_id := by rw [← comp_rightAdjointMate, e.inv_hom_id, rightAdjointMate_id] }
-  exact h.2.2.of_iso k eD
-
- 
-theorem IsTilting.tensor {A B : FDRep k (SL2 k)}
-    (hA : IsTilting k A) (hB : IsTilting k B) : IsTilting k (A ⊗ B) := by
-  refine ⟨IsRational.tensor k A.ρ B.ρ hA.1 hB.1, hA.2.1.tensor k hB.2.1, ?_⟩
-  exact (hB.2.2.tensor k hA.2.2).of_iso k (rightDualTensorIso A B).symm
-
- 
-theorem IsTilting.dual {A : FDRep k (SL2 k)} (hA : IsTilting k A) : IsTilting k Aᘁ :=
-  ⟨by rw [fdrep_dual_repr]; exact IsRational.dual k A.ρ hA.1,
-   hA.2.2, hA.2.1.of_iso k (fdrepBidual k A)⟩
-
-end FiberFunctorProof
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.RestrictedTilting.ExtensionSplitting
+import OAI.RepresentationTheory.RestrictedTilting.TensorLimits
+
+namespace OAI
+
+/-! Binary multiplication exact sequences and tensor closure of good filtrations. -/
+
+noncomputable section
+open CategoryTheory CategoryTheory.MonoidalCategory CategoryTheory.Limits
+open scoped BigOperators
+
+namespace FiberFunctorProof
+section
+
+universe u
+variable (k : Type u) [Field k]
+open scoped TensorProduct
+
+abbrev BiBinary (a b : ℕ) := Binary k a ⊗[k] Binary k b
+abbrev BiPolynomial := MvPolynomial (Fin 2) k ⊗[k] MvPolynomial (Fin 2) k
+
+noncomputable instance biBinaryAddCommGroup (a b : ℕ) : AddCommGroup (BiBinary k a b) :=
+  TensorProduct.addCommGroup
+
+noncomputable def binaryMulX (n : ℕ) (i : Fin 2) : Binary k n →ₗ[k] Binary k (n + 1) where
+  toFun f := ⟨MvPolynomial.X i * f.val,
+    by simpa [Nat.add_comm] using (MvPolynomial.isHomogeneous_X k i).mul f.property⟩
+  map_add' f g := by apply Subtype.ext; simp [mul_add]
+  map_smul' c f := by apply Subtype.ext; simp
+
+noncomputable def biBinaryEmbed (a b : ℕ) : BiBinary k a b →ₗ[k] BiPolynomial k :=
+  TensorProduct.map (Binary k a).subtype (Binary k b).subtype
+
+lemma biBinaryEmbed_injective (a b : ℕ) : Function.Injective (biBinaryEmbed k a b) :=
+  TensorProduct.map_injective_of_flat_flat _ _
+    (Binary k a).injective_subtype (Binary k b).injective_subtype
+
+ 
+noncomputable def determinantMap (a b : ℕ) : BiBinary k a b →ₗ[k] BiBinary k (a + 1) (b + 1) :=
+  TensorProduct.map (binaryMulX k a 0) (binaryMulX k b 1) -
+    TensorProduct.map (binaryMulX k a 1) (binaryMulX k b 0)
+
+noncomputable def biDeterminant : BiPolynomial k :=
+  MvPolynomial.X 0 ⊗ₜ[k] MvPolynomial.X 1 - MvPolynomial.X 1 ⊗ₜ[k] MvPolynomial.X 0
+
+lemma embed_determinantMap (a b : ℕ) (x : BiBinary k a b) :
+    biBinaryEmbed k (a + 1) (b + 1) (determinantMap k a b x) =
+      biDeterminant k * biBinaryEmbed k a b x := by
+  induction x using TensorProduct.inductionOn with
+  | tmul f g =>
+    simp only [biBinaryEmbed, determinantMap, LinearMap.sub_apply, TensorProduct.map_tmul,
+      map_sub, biDeterminant, sub_mul, Algebra.TensorProduct.tmul_mul_tmul]
+    rfl
+  | add x y hx hy => simp [map_add, hx, hy, mul_add]
+
+lemma biDeterminant_ne_zero :
+    MvPolynomial.tensorEquivSum k (Fin 2) (Fin 2) k (biDeterminant k) ≠ 0 := by
+  intro h
+  have hh := congrArg
+    (MvPolynomial.eval (fun i : Fin 2 ⊕ Fin 2 => match i with
+      | .inl i => if i = 0 then (1 : k) else 0
+      | .inr i => if i = 1 then (1 : k) else 0)) h
+  simp [biDeterminant] at hh
+
+lemma determinantMap_injective (a b : ℕ) : Function.Injective (determinantMap k a b) := by
+  intro x y hxy
+  apply biBinaryEmbed_injective k a b
+  apply (MvPolynomial.tensorEquivSum k (Fin 2) (Fin 2) k).injective
+  apply mul_left_cancel₀ (biDeterminant_ne_zero k)
+  rw [← map_mul, ← map_mul, ← embed_determinantMap, ← embed_determinantMap, hxy]
+
+noncomputable def binaryMultiply (a b : ℕ) : BiBinary k a b →ₗ[k] Binary k (a + b) :=
+  TensorProduct.lift
+    { toFun := fun f =>
+        { toFun := fun g => ⟨f.val * g.val, f.property.mul g.property⟩
+          map_add' := by intro g h; apply Subtype.ext; simp [mul_add]
+          map_smul' := by intro c g; apply Subtype.ext; simp }
+      map_add' := by intro f g; ext h : 1; apply Subtype.ext; simp [add_mul]
+      map_smul' := by intro c f; ext g : 1; apply Subtype.ext; simp }
+
+@[simp] lemma binaryMultiply_tmul (a b : ℕ) (f : Binary k a) (g : Binary k b) :
+    (binaryMultiply k a b (f ⊗ₜ[k] g)).val = f.val * g.val := rfl
+
+lemma binaryMultiply_determinantMap (a b : ℕ) :
+    (binaryMultiply k (a + 1) (b + 1)).comp (determinantMap k a b) = 0 := by
+  apply TensorProduct.ext'
+  intro f g
+  apply Subtype.ext
+  simp only [LinearMap.comp_apply, determinantMap, LinearMap.sub_apply, TensorProduct.map_tmul,
+    map_sub, Submodule.coe_sub, binaryMultiply_tmul]
+  change (MvPolynomial.X 0 * f.val) * (MvPolynomial.X 1 * g.val) -
+    (MvPolynomial.X 1 * f.val) * (MvPolynomial.X 0 * g.val) = 0
+  ring
+
+lemma binaryMultiply_surjective (a b : ℕ) : Function.Surjective (binaryMultiply k a b) := by
+  rw [← LinearMap.range_eq_top]
+  apply top_unique
+  rw [← (binaryBasis k (a + b)).span_eq]
+  apply Submodule.span_le.mpr
+  rintro _ ⟨i, rfl⟩
+  let j : Fin (a + 1) := ⟨min i.val a, by omega⟩
+  let l : Fin (b + 1) := ⟨i.val - j.val, by dsimp [j]; omega⟩
+  refine ⟨binaryBasis k a j ⊗ₜ[k] binaryBasis k b l, ?_⟩
+  apply Subtype.ext
+  simp only [binaryMultiply_tmul, binaryBasis_apply]
+  have h₁ : a - j.val + (b - l.val) = a + b - i.val := by dsimp [j, l]; omega
+  have h₂ : j.val + l.val = i.val := by dsimp [j, l]; omega
+  rw [mul_mul_mul_comm, ← pow_add, ← pow_add, h₁, h₂]
+
+lemma binaryMultiply_exact (a b : ℕ) :
+    LinearMap.range (determinantMap k a b) = LinearMap.ker (binaryMultiply k (a + 1) (b + 1)) := by
+  have hin : LinearMap.range (determinantMap k a b) ≤
+      LinearMap.ker (binaryMultiply k (a + 1) (b + 1)) := by
+    rw [LinearMap.range_le_ker_iff]
+    exact binaryMultiply_determinantMap k a b
+  have : FiniteDimensional k (Binary k a) := Module.Finite.of_basis (binaryBasis k a)
+  have : FiniteDimensional k (Binary k b) := Module.Finite.of_basis (binaryBasis k b)
+  have : FiniteDimensional k (Binary k (a + 1)) := Module.Finite.of_basis (binaryBasis k (a + 1))
+  have : FiniteDimensional k (Binary k (b + 1)) := Module.Finite.of_basis (binaryBasis k (b + 1))
+  apply Submodule.eq_of_le_of_finrank_eq hin
+  have hr := LinearMap.finrank_range_of_inj (determinantMap_injective k a b)
+  have hn := LinearMap.finrank_range_add_finrank_ker (binaryMultiply k (a + 1) (b + 1))
+  rw [LinearMap.range_eq_top.mpr (binaryMultiply_surjective k (a + 1) (b + 1)),
+    finrank_top] at hn
+  rw [Module.finrank_tensorProduct, binaryBasis_finrank, binaryBasis_finrank] at hr hn
+  rw [binaryBasis_finrank] at hn
+  nlinarith
+
+
+
+
+noncomputable def biLinearChange (A : Matrix (Fin 2) (Fin 2) k) :
+    BiPolynomial k →ₐ[k] BiPolynomial k :=
+  Algebra.TensorProduct.map (linearChange A) (linearChange A)
+
+lemma biLinearChange_determinant (A : Matrix (Fin 2) (Fin 2) k) :
+    biLinearChange k A (biDeterminant k) = A.det • biDeterminant k := by
+  apply (MvPolynomial.tensorEquivSum k (Fin 2) (Fin 2) k).injective
+  simp only [biLinearChange, biDeterminant, map_sub, Algebra.TensorProduct.map_tmul,
+    linearChange_X, Fin.sum_univ_two, ← MvPolynomial.smul_eq_C_mul,
+    TensorProduct.add_tmul, TensorProduct.tmul_add, TensorProduct.smul_tmul_smul,
+    map_add, map_smul, MvPolynomial.tensorEquivSum_X_tmul_X]
+  simp only [MvPolynomial.smul_eq_C_mul, Matrix.det_fin_two, map_mul, map_sub]
+  ring
+
+noncomputable def biBinaryAction (a b : ℕ) (A : Matrix (Fin 2) (Fin 2) k) :
+    BiBinary k a b →ₗ[k] BiBinary k a b :=
+  TensorProduct.map (homogeneousRepresentation a A) (homogeneousRepresentation b A)
+
+lemma biBinaryEmbed_action (a b : ℕ) (A : Matrix (Fin 2) (Fin 2) k) (x : BiBinary k a b) :
+    biBinaryEmbed k a b (biBinaryAction k a b A x) =
+      biLinearChange k A (biBinaryEmbed k a b x) := by
+  induction x using TensorProduct.inductionOn with
+  | tmul f g => rfl
+  | add x y hx hy => simp [hx, hy]
+
+lemma determinantMap_equivariant (a b : ℕ) (A : Matrix (Fin 2) (Fin 2) k)
+    (hA : A.det = 1) (x : BiBinary k a b) :
+    determinantMap k a b (biBinaryAction k a b A x) =
+      biBinaryAction k (a + 1) (b + 1) A (determinantMap k a b x) := by
+  apply biBinaryEmbed_injective k (a + 1) (b + 1)
+  rw [embed_determinantMap, biBinaryEmbed_action, biBinaryEmbed_action,
+    embed_determinantMap, map_mul, biLinearChange_determinant, hA, one_smul]
+
+lemma binaryMultiply_equivariant (a b : ℕ) (A : Matrix (Fin 2) (Fin 2) k)
+    (x : BiBinary k a b) :
+    binaryMultiply k a b (biBinaryAction k a b A x) =
+      homogeneousRepresentation (a + b) A (binaryMultiply k a b x) := by
+  induction x using TensorProduct.inductionOn with
+  | tmul f g =>
+    apply Subtype.ext
+    change linearChange A f.val * linearChange A g.val = linearChange A (f.val * g.val)
+    exact (map_mul _ _ _).symm
+  | add x y hx hy => simp [hx, hy]
+
+lemma dualMatrix_det (g : SL2 k) : (dualMatrix k g).det = 1 := by
+  change Matrix.det ((g⁻¹ : SL2 k).val.transpose) = 1
+  rw [Matrix.det_transpose]
+  exact (g⁻¹ : SL2 k).property
+
+end
+
+universe u
+variable (k : Type u) [Field k]
+noncomputable def costandardDeterminant (a b : ℕ) :
+    costandard k a ⊗ costandard k b ⟶ costandard k (a + 1) ⊗ costandard k (b + 1) :=
+  fdrepHom k (determinantMap k a b) (fun g x =>
+    determinantMap_equivariant k a b (dualMatrix k g) (dualMatrix_det k g) x)
+
+ 
+noncomputable def costandardMultiply (a b : ℕ) :
+    costandard k a ⊗ costandard k b ⟶ costandard k (a + b) :=
+  fdrepHom k (binaryMultiply k a b) (fun g x =>
+    binaryMultiply_equivariant k a b (dualMatrix k g) x)
+
+lemma costandardDeterminant_multiply (a b : ℕ) :
+    costandardDeterminant k a b ≫ costandardMultiply k (a + 1) (b + 1) = 0 := by
+  apply Action.hom_ext
+  apply FGModuleCat.hom_ext
+  exact binaryMultiply_determinantMap k a b
+
+ 
+noncomputable def costandardTensorSequence (a b : ℕ) : ShortComplex (FDRep k (SL2 k)) :=
+  ShortComplex.mk (costandardDeterminant k a b) (costandardMultiply k (a + 1) (b + 1))
+    (costandardDeterminant_multiply k a b)
+
+lemma costandardTensorSequence_shortExact (a b : ℕ) :
+    (costandardTensorSequence k a b).ShortExact :=
+  fdrep_shortExact k _ (determinantMap_injective k a b)
+    (binaryMultiply_surjective k (a + 1) (b + 1)) (binaryMultiply_exact k a b)
+
+
+lemma costandardMultiply_isIso_of_zero (a b : ℕ) (h : a = 0 ∨ b = 0) :
+    IsIso (costandardMultiply k a b) := by
+  let F := fdrepForget k (SL2 k)
+  have hr : Module.finrank k (BiBinary k a b) = Module.finrank k (Binary k (a + b)) := by
+    rw [Module.finrank_tensorProduct, binaryBasis_finrank, binaryBasis_finrank,
+      binaryBasis_finrank]
+    rcases h with rfl | rfl <;> simp
+  have hi : Function.Injective (binaryMultiply k a b) :=
+    (LinearMap.injective_iff_surjective_of_finrank_eq_finrank hr).mpr
+      (binaryMultiply_surjective k a b)
+  have : Mono (F.map (costandardMultiply k a b)) :=
+    (ModuleCat.mono_iff_injective _).mpr hi
+  have : Epi (F.map (costandardMultiply k a b)) :=
+    (ModuleCat.epi_iff_surjective _).mpr (binaryMultiply_surjective k a b)
+  have : Mono (costandardMultiply k a b) := F.mono_of_mono_map (by infer_instance)
+  have : Epi (costandardMultiply k a b) := F.epi_of_epi_map (by infer_instance)
+  exact isIso_of_mono_of_epi (costandardMultiply k a b)
+
+ 
+
+theorem costandard_tensor_hasGoodFiltration (a b : ℕ) :
+    HasGoodFiltration k (costandard k a ⊗ costandard k b) := by
+  induction a generalizing b with
+  | zero =>
+    have := costandardMultiply_isIso_of_zero k 0 b (Or.inl rfl)
+    exact (costandard_hasGoodFiltration k (0 + b)).of_iso k
+      (asIso (costandardMultiply k 0 b)).symm
+  | succ a ih =>
+    cases b with
+    | zero =>
+      have := costandardMultiply_isIso_of_zero k (a + 1) 0 (Or.inr rfl)
+      exact (costandard_hasGoodFiltration k (a + 1 + 0)).of_iso k
+        (asIso (costandardMultiply k (a + 1) 0)).symm
+    | succ b =>
+      exact .step _ (costandardDeterminant k a b)
+        (costandardMultiply k (a + 1) (b + 1)) (costandardDeterminant_multiply k a b)
+        (costandardTensorSequence_shortExact k a b) (ih b)
+
+
+lemma HasGoodFiltration.tensor_costandard {A : FDRep k (SL2 k)}
+    (hA : HasGoodFiltration k A) (n : ℕ) :
+    HasGoodFiltration k (A ⊗ costandard k n) := by
+  induction hA with
+  | zero A hA => exact .zero _ ((tensorRight (costandard k n)).map_isZero hA)
+  | @step A B m f g w hS hA ih =>
+    let := tensorRight_preservesFiniteLimits (costandard k n)
+    let := tensorRight_preservesFiniteColimits (costandard k n)
+    have hT := hS.map_of_exact (tensorRight (costandard k n))
+    exact ih.extension k (costandard_tensor_hasGoodFiltration k m n)
+      _ _ _ hT
+
+ 
+theorem HasGoodFiltration.tensor {A B : FDRep k (SL2 k)}
+    (hA : HasGoodFiltration k A) (hB : HasGoodFiltration k B) :
+    HasGoodFiltration k (A ⊗ B) := by
+  induction hB with
+  | zero B hB => exact .zero _ ((tensorLeft A).map_isZero hB)
+  | @step B C n f g w hS hB ih =>
+    let := tensorLeft_preservesFiniteLimits A
+    let := tensorLeft_preservesFiniteColimits A
+    have hT := hS.map_of_exact (tensorLeft A)
+    exact ih.extension k (hA.tensor_costandard k n) _ _ _ hT
+
+noncomputable def fdrepBidual (A : FDRep k (SL2 k)) : A ≅ (Aᘁ)ᘁ :=
+  (rightDualIso
+    (@HasRightDual.exact _ _ _ (Aᘁ) (RightRigidCategory.rightDual (Aᘁ)))
+    (BraidedCategory.exactPairing_swap A (Aᘁ))).symm
+
+lemma IsRational.of_iso {A B : FDRep k (SL2 k)}
+    (h : IsRational k A.ρ) (e : A ≅ B) : IsRational k B.ρ :=
+  h.of_surjective k A.ρ B.ρ (fdrepLinear k e.hom) (fdrep_surjective k e.hom)
+    (fdrepLinear_comm k e.hom)
+
+lemma IsTilting.of_iso {A B : FDRep k (SL2 k)}
+    (h : IsTilting k A) (e : A ≅ B) : IsTilting k B := by
+  refine ⟨h.1.of_iso k e, h.2.1.of_iso k e, ?_⟩
+  let eD : Aᘁ ≅ Bᘁ :=
+    { hom := e.invᘁ
+      inv := e.homᘁ
+      hom_inv_id := by rw [← comp_rightAdjointMate, e.hom_inv_id, rightAdjointMate_id]
+      inv_hom_id := by rw [← comp_rightAdjointMate, e.inv_hom_id, rightAdjointMate_id] }
+  exact h.2.2.of_iso k eD
+
+ 
+theorem IsTilting.tensor {A B : FDRep k (SL2 k)}
+    (hA : IsTilting k A) (hB : IsTilting k B) : IsTilting k (A ⊗ B) := by
+  refine ⟨IsRational.tensor k A.ρ B.ρ hA.1 hB.1, hA.2.1.tensor k hB.2.1, ?_⟩
+  exact (hB.2.2.tensor k hA.2.2).of_iso k (rightDualTensorIso A B).symm
+
+ 
+theorem IsTilting.dual {A : FDRep k (SL2 k)} (hA : IsTilting k A) : IsTilting k Aᘁ :=
+  ⟨by rw [fdrep_dual_repr]; exact IsRational.dual k A.ρ hA.1,
+   hA.2.2, hA.2.1.of_iso k (fdrepBidual k A)⟩
+
+end FiberFunctorProof
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/RestrictedTilting/BoundedOrigin.lean b/lean/OAI/RepresentationTheory/RestrictedTilting/BoundedOrigin.lean
index 08214849af0c464dd4601be4eacfb910c4a02bd0..0a1a621ade575ff9acdfe2398127a57e471e6884 100644
--- a/lean/OAI/RepresentationTheory/RestrictedTilting/BoundedOrigin.lean
+++ b/lean/OAI/RepresentationTheory/RestrictedTilting/BoundedOrigin.lean
@@ -1,53 +1,53 @@
-import OAI.RepresentationTheory.RestrictedTilting.AssociatedBundle
-import OAI.RepresentationTheory.RestrictedTilting.AssociatedFibre
-import OAI.RepresentationTheory.RestrictedTilting.BoundedWeightsFactors
-import OAI.RepresentationTheory.RestrictedTilting.FiniteSubgroupLift
-import OAI.RepresentationTheory.RestrictedTilting.CoordinateTensor
-
-namespace OAI
-
-namespace FiberFunctorProof
-universe u
-
- 
-
-theorem bounded_origin : BoundedOriginTheorem.{u} := by
-  intro k _ _ p _ _ r hr l hl D
-  classical
-  let ι := ULift.{u} (Fin (Module.finrank k D))
-  let f : D ≃ₗ[k] (ι → k) := ((Module.finBasis k D).reindex Equiv.ulift.symm).equivFun
-  obtain ⟨ρ,P,hP,hd,hρ⟩ := finite_subgroup_polynomial_lift k l hl
-    (coordinateRepresentation k D.ρ f)
-  have hc : Fintype.card (Fin r → ZMod p) = p^r := by
-    simp only [Fintype.card_fun, Fintype.card_fin, ZMod.card]
-  rw [hc] at hd
-  let B := associatedPlaneBundle k ι ρ P hP (p^r-1) hd
-  let E := associatedE1Equiv k ι ρ P hP (p^r-1) hd
-  refine ⟨B, ⟨E.trans f.symm, ?_⟩, ?_⟩
-  · intro g v
-    apply f.injective
-    change f (f.symm (E ((planeFibreRelations (k := k) _ ![1,0]).mkQ
-      (associatedSectionsRepresentation k ι ρ (unipotent k (l g.toAdd)) v)))) =
-      f (D.ρ g (f.symm (E ((planeFibreRelations (k := k) _ ![1,0]).mkQ v))))
-    rw [f.apply_symm_apply]
-    have h := associatedE1Equiv_unipotent k ι ρ P hP (p^r-1) hd (l g.toAdd) v
-    change E _ = _ at h
-    rw [h, hρ]
-    rfl
-  · have originData (R₀ : Submodule k (associatedSectionsSubmodule k ι ρ))
-        (he : R₀ = associatedOriginRelations k ι ρ) :
-        Module.Finite k ((associatedSectionsSubmodule k ι ρ) ⧸ R₀) ∧
-        ∃ σ₀ : Representation k (SL2 k) ((associatedSectionsSubmodule k ι ρ) ⧸ R₀),
-          (∀ g v, σ₀ g (R₀.mkQ v) =
-            R₀.mkQ (associatedSectionsRepresentation k ι ρ g v)) ∧
-          IsRational k σ₀ ∧ AllSimpleFactorsHighestBelow σ₀ (p^r) := by
-      subst R₀
-      refine ⟨associatedOrigin_finite k ι ρ P hP (p^r-1) hd,
-        (associatedOriginRep k ι ρ).ρ, (fun _ _ => rfl), associatedOrigin_rational k ι ρ, ?_⟩
-      exact allSimpleFactorsHighestBelow_of_weights k _ (associatedOrigin_rational k ι ρ)
-        (p^r-1) (p^r) (Nat.sub_lt (Nat.pow_pos (Fact.out : p.Prime).pos) (by decide))
-        (associatedOrigin_weights k ι ρ P hP (p^r-1) hd)
-    exact originData _ (associatedPlaneBundle_originRelation k ι ρ P hP (p^r-1) hd)
-end FiberFunctorProof
-
-end OAI
+import OAI.RepresentationTheory.RestrictedTilting.AssociatedBundle
+import OAI.RepresentationTheory.RestrictedTilting.AssociatedFibre
+import OAI.RepresentationTheory.RestrictedTilting.BoundedWeightsFactors
+import OAI.RepresentationTheory.RestrictedTilting.FiniteSubgroupLift
+import OAI.RepresentationTheory.RestrictedTilting.CoordinateTensor
+
+namespace OAI
+
+namespace FiberFunctorProof
+universe u
+
+ 
+
+theorem bounded_origin : BoundedOriginTheorem.{u} := by
+  intro k _ _ p _ _ r hr l hl D
+  classical
+  let ι := ULift.{u} (Fin (Module.finrank k D))
+  let f : D ≃ₗ[k] (ι → k) := ((Module.finBasis k D).reindex Equiv.ulift.symm).equivFun
+  obtain ⟨ρ,P,hP,hd,hρ⟩ := finite_subgroup_polynomial_lift k l hl
+    (coordinateRepresentation k D.ρ f)
+  have hc : Fintype.card (Fin r → ZMod p) = p^r := by
+    simp only [Fintype.card_fun, Fintype.card_fin, ZMod.card]
+  rw [hc] at hd
+  let B := associatedPlaneBundle k ι ρ P hP (p^r-1) hd
+  let E := associatedE1Equiv k ι ρ P hP (p^r-1) hd
+  refine ⟨B, ⟨E.trans f.symm, ?_⟩, ?_⟩
+  · intro g v
+    apply f.injective
+    change f (f.symm (E ((planeFibreRelations (k := k) _ ![1,0]).mkQ
+      (associatedSectionsRepresentation k ι ρ (unipotent k (l g.toAdd)) v)))) =
+      f (D.ρ g (f.symm (E ((planeFibreRelations (k := k) _ ![1,0]).mkQ v))))
+    rw [f.apply_symm_apply]
+    have h := associatedE1Equiv_unipotent k ι ρ P hP (p^r-1) hd (l g.toAdd) v
+    change E _ = _ at h
+    rw [h, hρ]
+    rfl
+  · have originData (R₀ : Submodule k (associatedSectionsSubmodule k ι ρ))
+        (he : R₀ = associatedOriginRelations k ι ρ) :
+        Module.Finite k ((associatedSectionsSubmodule k ι ρ) ⧸ R₀) ∧
+        ∃ σ₀ : Representation k (SL2 k) ((associatedSectionsSubmodule k ι ρ) ⧸ R₀),
+          (∀ g v, σ₀ g (R₀.mkQ v) =
+            R₀.mkQ (associatedSectionsRepresentation k ι ρ g v)) ∧
+          IsRational k σ₀ ∧ AllSimpleFactorsHighestBelow σ₀ (p^r) := by
+      subst R₀
+      refine ⟨associatedOrigin_finite k ι ρ P hP (p^r-1) hd,
+        (associatedOriginRep k ι ρ).ρ, (fun _ _ => rfl), associatedOrigin_rational k ι ρ, ?_⟩
+      exact allSimpleFactorsHighestBelow_of_weights k _ (associatedOrigin_rational k ι ρ)
+        (p^r-1) (p^r) (Nat.sub_lt (Nat.pow_pos (Fact.out : p.Prime).pos) (by decide))
+        (associatedOrigin_weights k ι ρ P hP (p^r-1) hd)
+    exact originData _ (associatedPlaneBundle_originRelation k ι ρ P hP (p^r-1) hd)
+end FiberFunctorProof
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/RestrictedTilting/BoundedOriginTarget.lean b/lean/OAI/RepresentationTheory/RestrictedTilting/BoundedOriginTarget.lean
index 47a7dd3f254f99e7f35a1c78c0d0a592c68caecb..f95b115c363f00db1f4c559cd7ce672eca35d649 100644
--- a/lean/OAI/RepresentationTheory/RestrictedTilting/BoundedOriginTarget.lean
+++ b/lean/OAI/RepresentationTheory/RestrictedTilting/BoundedOriginTarget.lean
@@ -1,80 +1,80 @@
-import OAI.RepresentationTheory.RestrictedTilting.Basic
-
-namespace OAI
-
-namespace FiberFunctorProof
-open CategoryTheory
-open scoped Pointwise
-universe u
-noncomputable section
-
- 
-
-structure EquivariantPlaneBundle (k : Type u) [Field k] where
-  Sections : Type u
-  [addCommGroup : AddCommGroup Sections]
-  [fieldModule : Module k Sections]
-  [planeModule : Module (MvPolynomial (Fin 2) k) Sections]
-  [scalarTower : IsScalarTower k (MvPolynomial (Fin 2) k) Sections]
-  finite : Module.Finite (MvPolynomial (Fin 2) k) Sections
-  projective : Module.Projective (MvPolynomial (Fin 2) k) Sections
-  action : Representation k (SL2 k) Sections
-  rational : IsRational k action
-  compatible : PlaneCompatible k action
-
-attribute [instance] EquivariantPlaneBundle.addCommGroup EquivariantPlaneBundle.fieldModule
-  EquivariantPlaneBundle.planeModule EquivariantPlaneBundle.scalarTower
-
-variable {k : Type u} [Field k]
-
- 
-
-def planePointIdeal (a : Fin 2 → k) : Ideal (MvPolynomial (Fin 2) k) :=
-  RingHom.ker (MvPolynomial.eval a)
-
-def EquivariantPlaneBundle.fibreRelation (B : EquivariantPlaneBundle k) (a : Fin 2 → k) :
-    Submodule k B.Sections :=
-  ((planePointIdeal a • (⊤ : Submodule (MvPolynomial (Fin 2) k) B.Sections))).restrictScalars k
-
-abbrev EquivariantPlaneBundle.fibre (B : EquivariantPlaneBundle k) (a : Fin 2 → k) :=
-  B.Sections ⧸ B.fibreRelation a
-
- 
-
-def HasHighestWeight {V : Type u} [AddCommGroup V] [Module k V]
-    (ρ : Representation k (SL2 k) V) (n : ℕ) : Prop :=
-  HasWeightsBetween ρ (-(n : ℤ)) n ∧ ∃ v : V, v ≠ 0 ∧
-    (∀ a, ρ (diagonalTorus k a) v = integerCharacter k n a • v) ∧
-    ∀ t, ρ (unipotent k t) v = v
-
- 
-
-def AllSimpleFactorsHighestBelow {M : Type u} [AddCommGroup M] [Module k M]
-    (ρ : Representation k (SL2 k) M) (q : ℕ) : Prop :=
-  ∀ (A B : FDRep k (SL2 k)) [Representation.IsIrreducible B.ρ]
-    (i : A →ₗ[k] M) (π : A ⟶ B), Function.Injective i →
-    Function.Surjective π.hom.hom.hom →
-    (∀ g v, i (A.ρ g v) = ρ g (i v)) →
-    ∃ n : ℕ, n < q ∧ HasHighestWeight B.ρ n
-
- 
-
-def BoundedOriginTheorem : Prop :=
-  ∀ (k : Type u) [Field k] [IsAlgClosed k] (p : ℕ) [Fact p.Prime] [CharP k p]
-    (r : ℕ), 1 ≤ r → ∀ (l : (Fin r → ZMod p) →+ k), Function.Injective l →
-    ∀ (D : FDRep k (Multiplicative (Fin r → ZMod p))),
-    ∃ B : EquivariantPlaneBundle k,
-      (∃ e : B.fibre ![1,0] ≃ₗ[k] D,
-        ∀ (g : Multiplicative (Fin r → ZMod p)) (v : B.Sections),
-          e ((B.fibreRelation ![1,0]).mkQ (B.action (elementaryEmbedding k p r l g) v)) =
-            D.ρ g (e ((B.fibreRelation ![1,0]).mkQ v))) ∧
-      Module.Finite k (B.fibre 0) ∧
-      ∃ σ₀ : Representation k (SL2 k) (B.fibre 0),
-        (∀ g v, σ₀ g ((B.fibreRelation 0).mkQ v) =
-          (B.fibreRelation 0).mkQ (B.action g v)) ∧
-        IsRational k σ₀ ∧ AllSimpleFactorsHighestBelow σ₀ (p^r)
-
-end
-end FiberFunctorProof
-
-end OAI
+import OAI.RepresentationTheory.RestrictedTilting.Basic
+
+namespace OAI
+
+namespace FiberFunctorProof
+open CategoryTheory
+open scoped Pointwise
+universe u
+noncomputable section
+
+ 
+
+structure EquivariantPlaneBundle (k : Type u) [Field k] where
+  Sections : Type u
+  [addCommGroup : AddCommGroup Sections]
+  [fieldModule : Module k Sections]
+  [planeModule : Module (MvPolynomial (Fin 2) k) Sections]
+  [scalarTower : IsScalarTower k (MvPolynomial (Fin 2) k) Sections]
+  finite : Module.Finite (MvPolynomial (Fin 2) k) Sections
+  projective : Module.Projective (MvPolynomial (Fin 2) k) Sections
+  action : Representation k (SL2 k) Sections
+  rational : IsRational k action
+  compatible : PlaneCompatible k action
+
+attribute [instance] EquivariantPlaneBundle.addCommGroup EquivariantPlaneBundle.fieldModule
+  EquivariantPlaneBundle.planeModule EquivariantPlaneBundle.scalarTower
+
+variable {k : Type u} [Field k]
+
+ 
+
+def planePointIdeal (a : Fin 2 → k) : Ideal (MvPolynomial (Fin 2) k) :=
+  RingHom.ker (MvPolynomial.eval a)
+
+def EquivariantPlaneBundle.fibreRelation (B : EquivariantPlaneBundle k) (a : Fin 2 → k) :
+    Submodule k B.Sections :=
+  ((planePointIdeal a • (⊤ : Submodule (MvPolynomial (Fin 2) k) B.Sections))).restrictScalars k
+
+abbrev EquivariantPlaneBundle.fibre (B : EquivariantPlaneBundle k) (a : Fin 2 → k) :=
+  B.Sections ⧸ B.fibreRelation a
+
+ 
+
+def HasHighestWeight {V : Type u} [AddCommGroup V] [Module k V]
+    (ρ : Representation k (SL2 k) V) (n : ℕ) : Prop :=
+  HasWeightsBetween ρ (-(n : ℤ)) n ∧ ∃ v : V, v ≠ 0 ∧
+    (∀ a, ρ (diagonalTorus k a) v = integerCharacter k n a • v) ∧
+    ∀ t, ρ (unipotent k t) v = v
+
+ 
+
+def AllSimpleFactorsHighestBelow {M : Type u} [AddCommGroup M] [Module k M]
+    (ρ : Representation k (SL2 k) M) (q : ℕ) : Prop :=
+  ∀ (A B : FDRep k (SL2 k)) [Representation.IsIrreducible B.ρ]
+    (i : A →ₗ[k] M) (π : A ⟶ B), Function.Injective i →
+    Function.Surjective π.hom.hom.hom →
+    (∀ g v, i (A.ρ g v) = ρ g (i v)) →
+    ∃ n : ℕ, n < q ∧ HasHighestWeight B.ρ n
+
+ 
+
+def BoundedOriginTheorem : Prop :=
+  ∀ (k : Type u) [Field k] [IsAlgClosed k] (p : ℕ) [Fact p.Prime] [CharP k p]
+    (r : ℕ), 1 ≤ r → ∀ (l : (Fin r → ZMod p) →+ k), Function.Injective l →
+    ∀ (D : FDRep k (Multiplicative (Fin r → ZMod p))),
+    ∃ B : EquivariantPlaneBundle k,
+      (∃ e : B.fibre ![1,0] ≃ₗ[k] D,
+        ∀ (g : Multiplicative (Fin r → ZMod p)) (v : B.Sections),
+          e ((B.fibreRelation ![1,0]).mkQ (B.action (elementaryEmbedding k p r l g) v)) =
+            D.ρ g (e ((B.fibreRelation ![1,0]).mkQ v))) ∧
+      Module.Finite k (B.fibre 0) ∧
+      ∃ σ₀ : Representation k (SL2 k) (B.fibre 0),
+        (∀ g v, σ₀ g ((B.fibreRelation 0).mkQ v) =
+          (B.fibreRelation 0).mkQ (B.action g v)) ∧
+        IsRational k σ₀ ∧ AllSimpleFactorsHighestBelow σ₀ (p^r)
+
+end
+end FiberFunctorProof
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/RestrictedTilting/BoundedStableFaithfulness.lean b/lean/OAI/RepresentationTheory/RestrictedTilting/BoundedStableFaithfulness.lean
index feb1c2426834b7e2eceeba641059e22992d8a38b..11789d7b050a4d3f9d977fd1d71163a1b297acd3 100644
--- a/lean/OAI/RepresentationTheory/RestrictedTilting/BoundedStableFaithfulness.lean
+++ b/lean/OAI/RepresentationTheory/RestrictedTilting/BoundedStableFaithfulness.lean
@@ -1,149 +1,149 @@
-import OAI.RepresentationTheory.RestrictedTilting.Basic
-import OAI.RepresentationTheory.RestrictedTilting.LowDegreeTrace
-import OAI.RepresentationTheory.RestrictedTilting.SmallGroupTrace
-import OAI.RepresentationTheory.RestrictedTilting.TorusWeightProjection
-import OAI.RepresentationTheory.RestrictedTilting.RationalHomWeights
-
-namespace OAI
-
-namespace FiberFunctorProof
-open scoped BigOperators
-open CategoryTheory Representation
-universe u v
-variable {k : Type u} [Field k] [Infinite k]
-variable {E : Type v} [AddGroup E] [Fintype E] [DecidableEq E]
-variable {V : Type u} [AddCommGroup V] [Module k V] [Module.Finite k V]
-
- 
-
-theorem bounded_rational_norm_zero (ρ : Representation k (SL2 k) V)
-    (hρ : IsRational k ρ) (N : ℕ) (hN : HasWeightsBetween ρ (-(N : ℤ)) N)
-    (l : E →+ k) (hl : Function.Injective l) (hE : N < Fintype.card E - 1) (v : V) :
-    ∑ e : E, ρ (unipotent k (l e)) v = 0 := by
-  classical
-  have heigen (n : ℤ) (w : V)
-      (hw : ∀ a, ρ (diagonalTorus k a) w = integerCharacter k n a • w) :
-      ∑ e : E, ρ (unipotent k (l e)) w = 0 := by
-    by_cases hz : w = 0
-    · simp [hz]
-    have hn : -(N : ℤ) ≤ n := by
-      by_contra h
-      exact hz (hN n w (Or.inl (by omega)) hw)
-    obtain ⟨c, hc⟩ := rational_unipotent_expansion k ρ hρ w
-    have hcz (i : ℕ) (hi : N < i) : c i = 0 :=
-      hN (n + 2 * i) (c i) (Or.inr (by omega))
-        (unipotent_coefficient_weight k ρ w n hw c hc i)
-    simp_rw [hc, Finsupp.sum]
-    rw [Finset.sum_comm]
-    apply Finset.sum_eq_zero
-    intro i hi
-    by_cases hiN : i ≤ N
-    · rw [← Finset.sum_smul, finite_additive_power_sum_zero l hl i (lt_of_le_of_lt hiN hE),
-        zero_smul]
-    · simp [hcz i (by omega)]
-  obtain ⟨c, hc⟩ := rational_torus_integerExpansions k ρ hρ v
-  have hs : v = ∑ n ∈ c.support, c n := by
-    simpa [Finsupp.sum, integerCharacter] using hc 1
-  rw [hs]
-  simp_rw [map_sum]
-  rw [Finset.sum_comm]
-  apply Finset.sum_eq_zero
-  intro n hn
-  exact heigen n (c n) (integer_coefficient_eigen k ρ v c hc n)
-
- 
-
-theorem bounded_norm_zero_weight (ρ : Representation k (SL2 k) V)
-    (hρ : IsRational k ρ) (N : ℕ) (hN : HasWeightsBetween ρ (-(N : ℤ)) N)
-    (l : E →+ k) (hl : Function.Injective l)
-    (hE : N < 2 * (Fintype.card E - 1)) (v : V) :
-    torusWeightProjection ρ hρ 0 (∑ e : E, ρ (unipotent k (l e)) v) = 0 := by
-  classical
-  have heigen (n : ℤ) (w : V)
-      (hw : ∀ a, ρ (diagonalTorus k a) w = integerCharacter k n a • w) :
-      torusWeightProjection ρ hρ 0 (∑ e : E, ρ (unipotent k (l e)) w) = 0 := by
-    by_cases hz : w = 0
-    · simp [hz]
-    have hn : -(N : ℤ) ≤ n := by
-      by_contra h
-      exact hz (hN n w (Or.inl (by omega)) hw)
-    obtain ⟨c, hc⟩ := rational_unipotent_expansion k ρ hρ w
-    simp_rw [hc, Finsupp.sum, map_sum, map_smul]
-    rw [Finset.sum_comm]
-    apply Finset.sum_eq_zero
-    intro i hi
-    by_cases hiE : i < Fintype.card E - 1
-    · rw [← Finset.sum_smul, finite_additive_power_sum_zero l hl i hiE, zero_smul]
-    · have hni : n + 2 * (i : ℤ) ≠ 0 := by omega
-      rw [torusWeightProjection_eigen ρ hρ 0 (n+2*i) (c i)
-        (unipotent_coefficient_weight k ρ w n hw c hc i), ite_eq_right hni]
-      simp
-  obtain ⟨c, hc⟩ := rational_torus_integerExpansions k ρ hρ v
-  have hs : v = ∑ n ∈ c.support, c n := by
-    simpa [Finsupp.sum, integerCharacter] using hc 1
-  rw [hs]
-  simp_rw [map_sum]
-  rw [Finset.sum_comm]
-  apply Finset.sum_eq_zero
-  intro n hn
-  have h := heigen n (c n) (integer_coefficient_eigen k ρ v c hc n)
-  simpa only [map_sum] using h
-
- 
-theorem bounded_fixed_norm_zero (ρ : Representation k (SL2 k) V)
-    (hρ : IsRational k ρ) (N : ℕ) (hN : HasWeightsBetween ρ (-(N : ℤ)) N)
-    (l : E →+ k) (hl : Function.Injective l)
-    (hE : N < 2 * (Fintype.card E - 1)) (v : V)
-    (hv : ∀ a, ρ (diagonalTorus k a) (∑ e : E, ρ (unipotent k (l e)) v) =
-      ∑ e : E, ρ (unipotent k (l e)) v) :
-    ∑ e : E, ρ (unipotent k (l e)) v = 0 := by
-  rw [← torusWeightProjection_zero_fixed ρ hρ _ hv]
-  exact bounded_norm_zero_weight ρ hρ N hN l hl hE v
-
- 
-
-theorem low_weight_projective_factor_zero (p r : ℕ) [NeZero p]
-    (l : (Fin r → ZMod p) →+ k) (hl : Function.Injective l)
-    (A B : FDRep k (SL2 k)) (hA : IsRational k A.ρ) (hB : IsRational k B.ρ)
-    (a b : ℕ) (ha : HasWeightsBetween A.ρ (-(a : ℤ)) a)
-    (hb : HasWeightsBetween B.ρ (-(b : ℤ)) b) (hab : a+b < 2*(p^r-1))
-    (f : A ⟶ B) (hf : FactorsThroughProjective ((restrictElementary k p r l).map f)) :
-    f = 0 := by
-  classical
-  let R := restrictElementary k p r l
-  obtain ⟨t, ht⟩ := SmallGroupTrace.relativeTrace_of_factors (R.obj A) (R.obj B) (R.map f) hf
-  let ρ := Representation.linHom A.ρ B.ρ
-  have he : (∑ e : Fin r → ZMod p, ρ (unipotent k (l e)) t) = f.hom.hom.hom := by
-    change relativeTrace (R.obj A).ρ (R.obj B).ρ t = f.hom.hom.hom at ht
-    rw [← ht]
-    simp only [relativeTrace, Representation.norm, LinearMap.sum_apply]
-    change _ = ∑ e : Multiplicative (Fin r → ZMod p),
-      (Representation.linHom (R.obj A).ρ (R.obj B).ρ) e t
-    rw [← Equiv.sum_comp Multiplicative.ofAdd]
-    apply Finset.sum_congr rfl
-    intro e he
-    change B.ρ (unipotent k (l e)) ∘ₗ t ∘ₗ A.ρ (unipotent k (l e))⁻¹ =
-      B.ρ (unipotent k (l e)) ∘ₗ t ∘ₗ A.ρ (unipotent k (l (-e)))
-    congr 2
-    have h := map_inv (elementaryEmbedding k p r l) (Multiplicative.ofAdd e)
-    exact congrArg A.ρ h.symm
-  have hr : IsRational k ρ := IsRational.linHom A.ρ B.ρ hA hB
-  have hw : HasWeightsBetween ρ (-((a+b : ℕ) : ℤ)) (a+b) := by
-    simpa only [Nat.cast_add] using hasWeightsBetween_linHom A.ρ B.ρ hA a b ha hb
-  have hz := bounded_fixed_norm_zero ρ hr (a+b) hw l hl
-    (by simpa [Fintype.card_fun, ZMod.card] using hab) t (fun u => by
-      rw [he]
-      change B.ρ (diagonalTorus k u) ∘ₗ f.hom.hom.hom ∘ₗ A.ρ (diagonalTorus k u)⁻¹ =
-        f.hom.hom.hom
-      ext x
-      change B.ρ (diagonalTorus k u) (f.hom.hom.hom (A.ρ (diagonalTorus k u)⁻¹ x)) = _
-      rw [← fdrepLinear_comm k f]
-      simp)
-  rw [he] at hz
-  apply Action.Hom.ext
-  apply InducedCategory.hom_ext
-  apply ModuleCat.hom_ext
-  exact hz
-end FiberFunctorProof
-
-end OAI
+import OAI.RepresentationTheory.RestrictedTilting.Basic
+import OAI.RepresentationTheory.RestrictedTilting.LowDegreeTrace
+import OAI.RepresentationTheory.RestrictedTilting.SmallGroupTrace
+import OAI.RepresentationTheory.RestrictedTilting.TorusWeightProjection
+import OAI.RepresentationTheory.RestrictedTilting.RationalHomWeights
+
+namespace OAI
+
+namespace FiberFunctorProof
+open scoped BigOperators
+open CategoryTheory Representation
+universe u v
+variable {k : Type u} [Field k] [Infinite k]
+variable {E : Type v} [AddGroup E] [Fintype E] [DecidableEq E]
+variable {V : Type u} [AddCommGroup V] [Module k V] [Module.Finite k V]
+
+ 
+
+theorem bounded_rational_norm_zero (ρ : Representation k (SL2 k) V)
+    (hρ : IsRational k ρ) (N : ℕ) (hN : HasWeightsBetween ρ (-(N : ℤ)) N)
+    (l : E →+ k) (hl : Function.Injective l) (hE : N < Fintype.card E - 1) (v : V) :
+    ∑ e : E, ρ (unipotent k (l e)) v = 0 := by
+  classical
+  have heigen (n : ℤ) (w : V)
+      (hw : ∀ a, ρ (diagonalTorus k a) w = integerCharacter k n a • w) :
+      ∑ e : E, ρ (unipotent k (l e)) w = 0 := by
+    by_cases hz : w = 0
+    · simp [hz]
+    have hn : -(N : ℤ) ≤ n := by
+      by_contra h
+      exact hz (hN n w (Or.inl (by omega)) hw)
+    obtain ⟨c, hc⟩ := rational_unipotent_expansion k ρ hρ w
+    have hcz (i : ℕ) (hi : N < i) : c i = 0 :=
+      hN (n + 2 * i) (c i) (Or.inr (by omega))
+        (unipotent_coefficient_weight k ρ w n hw c hc i)
+    simp_rw [hc, Finsupp.sum]
+    rw [Finset.sum_comm]
+    apply Finset.sum_eq_zero
+    intro i hi
+    by_cases hiN : i ≤ N
+    · rw [← Finset.sum_smul, finite_additive_power_sum_zero l hl i (lt_of_le_of_lt hiN hE),
+        zero_smul]
+    · simp [hcz i (by omega)]
+  obtain ⟨c, hc⟩ := rational_torus_integerExpansions k ρ hρ v
+  have hs : v = ∑ n ∈ c.support, c n := by
+    simpa [Finsupp.sum, integerCharacter] using hc 1
+  rw [hs]
+  simp_rw [map_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_eq_zero
+  intro n hn
+  exact heigen n (c n) (integer_coefficient_eigen k ρ v c hc n)
+
+ 
+
+theorem bounded_norm_zero_weight (ρ : Representation k (SL2 k) V)
+    (hρ : IsRational k ρ) (N : ℕ) (hN : HasWeightsBetween ρ (-(N : ℤ)) N)
+    (l : E →+ k) (hl : Function.Injective l)
+    (hE : N < 2 * (Fintype.card E - 1)) (v : V) :
+    torusWeightProjection ρ hρ 0 (∑ e : E, ρ (unipotent k (l e)) v) = 0 := by
+  classical
+  have heigen (n : ℤ) (w : V)
+      (hw : ∀ a, ρ (diagonalTorus k a) w = integerCharacter k n a • w) :
+      torusWeightProjection ρ hρ 0 (∑ e : E, ρ (unipotent k (l e)) w) = 0 := by
+    by_cases hz : w = 0
+    · simp [hz]
+    have hn : -(N : ℤ) ≤ n := by
+      by_contra h
+      exact hz (hN n w (Or.inl (by omega)) hw)
+    obtain ⟨c, hc⟩ := rational_unipotent_expansion k ρ hρ w
+    simp_rw [hc, Finsupp.sum, map_sum, map_smul]
+    rw [Finset.sum_comm]
+    apply Finset.sum_eq_zero
+    intro i hi
+    by_cases hiE : i < Fintype.card E - 1
+    · rw [← Finset.sum_smul, finite_additive_power_sum_zero l hl i hiE, zero_smul]
+    · have hni : n + 2 * (i : ℤ) ≠ 0 := by omega
+      rw [torusWeightProjection_eigen ρ hρ 0 (n+2*i) (c i)
+        (unipotent_coefficient_weight k ρ w n hw c hc i), ite_eq_right hni]
+      simp
+  obtain ⟨c, hc⟩ := rational_torus_integerExpansions k ρ hρ v
+  have hs : v = ∑ n ∈ c.support, c n := by
+    simpa [Finsupp.sum, integerCharacter] using hc 1
+  rw [hs]
+  simp_rw [map_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_eq_zero
+  intro n hn
+  have h := heigen n (c n) (integer_coefficient_eigen k ρ v c hc n)
+  simpa only [map_sum] using h
+
+ 
+theorem bounded_fixed_norm_zero (ρ : Representation k (SL2 k) V)
+    (hρ : IsRational k ρ) (N : ℕ) (hN : HasWeightsBetween ρ (-(N : ℤ)) N)
+    (l : E →+ k) (hl : Function.Injective l)
+    (hE : N < 2 * (Fintype.card E - 1)) (v : V)
+    (hv : ∀ a, ρ (diagonalTorus k a) (∑ e : E, ρ (unipotent k (l e)) v) =
+      ∑ e : E, ρ (unipotent k (l e)) v) :
+    ∑ e : E, ρ (unipotent k (l e)) v = 0 := by
+  rw [← torusWeightProjection_zero_fixed ρ hρ _ hv]
+  exact bounded_norm_zero_weight ρ hρ N hN l hl hE v
+
+ 
+
+theorem low_weight_projective_factor_zero (p r : ℕ) [NeZero p]
+    (l : (Fin r → ZMod p) →+ k) (hl : Function.Injective l)
+    (A B : FDRep k (SL2 k)) (hA : IsRational k A.ρ) (hB : IsRational k B.ρ)
+    (a b : ℕ) (ha : HasWeightsBetween A.ρ (-(a : ℤ)) a)
+    (hb : HasWeightsBetween B.ρ (-(b : ℤ)) b) (hab : a+b < 2*(p^r-1))
+    (f : A ⟶ B) (hf : FactorsThroughProjective ((restrictElementary k p r l).map f)) :
+    f = 0 := by
+  classical
+  let R := restrictElementary k p r l
+  obtain ⟨t, ht⟩ := SmallGroupTrace.relativeTrace_of_factors (R.obj A) (R.obj B) (R.map f) hf
+  let ρ := Representation.linHom A.ρ B.ρ
+  have he : (∑ e : Fin r → ZMod p, ρ (unipotent k (l e)) t) = f.hom.hom.hom := by
+    change relativeTrace (R.obj A).ρ (R.obj B).ρ t = f.hom.hom.hom at ht
+    rw [← ht]
+    simp only [relativeTrace, Representation.norm, LinearMap.sum_apply]
+    change _ = ∑ e : Multiplicative (Fin r → ZMod p),
+      (Representation.linHom (R.obj A).ρ (R.obj B).ρ) e t
+    rw [← Equiv.sum_comp Multiplicative.ofAdd]
+    apply Finset.sum_congr rfl
+    intro e he
+    change B.ρ (unipotent k (l e)) ∘ₗ t ∘ₗ A.ρ (unipotent k (l e))⁻¹ =
+      B.ρ (unipotent k (l e)) ∘ₗ t ∘ₗ A.ρ (unipotent k (l (-e)))
+    congr 2
+    have h := map_inv (elementaryEmbedding k p r l) (Multiplicative.ofAdd e)
+    exact congrArg A.ρ h.symm
+  have hr : IsRational k ρ := IsRational.linHom A.ρ B.ρ hA hB
+  have hw : HasWeightsBetween ρ (-((a+b : ℕ) : ℤ)) (a+b) := by
+    simpa only [Nat.cast_add] using hasWeightsBetween_linHom A.ρ B.ρ hA a b ha hb
+  have hz := bounded_fixed_norm_zero ρ hr (a+b) hw l hl
+    (by simpa [Fintype.card_fun, ZMod.card] using hab) t (fun u => by
+      rw [he]
+      change B.ρ (diagonalTorus k u) ∘ₗ f.hom.hom.hom ∘ₗ A.ρ (diagonalTorus k u)⁻¹ =
+        f.hom.hom.hom
+      ext x
+      change B.ρ (diagonalTorus k u) (f.hom.hom.hom (A.ρ (diagonalTorus k u)⁻¹ x)) = _
+      rw [← fdrepLinear_comm k f]
+      simp)
+  rw [he] at hz
+  apply Action.Hom.ext
+  apply InducedCategory.hom_ext
+  apply ModuleCat.hom_ext
+  exact hz
+end FiberFunctorProof
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/RestrictedTilting/BoundedUnipotentTilting.lean b/lean/OAI/RepresentationTheory/RestrictedTilting/BoundedUnipotentTilting.lean
index 219e12c631d53f984c980a46f5c25de9069127b3..fbe9072727dc32a5fba723ea5f4c3ee7d2cecc67 100644
--- a/lean/OAI/RepresentationTheory/RestrictedTilting/BoundedUnipotentTilting.lean
+++ b/lean/OAI/RepresentationTheory/RestrictedTilting/BoundedUnipotentTilting.lean
@@ -1,50 +1,50 @@
-import OAI.RepresentationTheory.RestrictedTilting.TensorAssociatedIso
-import OAI.RepresentationTheory.RestrictedTilting.TensorFormalComparison
-
-namespace OAI
-
-namespace FiberFunctorProof
-open CategoryTheory CategoryTheory.MonoidalCategory Matrix
-open scoped TensorProduct Kronecker
-universe u
-variable (k : Type u) [Field k] [Infinite k]
-attribute [local instance 1100] HahnSeries.instAlgebra
-variable (p : ℕ) [Fact p.Prime] [CharP k p]
-variable (β ι : Type u) [Fintype β] [DecidableEq β] [Fintype ι] [DecidableEq ι]
-variable (ρ : Representation k (Multiplicative k) (ι → k))
-variable (P : Matrix ι ι (Polynomial k))
-variable (hP : ∀ t : k, polynomialMatrixEval P t = LinearMap.toMatrix' (ρ (Multiplicative.ofAdd t)))
-variable (d : ℕ) (hd : ∀ i j, (P i j).natDegree ≤ d)
-
-include P hP hd in
- 
-
-theorem bounded_unipotent_tilting_coordinates (m : ℕ) (hm : d < p^(m+1))
-    (e : steinberg k p m ≃ₗ[k] (β → k)) :
-    ∃ T : FDRep k (SL2 k), IsTilting k T ∧
-      ∃ E : T ≃ₗ[k] (β × ι → k), ∀ t : k, ∀ v : T,
-        E (T.ρ (unipotent k t) v) =
-          (LinearMap.toMatrix' (coordinateRepresentation k (steinberg k p m).ρ e (unipotent k t)) ⊗ₖ
-            LinearMap.toMatrix' (ρ (Multiplicative.ofAdd t))) *ᵥ E v := by
-  let N := associatedSectionsSubmodule k ι ρ
-  let τ := coordinateRepresentation k (steinberg k p m).ρ e
-  let σ := matrixTensorRepresentation τ (associatedSectionsRepresentation k ι ρ)
-  have hσ : PlaneCompatible k σ := matrixTensor_planeCompatible k τ _ (associatedSections_planeCompatible k ι ρ)
-  let := associated_planeJet_one_finite k ι ρ P hP d hd
-  let := piPlaneOrigin_finite (β := β) (M := N) k
-  let T := planeJetFD k σ hσ 1
-  have hT : IsTilting k T := tensorAssociatedOrigin_tilting k ι ρ P hP d hd p m hm e
-  have hτ : IsRational k τ := coordinateRepresentation_rational k _ e (steinberg_tilting k p m).1
-  have hr : IsRational k σ := matrixTensor_rational k τ _ hτ (associatedSections_rational k ι ρ)
-  have hx := pi_regular_x k (β := β) (And.left (associatedSections_regular k ι ρ P hP))
-  have hy := pi_regular_y k (β := β) (And.right (associatedSections_regular k ι ρ P hP))
-  obtain ⟨s,hs0,hs⟩ := compatible_planeJet_sections k σ hσ hr hx hy hT
-  let F := tensorAssociatedFieldTrivialization k β ι ρ P hP d hd τ s hs0 hs
-  obtain ⟨E,hE⟩ := tensor_linear_formal_comparison T.ρ hT.1 τ hτ P
-    (formalLineParameter k) (formalLineParameter_ne_zero k) F
-    (tensorAssociatedFieldTrivialization_unipotent_one k β ι ρ P hP d hd τ s hs0 hs)
-  refine ⟨T,hT,E, fun t v => ?_⟩
-  simpa only [hP] using hE t v
-end FiberFunctorProof
-
-end OAI
+import OAI.RepresentationTheory.RestrictedTilting.TensorAssociatedIso
+import OAI.RepresentationTheory.RestrictedTilting.TensorFormalComparison
+
+namespace OAI
+
+namespace FiberFunctorProof
+open CategoryTheory CategoryTheory.MonoidalCategory Matrix
+open scoped TensorProduct Kronecker
+universe u
+variable (k : Type u) [Field k] [Infinite k]
+attribute [local instance 1100] HahnSeries.instAlgebra
+variable (p : ℕ) [Fact p.Prime] [CharP k p]
+variable (β ι : Type u) [Fintype β] [DecidableEq β] [Fintype ι] [DecidableEq ι]
+variable (ρ : Representation k (Multiplicative k) (ι → k))
+variable (P : Matrix ι ι (Polynomial k))
+variable (hP : ∀ t : k, polynomialMatrixEval P t = LinearMap.toMatrix' (ρ (Multiplicative.ofAdd t)))
+variable (d : ℕ) (hd : ∀ i j, (P i j).natDegree ≤ d)
+
+include P hP hd in
+ 
+
+theorem bounded_unipotent_tilting_coordinates (m : ℕ) (hm : d < p^(m+1))
+    (e : steinberg k p m ≃ₗ[k] (β → k)) :
+    ∃ T : FDRep k (SL2 k), IsTilting k T ∧
+      ∃ E : T ≃ₗ[k] (β × ι → k), ∀ t : k, ∀ v : T,
+        E (T.ρ (unipotent k t) v) =
+          (LinearMap.toMatrix' (coordinateRepresentation k (steinberg k p m).ρ e (unipotent k t)) ⊗ₖ
+            LinearMap.toMatrix' (ρ (Multiplicative.ofAdd t))) *ᵥ E v := by
+  let N := associatedSectionsSubmodule k ι ρ
+  let τ := coordinateRepresentation k (steinberg k p m).ρ e
+  let σ := matrixTensorRepresentation τ (associatedSectionsRepresentation k ι ρ)
+  have hσ : PlaneCompatible k σ := matrixTensor_planeCompatible k τ _ (associatedSections_planeCompatible k ι ρ)
+  let := associated_planeJet_one_finite k ι ρ P hP d hd
+  let := piPlaneOrigin_finite (β := β) (M := N) k
+  let T := planeJetFD k σ hσ 1
+  have hT : IsTilting k T := tensorAssociatedOrigin_tilting k ι ρ P hP d hd p m hm e
+  have hτ : IsRational k τ := coordinateRepresentation_rational k _ e (steinberg_tilting k p m).1
+  have hr : IsRational k σ := matrixTensor_rational k τ _ hτ (associatedSections_rational k ι ρ)
+  have hx := pi_regular_x k (β := β) (And.left (associatedSections_regular k ι ρ P hP))
+  have hy := pi_regular_y k (β := β) (And.right (associatedSections_regular k ι ρ P hP))
+  obtain ⟨s,hs0,hs⟩ := compatible_planeJet_sections k σ hσ hr hx hy hT
+  let F := tensorAssociatedFieldTrivialization k β ι ρ P hP d hd τ s hs0 hs
+  obtain ⟨E,hE⟩ := tensor_linear_formal_comparison T.ρ hT.1 τ hτ P
+    (formalLineParameter k) (formalLineParameter_ne_zero k) F
+    (tensorAssociatedFieldTrivialization_unipotent_one k β ι ρ P hP d hd τ s hs0 hs)
+  refine ⟨T,hT,E, fun t v => ?_⟩
+  simpa only [hP] using hE t v
+end FiberFunctorProof
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/RestrictedTilting/BoundedWeightsFactors.lean b/lean/OAI/RepresentationTheory/RestrictedTilting/BoundedWeightsFactors.lean
index 14a74448c49085567e22c7c7e6db2bcd54db7e22..babc7152ed15dd4d2a607e3dd5e5162ed8340c07 100644
--- a/lean/OAI/RepresentationTheory/RestrictedTilting/BoundedWeightsFactors.lean
+++ b/lean/OAI/RepresentationTheory/RestrictedTilting/BoundedWeightsFactors.lean
@@ -1,28 +1,28 @@
-import OAI.RepresentationTheory.RestrictedTilting.BoundedOriginTarget
-
-namespace OAI
-
-namespace FiberFunctorProof
-universe u
-variable (k : Type u) [Field k] [Infinite k]
-
- 
-
-lemma allSimpleFactorsHighestBelow_of_weights {M : Type u} [AddCommGroup M] [Module k M]
-    (ρ : Representation k (SL2 k) M) (hρ : IsRational k ρ) (d q : ℕ) (hdq : d < q)
-    (hb : HasWeightsBetween ρ (-(d : ℤ)) d) : AllSimpleFactorsHighestBelow ρ q := by
-  intro A B _ i π hi hπ he
-  let : Nontrivial (Representation.asModule B.ρ) :=
-    IsSimpleModule.nontrivial (MonoidAlgebra k (SL2 k)) (Representation.asModule B.ρ)
-  let : Nontrivial B := (Representation.asModuleEquiv B.ρ).symm.toEquiv.nontrivial
-  have hA := IsRational.of_injective k A.ρ ρ i hi he hρ
-  have hB := IsRational.of_surjective k A.ρ B.ρ (fdrepLinear k π) hπ (fdrepLinear_comm k π) hA
-  have hbA := hasWeightsBetween_of_injective k A.ρ ρ i hi he _ _ hb
-  have hbB := hasWeightsBetween_of_surjective k A.ρ B.ρ hA (fdrepLinear k π) hπ
-    (fdrepLinear_comm k π) _ _ hbA
-  obtain ⟨n,hn,hwn,v,hv,hwv⟩ := exists_highest_in_bound k B.ρ hB d hbB
-  refine ⟨n,hn.trans_lt hdq,hwn,v,hv,hwv,?_⟩
-  exact highest_unipotent_fixed B.ρ hB n v hwv (fun m w hm => hwn m w (Or.inr hm))
-end FiberFunctorProof
-
-end OAI
+import OAI.RepresentationTheory.RestrictedTilting.BoundedOriginTarget
+
+namespace OAI
+
+namespace FiberFunctorProof
+universe u
+variable (k : Type u) [Field k] [Infinite k]
+
+ 
+
+lemma allSimpleFactorsHighestBelow_of_weights {M : Type u} [AddCommGroup M] [Module k M]
+    (ρ : Representation k (SL2 k) M) (hρ : IsRational k ρ) (d q : ℕ) (hdq : d < q)
+    (hb : HasWeightsBetween ρ (-(d : ℤ)) d) : AllSimpleFactorsHighestBelow ρ q := by
+  intro A B _ i π hi hπ he
+  let : Nontrivial (Representation.asModule B.ρ) :=
+    IsSimpleModule.nontrivial (MonoidAlgebra k (SL2 k)) (Representation.asModule B.ρ)
+  let : Nontrivial B := (Representation.asModuleEquiv B.ρ).symm.toEquiv.nontrivial
+  have hA := IsRational.of_injective k A.ρ ρ i hi he hρ
+  have hB := IsRational.of_surjective k A.ρ B.ρ (fdrepLinear k π) hπ (fdrepLinear_comm k π) hA
+  have hbA := hasWeightsBetween_of_injective k A.ρ ρ i hi he _ _ hb
+  have hbB := hasWeightsBetween_of_surjective k A.ρ B.ρ hA (fdrepLinear k π) hπ
+    (fdrepLinear_comm k π) _ _ hbA
+  obtain ⟨n,hn,hwn,v,hv,hwv⟩ := exists_highest_in_bound k B.ρ hB d hbB
+  refine ⟨n,hn.trans_lt hdq,hwn,v,hv,hwv,?_⟩
+  exact highest_unipotent_fixed B.ρ hB n v hwv (fun m w hm => hwn m w (Or.inr hm))
+end FiberFunctorProof
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/RestrictedTilting/CharacterExpansions.lean b/lean/OAI/RepresentationTheory/RestrictedTilting/CharacterExpansions.lean
index 9840230398b14bc6b5c6c830b967fe90b912f026..8f315c6ac0ff919f644a340b41a9df56e40ca2f9 100644
--- a/lean/OAI/RepresentationTheory/RestrictedTilting/CharacterExpansions.lean
+++ b/lean/OAI/RepresentationTheory/RestrictedTilting/CharacterExpansions.lean
@@ -1,562 +1,562 @@
-import Mathlib
-import OAI.RepresentationTheory.RestrictedTilting.SL2Definitions
-
-namespace OAI
-
-/-! Character expansions on the diagonal torus and additive root group. -/
-
-noncomputable section
-open CategoryTheory CategoryTheory.MonoidalCategory CategoryTheory.Limits
-open scoped BigOperators
-
-namespace FiberFunctorProof
-section
-
-universe u v w
-variable {k : Type u} [Field k] {G : Type v} [Monoid G]
-variable {M : Type w} [AddCommGroup M] [Module k M]
-
- 
-lemma character_sum_eq_zero (s : Finset (G →* k)) (c : (G →* k) → M)
-    (h : ∀ g, ∑ χ ∈ s, χ g • c χ = 0) : ∀ χ ∈ s, c χ = 0 := by
-  intro χ hχ
-  apply (Module.forall_dual_apply_eq_zero_iff k (c χ)).mp
-  intro φ
-  apply (linearIndependent_iff'.mp (linearIndependent_monoidHom G k)) s
-    (fun ψ => φ (c ψ)) ?_ χ hχ
-  ext g
-  have hg := congrArg φ (h g)
-  simpa only [map_sum, map_smul, map_zero, Finset.sum_apply, Pi.smul_apply,
-    smul_eq_mul, Pi.zero_apply, mul_comm] using hg
-
- 
-def HasCharacterExpansions (ρ : Representation k G M) : Prop :=
-  ∀ v : M, ∃ c : (G →* k) →₀ M, ∀ g : G,
-    ρ g v = c.sum (fun χ x => χ g • x)
-
-lemma character_finsupp_eq_zero (c : (G →* k) →₀ M)
-    (h : ∀ g, c.sum (fun χ x => χ g • x) = 0) : c = 0 := by
-  classical
-  ext χ
-  by_cases hχ : χ ∈ c.support
-  · exact character_sum_eq_zero c.support c h χ hχ
-  · simpa using Finsupp.notMem_support_iff.mp hχ
-
-lemma character_finsupp_ext (c d : (G →* k) →₀ M)
-    (h : ∀ g, c.sum (fun χ x => χ g • x) = d.sum (fun χ x => χ g • x)) : c = d := by
-  apply sub_eq_zero.mp
-  apply character_finsupp_eq_zero
-  intro g
-  rw [Finsupp.sum_sub_index]
-  · exact sub_eq_zero.mpr (h g)
-  · intro χ x y
-    exact smul_sub _ _ _
-
- 
-lemma character_coefficient_eigen (ρ : Representation k G M) (v : M)
-    (c : (G →* k) →₀ M)
-    (hc : ∀ g, ρ g v = c.sum (fun χ x => χ g • x)) :
-    ∀ χ g, ρ g (c χ) = χ g • c χ := by
-  classical
-  intro χ t
-  by_cases hχ : χ ∈ c.support
-  · apply sub_eq_zero.mp
-    apply character_sum_eq_zero c.support
-      (fun ψ => ρ t (c ψ) - ψ t • c ψ) ?_ χ hχ
-    intro g
-    calc
-      ∑ ψ ∈ c.support, ψ g • (ρ t (c ψ) - ψ t • c ψ) =
-          ρ t (c.sum (fun ψ x => ψ g • x)) -
-            c.sum (fun ψ x => ψ (t * g) • x) := by
-        simp only [Finsupp.sum, map_sum, map_smul, smul_sub,
-          Finset.sum_sub_distrib, map_mul, mul_smul]
-        congr 1
-        apply Finset.sum_congr rfl
-        intro ψ hψ
-        exact smul_comm _ _ _
-      _ = 0 := by rw [← hc, ← hc, map_mul]; simp
-  · rw [Finsupp.notMem_support_iff.mp hχ, map_zero, smul_zero]
-
- 
-
-lemma eigenvector_lift {N : Type*} [AddCommGroup N] [Module k N]
-    (ρ : Representation k G M) (σ : Representation k G N)
-    (hρ : HasCharacterExpansions ρ) (f : M →ₗ[k] N)
-    (hf : ∀ g v, f (ρ g v) = σ g (f v)) (hs : Function.Surjective f)
-    (χ : G →* k) (w : N) (hw : ∀ g, σ g w = χ g • w) :
-    ∃ v : M, f v = w ∧ ∀ g, ρ g v = χ g • v := by
-  classical
-  obtain ⟨v, hv⟩ := hs w
-  obtain ⟨c, hc⟩ := hρ v
-  have hcf : c.mapRange f f.map_zero = Finsupp.single χ w := by
-    apply character_finsupp_ext
-    intro g
-    rw [Finsupp.sum_mapRange_index (fun _ => smul_zero _),
-      Finsupp.sum_single_index (h := fun ψ : G →* k => fun x : N => ψ g • x)
-        (smul_zero (χ g))]
-    calc
-      c.sum (fun ψ x => ψ g • f x) = f (c.sum (fun ψ x => ψ g • x)) := by
-        simp only [Finsupp.sum, map_sum, map_smul]
-      _ = χ g • w := by rw [← hc, hf, hv, hw]
-  refine ⟨c χ, ?_, character_coefficient_eigen ρ v c hc χ⟩
-  have := congrArg (fun d : (G →* k) →₀ N => d χ) hcf
-  simpa using this
-
- 
-def characterFunctions : Submonoid (G → k) where
-  carrier := Set.range (fun χ : G →* k => (χ : G → k))
-  one_mem' := ⟨1, rfl⟩
-  mul_mem' := by
-    rintro _ _ ⟨χ, rfl⟩ ⟨ψ, rfl⟩
-    exact ⟨χ * ψ, rfl⟩
-
-noncomputable def characterAlgebra : Subalgebra k (G → k) :=
-  Algebra.adjoin k (characterFunctions (k := k) (G := G) : Set (G → k))
-
-lemma characterAlgebra_toSubmodule :
-    (characterAlgebra (k := k) (G := G)).toSubmodule =
-      Submodule.span k (Set.range (fun χ : G →* k => (χ : G → k))) := by
-  rw [characterAlgebra, Algebra.adjoin_eq_span, Submonoid.closure_eq]
-  rfl
-
-lemma mem_characterAlgebra_iff (f : G → k) :
-    f ∈ characterAlgebra (k := k) (G := G) ↔ ∃ c : (G →* k) →₀ k,
-      ∀ g, f g = c.sum (fun χ x => χ g * x) := by
-  change f ∈ (characterAlgebra (k := k) (G := G)).toSubmodule ↔ _
-  rw [characterAlgebra_toSubmodule, Finsupp.mem_span_range_iff_exists_finsupp]
-  constructor
-  · rintro ⟨c, hc⟩
-    refine ⟨c, fun g => ?_⟩
-    have := congrFun hc g
-    simpa only [Finsupp.sum, Finset.sum_apply, Pi.smul_apply, smul_eq_mul, mul_comm] using this.symm
-  · rintro ⟨c, hc⟩
-    refine ⟨c, ?_⟩
-    ext g
-    simpa only [Finsupp.sum, Finset.sum_apply, Pi.smul_apply, smul_eq_mul, mul_comm] using (hc g).symm
-
-lemma character_mem_characterAlgebra (χ : G →* k) :
-    (χ : G → k) ∈ characterAlgebra (k := k) (G := G) :=
-  Algebra.subset_adjoin ⟨χ, rfl⟩
-
-noncomputable def characterEvaluation (g : G) : ((G →* k) →₀ M) →ₗ[k] M :=
-  Finsupp.lsum k (fun χ : G →* k => χ g • (LinearMap.id : M →ₗ[k] M))
-
-lemma characterEvaluation_apply (g : G) (c : (G →* k) →₀ M) :
-    characterEvaluation g c = c.sum (fun χ x => χ g • x) := rfl
-
- 
-
-lemma characterExpansions_of_coefficients [Module.Finite k M]
-    (ρ : Representation k G M)
-    (h : ∀ (v : M) (φ : M →ₗ[k] k),
-      (fun g => φ (ρ g v)) ∈ characterAlgebra (k := k) (G := G)) :
-    HasCharacterExpansions ρ := by
-  classical
-  intro v
-  let b := Module.finBasis k M
-  choose c hc using fun i => (mem_characterAlgebra_iff _).mp (h v (b.coord i))
-  let c' : (G →* k) →₀ M :=
-    ∑ i, (c i).mapRange (LinearMap.toSpanSingleton k M (b i)) (map_zero _)
-  refine ⟨c', fun g => ?_⟩
-  change ρ g v = characterEvaluation g c'
-  dsimp only [c']
-  rw [map_sum]
-  calc
-    ρ g v = ∑ i, b.repr (ρ g v) i • b i := (b.sum_repr (ρ g v)).symm
-    _ = ∑ i, characterEvaluation g
-        ((c i).mapRange (LinearMap.toSpanSingleton k M (b i)) (map_zero _)) := by
-      apply Finset.sum_congr rfl
-      intro i hi
-      rw [characterEvaluation_apply, Finsupp.sum_mapRange_index (fun _ => smul_zero _)]
-      simp only [LinearMap.toSpanSingleton_apply, ← mul_smul, Finsupp.sum,
-        ← Finset.sum_smul]
-      congr 1
-      exact hc i g
-
-end
-
-section
-
-universe u
-variable (k : Type u) [Field k]
-
- 
-def diagonalTorus : kˣ →* SL2 k where
-  toFun a := ⟨!![(a : k), 0; 0, (↑a⁻¹ : k)], by simp [Matrix.det_fin_two]⟩
-  map_one' := by
-    apply Subtype.ext
-    ext i j
-    fin_cases i <;> fin_cases j <;> simp
-  map_mul' a b := by
-    apply Subtype.ext
-    change !![(↑(a * b) : k), 0; 0, (↑(a * b)⁻¹ : k)] =
-      !![(a : k), 0; 0, (↑a⁻¹ : k)] * !![(b : k), 0; 0, (↑b⁻¹ : k)]
-    ext i j
-    fin_cases i <;> fin_cases j <;>
-      simp [Matrix.mul_apply, Fin.sum_univ_two, mul_comm]
-
-def inverseUnitCharacter : kˣ →* k where
-  toFun a := ↑a⁻¹
-  map_one' := by simp
-  map_mul' a b := by simp [mul_comm]
-
-lemma diagonal_coordinate_mem_characterAlgebra (i j : Fin 2) :
-    (fun a : kˣ => diagonalTorus k a i j) ∈
-      characterAlgebra (k := k) (G := kˣ) := by
-  fin_cases i <;> fin_cases j
-  · exact character_mem_characterAlgebra (Units.coeHom k)
-  · exact (characterAlgebra (k := k) (G := kˣ)).zero_mem
-  · exact (characterAlgebra (k := k) (G := kˣ)).zero_mem
-  · exact character_mem_characterAlgebra (inverseUnitCharacter k)
-
-lemma polynomial_on_torus_mem_characterAlgebra
-    (P : MvPolynomial (Fin 2 × Fin 2) k) :
-    (fun a : kˣ => MvPolynomial.eval (fun ij => diagonalTorus k a ij.1 ij.2) P) ∈
-      characterAlgebra (k := k) (G := kˣ) := by
-  induction P using MvPolynomial.induction_on with
-  | C c =>
-    simpa only [MvPolynomial.eval_C, Pi.algebraMap_def, Algebra.algebraMap_self, RingHom.id_apply] using
-      (characterAlgebra (k := k) (G := kˣ)).algebraMap_mem c
-  | add P Q hP hQ =>
-    simpa only [MvPolynomial.eval_add, Pi.add_def] using
-      (characterAlgebra (k := k) (G := kˣ)).add_mem hP hQ
-  | mul_X P ij hP =>
-    simpa only [MvPolynomial.eval_mul, MvPolynomial.eval_X, Pi.mul_def] using
-      (characterAlgebra (k := k) (G := kˣ)).mul_mem hP
-        (diagonal_coordinate_mem_characterAlgebra k ij.1 ij.2)
-
- 
-
-lemma rational_torus_characterExpansions {M : Type u} [AddCommGroup M]
-    [Module k M] [Module.Finite k M] (ρ : Representation k (SL2 k) M)
-    (hρ : IsRational k ρ) : HasCharacterExpansions (ρ.comp (diagonalTorus k)) := by
-  apply characterExpansions_of_coefficients
-  intro v φ
-  obtain ⟨P, hP⟩ := hρ v φ
-  have heq : (fun a : kˣ => φ ((ρ.comp (diagonalTorus k)) a v)) =
-      (fun a : kˣ => MvPolynomial.eval (fun ij => diagonalTorus k a ij.1 ij.2) P) := by
-    funext a
-    exact hP (diagonalTorus k a)
-  rw [heq]
-  exact polynomial_on_torus_mem_characterAlgebra k P
-
- 
-lemma rational_weight_lift {M N : Type u} [AddCommGroup M] [AddCommGroup N]
-    [Module k M] [Module k N] [Module.Finite k M]
-    (ρ : Representation k (SL2 k) M) (σ : Representation k (SL2 k) N)
-    (hρ : IsRational k ρ) (f : M →ₗ[k] N)
-    (hf : ∀ g v, f (ρ g v) = σ g (f v)) (hs : Function.Surjective f)
-    (χ : kˣ →* k) (w : N) (hw : ∀ a, σ (diagonalTorus k a) w = χ a • w) :
-    ∃ v : M, f v = w ∧ ∀ a, ρ (diagonalTorus k a) v = χ a • v :=
-  eigenvector_lift (ρ.comp (diagonalTorus k)) (σ.comp (diagonalTorus k))
-    (rational_torus_characterExpansions k ρ hρ) f
-    (fun a v => hf (diagonalTorus k a) v) hs χ w hw
-
- 
-def integerCharacter (n : ℤ) : kˣ →* k :=
-  (Units.coeHom k).comp (zpowGroupHom n)
-
-lemma unit_power_functions_injective [Infinite k] :
-    Function.Injective (fun n : ℕ => fun a : kˣ => (a : k) ^ n) := by
-  intro n m h
-  have hp : (Polynomial.X : Polynomial k) ^ n = Polynomial.X ^ m := by
-    apply Polynomial.eq_of_infinite_eval_eq
-    apply (Set.finite_singleton (0 : k)).infinite_compl.mono
-    intro a ha
-    have ha0 : a ≠ 0 := ha
-    simpa only [Polynomial.eval_pow, Polynomial.eval_X, Units.val_mk0, Set.mem_ofPred_eq] using
-      congrFun h (Units.mk0 a ha0)
-  have := congrArg Polynomial.natDegree hp
-  simpa using this
-
-lemma integerCharacter_injective [Infinite k] : Function.Injective (integerCharacter k) := by
-  intro n m h
-  have hz (a : kˣ) : a ^ (n - m) = 1 := by
-    rw [zpow_sub, mul_inv_eq_one]
-    apply Units.ext
-    exact DFunLike.congr_fun h a
-  have hn : (n - m).natAbs = 0 := by
-    apply unit_power_functions_injective k
-    funext a
-    have ht : a ^ (n - m).natAbs = 1 := (pow_natAbs_eq_one).mpr (hz a)
-    simpa using congrArg (fun x : kˣ => (x : k)) ht
-  exact sub_eq_zero.mp (Int.natAbs_eq_zero.mp hn)
-
-end
-
-section
-
-universe u v w
-variable {k : Type u} [Field k] {G : Type v}
-variable {M : Type w} [AddCommGroup M] [Module k M]
-
- 
-
-lemma vector_expansion_of_scalar_span [Module.Finite k M]
-    {I : Type*} (f : G → M) (e : I → G → k)
-    (h : ∀ φ : M →ₗ[k] k, (fun g => φ (f g)) ∈ Submodule.span k (Set.range e)) :
-    ∃ c : I →₀ M, ∀ g, f g = c.sum (fun i x => e i g • x) := by
-  classical
-  let b := Module.finBasis k M
-  choose c hc using fun i => Finsupp.mem_span_range_iff_exists_finsupp.mp (h (b.coord i))
-  let c' : I →₀ M :=
-    ∑ i, (c i).mapRange (LinearMap.toSpanSingleton k M (b i)) (map_zero _)
-  refine ⟨c', fun g => ?_⟩
-  let E : (I →₀ M) →ₗ[k] M :=
-    Finsupp.lsum k (fun i => e i g • (LinearMap.id : M →ₗ[k] M))
-  change f g = E c'
-  dsimp only [c']
-  rw [map_sum]
-  calc
-    f g = ∑ i, b.repr (f g) i • b i := (b.sum_repr (f g)).symm
-    _ = ∑ i, E ((c i).mapRange (LinearMap.toSpanSingleton k M (b i)) (map_zero _)) := by
-      apply Finset.sum_congr rfl
-      intro i hi
-      change _ = ((c i).mapRange _ _).sum (fun j x => e j g • x)
-      rw [Finsupp.sum_mapRange_index (fun _ => smul_zero _)]
-      simp only [LinearMap.toSpanSingleton_apply, ← mul_smul, Finsupp.sum,
-        ← Finset.sum_smul]
-      congr 1
-      have hi' := congrFun (hc i) g
-      simpa only [Finsupp.sum, Finset.sum_apply, Pi.smul_apply, smul_eq_mul,
-        mul_comm, Module.Basis.coord_apply] using hi'.symm
-
-end
-
-section
-
-universe u
-variable (k : Type u) [Field k]
-
- 
-def integerFunctions : Submonoid (kˣ → k) where
-  carrier := Set.range (fun n : ℤ => (integerCharacter k n : kˣ → k))
-  one_mem' := ⟨0, by ext a; simp [integerCharacter]⟩
-  mul_mem' := by
-    rintro _ _ ⟨n, rfl⟩ ⟨m, rfl⟩
-    refine ⟨n + m, ?_⟩
-    ext a
-    simp [integerCharacter, zpow_add]
-
-noncomputable def integerAlgebra : Subalgebra k (kˣ → k) :=
-  Algebra.adjoin k (integerFunctions k : Set (kˣ → k))
-
-lemma integerAlgebra_toSubmodule : (integerAlgebra k).toSubmodule =
-    Submodule.span k (Set.range (fun n : ℤ => (integerCharacter k n : kˣ → k))) := by
-  rw [integerAlgebra, Algebra.adjoin_eq_span, Submonoid.closure_eq]
-  rfl
-
-lemma integerCharacter_mem_integerAlgebra (n : ℤ) :
-    (integerCharacter k n : kˣ → k) ∈ integerAlgebra k :=
-  Algebra.subset_adjoin ⟨n, rfl⟩
-
-lemma diagonal_coordinate_mem_integerAlgebra (i j : Fin 2) :
-    (fun a : kˣ => diagonalTorus k a i j) ∈ integerAlgebra k := by
-  fin_cases i <;> fin_cases j
-  · convert integerCharacter_mem_integerAlgebra k 1 using 1
-    ext a
-    change (a : k) = integerCharacter k 1 a
-    simp [integerCharacter]
-  · exact (integerAlgebra k).zero_mem
-  · exact (integerAlgebra k).zero_mem
-  · convert integerCharacter_mem_integerAlgebra k (-1) using 1
-    ext a
-    change (↑a⁻¹ : k) = integerCharacter k (-1) a
-    simp [integerCharacter]
-
-lemma polynomial_on_torus_mem_integerAlgebra
-    (P : MvPolynomial (Fin 2 × Fin 2) k) :
-    (fun a : kˣ => MvPolynomial.eval (fun ij => diagonalTorus k a ij.1 ij.2) P) ∈
-      integerAlgebra k := by
-  induction P using MvPolynomial.induction_on with
-  | C c =>
-    simpa only [MvPolynomial.eval_C, Pi.algebraMap_def, Algebra.algebraMap_self,
-      RingHom.id_apply] using (integerAlgebra k).algebraMap_mem c
-  | add P Q hP hQ =>
-    simpa only [MvPolynomial.eval_add, Pi.add_def] using (integerAlgebra k).add_mem hP hQ
-  | mul_X P ij hP =>
-    simpa only [MvPolynomial.eval_mul, MvPolynomial.eval_X, Pi.mul_def] using
-      (integerAlgebra k).mul_mem hP (diagonal_coordinate_mem_integerAlgebra k ij.1 ij.2)
-
- 
-
-lemma rational_torus_integerExpansions {M : Type u} [AddCommGroup M]
-    [Module k M] [Module.Finite k M] (ρ : Representation k (SL2 k) M)
-    (hρ : IsRational k ρ) (v : M) : ∃ c : ℤ →₀ M, ∀ a : kˣ,
-      ρ (diagonalTorus k a) v = c.sum (fun n x => integerCharacter k n a • x) := by
-  apply vector_expansion_of_scalar_span
-  intro φ
-  obtain ⟨P, hP⟩ := hρ v φ
-  have heq : (fun a : kˣ => φ (ρ (diagonalTorus k a) v)) =
-      (fun a : kˣ => MvPolynomial.eval (fun ij => diagonalTorus k a ij.1 ij.2) P) := by
-    funext a
-    exact hP (diagonalTorus k a)
-  rw [heq, ← integerAlgebra_toSubmodule]
-  exact polynomial_on_torus_mem_integerAlgebra k P
-
-end
-
-section
-
-universe u v w
-variable {k : Type u} [Field k] {G : Type v}
-variable {M : Type w} [AddCommGroup M] [Module k M]
-
-lemma function_sum_eq_zero {I : Type*} (e : I → G → k)
-    (he : LinearIndependent k e) (s : Finset I) (c : I → M)
-    (h : ∀ g, ∑ i ∈ s, e i g • c i = 0) : ∀ i ∈ s, c i = 0 := by
-  intro i hi
-  apply (Module.forall_dual_apply_eq_zero_iff k (c i)).mp
-  intro φ
-  apply (linearIndependent_iff'.mp he) s (fun j => φ (c j)) ?_ i hi
-  ext g
-  have hg := congrArg φ (h g)
-  simpa only [map_sum, map_smul, map_zero, Finset.sum_apply, Pi.smul_apply,
-    smul_eq_mul, Pi.zero_apply, mul_comm] using hg
-
-lemma function_finsupp_eq_zero {I : Type*} (e : I → G → k)
-    (he : LinearIndependent k e) (c : I →₀ M)
-    (h : ∀ g, c.sum (fun i x => e i g • x) = 0) : c = 0 := by
-  classical
-  ext i
-  by_cases hi : i ∈ c.support
-  · exact function_sum_eq_zero e he c.support c h i hi
-  · simpa using Finsupp.notMem_support_iff.mp hi
-
-lemma function_finsupp_ext {I : Type*} (e : I → G → k)
-    (he : LinearIndependent k e) (c d : I →₀ M)
-    (h : ∀ g, c.sum (fun i x => e i g • x) = d.sum (fun i x => e i g • x)) : c = d := by
-  apply sub_eq_zero.mp
-  apply function_finsupp_eq_zero e he
-  intro g
-  rw [Finsupp.sum_sub_index]
-  · exact sub_eq_zero.mpr (h g)
-  · intro χ x y
-    exact smul_sub _ _ _
-
-variable (k)
-
-lemma linearIndependent_powerFunctions [Infinite k] :
-    LinearIndependent k (fun (n : ℕ) (x : k) => x ^ n) := by
-  classical
-  apply linearIndependent_iff'.mpr
-  intro s a ha i hi
-  have hp : (∑ j ∈ s, Polynomial.monomial j (a j)) = (0 : Polynomial k) := by
-    apply Polynomial.funext
-    intro x
-    have hx := congrFun ha x
-    simpa only [Polynomial.eval_finsetSum, Polynomial.eval_monomial,
-      Polynomial.eval_zero, Finset.sum_apply, Pi.smul_apply, Pi.zero_apply,
-      smul_eq_mul] using hx
-  have hc := congrArg (fun p : Polynomial k => p.coeff i) hp
-  simpa [Polynomial.coeff_monomial, hi] using hc
-
-end
-
-universe u
-variable (k : Type u) [Field k]
-
-def powerFunctions : Submonoid (k → k) where
-  carrier := Set.range (fun n : ℕ => fun t : k => t ^ n)
-  one_mem' := ⟨0, by ext t; simp⟩
-  mul_mem' := by
-    rintro _ _ ⟨n, rfl⟩ ⟨m, rfl⟩
-    exact ⟨n + m, by ext t; exact pow_add t n m⟩
-
-noncomputable def powerAlgebra : Subalgebra k (k → k) :=
-  Algebra.adjoin k (powerFunctions k : Set (k → k))
-
-lemma powerAlgebra_toSubmodule : (powerAlgebra k).toSubmodule =
-    Submodule.span k (Set.range (fun (n : ℕ) (t : k) => t ^ n)) := by
-  rw [powerAlgebra, Algebra.adjoin_eq_span, Submonoid.closure_eq]
-  rfl
-
-lemma power_mem_powerAlgebra (n : ℕ) :
-    (fun t : k => t ^ n) ∈ powerAlgebra k := Algebra.subset_adjoin ⟨n, rfl⟩
-
-lemma unipotent_coordinate_mem_powerAlgebra (i j : Fin 2) :
-    (fun t : k => unipotent k t i j) ∈ powerAlgebra k := by
-  fin_cases i <;> fin_cases j
-  · exact (powerAlgebra k).one_mem
-  · change (fun t : k => t) ∈ powerAlgebra k
-    simpa only [pow_one] using power_mem_powerAlgebra k 1
-  · exact (powerAlgebra k).zero_mem
-  · exact (powerAlgebra k).one_mem
-
-lemma polynomial_on_unipotent_mem_powerAlgebra
-    (P : MvPolynomial (Fin 2 × Fin 2) k) :
-    (fun t : k => MvPolynomial.eval (fun ij => unipotent k t ij.1 ij.2) P) ∈
-      powerAlgebra k := by
-  induction P using MvPolynomial.induction_on with
-  | C c =>
-    simpa only [MvPolynomial.eval_C, Pi.algebraMap_def, Algebra.algebraMap_self,
-      RingHom.id_apply] using (powerAlgebra k).algebraMap_mem c
-  | add P Q hP hQ =>
-    simpa only [MvPolynomial.eval_add, Pi.add_def] using (powerAlgebra k).add_mem hP hQ
-  | mul_X P ij hP =>
-    simpa only [MvPolynomial.eval_mul, MvPolynomial.eval_X, Pi.mul_def] using
-      (powerAlgebra k).mul_mem hP (unipotent_coordinate_mem_powerAlgebra k ij.1 ij.2)
-
-lemma rational_unipotent_expansion {M : Type u} [AddCommGroup M]
-    [Module k M] [Module.Finite k M] (ρ : Representation k (SL2 k) M)
-    (hρ : IsRational k ρ) (v : M) : ∃ c : ℕ →₀ M, ∀ t : k,
-      ρ (unipotent k t) v = c.sum (fun n x => t ^ n • x) := by
-  apply vector_expansion_of_scalar_span
-  intro φ
-  obtain ⟨P, hP⟩ := hρ v φ
-  have heq : (fun t : k => φ (ρ (unipotent k t) v)) =
-      (fun t : k => MvPolynomial.eval (fun ij => unipotent k t ij.1 ij.2) P) := by
-    funext t
-    exact hP (unipotent k t)
-  rw [heq, ← powerAlgebra_toSubmodule]
-  exact polynomial_on_unipotent_mem_powerAlgebra k P
-
-lemma diagonal_unipotent (a : kˣ) (t : k) :
-    diagonalTorus k a * unipotent k t =
-      unipotent k ((a : k) ^ 2 * t) * diagonalTorus k a := by
-  apply Subtype.ext
-  change !![(a : k), 0; 0, (↑a⁻¹ : k)] * !![1, t; 0, 1] =
-    !![1, (a : k) ^ 2 * t; 0, 1] * !![(a : k), 0; 0, (↑a⁻¹ : k)]
-  ext i j
-  fin_cases i <;> fin_cases j <;>
-    simp [Matrix.mul_apply, Fin.sum_univ_two, pow_two, mul_assoc, mul_comm]
-
- 
-
-lemma unipotent_coefficient_weight [Infinite k] {M : Type u}
-    [AddCommGroup M] [Module k M] (ρ : Representation k (SL2 k) M)
-    (v : M) (n : ℤ) (hv : ∀ a, ρ (diagonalTorus k a) v = integerCharacter k n a • v)
-    (c : ℕ →₀ M) (hc : ∀ t, ρ (unipotent k t) v = c.sum (fun i x => t ^ i • x)) :
-    ∀ i a, ρ (diagonalTorus k a) (c i) = integerCharacter k (n + 2 * i) a • c i := by
-  classical
-  intro i a
-  by_cases hi : i ∈ c.support
-  · apply sub_eq_zero.mp
-    apply function_sum_eq_zero (fun (j : ℕ) (t : k) => t ^ j)
-      (linearIndependent_powerFunctions k) c.support
-      (fun j => ρ (diagonalTorus k a) (c j) - integerCharacter k (n + 2 * j) a • c j)
-      ?_ i hi
-    intro t
-    calc
-      ∑ j ∈ c.support, t ^ j •
-          (ρ (diagonalTorus k a) (c j) - integerCharacter k (n + 2 * j) a • c j) =
-          ρ (diagonalTorus k a) (c.sum (fun j x => t ^ j • x)) -
-            integerCharacter k n a • c.sum (fun j x => ((a : k) ^ 2 * t) ^ j • x) := by
-        simp only [Finsupp.sum, map_sum, map_smul, smul_sub, Finset.sum_sub_distrib,
-          Finset.smul_sum, smul_smul]
-        congr 1
-        apply Finset.sum_congr rfl
-        intro j hj
-        congr 1
-        simp [integerCharacter, zpow_add, zpow_mul, mul_pow, ← pow_mul, mul_comm, mul_left_comm]
-      _ = 0 := by
-        rw [← hc, ← hc]
-        apply sub_eq_zero.mpr
-        have hh := congrArg (fun g => ρ g v) (diagonal_unipotent k a t)
-        simpa only [map_mul, Module.End.mul_apply, hv, map_smul] using hh
-  · rw [Finsupp.notMem_support_iff.mp hi, map_zero, smul_zero]
-
-end FiberFunctorProof
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.RestrictedTilting.SL2Definitions
+
+namespace OAI
+
+/-! Character expansions on the diagonal torus and additive root group. -/
+
+noncomputable section
+open CategoryTheory CategoryTheory.MonoidalCategory CategoryTheory.Limits
+open scoped BigOperators
+
+namespace FiberFunctorProof
+section
+
+universe u v w
+variable {k : Type u} [Field k] {G : Type v} [Monoid G]
+variable {M : Type w} [AddCommGroup M] [Module k M]
+
+ 
+lemma character_sum_eq_zero (s : Finset (G →* k)) (c : (G →* k) → M)
+    (h : ∀ g, ∑ χ ∈ s, χ g • c χ = 0) : ∀ χ ∈ s, c χ = 0 := by
+  intro χ hχ
+  apply (Module.forall_dual_apply_eq_zero_iff k (c χ)).mp
+  intro φ
+  apply (linearIndependent_iff'.mp (linearIndependent_monoidHom G k)) s
+    (fun ψ => φ (c ψ)) ?_ χ hχ
+  ext g
+  have hg := congrArg φ (h g)
+  simpa only [map_sum, map_smul, map_zero, Finset.sum_apply, Pi.smul_apply,
+    smul_eq_mul, Pi.zero_apply, mul_comm] using hg
+
+ 
+def HasCharacterExpansions (ρ : Representation k G M) : Prop :=
+  ∀ v : M, ∃ c : (G →* k) →₀ M, ∀ g : G,
+    ρ g v = c.sum (fun χ x => χ g • x)
+
+lemma character_finsupp_eq_zero (c : (G →* k) →₀ M)
+    (h : ∀ g, c.sum (fun χ x => χ g • x) = 0) : c = 0 := by
+  classical
+  ext χ
+  by_cases hχ : χ ∈ c.support
+  · exact character_sum_eq_zero c.support c h χ hχ
+  · simpa using Finsupp.notMem_support_iff.mp hχ
+
+lemma character_finsupp_ext (c d : (G →* k) →₀ M)
+    (h : ∀ g, c.sum (fun χ x => χ g • x) = d.sum (fun χ x => χ g • x)) : c = d := by
+  apply sub_eq_zero.mp
+  apply character_finsupp_eq_zero
+  intro g
+  rw [Finsupp.sum_sub_index]
+  · exact sub_eq_zero.mpr (h g)
+  · intro χ x y
+    exact smul_sub _ _ _
+
+ 
+lemma character_coefficient_eigen (ρ : Representation k G M) (v : M)
+    (c : (G →* k) →₀ M)
+    (hc : ∀ g, ρ g v = c.sum (fun χ x => χ g • x)) :
+    ∀ χ g, ρ g (c χ) = χ g • c χ := by
+  classical
+  intro χ t
+  by_cases hχ : χ ∈ c.support
+  · apply sub_eq_zero.mp
+    apply character_sum_eq_zero c.support
+      (fun ψ => ρ t (c ψ) - ψ t • c ψ) ?_ χ hχ
+    intro g
+    calc
+      ∑ ψ ∈ c.support, ψ g • (ρ t (c ψ) - ψ t • c ψ) =
+          ρ t (c.sum (fun ψ x => ψ g • x)) -
+            c.sum (fun ψ x => ψ (t * g) • x) := by
+        simp only [Finsupp.sum, map_sum, map_smul, smul_sub,
+          Finset.sum_sub_distrib, map_mul, mul_smul]
+        congr 1
+        apply Finset.sum_congr rfl
+        intro ψ hψ
+        exact smul_comm _ _ _
+      _ = 0 := by rw [← hc, ← hc, map_mul]; simp
+  · rw [Finsupp.notMem_support_iff.mp hχ, map_zero, smul_zero]
+
+ 
+
+lemma eigenvector_lift {N : Type*} [AddCommGroup N] [Module k N]
+    (ρ : Representation k G M) (σ : Representation k G N)
+    (hρ : HasCharacterExpansions ρ) (f : M →ₗ[k] N)
+    (hf : ∀ g v, f (ρ g v) = σ g (f v)) (hs : Function.Surjective f)
+    (χ : G →* k) (w : N) (hw : ∀ g, σ g w = χ g • w) :
+    ∃ v : M, f v = w ∧ ∀ g, ρ g v = χ g • v := by
+  classical
+  obtain ⟨v, hv⟩ := hs w
+  obtain ⟨c, hc⟩ := hρ v
+  have hcf : c.mapRange f f.map_zero = Finsupp.single χ w := by
+    apply character_finsupp_ext
+    intro g
+    rw [Finsupp.sum_mapRange_index (fun _ => smul_zero _),
+      Finsupp.sum_single_index (h := fun ψ : G →* k => fun x : N => ψ g • x)
+        (smul_zero (χ g))]
+    calc
+      c.sum (fun ψ x => ψ g • f x) = f (c.sum (fun ψ x => ψ g • x)) := by
+        simp only [Finsupp.sum, map_sum, map_smul]
+      _ = χ g • w := by rw [← hc, hf, hv, hw]
+  refine ⟨c χ, ?_, character_coefficient_eigen ρ v c hc χ⟩
+  have := congrArg (fun d : (G →* k) →₀ N => d χ) hcf
+  simpa using this
+
+ 
+def characterFunctions : Submonoid (G → k) where
+  carrier := Set.range (fun χ : G →* k => (χ : G → k))
+  one_mem' := ⟨1, rfl⟩
+  mul_mem' := by
+    rintro _ _ ⟨χ, rfl⟩ ⟨ψ, rfl⟩
+    exact ⟨χ * ψ, rfl⟩
+
+noncomputable def characterAlgebra : Subalgebra k (G → k) :=
+  Algebra.adjoin k (characterFunctions (k := k) (G := G) : Set (G → k))
+
+lemma characterAlgebra_toSubmodule :
+    (characterAlgebra (k := k) (G := G)).toSubmodule =
+      Submodule.span k (Set.range (fun χ : G →* k => (χ : G → k))) := by
+  rw [characterAlgebra, Algebra.adjoin_eq_span, Submonoid.closure_eq]
+  rfl
+
+lemma mem_characterAlgebra_iff (f : G → k) :
+    f ∈ characterAlgebra (k := k) (G := G) ↔ ∃ c : (G →* k) →₀ k,
+      ∀ g, f g = c.sum (fun χ x => χ g * x) := by
+  change f ∈ (characterAlgebra (k := k) (G := G)).toSubmodule ↔ _
+  rw [characterAlgebra_toSubmodule, Finsupp.mem_span_range_iff_exists_finsupp]
+  constructor
+  · rintro ⟨c, hc⟩
+    refine ⟨c, fun g => ?_⟩
+    have := congrFun hc g
+    simpa only [Finsupp.sum, Finset.sum_apply, Pi.smul_apply, smul_eq_mul, mul_comm] using this.symm
+  · rintro ⟨c, hc⟩
+    refine ⟨c, ?_⟩
+    ext g
+    simpa only [Finsupp.sum, Finset.sum_apply, Pi.smul_apply, smul_eq_mul, mul_comm] using (hc g).symm
+
+lemma character_mem_characterAlgebra (χ : G →* k) :
+    (χ : G → k) ∈ characterAlgebra (k := k) (G := G) :=
+  Algebra.subset_adjoin ⟨χ, rfl⟩
+
+noncomputable def characterEvaluation (g : G) : ((G →* k) →₀ M) →ₗ[k] M :=
+  Finsupp.lsum k (fun χ : G →* k => χ g • (LinearMap.id : M →ₗ[k] M))
+
+lemma characterEvaluation_apply (g : G) (c : (G →* k) →₀ M) :
+    characterEvaluation g c = c.sum (fun χ x => χ g • x) := rfl
+
+ 
+
+lemma characterExpansions_of_coefficients [Module.Finite k M]
+    (ρ : Representation k G M)
+    (h : ∀ (v : M) (φ : M →ₗ[k] k),
+      (fun g => φ (ρ g v)) ∈ characterAlgebra (k := k) (G := G)) :
+    HasCharacterExpansions ρ := by
+  classical
+  intro v
+  let b := Module.finBasis k M
+  choose c hc using fun i => (mem_characterAlgebra_iff _).mp (h v (b.coord i))
+  let c' : (G →* k) →₀ M :=
+    ∑ i, (c i).mapRange (LinearMap.toSpanSingleton k M (b i)) (map_zero _)
+  refine ⟨c', fun g => ?_⟩
+  change ρ g v = characterEvaluation g c'
+  dsimp only [c']
+  rw [map_sum]
+  calc
+    ρ g v = ∑ i, b.repr (ρ g v) i • b i := (b.sum_repr (ρ g v)).symm
+    _ = ∑ i, characterEvaluation g
+        ((c i).mapRange (LinearMap.toSpanSingleton k M (b i)) (map_zero _)) := by
+      apply Finset.sum_congr rfl
+      intro i hi
+      rw [characterEvaluation_apply, Finsupp.sum_mapRange_index (fun _ => smul_zero _)]
+      simp only [LinearMap.toSpanSingleton_apply, ← mul_smul, Finsupp.sum,
+        ← Finset.sum_smul]
+      congr 1
+      exact hc i g
+
+end
+
+section
+
+universe u
+variable (k : Type u) [Field k]
+
+ 
+def diagonalTorus : kˣ →* SL2 k where
+  toFun a := ⟨!![(a : k), 0; 0, (↑a⁻¹ : k)], by simp [Matrix.det_fin_two]⟩
+  map_one' := by
+    apply Subtype.ext
+    ext i j
+    fin_cases i <;> fin_cases j <;> simp
+  map_mul' a b := by
+    apply Subtype.ext
+    change !![(↑(a * b) : k), 0; 0, (↑(a * b)⁻¹ : k)] =
+      !![(a : k), 0; 0, (↑a⁻¹ : k)] * !![(b : k), 0; 0, (↑b⁻¹ : k)]
+    ext i j
+    fin_cases i <;> fin_cases j <;>
+      simp [Matrix.mul_apply, Fin.sum_univ_two, mul_comm]
+
+def inverseUnitCharacter : kˣ →* k where
+  toFun a := ↑a⁻¹
+  map_one' := by simp
+  map_mul' a b := by simp [mul_comm]
+
+lemma diagonal_coordinate_mem_characterAlgebra (i j : Fin 2) :
+    (fun a : kˣ => diagonalTorus k a i j) ∈
+      characterAlgebra (k := k) (G := kˣ) := by
+  fin_cases i <;> fin_cases j
+  · exact character_mem_characterAlgebra (Units.coeHom k)
+  · exact (characterAlgebra (k := k) (G := kˣ)).zero_mem
+  · exact (characterAlgebra (k := k) (G := kˣ)).zero_mem
+  · exact character_mem_characterAlgebra (inverseUnitCharacter k)
+
+lemma polynomial_on_torus_mem_characterAlgebra
+    (P : MvPolynomial (Fin 2 × Fin 2) k) :
+    (fun a : kˣ => MvPolynomial.eval (fun ij => diagonalTorus k a ij.1 ij.2) P) ∈
+      characterAlgebra (k := k) (G := kˣ) := by
+  induction P using MvPolynomial.induction_on with
+  | C c =>
+    simpa only [MvPolynomial.eval_C, Pi.algebraMap_def, Algebra.algebraMap_self, RingHom.id_apply] using
+      (characterAlgebra (k := k) (G := kˣ)).algebraMap_mem c
+  | add P Q hP hQ =>
+    simpa only [MvPolynomial.eval_add, Pi.add_def] using
+      (characterAlgebra (k := k) (G := kˣ)).add_mem hP hQ
+  | mul_X P ij hP =>
+    simpa only [MvPolynomial.eval_mul, MvPolynomial.eval_X, Pi.mul_def] using
+      (characterAlgebra (k := k) (G := kˣ)).mul_mem hP
+        (diagonal_coordinate_mem_characterAlgebra k ij.1 ij.2)
+
+ 
+
+lemma rational_torus_characterExpansions {M : Type u} [AddCommGroup M]
+    [Module k M] [Module.Finite k M] (ρ : Representation k (SL2 k) M)
+    (hρ : IsRational k ρ) : HasCharacterExpansions (ρ.comp (diagonalTorus k)) := by
+  apply characterExpansions_of_coefficients
+  intro v φ
+  obtain ⟨P, hP⟩ := hρ v φ
+  have heq : (fun a : kˣ => φ ((ρ.comp (diagonalTorus k)) a v)) =
+      (fun a : kˣ => MvPolynomial.eval (fun ij => diagonalTorus k a ij.1 ij.2) P) := by
+    funext a
+    exact hP (diagonalTorus k a)
+  rw [heq]
+  exact polynomial_on_torus_mem_characterAlgebra k P
+
+ 
+lemma rational_weight_lift {M N : Type u} [AddCommGroup M] [AddCommGroup N]
+    [Module k M] [Module k N] [Module.Finite k M]
+    (ρ : Representation k (SL2 k) M) (σ : Representation k (SL2 k) N)
+    (hρ : IsRational k ρ) (f : M →ₗ[k] N)
+    (hf : ∀ g v, f (ρ g v) = σ g (f v)) (hs : Function.Surjective f)
+    (χ : kˣ →* k) (w : N) (hw : ∀ a, σ (diagonalTorus k a) w = χ a • w) :
+    ∃ v : M, f v = w ∧ ∀ a, ρ (diagonalTorus k a) v = χ a • v :=
+  eigenvector_lift (ρ.comp (diagonalTorus k)) (σ.comp (diagonalTorus k))
+    (rational_torus_characterExpansions k ρ hρ) f
+    (fun a v => hf (diagonalTorus k a) v) hs χ w hw
+
+ 
+def integerCharacter (n : ℤ) : kˣ →* k :=
+  (Units.coeHom k).comp (zpowGroupHom n)
+
+lemma unit_power_functions_injective [Infinite k] :
+    Function.Injective (fun n : ℕ => fun a : kˣ => (a : k) ^ n) := by
+  intro n m h
+  have hp : (Polynomial.X : Polynomial k) ^ n = Polynomial.X ^ m := by
+    apply Polynomial.eq_of_infinite_eval_eq
+    apply (Set.finite_singleton (0 : k)).infinite_compl.mono
+    intro a ha
+    have ha0 : a ≠ 0 := ha
+    simpa only [Polynomial.eval_pow, Polynomial.eval_X, Units.val_mk0, Set.mem_ofPred_eq] using
+      congrFun h (Units.mk0 a ha0)
+  have := congrArg Polynomial.natDegree hp
+  simpa using this
+
+lemma integerCharacter_injective [Infinite k] : Function.Injective (integerCharacter k) := by
+  intro n m h
+  have hz (a : kˣ) : a ^ (n - m) = 1 := by
+    rw [zpow_sub, mul_inv_eq_one]
+    apply Units.ext
+    exact DFunLike.congr_fun h a
+  have hn : (n - m).natAbs = 0 := by
+    apply unit_power_functions_injective k
+    funext a
+    have ht : a ^ (n - m).natAbs = 1 := (pow_natAbs_eq_one).mpr (hz a)
+    simpa using congrArg (fun x : kˣ => (x : k)) ht
+  exact sub_eq_zero.mp (Int.natAbs_eq_zero.mp hn)
+
+end
+
+section
+
+universe u v w
+variable {k : Type u} [Field k] {G : Type v}
+variable {M : Type w} [AddCommGroup M] [Module k M]
+
+ 
+
+lemma vector_expansion_of_scalar_span [Module.Finite k M]
+    {I : Type*} (f : G → M) (e : I → G → k)
+    (h : ∀ φ : M →ₗ[k] k, (fun g => φ (f g)) ∈ Submodule.span k (Set.range e)) :
+    ∃ c : I →₀ M, ∀ g, f g = c.sum (fun i x => e i g • x) := by
+  classical
+  let b := Module.finBasis k M
+  choose c hc using fun i => Finsupp.mem_span_range_iff_exists_finsupp.mp (h (b.coord i))
+  let c' : I →₀ M :=
+    ∑ i, (c i).mapRange (LinearMap.toSpanSingleton k M (b i)) (map_zero _)
+  refine ⟨c', fun g => ?_⟩
+  let E : (I →₀ M) →ₗ[k] M :=
+    Finsupp.lsum k (fun i => e i g • (LinearMap.id : M →ₗ[k] M))
+  change f g = E c'
+  dsimp only [c']
+  rw [map_sum]
+  calc
+    f g = ∑ i, b.repr (f g) i • b i := (b.sum_repr (f g)).symm
+    _ = ∑ i, E ((c i).mapRange (LinearMap.toSpanSingleton k M (b i)) (map_zero _)) := by
+      apply Finset.sum_congr rfl
+      intro i hi
+      change _ = ((c i).mapRange _ _).sum (fun j x => e j g • x)
+      rw [Finsupp.sum_mapRange_index (fun _ => smul_zero _)]
+      simp only [LinearMap.toSpanSingleton_apply, ← mul_smul, Finsupp.sum,
+        ← Finset.sum_smul]
+      congr 1
+      have hi' := congrFun (hc i) g
+      simpa only [Finsupp.sum, Finset.sum_apply, Pi.smul_apply, smul_eq_mul,
+        mul_comm, Module.Basis.coord_apply] using hi'.symm
+
+end
+
+section
+
+universe u
+variable (k : Type u) [Field k]
+
+ 
+def integerFunctions : Submonoid (kˣ → k) where
+  carrier := Set.range (fun n : ℤ => (integerCharacter k n : kˣ → k))
+  one_mem' := ⟨0, by ext a; simp [integerCharacter]⟩
+  mul_mem' := by
+    rintro _ _ ⟨n, rfl⟩ ⟨m, rfl⟩
+    refine ⟨n + m, ?_⟩
+    ext a
+    simp [integerCharacter, zpow_add]
+
+noncomputable def integerAlgebra : Subalgebra k (kˣ → k) :=
+  Algebra.adjoin k (integerFunctions k : Set (kˣ → k))
+
+lemma integerAlgebra_toSubmodule : (integerAlgebra k).toSubmodule =
+    Submodule.span k (Set.range (fun n : ℤ => (integerCharacter k n : kˣ → k))) := by
+  rw [integerAlgebra, Algebra.adjoin_eq_span, Submonoid.closure_eq]
+  rfl
+
+lemma integerCharacter_mem_integerAlgebra (n : ℤ) :
+    (integerCharacter k n : kˣ → k) ∈ integerAlgebra k :=
+  Algebra.subset_adjoin ⟨n, rfl⟩
+
+lemma diagonal_coordinate_mem_integerAlgebra (i j : Fin 2) :
+    (fun a : kˣ => diagonalTorus k a i j) ∈ integerAlgebra k := by
+  fin_cases i <;> fin_cases j
+  · convert integerCharacter_mem_integerAlgebra k 1 using 1
+    ext a
+    change (a : k) = integerCharacter k 1 a
+    simp [integerCharacter]
+  · exact (integerAlgebra k).zero_mem
+  · exact (integerAlgebra k).zero_mem
+  · convert integerCharacter_mem_integerAlgebra k (-1) using 1
+    ext a
+    change (↑a⁻¹ : k) = integerCharacter k (-1) a
+    simp [integerCharacter]
+
+lemma polynomial_on_torus_mem_integerAlgebra
+    (P : MvPolynomial (Fin 2 × Fin 2) k) :
+    (fun a : kˣ => MvPolynomial.eval (fun ij => diagonalTorus k a ij.1 ij.2) P) ∈
+      integerAlgebra k := by
+  induction P using MvPolynomial.induction_on with
+  | C c =>
+    simpa only [MvPolynomial.eval_C, Pi.algebraMap_def, Algebra.algebraMap_self,
+      RingHom.id_apply] using (integerAlgebra k).algebraMap_mem c
+  | add P Q hP hQ =>
+    simpa only [MvPolynomial.eval_add, Pi.add_def] using (integerAlgebra k).add_mem hP hQ
+  | mul_X P ij hP =>
+    simpa only [MvPolynomial.eval_mul, MvPolynomial.eval_X, Pi.mul_def] using
+      (integerAlgebra k).mul_mem hP (diagonal_coordinate_mem_integerAlgebra k ij.1 ij.2)
+
+ 
+
+lemma rational_torus_integerExpansions {M : Type u} [AddCommGroup M]
+    [Module k M] [Module.Finite k M] (ρ : Representation k (SL2 k) M)
+    (hρ : IsRational k ρ) (v : M) : ∃ c : ℤ →₀ M, ∀ a : kˣ,
+      ρ (diagonalTorus k a) v = c.sum (fun n x => integerCharacter k n a • x) := by
+  apply vector_expansion_of_scalar_span
+  intro φ
+  obtain ⟨P, hP⟩ := hρ v φ
+  have heq : (fun a : kˣ => φ (ρ (diagonalTorus k a) v)) =
+      (fun a : kˣ => MvPolynomial.eval (fun ij => diagonalTorus k a ij.1 ij.2) P) := by
+    funext a
+    exact hP (diagonalTorus k a)
+  rw [heq, ← integerAlgebra_toSubmodule]
+  exact polynomial_on_torus_mem_integerAlgebra k P
+
+end
+
+section
+
+universe u v w
+variable {k : Type u} [Field k] {G : Type v}
+variable {M : Type w} [AddCommGroup M] [Module k M]
+
+lemma function_sum_eq_zero {I : Type*} (e : I → G → k)
+    (he : LinearIndependent k e) (s : Finset I) (c : I → M)
+    (h : ∀ g, ∑ i ∈ s, e i g • c i = 0) : ∀ i ∈ s, c i = 0 := by
+  intro i hi
+  apply (Module.forall_dual_apply_eq_zero_iff k (c i)).mp
+  intro φ
+  apply (linearIndependent_iff'.mp he) s (fun j => φ (c j)) ?_ i hi
+  ext g
+  have hg := congrArg φ (h g)
+  simpa only [map_sum, map_smul, map_zero, Finset.sum_apply, Pi.smul_apply,
+    smul_eq_mul, Pi.zero_apply, mul_comm] using hg
+
+lemma function_finsupp_eq_zero {I : Type*} (e : I → G → k)
+    (he : LinearIndependent k e) (c : I →₀ M)
+    (h : ∀ g, c.sum (fun i x => e i g • x) = 0) : c = 0 := by
+  classical
+  ext i
+  by_cases hi : i ∈ c.support
+  · exact function_sum_eq_zero e he c.support c h i hi
+  · simpa using Finsupp.notMem_support_iff.mp hi
+
+lemma function_finsupp_ext {I : Type*} (e : I → G → k)
+    (he : LinearIndependent k e) (c d : I →₀ M)
+    (h : ∀ g, c.sum (fun i x => e i g • x) = d.sum (fun i x => e i g • x)) : c = d := by
+  apply sub_eq_zero.mp
+  apply function_finsupp_eq_zero e he
+  intro g
+  rw [Finsupp.sum_sub_index]
+  · exact sub_eq_zero.mpr (h g)
+  · intro χ x y
+    exact smul_sub _ _ _
+
+variable (k)
+
+lemma linearIndependent_powerFunctions [Infinite k] :
+    LinearIndependent k (fun (n : ℕ) (x : k) => x ^ n) := by
+  classical
+  apply linearIndependent_iff'.mpr
+  intro s a ha i hi
+  have hp : (∑ j ∈ s, Polynomial.monomial j (a j)) = (0 : Polynomial k) := by
+    apply Polynomial.funext
+    intro x
+    have hx := congrFun ha x
+    simpa only [Polynomial.eval_finsetSum, Polynomial.eval_monomial,
+      Polynomial.eval_zero, Finset.sum_apply, Pi.smul_apply, Pi.zero_apply,
+      smul_eq_mul] using hx
+  have hc := congrArg (fun p : Polynomial k => p.coeff i) hp
+  simpa [Polynomial.coeff_monomial, hi] using hc
+
+end
+
+universe u
+variable (k : Type u) [Field k]
+
+def powerFunctions : Submonoid (k → k) where
+  carrier := Set.range (fun n : ℕ => fun t : k => t ^ n)
+  one_mem' := ⟨0, by ext t; simp⟩
+  mul_mem' := by
+    rintro _ _ ⟨n, rfl⟩ ⟨m, rfl⟩
+    exact ⟨n + m, by ext t; exact pow_add t n m⟩
+
+noncomputable def powerAlgebra : Subalgebra k (k → k) :=
+  Algebra.adjoin k (powerFunctions k : Set (k → k))
+
+lemma powerAlgebra_toSubmodule : (powerAlgebra k).toSubmodule =
+    Submodule.span k (Set.range (fun (n : ℕ) (t : k) => t ^ n)) := by
+  rw [powerAlgebra, Algebra.adjoin_eq_span, Submonoid.closure_eq]
+  rfl
+
+lemma power_mem_powerAlgebra (n : ℕ) :
+    (fun t : k => t ^ n) ∈ powerAlgebra k := Algebra.subset_adjoin ⟨n, rfl⟩
+
+lemma unipotent_coordinate_mem_powerAlgebra (i j : Fin 2) :
+    (fun t : k => unipotent k t i j) ∈ powerAlgebra k := by
+  fin_cases i <;> fin_cases j
+  · exact (powerAlgebra k).one_mem
+  · change (fun t : k => t) ∈ powerAlgebra k
+    simpa only [pow_one] using power_mem_powerAlgebra k 1
+  · exact (powerAlgebra k).zero_mem
+  · exact (powerAlgebra k).one_mem
+
+lemma polynomial_on_unipotent_mem_powerAlgebra
+    (P : MvPolynomial (Fin 2 × Fin 2) k) :
+    (fun t : k => MvPolynomial.eval (fun ij => unipotent k t ij.1 ij.2) P) ∈
+      powerAlgebra k := by
+  induction P using MvPolynomial.induction_on with
+  | C c =>
+    simpa only [MvPolynomial.eval_C, Pi.algebraMap_def, Algebra.algebraMap_self,
+      RingHom.id_apply] using (powerAlgebra k).algebraMap_mem c
+  | add P Q hP hQ =>
+    simpa only [MvPolynomial.eval_add, Pi.add_def] using (powerAlgebra k).add_mem hP hQ
+  | mul_X P ij hP =>
+    simpa only [MvPolynomial.eval_mul, MvPolynomial.eval_X, Pi.mul_def] using
+      (powerAlgebra k).mul_mem hP (unipotent_coordinate_mem_powerAlgebra k ij.1 ij.2)
+
+lemma rational_unipotent_expansion {M : Type u} [AddCommGroup M]
+    [Module k M] [Module.Finite k M] (ρ : Representation k (SL2 k) M)
+    (hρ : IsRational k ρ) (v : M) : ∃ c : ℕ →₀ M, ∀ t : k,
+      ρ (unipotent k t) v = c.sum (fun n x => t ^ n • x) := by
+  apply vector_expansion_of_scalar_span
+  intro φ
+  obtain ⟨P, hP⟩ := hρ v φ
+  have heq : (fun t : k => φ (ρ (unipotent k t) v)) =
+      (fun t : k => MvPolynomial.eval (fun ij => unipotent k t ij.1 ij.2) P) := by
+    funext t
+    exact hP (unipotent k t)
+  rw [heq, ← powerAlgebra_toSubmodule]
+  exact polynomial_on_unipotent_mem_powerAlgebra k P
+
+lemma diagonal_unipotent (a : kˣ) (t : k) :
+    diagonalTorus k a * unipotent k t =
+      unipotent k ((a : k) ^ 2 * t) * diagonalTorus k a := by
+  apply Subtype.ext
+  change !![(a : k), 0; 0, (↑a⁻¹ : k)] * !![1, t; 0, 1] =
+    !![1, (a : k) ^ 2 * t; 0, 1] * !![(a : k), 0; 0, (↑a⁻¹ : k)]
+  ext i j
+  fin_cases i <;> fin_cases j <;>
+    simp [Matrix.mul_apply, Fin.sum_univ_two, pow_two, mul_assoc, mul_comm]
+
+ 
+
+lemma unipotent_coefficient_weight [Infinite k] {M : Type u}
+    [AddCommGroup M] [Module k M] (ρ : Representation k (SL2 k) M)
+    (v : M) (n : ℤ) (hv : ∀ a, ρ (diagonalTorus k a) v = integerCharacter k n a • v)
+    (c : ℕ →₀ M) (hc : ∀ t, ρ (unipotent k t) v = c.sum (fun i x => t ^ i • x)) :
+    ∀ i a, ρ (diagonalTorus k a) (c i) = integerCharacter k (n + 2 * i) a • c i := by
+  classical
+  intro i a
+  by_cases hi : i ∈ c.support
+  · apply sub_eq_zero.mp
+    apply function_sum_eq_zero (fun (j : ℕ) (t : k) => t ^ j)
+      (linearIndependent_powerFunctions k) c.support
+      (fun j => ρ (diagonalTorus k a) (c j) - integerCharacter k (n + 2 * j) a • c j)
+      ?_ i hi
+    intro t
+    calc
+      ∑ j ∈ c.support, t ^ j •
+          (ρ (diagonalTorus k a) (c j) - integerCharacter k (n + 2 * j) a • c j) =
+          ρ (diagonalTorus k a) (c.sum (fun j x => t ^ j • x)) -
+            integerCharacter k n a • c.sum (fun j x => ((a : k) ^ 2 * t) ^ j • x) := by
+        simp only [Finsupp.sum, map_sum, map_smul, smul_sub, Finset.sum_sub_distrib,
+          Finset.smul_sum, smul_smul]
+        congr 1
+        apply Finset.sum_congr rfl
+        intro j hj
+        congr 1
+        simp [integerCharacter, zpow_add, zpow_mul, mul_pow, ← pow_mul, mul_comm, mul_left_comm]
+      _ = 0 := by
+        rw [← hc, ← hc]
+        apply sub_eq_zero.mpr
+        have hh := congrArg (fun g => ρ g v) (diagonal_unipotent k a t)
+        simpa only [map_mul, Module.End.mul_apply, hv, map_smul] using hh
+  · rw [Finsupp.notMem_support_iff.mp hi, map_zero, smul_zero]
+
+end FiberFunctorProof
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/RestrictedTilting/ClearedPolynomial.lean b/lean/OAI/RepresentationTheory/RestrictedTilting/ClearedPolynomial.lean
index 555133fe7aa5cc49d017953491e5ced1d99154b9..fd108f0aef98a6253db1587f3a312fe485698662 100644
--- a/lean/OAI/RepresentationTheory/RestrictedTilting/ClearedPolynomial.lean
+++ b/lean/OAI/RepresentationTheory/RestrictedTilting/ClearedPolynomial.lean
@@ -1,61 +1,61 @@
-import Mathlib
-
-namespace OAI
-
-namespace FiberFunctorProof
-open scoped BigOperators
-section
-variable {k R : Type*} [CommRing k] [CommRing R] [Algebra k R]
- 
-
-def clearedPolynomial (p : Polynomial k) (a b : R) (d : ℕ) : R :=
-  ∑ n ∈ Finset.range (d+1), algebraMap k R (p.coeff n) * a^n * b^(d-n)
-
-lemma map_clearedPolynomial {S : Type*} [CommRing S] [Algebra k S]
-    (f : R →ₐ[k] S) (p : Polynomial k) (a b : R) (d : ℕ) :
-    f (clearedPolynomial p a b d) = clearedPolynomial p (f a) (f b) d := by
-  simp only [clearedPolynomial, map_sum, map_mul, map_pow, AlgHom.commutes]
-
-lemma clearedPolynomial_eval {L : Type*} [Field L] [Algebra k L]
-    (f : R →ₐ[k] L) (p : Polynomial k) (a b : R) (d : ℕ)
-    (hd : p.natDegree ≤ d) (hb : f b ≠ 0) :
-    f (clearedPolynomial p a b d) = f b ^ d * Polynomial.aeval (f a * (f b)⁻¹) p := by
-  rw [Polynomial.aeval_eq_sum_range' (Nat.lt_succ_of_le hd), Finset.mul_sum]
-  simp only [clearedPolynomial, map_sum, map_mul, map_pow, AlgHom.commutes]
-  apply Finset.sum_congr rfl
-  intro n hn
-  rw [pow_sub₀ _ hb (Nat.lt_succ_iff.mp (Finset.mem_range.mp hn)), mul_pow, inv_pow]
-  simp only [Algebra.smul_def]
-  ring
-end
-
-section
-variable (k : Type*) [Field k]
-noncomputable def linePolynomial : MvPolynomial (Fin 2) k →ₐ[k] Polynomial k :=
-  MvPolynomial.aeval (fun i => if i = 0 then Polynomial.X else 0)
-
-@[simp] lemma linePolynomial_eval (a : k) (P : MvPolynomial (Fin 2) k) :
-    (linePolynomial k P).eval a = MvPolynomial.eval (fun i => if i = 0 then a else 0) P := by
-  change (Polynomial.aeval a).comp (linePolynomial k) P = _
-  have he : (Polynomial.aeval a).comp (linePolynomial k) =
-      MvPolynomial.aeval (fun i => if i = 0 then a else 0) := by
-    ext i
-    simp only [AlgHom.comp_apply, linePolynomial, MvPolynomial.aeval_X]
-    split_ifs <;> simp
-  rw [he]
-  rfl
-
-noncomputable def clearedLineAction (p : Polynomial k) (t : k) (d : ℕ) : Polynomial k :=
-  clearedPolynomial p (Polynomial.C t) ((Polynomial.X : Polynomial k)^2) d
-
-lemma clearedLineAction_eval (p : Polynomial k) (t a : k) (d : ℕ)
-    (hd : p.natDegree ≤ d) (ha : a ≠ 0) :
-    (clearedLineAction k p t d).eval a = a^(2*d) * p.eval (t / a^2) := by
-  have h := clearedPolynomial_eval (Polynomial.aeval a) p
-    (Polynomial.C t) ((Polynomial.X : Polynomial k)^2) d hd (by simpa using pow_ne_zero 2 ha)
-  simpa [clearedLineAction, Polynomial.aeval_def, Algebra.algebraMap_self,
-    ← pow_mul, div_eq_mul_inv] using h
-end
-end FiberFunctorProof
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+namespace FiberFunctorProof
+open scoped BigOperators
+section
+variable {k R : Type*} [CommRing k] [CommRing R] [Algebra k R]
+ 
+
+def clearedPolynomial (p : Polynomial k) (a b : R) (d : ℕ) : R :=
+  ∑ n ∈ Finset.range (d+1), algebraMap k R (p.coeff n) * a^n * b^(d-n)
+
+lemma map_clearedPolynomial {S : Type*} [CommRing S] [Algebra k S]
+    (f : R →ₐ[k] S) (p : Polynomial k) (a b : R) (d : ℕ) :
+    f (clearedPolynomial p a b d) = clearedPolynomial p (f a) (f b) d := by
+  simp only [clearedPolynomial, map_sum, map_mul, map_pow, AlgHom.commutes]
+
+lemma clearedPolynomial_eval {L : Type*} [Field L] [Algebra k L]
+    (f : R →ₐ[k] L) (p : Polynomial k) (a b : R) (d : ℕ)
+    (hd : p.natDegree ≤ d) (hb : f b ≠ 0) :
+    f (clearedPolynomial p a b d) = f b ^ d * Polynomial.aeval (f a * (f b)⁻¹) p := by
+  rw [Polynomial.aeval_eq_sum_range' (Nat.lt_succ_of_le hd), Finset.mul_sum]
+  simp only [clearedPolynomial, map_sum, map_mul, map_pow, AlgHom.commutes]
+  apply Finset.sum_congr rfl
+  intro n hn
+  rw [pow_sub₀ _ hb (Nat.lt_succ_iff.mp (Finset.mem_range.mp hn)), mul_pow, inv_pow]
+  simp only [Algebra.smul_def]
+  ring
+end
+
+section
+variable (k : Type*) [Field k]
+noncomputable def linePolynomial : MvPolynomial (Fin 2) k →ₐ[k] Polynomial k :=
+  MvPolynomial.aeval (fun i => if i = 0 then Polynomial.X else 0)
+
+@[simp] lemma linePolynomial_eval (a : k) (P : MvPolynomial (Fin 2) k) :
+    (linePolynomial k P).eval a = MvPolynomial.eval (fun i => if i = 0 then a else 0) P := by
+  change (Polynomial.aeval a).comp (linePolynomial k) P = _
+  have he : (Polynomial.aeval a).comp (linePolynomial k) =
+      MvPolynomial.aeval (fun i => if i = 0 then a else 0) := by
+    ext i
+    simp only [AlgHom.comp_apply, linePolynomial, MvPolynomial.aeval_X]
+    split_ifs <;> simp
+  rw [he]
+  rfl
+
+noncomputable def clearedLineAction (p : Polynomial k) (t : k) (d : ℕ) : Polynomial k :=
+  clearedPolynomial p (Polynomial.C t) ((Polynomial.X : Polynomial k)^2) d
+
+lemma clearedLineAction_eval (p : Polynomial k) (t a : k) (d : ℕ)
+    (hd : p.natDegree ≤ d) (ha : a ≠ 0) :
+    (clearedLineAction k p t d).eval a = a^(2*d) * p.eval (t / a^2) := by
+  have h := clearedPolynomial_eval (Polynomial.aeval a) p
+    (Polynomial.C t) ((Polynomial.X : Polynomial k)^2) d hd (by simpa using pow_ne_zero 2 ha)
+  simpa [clearedLineAction, Polynomial.aeval_def, Algebra.algebraMap_self,
+    ← pow_mul, div_eq_mul_inv] using h
+end
+end FiberFunctorProof
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/RestrictedTilting/CompatibleJets.lean b/lean/OAI/RepresentationTheory/RestrictedTilting/CompatibleJets.lean
index 9b0368a40a9770f5891427b1a78f98f196ff1b36..e27650edc88fbd433c583937ee4a3bd9762dba79 100644
--- a/lean/OAI/RepresentationTheory/RestrictedTilting/CompatibleJets.lean
+++ b/lean/OAI/RepresentationTheory/RestrictedTilting/CompatibleJets.lean
@@ -1,206 +1,206 @@
-import OAI.RepresentationTheory.RestrictedTilting.Basic
-
-namespace OAI
-
-namespace FiberFunctorProof
-open CategoryTheory CategoryTheory.MonoidalCategory CategoryTheory.Limits
-open scoped TensorProduct
-universe u
-variable (k : Type u) [Field k]
-
- 
-lemma IsRational.prod {M N : Type u} [AddCommGroup M] [Module k M]
-    [AddCommGroup N] [Module k N]
-    {ρ : Representation k (SL2 k) M} {σ : Representation k (SL2 k) N}
-    (hρ : IsRational k ρ) (hσ : IsRational k σ) :
-    IsRational k (Representation.prod ρ σ) := by
-  intro v l
-  obtain ⟨P,hP⟩ := hρ v.1 (l.comp (LinearMap.inl k M N))
-  obtain ⟨Q,hQ⟩ := hσ v.2 (l.comp (LinearMap.inr k M N))
-  refine ⟨P+Q, fun g => ?_⟩
-  rw [MvPolynomial.eval_add, ← hP, ← hQ]
-  change l _ = l (ρ g v.1, 0) + l (0, σ g v.2)
-  rw [← map_add]
-  congr 1
-  ext <;> simp [Representation.prod]
-
- 
-def representationPullback {G : Type u} [Group G] {B C T : FDRep k G}
-    (g : B ⟶ C) (h : T ⟶ C) : Subrepresentation (Representation.prod B.ρ T.ρ) where
-  toSubmodule := LinearMap.ker
-    ((fdrepLinear k g).comp (LinearMap.fst k B T) -
-      (fdrepLinear k h).comp (LinearMap.snd k B T))
-  apply_mem_toSubmodule a v hv := by
-    change fdrepLinear k g (B.ρ a v.1) - fdrepLinear k h (T.ρ a v.2) = 0
-    rw [fdrepLinear_comm, fdrepLinear_comm, ← map_sub]
-    change fdrepLinear k g v.1 - fdrepLinear k h v.2 = 0 at hv
-    rw [hv, map_zero]
-
- 
-
-theorem good_weyl_lift [Infinite k] {A B C T : FDRep k (SL2 k)}
-    (f : A ⟶ B) (g : B ⟶ C) (w : f ≫ g = 0)
-    (hS : (ShortComplex.mk f g w).ShortExact)
-    (hB : IsRational k B.ρ) (hT : IsTilting k T)
-    (hA : HasGoodFiltration k A) (h : T ⟶ C) :
-    ∃ s : T ⟶ B, s ≫ g = h := by
-  let PB := representationPullback k g h
-  let E : FDRep k (SL2 k) := FDRep.of PB.toRepresentation
-  have he (v : PB) : fdrepLinear k g v.val.1 = fdrepLinear k h v.val.2 :=
-    sub_eq_zero.mp v.property
-  let i : A ⟶ E := fdrepHom k
-    { toFun := fun a => ⟨(fdrepLinear k f a, 0), by
-        change fdrepLinear k g (fdrepLinear k f a) - fdrepLinear k h 0 = 0
-        rw [map_zero, sub_zero]
-        exact congrArg (fun z => fdrepLinear k z a) w⟩
-      map_add' := by
-        intro a b
-        apply Subtype.ext
-        change (fdrepLinear k f (a + b), (0 : T)) =
-          (fdrepLinear k f a, (0 : T)) + (fdrepLinear k f b, (0 : T))
-        simp
-      map_smul' := by
-        intro c a
-        apply Subtype.ext
-        change (fdrepLinear k f (c • a), (0 : T)) = c • (fdrepLinear k f a, (0 : T))
-        simp }
-    (by
-      intro a v
-      apply Subtype.ext
-      change (fdrepLinear k f (A.ρ a v), (0 : T)) = (B.ρ a (fdrepLinear k f v), T.ρ a 0)
-      exact Prod.ext (fdrepLinear_comm k f a v) (map_zero (T.ρ a)).symm)
-  let q : E ⟶ T := fdrepHom k ((LinearMap.snd k B T).comp PB.toSubmodule.subtype)
-    (by intros; rfl)
-  let r : E ⟶ B := fdrepHom k ((LinearMap.fst k B T).comp PB.toSubmodule.subtype)
-    (by intros; rfl)
-  have viq : i ≫ q = 0 := by apply fdrep_hom_ext k; intro a; rfl
-  have hi : Function.Injective (fdrepLinear k i) := by
-    have := hS.mono_f
-    intro a b hab
-    apply fdrep_injective k f
-    exact congrArg (fun x : PB => x.val.1) hab
-  have hq : Function.Surjective (fdrepLinear k q) := by
-    have := hS.epi_g
-    intro t
-    obtain ⟨b,hb⟩ := fdrep_surjective k g (fdrepLinear k h t)
-    exact ⟨⟨(b,t), sub_eq_zero.mpr hb⟩,rfl⟩
-  have hex : LinearMap.range (fdrepLinear k i) = LinearMap.ker (fdrepLinear k q) := by
-    ext v
-    constructor
-    · rintro ⟨a,rfl⟩
-      rfl
-    · intro hv
-      have hv2 : v.val.2 = 0 := hv
-      have hv1 : v.val.1 ∈ LinearMap.ker (fdrepLinear k g) := by
-        change fdrepLinear k g v.val.1 = 0
-        rw [he v, hv2, map_zero]
-      rw [← fdrep_exact k hS.exact] at hv1
-      obtain ⟨a,ha⟩ := hv1
-      refine ⟨a, ?_⟩
-      apply Subtype.ext
-      exact Prod.ext ha hv2.symm
-  have hE : IsRational k E.ρ :=
-    IsRational.of_injective k E.ρ (Representation.prod B.ρ T.ρ)
-      PB.toSubmodule.subtype Subtype.val_injective (by intros; rfl) (hB.prod k hT.1)
-  obtain ⟨s,hs⟩ := good_weyl_extension_split k hA hT.2.2 E hE i q viq
-    (fdrep_shortExact k (ShortComplex.mk i q viq) hi hq hex)
-  refine ⟨s ≫ r, ?_⟩
-  apply fdrep_hom_ext k
-  intro t
-  change fdrepLinear k g (fdrepLinear k s t).val.1 = fdrepLinear k h t
-  rw [he]
-  exact congrArg (fdrepLinear k h) (congrArg (fun z => fdrepLinear k z t) hs)
-
-variable {M : Type u} [AddCommGroup M] [Module k M]
-  [Module (MvPolynomial (Fin 2) k) M] [IsScalarTower k (MvPolynomial (Fin 2) k) M]
-  [Module.Finite k (M ⧸ planeJetRelation (M := M) k 1)]
-
- 
-noncomputable def planeJetFD (σ : Representation k (SL2 k) M)
-    (hσ : PlaneCompatible k σ) (j : ℕ) : FDRep k (SL2 k) := by
-  let := planeJet_finite (M := M) k j
-  exact FDRep.of (planeJetRep k σ hσ j)
-
-noncomputable def planeJetLayerFD (σ : Representation k (SL2 k) M)
-    (hσ : PlaneCompatible k σ) (j : ℕ) : FDRep k (SL2 k) := by
-  let := planeLayer_finite (M := M) k j
-  exact FDRep.of (planeJetLayerRep k σ hσ j)
-
-noncomputable def planeJetTransitionHom (σ : Representation k (SL2 k) M)
-    (hσ : PlaneCompatible k σ) (j : ℕ) :
-    planeJetFD k σ hσ (j+1) ⟶ planeJetFD k σ hσ j :=
-  fdrepHom k (planeJetTransition k j) (planeJetTransition_intertwines k σ hσ j)
-
-noncomputable def planeJetLayerInclusionHom (σ : Representation k (SL2 k) M)
-    (hσ : PlaneCompatible k σ) (j : ℕ) :
-    planeJetLayerFD k σ hσ j ⟶ planeJetFD k σ hσ (j+1) :=
-  fdrepHom k (planeJetLayerInclusion k j) (planeJetLayerInclusion_intertwines k σ hσ j)
-
-noncomputable def planeLayerMultiplyHom (σ : Representation k (SL2 k) M)
-    (hσ : PlaneCompatible k σ) (j : ℕ) :
-    costandard k j ⊗ planeJetFD k σ hσ 1 ⟶ planeJetLayerFD k σ hσ j :=
-  fdrepHom k (planeLayerMultiply k j) (planeLayerMultiply_intertwines k σ hσ j)
-
- 
-
-theorem planeJetLayer_good (σ : Representation k (SL2 k) M)
-    (hσ : PlaneCompatible k σ)
-    (hx : Function.Injective (fun m : M => (MvPolynomial.X 0 : MvPolynomial (Fin 2) k) • m))
-    (hy : ∀ v n : M, (MvPolynomial.X 1 : MvPolynomial (Fin 2) k) • v =
-      (MvPolynomial.X 0 : MvPolynomial (Fin 2) k) • n →
-      ∃ w : M, v = (MvPolynomial.X 0 : MvPolynomial (Fin 2) k) • w)
-    (hT : HasGoodFiltration k (planeJetFD k σ hσ 1)) (j : ℕ) :
-    HasGoodFiltration k (planeJetLayerFD k σ hσ j) := by
-  let f := planeLayerMultiplyHom k σ hσ j
-  let : IsIso f := fdrepHom_isIso_of_bijective k f
-    ⟨planeLayerMultiply_injective k hx hy j, planeLayerMultiply_surjective k j⟩
-  exact ((costandard_hasGoodFiltration k j).tensor k hT).of_iso k (asIso f)
-
- 
-
-theorem planeJet_lift [Infinite k] (σ : Representation k (SL2 k) M)
-    (hσ : PlaneCompatible k σ) (hr : IsRational k σ)
-    (hx : Function.Injective (fun m : M => (MvPolynomial.X 0 : MvPolynomial (Fin 2) k) • m))
-    (hy : ∀ v n : M, (MvPolynomial.X 1 : MvPolynomial (Fin 2) k) • v =
-      (MvPolynomial.X 0 : MvPolynomial (Fin 2) k) • n →
-      ∃ w : M, v = (MvPolynomial.X 0 : MvPolynomial (Fin 2) k) • w)
-    (hT : IsTilting k (planeJetFD k σ hσ 1)) (j : ℕ)
-    (h : planeJetFD k σ hσ 1 ⟶ planeJetFD k σ hσ j) :
-    ∃ s : planeJetFD k σ hσ 1 ⟶ planeJetFD k σ hσ (j+1),
-      s ≫ planeJetTransitionHom k σ hσ j = h := by
-  let f := planeJetLayerInclusionHom k σ hσ j
-  let g := planeJetTransitionHom k σ hσ j
-  have w : f ≫ g = 0 := by
-    apply fdrep_hom_ext k
-    intro v
-    change planeJetTransition (M := M) k j (planeJetLayerInclusion k j v) = 0
-    exact (planeJet_shortExact (M := M) k j (planeJetLayerInclusion k j v)).2 ⟨v,rfl⟩
-  have hS : (ShortComplex.mk f g w).ShortExact := by
-    apply fdrep_shortExact k _ (planeJetLayerInclusion_injective k j)
-      (planeJetTransition_surjective k j)
-    change LinearMap.range (planeJetLayerInclusion (M := M) k j) =
-      LinearMap.ker (planeJetTransition (M := M) k j)
-    ext v
-    exact (planeJet_shortExact (M := M) k j v).symm
-  exact good_weyl_lift k f g w hS (planeJetRep_rational k σ hσ hr (j+1)) hT
-    (planeJetLayer_good k σ hσ hx hy hT.2.1 j) h
-
- 
-
-theorem compatible_planeJet_sections [Infinite k] (σ : Representation k (SL2 k) M)
-    (hσ : PlaneCompatible k σ) (hr : IsRational k σ)
-    (hx : Function.Injective (fun m : M => (MvPolynomial.X 0 : MvPolynomial (Fin 2) k) • m))
-    (hy : ∀ v n : M, (MvPolynomial.X 1 : MvPolynomial (Fin 2) k) • v =
-      (MvPolynomial.X 0 : MvPolynomial (Fin 2) k) • n →
-      ∃ w : M, v = (MvPolynomial.X 0 : MvPolynomial (Fin 2) k) • w)
-    (hT : IsTilting k (planeJetFD k σ hσ 1)) :
-    ∃ s : (j : ℕ) → planeJetFD k σ hσ 1 ⟶ planeJetFD k σ hσ (j+1),
-      s 0 = 𝟙 _ ∧ ∀ j, s (j+1) ≫ planeJetTransitionHom k σ hσ (j+1) = s j := by
-  classical
-  let s : (j : ℕ) → planeJetFD k σ hσ 1 ⟶ planeJetFD k σ hσ (j+1) :=
-    Nat.rec (𝟙 _) (fun j t => Classical.choose (planeJet_lift k σ hσ hr hx hy hT (j+1) t))
-  refine ⟨s,rfl,fun j => ?_⟩
-  exact Classical.choose_spec (planeJet_lift k σ hσ hr hx hy hT (j+1) (s j))
-end FiberFunctorProof
-
-end OAI
+import OAI.RepresentationTheory.RestrictedTilting.Basic
+
+namespace OAI
+
+namespace FiberFunctorProof
+open CategoryTheory CategoryTheory.MonoidalCategory CategoryTheory.Limits
+open scoped TensorProduct
+universe u
+variable (k : Type u) [Field k]
+
+ 
+lemma IsRational.prod {M N : Type u} [AddCommGroup M] [Module k M]
+    [AddCommGroup N] [Module k N]
+    {ρ : Representation k (SL2 k) M} {σ : Representation k (SL2 k) N}
+    (hρ : IsRational k ρ) (hσ : IsRational k σ) :
+    IsRational k (Representation.prod ρ σ) := by
+  intro v l
+  obtain ⟨P,hP⟩ := hρ v.1 (l.comp (LinearMap.inl k M N))
+  obtain ⟨Q,hQ⟩ := hσ v.2 (l.comp (LinearMap.inr k M N))
+  refine ⟨P+Q, fun g => ?_⟩
+  rw [MvPolynomial.eval_add, ← hP, ← hQ]
+  change l _ = l (ρ g v.1, 0) + l (0, σ g v.2)
+  rw [← map_add]
+  congr 1
+  ext <;> simp [Representation.prod]
+
+ 
+def representationPullback {G : Type u} [Group G] {B C T : FDRep k G}
+    (g : B ⟶ C) (h : T ⟶ C) : Subrepresentation (Representation.prod B.ρ T.ρ) where
+  toSubmodule := LinearMap.ker
+    ((fdrepLinear k g).comp (LinearMap.fst k B T) -
+      (fdrepLinear k h).comp (LinearMap.snd k B T))
+  apply_mem_toSubmodule a v hv := by
+    change fdrepLinear k g (B.ρ a v.1) - fdrepLinear k h (T.ρ a v.2) = 0
+    rw [fdrepLinear_comm, fdrepLinear_comm, ← map_sub]
+    change fdrepLinear k g v.1 - fdrepLinear k h v.2 = 0 at hv
+    rw [hv, map_zero]
+
+ 
+
+theorem good_weyl_lift [Infinite k] {A B C T : FDRep k (SL2 k)}
+    (f : A ⟶ B) (g : B ⟶ C) (w : f ≫ g = 0)
+    (hS : (ShortComplex.mk f g w).ShortExact)
+    (hB : IsRational k B.ρ) (hT : IsTilting k T)
+    (hA : HasGoodFiltration k A) (h : T ⟶ C) :
+    ∃ s : T ⟶ B, s ≫ g = h := by
+  let PB := representationPullback k g h
+  let E : FDRep k (SL2 k) := FDRep.of PB.toRepresentation
+  have he (v : PB) : fdrepLinear k g v.val.1 = fdrepLinear k h v.val.2 :=
+    sub_eq_zero.mp v.property
+  let i : A ⟶ E := fdrepHom k
+    { toFun := fun a => ⟨(fdrepLinear k f a, 0), by
+        change fdrepLinear k g (fdrepLinear k f a) - fdrepLinear k h 0 = 0
+        rw [map_zero, sub_zero]
+        exact congrArg (fun z => fdrepLinear k z a) w⟩
+      map_add' := by
+        intro a b
+        apply Subtype.ext
+        change (fdrepLinear k f (a + b), (0 : T)) =
+          (fdrepLinear k f a, (0 : T)) + (fdrepLinear k f b, (0 : T))
+        simp
+      map_smul' := by
+        intro c a
+        apply Subtype.ext
+        change (fdrepLinear k f (c • a), (0 : T)) = c • (fdrepLinear k f a, (0 : T))
+        simp }
+    (by
+      intro a v
+      apply Subtype.ext
+      change (fdrepLinear k f (A.ρ a v), (0 : T)) = (B.ρ a (fdrepLinear k f v), T.ρ a 0)
+      exact Prod.ext (fdrepLinear_comm k f a v) (map_zero (T.ρ a)).symm)
+  let q : E ⟶ T := fdrepHom k ((LinearMap.snd k B T).comp PB.toSubmodule.subtype)
+    (by intros; rfl)
+  let r : E ⟶ B := fdrepHom k ((LinearMap.fst k B T).comp PB.toSubmodule.subtype)
+    (by intros; rfl)
+  have viq : i ≫ q = 0 := by apply fdrep_hom_ext k; intro a; rfl
+  have hi : Function.Injective (fdrepLinear k i) := by
+    have := hS.mono_f
+    intro a b hab
+    apply fdrep_injective k f
+    exact congrArg (fun x : PB => x.val.1) hab
+  have hq : Function.Surjective (fdrepLinear k q) := by
+    have := hS.epi_g
+    intro t
+    obtain ⟨b,hb⟩ := fdrep_surjective k g (fdrepLinear k h t)
+    exact ⟨⟨(b,t), sub_eq_zero.mpr hb⟩,rfl⟩
+  have hex : LinearMap.range (fdrepLinear k i) = LinearMap.ker (fdrepLinear k q) := by
+    ext v
+    constructor
+    · rintro ⟨a,rfl⟩
+      rfl
+    · intro hv
+      have hv2 : v.val.2 = 0 := hv
+      have hv1 : v.val.1 ∈ LinearMap.ker (fdrepLinear k g) := by
+        change fdrepLinear k g v.val.1 = 0
+        rw [he v, hv2, map_zero]
+      rw [← fdrep_exact k hS.exact] at hv1
+      obtain ⟨a,ha⟩ := hv1
+      refine ⟨a, ?_⟩
+      apply Subtype.ext
+      exact Prod.ext ha hv2.symm
+  have hE : IsRational k E.ρ :=
+    IsRational.of_injective k E.ρ (Representation.prod B.ρ T.ρ)
+      PB.toSubmodule.subtype Subtype.val_injective (by intros; rfl) (hB.prod k hT.1)
+  obtain ⟨s,hs⟩ := good_weyl_extension_split k hA hT.2.2 E hE i q viq
+    (fdrep_shortExact k (ShortComplex.mk i q viq) hi hq hex)
+  refine ⟨s ≫ r, ?_⟩
+  apply fdrep_hom_ext k
+  intro t
+  change fdrepLinear k g (fdrepLinear k s t).val.1 = fdrepLinear k h t
+  rw [he]
+  exact congrArg (fdrepLinear k h) (congrArg (fun z => fdrepLinear k z t) hs)
+
+variable {M : Type u} [AddCommGroup M] [Module k M]
+  [Module (MvPolynomial (Fin 2) k) M] [IsScalarTower k (MvPolynomial (Fin 2) k) M]
+  [Module.Finite k (M ⧸ planeJetRelation (M := M) k 1)]
+
+ 
+noncomputable def planeJetFD (σ : Representation k (SL2 k) M)
+    (hσ : PlaneCompatible k σ) (j : ℕ) : FDRep k (SL2 k) := by
+  let := planeJet_finite (M := M) k j
+  exact FDRep.of (planeJetRep k σ hσ j)
+
+noncomputable def planeJetLayerFD (σ : Representation k (SL2 k) M)
+    (hσ : PlaneCompatible k σ) (j : ℕ) : FDRep k (SL2 k) := by
+  let := planeLayer_finite (M := M) k j
+  exact FDRep.of (planeJetLayerRep k σ hσ j)
+
+noncomputable def planeJetTransitionHom (σ : Representation k (SL2 k) M)
+    (hσ : PlaneCompatible k σ) (j : ℕ) :
+    planeJetFD k σ hσ (j+1) ⟶ planeJetFD k σ hσ j :=
+  fdrepHom k (planeJetTransition k j) (planeJetTransition_intertwines k σ hσ j)
+
+noncomputable def planeJetLayerInclusionHom (σ : Representation k (SL2 k) M)
+    (hσ : PlaneCompatible k σ) (j : ℕ) :
+    planeJetLayerFD k σ hσ j ⟶ planeJetFD k σ hσ (j+1) :=
+  fdrepHom k (planeJetLayerInclusion k j) (planeJetLayerInclusion_intertwines k σ hσ j)
+
+noncomputable def planeLayerMultiplyHom (σ : Representation k (SL2 k) M)
+    (hσ : PlaneCompatible k σ) (j : ℕ) :
+    costandard k j ⊗ planeJetFD k σ hσ 1 ⟶ planeJetLayerFD k σ hσ j :=
+  fdrepHom k (planeLayerMultiply k j) (planeLayerMultiply_intertwines k σ hσ j)
+
+ 
+
+theorem planeJetLayer_good (σ : Representation k (SL2 k) M)
+    (hσ : PlaneCompatible k σ)
+    (hx : Function.Injective (fun m : M => (MvPolynomial.X 0 : MvPolynomial (Fin 2) k) • m))
+    (hy : ∀ v n : M, (MvPolynomial.X 1 : MvPolynomial (Fin 2) k) • v =
+      (MvPolynomial.X 0 : MvPolynomial (Fin 2) k) • n →
+      ∃ w : M, v = (MvPolynomial.X 0 : MvPolynomial (Fin 2) k) • w)
+    (hT : HasGoodFiltration k (planeJetFD k σ hσ 1)) (j : ℕ) :
+    HasGoodFiltration k (planeJetLayerFD k σ hσ j) := by
+  let f := planeLayerMultiplyHom k σ hσ j
+  let : IsIso f := fdrepHom_isIso_of_bijective k f
+    ⟨planeLayerMultiply_injective k hx hy j, planeLayerMultiply_surjective k j⟩
+  exact ((costandard_hasGoodFiltration k j).tensor k hT).of_iso k (asIso f)
+
+ 
+
+theorem planeJet_lift [Infinite k] (σ : Representation k (SL2 k) M)
+    (hσ : PlaneCompatible k σ) (hr : IsRational k σ)
+    (hx : Function.Injective (fun m : M => (MvPolynomial.X 0 : MvPolynomial (Fin 2) k) • m))
+    (hy : ∀ v n : M, (MvPolynomial.X 1 : MvPolynomial (Fin 2) k) • v =
+      (MvPolynomial.X 0 : MvPolynomial (Fin 2) k) • n →
+      ∃ w : M, v = (MvPolynomial.X 0 : MvPolynomial (Fin 2) k) • w)
+    (hT : IsTilting k (planeJetFD k σ hσ 1)) (j : ℕ)
+    (h : planeJetFD k σ hσ 1 ⟶ planeJetFD k σ hσ j) :
+    ∃ s : planeJetFD k σ hσ 1 ⟶ planeJetFD k σ hσ (j+1),
+      s ≫ planeJetTransitionHom k σ hσ j = h := by
+  let f := planeJetLayerInclusionHom k σ hσ j
+  let g := planeJetTransitionHom k σ hσ j
+  have w : f ≫ g = 0 := by
+    apply fdrep_hom_ext k
+    intro v
+    change planeJetTransition (M := M) k j (planeJetLayerInclusion k j v) = 0
+    exact (planeJet_shortExact (M := M) k j (planeJetLayerInclusion k j v)).2 ⟨v,rfl⟩
+  have hS : (ShortComplex.mk f g w).ShortExact := by
+    apply fdrep_shortExact k _ (planeJetLayerInclusion_injective k j)
+      (planeJetTransition_surjective k j)
+    change LinearMap.range (planeJetLayerInclusion (M := M) k j) =
+      LinearMap.ker (planeJetTransition (M := M) k j)
+    ext v
+    exact (planeJet_shortExact (M := M) k j v).symm
+  exact good_weyl_lift k f g w hS (planeJetRep_rational k σ hσ hr (j+1)) hT
+    (planeJetLayer_good k σ hσ hx hy hT.2.1 j) h
+
+ 
+
+theorem compatible_planeJet_sections [Infinite k] (σ : Representation k (SL2 k) M)
+    (hσ : PlaneCompatible k σ) (hr : IsRational k σ)
+    (hx : Function.Injective (fun m : M => (MvPolynomial.X 0 : MvPolynomial (Fin 2) k) • m))
+    (hy : ∀ v n : M, (MvPolynomial.X 1 : MvPolynomial (Fin 2) k) • v =
+      (MvPolynomial.X 0 : MvPolynomial (Fin 2) k) • n →
+      ∃ w : M, v = (MvPolynomial.X 0 : MvPolynomial (Fin 2) k) • w)
+    (hT : IsTilting k (planeJetFD k σ hσ 1)) :
+    ∃ s : (j : ℕ) → planeJetFD k σ hσ 1 ⟶ planeJetFD k σ hσ (j+1),
+      s 0 = 𝟙 _ ∧ ∀ j, s (j+1) ≫ planeJetTransitionHom k σ hσ (j+1) = s j := by
+  classical
+  let s : (j : ℕ) → planeJetFD k σ hσ 1 ⟶ planeJetFD k σ hσ (j+1) :=
+    Nat.rec (𝟙 _) (fun j t => Classical.choose (planeJet_lift k σ hσ hr hx hy hT (j+1) t))
+  refine ⟨s,rfl,fun j => ?_⟩
+  exact Classical.choose_spec (planeJet_lift k σ hσ hr hx hy hT (j+1) (s j))
+end FiberFunctorProof
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/RestrictedTilting/CompletedCoordinates.lean b/lean/OAI/RepresentationTheory/RestrictedTilting/CompletedCoordinates.lean
index 65d50c4b245d361256b5c9b0766f0dd67d6d0842..1e9126f34c55db5bb59d42190a445a58d0d314dd 100644
--- a/lean/OAI/RepresentationTheory/RestrictedTilting/CompletedCoordinates.lean
+++ b/lean/OAI/RepresentationTheory/RestrictedTilting/CompletedCoordinates.lean
@@ -1,61 +1,61 @@
-import OAI.RepresentationTheory.RestrictedTilting.SpecializeCompletion
-
-namespace OAI
-
-namespace FiberFunctorProof
-open scoped TensorProduct
-section
-variable {R M : Type*} [CommRing R] [AddCommGroup M] [Module R M]
-variable {ι : Type*} [Fintype ι] [DecidableEq ι]
-variable (I : Ideal R) (W : M →ₗ[R] (ι → R)) (Z : (ι → R) →ₗ[R] M) (a : R)
-variable (hWZ : ∀ v, W (Z v) = a • v) (hZW : ∀ v, Z (W v) = a • v)
-
-noncomputable def completedCoordinates : AdicCompletion I M →ₗ[AdicCompletion I R]
-    (ι → AdicCompletion I R) :=
-  (AdicCompletion.piEquivOfFintype I (fun _ : ι => R)).toLinearMap ∘ₗ AdicCompletion.map I W
-
-noncomputable def completedDenominator : (ι → AdicCompletion I R) →ₗ[AdicCompletion I R]
-    AdicCompletion I M :=
-  AdicCompletion.map I Z ∘ₗ (AdicCompletion.piEquivOfFintype I (fun _ : ι => R)).symm.toLinearMap
-
-lemma completedCoordinates_apply (v : AdicCompletion I M) (i : ι) :
-    completedCoordinates I W v i = AdicCompletion.map I ((LinearMap.proj i).comp W) v := by
-  change AdicCompletion.piEquivOfFintype I (fun _ : ι => R) (AdicCompletion.map I W v) i = _
-  rw [AdicCompletion.piEquivOfFintype_apply]
-  exact AdicCompletion.map_comp_apply I W (LinearMap.proj i) v
-
-include hWZ in
-lemma completedCoordinates_denominator (v : ι → AdicCompletion I R) :
-    completedCoordinates I W (completedDenominator I Z v) =
-      algebraMap R (AdicCompletion I R) a • v := by
-  have he : W.comp Z = a • LinearMap.id := LinearMap.ext hWZ
-  change AdicCompletion.piEquivOfFintype I (fun _ : ι => R)
-    (AdicCompletion.map I W (AdicCompletion.map I Z
-      ((AdicCompletion.piEquivOfFintype I (fun _ : ι => R)).symm v))) = _
-  rw [AdicCompletion.map_comp_apply, he, completion_map_scalar, map_smul,
-    LinearEquiv.apply_symm_apply]
-
-include hZW in
-lemma completedDenominator_coordinates (v : AdicCompletion I M) :
-    completedDenominator I Z (completedCoordinates I W v) =
-      algebraMap R (AdicCompletion I R) a • v := by
-  have he : Z.comp W = a • LinearMap.id := LinearMap.ext hZW
-  change AdicCompletion.map I Z
-    ((AdicCompletion.piEquivOfFintype I (fun _ : ι => R)).symm
-      ((AdicCompletion.piEquivOfFintype I (fun _ : ι => R)) (AdicCompletion.map I W v))) = _
-  rw [LinearEquiv.symm_apply_apply, AdicCompletion.map_comp_apply, he, completion_map_scalar]
-
-variable (L : Type*) [Field L] [Algebra (AdicCompletion I R) L]
-
-noncomputable def completedCoordinatesEquiv
-    (ha : algebraMap (AdicCompletion I R) L (algebraMap R (AdicCompletion I R) a) ≠ 0) :
-    (L ⊗[AdicCompletion I R] AdicCompletion I M) ≃ₗ[L]
-      (L ⊗[AdicCompletion I R] (ι → AdicCompletion I R)) :=
-  specializedScalarEquiv (completedCoordinates I W) (completedDenominator I Z)
-    (algebraMap R (AdicCompletion I R) a)
-    (completedCoordinates_denominator I W Z a hWZ)
-    (completedDenominator_coordinates I W Z a hZW) ha
-end
-end FiberFunctorProof
-
-end OAI
+import OAI.RepresentationTheory.RestrictedTilting.SpecializeCompletion
+
+namespace OAI
+
+namespace FiberFunctorProof
+open scoped TensorProduct
+section
+variable {R M : Type*} [CommRing R] [AddCommGroup M] [Module R M]
+variable {ι : Type*} [Fintype ι] [DecidableEq ι]
+variable (I : Ideal R) (W : M →ₗ[R] (ι → R)) (Z : (ι → R) →ₗ[R] M) (a : R)
+variable (hWZ : ∀ v, W (Z v) = a • v) (hZW : ∀ v, Z (W v) = a • v)
+
+noncomputable def completedCoordinates : AdicCompletion I M →ₗ[AdicCompletion I R]
+    (ι → AdicCompletion I R) :=
+  (AdicCompletion.piEquivOfFintype I (fun _ : ι => R)).toLinearMap ∘ₗ AdicCompletion.map I W
+
+noncomputable def completedDenominator : (ι → AdicCompletion I R) →ₗ[AdicCompletion I R]
+    AdicCompletion I M :=
+  AdicCompletion.map I Z ∘ₗ (AdicCompletion.piEquivOfFintype I (fun _ : ι => R)).symm.toLinearMap
+
+lemma completedCoordinates_apply (v : AdicCompletion I M) (i : ι) :
+    completedCoordinates I W v i = AdicCompletion.map I ((LinearMap.proj i).comp W) v := by
+  change AdicCompletion.piEquivOfFintype I (fun _ : ι => R) (AdicCompletion.map I W v) i = _
+  rw [AdicCompletion.piEquivOfFintype_apply]
+  exact AdicCompletion.map_comp_apply I W (LinearMap.proj i) v
+
+include hWZ in
+lemma completedCoordinates_denominator (v : ι → AdicCompletion I R) :
+    completedCoordinates I W (completedDenominator I Z v) =
+      algebraMap R (AdicCompletion I R) a • v := by
+  have he : W.comp Z = a • LinearMap.id := LinearMap.ext hWZ
+  change AdicCompletion.piEquivOfFintype I (fun _ : ι => R)
+    (AdicCompletion.map I W (AdicCompletion.map I Z
+      ((AdicCompletion.piEquivOfFintype I (fun _ : ι => R)).symm v))) = _
+  rw [AdicCompletion.map_comp_apply, he, completion_map_scalar, map_smul,
+    LinearEquiv.apply_symm_apply]
+
+include hZW in
+lemma completedDenominator_coordinates (v : AdicCompletion I M) :
+    completedDenominator I Z (completedCoordinates I W v) =
+      algebraMap R (AdicCompletion I R) a • v := by
+  have he : Z.comp W = a • LinearMap.id := LinearMap.ext hZW
+  change AdicCompletion.map I Z
+    ((AdicCompletion.piEquivOfFintype I (fun _ : ι => R)).symm
+      ((AdicCompletion.piEquivOfFintype I (fun _ : ι => R)) (AdicCompletion.map I W v))) = _
+  rw [LinearEquiv.symm_apply_apply, AdicCompletion.map_comp_apply, he, completion_map_scalar]
+
+variable (L : Type*) [Field L] [Algebra (AdicCompletion I R) L]
+
+noncomputable def completedCoordinatesEquiv
+    (ha : algebraMap (AdicCompletion I R) L (algebraMap R (AdicCompletion I R) a) ≠ 0) :
+    (L ⊗[AdicCompletion I R] AdicCompletion I M) ≃ₗ[L]
+      (L ⊗[AdicCompletion I R] (ι → AdicCompletion I R)) :=
+  specializedScalarEquiv (completedCoordinates I W) (completedDenominator I Z)
+    (algebraMap R (AdicCompletion I R) a)
+    (completedCoordinates_denominator I W Z a hWZ)
+    (completedDenominator_coordinates I W Z a hZW) ha
+end
+end FiberFunctorProof
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/RestrictedTilting/CompletionCoefficients.lean b/lean/OAI/RepresentationTheory/RestrictedTilting/CompletionCoefficients.lean
index 579ef87ac95cd696f23e103f3ef92fa49b4467bd..2cf24713eca365b8fad9a73c46fece2c800a2ff6 100644
--- a/lean/OAI/RepresentationTheory/RestrictedTilting/CompletionCoefficients.lean
+++ b/lean/OAI/RepresentationTheory/RestrictedTilting/CompletionCoefficients.lean
@@ -1,43 +1,43 @@
-import OAI.RepresentationTheory.RestrictedTilting.SpecializeCompletion
-
-namespace OAI
-
-namespace FiberFunctorProof
-variable {R M N : Type*} [CommRing R] [AddCommGroup M] [Module R M]
-  [AddCommGroup N] [Module R N] (I : Ideal R)
-
-attribute [local instance 1100] AdicCompletion.smul
-
-lemma completion_self_smul (a b : AdicCompletion I R) : a • b = a * b := by
-  apply AdicCompletion.ext
-  intro n
-  rw [AdicCompletion.smul_eval, AdicCompletion.val_mul]
-  induction a.val n, b.val n using Quotient.inductionOn₂' with
-  | h r s => rfl
-
-lemma completion_map_add (F G : M →ₗ[R] N) (x : AdicCompletion I M) :
-    AdicCompletion.map I (F+G) x = AdicCompletion.map I F x + AdicCompletion.map I G x := by
-  apply AdicCompletion.ext
-  intro n
-  obtain ⟨v,hv⟩ := (I^n • (⊤ : Submodule R M)).mkQ_surjective (x.val n)
-  simp only [AdicCompletion.map_val_apply]
-  change (F+G).reduceModIdeal (I^n) (x.val n) =
-    F.reduceModIdeal (I^n) (x.val n) + G.reduceModIdeal (I^n) (x.val n)
-  rw [← hv]
-  rfl
-
-lemma completion_map_sum {ι : Type*} (s : Finset ι) (F : ι → M →ₗ[R] N)
-    (x : AdicCompletion I M) :
-    AdicCompletion.map I (∑ i ∈ s, F i) x = ∑ i ∈ s, AdicCompletion.map I (F i) x := by
-  classical
-  induction s using Finset.induction_on with
-  | empty => simp
-  | @insert i s hi ih => simp only [Finset.sum_insert hi, completion_map_add, ih]
-
-lemma completion_map_smul (a : R) (F : M →ₗ[R] N) (x : AdicCompletion I M) :
-    AdicCompletion.map I (a • F) x = algebraMap R (AdicCompletion I R) a • AdicCompletion.map I F x := by
-  have he : a • F = (a • (LinearMap.id : N →ₗ[R] N)).comp F := rfl
-  rw [he, ← AdicCompletion.map_comp_apply, completion_map_scalar]
-end FiberFunctorProof
-
-end OAI
+import OAI.RepresentationTheory.RestrictedTilting.SpecializeCompletion
+
+namespace OAI
+
+namespace FiberFunctorProof
+variable {R M N : Type*} [CommRing R] [AddCommGroup M] [Module R M]
+  [AddCommGroup N] [Module R N] (I : Ideal R)
+
+attribute [local instance 1100] AdicCompletion.smul
+
+lemma completion_self_smul (a b : AdicCompletion I R) : a • b = a * b := by
+  apply AdicCompletion.ext
+  intro n
+  rw [AdicCompletion.smul_eval, AdicCompletion.val_mul]
+  induction a.val n, b.val n using Quotient.inductionOn₂' with
+  | h r s => rfl
+
+lemma completion_map_add (F G : M →ₗ[R] N) (x : AdicCompletion I M) :
+    AdicCompletion.map I (F+G) x = AdicCompletion.map I F x + AdicCompletion.map I G x := by
+  apply AdicCompletion.ext
+  intro n
+  obtain ⟨v,hv⟩ := (I^n • (⊤ : Submodule R M)).mkQ_surjective (x.val n)
+  simp only [AdicCompletion.map_val_apply]
+  change (F+G).reduceModIdeal (I^n) (x.val n) =
+    F.reduceModIdeal (I^n) (x.val n) + G.reduceModIdeal (I^n) (x.val n)
+  rw [← hv]
+  rfl
+
+lemma completion_map_sum {ι : Type*} (s : Finset ι) (F : ι → M →ₗ[R] N)
+    (x : AdicCompletion I M) :
+    AdicCompletion.map I (∑ i ∈ s, F i) x = ∑ i ∈ s, AdicCompletion.map I (F i) x := by
+  classical
+  induction s using Finset.induction_on with
+  | empty => simp
+  | @insert i s hi ih => simp only [Finset.sum_insert hi, completion_map_add, ih]
+
+lemma completion_map_smul (a : R) (F : M →ₗ[R] N) (x : AdicCompletion I M) :
+    AdicCompletion.map I (a • F) x = algebraMap R (AdicCompletion I R) a • AdicCompletion.map I F x := by
+  have he : a • F = (a • (LinearMap.id : N →ₗ[R] N)).comp F := rfl
+  rw [he, ← AdicCompletion.map_comp_apply, completion_map_scalar]
+end FiberFunctorProof
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/RestrictedTilting/CompletionTriviality.lean b/lean/OAI/RepresentationTheory/RestrictedTilting/CompletionTriviality.lean
index 6a8579ab12bf06b5de936501e099a55f120b4b1d..e58c766d8617b99645cb3a1f28251d27017949b6 100644
--- a/lean/OAI/RepresentationTheory/RestrictedTilting/CompletionTriviality.lean
+++ b/lean/OAI/RepresentationTheory/RestrictedTilting/CompletionTriviality.lean
@@ -1,63 +1,63 @@
-import OAI.RepresentationTheory.RestrictedTilting.AdicJets
-
-namespace OAI
-
-namespace FiberFunctorProof
-open scoped TensorProduct Pointwise
-section
-variable {R M N : Type*} [CommRing R] [AddCommGroup M] [Module R M]
-  [AddCommGroup N] [Module R N]
-variable (I : Ideal R)
-
- 
-def completionOfJetMaps
-    (F : (j : ℕ) → (M ⧸ (I^j • (⊤ : Submodule R M))) →ₗ[R]
-      (N ⧸ (I^j • (⊤ : Submodule R N))))
-    (hF : ∀ m n (h : m ≤ n) z,
-      AdicCompletion.transitionMap I N h (F n z) =
-        F m (AdicCompletion.transitionMap I M h z)) :
-    AdicCompletion I M →ₗ[AdicCompletion I R] AdicCompletion I N where
-  toFun x := ⟨fun j => F j (x.val j), by
-    intro m n h
-    rw [hF, x.property h]⟩
-  map_add' x y := by ext j; exact map_add (F j) _ _
-  map_smul' r x := by
-    apply AdicCompletion.ext
-    intro j
-    change F j (r.val j • x.val j) = r.val j • F j (x.val j)
-    obtain ⟨a,ha⟩ := (I^j • (⊤ : Submodule R R)).mkQ_surjective (r.val j)
-    rw [← ha]
-    exact map_smul (F j) a (x.val j)
-
- 
-
-theorem completionOfJetMaps_bijective
-    (F : (j : ℕ) → (M ⧸ (I^j • (⊤ : Submodule R M))) →ₗ[R]
-      (N ⧸ (I^j • (⊤ : Submodule R N))))
-    (hF : ∀ m n (h : m ≤ n) z,
-      AdicCompletion.transitionMap I N h (F n z) =
-        F m (AdicCompletion.transitionMap I M h z))
-    (hbij : ∀ j, Function.Bijective (F j)) :
-    Function.Bijective (completionOfJetMaps I F hF) := by
-  constructor
-  · intro x y h
-    apply AdicCompletion.ext
-    intro j
-    exact (hbij j).1 (congrArg (fun z : AdicCompletion I N => z.val j) h)
-  · intro y
-    let e (j : ℕ) := LinearEquiv.ofBijective (F j) (hbij j)
-    let x : AdicCompletion I M := ⟨fun j => (e j).symm (y.val j), by
-      intro m n h
-      apply (hbij m).1
-      rw [← hF]
-      change AdicCompletion.transitionMap I N h ((e n) ((e n).symm (y.val n))) =
-        (e m) ((e m).symm (y.val m))
-      rw [LinearEquiv.apply_symm_apply, LinearEquiv.apply_symm_apply, y.property h]⟩
-    refine ⟨x, ?_⟩
-    apply AdicCompletion.ext
-    intro j
-    exact (e j).apply_symm_apply _
-end
-end FiberFunctorProof
-
-end OAI
+import OAI.RepresentationTheory.RestrictedTilting.AdicJets
+
+namespace OAI
+
+namespace FiberFunctorProof
+open scoped TensorProduct Pointwise
+section
+variable {R M N : Type*} [CommRing R] [AddCommGroup M] [Module R M]
+  [AddCommGroup N] [Module R N]
+variable (I : Ideal R)
+
+ 
+def completionOfJetMaps
+    (F : (j : ℕ) → (M ⧸ (I^j • (⊤ : Submodule R M))) →ₗ[R]
+      (N ⧸ (I^j • (⊤ : Submodule R N))))
+    (hF : ∀ m n (h : m ≤ n) z,
+      AdicCompletion.transitionMap I N h (F n z) =
+        F m (AdicCompletion.transitionMap I M h z)) :
+    AdicCompletion I M →ₗ[AdicCompletion I R] AdicCompletion I N where
+  toFun x := ⟨fun j => F j (x.val j), by
+    intro m n h
+    rw [hF, x.property h]⟩
+  map_add' x y := by ext j; exact map_add (F j) _ _
+  map_smul' r x := by
+    apply AdicCompletion.ext
+    intro j
+    change F j (r.val j • x.val j) = r.val j • F j (x.val j)
+    obtain ⟨a,ha⟩ := (I^j • (⊤ : Submodule R R)).mkQ_surjective (r.val j)
+    rw [← ha]
+    exact map_smul (F j) a (x.val j)
+
+ 
+
+theorem completionOfJetMaps_bijective
+    (F : (j : ℕ) → (M ⧸ (I^j • (⊤ : Submodule R M))) →ₗ[R]
+      (N ⧸ (I^j • (⊤ : Submodule R N))))
+    (hF : ∀ m n (h : m ≤ n) z,
+      AdicCompletion.transitionMap I N h (F n z) =
+        F m (AdicCompletion.transitionMap I M h z))
+    (hbij : ∀ j, Function.Bijective (F j)) :
+    Function.Bijective (completionOfJetMaps I F hF) := by
+  constructor
+  · intro x y h
+    apply AdicCompletion.ext
+    intro j
+    exact (hbij j).1 (congrArg (fun z : AdicCompletion I N => z.val j) h)
+  · intro y
+    let e (j : ℕ) := LinearEquiv.ofBijective (F j) (hbij j)
+    let x : AdicCompletion I M := ⟨fun j => (e j).symm (y.val j), by
+      intro m n h
+      apply (hbij m).1
+      rw [← hF]
+      change AdicCompletion.transitionMap I N h ((e n) ((e n).symm (y.val n))) =
+        (e m) ((e m).symm (y.val m))
+      rw [LinearEquiv.apply_symm_apply, LinearEquiv.apply_symm_apply, y.property h]⟩
+    refine ⟨x, ?_⟩
+    apply AdicCompletion.ext
+    intro j
+    exact (e j).apply_symm_apply _
+end
+end FiberFunctorProof
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/RestrictedTilting/CoordinateTensor.lean b/lean/OAI/RepresentationTheory/RestrictedTilting/CoordinateTensor.lean
index 64225f34a00ffae35f399338b2c42fe0edab7c75..438bb9611ab01388eb1f686c1b03a38269800b62 100644
--- a/lean/OAI/RepresentationTheory/RestrictedTilting/CoordinateTensor.lean
+++ b/lean/OAI/RepresentationTheory/RestrictedTilting/CoordinateTensor.lean
@@ -1,63 +1,63 @@
-import OAI.RepresentationTheory.RestrictedTilting.PlaneMatrixTensor
-
-namespace OAI
-namespace FiberFunctorProof
-open CategoryTheory CategoryTheory.MonoidalCategory Matrix
-open scoped TensorProduct BigOperators
-universe u v
-variable (k : Type u) [Field k]
-variable {β : Type u} [Fintype β] [DecidableEq β]
-variable {G : Type u} [Monoid G]
-variable {M : Type u} [AddCommGroup M] [Module k M]
-
-noncomputable def coordinateTensorEquiv : ((β → k) ⊗[k] M) ≃ₗ[k] (β → M) :=
-  (TensorProduct.comm k (β → k) M).trans (TensorProduct.piScalarRight k k M β)
-
-lemma coordinateTensorEquiv_tmul (v : β → k) (w : M) :
-    coordinateTensorEquiv k (v ⊗ₜ[k] w) = fun i => v i • w := by
-  simp [coordinateTensorEquiv, TensorProduct.piScalarRight_apply,
-    TensorProduct.piScalarRightHom_tmul]
-
-lemma coordinateTensorEquiv_intertwines
-    (τ : Representation k G (β → k)) (σ : Representation k G M) (g : G)
-    (v : (β → k) ⊗[k] M) :
-    coordinateTensorEquiv k ((τ.tprod σ) g v) =
-      matrixTensorRepresentation τ σ g (coordinateTensorEquiv k v) := by
-  induction v using TensorProduct.inductionOn with
-  | add v w hv hw => simp only [map_add, hv, hw]
-  | tmul v w =>
-    rw [Representation.tprod_apply, TensorProduct.map_tmul, coordinateTensorEquiv_tmul,
-      coordinateTensorEquiv_tmul]
-    funext i
-    rw [matrixTensorRepresentation_apply]
-    simp only [map_smul, smul_smul, ← Finset.sum_smul]
-    congr 1
-    exact (congrFun (LinearMap.toMatrix'_mulVec (τ g) v) i).symm
-
-variable {V : Type u} [AddCommGroup V] [Module k V]
-variable {H : Type v} [Monoid H]
-noncomputable def coordinateRepresentation (ρ : Representation k H V) (e : V ≃ₗ[k] (β → k)) :
-    Representation k H (β → k) := {
-    toFun g := e.conj (ρ g)
-    map_one' := by ext v; simp [LinearEquiv.conj_apply]
-    map_mul' g h := by ext v; simp [LinearEquiv.conj_apply, Module.End.mul_apply]
-  }
-
-lemma coordinateRepresentation_equiv (ρ : Representation k H V) (e : V ≃ₗ[k] (β → k))
-    (g : H) (v : V) : e (ρ g v) = coordinateRepresentation k ρ e g (e v) := by
-  simp [coordinateRepresentation, LinearEquiv.conj_apply]
-
-lemma coordinateRepresentation_rational (ρ : Representation k (SL2 k) V)
-    (e : V ≃ₗ[k] (β → k)) (hρ : IsRational k ρ) :
-    IsRational k (coordinateRepresentation k ρ e) :=
-  IsRational.of_surjective k ρ _ e.toLinearMap e.surjective
-    (coordinateRepresentation_equiv k ρ e) hρ
-
-noncomputable def coordinateFDIso (S : FDRep k G) (e : S ≃ₗ[k] (β → k)) :
-    S ≅ FDRep.of (coordinateRepresentation k S.ρ e) := by
-  let f : S ⟶ FDRep.of (coordinateRepresentation k S.ρ e) := fdrepHom k e.toLinearMap (coordinateRepresentation_equiv k S.ρ e)
-  letI : IsIso f := fdrepHom_isIso_of_bijective k f e.bijective
-  exact asIso f
-end FiberFunctorProof
-
-end OAI
+import OAI.RepresentationTheory.RestrictedTilting.PlaneMatrixTensor
+
+namespace OAI
+namespace FiberFunctorProof
+open CategoryTheory CategoryTheory.MonoidalCategory Matrix
+open scoped TensorProduct BigOperators
+universe u v
+variable (k : Type u) [Field k]
+variable {β : Type u} [Fintype β] [DecidableEq β]
+variable {G : Type u} [Monoid G]
+variable {M : Type u} [AddCommGroup M] [Module k M]
+
+noncomputable def coordinateTensorEquiv : ((β → k) ⊗[k] M) ≃ₗ[k] (β → M) :=
+  (TensorProduct.comm k (β → k) M).trans (TensorProduct.piScalarRight k k M β)
+
+lemma coordinateTensorEquiv_tmul (v : β → k) (w : M) :
+    coordinateTensorEquiv k (v ⊗ₜ[k] w) = fun i => v i • w := by
+  simp [coordinateTensorEquiv, TensorProduct.piScalarRight_apply,
+    TensorProduct.piScalarRightHom_tmul]
+
+lemma coordinateTensorEquiv_intertwines
+    (τ : Representation k G (β → k)) (σ : Representation k G M) (g : G)
+    (v : (β → k) ⊗[k] M) :
+    coordinateTensorEquiv k ((τ.tprod σ) g v) =
+      matrixTensorRepresentation τ σ g (coordinateTensorEquiv k v) := by
+  induction v using TensorProduct.inductionOn with
+  | add v w hv hw => simp only [map_add, hv, hw]
+  | tmul v w =>
+    rw [Representation.tprod_apply, TensorProduct.map_tmul, coordinateTensorEquiv_tmul,
+      coordinateTensorEquiv_tmul]
+    funext i
+    rw [matrixTensorRepresentation_apply]
+    simp only [map_smul, smul_smul, ← Finset.sum_smul]
+    congr 1
+    exact (congrFun (LinearMap.toMatrix'_mulVec (τ g) v) i).symm
+
+variable {V : Type u} [AddCommGroup V] [Module k V]
+variable {H : Type v} [Monoid H]
+noncomputable def coordinateRepresentation (ρ : Representation k H V) (e : V ≃ₗ[k] (β → k)) :
+    Representation k H (β → k) := {
+    toFun g := e.conj (ρ g)
+    map_one' := by ext v; simp [LinearEquiv.conj_apply]
+    map_mul' g h := by ext v; simp [LinearEquiv.conj_apply, Module.End.mul_apply]
+  }
+
+lemma coordinateRepresentation_equiv (ρ : Representation k H V) (e : V ≃ₗ[k] (β → k))
+    (g : H) (v : V) : e (ρ g v) = coordinateRepresentation k ρ e g (e v) := by
+  simp [coordinateRepresentation, LinearEquiv.conj_apply]
+
+lemma coordinateRepresentation_rational (ρ : Representation k (SL2 k) V)
+    (e : V ≃ₗ[k] (β → k)) (hρ : IsRational k ρ) :
+    IsRational k (coordinateRepresentation k ρ e) :=
+  IsRational.of_surjective k ρ _ e.toLinearMap e.surjective
+    (coordinateRepresentation_equiv k ρ e) hρ
+
+noncomputable def coordinateFDIso (S : FDRep k G) (e : S ≃ₗ[k] (β → k)) :
+    S ≅ FDRep.of (coordinateRepresentation k S.ρ e) := by
+  let f : S ⟶ FDRep.of (coordinateRepresentation k S.ρ e) := fdrepHom k e.toLinearMap (coordinateRepresentation_equiv k S.ρ e)
+  letI : IsIso f := fdrepHom_isIso_of_bijective k f e.bijective
+  exact asIso f
+end FiberFunctorProof
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/RestrictedTilting/DualSplitting.lean b/lean/OAI/RepresentationTheory/RestrictedTilting/DualSplitting.lean
index 824ec41c4d39300ff395920836c9680fc1e028ba..0839e94ad1a0534f5529ad874b02377f808e1bf6 100644
--- a/lean/OAI/RepresentationTheory/RestrictedTilting/DualSplitting.lean
+++ b/lean/OAI/RepresentationTheory/RestrictedTilting/DualSplitting.lean
@@ -1,343 +1,343 @@
-import Mathlib
-import OAI.RepresentationTheory.RestrictedTilting.StandardLifts
-
-namespace OAI
-
-/-! Standard-costandard extension splitting and dual rational representations. -/
-
-noncomputable section
-open CategoryTheory CategoryTheory.MonoidalCategory CategoryTheory.Limits
-open scoped BigOperators
-
-namespace FiberFunctorProof
-section
-
-universe u
-variable (k : Type u) [Field k] [Infinite k]
-
-lemma standard_weight_bound (n : ℕ) :
-    HasWeightsBetween (standardRepresentation k n) (-(n : ℤ)) n := by
-  intro m v hm hv
-  apply eigenvector_eq_zero_of_basis k (standardBasis k n)
-    (fun i => (n : ℤ) - 2 * i) (standardRepresentation k n)
-    (standardBasis_weight k n) m ?_ v hv
-  intro i
-  have hi := i.isLt
-  rcases hm with hm | hm <;> omega
-
-omit [Infinite k] in
-lemma hasWeightsBetween_mono {M : Type*} [AddCommGroup M] [Module k M]
-    (ρ : Representation k (SL2 k) M) {l h l' h' : ℤ}
-    (hb : HasWeightsBetween ρ l h) (hl : l' ≤ l) (hh : h ≤ h') :
-    HasWeightsBetween ρ l' h' := by
-  intro m v hm hv
-  exact hb m v (by rcases hm with hm | hm <;> omega) hv
-
- 
-
-theorem standard_costandard_split_of_le {M : Type u} [AddCommGroup M]
-    [Module k M] [Module.Finite k M]
-    (ρ : Representation k (SL2 k) M) (hρ : IsRational k ρ)
-    (n m : ℕ) (hmn : m ≤ n)
-    (f : MvPolynomial.homogeneousSubmodule (Fin 2) k m →ₗ[k] M)
-    (g : M →ₗ[k] Module.Dual k (MvPolynomial.homogeneousSubmodule (Fin 2) k n))
-    (hf : Function.Injective f) (hg : Function.Surjective g)
-    (he : LinearMap.ker g = LinearMap.range f)
-    (hfi : ∀ a x, f ((costandard k m).ρ a x) = ρ a (f x))
-    (hgi : ∀ a x, g (ρ a x) = standardRepresentation k n a (g x)) :
-    ∃ s : Module.Dual k (MvPolynomial.homogeneousSubmodule (Fin 2) k n) →ₗ[k] M,
-      g.comp s = LinearMap.id ∧
-      ∀ a x, s (standardRepresentation k n a x) = ρ a (s x) := by
-  have hb : HasWeightsBetween ρ (-(n : ℤ)) n := by
-    apply hasWeightsBetween_extension (costandard k m).ρ ρ (standardRepresentation k n)
-      f g hf he (fun x a => hfi a x) (fun x a => hgi a x)
-    · apply hasWeightsBetween_mono k _ (fun a v hv hw => costandard_weight_bound k m a hv v hw)
-      <;> omega
-    · exact standard_weight_bound k n
-  obtain ⟨v, hgv, hv⟩ := rational_weight_lift k ρ (standardRepresentation k n) hρ g
-    hgi hg (integerCharacter k n) (standardHighest k n) (standardHighest_weight k n)
-  obtain ⟨s, hsv, hs⟩ := exists_standard_lift k ρ hρ n hb v hv
-  refine ⟨s, ?_, hs⟩
-  apply standard_lower_orbit_ext k n Set.univ Set.infinite_univ
-  intro t ht
-  simp only [LinearMap.comp_apply, LinearMap.id_apply, hs, hgi, hsv, hgv]
-
-end
-
-section
-
-universe u
-variable (k : Type u) [Field k]
-
-noncomputable def inverseCoordinatePolynomial (ij : Fin 2 × Fin 2) :
-    MvPolynomial (Fin 2 × Fin 2) k :=
-  !![MvPolynomial.X (1,1), -MvPolynomial.X (0,1);
-    -MvPolynomial.X (1,0), MvPolynomial.X (0,0)] ij.1 ij.2
-
-lemma eval_inverseCoordinatePolynomial (g : SL2 k) (ij : Fin 2 × Fin 2) :
-    MvPolynomial.eval (fun ij => g ij.1 ij.2) (inverseCoordinatePolynomial k ij) =
-      (g⁻¹ : SL2 k) ij.1 ij.2 := by
-  rw [Matrix.SpecialLinearGroup.coe_inv, Matrix.adjugate_fin_two]
-  rcases ij with ⟨i,j⟩
-  fin_cases i <;> fin_cases j <;> simp [inverseCoordinatePolynomial]
-
- 
-lemma IsRational.dual {M : Type u} [AddCommGroup M] [Module k M]
-    [Module.Finite k M] (ρ : Representation k (SL2 k) M) (hρ : IsRational k ρ) :
-    IsRational k (Representation.dual ρ) := by
-  intro φ ψ
-  obtain ⟨v, rfl⟩ := (Module.evalEquiv k M).surjective ψ
-  obtain ⟨P, hP⟩ := hρ v φ
-  refine ⟨MvPolynomial.bind₁ (inverseCoordinatePolynomial k) P, fun g => ?_⟩
-  change φ (ρ g⁻¹ v) = _
-  rw [hP]
-  change _ = MvPolynomial.eval₂Hom (RingHom.id k) (fun ij => g ij.1 ij.2)
-    (MvPolynomial.bind₁ (inverseCoordinatePolynomial k) P)
-  rw [MvPolynomial.eval₂Hom_bind₁]
-  change _ = MvPolynomial.eval (fun ij =>
-    MvPolynomial.eval (fun ij => g ij.1 ij.2) (inverseCoordinatePolynomial k ij)) P
-  simp only [eval_inverseCoordinatePolynomial]
-
-lemma dual_intertwines {M N : Type*} [AddCommGroup M] [Module k M]
-    [AddCommGroup N] [Module k N]
-    (ρ : Representation k (SL2 k) M) (σ : Representation k (SL2 k) N)
-    (f : M →ₗ[k] N) (hf : ∀ g x, f (ρ g x) = σ g (f x)) :
-    ∀ g φ, f.dualMap (Representation.dual σ g φ) =
-      Representation.dual ρ g (f.dualMap φ) := by
-  intro g φ
-  ext x
-  change φ (σ g⁻¹ (f x)) = φ (f (ρ g⁻¹ x))
-  rw [hf]
-
-lemma eval_intertwines {M : Type*} [AddCommGroup M] [Module k M]
-    (ρ : Representation k (SL2 k) M) (g : SL2 k) (x : M) :
-    Module.Dual.eval k M (ρ g x) =
-      Representation.dual (Representation.dual ρ) g (Module.Dual.eval k M x) := by
-  ext φ
-  change φ (ρ g x) = φ (ρ (g⁻¹)⁻¹ x)
-  rw [inv_inv]
-
-end
-
-section
-
-variable {k G A B C : Type*} [Field k] [Group G]
-  [AddCommGroup A] [Module k A] [AddCommGroup B] [Module k B]
-  [AddCommGroup C] [Module k C]
-
- 
-lemma equivariant_section_of_retraction
-    (ρA : Representation k G A) (ρB : Representation k G B)
-    (ρC : Representation k G C) (f : A →ₗ[k] B) (g : B →ₗ[k] C)
-    (hg : Function.Surjective g) (he : LinearMap.ker g = LinearMap.range f)
-    (hgρ : ∀ a x, g (ρB a x) = ρC a (g x))
-    (r : B →ₗ[k] A) (hr : r.comp f = LinearMap.id)
-    (hrρ : ∀ a x, r (ρB a x) = ρA a (r x)) :
-    ∃ s : C →ₗ[k] B, g.comp s = LinearMap.id ∧
-      ∀ a x, s (ρC a x) = ρB a (s x) := by
-  obtain ⟨s₀, hs₀⟩ := g.exists_rightInverse_of_surjective (LinearMap.range_eq_top.mpr hg)
-  have hgf (x : A) : g (f x) = 0 := by
-    exact LinearMap.mem_ker.mp (he.symm ▸ LinearMap.mem_range_self f x)
-  have hrf (x : A) : r (f x) = x := LinearMap.congr_fun hr x
-  let s : C →ₗ[k] B := s₀ - f.comp (r.comp s₀)
-  have hgs (x : C) : g (s x) = x := by
-    dsimp [s]
-    rw [map_sub, hgf, sub_zero]
-    exact LinearMap.congr_fun hs₀ x
-  have hrs (x : C) : r (s x) = 0 := by simp [s, hrf]
-  have hext {x y : B} (h₁ : g x = g y) (h₂ : r x = r y) : x = y := by
-    have hxy : x - y ∈ LinearMap.range f := by
-      rw [← he, LinearMap.mem_ker, map_sub, h₁, sub_self]
-    obtain ⟨z, hz⟩ := hxy
-    have hz0 : z = 0 := by
-      rw [← hrf z, hz, map_sub, h₂, sub_self]
-    exact sub_eq_zero.mp (by simpa [hz0] using hz.symm)
-  refine ⟨s, LinearMap.ext hgs, fun a x => hext ?_ ?_⟩
-  · rw [hgs, hgρ, hgs]
-  · rw [hrs, hrρ, hrs, map_zero]
-
- 
-lemma equivariant_retraction_of_dual_section [Module.Finite k A]
-    (ρA : Representation k G A) (ρB : Representation k G B) (f : A →ₗ[k] B)
-    (s : Module.Dual k A →ₗ[k] Module.Dual k B)
-    (hs : f.dualMap.comp s = LinearMap.id)
-    (hsρ : ∀ a φ, s (Representation.dual ρA a φ) = Representation.dual ρB a (s φ)) :
-    ∃ r : B →ₗ[k] A, r.comp f = LinearMap.id ∧
-      ∀ a x, r (ρB a x) = ρA a (r x) := by
-  let r : B →ₗ[k] A := (Module.evalEquiv k A).symm.toLinearMap.comp
-    (s.dualMap.comp (Module.Dual.eval k B))
-  have hr (x : B) (φ : Module.Dual k A) : φ (r x) = s φ x := by
-    exact Module.apply_evalEquiv_symm_apply k A φ (s.dualMap (Module.Dual.eval k B x))
-  refine ⟨r, ?_, ?_⟩
-  · apply LinearMap.ext
-    intro x
-    apply (Module.evalEquiv k A).injective
-    apply LinearMap.ext
-    intro φ
-    change φ (r (f x)) = φ x
-    rw [hr]
-    exact congrArg (fun ψ => ψ x) (LinearMap.congr_fun hs φ)
-  · intro a x
-    apply (Module.evalEquiv k A).injective
-    apply LinearMap.ext
-    intro φ
-    change φ (r (ρB a x)) = φ (ρA a (r x))
-    rw [hr]
-    have hhe := congrArg (fun ψ => ψ x) (hsρ a⁻¹ φ)
-    change s (Representation.dual ρA a⁻¹ φ) x = s φ (ρB (a⁻¹)⁻¹ x) at hhe
-    rw [inv_inv] at hhe
-    rw [← hhe, ← hr]
-    change φ (ρA (a⁻¹)⁻¹ (r x)) = _
-    rw [inv_inv]
-
-lemma dual_sequence_exact [Module.Finite k C]
-    (f : A →ₗ[k] B) (g : B →ₗ[k] Module.Dual k C)
-    (he : LinearMap.ker g = LinearMap.range f) :
-    LinearMap.ker f.dualMap =
-      LinearMap.range (g.dualMap.comp (Module.Dual.eval k C)) := by
-  rw [← Module.evalEquiv_toLinearMap,
-    LinearMap.range_comp_of_range_eq_top _ (LinearEquiv.range (Module.evalEquiv k C)),
-    LinearMap.ker_dualMap_eq_dualAnnihilator_range,
-    LinearMap.range_dualMap_eq_dualAnnihilator_ker, he]
-
-lemma dual_eval_intertwines
-    (ρB : Representation k G B) (ρC : Representation k G C)
-    (g : B →ₗ[k] Module.Dual k C)
-    (hg : ∀ a x, g (ρB a x) = Representation.dual ρC a (g x)) :
-    ∀ a x, (g.dualMap.comp (Module.Dual.eval k C)) (ρC a x) =
-      Representation.dual ρB a ((g.dualMap.comp (Module.Dual.eval k C)) x) := by
-  intro a x
-  apply LinearMap.ext
-  intro y
-  change g y (ρC a x) = g (ρB a⁻¹ y) x
-  rw [hg]
-  change _ = g y (ρC (a⁻¹)⁻¹ x)
-  rw [inv_inv]
-
-end
-
-section
-
-universe u
-variable (k : Type u) [Field k] [Infinite k]
-
- 
-theorem standard_costandard_split {M : Type u} [AddCommGroup M]
-    [Module k M] [Module.Finite k M]
-    (ρ : Representation k (SL2 k) M) (hρ : IsRational k ρ)
-    (n m : ℕ)
-    (f : MvPolynomial.homogeneousSubmodule (Fin 2) k m →ₗ[k] M)
-    (g : M →ₗ[k] Module.Dual k (MvPolynomial.homogeneousSubmodule (Fin 2) k n))
-    (hf : Function.Injective f) (hg : Function.Surjective g)
-    (he : LinearMap.ker g = LinearMap.range f)
-    (hfi : ∀ a x, f ((costandard k m).ρ a x) = ρ a (f x))
-    (hgi : ∀ a x, g (ρ a x) = standardRepresentation k n a (g x)) :
-    ∃ s : Module.Dual k (MvPolynomial.homogeneousSubmodule (Fin 2) k n) →ₗ[k] M,
-      g.comp s = LinearMap.id ∧
-      ∀ a x, s (standardRepresentation k n a x) = ρ a (s x) := by
-  by_cases hmn : m ≤ n
-  · exact standard_costandard_split_of_le k ρ hρ n m hmn f g hf hg he hfi hgi
-  let H := MvPolynomial.homogeneousSubmodule (Fin 2) k n
-  let f' : H →ₗ[k] Module.Dual k M := g.dualMap.comp (Module.Dual.eval k H)
-  have hf' : Function.Injective f' :=
-    (LinearMap.dualMap_injective_of_surjective hg).comp (Module.evalEquiv k H).injective
-  have hg' : Function.Surjective f.dualMap :=
-    LinearMap.dualMap_surjective_of_injective hf
-  have he' : LinearMap.ker f.dualMap = LinearMap.range f' :=
-    dual_sequence_exact f g he
-  have hfi' : ∀ a x, f' ((costandard k n).ρ a x) = Representation.dual ρ a (f' x) :=
-    dual_eval_intertwines ρ (costandard k n).ρ g hgi
-  have hgi' : ∀ a x, f.dualMap (Representation.dual ρ a x) =
-      standardRepresentation k m a (f.dualMap x) :=
-    dual_intertwines k (costandard k m).ρ ρ f hfi
-  obtain ⟨s', hs', hsρ'⟩ := standard_costandard_split_of_le k
-    (Representation.dual ρ) (IsRational.dual k ρ hρ) m n (by omega)
-    f' f.dualMap hf' hg' he' hfi' hgi'
-  obtain ⟨r, hrf, hrρ⟩ := equivariant_retraction_of_dual_section
-    (costandard k m).ρ ρ f s' hs' hsρ'
-  exact equivariant_section_of_retraction (costandard k m).ρ ρ
-    (standardRepresentation k n) f g hg he hgi r hrf hrρ
-
-end
-
-universe u
-variable (k : Type u) [Field k]
-
-lemma IsRational.of_injective {M N : Type u} [AddCommGroup M] [Module k M]
-    [AddCommGroup N] [Module k N]
-    (ρ : Representation k (SL2 k) M) (σ : Representation k (SL2 k) N)
-    (f : M →ₗ[k] N) (hf : Function.Injective f)
-    (hi : ∀ g x, f (ρ g x) = σ g (f x)) (hσ : IsRational k σ) :
-    IsRational k ρ := by
-  intro v l
-  obtain ⟨l', hl'⟩ := LinearMap.dualMap_surjective_of_injective hf l
-  obtain ⟨P, hP⟩ := hσ (f v) l'
-  refine ⟨P, fun g => ?_⟩
-  rw [← hP, ← hi]
-  exact congrArg (fun a => a (ρ g v)) hl'.symm
-
-lemma IsRational.of_surjective {M N : Type u} [AddCommGroup M] [Module k M]
-    [AddCommGroup N] [Module k N]
-    (ρ : Representation k (SL2 k) M) (σ : Representation k (SL2 k) N)
-    (f : M →ₗ[k] N) (hf : Function.Surjective f)
-    (hi : ∀ g x, f (ρ g x) = σ g (f x)) (hρ : IsRational k ρ) :
-    IsRational k σ := by
-  intro v l
-  obtain ⟨w, rfl⟩ := hf v
-  obtain ⟨P, hP⟩ := hρ w (l.comp f)
-  refine ⟨P, fun g => ?_⟩
-  rw [← hP]
-  change l (σ g (f w)) = l (f (ρ g w))
-  rw [hi]
-
- 
-
-lemma regular_linearChange_coeff {ι : Type*} [Fintype ι]
-    {S : Type*} {τ : Type*} (z : S → τ → k) (A : S → Matrix ι ι k)
-    (hA : ∀ i j, ∃ P : MvPolynomial τ k, ∀ g, A g i j = MvPolynomial.eval (z g) P)
-    (v : MvPolynomial ι k) :
-    ∀ l : MvPolynomial ι k →ₗ[k] k,
-      ∃ P : MvPolynomial τ k, ∀ g, l (linearChange (A g) v) = MvPolynomial.eval (z g) P := by
-  classical
-  induction v using MvPolynomial.induction_on with
-  | C a =>
-    intro l
-    exact ⟨MvPolynomial.C (l (MvPolynomial.C a)), by simp⟩
-  | add v w hv hw =>
-    intro l
-    obtain ⟨P, hP⟩ := hv l
-    obtain ⟨Q, hQ⟩ := hw l
-    exact ⟨P + Q, by simp [hP, hQ]⟩
-  | mul_X v i hv =>
-    intro l
-    choose P hP using fun j => hv (l.comp (LinearMap.mulRight k (MvPolynomial.X j)))
-    choose Q hQ using fun j => hA j i
-    refine ⟨∑ j, Q j * P j, fun g => ?_⟩
-    simp only [map_mul, linearChange_X, Finset.mul_sum, map_sum]
-    apply Finset.sum_congr rfl
-    intro j hj
-    rw [mul_comm (MvPolynomial.C _), ← mul_assoc, mul_comm _ (MvPolynomial.C _)]
-    rw [← MvPolynomial.smul_eq_C_mul, map_smul, hQ]
-    change _ = _ * MvPolynomial.eval (z g) (P j)
-    rw [← hP]
-    rfl
-
-lemma costandard_rational (n : ℕ) : IsRational k (costandard k n).ρ := by
-  intro v l
-  obtain ⟨l', hl'⟩ := LinearMap.dualMap_surjective_of_injective
-    (MvPolynomial.homogeneousSubmodule (Fin 2) k n).injective_subtype l
-  obtain ⟨P, hP⟩ := regular_linearChange_coeff k
-    (fun g : SL2 k => fun ij : Fin 2 × Fin 2 => g ij.1 ij.2)
-    (dualMatrix k) (fun i j => ⟨inverseCoordinatePolynomial k (j,i), fun g => by
-      change (g⁻¹ : SL2 k) j i = _
-      exact (eval_inverseCoordinatePolynomial k g (j,i)).symm⟩) v.val l'
-  refine ⟨P, fun g => ?_⟩
-  rw [← hP]
-  exact congrArg (fun a => a ((costandard k n).ρ g v)) hl'.symm
-
-end FiberFunctorProof
-
-open CategoryTheory CategoryTheory.Limits
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.RestrictedTilting.StandardLifts
+
+namespace OAI
+
+/-! Standard-costandard extension splitting and dual rational representations. -/
+
+noncomputable section
+open CategoryTheory CategoryTheory.MonoidalCategory CategoryTheory.Limits
+open scoped BigOperators
+
+namespace FiberFunctorProof
+section
+
+universe u
+variable (k : Type u) [Field k] [Infinite k]
+
+lemma standard_weight_bound (n : ℕ) :
+    HasWeightsBetween (standardRepresentation k n) (-(n : ℤ)) n := by
+  intro m v hm hv
+  apply eigenvector_eq_zero_of_basis k (standardBasis k n)
+    (fun i => (n : ℤ) - 2 * i) (standardRepresentation k n)
+    (standardBasis_weight k n) m ?_ v hv
+  intro i
+  have hi := i.isLt
+  rcases hm with hm | hm <;> omega
+
+omit [Infinite k] in
+lemma hasWeightsBetween_mono {M : Type*} [AddCommGroup M] [Module k M]
+    (ρ : Representation k (SL2 k) M) {l h l' h' : ℤ}
+    (hb : HasWeightsBetween ρ l h) (hl : l' ≤ l) (hh : h ≤ h') :
+    HasWeightsBetween ρ l' h' := by
+  intro m v hm hv
+  exact hb m v (by rcases hm with hm | hm <;> omega) hv
+
+ 
+
+theorem standard_costandard_split_of_le {M : Type u} [AddCommGroup M]
+    [Module k M] [Module.Finite k M]
+    (ρ : Representation k (SL2 k) M) (hρ : IsRational k ρ)
+    (n m : ℕ) (hmn : m ≤ n)
+    (f : MvPolynomial.homogeneousSubmodule (Fin 2) k m →ₗ[k] M)
+    (g : M →ₗ[k] Module.Dual k (MvPolynomial.homogeneousSubmodule (Fin 2) k n))
+    (hf : Function.Injective f) (hg : Function.Surjective g)
+    (he : LinearMap.ker g = LinearMap.range f)
+    (hfi : ∀ a x, f ((costandard k m).ρ a x) = ρ a (f x))
+    (hgi : ∀ a x, g (ρ a x) = standardRepresentation k n a (g x)) :
+    ∃ s : Module.Dual k (MvPolynomial.homogeneousSubmodule (Fin 2) k n) →ₗ[k] M,
+      g.comp s = LinearMap.id ∧
+      ∀ a x, s (standardRepresentation k n a x) = ρ a (s x) := by
+  have hb : HasWeightsBetween ρ (-(n : ℤ)) n := by
+    apply hasWeightsBetween_extension (costandard k m).ρ ρ (standardRepresentation k n)
+      f g hf he (fun x a => hfi a x) (fun x a => hgi a x)
+    · apply hasWeightsBetween_mono k _ (fun a v hv hw => costandard_weight_bound k m a hv v hw)
+      <;> omega
+    · exact standard_weight_bound k n
+  obtain ⟨v, hgv, hv⟩ := rational_weight_lift k ρ (standardRepresentation k n) hρ g
+    hgi hg (integerCharacter k n) (standardHighest k n) (standardHighest_weight k n)
+  obtain ⟨s, hsv, hs⟩ := exists_standard_lift k ρ hρ n hb v hv
+  refine ⟨s, ?_, hs⟩
+  apply standard_lower_orbit_ext k n Set.univ Set.infinite_univ
+  intro t ht
+  simp only [LinearMap.comp_apply, LinearMap.id_apply, hs, hgi, hsv, hgv]
+
+end
+
+section
+
+universe u
+variable (k : Type u) [Field k]
+
+noncomputable def inverseCoordinatePolynomial (ij : Fin 2 × Fin 2) :
+    MvPolynomial (Fin 2 × Fin 2) k :=
+  !![MvPolynomial.X (1,1), -MvPolynomial.X (0,1);
+    -MvPolynomial.X (1,0), MvPolynomial.X (0,0)] ij.1 ij.2
+
+lemma eval_inverseCoordinatePolynomial (g : SL2 k) (ij : Fin 2 × Fin 2) :
+    MvPolynomial.eval (fun ij => g ij.1 ij.2) (inverseCoordinatePolynomial k ij) =
+      (g⁻¹ : SL2 k) ij.1 ij.2 := by
+  rw [Matrix.SpecialLinearGroup.coe_inv, Matrix.adjugate_fin_two]
+  rcases ij with ⟨i,j⟩
+  fin_cases i <;> fin_cases j <;> simp [inverseCoordinatePolynomial]
+
+ 
+lemma IsRational.dual {M : Type u} [AddCommGroup M] [Module k M]
+    [Module.Finite k M] (ρ : Representation k (SL2 k) M) (hρ : IsRational k ρ) :
+    IsRational k (Representation.dual ρ) := by
+  intro φ ψ
+  obtain ⟨v, rfl⟩ := (Module.evalEquiv k M).surjective ψ
+  obtain ⟨P, hP⟩ := hρ v φ
+  refine ⟨MvPolynomial.bind₁ (inverseCoordinatePolynomial k) P, fun g => ?_⟩
+  change φ (ρ g⁻¹ v) = _
+  rw [hP]
+  change _ = MvPolynomial.eval₂Hom (RingHom.id k) (fun ij => g ij.1 ij.2)
+    (MvPolynomial.bind₁ (inverseCoordinatePolynomial k) P)
+  rw [MvPolynomial.eval₂Hom_bind₁]
+  change _ = MvPolynomial.eval (fun ij =>
+    MvPolynomial.eval (fun ij => g ij.1 ij.2) (inverseCoordinatePolynomial k ij)) P
+  simp only [eval_inverseCoordinatePolynomial]
+
+lemma dual_intertwines {M N : Type*} [AddCommGroup M] [Module k M]
+    [AddCommGroup N] [Module k N]
+    (ρ : Representation k (SL2 k) M) (σ : Representation k (SL2 k) N)
+    (f : M →ₗ[k] N) (hf : ∀ g x, f (ρ g x) = σ g (f x)) :
+    ∀ g φ, f.dualMap (Representation.dual σ g φ) =
+      Representation.dual ρ g (f.dualMap φ) := by
+  intro g φ
+  ext x
+  change φ (σ g⁻¹ (f x)) = φ (f (ρ g⁻¹ x))
+  rw [hf]
+
+lemma eval_intertwines {M : Type*} [AddCommGroup M] [Module k M]
+    (ρ : Representation k (SL2 k) M) (g : SL2 k) (x : M) :
+    Module.Dual.eval k M (ρ g x) =
+      Representation.dual (Representation.dual ρ) g (Module.Dual.eval k M x) := by
+  ext φ
+  change φ (ρ g x) = φ (ρ (g⁻¹)⁻¹ x)
+  rw [inv_inv]
+
+end
+
+section
+
+variable {k G A B C : Type*} [Field k] [Group G]
+  [AddCommGroup A] [Module k A] [AddCommGroup B] [Module k B]
+  [AddCommGroup C] [Module k C]
+
+ 
+lemma equivariant_section_of_retraction
+    (ρA : Representation k G A) (ρB : Representation k G B)
+    (ρC : Representation k G C) (f : A →ₗ[k] B) (g : B →ₗ[k] C)
+    (hg : Function.Surjective g) (he : LinearMap.ker g = LinearMap.range f)
+    (hgρ : ∀ a x, g (ρB a x) = ρC a (g x))
+    (r : B →ₗ[k] A) (hr : r.comp f = LinearMap.id)
+    (hrρ : ∀ a x, r (ρB a x) = ρA a (r x)) :
+    ∃ s : C →ₗ[k] B, g.comp s = LinearMap.id ∧
+      ∀ a x, s (ρC a x) = ρB a (s x) := by
+  obtain ⟨s₀, hs₀⟩ := g.exists_rightInverse_of_surjective (LinearMap.range_eq_top.mpr hg)
+  have hgf (x : A) : g (f x) = 0 := by
+    exact LinearMap.mem_ker.mp (he.symm ▸ LinearMap.mem_range_self f x)
+  have hrf (x : A) : r (f x) = x := LinearMap.congr_fun hr x
+  let s : C →ₗ[k] B := s₀ - f.comp (r.comp s₀)
+  have hgs (x : C) : g (s x) = x := by
+    dsimp [s]
+    rw [map_sub, hgf, sub_zero]
+    exact LinearMap.congr_fun hs₀ x
+  have hrs (x : C) : r (s x) = 0 := by simp [s, hrf]
+  have hext {x y : B} (h₁ : g x = g y) (h₂ : r x = r y) : x = y := by
+    have hxy : x - y ∈ LinearMap.range f := by
+      rw [← he, LinearMap.mem_ker, map_sub, h₁, sub_self]
+    obtain ⟨z, hz⟩ := hxy
+    have hz0 : z = 0 := by
+      rw [← hrf z, hz, map_sub, h₂, sub_self]
+    exact sub_eq_zero.mp (by simpa [hz0] using hz.symm)
+  refine ⟨s, LinearMap.ext hgs, fun a x => hext ?_ ?_⟩
+  · rw [hgs, hgρ, hgs]
+  · rw [hrs, hrρ, hrs, map_zero]
+
+ 
+lemma equivariant_retraction_of_dual_section [Module.Finite k A]
+    (ρA : Representation k G A) (ρB : Representation k G B) (f : A →ₗ[k] B)
+    (s : Module.Dual k A →ₗ[k] Module.Dual k B)
+    (hs : f.dualMap.comp s = LinearMap.id)
+    (hsρ : ∀ a φ, s (Representation.dual ρA a φ) = Representation.dual ρB a (s φ)) :
+    ∃ r : B →ₗ[k] A, r.comp f = LinearMap.id ∧
+      ∀ a x, r (ρB a x) = ρA a (r x) := by
+  let r : B →ₗ[k] A := (Module.evalEquiv k A).symm.toLinearMap.comp
+    (s.dualMap.comp (Module.Dual.eval k B))
+  have hr (x : B) (φ : Module.Dual k A) : φ (r x) = s φ x := by
+    exact Module.apply_evalEquiv_symm_apply k A φ (s.dualMap (Module.Dual.eval k B x))
+  refine ⟨r, ?_, ?_⟩
+  · apply LinearMap.ext
+    intro x
+    apply (Module.evalEquiv k A).injective
+    apply LinearMap.ext
+    intro φ
+    change φ (r (f x)) = φ x
+    rw [hr]
+    exact congrArg (fun ψ => ψ x) (LinearMap.congr_fun hs φ)
+  · intro a x
+    apply (Module.evalEquiv k A).injective
+    apply LinearMap.ext
+    intro φ
+    change φ (r (ρB a x)) = φ (ρA a (r x))
+    rw [hr]
+    have hhe := congrArg (fun ψ => ψ x) (hsρ a⁻¹ φ)
+    change s (Representation.dual ρA a⁻¹ φ) x = s φ (ρB (a⁻¹)⁻¹ x) at hhe
+    rw [inv_inv] at hhe
+    rw [← hhe, ← hr]
+    change φ (ρA (a⁻¹)⁻¹ (r x)) = _
+    rw [inv_inv]
+
+lemma dual_sequence_exact [Module.Finite k C]
+    (f : A →ₗ[k] B) (g : B →ₗ[k] Module.Dual k C)
+    (he : LinearMap.ker g = LinearMap.range f) :
+    LinearMap.ker f.dualMap =
+      LinearMap.range (g.dualMap.comp (Module.Dual.eval k C)) := by
+  rw [← Module.evalEquiv_toLinearMap,
+    LinearMap.range_comp_of_range_eq_top _ (LinearEquiv.range (Module.evalEquiv k C)),
+    LinearMap.ker_dualMap_eq_dualAnnihilator_range,
+    LinearMap.range_dualMap_eq_dualAnnihilator_ker, he]
+
+lemma dual_eval_intertwines
+    (ρB : Representation k G B) (ρC : Representation k G C)
+    (g : B →ₗ[k] Module.Dual k C)
+    (hg : ∀ a x, g (ρB a x) = Representation.dual ρC a (g x)) :
+    ∀ a x, (g.dualMap.comp (Module.Dual.eval k C)) (ρC a x) =
+      Representation.dual ρB a ((g.dualMap.comp (Module.Dual.eval k C)) x) := by
+  intro a x
+  apply LinearMap.ext
+  intro y
+  change g y (ρC a x) = g (ρB a⁻¹ y) x
+  rw [hg]
+  change _ = g y (ρC (a⁻¹)⁻¹ x)
+  rw [inv_inv]
+
+end
+
+section
+
+universe u
+variable (k : Type u) [Field k] [Infinite k]
+
+ 
+theorem standard_costandard_split {M : Type u} [AddCommGroup M]
+    [Module k M] [Module.Finite k M]
+    (ρ : Representation k (SL2 k) M) (hρ : IsRational k ρ)
+    (n m : ℕ)
+    (f : MvPolynomial.homogeneousSubmodule (Fin 2) k m →ₗ[k] M)
+    (g : M →ₗ[k] Module.Dual k (MvPolynomial.homogeneousSubmodule (Fin 2) k n))
+    (hf : Function.Injective f) (hg : Function.Surjective g)
+    (he : LinearMap.ker g = LinearMap.range f)
+    (hfi : ∀ a x, f ((costandard k m).ρ a x) = ρ a (f x))
+    (hgi : ∀ a x, g (ρ a x) = standardRepresentation k n a (g x)) :
+    ∃ s : Module.Dual k (MvPolynomial.homogeneousSubmodule (Fin 2) k n) →ₗ[k] M,
+      g.comp s = LinearMap.id ∧
+      ∀ a x, s (standardRepresentation k n a x) = ρ a (s x) := by
+  by_cases hmn : m ≤ n
+  · exact standard_costandard_split_of_le k ρ hρ n m hmn f g hf hg he hfi hgi
+  let H := MvPolynomial.homogeneousSubmodule (Fin 2) k n
+  let f' : H →ₗ[k] Module.Dual k M := g.dualMap.comp (Module.Dual.eval k H)
+  have hf' : Function.Injective f' :=
+    (LinearMap.dualMap_injective_of_surjective hg).comp (Module.evalEquiv k H).injective
+  have hg' : Function.Surjective f.dualMap :=
+    LinearMap.dualMap_surjective_of_injective hf
+  have he' : LinearMap.ker f.dualMap = LinearMap.range f' :=
+    dual_sequence_exact f g he
+  have hfi' : ∀ a x, f' ((costandard k n).ρ a x) = Representation.dual ρ a (f' x) :=
+    dual_eval_intertwines ρ (costandard k n).ρ g hgi
+  have hgi' : ∀ a x, f.dualMap (Representation.dual ρ a x) =
+      standardRepresentation k m a (f.dualMap x) :=
+    dual_intertwines k (costandard k m).ρ ρ f hfi
+  obtain ⟨s', hs', hsρ'⟩ := standard_costandard_split_of_le k
+    (Representation.dual ρ) (IsRational.dual k ρ hρ) m n (by omega)
+    f' f.dualMap hf' hg' he' hfi' hgi'
+  obtain ⟨r, hrf, hrρ⟩ := equivariant_retraction_of_dual_section
+    (costandard k m).ρ ρ f s' hs' hsρ'
+  exact equivariant_section_of_retraction (costandard k m).ρ ρ
+    (standardRepresentation k n) f g hg he hgi r hrf hrρ
+
+end
+
+universe u
+variable (k : Type u) [Field k]
+
+lemma IsRational.of_injective {M N : Type u} [AddCommGroup M] [Module k M]
+    [AddCommGroup N] [Module k N]
+    (ρ : Representation k (SL2 k) M) (σ : Representation k (SL2 k) N)
+    (f : M →ₗ[k] N) (hf : Function.Injective f)
+    (hi : ∀ g x, f (ρ g x) = σ g (f x)) (hσ : IsRational k σ) :
+    IsRational k ρ := by
+  intro v l
+  obtain ⟨l', hl'⟩ := LinearMap.dualMap_surjective_of_injective hf l
+  obtain ⟨P, hP⟩ := hσ (f v) l'
+  refine ⟨P, fun g => ?_⟩
+  rw [← hP, ← hi]
+  exact congrArg (fun a => a (ρ g v)) hl'.symm
+
+lemma IsRational.of_surjective {M N : Type u} [AddCommGroup M] [Module k M]
+    [AddCommGroup N] [Module k N]
+    (ρ : Representation k (SL2 k) M) (σ : Representation k (SL2 k) N)
+    (f : M →ₗ[k] N) (hf : Function.Surjective f)
+    (hi : ∀ g x, f (ρ g x) = σ g (f x)) (hρ : IsRational k ρ) :
+    IsRational k σ := by
+  intro v l
+  obtain ⟨w, rfl⟩ := hf v
+  obtain ⟨P, hP⟩ := hρ w (l.comp f)
+  refine ⟨P, fun g => ?_⟩
+  rw [← hP]
+  change l (σ g (f w)) = l (f (ρ g w))
+  rw [hi]
+
+ 
+
+lemma regular_linearChange_coeff {ι : Type*} [Fintype ι]
+    {S : Type*} {τ : Type*} (z : S → τ → k) (A : S → Matrix ι ι k)
+    (hA : ∀ i j, ∃ P : MvPolynomial τ k, ∀ g, A g i j = MvPolynomial.eval (z g) P)
+    (v : MvPolynomial ι k) :
+    ∀ l : MvPolynomial ι k →ₗ[k] k,
+      ∃ P : MvPolynomial τ k, ∀ g, l (linearChange (A g) v) = MvPolynomial.eval (z g) P := by
+  classical
+  induction v using MvPolynomial.induction_on with
+  | C a =>
+    intro l
+    exact ⟨MvPolynomial.C (l (MvPolynomial.C a)), by simp⟩
+  | add v w hv hw =>
+    intro l
+    obtain ⟨P, hP⟩ := hv l
+    obtain ⟨Q, hQ⟩ := hw l
+    exact ⟨P + Q, by simp [hP, hQ]⟩
+  | mul_X v i hv =>
+    intro l
+    choose P hP using fun j => hv (l.comp (LinearMap.mulRight k (MvPolynomial.X j)))
+    choose Q hQ using fun j => hA j i
+    refine ⟨∑ j, Q j * P j, fun g => ?_⟩
+    simp only [map_mul, linearChange_X, Finset.mul_sum, map_sum]
+    apply Finset.sum_congr rfl
+    intro j hj
+    rw [mul_comm (MvPolynomial.C _), ← mul_assoc, mul_comm _ (MvPolynomial.C _)]
+    rw [← MvPolynomial.smul_eq_C_mul, map_smul, hQ]
+    change _ = _ * MvPolynomial.eval (z g) (P j)
+    rw [← hP]
+    rfl
+
+lemma costandard_rational (n : ℕ) : IsRational k (costandard k n).ρ := by
+  intro v l
+  obtain ⟨l', hl'⟩ := LinearMap.dualMap_surjective_of_injective
+    (MvPolynomial.homogeneousSubmodule (Fin 2) k n).injective_subtype l
+  obtain ⟨P, hP⟩ := regular_linearChange_coeff k
+    (fun g : SL2 k => fun ij : Fin 2 × Fin 2 => g ij.1 ij.2)
+    (dualMatrix k) (fun i j => ⟨inverseCoordinatePolynomial k (j,i), fun g => by
+      change (g⁻¹ : SL2 k) j i = _
+      exact (eval_inverseCoordinatePolynomial k g (j,i)).symm⟩) v.val l'
+  refine ⟨P, fun g => ?_⟩
+  rw [← hP]
+  exact congrArg (fun a => a ((costandard k n).ρ g v)) hl'.symm
+
+end FiberFunctorProof
+
+open CategoryTheory CategoryTheory.Limits
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/RestrictedTilting/ExtensionSplitting.lean b/lean/OAI/RepresentationTheory/RestrictedTilting/ExtensionSplitting.lean
index 72f9edf5c3edfc43386c58227ead453f7f153099..47e4fc4a4652f2ba55842531fd577a66dd2d7728 100644
--- a/lean/OAI/RepresentationTheory/RestrictedTilting/ExtensionSplitting.lean
+++ b/lean/OAI/RepresentationTheory/RestrictedTilting/ExtensionSplitting.lean
@@ -1,440 +1,440 @@
-import Mathlib
-import OAI.RepresentationTheory.RestrictedTilting.DualSplitting
-import OAI.RepresentationTheory.RestrictedTilting.RepresentationMaps
-
-namespace OAI
-
-/-! Representation extension splitting, duality and good filtrations. -/
-
-noncomputable section
-open CategoryTheory CategoryTheory.MonoidalCategory CategoryTheory.Limits
-open scoped BigOperators
-
-namespace FiberFunctorProof
-universe u
-variable (k : Type u) [Field k]
-variable {G : Type u} [Monoid G]
-
-lemma fdrep_hom_ext {A B : FDRep k G} {f g : A ⟶ B}
-    (h : ∀ x, fdrepLinear k f x = fdrepLinear k g x) : f = g := by
-  apply Action.hom_ext
-  apply FGModuleCat.hom_ext
-  exact LinearMap.ext h
-
-@[simp] lemma fdrep_comp_apply {A B C : FDRep k G} (f : A ⟶ B) (g : B ⟶ C) (x : A) :
-    fdrepLinear k (f ≫ g) x = fdrepLinear k g (fdrepLinear k f x) := rfl
-
-@[simp] lemma fdrep_id_apply {A : FDRep k G} (x : A) :
-    fdrepLinear k (𝟙 A) x = x := rfl
-
-@[simp] lemma fdrep_zero_apply {A B : FDRep k G} (x : A) :
-    fdrepLinear k (0 : A ⟶ B) x = 0 := rfl
-
- 
-
-def RepExtensionSplit (P : FDRep k G → Prop) (A C : FDRep k G) : Prop :=
-  ∀ (B : FDRep k G), P B → ∀ (f : A ⟶ B) (g : B ⟶ C) (w : f ≫ g = 0),
-    (ShortComplex.mk f g w).ShortExact → ∃ s : C ⟶ B, s ≫ g = 𝟙 C
-
- 
-lemma fdrep_quotient_splice {A₁ A A₂ B C : FDRep k G}
-    (i : A₁ ⟶ A) (j : A ⟶ A₂) (v : i ≫ j = 0)
-    (hT : (ShortComplex.mk i j v).ShortExact)
-    (f : A ⟶ B) (g : B ⟶ C) (w : f ≫ g = 0)
-    (hS : (ShortComplex.mk f g w).ShortExact) :
-    ∃ (Q : FDRep k G) (a : A₂ ⟶ Q) (b : Q ⟶ C) (q : B ⟶ Q)
-      (wab : a ≫ b = 0),
-      Function.Surjective (fdrepLinear k q) ∧
-      f ≫ q = j ≫ a ∧ q ≫ b = g ∧
-      (ShortComplex.mk a b wab).ShortExact := by
-  have := hT.mono_f
-  have := hT.epi_g
-  have := hS.mono_f
-  have := hS.epi_g
-  have hi := fdrep_injective k i
-  have hj := fdrep_surjective k j
-  have hf := fdrep_injective k f
-  have hg := fdrep_surjective k g
-  have hT' := fdrep_exact k hT.exact
-  have hS' := fdrep_exact k hS.exact
-  let Q := fdrepQuotient k (i ≫ f)
-  let q : B ⟶ Q := fdrepQuotientπ k (i ≫ f)
-  have hq := fdrepQuotientπ_surjective k (i ≫ f)
-  have hqa : LinearMap.ker (fdrepLinear k j) ≤ LinearMap.ker (fdrepLinear k (f ≫ q)) := by
-    intro x hx
-    rw [← hT'] at hx
-    obtain ⟨y,rfl⟩ := hx
-    apply (fdrepQuotientπ_eq_zero k (i ≫ f) _).mpr
-    exact ⟨y,rfl⟩
-  let a := fdrepDesc k j hj (f ≫ q) hqa
-  have ha : j ≫ a = f ≫ q := fdrepDesc_comp k j hj (f ≫ q) hqa
-  have hqb : LinearMap.ker (fdrepLinear k q) ≤ LinearMap.ker (fdrepLinear k g) := by
-    intro x hx
-    obtain ⟨y,rfl⟩ := (fdrepQuotientπ_eq_zero k (i ≫ f) x).mp hx
-    have h := LinearMap.congr_fun (congrArg (fdrepLinear k) w) (fdrepLinear k i y)
-    exact h
-  let b := fdrepDesc k q hq g hqb
-  have hb : q ≫ b = g := fdrepDesc_comp k q hq g hqb
-  have hab : a ≫ b = 0 := by
-    apply fdrep_hom_ext k
-    intro x
-    obtain ⟨y,rfl⟩ := hj x
-    have hea := LinearMap.congr_fun (congrArg (fdrepLinear k) ha) y
-    have heb := LinearMap.congr_fun (congrArg (fdrepLinear k) hb) (fdrepLinear k f y)
-    change fdrepLinear k b (fdrepLinear k a (fdrepLinear k j y)) = 0
-    change fdrepLinear k a (fdrepLinear k j y) = fdrepLinear k q (fdrepLinear k f y) at hea
-    rw [hea]
-    rw [show fdrepLinear k b (fdrepLinear k q (fdrepLinear k f y)) =
-      fdrepLinear k g (fdrepLinear k f y) from heb]
-    exact LinearMap.congr_fun (congrArg (fdrepLinear k) w) y
-  refine ⟨Q,a,b,q,hab,hq,ha.symm,hb,fdrep_shortExact k _ ?_ ?_ ?_⟩
-  · apply LinearMap.ker_eq_bot.mp
-    apply LinearMap.ker_eq_bot'.mpr
-    intro x hx
-    obtain ⟨y,rfl⟩ := hj x
-    have hy : fdrepLinear k q (fdrepLinear k f y) = 0 := by
-      have := LinearMap.congr_fun (congrArg (fdrepLinear k) ha) y
-      exact this.symm.trans hx
-    obtain ⟨z,hz⟩ := (fdrepQuotientπ_eq_zero k (i ≫ f) _).mp hy
-    have hfz : fdrepLinear k i z = y := hf hz
-    rw [← hfz]
-    exact LinearMap.congr_fun (congrArg (fdrepLinear k) v) z
-  · intro c
-    obtain ⟨x,rfl⟩ := hg c
-    exact ⟨fdrepLinear k q x, LinearMap.congr_fun (congrArg (fdrepLinear k) hb) x⟩
-  · apply le_antisymm
-    · rw [LinearMap.range_le_ker_iff]
-      exact congrArg (fdrepLinear k) hab
-    · intro x hx
-      obtain ⟨y,rfl⟩ := hq x
-      have hy : fdrepLinear k g y = 0 := by
-        have := LinearMap.congr_fun (congrArg (fdrepLinear k) hb) y
-        exact this.symm.trans hx
-      have hyrange : y ∈ LinearMap.range (fdrepLinear k f) := by
-        rw [hS']
-        exact hy
-      obtain ⟨z,rfl⟩ := hyrange
-      exact ⟨fdrepLinear k j z,
-        LinearMap.congr_fun (congrArg (fdrepLinear k) ha) z⟩
-
-
- 
-
-lemma fdrep_kernel_splice {A₁ A A₂ B C : FDRep k G}
-    (i : A₁ ⟶ A) (j : A ⟶ A₂) (v : i ≫ j = 0)
-    (hT : (ShortComplex.mk i j v).ShortExact)
-    (f : A ⟶ B) (g : B ⟶ C) (w : f ≫ g = 0)
-    (hS : (ShortComplex.mk f g w).ShortExact)
-    (r : B ⟶ A₂) (hr : f ≫ r = j) :
-    ∃ (K : FDRep k G) (a : A₁ ⟶ K) (b : K ⟶ C) (t : K ⟶ B)
-      (wab : a ≫ b = 0),
-      Function.Injective (fdrepLinear k t) ∧ t ≫ g = b ∧
-      (ShortComplex.mk a b wab).ShortExact := by
-  have := hT.mono_f
-  have := hT.epi_g
-  have := hS.mono_f
-  have := hS.epi_g
-  have hi := fdrep_injective k i
-  have hj := fdrep_surjective k j
-  have hf := fdrep_injective k f
-  have hg := fdrep_surjective k g
-  have hT' := fdrep_exact k hT.exact
-  have hS' := fdrep_exact k hS.exact
-  have hif : (i ≫ f) ≫ r = 0 := by rw [Category.assoc, hr, v]
-  let K := fdrepKernel k r
-  let t : K ⟶ B := fdrepKernelι k r
-  let a : A₁ ⟶ K := fdrepKernelLift k (i ≫ f) r hif
-  let b : K ⟶ C := t ≫ g
-  have hab : a ≫ b = 0 := by
-    change a ≫ (t ≫ g) = 0
-    rw [← Category.assoc, fdrepKernelLift_ι, Category.assoc, w, comp_zero]
-  refine ⟨K,a,b,t,hab,Subtype.val_injective,rfl,fdrep_shortExact k _ ?_ ?_ ?_⟩
-  · intro x y hxy
-    apply hi
-    apply hf
-    exact congrArg Subtype.val hxy
-  · intro c
-    obtain ⟨x,hx⟩ := hg c
-    obtain ⟨y,hy⟩ := hj (fdrepLinear k r x)
-    have hy' : fdrepLinear k r (fdrepLinear k f y) = fdrepLinear k j y :=
-      LinearMap.congr_fun (congrArg (fdrepLinear k) hr) y
-    have hz : fdrepLinear k r (x-fdrepLinear k f y) = 0 := by
-      rw [map_sub, hy', hy, sub_self]
-    refine ⟨⟨x-fdrepLinear k f y,hz⟩, ?_⟩
-    change fdrepLinear k g (x-fdrepLinear k f y) = c
-    rw [map_sub, hx]
-    have hw := LinearMap.congr_fun (congrArg (fdrepLinear k) w) y
-    change fdrepLinear k g (fdrepLinear k f y) = 0 at hw
-    rw [hw, sub_zero]
-  · apply le_antisymm
-    · rw [LinearMap.range_le_ker_iff]
-      exact congrArg (fdrepLinear k) hab
-    · intro x hx
-      have hx' : x.val ∈ LinearMap.range (fdrepLinear k f) := by
-        rw [hS']
-        exact hx
-      obtain ⟨y,hy⟩ := hx'
-      have hyk : y ∈ LinearMap.ker (fdrepLinear k j) := by
-        have h := LinearMap.congr_fun (congrArg (fdrepLinear k) hr) y
-        change fdrepLinear k r (fdrepLinear k f y) = fdrepLinear k j y at h
-        change fdrepLinear k j y = 0
-        rw [← h, hy]
-        exact x.property
-      rw [← hT'] at hyk
-      obtain ⟨z,hz⟩ := hyk
-      refine ⟨z, ?_⟩
-      apply Subtype.ext
-      change fdrepLinear k f (fdrepLinear k i z) = x.val
-      rw [hz,hy]
-
- 
-
-theorem RepExtensionSplit.extension_left (P : FDRep k G → Prop)
-    (hSub : ∀ {X Y : FDRep k G} (f : X ⟶ Y),
-      Function.Injective (fdrepLinear k f) → P Y → P X)
-    (hQuot : ∀ {X Y : FDRep k G} (f : X ⟶ Y),
-      Function.Surjective (fdrepLinear k f) → P X → P Y)
-    {A₁ A A₂ C : FDRep k G}
-    (i : A₁ ⟶ A) (j : A ⟶ A₂) (v : i ≫ j = 0)
-    (hT : (ShortComplex.mk i j v).ShortExact)
-    (h₁ : RepExtensionSplit k P A₁ C) (h₂ : RepExtensionSplit k P A₂ C) :
-    RepExtensionSplit k P A C := by
-  intro B hB f g w hS
-  obtain ⟨Q,a,b,q,wab,hq,hfq,hqb,hQ⟩ :=
-    fdrep_quotient_splice k i j v hT f g w hS
-  obtain ⟨s,hs⟩ := h₂ Q (hQuot q hq hB) a b wab hQ
-  let e := ShortComplex.Splitting.ofExactOfSection (ShortComplex.mk a b wab)
-    hQ.exact s hs hQ.mono_f
-  have hr : f ≫ (q ≫ e.r) = j := by
-    rw [← Category.assoc, hfq, Category.assoc, e.f_r, Category.comp_id]
-  obtain ⟨K,a',b',t,wab',ht,htg,hK⟩ :=
-    fdrep_kernel_splice k i j v hT f g w hS (q ≫ e.r) hr
-  obtain ⟨s',hs'⟩ := h₁ K (hSub t ht hB) a' b' wab' hK
-  refine ⟨s' ≫ t, ?_⟩
-  rw [Category.assoc, htg, hs']
-
-end FiberFunctorProof
-
-open CategoryTheory CategoryTheory.MonoidalCategory
-open scoped TensorProduct
-namespace FiberFunctorProof
-section
-universe u
-variable (k : Type u) [Field k]
-lemma fgmodule_mate_apply {A B : FGModuleCat k} (f : A ⟶ B)
-    (l : Module.Dual k B) (x : A) :
-    (show Module.Dual k A from (fᘁ).hom.hom l) x = l (f.hom.hom x) := by
-  have h := rightAdjointMate_comp_evaluation f
-  have hx := ConcreteCategory.congr_hom h (l ⊗ₜ[k] x)
-  change contractLeft k A
-      (TensorProduct.map (fᘁ).hom.hom (LinearMap.id : A →ₗ[k] A) (l ⊗ₜ[k] x)) =
-    contractLeft k B
-      (TensorProduct.map (LinearMap.id : Module.Dual k B →ₗ[k] Module.Dual k B)
-        f.hom.hom (l ⊗ₜ[k] x)) at hx
-  change (contractLeft k A)
-      (((fᘁ).hom.hom : Module.Dual k B →ₗ[k] Module.Dual k A) l ⊗ₜ[k] x) =
-    (contractLeft k B) (l ⊗ₜ[k] f.hom.hom x) at hx
-  exact hx
-
-lemma fdrep_dual_repr {G : Type u} [Group G] (A : FDRep k G) :
-    (Aᘁ).ρ = Representation.dual A.ρ := by
-  ext g l : 2
-  apply LinearMap.ext
-  intro x
-  change (show Module.Dual k A from ((Action.ρ (Aᘁ)) g).hom.hom l) x = _
-  rw [Action.rightDual_ρ]
-  exact fgmodule_mate_apply k _ _ _
-
-end
-
-section
-open CategoryTheory CategoryTheory.MonoidalCategory
-universe u
-variable (k : Type u) [Field k]
-variable {G : Type u} [Group G]
-
- 
-noncomputable def fdrepDualMap {A B : FDRep k G} (f : A ⟶ B) : Bᘁ ⟶ Aᘁ :=
-  fdrepHom k (fdrepLinear k f).dualMap (by
-    intro g φ
-    change Module.Dual k B at φ
-    rw [fdrep_dual_repr, fdrep_dual_repr]
-    apply LinearMap.ext
-    intro x
-    change φ (B.ρ g⁻¹ (fdrepLinear k f x)) = φ (fdrepLinear k f (A.ρ g⁻¹ x))
-    exact congrArg φ (fdrepLinear_comm k f g⁻¹ x).symm)
-
-lemma fdrepDualMap_zero_comp {A B C : FDRep k G}
-    (f : A ⟶ B) (g : B ⟶ C) (w : f ≫ g = 0) :
-    fdrepDualMap k g ≫ fdrepDualMap k f = 0 := by
-  apply fdrep_hom_ext
-  intro φ
-  change Module.Dual k C at φ
-  apply LinearMap.ext
-  intro x
-  change φ (fdrepLinear k g (fdrepLinear k f x)) = 0
-  have h := LinearMap.congr_fun (congrArg (fdrepLinear k) w) x
-  change fdrepLinear k g (fdrepLinear k f x) = 0 at h
-  rw [h, map_zero]
-
-lemma fdrep_dual_shortExact {A B C : FDRep k G}
-    (f : A ⟶ B) (g : B ⟶ C) (w : f ≫ g = 0)
-    (hS : (ShortComplex.mk f g w).ShortExact) :
-    (ShortComplex.mk (fdrepDualMap k g) (fdrepDualMap k f)
-      (fdrepDualMap_zero_comp k f g w)).ShortExact := by
-  have := hS.mono_f
-  have := hS.epi_g
-  refine fdrep_shortExact k _
-    (LinearMap.dualMap_injective_of_surjective (fdrep_surjective k g))
-    (LinearMap.dualMap_surjective_of_injective (fdrep_injective k f)) ?_
-  change LinearMap.range (fdrepLinear k g).dualMap =
-    LinearMap.ker (fdrepLinear k f).dualMap
-  rw [LinearMap.range_dualMap_eq_dualAnnihilator_ker,
-    LinearMap.ker_dualMap_eq_dualAnnihilator_range, fdrep_exact k hS.exact]
-
- 
-
-theorem RepExtensionSplit.of_dual (P : FDRep k G → Prop)
-    (hDual : ∀ B, P B → P Bᘁ) {A C : FDRep k G}
-    (h : RepExtensionSplit k P Cᘁ Aᘁ) : RepExtensionSplit k P A C := by
-  intro B hB f g w hS
-  obtain ⟨s,hs⟩ := h Bᘁ (hDual B hB) (fdrepDualMap k g)
-    (fdrepDualMap k f) (fdrepDualMap_zero_comp k f g w)
-    (fdrep_dual_shortExact k f g w hS)
-  have hs' : (fdrepLinear k f).dualMap.comp (fdrepLinear k s) = LinearMap.id :=
-    congrArg (fdrepLinear k) hs
-  have hsρ : ∀ a φ, fdrepLinear k s (Representation.dual A.ρ a φ) =
-      Representation.dual B.ρ a (fdrepLinear k s φ) := by
-    intro a φ
-    rw [← fdrep_dual_repr k A, ← fdrep_dual_repr k B]
-    exact fdrepLinear_comm k s a φ
-  obtain ⟨r,hr,hrρ⟩ := equivariant_retraction_of_dual_section A.ρ B.ρ
-    (fdrepLinear k f) (fdrepLinear k s) hs' hsρ
-  let r' : B ⟶ A := fdrepHom k r hrρ
-  have hfr : f ≫ r' = 𝟙 A := by
-    apply fdrep_hom_ext
-    exact LinearMap.congr_fun hr
-  let spl := ShortComplex.Splitting.ofExactOfRetraction (ShortComplex.mk f g w)
-    hS.exact r' hfr hS.epi_g
-  exact ⟨spl.s,spl.s_g⟩
-
-end
-
-section
-open CategoryTheory CategoryTheory.MonoidalCategory
-universe u
-variable (k : Type u) [Field k]
-
- 
-abbrev RationalExtensionSplit (A C : FDRep k (SL2 k)) : Prop :=
-  RepExtensionSplit k (fun B => IsRational k B.ρ) A C
-
-lemma rationalExtensionSplit_extension_left {A₁ A A₂ C : FDRep k (SL2 k)}
-    (i : A₁ ⟶ A) (j : A ⟶ A₂) (v : i ≫ j = 0)
-    (hT : (ShortComplex.mk i j v).ShortExact)
-    (h₁ : RationalExtensionSplit k A₁ C) (h₂ : RationalExtensionSplit k A₂ C) :
-    RationalExtensionSplit k A C := by
-  apply RepExtensionSplit.extension_left k _ (fun f hf h => ?_) (fun f hf h => ?_)
-    i j v hT h₁ h₂
-  · exact IsRational.of_injective k _ _ (fdrepLinear k f) hf (fdrepLinear_comm k f) h
-  · exact IsRational.of_surjective k _ _ (fdrepLinear k f) hf (fdrepLinear_comm k f) h
-
-lemma rationalExtensionSplit_of_dual {A C : FDRep k (SL2 k)}
-    (h : RationalExtensionSplit k Cᘁ Aᘁ) : RationalExtensionSplit k A C := by
-  apply RepExtensionSplit.of_dual k _ (fun B hB => ?_) h
-  rw [fdrep_dual_repr]
-  exact IsRational.dual k B.ρ hB
-
-lemma rationalExtensionSplit_zero {A C : FDRep k (SL2 k)} (hA : IsZero A) :
-    RationalExtensionSplit k A C := by
-  intro B _ f g w hS
-  have : Mono g := hS.exact.mono_g (hA.eq_zero_of_src f)
-  have := hS.epi_g
-  have : IsIso g := isIso_of_mono_of_epi g
-  exact ⟨inv g, IsIso.inv_hom_id g⟩
-
-variable [Infinite k]
-
- 
-theorem costandard_standard_extension_split (m n : ℕ) :
-    RationalExtensionSplit k (costandard k m) (costandard k n)ᘁ := by
-  intro B hB f g w hS
-  have := hS.mono_f
-  have := hS.epi_g
-  obtain ⟨s,hs,hsρ⟩ := standard_costandard_split k B.ρ hB n m
-    (fdrepLinear k f) (fdrepLinear k g) (fdrep_injective k f) (fdrep_surjective k g)
-    (fdrep_exact k hS.exact).symm (fdrepLinear_comm k f) (by
-      intro a x
-      have h := fdrepLinear_comm k g a x
-      rw [fdrep_dual_repr] at h
-      exact h)
-  let s' : (costandard k n)ᘁ ⟶ B := fdrepHom k s (by
-    intro a x
-    rw [fdrep_dual_repr]
-    exact hsρ a x)
-  refine ⟨s', fdrep_hom_ext k (LinearMap.congr_fun hs)⟩
-
- 
-theorem good_standard_extension_split {A : FDRep k (SL2 k)}
-    (hA : HasGoodFiltration k A) (n : ℕ) :
-    RationalExtensionSplit k A (costandard k n)ᘁ := by
-  induction hA with
-  | zero A hA => exact rationalExtensionSplit_zero k hA
-  | @step A B m f g w hS hA ih =>
-    exact rationalExtensionSplit_extension_left k f g w hS ih
-      (costandard_standard_extension_split k m n)
-
- 
-
-theorem good_weyl_extension_split {A C : FDRep k (SL2 k)}
-    (hA : HasGoodFiltration k A) (hC : HasGoodFiltration k Cᘁ) :
-    RationalExtensionSplit k A C := by
-  induction hA with
-  | zero A hA => exact rationalExtensionSplit_zero k hA
-  | @step A B m f g w hS hA ih =>
-    apply rationalExtensionSplit_extension_left k f g w hS ih
-    exact rationalExtensionSplit_of_dual k (good_standard_extension_split k hC m)
-
-end
-
-section
-open scoped ZeroObject
-universe u
-variable (k : Type u) [Field k]
-lemma HasGoodFiltration.of_iso {A B : FDRep k (SL2 k)}
-    (h : HasGoodFiltration k A) (e : A ≅ B) : HasGoodFiltration k B := by
-  cases h with
-  | zero A hA => exact .zero B (hA.of_iso e.symm)
-  | @step A B' n f g w hS hA =>
-    let f' := f ≫ e.hom
-    let g' := e.inv ≫ g
-    have w' : f' ≫ g' = 0 := by simpa [f', g'] using w
-    have hS' : (ShortComplex.mk f' g' w').ShortExact :=
-      ShortComplex.shortExact_of_iso
-        (ShortComplex.isoMk (Iso.refl A) e (Iso.refl (costandard k n))
-          (by simp [f']) (by simp [g'])) hS
-    exact .step n f' g' w' hS' hA
-
-lemma costandard_hasGoodFiltration (n : ℕ) : HasGoodFiltration k (costandard k n) := by
-  let S : ShortComplex (FDRep k (SL2 k)) := ShortComplex.mk
-    (0 : 0 ⟶ costandard k n) (𝟙 _) (by simp)
-  have hS : S.ShortExact :=
-    (ShortComplex.Splitting.ofIsZeroOfIsIso S (isZero_zero _) inferInstance).shortExact
-  exact .step n S.f S.g S.zero hS (.zero _ (isZero_zero _))
-
-end
-
-universe u
-variable (k : Type u) [Field k]
-lemma fdrepHom_isIso_of_bijective {G : Type u} [Monoid G] {A B : FDRep k G}
-    (f : A ⟶ B) (hf : Function.Bijective (fdrepLinear k f)) : IsIso f := by
-  have : Mono ((fdrepForget k G).map f) := (ModuleCat.mono_iff_injective _).mpr hf.1
-  have : Epi ((fdrepForget k G).map f) := (ModuleCat.epi_iff_surjective _).mpr hf.2
-  have : Mono f := (fdrepForget k G).mono_of_mono_map (by infer_instance)
-  have : Epi f := (fdrepForget k G).epi_of_epi_map (by infer_instance)
-  exact isIso_of_mono_of_epi f
-
-end FiberFunctorProof
-end
-
-end OAI
+import Mathlib
+import OAI.RepresentationTheory.RestrictedTilting.DualSplitting
+import OAI.RepresentationTheory.RestrictedTilting.RepresentationMaps
+
+namespace OAI
+
+/-! Representation extension splitting, duality and good filtrations. -/
+
+noncomputable section
+open CategoryTheory CategoryTheory.MonoidalCategory CategoryTheory.Limits
+open scoped BigOperators
+
+namespace FiberFunctorProof
+universe u
+variable (k : Type u) [Field k]
+variable {G : Type u} [Monoid G]
+
+lemma fdrep_hom_ext {A B : FDRep k G} {f g : A ⟶ B}
+    (h : ∀ x, fdrepLinear k f x = fdrepLinear k g x) : f = g := by
+  apply Action.hom_ext
+  apply FGModuleCat.hom_ext
+  exact LinearMap.ext h
+
+@[simp] lemma fdrep_comp_apply {A B C : FDRep k G} (f : A ⟶ B) (g : B ⟶ C) (x : A) :
+    fdrepLinear k (f ≫ g) x = fdrepLinear k g (fdrepLinear k f x) := rfl
+
+@[simp] lemma fdrep_id_apply {A : FDRep k G} (x : A) :
+    fdrepLinear k (𝟙 A) x = x := rfl
+
+@[simp] lemma fdrep_zero_apply {A B : FDRep k G} (x : A) :
+    fdrepLinear k (0 : A ⟶ B) x = 0 := rfl
+
+ 
+
+def RepExtensionSplit (P : FDRep k G → Prop) (A C : FDRep k G) : Prop :=
+  ∀ (B : FDRep k G), P B → ∀ (f : A ⟶ B) (g : B ⟶ C) (w : f ≫ g = 0),
+    (ShortComplex.mk f g w).ShortExact → ∃ s : C ⟶ B, s ≫ g = 𝟙 C
+
+ 
+lemma fdrep_quotient_splice {A₁ A A₂ B C : FDRep k G}
+    (i : A₁ ⟶ A) (j : A ⟶ A₂) (v : i ≫ j = 0)
+    (hT : (ShortComplex.mk i j v).ShortExact)
+    (f : A ⟶ B) (g : B ⟶ C) (w : f ≫ g = 0)
+    (hS : (ShortComplex.mk f g w).ShortExact) :
+    ∃ (Q : FDRep k G) (a : A₂ ⟶ Q) (b : Q ⟶ C) (q : B ⟶ Q)
+      (wab : a ≫ b = 0),
+      Function.Surjective (fdrepLinear k q) ∧
+      f ≫ q = j ≫ a ∧ q ≫ b = g ∧
+      (ShortComplex.mk a b wab).ShortExact := by
+  have := hT.mono_f
+  have := hT.epi_g
+  have := hS.mono_f
+  have := hS.epi_g
+  have hi := fdrep_injective k i
+  have hj := fdrep_surjective k j
+  have hf := fdrep_injective k f
+  have hg := fdrep_surjective k g
+  have hT' := fdrep_exact k hT.exact
+  have hS' := fdrep_exact k hS.exact
+  let Q := fdrepQuotient k (i ≫ f)
+  let q : B ⟶ Q := fdrepQuotientπ k (i ≫ f)
+  have hq := fdrepQuotientπ_surjective k (i ≫ f)
+  have hqa : LinearMap.ker (fdrepLinear k j) ≤ LinearMap.ker (fdrepLinear k (f ≫ q)) := by
+    intro x hx
+    rw [← hT'] at hx
+    obtain ⟨y,rfl⟩ := hx
+    apply (fdrepQuotientπ_eq_zero k (i ≫ f) _).mpr
+    exact ⟨y,rfl⟩
+  let a := fdrepDesc k j hj (f ≫ q) hqa
+  have ha : j ≫ a = f ≫ q := fdrepDesc_comp k j hj (f ≫ q) hqa
+  have hqb : LinearMap.ker (fdrepLinear k q) ≤ LinearMap.ker (fdrepLinear k g) := by
+    intro x hx
+    obtain ⟨y,rfl⟩ := (fdrepQuotientπ_eq_zero k (i ≫ f) x).mp hx
+    have h := LinearMap.congr_fun (congrArg (fdrepLinear k) w) (fdrepLinear k i y)
+    exact h
+  let b := fdrepDesc k q hq g hqb
+  have hb : q ≫ b = g := fdrepDesc_comp k q hq g hqb
+  have hab : a ≫ b = 0 := by
+    apply fdrep_hom_ext k
+    intro x
+    obtain ⟨y,rfl⟩ := hj x
+    have hea := LinearMap.congr_fun (congrArg (fdrepLinear k) ha) y
+    have heb := LinearMap.congr_fun (congrArg (fdrepLinear k) hb) (fdrepLinear k f y)
+    change fdrepLinear k b (fdrepLinear k a (fdrepLinear k j y)) = 0
+    change fdrepLinear k a (fdrepLinear k j y) = fdrepLinear k q (fdrepLinear k f y) at hea
+    rw [hea]
+    rw [show fdrepLinear k b (fdrepLinear k q (fdrepLinear k f y)) =
+      fdrepLinear k g (fdrepLinear k f y) from heb]
+    exact LinearMap.congr_fun (congrArg (fdrepLinear k) w) y
+  refine ⟨Q,a,b,q,hab,hq,ha.symm,hb,fdrep_shortExact k _ ?_ ?_ ?_⟩
+  · apply LinearMap.ker_eq_bot.mp
+    apply LinearMap.ker_eq_bot'.mpr
+    intro x hx
+    obtain ⟨y,rfl⟩ := hj x
+    have hy : fdrepLinear k q (fdrepLinear k f y) = 0 := by
+      have := LinearMap.congr_fun (congrArg (fdrepLinear k) ha) y
+      exact this.symm.trans hx
+    obtain ⟨z,hz⟩ := (fdrepQuotientπ_eq_zero k (i ≫ f) _).mp hy
+    have hfz : fdrepLinear k i z = y := hf hz
+    rw [← hfz]
+    exact LinearMap.congr_fun (congrArg (fdrepLinear k) v) z
+  · intro c
+    obtain ⟨x,rfl⟩ := hg c
+    exact ⟨fdrepLinear k q x, LinearMap.congr_fun (congrArg (fdrepLinear k) hb) x⟩
+  · apply le_antisymm
+    · rw [LinearMap.range_le_ker_iff]
+      exact congrArg (fdrepLinear k) hab
+    · intro x hx
+      obtain ⟨y,rfl⟩ := hq x
+      have hy : fdrepLinear k g y = 0 := by
+        have := LinearMap.congr_fun (congrArg (fdrepLinear k) hb) y
+        exact this.symm.trans hx
+      have hyrange : y ∈ LinearMap.range (fdrepLinear k f) := by
+        rw [hS']
+        exact hy
+      obtain ⟨z,rfl⟩ := hyrange
+      exact ⟨fdrepLinear k j z,
+        LinearMap.congr_fun (congrArg (fdrepLinear k) ha) z⟩
+
+
+ 
+
+lemma fdrep_kernel_splice {A₁ A A₂ B C : FDRep k G}
+    (i : A₁ ⟶ A) (j : A ⟶ A₂) (v : i ≫ j = 0)
+    (hT : (ShortComplex.mk i j v).ShortExact)
+    (f : A ⟶ B) (g : B ⟶ C) (w : f ≫ g = 0)
+    (hS : (ShortComplex.mk f g w).ShortExact)
+    (r : B ⟶ A₂) (hr : f ≫ r = j) :
+    ∃ (K : FDRep k G) (a : A₁ ⟶ K) (b : K ⟶ C) (t : K ⟶ B)
+      (wab : a ≫ b = 0),
+      Function.Injective (fdrepLinear k t) ∧ t ≫ g = b ∧
+      (ShortComplex.mk a b wab).ShortExact := by
+  have := hT.mono_f
+  have := hT.epi_g
+  have := hS.mono_f
+  have := hS.epi_g
+  have hi := fdrep_injective k i
+  have hj := fdrep_surjective k j
+  have hf := fdrep_injective k f
+  have hg := fdrep_surjective k g
+  have hT' := fdrep_exact k hT.exact
+  have hS' := fdrep_exact k hS.exact
+  have hif : (i ≫ f) ≫ r = 0 := by rw [Category.assoc, hr, v]
+  let K := fdrepKernel k r
+  let t : K ⟶ B := fdrepKernelι k r
+  let a : A₁ ⟶ K := fdrepKernelLift k (i ≫ f) r hif
+  let b : K ⟶ C := t ≫ g
+  have hab : a ≫ b = 0 := by
+    change a ≫ (t ≫ g) = 0
+    rw [← Category.assoc, fdrepKernelLift_ι, Category.assoc, w, comp_zero]
+  refine ⟨K,a,b,t,hab,Subtype.val_injective,rfl,fdrep_shortExact k _ ?_ ?_ ?_⟩
+  · intro x y hxy
+    apply hi
+    apply hf
+    exact congrArg Subtype.val hxy
+  · intro c
+    obtain ⟨x,hx⟩ := hg c
+    obtain ⟨y,hy⟩ := hj (fdrepLinear k r x)
+    have hy' : fdrepLinear k r (fdrepLinear k f y) = fdrepLinear k j y :=
+      LinearMap.congr_fun (congrArg (fdrepLinear k) hr) y
+    have hz : fdrepLinear k r (x-fdrepLinear k f y) = 0 := by
+      rw [map_sub, hy', hy, sub_self]
+    refine ⟨⟨x-fdrepLinear k f y,hz⟩, ?_⟩
+    change fdrepLinear k g (x-fdrepLinear k f y) = c
+    rw [map_sub, hx]
+    have hw := LinearMap.congr_fun (congrArg (fdrepLinear k) w) y
+    change fdrepLinear k g (fdrepLinear k f y) = 0 at hw
+    rw [hw, sub_zero]
+  · apply le_antisymm
+    · rw [LinearMap.range_le_ker_iff]
+      exact congrArg (fdrepLinear k) hab
+    · intro x hx
+      have hx' : x.val ∈ LinearMap.range (fdrepLinear k f) := by
+        rw [hS']
+        exact hx
+      obtain ⟨y,hy⟩ := hx'
+      have hyk : y ∈ LinearMap.ker (fdrepLinear k j) := by
+        have h := LinearMap.congr_fun (congrArg (fdrepLinear k) hr) y
+        change fdrepLinear k r (fdrepLinear k f y) = fdrepLinear k j y at h
+        change fdrepLinear k j y = 0
+        rw [← h, hy]
+        exact x.property
+      rw [← hT'] at hyk
+      obtain ⟨z,hz⟩ := hyk
+      refine ⟨z, ?_⟩
+      apply Subtype.ext
+      change fdrepLinear k f (fdrepLinear k i z) = x.val
+      rw [hz,hy]
+
+ 
+
+theorem RepExtensionSplit.extension_left (P : FDRep k G → Prop)
+    (hSub : ∀ {X Y : FDRep k G} (f : X ⟶ Y),
+      Function.Injective (fdrepLinear k f) → P Y → P X)
+    (hQuot : ∀ {X Y : FDRep k G} (f : X ⟶ Y),
+      Function.Surjective (fdrepLinear k f) → P X → P Y)
+    {A₁ A A₂ C : FDRep k G}
+    (i : A₁ ⟶ A) (j : A ⟶ A₂) (v : i ≫ j = 0)
+    (hT : (ShortComplex.mk i j v).ShortExact)
+    (h₁ : RepExtensionSplit k P A₁ C) (h₂ : RepExtensionSplit k P A₂ C) :
+    RepExtensionSplit k P A C := by
+  intro B hB f g w hS
+  obtain ⟨Q,a,b,q,wab,hq,hfq,hqb,hQ⟩ :=
+    fdrep_quotient_splice k i j v hT f g w hS
+  obtain ⟨s,hs⟩ := h₂ Q (hQuot q hq hB) a b wab hQ
+  let e := ShortComplex.Splitting.ofExactOfSection (ShortComplex.mk a b wab)
+    hQ.exact s hs hQ.mono_f
+  have hr : f ≫ (q ≫ e.r) = j := by
+    rw [← Category.assoc, hfq, Category.assoc, e.f_r, Category.comp_id]
+  obtain ⟨K,a',b',t,wab',ht,htg,hK⟩ :=
+    fdrep_kernel_splice k i j v hT f g w hS (q ≫ e.r) hr
+  obtain ⟨s',hs'⟩ := h₁ K (hSub t ht hB) a' b' wab' hK
+  refine ⟨s' ≫ t, ?_⟩
+  rw [Category.assoc, htg, hs']
+
+end FiberFunctorProof
+
+open CategoryTheory CategoryTheory.MonoidalCategory
+open scoped TensorProduct
+namespace FiberFunctorProof
+section
+universe u
+variable (k : Type u) [Field k]
+lemma fgmodule_mate_apply {A B : FGModuleCat k} (f : A ⟶ B)
+    (l : Module.Dual k B) (x : A) :
+    (show Module.Dual k A from (fᘁ).hom.hom l) x = l (f.hom.hom x) := by
+  have h := rightAdjointMate_comp_evaluation f
+  have hx := ConcreteCategory.congr_hom h (l ⊗ₜ[k] x)
+  change contractLeft k A
+      (TensorProduct.map (fᘁ).hom.hom (LinearMap.id : A →ₗ[k] A) (l ⊗ₜ[k] x)) =
+    contractLeft k B
+      (TensorProduct.map (LinearMap.id : Module.Dual k B →ₗ[k] Module.Dual k B)
+        f.hom.hom (l ⊗ₜ[k] x)) at hx
+  change (contractLeft k A)
+      (((fᘁ).hom.hom : Module.Dual k B →ₗ[k] Module.Dual k A) l ⊗ₜ[k] x) =
+    (contractLeft k B) (l ⊗ₜ[k] f.hom.hom x) at hx
+  exact hx
+
+lemma fdrep_dual_repr {G : Type u} [Group G] (A : FDRep k G) :
+    (Aᘁ).ρ = Representation.dual A.ρ := by
+  ext g l : 2
+  apply LinearMap.ext
+  intro x
+  change (show Module.Dual k A from ((Action.ρ (Aᘁ)) g).hom.hom l) x = _
+  rw [Action.rightDual_ρ]
+  exact fgmodule_mate_apply k _ _ _
+
+end
+
+section
+open CategoryTheory CategoryTheory.MonoidalCategory
+universe u
+variable (k : Type u) [Field k]
+variable {G : Type u} [Group G]
+
+ 
+noncomputable def fdrepDualMap {A B : FDRep k G} (f : A ⟶ B) : Bᘁ ⟶ Aᘁ :=
+  fdrepHom k (fdrepLinear k f).dualMap (by
+    intro g φ
+    change Module.Dual k B at φ
+    rw [fdrep_dual_repr, fdrep_dual_repr]
+    apply LinearMap.ext
+    intro x
+    change φ (B.ρ g⁻¹ (fdrepLinear k f x)) = φ (fdrepLinear k f (A.ρ g⁻¹ x))
+    exact congrArg φ (fdrepLinear_comm k f g⁻¹ x).symm)
+
+lemma fdrepDualMap_zero_comp {A B C : FDRep k G}
+    (f : A ⟶ B) (g : B ⟶ C) (w : f ≫ g = 0) :
+    fdrepDualMap k g ≫ fdrepDualMap k f = 0 := by
+  apply fdrep_hom_ext
+  intro φ
+  change Module.Dual k C at φ
+  apply LinearMap.ext
+  intro x
+  change φ (fdrepLinear k g (fdrepLinear k f x)) = 0
+  have h := LinearMap.congr_fun (congrArg (fdrepLinear k) w) x
+  change fdrepLinear k g (fdrepLinear k f x) = 0 at h
+  rw [h, map_zero]
+
+lemma fdrep_dual_shortExact {A B C : FDRep k G}
+    (f : A ⟶ B) (g : B ⟶ C) (w : f ≫ g = 0)
+    (hS : (ShortComplex.mk f g w).ShortExact) :
+    (ShortComplex.mk (fdrepDualMap k g) (fdrepDualMap k f)
+      (fdrepDualMap_zero_comp k f g w)).ShortExact := by
+  have := hS.mono_f
+  have := hS.epi_g
+  refine fdrep_shortExact k _
+    (LinearMap.dualMap_injective_of_surjective (fdrep_surjective k g))
+    (LinearMap.dualMap_surjective_of_injective (fdrep_injective k f)) ?_
+  change LinearMap.range (fdrepLinear k g).dualMap =
+    LinearMap.ker (fdrepLinear k f).dualMap
+  rw [LinearMap.range_dualMap_eq_dualAnnihilator_ker,
+    LinearMap.ker_dualMap_eq_dualAnnihilator_range, fdrep_exact k hS.exact]
+
+ 
+
+theorem RepExtensionSplit.of_dual (P : FDRep k G → Prop)
+    (hDual : ∀ B, P B → P Bᘁ) {A C : FDRep k G}
+    (h : RepExtensionSplit k P Cᘁ Aᘁ) : RepExtensionSplit k P A C := by
+  intro B hB f g w hS
+  obtain ⟨s,hs⟩ := h Bᘁ (hDual B hB) (fdrepDualMap k g)
+    (fdrepDualMap k f) (fdrepDualMap_zero_comp k f g w)
+    (fdrep_dual_shortExact k f g w hS)
+  have hs' : (fdrepLinear k f).dualMap.comp (fdrepLinear k s) = LinearMap.id :=
+    congrArg (fdrepLinear k) hs
+  have hsρ : ∀ a φ, fdrepLinear k s (Representation.dual A.ρ a φ) =
+      Representation.dual B.ρ a (fdrepLinear k s φ) := by
+    intro a φ
+    rw [← fdrep_dual_repr k A, ← fdrep_dual_repr k B]
+    exact fdrepLinear_comm k s a φ
+  obtain ⟨r,hr,hrρ⟩ := equivariant_retraction_of_dual_section A.ρ B.ρ
+    (fdrepLinear k f) (fdrepLinear k s) hs' hsρ
+  let r' : B ⟶ A := fdrepHom k r hrρ
+  have hfr : f ≫ r' = 𝟙 A := by
+    apply fdrep_hom_ext
+    exact LinearMap.congr_fun hr
+  let spl := ShortComplex.Splitting.ofExactOfRetraction (ShortComplex.mk f g w)
+    hS.exact r' hfr hS.epi_g
+  exact ⟨spl.s,spl.s_g⟩
+
+end
+
+section
+open CategoryTheory CategoryTheory.MonoidalCategory
+universe u
+variable (k : Type u) [Field k]
+
+ 
+abbrev RationalExtensionSplit (A C : FDRep k (SL2 k)) : Prop :=
+  RepExtensionSplit k (fun B => IsRational k B.ρ) A C
+
+lemma rationalExtensionSplit_extension_left {A₁ A A₂ C : FDRep k (SL2 k)}
+    (i : A₁ ⟶ A) (j : A ⟶ A₂) (v : i ≫ j = 0)
+    (hT : (ShortComplex.mk i j v).ShortExact)
+    (h₁ : RationalExtensionSplit k A₁ C) (h₂ : RationalExtensionSplit k A₂ C) :
+    RationalExtensionSplit k A C := by
+  apply RepExtensionSplit.extension_left k _ (fun f hf h => ?_) (fun f hf h => ?_)
+    i j v hT h₁ h₂
+  · exact IsRational.of_injective k _ _ (fdrepLinear k f) hf (fdrepLinear_comm k f) h
+  · exact IsRational.of_surjective k _ _ (fdrepLinear k f) hf (fdrepLinear_comm k f) h
+
+lemma rationalExtensionSplit_of_dual {A C : FDRep k (SL2 k)}
+    (h : RationalExtensionSplit k Cᘁ Aᘁ) : RationalExtensionSplit k A C := by
+  apply RepExtensionSplit.of_dual k _ (fun B hB => ?_) h
+  rw [fdrep_dual_repr]
+  exact IsRational.dual k B.ρ hB
+
+lemma rationalExtensionSplit_zero {A C : FDRep k (SL2 k)} (hA : IsZero A) :
+    RationalExtensionSplit k A C := by
+  intro B _ f g w hS
+  have : Mono g := hS.exact.mono_g (hA.eq_zero_of_src f)
+  have := hS.epi_g
+  have : IsIso g := isIso_of_mono_of_epi g
+  exact ⟨inv g, IsIso.inv_hom_id g⟩
+
+variable [Infinite k]
+
+ 
+theorem costandard_standard_extension_split (m n : ℕ) :
+    RationalExtensionSplit k (costandard k m) (costandard k n)ᘁ := by
+  intro B hB f g w hS
+  have := hS.mono_f
+  have := hS.epi_g
+  obtain ⟨s,hs,hsρ⟩ := standard_costandard_split k B.ρ hB n m
+    (fdrepLinear k f) (fdrepLinear k g) (fdrep_injective k f) (fdrep_surjective k g)
+    (fdrep_exact k hS.exact).symm (fdrepLinear_comm k f) (by
+      intro a x
+      have h := fdrepLinear_comm k g a x
+      rw [fdrep_dual_repr] at h
+      exact h)
+  let s' : (costandard k n)ᘁ ⟶ B := fdrepHom k s (by
+    intro a x
+    rw [fdrep_dual_repr]
+    exact hsρ a x)
+  refine ⟨s', fdrep_hom_ext k (LinearMap.congr_fun hs)⟩
+
+ 
+theorem good_standard_extension_split {A : FDRep k (SL2 k)}
+    (hA : HasGoodFiltration k A) (n : ℕ) :
+    RationalExtensionSplit k A (costandard k n)ᘁ := by
+  induction hA with
+  | zero A hA => exact rationalExtensionSplit_zero k hA
+  | @step A B m f g w hS hA ih =>
+    exact rationalExtensionSplit_extension_left k f g w hS ih
+      (costandard_standard_extension_split k m n)
+
+ 
+
+theorem good_weyl_extension_split {A C : FDRep k (SL2 k)}
+    (hA : HasGoodFiltration k A) (hC : HasGoodFiltration k Cᘁ) :
+    RationalExtensionSplit k A C := by
+  induction hA with
+  | zero A hA => exact rationalExtensionSplit_zero k hA
+  | @step A B m f g w hS hA ih =>
+    apply rationalExtensionSplit_extension_left k f g w hS ih
+    exact rationalExtensionSplit_of_dual k (good_standard_extension_split k hC m)
+
+end
+
+section
+open scoped ZeroObject
+universe u
+variable (k : Type u) [Field k]
+lemma HasGoodFiltration.of_iso {A B : FDRep k (SL2 k)}
+    (h : HasGoodFiltration k A) (e : A ≅ B) : HasGoodFiltration k B := by
+  cases h with
+  | zero A hA => exact .zero B (hA.of_iso e.symm)
+  | @step A B' n f g w hS hA =>
+    let f' := f ≫ e.hom
+    let g' := e.inv ≫ g
+    have w' : f' ≫ g' = 0 := by simpa [f', g'] using w
+    have hS' : (ShortComplex.mk f' g' w').ShortExact :=
+      ShortComplex.shortExact_of_iso
+        (ShortComplex.isoMk (Iso.refl A) e (Iso.refl (costandard k n))
+          (by simp [f']) (by simp [g'])) hS
+    exact .step n f' g' w' hS' hA
+
+lemma costandard_hasGoodFiltration (n : ℕ) : HasGoodFiltration k (costandard k n) := by
+  let S : ShortComplex (FDRep k (SL2 k)) := ShortComplex.mk
+    (0 : 0 ⟶ costandard k n) (𝟙 _) (by simp)
+  have hS : S.ShortExact :=
+    (ShortComplex.Splitting.ofIsZeroOfIsIso S (isZero_zero _) inferInstance).shortExact
+  exact .step n S.f S.g S.zero hS (.zero _ (isZero_zero _))
+
+end
+
+universe u
+variable (k : Type u) [Field k]
+lemma fdrepHom_isIso_of_bijective {G : Type u} [Monoid G] {A B : FDRep k G}
+    (f : A ⟶ B) (hf : Function.Bijective (fdrepLinear k f)) : IsIso f := by
+  have : Mono ((fdrepForget k G).map f) := (ModuleCat.mono_iff_injective _).mpr hf.1
+  have : Epi ((fdrepForget k G).map f) := (ModuleCat.epi_iff_surjective _).mpr hf.2
+  have : Mono f := (fdrepForget k G).mono_of_mono_map (by infer_instance)
+  have : Epi f := (fdrepForget k G).epi_of_epi_map (by infer_instance)
+  exact isIso_of_mono_of_epi f
+
+end FiberFunctorProof
+end
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/RestrictedTilting/FaithfulTarget.lean b/lean/OAI/RepresentationTheory/RestrictedTilting/FaithfulTarget.lean
index 18c9740ff36ea8bcea9279416ca4b4df16943dca..d7103b42faaa30ade56dfa2829e7b6727e4dc3d9 100644
--- a/lean/OAI/RepresentationTheory/RestrictedTilting/FaithfulTarget.lean
+++ b/lean/OAI/RepresentationTheory/RestrictedTilting/FaithfulTarget.lean
@@ -1,284 +1,284 @@
-import Mathlib
-
-namespace OAI
-
- 
-
-open CategoryTheory CategoryTheory.MonoidalCategory CategoryTheory.Limits
-open scoped BigOperators
-
-namespace FiberFunctorProof.FaithfulTarget
-
-universe u
-
-section PolynomialAction
-variable {k : Type u} [CommRing k] {ι : Type*} [Fintype ι]
-
- 
-noncomputable def linearChange (A : Matrix ι ι k) :
-    MvPolynomial ι k →ₐ[k] MvPolynomial ι k :=
-  MvPolynomial.aeval (fun i => ∑ j, MvPolynomial.C (A j i) * MvPolynomial.X j)
-
-@[simp] lemma linearChange_X (A : Matrix ι ι k) (i : ι) :
-    linearChange A (MvPolynomial.X i) =
-      ∑ j, MvPolynomial.C (A j i) * MvPolynomial.X j := by
-  simp [linearChange]
-
-@[simp] lemma linearChange_C (A : Matrix ι ι k) (x : k) :
-    linearChange A (MvPolynomial.C x) = MvPolynomial.C x := by
-  simp [linearChange]
-
-variable [DecidableEq ι]
-
-@[simp] lemma linearChange_one : linearChange (1 : Matrix ι ι k) = AlgHom.id k _ := by
-  ext i : 1
-  simp [linearChange, Matrix.one_apply]
-
-omit [DecidableEq ι] in
-lemma linearChange_mul (A B : Matrix ι ι k) :
-    linearChange (A * B) = (linearChange A).comp (linearChange B) := by
-  ext i : 1
-  simp only [AlgHom.comp_apply, linearChange_X, map_sum, map_mul, linearChange_C,
-    Matrix.mul_apply, Finset.sum_mul, Finset.mul_sum]
-  rw [Finset.sum_comm]
-  apply Finset.sum_congr rfl
-  intro j hj
-  apply Finset.sum_congr rfl
-  intro l hl
-  ring
-
-omit [DecidableEq ι] in
-lemma linearChange_homogeneous (A : Matrix ι ι k) {f : MvPolynomial ι k} {n : ℕ}
-    (hf : f.IsHomogeneous n) : (linearChange A f).IsHomogeneous n := by
-  change (MvPolynomial.eval₂ MvPolynomial.C
-    (fun i => ∑ j, MvPolynomial.C (A j i) * MvPolynomial.X j) f).IsHomogeneous n
-  simpa only [one_mul] using
-    hf.eval₂ MvPolynomial.C
-      (fun i => ∑ j, MvPolynomial.C (A j i) * MvPolynomial.X j)
-      (fun x => MvPolynomial.isHomogeneous_C ι x)
-      (fun i => MvPolynomial.IsHomogeneous.sum _ _ _
-        (fun j _ => (MvPolynomial.isHomogeneous_X k j).C_mul (A j i)))
-
-noncomputable def homogeneousRepresentation (n : ℕ) :
-    Representation k (Matrix ι ι k) (MvPolynomial.homogeneousSubmodule ι k n) where
-  toFun A := (linearChange A).toLinearMap.restrict (fun _ hf => linearChange_homogeneous A hf)
-  map_one' := by
-    ext f : 1
-    apply Subtype.ext
-    change linearChange 1 f.val = f.val
-    rw [linearChange_one]
-    rfl
-  map_mul' A B := by
-    ext f : 1
-    apply Subtype.ext
-    change linearChange (A * B) f.val = linearChange A (linearChange B f.val)
-    rw [linearChange_mul]
-    rfl
-
-end PolynomialAction
-
-section SL2
-variable (k : Type u) [Field k]
-
-abbrev SL2 := Matrix.SpecialLinearGroup (Fin 2) k
-
- 
-def IsRational {M : Type u} [AddCommGroup M] [Module k M]
-    (ρ : Representation k (SL2 k) M) : Prop :=
-  ∀ (v : M) (l : M →ₗ[k] k),
-    ∃ P : MvPolynomial (Fin 2 × Fin 2) k,
-      ∀ g : SL2 k, l (ρ g v) = MvPolynomial.eval (fun ij => g ij.1 ij.2) P
-
- 
-noncomputable def dualMatrix : SL2 k →* Matrix (Fin 2) (Fin 2) k where
-  toFun g := ((g⁻¹ : SL2 k) : Matrix (Fin 2) (Fin 2) k).transpose
-  map_one' := by simp
-  map_mul' g h := by
-    simp [mul_inv_rev, Matrix.SpecialLinearGroup.coe_mul, Matrix.transpose_mul]
-
-instance homogeneous_finite (n : ℕ) :
-    Module.Finite k (MvPolynomial.homogeneousSubmodule (Fin 2) k n) :=
-  Module.Finite.of_fg (MvPolynomial.homogeneousSubmodule_fg (Fin 2) k n)
-
- 
-noncomputable def costandard (n : ℕ) : FDRep k (SL2 k) :=
-  FDRep.of ((homogeneousRepresentation n).comp (dualMatrix k))
-
- 
-inductive HasGoodFiltration : FDRep k (SL2 k) → Prop
-  | zero (V) (hV : IsZero V) : HasGoodFiltration V
-  | step {A B} (n : ℕ) (f : A ⟶ B) (g : B ⟶ costandard k n)
-      (w : f ≫ g = 0) (hS : (ShortComplex.mk f g w).ShortExact)
-      (hA : HasGoodFiltration A) : HasGoodFiltration B
-
- 
-
-def IsTilting (V : FDRep k (SL2 k)) : Prop :=
-  IsRational k V.ρ ∧ HasGoodFiltration k V ∧ HasGoodFiltration k Vᘁ
-
- 
-noncomputable def steinberg (p m : ℕ) : FDRep k (SL2 k) :=
-  costandard k (p ^ m - 1)
-
- 
-def unipotent (t : k) : SL2 k :=
-  ⟨!![1, t; 0, 1], by simp [Matrix.det_fin_two]⟩
-
-lemma unipotent_zero : unipotent k 0 = 1 := by
-  apply Subtype.ext
-  ext i j
-  fin_cases i <;> fin_cases j <;> simp [unipotent]
-
-lemma unipotent_add (t s : k) : unipotent k (t + s) = unipotent k t * unipotent k s := by
-  apply Subtype.ext
-  change !![1, t + s; 0, 1] = !![1, t; 0, 1] * !![1, s; 0, 1]
-  ext i j
-  fin_cases i <;> fin_cases j <;>
-    simp [Matrix.mul_apply, Fin.sum_univ_two, add_comm]
-
- 
-def elementaryEmbedding (p r : ℕ) (l : (Fin r → ZMod p) →+ k) :
-    Multiplicative (Fin r → ZMod p) →* SL2 k where
-  toFun a := unipotent k (l a.toAdd)
-  map_one' := by simp [unipotent_zero]
-  map_mul' a b := by simp [unipotent_add]
-
-noncomputable def restrictElementary (p r : ℕ) (l : (Fin r → ZMod p) →+ k) :
-    FDRep k (SL2 k) ⥤ FDRep k (Multiplicative (Fin r → ZMod p)) :=
-  Action.res (FGModuleCat k) (elementaryEmbedding k p r l)
-
-end SL2
-end FiberFunctorProof.FaithfulTarget
-
- 
-
-namespace FiberFunctorProof.FaithfulTarget
-
-universe u
-
- 
-structure FiniteSymmetricTensorCategory (k : Type u) [Field k] where
-  Obj : Type (u + 1)
-  [category : Category.{u} Obj]
-  [abelian : Abelian Obj]
-  [linear : Linear k Obj]
-  [monoidal : MonoidalCategory Obj]
-  [monoidalPreadditive : MonoidalPreadditive Obj]
-  [monoidalLinear : MonoidalLinear k Obj]
-  [symmetric : SymmetricCategory Obj]
-  [rigid : RigidCategory Obj]
-  essentiallySmall : EssentiallySmall.{u} Obj
-  finiteHom : ∀ X Y : Obj, Module.Finite k (X ⟶ Y)
-  artinian : ∀ X : Obj, IsArtinianObject X
-  noetherian : ∀ X : Obj, IsNoetherianObject X
-  finiteSimple : ∃ (m : ℕ) (S : Fin m → Obj),
-    (∀ i, Simple (S i)) ∧ ∀ X : Obj, Simple X → ∃ i, Nonempty (S i ≅ X)
-  enoughProjectives : EnoughProjectives Obj
-  unitScalars : Function.Bijective (algebraMap k (End (𝟙_ Obj)))
-  tensorLeftExact : ∀ X : Obj,
-    PreservesFiniteLimits (tensorLeft X) ∧ PreservesFiniteColimits (tensorLeft X)
-  tensorRightExact : ∀ X : Obj,
-    PreservesFiniteLimits (tensorRight X) ∧ PreservesFiniteColimits (tensorRight X)
-
-attribute [instance] FiniteSymmetricTensorCategory.category
-  FiniteSymmetricTensorCategory.abelian FiniteSymmetricTensorCategory.linear
-  FiniteSymmetricTensorCategory.monoidal FiniteSymmetricTensorCategory.monoidalPreadditive
-  FiniteSymmetricTensorCategory.monoidalLinear FiniteSymmetricTensorCategory.symmetric
-  FiniteSymmetricTensorCategory.rigid
-
- 
-abbrev Tilting (k : Type u) [Field k] :=
-  ObjectProperty.FullSubcategory (IsTilting k)
-
-abbrev tiltingInclusion (k : Type u) [Field k] : Tilting k ⥤ FDRep k (SL2 k) :=
-  ObjectProperty.ι (IsTilting k)
-
- 
-
-class TiltingTensorStructure (k : Type u) [Field k] where
-  [monoidal : MonoidalCategory (Tilting k)]
-  [monoidalPreadditive : MonoidalPreadditive (Tilting k)]
-  [monoidalLinear : MonoidalLinear k (Tilting k)]
-  [symmetric : SymmetricCategory (Tilting k)]
-  [rigid : RigidCategory (Tilting k)]
-  [inclusionBraided : (tiltingInclusion k).Braided]
-
-attribute [instance_reducible, instance] TiltingTensorStructure.monoidal
-  TiltingTensorStructure.symmetric TiltingTensorStructure.rigid
-  TiltingTensorStructure.inclusionBraided
-attribute [instance] TiltingTensorStructure.monoidalPreadditive
-  TiltingTensorStructure.monoidalLinear
-
- 
-
-noncomputable def steinbergIdeal (k : Type u) [Field k] (p n : ℕ) (A B : Tilting k) :
-    Submodule k (A ⟶ B) :=
-  Submodule.span k {f | ∃ X : Tilting k,
-    ∃ a : A.obj ⟶ steinberg k p n ⊗ X.obj,
-    ∃ b : steinberg k p n ⊗ X.obj ⟶ B.obj, f.hom = a ≫ b}
-
- 
-structure TiltingFunctor (k : Type u) [Field k] [TiltingTensorStructure k]
-    (D : FiniteSymmetricTensorCategory k) where
-  toFunctor : Tilting k ⥤ D.Obj
-  [additive : toFunctor.Additive]
-  [linear : toFunctor.Linear k]
-  [braided : toFunctor.Braided]
-
-attribute [instance] TiltingFunctor.additive TiltingFunctor.linear TiltingFunctor.braided
-
- 
-structure ExactTensorFunctor {k : Type u} [Field k]
-    (C D : FiniteSymmetricTensorCategory k) where
-  toFunctor : C.Obj ⥤ D.Obj
-  [additive : toFunctor.Additive]
-  [linear : toFunctor.Linear k]
-  [braided : toFunctor.Braided]
-  exact : ∀ S : ShortComplex C.Obj, S.ShortExact → (S.map toFunctor).ShortExact
-
-attribute [instance] ExactTensorFunctor.additive ExactTensorFunctor.linear
-  ExactTensorFunctor.braided
-
- 
-structure FiberFunctor {k : Type u} [Field k]
-    (C D : FiniteSymmetricTensorCategory k) extends ExactTensorFunctor C D where
-  faithful : toFunctor.Faithful
-
- 
-
-def MonoidallyIsomorphic {C D : Type*} [Category C] [Category D]
-    [MonoidalCategory C] [MonoidalCategory D]
-    (F G : C ⥤ D) [F.Monoidal] [G.Monoidal] : Prop :=
-  ∃ e : F ≅ G, e.hom.IsMonoidal
-
- 
-def HasSteinbergKernel (k : Type u) [Field k] [TiltingTensorStructure k]
-    (p n : ℕ) {D : FiniteSymmetricTensorCategory k} (F : TiltingFunctor k D) : Prop :=
-  ∀ (A B : Tilting k) (f : A ⟶ B), F.toFunctor.map f = 0 ↔ f ∈ steinbergIdeal k p n A B
-
- 
-
-structure VerlindePresentation (k : Type u) [Field k] [TiltingTensorStructure k]
-    (p n : ℕ) (W : FiniteSymmetricTensorCategory k) where
-  sigma : TiltingFunctor k W
-  full : sigma.toFunctor.Full
-  kernel : HasSteinbergKernel k p n sigma
-  extend : ∀ (D : FiniteSymmetricTensorCategory k) (F : TiltingFunctor k D),
-    HasSteinbergKernel k p n F → ∃ H : ExactTensorFunctor W D,
-      MonoidallyIsomorphic (sigma.toFunctor ⋙ H.toFunctor) F.toFunctor
-  unique : ∀ (D : FiniteSymmetricTensorCategory k) (F G : ExactTensorFunctor W D),
-    MonoidallyIsomorphic (sigma.toFunctor ⋙ F.toFunctor) (sigma.toFunctor ⋙ G.toFunctor) →
-      MonoidallyIsomorphic F.toFunctor G.toFunctor
-
- 
-
-def MainTheorem : Prop :=
-  ∀ (k : Type u) [Field k] [IsAlgClosed k]
-    (p : ℕ) [Fact p.Prime] [CharP k p] (C : FiniteSymmetricTensorCategory k),
-    ∃ s : TiltingTensorStructure k, letI := s
-      ∃ n : ℕ, 1 ≤ n ∧ ∃ W : FiniteSymmetricTensorCategory k,
-        Nonempty (VerlindePresentation k p n W) ∧ Nonempty (FiberFunctor C W)
-
-end FiberFunctorProof.FaithfulTarget
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+ 
+
+open CategoryTheory CategoryTheory.MonoidalCategory CategoryTheory.Limits
+open scoped BigOperators
+
+namespace FiberFunctorProof.FaithfulTarget
+
+universe u
+
+section PolynomialAction
+variable {k : Type u} [CommRing k] {ι : Type*} [Fintype ι]
+
+ 
+noncomputable def linearChange (A : Matrix ι ι k) :
+    MvPolynomial ι k →ₐ[k] MvPolynomial ι k :=
+  MvPolynomial.aeval (fun i => ∑ j, MvPolynomial.C (A j i) * MvPolynomial.X j)
+
+@[simp] lemma linearChange_X (A : Matrix ι ι k) (i : ι) :
+    linearChange A (MvPolynomial.X i) =
+      ∑ j, MvPolynomial.C (A j i) * MvPolynomial.X j := by
+  simp [linearChange]
+
+@[simp] lemma linearChange_C (A : Matrix ι ι k) (x : k) :
+    linearChange A (MvPolynomial.C x) = MvPolynomial.C x := by
+  simp [linearChange]
+
+variable [DecidableEq ι]
+
+@[simp] lemma linearChange_one : linearChange (1 : Matrix ι ι k) = AlgHom.id k _ := by
+  ext i : 1
+  simp [linearChange, Matrix.one_apply]
+
+omit [DecidableEq ι] in
+lemma linearChange_mul (A B : Matrix ι ι k) :
+    linearChange (A * B) = (linearChange A).comp (linearChange B) := by
+  ext i : 1
+  simp only [AlgHom.comp_apply, linearChange_X, map_sum, map_mul, linearChange_C,
+    Matrix.mul_apply, Finset.sum_mul, Finset.mul_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro j hj
+  apply Finset.sum_congr rfl
+  intro l hl
+  ring
+
+omit [DecidableEq ι] in
+lemma linearChange_homogeneous (A : Matrix ι ι k) {f : MvPolynomial ι k} {n : ℕ}
+    (hf : f.IsHomogeneous n) : (linearChange A f).IsHomogeneous n := by
+  change (MvPolynomial.eval₂ MvPolynomial.C
+    (fun i => ∑ j, MvPolynomial.C (A j i) * MvPolynomial.X j) f).IsHomogeneous n
+  simpa only [one_mul] using
+    hf.eval₂ MvPolynomial.C
+      (fun i => ∑ j, MvPolynomial.C (A j i) * MvPolynomial.X j)
+      (fun x => MvPolynomial.isHomogeneous_C ι x)
+      (fun i => MvPolynomial.IsHomogeneous.sum _ _ _
+        (fun j _ => (MvPolynomial.isHomogeneous_X k j).C_mul (A j i)))
+
+noncomputable def homogeneousRepresentation (n : ℕ) :
+    Representation k (Matrix ι ι k) (MvPolynomial.homogeneousSubmodule ι k n) where
+  toFun A := (linearChange A).toLinearMap.restrict (fun _ hf => linearChange_homogeneous A hf)
+  map_one' := by
+    ext f : 1
+    apply Subtype.ext
+    change linearChange 1 f.val = f.val
+    rw [linearChange_one]
+    rfl
+  map_mul' A B := by
+    ext f : 1
+    apply Subtype.ext
+    change linearChange (A * B) f.val = linearChange A (linearChange B f.val)
+    rw [linearChange_mul]
+    rfl
+
+end PolynomialAction
+
+section SL2
+variable (k : Type u) [Field k]
+
+abbrev SL2 := Matrix.SpecialLinearGroup (Fin 2) k
+
+ 
+def IsRational {M : Type u} [AddCommGroup M] [Module k M]
+    (ρ : Representation k (SL2 k) M) : Prop :=
+  ∀ (v : M) (l : M →ₗ[k] k),
+    ∃ P : MvPolynomial (Fin 2 × Fin 2) k,
+      ∀ g : SL2 k, l (ρ g v) = MvPolynomial.eval (fun ij => g ij.1 ij.2) P
+
+ 
+noncomputable def dualMatrix : SL2 k →* Matrix (Fin 2) (Fin 2) k where
+  toFun g := ((g⁻¹ : SL2 k) : Matrix (Fin 2) (Fin 2) k).transpose
+  map_one' := by simp
+  map_mul' g h := by
+    simp [mul_inv_rev, Matrix.SpecialLinearGroup.coe_mul, Matrix.transpose_mul]
+
+instance homogeneous_finite (n : ℕ) :
+    Module.Finite k (MvPolynomial.homogeneousSubmodule (Fin 2) k n) :=
+  Module.Finite.of_fg (MvPolynomial.homogeneousSubmodule_fg (Fin 2) k n)
+
+ 
+noncomputable def costandard (n : ℕ) : FDRep k (SL2 k) :=
+  FDRep.of ((homogeneousRepresentation n).comp (dualMatrix k))
+
+ 
+inductive HasGoodFiltration : FDRep k (SL2 k) → Prop
+  | zero (V) (hV : IsZero V) : HasGoodFiltration V
+  | step {A B} (n : ℕ) (f : A ⟶ B) (g : B ⟶ costandard k n)
+      (w : f ≫ g = 0) (hS : (ShortComplex.mk f g w).ShortExact)
+      (hA : HasGoodFiltration A) : HasGoodFiltration B
+
+ 
+
+def IsTilting (V : FDRep k (SL2 k)) : Prop :=
+  IsRational k V.ρ ∧ HasGoodFiltration k V ∧ HasGoodFiltration k Vᘁ
+
+ 
+noncomputable def steinberg (p m : ℕ) : FDRep k (SL2 k) :=
+  costandard k (p ^ m - 1)
+
+ 
+def unipotent (t : k) : SL2 k :=
+  ⟨!![1, t; 0, 1], by simp [Matrix.det_fin_two]⟩
+
+lemma unipotent_zero : unipotent k 0 = 1 := by
+  apply Subtype.ext
+  ext i j
+  fin_cases i <;> fin_cases j <;> simp [unipotent]
+
+lemma unipotent_add (t s : k) : unipotent k (t + s) = unipotent k t * unipotent k s := by
+  apply Subtype.ext
+  change !![1, t + s; 0, 1] = !![1, t; 0, 1] * !![1, s; 0, 1]
+  ext i j
+  fin_cases i <;> fin_cases j <;>
+    simp [Matrix.mul_apply, Fin.sum_univ_two, add_comm]
+
+ 
+def elementaryEmbedding (p r : ℕ) (l : (Fin r → ZMod p) →+ k) :
+    Multiplicative (Fin r → ZMod p) →* SL2 k where
+  toFun a := unipotent k (l a.toAdd)
+  map_one' := by simp [unipotent_zero]
+  map_mul' a b := by simp [unipotent_add]
+
+noncomputable def restrictElementary (p r : ℕ) (l : (Fin r → ZMod p) →+ k) :
+    FDRep k (SL2 k) ⥤ FDRep k (Multiplicative (Fin r → ZMod p)) :=
+  Action.res (FGModuleCat k) (elementaryEmbedding k p r l)
+
+end SL2
+end FiberFunctorProof.FaithfulTarget
+
+ 
+
+namespace FiberFunctorProof.FaithfulTarget
+
+universe u
+
+ 
+structure FiniteSymmetricTensorCategory (k : Type u) [Field k] where
+  Obj : Type (u + 1)
+  [category : Category.{u} Obj]
+  [abelian : Abelian Obj]
+  [linear : Linear k Obj]
+  [monoidal : MonoidalCategory Obj]
+  [monoidalPreadditive : MonoidalPreadditive Obj]
+  [monoidalLinear : MonoidalLinear k Obj]
+  [symmetric : SymmetricCategory Obj]
+  [rigid : RigidCategory Obj]
+  essentiallySmall : EssentiallySmall.{u} Obj
+  finiteHom : ∀ X Y : Obj, Module.Finite k (X ⟶ Y)
+  artinian : ∀ X : Obj, IsArtinianObject X
+  noetherian : ∀ X : Obj, IsNoetherianObject X
+  finiteSimple : ∃ (m : ℕ) (S : Fin m → Obj),
+    (∀ i, Simple (S i)) ∧ ∀ X : Obj, Simple X → ∃ i, Nonempty (S i ≅ X)
+  enoughProjectives : EnoughProjectives Obj
+  unitScalars : Function.Bijective (algebraMap k (End (𝟙_ Obj)))
+  tensorLeftExact : ∀ X : Obj,
+    PreservesFiniteLimits (tensorLeft X) ∧ PreservesFiniteColimits (tensorLeft X)
+  tensorRightExact : ∀ X : Obj,
+    PreservesFiniteLimits (tensorRight X) ∧ PreservesFiniteColimits (tensorRight X)
+
+attribute [instance] FiniteSymmetricTensorCategory.category
+  FiniteSymmetricTensorCategory.abelian FiniteSymmetricTensorCategory.linear
+  FiniteSymmetricTensorCategory.monoidal FiniteSymmetricTensorCategory.monoidalPreadditive
+  FiniteSymmetricTensorCategory.monoidalLinear FiniteSymmetricTensorCategory.symmetric
+  FiniteSymmetricTensorCategory.rigid
+
+ 
+abbrev Tilting (k : Type u) [Field k] :=
+  ObjectProperty.FullSubcategory (IsTilting k)
+
+abbrev tiltingInclusion (k : Type u) [Field k] : Tilting k ⥤ FDRep k (SL2 k) :=
+  ObjectProperty.ι (IsTilting k)
+
+ 
+
+class TiltingTensorStructure (k : Type u) [Field k] where
+  [monoidal : MonoidalCategory (Tilting k)]
+  [monoidalPreadditive : MonoidalPreadditive (Tilting k)]
+  [monoidalLinear : MonoidalLinear k (Tilting k)]
+  [symmetric : SymmetricCategory (Tilting k)]
+  [rigid : RigidCategory (Tilting k)]
+  [inclusionBraided : (tiltingInclusion k).Braided]
+
+attribute [instance_reducible, instance] TiltingTensorStructure.monoidal
+  TiltingTensorStructure.symmetric TiltingTensorStructure.rigid
+  TiltingTensorStructure.inclusionBraided
+attribute [instance] TiltingTensorStructure.monoidalPreadditive
+  TiltingTensorStructure.monoidalLinear
+
+ 
+
+noncomputable def steinbergIdeal (k : Type u) [Field k] (p n : ℕ) (A B : Tilting k) :
+    Submodule k (A ⟶ B) :=
+  Submodule.span k {f | ∃ X : Tilting k,
+    ∃ a : A.obj ⟶ steinberg k p n ⊗ X.obj,
+    ∃ b : steinberg k p n ⊗ X.obj ⟶ B.obj, f.hom = a ≫ b}
+
+ 
+structure TiltingFunctor (k : Type u) [Field k] [TiltingTensorStructure k]
+    (D : FiniteSymmetricTensorCategory k) where
+  toFunctor : Tilting k ⥤ D.Obj
+  [additive : toFunctor.Additive]
+  [linear : toFunctor.Linear k]
+  [braided : toFunctor.Braided]
+
+attribute [instance] TiltingFunctor.additive TiltingFunctor.linear TiltingFunctor.braided
+
+ 
+structure ExactTensorFunctor {k : Type u} [Field k]
+    (C D : FiniteSymmetricTensorCategory k) where
+  toFunctor : C.Obj ⥤ D.Obj
+  [additive : toFunctor.Additive]
+  [linear : toFunctor.Linear k]
+  [braided : toFunctor.Braided]
+  exact : ∀ S : ShortComplex C.Obj, S.ShortExact → (S.map toFunctor).ShortExact
+
+attribute [instance] ExactTensorFunctor.additive ExactTensorFunctor.linear
+  ExactTensorFunctor.braided
+
+ 
+structure FiberFunctor {k : Type u} [Field k]
+    (C D : FiniteSymmetricTensorCategory k) extends ExactTensorFunctor C D where
+  faithful : toFunctor.Faithful
+
+ 
+
+def MonoidallyIsomorphic {C D : Type*} [Category C] [Category D]
+    [MonoidalCategory C] [MonoidalCategory D]
+    (F G : C ⥤ D) [F.Monoidal] [G.Monoidal] : Prop :=
+  ∃ e : F ≅ G, e.hom.IsMonoidal
+
+ 
+def HasSteinbergKernel (k : Type u) [Field k] [TiltingTensorStructure k]
+    (p n : ℕ) {D : FiniteSymmetricTensorCategory k} (F : TiltingFunctor k D) : Prop :=
+  ∀ (A B : Tilting k) (f : A ⟶ B), F.toFunctor.map f = 0 ↔ f ∈ steinbergIdeal k p n A B
+
+ 
+
+structure VerlindePresentation (k : Type u) [Field k] [TiltingTensorStructure k]
+    (p n : ℕ) (W : FiniteSymmetricTensorCategory k) where
+  sigma : TiltingFunctor k W
+  full : sigma.toFunctor.Full
+  kernel : HasSteinbergKernel k p n sigma
+  extend : ∀ (D : FiniteSymmetricTensorCategory k) (F : TiltingFunctor k D),
+    HasSteinbergKernel k p n F → ∃ H : ExactTensorFunctor W D,
+      MonoidallyIsomorphic (sigma.toFunctor ⋙ H.toFunctor) F.toFunctor
+  unique : ∀ (D : FiniteSymmetricTensorCategory k) (F G : ExactTensorFunctor W D),
+    MonoidallyIsomorphic (sigma.toFunctor ⋙ F.toFunctor) (sigma.toFunctor ⋙ G.toFunctor) →
+      MonoidallyIsomorphic F.toFunctor G.toFunctor
+
+ 
+
+def MainTheorem : Prop :=
+  ∀ (k : Type u) [Field k] [IsAlgClosed k]
+    (p : ℕ) [Fact p.Prime] [CharP k p] (C : FiniteSymmetricTensorCategory k),
+    ∃ s : TiltingTensorStructure k, letI := s
+      ∃ n : ℕ, 1 ≤ n ∧ ∃ W : FiniteSymmetricTensorCategory k,
+        Nonempty (VerlindePresentation k p n W) ∧ Nonempty (FiberFunctor C W)
+
+end FiberFunctorProof.FaithfulTarget
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/RestrictedTilting/FiniteFreeTrace.lean b/lean/OAI/RepresentationTheory/RestrictedTilting/FiniteFreeTrace.lean
index 749cca6c081b6732261dbfd710248778e6cd776d..8f3c14cb5c4e16e50169f1c5fee32197690261a1 100644
--- a/lean/OAI/RepresentationTheory/RestrictedTilting/FiniteFreeTrace.lean
+++ b/lean/OAI/RepresentationTheory/RestrictedTilting/FiniteFreeTrace.lean
@@ -1,75 +1,75 @@
-import OAI.RepresentationTheory.RestrictedTilting.RelativeTrace
-import OAI.RepresentationTheory.RestrictedTilting.NormFree
-import OAI.RepresentationTheory.RestrictedTilting.ProjectiveFactor
-import OAI.RepresentationTheory.RestrictedTilting.ProjectiveRegular
-
-namespace OAI
-
-namespace FiberFunctorProof
-open CategoryTheory Representation
-universe u
-variable {k : Type u} [Field k] {G : Type u} [Group G] [Fintype G]
-
- 
-
-lemma free_relativeTrace (I : Type u) :
-    relativeTrace (Representation.free k G I) (Representation.free k G I)
-      (freeNormRepresentative I) = LinearMap.id := by
-  classical
-  apply LinearMap.ext
-  intro x
-  apply Finsupp.ext
-  intro i
-  apply MonoidAlgebra.ext
-  apply Finsupp.ext
-  intro g
-  simp only [relativeTrace_apply, Finsupp.finsetSum_apply,
-    free_action_apply, freeNormRepresentative_apply]
-  simp [Representation.coeff_ofMulAction]
-
- 
-noncomputable def finiteIntertwiner {V W : FDRep k G} (f : Representation.IntertwiningMap V.ρ W.ρ) : V ⟶ W :=
-  (FDRep.forget₂HomLinearEquiv V W) (Rep.ofHom f)
-
-omit [Fintype G] in
-@[simp] lemma finiteIntertwiner_hom {V W : FDRep k G} (f : Representation.IntertwiningMap V.ρ W.ρ) :
-    (finiteIntertwiner f).hom.hom.hom = f.toLinearMap := rfl
-
-omit [Fintype G] in
-lemma finiteIntertwiner_epi {V W : FDRep k G} (f : Representation.IntertwiningMap V.ρ W.ρ)
-    (hf : Function.Surjective f) : Epi (finiteIntertwiner f) := by
-  let F := forget₂ (FDRep k G) (Rep k G)
-  have : Epi (F.map (finiteIntertwiner f)) := (Rep.epi_iff_surjective _).2 hf
-  exact F.epi_of_epi_map this
-
- 
-
-lemma fdrep_free_projective (I : Type u) [Finite I] :
-    Projective (FDRep.of (Representation.free k G I)) := by
-  apply (forget₂ (FDRep k G) (Rep k G)).projective_of_map_projective
-  exact (Rep.toModuleMonoidAlgebra (k := k) (G := G)).projective_of_map_projective
-    (P := Rep.of (Representation.free k G I))
-    (ModuleCat.projective_of_free (Representation.freeAsModuleBasis k G I))
-
- 
-noncomputable def finiteFreeCover (V : FDRep k G) :
-    FDRep.of (Representation.free k G (ULift.{u} (Fin (Module.finrank k V)))) ⟶ V :=
-  finiteIntertwiner (Representation.freeLift V.ρ
-    (fun i : ULift.{u} (Fin (Module.finrank k V)) => Module.finBasis k V i.down))
-
-lemma finiteFreeCover_epi (V : FDRep k G) : Epi (finiteFreeCover V) := by
-  classical
-  apply finiteIntertwiner_epi
-  intro v
-  refine ⟨∑ i : Fin (Module.finrank k V),
-    Finsupp.single (ULift.up i) (MonoidAlgebra.single 1 ((Module.finBasis k V).repr v i)), ?_⟩
-  change (Representation.freeLift V.ρ (fun i : ULift.{u} (Fin (Module.finrank k V)) =>
-    Module.finBasis k V i.down)).toLinearMap _ = v
-  rw [map_sum]
-  simp only [IntertwiningMap.toLinearMap_apply, Representation.freeLift_single_single,
-    map_one, Module.End.one_apply]
-  exact (Module.finBasis k V).sum_repr v
-
-end FiberFunctorProof
-
-end OAI
+import OAI.RepresentationTheory.RestrictedTilting.RelativeTrace
+import OAI.RepresentationTheory.RestrictedTilting.NormFree
+import OAI.RepresentationTheory.RestrictedTilting.ProjectiveFactor
+import OAI.RepresentationTheory.RestrictedTilting.ProjectiveRegular
+
+namespace OAI
+
+namespace FiberFunctorProof
+open CategoryTheory Representation
+universe u
+variable {k : Type u} [Field k] {G : Type u} [Group G] [Fintype G]
+
+ 
+
+lemma free_relativeTrace (I : Type u) :
+    relativeTrace (Representation.free k G I) (Representation.free k G I)
+      (freeNormRepresentative I) = LinearMap.id := by
+  classical
+  apply LinearMap.ext
+  intro x
+  apply Finsupp.ext
+  intro i
+  apply MonoidAlgebra.ext
+  apply Finsupp.ext
+  intro g
+  simp only [relativeTrace_apply, Finsupp.finsetSum_apply,
+    free_action_apply, freeNormRepresentative_apply]
+  simp [Representation.coeff_ofMulAction]
+
+ 
+noncomputable def finiteIntertwiner {V W : FDRep k G} (f : Representation.IntertwiningMap V.ρ W.ρ) : V ⟶ W :=
+  (FDRep.forget₂HomLinearEquiv V W) (Rep.ofHom f)
+
+omit [Fintype G] in
+@[simp] lemma finiteIntertwiner_hom {V W : FDRep k G} (f : Representation.IntertwiningMap V.ρ W.ρ) :
+    (finiteIntertwiner f).hom.hom.hom = f.toLinearMap := rfl
+
+omit [Fintype G] in
+lemma finiteIntertwiner_epi {V W : FDRep k G} (f : Representation.IntertwiningMap V.ρ W.ρ)
+    (hf : Function.Surjective f) : Epi (finiteIntertwiner f) := by
+  let F := forget₂ (FDRep k G) (Rep k G)
+  have : Epi (F.map (finiteIntertwiner f)) := (Rep.epi_iff_surjective _).2 hf
+  exact F.epi_of_epi_map this
+
+ 
+
+lemma fdrep_free_projective (I : Type u) [Finite I] :
+    Projective (FDRep.of (Representation.free k G I)) := by
+  apply (forget₂ (FDRep k G) (Rep k G)).projective_of_map_projective
+  exact (Rep.toModuleMonoidAlgebra (k := k) (G := G)).projective_of_map_projective
+    (P := Rep.of (Representation.free k G I))
+    (ModuleCat.projective_of_free (Representation.freeAsModuleBasis k G I))
+
+ 
+noncomputable def finiteFreeCover (V : FDRep k G) :
+    FDRep.of (Representation.free k G (ULift.{u} (Fin (Module.finrank k V)))) ⟶ V :=
+  finiteIntertwiner (Representation.freeLift V.ρ
+    (fun i : ULift.{u} (Fin (Module.finrank k V)) => Module.finBasis k V i.down))
+
+lemma finiteFreeCover_epi (V : FDRep k G) : Epi (finiteFreeCover V) := by
+  classical
+  apply finiteIntertwiner_epi
+  intro v
+  refine ⟨∑ i : Fin (Module.finrank k V),
+    Finsupp.single (ULift.up i) (MonoidAlgebra.single 1 ((Module.finBasis k V).repr v i)), ?_⟩
+  change (Representation.freeLift V.ρ (fun i : ULift.{u} (Fin (Module.finrank k V)) =>
+    Module.finBasis k V i.down)).toLinearMap _ = v
+  rw [map_sum]
+  simp only [IntertwiningMap.toLinearMap_apply, Representation.freeLift_single_single,
+    map_one, Module.End.one_apply]
+  exact (Module.finBasis k V).sum_repr v
+
+end FiberFunctorProof
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/RestrictedTilting/FiniteGenerator.lean b/lean/OAI/RepresentationTheory/RestrictedTilting/FiniteGenerator.lean
index 4cdef872ae2f5dc753c53e1b2102cc89624cb851..f7c3a61c2c8988e2a58ba3bda0a1cbad03a72525 100644
--- a/lean/OAI/RepresentationTheory/RestrictedTilting/FiniteGenerator.lean
+++ b/lean/OAI/RepresentationTheory/RestrictedTilting/FiniteGenerator.lean
@@ -1,92 +1,92 @@
-import Mathlib
-
-namespace OAI
-
-open CategoryTheory CategoryTheory.Limits
-namespace FiberFunctorProof
-universe v u
-noncomputable section
-variable {C : Type u} [Category.{v} C] [Abelian C]
-
- 
-
-lemma separator_of_projective_weak_generator (P : C) [Projective P]
-    (hP : ∀ X : C, ¬ IsZero X → ∃ f : P ⟶ X, f ≠ 0) : IsSeparator P := by
-  apply (Preadditive.isSeparator_iff P).2
-  intro X Y f hf
-  by_contra hn
-  have hI : ¬ IsZero (image f) := by
-    intro hz
-    apply hn
-    rw [← image.fac f]
-    simpa using congrArg (fun g => factorThruImage f ≫ g) (hz.eq_of_src (image.ι f) 0)
-  obtain ⟨a,ha⟩ := hP (image f) hI
-  let b := Projective.factorThru a (factorThruImage f)
-  have hb : b ≫ factorThruImage f = a := Projective.factorThru_comp _ _
-  apply ha
-  apply (cancel_mono (image.ι f)).1
-  calc
-    a ≫ image.ι f = b ≫ f := by rw [← hb, Category.assoc, image.fac]
-    _ = 0 := hf b
-    _ = 0 ≫ image.ι f := by simp
-
- 
-
-theorem finite_projective_separator
-    (C : Type u) [Category.{v} C] [Abelian C] [EnoughProjectives C]
-    (hA : ∀ X : C, IsArtinianObject X)
-    (hS : ∃ (m : ℕ) (S : Fin m → C),
-      (∀ i, Simple (S i)) ∧ ∀ X : C, Simple X → ∃ i, Nonempty (S i ≅ X)) :
-    ∃ P : C, Projective P ∧ IsSeparator P := by
-  classical
-  obtain ⟨m,S,hS,hrep⟩ := hS
-  let J := ULift.{v} (Fin m)
-  let Q : J → C := fun i => Projective.over (S i.down)
-  let : HasFiniteBiproducts C := HasFiniteBiproducts.of_hasFiniteProducts
-  let P := ⨁ Q
-  have hproj : Projective P := inferInstance
-  refine ⟨P,hproj,separator_of_projective_weak_generator P ?_⟩
-  intro X hX
-  let := hA X
-  obtain ⟨T,hT⟩ := exists_simple_subobject hX
-  obtain ⟨i,⟨e⟩⟩ := hrep (T : C) hT
-  let a : Q ⟨i⟩ ⟶ X := Projective.π (S i) ≫ e.hom ≫ T.arrow
-  have ha : a ≠ 0 := by
-    intro hz
-    have ht : e.hom ≫ T.arrow = 0 := (cancel_epi (Projective.π (S i))).1
-      (by simpa [a] using hz)
-    have hzT : IsZero (T : C) := IsZero.of_mono_eq_zero T.arrow
-      ((cancel_epi e.hom).1 (by simpa using ht))
-    exact (Simple.not_isZero (T : C)) hzT
-  refine ⟨biproduct.π Q ⟨i⟩ ≫ a, ?_⟩
-  intro hz
-  apply ha
-  have he := congrArg (fun f => biproduct.ι Q ⟨i⟩ ≫ f) hz
-  simpa using he
-
-local instance : HasFiniteBiproducts C := HasFiniteBiproducts.of_hasFiniteProducts
-
- 
-
-theorem finite_sum_surjection (k : Type*) [Field k] [Linear k C]
-    (P X : C) (hP : IsSeparator P) [Module.Finite k (P ⟶ X)] :
-    ∃ (n : ℕ) (e : (⨁ fun _ : Fin n => P) ⟶ X), Epi e := by
-  classical
-  let b := Module.finBasis k (P ⟶ X)
-  let e : (⨁ fun _ : Fin (Module.finrank k (P ⟶ X)) => P) ⟶ X := biproduct.desc b
-  refine ⟨_, e, ?_⟩
-  apply Abelian.epi_of_cokernel_π_eq_zero
-  apply (Preadditive.isSeparator_iff P).1 hP
-  intro a
-  have hL : Linear.rightComp k P (cokernel.π e) = 0 := by
-    apply b.ext
-    intro i
-    change b i ≫ cokernel.π e = 0
-    have h := congrArg (fun f => biproduct.ι (fun _ : Fin (Module.finrank k (P ⟶ X)) => P) i ≫ f)
-      (cokernel.condition e)
-    simpa only [e, ← Category.assoc, biproduct.ι_desc, Limits.comp_zero] using h
-  exact congrArg (fun L : (P ⟶ X) →ₗ[k] (P ⟶ cokernel e) => L a) hL
-end
-end FiberFunctorProof
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+open CategoryTheory CategoryTheory.Limits
+namespace FiberFunctorProof
+universe v u
+noncomputable section
+variable {C : Type u} [Category.{v} C] [Abelian C]
+
+ 
+
+lemma separator_of_projective_weak_generator (P : C) [Projective P]
+    (hP : ∀ X : C, ¬ IsZero X → ∃ f : P ⟶ X, f ≠ 0) : IsSeparator P := by
+  apply (Preadditive.isSeparator_iff P).2
+  intro X Y f hf
+  by_contra hn
+  have hI : ¬ IsZero (image f) := by
+    intro hz
+    apply hn
+    rw [← image.fac f]
+    simpa using congrArg (fun g => factorThruImage f ≫ g) (hz.eq_of_src (image.ι f) 0)
+  obtain ⟨a,ha⟩ := hP (image f) hI
+  let b := Projective.factorThru a (factorThruImage f)
+  have hb : b ≫ factorThruImage f = a := Projective.factorThru_comp _ _
+  apply ha
+  apply (cancel_mono (image.ι f)).1
+  calc
+    a ≫ image.ι f = b ≫ f := by rw [← hb, Category.assoc, image.fac]
+    _ = 0 := hf b
+    _ = 0 ≫ image.ι f := by simp
+
+ 
+
+theorem finite_projective_separator
+    (C : Type u) [Category.{v} C] [Abelian C] [EnoughProjectives C]
+    (hA : ∀ X : C, IsArtinianObject X)
+    (hS : ∃ (m : ℕ) (S : Fin m → C),
+      (∀ i, Simple (S i)) ∧ ∀ X : C, Simple X → ∃ i, Nonempty (S i ≅ X)) :
+    ∃ P : C, Projective P ∧ IsSeparator P := by
+  classical
+  obtain ⟨m,S,hS,hrep⟩ := hS
+  let J := ULift.{v} (Fin m)
+  let Q : J → C := fun i => Projective.over (S i.down)
+  let : HasFiniteBiproducts C := HasFiniteBiproducts.of_hasFiniteProducts
+  let P := ⨁ Q
+  have hproj : Projective P := inferInstance
+  refine ⟨P,hproj,separator_of_projective_weak_generator P ?_⟩
+  intro X hX
+  let := hA X
+  obtain ⟨T,hT⟩ := exists_simple_subobject hX
+  obtain ⟨i,⟨e⟩⟩ := hrep (T : C) hT
+  let a : Q ⟨i⟩ ⟶ X := Projective.π (S i) ≫ e.hom ≫ T.arrow
+  have ha : a ≠ 0 := by
+    intro hz
+    have ht : e.hom ≫ T.arrow = 0 := (cancel_epi (Projective.π (S i))).1
+      (by simpa [a] using hz)
+    have hzT : IsZero (T : C) := IsZero.of_mono_eq_zero T.arrow
+      ((cancel_epi e.hom).1 (by simpa using ht))
+    exact (Simple.not_isZero (T : C)) hzT
+  refine ⟨biproduct.π Q ⟨i⟩ ≫ a, ?_⟩
+  intro hz
+  apply ha
+  have he := congrArg (fun f => biproduct.ι Q ⟨i⟩ ≫ f) hz
+  simpa using he
+
+local instance : HasFiniteBiproducts C := HasFiniteBiproducts.of_hasFiniteProducts
+
+ 
+
+theorem finite_sum_surjection (k : Type*) [Field k] [Linear k C]
+    (P X : C) (hP : IsSeparator P) [Module.Finite k (P ⟶ X)] :
+    ∃ (n : ℕ) (e : (⨁ fun _ : Fin n => P) ⟶ X), Epi e := by
+  classical
+  let b := Module.finBasis k (P ⟶ X)
+  let e : (⨁ fun _ : Fin (Module.finrank k (P ⟶ X)) => P) ⟶ X := biproduct.desc b
+  refine ⟨_, e, ?_⟩
+  apply Abelian.epi_of_cokernel_π_eq_zero
+  apply (Preadditive.isSeparator_iff P).1 hP
+  intro a
+  have hL : Linear.rightComp k P (cokernel.π e) = 0 := by
+    apply b.ext
+    intro i
+    change b i ≫ cokernel.π e = 0
+    have h := congrArg (fun f => biproduct.ι (fun _ : Fin (Module.finrank k (P ⟶ X)) => P) i ≫ f)
+      (cokernel.condition e)
+    simpa only [e, ← Category.assoc, biproduct.ι_desc, Limits.comp_zero] using h
+  exact congrArg (fun L : (P ⟶ X) →ₗ[k] (P ⟶ cokernel e) => L a) hL
+end
+end FiberFunctorProof
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/RestrictedTilting/FiniteJetMaps.lean b/lean/OAI/RepresentationTheory/RestrictedTilting/FiniteJetMaps.lean
index 3690e74b5a2ef1944ae0ebf10b0a53ad5af51247..44623cc6e136ced764062e5778459436ae4f969b 100644
--- a/lean/OAI/RepresentationTheory/RestrictedTilting/FiniteJetMaps.lean
+++ b/lean/OAI/RepresentationTheory/RestrictedTilting/FiniteJetMaps.lean
@@ -1,126 +1,126 @@
-import OAI.RepresentationTheory.RestrictedTilting.AdicJets
-
-namespace OAI
-
-namespace FiberFunctorProof
-open scoped TensorProduct Pointwise
-universe u
-variable {k : Type u} [Field k]
-variable {T N : Type u} [AddCommGroup T] [Module k T]
-  [AddCommGroup N] [Module k N]
-  [Module (MvPolynomial (Fin 2) k) N] [IsScalarTower k (MvPolynomial (Fin 2) k) N]
-
- 
-noncomputable def polynomialJetExtend (j : ℕ)
-    (s : T →ₗ[k] N ⧸ ((MvPolynomial.idealOfVars (Fin 2) k)^j •
-      (⊤ : Submodule (MvPolynomial (Fin 2) k) N))) :
-    (((MvPolynomial (Fin 2) k) ⊗[k] T) ⧸ ((MvPolynomial.idealOfVars (Fin 2) k)^j •
-      (⊤ : Submodule (MvPolynomial (Fin 2) k) ((MvPolynomial (Fin 2) k) ⊗[k] T)))) →ₗ[MvPolynomial (Fin 2) k]
-      (N ⧸ ((MvPolynomial.idealOfVars (Fin 2) k)^j •
-      (⊤ : Submodule (MvPolynomial (Fin 2) k) N))) :=
-  ((MvPolynomial.idealOfVars (Fin 2) k)^j • (⊤ : Submodule (MvPolynomial (Fin 2) k) _)).liftQ
-    (polynomialExtend s) (by
-      intro z hz
-      change polynomialExtend s z = 0
-      refine Submodule.smul_induction_on hz (fun a ha z _ => ?_) (fun z w hz hw => ?_)
-      ·
-        rw [map_smul]
-        obtain ⟨v,hv⟩ := ((MvPolynomial.idealOfVars (Fin 2) k)^j •
-          (⊤ : Submodule (MvPolynomial (Fin 2) k) N)).mkQ_surjective (polynomialExtend s z)
-        rw [← hv, ← map_smul]
-        exact (Submodule.Quotient.mk_eq_zero _).mpr (Submodule.smul_mem_smul ha (by trivial))
-      · rw [map_add, hz, hw, add_zero])
-
-@[simp] lemma polynomialJetExtend_mk (j : ℕ)
-    (s : T →ₗ[k] N ⧸ ((MvPolynomial.idealOfVars (Fin 2) k)^j •
-      (⊤ : Submodule (MvPolynomial (Fin 2) k) N)))
-    (z : (MvPolynomial (Fin 2) k) ⊗[k] T) :
-    polynomialJetExtend j s (Submodule.Quotient.mk z) = polynomialExtend s z := rfl
-
-@[simp] lemma polynomialJetExtend_mk_tmul (j : ℕ)
-    (s : T →ₗ[k] N ⧸ ((MvPolynomial.idealOfVars (Fin 2) k)^j •
-      (⊤ : Submodule (MvPolynomial (Fin 2) k) N)))
-    (P : MvPolynomial (Fin 2) k) (t : T) :
-    polynomialJetExtend j s (Submodule.Quotient.mk (P ⊗ₜ[k] t)) = P • s t := rfl
-
-lemma polynomialJetExtend_transition (m n : ℕ) (h : m ≤ n)
-    (s : T →ₗ[k] N ⧸ ((MvPolynomial.idealOfVars (Fin 2) k)^n •
-      (⊤ : Submodule (MvPolynomial (Fin 2) k) N)))
-    (t : T →ₗ[k] N ⧸ ((MvPolynomial.idealOfVars (Fin 2) k)^m •
-      (⊤ : Submodule (MvPolynomial (Fin 2) k) N)))
-    (ht : ∀ x, AdicCompletion.transitionMap (MvPolynomial.idealOfVars (Fin 2) k) N h (s x) = t x)
-    (z : ((MvPolynomial (Fin 2) k) ⊗[k] T) ⧸ ((MvPolynomial.idealOfVars (Fin 2) k)^n •
-      (⊤ : Submodule (MvPolynomial (Fin 2) k) ((MvPolynomial (Fin 2) k) ⊗[k] T)))) :
-    AdicCompletion.transitionMap (MvPolynomial.idealOfVars (Fin 2) k) N h (polynomialJetExtend n s z) =
-      polynomialJetExtend m t (AdicCompletion.transitionMap (MvPolynomial.idealOfVars (Fin 2) k) _ h z) := by
-  obtain ⟨w,rfl⟩ := ((MvPolynomial.idealOfVars (Fin 2) k)^n •
-    (⊤ : Submodule (MvPolynomial (Fin 2) k) ((MvPolynomial (Fin 2) k) ⊗[k] T))).mkQ_surjective z
-  change AdicCompletion.transitionMap _ N h (polynomialExtend s w) = polynomialExtend t w
-  induction w using TensorProduct.inductionOn with
-  | tmul P x => rw [polynomialExtend_tmul, polynomialExtend_tmul, map_smul, ht]
-  | add z w hz hw => simp [hz, hw]
-
-noncomputable def coefficientAdicEquiv (j : ℕ) :
-    (N ⧸ jetRelation (M := N) (MvPolynomial.X 0 : MvPolynomial (Fin 2) k) (MvPolynomial.X 1) j) ≃ₗ[MvPolynomial (Fin 2) k]
-      (N ⧸ ((MvPolynomial.idealOfVars (Fin 2) k)^j • (⊤ : Submodule (MvPolynomial (Fin 2) k) N))) :=
-  Submodule.quotEquivOfEq _ _ (by rw [planeIdeal_eq, jetRelation_eq_pow_smul])
-
-omit [Module k N] [IsScalarTower k (MvPolynomial (Fin 2) k) N] in
-@[simp] lemma coefficientAdicEquiv_mk (j : ℕ) (n : N) :
-    coefficientAdicEquiv (k := k) j (Submodule.Quotient.mk n) = Submodule.Quotient.mk n := rfl
-
- 
-
-theorem polynomialJetExtend_bijective (j : ℕ) (q : N →ₗ[k] T)
-    (s : T →ₗ[k] N ⧸ ((MvPolynomial.idealOfVars (Fin 2) k)^j •
-      (⊤ : Submodule (MvPolynomial (Fin 2) k) N)))
-    (r : (N ⧸ ((MvPolynomial.idealOfVars (Fin 2) k)^j •
-      (⊤ : Submodule (MvPolynomial (Fin 2) k) N))) →ₗ[k] T)
-    (hq : ∀ n, q n = 0 ↔ n ∈ pairRelation (MvPolynomial.X 0 : MvPolynomial (Fin 2) k) (MvPolynomial.X 1))
-    (hr : ∀ n, r (Submodule.Quotient.mk n) = q n)
-    (hs : r ∘ₗ s = LinearMap.id)
-    (hx : Function.Injective (fun n : N => (MvPolynomial.X 0 : MvPolynomial (Fin 2) k) • n))
-    (hy : ∀ v n : N, (MvPolynomial.X 1 : MvPolynomial (Fin 2) k) • v =
-      (MvPolynomial.X 0 : MvPolynomial (Fin 2) k) • n →
-      ∃ w, v = (MvPolynomial.X 0 : MvPolynomial (Fin 2) k) • w) :
-    Function.Bijective (polynomialJetExtend j s) := by
-  let Q := (MvPolynomial.idealOfVars (Fin 2) k)^j • (⊤ : Submodule (MvPolynomial (Fin 2) k) N)
-  obtain ⟨l,hl⟩ := (Q.mkQ.restrictScalars k).exists_rightInverse_of_surjective
-    (LinearMap.range_eq_top.mpr Q.mkQ_surjective)
-  let L : T →ₗ[k] N := l ∘ₗ s
-  have hL : ∀ t, Q.mkQ (L t) = s t := by
-    intro t
-    exact congrArg (fun f : (N ⧸ Q) →ₗ[k] (N ⧸ Q) => f (s t)) hl
-  have hqL : q ∘ₗ L = LinearMap.id := by
-    ext t
-    change q (L t) = t
-    rw [← hr]
-    change r (Q.mkQ (L t)) = t
-    rw [hL]
-    exact congrArg (fun f : T →ₗ[k] T => f t) hs
-  have he (z : (MvPolynomial (Fin 2) k) ⊗[k] T) :
-      polynomialExtend s z = Q.mkQ (polynomialExtend L z) := by
-    induction z using TensorProduct.inductionOn with
-    | tmul P t => simp only [polynomialExtend_tmul, map_smul, hL]
-    | add z w hz hw => simp [hz, hw]
-  let a := coefficientAdicEquiv (k := k) (N := (MvPolynomial (Fin 2) k) ⊗[k] T) j
-  let b := coefficientAdicEquiv (k := k) (N := N) j
-  let f := jetMap (polynomialExtend L) (MvPolynomial.X 0 : MvPolynomial (Fin 2) k) (MvPolynomial.X 1) j
-  have hbij : Function.Bijective f := polynomialExtend_jet_bijective q L hq hqL hx hy j
-  have hc (z) : polynomialJetExtend j s (a z) = b (f z) := by
-    obtain ⟨w,rfl⟩ := (jetRelation (M := (MvPolynomial (Fin 2) k) ⊗[k] T)
-      (MvPolynomial.X 0 : MvPolynomial (Fin 2) k) (MvPolynomial.X 1) j).mkQ_surjective z
-    exact he w
-  constructor
-  · intro x y h
-    obtain ⟨x,rfl⟩ := a.surjective x
-    obtain ⟨y,rfl⟩ := a.surjective y
-    rw [hc,hc] at h
-    exact congrArg a (hbij.1 (b.injective h))
-  · intro z
-    obtain ⟨w,hw⟩ := hbij.2 (b.symm z)
-    refine ⟨a w, ?_⟩
-    rw [hc, hw, b.apply_symm_apply]
-end FiberFunctorProof
-
-end OAI
+import OAI.RepresentationTheory.RestrictedTilting.AdicJets
+
+namespace OAI
+
+namespace FiberFunctorProof
+open scoped TensorProduct Pointwise
+universe u
+variable {k : Type u} [Field k]
+variable {T N : Type u} [AddCommGroup T] [Module k T]
+  [AddCommGroup N] [Module k N]
+  [Module (MvPolynomial (Fin 2) k) N] [IsScalarTower k (MvPolynomial (Fin 2) k) N]
+
+ 
+noncomputable def polynomialJetExtend (j : ℕ)
+    (s : T →ₗ[k] N ⧸ ((MvPolynomial.idealOfVars (Fin 2) k)^j •
+      (⊤ : Submodule (MvPolynomial (Fin 2) k) N))) :
+    (((MvPolynomial (Fin 2) k) ⊗[k] T) ⧸ ((MvPolynomial.idealOfVars (Fin 2) k)^j •
+      (⊤ : Submodule (MvPolynomial (Fin 2) k) ((MvPolynomial (Fin 2) k) ⊗[k] T)))) →ₗ[MvPolynomial (Fin 2) k]
+      (N ⧸ ((MvPolynomial.idealOfVars (Fin 2) k)^j •
+      (⊤ : Submodule (MvPolynomial (Fin 2) k) N))) :=
+  ((MvPolynomial.idealOfVars (Fin 2) k)^j • (⊤ : Submodule (MvPolynomial (Fin 2) k) _)).liftQ
+    (polynomialExtend s) (by
+      intro z hz
+      change polynomialExtend s z = 0
+      refine Submodule.smul_induction_on hz (fun a ha z _ => ?_) (fun z w hz hw => ?_)
+      ·
+        rw [map_smul]
+        obtain ⟨v,hv⟩ := ((MvPolynomial.idealOfVars (Fin 2) k)^j •
+          (⊤ : Submodule (MvPolynomial (Fin 2) k) N)).mkQ_surjective (polynomialExtend s z)
+        rw [← hv, ← map_smul]
+        exact (Submodule.Quotient.mk_eq_zero _).mpr (Submodule.smul_mem_smul ha (by trivial))
+      · rw [map_add, hz, hw, add_zero])
+
+@[simp] lemma polynomialJetExtend_mk (j : ℕ)
+    (s : T →ₗ[k] N ⧸ ((MvPolynomial.idealOfVars (Fin 2) k)^j •
+      (⊤ : Submodule (MvPolynomial (Fin 2) k) N)))
+    (z : (MvPolynomial (Fin 2) k) ⊗[k] T) :
+    polynomialJetExtend j s (Submodule.Quotient.mk z) = polynomialExtend s z := rfl
+
+@[simp] lemma polynomialJetExtend_mk_tmul (j : ℕ)
+    (s : T →ₗ[k] N ⧸ ((MvPolynomial.idealOfVars (Fin 2) k)^j •
+      (⊤ : Submodule (MvPolynomial (Fin 2) k) N)))
+    (P : MvPolynomial (Fin 2) k) (t : T) :
+    polynomialJetExtend j s (Submodule.Quotient.mk (P ⊗ₜ[k] t)) = P • s t := rfl
+
+lemma polynomialJetExtend_transition (m n : ℕ) (h : m ≤ n)
+    (s : T →ₗ[k] N ⧸ ((MvPolynomial.idealOfVars (Fin 2) k)^n •
+      (⊤ : Submodule (MvPolynomial (Fin 2) k) N)))
+    (t : T →ₗ[k] N ⧸ ((MvPolynomial.idealOfVars (Fin 2) k)^m •
+      (⊤ : Submodule (MvPolynomial (Fin 2) k) N)))
+    (ht : ∀ x, AdicCompletion.transitionMap (MvPolynomial.idealOfVars (Fin 2) k) N h (s x) = t x)
+    (z : ((MvPolynomial (Fin 2) k) ⊗[k] T) ⧸ ((MvPolynomial.idealOfVars (Fin 2) k)^n •
+      (⊤ : Submodule (MvPolynomial (Fin 2) k) ((MvPolynomial (Fin 2) k) ⊗[k] T)))) :
+    AdicCompletion.transitionMap (MvPolynomial.idealOfVars (Fin 2) k) N h (polynomialJetExtend n s z) =
+      polynomialJetExtend m t (AdicCompletion.transitionMap (MvPolynomial.idealOfVars (Fin 2) k) _ h z) := by
+  obtain ⟨w,rfl⟩ := ((MvPolynomial.idealOfVars (Fin 2) k)^n •
+    (⊤ : Submodule (MvPolynomial (Fin 2) k) ((MvPolynomial (Fin 2) k) ⊗[k] T))).mkQ_surjective z
+  change AdicCompletion.transitionMap _ N h (polynomialExtend s w) = polynomialExtend t w
+  induction w using TensorProduct.inductionOn with
+  | tmul P x => rw [polynomialExtend_tmul, polynomialExtend_tmul, map_smul, ht]
+  | add z w hz hw => simp [hz, hw]
+
+noncomputable def coefficientAdicEquiv (j : ℕ) :
+    (N ⧸ jetRelation (M := N) (MvPolynomial.X 0 : MvPolynomial (Fin 2) k) (MvPolynomial.X 1) j) ≃ₗ[MvPolynomial (Fin 2) k]
+      (N ⧸ ((MvPolynomial.idealOfVars (Fin 2) k)^j • (⊤ : Submodule (MvPolynomial (Fin 2) k) N))) :=
+  Submodule.quotEquivOfEq _ _ (by rw [planeIdeal_eq, jetRelation_eq_pow_smul])
+
+omit [Module k N] [IsScalarTower k (MvPolynomial (Fin 2) k) N] in
+@[simp] lemma coefficientAdicEquiv_mk (j : ℕ) (n : N) :
+    coefficientAdicEquiv (k := k) j (Submodule.Quotient.mk n) = Submodule.Quotient.mk n := rfl
+
+ 
+
+theorem polynomialJetExtend_bijective (j : ℕ) (q : N →ₗ[k] T)
+    (s : T →ₗ[k] N ⧸ ((MvPolynomial.idealOfVars (Fin 2) k)^j •
+      (⊤ : Submodule (MvPolynomial (Fin 2) k) N)))
+    (r : (N ⧸ ((MvPolynomial.idealOfVars (Fin 2) k)^j •
+      (⊤ : Submodule (MvPolynomial (Fin 2) k) N))) →ₗ[k] T)
+    (hq : ∀ n, q n = 0 ↔ n ∈ pairRelation (MvPolynomial.X 0 : MvPolynomial (Fin 2) k) (MvPolynomial.X 1))
+    (hr : ∀ n, r (Submodule.Quotient.mk n) = q n)
+    (hs : r ∘ₗ s = LinearMap.id)
+    (hx : Function.Injective (fun n : N => (MvPolynomial.X 0 : MvPolynomial (Fin 2) k) • n))
+    (hy : ∀ v n : N, (MvPolynomial.X 1 : MvPolynomial (Fin 2) k) • v =
+      (MvPolynomial.X 0 : MvPolynomial (Fin 2) k) • n →
+      ∃ w, v = (MvPolynomial.X 0 : MvPolynomial (Fin 2) k) • w) :
+    Function.Bijective (polynomialJetExtend j s) := by
+  let Q := (MvPolynomial.idealOfVars (Fin 2) k)^j • (⊤ : Submodule (MvPolynomial (Fin 2) k) N)
+  obtain ⟨l,hl⟩ := (Q.mkQ.restrictScalars k).exists_rightInverse_of_surjective
+    (LinearMap.range_eq_top.mpr Q.mkQ_surjective)
+  let L : T →ₗ[k] N := l ∘ₗ s
+  have hL : ∀ t, Q.mkQ (L t) = s t := by
+    intro t
+    exact congrArg (fun f : (N ⧸ Q) →ₗ[k] (N ⧸ Q) => f (s t)) hl
+  have hqL : q ∘ₗ L = LinearMap.id := by
+    ext t
+    change q (L t) = t
+    rw [← hr]
+    change r (Q.mkQ (L t)) = t
+    rw [hL]
+    exact congrArg (fun f : T →ₗ[k] T => f t) hs
+  have he (z : (MvPolynomial (Fin 2) k) ⊗[k] T) :
+      polynomialExtend s z = Q.mkQ (polynomialExtend L z) := by
+    induction z using TensorProduct.inductionOn with
+    | tmul P t => simp only [polynomialExtend_tmul, map_smul, hL]
+    | add z w hz hw => simp [hz, hw]
+  let a := coefficientAdicEquiv (k := k) (N := (MvPolynomial (Fin 2) k) ⊗[k] T) j
+  let b := coefficientAdicEquiv (k := k) (N := N) j
+  let f := jetMap (polynomialExtend L) (MvPolynomial.X 0 : MvPolynomial (Fin 2) k) (MvPolynomial.X 1) j
+  have hbij : Function.Bijective f := polynomialExtend_jet_bijective q L hq hqL hx hy j
+  have hc (z) : polynomialJetExtend j s (a z) = b (f z) := by
+    obtain ⟨w,rfl⟩ := (jetRelation (M := (MvPolynomial (Fin 2) k) ⊗[k] T)
+      (MvPolynomial.X 0 : MvPolynomial (Fin 2) k) (MvPolynomial.X 1) j).mkQ_surjective z
+    exact he w
+  constructor
+  · intro x y h
+    obtain ⟨x,rfl⟩ := a.surjective x
+    obtain ⟨y,rfl⟩ := a.surjective y
+    rw [hc,hc] at h
+    exact congrArg a (hbij.1 (b.injective h))
+  · intro z
+    obtain ⟨w,hw⟩ := hbij.2 (b.symm z)
+    refine ⟨a w, ?_⟩
+    rw [hc, hw, b.apply_symm_apply]
+end FiberFunctorProof
+
+end OAI
diff --git a/lean/OAI/RepresentationTheory/RestrictedTilting/FiniteSubgroupLift.lean b/lean/OAI/RepresentationTheory/RestrictedTilting/FiniteSubgroupLift.lean
index 9eeb5febc8fb17bb3df3c5f5976d62a26fa826a5..d59bc9f2b5a7a10f4b17a178b4f73bd2121e422c 100644
--- a/lean/OAI/RepresentationTheory/RestrictedTilting/FiniteSubgroupLift.lean
+++ b/lean/OAI/RepresentationTheory/RestrictedTilting/FiniteSubgroupLift.lean
@@ -1,46 +1,46 @@
-import OAI.RepresentationTheory.RestrictedTilting.RestrictedLift
-import OAI.RepresentationTheory.RestrictedTilting.CoordinateTensor
-
-namespace OAI
-
-namespace FiberFunctorProof
-universe u v
-variable (k : Type u) [Field k]
-variable {E : Type v} [AddGroup E] [Fintype E]
-variable {ι : Type u} [Fintype ι] [DecidableEq ι]
-
- 
-
-theorem finite_subgroup_polynomial_lift (l : E →+ k) (hl : Function.Injective l)
-    (σ : Representation k (Multiplicative E) (ι → k)) :
-    ∃ ρ : Representation k (Multiplicative k) (ι → k),
-      ∃ P : Matrix ι ι (Polynomial k),
-        (∀ t : k, polynomialMatrixEval P t = LinearMap.toMatrix' (ρ (Multiplicative.ofAdd t))) ∧
-        (∀ i j, (P i j).natDegree ≤ Fintype.card E - 1) ∧
-        (∀ a : E, ρ (Multiplicative.ofAdd (l a)) = σ (Multiplicative.ofAdd a)) := by
-  classical
-  let H := l.range
-  let e : E ≃+ H := AddMonoidHom.ofInjective hl
-  let σH := σ.comp (AddEquiv.toMultiplicative e.symm).toMonoidHom
-  let R : Multiplicative H →* Matrix ι ι k := LinearMap.toMatrixAlgEquiv'.toMonoidHom.comp σH
-  let P := liftEntry H R
-  let ρ : Representation k (Multiplicative k) (ι → k) :=
-    Matrix.toLinAlgEquiv'.toMonoidHom.comp (extendedRepresentation H R)
-  have heval (t : k) : LinearMap.toMatrix' (ρ (Multiplicative.ofAdd t)) = liftMatrix H R t := by
-    exact LinearMap.toMatrixAlgEquiv'_toLinAlgEquiv' _
-  refine ⟨ρ,P,fun t => (heval t).symm,fun i j => ?_,fun a => ?_⟩
-  · have hc : Fintype.card H = Fintype.card E := Fintype.card_congr e.toEquiv.symm
-    by_cases hz : P i j = 0
-    · simp only [hz, Polynomial.natDegree_zero, Nat.zero_le]
-    · have hd := (Polynomial.natDegree_lt_iff_degree_lt hz).mpr (liftEntry_degree H R i j)
-      rw [hc] at hd
-      omega
-  · apply LinearMap.toMatrix'.injective
-    rw [heval]
-    have hla : l a = ((e a : H) : k) := rfl
-    rw [hla, liftMatrix_at_subgroup]
-    change LinearMap.toMatrix' (σ (Multiplicative.ofAdd (e.symm (e a)))) = _
-    rw [e.symm_apply_apply]
-end FiberFunctorProof
-
-end OAI
+import OAI.RepresentationTheory.RestrictedTilting.RestrictedLift
+import OAI.RepresentationTheory.RestrictedTilting.CoordinateTensor
+
+namespace OAI
+
+namespace FiberFunctorProof
+universe u v
+variable (k : Type u) [Field k]
+variable {E : Type v} [AddGroup E] [Fintype E]
+variable {ι : Type u} [Fintype ι] [DecidableEq ι]
+
+ 
+
+theorem finite_subgroup_polynomial_lift (l : E →+ k) (hl : Function.Injective l)
+    (σ : Representation k (Multiplicative E) (ι → k)) :
+    ∃ ρ : Representation k (Multiplicative k) (ι → k),
+      ∃ P : Matrix ι ι (Polynomial k),
+        (∀ t : k, polynomialMatrixEval P t = LinearMap.toMatrix' (ρ (Multiplicative.ofAdd t))) ∧
+        (∀ i j, (P i j).natDegree ≤ Fintype.card E - 1) ∧
+        (∀ a : E, ρ (Multiplicative.ofAdd (l a)) = σ (Multiplicative.ofAdd a)) := by
+  classical
+  let H := l.range
+  let e : E ≃+ H := AddMonoidHom.ofInjective hl
+  let σH := σ.comp (AddEquiv.toMultiplicative e.symm).toMonoidHom
+  let R : Multiplicative H →* Matrix ι ι k := LinearMap.toMatrixAlgEquiv'.toMonoidHom.comp σH
+  let P := liftEntry H R
+  let ρ : Representation k (Multiplicative k) (ι → k) :=
+    Matrix.toLinAlgEquiv'.toMonoidHom.comp (extendedRepresentation H R)
+  have heval (t : k) : LinearMap.toMatrix' (ρ (Multiplicative.ofAdd t)) = liftMatrix H R t := by
+    exact LinearMap.toMatrixAlgEquiv'_toLinAlgEquiv' _
+  refine ⟨ρ,P,fun t => (heval t).symm,fun i j => ?_,fun a => ?_⟩
+  · have hc : Fintype.card H = Fintype.card E := Fintype.card_congr e.toEquiv.symm
+    by_cases hz : P i j = 0
+    · simp only [hz, Polynomial.natDegree_zero, Nat.zero_le]
+    · have hd := (Polynomial.natDegree_lt_iff_degree_lt hz).mpr (liftEntry_degree H R i j)
+      rw [hc] at hd
+      omega
+  · apply LinearMap.toMatrix'.injective
+    rw [heval]
+    have hla : l a = ((e a : H) : k) := rfl
+    rw [hla, liftMatrix_at_subgroup]
+    change LinearMap.toMatrix' (σ (Multiplicative.ofAdd (e.symm (e a)))) = _
+    rw [e.symm_apply_apply]
+end FiberFunctorProof
+
+end OAI